Bulletin of the
Korean Mathematical Society BKMS

In this article, we find bases for the spaces of modular forms $M_{2}(\Gamma _{0}(88),\big( \frac{d}{\cdot }\big) )$ for $d=1,8,44\text{ and }88$. We then derive formulas for the number of representations of a positive integer by the diagonal quaternary quadratic forms with coefficients $1,2,11$ and $ 22 $.

Keywords: Eta quotients, Dedekind eta function, theta products, Eisenstein series, modular forms, cusp forms, quadratic forms, representation numbers

MSC numbers: 11E25, 11E20, 11F11, 11F20, 11F27

Supported by: This material is based upon a work supported by the Simons Foundation Institute Grant Award ID 507536 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI.