LMFDB - Embedded newform 222.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [222,2,Mod(1,222)] 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("222.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(222, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 222 = 2 \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	222.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,1,1,0] 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:	yes
Analytic conductor:	$1.77267892487$
Analytic rank:	$0$
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	$\chi$	$=$	222.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -7.00000 q^{19} -1.00000 q^{21} +3.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} -7.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -5.00000 q^{50} -3.00000 q^{51} -1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -7.00000 q^{57} -10.0000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +2.00000 q^{67} -3.00000 q^{68} +3.00000 q^{69} +12.0000 q^{71} +1.00000 q^{72} +5.00000 q^{73} +1.00000 q^{74} -5.00000 q^{75} -7.00000 q^{76} -3.00000 q^{77} -1.00000 q^{78} +2.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +3.00000 q^{83} -1.00000 q^{84} -4.00000 q^{86} +3.00000 q^{88} -3.00000 q^{89} +1.00000 q^{91} +3.00000 q^{92} +2.00000 q^{93} +6.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	3.00000	0.639602
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	3.00000	0.522233
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−7.00000	−1.13555
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−1.00000	−0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	3.00000	0.442326
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−5.00000	−0.707107
$51$	−3.00000	−0.420084
$52$	−1.00000	−0.138675
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−7.00000	−0.927173
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	2.00000	0.254000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−3.00000	−0.363803
$69$	3.00000	0.361158
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000	0.117851
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	1.00000	0.116248
$75$	−5.00000	−0.577350
$76$	−7.00000	−0.802955
$77$	−3.00000	−0.341882
$78$	−1.00000	−0.113228
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	3.00000	0.319801
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	3.00000	0.312772
$93$	2.00000	0.207390
$94$	6.00000	0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−6.00000	−0.606092
$99$	3.00000	0.301511
$100$	−5.00000	−0.500000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−3.00000	−0.297044
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	9.00000	0.874157
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	1.00000	0.0962250
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−7.00000	−0.655610
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−10.0000	−0.905357
$123$	−6.00000	−0.541002
$124$	2.00000	0.179605
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	3.00000	0.261116
$133$	7.00000	0.606977
$134$	2.00000	0.172774
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	3.00000	0.255377
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	12.0000	1.00702
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	5.00000	0.413803
$147$	−6.00000	−0.494872
$148$	1.00000	0.0821995
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−5.00000	−0.408248
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−7.00000	−0.567775
$153$	−3.00000	−0.242536
$154$	−3.00000	−0.241747
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	2.00000	0.159111
$159$	9.00000	0.713746
$160$	0	0
$161$	−3.00000	−0.236433
$162$	1.00000	0.0785674
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	3.00000	0.232845
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	−1.00000	−0.0771517
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−7.00000	−0.535303
$172$	−4.00000	−0.304997
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	3.00000	0.226134
$177$	0	0
$178$	−3.00000	−0.224860
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	1.00000	0.0741249
$183$	−10.0000	−0.739221
$184$	3.00000	0.221163
$185$	0	0
$186$	2.00000	0.146647
$187$	−9.00000	−0.658145
$188$	6.00000	0.437595
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	1.00000	0.0721688
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	3.00000	0.213201
$199$	−28.0000	−1.98487	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−28.0000	−1.98487	−0.992434	+	0.122782i	$0.960818\pi$
$200$	−5.00000	−0.353553
$201$	2.00000	0.141069
$202$	6.00000	0.422159
$203$	0	0
$204$	−3.00000	−0.210042
$205$	0	0
$206$	14.0000	0.975426
$207$	3.00000	0.208514
$208$	−1.00000	−0.0693375
