LMFDB - Embedded newform 8470.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
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	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +5.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{18} +5.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +9.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -5.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +1.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -5.00000 q^{38} +5.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} -10.0000 q^{43} -2.00000 q^{45} -9.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +5.00000 q^{52} +6.00000 q^{53} +5.00000 q^{54} -1.00000 q^{56} +5.00000 q^{57} +6.00000 q^{58} -3.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} -10.0000 q^{67} +9.00000 q^{69} -1.00000 q^{70} +12.0000 q^{71} +2.00000 q^{72} -10.0000 q^{73} -8.00000 q^{74} +1.00000 q^{75} +5.00000 q^{76} -5.00000 q^{78} -13.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -3.00000 q^{83} +1.00000 q^{84} +10.0000 q^{86} -6.00000 q^{87} +12.0000 q^{89} +2.00000 q^{90} +5.00000 q^{91} +9.00000 q^{92} +8.00000 q^{93} +5.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.00000	0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	2.00000	0.471405
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	1.00000	0.223607
$21$	1.00000	0.218218
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−5.00000	−0.980581
$27$	−5.00000	−0.962250
$28$	1.00000	0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	−2.00000	−0.333333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−5.00000	−0.811107
$39$	5.00000	0.800641
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−1.00000	−0.154303
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−9.00000	−1.32698
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	5.00000	0.693375
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−1.00000	−0.133631
$57$	5.00000	0.662266
$58$	6.00000	0.787839
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	1.00000	0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−8.00000	−1.01600
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	5.00000	0.620174
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	9.00000	1.08347
$70$	−1.00000	−0.119523
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	2.00000	0.235702
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−8.00000	−0.929981
$75$	1.00000	0.115470
$76$	5.00000	0.573539
$77$	0	0
$78$	−5.00000	−0.566139
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	10.0000	1.07833
$87$	−6.00000	−0.643268
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	2.00000	0.210819
$91$	5.00000	0.524142
$92$	9.00000	0.938315
$93$	8.00000	0.829561
$94$	0	0
$95$	5.00000	0.512989
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−5.00000	−0.490290
$105$	1.00000	0.0975900
$106$	−6.00000	−0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−5.00000	−0.481125
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	1.00000	0.0944911
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	−5.00000	−0.468293
$115$	9.00000	0.839254
$116$	−6.00000	−0.557086
$117$	−10.0000	−0.924500
$118$	3.00000	0.276172
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	0	0
$122$	−2.00000	−0.181071
$123$	6.00000	0.541002
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	−19.0000	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	−5.00000	−0.438529
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	10.0000	0.863868
$135$	−5.00000	−0.430331
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−9.00000	−0.766131
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	−2.00000	−0.166667
$145$	−6.00000	−0.498273
$146$	10.0000	0.827606
