LMFDB - Embedded newform 3.68.a.b.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3,68,Mod(1,3)] 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("3.1"); S:= CuspForms(chi, 68); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 68, names="a")

Level:	$ N $	$=$	$ 3 $
Weight:	$ k $	$=$	$ 68 $
Character orbit:	$[\chi]$	$=$	3.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$85.2871055790$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3 x^{5} + \cdots - 80\!\cdots\!00 $ x^6 - 3x^5 - 18265801580559590892x^4 - 7523970115455178716247853536x^3 + 78501261395396048023472482629898922880x^2 + 29605717530431234535889524616359650095893868800*x - 80637078962263575025979556823830075867035962985404800000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{46}\cdot 3^{29}\cdot 5^{6}\cdot 7^{2}\cdot 11^{2}\cdot 13\cdot 17 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.58092e9$ of defining polynomial
Character	$\chi$	$=$	3.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.17747e10 q^{2} -5.55906e15 q^{3} -8.92992e18 q^{4} +8.66624e22 q^{5} -6.54564e25 q^{6} +3.38994e28 q^{7} -1.84279e30 q^{8} +3.09032e31 q^{9} +1.02043e33 q^{10} -2.84875e34 q^{11} +4.96420e34 q^{12} +5.77669e36 q^{13} +3.99156e38 q^{14} -4.81762e38 q^{15} -2.03805e40 q^{16} -1.58934e41 q^{17} +3.63876e41 q^{18} +1.30638e43 q^{19} -7.73889e41 q^{20} -1.88449e44 q^{21} -3.35432e44 q^{22} +5.95710e44 q^{23} +1.02442e46 q^{24} -6.02523e46 q^{25} +6.80189e46 q^{26} -1.71793e47 q^{27} -3.02719e47 q^{28} -1.88314e48 q^{29} -5.67261e48 q^{30} -9.76621e49 q^{31} +3.19730e49 q^{32} +1.58364e50 q^{33} -1.87140e51 q^{34} +2.93780e51 q^{35} -2.75963e50 q^{36} -2.00773e52 q^{37} +1.53823e53 q^{38} -3.21130e52 q^{39} -1.59701e53 q^{40} +1.76167e54 q^{41} -2.21893e54 q^{42} +5.65878e54 q^{43} +2.54391e53 q^{44} +2.67814e54 q^{45} +7.01432e54 q^{46} +1.13424e56 q^{47} +1.13296e56 q^{48} +7.30791e56 q^{49} -7.09453e56 q^{50} +8.83524e56 q^{51} -5.15854e55 q^{52} +5.57999e57 q^{53} -2.02281e57 q^{54} -2.46879e57 q^{55} -6.24694e58 q^{56} -7.26227e58 q^{57} -2.21734e58 q^{58} +2.54355e59 q^{59} +4.30210e57 q^{60} -4.65006e59 q^{61} -1.14994e60 q^{62} +1.04760e60 q^{63} +3.38410e60 q^{64} +5.00622e59 q^{65} +1.86469e60 q^{66} +7.13257e59 q^{67} +1.41927e60 q^{68} -3.31159e60 q^{69} +3.45918e61 q^{70} +1.41637e62 q^{71} -5.69480e61 q^{72} -2.86823e62 q^{73} -2.36405e62 q^{74} +3.34946e62 q^{75} -1.16659e62 q^{76} -9.65708e62 q^{77} -3.78121e62 q^{78} +2.81077e63 q^{79} -1.76622e63 q^{80} +9.55005e62 q^{81} +2.07431e64 q^{82} +3.57371e64 q^{83} +1.68283e63 q^{84} -1.37736e64 q^{85} +6.66306e64 q^{86} +1.04685e64 q^{87} +5.24964e64 q^{88} +2.86517e64 q^{89} +3.15344e64 q^{90} +1.95826e65 q^{91} -5.31964e63 q^{92} +5.42909e65 q^{93} +1.33553e66 q^{94} +1.13215e66 q^{95} -1.77740e65 q^{96} +4.69701e66 q^{97} +8.60485e66 q^{98} -8.80353e65 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 13735355166 q^{2} - 33\!\cdots\!38 q^{3} + 46\!\cdots\!52 q^{4} - 18\!\cdots\!00 q^{5} - 76\!\cdots\!18 q^{6} - 28\!\cdots\!08 q^{7} + 17\!\cdots\!92 q^{8} + 18\!\cdots\!74 q^{9} + 73\!\cdots\!00 q^{10}+ \cdots + 38\!\cdots\!96 q^{99}+O(q^{100}) $ 6 * q + 13735355166 * q^2 - 33354363399333138 * q^3 + 461137328518259214852 * q^4 - 180289059197853505182300 * q^5 - 76355671270945289663881818 * q^6 - 28067678610591277820235434208 * q^7 + 174428717775449559766832729592 * q^8 + 185418926295795674171523850821174 * q^9 + 7304190735814693245550459325330100 * q^10 + 1253554983811830341203783945028424 * q^11 - 2563490338732614404400323640044227596 * q^12 - 1032427018834954393824819765456935964 * q^13 - 447380929503565966859381489320173353952 * q^14 + 1002237799568181731536569743941186842900 * q^15 - 10777060270558286469269005744606427140080 * q^16 - 22303705201707705421583405421371675716596 * q^17 + 424465801195188392885382149078137607180814 * q^18 - 6709513794502265961382038704145157001302152 * q^19 + 46181758835098395119891481090112367056690200 * q^20 + 156029925358891883498812737570110890845530784 * q^21 + 1406045276258114363650986812146594678084267368 * q^22 + 12239031860950699209155226229708548540686529264 * q^23 - 969659806660344055206631047583378687513136616 * q^24 + 124984998689116279435075553443500491502742646250 * q^25 + 147728325197899719596176944836466085493373208692 * q^26 - 1030755041464022662072922259531242545404215044002 * q^27 - 14712260029031223829746951621139067346976206433344 * q^28 - 18873418589468921842047146695821329699174822684364 * q^29 - 40604438690067631054858099234477451658621953142300 * q^30 - 224044794880051525600176726995317503621741761218704 * q^31 - 647694914760137169573449462765066287313897316284448 * q^32 - 6968588078517493039340379725427973276022009185752 * q^33 + 693429763914603914127075176718731888866377974041404 * q^34 - 6444089679967517736201498858201618112886103095496000 * q^35 + 14250598054794536997029174176783125114340868782812708 * q^36 + 27414420131407696449125644786001907623181109991614452 * q^37 + 12552113658955455007053647987482106208988678055446104 * q^38 + 5739324328251871187904239839929767039225529061529172 * q^39 + 1194094866560066539991934179669093556164670510853198800 * q^40 + 2660548042321177320771169314506367536908331651920652796 * q^41 + 2487017683432229918848259981037750023330134118939476896 * q^42 - 1105961611152903745004520895209354208863582663043308600 * q^43 - 34817398461777279630054811602795668745922518275227441104 * q^44 - 5571500629890857041236058470500559035466867019928336700 * q^45 - 45061363378216725453974438646463842380148485881039366224 * q^46 + 58003762634770246900699775563009450632230835045286599360 * q^47 + 59910330773452765968526051655136782182831882309418661840 * q^48 + 985569916983315185097203394977137449656339808292459655670 * q^49 + 4791887184864825461869475209769884139573386495444648751250 * q^50 + 123987648074892602292236983996852174235365272762446559708 * q^51 - 772447381048143948901083488309399868612217101599766652776 * q^52 - 13674721974185597882029982508413224730588130474226494780636 * q^53 - 2359631097275567979107397504361474172402795777134631335722 * q^54 - 51238754618576674303553172437218137404754442562163094366800 * q^55 - 53382539406310631647595973723370754348378361469563798407040 * q^56 + 37298593555777863535226252514563917312860433886972407385496 * q^57 - 27334542479326918025552783071893481077859694969007500722812 * q^58 + 432173245375539353201786532564121354794334705181731449983112 * q^59 - 256727194434372614254225940479565827463386469783086909974600 * q^60 - 301681480554705175984062079070767927549030780778630557968540 * q^61 + 1319649268553974306953641963470020465251839304368009232497584 * q^62 - 867379805265217479240863693041421061959681291088645541720032 * q^63 - 3306273837825060723687391413275118884560206796975317864472512 * q^64 + 9144509139500418095807614952796559412618850636965585264371800 * q^65 - 7816290850038150086493121040028671432967827865802816749073464 * q^66 - 3659535002024051270225014883853538282826668950514180780948296 * q^67 + 62841982068874697565842994801948402899655187355391437927918088 * q^68 - 68037519391027690940233119583149188606433713409875206220325072 * q^69 - 17848574477080955030302460292958517135297977483696841850120000 * q^70 + 252155511033891103624460277167902994429076906738358010146768272 * q^71 + 5390397594179371118067083948912583358245775647599644848330168 * q^72 + 166779610709719144717689715238124518472612391435088881765181724 * q^73 + 1006325717889854285502211107525082701977251477599580185617852868 * q^74 - 694799177623660043822939209279000567990108449155940880572738750 * q^75 - 517707814526076465205777997582680722788148819042185613752993712 * q^76 + 352534620603367398775736559627568751080150267776772663929922944 * q^77 - 