LMFDB - Embedded newform 2006.2.a.f.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -3.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -3.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} +3.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} +6.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -6.00000 q^{18} +2.00000 q^{20} +12.0000 q^{21} +2.00000 q^{22} -1.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} +9.00000 q^{27} +4.00000 q^{28} -3.00000 q^{29} -6.00000 q^{30} -7.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -1.00000 q^{34} +8.00000 q^{35} +6.00000 q^{36} +6.00000 q^{37} +3.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} -12.0000 q^{42} +4.00000 q^{43} -2.00000 q^{44} +12.0000 q^{45} +1.00000 q^{46} -4.00000 q^{47} +3.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +3.00000 q^{51} +1.00000 q^{52} +4.00000 q^{53} -9.00000 q^{54} -4.00000 q^{55} -4.00000 q^{56} +3.00000 q^{58} -1.00000 q^{59} +6.00000 q^{60} -8.00000 q^{61} +7.00000 q^{62} +24.0000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +6.00000 q^{66} -11.0000 q^{67} +1.00000 q^{68} -3.00000 q^{69} -8.00000 q^{70} -14.0000 q^{71} -6.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} -3.00000 q^{75} -8.00000 q^{77} -3.00000 q^{78} +10.0000 q^{79} +2.00000 q^{80} +9.00000 q^{81} +6.00000 q^{82} -1.00000 q^{83} +12.0000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -9.00000 q^{87} +2.00000 q^{88} -4.00000 q^{89} -12.0000 q^{90} +4.00000 q^{91} -1.00000 q^{92} -21.0000 q^{93} +4.00000 q^{94} -3.00000 q^{96} +17.0000 q^{97} -9.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−3.00000	−1.22474
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	−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	$0.597491\pi$
$12$	3.00000	0.866025
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−4.00000	−1.06904
$15$	6.00000	1.54919
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−6.00000	−1.41421
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.00000	0.447214
$21$	12.0000	2.61861
$22$	2.00000	0.426401
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	−3.00000	−0.612372
$25$	−1.00000	−0.200000
$26$	−1.00000	−0.196116
$27$	9.00000	1.73205
$28$	4.00000	0.755929
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	−6.00000	−1.09545
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	−1.00000	−0.171499
$35$	8.00000	1.35225
$36$	6.00000	1.00000
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	−2.00000	−0.316228
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−12.0000	−1.85164
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−2.00000	−0.301511
$45$	12.0000	1.78885
$46$	1.00000	0.147442
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	3.00000	0.433013
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	3.00000	0.420084
$52$	1.00000	0.138675
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−9.00000	−1.22474
$55$	−4.00000	−0.539360
$56$	−4.00000	−0.534522
$57$	0	0
$58$	3.00000	0.393919
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	6.00000	0.774597
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	7.00000	0.889001
$63$	24.0000	3.02372
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	6.00000	0.738549
$67$	−11.0000	−1.34386	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000	−1.34386	−0.671932	+	0.740613i	$0.734535\pi$
$68$	1.00000	0.121268
$69$	−3.00000	−0.361158
$70$	−8.00000	−0.956183
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	−6.00000	−0.707107
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−6.00000	−0.697486
$75$	−3.00000	−0.346410
$76$	0	0
$77$	−8.00000	−0.911685
$78$	−3.00000	−0.339683
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	2.00000	0.223607
$81$	9.00000	1.00000
$82$	6.00000	0.662589
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	12.0000	1.30931