$147$	1.00000	0.0824786
$148$	8.00000	0.657596
$149$	−24.0000	−1.96616	−0.983078	−	0.183186i	$-0.941359\pi$
$149$	−24.0000	−1.96616	−0.983078	+	0.183186i	$0.941359\pi$
$150$	−1.00000	−0.0816497
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	5.00000	0.400320
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	13.0000	1.03422
$159$	6.00000	0.475831
$160$	−1.00000	−0.0790569
$161$	9.00000	0.709299
$162$	−1.00000	−0.0785674
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	3.00000	0.232845
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	−1.00000	−0.0771517
$169$	12.0000	0.923077
$170$	0	0
$171$	−10.0000	−0.764719
$172$	−10.0000	−0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	6.00000	0.454859
$175$	1.00000	0.0755929
$176$	0	0
$177$	−3.00000	−0.225494
$178$	−12.0000	−0.899438
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	−2.00000	−0.149071
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	−5.00000	−0.370625
$183$	2.00000	0.147844
$184$	−9.00000	−0.663489
$185$	8.00000	0.588172
$186$	−8.00000	−0.586588
$187$	0	0
$188$	0	0
$189$	−5.00000	−0.363696
$190$	−5.00000	−0.362738
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000	0.0721688
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	10.0000	0.717958
$195$	5.00000	0.358057
$196$	1.00000	0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−1.00000	−0.0707107
$201$	−10.0000	−0.705346
$202$	3.00000	0.211079
$203$	−6.00000	−0.421117
$204$	0	0
$205$	6.00000	0.419058
$206$	16.0000	1.11477
$207$	−18.0000	−1.25109
$208$	5.00000	0.346688
$209$	0	0
$210$	−1.00000	−0.0690066
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	6.00000	0.412082
$213$	12.0000	0.822226
$214$	−12.0000	−0.820303
$215$	−10.0000	−0.681994
$216$	5.00000	0.340207
$217$	8.00000	0.543075
$218$	16.0000	1.08366
$219$	−10.0000	−0.675737
$220$	0	0
$221$	0	0
$222$	−8.00000	−0.536925
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	−1.00000	−0.0668153
$225$	−2.00000	−0.133333
$226$	3.00000	0.199557
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	5.00000	0.331133
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−9.00000	−0.593442
$231$	0	0
$232$	6.00000	0.393919
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	10.0000	0.653720
$235$	0	0
$236$	−3.00000	−0.195283
$237$	−13.0000	−0.844441
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	1.00000	0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	2.00000	0.128037
$245$	1.00000	0.0638877
$246$	−6.00000	−0.382546
$247$	25.0000	1.59071
$248$	−8.00000	−0.508001
$249$	−3.00000	−0.190117
$250$	−1.00000	−0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	19.0000	1.19217
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	10.0000	0.622573
$259$	8.00000	0.497096
$260$	5.00000	0.310087
$261$	12.0000	0.742781
$262$	−9.00000	−0.556022
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−5.00000	−0.306570
$267$	12.0000	0.734388
$268$	−10.0000	−0.610847
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	5.00000	0.304290
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	5.00000	0.302614
$274$	−9.00000	−0.543710
$275$	0	0
$276$	9.00000	0.541736
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−5.00000	−0.299880
$279$	−16.0000	−0.957895
$280$	−1.00000	−0.0597614
$281$	33.0000	1.96861	0.984307	−	0.176462i	$-0.0564652\pi$
$281$	33.0000	1.96861	0.984307	+	0.176462i	$0.0564652\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	12.0000	0.712069
$285$	5.00000	0.296174
$286$	0	0
$287$	6.00000	0.354169
$288$	2.00000	0.117851
$289$	−17.0000	−1.00000
$290$	6.00000	0.352332
$291$	−10.0000	−0.586210
$292$	−10.0000	−0.585206
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	−1.00000	−0.0583212
$295$	−3.00000	−0.174667
$296$	−8.00000	−0.464991
$297$	0	0
