Atlas › Modular forms › Weight 2 › Level 798 › 798.2.a › 798.2.a.k

Newform orbit 798.2.a.k

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{23}$ , whose minimal polynomial is

$x^{2} - 8x - 1436$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
7	$+1$
19	$-1$

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
$\operatorname{tr} a_p$	−2	−2	0	−2	−4	0	8	2	−4	−12	0	20	−4	0	8	4	8	4	12	−8	20	−4	16	−4	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.877 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, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798]: [0, 1, 1, 3, 1, 4, 2, 4, 7, 12, 8, 24, 14, 44, 56, 168] (certified at ell = 2147483647)
T = [(5, 1), (11, 3), (13, 5), (17, 7), (23, 9)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 5 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(798, 2)[10]   # 798.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 798:2 1000