The 2-adic valuations of the algebraic central $L$-values for quadratic twists of weight 2 newforms

Let $f$ be a normalized newform of weight 2 on $\varGamma _0(N)$ whose coefficients lie in $\mathbb {Q}$ and let $\chi _M$ be a primitive quadratic Dirichlet character with conductor $M$. Under mild assumptions on $M$, we give a sharp lower bound for the 2-adic valuation of the algebraic part of the $L$-value $L(f, \chi _M, 1)$ and evaluate the 2-adic valuation for infinitely many $M$.