LMFDB - Embedded newform 189.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,4,Mod(1,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("189.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	189.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:	$11.1513609911$
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	$\chi$	$=$	189.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} +1.00000 q^{4} +12.0000 q^{5} +7.00000 q^{7} +21.0000 q^{8} +O(q^{10})$

\(q-3.00000 q^{2} +1.00000 q^{4} +12.0000 q^{5} +7.00000 q^{7} +21.0000 q^{8} -36.0000 q^{10} -12.0000 q^{11} -61.0000 q^{13} -21.0000 q^{14} -71.0000 q^{16} +117.000 q^{17} +2.00000 q^{19} +12.0000 q^{20} +36.0000 q^{22} +75.0000 q^{23} +19.0000 q^{25} +183.000 q^{26} +7.00000 q^{28} -3.00000 q^{29} +263.000 q^{31} +45.0000 q^{32} -351.000 q^{34} +84.0000 q^{35} +218.000 q^{37} -6.00000 q^{38} +252.000 q^{40} +246.000 q^{41} +515.000 q^{43} -12.0000 q^{44} -225.000 q^{46} -318.000 q^{47} +49.0000 q^{49} -57.0000 q^{50} -61.0000 q^{52} +459.000 q^{53} -144.000 q^{55} +147.000 q^{56} +9.00000 q^{58} +255.000 q^{59} -862.000 q^{61} -789.000 q^{62} +433.000 q^{64} -732.000 q^{65} +479.000 q^{67} +117.000 q^{68} -252.000 q^{70} +117.000 q^{71} -430.000 q^{73} -654.000 q^{74} +2.00000 q^{76} -84.0000 q^{77} -646.000 q^{79} -852.000 q^{80} -738.000 q^{82} +348.000 q^{83} +1404.00 q^{85} -1545.00 q^{86} -252.000 q^{88} +585.000 q^{89} -427.000 q^{91} +75.0000 q^{92} +954.000 q^{94} +24.0000 q^{95} -376.000 q^{97} -147.000 q^{98} +O(q^{100})\)

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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	21.0000	0.928078
$9$	0	0
$10$	−36.0000	−1.13842
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	−61.0000	−1.30141	−0.650706	−	0.759330i	$-0.725527\pi$
$13$	−61.0000	−1.30141	−0.650706	+	0.759330i	$0.725527\pi$
$14$	−21.0000	−0.400892
$15$	0	0
$16$	−71.0000	−1.10938
$17$	117.000	1.66922	0.834608	−	0.550845i	$-0.185694\pi$
$17$	117.000	1.66922	0.834608	+	0.550845i	$0.185694\pi$
$18$	0	0
$19$	2.00000	0.0241490	0.0120745	−	0.999927i	$-0.496156\pi$
$19$	2.00000	0.0241490	0.0120745	+	0.999927i	$0.496156\pi$
$20$	12.0000	0.134164
$21$	0	0
$22$	36.0000	0.348874
$23$	75.0000	0.679938	0.339969	−	0.940437i	$-0.389583\pi$
$23$	75.0000	0.679938	0.339969	+	0.940437i	$0.389583\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	183.000	1.38036
$27$	0	0
$28$	7.00000	0.0472456
$29$	−3.00000	−0.0192099	−0.00960493	−	0.999954i	$-0.503057\pi$
$29$	−3.00000	−0.0192099	−0.00960493	+	0.999954i	$0.503057\pi$
$30$	0	0
$31$	263.000	1.52375	0.761874	−	0.647725i	$-0.224279\pi$
$31$	263.000	1.52375	0.761874	+	0.647725i	$0.224279\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	−351.000	−1.77047
$35$	84.0000	0.405674
$36$	0	0
$37$	218.000	0.968621	0.484311	−	0.874896i	$-0.339070\pi$
$37$	218.000	0.968621	0.484311	+	0.874896i	$0.339070\pi$
$38$	−6.00000	−0.0256139
$39$	0	0
$40$	252.000	0.996117
$41$	246.000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	246.000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	515.000	1.82644	0.913218	−	0.407471i	$-0.133589\pi$
$43$	515.000	1.82644	0.913218	+	0.407471i	$0.133589\pi$
$44$	−12.0000	−0.0411152
$45$	0	0
$46$	−225.000	−0.721183
$47$	−318.000	−0.986916	−0.493458	−	0.869770i	$-0.664267\pi$
$47$	−318.000	−0.986916	−0.493458	+	0.869770i	$0.664267\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−57.0000	−0.161220
$51$	0	0
$52$	−61.0000	−0.162676
$53$	459.000	1.18959	0.594797	−	0.803876i	$-0.297232\pi$
$53$	459.000	1.18959	0.594797	+	0.803876i	$0.297232\pi$
$54$	0	0
$55$	−144.000	−0.353036
$56$	147.000	0.350780
$57$	0	0
$58$	9.00000	0.0203751
$59$	255.000	0.562681	0.281340	−	0.959608i	$-0.409221\pi$
$59$	255.000	0.562681	0.281340	+	0.959608i	$0.409221\pi$
$60$	0	0
$61$	−862.000	−1.80931	−0.904654	−	0.426147i	$-0.859871\pi$
$61$	−862.000	−1.80931	−0.904654	+	0.426147i	$0.859871\pi$
$62$	−789.000	−1.61618
$63$	0	0
$64$	433.000	0.845703
$65$	−732.000	−1.39682
$66$	0	0
$67$	479.000	0.873420	0.436710	−	0.899602i	$-0.356144\pi$
$67$	479.000	0.873420	0.436710	+	0.899602i	$0.356144\pi$
$68$	117.000	0.208652
$69$	0	0
$70$	−252.000	−0.430282
$71$	117.000	0.195568	0.0977841	−	0.995208i	$-0.468825\pi$
$71$	117.000	0.195568	0.0977841	+	0.995208i	$0.468825\pi$
$72$	0	0
$73$	−430.000	−0.689420	−0.344710	−	0.938709i	$-0.612023\pi$
$73$	−430.000	−0.689420	−0.344710	+	0.938709i	$0.612023\pi$
$74$	−654.000	−1.02738
$75$	0	0
$76$	2.00000	0.00301863
$77$	−84.0000	−0.124321
$78$	0	0
$79$	−646.000	−0.920009	−0.460004	−	0.887917i	$-0.652152\pi$
$79$	−646.000	−0.920009	−0.460004	+	0.887917i	$0.652152\pi$
