LMFDB - Embedded newform 4002.2.a.k.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +1.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -9.00000 q^{41} -1.00000 q^{42} -5.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +7.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} +3.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} -1.00000 q^{57} -1.00000 q^{58} -9.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -8.00000 q^{67} -3.00000 q^{68} -1.00000 q^{69} +1.00000 q^{70} -2.00000 q^{71} +1.00000 q^{72} +14.0000 q^{73} -1.00000 q^{74} +4.00000 q^{75} +1.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -8.00000 q^{83} -1.00000 q^{84} -3.00000 q^{85} -5.00000 q^{86} +1.00000 q^{87} -4.00000 q^{88} +18.0000 q^{89} +1.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} +2.00000 q^{93} +7.00000 q^{94} +1.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} -6.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−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	$0.589456\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	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
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	−4.00000	−0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−1.00000	−0.182574
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−3.00000	−0.514496
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	1.00000	0.162221
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−1.00000	−0.154303
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	−4.00000	−0.565685
$51$	3.00000	0.420084
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	−4.00000	−0.539360
$56$	1.00000	0.133631
$57$	−1.00000	−0.132453
$58$	−1.00000	−0.131306
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	−1.00000	−0.129099
$61$	−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
$65$	−2.00000	−0.248069
$66$	4.00000	0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−3.00000	−0.363803
$69$	−1.00000	−0.120386
$70$	1.00000	0.119523
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	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	$0.194369\pi$
$74$	−1.00000	−0.116248
$75$	4.00000	0.461880
$76$	1.00000	0.114708
$77$	−4.00000	−0.455842
$78$	2.00000	0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	−1.00000	−0.109109
$85$	−3.00000	−0.325396
$86$	−5.00000	−0.539164
$87$	1.00000	0.107211
$88$	−4.00000	−0.426401
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	1.00000	0.105409
$91$	−2.00000	−0.209657
$92$	1.00000	0.104257
$93$	2.00000	0.207390
$94$	7.00000	0.721995
$95$	1.00000	0.102598
$96$	−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	$0.633125\pi$
$98$	−6.00000	−0.606092
$99$	−4.00000	−0.402015
$100$	−4.00000	−0.400000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	3.00000	0.297044
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	−2.00000	−0.196116
$105$	−1.00000	−0.0975900
$106$	6.00000	0.582772
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	−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	$0.625154\pi$
$110$	−4.00000	−0.381385
$111$	1.00000	0.0949158
$112$	1.00000	0.0944911
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	−1.00000	−0.0936586
$115$	1.00000	0.0932505
$116$	−1.00000	−0.0928477
$117$	−2.00000	−0.184900
$118$	−9.00000	−0.828517
$119$	−3.00000	−0.275010
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	9.00000	0.811503
$124$	−2.00000	−0.179605
$125$	−9.00000	−0.804984
$126$	1.00000	0.0890871
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	5.00000	0.440225
$130$	−2.00000	−0.175412
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	4.00000	0.348155
$133$	1.00000	0.0867110
$134$	−8.00000	−0.691095
$135$	−1.00000	−0.0860663
$136$	−3.00000	−0.257248
