Eigenvalue of Fricke involution on newforms of level 4 and of trivial character

TL;DR

This paper shows that all newforms of level 4 with trivial character are actually level 1 forms with non-Dirichlet characters and are eigenfunctions of the Fricke involution with eigenvalue -1.

Contribution

It establishes a surprising equivalence between certain level 4 newforms and level 1 forms with non-Dirichlet characters, and characterizes their Fricke involution eigenvalues.

Findings

All level 4 newforms are eigenfunctions of the Fricke involution with eigenvalue -1.

Level 4 newforms are actually level 1 forms with non-Dirichlet characters.

The result links level 4 newforms to simpler level 1 forms.

Abstract

In this note, we consider the newforms of integral weight, level 4 and of trivial character, and prove that all of them are actually level 1 forms of some non-Dirichlet character. As a byproduct, we can prove that all of them are eigenfunctions of the Fricke involution with eigenvalue -1.