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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
DOI: https://doi.org/10.1090/S0025-5718-97-00872-7
MathSciNet review: 1423070
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Abstract: In this paper, we determine all modular forms of weights $36\leq k\leq 56$, $4\mid k$, for the full modular group $SL_2(\mathbb Z)$ which behave like theta series, i.e., which have in their Fourier expansions, the constant term $1$ and all other Fourier coefficients are non-negative rational integers. In fact, we give convex regions in ${\mathbb R}^3$ (resp. in ${\mathbb R}^4$) for the cases $k = 36, 40 \hbox {~and~} 44$ (resp. for the cases $k = 48, 52 \hbox {~and~} 56$). 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.


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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