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

Newform orbit 13.12.a.b

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 55q^{2} + 476q^{3} + 15309q^{4} + 3312q^{5} - 57797q^{6} - 4176q^{7} + 158493q^{8} + 876218q^{9} + 997497q^{10} + 275060q^{11} + 3949049q^{12} - 2227758q^{13} + 6462587q^{14} + 5951652q^{15} + 25038945q^{16} + 18470848q^{17} + 1544758q^{18} + 2382612q^{19} + 37821799q^{20} - 67640772q^{21} - 52649718q^{22} + 25001944q^{23} - 243039615q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 531x^{5} - 784571x^{4} + 368209683x^{3} + 170603219290x^{2} - 63992195659536x - 6815180829926288$

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 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
$\operatorname{tr} a_p$	55	476	3312	−4176	275060	−2227758	18470848	2382612	25001944	−142876028	−158397468	47994456	112037548	1399191924	−3383597640	546961604	10067834260	15731821572	50546073444	−2646136112	4198695060	−92124930312	20440296092	20540234076	−203942467020

$\operatorname{tr} a_n$ 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)[1]   # 13.12.a.b

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

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