LMFDB - Embedded newform 4002.2.a.i.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:	$1$
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	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -2.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{29} +2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} -8.00000 q^{43} -2.00000 q^{44} -2.00000 q^{45} +1.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +2.00000 q^{56} +4.00000 q^{57} -1.00000 q^{58} +2.00000 q^{60} +8.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{66} +8.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -4.00000 q^{70} +1.00000 q^{72} -10.0000 q^{73} -4.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} -14.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} -2.00000 q^{84} -8.00000 q^{85} -8.00000 q^{86} +1.00000 q^{87} -2.00000 q^{88} -12.0000 q^{89} -2.00000 q^{90} -4.00000 q^{91} +1.00000 q^{92} -4.00000 q^{93} +8.00000 q^{95} -1.00000 q^{96} +18.0000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−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
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	−2.00000	−0.436436
$22$	−2.00000	−0.426401
$23$	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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	−2.00000	−0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−2.00000	−0.308607
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−2.00000	−0.301511
$45$	−2.00000	−0.298142
$46$	1.00000	0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−4.00000	−0.560112
$52$	−2.00000	−0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.00000	−0.136083
$55$	4.00000	0.539360
$56$	2.00000	0.267261
$57$	4.00000	0.529813
$58$	−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
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.00000	0.508001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	2.00000	0.246183
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	−4.00000	−0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−4.00000	−0.464991
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	2.00000	0.226455
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−2.00000	−0.218218
$85$	−8.00000	−0.867722
$86$	−8.00000	−0.862662
$87$	1.00000	0.107211
$88$	−2.00000	−0.213201
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	−2.00000	−0.210819
$91$	−4.00000	−0.419314
$92$	1.00000	0.104257
$93$	−4.00000	−0.414781
$94$	0	0
$95$	8.00000	0.820783
$96$	−1.00000	−0.102062
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	−3.00000	−0.303046
$99$	−2.00000	−0.201008
$100$	−1.00000	−0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−4.00000	−0.396059
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	−2.00000	−0.196116
$105$	4.00000	0.390360
$106$	2.00000	0.194257
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	−1.00000	−0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	4.00000	0.381385
$111$	4.00000	0.379663
$112$	2.00000	0.188982
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	4.00000	0.374634
$115$	−2.00000	−0.186501
$116$	−1.00000	−0.0928477
$117$	−2.00000	−0.184900
$118$	0	0
$119$	8.00000	0.733359
$120$	2.00000	0.182574
$121$	−7.00000	−0.636364
$122$	8.00000	0.724286
$123$	2.00000	0.180334
$124$	4.00000	0.359211
$125$	12.0000	1.07331
$126$	2.00000	0.178174
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	4.00000	0.350823
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	2.00000	0.174078
$133$	−8.00000	−0.693688
$134$	8.00000	0.691095
$135$	2.00000	0.172133
$136$	4.00000	0.342997
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	−1.00000	−0.0851257
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	−10.0000	−0.827606
$147$	3.00000	0.247436
$148$	−4.00000	−0.328798
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	1.00000	0.0816497
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	−4.00000	−0.324443
$153$	4.00000	0.323381
$154$	−4.00000	−0.322329
$155$	−8.00000	−0.642575
$156$	2.00000	0.160128
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	−14.0000	−1.11378
$159$	−2.00000	−0.158610
$160$	−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	$0.449930\pi$
$164$	−2.00000	−0.156174
$165$	−4.00000	−0.311400
$166$	6.00000	0.465690
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	−8.00000	−0.613572
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	1.00000	0.0758098
