Atlas › Modular forms › Weight 12 › Level 5 › 5.12.a › 5.12.a.a

Newform orbit 5.12.a.a

$q$ -expansion

$q + 34q^{2} - 792q^{3} - 892q^{4} + 3125q^{5} - 26928q^{6} - 17556q^{7} - 99960q^{8} + 450117q^{9} + 106250q^{10} - 468788q^{11} + 706464q^{12} - 374042q^{13} - 596904q^{14} - 2475000q^{15} - 1571824q^{16} - 3724286q^{17} + 15303978q^{18} - 379460q^{19} - 2787500q^{20} + 13904352q^{21} - 15938792q^{22} - 32458092q^{23} + 79168320q^{24} + O(q^{25})$

Coefficient field

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

$x + 758$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$

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
$\operatorname{tr} a_p$	34	−792	3125	−17556	−468788	−374042	−3724286	−379460	−32458092	69696710	171448632	−291340546	191343242	−1759857392	1623469924	−644888642	925569220	−10898589338	3795674064	−22966943728	9880820458	−20768886240	3204862008	63176321130	126494473874

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

Sato–Tate

The normalized $a_p / 2p^{11/2}$ for the primes $p < 1000$ not dividing 5, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 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, 5]: [2, 7] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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(5, 12)[0]   # 5.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 5:12 1000