LMFDB - Embedded newform 99.6.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [99,6,Mod(1,99)] 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("99.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [1,4] 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:	$15.8779981615$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
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)

$f(q)$

$=$

\(q+4.00000 q^{2} -16.0000 q^{4} +19.0000 q^{5} +10.0000 q^{7} -192.000 q^{8} +76.0000 q^{10} +121.000 q^{11} -1148.00 q^{13} +40.0000 q^{14} -256.000 q^{16} -686.000 q^{17} -384.000 q^{19} -304.000 q^{20} +484.000 q^{22} -3709.00 q^{23} -2764.00 q^{25} -4592.00 q^{26} -160.000 q^{28} +5424.00 q^{29} -6443.00 q^{31} +5120.00 q^{32} -2744.00 q^{34} +190.000 q^{35} +12063.0 q^{37} -1536.00 q^{38} -3648.00 q^{40} +1528.00 q^{41} -4026.00 q^{43} -1936.00 q^{44} -14836.0 q^{46} -7168.00 q^{47} -16707.0 q^{49} -11056.0 q^{50} +18368.0 q^{52} +29862.0 q^{53} +2299.00 q^{55} -1920.00 q^{56} +21696.0 q^{58} +6461.00 q^{59} -16980.0 q^{61} -25772.0 q^{62} +28672.0 q^{64} -21812.0 q^{65} +29999.0 q^{67} +10976.0 q^{68} +760.000 q^{70} -31023.0 q^{71} +1924.00 q^{73} +48252.0 q^{74} +6144.00 q^{76} +1210.00 q^{77} +65138.0 q^{79} -4864.00 q^{80} +6112.00 q^{82} +102714. q^{83} -13034.0 q^{85} -16104.0 q^{86} -23232.0 q^{88} -17415.0 q^{89} -11480.0 q^{91} +59344.0 q^{92} -28672.0 q^{94} -7296.00 q^{95} +66905.0 q^{97} -66828.0 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	4.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	4.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−16.0000	−0.500000
$5$	19.0000	0.339882	0.169941	−	0.985454i	$-0.445642\pi$
$5$	19.0000	0.339882	0.169941	+	0.985454i	$0.445642\pi$
$6$	0	0
$7$	10.0000	0.0771356	0.0385678	−	0.999256i	$-0.487720\pi$
$7$	10.0000	0.0771356	0.0385678	+	0.999256i	$0.487720\pi$
$8$	−192.000	−1.06066
$9$	0	0
$10$	76.0000	0.240333
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−1148.00	−1.88401	−0.942006	−	0.335597i	$-0.891062\pi$
$13$	−1148.00	−1.88401	−0.942006	+	0.335597i	$0.891062\pi$
$14$	40.0000	0.0545431
$15$	0	0
$16$	−256.000	−0.250000
$17$	−686.000	−0.575707	−0.287854	−	0.957674i	$-0.592942\pi$
$17$	−686.000	−0.575707	−0.287854	+	0.957674i	$0.592942\pi$
$18$	0	0
$19$	−384.000	−0.244032	−0.122016	−	0.992528i	$-0.538936\pi$
$19$	−384.000	−0.244032	−0.122016	+	0.992528i	$0.538936\pi$
$20$	−304.000	−0.169941
$21$	0	0
$22$	484.000	0.213201
$23$	−3709.00	−1.46197	−0.730983	−	0.682396i	$-0.760938\pi$
$23$	−3709.00	−1.46197	−0.730983	+	0.682396i	$0.760938\pi$
$24$	0	0
$25$	−2764.00	−0.884480
$26$	−4592.00	−1.33220
$27$	0	0
$28$	−160.000	−0.0385678
$29$	5424.00	1.19764	0.598818	−	0.800885i	$-0.295637\pi$
$29$	5424.00	1.19764	0.598818	+	0.800885i	$0.295637\pi$
$30$	0	0
$31$	−6443.00	−1.20416	−0.602080	−	0.798436i	$-0.705661\pi$
$31$	−6443.00	−1.20416	−0.602080	+	0.798436i	$0.705661\pi$
$32$	5120.00	0.883883
$33$	0	0
$34$	−2744.00	−0.407087
$35$	190.000	0.0262170
$36$	0	0
$37$	12063.0	1.44861	0.724304	−	0.689481i	$-0.242161\pi$
$37$	12063.0	1.44861	0.724304	+	0.689481i	$0.242161\pi$
$38$	−1536.00	−0.172557
$39$	0	0
$40$	−3648.00	−0.360500
$41$	1528.00	0.141959	0.0709796	−	0.997478i	$-0.477387\pi$
$41$	1528.00	0.141959	0.0709796	+	0.997478i	$0.477387\pi$
$42$	0	0
$43$	−4026.00	−0.332049	−0.166025	−	0.986122i	$-0.553093\pi$
$43$	−4026.00	−0.332049	−0.166025	+	0.986122i	$0.553093\pi$
$44$	−1936.00	−0.150756
$45$	0	0
$46$	−14836.0	−1.03377
$47$	−7168.00	−0.473318	−0.236659	−	0.971593i	$-0.576052\pi$
$47$	−7168.00	−0.473318	−0.236659	+	0.971593i	$0.576052\pi$
$48$	0	0
$49$	−16707.0	−0.994050
$50$	−11056.0	−0.625422
$51$	0	0
$52$	18368.0	0.942006
$53$	29862.0	1.46026	0.730128	−	0.683310i	$-0.239460\pi$
$53$	29862.0	1.46026	0.730128	+	0.683310i	$0.239460\pi$
$54$	0	0
$55$	2299.00	0.102478
$56$	−1920.00	−0.0818147
$57$	0	0
$58$	21696.0	0.846856
$59$	6461.00	0.241640	0.120820	−	0.992674i	$-0.461448\pi$
$59$	6461.00	0.241640	0.120820	+	0.992674i	$0.461448\pi$
$60$	0	0
$61$	−16980.0	−0.584269	−0.292135	−	0.956377i	$-0.594366\pi$
$61$	−16980.0	−0.584269	−0.292135	+	0.956377i	$0.594366\pi$
$62$	−25772.0	−0.851469
$63$	0	0
$64$	28672.0	0.875000
$65$	−21812.0	−0.640342
$66$	0	0
$67$	29999.0	0.816432	0.408216	−	0.912885i	$-0.366151\pi$
$67$	29999.0	0.816432	0.408216	+	0.912885i	$0.366151\pi$
$68$	10976.0	0.287854
$69$	0	0
$70$	760.000	0.0185382
$71$	−31023.0	−0.730362	−0.365181	−	0.930937i	$-0.618993\pi$
$71$	−31023.0	−0.730362	−0.365181	+	0.930937i	$0.618993\pi$
$72$	0	0
$73$	1924.00	0.0422569	0.0211285	−	0.999777i	$-0.493274\pi$
$73$	1924.00	0.0422569	0.0211285	+	0.999777i	$0.493274\pi$
$74$	48252.0	1.02432
$75$	0	0
$76$	6144.00	0.122016
$77$	1210.00	0.0232573
$78$	0	0
$79$	65138.0	1.17427	0.587133	−	0.809490i	$-0.300256\pi$
$79$	65138.0	1.17427	0.587133	+	0.809490i	$0.300256\pi$
