LMFDB - Embedded newform 220.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [220,4,Mod(1,220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("220.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	220.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.9804202013$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	220.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+5.00000 q^{3} -5.00000 q^{5} -19.0000 q^{7} -2.00000 q^{9} -11.0000 q^{11} -62.0000 q^{13} -25.0000 q^{15} +19.0000 q^{17} -131.000 q^{19} -95.0000 q^{21} +138.000 q^{23} +25.0000 q^{25} -145.000 q^{27} -79.0000 q^{29} +217.000 q^{31} -55.0000 q^{33} +95.0000 q^{35} -91.0000 q^{37} -310.000 q^{39} +158.000 q^{41} +120.000 q^{43} +10.0000 q^{45} -546.000 q^{47} +18.0000 q^{49} +95.0000 q^{51} -439.000 q^{53} +55.0000 q^{55} -655.000 q^{57} +290.000 q^{59} -373.000 q^{61} +38.0000 q^{63} +310.000 q^{65} +728.000 q^{67} +690.000 q^{69} -709.000 q^{71} +850.000 q^{73} +125.000 q^{75} +209.000 q^{77} -1194.00 q^{79} -671.000 q^{81} +58.0000 q^{83} -95.0000 q^{85} -395.000 q^{87} +753.000 q^{89} +1178.00 q^{91} +1085.00 q^{93} +655.000 q^{95} +1228.00 q^{97} +22.0000 q^{99} +O(q^{100})$

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
$3$	5.00000	0.962250	0.481125	−	0.876652i	$-0.340228\pi$
$3$	5.00000	0.962250	0.481125	+	0.876652i	$0.340228\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−19.0000	−1.02590	−0.512952	−	0.858417i	$-0.671448\pi$
$7$	−19.0000	−1.02590	−0.512952	+	0.858417i	$0.671448\pi$
$8$	0	0
$9$	−2.00000	−0.0740741
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−62.0000	−1.32275	−0.661373	−	0.750057i	$-0.730026\pi$
$13$	−62.0000	−1.32275	−0.661373	+	0.750057i	$0.730026\pi$
$14$	0	0
$15$	−25.0000	−0.430331
$16$	0	0
$17$	19.0000	0.271069	0.135535	−	0.990773i	$-0.456725\pi$
$17$	19.0000	0.271069	0.135535	+	0.990773i	$0.456725\pi$
$18$	0	0
$19$	−131.000	−1.58176	−0.790881	−	0.611971i	$-0.790377\pi$
$19$	−131.000	−1.58176	−0.790881	+	0.611971i	$0.790377\pi$
$20$	0	0
$21$	−95.0000	−0.987176
$22$	0	0
$23$	138.000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	138.000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−145.000	−1.03353
$28$	0	0
$29$	−79.0000	−0.505860	−0.252930	−	0.967485i	$-0.581394\pi$
$29$	−79.0000	−0.505860	−0.252930	+	0.967485i	$0.581394\pi$
$30$	0	0
$31$	217.000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	217.000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	−55.0000	−0.290129
$34$	0	0
$35$	95.0000	0.458798
$36$	0	0
$37$	−91.0000	−0.404333	−0.202166	−	0.979351i	$-0.564798\pi$
$37$	−91.0000	−0.404333	−0.202166	+	0.979351i	$0.564798\pi$
$38$	0	0
$39$	−310.000	−1.27281
$40$	0	0
$41$	158.000	0.601840	0.300920	−	0.953649i	$-0.402706\pi$
$41$	158.000	0.601840	0.300920	+	0.953649i	$0.402706\pi$
$42$	0	0
$43$	120.000	0.425577	0.212789	−	0.977098i	$-0.431745\pi$
$43$	120.000	0.425577	0.212789	+	0.977098i	$0.431745\pi$
$44$	0	0
$45$	10.0000	0.0331269
$46$	0	0
$47$	−546.000	−1.69452	−0.847258	−	0.531181i	$-0.821748\pi$
$47$	−546.000	−1.69452	−0.847258	+	0.531181i	$0.821748\pi$
$48$	0	0
$49$	18.0000	0.0524781
$50$	0	0
$51$	95.0000	0.260836
$52$	0	0
$53$	−439.000	−1.13776	−0.568880	−	0.822420i	$-0.692623\pi$
$53$	−439.000	−1.13776	−0.568880	+	0.822420i	$0.692623\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	0	0
$57$	−655.000	−1.52205
$58$	0	0
$59$	290.000	0.639912	0.319956	−	0.947432i	$-0.396332\pi$
$59$	290.000	0.639912	0.319956	+	0.947432i	$0.396332\pi$
$60$	0	0
$61$	−373.000	−0.782914	−0.391457	−	0.920196i	$-0.628029\pi$
$61$	−373.000	−0.782914	−0.391457	+	0.920196i	$0.628029\pi$
$62$	0	0
$63$	38.0000	0.0759929
$64$	0	0
$65$	310.000	0.591550
$66$	0	0
$67$	728.000	1.32745	0.663727	−	0.747975i	$-0.268974\pi$
$67$	728.000	1.32745	0.663727	+	0.747975i	$0.268974\pi$
$68$	0	0
$69$	690.000	1.20386
$70$	0	0
$71$	−709.000	−1.18511	−0.592555	−	0.805530i	$-0.701881\pi$
$71$	−709.000	−1.18511	−0.592555	+	0.805530i	$0.701881\pi$
$72$	0	0
$73$	850.000	1.36281	0.681404	−	0.731908i	$-0.261370\pi$
$73$	850.000	1.36281	0.681404	+	0.731908i	$0.261370\pi$
$74$	0	0
$75$	125.000	0.192450
$76$	0	0
$77$	209.000	0.309322
$78$	0	0
$79$	−1194.00	−1.70045	−0.850225	−	0.526420i	$-0.823534\pi$
$79$	−1194.00	−1.70045	−0.850225	+	0.526420i	$0.823534\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	58.0000	0.0767027	0.0383514	−	0.999264i	$-0.487789\pi$
