Atlas › Modular forms › Weight 2 › Level 406 › 406.2.a › 406.2.a.e

Newform orbit 406.2.a.e

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ , whose minimal polynomial is

$x^{2} - 30x - 87498$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
7	$+1$
29	$+1$

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
$\operatorname{tr} a_p$	2	4	2	−2	0	−2	−2	0	−4	−2	6	−8	−2	8	2	−4	18	8	0	4	14	−12	2	22	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.43 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, 7, 14, 29, 58, 203, 406]: [0, 1, 1, 4, 3, 9, 22, 64] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 13 = 1 x 13 (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(406, 2)[4]   # 406.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 406:2 1000