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

Newform orbit 57.6.a.a

$q$ -expansion

$q - 2q^{2} + 9q^{3} - 28q^{4} - 98q^{5} - 18q^{6} + 240q^{7} + 120q^{8} + 81q^{9} + 196q^{10} + 336q^{11} - 252q^{12} + 342q^{13} - 480q^{14} - 882q^{15} + 656q^{16} - 6q^{17} - 162q^{18} - 361q^{19} + 2744q^{20} + 2160q^{21} - 672q^{22} + 2836q^{23} + 1080q^{24} + O(q^{25})$

Coefficient field

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

$x + 100$

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
$\operatorname{tr} a_p$	−2	9	−98	240	336	342	−6	−361	2836	−5902	2744	13670	10990	−4996	−17124	−4470	26292	29134	−42052	−26112	−49046	79056	9472	82894	39850

$\operatorname{tr} a_n$ 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)[0]   # 57.6.a.a

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

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