Atlas › Modular forms › Weight 2 › Level 375 › 375.2.a › 375.2.a.e

Newform orbit 375.2.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7}$ , whose minimal polynomial is

$x^{4} - 9x^{3} - 177x^{2} + 907x + 9479$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
5	— ( $5^2 \mid 375$ )

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
$\operatorname{tr} a_p$	−3	−4	0	4	6	8	−2	8	−8	6	4	14	24	10	0	2	2	10	8	−12	2	−20	−24	28	18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.322 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, 3, 5, 15, 25, 75, 125, 375]: [0, 1, 1, 4, 3, 12, 13, 52] (certified at ell = 2147482801)
T = [(2, 1), (7, 3)]; h of degree 16 = 1 x 16 (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(375, 2)[4]   # 375.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 375:2 1000