LMFDB - Embedded newform 8190.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8190,2,Mod(1,8190)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8190, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8190.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8190 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8190.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:	$65.3974792554$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 910)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8190.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{19} +1.00000 q^{20} +6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{35} -10.0000 q^{37} +2.00000 q^{38} +1.00000 q^{40} +12.0000 q^{41} -4.00000 q^{43} +6.00000 q^{46} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +12.0000 q^{53} +1.00000 q^{56} -6.00000 q^{58} -6.00000 q^{59} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -4.00000 q^{67} +1.00000 q^{70} +6.00000 q^{71} -10.0000 q^{73} -10.0000 q^{74} +2.00000 q^{76} -16.0000 q^{79} +1.00000 q^{80} +12.0000 q^{82} -4.00000 q^{86} +12.0000 q^{89} +1.00000 q^{91} +6.00000 q^{92} +2.00000 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	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	$0.688081\pi$
$30$	0	0
$31$	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
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	1.00000	0.158114
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	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
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	1.00000	0.119523
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	12.0000	1.32518
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	6.00000	0.625543
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−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
$105$	0	0
$106$	12.0000	1.16554
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	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
$115$	6.00000	0.559503
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	2.00000	0.181071
$123$	0	0
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	6.00000	0.503509
$143$	0	0
$144$	0	0
$145$	−6.00000	−0.498273
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−10.0000	−0.821995
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−16.0000	−1.27289
$159$	0	0
$160$	1.00000	0.0790569
$161$	6.00000	0.472866
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	12.0000	0.899438
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	1.00000	0.0741249
$183$	0	0
$184$	6.00000	0.442326
$185$	−10.0000	−0.735215
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	2.00000	0.145095
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	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
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	−6.00000	−0.421117
$204$	0	0
$205$	12.0000	0.838116
$206$	−4.00000	−0.278693
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	12.0000	0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	8.00000	0.543075
$218$	20.0000	1.35457
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	−6.00000	−0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	0	0
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	2.00000	0.128037
$245$	1.00000	0.0638877
$246$	0	0
$247$	2.00000	0.127257
$248$	8.00000	0.508001
$249$	0	0
$250$	1.00000	0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	1.00000	0.0620174
$261$	0	0
$262$	0	0
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	2.00000	0.122628
$267$	0	0
$268$	−4.00000	−0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−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
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	1.00000	0.0597614
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−6.00000	−0.352332
$291$	0	0
$292$	−10.0000	−0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	−10.0000	−0.581238
$297$	0	0
$298$	12.0000	0.695141
$299$	6.00000	0.346989
$300$	0	0
$301$	−4.00000	−0.230556
$302$	−22.0000	−1.26596
$303$	0	0
$304$	2.00000	0.114708
$305$	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
$309$	0	0
$310$	8.00000	0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−16.0000	−0.900070
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	6.00000	0.334367
$323$	0	0
$324$	0	0
$325$	1.00000	0.0554700
$326$	20.0000	1.10770
$327$	0	0
$328$	12.0000	0.662589
$329$	0	0
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	0	0
$333$	0	0
$334$	−24.0000	−1.31322
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	12.0000	0.635999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−10.0000	−0.525588
$363$	0	0
$364$	1.00000	0.0524142
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−10.0000	−0.519875
$371$	12.0000	0.623009
$372$	0	0
$373$	−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
$375$	0	0
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	12.0000	0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	2.00000	0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	0	0
$394$	18.0000	0.906827
$395$	−16.0000	−0.805047
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	0	0
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	−4.00000	−0.197066
$413$	−6.00000	−0.295241
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	12.0000	0.580042
$429$	0	0
$430$	−4.00000	−0.192897
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	−40.0000	−1.92228	−0.961139	−	0.276066i	$-0.910969\pi$
$433$	−40.0000	−1.92228	−0.961139	+	0.276066i	$0.910969\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	20.0000	0.957826
$437$	12.0000	0.574038
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	8.00000	0.378811
$447$	0	0
$448$	1.00000	0.0472456
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	6.00000	0.279751
$461$	−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
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	18.0000	0.833834
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	−6.00000	−0.276172
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	−18.0000	−0.823301
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	−16.0000	−0.728780
$483$	0	0
$484$	−11.0000	−0.500000
$485$	2.00000	0.0908153
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	1.00000	0.0451754
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	0	0
$494$	2.00000	0.0899843
$495$	0	0
$496$	8.00000	0.359211
$497$	6.00000	0.269137
$498$	0	0
$499$	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
$501$	0	0
$502$	−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	$0.796540\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	14.0000	0.621150
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	0	0
$518$	−10.0000	−0.439375
$519$	0	0
$520$	1.00000	0.0438529
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	0	0
$525$	0	0
$526$	6.00000	0.261612
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	12.0000	0.521247
$531$	0	0
$532$	2.00000	0.0867110
$533$	12.0000	0.519778
$534$	0	0
$535$	12.0000	0.518805
$536$	−4.00000	−0.172774
$537$	0	0
$538$	18.0000	0.776035
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−16.0000	−0.680389
$554$	8.00000	0.339887
$555$	0	0
$556$	8.00000	0.339276
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	1.00000	0.0422577
