LMFDB - Embedded newform 117.2.b.b.64.4

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			64.4
Root			$-0.835000i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.2.b.b.64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.75782i q^{2} -5.60555 q^{4} +1.67000i q^{5} -9.94345i q^{8} +O(q^{10})$	$q+2.75782i q^{2} -5.60555 q^{4} +1.67000i q^{5} -9.94345i q^{8} -4.60555 q^{10} +3.84563i q^{11} +3.60555 q^{13} +16.2111 q^{16} -9.36127i q^{20} -10.6056 q^{22} +2.21110 q^{25} +9.94345i q^{26} +24.8203i q^{32} +16.6056 q^{40} -12.7013i q^{41} -4.00000 q^{43} -21.5569i q^{44} -0.505635i q^{47} +7.00000 q^{49} +6.09781i q^{50} -20.2111 q^{52} -6.42221 q^{55} -10.5256i q^{59} +7.21110 q^{61} -36.0278 q^{64} +6.02127i q^{65} -14.8769i q^{71} -14.4222 q^{79} +27.0725i q^{80} +35.0278 q^{82} +18.2169i q^{83} -11.0313i q^{86} +38.2389 q^{88} +16.0413i q^{89} +1.39445 q^{94} +19.3047i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.75782i	1.95007i	0.222050	+	0.975035i	$0.428725\pi$
$2$	2.75782i	1.95007i	−0.222050	+	0.975035i	$0.571275\pi$
$3$	0	0
$3$	0	0
$4$	−5.60555	−2.80278
$5$	1.67000i	0.746846i	0.927661	+	0.373423i	$0.121816\pi$
$5$	1.67000i	0.746846i	−0.927661	+	0.373423i	$0.878184\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 9.94345i	− 3.51554i
$9$	0	0
$10$	−4.60555	−1.45640
$11$	3.84563i	1.15950i	0.814794	+	0.579751i	$0.196850\pi$
$11$	3.84563i	1.15950i	−0.814794	+	0.579751i	$0.803150\pi$
$12$	0	0
$13$	3.60555	1.00000
$13$	3.60555	1.00000
$14$	0	0
$15$	0	0
$16$	16.2111	4.05278
$17$	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			$\pi$
$20$	− 9.36127i	− 2.09324i
$21$	0	0
$22$	−10.6056	−2.26111
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	2.21110	0.442221
$26$	9.94345i	1.95007i
$27$	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			$\pi$
$32$	24.8203i	4.38766i
$33$	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
$40$	16.6056	2.62557
$41$	− 12.7013i	− 1.98360i	−0.127783	−	0.991802i	$-0.540786\pi$
$41$	− 12.7013i	− 1.98360i	0.127783	−	0.991802i	$-0.459214\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	− 21.5569i	− 3.24982i
$45$	0	0
$46$	0	0
$47$	− 0.505635i	− 0.0737545i	−0.999320	−	0.0368772i	$-0.988259\pi$
$47$	− 0.505635i	− 0.0737545i	0.999320	−	0.0368772i	$-0.0117410\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	6.09781i	0.862361i
$51$	0	0
$52$	−20.2111	−2.80278
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−6.42221	−0.865970
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.5256i	− 1.37032i	−0.728392	−	0.685160i	$-0.759732\pi$
$59$	− 10.5256i	− 1.37032i	0.728392	−	0.685160i	$-0.240268\pi$
$60$	0	0
$61$	7.21110	0.923287	0.461644	−	0.887066i	$-0.347260\pi$
$61$	7.21110	0.923287	0.461644	+	0.887066i	$0.347260\pi$
$62$	0	0
$63$	0	0
$64$	−36.0278	−4.50347
$65$	6.02127i	0.746846i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 14.8769i	− 1.76556i	−0.469784	−	0.882782i	$-0.655668\pi$
$71$	− 14.8769i	− 1.76556i	0.469784	−	0.882782i	$-0.344332\pi$
$72$	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	$-0.801246\pi$
$79$	−14.4222	−1.62262	−0.811312	+	0.584613i	$0.801246\pi$
$80$	27.0725i	3.02680i
$81$	0	0
$82$	35.0278	3.86817
$83$	18.2169i	1.99956i	0.0208726	+	0.999782i	$0.493356\pi$
