In book of Fred Diamond a first course in modular form in chapeter 9 theorem 9.6.3 it said that let $E$ be an elliptic curve over $Q$ then for some new from $f \in s_{2}(\gamma_{0}(N))$ with number field $Q$ Then $\rho_{E.p} \sim \rho_{f.p}$ for all $p$. And there is also another theorem of modularity in chapter 8 theorem 8.8.3 Let E be an elliptic curve over $Q$ with conductor N then for some newfrom f∈$s_{2}(\gamma_{0}(N))$ we have $L(E.s)=L(f.s)$ Question : is this two difintion are equivalente ? (if it is not .what is the diffrent between them)
$$\rho_{E.p} \sim \rho_{f.p} \iff L(E.s)=L(f.s)$$
Where $L(E.s)$ is the L_function of elliptic curve $E$ and $L(f.s)$ is the L-function of the newfrom $f$