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	−9.00000	−0.964901
$88$	2.00000	0.213201
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	−12.0000	−1.26491
$91$	4.00000	0.419314
$92$	−1.00000	−0.104257
$93$	−21.0000	−2.17760
$94$	4.00000	0.412568
$95$	0	0
$96$	−3.00000	−0.306186
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	−9.00000	−0.909137
$99$	−12.0000	−1.20605
$100$	−1.00000	−0.100000
$101$	−1.00000	−0.0995037	−0.0497519	−	0.998762i	$-0.515843\pi$
$101$	−1.00000	−0.0995037	−0.0497519	+	0.998762i	$0.515843\pi$
$102$	−3.00000	−0.297044
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−1.00000	−0.0980581
$105$	24.0000	2.34216
$106$	−4.00000	−0.388514
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	9.00000	0.866025
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	4.00000	0.381385
$111$	18.0000	1.70848
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	−3.00000	−0.278543
$117$	6.00000	0.554700
$118$	1.00000	0.0920575
$119$	4.00000	0.366679
$120$	−6.00000	−0.547723
$121$	−7.00000	−0.636364
$122$	8.00000	0.724286
$123$	−18.0000	−1.62301
$124$	−7.00000	−0.628619
$125$	−12.0000	−1.07331
$126$	−24.0000	−2.13809
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	12.0000	1.05654
$130$	−2.00000	−0.175412
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	11.0000	0.950255
$135$	18.0000	1.54919
$136$	−1.00000	−0.0857493
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	3.00000	0.255377
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	8.00000	0.676123
$141$	−12.0000	−1.01058
$142$	14.0000	1.17485
$143$	−2.00000	−0.167248
$144$	6.00000	0.500000
$145$	−6.00000	−0.498273
$146$	−2.00000	−0.165521
$147$	27.0000	2.22692
$148$	6.00000	0.493197
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	3.00000	0.244949
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	8.00000	0.644658
$155$	−14.0000	−1.12451
$156$	3.00000	0.240192
$157$	−11.0000	−0.877896	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000	−0.877896	−0.438948	+	0.898513i	$0.644649\pi$
$158$	−10.0000	−0.795557
$159$	12.0000	0.951662
$160$	−2.00000	−0.158114
$161$	−4.00000	−0.315244
$162$	−9.00000	−0.707107
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	−6.00000	−0.468521
$165$	−12.0000	−0.934199
$166$	1.00000	0.0776151
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	−12.0000	−0.925820
$169$	−12.0000	−0.923077
$170$	−2.00000	−0.153393
$171$	0	0
$172$	4.00000	0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	9.00000	0.682288
$175$	−4.00000	−0.302372
$176$	−2.00000	−0.150756
$177$	−3.00000	−0.225494
$178$	4.00000	0.299813
$179$	11.0000	0.822179	0.411089	−	0.911595i	$-0.365148\pi$
$179$	11.0000	0.822179	0.411089	+	0.911595i	$0.365148\pi$
$180$	12.0000	0.894427
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	−4.00000	−0.296500
$183$	−24.0000	−1.77413
$184$	1.00000	0.0737210
$185$	12.0000	0.882258
$186$	21.0000	1.53979
$187$	−2.00000	−0.146254
$188$	−4.00000	−0.291730
$189$	36.0000	2.61861
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	3.00000	0.216506
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−17.0000	−1.22053
$195$	6.00000	0.429669
$196$	9.00000	0.642857
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	12.0000	0.852803
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	1.00000	0.0707107
$201$	−33.0000	−2.32764
$202$	1.00000	0.0703598
$203$	−12.0000	−0.842235
$204$	3.00000	0.210042
$205$	−12.0000	−0.838116
$206$	4.00000	0.278693
$207$	−6.00000	−0.417029
$208$	1.00000	0.0693375
$209$	0	0
$210$	−24.0000	−1.65616
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	4.00000	0.274721
$213$	−42.0000	−2.87779
$214$	3.00000	0.205076
$215$	8.00000	0.545595
$216$	−9.00000	−0.612372
$217$	−28.0000	−1.90076
$218$	4.00000	0.270914
$219$	6.00000	0.405442
$220$	−4.00000	−0.269680
$221$	1.00000	0.0672673
$222$	−18.0000	−1.20808