$80$	−852.000	−1.19071
$81$	0	0
$82$	−738.000	−0.993884
$83$	348.000	0.460216	0.230108	−	0.973165i	$-0.426092\pi$
$83$	348.000	0.460216	0.230108	+	0.973165i	$0.426092\pi$
$84$	0	0
$85$	1404.00	1.79159
$86$	−1545.00	−1.93723
$87$	0	0
$88$	−252.000	−0.305265
$89$	585.000	0.696740	0.348370	−	0.937357i	$-0.386735\pi$
$89$	585.000	0.696740	0.348370	+	0.937357i	$0.386735\pi$
$90$	0	0
$91$	−427.000	−0.491888
$92$	75.0000	0.0849923
$93$	0	0
$94$	954.000	1.04678
$95$	24.0000	0.0259195
$96$	0	0
$97$	−376.000	−0.393577	−0.196789	−	0.980446i	$-0.563051\pi$
$97$	−376.000	−0.393577	−0.196789	+	0.980446i	$0.563051\pi$
$98$	−147.000	−0.151523
$99$	0	0
$100$	19.0000	0.0190000
$101$	978.000	0.963511	0.481756	−	0.876306i	$-0.339999\pi$
$101$	978.000	0.963511	0.481756	+	0.876306i	$0.339999\pi$
$102$	0	0
$103$	−655.000	−0.626593	−0.313296	−	0.949655i	$-0.601433\pi$
$103$	−655.000	−0.626593	−0.313296	+	0.949655i	$0.601433\pi$
$104$	−1281.00	−1.20781
$105$	0	0
$106$	−1377.00	−1.26176
$107$	108.000	0.0975771	0.0487886	−	0.998809i	$-0.484464\pi$
$107$	108.000	0.0975771	0.0487886	+	0.998809i	$0.484464\pi$
$108$	0	0
$109$	272.000	0.239017	0.119509	−	0.992833i	$-0.461868\pi$
$109$	272.000	0.239017	0.119509	+	0.992833i	$0.461868\pi$
$110$	432.000	0.374451
$111$	0	0
$112$	−497.000	−0.419304
$113$	−2256.00	−1.87811	−0.939056	−	0.343765i	$-0.888298\pi$
$113$	−2256.00	−1.87811	−0.939056	+	0.343765i	$0.888298\pi$
$114$	0	0
$115$	900.000	0.729786
$116$	−3.00000	−0.00240123
$117$	0	0
$118$	−765.000	−0.596813
$119$	819.000	0.630904
$120$	0	0
$121$	−1187.00	−0.891811
$122$	2586.00	1.91906
$123$	0	0
$124$	263.000	0.190469
$125$	−1272.00	−0.910169
$126$	0	0
$127$	1784.00	1.24649	0.623246	−	0.782026i	$-0.285814\pi$
$127$	1784.00	1.24649	0.623246	+	0.782026i	$0.285814\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	2196.00	1.48155
$131$	1749.00	1.16649	0.583247	−	0.812295i	$-0.301782\pi$
$131$	1749.00	1.16649	0.583247	+	0.812295i	$0.301782\pi$
$132$	0	0
$133$	14.0000	0.00912747
$134$	−1437.00	−0.926402
$135$	0	0
$136$	2457.00	1.54916
$137$	1374.00	0.856852	0.428426	−	0.903577i	$-0.359068\pi$
$137$	1374.00	0.856852	0.428426	+	0.903577i	$0.359068\pi$
$138$	0	0
$139$	2450.00	1.49501	0.747505	−	0.664257i	$-0.231252\pi$
$139$	2450.00	1.49501	0.747505	+	0.664257i	$0.231252\pi$
$140$	84.0000	0.0507093
$141$	0	0
$142$	−351.000	−0.207431
$143$	732.000	0.428062
$144$	0	0
$145$	−36.0000	−0.0206182
$146$	1290.00	0.731241
$147$	0	0
$148$	218.000	0.121078
$149$	921.000	0.506384	0.253192	−	0.967416i	$-0.418520\pi$
$149$	921.000	0.506384	0.253192	+	0.967416i	$0.418520\pi$
$150$	0	0
$151$	650.000	0.350306	0.175153	−	0.984541i	$-0.443958\pi$
$151$	650.000	0.350306	0.175153	+	0.984541i	$0.443958\pi$
$152$	42.0000	0.0224122
$153$	0	0
$154$	252.000	0.131862
$155$	3156.00	1.63546
$156$	0	0
$157$	−871.000	−0.442760	−0.221380	−	0.975188i	$-0.571056\pi$
$157$	−871.000	−0.442760	−0.221380	+	0.975188i	$0.571056\pi$
$158$	1938.00	0.975816
$159$	0	0
$160$	540.000	0.266817
$161$	525.000	0.256993
$162$	0	0
$163$	−3697.00	−1.77651	−0.888256	−	0.459349i	$-0.848083\pi$
$163$	−3697.00	−1.77651	−0.888256	+	0.459349i	$0.848083\pi$
$164$	246.000	0.117130
$165$	0	0
$166$	−1044.00	−0.488133
$167$	−1824.00	−0.845182	−0.422591	−	0.906321i	$-0.638879\pi$
$167$	−1824.00	−0.845182	−0.422591	+	0.906321i	$0.638879\pi$
$168$	0	0
$169$	1524.00	0.693673
$170$	−4212.00	−1.90027
$171$	0	0
$172$	515.000	0.228305
$173$	744.000	0.326967	0.163483	−	0.986546i	$-0.447727\pi$
$173$	744.000	0.326967	0.163483	+	0.986546i	$0.447727\pi$
$174$	0	0
$175$	133.000	0.0574506
$176$	852.000	0.364897
$177$	0	0
$178$	−1755.00	−0.739005
$179$	4572.00	1.90909	0.954546	−	0.298065i	$-0.0963412\pi$
$179$	4572.00	1.90909	0.954546	+	0.298065i	$0.0963412\pi$
$180$	0	0
$181$	−3481.00	−1.42951	−0.714753	−	0.699377i	$-0.753461\pi$
$181$	−3481.00	−1.42951	−0.714753	+	0.699377i	$0.753461\pi$
$182$	1281.00	0.521725
$183$	0	0
$184$	1575.00	0.631036
$185$	2616.00	1.03963
$186$	0	0
$187$	−1404.00	−0.549041
$188$	−318.000	−0.123365
$189$	0	0
$190$	−72.0000	−0.0274917
$191$	−3756.00	−1.42290	−0.711452	−	0.702735i	$-0.751962\pi$
$191$	−3756.00	−1.42290	−0.711452	+	0.702735i	$0.751962\pi$
$192$	0	0
$193$	−4867.00	−1.81520	−0.907602	−	0.419832i	$-0.862089\pi$
$193$	−4867.00	−1.81520	−0.907602	+	0.419832i	$0.862089\pi$
$194$	1128.00	0.417452
$195$	0	0
$196$	49.0000	0.0178571
$197$	18.0000	0.00650988	0.00325494	−	0.999995i	$-0.498964\pi$
$197$	18.0000	0.00650988	0.00325494	+	0.999995i	$0.498964\pi$
$198$	0	0
$199$	−1807.00	−0.643693	−0.321846	−	0.946792i	$-0.604303\pi$
$199$	−1807.00	−0.643693	−0.321846	+	0.946792i	$0.604303\pi$
$200$	399.000	0.141068
