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

Newform orbit 77.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 2q^{2} - 4q^{3} + 2q^{4} - 4q^{5} + 4q^{6} + 14q^{7} - 18q^{8} - 46q^{9} - 44q^{10} - 22q^{11} - 4q^{12} - 80q^{13} - 14q^{14} + 8q^{15} - 78q^{16} + 48q^{17} + 46q^{18} - 96q^{19} + 92q^{20} - 28q^{21} + 22q^{22} - 56q^{23} + 36q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 6x + 1$

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
$\operatorname{tr} a_p$	−2	−4	−4	14	−22	−80	48	−96	−56	4	60	−380	112	−632	−604	676	−516	−296	−1152	−1064	168	−368	1728	1524	1404

$\operatorname{tr} a_n$ 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)[1]   # 77.4.a.b

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

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