Atlas › Modular forms › Weight 2 › Level 477 › 477.2.a › 477.2.a.f

Newform orbit 477.2.a.f

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 10q^{4} + 4q^{7} + 8q^{10} - 2q^{11} + 8q^{13} - q^{14} + 12q^{16} + 2q^{19} + 7q^{20} - 18q^{22} + 6q^{23} + O(q^{25})$

Coefficient field

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

$x^{5} - 37x^{3} - 8x^{2} + 195x + 99$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 477$ )
53	$+1$

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	0	0	4	−2	8	0	2	6	−20	8	4	−18	12	4	−5	18	12	6	−4	8	−2	36	−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 0.295 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, 3, 9, 53, 159, 477]: [0, 1, 2, 5, 20, 56] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 21 = 1 x 21 (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(477, 2)[5]   # 477.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 477:2 1000