$201$	0	0
$202$	−2934.00	−1.02196
$203$	−21.0000	−0.00726065
$204$	0	0
$205$	2952.00	1.00574
$206$	1965.00	0.664602
$207$	0	0
$208$	4331.00	1.44375
$209$	−24.0000	−0.00794313
$210$	0	0
$211$	−4273.00	−1.39415	−0.697075	−	0.716999i	$-0.745515\pi$
$211$	−4273.00	−1.39415	−0.697075	+	0.716999i	$0.745515\pi$
$212$	459.000	0.148699
$213$	0	0
$214$	−324.000	−0.103496
$215$	6180.00	1.96034
$216$	0	0
$217$	1841.00	0.575923
$218$	−816.000	−0.253516
$219$	0	0
$220$	−144.000	−0.0441294
$221$	−7137.00	−2.17234
$222$	0	0
$223$	2648.00	0.795171	0.397586	−	0.917565i	$-0.369848\pi$
$223$	2648.00	0.795171	0.397586	+	0.917565i	$0.369848\pi$
$224$	315.000	0.0939590
$225$	0	0
$226$	6768.00	1.99204
$227$	−5493.00	−1.60609	−0.803047	−	0.595916i	$-0.796789\pi$
$227$	−5493.00	−1.60609	−0.803047	+	0.595916i	$0.796789\pi$
$228$	0	0
$229$	−790.000	−0.227968	−0.113984	−	0.993483i	$-0.536361\pi$
$229$	−790.000	−0.227968	−0.113984	+	0.993483i	$0.536361\pi$
$230$	−2700.00	−0.774055
$231$	0	0
$232$	−63.0000	−0.0178282
$233$	−6228.00	−1.75112	−0.875558	−	0.483114i	$-0.839506\pi$
$233$	−6228.00	−1.75112	−0.875558	+	0.483114i	$0.839506\pi$
$234$	0	0
$235$	−3816.00	−1.05927
$236$	255.000	0.0703351
$237$	0	0
$238$	−2457.00	−0.669175
$239$	−4812.00	−1.30235	−0.651177	−	0.758926i	$-0.725724\pi$
$239$	−4812.00	−1.30235	−0.651177	+	0.758926i	$0.725724\pi$
$240$	0	0
$241$	1622.00	0.433536	0.216768	−	0.976223i	$-0.430448\pi$
$241$	1622.00	0.433536	0.216768	+	0.976223i	$0.430448\pi$
$242$	3561.00	0.945908
$243$	0	0
$244$	−862.000	−0.226164
$245$	588.000	0.153330
$246$	0	0
$247$	−122.000	−0.0314278
$248$	5523.00	1.41416
$249$	0	0
$250$	3816.00	0.965380
$251$	2880.00	0.724239	0.362119	−	0.932132i	$-0.382053\pi$
$251$	2880.00	0.724239	0.362119	+	0.932132i	$0.382053\pi$
$252$	0	0
$253$	−900.000	−0.223646
$254$	−5352.00	−1.32210
$255$	0	0
$256$	1513.00	0.369385
$257$	2190.00	0.531550	0.265775	−	0.964035i	$-0.414372\pi$
$257$	2190.00	0.531550	0.265775	+	0.964035i	$0.414372\pi$
$258$	0	0
$259$	1526.00	0.366104
$260$	−732.000	−0.174603
$261$	0	0
$262$	−5247.00	−1.23725
$263$	1869.00	0.438203	0.219102	−	0.975702i	$-0.429687\pi$
$263$	1869.00	0.438203	0.219102	+	0.975702i	$0.429687\pi$
$264$	0	0
$265$	5508.00	1.27681
$266$	−42.0000	−0.00968115
$267$	0	0
$268$	479.000	0.109178
$269$	2268.00	0.514061	0.257030	−	0.966403i	$-0.417256\pi$
$269$	2268.00	0.514061	0.257030	+	0.966403i	$0.417256\pi$
$270$	0	0
$271$	2027.00	0.454360	0.227180	−	0.973853i	$-0.427049\pi$
$271$	2027.00	0.454360	0.227180	+	0.973853i	$0.427049\pi$
$272$	−8307.00	−1.85179
$273$	0	0
$274$	−4122.00	−0.908829
$275$	−228.000	−0.0499961
$276$	0	0
$277$	3944.00	0.855495	0.427747	−	0.903898i	$-0.359307\pi$
$277$	3944.00	0.855495	0.427747	+	0.903898i	$0.359307\pi$
$278$	−7350.00	−1.58570
$279$	0	0
$280$	1764.00	0.376497
$281$	−4602.00	−0.976983	−0.488492	−	0.872569i	$-0.662453\pi$
$281$	−4602.00	−0.976983	−0.488492	+	0.872569i	$0.662453\pi$
$282$	0	0
$283$	−3220.00	−0.676357	−0.338179	−	0.941082i	$-0.609811\pi$
$283$	−3220.00	−0.676357	−0.338179	+	0.941082i	$0.609811\pi$
$284$	117.000	0.0244460
$285$	0	0
$286$	−2196.00	−0.454029
$287$	1722.00	0.354169
$288$	0	0
$289$	8776.00	1.78628
$290$	108.000	0.0218689
$291$	0	0
$292$	−430.000	−0.0861776
$293$	−6810.00	−1.35783	−0.678915	−	0.734216i	$-0.737550\pi$
$293$	−6810.00	−1.35783	−0.678915	+	0.734216i	$0.737550\pi$
$294$	0	0
$295$	3060.00	0.603933
$296$	4578.00	0.898956
$297$	0	0
$298$	−2763.00	−0.537102
$299$	−4575.00	−0.884880
$300$	0	0
$301$	3605.00	0.690328
$302$	−1950.00	−0.371556
$303$	0	0
$304$	−142.000	−0.0267903
$305$	−10344.0	−1.94195
$306$	0	0
$307$	758.000	0.140916	0.0704582	−	0.997515i	$-0.477554\pi$
$307$	758.000	0.140916	0.0704582	+	0.997515i	$0.477554\pi$
$308$	−84.0000	−0.0155401
$309$	0	0
$310$	−9468.00	−1.73467
$311$	9480.00	1.72849	0.864247	−	0.503068i	$-0.167796\pi$
$311$	9480.00	1.72849	0.864247	+	0.503068i	$0.167796\pi$
$312$	0	0
$313$	3350.00	0.604962	0.302481	−	0.953155i	$-0.402185\pi$
$313$	3350.00	0.604962	0.302481	+	0.953155i	$0.402185\pi$
$314$	2613.00	0.469618
$315$	0	0
$316$	−646.000	−0.115001
$317$	−426.000	−0.0754781	−0.0377390	−	0.999288i	$-0.512016\pi$
$317$	−426.000	−0.0754781	−0.0377390	+	0.999288i	$0.512016\pi$
$318$	0	0
$319$	36.0000	0.00631854
$320$	5196.00	0.907704
$321$	0	0
$322$	−1575.00	−0.272582
$323$	234.000	0.0403099
$324$	0	0
$325$	−1159.00	−0.197815
$326$	11091.0	1.88428
$327$	0	0
$328$	5166.00	0.869648
$329$	−2226.00	−0.373019
$330$	0	0
$331$	−3103.00	−0.515276	−0.257638	−	0.966242i	$-0.582944\pi$
$331$	−3103.00	−0.515276	−0.257638	+	0.966242i	$0.582944\pi$
$332$	348.000	0.0575271
$333$	0	0
$334$	5472.00	0.896451
