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

Newform orbit 67.4.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 8q^{2} + 9q^{3} + 34q^{4} + 45q^{5} + 9q^{6} + 24q^{7} + 99q^{8} + 92q^{9} + 24q^{10} + 51q^{11} + 80q^{12} + 185q^{13} - 45q^{14} + 6q^{15} + 218q^{16} + 191q^{17} - 262q^{18} - 46q^{19} - 226q^{20} - 170q^{21} - 354q^{22} - 115q^{23} - 298q^{24} + O(q^{25})$

Coefficient field

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

$x^{9} - 17x^{8} - 85x^{7} + 2825x^{6} - 8544x^{5} - 99181x^{4} + 712884x^{3} - 1560903x^{2} + 1340766x - 383670$

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 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
$\operatorname{tr} a_p$	8	9	45	24	51	185	191	−46	−115	326	56	−20	500	−263	967	1919	702	1179	−603	179	971	−2534	1432	1549	−1888

$\operatorname{tr} a_n$ 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)[1]   # 67.4.a.b

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

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