LMFDB - Embedded newform 880.3.i.b.769.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,3,Mod(769,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.769"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	880.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$23.9782632637$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			769.1
Root			$0.500000 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	880.769
Dual form			880.3.i.b.769.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.31662i q^{3} +(0.500000 + 4.97494i) q^{5} -2.00000 q^{9} -11.0000 q^{11} +(16.5000 - 1.65831i) q^{15} +29.8496i q^{23} +(-24.5000 + 4.97494i) q^{25} -23.2164i q^{27} -37.0000 q^{31} +36.4829i q^{33} +69.6491i q^{37} +(-1.00000 - 9.94987i) q^{45} +79.5990i q^{47} -49.0000 q^{49} -79.5990i q^{53} +(-5.50000 - 54.7243i) q^{55} -107.000 q^{59} +129.348i q^{67} +99.0000 q^{69} +133.000 q^{71} +(16.5000 + 81.2573i) q^{75} -95.0000 q^{81} +97.0000 q^{89} +122.715i q^{93} +169.148i q^{97} +22.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - 4 q^{9} - 22 q^{11} + 33 q^{15} - 49 q^{25} - 74 q^{31} - 2 q^{45} - 98 q^{49} - 11 q^{55} - 214 q^{59} + 198 q^{69} + 266 q^{71} + 33 q^{75} - 190 q^{81} + 194 q^{89} + 44 q^{99}+O(q^{100}) $ 2 * q + q^5 - 4 * q^9 - 22 * q^11 + 33 * q^15 - 49 * q^25 - 74 * q^31 - 2 * q^45 - 98 * q^49 - 11 * q^55 - 214 * q^59 + 198 * q^69 + 266 * q^71 + 33 * q^75 - 190 * q^81 + 194 * q^89 + 44 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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			0
$2$		0			0
$3$		−	3.31662i		−	1.10554i	−0.833333	−	0.552771i	$-0.813571\pi$
$3$		−	3.31662i		−	1.10554i	0.833333	−	0.552771i	$-0.186429\pi$
$4$		0			0
$5$	0.500000	+	4.97494i	0.100000	+	0.994987i
$5$	0.500000	+	4.97494i	0.100000	+	0.994987i
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	−2.00000			−0.222222
$10$		0			0
$11$	−11.0000			−1.00000
$11$	−11.0000			−1.00000
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	16.5000	−	1.65831i	1.10000	−	0.110554i
$16$		0			0
$17$		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			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			29.8496i			1.29781i	0.760870	+	0.648905i	$0.224773\pi$
$23$			29.8496i			1.29781i	−0.760870	+	0.648905i	$0.775227\pi$
$24$		0			0
$25$	−24.5000	+	4.97494i	−0.980000	+	0.198997i
$26$		0			0
$27$		−	23.2164i		−	0.859866i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−37.0000			−1.19355			−0.596774	−	0.802409i	$-0.703551\pi$
$31$	−37.0000			−1.19355			−0.596774	+	0.802409i	$0.703551\pi$
$32$		0			0
$33$			36.4829i			1.10554i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			69.6491i			1.88241i	0.337838	+	0.941204i	$0.390304\pi$
$37$			69.6491i			1.88241i	−0.337838	+	0.941204i	$0.609696\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−1.00000	−	9.94987i	−0.0222222	−	0.221108i
$46$		0			0
$47$			79.5990i			1.69360i	0.531915	+	0.846798i	$0.321473\pi$
$47$			79.5990i			1.69360i	−0.531915	+	0.846798i	$0.678527\pi$
$48$		0			0
$49$	−49.0000			−1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	79.5990i		−	1.50187i	−0.660377	−	0.750934i	$-0.729604\pi$
$53$		−	79.5990i		−	1.50187i	0.660377	−	0.750934i	$-0.270396\pi$
$54$		0			0
$55$	−5.50000	−	54.7243i	−0.100000	−	0.994987i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−107.000			−1.81356			−0.906780	−	0.421605i	$-0.861467\pi$
$59$	−107.000			−1.81356			−0.906780	+	0.421605i	$0.861467\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			129.348i			1.93057i	0.261194	+	0.965286i	$0.415884\pi$
$67$			129.348i			1.93057i	−0.261194	+	0.965286i	$0.584116\pi$
$68$		0			0
$69$	99.0000			1.43478
$70$		0			0
$71$	133.000			1.87324			0.936620	−	0.350348i	$-0.113937\pi$
