Abstract: Holomorphic modular forms of (integral) weight k are analytic functions on the upper half-plane which are invariant under the action of certain discrete subgroups ? of SL2(R) and holomorphic at the cusps of ?. Hecke eigenforms are certain nice functions which span this finite-dimensional Hilbert Space and are simultaneous eigenfunctions corresponding to a class of nice operators on this space called Hecke Operators. The Fourier expansion of Hecke eigenforms around the cusps are arithmetically significant and demand the study of the Dirichlet L-series (and its analytic continuation) associated to it. Riemann Hypothesis in this context predicts that non-trivial zeros of their L function lie exactly on the line of symmetry <(s) = 1/2 and is a long-standing open problem in Mathematics. While Riemann Hypothesis is at present out of reach, one can of course ask simpler questions.
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