Atlas › Modular forms › Weight 2 › Level 791 › 791.2.a › 791.2.a.e

Newform orbit 791.2.a.e

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{8} + 227x^{7} + 18544x^{6} + 611082x^{5} + 2764412x^{4} - 272227029x^{3} - 4400129944x^{2} + 15838657232x + 484919542787$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
113	$+1$

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
$\operatorname{tr} a_p$	0	−1	−4	−8	−4	−8	−7	−15	10	−9	−17	−12	−16	−2	−6	5	−12	−23	11	−18	−15	−5	−5	−30	−45

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.367 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, 7, 113, 791]: [0, 1, 10, 78] (certified at ell = 2147480161)
T = [(2, 1), (3, 3), (13, 5), (17, 7), (19, 9)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 15 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(791, 2)[4]   # 791.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 791:2 1000