Atlas › Modular forms › Weight 4 › Level 5 › 5.4.a › 5.4.a.a

Newform orbit 5.4.a.a

$q$ -expansion

$q - 4q^{2} + 2q^{3} + 8q^{4} - 5q^{5} - 8q^{6} + 6q^{7} - 23q^{9} + 20q^{10} + 32q^{11} + 16q^{12} - 38q^{13} - 24q^{14} - 10q^{15} - 64q^{16} + 26q^{17} + 92q^{18} + 100q^{19} - 40q^{20} + 12q^{21} - 128q^{22} - 78q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x + 2$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$

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
$\operatorname{tr} a_p$	−4	2	−5	6	32	−38	26	100	−78	−50	−108	266	22	442	−514	2	500	−518	126	412	−878	600	282	−150	386

$\operatorname{tr} a_n$ 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 5, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.108 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, 5]: [1, 3] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 1 = 1 x 1 (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(5, 4)[0]   # 5.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 5:4 1000