Chakraborty, K.; Lal, A. K.; Ramakrishnan, B. Modular forms which behave like theta series. (English) Zbl 1036.11503 Math. Comput. 66, No. 219, 1169-1183 (1997). Summary: We determine all modular forms of weights \(36\leq k\leq 56\), \(4\mid k\), for the full modular group \(\text{SL}_2(\mathbb Z)\) 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. Cited in 3 Documents MSC: 11F11 Holomorphic modular forms of integral weight 11Y35 Analytic computations 11F27 Theta series; Weil representation; theta correspondences 11F30 Fourier coefficients of automorphic forms Keywords:Modular forms; theta series PDFBibTeX XMLCite \textit{K. Chakraborty} et al., Math. Comput. 66, No. 219, 1169--1183 (1997; Zbl 1036.11503) Full Text: DOI Online Encyclopedia of Integer Sequences: a(n) is the least integer such that every even unimodular lattice in dimension 8n contains some vectors of all even (squared) norm >= 2*a(n). References: [1] Serge Lang, Introduction to modular forms, Springer-Verlag, Berlin-New York, 1976. Grundlehren der mathematischen Wissenschaften, No. 222. · Zbl 0344.10011 [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 \(20, 24\) and \(28\) 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. · Zbl 0439.10015 · doi: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. · Zbl 0266.10021 · doi:10.1007/BF01429207 [6] J.-P. Serre, A course in arithmetic, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. · Zbl 0256.12001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.