$80$	−4864.00	−0.0849706
$81$	0	0
$82$	6112.00	0.100380
$83$	102714.	1.63657	0.818285	−	0.574813i	$-0.194925\pi$
$83$	102714.	1.63657	0.818285	+	0.574813i	$0.194925\pi$
$84$	0	0
$85$	−13034.0	−0.195673
$86$	−16104.0	−0.234794
$87$	0	0
$88$	−23232.0	−0.319801
$89$	−17415.0	−0.233050	−0.116525	−	0.993188i	$-0.537175\pi$
$89$	−17415.0	−0.233050	−0.116525	+	0.993188i	$0.537175\pi$
$90$	0	0
$91$	−11480.0	−0.145324
$92$	59344.0	0.730983
$93$	0	0
$94$	−28672.0	−0.334687
$95$	−7296.00	−0.0829422
$96$	0	0
$97$	66905.0	0.721987	0.360993	−	0.932568i	$-0.382438\pi$
$97$	66905.0	0.721987	0.360993	+	0.932568i	$0.382438\pi$
$98$	−66828.0	−0.702900
$99$	0	0
$100$	44224.0	0.442240
$101$	−96730.0	−0.943534	−0.471767	−	0.881723i	$-0.656384\pi$
$101$	−96730.0	−0.943534	−0.471767	+	0.881723i	$0.656384\pi$
$102$	0	0
$103$	−95704.0	−0.888868	−0.444434	−	0.895812i	$-0.646595\pi$
$103$	−95704.0	−0.888868	−0.444434	+	0.895812i	$0.646595\pi$
$104$	220416.	1.99830
$105$	0	0
$106$	119448.	1.03256
$107$	32658.0	0.275759	0.137880	−	0.990449i	$-0.455971\pi$
$107$	32658.0	0.275759	0.137880	+	0.990449i	$0.455971\pi$
$108$	0	0
$109$	−185438.	−1.49497	−0.747485	−	0.664279i	$-0.768739\pi$
$109$	−185438.	−1.49497	−0.747485	+	0.664279i	$0.768739\pi$
$110$	9196.00	0.0724632
$111$	0	0
$112$	−2560.00	−0.0192839
$113$	−72849.0	−0.536695	−0.268347	−	0.963322i	$-0.586478\pi$
$113$	−72849.0	−0.536695	−0.268347	+	0.963322i	$0.586478\pi$
$114$	0	0
$115$	−70471.0	−0.496896
$116$	−86784.0	−0.598818
$117$	0	0
$118$	25844.0	0.170866
$119$	−6860.00	−0.0444075
$120$	0	0
$121$	14641.0	0.0909091
$122$	−67920.0	−0.413141
$123$	0	0
$124$	103088.	0.602080
$125$	−111891.	−0.640501
$126$	0	0
$127$	−78184.0	−0.430139	−0.215069	−	0.976599i	$-0.568998\pi$
$127$	−78184.0	−0.430139	−0.215069	+	0.976599i	$0.568998\pi$
$128$	−49152.0	−0.265165
$129$	0	0
$130$	−87248.0	−0.452790
$131$	462.000	0.00235214	0.00117607	−	0.999999i	$-0.499626\pi$
$131$	462.000	0.00235214	0.00117607	+	0.999999i	$0.499626\pi$
$132$	0	0
$133$	−3840.00	−0.0188236
$134$	119996.	0.577304
$135$	0	0
$136$	131712.	0.610630
$137$	−296233.	−1.34844	−0.674221	−	0.738530i	$-0.735520\pi$
$137$	−296233.	−1.34844	−0.674221	+	0.738530i	$0.735520\pi$
$138$	0	0
$139$	−399818.	−1.75519	−0.877597	−	0.479398i	$-0.840855\pi$
$139$	−399818.	−1.75519	−0.877597	+	0.479398i	$0.840855\pi$
$140$	−3040.00	−0.0131085
$141$	0	0
$142$	−124092.	−0.516444
$143$	−138908.	−0.568051
$144$	0	0
$145$	103056.	0.407055
$146$	7696.00	0.0298802
$147$	0	0
$148$	−193008.	−0.724304
$149$	−72670.0	−0.268157	−0.134079	−	0.990971i	$-0.542807\pi$
$149$	−72670.0	−0.268157	−0.134079	+	0.990971i	$0.542807\pi$
$150$	0	0
$151$	−303082.	−1.08173	−0.540864	−	0.841110i	$-0.681902\pi$
$151$	−303082.	−1.08173	−0.540864	+	0.841110i	$0.681902\pi$
$152$	73728.0	0.258835
$153$	0	0
$154$	4840.00	0.0164454
$155$	−122417.	−0.409272
$156$	0	0
$157$	−532987.	−1.72571	−0.862854	−	0.505453i	$-0.831326\pi$
$157$	−532987.	−1.72571	−0.862854	+	0.505453i	$0.831326\pi$
$158$	260552.	0.830332
$159$	0	0
$160$	97280.0	0.300416
$161$	−37090.0	−0.112770
$162$	0	0
$163$	282076.	0.831567	0.415783	−	0.909464i	$-0.363507\pi$
$163$	282076.	0.831567	0.415783	+	0.909464i	$0.363507\pi$
$164$	−24448.0	−0.0709796
$165$	0	0
$166$	410856.	1.15723
$167$	573588.	1.59151	0.795754	−	0.605620i	$-0.207075\pi$
$167$	573588.	1.59151	0.795754	+	0.605620i	$0.207075\pi$
$168$	0	0
$169$	946611.	2.54950
$170$	−52136.0	−0.138362
$171$	0	0
$172$	64416.0	0.166025
$173$	386286.	0.981282	0.490641	−	0.871362i	$-0.336763\pi$
$173$	386286.	0.981282	0.490641	+	0.871362i	$0.336763\pi$
$174$	0	0
$175$	−27640.0	−0.0682249
$176$	−30976.0	−0.0753778
$177$	0	0
$178$	−69660.0	−0.164791
$179$	−545079.	−1.27153	−0.635765	−	0.771882i	$-0.719315\pi$
$179$	−545079.	−1.27153	−0.635765	+	0.771882i	$0.719315\pi$
$180$	0	0
$181$	−279485.	−0.634106	−0.317053	−	0.948408i	$-0.602693\pi$
$181$	−279485.	−0.634106	−0.317053	+	0.948408i	$0.602693\pi$
$182$	−45920.0	−0.102760
$183$	0	0
$184$	712128.	1.55065
$185$	229197.	0.492356
$186$	0	0
$187$	−83006.0	−0.173582
$188$	114688.	0.236659
$189$	0	0
$190$	−29184.0	−0.0586490
$191$	444437.	0.881509	0.440755	−	0.897628i	$-0.354711\pi$
$191$	444437.	0.881509	0.440755	+	0.897628i	$0.354711\pi$
$192$	0	0
$193$	−18476.0	−0.0357038	−0.0178519	−	0.999841i	$-0.505683\pi$
$193$	−18476.0	−0.0357038	−0.0178519	+	0.999841i	$0.505683\pi$
$194$	267620.	0.510522
$195$	0	0
$196$	267312.	0.497025
$197$	−270182.	−0.496010	−0.248005	−	0.968759i	$-0.579775\pi$
$197$	−270182.	−0.496010	−0.248005	+	0.968759i	$0.579775\pi$
$198$	0	0
$199$	43320.0	0.0775453	0.0387727	−	0.999248i	$-0.487655\pi$
$199$	43320.0	0.0775453	0.0387727	+	0.999248i	$0.487655\pi$
$200$	530688.	0.938133
$201$	0	0
$202$	−386920.	−0.667180
$203$	54240.0	0.0923803
$204$	0	0
$205$	29032.0	0.0482494
$206$	−382816.	−0.628524
$207$	0	0
$208$	293888.	0.471003
