LMFDB - Embedded newform 162.8.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,8,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, 8); 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, 8, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,8,0,64,-165] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$50.6063741284$
Analytic rank:	$1$
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)

$f(q)$

$=$

\(q+8.00000 q^{2} +64.0000 q^{4} -165.000 q^{5} -508.000 q^{7} +512.000 q^{8} -1320.00 q^{10} +3024.00 q^{11} +5039.00 q^{13} -4064.00 q^{14} +4096.00 q^{16} -3189.00 q^{17} +1508.00 q^{19} -10560.0 q^{20} +24192.0 q^{22} -75600.0 q^{23} -50900.0 q^{25} +40312.0 q^{26} -32512.0 q^{28} -82665.0 q^{29} -174892. q^{31} +32768.0 q^{32} -25512.0 q^{34} +83820.0 q^{35} -323569. q^{37} +12064.0 q^{38} -84480.0 q^{40} -308118. q^{41} +336680. q^{43} +193536. q^{44} -604800. q^{46} -383196. q^{47} -565479. q^{49} -407200. q^{50} +322496. q^{52} +760206. q^{53} -498960. q^{55} -260096. q^{56} -661320. q^{58} -2.22566e6 q^{59} +2.24482e6 q^{61} -1.39914e6 q^{62} +262144. q^{64} -831435. q^{65} +1.47319e6 q^{67} -204096. q^{68} +670560. q^{70} -5.00689e6 q^{71} -5.89830e6 q^{73} -2.58855e6 q^{74} +96512.0 q^{76} -1.53619e6 q^{77} +7.02877e6 q^{79} -675840. q^{80} -2.46494e6 q^{82} -2.65120e6 q^{83} +526185. q^{85} +2.69344e6 q^{86} +1.54829e6 q^{88} -6.77090e6 q^{89} -2.55981e6 q^{91} -4.83840e6 q^{92} -3.06557e6 q^{94} -248820. q^{95} +1.61764e7 q^{97} -4.52383e6 q^{98} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−165.000	−0.590322	−0.295161	−	0.955448i	$-0.595373\pi$
$5$	−165.000	−0.590322	−0.295161	+	0.955448i	$0.595373\pi$
$6$	0	0
$7$	−508.000	−0.559784	−0.279892	−	0.960031i	$-0.590299\pi$
$7$	−508.000	−0.559784	−0.279892	+	0.960031i	$0.590299\pi$
$8$	512.000	0.353553
$9$	0	0
$10$	−1320.00	−0.417421
$11$	3024.00	0.685027	0.342513	−	0.939513i	$-0.388722\pi$
$11$	3024.00	0.685027	0.342513	+	0.939513i	$0.388722\pi$
$12$	0	0
$13$	5039.00	0.636125	0.318063	−	0.948070i	$-0.396968\pi$
$13$	5039.00	0.636125	0.318063	+	0.948070i	$0.396968\pi$
$14$	−4064.00	−0.395827
$15$	0	0
$16$	4096.00	0.250000
$17$	−3189.00	−0.157428	−0.0787142	−	0.996897i	$-0.525081\pi$
$17$	−3189.00	−0.157428	−0.0787142	+	0.996897i	$0.525081\pi$
$18$	0	0
$19$	1508.00	0.0504387	0.0252193	−	0.999682i	$-0.491972\pi$
$19$	1508.00	0.0504387	0.0252193	+	0.999682i	$0.491972\pi$
$20$	−10560.0	−0.295161
$21$	0	0
$22$	24192.0	0.484387
$23$	−75600.0	−1.29561	−0.647805	−	0.761806i	$-0.724313\pi$
$23$	−75600.0	−1.29561	−0.647805	+	0.761806i	$0.724313\pi$
$24$	0	0
$25$	−50900.0	−0.651520
$26$	40312.0	0.449808
$27$	0	0
$28$	−32512.0	−0.279892
$29$	−82665.0	−0.629403	−0.314701	−	0.949191i	$-0.601904\pi$
$29$	−82665.0	−0.629403	−0.314701	+	0.949191i	$0.601904\pi$
$30$	0	0
$31$	−174892.	−1.05440	−0.527198	−	0.849742i	$-0.676758\pi$
$31$	−174892.	−1.05440	−0.527198	+	0.849742i	$0.676758\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	−25512.0	−0.111319
$35$	83820.0	0.330453
$36$	0	0
$37$	−323569.	−1.05017	−0.525087	−	0.851049i	$-0.675967\pi$
$37$	−323569.	−1.05017	−0.525087	+	0.851049i	$0.675967\pi$
$38$	12064.0	0.0356655
$39$	0	0
$40$	−84480.0	−0.208710
$41$	−308118.	−0.698190	−0.349095	−	0.937087i	$-0.613511\pi$
$41$	−308118.	−0.698190	−0.349095	+	0.937087i	$0.613511\pi$
$42$	0	0
$43$	336680.	0.645770	0.322885	−	0.946438i	$-0.395347\pi$
$43$	336680.	0.645770	0.322885	+	0.946438i	$0.395347\pi$
$44$	193536.	0.342513
$45$	0	0
$46$	−604800.	−0.916135
$47$	−383196.	−0.538367	−0.269184	−	0.963089i	$-0.586754\pi$
$47$	−383196.	−0.538367	−0.269184	+	0.963089i	$0.586754\pi$
$48$	0	0
$49$	−565479.	−0.686642
$50$	−407200.	−0.460694
$51$	0	0
$52$	322496.	0.318063
$53$	760206.	0.701400	0.350700	−	0.936488i	$-0.385944\pi$
$53$	760206.	0.701400	0.350700	+	0.936488i	$0.385944\pi$
$54$	0	0
$55$	−498960.	−0.404386
$56$	−260096.	−0.197914
$57$	0	0
$58$	−661320.	−0.445055
$59$	−2.22566e6	−1.41084	−0.705420	−	0.708790i	$-0.749241\pi$
$59$	−2.22566e6	−1.41084	−0.705420	+	0.708790i	$0.749241\pi$
$60$	0	0
$61$	2.24482e6	1.26627	0.633135	−	0.774042i	$-0.281768\pi$
$61$	2.24482e6	1.26627	0.633135	+	0.774042i	$0.281768\pi$
$62$	−1.39914e6	−0.745571
$63$	0	0
$64$	262144.	0.125000
$65$	−831435.	−0.375519
$66$	0	0
$67$	1.47319e6	0.598407	0.299203	−	0.954189i	$-0.403279\pi$
$67$	1.47319e6	0.598407	0.299203	+	0.954189i	$0.403279\pi$
$68$	−204096.	−0.0787142
$69$	0	0
$70$	670560.	0.233665
$71$	−5.00689e6	−1.66021	−0.830107	−	0.557604i	$-0.811721\pi$
$71$	−5.00689e6	−1.66021	−0.830107	+	0.557604i	$0.811721\pi$
$72$	0	0
$73$	−5.89830e6	−1.77459	−0.887293	−	0.461207i	$-0.847417\pi$
$73$	−5.89830e6	−1.77459	−0.887293	+	0.461207i	$0.847417\pi$
$74$	−2.58855e6	−0.742584
$75$	0	0
$76$	96512.0	0.0252193
$77$	−1.53619e6	−0.383467
$78$	0	0
$79$	7.02877e6	1.60393	0.801963	−	0.597374i	$-0.203789\pi$
$79$	7.02877e6	1.60393	0.801963	+	0.597374i	$0.203789\pi$
