LMFDB - Embedded newform 3.68.a.b.1.5

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			1.5
Root			$-2.19503e9$ 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)

$f(q)$	$=$	$q+1.54594e10 q^{2} -5.55906e15 q^{3} +9.14202e19 q^{4} -4.50800e23 q^{5} -8.59399e25 q^{6} -2.44076e28 q^{7} -8.68106e29 q^{8} +3.09032e31 q^{9} +O(q^{10})$	\(q+1.54594e10 q^{2} -5.55906e15 q^{3} +9.14202e19 q^{4} -4.50800e23 q^{5} -8.59399e25 q^{6} -2.44076e28 q^{7} -8.68106e29 q^{8} +3.09032e31 q^{9} -6.96912e33 q^{10} +7.10839e34 q^{11} -5.08210e35 q^{12} -3.02610e37 q^{13} -3.77328e38 q^{14} +2.50603e39 q^{15} -2.69117e40 q^{16} -2.20518e41 q^{17} +4.77745e41 q^{18} -9.99180e42 q^{19} -4.12122e43 q^{20} +1.35683e44 q^{21} +1.09892e45 q^{22} +7.61516e44 q^{23} +4.82585e45 q^{24} +1.35458e47 q^{25} -4.67817e47 q^{26} -1.71793e47 q^{27} -2.23135e48 q^{28} -1.27112e49 q^{29} +3.87417e49 q^{30} +3.96906e49 q^{31} -2.87929e50 q^{32} -3.95160e50 q^{33} -3.40908e51 q^{34} +1.10029e52 q^{35} +2.82517e51 q^{36} +3.96198e52 q^{37} -1.54468e53 q^{38} +1.68223e53 q^{39} +3.91342e53 q^{40} +9.03507e52 q^{41} +2.09759e54 q^{42} +5.47520e54 q^{43} +6.49850e54 q^{44} -1.39311e55 q^{45} +1.17726e55 q^{46} +1.13222e56 q^{47} +1.49604e56 q^{48} +1.77353e56 q^{49} +2.09411e57 q^{50} +1.22587e57 q^{51} -2.76646e57 q^{52} -2.14393e57 q^{53} -2.65581e57 q^{54} -3.20446e58 q^{55} +2.11884e58 q^{56} +5.55450e58 q^{57} -1.96508e59 q^{58} -2.48533e59 q^{59} +2.29101e59 q^{60} -5.29308e59 q^{61} +6.13595e59 q^{62} -7.54272e59 q^{63} -4.79763e59 q^{64} +1.36416e61 q^{65} -6.10895e60 q^{66} +7.51700e60 q^{67} -2.01598e61 q^{68} -4.23331e60 q^{69} +1.70099e62 q^{70} -7.74191e61 q^{71} -2.68272e61 q^{72} -3.21462e62 q^{73} +6.12500e62 q^{74} -7.53020e62 q^{75} -9.13452e62 q^{76} -1.73499e63 q^{77} +2.60063e63 q^{78} -1.09603e63 q^{79} +1.21318e64 q^{80} +9.55005e62 q^{81} +1.39677e63 q^{82} -1.53293e63 q^{83} +1.24042e64 q^{84} +9.94094e64 q^{85} +8.46435e64 q^{86} +7.06622e64 q^{87} -6.17084e64 q^{88} -3.16481e65 q^{89} -2.15368e65 q^{90} +7.38597e65 q^{91} +6.96179e64 q^{92} -2.20643e65 q^{93} +1.75034e66 q^{94} +4.50430e66 q^{95} +1.60062e66 q^{96} -3.41847e66 q^{97} +2.74178e66 q^{98} +2.19672e66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.54594e10	1.27259	0.636295	−	0.771446i	$-0.280466\pi$
$2$	1.54594e10	1.27259	0.636295	+	0.771446i	$0.280466\pi$
$3$	−5.55906e15	−0.577350
$3$	−5.55906e15	−0.577350
$4$	9.14202e19	0.619487
$5$	−4.50800e23	−1.73176	−0.865882	−	0.500248i	$-0.833242\pi$
$5$	−4.50800e23	−1.73176	−0.865882	+	0.500248i	$0.833242\pi$
$6$	−8.59399e25	−0.734731
$7$	−2.44076e28	−1.19328	−0.596638	−	0.802511i	$-0.703497\pi$
$7$	−2.44076e28	−1.19328	−0.596638	+	0.802511i	$0.703497\pi$
$8$	−8.68106e29	−0.484237
$9$	3.09032e31	0.333333
$10$	−6.96912e33	−2.20383
$11$	7.10839e34	0.922819	0.461409	−	0.887187i	$-0.347344\pi$
$11$	7.10839e34	0.922819	0.461409	+	0.887187i	$0.347344\pi$
$12$	−5.08210e35	−0.357661
$13$	−3.02610e37	−1.45808	−0.729039	−	0.684473i	$-0.760033\pi$
$13$	−3.02610e37	−1.45808	−0.729039	+	0.684473i	$0.760033\pi$
$14$	−3.77328e38	−1.51855
$15$	2.50603e39	0.999835
$16$	−2.69117e40	−1.23572
$17$	−2.20518e41	−1.32863	−0.664316	−	0.747452i	$-0.731277\pi$
$17$	−2.20518e41	−1.32863	−0.664316	+	0.747452i	$0.731277\pi$
$18$	4.77745e41	0.424197
$19$	−9.99180e42	−1.45010	−0.725050	−	0.688696i	$-0.758183\pi$
$19$	−9.99180e42	−1.45010	−0.725050	+	0.688696i	$0.758183\pi$
$20$	−4.12122e43	−1.07281
$21$	1.35683e44	0.688938
$22$	1.09892e45	1.17437
$23$	7.61516e44	0.183568	0.0917839	−	0.995779i	$-0.470743\pi$
$23$	7.61516e44	0.183568	0.0917839	+	0.995779i	$0.470743\pi$
$24$	4.82585e45	0.279574
$25$	1.35458e47	1.99901
$26$	−4.67817e47	−1.85554
$27$	−1.71793e47	−0.192450
$28$	−2.23135e48	−0.739219
$29$	−1.27112e49	−1.29973	−0.649865	−	0.760050i	$-0.725174\pi$
