LMFDB - Embedded newform 162.6.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,6,Mod(1,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,4,0,16,21,0,74] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$25.9821788097$
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	$\chi$	$=$	162.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+4.00000 q^{2} +16.0000 q^{4} +21.0000 q^{5} +74.0000 q^{7} +64.0000 q^{8} +84.0000 q^{10} +270.000 q^{11} -115.000 q^{13} +296.000 q^{14} +256.000 q^{16} -861.000 q^{17} +1850.00 q^{19} +336.000 q^{20} +1080.00 q^{22} +3618.00 q^{23} -2684.00 q^{25} -460.000 q^{26} +1184.00 q^{28} +1125.00 q^{29} +5228.00 q^{31} +1024.00 q^{32} -3444.00 q^{34} +1554.00 q^{35} +9917.00 q^{37} +7400.00 q^{38} +1344.00 q^{40} +10758.0 q^{41} -19714.0 q^{43} +4320.00 q^{44} +14472.0 q^{46} +9984.00 q^{47} -11331.0 q^{49} -10736.0 q^{50} -1840.00 q^{52} +36726.0 q^{53} +5670.00 q^{55} +4736.00 q^{56} +4500.00 q^{58} +26460.0 q^{59} -53779.0 q^{61} +20912.0 q^{62} +4096.00 q^{64} -2415.00 q^{65} -12934.0 q^{67} -13776.0 q^{68} +6216.00 q^{70} -4254.00 q^{71} -17521.0 q^{73} +39668.0 q^{74} +29600.0 q^{76} +19980.0 q^{77} -36946.0 q^{79} +5376.00 q^{80} +43032.0 q^{82} -76416.0 q^{83} -18081.0 q^{85} -78856.0 q^{86} +17280.0 q^{88} -45357.0 q^{89} -8510.00 q^{91} +57888.0 q^{92} +39936.0 q^{94} +38850.0 q^{95} +127574. q^{97} -45324.0 q^{98} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	21.0000	0.375659	0.187830	−	0.982202i	$-0.439855\pi$
$5$	21.0000	0.375659	0.187830	+	0.982202i	$0.439855\pi$
$6$	0	0
$7$	74.0000	0.570803	0.285402	−	0.958408i	$-0.407873\pi$
$7$	74.0000	0.570803	0.285402	+	0.958408i	$0.407873\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	84.0000	0.265631
$11$	270.000	0.672794	0.336397	−	0.941720i	$-0.390792\pi$
$11$	270.000	0.672794	0.336397	+	0.941720i	$0.390792\pi$
$12$	0	0
$13$	−115.000	−0.188729	−0.0943647	−	0.995538i	$-0.530082\pi$
$13$	−115.000	−0.188729	−0.0943647	+	0.995538i	$0.530082\pi$
$14$	296.000	0.403619
$15$	0	0
$16$	256.000	0.250000
$17$	−861.000	−0.722572	−0.361286	−	0.932455i	$-0.617662\pi$
$17$	−861.000	−0.722572	−0.361286	+	0.932455i	$0.617662\pi$
$18$	0	0
$19$	1850.00	1.17568	0.587838	−	0.808979i	$-0.299979\pi$
$19$	1850.00	1.17568	0.587838	+	0.808979i	$0.299979\pi$
$20$	336.000	0.187830
$21$	0	0
$22$	1080.00	0.475737
$23$	3618.00	1.42610	0.713048	−	0.701115i	$-0.247314\pi$
$23$	3618.00	1.42610	0.713048	+	0.701115i	$0.247314\pi$
$24$	0	0
$25$	−2684.00	−0.858880
$26$	−460.000	−0.133452
$27$	0	0
$28$	1184.00	0.285402
$29$	1125.00	0.248403	0.124202	−	0.992257i	$-0.460363\pi$
$29$	1125.00	0.248403	0.124202	+	0.992257i	$0.460363\pi$
$30$	0	0
$31$	5228.00	0.977083	0.488541	−	0.872541i	$-0.337529\pi$
$31$	5228.00	0.977083	0.488541	+	0.872541i	$0.337529\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−3444.00	−0.510935
$35$	1554.00	0.214428
$36$	0	0
$37$	9917.00	1.19090	0.595451	−	0.803392i	$-0.296973\pi$
$37$	9917.00	1.19090	0.595451	+	0.803392i	$0.296973\pi$
$38$	7400.00	0.831329
$39$	0	0
$40$	1344.00	0.132816
$41$	10758.0	0.999475	0.499737	−	0.866177i	$-0.333430\pi$
$41$	10758.0	0.999475	0.499737	+	0.866177i	$0.333430\pi$
$42$	0	0
$43$	−19714.0	−1.62594	−0.812968	−	0.582308i	$-0.802150\pi$
$43$	−19714.0	−1.62594	−0.812968	+	0.582308i	$0.802150\pi$
$44$	4320.00	0.336397
$45$	0	0
$46$	14472.0	1.00840
$47$	9984.00	0.659265	0.329632	−	0.944109i	$-0.393075\pi$
$47$	9984.00	0.659265	0.329632	+	0.944109i	$0.393075\pi$
$48$	0	0
$49$	−11331.0	−0.674183
$50$	−10736.0	−0.607320
$51$	0	0
$52$	−1840.00	−0.0943647
$53$	36726.0	1.79591	0.897954	−	0.440090i	$-0.145053\pi$
$53$	36726.0	1.79591	0.897954	+	0.440090i	$0.145053\pi$
$54$	0	0
$55$	5670.00	0.252741
$56$	4736.00	0.201810
$57$	0	0
$58$	4500.00	0.175648
$59$	26460.0	0.989600	0.494800	−	0.869007i	$-0.335241\pi$
$59$	26460.0	0.989600	0.494800	+	0.869007i	$0.335241\pi$
$60$	0	0
$61$	−53779.0	−1.85050	−0.925248	−	0.379363i	$-0.876143\pi$
$61$	−53779.0	−1.85050	−0.925248	+	0.379363i	$0.876143\pi$
$62$	20912.0	0.690902
$63$	0	0
$64$	4096.00	0.125000
$65$	−2415.00	−0.0708980
$66$	0	0
$67$	−12934.0	−0.352003	−0.176001	−	0.984390i	$-0.556316\pi$
$67$	−12934.0	−0.352003	−0.176001	+	0.984390i	$0.556316\pi$
$68$	−13776.0	−0.361286
$69$	0	0
$70$	6216.00	0.151623
$71$	−4254.00	−0.100150	−0.0500751	−	0.998745i	$-0.515946\pi$
$71$	−4254.00	−0.100150	−0.0500751	+	0.998745i	$0.515946\pi$
$72$	0	0
$73$	−17521.0	−0.384815	−0.192407	−	0.981315i	$-0.561630\pi$
$73$	−17521.0	−0.384815	−0.192407	+	0.981315i	$0.561630\pi$
$74$	39668.0	0.842095
$75$	0	0
$76$	29600.0	0.587838
$77$	19980.0	0.384033
$78$	0	0
$79$	−36946.0	−0.666039	−0.333020	−	0.942920i	$-0.608068\pi$
$79$	−36946.0	−0.666039	−0.333020	+	0.942920i	$0.608068\pi$
$80$	5376.00	0.0939149
$81$	0	0
$82$	43032.0	0.706735