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	−4.00000	−0.267261
$225$	−6.00000	−0.400000
$226$	−6.00000	−0.399114
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	2.00000	0.131876
$231$	−24.0000	−1.57908
$232$	3.00000	0.196960
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	−6.00000	−0.392232
$235$	−8.00000	−0.521862
$236$	−1.00000	−0.0650945
$237$	30.0000	1.94871
$238$	−4.00000	−0.259281
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	6.00000	0.387298
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−8.00000	−0.512148
$245$	18.0000	1.14998
$246$	18.0000	1.14764
$247$	0	0
$248$	7.00000	0.444500
$249$	−3.00000	−0.190117
$250$	12.0000	0.758947
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	24.0000	1.51186
$253$	2.00000	0.125739
$254$	−5.00000	−0.313728
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	−12.0000	−0.747087
$259$	24.0000	1.49129
$260$	2.00000	0.124035
$261$	−18.0000	−1.11417
$262$	8.00000	0.494242
$263$	27.0000	1.66489	0.832446	−	0.554107i	$-0.186940\pi$
$263$	27.0000	1.66489	0.832446	+	0.554107i	$0.186940\pi$
$264$	6.00000	0.369274
$265$	8.00000	0.491436
$266$	0	0
$267$	−12.0000	−0.734388
$268$	−11.0000	−0.671932
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	−18.0000	−1.09545
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.00000	0.0606339
$273$	12.0000	0.726273
$274$	−15.0000	−0.906183
$275$	2.00000	0.120605
$276$	−3.00000	−0.180579
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	−5.00000	−0.299880
$279$	−42.0000	−2.51447
$280$	−8.00000	−0.478091
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	12.0000	0.714590
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	2.00000	0.118262
$287$	−24.0000	−1.41668
$288$	−6.00000	−0.353553
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	51.0000	2.98967
$292$	2.00000	0.117041
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	−27.0000	−1.57467
$295$	−2.00000	−0.116445
$296$	−6.00000	−0.348743
$297$	−18.0000	−1.04447
$298$	15.0000	0.868927
$299$	−1.00000	−0.0578315
$300$	−3.00000	−0.173205
$301$	16.0000	0.922225
$302$	−10.0000	−0.575435
$303$	−3.00000	−0.172345
$304$	0	0
$305$	−16.0000	−0.916157
$306$	−6.00000	−0.342997
$307$	14.0000	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$308$	−8.00000	−0.455842
$309$	−12.0000	−0.682656
$310$	14.0000	0.795147
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	−3.00000	−0.169842
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	11.0000	0.620766
$315$	48.0000	2.70449
$316$	10.0000	0.562544
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	−12.0000	−0.672927
$319$	6.00000	0.335936
$320$	2.00000	0.111803
$321$	−9.00000	−0.502331
$322$	4.00000	0.222911
$323$	0	0
$324$	9.00000	0.500000
$325$	−1.00000	−0.0554700
$326$	−1.00000	−0.0553849
$327$	−12.0000	−0.663602
$328$	6.00000	0.331295
$329$	−16.0000	−0.882109
$330$	12.0000	0.660578
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−1.00000	−0.0548821
$333$	36.0000	1.97279
$334$	14.0000	0.766046
$335$	−22.0000	−1.20199
$336$	12.0000	0.654654
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	12.0000	0.652714
$339$	18.0000	0.977626
$340$	2.00000	0.108465
$341$	14.0000	0.758143
$342$	0	0
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	−6.00000	−0.323029
$346$	−6.00000	−0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−9.00000	−0.482451
$349$	21.0000	1.12410	0.562052	−	0.827102i	$-0.310012\pi$
$349$	21.0000	1.12410	0.562052	+	0.827102i	$0.310012\pi$
$350$	4.00000	0.213809
$351$	9.00000	0.480384
$352$	2.00000	0.106600
$353$	−36.0000	−1.91609	−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000	−1.91609	−0.958043	+	0.286623i	$0.907467\pi$
$354$	3.00000	0.159448
$355$	−28.0000	−1.48609
$356$	−4.00000	−0.212000
$357$	12.0000	0.635107
$358$	−11.0000	−0.581368
$359$	5.00000	0.263890	0.131945	−	0.991257i	$-0.457878\pi$
