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

Newform orbit 411.2.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{5} - 5x^{4} + 13x^{2} - 10x + 2$

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
$\operatorname{tr} a_p$	−1	5	6	−1	1	−1	4	−5	13	22	−3	1	24	3	1	2	−8	17	−31	5	−5	−22	14	−4	−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 0.195 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)[3]   # 411.2.a.d

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

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