$335$	5748.00	0.937453
$336$	0	0
$337$	−11725.0	−1.89526	−0.947628	−	0.319375i	$-0.896527\pi$
$337$	−11725.0	−1.89526	−0.947628	+	0.319375i	$0.896527\pi$
$338$	−4572.00	−0.735752
$339$	0	0
$340$	1404.00	0.223949
$341$	−3156.00	−0.501193
$342$	0	0
$343$	343.000	0.0539949
$344$	10815.0	1.69507
$345$	0	0
$346$	−2232.00	−0.346801
$347$	1506.00	0.232987	0.116493	−	0.993191i	$-0.462835\pi$
$347$	1506.00	0.232987	0.116493	+	0.993191i	$0.462835\pi$
$348$	0	0
$349$	10289.0	1.57810	0.789051	−	0.614328i	$-0.210573\pi$
$349$	10289.0	1.57810	0.789051	+	0.614328i	$0.210573\pi$
$350$	−399.000	−0.0609356
$351$	0	0
$352$	−540.000	−0.0817673
$353$	−12693.0	−1.91382	−0.956912	−	0.290376i	$-0.906219\pi$
$353$	−12693.0	−1.91382	−0.956912	+	0.290376i	$0.906219\pi$
$354$	0	0
$355$	1404.00	0.209906
$356$	585.000	0.0870925
$357$	0	0
$358$	−13716.0	−2.02490
$359$	−5031.00	−0.739627	−0.369813	−	0.929106i	$-0.620578\pi$
$359$	−5031.00	−0.739627	−0.369813	+	0.929106i	$0.620578\pi$
$360$	0	0
$361$	−6855.00	−0.999417
$362$	10443.0	1.51622
$363$	0	0
$364$	−427.000	−0.0614859
$365$	−5160.00	−0.739964
$366$	0	0
$367$	9479.00	1.34823	0.674114	−	0.738627i	$-0.264526\pi$
$367$	9479.00	1.34823	0.674114	+	0.738627i	$0.264526\pi$
$368$	−5325.00	−0.754307
$369$	0	0
$370$	−7848.00	−1.10270
$371$	3213.00	0.449624
$372$	0	0
$373$	−8566.00	−1.18909	−0.594545	−	0.804062i	$-0.702668\pi$
$373$	−8566.00	−1.18909	−0.594545	+	0.804062i	$0.702668\pi$
$374$	4212.00	0.582346
$375$	0	0
$376$	−6678.00	−0.915935
$377$	183.000	0.0249999
$378$	0	0
$379$	−2320.00	−0.314434	−0.157217	−	0.987564i	$-0.550252\pi$
$379$	−2320.00	−0.314434	−0.157217	+	0.987564i	$0.550252\pi$
$380$	24.0000	0.00323993
$381$	0	0
$382$	11268.0	1.50922
$383$	1092.00	0.145688	0.0728441	−	0.997343i	$-0.476792\pi$
$383$	1092.00	0.145688	0.0728441	+	0.997343i	$0.476792\pi$
$384$	0	0
$385$	−1008.00	−0.133435
$386$	14601.0	1.92531
$387$	0	0
$388$	−376.000	−0.0491972
$389$	12570.0	1.63837	0.819183	−	0.573532i	$-0.194427\pi$
$389$	12570.0	1.63837	0.819183	+	0.573532i	$0.194427\pi$
$390$	0	0
$391$	8775.00	1.13496
$392$	1029.00	0.132583
$393$	0	0
$394$	−54.0000	−0.00690477
$395$	−7752.00	−0.987457
$396$	0	0
$397$	−6802.00	−0.859906	−0.429953	−	0.902851i	$-0.641470\pi$
$397$	−6802.00	−0.859906	−0.429953	+	0.902851i	$0.641470\pi$
$398$	5421.00	0.682739
$399$	0	0
$400$	−1349.00	−0.168625
$401$	8526.00	1.06177	0.530883	−	0.847445i	$-0.321860\pi$
$401$	8526.00	1.06177	0.530883	+	0.847445i	$0.321860\pi$
$402$	0	0
$403$	−16043.0	−1.98302
$404$	978.000	0.120439
$405$	0	0
$406$	63.0000	0.00770108
$407$	−2616.00	−0.318600
$408$	0	0
$409$	13466.0	1.62800	0.813999	−	0.580867i	$-0.197286\pi$
$409$	13466.0	1.62800	0.813999	+	0.580867i	$0.197286\pi$
$410$	−8856.00	−1.06675
$411$	0	0
$412$	−655.000	−0.0783241
$413$	1785.00	0.212673
$414$	0	0
$415$	4176.00	0.493956
$416$	−2745.00	−0.323521
$417$	0	0
$418$	72.0000	0.00842496
$419$	15339.0	1.78845	0.894223	−	0.447621i	$-0.147729\pi$
$419$	15339.0	1.78845	0.894223	+	0.447621i	$0.147729\pi$
$420$	0	0
$421$	−4102.00	−0.474868	−0.237434	−	0.971404i	$-0.576306\pi$
$421$	−4102.00	−0.474868	−0.237434	+	0.971404i	$0.576306\pi$
$422$	12819.0	1.47872
$423$	0	0
$424$	9639.00	1.10404
$425$	2223.00	0.253721
$426$	0	0
$427$	−6034.00	−0.683854
$428$	108.000	0.0121971
$429$	0	0
$430$	−18540.0	−2.07925
$431$	2592.00	0.289680	0.144840	−	0.989455i	$-0.453733\pi$
$431$	2592.00	0.289680	0.144840	+	0.989455i	$0.453733\pi$
$432$	0	0
$433$	−9448.00	−1.04860	−0.524298	−	0.851535i	$-0.675672\pi$
$433$	−9448.00	−1.04860	−0.524298	+	0.851535i	$0.675672\pi$
$434$	−5523.00	−0.610858
$435$	0	0
$436$	272.000	0.0298772
$437$	150.000	0.0164198
$438$	0	0
$439$	11369.0	1.23602	0.618010	−	0.786170i	$-0.287939\pi$
$439$	11369.0	1.23602	0.618010	+	0.786170i	$0.287939\pi$
$440$	−3024.00	−0.327644
$441$	0	0
$442$	21411.0	2.30411
$443$	10842.0	1.16280	0.581398	−	0.813619i	$-0.302506\pi$
$443$	10842.0	1.16280	0.581398	+	0.813619i	$0.302506\pi$
$444$	0	0
$445$	7020.00	0.747820
$446$	−7944.00	−0.843407
$447$	0	0
$448$	3031.00	0.319646
$449$	−13140.0	−1.38110	−0.690551	−	0.723284i	$-0.742632\pi$
$449$	−13140.0	−1.38110	−0.690551	+	0.723284i	$0.742632\pi$
$450$	0	0
$451$	−2952.00	−0.308213
$452$	−2256.00	−0.234764
$453$	0	0
$454$	16479.0	1.70352
$455$	−5124.00	−0.527949
$456$	0	0
$457$	1379.00	0.141153	0.0705765	−	0.997506i	$-0.477516\pi$
$457$	1379.00	0.141153	0.0705765	+	0.997506i	$0.477516\pi$
$458$	2370.00	0.241797
$459$	0	0
$460$	900.000	0.0912233
$461$	−678.000	−0.0684981	−0.0342490	−	0.999413i	$-0.510904\pi$
$461$	−678.000	−0.0684981	−0.0342490	+	0.999413i	$0.510904\pi$
$462$	0	0
$463$	15734.0	1.57931	0.789655	−	0.613550i	$-0.210259\pi$
