Atlas › Modular forms › Weight 2 › Level 476 › 476.2.a › 476.2.a.a

Newform orbit 476.2.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 3q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + 2q^{11} - 6q^{13} - 2q^{15} - 2q^{17} - 12q^{19} - 3q^{21} + 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 12x + 23$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 476$ )
7	$-1$
17	$+1$

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	−3	2	2	−6	−2	−12	2	−10	−17	0	1	−5	20	5	−4	−5	9	−4	−9	−12	4	−4	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.251 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, 7, 14, 17, 28, 34, 68, 119, 238, 476]: [0, 1, 2, 1, 4, 2, 7, 6, 12, 14, 40, 78] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(476, 2)[0]   # 476.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 476:2 1000