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

Newform orbit 812.2.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 2q^{3} - q^{5} - 2q^{7} - 4q^{9} + 3q^{11} + q^{13} + q^{15} + 4q^{17} + 9q^{19} + 2q^{21} + 5q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . 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

$x^{2} - 333x + 26724$

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	−2	−1	−2	3	1	4	9	5	−2	3	−8	7	11	8	8	6	−4	9	−5	17	−3	20	−11	−7

$\operatorname{tr} a_n$ 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)[2]   # 812.2.a.c

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

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