LMFDB - Embedded newform 33.6.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [33,6,Mod(1,33)] 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("33.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(33, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	33.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.29266605383$
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	$\chi$	$=$	33.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} +46.0000 q^{5} +18.0000 q^{6} +148.000 q^{7} +120.000 q^{8} +81.0000 q^{9} -92.0000 q^{10} +121.000 q^{11} +252.000 q^{12} +574.000 q^{13} -296.000 q^{14} -414.000 q^{15} +656.000 q^{16} -722.000 q^{17} -162.000 q^{18} +2160.00 q^{19} -1288.00 q^{20} -1332.00 q^{21} -242.000 q^{22} -2536.00 q^{23} -1080.00 q^{24} -1009.00 q^{25} -1148.00 q^{26} -729.000 q^{27} -4144.00 q^{28} +4650.00 q^{29} +828.000 q^{30} +5032.00 q^{31} -5152.00 q^{32} -1089.00 q^{33} +1444.00 q^{34} +6808.00 q^{35} -2268.00 q^{36} +8118.00 q^{37} -4320.00 q^{38} -5166.00 q^{39} +5520.00 q^{40} -5138.00 q^{41} +2664.00 q^{42} +8304.00 q^{43} -3388.00 q^{44} +3726.00 q^{45} +5072.00 q^{46} +24728.0 q^{47} -5904.00 q^{48} +5097.00 q^{49} +2018.00 q^{50} +6498.00 q^{51} -16072.0 q^{52} -28746.0 q^{53} +1458.00 q^{54} +5566.00 q^{55} +17760.0 q^{56} -19440.0 q^{57} -9300.00 q^{58} -5860.00 q^{59} +11592.0 q^{60} -53658.0 q^{61} -10064.0 q^{62} +11988.0 q^{63} -10688.0 q^{64} +26404.0 q^{65} +2178.00 q^{66} +30908.0 q^{67} +20216.0 q^{68} +22824.0 q^{69} -13616.0 q^{70} -69648.0 q^{71} +9720.00 q^{72} -18446.0 q^{73} -16236.0 q^{74} +9081.00 q^{75} -60480.0 q^{76} +17908.0 q^{77} +10332.0 q^{78} -25300.0 q^{79} +30176.0 q^{80} +6561.00 q^{81} +10276.0 q^{82} -17556.0 q^{83} +37296.0 q^{84} -33212.0 q^{85} -16608.0 q^{86} -41850.0 q^{87} +14520.0 q^{88} +132570. q^{89} -7452.00 q^{90} +84952.0 q^{91} +71008.0 q^{92} -45288.0 q^{93} -49456.0 q^{94} +99360.0 q^{95} +46368.0 q^{96} +70658.0 q^{97} -10194.0 q^{98} +9801.00 q^{99} +O(q^{100})\)

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	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−2.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−28.0000	−0.875000
$5$	46.0000	0.822873	0.411437	−	0.911438i	$-0.365027\pi$
$5$	46.0000	0.822873	0.411437	+	0.911438i	$0.365027\pi$
$6$	18.0000	0.204124
$7$	148.000	1.14161	0.570803	−	0.821087i	$-0.306632\pi$
$7$	148.000	1.14161	0.570803	+	0.821087i	$0.306632\pi$
$8$	120.000	0.662913
$9$	81.0000	0.333333
$10$	−92.0000	−0.290930
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	252.000	0.505181
$13$	574.000	0.942006	0.471003	−	0.882132i	$-0.343892\pi$
$13$	574.000	0.942006	0.471003	+	0.882132i	$0.343892\pi$
$14$	−296.000	−0.403619
$15$	−414.000	−0.475086
$16$	656.000	0.640625
$17$	−722.000	−0.605919	−0.302960	−	0.953003i	$-0.597975\pi$
$17$	−722.000	−0.605919	−0.302960	+	0.953003i	$0.597975\pi$
$18$	−162.000	−0.117851
$19$	2160.00	1.37268	0.686341	−	0.727280i	$-0.259216\pi$
$19$	2160.00	1.37268	0.686341	+	0.727280i	$0.259216\pi$
$20$	−1288.00	−0.720014
$21$	−1332.00	−0.659107
$22$	−242.000	−0.106600
$23$	−2536.00	−0.999608	−0.499804	−	0.866139i	$-0.666595\pi$
$23$	−2536.00	−0.999608	−0.499804	+	0.866139i	$0.666595\pi$
$24$	−1080.00	−0.382733
$25$	−1009.00	−0.322880
$26$	−1148.00	−0.333049
$27$	−729.000	−0.192450
$28$	−4144.00	−0.998906
$29$	4650.00	1.02673	0.513367	−	0.858169i	$-0.328398\pi$
$29$	4650.00	1.02673	0.513367	+	0.858169i	$0.328398\pi$
$30$	828.000	0.167968
$31$	5032.00	0.940451	0.470226	−	0.882546i	$-0.344172\pi$
$31$	5032.00	0.940451	0.470226	+	0.882546i	$0.344172\pi$
$32$	−5152.00	−0.889408
$33$	−1089.00	−0.174078
$34$	1444.00	0.214225
$35$	6808.00	0.939398
$36$	−2268.00	−0.291667
$37$	8118.00	0.974866	0.487433	−	0.873161i	$-0.337933\pi$
$37$	8118.00	0.974866	0.487433	+	0.873161i	$0.337933\pi$
$38$	−4320.00	−0.485316
$39$	−5166.00	−0.543867
$40$	5520.00	0.545493
$41$	−5138.00	−0.477347	−0.238674	−	0.971100i	$-0.576713\pi$
$41$	−5138.00	−0.477347	−0.238674	+	0.971100i	$0.576713\pi$
$42$	2664.00	0.233030
$43$	8304.00	0.684883	0.342441	−	0.939539i	$-0.388746\pi$
$43$	8304.00	0.684883	0.342441	+	0.939539i	$0.388746\pi$
$44$	−3388.00	−0.263822
$45$	3726.00	0.274291
$46$	5072.00	0.353415
$47$	24728.0	1.63284	0.816421	−	0.577457i	$-0.195955\pi$
$47$	24728.0	1.63284	0.816421	+	0.577457i	$0.195955\pi$
$48$	−5904.00	−0.369865
$49$	5097.00	0.303266
$50$	2018.00	0.114155
$51$	6498.00	0.349828
$52$	−16072.0	−0.824255
$53$	−28746.0	−1.40568	−0.702842	−	0.711346i	$-0.748086\pi$
$53$	−28746.0	−1.40568	−0.702842	+	0.711346i	$0.748086\pi$
$54$	1458.00	0.0680414
$55$	5566.00	0.248106
$56$	17760.0	0.756786
$57$	−19440.0	−0.792518
$58$	−9300.00	−0.363005
$59$	−5860.00	−0.219163	−0.109582	−	0.993978i	$-0.534951\pi$
$59$	−5860.00	−0.219163	−0.109582	+	0.993978i	$0.534951\pi$
$60$	11592.0	0.415700
$61$	−53658.0	−1.84633	−0.923166	−	0.384401i	$-0.874408\pi$
$61$	−53658.0	−1.84633	−0.923166	+	0.384401i	$0.874408\pi$
$62$	−10064.0	−0.332500
$63$	11988.0	0.380536
$64$	−10688.0	−0.326172
$65$	26404.0	0.775151
$66$	2178.00	0.0615457
$67$	30908.0	0.841170	0.420585	−	0.907253i	$-0.361825\pi$
$67$	30908.0	0.841170	0.420585	+	0.907253i	$0.361825\pi$
$68$	20216.0	0.530180
$69$	22824.0	0.577124
$70$	−13616.0	−0.332127
$71$	−69648.0	−1.63969	−0.819847	−	0.572583i	$-0.805942\pi$
$71$	−69648.0	−1.63969	−0.819847	+	0.572583i	$0.805942\pi$
$72$	9720.00	0.220971
$73$	−18446.0	−0.405131	−0.202565	−	0.979269i	$-0.564928\pi$
$73$	−18446.0	−0.405131	−0.202565	+	0.979269i	$0.564928\pi$
