Atlas › Modular forms › Weight 2 › Level 989 › 989.2.a › 989.2.a.d

Newform orbit 989.2.a.d

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q - 2q^{2} - 11q^{3} + 8q^{4} - 6q^{6} - 8q^{7} - 6q^{8} + 11q^{9} - 10q^{10} - 10q^{11} - 11q^{12} - 9q^{13} - 14q^{14} - 7q^{15} + 8q^{16} + 3q^{17} - 3q^{18} - 28q^{19} + 16q^{20} - 7q^{21} - q^{22} - 16q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} + 13x^{15} + 41x^{14} - 118x^{13} - 786x^{12} - 116x^{11} + 4963x^{10} + 4273x^{9} - 15128x^{8} - 16737x^{7} + 24057x^{6} + 26196x^{5} - 18910x^{4} - 14776x^{3} + 5655x^{2} - 99x - 61$

Atkin–Lehner signs

$p$	$w_p$
23	$+1$
43	$+1$

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	−11	0	−8	−10	−9	3	−28	−16	−16	−43	−17	−23	−16	−2	23	−23	−32	−40	−18	−29	−14	9	10	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 23, 43, 989]: [0, 3, 4, 90] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 77 = 1 x 77 (phi(ord eps) x dim S_k^new); 8 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(989, 2)[3]   # 989.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 989:2 1000