$175$	−2.00000	−0.151186
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−12.0000	−0.899438
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−2.00000	−0.149071
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	−4.00000	−0.296500
$183$	−8.00000	−0.591377
$184$	1.00000	0.0737210
$185$	8.00000	0.588172
$186$	−4.00000	−0.293294
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−2.00000	−0.145479
$190$	8.00000	0.580381
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	18.0000	1.29232
$195$	−4.00000	−0.286446
$196$	−3.00000	−0.214286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−2.00000	−0.142134
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	−1.00000	−0.0707107
$201$	−8.00000	−0.564276
$202$	−18.0000	−1.26648
$203$	−2.00000	−0.140372
$204$	−4.00000	−0.280056
$205$	4.00000	0.279372
$206$	−14.0000	−0.975426
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	8.00000	0.553372
$210$	4.00000	0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−18.0000	−1.23045
$215$	16.0000	1.09119
$216$	−1.00000	−0.0680414
$217$	8.00000	0.543075
$218$	−16.0000	−1.08366
$219$	10.0000	0.675737
$220$	4.00000	0.269680
$221$	−8.00000	−0.538138
$222$	4.00000	0.268462
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	2.00000	0.133631
$225$	−1.00000	−0.0666667
$226$	4.00000	0.266076
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	4.00000	0.264906
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	−2.00000	−0.131876
$231$	4.00000	0.263181
$232$	−1.00000	−0.0656532
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	14.0000	0.909398
$238$	8.00000	0.518563
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	2.00000	0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−7.00000	−0.449977
$243$	−1.00000	−0.0641500
$244$	8.00000	0.512148
$245$	6.00000	0.383326
$246$	2.00000	0.127515
$247$	8.00000	0.509028
$248$	4.00000	0.254000
$249$	−6.00000	−0.380235
$250$	12.0000	0.758947
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	2.00000	0.125988
$253$	−2.00000	−0.125739
$254$	8.00000	0.501965
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	8.00000	0.498058
$259$	−8.00000	−0.497096
$260$	4.00000	0.248069
$261$	−1.00000	−0.0618984
$262$	8.00000	0.494242
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	2.00000	0.123091
$265$	−4.00000	−0.245718
$266$	−8.00000	−0.490511
$267$	12.0000	0.734388
$268$	8.00000	0.488678
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	2.00000	0.121716
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	4.00000	0.242536
$273$	4.00000	0.242091
$274$	0	0
$275$	2.00000	0.120605
$276$	−1.00000	−0.0601929
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	−12.0000	−0.719712
$279$	4.00000	0.239474
$280$	−4.00000	−0.239046
$281$	32.0000	1.90896	0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000	1.90896	0.954480	+	0.298275i	$0.0964112\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	4.00000	0.236525
$287$	−4.00000	−0.236113
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	2.00000	0.117444
$291$	−18.0000	−1.05518
$292$	−10.0000	−0.585206
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−4.00000	−0.232495
$297$	2.00000	0.116052
$298$	14.0000	0.810998
$299$	−2.00000	−0.115663
$300$	1.00000	0.0577350
$301$	−16.0000	−0.922225
$302$	−20.0000	−1.15087
$303$	18.0000	1.03407
$304$	−4.00000	−0.229416
$305$	−16.0000	−0.916157
$306$	4.00000	0.228665
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−4.00000	−0.227921
$309$	14.0000	0.796432
$310$	−8.00000	−0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	2.00000	0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−20.0000	−1.12867
$315$	−4.00000	−0.225374
$316$	−14.0000	−0.787562
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	−2.00000	−0.112154
$319$	2.00000	0.111979
$320$	−2.00000	−0.111803
$321$	18.0000	1.00466
$322$	2.00000	0.111456
$323$	−16.0000	−0.890264
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	4.00000	0.221540
$327$	16.0000	0.884802
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−4.00000	−0.220193
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	6.00000	0.329293
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−16.0000	−0.874173
$336$	−2.00000	−0.109109
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−9.00000	−0.489535
$339$	−4.00000	−0.217250
$340$	−8.00000	−0.433861