$71$	133.000			1.87324			0.936620	+	0.350348i	$0.113937\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	16.5000	+	81.2573i	0.220000	+	1.08343i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−95.0000			−1.17284
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	97.0000			1.08989			0.544944	−	0.838473i	$-0.316551\pi$
$89$	97.0000			1.08989			0.544944	+	0.838473i	$0.316551\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			122.715i			1.31952i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			169.148i			1.74379i	0.489691	+	0.871896i	$0.337110\pi$
$97$			169.148i			1.74379i	−0.489691	+	0.871896i	$0.662890\pi$
$98$		0			0
$99$	22.0000			0.222222
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$			79.5990i			0.772806i	0.922330	+	0.386403i	$0.126283\pi$
$103$			79.5990i			0.772806i	−0.922330	+	0.386403i	$0.873717\pi$
$104$		0			0
$105$		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			$\pi$
$110$		0			0
$111$	231.000			2.08108
$112$		0			0
$113$			69.6491i			0.616364i	0.951327	+	0.308182i	$0.0997206\pi$
$113$			69.6491i			0.616364i	−0.951327	+	0.308182i	$0.900279\pi$
$114$		0			0
$115$	−148.500	+	14.9248i	−1.29130	+	0.129781i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	121.000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−37.0000	−	119.398i	−0.296000	−	0.955188i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	115.500	−	11.6082i	0.855556	−	0.0859866i
$136$		0			0
$137$			69.6491i			0.508388i	0.967153	+	0.254194i	$0.0818101\pi$
$137$			69.6491i			0.508388i	−0.967153	+	0.254194i	$0.918190\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	264.000			1.87234
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$			162.515i			1.10554i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−18.5000	−	184.073i	−0.119355	−	1.18757i
$156$		0			0
$157$		−	228.847i		−	1.45762i	−0.684713	−	0.728812i	$-0.740073\pi$
$157$		−	228.847i		−	1.45762i	0.684713	−	0.728812i	$-0.259927\pi$
$158$		0			0
$159$	−264.000			−1.66038
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	318.396i		−	1.95335i	−0.214724	−	0.976675i	$-0.568885\pi$
$163$		−	318.396i		−	1.95335i	0.214724	−	0.976675i	$-0.431115\pi$
$164$		0			0
$165$	−181.500	+	18.2414i	−1.10000	+	0.110554i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			354.879i			2.00497i
$178$		0			0
$179$	83.0000			0.463687			0.231844	−	0.972753i	$-0.425524\pi$
$179$	83.0000			0.463687			0.231844	+	0.972753i	$0.425524\pi$
$180$		0			0
$181$	−263.000			−1.45304			−0.726519	−	0.687146i	$-0.758863\pi$
$181$	−263.000			−1.45304			−0.726519	+	0.687146i	$0.758863\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−346.500	+	34.8246i	−1.87297	+	0.188241i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−157.000			−0.821990			−0.410995	−	0.911638i	$-0.634819\pi$
$191$	−157.000			−0.821990			−0.410995	+	0.911638i	$0.634819\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−2.00000			−0.0100503			−0.00502513	−	0.999987i	$-0.501600\pi$
$199$	−2.00000			−0.0100503			−0.00502513	+	0.999987i	$0.501600\pi$
$200$		0			0
$201$	429.000			2.13433
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	59.6992i		−	0.288402i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		−	441.111i		−	2.07094i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			29.8496i			0.133855i	0.997758	+	0.0669274i	$0.0213196\pi$
$223$			29.8496i			0.133855i	−0.997758	+	0.0669274i	$0.978680\pi$
$224$		0			0
$225$	49.0000	−	9.94987i	0.217778	−	0.0442217i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−433.000			−1.89083			−0.945415	−	0.325869i	$-0.894343\pi$
$229$	−433.000			−1.89083			−0.945415	+	0.325869i	$0.894343\pi$
$230$		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
