LMFDB - Embedded newform 102.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [102,6,Mod(1,102)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, 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("102.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 102 = 2 \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	102.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:	$16.3591496209$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	102.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -81.0000 q^{5} +36.0000 q^{6} -88.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -81.0000 q^{5} +36.0000 q^{6} -88.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -324.000 q^{10} -399.000 q^{11} +144.000 q^{12} -481.000 q^{13} -352.000 q^{14} -729.000 q^{15} +256.000 q^{16} -289.000 q^{17} +324.000 q^{18} -2569.00 q^{19} -1296.00 q^{20} -792.000 q^{21} -1596.00 q^{22} +5013.00 q^{23} +576.000 q^{24} +3436.00 q^{25} -1924.00 q^{26} +729.000 q^{27} -1408.00 q^{28} -378.000 q^{29} -2916.00 q^{30} -7606.00 q^{31} +1024.00 q^{32} -3591.00 q^{33} -1156.00 q^{34} +7128.00 q^{35} +1296.00 q^{36} +5756.00 q^{37} -10276.0 q^{38} -4329.00 q^{39} -5184.00 q^{40} -16167.0 q^{41} -3168.00 q^{42} +12641.0 q^{43} -6384.00 q^{44} -6561.00 q^{45} +20052.0 q^{46} -7002.00 q^{47} +2304.00 q^{48} -9063.00 q^{49} +13744.0 q^{50} -2601.00 q^{51} -7696.00 q^{52} +17598.0 q^{53} +2916.00 q^{54} +32319.0 q^{55} -5632.00 q^{56} -23121.0 q^{57} -1512.00 q^{58} +23094.0 q^{59} -11664.0 q^{60} -13876.0 q^{61} -30424.0 q^{62} -7128.00 q^{63} +4096.00 q^{64} +38961.0 q^{65} -14364.0 q^{66} +42788.0 q^{67} -4624.00 q^{68} +45117.0 q^{69} +28512.0 q^{70} +20772.0 q^{71} +5184.00 q^{72} -56626.0 q^{73} +23024.0 q^{74} +30924.0 q^{75} -41104.0 q^{76} +35112.0 q^{77} -17316.0 q^{78} -85450.0 q^{79} -20736.0 q^{80} +6561.00 q^{81} -64668.0 q^{82} +76374.0 q^{83} -12672.0 q^{84} +23409.0 q^{85} +50564.0 q^{86} -3402.00 q^{87} -25536.0 q^{88} +3468.00 q^{89} -26244.0 q^{90} +42328.0 q^{91} +80208.0 q^{92} -68454.0 q^{93} -28008.0 q^{94} +208089. q^{95} +9216.00 q^{96} +48260.0 q^{97} -36252.0 q^{98} -32319.0 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	−81.0000	−1.44897	−0.724486	−	0.689289i	$-0.757923\pi$
$5$	−81.0000	−1.44897	−0.724486	+	0.689289i	$0.757923\pi$
$6$	36.0000	0.408248
$7$	−88.0000	−0.678793	−0.339397	−	0.940643i	$-0.610223\pi$
$7$	−88.0000	−0.678793	−0.339397	+	0.940643i	$0.610223\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−324.000	−1.02458
$11$	−399.000	−0.994240	−0.497120	−	0.867682i	$-0.665609\pi$
$11$	−399.000	−0.994240	−0.497120	+	0.867682i	$0.665609\pi$
$12$	144.000	0.288675
$13$	−481.000	−0.789381	−0.394691	−	0.918814i	$-0.629148\pi$
$13$	−481.000	−0.789381	−0.394691	+	0.918814i	$0.629148\pi$
$14$	−352.000	−0.479979
$15$	−729.000	−0.836564
$16$	256.000	0.250000
$17$	−289.000	−0.242536
$17$	−289.000	−0.242536
$18$	324.000	0.235702
$19$	−2569.00	−1.63260	−0.816301	−	0.577627i	$-0.803979\pi$
$19$	−2569.00	−1.63260	−0.816301	+	0.577627i	$0.803979\pi$
$20$	−1296.00	−0.724486
$21$	−792.000	−0.391902
$22$	−1596.00	−0.703034
$23$	5013.00	1.97596	0.987980	−	0.154582i	$-0.0494032\pi$
$23$	5013.00	1.97596	0.987980	+	0.154582i	$0.0494032\pi$
$24$	576.000	0.204124
$25$	3436.00	1.09952
$26$	−1924.00	−0.558177
$27$	729.000	0.192450
$28$	−1408.00	−0.339397
$29$	−378.000	−0.0834635	−0.0417318	−	0.999129i	$-0.513287\pi$
$29$	−378.000	−0.0834635	−0.0417318	+	0.999129i	$0.513287\pi$
$30$	−2916.00	−0.591540
$31$	−7606.00	−1.42152	−0.710759	−	0.703436i	$-0.751648\pi$
$31$	−7606.00	−1.42152	−0.710759	+	0.703436i	$0.751648\pi$
$32$	1024.00	0.176777
$33$	−3591.00	−0.574025
$34$	−1156.00	−0.171499
$35$	7128.00	0.983553
$36$	1296.00	0.166667
$37$	5756.00	0.691220	0.345610	−	0.938378i	$-0.387672\pi$
$37$	5756.00	0.691220	0.345610	+	0.938378i	$0.387672\pi$
$38$	−10276.0	−1.15442
$39$	−4329.00	−0.455749
$40$	−5184.00	−0.512289
$41$	−16167.0	−1.50200	−0.751000	−	0.660302i	$-0.770428\pi$
$41$	−16167.0	−1.50200	−0.751000	+	0.660302i	$0.770428\pi$
$42$	−3168.00	−0.277116
$43$	12641.0	1.04258	0.521291	−	0.853379i	$-0.325451\pi$
$43$	12641.0	1.04258	0.521291	+	0.853379i	$0.325451\pi$
$44$	−6384.00	−0.497120
$45$	−6561.00	−0.482991
$46$	20052.0	1.39721
$47$	−7002.00	−0.462357	−0.231179	−	0.972911i	$-0.574258\pi$
$47$	−7002.00	−0.462357	−0.231179	+	0.972911i	$0.574258\pi$
$48$	2304.00	0.144338
$49$	−9063.00	−0.539240
$50$	13744.0	0.777478
$51$	−2601.00	−0.140028
$52$	−7696.00	−0.394691
$53$	17598.0	0.860545	0.430273	−	0.902699i	$-0.358417\pi$
$53$	17598.0	0.860545	0.430273	+	0.902699i	$0.358417\pi$
$54$	2916.00	0.136083
$55$	32319.0	1.44063
$56$	−5632.00	−0.239990
$57$	−23121.0	−0.942583
$58$	−1512.00	−0.0590176
$59$	23094.0	0.863712	0.431856	−	0.901942i	$-0.357859\pi$
$59$	23094.0	0.863712	0.431856	+	0.901942i	$0.357859\pi$
$60$	−11664.0	−0.418282
$61$	−13876.0	−0.477463	−0.238731	−	0.971086i	$-0.576732\pi$
$61$	−13876.0	−0.477463	−0.238731	+	0.971086i	$0.576732\pi$
$62$	−30424.0	−1.00516
$63$	−7128.00	−0.226264
$64$	4096.00	0.125000
$65$	38961.0	1.14379
$66$	−14364.0	−0.405897
$67$	42788.0	1.16449	0.582244	−	0.813014i	$-0.302175\pi$
$67$	42788.0	1.16449	0.582244	+	0.813014i	$0.302175\pi$
$68$	−4624.00	−0.121268
$69$	45117.0	1.14082
$70$	28512.0	0.695477
$71$	20772.0	0.489027	0.244513	−	0.969646i	$-0.421372\pi$
$71$	20772.0	0.489027	0.244513	+	0.969646i	$0.421372\pi$
$72$	5184.00	0.117851
$73$	−56626.0	−1.24368	−0.621840	−	0.783144i	$-0.713615\pi$
