Atlas › Modular forms › Weight 2 › Level 889 › 889.2.a › 889.2.a.d

Newform orbit 889.2.a.d

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 8q^{2} + 24q^{4} + 3q^{5} + 6q^{6} + 20q^{7} + 24q^{8} + 30q^{9} - 8q^{10} + 26q^{11} - 4q^{12} - 4q^{13} + 8q^{14} + 10q^{15} + 24q^{16} + 4q^{17} + 5q^{18} + q^{19} - 2q^{20} + q^{22} + 31q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} - 8x^{19} - 51x^{18} + 541x^{17} + 547x^{16} - 13299x^{15} + 7394x^{14} + 158777x^{13} - 196010x^{12} - 1010024x^{11} + 1579950x^{10} + 3487473x^{9} - 5931582x^{8} - 6432909x^{7} + 10767695x^{6} + 6439893x^{5} - 9049055x^{4} - 3575206x^{3} + 3089232x^{2} + 794926x - 302981$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
127	$+1$

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
$\operatorname{tr} a_p$	8	0	3	20	26	−4	4	1	31	16	6	2	25	13	19	24	23	−27	9	63	−21	18	−1	−16	−32

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.258 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, 7, 127, 889]: [0, 1, 11, 86] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 7 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(889, 2)[3]   # 889.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 889:2 1000