LMFDB - Embedded newform 4002.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$31.9561308889$
Analytic rank:	$2$
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	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} -2.00000 q^{20} +2.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -1.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} -2.00000 q^{42} -2.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} +2.00000 q^{56} +2.00000 q^{57} +1.00000 q^{58} +2.00000 q^{60} -10.0000 q^{61} +8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{66} +16.0000 q^{67} -6.00000 q^{68} -1.00000 q^{69} -4.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} +8.00000 q^{77} -2.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -14.0000 q^{83} +2.00000 q^{84} +12.0000 q^{85} +2.00000 q^{86} +1.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +4.00000 q^{91} +1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} +3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−2.00000	−0.447214
$21$	2.00000	0.436436
$22$	4.00000	0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−2.00000	−0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	6.00000	1.02899
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	2.00000	0.320256
$40$	2.00000	0.316228
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−2.00000	−0.308607
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	−1.00000	−0.147442
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	8.00000	1.07872
$56$	2.00000	0.267261
$57$	2.00000	0.264906
$58$	1.00000	0.131306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	8.00000	1.01600
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	−4.00000	−0.492366
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	−6.00000	−0.727607
$69$	−1.00000	−0.120386
$70$	−4.00000	−0.478091
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−2.00000	−0.229416
$77$	8.00000	0.911685
$78$	−2.00000	−0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	2.00000	0.218218
$85$	12.0000	1.30158
$86$	2.00000	0.215666
$87$	1.00000	0.107211
$88$	4.00000	0.426401
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	4.00000	0.419314
$92$	1.00000	0.104257
$93$	8.00000	0.829561
$94$	8.00000	0.825137
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	3.00000	0.303046
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−6.00000	−0.594089
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	2.00000	0.196116
$105$	−4.00000	−0.390360
$106$	6.00000	0.582772
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	−1.00000	−0.0962250
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−8.00000	−0.762770
$111$	−2.00000	−0.189832
$112$	−2.00000	−0.188982
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	−2.00000	−0.187317
$115$	−2.00000	−0.186501
$116$	−1.00000	−0.0928477
$117$	−2.00000	−0.184900
$118$	0	0
$119$	12.0000	1.10004
$120$	−2.00000	−0.182574
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	6.00000	0.541002
$124$	−8.00000	−0.718421
$125$	12.0000	1.07331
$126$	2.00000	0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	−4.00000	−0.350823
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	4.00000	0.348155
$133$	4.00000	0.346844
$134$	−16.0000	−1.38219
$135$	2.00000	0.172133
$136$	6.00000	0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	1.00000	0.0851257
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	4.00000	0.338062
$141$	8.00000	0.673722
$142$	8.00000	0.671345
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	−14.0000	−1.15865
$147$	3.00000	0.247436
$148$	2.00000	0.164399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−1.00000	−0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	2.00000	0.162221
$153$	−6.00000	−0.485071
$154$	−8.00000	−0.644658
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−8.00000	−0.636446
