On the order of vanishing of newforms at cusps

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$. We obtain an explicit formula, as a product of local terms, for the ramification index at each cusp of a modular parametrization of $E$ by $X_0(N)$. Our formula shows that the ramification index always divides 24, a fact that had been previously conjectured by Brunault as a result of numerical computations. In fact, we prove a more general result which gives the order of vanishing at each cusp of a holomorphic newform of arbitary level, weight and character, provided its field of rationality satisfies a certain condition. The above result relies on a purely $p$-adic computation of possibly independent interest. Let $F$ be a non-archimedean local field and $\pi$ an irreducible, admissible, generic representation of $\mathrm{GL}_2(F)$. We introduce a new integral invariant, which we call the \emph{vanishing index} and denote $e_\pi(l)$, that measures the degree of "extra vanishing" at matrices of level $l$ of the Whittaker function associated to the newvector of $\pi$. Our main local result writes down the value of $e_\pi(l)$ in every case.