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For a newspace Sknew(N,χ)S^\mathrm{new}_k(N,\chi) we say that a set of Hecke operators T:={Tp1,,Tpr}\mathcal T:=\{T_{p_1},\ldots,T_{p_r}\} distinguishes the newforms in the space if the sets Xf(T)X_f(\mathcal T) of characteristic polynomials of the TpTT_p\in \mathcal T acting on the subspace VfV_f spanned by the Galois orbit of ff in Sknew(N,χ)S_k^\mathrm{new}(N,\chi) are distinct as ff ranges over (non-conjugate) newforms in Sknew(N,χ)S_k^\mathrm{new}(N,\chi).

The set T\mathcal T can be identified by a list of primes pp. For convenience we restrict to primes pp that do not divide the level NN and list the unique ordered sequence of primes p1,,pnp_1,\ldots,p_n for which the sequence of integers c1,,cnc_1,\ldots,c_n defined by cm:=#{Xf({Tpi:i<m}):newforms fSknew(N,χ)} c_m := \#\bigl\{X_f(\{T_{p_i}:i < m\}): \mathrm{newforms}\ f\in S_k^\mathrm{new}(N,\chi)\bigr\} is strictly increasing. The length of the sequence p1,pnp_1,\ldots p_n is always less then the number of newforms in Sknew(N,χ)S_k^\mathrm{new}(N,\chi) and we obtain the empty sequence when Sknew(N,χ)S_k^\mathrm{new}(N,\chi) contains just one newform.

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  • Last edited by Andrew Sutherland on 2024-03-15 16:13:31
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