LMFDB - Embedded newform 187.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [187,2,Mod(1,187)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(187, 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("187.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 187 = 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	187.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:	$1.49320251780$
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	$\chi$	$=$	187.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} +3.00000 q^{15} +4.00000 q^{16} -1.00000 q^{17} +2.00000 q^{19} -6.00000 q^{20} +2.00000 q^{21} -3.00000 q^{23} +4.00000 q^{25} -5.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} -7.00000 q^{31} +1.00000 q^{33} +6.00000 q^{35} +4.00000 q^{36} -7.00000 q^{37} +2.00000 q^{39} +12.0000 q^{41} -10.0000 q^{43} -2.00000 q^{44} -6.00000 q^{45} +4.00000 q^{48} -3.00000 q^{49} -1.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} +3.00000 q^{55} +2.00000 q^{57} -3.00000 q^{59} -6.00000 q^{60} +8.00000 q^{61} -4.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} -7.00000 q^{67} +2.00000 q^{68} -3.00000 q^{69} -9.00000 q^{71} +2.00000 q^{73} +4.00000 q^{75} -4.00000 q^{76} +2.00000 q^{77} +8.00000 q^{79} +12.0000 q^{80} +1.00000 q^{81} +6.00000 q^{83} -4.00000 q^{84} -3.00000 q^{85} -6.00000 q^{87} +15.0000 q^{89} +4.00000 q^{91} +6.00000 q^{92} -7.00000 q^{93} +6.00000 q^{95} +11.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−2.00000	−1.00000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	4.00000	1.00000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−6.00000	−1.34164
$21$	2.00000	0.436436
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	−4.00000	−0.755929
$29$	−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
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	6.00000	1.01419
$36$	4.00000	0.666667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	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
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−2.00000	−0.301511
$45$	−6.00000	−0.894427
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	4.00000	0.577350
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−6.00000	−0.774597
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	−8.00000	−1.00000
$65$	6.00000	0.744208
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	2.00000	0.242536
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	−4.00000	−0.458831
$77$	2.00000	0.227921
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	12.0000	1.34164
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−4.00000	−0.436436
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	6.00000	0.625543
$93$	−7.00000	−0.725866
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	11.0000	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	−8.00000	−0.800000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	10.0000	0.962250
$109$	−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
$111$	−7.00000	−0.664411
$112$	8.00000	0.755929
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	12.0000	1.11417
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.0000	1.08200
$124$	14.0000	1.25724
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−2.00000	−0.174078
$133$	4.00000	0.346844
$134$	0	0
$135$	−15.0000	−1.29099
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−12.0000	−1.01419
$141$	0	0
$142$	0	0
$143$	2.00000	0.167248
$144$	−8.00000	−0.666667
$145$	−18.0000	−1.49482
$146$	0	0
$147$	−3.00000	−0.247436
$148$	14.0000	1.15079
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−21.0000	−1.68676
$156$	−4.00000	−0.320256
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−24.0000	−1.87409
$165$	3.00000	0.233550
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	20.0000	1.52499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	4.00000	0.301511
$177$	−3.00000	−0.225494
$178$	0	0
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	12.0000	0.894427
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	−21.0000	−1.54395
$186$	0	0
$187$	−1.00000	−0.0731272
$188$	0	0
$189$	−10.0000	−0.727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−8.00000	−0.577350
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	6.00000	0.428571
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	−12.0000	−0.842235
$204$	2.00000	0.140028
