Theta Products and Eta Quotients of Level 24 and Weight 2

2016/07/14 by Ayşe Alaca, Alaca, Ayşe, Şaban Alaca +3
Mathematics · #11E20 #11F11 #11F20 #11F27 #11F30 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11E20 #msc:11F11 #msc:11F20 #msc:11F27 #msc:11F30

paper · pdf · doi:10.48550/arxiv.1607.03997

27 pages

arxiv created 2016/07/14 · arxiv updated 2016/07/15

Abstract

We find bases for the spaces M₂(Γ₀(24),((d)/(⋅))) (d=1,8,12, 24) of modular forms. We determine the Fourier coefficients of all 35 theta products φ[a₁,a₂,a₃,a₄](z) in these spaces. We then deduce formulas for the number of representations of a positive integer n by diagonal quaternary quadratic forms with coefficients 1, 2, 3 or 6 in a uniform manner, of which 14 are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces E₂(Γ₀(24),((d)/(⋅))) (d=1,8,12,24) and give their Fourier coefficients.

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