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

Newform orbit 66.6.a.b

$q$ -expansion

$q - 4q^{2} - 9q^{3} + 16q^{4} - 14q^{5} + 36q^{6} + 130q^{7} - 64q^{8} + 81q^{9} + 56q^{10} - 121q^{11} - 144q^{12} - 122q^{13} - 520q^{14} + 126q^{15} + 256q^{16} - 1108q^{17} - 324q^{18} - 1314q^{19} - 224q^{20} - 1170q^{21} + 484q^{22} - 3000q^{23} + 576q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + 3T_{7}$ is a root of

$x - 376$

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 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
$\operatorname{tr} a_p$	−4	−9	−14	130	−121	−122	−1108	−1314	−3000	−4432	880	11818	−5648	778	−10672	9086	−12012	39826	−56316	−51920	−10266	−79646	30224	−75310	−43778

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 66, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[1]   # 66.6.a.b

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

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