LMFDB - Embedded newform 162.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,4,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, 4); 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, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [1,-2,0,4,21] 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:	$9.55830942093$
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)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} +21.0000 q^{5} +8.00000 q^{7} -8.00000 q^{8} -42.0000 q^{10} +36.0000 q^{11} -49.0000 q^{13} -16.0000 q^{14} +16.0000 q^{16} +21.0000 q^{17} -112.000 q^{19} +84.0000 q^{20} -72.0000 q^{22} +180.000 q^{23} +316.000 q^{25} +98.0000 q^{26} +32.0000 q^{28} -135.000 q^{29} +308.000 q^{31} -32.0000 q^{32} -42.0000 q^{34} +168.000 q^{35} -1.00000 q^{37} +224.000 q^{38} -168.000 q^{40} -42.0000 q^{41} +20.0000 q^{43} +144.000 q^{44} -360.000 q^{46} +84.0000 q^{47} -279.000 q^{49} -632.000 q^{50} -196.000 q^{52} -174.000 q^{53} +756.000 q^{55} -64.0000 q^{56} +270.000 q^{58} +504.000 q^{59} -385.000 q^{61} -616.000 q^{62} +64.0000 q^{64} -1029.00 q^{65} +272.000 q^{67} +84.0000 q^{68} -336.000 q^{70} -888.000 q^{71} +371.000 q^{73} +2.00000 q^{74} -448.000 q^{76} +288.000 q^{77} -652.000 q^{79} +336.000 q^{80} +84.0000 q^{82} +84.0000 q^{83} +441.000 q^{85} -40.0000 q^{86} -288.000 q^{88} +21.0000 q^{89} -392.000 q^{91} +720.000 q^{92} -168.000 q^{94} -2352.00 q^{95} -1246.00 q^{97} +558.000 q^{98} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	21.0000	1.87830	0.939149	−	0.343511i	$-0.111616\pi$
$5$	21.0000	1.87830	0.939149	+	0.343511i	$0.111616\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−42.0000	−1.32816
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	−49.0000	−1.04540	−0.522698	−	0.852518i	$-0.675075\pi$
$13$	−49.0000	−1.04540	−0.522698	+	0.852518i	$0.675075\pi$
$14$	−16.0000	−0.305441
$15$	0	0
$16$	16.0000	0.250000
$17$	21.0000	0.299603	0.149801	−	0.988716i	$-0.452137\pi$
$17$	21.0000	0.299603	0.149801	+	0.988716i	$0.452137\pi$
$18$	0	0
$19$	−112.000	−1.35235	−0.676173	−	0.736743i	$-0.736363\pi$
$19$	−112.000	−1.35235	−0.676173	+	0.736743i	$0.736363\pi$
$20$	84.0000	0.939149
$21$	0	0
$22$	−72.0000	−0.697748
$23$	180.000	1.63185	0.815926	−	0.578156i	$-0.196228\pi$
$23$	180.000	1.63185	0.815926	+	0.578156i	$0.196228\pi$
$24$	0	0
$25$	316.000	2.52800
$26$	98.0000	0.739207
$27$	0	0
$28$	32.0000	0.215980
$29$	−135.000	−0.864444	−0.432222	−	0.901767i	$-0.642270\pi$
$29$	−135.000	−0.864444	−0.432222	+	0.901767i	$0.642270\pi$
$30$	0	0
$31$	308.000	1.78447	0.892233	−	0.451576i	$-0.149138\pi$
$31$	308.000	1.78447	0.892233	+	0.451576i	$0.149138\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−42.0000	−0.211851
$35$	168.000	0.811348
$36$	0	0
$37$	−1.00000	−0.00444322	−0.00222161	−	0.999998i	$-0.500707\pi$
$37$	−1.00000	−0.00444322	−0.00222161	+	0.999998i	$0.500707\pi$
$38$	224.000	0.956253
$39$	0	0
$40$	−168.000	−0.664078
$41$	−42.0000	−0.159983	−0.0799914	−	0.996796i	$-0.525489\pi$
$41$	−42.0000	−0.159983	−0.0799914	+	0.996796i	$0.525489\pi$
$42$	0	0
$43$	20.0000	0.0709296	0.0354648	−	0.999371i	$-0.488709\pi$
$43$	20.0000	0.0709296	0.0354648	+	0.999371i	$0.488709\pi$
$44$	144.000	0.493382
$45$	0	0
$46$	−360.000	−1.15389
$47$	84.0000	0.260695	0.130347	−	0.991468i	$-0.458391\pi$
$47$	84.0000	0.260695	0.130347	+	0.991468i	$0.458391\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	−632.000	−1.78757
$51$	0	0
$52$	−196.000	−0.522698
$53$	−174.000	−0.450957	−0.225479	−	0.974248i	$-0.572395\pi$
$53$	−174.000	−0.450957	−0.225479	+	0.974248i	$0.572395\pi$
$54$	0	0
$55$	756.000	1.85344
$56$	−64.0000	−0.152721
$57$	0	0
$58$	270.000	0.611254
$59$	504.000	1.11212	0.556061	−	0.831141i	$-0.312312\pi$
$59$	504.000	1.11212	0.556061	+	0.831141i	$0.312312\pi$
$60$	0	0
$61$	−385.000	−0.808102	−0.404051	−	0.914737i	$-0.632398\pi$
$61$	−385.000	−0.808102	−0.404051	+	0.914737i	$0.632398\pi$
$62$	−616.000	−1.26181
$63$	0	0
$64$	64.0000	0.125000
$65$	−1029.00	−1.96357
$66$	0	0
$67$	272.000	0.495971	0.247986	−	0.968764i	$-0.420231\pi$
$67$	272.000	0.495971	0.247986	+	0.968764i	$0.420231\pi$
$68$	84.0000	0.149801
$69$	0	0
$70$	−336.000	−0.573710
$71$	−888.000	−1.48431	−0.742156	−	0.670227i	$-0.766197\pi$
$71$	−888.000	−1.48431	−0.742156	+	0.670227i	$0.766197\pi$
$72$	0	0
$73$	371.000	0.594826	0.297413	−	0.954749i	$-0.403876\pi$
$73$	371.000	0.594826	0.297413	+	0.954749i	$0.403876\pi$
$74$	2.00000	0.00314183
$75$	0	0
$76$	−448.000	−0.676173
$77$	288.000	0.426242
$78$	0	0
$79$	−652.000	−0.928554	−0.464277	−	0.885690i	$-0.653686\pi$
$79$	−652.000	−0.928554	−0.464277	+	0.885690i	$0.653686\pi$
$80$	336.000	0.469574
