Atlas › Modular forms › Weight 2 › Level 836 › 836.2.a › 836.2.a.d

Newform orbit 836.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 2q^{3} + 2q^{7} + 10q^{9} + 6q^{11} + 2q^{13} - 12q^{15} + 12q^{17} + 6q^{19} + 2q^{21} - 2q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{17} + 9T_{23} + 11T_{29}$ , whose minimal polynomial is

$x^{6} - 118x^{5} - 10501x^{4} + 1124924x^{3} + 44895719x^{2} - 2578090150x - 92100848579$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 836$ )
11	$-1$
19	$-1$

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	−2	0	2	6	2	12	6	−2	4	−20	22	−2	10	16	12	−14	12	−12	−30	24	0	−14	−14	−46

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.462 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, 11, 19, 22, 38, 44, 76, 209, 418, 836]: [0, 1, 2, 2, 2, 5, 7, 9, 13, 22, 64, 126] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (17, 7), (23, 9), (29, 11)]; h of degree 16 = 1 x 16 (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(836, 2)[3]   # 836.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 836:2 1000