Atlas › Modular forms › Weight 10 › Level 22 › 22.10.a › 22.10.a.e

Newform orbit 22.10.a.e

Trace form

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

$2q - 32q^{2} + 34q^{3} + 512q^{4} - 1478q^{5} - 544q^{6} + 8196q^{7} - 8192q^{8} + 20476q^{9} + 23648q^{10} - 29282q^{11} + 8704q^{12} - 196296q^{13} - 131136q^{14} - 617766q^{15} + 131072q^{16} - 548788q^{17} - 327616q^{18} - 90928q^{19} - 378368q^{20} - 512572q^{21} + 468512q^{22} - 3971046q^{23} - 139264q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{463})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 1444x - 1878908$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
11	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

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
$\operatorname{tr} a_p$	−32	34	−1478	8196	−29282	−196296	−548788	−90928	−3971046	−656128	−872214	−190722	3350080	3876012	38884544	−134346316	−125273754	114821880	−91519714	411397438	506142392	810072516	−718904884	−810786322	−230208654

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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 engine's checks

dim M^+ at levels [1, 2, 11, 22]: [1, 3, 10, 29] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 3 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(22, 10)[4]   # 22.10.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 22:10 1000