$209$	−46464.0	−0.0735785
$210$	0	0
$211$	1.02968e6	1.59220	0.796100	−	0.605165i	$-0.206893\pi$
$211$	1.02968e6	1.59220	0.796100	+	0.605165i	$0.206893\pi$
$212$	−477792.	−0.730128
$213$	0	0
$214$	130632.	0.194991
$215$	−76494.0	−0.112858
$216$	0	0
$217$	−64430.0	−0.0928835
$218$	−741752.	−1.05710
$219$	0	0
$220$	−36784.0	−0.0512392
$221$	787528.	1.08464
$222$	0	0
$223$	461281.	0.621160	0.310580	−	0.950547i	$-0.399477\pi$
$223$	461281.	0.621160	0.310580	+	0.950547i	$0.399477\pi$
$224$	51200.0	0.0681789
$225$	0	0
$226$	−291396.	−0.379501
$227$	855570.	1.10202	0.551012	−	0.834497i	$-0.314242\pi$
$227$	855570.	1.10202	0.551012	+	0.834497i	$0.314242\pi$
$228$	0	0
$229$	−665805.	−0.838993	−0.419497	−	0.907757i	$-0.637793\pi$
$229$	−665805.	−0.838993	−0.419497	+	0.907757i	$0.637793\pi$
$230$	−281884.	−0.351359
$231$	0	0
$232$	−1.04141e6	−1.27028
$233$	−1.20934e6	−1.45934	−0.729671	−	0.683798i	$-0.760327\pi$
$233$	−1.20934e6	−1.45934	−0.729671	+	0.683798i	$0.760327\pi$
$234$	0	0
$235$	−136192.	−0.160873
$236$	−103376.	−0.120820
$237$	0	0
$238$	−27440.0	−0.0314009
$239$	571482.	0.647154	0.323577	−	0.946202i	$-0.395114\pi$
$239$	571482.	0.647154	0.323577	+	0.946202i	$0.395114\pi$
$240$	0	0
$241$	−267080.	−0.296209	−0.148105	−	0.988972i	$-0.547317\pi$
$241$	−267080.	−0.296209	−0.148105	+	0.988972i	$0.547317\pi$
$242$	58564.0	0.0642824
$243$	0	0
$244$	271680.	0.292135
$245$	−317433.	−0.337860
$246$	0	0
$247$	440832.	0.459760
$248$	1.23706e6	1.27720
$249$	0	0
$250$	−447564.	−0.452903
$251$	−1.38737e6	−1.38998	−0.694988	−	0.719022i	$-0.744590\pi$
$251$	−1.38737e6	−1.38998	−0.694988	+	0.719022i	$0.744590\pi$
$252$	0	0
$253$	−448789.	−0.440799
$254$	−312736.	−0.304154
$255$	0	0
$256$	−1.11411e6	−1.06250
$257$	885922.	0.836686	0.418343	−	0.908289i	$-0.362611\pi$
$257$	885922.	0.836686	0.418343	+	0.908289i	$0.362611\pi$
$258$	0	0
$259$	120630.	0.111739
$260$	348992.	0.320171
$261$	0	0
$262$	1848.00	0.00166322
$263$	−1.44687e6	−1.28986	−0.644928	−	0.764243i	$-0.723113\pi$
$263$	−1.44687e6	−1.28986	−0.644928	+	0.764243i	$0.723113\pi$
$264$	0	0
$265$	567378.	0.496315
$266$	−15360.0	−0.0133103
$267$	0	0
$268$	−479984.	−0.408216
$269$	353878.	0.298176	0.149088	−	0.988824i	$-0.452366\pi$
$269$	353878.	0.298176	0.149088	+	0.988824i	$0.452366\pi$
$270$	0	0
$271$	525260.	0.434461	0.217231	−	0.976120i	$-0.430298\pi$
$271$	525260.	0.434461	0.217231	+	0.976120i	$0.430298\pi$
$272$	175616.	0.143927
$273$	0	0
$274$	−1.18493e6	−0.953492
$275$	−334444.	−0.266681
$276$	0	0
$277$	−595610.	−0.466404	−0.233202	−	0.972428i	$-0.574920\pi$
$277$	−595610.	−0.466404	−0.233202	+	0.972428i	$0.574920\pi$
$278$	−1.59927e6	−1.24111
$279$	0	0
$280$	−36480.0	−0.0278074
$281$	−732318.	−0.553266	−0.276633	−	0.960976i	$-0.589219\pi$
$281$	−732318.	−0.553266	−0.276633	+	0.960976i	$0.589219\pi$
$282$	0	0
$283$	2.23380e6	1.65798	0.828989	−	0.559264i	$-0.188916\pi$
$283$	2.23380e6	1.65798	0.828989	+	0.559264i	$0.188916\pi$
$284$	496368.	0.365181
$285$	0	0
$286$	−555632.	−0.401673
$287$	15280.0	0.0109501
$288$	0	0
$289$	−949261.	−0.668561
$290$	412224.	0.287831
$291$	0	0
$292$	−30784.0	−0.0211285
$293$	1.53108e6	1.04191	0.520953	−	0.853585i	$-0.325577\pi$
$293$	1.53108e6	1.04191	0.520953	+	0.853585i	$0.325577\pi$
$294$	0	0
$295$	122759.	0.0821293
$296$	−2.31610e6	−1.53648
$297$	0	0
$298$	−290680.	−0.189616
$299$	4.25793e6	2.75436
$300$	0	0
$301$	−40260.0	−0.0256128
$302$	−1.21233e6	−0.764897
$303$	0	0
$304$	98304.0	0.0610081
$305$	−322620.	−0.198583
$306$	0	0
$307$	−1.14268e6	−0.691956	−0.345978	−	0.938243i	$-0.612453\pi$
$307$	−1.14268e6	−0.691956	−0.345978	+	0.938243i	$0.612453\pi$
$308$	−19360.0	−0.0116286
$309$	0	0
$310$	−489668.	−0.289399
$311$	−586956.	−0.344116	−0.172058	−	0.985087i	$-0.555042\pi$
$311$	−586956.	−0.344116	−0.172058	+	0.985087i	$0.555042\pi$
$312$	0	0
$313$	−233857.	−0.134924	−0.0674621	−	0.997722i	$-0.521490\pi$
$313$	−233857.	−0.134924	−0.0674621	+	0.997722i	$0.521490\pi$
$314$	−2.13195e6	−1.22026
$315$	0	0
$316$	−1.04221e6	−0.587133
$317$	935503.	0.522874	0.261437	−	0.965221i	$-0.415804\pi$
$317$	935503.	0.522874	0.261437	+	0.965221i	$0.415804\pi$
$318$	0	0
$319$	656304.	0.361101
$320$	544768.	0.297397
$321$	0	0
$322$	−148360.	−0.0797402
$323$	263424.	0.140491
$324$	0	0
$325$	3.17307e6	1.66637
$326$	1.12830e6	0.588007
$327$	0	0
$328$	−293376.	−0.150571
$329$	−71680.0	−0.0365097
$330$	0	0
$331$	−1.05823e6	−0.530897	−0.265449	−	0.964125i	$-0.585520\pi$
$331$	−1.05823e6	−0.530897	−0.265449	+	0.964125i	$0.585520\pi$
$332$	−1.64342e6	−0.818285
$333$	0	0
$334$	2.29435e6	1.12537
$335$	569981.	0.277491
$336$	0	0
$337$	506186.	0.242793	0.121396	−	0.992604i	$-0.461263\pi$
$337$	506186.	0.242793	0.121396	+	0.992604i	$0.461263\pi$
$338$	3.78644e6	1.80277
$339$	0	0
$340$	208544.	0.0978364
$341$	−779603.	−0.363068
$342$	0	0
$343$	−335140.	−0.153812
$344$	772992.	0.352192