$83$	58.0000	0.0767027	0.0383514	+	0.999264i	$0.487789\pi$
$84$	0	0
$85$	−95.0000	−0.121226
$86$	0	0
$87$	−395.000	−0.486764
$88$	0	0
$89$	753.000	0.896830	0.448415	−	0.893826i	$-0.351989\pi$
$89$	753.000	0.896830	0.448415	+	0.893826i	$0.351989\pi$
$90$	0	0
$91$	1178.00	1.35701
$92$	0	0
$93$	1085.00	1.20978
$94$	0	0
$95$	655.000	0.707385
$96$	0	0
$97$	1228.00	1.28541	0.642704	−	0.766115i	$-0.277813\pi$
$97$	1228.00	1.28541	0.642704	+	0.766115i	$0.277813\pi$
$98$	0	0
$99$	22.0000	0.0223342
$100$	0	0
$101$	1266.00	1.24724	0.623622	−	0.781726i	$-0.285660\pi$
$101$	1266.00	1.24724	0.623622	+	0.781726i	$0.285660\pi$
$102$	0	0
$103$	−76.0000	−0.0727039	−0.0363520	−	0.999339i	$-0.511574\pi$
$103$	−76.0000	−0.0727039	−0.0363520	+	0.999339i	$0.511574\pi$
$104$	0	0
$105$	475.000	0.441479
$106$	0	0
$107$	−1456.00	−1.31548	−0.657742	−	0.753243i	$-0.728488\pi$
$107$	−1456.00	−1.31548	−0.657742	+	0.753243i	$0.728488\pi$
$108$	0	0
$109$	1142.00	1.00352	0.501760	−	0.865007i	$-0.332686\pi$
$109$	1142.00	1.00352	0.501760	+	0.865007i	$0.332686\pi$
$110$	0	0
$111$	−455.000	−0.389069
$112$	0	0
$113$	1764.00	1.46852	0.734262	−	0.678866i	$-0.237528\pi$
$113$	1764.00	1.46852	0.734262	+	0.678866i	$0.237528\pi$
$114$	0	0
$115$	−690.000	−0.559503
$116$	0	0
$117$	124.000	0.0979812
$118$	0	0
$119$	−361.000	−0.278091
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	790.000	0.579121
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	144.000	0.100614	0.0503068	−	0.998734i	$-0.483980\pi$
$127$	144.000	0.100614	0.0503068	+	0.998734i	$0.483980\pi$
$128$	0	0
$129$	600.000	0.409512
$130$	0	0
$131$	−2079.00	−1.38659	−0.693294	−	0.720655i	$-0.743841\pi$
$131$	−2079.00	−1.38659	−0.693294	+	0.720655i	$0.743841\pi$
$132$	0	0
$133$	2489.00	1.62273
$134$	0	0
$135$	725.000	0.462208
$136$	0	0
$137$	−636.000	−0.396622	−0.198311	−	0.980139i	$-0.563546\pi$
$137$	−636.000	−0.396622	−0.198311	+	0.980139i	$0.563546\pi$
$138$	0	0
$139$	−468.000	−0.285577	−0.142789	−	0.989753i	$-0.545607\pi$
$139$	−468.000	−0.285577	−0.142789	+	0.989753i	$0.545607\pi$
$140$	0	0
$141$	−2730.00	−1.63055
$142$	0	0
$143$	682.000	0.398823
$144$	0	0
$145$	395.000	0.226227
$146$	0	0
$147$	90.0000	0.0504971
$148$	0	0
$149$	−2797.00	−1.53785	−0.768923	−	0.639341i	$-0.779207\pi$
$149$	−2797.00	−1.53785	−0.768923	+	0.639341i	$0.779207\pi$
$150$	0	0
$151$	−2150.00	−1.15871	−0.579353	−	0.815077i	$-0.696695\pi$
$151$	−2150.00	−1.15871	−0.579353	+	0.815077i	$0.696695\pi$
$152$	0	0
$153$	−38.0000	−0.0200792
$154$	0	0
$155$	−1085.00	−0.562254
$156$	0	0
$157$	253.000	0.128609	0.0643045	−	0.997930i	$-0.479517\pi$
$157$	253.000	0.128609	0.0643045	+	0.997930i	$0.479517\pi$
$158$	0	0
$159$	−2195.00	−1.09481
$160$	0	0
$161$	−2622.00	−1.28349
$162$	0	0
$163$	2649.00	1.27292	0.636459	−	0.771310i	$-0.280398\pi$
$163$	2649.00	1.27292	0.636459	+	0.771310i	$0.280398\pi$
$164$	0	0
$165$	275.000	0.129750
$166$	0	0
$167$	−723.000	−0.335014	−0.167507	−	0.985871i	$-0.553572\pi$
$167$	−723.000	−0.335014	−0.167507	+	0.985871i	$0.553572\pi$
$168$	0	0
$169$	1647.00	0.749659
$170$	0	0
$171$	262.000	0.117167
$172$	0	0
$173$	−334.000	−0.146784	−0.0733918	−	0.997303i	$-0.523382\pi$
$173$	−334.000	−0.146784	−0.0733918	+	0.997303i	$0.523382\pi$
$174$	0	0
$175$	−475.000	−0.205181
$176$	0	0
$177$	1450.00	0.615755
$178$	0	0
$179$	−1092.00	−0.455977	−0.227989	−	0.973664i	$-0.573215\pi$
$179$	−1092.00	−0.455977	−0.227989	+	0.973664i	$0.573215\pi$
$180$	0	0
$181$	−4230.00	−1.73709	−0.868545	−	0.495610i	$-0.834945\pi$
$181$	−4230.00	−1.73709	−0.868545	+	0.495610i	$0.834945\pi$
$182$	0	0
$183$	−1865.00	−0.753359
$184$	0	0
$185$	455.000	0.180823
$186$	0	0
$187$	−209.000	−0.0817304
$188$	0	0
$189$	2755.00	1.06030
$190$	0	0
$191$	−2380.00	−0.901627	−0.450813	−	0.892618i	$-0.648866\pi$
$191$	−2380.00	−0.901627	−0.450813	+	0.892618i	$0.648866\pi$
$192$	0	0
$193$	343.000	0.127926	0.0639629	−	0.997952i	$-0.479626\pi$
$193$	343.000	0.127926	0.0639629	+	0.997952i	$0.479626\pi$
$194$	0	0
$195$	1550.00	0.569220
$196$	0	0
$197$	2336.00	0.844838	0.422419	−	0.906401i	$-0.361181\pi$
$197$	2336.00	0.844838	0.422419	+	0.906401i	$0.361181\pi$
$198$	0	0
$199$	−2445.00	−0.870962	−0.435481	−	0.900198i	$-0.643422\pi$
$199$	−2445.00	−0.870962	−0.435481	+	0.900198i	$0.643422\pi$
$200$	0	0
$201$	3640.00	1.27734
$202$	0	0
$203$	1501.00	0.518963