$209$	−21.0000	−1.45260
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	9.00000	0.618123
$213$	12.0000	0.822226
$214$	9.00000	0.615227
$215$	0	0
$216$	1.00000	0.0680414
$217$	−2.00000	−0.135769
$218$	5.00000	0.338643
$219$	5.00000	0.337869
$220$	0	0
$221$	3.00000	0.201802
$222$	1.00000	0.0671156
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−1.00000	−0.0668153
$225$	−5.00000	−0.333333
$226$	−6.00000	−0.399114
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−7.00000	−0.463586
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	30.0000	1.96537	0.982683	−	0.185296i	$-0.0593245\pi$
$233$	30.0000	1.96537	0.982683	+	0.185296i	$0.0593245\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	0	0
$237$	2.00000	0.129914
$238$	3.00000	0.194461
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	−2.00000	−0.128565
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−6.00000	−0.382546
$247$	7.00000	0.445399
$248$	2.00000	0.127000
$249$	3.00000	0.190117
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−1.00000	−0.0629941
$253$	9.00000	0.565825
$254$	5.00000	0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	−4.00000	−0.249029
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	0	0
$262$	18.0000	1.11204
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	7.00000	0.429198
$267$	−3.00000	−0.183597
$268$	2.00000	0.122169
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−3.00000	−0.181902
$273$	1.00000	0.0605228
$274$	−12.0000	−0.724947
$275$	−15.0000	−0.904534
$276$	3.00000	0.180579
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	14.0000	0.839664
$279$	2.00000	0.119737
$280$	0	0
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	6.00000	0.357295
$283$	−1.00000	−0.0594438	−0.0297219	−	0.999558i	$-0.509462\pi$
$283$	−1.00000	−0.0594438	−0.0297219	+	0.999558i	$0.509462\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−3.00000	−0.177394
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	2.00000	0.117242
$292$	5.00000	0.292603
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	1.00000	0.0581238
$297$	3.00000	0.174078
$298$	−18.0000	−1.04271
$299$	−3.00000	−0.173494
$300$	−5.00000	−0.288675
$301$	4.00000	0.230556
$302$	5.00000	0.287718
$303$	6.00000	0.344691
$304$	−7.00000	−0.401478
$305$	0	0
$306$	−3.00000	−0.171499
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	−3.00000	−0.170941
$309$	14.0000	0.796432
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−1.00000	−0.0566139
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	8.00000	0.451466
$315$	0	0
$316$	2.00000	0.112509
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	9.00000	0.504695
$319$	0	0
$320$	0	0
$321$	9.00000	0.502331
$322$	−3.00000	−0.167183
$323$	21.0000	1.16847
$324$	1.00000	0.0555556
$325$	5.00000	0.277350
$326$	11.0000	0.609234
$327$	5.00000	0.276501
$328$	−6.00000	−0.331295
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	3.00000	0.164646
$333$	1.00000	0.0547997
$334$	−15.0000	−0.820763
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	−12.0000	−0.652714
$339$	−6.00000	−0.325875
$340$	0	0
$341$	6.00000	0.324918
$342$	−7.00000	−0.378517
$343$	13.0000	0.701934
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−9.00000	−0.483843
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	5.00000	0.267261
$351$	−1.00000	−0.0533761
$352$	3.00000	0.159901
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	−3.00000	−0.159000
$357$	3.00000	0.158777
$358$	12.0000	0.634220
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	20.0000	1.05118
$363$	−2.00000	−0.104973
$364$	1.00000	0.0524142
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	3.00000	0.156386
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−9.00000	−0.467257
$372$	2.00000	0.103695
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	6.00000	0.309426
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	5.00000	0.256158
$382$	−9.00000	−0.460480
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	26.0000	1.32337
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	−6.00000	−0.303046
$393$	18.0000	0.907980
$394$	−21.0000	−1.05796
$395$	0	0
$396$	3.00000	0.150756
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	−28.0000	−1.40351
$399$	7.00000	0.350438
$400$	−5.00000	−0.250000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	2.00000	0.0997509
$403$	−2.00000	−0.0996271
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	3.00000	0.148704
$408$	−3.00000	−0.148522
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	14.0000	0.689730
$413$	0	0
$414$	3.00000	0.147442
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	14.0000	0.685583
$418$	−21.0000	−1.02714
