On the number of representations of integers by quadratic forms with congruence conditions

TL;DR

This paper characterizes the generating function for counting integer representations by quadratic forms with congruence conditions using modular forms, leading to explicit formulas for these representation numbers.

Contribution

It introduces a modular form-based approach to explicitly compute representation numbers for quadratic forms with congruence conditions.

Findings

Explicit formulas for representation numbers derived

Generating functions characterized as modular forms

Application of modular form theory to quadratic forms

Abstract

We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as examples.