$204$	0	0
$205$	−790.000	−0.269151
$206$	0	0
$207$	−276.000	−0.0926731
$208$	0	0
$209$	1441.00	0.476919
$210$	0	0
$211$	−5895.00	−1.92336	−0.961679	−	0.274178i	$-0.911594\pi$
$211$	−5895.00	−1.92336	−0.961679	+	0.274178i	$0.911594\pi$
$212$	0	0
$213$	−3545.00	−1.14037
$214$	0	0
$215$	−600.000	−0.190324
$216$	0	0
$217$	−4123.00	−1.28980
$218$	0	0
$219$	4250.00	1.31136
$220$	0	0
$221$	−1178.00	−0.358556
$222$	0	0
$223$	5470.00	1.64259	0.821297	−	0.570501i	$-0.193251\pi$
$223$	5470.00	1.64259	0.821297	+	0.570501i	$0.193251\pi$
$224$	0	0
$225$	−50.0000	−0.0148148
$226$	0	0
$227$	−3914.00	−1.14441	−0.572206	−	0.820110i	$-0.693912\pi$
$227$	−3914.00	−1.14441	−0.572206	+	0.820110i	$0.693912\pi$
$228$	0	0
$229$	1886.00	0.544238	0.272119	−	0.962264i	$-0.412276\pi$
$229$	1886.00	0.544238	0.272119	+	0.962264i	$0.412276\pi$
$230$	0	0
$231$	1045.00	0.297645
$232$	0	0
$233$	−749.000	−0.210595	−0.105297	−	0.994441i	$-0.533579\pi$
$233$	−749.000	−0.210595	−0.105297	+	0.994441i	$0.533579\pi$
$234$	0	0
$235$	2730.00	0.757811
$236$	0	0
$237$	−5970.00	−1.63626
$238$	0	0
$239$	−5010.00	−1.35594	−0.677971	−	0.735089i	$-0.737140\pi$
$239$	−5010.00	−1.35594	−0.677971	+	0.735089i	$0.737140\pi$
$240$	0	0
$241$	602.000	0.160906	0.0804528	−	0.996758i	$-0.474363\pi$
$241$	602.000	0.160906	0.0804528	+	0.996758i	$0.474363\pi$
$242$	0	0
$243$	560.000	0.147835
$244$	0	0
$245$	−90.0000	−0.0234689
$246$	0	0
$247$	8122.00	2.09227
$248$	0	0
$249$	290.000	0.0738072
$250$	0	0
$251$	−462.000	−0.116180	−0.0580900	−	0.998311i	$-0.518501\pi$
$251$	−462.000	−0.116180	−0.0580900	+	0.998311i	$0.518501\pi$
$252$	0	0
$253$	−1518.00	−0.377217
$254$	0	0
$255$	−475.000	−0.116650
$256$	0	0
$257$	−6426.00	−1.55970	−0.779850	−	0.625967i	$-0.784705\pi$
$257$	−6426.00	−1.55970	−0.779850	+	0.625967i	$0.784705\pi$
$258$	0	0
$259$	1729.00	0.414806
$260$	0	0
$261$	158.000	0.0374711
$262$	0	0
$263$	1151.00	0.269862	0.134931	−	0.990855i	$-0.456919\pi$
$263$	1151.00	0.269862	0.134931	+	0.990855i	$0.456919\pi$
$264$	0	0
$265$	2195.00	0.508822
$266$	0	0
$267$	3765.00	0.862975
$268$	0	0
$269$	860.000	0.194926	0.0974631	−	0.995239i	$-0.468927\pi$
$269$	860.000	0.194926	0.0974631	+	0.995239i	$0.468927\pi$
$270$	0	0
$271$	−916.000	−0.205325	−0.102662	−	0.994716i	$-0.532736\pi$
$271$	−916.000	−0.205325	−0.102662	+	0.994716i	$0.532736\pi$
$272$	0	0
$273$	5890.00	1.30578
$274$	0	0
$275$	−275.000	−0.0603023
$276$	0	0
$277$	1196.00	0.259425	0.129712	−	0.991552i	$-0.458595\pi$
$277$	1196.00	0.259425	0.129712	+	0.991552i	$0.458595\pi$
$278$	0	0
$279$	−434.000	−0.0931287
$280$	0	0
$281$	9078.00	1.92722	0.963609	−	0.267317i	$-0.0861370\pi$
$281$	9078.00	1.92722	0.963609	+	0.267317i	$0.0861370\pi$
$282$	0	0
$283$	−4750.00	−0.997732	−0.498866	−	0.866679i	$-0.666250\pi$
$283$	−4750.00	−0.997732	−0.498866	+	0.866679i	$0.666250\pi$
$284$	0	0
$285$	3275.00	0.680682
$286$	0	0
$287$	−3002.00	−0.617430
$288$	0	0
$289$	−4552.00	−0.926521
$290$	0	0
$291$	6140.00	1.23688
$292$	0	0
$293$	−2718.00	−0.541936	−0.270968	−	0.962588i	$-0.587344\pi$
$293$	−2718.00	−0.541936	−0.270968	+	0.962588i	$0.587344\pi$
$294$	0	0
$295$	−1450.00	−0.286177
$296$	0	0
$297$	1595.00	0.311620
$298$	0	0
$299$	−8556.00	−1.65487
$300$	0	0
$301$	−2280.00	−0.436601
$302$	0	0
$303$	6330.00	1.20016
$304$	0	0
$305$	1865.00	0.350130
$306$	0	0
$307$	3870.00	0.719455	0.359727	−	0.933057i	$-0.382870\pi$
$307$	3870.00	0.719455	0.359727	+	0.933057i	$0.382870\pi$
$308$	0	0
$309$	−380.000	−0.0699594
$310$	0	0
$311$	1433.00	0.261280	0.130640	−	0.991430i	$-0.458297\pi$
$311$	1433.00	0.261280	0.130640	+	0.991430i	$0.458297\pi$
$312$	0	0
$313$	1430.00	0.258238	0.129119	−	0.991629i	$-0.458785\pi$
$313$	1430.00	0.258238	0.129119	+	0.991629i	$0.458785\pi$
$314$	0	0
$315$	−190.000	−0.0339850
$316$	0	0
$317$	−2265.00	−0.401309	−0.200655	−	0.979662i	$-0.564307\pi$
$317$	−2265.00	−0.401309	−0.200655	+	0.979662i	$0.564307\pi$
$318$	0	0
$319$	869.000	0.152522
$320$	0	0
$321$	−7280.00	−1.26583
$322$	0	0
$323$	−2489.00	−0.428767
$324$	0	0
$325$	−1550.00	−0.264549
$326$	0	0
$327$	5710.00	0.965638
$328$	0	0
$329$	10374.0	1.73841
$330$	0	0
$331$	8280.00	1.37495	0.687477	−	0.726206i	$-0.258718\pi$
$331$	8280.00	1.37495	0.687477	+	0.726206i	$0.258718\pi$
$332$	0	0
$333$	182.000	0.0299506
$334$	0	0
$335$	−3640.00	−0.593655
$336$	0	0
$337$	−8731.00	−1.41130	−0.705650	−	0.708561i	$-0.749345\pi$
