Atlas › Modular forms › Weight 4 › Level 30 › 30.4.a › 30.4.a.b

Newform orbit 30.4.a.b

$q$ -expansion

$q + 2q^{2} + 3q^{3} + 4q^{4} - 5q^{5} + 6q^{6} - 4q^{7} + 8q^{8} + 9q^{9} - 10q^{10} - 48q^{11} + 12q^{12} + 2q^{13} - 8q^{14} - 15q^{15} + 16q^{16} - 114q^{17} + 18q^{18} + 140q^{19} - 20q^{20} - 12q^{21} - 96q^{22} + 72q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

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

$x + 52$

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
$\operatorname{tr} a_p$	2	3	−5	−4	−48	2	−114	140	72	210	272	−334	−198	−268	216	−78	240	302	596	−768	−478	−640	−348	210	−1534

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ 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 0.109 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, 2, 3, 5, 7, 8, 22] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; 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, 4)[1]   # 30.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 30:4 1000