# The Bruinier--Funke pairing and the orthogonal complement of unary theta functions

## TL;DR

This paper introduces an efficient algorithm to compute the inner product between holomorphic modular forms and unary theta functions, enabling orthogonality testing without full basis decomposition.

## Contribution

It presents a novel algorithm that simplifies orthogonality checks between modular forms and unary theta functions, avoiding extensive basis computations.

## Key findings

- Algorithm accurately computes inner products
- Enables orthogonality testing without full basis decomposition
- Reduces computational complexity in modular form analysis

## Abstract

We describe an algorithm for computing the inner product between a holomorphic modular form and a unary theta function, in order to determine whether the form is orthogonal to unary theta functions without needing a basis of the entire space of modular forms and without needing to use linear algebra to decompose this space completely.