Atlas › Modular forms › Weight 6 › Level 13 › 13.6.a › 13.6.a.a

Newform orbit 13.6.a.a

Trace form

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

$2q - 5q^{2} - 28q^{3} - 43q^{4} - 42q^{5} + 19q^{6} - 36q^{7} + 225q^{8} + 212q^{9} + 445q^{10} - 376q^{11} + 857q^{12} - 338q^{13} - 505q^{14} - 1452q^{15} + 465q^{16} - 2630q^{17} + 898q^{18} - 312q^{19} - 797q^{20} + 4074q^{21} + 226q^{22} - 2624q^{23} - 1059q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 33x + 166$

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
$\operatorname{tr} a_p$	−5	−28	−42	−36	−376	−338	−2630	−312	−2624	−812	7720	−16858	7840	2420	9972	−43720	−38936	1984	−69928	67396	74412	−55296	−75712	116508	−34756

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 7] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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, 6)[0]   # 13.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 13:6 1000