LMFDB - Embedded newform 99.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,6,Mod(1,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	99.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$15.8779981615$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -28.0000 q^{4} -46.0000 q^{5} +148.000 q^{7} -120.000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} -28.0000 q^{4} -46.0000 q^{5} +148.000 q^{7} -120.000 q^{8} -92.0000 q^{10} -121.000 q^{11} +574.000 q^{13} +296.000 q^{14} +656.000 q^{16} +722.000 q^{17} +2160.00 q^{19} +1288.00 q^{20} -242.000 q^{22} +2536.00 q^{23} -1009.00 q^{25} +1148.00 q^{26} -4144.00 q^{28} -4650.00 q^{29} +5032.00 q^{31} +5152.00 q^{32} +1444.00 q^{34} -6808.00 q^{35} +8118.00 q^{37} +4320.00 q^{38} +5520.00 q^{40} +5138.00 q^{41} +8304.00 q^{43} +3388.00 q^{44} +5072.00 q^{46} -24728.0 q^{47} +5097.00 q^{49} -2018.00 q^{50} -16072.0 q^{52} +28746.0 q^{53} +5566.00 q^{55} -17760.0 q^{56} -9300.00 q^{58} +5860.00 q^{59} -53658.0 q^{61} +10064.0 q^{62} -10688.0 q^{64} -26404.0 q^{65} +30908.0 q^{67} -20216.0 q^{68} -13616.0 q^{70} +69648.0 q^{71} -18446.0 q^{73} +16236.0 q^{74} -60480.0 q^{76} -17908.0 q^{77} -25300.0 q^{79} -30176.0 q^{80} +10276.0 q^{82} +17556.0 q^{83} -33212.0 q^{85} +16608.0 q^{86} +14520.0 q^{88} -132570. q^{89} +84952.0 q^{91} -71008.0 q^{92} -49456.0 q^{94} -99360.0 q^{95} +70658.0 q^{97} +10194.0 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−28.0000	−0.875000
$5$	−46.0000	−0.822873	−0.411437	−	0.911438i	$-0.634973\pi$
$5$	−46.0000	−0.822873	−0.411437	+	0.911438i	$0.634973\pi$
$6$	0	0
$7$	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
$9$	0	0
$10$	−92.0000	−0.290930
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	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
$15$	0	0
$16$	656.000	0.640625
$17$	722.000	0.605919	0.302960	−	0.953003i	$-0.402025\pi$
$17$	722.000	0.605919	0.302960	+	0.953003i	$0.402025\pi$
$18$	0	0
$19$	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
$21$	0	0
$22$	−242.000	−0.106600
$23$	2536.00	0.999608	0.499804	−	0.866139i	$-0.333405\pi$
$23$	2536.00	0.999608	0.499804	+	0.866139i	$0.333405\pi$
$24$	0	0
$25$	−1009.00	−0.322880
$26$	1148.00	0.333049
$27$	0	0
$28$	−4144.00	−0.998906
$29$	−4650.00	−1.02673	−0.513367	−	0.858169i	$-0.671602\pi$
$29$	−4650.00	−1.02673	−0.513367	+	0.858169i	$0.671602\pi$
$30$	0	0
$31$	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
$33$	0	0
$34$	1444.00	0.214225
$35$	−6808.00	−0.939398
$36$	0	0
$37$	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
$39$	0	0
$40$	5520.00	0.545493
$41$	5138.00	0.477347	0.238674	−	0.971100i	$-0.423287\pi$
$41$	5138.00	0.477347	0.238674	+	0.971100i	$0.423287\pi$
$42$	0	0
$43$	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
$45$	0	0
$46$	5072.00	0.353415
$47$	−24728.0	−1.63284	−0.816421	−	0.577457i	$-0.804045\pi$
$47$	−24728.0	−1.63284	−0.816421	+	0.577457i	$0.804045\pi$
$48$	0	0
$49$	5097.00	0.303266
$50$	−2018.00	−0.114155
$51$	0	0
$52$	−16072.0	−0.824255
$53$	28746.0	1.40568	0.702842	−	0.711346i	$-0.251914\pi$
$53$	28746.0	1.40568	0.702842	+	0.711346i	$0.251914\pi$
$54$	0	0
$55$	5566.00	0.248106
$56$	−17760.0	−0.756786
$57$	0	0
$58$	−9300.00	−0.363005
$59$	5860.00	0.219163	0.109582	−	0.993978i	$-0.465049\pi$
$59$	5860.00	0.219163	0.109582	+	0.993978i	$0.465049\pi$
$60$	0	0
$61$	−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
$63$	0	0
$64$	−10688.0	−0.326172
$65$	−26404.0	−0.775151
$66$	0	0
$67$	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
$69$	0	0
$70$	−13616.0	−0.332127
$71$	69648.0	1.63969	0.819847	−	0.572583i	$-0.194058\pi$
$71$	69648.0	1.63969	0.819847	+	0.572583i	$0.194058\pi$
$72$	0	0
$73$	−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
$75$	0	0
$76$	−60480.0	−1.20110
$77$	−17908.0	−0.344207
$78$	0	0