$81$	0	0
$82$	84.0000	0.113125
$83$	84.0000	0.111087	0.0555434	−	0.998456i	$-0.482311\pi$
$83$	84.0000	0.111087	0.0555434	+	0.998456i	$0.482311\pi$
$84$	0	0
$85$	441.000	0.562743
$86$	−40.0000	−0.0501548
$87$	0	0
$88$	−288.000	−0.348874
$89$	21.0000	0.0250112	0.0125056	−	0.999922i	$-0.496019\pi$
$89$	21.0000	0.0250112	0.0125056	+	0.999922i	$0.496019\pi$
$90$	0	0
$91$	−392.000	−0.451569
$92$	720.000	0.815926
$93$	0	0
$94$	−168.000	−0.184339
$95$	−2352.00	−2.54011
$96$	0	0
$97$	−1246.00	−1.30425	−0.652124	−	0.758112i	$-0.726122\pi$
$97$	−1246.00	−1.30425	−0.652124	+	0.758112i	$0.726122\pi$
$98$	558.000	0.575168
$99$	0	0
$100$	1264.00	1.26400
$101$	546.000	0.537911	0.268956	−	0.963153i	$-0.413322\pi$
$101$	546.000	0.537911	0.268956	+	0.963153i	$0.413322\pi$
$102$	0	0
$103$	−196.000	−0.187500	−0.0937498	−	0.995596i	$-0.529885\pi$
$103$	−196.000	−0.187500	−0.0937498	+	0.995596i	$0.529885\pi$
$104$	392.000	0.369603
$105$	0	0
$106$	348.000	0.318875
$107$	−300.000	−0.271048	−0.135524	−	0.990774i	$-0.543272\pi$
$107$	−300.000	−0.271048	−0.135524	+	0.990774i	$0.543272\pi$
$108$	0	0
$109$	−1069.00	−0.939373	−0.469686	−	0.882833i	$-0.655633\pi$
$109$	−1069.00	−0.939373	−0.469686	+	0.882833i	$0.655633\pi$
$110$	−1512.00	−1.31058
$111$	0	0
$112$	128.000	0.107990
$113$	897.000	0.746749	0.373375	−	0.927681i	$-0.378201\pi$
$113$	897.000	0.746749	0.373375	+	0.927681i	$0.378201\pi$
$114$	0	0
$115$	3780.00	3.06510
$116$	−540.000	−0.432222
$117$	0	0
$118$	−1008.00	−0.786389
$119$	168.000	0.129416
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	770.000	0.571414
$123$	0	0
$124$	1232.00	0.892233
$125$	4011.00	2.87004
$126$	0	0
$127$	1532.00	1.07042	0.535209	−	0.844720i	$-0.320233\pi$
$127$	1532.00	1.07042	0.535209	+	0.844720i	$0.320233\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	2058.00	1.38845
$131$	840.000	0.560238	0.280119	−	0.959965i	$-0.409626\pi$
$131$	840.000	0.560238	0.280119	+	0.959965i	$0.409626\pi$
$132$	0	0
$133$	−896.000	−0.584158
$134$	−544.000	−0.350705
$135$	0	0
$136$	−168.000	−0.105926
$137$	729.000	0.454618	0.227309	−	0.973823i	$-0.427007\pi$
$137$	729.000	0.454618	0.227309	+	0.973823i	$0.427007\pi$
$138$	0	0
$139$	−2044.00	−1.24726	−0.623632	−	0.781718i	$-0.714344\pi$
$139$	−2044.00	−1.24726	−0.623632	+	0.781718i	$0.714344\pi$
$140$	672.000	0.405674
$141$	0	0
$142$	1776.00	1.04957
$143$	−1764.00	−1.03156
$144$	0	0
$145$	−2835.00	−1.62368
$146$	−742.000	−0.420605
$147$	0	0
$148$	−4.00000	−0.00222161
$149$	−1287.00	−0.707618	−0.353809	−	0.935318i	$-0.615114\pi$
$149$	−1287.00	−0.707618	−0.353809	+	0.935318i	$0.615114\pi$
$150$	0	0
$151$	−736.000	−0.396655	−0.198327	−	0.980136i	$-0.563551\pi$
$151$	−736.000	−0.396655	−0.198327	+	0.980136i	$0.563551\pi$
$152$	896.000	0.478126
$153$	0	0
$154$	−576.000	−0.301399
$155$	6468.00	3.35176
$156$	0	0
$157$	−2149.00	−1.09241	−0.546207	−	0.837650i	$-0.683929\pi$
$157$	−2149.00	−1.09241	−0.546207	+	0.837650i	$0.683929\pi$
$158$	1304.00	0.656587
$159$	0	0
$160$	−672.000	−0.332039
$161$	1440.00	0.704894
$162$	0	0
$163$	−3088.00	−1.48387	−0.741935	−	0.670472i	$-0.766092\pi$
$163$	−3088.00	−1.48387	−0.741935	+	0.670472i	$0.766092\pi$
$164$	−168.000	−0.0799914
$165$	0	0
$166$	−168.000	−0.0785502
$167$	168.000	0.0778457	0.0389228	−	0.999242i	$-0.487607\pi$
$167$	168.000	0.0778457	0.0389228	+	0.999242i	$0.487607\pi$
$168$	0	0
$169$	204.000	0.0928539
$170$	−882.000	−0.397919
$171$	0	0
$172$	80.0000	0.0354648
$173$	−3003.00	−1.31973	−0.659867	−	0.751383i	$-0.729387\pi$
$173$	−3003.00	−1.31973	−0.659867	+	0.751383i	$0.729387\pi$
$174$	0	0
$175$	2528.00	1.09199
$176$	576.000	0.246691
$177$	0	0
$178$	−42.0000	−0.0176856
$179$	−1164.00	−0.486042	−0.243021	−	0.970021i	$-0.578138\pi$
$179$	−1164.00	−0.486042	−0.243021	+	0.970021i	$0.578138\pi$
$180$	0	0
$181$	−1666.00	−0.684159	−0.342080	−	0.939671i	$-0.611131\pi$
$181$	−1666.00	−0.684159	−0.342080	+	0.939671i	$0.611131\pi$
$182$	784.000	0.319307
$183$	0	0
$184$	−1440.00	−0.576947
$185$	−21.0000	−0.00834568
$186$	0	0
$187$	756.000	0.295637
$188$	336.000	0.130347
$189$	0	0
$190$	4704.00	1.79613
$191$	−2064.00	−0.781915	−0.390958	−	0.920409i	$-0.627856\pi$
$191$	−2064.00	−0.781915	−0.390958	+	0.920409i	$0.627856\pi$
$192$	0	0
$193$	−565.000	−0.210723	−0.105362	−	0.994434i	$-0.533600\pi$
$193$	−565.000	−0.210723	−0.105362	+	0.994434i	$0.533600\pi$
$194$	2492.00	0.922243
$195$	0	0
$196$	−1116.00	−0.406706
$197$	−4731.00	−1.71101	−0.855507	−	0.517791i	$-0.826754\pi$
$197$	−4731.00	−1.71101	−0.855507	+	0.517791i	$0.826754\pi$
$198$	0	0
$199$	4676.00	1.66569	0.832846	−	0.553504i	$-0.186710\pi$
$199$	4676.00	1.66569	0.832846	+	0.553504i	$0.186710\pi$
$200$	−2528.00	−0.893783