$345$	0	0
$346$	1.54514e6	0.693871
$347$	−467636.	−0.208490	−0.104245	−	0.994552i	$-0.533243\pi$
$347$	−467636.	−0.208490	−0.104245	+	0.994552i	$0.533243\pi$
$348$	0	0
$349$	304470.	0.133808	0.0669038	−	0.997759i	$-0.478688\pi$
$349$	304470.	0.133808	0.0669038	+	0.997759i	$0.478688\pi$
$350$	−110560.	−0.0482423
$351$	0	0
$352$	619520.	0.266501
$353$	−2.51868e6	−1.07581	−0.537906	−	0.843005i	$-0.680785\pi$
$353$	−2.51868e6	−1.07581	−0.537906	+	0.843005i	$0.680785\pi$
$354$	0	0
$355$	−589437.	−0.248237
$356$	278640.	0.116525
$357$	0	0
$358$	−2.18032e6	−0.899108
$359$	3.01841e6	1.23607	0.618034	−	0.786151i	$-0.287929\pi$
$359$	3.01841e6	1.23607	0.618034	+	0.786151i	$0.287929\pi$
$360$	0	0
$361$	−2.32864e6	−0.940448
$362$	−1.11794e6	−0.448381
$363$	0	0
$364$	183680.	0.0726622
$365$	36556.0	0.0143624
$366$	0	0
$367$	994429.	0.385397	0.192699	−	0.981258i	$-0.438276\pi$
$367$	994429.	0.385397	0.192699	+	0.981258i	$0.438276\pi$
$368$	949504.	0.365491
$369$	0	0
$370$	916788.	0.348149
$371$	298620.	0.112638
$372$	0	0
$373$	1.72896e6	0.643446	0.321723	−	0.946834i	$-0.395738\pi$
$373$	1.72896e6	0.643446	0.321723	+	0.946834i	$0.395738\pi$
$374$	−332024.	−0.122741
$375$	0	0
$376$	1.37626e6	0.502030
$377$	−6.22675e6	−2.25636
$378$	0	0
$379$	454765.	0.162626	0.0813128	−	0.996689i	$-0.474089\pi$
$379$	454765.	0.162626	0.0813128	+	0.996689i	$0.474089\pi$
$380$	116736.	0.0414711
$381$	0	0
$382$	1.77775e6	0.623321
$383$	−2.27557e6	−0.792673	−0.396336	−	0.918105i	$-0.629719\pi$
$383$	−2.27557e6	−0.792673	−0.396336	+	0.918105i	$0.629719\pi$
$384$	0	0
$385$	22990.0	0.00790473
$386$	−73904.0	−0.0252464
$387$	0	0
$388$	−1.07048e6	−0.360993
$389$	−389781.	−0.130601	−0.0653005	−	0.997866i	$-0.520801\pi$
$389$	−389781.	−0.130601	−0.0653005	+	0.997866i	$0.520801\pi$
$390$	0	0
$391$	2.54437e6	0.841665
$392$	3.20774e6	1.05435
$393$	0	0
$394$	−1.08073e6	−0.350732
$395$	1.23762e6	0.399112
$396$	0	0
$397$	−1.61933e6	−0.515655	−0.257827	−	0.966191i	$-0.583007\pi$
$397$	−1.61933e6	−0.515655	−0.257827	+	0.966191i	$0.583007\pi$
$398$	173280.	0.0548328
$399$	0	0
$400$	707584.	0.221120
$401$	5.54368e6	1.72162	0.860810	−	0.508927i	$-0.169958\pi$
$401$	5.54368e6	1.72162	0.860810	+	0.508927i	$0.169958\pi$
$402$	0	0
$403$	7.39656e6	2.26865
$404$	1.54768e6	0.471767
$405$	0	0
$406$	216960.	0.0653228
$407$	1.45962e6	0.436772
$408$	0	0
$409$	−2.70493e6	−0.799553	−0.399776	−	0.916613i	$-0.630912\pi$
$409$	−2.70493e6	−0.799553	−0.399776	+	0.916613i	$0.630912\pi$
$410$	116128.	0.0341175
$411$	0	0
$412$	1.53126e6	0.444434
$413$	64610.0	0.0186391
$414$	0	0
$415$	1.95157e6	0.556241
$416$	−5.87776e6	−1.66525
$417$	0	0
$418$	−185856.	−0.0520279
$419$	−3.37337e6	−0.938705	−0.469353	−	0.883011i	$-0.655513\pi$
$419$	−3.37337e6	−0.938705	−0.469353	+	0.883011i	$0.655513\pi$
$420$	0	0
$421$	−4.52551e6	−1.24441	−0.622204	−	0.782855i	$-0.713762\pi$
$421$	−4.52551e6	−1.24441	−0.622204	+	0.782855i	$0.713762\pi$
$422$	4.11874e6	1.12586
$423$	0	0
$424$	−5.73350e6	−1.54884
$425$	1.89610e6	0.509202
$426$	0	0
$427$	−169800.	−0.0450680
$428$	−522528.	−0.137880
$429$	0	0
$430$	−305976.	−0.0798024
$431$	684534.	0.177501	0.0887507	−	0.996054i	$-0.471713\pi$
$431$	684534.	0.177501	0.0887507	+	0.996054i	$0.471713\pi$
$432$	0	0
$433$	−4.22591e6	−1.08318	−0.541589	−	0.840643i	$-0.682177\pi$
$433$	−4.22591e6	−1.08318	−0.541589	+	0.840643i	$0.682177\pi$
$434$	−257720.	−0.0656786
$435$	0	0
$436$	2.96701e6	0.747485
$437$	1.42426e6	0.356767
$438$	0	0
$439$	−2.09185e6	−0.518047	−0.259023	−	0.965871i	$-0.583401\pi$
$439$	−2.09185e6	−0.518047	−0.259023	+	0.965871i	$0.583401\pi$
$440$	−441408.	−0.108695
$441$	0	0
$442$	3.15011e6	0.766956
$443$	−1.56284e6	−0.378361	−0.189180	−	0.981942i	$-0.560583\pi$
$443$	−1.56284e6	−0.378361	−0.189180	+	0.981942i	$0.560583\pi$
$444$	0	0
$445$	−330885.	−0.0792095
$446$	1.84512e6	0.439226
$447$	0	0
$448$	286720.	0.0674937
$449$	3.00449e6	0.703324	0.351662	−	0.936127i	$-0.385617\pi$
$449$	3.00449e6	0.703324	0.351662	+	0.936127i	$0.385617\pi$
$450$	0	0
$451$	184888.	0.0428023
$452$	1.16558e6	0.268347
$453$	0	0
$454$	3.42228e6	0.779248
$455$	−218120.	−0.0493932
$456$	0	0
$457$	−2.44552e6	−0.547747	−0.273874	−	0.961766i	$-0.588305\pi$
$457$	−2.44552e6	−0.547747	−0.273874	+	0.961766i	$0.588305\pi$
$458$	−2.66322e6	−0.593258
$459$	0	0
$460$	1.12754e6	0.248448
$461$	−7.79104e6	−1.70743	−0.853715	−	0.520741i	$-0.825656\pi$
$461$	−7.79104e6	−1.70743	−0.853715	+	0.520741i	$0.825656\pi$
$462$	0	0
$463$	−1.05196e6	−0.228059	−0.114029	−	0.993477i	$-0.536376\pi$
$463$	−1.05196e6	−0.228059	−0.114029	+	0.993477i	$0.536376\pi$
$464$	−1.38854e6	−0.299409
$465$	0	0
$466$	−4.83734e6	−1.03191
$467$	−3.97003e6	−0.842369	−0.421184	−	0.906975i	$-0.638385\pi$
$467$	−3.97003e6	−0.842369	−0.421184	+	0.906975i	$0.638385\pi$
$468$	0	0
$469$	299990.	0.0629759
$470$	−544768.	−0.113754
$471$	0	0