$79$	−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
$81$	0	0
$82$	10276.0	0.168768
$83$	17556.0	0.279724	0.139862	−	0.990171i	$-0.455334\pi$
$83$	17556.0	0.279724	0.139862	+	0.990171i	$0.455334\pi$
$84$	0	0
$85$	−33212.0	−0.498595
$86$	16608.0	0.242143
$87$	0	0
$88$	14520.0	0.199876
$89$	−132570.	−1.77407	−0.887034	−	0.461704i	$-0.847238\pi$
$89$	−132570.	−1.77407	−0.887034	+	0.461704i	$0.847238\pi$
$90$	0	0
$91$	84952.0	1.07540
$92$	−71008.0	−0.874657
$93$	0	0
$94$	−49456.0	−0.577297
$95$	−99360.0	−1.12954
$96$	0	0
$97$	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
$99$	0	0
$100$	28252.0	0.282520
$101$	101998.	0.994920	0.497460	−	0.867487i	$-0.334266\pi$
$101$	101998.	0.994920	0.497460	+	0.867487i	$0.334266\pi$
$102$	0	0
$103$	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
$105$	0	0
$106$	57492.0	0.496984
$107$	141612.	1.19575	0.597875	−	0.801589i	$-0.296012\pi$
$107$	141612.	1.19575	0.597875	+	0.801589i	$0.296012\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	97088.0	0.731342
$113$	42726.0	0.314772	0.157386	−	0.987537i	$-0.449693\pi$
$113$	42726.0	0.314772	0.157386	+	0.987537i	$0.449693\pi$
$114$	0	0
$115$	−116656.	−0.822550
$116$	130200.	0.898392
$117$	0	0
$118$	11720.0	0.0774859
$119$	106856.	0.691722
$120$	0	0
$121$	14641.0	0.0909091
$122$	−107316.	−0.652777
$123$	0	0
$124$	−140896.	−0.822895
$125$	190164.	1.08856
$126$	0	0
$127$	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
$129$	0	0
$130$	−52808.0	−0.274057
$131$	−53652.0	−0.273154	−0.136577	−	0.990629i	$-0.543610\pi$
$131$	−53652.0	−0.273154	−0.136577	+	0.990629i	$0.543610\pi$
$132$	0	0
$133$	319680.	1.56706
$134$	61816.0	0.297399
$135$	0	0
$136$	−86640.0	−0.401672
$137$	228862.	1.04177	0.520886	−	0.853627i	$-0.325602\pi$
$137$	228862.	1.04177	0.520886	+	0.853627i	$0.325602\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	139296.	0.579719
$143$	−69454.0	−0.284025
$144$	0	0
$145$	213900.	0.844872
$146$	−36892.0	−0.143235
$147$	0	0
$148$	−227304.	−0.853007
$149$	65830.0	0.242917	0.121459	−	0.992597i	$-0.461243\pi$
$149$	65830.0	0.242917	0.121459	+	0.992597i	$0.461243\pi$
$150$	0	0
$151$	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
$153$	0	0
$154$	−35816.0	−0.121696
$155$	−231472.	−0.773872
$156$	0	0
$157$	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
$159$	0	0
$160$	−236992.	−0.731870
$161$	375328.	1.14116
$162$	0	0
$163$	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
$165$	0	0
$166$	35112.0	0.0988975
$167$	−150528.	−0.417663	−0.208832	−	0.977952i	$-0.566966\pi$
$167$	−150528.	−0.417663	−0.208832	+	0.977952i	$0.566966\pi$
$168$	0	0
$169$	−41817.0	−0.112625
$170$	−66424.0	−0.176280
$171$	0	0
$172$	−232512.	−0.599272
$173$	2166.00	0.00550229	0.00275114	−	0.999996i	$-0.499124\pi$
$173$	2166.00	0.00550229	0.00275114	+	0.999996i	$0.499124\pi$
$174$	0	0
$175$	−149332.	−0.368602
$176$	−79376.0	−0.193156
$177$	0	0
$178$	−265140.	−0.627228
$179$	−672780.	−1.56942	−0.784712	−	0.619860i	$-0.787189\pi$
$179$	−672780.	−1.56942	−0.784712	+	0.619860i	$0.787189\pi$
$180$	0	0
$181$	−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
$183$	0	0
$184$	−304320.	−0.662653
$185$	−373428.	−0.802191
$186$	0	0
$187$	−87362.0	−0.182692
$188$	692384.	1.42874
$189$	0	0
$190$	−198720.	−0.399354
$191$	305608.	0.606152	0.303076	−	0.952966i	$-0.401986\pi$
$191$	305608.	0.606152	0.303076	+	0.952966i	$0.401986\pi$
$192$	0	0
$193$	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
$195$	0	0
$196$	−142716.	−0.265358
$197$	247742.	0.454814	0.227407	−	0.973800i	$-0.426975\pi$
$197$	247742.	0.454814	0.227407	+	0.973800i	$0.426975\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	203996.	0.351757
$203$	−688200.	−1.17213
$204$	0	0
$205$	−236348.	−0.392796
$206$	261808.	0.429848
$207$	0	0
$208$	376544.	0.603472
$209$	−261360.	−0.413879
