LMFDB - Embedded newform 121.6.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [121,6,Mod(1,121)] 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("121.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	121.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:	$19.4064421974$
Analytic rank:	$0$
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	$\chi$	$=$	121.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+4.00000 q^{2} -15.0000 q^{3} -16.0000 q^{4} -19.0000 q^{5} -60.0000 q^{6} -10.0000 q^{7} -192.000 q^{8} -18.0000 q^{9} -76.0000 q^{10} +240.000 q^{12} +1148.00 q^{13} -40.0000 q^{14} +285.000 q^{15} -256.000 q^{16} -686.000 q^{17} -72.0000 q^{18} +384.000 q^{19} +304.000 q^{20} +150.000 q^{21} +3709.00 q^{23} +2880.00 q^{24} -2764.00 q^{25} +4592.00 q^{26} +3915.00 q^{27} +160.000 q^{28} +5424.00 q^{29} +1140.00 q^{30} -6443.00 q^{31} +5120.00 q^{32} -2744.00 q^{34} +190.000 q^{35} +288.000 q^{36} +12063.0 q^{37} +1536.00 q^{38} -17220.0 q^{39} +3648.00 q^{40} +1528.00 q^{41} +600.000 q^{42} +4026.00 q^{43} +342.000 q^{45} +14836.0 q^{46} +7168.00 q^{47} +3840.00 q^{48} -16707.0 q^{49} -11056.0 q^{50} +10290.0 q^{51} -18368.0 q^{52} -29862.0 q^{53} +15660.0 q^{54} +1920.00 q^{56} -5760.00 q^{57} +21696.0 q^{58} -6461.00 q^{59} -4560.00 q^{60} +16980.0 q^{61} -25772.0 q^{62} +180.000 q^{63} +28672.0 q^{64} -21812.0 q^{65} +29999.0 q^{67} +10976.0 q^{68} -55635.0 q^{69} +760.000 q^{70} +31023.0 q^{71} +3456.00 q^{72} -1924.00 q^{73} +48252.0 q^{74} +41460.0 q^{75} -6144.00 q^{76} -68880.0 q^{78} -65138.0 q^{79} +4864.00 q^{80} -54351.0 q^{81} +6112.00 q^{82} +102714. q^{83} -2400.00 q^{84} +13034.0 q^{85} +16104.0 q^{86} -81360.0 q^{87} +17415.0 q^{89} +1368.00 q^{90} -11480.0 q^{91} -59344.0 q^{92} +96645.0 q^{93} +28672.0 q^{94} -7296.00 q^{95} -76800.0 q^{96} +66905.0 q^{97} -66828.0 q^{98} +O(q^{100})\)

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	$0.384973\pi$
$3$	−15.0000	−0.962250	−0.481125	−	0.876652i	$-0.659772\pi$
$3$	−15.0000	−0.962250	−0.481125	+	0.876652i	$0.659772\pi$
$4$	−16.0000	−0.500000
$5$	−19.0000	−0.339882	−0.169941	−	0.985454i	$-0.554358\pi$
$5$	−19.0000	−0.339882	−0.169941	+	0.985454i	$0.554358\pi$
$6$	−60.0000	−0.680414
$7$	−10.0000	−0.0771356	−0.0385678	−	0.999256i	$-0.512280\pi$
$7$	−10.0000	−0.0771356	−0.0385678	+	0.999256i	$0.512280\pi$
$8$	−192.000	−1.06066
$9$	−18.0000	−0.0740741
$10$	−76.0000	−0.240333
$11$	0	0
$11$	0	0
$12$	240.000	0.481125
$13$	1148.00	1.88401	0.942006	−	0.335597i	$-0.108938\pi$
$13$	1148.00	1.88401	0.942006	+	0.335597i	$0.108938\pi$
$14$	−40.0000	−0.0545431
$15$	285.000	0.327052
$16$	−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	$0.592942\pi$
$18$	−72.0000	−0.0523783
$19$	384.000	0.244032	0.122016	−	0.992528i	$-0.461064\pi$
$19$	384.000	0.244032	0.122016	+	0.992528i	$0.461064\pi$
$20$	304.000	0.169941
$21$	150.000	0.0742238
$22$	0	0
$23$	3709.00	1.46197	0.730983	−	0.682396i	$-0.239062\pi$
$23$	3709.00	1.46197	0.730983	+	0.682396i	$0.239062\pi$
$24$	2880.00	1.02062
$25$	−2764.00	−0.884480
$26$	4592.00	1.33220
$27$	3915.00	1.03353
$28$	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	$0.295637\pi$
$30$	1140.00	0.231261
$31$	−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
$36$	288.000	0.0370370
$37$	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
$39$	−17220.0	−1.81289
$40$	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	$0.477387\pi$
$42$	600.000	0.0524841
$43$	4026.00	0.332049	0.166025	−	0.986122i	$-0.446907\pi$
$43$	4026.00	0.332049	0.166025	+	0.986122i	$0.446907\pi$
$44$	0	0
$45$	342.000	0.0251765
$46$	14836.0	1.03377
$47$	7168.00	0.473318	0.236659	−	0.971593i	$-0.423948\pi$
$47$	7168.00	0.473318	0.236659	+	0.971593i	$0.423948\pi$
$48$	3840.00	0.240563
$49$	−16707.0	−0.994050
$50$	−11056.0	−0.625422
$51$	10290.0	0.553975
$52$	−18368.0	−0.942006
$53$	−29862.0	−1.46026	−0.730128	−	0.683310i	$-0.760540\pi$
$53$	−29862.0	−1.46026	−0.730128	+	0.683310i	$0.760540\pi$
$54$	15660.0	0.730815
$55$	0	0
$56$	1920.00	0.0818147
$57$	−5760.00	−0.234820
$58$	21696.0	0.846856
$59$	−6461.00	−0.241640	−0.120820	−	0.992674i	$-0.538552\pi$
$59$	−6461.00	−0.241640	−0.120820	+	0.992674i	$0.538552\pi$
$60$	−4560.00	−0.163526
$61$	16980.0	0.584269	0.292135	−	0.956377i	$-0.405634\pi$
$61$	16980.0	0.584269	0.292135	+	0.956377i	$0.405634\pi$
$62$	−25772.0	−0.851469
$63$	180.000	0.00571375
$64$	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
$69$	−55635.0	−1.40678
$70$	760.000	0.0185382