$29$	−1.27112e49	−1.29973	−0.649865	+	0.760050i	$0.725174\pi$
$30$	3.87417e49	1.27238
$31$	3.96906e49	0.434582	0.217291	−	0.976107i	$-0.430278\pi$
$31$	3.96906e49	0.434582	0.217291	+	0.976107i	$0.430278\pi$
$32$	−2.87929e50	−1.08833
$33$	−3.95160e50	−0.532790
$34$	−3.40908e51	−1.69080
$35$	1.10029e52	2.06647
$36$	2.82517e51	0.206496
$37$	3.96198e52	1.15652	0.578262	−	0.815851i	$-0.303731\pi$
$37$	3.96198e52	1.15652	0.578262	+	0.815851i	$0.303731\pi$
$38$	−1.54468e53	−1.84538
$39$	1.68223e53	0.841821
$40$	3.91342e53	0.838585
$41$	9.03507e52	0.0846588	0.0423294	−	0.999104i	$-0.486522\pi$
$41$	9.03507e52	0.0846588	0.0423294	+	0.999104i	$0.486522\pi$
$42$	2.09759e54	0.876736
$43$	5.47520e54	1.04042	0.520208	−	0.854040i	$-0.325855\pi$
$43$	5.47520e54	1.04042	0.520208	+	0.854040i	$0.325855\pi$
$44$	6.49850e54	0.571674
$45$	−1.39311e55	−0.577255
$46$	1.17726e55	0.233607
$47$	1.13222e56	1.09308	0.546539	−	0.837434i	$-0.315945\pi$
$47$	1.13222e56	1.09308	0.546539	+	0.837434i	$0.315945\pi$
$48$	1.49604e56	0.713445
$49$	1.77353e56	0.423906
$50$	2.09411e57	2.54392
$51$	1.22587e57	0.767086
$52$	−2.76646e57	−0.903260
$53$	−2.14393e57	−0.369802	−0.184901	−	0.982757i	$-0.559196\pi$
$53$	−2.14393e57	−0.369802	−0.184901	+	0.982757i	$0.559196\pi$
$54$	−2.65581e57	−0.244910
$55$	−3.20446e58	−1.59810
$56$	2.11884e58	0.577828
$57$	5.55450e58	0.837216
$58$	−1.96508e59	−1.65402
$59$	−2.48533e59	−1.17989	−0.589945	−	0.807444i	$-0.700850\pi$
$59$	−2.48533e59	−1.17989	−0.589945	+	0.807444i	$0.700850\pi$
$60$	2.29101e59	0.619385
$61$	−5.29308e59	−0.822544	−0.411272	−	0.911513i	$-0.634915\pi$
$61$	−5.29308e59	−0.822544	−0.411272	+	0.911513i	$0.634915\pi$
$62$	6.13595e59	0.553045
$63$	−7.54272e59	−0.397758
$64$	−4.79763e59	−0.149279
$65$	1.36416e61	2.52505
$66$	−6.10895e60	−0.678023
$67$	7.51700e60	0.504125	0.252063	−	0.967711i	$-0.418891\pi$
$67$	7.51700e60	0.504125	0.252063	+	0.967711i	$0.418891\pi$
$68$	−2.01598e61	−0.823070
$69$	−4.23331e60	−0.105983
$70$	1.70099e62	2.62977
$71$	−7.74191e61	−0.744207	−0.372103	−	0.928191i	$-0.621363\pi$
$71$	−7.74191e61	−0.744207	−0.372103	+	0.928191i	$0.621363\pi$
$72$	−2.68272e61	−0.161412
$73$	−3.21462e62	−1.21847	−0.609234	−	0.792990i	$-0.708523\pi$
$73$	−3.21462e62	−1.21847	−0.609234	+	0.792990i	$0.708523\pi$
$74$	6.12500e62	1.47178
$75$	−7.53020e62	−1.15413
$76$	−9.13452e62	−0.898318
$77$	−1.73499e63	−1.10118
$78$	2.60063e63	1.07129
$79$	−1.09603e63	−0.294656	−0.147328	−	0.989088i	$-0.547067\pi$
$79$	−1.09603e63	−0.294656	−0.147328	+	0.989088i	$0.547067\pi$
$80$	1.21318e64	2.13998
$81$	9.55005e62	0.111111
$82$	1.39677e63	0.107736
$83$	−1.53293e63	−0.0787783	−0.0393891	−	0.999224i	$-0.512541\pi$
$83$	−1.53293e63	−0.0787783	−0.0393891	+	0.999224i	$0.512541\pi$
$84$	1.24042e64	0.426788
$85$	9.94094e64	2.30088
$86$	8.46435e64	1.32402
$87$	7.06622e64	0.750399
$88$	−6.17084e64	−0.446863
$89$	−3.16481e65	−1.56958	−0.784788	−	0.619764i	$-0.787228\pi$
$89$	−3.16481e65	−1.56958	−0.784788	+	0.619764i	$0.787228\pi$
$90$	−2.15368e65	−0.734609
$91$	7.38597e65	1.73989
$92$	6.96179e64	0.113718
$93$	−2.20643e65	−0.250906
$94$	1.75034e66	1.39104
$95$	4.50430e66	2.51123
$96$	1.60062e66	0.628349
$97$	−3.41847e66	−0.948372	−0.474186	−	0.880425i	$-0.657257\pi$
$97$	−3.41847e66	−0.948372	−0.474186	+	0.880425i	$0.657257\pi$
$98$	2.74178e66	0.539459
$99$	2.19672e66	0.307606
$100$	1.23836e67	1.23836
$101$	1.19802e67	0.858415	0.429208	−	0.903206i	$-0.358793\pi$
$101$	1.19802e67	0.858415	0.429208	+	0.903206i	$0.358793\pi$
$102$	1.89513e67	0.976186
$103$	6.50626e66	0.241704	0.120852	−	0.992671i	$-0.461437\pi$
$103$	6.50626e66	0.241704	0.120852	+	0.992671i	$0.461437\pi$
$104$	2.62697e67	0.706055
$105$	−6.11661e67	−1.19308
$106$	−3.31439e67	−0.470606
$107$	−2.53568e66	−0.0262868	−0.0131434	−	0.999914i	$-0.504184\pi$
$107$	−2.53568e66	−0.0262868	−0.0131434	+	0.999914i	$0.504184\pi$
$108$	−1.57053e67	−0.119220
