Atlas › Modular forms › Weight 2 › Level 872 › 872.2.a › 872.2.a.f

Newform orbit 872.2.a.f

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{9} - 26x^{8} + 43x^{7} + 3879x^{6} - 30662x^{5} - 56057x^{4} + 1203174x^{3} - 2071872x^{2} - 10220000x + 27668608$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 872$ )
109	$-1$

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
$\operatorname{tr} a_p$	0	5	7	5	4	12	13	6	−3	15	−1	17	14	18	−14	20	−7	23	−1	−32	26	−5	−8	13	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 0.368 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, 2, 4, 8, 109, 218, 436, 872]: [0, 1, 2, 3, 9, 29, 58, 114] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 27 = 1 x 27 (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(872, 2)[5]   # 872.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 872:2 1000