$71$	31023.0	0.730362	0.365181	−	0.930937i	$-0.381007\pi$
$71$	31023.0	0.730362	0.365181	+	0.930937i	$0.381007\pi$
$72$	3456.00	0.0785674
$73$	−1924.00	−0.0422569	−0.0211285	−	0.999777i	$-0.506726\pi$
$73$	−1924.00	−0.0422569	−0.0211285	+	0.999777i	$0.506726\pi$
$74$	48252.0	1.02432
$75$	41460.0	0.851091
$76$	−6144.00	−0.122016
$77$	0	0
$78$	−68880.0	−1.28191
$79$	−65138.0	−1.17427	−0.587133	−	0.809490i	$-0.699744\pi$
$79$	−65138.0	−1.17427	−0.587133	+	0.809490i	$0.699744\pi$
$80$	4864.00	0.0849706
$81$	−54351.0	−0.920439
$82$	6112.00	0.100380
$83$	102714.	1.63657	0.818285	−	0.574813i	$-0.194925\pi$
$83$	102714.	1.63657	0.818285	+	0.574813i	$0.194925\pi$
$84$	−2400.00	−0.0371119
$85$	13034.0	0.195673
$86$	16104.0	0.234794
$87$	−81360.0	−1.15243
$88$	0	0
$89$	17415.0	0.233050	0.116525	−	0.993188i	$-0.462825\pi$
$89$	17415.0	0.233050	0.116525	+	0.993188i	$0.462825\pi$
$90$	1368.00	0.0178025
$91$	−11480.0	−0.145324
$92$	−59344.0	−0.730983
$93$	96645.0	1.15870
$94$	28672.0	0.334687
$95$	−7296.00	−0.0829422
$96$	−76800.0	−0.850517
$97$	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	$0.656384\pi$
$102$	41160.0	0.391719
$103$	−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
$105$	−2850.00	−0.0252273
$106$	−119448.	−1.03256
$107$	32658.0	0.275759	0.137880	−	0.990449i	$-0.455971\pi$
$107$	32658.0	0.275759	0.137880	+	0.990449i	$0.455971\pi$
$108$	−62640.0	−0.516764
$109$	185438.	1.49497	0.747485	−	0.664279i	$-0.231261\pi$
$109$	185438.	1.49497	0.747485	+	0.664279i	$0.231261\pi$
$110$	0	0
$111$	−180945.	−1.39392
$112$	2560.00	0.0192839
$113$	72849.0	0.536695	0.268347	−	0.963322i	$-0.413522\pi$
$113$	72849.0	0.536695	0.268347	+	0.963322i	$0.413522\pi$
$114$	−23040.0	−0.166043
$115$	−70471.0	−0.496896
$116$	−86784.0	−0.598818
$117$	−20664.0	−0.139556
$118$	−25844.0	−0.170866
$119$	6860.00	0.0444075
$120$	−54720.0	−0.346891
$121$	0	0
$122$	67920.0	0.413141
$123$	−22920.0	−0.136600
$124$	103088.	0.602080
$125$	111891.	0.640501
$126$	720.000	0.00404023
$127$	78184.0	0.430139	0.215069	−	0.976599i	$-0.431002\pi$
$127$	78184.0	0.430139	0.215069	+	0.976599i	$0.431002\pi$
$128$	−49152.0	−0.265165
$129$	−60390.0	−0.319515
$130$	−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
$135$	−74385.0	−0.351278
$136$	131712.	0.610630
$137$	296233.	1.34844	0.674221	−	0.738530i	$-0.264480\pi$
$137$	296233.	1.34844	0.674221	+	0.738530i	$0.264480\pi$
$138$	−222540.	−0.994742
$139$	399818.	1.75519	0.877597	−	0.479398i	$-0.159145\pi$
$139$	399818.	1.75519	0.877597	+	0.479398i	$0.159145\pi$
$140$	−3040.00	−0.0131085
$141$	−107520.	−0.455451
$142$	124092.	0.516444
$143$	0	0
$144$	4608.00	0.0185185
$145$	−103056.	−0.407055
$146$	−7696.00	−0.0298802
$147$	250605.	0.956525
$148$	−193008.	−0.724304
$149$	−72670.0	−0.268157	−0.134079	−	0.990971i	$-0.542807\pi$
$149$	−72670.0	−0.268157	−0.134079	+	0.990971i	$0.542807\pi$
$150$	165840.	0.601812
$151$	303082.	1.08173	0.540864	−	0.841110i	$-0.318098\pi$
$151$	303082.	1.08173	0.540864	+	0.841110i	$0.318098\pi$
$152$	−73728.0	−0.258835
$153$	12348.0	0.0426450
$154$	0	0
$155$	122417.	0.409272
$156$	275520.	0.906445
$157$	−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
$159$	447930.	1.40513
$160$	−97280.0	−0.300416
$161$	−37090.0	−0.112770
$162$	−217404.	−0.650849
$163$	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	$0.207075\pi$
$168$	−28800.0	−0.0787262
$169$	946611.	2.54950
$170$	52136.0	0.138362
$171$	−6912.00	−0.0180765
$172$	−64416.0	−0.166025
$173$	386286.	0.981282	0.490641	−	0.871362i	$-0.336763\pi$
$173$	386286.	0.981282	0.490641	+	0.871362i	$0.336763\pi$
$174$	−325440.	−0.814888
$175$	27640.0	0.0682249
$176$	0	0
$177$	96915.0	0.232519
$178$	69660.0	0.164791
$179$	545079.	1.27153	0.635765	−	0.771882i	$-0.280685\pi$
$179$	545079.	1.27153	0.635765	+	0.771882i	$0.280685\pi$
$180$	−5472.00	−0.0125882
$181$	−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
$183$	−254700.	−0.562213
$184$	−712128.	−1.55065
$185$	−229197.	−0.492356
$186$	386580.	0.819327
$187$	0	0
$188$	−114688.	−0.236659
$189$	−39150.0	−0.0797218
$190$	−29184.0	−0.0586490
$191$	−444437.	−0.881509	−0.440755	−	0.897628i	$-0.645289\pi$
$191$	−444437.	−0.881509	−0.440755	+	0.897628i	$0.645289\pi$
$192$	−430080.	−0.841969
$193$	18476.0	0.0357038	0.0178519	−	0.999841i	$-0.494317\pi$
$193$	18476.0	0.0357038	0.0178519	+	0.999841i	$0.494317\pi$
$194$	267620.	0.510522
$195$	327180.	0.616170
$196$	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
$201$	−449985.	−0.785612
$202$	−386920.	−0.667180