$80$	−675840.	−0.147580
$81$	0	0
$82$	−2.46494e6	−0.493695
$83$	−2.65120e6	−0.508942	−0.254471	−	0.967080i	$-0.581901\pi$
$83$	−2.65120e6	−0.508942	−0.254471	+	0.967080i	$0.581901\pi$
$84$	0	0
$85$	526185.	0.0929335
$86$	2.69344e6	0.456628
$87$	0	0
$88$	1.54829e6	0.242193
$89$	−6.77090e6	−1.01808	−0.509039	−	0.860743i	$-0.669999\pi$
$89$	−6.77090e6	−1.01808	−0.509039	+	0.860743i	$0.669999\pi$
$90$	0	0
$91$	−2.55981e6	−0.356093
$92$	−4.83840e6	−0.647805
$93$	0	0
$94$	−3.06557e6	−0.380683
$95$	−248820.	−0.0297751
$96$	0	0
$97$	1.61764e7	1.79962	0.899809	−	0.436284i	$-0.143706\pi$
$97$	1.61764e7	1.79962	0.899809	+	0.436284i	$0.143706\pi$
$98$	−4.52383e6	−0.485529
$99$	0	0
$100$	−3.25760e6	−0.325760
$101$	2.70161e6	0.260915	0.130457	−	0.991454i	$-0.458355\pi$
$101$	2.70161e6	0.260915	0.130457	+	0.991454i	$0.458355\pi$
$102$	0	0
$103$	2.06892e6	0.186557	0.0932787	−	0.995640i	$-0.470265\pi$
$103$	2.06892e6	0.186557	0.0932787	+	0.995640i	$0.470265\pi$
$104$	2.57997e6	0.224904
$105$	0	0
$106$	6.08165e6	0.495965
$107$	−9.15450e6	−0.722423	−0.361211	−	0.932484i	$-0.617637\pi$
$107$	−9.15450e6	−0.722423	−0.361211	+	0.932484i	$0.617637\pi$
$108$	0	0
$109$	8.68645e6	0.642465	0.321233	−	0.947000i	$-0.395903\pi$
$109$	8.68645e6	0.642465	0.321233	+	0.947000i	$0.395903\pi$
$110$	−3.99168e6	−0.285944
$111$	0	0
$112$	−2.08077e6	−0.139946
$113$	−2.49776e7	−1.62845	−0.814227	−	0.580547i	$-0.802839\pi$
$113$	−2.49776e7	−1.62845	−0.814227	+	0.580547i	$0.802839\pi$
$114$	0	0
$115$	1.24740e7	0.764827
$116$	−5.29056e6	−0.314701
$117$	0	0
$118$	−1.78053e7	−0.997614
$119$	1.62001e6	0.0881260
$120$	0	0
$121$	−1.03426e7	−0.530739
$122$	1.79585e7	0.895388
$123$	0	0
$124$	−1.11931e7	−0.527198
$125$	2.12891e7	0.974929
$126$	0	0
$127$	−4.98659e6	−0.216018	−0.108009	−	0.994150i	$-0.534448\pi$
$127$	−4.98659e6	−0.216018	−0.108009	+	0.994150i	$0.534448\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	−6.65148e6	−0.265532
$131$	−2.23405e7	−0.868246	−0.434123	−	0.900854i	$-0.642942\pi$
$131$	−2.23405e7	−0.868246	−0.434123	+	0.900854i	$0.642942\pi$
$132$	0	0
$133$	−766064.	−0.0282348
$134$	1.17855e7	0.423137
$135$	0	0
$136$	−1.63277e6	−0.0556594
$137$	1.28631e7	0.427389	0.213694	−	0.976901i	$-0.431450\pi$
$137$	1.28631e7	0.427389	0.213694	+	0.976901i	$0.431450\pi$
$138$	0	0
$139$	4.41902e7	1.39564	0.697822	−	0.716271i	$-0.254153\pi$
$139$	4.41902e7	1.39564	0.697822	+	0.716271i	$0.254153\pi$
$140$	5.36448e6	0.165226
$141$	0	0
$142$	−4.00551e7	−1.17395
$143$	1.52379e7	0.435763
$144$	0	0
$145$	1.36397e7	0.371550
$146$	−4.71864e7	−1.25482
$147$	0	0
$148$	−2.07084e7	−0.525087
$149$	−2.62887e6	−0.0651055	−0.0325527	−	0.999470i	$-0.510364\pi$
$149$	−2.62887e6	−0.0651055	−0.0325527	+	0.999470i	$0.510364\pi$
$150$	0	0
$151$	−5.53019e7	−1.30714	−0.653568	−	0.756868i	$-0.726729\pi$
$151$	−5.53019e7	−1.30714	−0.653568	+	0.756868i	$0.726729\pi$
$152$	772096.	0.0178328
$153$	0	0
$154$	−1.22895e7	−0.271152
$155$	2.88572e7	0.622433
$156$	0	0
$157$	−2.23711e7	−0.461358	−0.230679	−	0.973030i	$-0.574095\pi$
$157$	−2.23711e7	−0.461358	−0.230679	+	0.973030i	$0.574095\pi$
$158$	5.62301e7	1.13415
$159$	0	0
$160$	−5.40672e6	−0.104355
$161$	3.84048e7	0.725262
$162$	0	0
$163$	5.23606e7	0.946995	0.473498	−	0.880795i	$-0.342991\pi$
$163$	5.23606e7	0.946995	0.473498	+	0.880795i	$0.342991\pi$
$164$	−1.97196e7	−0.349095
$165$	0	0
$166$	−2.12096e7	−0.359877
$167$	−5.47957e7	−0.910414	−0.455207	−	0.890386i	$-0.650435\pi$
$167$	−5.47957e7	−0.910414	−0.455207	+	0.890386i	$0.650435\pi$
$168$	0	0
$169$	−3.73570e7	−0.595345
$170$	4.20948e6	0.0657139
$171$	0	0
$172$	2.15475e7	0.322885
$173$	1.23645e8	1.81558	0.907791	−	0.419424i	$-0.137768\pi$
$173$	1.23645e8	1.81558	0.907791	+	0.419424i	$0.137768\pi$
$174$	0	0
$175$	2.58572e7	0.364711
$176$	1.23863e7	0.171257
$177$	0	0
$178$	−5.41672e7	−0.719891
$179$	9.87297e7	1.28666	0.643328	−	0.765591i	$-0.277553\pi$
$179$	9.87297e7	1.28666	0.643328	+	0.765591i	$0.277553\pi$
$180$	0	0
$181$	9.19855e7	1.15304	0.576520	−	0.817083i	$-0.304410\pi$
$181$	9.19855e7	1.15304	0.576520	+	0.817083i	$0.304410\pi$
$182$	−2.04785e7	−0.251796
$183$	0	0
$184$	−3.87072e7	−0.458067
$185$	5.33889e7	0.619940
$186$	0	0
$187$	−9.64354e6	−0.107843
$188$	−2.45245e7	−0.269184
$189$	0	0
$190$	−1.99056e6	−0.0210541
$191$	1.31979e8	1.37052	0.685262	−	0.728296i	$-0.259688\pi$
$191$	1.31979e8	1.37052	0.685262	+	0.728296i	$0.259688\pi$
$192$	0	0
$193$	1.47907e8	1.48095	0.740473	−	0.672086i	$-0.234602\pi$
$193$	1.47907e8	1.48095	0.740473	+	0.672086i	$0.234602\pi$
$194$	1.29411e8	1.27252
$195$	0	0
$196$	−3.61907e7	−0.343321
$197$	−2.69561e7	−0.251203	−0.125602	−	0.992081i	$-0.540086\pi$
$197$	−2.69561e7	−0.251203	−0.125602	+	0.992081i	$0.540086\pi$
$198$	0	0
$199$	1.35831e8	1.22183	0.610916	−	0.791695i	$-0.290801\pi$
$199$	1.35831e8	1.22183	0.610916	+	0.791695i	$0.290801\pi$
$200$	−2.60608e7	−0.230347
$201$	0	0
$202$	2.16129e7	0.184495
