On the traces of Hecke operators
For a positive divisor \(D\) of positive integer \(N\), the author introduces the concept of the space \(S_2(\Gamma_0(N))^{D\text{-old}}\) of \(D\)-oldforms of weight 2 and level \(N\), and its orthogonal complement \(S_2(\Gamma_0(N))^{D\text{-new}}\), the space of \(D\)-newforms. If \(N\) is squarefree and if \(l\) is a prime not dividing \(N\), the author derives a formula for \(\text{tr}(T(l)\mid S_2(\Gamma_0(N))^{D\text{-new}})\), the trace of the Hecke operator \(T(l)\) on the space of \(D\)-newforms of weight 2. Let \(\Gamma(D, M)\), with \(M={N\over D}\), denote the group of units of norm 1 of an Eichler order of level \(M\) in an indefinite rational quaternion algebra of discriminant \(D\neq 1\). The trace of \(T(l)\) on \(S_2(\Gamma(D, M))\) was calculated by \textit{M. Eichler} [J. Indian Math. Soc., New Ser. 20, 163-206 (1956; Zbl 0073.26501)]. The author observes that \(\text{tr}(T(l)\mid S_2(\Gamma(D, M)))= \text{tr}(T(l)\mid S_2(\Gamma_0(N))^{D\text{-new}})\).
scientific article; zbMATH DE number 1934770
scientific article