$205$	36.0000	2.51435
$206$	0	0
$207$	6.00000	0.417029
$208$	8.00000	0.554700
$209$	2.00000	0.138343
$210$	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
$213$	−9.00000	−0.616670
$214$	0	0
$215$	−30.0000	−2.04598
$216$	0	0
$217$	−14.0000	−0.950382
$218$	0	0
$219$	2.00000	0.135147
$220$	−6.00000	−0.404520
$221$	−2.00000	−0.134535
$222$	0	0
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−4.00000	−0.264906
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	8.00000	0.519656
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	12.0000	0.774597
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	−16.0000	−1.02430
$245$	−9.00000	−0.574989
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	8.00000	0.503953
$253$	−3.00000	−0.188608
$254$	0	0
$255$	−3.00000	−0.187867
$256$	16.0000	1.00000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−14.0000	−0.869918
$260$	−12.0000	−0.744208
$261$	12.0000	0.742781
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	15.0000	0.917985
$268$	14.0000	0.855186
$269$	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
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−4.00000	−0.242536
$273$	4.00000	0.242091
$274$	0	0
$275$	4.00000	0.241209
$276$	6.00000	0.361158
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	14.0000	0.838158
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	18.0000	1.06810
$285$	6.00000	0.355409
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	11.0000	0.644831
$292$	−4.00000	−0.234082
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	−6.00000	−0.346989
$300$	−8.00000	−0.461880
$301$	−20.0000	−1.15278
$302$	0	0
$303$	12.0000	0.689382
$304$	8.00000	0.458831
$305$	24.0000	1.37424
$306$	0	0
$307$	−34.0000	−1.94048	−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000	−1.94048	−0.970241	+	0.242140i	$0.922151\pi$
$308$	−4.00000	−0.227921
$309$	8.00000	0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	−16.0000	−0.900070
$317$	27.0000	1.51647	0.758236	−	0.651981i	$-0.226062\pi$
$317$	27.0000	1.51647	0.758236	+	0.651981i	$0.226062\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	−24.0000	−1.34164
$321$	12.0000	0.669775
$322$	0	0
$323$	−2.00000	−0.111283
$324$	−2.00000	−0.111111
$325$	8.00000	0.443760
$326$	0	0
$327$	−16.0000	−0.884802
$328$	0	0
$329$	0	0
$330$	0	0
$331$	23.0000	1.26419	0.632097	−	0.774889i	$-0.282194\pi$
$331$	23.0000	1.26419	0.632097	+	0.774889i	$0.282194\pi$
$332$	−12.0000	−0.658586
$333$	14.0000	0.767195
$334$	0	0
$335$	−21.0000	−1.14735
$336$	8.00000	0.436436
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−9.00000	−0.488813
$340$	6.00000	0.325396
$341$	−7.00000	−0.379071
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−9.00000	−0.484544
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	12.0000	0.643268
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	0	0
$353$	27.0000	1.43706	0.718532	−	0.695493i	$-0.244814\pi$
$353$	27.0000	1.43706	0.718532	+	0.695493i	$0.244814\pi$
$354$	0	0
$355$	−27.0000	−1.43301
$356$	−30.0000	−1.59000
$357$	−2.00000	−0.105851
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	1.00000	0.0524864
$364$	−8.00000	−0.419314
$365$	6.00000	0.314054
$366$	0	0
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	−12.0000	−0.625543
$369$	−24.0000	−1.24939
$370$	0	0
$371$	12.0000	0.623009
$372$	14.0000	0.725866
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	−12.0000	−0.615587
$381$	−4.00000	−0.204926
$382$	0	0
$383$	27.0000	1.37964	0.689818	−	0.723983i	$-0.257691\pi$
$383$	27.0000	1.37964	0.689818	+	0.723983i	$0.257691\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	20.0000	1.01666
$388$	−22.0000	−1.11688
$389$	33.0000	1.67317	0.836583	−	0.547840i	$-0.184550\pi$
$389$	33.0000	1.67317	0.836583	+	0.547840i	$0.184550\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	24.0000	1.20757
$396$	4.00000	0.201008
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	16.0000	0.800000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	−14.0000	−0.697390
$404$	−24.0000	−1.19404
$405$	3.00000	0.149071
$406$	0	0
$407$	−7.00000	−0.346977
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	−16.0000	−0.788263
$413$	−6.00000	−0.295241
$414$	0	0
$415$	18.0000	0.883585
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	−12.0000	−0.585540
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	16.0000	0.774294
$428$	−24.0000	−1.16008
$429$	2.00000	0.0965609
