TITLE: How many L-values determine a modular form?
QUESTION [9 upvotes]: Suppose $f$ and $g$ are two newforms of certain levels, weights etc. If we know that
L(f,n)=L(g,n) for all sufficiently large $n$, can we conclude that $f=g$? 
Same question when the forms have the same weight and $n$ runs over critical points.

REPLY [5 votes]: The answer to the first question is "yes".  The standard proof of the uniqueness of a Dirichlet series expansion actually generalizes to show the following.
Theorem.  Suppose that $A(s) = \sum_n a_n n^{-s}$ and $B(s) = \sum_n b_n n^{-s}$ are Dirichlet series with coefficients $a_n, b_n$ bounded by a polynomial.  If there exists a sequence of complex numbers $s_k$ with real part approaching infinity such that $A(s_k) = B(s_k)$ for all $k$, then $a_n = b_n$ for all $n$.
Proof (sketch).  Proceed by induction.  For $k$ big we have $A(s_k) = a_1 + O(2^{-\sigma_k})$ where $\sigma_k$ is the real part of $s_k$.  Similarly, $B(s_k) = b_1 + O(2^{-\sigma_k})$.  Since $A(s_k) = B(s_k)$, we conclude that $a_1 = b_1$.  A similar argument shows $a_2 = b_2$, $a_3 = b_3$, etc.