$203$	4.19938e7	0.352330
$204$	0	0
$205$	5.08395e7	0.412157
$206$	1.65513e7	0.131916
$207$	0	0
$208$	2.06397e7	0.159031
$209$	4.56019e6	0.0345518
$210$	0	0
$211$	−2.19340e6	−0.0160742	−0.00803710	−	0.999968i	$-0.502558\pi$
$211$	−2.19340e6	−0.0160742	−0.00803710	+	0.999968i	$0.502558\pi$
$212$	4.86532e7	0.350700
$213$	0	0
$214$	−7.32360e7	−0.510830
$215$	−5.55522e7	−0.381212
$216$	0	0
$217$	8.88451e7	0.590234
$218$	6.94916e7	0.454292
$219$	0	0
$220$	−3.19334e7	−0.202193
$221$	−1.60694e7	−0.100144
$222$	0	0
$223$	1.99443e7	0.120435	0.0602174	−	0.998185i	$-0.480821\pi$
$223$	1.99443e7	0.120435	0.0602174	+	0.998185i	$0.480821\pi$
$224$	−1.66461e7	−0.0989568
$225$	0	0
$226$	−1.99820e8	−1.15149
$227$	−1.25985e8	−0.714872	−0.357436	−	0.933938i	$-0.616349\pi$
$227$	−1.25985e8	−0.714872	−0.357436	+	0.933938i	$0.616349\pi$
$228$	0	0
$229$	2.35046e7	0.129339	0.0646693	−	0.997907i	$-0.479401\pi$
$229$	2.35046e7	0.129339	0.0646693	+	0.997907i	$0.479401\pi$
$230$	9.97920e7	0.540814
$231$	0	0
$232$	−4.23245e7	−0.222527
$233$	−1.27757e8	−0.661665	−0.330833	−	0.943689i	$-0.607330\pi$
$233$	−1.27757e8	−0.661665	−0.330833	+	0.943689i	$0.607330\pi$
$234$	0	0
$235$	6.32273e7	0.317810
$236$	−1.42442e8	−0.705420
$237$	0	0
$238$	1.29601e7	0.0623145
$239$	−9.34878e7	−0.442958	−0.221479	−	0.975165i	$-0.571088\pi$
$239$	−9.34878e7	−0.442958	−0.221479	+	0.975165i	$0.571088\pi$
$240$	0	0
$241$	−1.34780e8	−0.620250	−0.310125	−	0.950696i	$-0.600371\pi$
$241$	−1.34780e8	−0.620250	−0.310125	+	0.950696i	$0.600371\pi$
$242$	−8.27408e7	−0.375289
$243$	0	0
$244$	1.43668e8	0.633135
$245$	9.33040e7	0.405340
$246$	0	0
$247$	7.59881e6	0.0320853
$248$	−8.95447e7	−0.372786
$249$	0	0
$250$	1.70313e8	0.689379
$251$	1.31470e7	0.0524771	0.0262385	−	0.999656i	$-0.491647\pi$
$251$	1.31470e7	0.0524771	0.0262385	+	0.999656i	$0.491647\pi$
$252$	0	0
$253$	−2.28614e8	−0.887527
$254$	−3.98927e7	−0.152748
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−1.28471e8	−0.472105	−0.236053	−	0.971740i	$-0.575854\pi$
$257$	−1.28471e8	−0.472105	−0.236053	+	0.971740i	$0.575854\pi$
$258$	0	0
$259$	1.64373e8	0.587870
$260$	−5.32118e7	−0.187759
$261$	0	0
$262$	−1.78724e8	−0.613942
$263$	−9.77517e7	−0.331344	−0.165672	−	0.986181i	$-0.552979\pi$
$263$	−9.77517e7	−0.331344	−0.165672	+	0.986181i	$0.552979\pi$
$264$	0	0
$265$	−1.25434e8	−0.414052
$266$	−6.12851e6	−0.0199650
$267$	0	0
$268$	9.42840e7	0.299203
$269$	3.96851e7	0.124307	0.0621534	−	0.998067i	$-0.480203\pi$
$269$	3.96851e7	0.124307	0.0621534	+	0.998067i	$0.480203\pi$
$270$	0	0
$271$	−3.50733e8	−1.07049	−0.535246	−	0.844696i	$-0.679781\pi$
$271$	−3.50733e8	−1.07049	−0.535246	+	0.844696i	$0.679781\pi$
$272$	−1.30621e7	−0.0393571
$273$	0	0
$274$	1.02905e8	0.302209
$275$	−1.53922e8	−0.446308
$276$	0	0
$277$	5.39602e8	1.52544	0.762719	−	0.646730i	$-0.223864\pi$
$277$	5.39602e8	1.52544	0.762719	+	0.646730i	$0.223864\pi$
$278$	3.53522e8	0.986869
$279$	0	0
$280$	4.29158e7	0.116833
$281$	−1.00852e8	−0.271151	−0.135576	−	0.990767i	$-0.543288\pi$
$281$	−1.00852e8	−0.271151	−0.135576	+	0.990767i	$0.543288\pi$
$282$	0	0
$283$	−2.54112e8	−0.666458	−0.333229	−	0.942846i	$-0.608138\pi$
$283$	−2.54112e8	−0.666458	−0.333229	+	0.942846i	$0.608138\pi$
$284$	−3.20441e8	−0.830107
$285$	0	0
$286$	1.21903e8	0.308131
$287$	1.56524e8	0.390836
$288$	0	0
$289$	−4.00169e8	−0.975216
$290$	1.09118e8	0.262726
$291$	0	0
$292$	−3.77491e8	−0.887293
$293$	−5.43084e8	−1.26134	−0.630668	−	0.776053i	$-0.717219\pi$
$293$	−5.43084e8	−1.26134	−0.630668	+	0.776053i	$0.717219\pi$
$294$	0	0
$295$	3.67235e8	0.832849
$296$	−1.65667e8	−0.371292
$297$	0	0
$298$	−2.10310e7	−0.0460365
$299$	−3.80948e8	−0.824170
$300$	0	0
$301$	−1.71033e8	−0.361492
$302$	−4.42415e8	−0.924285
$303$	0	0
$304$	6.17677e6	0.0126097
$305$	−3.70394e8	−0.747507
$306$	0	0
$307$	−7.29877e8	−1.43968	−0.719839	−	0.694141i	$-0.755784\pi$
$307$	−7.29877e8	−1.43968	−0.719839	+	0.694141i	$0.755784\pi$
$308$	−9.83163e7	−0.191733
$309$	0	0
$310$	2.30857e8	0.440127
$311$	6.38770e8	1.20416	0.602078	−	0.798437i	$-0.294339\pi$
$311$	6.38770e8	1.20416	0.602078	+	0.798437i	$0.294339\pi$
$312$	0	0
$313$	−6.37761e8	−1.17558	−0.587790	−	0.809013i	$-0.700002\pi$
$313$	−6.37761e8	−1.17558	−0.587790	+	0.809013i	$0.700002\pi$
$314$	−1.78968e8	−0.326229
$315$	0	0
$316$	4.49841e8	0.801963
$317$	−6.54377e8	−1.15377	−0.576887	−	0.816824i	$-0.695733\pi$
$317$	−6.54377e8	−1.15377	−0.576887	+	0.816824i	$0.695733\pi$
$318$	0	0
$319$	−2.49979e8	−0.431158
$320$	−4.32538e7	−0.0737902
$321$	0	0
$322$	3.07238e8	0.512838
$323$	−4.80901e6	−0.00794049
$324$	0	0
$325$	−2.56485e8	−0.414448
$326$	4.18885e8	0.669627
$327$	0	0
$328$	−1.57756e8	−0.246847
$329$	1.94664e8	0.301369
$330$	0	0
$331$	8.07689e8	1.22418	0.612091	−	0.790787i	$-0.290329\pi$
$331$	8.07689e8	1.22418	0.612091	+	0.790787i	$0.290329\pi$
$332$	−1.69677e8	−0.254471
$333$	0	0
$334$	−4.38366e8	−0.643760
$335$	−2.43076e8	−0.353253
$336$	0	0
