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

Newform orbit 799.2.a.d

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{8} + 151x^{7} + 4438x^{6} - 317748x^{5} - 21040182x^{4} - 309469901x^{3} + 3896635431x^{2} + 134292769928x + 890716588271$

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 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
$\operatorname{tr} a_p$	1	−7	−10	−9	−2	−3	8	−8	−9	−2	−7	−13	−39	35	8	−6	−19	−15	15	−7	−22	−43	−3	−53	−40

$\operatorname{tr} a_n$ 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)[3]   # 799.2.a.d

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

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