$203$	−54240.0	−0.0923803
$204$	−164640.	−0.276987
$205$	−29032.0	−0.0482494
$206$	−382816.	−0.628524
$207$	−66762.0	−0.108294
$208$	−293888.	−0.471003
$209$	0	0
$210$	−11400.0	−0.0178384
$211$	−1.02968e6	−1.59220	−0.796100	−	0.605165i	$-0.793107\pi$
$211$	−1.02968e6	−1.59220	−0.796100	+	0.605165i	$0.793107\pi$
$212$	477792.	0.730128
$213$	−465345.	−0.702791
$214$	130632.	0.194991
$215$	−76494.0	−0.112858
$216$	−751680.	−1.09622
$217$	64430.0	0.0928835
$218$	741752.	1.05710
$219$	28860.0	0.0406617
$220$	0	0
$221$	−787528.	−1.08464
$222$	−723780.	−0.985653
$223$	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
$225$	49752.0	0.0655170
$226$	291396.	0.379501
$227$	855570.	1.10202	0.551012	−	0.834497i	$-0.314242\pi$
$227$	855570.	1.10202	0.551012	+	0.834497i	$0.314242\pi$
$228$	92160.0	0.117410
$229$	−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	$0.760327\pi$
$234$	−82656.0	−0.0986813
$235$	−136192.	−0.160873
$236$	103376.	0.120820
$237$	977070.	1.12994
$238$	27440.0	0.0314009
$239$	571482.	0.647154	0.323577	−	0.946202i	$-0.395114\pi$
$239$	571482.	0.647154	0.323577	+	0.946202i	$0.395114\pi$
$240$	−72960.0	−0.0817630
$241$	267080.	0.296209	0.148105	−	0.988972i	$-0.452683\pi$
$241$	267080.	0.296209	0.148105	+	0.988972i	$0.452683\pi$
$242$	0	0
$243$	−136080.	−0.147835
$244$	−271680.	−0.292135
$245$	317433.	0.337860
$246$	−91680.0	−0.0965910
$247$	440832.	0.459760
$248$	1.23706e6	1.27720
$249$	−1.54071e6	−1.57479
$250$	447564.	0.452903
$251$	1.38737e6	1.38998	0.694988	−	0.719022i	$-0.255410\pi$
$251$	1.38737e6	1.38998	0.694988	+	0.719022i	$0.255410\pi$
$252$	−2880.00	−0.00285687
$253$	0	0
$254$	312736.	0.304154
$255$	−195510.	−0.188286
$256$	−1.11411e6	−1.06250
$257$	−885922.	−0.836686	−0.418343	−	0.908289i	$-0.637389\pi$
$257$	−885922.	−0.836686	−0.418343	+	0.908289i	$0.637389\pi$
$258$	−241560.	−0.225931
$259$	−120630.	−0.111739
$260$	348992.	0.320171
$261$	−97632.0	−0.0887137
$262$	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
$267$	−261225.	−0.224252
$268$	−479984.	−0.408216
$269$	−353878.	−0.298176	−0.149088	−	0.988824i	$-0.547634\pi$
$269$	−353878.	−0.298176	−0.149088	+	0.988824i	$0.547634\pi$
$270$	−297540.	−0.248391
$271$	−525260.	−0.434461	−0.217231	−	0.976120i	$-0.569702\pi$
$271$	−525260.	−0.434461	−0.217231	+	0.976120i	$0.569702\pi$
$272$	175616.	0.143927
$273$	172200.	0.139838
$274$	1.18493e6	0.953492
$275$	0	0
$276$	890160.	0.703389
$277$	595610.	0.466404	0.233202	−	0.972428i	$-0.425080\pi$
$277$	595610.	0.466404	0.233202	+	0.972428i	$0.425080\pi$
$278$	1.59927e6	1.24111
$279$	115974.	0.0891970
$280$	−36480.0	−0.0278074
$281$	−732318.	−0.553266	−0.276633	−	0.960976i	$-0.589219\pi$
$281$	−732318.	−0.553266	−0.276633	+	0.960976i	$0.589219\pi$
$282$	−430080.	−0.322052
$283$	−2.23380e6	−1.65798	−0.828989	−	0.559264i	$-0.811084\pi$
$283$	−2.23380e6	−1.65798	−0.828989	+	0.559264i	$0.811084\pi$
$284$	−496368.	−0.365181
$285$	109440.	0.0798112
$286$	0	0
$287$	−15280.0	−0.0109501
$288$	−92160.0	−0.0654729
$289$	−949261.	−0.668561
$290$	−412224.	−0.287831
$291$	−1.00358e6	−0.694732
$292$	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	$0.325577\pi$
$294$	1.00242e6	0.676365
$295$	122759.	0.0821293
$296$	−2.31610e6	−1.53648
$297$	0	0
$298$	−290680.	−0.189616
$299$	4.25793e6	2.75436
$300$	−663360.	−0.425546
$301$	−40260.0	−0.0256128
$302$	1.21233e6	0.764897
$303$	1.45095e6	0.907916
$304$	−98304.0	−0.0610081
$305$	−322620.	−0.198583
$306$	49392.0	0.0301546
$307$	1.14268e6	0.691956	0.345978	−	0.938243i	$-0.387547\pi$
$307$	1.14268e6	0.691956	0.345978	+	0.938243i	$0.387547\pi$
$308$	0	0
$309$	1.43556e6	0.855313
$310$	489668.	0.289399
$311$	586956.	0.344116	0.172058	−	0.985087i	$-0.444958\pi$
$311$	586956.	0.344116	0.172058	+	0.985087i	$0.444958\pi$
$312$	3.30624e6	1.92286
$313$	−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
$315$	−3420.00	−0.00194200
$316$	1.04221e6	0.587133
$317$	−935503.	−0.522874	−0.261437	−	0.965221i	$-0.584196\pi$
$317$	−935503.	−0.522874	−0.261437	+	0.965221i	$0.584196\pi$
$318$	1.79172e6	0.993579
$319$	0	0
$320$	−544768.	−0.297397
$321$	−489870.	−0.265349
$322$	−148360.	−0.0797402
$323$	−263424.	−0.140491
$324$	869616.	0.460219
$325$	−3.17307e6	−1.66637
$326$	1.12830e6	0.588007
$327$	−2.78157e6	−1.43854
$328$	−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
$333$	−217134.	−0.107304
$334$	2.29435e6	1.12537
$335$	−569981.	−0.277491
$336$	−38400.0	−0.0185559
$337$	−506186.	−0.242793	−0.121396	−	0.992604i	$-0.538737\pi$
