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

Newform orbit 799.2.a.g

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 4q^{2} + 3q^{3} + 24q^{4} + 13q^{5} + 11q^{6} - 3q^{7} + 12q^{8} + 37q^{9} - 2q^{10} + 10q^{11} + 8q^{12} + q^{13} + 13q^{14} + q^{15} + 28q^{16} - 20q^{17} + 15q^{18} + 5q^{19} + 37q^{20} + 6q^{21} - 19q^{22} + 4q^{23} + 30q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{20} - 147x^{19} - 8278x^{18} + 2267100x^{17} - 42653475x^{16} - 10845599284x^{15} + 541699343743x^{14} + 15206243901214x^{13} - 1534665254050088x^{12} + 10703030337182072x^{11} + 1502021849942256205x^{10} - 31343730664727996104x^{9} - 507937072213646181294x^{8} + 19368722563226758295436x^{7} - 21804468886010623471035x^{6} - 4204366297860108705480446x^{5} + 36606586740145421618785723x^{4} + 163847564754158241267420763x^{3} - 3314848479908423738976562643x^{2} + 12589682422287260129246873108x - 8395243417324645960907831403$

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
$\operatorname{tr} a_p$	4	3	13	−3	10	1	−20	5	4	26	6	−8	69	−7	20	29	55	−17	−6	47	−32	−26	3	119	−13

$\operatorname{tr} a_n$ 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)[6]   # 799.2.a.g

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

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