$337$	−1.87280e8	−0.266555	−0.133278	−	0.991079i	$-0.542550\pi$
$337$	−1.87280e8	−0.266555	−0.133278	+	0.991079i	$0.542550\pi$
$338$	−2.98856e8	−0.420972
$339$	0	0
$340$	3.36758e7	0.0464667
$341$	−5.28873e8	−0.722290
$342$	0	0
$343$	7.05623e8	0.944155
$344$	1.72380e8	0.228314
$345$	0	0
$346$	9.89161e8	1.28381
$347$	2.82556e8	0.363037	0.181518	−	0.983388i	$-0.441899\pi$
$347$	2.82556e8	0.363037	0.181518	+	0.983388i	$0.441899\pi$
$348$	0	0
$349$	−7.60612e8	−0.957799	−0.478899	−	0.877870i	$-0.658964\pi$
$349$	−7.60612e8	−0.957799	−0.478899	+	0.877870i	$0.658964\pi$
$350$	2.06858e8	0.257889
$351$	0	0
$352$	9.90904e7	0.121097
$353$	−4.68893e8	−0.567365	−0.283683	−	0.958918i	$-0.591556\pi$
$353$	−4.68893e8	−0.567365	−0.283683	+	0.958918i	$0.591556\pi$
$354$	0	0
$355$	8.26137e8	0.980061
$356$	−4.33338e8	−0.509039
$357$	0	0
$358$	7.89838e8	0.909803
$359$	1.53906e8	0.175560	0.0877802	−	0.996140i	$-0.472023\pi$
$359$	1.53906e8	0.175560	0.0877802	+	0.996140i	$0.472023\pi$
$360$	0	0
$361$	−8.91598e8	−0.997456
$362$	7.35884e8	0.815323
$363$	0	0
$364$	−1.63828e8	−0.178046
$365$	9.73220e8	1.04758
$366$	0	0
$367$	−7.49528e8	−0.791510	−0.395755	−	0.918356i	$-0.629517\pi$
$367$	−7.49528e8	−0.791510	−0.395755	+	0.918356i	$0.629517\pi$
$368$	−3.09658e8	−0.323903
$369$	0	0
$370$	4.27111e8	0.438364
$371$	−3.86185e8	−0.392633
$372$	0	0
$373$	−5.41728e8	−0.540506	−0.270253	−	0.962789i	$-0.587107\pi$
$373$	−5.41728e8	−0.540506	−0.270253	+	0.962789i	$0.587107\pi$
$374$	−7.71483e7	−0.0762563
$375$	0	0
$376$	−1.96196e8	−0.190341
$377$	−4.16549e8	−0.400379
$378$	0	0
$379$	1.32974e9	1.25467	0.627334	−	0.778750i	$-0.284146\pi$
$379$	1.32974e9	1.25467	0.627334	+	0.778750i	$0.284146\pi$
$380$	−1.59245e7	−0.0148875
$381$	0	0
$382$	1.05583e9	0.969107
$383$	7.19949e8	0.654796	0.327398	−	0.944887i	$-0.393828\pi$
$383$	7.19949e8	0.654796	0.327398	+	0.944887i	$0.393828\pi$
$384$	0	0
$385$	2.53472e8	0.226369
$386$	1.18326e9	1.04719
$387$	0	0
$388$	1.03529e9	0.899809
$389$	2.32067e9	1.99889	0.999447	−	0.0332482i	$-0.0105852\pi$
$389$	2.32067e9	1.99889	0.999447	+	0.0332482i	$0.0105852\pi$
$390$	0	0
$391$	2.41088e8	0.203966
$392$	−2.89525e8	−0.242765
$393$	0	0
$394$	−2.15649e8	−0.177628
$395$	−1.15975e9	−0.946833
$396$	0	0
$397$	1.18373e9	0.949477	0.474739	−	0.880127i	$-0.342543\pi$
$397$	1.18373e9	0.949477	0.474739	+	0.880127i	$0.342543\pi$
$398$	1.08664e9	0.863966
$399$	0	0
$400$	−2.08486e8	−0.162880
$401$	1.26764e9	0.981726	0.490863	−	0.871237i	$-0.336681\pi$
$401$	1.26764e9	0.981726	0.490863	+	0.871237i	$0.336681\pi$
$402$	0	0
$403$	−8.81281e8	−0.670728
$404$	1.72903e8	0.130457
$405$	0	0
$406$	3.35951e8	0.249135
$407$	−9.78473e8	−0.719396
$408$	0	0
$409$	−1.01949e9	−0.736802	−0.368401	−	0.929667i	$-0.620095\pi$
$409$	−1.01949e9	−0.736802	−0.368401	+	0.929667i	$0.620095\pi$
$410$	4.06716e8	0.291439
$411$	0	0
$412$	1.32411e8	0.0932787
$413$	1.13064e9	0.789765
$414$	0	0
$415$	4.37447e8	0.300440
$416$	1.65118e8	0.112452
$417$	0	0
$418$	3.64815e7	0.0244318
$419$	1.04129e9	0.691546	0.345773	−	0.938318i	$-0.387617\pi$
$419$	1.04129e9	0.691546	0.345773	+	0.938318i	$0.387617\pi$
$420$	0	0
$421$	−1.95845e9	−1.27916	−0.639580	−	0.768725i	$-0.720892\pi$
$421$	−1.95845e9	−1.27916	−0.639580	+	0.768725i	$0.720892\pi$
$422$	−1.75472e7	−0.0113662
$423$	0	0
$424$	3.89225e8	0.247982
$425$	1.62320e8	0.102568
$426$	0	0
$427$	−1.14037e9	−0.708838
$428$	−5.85888e8	−0.361211
$429$	0	0
$430$	−4.44418e8	−0.269558
$431$	2.84256e9	1.71017	0.855084	−	0.518489i	$-0.173505\pi$
$431$	2.84256e9	1.71017	0.855084	+	0.518489i	$0.173505\pi$
$432$	0	0
$433$	−8.41292e7	−0.0498011	−0.0249006	−	0.999690i	$-0.507927\pi$
$433$	−8.41292e7	−0.0498011	−0.0249006	+	0.999690i	$0.507927\pi$
$434$	7.10761e8	0.417359
$435$	0	0
$436$	5.55933e8	0.321233
$437$	−1.14005e8	−0.0653489
$438$	0	0
$439$	−1.87047e9	−1.05517	−0.527587	−	0.849501i	$-0.676903\pi$
$439$	−1.87047e9	−1.05517	−0.527587	+	0.849501i	$0.676903\pi$
$440$	−2.55468e8	−0.142972
$441$	0	0
$442$	−1.28555e8	−0.0708127
$443$	−1.01333e9	−0.553779	−0.276889	−	0.960902i	$-0.589304\pi$
$443$	−1.01333e9	−0.553779	−0.276889	+	0.960902i	$0.589304\pi$
$444$	0	0
$445$	1.11720e9	0.600994
$446$	1.59554e8	0.0851602
$447$	0	0
$448$	−1.33169e8	−0.0699730
$449$	−1.65756e8	−0.0864183	−0.0432092	−	0.999066i	$-0.513758\pi$
$449$	−1.65756e8	−0.0864183	−0.0432092	+	0.999066i	$0.513758\pi$
$450$	0	0
$451$	−9.31749e8	−0.478279
$452$	−1.59856e9	−0.814227
$453$	0	0
$454$	−1.00788e9	−0.505491
$455$	4.22369e8	0.210209
$456$	0	0
$457$	−1.38080e9	−0.676742	−0.338371	−	0.941013i	$-0.609876\pi$
$457$	−1.38080e9	−0.676742	−0.338371	+	0.941013i	$0.609876\pi$
$458$	1.88037e8	0.0914562
$459$	0	0
$460$	7.98336e8	0.382414
$461$	2.78491e9	1.32391	0.661955	−	0.749544i	$-0.269727\pi$
$461$	2.78491e9	1.32391	0.661955	+	0.749544i	$0.269727\pi$
$462$	0	0
$463$	2.78919e9	1.30600	0.653001	−	0.757357i	$-0.273510\pi$
$463$	2.78919e9	1.30600	0.653001	+	0.757357i	$0.273510\pi$