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−1.00000	−0.0851257
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	1.00000	0.0845154
$141$	−7.00000	−0.589506
$142$	−2.00000	−0.167836
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	−1.00000	−0.0830455
$146$	14.0000	1.15865
$147$	6.00000	0.494872
$148$	−1.00000	−0.0821995
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	4.00000	0.326599
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	1.00000	0.0811107
$153$	−3.00000	−0.242536
$154$	−4.00000	−0.322329
$155$	−2.00000	−0.160644
$156$	2.00000	0.160128
$157$	19.0000	1.51637	0.758183	−	0.652042i	$-0.226088\pi$
$157$	19.0000	1.51637	0.758183	+	0.652042i	$0.226088\pi$
$158$	−4.00000	−0.318223
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	1.00000	0.0788110
$162$	1.00000	0.0785674
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	−9.00000	−0.702782
$165$	4.00000	0.311400
$166$	−8.00000	−0.620920
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	−3.00000	−0.230089
$171$	1.00000	0.0764719
$172$	−5.00000	−0.381246
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	1.00000	0.0758098
$175$	−4.00000	−0.302372
$176$	−4.00000	−0.301511
$177$	9.00000	0.676481
$178$	18.0000	1.34916
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	1.00000	0.0745356
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	−2.00000	−0.148250
$183$	10.0000	0.739221
$184$	1.00000	0.0737210
$185$	−1.00000	−0.0735215
$186$	2.00000	0.146647
$187$	12.0000	0.877527
$188$	7.00000	0.510527
$189$	−1.00000	−0.0727393
$190$	1.00000	0.0725476
$191$	19.0000	1.37479	0.687396	−	0.726283i	$-0.258754\pi$
$191$	19.0000	1.37479	0.687396	+	0.726283i	$0.258754\pi$
$192$	−1.00000	−0.0721688
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	−8.00000	−0.574367
$195$	2.00000	0.143223
$196$	−6.00000	−0.428571
$197$	−17.0000	−1.21120	−0.605600	−	0.795769i	$-0.707067\pi$
$197$	−17.0000	−1.21120	−0.605600	+	0.795769i	$0.707067\pi$
$198$	−4.00000	−0.284268
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−4.00000	−0.282843
$201$	8.00000	0.564276
$202$	2.00000	0.140720
$203$	−1.00000	−0.0701862
$204$	3.00000	0.210042
$205$	−9.00000	−0.628587
$206$	−1.00000	−0.0696733
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	−4.00000	−0.276686
$210$	−1.00000	−0.0690066
$211$	3.00000	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$212$	6.00000	0.412082
$213$	2.00000	0.137038
$214$	13.0000	0.888662
$215$	−5.00000	−0.340997
$216$	−1.00000	−0.0680414
$217$	−2.00000	−0.135769
$218$	−8.00000	−0.541828
$219$	−14.0000	−0.946032
$220$	−4.00000	−0.269680
$221$	6.00000	0.403604
$222$	1.00000	0.0671156
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	1.00000	0.0668153
$225$	−4.00000	−0.266667
$226$	11.0000	0.731709
$227$	−11.0000	−0.730096	−0.365048	−	0.930989i	$-0.618947\pi$
$227$	−11.0000	−0.730096	−0.365048	+	0.930989i	$0.618947\pi$
$228$	−1.00000	−0.0662266
$229$	−19.0000	−1.25556	−0.627778	−	0.778393i	$-0.716035\pi$
$229$	−19.0000	−1.25556	−0.627778	+	0.778393i	$0.716035\pi$
$230$	1.00000	0.0659380
$231$	4.00000	0.263181
$232$	−1.00000	−0.0656532
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	−2.00000	−0.130744
$235$	7.00000	0.456630
$236$	−9.00000	−0.585850
$237$	4.00000	0.259828
$238$	−3.00000	−0.194461
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−1.00000	−0.0645497
$241$	3.00000	0.193247	0.0966235	−	0.995321i	$-0.469196\pi$
$241$	3.00000	0.193247	0.0966235	+	0.995321i	$0.469196\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−6.00000	−0.383326
$246$	9.00000	0.573819
$247$	−2.00000	−0.127257
$248$	−2.00000	−0.127000
$249$	8.00000	0.506979
$250$	−9.00000	−0.569210
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	1.00000	0.0629941
$253$	−4.00000	−0.251478
