LMFDB - Embedded newform 3.68.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3,68,Mod(1,3)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 3 $
Weight:	$ k $	$=$	$ 68 $
Character orbit:	$[\chi]$	$=$	3.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:	$85.2871055790$
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			1.3
Root			$9.72648e8$ 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)

$f(q)$	$=$	$q-3.54666e9 q^{2} -5.55906e15 q^{3} -1.34995e20 q^{4} +1.66367e23 q^{5} +1.97161e25 q^{6} -1.26991e28 q^{7} +1.00218e30 q^{8} +3.09032e31 q^{9} +O(q^{10})$	\(q-3.54666e9 q^{2} -5.55906e15 q^{3} -1.34995e20 q^{4} +1.66367e23 q^{5} +1.97161e25 q^{6} -1.26991e28 q^{7} +1.00218e30 q^{8} +3.09032e31 q^{9} -5.90048e32 q^{10} +3.05496e34 q^{11} +7.50446e35 q^{12} -2.14767e37 q^{13} +4.50395e37 q^{14} -9.24845e38 q^{15} +1.63674e40 q^{16} +1.02290e41 q^{17} -1.09603e41 q^{18} +2.26035e42 q^{19} -2.24588e43 q^{20} +7.05952e43 q^{21} -1.08349e44 q^{22} -7.58105e45 q^{23} -5.57116e45 q^{24} -4.00846e46 q^{25} +7.61705e46 q^{26} -1.71793e47 q^{27} +1.71432e48 q^{28} -4.33411e48 q^{29} +3.28011e48 q^{30} -3.83579e49 q^{31} -2.05945e50 q^{32} -1.69827e50 q^{33} -3.62787e50 q^{34} -2.11272e51 q^{35} -4.17178e51 q^{36} +3.81245e52 q^{37} -8.01669e51 q^{38} +1.19390e53 q^{39} +1.66729e53 q^{40} -6.60130e53 q^{41} -2.50377e53 q^{42} -5.55298e54 q^{43} -4.12405e54 q^{44} +5.14127e54 q^{45} +2.68874e55 q^{46} -2.64005e55 q^{47} -9.09873e55 q^{48} -2.57110e56 q^{49} +1.42167e56 q^{50} -5.68634e56 q^{51} +2.89925e57 q^{52} -1.96065e57 q^{53} +6.09290e56 q^{54} +5.08245e57 q^{55} -1.27268e58 q^{56} -1.25654e58 q^{57} +1.53716e58 q^{58} -1.16514e59 q^{59} +1.24850e59 q^{60} +1.04181e60 q^{61} +1.36043e59 q^{62} -3.92443e59 q^{63} -1.68498e60 q^{64} -3.57302e60 q^{65} +6.02320e59 q^{66} -5.71388e60 q^{67} -1.38086e61 q^{68} +4.21435e61 q^{69} +7.49308e60 q^{70} -3.93949e61 q^{71} +3.09704e61 q^{72} +2.89309e62 q^{73} -1.35215e62 q^{74} +2.22833e62 q^{75} -3.05136e62 q^{76} -3.87953e62 q^{77} -4.23437e62 q^{78} +3.53480e63 q^{79} +2.72299e63 q^{80} +9.55005e62 q^{81} +2.34126e63 q^{82} +1.40445e64 q^{83} -9.53001e63 q^{84} +1.70176e64 q^{85} +1.96945e64 q^{86} +2.40936e64 q^{87} +3.06161e64 q^{88} +3.15803e65 q^{89} -1.82343e64 q^{90} +2.72735e65 q^{91} +1.02341e66 q^{92} +2.13234e65 q^{93} +9.36338e64 q^{94} +3.76047e65 q^{95} +1.14486e66 q^{96} +7.03827e66 q^{97} +9.11883e65 q^{98} +9.44080e65 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 13735355166 q^{2} - 33\!\cdots\!38 q^{3}+ \cdots + 18\!\cdots\!74 q^{9}+O(q^{10}) $ 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	$ 6 q + 13735355166 q^{2} - 33\!\cdots\!38 q^{3}+ \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

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 $

$2$	−3.54666e9	−0.291954	−0.145977	−	0.989288i	$-0.546633\pi$
$2$	−3.54666e9	−0.291954	−0.145977	+	0.989288i	$0.546633\pi$
$3$	−5.55906e15	−0.577350
$3$	−5.55906e15	−0.577350
$4$	−1.34995e20	−0.914763
$5$	1.66367e23	0.639105	0.319553	−	0.947569i	$-0.396467\pi$
$5$	1.66367e23	0.639105	0.319553	+	0.947569i	$0.396467\pi$
$6$	1.97161e25	0.168560
$7$	−1.26991e28	−0.620854	−0.310427	−	0.950597i	$-0.600472\pi$
$7$	−1.26991e28	−0.620854	−0.310427	+	0.950597i	$0.600472\pi$
$8$	1.00218e30	0.559023
$9$	3.09032e31	0.333333
$10$	−5.90048e32	−0.186589
$11$	3.05496e34	0.396598	0.198299	−	0.980142i	$-0.436458\pi$
$11$	3.05496e34	0.396598	0.198299	+	0.980142i	$0.436458\pi$
$12$	7.50446e35	0.528139
$13$	−2.14767e37	−1.03482	−0.517410	−	0.855737i	$-0.673104\pi$
$13$	−2.14767e37	−1.03482	−0.517410	+	0.855737i	$0.673104\pi$
$14$	4.50395e37	0.181261
$15$	−9.24845e38	−0.368987
$16$	1.63674e40	0.751554
$17$	1.02290e41	0.616301	0.308151	−	0.951338i	$-0.400290\pi$
$17$	1.02290e41	0.616301	0.308151	+	0.951338i	$0.400290\pi$
$18$	−1.09603e41	−0.0973181
$19$	2.26035e42	0.328042	0.164021	−	0.986457i	$-0.447553\pi$
$19$	2.26035e42	0.328042	0.164021	+	0.986457i	$0.447553\pi$
$20$	−2.24588e43	−0.584630
$21$	7.05952e43	0.358450
$22$	−1.08349e44	−0.115789
$23$	−7.58105e45	−1.82746	−0.913728	−	0.406327i	$-0.866810\pi$
$23$	−7.58105e45	−1.82746	−0.913728	+	0.406327i	$0.866810\pi$
$24$	−5.57116e45	−0.322752
$25$	−4.00846e46	−0.591545
$26$	7.61705e46	0.302120
$27$	−1.71793e47	−0.192450
$28$	1.71432e48	0.567934
$29$	−4.33411e48	−0.443166	−0.221583	−	0.975141i	$-0.571122\pi$
$29$	−4.33411e48	−0.443166	−0.221583	+	0.975141i	$0.571122\pi$
$30$	3.28011e48	0.107727
$31$	−3.83579e49	−0.419989	−0.209995	−	0.977703i	$-0.567345\pi$
$31$	−3.83579e49	−0.419989	−0.209995	+	0.977703i	$0.567345\pi$
$32$	−2.05945e50	−0.778442
$33$	−1.69827e50	−0.228976
$34$	−3.62787e50	−0.179932
$35$	−2.11272e51	−0.396791
$36$	−4.17178e51	−0.304921
$37$	3.81245e52	1.11288	0.556438	−	0.830889i	$-0.312168\pi$
$37$	3.81245e52	1.11288	0.556438	+	0.830889i	$0.312168\pi$
$38$	−8.01669e51	−0.0957733
$39$	1.19390e53	0.597454
$40$	1.66729e53	0.357274
$41$	−6.60130e53	−0.618544	−0.309272	−	0.950974i	$-0.600085\pi$
$41$	−6.60130e53	−0.618544	−0.309272	+	0.950974i	$0.600085\pi$
$42$	−2.50377e53	−0.104651
$43$	−5.55298e54	−1.05520	−0.527598	−	0.849494i	$-0.676907\pi$
$43$	−5.55298e54	−1.05520	−0.527598	+	0.849494i	$0.676907\pi$
$44$	−4.12405e54	−0.362793
$45$	5.14127e54	0.213035
$46$	2.68874e55	0.533533
$47$	−2.64005e55	−0.254879	−0.127440	−	0.991846i	$-0.540676\pi$
$47$	−2.64005e55	−0.254879	−0.127440	+	0.991846i	$0.540676\pi$
$48$	−9.09873e55	−0.433910
$49$	−2.57110e56	−0.614541
$50$	1.42167e56	0.172704
$51$	−5.68634e56	−0.355822
$52$	2.89925e57	0.946616
$53$	−1.96065e57	−0.338188	−0.169094	−	0.985600i	$-0.554084\pi$
$53$	−1.96065e57	−0.338188	−0.169094	+	0.985600i	$0.554084\pi$
$54$	6.09290e56	0.0561866
$55$	5.08245e57	0.253468
$56$	−1.27268e58	−0.347071
$57$	−1.25654e58	−0.189395
$58$	1.53716e58	0.129384
$59$	−1.16514e59	−0.553139	−0.276570	−	0.960994i	$-0.589198\pi$
$59$	−1.16514e59	−0.553139	−0.276570	+	0.960994i	$0.589198\pi$
$60$	1.24850e59	0.337536
$61$	1.04181e60	1.61898	0.809488	−	0.587137i	$-0.199745\pi$
$61$	1.04181e60	1.61898	0.809488	+	0.587137i	$0.199745\pi$
$62$	1.36043e59	0.122618
$63$	−3.92443e59	−0.206951
