Atlas › Modular forms › Weight 8 › Level 30 › 30.8.a › 30.8.a.d

Newform orbit 30.8.a.d

$q$ -expansion

$q + 8q^{2} - 27q^{3} + 64q^{4} - 125q^{5} - 216q^{6} - 988q^{7} + 512q^{8} + 729q^{9} - 1000q^{10} - 8040q^{11} - 1728q^{12} - 3334q^{13} - 7904q^{14} + 3375q^{15} + 4096q^{16} + 6582q^{17} + 5832q^{18} - 27436q^{19} - 8000q^{20} + 26676q^{21} - 64320q^{22} + 48600q^{23} - 13824q^{24} + O(q^{25})$

Coefficient field

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

$x + 9028$

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 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
$\operatorname{tr} a_p$	8	−27	−125	−988	−8040	−3334	6582	−27436	48600	−132414	254408	519434	92394	−234532	−1277640	−835278	−3068760	−1009330	3082172	−3666720	1122866	−4128808	4586556	−5763678	6747554

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.299 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, 3, 3, 5, 9, 13, 16, 46] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; h of degree 6 = 1 x 6 (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(30, 8)[3]   # 30.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 30:8 1000