$337$	−506186.	−0.242793	−0.121396	+	0.992604i	$0.538737\pi$
$338$	3.78644e6	1.80277
$339$	−1.09274e6	−0.516435
$340$	−208544.	−0.0978364
$341$	0	0
$342$	−27648.0	−0.0127820
$343$	335140.	0.153812
$344$	−772992.	−0.352192
$345$	1.05706e6	0.478139
$346$	1.54514e6	0.693871
$347$	−467636.	−0.208490	−0.104245	−	0.994552i	$-0.533243\pi$
$347$	−467636.	−0.208490	−0.104245	+	0.994552i	$0.533243\pi$
$348$	1.30176e6	0.576213
$349$	−304470.	−0.133808	−0.0669038	−	0.997759i	$-0.521312\pi$
$349$	−304470.	−0.133808	−0.0669038	+	0.997759i	$0.521312\pi$
$350$	110560.	0.0482423
$351$	4.49442e6	1.94718
$352$	0	0
$353$	2.51868e6	1.07581	0.537906	−	0.843005i	$-0.319215\pi$
$353$	2.51868e6	1.07581	0.537906	+	0.843005i	$0.319215\pi$
$354$	387660.	0.164416
$355$	−589437.	−0.248237
$356$	−278640.	−0.116525
$357$	−102900.	−0.0427312
$358$	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	$0.287929\pi$
$360$	−65664.0	−0.0267037
$361$	−2.32864e6	−0.940448
$362$	−1.11794e6	−0.448381
$363$	0	0
$364$	183680.	0.0726622
$365$	36556.0	0.0143624
$366$	−1.01880e6	−0.397545
$367$	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
$369$	−27504.0	−0.0105155
$370$	−916788.	−0.348149
$371$	298620.	0.112638
$372$	−1.54632e6	−0.579351
$373$	−1.72896e6	−0.643446	−0.321723	−	0.946834i	$-0.604262\pi$
$373$	−1.72896e6	−0.643446	−0.321723	+	0.946834i	$0.604262\pi$
$374$	0	0
$375$	−1.67836e6	−0.616323
$376$	−1.37626e6	−0.502030
$377$	6.22675e6	2.25636
$378$	−156600.	−0.0563718
$379$	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
$381$	−1.17276e6	−0.413901
$382$	−1.77775e6	−0.623321
$383$	2.27557e6	0.792673	0.396336	−	0.918105i	$-0.370281\pi$
$383$	2.27557e6	0.792673	0.396336	+	0.918105i	$0.370281\pi$
$384$	737280.	0.255155
$385$	0	0
$386$	73904.0	0.0252464
$387$	−72468.0	−0.0245962
$388$	−1.07048e6	−0.360993
$389$	389781.	0.130601	0.0653005	−	0.997866i	$-0.479199\pi$
$389$	389781.	0.130601	0.0653005	+	0.997866i	$0.479199\pi$
$390$	1.30872e6	0.435698
$391$	−2.54437e6	−0.841665
$392$	3.20774e6	1.05435
$393$	−6930.00	−0.00226335
$394$	−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
$399$	57600.0	0.0181130
$400$	707584.	0.221120
$401$	−5.54368e6	−1.72162	−0.860810	−	0.508927i	$-0.830042\pi$
$401$	−5.54368e6	−1.72162	−0.860810	+	0.508927i	$0.830042\pi$
$402$	−1.79994e6	−0.555511
$403$	−7.39656e6	−2.26865
$404$	1.54768e6	0.471767
$405$	1.03267e6	0.312841
$406$	−216960.	−0.0653228
$407$	0	0
$408$	−1.97568e6	−0.587579
$409$	2.70493e6	0.799553	0.399776	−	0.916613i	$-0.369088\pi$
$409$	2.70493e6	0.799553	0.399776	+	0.916613i	$0.369088\pi$
$410$	−116128.	−0.0341175
$411$	−4.44350e6	−1.29754
$412$	1.53126e6	0.444434
$413$	64610.0	0.0186391
$414$	−267048.	−0.0765753
$415$	−1.95157e6	−0.556241
$416$	5.87776e6	1.66525
$417$	−5.99727e6	−1.68894
$418$	0	0
$419$	3.37337e6	0.938705	0.469353	−	0.883011i	$-0.344487\pi$
$419$	3.37337e6	0.938705	0.469353	+	0.883011i	$0.344487\pi$
$420$	45600.0	0.0126137
$421$	−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
$423$	−129024.	−0.0350606
$424$	5.73350e6	1.54884
$425$	1.89610e6	0.509202
$426$	−1.86138e6	−0.496948
$427$	−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	$0.471713\pi$
$432$	−1.00224e6	−0.258382
$433$	−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
$435$	1.54584e6	0.391689
$436$	−2.96701e6	−0.747485
$437$	1.42426e6	0.356767
$438$	115440.	0.0287522
$439$	2.09185e6	0.518047	0.259023	−	0.965871i	$-0.416599\pi$
$439$	2.09185e6	0.518047	0.259023	+	0.965871i	$0.416599\pi$
$440$	0	0
$441$	300726.	0.0736333
$442$	−3.15011e6	−0.766956
$443$	1.56284e6	0.378361	0.189180	−	0.981942i	$-0.439417\pi$
$443$	1.56284e6	0.378361	0.189180	+	0.981942i	$0.439417\pi$
$444$	2.89512e6	0.696962
$445$	−330885.	−0.0792095
$446$	1.84512e6	0.439226
$447$	1.09005e6	0.258034
$448$	−286720.	−0.0674937
$449$	−3.00449e6	−0.703324	−0.351662	−	0.936127i	$-0.614383\pi$
$449$	−3.00449e6	−0.703324	−0.351662	+	0.936127i	$0.614383\pi$
$450$	199008.	0.0463275
$451$	0	0
$452$	−1.16558e6	−0.268347
$453$	−4.54623e6	−1.04089
$454$	3.42228e6	0.779248
$455$	218120.	0.0493932
$456$	1.10592e6	0.249064
$457$	2.44552e6	0.547747	0.273874	−	0.961766i	$-0.411695\pi$
$457$	2.44552e6	0.547747	0.273874	+	0.961766i	$0.411695\pi$
$458$	−2.66322e6	−0.593258
$459$	−2.68569e6	−0.595010
$460$	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
$465$	−1.83626e6	−0.393823
$466$	−4.83734e6	−1.03191
$467$	3.97003e6	0.842369	0.421184	−	0.906975i	$-0.361615\pi$
$467$	3.97003e6	0.842369	0.421184	+	0.906975i	$0.361615\pi$