$64$	−1.68498e60	−0.524284
$65$	−3.57302e60	−0.661359
$66$	6.02320e59	0.0668506
$67$	−5.71388e60	−0.383200	−0.191600	−	0.981473i	$-0.561368\pi$
$67$	−5.71388e60	−0.383200	−0.191600	+	0.981473i	$0.561368\pi$
$68$	−1.38086e61	−0.563769
$69$	4.21435e61	1.05508
$70$	7.49308e60	0.115845
$71$	−3.93949e61	−0.378691	−0.189345	−	0.981911i	$-0.560637\pi$
$71$	−3.93949e61	−0.378691	−0.189345	+	0.981911i	$0.560637\pi$
$72$	3.09704e61	0.186341
$73$	2.89309e62	1.09660	0.548298	−	0.836283i	$-0.315276\pi$
$73$	2.89309e62	1.09660	0.548298	+	0.836283i	$0.315276\pi$
$74$	−1.35215e62	−0.324909
$75$	2.22833e62	0.341528
$76$	−3.05136e62	−0.300081
$77$	−3.87953e62	−0.246230
$78$	−4.23437e62	−0.174429
$79$	3.53480e63	0.950292	0.475146	−	0.879907i	$-0.342395\pi$
$79$	3.53480e63	0.950292	0.475146	+	0.879907i	$0.342395\pi$
$80$	2.72299e63	0.480322
$81$	9.55005e62	0.111111
$82$	2.34126e63	0.180587
$83$	1.40445e64	0.721754	0.360877	−	0.932613i	$-0.382477\pi$
$83$	1.40445e64	0.721754	0.360877	+	0.932613i	$0.382477\pi$
$84$	−9.53001e63	−0.327897
$85$	1.70176e64	0.393881
$86$	1.96945e64	0.308069
$87$	2.40936e64	0.255862
$88$	3.06161e64	0.221708
$89$	3.15803e65	1.56622	0.783108	−	0.621886i	$-0.213633\pi$
$89$	3.15803e65	1.56622	0.783108	+	0.621886i	$0.213633\pi$
$90$	−1.82343e64	−0.0621965
$91$	2.72735e65	0.642472
$92$	1.02341e66	1.67169
$93$	2.13234e65	0.242481
$94$	9.36338e64	0.0744131
$95$	3.76047e65	0.209653
$96$	1.14486e66	0.449434
$97$	7.03827e66	1.95260	0.976298	−	0.216429i	$-0.0694410\pi$
$97$	7.03827e66	1.95260	0.976298	+	0.216429i	$0.0694410\pi$
$98$	9.11883e65	0.179418
$99$	9.44080e65	0.132199
$100$	5.41123e66	0.541123
$101$	−2.82872e66	−0.202686	−0.101343	−	0.994852i	$-0.532314\pi$
$101$	−2.82872e66	−0.202686	−0.101343	+	0.994852i	$0.532314\pi$
$102$	2.01675e66	0.103884
$103$	1.44038e67	0.535095	0.267547	−	0.963545i	$-0.413787\pi$
$103$	1.44038e67	0.535095	0.267547	+	0.963545i	$0.413787\pi$
$104$	−2.15234e67	−0.578489
$105$	1.17447e67	0.229087
$106$	6.95376e66	0.0987355
$107$	1.65663e68	1.71739	0.858694	−	0.512488i	$-0.171276\pi$
$107$	1.65663e68	1.71739	0.858694	+	0.512488i	$0.171276\pi$
$108$	2.31912e67	0.176046
$109$	−1.71048e68	−0.953523	−0.476761	−	0.879033i	$-0.658189\pi$
$109$	−1.71048e68	−0.953523	−0.476761	+	0.879033i	$0.658189\pi$
$110$	−1.80257e67	−0.0740011
$111$	−2.11937e68	−0.642520
$112$	−2.07851e68	−0.466605
$113$	−9.98863e68	−1.66486	−0.832431	−	0.554129i	$-0.813052\pi$
$113$	−9.98863e68	−1.66486	−0.832431	+	0.554129i	$0.813052\pi$
$114$	4.45652e67	0.0552947
$115$	−1.26124e69	−1.16794
$116$	5.85083e68	0.405392
$117$	−6.63698e68	−0.344940
$118$	4.13234e68	0.161491
$119$	−1.29899e69	−0.382633
$120$	−9.26858e68	−0.206273
$121$	−5.00021e69	−0.842710
$122$	−3.69496e69	−0.472667
$123$	3.66971e69	0.357117
$124$	5.17813e69	0.384191
$125$	−1.79422e70	−1.01716
$126$	1.39186e69	0.0604203
$127$	6.32448e69	0.210669	0.105334	−	0.994437i	$-0.466409\pi$
$127$	6.32448e69	0.210669	0.105334	+	0.994437i	$0.466409\pi$
$128$	3.63681e70	0.931509
$129$	3.08694e70	0.609217
$130$	1.26723e70	0.193087
$131$	7.81112e70	0.920714	0.460357	−	0.887734i	$-0.347721\pi$
$131$	7.81112e70	0.920714	0.460357	+	0.887734i	$0.347721\pi$
$132$	2.29259e70	0.209459
$133$	−2.87044e70	−0.203666
$134$	2.02652e70	0.111877
$135$	−2.85806e70	−0.122996
$136$	1.02512e71	0.344527
$137$	5.13243e71	1.34954	0.674768	−	0.738030i	$-0.264244\pi$
$137$	5.13243e71	1.34954	0.674768	+	0.738030i	$0.264244\pi$
$138$	−1.49469e71	−0.308036
$139$	1.33977e70	0.0216787	0.0108394	−	0.999941i	$-0.496550\pi$
$139$	1.33977e70	0.0216787	0.0108394	+	0.999941i	$0.496550\pi$
$140$	2.85206e71	0.362969
$141$	1.46762e71	0.147155
$142$	1.39720e71	0.110560
$143$	−6.56105e71	−0.410408
$144$	5.05804e71	0.250518
$145$	−7.21053e71	−0.283230
$146$	−1.02608e72	−0.320156
$147$	1.42929e72	0.354805
$148$	−5.14663e72	−1.01802
$149$	−7.90185e72	−1.24735	−0.623676	−	0.781683i	$-0.714362\pi$
$149$	−7.90185e72	−1.24735	−0.623676	+	0.781683i	$0.714362\pi$
$150$	−7.90312e71	−0.0997107
$151$	−1.21590e73	−1.22792	−0.613959	−	0.789338i	$-0.710424\pi$
$151$	−1.21590e73	−1.22792	−0.613959	+	0.789338i	$0.710424\pi$
$152$	2.26527e72	0.183383
$153$	3.16107e72	0.205434
$154$	1.37594e72	0.0718878
$155$	−6.38150e72	−0.268417
$156$	−1.61171e73	−0.546529
$157$	2.23950e73	0.613074	0.306537	−	0.951859i	$-0.400830\pi$
$157$	2.23950e73	0.613074	0.306537	+	0.951859i	$0.400830\pi$
$158$	−1.25367e73	−0.277442
$159$	1.08994e73	0.195253
$160$	−3.42624e73	−0.497506
$161$	9.62727e73	1.13458
$162$	−3.38708e72	−0.0324394
$163$	1.46906e74	1.14487	0.572434	−	0.819951i	$-0.305999\pi$
$163$	1.46906e74	1.14487	0.572434	+	0.819951i	$0.305999\pi$
$164$	8.91144e73	0.565821
$165$	−2.82537e73	−0.146340
$166$	−4.98110e73	−0.210719
$167$	6.76089e73	0.233885	0.116943	−	0.993139i	$-0.462691\pi$
$167$	6.76089e73	0.233885	0.116943	+	0.993139i	$0.462691\pi$
$168$	7.07488e73	0.200382
$169$	3.05190e73	0.0708542
$170$	−6.03558e73	−0.114995
$171$	6.98519e73	0.109347
$172$	7.49626e74	0.965254
$173$	−1.95747e74	−0.207563	−0.103782	−	0.994600i	$-0.533094\pi$
$173$	−1.95747e74	−0.207563	−0.103782	+	0.994600i	$0.533094\pi$
$174$	−8.54517e73	−0.0747000
$175$	5.09039e74	0.367263
$176$	5.00018e74	0.298065
$177$	6.47706e74	0.319355
$178$	−1.12005e75	−0.457263
$179$	−5.44647e75	−1.84306	−0.921528	−	0.388311i	$-0.873059\pi$
$179$	−5.44647e75	−1.84306	−0.921528	+	0.388311i	$0.873059\pi$
$180$	−6.94046e74	−0.194877
$181$	−1.80801e74	−0.0421666	−0.0210833	−	0.999778i	$-0.506712\pi$
$181$	−1.80801e74	−0.0421666	−0.0210833	+	0.999778i	$0.506712\pi$
$182$	−9.67299e74	−0.187572
$183$	−5.79150e75	−0.934716
$184$	−7.59755e75	−1.02159
$185$	6.34267e75	0.711245
$186$	−7.56269e74	−0.0707933
$187$	3.12491e75	0.244424
$188$	3.56395e75	0.233154
$189$	2.18161e75	0.119483
$190$	−1.33371e75	−0.0612092
$191$	−1.51865e75	−0.0584575	−0.0292287	−	0.999573i	$-0.509305\pi$
$191$	−1.51865e75	−0.0584575	−0.0292287	+	0.999573i	$0.509305\pi$
$192$	9.36693e75	0.302696
$193$	−3.48124e76	−0.945288	−0.472644	−	0.881253i	$-0.656700\pi$
$193$	−3.48124e76	−0.945288	−0.472644	+	0.881253i	$0.656700\pi$
$194$	−2.49623e76	−0.570069
$195$	1.98626e76	0.381836
$196$	3.47086e76	0.562159