$254$	2.00000	0.125491
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	5.00000	0.311286
$259$	−1.00000	−0.0621370
$260$	−2.00000	−0.124035
$261$	−1.00000	−0.0618984
$262$	6.00000	0.370681
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	4.00000	0.246183
$265$	6.00000	0.368577
$266$	1.00000	0.0613139
$267$	−18.0000	−1.10158
$268$	−8.00000	−0.488678
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	−1.00000	−0.0608581
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−3.00000	−0.181902
$273$	2.00000	0.121046
$274$	−18.0000	−1.08742
$275$	16.0000	0.964836
$276$	−1.00000	−0.0601929
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	6.00000	0.359856
$279$	−2.00000	−0.119737
$280$	1.00000	0.0597614
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	−7.00000	−0.416844
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−2.00000	−0.118678
$285$	−1.00000	−0.0592349
$286$	8.00000	0.473050
$287$	−9.00000	−0.531253
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−1.00000	−0.0587220
$291$	8.00000	0.468968
$292$	14.0000	0.819288
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	6.00000	0.349927
$295$	−9.00000	−0.524000
$296$	−1.00000	−0.0581238
$297$	4.00000	0.232104
$298$	−9.00000	−0.521356
$299$	−2.00000	−0.115663
$300$	4.00000	0.230940
$301$	−5.00000	−0.288195
$302$	−13.0000	−0.748066
$303$	−2.00000	−0.114897
$304$	1.00000	0.0573539
$305$	−10.0000	−0.572598
$306$	−3.00000	−0.171499
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	−4.00000	−0.227921
$309$	1.00000	0.0568880
$310$	−2.00000	−0.113592
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	2.00000	0.113228
$313$	17.0000	0.960897	0.480448	−	0.877023i	$-0.340474\pi$
$313$	17.0000	0.960897	0.480448	+	0.877023i	$0.340474\pi$
$314$	19.0000	1.07223
$315$	1.00000	0.0563436
$316$	−4.00000	−0.225018
$317$	−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
$320$	1.00000	0.0559017
$321$	−13.0000	−0.725589
$322$	1.00000	0.0557278
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	1.00000	0.0553849
$327$	8.00000	0.442401
$328$	−9.00000	−0.496942
$329$	7.00000	0.385922
$330$	4.00000	0.220193
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	−8.00000	−0.439057
$333$	−1.00000	−0.0547997
$334$	−14.0000	−0.766046
$335$	−8.00000	−0.437087
$336$	−1.00000	−0.0545545
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	−9.00000	−0.489535
$339$	−11.0000	−0.597438
$340$	−3.00000	−0.162698
$341$	8.00000	0.433224
$342$	1.00000	0.0540738
$343$	−13.0000	−0.701934
$344$	−5.00000	−0.269582
$345$	−1.00000	−0.0538382
$346$	−15.0000	−0.806405
$347$	−11.0000	−0.590511	−0.295255	−	0.955418i	$-0.595405\pi$
$347$	−11.0000	−0.590511	−0.295255	+	0.955418i	$0.595405\pi$
$348$	1.00000	0.0536056
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	−4.00000	−0.213809
$351$	2.00000	0.106752
$352$	−4.00000	−0.213201
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	9.00000	0.478345
$355$	−2.00000	−0.106149
$356$	18.0000	0.953998
$357$	3.00000	0.158777
$358$	20.0000	1.05703
$359$	−31.0000	−1.63612	−0.818059	−	0.575135i	$-0.804950\pi$
$359$	−31.0000	−1.63612	−0.818059	+	0.575135i	$0.804950\pi$
$360$	1.00000	0.0527046
$361$	−18.0000	−0.947368
$362$	−18.0000	−0.946059
$363$	−5.00000	−0.262432
$364$	−2.00000	−0.104828
$365$	14.0000	0.732793
$366$	10.0000	0.522708
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	1.00000	0.0521286
$369$	−9.00000	−0.468521
$370$	−1.00000	−0.0519875
$371$	6.00000	0.311504
$372$	2.00000	0.103695
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	12.0000	0.620505
$375$	9.00000	0.464758
$376$	7.00000	0.360997
$377$	2.00000	0.103005
$378$	−1.00000	−0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	1.00000	0.0512989
$381$	−2.00000	−0.102463
$382$	19.0000	0.972125
