Atlas › Modular forms › Weight 2 › Level 711 › 711.2.a › 711.2.a.g

Newform orbit 711.2.a.g

Trace form

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

$7q - 2q^{2} + 12q^{4} + 2q^{5} + 4q^{7} - 4q^{10} - 2q^{11} + 6q^{13} + 16q^{14} + 10q^{16} + 8q^{17} + 4q^{19} + 25q^{20} - 5q^{22} - 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} - 36x^{5} - 50x^{4} + 104x^{3} - 41x^{2} + x + 1$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 711$ )
79	$+1$

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
$\operatorname{tr} a_p$	−2	0	2	4	−2	6	8	4	−8	10	4	10	20	22	10	0	2	4	20	20	2	−7	−10	20	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.429 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, 79, 237, 711]: [0, 1, 2, 7, 28, 82] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 33 = 1 x 33 (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(711, 2)[6]   # 711.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 711:2 1000