$298$	24.0000	1.39028
$299$	45.0000	2.60242
$300$	1.00000	0.0577350
$301$	−10.0000	−0.576390
$302$	−17.0000	−0.978240
$303$	−3.00000	−0.172345
$304$	5.00000	0.286770
$305$	2.00000	0.114520
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	−8.00000	−0.454369
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	−5.00000	−0.283069
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−17.0000	−0.959366
$315$	−2.00000	−0.112687
$316$	−13.0000	−0.731307
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	1.00000	0.0559017
$321$	12.0000	0.669775
$322$	−9.00000	−0.501550
$323$	0	0
$324$	1.00000	0.0555556
$325$	5.00000	0.277350
$326$	−14.0000	−0.775388
$327$	−16.0000	−0.884802
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	−3.00000	−0.164646
$333$	−16.0000	−0.876795
$334$	6.00000	0.328305
$335$	−10.0000	−0.546358
$336$	1.00000	0.0545545
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	−12.0000	−0.652714
$339$	−3.00000	−0.162938
$340$	0	0
$341$	0	0
$342$	10.0000	0.540738
$343$	1.00000	0.0539949
$344$	10.0000	0.539164
$345$	9.00000	0.484544
$346$	−6.00000	−0.322562
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	−6.00000	−0.321634
$349$	−13.0000	−0.695874	−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000	−0.695874	−0.347937	+	0.937518i	$0.613118\pi$
$350$	−1.00000	−0.0534522
$351$	−25.0000	−1.33440
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	3.00000	0.159448
$355$	12.0000	0.636894
$356$	12.0000	0.635999
$357$	0	0
$358$	−18.0000	−0.951330
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	2.00000	0.105409
$361$	6.00000	0.315789
$362$	13.0000	0.683265
$363$	0	0
$364$	5.00000	0.262071
$365$	−10.0000	−0.523424
$366$	−2.00000	−0.104542
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	9.00000	0.469157
$369$	−12.0000	−0.624695
$370$	−8.00000	−0.415900
$371$	6.00000	0.311504
$372$	8.00000	0.414781
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−30.0000	−1.54508
$378$	5.00000	0.257172
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	5.00000	0.256495
$381$	−19.0000	−0.973399
$382$	−3.00000	−0.153493
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	1.00000	0.0508987
$387$	20.0000	1.01666
$388$	−10.0000	−0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−5.00000	−0.253185
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	9.00000	0.453990
$394$	−18.0000	−0.906827
$395$	−13.0000	−0.654101
$396$	0	0
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	−2.00000	−0.100251
$399$	5.00000	0.250313
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	10.0000	0.498755
$403$	40.0000	1.99254
$404$	−3.00000	−0.149256
$405$	1.00000	0.0496904
$406$	6.00000	0.297775
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	−6.00000	−0.296319
$411$	9.00000	0.443937
$412$	−16.0000	−0.788263
$413$	−3.00000	−0.147620
$414$	18.0000	0.884652
$415$	−3.00000	−0.147264
$416$	−5.00000	−0.245145
$417$	5.00000	0.244851
$418$	0	0
$419$	−27.0000	−1.31904	−0.659518	−	0.751689i	$-0.729240\pi$
$419$	−27.0000	−1.31904	−0.659518	+	0.751689i	$0.729240\pi$
$420$	1.00000	0.0487950
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−14.0000	−0.681509
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−12.0000	−0.581402
$427$	2.00000	0.0967868
$428$	12.0000	0.580042
$429$	0	0
$430$	10.0000	0.482243
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	−5.00000	−0.240563
$433$	32.0000	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$434$	−8.00000	−0.384012
$435$	−6.00000	−0.287678
$436$	−16.0000	−0.766261
$437$	45.0000	2.15264
$438$	10.0000	0.477818
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	8.00000	0.379663
$445$	12.0000	0.568855
$446$	−2.00000	−0.0947027
$447$	−24.0000	−1.13516
$448$	1.00000	0.0472456
