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

Newform orbit 767.2.a.e

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 3q^{2} + 5q^{3} + 15q^{4} + 4q^{5} + 6q^{6} + 9q^{7} + 9q^{8} + 24q^{9} + 6q^{10} + 7q^{11} + 15q^{12} - 17q^{13} + 10q^{14} + 11q^{15} + 7q^{16} + 9q^{17} + 5q^{18} + 8q^{19} + 12q^{20} + 2q^{21} - 14q^{22} + 8q^{23} + 23q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 8x^{16} - 36x^{15} + 399x^{14} + 208x^{13} - 6817x^{12} + 2145x^{11} + 58409x^{10} - 26020x^{9} - 283010x^{8} + 80825x^{7} + 789364x^{6} + 9914x^{5} - 1172712x^{4} - 424769x^{3} + 676118x^{2} + 506904x + 87206$

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 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
$\operatorname{tr} a_p$	3	5	4	9	7	−17	9	8	8	−12	23	35	10	7	17	−1	17	−29	44	−25	47	−20	40	13	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)[4]   # 767.2.a.e

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

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