Atlas › Modular forms › Weight 2 › Level 799 › 799.2.a › 799.2.a.f

Newform orbit 799.2.a.f

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 3q^{2} + q^{3} + 19q^{4} + 11q^{5} + 5q^{6} + q^{7} + 3q^{8} + 30q^{9} + 12q^{10} + 8q^{11} + 12q^{12} + q^{13} + 11q^{14} - 7q^{15} + 15q^{16} + 17q^{17} - 26q^{18} - q^{19} + 13q^{20} + 14q^{21} + 3q^{22} + 12q^{23} - 24q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{17} - 140x^{16} - 3803x^{15} + 1326062x^{14} - 26531600x^{13} - 4453271133x^{12} + 186288292084x^{11} + 5816514427435x^{10} - 419448589691954x^{9} - 421675032020617x^{8} + 410004844558822969x^{7} - 5102096086265869552x^{6} - 156513792034819904472x^{5} + 3450591291917968727165x^{4} + 11022219367597009713506x^{3} - 616516351134528977510381x^{2} + 1860397705981978243248869x + 17733328387528411007449913$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$
47	$+1$

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	1	11	1	8	1	17	−1	12	10	2	10	65	−3	−17	−13	39	23	−14	21	24	−26	−5	61	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.41 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, 47, 799]: [0, 2, 5, 74] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 61 = 1 x 61 (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(799, 2)[5]   # 799.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 799:2 1000