LMFDB - Embedded newform 117.2.b.b.64.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(64,117)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.8112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 5x^{2} + 3 $ x^4 + 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			64.2
Root			$-2.07431i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.2.b.b.64.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.628052i q^{2} +1.60555 q^{4} +4.14863i q^{5} -2.26447i q^{8} +O(q^{10})$	$q-0.628052i q^{2} +1.60555 q^{4} +4.14863i q^{5} -2.26447i q^{8} +2.60555 q^{10} -5.40473i q^{11} -3.60555 q^{13} +1.78890 q^{16} +6.66083i q^{20} -3.39445 q^{22} -12.2111 q^{25} +2.26447i q^{26} -5.65246i q^{32} +9.39445 q^{40} -1.63642i q^{41} -4.00000 q^{43} -8.67757i q^{44} +13.7020i q^{47} +7.00000 q^{49} +7.66920i q^{50} -5.78890 q^{52} +22.4222 q^{55} -11.1898i q^{59} -7.21110 q^{61} +0.0277564 q^{64} -14.9581i q^{65} +7.91694i q^{71} +14.4222 q^{79} +7.42147i q^{80} -1.02776 q^{82} +0.380317i q^{83} +2.51221i q^{86} -12.2389 q^{88} +9.93367i q^{89} +8.60555 q^{94} -4.39636i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} - 4 q^{10} + 36 q^{16} - 28 q^{22} - 20 q^{25} + 52 q^{40} - 16 q^{43} + 28 q^{49} - 52 q^{52} + 32 q^{55} - 72 q^{64} + 68 q^{82} + 52 q^{88} + 20 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^10 + 36 * q^16 - 28 * q^22 - 20 * q^25 + 52 * q^40 - 16 * q^43 + 28 * q^49 - 52 * q^52 + 32 * q^55 - 72 * q^64 + 68 * q^82 + 52 * q^88 + 20 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 0.628052i	− 0.444099i	−0.975035	−	0.222050i	$-0.928725\pi$
$2$	− 0.628052i	− 0.444099i	0.975035	−	0.222050i	$-0.0712747\pi$
$3$	0	0
$3$	0	0
$4$	1.60555	0.802776
$5$	4.14863i	1.85532i	0.373423	+	0.927661i	$0.378184\pi$
$5$	4.14863i	1.85532i	−0.373423	+	0.927661i	$0.621816\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 2.26447i	− 0.800612i
$9$	0	0
$10$	2.60555	0.823948
$11$	− 5.40473i	− 1.62959i	−0.579751	−	0.814794i	$-0.696850\pi$
$11$	− 5.40473i	− 1.62959i	0.579751	−	0.814794i	$-0.303150\pi$
$12$	0	0
$13$	−3.60555	−1.00000
$13$	−3.60555	−1.00000
$14$	0	0
$15$	0	0
$16$	1.78890	0.447224
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	6.66083i	1.48941i
$21$	0	0
$22$	−3.39445	−0.723699
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−12.2111	−2.44222
$26$	2.26447i	0.444099i
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 5.65246i	− 0.999224i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	9.39445	1.48539
$41$	− 1.63642i	− 0.255566i	−0.991802	−	0.127783i	$-0.959214\pi$
$41$	− 1.63642i	− 0.255566i	0.991802	−	0.127783i	$-0.0407861\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	− 8.67757i	− 1.30819i
$45$	0	0
$46$	0	0
$47$	13.7020i	1.99864i	0.0368772	+	0.999320i	$0.488259\pi$
$47$	13.7020i	1.99864i	−0.0368772	+	0.999320i	$0.511741\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	7.66920i	1.08459i
$51$	0	0
$52$	−5.78890	−0.802776
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	22.4222	3.02341
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 11.1898i	− 1.45678i	−0.685160	−	0.728392i	$-0.740268\pi$
$59$	− 11.1898i	− 1.45678i	0.685160	−	0.728392i	$-0.259732\pi$
$60$	0	0
$61$	−7.21110	−0.923287	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	−7.21110	−0.923287	−0.461644	+	0.887066i	$0.652740\pi$