$235$	−396.000	+	39.7995i	−1.68511	+	0.169360i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$			106.132i			0.436757i
$244$		0			0
$245$	−24.5000	−	243.772i	−0.100000	−	0.994987i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−227.000			−0.904382			−0.452191	−	0.891921i	$-0.649358\pi$
$251$	−227.000			−0.904382			−0.452191	+	0.891921i	$0.649358\pi$
$252$		0			0
$253$		−	328.346i		−	1.29781i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	477.594i		−	1.85834i	−0.369650	−	0.929171i	$-0.620522\pi$
$257$		−	477.594i		−	1.85834i	0.369650	−	0.929171i	$-0.379478\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	396.000	−	39.7995i	1.49434	−	0.150187i
$266$		0			0
$267$		−	321.713i		−	1.20492i
$268$		0			0
$269$	362.000			1.34572			0.672862	−	0.739768i	$-0.265065\pi$
$269$	362.000			1.34572			0.672862	+	0.739768i	$0.265065\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	269.500	−	54.7243i	0.980000	−	0.198997i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$	74.0000			0.265233
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−289.000			−1.00000
$290$		0			0
$291$	561.000			1.92784
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	−53.5000	−	532.318i	−0.181356	−	1.80447i
$296$		0			0
$297$			255.380i			0.859866i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	264.000			0.854369
$310$		0			0
$311$	478.000			1.53698			0.768489	−	0.639863i	$-0.221009\pi$
$311$	478.000			1.53698			0.768489	+	0.639863i	$0.221009\pi$
$312$		0			0
$313$			567.143i			1.81196i	0.423323	+	0.905979i	$0.360864\pi$
$313$			567.143i			1.81196i	−0.423323	+	0.905979i	$0.639136\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	626.842i		−	1.97742i	−0.149842	−	0.988710i	$-0.547877\pi$
$317$		−	626.842i		−	1.97742i	0.149842	−	0.988710i	$-0.452123\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	563.000			1.70091			0.850453	−	0.526051i	$-0.176328\pi$
$331$	563.000			1.70091			0.850453	+	0.526051i	$0.176328\pi$
$332$		0			0
$333$		−	139.298i		−	0.418313i
$334$		0			0
$335$	−643.500	+	64.6742i	−1.92090	+	0.193057i
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$	231.000			0.681416
$340$		0			0
$341$	407.000			1.19355
$342$		0			0
$343$		0			0
$344$		0			0
$345$	49.5000	+	492.519i	0.143478	+	1.42759i
$346$		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
$352$		0			0
$353$		−	328.346i		−	0.930158i	−0.885269	−	0.465079i	$-0.846026\pi$
$353$		−	328.346i		−	0.930158i	0.885269	−	0.465079i	$-0.153974\pi$
$354$		0			0
$355$	66.5000	+	661.667i	0.187324	+	1.86385i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	361.000			1.00000
$362$		0			0
$363$		−	401.312i		−	1.10554i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	368.145i		−	1.00312i	−0.865123	−	0.501560i	$-0.832759\pi$
$367$		−	368.145i		−	1.00312i	0.865123	−	0.501560i	$-0.167241\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−396.000	+	122.715i	−1.05600	+	0.327240i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	133.000			0.350923			0.175462	−	0.984486i	$-0.443858\pi$
$379$	133.000			0.350923			0.175462	+	0.984486i	$0.443858\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			129.348i			0.337724i	0.985640	+	0.168862i	$0.0540093\pi$
$383$			129.348i			0.337724i	−0.985640	+	0.168862i	$0.945991\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−553.000			−1.42159			−0.710797	−	0.703397i	$-0.751666\pi$
$389$	−553.000			−1.42159			−0.710797	+	0.703397i	$0.751666\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	79.5990i		−	0.200501i	−0.994962	−	0.100251i	$-0.968036\pi$
$397$		−	79.5990i		−	0.200501i	0.994962	−	0.100251i	$-0.0319645\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−98.0000			−0.244389			−0.122195	−	0.992506i	$-0.538993\pi$