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	2.00000	0.0942809
$451$	0	0
$452$	−3.00000	−0.141108
$453$	17.0000	0.798730
$454$	−24.0000	−1.12638
$455$	5.00000	0.234404
$456$	−5.00000	−0.234146
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	9.00000	0.419627
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	−6.00000	−0.278543
$465$	8.00000	0.370991
$466$	−3.00000	−0.138972
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	−10.0000	−0.462250
$469$	−10.0000	−0.461757
$470$	0	0
$471$	17.0000	0.783319
$472$	3.00000	0.138086
$473$	0	0
$474$	13.0000	0.597110
$475$	5.00000	0.229416
$476$	0	0
$477$	−12.0000	−0.549442
$478$	−21.0000	−0.960518
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	−1.00000	−0.0456435
$481$	40.0000	1.82384
$482$	10.0000	0.455488
$483$	9.00000	0.409514
$484$	0	0
$485$	−10.0000	−0.454077
$486$	−16.0000	−0.725775
$487$	−25.0000	−1.13286	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−25.0000	−1.13286	−0.566429	+	0.824110i	$0.691675\pi$
$488$	−2.00000	−0.0905357
$489$	14.0000	0.633102
$490$	−1.00000	−0.0451754
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	−25.0000	−1.12480
$495$	0	0
$496$	8.00000	0.359211
$497$	12.0000	0.538274
$498$	3.00000	0.134433
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	1.00000	0.0447214
$501$	−6.00000	−0.268060
$502$	−12.0000	−0.535586
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	2.00000	0.0890871
$505$	−3.00000	−0.133498
$506$	0	0
$507$	12.0000	0.532939
$508$	−19.0000	−0.842989
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	−1.00000	−0.0441942
$513$	−25.0000	−1.10378
$514$	6.00000	0.264649
$515$	−16.0000	−0.705044
$516$	−10.0000	−0.440225
$517$	0	0
$518$	−8.00000	−0.351500
$519$	6.00000	0.263371
$520$	−5.00000	−0.219265
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−12.0000	−0.525226
$523$	−31.0000	−1.35554	−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000	−1.35554	−0.677768	+	0.735276i	$0.737052\pi$
$524$	9.00000	0.393167
$525$	1.00000	0.0436436
$526$	−3.00000	−0.130806
$527$	0	0
$528$	0	0
$529$	58.0000	2.52174
$530$	−6.00000	−0.260623
$531$	6.00000	0.260378
$532$	5.00000	0.216777
$533$	30.0000	1.29944
$534$	−12.0000	−0.519291
$535$	12.0000	0.518805
$536$	10.0000	0.431934
$537$	18.0000	0.776757
$538$	−3.00000	−0.129339
$539$	0	0
$540$	−5.00000	−0.215166
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	16.0000	0.687259
$543$	−13.0000	−0.557883
$544$	0	0
$545$	−16.0000	−0.685365
$546$	−5.00000	−0.213980
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	9.00000	0.384461
$549$	−4.00000	−0.170716
$550$	0	0
$551$	−30.0000	−1.27804
$552$	−9.00000	−0.383065
$553$	−13.0000	−0.552816
$554$	−2.00000	−0.0849719
$555$	8.00000	0.339581
$556$	5.00000	0.212047
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	16.0000	0.677334
$559$	−50.0000	−2.11477
$560$	1.00000	0.0422577
$561$	0	0
$562$	−33.0000	−1.39202
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	−3.00000	−0.126211
$566$	−17.0000	−0.714563
$567$	1.00000	0.0419961
$568$	−12.0000	−0.503509
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	−5.00000	−0.209427
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	3.00000	0.125327
$574$	−6.00000	−0.250435
$575$	9.00000	0.375326
$576$	−2.00000	−0.0833333
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	17.0000	0.707107
$579$	−1.00000	−0.0415586
$580$	−6.00000	−0.249136
$581$	−3.00000	−0.124461
$582$	10.0000	0.414513
$583$	0	0
$584$	10.0000	0.413803
$585$	−10.0000	−0.413449
$586$	9.00000	0.371787
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	1.00000	0.0412393
$589$	40.0000	1.64817
$590$	3.00000	0.123508
$591$	18.0000	0.740421
$592$	8.00000	0.328798
$593$	−48.0000	−1.97112	−0.985562	−	0.169316i	$-0.945844\pi$
$593$	−48.0000	−1.97112	−0.985562	+	0.169316i	$0.945844\pi$
