Atlas › Modular forms › Weight 6 › Level 6 › 6.6.a › 6.6.a.a

Newform orbit 6.6.a.a

$q$ -expansion

$q + 4q^{2} - 9q^{3} + 16q^{4} - 66q^{5} - 36q^{6} + 176q^{7} + 64q^{8} + 81q^{9} - 264q^{10} - 60q^{11} - 144q^{12} - 658q^{13} + 704q^{14} + 594q^{15} + 256q^{16} - 414q^{17} + 324q^{18} + 956q^{19} - 1056q^{20} - 1584q^{21} - 240q^{22} + 600q^{23} - 576q^{24} + O(q^{25})$

Coefficient field

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

$x - 110$

Atkin–Lehner signs

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

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	−66	176	−60	−658	−414	956	600	5574	−3592	−8458	19194	13316	−19680	−31266	26340	−31090	−16804	6120	−25558	74408	−6468	−32742	166082

$\operatorname{tr} a_n$ 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 6, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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]: [1, 2, 3, 7] (certified at ell = 2147483647)
T = [(5, 1), (7, 1)]; h of degree 1 = 1 x 1 (phi(ord eps) x dim S_k^new); 1 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(6, 6)[0]   # 6.6.a.a

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

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