$341$	−8.00000	−0.433224
$342$	−4.00000	−0.216295
$343$	−20.0000	−1.07990
$344$	−8.00000	−0.431331
$345$	2.00000	0.107676
$346$	2.00000	0.107521
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	1.00000	0.0536056
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−2.00000	−0.106904
$351$	2.00000	0.106752
$352$	−2.00000	−0.106600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	−12.0000	−0.635999
$357$	−8.00000	−0.423405
$358$	4.00000	0.211407
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	0	0
$363$	7.00000	0.367405
$364$	−4.00000	−0.209657
$365$	20.0000	1.04685
$366$	−8.00000	−0.418167
$367$	34.0000	1.77479	0.887393	−	0.461014i	$-0.152514\pi$
$367$	34.0000	1.77479	0.887393	+	0.461014i	$0.152514\pi$
$368$	1.00000	0.0521286
$369$	−2.00000	−0.104116
$370$	8.00000	0.415900
$371$	4.00000	0.207670
$372$	−4.00000	−0.207390
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	−8.00000	−0.413670
$375$	−12.0000	−0.619677
$376$	0	0
$377$	2.00000	0.103005
$378$	−2.00000	−0.102869
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	8.00000	0.410391
$381$	−8.00000	−0.409852
$382$	24.0000	1.22795
$383$	28.0000	1.43073	0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000	1.43073	0.715367	+	0.698749i	$0.246260\pi$
$384$	−1.00000	−0.0510310
$385$	8.00000	0.407718
$386$	14.0000	0.712581
$387$	−8.00000	−0.406663
$388$	18.0000	0.913812
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	−4.00000	−0.202548
$391$	4.00000	0.202289
$392$	−3.00000	−0.151523
$393$	−8.00000	−0.403547
$394$	−18.0000	−0.906827
$395$	28.0000	1.40883
$396$	−2.00000	−0.100504
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	−26.0000	−1.30326
$399$	8.00000	0.400501
$400$	−1.00000	−0.0500000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	−8.00000	−0.399004
$403$	−8.00000	−0.398508
$404$	−18.0000	−0.895533
$405$	−2.00000	−0.0993808
$406$	−2.00000	−0.0992583
$407$	8.00000	0.396545
$408$	−4.00000	−0.198030
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	−14.0000	−0.689730
$413$	0	0
$414$	1.00000	0.0491473
$415$	−12.0000	−0.589057
$416$	−2.00000	−0.0980581
$417$	12.0000	0.587643
$418$	8.00000	0.391293
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	4.00000	0.195180
$421$	40.0000	1.94948	0.974740	−	0.223341i	$-0.0716964\pi$
$421$	40.0000	1.94948	0.974740	+	0.223341i	$0.0716964\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	2.00000	0.0971286
$425$	−4.00000	−0.194029
$426$	0	0
$427$	16.0000	0.774294
$428$	−18.0000	−0.870063
$429$	−4.00000	−0.193122
$430$	16.0000	0.771589
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	−1.00000	−0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	8.00000	0.384012
$435$	−2.00000	−0.0958927
$436$	−16.0000	−0.766261
$437$	−4.00000	−0.191346
$438$	10.0000	0.477818
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	4.00000	0.190693
$441$	−3.00000	−0.142857
$442$	−8.00000	−0.380521
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	4.00000	0.189832
$445$	24.0000	1.13771
$446$	−4.00000	−0.189405
$447$	−14.0000	−0.662177
$448$	2.00000	0.0944911
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−1.00000	−0.0471405
$451$	4.00000	0.188353
$452$	4.00000	0.188144
$453$	20.0000	0.939682
$454$	−18.0000	−0.844782
$455$	8.00000	0.375046
$456$	4.00000	0.187317
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−20.0000	−0.934539
$459$	−4.00000	−0.186704
$460$	−2.00000	−0.0932505
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	4.00000	0.186097
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−1.00000	−0.0464238
$465$	8.00000	0.370991
$466$	−6.00000	−0.277945
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	−2.00000	−0.0924500
$469$	16.0000	0.738811
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	16.0000	0.735681
$474$	14.0000	0.643041
$475$	4.00000	0.183533
$476$	8.00000	0.366679
$477$	2.00000	0.0915737
$478$	−8.00000	−0.365911
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	2.00000	0.0912871
$481$	8.00000	0.364769
$482$	10.0000	0.455488
$483$	−2.00000	−0.0910032
$484$	−7.00000	−0.318182
$485$	−36.0000	−1.63468
$486$	−1.00000	−0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	8.00000	0.362143
$489$	−4.00000	−0.180886
$490$	6.00000	0.271052
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	2.00000	0.0901670
$493$	−4.00000	−0.180151
$494$	8.00000	0.359937
$495$	4.00000	0.179787
$496$	4.00000	0.179605
$497$	0	0