$83$	−76416.0	−1.21756	−0.608778	−	0.793340i	$-0.708340\pi$
$83$	−76416.0	−1.21756	−0.608778	+	0.793340i	$0.708340\pi$
$84$	0	0
$85$	−18081.0	−0.271441
$86$	−78856.0	−1.14971
$87$	0	0
$88$	17280.0	0.237869
$89$	−45357.0	−0.606973	−0.303486	−	0.952836i	$-0.598151\pi$
$89$	−45357.0	−0.606973	−0.303486	+	0.952836i	$0.598151\pi$
$90$	0	0
$91$	−8510.00	−0.107727
$92$	57888.0	0.713048
$93$	0	0
$94$	39936.0	0.466171
$95$	38850.0	0.441654
$96$	0	0
$97$	127574.	1.37668	0.688340	−	0.725388i	$-0.258340\pi$
$97$	127574.	1.37668	0.688340	+	0.725388i	$0.258340\pi$
$98$	−45324.0	−0.476720
$99$	0	0
$100$	−42944.0	−0.429440
$101$	−78870.0	−0.769322	−0.384661	−	0.923058i	$-0.625682\pi$
$101$	−78870.0	−0.769322	−0.384661	+	0.923058i	$0.625682\pi$
$102$	0	0
$103$	−17488.0	−0.162423	−0.0812114	−	0.996697i	$-0.525879\pi$
$103$	−17488.0	−0.162423	−0.0812114	+	0.996697i	$0.525879\pi$
$104$	−7360.00	−0.0667259
$105$	0	0
$106$	146904.	1.26990
$107$	−134364.	−1.13455	−0.567275	−	0.823529i	$-0.692002\pi$
$107$	−134364.	−1.13455	−0.567275	+	0.823529i	$0.692002\pi$
$108$	0	0
$109$	−123487.	−0.995531	−0.497766	−	0.867312i	$-0.665846\pi$
$109$	−123487.	−0.995531	−0.497766	+	0.867312i	$0.665846\pi$
$110$	22680.0	0.178715
$111$	0	0
$112$	18944.0	0.142701
$113$	174039.	1.28218	0.641092	−	0.767464i	$-0.278482\pi$
$113$	174039.	1.28218	0.641092	+	0.767464i	$0.278482\pi$
$114$	0	0
$115$	75978.0	0.535727
$116$	18000.0	0.124202
$117$	0	0
$118$	105840.	0.699753
$119$	−63714.0	−0.412446
$120$	0	0
$121$	−88151.0	−0.547348
$122$	−215116.	−1.30850
$123$	0	0
$124$	83648.0	0.488541
$125$	−121989.	−0.698306
$126$	0	0
$127$	−312982.	−1.72191	−0.860954	−	0.508682i	$-0.830133\pi$
$127$	−312982.	−1.72191	−0.860954	+	0.508682i	$0.830133\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−9660.00	−0.0501324
$131$	−105810.	−0.538702	−0.269351	−	0.963042i	$-0.586809\pi$
$131$	−105810.	−0.538702	−0.269351	+	0.963042i	$0.586809\pi$
$132$	0	0
$133$	136900.	0.671080
$134$	−51736.0	−0.248903
$135$	0	0
$136$	−55104.0	−0.255468
$137$	−284481.	−1.29495	−0.647473	−	0.762088i	$-0.724174\pi$
$137$	−284481.	−1.29495	−0.647473	+	0.762088i	$0.724174\pi$
$138$	0	0
$139$	−162328.	−0.712617	−0.356309	−	0.934368i	$-0.615965\pi$
$139$	−162328.	−0.712617	−0.356309	+	0.934368i	$0.615965\pi$
$140$	24864.0	0.107214
$141$	0	0
$142$	−17016.0	−0.0708169
$143$	−31050.0	−0.126976
$144$	0	0
$145$	23625.0	0.0933151
$146$	−70084.0	−0.272105
$147$	0	0
$148$	158672.	0.595451
$149$	−270711.	−0.998942	−0.499471	−	0.866331i	$-0.666472\pi$
$149$	−270711.	−0.998942	−0.499471	+	0.866331i	$0.666472\pi$
$150$	0	0
$151$	−16852.0	−0.0601463	−0.0300732	−	0.999548i	$-0.509574\pi$
$151$	−16852.0	−0.0601463	−0.0300732	+	0.999548i	$0.509574\pi$
$152$	118400.	0.415664
$153$	0	0
$154$	79920.0	0.271552
$155$	109788.	0.367050
$156$	0	0
$157$	247373.	0.800946	0.400473	−	0.916309i	$-0.368846\pi$
$157$	247373.	0.800946	0.400473	+	0.916309i	$0.368846\pi$
$158$	−147784.	−0.470961
$159$	0	0
$160$	21504.0	0.0664078
$161$	267732.	0.814021
$162$	0	0
$163$	−200116.	−0.589947	−0.294973	−	0.955505i	$-0.595311\pi$
$163$	−200116.	−0.589947	−0.294973	+	0.955505i	$0.595311\pi$
$164$	172128.	0.499737
$165$	0	0
$166$	−305664.	−0.860942
$167$	14118.0	0.0391726	0.0195863	−	0.999808i	$-0.493765\pi$
$167$	14118.0	0.0391726	0.0195863	+	0.999808i	$0.493765\pi$
$168$	0	0
$169$	−358068.	−0.964381
$170$	−72324.0	−0.191938
$171$	0	0
$172$	−315424.	−0.812968
$173$	−48399.0	−0.122948	−0.0614740	−	0.998109i	$-0.519580\pi$
$173$	−48399.0	−0.122948	−0.0614740	+	0.998109i	$0.519580\pi$
$174$	0	0
$175$	−198616.	−0.490252
$176$	69120.0	0.168198
$177$	0	0
$178$	−181428.	−0.429195
$179$	375396.	0.875703	0.437852	−	0.899047i	$-0.355739\pi$
$179$	375396.	0.875703	0.437852	+	0.899047i	$0.355739\pi$
$180$	0	0
$181$	440702.	0.999882	0.499941	−	0.866060i	$-0.333355\pi$
$181$	440702.	0.999882	0.499941	+	0.866060i	$0.333355\pi$
$182$	−34040.0	−0.0761748
$183$	0	0
$184$	231552.	0.504201
$185$	208257.	0.447374
$186$	0	0
$187$	−232470.	−0.486142
$188$	159744.	0.329632
$189$	0	0
$190$	155400.	0.312296
$191$	486978.	0.965886	0.482943	−	0.875652i	$-0.339568\pi$
$191$	486978.	0.965886	0.482943	+	0.875652i	$0.339568\pi$
$192$	0	0
$193$	−104185.	−0.201332	−0.100666	−	0.994920i	$-0.532097\pi$
$193$	−104185.	−0.201332	−0.100666	+	0.994920i	$0.532097\pi$
$194$	510296.	0.973459
$195$	0	0
$196$	−181296.	−0.337092
$197$	39369.0	0.0722751	0.0361376	−	0.999347i	$-0.488495\pi$
$197$	39369.0	0.0722751	0.0361376	+	0.999347i	$0.488495\pi$
$198$	0	0
$199$	952484.	1.70500	0.852501	−	0.522725i	$-0.175085\pi$
$199$	952484.	1.70500	0.852501	+	0.522725i	$0.175085\pi$
$200$	−171776.	−0.303660
$201$	0	0
$202$	−315480.	−0.543993
$203$	83250.0	0.141790
$204$	0	0
$205$	225918.	0.375462
$206$	−69952.0	−0.114850
$207$	0	0
