Atlas › Modular forms › Weight 12 › Level 12 › 12.12.a › 12.12.a.b

Newform orbit 12.12.a.b

$q$ -expansion

$q + 243q^{3} + 2862q^{5} + 9128q^{7} + 59049q^{9} + 668196q^{11} + 2052950q^{13} + 695466q^{15} + 1604178q^{17} - 230500q^{19} + 2218104q^{21} - 43012728q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + T_{7}$ is a root of

$x - 11990$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 12$ )
3	$-1$

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
$\operatorname{tr} a_p$	0	243	2862	9128	668196	2052950	1604178	−230500	−43012728	−141745194	233221904	278269694	−1181577510	856975172	−1664054928	−3851181666	10339000596	185948102	2915010572	12662314200	−15201270694	−36644027488	−9217637028	30573828810	145701815906

$\operatorname{tr} a_n$ 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 12, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.338 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, 3, 4, 6, 12]: [2, 4, 5, 7, 13, 25] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 2 = 1 x 2 (phi(ord eps) x dim S_k^new); 2 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(12, 12)[1]   # 12.12.a.b

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

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