$383$	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
$385$	−4.00000	−0.203859
$386$	−12.0000	−0.610784
$387$	−5.00000	−0.254164
$388$	−8.00000	−0.406138
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	2.00000	0.101274
$391$	−3.00000	−0.151717
$392$	−6.00000	−0.303046
$393$	−6.00000	−0.302660
$394$	−17.0000	−0.856448
$395$	−4.00000	−0.201262
$396$	−4.00000	−0.201008
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	4.00000	0.200502
$399$	−1.00000	−0.0500626
$400$	−4.00000	−0.200000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	8.00000	0.399004
$403$	4.00000	0.199254
$404$	2.00000	0.0995037
$405$	1.00000	0.0496904
$406$	−1.00000	−0.0496292
$407$	4.00000	0.198273
$408$	3.00000	0.148522
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	−9.00000	−0.444478
$411$	18.0000	0.887875
$412$	−1.00000	−0.0492665
$413$	−9.00000	−0.442861
$414$	1.00000	0.0491473
$415$	−8.00000	−0.392705
$416$	−2.00000	−0.0980581
$417$	−6.00000	−0.293821
$418$	−4.00000	−0.195646
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	−1.00000	−0.0487950
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	3.00000	0.146038
$423$	7.00000	0.340352
$424$	6.00000	0.291386
$425$	12.0000	0.582086
$426$	2.00000	0.0969003
$427$	−10.0000	−0.483934
$428$	13.0000	0.628379
$429$	−8.00000	−0.386244
$430$	−5.00000	−0.241121
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	−1.00000	−0.0481125
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	−2.00000	−0.0960031
$435$	1.00000	0.0479463
$436$	−8.00000	−0.383131
$437$	1.00000	0.0478365
$438$	−14.0000	−0.668946
$439$	33.0000	1.57500	0.787502	−	0.616312i	$-0.211374\pi$
$439$	33.0000	1.57500	0.787502	+	0.616312i	$0.211374\pi$
$440$	−4.00000	−0.190693
$441$	−6.00000	−0.285714
$442$	6.00000	0.285391
$443$	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
$445$	18.0000	0.853282
$446$	16.0000	0.757622
$447$	9.00000	0.425685
$448$	1.00000	0.0472456
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	−4.00000	−0.188562
$451$	36.0000	1.69517
$452$	11.0000	0.517396
$453$	13.0000	0.610793
$454$	−11.0000	−0.516256
$455$	−2.00000	−0.0937614
$456$	−1.00000	−0.0468293
$457$	37.0000	1.73079	0.865393	−	0.501093i	$-0.167069\pi$
$457$	37.0000	1.73079	0.865393	+	0.501093i	$0.167069\pi$
$458$	−19.0000	−0.887812
$459$	3.00000	0.140028
$460$	1.00000	0.0466252
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	4.00000	0.186097
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−1.00000	−0.0464238
$465$	2.00000	0.0927478
$466$	4.00000	0.185296
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	−2.00000	−0.0924500
$469$	−8.00000	−0.369406
$470$	7.00000	0.322886
$471$	−19.0000	−0.875474
$472$	−9.00000	−0.414259
$473$	20.0000	0.919601
$474$	4.00000	0.183726
$475$	−4.00000	−0.183533
$476$	−3.00000	−0.137505
$477$	6.00000	0.274721
$478$	−6.00000	−0.274434
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−1.00000	−0.0456435
$481$	2.00000	0.0911922
$482$	3.00000	0.136646
$483$	−1.00000	−0.0455016
$484$	5.00000	0.227273
$485$	−8.00000	−0.363261
$486$	−1.00000	−0.0453609
$487$	−3.00000	−0.135943	−0.0679715	−	0.997687i	$-0.521653\pi$
$487$	−3.00000	−0.135943	−0.0679715	+	0.997687i	$0.521653\pi$
$488$	−10.0000	−0.452679
$489$	−1.00000	−0.0452216
$490$	−6.00000	−0.271052
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	9.00000	0.405751
$493$	3.00000	0.135113
$494$	−2.00000	−0.0899843
$495$	−4.00000	−0.179787
$496$	−2.00000	−0.0898027
$497$	−2.00000	−0.0897123
$498$	8.00000	0.358489
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−9.00000	−0.402492
$501$	14.0000	0.625474
$502$	8.00000	0.357057
$503$	27.0000	1.20387	0.601935	−	0.798545i	$-0.294397\pi$
$503$	27.0000	1.20387	0.601935	+	0.798545i	$0.294397\pi$
$504$	1.00000	0.0445435
$505$	2.00000	0.0889988