$62$	0	0
$63$	0	0
$64$	0.0277564	0.00346955
$65$	− 14.9581i	− 1.85532i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.91694i	0.939567i	0.882782	+	0.469784i	$0.155668\pi$
$71$	7.91694i	0.939567i	−0.882782	+	0.469784i	$0.844332\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.4222	1.62262	0.811312	−	0.584613i	$-0.198754\pi$
$79$	14.4222	1.62262	0.811312	+	0.584613i	$0.198754\pi$
$80$	7.42147i	0.829745i
$81$	0	0
$82$	−1.02776	−0.113497
$83$	0.380317i	0.0417453i	0.999782	+	0.0208726i	$0.00664445\pi$
$83$	0.380317i	0.0417453i	−0.999782	+	0.0208726i	$0.993356\pi$
$84$	0	0
$85$	0	0
$86$	2.51221i	0.270898i
$87$	0	0
$88$	−12.2389	−1.30467
$89$	9.93367i	1.05297i	0.850185	+	0.526484i	$0.176490\pi$
$89$	9.93367i	1.05297i	−0.850185	+	0.526484i	$0.823510\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	8.60555	0.887595
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 4.39636i	− 0.444099i
$99$	0	0
$100$	−19.6056	−1.96056
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	8.16467i	0.800612i
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	− 14.0823i	− 1.34269i
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−7.02776	−0.646957
$119$	0	0
$120$	0	0
$121$	−18.2111	−1.65555
$122$	4.52894i	0.410031i
$123$	0	0
$124$	0	0
$125$	− 29.9162i	− 2.67578i
$126$	0	0
$127$	14.4222	1.27976	0.639882	−	0.768473i	$-0.278983\pi$
$127$	14.4222	1.27976	0.639882	+	0.768473i	$0.278983\pi$
$128$	− 11.3224i	− 1.00076i
$129$	0	0
$130$	−9.39445	−0.823948
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	23.2553i	1.98684i	0.114538	+	0.993419i	$0.463461\pi$
$137$	23.2553i	1.98684i	−0.114538	+	0.993419i	$0.536539\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	4.97224	0.417261
$143$	19.4870i	1.62959i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.4703i	1.43122i	0.698499	+	0.715611i	$0.253852\pi$
$149$	17.4703i	1.43122i	−0.698499	+	0.715611i	$0.746148\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	− 9.05789i	− 0.720607i
$159$	0	0
$160$	23.4500	1.85388
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 2.62736i	− 0.205162i
$165$	0	0
$166$	0.238859	0.0185390
$167$	− 24.5114i	− 1.89675i	−0.317148	−	0.948376i	$-0.602725\pi$
$167$	− 24.5114i	− 1.89675i	0.317148	−	0.948376i	$-0.397275\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	−6.42221	−0.489689
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	− 9.66851i	− 0.728791i
$177$	0	0
$178$	6.23886	0.467622
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	21.9992i	1.60446i
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	11.2389	0.802776
$197$	− 7.42147i	− 0.528758i	−0.964419	−	0.264379i	$-0.914833\pi$
$197$	− 7.42147i	− 0.528758i	0.964419	−	0.264379i	$-0.0851669\pi$
$198$	0	0
$199$	14.4222	1.02236	0.511182	−	0.859473i	$-0.329208\pi$
$199$	14.4222	1.02236	0.511182	+	0.859473i	$0.329208\pi$
$200$	27.6517i	1.95527i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.78890	0.474157
$206$	10.0488i	0.700135i
$207$	0	0
$208$	−6.44996	−0.447224
$209$	0	0
$210$	0	0
$211$	−28.8444	−1.98573	−0.992866	−	0.119239i	$-0.961954\pi$
$211$	−28.8444	−1.98573	−0.992866	+	0.119239i	$0.961954\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 16.5945i	− 1.13174i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	36.0000	2.42712
