Atlas › Modular forms › Weight 12 › Level 22 › 22.12.a › 22.12.a.a

Newform orbit 22.12.a.a

Trace form

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

$2q + 64q^{2} - 426q^{3} + 2048q^{4} + 2290q^{5} - 13632q^{6} - 86324q^{7} + 65536q^{8} + 117756q^{9} + 73280q^{10} - 322102q^{11} - 436224q^{12} - 2100184q^{13} - 2762368q^{14} - 6842970q^{15} + 2097152q^{16} - 2882276q^{17} + 3768192q^{18} - 19571712q^{19} + 2344960q^{20} - 2585148q^{21} - 10307264q^{22} + 12680534q^{23} - 13959168q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{331})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 1864x - 45926832$

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
$\operatorname{tr} a_p$	64	−426	2290	−86324	−322102	−2100184	−2882276	−19571712	12680534	45662496	−506504170	402672518	608864016	1100094564	1012342272	−68189276	−6791617518	5704046520	−36514311702	20672196594	3082870856	−28681382180	29532640772	85063742462	−180832449678

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.359 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]: [2, 4, 12, 35] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (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(22, 12)[0]   # 22.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 22:12 1000