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

Newform orbit 22.10.a.d

Trace form

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

$2q - 32q^{2} - 21q^{3} + 512q^{4} - 521q^{5} + 336q^{6} - 7490q^{7} - 8192q^{8} - 3141q^{9} + 8336q^{10} + 29282q^{11} - 5376q^{12} + 150314q^{13} + 119840q^{14} + 393519q^{15} + 131072q^{16} + 690472q^{17} + 50256q^{18} + 511212q^{19} - 133376q^{20} + 1398810q^{21} - 468512q^{22} + 874751q^{23} + 86016q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{889})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 542x - 2423760$

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 even 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	−21	−521	−7490	29282	150314	690472	511212	874751	−2951058	−5818705	2658905	13427994	−17820762	56044104	96842752	−119136183	−90424326	−295944891	−322953267	−255975514	−889658	−277699042	1363672217	1398434043

$\operatorname{tr} a_n$ 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)[3]   # 22.10.a.d

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

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