Atlas › Modular forms › Weight 6 › Level 66 › 66.6.a › 66.6.a.e

Newform orbit 66.6.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 8q^{2} + 18q^{3} + 32q^{4} - 14q^{5} - 72q^{6} + 210q^{7} - 128q^{8} + 162q^{9} + 56q^{10} - 242q^{11} + 288q^{12} + 246q^{13} - 840q^{14} - 126q^{15} + 512q^{16} + 1424q^{17} - 648q^{18} + 3308q^{19} - 224q^{20} + 1890q^{21} + 968q^{22} + 4890q^{23} - 1152q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{8761})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{2} - 616x + 59820$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$-1$
11	$+1$

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	18	−14	210	−242	246	1424	3308	4890	3152	0	13356	−21572	−10368	2534	−31414	−132	−24654	46472	12994	46724	−106770	−109792	88340	−66096

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.279 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, 2, 3, 6, 11, 22, 33, 66]: [1, 2, 3, 7, 6, 17, 22, 64] (certified at ell = 2147483171)
T = [(5, 1), (7, 3)]; h of degree 8 = 1 x 8 (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(66, 6)[4]   # 66.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 66:6 1000