Atlas › Modular forms › Weight 2 › Level 411 › 411.2.a › 411.2.a.e

Newform orbit 411.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 9q^{3} + 14q^{4} + 6q^{5} + 7q^{7} - 3q^{8} + 9q^{9} + 5q^{10} + 7q^{11} - 14q^{12} + 7q^{13} - 6q^{14} - 6q^{15} + 28q^{16} + 11q^{19} + 6q^{20} - 7q^{21} + 4q^{22} - 3q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{9} - 6x^{8} - 36x^{7} + 230x^{6} + 246x^{5} - 2262x^{4} + 1502x^{3} + 4157x^{2} - 5590x + 1790$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
137	$-1$

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
$\operatorname{tr} a_p$	0	−9	6	7	7	7	0	11	−3	22	15	27	5	3	−37	1	−4	19	−7	−2	25	2	−44	−4	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 3, 137, 411]: [0, 1, 12, 48] (certified at ell = 2147483249)
T = [(2, 1), (5, 1)]; h of degree 23 = 1 x 23 (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(411, 2)[4]   # 411.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 411:2 1000