$468$	330624.	0.0697782
$469$	−299990.	−0.0629759
$470$	−544768.	−0.113754
$471$	7.99480e6	1.66056
$472$	1.24051e6	0.256298
$473$	0	0
$474$	3.90828e6	0.798987
$475$	−1.06138e6	−0.215842
$476$	−109760.	−0.0222038
$477$	537516.	0.108167
$478$	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	$0.176456\pi$
$480$	1.45920e6	0.289076
$481$	1.38483e7	2.72919
$482$	1.06832e6	0.209452
$483$	556350.	0.108513
$484$	0	0
$485$	−1.27120e6	−0.245391
$486$	−544320.	−0.104535
$487$	−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
$489$	−4.23114e6	−0.800175
$490$	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	$0.660347\pi$
$492$	366720.	0.0683002
$493$	−3.72086e6	−0.689488
$494$	1.76333e6	0.325099
$495$	0	0
$496$	1.64941e6	0.301040
$497$	−310230.	−0.0563369
$498$	−6.16284e6	−1.11354
$499$	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
$501$	−8.60382e6	−1.53143
$502$	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	$0.548192\pi$
$504$	−34560.0	−0.00606035
$505$	1.83787e6	0.320691
$506$	0	0
$507$	−1.41992e7	−2.45326
$508$	−1.25094e6	−0.215069
$509$	9.73822e6	1.66604	0.833019	−	0.553244i	$-0.186610\pi$
$509$	9.73822e6	1.66604	0.833019	+	0.553244i	$0.186610\pi$
$510$	−782040.	−0.133138
$511$	19240.0	0.00325951
$512$	−2.88358e6	−0.486136
$513$	1.50336e6	0.252214
$514$	−3.54369e6	−0.591627
$515$	1.81838e6	0.302110
$516$	966240.	0.159757
$517$	0	0
$518$	−482520.	−0.0790116
$519$	−5.79429e6	−0.944239
$520$	4.18790e6	0.679185
$521$	4.30279e6	0.694474	0.347237	−	0.937777i	$-0.387120\pi$
$521$	4.30279e6	0.694474	0.347237	+	0.937777i	$0.387120\pi$
$522$	−390528.	−0.0627301
$523$	−2.62280e6	−0.419287	−0.209643	−	0.977778i	$-0.567230\pi$
$523$	−2.62280e6	−0.419287	−0.209643	+	0.977778i	$0.567230\pi$
$524$	−7392.00	−0.00117607
$525$	−414600.	−0.0656494
$526$	−5.78750e6	−0.912066
$527$	4.41990e6	0.693243
$528$	0	0
$529$	7.32034e6	1.13734
$530$	2.26951e6	0.350948
$531$	116298.	0.0178993
$532$	61440.0	0.00941179
$533$	1.75414e6	0.267453
$534$	−1.04490e6	−0.158570
$535$	−620502.	−0.0937257
$536$	−5.75981e6	−0.865956
$537$	−8.17618e6	−1.22353
$538$	−1.41551e6	−0.210842
$539$	0	0
$540$	1.19016e6	0.175639
$541$	2.49634e6	0.366700	0.183350	−	0.983048i	$-0.441306\pi$
$541$	2.49634e6	0.366700	0.183350	+	0.983048i	$0.441306\pi$
$542$	−2.10104e6	−0.307211
$543$	4.19228e6	0.610169
$544$	−3.51232e6	−0.508858
$545$	−3.52332e6	−0.508114
$546$	688800.	0.0988807
$547$	−1.14323e7	−1.63368	−0.816838	−	0.576868i	$-0.804275\pi$
$547$	−1.14323e7	−1.63368	−0.816838	+	0.576868i	$0.804275\pi$
$548$	−4.73973e6	−0.674221
$549$	−305640.	−0.0432792
$550$	0	0
$551$	2.08282e6	0.292262
$552$	1.06819e7	1.49211
$553$	651380.	0.0905778
$554$	2.38244e6	0.329798
$555$	3.43796e6	0.473770
$556$	−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	$0.266188\pi$
$558$	463896.	0.0630718
$559$	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	$0.316678\pi$
$564$	1.72032e6	0.227725
$565$	−1.38413e6	−0.182413
$566$	−8.93522e6	−1.17237
$567$	543510.	0.0709986
$568$	−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	$0.337601\pi$
$570$	437760.	0.0564351
$571$	8.69400e6	1.11591	0.557956	−	0.829871i	$-0.311586\pi$
$571$	8.69400e6	1.11591	0.557956	+	0.829871i	$0.311586\pi$
$572$	0	0
$573$	6.66656e6	0.848233
$574$	−61120.0	−0.00774290
$575$	−1.02517e7	−1.29308
$576$	−516096.	−0.0648148
$577$	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
$579$	−277140.	−0.0343560
$580$	1.64890e6	0.203528
$581$	−1.02714e6	−0.126238
$582$	−4.01430e6	−0.491250
$583$	0	0
$584$	369408.	0.0448202
$585$	392616.	0.0474328
$586$	6.12432e6	0.736739
$587$	3.51780e6	0.421381	0.210691	−	0.977553i	$-0.432429\pi$
$587$	3.51780e6	0.421381	0.210691	+	0.977553i	$0.432429\pi$
$588$	−4.00968e6	−0.478263
$589$	−2.47411e6	−0.293854
$590$	491036.	0.0580742
$591$	4.05273e6	0.477286
$592$	−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
$597$	−649800.	−0.0746180
$598$	1.70317e7	1.94763
$599$	6.15022e6	0.700364	0.350182	−	0.936682i	$-0.386120\pi$
$599$	6.15022e6	0.700364	0.350182	+	0.936682i	$0.386120\pi$
$600$	−7.96032e6	−0.902719
$601$	6.86232e6	0.774970	0.387485	−	0.921876i	$-0.373344\pi$
$601$	6.86232e6	0.774970	0.387485	+	0.921876i	$0.373344\pi$
$602$	−161040.	−0.0181110
$603$	−539982.	−0.0604764
$604$	−4.84931e6	−0.540864
$605$	0	0
$606$	5.80380e6	0.641994
$607$	9.45536e6	1.04161	0.520807	−	0.853675i	$-0.325631\pi$
$607$	9.45536e6	1.04161	0.520807	+	0.853675i	$0.325631\pi$
$608$	1.96608e6	0.215696
$609$	813600.	0.0888930
$610$	−1.29048e6	−0.140419
