Atlas › Modular forms › Weight 2 › Level 471 › 471.2.a › 471.2.a.e

Newform orbit 471.2.a.e

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{12} - 3x^{11} - 66x^{10} + 207x^{9} + 1441x^{8} - 5006x^{7} - 10184x^{6} + 46490x^{5} - 10091x^{4} - 100439x^{3} + 110924x^{2} - 32453x - 565$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
157	$+1$

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	12	4	8	−1	15	10	14	−8	5	5	4	18	28	−3	−18	0	20	27	7	15	4	−13	−9	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.199 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, 157, 471]: [0, 1, 13, 54] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 27 = 1 x 27 (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(471, 2)[4]   # 471.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 471:2 1000