821230707170934959628477544327982939588983807972120804684205916 * q^78 - 4778664108337215079572742262604077882794836667809906425462078640 * q^79 + 22035563788144287210408960904426824039627000323030751017638600800 * q^80 + 5730029704780951421359144210646484071519610833846058779021123046 * q^81 + 64371214133499796977109337023506169313379178165909274002938946284 * q^82 + 36651193288697208236374412380857358965233031511415957283935392632 * q^83 + 81786344572298490402773289071356477096023706633672401039174558912 * q^84 + 251284152586026236911243321213724923566970783996009477585132585800 * q^85 + 109459654412686722699057856511584566836933916033705406023221718504 * q^86 + 104918477036812644409737917778017107986270152063164872212109942372 * q^87 + 294858564158317243888419031127963434466163056120837408554388440992 * q^88 - 24716082341461782003245292062820103472910477446347105193539129092 * q^89 + 225722533949076363264521082786586324992651210993012359817269922900 * q^90 + 1637449943486433383394466120659741425501433624430401587862051946176 * q^91 + 4192506910423434290693315227336082767370233979633693670230439475872 * q^92 + 1245478584359715172598232789326604176943806570590746356443962102192 * q^93 + 10776123537142027626725320636495971787147507990566237758042792171712 * q^94 + 8892438758299572900094061329818154779583617585762349202154337045200 * q^95 + 3600575259801619310258913629904116686086350553857582694711853406304 * q^96 + 3864203546244162410999773872217768224094576263762936458169046609164 * q^97 + 26255683859954817683324025976879398726779583524717348419525625709774 * q^98 + 38738803191855518251318864868786039478244708610819286864528508296 * q^99

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 $

$2$	1.17747e10	0.969272	0.484636	−	0.874716i	$-0.338952\pi$
$2$	1.17747e10	0.969272	0.484636	+	0.874716i	$0.338952\pi$
$3$	−5.55906e15	−0.577350
$3$	−5.55906e15	−0.577350
$4$	−8.92992e18	−0.0605115
$5$	8.66624e22	0.332917	0.166458	−	0.986048i	$-0.446767\pi$
$5$	8.66624e22	0.332917	0.166458	+	0.986048i	$0.446767\pi$
$6$	−6.54564e25	−0.559610
$7$	3.38994e28	1.65732	0.828662	−	0.559749i	$-0.189102\pi$
$7$	3.38994e28	1.65732	0.828662	+	0.559749i	$0.189102\pi$
$8$	−1.84279e30	−1.02792
$9$	3.09032e31	0.333333
$10$	1.02043e33	0.322687
$11$	−2.84875e34	−0.369827	−0.184914	−	0.982755i	$-0.559201\pi$
$11$	−2.84875e34	−0.369827	−0.184914	+	0.982755i	$0.559201\pi$
$12$	4.96420e34	0.0349363
$13$	5.77669e36	0.278341	0.139170	−	0.990268i	$-0.455556\pi$
$13$	5.77669e36	0.278341	0.139170	+	0.990268i	$0.455556\pi$
$14$	3.99156e38	1.60640
$15$	−4.81762e38	−0.192210
$16$	−2.03805e40	−0.935827
$17$	−1.58934e41	−0.957587	−0.478793	−	0.877928i	$-0.658926\pi$
$17$	−1.58934e41	−0.957587	−0.478793	+	0.877928i	$0.658926\pi$
$18$	3.63876e41	0.323091
$19$	1.30638e43	1.89594	0.947972	−	0.318354i	$-0.103130\pi$
$19$	1.30638e43	1.89594	0.947972	+	0.318354i	$0.103130\pi$
$20$	−7.73889e41	−0.0201453
$21$	−1.88449e44	−0.956857
$22$	−3.35432e44	−0.358463
$23$	5.95710e44	0.143599	0.0717997	−	0.997419i	$-0.477126\pi$
$23$	5.95710e44	0.143599	0.0717997	+	0.997419i	$0.477126\pi$
$24$	1.02442e46	0.593472
$25$	−6.02523e46	−0.889166
$26$	6.80189e46	0.269788
$27$	−1.71793e47	−0.192450
$28$	−3.02719e47	−0.100287
$29$	−1.88314e48	−0.192553	−0.0962763	−	0.995355i	$-0.530693\pi$
$29$	−1.88314e48	−0.192553	−0.0962763	+	0.995355i	$0.530693\pi$
$30$	−5.67261e48	−0.186303
$31$	−9.76621e49	−1.06932	−0.534662	−	0.845066i	$-0.679561\pi$
$31$	−9.76621e49	−1.06932	−0.534662	+	0.845066i	$0.679561\pi$
$32$	3.19730e49	0.120853
$33$	1.58364e50	0.213520
$34$	−1.87140e51	−0.928162
$35$	2.93780e51	0.551751
$36$	−2.75963e50	−0.0201705
$37$	−2.00773e52	−0.586067	−0.293034	−	0.956102i	$-0.594665\pi$
$37$	−2.00773e52	−0.586067	−0.293034	+	0.956102i	$0.594665\pi$
$38$	1.53823e53	1.83769
$39$	−3.21130e52	−0.160700
$40$	−1.59701e53	−0.342213
$41$	1.76167e54	1.65069	0.825344	−	0.564631i	$-0.190981\pi$
$41$	1.76167e54	1.65069	0.825344	+	0.564631i	$0.190981\pi$
$42$	−2.21893e54	−0.927454
$43$	5.65878e54	1.07530	0.537650	−	0.843168i	$-0.319312\pi$
$43$	5.65878e54	1.07530	0.537650	+	0.843168i	$0.319312\pi$
$44$	2.54391e53	0.0223788
$45$	2.67814e54	0.110972
$46$	7.01432e54	0.139187
$47$	1.13424e56	1.09503	0.547515	−	0.836796i	$-0.315574\pi$
$47$	1.13424e56	1.09503	0.547515	+	0.836796i	$0.315574\pi$
$48$	1.13296e56	0.540300
$49$	7.30791e56	1.74672
$50$	−7.09453e56	−0.861844
$51$	8.83524e56	0.552863
$52$	−5.15854e55	−0.0168428
$53$	5.57999e57	0.962481	0.481240	−	0.876589i	$-0.340186\pi$
$53$	5.57999e57	0.962481	0.481240	+	0.876589i	$0.340186\pi$
$54$	−2.02281e57	−0.186537
$55$	−2.46879e57	−0.123122
$56$	−6.24694e58	−1.70360
$57$	−7.26227e58	−1.09462
$58$	−2.21734e58	−0.186636
$59$	2.54355e59	1.20753	0.603764	−	0.797163i	$-0.293667\pi$
$59$	2.54355e59	1.20753	0.603764	+	0.797163i	$0.293667\pi$
$60$	4.30210e57	0.0116309
$61$	−4.65006e59	−0.722619	−0.361309	−	0.932446i	$-0.617670\pi$
$61$	−4.65006e59	−0.722619	−0.361309	+	0.932446i	$0.617670\pi$
$62$	−1.14994e60	−1.03647
$63$	1.04760e60	0.552441
$64$	3.38410e60	1.05297
$65$	5.00622e59	0.0926643
$66$	1.86469e60	0.206959
$67$	7.13257e59	0.0478344	0.0239172	−	0.999714i	$-0.492386\pi$
$67$	7.13257e59	0.0478344	0.0239172	+	0.999714i	$0.492386\pi$
$68$	1.41927e60	0.0579450
$69$	−3.31159e60	−0.0829071
$70$	3.45918e61	0.534797
$71$	1.41637e62	1.36152	0.680759	−	0.732507i	$-0.261650\pi$
$71$	1.41637e62	1.36152	0.680759	+	0.732507i	$0.261650\pi$
$72$	−5.69480e61	−0.342641
$73$	−2.86823e62	−1.08717	−0.543587	−	0.839353i	$-0.682934\pi$
$73$	−2.86823e62	−1.08717	−0.543587	+	0.839353i	$0.682934\pi$
$74$	−2.36405e62	−0.568059
$75$	3.34946e62	0.513360
$76$	−1.16659e62	−0.114726
$77$	−9.65708e62	−0.612924
$78$	−3.78121e62	−0.155762
$79$	2.81077e63	0.755646	0.377823	−	0.925878i	$-0.376673\pi$
$79$	2.81077e63	0.755646	0.377823	+	0.925878i	$0.376673\pi$
$80$	−1.76622e63	−0.311553
$81$	9.55005e62	0.111111
$82$	2.07431e64	1.59997
$83$	3.57371e64	1.83655	0.918275	−	0.395942i	$-0.129582\pi$
$83$	3.57371e64	1.83655	0.918275	+	0.395942i	$0.129582\pi$
$84$	1.68283e63	0.0579008
$85$	−1.37736e64	−0.318797
$86$	6.66306e64	1.04226
$87$	1.04685e64	0.111170
$88$	5.24964e64	0.380154
$89$	2.86517e64	0.142097	0.0710486	−	0.997473i	$-0.477365\pi$
$89$	2.86517e64	0.142097	0.0710486	+	0.997473i	$0.477365\pi$
$90$	3.15344e64	0.107562
$91$	1.95826e65	0.461301
$92$	−5.31964e63	−0.00868941
$93$	5.42909e65	0.617374
$94$	1.33553e66	1.06138
$95$	1.13215e66	0.631192
$96$	−1.77740e65	−0.0697747
$97$	4.69701e66	1.30307	0.651536	−	0.758618i	$-0.274125\pi$
$97$	4.69701e66	1.30307	0.651536	+	0.758618i	$0.274125\pi$
$98$	8.60485e66	1.69305
$99$	−8.80353e65	−0.123276
$100$	5.38048e65	0.0538048
$101$	1.32619e67	0.950253	0.475127	−	0.879917i	$-0.342402\pi$
$101$	1.32619e67	0.950253	0.475127	+	0.879917i	$0.342402\pi$
$102$	1.04032e67	0.535875
$103$	1.08115e67	0.401643	0.200821	−	0.979628i	$-0.435639\pi$
$103$	1.08115e67	0.401643	0.200821	+	0.979628i	$0.435639\pi$
$104$	−1.06452e67	−0.286113
$105$	−1.63314e67	−0.318554
$106$	6.57029e67	0.932906
$107$	−1.18331e68	−1.22671	−0.613356	−	0.789806i	$-0.710181\pi$
$107$	−1.18331e68	−1.22671	−0.613356	+	0.789806i	$0.710181\pi$
$108$	1.53409e66	0.0116454
$109$	−1.33578e68	−0.744643	−0.372321	−	0.928104i	$-0.621438\pi$
$109$	−1.33578e68	−0.744643	−0.372321	+	0.928104i	$0.621438\pi$
$110$	−2.90694e67	−0.119338
$111$	1.11611e68	0.338366
$112$	−6.90887e68	−1.55097
$113$	1.37803e68	0.229683	0.114842	−	0.993384i	$-0.463364\pi$
$113$	1.37803e68	0.229683	0.114842	+	0.993384i	$0.463364\pi$
$114$	−8.55112e68	−1.06099
$115$	5.16257e67	0.0478066
$116$	1.68163e67	0.0116516
$117$	1.78518e68	0.0927802
$118$	2.99496e69	1.17042
$119$	−5.38777e69	−1.58703
$120$	8.87785e68	0.197577
$121$	−5.12195e69	−0.863228
$122$	−5.47532e69	−0.700414
$123$	−9.79322e69	−0.953025
