Modular forms which behave like theta series

Authors:
K. Chakraborty, A. K. Lal and B. Ramakrishnan

Journal:
Math. Comp. **66** (1997), 1169-1183

MSC (1991):
Primary 11F11, 11F12, 11F27, 11F30

MathSciNet review:
1423070

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Abstract: In this paper, we determine all modular forms of weights , , for the full modular group which behave like theta series, i.e., which have in their Fourier expansions, the constant term and all other Fourier coefficients are non-negative rational integers. In fact, we give convex regions in (resp. in ) for the cases (resp. for the cases ). Corresponding to each lattice point in these regions, we get a modular form with the above property. As an application, we determine the possible exceptions of quadratic forms in the respective dimensions.

**1.**Serge Lang,*Introduction to modular forms*, Springer-Verlag, Berlin-New York, 1976. Grundlehren der mathematischen Wissenschaften, No. 222. MR**0429740****2.**M. Manickam,*Newforms of half-integral weight and some problems on modular forms*, Ph. D. Thesis, Univ. of Madras 1989.**3.**M. Manickam and B. Ramakrishnan,*On normalized modular forms of weights and with non-negative integral Fourier coefficients*, preprint 1985.**4.**A. M. Odlyzko and N. J. A. Sloane,*On exceptions of integral quadratic forms*, J. Reine Angew. Math.**321**(1981), 212–216. MR**597991**, 10.1515/crll.1981.321.212**5.**Michio Ozeki,*On modular forms whose Fourier coefficients are non-negative integers with the constant term unity*, Math. Ann.**206**(1973), 187–203. MR**0330053****6.**J.-P. Serre,*A course in arithmetic*, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR**0344216**

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Additional Information

**K. Chakraborty**

Affiliation:
Mehta Research Institute of Mathematics and Mathematical Physics, 10, Kasturba Gandhi Marg (Old Kutchery Road), Allahabad 211 002, India

Email:
kalyan@mri.ernet.in

**A. K. Lal**

Affiliation:
Mehta Research Institute of Mathematics and Mathematical Physics, 10, Kasturba Gandhi Marg (Old Kutchery Road), Allahabad 211 002, India

Address at time of publication:
Department of Mathematics, Indian Institute of Technology, Kanpur 208 016, India

Email:
arlal@iitk.ernet.in

**B. Ramakrishnan**

Affiliation:
Mehta Research Institute of Mathematics and Mathematical Physics, 10, Kasturba Gandhi Marg (Old Kutchery Road), Allahabad 211 002, India

Address at time of publication:
Department of Mathematics, Indian Institute of Technology, Kanpur 208 016, India

Email:
ramki@mri.ernet.in

DOI:
https://doi.org/10.1090/S0025-5718-97-00872-7

Keywords:
Modular forms,
theta series

Received by editor(s):
May 10, 1995

Received by editor(s) in revised form:
October 16, 1995, and March 8, 1996

Article copyright:
© Copyright 1997
American Mathematical Society