$159$	6.00000	0.475831
$160$	2.00000	0.158114
$161$	−2.00000	−0.157622
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	−8.00000	−0.622799
$166$	14.0000	1.08661
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	−2.00000	−0.152944
$172$	−2.00000	−0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−1.00000	−0.0758098
$175$	2.00000	0.151186
$176$	−4.00000	−0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	−2.00000	−0.149071
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	−4.00000	−0.296500
$183$	10.0000	0.739221
$184$	−1.00000	−0.0737210
$185$	−4.00000	−0.294086
$186$	−8.00000	−0.586588
$187$	24.0000	1.75505
$188$	−8.00000	−0.583460
$189$	2.00000	0.145479
$190$	−4.00000	−0.290191
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	−1.00000	−0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	8.00000	0.574367
$195$	−4.00000	−0.286446
$196$	−3.00000	−0.214286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	4.00000	0.284268
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	1.00000	0.0707107
$201$	−16.0000	−1.12855
$202$	10.0000	0.703598
$203$	2.00000	0.140372
$204$	6.00000	0.420084
$205$	12.0000	0.838116
$206$	−2.00000	−0.139347
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	8.00000	0.553372
$210$	4.00000	0.276026
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−6.00000	−0.412082
$213$	8.00000	0.548151
$214$	2.00000	0.136717
$215$	4.00000	0.272798
$216$	1.00000	0.0680414
$217$	16.0000	1.08615
$218$	8.00000	0.541828
$219$	−14.0000	−0.946032
$220$	8.00000	0.539360
$221$	12.0000	0.807207
$222$	2.00000	0.134231
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	2.00000	0.133631
$225$	−1.00000	−0.0666667
$226$	10.0000	0.665190
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	2.00000	0.132453
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	2.00000	0.131876
$231$	−8.00000	−0.526361
$232$	1.00000	0.0656532
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	2.00000	0.130744
$235$	16.0000	1.04372
$236$	0	0
$237$	−8.00000	−0.519656
$238$	−12.0000	−0.777844
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	2.00000	0.129099
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	6.00000	0.383326
$246$	−6.00000	−0.382546
$247$	4.00000	0.254514
$248$	8.00000	0.508001
$249$	14.0000	0.887214
$250$	−12.0000	−0.758947
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	−2.00000	−0.125988
$253$	−4.00000	−0.251478
$254$	16.0000	1.00393
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	−2.00000	−0.124515
$259$	−4.00000	−0.248548
$260$	4.00000	0.248069
$261$	−1.00000	−0.0618984
$262$	12.0000	0.741362
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	−4.00000	−0.246183
$265$	12.0000	0.737154
$266$	−4.00000	−0.245256
$267$	6.00000	0.367194
$268$	16.0000	0.977356
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−2.00000	−0.121716
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−6.00000	−0.363803
$273$	−4.00000	−0.242091
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	−1.00000	−0.0601929
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−12.0000	−0.719712
$279$	−8.00000	−0.478947
$280$	−4.00000	−0.239046
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	−8.00000	−0.476393
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	−8.00000	−0.474713
$285$	−4.00000	−0.236940
$286$	−8.00000	−0.473050
$287$	12.0000	0.708338
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	−2.00000	−0.117444
$291$	8.00000	0.468968
$292$	14.0000	0.819288
$293$	−20.0000	−1.16841	−0.584206	−	0.811605i	$-0.698594\pi$
$293$	−20.0000	−1.16841	−0.584206	+	0.811605i	$0.698594\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	−18.0000	−1.04271
$299$	−2.00000	−0.115663
$300$	1.00000	0.0577350
$301$	4.00000	0.230556
$302$	−8.00000	−0.460348
$303$	10.0000	0.574485
$304$	−2.00000	−0.114708
$305$	20.0000	1.14520
