Atlas › Modular forms › Weight 2 › Level 681 › 681.2.a › 681.2.a.f

Newform orbit 681.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{6} + 90x^{5} - 4662x^{4} - 608383x^{3} - 12147443x^{2} + 104458612x + 3015654707$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
227	$-1$

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
$\operatorname{tr} a_p$	−5	6	−10	−8	−13	4	−2	−15	−10	−7	−29	−7	−5	4	−32	−10	−20	14	2	−29	−6	−28	−21	−30	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 0.404 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, 227, 681]: [0, 1, 20, 78] (certified at ell = 2147479573)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 10 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(681, 2)[5]   # 681.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 681:2 1000