$74$	−16236.0	−0.344667
$75$	9081.00	0.186415
$76$	−60480.0	−1.20110
$77$	17908.0	0.344207
$78$	10332.0	0.192286
$79$	−25300.0	−0.456092	−0.228046	−	0.973650i	$-0.573234\pi$
$79$	−25300.0	−0.456092	−0.228046	+	0.973650i	$0.573234\pi$
$80$	30176.0	0.527153
$81$	6561.00	0.111111
$82$	10276.0	0.168768
$83$	−17556.0	−0.279724	−0.139862	−	0.990171i	$-0.544666\pi$
$83$	−17556.0	−0.279724	−0.139862	+	0.990171i	$0.544666\pi$
$84$	37296.0	0.576719
$85$	−33212.0	−0.498595
$86$	−16608.0	−0.242143
$87$	−41850.0	−0.592785
$88$	14520.0	0.199876
$89$	132570.	1.77407	0.887034	−	0.461704i	$-0.152762\pi$
$89$	132570.	1.77407	0.887034	+	0.461704i	$0.152762\pi$
$90$	−7452.00	−0.0969765
$91$	84952.0	1.07540
$92$	71008.0	0.874657
$93$	−45288.0	−0.542970
$94$	−49456.0	−0.577297
$95$	99360.0	1.12954
$96$	46368.0	0.513500
$97$	70658.0	0.762486	0.381243	−	0.924475i	$-0.375496\pi$
$97$	70658.0	0.762486	0.381243	+	0.924475i	$0.375496\pi$
$98$	−10194.0	−0.107221
$99$	9801.00	0.100504
$100$	28252.0	0.282520
$101$	−101998.	−0.994920	−0.497460	−	0.867487i	$-0.665734\pi$
$101$	−101998.	−0.994920	−0.497460	+	0.867487i	$0.665734\pi$
$102$	−12996.0	−0.123683
$103$	130904.	1.21579	0.607897	−	0.794016i	$-0.292013\pi$
$103$	130904.	1.21579	0.607897	+	0.794016i	$0.292013\pi$
$104$	68880.0	0.624467
$105$	−61272.0	−0.542361
$106$	57492.0	0.496984
$107$	−141612.	−1.19575	−0.597875	−	0.801589i	$-0.703988\pi$
$107$	−141612.	−1.19575	−0.597875	+	0.801589i	$0.703988\pi$
$108$	20412.0	0.168394
$109$	−239810.	−1.93331	−0.966654	−	0.256086i	$-0.917567\pi$
$109$	−239810.	−1.93331	−0.966654	+	0.256086i	$0.917567\pi$
$110$	−11132.0	−0.0877186
$111$	−73062.0	−0.562839
$112$	97088.0	0.731342
$113$	−42726.0	−0.314772	−0.157386	−	0.987537i	$-0.550307\pi$
$113$	−42726.0	−0.314772	−0.157386	+	0.987537i	$0.550307\pi$
$114$	38880.0	0.280197
$115$	−116656.	−0.822550
$116$	−130200.	−0.898392
$117$	46494.0	0.314002
$118$	11720.0	0.0774859
$119$	−106856.	−0.691722
$120$	−49680.0	−0.314940
$121$	14641.0	0.0909091
$122$	107316.	0.652777
$123$	46242.0	0.275597
$124$	−140896.	−0.822895
$125$	−190164.	−1.08856
$126$	−23976.0	−0.134540
$127$	51788.0	0.284918	0.142459	−	0.989801i	$-0.454499\pi$
$127$	51788.0	0.284918	0.142459	+	0.989801i	$0.454499\pi$
$128$	186240.	1.00473
$129$	−74736.0	−0.395417
$130$	−52808.0	−0.274057
$131$	53652.0	0.273154	0.136577	−	0.990629i	$-0.456390\pi$
$131$	53652.0	0.273154	0.136577	+	0.990629i	$0.456390\pi$
$132$	30492.0	0.152318
$133$	319680.	1.56706
$134$	−61816.0	−0.297399
$135$	−33534.0	−0.158362
$136$	−86640.0	−0.401672
$137$	−228862.	−1.04177	−0.520886	−	0.853627i	$-0.674398\pi$
$137$	−228862.	−1.04177	−0.520886	+	0.853627i	$0.674398\pi$
$138$	−45648.0	−0.204044
$139$	374920.	1.64589	0.822947	−	0.568119i	$-0.192329\pi$
$139$	374920.	1.64589	0.822947	+	0.568119i	$0.192329\pi$
$140$	−190624.	−0.821973
$141$	−222552.	−0.942722
$142$	139296.	0.579719
$143$	69454.0	0.284025
$144$	53136.0	0.213542
$145$	213900.	0.844872
$146$	36892.0	0.143235
$147$	−45873.0	−0.175091
$148$	−227304.	−0.853007
$149$	−65830.0	−0.242917	−0.121459	−	0.992597i	$-0.538757\pi$
$149$	−65830.0	−0.242917	−0.121459	+	0.992597i	$0.538757\pi$
$150$	−18162.0	−0.0659076
$151$	154052.	0.549826	0.274913	−	0.961469i	$-0.411351\pi$
$151$	154052.	0.549826	0.274913	+	0.961469i	$0.411351\pi$
$152$	259200.	0.909968
$153$	−58482.0	−0.201973
$154$	−35816.0	−0.121696
$155$	231472.	0.773872
$156$	144648.	0.475884
$157$	287678.	0.931446	0.465723	−	0.884931i	$-0.345794\pi$
$157$	287678.	0.931446	0.465723	+	0.884931i	$0.345794\pi$
$158$	50600.0	0.161253
$159$	258714.	0.811572
$160$	−236992.	−0.731870
$161$	−375328.	−1.14116
$162$	−13122.0	−0.0392837
$163$	105124.	0.309908	0.154954	−	0.987922i	$-0.450477\pi$
$163$	105124.	0.309908	0.154954	+	0.987922i	$0.450477\pi$
$164$	143864.	0.417679
$165$	−50094.0	−0.143244
$166$	35112.0	0.0988975
$167$	150528.	0.417663	0.208832	−	0.977952i	$-0.433034\pi$
$167$	150528.	0.417663	0.208832	+	0.977952i	$0.433034\pi$
$168$	−159840.	−0.436930
$169$	−41817.0	−0.112625
$170$	66424.0	0.176280
$171$	174960.	0.457560
$172$	−232512.	−0.599272
$173$	−2166.00	−0.00550229	−0.00275114	−	0.999996i	$-0.500876\pi$
$173$	−2166.00	−0.00550229	−0.00275114	+	0.999996i	$0.500876\pi$
$174$	83700.0	0.209581
$175$	−149332.	−0.368602
$176$	79376.0	0.193156
$177$	52740.0	0.126534
$178$	−265140.	−0.627228
$179$	672780.	1.56942	0.784712	−	0.619860i	$-0.212811\pi$
$179$	672780.	1.56942	0.784712	+	0.619860i	$0.212811\pi$
$180$	−104328.	−0.240005
$181$	−526778.	−1.19517	−0.597587	−	0.801804i	$-0.703874\pi$
$181$	−526778.	−1.19517	−0.597587	+	0.801804i	$0.703874\pi$
$182$	−169904.	−0.380211
$183$	482922.	1.06598
$184$	−304320.	−0.662653
$185$	373428.	0.802191
$186$	90576.0	0.191969
$187$	−87362.0	−0.182692
$188$	−692384.	−1.42874
$189$	−107892.	−0.219702
$190$	−198720.	−0.399354
$191$	−305608.	−0.606152	−0.303076	−	0.952966i	$-0.598014\pi$
$191$	−305608.	−0.606152	−0.303076	+	0.952966i	$0.598014\pi$
$192$	96192.0	0.188315
$193$	116434.	0.225002	0.112501	−	0.993652i	$-0.464114\pi$
$193$	116434.	0.225002	0.112501	+	0.993652i	$0.464114\pi$
$194$	−141316.	−0.269580
$195$	−237636.	−0.447534
$196$	−142716.	−0.265358
$197$	−247742.	−0.454814	−0.227407	−	0.973800i	$-0.573025\pi$
$197$	−247742.	−0.454814	−0.227407	+	0.973800i	$0.573025\pi$
$198$	−19602.0	−0.0355335
$199$	−513360.	−0.918945	−0.459472	−	0.888192i	$-0.651961\pi$
$199$	−513360.	−0.918945	−0.459472	+	0.888192i	$0.651961\pi$
$200$	−121080.	−0.214041
$201$	−278172.	−0.485650