$561$	0	0
$562$	18.0000	0.759284
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	14.0000	0.588464
$567$	0	0
$568$	6.00000	0.251754
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	6.00000	0.250217
$576$	0	0
$577$	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
$579$	0	0
$580$	−6.00000	−0.249136
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−10.0000	−0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−6.00000	−0.247016
$591$	0	0
$592$	−10.0000	−0.410997
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	0	0
$598$	6.00000	0.245358
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−22.0000	−0.895167
$605$	−11.0000	−0.447214
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	2.00000	0.0809776
$611$	0	0
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	0	0
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	−16.0000	−0.639489
$627$	0	0
$628$	−10.0000	−0.399043
$629$	0	0
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	−16.0000	−0.636446
$633$	0	0
$634$	−18.0000	−0.714871
$635$	14.0000	0.555573
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	1.00000	0.0392232
$651$	0	0
$652$	20.0000	0.783260
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	0	0
$656$	12.0000	0.468521
$657$	0	0
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	32.0000	1.24372
$663$	0	0
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	−36.0000	−1.39393
$668$	−24.0000	−0.928588
$669$	0	0
$670$	−4.00000	−0.154533
$671$	0	0
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	1.00000	0.0384615
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	1.00000	0.0381802
$687$	0	0
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	38.0000	1.44559	0.722794	−	0.691063i	$-0.242858\pi$
$691$	38.0000	1.44559	0.722794	+	0.691063i	$0.242858\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	0	0
$698$	−10.0000	−0.378506
$699$	0	0
$700$	1.00000	0.0377964
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	0	0
$705$	0	0
$706$	−18.0000	−0.677439
$707$	6.00000	0.225653
$708$	0	0
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	6.00000	0.225176
$711$	0	0
$712$	12.0000	0.449719
$713$	48.0000	1.79761
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	6.00000	0.223918
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	−15.0000	−0.558242
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−6.00000	−0.222834
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	1.00000	0.0370625
$729$	0	0
$730$	−10.0000	−0.370117
$731$	0	0
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−10.0000	−0.367607
$741$	0	0
$742$	12.0000	0.440534
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	−4.00000	−0.146450
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	0	0
$754$	−6.00000	−0.218507
$755$	−22.0000	−0.800662
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	2.00000	0.0725476
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	12.0000	0.434145
$765$	0	0
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−52.0000	−1.87517	−0.937584	−	0.347759i	$-0.886943\pi$
$769$	−52.0000	−1.87517	−0.937584	+	0.347759i	$0.886943\pi$
$770$	0	0
$771$	0	0
$772$	14.0000	0.503871
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	2.00000	0.0717958
$777$	0	0
$778$	−6.00000	−0.215110
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−16.0000	−0.569254
$791$	6.00000	0.213335
$792$	0	0
$793$	2.00000	0.0710221
$794$	26.0000	0.922705
$795$	0	0
$796$	8.00000	0.283552
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	6.00000	0.211867
$803$	0	0
$804$	0	0
$805$	6.00000	0.211472
$806$	8.00000	0.281788
$807$	0	0
$808$	6.00000	0.211079
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	50.0000	1.75574	0.877869	−	0.478901i	$-0.158965\pi$
$811$	50.0000	1.75574	0.877869	+	0.478901i	$0.158965\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	−8.00000	−0.279885
$818$	−4.00000	−0.139857
$819$	0	0
$820$	12.0000	0.419058
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−6.00000	−0.208767
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	0	0
$838$	24.0000	0.829066
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−28.0000	−0.964944
$843$	0	0
$844$	−4.00000	−0.137686
$845$	1.00000	0.0344010
$846$	0	0
$847$	−11.0000	−0.377964
$848$	12.0000	0.412082
$849$	0	0
$850$	0	0
$851$	−60.0000	−2.05677
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	12.0000	0.410152
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	18.0000	0.613082
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	−40.0000	−1.35926
$867$	0	0
$868$	8.00000	0.271538
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	20.0000	0.677285
$873$	0	0
$874$	12.0000	0.405906
$875$	1.00000	0.0338062
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−16.0000	−0.539974
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−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
$885$	0	0
$886$	36.0000	1.20944
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	12.0000	0.402241
$891$	0	0
$892$	8.00000	0.267860
$893$	0	0
$894$	0	0
$895$	−12.0000	−0.401116
$896$	1.00000	0.0334077
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−48.0000	−1.60089
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	0	0
$909$	0	0
$910$	1.00000	0.0331497
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	0	0
$914$	−22.0000	−0.727695
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	6.00000	0.197814
$921$	0	0
$922$	−6.00000	−0.197599
$923$	6.00000	0.197492
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−16.0000	−0.525793
$927$	0	0
$928$	−6.00000	−0.196960
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	18.0000	0.589610
$933$	0	0
$934$	−6.00000	−0.196326
$935$	0	0
$936$	0	0
$937$	−4.00000	−0.130674	−0.0653372	−	0.997863i	$-0.520812\pi$
$937$	−4.00000	−0.130674	−0.0653372	+	0.997863i	$0.520812\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	72.0000	2.34464
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	2.00000	0.0648886
$951$	0	0
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	−18.0000	−0.582162
$957$	0	0
$958$	36.0000	1.16311
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	−10.0000	−0.322413
$963$	0	0
$964$	−16.0000	−0.515325
$965$	14.0000	0.450676
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	2.00000	0.0642161
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	8.00000	0.256337
$975$	0	0
$976$	2.00000	0.0640184
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	1.00000	0.0319438
$981$	0	0
$982$	12.0000	0.382935
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	2.00000	0.0636285
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	6.00000	0.190308
$995$	8.00000	0.253617
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8190.2.a.bw.1.1		1
3.2	odd	2		910.2.a.a.1.1	✓	1
12.11	even	2		7280.2.a.t.1.1		1
15.14	odd	2		4550.2.a.z.1.1		1
21.20	even	2		6370.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
910.2.a.a.1.1	✓	1	3.2	odd	2
4550.2.a.z.1.1		1	15.14	odd	2
6370.2.a.j.1.1		1	21.20	even	2
7280.2.a.t.1.1		1	12.11	even	2
8190.2.a.bw.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:	\( N \)	\(=\)	\( 8190 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8190.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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