$463$	15734.0	1.57931	0.789655	+	0.613550i	$0.210259\pi$
$464$	213.000	0.0213109
$465$	0	0
$466$	18684.0	1.85734
$467$	7416.00	0.734843	0.367421	−	0.930055i	$-0.380241\pi$
$467$	7416.00	0.734843	0.367421	+	0.930055i	$0.380241\pi$
$468$	0	0
$469$	3353.00	0.330122
$470$	11448.0	1.12353
$471$	0	0
$472$	5355.00	0.522212
$473$	−6180.00	−0.600754
$474$	0	0
$475$	38.0000	0.00367065
$476$	819.000	0.0788630
$477$	0	0
$478$	14436.0	1.38135
$479$	1248.00	0.119045	0.0595225	−	0.998227i	$-0.481042\pi$
$479$	1248.00	0.119045	0.0595225	+	0.998227i	$0.481042\pi$
$480$	0	0
$481$	−13298.0	−1.26058
$482$	−4866.00	−0.459834
$483$	0	0
$484$	−1187.00	−0.111476
$485$	−4512.00	−0.422432
$486$	0	0
$487$	−20950.0	−1.94935	−0.974677	−	0.223619i	$-0.928213\pi$
$487$	−20950.0	−1.94935	−0.974677	+	0.223619i	$0.928213\pi$
$488$	−18102.0	−1.67918
$489$	0	0
$490$	−1764.00	−0.162631
$491$	768.000	0.0705893	0.0352947	−	0.999377i	$-0.488763\pi$
$491$	768.000	0.0705893	0.0352947	+	0.999377i	$0.488763\pi$
$492$	0	0
$493$	−351.000	−0.0320654
$494$	366.000	0.0333342
$495$	0	0
$496$	−18673.0	−1.69041
$497$	819.000	0.0739178
$498$	0	0
$499$	4772.00	0.428104	0.214052	−	0.976822i	$-0.431334\pi$
$499$	4772.00	0.428104	0.214052	+	0.976822i	$0.431334\pi$
$500$	−1272.00	−0.113771
$501$	0	0
$502$	−8640.00	−0.768171
$503$	−11088.0	−0.982882	−0.491441	−	0.870911i	$-0.663530\pi$
$503$	−11088.0	−0.982882	−0.491441	+	0.870911i	$0.663530\pi$
$504$	0	0
$505$	11736.0	1.03415
$506$	2700.00	0.237213
$507$	0	0
$508$	1784.00	0.155811
$509$	−16854.0	−1.46766	−0.733831	−	0.679332i	$-0.762270\pi$
$509$	−16854.0	−1.46766	−0.733831	+	0.679332i	$0.762270\pi$
$510$	0	0
$511$	−3010.00	−0.260576
$512$	8733.00	0.753804
$513$	0	0
$514$	−6570.00	−0.563794
$515$	−7860.00	−0.672530
$516$	0	0
$517$	3816.00	0.324618
$518$	−4578.00	−0.388312
$519$	0	0
$520$	−15372.0	−1.29636
$521$	−3195.00	−0.268667	−0.134333	−	0.990936i	$-0.542889\pi$
$521$	−3195.00	−0.268667	−0.134333	+	0.990936i	$0.542889\pi$
$522$	0	0
$523$	920.000	0.0769193	0.0384596	−	0.999260i	$-0.487755\pi$
$523$	920.000	0.0769193	0.0384596	+	0.999260i	$0.487755\pi$
$524$	1749.00	0.145812
$525$	0	0
$526$	−5607.00	−0.464785
$527$	30771.0	2.54346
$528$	0	0
$529$	−6542.00	−0.537684
$530$	−16524.0	−1.35426
$531$	0	0
$532$	14.0000	0.00114093
$533$	−15006.0	−1.21948
$534$	0	0
$535$	1296.00	0.104731
$536$	10059.0	0.810602
$537$	0	0
$538$	−6804.00	−0.545244
$539$	−588.000	−0.0469888
$540$	0	0
$541$	−7180.00	−0.570596	−0.285298	−	0.958439i	$-0.592093\pi$
$541$	−7180.00	−0.570596	−0.285298	+	0.958439i	$0.592093\pi$
$542$	−6081.00	−0.481921
$543$	0	0
$544$	5265.00	0.414954
$545$	3264.00	0.256540
$546$	0	0
$547$	11612.0	0.907666	0.453833	−	0.891087i	$-0.350056\pi$
$547$	11612.0	0.907666	0.453833	+	0.891087i	$0.350056\pi$
$548$	1374.00	0.107107
$549$	0	0
$550$	684.000	0.0530288
$551$	−6.00000	−0.000463899 0
$552$	0	0
$553$	−4522.00	−0.347731
$554$	−11832.0	−0.907389
$555$	0	0
$556$	2450.00	0.186876
$557$	−9819.00	−0.746938	−0.373469	−	0.927643i	$-0.621832\pi$
$557$	−9819.00	−0.746938	−0.373469	+	0.927643i	$0.621832\pi$
$558$	0	0
$559$	−31415.0	−2.37695
$560$	−5964.00	−0.450045
$561$	0	0
$562$	13806.0	1.03625
$563$	−15801.0	−1.18283	−0.591415	−	0.806368i	$-0.701430\pi$
$563$	−15801.0	−1.18283	−0.591415	+	0.806368i	$0.701430\pi$
$564$	0	0
$565$	−27072.0	−2.01580
$566$	9660.00	0.717385
$567$	0	0
$568$	2457.00	0.181503
$569$	8340.00	0.614466	0.307233	−	0.951634i	$-0.400597\pi$
$569$	8340.00	0.614466	0.307233	+	0.951634i	$0.400597\pi$
$570$	0	0
$571$	1073.00	0.0786404	0.0393202	−	0.999227i	$-0.487481\pi$
$571$	1073.00	0.0786404	0.0393202	+	0.999227i	$0.487481\pi$
$572$	732.000	0.0535078
$573$	0	0
$574$	−5166.00	−0.375653
$575$	1425.00	0.103351
$576$	0	0
$577$	22466.0	1.62092	0.810461	−	0.585793i	$-0.199217\pi$
$577$	22466.0	1.62092	0.810461	+	0.585793i	$0.199217\pi$
$578$	−26328.0	−1.89464
$579$	0	0
$580$	−36.0000	−0.00257727
$581$	2436.00	0.173945
$582$	0	0
$583$	−5508.00	−0.391283
$584$	−9030.00	−0.639836
$585$	0	0
$586$	20430.0	1.44020
$587$	3219.00	0.226341	0.113171	−	0.993576i	$-0.463899\pi$
$587$	3219.00	0.226341	0.113171	+	0.993576i	$0.463899\pi$
$588$	0	0
$589$	526.000	0.0367970
$590$	−9180.00	−0.640567
$591$	0	0
$592$	−15478.0	−1.07456
$593$	9342.00	0.646931	0.323465	−	0.946240i	$-0.395152\pi$
$593$	9342.00	0.646931	0.323465	+	0.946240i	$0.395152\pi$
$594$	0	0
$595$	9828.00	0.677158
$596$	921.000	0.0632980
$597$	0	0
$598$	13725.0	0.938557
$599$	−6927.00	−0.472503	−0.236252	−	0.971692i	$-0.575919\pi$
$599$	−6927.00	−0.472503	−0.236252	+	0.971692i	$0.575919\pi$
$600$	0	0
$601$	−15172.0	−1.02975	−0.514874	−	0.857266i	$-0.672161\pi$