$109$	−6.11155e67	−0.340694	−0.170347	−	0.985384i	$-0.554489\pi$
$109$	−6.11155e67	−0.340694	−0.170347	+	0.985384i	$0.554489\pi$
$110$	−4.95392e68	−2.03373
$111$	−2.20249e68	−0.667719
$112$	6.56849e68	1.47456
$113$	−4.79537e68	−0.799271	−0.399635	−	0.916674i	$-0.630863\pi$
$113$	−4.79537e68	−0.799271	−0.399635	+	0.916674i	$0.630863\pi$
$114$	8.58694e68	1.06543
$115$	−3.43291e68	−0.317896
$116$	−1.16206e69	−0.805166
$117$	−9.35159e68	−0.486026
$118$	−3.84218e69	−1.50152
$119$	5.38231e69	1.58542
$120$	−2.17550e69	−0.484157
$121$	−8.80563e68	−0.148406
$122$	−8.18280e69	−1.04676
$123$	−5.02265e68	−0.0488778
$124$	3.62853e69	0.269218
$125$	−3.05171e70	−1.73005
$126$	−1.16606e70	−0.506184
$127$	−4.17481e69	−0.139063	−0.0695315	−	0.997580i	$-0.522150\pi$
$127$	−4.17481e69	−0.139063	−0.0695315	+	0.997580i	$0.522150\pi$
$128$	3.50740e70	0.898362
$129$	−3.04370e70	−0.600684
$130$	2.10892e71	3.21335
$131$	2.60226e70	0.306734	0.153367	−	0.988169i	$-0.450988\pi$
$131$	2.60226e70	0.306734	0.153367	+	0.988169i	$0.450988\pi$
$132$	−3.61256e70	−0.330056
$133$	2.43876e71	1.73037
$134$	1.16209e71	0.641545
$135$	7.74441e70	0.333278
$136$	1.91433e71	0.643373
$137$	−6.00522e71	−1.57903	−0.789514	−	0.613732i	$-0.789667\pi$
$137$	−6.00522e71	−1.57903	−0.789514	+	0.613732i	$0.789667\pi$
$138$	−6.54446e70	−0.134873
$139$	2.87650e71	0.465446	0.232723	−	0.972543i	$-0.425236\pi$
$139$	2.87650e71	0.465446	0.232723	+	0.972543i	$0.425236\pi$
$140$	1.00589e72	1.28015
$141$	−6.29406e71	−0.631089
$142$	−1.19686e72	−0.947070
$143$	−2.15107e72	−1.34554
$144$	−8.31655e71	−0.411908
$145$	5.73020e72	2.25083
$146$	−4.96963e72	−1.55061
$147$	−9.85916e71	−0.244742
$148$	3.62205e72	0.716451
$149$	−3.32108e72	−0.524250	−0.262125	−	0.965034i	$-0.584423\pi$
$149$	−3.32108e72	−0.524250	−0.262125	+	0.965034i	$0.584423\pi$
$150$	−1.16413e73	−1.46873
$151$	1.17411e73	1.18571	0.592857	−	0.805308i	$-0.298000\pi$
$151$	1.17411e73	1.18571	0.592857	+	0.805308i	$0.298000\pi$
$152$	8.67394e72	0.702192
$153$	−6.81469e72	−0.442877
$154$	−2.68219e73	−1.40135
$155$	−1.78926e73	−0.752593
$156$	1.53789e73	0.521497
$157$	−3.36199e73	−0.920362	−0.460181	−	0.887825i	$-0.652215\pi$
$157$	−3.36199e73	−0.920362	−0.460181	+	0.887825i	$0.652215\pi$
$158$	−1.69440e73	−0.374976
$159$	1.19182e73	0.213505
$160$	1.29799e74	1.88474
$161$	−1.85868e73	−0.219047
$162$	1.47638e73	0.141399
$163$	−1.45484e74	−1.13379	−0.566893	−	0.823792i	$-0.691855\pi$
$163$	−1.45484e74	−1.13379	−0.566893	+	0.823792i	$0.691855\pi$
$164$	8.25987e72	0.0524451
$165$	1.78138e74	0.922666
$166$	−2.36982e73	−0.100252
$167$	3.06761e74	1.06120	0.530601	−	0.847622i	$-0.321966\pi$
$167$	3.06761e74	1.06120	0.530601	+	0.847622i	$0.321966\pi$
$168$	−1.17787e74	−0.333609
$169$	4.84997e74	1.12599
$170$	1.53681e75	2.92808
$171$	−3.08778e74	−0.483367
$172$	5.00544e74	0.644524
$173$	−9.30738e74	−0.986923	−0.493461	−	0.869768i	$-0.664269\pi$
$173$	−9.30738e74	−0.986923	−0.493461	+	0.869768i	$0.664269\pi$
$174$	1.09240e75	0.954951
$175$	−3.30621e75	−2.38537
$176$	−1.91299e75	−1.14035
$177$	1.38161e75	0.681210
$178$	−4.89262e75	−1.99743
$179$	−8.04736e74	−0.272318	−0.136159	−	0.990687i	$-0.543476\pi$
$179$	−8.04736e74	−0.272318	−0.136159	+	0.990687i	$0.543476\pi$
$180$	−1.27359e75	−0.357602
$181$	−2.73278e75	−0.637342	−0.318671	−	0.947865i	$-0.603237\pi$
$181$	−2.73278e75	−0.637342	−0.318671	+	0.947865i	$0.603237\pi$
$182$	1.14183e76	2.21416
$183$	2.94245e75	0.474896
$184$	−6.61076e74	−0.0888903
$185$	−1.78606e76	−2.00283
$186$	−3.41101e75	−0.319300
$187$	−1.56753e76	−1.22609
$188$	1.03507e76	0.677147
$189$	4.19304e75	0.229646
$190$	6.96340e76	3.19577
$191$	−1.46165e76	−0.562632	−0.281316	−	0.959615i	$-0.590771\pi$
$191$	−1.46165e76	−0.562632	−0.281316	+	0.959615i	$0.590771\pi$
$192$	2.66703e75	0.0861860
$193$	−7.27272e74	−0.0197482	−0.00987409	−	0.999951i	$-0.503143\pi$
