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

Newform orbit 77.4.a.a

$q$ -expansion

$q + 3q^{2} + 4q^{3} + q^{4} + 12q^{5} + 12q^{6} + 7q^{7} - 21q^{8} - 11q^{9} + 36q^{10} + 11q^{11} + 4q^{12} + 38q^{13} + 21q^{14} + 48q^{15} - 71q^{16} - 48q^{17} - 33q^{18} - 70q^{19} + 12q^{20} + 28q^{21} + 33q^{22} + 12q^{23} - 84q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x - 7$

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
$\operatorname{tr} a_p$	3	4	12	7	11	38	−48	−70	12	126	−70	−358	−216	344	390	438	−552	830	−196	648	−16	1352	90	1146	−70

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 77, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 77.4.a.a

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

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