$401$	−98.0000			−0.244389			−0.122195	+	0.992506i	$0.538993\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−47.5000	−	472.619i	−0.117284	−	1.16696i
$406$		0			0
$407$		−	766.140i		−	1.88241i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	231.000			0.562044
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−262.000			−0.625298			−0.312649	−	0.949869i	$-0.601216\pi$
$419$	−262.000			−0.625298			−0.312649	+	0.949869i	$0.601216\pi$
$420$		0			0
$421$	742.000			1.76247			0.881235	−	0.472678i	$-0.156713\pi$
$421$	742.000			1.76247			0.881235	+	0.472678i	$0.156713\pi$
$422$		0			0
$423$		−	159.198i		−	0.376355i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$			865.639i			1.99917i	0.0288684	+	0.999583i	$0.490810\pi$
$433$			865.639i			1.99917i	−0.0288684	+	0.999583i	$0.509190\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	98.0000			0.222222
$442$		0			0
$443$		−	766.140i		−	1.72944i	−0.502257	−	0.864718i	$-0.667497\pi$
$443$		−	766.140i		−	1.72944i	0.502257	−	0.864718i	$-0.332503\pi$
$444$		0			0
$445$	48.5000	+	482.569i	0.108989	+	1.08442i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−623.000			−1.38753			−0.693764	−	0.720202i	$-0.744049\pi$
$449$	−623.000			−1.38753			−0.693764	+	0.720202i	$0.744049\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$			925.338i			1.99857i	0.0377970	+	0.999285i	$0.487966\pi$
$463$			925.338i			1.99857i	−0.0377970	+	0.999285i	$0.512034\pi$
$464$		0			0
$465$	−610.500	+	61.3576i	−1.31290	+	0.131952i
$466$		0			0
$467$			129.348i			0.276977i	0.990364	+	0.138489i	$0.0442244\pi$
$467$			129.348i			0.276977i	−0.990364	+	0.138489i	$0.955776\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−759.000			−1.61146
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$			159.198i			0.333748i
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−841.500	+	84.5739i	−1.73505	+	0.174379i
$486$		0			0
$487$			427.845i			0.878531i	0.898357	+	0.439266i	$0.144761\pi$
$487$			427.845i			0.878531i	−0.898357	+	0.439266i	$0.855239\pi$
$488$		0			0
$489$	−1056.00			−2.15951
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	11.0000	+	109.449i	0.0222222	+	0.221108i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−602.000			−1.20641			−0.603206	−	0.797585i	$-0.706110\pi$
$499$	−602.000			−1.20641			−0.603206	+	0.797585i	$0.706110\pi$
$500$		0			0
$501$		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
$507$			560.510i			1.10554i
$508$		0			0
$509$	1007.00			1.97839			0.989194	−	0.146609i	$-0.0468360\pi$
$509$	1007.00			1.97839			0.989194	+	0.146609i	$0.0468360\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−396.000	+	39.7995i	−0.768932	+	0.0772806i
$516$		0			0
$517$		−	875.589i		−	1.69360i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	817.000			1.56814			0.784069	−	0.620674i	$-0.213141\pi$
$521$	817.000			1.56814			0.784069	+	0.620674i	$0.213141\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−362.000			−0.684310
$530$		0			0
$531$	214.000			0.403013
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	275.280i		−	0.512625i
$538$		0			0
$539$	539.000			1.00000
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$			872.272i			1.60639i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	115.500	+	1149.21i	0.208108	+	2.07065i
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$	−346.500	+	34.8246i	−0.613274	+	0.0616364i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$			520.710i			0.908744i
$574$		0			0
$575$	−148.500	−	731.316i	−0.258261	−	1.27185i
$576$		0			0
$577$			467.644i			0.810475i	0.914211	+	0.405238i	$0.132811\pi$
$577$			467.644i			0.810475i	−0.914211	+	0.405238i	$0.867189\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			875.589i			1.50187i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	318.396i		−	0.542412i	−0.962521	−	0.271206i	$-0.912577\pi$
