Abstract
We describe a method for understanding averages over newforms on \(\Gamma _0(q)\) in terms of averages over all forms of some level. The method is simplest when q is divisible by the cubes of its prime divisors.
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References
A. Abbes and E. Ullmo, Comparaison des métriques d’Arakelov et de Poincaré sur \(X_0(N)\), Duke Math. J. 80 (1995), 95–307.
A. O. L. Atkin and J. Lehner, Hecke operators on \(\Gamma _{0}(m)\), Math. Ann. 185 (1970), 134–160.
O. Barrett, P. Burkhardt, J. De Witt, R. Dorward, and S. J. Miller, One-Level density for holomorphic cusp forms of arbitrary level, ArXiv e-prints, April 2016.
V. Blomer and D. Milićević, The second moment of twisted modular \(L\)-functions, Geom. Funct. Anal. 25 (2015), 453–516.
W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314.
M. Dickson, Local spectral equidistribution for degree two Siegel modular forms in level and weight aspects, Int. J. Number Theory 11 (2015), 341–396.
M. Dimitrov and L. Nyssen, Test vectors for trilinear forms when at least one representation is not supercuspidal, Manuscripta Math. 133 (2010), 479–504.
S. Gelbart, Automorphic Forms on Adèle Groups, Annals of Mathematics Studies, No. 83, Princeton University Press, Princeton, N.J., 1975.
C. Hamer, A formula for the traces of the Hecke operators on certain spaces of newforms, Arch. Math. (Basel) 70 (1998), 204–210.
H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of \(L\)-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 55–131.
H. Jacquet, I. I. Piatetski-Shapiro, and J. Shalika, Conducteur des représentations du groupe linéaire, Math. Ann. 256 (1981), 199–214.
G. Martin, Dimensions of the spaces of cusp forms and newforms on \(\Gamma _0(N)\) and \(\Gamma _1(N)\), J. Number Theory 112 (2005), 298–331.
P. D. Nelson, Quantum variance on quaternion algebras, I, preprint, 2016.
I. Petrow and M. P. Young, A generalized cubic moment and the Petersson formula for newforms, ArXiv e-prints, August 2016.
D. Rouymi, Formules de trace et non-annulation de fonctions \(L\) automorphes au niveau \(\mathfrak{p}^{\nu } \), Acta Arith. 147 (2011), 1–32.
R. Schmidt, Some remarks on local newforms for \({\rm GL}(2)\), J. Ramanujan Math. Soc. 17 (2002), 115–147.
J.-P. Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke \(T_p\), J. Amer. Math. Soc. 10 (1997), 75–102.
J.-P. Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation.
M.-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800, Springer, Berlin, 1980.
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Nelson, P.D. Analytic isolation of newforms of given level. Arch. Math. 108, 555–568 (2017). https://doi.org/10.1007/s00013-017-1039-y
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DOI: https://doi.org/10.1007/s00013-017-1039-y