Atlas › Modular forms › Weight 2 › Level 404 › 404.2.a › 404.2.a.c

Newform orbit 404.2.a.c

Trace form

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

$7q + 2q^{3} + 2q^{7} + 17q^{9} + 4q^{11} + 4q^{13} + 18q^{15} + 4q^{17} - 4q^{19} + 24q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

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

$x^{7} - 2x^{6} - 66x^{5} + 206x^{4} + 729x^{3} - 2856x^{2} + 2253x + 24$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 404$ )
101	$+1$

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	2	0	2	4	4	4	−4	8	6	4	32	−14	−24	−4	6	−30	10	−2	0	−14	24	−24	−6	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 101, 202, 404]: [0, 1, 2, 9, 27, 54] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 1 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(404, 2)[2]   # 404.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 404:2 1000