Atlas › Modular forms › Weight 12 › Level 11 › 11.12.a › 11.12.a.b

Newform orbit 11.12.a.b

Trace form

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

$5q + 32q^{2} + 160q^{3} + 6220q^{4} - 8398q^{5} + 4078q^{6} + 79040q^{7} + 219996q^{8} + 505901q^{9} + 1098950q^{10} - 805255q^{11} + 3191960q^{12} + 4398926q^{13} + 6003692q^{14} + 2987476q^{15} + 5631688q^{16} + 4282922q^{17} - 25570654q^{18} - 10999932q^{19} - 32692832q^{20} - 10622888q^{21} - 5153632q^{22} - 91304228q^{23} - 122458836q^{24} + O(q^{25})$

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

$x^{5} - 192x^{4} - 689694x^{3} + 87529598x^{2} + 118411313409x - 7587187401978$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$

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
$\operatorname{tr} a_p$	32	160	−8398	79040	−805255	4398926	4282922	−10999932	−91304228	−99658578	684755684	877059786	542414234	−211657308	−1172877944	7728362934	7164217984	1624665870	23150623288	−52872507708	−3744118918	44261061544	−67350622692	−106478658714	170845613350

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.484 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, 11]: [2, 12] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(11, 12)[1]   # 11.12.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 11:12 1000