$464$	−3.38596e8	−0.157351
$465$	0	0
$466$	−1.02205e9	−0.467868
$467$	3.97258e9	1.80494	0.902471	−	0.430751i	$-0.141751\pi$
$467$	3.97258e9	1.80494	0.902471	+	0.430751i	$0.141751\pi$
$468$	0	0
$469$	−7.48380e8	−0.334979
$470$	5.05819e8	0.224726
$471$	0	0
$472$	−1.13954e9	−0.498807
$473$	1.01812e9	0.442369
$474$	0	0
$475$	−7.67572e7	−0.0328618
$476$	1.03681e8	0.0440630
$477$	0	0
$478$	−7.47902e8	−0.313218
$479$	−2.34467e9	−0.974782	−0.487391	−	0.873184i	$-0.662051\pi$
$479$	−2.34467e9	−0.974782	−0.487391	+	0.873184i	$0.662051\pi$
$480$	0	0
$481$	−1.63046e9	−0.668042
$482$	−1.07824e9	−0.438583
$483$	0	0
$484$	−6.61926e8	−0.265369
$485$	−2.66910e9	−1.06235
$486$	0	0
$487$	3.08381e9	1.20986	0.604931	−	0.796278i	$-0.293201\pi$
$487$	3.08381e9	1.20986	0.604931	+	0.796278i	$0.293201\pi$
$488$	1.14935e9	0.447694
$489$	0	0
$490$	7.46432e8	0.286618
$491$	−3.71851e9	−1.41770	−0.708848	−	0.705361i	$-0.750785\pi$
$491$	−3.71851e9	−1.41770	−0.708848	+	0.705361i	$0.750785\pi$
$492$	0	0
$493$	2.63619e8	0.0990859
$494$	6.07905e7	0.0226877
$495$	0	0
$496$	−7.16358e8	−0.263599
$497$	2.54350e9	0.929361
$498$	0	0
$499$	4.36426e8	0.157238	0.0786192	−	0.996905i	$-0.474949\pi$
$499$	4.36426e8	0.157238	0.0786192	+	0.996905i	$0.474949\pi$
$500$	1.36250e9	0.487464
$501$	0	0
$502$	1.05176e8	0.0371069
$503$	−5.04603e8	−0.176792	−0.0883958	−	0.996085i	$-0.528174\pi$
$503$	−5.04603e8	−0.176792	−0.0883958	+	0.996085i	$0.528174\pi$
$504$	0	0
$505$	−4.45766e8	−0.154024
$506$	−1.82892e9	−0.627577
$507$	0	0
$508$	−3.19142e8	−0.108009
$509$	−4.60085e9	−1.54642	−0.773208	−	0.634153i	$-0.781349\pi$
$509$	−4.60085e9	−1.54642	−0.773208	+	0.634153i	$0.781349\pi$
$510$	0	0
$511$	2.99634e9	0.993385
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	−1.02777e9	−0.333829
$515$	−3.41371e8	−0.110129
$516$	0	0
$517$	−1.15878e9	−0.368796
$518$	1.31498e9	0.415687
$519$	0	0
$520$	−4.25695e8	−0.132766
$521$	7.14937e7	0.0221481	0.0110740	−	0.999939i	$-0.496475\pi$
$521$	7.14937e7	0.0221481	0.0110740	+	0.999939i	$0.496475\pi$
$522$	0	0
$523$	−2.12182e9	−0.648563	−0.324281	−	0.945961i	$-0.605122\pi$
$523$	−2.12182e9	−0.648563	−0.324281	+	0.945961i	$0.605122\pi$
$524$	−1.42979e9	−0.434123
$525$	0	0
$526$	−7.82013e8	−0.234296
$527$	5.57731e8	0.165992
$528$	0	0
$529$	2.31053e9	0.678606
$530$	−1.00347e9	−0.292779
$531$	0	0
$532$	−4.90281e7	−0.0141174
$533$	−1.55261e9	−0.444136
$534$	0	0
$535$	1.51049e9	0.426462
$536$	7.54272e8	0.211569
$537$	0	0
$538$	3.17481e8	0.0878981
$539$	−1.71001e9	−0.470368
$540$	0	0
$541$	−2.11351e9	−0.573870	−0.286935	−	0.957950i	$-0.592636\pi$
$541$	−2.11351e9	−0.573870	−0.286935	+	0.957950i	$0.592636\pi$
$542$	−2.80586e9	−0.756953
$543$	0	0
$544$	−1.04497e8	−0.0278297
$545$	−1.43326e9	−0.379261
$546$	0	0
$547$	6.25553e9	1.63421	0.817107	−	0.576487i	$-0.195577\pi$
$547$	6.25553e9	1.63421	0.817107	+	0.576487i	$0.195577\pi$
$548$	8.23237e8	0.213694
$549$	0	0
$550$	−1.23137e9	−0.315588
$551$	−1.24659e8	−0.0317462
$552$	0	0
$553$	−3.57061e9	−0.897852
$554$	4.31682e9	1.07865
$555$	0	0
$556$	2.82818e9	0.697822
$557$	−6.50846e9	−1.59583	−0.797913	−	0.602773i	$-0.794062\pi$
$557$	−6.50846e9	−1.59583	−0.797913	+	0.602773i	$0.794062\pi$
$558$	0	0
$559$	1.69653e9	0.410790
$560$	3.43327e8	0.0826132
$561$	0	0
$562$	−8.06815e8	−0.191733
$563$	−1.73249e8	−0.0409158	−0.0204579	−	0.999791i	$-0.506512\pi$
$563$	−1.73249e8	−0.0409158	−0.0204579	+	0.999791i	$0.506512\pi$
$564$	0	0
$565$	4.12130e9	0.961312
$566$	−2.03290e9	−0.471257
$567$	0	0
$568$	−2.56353e9	−0.586974
$569$	7.28881e9	1.65869	0.829343	−	0.558740i	$-0.188715\pi$
$569$	7.28881e9	1.65869	0.829343	+	0.558740i	$0.188715\pi$
$570$	0	0
$571$	4.29036e9	0.964423	0.482212	−	0.876055i	$-0.339834\pi$
$571$	4.29036e9	0.964423	0.482212	+	0.876055i	$0.339834\pi$
$572$	9.75228e8	0.217881
$573$	0	0
$574$	1.25219e9	0.276363
$575$	3.84804e9	0.844116
$576$	0	0
$577$	6.28378e8	0.136178	0.0680888	−	0.997679i	$-0.478310\pi$
$577$	6.28378e8	0.136178	0.0680888	+	0.997679i	$0.478310\pi$
$578$	−3.20135e9	−0.689582
$579$	0	0
$580$	8.72942e8	0.185775
$581$	1.34681e9	0.284898
$582$	0	0
$583$	2.29886e9	0.480478
$584$	−3.01993e9	−0.627411
$585$	0	0
$586$	−4.34467e9	−0.891899
$587$	−8.96802e9	−1.83005	−0.915026	−	0.403395i	$-0.867830\pi$
$587$	−8.96802e9	−1.83005	−0.915026	+	0.403395i	$0.867830\pi$
$588$	0	0
$589$	−2.63737e8	−0.0531824
$590$	2.93788e9	0.588913
$591$	0	0
$592$	−1.32534e9	−0.262543
$593$	−7.57484e9	−1.49170	−0.745851	−	0.666112i	$-0.767957\pi$
$593$	−7.57484e9	−1.49170	−0.745851	+	0.666112i	$0.767957\pi$
$594$	0	0
$595$	−2.67302e8	−0.0520227
$596$	−1.68248e8	−0.0325527
$597$	0	0
$598$	−3.04759e9	−0.582776
$599$	−6.05946e9	−1.15197	−0.575984	−	0.817461i	$-0.695381\pi$
$599$	−6.05946e9	−1.15197	−0.575984	+	0.817461i	$0.695381\pi$
$600$	0	0
$601$	−6.55797e8	−0.123228	−0.0616139	−	0.998100i	$-0.519625\pi$
$601$	−6.55797e8	−0.123228	−0.0616139	+	0.998100i	$0.519625\pi$
