On l-adic representations for a space of noncongruence cuspforms

This paper is concerned with a compatible family of 4-dimensional \ell-adic representations \rho_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of weight 3 cuspforms S_3 (\Gamma) on a noncongruence subgroup \Gamma \subset \SL. For this representation we prove that: 1.)It is automorphic: the L-function L(s, \rho_{\ell}^{\vee}) agrees with the L-function for an automorphic form for \text{GL}_4(\mathbb A_{\Q}), where \rho_{\ell}^{\vee} is the dual of \rho_{\ell}. 2.) For each prime p \ge 5 there is a basis h_p = \{h_p ^+, h_p ^- \} of S_3 (\Gamma) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12. The key point is that the representation $\rho_{\ell}$ admits a quaternion multiplication structure in the sense of a recent work of Atkin, Li, Liu and Long.

1. Introduction 1.1. Recall that a subgroup of finite index Γ ⊂ SL 2 (Z) is a congruence subgroup if Γ ⊃ Γ(N ) for some integer N ≥ 1, where Γ(N ) ⊂ SL 2 (Z) is the normal subgroup consisting of matrices congruent to the identity modulo N ; Γ is a noncongruence subgroup if it is not a congruence subgroup. There is a vast theory of modular forms on congruence subgroups (general reference for facts and notation: [Shi71], [DS05]). By contrast, modular forms on noncongruence subgroups are less wellunderstood, and they exhibit qualitatively different behavior. It is well known that S k (Γ 0 (N ), χ) has a basis of Hecke eigenforms, which have q-expansions f (z) = n≥1 a n (f )q n , where q = exp(2πiz), with a n satisfying the relations (1) a np − a p a n + χ(p)p k−1 a n/p = 0, a n = a n (f ) for all positive integers n and primes p ∤ N , taking a n/p = 0 if p ∤ n. Moreover, a p is the trace of Frobenius for a two-dimensional λ-adic representation ρ f of G Q := Gal(Q/Q) ([Del68, DS75,Lan72]).
Date: today. The second author was supported in part by the NSA grant #H98230-08-1-0076. Part of the work was done during Long's visit to University of California at Santa Cruz. This research was initiated during an REU summer program at LSU, supported by the National Science  1.2. If Γ is a noncongruence subgroup, there is in general no Hecke eigenbasis for S k (Γ), the space of weight k cuspforms for Γ, but rather it is conjectured that, at least in certain circumstances, for almost all primes p there is a basis {h j = h p,j } such that the q-expansion coefficients satisfy 3-term Atkin-Swinnerton-Dyer (ASD) congruences in the general shape: (2) a np (h j ) − α p (j)a n (h j ) + χ j (p)p k−1 a n/p (h j ) ≡ 0 mod (np) k−1 where |α p (j)| ≤ 2p (k−1)/2 and χ j (p) is a root of unity. In [Sch85i,Sch85ii], A. J. Scholl proved the existence of (2d + 1)-term ASD congruences (d = dim S k (Γ)), under some standard assumptions such as the modular curve being defined over Q with infinity as a Q-rational point. In fact, for every prime ℓ, Scholl proved the existence of a G Q -representation ρ ℓ = ρ Γ,k,ℓ acting on an ℓ-adic space W ℓ (Γ) of dimension 2d analogous to Deligne's construction for congruence subgroups. He also constructed 2d-dimensional Q p -vector spaces, V p (Γ), with an action of a Frobenius operator and containing the subspace S k (Γ) ⊗ Q p , which are the analogs in crystalline cohomology of the ℓ-adic space W ℓ (Γ). Scholl achieved the (2d + 1)-term ASD congruences via a comparison theorem. He managed to refine this to obtain 3-term congruences when the characteristic polynomials of those (2d + 1)-term recursions have d distinct p-adic roots. In special cases involving extra symmetries, such as 4-dimensional Scholl representations satisfying Quaternion Multiplications (see [ALLL10]), we can find in a systematic way a basis of the noncongruence modular forms whose members satisfy 3-term congruences, see Section 6.
In recent studies ( [LLY05], [Lon08], [ALL08], [FHL + 08], [ALLL10], to which we send the reader for more background and precise conjectures) the α p (j), up to multiplying by roots of unity in clear patterns, are the pth q-expansion coefficients of newforms f j on congruence subgroups. To be more precise, there is a quadratic field K such that T 2 − a p (j)T + χ j (p)p k−1 is the Hecke polynomial at some place of K over p for some automorphic form for GL 2 over K (see [ALLL10,Thm. 4.3.2] for details).