$359$	5.00000	0.263890	0.131945	+	0.991257i	$0.457878\pi$
$360$	−12.0000	−0.632456
$361$	−19.0000	−1.00000
$362$	−7.00000	−0.367912
$363$	−21.0000	−1.10221
$364$	4.00000	0.209657
$365$	4.00000	0.209370
$366$	24.0000	1.25450
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	−1.00000	−0.0521286
$369$	−36.0000	−1.87409
$370$	−12.0000	−0.623850
$371$	16.0000	0.830679
$372$	−21.0000	−1.08880
$373$	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
$375$	−36.0000	−1.85903
$376$	4.00000	0.206284
$377$	−3.00000	−0.154508
$378$	−36.0000	−1.85164
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	15.0000	0.768473
$382$	18.0000	0.920960
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−3.00000	−0.153093
$385$	−16.0000	−0.815436
$386$	6.00000	0.305392
$387$	24.0000	1.21999
$388$	17.0000	0.863044
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	−6.00000	−0.303822
$391$	−1.00000	−0.0505722
$392$	−9.00000	−0.454569
$393$	−24.0000	−1.21064
$394$	−15.0000	−0.755689
$395$	20.0000	1.00631
$396$	−12.0000	−0.603023
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	−10.0000	−0.501255
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	31.0000	1.54807	0.774033	−	0.633145i	$-0.218236\pi$
$401$	31.0000	1.54807	0.774033	+	0.633145i	$0.218236\pi$
$402$	33.0000	1.64589
$403$	−7.00000	−0.348695
$404$	−1.00000	−0.0497519
$405$	18.0000	0.894427
$406$	12.0000	0.595550
$407$	−12.0000	−0.594818
$408$	−3.00000	−0.148522
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	12.0000	0.592638
$411$	45.0000	2.21969
$412$	−4.00000	−0.197066
$413$	−4.00000	−0.196827
$414$	6.00000	0.294884
$415$	−2.00000	−0.0981761
$416$	−1.00000	−0.0490290
$417$	15.0000	0.734553
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	24.0000	1.17108
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	−24.0000	−1.16692
$424$	−4.00000	−0.194257
$425$	−1.00000	−0.0485071
$426$	42.0000	2.03491
$427$	−32.0000	−1.54859
$428$	−3.00000	−0.145010
$429$	−6.00000	−0.289683
$430$	−8.00000	−0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	9.00000	0.433013
$433$	33.0000	1.58588	0.792939	−	0.609301i	$-0.208550\pi$
$433$	33.0000	1.58588	0.792939	+	0.609301i	$0.208550\pi$
$434$	28.0000	1.34404
$435$	−18.0000	−0.863034
$436$	−4.00000	−0.191565
$437$	0	0
$438$	−6.00000	−0.286691
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	4.00000	0.190693
$441$	54.0000	2.57143
$442$	−1.00000	−0.0475651
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	18.0000	0.854242
$445$	−8.00000	−0.379236
$446$	−21.0000	−0.994379
$447$	−45.0000	−2.12843
$448$	4.00000	0.188982
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	6.00000	0.282843
$451$	12.0000	0.565058
$452$	6.00000	0.282216
$453$	30.0000	1.40952
$454$	−24.0000	−1.12638
$455$	8.00000	0.375046
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	−13.0000	−0.607450
$459$	9.00000	0.420084
$460$	−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	$0.637680\pi$
$462$	24.0000	1.11658
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−3.00000	−0.139272
$465$	−42.0000	−1.94770
$466$	−10.0000	−0.463241
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	6.00000	0.277350
$469$	−44.0000	−2.03173
$470$	8.00000	0.369012
$471$	−33.0000	−1.52056
$472$	1.00000	0.0460287
$473$	−8.00000	−0.367840
$474$	−30.0000	−1.37795
$475$	0	0
$476$	4.00000	0.183340
$477$	24.0000	1.09888
$478$	−8.00000	−0.365911
$479$	−34.0000	−1.55350	−0.776750	−	0.629809i	$-0.783133\pi$
$479$	−34.0000	−1.55350	−0.776750	+	0.629809i	$0.783133\pi$
$480$	−6.00000	−0.273861
$481$	6.00000	0.273576
$482$	−22.0000	−1.00207
$483$	−12.0000	−0.546019
$484$	−7.00000	−0.318182
$485$	34.0000	1.54386
$486$	0	0
$487$	14.0000	0.634401	0.317200	−	0.948359i	$-0.397257\pi$
$487$	14.0000	0.634401	0.317200	+	0.948359i	$0.397257\pi$
$488$	8.00000	0.362143
$489$	3.00000	0.135665
$490$	−18.0000	−0.813157
