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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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Modular forms which behave like theta series
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by K. Chakraborty, A. K. Lal and B. Ramakrishnan PDF
Math. Comp. 66 (1997), 1169-1183 Request permission

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, and 44 (resp. for the cases $k = 48$, 52, 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.
References
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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
  • Received by editor(s): May 10, 1995
  • Received by editor(s) in revised form: October 16, 1995, and March 8, 1996
  • © Copyright 1997 American Mathematical Society
  • 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