1.3. In this paper, we have a particular noncongruence group Γ and k = 3, d = 2. Experimentally it was discovered that the degree 4 polynomials of the geometric Frobenii under the corresponding Scholl representation (denoted by ρ ℓ,2 below) factor into quadratic pieces with coefficients in Q( √ −3), Q( √ −2) or Q( √ −6), respectively as p is congruent to 1 mod 3, 5 mod 12, or 11 mod 12, and moreover, the linear terms in these polynomials matched the pth Fourier coefficients of a newform f of level 432, up to multiplication by a twelfth root of unity. This trichotomy is explained by the existence of an order-8 quaternion group acting on ρ ℓ,2 . This falls into the general framework of 4-dimensional Galois representations admitting quaternion multiplications that is studied in detail in [ALLL10], except that the action of the quaternion group is not defined over a quadratic or biquadratic field. To overcome the extra complication, we use an auxiliary 4-dimensional G Q representation ρ ℓ,4 , which is a 4-dimensional Galois representation admitting quaternion multiplication defined over a biquadratic field. The automorphy of ρ ℓ,4 is known due to [ALLL10]. It requires additional work to link ρ ℓ,4 to the level 432 newform f . We use ρ f to denote the 2-dimensional Deligne representation of G Q attached to the Hecke newform f . The relations between ρ ℓ,2 , ρ ℓ,4 , and ρ f are depicted by the following diagram. For the precise statements, see Corollary 1, Theorems 1, 2, and 3.
For a number field K, let G K denote Gal(Q/K). In the diagram above, both σ λ,2,−3 and σ λ,4,−3 are 2-dimensional representations of G Q( 1.4. For each prime p ≥ 5 we prove that there exists a basis h p = {h + p , h − p } of S 3 (Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations. To be more precise, there exists a finite extension E of Q p such that the coefficients of where O E is the ring of integers of E, A p,± , B p,± ∈ O E and the weight k = 3. We say that we have ASD congruences relative to the polynomial X 2 − A p,± X + B p,± . The structure of h p depends only on the class of p mod 12. As an application of the modularity result mentioned above, A p,± , B p,± are determined by the p-coefficient of f and the characters ψ and χ. See Propositions 4, 5, and 6.
1.5. The paper is organized as follows. In §2, we describe the group Γ and the family of elliptic curves E(Γ) → X(Γ) associated to it. In §3 we define some correspondences B j of the elliptic surface E(Γ). The main point is that these define a quaternion multiplication structure on associated cohomology spaces. In §4 we show that the 4-dimensional ℓ-adic representations ρ ℓ,2 , ρ ℓ,4 are induced in several ways from subgroups of index two in G Q . §5 proves the main modularity theorem: the L-function of the representations ρ ∨ ℓ,2 (resp. ρ ∨ ℓ,4 ) can be expressed in terms of the Hecke L-function of a newform f of level 432 and some explicit characters. This is done by the method of Faltings-Serre. The ASD congruences are proved in §6. Tables of experimental data which form the basis of this paper appear in §7. These computations began in an REU project in summer 2005. The software systems Magma, Mathematica, and pari/gp were used.
2. The group and the space of noncongruence cuspforms 2.1. If Γ 0 ⊂ SL 2 (Z) is a torsion-free subgroup of finite index we let Y (Γ 0 ) = Γ 0 \H be the quotient of the upper half plane of complex numbers, and j : Y (Γ 0 ) → X(Γ 0 ) be the compactification by adding cusps. It is known that these are the C-points of algebraic curves defined over number fields; in this paper, they will have models over Q, which we will denote by the same symbols. Define the analytic space E(Γ 0 ) as the quotient of C × H by the equivalence relation is a fibration of elliptic curves. When Γ 0 is a congruence subgroup, these are the C-points of schemes defined over number fields and represent (at least coarsely) moduli problems for elliptic curves. In this paper, they will be defined over Q and designated by the same symbols.