$587$		−	318.396i		−	0.542412i	0.962521	−	0.271206i	$-0.0874225\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			6.63325i			0.0111110i
$598$		0			0
$599$	98.0000			0.163606			0.0818030	−	0.996649i	$-0.473932\pi$
$599$	98.0000			0.163606			0.0818030	+	0.996649i	$0.473932\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		−	258.697i		−	0.429016i
$604$		0			0
$605$	60.5000	+	601.967i	0.100000	+	0.994987i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1114.39i			1.80614i	0.429498	+	0.903068i	$0.358691\pi$
$617$			1114.39i			1.80614i	−0.429498	+	0.903068i	$0.641309\pi$
$618$		0			0
$619$	−1237.00			−1.99838			−0.999192	−	0.0401853i	$-0.987205\pi$
$619$	−1237.00			−1.99838			−0.999192	+	0.0401853i	$0.987205\pi$
$620$		0			0
$621$	693.000			1.11594
$622$		0			0
$623$		0			0
$624$		0			0
$625$	575.500	−	243.772i	0.920800	−	0.390035i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1213.00			1.92235			0.961173	−	0.275947i	$-0.0889916\pi$
$631$	1213.00			1.92235			0.961173	+	0.275947i	$0.0889916\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−266.000			−0.416275
$640$		0			0
$641$	−743.000			−1.15913			−0.579563	−	0.814927i	$-0.696777\pi$
$641$	−743.000			−1.15913			−0.579563	+	0.814927i	$0.696777\pi$
$642$		0			0
$643$			1223.83i			1.90332i	0.307154	+	0.951660i	$0.400623\pi$
$643$			1223.83i			1.90332i	−0.307154	+	0.951660i	$0.599377\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1164.14i		−	1.79928i	−0.436631	−	0.899641i	$-0.643828\pi$
$647$		−	1164.14i		−	1.79928i	0.436631	−	0.899641i	$-0.356172\pi$
$648$		0			0
$649$	1177.00			1.81356
$650$		0			0
$651$		0			0
$652$		0			0
$653$			169.148i			0.259032i	0.991577	+	0.129516i	$0.0413424\pi$
$653$			169.148i			0.259032i	−0.991577	+	0.129516i	$0.958658\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1153.00			−1.74433			−0.872163	−	0.489215i	$-0.837283\pi$
$661$	−1153.00			−1.74433			−0.872163	+	0.489215i	$0.837283\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	99.0000			0.147982
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$	115.500	+	568.801i	0.171111	+	0.842668i
$676$		0			0
$677$		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
$683$		−	1114.39i		−	1.63160i	−0.578331	−	0.815802i	$-0.696296\pi$
$683$		−	1114.39i		−	1.63160i	0.578331	−	0.815802i	$-0.303704\pi$
$684$		0			0
$685$	−346.500	+	34.8246i	−0.505839	+	0.0508388i
$686$		0			0
$687$			1436.10i			2.09039i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1093.00			1.58177			0.790883	−	0.611968i	$-0.209622\pi$
$691$	1093.00			1.58177			0.790883	+	0.611968i	$0.209622\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	132.000	+	1313.38i	0.187234	+	1.86296i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1057.00			1.49083			0.745416	−	0.666599i	$-0.232251\pi$
$709$	1057.00			1.49083			0.745416	+	0.666599i	$0.232251\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1104.44i		−	1.54900i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1163.00			1.61752			0.808762	−	0.588136i	$-0.200138\pi$
$719$	1163.00			1.61752			0.808762	+	0.588136i	$0.200138\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			527.343i			0.725369i	0.931912	+	0.362685i	$0.118140\pi$
$727$			527.343i			0.725369i	−0.931912	+	0.362685i	$0.881860\pi$
$728$		0			0
$729$	−503.000			−0.689986
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	−808.500	+	81.2573i	−1.10000	+	0.110554i
$736$		0			0
$737$		−	1422.83i		−	1.93057i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	973.000			1.29561			0.647803	−	0.761808i	$-0.275688\pi$
$751$	973.000			1.29561			0.647803	+	0.761808i	$0.275688\pi$
$752$		0			0
