Ummm. Just so you know, the same type of conclusion holds for $$ x^2 - p y^2 $$ for prime $p,$ $$ 5 \leq p \leq 197, \; \; \; \; p \equiv 1 \pmod 4. $$ For these forms, a number $n$ is represented if and only if $-n$ is represented. There is a solution to $x^2 - p y^2 = -1,$ a result in Mordell's book. Since every odd prime $q$ that satisfies $(p|q) = 1$ is represented by some form of the discriminant, and there is only one class of this discriminant, then all odd primes $q $ with $(q|p) = 1$ are integrally represented. Representation of the prime $2$ is a different matter, as we need $(2|p) = 1,$ so this works only when we further demand $p \equiv 1 \pmod 8.$

Ummm. Just so you know, the same type of conclusion holds for $$ x^2 - p y^2 $$ for prime $p,$ $$ 5 \leq p \leq 197, \; \; \; \; p \equiv 1 \pmod 4. $$ For that matter, one may switch to the forms $$ x^2 + xy - \left( \frac{p-1}{4} \right) y^2 $$

For these forms, a number $n$ is represented if and only if $-n$ is represented. There is a solution to $x^2 - p y^2 = -1,$ a result in Mordell's book. Since every odd prime $q$ that satisfies $(p|q) = 1$ is represented by some form of the discriminant, and there is only one class of this discriminant, then all odd primes $q $ with $(q|p) = 1$ are integrally represented. Representation of the prime $2$ is a different matter, as we need $(2|p) = 1,$ so this works only when we further demand $p \equiv 1 \pmod 8.$

Things start to go sideways for primes $229, 257, 401.$ These have class numbers $3,3,5.$ I found a way I really like for choosing one "reduced" form for each $SL_2 \mathbb Z$ class; here $\langle a,b,c \rangle \mapsto ax^2 + b xy + c y^2$ is called reduced precisely when the discriminant is correct and both $$ ac <0, \; \; \; b > |a+c|. $$ This is equivalent to the Gauss-lagrange-Legendre definition of reduced, most people do not know that.

229    factored    229

    1.             1          15          -1   cycle length             2
    2.             3          13          -5   cycle length             6
    3.             5          13          -3   cycle length             6

  form class number is   3


257    factored    257

    1.             1          15          -8   cycle length             6
    2.             2          15          -4   cycle length             6
    3.             4          15          -2   cycle length             6

  form class number is   3


401    factored    401

    1.             1          19         -10   cycle length             6
    2.             2          19          -5   cycle length             6
    3.             5          19          -2   cycle length             6
    4.             4          17          -7   cycle length            10
    5.             7          17          -4   cycle length            10

  form class number is   5