$208$	−29440.0	−0.0471823
$209$	499500.	0.790988
$210$	0	0
$211$	441110.	0.682089	0.341044	−	0.940047i	$-0.389219\pi$
$211$	441110.	0.682089	0.341044	+	0.940047i	$0.389219\pi$
$212$	587616.	0.897954
$213$	0	0
$214$	−537456.	−0.802248
$215$	−413994.	−0.610798
$216$	0	0
$217$	386872.	0.557722
$218$	−493948.	−0.703947
$219$	0	0
$220$	90720.0	0.126371
$221$	99015.0	0.136370
$222$	0	0
$223$	−825106.	−1.11109	−0.555543	−	0.831488i	$-0.687490\pi$
$223$	−825106.	−1.11109	−0.555543	+	0.831488i	$0.687490\pi$
$224$	75776.0	0.100905
$225$	0	0
$226$	696156.	0.906641
$227$	1.15570e6	1.48861	0.744305	−	0.667839i	$-0.232781\pi$
$227$	1.15570e6	1.48861	0.744305	+	0.667839i	$0.232781\pi$
$228$	0	0
$229$	1.34336e6	1.69279	0.846394	−	0.532557i	$-0.178769\pi$
$229$	1.34336e6	1.69279	0.846394	+	0.532557i	$0.178769\pi$
$230$	303912.	0.378816
$231$	0	0
$232$	72000.0	0.0878239
$233$	1.07555e6	1.29790	0.648950	−	0.760831i	$-0.275208\pi$
$233$	1.07555e6	1.29790	0.648950	+	0.760831i	$0.275208\pi$
$234$	0	0
$235$	209664.	0.247659
$236$	423360.	0.494800
$237$	0	0
$238$	−254856.	−0.291644
$239$	36096.0	0.0408756	0.0204378	−	0.999791i	$-0.493494\pi$
$239$	36096.0	0.0408756	0.0204378	+	0.999791i	$0.493494\pi$
$240$	0	0
$241$	72875.0	0.0808232	0.0404116	−	0.999183i	$-0.487133\pi$
$241$	72875.0	0.0808232	0.0404116	+	0.999183i	$0.487133\pi$
$242$	−352604.	−0.387034
$243$	0	0
$244$	−860464.	−0.925248
$245$	−237951.	−0.253263
$246$	0	0
$247$	−212750.	−0.221885
$248$	334592.	0.345451
$249$	0	0
$250$	−487956.	−0.493777
$251$	−1.65869e6	−1.66181	−0.830906	−	0.556413i	$-0.812177\pi$
$251$	−1.65869e6	−1.66181	−0.830906	+	0.556413i	$0.812177\pi$
$252$	0	0
$253$	976860.	0.959469
$254$	−1.25193e6	−1.21757
$255$	0	0
$256$	65536.0	0.0625000
$257$	−794181.	−0.750044	−0.375022	−	0.927016i	$-0.622365\pi$
$257$	−794181.	−0.750044	−0.375022	+	0.927016i	$0.622365\pi$
$258$	0	0
$259$	733858.	0.679771
$260$	−38640.0	−0.0354490
$261$	0	0
$262$	−423240.	−0.380920
$263$	1.59079e6	1.41816	0.709078	−	0.705130i	$-0.249111\pi$
$263$	1.59079e6	1.41816	0.709078	+	0.705130i	$0.249111\pi$
$264$	0	0
$265$	771246.	0.674650
$266$	547600.	0.474525
$267$	0	0
$268$	−206944.	−0.176001
$269$	1.57975e6	1.33109	0.665545	−	0.746358i	$-0.268199\pi$
$269$	1.57975e6	1.33109	0.665545	+	0.746358i	$0.268199\pi$
$270$	0	0
$271$	−415762.	−0.343892	−0.171946	−	0.985106i	$-0.555005\pi$
$271$	−415762.	−0.343892	−0.171946	+	0.985106i	$0.555005\pi$
$272$	−220416.	−0.180643
$273$	0	0
$274$	−1.13792e6	−0.915665
$275$	−724680.	−0.577849
$276$	0	0
$277$	562406.	0.440403	0.220202	−	0.975454i	$-0.429328\pi$
$277$	562406.	0.440403	0.220202	+	0.975454i	$0.429328\pi$
$278$	−649312.	−0.503897
$279$	0	0
$280$	99456.0	0.0758116
$281$	−2.42663e6	−1.83331	−0.916657	−	0.399675i	$-0.869123\pi$
$281$	−2.42663e6	−1.83331	−0.916657	+	0.399675i	$0.869123\pi$
$282$	0	0
$283$	−1.20293e6	−0.892843	−0.446421	−	0.894823i	$-0.647302\pi$
$283$	−1.20293e6	−0.892843	−0.446421	+	0.894823i	$0.647302\pi$
$284$	−68064.0	−0.0500751
$285$	0	0
$286$	−124200.	−0.0897856
$287$	796092.	0.570504
$288$	0	0
$289$	−678536.	−0.477890
$290$	94500.0	0.0659837
$291$	0	0
$292$	−280336.	−0.192407
$293$	−672471.	−0.457619	−0.228810	−	0.973471i	$-0.573483\pi$
$293$	−672471.	−0.457619	−0.228810	+	0.973471i	$0.573483\pi$
$294$	0	0
$295$	555660.	0.371753
$296$	634688.	0.421047
$297$	0	0
$298$	−1.08284e6	−0.706359
$299$	−416070.	−0.269146
$300$	0	0
$301$	−1.45884e6	−0.928090
$302$	−67408.0	−0.0425299
$303$	0	0
$304$	473600.	0.293919
$305$	−1.12936e6	−0.695156
$306$	0	0
$307$	−2.79488e6	−1.69245	−0.846226	−	0.532823i	$-0.821131\pi$
$307$	−2.79488e6	−1.69245	−0.846226	+	0.532823i	$0.821131\pi$
$308$	319680.	0.192017
$309$	0	0
$310$	439152.	0.259544
$311$	−3.05959e6	−1.79375	−0.896876	−	0.442281i	$-0.854169\pi$
$311$	−3.05959e6	−1.79375	−0.896876	+	0.442281i	$0.854169\pi$
$312$	0	0
$313$	2.63474e6	1.52012	0.760059	−	0.649854i	$-0.225170\pi$
$313$	2.63474e6	1.52012	0.760059	+	0.649854i	$0.225170\pi$
$314$	989492.	0.566354
$315$	0	0
$316$	−591136.	−0.333020
$317$	808377.	0.451820	0.225910	−	0.974148i	$-0.427464\pi$
$317$	808377.	0.451820	0.225910	+	0.974148i	$0.427464\pi$
$318$	0	0
$319$	303750.	0.167124
$320$	86016.0	0.0469574
$321$	0	0
$322$	1.07093e6	0.575600
$323$	−1.59285e6	−0.849510
$324$	0	0
$325$	308660.	0.162096
$326$	−800464.	−0.417155
$327$	0	0
$328$	688512.	0.353368
$329$	738816.	0.376311
$330$	0	0
$331$	−853678.	−0.428276	−0.214138	−	0.976803i	$-0.568694\pi$
$331$	−853678.	−0.428276	−0.214138	+	0.976803i	$0.568694\pi$
$332$	−1.22266e6	−0.608778
$333$	0	0
$334$	56472.0	0.0276992
$335$	−271614.	−0.132233
$336$	0	0
$337$	297878.	0.142877	0.0714387	−	0.997445i	$-0.477241\pi$
$337$	297878.	0.142877	0.0714387	+	0.997445i	$0.477241\pi$
$338$	−1.43227e6	−0.681921
$339$	0	0
$340$	−289296.	−0.135720