$210$	0	0
$211$	−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
$213$	0	0
$214$	283224.	0.422762
$215$	−381984.	−0.563571
$216$	0	0
$217$	744736.	1.07363
$218$	−479620.	−0.683528
$219$	0	0
$220$	−155848.	−0.217092
$221$	414428.	0.570780
$222$	0	0
$223$	−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
$225$	0	0
$226$	85452.0	0.111289
$227$	887412.	1.14304	0.571519	−	0.820589i	$-0.306354\pi$
$227$	887412.	1.14304	0.571519	+	0.820589i	$0.306354\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	558000.	0.680635
$233$	914706.	1.10380	0.551902	−	0.833909i	$-0.313902\pi$
$233$	914706.	1.10380	0.551902	+	0.833909i	$0.313902\pi$
$234$	0	0
$235$	1.13749e6	1.34362
$236$	−164080.	−0.191768
$237$	0	0
$238$	213712.	0.244561
$239$	−1.40892e6	−1.59548	−0.797740	−	0.603001i	$-0.793971\pi$
$239$	−1.40892e6	−1.59548	−0.797740	+	0.603001i	$0.793971\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	1.50242e6	1.61554
$245$	−234462.	−0.249550
$246$	0	0
$247$	1.23984e6	1.29307
$248$	−603840.	−0.623437
$249$	0	0
$250$	380328.	0.384865
$251$	1.60387e6	1.60688	0.803442	−	0.595384i	$-0.203000\pi$
$251$	1.60387e6	1.60688	0.803442	+	0.595384i	$0.203000\pi$
$252$	0	0
$253$	−306856.	−0.301393
$254$	103576.	0.100734
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	−397618.	−0.375520	−0.187760	−	0.982215i	$-0.560123\pi$
$257$	−397618.	−0.375520	−0.187760	+	0.982215i	$0.560123\pi$
$258$	0	0
$259$	1.20146e6	1.11291
$260$	739312.	0.678257
$261$	0	0
$262$	−107304.	−0.0965745
$263$	−2.13166e6	−1.90033	−0.950166	−	0.311745i	$-0.899087\pi$
$263$	−2.13166e6	−1.90033	−0.950166	+	0.311745i	$0.899087\pi$
$264$	0	0
$265$	−1.32232e6	−1.15670
$266$	639360.	0.554040
$267$	0	0
$268$	−865424.	−0.736024
$269$	725810.	0.611564	0.305782	−	0.952101i	$-0.401082\pi$
$269$	725810.	0.611564	0.305782	+	0.952101i	$0.401082\pi$
$270$	0	0
$271$	−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
$273$	0	0
$274$	457724.	0.368322
$275$	122089.	0.0973520
$276$	0	0
$277$	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
$279$	0	0
$280$	816960.	0.622738
$281$	−464382.	−0.350840	−0.175420	−	0.984494i	$-0.556128\pi$
$281$	−464382.	−0.350840	−0.175420	+	0.984494i	$0.556128\pi$
$282$	0	0
$283$	−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
$285$	0	0
$286$	−138908.	−0.100418
$287$	760424.	0.544943
$288$	0	0
$289$	−898573.	−0.632862
$290$	427800.	0.298707
$291$	0	0
$292$	516488.	0.354489
$293$	2.59321e6	1.76469	0.882344	−	0.470605i	$-0.155964\pi$
$293$	2.59321e6	1.76469	0.882344	+	0.470605i	$0.155964\pi$
$294$	0	0
$295$	−269560.	−0.180343
$296$	−974160.	−0.646251
$297$	0	0
$298$	131660.	0.0858842
$299$	1.45566e6	0.941636
$300$	0	0
$301$	1.22899e6	0.781867
$302$	308104.	0.194393
$303$	0	0
$304$	1.41696e6	0.879374
$305$	2.46827e6	1.51930
$306$	0	0
$307$	−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
$309$	0	0
$310$	−462944.	−0.273605
$311$	−2.48527e6	−1.45704	−0.728522	−	0.685022i	$-0.759793\pi$
$311$	−2.48527e6	−1.45704	−0.728522	+	0.685022i	$0.759793\pi$
$312$	0	0
$313$	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
$315$	0	0
$316$	708400.	0.399081
$317$	−2.25540e6	−1.26059	−0.630297	−	0.776354i	$-0.717067\pi$
$317$	−2.25540e6	−1.26059	−0.630297	+	0.776354i	$0.717067\pi$
$318$	0	0
$319$	562650.	0.309572
$320$	491648.	0.268398
$321$	0	0
$322$	750656.	0.403461
$323$	1.55952e6	0.831734
$324$	0	0
$325$	−579166.	−0.304155
$326$	210248.	0.109569
$327$	0	0
$328$	−616560.	−0.316440
$329$	−3.65974e6	−1.86406
$330$	0	0
$331$	−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
$333$	0	0
$334$	−301056.	−0.147666
$335$	−1.42177e6	−0.692176
$336$	0	0
$337$	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
$339$	0	0
$340$	929936.	0.436270
$341$	−608872.	−0.283557
$342$	0	0
$343$	−1.73308e6	−0.795396
$344$	−996480.	−0.454017
$345$	0	0
