Atlas › Modular forms › Weight 2 › Level 431 › 431.2.a › 431.2.a.f

Newform orbit 431.2.a.f

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + q^{2} + q^{3} + 33q^{4} + 13q^{5} + 17q^{6} + 8q^{7} - 3q^{8} + 31q^{9} - 6q^{10} + 15q^{11} - 12q^{12} + 11q^{13} + 16q^{14} - 5q^{15} + 43q^{16} + 6q^{17} - 8q^{18} + 18q^{19} - 10q^{20} - 7q^{21} - 5q^{22} - 8q^{23} + 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{24} - 4x^{23} - 547x^{22} + 2113x^{21} + 126101x^{20} - 451086x^{19} - 16156477x^{18} + 50965066x^{17} + 1276409252x^{16} - 3327977393x^{15} - 65142631706x^{14} + 127769456159x^{13} + 2179977658082x^{12} - 2753837252287x^{11} - 47219887506411x^{10} + 27322019836141x^{9} + 631086685409461x^{8} + 7488142619012x^{7} - 4690129691489764x^{6} - 1862090641190448x^{5} + 15762907809605344x^{4} + 5412692447679040x^{3} - 19513376951187008x^{2} + 2819863398264832x + 730731043717120$

Atkin–Lehner signs

$p$	$w_p$
431	$-1$

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	1	13	8	15	11	6	18	−8	17	37	7	24	10	−6	−22	21	46	−25	−6	86	−6	−49	61	88

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 431]: [0, 37] (certified at ell = 2147481491)
T = [(2, 1), (3, 3)]; h of degree 36 = 1 x 36 (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(431, 2)[5]   # 431.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 431:2 1000