$337$	−8731.00	−1.41130	−0.705650	+	0.708561i	$0.749345\pi$
$338$	0	0
$339$	8820.00	1.41309
$340$	0	0
$341$	−2387.00	−0.379071
$342$	0	0
$343$	6175.00	0.972066
$344$	0	0
$345$	−3450.00	−0.538382
$346$	0	0
$347$	−202.000	−0.0312505	−0.0156253	−	0.999878i	$-0.504974\pi$
$347$	−202.000	−0.0312505	−0.0156253	+	0.999878i	$0.504974\pi$
$348$	0	0
$349$	2494.00	0.382524	0.191262	−	0.981539i	$-0.438742\pi$
$349$	2494.00	0.382524	0.191262	+	0.981539i	$0.438742\pi$
$350$	0	0
$351$	8990.00	1.36710
$352$	0	0
$353$	−26.0000	−0.00392023	−0.00196011	−	0.999998i	$-0.500624\pi$
$353$	−26.0000	−0.00392023	−0.00196011	+	0.999998i	$0.500624\pi$
$354$	0	0
$355$	3545.00	0.529997
$356$	0	0
$357$	−1805.00	−0.267593
$358$	0	0
$359$	−10524.0	−1.54717	−0.773587	−	0.633690i	$-0.781539\pi$
$359$	−10524.0	−1.54717	−0.773587	+	0.633690i	$0.781539\pi$
$360$	0	0
$361$	10302.0	1.50197
$362$	0	0
$363$	605.000	0.0874773
$364$	0	0
$365$	−4250.00	−0.609466
$366$	0	0
$367$	12276.0	1.74605	0.873027	−	0.487671i	$-0.162154\pi$
$367$	12276.0	1.74605	0.873027	+	0.487671i	$0.162154\pi$
$368$	0	0
$369$	−316.000	−0.0445808
$370$	0	0
$371$	8341.00	1.16723
$372$	0	0
$373$	6266.00	0.869816	0.434908	−	0.900475i	$-0.356781\pi$
$373$	6266.00	0.869816	0.434908	+	0.900475i	$0.356781\pi$
$374$	0	0
$375$	−625.000	−0.0860663
$376$	0	0
$377$	4898.00	0.669124
$378$	0	0
$379$	−8654.00	−1.17289	−0.586446	−	0.809988i	$-0.699473\pi$
$379$	−8654.00	−1.17289	−0.586446	+	0.809988i	$0.699473\pi$
$380$	0	0
$381$	720.000	0.0968155
$382$	0	0
$383$	−5226.00	−0.697222	−0.348611	−	0.937267i	$-0.613347\pi$
$383$	−5226.00	−0.697222	−0.348611	+	0.937267i	$0.613347\pi$
$384$	0	0
$385$	−1045.00	−0.138333
$386$	0	0
$387$	−240.000	−0.0315243
$388$	0	0
$389$	−6998.00	−0.912115	−0.456057	−	0.889950i	$-0.650739\pi$
$389$	−6998.00	−0.912115	−0.456057	+	0.889950i	$0.650739\pi$
$390$	0	0
$391$	2622.00	0.339131
$392$	0	0
$393$	−10395.0	−1.33425
$394$	0	0
$395$	5970.00	0.760464
$396$	0	0
$397$	−9150.00	−1.15674	−0.578369	−	0.815775i	$-0.696311\pi$
$397$	−9150.00	−1.15674	−0.578369	+	0.815775i	$0.696311\pi$
$398$	0	0
$399$	12445.0	1.56148
$400$	0	0
$401$	−4781.00	−0.595391	−0.297695	−	0.954661i	$-0.596218\pi$
$401$	−4781.00	−0.595391	−0.297695	+	0.954661i	$0.596218\pi$
$402$	0	0
$403$	−13454.0	−1.66301
$404$	0	0
$405$	3355.00	0.411633
$406$	0	0
$407$	1001.00	0.121911
$408$	0	0
$409$	10228.0	1.23653	0.618267	−	0.785968i	$-0.287835\pi$
$409$	10228.0	1.23653	0.618267	+	0.785968i	$0.287835\pi$
$410$	0	0
$411$	−3180.00	−0.381649
$412$	0	0
$413$	−5510.00	−0.656488
$414$	0	0
$415$	−290.000	−0.0343025
$416$	0	0
$417$	−2340.00	−0.274797
$418$	0	0
$419$	−7752.00	−0.903842	−0.451921	−	0.892058i	$-0.649261\pi$
$419$	−7752.00	−0.903842	−0.451921	+	0.892058i	$0.649261\pi$
$420$	0	0
$421$	12112.0	1.40214	0.701072	−	0.713090i	$-0.252705\pi$
$421$	12112.0	1.40214	0.701072	+	0.713090i	$0.252705\pi$
$422$	0	0
$423$	1092.00	0.125520
$424$	0	0
$425$	475.000	0.0542138
$426$	0	0
$427$	7087.00	0.803194
$428$	0	0
$429$	3410.00	0.383768
$430$	0	0
$431$	−16032.0	−1.79173	−0.895863	−	0.444330i	$-0.853442\pi$
$431$	−16032.0	−1.79173	−0.895863	+	0.444330i	$0.853442\pi$
$432$	0	0
$433$	−13412.0	−1.48854	−0.744272	−	0.667877i	$-0.767203\pi$
$433$	−13412.0	−1.48854	−0.744272	+	0.667877i	$0.767203\pi$
$434$	0	0
$435$	1975.00	0.217687
$436$	0	0
$437$	−18078.0	−1.97892
$438$	0	0
$439$	−1768.00	−0.192214	−0.0961071	−	0.995371i	$-0.530639\pi$
$439$	−1768.00	−0.192214	−0.0961071	+	0.995371i	$0.530639\pi$
$440$	0	0
$441$	−36.0000	−0.00388727
$442$	0	0
$443$	−5284.00	−0.566705	−0.283353	−	0.959016i	$-0.591447\pi$
$443$	−5284.00	−0.566705	−0.283353	+	0.959016i	$0.591447\pi$
$444$	0	0
$445$	−3765.00	−0.401074
$446$	0	0
$447$	−13985.0	−1.47979
$448$	0	0
$449$	−10814.0	−1.13662	−0.568312	−	0.822813i	$-0.692403\pi$
$449$	−10814.0	−1.13662	−0.568312	+	0.822813i	$0.692403\pi$
$450$	0	0
$451$	−1738.00	−0.181462
$452$	0	0
$453$	−10750.0	−1.11496
$454$	0	0
$455$	−5890.00	−0.606874
$456$	0	0
$457$	14665.0	1.50109	0.750547	−	0.660817i	$-0.229790\pi$
$457$	14665.0	1.50109	0.750547	+	0.660817i	$0.229790\pi$
$458$	0	0
$459$	−2755.00	−0.280158
$460$	0	0
$461$	−481.000	−0.0485952	−0.0242976	−	0.999705i	$-0.507735\pi$
$461$	−481.000	−0.0485952	−0.0242976	+	0.999705i	$0.507735\pi$
$462$	0	0
$463$	−11998.0	−1.20431	−0.602154	−	0.798380i	$-0.705691\pi$
