Atlas › Modular forms › Weight 2 › Level 797 › 797.2.a › 797.2.a.b

Newform orbit 797.2.a.b

Trace form

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

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

Coefficient field

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

$x^{2} - 2$

Atkin–Lehner signs

$p$	$w_p$
797	$-1$

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	4	8	−4	10	0	2	−2	0	2	0	−2	−4	14	0	2	6	12	4	−16	10	0	6	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.369 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, 797]: [0, 67] (certified at ell = 2147483029)
T = [(2, 1), (3, 1)]; h of degree 66 = 1 x 66 (phi(ord eps) x dim S_k^new); 7 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(797, 2)[1]   # 797.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 797:2 1000