Atlas › Modular forms › Weight 2 › Level 812 › 812.2.a › 812.2.a.e

Newform orbit 812.2.a.e

Trace form

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

$6q + 3q^{3} - q^{5} + 6q^{7} + 11q^{9} + q^{11} - 3q^{13} + 13q^{15} + 12q^{19} + 3q^{21} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{11} + 7T_{13} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ , whose minimal polynomial is

$x^{6} - 645x^{5} + 104383x^{4} - 988449x^{3} - 342648472x^{2} - 4933703822x + 30023822908$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 812$ )
7	$-1$
29	$-1$

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	3	−1	6	1	−3	0	12	0	6	3	4	−14	29	31	−7	−2	−6	6	0	10	19	−14	−16	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.016 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, 7, 14, 28, 29, 58, 116, 203, 406, 812]: [0, 1, 2, 1, 4, 7, 3, 9, 18, 22, 64, 126] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (17, 3), (19, 4), (23, 5), (11, 6), (13, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 14 = 1 x 14 (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(812, 2)[4]   # 812.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 812:2 1000