Atlas › Modular forms › Weight 2 › Level 689 › 689.2.a › 689.2.a.f

Newform orbit 689.2.a.f

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 2q^{2} + q^{3} + 26q^{4} + q^{5} + 5q^{6} + 19q^{7} + 19q^{9} - 7q^{10} + 9q^{11} - 9q^{12} + 18q^{13} - q^{14} + 27q^{15} + 38q^{16} - q^{17} + 17q^{18} + 24q^{19} - 19q^{20} - 8q^{21} - 7q^{22} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{18} - 5x^{17} - 362x^{16} + 1681x^{15} + 50080x^{14} - 196399x^{13} - 3576555x^{12} + 11028396x^{11} + 145234150x^{10} - 336976478x^{9} - 3441005218x^{8} + 5943203460x^{7} + 46765949293x^{6} - 61393728369x^{5} - 337180309360x^{4} + 356793961763x^{3} + 1018606803678x^{2} - 948764433015x - 261538730199$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
53	$+1$

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
$\operatorname{tr} a_p$	2	1	1	19	9	18	−1	24	0	13	38	18	1	0	17	−18	28	0	48	22	23	53	2	−22	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.237 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, 13, 53, 689]: [0, 1, 5, 64] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 53 = 1 x 53 (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(689, 2)[5]   # 689.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 689:2 1000