Atlas › Modular forms › Weight 2 › Level 781 › 781.2.a › 781.2.a.c

Newform orbit 781.2.a.c

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{3} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{5} + 7T_{19} + 8T_{23}$ , whose minimal polynomial is

$x^{2} + 165x + 5373$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
71	$-1$

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	−1	−3	1	2	−10	−2	−5	−8	7	−10	4	−13	16	4	−6	0	−8	−14	2	17	17	−4	−12	−11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.407 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, 11, 71, 781]: [0, 2, 7, 74] (certified at ell = 2147483171)
T = [(2, 1), (3, 2), (7, 3), (13, 4), (17, 5), (5, 6), (19, 7), (23, 8)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 16 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(781, 2)[2]   # 781.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 781:2 1000