LMFDB - Embedded newform 2006.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2006,2,Mod(1,2006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2006 = 2 \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2006.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:	$16.0179906455$
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	$\chi$	$=$	2006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +2.00000 q^{18} +7.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} +3.00000 q^{29} +1.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} -1.00000 q^{35} -2.00000 q^{36} -4.00000 q^{37} -7.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} +3.00000 q^{41} -1.00000 q^{42} -2.00000 q^{43} -2.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +4.00000 q^{50} +1.00000 q^{51} +2.00000 q^{52} -11.0000 q^{53} -5.00000 q^{54} -2.00000 q^{55} +1.00000 q^{56} -7.00000 q^{57} -3.00000 q^{58} -1.00000 q^{59} -1.00000 q^{60} +8.00000 q^{61} +10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -2.00000 q^{66} -14.0000 q^{67} -1.00000 q^{68} -4.00000 q^{69} +1.00000 q^{70} -12.0000 q^{71} +2.00000 q^{72} +4.00000 q^{73} +4.00000 q^{74} +4.00000 q^{75} +7.00000 q^{76} +2.00000 q^{77} +2.00000 q^{78} +1.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -3.00000 q^{82} +18.0000 q^{83} +1.00000 q^{84} -1.00000 q^{85} +2.00000 q^{86} -3.00000 q^{87} +2.00000 q^{88} -8.00000 q^{89} +2.00000 q^{90} -2.00000 q^{91} +4.00000 q^{92} +10.0000 q^{93} -6.00000 q^{94} +7.00000 q^{95} +1.00000 q^{96} -12.0000 q^{97} +6.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	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	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	−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	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.00000	0.471405
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	1.00000	0.223607
$21$	1.00000	0.218218
$22$	2.00000	0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	1.00000	0.182574
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	1.00000	0.171499
$35$	−1.00000	−0.169031
$36$	−2.00000	−0.333333
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−7.00000	−1.13555
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	−1.00000	−0.154303
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−2.00000	−0.301511
$45$	−2.00000	−0.298142
$46$	−4.00000	−0.589768
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	4.00000	0.565685
$51$	1.00000	0.140028
$52$	2.00000	0.277350
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−5.00000	−0.680414
$55$	−2.00000	−0.269680
$56$	1.00000	0.133631
$57$	−7.00000	−0.927173
$58$	−3.00000	−0.393919
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−1.00000	−0.129099
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	10.0000	1.27000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−2.00000	−0.246183
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	−1.00000	−0.121268
$69$	−4.00000	−0.481543
$70$	1.00000	0.119523
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	2.00000	0.235702
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	4.00000	0.464991
$75$	4.00000	0.461880
$76$	7.00000	0.802955
$77$	2.00000	0.227921
$78$	2.00000	0.226455
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−3.00000	−0.331295
$83$	18.0000	1.97576	0.987878	−	0.155230i	$-0.0496119\pi$
$83$	18.0000	1.97576	0.987878	+	0.155230i	$0.0496119\pi$
$84$	1.00000	0.109109
$85$	−1.00000	−0.108465
$86$	2.00000	0.215666
$87$	−3.00000	−0.321634
$88$	2.00000	0.213201
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	2.00000	0.210819
$91$	−2.00000	−0.209657
$92$	4.00000	0.417029
$93$	10.0000	1.03695
$94$	−6.00000	−0.618853
$95$	7.00000	0.718185
$96$	1.00000	0.102062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	6.00000	0.606092
$99$	4.00000	0.402015
$100$	−4.00000	−0.400000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−1.00000	−0.0990148