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−20.0000	−0.962250
$433$	−31.0000	−1.48976	−0.744882	−	0.667196i	$-0.767494\pi$
$433$	−31.0000	−1.48976	−0.744882	+	0.667196i	$0.767494\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	32.0000	1.53252
$437$	−6.00000	−0.287019
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−39.0000	−1.85295	−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000	−1.85295	−0.926473	+	0.376361i	$0.877175\pi$
$444$	14.0000	0.664411
$445$	45.0000	2.13320
$446$	0	0
$447$	0	0
$448$	−16.0000	−0.755929
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	18.0000	0.846649
$453$	−16.0000	−0.751746
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	18.0000	0.839254
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	29.0000	1.34774	0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000	1.34774	0.673872	+	0.738848i	$0.264630\pi$
$464$	−24.0000	−1.11417
$465$	−21.0000	−0.973852
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	8.00000	0.369800
$469$	−14.0000	−0.646460
$470$	0	0
$471$	17.0000	0.783319
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	8.00000	0.367065
$476$	4.00000	0.183340
$477$	−12.0000	−0.549442
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	0	0
$483$	−6.00000	−0.273009
$484$	−2.00000	−0.0909091
$485$	33.0000	1.49845
$486$	0	0
$487$	17.0000	0.770344	0.385172	−	0.922845i	$-0.374142\pi$
$487$	17.0000	0.770344	0.385172	+	0.922845i	$0.374142\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	−24.0000	−1.08200
$493$	6.00000	0.270226
$494$	0	0
$495$	−6.00000	−0.269680
$496$	−28.0000	−1.25724
$497$	−18.0000	−0.807410
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	6.00000	0.268328
$501$	18.0000	0.804181
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	−9.00000	−0.399704
$508$	8.00000	0.354943
$509$	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
$511$	4.00000	0.176950
$512$	0	0
$513$	−10.0000	−0.441511
$514$	0	0
$515$	24.0000	1.05757
$516$	20.0000	0.880451
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−21.0000	−0.920027	−0.460013	−	0.887912i	$-0.652155\pi$
$521$	−21.0000	−0.920027	−0.460013	+	0.887912i	$0.652155\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−12.0000	−0.524222
$525$	8.00000	0.349149
$526$	0	0
$527$	7.00000	0.304925
$528$	4.00000	0.174078
$529$	−14.0000	−0.608696
$530$	0	0
$531$	6.00000	0.260378
$532$	−8.00000	−0.346844
$533$	24.0000	1.03956
$534$	0	0
$535$	36.0000	1.55642
$536$	0	0
$537$	9.00000	0.388379
$538$	0	0
$539$	−3.00000	−0.129219
$540$	30.0000	1.29099
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	17.0000	0.729540
$544$	0	0
$545$	−48.0000	−2.05609
$546$	0	0
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	6.00000	0.256307
$549$	−16.0000	−0.682863
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	−21.0000	−0.891400
$556$	32.0000	1.35710
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	24.0000	1.01419
$561$	−1.00000	−0.0422200
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	−27.0000	−1.13590
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	−4.00000	−0.167248
$573$	−3.00000	−0.125327
$574$	0	0
$575$	−12.0000	−0.500435
$576$	16.0000	0.666667
$577$	29.0000	1.20729	0.603643	−	0.797255i	$-0.293715\pi$
$577$	29.0000	1.20729	0.603643	+	0.797255i	$0.293715\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	36.0000	1.49482
$581$	12.0000	0.497844
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	6.00000	0.247436
$589$	−14.0000	−0.576860
$590$	0	0
$591$	12.0000	0.493614
$592$	−28.0000	−1.15079
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	14.0000	0.570124
$604$	32.0000	1.30206
$605$	3.00000	0.121967
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	0	0
$612$	−4.00000	−0.161690
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	0	0
$615$	36.0000	1.45166
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−19.0000	−0.763674	−0.381837	−	0.924230i	$-0.624709\pi$
$619$	−19.0000	−0.763674	−0.381837	+	0.924230i	$0.624709\pi$
$620$	42.0000	1.68676
$621$	15.0000	0.601929
$622$	0	0
$623$	30.0000	1.20192
$624$	8.00000	0.320256
$625$	−29.0000	−1.16000
$626$	0	0
$627$	2.00000	0.0798723
$628$	−34.0000	−1.35675
$629$	7.00000	0.279108
$630$	0	0
$631$	−37.0000	−1.47295	−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000	−1.47295	−0.736473	+	0.676467i	$0.763510\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−12.0000	−0.476205
