As GH suggests, here the relevant Eisenstein and cusp spaces are small enough that everything can be done explicitly. It's even a bit better than the dimensions $5+4$ suggest, because our quadratic form is isodual, which puts its theta series in an eigenspace for the Atkin-Lehner involution $w_{44}$. The resulting formula is particularly nice for $n$ prime, and immediately shows that every prime other than $2$ and $11$ is represented, and indeed the number of representations is proportional to the number of points modulo the prime of an elliptic curve of conductor $11$.
Namely: let $$ E_2(q) = 1 - 24 \sum_{n=1}^\infty \frac{nq^n}{1-q^n}; $$ this is not a modular form, but for every factor $d|44$ the combination $$ \varepsilon^{(d)}_2(q) := d \cdot E_2(q^d) - \frac{44}{d} E_2(q^{44/d}) $$ is a weight-2 form for $\Gamma_0(44)$. Let $$ \phi(q) = q \prod_{n=1}^\infty \bigl( (1-q^n)(1-q^{11n}) \bigr)^2 = q - 2 q^2 - q^3 + 2 q^4 + q^5 + 2 q^6 - 2 q^7 \cdots $$ be the unique eigen-cuspform for $\Gamma_0(11)$, associated to the elliptic curve $E: y^2+y=x^3-x^2$ of discriminant $-11$. Then the theta function $\sum_{n=0}^\infty r(n) q^n$ is $$ \frac -\frac {\varepsilon^{(1)}_2(q) - \varepsilon^{(2)}_2(q) + \varepsilon^{(4)}_2(q)} {30} + - \frac45\bigl(\phi(q)+3\phi(q^2)+4\phi(q^4)\bigr). $$ The coefficients are obtained by matching $q$-expansions to $O(q^{125})$, which is more than enough to prove that two weight-$2$ forms on $\Gamma_0(44)$ coincide. In particular, for the number of representations of a prime $p$ other than $2$ and $11$ we have $$ r(p) = \frac45 (p + 1 - a_p) $$ which is positive because $p+1 - a_p$ is the number of points on $E \bmod p$ (which is indeed divisible by $5$ because $E$ has a rational $5$-torsion point $x=y=0$).