$611$	8.22886e6	0.891737
$612$	−197568.	−0.0213225
$613$	4.63658e6	0.498363	0.249182	−	0.968457i	$-0.419838\pi$
$613$	4.63658e6	0.498363	0.249182	+	0.968457i	$0.419838\pi$
$614$	4.57072e6	0.489287
$615$	435480.	0.0464280
$616$	0	0
$617$	6.05704e6	0.640542	0.320271	−	0.947326i	$-0.396226\pi$
$617$	6.05704e6	0.640542	0.320271	+	0.947326i	$0.396226\pi$
$618$	5.74224e6	0.604798
$619$	−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
$621$	1.45207e7	1.51098
$622$	2.34782e6	0.243327
$623$	−174150.	−0.0179764
$624$	4.40832e6	0.453223
$625$	6.51157e6	0.666785
$626$	−935428.	−0.0954057
$627$	0	0
$628$	8.52779e6	0.862854
$629$	−8.27522e6	−0.833975
$630$	−13680.0	−0.00137320
$631$	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
$633$	1.54453e7	1.53210
$634$	−3.74201e6	−0.369728
$635$	−1.48550e6	−0.146197
$636$	−7.16688e6	−0.702566
$637$	−1.91796e7	−1.87280
$638$	0	0
$639$	−558414.	−0.0541009
$640$	933888.	0.0901249
$641$	−1.42020e7	−1.36522	−0.682611	−	0.730782i	$-0.739156\pi$
$641$	−1.42020e7	−1.36522	−0.682611	+	0.730782i	$0.739156\pi$
$642$	−1.95948e6	−0.187630
$643$	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
$645$	1.14741e6	0.108597
$646$	−1.05370e6	−0.0993423
$647$	3.10236e6	0.291361	0.145680	−	0.989332i	$-0.453463\pi$
$647$	3.10236e6	0.291361	0.145680	+	0.989332i	$0.453463\pi$
$648$	1.04354e7	0.976273
$649$	0	0
$650$	−1.26923e7	−1.17830
$651$	−966450.	−0.0893772
$652$	−4.51322e6	−0.415783
$653$	6.88852e6	0.632183	0.316091	−	0.948729i	$-0.397629\pi$
$653$	6.88852e6	0.632183	0.316091	+	0.948729i	$0.397629\pi$
$654$	−1.11263e7	−1.01720
$655$	−8778.00	−0.000799452 0
$656$	−391168.	−0.0354898
$657$	34632.0	0.00313014
$658$	−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
$663$	1.18129e7	1.04369
$664$	−1.97211e7	−1.73584
$665$	72960.0	0.00639780
$666$	−868536.	−0.0758756
$667$	2.01176e7	1.75090
$668$	−9.17741e6	−0.795754
$669$	−6.91922e6	−0.597711
$670$	−2.27992e6	−0.196216
$671$	0	0
$672$	768000.	0.0656052
$673$	−1.78063e7	−1.51543	−0.757717	−	0.652584i	$-0.773685\pi$
$673$	−1.78063e7	−1.51543	−0.757717	+	0.652584i	$0.773685\pi$
$674$	−2.02474e6	−0.171680
$675$	−1.08211e7	−0.914135
$676$	−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	$0.725493\pi$
$678$	−4.37094e6	−0.365175
$679$	−669050.	−0.0556909
$680$	−2.50253e6	−0.207542
$681$	−1.28336e7	−1.06042
$682$	0	0
$683$	−2.18106e6	−0.178902	−0.0894510	−	0.995991i	$-0.528511\pi$
$683$	−2.18106e6	−0.178902	−0.0894510	+	0.995991i	$0.528511\pi$
$684$	110592.	0.00903823
$685$	−5.62843e6	−0.458311
$686$	1.34056e6	0.108762
$687$	9.98708e6	0.807321
$688$	−1.03066e6	−0.0830123
$689$	−3.42816e7	−2.75114
$690$	4.22826e6	0.338095
$691$	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
$696$	1.56211e7	1.22233
$697$	−1.04821e6	−0.0817270
$698$	−1.21788e6	−0.0946163
$699$	1.81400e7	1.40425
$700$	−442240.	−0.0341125
$701$	−2.34092e6	−0.179925	−0.0899626	−	0.995945i	$-0.528675\pi$
$701$	−2.34092e6	−0.179925	−0.0899626	+	0.995945i	$0.528675\pi$
$702$	1.79777e7	1.37686
$703$	4.63219e6	0.353507
$704$	0	0
$705$	2.04288e6	0.154800
$706$	1.00747e7	0.760715
$707$	967300.	0.0727801
$708$	−1.55064e6	−0.116259
$709$	−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
$711$	1.17248e6	0.0869827
$712$	−3.34368e6	−0.247186
$713$	−2.38971e7	−1.76044
$714$	−411600.	−0.0302155
$715$	0	0
$716$	−8.72126e6	−0.635765
$717$	−8.57223e6	−0.622724
$718$	1.20736e7	0.874032
$719$	−2.14665e7	−1.54860	−0.774300	−	0.632819i	$-0.781898\pi$
$719$	−2.14665e7	−1.54860	−0.774300	+	0.632819i	$0.781898\pi$
$720$	−87552.0	−0.00629412
$721$	957040.	0.0685633
$722$	−9.31457e6	−0.664997
$723$	−4.00620e6	−0.285028
$724$	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
$729$	1.52485e7	1.06269
$730$	146224.	0.0101557
$731$	−2.76184e6	−0.191163
$732$	4.07520e6	0.281107
$733$	1.75373e7	1.20560	0.602798	−	0.797894i	$-0.294052\pi$
$733$	1.75373e7	1.20560	0.602798	+	0.797894i	$0.294052\pi$
$734$	3.97772e6	0.272517
$735$	−4.76150e6	−0.325106
$736$	1.89901e7	1.29221
$737$	0	0
$738$	−110016.	−0.00743558
$739$	−1.47387e7	−0.992766	−0.496383	−	0.868104i	$-0.665339\pi$
$739$	−1.47387e7	−0.992766	−0.496383	+	0.868104i	$0.665339\pi$
$740$	3.66715e6	0.246178
$741$	−6.61248e6	−0.442404
$742$	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	$0.448913\pi$
$744$	−1.85558e7	−1.22899
$745$	1.38073e6	0.0911419
$746$	−6.91583e6	−0.454985
$747$	−1.84885e6	−0.121227
$748$	0	0
$749$	−326580.	−0.0212709
$750$	−6.71346e6	−0.435806
$751$	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