$73$	−56626.0	−1.24368	−0.621840	+	0.783144i	$0.713615\pi$
$74$	23024.0	0.488767
$75$	30924.0	0.634808
$76$	−41104.0	−0.816301
$77$	35112.0	0.674883
$78$	−17316.0	−0.322263
$79$	−85450.0	−1.54044	−0.770219	−	0.637779i	$-0.779853\pi$
$79$	−85450.0	−1.54044	−0.770219	+	0.637779i	$0.779853\pi$
$80$	−20736.0	−0.362243
$81$	6561.00	0.111111
$82$	−64668.0	−1.06207
$83$	76374.0	1.21689	0.608444	−	0.793597i	$-0.291794\pi$
$83$	76374.0	1.21689	0.608444	+	0.793597i	$0.291794\pi$
$84$	−12672.0	−0.195951
$85$	23409.0	0.351427
$86$	50564.0	0.737217
$87$	−3402.00	−0.0481877
$88$	−25536.0	−0.351517
$89$	3468.00	0.0464092	0.0232046	−	0.999731i	$-0.492613\pi$
$89$	3468.00	0.0464092	0.0232046	+	0.999731i	$0.492613\pi$
$90$	−26244.0	−0.341526
$91$	42328.0	0.535827
$92$	80208.0	0.987980
$93$	−68454.0	−0.820713
$94$	−28008.0	−0.326936
$95$	208089.	2.36559
$96$	9216.00	0.102062
$97$	48260.0	0.520784	0.260392	−	0.965503i	$-0.416148\pi$
$97$	48260.0	0.520784	0.260392	+	0.965503i	$0.416148\pi$
$98$	−36252.0	−0.381300
$99$	−32319.0	−0.331413
$100$	54976.0	0.549760
$101$	149688.	1.46010	0.730052	−	0.683392i	$-0.239496\pi$
$101$	149688.	1.46010	0.730052	+	0.683392i	$0.239496\pi$
$102$	−10404.0	−0.0990148
$103$	52853.0	0.490881	0.245441	−	0.969412i	$-0.421067\pi$
$103$	52853.0	0.490881	0.245441	+	0.969412i	$0.421067\pi$
$104$	−30784.0	−0.279088
$105$	64152.0	0.567854
$106$	70392.0	0.608497
$107$	−155643.	−1.31423	−0.657113	−	0.753792i	$-0.728223\pi$
$107$	−155643.	−1.31423	−0.657113	+	0.753792i	$0.728223\pi$
$108$	11664.0	0.0962250
$109$	−39004.0	−0.314444	−0.157222	−	0.987563i	$-0.550254\pi$
$109$	−39004.0	−0.314444	−0.157222	+	0.987563i	$0.550254\pi$
$110$	129276.	1.01868
$111$	51804.0	0.399076
$112$	−22528.0	−0.169698
$113$	−90429.0	−0.666211	−0.333105	−	0.942890i	$-0.608096\pi$
$113$	−90429.0	−0.666211	−0.333105	+	0.942890i	$0.608096\pi$
$114$	−92484.0	−0.666507
$115$	−406053.	−2.86311
$116$	−6048.00	−0.0417318
$117$	−38961.0	−0.263127
$118$	92376.0	0.610737
$119$	25432.0	0.164632
$120$	−46656.0	−0.295770
$121$	−1850.00	−0.0114870
$122$	−55504.0	−0.337617
$123$	−145503.	−0.867180
$124$	−121696.	−0.710759
$125$	−25191.0	−0.144202
$126$	−28512.0	−0.159993
$127$	−307789.	−1.69334	−0.846669	−	0.532119i	$-0.821396\pi$
$127$	−307789.	−1.69334	−0.846669	+	0.532119i	$0.821396\pi$
$128$	16384.0	0.0883883
$129$	113769.	0.601935
$130$	155844.	0.808782
$131$	−273561.	−1.39276	−0.696379	−	0.717674i	$-0.745207\pi$
$131$	−273561.	−1.39276	−0.696379	+	0.717674i	$0.745207\pi$
$132$	−57456.0	−0.287012
$133$	226072.	1.10820
$134$	171152.	0.823417
$135$	−59049.0	−0.278855
$136$	−18496.0	−0.0857493
$137$	−315066.	−1.43417	−0.717084	−	0.696987i	$-0.754524\pi$
$137$	−315066.	−1.43417	−0.717084	+	0.696987i	$0.754524\pi$
$138$	180468.	0.806682
$139$	−249610.	−1.09578	−0.547892	−	0.836549i	$-0.684570\pi$
$139$	−249610.	−1.09578	−0.547892	+	0.836549i	$0.684570\pi$
$140$	114048.	0.491776
$141$	−63018.0	−0.266942
$142$	83088.0	0.345794
$143$	191919.	0.784834
$144$	20736.0	0.0833333
$145$	30618.0	0.120936
$146$	−226504.	−0.879415
$147$	−81567.0	−0.311330
$148$	92096.0	0.345610
$149$	342186.	1.26269	0.631345	−	0.775502i	$-0.282503\pi$
$149$	342186.	1.26269	0.631345	+	0.775502i	$0.282503\pi$
$150$	123696.	0.448877
$151$	−454168.	−1.62097	−0.810483	−	0.585761i	$-0.800796\pi$
$151$	−454168.	−1.62097	−0.810483	+	0.585761i	$0.800796\pi$
$152$	−164416.	−0.577212
$153$	−23409.0	−0.0808452
$154$	140448.	0.477215
$155$	616086.	2.05974
$156$	−69264.0	−0.227875
$157$	−94429.0	−0.305743	−0.152871	−	0.988246i	$-0.548852\pi$
$157$	−94429.0	−0.305743	−0.152871	+	0.988246i	$0.548852\pi$
$158$	−341800.	−1.08925
$159$	158382.	0.496836
$160$	−82944.0	−0.256144
$161$	−441144.	−1.34127
$162$	26244.0	0.0785674
$163$	−214918.	−0.633583	−0.316792	−	0.948495i	$-0.602606\pi$
$163$	−214918.	−0.633583	−0.316792	+	0.948495i	$0.602606\pi$
$164$	−258672.	−0.751000
$165$	290871.	0.831746
$166$	305496.	0.860469
$167$	−256047.	−0.710442	−0.355221	−	0.934782i	$-0.615594\pi$
$167$	−256047.	−0.710442	−0.355221	+	0.934782i	$0.615594\pi$
$168$	−50688.0	−0.138558
$169$	−139932.	−0.376878
$170$	93636.0	0.248497
$171$	−208089.	−0.544200
$172$	202256.	0.521291
$173$	291867.	0.741429	0.370715	−	0.928747i	$-0.379113\pi$
$173$	291867.	0.741429	0.370715	+	0.928747i	$0.379113\pi$
$174$	−13608.0	−0.0340738
$175$	−302368.	−0.746347
$176$	−102144.	−0.248560
$177$	207846.	0.498665
$178$	13872.0	0.0328163
$179$	118866.	0.277284	0.138642	−	0.990343i	$-0.455726\pi$
$179$	118866.	0.277284	0.138642	+	0.990343i	$0.455726\pi$
$180$	−104976.	−0.241495
$181$	209426.	0.475154	0.237577	−	0.971369i	$-0.423647\pi$
$181$	209426.	0.475154	0.237577	+	0.971369i	$0.423647\pi$
$182$	169312.	0.378887
$183$	−124884.	−0.275663
$184$	320832.	0.698607
$185$	−466236.	−1.00156
$186$	−273816.	−0.580332
$187$	115311.	0.241139
$188$	−112032.	−0.231179
$189$	−64152.0	−0.130634
$190$	832356.	1.67273
$191$	631974.	1.25348	0.626738	−	0.779230i	$-0.284390\pi$
$191$	631974.	1.25348	0.626738	+	0.779230i	$0.284390\pi$
$192$	36864.0	0.0721688
$193$	−89566.0	−0.173081	−0.0865406	−	0.996248i	$-0.527581\pi$
$193$	−89566.0	−0.173081	−0.0865406	+	0.996248i	$0.527581\pi$
$194$	193040.	0.368250
$195$	350649.	0.660368
$196$	−145008.	−0.269620
$197$	−224121.	−0.411450	−0.205725	−	0.978610i	$-0.565955\pi$
$197$	−224121.	−0.411450	−0.205725	+	0.978610i	$0.565955\pi$
$198$	−129276.	−0.234345
$199$	27020.0	0.0483674	0.0241837	−	0.999708i	$-0.492301\pi$
$199$	27020.0	0.0483674	0.0241837	+	0.999708i	$0.492301\pi$
$200$	219904.	0.388739
