Atlas › Modular forms › Weight 6 › Level 57 › 57.6.a › 57.6.a.b

Newform orbit 57.6.a.b

$q$ -expansion

$q + 11q^{2} + 9q^{3} + 89q^{4} + 6q^{5} + 99q^{6} - 176q^{7} + 627q^{8} + 81q^{9} + 66q^{10} - 496q^{11} + 801q^{12} - 178q^{13} - 1936q^{14} + 54q^{15} + 4049q^{16} + 202q^{17} + 891q^{18} - 361q^{19} + 534q^{20} - 1584q^{21} - 5456q^{22} + 4396q^{23} + 5643q^{24} + O(q^{25})$

Coefficient field

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

$x - 17$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
19	$+1$

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
$\operatorname{tr} a_p$	11	9	6	−176	−496	−178	202	−361	4396	−5902	5760	−3906	15774	−7492	−7452	−29014	13604	−12466	43436	28800	80746	76456	−56880	−103266	82490

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ for the primes $p < 1000$ not dividing 57, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 19, 57]: [1, 3, 10, 36] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (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(57, 6)[1]   # 57.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 57:6 1000