$753$	−2.08105e7	−1.33750
$754$	2.49070e7	1.59549
$755$	−5.75856e6	−0.367660
$756$	626400.	0.0398609
$757$	−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	$0.677387\pi$
$762$	−4.69104e6	−0.292672
$763$	−1.85438e6	−0.115315
$764$	7.11099e6	0.440755
$765$	−234612.	−0.0144943
$766$	9.10229e6	0.560504
$767$	−7.41723e6	−0.455253
$768$	1.67117e7	1.02239
$769$	36652.0	0.00223502	0.00111751	−	0.999999i	$-0.499644\pi$
$769$	36652.0	0.00223502	0.00111751	+	0.999999i	$0.499644\pi$
$770$	0	0
$771$	1.32888e7	0.805102
$772$	−295616.	−0.0178519
$773$	−3.17462e7	−1.91093	−0.955463	−	0.295113i	$-0.904643\pi$
$773$	−3.17462e7	−1.91093	−0.955463	+	0.295113i	$0.904643\pi$
$774$	−289872.	−0.0173922
$775$	1.78085e7	1.06505
$776$	−1.28458e7	−0.765783
$777$	1.80945e6	0.107521
$778$	1.55912e6	0.0923489
$779$	586752.	0.0346426
$780$	−5.23488e6	−0.308085
$781$	0	0
$782$	−1.01775e7	−0.595147
$783$	2.12350e7	1.23779
$784$	4.27699e6	0.248513
$785$	1.01268e7	0.586538
$786$	−27720.0	−0.00160043
$787$	−2.01985e7	−1.16247	−0.581236	−	0.813735i	$-0.697431\pi$
$787$	−2.01985e7	−1.16247	−0.581236	+	0.813735i	$0.697431\pi$
$788$	4.32291e6	0.248005
$789$	2.17031e7	1.24116
$790$	4.95049e6	0.282215
$791$	−728490.	−0.0413983
$792$	0	0
$793$	1.94930e7	1.10077
$794$	−6.47732e6	−0.364623
$795$	−8.51067e6	−0.477580
$796$	−693120.	−0.0387727
$797$	1.55660e7	0.868023	0.434011	−	0.900907i	$-0.357098\pi$
$797$	1.55660e7	0.868023	0.434011	+	0.900907i	$0.357098\pi$
$798$	230400.	0.0128078
$799$	−4.91725e6	−0.272493
$800$	−1.41517e7	−0.781777
$801$	−313470.	−0.0172629
$802$	−2.21747e7	−1.21737
$803$	0	0
$804$	7.19976e6	0.392806
$805$	704710.	0.0383284
$806$	−2.95863e7	−1.60418
$807$	5.30817e6	0.286920
$808$	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	$0.213482\pi$
$810$	4.13068e6	0.221212
$811$	1.65215e7	0.882057	0.441029	−	0.897493i	$-0.354614\pi$
$811$	1.65215e7	0.882057	0.441029	+	0.897493i	$0.354614\pi$
$812$	867840.	0.0461902
$813$	7.87890e6	0.418061
$814$	0	0
$815$	−5.35944e6	−0.282635
$816$	−2.63424e6	−0.138494
$817$	1.54598e6	0.0810307
$818$	1.08197e7	0.565369
$819$	206640.	0.0107648
$820$	464512.	0.0241247
$821$	−5.56614e6	−0.288202	−0.144101	−	0.989563i	$-0.546029\pi$
$821$	−5.56614e6	−0.288202	−0.144101	+	0.989563i	$0.546029\pi$
$822$	−1.77740e7	−0.917498
$823$	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	$0.390336\pi$
$828$	1.06819e6	0.0541469
$829$	−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
$831$	−8.93415e6	−0.448798
$832$	3.29155e7	1.64851
$833$	1.14610e7	0.572282
$834$	−2.39891e7	−1.19426
$835$	−1.08982e7	−0.540926
$836$	0	0
$837$	−2.52243e7	−1.24453
$838$	1.34935e7	0.663765
$839$	2.47747e7	1.21508	0.607538	−	0.794290i	$-0.292157\pi$
$839$	2.47747e7	1.21508	0.607538	+	0.794290i	$0.292157\pi$
$840$	547200.	0.0267576
$841$	8.90863e6	0.434331
$842$	−1.81021e7	−0.879929
$843$	1.09848e7	0.532380
$844$	1.64749e7	0.796100
$845$	−1.79856e7	−0.866530
$846$	−516096.	−0.0247916
$847$	0	0
$848$	7.64467e6	0.365064
$849$	3.35071e7	1.59539
$850$	7.58442e6	0.360060
$851$	4.47417e7	2.11782
$852$	7.44552e6	0.351395
$853$	−2.71291e7	−1.27662	−0.638311	−	0.769779i	$-0.720367\pi$
$853$	−2.71291e7	−1.27662	−0.638311	+	0.769779i	$0.720367\pi$
$854$	−679200.	−0.0318679
$855$	131328.	0.00614387
$856$	−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
$861$	229200.	0.0105368
$862$	2.73814e6	0.125512
$863$	−2.22500e7	−1.01696	−0.508479	−	0.861074i	$-0.669792\pi$
$863$	−2.22500e7	−1.01696	−0.508479	+	0.861074i	$0.669792\pi$
$864$	2.00448e7	0.913519
$865$	−7.33943e6	−0.333520
$866$	−1.69036e7	−0.765923
$867$	1.42389e7	0.643323
$868$	−1.03088e6	−0.0464418
$869$	0	0
$870$	6.18336e6	0.276966
$871$	3.44389e7	1.53817
$872$	−3.56041e7	−1.58566
$873$	−1.20429e6	−0.0534805
$874$	5.69702e6	0.252272
$875$	−1.11891e6	−0.0494055
$876$	−461760.	−0.0203309
$877$	−2.83428e7	−1.24435	−0.622176	−	0.782877i	$-0.713751\pi$
$877$	−2.83428e7	−1.24435	−0.622176	+	0.782877i	$0.713751\pi$
$878$	8.36739e6	0.366314
$879$	−2.29662e7	−1.00258
$880$	0	0
$881$	3.66445e7	1.59063	0.795315	−	0.606196i	$-0.207305\pi$
$881$	3.66445e7	1.59063	0.795315	+	0.606196i	$0.207305\pi$
$882$	1.20290e6	0.0520666
$883$	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
$885$	−1.84138e6	−0.0790290
$886$	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	$0.518617\pi$
$888$	3.47414e7	1.47848
$889$	−781840.	−0.0331790
$890$	−1.32354e6	−0.0560095
$891$	0	0
$892$	−7.38050e6	−0.310580
$893$	2.75251e6	0.115505
$894$	4.36020e6	0.182458