$83$	18.2169i	1.99956i	−0.0208726	+	0.999782i	$0.506644\pi$
$84$	0	0
$85$	0	0
$86$	− 11.0313i	− 1.18953i
$87$	0	0
$88$	38.2389	4.07628
$89$	16.0413i	1.70037i	0.526484	+	0.850185i	$0.323510\pi$
$89$	16.0413i	1.70037i	−0.526484	+	0.850185i	$0.676490\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	1.39445	0.143826
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	19.3047i	1.95007i
$99$	0	0
$100$	−12.3944	−1.23944
$101$	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	$0.789020\pi$
$104$	− 35.8516i	− 3.51554i
$105$	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			$\pi$
$110$	− 17.7113i	− 1.68870i
$111$	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
$118$	29.0278	2.67222
$119$	0	0
$120$	0	0
$121$	−3.78890	−0.344445
$122$	19.8869i	1.80048i
$123$	0	0
$124$	0	0
$125$	12.0425i	1.07712i
$126$	0	0
$127$	−14.4222	−1.27976	−0.639882	−	0.768473i	$-0.721017\pi$
$127$	−14.4222	−1.27976	−0.639882	+	0.768473i	$0.721017\pi$
$128$	− 49.7172i	− 4.39442i
$129$	0	0
$130$	−16.6056	−1.45640
$131$	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
$137$	− 2.68127i	− 0.229076i	−0.993419	−	0.114538i	$-0.963461\pi$
$137$	− 2.68127i	− 0.229076i	0.993419	−	0.114538i	$-0.0365388\pi$
$138$	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
$142$	41.0278	3.44297
$143$	13.8656i	1.15950i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 17.0525i	− 1.39700i	−0.715611	−	0.698499i	$-0.753852\pi$
$149$	− 17.0525i	− 1.39700i	0.715611	−	0.698499i	$-0.246148\pi$
$150$	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	$0.474569\pi$
$158$	− 39.7738i	− 3.16423i
$159$	0	0
$160$	−41.4500	−3.27691
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	71.1976i	5.55960i
$165$	0	0
$166$	−50.2389	−3.89929
$167$	8.19690i	0.634295i	0.948376	+	0.317148i	$0.102725\pi$
$167$	8.19690i	0.634295i	−0.948376	+	0.317148i	$0.897275\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	22.4222	1.70968
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	62.3420i	4.69920i
$177$	0	0
$178$	−44.2389	−3.31584
$179$	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
$188$	2.83436i	0.206717i
$189$	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
$196$	−39.2389	−2.80278
$197$	− 27.0725i	− 1.92884i	−0.264379	−	0.964419i	$-0.585167\pi$
$197$	− 27.0725i	− 1.92884i	0.264379	−	0.964419i	$-0.414833\pi$
$198$	0	0
$199$	−14.4222	−1.02236	−0.511182	−	0.859473i	$-0.670792\pi$
$199$	−14.4222	−1.02236	−0.511182	+	0.859473i	$0.670792\pi$
$200$	− 21.9860i	− 1.55464i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	21.2111	1.48145
$206$	− 44.1251i	− 3.07434i
$207$	0	0
$208$	58.4500	4.05278
$209$	0	0
$210$	0	0
$211$	28.8444	1.98573	0.992866	−	0.119239i	$-0.0380456\pi$
$211$	28.8444	1.98573	0.992866	+	0.119239i	$0.0380456\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 6.68000i	− 0.455572i
$216$	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
$227$	− 24.8969i	− 1.65246i	−0.563329	−	0.826232i	$-0.690480\pi$
$227$	− 24.8969i	− 1.65246i	0.563329	−	0.826232i	$-0.309520\pi$
$228$	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
$235$	0.844410	0.0550832
$236$	59.0020i	3.84070i
$237$	0	0
$238$	0	0
$239$	− 6.17436i	− 0.399386i	−0.979858	−	0.199693i	$-0.936005\pi$
