Atlas › Modular forms › Weight 6 › Level 25 › 25.6.a › 25.6.a.a

Newform orbit 25.6.a.a

$q$ -expansion

$q - 2q^{2} + 4q^{3} - 28q^{4} - 8q^{6} - 192q^{7} + 120q^{8} - 227q^{9} - 148q^{11} - 112q^{12} - 286q^{13} + 384q^{14} + 656q^{16} + 1678q^{17} + 454q^{18} + 1060q^{19} - 768q^{21} + 296q^{22} - 2976q^{23} + 480q^{24} + O(q^{25})$

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
$\operatorname{tr} a_p$	−2	4	0	−192	−148	−286	1678	1060	−2976	−3410	−2448	−182	−9398	1244	12088	−23846	−20020	32302	−60972	−32648	38774	−33360	−16716	101370	119038

$\operatorname{tr} a_n$ 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 25, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.217 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, 13] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (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(25, 6)[0]   # 25.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 25:6 1000