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

Newform orbit 66.6.a.a

$q$ -expansion

$q - 4q^{2} - 9q^{3} + 16q^{4} - 14q^{5} + 36q^{6} - 112q^{7} - 64q^{8} + 81q^{9} + 56q^{10} + 121q^{11} - 144q^{12} - 606q^{13} + 448q^{14} + 126q^{15} + 256q^{16} + 2038q^{17} - 324q^{18} + 2800q^{19} - 224q^{20} + 1008q^{21} - 484q^{22} + 2324q^{23} + 576q^{24} + O(q^{25})$

Coefficient field

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

$x + 350$

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
$\operatorname{tr} a_p$	−4	−9	−14	−112	121	−606	2038	2800	2324	5490	4752	−282	−6858	−13016	15948	2794	11220	22402	−48572	46332	26034	100160	81044	−118870	−10382

$\operatorname{tr} a_n$ 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)[0]   # 66.6.a.a

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

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