Atlas › Modular forms › Weight 2 › Level 731 › 731.2.a › 731.2.a.d

Newform orbit 731.2.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} + 2x^{7} - 21x^{6} - 24x^{5} + 131x^{4} + 69x^{3} - 215x^{2} - 50x + 46$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$
43	$-1$

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	−7	−5	−4	−12	8	0	−9	−27	−12	−24	−8	8	15	−23	16	−34	0	−3	−3	−24	−8	23	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.236 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, 17, 43, 731]: [0, 2, 4, 68] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 6 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(731, 2)[3]   # 731.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 731:2 1000