$594$	0	0
$595$	0	0
$596$	−24.0000	−0.983078
$597$	2.00000	0.0818546
$598$	−45.0000	−1.84019
$599$	−21.0000	−0.858037	−0.429018	−	0.903296i	$-0.641140\pi$
$599$	−21.0000	−0.858037	−0.429018	+	0.903296i	$0.641140\pi$
$600$	−1.00000	−0.0408248
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	10.0000	0.407570
$603$	20.0000	0.814463
$604$	17.0000	0.691720
$605$	0	0
$606$	3.00000	0.121867
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−5.00000	−0.202777
$609$	−6.00000	−0.243132
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	4.00000	0.161427
$615$	6.00000	0.241943
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	16.0000	0.643614
$619$	−1.00000	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$620$	8.00000	0.321288
$621$	−45.0000	−1.80579
$622$	−24.0000	−0.962312
$623$	12.0000	0.480770
$624$	5.00000	0.200160
$625$	1.00000	0.0400000
$626$	−26.0000	−1.03917
$627$	0	0
$628$	17.0000	0.678374
$629$	0	0
$630$	2.00000	0.0796819
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	13.0000	0.517112
$633$	14.0000	0.556450
$634$	0	0
$635$	−19.0000	−0.753992
$636$	6.00000	0.237915
$637$	5.00000	0.198107
$638$	0	0
$639$	−24.0000	−0.949425
$640$	−1.00000	−0.0395285
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	−12.0000	−0.473602
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	9.00000	0.354650
$645$	−10.0000	−0.393750
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−5.00000	−0.196116
$651$	8.00000	0.313545
$652$	14.0000	0.548282
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	16.0000	0.625650
$655$	9.00000	0.351659
$656$	6.00000	0.234261
$657$	20.0000	0.780274
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	22.0000	0.855054
$663$	0	0
$664$	3.00000	0.116423
$665$	5.00000	0.193892
$666$	16.0000	0.619987
$667$	−54.0000	−2.09089
$668$	−6.00000	−0.232147
$669$	2.00000	0.0773245
$670$	10.0000	0.386334
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	41.0000	1.58043	0.790217	−	0.612827i	$-0.209968\pi$
$673$	41.0000	1.58043	0.790217	+	0.612827i	$0.209968\pi$
$674$	−23.0000	−0.885927
$675$	−5.00000	−0.192450
$676$	12.0000	0.461538
$677$	45.0000	1.72949	0.864745	−	0.502211i	$-0.167480\pi$
$677$	45.0000	1.72949	0.864745	+	0.502211i	$0.167480\pi$
$678$	3.00000	0.115214
$679$	−10.0000	−0.383765
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	−10.0000	−0.382360
$685$	9.00000	0.343872
$686$	−1.00000	−0.0381802
$687$	−22.0000	−0.839352
$688$	−10.0000	−0.381246
$689$	30.0000	1.14291
$690$	−9.00000	−0.342624
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−18.0000	−0.683271
$695$	5.00000	0.189661
$696$	6.00000	0.227429
$697$	0	0
$698$	13.0000	0.492057
$699$	3.00000	0.113470
$700$	1.00000	0.0377964
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	25.0000	0.943564
$703$	40.0000	1.50863
$704$	0	0
$705$	0	0
$706$	30.0000	1.12906
$707$	−3.00000	−0.112827
$708$	−3.00000	−0.112747
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	−12.0000	−0.450352
$711$	26.0000	0.975076
$712$	−12.0000	−0.449719
$713$	72.0000	2.69642
$714$	0	0
$715$	0	0
$716$	18.0000	0.672692
$717$	21.0000	0.784259
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−2.00000	−0.0745356
$721$	−16.0000	−0.595871
$722$	−6.00000	−0.223297
$723$	−10.0000	−0.371904
$724$	−13.0000	−0.483141
$725$	−6.00000	−0.222834
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	−5.00000	−0.185312
$729$	13.0000	0.481481
$730$	10.0000	0.370117
$731$	0	0
$732$	2.00000	0.0739221
$733$	23.0000	0.849524	0.424762	−	0.905305i	$-0.360358\pi$
$733$	23.0000	0.849524	0.424762	+	0.905305i	$0.360358\pi$
$734$	−26.0000	−0.959678
$735$	1.00000	0.0368856
$736$	−9.00000	−0.331744
$737$	0	0
$738$	12.0000	0.441726
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	8.00000	0.294086