$341$	1.41156e6	0.657375
$342$	0	0
$343$	−2.08221e6	−0.955630
$344$	−1.26170e6	−0.574855
$345$	0	0
$346$	−193596.	−0.0869373
$347$	88746.0	0.0395663	0.0197831	−	0.999804i	$-0.493702\pi$
$347$	88746.0	0.0395663	0.0197831	+	0.999804i	$0.493702\pi$
$348$	0	0
$349$	2.36175e6	1.03793	0.518967	−	0.854794i	$-0.326317\pi$
$349$	2.36175e6	1.03793	0.518967	+	0.854794i	$0.326317\pi$
$350$	−794464.	−0.346660
$351$	0	0
$352$	276480.	0.118934
$353$	−3.86077e6	−1.64906	−0.824530	−	0.565818i	$-0.808561\pi$
$353$	−3.86077e6	−1.64906	−0.824530	+	0.565818i	$0.808561\pi$
$354$	0	0
$355$	−89334.0	−0.0376223
$356$	−725712.	−0.303486
$357$	0	0
$358$	1.50158e6	0.619216
$359$	−3.74852e6	−1.53505	−0.767527	−	0.641017i	$-0.778513\pi$
$359$	−3.74852e6	−1.53505	−0.767527	+	0.641017i	$0.778513\pi$
$360$	0	0
$361$	946401.	0.382215
$362$	1.76281e6	0.707023
$363$	0	0
$364$	−136160.	−0.0538637
$365$	−367941.	−0.144559
$366$	0	0
$367$	−2.66339e6	−1.03221	−0.516107	−	0.856524i	$-0.672619\pi$
$367$	−2.66339e6	−1.03221	−0.516107	+	0.856524i	$0.672619\pi$
$368$	926208.	0.356524
$369$	0	0
$370$	833028.	0.316341
$371$	2.71772e6	1.02511
$372$	0	0
$373$	959870.	0.357224	0.178612	−	0.983920i	$-0.442839\pi$
$373$	959870.	0.357224	0.178612	+	0.983920i	$0.442839\pi$
$374$	−929880.	−0.343754
$375$	0	0
$376$	638976.	0.233085
$377$	−129375.	−0.0468810
$378$	0	0
$379$	−193780.	−0.0692964	−0.0346482	−	0.999400i	$-0.511031\pi$
$379$	−193780.	−0.0692964	−0.0346482	+	0.999400i	$0.511031\pi$
$380$	621600.	0.220827
$381$	0	0
$382$	1.94791e6	0.682985
$383$	5.55393e6	1.93465	0.967327	−	0.253531i	$-0.0815919\pi$
$383$	5.55393e6	1.93465	0.967327	+	0.253531i	$0.0815919\pi$
$384$	0	0
$385$	419580.	0.144266
$386$	−416740.	−0.142363
$387$	0	0
$388$	2.04118e6	0.688340
$389$	−4.82795e6	−1.61767	−0.808833	−	0.588038i	$-0.799901\pi$
$389$	−4.82795e6	−1.61767	−0.808833	+	0.588038i	$0.799901\pi$
$390$	0	0
$391$	−3.11510e6	−1.03046
$392$	−725184.	−0.238360
$393$	0	0
$394$	157476.	0.0511062
$395$	−775866.	−0.250204
$396$	0	0
$397$	313409.	0.0998011	0.0499005	−	0.998754i	$-0.484110\pi$
$397$	313409.	0.0998011	0.0499005	+	0.998754i	$0.484110\pi$
$398$	3.80994e6	1.20562
$399$	0	0
$400$	−687104.	−0.214720
$401$	5.28290e6	1.64063	0.820316	−	0.571911i	$-0.193798\pi$
$401$	5.28290e6	1.64063	0.820316	+	0.571911i	$0.193798\pi$
$402$	0	0
$403$	−601220.	−0.184404
$404$	−1.26192e6	−0.384661
$405$	0	0
$406$	333000.	0.100260
$407$	2.67759e6	0.801232
$408$	0	0
$409$	4.58812e6	1.35621	0.678104	−	0.734966i	$-0.262802\pi$
$409$	4.58812e6	1.35621	0.678104	+	0.734966i	$0.262802\pi$
$410$	903672.	0.265492
$411$	0	0
$412$	−279808.	−0.0812114
$413$	1.95804e6	0.564867
$414$	0	0
$415$	−1.60474e6	−0.457387
$416$	−117760.	−0.0333630
$417$	0	0
$418$	1.99800e6	0.559313
$419$	−2.36698e6	−0.658656	−0.329328	−	0.944216i	$-0.606822\pi$
$419$	−2.36698e6	−0.658656	−0.329328	+	0.944216i	$0.606822\pi$
$420$	0	0
$421$	−454795.	−0.125058	−0.0625289	−	0.998043i	$-0.519917\pi$
$421$	−454795.	−0.125058	−0.0625289	+	0.998043i	$0.519917\pi$
$422$	1.76444e6	0.482309
$423$	0	0
$424$	2.35046e6	0.634949
$425$	2.31092e6	0.620602
$426$	0	0
$427$	−3.97965e6	−1.05627
$428$	−2.14982e6	−0.567275
$429$	0	0
$430$	−1.65598e6	−0.431900
$431$	1.26286e6	0.327462	0.163731	−	0.986505i	$-0.447647\pi$
$431$	1.26286e6	0.327462	0.163731	+	0.986505i	$0.447647\pi$
$432$	0	0
$433$	5.48900e6	1.40693	0.703467	−	0.710728i	$-0.251634\pi$
$433$	5.48900e6	1.40693	0.703467	+	0.710728i	$0.251634\pi$
$434$	1.54749e6	0.394369
$435$	0	0
$436$	−1.97579e6	−0.497766
$437$	6.69330e6	1.67663
$438$	0	0
$439$	−6.11644e6	−1.51474	−0.757369	−	0.652987i	$-0.773515\pi$
$439$	−6.11644e6	−1.51474	−0.757369	+	0.652987i	$0.773515\pi$
$440$	362880.	0.0893576
$441$	0	0
$442$	396060.	0.0964285
$443$	739200.	0.178959	0.0894793	−	0.995989i	$-0.471480\pi$
$443$	739200.	0.178959	0.0894793	+	0.995989i	$0.471480\pi$
$444$	0	0
$445$	−952497.	−0.228015
$446$	−3.30042e6	−0.785656
$447$	0	0
$448$	303104.	0.0713504
$449$	7.31432e6	1.71221	0.856107	−	0.516799i	$-0.172876\pi$
$449$	7.31432e6	1.71221	0.856107	+	0.516799i	$0.172876\pi$
$450$	0	0
$451$	2.90466e6	0.672441
$452$	2.78462e6	0.641092
$453$	0	0
$454$	4.62281e6	1.05261
$455$	−178710.	−0.0404688
$456$	0	0
$457$	−7.68904e6	−1.72219	−0.861096	−	0.508443i	$-0.830221\pi$
$457$	−7.68904e6	−1.72219	−0.861096	+	0.508443i	$0.830221\pi$
$458$	5.37343e6	1.19698
$459$	0	0
$460$	1.21565e6	0.267863
$461$	−5.93937e6	−1.30163	−0.650816	−	0.759236i	$-0.725573\pi$
$461$	−5.93937e6	−1.30163	−0.650816	+	0.759236i	$0.725573\pi$
$462$	0	0
$463$	−455956.	−0.0988486	−0.0494243	−	0.998778i	$-0.515739\pi$
$463$	−455956.	−0.0988486	−0.0494243	+	0.998778i	$0.515739\pi$
$464$	288000.	0.0621008
$465$	0	0
$466$	4.30220e6	0.917754
$467$	−5.97097e6	−1.26693	−0.633465	−	0.773771i	$-0.718368\pi$
