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

Newform orbit 781.2.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. 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

$x^{6} - 7x^{5} - 4975x^{4} - 27764x^{3} + 623055x^{2} - 601100x - 933057$

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
$\operatorname{tr} a_p$	−2	−6	−5	−6	6	1	2	−7	13	−14	−12	−11	0	3	−19	6	−41	3	7	6	−3	−17	6	−24	−22

$\operatorname{tr} a_n$ 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)[3]   # 781.2.a.d

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

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