  Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, and let $a$ be a positive
integer coprime with $q$. For any $\epsilon>0$, there exists $\alpha>0$
(computable), such that for all $x\ge \alpha (\log q)^2$, the interval $\left[
e^x,e^{x+\epsilon }\right]$ contains a prime $p$ in the arithmetic progression
$a \bmod q$. This gives the bound for the least prime in this arithmetic
progression: $P(a,q) \le e^{\alpha (\log q)^2}$. For instance for all $q\ge
10^{30}$, $P(a,q) \le e^{4.401(\log q)^2}$. Finally, we apply this result to
establish that every integer larger than $e^{71\,000}$ is a sum of seven cubes.