$239$	− 6.17436i	− 0.399386i	0.979858	−	0.199693i	$-0.0639945\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	− 10.4491i	− 0.671693i
$243$	0	0
$244$	−40.4222	−2.58777
$245$	11.6900i	0.746846i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−33.2111	−2.10045
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	− 39.7738i	− 2.49563i
$255$	0	0
$256$	65.0555	4.06597
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	− 33.7525i	− 2.09324i
$261$	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
$274$	7.39445	0.446715
$275$	8.50309i	0.512756i
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	55.1563i	3.30806i
$279$	0	0
$280$	0	0
$281$	− 31.4238i	− 1.87459i	−0.348542	−	0.937293i	$-0.613323\pi$
$281$	− 31.4238i	− 1.87459i	0.348542	−	0.937293i	$-0.386677\pi$
$282$	0	0
$283$	28.8444	1.71462	0.857311	−	0.514799i	$-0.172133\pi$
$283$	28.8444	1.71462	0.857311	+	0.514799i	$0.172133\pi$
$284$	83.3932i	4.94848i
$285$	0	0
$286$	−38.2389	−2.26111
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.4125i	1.77672i	0.459149	+	0.888359i	$0.348154\pi$
$293$	30.4125i	1.77672i	−0.459149	+	0.888359i	$0.651846\pi$
$294$	0	0
$295$	17.5778	1.02342
$296$	0	0
$297$	0	0
$298$	47.0278	2.72424
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.0425i	0.689554i
$306$	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	$-0.434671\pi$
$313$	7.21110	0.407596	0.203798	+	0.979013i	$0.434671\pi$
$314$	5.51563i	0.311265i
$315$	0	0
$316$	80.8444	4.54785
$317$	10.3725i	0.582580i	0.956635	+	0.291290i	$0.0940844\pi$
$317$	10.3725i	0.582580i	−0.956635	+	0.291290i	$0.905916\pi$
$318$	0	0
$319$	0	0
$320$	− 60.1663i	− 3.36340i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	7.97224	0.442221
$326$	0	0
$327$	0	0
$328$	−126.294	−6.97344
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 102.116i	− 5.60433i
$333$	0	0
$334$	−22.6056	−1.23692
$335$	0	0
$336$	0	0
$337$	−36.0555	−1.96407	−0.982034	−	0.188702i	$-0.939572\pi$
$337$	−36.0555	−1.96407	−0.982034	+	0.188702i	$0.939572\pi$
$338$	35.8516i	1.95007i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	39.7738i	2.14446i
$345$	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
$352$	−95.4500	−5.08750
$353$	24.7438i	1.31698i	0.752590	+	0.658490i	$0.228804\pi$
$353$	24.7438i	1.31698i	−0.752590	+	0.658490i	$0.771196\pi$
$354$	0	0
$355$	24.8444	1.31860
$356$	− 89.9201i	− 4.76576i
$357$	0	0
$358$	0	0
$359$	28.2369i	1.49029i	0.666905	+	0.745143i	$0.267619\pi$
$359$	28.2369i	1.49029i	−0.666905	+	0.745143i	$0.732381\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	− 27.5782i	− 1.44948i
$363$	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	$-0.883210\pi$
$373$	−36.0555	−1.86688	−0.933442	+	0.358729i	$0.883210\pi$
$374$	0	0
$375$	0	0
$376$	−5.02776	−0.259287
$377$	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
$383$	36.9394i	1.88752i	0.330636	+	0.943758i	$0.392737\pi$
$383$	36.9394i	1.88752i	−0.330636	+	0.943758i	$0.607263\pi$
$384$	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
$392$	− 69.6041i	− 3.51554i
$393$	0	0
$394$	74.6611	3.76137
$395$	− 24.0851i	− 1.21185i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	− 39.7738i	− 1.99368i
$399$	0	0
$400$	35.8444	1.79222
$401$	− 3.99873i	− 0.199687i	−0.995003	−	0.0998435i	$-0.968166\pi$