$202$	203996.	0.351757
$203$	688200.	1.17213
$204$	−181944.	−0.306099
$205$	−236348.	−0.392796
$206$	−261808.	−0.429848
$207$	−205416.	−0.333203
$208$	376544.	0.603472
$209$	261360.	0.413879
$210$	122544.	0.191754
$211$	−620688.	−0.959770	−0.479885	−	0.877331i	$-0.659322\pi$
$211$	−620688.	−0.959770	−0.479885	+	0.877331i	$0.659322\pi$
$212$	804888.	1.22997
$213$	626832.	0.946678
$214$	283224.	0.422762
$215$	381984.	0.563571
$216$	−87480.0	−0.127578
$217$	744736.	1.07363
$218$	479620.	0.683528
$219$	166014.	0.233902
$220$	−155848.	−0.217092
$221$	−414428.	−0.570780
$222$	146124.	0.198994
$223$	−1.31802e6	−1.77484	−0.887419	−	0.460964i	$-0.847504\pi$
$223$	−1.31802e6	−1.77484	−0.887419	+	0.460964i	$0.847504\pi$
$224$	−762496.	−1.01535
$225$	−81729.0	−0.107627
$226$	85452.0	0.111289
$227$	−887412.	−1.14304	−0.571519	−	0.820589i	$-0.693646\pi$
$227$	−887412.	−1.14304	−0.571519	+	0.820589i	$0.693646\pi$
$228$	544320.	0.693453
$229$	−237450.	−0.299215	−0.149608	−	0.988745i	$-0.547801\pi$
$229$	−237450.	−0.299215	−0.149608	+	0.988745i	$0.547801\pi$
$230$	233312.	0.290815
$231$	−161172.	−0.198728
$232$	558000.	0.680635
$233$	−914706.	−1.10380	−0.551902	−	0.833909i	$-0.686098\pi$
$233$	−914706.	−1.10380	−0.551902	+	0.833909i	$0.686098\pi$
$234$	−92988.0	−0.111016
$235$	1.13749e6	1.34362
$236$	164080.	0.191768
$237$	227700.	0.263325
$238$	213712.	0.244561
$239$	1.40892e6	1.59548	0.797740	−	0.603001i	$-0.206029\pi$
$239$	1.40892e6	1.59548	0.797740	+	0.603001i	$0.206029\pi$
$240$	−271584.	−0.304352
$241$	−826358.	−0.916486	−0.458243	−	0.888827i	$-0.651521\pi$
$241$	−826358.	−0.916486	−0.458243	+	0.888827i	$0.651521\pi$
$242$	−29282.0	−0.0321412
$243$	−59049.0	−0.0641500
$244$	1.50242e6	1.61554
$245$	234462.	0.249550
$246$	−92484.0	−0.0974381
$247$	1.23984e6	1.29307
$248$	603840.	0.623437
$249$	158004.	0.161499
$250$	380328.	0.384865
$251$	−1.60387e6	−1.60688	−0.803442	−	0.595384i	$-0.797000\pi$
$251$	−1.60387e6	−1.60688	−0.803442	+	0.595384i	$0.797000\pi$
$252$	−335664.	−0.332969
$253$	−306856.	−0.301393
$254$	−103576.	−0.100734
$255$	298908.	0.287864
$256$	−30464.0	−0.0290527
$257$	397618.	0.375520	0.187760	−	0.982215i	$-0.439877\pi$
$257$	397618.	0.375520	0.187760	+	0.982215i	$0.439877\pi$
$258$	149472.	0.139801
$259$	1.20146e6	1.11291
$260$	−739312.	−0.678257
$261$	376650.	0.342245
$262$	−107304.	−0.0965745
$263$	2.13166e6	1.90033	0.950166	−	0.311745i	$-0.100913\pi$
$263$	2.13166e6	1.90033	0.950166	+	0.311745i	$0.100913\pi$
$264$	−130680.	−0.115398
$265$	−1.32232e6	−1.15670
$266$	−639360.	−0.554040
$267$	−1.19313e6	−1.02426
$268$	−865424.	−0.736024
$269$	−725810.	−0.611564	−0.305782	−	0.952101i	$-0.598918\pi$
$269$	−725810.	−0.611564	−0.305782	+	0.952101i	$0.598918\pi$
$270$	67068.0	0.0559894
$271$	−1.46787e6	−1.21413	−0.607063	−	0.794654i	$-0.707652\pi$
$271$	−1.46787e6	−1.21413	−0.607063	+	0.794654i	$0.707652\pi$
$272$	−473632.	−0.388167
$273$	−764568.	−0.620883
$274$	457724.	0.368322
$275$	−122089.	−0.0973520
$276$	−639072.	−0.504983
$277$	1.52100e6	1.19105	0.595524	−	0.803338i	$-0.296944\pi$
$277$	1.52100e6	1.19105	0.595524	+	0.803338i	$0.296944\pi$
$278$	−749840.	−0.581911
$279$	407592.	0.313484
$280$	816960.	0.622738
$281$	464382.	0.350840	0.175420	−	0.984494i	$-0.443872\pi$
$281$	464382.	0.350840	0.175420	+	0.984494i	$0.443872\pi$
$282$	445104.	0.333303
$283$	−415136.	−0.308123	−0.154062	−	0.988061i	$-0.549235\pi$
$283$	−415136.	−0.308123	−0.154062	+	0.988061i	$0.549235\pi$
$284$	1.95014e6	1.43473
$285$	−894240.	−0.652142
$286$	−138908.	−0.100418
$287$	−760424.	−0.544943
$288$	−417312.	−0.296469
$289$	−898573.	−0.632862
$290$	−427800.	−0.298707
$291$	−635922.	−0.440222
$292$	516488.	0.354489
$293$	−2.59321e6	−1.76469	−0.882344	−	0.470605i	$-0.844036\pi$
$293$	−2.59321e6	−1.76469	−0.882344	+	0.470605i	$0.844036\pi$
$294$	91746.0	0.0619040
$295$	−269560.	−0.180343
$296$	974160.	0.646251
$297$	−88209.0	−0.0580259
$298$	131660.	0.0858842
$299$	−1.45566e6	−0.941636
$300$	−254268.	−0.163113
$301$	1.22899e6	0.781867
$302$	−308104.	−0.194393
$303$	917982.	0.574417
$304$	1.41696e6	0.879374
$305$	−2.46827e6	−1.51930
$306$	116964.	0.0714083
$307$	−930832.	−0.563671	−0.281835	−	0.959463i	$-0.590943\pi$
$307$	−930832.	−0.563671	−0.281835	+	0.959463i	$0.590943\pi$
$308$	−501424.	−0.301182
$309$	−1.17814e6	−0.701939
$310$	−462944.	−0.273605
$311$	2.48527e6	1.45704	0.728522	−	0.685022i	$-0.240207\pi$
$311$	2.48527e6	1.45704	0.728522	+	0.685022i	$0.240207\pi$
$312$	−619920.	−0.360536
$313$	1.31719e6	0.759957	0.379978	−	0.924995i	$-0.375931\pi$
$313$	1.31719e6	0.759957	0.379978	+	0.924995i	$0.375931\pi$
$314$	−575356.	−0.329316
$315$	551448.	0.313133
$316$	708400.	0.399081
$317$	2.25540e6	1.26059	0.630297	−	0.776354i	$-0.282933\pi$
$317$	2.25540e6	1.26059	0.630297	+	0.776354i	$0.282933\pi$
$318$	−517428.	−0.286934
$319$	562650.	0.309572
$320$	−491648.	−0.268398
$321$	1.27451e6	0.690367
$322$	750656.	0.403461
$323$	−1.55952e6	−0.831734
$324$	−183708.	−0.0972222
$325$	−579166.	−0.304155
$326$	−210248.	−0.109569
$327$	2.15829e6	1.11620
$328$	−616560.	−0.316440
$329$	3.65974e6	1.86406
$330$	100188.	0.0506443
$331$	−3.17071e6	−1.59069	−0.795346	−	0.606155i	$-0.792711\pi$
$331$	−3.17071e6	−1.59069	−0.795346	+	0.606155i	$0.792711\pi$
$332$	491568.	0.244759
$333$	657558.	0.324955
$334$	−301056.	−0.147666
$335$	1.42177e6	0.692176
$336$	−873792.	−0.422240
$337$	1.27630e6	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$337$	1.27630e6	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$338$	83634.0	0.0398191
