Atlas › Modular forms › Weight 10 › Level 30 › 30.10.a › 30.10.a.e

Newform orbit 30.10.a.e

$q$ -expansion

$q + 16q^{2} - 81q^{3} + 256q^{4} + 625q^{5} - 1296q^{6} - 10336q^{7} + 4096q^{8} + 6561q^{9} + 10000q^{10} + 27420q^{11} - 20736q^{12} - 169762q^{13} - 165376q^{14} - 50625q^{15} + 65536q^{16} - 385086q^{17} + 104976q^{18} - 637084q^{19} + 160000q^{20} + 837216q^{21} + 438720q^{22} - 1298400q^{23} - 331776q^{24} + O(q^{25})$

Coefficient field

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

$x - 17084$

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
$\operatorname{tr} a_p$	16	−81	625	−10336	27420	−169762	−385086	−637084	−1298400	7162974	−7031872	1926038	8896074	32429444	17206440	−20642154	−63193380	−63758050	145261964	−367656840	252486218	−185523712	−467897652	579096378	−1314516862

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

Sato–Tate

The normalized $a_p / 2p^{9/2}$ 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 0.362 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, 4, 5, 11, 15, 20, 58] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 4 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, 10)[4]   # 30.10.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 30:10 1000