Newform orbit 529.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$; the sum of their $q$-expansions is

$3q + 6q^{4} + 3q^{6} + 9q^{8} + 9q^{9} - 15q^{12} + 12q^{16} - 21q^{18} + 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$, whose minimal polynomial is

$x^{3} - 96x + 123$

Complex multiplication by $\mathbb{Q}(\sqrt{-23})$: $a_p = 0$ for every conjugate at the 5 primes $p \le 31$ inert in it (exactly, from $\operatorname{tr} a_{p^2} = -\dim \cdot p^{k-1}$), and $\operatorname{tr} a_p = 0$ at all 87 inert primes $p < 1000$. This is evidence from the stored coefficients, not a proof.

Atkin–Lehner signs

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$.

Traces of Hecke eigenvalues

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

newform_orbits(529, 2)[5]   # 529.2.a.f

cargo run --release -p sagebrush-web --example atlas -- 529:2 1000