$197$	1.09636e77	1.49738	0.748691	−	0.662919i	$-0.230682\pi$
$197$	1.09636e77	1.49738	0.748691	+	0.662919i	$0.230682\pi$
$198$	−3.34833e75	−0.0385962
$199$	1.76321e77	1.71683	0.858413	−	0.512959i	$-0.171451\pi$
$199$	1.76321e77	1.71683	0.858413	+	0.512959i	$0.171451\pi$
$200$	−4.01719e76	−0.330687
$201$	3.17638e76	0.221241
$202$	1.00325e76	0.0591751
$203$	5.50393e76	0.275141
$204$	7.67629e76	0.325492
$205$	−1.09824e77	−0.395315
$206$	−5.10855e76	−0.156223
$207$	−2.34278e77	−0.609152
$208$	−3.51517e77	−0.777723
$209$	6.90528e76	0.130101
$210$	−4.16545e76	−0.0668830
$211$	−4.62226e77	−0.632983	−0.316491	−	0.948595i	$-0.602505\pi$
$211$	−4.62226e77	−0.632983	−0.316491	+	0.948595i	$0.602505\pi$
$212$	2.64678e77	0.309362
$213$	2.18998e77	0.218637
$214$	−5.87550e77	−0.501399
$215$	−9.23833e77	−0.674381
$216$	−1.72166e77	−0.107584
$217$	4.87112e77	0.260752
$218$	6.06650e77	0.278385
$219$	−1.60829e78	−0.633120
$220$	−6.86107e77	−0.231863
$221$	−2.19684e78	−0.637761
$222$	7.51667e77	0.187586
$223$	4.30563e77	0.0924324	0.0462162	−	0.998931i	$-0.485284\pi$
$223$	4.30563e77	0.0924324	0.0462162	+	0.998931i	$0.485284\pi$
$224$	2.61532e78	0.483299
$225$	−1.23874e78	−0.197182
$226$	3.54263e78	0.486063
$227$	−5.95123e78	−0.704272	−0.352136	−	0.935949i	$-0.614545\pi$
$227$	−5.95123e78	−0.704272	−0.352136	+	0.935949i	$0.614545\pi$
$228$	1.69627e78	0.173252
$229$	1.26941e79	1.11973	0.559863	−	0.828585i	$-0.310854\pi$
$229$	1.26941e79	1.11973	0.559863	+	0.828585i	$0.310854\pi$
$230$	4.47318e78	0.340984
$231$	2.15666e78	0.142161
$232$	−4.34354e78	−0.247740
$233$	2.79863e79	1.38204	0.691022	−	0.722834i	$-0.257161\pi$
$233$	2.79863e79	1.38204	0.691022	+	0.722834i	$0.257161\pi$
$234$	2.35391e78	0.100707
$235$	−4.39218e78	−0.162895
$236$	1.57288e79	0.505991
$237$	−1.96502e79	−0.548651
$238$	4.60707e78	0.111711
$239$	3.49525e79	0.736458	0.368229	−	0.929735i	$-0.379964\pi$
$239$	3.49525e79	0.736458	0.368229	+	0.929735i	$0.379964\pi$
$240$	−1.51373e79	−0.277314
$241$	−1.86931e79	−0.297928	−0.148964	−	0.988843i	$-0.547594\pi$
$241$	−1.86931e79	−0.297928	−0.148964	+	0.988843i	$0.547594\pi$
$242$	1.77340e79	0.246033
$243$	−5.30893e78	−0.0641500
$244$	−1.40640e80	−1.48098
$245$	−4.27747e79	−0.392756
$246$	−1.30152e79	−0.104262
$247$	−4.85448e79	−0.339465
$248$	−3.84414e79	−0.234784
$249$	−7.80741e79	−0.416705
$250$	6.36350e79	0.296965
$251$	−7.13517e79	−0.291296	−0.145648	−	0.989336i	$-0.546527\pi$
$251$	−7.13517e79	−0.291296	−0.145648	+	0.989336i	$0.546527\pi$
$252$	5.29779e79	0.189311
$253$	−2.31598e80	−0.724766
$254$	−2.24308e79	−0.0615056
$255$	−9.46021e79	−0.227407
$256$	1.19674e80	0.252326
$257$	7.32793e80	1.35588	0.677940	−	0.735117i	$-0.262873\pi$
$257$	7.32793e80	1.35588	0.677940	+	0.735117i	$0.262873\pi$
$258$	−1.09483e80	−0.177864
$259$	−4.84148e80	−0.690933
$260$	4.82340e80	0.604987
$261$	−1.33938e80	−0.147722
$262$	−2.77034e80	−0.268806
$263$	2.32553e81	1.98612	0.993058	−	0.117629i	$-0.0375292\pi$
$263$	2.32553e81	1.98612	0.993058	+	0.117629i	$0.0375292\pi$
$264$	−1.70197e80	−0.128003
$265$	−3.26188e80	−0.216138
$266$	1.01805e80	0.0594612
$267$	−1.75557e81	−0.904255
$268$	7.71346e80	0.350537
$269$	3.59341e81	1.44147	0.720734	−	0.693211i	$-0.243805\pi$
$269$	3.59341e81	1.44147	0.720734	+	0.693211i	$0.243805\pi$
$270$	1.01366e80	0.0359092
$271$	2.85431e81	0.893367	0.446684	−	0.894692i	$-0.352605\pi$
$271$	2.85431e81	0.893367	0.446684	+	0.894692i	$0.352605\pi$
$272$	1.67421e81	0.463183
$273$	−1.51615e81	−0.370932
$274$	−1.82030e81	−0.394003
$275$	−1.22457e81	−0.234606
$276$	−5.68917e81	−0.965150
$277$	−1.00709e82	−1.51354	−0.756771	−	0.653680i	$-0.773224\pi$
$277$	−1.00709e82	−1.51354	−0.756771	+	0.653680i	$0.773224\pi$
$278$	−4.75170e79	−0.00632919
$279$	−1.18538e81	−0.139996
$280$	−2.11731e81	−0.221815
$281$	1.13532e82	1.05549	0.527746	−	0.849402i	$-0.323037\pi$
$281$	1.13532e82	1.05549	0.527746	+	0.849402i	$0.323037\pi$
$282$	−5.20516e80	−0.0429624
$283$	−1.14040e82	−0.836013	−0.418006	−	0.908444i	$-0.637271\pi$
$283$	−1.14040e82	−0.836013	−0.418006	+	0.908444i	$0.637271\pi$
$284$	5.31811e81	0.346412
$285$	−2.09047e81	−0.121043
$286$	2.32698e81	0.119820
$287$	8.38307e81	0.384025
$288$	−6.36434e81	−0.259481
$289$	−1.70840e82	−0.620173
$290$	2.55733e81	0.0826901
$291$	−3.91261e82	−1.12733
$292$	−3.90554e82	−1.00313
$293$	6.70401e82	1.53557	0.767785	−	0.640708i	$-0.221359\pi$
$293$	6.70401e82	1.53557	0.767785	+	0.640708i	$0.221359\pi$
$294$	−5.06921e81	−0.103587
$295$	−1.93840e82	−0.353514
$296$	3.82075e82	0.622124
$297$	−5.24820e81	−0.0763254
$298$	2.80252e82	0.364170
$299$	1.62816e83	1.89109
$300$	−3.00814e82	−0.312418
$301$	7.05180e82	0.655122
$302$	4.31237e82	0.358496
$303$	1.57251e82	0.117021
$304$	3.69960e82	0.246541
$305$	1.73323e83	1.03470
$306$	−1.12113e82	−0.0599772
$307$	−3.19616e83	−1.53283	−0.766414	−	0.642347i	$-0.777961\pi$
$307$	−3.19616e83	−1.53283	−0.766414	+	0.642347i	$0.777961\pi$
$308$	5.23718e82	0.225242
$309$	−8.00717e82	−0.308937
$310$	2.26330e82	0.0783656
$311$	−5.23816e83	−1.62819	−0.814097	−	0.580730i	$-0.802767\pi$
$311$	−5.23816e83	−1.62819	−0.814097	+	0.580730i	$0.802767\pi$
$312$	1.19650e83	0.333991
$313$	2.07664e83	0.520745	0.260372	−	0.965508i	$-0.416155\pi$
$313$	2.07664e83	0.520745	0.260372	+	0.965508i	$0.416155\pi$
$314$	−7.94275e82	−0.178989
$315$	−6.52896e82	−0.132264
$316$	−4.77180e83	−0.869291
$317$	3.07775e83	0.504369	0.252185	−	0.967679i	$-0.418851\pi$
$317$	3.07775e83	0.504369	0.252185	+	0.967679i	$0.418851\pi$
$318$	−3.86564e82	−0.0570050
$319$	−1.32405e83	−0.175759
$320$	−2.80326e83	−0.335073
$321$	−9.20930e83	−0.991535
$322$	−3.41446e83	−0.331246
$323$	2.31210e83	0.202173
$324$	−1.28921e83	−0.101640
$325$	8.60885e83	0.612143
$326$	−5.21027e83	−0.334249
$327$	9.50868e83	0.550517
$328$	−6.61567e83	−0.345780
$329$	3.35264e83	0.158243
$330$	1.00206e83	0.0427245
$331$	4.04685e84	1.55912	0.779559	−	0.626328i	$-0.215443\pi$
$331$	4.04685e84	1.55912	0.779559	+	0.626328i	$0.215443\pi$
$332$	−1.89594e84	−0.660234
$333$	1.17817e84	0.370959
$334$	−2.39786e83	−0.0682837
$335$	−9.50602e83	−0.244905
$336$	1.15546e84	0.269394
$337$	2.49688e84	0.526983	0.263491	−	0.964662i	$-0.415126\pi$