$463$	−11998.0	−1.20431	−0.602154	+	0.798380i	$0.705691\pi$
$464$	0	0
$465$	−5425.00	−0.541029
$466$	0	0
$467$	8045.00	0.797170	0.398585	−	0.917131i	$-0.369501\pi$
$467$	8045.00	0.797170	0.398585	+	0.917131i	$0.369501\pi$
$468$	0	0
$469$	−13832.0	−1.36184
$470$	0	0
$471$	1265.00	0.123754
$472$	0	0
$473$	−1320.00	−0.128316
$474$	0	0
$475$	−3275.00	−0.316352
$476$	0	0
$477$	878.000	0.0842785
$478$	0	0
$479$	10636.0	1.01455	0.507277	−	0.861783i	$-0.330652\pi$
$479$	10636.0	1.01455	0.507277	+	0.861783i	$0.330652\pi$
$480$	0	0
$481$	5642.00	0.534830
$482$	0	0
$483$	−13110.0	−1.23504
$484$	0	0
$485$	−6140.00	−0.574852
$486$	0	0
$487$	4456.00	0.414621	0.207311	−	0.978275i	$-0.433529\pi$
$487$	4456.00	0.414621	0.207311	+	0.978275i	$0.433529\pi$
$488$	0	0
$489$	13245.0	1.22487
$490$	0	0
$491$	−2563.00	−0.235573	−0.117787	−	0.993039i	$-0.537580\pi$
$491$	−2563.00	−0.235573	−0.117787	+	0.993039i	$0.537580\pi$
$492$	0	0
$493$	−1501.00	−0.137123
$494$	0	0
$495$	−110.000	−0.00998815
$496$	0	0
$497$	13471.0	1.21581
$498$	0	0
$499$	11204.0	1.00513	0.502565	−	0.864539i	$-0.332390\pi$
$499$	11204.0	1.00513	0.502565	+	0.864539i	$0.332390\pi$
$500$	0	0
$501$	−3615.00	−0.322368
$502$	0	0
$503$	13584.0	1.20414	0.602068	−	0.798445i	$-0.294344\pi$
$503$	13584.0	1.20414	0.602068	+	0.798445i	$0.294344\pi$
$504$	0	0
$505$	−6330.00	−0.557785
$506$	0	0
$507$	8235.00	0.721359
$508$	0	0
$509$	13020.0	1.13379	0.566897	−	0.823789i	$-0.308144\pi$
$509$	13020.0	1.13379	0.566897	+	0.823789i	$0.308144\pi$
$510$	0	0
$511$	−16150.0	−1.39811
$512$	0	0
$513$	18995.0	1.63479
$514$	0	0
$515$	380.000	0.0325142
$516$	0	0
$517$	6006.00	0.510916
$518$	0	0
$519$	−1670.00	−0.141243
$520$	0	0
$521$	−6514.00	−0.547761	−0.273881	−	0.961764i	$-0.588307\pi$
$521$	−6514.00	−0.547761	−0.273881	+	0.961764i	$0.588307\pi$
$522$	0	0
$523$	−11048.0	−0.923700	−0.461850	−	0.886958i	$-0.652814\pi$
$523$	−11048.0	−0.923700	−0.461850	+	0.886958i	$0.652814\pi$
$524$	0	0
$525$	−2375.00	−0.197435
$526$	0	0
$527$	4123.00	0.340798
$528$	0	0
$529$	6877.00	0.565217
$530$	0	0
$531$	−580.000	−0.0474009
$532$	0	0
$533$	−9796.00	−0.796082
$534$	0	0
$535$	7280.00	0.588303
$536$	0	0
$537$	−5460.00	−0.438764
$538$	0	0
$539$	−198.000	−0.0158228
$540$	0	0
$541$	941.000	0.0747814	0.0373907	−	0.999301i	$-0.488095\pi$
$541$	941.000	0.0747814	0.0373907	+	0.999301i	$0.488095\pi$
$542$	0	0
$543$	−21150.0	−1.67152
$544$	0	0
$545$	−5710.00	−0.448788
$546$	0	0
$547$	−9208.00	−0.719754	−0.359877	−	0.933000i	$-0.617181\pi$
$547$	−9208.00	−0.719754	−0.359877	+	0.933000i	$0.617181\pi$
$548$	0	0
$549$	746.000	0.0579936
$550$	0	0
$551$	10349.0	0.800149
$552$	0	0
$553$	22686.0	1.74450
$554$	0	0
$555$	2275.00	0.173997
$556$	0	0
$557$	11362.0	0.864315	0.432157	−	0.901798i	$-0.357752\pi$
$557$	11362.0	0.864315	0.432157	+	0.901798i	$0.357752\pi$
$558$	0	0
$559$	−7440.00	−0.562931
$560$	0	0
$561$	−1045.00	−0.0786452
$562$	0	0
$563$	−630.000	−0.0471605	−0.0235802	−	0.999722i	$-0.507507\pi$
$563$	−630.000	−0.0471605	−0.0235802	+	0.999722i	$0.507507\pi$
$564$	0	0
$565$	−8820.00	−0.656744
$566$	0	0
$567$	12749.0	0.944282
$568$	0	0
$569$	−3720.00	−0.274078	−0.137039	−	0.990566i	$-0.543759\pi$
$569$	−3720.00	−0.274078	−0.137039	+	0.990566i	$0.543759\pi$
$570$	0	0
$571$	16811.0	1.23208	0.616041	−	0.787714i	$-0.288736\pi$
$571$	16811.0	1.23208	0.616041	+	0.787714i	$0.288736\pi$
$572$	0	0
$573$	−11900.0	−0.867591
$574$	0	0
$575$	3450.00	0.250217
$576$	0	0
$577$	814.000	0.0587301	0.0293650	−	0.999569i	$-0.490651\pi$
$577$	814.000	0.0587301	0.0293650	+	0.999569i	$0.490651\pi$
$578$	0	0
$579$	1715.00	0.123097
$580$	0	0
$581$	−1102.00	−0.0786896
$582$	0	0
$583$	4829.00	0.343048
$584$	0	0
$585$	−620.000	−0.0438185
$586$	0	0
$587$	2231.00	0.156871	0.0784355	−	0.996919i	$-0.475008\pi$
$587$	2231.00	0.156871	0.0784355	+	0.996919i	$0.475008\pi$
$588$	0	0
$589$	−28427.0	−1.98865
$590$	0	0
$591$	11680.0	0.812946
$592$	0	0
$593$	27306.0	1.89093	0.945466	−	0.325720i	$-0.105607\pi$
$593$	27306.0	1.89093	0.945466	+	0.325720i	$0.105607\pi$
$594$	0	0
$595$	1805.00	0.124366
$596$	0	0
$597$	−12225.0	−0.838084
$598$	0	0
$599$	21681.0	1.47890	0.739450	−	0.673211i	$-0.235085\pi$
$599$	21681.0	1.47890	0.739450	+	0.673211i	$0.235085\pi$
$600$	0	0
$601$	3274.00	0.222212	0.111106	−	0.993809i	$-0.464561\pi$
$601$	3274.00	0.222212	0.111106	+	0.993809i	$0.464561\pi$
$602$	0	0
$603$	−1456.00	−0.0983299
