Atlas › Modular forms › Weight 2 › Level 682 › 682.2.a › 682.2.a.i

Newform orbit 682.2.a.i

Trace form

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

$6q + 6q^{2} + q^{3} + 6q^{4} + 5q^{5} + q^{6} + q^{7} + 6q^{8} + 15q^{9} + 5q^{10} - 6q^{11} + q^{12} + 3q^{13} + q^{14} + 12q^{15} + 6q^{16} + q^{17} + 15q^{18} - 9q^{19} + 5q^{20} - 3q^{21} - 6q^{22} + 5q^{23} + q^{24} + O(q^{25})$

Coefficient field

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

$x^{6} - 51x^{5} - 7700x^{4} + 185787x^{3} + 10273516x^{2} - 199187170x + 873301916$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
11	$+1$
31	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	6	1	5	1	−6	3	1	−9	5	7	−6	14	14	−30	14	6	14	−3	16	2	−7	−14	−34	10	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.448 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, 11, 22, 31, 62, 341, 682]: [0, 1, 2, 5, 3, 10, 34, 100] (certified at ell = 2147482951)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (17, 9)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 7 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(682, 2)[8]   # 682.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 682:2 1000