$506$	−4.00000	−0.177822
$507$	9.00000	0.399704
$508$	2.00000	0.0887357
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	3.00000	0.132842
$511$	14.0000	0.619324
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−2.00000	−0.0882162
$515$	−1.00000	−0.0440653
$516$	5.00000	0.220113
$517$	−28.0000	−1.23144
$518$	−1.00000	−0.0439375
$519$	15.0000	0.658427
$520$	−2.00000	−0.0877058
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	−1.00000	−0.0437688
$523$	−30.0000	−1.31181	−0.655904	−	0.754844i	$-0.727712\pi$
$523$	−30.0000	−1.31181	−0.655904	+	0.754844i	$0.727712\pi$
$524$	6.00000	0.262111
$525$	4.00000	0.174574
$526$	21.0000	0.915644
$527$	6.00000	0.261364
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	6.00000	0.260623
$531$	−9.00000	−0.390567
$532$	1.00000	0.0433555
$533$	18.0000	0.779667
$534$	−18.0000	−0.778936
$535$	13.0000	0.562039
$536$	−8.00000	−0.345547
$537$	−20.0000	−0.863064
$538$	12.0000	0.517357
$539$	24.0000	1.03375
$540$	−1.00000	−0.0430331
$541$	45.0000	1.93470	0.967351	−	0.253442i	$-0.0815627\pi$
$541$	45.0000	1.93470	0.967351	+	0.253442i	$0.0815627\pi$
$542$	−20.0000	−0.859074
$543$	18.0000	0.772454
$544$	−3.00000	−0.128624
$545$	−8.00000	−0.342682
$546$	2.00000	0.0855921
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−18.0000	−0.768922
$549$	−10.0000	−0.426790
$550$	16.0000	0.682242
$551$	−1.00000	−0.0426014
$552$	−1.00000	−0.0425628
$553$	−4.00000	−0.170097
$554$	−24.0000	−1.01966
$555$	1.00000	0.0424476
$556$	6.00000	0.254457
$557$	1.00000	0.0423714	0.0211857	−	0.999776i	$-0.493256\pi$
$557$	1.00000	0.0423714	0.0211857	+	0.999776i	$0.493256\pi$
$558$	−2.00000	−0.0846668
$559$	10.0000	0.422955
$560$	1.00000	0.0422577
$561$	−12.0000	−0.506640
$562$	−22.0000	−0.928014
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	−7.00000	−0.294753
$565$	11.0000	0.462773
$566$	0	0
$567$	1.00000	0.0419961
$568$	−2.00000	−0.0839181
$569$	39.0000	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$570$	−1.00000	−0.0418854
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	8.00000	0.334497
$573$	−19.0000	−0.793736
$574$	−9.00000	−0.375653
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	−8.00000	−0.332756
$579$	12.0000	0.498703
$580$	−1.00000	−0.0415227
$581$	−8.00000	−0.331896
$582$	8.00000	0.331611
$583$	−24.0000	−0.993978
$584$	14.0000	0.579324
$585$	−2.00000	−0.0826898
$586$	−26.0000	−1.07405
$587$	−7.00000	−0.288921	−0.144460	−	0.989511i	$-0.546145\pi$
$587$	−7.00000	−0.288921	−0.144460	+	0.989511i	$0.546145\pi$
$588$	6.00000	0.247436
$589$	−2.00000	−0.0824086
$590$	−9.00000	−0.370524
$591$	17.0000	0.699287
$592$	−1.00000	−0.0410997
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	4.00000	0.164122
$595$	−3.00000	−0.122988
$596$	−9.00000	−0.368654
$597$	−4.00000	−0.163709
$598$	−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	$0.552258\pi$
$600$	4.00000	0.163299
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	−5.00000	−0.203785
$603$	−8.00000	−0.325785
$604$	−13.0000	−0.528962
$605$	5.00000	0.203279
$606$	−2.00000	−0.0812444
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	1.00000	0.0405554
$609$	1.00000	0.0405220
$610$	−10.0000	−0.404888
$611$	−14.0000	−0.566379
$612$	−3.00000	−0.121268
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	20.0000	0.807134
$615$	9.00000	0.362915
$616$	−4.00000	−0.161165
$617$	−45.0000	−1.81163	−0.905816	−	0.423672i	$-0.860741\pi$
$617$	−45.0000	−1.81163	−0.905816	+	0.423672i	$0.860741\pi$
$618$	1.00000	0.0402259
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	−2.00000	−0.0803219
$621$	−1.00000	−0.0401286
$622$	15.0000	0.601445
$623$	18.0000	0.721155
$624$	2.00000	0.0800641
$625$	11.0000	0.440000
$626$	17.0000	0.679457
$627$	4.00000	0.159745
