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

Newform orbit 986.2.a.l

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. 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

$x^{9} - 34x^{8} - 69240x^{7} - 1919742x^{6} + 1090632009x^{5} + 36944551571x^{4} - 5614845303285x^{3} - 150108531924244x^{2} + 5816030905801728x + 910095759341568$

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
$\operatorname{tr} a_p$	9	0	0	1	0	9	9	10	−3	9	15	6	3	−19	−4	21	−1	6	3	9	9	7	−23	20	−29

$\operatorname{tr} a_n$ 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)[11]   # 986.2.a.l

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

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