$306$	6.00000	0.342997
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	8.00000	0.455842
$309$	−2.00000	−0.113776
$310$	−16.0000	−0.908739
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	−2.00000	−0.113228
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	14.0000	0.790066
$315$	4.00000	0.225374
$316$	8.00000	0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−6.00000	−0.336463
$319$	4.00000	0.223957
$320$	−2.00000	−0.111803
$321$	2.00000	0.111629
$322$	2.00000	0.111456
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	8.00000	0.442401
$328$	6.00000	0.331295
$329$	16.0000	0.882109
$330$	8.00000	0.440386
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−14.0000	−0.768350
$333$	2.00000	0.109599
$334$	−16.0000	−0.875481
$335$	−32.0000	−1.74835
$336$	2.00000	0.109109
$337$	36.0000	1.96104	0.980522	−	0.196407i	$-0.0629273\pi$
$337$	36.0000	1.96104	0.980522	+	0.196407i	$0.0629273\pi$
$338$	9.00000	0.489535
$339$	10.0000	0.543125
$340$	12.0000	0.650791
$341$	32.0000	1.73290
$342$	2.00000	0.108148
$343$	20.0000	1.07990
$344$	2.00000	0.107833
$345$	2.00000	0.107676
$346$	−6.00000	−0.322562
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	1.00000	0.0536056
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	−2.00000	−0.106904
$351$	2.00000	0.106752
$352$	4.00000	0.213201
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	−6.00000	−0.317999
$357$	−12.0000	−0.635107
$358$	16.0000	0.845626
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	2.00000	0.105409
$361$	−15.0000	−0.789474
$362$	12.0000	0.630706
$363$	−5.00000	−0.262432
$364$	4.00000	0.209657
$365$	−28.0000	−1.46559
$366$	−10.0000	−0.522708
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	1.00000	0.0521286
$369$	−6.00000	−0.312348
$370$	4.00000	0.207950
$371$	12.0000	0.623009
$372$	8.00000	0.414781
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−24.0000	−1.24101
$375$	−12.0000	−0.619677
$376$	8.00000	0.412568
$377$	2.00000	0.103005
$378$	−2.00000	−0.102869
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	4.00000	0.205196
$381$	16.0000	0.819705
$382$	−10.0000	−0.511645
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	1.00000	0.0510310
$385$	−16.0000	−0.815436
$386$	−6.00000	−0.305392
$387$	−2.00000	−0.101666
$388$	−8.00000	−0.406138
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	4.00000	0.202548
$391$	−6.00000	−0.303433
$392$	3.00000	0.151523
$393$	12.0000	0.605320
$394$	−10.0000	−0.503793
$395$	−16.0000	−0.805047
$396$	−4.00000	−0.201008
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−22.0000	−1.10276
$399$	−4.00000	−0.200250
$400$	−1.00000	−0.0500000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	16.0000	0.798007
$403$	16.0000	0.797017
$404$	−10.0000	−0.497519
$405$	−2.00000	−0.0993808
$406$	−2.00000	−0.0992583
$407$	−8.00000	−0.396545
$408$	−6.00000	−0.297044
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	−12.0000	−0.592638
$411$	−6.00000	−0.295958
$412$	2.00000	0.0985329
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	28.0000	1.37447
$416$	2.00000	0.0980581
$417$	−12.0000	−0.587643
$418$	−8.00000	−0.391293
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	−4.00000	−0.195180
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	−8.00000	−0.387601
$427$	20.0000	0.967868
$428$	−2.00000	−0.0966736
$429$	−8.00000	−0.386244
$430$	−4.00000	−0.192897
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	−24.0000	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−16.0000	−0.768025
$435$	−2.00000	−0.0958927
$436$	−8.00000	−0.383131
$437$	−2.00000	−0.0956730
$438$	14.0000	0.668946
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	−8.00000	−0.381385
$441$	−3.00000	−0.142857
$442$	−12.0000	−0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−2.00000	−0.0949158
$445$	12.0000	0.568855
$446$	−4.00000	−0.189405
