Atlas › Modular forms › Weight 6 › Level 77 › 77.6.a › 77.6.a.c

Newform orbit 77.6.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 10q^{2} + 17q^{3} + 114q^{4} + 119q^{5} - 112q^{6} - 245q^{7} + 714q^{8} + 934q^{9} + 616q^{10} + 605q^{11} + 700q^{12} + 558q^{13} - 490q^{14} - 2635q^{15} + 306q^{16} + 584q^{17} - 14q^{18} + 7482q^{19} + 8664q^{20} - 833q^{21} + 1210q^{22} + 4117q^{23} + 9300q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 27x^{4} - 735x^{3} + 14031x^{2} + 151998x - 56628$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
11	$-1$

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	17	119	−245	605	558	584	7482	4117	1454	−6403	−18317	12468	51568	34626	−72330	−16765	2380	34799	−27961	126260	2066	148080	191301	183723

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.251 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 engine's checks

dim M^+ at levels [1, 7, 11, 77]: [1, 5, 6, 42] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 5 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(77, 6)[2]   # 77.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 77:6 1000