$472$	−1.24051e6	−0.256298
$473$	−487146.	−0.100117
$474$	0	0
$475$	1.06138e6	0.215842
$476$	109760.	0.0222038
$477$	0	0
$478$	2.28593e6	0.457607
$479$	8.53908e6	1.70048	0.850241	−	0.526393i	$-0.176456\pi$
$479$	8.53908e6	1.70048	0.850241	+	0.526393i	$0.176456\pi$
$480$	0	0
$481$	−1.38483e7	−2.72919
$482$	−1.06832e6	−0.209452
$483$	0	0
$484$	−234256.	−0.0454545
$485$	1.27120e6	0.245391
$486$	0	0
$487$	−1.86487e6	−0.356308	−0.178154	−	0.984003i	$-0.557013\pi$
$487$	−1.86487e6	−0.356308	−0.178154	+	0.984003i	$0.557013\pi$
$488$	3.26016e6	0.619711
$489$	0	0
$490$	−1.26973e6	−0.238903
$491$	−5.15727e6	−0.965420	−0.482710	−	0.875780i	$-0.660347\pi$
$491$	−5.15727e6	−0.965420	−0.482710	+	0.875780i	$0.660347\pi$
$492$	0	0
$493$	−3.72086e6	−0.689488
$494$	1.76333e6	0.325099
$495$	0	0
$496$	1.64941e6	0.301040
$497$	−310230.	−0.0563369
$498$	0	0
$499$	4.53340e6	0.815029	0.407514	−	0.913199i	$-0.366396\pi$
$499$	4.53340e6	0.815029	0.407514	+	0.913199i	$0.366396\pi$
$500$	1.79026e6	0.320251
$501$	0	0
$502$	−5.54947e6	−0.982861
$503$	−1.71163e6	−0.301641	−0.150821	−	0.988561i	$-0.548192\pi$
$503$	−1.71163e6	−0.301641	−0.150821	+	0.988561i	$0.548192\pi$
$504$	0	0
$505$	−1.83787e6	−0.320691
$506$	−1.79516e6	−0.311692
$507$	0	0
$508$	1.25094e6	0.215069
$509$	−9.73822e6	−1.66604	−0.833019	−	0.553244i	$-0.813390\pi$
$509$	−9.73822e6	−1.66604	−0.833019	+	0.553244i	$0.813390\pi$
$510$	0	0
$511$	19240.0	0.00325951
$512$	−2.88358e6	−0.486136
$513$	0	0
$514$	3.54369e6	0.591627
$515$	−1.81838e6	−0.302110
$516$	0	0
$517$	−867328.	−0.142711
$518$	482520.	0.0790116
$519$	0	0
$520$	4.18790e6	0.679185
$521$	−4.30279e6	−0.694474	−0.347237	−	0.937777i	$-0.612880\pi$
$521$	−4.30279e6	−0.694474	−0.347237	+	0.937777i	$0.612880\pi$
$522$	0	0
$523$	2.62280e6	0.419287	0.209643	−	0.977778i	$-0.432770\pi$
$523$	2.62280e6	0.419287	0.209643	+	0.977778i	$0.432770\pi$
$524$	−7392.00	−0.00117607
$525$	0	0
$526$	−5.78750e6	−0.912066
$527$	4.41990e6	0.693243
$528$	0	0
$529$	7.32034e6	1.13734
$530$	2.26951e6	0.350948
$531$	0	0
$532$	61440.0	0.00941179
$533$	−1.75414e6	−0.267453
$534$	0	0
$535$	620502.	0.0937257
$536$	−5.75981e6	−0.865956
$537$	0	0
$538$	1.41551e6	0.210842
$539$	−2.02155e6	−0.299717
$540$	0	0
$541$	−2.49634e6	−0.366700	−0.183350	−	0.983048i	$-0.558694\pi$
$541$	−2.49634e6	−0.366700	−0.183350	+	0.983048i	$0.558694\pi$
$542$	2.10104e6	0.307211
$543$	0	0
$544$	−3.51232e6	−0.508858
$545$	−3.52332e6	−0.508114
$546$	0	0
$547$	1.14323e7	1.63368	0.816838	−	0.576868i	$-0.195725\pi$
$547$	1.14323e7	1.63368	0.816838	+	0.576868i	$0.195725\pi$
$548$	4.73973e6	0.674221
$549$	0	0
$550$	−1.33778e6	−0.188572
$551$	−2.08282e6	−0.292262
$552$	0	0
$553$	651380.	0.0905778
$554$	−2.38244e6	−0.329798
$555$	0	0
$556$	6.39709e6	0.877597
$557$	9.81529e6	1.34049	0.670247	−	0.742138i	$-0.266188\pi$
$557$	9.81529e6	1.34049	0.670247	+	0.742138i	$0.266188\pi$
$558$	0	0
$559$	4.62185e6	0.625585
$560$	−48640.0	−0.00655426
$561$	0	0
$562$	−2.92927e6	−0.391218
$563$	8.19192e6	1.08922	0.544609	−	0.838690i	$-0.316678\pi$
$563$	8.19192e6	1.08922	0.544609	+	0.838690i	$0.316678\pi$
$564$	0	0
$565$	−1.38413e6	−0.182413
$566$	8.93522e6	1.17237
$567$	0	0
$568$	5.95642e6	0.774665
$569$	7.54286e6	0.976687	0.488344	−	0.872651i	$-0.337601\pi$
$569$	7.54286e6	0.976687	0.488344	+	0.872651i	$0.337601\pi$
$570$	0	0
$571$	−8.69400e6	−1.11591	−0.557956	−	0.829871i	$-0.688414\pi$
$571$	−8.69400e6	−1.11591	−0.557956	+	0.829871i	$0.688414\pi$
$572$	2.22253e6	0.284025
$573$	0	0
$574$	61120.0	0.00774290
$575$	1.02517e7	1.29308
$576$	0	0
$577$	2.03379e6	0.254312	0.127156	−	0.991883i	$-0.459415\pi$
$577$	2.03379e6	0.254312	0.127156	+	0.991883i	$0.459415\pi$
$578$	−3.79704e6	−0.472744
$579$	0	0
$580$	−1.64890e6	−0.203528
$581$	1.02714e6	0.126238
$582$	0	0
$583$	3.61330e6	0.440284
$584$	−369408.	−0.0448202
$585$	0	0
$586$	6.12432e6	0.736739
$587$	−3.51780e6	−0.421381	−0.210691	−	0.977553i	$-0.567571\pi$
$587$	−3.51780e6	−0.421381	−0.210691	+	0.977553i	$0.567571\pi$
$588$	0	0
$589$	2.47411e6	0.293854
$590$	491036.	0.0580742
$591$	0	0
$592$	−3.08813e6	−0.362152
$593$	8.34535e6	0.974558	0.487279	−	0.873246i	$-0.337989\pi$
$593$	8.34535e6	0.974558	0.487279	+	0.873246i	$0.337989\pi$
$594$	0	0
$595$	−130340.	−0.0150933
$596$	1.16272e6	0.134079
$597$	0	0
$598$	1.70317e7	1.94763
$599$	−6.15022e6	−0.700364	−0.350182	−	0.936682i	$-0.613880\pi$
$599$	−6.15022e6	−0.700364	−0.350182	+	0.936682i	$0.613880\pi$
$600$	0	0
$601$	−6.86232e6	−0.774970	−0.387485	−	0.921876i	$-0.626656\pi$
$601$	−6.86232e6	−0.774970	−0.387485	+	0.921876i	$0.626656\pi$
$602$	−161040.	−0.0181110
$603$	0	0
$604$	4.84931e6	0.540864
$605$	278179.	0.0308984
$606$	0	0
$607$	−9.45536e6	−1.04161	−0.520807	−	0.853675i	$-0.674369\pi$
$607$	−9.45536e6	−1.04161	−0.520807	+	0.853675i	$0.674369\pi$
$608$	−1.96608e6	−0.215696
$609$	0	0
$610$	−1.29048e6	−0.140419