$201$	385092.	0.672317
$202$	598752.	1.03245
$203$	33264.0	0.0566545
$204$	−41616.0	−0.0700140
$205$	1.30953e6	2.17636
$206$	211412.	0.347106
$207$	406053.	0.658653
$208$	−123136.	−0.197345
$209$	1.02503e6	1.62320
$210$	256608.	0.401534
$211$	111674.	0.172682	0.0863408	−	0.996266i	$-0.472483\pi$
$211$	111674.	0.172682	0.0863408	+	0.996266i	$0.472483\pi$
$212$	281568.	0.430273
$213$	186948.	0.282340
$214$	−622572.	−0.929298
$215$	−1.02392e6	−1.51067
$216$	46656.0	0.0680414
$217$	669328.	0.964916
$218$	−156016.	−0.222345
$219$	−509634.	−0.718039
$220$	517104.	0.720313
$221$	139009.	0.191453
$222$	207216.	0.282189
$223$	401423.	0.540555	0.270278	−	0.962782i	$-0.412885\pi$
$223$	401423.	0.540555	0.270278	+	0.962782i	$0.412885\pi$
$224$	−90112.0	−0.119995
$225$	278316.	0.366507
$226$	−361716.	−0.471082
$227$	−906297.	−1.16736	−0.583681	−	0.811983i	$-0.698388\pi$
$227$	−906297.	−1.16736	−0.583681	+	0.811983i	$0.698388\pi$
$228$	−369936.	−0.471291
$229$	924686.	1.16521	0.582607	−	0.812754i	$-0.302033\pi$
$229$	924686.	1.16521	0.582607	+	0.812754i	$0.302033\pi$
$230$	−1.62421e6	−2.02452
$231$	316008.	0.389644
$232$	−24192.0	−0.0295088
$233$	−1.27605e6	−1.53984	−0.769922	−	0.638138i	$-0.779705\pi$
$233$	−1.27605e6	−1.53984	−0.769922	+	0.638138i	$0.779705\pi$
$234$	−155844.	−0.186059
$235$	567162.	0.669942
$236$	369504.	0.431856
$237$	−769050.	−0.889373
$238$	101728.	0.116412
$239$	−1.57872e6	−1.78776	−0.893882	−	0.448302i	$-0.852029\pi$
$239$	−1.57872e6	−1.78776	−0.893882	+	0.448302i	$0.852029\pi$
$240$	−186624.	−0.209141
$241$	−741244.	−0.822089	−0.411044	−	0.911615i	$-0.634836\pi$
$241$	−741244.	−0.822089	−0.411044	+	0.911615i	$0.634836\pi$
$242$	−7400.00	−0.00812257
$243$	59049.0	0.0641500
$244$	−222016.	−0.238731
$245$	734103.	0.781343
$246$	−582012.	−0.613189
$247$	1.23569e6	1.28874
$248$	−486784.	−0.502582
$249$	687366.	0.702570
$250$	−100764.	−0.101966
$251$	−788064.	−0.789546	−0.394773	−	0.918779i	$-0.629177\pi$
$251$	−788064.	−0.789546	−0.394773	+	0.918779i	$0.629177\pi$
$252$	−114048.	−0.113132
$253$	−2.00019e6	−1.96458
$254$	−1.23116e6	−1.19737
$255$	210681.	0.202897
$256$	65536.0	0.0625000
$257$	497472.	0.469825	0.234912	−	0.972017i	$-0.424520\pi$
$257$	497472.	0.469825	0.234912	+	0.972017i	$0.424520\pi$
$258$	455076.	0.425632
$259$	−506528.	−0.469196
$260$	623376.	0.571896
$261$	−30618.0	−0.0278212
$262$	−1.09424e6	−0.984829
$263$	1.66211e6	1.48173	0.740866	−	0.671652i	$-0.234415\pi$
$263$	1.66211e6	1.48173	0.740866	+	0.671652i	$0.234415\pi$
$264$	−229824.	−0.202948
$265$	−1.42544e6	−1.24691
$266$	904288.	0.783615
$267$	31212.0	0.0267944
$268$	684608.	0.582244
$269$	376005.	0.316820	0.158410	−	0.987373i	$-0.449363\pi$
$269$	376005.	0.316820	0.158410	+	0.987373i	$0.449363\pi$
$270$	−236196.	−0.197180
$271$	−326629.	−0.270167	−0.135083	−	0.990834i	$-0.543130\pi$
$271$	−326629.	−0.270167	−0.135083	+	0.990834i	$0.543130\pi$
$272$	−73984.0	−0.0606339
$273$	380952.	0.309360
$274$	−1.26026e6	−1.01411
$275$	−1.37096e6	−1.09319
$276$	721872.	0.570410
$277$	2.05518e6	1.60935	0.804676	−	0.593715i	$-0.202339\pi$
$277$	2.05518e6	1.60935	0.804676	+	0.593715i	$0.202339\pi$
$278$	−998440.	−0.774836
$279$	−616086.	−0.473839
$280$	456192.	0.347738
$281$	546288.	0.412720	0.206360	−	0.978476i	$-0.433838\pi$
$281$	546288.	0.412720	0.206360	+	0.978476i	$0.433838\pi$
$282$	−252072.	−0.188756
$283$	−1.56565e6	−1.16206	−0.581030	−	0.813882i	$-0.697350\pi$
$283$	−1.56565e6	−1.16206	−0.581030	+	0.813882i	$0.697350\pi$
$284$	332352.	0.244513
$285$	1.87280e6	1.36578
$286$	767676.	0.554962
$287$	1.42270e6	1.01955
$288$	82944.0	0.0589256
$289$	83521.0	0.0588235
$290$	122472.	0.0855149
$291$	434340.	0.300675
$292$	−906016.	−0.621840
$293$	−982344.	−0.668489	−0.334245	−	0.942486i	$-0.608481\pi$
$293$	−982344.	−0.668489	−0.334245	+	0.942486i	$0.608481\pi$
$294$	−326268.	−0.220144
$295$	−1.87061e6	−1.25150
$296$	368384.	0.244383
$297$	−290871.	−0.191342
$298$	1.36874e6	0.892856
$299$	−2.41125e6	−1.55979
$300$	494784.	0.317404
$301$	−1.11241e6	−0.707698
$302$	−1.81667e6	−1.14620
$303$	1.34719e6	0.842991
$304$	−657664.	−0.408150
$305$	1.12396e6	0.691831
$306$	−93636.0	−0.0571662
$307$	−659500.	−0.399364	−0.199682	−	0.979861i	$-0.563991\pi$
$307$	−659500.	−0.399364	−0.199682	+	0.979861i	$0.563991\pi$
$308$	561792.	0.337442
$309$	475677.	0.283411
$310$	2.46434e6	1.45646
$311$	−15936.0	−0.00934283	−0.00467141	−	0.999989i	$-0.501487\pi$
$311$	−15936.0	−0.00934283	−0.00467141	+	0.999989i	$0.501487\pi$
$312$	−277056.	−0.161132
$313$	2.61840e6	1.51069	0.755343	−	0.655329i	$-0.227470\pi$
$313$	2.61840e6	1.51069	0.755343	+	0.655329i	$0.227470\pi$
$314$	−377716.	−0.216193
$315$	577368.	0.327851
$316$	−1.36720e6	−0.770219
$317$	2.16854e6	1.21205	0.606023	−	0.795447i	$-0.292764\pi$
$317$	2.16854e6	1.21205	0.606023	+	0.795447i	$0.292764\pi$
$318$	633528.	0.351316
$319$	150822.	0.0829828
$320$	−331776.	−0.181122
$321$	−1.40079e6	−0.758769
$322$	−1.76458e6	−0.948420
$323$	742441.	0.395964
$324$	104976.	0.0555556
$325$	−1.65272e6	−0.867940
$326$	−859672.	−0.448011
$327$	−351036.	−0.181544
$328$	−1.03469e6	−0.531037
$329$	616176.	0.313845
$330$	1.16348e6	0.588133
$331$	−2.42089e6	−1.21452	−0.607261	−	0.794502i	$-0.707732\pi$
$331$	−2.42089e6	−1.21452	−0.607261	+	0.794502i	$0.707732\pi$
$332$	1.22198e6	0.608444
$333$	466236.	0.230407
$334$	−1.02419e6	−0.502358
$335$	−3.46583e6	−1.68731
$336$	−202752.	−0.0979754
$337$	951650.	0.456460	0.228230	−	0.973607i	$-0.426706\pi$