$636$	−12.0000	−0.475831
$637$	−6.00000	−0.237729
$638$	0	0
$639$	18.0000	0.712069
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	12.0000	0.472866
$645$	−30.0000	−1.18125
$646$	0	0
$647$	−27.0000	−1.06148	−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000	−1.06148	−0.530740	+	0.847535i	$0.678086\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	−14.0000	−0.548703
$652$	32.0000	1.25322
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	0	0
$655$	18.0000	0.703318
$656$	48.0000	1.87409
$657$	−4.00000	−0.156055
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	−6.00000	−0.233550
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	18.0000	0.696963
$668$	−36.0000	−1.39288
$669$	−1.00000	−0.0386622
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	−40.0000	−1.54189	−0.770943	−	0.636904i	$-0.780215\pi$
$673$	−40.0000	−1.54189	−0.770943	+	0.636904i	$0.780215\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	18.0000	0.692308
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	0	0
$679$	22.0000	0.844283
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	8.00000	0.305888
$685$	−9.00000	−0.343872
$686$	0	0
$687$	−13.0000	−0.495981
$688$	−40.0000	−1.52499
$689$	12.0000	0.457164
$690$	0	0
$691$	11.0000	0.418460	0.209230	−	0.977866i	$-0.432904\pi$
$691$	11.0000	0.418460	0.209230	+	0.977866i	$0.432904\pi$
$692$	−12.0000	−0.456172
$693$	−4.00000	−0.151947
$694$	0	0
$695$	−48.0000	−1.82074
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	−12.0000	−0.453882
$700$	−16.0000	−0.604743
$701$	−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
$703$	−14.0000	−0.528020
$704$	−8.00000	−0.301511
$705$	0	0
$706$	0	0
$707$	24.0000	0.902613
$708$	6.00000	0.225494
$709$	29.0000	1.08912	0.544559	−	0.838723i	$-0.316697\pi$
$709$	29.0000	1.08912	0.544559	+	0.838723i	$0.316697\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	21.0000	0.786456
$714$	0	0
$715$	6.00000	0.224387
$716$	−18.0000	−0.672692
$717$	24.0000	0.896296
$718$	0	0
$719$	9.00000	0.335643	0.167822	−	0.985817i	$-0.446327\pi$
$719$	9.00000	0.335643	0.167822	+	0.985817i	$0.446327\pi$
$720$	−24.0000	−0.894427
$721$	16.0000	0.595871
$722$	0	0
$723$	8.00000	0.297523
$724$	−34.0000	−1.26360
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−37.0000	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	10.0000	0.369863
$732$	−16.0000	−0.591377
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	−7.00000	−0.257848
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	42.0000	1.54395
$741$	4.00000	0.146944
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.0000	−0.439057
$748$	2.00000	0.0731272
$749$	24.0000	0.876941
$750$	0	0
$751$	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
$753$	−27.0000	−0.983935
$754$	0	0
$755$	−48.0000	−1.74690
$756$	20.0000	0.727393
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	−3.00000	−0.108893
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	−32.0000	−1.15848
$764$	6.00000	0.217072
$765$	6.00000	0.216930
$766$	0	0
$767$	−6.00000	−0.216647
$768$	16.0000	0.577350
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−28.0000	−1.00774
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−28.0000	−1.00579
$776$	0	0
$777$	−14.0000	−0.502247
$778$	0	0
$779$	24.0000	0.859889
$780$	−12.0000	−0.429669
$781$	−9.00000	−0.322045
$782$	0	0
$783$	30.0000	1.07211
$784$	−12.0000	−0.428571
$785$	51.0000	1.82027
$786$	0	0
$787$	−46.0000	−1.63972	−0.819861	−	0.572562i	$-0.805950\pi$
$787$	−46.0000	−1.63972	−0.819861	+	0.572562i	$0.805950\pi$
$788$	−24.0000	−0.854965
$789$	12.0000	0.427211
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	16.0000	0.568177
$794$	0	0
$795$	18.0000	0.638394
$796$	32.0000	1.13421
$797$	−15.0000	−0.531327	−0.265664	−	0.964066i	$-0.585591\pi$
$797$	−15.0000	−0.531327	−0.265664	+	0.964066i	$0.585591\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	2.00000	0.0705785
$804$	14.0000	0.493742
$805$	−18.0000	−0.634417
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	24.0000	0.842235
$813$	8.00000	0.280572
$814$	0	0
$815$	−48.0000	−1.68137
$816$	−4.00000	−0.140028
$817$	−20.0000	−0.699711
$818$	0	0
$819$	−8.00000	−0.279543
$820$	−72.0000	−2.51435
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	54.0000	1.87776	0.938882	−	0.344239i	$-0.111863\pi$
$827$	54.0000	1.87776	0.938882	+	0.344239i	$0.111863\pi$
$828$	−12.0000	−0.417029
