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

Newform orbit 422.2.a.e

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} + 94x^{2} + 1459x - 27103$

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 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
$\operatorname{tr} a_p$	3	−5	−3	−9	−5	−7	−7	−4	−7	10	−7	−6	−3	5	−4	17	−13	9	−1	1	−31	−4	−4	1	−18

$\operatorname{tr} a_n$ 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)[4]   # 422.2.a.e

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

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