Atlas › Modular forms › Weight 2 › Level 492 › 492.2.a › 492.2.a.c

Newform orbit 492.2.a.c

Trace form

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

$2q - 2q^{3} - 4q^{5} + 4q^{7} + 2q^{9} - 2q^{11} + 4q^{13} + 4q^{15} + 2q^{17} + 8q^{19} - 4q^{21} + 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{6})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{2} - 114x + 3243$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 492$ )
3	$+1$
41	$+1$

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	−4	4	−2	4	2	8	8	2	14	−2	−2	2	14	−16	0	−6	−4	−6	2	20	20	−12	8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.301 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, 3, 4, 6, 12, 41, 82, 123, 164, 246, 492]: [0, 1, 1, 2, 3, 5, 4, 12, 16, 24, 46, 90] (certified at ell = 2147482921)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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(492, 2)[2]   # 492.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 492:2 1000