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

Newform orbit 431.2.a.e

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} + 10x^{3} - 9x^{2} - 190x + 109$

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 odd 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	−3	−5	2	1	−5	2	−6	4	−5	−25	−1	2	−16	−4	4	5	0	9	10	−50	8	−15	−35	−6

$\operatorname{tr} a_n$ 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)[4]   # 431.2.a.e

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

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