$498$	−6.00000	−0.268866
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	30.0000	1.33897
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	2.00000	0.0890871
$505$	36.0000	1.60198
$506$	−2.00000	−0.0889108
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	8.00000	0.354246
$511$	−20.0000	−0.884748
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	−6.00000	−0.264649
$515$	28.0000	1.23383
$516$	8.00000	0.352180
$517$	0	0
$518$	−8.00000	−0.351500
$519$	−2.00000	−0.0877903
$520$	4.00000	0.175412
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	−1.00000	−0.0437688
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	8.00000	0.349482
$525$	2.00000	0.0872872
$526$	−12.0000	−0.523225
$527$	16.0000	0.696971
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	−4.00000	−0.173749
$531$	0	0
$532$	−8.00000	−0.346844
$533$	4.00000	0.173259
$534$	12.0000	0.519291
$535$	36.0000	1.55642
$536$	8.00000	0.345547
$537$	−4.00000	−0.172613
$538$	14.0000	0.603583
$539$	6.00000	0.258438
$540$	2.00000	0.0860663
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	24.0000	1.03089
$543$	0	0
$544$	4.00000	0.171499
$545$	32.0000	1.37073
$546$	4.00000	0.171184
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	2.00000	0.0852803
$551$	4.00000	0.170406
$552$	−1.00000	−0.0425628
$553$	−28.0000	−1.19068
$554$	−6.00000	−0.254916
$555$	−8.00000	−0.339581
$556$	−12.0000	−0.508913
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	4.00000	0.169334
$559$	16.0000	0.676728
$560$	−4.00000	−0.169031
$561$	8.00000	0.337760
$562$	32.0000	1.34984
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	12.0000	0.504398
$567$	2.00000	0.0839921
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	−8.00000	−0.335083
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	4.00000	0.167248
$573$	−24.0000	−1.00261
$574$	−4.00000	−0.166957
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−1.00000	−0.0415945
$579$	−14.0000	−0.581820
$580$	2.00000	0.0830455
$581$	12.0000	0.497844
$582$	−18.0000	−0.746124
$583$	−4.00000	−0.165663
$584$	−10.0000	−0.413803
$585$	4.00000	0.165380
$586$	−14.0000	−0.578335
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	3.00000	0.123718
$589$	−16.0000	−0.659269
$590$	0	0
$591$	18.0000	0.740421
$592$	−4.00000	−0.164399
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	2.00000	0.0820610
$595$	−16.0000	−0.655936
$596$	14.0000	0.573462
$597$	26.0000	1.06411
$598$	−2.00000	−0.0817861
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	1.00000	0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−16.0000	−0.652111
$603$	8.00000	0.325785
$604$	−20.0000	−0.813788
$605$	14.0000	0.569181
$606$	18.0000	0.731200
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	−4.00000	−0.162221
$609$	2.00000	0.0810441
$610$	−16.0000	−0.647821
$611$	0	0
$612$	4.00000	0.161690
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	−20.0000	−0.807134
$615$	−4.00000	−0.161296
$616$	−4.00000	−0.161165
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	14.0000	0.563163
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−8.00000	−0.321288
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−24.0000	−0.961540
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	14.0000	0.559553
$627$	−8.00000	−0.319489
$628$	−20.0000	−0.798087
$629$	−16.0000	−0.637962
$630$	−4.00000	−0.159364
$631$	42.0000	1.67199	0.835997	−	0.548734i	$-0.184890\pi$
$631$	42.0000	1.67199	0.835997	+	0.548734i	$0.184890\pi$
$632$	−14.0000	−0.556890
$633$	12.0000	0.476957
$634$	−30.0000	−1.19145
$635$	−16.0000	−0.634941
$636$	−2.00000	−0.0793052
$637$	6.00000	0.237729
$638$	2.00000	0.0791808
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	18.0000	0.710403
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	2.00000	0.0788110
$645$	−16.0000	−0.629999
$646$	−16.0000	−0.629512
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	2.00000	0.0784465
$651$	−8.00000	−0.313545
$652$	4.00000	0.156652
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	16.0000	0.625650
$655$	−16.0000	−0.625172
$656$	−2.00000	−0.0780869
$657$	−10.0000	−0.390137
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	−4.00000	−0.155700
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	−4.00000	−0.155464
$663$	8.00000	0.310694
$664$	6.00000	0.232845
$665$	16.0000	0.620453
$666$	−4.00000	−0.154997