$628$	19.0000	0.758183
$629$	3.00000	0.119618
$630$	1.00000	0.0398410
$631$	39.0000	1.55257	0.776283	−	0.630385i	$-0.217103\pi$
$631$	39.0000	1.55257	0.776283	+	0.630385i	$0.217103\pi$
$632$	−4.00000	−0.159111
$633$	−3.00000	−0.119239
$634$	−6.00000	−0.238290
$635$	2.00000	0.0793676
$636$	−6.00000	−0.237915
$637$	12.0000	0.475457
$638$	4.00000	0.158362
$639$	−2.00000	−0.0791188
$640$	1.00000	0.0395285
$641$	−29.0000	−1.14543	−0.572716	−	0.819754i	$-0.694110\pi$
$641$	−29.0000	−1.14543	−0.572716	+	0.819754i	$0.694110\pi$
$642$	−13.0000	−0.513069
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	1.00000	0.0394055
$645$	5.00000	0.196875
$646$	−3.00000	−0.118033
$647$	44.0000	1.72982	0.864909	−	0.501928i	$-0.167376\pi$
$647$	44.0000	1.72982	0.864909	+	0.501928i	$0.167376\pi$
$648$	1.00000	0.0392837
$649$	36.0000	1.41312
$650$	8.00000	0.313786
$651$	2.00000	0.0783862
$652$	1.00000	0.0391630
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	8.00000	0.312825
$655$	6.00000	0.234439
$656$	−9.00000	−0.351391
$657$	14.0000	0.546192
$658$	7.00000	0.272888
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	4.00000	0.155700
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	17.0000	0.660724
$663$	−6.00000	−0.233021
$664$	−8.00000	−0.310460
$665$	1.00000	0.0387783
$666$	−1.00000	−0.0387492
$667$	−1.00000	−0.0387202
$668$	−14.0000	−0.541676
$669$	−16.0000	−0.618596
$670$	−8.00000	−0.309067
$671$	40.0000	1.54418
$672$	−1.00000	−0.0385758
$673$	51.0000	1.96591	0.982953	−	0.183858i	$-0.0588587\pi$
$673$	51.0000	1.96591	0.982953	+	0.183858i	$0.0588587\pi$
$674$	30.0000	1.15556
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−11.0000	−0.422452
$679$	−8.00000	−0.307012
$680$	−3.00000	−0.115045
$681$	11.0000	0.421521
$682$	8.00000	0.306336
$683$	35.0000	1.33924	0.669619	−	0.742705i	$-0.266457\pi$
$683$	35.0000	1.33924	0.669619	+	0.742705i	$0.266457\pi$
$684$	1.00000	0.0382360
$685$	−18.0000	−0.687745
$686$	−13.0000	−0.496342
$687$	19.0000	0.724895
$688$	−5.00000	−0.190623
$689$	−12.0000	−0.457164
$690$	−1.00000	−0.0380693
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	−15.0000	−0.570214
$693$	−4.00000	−0.151947
$694$	−11.0000	−0.417554
$695$	6.00000	0.227593
$696$	1.00000	0.0379049
$697$	27.0000	1.02270
$698$	6.00000	0.227103
$699$	−4.00000	−0.151294
$700$	−4.00000	−0.151186
$701$	9.00000	0.339925	0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000	0.339925	0.169963	+	0.985451i	$0.445635\pi$
$702$	2.00000	0.0754851
$703$	−1.00000	−0.0377157
$704$	−4.00000	−0.150756
$705$	−7.00000	−0.263635
$706$	−18.0000	−0.677439
$707$	2.00000	0.0752177
$708$	9.00000	0.338241
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	−2.00000	−0.0750587
$711$	−4.00000	−0.150012
$712$	18.0000	0.674579
$713$	−2.00000	−0.0749006
$714$	3.00000	0.112272
$715$	8.00000	0.299183
$716$	20.0000	0.747435
$717$	6.00000	0.224074
$718$	−31.0000	−1.15691
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	1.00000	0.0372678
$721$	−1.00000	−0.0372419
$722$	−18.0000	−0.669891
$723$	−3.00000	−0.111571
$724$	−18.0000	−0.668965
$725$	4.00000	0.148556
$726$	−5.00000	−0.185567
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	14.0000	0.518163
$731$	15.0000	0.554795
$732$	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	$0.686914\pi$
$734$	−4.00000	−0.147643
$735$	6.00000	0.221313
$736$	1.00000	0.0368605
$737$	32.0000	1.17874
$738$	−9.00000	−0.331295
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−1.00000	−0.0367607
$741$	2.00000	0.0734718
$742$	6.00000	0.220267
$743$	27.0000	0.990534	0.495267	−	0.868741i	$-0.335070\pi$
$743$	27.0000	0.990534	0.495267	+	0.868741i	$0.335070\pi$
$744$	2.00000	0.0733236
$745$	−9.00000	−0.329734
$746$	−4.00000	−0.146450
$747$	−8.00000	−0.292705
$748$	12.0000	0.438763