$741$	25.0000	0.918398
$742$	−6.00000	−0.220267
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−8.00000	−0.293294
$745$	−24.0000	−0.879292
$746$	4.00000	0.146450
$747$	6.00000	0.219529
$748$	0	0
$749$	12.0000	0.438470
$750$	−1.00000	−0.0365148
$751$	11.0000	0.401396	0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000	0.401396	0.200698	+	0.979653i	$0.435679\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	30.0000	1.09254
$755$	17.0000	0.618693
$756$	−5.00000	−0.181848
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	−5.00000	−0.181369
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	19.0000	0.688297
$763$	−16.0000	−0.579239
$764$	3.00000	0.108536
$765$	0	0
$766$	24.0000	0.867155
$767$	−15.0000	−0.541619
$768$	1.00000	0.0360844
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	−1.00000	−0.0359908
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	−20.0000	−0.718885
$775$	8.00000	0.287368
$776$	10.0000	0.358979
$777$	8.00000	0.286998
$778$	−6.00000	−0.215110
$779$	30.0000	1.07486
$780$	5.00000	0.179029
$781$	0	0
$782$	0	0
$783$	30.0000	1.07211
$784$	1.00000	0.0357143
$785$	17.0000	0.606756
$786$	−9.00000	−0.321019
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	18.0000	0.641223
$789$	3.00000	0.106803
$790$	13.0000	0.462519
$791$	−3.00000	−0.106668
$792$	0	0
$793$	10.0000	0.355110
$794$	−38.0000	−1.34857
$795$	6.00000	0.212798
$796$	2.00000	0.0708881
$797$	9.00000	0.318796	0.159398	−	0.987214i	$-0.449045\pi$
$797$	9.00000	0.318796	0.159398	+	0.987214i	$0.449045\pi$
$798$	−5.00000	−0.176998
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−24.0000	−0.847998
$802$	18.0000	0.635602
$803$	0	0
$804$	−10.0000	−0.352673
$805$	9.00000	0.317208
$806$	−40.0000	−1.40894
$807$	3.00000	0.105605
$808$	3.00000	0.105540
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−1.00000	−0.0351364
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	−6.00000	−0.210559
$813$	−16.0000	−0.561144
$814$	0	0
$815$	14.0000	0.490399
$816$	0	0
$817$	−50.0000	−1.74928
$818$	−14.0000	−0.489499
$819$	−10.0000	−0.349428
$820$	6.00000	0.209529
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−9.00000	−0.313911
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	3.00000	0.104383
$827$	−42.0000	−1.46048	−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000	−1.46048	−0.730242	+	0.683189i	$0.760592\pi$
$828$	−18.0000	−0.625543
$829$	−55.0000	−1.91023	−0.955114	−	0.296237i	$-0.904268\pi$
$829$	−55.0000	−1.91023	−0.955114	+	0.296237i	$0.904268\pi$
$830$	3.00000	0.104132
$831$	2.00000	0.0693792
$832$	5.00000	0.173344
$833$	0	0
$834$	−5.00000	−0.173136
$835$	−6.00000	−0.207639
$836$	0	0
$837$	−40.0000	−1.38260
$838$	27.0000	0.932700
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	−1.00000	−0.0345033
$841$	7.00000	0.241379
$842$	−8.00000	−0.275698
$843$	33.0000	1.13658
$844$	14.0000	0.481900
$845$	12.0000	0.412813
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	17.0000	0.583438
$850$	0	0
$851$	72.0000	2.46813
$852$	12.0000	0.411113
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	−2.00000	−0.0684386
$855$	−10.0000	−0.341993
$856$	−12.0000	−0.410152
$857$	−36.0000	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$857$	−36.0000	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$858$	0	0
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	−10.0000	−0.340997
$861$	6.00000	0.204479
$862$	3.00000	0.102180
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	5.00000	0.170103
$865$	6.00000	0.204006
$866$	−32.0000	−1.08740
$867$	−17.0000	−0.577350
$868$	8.00000	0.271538
$869$	0	0
$870$	6.00000	0.203419
$871$	−50.0000	−1.69419
$872$	16.0000	0.541828
$873$	20.0000	0.676897
$874$	−45.0000	−1.52215
$875$	1.00000	0.0338062