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−2.00000	−0.196116
$105$	1.00000	0.0975900
$106$	11.0000	1.06841
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	5.00000	0.481125
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	2.00000	0.190693
$111$	4.00000	0.379663
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	7.00000	0.655610
$115$	4.00000	0.373002
$116$	3.00000	0.278543
$117$	−4.00000	−0.369800
$118$	1.00000	0.0920575
$119$	1.00000	0.0916698
$120$	1.00000	0.0912871
$121$	−7.00000	−0.636364
$122$	−8.00000	−0.724286
$123$	−3.00000	−0.270501
$124$	−10.0000	−0.898027
$125$	−9.00000	−0.804984
$126$	−2.00000	−0.178174
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	−2.00000	−0.175412
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	2.00000	0.174078
$133$	−7.00000	−0.606977
$134$	14.0000	1.20942
$135$	5.00000	0.430331
$136$	1.00000	0.0857493
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	4.00000	0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−1.00000	−0.0845154
$141$	−6.00000	−0.505291
$142$	12.0000	1.00702
$143$	−4.00000	−0.334497
$144$	−2.00000	−0.166667
$145$	3.00000	0.249136
$146$	−4.00000	−0.331042
$147$	6.00000	0.494872
$148$	−4.00000	−0.328798
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−4.00000	−0.326599
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	−7.00000	−0.567775
$153$	2.00000	0.161690
$154$	−2.00000	−0.161165
$155$	−10.0000	−0.803219
$156$	−2.00000	−0.160128
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	−1.00000	−0.0795557
$159$	11.0000	0.872357
$160$	−1.00000	−0.0790569
$161$	−4.00000	−0.315244
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	3.00000	0.234261
$165$	2.00000	0.155700
$166$	−18.0000	−1.39707
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	1.00000	0.0766965
$171$	−14.0000	−1.07061
$172$	−2.00000	−0.152499
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	3.00000	0.227429
$175$	4.00000	0.302372
$176$	−2.00000	−0.150756
$177$	1.00000	0.0751646
$178$	8.00000	0.599625
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−2.00000	−0.149071
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	2.00000	0.148250
$183$	−8.00000	−0.591377
$184$	−4.00000	−0.294884
$185$	−4.00000	−0.294086
$186$	−10.0000	−0.733236
$187$	2.00000	0.146254
$188$	6.00000	0.437595
$189$	−5.00000	−0.363696
$190$	−7.00000	−0.507833
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	−17.0000	−1.22369	−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000	−1.22369	−0.611843	+	0.790979i	$0.709572\pi$
$194$	12.0000	0.861550
$195$	−2.00000	−0.143223
$196$	−6.00000	−0.428571
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−4.00000	−0.284268
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	4.00000	0.282843
$201$	14.0000	0.987484
$202$	2.00000	0.140720
$203$	−3.00000	−0.210559
$204$	1.00000	0.0700140
$205$	3.00000	0.209529
$206$	0	0
$207$	−8.00000	−0.556038
$208$	2.00000	0.138675
$209$	−14.0000	−0.968400
$210$	−1.00000	−0.0690066
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−11.0000	−0.755483
$213$	12.0000	0.822226
$214$	−3.00000	−0.205076
$215$	−2.00000	−0.136399
$216$	−5.00000	−0.340207
$217$	10.0000	0.678844
$218$	4.00000	0.270914
$219$	−4.00000	−0.270295
$220$	−2.00000	−0.134840
$221$	−2.00000	−0.134535
$222$	−4.00000	−0.268462
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	1.00000	0.0668153
$225$	8.00000	0.533333
$226$	6.00000	0.399114
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	−7.00000	−0.463586
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−4.00000	−0.263752
$231$	−2.00000	−0.131590
$232$	−3.00000	−0.196960
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	4.00000	0.261488
$235$	6.00000	0.391397
$236$	−1.00000	−0.0650945
$237$	−1.00000	−0.0649570
$238$	−1.00000	−0.0648204
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	−1.00000	−0.0645497