$124$	8.72115e68	0.0647064
$125$	−1.10941e70	−0.628935
$126$	1.23352e70	0.535466
$127$	1.20969e70	0.402949	0.201475	−	0.979494i	$-0.435427\pi$
$127$	1.20969e70	0.402949	0.201475	+	0.979494i	$0.435427\pi$
$128$	3.51285e70	0.899758
$129$	−3.14575e70	−0.620825
$130$	5.89468e69	0.0898169
$131$	7.83782e70	0.923862	0.461931	−	0.886916i	$-0.347157\pi$
$131$	7.83782e70	0.923862	0.461931	+	0.886916i	$0.347157\pi$
$132$	−1.41417e69	−0.0129204
$133$	4.42856e71	3.14219
$134$	8.39840e69	0.0463645
$135$	−1.48880e70	−0.0640699
$136$	2.92882e71	0.984326
$137$	−3.50919e71	−0.922716	−0.461358	−	0.887214i	$-0.652638\pi$
$137$	−3.50919e71	−0.922716	−0.461358	+	0.887214i	$0.652638\pi$
$138$	−3.89930e70	−0.0803595
$139$	−3.51334e71	−0.568492	−0.284246	−	0.958751i	$-0.591743\pi$
$139$	−3.51334e71	−0.568492	−0.284246	+	0.958751i	$0.591743\pi$
$140$	−2.62344e70	−0.0333873
$141$	−6.30530e71	−0.632216
$142$	1.66774e72	1.31968
$143$	−1.64563e71	−0.102938
$144$	−6.29822e71	−0.311942
$145$	−1.63197e71	−0.0641040
$146$	−3.37727e72	−1.05377
$147$	−4.06251e72	−1.00847
$148$	1.79289e71	0.0354638
$149$	2.55470e72	0.403273	0.201636	−	0.979460i	$-0.435374\pi$
$149$	2.55470e72	0.403273	0.201636	+	0.979460i	$0.435374\pi$
$150$	3.94389e72	0.497586
$151$	9.10289e72	0.919288	0.459644	−	0.888103i	$-0.347977\pi$
$151$	9.10289e72	0.919288	0.459644	+	0.888103i	$0.347977\pi$
$152$	−2.40739e73	−1.94889
$153$	−4.91156e72	−0.319196
$154$	−1.13709e73	−0.594090
$155$	−8.46364e72	−0.355996
$156$	2.86766e71	0.00972421
$157$	4.13316e72	0.113147	0.0565736	−	0.998398i	$-0.481982\pi$
$157$	4.13316e72	0.113147	0.0565736	+	0.998398i	$0.481982\pi$
$158$	3.30961e73	0.732426
$159$	−3.10195e73	−0.555689
$160$	2.77086e72	0.0402341
$161$	2.01942e73	0.237991
$162$	1.12449e73	0.107697
$163$	−6.44265e73	−0.502088	−0.251044	−	0.967976i	$-0.580774\pi$
$163$	−6.44265e73	−0.502088	−0.251044	+	0.967976i	$0.580774\pi$
$164$	−1.57316e73	−0.0998856
$165$	1.37242e73	0.0710844
$166$	4.20794e74	1.78012
$167$	4.51496e74	1.56190	0.780949	−	0.624595i	$-0.214736\pi$
$167$	4.51496e74	1.56190	0.780949	+	0.624595i	$0.214736\pi$
$168$	3.47271e74	0.983576
$169$	−3.97359e74	−0.922526
$170$	−1.62180e74	−0.309001
$171$	4.03714e74	0.631981
$172$	−5.05325e73	−0.0650680
$173$	1.01593e75	1.07726	0.538629	−	0.842543i	$-0.318942\pi$
$173$	1.01593e75	1.07726	0.538629	+	0.842543i	$0.318942\pi$
$174$	1.23263e74	0.107754
$175$	−2.04251e75	−1.47364
$176$	5.80589e74	0.346094
$177$	−1.41397e75	−0.697167
$178$	3.37366e74	0.137731
$179$	−5.16237e75	−1.74692	−0.873459	−	0.486897i	$-0.838129\pi$
$179$	−5.16237e75	−1.74692	−0.873459	+	0.486897i	$0.838129\pi$
$180$	−2.39156e73	−0.00671510
$181$	−7.57317e75	−1.76622	−0.883112	−	0.469163i	$-0.844556\pi$
$181$	−7.57317e75	−1.76622	−0.883112	+	0.469163i	$0.844556\pi$
$182$	2.30580e75	0.447126
$183$	2.58500e75	0.417204
$184$	−1.09777e75	−0.147609
$185$	−1.73995e75	−0.195112
$186$	6.39261e75	0.598404
$187$	4.52763e75	0.354142
$188$	−1.01287e75	−0.0662620
$189$	−5.82366e75	−0.318952
$190$	1.33307e76	0.611797
$191$	−3.59630e75	−0.138432	−0.0692162	−	0.997602i	$-0.522050\pi$
$191$	−3.59630e75	−0.138432	−0.0692162	+	0.997602i	$0.522050\pi$
$192$	−1.88124e76	−0.607931
$193$	−6.38661e75	−0.173420	−0.0867102	−	0.996234i	$-0.527635\pi$
$193$	−6.38661e75	−0.173420	−0.0867102	+	0.996234i	$0.527635\pi$
$194$	5.53060e76	1.26303
$195$	−2.78299e75	−0.0534998
$196$	−6.52590e75	−0.105697
$197$	−1.09438e77	−1.49468	−0.747340	−	0.664442i	$-0.768669\pi$
$197$	−1.09438e77	−1.49468	−0.747340	+	0.664442i	$0.768669\pi$
$198$	−1.03659e76	−0.119488
$199$	1.77255e77	1.72592	0.862960	−	0.505272i	$-0.168608\pi$
$199$	1.77255e77	1.72592	0.862960	+	0.505272i	$0.168608\pi$
$200$	1.11032e77	0.913996
$201$	−3.96504e75	−0.0276172
$202$	1.56155e77	0.921054
$203$	−6.38372e76	−0.319122
$204$	−7.88980e75	−0.0334546
$205$	1.52670e77	0.549542
$206$	1.27303e77	0.389301
$207$	1.84093e76	0.0478664
$208$	−1.17732e77	−0.260479
$209$	−3.72156e77	−0.701172
$210$	−1.92298e77	−0.308765
$211$	1.93889e77	0.265516	0.132758	−	0.991148i	$-0.457617\pi$
$211$	1.93889e77	0.265516	0.132758	+	0.991148i	$0.457617\pi$
$212$	−4.98289e76	−0.0582412
$213$	−7.87371e77	−0.786073
$214$	−1.39332e78	−1.18902
$215$	4.90404e77	0.357986
$216$	3.16577e77	0.197824
$217$	−3.31069e78	−1.77222
$218$	−1.57285e78	−0.721762
$219$	1.59447e78	0.627680
$220$	2.20461e76	0.00745028
$221$	−9.18113e77	−0.266535
$222$	1.31419e78	0.327969
$223$	2.20956e78	0.474343	0.237172	−	0.971468i	$-0.423780\pi$
$223$	2.20956e78	0.474343	0.237172	+	0.971468i	$0.423780\pi$
$224$	1.08386e78	0.200293
$225$	−1.86198e78	−0.296389
$226$	1.62259e78	0.222626
$227$	1.52442e79	1.80401	0.902006	−	0.431724i	$-0.142094\pi$
$227$	1.52442e79	1.80401	0.902006	+	0.431724i	$0.142094\pi$
$228$	6.48515e77	0.0662373
$229$	−8.58294e78	−0.757090	−0.378545	−	0.925583i	$-0.623575\pi$
$229$	−8.58294e78	−0.757090	−0.378545	+	0.925583i	$0.623575\pi$
$230$	6.07878e77	0.0463376
$231$	5.36843e78	0.353872
$232$	3.47023e78	0.197929
$233$	3.80136e79	1.87722	0.938611	−	0.344976i	$-0.112113\pi$
$233$	3.80136e79	1.87722	0.938611	+	0.344976i	$0.112113\pi$
$234$	2.10200e78	0.0899293
$235$	9.82959e78	0.364554
$236$	−2.27137e78	−0.0730694
$237$	−1.56253e79	−0.436272
$238$	−6.34394e79	−1.53827
$239$	−3.96682e79	−0.835820	−0.417910	−	0.908489i	$-0.637237\pi$
$239$	−3.96682e79	−0.835820	−0.417910	+	0.908489i	$0.637237\pi$
$240$	9.81855e78	0.179875
$241$	5.56846e79	0.887492	0.443746	−	0.896153i	$-0.353649\pi$
$241$	5.56846e79	0.887492	0.443746	+	0.896153i	$0.353649\pi$
$242$	−6.03095e79	−0.836703
$243$	−5.30893e78	−0.0641500
$244$	4.15247e78	0.0437267
$245$	6.33321e79	0.581514
$246$	−1.15312e80	−0.923740
$247$	7.54658e79	0.527718
$248$	1.79971e80	1.09918
$249$	−1.98665e80	−1.06033
$250$	−1.30630e80	−0.609609
$251$	−1.96582e80	−0.802553	−0.401277	−	0.915957i	$-0.631433\pi$
$251$	−1.96582e80	−0.802553	−0.401277	+	0.915957i	$0.631433\pi$
$252$	−9.35497e78	−0.0334291
$253$	−1.69703e79	−0.0531069
$254$	1.42438e80	0.390567
$255$	7.65683e79	0.184057
$256$	−8.57774e79	−0.180856
$257$	1.75539e80	0.324799	0.162399	−	0.986725i	$-0.448077\pi$
$257$	1.75539e80	0.324799	0.162399	+	0.986725i	$0.448077\pi$
$258$	−3.70403e80	−0.601748
$259$	−6.80608e80	−0.971304
$260$	−4.47052e78	−0.00560726
$261$	−5.81949e79	−0.0641842
$262$	9.22882e80	0.895473
$263$	−3.81404e79	−0.0325737	−0.0162869	−	0.999867i	$-0.505184\pi$
$263$	−3.81404e79	−0.0325737	−0.0162869	+	0.999867i	$0.505184\pi$
$264$	−2.91831e80	−0.219482
$265$	4.83576e80	0.320426
$266$	5.21451e81	3.04564
$267$	−1.59277e80	−0.0820398
$268$	−6.36933e78	−0.00289453
$269$	3.78256e81	1.51735	0.758673	−	0.651472i	$-0.225848\pi$
$269$	3.78256e81	1.51735	0.758673	+	0.651472i	$0.225848\pi$
$270$	−1.75302e80	−0.0621011
$271$	−5.36695e81	−1.67979	−0.839897	−	0.542746i	$-0.817385\pi$
$271$	−5.36695e81	−1.67979	−0.839897	+	0.542746i	$0.817385\pi$
$272$	3.23916e81	0.896135
$273$	−1.08861e81	−0.266332
$274$	−4.13197e81	−0.894363
$275$	1.71643e81	0.328838
$276$	2.95722e79	0.00501683
$277$	−1.61409e81	−0.242580	−0.121290	−	0.992617i	$-0.538703\pi$
$277$	−1.61409e81	−0.242580	−0.121290	+	0.992617i	$0.538703\pi$
$278$	−4.13686e81	−0.551023
$279$	−3.01807e81	−0.356441
$280$	−5.41375e81	−0.567158
$281$	4.09174e80	0.0380405	0.0190203	−	0.999819i	$-0.493945\pi$
$281$	4.09174e80	0.0380405	0.0190203	+	0.999819i	$0.493945\pi$
$282$	−7.42432e81	−0.612790
$283$	−2.49274e82	−1.82739	−0.913696	−	0.406399i	$-0.866784\pi$
$283$	−2.49274e82	−1.82739	−0.913696	+	0.406399i	$0.866784\pi$
$284$	−1.26481e81	−0.0823875
$285$	−6.29366e81	−0.364419
$286$	−1.93769e81	−0.0997749
$287$	5.97195e82	2.73572
$288$	9.88066e80	0.0402845
$289$	−2.28719e81	−0.0830281
$290$	−1.92160e81	−0.0621342
$291$	−2.61110e82	−0.752329
$292$	2.56131e81	0.0657865
$293$	3.33237e82	0.763286	0.381643	−	0.924310i	$-0.375358\pi$