$339$	384534.	0.181734
$340$	929936.	0.436270
$341$	608872.	0.283557
$342$	−349920.	−0.161772
$343$	−1.73308e6	−0.795396
$344$	996480.	0.454017
$345$	1.04990e6	0.474900
$346$	4332.00	0.00194535
$347$	3.69303e6	1.64649	0.823245	−	0.567687i	$-0.192162\pi$
$347$	3.69303e6	1.64649	0.823245	+	0.567687i	$0.192162\pi$
$348$	1.17180e6	0.518687
$349$	1.70919e6	0.751150	0.375575	−	0.926792i	$-0.377445\pi$
$349$	1.70919e6	0.751150	0.375575	+	0.926792i	$0.377445\pi$
$350$	298664.	0.130321
$351$	−418446.	−0.181289
$352$	−623392.	−0.268167
$353$	4.36859e6	1.86597	0.932986	−	0.359914i	$-0.117194\pi$
$353$	4.36859e6	1.86597	0.932986	+	0.359914i	$0.117194\pi$
$354$	−105480.	−0.0447365
$355$	−3.20381e6	−1.34926
$356$	−3.71196e6	−1.55231
$357$	961704.	0.399366
$358$	−1.34556e6	−0.554875
$359$	−3.51284e6	−1.43854	−0.719271	−	0.694730i	$-0.755524\pi$
$359$	−3.51284e6	−1.43854	−0.719271	+	0.694730i	$0.755524\pi$
$360$	447120.	0.181831
$361$	2.18950e6	0.884254
$362$	1.05356e6	0.422558
$363$	−131769.	−0.0524864
$364$	−2.37866e6	−0.940975
$365$	−848516.	−0.333371
$366$	−965844.	−0.376881
$367$	−2.15259e6	−0.834251	−0.417125	−	0.908849i	$-0.636962\pi$
$367$	−2.15259e6	−0.834251	−0.417125	+	0.908849i	$0.636962\pi$
$368$	−1.66362e6	−0.640374
$369$	−416178.	−0.159116
$370$	−746856.	−0.283617
$371$	−4.25441e6	−1.60474
$372$	1.26806e6	0.475099
$373$	−2.24247e6	−0.834553	−0.417276	−	0.908780i	$-0.637015\pi$
$373$	−2.24247e6	−0.834553	−0.417276	+	0.908780i	$0.637015\pi$
$374$	174724.	0.0645912
$375$	1.71148e6	0.628482
$376$	2.96736e6	1.08243
$377$	2.66910e6	0.967189
$378$	215784.	0.0776765
$379$	−2.40986e6	−0.861775	−0.430887	−	0.902406i	$-0.641799\pi$
$379$	−2.40986e6	−0.861775	−0.430887	+	0.902406i	$0.641799\pi$
$380$	−2.78208e6	−0.988350
$381$	−466092.	−0.164497
$382$	611216.	0.214307
$383$	−1.01066e6	−0.352052	−0.176026	−	0.984386i	$-0.556324\pi$
$383$	−1.01066e6	−0.352052	−0.176026	+	0.984386i	$0.556324\pi$
$384$	−1.67616e6	−0.580079
$385$	823768.	0.283239
$386$	−232868.	−0.0795503
$387$	672624.	0.228294
$388$	−1.97842e6	−0.667175
$389$	1.27779e6	0.428140	0.214070	−	0.976818i	$-0.431328\pi$
$389$	1.27779e6	0.428140	0.214070	+	0.976818i	$0.431328\pi$
$390$	475272.	0.158227
$391$	1.83099e6	0.605682
$392$	611640.	0.201039
$393$	−482868.	−0.157706
$394$	495484.	0.160801
$395$	−1.16380e6	−0.375306
$396$	−274428.	−0.0879408
$397$	5.45400e6	1.73676	0.868378	−	0.495903i	$-0.165163\pi$
$397$	5.45400e6	1.73676	0.868378	+	0.495903i	$0.165163\pi$
$398$	1.02672e6	0.324896
$399$	−2.87712e6	−0.904744
$400$	−661904.	−0.206845
$401$	−1.48980e6	−0.462665	−0.231332	−	0.972875i	$-0.574308\pi$
$401$	−1.48980e6	−0.462665	−0.231332	+	0.972875i	$0.574308\pi$
$402$	556344.	0.171703
$403$	2.88837e6	0.885911
$404$	2.85594e6	0.870555
$405$	301806.	0.0914303
$406$	−1.37640e6	−0.414409
$407$	982278.	0.293933
$408$	779760.	0.231905
$409$	−4.39899e6	−1.30030	−0.650152	−	0.759804i	$-0.725295\pi$
$409$	−4.39899e6	−1.30030	−0.650152	+	0.759804i	$0.725295\pi$
$410$	472696.	0.138874
$411$	2.05976e6	0.601467
$412$	−3.66531e6	−1.06382
$413$	−867280.	−0.250198
$414$	410832.	0.117805
$415$	−807576.	−0.230178
$416$	−2.95725e6	−0.837827
$417$	−3.37428e6	−0.950257
$418$	−522720.	−0.146328
$419$	−280420.	−0.0780322	−0.0390161	−	0.999239i	$-0.512422\pi$
$419$	−280420.	−0.0780322	−0.0390161	+	0.999239i	$0.512422\pi$
$420$	1.71562e6	0.474566
$421$	817462.	0.224782	0.112391	−	0.993664i	$-0.464149\pi$
$421$	817462.	0.224782	0.112391	+	0.993664i	$0.464149\pi$
$422$	1.24138e6	0.339330
$423$	2.00297e6	0.544281
$424$	−3.44952e6	−0.931846
$425$	728498.	0.195639
$426$	−1.25366e6	−0.334701
$427$	−7.94138e6	−2.10779
$428$	3.96514e6	1.04628
$429$	−625086.	−0.163982
$430$	−763968.	−0.199253
$431$	1.88599e6	0.489043	0.244521	−	0.969644i	$-0.421369\pi$
$431$	1.88599e6	0.489043	0.244521	+	0.969644i	$0.421369\pi$
$432$	−478224.	−0.123288
$433$	5.84067e6	1.49707	0.748537	−	0.663093i	$-0.230757\pi$
$433$	5.84067e6	1.49707	0.748537	+	0.663093i	$0.230757\pi$
$434$	−1.48947e6	−0.379584
$435$	−1.92510e6	−0.487787
$436$	6.71468e6	1.69164
$437$	−5.47776e6	−1.37214
$438$	−332028.	−0.0826969
$439$	−509540.	−0.126188	−0.0630938	−	0.998008i	$-0.520097\pi$
$439$	−509540.	−0.126188	−0.0630938	+	0.998008i	$0.520097\pi$
$440$	667920.	0.164472
$441$	412857.	0.101089
$442$	828856.	0.201801
$443$	4.10268e6	0.993250	0.496625	−	0.867965i	$-0.334572\pi$
$443$	4.10268e6	0.993250	0.496625	+	0.867965i	$0.334572\pi$
$444$	2.04574e6	0.492484
$445$	6.09822e6	1.45983
$446$	2.63603e6	0.627500
$447$	592470.	0.140248
$448$	−1.58182e6	−0.372360
$449$	513410.	0.120185	0.0600923	−	0.998193i	$-0.480861\pi$
$449$	513410.	0.120185	0.0600923	+	0.998193i	$0.480861\pi$
$450$	163458.	0.0380518
$451$	−621698.	−0.143926
$452$	1.19633e6	0.275426
$453$	−1.38647e6	−0.317442
$454$	1.77482e6	0.404125
$455$	3.90779e6	0.884918
$456$	−2.33280e6	−0.525370
$457$	1.22738e6	0.274908	0.137454	−	0.990508i	$-0.456108\pi$
$457$	1.22738e6	0.274908	0.137454	+	0.990508i	$0.456108\pi$
$458$	474900.	0.105789
$459$	526338.	0.116609
$460$	3.26637e6	0.719732
$461$	−6.41000e6	−1.40477	−0.702386	−	0.711797i	$-0.747882\pi$
$461$	−6.41000e6	−1.40477	−0.702386	+	0.711797i	$0.747882\pi$
$462$	322344.	0.0702611
$463$	6.63030e6	1.43741	0.718705	−	0.695315i	$-0.244735\pi$
$463$	6.63030e6	1.43741	0.718705	+	0.695315i	$0.244735\pi$
$464$	3.05040e6	0.657751
$465$	−2.08325e6	−0.446795
$466$	1.82941e6	0.390253
$467$	−4.14769e6	−0.880064	−0.440032	−	0.897982i	$-0.645033\pi$