$337$	2.49688e84	0.526983	0.263491	+	0.964662i	$0.415126\pi$
$338$	−1.08241e83	−0.0206862
$339$	5.55274e84	0.961209
$340$	−2.29730e84	−0.360308
$341$	−1.17182e84	−0.166567
$342$	−2.47741e83	−0.0319244
$343$	8.57810e84	1.00239
$344$	−5.56507e84	−0.589879
$345$	7.01129e84	0.674308
$346$	6.94247e83	0.0605989
$347$	−1.62541e85	−1.28803	−0.644013	−	0.765015i	$-0.722732\pi$
$347$	−1.62541e85	−1.28803	−0.644013	+	0.765015i	$0.722732\pi$
$348$	−3.25251e84	−0.234053
$349$	2.30166e84	0.150449	0.0752244	−	0.997167i	$-0.476033\pi$
$349$	2.30166e84	0.150449	0.0752244	+	0.997167i	$0.476033\pi$
$350$	−1.80539e84	−0.107224
$351$	3.68954e84	0.199151
$352$	−6.29154e84	−0.308729
$353$	−9.24470e84	−0.412515	−0.206258	−	0.978498i	$-0.566129\pi$
$353$	−9.24470e84	−0.412515	−0.206258	+	0.978498i	$0.566129\pi$
$354$	−2.29719e84	−0.0932370
$355$	−6.55401e84	−0.242023
$356$	−4.26319e85	−1.43272
$357$	7.22116e84	0.220913
$358$	1.93168e85	0.538088
$359$	−1.12490e85	−0.285397	−0.142699	−	0.989766i	$-0.545578\pi$
$359$	−1.12490e85	−0.285397	−0.142699	+	0.989766i	$0.545578\pi$
$360$	5.15246e84	0.119091
$361$	−4.23687e85	−0.892388
$362$	6.41240e83	0.0123107
$363$	2.77964e85	0.486539
$364$	−3.68179e85	−0.587710
$365$	4.81316e85	0.700840
$366$	2.05405e85	0.272894
$367$	1.48969e86	1.80627	0.903134	−	0.429358i	$-0.141260\pi$
$367$	1.48969e86	1.80627	0.903134	+	0.429358i	$0.141260\pi$
$368$	−1.24082e86	−1.37343
$369$	−2.04001e85	−0.206181
$370$	−2.24953e85	−0.207651
$371$	2.48985e85	0.209965
$372$	−2.87856e85	−0.221813
$373$	2.32298e86	1.63606	0.818031	−	0.575175i	$-0.195066\pi$
$373$	2.32298e86	1.63606	0.818031	+	0.575175i	$0.195066\pi$
$374$	−1.10830e85	−0.0713606
$375$	9.97420e85	0.587260
$376$	−2.64580e85	−0.142483
$377$	9.30823e85	0.458598
$378$	−7.73744e84	−0.0348837
$379$	6.80738e85	0.280909	0.140455	−	0.990087i	$-0.455144\pi$
$379$	6.80738e85	0.280909	0.140455	+	0.990087i	$0.455144\pi$
$380$	−5.07646e85	−0.191783
$381$	−3.51582e85	−0.121630
$382$	5.38614e84	0.0170669
$383$	4.50234e86	1.30701	0.653505	−	0.756922i	$-0.273298\pi$
$383$	4.50234e86	1.30701	0.653505	+	0.756922i	$0.273298\pi$
$384$	−2.02173e86	−0.537807
$385$	−6.45427e85	−0.157367
$386$	1.23468e86	0.275981
$387$	−1.71605e86	−0.351732
$388$	−9.50132e86	−1.78616
$389$	1.57388e86	0.271431	0.135715	−	0.990748i	$-0.456667\pi$
$389$	1.57388e86	0.271431	0.135715	+	0.990748i	$0.456667\pi$
$390$	−7.04459e85	−0.111479
$391$	−7.75463e86	−1.12626
$392$	−2.57670e86	−0.343542
$393$	−4.34225e86	−0.531575
$394$	−3.88840e86	−0.437167
$395$	5.88074e86	0.607336
$396$	−1.27446e86	−0.120931
$397$	4.05321e86	0.353441	0.176721	−	0.984261i	$-0.443451\pi$
$397$	4.05321e86	0.353441	0.176721	+	0.984261i	$0.443451\pi$
$398$	−6.25351e86	−0.501235
$399$	1.59570e86	0.117587
$400$	−6.56081e86	−0.444578
$401$	2.28951e87	1.42694	0.713470	−	0.700686i	$-0.247122\pi$
$401$	2.28951e87	1.42694	0.713470	+	0.700686i	$0.247122\pi$
$402$	−1.12655e86	−0.0645921
$403$	8.23801e86	0.434614
$404$	3.81864e86	0.185410
$405$	1.58881e86	0.0710117
$406$	−1.95206e86	−0.0803287
$407$	1.16469e87	0.441365
$408$	−5.69872e86	−0.198912
$409$	2.15532e86	0.0693075	0.0346537	−	0.999399i	$-0.488967\pi$
$409$	2.15532e86	0.0693075	0.0346537	+	0.999399i	$0.488967\pi$
$410$	3.89508e86	0.115414
$411$	−2.85315e87	−0.779155
$412$	−1.94445e87	−0.489485
$413$	1.47962e87	0.343418
$414$	8.30906e86	0.177844
$415$	2.33654e87	0.461277
$416$	4.42301e87	0.805548
$417$	−7.44784e85	−0.0125162
$418$	−2.44907e86	−0.0379835
$419$	1.08084e88	1.54736	0.773681	−	0.633575i	$-0.218413\pi$
$419$	1.08084e88	1.54736	0.773681	+	0.633575i	$0.218413\pi$
$420$	−1.58548e87	−0.209560
$421$	2.71807e87	0.331750	0.165875	−	0.986147i	$-0.446955\pi$
$421$	2.71807e87	0.331750	0.165875	+	0.986147i	$0.446955\pi$
$422$	1.63936e87	0.184802
$423$	−8.15860e86	−0.0849598
$424$	−1.96492e87	−0.189055
$425$	−4.10024e87	−0.364570
$426$	−7.76713e86	−0.0638321
$427$	−1.32301e88	−1.00515
$428$	−2.23637e88	−1.57100
$429$	3.64733e87	0.236949
$430$	3.27652e87	0.196888
$431$	−1.82590e88	−1.01505	−0.507527	−	0.861636i	$-0.669440\pi$
$431$	−1.82590e88	−1.01505	−0.507527	+	0.861636i	$0.669440\pi$
$432$	−2.81179e87	−0.144637
$433$	9.88211e87	0.470441	0.235220	−	0.971942i	$-0.424419\pi$
$433$	9.88211e87	0.470441	0.235220	+	0.971942i	$0.424419\pi$
$434$	−1.72762e87	−0.0761276
$435$	4.00838e87	0.163523
$436$	2.30907e88	0.872247
$437$	−1.71358e88	−0.599482
$438$	5.70405e87	0.184842
$439$	−3.91460e88	−1.17524	−0.587619	−	0.809138i	$-0.699934\pi$
$439$	−3.91460e88	−1.17524	−0.587619	+	0.809138i	$0.699934\pi$
$440$	5.09352e87	0.141694
$441$	−7.94552e87	−0.204847
$442$	7.79146e87	0.186197
$443$	−5.04930e88	−1.11868	−0.559341	−	0.828938i	$-0.688946\pi$
$443$	−5.04930e88	−1.11868	−0.559341	+	0.828938i	$0.688946\pi$
$444$	2.86104e88	0.587753
$445$	5.25393e88	1.00098
$446$	−1.52706e87	−0.0269860
$447$	4.39269e88	0.720159
$448$	2.13978e88	0.325504
$449$	−6.56038e88	−0.926139	−0.463070	−	0.886322i	$-0.653252\pi$
$449$	−6.56038e88	−0.926139	−0.463070	+	0.886322i	$0.653252\pi$
$450$	4.39340e87	0.0575680
$451$	−2.01667e88	−0.245314
$452$	1.34842e89	1.52295
$453$	6.75925e88	0.708938
$454$	2.11070e88	0.205615
$455$	4.53741e88	0.410607
$456$	−1.25928e88	−0.105876
$457$	−1.05473e88	−0.0824039	−0.0412020	−	0.999151i	$-0.513119\pi$
$457$	−1.05473e88	−0.0824039	−0.0412020	+	0.999151i	$0.513119\pi$
$458$	−4.50215e88	−0.326909
$459$	−1.75726e88	−0.118607
$460$	1.70261e89	1.06838
$461$	−7.33038e88	−0.427706	−0.213853	−	0.976866i	$-0.568601\pi$
$461$	−7.33038e88	−0.427706	−0.213853	+	0.976866i	$0.568601\pi$
$462$	−7.64893e87	−0.0415044
$463$	6.14153e88	0.309966	0.154983	−	0.987917i	$-0.450468\pi$
$463$	6.14153e88	0.309966	0.154983	+	0.987917i	$0.450468\pi$
$464$	−7.09380e88	−0.333063
$465$	3.54751e88	0.154971
$466$	−9.92578e88	−0.403493
$467$	−4.75062e89	−1.79736	−0.898679	−	0.438607i	$-0.855472\pi$
$467$	−4.75062e89	−1.79736	−0.898679	+	0.438607i	$0.855472\pi$
$468$	8.95960e88	0.315539
$469$	7.25613e88	0.237911
$470$	1.55776e88	0.0475578
$471$	−1.24495e89	−0.353958
$472$	−1.16767e89	−0.309218
$473$	−1.69642e89	−0.418489
$474$	6.96924e88	0.160181
$475$	−9.06052e88	−0.194052
$476$	1.75357e89	0.350018
