Atlas › Modular forms › Weight 2 › Level 625 › 625.2.a › 625.2.a.a

Newform orbit 625.2.a.a

Trace form

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

$2q - q^{2} - 3q^{3} - q^{4} + 4q^{6} - q^{7} + q^{9} - q^{11} - q^{12} + 2q^{13} + 8q^{14} - 3q^{16} - 6q^{17} - 8q^{18} + 5q^{19} + 9q^{21} - 2q^{22} - 3q^{23} - 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 4x - 1$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 625$ )

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
$\operatorname{tr} a_p$	−1	−3	0	−1	−1	2	−6	5	−3	−10	4	−6	4	−8	−21	−8	−10	−11	14	9	7	0	−8	5	19

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.206 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 25, 125, 625]: [0, 1, 3, 13, 63] (certified at ell = 2147482501)
T = [(2, 1), (3, 1)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 4 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(625, 2)[0]   # 625.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 625:2 1000