$337$	951650.	0.456460	0.228230	+	0.973607i	$0.426706\pi$
$338$	−559728.	−0.266493
$339$	−813861.	−0.384637
$340$	374544.	0.175714
$341$	3.03479e6	1.41333
$342$	−832356.	−0.384808
$343$	2.27656e6	1.04483
$344$	809024.	0.368608
$345$	−3.65448e6	−1.65302
$346$	1.16747e6	0.524270
$347$	4.27777e6	1.90719	0.953595	−	0.301092i	$-0.0973513\pi$
$347$	4.27777e6	1.90719	0.953595	+	0.301092i	$0.0973513\pi$
$348$	−54432.0	−0.0240938
$349$	2.28514e6	1.00427	0.502134	−	0.864790i	$-0.332548\pi$
$349$	2.28514e6	1.00427	0.502134	+	0.864790i	$0.332548\pi$
$350$	−1.20947e6	−0.527747
$351$	−350649.	−0.151916
$352$	−408576.	−0.175758
$353$	−2.62245e6	−1.12014	−0.560068	−	0.828447i	$-0.689225\pi$
$353$	−2.62245e6	−1.12014	−0.560068	+	0.828447i	$0.689225\pi$
$354$	831384.	0.352609
$355$	−1.68253e6	−0.708586
$356$	55488.0	0.0232046
$357$	228888.	0.0950501
$358$	475464.	0.196070
$359$	−706584.	−0.289353	−0.144676	−	0.989479i	$-0.546214\pi$
$359$	−706584.	−0.289353	−0.144676	+	0.989479i	$0.546214\pi$
$360$	−419904.	−0.170763
$361$	4.12366e6	1.66539
$362$	837704.	0.335984
$363$	−16650.0	−0.00663205
$364$	677248.	0.267913
$365$	4.58671e6	1.80206
$366$	−499536.	−0.194923
$367$	1.01355e6	0.392808	0.196404	−	0.980523i	$-0.437074\pi$
$367$	1.01355e6	0.392808	0.196404	+	0.980523i	$0.437074\pi$
$368$	1.28333e6	0.493990
$369$	−1.30953e6	−0.500667
$370$	−1.86494e6	−0.708209
$371$	−1.54862e6	−0.584132
$372$	−1.09526e6	−0.410357
$373$	−2.91569e6	−1.08510	−0.542549	−	0.840024i	$-0.682541\pi$
$373$	−2.91569e6	−1.08510	−0.542549	+	0.840024i	$0.682541\pi$
$374$	461244.	0.170511
$375$	−226719.	−0.0832549
$376$	−448128.	−0.163468
$377$	181818.	0.0658845
$378$	−256608.	−0.0923721
$379$	−1.39464e6	−0.498730	−0.249365	−	0.968410i	$-0.580222\pi$
$379$	−1.39464e6	−0.498730	−0.249365	+	0.968410i	$0.580222\pi$
$380$	3.32942e6	1.18280
$381$	−2.77010e6	−0.977649
$382$	2.52790e6	0.886341
$383$	−642552.	−0.223826	−0.111913	−	0.993718i	$-0.535698\pi$
$383$	−642552.	−0.223826	−0.111913	+	0.993718i	$0.535698\pi$
$384$	147456.	0.0510310
$385$	−2.84407e6	−0.977887
$386$	−358264.	−0.122387
$387$	1.02392e6	0.347527
$388$	772160.	0.260392
$389$	796404.	0.266845	0.133423	−	0.991059i	$-0.457403\pi$
$389$	796404.	0.266845	0.133423	+	0.991059i	$0.457403\pi$
$390$	1.40260e6	0.466951
$391$	−1.44876e6	−0.479241
$392$	−580032.	−0.190650
$393$	−2.46205e6	−0.804110
$394$	−896484.	−0.290939
$395$	6.92145e6	2.23205
$396$	−517104.	−0.165707
$397$	3.12378e6	0.994726	0.497363	−	0.867542i	$-0.334302\pi$
$397$	3.12378e6	0.994726	0.497363	+	0.867542i	$0.334302\pi$
$398$	108080.	0.0342009
$399$	2.03465e6	0.639819
$400$	879616.	0.274880
$401$	2.25297e6	0.699673	0.349836	−	0.936811i	$-0.386237\pi$
$401$	2.25297e6	0.699673	0.349836	+	0.936811i	$0.386237\pi$
$402$	1.54037e6	0.475400
$403$	3.65849e6	1.12212
$404$	2.39501e6	0.730052
$405$	−531441.	−0.160997
$406$	133056.	0.0400608
$407$	−2.29664e6	−0.687239
$408$	−166464.	−0.0495074
$409$	2.22051e6	0.656363	0.328182	−	0.944615i	$-0.393564\pi$
$409$	2.22051e6	0.656363	0.328182	+	0.944615i	$0.393564\pi$
$410$	5.23811e6	1.53892
$411$	−2.83559e6	−0.828017
$412$	845648.	0.245441
$413$	−2.03227e6	−0.586282
$414$	1.62421e6	0.465738
$415$	−6.18629e6	−1.76324
$416$	−492544.	−0.139544
$417$	−2.24649e6	−0.632651
$418$	4.10012e6	1.14777
$419$	−385140.	−0.107173	−0.0535863	−	0.998563i	$-0.517065\pi$
$419$	−385140.	−0.107173	−0.0535863	+	0.998563i	$0.517065\pi$
$420$	1.02643e6	0.283927
$421$	−4.17031e6	−1.14674	−0.573368	−	0.819298i	$-0.694363\pi$
$421$	−4.17031e6	−1.14674	−0.573368	+	0.819298i	$0.694363\pi$
$422$	446696.	0.122104
$423$	−567162.	−0.154119
$424$	1.12627e6	0.304249
$425$	−993004.	−0.266673
$426$	747792.	0.199644
$427$	1.22109e6	0.324099
$428$	−2.49029e6	−0.657113
$429$	1.72727e6	0.453124
$430$	−4.09568e6	−1.06821
$431$	−2.82346e6	−0.732129	−0.366065	−	0.930589i	$-0.619295\pi$
$431$	−2.82346e6	−0.732129	−0.366065	+	0.930589i	$0.619295\pi$
$432$	186624.	0.0481125
$433$	2.45424e6	0.629067	0.314534	−	0.949246i	$-0.398152\pi$
$433$	2.45424e6	0.629067	0.314534	+	0.949246i	$0.398152\pi$
$434$	2.67731e6	0.682299
$435$	275562.	0.0698226
$436$	−624064.	−0.157222
$437$	−1.28784e7	−3.22595
$438$	−2.03854e6	−0.507730
$439$	5.73552e6	1.42040	0.710201	−	0.703999i	$-0.248604\pi$
$439$	5.73552e6	1.42040	0.710201	+	0.703999i	$0.248604\pi$
$440$	2.06842e6	0.509338
$441$	−734103.	−0.179747
$442$	556036.	0.135378
$443$	1.89899e6	0.459742	0.229871	−	0.973221i	$-0.426170\pi$
$443$	1.89899e6	0.459742	0.229871	+	0.973221i	$0.426170\pi$
$444$	828864.	0.199538
$445$	−280908.	−0.0672456
$446$	1.60569e6	0.382230
$447$	3.07967e6	0.729014
$448$	−360448.	−0.0848492
$449$	−3.04666e6	−0.713195	−0.356598	−	0.934258i	$-0.616063\pi$
$449$	−3.04666e6	−0.713195	−0.356598	+	0.934258i	$0.616063\pi$
$450$	1.11326e6	0.259159
$451$	6.45063e6	1.49335
$452$	−1.44686e6	−0.333105
$453$	−4.08751e6	−0.935866
$454$	−3.62519e6	−0.825450
$455$	−3.42857e6	−0.776398
$456$	−1.47974e6	−0.333253
$457$	7.05701e6	1.58063	0.790315	−	0.612700i	$-0.209917\pi$
$457$	7.05701e6	1.58063	0.790315	+	0.612700i	$0.209917\pi$
$458$	3.69874e6	0.823930
$459$	−210681.	−0.0466760
$460$	−6.49685e6	−1.43156
$461$	7.31674e6	1.60349	0.801743	−	0.597668i	$-0.203906\pi$
$461$	7.31674e6	1.60349	0.801743	+	0.597668i	$0.203906\pi$
$462$	1.26403e6	0.275520
$463$	−2.50622e6	−0.543333	−0.271666	−	0.962391i	$-0.587575\pi$
$463$	−2.50622e6	−0.543333	−0.271666	+	0.962391i	$0.587575\pi$
$464$	−96768.0	−0.0208659
$465$	5.54477e6	1.18919
$466$	−5.10419e6	−1.08883
$467$	6.86264e6	1.45613	0.728063	−	0.685510i	$-0.240421\pi$