$895$	−1.03565e7	−0.432171
$896$	491520.	0.0204537
$897$	−6.38690e7	−2.65038
$898$	−1.20180e7	−0.497325
$899$	−3.49468e7	−1.44214
$900$	−796032.	−0.0327585
$901$	2.04853e7	0.840681
$902$	0	0
$903$	603900.	0.0246460
$904$	−1.39870e7	−0.569251
$905$	5.31022e6	0.215522
$906$	−1.81849e7	−0.736022
$907$	−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
$909$	1.74114e6	0.0698914
$910$	872480.	0.0349263
$911$	−2.49762e7	−0.997081	−0.498541	−	0.866866i	$-0.666131\pi$
$911$	−2.49762e7	−0.997081	−0.498541	+	0.866866i	$0.666131\pi$
$912$	1.47456e6	0.0587050
$913$	0	0
$914$	9.78206e6	0.387316
$915$	4.83930e6	0.191086
$916$	1.06529e7	0.419497
$917$	−4620.00	−0.000181434 0
$918$	−1.07428e7	−0.420736
$919$	1.10613e7	0.432032	0.216016	−	0.976390i	$-0.430694\pi$
$919$	1.10613e7	0.432032	0.216016	+	0.976390i	$0.430694\pi$
$920$	1.35304e7	0.527038
$921$	−1.71402e7	−0.665835
$922$	−3.11641e7	−1.20734
$923$	3.56144e7	1.37601
$924$	0	0
$925$	−3.33421e7	−1.28127
$926$	−4.20784e6	−0.161262
$927$	1.72267e6	0.0658420
$928$	2.77709e7	1.05857
$929$	−2.01739e7	−0.766919	−0.383460	−	0.923558i	$-0.625267\pi$
$929$	−2.01739e7	−0.766919	−0.383460	+	0.923558i	$0.625267\pi$
$930$	−7.34502e6	−0.278475
$931$	−6.41549e6	−0.242580
$932$	1.93494e7	0.729671
$933$	−8.80434e6	−0.331126
$934$	1.58801e7	0.595645
$935$	0	0
$936$	3.96749e6	0.148022
$937$	−9.10734e6	−0.338877	−0.169439	−	0.985541i	$-0.554195\pi$
$937$	−9.10734e6	−0.338877	−0.169439	+	0.985541i	$0.554195\pi$
$938$	−1.19996e6	−0.0445307
$939$	3.50785e6	0.129831
$940$	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	$0.263340\pi$
$942$	3.19792e7	1.17420
$943$	5.66735e6	0.207540
$944$	1.65402e6	0.0604101
$945$	743850.	0.0270960
$946$	0	0
$947$	−4.95743e7	−1.79631	−0.898156	−	0.439677i	$-0.855093\pi$
$947$	−4.95743e7	−1.79631	−0.898156	+	0.439677i	$0.855093\pi$
$948$	−1.56331e7	−0.564969
$949$	−2.20875e6	−0.0796125
$950$	−4.24550e6	−0.152623
$951$	1.40325e7	0.503136
$952$	−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	$0.717326\pi$
$954$	2.15006e6	0.0764857
$955$	8.44430e6	0.299609
$956$	−9.14371e6	−0.323577
$957$	0	0
$958$	3.41563e7	1.20242
$959$	−2.96233e6	−0.104013
$960$	8.17152e6	0.286170
$961$	1.28831e7	0.449999
$962$	5.53933e7	1.92983
$963$	−587844.	−0.0204266
$964$	−4.27328e6	−0.148105
$965$	−351044.	−0.0121351
$966$	2.22540e6	0.0767300
$967$	−2.78059e7	−0.956248	−0.478124	−	0.878292i	$-0.658683\pi$
$967$	−2.78059e7	−0.956248	−0.478124	+	0.878292i	$0.658683\pi$
$968$	0	0
$969$	3.95136e6	0.135188
$970$	−5.08478e6	−0.173517
$971$	1.56835e7	0.533821	0.266910	−	0.963721i	$-0.413997\pi$
$971$	1.56835e7	0.533821	0.266910	+	0.963721i	$0.413997\pi$
$972$	2.17728e6	0.0739177
$973$	−3.99818e6	−0.135388
$974$	−7.45947e6	−0.251948
$975$	4.75961e7	1.60347
$976$	−4.34688e6	−0.146067
$977$	2.01140e7	0.674157	0.337079	−	0.941476i	$-0.390561\pi$
$977$	2.01140e7	0.674157	0.337079	+	0.941476i	$0.390561\pi$
$978$	−1.69246e7	−0.565810
$979$	0	0
$980$	−5.07893e6	−0.168930
$981$	−3.33788e6	−0.110739
$982$	−2.06291e7	−0.682655
$983$	−2.09269e7	−0.690750	−0.345375	−	0.938465i	$-0.612248\pi$
$983$	−2.09269e7	−0.690750	−0.345375	+	0.938465i	$0.612248\pi$
$984$	4.40064e6	0.144887
$985$	5.13346e6	0.168585
$986$	−1.48835e7	−0.487541
$987$	1.07520e6	0.0351315
$988$	−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
$993$	1.58735e7	0.510856
$994$	−1.24092e6	−0.0398362
$995$	−823080.	−0.0263563
$996$	2.46514e7	0.787395
$997$	−1.38913e6	−0.0442595	−0.0221297	−	0.999755i	$-0.507045\pi$
$997$	−1.38913e6	−0.0442595	−0.0221297	+	0.999755i	$0.507045\pi$
$998$	1.81336e7	0.576313
$999$	4.72266e7	1.49718

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.6.a.b.1.1		1
3.2	odd	2		1089.6.a.c.1.1		1
11.10	odd	2		11.6.a.a.1.1	✓	1
33.32	even	2		99.6.a.c.1.1		1
44.43	even	2		176.6.a.c.1.1		1
55.32	even	4		275.6.b.a.199.1		2
55.43	even	4		275.6.b.a.199.2		2
55.54	odd	2		275.6.a.a.1.1		1
77.76	even	2		539.6.a.c.1.1		1
88.21	odd	2		704.6.a.h.1.1		1
88.43	even	2		704.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.a.a.1.1	✓	1	11.10	odd	2
99.6.a.c.1.1		1	33.32	even	2
121.6.a.b.1.1		1	1.1	even	1	trivial
176.6.a.c.1.1		1	44.43	even	2
275.6.a.a.1.1		1	55.54	odd	2
275.6.b.a.199.1		2	55.32	even	4
275.6.b.a.199.2		2	55.43	even	4
539.6.a.c.1.1		1	77.76	even	2
704.6.a.c.1.1		1	88.43	even	2
704.6.a.h.1.1		1	88.21	odd	2
1089.6.a.c.1.1		1	3.2	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:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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