$346$	4332.00	0.00194535
$347$	−3.69303e6	−1.64649	−0.823245	−	0.567687i	$-0.807838\pi$
$347$	−3.69303e6	−1.64649	−0.823245	+	0.567687i	$0.807838\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	−623392.	−0.268167
$353$	−4.36859e6	−1.86597	−0.932986	−	0.359914i	$-0.882806\pi$
$353$	−4.36859e6	−1.86597	−0.932986	+	0.359914i	$0.882806\pi$
$354$	0	0
$355$	−3.20381e6	−1.34926
$356$	3.71196e6	1.55231
$357$	0	0
$358$	−1.34556e6	−0.554875
$359$	3.51284e6	1.43854	0.719271	−	0.694730i	$-0.244476\pi$
$359$	3.51284e6	1.43854	0.719271	+	0.694730i	$0.244476\pi$
$360$	0	0
$361$	2.18950e6	0.884254
$362$	−1.05356e6	−0.422558
$363$	0	0
$364$	−2.37866e6	−0.940975
$365$	848516.	0.333371
$366$	0	0
$367$	−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
$369$	0	0
$370$	−746856.	−0.283617
$371$	4.25441e6	1.60474
$372$	0	0
$373$	−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
$375$	0	0
$376$	2.96736e6	1.08243
$377$	−2.66910e6	−0.967189
$378$	0	0
$379$	−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
$381$	0	0
$382$	611216.	0.214307
$383$	1.01066e6	0.352052	0.176026	−	0.984386i	$-0.443676\pi$
$383$	1.01066e6	0.352052	0.176026	+	0.984386i	$0.443676\pi$
$384$	0	0
$385$	823768.	0.283239
$386$	232868.	0.0795503
$387$	0	0
$388$	−1.97842e6	−0.667175
$389$	−1.27779e6	−0.428140	−0.214070	−	0.976818i	$-0.568672\pi$
$389$	−1.27779e6	−0.428140	−0.214070	+	0.976818i	$0.568672\pi$
$390$	0	0
$391$	1.83099e6	0.605682
$392$	−611640.	−0.201039
$393$	0	0
$394$	495484.	0.160801
$395$	1.16380e6	0.375306
$396$	0	0
$397$	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
$399$	0	0
$400$	−661904.	−0.206845
$401$	1.48980e6	0.462665	0.231332	−	0.972875i	$-0.425692\pi$
$401$	1.48980e6	0.462665	0.231332	+	0.972875i	$0.425692\pi$
$402$	0	0
$403$	2.88837e6	0.885911
$404$	−2.85594e6	−0.870555
$405$	0	0
$406$	−1.37640e6	−0.414409
$407$	−982278.	−0.293933
$408$	0	0
$409$	−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
$411$	0	0
$412$	−3.66531e6	−1.06382
$413$	867280.	0.250198
$414$	0	0
$415$	−807576.	−0.230178
$416$	2.95725e6	0.837827
$417$	0	0
$418$	−522720.	−0.146328
$419$	280420.	0.0780322	0.0390161	−	0.999239i	$-0.487578\pi$
$419$	280420.	0.0780322	0.0390161	+	0.999239i	$0.487578\pi$
$420$	0	0
$421$	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
$423$	0	0
$424$	−3.44952e6	−0.931846
$425$	−728498.	−0.195639
$426$	0	0
$427$	−7.94138e6	−2.10779
$428$	−3.96514e6	−1.04628
$429$	0	0
$430$	−763968.	−0.199253
$431$	−1.88599e6	−0.489043	−0.244521	−	0.969644i	$-0.578631\pi$
$431$	−1.88599e6	−0.489043	−0.244521	+	0.969644i	$0.578631\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	6.71468e6	1.69164
$437$	5.47776e6	1.37214
$438$	0	0
$439$	−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
$441$	0	0
$442$	828856.	0.201801
$443$	−4.10268e6	−0.993250	−0.496625	−	0.867965i	$-0.665428\pi$
$443$	−4.10268e6	−0.993250	−0.496625	+	0.867965i	$0.665428\pi$
$444$	0	0
$445$	6.09822e6	1.45983
$446$	−2.63603e6	−0.627500
$447$	0	0
$448$	−1.58182e6	−0.372360
$449$	−513410.	−0.120185	−0.0600923	−	0.998193i	$-0.519139\pi$
$449$	−513410.	−0.120185	−0.0600923	+	0.998193i	$0.519139\pi$
$450$	0	0
$451$	−621698.	−0.143926
$452$	−1.19633e6	−0.275426
$453$	0	0
$454$	1.77482e6	0.404125
$455$	−3.90779e6	−0.884918
$456$	0	0
$457$	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
$459$	0	0
$460$	3.26637e6	0.719732
$461$	6.41000e6	1.40477	0.702386	−	0.711797i	$-0.252118\pi$
$461$	6.41000e6	1.40477	0.702386	+	0.711797i	$0.252118\pi$
$462$	0	0
$463$	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
$465$	0	0
$466$	1.82941e6	0.390253
$467$	4.14769e6	0.880064	0.440032	−	0.897982i	$-0.354967\pi$
$467$	4.14769e6	0.880064	0.440032	+	0.897982i	$0.354967\pi$
$468$	0	0
$469$	4.57438e6	0.960286
$470$	2.27498e6	0.475042
$471$	0	0
$472$	−703200.	−0.145286
$473$	−1.00478e6	−0.206500
