Atlas › Modular forms › Weight 4 › Level 25 › 25.4.a › 25.4.a.c

Newform orbit 25.4.a.c

$q$ -expansion

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

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	— ( $5^2 \mid 25$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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	0	−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 25, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.097 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, 25]: [1, 3, 9] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (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(25, 4)[2]   # 25.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 25:4 1000