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	7.00000	0.449977
$243$	−16.0000	−1.02640
$244$	8.00000	0.512148
$245$	−6.00000	−0.383326
$246$	3.00000	0.191273
$247$	14.0000	0.890799
$248$	10.0000	0.635001
$249$	−18.0000	−1.14070
$250$	9.00000	0.569210
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	2.00000	0.125988
$253$	−8.00000	−0.502956
$254$	11.0000	0.690201
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	5.00000	0.311891	0.155946	−	0.987766i	$-0.450158\pi$
$257$	5.00000	0.311891	0.155946	+	0.987766i	$0.450158\pi$
$258$	−2.00000	−0.124515
$259$	4.00000	0.248548
$260$	2.00000	0.124035
$261$	−6.00000	−0.371391
$262$	−2.00000	−0.123560
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	−2.00000	−0.123091
$265$	−11.0000	−0.675725
$266$	7.00000	0.429198
$267$	8.00000	0.489592
$268$	−14.0000	−0.855186
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−5.00000	−0.304290
$271$	9.00000	0.546711	0.273356	−	0.961913i	$-0.411866\pi$
$271$	9.00000	0.546711	0.273356	+	0.961913i	$0.411866\pi$
$272$	−1.00000	−0.0606339
$273$	2.00000	0.121046
$274$	21.0000	1.26866
$275$	8.00000	0.482418
$276$	−4.00000	−0.240772
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	4.00000	0.239904
$279$	20.0000	1.19737
$280$	1.00000	0.0597614
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	6.00000	0.357295
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−12.0000	−0.712069
$285$	−7.00000	−0.414644
$286$	4.00000	0.236525
$287$	−3.00000	−0.177084
$288$	2.00000	0.117851
$289$	1.00000	0.0588235
$290$	−3.00000	−0.176166
$291$	12.0000	0.703452
$292$	4.00000	0.234082
$293$	7.00000	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$293$	7.00000	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$294$	−6.00000	−0.349927
$295$	−1.00000	−0.0582223
$296$	4.00000	0.232495
$297$	−10.0000	−0.580259
$298$	−2.00000	−0.115857
$299$	8.00000	0.462652
$300$	4.00000	0.230940
$301$	2.00000	0.115278
$302$	14.0000	0.805609
$303$	2.00000	0.114897
$304$	7.00000	0.401478
$305$	8.00000	0.458079
$306$	−2.00000	−0.114332
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	10.0000	0.567962
$311$	−13.0000	−0.737162	−0.368581	−	0.929596i	$-0.620156\pi$
$311$	−13.0000	−0.737162	−0.368581	+	0.929596i	$0.620156\pi$
$312$	2.00000	0.113228
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	8.00000	0.451466
$315$	2.00000	0.112687
$316$	1.00000	0.0562544
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−11.0000	−0.616849
$319$	−6.00000	−0.335936
$320$	1.00000	0.0559017
$321$	−3.00000	−0.167444
$322$	4.00000	0.222911
$323$	−7.00000	−0.389490
$324$	1.00000	0.0555556
$325$	−8.00000	−0.443760
$326$	−12.0000	−0.664619
$327$	4.00000	0.221201
$328$	−3.00000	−0.165647
$329$	−6.00000	−0.330791
$330$	−2.00000	−0.110096
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	18.0000	0.987878
$333$	8.00000	0.438397
$334$	−15.0000	−0.820763
$335$	−14.0000	−0.764902
$336$	1.00000	0.0545545
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	9.00000	0.489535
$339$	6.00000	0.325875
$340$	−1.00000	−0.0542326
$341$	20.0000	1.08306
$342$	14.0000	0.757033
$343$	13.0000	0.701934
$344$	2.00000	0.107833
$345$	−4.00000	−0.215353
$346$	8.00000	0.430083
$347$	22.0000	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$348$	−3.00000	−0.160817
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−4.00000	−0.213809
$351$	10.0000	0.533761
$352$	2.00000	0.106600
$353$	−16.0000	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$354$	−1.00000	−0.0531494
$355$	−12.0000	−0.636894
$356$	−8.00000	−0.423999
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	17.0000	0.897226	0.448613	−	0.893726i	$-0.351918\pi$
$359$	17.0000	0.897226	0.448613	+	0.893726i	$0.351918\pi$
$360$	2.00000	0.105409
$361$	30.0000	1.57895
$362$	5.00000	0.262794
$363$	7.00000	0.367405
$364$	−2.00000	−0.104828
$365$	4.00000	0.209370
$366$	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	$0.152514\pi$