$604$	0	0
$605$	−605.000	−0.0406558
$606$	0	0
$607$	16863.0	1.12759	0.563796	−	0.825914i	$-0.309341\pi$
$607$	16863.0	1.12759	0.563796	+	0.825914i	$0.309341\pi$
$608$	0	0
$609$	7505.00	0.499373
$610$	0	0
$611$	33852.0	2.24142
$612$	0	0
$613$	−15758.0	−1.03827	−0.519135	−	0.854692i	$-0.673746\pi$
$613$	−15758.0	−1.03827	−0.519135	+	0.854692i	$0.673746\pi$
$614$	0	0
$615$	−3950.00	−0.258991
$616$	0	0
$617$	6264.00	0.408718	0.204359	−	0.978896i	$-0.434489\pi$
$617$	6264.00	0.408718	0.204359	+	0.978896i	$0.434489\pi$
$618$	0	0
$619$	−19828.0	−1.28749	−0.643744	−	0.765241i	$-0.722620\pi$
$619$	−19828.0	−1.28749	−0.643744	+	0.765241i	$0.722620\pi$
$620$	0	0
$621$	−20010.0	−1.29303
$622$	0	0
$623$	−14307.0	−0.920061
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	7205.00	0.458915
$628$	0	0
$629$	−1729.00	−0.109602
$630$	0	0
$631$	14683.0	0.926341	0.463170	−	0.886269i	$-0.346712\pi$
$631$	14683.0	0.926341	0.463170	+	0.886269i	$0.346712\pi$
$632$	0	0
$633$	−29475.0	−1.85075
$634$	0	0
$635$	−720.000	−0.0449958
$636$	0	0
$637$	−1116.00	−0.0694153
$638$	0	0
$639$	1418.00	0.0877859
$640$	0	0
$641$	26335.0	1.62273	0.811365	−	0.584540i	$-0.198725\pi$
$641$	26335.0	1.62273	0.811365	+	0.584540i	$0.198725\pi$
$642$	0	0
$643$	1225.00	0.0751311	0.0375655	−	0.999294i	$-0.488040\pi$
$643$	1225.00	0.0751311	0.0375655	+	0.999294i	$0.488040\pi$
$644$	0	0
$645$	−3000.00	−0.183139
$646$	0	0
$647$	19342.0	1.17529	0.587645	−	0.809119i	$-0.300055\pi$
$647$	19342.0	1.17529	0.587645	+	0.809119i	$0.300055\pi$
$648$	0	0
$649$	−3190.00	−0.192941
$650$	0	0
$651$	−20615.0	−1.24111
$652$	0	0
$653$	28207.0	1.69039	0.845195	−	0.534457i	$-0.179484\pi$
$653$	28207.0	1.69039	0.845195	+	0.534457i	$0.179484\pi$
$654$	0	0
$655$	10395.0	0.620101
$656$	0	0
$657$	−1700.00	−0.100949
$658$	0	0
$659$	−9705.00	−0.573677	−0.286838	−	0.957979i	$-0.592604\pi$
$659$	−9705.00	−0.573677	−0.286838	+	0.957979i	$0.592604\pi$
$660$	0	0
$661$	26510.0	1.55994	0.779969	−	0.625818i	$-0.215235\pi$
$661$	26510.0	1.55994	0.779969	+	0.625818i	$0.215235\pi$
$662$	0	0
$663$	−5890.00	−0.345021
$664$	0	0
$665$	−12445.0	−0.725709
$666$	0	0
$667$	−10902.0	−0.632874
$668$	0	0
$669$	27350.0	1.58059
$670$	0	0
$671$	4103.00	0.236057
$672$	0	0
$673$	18223.0	1.04375	0.521876	−	0.853021i	$-0.325232\pi$
$673$	18223.0	1.04375	0.521876	+	0.853021i	$0.325232\pi$
$674$	0	0
$675$	−3625.00	−0.206706
$676$	0	0
$677$	−30868.0	−1.75237	−0.876184	−	0.481976i	$-0.839919\pi$
$677$	−30868.0	−1.75237	−0.876184	+	0.481976i	$0.839919\pi$
$678$	0	0
$679$	−23332.0	−1.31870
$680$	0	0
$681$	−19570.0	−1.10121
$682$	0	0
$683$	1179.00	0.0660515	0.0330258	−	0.999455i	$-0.489486\pi$
$683$	1179.00	0.0660515	0.0330258	+	0.999455i	$0.489486\pi$
$684$	0	0
$685$	3180.00	0.177375
$686$	0	0
$687$	9430.00	0.523693
$688$	0	0
$689$	27218.0	1.50497
$690$	0	0
$691$	−28242.0	−1.55481	−0.777407	−	0.628998i	$-0.783465\pi$
$691$	−28242.0	−1.55481	−0.777407	+	0.628998i	$0.783465\pi$
$692$	0	0
$693$	−418.000	−0.0229127
$694$	0	0
$695$	2340.00	0.127714
$696$	0	0
$697$	3002.00	0.163140
$698$	0	0
$699$	−3745.00	−0.202645
$700$	0	0
$701$	−2253.00	−0.121390	−0.0606952	−	0.998156i	$-0.519332\pi$
$701$	−2253.00	−0.121390	−0.0606952	+	0.998156i	$0.519332\pi$
$702$	0	0
$703$	11921.0	0.639558
$704$	0	0
$705$	13650.0	0.729204
$706$	0	0
$707$	−24054.0	−1.27955
$708$	0	0
$709$	974.000	0.0515929	0.0257964	−	0.999667i	$-0.491788\pi$
$709$	974.000	0.0515929	0.0257964	+	0.999667i	$0.491788\pi$
$710$	0	0
$711$	2388.00	0.125959
$712$	0	0
$713$	29946.0	1.57291
$714$	0	0
$715$	−3410.00	−0.178359
$716$	0	0
$717$	−25050.0	−1.30476
$718$	0	0
$719$	−1603.00	−0.0831458	−0.0415729	−	0.999135i	$-0.513237\pi$
$719$	−1603.00	−0.0831458	−0.0415729	+	0.999135i	$0.513237\pi$
$720$	0	0
$721$	1444.00	0.0745872
$722$	0	0
$723$	3010.00	0.154831
$724$	0	0
$725$	−1975.00	−0.101172
$726$	0	0
$727$	30886.0	1.57565	0.787826	−	0.615898i	$-0.211207\pi$
$727$	30886.0	1.57565	0.787826	+	0.615898i	$0.211207\pi$
$728$	0	0
$729$	20917.0	1.06269
$730$	0	0
$731$	2280.00	0.115361
$732$	0	0
$733$	3212.00	0.161853	0.0809263	−	0.996720i	$-0.474212\pi$
$733$	3212.00	0.161853	0.0809263	+	0.996720i	$0.474212\pi$
$734$	0	0
$735$	−450.000	−0.0225830
$736$	0	0
$737$	−8008.00	−0.400242
$738$	0	0
$739$	17728.0	0.882456	0.441228	−	0.897395i	$-0.354543\pi$
$739$	17728.0	0.882456	0.441228	+	0.897395i	$0.354543\pi$
$740$	0	0
