Atlas › Modular forms › Weight 4 › Level 67 › 67.4.a › 67.4.a.a

Newform orbit 67.4.a.a

Trace form

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

$7q - 6q^{2} - 9q^{3} + 14q^{4} - 45q^{5} - 39q^{6} - 32q^{7} - 69q^{8} + 2q^{9} - 36q^{10} - 59q^{11} - 136q^{12} - 101q^{13} - 157q^{14} - 114q^{15} + 42q^{16} - 303q^{17} + 242q^{18} + 4q^{19} + 88q^{20} + 62q^{21} + 550q^{22} - 221q^{23} + 502q^{24} + O(q^{25})$

Coefficient field

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

$x^{7} + 15x^{6} + 21x^{5} - 403x^{4} - 1666x^{3} - 2425x^{2} - 1452x - 303$

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
$\operatorname{tr} a_p$	−6	−9	−45	−32	−59	−101	−303	4	−221	−260	384	−398	−612	−481	−747	−819	−948	−483	469	207	961	3082	−424	−233	−212

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 18] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 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, 4)[0]   # 67.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 67:4 1000