$201$	0	0
$202$	−1092.00	−0.380361
$203$	−1080.00	−0.373405
$204$	0	0
$205$	−882.000	−0.300495
$206$	392.000	0.132582
$207$	0	0
$208$	−784.000	−0.261349
$209$	−4032.00	−1.33445
$210$	0	0
$211$	3380.00	1.10279	0.551395	−	0.834244i	$-0.314096\pi$
$211$	3380.00	1.10279	0.551395	+	0.834244i	$0.314096\pi$
$212$	−696.000	−0.225479
$213$	0	0
$214$	600.000	0.191660
$215$	420.000	0.133227
$216$	0	0
$217$	2464.00	0.770817
$218$	2138.00	0.664237
$219$	0	0
$220$	3024.00	0.926718
$221$	−1029.00	−0.313204
$222$	0	0
$223$	−5236.00	−1.57233	−0.786163	−	0.618020i	$-0.787935\pi$
$223$	−5236.00	−1.57233	−0.786163	+	0.618020i	$0.787935\pi$
$224$	−256.000	−0.0763604
$225$	0	0
$226$	−1794.00	−0.528031
$227$	3864.00	1.12979	0.564896	−	0.825162i	$-0.308916\pi$
$227$	3864.00	1.12979	0.564896	+	0.825162i	$0.308916\pi$
$228$	0	0
$229$	−3913.00	−1.12916	−0.564581	−	0.825377i	$-0.690962\pi$
$229$	−3913.00	−1.12916	−0.564581	+	0.825377i	$0.690962\pi$
$230$	−7560.00	−2.16735
$231$	0	0
$232$	1080.00	0.305627
$233$	6333.00	1.78064	0.890319	−	0.455337i	$-0.150481\pi$
$233$	6333.00	1.78064	0.890319	+	0.455337i	$0.150481\pi$
$234$	0	0
$235$	1764.00	0.489662
$236$	2016.00	0.556061
$237$	0	0
$238$	−336.000	−0.0915111
$239$	−3828.00	−1.03604	−0.518018	−	0.855370i	$-0.673330\pi$
$239$	−3828.00	−1.03604	−0.518018	+	0.855370i	$0.673330\pi$
$240$	0	0
$241$	−1477.00	−0.394780	−0.197390	−	0.980325i	$-0.563246\pi$
$241$	−1477.00	−0.394780	−0.197390	+	0.980325i	$0.563246\pi$
$242$	70.0000	0.0185941
$243$	0	0
$244$	−1540.00	−0.404051
$245$	−5859.00	−1.52783
$246$	0	0
$247$	5488.00	1.41374
$248$	−2464.00	−0.630904
$249$	0	0
$250$	−8022.00	−2.02942
$251$	−3612.00	−0.908316	−0.454158	−	0.890921i	$-0.650060\pi$
$251$	−3612.00	−0.908316	−0.454158	+	0.890921i	$0.650060\pi$
$252$	0	0
$253$	6480.00	1.61025
$254$	−3064.00	−0.756899
$255$	0	0
$256$	256.000	0.0625000
$257$	−399.000	−0.0968441	−0.0484221	−	0.998827i	$-0.515419\pi$
$257$	−399.000	−0.0968441	−0.0484221	+	0.998827i	$0.515419\pi$
$258$	0	0
$259$	−8.00000	−0.00191929
$260$	−4116.00	−0.981783
$261$	0	0
$262$	−1680.00	−0.396148
$263$	3228.00	0.756833	0.378416	−	0.925635i	$-0.376469\pi$
$263$	3228.00	0.756833	0.378416	+	0.925635i	$0.376469\pi$
$264$	0	0
$265$	−3654.00	−0.847032
$266$	1792.00	0.413062
$267$	0	0
$268$	1088.00	0.247986
$269$	−147.000	−0.0333188	−0.0166594	−	0.999861i	$-0.505303\pi$
$269$	−147.000	−0.0333188	−0.0166594	+	0.999861i	$0.505303\pi$
$270$	0	0
$271$	3332.00	0.746880	0.373440	−	0.927654i	$-0.378178\pi$
$271$	3332.00	0.746880	0.373440	+	0.927654i	$0.378178\pi$
$272$	336.000	0.0749007
$273$	0	0
$274$	−1458.00	−0.321464
$275$	11376.0	2.49454
$276$	0	0
$277$	2414.00	0.523622	0.261811	−	0.965119i	$-0.415680\pi$
$277$	2414.00	0.523622	0.261811	+	0.965119i	$0.415680\pi$
$278$	4088.00	0.881949
$279$	0	0
$280$	−1344.00	−0.286855
$281$	−3555.00	−0.754710	−0.377355	−	0.926069i	$-0.623166\pi$
$281$	−3555.00	−0.754710	−0.377355	+	0.926069i	$0.623166\pi$
$282$	0	0
$283$	5348.00	1.12334	0.561671	−	0.827361i	$-0.310159\pi$
$283$	5348.00	1.12334	0.561671	+	0.827361i	$0.310159\pi$
$284$	−3552.00	−0.742156
$285$	0	0
$286$	3528.00	0.729423
$287$	−336.000	−0.0691061
$288$	0	0
$289$	−4472.00	−0.910238
$290$	5670.00	1.14812
$291$	0	0
$292$	1484.00	0.297413
$293$	6489.00	1.29383	0.646914	−	0.762563i	$-0.276059\pi$
$293$	6489.00	1.29383	0.646914	+	0.762563i	$0.276059\pi$
$294$	0	0
$295$	10584.0	2.08890
$296$	8.00000	0.00157091
$297$	0	0
$298$	2574.00	0.500362
$299$	−8820.00	−1.70593
$300$	0	0
$301$	160.000	0.0306387
$302$	1472.00	0.280477
$303$	0	0
$304$	−1792.00	−0.338086
$305$	−8085.00	−1.51785
$306$	0	0
$307$	−1204.00	−0.223830	−0.111915	−	0.993718i	$-0.535698\pi$
$307$	−1204.00	−0.223830	−0.111915	+	0.993718i	$0.535698\pi$
$308$	1152.00	0.213121
$309$	0	0
$310$	−12936.0	−2.37005
$311$	−3192.00	−0.581999	−0.291000	−	0.956723i	$-0.593988\pi$
$311$	−3192.00	−0.581999	−0.291000	+	0.956723i	$0.593988\pi$
$312$	0	0
$313$	−3241.00	−0.585278	−0.292639	−	0.956223i	$-0.594533\pi$
$313$	−3241.00	−0.585278	−0.292639	+	0.956223i	$0.594533\pi$
$314$	4298.00	0.772453
$315$	0	0
$316$	−2608.00	−0.464277
$317$	5325.00	0.943476	0.471738	−	0.881739i	$-0.343627\pi$
$317$	5325.00	0.943476	0.471738	+	0.881739i	$0.343627\pi$
$318$	0	0
$319$	−4860.00	−0.853002
$320$	1344.00	0.234787
$321$	0	0
$322$	−2880.00	−0.498435
$323$	−2352.00	−0.405167
$324$	0	0
$325$	−15484.0	−2.64276
$326$	6176.00	1.04925
$327$	0	0
$328$	336.000	0.0565625
$329$	672.000	0.112610
$330$	0	0
$331$	968.000	0.160743	0.0803717	−	0.996765i	$-0.474389\pi$
$331$	968.000	0.160743	0.0803717	+	0.996765i	$0.474389\pi$
$332$	336.000	0.0555434
$333$	0	0
$334$	−336.000	−0.0550452
$335$	5712.00	0.931582