$741$	40610.0	2.01329
$742$	0	0
$743$	−20171.0	−0.995965	−0.497983	−	0.867187i	$-0.665926\pi$
$743$	−20171.0	−0.995965	−0.497983	+	0.867187i	$0.665926\pi$
$744$	0	0
$745$	13985.0	0.687746
$746$	0	0
$747$	−116.000	−0.00568168
$748$	0	0
$749$	27664.0	1.34956
$750$	0	0
$751$	−8779.00	−0.426565	−0.213282	−	0.976991i	$-0.568415\pi$
$751$	−8779.00	−0.426565	−0.213282	+	0.976991i	$0.568415\pi$
$752$	0	0
$753$	−2310.00	−0.111794
$754$	0	0
$755$	10750.0	0.518189
$756$	0	0
$757$	−3630.00	−0.174286	−0.0871431	−	0.996196i	$-0.527774\pi$
$757$	−3630.00	−0.174286	−0.0871431	+	0.996196i	$0.527774\pi$
$758$	0	0
$759$	−7590.00	−0.362977
$760$	0	0
$761$	−6582.00	−0.313531	−0.156766	−	0.987636i	$-0.550107\pi$
$761$	−6582.00	−0.313531	−0.156766	+	0.987636i	$0.550107\pi$
$762$	0	0
$763$	−21698.0	−1.02952
$764$	0	0
$765$	190.000	0.00897969
$766$	0	0
$767$	−17980.0	−0.846441
$768$	0	0
$769$	−40130.0	−1.88183	−0.940913	−	0.338647i	$-0.890031\pi$
$769$	−40130.0	−1.88183	−0.940913	+	0.338647i	$0.890031\pi$
$770$	0	0
$771$	−32130.0	−1.50082
$772$	0	0
$773$	−31627.0	−1.47160	−0.735798	−	0.677201i	$-0.763193\pi$
$773$	−31627.0	−1.47160	−0.735798	+	0.677201i	$0.763193\pi$
$774$	0	0
$775$	5425.00	0.251447
$776$	0	0
$777$	8645.00	0.399148
$778$	0	0
$779$	−20698.0	−0.951968
$780$	0	0
$781$	7799.00	0.357324
$782$	0	0
$783$	11455.0	0.522820
$784$	0	0
$785$	−1265.00	−0.0575157
$786$	0	0
$787$	−25408.0	−1.15082	−0.575411	−	0.817864i	$-0.695158\pi$
$787$	−25408.0	−1.15082	−0.575411	+	0.817864i	$0.695158\pi$
$788$	0	0
$789$	5755.00	0.259675
$790$	0	0
$791$	−33516.0	−1.50656
$792$	0	0
$793$	23126.0	1.03560
$794$	0	0
$795$	10975.0	0.489614
$796$	0	0
$797$	33254.0	1.47794	0.738969	−	0.673739i	$-0.235313\pi$
$797$	33254.0	1.47794	0.738969	+	0.673739i	$0.235313\pi$
$798$	0	0
$799$	−10374.0	−0.459331
$800$	0	0
$801$	−1506.00	−0.0664318
$802$	0	0
$803$	−9350.00	−0.410902
$804$	0	0
$805$	13110.0	0.573996
$806$	0	0
$807$	4300.00	0.187568
$808$	0	0
$809$	27882.0	1.21172	0.605858	−	0.795572i	$-0.292830\pi$
$809$	27882.0	1.21172	0.605858	+	0.795572i	$0.292830\pi$
$810$	0	0
$811$	29319.0	1.26946	0.634728	−	0.772735i	$-0.281112\pi$
$811$	29319.0	1.26946	0.634728	+	0.772735i	$0.281112\pi$
$812$	0	0
$813$	−4580.00	−0.197574
$814$	0	0
$815$	−13245.0	−0.569266
$816$	0	0
$817$	−15720.0	−0.673162
$818$	0	0
$819$	−2356.00	−0.100519
$820$	0	0
$821$	16694.0	0.709652	0.354826	−	0.934932i	$-0.384540\pi$
$821$	16694.0	0.709652	0.354826	+	0.934932i	$0.384540\pi$
$822$	0	0
$823$	1596.00	0.0675979	0.0337989	−	0.999429i	$-0.489239\pi$
$823$	1596.00	0.0675979	0.0337989	+	0.999429i	$0.489239\pi$
$824$	0	0
$825$	−1375.00	−0.0580259
$826$	0	0
$827$	−3228.00	−0.135730	−0.0678649	−	0.997695i	$-0.521619\pi$
$827$	−3228.00	−0.135730	−0.0678649	+	0.997695i	$0.521619\pi$
$828$	0	0
$829$	−42716.0	−1.78961	−0.894806	−	0.446456i	$-0.852686\pi$
$829$	−42716.0	−1.78961	−0.894806	+	0.446456i	$0.852686\pi$
$830$	0	0
$831$	5980.00	0.249632
$832$	0	0
$833$	342.000	0.0142252
$834$	0	0
$835$	3615.00	0.149823
$836$	0	0
$837$	−31465.0	−1.29939
$838$	0	0
$839$	−20032.0	−0.824293	−0.412146	−	0.911118i	$-0.635221\pi$
$839$	−20032.0	−0.824293	−0.412146	+	0.911118i	$0.635221\pi$
$840$	0	0
$841$	−18148.0	−0.744106
$842$	0	0
$843$	45390.0	1.85447
$844$	0	0
$845$	−8235.00	−0.335258
$846$	0	0
$847$	−2299.00	−0.0932640
$848$	0	0
$849$	−23750.0	−0.960068
$850$	0	0
$851$	−12558.0	−0.505855
$852$	0	0
$853$	−1834.00	−0.0736166	−0.0368083	−	0.999322i	$-0.511719\pi$
$853$	−1834.00	−0.0736166	−0.0368083	+	0.999322i	$0.511719\pi$
$854$	0	0
$855$	−1310.00	−0.0523989
$856$	0	0
$857$	−9621.00	−0.383486	−0.191743	−	0.981445i	$-0.561414\pi$
$857$	−9621.00	−0.383486	−0.191743	+	0.981445i	$0.561414\pi$
$858$	0	0
$859$	8950.00	0.355495	0.177747	−	0.984076i	$-0.443119\pi$
$859$	8950.00	0.355495	0.177747	+	0.984076i	$0.443119\pi$
$860$	0	0
$861$	−15010.0	−0.594122
$862$	0	0
$863$	−42870.0	−1.69098	−0.845488	−	0.533995i	$-0.820690\pi$
$863$	−42870.0	−1.69098	−0.845488	+	0.533995i	$0.820690\pi$
$864$	0	0
$865$	1670.00	0.0656436
$866$	0	0
$867$	−22760.0	−0.891546
$868$	0	0
$869$	13134.0	0.512705
$870$	0	0
$871$	−45136.0	−1.75588
$872$	0	0
$873$	−2456.00	−0.0952154
$874$	0	0
$875$	2375.00	0.0917596
$876$	0	0
$877$	−15126.0	−0.582404	−0.291202	−	0.956662i	$-0.594055\pi$
$877$	−15126.0	−0.582404	−0.291202	+	0.956662i	$0.594055\pi$
$878$	0	0
$879$	−13590.0	−0.521478