$753$			752.874i			0.999832i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			1512.38i			1.99786i	0.0462351	+	0.998931i	$0.485278\pi$
$757$			1512.38i			1.99786i	−0.0462351	+	0.998931i	$0.514722\pi$
$758$		0			0
$759$	−1089.00			−1.43478
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−1584.00			−2.05447
$772$		0			0
$773$			716.391i			0.926767i	0.886158	+	0.463384i	$0.153365\pi$
$773$			716.391i			0.926767i	−0.886158	+	0.463384i	$0.846635\pi$
$774$		0			0
$775$	906.500	−	184.073i	1.16968	−	0.237513i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−1463.00			−1.87324
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1138.50	−	114.424i	1.45032	−	0.145762i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−132.000	−	1313.38i	−0.166038	−	1.65205i
$796$		0			0
$797$			169.148i			0.212231i	0.994354	+	0.106115i	$0.0338413\pi$
$797$			169.148i			0.212231i	−0.994354	+	0.106115i	$0.966159\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−194.000			−0.242197
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	1200.62i		−	1.48775i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1584.00	−	159.198i	1.94356	−	0.195335i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		−	1462.63i		−	1.77720i	−0.458688	−	0.888598i	$-0.651680\pi$
$823$		−	1462.63i		−	1.77720i	0.458688	−	0.888598i	$-0.348320\pi$
$824$		0			0
$825$	−181.500	−	893.830i	−0.220000	−	1.08343i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	817.000			0.985525			0.492762	−	0.870164i	$-0.335987\pi$
$829$	817.000			0.985525			0.492762	+	0.870164i	$0.335987\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$			859.006i			1.02629i
$838$		0			0
$839$	−347.000			−0.413588			−0.206794	−	0.978385i	$-0.566303\pi$
$839$	−347.000			−0.413588			−0.206794	+	0.978385i	$0.566303\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−84.5000	−	840.764i	−0.100000	−	0.994987i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2079.00			−2.44301
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		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
$859$	−757.000			−0.881257			−0.440629	−	0.897689i	$-0.645244\pi$
$859$	−757.000			−0.881257			−0.440629	+	0.897689i	$0.645244\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1671.58i			1.93694i	0.249131	+	0.968470i	$0.419855\pi$
$863$			1671.58i			1.93694i	−0.249131	+	0.968470i	$0.580145\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			958.505i			1.10554i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	338.296i		−	0.387509i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1487.00			1.68785			0.843927	−	0.536457i	$-0.180238\pi$
$881$	1487.00			1.68785			0.843927	+	0.536457i	$0.180238\pi$
$882$		0			0
$883$		−	1114.39i		−	1.26205i	−0.775764	−	0.631023i	$-0.782636\pi$
$883$		−	1114.39i		−	1.26205i	0.775764	−	0.631023i	$-0.217364\pi$
$884$		0			0
$885$	−1765.50	+	177.439i	−1.99492	+	0.200497i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1045.00			1.17284
$892$		0			0
$893$		0			0
$894$		0			0
$895$	41.5000	+	412.920i	0.0463687	+	0.461363i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−131.500	−	1308.41i	−0.145304	−	1.44576i
$906$		0			0
$907$			477.594i			0.526564i	0.964719	+	0.263282i	$0.0848050\pi$
$907$			477.594i			0.526564i	−0.964719	+	0.263282i	$0.915195\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1778.00			1.95170			0.975851	−	0.218439i	$-0.0700963\pi$
$911$	1778.00			1.95170			0.975851	+	0.218439i	$0.0700963\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−346.500	−	1706.40i	−0.374595	−	1.84476i
$926$		0			0
$927$		−	159.198i		−	0.171735i
$928$		0			0
$929$	−958.000			−1.03122			−0.515608	−	0.856824i	$-0.672434\pi$
