Representations of integers belonging to subsequences of the positive integers by binary quadratic forms

We consider positive-definite primitive binary quadratic forms of fundamental discriminant $d<0$; $R$ is the genus and $C$ is the class of such forms. We obtain asymptotics for the sum of absolute values of the Fourier coefficients for the Hecke eigenforms of weight 1 and of dihedral type. In an earlier paper (Zap. Nauchn. Semin. POMI, 226 (1996)), the author showed that if $C\in R$, then almost all $R$-representable positive integers are $C$-representable. We extend this result to certain subsequences of $\mathbb N$ such as $\{a_n=p_n+l\}$, $\{a_n=n(n+1)\}$, etc. Finally, for certain genera $R$ with class number greater than one, we prove an asymptotics $(x\to\infty)$ for the sum
$$ \sum_{\substack{n\le x\\ r(n;C)>0}}\frac1{r(n;C)}, $$
where $C$ is a class in $R$ and $r(n;C)$ is the number of representations of a positive integer $n$ by the class $C$.