$401$	− 3.99873i	− 0.199687i	0.995003	−	0.0998435i	$-0.0318342\pi$
$402$	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			$\pi$
$410$	58.4963i	2.88893i
$411$	0	0
$412$	89.6888	4.41865
$413$	0	0
$414$	0	0
$415$	−30.4222	−1.49337
$416$	89.4910i	4.38766i
$417$	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			$\pi$
$422$	79.5476i	3.87232i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	18.4222	0.888398
$431$	− 20.5456i	− 0.989648i	−0.868993	−	0.494824i	$-0.835233\pi$
$431$	− 20.5456i	− 0.989648i	0.868993	−	0.494824i	$-0.164767\pi$
$432$	0	0
$433$	−36.0555	−1.73272	−0.866359	−	0.499422i	$-0.833546\pi$
$433$	−36.0555	−1.73272	−0.866359	+	0.499422i	$0.833546\pi$
$434$	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	$0.903661\pi$
$440$	63.8589i	3.04435i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−26.7889	−1.26992
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 21.4038i	− 1.01011i	−0.863088	−	0.505054i	$-0.831473\pi$
$449$	− 21.4038i	− 1.01011i	0.863088	−	0.505054i	$-0.168527\pi$
$450$	0	0
$451$	48.8444	2.29999
$452$	0	0
$453$	0	0
$454$	68.6611	3.22242
$455$	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
$461$	− 7.03254i	− 0.327538i	−0.986499	−	0.163769i	$-0.947635\pi$
$461$	− 7.03254i	− 0.327538i	0.986499	−	0.163769i	$-0.0523652\pi$
$462$	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
$470$	2.32873i	0.107416i
$471$	0	0
$472$	−104.661	−4.81742
$473$	− 15.3825i	− 0.707290i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	17.0278	0.778831
$479$	− 43.6194i	− 1.99302i	−0.0834585	−	0.996511i	$-0.526597\pi$
$479$	− 43.6194i	− 1.99302i	0.0834585	−	0.996511i	$-0.473403\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	21.2389	0.965403
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	− 71.7032i	− 3.24585i
$489$	0	0
$490$	−32.2389	−1.45640
$491$	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			$\pi$
$500$	− 67.5051i	− 3.01892i
$501$	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
$508$	80.8444	3.58689
$509$	39.1151i	1.73374i	0.498530	+	0.866872i	$0.333873\pi$
$509$	39.1151i	1.73374i	−0.498530	+	0.866872i	$0.666127\pi$
$510$	0	0
$511$	0	0
$512$	79.9767i	3.53450i
$513$	0	0
$514$	0	0
$515$	− 26.7200i	− 1.17742i
$516$	0	0
$517$	1.94449	0.0855185
$518$	0	0
$519$	0	0
$520$	59.8722	2.62557
$521$	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
$533$	− 45.7951i	− 1.98360i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	26.9194i	1.15950i
$540$	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	$0.704275\pi$
$548$	15.0300i	0.642049i
$549$	0	0
$550$	−23.4500	−0.999910
$551$	0	0
$552$	0	0
$553$	0	0
$554$	− 60.6720i	− 2.57770i
$555$	0	0
$556$	−112.111	−4.75457
$557$	− 35.7751i	− 1.51584i	−0.652349	−	0.757919i	$-0.726216\pi$
$557$	− 35.7751i	− 1.51584i	0.652349	−	0.757919i	$-0.273784\pi$
$558$	0	0
$559$	−14.4222	−0.609994
$560$	0	0
$561$	0	0
$562$	86.6611	3.65558
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	79.5476i	3.34363i
$567$	0	0
$568$	−147.928	−6.20691
$569$	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	$-0.293752\pi$
$571$	28.8444	1.20710	0.603550	+	0.797325i	$0.293752\pi$
$572$	− 77.7245i	− 3.24982i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 46.8829i	− 1.95007i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−83.8722	−3.46473