2.3. The stack [Γ 1 (4)] classifies pairs (E, P ) of elliptic curves E together with a point P of exact order 4. This time [Γ 1 (4)] = M (Γ 1 (4)). One knows that there are two connected components defined over Q(i) each of which is isomorphic to P 1 Q(i) .
2.4. The stack [Γ 0 (8) ∩ Γ 1 (4)] classifies triplets (E, C, P ) of elliptic curves E together with P ∈ C a point of exact order 4 in a cyclic subgroup of order 8. We It has a fine moduli interpretation as the moduli space of triples (E, C, P ) where E is an elliptic curve, C ⊂ E is a subgroup scheme of order 8, and P ∈ E is a point of order 4. A model for its universal elliptic curve is The modular function t is a Hauptmodul of Γ 0 (8) ∩ Γ 1 (4): a generator of the function field of the modular curve. Let Γ be a special index 3 normal subgroup of Γ 0 := Γ 0 (8) ∩ Γ 1 (4) whose modular curve X(Γ) is a cubic cover of X(Γ 0 (8) ∩ Γ 1 (4)) unramified everywhere except the cusps 1 2 and 1 4 (whose t-values are ∞ and −1 respectively), with ramification degree 3. It is easy to see that the genus of X(Γ) is also 0. To facilitate our calculation, we need to find an algebraic map between the two modular curves, in other words, we need to describe a relation between a Hauptmodul r a of Γ and t. By our construction, ar 3 a = t + 1 for some nonzero constant a. Here a is a rational number written in lowest form. As we look for a model so that the q-expansion of r a is in a number field as small as possible, we take a = 2 in which case the coefficients of r a can be chosen in Q. The Riemann surface Y (Γ) := Γ\H has the structure of an algebraic curve over Q. This group Γ is labeled by Γ 8 3 ·6·3·1 3 in [FHL + 08]. The cusp widths of Γ are 8-8-8-6-3-1-1-1 from which we know Γ is a noncongruence subgroup. The quotient Γ 0 /Γ is generated by Both matrices ζ and A induce actions on X(Γ) as well as the spaces of cuspforms for Γ.
2.5. Note that in [FHL + 08], different choices of a are picked. The reason for varying a is that for some choices, the operators to be defined below had smaller fields of definition, whereas for other choices, the Galois representation corresponding to S 3 (Γ) and parameter a (see §4) was easier to analyze. Only two choices a = 2, 4 are relevant to this paper; the corresponding Hauptmoduls are denoted r 2 and r 4 . We let f a : E(Γ) → Y (Γ) be the pull-back of the universal elliptic curve in the previous section via the covering of degree three: r a → ar 3 a − 1 We let E Γ be the complex elliptic surface obtained by completing and desingularizing f : E(Γ) → Y (Γ) over the compact curve X(Γ) := (Γ\H) * , which is independent of the choice of a. The nonzero Hodge numbers of this surface are h 0,0 = h 2,2 = 1, h 1,1 = 30, h 2,0 = h 0,2 = 2. In particular the space of weight three cuspforms is two dimensional. This S 3 (Γ) has a basis (see [FHL + The first arrow sends τ ∈ H to τ and the second arrow sends τ to α.τ . If Γ 2 is decomposed into cosets for the subgroup Γ so that the ε i .τ are the elements of Γ\H projecting to τ ∈ Γ 2 \H, then the double coset decomposes as

This diagram lifts to morphisms
The arrow on the left is induced by (z, τ ) → (z, τ ). This is a fiberwise isomorphism. The map on the right is induced from This is a fiberwise isogeny.
3.5. The involution A of the modular curve X(Γ 0 (8) ∩ Γ 1 (4)) lifts to an involution of the curve X(Γ) where Γ ⊂ Γ 0 (8) ∩ Γ 1 (4) is defined in §2.4. Let r 4 be the Hauptmodul for this curve with t = 4r 3 4 − 1. Under the action of A, r 4 → 1/2r 4 . By base-change, the isogeny A defined in the above proposition lifts to an isogeny of the elliptic curve E(Γ), which we will denote by E(r 4 ). This map defined over Q(i). For the purposes of the ASD congruences, we need that the curve X(Γ) has a Q-rational cusp corresponding to τ = i∞; unfortunately, this is not the case for this model: the point τ = i∞ corresponds to t = 1 and there is no Q -rational r 4 with t = 1 = 4r 3 4 − 1. In the model with t = 2r 3 2 − 1 corresponding to the representation ρ ℓ,2 (details see §4), there is a Q-rational point over t = 1, namely r 2 = 1. But in this model, the involution A is now given by r 2 → 1/ 3 √ 2r 2 , and this gives an isogeny of E(Γ) defined over the larger field Q(i, 3 √ 2). In either of these models, ζ acts on the Hauptmodul by r a → exp (2πi/3) r a .