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	4.00000	0.154649
$670$	−16.0000	−0.618134
$671$	−16.0000	−0.617673
$672$	−2.00000	−0.0771517
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−18.0000	−0.693334
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−4.00000	−0.153619
$679$	36.0000	1.38155
$680$	−8.00000	−0.306786
$681$	18.0000	0.689761
$682$	−8.00000	−0.306336
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−20.0000	−0.763604
$687$	20.0000	0.763048
$688$	−8.00000	−0.304997
$689$	−4.00000	−0.152388
$690$	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	$0.624220\pi$
$692$	2.00000	0.0760286
$693$	−4.00000	−0.151947
$694$	12.0000	0.455514
$695$	24.0000	0.910372
$696$	1.00000	0.0379049
$697$	−8.00000	−0.303022
$698$	−6.00000	−0.227103
$699$	6.00000	0.226941
$700$	−2.00000	−0.0755929
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	2.00000	0.0754851
$703$	16.0000	0.603451
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	14.0000	0.526897
$707$	−36.0000	−1.35392
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	−14.0000	−0.525041
$712$	−12.0000	−0.449719
$713$	4.00000	0.149801
$714$	−8.00000	−0.299392
$715$	−8.00000	−0.299183
$716$	4.00000	0.149487
$717$	8.00000	0.298765
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−2.00000	−0.0745356
$721$	−28.0000	−1.04277
$722$	−3.00000	−0.111648
$723$	−10.0000	−0.371904
$724$	0	0
$725$	1.00000	0.0371391
$726$	7.00000	0.259794
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	20.0000	0.740233
$731$	−32.0000	−1.18356
$732$	−8.00000	−0.295689
$733$	48.0000	1.77292	0.886460	−	0.462805i	$-0.153157\pi$
$733$	48.0000	1.77292	0.886460	+	0.462805i	$0.153157\pi$
$734$	34.0000	1.25496
$735$	−6.00000	−0.221313
$736$	1.00000	0.0368605
$737$	−16.0000	−0.589368
$738$	−2.00000	−0.0736210
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	8.00000	0.294086
$741$	−8.00000	−0.293887
$742$	4.00000	0.146845
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	−4.00000	−0.146647
$745$	−28.0000	−1.02584
$746$	−24.0000	−0.878702
$747$	6.00000	0.219529
$748$	−8.00000	−0.292509
$749$	−36.0000	−1.31541
$750$	−12.0000	−0.438178
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	2.00000	0.0728357
$755$	40.0000	1.45575
$756$	−2.00000	−0.0727393
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	−4.00000	−0.145287
$759$	2.00000	0.0725954
$760$	8.00000	0.290191
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	−8.00000	−0.289809
$763$	−32.0000	−1.15848
$764$	24.0000	0.868290
$765$	−8.00000	−0.289241
$766$	28.0000	1.01168
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	8.00000	0.288300
$771$	6.00000	0.216085
$772$	14.0000	0.503871
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−8.00000	−0.287554
$775$	−4.00000	−0.143684
$776$	18.0000	0.646162
$777$	8.00000	0.286998
$778$	−26.0000	−0.932145
$779$	8.00000	0.286630
$780$	−4.00000	−0.143223
$781$	0	0
$782$	4.00000	0.143040
$783$	1.00000	0.0357371
$784$	−3.00000	−0.107143
$785$	40.0000	1.42766
$786$	−8.00000	−0.285351
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	−18.0000	−0.641223
$789$	12.0000	0.427211
$790$	28.0000	0.996195
$791$	8.00000	0.284447
$792$	−2.00000	−0.0710669
$793$	−16.0000	−0.568177
$794$	38.0000	1.34857
$795$	4.00000	0.141865
$796$	−26.0000	−0.921546
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	8.00000	0.283197
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−12.0000	−0.423999
$802$	32.0000	1.12996
$803$	20.0000	0.705785
$804$	−8.00000	−0.282138
$805$	−4.00000	−0.140981
$806$	−8.00000	−0.281788
$807$	−14.0000	−0.492823
$808$	−18.0000	−0.633238
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	−2.00000	−0.0702728
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	−2.00000	−0.0701862
$813$	−24.0000	−0.841717
$814$	8.00000	0.280400
$815$	−8.00000	−0.280228
$816$	−4.00000	−0.140028
$817$	32.0000	1.11954
$818$	−18.0000	−0.629355
$819$	−4.00000	−0.139771
$820$	4.00000	0.139686
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	−14.0000	−0.487713
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	1.00000	0.0347524
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−12.0000	−0.416526
$831$	6.00000	0.208138
$832$	−2.00000	−0.0693375
$833$	−12.0000	−0.415775
$834$	12.0000	0.415526
$835$	0	0
$836$	8.00000	0.276686
$837$	−4.00000	−0.138260
$838$	−26.0000	−0.898155
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	4.00000	0.138013
