Something came up yesterday in a referee request and I was surprised to find that I did not know the facts in full generality. This is about positive quadratic forms in three variables with integer coefficients, primitive, meaning the GCD of coefficients is 1.

In the case of the sum of three squares, if $R(n)$ is the total number of integral representations of $n,$ and $p$ an odd prime, it follows (induction on the highest power of $p$ dividing $n$) from the result in Hirschhorn and Sellers that $$ R(n p^2) \geq p R(n). $$

The relation is $$ R(n p^2) = (p+1 - (-n|p)) R(n) - p R(n/p^2), $$ where $(-n|p) $ is taken as $0$ when $p$ divides $n,$ and $R(n/p^2)$ is taken as $0$ when $p^2$ does not divide $n.$ See Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?

On the other hand, $$ R(4n) = R(n). $$

So, very simple from a modular forms perspective, I imagine, given some other positive ternary form $f,$ representation count function $R(n),$ and a prime $p$ (possibly $2$) for which $f(x,y,z)$ is isotropic in $\mathbb Q_p,$ is it true that

$$ R(n p^2) \geq p R(n)? $$

Note: I've been running experiments, it seems to be alright if $p$ divides the discriminant of $f,$ as long as isotropic as I said. I suspect this is trivial, with key words such as Eisenstein series, Hecke operators, other things the kids enjoy.