Atlas › Modular forms › Weight 2 › Level 793 › 793.2.a › 793.2.a.f

Newform orbit 793.2.a.f

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 11x^{15} + 4x^{14} + 331x^{13} - 895x^{12} - 2843x^{11} + 13302x^{10} + 375x^{9} - 64116x^{8} + 74514x^{7} + 60360x^{6} - 164495x^{5} + 94645x^{4} + 1080x^{3} - 14608x^{2} + 2041x + 413$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
61	$+1$

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
$\operatorname{tr} a_p$	5	6	10	10	18	16	12	9	24	2	8	−13	16	−4	57	0	24	−16	−13	47	0	15	68	4	−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 0.243 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, 13, 61, 793]: [0, 1, 5, 72] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (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(793, 2)[5]   # 793.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 793:2 1000