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	−18.0000	−0.811503
$493$	−3.00000	−0.135113
$494$	0	0
$495$	−24.0000	−1.07872
$496$	−7.00000	−0.314309
$497$	−56.0000	−2.51194
$498$	3.00000	0.134433
$499$	−43.0000	−1.92494	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−43.0000	−1.92494	−0.962472	+	0.271380i	$0.912520\pi$
$500$	−12.0000	−0.536656
$501$	−42.0000	−1.87642
$502$	6.00000	0.267793
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	−24.0000	−1.06904
$505$	−2.00000	−0.0889988
$506$	−2.00000	−0.0889108
$507$	−36.0000	−1.59882
$508$	5.00000	0.221839
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	−6.00000	−0.265684
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.0000	−0.926270
$515$	−8.00000	−0.352522
$516$	12.0000	0.528271
$517$	8.00000	0.351840
$518$	−24.0000	−1.05450
$519$	18.0000	0.790112
$520$	−2.00000	−0.0877058
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	18.0000	0.787839
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−8.00000	−0.349482
$525$	−12.0000	−0.523723
$526$	−27.0000	−1.17726
$527$	−7.00000	−0.304925
$528$	−6.00000	−0.261116
$529$	−22.0000	−0.956522
$530$	−8.00000	−0.347498
$531$	−6.00000	−0.260378
$532$	0	0
$533$	−6.00000	−0.259889
$534$	12.0000	0.519291
$535$	−6.00000	−0.259403
$536$	11.0000	0.475128
$537$	33.0000	1.42406
$538$	12.0000	0.517357
$539$	−18.0000	−0.775315
$540$	18.0000	0.774597
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	−8.00000	−0.343629
$543$	21.0000	0.901196
$544$	−1.00000	−0.0428746
$545$	−8.00000	−0.342682
$546$	−12.0000	−0.513553
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	15.0000	0.640768
$549$	−48.0000	−2.04859
$550$	−2.00000	−0.0852803
$551$	0	0
$552$	3.00000	0.127688
$553$	40.0000	1.70097
$554$	−23.0000	−0.977176
$555$	36.0000	1.52811
$556$	5.00000	0.212047
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	42.0000	1.77800
$559$	4.00000	0.169182
$560$	8.00000	0.338062
$561$	−6.00000	−0.253320
$562$	27.0000	1.13893
$563$	−23.0000	−0.969334	−0.484667	−	0.874699i	$-0.661059\pi$
$563$	−23.0000	−0.969334	−0.484667	+	0.874699i	$0.661059\pi$
$564$	−12.0000	−0.505291
$565$	12.0000	0.504844
$566$	−14.0000	−0.588464
$567$	36.0000	1.51186
$568$	14.0000	0.587427
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−2.00000	−0.0836242
$573$	−54.0000	−2.25588
$574$	24.0000	1.00174
$575$	1.00000	0.0417029
$576$	6.00000	0.250000
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	−1.00000	−0.0415945
$579$	−18.0000	−0.748054
$580$	−6.00000	−0.249136
$581$	−4.00000	−0.165948
$582$	−51.0000	−2.11402
$583$	−8.00000	−0.331326
$584$	−2.00000	−0.0827606
$585$	12.0000	0.496139
$586$	2.00000	0.0826192
$587$	19.0000	0.784214	0.392107	−	0.919920i	$-0.371746\pi$
$587$	19.0000	0.784214	0.392107	+	0.919920i	$0.371746\pi$
$588$	27.0000	1.11346
$589$	0	0
$590$	2.00000	0.0823387
$591$	45.0000	1.85105
$592$	6.00000	0.246598
$593$	−17.0000	−0.698106	−0.349053	−	0.937103i	$-0.613497\pi$
$593$	−17.0000	−0.698106	−0.349053	+	0.937103i	$0.613497\pi$
$594$	18.0000	0.738549
$595$	8.00000	0.327968
$596$	−15.0000	−0.614424
$597$	30.0000	1.22782
$598$	1.00000	0.0408930
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	3.00000	0.122474
$601$	23.0000	0.938190	0.469095	−	0.883148i	$-0.344580\pi$
$601$	23.0000	0.938190	0.469095	+	0.883148i	$0.344580\pi$
$602$	−16.0000	−0.652111
$603$	−66.0000	−2.68773
$604$	10.0000	0.406894
$605$	−14.0000	−0.569181
$606$	3.00000	0.121867
$607$	−38.0000	−1.54237	−0.771186	−	0.636610i	$-0.780336\pi$
$607$	−38.0000	−1.54237	−0.771186	+	0.636610i	$0.780336\pi$
$608$	0	0
$609$	−36.0000	−1.45879
$610$	16.0000	0.647821
$611$	−4.00000	−0.161823
$612$	6.00000	0.242536
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	−14.0000	−0.564994
$615$	−36.0000	−1.45166
$616$	8.00000	0.322329
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	12.0000	0.482711