$368$	4.00000	0.208514
$369$	−6.00000	−0.312348
$370$	4.00000	0.207950
$371$	11.0000	0.571092
$372$	10.0000	0.518476
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−2.00000	−0.103418
$375$	9.00000	0.464758
$376$	−6.00000	−0.309426
$377$	6.00000	0.309016
$378$	5.00000	0.257172
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	7.00000	0.359092
$381$	11.0000	0.563547
$382$	6.00000	0.306987
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	1.00000	0.0510310
$385$	2.00000	0.101929
$386$	17.0000	0.865277
$387$	4.00000	0.203331
$388$	−12.0000	−0.609208
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	2.00000	0.101274
$391$	−4.00000	−0.202289
$392$	6.00000	0.303046
$393$	−2.00000	−0.100887
$394$	−2.00000	−0.100759
$395$	1.00000	0.0503155
$396$	4.00000	0.201008
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	9.00000	0.451129
$399$	7.00000	0.350438
$400$	−4.00000	−0.200000
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	−14.0000	−0.698257
$403$	−20.0000	−0.996271
$404$	−2.00000	−0.0995037
$405$	1.00000	0.0496904
$406$	3.00000	0.148888
$407$	8.00000	0.396545
$408$	−1.00000	−0.0495074
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−3.00000	−0.148159
$411$	21.0000	1.03585
$412$	0	0
$413$	1.00000	0.0492068
$414$	8.00000	0.393179
$415$	18.0000	0.883585
$416$	−2.00000	−0.0980581
$417$	4.00000	0.195881
$418$	14.0000	0.684762
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	1.00000	0.0487950
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	8.00000	0.389434
$423$	−12.0000	−0.583460
$424$	11.0000	0.534207
$425$	4.00000	0.194029
$426$	−12.0000	−0.581402
$427$	−8.00000	−0.387147
$428$	3.00000	0.145010
$429$	4.00000	0.193122
$430$	2.00000	0.0964486
$431$	22.0000	1.05970	0.529851	−	0.848091i	$-0.322248\pi$
$431$	22.0000	1.05970	0.529851	+	0.848091i	$0.322248\pi$
$432$	5.00000	0.240563
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	−10.0000	−0.480015
$435$	−3.00000	−0.143839
$436$	−4.00000	−0.191565
$437$	28.0000	1.33942
$438$	4.00000	0.191127
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	2.00000	0.0953463
$441$	12.0000	0.571429
$442$	2.00000	0.0951303
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	4.00000	0.189832
$445$	−8.00000	−0.379236
$446$	−24.0000	−1.13643
$447$	−2.00000	−0.0945968
$448$	−1.00000	−0.0472456
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	−8.00000	−0.377124
$451$	−6.00000	−0.282529
$452$	−6.00000	−0.282216
$453$	14.0000	0.657777
$454$	−28.0000	−1.31411
$455$	−2.00000	−0.0937614
$456$	7.00000	0.327805
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	14.0000	0.654177
$459$	−5.00000	−0.233380
$460$	4.00000	0.186501
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	2.00000	0.0930484
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	3.00000	0.139272
$465$	10.0000	0.463739
$466$	10.0000	0.463241
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−4.00000	−0.184900
$469$	14.0000	0.646460
$470$	−6.00000	−0.276759
$471$	8.00000	0.368621
$472$	1.00000	0.0460287
$473$	4.00000	0.183920
$474$	1.00000	0.0459315
$475$	−28.0000	−1.28473
$476$	1.00000	0.0458349
$477$	22.0000	1.00731
$478$	9.00000	0.411650
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	1.00000	0.0456435
$481$	−8.00000	−0.364769
$482$	19.0000	0.865426
$483$	4.00000	0.182006
$484$	−7.00000	−0.318182
$485$	−12.0000	−0.544892
$486$	16.0000	0.725775
$487$	3.00000	0.135943	0.0679715	−	0.997687i	$-0.478347\pi$
$487$	3.00000	0.135943	0.0679715	+	0.997687i	$0.478347\pi$
$488$	−8.00000	−0.362143
$489$	−12.0000	−0.542659
$490$	6.00000	0.271052
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	−3.00000	−0.135250
$493$	−3.00000	−0.135113
$494$	−14.0000	−0.629890
$495$	4.00000	0.179787
$496$	−10.0000	−0.449013
$497$	12.0000	0.538274
$498$	18.0000	0.806599
$499$	−17.0000	−0.761025	−0.380512	−	0.924776i	$-0.624252\pi$
$499$	−17.0000	−0.761025	−0.380512	+	0.924776i	$0.624252\pi$
$500$	−9.00000	−0.402492
