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I apologize in advance for what is probably a very naive question: I'd like to understand the Fourier coefficients of newforms, and so I was wondering what exactly was known about them (I do know that the situation isn't as straightforward as for Eisenstein series). I have looked at the algorithms in Modular Forms a Computational Approach, but I was hoping for more explicit expressions.

In particular, when I run the command Newforms(CuspForms(N,k)) for lowish weight and level in Magma, the q-expansions that are outputted usually look "nice" (for example one q-expansion will differ from another by a quadratic character). I was interested in more information on this, as well as any explanation for why Magma outputs the expansions in the form that they do. Thanks!

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When you say differ by a quadratic character, how do you mean? When I think of twisting a newform (say of level N and weight 2) by a quadratic character (of conductor D with D squarefree and coprime to N) the resulting newform should no longer be of level N, but rather N(D^2). – stankewicz Mar 22 '11 at 16:05
I mean, for example (and I could be totally misunderstanding any number of things including what is output), if I type: Newforms(CuspForms(25,4)); [* [* q + q^2 + 7*q^3 - 7*q^4 + 7*q^6 + 6*q^7 - 15*q^8 + 22*q^9 - 43*q^11 + O(q^12) ], [ q - q^2 - 7*q^3 - 7*q^4 + 7*q^6 - 6*q^7 + 15*q^8 + 22*q^9 - 43*q^11 + O(q^12) ], [ q + 4*q^2 - 2*q^3 + 8*q^4 - 8*q^6 - 6*q^7 - 23*q^9 + 32*q^11 + O(q^12) *] *] Now let $f_1=\sum a(n)q^n$ be the first series, and $f_2=\sum b(n)q^n$ the second. Then, looking at their q-expansion, it seems that $$a(p)=\phi(p)b(p)$$ where $\phi$ is quadratic mod 5. – Jill Mar 22 '11 at 17:08
Jill, this has to do with the fact that $25=5^2$; a twist of a level $d^2$ form by a quadratic character of conductor $d$ will also have level $d^2$. However, for e.g. squarefree levels this kind of thing will never happen. – David Hansen Mar 22 '11 at 18:21
Excuse me, should've written "...will SOMETIMES also have level $d^2$." – David Hansen Mar 22 '11 at 18:22
As for your Magma output question, it appears (from your singular example) that it returns a normalized Fourier expansion (first coefficient is 1), which is typically what one does. – Kimball Mar 23 '11 at 14:32

I'm not quite sure what kind of information you're expecting, but there is a basis of the space of newforms which consists of eigenfunctions of the Hecke operators, which means there is an Euler product expression ; this is Atkin-Lehner theory.

One could also mention growth conditions...

Serre's "Cours d'arithmétique" is a nice reference, and there are english translations.

EDIT. Yes, in the reference I gave, Serre limits himself to level one -- but he does cover Hecke operators and their eigenfunctions and discuss coefficients growth and Euler product, so the basics are nicely laid out.

Other references : Miyake's "Modular forms", Hida's "Elementary theory of $L$-functions and Eisenstein series", Bump's "Automorphic forms and representations"... and of course, there's Shimura's "Introduction to the arithmetic theory of automorphic functions"!

EDIT2. Manin&Panchishkin's "Introduction to modern number theory" has a nice exposition of the Atkin-Lehner theory too. I can't help but notice that I'm still not quite sure what type of information was expected... and mostly keep on piling on references...

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Well, there's at least 1 English translation. – Kimball Mar 23 '11 at 14:30
Doesn't Serre only address forms of level 1 in that book? I think that will preclude a nuanced discussion of newforms. – Ramsey Mar 23 '11 at 15:15
In my opinion, Iwaniec's "Topics in classical automorphic forms" book is infinitely more readable than any of the ones you've listed (Serre excluded, of course). – David Hansen Mar 24 '11 at 13:53
That is the reason why I mentioned Serre first ;-) – Julien Puydt Mar 24 '11 at 14:18

You should have a look at Ken Ono's nice book: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series. Theorem 2.27 in this book is a theorem of Atkin and Lehner that "captures the essential properties of a newform" in the integer weight case. Read section 3.4 for Kohnen's theory for the half-integer weight case.

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