$474$	0	0
$475$	−2.17944e6	−0.443211
$476$	−2.99197e6	−0.605257
$477$	0	0
$478$	−2.81784e6	−0.564088
$479$	5.05132e6	1.00593	0.502963	−	0.864308i	$-0.332243\pi$
$479$	5.05132e6	1.00593	0.502963	+	0.864308i	$0.332243\pi$
$480$	0	0
$481$	4.65973e6	0.918329
$482$	−1.65272e6	−0.324027
$483$	0	0
$484$	−409948.	−0.0795455
$485$	−3.25027e6	−0.627429
$486$	0	0
$487$	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
$489$	0	0
$490$	−468924.	−0.0882292
$491$	5.54659e6	1.03830	0.519149	−	0.854684i	$-0.326249\pi$
$491$	5.54659e6	1.03830	0.519149	+	0.854684i	$0.326249\pi$
$492$	0	0
$493$	−3.35730e6	−0.622118
$494$	2.47968e6	0.457171
$495$	0	0
$496$	3.30099e6	0.602477
$497$	1.03079e7	1.87189
$498$	0	0
$499$	−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
$501$	0	0
$502$	3.20774e6	0.568119
$503$	451136.	0.0795037	0.0397519	−	0.999210i	$-0.487343\pi$
$503$	451136.	0.0795037	0.0397519	+	0.999210i	$0.487343\pi$
$504$	0	0
$505$	−4.69191e6	−0.818693
$506$	−613712.	−0.106559
$507$	0	0
$508$	−1.45006e6	−0.249303
$509$	−393390.	−0.0673021	−0.0336511	−	0.999434i	$-0.510713\pi$
$509$	−393390.	−0.0673021	−0.0336511	+	0.999434i	$0.510713\pi$
$510$	0	0
$511$	−2.73001e6	−0.462500
$512$	5.89875e6	0.994455
$513$	0	0
$514$	−795236.	−0.132766
$515$	−6.02158e6	−1.00044
$516$	0	0
$517$	2.99209e6	0.492321
$518$	2.40293e6	0.393474
$519$	0	0
$520$	3.16848e6	0.513857
$521$	−3.28432e6	−0.530092	−0.265046	−	0.964236i	$-0.585387\pi$
$521$	−3.28432e6	−0.530092	−0.265046	+	0.964236i	$0.585387\pi$
$522$	0	0
$523$	−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
$525$	0	0
$526$	−4.26333e6	−0.671869
$527$	3.63310e6	0.569838
$528$	0	0
$529$	−5047.00	−0.000784141 0
$530$	−2.64463e6	−0.408955
$531$	0	0
$532$	−8.95104e6	−1.37118
$533$	2.94921e6	0.449664
$534$	0	0
$535$	−6.51415e6	−0.983951
$536$	−3.70896e6	−0.557622
$537$	0	0
$538$	1.45162e6	0.216221
$539$	−616737.	−0.0914383
$540$	0	0
$541$	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
$543$	0	0
$544$	3.71974e6	0.538909
$545$	1.10313e7	1.59087
$546$	0	0
$547$	−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
$549$	0	0
$550$	244178.	0.0344191
$551$	−1.00440e7	−1.40938
$552$	0	0
$553$	−3.74440e6	−0.520678
$554$	3.04200e6	0.421099
$555$	0	0
$556$	−1.04978e7	−1.44016
$557$	−8.49594e6	−1.16031	−0.580154	−	0.814507i	$-0.697008\pi$
$557$	−8.49594e6	−1.16031	−0.580154	+	0.814507i	$0.697008\pi$
$558$	0	0
$559$	4.76650e6	0.645163
$560$	−4.46605e6	−0.601802
$561$	0	0
$562$	−928764.	−0.124041
$563$	7.02216e6	0.933683	0.466842	−	0.884341i	$-0.345392\pi$
$563$	7.02216e6	0.933683	0.466842	+	0.884341i	$0.345392\pi$
$564$	0	0
$565$	−1.96540e6	−0.259017
$566$	−830272.	−0.108938
$567$	0	0
$568$	−8.35776e6	−1.08697
$569$	−9.41847e6	−1.21955	−0.609775	−	0.792574i	$-0.708740\pi$
$569$	−9.41847e6	−1.21955	−0.609775	+	0.792574i	$0.708740\pi$
$570$	0	0
$571$	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
$573$	0	0
$574$	1.52085e6	0.192666
$575$	−2.55882e6	−0.322753
$576$	0	0
$577$	−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
$579$	0	0
$580$	−5.98920e6	−0.739263
$581$	2.59829e6	0.319335
$582$	0	0
$583$	−3.47827e6	−0.423830
$584$	2.21352e6	0.268566
$585$	0	0
$586$	5.18641e6	0.623911
$587$	−4.39827e6	−0.526849	−0.263425	−	0.964680i	$-0.584852\pi$
$587$	−4.39827e6	−0.526849	−0.263425	+	0.964680i	$0.584852\pi$
$588$	0	0
$589$	1.08691e7	1.29094
$590$	−539120.	−0.0637610
$591$	0	0
$592$	5.32541e6	0.624523
$593$	−9.21781e6	−1.07644	−0.538222	−	0.842803i	$-0.680904\pi$
$593$	−9.21781e6	−1.07644	−0.538222	+	0.842803i	$0.680904\pi$
$594$	0	0
$595$	−4.91538e6	−0.569199
$596$	−1.84324e6	−0.212553
$597$	0	0
$598$	2.91133e6	0.332919
$599$	−3.77140e6	−0.429473	−0.214736	−	0.976672i	$-0.568889\pi$
$599$	−3.77140e6	−0.429473	−0.214736	+	0.976672i	$0.568889\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	−4.31346e6	−0.481097
