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

Newform orbit 77.6.a.e

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 4q^{2} + 42q^{3} + 166q^{4} + 50q^{5} + 238q^{6} + 392q^{7} + 168q^{8} + 538q^{9} - 160q^{10} - 968q^{11} + 1582q^{12} + 1580q^{13} + 196q^{14} + 3382q^{15} + 3066q^{16} - 1304q^{17} + 1328q^{18} + 2116q^{19} + 4760q^{20} + 2058q^{21} - 484q^{22} + 2774q^{23} + 15530q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 46x^{7} - 632x^{6} + 47440x^{5} - 126953x^{4} - 13089262x^{3} + 98203060x^{2} + 444736128x - 1190863872$

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
$\operatorname{tr} a_p$	4	42	50	392	−968	1580	−1304	2116	2774	−2264	19026	19242	−18672	15576	−33336	29304	49238	25216	54050	−111806	−140280	109244	−51368	427670	−74262

$\operatorname{tr} a_n$ 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)[4]   # 77.6.a.e

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

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