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−14.0000	−0.562254
$621$	−9.00000	−0.361158
$622$	−26.0000	−1.04251
$623$	−16.0000	−0.641026
$624$	3.00000	0.120096
$625$	−19.0000	−0.760000
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	−11.0000	−0.438948
$629$	6.00000	0.239236
$630$	−48.0000	−1.91237
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	−10.0000	−0.397779
$633$	0	0
$634$	−3.00000	−0.119145
$635$	10.0000	0.396838
$636$	12.0000	0.475831
$637$	9.00000	0.356593
$638$	−6.00000	−0.237542
$639$	−84.0000	−3.32299
$640$	−2.00000	−0.0790569
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	9.00000	0.355202
$643$	31.0000	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$644$	−4.00000	−0.157622
$645$	24.0000	0.944999
$646$	0	0
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	−9.00000	−0.353553
$649$	2.00000	0.0785069
$650$	1.00000	0.0392232
$651$	−84.0000	−3.29222
$652$	1.00000	0.0391630
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	12.0000	0.469237
$655$	−16.0000	−0.625172
$656$	−6.00000	−0.234261
$657$	12.0000	0.468165
$658$	16.0000	0.623745
$659$	25.0000	0.973862	0.486931	−	0.873441i	$-0.338116\pi$
$659$	25.0000	0.973862	0.486931	+	0.873441i	$0.338116\pi$
$660$	−12.0000	−0.467099
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	28.0000	1.08825
$663$	3.00000	0.116510
$664$	1.00000	0.0388075
$665$	0	0
$666$	−36.0000	−1.39497
$667$	3.00000	0.116160
$668$	−14.0000	−0.541676
$669$	63.0000	2.43572
$670$	22.0000	0.849934
$671$	16.0000	0.617673
$672$	−12.0000	−0.462910
$673$	43.0000	1.65753	0.828764	−	0.559598i	$-0.189045\pi$
$673$	43.0000	1.65753	0.828764	+	0.559598i	$0.189045\pi$
$674$	13.0000	0.500741
$675$	−9.00000	−0.346410
$676$	−12.0000	−0.461538
$677$	31.0000	1.19143	0.595713	−	0.803197i	$-0.296869\pi$
$677$	31.0000	1.19143	0.595713	+	0.803197i	$0.296869\pi$
$678$	−18.0000	−0.691286
$679$	68.0000	2.60960
$680$	−2.00000	−0.0766965
$681$	72.0000	2.75905
$682$	−14.0000	−0.536088
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	30.0000	1.14624
$686$	−8.00000	−0.305441
$687$	39.0000	1.48794
$688$	4.00000	0.152499
$689$	4.00000	0.152388
$690$	6.00000	0.228416
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000	0.228086
$693$	−48.0000	−1.82337
$694$	−12.0000	−0.455514
$695$	10.0000	0.379322
$696$	9.00000	0.341144
$697$	−6.00000	−0.227266
$698$	−21.0000	−0.794862
$699$	30.0000	1.13470
$700$	−4.00000	−0.151186
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	−9.00000	−0.339683
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	−24.0000	−0.903892
$706$	36.0000	1.35488
$707$	−4.00000	−0.150435
$708$	−3.00000	−0.112747
$709$	−11.0000	−0.413114	−0.206557	−	0.978435i	$-0.566226\pi$
$709$	−11.0000	−0.413114	−0.206557	+	0.978435i	$0.566226\pi$
$710$	28.0000	1.05082
$711$	60.0000	2.25018
$712$	4.00000	0.149906
$713$	7.00000	0.262152
$714$	−12.0000	−0.449089
$715$	−4.00000	−0.149592
$716$	11.0000	0.411089
$717$	24.0000	0.896296
$718$	−5.00000	−0.186598
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	12.0000	0.447214
$721$	−16.0000	−0.595871
$722$	19.0000	0.707107
$723$	66.0000	2.45457
$724$	7.00000	0.260153
$725$	3.00000	0.111417
$726$	21.0000	0.779383
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	−4.00000	−0.148250
$729$	−27.0000	−1.00000
$730$	−4.00000	−0.148047
$731$	4.00000	0.147945
$732$	−24.0000	−0.887066
$733$	12.0000	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$734$	7.00000	0.258375
$735$	54.0000	1.99182
$736$	1.00000	0.0368605
$737$	22.0000	0.810380
$738$	36.0000	1.32518
$739$	−23.0000	−0.846069	−0.423034	−	0.906114i	$-0.639035\pi$
$739$	−23.0000	−0.846069	−0.423034	+	0.906114i	$0.639035\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	−16.0000	−0.587378
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	21.0000	0.769897
$745$	−30.0000	−1.09911
$746$	−14.0000	−0.512576
$747$	−6.00000	−0.219529