$477$	−6.05903e88	−0.112729
$478$	−1.23965e89	−0.215012
$479$	−4.59749e89	−0.743501	−0.371750	−	0.928333i	$-0.621242\pi$
$479$	−4.59749e89	−0.743501	−0.371750	+	0.928333i	$0.621242\pi$
$480$	1.90467e89	0.287235
$481$	−8.18789e89	−1.15163
$482$	6.62981e88	0.0869812
$483$	−5.35186e89	−0.655052
$484$	6.75004e89	0.770879
$485$	1.17094e90	1.24791
$486$	1.88290e88	0.0187289
$487$	2.89512e89	0.268810	0.134405	−	0.990926i	$-0.457088\pi$
$487$	2.89512e89	0.268810	0.134405	+	0.990926i	$0.457088\pi$
$488$	1.04408e90	0.905045
$489$	−8.16661e89	−0.660990
$490$	1.51707e89	0.114667
$491$	2.28241e90	1.61126	0.805628	−	0.592422i	$-0.201828\pi$
$491$	2.28241e90	1.61126	0.805628	+	0.592422i	$0.201828\pi$
$492$	−4.95392e89	−0.326677
$493$	−4.43334e89	−0.273124
$494$	1.72172e89	0.0991082
$495$	1.57064e89	0.0844894
$496$	−6.27819e89	−0.315645
$497$	5.00280e89	0.235112
$498$	2.76902e89	0.121659
$499$	−7.84213e88	−0.0322156	−0.0161078	−	0.999870i	$-0.505127\pi$
$499$	−7.84213e88	−0.0322156	−0.0161078	+	0.999870i	$0.505127\pi$
$500$	2.42211e90	0.930464
$501$	−3.75842e89	−0.135034
$502$	2.53060e89	0.0850452
$503$	−3.29209e90	−1.03501	−0.517505	−	0.855680i	$-0.673139\pi$
$503$	−3.29209e90	−1.03501	−0.517505	+	0.855680i	$0.673139\pi$
$504$	−3.93297e89	−0.115690
$505$	−4.70607e89	−0.129538
$506$	8.21400e89	0.211599
$507$	−1.69657e89	−0.0409077
$508$	−8.53774e89	−0.192712
$509$	6.95196e90	1.46914	0.734568	−	0.678536i	$-0.237385\pi$
$509$	6.95196e90	1.46914	0.734568	+	0.678536i	$0.237385\pi$
$510$	3.35521e89	0.0663925
$511$	−3.67397e90	−0.680826
$512$	−5.79144e90	−1.00518
$513$	−3.88311e89	−0.0631317
$514$	−2.59897e90	−0.395855
$515$	2.39632e90	0.341982
$516$	−4.16721e90	−0.557289
$517$	−8.06527e89	−0.101085
$518$	1.71711e90	0.201721
$519$	1.08817e90	0.119837
$520$	−3.58079e90	−0.369715
$521$	−9.53203e89	−0.0922830	−0.0461415	−	0.998935i	$-0.514693\pi$
$521$	−9.53203e89	−0.0922830	−0.0461415	+	0.998935i	$0.514693\pi$
$522$	4.75031e89	0.0431281
$523$	−9.89851e90	−0.842875	−0.421438	−	0.906857i	$-0.638474\pi$
$523$	−9.89851e90	−0.842875	−0.421438	+	0.906857i	$0.638474\pi$
$524$	−1.05446e91	−0.842235
$525$	−2.82978e90	−0.212039
$526$	−8.24787e90	−0.579855
$527$	−3.92362e90	−0.258840
$528$	−2.77963e90	−0.172088
$529$	4.02630e91	2.33959
$530$	1.15688e90	0.0631024
$531$	−3.60064e90	−0.184380
$532$	3.87496e90	0.186306
$533$	1.41774e91	0.640082
$534$	6.22641e90	0.264001
$535$	2.75608e91	1.09759
$536$	−5.72632e90	−0.214218
$537$	3.02773e91	1.06409
$538$	−1.27446e91	−0.420843
$539$	−7.85462e90	−0.243726
$540$	3.85825e90	0.112512
$541$	5.31156e91	1.45584	0.727921	−	0.685661i	$-0.240487\pi$
$541$	5.31156e91	1.45584	0.727921	+	0.685661i	$0.240487\pi$
$542$	−1.01233e91	−0.260822
$543$	1.00508e90	0.0243449
$544$	−2.10660e91	−0.479755
$545$	−2.84568e91	−0.609401
$546$	5.37727e90	0.108295
$547$	−9.06405e90	−0.171691	−0.0858454	−	0.996308i	$-0.527359\pi$
$547$	−9.06405e90	−0.171691	−0.0858454	+	0.996308i	$0.527359\pi$
$548$	−6.92854e91	−1.23450
$549$	3.21953e91	0.539659
$550$	4.34314e90	0.0684941
$551$	−9.79659e90	−0.145377
$552$	4.22353e91	0.589815
$553$	−4.48888e91	−0.589992
$554$	3.57179e91	0.441885
$555$	−3.52593e91	−0.410637
$556$	−1.80862e90	−0.0198309
$557$	−1.15401e92	−1.19141	−0.595706	−	0.803203i	$-0.703128\pi$
$557$	−1.15401e92	−1.19141	−0.595706	+	0.803203i	$0.703128\pi$
$558$	4.20414e90	0.0408726
$559$	1.19260e92	1.09194
$560$	−3.45796e91	−0.298209
$561$	−1.73716e91	−0.141118
$562$	−4.02658e91	−0.308155
$563$	−6.53348e91	−0.471100	−0.235550	−	0.971862i	$-0.575689\pi$
$563$	−6.53348e91	−0.471100	−0.235550	+	0.971862i	$0.575689\pi$
$564$	−1.98122e91	−0.134612
$565$	−1.66178e92	−1.06402
$566$	4.04462e91	0.244077
$567$	−1.21277e91	−0.0689837
$568$	−3.94806e91	−0.211697
$569$	6.96531e90	0.0352111	0.0176055	−	0.999845i	$-0.494396\pi$
$569$	6.96531e90	0.0352111	0.0176055	+	0.999845i	$0.494396\pi$
$570$	7.41419e90	0.0353391
$571$	−9.46031e91	−0.425202	−0.212601	−	0.977139i	$-0.568193\pi$
$571$	−9.46031e91	−0.425202	−0.212601	+	0.977139i	$0.568193\pi$
$572$	8.85710e91	0.375426
$573$	8.44227e90	0.0337504
$574$	−2.97319e91	−0.112118
$575$	3.03884e92	1.08102
$576$	−5.20713e91	−0.174761
$577$	5.08231e92	1.60943	0.804714	−	0.593663i	$-0.202319\pi$
$577$	5.08231e92	1.60943	0.804714	+	0.593663i	$0.202319\pi$
$578$	6.05913e91	0.181062
$579$	1.93524e92	0.545762
$580$	9.73386e91	0.259088
$581$	−1.78352e92	−0.448104
$582$	1.38767e92	0.329129
$583$	−5.98972e91	−0.134125
$584$	2.89939e92	0.613022
$585$	−1.10417e92	−0.220453
$586$	−2.37769e92	−0.448316
$587$	4.46800e91	0.0795678	0.0397839	−	0.999208i	$-0.487333\pi$
$587$	4.46800e91	0.0795678	0.0397839	+	0.999208i	$0.487333\pi$
$588$	−1.92947e92	−0.324563
$589$	−8.67022e91	−0.137774
$590$	6.87486e91	0.103210
$591$	−6.09471e92	−0.864514
$592$	6.23999e92	0.836386
$593$	2.14083e92	0.271175	0.135588	−	0.990765i	$-0.456708\pi$
$593$	2.14083e92	0.271175	0.135588	+	0.990765i	$0.456708\pi$
$594$	1.86136e91	0.0222835
$595$	−2.16109e92	−0.244543
$596$	1.06671e93	1.14103
$597$	−9.80180e92	−0.991210
$598$	−5.77453e92	−0.552112
$599$	1.12807e93	1.01985	0.509926	−	0.860219i	$-0.329673\pi$
$599$	1.12807e93	1.01985	0.509926	+	0.860219i	$0.329673\pi$
$600$	2.23318e92	0.190922
$601$	−1.96269e93	−1.58693	−0.793464	−	0.608617i	$-0.791725\pi$
$601$	−1.96269e93	−1.58693	−0.793464	+	0.608617i	$0.791725\pi$
$602$	−2.50103e92	−0.191266
$603$	−1.76577e92	−0.127733
$604$	1.64140e93	1.12325
$605$	−8.31870e92	−0.538580
$606$	−5.57714e91	−0.0341648
$607$	−2.93504e93	−1.70135	−0.850673	−	0.525695i	$-0.823805\pi$
$607$	−2.93504e93	−1.70135	−0.850673	+	0.525695i	$0.823805\pi$
$608$	−4.65507e92	−0.255362
$609$	−3.05967e92	−0.158853
$610$	−6.14719e92	−0.302084
$611$	5.66997e92	0.263755
$612$	−4.26730e92	−0.187923
$613$	3.78455e92	0.157793	0.0788966	−	0.996883i	$-0.474860\pi$
$613$	3.78455e92	0.157793	0.0788966	+	0.996883i	$0.474860\pi$
$614$	1.13357e93	0.447516
$615$	6.10518e92	0.228235
$616$	−3.88798e92	−0.137648
$617$	−8.57063e92	−0.287382	−0.143691	−	0.989623i	$-0.545897\pi$
$617$	−8.57063e92	−0.287382	−0.143691	+	0.989623i	$0.545897\pi$
$618$	2.83987e92	0.0901955
$619$	2.20381e93	0.663038	0.331519	−	0.943449i	$-0.392439\pi$
