Atlas › Modular forms › Weight 6 › Level 67 › 67.6.a › 67.6.a.a

Newform orbit 67.6.a.a

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 16q^{2} - 27q^{3} + 202q^{4} - 222q^{5} - 141q^{6} - 293q^{7} - 981q^{8} + 920q^{9} - 66q^{10} - 864q^{11} - 1540q^{12} - 2265q^{13} - 717q^{14} - 411q^{15} + 1834q^{16} - 3769q^{17} - 15478q^{18} - 5666q^{19} - 18448q^{20} - 12206q^{21} - 7804q^{22} - 7327q^{23} - 14434q^{24} + O(q^{25})$

Coefficient field

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

$x^{13} + 43x^{12} - 1283x^{11} - 51929x^{10} + 794183x^{9} + 21130939x^{8} - 262463638x^{7} - 3127147533x^{6} + 33994956891x^{5} + 197973067973x^{4} - 1714241302815x^{3} - 5259059192055x^{2} + 26073066653685x + 48504043340550$

Atkin–Lehner signs

$p$	$w_p$
67	$+1$

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
$\operatorname{tr} a_p$	−16	−27	−222	−293	−864	−2265	−3769	−5666	−7327	−7015	1702	−772	−16981	6979	−23066	−43462	4635	−13073	−58357	−119212	−159437	−203647	−291287	−238427	39900

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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 engine's checks

dim M^+ at levels [1, 67]: [1, 30] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (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(67, 6)[0]   # 67.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 67:6 1000