$602$	−1.36827e9	−0.255613
$603$	0	0
$604$	−3.53932e9	−0.653568
$605$	1.70653e9	0.313307
$606$	0	0
$607$	3.85611e9	0.699825	0.349913	−	0.936782i	$-0.386211\pi$
$607$	3.85611e9	0.699825	0.349913	+	0.936782i	$0.386211\pi$
$608$	4.94141e7	0.00891638
$609$	0	0
$610$	−2.96316e9	−0.528567
$611$	−1.93092e9	−0.342469
$612$	0	0
$613$	−4.89889e9	−0.858986	−0.429493	−	0.903070i	$-0.641308\pi$
$613$	−4.89889e9	−0.858986	−0.429493	+	0.903070i	$0.641308\pi$
$614$	−5.83902e9	−1.01801
$615$	0	0
$616$	−7.86530e8	−0.135576
$617$	−6.65456e9	−1.14057	−0.570284	−	0.821448i	$-0.693167\pi$
$617$	−6.65456e9	−1.14057	−0.570284	+	0.821448i	$0.693167\pi$
$618$	0	0
$619$	1.05531e10	1.78840	0.894199	−	0.447669i	$-0.147746\pi$
$619$	1.05531e10	1.78840	0.894199	+	0.447669i	$0.147746\pi$
$620$	1.84686e9	0.311217
$621$	0	0
$622$	5.11016e9	0.851468
$623$	3.43962e9	0.569904
$624$	0	0
$625$	4.63857e8	0.0759983
$626$	−5.10208e9	−0.831261
$627$	0	0
$628$	−1.43175e9	−0.230679
$629$	1.03186e9	0.165327
$630$	0	0
$631$	6.37775e9	1.01057	0.505283	−	0.862954i	$-0.331388\pi$
$631$	6.37775e9	1.01057	0.505283	+	0.862954i	$0.331388\pi$
$632$	3.59873e9	0.567074
$633$	0	0
$634$	−5.23502e9	−0.815842
$635$	8.22788e8	0.127520
$636$	0	0
$637$	−2.84945e9	−0.436790
$638$	−1.99983e9	−0.304874
$639$	0	0
$640$	−3.46030e8	−0.0521776
$641$	6.68703e8	0.100284	0.0501418	−	0.998742i	$-0.484033\pi$
$641$	6.68703e8	0.100284	0.0501418	+	0.998742i	$0.484033\pi$
$642$	0	0
$643$	3.24632e9	0.481564	0.240782	−	0.970579i	$-0.422596\pi$
$643$	3.24632e9	0.481564	0.240782	+	0.970579i	$0.422596\pi$
$644$	2.45791e9	0.362631
$645$	0	0
$646$	−3.84721e7	−0.00561477
$647$	7.46321e9	1.08333	0.541665	−	0.840595i	$-0.317794\pi$
$647$	7.46321e9	1.08333	0.541665	+	0.840595i	$0.317794\pi$
$648$	0	0
$649$	−6.73041e9	−0.966462
$650$	−2.05188e9	−0.293059
$651$	0	0
$652$	3.35108e9	0.473498
$653$	−1.01630e10	−1.42832	−0.714160	−	0.699982i	$-0.753191\pi$
$653$	−1.01630e10	−1.42832	−0.714160	+	0.699982i	$0.753191\pi$
$654$	0	0
$655$	3.68618e9	0.512544
$656$	−1.26205e9	−0.174547
$657$	0	0
$658$	1.55731e9	0.213100
$659$	−2.83068e9	−0.385294	−0.192647	−	0.981268i	$-0.561707\pi$
$659$	−2.83068e9	−0.385294	−0.192647	+	0.981268i	$0.561707\pi$
$660$	0	0
$661$	3.96275e9	0.533693	0.266846	−	0.963739i	$-0.414018\pi$
$661$	3.96275e9	0.533693	0.266846	+	0.963739i	$0.414018\pi$
$662$	6.46151e9	0.865628
$663$	0	0
$664$	−1.35741e9	−0.179938
$665$	1.26401e8	0.0166676
$666$	0	0
$667$	6.24947e9	0.815461
$668$	−3.50693e9	−0.455207
$669$	0	0
$670$	−1.94461e9	−0.249787
$671$	6.78832e9	0.867428
$672$	0	0
$673$	−1.18615e10	−1.49999	−0.749995	−	0.661443i	$-0.769944\pi$
$673$	−1.18615e10	−1.49999	−0.749995	+	0.661443i	$0.769944\pi$
$674$	−1.49824e9	−0.188483
$675$	0	0
$676$	−2.39085e9	−0.297672
$677$	9.86836e9	1.22232	0.611159	−	0.791508i	$-0.290703\pi$
$677$	9.86836e9	1.22232	0.611159	+	0.791508i	$0.290703\pi$
$678$	0	0
$679$	−8.21760e9	−1.00740
$680$	2.69407e8	0.0328569
$681$	0	0
$682$	−4.23099e9	−0.510736
$683$	1.36287e10	1.63674	0.818372	−	0.574689i	$-0.194877\pi$
$683$	1.36287e10	1.63674	0.818372	+	0.574689i	$0.194877\pi$
$684$	0	0
$685$	−2.12241e9	−0.252297
$686$	5.64499e9	0.667619
$687$	0	0
$688$	1.37904e9	0.161442
$689$	3.83068e9	0.446178
$690$	0	0
$691$	−2.16270e9	−0.249357	−0.124679	−	0.992197i	$-0.539790\pi$
$691$	−2.16270e9	−0.249357	−0.124679	+	0.992197i	$0.539790\pi$
$692$	7.91329e9	0.907791
$693$	0	0
$694$	2.26045e9	0.256706
$695$	−7.29139e9	−0.823879
$696$	0	0
$697$	9.82588e8	0.109915
$698$	−6.08490e9	−0.677266
$699$	0	0
$700$	1.65486e9	0.182355
$701$	2.93914e9	0.322260	0.161130	−	0.986933i	$-0.448486\pi$
$701$	2.93914e9	0.322260	0.161130	+	0.986933i	$0.448486\pi$
$702$	0	0
$703$	−4.87942e8	−0.0529693
$704$	7.92723e8	0.0856283
$705$	0	0
$706$	−3.75115e9	−0.401188
$707$	−1.37242e9	−0.146056
$708$	0	0
$709$	−1.30914e10	−1.37951	−0.689756	−	0.724041i	$-0.742282\pi$
$709$	−1.30914e10	−1.37951	−0.689756	+	0.724041i	$0.742282\pi$
$710$	6.60910e9	0.693007
$711$	0	0
$712$	−3.46670e9	−0.359945
$713$	1.32218e10	1.36609
$714$	0	0
$715$	−2.51426e9	−0.257240
$716$	6.31870e9	0.643328
$717$	0	0
$718$	1.23125e9	0.124140
$719$	−3.03015e9	−0.304027	−0.152014	−	0.988378i	$-0.548576\pi$
$719$	−3.03015e9	−0.304027	−0.152014	+	0.988378i	$0.548576\pi$
$720$	0	0
$721$	−1.05101e9	−0.104432
$722$	−7.13278e9	−0.705308
$723$	0	0
$724$	5.88707e9	0.576520
$725$	4.20765e9	0.410069
$726$	0	0
$727$	−1.06848e10	−1.03132	−0.515661	−	0.856793i	$-0.672454\pi$
$727$	−1.06848e10	−1.03132	−0.515661	+	0.856793i	$0.672454\pi$
$728$	−1.31062e9	−0.125898
$729$	0	0
$730$	7.78576e9	0.740749
$731$	−1.07367e9	−0.101663
$732$	0	0
$733$	−1.67705e10	−1.57283	−0.786414	−	0.617699i	$-0.788065\pi$
$733$	−1.67705e10	−1.57283	−0.786414	+	0.617699i	$0.788065\pi$
$734$	−5.99622e9	−0.559682
$735$	0	0
$736$	−2.47726e9	−0.229034
$737$	4.45492e9	0.409924
$738$	0	0
$739$	1.68052e10	1.53175	0.765877	−	0.642987i	$-0.222305\pi$