$193$	−7.27272e74	−0.0197482	−0.00987409	+	0.999951i	$0.503143\pi$
$194$	−5.28476e76	−1.20689
$195$	−7.58347e76	−1.45784
$196$	1.62136e76	0.262604
$197$	1.25043e77	1.70782	0.853909	−	0.520422i	$-0.174225\pi$
$197$	1.25043e77	1.70782	0.853909	+	0.520422i	$0.174225\pi$
$198$	3.39600e76	0.391457
$199$	−2.48331e75	−0.0241798	−0.0120899	−	0.999927i	$-0.503848\pi$
$199$	−2.48331e75	−0.0241798	−0.0120899	+	0.999927i	$0.503848\pi$
$200$	−1.17592e77	−0.967995
$201$	−4.17874e76	−0.291057
$202$	1.85207e77	1.09241
$203$	3.10249e77	1.55094
$204$	1.12069e77	0.475200
$205$	−4.07301e76	−0.146609
$206$	1.00583e77	0.307591
$207$	2.35332e76	0.0611893
$208$	8.14373e77	1.80178
$209$	−7.10256e77	−1.33818
$210$	−9.45593e77	−1.51830
$211$	−7.12060e77	−0.975111	−0.487555	−	0.873092i	$-0.662111\pi$
$211$	−7.12060e77	−0.975111	−0.487555	+	0.873092i	$0.662111\pi$
$212$	−1.95998e77	−0.229087
$213$	4.30378e77	0.429668
$214$	−3.92002e76	−0.0334524
$215$	−2.46822e78	−1.80175
$216$	1.49134e77	0.0931915
$217$	−9.68753e77	−0.518576
$218$	−9.44812e77	−0.433563
$219$	1.78703e78	0.703483
$220$	−2.92953e78	−0.990005
$221$	6.67308e78	1.93725
$222$	−3.40492e78	−0.849733
$223$	−8.54795e78	−1.83505	−0.917527	−	0.397673i	$-0.869818\pi$
$223$	−8.54795e78	−1.83505	−0.917527	+	0.397673i	$0.869818\pi$
$224$	7.02766e78	1.29868
$225$	4.18608e78	0.666336
$226$	−7.41337e78	−1.01714
$227$	1.20523e79	1.42628	0.713140	−	0.701022i	$-0.247273\pi$
$227$	1.20523e79	1.42628	0.713140	+	0.701022i	$0.247273\pi$
$228$	5.07793e78	0.518644
$229$	2.13231e79	1.88088	0.940441	−	0.339956i	$-0.110412\pi$
$229$	2.13231e79	1.88088	0.940441	+	0.339956i	$0.110412\pi$
$230$	−5.30709e78	−0.404552
$231$	9.64490e78	0.635765
$232$	1.10347e79	0.629377
$233$	8.68622e78	0.428951	0.214475	−	0.976729i	$-0.431196\pi$
$233$	8.68622e78	0.428951	0.214475	+	0.976729i	$0.431196\pi$
$234$	−1.44570e79	−0.618512
$235$	−5.10403e79	−1.89295
$236$	−2.27209e79	−0.730926
$237$	6.09290e78	0.170120
$238$	8.32074e79	2.01760
$239$	5.11341e79	1.07741	0.538705	−	0.842494i	$-0.318914\pi$
$239$	5.11341e79	1.07741	0.538705	+	0.842494i	$0.318914\pi$
$240$	−6.74413e79	−1.23552
$241$	−7.96115e79	−1.26883	−0.634417	−	0.772991i	$-0.718760\pi$
$241$	−7.96115e79	−1.26883	−0.634417	+	0.772991i	$0.718760\pi$
$242$	−1.36130e79	−0.188860
$243$	−5.30893e78	−0.0641500
$244$	−4.83894e79	−0.509555
$245$	−7.99507e79	−0.734106
$246$	−7.76473e78	−0.0622014
$247$	3.02361e80	2.11436
$248$	−3.44557e79	−0.210441
$249$	8.52165e78	0.0454826
$250$	−4.71778e80	−2.20164
$251$	4.24515e80	1.73310	0.866550	−	0.499089i	$-0.166332\pi$
$251$	4.24515e80	1.73310	0.866550	+	0.499089i	$0.166332\pi$
$252$	−6.89556e79	−0.246406
$253$	5.41315e79	0.169400
$254$	−6.45401e79	−0.176970
$255$	−5.52623e80	−1.32841
$256$	6.13025e80	1.29253
$257$	−7.49023e80	−1.38591	−0.692955	−	0.720981i	$-0.743692\pi$
$257$	−7.49023e80	−1.38591	−0.692955	+	0.720981i	$0.743692\pi$
$258$	−4.70538e80	−0.764425
$259$	−9.67024e80	−1.38005
$260$	1.24712e81	1.56423
$261$	−3.92816e80	−0.433243
$262$	4.02294e80	0.390347
$263$	6.83991e80	0.584161	0.292081	−	0.956394i	$-0.405652\pi$
$263$	6.83991e80	0.584161	0.292081	+	0.956394i	$0.405652\pi$
$264$	3.43041e80	0.257997
$265$	9.66483e80	0.640409
$266$	3.77018e81	2.20205
$267$	1.75934e81	0.906196
$268$	6.87205e80	0.312299
$269$	−1.13439e81	−0.455052	−0.227526	−	0.973772i	$-0.573064\pi$
$269$	−1.13439e81	−0.455052	−0.227526	+	0.973772i	$0.573064\pi$
$270$	1.19724e81	0.424127
$271$	−3.90095e81	−1.22095	−0.610477	−	0.792034i	$-0.709022\pi$
$271$	−3.90095e81	−1.22095	−0.610477	+	0.792034i	$0.709022\pi$
$272$	5.93450e81	1.64182
$273$	−4.10591e81	−1.00452
$274$	−9.28373e81	−2.00946
$275$	9.62889e81	1.84472
$276$	−3.87010e80	−0.0656550
$277$	−1.27565e82	−1.91716	−0.958580	−	0.284822i	$-0.908065\pi$
$277$	−1.27565e82	−1.91716	−0.958580	+	0.284822i	$0.908065\pi$
$278$	4.44691e81	0.592322
