What is the best reference on Eisenstein Series for $\Gamma_0(N)$ in $\mathrm{GL}(2,\mathbb{R})$?
For fixed $\Gamma_0(N)$, should there be several Eisenstein series (corresponding to each cusp)?
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$\begingroup$ Are you assuming trivial central character? The given answers are only for $Psl(2, \mathbb{R})$, and which is fine, depending upon your definition of $\Gamma_0(N)$. $\endgroup$– Marc PalmCommented Apr 15, 2012 at 9:09
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2 Answers
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Miyake, already recommended by GH, is a very complete reference, which is perhaps the only place to contain complete proofs about the subject. However, for that reason, and also because he works with weird modular groups $\Gamma_1(a,b)$ conjugate to but not equal to the more familiar $\Gamma_1(ab)$, it is difficult to read.
So I propose another reference which contains a clear complete basis of Eisenstein series, which are eigenforms for $\Gamma_1(N)$, together with their nebentypus, and their Hecke eigenvalues : W. Stein, Modular Forms, A Computational Approach, available at http://wstein.org/books/modform/modform/index.html . More precisely, see here.
To get Eisenstein series for $\Gamma_0(N)$ from those for $\Gamma_1(N)$, just restrict to those with trivial nebentypus, that is $\psi \chi=1$ in the notation of the reference given.
And about your question: in weight $>2$, there are as many independent Eisenstein series as cusps; in weight $2$, the dimension of the Eisenstein subspace is the number of cusp minus 1, because "$E_2$ is missing".
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2$\begingroup$ Stein's book is a good reference, but beware of a typo in Theorem 5.9 which means that some formulae for Eisenstein series with character are not quite correct (some $\psi$'s should be $\bar{\psi}$'s). See trac.sagemath.org/sage_trac/ticket/4062 (a bug in the Sage computer algebra system which was traced to an incorrect formula copied from Stein's book). $\endgroup$ Commented Apr 15, 2012 at 20:33
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For detailed information I recommend "Chapter 7. Eisenstein series" in Miyake: Modular Forms (Springer Verlag, 2006).
answered Apr 14, 2012 at 16:59