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	−16.0000	−0.554700
$833$	3.00000	0.103944
$834$	0	0
$835$	54.0000	1.86875
$836$	−4.00000	−0.138343
$837$	35.0000	1.20978
$838$	0	0
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−24.0000	−0.826604
$844$	8.00000	0.275371
$845$	−27.0000	−0.928828
$846$	0	0
$847$	2.00000	0.0687208
$848$	24.0000	0.824163
$849$	26.0000	0.892318
$850$	0	0
$851$	21.0000	0.719871
$852$	18.0000	0.616670
$853$	56.0000	1.91740	0.958702	−	0.284413i	$-0.0917988\pi$
$853$	56.0000	1.91740	0.958702	+	0.284413i	$0.0917988\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	41.0000	1.39890	0.699451	−	0.714681i	$-0.253428\pi$
$859$	41.0000	1.39890	0.699451	+	0.714681i	$0.253428\pi$
$860$	60.0000	2.04598
$861$	24.0000	0.817918
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	0	0
$867$	1.00000	0.0339618
$868$	28.0000	0.950382
$869$	8.00000	0.271381
$870$	0	0
$871$	−14.0000	−0.474372
$872$	0	0
$873$	−22.0000	−0.744587
$874$	0	0
$875$	−6.00000	−0.202837
$876$	−4.00000	−0.135147
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	0	0
$880$	12.0000	0.404520
$881$	−57.0000	−1.92038	−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000	−1.92038	−0.960189	+	0.279350i	$0.909881\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	4.00000	0.134535
$885$	−9.00000	−0.302532
$886$	0	0
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	1.00000	0.0335013
$892$	2.00000	0.0669650
$893$	0	0
$894$	0	0
$895$	27.0000	0.902510
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	42.0000	1.40078
$900$	16.0000	0.533333
$901$	−6.00000	−0.199889
$902$	0	0
$903$	−20.0000	−0.665558
$904$	0	0
$905$	51.0000	1.69530
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	48.0000	1.59294
$909$	−24.0000	−0.796030
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	8.00000	0.264906
$913$	6.00000	0.198571
$914$	0	0
$915$	24.0000	0.793416
$916$	26.0000	0.859064
$917$	12.0000	0.396275
$918$	0	0
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	0	0
$921$	−34.0000	−1.12034
$922$	0	0
$923$	−18.0000	−0.592477
$924$	−4.00000	−0.131590
$925$	−28.0000	−0.920634
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	24.0000	0.786146
$933$	0	0
$934$	0	0
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	29.0000	0.946379
$940$	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
$943$	−36.0000	−1.17232
$944$	−12.0000	−0.390567
$945$	−30.0000	−0.975900
$946$	0	0
$947$	−21.0000	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$947$	−21.0000	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$948$	−16.0000	−0.519656
$949$	4.00000	0.129845
$950$	0	0
$951$	27.0000	0.875535
$952$	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
$955$	−9.00000	−0.291233
$956$	−48.0000	−1.55243
$957$	−6.00000	−0.193952
$958$	0	0
$959$	−6.00000	−0.193750
$960$	−24.0000	−0.774597
$961$	18.0000	0.580645
$962$	0	0
$963$	−24.0000	−0.773389
$964$	−16.0000	−0.515325
$965$	42.0000	1.35203
$966$	0	0
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	0	0
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−33.0000	−1.05902	−0.529510	−	0.848304i	$-0.677624\pi$
$971$	−33.0000	−1.05902	−0.529510	+	0.848304i	$0.677624\pi$
$972$	−32.0000	−1.02640
$973$	−32.0000	−1.02587
$974$	0	0
$975$	8.00000	0.256205
$976$	32.0000	1.02430
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	18.0000	0.574989
$981$	32.0000	1.02168
$982$	0	0
$983$	−51.0000	−1.62665	−0.813324	−	0.581811i	$-0.802344\pi$
$983$	−51.0000	−1.62665	−0.813324	+	0.581811i	$0.802344\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	0	0
$987$	0	0
$988$	−8.00000	−0.254514
$989$	30.0000	0.953945
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	23.0000	0.729883
$994$	0	0
$995$	−48.0000	−1.52170
$996$	−12.0000	−0.380235
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	0	0
$999$	35.0000	1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.a.a.1.1	✓	1
3.2	odd	2		1683.2.a.d.1.1		1
4.3	odd	2		2992.2.a.c.1.1		1
5.4	even	2		4675.2.a.j.1.1		1
7.6	odd	2		9163.2.a.d.1.1		1
11.10	odd	2		2057.2.a.b.1.1		1
17.16	even	2		3179.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.a.a.1.1	✓	1	1.1	even	1	trivial
1683.2.a.d.1.1		1	3.2	odd	2
2057.2.a.b.1.1		1	11.10	odd	2
2992.2.a.c.1.1		1	4.3	odd	2
3179.2.a.b.1.1		1	17.16	even	2
4675.2.a.j.1.1		1	5.4	even	2
9163.2.a.d.1.1		1	7.6	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:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more