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!

], [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