Atlas › Modular forms › Weight 2 › Level 767 › 767.2.a › 767.2.a.d

Newform orbit 767.2.a.d

Trace form

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

$10q - 3q^{3} + 6q^{4} - 6q^{5} - 8q^{6} - 11q^{7} + 9q^{9} - 14q^{10} - q^{11} - 5q^{12} - 10q^{13} - 4q^{14} - 8q^{15} - 10q^{16} - 10q^{17} - q^{18} - 14q^{19} - 8q^{20} - 17q^{21} + 6q^{22} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 3x^{9} - 20x^{8} - 49x^{7} + 123x^{6} + 168x^{5} - 350x^{4} - 41x^{3} + 214x^{2} - 52x + 2$

Atkin–Lehner signs

$p$	$w_p$
13	$+1$
59	$+1$

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
$\operatorname{tr} a_p$	0	−3	−6	−11	−1	−10	−10	−14	0	−14	−5	−21	−2	9	−13	−3	−10	5	−20	18	−53	4	24	−25	−79

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 13, 59, 767]: [0, 1, 6, 72] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 59 = 1 x 59 (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(767, 2)[3]   # 767.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 767:2 1000