$279$	1.22657e81	0.144861
$280$	−9.55172e81	−1.00066
$281$	−5.22942e81	−0.486174	−0.243087	−	0.970004i	$-0.578160\pi$
$281$	−5.22942e81	−0.486174	−0.243087	+	0.970004i	$0.578160\pi$
$282$	−9.73025e81	−0.803117
$283$	2.21940e82	1.62701	0.813504	−	0.581560i	$-0.197557\pi$
$283$	2.21940e82	1.62701	0.813504	+	0.581560i	$0.197557\pi$
$284$	−7.07767e81	−0.461026
$285$	−2.50397e82	−1.44986
$286$	−3.32543e82	−1.71232
$287$	−2.20524e81	−0.101021
$288$	−8.89792e81	−0.362777
$289$	2.10809e82	0.765263
$290$	8.85857e82	2.86438
$291$	1.90035e82	0.547543
$292$	−2.93881e82	−0.754826
$293$	2.74650e82	0.629092	0.314546	−	0.949242i	$-0.398148\pi$
$293$	2.74650e82	0.629092	0.314546	+	0.949242i	$0.398148\pi$
$294$	−1.52417e82	−0.311457
$295$	1.12039e83	2.04329
$296$	−3.43942e82	−0.560032
$297$	−1.22117e82	−0.177597
$298$	−5.13420e82	−0.667156
$299$	−2.30442e82	−0.267656
$300$	−6.88412e82	−0.714968
$301$	−1.33636e83	−1.24150
$302$	1.81510e83	1.50893
$303$	−6.65986e82	−0.495606
$304$	2.68896e83	1.79192
$305$	2.38612e83	1.42445
$306$	−1.05351e83	−0.563602
$307$	5.21767e82	0.250231	0.125116	−	0.992142i	$-0.460070\pi$
$307$	5.21767e82	0.250231	0.125116	+	0.992142i	$0.460070\pi$
$308$	−1.58613e83	−0.682165
$309$	−3.61687e82	−0.139548
$310$	−2.76609e83	−0.957743
$311$	1.99778e83	0.620974	0.310487	−	0.950578i	$-0.399508\pi$
$311$	1.99778e83	0.620974	0.310487	+	0.950578i	$0.399508\pi$
$312$	−1.46035e83	−0.407641
$313$	4.74161e82	0.118902	0.0594512	−	0.998231i	$-0.481065\pi$
$313$	4.74161e82	0.118902	0.0594512	+	0.998231i	$0.481065\pi$
$314$	−5.19745e83	−1.17124
$315$	3.40026e83	0.688824
$316$	−1.00199e83	−0.182536
$317$	1.74578e83	0.286091	0.143046	−	0.989716i	$-0.454310\pi$
$317$	1.74578e83	0.286091	0.143046	+	0.989716i	$0.454310\pi$
$318$	1.84249e83	0.271704
$319$	−9.03560e83	−1.19941
$320$	2.16277e83	0.258515
$321$	1.40960e82	0.0151767
$322$	−2.87341e83	−0.278757
$323$	2.20337e84	1.92665
$324$	8.73067e82	0.0688319
$325$	−4.09909e84	−2.91471
$326$	−2.24910e84	−1.44284
$327$	3.39745e83	0.196700
$328$	−7.84340e82	−0.0409950
$329$	−2.76347e84	−1.30434
$330$	2.75391e84	1.17418
$331$	5.00001e84	1.92634	0.963170	−	0.268892i	$-0.0866575\pi$
$331$	5.00001e84	1.92634	0.963170	+	0.268892i	$0.0866575\pi$
$332$	−1.40141e83	−0.0488021
$333$	1.22438e84	0.385508
$334$	4.74235e84	1.35048
$335$	−3.38866e84	−0.873026
$336$	−3.65146e84	−0.851336
$337$	3.86569e84	0.815879	0.407940	−	0.913009i	$-0.366247\pi$
$337$	3.86569e84	0.815879	0.407940	+	0.913009i	$0.366247\pi$
$338$	7.49777e84	1.43292
$339$	2.66577e84	0.461459
$340$	9.08803e84	1.42536
$341$	2.82137e84	0.401040
$342$	−4.77353e84	−0.615128
$343$	5.88284e84	0.687439
$344$	−4.75305e84	−0.503808
$345$	1.90838e84	0.183537
$346$	−1.43887e85	−1.25595
$347$	−1.81272e85	−1.43646	−0.718230	−	0.695805i	$-0.755048\pi$
$347$	−1.81272e85	−1.43646	−0.718230	+	0.695805i	$0.755048\pi$
$348$	6.45995e84	0.464863
$349$	1.08268e85	0.707701	0.353851	−	0.935302i	$-0.384872\pi$
$349$	1.08268e85	0.707701	0.353851	+	0.935302i	$0.384872\pi$
$350$	−5.11121e85	−3.03560
$351$	5.19861e84	0.280607
$352$	−2.04671e85	−1.00433
$353$	3.31305e84	0.147834	0.0739172	−	0.997264i	$-0.476450\pi$
$353$	3.31305e84	0.147834	0.0739172	+	0.997264i	$0.476450\pi$
$354$	2.13589e85	0.866901
$355$	3.49005e85	1.28879
$356$	−2.89328e85	−0.972332
$357$	−2.99206e85	−0.915345
$358$	−1.24408e85	−0.346550
$359$	−2.51209e85	−0.637339	−0.318670	−	0.947866i	$-0.603236\pi$
$359$	−2.51209e85	−0.637339	−0.318670	+	0.947866i	$0.603236\pi$
$360$	1.20937e85	0.279528
$361$	5.23582e85	1.10279
$362$	−4.22472e85	−0.811075
$363$	4.89510e84	0.0856821
$364$	6.75227e85	1.07784
$365$	1.44915e86	2.11010
$366$	4.54887e85	0.604348
$367$	−3.73247e85	−0.452567	−0.226284	−	0.974061i	$-0.572658\pi$
$367$	−3.73247e85	−0.452567	−0.226284	+	0.974061i	$0.572658\pi$
$368$	−2.04937e85	−0.226839
$369$	2.79212e84	0.0282196