$467$	−4.14769e6	−0.880064	−0.440032	+	0.897982i	$0.645033\pi$
$468$	−1.30183e6	−0.274752
$469$	4.57438e6	0.960286
$470$	−2.27498e6	−0.475042
$471$	−2.58910e6	−0.537770
$472$	−703200.	−0.145286
$473$	1.00478e6	0.206500
$474$	−455400.	−0.0930995
$475$	−2.17944e6	−0.443211
$476$	2.99197e6	0.605257
$477$	−2.32843e6	−0.468561
$478$	−2.81784e6	−0.564088
$479$	−5.05132e6	−1.00593	−0.502963	−	0.864308i	$-0.667757\pi$
$479$	−5.05132e6	−1.00593	−0.502963	+	0.864308i	$0.667757\pi$
$480$	2.13293e6	0.422545
$481$	4.65973e6	0.918329
$482$	1.65272e6	0.324027
$483$	3.37795e6	0.658849
$484$	−409948.	−0.0795455
$485$	3.25027e6	0.627429
$486$	118098.	0.0226805
$487$	2.66221e6	0.508651	0.254325	−	0.967119i	$-0.418147\pi$
$487$	2.66221e6	0.508651	0.254325	+	0.967119i	$0.418147\pi$
$488$	−6.43896e6	−1.22396
$489$	−946116.	−0.178925
$490$	−468924.	−0.0882292
$491$	−5.54659e6	−1.03830	−0.519149	−	0.854684i	$-0.673751\pi$
$491$	−5.54659e6	−1.03830	−0.519149	+	0.854684i	$0.673751\pi$
$492$	−1.29478e6	−0.241147
$493$	−3.35730e6	−0.622118
$494$	−2.47968e6	−0.457171
$495$	450846.	0.0827018
$496$	3.30099e6	0.602477
$497$	−1.03079e7	−1.87189
$498$	−316008.	−0.0570985
$499$	−6820.00	−0.00122612	−0.000613060	−	1.00000i	$-0.500195\pi$
$499$	−6820.00	−0.00122612	−0.000613060		1.00000i	$0.500195\pi$
$500$	5.32459e6	0.952492
$501$	−1.35475e6	−0.241138
$502$	3.20774e6	0.568119
$503$	−451136.	−0.0795037	−0.0397519	−	0.999210i	$-0.512657\pi$
$503$	−451136.	−0.0795037	−0.0397519	+	0.999210i	$0.512657\pi$
$504$	1.43856e6	0.252262
$505$	−4.69191e6	−0.818693
$506$	613712.	0.106559
$507$	376353.	0.0650243
$508$	−1.45006e6	−0.249303
$509$	393390.	0.0673021	0.0336511	−	0.999434i	$-0.489287\pi$
$509$	393390.	0.0673021	0.0336511	+	0.999434i	$0.489287\pi$
$510$	−597816.	−0.101775
$511$	−2.73001e6	−0.462500
$512$	−5.89875e6	−0.994455
$513$	−1.57464e6	−0.264173
$514$	−795236.	−0.132766
$515$	6.02158e6	1.00044
$516$	2.09261e6	0.345990
$517$	2.99209e6	0.492321
$518$	−2.40293e6	−0.393474
$519$	19494.0	0.00317675
$520$	3.16848e6	0.513857
$521$	3.28432e6	0.530092	0.265046	−	0.964236i	$-0.414613\pi$
$521$	3.28432e6	0.530092	0.265046	+	0.964236i	$0.414613\pi$
$522$	−753300.	−0.121002
$523$	−1.68266e6	−0.268993	−0.134497	−	0.990914i	$-0.542942\pi$
$523$	−1.68266e6	−0.268993	−0.134497	+	0.990914i	$0.542942\pi$
$524$	−1.50226e6	−0.239010
$525$	1.34399e6	0.212812
$526$	−4.26333e6	−0.671869
$527$	−3.63310e6	−0.569838
$528$	−714384.	−0.111518
$529$	−5047.00	−0.000784141 0
$530$	2.64463e6	0.408955
$531$	−474660.	−0.0730544
$532$	−8.95104e6	−1.37118
$533$	−2.94921e6	−0.449664
$534$	2.38626e6	0.362130
$535$	−6.51415e6	−0.983951
$536$	3.70896e6	0.557622
$537$	−6.05502e6	−0.906108
$538$	1.45162e6	0.216221
$539$	616737.	0.0914383
$540$	938952.	0.138567
$541$	9.48158e6	1.39280	0.696398	−	0.717656i	$-0.254785\pi$
$541$	9.48158e6	1.39280	0.696398	+	0.717656i	$0.254785\pi$
$542$	2.93574e6	0.429258
$543$	4.74100e6	0.690034
$544$	3.71974e6	0.538909
$545$	−1.10313e7	−1.59087
$546$	1.52914e6	0.219515
$547$	−6.09239e6	−0.870602	−0.435301	−	0.900285i	$-0.643358\pi$
$547$	−6.09239e6	−0.870602	−0.435301	+	0.900285i	$0.643358\pi$
$548$	6.40814e6	0.911550
$549$	−4.34630e6	−0.615444
$550$	244178.	0.0344191
$551$	1.00440e7	1.40938
$552$	2.73888e6	0.382583
$553$	−3.74440e6	−0.520678
$554$	−3.04200e6	−0.421099
$555$	−3.36085e6	−0.463145
$556$	−1.04978e7	−1.44016
$557$	8.49594e6	1.16031	0.580154	−	0.814507i	$-0.302992\pi$
$557$	8.49594e6	1.16031	0.580154	+	0.814507i	$0.302992\pi$
$558$	−815184.	−0.110833
$559$	4.76650e6	0.645163
$560$	4.46605e6	0.601802
$561$	786258.	0.105477
$562$	−928764.	−0.124041
$563$	−7.02216e6	−0.933683	−0.466842	−	0.884341i	$-0.654608\pi$
$563$	−7.02216e6	−0.933683	−0.466842	+	0.884341i	$0.654608\pi$
$564$	6.23146e6	0.824882
$565$	−1.96540e6	−0.259017
$566$	830272.	0.108938
$567$	971028.	0.126845
$568$	−8.35776e6	−1.08697
$569$	9.41847e6	1.21955	0.609775	−	0.792574i	$-0.291260\pi$
$569$	9.41847e6	1.21955	0.609775	+	0.792574i	$0.291260\pi$
$570$	1.78848e6	0.230567
$571$	7.29699e6	0.936599	0.468299	−	0.883570i	$-0.344867\pi$
$571$	7.29699e6	0.936599	0.468299	+	0.883570i	$0.344867\pi$
$572$	−1.94471e6	−0.248522
$573$	2.75047e6	0.349962
$574$	1.52085e6	0.192666
$575$	2.55882e6	0.322753
$576$	−865728.	−0.108724
$577$	−3.29590e6	−0.412131	−0.206065	−	0.978538i	$-0.566066\pi$
$577$	−3.29590e6	−0.412131	−0.206065	+	0.978538i	$0.566066\pi$
$578$	1.79715e6	0.223750
$579$	−1.04791e6	−0.129905
$580$	−5.98920e6	−0.739263
$581$	−2.59829e6	−0.319335
$582$	1.27184e6	0.155642
$583$	−3.47827e6	−0.423830
$584$	−2.21352e6	−0.268566
$585$	2.13872e6	0.258384
$586$	5.18641e6	0.623911
$587$	4.39827e6	0.526849	0.263425	−	0.964680i	$-0.415148\pi$
$587$	4.39827e6	0.526849	0.263425	+	0.964680i	$0.415148\pi$
$588$	1.28444e6	0.153205
$589$	1.08691e7	1.29094
$590$	539120.	0.0637610
$591$	2.22968e6	0.262587
$592$	5.32541e6	0.624523
$593$	9.21781e6	1.07644	0.538222	−	0.842803i	$-0.319096\pi$
$593$	9.21781e6	1.07644	0.538222	+	0.842803i	$0.319096\pi$
$594$	176418.	0.0205152
$595$	−4.91538e6	−0.569199
$596$	1.84324e6	0.212553
$597$	4.62024e6	0.530553
$598$	2.91133e6	0.332919
$599$	3.77140e6	0.429473	0.214736	−	0.976672i	$-0.431111\pi$
$599$	3.77140e6	0.429473	0.214736	+	0.976672i	$0.431111\pi$
$600$	1.08972e6	0.123577
$601$	4.19724e6	0.473999	0.237000	−	0.971510i	$-0.423836\pi$
$601$	4.19724e6	0.473999	0.237000	+	0.971510i	$0.423836\pi$
$602$	−2.45798e6	−0.276432
$603$	2.50355e6	0.280390
$604$	−4.31346e6	−0.481097
$605$	673486.	0.0748066
$606$	−1.83596e6	−0.203087