3.6. On the p-adic space V p (see sections 1.2 and 6.1), we define operators where A and ζ are given in §2.4.
To prove the identities, it suffices to prove them for the corresponding operators on S 3 (Γ). The reason is that this is the Hodge (2, 0) component of H DR ( 3 Γ W) for Scholl's motive (see [Sch90]), these operators act on this motive, which is a natural factor of H 2 (E Γ ), and V p = H cris ( 3 Γ W). 1 The effect of ζ on the basis h 1 , h 2 1 Scholl only constructs k n W for principal congruence subgroups of level n. But the construction also works here: it is the Grothendieck motive which is formal image of the projector denoted Πε in loc. cit. acting on the elliptic surface E Γ,a , now regarded as a scheme over Q.
(see (4)) is given by the matrix where ω 3 is a primitive cubic root of unity. To compute the effect of A we use the stroke operator defined in section (3.3) on H 1 = h 3 1 , H 2 = h 3 2 , and use properties of the η-function. The result is, in the basis h 1 , h 2 A := 0 i2 4/3 i2 5/3 0 These identities follow immediately.

ℓ-adic representations
In this paper, for any place v of a number field K, we use Fr v and Frob v := Fr −1 v to denote the corresponding arithmetic Frobenius and geometric Frobenius, respectively. We use ℓ to denote an arbitrary prime, unless it is specified.
which is a 4-dimensional Q ℓ -space, whereétale cohomology is understood, and we are using the symbols Y (Γ) and X(Γ) to denote the schemes over Q whose C-valued points were previously denoted by these letters. We thus obtain a continuous ℓ-adic representation ρ ℓ,a : G Q → GL(W ℓ,a ) = GL 4 (Q ℓ ).
Let N (a) be the least common multiple of the numerator and denominator of a. This representation is unramified outside of 2, 3, ℓ, N (a), and for all primes p ∤ 6ℓN (a), we have the characteristic polynomial of Frobenius det(X − ρ ℓ,a (Frob p )) = Char(Frob p , W ℓ,a , X) = H p,a (X) ∈ Z[X].
A useful fact is that the roots of H p,a (X) have the same absolute value (cf. [Sch85ii]), which is p in this case.
4.2. When a = 4, it was first observed experimentally that there are factorizations H p,4 (X) = g p,4 (X)g p,4 (X), where the bar notation stands for complex conjugation and the coefficients of the quadratic polynomials g p,4 (X) lie respectively in the fields Q, Q( √ −3), Q( √ −2), Q( √ −6) as p ≡ 1, 7, 5, 11 mod 12. This is a property that follows from ρ ℓ,4 satisfying the so-called quaternion multiplication over the biquadratic field Q( √ −3, √ −2) in the sense of [ALLL10,§3]. To be more precise, we consider the following maps on W ℓ,4 ⊗ Q ℓ Q ℓ . Let ζ * be the map on W ℓ,4 induced by ζ and A * be the map induced by A (see §3.4 and a related discussion in [ALLL10,§5]). It sends E(Γ) to an isogenous elliptic curve over Q(i, r 4 ). It is obvious that ζ * is defined over Q( √ −3). By §3.4, A * is defined over Q(i). We define the following operators on W ℓ,4 ⊗ Q ℓ Q ℓ : B * −1 = A * , B * −3 = ζ * − (ζ * ) 2 ; as well as These depend on choices of embeddings √ 8, √ 3 ∈ Q ℓ , but the decompositions that follow below do not depend on these choices. Proof.
(1) In the proof of Proposition 2 these identities were shown to hold on H DR ( 3 Γ W). By comparison isomorphisms, these also hold on H ℓ ( 3 Γ W) = W ℓ,a .
