The trace of $T_2$ takes no repeated values

TL;DR

This paper proves that the trace of the Hecke operator T_2 on cusp forms of level one is injective, taking each value at most once, except for zero in trivial cases.

Contribution

It establishes a new property of the trace of T_2, showing it has no repeated values except zero in trivial spaces.

Findings

Trace of T_2 is injective on cusp forms of level one.

Zero occurs only when the space of cusp forms is trivial.

The result characterizes the distribution of trace values for T_2.

Abstract

We prove that the trace of the Hecke operator $T_2$ acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.