Atlas › Modular forms › Weight 2 › Level 925 › 925.2.a › 925.2.a.k

Newform orbit 925.2.a.k

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + q^{2} - q^{3} + 7q^{4} + 6q^{6} + 10q^{9} + 16q^{11} + 11q^{12} - q^{13} - 3q^{14} + 3q^{16} + 4q^{17} - 23q^{18} + 9q^{19} + 2q^{21} + q^{22} - q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{7} - 110x^{6} + 3876x^{5} - 25428x^{4} - 1312123x^{3} + 28868671x^{2} - 202488108x + 393056163$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 925$ )
37	$-1$

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
$\operatorname{tr} a_p$	1	−1	0	0	16	−1	4	9	−1	3	11	7	29	3	−2	−6	42	−11	3	10	−14	−9	−6	12	−1

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.457 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 25, 37, 185, 925]: [0, 1, 3, 3, 20, 96] (certified at ell = 2147482801)
T = [(2, 1), (3, 3), (7, 5), (11, 7)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 13 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(925, 2)[10]   # 925.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 925:2 1000