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.9748i	− 1.12666i	−0.826232	−	0.563329i	$-0.809520\pi$
$227$	− 16.9748i	− 1.12666i	0.826232	−	0.563329i	$-0.190480\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−56.8444	−3.70812
$236$	− 17.9658i	− 1.16947i
$237$	0	0
$238$	0	0
$239$	− 30.2965i	− 1.95972i	−0.199693	−	0.979858i	$-0.563995\pi$
$239$	− 30.2965i	− 1.95972i	0.199693	−	0.979858i	$-0.436005\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	11.4375i	0.735231i
$243$	0	0
$244$	−11.5778	−0.741192
$245$	29.0404i	1.85532i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−18.7889	−1.18831
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	− 9.05789i	− 0.568342i
$255$	0	0
$256$	−7.05551	−0.440970
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	− 24.0160i	− 1.48941i
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	14.6056	0.882354
$275$	65.9977i	3.97981i
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	− 12.5610i	− 0.753361i
$279$	0	0
$280$	0	0
$281$	11.6852i	0.697083i	0.937293	+	0.348542i	$0.113323\pi$
$281$	11.6852i	0.697083i	−0.937293	+	0.348542i	$0.886677\pi$
$282$	0	0
$283$	−28.8444	−1.71462	−0.857311	−	0.514799i	$-0.827867\pi$
$283$	−28.8444	−1.71462	−0.857311	+	0.514799i	$0.827867\pi$
$284$	12.7110i	0.754262i
$285$	0	0
$286$	12.2389	0.723699
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.7187i	0.918297i	0.888359	+	0.459149i	$0.151846\pi$
$293$	15.7187i	0.918297i	−0.888359	+	0.459149i	$0.848154\pi$
$294$	0	0
$295$	46.4222	2.70281
$296$	0	0
$297$	0	0
$298$	10.9722	0.635605
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 29.9162i	− 1.71300i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−7.21110	−0.407596	−0.203798	−	0.979013i	$-0.565329\pi$
$313$	−7.21110	−0.407596	−0.203798	+	0.979013i	$0.565329\pi$
$314$	− 1.25610i	− 0.0708860i
$315$	0	0
$316$	23.1556	1.30260
$317$	− 34.0648i	− 1.91327i	−0.291290	−	0.956635i	$-0.594084\pi$
$317$	− 34.0648i	− 1.91327i	0.291290	−	0.956635i	$-0.405916\pi$
$318$	0	0
$319$	0	0
$320$	0.115151i	0.00643713i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	44.0278	2.44222
$326$	0	0
$327$	0	0
$328$	−3.70563	−0.204609
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0.610619i	0.0335121i
$333$	0	0
$334$	−15.3944	−0.842347
$335$	0	0
$336$	0	0
$337$	36.0555	1.96407	0.982034	−	0.188702i	$-0.0604279\pi$
$337$	36.0555	1.96407	0.982034	+	0.188702i	$0.0604279\pi$
$338$	− 8.16467i	− 0.444099i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	9.05789i	0.488369i
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−30.5500	−1.62832
$353$	− 28.2798i	− 1.50518i	−0.658490	−	0.752590i	$-0.728804\pi$
$353$	− 28.2798i	− 1.50518i	0.658490	−	0.752590i	$-0.271196\pi$
$354$	0	0
$355$	−32.8444	−1.74320
$356$	15.9490i	0.845296i
$357$	0	0
$358$	0	0
$359$	25.2721i	1.33381i	0.745143	+	0.666905i	$0.232381\pi$
$359$	25.2721i	1.33381i	−0.745143	+	0.666905i	$0.767619\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	6.28052i	0.330097i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	36.0555	1.86688	0.933442	−	0.358729i	$-0.116790\pi$
$373$	36.0555	1.86688	0.933442	+	0.358729i	$0.116790\pi$
$374$	0	0
$375$	0	0
$376$	31.0278	1.60013
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 12.9413i	− 0.661272i	−0.943758	−	0.330636i	$-0.892737\pi$
$383$	− 12.9413i	− 0.661272i	0.943758	−	0.330636i	$-0.107263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	− 15.8513i	− 0.800612i
