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

Newform orbit 689.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} + 12x^{8} - 72x^{7} - 1043x^{6} + 1584x^{5} + 28218x^{4} - 10953x^{3} - 235446x^{2} + 3660x + 302525$

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 odd 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
$\operatorname{tr} a_p$	−3	−3	−3	−13	−3	9	5	−22	−12	−5	−42	0	−3	−12	−3	9	4	−4	−38	−38	−19	−25	−22	16	−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 0.237 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)[4]   # 689.2.a.e

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

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