$467$	6.86264e6	1.45613	0.728063	+	0.685510i	$0.240421\pi$
$468$	−623376.	−0.131564
$469$	−3.76534e6	−0.790447
$470$	2.26865e6	0.473721
$471$	−849861.	−0.176521
$472$	1.47802e6	0.305368
$473$	−5.04376e6	−1.03658
$474$	−3.07620e6	−0.628881
$475$	−8.82708e6	−1.79508
$476$	406912.	0.0823158
$477$	1.42544e6	0.286848
$478$	−6.31488e6	−1.26414
$479$	6.18411e6	1.23151	0.615756	−	0.787937i	$-0.288851\pi$
$479$	6.18411e6	1.23151	0.615756	+	0.787937i	$0.288851\pi$
$480$	−746496.	−0.147885
$481$	−2.76864e6	−0.545636
$482$	−2.96498e6	−0.581304
$483$	−3.97030e6	−0.774382
$484$	−29600.0	−0.00574352
$485$	−3.90906e6	−0.754602
$486$	236196.	0.0453609
$487$	−1.16702e6	−0.222975	−0.111488	−	0.993766i	$-0.535562\pi$
$487$	−1.16702e6	−0.222975	−0.111488	+	0.993766i	$0.535562\pi$
$488$	−888064.	−0.168809
$489$	−1.93426e6	−0.365800
$490$	2.93641e6	0.552493
$491$	973302.	0.182198	0.0910991	−	0.995842i	$-0.470962\pi$
$491$	973302.	0.182198	0.0910991	+	0.995842i	$0.470962\pi$
$492$	−2.32805e6	−0.433590
$493$	109242.	0.0202429
$494$	4.94276e6	0.911280
$495$	2.61784e6	0.480209
$496$	−1.94714e6	−0.355379
$497$	−1.82794e6	−0.331948
$498$	2.74946e6	0.496792
$499$	8.18185e6	1.47096	0.735479	−	0.677547i	$-0.236957\pi$
$499$	8.18185e6	1.47096	0.735479	+	0.677547i	$0.236957\pi$
$500$	−403056.	−0.0721008
$501$	−2.30442e6	−0.410174
$502$	−3.15226e6	−0.558293
$503$	−3.58718e6	−0.632169	−0.316085	−	0.948731i	$-0.602368\pi$
$503$	−3.58718e6	−0.632169	−0.316085	+	0.948731i	$0.602368\pi$
$504$	−456192.	−0.0799966
$505$	−1.21247e7	−2.11565
$506$	−8.00075e6	−1.38917
$507$	−1.25939e6	−0.217590
$508$	−4.92462e6	−0.846669
$509$	−6.70477e6	−1.14707	−0.573535	−	0.819181i	$-0.694428\pi$
$509$	−6.70477e6	−1.14707	−0.573535	+	0.819181i	$0.694428\pi$
$510$	842724.	0.143470
$511$	4.98309e6	0.844202
$512$	262144.	0.0441942
$513$	−1.87280e6	−0.314194
$514$	1.98989e6	0.332216
$515$	−4.28109e6	−0.711273
$516$	1.82030e6	0.300968
$517$	2.79380e6	0.459694
$518$	−2.02611e6	−0.331771
$519$	2.62680e6	0.428064
$520$	2.49350e6	0.404391
$521$	−9.37187e6	−1.51263	−0.756314	−	0.654209i	$-0.773002\pi$
$521$	−9.37187e6	−1.51263	−0.756314	+	0.654209i	$0.773002\pi$
$522$	−122472.	−0.0196725
$523$	−8.19156e6	−1.30952	−0.654761	−	0.755836i	$-0.727231\pi$
$523$	−8.19156e6	−1.30952	−0.654761	+	0.755836i	$0.727231\pi$
$524$	−4.37698e6	−0.696379
$525$	−2.72131e6	−0.430904
$526$	6.64843e6	1.04774
$527$	2.19813e6	0.344769
$528$	−919296.	−0.143506
$529$	1.86938e7	2.90442
$530$	−5.70175e6	−0.881696
$531$	1.87061e6	0.287904
$532$	3.61715e6	0.554099
$533$	7.77633e6	1.18565
$534$	124848.	0.0189465
$535$	1.26071e7	1.90428
$536$	2.73843e6	0.411709
$537$	1.06979e6	0.160090
$538$	1.50402e6	0.224026
$539$	3.61614e6	0.536134
$540$	−944784.	−0.139427
$541$	−1.51684e6	−0.222816	−0.111408	−	0.993775i	$-0.535536\pi$
$541$	−1.51684e6	−0.222816	−0.111408	+	0.993775i	$0.535536\pi$
$542$	−1.30652e6	−0.191037
$543$	1.88483e6	0.274330
$544$	−295936.	−0.0428746
$545$	3.15932e6	0.455620
$546$	1.52381e6	0.218750
$547$	−1.19516e7	−1.70788	−0.853940	−	0.520371i	$-0.825794\pi$
$547$	−1.19516e7	−1.70788	−0.853940	+	0.520371i	$0.825794\pi$
$548$	−5.04106e6	−0.717084
$549$	−1.12396e6	−0.159154
$550$	−5.48386e6	−0.773000
$551$	971082.	0.136263
$552$	2.88749e6	0.403341
$553$	7.51960e6	1.04564
$554$	8.22073e6	1.13798
$555$	−4.19612e6	−0.578250
$556$	−3.99376e6	−0.547892
$557$	6.26246e6	0.855277	0.427639	−	0.903950i	$-0.359346\pi$
$557$	6.26246e6	0.855277	0.427639	+	0.903950i	$0.359346\pi$
$558$	−2.46434e6	−0.335055
$559$	−6.08032e6	−0.822995
$560$	1.82477e6	0.245888
$561$	1.03780e6	0.139221
$562$	2.18515e6	0.291837
$563$	−9.21089e6	−1.22470	−0.612352	−	0.790586i	$-0.709776\pi$
$563$	−9.21089e6	−1.22470	−0.612352	+	0.790586i	$0.709776\pi$
$564$	−1.00829e6	−0.133471
$565$	7.32475e6	0.965321
$566$	−6.26260e6	−0.821701
$567$	−577368.	−0.0754215
$568$	1.32941e6	0.172897
$569$	3.39745e6	0.439919	0.219959	−	0.975509i	$-0.429407\pi$
$569$	3.39745e6	0.439919	0.219959	+	0.975509i	$0.429407\pi$
$570$	7.49120e6	0.965749
$571$	−7.73357e6	−0.992636	−0.496318	−	0.868141i	$-0.665315\pi$
$571$	−7.73357e6	−0.992636	−0.496318	+	0.868141i	$0.665315\pi$
$572$	3.07070e6	0.392417
$573$	5.68777e6	0.723694
$574$	5.69078e6	0.720929
$575$	1.72247e7	2.17261
$576$	331776.	0.0416667
$577$	−3.49268e6	−0.436736	−0.218368	−	0.975866i	$-0.570073\pi$
$577$	−3.49268e6	−0.436736	−0.218368	+	0.975866i	$0.570073\pi$
$578$	334084.	0.0415945
$579$	−806094.	−0.0999285
$580$	489888.	0.0604682
$581$	−6.72091e6	−0.826015
$582$	1.73736e6	0.212609
$583$	−7.02160e6	−0.855588
$584$	−3.62406e6	−0.439707
$585$	3.15584e6	0.381264
$586$	−3.92938e6	−0.472693
$587$	1.36571e6	0.163592	0.0817961	−	0.996649i	$-0.473934\pi$
$587$	1.36571e6	0.163592	0.0817961	+	0.996649i	$0.473934\pi$
$588$	−1.30507e6	−0.155665
$589$	1.95398e7	2.32077
$590$	−7.48246e6	−0.884941
$591$	−2.01709e6	−0.237551
$592$	1.47354e6	0.172805
$593$	1.32700e6	0.154965	0.0774827	−	0.996994i	$-0.475312\pi$
$593$	1.32700e6	0.154965	0.0774827	+	0.996994i	$0.475312\pi$
$594$	−1.16348e6	−0.135299
$595$	−2.05999e6	−0.238547
$596$	5.47498e6	0.631345
$597$	243180.	0.0279249
$598$	−9.64501e6	−1.10293
$599$	−1.31698e7	−1.49972	−0.749862	−	0.661594i	$-0.769880\pi$
$599$	−1.31698e7	−1.49972	−0.749862	+	0.661594i	$0.769880\pi$
$600$	1.97914e6	0.224439
$601$	4.30223e6	0.485856	0.242928	−	0.970044i	$-0.421892\pi$
$601$	4.30223e6	0.485856	0.242928	+	0.970044i	$0.421892\pi$
$602$	−4.44963e6	−0.500418
$603$	3.46583e6	0.388163
$604$	−7.26669e6	−0.810483