$587$	46.9594i	1.93822i	0.246623	+	0.969111i	$0.420679\pi$
$587$	46.9594i	1.93822i	−0.246623	+	0.969111i	$0.579321\pi$
$588$	0	0
$589$	0	0
$590$	48.4763i	1.99574i
$591$	0	0
$592$	0	0
$593$	7.33873i	0.301365i	0.988582	+	0.150683i	$0.0481472\pi$
$593$	7.33873i	0.301365i	−0.988582	+	0.150683i	$0.951853\pi$
$594$	0	0
$595$	0	0
$596$	95.5889i	3.91547i
$597$	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	$-0.453015\pi$
$601$	7.21110	0.294147	0.147074	+	0.989126i	$0.453015\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.32746i	− 0.257248i
$606$	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
$610$	−33.2111	−1.34468
$611$	− 1.82309i	− 0.0737545i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	44.7838i	1.80293i	0.432855	+	0.901464i	$0.357506\pi$
$617$	44.7838i	1.80293i	−0.432855	+	0.901464i	$0.642494\pi$
$618$	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
$625$	−9.05551	−0.362221
$626$	19.8869i	0.794840i
$627$	0	0
$628$	−11.2111	−0.447372
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	143.406i	5.70440i
$633$	0	0
$634$	−28.6056	−1.13607
$635$	− 24.0851i	− 0.955787i
$636$	0	0
$637$	25.2389	1.00000
$638$	0	0
$639$	0	0
$640$	83.0278	3.28196
$641$	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
$649$	40.4777	1.58889
$650$	21.9860i	0.862361i
$651$	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
$656$	− 205.902i	− 8.03910i
$657$	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
$664$	181.139	7.02955
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 45.9482i	− 1.77779i
$669$	0	0
$670$	0	0
$671$	27.7313i	1.07055i
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	− 99.4345i	− 3.83007i
$675$	0	0
$676$	−72.8722	−2.80278
$677$	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
$683$	41.2907i	1.57994i	0.613142	+	0.789972i	$0.289905\pi$
$683$	41.2907i	1.57994i	−0.613142	+	0.789972i	$0.710095\pi$
$684$	0	0
$685$	4.47772	0.171085
$686$	0	0
$687$	0	0
$688$	−64.8444	−2.47217
$689$	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
$695$	33.4000i	1.26693i
$696$	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
$704$	− 138.550i	− 5.22178i
$705$	0	0
$706$	−68.2389	−2.56820
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	68.5163i	2.57137i
$711$	0	0
$712$	159.505	5.97772
$713$	0	0
$714$	0	0
$715$	−23.1556	−0.865970
$716$	0	0
$717$	0	0
$718$	−77.8722	−2.90616
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	52.3985i	1.95007i
$723$	0	0
$724$	56.0555	2.08329
$725$	0	0
$726$	0	0
$727$	−14.4222	−0.534890	−0.267445	−	0.963573i	$-0.586179\pi$
$727$	−14.4222	−0.534890	−0.267445	+	0.963573i	$0.586179\pi$
$728$	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			$\pi$
$734$	22.0625i	0.814343i
$735$	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
$743$	− 34.9169i	− 1.28098i	−0.767968	−	0.640488i	$-0.778732\pi$
$743$	− 34.9169i	− 1.28098i	0.767968	−	0.640488i	$-0.221268\pi$
$744$	0	0
$745$	28.4777	1.04334
$746$	− 99.4345i	− 3.64056i
$747$	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	$0.760392\pi$
$752$	− 8.19690i	− 0.298910i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−36.0555	−1.31046	−0.655230	−	0.755429i	$-0.727428\pi$
