Exponents of Mersenne primes represented by the quadratic form $x^2+dy^2$

List of Mersenne prime exponents:

[2,3,5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161,74207281,77232917,82589933,136279841];

Looking at the list of Mersenne prime exponents, one can notice a tendency of the exponents towards the $8k+1$ form; 20 of the 51 odd exponents have this form. If we observe the exponents of Mersenne primes through the prism of the quadratic form $x^2+dy^2$, it can be seen that 11 exponents take the form $x^2+40y^2$, 12 exponents take the form $x^2+112y^2$, and 13 exponents take the form $x^2+88y^2$. For an exponent to be representable in these three forms, it must be of the form $8k+1$. These three forms have the highest $z$-scores for $d$ in the range from 1 to 1000, respectively: the form $x^2+40y^2$ has a $z$-score of $1.958\sigma$, the form $x^2+112y^2$ has a $z$-score of $2.382\sigma$, and the form $x^2+88y^2$ has a $z$-score of $2.805\sigma$, where $\sigma$ is the standard deviation. Note that the numbers 40, 88, and 112 belong to the set of idoneal numbers.

Suppose $d$ is idoneal number of the form $2^{k+2}q$, where $k \geq 1$ and $q$ is a prime number . My question is whether there is a theoretical indication that this idoneal number influence the maximization of the $z$-score of the quadratic form $x^2+dy^2$ by which the exponents of a Mersenne prime numbers are represented?