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

Newform orbit 30.10.a.a

$q$ -expansion

$q - 16q^{2} - 81q^{3} + 256q^{4} - 625q^{5} + 1296q^{6} + 6332q^{7} - 4096q^{8} + 6561q^{9} + 10000q^{10} + 7752q^{11} - 20736q^{12} + 72626q^{13} - 101312q^{14} + 50625q^{15} + 65536q^{16} - 334698q^{17} - 104976q^{18} - 934660q^{19} - 160000q^{20} - 512892q^{21} - 124032q^{22} - 991704q^{23} + 331776q^{24} + O(q^{25})$

Coefficient field

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

$x - 14084$

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	6332	7752	72626	−334698	−934660	−991704	−3638790	−6063688	12489842	−5035398	30163316	−743928	−102388134	−49464840	−130545898	102905012	−190423008	−367621054	175880360	−100482444	−660904110	1321991522

$\operatorname{tr} a_n$ 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)[0]   # 30.10.a.a

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

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