Atlas › Modular forms › Weight 12 › Level 17 › 17.12.a › 17.12.a.b

Newform orbit 17.12.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 55q^{2} + 496q^{3} + 10757q^{4} + 8592q^{5} + 17194q^{6} + 95288q^{7} - 247863q^{8} + 500648q^{9} + 2824q^{10} + 435256q^{11} + 4295182q^{12} + 4193784q^{13} + 4377132q^{14} + 10362648q^{15} + 20722897q^{16} - 11358856q^{17} + 28881887q^{18} + 15158192q^{19} + 91612472q^{20} + 98415768q^{21} + 81767186q^{22} + 22374432q^{23} - 41056218q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 551x^{7} - 837876x^{6} + 391948436x^{5} + 192257457996x^{4} - 77857488689676x^{3} - 10513316544768032x^{2} + 2989821396207767984x + 318453699583382083008$

Atkin–Lehner signs

$p$	$w_p$
17	$+1$

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	496	8592	95288	435256	4193784	−11358856	15158192	22374432	−424656432	−172323152	−262792640	−1283308512	2219398472	260684408	9402026896	14325543480	−9811064576	52928023248	−34868356504	1248764080	18209008736	169643760088	201694397904	163430440672

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.504 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, 17]: [2, 18] (certified at ell = 2147483489)
T = [(2, 1), (3, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 5 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(17, 12)[1]   # 17.12.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 17:12 1000