$293$	3.33237e82	0.763286	0.381643	+	0.924310i	$0.375358\pi$
$294$	−4.78349e82	−0.977483
$295$	2.20430e82	0.402007
$296$	3.69982e82	0.602433
$297$	4.89393e81	0.0711733
$298$	3.00808e82	0.390881
$299$	3.44123e81	0.0399695
$300$	−2.99104e81	−0.0310642
$301$	1.91829e83	1.78212
$302$	1.07184e83	0.891040
$303$	−7.37236e82	−0.548629
$304$	−2.66248e83	−1.77428
$305$	−4.02986e82	−0.240572
$306$	−5.78323e82	−0.309387
$307$	1.72597e83	0.827749	0.413875	−	0.910334i	$-0.364175\pi$
$307$	1.72597e83	0.827749	0.413875	+	0.910334i	$0.364175\pi$
$308$	8.62370e81	0.0370889
$309$	−6.01019e82	−0.231889
$310$	−9.96569e82	−0.345057
$311$	6.02046e83	1.87136	0.935678	−	0.352854i	$-0.114789\pi$
$311$	6.02046e83	1.87136	0.935678	+	0.352854i	$0.114789\pi$
$312$	5.91774e82	0.165188
$313$	−7.50740e83	−1.88258	−0.941291	−	0.337595i	$-0.890386\pi$
$313$	−7.50740e83	−1.88258	−0.941291	+	0.337595i	$0.890386\pi$
$314$	4.86668e82	0.109670
$315$	9.07874e82	0.183917
$316$	−2.51000e82	−0.0457253
$317$	−5.32909e83	−0.873309	−0.436654	−	0.899629i	$-0.643837\pi$
$317$	−5.32909e83	−0.873309	−0.436654	+	0.899629i	$0.643837\pi$
$318$	−3.65246e83	−0.538613
$319$	5.36458e82	0.0712112
$320$	2.93275e83	0.350550
$321$	6.57810e83	0.708243
$322$	2.37781e83	0.230678
$323$	−2.07629e84	−1.81553
$324$	−8.52812e81	−0.00672350
$325$	−3.48059e83	−0.247491
$326$	−7.58605e83	−0.486660
$327$	7.42570e83	0.429920
$328$	−3.24638e84	−1.69678
$329$	3.84500e84	1.81482
$330$	1.61598e83	0.0689001
$331$	−5.64253e83	−0.217388	−0.108694	−	0.994075i	$-0.534667\pi$
$331$	−5.64253e83	−0.217388	−0.108694	+	0.994075i	$0.534667\pi$
$332$	−3.19129e83	−0.111132
$333$	−6.20452e83	−0.195356
$334$	5.31624e84	1.51390
$335$	6.18126e82	0.0159249
$336$	3.84068e84	0.895452
$337$	8.29452e84	1.75061	0.875305	−	0.483571i	$-0.160660\pi$
$337$	8.29452e84	1.75061	0.875305	+	0.483571i	$0.160660\pi$
$338$	−4.67880e84	−0.894179
$339$	−7.66053e83	−0.132608
$340$	1.22997e83	0.0192909
$341$	2.78215e84	0.395465
$342$	4.75362e84	0.612562
$343$	1.05906e85	1.23756
$344$	−1.04279e85	−1.10533
$345$	−2.86990e83	−0.0276012
$346$	1.19623e85	1.04416
$347$	−2.06934e85	−1.63981	−0.819907	−	0.572497i	$-0.805975\pi$
$347$	−2.06934e85	−1.63981	−0.819907	+	0.572497i	$0.805975\pi$
$348$	−9.34827e82	−0.00672708
$349$	−1.67071e85	−1.09207	−0.546035	−	0.837762i	$-0.683863\pi$
$349$	−1.67071e85	−1.09207	−0.546035	+	0.837762i	$0.683863\pi$
$350$	−2.40500e85	−1.42836
$351$	−9.92392e83	−0.0535667
$352$	−9.10829e83	−0.0446949
$353$	−1.33918e85	−0.597565	−0.298783	−	0.954321i	$-0.596581\pi$
$353$	−1.33918e85	−0.597565	−0.298783	+	0.954321i	$0.596581\pi$
$354$	−1.66491e85	−0.675745
$355$	1.22746e85	0.453272
$356$	−2.55858e83	−0.00859851
$357$	2.99509e85	0.916273
$358$	−6.07855e85	−1.69324
$359$	2.93947e85	0.745770	0.372885	−	0.927878i	$-0.378369\pi$
$359$	2.93947e85	0.745770	0.372885	+	0.927878i	$0.378369\pi$
$360$	−4.93525e84	−0.114071
$361$	1.23186e86	2.59460
$362$	−8.91720e85	−1.71195
$363$	2.84732e85	0.498385
$364$	−1.74871e84	−0.0279140
$365$	−2.48568e85	−0.361938
$366$	3.04376e85	0.404384
$367$	−3.43979e85	−0.417079	−0.208540	−	0.978014i	$-0.566871\pi$
$367$	−3.43979e85	−0.417079	−0.208540	+	0.978014i	$0.566871\pi$
$368$	−1.21409e85	−0.134384
$369$	5.44411e85	0.550229
$370$	−2.04874e85	−0.189116
$371$	1.89158e86	1.59514
$372$	−4.84814e84	−0.0373583
$373$	−8.47551e85	−0.596926	−0.298463	−	0.954421i	$-0.596474\pi$
$373$	−8.47551e85	−0.596926	−0.298463	+	0.954421i	$0.596474\pi$
$374$	5.33115e85	0.343260
$375$	6.16727e85	0.363116
$376$	−2.09016e86	−1.12561
$377$	−1.08783e85	−0.0535952
$378$	−6.85720e85	−0.309151
$379$	3.80867e86	1.57167	0.785833	−	0.618439i	$-0.212234\pi$
$379$	3.80867e86	1.57167	0.785833	+	0.618439i	$0.212234\pi$
$380$	−1.01100e85	−0.0381944
$381$	−6.72475e85	−0.232643
$382$	−4.23454e85	−0.134179
$383$	3.86709e86	1.12260	0.561301	−	0.827612i	$-0.310301\pi$
$383$	3.86709e86	1.12260	0.561301	+	0.827612i	$0.310301\pi$
$384$	−1.95281e86	−0.519475
$385$	−8.36906e85	−0.204053
$386$	−7.52005e85	−0.168092
$387$	1.74874e86	0.358433
$388$	−4.19439e85	−0.0788509
$389$	3.99336e86	0.688694	0.344347	−	0.938842i	$-0.388100\pi$
$389$	3.99336e86	0.688694	0.344347	+	0.938842i	$0.388100\pi$
$390$	−3.27689e85	−0.0518558
$391$	−9.46786e85	−0.137509
$392$	−1.34669e87	−1.79550
$393$	−4.35709e86	−0.533392
$394$	−1.28860e87	−1.44875
$395$	2.43588e86	0.251567
$396$	7.86148e84	0.00745960
$397$	−3.04225e85	−0.0265285	−0.0132642	−	0.999912i	$-0.504222\pi$
$397$	−3.04225e85	−0.0265285	−0.0132642	+	0.999912i	$0.504222\pi$
$398$	2.08713e87	1.67289
$399$	−2.46187e87	−1.81415
$400$	1.22797e87	0.832106
$401$	−9.25408e84	−0.00576762	−0.00288381	−	0.999996i	$-0.500918\pi$
$401$	−9.25408e84	−0.00576762	−0.00288381	+	0.999996i	$0.500918\pi$
$402$	−4.66872e85	−0.0267686
$403$	−5.64164e86	−0.297636
$404$	−1.18428e86	−0.0575013
$405$	8.27631e85	0.0369908
$406$	−7.51666e86	−0.309316
$407$	5.71951e86	0.216744
$408$	−1.62815e87	−0.568301
$409$	2.19777e87	0.706725	0.353363	−	0.935486i	$-0.385038\pi$
$409$	2.19777e87	0.706725	0.353363	+	0.935486i	$0.385038\pi$
$410$	1.79765e87	0.532655
$411$	1.95078e87	0.532730
$412$	−9.65461e85	−0.0243040
$413$	8.62247e87	2.00127
$414$	2.16765e86	0.0463956
$415$	3.09706e87	0.611419
$416$	1.84698e86	0.0336384
$417$	1.95309e87	0.328219
$418$	−4.38203e87	−0.679626
$419$	−7.19150e86	−0.102956	−0.0514778	−	0.998674i	$-0.516393\pi$
$419$	−7.19150e86	−0.102956	−0.0514778	+	0.998674i	$0.516393\pi$
$420$	1.45838e86	0.0192762
$421$	−2.30697e87	−0.281574	−0.140787	−	0.990040i	$-0.544963\pi$
$421$	−2.30697e87	−0.281574	−0.140787	+	0.990040i	$0.544963\pi$
$422$	2.28299e87	0.257357
$423$	3.50516e87	0.365010
$424$	−1.02828e88	−0.989357
$425$	9.57613e87	0.851454
$426$	−9.27108e87	−0.761919
$427$	−1.57634e88	−1.19761
$428$	1.05669e87	0.0742302
$429$	9.14817e86	0.0594313
$430$	5.77437e87	0.346985
$431$	−3.10236e88	−1.72466	−0.862329	−	0.506349i	$-0.830995\pi$
$431$	−3.10236e88	−1.72466	−0.862329	+	0.506349i	$0.830995\pi$
$432$	3.50122e87	0.180100
$433$	−1.16319e88	−0.553739	−0.276870	−	0.960908i	$-0.589297\pi$
$433$	−1.16319e88	−0.553739	−0.276870	+	0.960908i	$0.589297\pi$
$434$	−3.89824e88	−1.71776
$435$	9.07224e86	0.0370105
$436$	1.19284e87	0.0450595
$437$	7.78227e87	0.272256
$438$	1.87744e88	0.608393
$439$	6.19420e88	1.85962	0.929810	−	0.368041i	$-0.119971\pi$
$439$	6.19420e88	1.85962	0.929810	+	0.368041i	$0.119971\pi$
$440$	4.54947e87	0.126560
$441$	2.25837e88	0.582241
$442$	−1.08105e88	−0.258345
$443$	−2.39920e88	−0.531548	−0.265774	−	0.964035i	$-0.585627\pi$
$443$	−2.39920e88	−0.531548	−0.265774	+	0.964035i	$0.585627\pi$
$444$	−9.96677e86	−0.0204750
$445$	2.48303e87	0.0473065
$446$	2.60169e88	0.459768
$447$	−1.42017e88	−0.232830
$448$	1.14719e89	1.74511
$449$	−9.08623e88	−1.28272	−0.641359	−	0.767241i	$-0.721629\pi$
$449$	−9.08623e88	−1.28272	−0.641359	+	0.767241i	$0.721629\pi$
$450$	−2.19243e88	−0.287281
$451$	−5.01855e88	−0.610469
$452$	−1.23057e87	−0.0138985
$453$	−5.06035e88	−0.530751
$454$	1.79497e89	1.74858
$455$	1.69708e88	0.153575
$456$	1.33828e89	1.12519
$457$	−1.32640e88	−0.103629	−0.0518144	−	0.998657i	$-0.516500\pi$
$457$	−1.32640e88	−0.103629	−0.0518144	+	0.998657i	$0.516500\pi$
$458$	−1.01062e89	−0.733826
$459$	2.73037e88	0.184288
$460$	−4.61013e86	−0.00289285
$461$	−2.12429e89	−1.23946	−0.619729	−	0.784816i	$-0.712758\pi$
$461$	−2.12429e89	−1.23946	−0.619729	+	0.784816i	$0.712758\pi$
$462$	6.32117e88	0.342998
$463$	−1.44696e89	−0.730288	−0.365144	−	0.930951i	$-0.618980\pi$
$463$	−1.44696e89	−0.730288	−0.365144	+	0.930951i	$0.618980\pi$
$464$	3.83793e88	0.180196
$465$	4.70499e88	0.205534
$466$	4.47600e89	1.81954
$467$	5.46311e88	0.206692	0.103346	−	0.994645i	$-0.467045\pi$
$467$	5.46311e88	0.206692	0.103346	+	0.994645i	$0.467045\pi$
$468$	−1.59415e87	−0.00561427
$469$	2.41790e88	0.0792771
