Atlas › Modular forms › Weight 6 › Level 22 › 22.6.a › 22.6.a.d

Newform orbit 22.6.a.d

Trace form

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

$2q + 8q^{2} + 29q^{3} + 32q^{4} - 13q^{5} + 116q^{6} - 14q^{7} + 128q^{8} + 331q^{9} - 52q^{10} - 242q^{11} + 464q^{12} - 646q^{13} - 56q^{14} - 2171q^{15} + 512q^{16} - 208q^{17} + 1324q^{18} - 2148q^{19} - 208q^{20} - 2582q^{21} - 968q^{22} + 349q^{23} + 1856q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{793})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} - 16x - 3108$

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
$\operatorname{tr} a_p$	8	29	−13	−14	−242	−646	−208	−2148	349	4422	14381	−4267	−10110	−9798	21144	39584	90951	−29550	−64149	23583	−39058	−54974	29986	−18047	−30309

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 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, 2, 6, 17] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (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(22, 6)[3]   # 22.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 22:6 1000