Atlas › Modular forms › Weight 2 › Level 986 › 986.2.a › 986.2.a.k

Newform orbit 986.2.a.k

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ , whose minimal polynomial is

$x^{7} - 567x^{6} + 27591x^{5} + 24622543x^{4} - 2598123538x^{3} - 203065643385x^{2} + 34988269315552x - 1141621776419072$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
17	$+1$
29	$+1$

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
$\operatorname{tr} a_p$	7	3	5	9	−1	0	−7	2	13	−7	14	18	−1	16	−1	14	−11	20	−9	5	−9	−2	−33	−12	25

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.466 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, 17, 29, 34, 58, 493, 986]: [0, 1, 2, 3, 6, 9, 46, 138] (certified at ell = 2147480161)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (23, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 13 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(986, 2)[10]   # 986.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 986:2 1000