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

Newform orbit 3.10.a.b

$q$ -expansion

$q + 18q^{2} + 81q^{3} - 188q^{4} - 1530q^{5} + 1458q^{6} + 9128q^{7} - 12600q^{8} + 6561q^{9} - 27540q^{10} + 21132q^{11} - 15228q^{12} + 31214q^{13} + 164304q^{14} - 123930q^{15} - 130544q^{16} - 279342q^{17} + 118098q^{18} + 144020q^{19} + 287640q^{20} + 739368q^{21} + 380376q^{22} - 1763496q^{23} - 1020600q^{24} + O(q^{25})$

Coefficient field

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

$x + 1512$

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 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
$\operatorname{tr} a_p$	18	81	−1530	9128	21132	31214	−279342	144020	−1763496	4692510	−369088	9347078	−7226838	−23147476	22971888	78477174	−20310660	−179339938	274528388	−36342648	−247089526	191874800	−276159276	−678997350	−567657502

$\operatorname{tr} a_n$ 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)[1]   # 3.10.a.b

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

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