$929$	−958.000			−1.03122			−0.515608	+	0.856824i	$0.672434\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	1585.35i		−	1.69919i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$	1881.00			2.00319
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1323.33i			1.39740i	0.715417	+	0.698698i	$0.246237\pi$
$947$			1323.33i			1.39740i	−0.715417	+	0.698698i	$0.753763\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−2079.00			−2.18612
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−78.5000	−	781.065i	−0.0821990	−	0.817869i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	408.000			0.424558
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	83.0000			0.0854789			0.0427394	−	0.999086i	$-0.486391\pi$
$971$	83.0000			0.0854789			0.0427394	+	0.999086i	$0.486391\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1522.33i		−	1.55817i	−0.626919	−	0.779084i	$-0.715684\pi$
$977$		−	1522.33i		−	1.55817i	0.626919	−	0.779084i	$-0.284316\pi$
$978$		0			0
$979$	−1067.00			−1.08989
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1860.63i		−	1.89280i	−0.322991	−	0.946402i	$-0.604688\pi$
$983$		−	1860.63i		−	1.89280i	0.322991	−	0.946402i	$-0.395312\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1582.00			−1.59637			−0.798184	−	0.602414i	$-0.794206\pi$
$991$	−1582.00			−1.59637			−0.798184	+	0.602414i	$0.794206\pi$
$992$		0			0
$993$		−	1867.26i		−	1.88042i
$994$		0			0
$995$	−1.00000	−	9.94987i	−0.00100503	−	0.00999987i
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$	1617.00			1.61862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.3.i.b.769.1		2
4.3	odd	2		55.3.d.a.54.2	yes	2
5.4	even	2	inner	880.3.i.b.769.2		2
11.10	odd	2	CM	880.3.i.b.769.1		2
12.11	even	2		495.3.h.a.109.1		2
20.3	even	4		275.3.c.d.76.1		2
20.7	even	4		275.3.c.d.76.2		2
20.19	odd	2		55.3.d.a.54.1	✓	2
44.43	even	2		55.3.d.a.54.2	yes	2
55.54	odd	2	inner	880.3.i.b.769.2		2
60.59	even	2		495.3.h.a.109.2		2
132.131	odd	2		495.3.h.a.109.1		2
220.43	odd	4		275.3.c.d.76.1		2
220.87	odd	4		275.3.c.d.76.2		2
220.219	even	2		55.3.d.a.54.1	✓	2
660.659	odd	2		495.3.h.a.109.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.3.d.a.54.1	✓	2	20.19	odd	2
55.3.d.a.54.1	✓	2	220.219	even	2
55.3.d.a.54.2	yes	2	4.3	odd	2
55.3.d.a.54.2	yes	2	44.43	even	2
275.3.c.d.76.1		2	20.3	even	4
275.3.c.d.76.1		2	220.43	odd	4
275.3.c.d.76.2		2	20.7	even	4
275.3.c.d.76.2		2	220.87	odd	4
495.3.h.a.109.1		2	12.11	even	2
495.3.h.a.109.1		2	132.131	odd	2
495.3.h.a.109.2		2	60.59	even	2
495.3.h.a.109.2		2	660.659	odd	2
880.3.i.b.769.1		2	1.1	even	1	trivial
880.3.i.b.769.1		2	11.10	odd	2	CM
880.3.i.b.769.2		2	5.4	even	2	inner
880.3.i.b.769.2		2	55.54	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	880.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.31662i q^{3} +(0.500000 + 4.97494i) q^{5} -2.00000 q^{9} -11.0000 q^{11} +(16.5000 - 1.65831i) q^{15} +29.8496i q^{23} +(-24.5000 + 4.97494i) q^{25} -23.2164i q^{27} -37.0000 q^{31} +36.4829i q^{33} +69.6491i q^{37} +(-1.00000 - 9.94987i) q^{45} +79.5990i q^{47} -49.0000 q^{49} -79.5990i q^{53} +(-5.50000 - 54.7243i) q^{55} -107.000 q^{59} +129.348i q^{67} +99.0000 q^{69} +133.000 q^{71} +(16.5000 + 81.2573i) q^{75} -95.0000 q^{81} +97.0000 q^{89} +122.715i q^{93} +169.148i q^{97} +22.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - 4 q^{9} - 22 q^{11} + 33 q^{15} - 49 q^{25} - 74 q^{31} - 2 q^{45} - 98 q^{49} - 11 q^{55} - 214 q^{59} + 198 q^{69} + 266 q^{71} + 33 q^{75} - 190 q^{81} + 194 q^{89} + 44 q^{99}+O(q^{100}) \) 2 * q + q^5 - 4 * q^9 - 22 * q^11 + 33 * q^15 - 49 * q^25 - 74 * q^31 - 2 * q^45 - 98 * q^49 - 11 * q^55 - 214 * q^59 + 198 * q^69 + 266 * q^71 + 33 * q^75 - 190 * q^81 + 194 * q^89 + 44 * q^99

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