$470$	1.15741e89	0.353352
$471$	−2.29765e88	−0.0653256
$472$	−4.68722e89	−1.24125
$473$	−1.61204e89	−0.397675
$474$	−1.83983e89	−0.422867
$475$	−7.87126e89	−1.68581
$476$	4.81123e88	0.0960337
$477$	1.72439e89	0.320827
$478$	−4.67082e89	−0.810137
$479$	−4.23497e89	−0.684873	−0.342437	−	0.939541i	$-0.611252\pi$
$479$	−4.23497e89	−0.684873	−0.342437	+	0.939541i	$0.611252\pi$
$480$	−1.54034e88	−0.0232292
$481$	−1.15980e89	−0.163126
$482$	6.55671e89	0.860221
$483$	−1.12261e89	−0.137404
$484$	4.57386e88	0.0522352
$485$	4.07054e89	0.433815
$486$	−6.25112e88	−0.0621788
$487$	−4.84541e89	−0.449894	−0.224947	−	0.974371i	$-0.572221\pi$
$487$	−4.84541e89	−0.449894	−0.224947	+	0.974371i	$0.572221\pi$
$488$	8.56908e89	0.742797
$489$	3.58151e89	0.289881
$490$	7.45718e89	0.563645
$491$	−2.13083e89	−0.150424	−0.0752121	−	0.997168i	$-0.523963\pi$
$491$	−2.13083e89	−0.150424	−0.0752121	+	0.997168i	$0.523963\pi$
$492$	8.74527e88	0.0576690
$493$	2.99295e89	0.184386
$494$	8.88589e89	0.511503
$495$	−7.62935e88	−0.0410406
$496$	1.99040e90	1.00070
$497$	4.80142e90	2.25648
$498$	−2.33922e90	−1.02775
$499$	1.80425e90	0.741186	0.370593	−	0.928795i	$-0.379154\pi$
$499$	1.80425e90	0.741186	0.370593	+	0.928795i	$0.379154\pi$
$500$	9.90693e88	0.0380578
$501$	−2.50990e90	−0.901762
$502$	−2.31470e90	−0.777893
$503$	−1.06464e90	−0.334714	−0.167357	−	0.985896i	$-0.553523\pi$
$503$	−1.06464e90	−0.334714	−0.167357	+	0.985896i	$0.553523\pi$
$504$	−1.93050e90	−0.567868
$505$	1.14931e90	0.316355
$506$	−1.99820e89	−0.0514751
$507$	2.20894e90	0.532621
$508$	−1.08025e89	−0.0243831
$509$	3.06397e90	0.647498	0.323749	−	0.946143i	$-0.395057\pi$
$509$	3.06397e90	0.647498	0.323749	+	0.946143i	$0.395057\pi$
$510$	9.01571e89	0.178402
$511$	−9.72314e90	−1.80180
$512$	−6.19406e90	−1.07506
$513$	−2.24427e90	−0.364875
$514$	2.06692e90	0.314818
$515$	9.36953e89	0.133714
$516$	2.80913e89	0.0375671
$517$	−3.23116e90	−0.404972
$518$	−8.01397e90	−0.941458
$519$	−5.64762e90	−0.621955
$520$	−9.22541e89	−0.0952519
$521$	2.01669e90	0.195243	0.0976213	−	0.995224i	$-0.468877\pi$
$521$	2.01669e90	0.195243	0.0976213	+	0.995224i	$0.468877\pi$
$522$	−6.85229e89	−0.0622119
$523$	1.17519e91	1.00069	0.500346	−	0.865825i	$-0.333206\pi$
$523$	1.17519e91	1.00069	0.500346	+	0.865825i	$0.333206\pi$
$524$	−6.99911e89	−0.0559043
$525$	1.13545e91	0.850805
$526$	−4.49092e89	−0.0315728
$527$	1.55218e91	1.02397
$528$	−3.22753e90	−0.199818
$529$	−1.68545e91	−0.979379
$530$	5.69397e90	0.310580
$531$	7.86036e90	0.402510
$532$	−3.95467e90	−0.190139
$533$	1.01766e91	0.459453
$534$	−1.87544e90	−0.0795189
$535$	−1.02549e91	−0.408393
$536$	−1.31438e90	−0.0491701
$537$	2.86979e91	1.00858
$538$	4.45386e91	1.47072
$539$	−2.08184e91	−0.645986
$540$	1.32948e89	0.00387697
$541$	2.06689e91	0.566512	0.283256	−	0.959044i	$-0.408585\pi$
$541$	2.06689e91	0.566512	0.283256	+	0.959044i	$0.408585\pi$
$542$	−6.31943e91	−1.62818
$543$	4.20997e91	1.01973
$544$	−5.08159e90	−0.115728
$545$	−1.15762e91	−0.247904
$546$	−1.28181e91	−0.258148
$547$	−3.29187e91	−0.623545	−0.311773	−	0.950157i	$-0.600923\pi$
$547$	−3.29187e91	−0.623545	−0.311773	+	0.950157i	$0.600923\pi$
$548$	3.13368e90	0.0558349
$549$	−1.43702e91	−0.240873
$550$	2.02105e91	0.318733
$551$	−2.46010e91	−0.365069
$552$	6.10256e90	0.0852222
$553$	9.52835e91	1.25235
$554$	−1.90054e91	−0.235126
$555$	9.67248e90	0.112648
$556$	3.13738e90	0.0344003
$557$	9.99717e90	0.103211	0.0516057	−	0.998668i	$-0.483566\pi$
$557$	9.99717e90	0.103211	0.0516057	+	0.998668i	$0.483566\pi$
$558$	−3.55369e91	−0.345489
$559$	3.26890e91	0.299300
$560$	−5.98739e91	−0.516344
$561$	−2.51694e91	−0.204464
$562$	4.81791e90	0.0368716
$563$	−9.45819e91	−0.681988	−0.340994	−	0.940065i	$-0.610764\pi$
$563$	−9.45819e91	−0.681988	−0.340994	+	0.940065i	$0.610764\pi$
$564$	5.63059e90	0.0382564
$565$	1.19423e91	0.0764654
$566$	−2.93513e92	−1.77124
$567$	3.23741e91	0.184147
$568$	−2.61008e92	−1.39954
$569$	6.37286e91	0.322161	0.161081	−	0.986941i	$-0.448502\pi$
$569$	6.37286e91	0.322161	0.161081	+	0.986941i	$0.448502\pi$
$570$	−7.41061e91	−0.353221
$571$	8.06623e89	0.00362544	0.00181272	−	0.999998i	$-0.499423\pi$
$571$	8.06623e89	0.00362544	0.00181272	+	0.999998i	$0.499423\pi$
$572$	1.46954e90	0.00622893
$573$	1.99920e91	0.0799240
$574$	7.03180e92	2.65166
$575$	−3.58929e91	−0.127684
$576$	1.04579e92	0.350989
$577$	−4.61473e92	−1.46136	−0.730679	−	0.682721i	$-0.760796\pi$
$577$	−4.61473e92	−1.46136	−0.730679	+	0.682721i	$0.760796\pi$
$578$	−2.69311e91	−0.0804768
$579$	3.55035e91	0.100124
$580$	1.45734e90	0.00387903
$581$	1.21147e93	3.04376
$582$	−3.07449e92	−0.729211
$583$	−1.58960e92	−0.355952
$584$	5.28555e92	1.11753
$585$	1.54708e91	0.0308881
$586$	3.92377e92	0.739832
$587$	5.69889e92	1.01488	0.507439	−	0.861687i	$-0.330592\pi$
$587$	5.69889e92	1.01488	0.507439	+	0.861687i	$0.330592\pi$
$588$	3.62779e91	0.0610241
$589$	−1.27584e93	−2.02738
$590$	2.59550e92	0.389654
$591$	6.08370e92	0.862953
$592$	4.09185e92	0.548458
$593$	−3.99789e92	−0.506406	−0.253203	−	0.967413i	$-0.581484\pi$
$593$	−3.99789e92	−0.506406	−0.253203	+	0.967413i	$0.581484\pi$
$594$	5.76247e91	0.0689863
$595$	−4.66917e92	−0.528350
$596$	−2.28132e91	−0.0244027
$597$	−9.85372e92	−0.996460
$598$	4.05195e91	0.0387414
$599$	−2.15668e93	−1.94979	−0.974893	−	0.222672i	$-0.928522\pi$
$599$	−2.15668e93	−1.94979	−0.974893	+	0.222672i	$0.928522\pi$
$600$	−6.17235e92	−0.527696
$601$	−1.79477e93	−1.45115	−0.725577	−	0.688141i	$-0.758427\pi$
$601$	−1.79477e93	−1.45115	−0.725577	+	0.688141i	$0.758427\pi$
$602$	2.25874e93	1.72736
$603$	2.20419e91	0.0159448
$604$	−8.12881e91	−0.0556275
$605$	−4.43881e92	−0.287383
$606$	−8.68075e92	−0.531771
$607$	6.96333e92	0.403641	0.201821	−	0.979422i	$-0.435314\pi$
$607$	6.96333e92	0.403641	0.201821	+	0.979422i	$0.435314\pi$
$608$	4.17690e92	0.229131
$609$	3.54875e92	0.184245
$610$	−4.74504e92	−0.233180
$611$	6.55215e92	0.304792
$612$	4.38599e91	0.0193150
$613$	−2.53478e93	−1.05685	−0.528427	−	0.848979i	$-0.677218\pi$
$613$	−2.53478e93	−1.05685	−0.528427	+	0.848979i	$0.677218\pi$
$614$	2.03229e93	0.802314
$615$	−8.48704e92	−0.317278
$616$	1.77960e93	0.630039
$617$	−1.32260e93	−0.443480	−0.221740	−	0.975106i	$-0.571174\pi$
$617$	−1.32260e93	−0.443480	−0.221740	+	0.975106i	$0.571174\pi$
$618$	−7.07683e92	−0.224763
$619$	1.63639e93	0.492324	0.246162	−	0.969229i	$-0.420830\pi$
$619$	1.63639e93	0.492324	0.246162	+	0.969229i	$0.420830\pi$
$620$	7.55796e91	0.0215419
$621$	−1.02339e92	−0.0276357
$622$	7.08892e93	1.81385
$623$	9.71275e92	0.235501
$624$	6.54478e92	0.150387
$625$	3.12141e93	0.679783
$626$	−8.83975e93	−1.82473
$627$	2.06884e93	0.404822
$628$	−3.69088e91	−0.00684671
$629$	3.19097e93	0.561210
$630$	1.06900e93	0.178266
$631$	3.46704e93	0.548247	0.274123	−	0.961695i	$-0.411612\pi$
$631$	3.46704e93	0.548247	0.274123	+	0.961695i	$0.411612\pi$
$632$	−5.17966e93	−0.776747
$633$	−1.07784e93	−0.153296
$634$	−6.27486e93	−0.846474
$635$	1.04835e93	0.134149
$636$	2.77002e92	0.0336256
$637$	4.22155e93	0.486184
$638$	6.31665e92	0.0690230
$639$	4.37704e93	0.453839
$640$	3.04432e93	0.299545
$641$	5.70835e93	0.533048	0.266524	−	0.963828i	$-0.414125\pi$
$641$	5.70835e93	0.533048	0.266524	+	0.963828i	$0.414125\pi$
$642$	7.74553e93	0.686480
$643$	−1.13110e93	−0.0951551	−0.0475776	−	0.998868i	$-0.515150\pi$
$643$	−1.13110e93	−0.0951551	−0.0475776	+	0.998868i	$0.515150\pi$
$644$	−1.80333e92	−0.0144012
$645$	−2.72618e93	−0.206683
$646$	−2.44477e94	−1.75974
$647$	−2.31289e94	−1.58074	−0.790371	−	0.612629i	$-0.790112\pi$
$647$	−2.31289e94	−1.58074	−0.790371	+	0.612629i	$0.790112\pi$
$648$	−1.75987e93	−0.114214
$649$	−7.24592e93	−0.446577
$650$	−4.09829e93	−0.239886
$651$	1.84043e94	1.02319
$652$	5.75324e92	0.0303821
$653$	1.79330e94	0.899627	0.449813	−	0.893123i	$-0.351491\pi$
