Atlas › Modular forms › Weight 6 › Level 30 › 30.6.a › 30.6.a.b

Newform orbit 30.6.a.b

$q$ -expansion

$q + 4q^{2} + 9q^{3} + 16q^{4} + 25q^{5} + 36q^{6} + 32q^{7} + 64q^{8} + 81q^{9} + 100q^{10} + 12q^{11} + 144q^{12} - 154q^{13} + 128q^{14} + 225q^{15} + 256q^{16} - 918q^{17} + 324q^{18} - 1060q^{19} + 400q^{20} + 288q^{21} + 48q^{22} - 4224q^{23} + 576q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{7} + 3T_{11}$ is a root of

$x - 68$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
5	$-1$

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	25	32	12	−154	−918	−1060	−4224	−7890	5192	16382	3642	15116	23592	−16074	−14340	−47938	33092	51912	12026	25160	35796	−75510	−44158

$\operatorname{tr} a_n$ 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 30, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.123 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, 5, 6, 10, 15, 30]: [1, 2, 3, 3, 7, 9, 12, 34] (certified at ell = 2147483629)
T = [(7, 1), (11, 3)]; h of degree 2 = 1 x 2 (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(30, 6)[1]   # 30.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 30:6 1000