$605$	149850.	0.0166444
$606$	5.38877e6	0.596085
$607$	1.12581e7	1.24021	0.620103	−	0.784520i	$-0.287091\pi$
$607$	1.12581e7	1.24021	0.620103	+	0.784520i	$0.287091\pi$
$608$	−2.63066e6	−0.288606
$609$	299376.	0.0327095
$610$	4.49582e6	0.489198
$611$	3.36796e6	0.364976
$612$	−374544.	−0.0404226
$613$	−1.52743e7	−1.64176	−0.820881	−	0.571099i	$-0.806517\pi$
$613$	−1.52743e7	−1.64176	−0.820881	+	0.571099i	$0.806517\pi$
$614$	−2.63800e6	−0.282393
$615$	1.17857e7	1.25652
$616$	2.24717e6	0.238607
$617$	1.27963e7	1.35323	0.676616	−	0.736336i	$-0.263446\pi$
$617$	1.27963e7	1.35323	0.676616	+	0.736336i	$0.263446\pi$
$618$	1.90271e6	0.200401
$619$	−1.43991e6	−0.151046	−0.0755231	−	0.997144i	$-0.524063\pi$
$619$	−1.43991e6	−0.151046	−0.0755231	+	0.997144i	$0.524063\pi$
$620$	9.85738e6	1.02987
$621$	3.65448e6	0.380274
$622$	−63744.0	−0.00660638
$623$	−305184.	−0.0315023
$624$	−1.10822e6	−0.113937
$625$	−8.69703e6	−0.890576
$626$	1.04736e7	1.06822
$627$	9.22528e6	0.937153
$628$	−1.51086e6	−0.152871
$629$	−1.66348e6	−0.167646
$630$	2.30947e6	0.231826
$631$	−1.59066e6	−0.159039	−0.0795196	−	0.996833i	$-0.525339\pi$
$631$	−1.59066e6	−0.159039	−0.0795196	+	0.996833i	$0.525339\pi$
$632$	−5.46880e6	−0.544627
$633$	1.00507e6	0.0996977
$634$	8.67415e6	0.857046
$635$	2.49309e7	2.45360
$636$	2.53411e6	0.248418
$637$	4.35930e6	0.425666
$638$	603288.	0.0586777
$639$	1.68253e6	0.163009
$640$	−1.32710e6	−0.128072
$641$	−1.32235e7	−1.27116	−0.635581	−	0.772034i	$-0.719240\pi$
$641$	−1.32235e7	−1.27116	−0.635581	+	0.772034i	$0.719240\pi$
$642$	−5.60315e6	−0.536531
$643$	1.71707e7	1.63780	0.818898	−	0.573939i	$-0.194585\pi$
$643$	1.71707e7	1.63780	0.818898	+	0.573939i	$0.194585\pi$
$644$	−7.05830e6	−0.670634
$645$	−9.21529e6	−0.872187
$646$	2.96976e6	0.279989
$647$	−5.94655e6	−0.558475	−0.279238	−	0.960222i	$-0.590082\pi$
$647$	−5.94655e6	−0.558475	−0.279238	+	0.960222i	$0.590082\pi$
$648$	419904.	0.0392837
$649$	−9.21451e6	−0.858737
$650$	−6.61086e6	−0.613726
$651$	6.02395e6	0.557095
$652$	−3.43869e6	−0.316792
$653$	−2.03756e7	−1.86994	−0.934971	−	0.354725i	$-0.884574\pi$
$653$	−2.03756e7	−1.86994	−0.934971	+	0.354725i	$0.884574\pi$
$654$	−1.40414e6	−0.128371
$655$	2.21584e7	2.01807
$656$	−4.13875e6	−0.375500
$657$	−4.58671e6	−0.414560
$658$	2.46470e6	0.221922
$659$	1.57696e7	1.41451	0.707255	−	0.706959i	$-0.249933\pi$
$659$	1.57696e7	1.41451	0.707255	+	0.706959i	$0.249933\pi$
$660$	4.65394e6	0.415873
$661$	6.25946e6	0.557228	0.278614	−	0.960403i	$-0.410125\pi$
$661$	6.25946e6	0.557228	0.278614	+	0.960403i	$0.410125\pi$
$662$	−9.68357e6	−0.858797
$663$	1.25108e6	0.110535
$664$	4.88794e6	0.430235
$665$	−1.83118e7	−1.60575
$666$	1.86494e6	0.162922
$667$	−1.89491e6	−0.164921
$668$	−4.09675e6	−0.355221
$669$	3.61281e6	0.312090
$670$	−1.38633e7	−1.19311
$671$	5.53652e6	0.474713
$672$	−811008.	−0.0692791
$673$	−5.10006e6	−0.434048	−0.217024	−	0.976166i	$-0.569635\pi$
$673$	−5.10006e6	−0.434048	−0.217024	+	0.976166i	$0.569635\pi$
$674$	3.80660e6	0.322766
$675$	2.50484e6	0.211603
$676$	−2.23891e6	−0.188439
$677$	1.83463e7	1.53843	0.769213	−	0.638993i	$-0.220649\pi$
$677$	1.83463e7	1.53843	0.769213	+	0.638993i	$0.220649\pi$
$678$	−3.25544e6	−0.271979
$679$	−4.24688e6	−0.353505
$680$	1.49818e6	0.124248
$681$	−8.15667e6	−0.673977
$682$	1.21392e7	0.999375
$683$	−2.11430e7	−1.73427	−0.867133	−	0.498076i	$-0.834040\pi$
$683$	−2.11430e7	−1.73427	−0.867133	+	0.498076i	$0.834040\pi$
$684$	−3.32942e6	−0.272100
$685$	2.55203e7	2.07807
$686$	9.10624e6	0.738803
$687$	8.32217e6	0.672736
$688$	3.23610e6	0.260646
$689$	−8.46464e6	−0.679298
$690$	−1.46179e7	−1.16886
$691$	−1.79912e7	−1.43339	−0.716697	−	0.697385i	$-0.754347\pi$
$691$	−1.79912e7	−1.43339	−0.716697	+	0.697385i	$0.754347\pi$
$692$	4.66987e6	0.370715
$693$	2.84407e6	0.224961
$694$	1.71111e7	1.34859
$695$	2.02184e7	1.58776
$696$	−217728.	−0.0170369
$697$	4.67226e6	0.364288
$698$	9.14056e6	0.710124
$699$	−1.14844e7	−0.889030
$700$	−4.83789e6	−0.373173
$701$	−1.80797e7	−1.38962	−0.694810	−	0.719193i	$-0.744512\pi$
$701$	−1.80797e7	−1.38962	−0.694810	+	0.719193i	$0.744512\pi$
$702$	−1.40260e6	−0.107421
$703$	−1.47872e7	−1.12849
$704$	−1.63430e6	−0.124280
$705$	5.10446e6	0.386791
$706$	−1.04898e7	−0.792055
$707$	−1.31725e7	−0.991108
$708$	3.32554e6	0.249332
$709$	1.01375e7	0.757381	0.378691	−	0.925523i	$-0.376374\pi$
$709$	1.01375e7	0.757381	0.378691	+	0.925523i	$0.376374\pi$
$710$	−6.73013e6	−0.501046
$711$	−6.92145e6	−0.513479
$712$	221952.	0.0164081
$713$	−3.81289e7	−2.80886
$714$	915552.	0.0672106
$715$	−1.55454e7	−1.13720
$716$	1.90186e6	0.138642
$717$	−1.42085e7	−1.03217
$718$	−2.82634e6	−0.204603
$719$	−1.73064e7	−1.24849	−0.624244	−	0.781230i	$-0.714593\pi$
$719$	−1.73064e7	−1.24849	−0.624244	+	0.781230i	$0.714593\pi$
$720$	−1.67962e6	−0.120748
$721$	−4.65106e6	−0.333207
$722$	1.64946e7	1.17761
$723$	−6.67120e6	−0.474633
$724$	3.35082e6	0.237577
$725$	−1.29881e6	−0.0917698
$726$	−66600.0	−0.00468957
$727$	−2.19084e7	−1.53736	−0.768679	−	0.639635i	$-0.779086\pi$
$727$	−2.19084e7	−1.53736	−0.768679	+	0.639635i	$0.779086\pi$
$728$	2.70899e6	0.189443
$729$	531441.	0.0370370
$730$	1.83468e7	1.27425
$731$	−3.65325e6	−0.252863
$732$	−1.99814e6	−0.137832
$733$	−3.13622e6	−0.215599	−0.107800	−	0.994173i	$-0.534380\pi$
$733$	−3.13622e6	−0.215599	−0.107800	+	0.994173i	$0.534380\pi$
$734$	4.05421e6	0.277758
$735$	6.60693e6	0.451109
$736$	5.13331e6	0.349304
$737$	−1.70724e7	−1.15778
$738$	−5.23811e6	−0.354025
$739$	8.65798e6	0.583184	0.291592	−	0.956543i	$-0.405815\pi$