$501$	−15.0000	−0.670151
$502$	27.0000	1.20507
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	−2.00000	−0.0890871
$505$	−2.00000	−0.0889988
$506$	8.00000	0.355643
$507$	9.00000	0.399704
$508$	−11.0000	−0.488046
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	−1.00000	−0.0442807
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	35.0000	1.54529
$514$	−5.00000	−0.220541
$515$	0	0
$516$	2.00000	0.0880451
$517$	−12.0000	−0.527759
$518$	−4.00000	−0.175750
$519$	8.00000	0.351161
$520$	−2.00000	−0.0877058
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	6.00000	0.262613
$523$	−15.0000	−0.655904	−0.327952	−	0.944694i	$-0.606358\pi$
$523$	−15.0000	−0.655904	−0.327952	+	0.944694i	$0.606358\pi$
$524$	2.00000	0.0873704
$525$	−4.00000	−0.174574
$526$	21.0000	0.915644
$527$	10.0000	0.435607
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	11.0000	0.477809
$531$	2.00000	0.0867926
$532$	−7.00000	−0.303488
$533$	6.00000	0.259889
$534$	−8.00000	−0.346194
$535$	3.00000	0.129701
$536$	14.0000	0.604708
$537$	0	0
$538$	−14.0000	−0.603583
$539$	12.0000	0.516877
$540$	5.00000	0.215166
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	−9.00000	−0.386583
$543$	5.00000	0.214571
$544$	1.00000	0.0428746
$545$	−4.00000	−0.171341
$546$	−2.00000	−0.0855921
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−21.0000	−0.897076
$549$	−16.0000	−0.682863
$550$	−8.00000	−0.341121
$551$	21.0000	0.894630
$552$	4.00000	0.170251
$553$	−1.00000	−0.0425243
$554$	−13.0000	−0.552317
$555$	4.00000	0.169791
$556$	−4.00000	−0.169638
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	−20.0000	−0.846668
$559$	−4.00000	−0.169182
$560$	−1.00000	−0.0422577
$561$	−2.00000	−0.0844401
$562$	−3.00000	−0.126547
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	−6.00000	−0.252646
$565$	−6.00000	−0.252422
$566$	−4.00000	−0.168133
$567$	−1.00000	−0.0419961
$568$	12.0000	0.503509
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	7.00000	0.293198
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	−4.00000	−0.167248
$573$	6.00000	0.250654
$574$	3.00000	0.125218
$575$	−16.0000	−0.667246
$576$	−2.00000	−0.0833333
$577$	41.0000	1.70685	0.853426	−	0.521214i	$-0.174521\pi$
$577$	41.0000	1.70685	0.853426	+	0.521214i	$0.174521\pi$
$578$	−1.00000	−0.0415945
$579$	17.0000	0.706496
$580$	3.00000	0.124568
$581$	−18.0000	−0.746766
$582$	−12.0000	−0.497416
$583$	22.0000	0.911147
$584$	−4.00000	−0.165521
$585$	−4.00000	−0.165380
$586$	−7.00000	−0.289167
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	6.00000	0.247436
$589$	−70.0000	−2.88430
$590$	1.00000	0.0411693
$591$	−2.00000	−0.0822690
$592$	−4.00000	−0.164399
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	10.0000	0.410305
$595$	1.00000	0.0409960
$596$	2.00000	0.0819232
$597$	9.00000	0.368345
$598$	−8.00000	−0.327144
$599$	33.0000	1.34834	0.674172	−	0.738575i	$-0.264501\pi$
$599$	33.0000	1.34834	0.674172	+	0.738575i	$0.264501\pi$
$600$	−4.00000	−0.163299
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	−2.00000	−0.0815139
$603$	28.0000	1.14025
$604$	−14.0000	−0.569652
$605$	−7.00000	−0.284590
$606$	−2.00000	−0.0812444
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	−7.00000	−0.283887
$609$	3.00000	0.121566
$610$	−8.00000	−0.323911
$611$	12.0000	0.485468
$612$	2.00000	0.0808452
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−25.0000	−1.00892
$615$	−3.00000	−0.120972
$616$	−2.00000	−0.0805823
$617$	29.0000	1.16750	0.583748	−	0.811935i	$-0.301586\pi$
$617$	29.0000	1.16750	0.583748	+	0.811935i	$0.301586\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	−10.0000	−0.401610
$621$	20.0000	0.802572
$622$	13.0000	0.521253
$623$	8.00000	0.320513
$624$	−2.00000	−0.0800641
$625$	11.0000	0.440000
$626$	0	0
$627$	14.0000	0.559106
$628$	−8.00000	−0.319235
$629$	4.00000	0.159490
$630$	−2.00000	−0.0796819
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−1.00000	−0.0397779