$336$	0	0
$337$	9890.00	1.59864	0.799321	−	0.600904i	$-0.205192\pi$
$337$	9890.00	1.59864	0.799321	+	0.600904i	$0.205192\pi$
$338$	−408.000	−0.0656576
$339$	0	0
$340$	1764.00	0.281372
$341$	11088.0	1.76085
$342$	0	0
$343$	−4976.00	−0.783320
$344$	−160.000	−0.0250774
$345$	0	0
$346$	6006.00	0.933192
$347$	−1560.00	−0.241341	−0.120670	−	0.992693i	$-0.538504\pi$
$347$	−1560.00	−0.241341	−0.120670	+	0.992693i	$0.538504\pi$
$348$	0	0
$349$	2870.00	0.440194	0.220097	−	0.975478i	$-0.429363\pi$
$349$	2870.00	0.440194	0.220097	+	0.975478i	$0.429363\pi$
$350$	−5056.00	−0.772156
$351$	0	0
$352$	−1152.00	−0.174437
$353$	7182.00	1.08289	0.541444	−	0.840737i	$-0.317878\pi$
$353$	7182.00	1.08289	0.541444	+	0.840737i	$0.317878\pi$
$354$	0	0
$355$	−18648.0	−2.78798
$356$	84.0000	0.0125056
$357$	0	0
$358$	2328.00	0.343683
$359$	8100.00	1.19081	0.595406	−	0.803425i	$-0.296991\pi$
$359$	8100.00	1.19081	0.595406	+	0.803425i	$0.296991\pi$
$360$	0	0
$361$	5685.00	0.828838
$362$	3332.00	0.483774
$363$	0	0
$364$	−1568.00	−0.225784
$365$	7791.00	1.11726
$366$	0	0
$367$	11144.0	1.58505	0.792523	−	0.609842i	$-0.208767\pi$
$367$	11144.0	1.58505	0.792523	+	0.609842i	$0.208767\pi$
$368$	2880.00	0.407963
$369$	0	0
$370$	42.0000	0.00590129
$371$	−1392.00	−0.194795
$372$	0	0
$373$	13838.0	1.92092	0.960462	−	0.278412i	$-0.0898080\pi$
$373$	13838.0	1.92092	0.960462	+	0.278412i	$0.0898080\pi$
$374$	−1512.00	−0.209047
$375$	0	0
$376$	−672.000	−0.0921696
$377$	6615.00	0.903687
$378$	0	0
$379$	1196.00	0.162096	0.0810480	−	0.996710i	$-0.474173\pi$
$379$	1196.00	0.162096	0.0810480	+	0.996710i	$0.474173\pi$
$380$	−9408.00	−1.27005
$381$	0	0
$382$	4128.00	0.552898
$383$	3864.00	0.515512	0.257756	−	0.966210i	$-0.417017\pi$
$383$	3864.00	0.515512	0.257756	+	0.966210i	$0.417017\pi$
$384$	0	0
$385$	6048.00	0.800609
$386$	1130.00	0.149004
$387$	0	0
$388$	−4984.00	−0.652124
$389$	−5070.00	−0.660821	−0.330410	−	0.943837i	$-0.607187\pi$
$389$	−5070.00	−0.660821	−0.330410	+	0.943837i	$0.607187\pi$
$390$	0	0
$391$	3780.00	0.488907
$392$	2232.00	0.287584
$393$	0	0
$394$	9462.00	1.20987
$395$	−13692.0	−1.74410
$396$	0	0
$397$	15239.0	1.92651	0.963254	−	0.268593i	$-0.0865587\pi$
$397$	15239.0	1.92651	0.963254	+	0.268593i	$0.0865587\pi$
$398$	−9352.00	−1.17782
$399$	0	0
$400$	5056.00	0.632000
$401$	−1707.00	−0.212577	−0.106289	−	0.994335i	$-0.533897\pi$
$401$	−1707.00	−0.212577	−0.106289	+	0.994335i	$0.533897\pi$
$402$	0	0
$403$	−15092.0	−1.86547
$404$	2184.00	0.268956
$405$	0	0
$406$	2160.00	0.264037
$407$	−36.0000	−0.00438441
$408$	0	0
$409$	−13321.0	−1.61047	−0.805234	−	0.592958i	$-0.797960\pi$
$409$	−13321.0	−1.61047	−0.805234	+	0.592958i	$0.797960\pi$
$410$	1764.00	0.212482
$411$	0	0
$412$	−784.000	−0.0937498
$413$	4032.00	0.480392
$414$	0	0
$415$	1764.00	0.208654
$416$	1568.00	0.184802
$417$	0	0
$418$	8064.00	0.943596
$419$	−13944.0	−1.62580	−0.812899	−	0.582405i	$-0.802112\pi$
$419$	−13944.0	−1.62580	−0.812899	+	0.582405i	$0.802112\pi$
$420$	0	0
$421$	−10837.0	−1.25454	−0.627272	−	0.778800i	$-0.715829\pi$
$421$	−10837.0	−1.25454	−0.627272	+	0.778800i	$0.715829\pi$
$422$	−6760.00	−0.779791
$423$	0	0
$424$	1392.00	0.159437
$425$	6636.00	0.757396
$426$	0	0
$427$	−3080.00	−0.349067
$428$	−1200.00	−0.135524
$429$	0	0
$430$	−840.000	−0.0942056
$431$	12612.0	1.40951	0.704755	−	0.709451i	$-0.251057\pi$
$431$	12612.0	1.40951	0.704755	+	0.709451i	$0.251057\pi$
$432$	0	0
$433$	−9709.00	−1.07756	−0.538781	−	0.842446i	$-0.681115\pi$
$433$	−9709.00	−1.07756	−0.538781	+	0.842446i	$0.681115\pi$
$434$	−4928.00	−0.545050
$435$	0	0
$436$	−4276.00	−0.469686
$437$	−20160.0	−2.20683
$438$	0	0
$439$	10388.0	1.12937	0.564684	−	0.825307i	$-0.308998\pi$
$439$	10388.0	1.12937	0.564684	+	0.825307i	$0.308998\pi$
$440$	−6048.00	−0.655289
$441$	0	0
$442$	2058.00	0.221469
$443$	−2508.00	−0.268981	−0.134491	−	0.990915i	$-0.542940\pi$
$443$	−2508.00	−0.268981	−0.134491	+	0.990915i	$0.542940\pi$
$444$	0	0
$445$	441.000	0.0469784
$446$	10472.0	1.11180
$447$	0	0
$448$	512.000	0.0539949
$449$	−13698.0	−1.43975	−0.719876	−	0.694103i	$-0.755801\pi$
$449$	−13698.0	−1.43975	−0.719876	+	0.694103i	$0.755801\pi$
$450$	0	0
$451$	−1512.00	−0.157865
$452$	3588.00	0.373375
$453$	0	0
$454$	−7728.00	−0.798883
$455$	−8232.00	−0.848180
$456$	0	0
$457$	−9745.00	−0.997488	−0.498744	−	0.866749i	$-0.666205\pi$
$457$	−9745.00	−0.997488	−0.498744	+	0.866749i	$0.666205\pi$
$458$	7826.00	0.798439
$459$	0	0
$460$	15120.0	1.53255
$461$	17514.0	1.76943	0.884716	−	0.466130i	$-0.154352\pi$
$461$	17514.0	1.76943	0.884716	+	0.466130i	$0.154352\pi$
$462$	0	0
$463$	4640.00	0.465743	0.232872	−	0.972507i	$-0.425188\pi$
$463$	4640.00	0.465743	0.232872	+	0.972507i	$0.425188\pi$