$880$	0	0
$881$	3282.00	0.125509	0.0627545	−	0.998029i	$-0.480011\pi$
$881$	3282.00	0.125509	0.0627545	+	0.998029i	$0.480011\pi$
$882$	0	0
$883$	−29845.0	−1.13745	−0.568723	−	0.822529i	$-0.692562\pi$
$883$	−29845.0	−1.13745	−0.568723	+	0.822529i	$0.692562\pi$
$884$	0	0
$885$	−7250.00	−0.275374
$886$	0	0
$887$	−18816.0	−0.712265	−0.356133	−	0.934435i	$-0.615905\pi$
$887$	−18816.0	−0.712265	−0.356133	+	0.934435i	$0.615905\pi$
$888$	0	0
$889$	−2736.00	−0.103220
$890$	0	0
$891$	7381.00	0.277523
$892$	0	0
$893$	71526.0	2.68032
$894$	0	0
$895$	5460.00	0.203919
$896$	0	0
$897$	−42780.0	−1.59240
$898$	0	0
$899$	−17143.0	−0.635986
$900$	0	0
$901$	−8341.00	−0.308412
$902$	0	0
$903$	−11400.0	−0.420120
$904$	0	0
$905$	21150.0	0.776851
$906$	0	0
$907$	−131.000	−0.00479579	−0.00239790	−	0.999997i	$-0.500763\pi$
$907$	−131.000	−0.00479579	−0.00239790	+	0.999997i	$0.500763\pi$
$908$	0	0
$909$	−2532.00	−0.0923885
$910$	0	0
$911$	−45025.0	−1.63748	−0.818740	−	0.574164i	$-0.805327\pi$
$911$	−45025.0	−1.63748	−0.818740	+	0.574164i	$0.805327\pi$
$912$	0	0
$913$	−638.000	−0.0231267
$914$	0	0
$915$	9325.00	0.336913
$916$	0	0
$917$	39501.0	1.42251
$918$	0	0
$919$	3820.00	0.137117	0.0685583	−	0.997647i	$-0.478160\pi$
$919$	3820.00	0.137117	0.0685583	+	0.997647i	$0.478160\pi$
$920$	0	0
$921$	19350.0	0.692296
$922$	0	0
$923$	43958.0	1.56760
$924$	0	0
$925$	−2275.00	−0.0808665
$926$	0	0
$927$	152.000	0.00538547
$928$	0	0
$929$	32163.0	1.13588	0.567941	−	0.823069i	$-0.307740\pi$
$929$	32163.0	1.13588	0.567941	+	0.823069i	$0.307740\pi$
$930$	0	0
$931$	−2358.00	−0.0830079
$932$	0	0
$933$	7165.00	0.251416
$934$	0	0
$935$	1045.00	0.0365510
$936$	0	0
$937$	−9878.00	−0.344397	−0.172199	−	0.985062i	$-0.555087\pi$
$937$	−9878.00	−0.344397	−0.172199	+	0.985062i	$0.555087\pi$
$938$	0	0
$939$	7150.00	0.248489
$940$	0	0
$941$	22629.0	0.783937	0.391968	−	0.919979i	$-0.371794\pi$
$941$	22629.0	0.783937	0.391968	+	0.919979i	$0.371794\pi$
$942$	0	0
$943$	21804.0	0.752954
$944$	0	0
$945$	−13775.0	−0.474181
$946$	0	0
$947$	−18561.0	−0.636908	−0.318454	−	0.947938i	$-0.603164\pi$
$947$	−18561.0	−0.636908	−0.318454	+	0.947938i	$0.603164\pi$
$948$	0	0
$949$	−52700.0	−1.80265
$950$	0	0
$951$	−11325.0	−0.386160
$952$	0	0
$953$	52461.0	1.78319	0.891594	−	0.452835i	$-0.149587\pi$
$953$	52461.0	1.78319	0.891594	+	0.452835i	$0.149587\pi$
$954$	0	0
$955$	11900.0	0.403220
$956$	0	0
$957$	4345.00	0.146765
$958$	0	0
$959$	12084.0	0.406895
$960$	0	0
$961$	17298.0	0.580645
$962$	0	0
$963$	2912.00	0.0974433
$964$	0	0
$965$	−1715.00	−0.0572102
$966$	0	0
$967$	7227.00	0.240336	0.120168	−	0.992754i	$-0.461657\pi$
$967$	7227.00	0.240336	0.120168	+	0.992754i	$0.461657\pi$
$968$	0	0
$969$	−12445.0	−0.412581
$970$	0	0
$971$	16560.0	0.547308	0.273654	−	0.961828i	$-0.411768\pi$
$971$	16560.0	0.547308	0.273654	+	0.961828i	$0.411768\pi$
$972$	0	0
$973$	8892.00	0.292975
$974$	0	0
$975$	−7750.00	−0.254563
$976$	0	0
$977$	−51252.0	−1.67830	−0.839149	−	0.543902i	$-0.816946\pi$
$977$	−51252.0	−1.67830	−0.839149	+	0.543902i	$0.816946\pi$
$978$	0	0
$979$	−8283.00	−0.270404
$980$	0	0
$981$	−2284.00	−0.0743349
$982$	0	0
$983$	21202.0	0.687934	0.343967	−	0.938982i	$-0.388229\pi$
$983$	21202.0	0.687934	0.343967	+	0.938982i	$0.388229\pi$
$984$	0	0
$985$	−11680.0	−0.377823
$986$	0	0
$987$	51870.0	1.67279
$988$	0	0
$989$	16560.0	0.532434
$990$	0	0
$991$	−61736.0	−1.97892	−0.989459	−	0.144810i	$-0.953743\pi$
$991$	−61736.0	−1.97892	−0.989459	+	0.144810i	$0.953743\pi$
$992$	0	0
$993$	41400.0	1.32305
$994$	0	0
$995$	12225.0	0.389506
$996$	0	0
$997$	37076.0	1.17774	0.588871	−	0.808227i	$-0.299573\pi$
$997$	37076.0	1.17774	0.588871	+	0.808227i	$0.299573\pi$
$998$	0	0
$999$	13195.0	0.417889

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.4.a.b.1.1	✓	1
3.2	odd	2		1980.4.a.d.1.1		1
4.3	odd	2		880.4.a.e.1.1		1
5.2	odd	4		1100.4.b.d.749.1		2
5.3	odd	4		1100.4.b.d.749.2		2
5.4	even	2		1100.4.a.b.1.1		1
11.10	odd	2		2420.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.b.1.1	✓	1	1.1	even	1	trivial
880.4.a.e.1.1		1	4.3	odd	2
1100.4.a.b.1.1		1	5.4	even	2
1100.4.b.d.749.1		2	5.2	odd	4
1100.4.b.d.749.2		2	5.3	odd	4
1980.4.a.d.1.1		1	3.2	odd	2
2420.4.a.e.1.1		1	11.10	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	220.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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