$633$	8.00000	0.317971
$634$	2.00000	0.0794301
$635$	−11.0000	−0.436522
$636$	11.0000	0.436178
$637$	−12.0000	−0.475457
$638$	6.00000	0.237542
$639$	24.0000	0.949425
$640$	−1.00000	−0.0395285
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	3.00000	0.118401
$643$	27.0000	1.06478	0.532388	−	0.846500i	$-0.321295\pi$
$643$	27.0000	1.06478	0.532388	+	0.846500i	$0.321295\pi$
$644$	−4.00000	−0.157622
$645$	2.00000	0.0787499
$646$	7.00000	0.275411
$647$	47.0000	1.84776	0.923880	−	0.382682i	$-0.124999\pi$
$647$	47.0000	1.84776	0.923880	+	0.382682i	$0.124999\pi$
$648$	−1.00000	−0.0392837
$649$	2.00000	0.0785069
$650$	8.00000	0.313786
$651$	−10.0000	−0.391931
$652$	12.0000	0.469956
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	−4.00000	−0.156412
$655$	2.00000	0.0781465
$656$	3.00000	0.117130
$657$	−8.00000	−0.312110
$658$	6.00000	0.233904
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	2.00000	0.0778499
$661$	37.0000	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$661$	37.0000	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$662$	17.0000	0.660724
$663$	2.00000	0.0776736
$664$	−18.0000	−0.698535
$665$	−7.00000	−0.271448
$666$	−8.00000	−0.309994
$667$	12.0000	0.464642
$668$	15.0000	0.580367
$669$	−24.0000	−0.927894
$670$	14.0000	0.540867
$671$	−16.0000	−0.617673
$672$	−1.00000	−0.0385758
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	12.0000	0.462223
$675$	−20.0000	−0.769800
$676$	−9.00000	−0.346154
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	−6.00000	−0.230429
$679$	12.0000	0.460518
$680$	1.00000	0.0383482
$681$	−28.0000	−1.07296
$682$	−20.0000	−0.765840
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	−14.0000	−0.535303
$685$	−21.0000	−0.802369
$686$	−13.0000	−0.496342
$687$	14.0000	0.534133
$688$	−2.00000	−0.0762493
$689$	−22.0000	−0.838133
$690$	4.00000	0.152277
$691$	−24.0000	−0.913003	−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000	−0.913003	−0.456502	+	0.889723i	$0.650898\pi$
$692$	−8.00000	−0.304114
$693$	−4.00000	−0.151947
$694$	−22.0000	−0.835109
$695$	−4.00000	−0.151729
$696$	3.00000	0.113715
$697$	−3.00000	−0.113633
$698$	14.0000	0.529908
$699$	10.0000	0.378235
$700$	4.00000	0.151186
$701$	32.0000	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$702$	−10.0000	−0.377426
$703$	−28.0000	−1.05604
$704$	−2.00000	−0.0753778
$705$	−6.00000	−0.225973
$706$	16.0000	0.602168
$707$	2.00000	0.0752177
$708$	1.00000	0.0375823
$709$	−29.0000	−1.08912	−0.544559	−	0.838723i	$-0.683303\pi$
$709$	−29.0000	−1.08912	−0.544559	+	0.838723i	$0.683303\pi$
$710$	12.0000	0.450352
$711$	−2.00000	−0.0750059
$712$	8.00000	0.299813
$713$	−40.0000	−1.49801
$714$	1.00000	0.0374241
$715$	−4.00000	−0.149592
$716$	0	0
$717$	9.00000	0.336111
$718$	−17.0000	−0.634434
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−30.0000	−1.11648
$723$	19.0000	0.706618
$724$	−5.00000	−0.185824
$725$	−12.0000	−0.445669
$726$	−7.00000	−0.259794
$727$	48.0000	1.78022	0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000	1.78022	0.890111	+	0.455744i	$0.150627\pi$
$728$	2.00000	0.0741249
$729$	13.0000	0.481481
$730$	−4.00000	−0.148047
$731$	2.00000	0.0739727
$732$	−8.00000	−0.295689
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	−34.0000	−1.25496
$735$	6.00000	0.221313
$736$	−4.00000	−0.147442
$737$	28.0000	1.03139
$738$	6.00000	0.220863
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−4.00000	−0.147043
$741$	−14.0000	−0.514303
$742$	−11.0000	−0.403823
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−10.0000	−0.366618
$745$	2.00000	0.0732743
$746$	−14.0000	−0.512576
$747$	−36.0000	−1.31717
$748$	2.00000	0.0731272
$749$	−3.00000	−0.109618
$750$	−9.00000	−0.328634
$751$	−18.0000	−0.656829	−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000	−0.656829	−0.328415	+	0.944534i	$0.606514\pi$
$752$	6.00000	0.218797
$753$	27.0000	0.983935
$754$	−6.00000	−0.218507
$755$	−14.0000	−0.509512