$467$	−5.97097e6	−1.26693	−0.633465	+	0.773771i	$0.718368\pi$
$468$	0	0
$469$	−957116.	−0.200924
$470$	838656.	0.175121
$471$	0	0
$472$	1.69344e6	0.349877
$473$	−5.32278e6	−1.09392
$474$	0	0
$475$	−4.96540e6	−1.00976
$476$	−1.01942e6	−0.206223
$477$	0	0
$478$	144384.	0.0289034
$479$	4.24839e6	0.846030	0.423015	−	0.906123i	$-0.360972\pi$
$479$	4.24839e6	0.846030	0.423015	+	0.906123i	$0.360972\pi$
$480$	0	0
$481$	−1.14046e6	−0.224758
$482$	291500.	0.0571506
$483$	0	0
$484$	−1.41042e6	−0.273674
$485$	2.67905e6	0.517163
$486$	0	0
$487$	−1.04048e6	−0.198797	−0.0993985	−	0.995048i	$-0.531692\pi$
$487$	−1.04048e6	−0.198797	−0.0993985	+	0.995048i	$0.531692\pi$
$488$	−3.44186e6	−0.654249
$489$	0	0
$490$	−951804.	−0.179084
$491$	5.55698e6	1.04024	0.520122	−	0.854092i	$-0.325887\pi$
$491$	5.55698e6	1.04024	0.520122	+	0.854092i	$0.325887\pi$
$492$	0	0
$493$	−968625.	−0.179489
$494$	−851000.	−0.156896
$495$	0	0
$496$	1.33837e6	0.244271
$497$	−314796.	−0.0571661
$498$	0	0
$499$	−353278.	−0.0635134	−0.0317567	−	0.999496i	$-0.510110\pi$
$499$	−353278.	−0.0635134	−0.0317567	+	0.999496i	$0.510110\pi$
$500$	−1.95182e6	−0.349153
$501$	0	0
$502$	−6.63478e6	−1.17508
$503$	4.21978e6	0.743651	0.371826	−	0.928303i	$-0.378732\pi$
$503$	4.21978e6	0.743651	0.371826	+	0.928303i	$0.378732\pi$
$504$	0	0
$505$	−1.65627e6	−0.289003
$506$	3.90744e6	0.678447
$507$	0	0
$508$	−5.00771e6	−0.860954
$509$	346134.	0.0592175	0.0296087	−	0.999562i	$-0.490574\pi$
$509$	346134.	0.0592175	0.0296087	+	0.999562i	$0.490574\pi$
$510$	0	0
$511$	−1.29655e6	−0.219654
$512$	262144.	0.0441942
$513$	0	0
$514$	−3.17672e6	−0.530361
$515$	−367248.	−0.0610157
$516$	0	0
$517$	2.69568e6	0.443549
$518$	2.93543e6	0.480671
$519$	0	0
$520$	−154560.	−0.0250662
$521$	1.90025e6	0.306703	0.153351	−	0.988172i	$-0.450993\pi$
$521$	1.90025e6	0.306703	0.153351	+	0.988172i	$0.450993\pi$
$522$	0	0
$523$	−5.16589e6	−0.825831	−0.412915	−	0.910769i	$-0.635489\pi$
$523$	−5.16589e6	−0.825831	−0.412915	+	0.910769i	$0.635489\pi$
$524$	−1.69296e6	−0.269351
$525$	0	0
$526$	6.36317e6	1.00279
$527$	−4.50131e6	−0.706012
$528$	0	0
$529$	6.65358e6	1.03375
$530$	3.08498e6	0.477049
$531$	0	0
$532$	2.19040e6	0.335540
$533$	−1.23717e6	−0.188630
$534$	0	0
$535$	−2.82164e6	−0.426204
$536$	−827776.	−0.124452
$537$	0	0
$538$	6.31900e6	0.941223
$539$	−3.05937e6	−0.453586
$540$	0	0
$541$	4.15125e6	0.609797	0.304899	−	0.952385i	$-0.401377\pi$
$541$	4.15125e6	0.609797	0.304899	+	0.952385i	$0.401377\pi$
$542$	−1.66305e6	−0.243168
$543$	0	0
$544$	−881664.	−0.127734
$545$	−2.59323e6	−0.373981
$546$	0	0
$547$	9.12524e6	1.30400	0.651998	−	0.758221i	$-0.273931\pi$
$547$	9.12524e6	1.30400	0.651998	+	0.758221i	$0.273931\pi$
$548$	−4.55170e6	−0.647473
$549$	0	0
$550$	−2.89872e6	−0.408601
$551$	2.08125e6	0.292042
$552$	0	0
$553$	−2.73400e6	−0.380177
$554$	2.24962e6	0.311412
$555$	0	0
$556$	−2.59725e6	−0.356309
$557$	−3.78858e6	−0.517415	−0.258707	−	0.965956i	$-0.583297\pi$
$557$	−3.78858e6	−0.517415	−0.258707	+	0.965956i	$0.583297\pi$
$558$	0	0
$559$	2.26711e6	0.306862
$560$	397824.	0.0536069
$561$	0	0
$562$	−9.70650e6	−1.29635
$563$	5.68666e6	0.756112	0.378056	−	0.925783i	$-0.376593\pi$
$563$	5.68666e6	0.756112	0.378056	+	0.925783i	$0.376593\pi$
$564$	0	0
$565$	3.65482e6	0.481665
$566$	−4.81173e6	−0.631335
$567$	0	0
$568$	−272256.	−0.0354084
$569$	−1.05039e7	−1.36010	−0.680048	−	0.733168i	$-0.738041\pi$
$569$	−1.05039e7	−1.36010	−0.680048	+	0.733168i	$0.738041\pi$
$570$	0	0
$571$	7.00992e6	0.899751	0.449876	−	0.893091i	$-0.351468\pi$
$571$	7.00992e6	0.899751	0.449876	+	0.893091i	$0.351468\pi$
$572$	−496800.	−0.0634880
$573$	0	0
$574$	3.18437e6	0.403407
$575$	−9.71071e6	−1.22485
$576$	0	0
$577$	−3.54196e6	−0.442898	−0.221449	−	0.975172i	$-0.571079\pi$
$577$	−3.54196e6	−0.442898	−0.221449	+	0.975172i	$0.571079\pi$
$578$	−2.71414e6	−0.337920
$579$	0	0
$580$	378000.	0.0466575
$581$	−5.65478e6	−0.694985
$582$	0	0
$583$	9.91602e6	1.20828
$584$	−1.12134e6	−0.136053
$585$	0	0
$586$	−2.68988e6	−0.323586
$587$	1.52778e7	1.83006	0.915032	−	0.403381i	$-0.132165\pi$
$587$	1.52778e7	1.83006	0.915032	+	0.403381i	$0.132165\pi$
$588$	0	0
$589$	9.67180e6	1.14873
$590$	2.22264e6	0.262869
$591$	0	0
$592$	2.53875e6	0.297725
$593$	−278457.	−0.0325178	−0.0162589	−	0.999868i	$-0.505176\pi$
$593$	−278457.	−0.0325178	−0.0162589	+	0.999868i	$0.505176\pi$
$594$	0	0
$595$	−1.33799e6	−0.154939
$596$	−4.33138e6	−0.499471
$597$	0	0
$598$	−1.66428e6	−0.190315
$599$	−1.00636e7	−1.14601	−0.573003	−	0.819553i	$-0.694222\pi$
$599$	−1.00636e7	−1.14601	−0.573003	+	0.819553i	$0.694222\pi$
$600$	0	0
$601$	1.36578e7	1.54239	0.771194	−	0.636600i	$-0.219660\pi$
$601$	1.36578e7	1.54239	0.771194	+	0.636600i	$0.219660\pi$
$602$	−5.83534e6	−0.656259
$603$	0	0
$604$	−269632.	−0.0300732
$605$	−1.85117e6	−0.205617
$606$	0	0
