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

Newform orbit 25.8.a.c

Trace form

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

$2q - 15q^{2} - 40q^{3} + 181q^{4} + 949q^{6} + 600q^{7} - 4305q^{8} - 2276q^{9} + 4344q^{11} - 13355q^{12} - 17680q^{13} - 31758q^{14} + 33457q^{16} - 6870q^{17} - 8890q^{18} + 18200q^{19} - 66516q^{21} + 48545q^{22} - 21120q^{23} + 134775q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{649})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 55x - 704$

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
$\operatorname{tr} a_p$	−15	−40	0	600	4344	−17680	−6870	18200	−21120	55800	−301776	−609860	−108486	−966400	−1787880	−130740	2067600	582044	−255720	−4728216	1339430	−7186200	−12049560	−5990850	17120020

$\operatorname{tr} a_n$ 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)[2]   # 25.8.a.c

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

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