Atlas › Modular forms › Weight 12 › Level 13 › 13.12.a › 13.12.a.a

Newform orbit 13.12.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 41q^{2} - 496q^{3} + 2993q^{4} - 2542q^{5} + 35441q^{6} - 36296q^{7} - 200037q^{8} + 172645q^{9} - 1004055q^{10} - 1718912q^{11} - 3670067q^{12} + 1856465q^{13} - 3264661q^{14} - 15356636q^{15} - 9077455q^{16} - 6022762q^{17} - 25785896q^{18} - 6984968q^{19} + 38011501q^{20} + 30001972q^{21} + 35725110q^{22} - 26355744q^{23} + 88388907q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 537x^{4} - 427063x^{3} - 222063897x^{2} - 31352452530x - 1392053521176$

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
$\operatorname{tr} a_p$	−41	−496	−2542	−36296	−1718912	1856465	−6022762	−6984968	−26355744	−81933754	−13953380	−417857846	−2690154174	−2194005968	−4632149016	3964085286	−6213900336	−13653194690	−11630839736	−55420684056	−10807393382	−4898325368	−17839206992	65706244882	−66619160654

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.49 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]: [2, 15] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 4 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, 12)[0]   # 13.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 13:12 1000