$607$	−8.95938e6	−0.986975	−0.493487	−	0.869753i	$-0.664278\pi$
$607$	−8.95938e6	−0.986975	−0.493487	+	0.869753i	$0.664278\pi$
$608$	1.89440e6	0.207832
$609$	0	0
$610$	−4.51744e6	−0.491550
$611$	−1.14816e6	−0.124423
$612$	0	0
$613$	529958.	0.0569627	0.0284813	−	0.999594i	$-0.490933\pi$
$613$	529958.	0.0569627	0.0284813	+	0.999594i	$0.490933\pi$
$614$	−1.11795e7	−1.19674
$615$	0	0
$616$	1.27872e6	0.135776
$617$	−5.97027e6	−0.631366	−0.315683	−	0.948865i	$-0.602234\pi$
$617$	−5.97027e6	−0.631366	−0.315683	+	0.948865i	$0.602234\pi$
$618$	0	0
$619$	−1.17134e6	−0.122873	−0.0614363	−	0.998111i	$-0.519568\pi$
$619$	−1.17134e6	−0.122873	−0.0614363	+	0.998111i	$0.519568\pi$
$620$	1.75661e6	0.183525
$621$	0	0
$622$	−1.22384e7	−1.26837
$623$	−3.35642e6	−0.346462
$624$	0	0
$625$	5.82573e6	0.596555
$626$	1.05390e7	1.07489
$627$	0	0
$628$	3.95797e6	0.400473
$629$	−8.53854e6	−0.860512
$630$	0	0
$631$	−1.49126e7	−1.49101	−0.745504	−	0.666501i	$-0.767791\pi$
$631$	−1.49126e7	−1.49101	−0.745504	+	0.666501i	$0.767791\pi$
$632$	−2.36454e6	−0.235480
$633$	0	0
$634$	3.23351e6	0.319485
$635$	−6.57262e6	−0.646851
$636$	0	0
$637$	1.30306e6	0.127238
$638$	1.21500e6	0.118175
$639$	0	0
$640$	344064.	0.0332039
$641$	7.88504e6	0.757981	0.378991	−	0.925400i	$-0.376271\pi$
$641$	7.88504e6	0.757981	0.378991	+	0.925400i	$0.376271\pi$
$642$	0	0
$643$	1.85804e6	0.177226	0.0886130	−	0.996066i	$-0.471757\pi$
$643$	1.85804e6	0.177226	0.0886130	+	0.996066i	$0.471757\pi$
$644$	4.28371e6	0.407010
$645$	0	0
$646$	−6.37140e6	−0.600694
$647$	−1.54147e6	−0.144768	−0.0723841	−	0.997377i	$-0.523061\pi$
$647$	−1.54147e6	−0.144768	−0.0723841	+	0.997377i	$0.523061\pi$
$648$	0	0
$649$	7.14420e6	0.665797
$650$	1.23464e6	0.114619
$651$	0	0
$652$	−3.20186e6	−0.294973
$653$	4.13336e6	0.379333	0.189666	−	0.981849i	$-0.439259\pi$
$653$	4.13336e6	0.379333	0.189666	+	0.981849i	$0.439259\pi$
$654$	0	0
$655$	−2.22201e6	−0.202368
$656$	2.75405e6	0.249869
$657$	0	0
$658$	2.95526e6	0.266092
$659$	−1.70820e7	−1.53224	−0.766118	−	0.642699i	$-0.777814\pi$
$659$	−1.70820e7	−1.53224	−0.766118	+	0.642699i	$0.777814\pi$
$660$	0	0
$661$	−6.14644e6	−0.547167	−0.273584	−	0.961848i	$-0.588209\pi$
$661$	−6.14644e6	−0.547167	−0.273584	+	0.961848i	$0.588209\pi$
$662$	−3.41471e6	−0.302837
$663$	0	0
$664$	−4.89062e6	−0.430471
$665$	2.87490e6	0.252098
$666$	0	0
$667$	4.07025e6	0.354247
$668$	225888.	0.0195863
$669$	0	0
$670$	−1.08646e6	−0.0935029
$671$	−1.45203e7	−1.24500
$672$	0	0
$673$	8.72496e6	0.742550	0.371275	−	0.928523i	$-0.378921\pi$
$673$	8.72496e6	0.742550	0.371275	+	0.928523i	$0.378921\pi$
$674$	1.19151e6	0.101030
$675$	0	0
$676$	−5.72909e6	−0.482191
$677$	1.18449e7	0.993250	0.496625	−	0.867965i	$-0.334572\pi$
$677$	1.18449e7	0.993250	0.496625	+	0.867965i	$0.334572\pi$
$678$	0	0
$679$	9.44048e6	0.785813
$680$	−1.15718e6	−0.0959688
$681$	0	0
$682$	5.64624e6	0.464835
$683$	4.03085e6	0.330632	0.165316	−	0.986241i	$-0.447136\pi$
$683$	4.03085e6	0.330632	0.165316	+	0.986241i	$0.447136\pi$
$684$	0	0
$685$	−5.97410e6	−0.486459
$686$	−8.32885e6	−0.675732
$687$	0	0
$688$	−5.04678e6	−0.406484
$689$	−4.22349e6	−0.338940
$690$	0	0
$691$	3.22921e6	0.257277	0.128639	−	0.991692i	$-0.458939\pi$
$691$	3.22921e6	0.257277	0.128639	+	0.991692i	$0.458939\pi$
$692$	−774384.	−0.0614740
$693$	0	0
$694$	354984.	0.0279776
$695$	−3.40889e6	−0.267701
$696$	0	0
$697$	−9.26264e6	−0.722192
$698$	9.44698e6	0.733930
$699$	0	0
$700$	−3.17786e6	−0.245126
$701$	1.34398e7	1.03300	0.516499	−	0.856288i	$-0.327235\pi$
$701$	1.34398e7	1.03300	0.516499	+	0.856288i	$0.327235\pi$
$702$	0	0
$703$	1.83464e7	1.40012
$704$	1.10592e6	0.0840992
$705$	0	0
$706$	−1.54431e7	−1.16606
$707$	−5.83638e6	−0.439132
$708$	0	0
$709$	−2.08300e7	−1.55623	−0.778114	−	0.628123i	$-0.783823\pi$
$709$	−2.08300e7	−1.55623	−0.778114	+	0.628123i	$0.783823\pi$
$710$	−357336.	−0.0266030
$711$	0	0
$712$	−2.90285e6	−0.214597
$713$	1.89149e7	1.39341
$714$	0	0
$715$	−652050.	−0.0476997
$716$	6.00634e6	0.437852
$717$	0	0
$718$	−1.49941e7	−1.08545
$719$	2.57779e7	1.85963	0.929813	−	0.368032i	$-0.119968\pi$
$719$	2.57779e7	1.85963	0.929813	+	0.368032i	$0.119968\pi$
$720$	0	0
$721$	−1.29411e6	−0.0927115
$722$	3.78560e6	0.270266
$723$	0	0
$724$	7.05123e6	0.499941
$725$	−3.01950e6	−0.213349
$726$	0	0
$727$	−7.24491e6	−0.508390	−0.254195	−	0.967153i	$-0.581811\pi$
$727$	−7.24491e6	−0.508390	−0.254195	+	0.967153i	$0.581811\pi$
$728$	−544640.	−0.0380874
$729$	0	0
$730$	−1.47176e6	−0.102219
$731$	1.69738e7	1.17486
$732$	0	0
$733$	−5.64272e6	−0.387908	−0.193954	−	0.981011i	$-0.562131\pi$
$733$	−5.64272e6	−0.387908	−0.193954	+	0.981011i	$0.562131\pi$
$734$	−1.06536e7	−0.729886
$735$	0	0
$736$	3.70483e6	0.252101
$737$	−3.49218e6	−0.236825
$738$	0	0
$739$	1.72501e7	1.16193	0.580967	−	0.813927i	$-0.302674\pi$
$739$	1.72501e7	1.16193	0.580967	+	0.813927i	$0.302674\pi$