$757$	−36.0555	−1.31046	−0.655230	+	0.755429i	$0.727428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.7238i	0.533738i	0.963733	+	0.266869i	$0.0859890\pi$
$761$	14.7238i	0.533738i	−0.963733	+	0.266869i	$0.914011\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−101.872	−3.68079
$767$	− 37.9507i	− 1.37032i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29.0951i	1.04648i	0.852186	+	0.523238i	$0.175276\pi$
$773$	29.0951i	1.04648i	−0.852186	+	0.523238i	$0.824724\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	57.2111	2.04717
$782$	0	0
$783$	0	0
$784$	113.478	4.05278
$785$	3.34000i	0.119210i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	151.756i	5.40610i
$789$	0	0
$790$	66.4222	2.36320
$791$	0	0
$792$	0	0
$793$	26.0000	0.923287
$794$	0	0
$795$	0	0
$796$	80.8444	2.86545
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	54.8803i	1.94031i
$801$	0	0
$802$	11.0278	0.389404
$803$	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
$820$	−118.900	−4.15217
$821$	40.4325i	1.41110i	0.708658	+	0.705552i	$0.249301\pi$
$821$	40.4325i	1.41110i	−0.708658	+	0.705552i	$0.750699\pi$
$822$	0	0
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	159.095i	5.54234i
$825$	0	0
$826$	0	0
$827$	− 16.1944i	− 0.563133i	−0.959542	−	0.281566i	$-0.909146\pi$
$827$	− 16.1944i	− 0.563133i	0.959542	−	0.281566i	$-0.0908540\pi$
$828$	0	0
$829$	50.4777	1.75316	0.876582	−	0.481253i	$-0.159818\pi$
$829$	50.4777	1.75316	0.876582	+	0.481253i	$0.159818\pi$
$830$	− 83.8989i	− 2.91217i
$831$	0	0
$832$	−129.900	−4.50347
$833$	0	0
$834$	0	0
$835$	−13.6888	−0.473721
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.16309i	0.178250i	0.996020	+	0.0891249i	$0.0284070\pi$
$839$	5.16309i	0.178250i	−0.996020	+	0.0891249i	$0.971593\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	−161.689	−5.56556
$845$	21.7100i	0.746846i
$846$	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	$-0.943267\pi$
$859$	−57.6888	−1.96832	−0.984159	+	0.177290i	$0.943267\pi$
$860$	37.4451i	1.27687i
$861$	0	0
$862$	56.6611	1.92988
$863$	− 57.9907i	− 1.97403i	−0.160640	−	0.987013i	$-0.551356\pi$
$863$	− 57.9907i	− 1.97403i	0.160640	−	0.987013i	$-0.448644\pi$
$864$	0	0
$865$	0	0
$866$	− 99.4345i	− 3.37892i
$867$	0	0
$868$	0	0
$869$	− 55.4625i	− 1.88144i
$870$	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			$\pi$
$878$	− 110.313i	− 3.72287i
$879$	0	0
$880$	−104.111	−3.50958
$881$	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	$-0.338694\pi$
$883$	28.8444	0.970692	0.485346	+	0.874322i	$0.338694\pi$
$884$	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
$890$	− 73.8789i	− 2.47643i
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	59.0278	1.96978
$899$	0	0
$900$	0	0
$901$	0	0
$902$	134.704i	4.48515i
$903$	0	0
$904$	0	0
$905$	− 16.7000i	− 0.555127i
$906$	0	0
$907$	−57.6888	−1.91553	−0.957763	−	0.287559i	$-0.907156\pi$
$907$	−57.6888	−1.91553	−0.957763	+	0.287559i	$0.907156\pi$
$908$	139.561i	4.63149i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−70.0555	−2.31850
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.4222	−0.475745	−0.237872	−	0.971296i	$-0.576450\pi$
$919$	−14.4222	−0.475745	−0.237872	+	0.971296i	$0.576450\pi$
$920$	0	0
$921$	0	0
$922$	19.3944	0.638722