$611$	8.22886e6	0.891737
$612$	0	0
$613$	−4.63658e6	−0.498363	−0.249182	−	0.968457i	$-0.580162\pi$
$613$	−4.63658e6	−0.498363	−0.249182	+	0.968457i	$0.580162\pi$
$614$	−4.57072e6	−0.489287
$615$	0	0
$616$	−232320.	−0.0246680
$617$	−6.05704e6	−0.640542	−0.320271	−	0.947326i	$-0.603774\pi$
$617$	−6.05704e6	−0.640542	−0.320271	+	0.947326i	$0.603774\pi$
$618$	0	0
$619$	−5.63994e6	−0.591626	−0.295813	−	0.955246i	$-0.595591\pi$
$619$	−5.63994e6	−0.591626	−0.295813	+	0.955246i	$0.595591\pi$
$620$	1.95867e6	0.204636
$621$	0	0
$622$	−2.34782e6	−0.243327
$623$	−174150.	−0.0179764
$624$	0	0
$625$	6.51157e6	0.666785
$626$	−935428.	−0.0954057
$627$	0	0
$628$	8.52779e6	0.862854
$629$	−8.27522e6	−0.833975
$630$	0	0
$631$	1.12616e6	0.112597	0.0562987	−	0.998414i	$-0.482070\pi$
$631$	1.12616e6	0.112597	0.0562987	+	0.998414i	$0.482070\pi$
$632$	−1.25065e7	−1.24550
$633$	0	0
$634$	3.74201e6	0.369728
$635$	−1.48550e6	−0.146197
$636$	0	0
$637$	1.91796e7	1.87280
$638$	2.62522e6	0.255337
$639$	0	0
$640$	−933888.	−0.0901249
$641$	1.42020e7	1.36522	0.682611	−	0.730782i	$-0.260844\pi$
$641$	1.42020e7	1.36522	0.682611	+	0.730782i	$0.260844\pi$
$642$	0	0
$643$	1.60794e6	0.153371	0.0766853	−	0.997055i	$-0.475566\pi$
$643$	1.60794e6	0.153371	0.0766853	+	0.997055i	$0.475566\pi$
$644$	593440.	0.0563848
$645$	0	0
$646$	1.05370e6	0.0993423
$647$	−3.10236e6	−0.291361	−0.145680	−	0.989332i	$-0.546537\pi$
$647$	−3.10236e6	−0.291361	−0.145680	+	0.989332i	$0.546537\pi$
$648$	0	0
$649$	781781.	0.0728573
$650$	1.26923e7	1.17830
$651$	0	0
$652$	−4.51322e6	−0.415783
$653$	−6.88852e6	−0.632183	−0.316091	−	0.948729i	$-0.602371\pi$
$653$	−6.88852e6	−0.632183	−0.316091	+	0.948729i	$0.602371\pi$
$654$	0	0
$655$	8778.00	0.000799452 0
$656$	−391168.	−0.0354898
$657$	0	0
$658$	−286720.	−0.0258163
$659$	1.24134e7	1.11347	0.556735	−	0.830690i	$-0.312054\pi$
$659$	1.24134e7	1.11347	0.556735	+	0.830690i	$0.312054\pi$
$660$	0	0
$661$	−8.10994e6	−0.721961	−0.360980	−	0.932573i	$-0.617558\pi$
$661$	−8.10994e6	−0.721961	−0.360980	+	0.932573i	$0.617558\pi$
$662$	−4.23292e6	−0.375401
$663$	0	0
$664$	−1.97211e7	−1.73584
$665$	−72960.0	−0.00639780
$666$	0	0
$667$	−2.01176e7	−1.75090
$668$	−9.17741e6	−0.795754
$669$	0	0
$670$	2.27992e6	0.196216
$671$	−2.05458e6	−0.176164
$672$	0	0
$673$	1.78063e7	1.51543	0.757717	−	0.652584i	$-0.226315\pi$
$673$	1.78063e7	1.51543	0.757717	+	0.652584i	$0.226315\pi$
$674$	2.02474e6	0.171680
$675$	0	0
$676$	−1.51458e7	−1.27475
$677$	−1.55179e7	−1.30125	−0.650626	−	0.759398i	$-0.725493\pi$
$677$	−1.55179e7	−1.30125	−0.650626	+	0.759398i	$0.725493\pi$
$678$	0	0
$679$	669050.	0.0556909
$680$	2.50253e6	0.207542
$681$	0	0
$682$	−3.11841e6	−0.256728
$683$	2.18106e6	0.178902	0.0894510	−	0.995991i	$-0.471489\pi$
$683$	2.18106e6	0.178902	0.0894510	+	0.995991i	$0.471489\pi$
$684$	0	0
$685$	−5.62843e6	−0.458311
$686$	−1.34056e6	−0.108762
$687$	0	0
$688$	1.03066e6	0.0830123
$689$	−3.42816e7	−2.75114
$690$	0	0
$691$	2.29892e7	1.83159	0.915795	−	0.401647i	$-0.131562\pi$
$691$	2.29892e7	1.83159	0.915795	+	0.401647i	$0.131562\pi$
$692$	−6.18058e6	−0.490641
$693$	0	0
$694$	−1.87054e6	−0.147424
$695$	−7.59654e6	−0.596560
$696$	0	0
$697$	−1.04821e6	−0.0817270
$698$	1.21788e6	0.0946163
$699$	0	0
$700$	442240.	0.0341125
$701$	−2.34092e6	−0.179925	−0.0899626	−	0.995945i	$-0.528675\pi$
$701$	−2.34092e6	−0.179925	−0.0899626	+	0.995945i	$0.528675\pi$
$702$	0	0
$703$	−4.63219e6	−0.353507
$704$	3.46931e6	0.263822
$705$	0	0
$706$	−1.00747e7	−0.760715
$707$	−967300.	−0.0727801
$708$	0	0
$709$	−1.92694e7	−1.43964	−0.719820	−	0.694161i	$-0.755775\pi$
$709$	−1.92694e7	−1.43964	−0.719820	+	0.694161i	$0.755775\pi$
$710$	−2.35775e6	−0.175530
$711$	0	0
$712$	3.34368e6	0.247186
$713$	2.38971e7	1.76044
$714$	0	0
$715$	−2.63925e6	−0.193070
$716$	8.72126e6	0.635765
$717$	0	0
$718$	1.20736e7	0.874032
$719$	2.14665e7	1.54860	0.774300	−	0.632819i	$-0.218102\pi$
$719$	2.14665e7	1.54860	0.774300	+	0.632819i	$0.218102\pi$
$720$	0	0
$721$	−957040.	−0.0685633
$722$	−9.31457e6	−0.664997
$723$	0	0
$724$	4.47176e6	0.317053
$725$	−1.49919e7	−1.05928
$726$	0	0
$727$	−1.67705e7	−1.17682	−0.588411	−	0.808562i	$-0.700246\pi$
$727$	−1.67705e7	−1.17682	−0.588411	+	0.808562i	$0.700246\pi$
$728$	2.20416e6	0.154140
$729$	0	0
$730$	146224.	0.0101557
$731$	2.76184e6	0.191163
$732$	0	0
$733$	−1.75373e7	−1.20560	−0.602798	−	0.797894i	$-0.705948\pi$
$733$	−1.75373e7	−1.20560	−0.602798	+	0.797894i	$0.705948\pi$
$734$	3.97772e6	0.272517
$735$	0	0
$736$	−1.89901e7	−1.29221
$737$	3.62988e6	0.246163
$738$	0	0
$739$	1.47387e7	0.992766	0.496383	−	0.868104i	$-0.334661\pi$
$739$	1.47387e7	0.992766	0.496383	+	0.868104i	$0.334661\pi$
$740$	−3.66715e6	−0.246178
$741$	0	0
$742$	1.19448e6	0.0796469
$743$	4.80946e6	0.319613	0.159806	−	0.987148i	$-0.448913\pi$
$743$	4.80946e6	0.319613	0.159806	+	0.987148i	$0.448913\pi$
$744$	0	0