$393$	0	0
$394$	−4.66106	−0.234821
$395$	59.8323i	3.01049i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	− 9.05789i	− 0.454031i
$399$	0	0
$400$	−21.8444	−1.09222
$401$	− 39.8498i	− 1.99001i	−0.0998435	−	0.995003i	$-0.531834\pi$
$401$	− 39.8498i	− 1.99001i	0.0998435	−	0.995003i	$-0.468166\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	− 4.26378i	− 0.210573i
$411$	0	0
$412$	−25.6888	−1.26560
$413$	0	0
$414$	0	0
$415$	−1.57779	−0.0774509
$416$	20.3802i	0.999224i
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	18.1158i	0.881862i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−10.4222	−0.502603
$431$	− 36.0815i	− 1.73799i	−0.494824	−	0.868993i	$-0.664767\pi$
$431$	− 36.0815i	− 1.73799i	0.494824	−	0.868993i	$-0.335233\pi$
$432$	0	0
$433$	36.0555	1.73272	0.866359	−	0.499422i	$-0.166454\pi$
$433$	36.0555	1.73272	0.866359	+	0.499422i	$0.166454\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	− 50.7745i	− 2.42058i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−41.2111	−1.95359
$446$	0	0
$447$	0	0
$448$	0	0
$449$	36.5770i	1.72618i	0.505054	+	0.863088i	$0.331473\pi$
$449$	36.5770i	1.72618i	−0.505054	+	0.863088i	$0.668527\pi$
$450$	0	0
$451$	−8.84441	−0.416467
$452$	0	0
$453$	0	0
$454$	−10.6611	−0.500349
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	42.3621i	1.97300i	0.163769	+	0.986499i	$0.447635\pi$
$461$	42.3621i	1.97300i	−0.163769	+	0.986499i	$0.552365\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	35.7012i	1.64677i
$471$	0	0
$472$	−25.3389	−1.16632
$473$	21.6189i	0.994039i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−19.0278	−0.870309
$479$	− 3.65316i	− 0.166917i	−0.996511	−	0.0834585i	$-0.973403\pi$
$479$	− 3.65316i	− 0.166917i	0.996511	−	0.0834585i	$-0.0265966\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−29.2389	−1.32904
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	16.3293i	0.739194i
$489$	0	0
$490$	18.2389	0.823948
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 48.0319i	− 2.14805i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	23.1556	1.02736
$509$	− 22.4947i	− 0.997060i	−0.866872	−	0.498530i	$-0.833873\pi$
$509$	− 22.4947i	− 0.997060i	0.866872	−	0.498530i	$-0.166127\pi$
$510$	0	0
$511$	0	0
$512$	− 18.2135i	− 0.804930i
$513$	0	0
$514$	0	0
$515$	− 66.3780i	− 2.92497i
$516$	0	0
$517$	74.0555	3.25696
$518$	0	0
$519$	0	0
$520$	−33.8722	−1.48539
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.90020i	0.255566i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 37.8331i	− 1.62959i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	37.3376i	1.59498i
$549$	0	0
$550$	41.4500	1.76743
$551$	0	0
$552$	0	0
$553$	0	0
$554$	13.8171i	0.587034i
$555$	0	0
$556$	32.1110	1.36181
$557$	30.7920i	1.30470i	0.757919	+	0.652349i	$0.226216\pi$
$557$	30.7920i	1.30470i	−0.757919	+	0.652349i	$0.773784\pi$
$558$	0	0
$559$	14.4222	0.609994
$560$	0	0
$561$	0	0
$562$	7.33894	0.309574
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	18.1158i	0.761463i
$567$	0	0
$568$	17.9277	0.752229
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−28.8444	−1.20710	−0.603550	−	0.797325i	$-0.706248\pi$
$571$	−28.8444	−1.20710	−0.603550	+	0.797325i	$0.706248\pi$