$370$	−2.76115e86	−2.54878
$371$	5.23281e85	0.441275
$372$	−2.01712e85	−0.155433
$373$	2.06990e86	1.45782	0.728909	−	0.684611i	$-0.240028\pi$
$373$	2.06990e86	1.45782	0.728909	+	0.684611i	$0.240028\pi$
$374$	−2.42331e86	−1.56031
$375$	1.69647e86	0.998844
$376$	−9.82883e85	−0.529309
$377$	3.84653e86	1.89511
$378$	6.48221e85	0.292245
$379$	−3.00596e86	−1.24042	−0.620211	−	0.784435i	$-0.712953\pi$
$379$	−3.00596e86	−1.24042	−0.620211	+	0.784435i	$0.712953\pi$
$380$	4.11784e86	1.55568
$381$	2.32080e85	0.0802881
$382$	−2.25962e86	−0.716000
$383$	−1.84182e86	−0.534672	−0.267336	−	0.963603i	$-0.586143\pi$
$383$	−1.84182e86	−0.534672	−0.267336	+	0.963603i	$0.586143\pi$
$384$	−1.94978e86	−0.518669
$385$	7.82133e86	1.90698
$386$	−1.12432e85	−0.0251314
$387$	1.69201e86	0.346805
$388$	−3.12517e86	−0.587504
$389$	−4.23178e86	−0.729813	−0.364907	−	0.931044i	$-0.618899\pi$
$389$	−4.23178e86	−0.729813	−0.364907	+	0.931044i	$0.618899\pi$
$390$	−1.17236e87	−1.85523
$391$	−1.67928e86	−0.243894
$392$	−1.53961e86	−0.205271
$393$	−1.44661e86	−0.177093
$394$	1.93310e87	2.17335
$395$	4.94091e86	0.510275
$396$	2.00824e86	0.190558
$397$	−1.01728e87	−0.887071	−0.443536	−	0.896257i	$-0.646276\pi$
$397$	−1.01728e87	−0.887071	−0.443536	+	0.896257i	$0.646276\pi$
$398$	−3.83906e85	−0.0307710
$399$	−1.35572e87	−0.999029
$400$	−3.64540e87	−2.47022
$401$	3.63635e86	0.226636	0.113318	−	0.993559i	$-0.463852\pi$
$401$	3.63635e86	0.226636	0.113318	+	0.993559i	$0.463852\pi$
$402$	−6.46010e86	−0.370396
$403$	−1.20108e87	−0.633654
$404$	1.09523e87	0.531777
$405$	−4.30516e86	−0.192418
$406$	4.79628e87	1.97371
$407$	2.81633e87	1.06726
$408$	−1.06419e87	−0.371452
$409$	2.04435e86	0.0657391	0.0328696	−	0.999460i	$-0.489535\pi$
$409$	2.04435e86	0.0657391	0.0328696	+	0.999460i	$0.489535\pi$
$410$	−6.29664e86	−0.186573
$411$	3.33834e87	0.911652
$412$	5.94803e86	0.149733
$413$	6.06609e87	1.40793
$414$	3.63811e86	0.0778689
$415$	6.91045e86	0.136425
$416$	8.71302e87	1.58687
$417$	−1.59907e87	−0.268725
$418$	−1.09802e88	−1.70295
$419$	8.61912e87	1.23394	0.616969	−	0.786988i	$-0.288360\pi$
$419$	8.61912e87	1.23394	0.616969	+	0.786988i	$0.288360\pi$
$420$	−5.59181e87	−0.739097
$421$	5.00181e87	0.610488	0.305244	−	0.952274i	$-0.401262\pi$
$421$	5.00181e87	0.610488	0.305244	+	0.952274i	$0.401262\pi$
$422$	−1.10080e88	−1.24092
$423$	3.49890e87	0.364359
$424$	1.86116e87	0.179072
$425$	−2.98709e88	−2.65595
$426$	6.65339e87	0.546791
$427$	1.29191e88	0.981521
$428$	−2.31812e86	−0.0162844
$429$	1.19579e88	0.776848
$430$	−3.81573e88	−2.29290
$431$	2.13004e88	1.18413	0.592066	−	0.805890i	$-0.298313\pi$
$431$	2.13004e88	1.18413	0.592066	+	0.805890i	$0.298313\pi$
$432$	4.62322e87	0.237815
$433$	−3.49897e88	−1.66570	−0.832848	−	0.553502i	$-0.813291\pi$
$433$	−3.49897e88	−1.66570	−0.832848	+	0.553502i	$0.813291\pi$
$434$	−1.49764e88	−0.659935
$435$	−3.18545e88	−1.29951
$436$	−5.58719e87	−0.211055
$437$	−7.60891e87	−0.266192
$438$	2.76265e88	0.895246
$439$	−2.73112e88	−0.819935	−0.409968	−	0.912100i	$-0.634460\pi$
$439$	−2.73112e88	−0.819935	−0.409968	+	0.912100i	$0.634460\pi$
$440$	2.78181e88	0.773862
$441$	5.48076e87	0.141302
$442$	1.03162e89	2.46532
$443$	4.66123e88	1.03270	0.516351	−	0.856377i	$-0.327290\pi$
$443$	4.66123e88	1.03270	0.516351	+	0.856377i	$0.327290\pi$
$444$	−2.01352e88	−0.413643
$445$	1.42670e89	2.71814
$446$	−1.32146e89	−2.33527
$447$	1.84621e88	0.302676
$448$	1.17099e88	0.178130
$449$	−1.15079e89	−1.62458	−0.812291	−	0.583252i	$-0.801780\pi$
$449$	−1.15079e89	−1.62458	−0.812291	+	0.583252i	$0.801780\pi$
$450$	6.47145e88	0.847974
$451$	6.42248e87	0.0781248
$452$	−4.38393e88	−0.495138
$453$	−6.52693e88	−0.684572
$454$	1.86322e89	1.81507
$455$	−3.32960e89	−3.01308
$456$	−4.82190e88	−0.405411
$457$	−1.02016e88	−0.0797032	−0.0398516	−	0.999206i	$-0.512689\pi$
$457$	−1.02016e88	−0.0797032	−0.0398516	+	0.999206i	$0.512689\pi$
