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

Newform orbit 77.6.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 8q^{2} - 43q^{3} - 10q^{4} + 57q^{5} + 166q^{6} + 196q^{7} + 66q^{8} + 41q^{9} + 130q^{10} + 484q^{11} - 34q^{12} - 888q^{13} - 392q^{14} - 979q^{15} - 2126q^{16} - 2434q^{17} - 3068q^{18} - 1814q^{19} - 4246q^{20} - 2107q^{21} - 968q^{22} - 6115q^{23} - 642q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 51x^{3} + 569x^{2} - 975x - 4454$

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 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
$\operatorname{tr} a_p$	−8	−43	57	196	484	−888	−2434	−1814	−6115	−3832	1109	−20827	10714	−20572	17446	−18880	16831	31206	−35077	−113993	−85278	−171734	−64384	−150625	−83779

$\operatorname{tr} a_n$ 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)[1]   # 77.6.a.b

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

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