$653$	1.79330e94	0.899627	0.449813	+	0.893123i	$0.351491\pi$
$654$	8.74355e93	0.416709
$655$	6.79245e93	0.307569
$656$	−3.59037e94	−1.54476
$657$	−8.86375e93	−0.362391
$658$	4.52738e94	1.75906
$659$	−3.65704e94	−1.35042	−0.675210	−	0.737626i	$-0.735947\pi$
$659$	−3.65704e94	−1.35042	−0.675210	+	0.737626i	$0.735947\pi$
$660$	−1.22556e92	−0.00430142
$661$	−1.70005e93	−0.0567170	−0.0283585	−	0.999598i	$-0.509028\pi$
$661$	−1.70005e93	−0.0567170	−0.0283585	+	0.999598i	$0.509028\pi$
$662$	−6.64392e93	−0.210708
$663$	5.10384e93	0.153884
$664$	−6.58559e94	−1.88784
$665$	3.83790e94	1.04609
$666$	−7.30565e93	−0.189353
$667$	−1.12180e93	−0.0276504
$668$	−4.03183e93	−0.0945128
$669$	−1.22831e94	−0.273862
$670$	7.27826e92	0.0154355
$671$	1.32468e94	0.267244
$672$	−6.02527e93	−0.115639
$673$	4.91904e94	0.898205	0.449103	−	0.893480i	$-0.351744\pi$
$673$	4.91904e94	0.898205	0.449103	+	0.893480i	$0.351744\pi$
$674$	9.76656e94	1.69682
$675$	1.03509e94	0.171120
$676$	3.54839e93	0.0558235
$677$	−9.40701e93	−0.140842	−0.0704208	−	0.997517i	$-0.522434\pi$
$677$	−9.40701e93	−0.140842	−0.0704208	+	0.997517i	$0.522434\pi$
$678$	−9.02006e93	−0.128533
$679$	1.59226e95	2.15961
$680$	2.53819e94	0.327699
$681$	−8.47436e94	−1.04155
$682$	3.27590e94	0.383313
$683$	−6.94762e94	−0.774003	−0.387001	−	0.922079i	$-0.626489\pi$
$683$	−6.94762e94	−0.774003	−0.387001	+	0.922079i	$0.626489\pi$
$684$	−3.60514e93	−0.0382421
$685$	−3.04115e94	−0.307188
$686$	1.24701e95	1.19954
$687$	4.77131e94	0.437106
$688$	−1.15329e95	−1.00629
$689$	3.22339e94	0.267898
$690$	−3.37923e93	−0.0267530
$691$	1.28074e95	0.965929	0.482965	−	0.875640i	$-0.339560\pi$
$691$	1.28074e95	0.965929	0.482965	+	0.875640i	$0.339560\pi$
$692$	−9.07218e93	−0.0651865
$693$	−2.98434e94	−0.204308
$694$	−2.43659e95	−1.58943
$695$	−3.04474e94	−0.189260
$696$	−1.92912e94	−0.114275
$697$	−2.79989e95	−1.58068
$698$	−1.96722e95	−1.05851
$699$	−2.11320e95	−1.08382
$700$	1.82395e94	0.0891720
$701$	2.80240e95	1.30610	0.653049	−	0.757316i	$-0.273490\pi$
$701$	2.80240e95	1.30610	0.653049	+	0.757316i	$0.273490\pi$
$702$	−1.16851e94	−0.0519207
$703$	−2.62287e95	−1.11115
$704$	−9.64046e94	−0.389416
$705$	−5.46433e94	−0.210475
$706$	−1.57684e95	−0.579203
$707$	4.49570e95	1.57488
$708$	1.26267e94	0.0421866
$709$	2.25694e95	0.719235	0.359618	−	0.933100i	$-0.382907\pi$
$709$	2.25694e95	0.719235	0.359618	+	0.933100i	$0.382907\pi$
$710$	1.44531e95	0.439344
$711$	8.68618e94	0.251882
$712$	−5.27991e94	−0.146065
$713$	−5.81783e94	−0.153554
$714$	3.52664e95	0.888118
$715$	−1.42615e94	−0.0342698
$716$	4.60996e94	0.105709
$717$	2.20518e95	0.482561
$718$	3.46114e95	0.722854
$719$	−9.75088e95	−1.94368	−0.971842	−	0.235632i	$-0.924284\pi$
$719$	−9.75088e95	−1.94368	−0.971842	+	0.235632i	$0.924284\pi$
$720$	−5.45819e94	−0.103851
$721$	3.66504e95	0.665652
$722$	1.45048e96	2.51488
$723$	−3.09554e95	−0.512393
$724$	6.76278e94	0.106877
$725$	1.13463e95	0.171211
$726$	3.35264e95	0.483071
$727$	−5.69861e95	−0.784090	−0.392045	−	0.919946i	$-0.628232\pi$
$727$	−5.69861e95	−0.784090	−0.392045	+	0.919946i	$0.628232\pi$
$728$	−3.60866e95	−0.474182
$729$	2.95127e94	0.0370370
$730$	−2.92682e95	−0.350817
$731$	−8.99373e95	−1.02969
$732$	−2.30838e94	−0.0252457
$733$	5.44503e95	0.568876	0.284438	−	0.958694i	$-0.408193\pi$
$733$	5.44503e95	0.568876	0.284438	+	0.958694i	$0.408193\pi$
$734$	−4.05025e95	−0.404263
$735$	−3.52067e95	−0.335737
$736$	1.90466e94	0.0173545
$737$	−2.03189e94	−0.0176905
$738$	6.41029e95	0.533322
$739$	1.49500e96	1.18864	0.594321	−	0.804228i	$-0.297421\pi$
$739$	1.49500e96	1.18864	0.594321	+	0.804228i	$0.297421\pi$
$740$	1.55376e94	0.0118065
$741$	−4.19519e95	−0.304678
$742$	2.22729e96	1.54613
$743$	−1.54879e96	−1.02770	−0.513852	−	0.857879i	$-0.671782\pi$
$743$	−1.54879e96	−1.02770	−0.513852	+	0.857879i	$0.671782\pi$
$744$	−1.00047e96	−0.634614
$745$	2.21396e95	0.134256
$746$	−9.97968e95	−0.578583
$747$	1.10439e96	0.612184
$748$	−4.04314e94	−0.0214296
$749$	−4.01136e96	−2.03306
$750$	7.26179e95	0.351958
$751$	−8.73924e95	−0.405075	−0.202538	−	0.979274i	$-0.564919\pi$
$751$	−8.73924e95	−0.405075	−0.202538	+	0.979274i	$0.564919\pi$
$752$	−2.31164e96	−1.02476
$753$	1.09281e96	0.463354
$754$	−1.28089e95	−0.0519483
$755$	7.88878e95	0.306046
$756$	5.20048e94	0.0193003
$757$	−4.73020e95	−0.167945	−0.0839726	−	0.996468i	$-0.526761\pi$
$757$	−4.73020e95	−0.167945	−0.0839726	+	0.996468i	$0.526761\pi$
$758$	4.48461e96	1.52337
$759$	9.43388e94	0.0306613
$760$	−2.08630e96	−0.648817
$761$	−8.82131e95	−0.262511	−0.131255	−	0.991349i	$-0.541901\pi$
$761$	−8.82131e95	−0.262511	−0.131255	+	0.991349i	$0.541901\pi$
$762$	−7.91821e95	−0.225494
$763$	−4.52822e96	−1.23411
$764$	3.21147e94	0.00837675
$765$	−4.25648e95	−0.106266
$766$	4.55339e96	1.08811
$767$	1.46933e96	0.336104
$768$	4.76842e95	0.104418
$769$	−2.00203e96	−0.419699	−0.209849	−	0.977734i	$-0.567297\pi$
$769$	−2.00203e96	−0.419699	−0.209849	+	0.977734i	$0.567297\pi$
$770$	−9.85433e95	−0.197783
$771$	−9.75832e95	−0.187523
$772$	5.70319e94	0.0104939
$773$	4.95099e96	0.872326	0.436163	−	0.899868i	$-0.356337\pi$
$773$	4.95099e96	0.872326	0.436163	+	0.899868i	$0.356337\pi$
$774$	2.05909e96	0.347419
$775$	5.88436e96	0.950807
$776$	−8.65560e96	−1.33946
$777$	3.78354e96	0.560782
$778$	4.70206e96	0.667532
$779$	2.30142e97	3.12961
$780$	2.48519e94	0.00323735
$781$	−4.03489e96	−0.503526
$782$	−1.11481e96	−0.133283
$783$	3.23509e95	0.0370568
$784$	−1.48939e97	−1.63463
$785$	3.58190e95	0.0376686
$786$	−5.13035e96	−0.517002
$787$	2.38948e96	0.230754	0.115377	−	0.993322i	$-0.463192\pi$
$787$	2.38948e96	0.230754	0.115377	+	0.993322i	$0.463192\pi$
$788$	9.77269e95	0.0904453
$789$	2.12025e95	0.0188064
$790$	2.86819e96	0.243837
$791$	4.67142e96	0.380660
$792$	1.62230e96	0.126718
$793$	−2.68620e96	−0.201134
$794$	−3.58216e95	−0.0257133
$795$	−2.68823e96	−0.184998
$796$	−1.58287e96	−0.104438
$797$	1.68047e96	0.106311	0.0531555	−	0.998586i	$-0.483072\pi$
$797$	1.68047e96	0.106311	0.0531555	+	0.998586i	$0.483072\pi$
$798$	−2.89878e97	−1.75840
$799$	−1.80269e97	−1.04859
$800$	−1.92644e96	−0.107459
$801$	8.85428e95	0.0473657
$802$	−1.08964e95	−0.00559039
$803$	8.17087e96	0.402066
$804$	3.54075e94	0.00167116
$805$	1.75008e96	0.0792311
$806$	−6.64287e96	−0.288491
$807$	−2.10275e97	−0.876040
$808$	−2.44389e97	−0.976788
$809$	−2.72314e97	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$809$	−2.72314e97	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$810$	9.74512e95	0.0358541
$811$	2.37416e97	0.838129	0.419064	−	0.907957i	$-0.362358\pi$
$811$	2.37416e97	0.838129	0.419064	+	0.907957i	$0.362358\pi$
$812$	5.70062e95	0.0193106
$813$	2.98352e97	0.969830
$814$	6.73457e96	0.210084
$815$	−5.58336e96	−0.167154
$816$	−1.80067e97	−0.517384
$817$	7.39255e97	2.03871
$818$	2.58781e97	0.685009
$819$	6.05165e96	0.153767
$820$	−1.36334e96	−0.0332536
$821$	3.28742e97	0.769768	0.384884	−	0.922965i	$-0.374241\pi$
$821$	3.28742e97	0.769768	0.384884	+	0.922965i	$0.374241\pi$
$822$	2.29699e97	0.516361
$823$	−6.10619e97	−1.31788	−0.658940	−	0.752196i	$-0.728995\pi$
$823$	−6.10619e97	−1.31788	−0.658940	+	0.752196i	$0.728995\pi$
$824$	−1.99234e97	−0.412858
$825$	−9.54176e96	−0.189855
$826$	1.01527e98	1.93977
$827$	8.44596e97	1.54958	0.774791	−	0.632217i	$-0.217855\pi$
$827$	8.44596e97	1.54958	0.774791	+	0.632217i	$0.217855\pi$
$828$	−1.64394e95	−0.00289647
$829$	−9.39242e97	−1.58928	−0.794641	−	0.607080i	$-0.792341\pi$
$829$	−9.39242e97	−1.58928	−0.794641	+	0.607080i	$0.792341\pi$
$830$	3.64670e97	0.592631
$831$	8.97282e96	0.140054
$832$	1.95489e97	0.293083
$833$	−1.16147e98	−1.67264
$834$	2.29970e97	0.318133
$835$	3.91278e97	0.519982
$836$	3.32332e96	0.0424290
$837$	1.67776e97	0.205791
$838$	−8.46780e96	−0.0997920
$839$	−7.34342e97	−0.831520	−0.415760	−	0.909474i	$-0.636484\pi$
$839$	−7.34342e97	−0.831520	−0.415760	+	0.909474i	$0.636484\pi$
