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

Newform orbit 889.2.a.b

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q + 4q^{3} + 14q^{4} + 7q^{5} + 8q^{6} - 15q^{7} + 9q^{9} + 10q^{10} + 14q^{11} + 10q^{12} + 6q^{13} + 6q^{15} + 20q^{16} + 10q^{17} + q^{18} + 13q^{19} + 8q^{20} - 4q^{21} - 11q^{22} + 15q^{23} + 34q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} - 4x^{14} - 49x^{13} + 183x^{12} + 795x^{11} - 2534x^{10} - 5816x^{9} + 12232x^{8} + 23195x^{7} - 18713x^{6} - 41176x^{5} + 1205x^{4} + 25916x^{3} + 10258x^{2} - 854x - 691$

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
$\operatorname{tr} a_p$	0	4	7	−15	14	6	10	13	15	16	22	−14	19	−1	49	−28	43	27	3	55	−3	18	17	36	−2

$\operatorname{tr} a_n$ 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)[1]   # 889.2.a.b

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

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