$619$	2.20381e93	0.663038	0.331519	+	0.943449i	$0.392439\pi$
$620$	8.61471e92	0.245538
$621$	1.30237e93	0.351694
$622$	1.85780e93	0.475358
$623$	−4.01042e93	−0.972390
$624$	1.95411e93	0.449019
$625$	−2.68757e92	−0.0585302
$626$	−7.36512e92	−0.152034
$627$	−3.83869e92	−0.0751138
$628$	−3.02322e93	−0.560817
$629$	3.89975e93	0.685867
$630$	2.31560e92	0.0386149
$631$	1.10304e94	1.74425	0.872123	−	0.489286i	$-0.162743\pi$
$631$	1.10304e94	1.74425	0.872123	+	0.489286i	$0.162743\pi$
$632$	3.54249e93	0.531235
$633$	2.56954e93	0.365453
$634$	−1.09157e93	−0.147253
$635$	1.05219e93	0.134639
$636$	−1.47136e93	−0.178610
$637$	5.52188e93	0.635940
$638$	4.69597e92	0.0513136
$639$	−1.21743e93	−0.126230
$640$	6.05046e93	0.595332
$641$	−1.71036e94	−1.59714	−0.798569	−	0.601903i	$-0.794409\pi$
$641$	−1.71036e94	−1.59714	−0.798569	+	0.601903i	$0.794409\pi$
$642$	3.26623e93	0.289483
$643$	−9.74742e93	−0.820016	−0.410008	−	0.912082i	$-0.634474\pi$
$643$	−9.74742e93	−0.820016	−0.410008	+	0.912082i	$0.634474\pi$
$644$	−1.29963e94	−1.03787
$645$	5.13565e93	0.389354
$646$	−8.20024e92	−0.0590252
$647$	−2.06058e93	−0.140830	−0.0704150	−	0.997518i	$-0.522432\pi$
$647$	−2.06058e93	−0.140830	−0.0704150	+	0.997518i	$0.522432\pi$
$648$	9.57084e92	0.0621137
$649$	−3.55945e93	−0.219374
$650$	−3.05327e93	−0.178718
$651$	−2.70788e93	−0.150545
$652$	−1.98316e94	−1.04728
$653$	−2.00922e94	−1.00794	−0.503970	−	0.863721i	$-0.668128\pi$
$653$	−2.00922e94	−1.00794	−0.503970	+	0.863721i	$0.668128\pi$
$654$	−3.37241e93	−0.160726
$655$	1.29951e94	0.588433
$656$	−1.08046e94	−0.464869
$657$	8.94057e93	0.365532
$658$	−1.18907e93	−0.0461996
$659$	8.21097e93	0.303203	0.151601	−	0.988442i	$-0.451557\pi$
$659$	8.21097e93	0.303203	0.151601	+	0.988442i	$0.451557\pi$
$660$	3.81411e93	0.133866
$661$	2.48802e94	0.830052	0.415026	−	0.909810i	$-0.363772\pi$
$661$	2.48802e94	0.830052	0.415026	+	0.909810i	$0.363772\pi$
$662$	−1.43528e94	−0.455191
$663$	1.22124e94	0.368212
$664$	1.40750e94	0.403477
$665$	−4.77547e93	−0.130164
$666$	−4.17856e93	−0.108303
$667$	3.28571e94	0.809867
$668$	−9.12688e93	−0.213949
$669$	−2.39353e93	−0.0533659
$670$	3.37146e93	0.0715010
$671$	3.18270e94	0.642083
$672$	−1.45387e94	−0.279033
$673$	−2.98741e94	−0.545495	−0.272747	−	0.962086i	$-0.587932\pi$
$673$	−2.98741e94	−0.545495	−0.272747	+	0.962086i	$0.587932\pi$
$674$	−8.85559e93	−0.153855
$675$	6.88624e93	0.113843
$676$	−4.11992e93	−0.0648148
$677$	1.11284e95	1.66615	0.833073	−	0.553164i	$-0.186580\pi$
$677$	1.11284e95	1.66615	0.833073	+	0.553164i	$0.186580\pi$
$678$	−1.96937e94	−0.280629
$679$	−8.93798e94	−1.21228
$680$	1.70547e94	0.220189
$681$	3.30832e94	0.406612
$682$	4.15605e93	0.0486300
$683$	1.62723e95	1.81282	0.906409	−	0.422401i	$-0.138813\pi$
$683$	1.62723e95	1.81282	0.906409	+	0.422401i	$0.138813\pi$
$684$	−9.42966e93	−0.100027
$685$	8.53868e94	0.862495
$686$	−3.04236e94	−0.292653
$687$	−7.05671e94	−0.646474
$688$	−9.08878e94	−0.793036
$689$	4.21083e94	0.349964
$690$	−2.48667e94	−0.196867
$691$	−1.41176e95	−1.06474	−0.532372	−	0.846510i	$-0.678699\pi$
$691$	−1.41176e95	−1.06474	−0.532372	+	0.846510i	$0.678699\pi$
$692$	2.64249e94	0.189871
$693$	−1.19890e94	−0.0820765
$694$	5.76476e94	0.376045
$695$	2.22893e93	0.0138550
$696$	2.41460e94	0.143033
$697$	−6.75245e94	−0.381209
$698$	−8.16319e93	−0.0439242
$699$	−1.55577e95	−0.797923
$700$	−6.87179e94	−0.335958
$701$	−2.96929e95	−1.38388	−0.691941	−	0.721954i	$-0.743244\pi$
$701$	−2.96929e95	−1.38388	−0.691941	+	0.721954i	$0.743244\pi$
$702$	−1.30855e94	−0.0581431
$703$	8.61747e94	0.365070
$704$	−5.14756e94	−0.207930
$705$	2.44164e94	0.0940473
$706$	3.27878e94	0.120436
$707$	3.59223e94	0.125839
$708$	−8.74372e94	−0.292134
$709$	−2.38690e95	−0.760650	−0.380325	−	0.924853i	$-0.624188\pi$
$709$	−2.38690e95	−0.760650	−0.380325	+	0.924853i	$0.624188\pi$
$710$	2.32448e94	0.0706597
$711$	1.09236e95	0.316764
$712$	3.16491e95	0.875550
$713$	2.90793e95	0.767512
$714$	−2.56110e94	−0.0644965
$715$	−1.09154e95	−0.262294
$716$	7.35247e95	1.68596
$717$	−1.94303e95	−0.425194
$718$	3.98964e94	0.0833229
$719$	−4.92960e95	−0.982638	−0.491319	−	0.870980i	$-0.663485\pi$
$719$	−4.92960e95	−0.982638	−0.491319	+	0.870980i	$0.663485\pi$
$720$	8.41491e94	0.160107
$721$	−1.82916e95	−0.332215
$722$	1.50267e95	0.260537
$723$	1.03916e95	0.172009
$724$	2.44073e94	0.0385725
$725$	1.73731e95	0.262153
$726$	−9.85846e94	−0.142047
$727$	2.83726e95	0.390388	0.195194	−	0.980765i	$-0.437466\pi$
$727$	2.83726e95	0.390388	0.195194	+	0.980765i	$0.437466\pi$
$728$	2.73329e95	0.359157
$729$	2.95127e94	0.0370370
$730$	−1.70706e95	−0.204613
$731$	−5.68013e95	−0.650318
$732$	7.81825e95	0.855043
$733$	1.53710e96	1.60590	0.802950	−	0.596046i	$-0.203262\pi$
$733$	1.53710e96	1.60590	0.802950	+	0.596046i	$0.203262\pi$
$734$	−5.28342e95	−0.527348
$735$	2.37787e95	0.226758
$736$	1.56128e96	1.42257
$737$	−1.74557e95	−0.151976
$738$	7.23523e94	0.0601955
$739$	−1.56801e96	−1.24670	−0.623349	−	0.781944i	$-0.714228\pi$
$739$	−1.56801e96	−1.24670	−0.623349	+	0.781944i	$0.714228\pi$
$740$	−8.56230e95	−0.650620
$741$	2.69863e95	0.195990
$742$	−8.83066e94	−0.0613003
$743$	−1.24019e96	−0.822927	−0.411464	−	0.911426i	$-0.634982\pi$
$743$	−1.24019e96	−0.822927	−0.411464	+	0.911426i	$0.634982\pi$
$744$	2.13698e95	0.135552
$745$	−1.31461e96	−0.797189
$746$	−8.23882e95	−0.477655
$747$	4.34018e95	0.240585
$748$	−4.21848e95	−0.223590
$749$	−2.10377e96	−1.06625
$750$	−3.53751e95	−0.171453
$751$	2.53507e95	0.117504	0.0587519	−	0.998273i	$-0.481288\pi$
$751$	2.53507e95	0.117504	0.0587519	+	0.998273i	$0.481288\pi$
$752$	−4.32108e95	−0.191556
$753$	3.96649e95	0.168180
$754$	−3.30131e95	−0.133890
$755$	−2.02285e96	−0.784768
$756$	−2.94507e95	−0.109299
$757$	1.18981e94	0.00422440	0.00211220	−	0.999998i	$-0.499328\pi$
$757$	1.18981e94	0.00422440	0.00211220	+	0.999998i	$0.499328\pi$
$758$	−2.41435e95	−0.0820127
$759$	1.28747e96	0.418444
$760$	3.76866e95	0.117201
$761$	3.12153e96	0.928926	0.464463	−	0.885593i	$-0.346247\pi$
$761$	3.12153e96	0.928926	0.464463	+	0.885593i	$0.346247\pi$
$762$	1.24694e95	0.0355103
$763$	2.17216e96	0.591998
$764$	2.05010e95	0.0534747
$765$	5.25899e95	0.131294
