Atlas › Modular forms › Weight 12 › Level 10 › 10.12.a › 10.12.a.a

Newform orbit 10.12.a.a

$q$ -expansion

$q - 32q^{2} - 12q^{3} + 1024q^{4} + 3125q^{5} + 384q^{6} - 14176q^{7} - 32768q^{8} - 177003q^{9} - 100000q^{10} - 756348q^{11} - 12288q^{12} - 905482q^{13} + 453632q^{14} - 37500q^{15} + 1048576q^{16} + 2803794q^{17} + 5664096q^{18} - 5428660q^{19} + 3200000q^{20} + 170112q^{21} + 24203136q^{22} - 10236672q^{23} + 393216q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{7}$ is a root of

$x + 14188$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
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
$\operatorname{tr} a_p$	−32	−12	3125	−14176	−756348	−905482	2803794	−5428660	−10236672	−197498010	−44362288	576737054	930058362	1605598988	−1803684456	1558674798	−9501997020	6736320422	8402906564	−4806306168	7462713338	−20644540720	−68013349212	69871323210	39960952514

$\operatorname{tr} a_n$ 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 10, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.332 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, 2, 5, 10]: [2, 4, 7, 19] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (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(10, 12)[0]   # 10.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 10:12 1000