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2 Showing the q expansion relates to the eignevalues.

It's not true for any old modular form. Since the forms live in a vector space over $\mathbb C$, you can achieve any complex number as a coefficient.

Here's a partial answer to what is true. You need to have a cusp form that is an eigenfunction of the Hecke operators, normalized so the leading coefficient is $1$. Since the Hecke operators are self adjoint in the Peterson (sp?) inner product, the eigenvalues are real, and one can show these are the coefficients in the $q$ expansion as follows: for $p$ prime, the $m$th coefficient of $T_p f$ is $a_{mp}$, for all $m$, more or less from the definition of $T_p$. This is also $\lambda_p a_m$, and from this and $a_1=1$ one deduces $a_p=\lambda_p$ (take $m=1$.) The general case follows from the recursion for powers of primes, and multiplicativity.

This answer is not quite right because it doesn't explain how CM extensions can arise, but it's a start.

1

It's not true for any old modular form. Since the forms live in a vector space over $\mathbb C$, you can achieve any complex number as a coefficient.

Here's a partial answer to what is true. You need to have a cusp form that is an eigenfunction of the Hecke operators, normalized so the leading coefficient is $1$. Since the Hecke operators are self adjoint in the Peterson (sp?) inner product, the eigenvalues are real, and one can show these are the coefficients in the $q$ expansion.

This answer is not quite right because it doesn't explain how CM extensions can arise, but it's a start.