Atlas › Modular forms › Weight 10 › Level 3 › 3.10.a › 3.10.a.a

Newform orbit 3.10.a.a

$q$ -expansion

$q - 36q^{2} - 81q^{3} + 784q^{4} - 1314q^{5} + 2916q^{6} - 4480q^{7} - 9792q^{8} + 6561q^{9} + 47304q^{10} + 1476q^{11} - 63504q^{12} - 151522q^{13} + 161280q^{14} + 106434q^{15} - 48896q^{16} + 108162q^{17} - 236196q^{18} + 593084q^{19} - 1030176q^{20} + 362880q^{21} - 53136q^{22} - 969480q^{23} + 793152q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 1350$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$

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
$\operatorname{tr} a_p$	−36	−81	−1314	−4480	1476	−151522	108162	593084	−969480	−6642522	7070600	−7472410	−4350150	−4358716	28309248	16111710	−86075964	32213918	99531452	−44170488	−23560630	−401754760	−744528708	769871034	907130882

$\operatorname{tr} a_n$ 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 3, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 3]: [1, 4] (certified at ell = 2147483647)
T = [(2, 1), (5, 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(3, 10)[0]   # 3.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 3:10 1000