$572$	31.2874i	1.30819i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	10.6769i	0.444099i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	9.87217	0.407815
$587$	11.9504i	0.493246i	0.969111	+	0.246623i	$0.0793210\pi$
$587$	11.9504i	0.493246i	−0.969111	+	0.246623i	$0.920679\pi$
$588$	0	0
$589$	0	0
$590$	− 29.1555i	− 1.20031i
$591$	0	0
$592$	0	0
$593$	48.1471i	1.97716i	0.150683	+	0.988582i	$0.451853\pi$
$593$	48.1471i	1.97716i	−0.150683	+	0.988582i	$0.548147\pi$
$594$	0	0
$595$	0	0
$596$	28.0494i	1.14895i
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−7.21110	−0.294147	−0.147074	−	0.989126i	$-0.546985\pi$
$601$	−7.21110	−0.294147	−0.147074	+	0.989126i	$0.546985\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 75.5511i	− 3.07159i
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	0	0
$610$	−18.7889	−0.760740
$611$	− 49.4032i	− 1.99864i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.5038i	0.865709i	0.901464	+	0.432855i	$0.142494\pi$
$617$	21.5038i	0.865709i	−0.901464	+	0.432855i	$0.857506\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	63.0555	2.52222
$626$	4.52894i	0.181013i
$627$	0	0
$628$	3.21110	0.128137
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	− 32.6587i	− 1.29909i
$633$	0	0
$634$	−21.3944	−0.849682
$635$	59.8323i	2.37437i
$636$	0	0
$637$	−25.2389	−1.00000
$638$	0	0
$639$	0	0
$640$	46.9722	1.85674
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−60.4777	−2.37396
$650$	− 27.6517i	− 1.08459i
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	− 2.92739i	− 0.114295i
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0.861218	0.0334217
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 39.3544i	− 1.52267i
$669$	0	0
$670$	0	0
$671$	38.9741i	1.50458i
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	− 22.6447i	− 0.872242i
$675$	0	0
$676$	20.8722	0.802776
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 32.0481i	− 1.22628i	−0.789972	−	0.613142i	$-0.789905\pi$
$683$	− 32.0481i	− 1.22628i	0.789972	−	0.613142i	$-0.210095\pi$
$684$	0	0
$685$	−96.4777	−3.68622
$686$	0	0
$687$	0	0
$688$	−7.15559	−0.272804
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	82.9725i	3.14733i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	− 0.150016i	− 0.00565393i
$705$	0	0
$706$	−17.7611	−0.668449
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	20.6280i	0.774154i
$711$	0	0
$712$	22.4945	0.843018
$713$	0	0
$714$	0	0
$715$	−80.8444	−3.02341
$716$	0	0
$717$	0	0
$718$	15.8722	0.592344
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 11.9330i	− 0.444099i
$723$	0	0
$724$	−16.0555	−0.596698
$725$	0	0
$726$	0	0
$727$	14.4222	0.534890	0.267445	−	0.963573i	$-0.413821\pi$
$727$	14.4222	0.534890	0.267445	+	0.963573i	$0.413821\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	− 5.02441i	− 0.185454i
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 41.8666i	− 1.53594i	−0.640488	−	0.767968i	$-0.721268\pi$
$743$	− 41.8666i	− 1.53594i	0.640488	−	0.767968i	$-0.278732\pi$
$744$	0	0
$745$	−72.4777	−2.65538
$746$	− 22.6447i	− 0.829082i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	24.5114i	0.893840i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	36.0555	1.31046	0.655230	−	0.755429i	$-0.272572\pi$
$757$	36.0555	1.31046	0.655230	+	0.755429i	$0.272572\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 53.1715i	− 1.92747i	−0.266869	−	0.963733i	$-0.585989\pi$