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−28.0000	−1.36302
$423$	6.00000	0.291730
$424$	9.00000	0.437079
$425$	15.0000	0.727607
$426$	12.0000	0.581402
$427$	10.0000	0.483934
$428$	9.00000	0.435031
$429$	−3.00000	−0.144841
$430$	0	0
$431$	−39.0000	−1.87856	−0.939282	−	0.343146i	$-0.888507\pi$
$431$	−39.0000	−1.87856	−0.939282	+	0.343146i	$0.888507\pi$
$432$	1.00000	0.0481125
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	5.00000	0.239457
$437$	−21.0000	−1.00457
$438$	5.00000	0.238909
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	3.00000	0.142695
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	−16.0000	−0.757622
$447$	−18.0000	−0.851371
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−5.00000	−0.235702
$451$	−18.0000	−0.847587
$452$	−6.00000	−0.282216
$453$	5.00000	0.234920
$454$	−24.0000	−1.12638
$455$	0	0
$456$	−7.00000	−0.327805
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−16.0000	−0.747631
$459$	−3.00000	−0.140028
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	−3.00000	−0.139573
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	0	0
$465$	0	0
$466$	30.0000	1.38972
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	−1.00000	−0.0462250
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	8.00000	0.368621
$472$	0	0
$473$	−12.0000	−0.551761
$474$	2.00000	0.0918630
$475$	35.0000	1.60591
$476$	3.00000	0.137505
$477$	9.00000	0.412082
$478$	−24.0000	−1.09773
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	−28.0000	−1.27537
$483$	−3.00000	−0.136505
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	−10.0000	−0.452679
$489$	11.0000	0.497437
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	7.00000	0.314945
$495$	0	0
$496$	2.00000	0.0898027
$497$	−12.0000	−0.538274
$498$	3.00000	0.134433
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	−15.0000	−0.670151
$502$	18.0000	0.803379
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	9.00000	0.400099
$507$	−12.0000	−0.532939
$508$	5.00000	0.221839
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	−5.00000	−0.221187
$512$	1.00000	0.0441942
$513$	−7.00000	−0.309058
$514$	−27.0000	−1.19092
$515$	0	0
$516$	−4.00000	−0.176090
$517$	18.0000	0.791639
$518$	−1.00000	−0.0439375
$519$	−9.00000	−0.395056
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	18.0000	0.786334
$525$	5.00000	0.218218
$526$	6.00000	0.261612
$527$	−6.00000	−0.261364
$528$	3.00000	0.130558
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	7.00000	0.303488
$533$	6.00000	0.259889
$534$	−3.00000	−0.129823
$535$	0	0
$536$	2.00000	0.0863868
$537$	12.0000	0.517838
$538$	15.0000	0.646696
$539$	−18.0000	−0.775315
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	8.00000	0.343629
$543$	20.0000	0.858282
$544$	−3.00000	−0.128624
$545$	0	0
$546$	1.00000	0.0427960
$547$	−13.0000	−0.555840	−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000	−0.555840	−0.277920	+	0.960604i	$0.589645\pi$
$548$	−12.0000	−0.512615
$549$	−10.0000	−0.426790
$550$	−15.0000	−0.639602
$551$	0	0
$552$	3.00000	0.127688
$553$	−2.00000	−0.0850487
$554$	17.0000	0.722261
$555$	0	0
$556$	14.0000	0.593732
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	2.00000	0.0846668
$559$	4.00000	0.169182
$560$	0	0
$561$	−9.00000	−0.379980
$562$	9.00000	0.379642
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	−1.00000	−0.0420331
$567$	−1.00000	−0.0419961
$568$	12.0000	0.503509
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−3.00000	−0.125436
$573$	−9.00000	−0.375980
$574$	6.00000	0.250435
$575$	−15.0000	−0.625543
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	26.0000	1.08052
$580$	0	0
$581$	−3.00000	−0.124461
$582$	2.00000	0.0829027
$583$	27.0000	1.11823
$584$	5.00000	0.206901
$585$	0	0
$586$	9.00000	0.371787
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−6.00000	−0.247436
$589$	−14.0000	−0.576860
$590$	0	0
$591$	−21.0000	−0.863825
$592$	1.00000	0.0410997
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	−18.0000	−0.737309
$597$	−28.0000	−1.14596
$598$	−3.00000	−0.122679
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	−5.00000	−0.204124
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	4.00000	0.163028
$603$	2.00000	0.0814463
$604$	5.00000	0.203447
$605$	0	0
$606$	6.00000	0.243733
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	0	0
$611$	−6.00000	−0.242734
$612$	−3.00000	−0.121268
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	−3.00000	−0.120873
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	14.0000	0.563163
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	3.00000	0.120192
$624$	−1.00000	−0.0400320
$625$	25.0000	1.00000
$626$	−10.0000	−0.399680
$627$	−21.0000	−0.838659
$628$	8.00000	0.319235
$629$	−3.00000	−0.119618
