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

Newform orbit 989.2.a.f

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + 8q^{3} + 36q^{4} - 6q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + 32q^{9} + 16q^{10} + 9q^{11} + 22q^{12} + 18q^{13} - 4q^{14} - 9q^{15} + 56q^{16} - 7q^{17} + 6q^{18} + 40q^{19} - 2q^{20} - q^{21} - 22q^{22} - 24q^{23} - 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} - 8x^{23} - 67x^{22} + 635x^{21} + 1492x^{20} - 19315x^{19} - 11986x^{18} + 300142x^{17} - 20480x^{16} - 2622040x^{15} + 914225x^{14} + 13203652x^{13} - 5814579x^{12} - 37607009x^{11} + 16791472x^{10} + 57636021x^{9} - 22235579x^{8} - 43526668x^{7} + 9733288x^{6} + 15276009x^{5} - 502487x^{4} - 1572304x^{3} + 38996x^{2} + 40041x - 2372$

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 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
$\operatorname{tr} a_p$	0	8	−6	2	9	18	−7	40	−24	10	57	0	19	24	6	−35	67	27	29	−19	−1	63	6	−25	28

$\operatorname{tr} a_n$ 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)[5]   # 989.2.a.f

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

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