Atlas › Modular forms › Weight 8 › Level 25 › 25.8.a › 25.8.a.b

Newform orbit 25.8.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 20q^{2} - 20q^{3} + 96q^{4} - 1016q^{6} + 100q^{7} - 1440q^{8} + 5554q^{9} + 4544q^{11} + 23360q^{12} - 3540q^{13} + 7512q^{14} + 20352q^{16} + 27340q^{17} - 31220q^{18} + 38760q^{19} - 69096q^{21} - 106240q^{22} + 124140q^{23} - 131520q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{19})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 40x - 3324$

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
$\operatorname{tr} a_p$	−20	−20	0	100	4544	−3540	27340	38760	124140	−72260	306824	123020	264364	−423300	105460	2391580	−1120120	2257044	−4516460	621784	−4569060	4333040	9793020	6025620	−4609540

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.259 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, 5, 19] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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, 8)[1]   # 25.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 25:8 1000