$464$	−2160.00	−0.216111
$465$	0	0
$466$	−12666.0	−1.25910
$467$	−4368.00	−0.432820	−0.216410	−	0.976303i	$-0.569435\pi$
$467$	−4368.00	−0.432820	−0.216410	+	0.976303i	$0.569435\pi$
$468$	0	0
$469$	2176.00	0.214240
$470$	−3528.00	−0.346244
$471$	0	0
$472$	−4032.00	−0.393195
$473$	720.000	0.0699908
$474$	0	0
$475$	−35392.0	−3.41873
$476$	672.000	0.0647081
$477$	0	0
$478$	7656.00	0.732588
$479$	−18816.0	−1.79483	−0.897416	−	0.441184i	$-0.854558\pi$
$479$	−18816.0	−1.79483	−0.897416	+	0.441184i	$0.854558\pi$
$480$	0	0
$481$	49.0000	0.00464492
$482$	2954.00	0.279151
$483$	0	0
$484$	−140.000	−0.0131480
$485$	−26166.0	−2.44977
$486$	0	0
$487$	−13756.0	−1.27997	−0.639983	−	0.768389i	$-0.721059\pi$
$487$	−13756.0	−1.27997	−0.639983	+	0.768389i	$0.721059\pi$
$488$	3080.00	0.285707
$489$	0	0
$490$	11718.0	1.08034
$491$	7740.00	0.711408	0.355704	−	0.934599i	$-0.384241\pi$
$491$	7740.00	0.711408	0.355704	+	0.934599i	$0.384241\pi$
$492$	0	0
$493$	−2835.00	−0.258990
$494$	−10976.0	−0.999663
$495$	0	0
$496$	4928.00	0.446116
$497$	−7104.00	−0.641163
$498$	0	0
$499$	2396.00	0.214949	0.107475	−	0.994208i	$-0.465724\pi$
$499$	2396.00	0.214949	0.107475	+	0.994208i	$0.465724\pi$
$500$	16044.0	1.43502
$501$	0	0
$502$	7224.00	0.642277
$503$	12096.0	1.07223	0.536117	−	0.844144i	$-0.319890\pi$
$503$	12096.0	1.07223	0.536117	+	0.844144i	$0.319890\pi$
$504$	0	0
$505$	11466.0	1.01036
$506$	−12960.0	−1.13862
$507$	0	0
$508$	6128.00	0.535209
$509$	1722.00	0.149953	0.0749767	−	0.997185i	$-0.476112\pi$
$509$	1722.00	0.149953	0.0749767	+	0.997185i	$0.476112\pi$
$510$	0	0
$511$	2968.00	0.256940
$512$	−512.000	−0.0441942
$513$	0	0
$514$	798.000	0.0684791
$515$	−4116.00	−0.352180
$516$	0	0
$517$	3024.00	0.257244
$518$	16.0000	0.00135714
$519$	0	0
$520$	8232.00	0.694225
$521$	2982.00	0.250756	0.125378	−	0.992109i	$-0.459986\pi$
$521$	2982.00	0.250756	0.125378	+	0.992109i	$0.459986\pi$
$522$	0	0
$523$	812.000	0.0678896	0.0339448	−	0.999424i	$-0.489193\pi$
$523$	812.000	0.0678896	0.0339448	+	0.999424i	$0.489193\pi$
$524$	3360.00	0.280119
$525$	0	0
$526$	−6456.00	−0.535162
$527$	6468.00	0.534631
$528$	0	0
$529$	20233.0	1.66294
$530$	7308.00	0.598942
$531$	0	0
$532$	−3584.00	−0.292079
$533$	2058.00	0.167246
$534$	0	0
$535$	−6300.00	−0.509108
$536$	−2176.00	−0.175352
$537$	0	0
$538$	294.000	0.0235599
$539$	−10044.0	−0.802645
$540$	0	0
$541$	7055.00	0.560662	0.280331	−	0.959903i	$-0.409556\pi$
$541$	7055.00	0.560662	0.280331	+	0.959903i	$0.409556\pi$
$542$	−6664.00	−0.528124
$543$	0	0
$544$	−672.000	−0.0529628
$545$	−22449.0	−1.76442
$546$	0	0
$547$	−14596.0	−1.14091	−0.570457	−	0.821328i	$-0.693234\pi$
$547$	−14596.0	−1.14091	−0.570457	+	0.821328i	$0.693234\pi$
$548$	2916.00	0.227309
$549$	0	0
$550$	−22752.0	−1.76391
$551$	15120.0	1.16903
$552$	0	0
$553$	−5216.00	−0.401097
$554$	−4828.00	−0.370256
$555$	0	0
$556$	−8176.00	−0.623632
$557$	−7755.00	−0.589928	−0.294964	−	0.955508i	$-0.595308\pi$
$557$	−7755.00	−0.589928	−0.294964	+	0.955508i	$0.595308\pi$
$558$	0	0
$559$	−980.000	−0.0741495
$560$	2688.00	0.202837
$561$	0	0
$562$	7110.00	0.533661
$563$	16044.0	1.20102	0.600510	−	0.799617i	$-0.294964\pi$
$563$	16044.0	1.20102	0.600510	+	0.799617i	$0.294964\pi$
$564$	0	0
$565$	18837.0	1.40262
$566$	−10696.0	−0.794322
$567$	0	0
$568$	7104.00	0.524784
$569$	17025.0	1.25435	0.627175	−	0.778878i	$-0.284211\pi$
$569$	17025.0	1.25435	0.627175	+	0.778878i	$0.284211\pi$
$570$	0	0
$571$	3320.00	0.243323	0.121662	−	0.992572i	$-0.461178\pi$
$571$	3320.00	0.243323	0.121662	+	0.992572i	$0.461178\pi$
$572$	−7056.00	−0.515780
$573$	0	0
$574$	672.000	0.0488654
$575$	56880.0	4.12532
$576$	0	0
$577$	1127.00	0.0813130	0.0406565	−	0.999173i	$-0.487055\pi$
$577$	1127.00	0.0813130	0.0406565	+	0.999173i	$0.487055\pi$
$578$	8944.00	0.643636
$579$	0	0
$580$	−11340.0	−0.811841
$581$	672.000	0.0479850
$582$	0	0
$583$	−6264.00	−0.444989
$584$	−2968.00	−0.210303
$585$	0	0
$586$	−12978.0	−0.914874
$587$	−84.0000	−0.00590639	−0.00295320	−	0.999996i	$-0.500940\pi$
$587$	−84.0000	−0.00590639	−0.00295320	+	0.999996i	$0.500940\pi$
$588$	0	0
$589$	−34496.0	−2.41321
$590$	−21168.0	−1.47707
$591$	0	0
$592$	−16.0000	−0.00111080
$593$	−1743.00	−0.120702	−0.0603511	−	0.998177i	$-0.519222\pi$
$593$	−1743.00	−0.120702	−0.0603511	+	0.998177i	$0.519222\pi$
$594$	0	0
$595$	3528.00	0.243082
$596$	−5148.00	−0.353809
$597$	0	0
$598$	17640.0	1.20628
$599$	−16092.0	−1.09766	−0.548832	−	0.835932i	$-0.684928\pi$
$599$	−16092.0	−1.09766	−0.548832	+	0.835932i	$0.684928\pi$
$600$	0	0
$601$	21035.0	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	21035.0	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	−320.000	−0.0216648