(2) The Galois group G Q interacts with the operators J ±3 , J −1 in the same way that it interacts with the B * −1 , B * −3 , B * 3 , in other words the irrationalities √ 8, √ 3 do not affect the Galois action: the reason is that one treats the second factor in W ℓ,4 ⊗ Q ℓ Q ℓ as the trivial Galois module. Thus, when a = 4, by the discussion right before the proposition, is the minimal field of definition of J −1 (resp. J −3 , J 3 ). Thus the commutativity of G Q and J −1 , J −3 , J 3 is as claimed.
Proof. The polynomial X 3 − 2 has exactly one root in F p when p ≡ 2 mod 3 as X → X 3 is a bijection in F p . Thus for any prime p ≡ 2 mod 3, which is inert in ], it splits completely in the Galois extension K of Q( √ −3). This means ψ(p) = 1.

Modularity
Our goal is to prove Theorem 1. Since ρ ℓ,4 satisfies quaternion multiplication over Q(i, √ −3), by the main result of [ALLL10] the L-function of ρ ∨ ℓ,4 coincides with the L-function of a GL 4 (A Q ) automorphic form. Here we will prove this claim directly. 2 We identified using William Stein's Magma package the following Hecke eigenform More concretely, where E 6 (z) = 1 + 12 n≥1 (σ(3n) − 3σ(n))q n , and σ(n) = d|n d. Let ρ f be Deligne's 2-dimensional λ-adic representation of G Q attached to f , where λ is the place of Q ℓ ( √ 2, √ −3) lying over ℓ (see [Del68], [Lan72]). In particular the trace of arithmetic Frobenius Fr p under ρ f agrees with c p (f ), the p-coefficient of f . These are the conventions of [DS75]. This is at variance with the conventions in [Del68] and in Scholl's papers, which match the characteristic polynomials of the geometric Frobenius with the Hecke polynomials Therefore we must take duals in the statement of our main results.
2 Using the approach of [ALLL10], one can also derive that there exists a quadratic character where ρg is the Deligne representation attached to a certain new form g, that is different from the new form f below.
We will now apply the method of Faltings-Serre (see [Ser84]) to prove the result below. Briefly, the idea is that given two nonisomorphic semisimple Galois representations ρ 1 , ρ 2 : G K → GL 2 (Q ℓ ) there is a finite list of (G, t) that captures the difference between ρ 1 , ρ 2 , where eachG is a finite Galois group referred to as a deviation group, and t :G → F ℓ is a function with certain properties that can be computed from ρ 1 , ρ 2 . To establish the isomorphism between the semisimplifications of ρ 1 and ρ 2 we need to eliminate each (G, t) by explicit computation. The idea of the criterion is recast in [LLY05], right below the statement of Theorem 6.2.
Let H = G Q(i) . By the explicit form of f , ρ f | H is a 2-dimensional representation of H over Q ℓ ( √ 2). For any places v above p ≡ 5 mod 12, the character χ takes values ±i and the characteristic polynomial of σ λ,4,−1 (Frob v ) is of the form X 2 + a √ −2X − p 2 for some a ∈ Z. For any places v above p ≡ 1 mod 12, the character χ takes values ±1 and the characteristic polynomial of σ λ,4,−1 (Frob v ) is of the form X 2 + aX + p 2 for some a ∈ Z. Thus, the representation ρ f | H ⊗χ takes values in Q ℓ ( √ −2), as does the representation (σ λ,4,−1 ) ∨ . In the proof below, we will see that it suffices to compare c p (f )χ(v) with the trace of σ λ,4,−1 (Frob v ), that is, the trace of (σ λ,4,−1 ) ∨ (Fr v ) for every place v of Q(i) above the primes p = 5 and 13. Now we can use Faltings-Serre modularity criterion effectively. We take the representation with coefficients in Q 2 ( √ −2). Let ℘ = ( √ −2). We now consider both representations modulo ℘ with images in SL 2 (F 2 ). For simplicity, useρ to denote ρ ℓ,f | G Q( √ −1) . The trace ofρ(Fr 2+3i ) = 1 so the image has order 3 elements. If the image is C 3 , then it gives rise to a cubic extension of Q(i) which is unramified outside of 1 + i and 3. Like Lemma 19 of [Lon08], we know such a cubic field is the splitting field of x 3 − 3x + 1. It is irreducible mod 2 + i, but the characteristic polynomial ofρ(Fr 1+2i ) is T 2 + 1, hence it is of order 1 or 2 which leads to a contradiction. So kerρ corresponds an S 3 -extension of Q(i) which can be identified as the splitting field of x 3 − 2 over Q(i).
