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

Newform orbit 77.6.a.a

$q$ -expansion

$q - 2q^{2} - 6q^{3} - 28q^{4} - 74q^{5} + 12q^{6} - 49q^{7} + 120q^{8} - 207q^{9} + 148q^{10} + 121q^{11} + 168q^{12} + 364q^{13} + 98q^{14} + 444q^{15} + 656q^{16} + 148q^{17} + 414q^{18} - 1320q^{19} + 2072q^{20} + 294q^{21} - 242q^{22} - 436q^{23} - 720q^{24} + O(q^{25})$

Coefficient field

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

$x + 8$

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
$\operatorname{tr} a_p$	−2	−6	−74	−49	121	364	148	−1320	−436	2970	8842	138	532	−20676	−11722	5274	−27670	19512	64088	−3708	−24296	−2200	74424	34170	151718

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.251 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)[0]   # 77.6.a.a

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

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