$607$	−1.00133e6	−0.110308	−0.0551539	−	0.998478i	$-0.517565\pi$
$607$	−1.00133e6	−0.110308	−0.0551539	+	0.998478i	$0.517565\pi$
$608$	−1.11283e7	−1.22087
$609$	−6.19380e6	−0.676728
$610$	4.93654e6	0.537153
$611$	1.41939e7	1.53815
$612$	1.63750e6	0.176727
$613$	−7.38239e6	−0.793498	−0.396749	−	0.917927i	$-0.629862\pi$
$613$	−7.38239e6	−0.793498	−0.396749	+	0.917927i	$0.629862\pi$
$614$	1.86166e6	0.199288
$615$	2.12713e6	0.226781
$616$	2.14896e6	0.228179
$617$	−1.54025e7	−1.62884	−0.814418	−	0.580279i	$-0.802944\pi$
$617$	−1.54025e7	−1.62884	−0.814418	+	0.580279i	$0.802944\pi$
$618$	2.35627e6	0.248173
$619$	−7.12402e6	−0.747306	−0.373653	−	0.927569i	$-0.621895\pi$
$619$	−7.12402e6	−0.747306	−0.373653	+	0.927569i	$0.621895\pi$
$620$	−6.48122e6	−0.677138
$621$	1.84874e6	0.192375
$622$	−4.97054e6	−0.515143
$623$	1.96204e7	2.02529
$624$	−3.38890e6	−0.348415
$625$	−5.59442e6	−0.572869
$626$	−2.63439e6	−0.268685
$627$	−2.35224e6	−0.238953
$628$	−8.05498e6	−0.815015
$629$	−5.86120e6	−0.590690
$630$	−1.10290e6	−0.110709
$631$	1.16696e7	1.16677	0.583383	−	0.812197i	$-0.301729\pi$
$631$	1.16696e7	1.16677	0.583383	+	0.812197i	$0.301729\pi$
$632$	−3.03600e6	−0.302349
$633$	5.58619e6	0.554124
$634$	−4.51080e6	−0.445687
$635$	2.38225e6	0.234451
$636$	−7.24399e6	−0.710126
$637$	2.92568e6	0.285679
$638$	−1.12530e6	−0.109450
$639$	−5.64149e6	−0.546565
$640$	8.56704e6	0.826763
$641$	−1.10271e7	−1.06003	−0.530014	−	0.847989i	$-0.677813\pi$
$641$	−1.10271e7	−1.06003	−0.530014	+	0.847989i	$0.677813\pi$
$642$	−2.54902e6	−0.244082
$643$	−9.56024e6	−0.911887	−0.455944	−	0.890009i	$-0.650698\pi$
$643$	−9.56024e6	−0.911887	−0.455944	+	0.890009i	$0.650698\pi$
$644$	1.05092e7	0.998514
$645$	−3.43786e6	−0.325378
$646$	3.11904e6	0.294063
$647$	−1.09942e7	−1.03253	−0.516263	−	0.856430i	$-0.672677\pi$
$647$	−1.09942e7	−1.03253	−0.516263	+	0.856430i	$0.672677\pi$
$648$	787320.	0.0736570
$649$	−709060.	−0.0660802
$650$	1.15833e6	0.107535
$651$	−6.70262e6	−0.619858
$652$	−2.94347e6	−0.271170
$653$	−295346.	−0.0271049	−0.0135525	−	0.999908i	$-0.504314\pi$
$653$	−295346.	−0.0271049	−0.0135525	+	0.999908i	$0.504314\pi$
$654$	−4.31658e6	−0.394635
$655$	2.46799e6	0.224771
$656$	−3.37053e6	−0.305801
$657$	−1.49413e6	−0.135044
$658$	−7.31949e6	−0.659046
$659$	−1.65613e7	−1.48553	−0.742766	−	0.669551i	$-0.766486\pi$
$659$	−1.65613e7	−1.48553	−0.742766	+	0.669551i	$0.766486\pi$
$660$	1.40263e6	0.125338
$661$	1.97042e6	0.175411	0.0877053	−	0.996146i	$-0.472047\pi$
$661$	1.97042e6	0.175411	0.0877053	+	0.996146i	$0.472047\pi$
$662$	6.34142e6	0.562395
$663$	3.72985e6	0.329540
$664$	−2.10672e6	−0.185433
$665$	1.47053e7	1.28949
$666$	−1.31512e6	−0.114889
$667$	−1.17924e7	−1.02633
$668$	−4.21478e6	−0.365455
$669$	1.18621e7	1.02470
$670$	−2.84354e6	−0.244721
$671$	−6.49262e6	−0.556690
$672$	6.86246e6	0.586215
$673$	−1.63733e6	−0.139347	−0.0696735	−	0.997570i	$-0.522196\pi$
$673$	−1.63733e6	−0.139347	−0.0696735	+	0.997570i	$0.522196\pi$
$674$	−2.55260e6	−0.216437
$675$	735561.	0.0621383
$676$	1.17088e6	0.0985472
$677$	−6.35878e6	−0.533215	−0.266607	−	0.963805i	$-0.585903\pi$
$677$	−6.35878e6	−0.533215	−0.266607	+	0.963805i	$0.585903\pi$
$678$	−769068.	−0.0642526
$679$	1.04574e7	0.870460
$680$	−3.98544e6	−0.330525
$681$	7.98671e6	0.659933
$682$	−1.21774e6	−0.100252
$683$	1.11033e7	0.910751	0.455376	−	0.890299i	$-0.349505\pi$
$683$	1.11033e7	0.910751	0.455376	+	0.890299i	$0.349505\pi$
$684$	−4.89888e6	−0.400365
$685$	−1.05277e7	−0.857245
$686$	3.46616e6	0.281215
$687$	2.13705e6	0.172752
$688$	5.44742e6	0.438753
$689$	−1.65002e7	−1.32416
$690$	−2.09981e6	−0.167902
$691$	1.70189e7	1.35592	0.677962	−	0.735097i	$-0.262864\pi$
$691$	1.70189e7	1.35592	0.677962	+	0.735097i	$0.262864\pi$
$692$	60648.0	0.00481450
$693$	1.45055e6	0.114736
$694$	−7.38606e6	−0.582122
$695$	1.72463e7	1.35436
$696$	−5.02200e6	−0.392965
$697$	3.70964e6	0.289234
$698$	−3.41838e6	−0.265572
$699$	8.23235e6	0.637281
$700$	4.18130e6	0.322527
$701$	1.58021e7	1.21456	0.607280	−	0.794488i	$-0.292260\pi$
$701$	1.58021e7	1.21456	0.607280	+	0.794488i	$0.292260\pi$
$702$	836892.	0.0640954
$703$	1.75349e7	1.33818
$704$	−1.29325e6	−0.0983445
$705$	−1.02374e7	−0.775741
$706$	−8.73719e6	−0.659720
$707$	−1.50957e7	−1.13581
$708$	−1.47672e6	−0.110717
$709$	1.24834e7	0.932643	0.466322	−	0.884615i	$-0.345579\pi$
$709$	1.24834e7	0.932643	0.466322	+	0.884615i	$0.345579\pi$
$710$	6.40762e6	0.477035
$711$	−2.04930e6	−0.152031
$712$	1.59084e7	1.17605
$713$	−1.27612e7	−0.940083
$714$	−1.92341e6	−0.141197
$715$	3.19488e6	0.233717
$716$	−1.88378e7	−1.37325
$717$	−1.26803e7	−0.921151
$718$	7.02568e6	0.508601
$719$	2.00724e6	0.144803	0.0724014	−	0.997376i	$-0.476934\pi$
$719$	2.00724e6	0.144803	0.0724014	+	0.997376i	$0.476934\pi$
$720$	2.44426e6	0.175718
$721$	1.93738e7	1.38796
$722$	−4.37900e6	−0.312631
$723$	7.43722e6	0.529133
$724$	1.47498e7	1.04578
$725$	−4.69185e6	−0.331512
$726$	263538.	0.0185567
$727$	6.97301e6	0.489310	0.244655	−	0.969610i	$-0.421325\pi$
$727$	6.97301e6	0.489310	0.244655	+	0.969610i	$0.421325\pi$
$728$	1.01942e7	0.712896
$729$	531441.	0.0370370
$730$	1.69703e6	0.117864
$731$	−5.99549e6	−0.414984
$732$	−1.35218e7	−0.932733
$733$	−2.34965e7	−1.61527	−0.807633	−	0.589685i	$-0.799252\pi$
$733$	−2.34965e7	−1.61527	−0.807633	+	0.589685i	$0.799252\pi$
$734$	4.30518e6	0.294952
$735$	−2.11016e6	−0.144078
$736$	1.30655e7	0.889059
$737$	3.73987e6	0.253622
$738$	832356.	0.0562559
$739$	−1.39901e7	−0.942346	−0.471173	−	0.882041i	$-0.656169\pi$
