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

Newform orbit 989.2.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 6x^{9} - 15x^{8} - 112x^{7} + 84x^{6} + 666x^{5} - 266x^{4} - 1276x^{3} + 191x^{2} + 644x - 71$

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
$\operatorname{tr} a_p$	−1	−5	−1	−2	−24	−18	−2	−22	10	8	−21	2	13	10	−8	−12	−1	−38	9	1	−8	−47	−13	−4	−76

$\operatorname{tr} a_n$ 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)[2]   # 989.2.a.c

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

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