Atlas › Modular forms › Weight 2 › Level 422 › 422.2.a › 422.2.a.f

Newform orbit 422.2.a.f

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 6q^{2} + 4q^{3} + 6q^{4} - 2q^{5} + 4q^{6} + 7q^{7} + 6q^{8} + 6q^{9} - 2q^{10} + 4q^{11} + 4q^{12} + 8q^{13} + 7q^{14} - 6q^{15} + 6q^{16} + q^{17} + 6q^{18} + 11q^{19} - 2q^{20} - 3q^{21} + 4q^{22} - 3q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{6} - 61x^{5} + 470x^{4} + 24232x^{3} - 263344x^{2} - 2609751x + 23349152$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
211	$+1$

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
$\operatorname{tr} a_p$	6	4	−2	7	4	8	1	11	−3	−9	1	−5	−9	2	−12	−17	−7	2	2	−9	26	8	14	1	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 211, 422]: [0, 1, 18, 55] (certified at ell = 2147482681)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 4 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(422, 2)[5]   # 422.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 422:2 1000