Atlas › Modular forms › Weight 2 › Level 875 › 875.2.a › 875.2.a.i

Newform orbit 875.2.a.i

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - q^{2} + 8q^{3} + 13q^{4} + 2q^{6} + 8q^{7} - 12q^{8} + 18q^{9} - 5q^{11} + 20q^{12} + 6q^{13} - q^{14} + 35q^{16} - 13q^{17} - 3q^{18} + 13q^{19} + 8q^{21} - 22q^{22} + 5q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 7x^{7} - 13x^{6} + 155x^{5} - 104x^{4} - 625x^{3} + 543x^{2} + 274x - 199$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 875$ )
7	$-1$

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
$\operatorname{tr} a_p$	−1	8	0	8	−5	6	−13	13	5	22	3	−7	12	1	2	−10	−11	45	−6	−15	32	12	15	12	−20

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 7, 25, 35, 125, 175, 875]: [0, 1, 1, 3, 6, 13, 22, 102] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 48 = 1 x 48 (phi(ord eps) x dim S_k^new); 5 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(875, 2)[8]   # 875.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 875:2 1000