$458$	3.29643e89	2.39359
$459$	3.78833e88	0.255695
$460$	−3.13838e88	−0.196933
$461$	9.99507e88	0.583183	0.291591	−	0.956543i	$-0.405815\pi$
$461$	9.99507e88	0.583183	0.291591	+	0.956543i	$0.405815\pi$
$462$	1.49105e89	0.809068
$463$	−2.05937e89	−1.03937	−0.519686	−	0.854357i	$-0.673951\pi$
$463$	−2.05937e89	−1.03937	−0.519686	+	0.854357i	$0.673951\pi$
$464$	3.42079e89	1.60611
$465$	9.94658e88	0.434510
$466$	1.34284e89	0.545879
$467$	−2.71368e89	−1.02670	−0.513350	−	0.858179i	$-0.671596\pi$
$467$	−2.71368e89	−1.02670	−0.513350	+	0.858179i	$0.671596\pi$
$468$	−8.54924e88	−0.301087
$469$	−1.83472e89	−0.601560
$470$	−7.89054e89	−2.40895
$471$	1.86895e89	0.531371
$472$	2.15753e89	0.571346
$473$	3.89199e89	0.960114
$474$	9.41928e88	0.216493
$475$	−1.35347e90	−2.89876
$476$	4.92051e89	0.982149
$477$	−6.62542e88	−0.123267
$478$	7.90504e89	1.37110
$479$	−5.26009e88	−0.0850655	−0.0425328	−	0.999095i	$-0.513543\pi$
$479$	−5.26009e88	−0.0850655	−0.0425328	+	0.999095i	$0.513543\pi$
$480$	−7.21558e89	−1.08815
$481$	−1.19893e90	−1.68630
$482$	−1.23075e90	−1.61471
$483$	1.03325e89	0.126467
$484$	−8.05012e88	−0.0919354
$485$	1.54105e90	1.64236
$486$	−8.20731e88	−0.0816367
$487$	1.02581e90	0.952464	0.476232	−	0.879320i	$-0.342002\pi$
$487$	1.02581e90	0.952464	0.476232	+	0.879320i	$0.342002\pi$
$488$	4.59495e89	0.398306
$489$	8.08755e89	0.654591
$490$	−1.23599e90	−0.934216
$491$	−4.65116e89	−0.328346	−0.164173	−	0.986432i	$-0.552495\pi$
$491$	−4.65116e89	−0.328346	−0.164173	+	0.986432i	$0.552495\pi$
$492$	−4.59171e88	−0.0302792
$493$	2.80304e90	1.72686
$494$	4.67434e90	2.69071
$495$	−9.90280e89	−0.532702
$496$	−1.06814e90	−0.537023
$497$	1.88961e90	0.888043
$498$	1.31740e89	0.0578808
$499$	−2.05191e90	−0.842926	−0.421463	−	0.906846i	$-0.638483\pi$
$499$	−2.05191e90	−0.842926	−0.421463	+	0.906846i	$0.638483\pi$
$500$	−2.78988e90	−1.07174
$501$	−1.70530e90	−0.612685
$502$	6.56277e90	2.20553
$503$	2.91730e90	0.917178	0.458589	−	0.888649i	$-0.348355\pi$
$503$	2.91730e90	0.917178	0.458589	+	0.888649i	$0.348355\pi$
$504$	6.54788e89	0.192609
$505$	−5.40067e90	−1.48657
$506$	8.36843e89	0.215577
$507$	−2.69613e90	−0.650090
$508$	−3.81661e89	−0.0861477
$509$	−6.48428e90	−1.37030	−0.685151	−	0.728401i	$-0.740264\pi$
$509$	−6.48428e90	−1.37030	−0.685151	+	0.728401i	$0.740264\pi$
$510$	−8.54324e90	−1.69053
$511$	7.84613e90	1.45397
$512$	4.30101e90	0.746494
$513$	1.71652e90	0.279072
$514$	−1.15795e91	−1.76370
$515$	−2.93302e90	−0.418575
$516$	−2.78255e90	−0.372116
$517$	8.04823e90	1.00871
$518$	−1.49496e91	−1.75624
$519$	5.17403e90	0.569800
$520$	−1.18424e91	−1.22272
$521$	1.93294e91	1.87135	0.935676	−	0.352861i	$-0.114791\pi$
$521$	1.93294e91	1.87135	0.935676	+	0.352861i	$0.114791\pi$
$522$	−6.07271e90	−0.551341
$523$	−8.04944e90	−0.685423	−0.342712	−	0.939441i	$-0.611345\pi$
$523$	−8.04944e90	−0.685423	−0.342712	+	0.939441i	$0.611345\pi$
$524$	2.37899e90	0.190018
$525$	1.83794e91	1.37719
$526$	1.05741e91	0.743398
$527$	−8.75249e90	−0.577399
$528$	1.06344e91	0.658380
$529$	−1.66295e91	−0.966303
$530$	1.49413e91	0.814979
$531$	−7.68045e90	−0.393297
$532$	2.22952e91	1.07194
$533$	−2.73410e90	−0.123439
$534$	2.71984e91	1.15322
$535$	1.14309e90	0.0455226
$536$	−6.52555e90	−0.244116
$537$	4.47358e90	0.157223
$538$	−1.75370e91	−0.579096
$539$	1.26069e91	0.391188
$540$	7.07995e90	0.206462
$541$	−4.40128e91	−1.20634	−0.603171	−	0.797612i	$-0.706096\pi$
$541$	−4.40128e91	−1.20634	−0.603171	+	0.797612i	$0.706096\pi$
$542$	−6.03065e91	−1.55377
$543$	1.51917e91	0.367970
$544$	6.34935e91	1.44599
$545$	2.75509e91	0.590001
$546$	−6.34750e91	−1.27835
$547$	1.39402e90	0.0264055	0.0132027	−	0.999913i	$-0.495797\pi$
$547$	1.39402e90	0.0264055	0.0132027	+	0.999913i	$0.495797\pi$
$548$	−5.48998e91	−0.978188
$549$	−1.63573e91	−0.274181
$550$	1.48857e92	2.34758
$551$	1.27008e92	1.88474
