Explicit calculations with Eisenstein series

Abstract

We find explicit change-of-basis formulas between Eisenstein series attached to cusps, and newform Eisenstein series attached to pairs of primitive Dirichlet characters. As a consequence, we prove a Bruggeman-Kuznetsov formula for newforms of square-free level and trivial nebentypus. We also derive, in explicit form, the Fourier expansion, Hecke eigenvalues, Atkin-Lehner pseudo-eigenvalues, and other data associated to these Eisenstein series, with arbitrary integer weight, level, and nebentypus.

37 pages. v2: implements some corrections and improvements to exposition

Cited by in corpus (24)

• The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound
• Motohashi's fourth moment identity for non-archimedean test functions and applications
• On the Random Wave Conjecture for Dihedral Maaß Forms
• Dedekind sums arising from newform Eisenstein series
• Twist-minimal trace formulas and the Selberg eigenvalue conjecture
• Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field
• The twisted second moment of modular half integral weight $L$--functions
• Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications
• The kernel of newform Dedekind sums
• Quantum Unique Ergodicity for Eisenstein Series in the Level Aspect
• Uniform bounds for Kloosterman sums of half-integral weight with applications
• Archimedean Newform Theory for ${\mathrm{GL}}_n$
• Correction to: Equidistribution of Eisenstein Series in the Level Aspect
• Uniform Titchmarsh divisor problems
• Reciprocity and the Kernel of Dedekind Sums
• Self-Correlations of Hurwitz Class Numbers
• Subconvexity bounds for twisted L-functions, II
• On the Waldspurger Formula and the Metaplectic Ramanujan Conjecture over Number Fields
• First moments of Rankin-Selberg convolution of Automorphic Forms on $GL(2)$
• Second moments of Rankin-Selberg convolution and shifted Dirichlet series
• On the regularized $L^4$-norm for Eisenstein series in the level aspect, Part II
• Strong Hybrid Subconvexity for Twisted Selfdual ${\mathrm{GL}}_3$ $L$-Functions
• Non-Random Behaviour in Sums of Modular Symbols
• On the regularized $L^4$-norm of Eisenstein series in the level aspect