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.



How many L-values determine a modular form?
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.