$739$	−1.39901e7	−0.942346	−0.471173	+	0.882041i	$0.656169\pi$
$740$	−1.04560e7	−0.701917
$741$	−1.11586e7	−0.746556
$742$	8.50882e6	0.567361
$743$	2.42745e7	1.61316	0.806582	−	0.591123i	$-0.201315\pi$
$743$	2.42745e7	1.61316	0.806582	+	0.591123i	$0.201315\pi$
$744$	−5.43456e6	−0.359942
$745$	−3.02818e6	−0.199890
$746$	4.48493e6	0.295059
$747$	−1.42204e6	−0.0932415
$748$	2.44614e6	0.159855
$749$	−2.09586e7	−1.36508
$750$	−3.42295e6	−0.222202
$751$	1.53660e7	0.994170	0.497085	−	0.867702i	$-0.334404\pi$
$751$	1.53660e7	0.994170	0.497085	+	0.867702i	$0.334404\pi$
$752$	1.62216e7	1.04604
$753$	1.44348e7	0.927734
$754$	−5.33820e6	−0.341953
$755$	7.08639e6	0.452437
$756$	3.02098e6	0.192240
$757$	2.07605e7	1.31674	0.658368	−	0.752697i	$-0.271247\pi$
$757$	2.07605e7	1.31674	0.658368	+	0.752697i	$0.271247\pi$
$758$	4.81972e6	0.304683
$759$	2.76170e6	0.174009
$760$	1.19232e7	0.748788
$761$	5.83810e6	0.365435	0.182717	−	0.983165i	$-0.441511\pi$
$761$	5.83810e6	0.365435	0.182717	+	0.983165i	$0.441511\pi$
$762$	932184.	0.0581586
$763$	−3.54919e7	−2.20708
$764$	8.55702e6	0.530383
$765$	−2.69017e6	−0.166198
$766$	2.02131e6	0.124469
$767$	−3.36364e6	−0.206453
$768$	274176.	0.0167736
$769$	−1.39197e7	−0.848818	−0.424409	−	0.905471i	$-0.639518\pi$
$769$	−1.39197e7	−0.848818	−0.424409	+	0.905471i	$0.639518\pi$
$770$	−1.64754e6	−0.100140
$771$	−3.57856e6	−0.216807
$772$	−3.26015e6	−0.196877
$773$	−4.17883e6	−0.251539	−0.125770	−	0.992059i	$-0.540140\pi$
$773$	−4.17883e6	−0.251539	−0.125770	+	0.992059i	$0.540140\pi$
$774$	−1.34525e6	−0.0807142
$775$	−5.07729e6	−0.303653
$776$	8.47896e6	0.505462
$777$	−1.08132e7	−0.642541
$778$	−2.55558e6	−0.151370
$779$	−1.10981e7	−0.655246
$780$	6.65381e6	0.391592
$781$	−8.42741e6	−0.494386
$782$	−3.66198e6	−0.214141
$783$	−3.38985e6	−0.197595
$784$	3.34363e6	0.194280
$785$	1.32332e7	0.766461
$786$	965736.	0.0557573
$787$	9.66705e6	0.556361	0.278181	−	0.960529i	$-0.410269\pi$
$787$	9.66705e6	0.556361	0.278181	+	0.960529i	$0.410269\pi$
$788$	6.93678e6	0.397962
$789$	−1.91850e7	−1.09716
$790$	2.32760e6	0.132691
$791$	−6.32345e6	−0.359346
$792$	1.17612e6	0.0666252
$793$	−3.07997e7	−1.73926
$794$	−1.09080e7	−0.614036
$795$	1.19008e7	0.667821
$796$	1.43741e7	0.804077
$797$	−5.79884e6	−0.323367	−0.161683	−	0.986843i	$-0.551692\pi$
$797$	−5.79884e6	−0.323367	−0.161683	+	0.986843i	$0.551692\pi$
$798$	5.75424e6	0.319875
$799$	−1.78536e7	−0.989371
$800$	5.19837e6	0.287172
$801$	1.07382e7	0.591356
$802$	2.97960e6	0.163577
$803$	−2.23197e6	−0.122151
$804$	7.78882e6	0.424944
$805$	−1.72651e7	−0.939029
$806$	−5.77674e6	−0.313217
$807$	6.53229e6	0.353087
$808$	−1.22398e7	−0.659545
$809$	−1.92543e7	−1.03433	−0.517163	−	0.855887i	$-0.673012\pi$
$809$	−1.92543e7	−1.03433	−0.517163	+	0.855887i	$0.673012\pi$
$810$	−603612.	−0.0323255
$811$	−1.31938e7	−0.704396	−0.352198	−	0.935926i	$-0.614566\pi$
$811$	−1.31938e7	−0.704396	−0.352198	+	0.935926i	$0.614566\pi$
$812$	−1.92696e7	−1.02561
$813$	1.32108e7	0.700976
$814$	−1.96456e6	−0.103921
$815$	4.83570e6	0.255015
$816$	4.26269e6	0.224108
$817$	1.79366e7	0.940126
$818$	8.79798e6	0.459727
$819$	6.88111e6	0.358467
$820$	6.61774e6	0.343697
$821$	1.33779e7	0.692677	0.346338	−	0.938110i	$-0.387425\pi$
$821$	1.33779e7	0.692677	0.346338	+	0.938110i	$0.387425\pi$
$822$	−4.11952e6	−0.212651
$823$	−1.88613e7	−0.970673	−0.485336	−	0.874327i	$-0.661303\pi$
$823$	−1.88613e7	−0.970673	−0.485336	+	0.874327i	$0.661303\pi$
$824$	1.57085e7	0.805965
$825$	1.09880e6	0.0562062
$826$	1.73456e6	0.0884584
$827$	1.62680e7	0.827123	0.413561	−	0.910476i	$-0.364285\pi$
$827$	1.62680e7	0.827123	0.413561	+	0.910476i	$0.364285\pi$
$828$	5.75165e6	0.291552
$829$	−2.18098e7	−1.10221	−0.551107	−	0.834435i	$-0.685794\pi$
$829$	−2.18098e7	−1.10221	−0.551107	+	0.834435i	$0.685794\pi$
$830$	1.61515e6	0.0813801
$831$	−1.36890e7	−0.687652
$832$	−6.13491e6	−0.307256
$833$	−3.68003e6	−0.183755
$834$	6.74856e6	0.335967
$835$	6.92429e6	0.343684
$836$	−7.31808e6	−0.362144
$837$	−3.66833e6	−0.180990
$838$	560840.	0.0275886
$839$	1.17771e7	0.577607	0.288804	−	0.957388i	$-0.406743\pi$
$839$	1.17771e7	0.577607	0.288804	+	0.957388i	$0.406743\pi$
$840$	−7.35264e6	−0.359538
$841$	1.11135e6	0.0541828
$842$	−1.63492e6	−0.0794726
$843$	−4.17944e6	−0.202558
$844$	1.73793e7	0.839799
$845$	−1.92358e6	−0.0926764
$846$	−4.00594e6	−0.192432
$847$	2.16687e6	0.103782
$848$	−1.88574e7	−0.900516
$849$	3.73622e6	0.177895
$850$	−1.45700e6	−0.0691689
$851$	−2.05872e7	−0.974483
$852$	−1.75513e7	−0.828343
$853$	−1.43993e7	−0.677591	−0.338796	−	0.940860i	$-0.610020\pi$
$853$	−1.43993e7	−0.677591	−0.338796	+	0.940860i	$0.610020\pi$
$854$	1.58828e7	0.745215
$855$	8.04816e6	0.376514
$856$	−1.69934e7	−0.792678
$857$	6.27604e6	0.291900	0.145950	−	0.989292i	$-0.453376\pi$
$857$	6.27604e6	0.291900	0.145950	+	0.989292i	$0.453376\pi$
$858$	1.25017e6	0.0579764
$859$	−4.71738e6	−0.218131	−0.109066	−	0.994035i	$-0.534786\pi$
$859$	−4.71738e6	−0.218131	−0.109066	+	0.994035i	$0.534786\pi$
$860$	−1.06956e7	−0.493125
$861$	6.84382e6	0.314623
$862$	−3.77198e6	−0.172903
$863$	7.53926e6	0.344589	0.172295	−	0.985045i	$-0.444882\pi$
$863$	7.53926e6	0.344589	0.172295	+	0.985045i	$0.444882\pi$
$864$	3.75581e6	0.171167
$865$	−99636.0	−0.00452768
$866$	−1.16813e7	−0.529296
$867$	8.08716e6	0.365383
$868$	−2.08526e7	−0.939423
$869$	−3.06130e6	−0.137517
$870$	3.85020e6	0.172459
$871$	1.77412e7	0.792387
$872$	−2.87772e7	−1.28161
$873$	5.72330e6	0.254162
$874$	1.09555e7	0.485126
$875$	−2.81443e7	−1.24271
$876$	−4.64839e6	−0.204664
