Atlas › Modular forms › Weight 2 › Level 786 › 786.2.a › 786.2.a.n

Newform orbit 786.2.a.n

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 2T_{7} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{11} + 7T_{13} + 8T_{29}$ , whose minimal polynomial is

$x^{3} - 74x^{2} - 1974x - 9136$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$-1$
131	$+1$

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
$\operatorname{tr} a_p$	−3	3	2	3	−2	6	2	11	2	−3	8	9	−9	0	−8	−8	−4	16	19	14	6	10	−3	−7	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 0.886 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, 131, 262, 393, 786]: [0, 1, 1, 3, 12, 35, 46, 136] (certified at ell = 2147482481)
T = [(5, 1), (7, 2), (17, 3), (19, 4), (23, 5), (11, 6), (13, 7), (29, 8)]; h of degree 23 = 1 x 23 (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(786, 2)[13]   # 786.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 786:2 1000