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

Newform orbit 77.6.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 4q^{2} - 12q^{3} + 70q^{4} - 62q^{5} - 172q^{6} - 294q^{7} + 168q^{8} + 252q^{9} - 1126q^{10} - 726q^{11} - 972q^{12} + 10q^{13} - 196q^{14} - 1430q^{15} - 1014q^{16} - 1606q^{17} - 1684q^{18} - 1460q^{19} - 5566q^{20} + 588q^{21} - 484q^{22} - 5054q^{23} - 12912q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 8x^{5} - 782x^{4} + 1790x^{3} + 74177x^{2} - 173142x - 602052$

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
$\operatorname{tr} a_p$	4	−12	−62	−294	−726	10	−1606	−1460	−5054	−1304	544	−11926	8106	−31608	−37390	25872	−8240	−17230	−126074	107102	−55202	−146332	−92280	−67798	−392046

$\operatorname{tr} a_n$ 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)[3]   # 77.6.a.d

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

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