$923$	− 53.6394i	− 1.76556i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 32.7413i	− 1.07421i	−0.843517	−	0.537103i	$-0.819519\pi$
$929$	− 32.7413i	− 1.07421i	0.843517	−	0.537103i	$-0.180481\pi$
$930$	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	$-0.191447\pi$
$937$	50.4777	1.64904	0.824518	+	0.565836i	$0.191447\pi$
$938$	0	0
$939$	0	0
$940$	−4.73338	−0.154386
$941$	− 25.7551i	− 0.839591i	−0.907619	−	0.419796i	$-0.862102\pi$
$941$	− 25.7551i	− 0.839591i	0.907619	−	0.419796i	$-0.137898\pi$
$942$	0	0
$943$	0	0
$944$	− 170.632i	− 5.55360i
$945$	0	0
$946$	42.4222	1.37926
$947$	55.6620i	1.80877i	0.426717	+	0.904385i	$0.359670\pi$
$947$	55.6620i	1.80877i	−0.426717	+	0.904385i	$0.640330\pi$
$948$	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
$956$	34.6107i	1.11939i
$957$	0	0
$958$	120.294	3.88653
$959$	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			$\pi$
$968$	37.6747i	1.21091i
$969$	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
$976$	116.900	3.74188
$977$	− 11.3838i	− 0.364200i	−0.983280	−	0.182100i	$-0.941711\pi$
$977$	− 11.3838i	− 0.364200i	0.983280	−	0.182100i	$-0.0582895\pi$
$978$	0	0
$979$	−61.6888	−1.97158
$980$	− 65.5289i	− 2.09324i
$981$	0	0
$982$	0	0
$983$	− 52.3220i	− 1.66881i	−0.551151	−	0.834406i	$-0.685811\pi$
$983$	− 52.3220i	− 1.66881i	0.551151	−	0.834406i	$-0.314189\pi$
$984$	0	0
$985$	45.2111	1.44055
$986$	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
$995$	− 24.0851i	− 0.763548i
$996$	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.64.4	yes	4
3.2	odd	2	inner	117.2.b.b.64.1	✓	4
4.3	odd	2		1872.2.c.k.1585.3		4
12.11	even	2		1872.2.c.k.1585.2		4
13.5	odd	4		1521.2.a.t.1.4		4
13.8	odd	4		1521.2.a.t.1.1		4
13.12	even	2	inner	117.2.b.b.64.1	✓	4
39.5	even	4		1521.2.a.t.1.1		4
39.8	even	4		1521.2.a.t.1.4		4
39.38	odd	2	CM	117.2.b.b.64.4	yes	4
52.51	odd	2		1872.2.c.k.1585.2		4
156.155	even	2		1872.2.c.k.1585.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.b.b.64.1	✓	4	3.2	odd	2	inner
117.2.b.b.64.1	✓	4	13.12	even	2	inner
117.2.b.b.64.4	yes	4	1.1	even	1	trivial
117.2.b.b.64.4	yes	4	39.38	odd	2	CM
1521.2.a.t.1.1		4	13.8	odd	4
1521.2.a.t.1.1		4	39.5	even	4
1521.2.a.t.1.4		4	13.5	odd	4
1521.2.a.t.1.4		4	39.8	even	4
1872.2.c.k.1585.2		4	12.11	even	2
1872.2.c.k.1585.2		4	52.51	odd	2
1872.2.c.k.1585.3		4	4.3	odd	2
1872.2.c.k.1585.3		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)

\(f(q)\)	\(=\)	\(q+2.75782i q^{2} -5.60555 q^{4} +1.67000i q^{5} -9.94345i q^{8} +O(q^{10})\)	\(q+2.75782i q^{2} -5.60555 q^{4} +1.67000i q^{5} -9.94345i q^{8} -4.60555 q^{10} +3.84563i q^{11} +3.60555 q^{13} +16.2111 q^{16} -9.36127i q^{20} -10.6056 q^{22} +2.21110 q^{25} +9.94345i q^{26} +24.8203i q^{32} +16.6056 q^{40} -12.7013i q^{41} -4.00000 q^{43} -21.5569i q^{44} -0.505635i q^{47} +7.00000 q^{49} +6.09781i q^{50} -20.2111 q^{52} -6.42221 q^{55} -10.5256i q^{59} +7.21110 q^{61} -36.0278 q^{64} +6.02127i q^{65} -14.8769i q^{71} -14.4222 q^{79} +27.0725i q^{80} +35.0278 q^{82} +18.2169i q^{83} -11.0313i q^{86} +38.2389 q^{88} +16.0413i q^{89} +1.39445 q^{94} +19.3047i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 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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