$603$	0	0
$604$	−2944.00	−0.198327
$605$	−735.000	−0.0493917
$606$	0	0
$607$	6776.00	0.453096	0.226548	−	0.974000i	$-0.427256\pi$
$607$	6776.00	0.453096	0.226548	+	0.974000i	$0.427256\pi$
$608$	3584.00	0.239063
$609$	0	0
$610$	16170.0	1.07329
$611$	−4116.00	−0.272530
$612$	0	0
$613$	−23794.0	−1.56775	−0.783875	−	0.620919i	$-0.786760\pi$
$613$	−23794.0	−1.56775	−0.783875	+	0.620919i	$0.786760\pi$
$614$	2408.00	0.158272
$615$	0	0
$616$	−2304.00	−0.150699
$617$	21621.0	1.41074	0.705372	−	0.708838i	$-0.250780\pi$
$617$	21621.0	1.41074	0.705372	+	0.708838i	$0.250780\pi$
$618$	0	0
$619$	22232.0	1.44359	0.721793	−	0.692109i	$-0.243318\pi$
$619$	22232.0	1.44359	0.721793	+	0.692109i	$0.243318\pi$
$620$	25872.0	1.67588
$621$	0	0
$622$	6384.00	0.411535
$623$	168.000	0.0108038
$624$	0	0
$625$	44731.0	2.86278
$626$	6482.00	0.413854
$627$	0	0
$628$	−8596.00	−0.546207
$629$	−21.0000	−0.00133120
$630$	0	0
$631$	−9280.00	−0.585469	−0.292735	−	0.956194i	$-0.594565\pi$
$631$	−9280.00	−0.585469	−0.292735	+	0.956194i	$0.594565\pi$
$632$	5216.00	0.328293
$633$	0	0
$634$	−10650.0	−0.667138
$635$	32172.0	2.01056
$636$	0	0
$637$	13671.0	0.850337
$638$	9720.00	0.603164
$639$	0	0
$640$	−2688.00	−0.166020
$641$	−19179.0	−1.18179	−0.590893	−	0.806750i	$-0.701225\pi$
$641$	−19179.0	−1.18179	−0.590893	+	0.806750i	$0.701225\pi$
$642$	0	0
$643$	−3220.00	−0.197487	−0.0987437	−	0.995113i	$-0.531482\pi$
$643$	−3220.00	−0.197487	−0.0987437	+	0.995113i	$0.531482\pi$
$644$	5760.00	0.352447
$645$	0	0
$646$	4704.00	0.286496
$647$	14112.0	0.857496	0.428748	−	0.903424i	$-0.358955\pi$
$647$	14112.0	0.857496	0.428748	+	0.903424i	$0.358955\pi$
$648$	0	0
$649$	18144.0	1.09740
$650$	30968.0	1.86872
$651$	0	0
$652$	−12352.0	−0.741935
$653$	22842.0	1.36888	0.684438	−	0.729071i	$-0.260047\pi$
$653$	22842.0	1.36888	0.684438	+	0.729071i	$0.260047\pi$
$654$	0	0
$655$	17640.0	1.05229
$656$	−672.000	−0.0399957
$657$	0	0
$658$	−1344.00	−0.0796270
$659$	−21720.0	−1.28390	−0.641951	−	0.766746i	$-0.721875\pi$
$659$	−21720.0	−1.28390	−0.641951	+	0.766746i	$0.721875\pi$
$660$	0	0
$661$	26327.0	1.54917	0.774585	−	0.632470i	$-0.217959\pi$
$661$	26327.0	1.54917	0.774585	+	0.632470i	$0.217959\pi$
$662$	−1936.00	−0.113663
$663$	0	0
$664$	−672.000	−0.0392751
$665$	−18816.0	−1.09722
$666$	0	0
$667$	−24300.0	−1.41064
$668$	672.000	0.0389228
$669$	0	0
$670$	−11424.0	−0.658728
$671$	−13860.0	−0.797406
$672$	0	0
$673$	−19741.0	−1.13070	−0.565349	−	0.824852i	$-0.691258\pi$
$673$	−19741.0	−1.13070	−0.565349	+	0.824852i	$0.691258\pi$
$674$	−19780.0	−1.13041
$675$	0	0
$676$	816.000	0.0464269
$677$	12642.0	0.717683	0.358842	−	0.933398i	$-0.383172\pi$
$677$	12642.0	0.717683	0.358842	+	0.933398i	$0.383172\pi$
$678$	0	0
$679$	−9968.00	−0.563383
$680$	−3528.00	−0.198960
$681$	0	0
$682$	−22176.0	−1.24511
$683$	−26172.0	−1.46624	−0.733121	−	0.680098i	$-0.761937\pi$
$683$	−26172.0	−1.46624	−0.733121	+	0.680098i	$0.761937\pi$
$684$	0	0
$685$	15309.0	0.853908
$686$	9952.00	0.553891
$687$	0	0
$688$	320.000	0.0177324
$689$	8526.00	0.471429
$690$	0	0
$691$	−9520.00	−0.524107	−0.262053	−	0.965053i	$-0.584400\pi$
$691$	−9520.00	−0.524107	−0.262053	+	0.965053i	$0.584400\pi$
$692$	−12012.0	−0.659867
$693$	0	0
$694$	3120.00	0.170654
$695$	−42924.0	−2.34273
$696$	0	0
$697$	−882.000	−0.0479313
$698$	−5740.00	−0.311264
$699$	0	0
$700$	10112.0	0.545997
$701$	16773.0	0.903720	0.451860	−	0.892089i	$-0.350761\pi$
$701$	16773.0	0.903720	0.451860	+	0.892089i	$0.350761\pi$
$702$	0	0
$703$	112.000	0.00600876
$704$	2304.00	0.123346
$705$	0	0
$706$	−14364.0	−0.765717
$707$	4368.00	0.232356
$708$	0	0
$709$	12767.0	0.676269	0.338135	−	0.941098i	$-0.390204\pi$
$709$	12767.0	0.676269	0.338135	+	0.941098i	$0.390204\pi$
$710$	37296.0	1.97140
$711$	0	0
$712$	−168.000	−0.00884279
$713$	55440.0	2.91198
$714$	0	0
$715$	−37044.0	−1.93758
$716$	−4656.00	−0.243021
$717$	0	0
$718$	−16200.0	−0.842032
$719$	−24948.0	−1.29402	−0.647012	−	0.762480i	$-0.723982\pi$
$719$	−24948.0	−1.29402	−0.647012	+	0.762480i	$0.723982\pi$
$720$	0	0
$721$	−1568.00	−0.0809922
$722$	−11370.0	−0.586077
$723$	0	0
$724$	−6664.00	−0.342080
$725$	−42660.0	−2.18531
$726$	0	0
$727$	56.0000	0.00285684	0.00142842	−	0.999999i	$-0.499545\pi$
$727$	56.0000	0.00285684	0.00142842	+	0.999999i	$0.499545\pi$
$728$	3136.00	0.159654
$729$	0	0
$730$	−15582.0	−0.790021
$731$	420.000	0.0212507
$732$	0	0
$733$	1190.00	0.0599641	0.0299820	−	0.999550i	$-0.490455\pi$
$733$	1190.00	0.0599641	0.0299820	+	0.999550i	$0.490455\pi$
$734$	−22288.0	−1.12080
$735$	0	0
$736$	−5760.00	−0.288473
$737$	9792.00	0.489407
$738$	0	0
$739$	−26692.0	−1.32866	−0.664331	−	0.747439i	$-0.731283\pi$