$601$	−15172.0	−1.02975	−0.514874	+	0.857266i	$0.672161\pi$
$602$	−10815.0	−0.732203
$603$	0	0
$604$	650.000	0.0437883
$605$	−14244.0	−0.957192
$606$	0	0
$607$	−4921.00	−0.329056	−0.164528	−	0.986372i	$-0.552610\pi$
$607$	−4921.00	−0.329056	−0.164528	+	0.986372i	$0.552610\pi$
$608$	90.0000	0.00600326
$609$	0	0
$610$	31032.0	2.05975
$611$	19398.0	1.28438
$612$	0	0
$613$	218.000	0.0143637	0.00718184	−	0.999974i	$-0.497714\pi$
$613$	218.000	0.0143637	0.00718184	+	0.999974i	$0.497714\pi$
$614$	−2274.00	−0.149464
$615$	0	0
$616$	−1764.00	−0.115379
$617$	22350.0	1.45831	0.729155	−	0.684349i	$-0.239913\pi$
$617$	22350.0	1.45831	0.729155	+	0.684349i	$0.239913\pi$
$618$	0	0
$619$	9290.00	0.603226	0.301613	−	0.953431i	$-0.402475\pi$
$619$	9290.00	0.603226	0.301613	+	0.953431i	$0.402475\pi$
$620$	3156.00	0.204432
$621$	0	0
$622$	−28440.0	−1.83334
$623$	4095.00	0.263343
$624$	0	0
$625$	−17639.0	−1.12890
$626$	−10050.0	−0.641659
$627$	0	0
$628$	−871.000	−0.0553450
$629$	25506.0	1.61684
$630$	0	0
$631$	13988.0	0.882494	0.441247	−	0.897386i	$-0.354536\pi$
$631$	13988.0	0.882494	0.441247	+	0.897386i	$0.354536\pi$
$632$	−13566.0	−0.853839
$633$	0	0
$634$	1278.00	0.0800566
$635$	21408.0	1.33787
$636$	0	0
$637$	−2989.00	−0.185916
$638$	−108.000	−0.00670182
$639$	0	0
$640$	−19908.0	−1.22958
$641$	−11388.0	−0.701714	−0.350857	−	0.936429i	$-0.614110\pi$
$641$	−11388.0	−0.701714	−0.350857	+	0.936429i	$0.614110\pi$
$642$	0	0
$643$	9956.00	0.610616	0.305308	−	0.952254i	$-0.401241\pi$
$643$	9956.00	0.610616	0.305308	+	0.952254i	$0.401241\pi$
$644$	525.000	0.0321241
$645$	0	0
$646$	−702.000	−0.0427551
$647$	−7830.00	−0.475779	−0.237890	−	0.971292i	$-0.576456\pi$
$647$	−7830.00	−0.475779	−0.237890	+	0.971292i	$0.576456\pi$
$648$	0	0
$649$	−3060.00	−0.185078
$650$	3477.00	0.209814
$651$	0	0
$652$	−3697.00	−0.222064
$653$	−3633.00	−0.217719	−0.108859	−	0.994057i	$-0.534720\pi$
$653$	−3633.00	−0.217719	−0.108859	+	0.994057i	$0.534720\pi$
$654$	0	0
$655$	20988.0	1.25201
$656$	−17466.0	−1.03953
$657$	0	0
$658$	6678.00	0.395647
$659$	6036.00	0.356797	0.178398	−	0.983958i	$-0.442908\pi$
$659$	6036.00	0.356797	0.178398	+	0.983958i	$0.442908\pi$
$660$	0	0
$661$	−5110.00	−0.300690	−0.150345	−	0.988634i	$-0.548038\pi$
$661$	−5110.00	−0.300690	−0.150345	+	0.988634i	$0.548038\pi$
$662$	9309.00	0.546533
$663$	0	0
$664$	7308.00	0.427117
$665$	168.000	0.00979663
$666$	0	0
$667$	−225.000	−0.0130615
$668$	−1824.00	−0.105648
$669$	0	0
$670$	−17244.0	−0.994319
$671$	10344.0	0.595120
$672$	0	0
$673$	−12553.0	−0.718993	−0.359497	−	0.933146i	$-0.617052\pi$
$673$	−12553.0	−0.718993	−0.359497	+	0.933146i	$0.617052\pi$
$674$	35175.0	2.01022
$675$	0	0
$676$	1524.00	0.0867091
$677$	−22872.0	−1.29844	−0.649219	−	0.760602i	$-0.724904\pi$
$677$	−22872.0	−1.29844	−0.649219	+	0.760602i	$0.724904\pi$
$678$	0	0
$679$	−2632.00	−0.148758
$680$	29484.0	1.66273
$681$	0	0
$682$	9468.00	0.531596
$683$	21672.0	1.21414	0.607069	−	0.794649i	$-0.292345\pi$
$683$	21672.0	1.21414	0.607069	+	0.794649i	$0.292345\pi$
$684$	0	0
$685$	16488.0	0.919670
$686$	−1029.00	−0.0572703
$687$	0	0
$688$	−36565.0	−2.02620
$689$	−27999.0	−1.54815
$690$	0	0
$691$	17120.0	0.942512	0.471256	−	0.881997i	$-0.343801\pi$
$691$	17120.0	0.942512	0.471256	+	0.881997i	$0.343801\pi$
$692$	744.000	0.0408709
$693$	0	0
$694$	−4518.00	−0.247120
$695$	29400.0	1.60461
$696$	0	0
$697$	28782.0	1.56413
$698$	−30867.0	−1.67383
$699$	0	0
$700$	133.000	0.00718132
$701$	−5562.00	−0.299677	−0.149839	−	0.988710i	$-0.547875\pi$
$701$	−5562.00	−0.299677	−0.149839	+	0.988710i	$0.547875\pi$
$702$	0	0
$703$	436.000	0.0233913
$704$	−5196.00	−0.278170
$705$	0	0
$706$	38079.0	2.02992
$707$	6846.00	0.364173
$708$	0	0
$709$	16256.0	0.861082	0.430541	−	0.902571i	$-0.358323\pi$
$709$	16256.0	0.861082	0.430541	+	0.902571i	$0.358323\pi$
$710$	−4212.00	−0.222639
$711$	0	0
$712$	12285.0	0.646629
$713$	19725.0	1.03605
$714$	0	0
$715$	8784.00	0.459445
$716$	4572.00	0.238636
$717$	0	0
$718$	15093.0	0.784493
$719$	20016.0	1.03821	0.519104	−	0.854711i	$-0.326266\pi$
$719$	20016.0	1.03821	0.519104	+	0.854711i	$0.326266\pi$
$720$	0	0
$721$	−4585.00	−0.236830
$722$	20565.0	1.06004
$723$	0	0
$724$	−3481.00	−0.178688
$725$	−57.0000	−0.00291990
$726$	0	0
$727$	3539.00	0.180542	0.0902711	−	0.995917i	$-0.471227\pi$
$727$	3539.00	0.180542	0.0902711	+	0.995917i	$0.471227\pi$
$728$	−8967.00	−0.456510
$729$	0	0
$730$	15480.0	0.784850
$731$	60255.0	3.04872
$732$	0	0
$733$	−19663.0	−0.990818	−0.495409	−	0.868660i	$-0.664982\pi$
$733$	−19663.0	−0.990818	−0.495409	+	0.868660i	$0.664982\pi$
$734$	−28437.0	−1.43001
$735$	0	0
$736$	3375.00	0.169027
$737$	−5748.00	−0.287287