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	2.00000	0.0795557
$633$	−28.0000	−1.11290
$634$	30.0000	1.19145
$635$	0	0
$636$	9.00000	0.356873
$637$	6.00000	0.237729
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	9.00000	0.355202
$643$	5.00000	0.197181	0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000	0.197181	0.0985904	+	0.995128i	$0.468567\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	21.0000	0.826234
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	5.00000	0.196116
$651$	−2.00000	−0.0783862
$652$	11.0000	0.430793
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	5.00000	0.195515
$655$	0	0
$656$	−6.00000	−0.234261
$657$	5.00000	0.195069
$658$	−6.00000	−0.233904
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	−4.00000	−0.155464
$663$	3.00000	0.116510
$664$	3.00000	0.116423
$665$	0	0
$666$	1.00000	0.0387492
$667$	0	0
$668$	−15.0000	−0.580367
$669$	−16.0000	−0.618596
$670$	0	0
$671$	−30.0000	−1.15814
$672$	−1.00000	−0.0385758
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	5.00000	0.192593
$675$	−5.00000	−0.192450
$676$	−12.0000	−0.461538
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	−6.00000	−0.230429
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−24.0000	−0.919682
$682$	6.00000	0.229752
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	−7.00000	−0.267652
$685$	0	0
$686$	13.0000	0.496342
$687$	−16.0000	−0.610438
$688$	−4.00000	−0.152499
$689$	−9.00000	−0.342873
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	−9.00000	−0.342129
$693$	−3.00000	−0.113961
$694$	6.00000	0.227757
$695$	0	0
$696$	0	0
$697$	18.0000	0.681799
$698$	26.0000	0.984115
$699$	30.0000	1.13470
$700$	5.00000	0.188982
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	−1.00000	−0.0377426
$703$	−7.00000	−0.264010
$704$	3.00000	0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	−3.00000	−0.112430
$713$	6.00000	0.224702
$714$	3.00000	0.112272
$715$	0	0
$716$	12.0000	0.448461
$717$	−24.0000	−0.896296
$718$	30.0000	1.11959
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	30.0000	1.11648
$723$	−28.0000	−1.04133
$724$	20.0000	0.743294
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	−10.0000	−0.369611
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−25.0000	−0.922767
$735$	0	0
$736$	3.00000	0.110581
$737$	6.00000	0.221013
$738$	−6.00000	−0.220863
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	0	0
$741$	7.00000	0.257151
$742$	−9.00000	−0.330400
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	20.0000	0.732252
$747$	3.00000	0.109764
$748$	−9.00000	−0.329073
$749$	−9.00000	−0.328853
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	6.00000	0.218797
$753$	18.0000	0.655956
$754$	0	0
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	−16.0000	−0.581146
$759$	9.00000	0.326679
$760$	0	0
$761$	54.0000	1.95750	0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000	1.95750	0.978749	+	0.205061i	$0.0657392\pi$
$762$	5.00000	0.181131
$763$	−5.00000	−0.181012
$764$	−9.00000	−0.325609
$765$	0	0
$766$	21.0000	0.758761
$767$	0	0
$768$	1.00000	0.0360844
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	26.0000	0.935760
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	−4.00000	−0.143777
$775$	−10.0000	−0.359211
$776$	2.00000	0.0717958
$777$	−1.00000	−0.0358748
$778$	−6.00000	−0.215110
$779$	42.0000	1.50481
$780$	0	0
$781$	36.0000	1.28818
$782$	−9.00000	−0.321839
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	18.0000	0.642039
$787$	−34.0000	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$788$	−21.0000	−0.748094
$789$	6.00000	0.213606
$790$	0	0
$791$	6.00000	0.213335
$792$	3.00000	0.106600
$793$	10.0000	0.355110
$794$	8.00000	0.283909
$795$	0	0
$796$	−28.0000	−0.992434
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	7.00000	0.247797
$799$	−18.0000	−0.636794
$800$	−5.00000	−0.176777
$801$	−3.00000	−0.106000
$802$	−3.00000	−0.105934
$803$	15.0000	0.529339
$804$	2.00000	0.0705346
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	15.0000	0.528025
$808$	6.00000	0.211079
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	3.00000	0.105150
$815$	0	0
$816$	−3.00000	−0.105021
$817$	28.0000	0.979596
$818$	−22.0000	−0.769212
$819$	1.00000	0.0349428
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	−12.0000	−0.418548
$823$	17.0000	0.592583	0.296291	−	0.955098i	$-0.404250\pi$
$823$	17.0000	0.592583	0.296291	+	0.955098i	$0.404250\pi$
$824$	14.0000	0.487713
$825$	−15.0000	−0.522233
$826$	0	0
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	3.00000	0.104257
$829$	17.0000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$829$	17.0000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$830$	0	0
$831$	17.0000	0.589723