$605$	−673486.	−0.0748066
$606$	0	0
$607$	−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
$609$	0	0
$610$	4.93654e6	0.537153
$611$	−1.41939e7	−1.53815
$612$	0	0
$613$	−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
$615$	0	0
$616$	2.14896e6	0.228179
$617$	1.54025e7	1.62884	0.814418	−	0.580279i	$-0.197056\pi$
$617$	1.54025e7	1.62884	0.814418	+	0.580279i	$0.197056\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	−4.97054e6	−0.515143
$623$	−1.96204e7	−2.02529
$624$	0	0
$625$	−5.59442e6	−0.572869
$626$	2.63439e6	0.268685
$627$	0	0
$628$	−8.05498e6	−0.815015
$629$	5.86120e6	0.590690
$630$	0	0
$631$	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
$633$	0	0
$634$	−4.51080e6	−0.445687
$635$	−2.38225e6	−0.234451
$636$	0	0
$637$	2.92568e6	0.285679
$638$	1.12530e6	0.109450
$639$	0	0
$640$	8.56704e6	0.826763
$641$	1.10271e7	1.06003	0.530014	−	0.847989i	$-0.322187\pi$
$641$	1.10271e7	1.06003	0.530014	+	0.847989i	$0.322187\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	3.11904e6	0.294063
$647$	1.09942e7	1.03253	0.516263	−	0.856430i	$-0.327323\pi$
$647$	1.09942e7	1.03253	0.516263	+	0.856430i	$0.327323\pi$
$648$	0	0
$649$	−709060.	−0.0660802
$650$	−1.15833e6	−0.107535
$651$	0	0
$652$	−2.94347e6	−0.271170
$653$	295346.	0.0271049	0.0135525	−	0.999908i	$-0.495686\pi$
$653$	295346.	0.0271049	0.0135525	+	0.999908i	$0.495686\pi$
$654$	0	0
$655$	2.46799e6	0.224771
$656$	3.37053e6	0.305801
$657$	0	0
$658$	−7.31949e6	−0.659046
$659$	1.65613e7	1.48553	0.742766	−	0.669551i	$-0.233514\pi$
$659$	1.65613e7	1.48553	0.742766	+	0.669551i	$0.233514\pi$
$660$	0	0
$661$	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
$663$	0	0
$664$	−2.10672e6	−0.185433
$665$	−1.47053e7	−1.28949
$666$	0	0
$667$	−1.17924e7	−1.02633
$668$	4.21478e6	0.365455
$669$	0	0
$670$	−2.84354e6	−0.244721
$671$	6.49262e6	0.556690
$672$	0	0
$673$	−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
$675$	0	0
$676$	1.17088e6	0.0985472
$677$	6.35878e6	0.533215	0.266607	−	0.963805i	$-0.414097\pi$
$677$	6.35878e6	0.533215	0.266607	+	0.963805i	$0.414097\pi$
$678$	0	0
$679$	1.04574e7	0.870460
$680$	3.98544e6	0.330525
$681$	0	0
$682$	−1.21774e6	−0.100252
$683$	−1.11033e7	−0.910751	−0.455376	−	0.890299i	$-0.650495\pi$
$683$	−1.11033e7	−0.910751	−0.455376	+	0.890299i	$0.650495\pi$
$684$	0	0
$685$	−1.05277e7	−0.857245
$686$	−3.46616e6	−0.281215
$687$	0	0
$688$	5.44742e6	0.438753
$689$	1.65002e7	1.32416
$690$	0	0
$691$	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
$693$	0	0
$694$	−7.38606e6	−0.582122
$695$	−1.72463e7	−1.35436
$696$	0	0
$697$	3.70964e6	0.289234
$698$	3.41838e6	0.265572
$699$	0	0
$700$	4.18130e6	0.322527
$701$	−1.58021e7	−1.21456	−0.607280	−	0.794488i	$-0.707740\pi$
$701$	−1.58021e7	−1.21456	−0.607280	+	0.794488i	$0.707740\pi$
$702$	0	0
$703$	1.75349e7	1.33818
$704$	1.29325e6	0.0983445
$705$	0	0
$706$	−8.73719e6	−0.659720
$707$	1.50957e7	1.13581
$708$	0	0
$709$	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
$711$	0	0
$712$	1.59084e7	1.17605
$713$	1.27612e7	0.940083
$714$	0	0
$715$	3.19488e6	0.233717
$716$	1.88378e7	1.37325
$717$	0	0
$718$	7.02568e6	0.508601
$719$	−2.00724e6	−0.144803	−0.0724014	−	0.997376i	$-0.523066\pi$
$719$	−2.00724e6	−0.144803	−0.0724014	+	0.997376i	$0.523066\pi$
$720$	0	0
$721$	1.93738e7	1.38796
$722$	4.37900e6	0.312631
$723$	0	0
$724$	1.47498e7	1.04578
$725$	4.69185e6	0.331512
$726$	0	0
$727$	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
$729$	0	0
$730$	1.69703e6	0.117864
$731$	5.99549e6	0.414984
$732$	0	0
$733$	−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
$735$	0	0
$736$	1.30655e7	0.889059
$737$	−3.73987e6	−0.253622
$738$	0	0
$739$	−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
$741$	0	0
$742$	8.50882e6	0.567361
$743$	−2.42745e7	−1.61316	−0.806582	−	0.591123i	$-0.798685\pi$