$766$	−1.59683e96	−0.381587
$767$	2.50233e96	0.572400
$768$	−6.65277e95	−0.145681
$769$	2.83694e95	0.0594726	0.0297363	−	0.999558i	$-0.490533\pi$
$769$	2.83694e95	0.0594726	0.0297363	+	0.999558i	$0.490533\pi$
$770$	2.28911e95	0.0459438
$771$	−4.07364e96	−0.782818
$772$	4.69951e96	0.864714
$773$	4.89330e95	0.0862162	0.0431081	−	0.999070i	$-0.486274\pi$
$773$	4.89330e95	0.0862162	0.0431081	+	0.999070i	$0.486274\pi$
$774$	6.08623e95	0.102690
$775$	1.53756e96	0.248443
$776$	7.05359e96	1.09155
$777$	2.69141e96	0.398911
$778$	−5.58201e95	−0.0792454
$779$	−1.49212e96	−0.202908
$780$	−2.68136e96	−0.349289
$781$	−1.20350e96	−0.150188
$782$	2.75030e96	0.328817
$783$	7.44567e95	0.0852874
$784$	−4.20822e96	−0.461860
$785$	3.72579e96	0.391819
$786$	1.54005e96	0.155195
$787$	1.83342e97	1.77055	0.885276	−	0.465065i	$-0.153969\pi$
$787$	1.83342e97	1.77055	0.885276	+	0.465065i	$0.153969\pi$
$788$	−1.48003e97	−1.36975
$789$	−1.29278e97	−1.14668
$790$	−2.08570e96	−0.177314
$791$	1.26847e97	1.03364
$792$	9.46135e95	0.0739026
$793$	−2.23747e97	−1.67535
$794$	−1.43754e96	−0.103189
$795$	1.81330e96	0.124787
$796$	−2.38025e97	−1.57049
$797$	6.71228e96	0.424636	0.212318	−	0.977201i	$-0.431899\pi$
$797$	6.71228e96	0.424636	0.212318	+	0.977201i	$0.431899\pi$
$798$	−5.65939e95	−0.0343299
$799$	−2.70050e96	−0.157082
$800$	8.25522e96	0.460483
$801$	9.75932e96	0.522072
$802$	−8.12011e96	−0.416601
$803$	8.83829e96	0.434908
$804$	−4.28796e96	−0.202383
$805$	1.60166e97	0.725117
$806$	−2.92174e96	−0.126887
$807$	−1.99760e97	−0.832232
$808$	−2.83488e96	−0.113306
$809$	1.23268e97	0.472688	0.236344	−	0.971669i	$-0.424051\pi$
$809$	1.23268e97	0.472688	0.236344	+	0.971669i	$0.424051\pi$
$810$	−5.63498e95	−0.0207322
$811$	−2.90320e97	−1.02489	−0.512447	−	0.858719i	$-0.671261\pi$
$811$	−2.90320e97	−1.02489	−0.512447	+	0.858719i	$0.671261\pi$
$812$	−7.43004e96	−0.251689
$813$	−1.58673e97	−0.515786
$814$	−4.13076e96	−0.128858
$815$	2.44404e97	0.731692
$816$	−9.30706e96	−0.267419
$817$	−1.25517e97	−0.346149
$818$	−7.64418e95	−0.0202346
$819$	8.42837e96	0.214157
$820$	1.48257e97	0.361619
$821$	−1.62725e97	−0.381029	−0.190515	−	0.981684i	$-0.561016\pi$
$821$	−1.62725e97	−0.381029	−0.190515	+	0.981684i	$0.561016\pi$
$822$	1.01192e97	0.227477
$823$	7.54690e97	1.62882	0.814412	−	0.580287i	$-0.197059\pi$
$823$	7.54690e97	1.62882	0.814412	+	0.580287i	$0.197059\pi$
$824$	1.44352e97	0.299130
$825$	6.80746e96	0.135450
$826$	−5.24771e96	−0.100262
$827$	−4.70909e96	−0.0863977	−0.0431988	−	0.999066i	$-0.513755\pi$
$827$	−4.70909e96	−0.0863977	−0.0431988	+	0.999066i	$0.513755\pi$
$828$	3.16264e97	0.557230
$829$	5.63395e97	0.953315	0.476657	−	0.879089i	$-0.341848\pi$
$829$	5.63395e97	0.953315	0.476657	+	0.879089i	$0.341848\pi$
$830$	−8.28690e96	−0.134672
$831$	5.59845e97	0.873844
$832$	3.61879e97	0.542540
$833$	−2.62997e97	−0.378742
$834$	2.64150e95	0.00365416
$835$	1.12479e97	0.149477
$836$	−9.32179e96	−0.119012
$837$	6.58960e96	0.0808270
$838$	−3.83338e97	−0.451759
$839$	−9.35016e97	−1.05875	−0.529375	−	0.848388i	$-0.677573\pi$
$839$	−9.35016e97	−1.05875	−0.529375	+	0.848388i	$0.677573\pi$
$840$	1.17703e97	0.128065
$841$	−7.68613e97	−0.803604
$842$	−9.64007e96	−0.0968557
$843$	−6.31129e97	−0.609388
$844$	6.23982e97	0.579029
$845$	5.07736e96	0.0452833
$846$	2.89358e96	0.0248044
$847$	6.34982e97	0.523199
$848$	−3.20907e97	−0.254167
$849$	6.33957e97	0.482672
$850$	1.45422e97	0.106438
$851$	−2.89024e98	−2.03373
$852$	−2.95637e97	−0.200001
$853$	−1.53678e98	−0.999586	−0.499793	−	0.866145i	$-0.666591\pi$
$853$	−1.53678e98	−0.999586	−0.499793	+	0.866145i	$0.666591\pi$
$854$	4.69227e97	0.293457
$855$	1.16211e97	0.0698845
$856$	1.66023e98	0.960060
$857$	3.74053e97	0.208006	0.104003	−	0.994577i	$-0.466835\pi$
$857$	3.74053e97	0.208006	0.104003	+	0.994577i	$0.466835\pi$
$858$	−1.29358e97	−0.0691784
$859$	−1.70881e98	−0.878865	−0.439433	−	0.898276i	$-0.644820\pi$
$859$	−1.70881e98	−0.878865	−0.439433	+	0.898276i	$0.644820\pi$
$860$	1.24713e98	0.616898
$861$	−4.66020e97	−0.221717
$862$	6.47586e97	0.296349
$863$	−2.97133e98	−1.30794	−0.653972	−	0.756519i	$-0.726898\pi$
$863$	−2.97133e98	−1.30794	−0.653972	+	0.756519i	$0.726898\pi$
$864$	3.53798e97	0.149811
$865$	−3.25658e97	−0.132655
$866$	−3.50485e97	−0.137347
$867$	9.49712e97	0.358057
$868$	−6.57577e97	−0.238526
$869$	1.07987e98	0.376884
$870$	−1.42163e97	−0.0477412
$871$	1.22715e98	0.396543
$872$	−1.71421e98	−0.533041
$873$	2.17505e98	0.650866
$874$	6.07749e97	0.175021
$875$	2.27851e98	0.631510
$876$	2.17111e98	0.579155
$877$	4.18737e97	0.107512	0.0537558	−	0.998554i	$-0.482881\pi$
$877$	4.18737e97	0.107512	0.0537558	+	0.998554i	$0.482881\pi$
$878$	1.38837e98	0.343115
$879$	−3.72680e98	−0.886562
$880$	8.31865e97	0.190495
$881$	3.25627e98	0.717839	0.358919	−	0.933369i	$-0.383145\pi$
$881$	3.25627e98	0.717839	0.358919	+	0.933369i	$0.383145\pi$
$882$	2.81801e97	0.0598059
$883$	1.02307e97	0.0209037	0.0104518	−	0.999945i	$-0.496673\pi$
$883$	1.02307e97	0.0209037	0.0104518	+	0.999945i	$0.496673\pi$
$884$	2.96563e98	0.583400
$885$	1.07757e98	0.204101
$886$	1.79082e98	0.326604
$887$	8.62313e98	1.51434	0.757171	−	0.653217i	$-0.226581\pi$
$887$	8.62313e98	1.51434	0.757171	+	0.653217i	$0.226581\pi$
$888$	−2.12398e98	−0.359183
$889$	−8.03153e97	−0.130794
$890$	−1.86339e98	−0.292239
$891$	2.91750e97	0.0440665
$892$	−5.81240e97	−0.0845537
$893$	−5.96744e97	−0.0836112
$894$	−1.55794e98	−0.210253
$895$	−9.06114e98	−1.17791
$896$	−4.61843e98	−0.578331
$897$	−9.05104e98	−1.09182
$898$	2.32674e98	0.270390
$899$	1.66247e98	0.186125
$900$	1.67224e98	0.180374
$901$	−2.00554e98	−0.208426
$902$	7.15246e97	0.0716203
$903$	−3.92014e98	−0.378235
$904$	−1.00104e99	−0.930696
$905$	−3.00794e97	−0.0269489
$906$	−2.39727e98	−0.206978
$907$	4.77849e97	0.0397601	0.0198800	−	0.999802i	$-0.493672\pi$
$907$	4.77849e97	0.0397601	0.0198800	+	0.999802i	$0.493672\pi$
$908$	8.03387e98	0.644242
$909$	−8.74165e97	−0.0675621
$910$	−1.60927e98	−0.119879
$911$	1.36204e99	0.977964	0.488982	−	0.872294i	$-0.337368\pi$
$911$	1.36204e99	0.977964	0.488982	+	0.872294i	$0.337368\pi$
$912$	−2.05663e98	−0.142341
$913$	4.29053e98	0.286247
$914$	3.74076e97	0.0240582