$749$	13.0000	0.475010
$750$	9.00000	0.328634
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	7.00000	0.255264
$753$	−8.00000	−0.291536
$754$	2.00000	0.0728357
$755$	−13.0000	−0.473118
$756$	−1.00000	−0.0363696
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	−20.0000	−0.726433
$759$	4.00000	0.145191
$760$	1.00000	0.0362738
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	−2.00000	−0.0724524
$763$	−8.00000	−0.289619
$764$	19.0000	0.687396
$765$	−3.00000	−0.108465
$766$	8.00000	0.289052
$767$	18.0000	0.649942
$768$	−1.00000	−0.0360844
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	−4.00000	−0.144150
$771$	2.00000	0.0720282
$772$	−12.0000	−0.431889
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	−5.00000	−0.179721
$775$	8.00000	0.287368
$776$	−8.00000	−0.287183
$777$	1.00000	0.0358748
$778$	24.0000	0.860442
$779$	−9.00000	−0.322458
$780$	2.00000	0.0716115
$781$	8.00000	0.286263
$782$	−3.00000	−0.107280
$783$	1.00000	0.0357371
$784$	−6.00000	−0.214286
$785$	19.0000	0.678139
$786$	−6.00000	−0.214013
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	−17.0000	−0.605600
$789$	−21.0000	−0.747620
$790$	−4.00000	−0.142314
$791$	11.0000	0.391115
$792$	−4.00000	−0.142134
$793$	20.0000	0.710221
$794$	−20.0000	−0.709773
$795$	−6.00000	−0.212798
$796$	4.00000	0.141776
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	−1.00000	−0.0353996
$799$	−21.0000	−0.742927
$800$	−4.00000	−0.141421
$801$	18.0000	0.635999
$802$	6.00000	0.211867
$803$	−56.0000	−1.97620
$804$	8.00000	0.282138
$805$	1.00000	0.0352454
$806$	4.00000	0.140894
$807$	−12.0000	−0.422420
$808$	2.00000	0.0703598
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	1.00000	0.0351364
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	−1.00000	−0.0350931
$813$	20.0000	0.701431
$814$	4.00000	0.140200
$815$	1.00000	0.0350285
$816$	3.00000	0.105021
$817$	−5.00000	−0.174928
$818$	−6.00000	−0.209785
$819$	−2.00000	−0.0698857
$820$	−9.00000	−0.314294
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	18.0000	0.627822
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−1.00000	−0.0348367
$825$	−16.0000	−0.557048
$826$	−9.00000	−0.313150
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	1.00000	0.0347524
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	−8.00000	−0.277684
$831$	24.0000	0.832551
$832$	−2.00000	−0.0693375
$833$	18.0000	0.623663
$834$	−6.00000	−0.207763
$835$	−14.0000	−0.484490
$836$	−4.00000	−0.138343
$837$	2.00000	0.0691301
$838$	−5.00000	−0.172722
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	−1.00000	−0.0345033
$841$	1.00000	0.0344828
$842$	−2.00000	−0.0689246
$843$	22.0000	0.757720
$844$	3.00000	0.103264
$845$	−9.00000	−0.309609
$846$	7.00000	0.240665
$847$	5.00000	0.171802
$848$	6.00000	0.206041
$849$	0	0
$850$	12.0000	0.411597
$851$	−1.00000	−0.0342796
$852$	2.00000	0.0685189
$853$	−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
$855$	1.00000	0.0341993
$856$	13.0000	0.444331
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	−8.00000	−0.273115
$859$	−7.00000	−0.238837	−0.119418	−	0.992844i	$-0.538103\pi$
$859$	−7.00000	−0.238837	−0.119418	+	0.992844i	$0.538103\pi$
$860$	−5.00000	−0.170499
$861$	9.00000	0.306719
$862$	30.0000	1.02180
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	−1.00000	−0.0340207
$865$	−15.0000	−0.510015
$866$	0	0
$867$	8.00000	0.271694
$868$	−2.00000	−0.0678844
$869$	16.0000	0.542763
$870$	1.00000	0.0339032
$871$	16.0000	0.542139
$872$	−8.00000	−0.270914
$873$	−8.00000	−0.270759
$874$	1.00000	0.0338255
$875$	−9.00000	−0.304256
$876$	−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	$0.621137\pi$
$878$	33.0000	1.11370
$879$	26.0000	0.876958
$880$	−4.00000	−0.134840
