Atlas › Modular forms › Weight 4 › Level 51 › 51.4.a › 51.4.a.a

Newform orbit 51.4.a.a

$q$ -expansion

$q - q^{2} - 3q^{3} - 7q^{4} + 16q^{5} + 3q^{6} + 34q^{7} + 15q^{8} + 9q^{9} - 16q^{10} - 48q^{11} + 21q^{12} + 58q^{13} - 34q^{14} - 48q^{15} + 41q^{16} - 17q^{17} - 9q^{18} + 20q^{19} - 112q^{20} - 102q^{21} + 48q^{22} + 58q^{23} - 45q^{24} + O(q^{25})$

Coefficient field

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

$x - 15$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
17	$+1$

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
$\operatorname{tr} a_p$	−1	−3	16	34	−48	58	−17	20	58	0	−218	184	−138	148	−516	−162	−180	152	−956	−538	−462	390	1268	−770	494

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 51, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 17, 51]: [1, 2, 6, 20] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(51, 4)[0]   # 51.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 51:4 1000