$740$	3.33211e6	0.223687
$741$	0	0
$742$	1.08709e7	0.724862
$743$	1.29346e7	0.859572	0.429786	−	0.902931i	$-0.358589\pi$
$743$	1.29346e7	0.859572	0.429786	+	0.902931i	$0.358589\pi$
$744$	0	0
$745$	−5.68493e6	−0.375262
$746$	3.83948e6	0.252595
$747$	0	0
$748$	−3.71952e6	−0.243071
$749$	−9.94294e6	−0.647605
$750$	0	0
$751$	5.69809e6	0.368663	0.184332	−	0.982864i	$-0.440988\pi$
$751$	5.69809e6	0.368663	0.184332	+	0.982864i	$0.440988\pi$
$752$	2.55590e6	0.164816
$753$	0	0
$754$	−517500.	−0.0331499
$755$	−353892.	−0.0225945
$756$	0	0
$757$	3.72388e6	0.236187	0.118093	−	0.993002i	$-0.462322\pi$
$757$	3.72388e6	0.236187	0.118093	+	0.993002i	$0.462322\pi$
$758$	−775120.	−0.0490000
$759$	0	0
$760$	2.48640e6	0.156148
$761$	1.76947e7	1.10759	0.553797	−	0.832652i	$-0.313178\pi$
$761$	1.76947e7	1.10759	0.553797	+	0.832652i	$0.313178\pi$
$762$	0	0
$763$	−9.13804e6	−0.568253
$764$	7.79165e6	0.482943
$765$	0	0
$766$	2.22157e7	1.36801
$767$	−3.04290e6	−0.186767
$768$	0	0
$769$	1.66580e7	1.01580	0.507900	−	0.861416i	$-0.330422\pi$
$769$	1.66580e7	1.01580	0.507900	+	0.861416i	$0.330422\pi$
$770$	1.67832e6	0.102011
$771$	0	0
$772$	−1.66696e6	−0.100666
$773$	2.04852e6	0.123308	0.0616539	−	0.998098i	$-0.480362\pi$
$773$	2.04852e6	0.123308	0.0616539	+	0.998098i	$0.480362\pi$
$774$	0	0
$775$	−1.40320e7	−0.839197
$776$	8.16474e6	0.486730
$777$	0	0
$778$	−1.93118e7	−1.14386
$779$	1.99023e7	1.17506
$780$	0	0
$781$	−1.14858e6	−0.0673804
$782$	−1.24604e7	−0.728643
$783$	0	0
$784$	−2.90074e6	−0.168546
$785$	5.19483e6	0.300883
$786$	0	0
$787$	7.56597e6	0.435439	0.217720	−	0.976011i	$-0.430138\pi$
$787$	7.56597e6	0.435439	0.217720	+	0.976011i	$0.430138\pi$
$788$	629904.	0.0361376
$789$	0	0
$790$	−3.10346e6	−0.176921
$791$	1.28789e7	0.731875
$792$	0	0
$793$	6.18458e6	0.349243
$794$	1.25364e6	0.0705700
$795$	0	0
$796$	1.52397e7	0.852501
$797$	1.00735e6	0.0561738	0.0280869	−	0.999605i	$-0.491058\pi$
$797$	1.00735e6	0.0561738	0.0280869	+	0.999605i	$0.491058\pi$
$798$	0	0
$799$	−8.59622e6	−0.476366
$800$	−2.74842e6	−0.151830
$801$	0	0
$802$	2.11316e7	1.16010
$803$	−4.73067e6	−0.258901
$804$	0	0
$805$	5.62237e6	0.305795
$806$	−2.40488e6	−0.130393
$807$	0	0
$808$	−5.04768e6	−0.271997
$809$	−2.37253e7	−1.27450	−0.637251	−	0.770657i	$-0.719928\pi$
$809$	−2.37253e7	−1.27450	−0.637251	+	0.770657i	$0.719928\pi$
$810$	0	0
$811$	2.09278e7	1.11730	0.558652	−	0.829402i	$-0.311319\pi$
$811$	2.09278e7	1.11730	0.558652	+	0.829402i	$0.311319\pi$
$812$	1.33200e6	0.0708948
$813$	0	0
$814$	1.07104e7	0.566556
$815$	−4.20244e6	−0.221619
$816$	0	0
$817$	−3.64709e7	−1.91157
$818$	1.83525e7	0.958983
$819$	0	0
$820$	3.61469e6	0.187731
$821$	1.39287e7	0.721194	0.360597	−	0.932722i	$-0.382573\pi$
$821$	1.39287e7	0.721194	0.360597	+	0.932722i	$0.382573\pi$
$822$	0	0
$823$	−1.99553e7	−1.02697	−0.513486	−	0.858098i	$-0.671646\pi$
$823$	−1.99553e7	−1.02697	−0.513486	+	0.858098i	$0.671646\pi$
$824$	−1.11923e6	−0.0574251
$825$	0	0
$826$	7.83216e6	0.399421
$827$	−2.48097e7	−1.26141	−0.630706	−	0.776022i	$-0.717235\pi$
$827$	−2.48097e7	−1.26141	−0.630706	+	0.776022i	$0.717235\pi$
$828$	0	0
$829$	1.22092e6	0.0617021	0.0308511	−	0.999524i	$-0.490178\pi$
$829$	1.22092e6	0.0617021	0.0308511	+	0.999524i	$0.490178\pi$
$830$	−6.41894e6	−0.323421
$831$	0	0
$832$	−471040.	−0.0235912
$833$	9.75599e6	0.487146
$834$	0	0
$835$	296478.	0.0147155
$836$	7.99200e6	0.395494
$837$	0	0
$838$	−9.46790e6	−0.465740
$839$	−1.76184e7	−0.864095	−0.432048	−	0.901851i	$-0.642209\pi$
$839$	−1.76184e7	−0.864095	−0.432048	+	0.901851i	$0.642209\pi$
$840$	0	0
$841$	−1.92455e7	−0.938296
$842$	−1.81918e6	−0.0884291
$843$	0	0
$844$	7.05776e6	0.341044
$845$	−7.51943e6	−0.362279
$846$	0	0
$847$	−6.52317e6	−0.312428
$848$	9.40186e6	0.448977
$849$	0	0
$850$	9.24370e6	0.438832
$851$	3.58797e7	1.69834
$852$	0	0
$853$	3.43560e7	1.61670	0.808350	−	0.588702i	$-0.200361\pi$
$853$	3.43560e7	1.61670	0.808350	+	0.588702i	$0.200361\pi$
$854$	−1.59186e7	−0.746895
$855$	0	0
$856$	−8.59930e6	−0.401124
$857$	−987261.	−0.0459177	−0.0229588	−	0.999736i	$-0.507309\pi$
$857$	−987261.	−0.0459177	−0.0229588	+	0.999736i	$0.507309\pi$
$858$	0	0
$859$	4.11853e7	1.90440	0.952202	−	0.305471i	$-0.0988138\pi$
$859$	4.11853e7	1.90440	0.952202	+	0.305471i	$0.0988138\pi$
$860$	−6.62390e6	−0.305399
$861$	0	0
$862$	5.05142e6	0.231550
$863$	2.07182e7	0.946946	0.473473	−	0.880808i	$-0.343000\pi$
$863$	2.07182e7	0.946946	0.473473	+	0.880808i	$0.343000\pi$
$864$	0	0
$865$	−1.01638e6	−0.0461865
$866$	2.19560e7	0.994853
$867$	0	0
$868$	6.18995e6	0.278861
$869$	−9.97542e6	−0.448107
$870$	0	0
$871$	1.48741e6	0.0664332
$872$	−7.90317e6	−0.351974
$873$	0	0
$874$	2.67732e7	1.18555
$875$	−9.02719e6	−0.398595
$876$	0	0
$877$	−3.70347e7	−1.62596	−0.812979	−	0.582293i	$-0.802156\pi$