$745$	−1.38073e6	−0.0911419
$746$	6.91583e6	0.454985
$747$	0	0
$748$	1.32810e6	0.0867912
$749$	326580.	0.0212709
$750$	0	0
$751$	8.29317e6	0.536563	0.268282	−	0.963341i	$-0.413544\pi$
$751$	8.29317e6	0.536563	0.268282	+	0.963341i	$0.413544\pi$
$752$	1.83501e6	0.118330
$753$	0	0
$754$	−2.49070e7	−1.59549
$755$	−5.75856e6	−0.367660
$756$	0	0
$757$	−352294.	−0.0223442	−0.0111721	−	0.999938i	$-0.503556\pi$
$757$	−352294.	−0.0223442	−0.0111721	+	0.999938i	$0.503556\pi$
$758$	1.81906e6	0.114994
$759$	0	0
$760$	1.40083e6	0.0879735
$761$	−1.68985e7	−1.05776	−0.528878	−	0.848698i	$-0.677387\pi$
$761$	−1.68985e7	−1.05776	−0.528878	+	0.848698i	$0.677387\pi$
$762$	0	0
$763$	−1.85438e6	−0.115315
$764$	−7.11099e6	−0.440755
$765$	0	0
$766$	−9.10229e6	−0.560504
$767$	−7.41723e6	−0.455253
$768$	0	0
$769$	−36652.0	−0.00223502	−0.00111751	−	0.999999i	$-0.500356\pi$
$769$	−36652.0	−0.00223502	−0.00111751	+	0.999999i	$0.500356\pi$
$770$	91960.0	0.00558949
$771$	0	0
$772$	295616.	0.0178519
$773$	3.17462e7	1.91093	0.955463	−	0.295113i	$-0.0953571\pi$
$773$	3.17462e7	1.91093	0.955463	+	0.295113i	$0.0953571\pi$
$774$	0	0
$775$	1.78085e7	1.06505
$776$	−1.28458e7	−0.765783
$777$	0	0
$778$	−1.55912e6	−0.0923489
$779$	−586752.	−0.0346426
$780$	0	0
$781$	−3.75378e6	−0.220212
$782$	1.01775e7	0.595147
$783$	0	0
$784$	4.27699e6	0.248513
$785$	−1.01268e7	−0.586538
$786$	0	0
$787$	2.01985e7	1.16247	0.581236	−	0.813735i	$-0.302569\pi$
$787$	2.01985e7	1.16247	0.581236	+	0.813735i	$0.302569\pi$
$788$	4.32291e6	0.248005
$789$	0	0
$790$	4.95049e6	0.282215
$791$	−728490.	−0.0413983
$792$	0	0
$793$	1.94930e7	1.10077
$794$	−6.47732e6	−0.364623
$795$	0	0
$796$	−693120.	−0.0387727
$797$	−1.55660e7	−0.868023	−0.434011	−	0.900907i	$-0.642902\pi$
$797$	−1.55660e7	−0.868023	−0.434011	+	0.900907i	$0.642902\pi$
$798$	0	0
$799$	4.91725e6	0.272493
$800$	−1.41517e7	−0.781777
$801$	0	0
$802$	2.21747e7	1.21737
$803$	232804.	0.0127409
$804$	0	0
$805$	−704710.	−0.0383284
$806$	2.95863e7	1.60418
$807$	0	0
$808$	1.85722e7	1.00077
$809$	2.91667e7	1.56681	0.783404	−	0.621513i	$-0.213482\pi$
$809$	2.91667e7	1.56681	0.783404	+	0.621513i	$0.213482\pi$
$810$	0	0
$811$	−1.65215e7	−0.882057	−0.441029	−	0.897493i	$-0.645386\pi$
$811$	−1.65215e7	−0.882057	−0.441029	+	0.897493i	$0.645386\pi$
$812$	−867840.	−0.0461902
$813$	0	0
$814$	5.83849e6	0.308844
$815$	5.35944e6	0.282635
$816$	0	0
$817$	1.54598e6	0.0810307
$818$	−1.08197e7	−0.565369
$819$	0	0
$820$	−464512.	−0.0241247
$821$	−5.56614e6	−0.288202	−0.144101	−	0.989563i	$-0.546029\pi$
$821$	−5.56614e6	−0.288202	−0.144101	+	0.989563i	$0.546029\pi$
$822$	0	0
$823$	1.18801e7	0.611391	0.305696	−	0.952129i	$-0.401111\pi$
$823$	1.18801e7	0.611391	0.305696	+	0.952129i	$0.401111\pi$
$824$	1.83752e7	0.942786
$825$	0	0
$826$	258440.	0.0131798
$827$	1.32856e7	0.675489	0.337745	−	0.941238i	$-0.390336\pi$
$827$	1.32856e7	0.675489	0.337745	+	0.941238i	$0.390336\pi$
$828$	0	0
$829$	−653987.	−0.0330509	−0.0165254	−	0.999863i	$-0.505260\pi$
$829$	−653987.	−0.0330509	−0.0165254	+	0.999863i	$0.505260\pi$
$830$	7.80626e6	0.393322
$831$	0	0
$832$	−3.29155e7	−1.64851
$833$	1.14610e7	0.572282
$834$	0	0
$835$	1.08982e7	0.540926
$836$	743424.	0.0367892
$837$	0	0
$838$	−1.34935e7	−0.663765
$839$	−2.47747e7	−1.21508	−0.607538	−	0.794290i	$-0.707843\pi$
$839$	−2.47747e7	−1.21508	−0.607538	+	0.794290i	$0.707843\pi$
$840$	0	0
$841$	8.90863e6	0.434331
$842$	−1.81021e7	−0.879929
$843$	0	0
$844$	−1.64749e7	−0.796100
$845$	1.79856e7	0.866530
$846$	0	0
$847$	146410.	0.00701233
$848$	−7.64467e6	−0.365064
$849$	0	0
$850$	7.58442e6	0.360060
$851$	−4.47417e7	−2.11782
$852$	0	0
$853$	2.71291e7	1.27662	0.638311	−	0.769779i	$-0.279633\pi$
$853$	2.71291e7	1.27662	0.638311	+	0.769779i	$0.279633\pi$
$854$	−679200.	−0.0318679
$855$	0	0
$856$	−6.27034e6	−0.292487
$857$	2.84232e7	1.32197	0.660984	−	0.750400i	$-0.270139\pi$
$857$	2.84232e7	1.32197	0.660984	+	0.750400i	$0.270139\pi$
$858$	0	0
$859$	2.65922e7	1.22962	0.614810	−	0.788675i	$-0.289233\pi$
$859$	2.65922e7	1.22962	0.614810	+	0.788675i	$0.289233\pi$
$860$	1.22390e6	0.0564289
$861$	0	0
$862$	2.73814e6	0.125512
$863$	2.22500e7	1.01696	0.508479	−	0.861074i	$-0.330208\pi$
$863$	2.22500e7	1.01696	0.508479	+	0.861074i	$0.330208\pi$
$864$	0	0
$865$	7.33943e6	0.333520
$866$	−1.69036e7	−0.765923
$867$	0	0
$868$	1.03088e6	0.0464418
$869$	7.88170e6	0.354055
$870$	0	0
$871$	−3.44389e7	−1.53817
$872$	3.56041e7	1.58566
$873$	0	0
$874$	5.69702e6	0.252272
$875$	−1.11891e6	−0.0494055
$876$	0	0
$877$	2.83428e7	1.24435	0.622176	−	0.782877i	$-0.286249\pi$
$877$	2.83428e7	1.24435	0.622176	+	0.782877i	$0.286249\pi$
$878$	−8.36739e6	−0.366314
$879$	0	0
$880$	−588544.	−0.0256196
$881$	−3.66445e7	−1.59063	−0.795315	−	0.606196i	$-0.792695\pi$
$881$	−3.66445e7	−1.59063	−0.795315	+	0.606196i	$0.792695\pi$
$882$	0	0
$883$	1.68772e7	0.728447	0.364223	−	0.931312i	$-0.381334\pi$
