Atlas › Modular forms › Weight 2 › Level 766 › 766.2.a › 766.2.a.f

Newform orbit 766.2.a.f

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 10q^{2} + 10q^{4} + 3q^{5} - 12q^{7} - 10q^{8} + 26q^{9} - 3q^{10} + 15q^{11} - 7q^{13} + 12q^{14} + 3q^{15} + 10q^{16} + q^{17} - 26q^{18} - 2q^{19} + 3q^{20} + q^{21} - 15q^{22} + 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{10} - 3x^{9} - 64x^{8} + 111x^{7} + 1223x^{6} - 943x^{5} - 7486x^{4} - 1282x^{3} + 7328x^{2} + 4872x + 872$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
383	$-1$

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 even 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
$\operatorname{tr} a_p$	−10	0	3	−12	15	−7	1	−2	3	26	7	5	37	−25	−4	14	37	11	−29	35	12	−13	10	70	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 2, 383, 766]: [0, 1, 33, 98] (certified at ell = 2147476031)
T = [(3, 1), (5, 1)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 4 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(766, 2)[5]   # 766.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 766:2 1000