$739$	1.68052e10	1.53175	0.765877	+	0.642987i	$0.222305\pi$
$740$	3.41689e9	0.309970
$741$	0	0
$742$	−3.08948e9	−0.277633
$743$	2.07834e10	1.85890	0.929448	−	0.368952i	$-0.120283\pi$
$743$	2.07834e10	1.85890	0.929448	+	0.368952i	$0.120283\pi$
$744$	0	0
$745$	4.33764e8	0.0384332
$746$	−4.33382e9	−0.382195
$747$	0	0
$748$	−6.17186e8	−0.0539213
$749$	4.65049e9	0.404401
$750$	0	0
$751$	1.00601e10	0.866689	0.433344	−	0.901228i	$-0.357333\pi$
$751$	1.00601e10	0.866689	0.433344	+	0.901228i	$0.357333\pi$
$752$	−1.56957e9	−0.134592
$753$	0	0
$754$	−3.33239e9	−0.283111
$755$	9.12481e9	0.771631
$756$	0	0
$757$	6.59893e9	0.552889	0.276445	−	0.961030i	$-0.410844\pi$
$757$	6.59893e9	0.552889	0.276445	+	0.961030i	$0.410844\pi$
$758$	1.06379e10	0.887185
$759$	0	0
$760$	−1.27396e8	−0.0105271
$761$	−6.07925e9	−0.500039	−0.250019	−	0.968241i	$-0.580437\pi$
$761$	−6.07925e9	−0.500039	−0.250019	+	0.968241i	$0.580437\pi$
$762$	0	0
$763$	−4.41272e9	−0.359642
$764$	8.44663e9	0.685262
$765$	0	0
$766$	5.75959e9	0.463011
$767$	−1.12151e10	−0.897471
$768$	0	0
$769$	4.85049e9	0.384630	0.192315	−	0.981333i	$-0.438400\pi$
$769$	4.85049e9	0.384630	0.192315	+	0.981333i	$0.438400\pi$
$770$	2.02777e9	0.160067
$771$	0	0
$772$	9.46607e9	0.740473
$773$	−2.33466e10	−1.81800	−0.909002	−	0.416792i	$-0.863154\pi$
$773$	−2.33466e10	−1.81800	−0.909002	+	0.416792i	$0.863154\pi$
$774$	0	0
$775$	8.90200e9	0.686961
$776$	8.28231e9	0.636261
$777$	0	0
$778$	1.85654e10	1.41343
$779$	−4.64642e8	−0.0352158
$780$	0	0
$781$	−1.51408e10	−1.13729
$782$	1.92871e9	0.144226
$783$	0	0
$784$	−2.31620e9	−0.171660
$785$	3.69123e9	0.272350
$786$	0	0
$787$	9.36056e9	0.684526	0.342263	−	0.939604i	$-0.388807\pi$
$787$	9.36056e9	0.684526	0.342263	+	0.939604i	$0.388807\pi$
$788$	−1.72519e9	−0.125602
$789$	0	0
$790$	−9.27797e9	−0.669512
$791$	1.26886e10	0.911582
$792$	0	0
$793$	1.13116e10	0.805506
$794$	9.46981e9	0.671382
$795$	0	0
$796$	8.69315e9	0.610916
$797$	2.62373e10	1.83576	0.917878	−	0.396864i	$-0.129901\pi$
$797$	2.62373e10	1.83576	0.917878	+	0.396864i	$0.129901\pi$
$798$	0	0
$799$	1.22201e9	0.0847543
$800$	−1.66789e9	−0.115174
$801$	0	0
$802$	1.01411e10	0.694185
$803$	−1.78365e10	−1.21564
$804$	0	0
$805$	−6.33679e9	−0.428138
$806$	−7.05025e9	−0.474277
$807$	0	0
$808$	1.38323e9	0.0922473
$809$	−1.59304e10	−1.05780	−0.528902	−	0.848683i	$-0.677396\pi$
$809$	−1.59304e10	−1.05780	−0.528902	+	0.848683i	$0.677396\pi$
$810$	0	0
$811$	−1.69281e10	−1.11438	−0.557191	−	0.830384i	$-0.688121\pi$
$811$	−1.69281e10	−1.11438	−0.557191	+	0.830384i	$0.688121\pi$
$812$	2.68760e9	0.176165
$813$	0	0
$814$	−7.82778e9	−0.508690
$815$	−8.63949e9	−0.559032
$816$	0	0
$817$	5.07713e8	0.0325718
$818$	−8.15591e9	−0.520997
$819$	0	0
$820$	3.25373e9	0.206078
$821$	8.73592e9	0.550944	0.275472	−	0.961309i	$-0.411166\pi$
$821$	8.73592e9	0.550944	0.275472	+	0.961309i	$0.411166\pi$
$822$	0	0
$823$	1.44369e10	0.902767	0.451384	−	0.892330i	$-0.350931\pi$
$823$	1.44369e10	0.902767	0.451384	+	0.892330i	$0.350931\pi$
$824$	1.05928e9	0.0659580
$825$	0	0
$826$	9.04510e9	0.558449
$827$	5.56666e9	0.342236	0.171118	−	0.985251i	$-0.445262\pi$
$827$	5.56666e9	0.342236	0.171118	+	0.985251i	$0.445262\pi$
$828$	0	0
$829$	−1.50766e10	−0.919101	−0.459550	−	0.888152i	$-0.651989\pi$
$829$	−1.50766e10	−0.919101	−0.459550	+	0.888152i	$0.651989\pi$
$830$	3.49958e9	0.212443
$831$	0	0
$832$	1.32094e9	0.0795157
$833$	1.80331e9	0.108097
$834$	0	0
$835$	9.04130e9	0.537438
$836$	2.91852e8	0.0172759
$837$	0	0
$838$	8.33029e9	0.488997
$839$	1.20004e10	0.701502	0.350751	−	0.936469i	$-0.385926\pi$
$839$	1.20004e10	0.701502	0.350751	+	0.936469i	$0.385926\pi$
$840$	0	0
$841$	−1.04164e10	−0.603852
$842$	−1.56676e10	−0.904503
$843$	0	0
$844$	−1.40378e8	−0.00803710
$845$	6.16390e9	0.351445
$846$	0	0
$847$	5.25404e9	0.297099
$848$	3.11380e9	0.175350
$849$	0	0
$850$	1.29856e9	0.0725264
$851$	2.44618e10	1.36061
$852$	0	0
$853$	−9.01661e8	−0.0497418	−0.0248709	−	0.999691i	$-0.507917\pi$
$853$	−9.01661e8	−0.0497418	−0.0248709	+	0.999691i	$0.507917\pi$
$854$	−9.12293e9	−0.501224
$855$	0	0
$856$	−4.68710e9	−0.255415
$857$	−2.81436e10	−1.52738	−0.763690	−	0.645583i	$-0.776614\pi$
$857$	−2.81436e10	−1.52738	−0.763690	+	0.645583i	$0.776614\pi$
$858$	0	0
$859$	−9.51543e9	−0.512215	−0.256107	−	0.966648i	$-0.582440\pi$
$859$	−9.51543e9	−0.512215	−0.256107	+	0.966648i	$0.582440\pi$
$860$	−3.55534e9	−0.190606
$861$	0	0
$862$	2.27405e10	1.20927
$863$	−1.24682e10	−0.660336	−0.330168	−	0.943922i	$-0.607105\pi$
$863$	−1.24682e10	−0.660336	−0.330168	+	0.943922i	$0.607105\pi$
$864$	0	0
$865$	−2.04014e10	−1.07178
$866$	−6.73033e8	−0.0352147
$867$	0	0
$868$	5.68609e9	0.295117
$869$	2.12550e10	1.09873
$870$	0	0
$871$	7.42339e9	0.380662
$872$	4.44746e9	0.227146
$873$	0	0
$874$	−9.12038e8	−0.0462086
$875$	−1.08149e10	−0.545749
$876$	0	0
$877$	−6.71074e9	−0.335948	−0.167974	−	0.985791i	$-0.553722\pi$
$877$	−6.71074e9	−0.335948	−0.167974	+	0.985791i	$0.553722\pi$