$756$	−5.00000	−0.181848
$757$	−29.0000	−1.05402	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−29.0000	−1.05402	−0.527011	+	0.849858i	$0.676688\pi$
$758$	11.0000	0.399538
$759$	8.00000	0.290382
$760$	−7.00000	−0.253917
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	−11.0000	−0.398488
$763$	4.00000	0.144810
$764$	−6.00000	−0.217072
$765$	2.00000	0.0723102
$766$	8.00000	0.289052
$767$	−2.00000	−0.0722158
$768$	−1.00000	−0.0360844
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	−2.00000	−0.0720750
$771$	−5.00000	−0.180071
$772$	−17.0000	−0.611843
$773$	−28.0000	−1.00709	−0.503545	−	0.863969i	$-0.667971\pi$
$773$	−28.0000	−1.00709	−0.503545	+	0.863969i	$0.667971\pi$
$774$	−4.00000	−0.143777
$775$	40.0000	1.43684
$776$	12.0000	0.430775
$777$	−4.00000	−0.143499
$778$	−30.0000	−1.07555
$779$	21.0000	0.752403
$780$	−2.00000	−0.0716115
$781$	24.0000	0.858788
$782$	4.00000	0.143040
$783$	15.0000	0.536056
$784$	−6.00000	−0.214286
$785$	−8.00000	−0.285532
$786$	2.00000	0.0713376
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	2.00000	0.0712470
$789$	21.0000	0.747620
$790$	−1.00000	−0.0355784
$791$	6.00000	0.213335
$792$	−4.00000	−0.142134
$793$	16.0000	0.568177
$794$	10.0000	0.354887
$795$	11.0000	0.390130
$796$	−9.00000	−0.318997
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	−7.00000	−0.247797
$799$	−6.00000	−0.212265
$800$	4.00000	0.141421
$801$	16.0000	0.565332
$802$	−28.0000	−0.988714
$803$	−8.00000	−0.282314
$804$	14.0000	0.493742
$805$	−4.00000	−0.140981
$806$	20.0000	0.704470
$807$	−14.0000	−0.492823
$808$	2.00000	0.0703598
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	−1.00000	−0.0351364
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	−3.00000	−0.105279
$813$	−9.00000	−0.315644
$814$	−8.00000	−0.280400
$815$	12.0000	0.420342
$816$	1.00000	0.0350070
$817$	−14.0000	−0.489798
$818$	4.00000	0.139857
$819$	4.00000	0.139771
$820$	3.00000	0.104765
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	−21.0000	−0.732459
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	−1.00000	−0.0347945
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	−8.00000	−0.278019
$829$	−1.00000	−0.0347314	−0.0173657	−	0.999849i	$-0.505528\pi$
$829$	−1.00000	−0.0347314	−0.0173657	+	0.999849i	$0.505528\pi$
$830$	−18.0000	−0.624789
$831$	−13.0000	−0.450965
$832$	2.00000	0.0693375
$833$	6.00000	0.207888
$834$	−4.00000	−0.138509
$835$	15.0000	0.519096
$836$	−14.0000	−0.484200
$837$	−50.0000	−1.72825
$838$	−16.0000	−0.552711
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	−1.00000	−0.0345033
$841$	−20.0000	−0.689655
$842$	20.0000	0.689246
$843$	−3.00000	−0.103325
$844$	−8.00000	−0.275371
$845$	−9.00000	−0.309609
$846$	12.0000	0.412568
$847$	7.00000	0.240523
$848$	−11.0000	−0.377742
$849$	−4.00000	−0.137280
$850$	−4.00000	−0.137199
$851$	−16.0000	−0.548473
$852$	12.0000	0.411113
$853$	43.0000	1.47229	0.736146	−	0.676823i	$-0.236644\pi$
$853$	43.0000	1.47229	0.736146	+	0.676823i	$0.236644\pi$
$854$	8.00000	0.273754
$855$	−14.0000	−0.478790
$856$	−3.00000	−0.102538
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	−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	$0.478262\pi$
$860$	−2.00000	−0.0681994
$861$	3.00000	0.102240
$862$	−22.0000	−0.749323
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	−5.00000	−0.170103
$865$	−8.00000	−0.272008
$866$	−11.0000	−0.373795
$867$	−1.00000	−0.0339618
$868$	10.0000	0.339422
$869$	−2.00000	−0.0678454
$870$	3.00000	0.101710
$871$	−28.0000	−0.948744
$872$	4.00000	0.135457
$873$	24.0000	0.812277
$874$	−28.0000	−0.947114
$875$	9.00000	0.304256
$876$	−4.00000	−0.135147
$877$	−21.0000	−0.709120	−0.354560	−	0.935033i	$-0.615369\pi$
$877$	−21.0000	−0.709120	−0.354560	+	0.935033i	$0.615369\pi$
$878$	36.0000	1.21494
$879$	−7.00000	−0.236104
$880$	−2.00000	−0.0674200
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	−12.0000	−0.404061
