Atlas › Modular forms › Weight 4 › Level 77 › 77.4.a › 77.4.a.e

Newform orbit 77.4.a.e

Trace form

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

$5q + q^{2} + 2q^{3} + 45q^{4} - 24q^{5} + 4q^{6} + 35q^{7} + 57q^{8} + 63q^{9} - 10q^{10} + 55q^{11} + 24q^{12} - 50q^{13} + 7q^{14} - 146q^{15} + 433q^{16} + 222q^{17} + 245q^{18} + 160q^{19} - 430q^{20} + 14q^{21} + 11q^{22} + 54q^{23} - 220q^{24} + O(q^{25})$

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

$x^{5} - 3x^{4} - 141x^{3} + 23x^{2} + 2634x - 1002$

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
$\operatorname{tr} a_p$	1	2	−24	35	55	−50	222	160	54	14	−34	1044	−114	672	−292	−710	270	138	1942	−278	−338	576	1644	−3656	692

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 3, 4, 26] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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, 4)[4]   # 77.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 77:4 1000