$840$	3.00954e97	0.327449
$841$	−9.20996e97	−0.962924
$842$	−2.71639e97	−0.272921
$843$	−2.27462e96	−0.0219627
$844$	−1.73141e96	−0.0160668
$845$	−3.44361e97	−0.307125
$846$	4.12722e97	0.353794
$847$	−1.73631e98	−1.43065
$848$	−1.13723e98	−0.900715
$849$	1.38573e98	1.05504
$850$	1.12756e98	0.825290
$851$	−1.19602e97	−0.0841589
$852$	7.03116e96	0.0475665
$853$	−1.13590e97	−0.0738833	−0.0369417	−	0.999317i	$-0.511762\pi$
$853$	−1.13590e97	−0.0738833	−0.0369417	+	0.999317i	$0.511762\pi$
$854$	−1.85610e98	−1.16081
$855$	3.49869e97	0.210397
$856$	2.18059e98	1.26097
$857$	1.84882e98	1.02810	0.514052	−	0.857759i	$-0.328144\pi$
$857$	1.84882e98	1.02810	0.514052	+	0.857759i	$0.328144\pi$
$858$	1.07717e97	0.0576051
$859$	2.66736e98	1.37186	0.685932	−	0.727666i	$-0.259395\pi$
$859$	2.66736e98	1.37186	0.685932	+	0.727666i	$0.259395\pi$
$860$	−4.37927e96	−0.0216622
$861$	−3.31984e98	−1.57947
$862$	−3.65294e98	−1.67166
$863$	−1.57014e98	−0.691158	−0.345579	−	0.938390i	$-0.612317\pi$
$863$	−1.57014e98	−0.691158	−0.345579	+	0.938390i	$0.612317\pi$
$864$	−5.49272e96	−0.0232582
$865$	8.80431e97	0.358637
$866$	−1.36962e98	−0.536724
$867$	1.27146e97	0.0479363
$868$	2.95642e97	0.107240
$869$	−8.00718e97	−0.279458
$870$	1.06823e97	0.0358732
$871$	4.12026e96	0.0133143
$872$	2.46157e98	0.765436
$873$	1.45152e98	0.434357
$874$	9.16340e97	0.263890
$875$	−3.76083e98	−1.04235
$876$	−1.42385e97	−0.0379819
$877$	3.11089e97	0.0798726	0.0399363	−	0.999202i	$-0.487284\pi$
$877$	3.11089e97	0.0798726	0.0399363	+	0.999202i	$0.487284\pi$
$878$	7.29350e98	1.80248
$879$	−1.85248e98	−0.440684
$880$	5.03153e97	0.115221
$881$	−1.64669e98	−0.363010	−0.181505	−	0.983390i	$-0.558097\pi$
$881$	−1.64669e98	−0.363010	−0.181505	+	0.983390i	$0.558097\pi$
$882$	2.65917e98	0.564350
$883$	7.37516e98	1.50691	0.753455	−	0.657499i	$-0.228386\pi$
$883$	7.37516e98	1.50691	0.753455	+	0.657499i	$0.228386\pi$
$884$	8.19867e96	0.0161285
$885$	−1.22538e98	−0.232099
$886$	−2.82500e98	−0.515215
$887$	−1.04918e98	−0.184250	−0.0921251	−	0.995747i	$-0.529366\pi$
$887$	−1.04918e98	−0.184250	−0.0921251	+	0.995747i	$0.529366\pi$
$888$	−2.05675e98	−0.347815
$889$	4.10078e98	0.667817
$890$	2.92369e97	0.0458529
$891$	−2.72057e97	−0.0410919
$892$	−1.97312e97	−0.0287032
$893$	1.48175e99	2.07612
$894$	−1.67221e98	−0.225675
$895$	−4.47384e98	−0.581579
$896$	1.19083e99	1.49119
$897$	−1.91300e97	−0.0230764
$898$	−1.06988e99	−1.24330
$899$	1.83911e98	0.205901
$900$	1.66274e97	0.0179349
$901$	−8.86851e98	−0.921659
$902$	−5.90920e98	−0.591711
$903$	−1.06639e99	−1.02891
$904$	−2.53941e98	−0.236097
$905$	−6.56310e98	−0.588005
$906$	−5.95842e98	−0.514442
$907$	−1.65700e99	−1.37873	−0.689365	−	0.724414i	$-0.742110\pi$
$907$	−1.65700e99	−1.37873	−0.689365	+	0.724414i	$0.742110\pi$
$908$	−1.36130e98	−0.109163
$909$	4.09834e98	0.316751
$910$	1.99826e98	0.148856
$911$	−1.04193e99	−0.748120	−0.374060	−	0.927405i	$-0.622035\pi$
$911$	−1.04193e99	−0.748120	−0.374060	+	0.927405i	$0.622035\pi$
$912$	1.48009e99	1.02438
$913$	−1.01806e99	−0.679207
$914$	−1.56179e98	−0.100445
$915$	2.24022e98	0.138894
$916$	7.66450e97	0.0458127
$917$	2.65697e99	1.53114
$918$	3.21493e98	0.178625
$919$	3.25435e98	0.174339	0.0871694	−	0.996194i	$-0.472218\pi$
$919$	3.25435e98	0.174339	0.0871694	+	0.996194i	$0.472218\pi$
$920$	−9.51353e97	−0.0491416
$921$	−9.59480e98	−0.477901
$922$	−2.50129e99	−1.20137
$923$	8.18196e98	0.378966
$924$	−4.79396e97	−0.0214133
$925$	1.20970e99	0.521111
$926$	−1.70376e99	−0.707848
$927$	3.34110e98	0.133881
$928$	−6.02095e97	−0.0232706
$929$	1.48846e99	0.554896	0.277448	−	0.960741i	$-0.410511\pi$
$929$	1.48846e99	0.554896	0.277448	+	0.960741i	$0.410511\pi$
$930$	5.53999e98	0.199219
$931$	9.54694e99	3.31169
$932$	−3.39459e98	−0.113594
$933$	−3.34681e99	−1.08043
$934$	6.43265e98	0.200341
$935$	3.92375e98	0.117900
$936$	−3.28971e98	−0.0953711
$937$	7.03087e98	0.196667	0.0983337	−	0.995153i	$-0.468649\pi$
$937$	7.03087e98	0.196667	0.0983337	+	0.995153i	$0.468649\pi$
$938$	2.84701e98	0.0768411
$939$	4.17341e99	1.08691
$940$	−8.77775e97	−0.0220597
$941$	2.70457e99	0.655913	0.327957	−	0.944693i	$-0.393640\pi$
$941$	2.70457e99	0.655913	0.327957	+	0.944693i	$0.393640\pi$
$942$	−2.70542e98	−0.0633182
$943$	1.04944e99	0.237038
$944$	−5.18388e99	−1.13004
$945$	−5.04693e98	−0.106185
$946$	−1.89814e99	−0.385456
$947$	−2.68543e99	−0.526367	−0.263184	−	0.964746i	$-0.584773\pi$
$947$	−2.68543e99	−0.526367	−0.263184	+	0.964746i	$0.584773\pi$
$948$	1.39532e98	0.0263995
$949$	−1.65689e99	−0.302605
$950$	−9.26819e99	−1.63401
$951$	2.96247e99	0.504205
$952$	9.92852e99	1.63135
$953$	−3.67501e99	−0.582970	−0.291485	−	0.956575i	$-0.594149\pi$
$953$	−3.67501e99	−0.582970	−0.291485	+	0.956575i	$0.594149\pi$
$954$	2.03043e99	0.310969
$955$	−3.11664e98	−0.0460865
$956$	3.54234e98	0.0505767
$957$	−2.98220e98	−0.0411138
$958$	−4.98656e99	−0.663829
$959$	−1.18959e100	−1.52924
$960$	−1.63033e99	−0.202390
$961$	1.19659e99	0.143454
$962$	−1.36564e99	−0.158114
$963$	−3.65681e99	−0.408904
$964$	−4.97259e98	−0.0537035
$965$	−5.53479e98	−0.0577346
$966$	−1.32184e99	−0.133182
$967$	−1.23398e100	−1.20094	−0.600471	−	0.799647i	$-0.705020\pi$
$967$	−1.23398e100	−1.20094	−0.600471	+	0.799647i	$0.705020\pi$
$968$	9.43867e99	0.887333
$969$	1.15422e100	1.04820
$970$	4.79295e99	0.420484
$971$	1.44651e100	1.22596	0.612982	−	0.790097i	$-0.289970\pi$
$971$	1.44651e100	1.22596	0.612982	+	0.790097i	$0.289970\pi$
$972$	4.74083e97	0.00388182
$973$	−1.19100e100	−0.942175
$974$	−5.70534e99	−0.436070
$975$	1.93488e99	0.142889
$976$	9.47706e99	0.676246
$977$	−5.88645e99	−0.405869	−0.202934	−	0.979192i	$-0.565048\pi$
$977$	−5.88645e99	−0.405869	−0.202934	+	0.979192i	$0.565048\pi$
$978$	4.21713e99	0.280973
$979$	−8.16215e98	−0.0525514
$980$	−5.65551e98	−0.0351883
$981$	−4.12799e99	−0.248214
$982$	−2.50899e99	−0.145802
$983$	7.65934e99	0.430179	0.215089	−	0.976594i	$-0.430996\pi$
$983$	7.65934e99	0.430179	0.215089	+	0.976594i	$0.430996\pi$
$984$	1.80468e100	0.979637
$985$	−9.48413e99	−0.497604
$986$	3.52411e99	0.178720
$987$	−2.13746e100	−1.04779
$988$	−6.73904e98	−0.0319330
$989$	3.37099e99	0.154412
$990$	−8.98335e98	−0.0397795
$991$	−1.25075e100	−0.535432	−0.267716	−	0.963498i	$-0.586269\pi$
$991$	−1.25075e100	−0.535432	−0.267716	+	0.963498i	$0.586269\pi$
$992$	−3.12255e99	−0.129231
$993$	3.13672e99	0.125509
$994$	5.65354e100	2.18714
$995$	1.53614e100	0.574588
$996$	1.77406e99	0.0641624
$997$	−2.47075e100	−0.864055	−0.432027	−	0.901861i	$-0.642202\pi$
$997$	−2.47075e100	−0.864055	−0.432027	+	0.901861i	$0.642202\pi$
$998$	2.12445e100	0.718411
$999$	3.44913e99	0.112789

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3.68.a.b.1.4	✓	6
3.2	odd	2		9.68.a.c.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.68.a.b.1.4	✓	6	1.1	even	1	trivial
9.68.a.c.1.3		6	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:	\( N \)	\(=\)	\( 3 \)
Weight:	\( k \)	\(=\)	\( 68 \)
