Tunnell-type criteria for variants of the congruent number problem

We study the $\theta$-congruent number problem for $\cos \theta =\pm 3\mathrm{/}5$ and $\pm 4\mathrm{/}5$ using the generalized theta series construction of Sirolli–Tornaría. We describe its specialization to newforms of weight $2$ over $\mathbb{Q}$ with nontrivial square-free odd part of the level, and explain the reduction of quadratic twists to odd fundamental discriminants. The same construction gives an effective procedure for every $\theta$-congruent number problem with nonzero rational cosine. For the four angles, we construct explicit forms of weight $3\mathrm{/}2$ whose Fourier coefficients determine the central $L$-values of the associated elliptic curves. This gives Tunnell-type criteria for every positive square-free integer: a nonzero coefficient implies non-$\theta$-congruence unconditionally, and the converse holds assuming the Birch–Swinnerton-Dyer conjecture. We also prove unconditional non-$\theta$-congruence for primes in explicit arithmetic progressions.