Atlas › Modular forms › Weight 8 › Level 13 › 13.8.a › 13.8.a.b

Newform orbit 13.8.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 19q^{2} + 45q^{3} + 93q^{4} - 353q^{5} - 933q^{6} - 2009q^{7} - 1653q^{8} - 1845q^{9} + 5207q^{10} - 1810q^{11} + 11697q^{12} + 4394q^{13} + 17569q^{14} - 13503q^{15} + 20481q^{16} - 25361q^{17} - 5220q^{18} + 22106q^{19} - 51631q^{20} - 40653q^{21} + 122002q^{22} - 26424q^{23} - 87237q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{337})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 26x - 168$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$

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
$\operatorname{tr} a_p$	−19	45	−353	−2009	−1810	4394	−25361	22106	−26424	−5804	39744	163299	−330870	229307	−1638525	1046382	−370158	4675422	−1821402	−1135611	−6459284	−73808	−12100972	9815060	−17591688

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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 engine's checks

dim M^+ at levels [1, 13]: [1, 9] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(13, 8)[1]   # 13.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 13:8 1000