Atlas › Modular forms › Weight 2 › Level 837 › 837.2.a › 837.2.a.k

Newform orbit 837.2.a.k

Trace form

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

$5q + 3q^{2} + 5q^{4} + 5q^{5} + 12q^{8} + 8q^{10} + 4q^{11} - 7q^{13} + 5q^{14} + 9q^{16} + 8q^{17} - q^{19} + 12q^{20} - 19q^{22} + 4q^{23} + O(q^{25})$

Coefficient field

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

$x^{5} - 18x^{4} + 1141x^{2} - 5707x + 8161$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 837$ )
31	$+1$

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
$\operatorname{tr} a_p$	3	0	5	0	4	−7	8	−1	4	2	−5	−1	11	2	20	−1	17	−13	15	14	−7	−23	26	19	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 9, 27, 31, 93, 279, 837]: [0, 1, 2, 4, 3, 12, 34, 98] (certified at ell = 2147482801)
T = [(2, 1), (5, 3)]; h of degree 40 = 1 x 40 (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(837, 2)[10]   # 837.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 837:2 1000