$877$	−1.04331e7	−0.458051	−0.229025	−	0.973420i	$-0.573554\pi$
$877$	−1.04331e7	−0.458051	−0.229025	+	0.973420i	$0.573554\pi$
$878$	1.01908e6	0.0446141
$879$	2.33389e7	1.01884
$880$	3.65130e6	0.158943
$881$	3.91076e7	1.69755	0.848774	−	0.528756i	$-0.177342\pi$
$881$	3.91076e7	1.69755	0.848774	+	0.528756i	$0.177342\pi$
$882$	−825714.	−0.0357403
$883$	1.29282e7	0.558003	0.279001	−	0.960291i	$-0.409997\pi$
$883$	1.29282e7	0.558003	0.279001	+	0.960291i	$0.409997\pi$
$884$	1.16040e7	0.499432
$885$	2.42604e6	0.104121
$886$	−8.20537e6	−0.351167
$887$	3.36466e7	1.43592	0.717962	−	0.696082i	$-0.245075\pi$
$887$	3.36466e7	1.43592	0.717962	+	0.696082i	$0.245075\pi$
$888$	−8.76744e6	−0.373113
$889$	7.66462e6	0.325264
$890$	−1.21964e7	−0.516129
$891$	793881.	0.0335013
$892$	3.69044e7	1.55298
$893$	5.34125e7	2.24137
$894$	−1.18494e6	−0.0495853
$895$	3.09479e7	1.29144
$896$	2.75635e7	1.14700
$897$	1.31010e7	0.543654
$898$	−1.02682e6	−0.0424916
$899$	2.33988e7	0.965594
$900$	2.28841e6	0.0941733
$901$	2.07546e7	0.851731
$902$	1.24340e6	0.0508854
$903$	−1.10609e7	−0.451411
$904$	−5.12712e6	−0.208666
$905$	−2.42318e7	−0.983477
$906$	2.77294e6	0.112233
$907$	−4.19629e7	−1.69374	−0.846872	−	0.531797i	$-0.821517\pi$
$907$	−4.19629e7	−1.69374	−0.846872	+	0.531797i	$0.821517\pi$
$908$	2.48475e7	1.00016
$909$	−8.26184e6	−0.331640
$910$	−7.81558e6	−0.312866
$911$	−1.92521e6	−0.0768567	−0.0384283	−	0.999261i	$-0.512235\pi$
$911$	−1.92521e6	−0.0768567	−0.0384283	+	0.999261i	$0.512235\pi$
$912$	−1.27526e7	−0.507707
$913$	−2.12428e6	−0.0843401
$914$	−2.45476e6	−0.0971948
$915$	2.22144e7	0.877167
$916$	6.64860e6	0.261813
$917$	7.94050e6	0.311835
$918$	−1.05268e6	−0.0412276
$919$	−1.72481e7	−0.673678	−0.336839	−	0.941562i	$-0.609358\pi$
$919$	−1.72481e7	−0.673678	−0.336839	+	0.941562i	$0.609358\pi$
$920$	−1.39987e7	−0.545279
$921$	8.37749e6	0.325435
$922$	1.28200e7	0.496662
$923$	−3.99780e7	−1.54460
$924$	4.51282e6	0.173887
$925$	−8.19106e6	−0.314765
$926$	−1.32606e7	−0.508202
$927$	1.06032e7	0.405265
$928$	−2.39568e7	−0.913185
$929$	2.51145e6	0.0954740	0.0477370	−	0.998860i	$-0.484799\pi$
$929$	2.51145e6	0.0954740	0.0477370	+	0.998860i	$0.484799\pi$
$930$	4.16650e6	0.157966
$931$	1.10095e7	0.416288
$932$	2.56118e7	0.965828
$933$	−2.23674e7	−0.841225
$934$	8.29538e6	0.311150
$935$	−4.01865e6	−0.150332
$936$	5.57928e6	0.208156
$937$	1.79853e7	0.669221	0.334611	−	0.942357i	$-0.391395\pi$
$937$	1.79853e7	0.669221	0.334611	+	0.942357i	$0.391395\pi$
$938$	−9.14877e6	−0.339512
$939$	−1.18547e7	−0.438761
$940$	−3.18497e7	−1.17567
$941$	−3.22586e7	−1.18760	−0.593802	−	0.804611i	$-0.702374\pi$
$941$	−3.22586e7	−1.18760	−0.593802	+	0.804611i	$0.702374\pi$
$942$	5.17820e6	0.190131
$943$	1.30300e7	0.477160
$944$	−3.84416e6	−0.140401
$945$	−4.96303e6	−0.180787
$946$	−2.00957e6	−0.0730087
$947$	4.41659e7	1.60034	0.800169	−	0.599774i	$-0.204743\pi$
$947$	4.41659e7	1.60034	0.800169	+	0.599774i	$0.204743\pi$
$948$	−6.37560e6	−0.230409
$949$	−1.05880e7	−0.381635
$950$	4.35888e6	0.156699
$951$	−2.02986e7	−0.727804
$952$	−1.28227e7	−0.458551
$953$	1.87488e7	0.668714	0.334357	−	0.942446i	$-0.391481\pi$
$953$	1.87488e7	0.668714	0.334357	+	0.942446i	$0.391481\pi$
$954$	4.65685e6	0.165661
$955$	−1.40580e7	−0.498786
$956$	−3.94498e7	−1.39605
$957$	−5.06385e6	−0.178731
$958$	1.01026e7	0.355649
$959$	−3.38716e7	−1.18929
$960$	4.42483e6	0.154960
$961$	−3.30813e6	−0.115551
$962$	−9.31946e6	−0.324678
$963$	−1.14706e7	−0.398584
$964$	2.31380e7	0.801925
$965$	5.35596e6	0.185148
$966$	−6.75590e6	−0.232938
$967$	1.08673e7	0.373730	0.186865	−	0.982386i	$-0.440167\pi$
$967$	1.08673e7	0.373730	0.186865	+	0.982386i	$0.440167\pi$
$968$	1.75692e6	0.0602648
$969$	1.40357e7	0.480202
$970$	−6.50054e6	−0.221830
$971$	−4.79123e7	−1.63079	−0.815397	−	0.578902i	$-0.803481\pi$
$971$	−4.79123e7	−1.63079	−0.815397	+	0.578902i	$0.803481\pi$
$972$	1.65337e6	0.0561313
$973$	5.54882e7	1.87896
$974$	−5.32442e6	−0.179835
$975$	5.21249e6	0.175604
$976$	−3.51996e7	−1.18281
$977$	4.01385e7	1.34532	0.672658	−	0.739954i	$-0.265153\pi$
$977$	4.01385e7	1.34532	0.672658	+	0.739954i	$0.265153\pi$
$978$	1.89223e6	0.0632597
$979$	1.60410e7	0.534902
$980$	−6.56494e6	−0.218356
$981$	−1.94246e7	−0.644436
$982$	1.10932e7	0.367094
$983$	3.22682e6	0.106510	0.0532551	−	0.998581i	$-0.483040\pi$
$983$	3.22682e6	0.106510	0.0532551	+	0.998581i	$0.483040\pi$
$984$	5.54904e6	0.182696
$985$	−1.13961e7	−0.374254
$986$	6.71460e6	0.219952
$987$	−3.29377e7	−1.07622
$988$	−3.47155e7	−1.13144
$989$	−2.10589e7	−0.684614
$990$	−901692.	−0.0292395
$991$	−5.95345e6	−0.192568	−0.0962841	−	0.995354i	$-0.530696\pi$
$991$	−5.95345e6	−0.192568	−0.0962841	+	0.995354i	$0.530696\pi$
$992$	−2.59249e7	−0.836445
$993$	2.85364e7	0.918387
$994$	2.06158e7	0.661812
$995$	−2.36146e7	−0.756175
$996$	−4.42411e6	−0.141312
$997$	3.20783e7	1.02205	0.511027	−	0.859565i	$-0.329265\pi$
$997$	3.20783e7	1.02205	0.511027	+	0.859565i	$0.329265\pi$
$998$	13640.0	0.000433499 0
$999$	−5.91802e6	−0.187613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.a.a.1.1	✓	1
3.2	odd	2		99.6.a.b.1.1		1
4.3	odd	2		528.6.a.i.1.1		1
5.4	even	2		825.6.a.b.1.1		1
11.10	odd	2		363.6.a.c.1.1		1
33.32	even	2		1089.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.a.1.1	✓	1	1.1	even	1	trivial
99.6.a.b.1.1		1	3.2	odd	2
363.6.a.c.1.1		1	11.10	odd	2
528.6.a.i.1.1		1	4.3	odd	2
825.6.a.b.1.1		1	5.4	even	2
1089.6.a.d.1.1		1	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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