$876$	−10.0000	−0.337869
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	28.0000	0.944954
$879$	−9.00000	−0.303562
$880$	0	0
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	2.00000	0.0673435
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	−3.00000	−0.100844
$886$	12.0000	0.403148
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	−8.00000	−0.268462
$889$	−19.0000	−0.637240
$890$	−12.0000	−0.402241
$891$	0	0
$892$	2.00000	0.0669650
$893$	0	0
$894$	24.0000	0.802680
$895$	18.0000	0.601674
$896$	−1.00000	−0.0334077
$897$	45.0000	1.50251
$898$	−9.00000	−0.300334
$899$	−48.0000	−1.60089
$900$	−2.00000	−0.0666667
$901$	0	0
$902$	0	0
$903$	−10.0000	−0.332779
$904$	3.00000	0.0997785
$905$	−13.0000	−0.432135
$906$	−17.0000	−0.564787
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	24.0000	0.796468
$909$	6.00000	0.199007
$910$	−5.00000	−0.165748
$911$	45.0000	1.49092	0.745458	−	0.666552i	$-0.232231\pi$
$911$	45.0000	1.49092	0.745458	+	0.666552i	$0.232231\pi$
$912$	5.00000	0.165567
$913$	0	0
$914$	37.0000	1.22385
$915$	2.00000	0.0661180
$916$	−22.0000	−0.726900
$917$	9.00000	0.297206
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−9.00000	−0.296721
$921$	−4.00000	−0.131804
$922$	6.00000	0.197599
$923$	60.0000	1.97492
$924$	0	0
$925$	8.00000	0.263038
$926$	−5.00000	−0.164310
$927$	32.0000	1.05102
$928$	6.00000	0.196960
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	−8.00000	−0.262330
$931$	5.00000	0.163868
$932$	3.00000	0.0982683
$933$	24.0000	0.785725
$934$	27.0000	0.883467
$935$	0	0
$936$	10.0000	0.326860
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	10.0000	0.326512
$939$	26.0000	0.848478
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	−17.0000	−0.553890
$943$	54.0000	1.75848
$944$	−3.00000	−0.0976417
$945$	−5.00000	−0.162650
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	−13.0000	−0.422220
$949$	−50.0000	−1.62307
$950$	−5.00000	−0.162221
$951$	0	0
$952$	0	0
$953$	39.0000	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$954$	12.0000	0.388514
$955$	3.00000	0.0970777
$956$	21.0000	0.679189
$957$	0	0
$958$	30.0000	0.969256
$959$	9.00000	0.290625
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	−40.0000	−1.28965
$963$	−24.0000	−0.773389
$964$	−10.0000	−0.322078
$965$	−1.00000	−0.0321911
$966$	−9.00000	−0.289570
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	0	0
$969$	0	0
$970$	10.0000	0.321081
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	16.0000	0.513200
$973$	5.00000	0.160293
$974$	25.0000	0.801052
$975$	5.00000	0.160128
$976$	2.00000	0.0640184
$977$	−51.0000	−1.63163	−0.815817	−	0.578310i	$-0.803713\pi$
$977$	−51.0000	−1.63163	−0.815817	+	0.578310i	$0.803713\pi$
$978$	−14.0000	−0.447671
$979$	0	0
$980$	1.00000	0.0319438
$981$	32.0000	1.02168
$982$	−18.0000	−0.574403
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	−6.00000	−0.191273
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	25.0000	0.795356
$989$	−90.0000	−2.86183
$990$	0	0
$991$	11.0000	0.349427	0.174713	−	0.984619i	$-0.444100\pi$
$991$	11.0000	0.349427	0.174713	+	0.984619i	$0.444100\pi$
$992$	−8.00000	−0.254000
$993$	−22.0000	−0.698149
$994$	−12.0000	−0.380617
$995$	2.00000	0.0634043
$996$	−3.00000	−0.0950586
$997$	−49.0000	−1.55185	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	−49.0000	−1.55185	−0.775923	+	0.630828i	$0.782715\pi$
$998$	−32.0000	−1.01294
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.o.1.1	✓	1
11.10	odd	2		8470.2.a.be.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.o.1.1	✓	1	1.1	even	1	trivial
8470.2.a.be.1.1	yes	1	11.10	odd	2

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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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