$447$	−18.0000	−0.851371
$448$	−2.00000	−0.0944911
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	1.00000	0.0471405
$451$	24.0000	1.13012
$452$	−10.0000	−0.470360
$453$	−8.00000	−0.375873
$454$	14.0000	0.657053
$455$	−8.00000	−0.375046
$456$	−2.00000	−0.0936586
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	10.0000	0.467269
$459$	6.00000	0.280056
$460$	−2.00000	−0.0932505
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	8.00000	0.372194
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	−1.00000	−0.0464238
$465$	−16.0000	−0.741982
$466$	14.0000	0.648537
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	−2.00000	−0.0924500
$469$	−32.0000	−1.47762
$470$	−16.0000	−0.738025
$471$	14.0000	0.645086
$472$	0	0
$473$	8.00000	0.367840
$474$	8.00000	0.367452
$475$	2.00000	0.0917663
$476$	12.0000	0.550019
$477$	−6.00000	−0.274721
$478$	24.0000	1.09773
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	−2.00000	−0.0912871
$481$	−4.00000	−0.182384
$482$	18.0000	0.819878
$483$	2.00000	0.0910032
$484$	5.00000	0.227273
$485$	16.0000	0.726523
$486$	1.00000	0.0453609
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	10.0000	0.452679
$489$	−4.00000	−0.180886
$490$	−6.00000	−0.271052
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	6.00000	0.270501
$493$	6.00000	0.270226
$494$	−4.00000	−0.179969
$495$	8.00000	0.359573
$496$	−8.00000	−0.359211
$497$	16.0000	0.717698
$498$	−14.0000	−0.627355
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	12.0000	0.536656
$501$	−16.0000	−0.714827
$502$	4.00000	0.178529
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	2.00000	0.0890871
$505$	20.0000	0.889988
$506$	4.00000	0.177822
$507$	9.00000	0.399704
$508$	−16.0000	−0.709885
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	12.0000	0.531369
$511$	−28.0000	−1.23865
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	−10.0000	−0.441081
$515$	−4.00000	−0.176261
$516$	2.00000	0.0880451
$517$	32.0000	1.40736
$518$	4.00000	0.175750
$519$	−6.00000	−0.263371
$520$	−4.00000	−0.175412
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	1.00000	0.0437688
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	−12.0000	−0.524222
$525$	−2.00000	−0.0872872
$526$	−30.0000	−1.30806
$527$	48.0000	2.09091
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	−12.0000	−0.521247
$531$	0	0
$532$	4.00000	0.173422
$533$	12.0000	0.519778
$534$	−6.00000	−0.259645
$535$	4.00000	0.172935
$536$	−16.0000	−0.691095
$537$	16.0000	0.690451
$538$	6.00000	0.258678
$539$	12.0000	0.516877
$540$	2.00000	0.0860663
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	−16.0000	−0.687259
$543$	12.0000	0.514969
$544$	6.00000	0.257248
$545$	16.0000	0.685365
$546$	4.00000	0.171184
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	6.00000	0.256307
$549$	−10.0000	−0.426790
$550$	−4.00000	−0.170561
$551$	2.00000	0.0852029
$552$	1.00000	0.0425628
$553$	−16.0000	−0.680389
$554$	−18.0000	−0.764747
$555$	4.00000	0.169791
$556$	12.0000	0.508913
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	8.00000	0.338667
$559$	4.00000	0.169182
$560$	4.00000	0.169031
$561$	−24.0000	−1.01328
$562$	−8.00000	−0.337460
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	8.00000	0.336861
$565$	20.0000	0.841406
$566$	24.0000	1.00880
$567$	−2.00000	−0.0839921
$568$	8.00000	0.335673
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	4.00000	0.167542
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	8.00000	0.334497
$573$	−10.0000	−0.417756
$574$	−12.0000	−0.500870
$575$	−1.00000	−0.0417029
$576$	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	$0.209571\pi$
$578$	−19.0000	−0.790296
$579$	−6.00000	−0.249351
$580$	2.00000	0.0830455
$581$	28.0000	1.16164
$582$	−8.00000	−0.331611
$583$	24.0000	0.993978
$584$	−14.0000	−0.579324
$585$	4.00000	0.165380
$586$	20.0000	0.826192