$878$	−1.49637e10	−0.746120
$879$	0	0
$880$	−2.04374e9	−0.101097
$881$	−3.59350e10	−1.77053	−0.885263	−	0.465091i	$-0.846022\pi$
$881$	−3.59350e10	−1.77053	−0.885263	+	0.465091i	$0.846022\pi$
$882$	0	0
$883$	−1.40241e9	−0.0685509	−0.0342754	−	0.999412i	$-0.510912\pi$
$883$	−1.40241e9	−0.0685509	−0.0342754	+	0.999412i	$0.510912\pi$
$884$	−1.02844e9	−0.0500721
$885$	0	0
$886$	−8.10661e9	−0.391581
$887$	−8.19690e8	−0.0394382	−0.0197191	−	0.999806i	$-0.506277\pi$
$887$	−8.19690e8	−0.0394382	−0.0197191	+	0.999806i	$0.506277\pi$
$888$	0	0
$889$	2.53319e9	0.120924
$890$	8.93759e9	0.424967
$891$	0	0
$892$	1.27643e9	0.0602174
$893$	−5.77860e8	−0.0271545
$894$	0	0
$895$	−1.62904e10	−0.759541
$896$	−1.06535e9	−0.0494784
$897$	0	0
$898$	−1.32604e9	−0.0611070
$899$	1.44574e10	0.663640
$900$	0	0
$901$	−2.42430e9	−0.110420
$902$	−7.45399e9	−0.338194
$903$	0	0
$904$	−1.27885e10	−0.575745
$905$	−1.51776e10	−0.680665
$906$	0	0
$907$	2.84506e9	0.126609	0.0633047	−	0.997994i	$-0.479836\pi$
$907$	2.84506e9	0.126609	0.0633047	+	0.997994i	$0.479836\pi$
$908$	−8.06304e9	−0.357436
$909$	0	0
$910$	3.37895e9	0.148640
$911$	−4.32996e10	−1.89745	−0.948723	−	0.316108i	$-0.897624\pi$
$911$	−4.32996e10	−1.89745	−0.948723	+	0.316108i	$0.897624\pi$
$912$	0	0
$913$	−8.01722e9	−0.348639
$914$	−1.10464e10	−0.478529
$915$	0	0
$916$	1.50429e9	0.0646693
$917$	1.13490e10	0.486030
$918$	0	0
$919$	−3.26317e10	−1.38687	−0.693435	−	0.720520i	$-0.743903\pi$
$919$	−3.26317e10	−1.38687	−0.693435	+	0.720520i	$0.743903\pi$
$920$	6.38669e9	0.270407
$921$	0	0
$922$	2.22793e10	0.936146
$923$	−2.52297e10	−1.05610
$924$	0	0
$925$	1.64697e10	0.684209
$926$	2.23135e10	0.923483
$927$	0	0
$928$	−2.70877e9	−0.111264
$929$	4.69546e10	1.92142	0.960712	−	0.277546i	$-0.0895211\pi$
$929$	4.69546e10	1.92142	0.960712	+	0.277546i	$0.0895211\pi$
$930$	0	0
$931$	−8.52742e8	−0.0346333
$932$	−8.17644e9	−0.330833
$933$	0	0
$934$	3.17806e10	1.27629
$935$	1.59118e9	0.0636619
$936$	0	0
$937$	−2.80951e10	−1.11569	−0.557843	−	0.829947i	$-0.688371\pi$
$937$	−2.80951e10	−1.11569	−0.557843	+	0.829947i	$0.688371\pi$
$938$	−5.98704e9	−0.236866
$939$	0	0
$940$	4.04655e9	0.158905
$941$	−2.72197e10	−1.06493	−0.532464	−	0.846453i	$-0.678734\pi$
$941$	−2.72197e10	−1.06493	−0.532464	+	0.846453i	$0.678734\pi$
$942$	0	0
$943$	2.32937e10	0.904582
$944$	−9.11632e9	−0.352710
$945$	0	0
$946$	8.14496e9	0.312802
$947$	−2.30441e10	−0.881729	−0.440864	−	0.897574i	$-0.645328\pi$
$947$	−2.30441e10	−0.881729	−0.440864	+	0.897574i	$0.645328\pi$
$948$	0	0
$949$	−2.97215e10	−1.12886
$950$	−6.14058e8	−0.0232368
$951$	0	0
$952$	8.29446e8	0.0311572
$953$	2.22131e10	0.831352	0.415676	−	0.909513i	$-0.363545\pi$
$953$	2.22131e10	0.831352	0.415676	+	0.909513i	$0.363545\pi$
$954$	0	0
$955$	−2.17765e10	−0.809051
$956$	−5.98322e9	−0.221479
$957$	0	0
$958$	−1.87574e10	−0.689275
$959$	−6.53444e9	−0.239245
$960$	0	0
$961$	3.07460e9	0.111752
$962$	−1.30437e10	−0.472377
$963$	0	0
$964$	−8.62593e9	−0.310125
$965$	−2.44047e10	−0.874235
$966$	0	0
$967$	2.70771e10	0.962962	0.481481	−	0.876457i	$-0.340099\pi$
$967$	2.70771e10	0.962962	0.481481	+	0.876457i	$0.340099\pi$
$968$	−5.29541e9	−0.187644
$969$	0	0
$970$	−2.13528e10	−0.751198
$971$	−2.57664e10	−0.903204	−0.451602	−	0.892220i	$-0.649147\pi$
$971$	−2.57664e10	−0.903204	−0.451602	+	0.892220i	$0.649147\pi$
$972$	0	0
$973$	−2.24486e10	−0.781259
$974$	2.46705e10	0.855502
$975$	0	0
$976$	9.19476e9	0.316567
$977$	−9.51873e9	−0.326549	−0.163274	−	0.986581i	$-0.552206\pi$
$977$	−9.51873e9	−0.326549	−0.163274	+	0.986581i	$0.552206\pi$
$978$	0	0
$979$	−2.04752e10	−0.697411
$980$	5.97146e9	0.202670
$981$	0	0
$982$	−2.97481e10	−1.00246
$983$	−5.51917e9	−0.185326	−0.0926631	−	0.995698i	$-0.529538\pi$
$983$	−5.51917e9	−0.185326	−0.0926631	+	0.995698i	$0.529538\pi$
$984$	0	0
$985$	4.44776e9	0.148291
$986$	2.10895e9	0.0700643
$987$	0	0
$988$	4.86324e8	0.0160427
$989$	−2.54530e10	−0.836666
$990$	0	0
$991$	1.65440e10	0.539987	0.269994	−	0.962862i	$-0.412978\pi$
$991$	1.65440e10	0.539987	0.269994	+	0.962862i	$0.412978\pi$
$992$	−5.73086e9	−0.186393
$993$	0	0
$994$	2.03480e10	0.657158
$995$	−2.24120e10	−0.721274
$996$	0	0
$997$	4.10055e10	1.31042	0.655208	−	0.755448i	$-0.272581\pi$
$997$	4.10055e10	1.31042	0.655208	+	0.755448i	$0.272581\pi$
$998$	3.49141e9	0.111184
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.a.b.1.1	yes	1
3.2	odd	2		162.8.a.a.1.1	✓	1
9.2	odd	6		162.8.c.h.109.1		2
9.4	even	3		162.8.c.e.55.1		2
9.5	odd	6		162.8.c.h.55.1		2
9.7	even	3		162.8.c.e.109.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.8.a.a.1.1	✓	1	3.2	odd	2
162.8.a.b.1.1	yes	1	1.1	even	1	trivial
162.8.c.e.55.1		2	9.4	even	3
162.8.c.e.109.1		2	9.7	even	3
162.8.c.h.55.1		2	9.5	odd	6
162.8.c.h.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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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