$832$	−1.00000	−0.0346688
$833$	18.0000	0.623663
$834$	14.0000	0.484780
$835$	0	0
$836$	−21.0000	−0.726300
$837$	2.00000	0.0691301
$838$	−15.0000	−0.518166
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−10.0000	−0.344623
$843$	9.00000	0.309976
$844$	−28.0000	−0.963800
$845$	0	0
$846$	6.00000	0.206284
$847$	2.00000	0.0687208
$848$	9.00000	0.309061
$849$	−1.00000	−0.0343199
$850$	15.0000	0.514496
$851$	3.00000	0.102839
$852$	12.0000	0.411113
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	9.00000	0.307614
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	−3.00000	−0.102418
$859$	−13.0000	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	−39.0000	−1.32835
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−25.0000	−0.849535
$867$	−8.00000	−0.271694
$868$	−2.00000	−0.0678844
$869$	6.00000	0.203536
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	5.00000	0.169321
$873$	2.00000	0.0676897
$874$	−21.0000	−0.710336
$875$	0	0
$876$	5.00000	0.168934
$877$	−58.0000	−1.95852	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−58.0000	−1.95852	−0.979260	+	0.202606i	$0.935059\pi$
$878$	−10.0000	−0.337484
$879$	9.00000	0.303562
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	−6.00000	−0.202031
$883$	−25.0000	−0.841317	−0.420658	−	0.907219i	$-0.638201\pi$
$883$	−25.0000	−0.841317	−0.420658	+	0.907219i	$0.638201\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	36.0000	1.20944
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	1.00000	0.0335578
$889$	−5.00000	−0.167695
$890$	0	0
$891$	3.00000	0.100504
$892$	−16.0000	−0.535720
$893$	−42.0000	−1.40548
$894$	−18.0000	−0.602010
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−3.00000	−0.100167
$898$	−18.0000	−0.600668
$899$	0	0
$900$	−5.00000	−0.166667
$901$	−27.0000	−0.899500
$902$	−18.0000	−0.599334
$903$	4.00000	0.133112
$904$	−6.00000	−0.199557
$905$	0	0
$906$	5.00000	0.166114
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	−24.0000	−0.796468
$909$	6.00000	0.199007
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	−7.00000	−0.231793
$913$	9.00000	0.297857
$914$	8.00000	0.264616
$915$	0	0
$916$	−16.0000	−0.528655
$917$	−18.0000	−0.594412
$918$	−3.00000	−0.0990148
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	−6.00000	−0.197599
$923$	−12.0000	−0.394985
$924$	−3.00000	−0.0986928
$925$	−5.00000	−0.164399
$926$	−34.0000	−1.11731
$927$	14.0000	0.459820
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	42.0000	1.37649
$932$	30.0000	0.982683
$933$	0	0
$934$	6.00000	0.196326
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	−2.00000	−0.0653023
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	8.00000	0.260654
$943$	−18.0000	−0.586161
$944$	0	0
$945$	0	0
$946$	−12.0000	−0.390154
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	2.00000	0.0649570
$949$	−5.00000	−0.162307
$950$	35.0000	1.13555
$951$	30.0000	0.972817
$952$	3.00000	0.0972306
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−39.0000	−1.26003
$959$	12.0000	0.387500
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−1.00000	−0.0322413
$963$	9.00000	0.290021
$964$	−28.0000	−0.901819
$965$	0	0
$966$	−3.00000	−0.0965234
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	−2.00000	−0.0642824
$969$	21.0000	0.674617
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000	0.0320750
$973$	−14.0000	−0.448819
$974$	38.0000	1.21760
$975$	5.00000	0.160128
$976$	−10.0000	−0.320092
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	11.0000	0.351741
$979$	−9.00000	−0.287641
$980$	0	0
$981$	5.00000	0.159638
$982$	15.0000	0.478669
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	0	0
$987$	−6.00000	−0.190982
$988$	7.00000	0.222700
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	2.00000	0.0635001
$993$	−4.00000	−0.126936
$994$	−12.0000	−0.380617
$995$	0	0
$996$	3.00000	0.0950586
$997$	53.0000	1.67853	0.839263	−	0.543725i	$-0.182987\pi$
$997$	53.0000	1.67853	0.839263	+	0.543725i	$0.182987\pi$
$998$	−13.0000	−0.411508
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	222.2.a.e.1.1	✓	1
3.2	odd	2		666.2.a.a.1.1		1
4.3	odd	2		1776.2.a.c.1.1		1
5.4	even	2		5550.2.a.h.1.1		1
8.3	odd	2		7104.2.a.u.1.1		1
8.5	even	2		7104.2.a.g.1.1		1
12.11	even	2		5328.2.a.l.1.1		1
37.36	even	2		8214.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.a.e.1.1	✓	1	1.1	even	1	trivial
666.2.a.a.1.1		1	3.2	odd	2
1776.2.a.c.1.1		1	4.3	odd	2
5328.2.a.l.1.1		1	12.11	even	2
5550.2.a.h.1.1		1	5.4	even	2
7104.2.a.g.1.1		1	8.5	even	2
7104.2.a.u.1.1		1	8.3	odd	2
8214.2.a.d.1.1		1	37.36	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	222.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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