$738$	0	0
$739$	−17116.0	−0.851992	−0.425996	−	0.904725i	$-0.640076\pi$
$739$	−17116.0	−0.851992	−0.425996	+	0.904725i	$0.640076\pi$
$740$	2616.00	0.129954
$741$	0	0
$742$	−9639.00	−0.476899
$743$	−31641.0	−1.56231	−0.781155	−	0.624338i	$-0.785369\pi$
$743$	−31641.0	−1.56231	−0.781155	+	0.624338i	$0.785369\pi$
$744$	0	0
$745$	11052.0	0.543509
$746$	25698.0	1.26122
$747$	0	0
$748$	−1404.00	−0.0686301
$749$	756.000	0.0368807
$750$	0	0
$751$	−17314.0	−0.841274	−0.420637	−	0.907229i	$-0.638193\pi$
$751$	−17314.0	−0.841274	−0.420637	+	0.907229i	$0.638193\pi$
$752$	22578.0	1.09486
$753$	0	0
$754$	−549.000	−0.0265164
$755$	7800.00	0.375988
$756$	0	0
$757$	12476.0	0.599007	0.299503	−	0.954095i	$-0.403179\pi$
$757$	12476.0	0.599007	0.299503	+	0.954095i	$0.403179\pi$
$758$	6960.00	0.333507
$759$	0	0
$760$	504.000	0.0240553
$761$	21855.0	1.04106	0.520528	−	0.853845i	$-0.325735\pi$
$761$	21855.0	1.04106	0.520528	+	0.853845i	$0.325735\pi$
$762$	0	0
$763$	1904.00	0.0903400
$764$	−3756.00	−0.177863
$765$	0	0
$766$	−3276.00	−0.154526
$767$	−15555.0	−0.732280
$768$	0	0
$769$	−5812.00	−0.272544	−0.136272	−	0.990671i	$-0.543512\pi$
$769$	−5812.00	−0.272544	−0.136272	+	0.990671i	$0.543512\pi$
$770$	3024.00	0.141529
$771$	0	0
$772$	−4867.00	−0.226900
$773$	252.000	0.0117255	0.00586275	−	0.999983i	$-0.498134\pi$
$773$	252.000	0.0117255	0.00586275	+	0.999983i	$0.498134\pi$
$774$	0	0
$775$	4997.00	0.231610
$776$	−7896.00	−0.365270
$777$	0	0
$778$	−37710.0	−1.73775
$779$	492.000	0.0226287
$780$	0	0
$781$	−1404.00	−0.0643266
$782$	−26325.0	−1.20381
$783$	0	0
$784$	−3479.00	−0.158482
$785$	−10452.0	−0.475220
$786$	0	0
$787$	−12886.0	−0.583655	−0.291827	−	0.956471i	$-0.594263\pi$
$787$	−12886.0	−0.583655	−0.291827	+	0.956471i	$0.594263\pi$
$788$	18.0000	0.000813735 0
$789$	0	0
$790$	23256.0	1.04736
$791$	−15792.0	−0.709860
$792$	0	0
$793$	52582.0	2.35466
$794$	20406.0	0.912068
$795$	0	0
$796$	−1807.00	−0.0804616
$797$	39342.0	1.74851	0.874257	−	0.485464i	$-0.161349\pi$
$797$	39342.0	1.74851	0.874257	+	0.485464i	$0.161349\pi$
$798$	0	0
$799$	−37206.0	−1.64738
$800$	855.000	0.0377860
$801$	0	0
$802$	−25578.0	−1.12617
$803$	5160.00	0.226765
$804$	0	0
$805$	6300.00	0.275833
$806$	48129.0	2.10331
$807$	0	0
$808$	20538.0	0.894213
$809$	−33264.0	−1.44561	−0.722806	−	0.691051i	$-0.757148\pi$
$809$	−33264.0	−1.44561	−0.722806	+	0.691051i	$0.757148\pi$
$810$	0	0
$811$	−10258.0	−0.444152	−0.222076	−	0.975029i	$-0.571283\pi$
$811$	−10258.0	−0.444152	−0.222076	+	0.975029i	$0.571283\pi$
$812$	−21.0000	−0.000907581 0
$813$	0	0
$814$	7848.00	0.337927
$815$	−44364.0	−1.90675
$816$	0	0
$817$	1030.00	0.0441067
$818$	−40398.0	−1.72675
$819$	0	0
$820$	2952.00	0.125717
$821$	33063.0	1.40549	0.702745	−	0.711442i	$-0.251958\pi$
$821$	33063.0	1.40549	0.702745	+	0.711442i	$0.251958\pi$
$822$	0	0
$823$	−13102.0	−0.554930	−0.277465	−	0.960736i	$-0.589494\pi$
$823$	−13102.0	−0.554930	−0.277465	+	0.960736i	$0.589494\pi$
$824$	−13755.0	−0.581527
$825$	0	0
$826$	−5355.00	−0.225574
$827$	−25182.0	−1.05884	−0.529422	−	0.848359i	$-0.677591\pi$
$827$	−25182.0	−1.05884	−0.529422	+	0.848359i	$0.677591\pi$
$828$	0	0
$829$	−17170.0	−0.719347	−0.359674	−	0.933078i	$-0.617112\pi$
$829$	−17170.0	−0.719347	−0.359674	+	0.933078i	$0.617112\pi$
$830$	−12528.0	−0.523920
$831$	0	0
$832$	−26413.0	−1.10061
$833$	5733.00	0.238459
$834$	0	0
$835$	−21888.0	−0.907144
$836$	−24.0000	−0.000992892 0
$837$	0	0
$838$	−46017.0	−1.89693
$839$	−4458.00	−0.183441	−0.0917207	−	0.995785i	$-0.529237\pi$
$839$	−4458.00	−0.183441	−0.0917207	+	0.995785i	$0.529237\pi$
$840$	0	0
$841$	−24380.0	−0.999631
$842$	12306.0	0.503673
$843$	0	0
$844$	−4273.00	−0.174269
$845$	18288.0	0.744528
$846$	0	0
$847$	−8309.00	−0.337073
$848$	−32589.0	−1.31971
$849$	0	0
$850$	−6669.00	−0.269112
$851$	16350.0	0.658603
$852$	0	0
$853$	39773.0	1.59648	0.798242	−	0.602336i	$-0.205763\pi$
$853$	39773.0	1.59648	0.798242	+	0.602336i	$0.205763\pi$
$854$	18102.0	0.725337
$855$	0	0
$856$	2268.00	0.0905592
$857$	4191.00	0.167050	0.0835250	−	0.996506i	$-0.473382\pi$
$857$	4191.00	0.167050	0.0835250	+	0.996506i	$0.473382\pi$
$858$	0	0
$859$	19946.0	0.792257	0.396128	−	0.918195i	$-0.370353\pi$
$859$	19946.0	0.792257	0.396128	+	0.918195i	$0.370353\pi$
$860$	6180.00	0.245042
$861$	0	0
$862$	−7776.00	−0.307252
$863$	−41067.0	−1.61986	−0.809929	−	0.586528i	$-0.800494\pi$
$863$	−41067.0	−1.61986	−0.809929	+	0.586528i	$0.800494\pi$
$864$	0	0
$865$	8928.00	0.350938
$866$	28344.0	1.11220
$867$	0	0
$868$	1841.00	0.0719903
$869$	7752.00	0.302611
$870$	0	0
$871$	−29219.0	−1.13668
$872$	5712.00	0.221827
$873$	0	0
$874$	−450.000	−0.0174159
$875$	−8904.00	−0.344012