Now we are going to consider all possible deviation groups G measuring the difference between (σ λ,4,−1 ) ∨ and (ρ f | G Q(i) ) ⊗ χ if they are not isomorphic up to semisimplification.
Let F ′ be the splitting field of x 3 − 2 over Q. Similar to the proof of Lemma 19 of [Lon08], we first look for S 4 -extensions L ′ of Q containing F ′ but not Q(i) and unramified outside of 2 and 3. By using the fact that their cubic resolvent is x 3 − 2 we conclude that such S 4 extensions are the splitting fields of irreducible polynomials of the form x 4 + ux 2 + vx − u 2 /12 where u, v ∈ Z[1/6] such that 8u 3 /27 + v 2 = ±2. The possible polynomials giving non-isomorphic fields are Fr 5 has order 4 in these fields. By comparing the traces of Frob 1±2i (or Fr 1±2i ) for both representations, we eliminate the possibility that G contains S 4 as a subgroup.
The remaining possibility is G = S 3 × Z/2. Among all the quadratic extensions Q(i, √ d) of Q(i) unramified at 1 + i and 3: d = i, 1 + i, 1 − i, 3 + 3i, 3 − 3i, we have that Fr v for v = 3 + 2i, 3 + 2i, 3 − 2i, 3 + 2i, 3 − 2i are inert in these fields respectively. Meanwhile these Frobenius elements map underρ to an element of order 3. Such an element in the conjugacy class of Fr v has order 6 as it has order 3 in the S 3 component and order 2 in the Z/2 component. Since the characteristic polynomials of Frob 3±2i (or Fr 3±2i ) under both representations agree, we know G cannot be S 3 × Z/2 either, because if the two representations were different, they would have different traces at elements of order 4 or higher in the deviation group, see [LLY05]. 6. Atkin-Swinnerton-Dyer congruences 6.1. Let V p,a be the 4-dimensional F -crystal (Q p vector space with Frobenius action) associated with S 3 (Γ) in the model of the curve X(Γ) with Hauptmodul r a . Note that the only difference among the V p,a is the action of the Frobenius F . We need to consider both a = 2, 4: for a = 4 it is easier to describe the modularity of the ℓ-adic counterpart ρ ℓ,4 (see Theorem 3) and the factorization of H p,4 (X), but to prove 3-term ASD congruences in the simplest form for the expansion coefficients of the cuspforms h 1 , h 2 relative to the q-parameter via the results of Scholl's paper [Sch85i], [Sch85ii], we need a = 2. We could derive ASD congruences using the F -crystal V p,4 but these would be expressed in a parameter γq for some algebraic number γ. Therefore we only consider the F -crystal V p = V p,2 .
6.2. The method is the following. We use the operators B s (resp. B * s ) s = −1, ±3 to decompose both V p (resp. W ℓ := W ℓ,2 ) into eigenspaces V ± p,s (resp. W ± λ,s ). By the way B s are defined, we know that B s is defined over the field L −1 = Q(i) · Q( 3 √ 2), L −3 = Q( √ −3) and L 3 = Q( √ 3) · Q( 3 √ 2) for s = −1, −3, 3 respectively (see §3.5). We show that for each prime p ≥ 5 there is an s such that the Frobenius F acts on the 2-dimensional eigenspaces V ± p,s which corresponds to an Frob p in G Ls acting on W ± λ,s . For p ≡ 1 mod 3 we use s = −3; for p ≡ 5 mod 12 we use s = −1; for p ≡ 11 mod 12 we use s = 3. Following the approach of Scholl [Sch85ii], there is a comparison theorem between Frobenius action on both ℓ-adic and p-adic spaces that implies (8) Char(F, V ± p,s , X) = Char(Frob p , W ± λ,s , X), where the right hand side can be computed from f , ψ and χ. (Also the Frobenius Frob p needs further justification to ensure it commutes with B * s .) In particular, for primes p ≡ 2 mod 3, the right hand side can be computed from f and χ only by using Lemma 1. We find the ASD basis as linear combinations h 1 ± αh 2 which span the 1-dimensional spaces V ± p ∩ (S 3 (Γ) ⊗ Q p ).