I'm interested in knowing if there exists a bound (possibly similar to the one found by Baker, Harman, and Pintz) for gaps between primes in arithmetic progression given a fixed quadratic character.

For example, does there exist an $\epsilon > 0$ depending on $d$ and $m$ such that, whenever $\gcd(d,m) = 1$, $\chi$ is the quadratic character of conductor $d$, and $x$ is sufficiently large, there is a prime $p$ congruent to $a \pmod m$ such that $x < p < x + x^{1-\epsilon}$ and $\chi(p) = 1$?

Uber den Primzahlsatz von Herrn Hoheisel(Math Zeitschrift, 1933). For more recent references and further improvements you may consult pages 141 and 142 of Narkiewicz'sRational Number Theory in the 20th Century. $\endgroup$ – Vesselin Dimitrov Jun 27 '17 at 15:34