$841$	1.00000	0.0344828
$842$	40.0000	1.37849
$843$	−32.0000	−1.10214
$844$	−12.0000	−0.413057
$845$	18.0000	0.619219
$846$	0	0
$847$	−14.0000	−0.481046
$848$	2.00000	0.0686803
$849$	−12.0000	−0.411839
$850$	−4.00000	−0.137199
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	16.0000	0.547509
$855$	8.00000	0.273594
$856$	−18.0000	−0.615227
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	−4.00000	−0.136558
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	16.0000	0.545595
$861$	4.00000	0.136320
$862$	4.00000	0.136241
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−1.00000	−0.0340207
$865$	−4.00000	−0.136004
$866$	6.00000	0.203888
$867$	1.00000	0.0339618
$868$	8.00000	0.271538
$869$	28.0000	0.949835
$870$	−2.00000	−0.0678064
$871$	−16.0000	−0.542139
$872$	−16.0000	−0.541828
$873$	18.0000	0.609208
$874$	−4.00000	−0.135302
$875$	24.0000	0.811348
$876$	10.0000	0.337869
$877$	−30.0000	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$878$	24.0000	0.809961
$879$	14.0000	0.472208
$880$	4.00000	0.134840
$881$	4.00000	0.134763	0.0673817	−	0.997727i	$-0.478535\pi$
$881$	4.00000	0.134763	0.0673817	+	0.997727i	$0.478535\pi$
$882$	−3.00000	−0.101015
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	4.00000	0.134231
$889$	16.0000	0.536623
$890$	24.0000	0.804482
$891$	−2.00000	−0.0670025
$892$	−4.00000	−0.133930
$893$	0	0
$894$	−14.0000	−0.468230
$895$	−8.00000	−0.267411
$896$	2.00000	0.0668153
$897$	2.00000	0.0667781
$898$	6.00000	0.200223
$899$	−4.00000	−0.133407
$900$	−1.00000	−0.0333333
$901$	8.00000	0.266519
$902$	4.00000	0.133185
$903$	16.0000	0.532447
$904$	4.00000	0.133038
$905$	0	0
$906$	20.0000	0.664455
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−18.0000	−0.597351
$909$	−18.0000	−0.597022
$910$	8.00000	0.265197
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	4.00000	0.132453
$913$	−12.0000	−0.397142
$914$	−22.0000	−0.727695
$915$	16.0000	0.528944
$916$	−20.0000	−0.660819
$917$	16.0000	0.528367
$918$	−4.00000	−0.132020
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	−2.00000	−0.0659380
$921$	20.0000	0.659022
$922$	−18.0000	−0.592798
$923$	0	0
$924$	4.00000	0.131590
$925$	4.00000	0.131519
$926$	24.0000	0.788689
$927$	−14.0000	−0.459820
$928$	−1.00000	−0.0328266
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	8.00000	0.262330
$931$	12.0000	0.393284
$932$	−6.00000	−0.196537
$933$	0	0
$934$	6.00000	0.196326
$935$	16.0000	0.523256
$936$	−2.00000	−0.0653720
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	16.0000	0.522419
$939$	−14.0000	−0.456873
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	20.0000	0.651635
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	4.00000	0.130120
$946$	16.0000	0.520205
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	14.0000	0.454699
$949$	20.0000	0.649227
$950$	4.00000	0.129777
$951$	30.0000	0.972817
$952$	8.00000	0.259281
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	2.00000	0.0647524
$955$	−48.0000	−1.55324
$956$	−8.00000	−0.258738
$957$	−2.00000	−0.0646508
$958$	−24.0000	−0.775405
$959$	0	0
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	8.00000	0.257930
$963$	−18.0000	−0.580042
$964$	10.0000	0.322078
$965$	−28.0000	−0.901352
$966$	−2.00000	−0.0643489
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	−7.00000	−0.224989
$969$	16.0000	0.513994
$970$	−36.0000	−1.15589
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	−1.00000	−0.0320750
$973$	−24.0000	−0.769405
$974$	24.0000	0.769010
$975$	−2.00000	−0.0640513
$976$	8.00000	0.256074
$977$	−44.0000	−1.40768	−0.703842	−	0.710356i	$-0.748534\pi$
$977$	−44.0000	−1.40768	−0.703842	+	0.710356i	$0.748534\pi$
$978$	−4.00000	−0.127906
$979$	24.0000	0.767043
$980$	6.00000	0.191663
$981$	−16.0000	−0.510841
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	2.00000	0.0637577
$985$	36.0000	1.14706
$986$	−4.00000	−0.127386
$987$	0	0
$988$	8.00000	0.254514
$989$	−8.00000	−0.254385
$990$	4.00000	0.127128
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	4.00000	0.127000
$993$	4.00000	0.126936
$994$	0	0
$995$	52.0000	1.64851
$996$	−6.00000	−0.190117
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	12.0000	0.379853
$999$	4.00000	0.126554

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.i.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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