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

Newform orbit 989.2.a.e

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q + q^{2} + 11q^{3} + 29q^{4} + 10q^{5} + q^{6} + 8q^{7} - 6q^{8} + 21q^{9} + 12q^{11} + 12q^{12} - q^{13} + 6q^{14} + 13q^{15} + 43q^{16} + 3q^{17} - 15q^{18} + 46q^{19} + 18q^{20} + 13q^{21} + 31q^{22} + 22q^{23} + 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} - 12x^{21} - 9x^{20} + 628x^{19} - 1610x^{18} - 11128x^{17} + 51124x^{16} + 61536x^{15} - 613838x^{14} + 294382x^{13} + 3366633x^{12} - 4439168x^{11} - 8332873x^{10} + 16942608x^{9} + 7821960x^{8} - 28262018x^{7} + 1864943x^{6} + 20340658x^{5} - 6928780x^{4} - 4128574x^{3} + 2502469x^{2} - 412176x + 21024$

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
$\operatorname{tr} a_p$	1	11	10	8	12	−1	3	46	22	0	33	17	−3	−22	−18	17	−3	30	18	26	25	14	31	36	−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)[4]   # 989.2.a.e

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

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