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

Newform orbit 793.2.a.e

Trace form

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

$16q + 3q^{2} + 6q^{3} + 17q^{4} + 4q^{5} + 8q^{6} + 10q^{7} + 6q^{8} + 20q^{9} + 3q^{10} + 23q^{12} - 16q^{13} + 14q^{14} - 3q^{15} + 23q^{16} + 22q^{17} - 5q^{18} - q^{19} + 5q^{20} + 15q^{21} - 7q^{22} + 38q^{23} + 12q^{24} + O(q^{25})$

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

$x^{16} - 9x^{15} - 26x^{14} + 417x^{13} - 249x^{12} - 6189x^{11} + 8302x^{10} + 46039x^{9} - 66712x^{8} - 198554x^{7} + 235140x^{6} + 502351x^{5} - 345777x^{4} - 638838x^{3} + 111306x^{2} + 207797x - 20741$

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
$\operatorname{tr} a_p$	3	6	4	10	0	−16	22	−1	38	10	10	−3	14	24	53	14	−16	16	−3	−15	16	−15	38	−6	−14

$\operatorname{tr} a_n$ 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)[4]   # 793.2.a.e

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

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