$748$	−2.00000	−0.0731272
$749$	−12.0000	−0.438470
$750$	36.0000	1.31453
$751$	49.0000	1.78804	0.894018	−	0.448032i	$-0.147875\pi$
$751$	49.0000	1.78804	0.894018	+	0.448032i	$0.147875\pi$
$752$	−4.00000	−0.145865
$753$	−18.0000	−0.655956
$754$	3.00000	0.109254
$755$	20.0000	0.727875
$756$	36.0000	1.30931
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	20.0000	0.726433
$759$	6.00000	0.217786
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	−15.0000	−0.543393
$763$	−16.0000	−0.579239
$764$	−18.0000	−0.651217
$765$	12.0000	0.433861
$766$	−24.0000	−0.867155
$767$	−1.00000	−0.0361079
$768$	3.00000	0.108253
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	16.0000	0.576600
$771$	63.0000	2.26889
$772$	−6.00000	−0.215945
$773$	−11.0000	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$773$	−11.0000	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$774$	−24.0000	−0.862662
$775$	7.00000	0.251447
$776$	−17.0000	−0.610264
$777$	72.0000	2.58299
$778$	8.00000	0.286814
$779$	0	0
$780$	6.00000	0.214834
$781$	28.0000	1.00192
$782$	1.00000	0.0357599
$783$	−27.0000	−0.964901
$784$	9.00000	0.321429
$785$	−22.0000	−0.785214
$786$	24.0000	0.856052
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	15.0000	0.534353
$789$	81.0000	2.88368
$790$	−20.0000	−0.711568
$791$	24.0000	0.853342
$792$	12.0000	0.426401
$793$	−8.00000	−0.284088
$794$	8.00000	0.283909
$795$	24.0000	0.851192
$796$	10.0000	0.354441
$797$	15.0000	0.531327	0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000	0.531327	0.265664	+	0.964066i	$0.414409\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	1.00000	0.0353553
$801$	−24.0000	−0.847998
$802$	−31.0000	−1.09465
$803$	−4.00000	−0.141157
$804$	−33.0000	−1.16382
$805$	−8.00000	−0.281963
$806$	7.00000	0.246564
$807$	−36.0000	−1.26726
$808$	1.00000	0.0351799
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	−18.0000	−0.632456
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	−12.0000	−0.421117
$813$	24.0000	0.841717
$814$	12.0000	0.420600
$815$	2.00000	0.0700569
$816$	3.00000	0.105021
$817$	0	0
$818$	20.0000	0.699284
$819$	24.0000	0.838628
$820$	−12.0000	−0.419058
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	−45.0000	−1.56956
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	4.00000	0.139347
$825$	6.00000	0.208893
$826$	4.00000	0.139178
$827$	−1.00000	−0.0347734	−0.0173867	−	0.999849i	$-0.505535\pi$
$827$	−1.00000	−0.0347734	−0.0173867	+	0.999849i	$0.505535\pi$
$828$	−6.00000	−0.208514
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	2.00000	0.0694210
$831$	69.0000	2.39358
$832$	1.00000	0.0346688
$833$	9.00000	0.311832
$834$	−15.0000	−0.519408
$835$	−28.0000	−0.968980
$836$	0	0
$837$	−63.0000	−2.17760
$838$	30.0000	1.03633
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	−24.0000	−0.828079
$841$	−20.0000	−0.689655
$842$	−26.0000	−0.896019
$843$	−81.0000	−2.78979
$844$	0	0
$845$	−24.0000	−0.825625
$846$	24.0000	0.825137
$847$	−28.0000	−0.962091
$848$	4.00000	0.137361
$849$	42.0000	1.44144
$850$	1.00000	0.0342997
$851$	−6.00000	−0.205677
$852$	−42.0000	−1.43890
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	32.0000	1.09502
$855$	0	0
$856$	3.00000	0.102538
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	6.00000	0.204837
$859$	51.0000	1.74010	0.870049	−	0.492966i	$-0.164087\pi$
$859$	51.0000	1.74010	0.870049	+	0.492966i	$0.164087\pi$
$860$	8.00000	0.272798
$861$	−72.0000	−2.45375
$862$	−24.0000	−0.817443
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−9.00000	−0.306186
$865$	12.0000	0.408012
$866$	−33.0000	−1.12139
$867$	3.00000	0.101885
$868$	−28.0000	−0.950382
$869$	−20.0000	−0.678454
$870$	18.0000	0.610257
$871$	−11.0000	−0.372721
$872$	4.00000	0.135457
$873$	102.000	3.45218
$874$	0	0
$875$	−48.0000	−1.62270
$876$	6.00000	0.202721
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	36.0000	1.21494
