Atlas › Modular forms › Weight 2 › Level 725 › 725.2.a › 725.2.a.k

Newform orbit 725.2.a.k

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 2q^{2} + 6q^{3} + 4q^{4} - q^{6} + 6q^{7} + 3q^{8} + 11q^{9} - 2q^{11} + 6q^{12} + 4q^{13} + 11q^{14} - 10q^{16} + 9q^{17} + 2q^{19} - q^{21} + 4q^{22} + q^{23} + 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{5} - 20x^{4} + 79x^{3} + 408x^{2} - 2296x + 2285$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 725$ )
29	$-1$

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	6	0	6	−2	4	9	2	1	5	−1	14	5	28	15	−8	−11	−5	23	−5	16	−10	9	−18	23

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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, 29, 145, 725]: [0, 1, 3, 3, 16, 76] (certified at ell = 2147482681)
T = [(2, 1), (3, 3)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 6 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(725, 2)[10]   # 725.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 725:2 1000