Cusp forms in $S_{4}\left( \Gamma _{0}\left( 79\right) \right) $ and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant $-79$

Bari\c{s} Kend\.{i}rl\.{i}

doi:10.4134/BKMS.2012.49.3.529

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2012; 49(3): 529-572

Printed May 1, 2012

https://doi.org/10.4134/BKMS.2012.49.3.529

Cusp forms in $S_{4}\left( \Gamma _{0}\left( 79\right) \right) $ and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant $-79$

Bari\c{s} Kend\.{i}rl\.{i}

Fatih University

Abstract

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Abstract

A basis of a subspace of $S_{4}\left( \Gamma _{0}\left( 79\right) \right) $ is given and the formulas for the number of representations of positive integers by some direct sums of the quadratic forms $x_{1}^{2}+x_{1}x_{2}+20x_{2}^{2},$ $ 4x_{1}^{2}\pm x_{1}x_{2}+5x_{2}^{2},$ $2x_{1}^{2}\pm x_{1}x_{2}+10x_{2}^{2}$ are determined.

Keywords: cusp forms, representation number, theta series

MSC numbers: 11E25, 11E76