Geodesic
The Eisenstein-Hecke series and their properties
Dalʹnevostočnyj matematičeskij žurnal, Tome 16 (2016) no. 1, pp. 3-8.

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Let $\Gamma_0(N)$ be the congruence subgroup of level N. If N is not a square-free number then the Fourier coefficients of the classical Eisenstein series are not multiplicative. In the paper we construct the modified Eisenstein-Hecke series with the desired property of multiplicativity. This result is of great importance for investigating trace formulas on the space of cusp forms. Similar results were obtained earlier by S. Gelbart and H. Jacquet using the theory of adeles. 
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V. A. Bykovskii. The Eisenstein-Hecke series and their properties. Dalʹnevostočnyj matematičeskij žurnal, Tome 16 (2016) no. 1, pp. 3-8. http://geodesic.mathdoc.fr/item/DVMG_2016_16_1_a0/

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