$879$	−6.00000	−0.202375
$880$	−4.00000	−0.134840
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	−54.0000	−1.81827
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	1.00000	0.0336336
$885$	−6.00000	−0.201688
$886$	0	0
$887$	−21.0000	−0.705111	−0.352555	−	0.935791i	$-0.614687\pi$
$887$	−21.0000	−0.705111	−0.352555	+	0.935791i	$0.614687\pi$
$888$	−18.0000	−0.604040
$889$	20.0000	0.670778
$890$	8.00000	0.268161
$891$	−18.0000	−0.603023
$892$	21.0000	0.703132
$893$	0	0
$894$	45.0000	1.50503
$895$	22.0000	0.735379
$896$	−4.00000	−0.133631
$897$	−3.00000	−0.100167
$898$	30.0000	1.00111
$899$	21.0000	0.700389
$900$	−6.00000	−0.200000
$901$	4.00000	0.133259
$902$	−12.0000	−0.399556
$903$	48.0000	1.59734
$904$	−6.00000	−0.199557
$905$	14.0000	0.465376
$906$	−30.0000	−0.996683
$907$	−49.0000	−1.62702	−0.813509	−	0.581552i	$-0.802446\pi$
$907$	−49.0000	−1.62702	−0.813509	+	0.581552i	$0.802446\pi$
$908$	24.0000	0.796468
$909$	−6.00000	−0.199007
$910$	−8.00000	−0.265197
$911$	−58.0000	−1.92163	−0.960813	−	0.277198i	$-0.910594\pi$
$911$	−58.0000	−1.92163	−0.960813	+	0.277198i	$0.910594\pi$
$912$	0	0
$913$	2.00000	0.0661903
$914$	−38.0000	−1.25693
$915$	−48.0000	−1.58683
$916$	13.0000	0.429532
$917$	−32.0000	−1.05673
$918$	−9.00000	−0.297044
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	2.00000	0.0659380
$921$	42.0000	1.38395
$922$	18.0000	0.592798
$923$	−14.0000	−0.460816
$924$	−24.0000	−0.789542
$925$	−6.00000	−0.197279
$926$	16.0000	0.525793
$927$	−24.0000	−0.788263
$928$	3.00000	0.0984798
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	42.0000	1.37723
$931$	0	0
$932$	10.0000	0.327561
$933$	78.0000	2.55361
$934$	21.0000	0.687141
$935$	−4.00000	−0.130814
$936$	−6.00000	−0.196116
$937$	12.0000	0.392023	0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000	0.392023	0.196011	+	0.980602i	$0.437201\pi$
$938$	44.0000	1.43665
$939$	3.00000	0.0979013
$940$	−8.00000	−0.260931
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	33.0000	1.07520
$943$	6.00000	0.195387
$944$	−1.00000	−0.0325472
$945$	72.0000	2.34216
$946$	8.00000	0.260102
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	30.0000	0.974355
$949$	2.00000	0.0649227
$950$	0	0
$951$	9.00000	0.291845
$952$	−4.00000	−0.129641
$953$	45.0000	1.45769	0.728846	−	0.684677i	$-0.240057\pi$
$953$	45.0000	1.45769	0.728846	+	0.684677i	$0.240057\pi$
$954$	−24.0000	−0.777029
$955$	−36.0000	−1.16493
$956$	8.00000	0.258738
$957$	18.0000	0.581857
$958$	34.0000	1.09849
$959$	60.0000	1.93750
$960$	6.00000	0.193649
$961$	18.0000	0.580645
$962$	−6.00000	−0.193448
$963$	−18.0000	−0.580042
$964$	22.0000	0.708572
$965$	−12.0000	−0.386294
$966$	12.0000	0.386094
$967$	−6.00000	−0.192947	−0.0964735	−	0.995336i	$-0.530756\pi$
$967$	−6.00000	−0.192947	−0.0964735	+	0.995336i	$0.530756\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−34.0000	−1.09167
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	−14.0000	−0.448589
$975$	−3.00000	−0.0960769
$976$	−8.00000	−0.256074
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	−3.00000	−0.0959294
$979$	8.00000	0.255681
$980$	18.0000	0.574989
$981$	−24.0000	−0.766261
$982$	6.00000	0.191468
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	18.0000	0.573819
$985$	30.0000	0.955879
$986$	3.00000	0.0955395
$987$	−48.0000	−1.52786
$988$	0	0
$989$	−4.00000	−0.127193
$990$	24.0000	0.762770
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	7.00000	0.222250
$993$	−84.0000	−2.66566
$994$	56.0000	1.77621
$995$	20.0000	0.634043
$996$	−3.00000	−0.0950586
$997$	55.0000	1.74187	0.870934	−	0.491400i	$-0.163515\pi$
$997$	55.0000	1.74187	0.870934	+	0.491400i	$0.163515\pi$
$998$	43.0000	1.36114
$999$	54.0000	1.70848

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2006.2.a.f.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

Downloads

Learn more