$739$	−26692.0	−1.32866	−0.664331	+	0.747439i	$0.731283\pi$
$740$	−84.0000	−0.00417284
$741$	0	0
$742$	2784.00	0.137741
$743$	18852.0	0.930838	0.465419	−	0.885090i	$-0.345904\pi$
$743$	18852.0	0.930838	0.465419	+	0.885090i	$0.345904\pi$
$744$	0	0
$745$	−27027.0	−1.32912
$746$	−27676.0	−1.35830
$747$	0	0
$748$	3024.00	0.147819
$749$	−2400.00	−0.117082
$750$	0	0
$751$	22616.0	1.09889	0.549447	−	0.835528i	$-0.314838\pi$
$751$	22616.0	1.09889	0.549447	+	0.835528i	$0.314838\pi$
$752$	1344.00	0.0651737
$753$	0	0
$754$	−13230.0	−0.639003
$755$	−15456.0	−0.745035
$756$	0	0
$757$	9326.00	0.447766	0.223883	−	0.974616i	$-0.428127\pi$
$757$	9326.00	0.447766	0.223883	+	0.974616i	$0.428127\pi$
$758$	−2392.00	−0.114619
$759$	0	0
$760$	18816.0	0.898063
$761$	20769.0	0.989324	0.494662	−	0.869085i	$-0.335292\pi$
$761$	20769.0	0.989324	0.494662	+	0.869085i	$0.335292\pi$
$762$	0	0
$763$	−8552.00	−0.405771
$764$	−8256.00	−0.390958
$765$	0	0
$766$	−7728.00	−0.364522
$767$	−24696.0	−1.16261
$768$	0	0
$769$	−301.000	−0.0141149	−0.00705744	−	0.999975i	$-0.502246\pi$
$769$	−301.000	−0.0141149	−0.00705744	+	0.999975i	$0.502246\pi$
$770$	−12096.0	−0.566116
$771$	0	0
$772$	−2260.00	−0.105362
$773$	−17955.0	−0.835442	−0.417721	−	0.908575i	$-0.637171\pi$
$773$	−17955.0	−0.835442	−0.417721	+	0.908575i	$0.637171\pi$
$774$	0	0
$775$	97328.0	4.51113
$776$	9968.00	0.461122
$777$	0	0
$778$	10140.0	0.467271
$779$	4704.00	0.216352
$780$	0	0
$781$	−31968.0	−1.46467
$782$	−7560.00	−0.345710
$783$	0	0
$784$	−4464.00	−0.203353
$785$	−45129.0	−2.05188
$786$	0	0
$787$	−27412.0	−1.24159	−0.620796	−	0.783972i	$-0.713190\pi$
$787$	−27412.0	−1.24159	−0.620796	+	0.783972i	$0.713190\pi$
$788$	−18924.0	−0.855507
$789$	0	0
$790$	27384.0	1.23326
$791$	7176.00	0.322565
$792$	0	0
$793$	18865.0	0.844787
$794$	−30478.0	−1.36225
$795$	0	0
$796$	18704.0	0.832846
$797$	22533.0	1.00146	0.500728	−	0.865605i	$-0.333066\pi$
$797$	22533.0	1.00146	0.500728	+	0.865605i	$0.333066\pi$
$798$	0	0
$799$	1764.00	0.0781049
$800$	−10112.0	−0.446891
$801$	0	0
$802$	3414.00	0.150315
$803$	13356.0	0.586953
$804$	0	0
$805$	30240.0	1.32400
$806$	30184.0	1.31909
$807$	0	0
$808$	−4368.00	−0.190180
$809$	6105.00	0.265316	0.132658	−	0.991162i	$-0.457649\pi$
$809$	6105.00	0.265316	0.132658	+	0.991162i	$0.457649\pi$
$810$	0	0
$811$	−3472.00	−0.150331	−0.0751655	−	0.997171i	$-0.523949\pi$
$811$	−3472.00	−0.150331	−0.0751655	+	0.997171i	$0.523949\pi$
$812$	−4320.00	−0.186702
$813$	0	0
$814$	72.0000	0.00310024
$815$	−64848.0	−2.78715
$816$	0	0
$817$	−2240.00	−0.0959213
$818$	26642.0	1.13877
$819$	0	0
$820$	−3528.00	−0.150248
$821$	4929.00	0.209529	0.104764	−	0.994497i	$-0.466591\pi$
$821$	4929.00	0.209529	0.104764	+	0.994497i	$0.466591\pi$
$822$	0	0
$823$	39524.0	1.67402	0.837011	−	0.547186i	$-0.184301\pi$
$823$	39524.0	1.67402	0.837011	+	0.547186i	$0.184301\pi$
$824$	1568.00	0.0662911
$825$	0	0
$826$	−8064.00	−0.339688
$827$	38676.0	1.62623	0.813117	−	0.582100i	$-0.197769\pi$
$827$	38676.0	1.62623	0.813117	+	0.582100i	$0.197769\pi$
$828$	0	0
$829$	16646.0	0.697394	0.348697	−	0.937236i	$-0.386624\pi$
$829$	16646.0	0.697394	0.348697	+	0.937236i	$0.386624\pi$
$830$	−3528.00	−0.147541
$831$	0	0
$832$	−3136.00	−0.130675
$833$	−5859.00	−0.243700
$834$	0	0
$835$	3528.00	0.146217
$836$	−16128.0	−0.667223
$837$	0	0
$838$	27888.0	1.14961
$839$	6636.00	0.273063	0.136532	−	0.990636i	$-0.456404\pi$
$839$	6636.00	0.273063	0.136532	+	0.990636i	$0.456404\pi$
$840$	0	0
$841$	−6164.00	−0.252737
$842$	21674.0	0.887097
$843$	0	0
$844$	13520.0	0.551395
$845$	4284.00	0.174407
$846$	0	0
$847$	−280.000	−0.0113588
$848$	−2784.00	−0.112739
$849$	0	0
$850$	−13272.0	−0.535560
$851$	−180.000	−0.00725067
$852$	0	0
$853$	−9394.00	−0.377074	−0.188537	−	0.982066i	$-0.560375\pi$
$853$	−9394.00	−0.377074	−0.188537	+	0.982066i	$0.560375\pi$
$854$	6160.00	0.246828
$855$	0	0
$856$	2400.00	0.0958298
$857$	−7287.00	−0.290454	−0.145227	−	0.989398i	$-0.546391\pi$
$857$	−7287.00	−0.290454	−0.145227	+	0.989398i	$0.546391\pi$
$858$	0	0
$859$	−25732.0	−1.02208	−0.511039	−	0.859558i	$-0.670739\pi$
$859$	−25732.0	−1.02208	−0.511039	+	0.859558i	$0.670739\pi$
$860$	1680.00	0.0666134
$861$	0	0
$862$	−25224.0	−0.996674
$863$	−7188.00	−0.283525	−0.141763	−	0.989901i	$-0.545277\pi$
$863$	−7188.00	−0.283525	−0.141763	+	0.989901i	$0.545277\pi$
$864$	0	0
$865$	−63063.0	−2.47885
$866$	19418.0	0.761952
$867$	0	0
$868$	9856.00	0.385408
$869$	−23472.0	−0.916264
$870$	0	0
$871$	−13328.0	−0.518487
$872$	8552.00	0.332118
$873$	0	0
$874$	40320.0	1.56046
$875$	32088.0	1.23974
$876$	0	0
$877$	−12601.0	−0.485183	−0.242592	−	0.970129i	$-0.577997\pi$
$877$	−12601.0	−0.485183	−0.242592	+	0.970129i	$0.577997\pi$
