What is the most complete binary quadratic form table? Is there a free, easily accessible quadratic form resource — maybe even just a big web-accessible table — which shows “all known results” [reasonably speaking] about numbers of the form $mx^2+ny^2$ for various $m,n$?

Immediate motivation: Given integers $a$ and $x$ with $a^3 \mid (x^2+1)(3x^2+5)$, I’m trying to characterize $a$. I know that if [odd] prime $\phi \mid (x^2+1)$, then $\phi \equiv 1\!\pmod{4}$… but what about $\phi \mid (3x^2+5)$?

Ultimately, of course, this kind of question really comes down to characterizing the primes that can divide a given quadratic form. I know about scattered resources like this Mathworld page and the (very short!) table on this page — I just can’t seem to find comprehensive information on that subject presented in a nice compact form.

Particular case where $(x, 3)=1$
For $3x^2+5$ we may write:
$x^2\equiv 1\bmod 3\Rightarrow 3x^2+5\equiv 8\equiv 2\bmod 3$
We may conclude that primes of the form $p=3m+2$ can divide $3x^2+5$ , for example:
$x=1\rightarrow 3x^2+5=8, k=0\rightarrow p=2; 2|8$,
$x=6\rightarrow 3x^2+5=113=3\times 37+2$
$x=4\rightarrow 3x^2+5=53=3\times 17+2$
$x=8\rightarrow 3\times 64+5=197=3\times 65+2$
$x=17\rightarrow 3x^2+5=872=2\times 436; 2=3\times 0+2$
$x=18\rightarrow 3x^2+5=977=3\times 325+2$
For $mx^2+ny^2$ we may use the same method, suppose $(m, n, 3)=1$ then:
$mx^2+ny^2\equiv (m+n)\bmod 3$