$587$	32.0000	1.32078	0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$588$	3.00000	0.123718
$589$	16.0000	0.659269
$590$	0	0
$591$	−10.0000	−0.411345
$592$	2.00000	0.0821995
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−4.00000	−0.164122
$595$	−24.0000	−0.983904
$596$	18.0000	0.737309
$597$	−22.0000	−0.900400
$598$	2.00000	0.0817861
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	−1.00000	−0.0408248
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	−4.00000	−0.163028
$603$	16.0000	0.651570
$604$	8.00000	0.325515
$605$	−10.0000	−0.406558
$606$	−10.0000	−0.406222
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	2.00000	0.0811107
$609$	−2.00000	−0.0810441
$610$	−20.0000	−0.809776
$611$	16.0000	0.647291
$612$	−6.00000	−0.242536
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	−20.0000	−0.807134
$615$	−12.0000	−0.483887
$616$	−8.00000	−0.322329
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	2.00000	0.0804518
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	16.0000	0.642575
$621$	−1.00000	−0.0401286
$622$	12.0000	0.481156
$623$	12.0000	0.480770
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	−2.00000	−0.0799361
$627$	−8.00000	−0.319489
$628$	−14.0000	−0.558661
$629$	−12.0000	−0.478471
$630$	−4.00000	−0.159364
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	6.00000	0.238290
$635$	32.0000	1.26988
$636$	6.00000	0.237915
$637$	6.00000	0.237729
$638$	−4.00000	−0.158362
$639$	−8.00000	−0.316475
$640$	2.00000	0.0790569
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	−2.00000	−0.0789337
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	−2.00000	−0.0788110
$645$	−4.00000	−0.157500
$646$	−12.0000	−0.472134
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	−16.0000	−0.627089
$652$	4.00000	0.156652
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−8.00000	−0.312825
$655$	24.0000	0.937758
$656$	−6.00000	−0.234261
$657$	14.0000	0.546192
$658$	−16.0000	−0.623745
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	−8.00000	−0.311400
$661$	−36.0000	−1.40024	−0.700119	−	0.714026i	$-0.746870\pi$
$661$	−36.0000	−1.40024	−0.700119	+	0.714026i	$0.746870\pi$
$662$	4.00000	0.155464
$663$	−12.0000	−0.466041
$664$	14.0000	0.543305
$665$	−8.00000	−0.310227
$666$	−2.00000	−0.0774984
$667$	−1.00000	−0.0387202
$668$	16.0000	0.619059
$669$	−4.00000	−0.154649
$670$	32.0000	1.23627
$671$	40.0000	1.54418
$672$	−2.00000	−0.0771517
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−36.0000	−1.38667
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−10.0000	−0.384048
$679$	16.0000	0.614024
$680$	−12.0000	−0.460179
$681$	14.0000	0.536481
$682$	−32.0000	−1.22534
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	−2.00000	−0.0764719
$685$	−12.0000	−0.458496
$686$	−20.0000	−0.763604
$687$	10.0000	0.381524
$688$	−2.00000	−0.0762493
$689$	12.0000	0.457164
$690$	−2.00000	−0.0761387
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	6.00000	0.228086
$693$	8.00000	0.303895
$694$	−28.0000	−1.06287
$695$	−24.0000	−0.910372
$696$	−1.00000	−0.0379049
$697$	36.0000	1.36360
$698$	30.0000	1.13552
$699$	14.0000	0.529529
$700$	2.00000	0.0755929
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−2.00000	−0.0754851
$703$	−4.00000	−0.150863
$704$	−4.00000	−0.150756
$705$	−16.0000	−0.602595
$706$	−6.00000	−0.225813
$707$	20.0000	0.752177
$708$	0	0
$709$	32.0000	1.20179	0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000	1.20179	0.600893	+	0.799330i	$0.294812\pi$
$710$	−16.0000	−0.600469
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	−8.00000	−0.299602
$714$	12.0000	0.449089
$715$	−16.0000	−0.598366
$716$	−16.0000	−0.597948
$717$	24.0000	0.896296
$718$	10.0000	0.373197
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−2.00000	−0.0745356
$721$	−4.00000	−0.148968
$722$	15.0000	0.558242
$723$	18.0000	0.669427
$724$	−12.0000	−0.445976