$761$	− 53.1715i	− 1.92747i	0.266869	−	0.963733i	$-0.414011\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−8.12783	−0.293671
$767$	40.3453i	1.45678i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 47.3865i	− 1.70437i	−0.523238	−	0.852186i	$-0.675276\pi$
$773$	− 47.3865i	− 1.70437i	0.523238	−	0.852186i	$-0.324724\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	42.7889	1.53111
$782$	0	0
$783$	0	0
$784$	12.5223	0.447224
$785$	8.29725i	0.296142i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 11.9155i	− 0.424474i
$789$	0	0
$790$	37.5778	1.33696
$791$	0	0
$792$	0	0
$793$	26.0000	0.923287
$794$	0	0
$795$	0	0
$796$	23.1556	0.820728
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	69.0228i	2.44033i
$801$	0	0
$802$	−25.0278	−0.883761
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	10.8999	0.380642
$821$	40.6105i	1.41732i	0.705552	+	0.708658i	$0.250699\pi$
$821$	40.6105i	1.41732i	−0.705552	+	0.708658i	$0.749301\pi$
$822$	0	0
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	36.2316i	1.26219i
$825$	0	0
$826$	0	0
$827$	− 55.1882i	− 1.91908i	−0.281566	−	0.959542i	$-0.590854\pi$
$827$	− 55.1882i	− 1.91908i	0.281566	−	0.959542i	$-0.409146\pi$
$828$	0	0
$829$	−50.4777	−1.75316	−0.876582	−	0.481253i	$-0.840182\pi$
$829$	−50.4777	−1.75316	−0.876582	+	0.481253i	$0.840182\pi$
$830$	0.990937i	0.0343959i
$831$	0	0
$832$	−0.100077	−0.00346955
$833$	0	0
$834$	0	0
$835$	101.689	3.51909
$836$	0	0
$837$	0	0
$838$	0	0
$839$	57.7005i	1.99204i	0.0891249	+	0.996020i	$0.471593\pi$
$839$	57.7005i	1.99204i	−0.0891249	+	0.996020i	$0.528407\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	−46.3112	−1.59410
$845$	53.9321i	1.85532i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	57.6888	1.96832	0.984159	−	0.177290i	$-0.0567332\pi$
$859$	57.6888	1.96832	0.984159	+	0.177290i	$0.0567332\pi$
$860$	− 26.6433i	− 0.908530i
$861$	0	0
$862$	−22.6611	−0.771839
$863$	− 9.43821i	− 0.321280i	−0.987013	−	0.160640i	$-0.948644\pi$
$863$	− 9.43821i	− 0.321280i	0.987013	−	0.160640i	$-0.0513559\pi$
$864$	0	0
$865$	0	0
$866$	− 22.6447i	− 0.769499i
$867$	0	0
$868$	0	0
$869$	− 77.9481i	− 2.64421i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	25.1221i	0.847829i
$879$	0	0
$880$	40.1110	1.35214
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−28.8444	−0.970692	−0.485346	−	0.874322i	$-0.661306\pi$
$883$	−28.8444	−0.970692	−0.485346	+	0.874322i	$0.661306\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	25.8827i	0.867590i
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	22.9722	0.766594
$899$	0	0
$900$	0	0
$901$	0	0
$902$	5.55475i	0.184953i
$903$	0	0
$904$	0	0
$905$	− 41.4863i	− 1.37905i
$906$	0	0
$907$	57.6888	1.91553	0.957763	−	0.287559i	$-0.0928437\pi$
$907$	57.6888	1.91553	0.957763	+	0.287559i	$0.0928437\pi$
$908$	− 27.2539i	− 0.904454i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	2.05551	0.0680275
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	14.4222	0.475745	0.237872	−	0.971296i	$-0.423550\pi$
$919$	14.4222	0.475745	0.237872	+	0.971296i	$0.423550\pi$
$920$	0	0
$921$	0	0
$922$	26.6056	0.876207
$923$	− 28.5449i	− 0.939567i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 51.4199i	− 1.68703i	−0.537103	−	0.843517i	$-0.680481\pi$
