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

Newform orbit 67.6.a.b

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 12q^{2} + 27q^{3} + 282q^{4} + 228q^{5} + 147q^{6} + 99q^{7} + 363q^{8} + 1730q^{9} + 534q^{10} + 346q^{11} + 1052q^{12} + 1453q^{13} + 851q^{14} + 1389q^{15} + 4650q^{16} + 6840q^{17} + 12836q^{18} + 3099q^{19} + 15070q^{20} + 9722q^{21} + 9292q^{22} + 4960q^{23} + 7102q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 39x^{14} - 2455x^{13} + 91451x^{12} + 2586183x^{11} - 84207393x^{10} - 1545307782x^{9} + 38652414207x^{8} + 565050235631x^{7} - 9221170524861x^{6} - 124335483896261x^{5} + 1046255872022023x^{4} + 14936043406349895x^{3} - 33860482772343282x^{2} - 743628283091088750x - 1560515808302001900$

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
$\operatorname{tr} a_p$	12	27	228	99	346	1453	6840	3099	4960	15406	−3106	5241	14541	−4015	33975	38940	50730	−14871	67335	95880	179174	292029	278429	179662	205628

$\operatorname{tr} a_n$ 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)[1]   # 67.6.a.b

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

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