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

Newform orbit 793.2.a.d

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 8q^{2} - 6q^{3} + 18q^{4} - 8q^{5} - 6q^{6} - 12q^{7} - 21q^{8} + 15q^{9} + 5q^{10} - 28q^{11} - 9q^{12} + 15q^{13} + 2q^{14} - 9q^{15} + 24q^{16} - 20q^{17} - 30q^{18} - 11q^{19} - 31q^{20} - 5q^{21} + 13q^{22} - 34q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} + 14x^{14} + 50x^{13} - 105x^{12} - 872x^{11} - 493x^{10} + 4525x^{9} + 6356x^{8} - 7628x^{7} - 18048x^{6} - 2348x^{5} + 14103x^{4} + 10392x^{3} + 2388x^{2} + 94x + 1$

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 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
$\operatorname{tr} a_p$	−8	−6	−8	−12	−28	15	−20	−11	−34	4	−28	−5	−22	−8	−51	−12	−48	15	−23	−41	6	−11	−36	−12	2

$\operatorname{tr} a_n$ 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)[3]   # 793.2.a.d

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

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