$739$	8.65798e6	0.583184	0.291592	+	0.956543i	$0.405815\pi$
$740$	−7.45978e6	−0.500779
$741$	1.11212e7	0.744057
$742$	−6.19450e6	−0.413044
$743$	−1.45778e7	−0.968771	−0.484386	−	0.874855i	$-0.660957\pi$
$743$	−1.45778e7	−0.968771	−0.484386	+	0.874855i	$0.660957\pi$
$744$	−4.38106e6	−0.290166
$745$	−2.77171e7	−1.82960
$746$	−1.16627e7	−0.767280
$747$	6.18629e6	0.405629
$748$	1.84498e6	0.120569
$749$	1.36966e7	0.892088
$750$	−906876.	−0.0588701
$751$	2.78641e7	1.80279	0.901397	−	0.432994i	$-0.142543\pi$
$751$	2.78641e7	1.80279	0.901397	+	0.432994i	$0.142543\pi$
$752$	−1.79251e6	−0.115589
$753$	−7.09258e6	−0.455844
$754$	727272.	0.0465874
$755$	3.67876e7	2.34874
$756$	−1.02643e6	−0.0653169
$757$	−8.57428e6	−0.543824	−0.271912	−	0.962322i	$-0.587656\pi$
$757$	−8.57428e6	−0.543824	−0.271912	+	0.962322i	$0.587656\pi$
$758$	−5.57858e6	−0.352655
$759$	−1.80017e7	−1.13425
$760$	1.33177e7	0.836364
$761$	1.46721e7	0.918397	0.459199	−	0.888334i	$-0.348137\pi$
$761$	1.46721e7	0.918397	0.459199	+	0.888334i	$0.348137\pi$
$762$	−1.10804e7	−0.691303
$763$	3.43235e6	0.213442
$764$	1.01116e7	0.626738
$765$	1.89613e6	0.117142
$766$	−2.57021e6	−0.158269
$767$	−1.11082e7	−0.681798
$768$	589824.	0.0360844
$769$	−1.71014e7	−1.04283	−0.521416	−	0.853302i	$-0.674596\pi$
$769$	−1.71014e7	−1.04283	−0.521416	+	0.853302i	$0.674596\pi$
$770$	−1.13763e7	−0.691471
$771$	4.47725e6	0.271253
$772$	−1.43306e6	−0.0865406
$773$	2.66279e6	0.160283	0.0801416	−	0.996783i	$-0.474463\pi$
$773$	2.66279e6	0.160283	0.0801416	+	0.996783i	$0.474463\pi$
$774$	4.09568e6	0.245739
$775$	−2.61342e7	−1.56299
$776$	3.08864e6	0.184125
$777$	−4.55875e6	−0.270890
$778$	3.18562e6	0.188688
$779$	4.15330e7	2.45217
$780$	5.61038e6	0.330184
$781$	−8.28803e6	−0.486210
$782$	−5.79503e6	−0.338874
$783$	−275562.	−0.0160626
$784$	−2.32013e6	−0.134810
$785$	7.64875e6	0.443013
$786$	−9.84820e6	−0.568591
$787$	9.90628e6	0.570130	0.285065	−	0.958508i	$-0.407985\pi$
$787$	9.90628e6	0.570130	0.285065	+	0.958508i	$0.407985\pi$
$788$	−3.58594e6	−0.205725
$789$	1.49590e7	0.855479
$790$	2.76858e7	1.57830
$791$	7.95775e6	0.452219
$792$	−2.06842e6	−0.117172
$793$	6.67436e6	0.376900
$794$	1.24951e7	0.703378
$795$	−1.28289e7	−0.719901
$796$	432320.	0.0241837
$797$	−3.29993e7	−1.84017	−0.920087	−	0.391714i	$-0.871882\pi$
$797$	−3.29993e7	−1.84017	−0.920087	+	0.391714i	$0.871882\pi$
$798$	8.13859e6	0.452420
$799$	2.02358e6	0.112138
$800$	3.51846e6	0.194370
$801$	280908.	0.0154697
$802$	9.01189e6	0.494743
$803$	2.25938e7	1.23652
$804$	6.16147e6	0.336159
$805$	3.57327e7	1.94346
$806$	1.46339e7	0.793458
$807$	3.38404e6	0.182916
$808$	9.58003e6	0.516224
$809$	1.70921e7	0.918174	0.459087	−	0.888391i	$-0.348177\pi$
$809$	1.70921e7	0.918174	0.459087	+	0.888391i	$0.348177\pi$
$810$	−2.12576e6	−0.113842
$811$	−3.37267e7	−1.80062	−0.900308	−	0.435252i	$-0.856659\pi$
$811$	−3.37267e7	−1.80062	−0.900308	+	0.435252i	$0.856659\pi$
$812$	532224.	0.0283272
$813$	−2.93966e6	−0.155981
$814$	−9.18658e6	−0.485951
$815$	1.74084e7	0.918045
$816$	−665856.	−0.0350070
$817$	−3.24747e7	−1.70212
$818$	8.88204e6	0.464119
$819$	3.42857e6	0.178609
$820$	2.09524e7	1.08818
$821$	−2.20851e7	−1.14351	−0.571756	−	0.820424i	$-0.693738\pi$
$821$	−2.20851e7	−1.14351	−0.571756	+	0.820424i	$0.693738\pi$
$822$	−1.13424e7	−0.585497
$823$	−2.83951e7	−1.46131	−0.730657	−	0.682745i	$-0.760786\pi$
$823$	−2.83951e7	−1.46131	−0.730657	+	0.682745i	$0.760786\pi$
$824$	3.38259e6	0.173553
$825$	−1.23387e7	−0.631152
$826$	−8.12909e6	−0.414564
$827$	−2.82908e6	−0.143841	−0.0719203	−	0.997410i	$-0.522913\pi$
$827$	−2.82908e6	−0.143841	−0.0719203	+	0.997410i	$0.522913\pi$
$828$	6.49685e6	0.329327
$829$	−6.99719e6	−0.353620	−0.176810	−	0.984245i	$-0.556578\pi$
$829$	−6.99719e6	−0.353620	−0.176810	+	0.984245i	$0.556578\pi$
$830$	−2.47452e7	−1.24680
$831$	1.84966e7	0.929159
$832$	−1.97018e6	−0.0986726
$833$	2.61921e6	0.130785
$834$	−8.98596e6	−0.447352
$835$	2.07398e7	1.02941
$836$	1.64005e7	0.811599
$837$	−5.54477e6	−0.273571
$838$	−1.54056e6	−0.0757824
$839$	1.22417e6	0.0600395	0.0300198	−	0.999549i	$-0.490443\pi$
$839$	1.22417e6	0.0600395	0.0300198	+	0.999549i	$0.490443\pi$
$840$	4.10573e6	0.200767
$841$	−2.03683e7	−0.993034
$842$	−1.66813e7	−0.810865
$843$	4.91659e6	0.238284
$844$	1.78678e6	0.0863408
$845$	1.13345e7	0.546085
$846$	−2.26865e6	−0.108979
$847$	162800.	0.00779733
$848$	4.50509e6	0.215136
$849$	−1.40908e7	−0.670916
$850$	−3.97202e6	−0.188566
$851$	2.88548e7	1.36582
$852$	2.99117e6	0.141170
$853$	2.03234e7	0.956367	0.478184	−	0.878260i	$-0.341295\pi$
$853$	2.03234e7	0.956367	0.478184	+	0.878260i	$0.341295\pi$
$854$	4.88435e6	0.229172
$855$	1.68552e7	0.788531
$856$	−9.96115e6	−0.464649
$857$	3.40476e7	1.58356	0.791779	−	0.610808i	$-0.209155\pi$
$857$	3.40476e7	1.58356	0.791779	+	0.610808i	$0.209155\pi$
$858$	6.90908e6	0.320407
$859$	1.00459e7	0.464520	0.232260	−	0.972654i	$-0.425388\pi$
$859$	1.00459e7	0.464520	0.232260	+	0.972654i	$0.425388\pi$
$860$	−1.63827e7	−0.755336
$861$	1.28043e7	0.588636
$862$	−1.12938e7	−0.517694
$863$	−2.12370e7	−0.970660	−0.485330	−	0.874331i	$-0.661300\pi$
$863$	−2.12370e7	−0.970660	−0.485330	+	0.874331i	$0.661300\pi$
$864$	746496.	0.0340207
$865$	−2.36412e7	−1.07431
$866$	9.81696e6	0.444818
$867$	751689.	0.0339618
$868$	1.07092e7	0.482458
$869$	3.40946e7	1.53157
$870$	1.10225e6	0.0493721
$871$	−2.05810e7	−0.919225
$872$	−2.49626e6	−0.111173
$873$	3.90906e6	0.173595
$874$	−5.15136e7	−2.28109
$875$	2.21681e6	0.0978832