$883$	1.68772e7	0.728447	0.364223	+	0.931312i	$0.381334\pi$
$884$	−1.26004e7	−0.542320
$885$	0	0
$886$	−6.25137e6	−0.267541
$887$	−2.73941e6	−0.116909	−0.0584544	−	0.998290i	$-0.518617\pi$
$887$	−2.73941e6	−0.116909	−0.0584544	+	0.998290i	$0.518617\pi$
$888$	0	0
$889$	−781840.	−0.0331790
$890$	−1.32354e6	−0.0560095
$891$	0	0
$892$	−7.38050e6	−0.310580
$893$	2.75251e6	0.115505
$894$	0	0
$895$	−1.03565e7	−0.432171
$896$	−491520.	−0.0204537
$897$	0	0
$898$	1.20180e7	0.497325
$899$	−3.49468e7	−1.44214
$900$	0	0
$901$	−2.04853e7	−0.840681
$902$	739552.	0.0302658
$903$	0	0
$904$	1.39870e7	0.569251
$905$	−5.31022e6	−0.215522
$906$	0	0
$907$	−3.13286e7	−1.26451	−0.632255	−	0.774760i	$-0.717871\pi$
$907$	−3.13286e7	−1.26451	−0.632255	+	0.774760i	$0.717871\pi$
$908$	−1.36891e7	−0.551012
$909$	0	0
$910$	−872480.	−0.0349263
$911$	2.49762e7	0.997081	0.498541	−	0.866866i	$-0.333869\pi$
$911$	2.49762e7	0.997081	0.498541	+	0.866866i	$0.333869\pi$
$912$	0	0
$913$	1.24284e7	0.493444
$914$	−9.78206e6	−0.387316
$915$	0	0
$916$	1.06529e7	0.419497
$917$	4620.00	0.000181434 0
$918$	0	0
$919$	−1.10613e7	−0.432032	−0.216016	−	0.976390i	$-0.569306\pi$
$919$	−1.10613e7	−0.432032	−0.216016	+	0.976390i	$0.569306\pi$
$920$	1.35304e7	0.527038
$921$	0	0
$922$	−3.11641e7	−1.20734
$923$	3.56144e7	1.37601
$924$	0	0
$925$	−3.33421e7	−1.28127
$926$	−4.20784e6	−0.161262
$927$	0	0
$928$	2.77709e7	1.05857
$929$	2.01739e7	0.766919	0.383460	−	0.923558i	$-0.374733\pi$
$929$	2.01739e7	0.766919	0.383460	+	0.923558i	$0.374733\pi$
$930$	0	0
$931$	6.41549e6	0.242580
$932$	1.93494e7	0.729671
$933$	0	0
$934$	−1.58801e7	−0.595645
$935$	−1.57711e6	−0.0589976
$936$	0	0
$937$	9.10734e6	0.338877	0.169439	−	0.985541i	$-0.445805\pi$
$937$	9.10734e6	0.338877	0.169439	+	0.985541i	$0.445805\pi$
$938$	1.19996e6	0.0445307
$939$	0	0
$940$	2.17907e6	0.0804363
$941$	3.67709e7	1.35372	0.676861	−	0.736110i	$-0.263340\pi$
$941$	3.67709e7	1.35372	0.676861	+	0.736110i	$0.263340\pi$
$942$	0	0
$943$	−5.66735e6	−0.207540
$944$	−1.65402e6	−0.0604101
$945$	0	0
$946$	−1.94858e6	−0.0707932
$947$	4.95743e7	1.79631	0.898156	−	0.439677i	$-0.144907\pi$
$947$	4.95743e7	1.79631	0.898156	+	0.439677i	$0.144907\pi$
$948$	0	0
$949$	−2.20875e6	−0.0796125
$950$	4.24550e6	0.152623
$951$	0	0
$952$	1.31712e6	0.0471013
$953$	−3.53787e7	−1.26186	−0.630928	−	0.775841i	$-0.717326\pi$
$953$	−3.53787e7	−1.26186	−0.630928	+	0.775841i	$0.717326\pi$
$954$	0	0
$955$	8.44430e6	0.299609
$956$	−9.14371e6	−0.323577
$957$	0	0
$958$	3.41563e7	1.20242
$959$	−2.96233e6	−0.104013
$960$	0	0
$961$	1.28831e7	0.449999
$962$	−5.53933e7	−1.92983
$963$	0	0
$964$	4.27328e6	0.148105
$965$	−351044.	−0.0121351
$966$	0	0
$967$	2.78059e7	0.956248	0.478124	−	0.878292i	$-0.341317\pi$
$967$	2.78059e7	0.956248	0.478124	+	0.878292i	$0.341317\pi$
$968$	−2.81107e6	−0.0964237
$969$	0	0
$970$	5.08478e6	0.173517
$971$	−1.56835e7	−0.533821	−0.266910	−	0.963721i	$-0.586003\pi$
$971$	−1.56835e7	−0.533821	−0.266910	+	0.963721i	$0.586003\pi$
$972$	0	0
$973$	−3.99818e6	−0.135388
$974$	−7.45947e6	−0.251948
$975$	0	0
$976$	4.34688e6	0.146067
$977$	−2.01140e7	−0.674157	−0.337079	−	0.941476i	$-0.609439\pi$
$977$	−2.01140e7	−0.674157	−0.337079	+	0.941476i	$0.609439\pi$
$978$	0	0
$979$	−2.10722e6	−0.0702671
$980$	5.07893e6	0.168930
$981$	0	0
$982$	−2.06291e7	−0.682655
$983$	2.09269e7	0.690750	0.345375	−	0.938465i	$-0.387752\pi$
$983$	2.09269e7	0.690750	0.345375	+	0.938465i	$0.387752\pi$
$984$	0	0
$985$	−5.13346e6	−0.168585
$986$	−1.48835e7	−0.487541
$987$	0	0
$988$	−7.05331e6	−0.229880
$989$	1.49324e7	0.485445
$990$	0	0
$991$	−3.18663e7	−1.03074	−0.515368	−	0.856969i	$-0.672345\pi$
$991$	−3.18663e7	−1.03074	−0.515368	+	0.856969i	$0.672345\pi$
$992$	−3.29882e7	−1.06434
$993$	0	0
$994$	−1.24092e6	−0.0398362
$995$	823080.	0.0263563
$996$	0	0
$997$	1.38913e6	0.0442595	0.0221297	−	0.999755i	$-0.492955\pi$
$997$	1.38913e6	0.0442595	0.0221297	+	0.999755i	$0.492955\pi$
$998$	1.81336e7	0.576313
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.6.a.c.1.1		1
3.2	odd	2		11.6.a.a.1.1	✓	1
11.10	odd	2		1089.6.a.c.1.1		1
12.11	even	2		176.6.a.c.1.1		1
15.2	even	4		275.6.b.a.199.1		2
15.8	even	4		275.6.b.a.199.2		2
15.14	odd	2		275.6.a.a.1.1		1
21.20	even	2		539.6.a.c.1.1		1
24.5	odd	2		704.6.a.h.1.1		1
24.11	even	2		704.6.a.c.1.1		1
33.32	even	2		121.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.a.a.1.1	✓	1	3.2	odd	2
99.6.a.c.1.1		1	1.1	even	1	trivial
121.6.a.b.1.1		1	33.32	even	2
176.6.a.c.1.1		1	12.11	even	2
275.6.a.a.1.1		1	15.14	odd	2
275.6.b.a.199.1		2	15.2	even	4
275.6.b.a.199.2		2	15.8	even	4
539.6.a.c.1.1		1	21.20	even	2
704.6.a.c.1.1		1	24.11	even	2
704.6.a.h.1.1		1	24.5	odd	2
1089.6.a.c.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 $\F_{q}$
Belyi maps

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

Related objects

Downloads

Learn more