$877$	−3.70347e7	−1.62596	−0.812979	+	0.582293i	$0.802156\pi$
$878$	−2.44658e7	−1.07108
$879$	0	0
$880$	1.45152e6	0.0631853
$881$	−7.00502e6	−0.304067	−0.152034	−	0.988375i	$-0.548582\pi$
$881$	−7.00502e6	−0.304067	−0.152034	+	0.988375i	$0.548582\pi$
$882$	0	0
$883$	5.47102e6	0.236139	0.118069	−	0.993005i	$-0.462330\pi$
$883$	5.47102e6	0.236139	0.118069	+	0.993005i	$0.462330\pi$
$884$	1.58424e6	0.0681852
$885$	0	0
$886$	2.95680e6	0.126543
$887$	2.18980e6	0.0934535	0.0467268	−	0.998908i	$-0.485121\pi$
$887$	2.18980e6	0.0934535	0.0467268	+	0.998908i	$0.485121\pi$
$888$	0	0
$889$	−2.31607e7	−0.982871
$890$	−3.80999e6	−0.161231
$891$	0	0
$892$	−1.32017e7	−0.555543
$893$	1.84704e7	0.775082
$894$	0	0
$895$	7.88332e6	0.328966
$896$	1.21242e6	0.0504524
$897$	0	0
$898$	2.92573e7	1.21072
$899$	5.88150e6	0.242711
$900$	0	0
$901$	−3.16211e7	−1.29767
$902$	1.16186e7	0.475487
$903$	0	0
$904$	1.11385e7	0.453321
$905$	9.25474e6	0.375615
$906$	0	0
$907$	4.86544e7	1.96383	0.981916	−	0.189318i	$-0.0606277\pi$
$907$	4.86544e7	1.96383	0.981916	+	0.189318i	$0.0606277\pi$
$908$	1.84912e7	0.744305
$909$	0	0
$910$	−714840.	−0.0286158
$911$	−4.70792e7	−1.87946	−0.939730	−	0.341918i	$-0.888924\pi$
$911$	−4.70792e7	−1.87946	−0.939730	+	0.341918i	$0.888924\pi$
$912$	0	0
$913$	−2.06323e7	−0.819164
$914$	−3.07561e7	−1.21777
$915$	0	0
$916$	2.14937e7	0.846394
$917$	−7.82994e6	−0.307493
$918$	0	0
$919$	−2.23428e7	−0.872667	−0.436333	−	0.899785i	$-0.643723\pi$
$919$	−2.23428e7	−0.872667	−0.436333	+	0.899785i	$0.643723\pi$
$920$	4.86259e6	0.189408
$921$	0	0
$922$	−2.37575e7	−0.920392
$923$	489210.	0.0189013
$924$	0	0
$925$	−2.66172e7	−1.02284
$926$	−1.82382e6	−0.0698965
$927$	0	0
$928$	1.15200e6	0.0439119
$929$	−2.02185e7	−0.768618	−0.384309	−	0.923205i	$-0.625560\pi$
$929$	−2.02185e7	−0.768618	−0.384309	+	0.923205i	$0.625560\pi$
$930$	0	0
$931$	−2.09624e7	−0.792621
$932$	1.72088e7	0.648950
$933$	0	0
$934$	−2.38839e7	−0.895855
$935$	−4.88187e6	−0.182624
$936$	0	0
$937$	−3.88053e7	−1.44392	−0.721959	−	0.691936i	$-0.756758\pi$
$937$	−3.88053e7	−1.44392	−0.721959	+	0.691936i	$0.756758\pi$
$938$	−3.82846e6	−0.142075
$939$	0	0
$940$	3.35462e6	0.123830
$941$	−2.78114e7	−1.02388	−0.511940	−	0.859021i	$-0.671073\pi$
$941$	−2.78114e7	−1.02388	−0.511940	+	0.859021i	$0.671073\pi$
$942$	0	0
$943$	3.89224e7	1.42535
$944$	6.77376e6	0.247400
$945$	0	0
$946$	−2.12911e7	−0.773518
$947$	−3.82392e7	−1.38559	−0.692794	−	0.721135i	$-0.743621\pi$
$947$	−3.82392e7	−1.38559	−0.692794	+	0.721135i	$0.743621\pi$
$948$	0	0
$949$	2.01491e6	0.0726258
$950$	−1.98616e7	−0.714012
$951$	0	0
$952$	−4.07770e6	−0.145822
$953$	1.01512e7	0.362065	0.181033	−	0.983477i	$-0.442056\pi$
$953$	1.01512e7	0.362065	0.181033	+	0.983477i	$0.442056\pi$
$954$	0	0
$955$	1.02265e7	0.362844
$956$	577536.	0.0204378
$957$	0	0
$958$	1.69936e7	0.598233
$959$	−2.10516e7	−0.739160
$960$	0	0
$961$	−1.29717e6	−0.0453093
$962$	−4.56182e6	−0.158928
$963$	0	0
$964$	1.16600e6	0.0404116
$965$	−2.18788e6	−0.0756321
$966$	0	0
$967$	−1.41574e7	−0.486876	−0.243438	−	0.969916i	$-0.578275\pi$
$967$	−1.41574e7	−0.486876	−0.243438	+	0.969916i	$0.578275\pi$
$968$	−5.64166e6	−0.193517
$969$	0	0
$970$	1.07162e7	0.365689
$971$	1.50291e6	0.0511546	0.0255773	−	0.999673i	$-0.491858\pi$
$971$	1.50291e6	0.0511546	0.0255773	+	0.999673i	$0.491858\pi$
$972$	0	0
$973$	−1.20123e7	−0.406765
$974$	−4.16190e6	−0.140571
$975$	0	0
$976$	−1.37674e7	−0.462624
$977$	−2.88920e7	−0.968371	−0.484185	−	0.874965i	$-0.660884\pi$
$977$	−2.88920e7	−0.968371	−0.484185	+	0.874965i	$0.660884\pi$
$978$	0	0
$979$	−1.22464e7	−0.408368
$980$	−3.80722e6	−0.126632
$981$	0	0
$982$	2.22279e7	0.735563
$983$	7.71598e6	0.254687	0.127344	−	0.991859i	$-0.459355\pi$
$983$	7.71598e6	0.254687	0.127344	+	0.991859i	$0.459355\pi$
$984$	0	0
$985$	826749.	0.0271508
$986$	−3.87450e6	−0.126918
$987$	0	0
$988$	−3.40400e6	−0.110942
$989$	−7.13253e7	−2.31874
$990$	0	0
$991$	−1.87209e7	−0.605538	−0.302769	−	0.953064i	$-0.597911\pi$
$991$	−1.87209e7	−0.605538	−0.302769	+	0.953064i	$0.597911\pi$
$992$	5.35347e6	0.172725
$993$	0	0
$994$	−1.25918e6	−0.0404225
$995$	2.00022e7	0.640500
$996$	0	0
$997$	5.37876e6	0.171374	0.0856869	−	0.996322i	$-0.472692\pi$
$997$	5.37876e6	0.171374	0.0856869	+	0.996322i	$0.472692\pi$
$998$	−1.41311e6	−0.0449107
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.6.a.b.1.1	yes	1
3.2	odd	2		162.6.a.a.1.1	✓	1
9.2	odd	6		162.6.c.i.109.1		2
9.4	even	3		162.6.c.d.55.1		2
9.5	odd	6		162.6.c.i.55.1		2
9.7	even	3		162.6.c.d.109.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.6.a.a.1.1	✓	1	3.2	odd	2
162.6.a.b.1.1	yes	1	1.1	even	1	trivial
162.6.c.d.55.1		2	9.4	even	3
162.6.c.d.109.1		2	9.7	even	3
162.6.c.i.55.1		2	9.5	odd	6
162.6.c.i.109.1		2	9.2	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more