$725$	1.00000	0.0371391
$726$	5.00000	0.185567
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	28.0000	1.03633
$731$	12.0000	0.443836
$732$	10.0000	0.369611
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−8.00000	−0.295285
$735$	−6.00000	−0.221313
$736$	−1.00000	−0.0368605
$737$	−64.0000	−2.35747
$738$	6.00000	0.220863
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	−4.00000	−0.147043
$741$	−4.00000	−0.146944
$742$	−12.0000	−0.440534
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	−8.00000	−0.293294
$745$	−36.0000	−1.31894
$746$	4.00000	0.146450
$747$	−14.0000	−0.512233
$748$	24.0000	0.877527
$749$	4.00000	0.146157
$750$	12.0000	0.438178
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−8.00000	−0.291730
$753$	4.00000	0.145768
$754$	−2.00000	−0.0728357
$755$	−16.0000	−0.582300
$756$	2.00000	0.0727393
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	2.00000	0.0726433
$759$	4.00000	0.145191
$760$	−4.00000	−0.145095
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	−16.0000	−0.579619
$763$	16.0000	0.579239
$764$	10.0000	0.361787
$765$	12.0000	0.433861
$766$	−8.00000	−0.289052
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	16.0000	0.576600
$771$	−10.0000	−0.360141
$772$	6.00000	0.215945
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	2.00000	0.0718885
$775$	8.00000	0.287368
$776$	8.00000	0.287183
$777$	4.00000	0.143499
$778$	24.0000	0.860442
$779$	12.0000	0.429945
$780$	−4.00000	−0.143223
$781$	32.0000	1.14505
$782$	6.00000	0.214560
$783$	1.00000	0.0357371
$784$	−3.00000	−0.107143
$785$	28.0000	0.999363
$786$	−12.0000	−0.428026
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	10.0000	0.356235
$789$	−30.0000	−1.06803
$790$	16.0000	0.569254
$791$	20.0000	0.711118
$792$	4.00000	0.142134
$793$	20.0000	0.710221
$794$	−22.0000	−0.780751
$795$	−12.0000	−0.425596
$796$	22.0000	0.779769
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	4.00000	0.141598
$799$	48.0000	1.69812
$800$	1.00000	0.0353553
$801$	−6.00000	−0.212000
$802$	24.0000	0.847469
$803$	−56.0000	−1.97620
$804$	−16.0000	−0.564276
$805$	4.00000	0.140981
$806$	−16.0000	−0.563576
$807$	6.00000	0.211210
$808$	10.0000	0.351799
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	2.00000	0.0702728
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	2.00000	0.0701862
$813$	−16.0000	−0.561144
$814$	8.00000	0.280400
$815$	−8.00000	−0.280228
$816$	6.00000	0.210042
$817$	4.00000	0.139942
$818$	−30.0000	−1.04893
$819$	4.00000	0.139771
$820$	12.0000	0.419058
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	6.00000	0.209274
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−2.00000	−0.0696733
$825$	−4.00000	−0.139262
$826$	0	0
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	1.00000	0.0347524
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	−28.0000	−0.971894
$831$	−18.0000	−0.624413
$832$	−2.00000	−0.0693375
$833$	18.0000	0.623663
$834$	12.0000	0.415526
$835$	−32.0000	−1.10741
$836$	8.00000	0.276686
$837$	8.00000	0.276520
$838$	14.0000	0.483622
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	4.00000	0.138013
$841$	1.00000	0.0344828
$842$	2.00000	0.0689246
$843$	−8.00000	−0.275535
$844$	0	0
$845$	18.0000	0.619219
$846$	8.00000	0.275046
$847$	−10.0000	−0.343604
$848$	−6.00000	−0.206041
$849$	24.0000	0.823678
$850$	−6.00000	−0.205798
$851$	2.00000	0.0685591
$852$	8.00000	0.274075
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−20.0000	−0.684386
$855$	4.00000	0.136797
$856$	2.00000	0.0683586
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	8.00000	0.273115
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	4.00000	0.136399
$861$	−12.0000	−0.408959
$862$	−12.0000	−0.408722
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	1.00000	0.0340207
$865$	−12.0000	−0.408012
$866$	24.0000	0.815553
$867$	−19.0000	−0.645274
