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

Newform orbit 25.12.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 72q^{4} - 1752q^{6} - 25452q^{9} - 326352q^{11} - 1080696q^{14} - 9834976q^{16} - 15460880q^{19} - 81019872q^{21} - 74415840q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} - 343948x^{2} + 12206704576$

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
$\operatorname{tr} a_p$	0	0	0	0	−326352	0	0	−15460880	0	−216242520	−684043072	0	1012873368	0	0	0	−3200971440	−2310471352	0	−60335466912	0	−74637768320	0	−118272499560	0

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

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)[4]   # 25.12.a.e

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

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