$876$	0	0
$877$	−32056.0	−1.23427	−0.617135	−	0.786858i	$-0.711707\pi$
$877$	−32056.0	−1.23427	−0.617135	+	0.786858i	$0.711707\pi$
$878$	−34107.0	−1.31100
$879$	0	0
$880$	10224.0	0.391649
$881$	1899.00	0.0726208	0.0363104	−	0.999341i	$-0.488439\pi$
$881$	1899.00	0.0726208	0.0363104	+	0.999341i	$0.488439\pi$
$882$	0	0
$883$	8021.00	0.305694	0.152847	−	0.988250i	$-0.451156\pi$
$883$	8021.00	0.305694	0.152847	+	0.988250i	$0.451156\pi$
$884$	−7137.00	−0.271542
$885$	0	0
$886$	−32526.0	−1.23333
$887$	−6540.00	−0.247567	−0.123783	−	0.992309i	$-0.539503\pi$
$887$	−6540.00	−0.247567	−0.123783	+	0.992309i	$0.539503\pi$
$888$	0	0
$889$	12488.0	0.471129
$890$	−21060.0	−0.793183
$891$	0	0
$892$	2648.00	0.0993964
$893$	−636.000	−0.0238331
$894$	0	0
$895$	54864.0	2.04905
$896$	−11613.0	−0.432995
$897$	0	0
$898$	39420.0	1.46488
$899$	−789.000	−0.0292710
$900$	0	0
$901$	53703.0	1.98569
$902$	8856.00	0.326910
$903$	0	0
$904$	−47376.0	−1.74303
$905$	−41772.0	−1.53431
$906$	0	0
$907$	5780.00	0.211601	0.105800	−	0.994387i	$-0.466260\pi$
$907$	5780.00	0.211601	0.105800	+	0.994387i	$0.466260\pi$
$908$	−5493.00	−0.200762
$909$	0	0
$910$	15372.0	0.559975
$911$	−27480.0	−0.999400	−0.499700	−	0.866199i	$-0.666556\pi$
$911$	−27480.0	−0.999400	−0.499700	+	0.866199i	$0.666556\pi$
$912$	0	0
$913$	−4176.00	−0.151375
$914$	−4137.00	−0.149715
$915$	0	0
$916$	−790.000	−0.0284960
$917$	12243.0	0.440894
$918$	0	0
$919$	−7450.00	−0.267413	−0.133707	−	0.991021i	$-0.542688\pi$
$919$	−7450.00	−0.267413	−0.133707	+	0.991021i	$0.542688\pi$
$920$	18900.0	0.677298
$921$	0	0
$922$	2034.00	0.0726532
$923$	−7137.00	−0.254515
$924$	0	0
$925$	4142.00	0.147230
$926$	−47202.0	−1.67511
$927$	0	0
$928$	−135.000	−0.00477542
$929$	7602.00	0.268475	0.134238	−	0.990949i	$-0.457141\pi$
$929$	7602.00	0.268475	0.134238	+	0.990949i	$0.457141\pi$
$930$	0	0
$931$	98.0000	0.00344986
$932$	−6228.00	−0.218889
$933$	0	0
$934$	−22248.0	−0.779418
$935$	−16848.0	−0.589293
$936$	0	0
$937$	−6316.00	−0.220208	−0.110104	−	0.993920i	$-0.535118\pi$
$937$	−6316.00	−0.220208	−0.110104	+	0.993920i	$0.535118\pi$
$938$	−10059.0	−0.350147
$939$	0	0
$940$	−3816.00	−0.132409
$941$	−25170.0	−0.871964	−0.435982	−	0.899955i	$-0.643599\pi$
$941$	−25170.0	−0.871964	−0.435982	+	0.899955i	$0.643599\pi$
$942$	0	0
$943$	18450.0	0.637131
$944$	−18105.0	−0.624224
$945$	0	0
$946$	18540.0	0.637196
$947$	−20874.0	−0.716277	−0.358138	−	0.933669i	$-0.616588\pi$
$947$	−20874.0	−0.716277	−0.358138	+	0.933669i	$0.616588\pi$
$948$	0	0
$949$	26230.0	0.897220
$950$	−114.000	−0.00389331
$951$	0	0
$952$	17199.0	0.585528
$953$	31824.0	1.08172	0.540861	−	0.841112i	$-0.318099\pi$
$953$	31824.0	1.08172	0.540861	+	0.841112i	$0.318099\pi$
$954$	0	0
$955$	−45072.0	−1.52722
$956$	−4812.00	−0.162794
$957$	0	0
$958$	−3744.00	−0.126266
$959$	9618.00	0.323860
$960$	0	0
$961$	39378.0	1.32181
$962$	39894.0	1.33704
$963$	0	0
$964$	1622.00	0.0541920
$965$	−58404.0	−1.94828
$966$	0	0
$967$	7040.00	0.234117	0.117058	−	0.993125i	$-0.462653\pi$
$967$	7040.00	0.234117	0.117058	+	0.993125i	$0.462653\pi$
$968$	−24927.0	−0.827670
$969$	0	0
$970$	13536.0	0.448056
$971$	27189.0	0.898596	0.449298	−	0.893382i	$-0.351674\pi$
$971$	27189.0	0.898596	0.449298	+	0.893382i	$0.351674\pi$
$972$	0	0
$973$	17150.0	0.565060
$974$	62850.0	2.06760
$975$	0	0
$976$	61202.0	2.00720
$977$	33438.0	1.09496	0.547480	−	0.836819i	$-0.315587\pi$
$977$	33438.0	1.09496	0.547480	+	0.836819i	$0.315587\pi$
$978$	0	0
$979$	−7020.00	−0.229173
$980$	588.000	0.0191663
$981$	0	0
$982$	−2304.00	−0.0748713
$983$	−16212.0	−0.526025	−0.263012	−	0.964792i	$-0.584716\pi$
$983$	−16212.0	−0.526025	−0.263012	+	0.964792i	$0.584716\pi$
$984$	0	0
$985$	216.000	0.00698714
$986$	1053.00	0.0340105
$987$	0	0
$988$	−122.000	−0.00392848
$989$	38625.0	1.24186
$990$	0	0
$991$	24194.0	0.775527	0.387764	−	0.921759i	$-0.373248\pi$
$991$	24194.0	0.775527	0.387764	+	0.921759i	$0.373248\pi$
$992$	11835.0	0.378792
$993$	0	0
$994$	−2457.00	−0.0784017
$995$	−21684.0	−0.690883
$996$	0	0
$997$	−27565.0	−0.875619	−0.437810	−	0.899068i	$-0.644246\pi$
$997$	−27565.0	−0.875619	−0.437810	+	0.899068i	$0.644246\pi$
$998$	−14316.0	−0.454073
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.4.a.a.1.1	✓	1
3.2	odd	2		189.4.a.d.1.1	yes	1
7.6	odd	2		1323.4.a.a.1.1		1
21.20	even	2		1323.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.a.1.1	✓	1	1.1	even	1	trivial
189.4.a.d.1.1	yes	1	3.2	odd	2
1323.4.a.a.1.1		1	7.6	odd	2
1323.4.a.n.1.1		1	21.20	even	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	189.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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