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

Newform orbit 682.2.a.h

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{4} + 121x^{3} - 442x^{2} - 227539x + 3512596$

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 odd 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
$\operatorname{tr} a_p$	−4	−3	−1	−6	4	−7	−4	−1	−10	−9	4	−5	0	−11	−4	−14	−5	7	−21	−2	−15	−8	−35	−24	−4

$\operatorname{tr} a_n$ 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)[7]   # 682.2.a.h

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

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