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

Newform orbit 799.2.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{2} + 76x + 1199$

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
$\operatorname{tr} a_p$	−3	0	−1	−2	−6	8	2	9	−5	−10	1	−11	−14	−10	2	−9	6	−4	−21	−24	−14	25	18	−12	5

$\operatorname{tr} a_n$ 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)[2]   # 799.2.a.c

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

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