$883$	−7.00000	−0.235569	−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000	−0.235569	−0.117784	+	0.993039i	$0.537579\pi$
$884$	−2.00000	−0.0672673
$885$	1.00000	0.0336146
$886$	−4.00000	−0.134383
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−4.00000	−0.134231
$889$	11.0000	0.368928
$890$	8.00000	0.268161
$891$	−2.00000	−0.0670025
$892$	24.0000	0.803579
$893$	42.0000	1.40548
$894$	2.00000	0.0668900
$895$	0	0
$896$	1.00000	0.0334077
$897$	−8.00000	−0.267112
$898$	15.0000	0.500556
$899$	−30.0000	−1.00056
$900$	8.00000	0.266667
$901$	11.0000	0.366463
$902$	6.00000	0.199778
$903$	−2.00000	−0.0665558
$904$	6.00000	0.199557
$905$	−5.00000	−0.166206
$906$	−14.0000	−0.465119
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	28.0000	0.929213
$909$	4.00000	0.132672
$910$	2.00000	0.0662994
$911$	−25.0000	−0.828287	−0.414143	−	0.910212i	$-0.635919\pi$
$911$	−25.0000	−0.828287	−0.414143	+	0.910212i	$0.635919\pi$
$912$	−7.00000	−0.231793
$913$	−36.0000	−1.19143
$914$	2.00000	0.0661541
$915$	−8.00000	−0.264472
$916$	−14.0000	−0.462573
$917$	−2.00000	−0.0660458
$918$	5.00000	0.165025
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	−4.00000	−0.131876
$921$	−25.0000	−0.823778
$922$	18.0000	0.592798
$923$	−24.0000	−0.789970
$924$	−2.00000	−0.0657952
$925$	16.0000	0.526077
$926$	−4.00000	−0.131448
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	−10.0000	−0.327913
$931$	−42.0000	−1.37649
$932$	−10.0000	−0.327561
$933$	13.0000	0.425601
$934$	28.0000	0.916188
$935$	2.00000	0.0654070
$936$	4.00000	0.130744
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	−14.0000	−0.457116
$939$	0	0
$940$	6.00000	0.195698
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	−8.00000	−0.260654
$943$	12.0000	0.390774
$944$	−1.00000	−0.0325472
$945$	−5.00000	−0.162650
$946$	−4.00000	−0.130051
$947$	−21.0000	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$947$	−21.0000	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$948$	−1.00000	−0.0324785
$949$	8.00000	0.259691
$950$	28.0000	0.908440
$951$	2.00000	0.0648544
$952$	−1.00000	−0.0324102
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−22.0000	−0.712276
$955$	−6.00000	−0.194155
$956$	−9.00000	−0.291081
$957$	6.00000	0.193952
$958$	24.0000	0.775405
$959$	21.0000	0.678125
$960$	−1.00000	−0.0322749
$961$	69.0000	2.22581
$962$	8.00000	0.257930
$963$	−6.00000	−0.193347
$964$	−19.0000	−0.611949
$965$	−17.0000	−0.547249
$966$	−4.00000	−0.128698
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	7.00000	0.224989
$969$	7.00000	0.224872
$970$	12.0000	0.385297
$971$	−7.00000	−0.224641	−0.112320	−	0.993672i	$-0.535828\pi$
$971$	−7.00000	−0.224641	−0.112320	+	0.993672i	$0.535828\pi$
$972$	−16.0000	−0.513200
$973$	4.00000	0.128234
$974$	−3.00000	−0.0961262
$975$	8.00000	0.256205
$976$	8.00000	0.256074
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	12.0000	0.383718
$979$	16.0000	0.511362
$980$	−6.00000	−0.191663
$981$	8.00000	0.255420
$982$	15.0000	0.478669
$983$	26.0000	0.829271	0.414636	−	0.909988i	$-0.363909\pi$
$983$	26.0000	0.829271	0.414636	+	0.909988i	$0.363909\pi$
$984$	3.00000	0.0956365
$985$	2.00000	0.0637253
$986$	3.00000	0.0955395
$987$	6.00000	0.190982
$988$	14.0000	0.445399
$989$	−8.00000	−0.254385
$990$	−4.00000	−0.127128
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	10.0000	0.317500
$993$	17.0000	0.539479
$994$	−12.0000	−0.380617
$995$	−9.00000	−0.285319
$996$	−18.0000	−0.570352
$997$	55.0000	1.74187	0.870934	−	0.491400i	$-0.163515\pi$
$997$	55.0000	1.74187	0.870934	+	0.491400i	$0.163515\pi$
$998$	17.0000	0.538126
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2006.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2006.2.a.b.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:	\( N \)	\(=\)	\( 2006 = 2 \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2006.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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