Atlas › Modular forms › Weight 2 › Level 657 › 657.2.a › 657.2.a.i

Newform orbit 657.2.a.i

Trace form

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

$6q + q^{2} + 7q^{4} - 5q^{5} + 8q^{7} + 9q^{8} - 6q^{10} + 2q^{11} + 4q^{13} + 14q^{14} + 9q^{16} + 3q^{17} + 5q^{19} - 6q^{20} + 6q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{6} + 14x^{5} - 69x^{4} - 1306x^{3} + 125x^{2} + 20684x - 21961$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 657$ )
73	$+1$

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
$\operatorname{tr} a_p$	1	0	−5	8	2	4	3	5	6	4	4	13	6	−12	19	5	3	11	5	8	−6	5	5	−20	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.337 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, 73, 219, 657]: [0, 1, 2, 6, 26, 76] (certified at ell = 2147482873)
T = [(2, 1), (5, 3)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 5 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(657, 2)[8]   # 657.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 657:2 1000