$915$	−9.63515e98	−0.597382
$916$	−1.71364e99	−1.02428
$917$	−9.91943e98	−0.571629
$918$	6.23240e97	0.0346279
$919$	1.87215e99	1.00293	0.501466	−	0.865177i	$-0.332794\pi$
$919$	1.87215e99	1.00293	0.501466	+	0.865177i	$0.332794\pi$
$920$	−1.26398e99	−0.652903
$921$	1.77677e99	0.884979
$922$	2.59984e98	0.124871
$923$	8.46071e98	0.391877
$924$	−2.91138e98	−0.130043
$925$	−1.52821e99	−0.658316
$926$	−2.17819e98	−0.0904958
$927$	4.45123e98	0.178365
$928$	8.92587e98	0.344979
$929$	1.09697e99	0.408947	0.204474	−	0.978872i	$-0.434452\pi$
$929$	1.09697e99	0.408947	0.204474	+	0.978872i	$0.434452\pi$
$930$	−1.25818e98	−0.0452444
$931$	−5.81159e98	−0.201595
$932$	−3.77801e99	−1.26424
$933$	2.91193e99	0.940038
$934$	1.68488e99	0.524746
$935$	5.19882e98	0.156213
$936$	−6.65142e98	−0.192830
$937$	3.10631e99	0.868898	0.434449	−	0.900696i	$-0.356943\pi$
$937$	3.10631e99	0.868898	0.434449	+	0.900696i	$0.356943\pi$
$938$	−2.57350e98	−0.0694591
$939$	−1.15441e99	−0.300652
$940$	5.92923e98	0.149010
$941$	7.31696e99	1.77451	0.887255	−	0.461280i	$-0.152610\pi$
$941$	7.31696e99	1.77451	0.887255	+	0.461280i	$0.152610\pi$
$942$	4.41542e98	0.103340
$943$	5.00448e99	1.13036
$944$	−1.90702e99	−0.415714
$945$	3.62949e98	0.0763624
$946$	6.01661e98	0.122180
$947$	−3.58770e99	−0.703221	−0.351610	−	0.936146i	$-0.614366\pi$
$947$	−3.58770e99	−0.703221	−0.351610	+	0.936146i	$0.614366\pi$
$948$	2.65268e99	0.501886
$949$	−6.21341e99	−1.13478
$950$	3.21346e98	0.0566542
$951$	−1.71094e99	−0.291198
$952$	−1.30182e99	−0.213901
$953$	−3.77034e99	−0.598093	−0.299047	−	0.954239i	$-0.596669\pi$
$953$	−3.77034e99	−0.598093	−0.299047	+	0.954239i	$0.596669\pi$
$954$	2.14893e98	0.0329118
$955$	−2.52653e98	−0.0373605
$956$	−4.71841e99	−0.673685
$957$	7.36049e98	0.101475
$958$	1.63058e99	0.217068
$959$	−6.51774e99	−0.837864
$960$	1.55835e99	0.193454
$961$	−6.86996e99	−0.823609
$962$	2.90397e99	0.336223
$963$	5.11951e99	0.572463
$964$	2.52348e99	0.272533
$965$	−5.79164e99	−0.604138
$966$	1.89812e99	0.191245
$967$	−3.34294e98	−0.0325344	−0.0162672	−	0.999868i	$-0.505178\pi$
$967$	−3.34294e98	−0.0325344	−0.0162672	+	0.999868i	$0.505178\pi$
$968$	−5.01109e99	−0.471094
$969$	−1.28531e99	−0.116724
$970$	−4.15291e99	−0.364334
$971$	−1.09912e100	−0.931541	−0.465771	−	0.884905i	$-0.654223\pi$
$971$	−1.09912e100	−0.931541	−0.465771	+	0.884905i	$0.654223\pi$
$972$	7.16680e98	0.0586821
$973$	−1.70139e98	−0.0134593
$974$	−1.02680e99	−0.0784803
$975$	−4.78571e99	−0.353421
$976$	1.70518e100	1.21675
$977$	−1.76276e100	−1.21542	−0.607710	−	0.794159i	$-0.707912\pi$
$977$	−1.76276e100	−1.21542	−0.607710	+	0.794159i	$0.707912\pi$
$978$	2.89642e99	0.192979
$979$	9.64767e99	0.621158
$980$	5.77438e99	0.359279
$981$	−5.28593e99	−0.317841
$982$	−8.09495e99	−0.470413
$983$	1.83321e100	1.02960	0.514801	−	0.857310i	$-0.327866\pi$
$983$	1.83321e100	1.02960	0.514801	+	0.857310i	$0.327866\pi$
$984$	3.67769e99	0.199636
$985$	1.82397e100	0.956985
$986$	1.57236e99	0.0797397
$987$	−1.86375e99	−0.0913615
$988$	6.55331e99	0.310530
$989$	4.20974e100	1.92832
$990$	−5.57052e98	−0.0246670
$991$	−9.79963e99	−0.419510	−0.209755	−	0.977754i	$-0.567267\pi$
$991$	−9.79963e99	−0.419510	−0.209755	+	0.977754i	$0.567267\pi$
$992$	7.89961e99	0.326938
$993$	−2.24967e100	−0.900158
$994$	−1.77432e99	−0.0686418
$995$	2.93340e100	1.09723
$996$	1.05396e100	0.381186
$997$	−2.14429e100	−0.749888	−0.374944	−	0.927047i	$-0.622338\pi$
$997$	−2.14429e100	−0.749888	−0.374944	+	0.927047i	$0.622338\pi$
$998$	2.78134e98	0.00940547
$999$	−6.54951e99	−0.214173

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.68.a.b.1.3	✓	6	1.1	even	1	trivial
9.68.a.c.1.4		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.54666e9 q^{2} -5.55906e15 q^{3} -1.34995e20 q^{4} +1.66367e23 q^{5} +1.97161e25 q^{6} -1.26991e28 q^{7} +1.00218e30 q^{8} +3.09032e31 q^{9} +O(q^{10})\)	\(q-3.54666e9 q^{2} -5.55906e15 q^{3} -1.34995e20 q^{4} +1.66367e23 q^{5} +1.97161e25 q^{6} -1.26991e28 q^{7} +1.00218e30 q^{8} +3.09032e31 q^{9} -5.90048e32 q^{10} +3.05496e34 q^{11} +7.50446e35 q^{12} -2.14767e37 q^{13} +4.50395e37 q^{14} -9.24845e38 q^{15} +1.63674e40 q^{16} +1.02290e41 q^{17} -1.09603e41 q^{18} +2.26035e42 q^{19} -2.24588e43 q^{20} +7.05952e43 q^{21} -1.08349e44 q^{22} -7.58105e45 q^{23} -5.57116e45 q^{24} -4.00846e46 q^{25} +7.61705e46 q^{26} -1.71793e47 q^{27} +1.71432e48 q^{28} -4.33411e48 q^{29} +3.28011e48 q^{30} -3.83579e49 q^{31} -2.05945e50 q^{32} -1.69827e50 q^{33} -3.62787e50 q^{34} -2.11272e51 q^{35} -4.17178e51 q^{36} +3.81245e52 q^{37} -8.01669e51 q^{38} +1.19390e53 q^{39} +1.66729e53 q^{40} -6.60130e53 q^{41} -2.50377e53 q^{42} -5.55298e54 q^{43} -4.12405e54 q^{44} +5.14127e54 q^{45} +2.68874e55 q^{46} -2.64005e55 q^{47} -9.09873e55 q^{48} -2.57110e56 q^{49} +1.42167e56 q^{50} -5.68634e56 q^{51} +2.89925e57 q^{52} -1.96065e57 q^{53} +6.09290e56 q^{54} +5.08245e57 q^{55} -1.27268e58 q^{56} -1.25654e58 q^{57} +1.53716e58 q^{58} -1.16514e59 q^{59} +1.24850e59 q^{60} +1.04181e60 q^{61} +1.36043e59 q^{62} -3.92443e59 q^{63} -1.68498e60 q^{64} -3.57302e60 q^{65} +6.02320e59 q^{66} -5.71388e60 q^{67} -1.38086e61 q^{68} +4.21435e61 q^{69} +7.49308e60 q^{70} -3.93949e61 q^{71} +3.09704e61 q^{72} +2.89309e62 q^{73} -1.35215e62 q^{74} +2.22833e62 q^{75} -3.05136e62 q^{76} -3.87953e62 q^{77} -4.23437e62 q^{78} +3.53480e63 q^{79} +2.72299e63 q^{80} +9.55005e62 q^{81} +2.34126e63 q^{82} +1.40445e64 q^{83} -9.53001e63 q^{84} +1.70176e64 q^{85} +1.96945e64 q^{86} +2.40936e64 q^{87} +3.06161e64 q^{88} +3.15803e65 q^{89} -1.82343e64 q^{90} +2.72735e65 q^{91} +1.02341e66 q^{92} +2.13234e65 q^{93} +9.36338e64 q^{94} +3.76047e65 q^{95} +1.14486e66 q^{96} +7.03827e66 q^{97} +9.11883e65 q^{98} +9.44080e65 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 13735355166 q^{2} - 33\!\cdots\!38 q^{3}+ \cdots + 18\!\cdots\!74 q^{9}+O(q^{10}) \) 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	\( 6 q + 13735355166 q^{2} - 33\!\cdots\!38 q^{3}+ \cdots + 38\!\cdots\!96 q^{99}+O(q^{100}) \) 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

Representations

Groups

Database

Properties

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