Atlas › Modular forms › Weight 2 › Level 904 › 904.2.a › 904.2.a.e

Newform orbit 904.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 3q^{3} + 9q^{5} + 2q^{7} + 14q^{9} + 4q^{11} + 8q^{13} - q^{15} + 10q^{17} - 8q^{19} + 11q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 6x^{8} - 35x^{7} + 221x^{6} + 198x^{5} - 2414x^{4} + 1884x^{3} + 7452x^{2} - 14024x + 6824$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 904$ )
113	$-1$

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	−3	9	2	4	8	10	−8	2	17	9	20	25	−18	19	27	13	23	−5	10	23	−3	0	26	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 2, 4, 8, 113, 226, 452, 904]: [0, 1, 2, 3, 10, 30, 60, 118] (certified at ell = 2147480161)
T = [(3, 1), (5, 1)]; h of degree 28 = 1 x 28 (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(904, 2)[4]   # 904.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 904:2 1000