Atlas › Modular forms › Weight 2 › Level 472 › 472.2.a › 472.2.a.g

Newform orbit 472.2.a.g

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - q^{3} + 7q^{5} - 4q^{7} + 13q^{9} - 3q^{11} + 9q^{13} + 14q^{15} + 14q^{17} - 3q^{19} - 2q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{6} - 194x^{5} + 8963x^{4} + 65099x^{3} - 11378297x^{2} + 220947332x - 1277304748$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 472$ )
59	$+1$

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	−1	7	−4	−3	9	14	−3	2	11	−8	3	10	1	0	25	−6	−6	0	2	14	−36	19	22	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.359 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, 59, 118, 236, 472]: [0, 1, 2, 3, 6, 17, 33, 64] (certified at ell = 2147482481)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 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(472, 2)[6]   # 472.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 472:2 1000