Character orbit:	\([\chi]\)	\(=\)	3.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17747e10 q^{2} -5.55906e15 q^{3} -8.92992e18 q^{4} +8.66624e22 q^{5} -6.54564e25 q^{6} +3.38994e28 q^{7} -1.84279e30 q^{8} +3.09032e31 q^{9} +1.02043e33 q^{10} -2.84875e34 q^{11} +4.96420e34 q^{12} +5.77669e36 q^{13} +3.99156e38 q^{14} -4.81762e38 q^{15} -2.03805e40 q^{16} -1.58934e41 q^{17} +3.63876e41 q^{18} +1.30638e43 q^{19} -7.73889e41 q^{20} -1.88449e44 q^{21} -3.35432e44 q^{22} +5.95710e44 q^{23} +1.02442e46 q^{24} -6.02523e46 q^{25} +6.80189e46 q^{26} -1.71793e47 q^{27} -3.02719e47 q^{28} -1.88314e48 q^{29} -5.67261e48 q^{30} -9.76621e49 q^{31} +3.19730e49 q^{32} +1.58364e50 q^{33} -1.87140e51 q^{34} +2.93780e51 q^{35} -2.75963e50 q^{36} -2.00773e52 q^{37} +1.53823e53 q^{38} -3.21130e52 q^{39} -1.59701e53 q^{40} +1.76167e54 q^{41} -2.21893e54 q^{42} +5.65878e54 q^{43} +2.54391e53 q^{44} +2.67814e54 q^{45} +7.01432e54 q^{46} +1.13424e56 q^{47} +1.13296e56 q^{48} +7.30791e56 q^{49} -7.09453e56 q^{50} +8.83524e56 q^{51} -5.15854e55 q^{52} +5.57999e57 q^{53} -2.02281e57 q^{54} -2.46879e57 q^{55} -6.24694e58 q^{56} -7.26227e58 q^{57} -2.21734e58 q^{58} +2.54355e59 q^{59} +4.30210e57 q^{60} -4.65006e59 q^{61} -1.14994e60 q^{62} +1.04760e60 q^{63} +3.38410e60 q^{64} +5.00622e59 q^{65} +1.86469e60 q^{66} +7.13257e59 q^{67} +1.41927e60 q^{68} -3.31159e60 q^{69} +3.45918e61 q^{70} +1.41637e62 q^{71} -5.69480e61 q^{72} -2.86823e62 q^{73} -2.36405e62 q^{74} +3.34946e62 q^{75} -1.16659e62 q^{76} -9.65708e62 q^{77} -3.78121e62 q^{78} +2.81077e63 q^{79} -1.76622e63 q^{80} +9.55005e62 q^{81} +2.07431e64 q^{82} +3.57371e64 q^{83} +1.68283e63 q^{84} -1.37736e64 q^{85} +6.66306e64 q^{86} +1.04685e64 q^{87} +5.24964e64 q^{88} +2.86517e64 q^{89} +3.15344e64 q^{90} +1.95826e65 q^{91} -5.31964e63 q^{92} +5.42909e65 q^{93} +1.33553e66 q^{94} +1.13215e66 q^{95} -1.77740e65 q^{96} +4.69701e66 q^{97} +8.60485e66 q^{98} -8.80353e65 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 13735355166 q^{2} - 33\!\cdots\!38 q^{3} + 46\!\cdots\!52 q^{4} - 18\!\cdots\!00 q^{5} - 76\!\cdots\!18 q^{6} - 28\!\cdots\!08 q^{7} + 17\!\cdots\!92 q^{8} + 18\!\cdots\!74 q^{9} + 73\!\cdots\!00 q^{10}+ \cdots + 38\!\cdots\!96 q^{99}+O(q^{100}) \) 6 * q + 13735355166 * q^2 - 33354363399333138 * q^3 + 461137328518259214852 * q^4 - 180289059197853505182300 * q^5 - 76355671270945289663881818 * q^6 - 28067678610591277820235434208 * q^7 + 174428717775449559766832729592 * q^8 + 185418926295795674171523850821174 * q^9 + 7304190735814693245550459325330100 * q^10 + 1253554983811830341203783945028424 * q^11 - 2563490338732614404400323640044227596 * q^12 - 1032427018834954393824819765456935964 * q^13 - 447380929503565966859381489320173353952 * q^14 + 1002237799568181731536569743941186842900 * q^15 - 10777060270558286469269005744606427140080 * q^16 - 22303705201707705421583405421371675716596 * q^17 + 424465801195188392885382149078137607180814 * q^18 - 6709513794502265961382038704145157001302152 * q^19 + 46181758835098395119891481090112367056690200 * q^20 + 156029925358891883498812737570110890845530784 * q^21 + 1406045276258114363650986812146594678084267368 * q^22 + 12239031860950699209155226229708548540686529264 * q^23 - 969659806660344055206631047583378687513136616 * q^24 + 124984998689116279435075553443500491502742646250 * q^25 + 147728325197899719596176944836466085493373208692 * q^26 - 1030755041464022662072922259531242545404215044002 * q^27 - 14712260029031223829746951621139067346976206433344 * q^28 - 18873418589468921842047146695821329699174822684364 * q^29 - 40604438690067631054858099234477451658621953142300 * q^30 - 224044794880051525600176726995317503621741761218704 * q^31 - 647694914760137169573449462765066287313897316284448 * q^32 - 6968588078517493039340379725427973276022009185752 * q^33 + 693429763914603914127075176718731888866377974041404 * q^34 - 6444089679967517736201498858201618112886103095496000 * q^35 + 14250598054794536997029174176783125114340868782812708 * q^36 + 27414420131407696449125644786001907623181109991614452 * q^37 + 12552113658955455007053647987482106208988678055446104 * q^38 + 5739324328251871187904239839929767039225529061529172 * q^39 + 1194094866560066539991934179669093556164670510853198800 * q^40 + 2660548042321177320771169314506367536908331651920652796 * q^41 + 2487017683432229918848259981037750023330134118939476896 * q^42 - 1105961611152903745004520895209354208863582663043308600 * q^43 - 34817398461777279630054811602795668745922518275227441104 * q^44 - 5571500629890857041236058470500559035466867019928336700 * q^45 - 45061363378216725453974438646463842380148485881039366224 * q^46 + 58003762634770246900699775563009450632230835045286599360 * q^47 + 59910330773452765968526051655136782182831882309418661840 * q^48 + 985569916983315185097203394977137449656339808292459655670 * q^49 + 4791887184864825461869475209769884139573386495444648751250 * q^50 + 123987648074892602292236983996852174235365272762446559708 * q^51 - 772447381048143948901083488309399868612217101599766652776 * q^52 - 13674721974185597882029982508413224730588130474226494780636 * q^53 - 2359631097275567979107397504361474172402795777134631335722 * q^54 - 51238754618576674303553172437218137404754442562163094366800 * q^55 - 53382539406310631647595973723370754348378361469563798407040 * q^56 + 37298593555777863535226252514563917312860433886972407385496 * q^57 - 27334542479326918025552783071893481077859694969007500722812 * q^58 + 432173245375539353201786532564121354794334705181731449983112 * q^59 - 256727194434372614254225940479565827463386469783086909974600 * q^60 - 301681480554705175984062079070767927549030780778630557968540 * q^61 + 1319649268553974306953641963470020465251839304368009232497584 * q^62 - 867379805265217479240863693041421061959681291088645541720032 * q^63 - 3306273837825060723687391413275118884560206796975317864472512 * q^64 + 9144509139500418095807614952796559412618850636965585264371800 * q^65 - 7816290850038150086493121040028671432967827865802816749073464 * q^66 - 3659535002024051270225014883853538282826668950514180780948296 * q^67 + 62841982068874697565842994801948402899655187355391437927918088 * q^68 - 68037519391027690940233119583149188606433713409875206220325072 * q^69 - 17848574477080955030302460292958517135297977483696841850120000 * q^70 + 252155511033891103624460277167902994429076906738358010146768272 * q^71 + 5390397594179371118067083948912583358245775647599644848330168 * q^72 + 166779610709719144717689715238124518472612391435088881765181724 * q^73 + 1006325717889854285502211107525082701977251477599580185617852868 * q^74 - 694799177623660043822939209279000567990108449155940880572738750 * q^75 - 517707814526076465205777997582680722788148819042185613752993712 * q^76 + 352534620603367398775736559627568751080150267776772663929922944 * q^77 - 821230707170934959628477544327982939588983807972120804684205916 * q^78 - 4778664108337215079572742262604077882794836667809906425462078640 * q^79 + 22035563788144287210408960904426824039627000323030751017638600800 * q^80 + 5730029704780951421359144210646484071519610833846058779021123046 * q^81 + 64371214133499796977109337023506169313379178165909274002938946284 * q^82 + 36651193288697208236374412380857358965233031511415957283935392632 * q^83 + 81786344572298490402773289071356477096023706633672401039174558912 * q^84 + 251284152586026236911243321213724923566970783996009477585132585800 * q^85 + 109459654412686722699057856511584566836933916033705406023221718504 * q^86 + 104918477036812644409737917778017107986270152063164872212109942372 * q^87 + 294858564158317243888419031127963434466163056120837408554388440992 * q^88 - 24716082341461782003245292062820103472910477446347105193539129092 * q^89 + 225722533949076363264521082786586324992651210993012359817269922900 * q^90 + 1637449943486433383394466120659741425501433624430401587862051946176 * q^91 + 4192506910423434290693315227336082767370233979633693670230439475872 * q^92 + 1245478584359715172598232789326604176943806570590746356443962102192 * q^93 + 10776123537142027626725320636495971787147507990566237758042792171712 * q^94 + 8892438758299572900094061329818154779583617585762349202154337045200 * q^95 + 3600575259801619310258913629904116686086350553857582694711853406304 * q^96 + 3864203546244162410999773872217768224094576263762936458169046609164 * q^97 + 26255683859954817683324025976879398726779583524717348419525625709774 * q^98 + 38738803191855518251318864868786039478244708610819286864528508296 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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