Atlas › Modular forms › Weight 2 › Level 865 › 865.2.a › 865.2.a.a

Newform orbit 865.2.a.a

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 3q^{2} - 4q^{3} + q^{4} + 9q^{5} - 3q^{6} - 6q^{7} - 3q^{8} - 5q^{9} - 3q^{10} - 18q^{11} - q^{13} - 3q^{14} - 4q^{15} - 7q^{16} - 4q^{17} + q^{18} - 17q^{19} + q^{20} - 8q^{21} - 14q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

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

$x^{9} + 7x^{8} + 7x^{7} - 36x^{6} - 47x^{5} + 73x^{4} + 53x^{3} - 72x^{2} + 16x - 1$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
173	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−3	−4	9	−6	−18	−1	−4	−17	−14	−14	−27	11	−14	−6	−33	−6	−28	−15	−7	−56	9	−10	−28	−7	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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.

The engine's checks

dim M^+ at levels [1, 5, 173, 865]: [0, 1, 15, 88] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 6 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(865, 2)[0]   # 865.2.a.a

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 865:2 1000