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−6.00000	−0.202031
$883$	−18.0000	−0.605748	−0.302874	−	0.953031i	$-0.597946\pi$
$883$	−18.0000	−0.605748	−0.302874	+	0.953031i	$0.597946\pi$
$884$	6.00000	0.201802
$885$	9.00000	0.302532
$886$	36.0000	1.20944
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	1.00000	0.0335578
$889$	2.00000	0.0670778
$890$	18.0000	0.603361
$891$	−4.00000	−0.134005
$892$	16.0000	0.535720
$893$	7.00000	0.234246
$894$	9.00000	0.301005
$895$	20.0000	0.668526
$896$	1.00000	0.0334077
$897$	2.00000	0.0667781
$898$	−9.00000	−0.300334
$899$	2.00000	0.0667037
$900$	−4.00000	−0.133333
$901$	−18.0000	−0.599667
$902$	36.0000	1.19867
$903$	5.00000	0.166390
$904$	11.0000	0.365855
$905$	−18.0000	−0.598340
$906$	13.0000	0.431896
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−11.0000	−0.365048
$909$	2.00000	0.0663358
$910$	−2.00000	−0.0662994
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−1.00000	−0.0331133
$913$	32.0000	1.05905
$914$	37.0000	1.22385
$915$	10.0000	0.330590
$916$	−19.0000	−0.627778
$917$	6.00000	0.198137
$918$	3.00000	0.0990148
$919$	−19.0000	−0.626752	−0.313376	−	0.949629i	$-0.601460\pi$
$919$	−19.0000	−0.626752	−0.313376	+	0.949629i	$0.601460\pi$
$920$	1.00000	0.0329690
$921$	−20.0000	−0.659022
$922$	14.0000	0.461065
$923$	4.00000	0.131662
$924$	4.00000	0.131590
$925$	4.00000	0.131519
$926$	−24.0000	−0.788689
$927$	−1.00000	−0.0328443
$928$	−1.00000	−0.0328266
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	2.00000	0.0655826
$931$	−6.00000	−0.196642
$932$	4.00000	0.131024
$933$	−15.0000	−0.491078
$934$	10.0000	0.327210
$935$	12.0000	0.392442
$936$	−2.00000	−0.0653720
$937$	5.00000	0.163343	0.0816714	−	0.996659i	$-0.473974\pi$
$937$	5.00000	0.163343	0.0816714	+	0.996659i	$0.473974\pi$
$938$	−8.00000	−0.261209
$939$	−17.0000	−0.554774
$940$	7.00000	0.228315
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	−19.0000	−0.619053
$943$	−9.00000	−0.293080
$944$	−9.00000	−0.292925
$945$	−1.00000	−0.0325300
$946$	20.0000	0.650256
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	4.00000	0.129914
$949$	−28.0000	−0.908918
$950$	−4.00000	−0.129777
$951$	6.00000	0.194563
$952$	−3.00000	−0.0972306
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	6.00000	0.194257
$955$	19.0000	0.614826
$956$	−6.00000	−0.194054
$957$	−4.00000	−0.129302
$958$	24.0000	0.775405
$959$	−18.0000	−0.581250
$960$	−1.00000	−0.0322749
$961$	−27.0000	−0.870968
$962$	2.00000	0.0644826
$963$	13.0000	0.418919
$964$	3.00000	0.0966235
$965$	−12.0000	−0.386294
$966$	−1.00000	−0.0321745
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	5.00000	0.160706
$969$	3.00000	0.0963739
$970$	−8.00000	−0.256865
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	−1.00000	−0.0320750
$973$	6.00000	0.192351
$974$	−3.00000	−0.0961262
$975$	−8.00000	−0.256205
$976$	−10.0000	−0.320092
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	−1.00000	−0.0319765
$979$	−72.0000	−2.30113
$980$	−6.00000	−0.191663
$981$	−8.00000	−0.255420
$982$	36.0000	1.14881
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	9.00000	0.286910
$985$	−17.0000	−0.541665
$986$	3.00000	0.0955395
$987$	−7.00000	−0.222812
$988$	−2.00000	−0.0636285
$989$	−5.00000	−0.158991
$990$	−4.00000	−0.127128
$991$	49.0000	1.55654	0.778268	−	0.627932i	$-0.216098\pi$
$991$	49.0000	1.55654	0.778268	+	0.627932i	$0.216098\pi$
$992$	−2.00000	−0.0635001
$993$	−17.0000	−0.539479
$994$	−2.00000	−0.0634361
$995$	4.00000	0.126809
$996$	8.00000	0.253490
$997$	−53.0000	−1.67853	−0.839263	−	0.543725i	$-0.817013\pi$
$997$	−53.0000	−1.67853	−0.839263	+	0.543725i	$0.817013\pi$
$998$	−24.0000	−0.759707
$999$	1.00000	0.0316386

Twists

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

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