$878$	−20776.0	−0.798583
$879$	0	0
$880$	12096.0	0.463359
$881$	−210.000	−0.00803074	−0.00401537	−	0.999992i	$-0.501278\pi$
$881$	−210.000	−0.00803074	−0.00401537	+	0.999992i	$0.501278\pi$
$882$	0	0
$883$	−47524.0	−1.81122	−0.905612	−	0.424108i	$-0.860588\pi$
$883$	−47524.0	−1.81122	−0.905612	+	0.424108i	$0.860588\pi$
$884$	−4116.00	−0.156602
$885$	0	0
$886$	5016.00	0.190198
$887$	−41748.0	−1.58034	−0.790169	−	0.612888i	$-0.790008\pi$
$887$	−41748.0	−1.58034	−0.790169	+	0.612888i	$0.790008\pi$
$888$	0	0
$889$	12256.0	0.462377
$890$	−882.000	−0.0332188
$891$	0	0
$892$	−20944.0	−0.786163
$893$	−9408.00	−0.352550
$894$	0	0
$895$	−24444.0	−0.912931
$896$	−1024.00	−0.0381802
$897$	0	0
$898$	27396.0	1.01806
$899$	−41580.0	−1.54257
$900$	0	0
$901$	−3654.00	−0.135108
$902$	3024.00	0.111628
$903$	0	0
$904$	−7176.00	−0.264016
$905$	−34986.0	−1.28505
$906$	0	0
$907$	14720.0	0.538886	0.269443	−	0.963016i	$-0.413160\pi$
$907$	14720.0	0.538886	0.269443	+	0.963016i	$0.413160\pi$
$908$	15456.0	0.564896
$909$	0	0
$910$	16464.0	0.599754
$911$	17652.0	0.641972	0.320986	−	0.947084i	$-0.395986\pi$
$911$	17652.0	0.641972	0.320986	+	0.947084i	$0.395986\pi$
$912$	0	0
$913$	3024.00	0.109616
$914$	19490.0	0.705330
$915$	0	0
$916$	−15652.0	−0.564581
$917$	6720.00	0.242000
$918$	0	0
$919$	17156.0	0.615804	0.307902	−	0.951418i	$-0.400373\pi$
$919$	17156.0	0.615804	0.307902	+	0.951418i	$0.400373\pi$
$920$	−30240.0	−1.08368
$921$	0	0
$922$	−35028.0	−1.25118
$923$	43512.0	1.55170
$924$	0	0
$925$	−316.000	−0.0112324
$926$	−9280.00	−0.329330
$927$	0	0
$928$	4320.00	0.152814
$929$	24213.0	0.855116	0.427558	−	0.903988i	$-0.359374\pi$
$929$	24213.0	0.855116	0.427558	+	0.903988i	$0.359374\pi$
$930$	0	0
$931$	31248.0	1.10001
$932$	25332.0	0.890319
$933$	0	0
$934$	8736.00	0.306050
$935$	15876.0	0.555295
$936$	0	0
$937$	−15085.0	−0.525940	−0.262970	−	0.964804i	$-0.584702\pi$
$937$	−15085.0	−0.525940	−0.262970	+	0.964804i	$0.584702\pi$
$938$	−4352.00	−0.151490
$939$	0	0
$940$	7056.00	0.244831
$941$	5229.00	0.181148	0.0905741	−	0.995890i	$-0.471130\pi$
$941$	5229.00	0.181148	0.0905741	+	0.995890i	$0.471130\pi$
$942$	0	0
$943$	−7560.00	−0.261068
$944$	8064.00	0.278031
$945$	0	0
$946$	−1440.00	−0.0494909
$947$	28488.0	0.977546	0.488773	−	0.872411i	$-0.337445\pi$
$947$	28488.0	0.977546	0.488773	+	0.872411i	$0.337445\pi$
$948$	0	0
$949$	−18179.0	−0.621829
$950$	70784.0	2.41741
$951$	0	0
$952$	−1344.00	−0.0457556
$953$	−1143.00	−0.0388514	−0.0194257	−	0.999811i	$-0.506184\pi$
$953$	−1143.00	−0.0388514	−0.0194257	+	0.999811i	$0.506184\pi$
$954$	0	0
$955$	−43344.0	−1.46867
$956$	−15312.0	−0.518018
$957$	0	0
$958$	37632.0	1.26914
$959$	5832.00	0.196377
$960$	0	0
$961$	65073.0	2.18432
$962$	−98.0000	−0.00328446
$963$	0	0
$964$	−5908.00	−0.197390
$965$	−11865.0	−0.395801
$966$	0	0
$967$	36416.0	1.21102	0.605512	−	0.795836i	$-0.292968\pi$
$967$	36416.0	1.21102	0.605512	+	0.795836i	$0.292968\pi$
$968$	280.000	0.00929705
$969$	0	0
$970$	52332.0	1.73225
$971$	25200.0	0.832859	0.416430	−	0.909168i	$-0.363281\pi$
$971$	25200.0	0.832859	0.416430	+	0.909168i	$0.363281\pi$
$972$	0	0
$973$	−16352.0	−0.538768
$974$	27512.0	0.905073
$975$	0	0
$976$	−6160.00	−0.202025
$977$	23742.0	0.777455	0.388728	−	0.921353i	$-0.372915\pi$
$977$	23742.0	0.777455	0.388728	+	0.921353i	$0.372915\pi$
$978$	0	0
$979$	756.000	0.0246801
$980$	−23436.0	−0.763914
$981$	0	0
$982$	−15480.0	−0.503041
$983$	4704.00	0.152629	0.0763145	−	0.997084i	$-0.475685\pi$
$983$	4704.00	0.152629	0.0763145	+	0.997084i	$0.475685\pi$
$984$	0	0
$985$	−99351.0	−3.21379
$986$	5670.00	0.183133
$987$	0	0
$988$	21952.0	0.706869
$989$	3600.00	0.115747
$990$	0	0
$991$	−6280.00	−0.201302	−0.100651	−	0.994922i	$-0.532093\pi$
$991$	−6280.00	−0.201302	−0.100651	+	0.994922i	$0.532093\pi$
$992$	−9856.00	−0.315452
$993$	0	0
$994$	14208.0	0.453371
$995$	98196.0	3.12867
$996$	0	0
$997$	−8701.00	−0.276393	−0.138196	−	0.990405i	$-0.544130\pi$
$997$	−8701.00	−0.276393	−0.138196	+	0.990405i	$0.544130\pi$
$998$	−4792.00	−0.151992
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.a.b.1.1	✓	1
3.2	odd	2		162.4.a.c.1.1	yes	1
4.3	odd	2		1296.4.a.h.1.1		1
9.2	odd	6		162.4.c.d.109.1		2
9.4	even	3		162.4.c.e.55.1		2
9.5	odd	6		162.4.c.d.55.1		2
9.7	even	3		162.4.c.e.109.1		2
12.11	even	2		1296.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.a.b.1.1	✓	1	1.1	even	1	trivial
162.4.a.c.1.1	yes	1	3.2	odd	2
162.4.c.d.55.1		2	9.5	odd	6
162.4.c.d.109.1		2	9.2	odd	6
162.4.c.e.55.1		2	9.4	even	3
162.4.c.e.109.1		2	9.7	even	3
1296.4.a.a.1.1		1	12.11	even	2
1296.4.a.h.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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