$876$	−8.15414e6	−0.359020
$877$	3.22965e7	1.41794	0.708968	−	0.705241i	$-0.249161\pi$
$877$	3.22965e7	1.41794	0.708968	+	0.705241i	$0.249161\pi$
$878$	2.29421e7	1.00438
$879$	−8.84110e6	−0.385953
$880$	8.27366e6	0.360156
$881$	1.38409e7	0.600791	0.300396	−	0.953815i	$-0.402881\pi$
$881$	1.38409e7	0.600791	0.300396	+	0.953815i	$0.402881\pi$
$882$	−2.93641e6	−0.127100
$883$	−2.39082e6	−0.103192	−0.0515959	−	0.998668i	$-0.516431\pi$
$883$	−2.39082e6	−0.103192	−0.0515959	+	0.998668i	$0.516431\pi$
$884$	2.22414e6	0.0957265
$885$	−1.68355e7	−0.722551
$886$	7.59598e6	0.325087
$887$	−7.03672e6	−0.300304	−0.150152	−	0.988663i	$-0.547976\pi$
$887$	−7.03672e6	−0.300304	−0.150152	+	0.988663i	$0.547976\pi$
$888$	3.31546e6	0.141095
$889$	2.70854e7	1.14943
$890$	−1.12363e6	−0.0475498
$891$	−2.61784e6	−0.110471
$892$	6.42277e6	0.270278
$893$	1.79881e7	0.754845
$894$	1.23187e7	0.515491
$895$	−9.62815e6	−0.401777
$896$	−1.44179e6	−0.0599974
$897$	−2.17013e7	−0.900542
$898$	−1.21866e7	−0.504305
$899$	2.87507e6	0.118645
$900$	4.45306e6	0.183253
$901$	−5.08582e6	−0.208713
$902$	2.58025e7	1.05596
$903$	−1.00117e7	−0.408590
$904$	−5.78746e6	−0.235541
$905$	−1.69635e7	−0.688485
$906$	−1.63500e7	−0.661757
$907$	−1.12316e7	−0.453339	−0.226670	−	0.973972i	$-0.572784\pi$
$907$	−1.12316e7	−0.453339	−0.226670	+	0.973972i	$0.572784\pi$
$908$	−1.45008e7	−0.583681
$909$	1.21247e7	0.486701
$910$	−1.37143e7	−0.548996
$911$	1.52186e7	0.607546	0.303773	−	0.952744i	$-0.401753\pi$
$911$	1.52186e7	0.607546	0.303773	+	0.952744i	$0.401753\pi$
$912$	−5.91898e6	−0.235646
$913$	−3.04732e7	−1.20988
$914$	2.82281e7	1.11767
$915$	1.01156e7	0.399429
$916$	1.47950e7	0.582607
$917$	2.40734e7	0.945395
$918$	−842724.	−0.0330049
$919$	899507.	0.0351330	0.0175665	−	0.999846i	$-0.494408\pi$
$919$	899507.	0.0351330	0.0175665	+	0.999846i	$0.494408\pi$
$920$	−2.59874e7	−1.01226
$921$	−5.93550e6	−0.230573
$922$	2.92670e7	1.13384
$923$	−9.99133e6	−0.386028
$924$	5.05613e6	0.194822
$925$	1.97776e7	0.760011
$926$	−1.00249e7	−0.384194
$927$	4.28109e6	0.163627
$928$	−387072.	−0.0147544
$929$	1.79791e7	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$929$	1.79791e7	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$930$	2.21791e7	0.840885
$931$	2.32828e7	0.880363
$932$	−2.04168e7	−0.769922
$933$	−143424.	−0.00539408
$934$	2.74506e7	1.02964
$935$	−9.34019e6	−0.349403
$936$	−2.49350e6	−0.0930294
$937$	939506.	0.0349583	0.0174792	−	0.999847i	$-0.494436\pi$
$937$	939506.	0.0349583	0.0174792	+	0.999847i	$0.494436\pi$
$938$	−1.50614e7	−0.558930
$939$	2.35656e7	0.872196
$940$	9.07459e6	0.334971
$941$	4.75596e7	1.75091	0.875456	−	0.483298i	$-0.160561\pi$
$941$	4.75596e7	1.75091	0.875456	+	0.483298i	$0.160561\pi$
$942$	−3.39944e6	−0.124819
$943$	−8.10452e7	−2.96789
$944$	5.91206e6	0.215928
$945$	5.19631e6	0.189285
$946$	−2.01750e7	−0.732970
$947$	−2.35875e7	−0.854688	−0.427344	−	0.904089i	$-0.640551\pi$
$947$	−2.35875e7	−0.854688	−0.427344	+	0.904089i	$0.640551\pi$
$948$	−1.23048e7	−0.444686
$949$	2.72371e7	0.981738
$950$	−3.53083e7	−1.26931
$951$	1.95168e7	0.699775
$952$	1.62765e6	0.0582060
$953$	−5.97731e6	−0.213193	−0.106597	−	0.994302i	$-0.533995\pi$
$953$	−5.97731e6	−0.213193	−0.106597	+	0.994302i	$0.533995\pi$
$954$	5.70175e6	0.202832
$955$	−5.11899e7	−1.81625
$956$	−2.52595e7	−0.893882
$957$	1.35740e6	0.0479101
$958$	2.47365e7	0.870810
$959$	2.77258e7	0.973504
$960$	−2.98598e6	−0.104571
$961$	2.92221e7	1.02071
$962$	−1.10745e7	−0.385823
$963$	−1.26071e7	−0.438075
$964$	−1.18599e7	−0.411044
$965$	7.25485e6	0.250790
$966$	−1.58812e7	−0.547571
$967$	4.33006e7	1.48911	0.744557	−	0.667559i	$-0.232661\pi$
$967$	4.33006e7	1.48911	0.744557	+	0.667559i	$0.232661\pi$
$968$	−118400.	−0.00406128
$969$	6.68197e6	0.228610
$970$	−1.56362e7	−0.533584
$971$	−3.60511e7	−1.22707	−0.613536	−	0.789666i	$-0.710254\pi$
$971$	−3.60511e7	−1.22707	−0.613536	+	0.789666i	$0.710254\pi$
$972$	944784.	0.0320750
$973$	2.19657e7	0.743811
$974$	−4.66809e6	−0.157667
$975$	−1.48744e7	−0.501106
$976$	−3.55226e6	−0.119366
$977$	−2.56119e7	−0.858432	−0.429216	−	0.903202i	$-0.641210\pi$
$977$	−2.56119e7	−0.858432	−0.429216	+	0.903202i	$0.641210\pi$
$978$	−7.73705e6	−0.258659
$979$	−1.38373e6	−0.0461419
$980$	1.17456e7	0.390672
$981$	−3.15932e6	−0.104815
$982$	3.89321e6	0.128834
$983$	−2.52358e7	−0.832976	−0.416488	−	0.909141i	$-0.636739\pi$
$983$	−2.52358e7	−0.832976	−0.416488	+	0.909141i	$0.636739\pi$
$984$	−9.31219e6	−0.306594
$985$	1.81538e7	0.596179
$986$	436968.	0.0143139
$987$	5.54558e6	0.181198
$988$	1.97710e7	0.644372
$989$	6.33693e7	2.06010
$990$	1.04714e7	0.339559
$991$	−2.16758e7	−0.701117	−0.350558	−	0.936541i	$-0.614008\pi$
$991$	−2.16758e7	−0.701117	−0.350558	+	0.936541i	$0.614008\pi$
$992$	−7.78854e6	−0.251291
$993$	−2.17880e7	−0.701205
$994$	−7.31174e6	−0.234723
$995$	−2.18862e6	−0.0700830
$996$	1.09979e7	0.351285
$997$	8.28851e6	0.264082	0.132041	−	0.991244i	$-0.457847\pi$
$997$	8.28851e6	0.264082	0.132041	+	0.991244i	$0.457847\pi$
$998$	3.27274e7	1.04012
$999$	4.19612e6	0.133025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.6.a.e.1.1	✓	1
3.2	odd	2		306.6.a.f.1.1		1
4.3	odd	2		816.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.6.a.e.1.1	✓	1	1.1	even	1	trivial
306.6.a.f.1.1		1	3.2	odd	2
816.6.a.a.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 102 = 2 \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	102.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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