$743$	−2.42745e7	−1.61316	−0.806582	+	0.591123i	$0.798685\pi$
$744$	0	0
$745$	−3.02818e6	−0.199890
$746$	−4.48493e6	−0.295059
$747$	0	0
$748$	2.44614e6	0.159855
$749$	2.09586e7	1.36508
$750$	0	0
$751$	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
$753$	0	0
$754$	−5.33820e6	−0.341953
$755$	−7.08639e6	−0.452437
$756$	0	0
$757$	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
$759$	0	0
$760$	1.19232e7	0.748788
$761$	−5.83810e6	−0.365435	−0.182717	−	0.983165i	$-0.558489\pi$
$761$	−5.83810e6	−0.365435	−0.182717	+	0.983165i	$0.558489\pi$
$762$	0	0
$763$	−3.54919e7	−2.20708
$764$	−8.55702e6	−0.530383
$765$	0	0
$766$	2.02131e6	0.124469
$767$	3.36364e6	0.206453
$768$	0	0
$769$	−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
$771$	0	0
$772$	−3.26015e6	−0.196877
$773$	4.17883e6	0.251539	0.125770	−	0.992059i	$-0.459860\pi$
$773$	4.17883e6	0.251539	0.125770	+	0.992059i	$0.459860\pi$
$774$	0	0
$775$	−5.07729e6	−0.303653
$776$	−8.47896e6	−0.505462
$777$	0	0
$778$	−2.55558e6	−0.151370
$779$	1.10981e7	0.655246
$780$	0	0
$781$	−8.42741e6	−0.494386
$782$	3.66198e6	0.214141
$783$	0	0
$784$	3.34363e6	0.194280
$785$	−1.32332e7	−0.766461
$786$	0	0
$787$	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
$789$	0	0
$790$	2.32760e6	0.132691
$791$	6.32345e6	0.359346
$792$	0	0
$793$	−3.07997e7	−1.73926
$794$	1.09080e7	0.614036
$795$	0	0
$796$	1.43741e7	0.804077
$797$	5.79884e6	0.323367	0.161683	−	0.986843i	$-0.448308\pi$
$797$	5.79884e6	0.323367	0.161683	+	0.986843i	$0.448308\pi$
$798$	0	0
$799$	−1.78536e7	−0.989371
$800$	−5.19837e6	−0.287172
$801$	0	0
$802$	2.97960e6	0.163577
$803$	2.23197e6	0.122151
$804$	0	0
$805$	−1.72651e7	−0.939029
$806$	5.77674e6	0.313217
$807$	0	0
$808$	−1.22398e7	−0.659545
$809$	1.92543e7	1.03433	0.517163	−	0.855887i	$-0.326988\pi$
$809$	1.92543e7	1.03433	0.517163	+	0.855887i	$0.326988\pi$
$810$	0	0
$811$	−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
$813$	0	0
$814$	−1.96456e6	−0.103921
$815$	−4.83570e6	−0.255015
$816$	0	0
$817$	1.79366e7	0.940126
$818$	−8.79798e6	−0.459727
$819$	0	0
$820$	6.61774e6	0.343697
$821$	−1.33779e7	−0.692677	−0.346338	−	0.938110i	$-0.612575\pi$
$821$	−1.33779e7	−0.692677	−0.346338	+	0.938110i	$0.612575\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	1.73456e6	0.0884584
$827$	−1.62680e7	−0.827123	−0.413561	−	0.910476i	$-0.635715\pi$
$827$	−1.62680e7	−0.827123	−0.413561	+	0.910476i	$0.635715\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	−6.13491e6	−0.307256
$833$	3.68003e6	0.183755
$834$	0	0
$835$	6.92429e6	0.343684
$836$	7.31808e6	0.362144
$837$	0	0
$838$	560840.	0.0275886
$839$	−1.17771e7	−0.577607	−0.288804	−	0.957388i	$-0.593257\pi$
$839$	−1.17771e7	−0.577607	−0.288804	+	0.957388i	$0.593257\pi$
$840$	0	0
$841$	1.11135e6	0.0541828
$842$	1.63492e6	0.0794726
$843$	0	0
$844$	1.73793e7	0.839799
$845$	1.92358e6	0.0926764
$846$	0	0
$847$	2.16687e6	0.103782
$848$	1.88574e7	0.900516
$849$	0	0
$850$	−1.45700e6	−0.0691689
$851$	2.05872e7	0.974483
$852$	0	0
$853$	−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
$855$	0	0
$856$	−1.69934e7	−0.792678
$857$	−6.27604e6	−0.291900	−0.145950	−	0.989292i	$-0.546624\pi$
$857$	−6.27604e6	−0.291900	−0.145950	+	0.989292i	$0.546624\pi$
$858$	0	0
$859$	−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
$861$	0	0
$862$	−3.77198e6	−0.172903
$863$	−7.53926e6	−0.344589	−0.172295	−	0.985045i	$-0.555118\pi$
$863$	−7.53926e6	−0.344589	−0.172295	+	0.985045i	$0.555118\pi$
$864$	0	0
$865$	−99636.0	−0.00452768
$866$	1.16813e7	0.529296
$867$	0	0
$868$	−2.08526e7	−0.939423
$869$	3.06130e6	0.137517
$870$	0	0
$871$	1.77412e7	0.792387
$872$	2.87772e7	1.28161
$873$	0	0
$874$	1.09555e7	0.485126
$875$	2.81443e7	1.24271
$876$	0	0
$877$	−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
$879$	0	0
