Atlas › Modular forms › Weight 12 › Level 25 › 25.12.a › 25.12.a.a

Newform orbit 25.12.a.a

$q$ -expansion

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

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	— ( $5^2 \mid 25$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	0	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 25, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.524 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, 25]: [2, 7, 29] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 6 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(25, 12)[0]   # 25.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 25:12 1000