$929$	− 51.4199i	− 1.68703i	0.537103	−	0.843517i	$-0.319519\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−50.4777	−1.64904	−0.824518	−	0.565836i	$-0.808553\pi$
$937$	−50.4777	−1.64904	−0.824518	+	0.565836i	$0.808553\pi$
$938$	0	0
$939$	0	0
$940$	−91.2666	−2.97679
$941$	55.6837i	1.81524i	0.419796	+	0.907619i	$0.362102\pi$
$941$	55.6837i	1.81524i	−0.419796	+	0.907619i	$0.637898\pi$
$942$	0	0
$943$	0	0
$944$	− 20.0174i	− 0.651510i
$945$	0	0
$946$	13.5778	0.441452
$947$	− 26.2630i	− 0.853433i	−0.904385	−	0.426717i	$-0.859670\pi$
$947$	− 26.2630i	− 0.853433i	0.904385	−	0.426717i	$-0.140330\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	− 48.6426i	− 1.57321i
$957$	0	0
$958$	−2.29437	−0.0741278
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	41.2385i	1.32546i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−12.8999	−0.412916
$977$	61.4688i	1.96656i	0.182100	+	0.983280i	$0.441711\pi$
$977$	61.4688i	1.96656i	−0.182100	+	0.983280i	$0.558289\pi$
$978$	0	0
$979$	53.6888	1.71590
$980$	46.6258i	1.48941i
$981$	0	0
$982$	0	0
$983$	34.5603i	1.10230i	0.834406	+	0.551151i	$0.185811\pi$
$983$	34.5603i	1.10230i	−0.834406	+	0.551151i	$0.814189\pi$
$984$	0	0
$985$	30.7889	0.981016
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	59.8323i	1.89681i
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.b.b.64.2	✓	4
3.2	odd	2	inner	117.2.b.b.64.3	yes	4
4.3	odd	2		1872.2.c.k.1585.4		4
12.11	even	2		1872.2.c.k.1585.1		4
13.5	odd	4		1521.2.a.t.1.2		4
13.8	odd	4		1521.2.a.t.1.3		4
13.12	even	2	inner	117.2.b.b.64.3	yes	4
39.5	even	4		1521.2.a.t.1.3		4
39.8	even	4		1521.2.a.t.1.2		4
39.38	odd	2	CM	117.2.b.b.64.2	✓	4
52.51	odd	2		1872.2.c.k.1585.1		4
156.155	even	2		1872.2.c.k.1585.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.b.b.64.2	✓	4	1.1	even	1	trivial
117.2.b.b.64.2	✓	4	39.38	odd	2	CM
117.2.b.b.64.3	yes	4	3.2	odd	2	inner
117.2.b.b.64.3	yes	4	13.12	even	2	inner
1521.2.a.t.1.2		4	13.5	odd	4
1521.2.a.t.1.2		4	39.8	even	4
1521.2.a.t.1.3		4	13.8	odd	4
1521.2.a.t.1.3		4	39.5	even	4
1872.2.c.k.1585.1		4	12.11	even	2
1872.2.c.k.1585.1		4	52.51	odd	2
1872.2.c.k.1585.4		4	4.3	odd	2
1872.2.c.k.1585.4		4	156.155	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.628052i q^{2} +1.60555 q^{4} +4.14863i q^{5} -2.26447i q^{8} +O(q^{10})\)	\(q-0.628052i q^{2} +1.60555 q^{4} +4.14863i q^{5} -2.26447i q^{8} +2.60555 q^{10} -5.40473i q^{11} -3.60555 q^{13} +1.78890 q^{16} +6.66083i q^{20} -3.39445 q^{22} -12.2111 q^{25} +2.26447i q^{26} -5.65246i q^{32} +9.39445 q^{40} -1.63642i q^{41} -4.00000 q^{43} -8.67757i q^{44} +13.7020i q^{47} +7.00000 q^{49} +7.66920i q^{50} -5.78890 q^{52} +22.4222 q^{55} -11.1898i q^{59} -7.21110 q^{61} +0.0277564 q^{64} -14.9581i q^{65} +7.91694i q^{71} +14.4222 q^{79} +7.42147i q^{80} -1.02776 q^{82} +0.380317i q^{83} +2.51221i q^{86} -12.2389 q^{88} +9.93367i q^{89} +8.60555 q^{94} -4.39636i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} - 4 q^{10} + 36 q^{16} - 28 q^{22} - 20 q^{25} + 52 q^{40} - 16 q^{43} + 28 q^{49} - 52 q^{52} + 32 q^{55} - 72 q^{64} + 68 q^{82} + 52 q^{88} + 20 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^10 + 36 * q^16 - 28 * q^22 - 20 * q^25 + 52 * q^40 - 16 * q^43 + 28 * q^49 - 52 * q^52 + 32 * q^55 - 72 * q^64 + 68 * q^82 + 52 * q^88 + 20 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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