Atlas › Modular forms › Weight 2 › Level 856 › 856.2.a › 856.2.a.i

Newform orbit 856.2.a.i

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - q^{3} + 5q^{5} - 4q^{7} + 17q^{9} + 6q^{11} + 4q^{13} + q^{15} + 17q^{17} + 4q^{19} + 13q^{21} - 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{10} - 189x^{9} - 87x^{8} + 2097560x^{7} - 124345253x^{6} - 2997812037x^{5} + 493432809735x^{4} - 15305624411606x^{3} + 109006634830372x^{2} + 1988306051323104x - 25009124866828048$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 856$ )
107	$+1$

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	−1	5	−4	6	4	17	4	−4	21	6	5	36	−1	1	8	23	3	−28	9	34	−9	2	34	26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.53 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, 107, 214, 428, 856]: [0, 1, 2, 3, 10, 29, 57, 112] (certified at ell = 2147482621)
T = [(3, 1), (5, 3), (13, 5), (17, 7), (19, 9)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 7 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(856, 2)[8]   # 856.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 856:2 1000