$880$	3.65130e6	0.158943
$881$	−3.91076e7	−1.69755	−0.848774	−	0.528756i	$-0.822658\pi$
$881$	−3.91076e7	−1.69755	−0.848774	+	0.528756i	$0.822658\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	−8.20537e6	−0.351167
$887$	−3.36466e7	−1.43592	−0.717962	−	0.696082i	$-0.754925\pi$
$887$	−3.36466e7	−1.43592	−0.717962	+	0.696082i	$0.754925\pi$
$888$	0	0
$889$	7.66462e6	0.325264
$890$	1.21964e7	0.516129
$891$	0	0
$892$	3.69044e7	1.55298
$893$	−5.34125e7	−2.24137
$894$	0	0
$895$	3.09479e7	1.29144
$896$	−2.75635e7	−1.14700
$897$	0	0
$898$	−1.02682e6	−0.0424916
$899$	−2.33988e7	−0.965594
$900$	0	0
$901$	2.07546e7	0.851731
$902$	−1.24340e6	−0.0508854
$903$	0	0
$904$	−5.12712e6	−0.208666
$905$	2.42318e7	0.983477
$906$	0	0
$907$	−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
$909$	0	0
$910$	−7.81558e6	−0.312866
$911$	1.92521e6	0.0768567	0.0384283	−	0.999261i	$-0.487765\pi$
$911$	1.92521e6	0.0768567	0.0384283	+	0.999261i	$0.487765\pi$
$912$	0	0
$913$	−2.12428e6	−0.0843401
$914$	2.45476e6	0.0971948
$915$	0	0
$916$	6.64860e6	0.261813
$917$	−7.94050e6	−0.311835
$918$	0	0
$919$	−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
$921$	0	0
$922$	1.28200e7	0.496662
$923$	3.99780e7	1.54460
$924$	0	0
$925$	−8.19106e6	−0.314765
$926$	1.32606e7	0.508202
$927$	0	0
$928$	−2.39568e7	−0.913185
$929$	−2.51145e6	−0.0954740	−0.0477370	−	0.998860i	$-0.515201\pi$
$929$	−2.51145e6	−0.0954740	−0.0477370	+	0.998860i	$0.515201\pi$
$930$	0	0
$931$	1.10095e7	0.416288
$932$	−2.56118e7	−0.965828
$933$	0	0
$934$	8.29538e6	0.311150
$935$	4.01865e6	0.150332
$936$	0	0
$937$	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
$939$	0	0
$940$	−3.18497e7	−1.17567
$941$	3.22586e7	1.18760	0.593802	−	0.804611i	$-0.297626\pi$
$941$	3.22586e7	1.18760	0.593802	+	0.804611i	$0.297626\pi$
$942$	0	0
$943$	1.30300e7	0.477160
$944$	3.84416e6	0.140401
$945$	0	0
$946$	−2.00957e6	−0.0730087
$947$	−4.41659e7	−1.60034	−0.800169	−	0.599774i	$-0.795257\pi$
$947$	−4.41659e7	−1.60034	−0.800169	+	0.599774i	$0.795257\pi$
$948$	0	0
$949$	−1.05880e7	−0.381635
$950$	−4.35888e6	−0.156699
$951$	0	0
$952$	−1.28227e7	−0.458551
$953$	−1.87488e7	−0.668714	−0.334357	−	0.942446i	$-0.608519\pi$
$953$	−1.87488e7	−0.668714	−0.334357	+	0.942446i	$0.608519\pi$
$954$	0	0
$955$	−1.40580e7	−0.498786
$956$	3.94498e7	1.39605
$957$	0	0
$958$	1.01026e7	0.355649
$959$	3.38716e7	1.18929
$960$	0	0
$961$	−3.30813e6	−0.115551
$962$	9.31946e6	0.324678
$963$	0	0
$964$	2.31380e7	0.801925
$965$	−5.35596e6	−0.185148
$966$	0	0
$967$	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
$969$	0	0
$970$	−6.50054e6	−0.221830
$971$	4.79123e7	1.63079	0.815397	−	0.578902i	$-0.196519\pi$
$971$	4.79123e7	1.63079	0.815397	+	0.578902i	$0.196519\pi$
$972$	0	0
$973$	5.54882e7	1.87896
$974$	5.32442e6	0.179835
$975$	0	0
$976$	−3.51996e7	−1.18281
$977$	−4.01385e7	−1.34532	−0.672658	−	0.739954i	$-0.734847\pi$
$977$	−4.01385e7	−1.34532	−0.672658	+	0.739954i	$0.734847\pi$
$978$	0	0
$979$	1.60410e7	0.534902
$980$	6.56494e6	0.218356
$981$	0	0
$982$	1.10932e7	0.367094
$983$	−3.22682e6	−0.106510	−0.0532551	−	0.998581i	$-0.516960\pi$
$983$	−3.22682e6	−0.106510	−0.0532551	+	0.998581i	$0.516960\pi$
$984$	0	0
$985$	−1.13961e7	−0.374254
$986$	−6.71460e6	−0.219952
$987$	0	0
$988$	−3.47155e7	−1.13144
$989$	2.10589e7	0.684614
$990$	0	0
$991$	−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
$993$	0	0
$994$	2.06158e7	0.661812
$995$	2.36146e7	0.756175
$996$	0	0
$997$	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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.a.1.1	✓	1	3.2	odd	2
99.6.a.b.1.1		1	1.1	even	1	trivial
363.6.a.c.1.1		1	33.32	even	2
528.6.a.i.1.1		1	12.11	even	2
825.6.a.b.1.1		1	15.14	odd	2
1089.6.a.d.1.1		1	11.10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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