$868$	16.0000	0.543075
$869$	−32.0000	−1.08553
$870$	2.00000	0.0678064
$871$	−32.0000	−1.08428
$872$	8.00000	0.270914
$873$	−8.00000	−0.270759
$874$	2.00000	0.0676510
$875$	−24.0000	−0.811348
$876$	−14.0000	−0.473016
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	20.0000	0.674583
$880$	8.00000	0.269680
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	3.00000	0.101015
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−12.0000	−0.403148
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	2.00000	0.0671156
$889$	32.0000	1.07325
$890$	−12.0000	−0.402241
$891$	−4.00000	−0.134005
$892$	4.00000	0.133930
$893$	16.0000	0.535420
$894$	18.0000	0.602010
$895$	32.0000	1.06964
$896$	2.00000	0.0668153
$897$	2.00000	0.0667781
$898$	−18.0000	−0.600668
$899$	8.00000	0.266815
$900$	−1.00000	−0.0333333
$901$	36.0000	1.19933
$902$	−24.0000	−0.799113
$903$	−4.00000	−0.133112
$904$	10.0000	0.332595
$905$	24.0000	0.797787
$906$	8.00000	0.265782
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	−14.0000	−0.464606
$909$	−10.0000	−0.331679
$910$	8.00000	0.265197
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	2.00000	0.0662266
$913$	56.0000	1.85333
$914$	14.0000	0.463079
$915$	−20.0000	−0.661180
$916$	−10.0000	−0.330409
$917$	24.0000	0.792550
$918$	−6.00000	−0.198030
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	2.00000	0.0659380
$921$	−20.0000	−0.659022
$922$	−2.00000	−0.0658665
$923$	16.0000	0.526646
$924$	−8.00000	−0.263181
$925$	−2.00000	−0.0657596
$926$	−36.0000	−1.18303
$927$	2.00000	0.0656886
$928$	1.00000	0.0328266
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	16.0000	0.524661
$931$	6.00000	0.196642
$932$	−14.0000	−0.458585
$933$	12.0000	0.392862
$934$	32.0000	1.04707
$935$	−48.0000	−1.56977
$936$	2.00000	0.0653720
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	32.0000	1.04484
$939$	−2.00000	−0.0652675
$940$	16.0000	0.521862
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	−14.0000	−0.456145
$943$	−6.00000	−0.195387
$944$	0	0
$945$	−4.00000	−0.130120
$946$	−8.00000	−0.260102
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−8.00000	−0.259828
$949$	−28.0000	−0.908918
$950$	−2.00000	−0.0648886
$951$	6.00000	0.194563
$952$	−12.0000	−0.388922
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	6.00000	0.194257
$955$	−20.0000	−0.647185
$956$	−24.0000	−0.776215
$957$	−4.00000	−0.129302
$958$	18.0000	0.581554
$959$	−12.0000	−0.387500
$960$	2.00000	0.0645497
$961$	33.0000	1.06452
$962$	4.00000	0.128965
$963$	−2.00000	−0.0644491
$964$	−18.0000	−0.579741
$965$	−12.0000	−0.386294
$966$	−2.00000	−0.0643489
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−5.00000	−0.160706
$969$	−12.0000	−0.385496
$970$	−16.0000	−0.513729
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	−1.00000	−0.0320750
$973$	−24.0000	−0.769405
$974$	12.0000	0.384505
$975$	−2.00000	−0.0640513
$976$	−10.0000	−0.320092
$977$	−16.0000	−0.511885	−0.255943	−	0.966692i	$-0.582386\pi$
$977$	−16.0000	−0.511885	−0.255943	+	0.966692i	$0.582386\pi$
$978$	4.00000	0.127906
$979$	24.0000	0.767043
$980$	6.00000	0.191663
$981$	−8.00000	−0.255420
$982$	−12.0000	−0.382935
$983$	26.0000	0.829271	0.414636	−	0.909988i	$-0.363909\pi$
$983$	26.0000	0.829271	0.414636	+	0.909988i	$0.363909\pi$
$984$	−6.00000	−0.191273
$985$	−20.0000	−0.637253
$986$	−6.00000	−0.191079
$987$	−16.0000	−0.509286
$988$	4.00000	0.127257
$989$	−2.00000	−0.0635963
$990$	−8.00000	−0.254257
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	8.00000	0.254000
$993$	4.00000	0.126936
$994$	−16.0000	−0.507489
$995$	−44.0000	−1.39489
$996$	14.0000	0.443607
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−36.0000	−1.13956
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.a.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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