$552$	3.67496e90	0.0513209
$553$	2.67515e91	0.351606
$554$	−1.97208e92	−2.43976
$555$	9.92882e91	1.15633
$556$	2.62970e91	0.288338
$557$	−7.45948e91	−0.770122	−0.385061	−	0.922891i	$-0.625820\pi$
$557$	−7.45948e91	−0.770122	−0.385061	+	0.922891i	$0.625820\pi$
$558$	1.89620e91	0.184348
$559$	−1.65685e92	−1.51701
$560$	−2.96108e92	−2.55359
$561$	8.71397e91	0.707881
$562$	−8.08439e91	−0.618701
$563$	6.11363e91	0.440826	0.220413	−	0.975407i	$-0.429259\pi$
$563$	6.11363e91	0.440826	0.220413	+	0.975407i	$0.429259\pi$
$564$	−5.75404e91	−0.390951
$565$	2.16175e92	1.38415
$566$	3.43106e92	2.07051
$567$	−2.33094e91	−0.132586
$568$	6.72080e91	0.360373
$569$	−2.02202e92	−1.02217	−0.511086	−	0.859530i	$-0.670757\pi$
$569$	−2.02202e92	−1.02217	−0.511086	+	0.859530i	$0.670757\pi$
$570$	−3.87100e92	−1.84508
$571$	−2.54392e92	−1.14339	−0.571693	−	0.820468i	$-0.693713\pi$
$571$	−2.54392e92	−1.14339	−0.571693	+	0.820468i	$0.693713\pi$
$572$	−1.96651e92	−0.833545
$573$	8.12538e91	0.324836
$574$	−3.40918e91	−0.128559
$575$	1.03154e92	0.366954
$576$	−1.48262e91	−0.0497595
$577$	9.66765e91	0.306148	0.153074	−	0.988215i	$-0.451083\pi$
$577$	9.66765e91	0.306148	0.153074	+	0.988215i	$0.451083\pi$
$578$	3.25898e92	0.973866
$579$	4.04295e90	0.0114016
$580$	5.23856e92	1.39436
$581$	3.74151e91	0.0940041
$582$	2.93783e92	0.696798
$583$	−1.52399e92	−0.341260
$584$	2.79063e92	0.590028
$585$	4.21570e92	0.841682
$586$	4.24593e92	0.800577
$587$	7.52251e92	1.33964	0.669818	−	0.742526i	$-0.266372\pi$
$587$	7.52251e92	1.33964	0.669818	+	0.742526i	$0.266372\pi$
$588$	−9.01326e91	−0.151615
$589$	−3.96581e92	−0.630187
$590$	1.73205e93	2.60027
$591$	−6.95123e92	−0.986010
$592$	−1.06623e93	−1.42914
$593$	1.43627e92	0.181930	0.0909649	−	0.995854i	$-0.471005\pi$
$593$	1.43627e92	0.181930	0.0909649	+	0.995854i	$0.471005\pi$
$594$	−1.88786e92	−0.226008
$595$	−2.42635e93	−2.74558
$596$	−3.03613e92	−0.324766
$597$	1.38049e91	0.0139602
$598$	−3.56250e92	−0.340616
$599$	−5.33863e92	−0.482649	−0.241324	−	0.970444i	$-0.577582\pi$
$599$	−5.33863e92	−0.482649	−0.241324	+	0.970444i	$0.577582\pi$
$600$	6.53701e92	0.558872
$601$	−1.40492e93	−1.13594	−0.567972	−	0.823048i	$-0.692272\pi$
$601$	−1.40492e93	−1.13594	−0.567972	+	0.823048i	$0.692272\pi$
$602$	−2.06594e93	−1.57992
$603$	2.32299e92	0.168042
$604$	1.07337e93	0.734534
$605$	3.96958e92	0.257004
$606$	−1.02958e93	−0.630704
$607$	1.52943e93	0.886560	0.443280	−	0.896383i	$-0.353815\pi$
$607$	1.52943e93	0.886560	0.443280	+	0.896383i	$0.353815\pi$
$608$	2.87693e93	1.57819
$609$	−1.72470e93	−0.895433
$610$	3.68881e93	1.81274
$611$	−3.42619e93	−1.59379
$612$	−6.23000e92	−0.274357
$613$	−2.74248e93	−1.14345	−0.571725	−	0.820446i	$-0.693725\pi$
$613$	−2.74248e93	−1.14345	−0.571725	+	0.820446i	$0.693725\pi$
$614$	8.06623e92	0.318442
$615$	2.26421e92	0.0846449
$616$	1.50615e93	0.533231
$617$	3.30502e91	0.0110821	0.00554104	−	0.999985i	$-0.498236\pi$
$617$	3.30502e91	0.0110821	0.00554104	+	0.999985i	$0.498236\pi$
$618$	−5.59147e92	−0.177587
$619$	6.32390e92	0.190260	0.0951301	−	0.995465i	$-0.469673\pi$
$619$	6.32390e92	0.190260	0.0951301	+	0.995465i	$0.469673\pi$
$620$	−1.63574e93	−0.466222
$621$	−1.30823e92	−0.0353276
$622$	3.08845e93	0.790246
$623$	7.72454e93	1.87294
$624$	−4.52715e93	−1.04026
$625$	4.57813e93	0.997029
$626$	7.33026e92	0.151314
$627$	3.94836e93	0.772598
$628$	−3.07354e93	−0.570152
$629$	−8.73687e93	−1.53659
$630$	5.25661e93	0.876591
$631$	5.87711e92	0.0929352	0.0464676	−	0.998920i	$-0.485204\pi$
$631$	5.87711e92	0.0929352	0.0464676	+	0.998920i	$0.485204\pi$
$632$	9.51471e92	0.142683
$633$	3.95838e93	0.562981
$634$	2.69888e93	0.364077
$635$	1.88200e93	0.240824
$636$	1.08957e93	0.132264
$637$	−5.36687e93	−0.618088
$638$	−1.39685e94	−1.52636
$639$	−2.39249e93	−0.248069
$640$	−1.58114e94	−1.55575
$641$	−3.94793e93	−0.368660	−0.184330	−	0.982864i	$-0.559011\pi$
$641$	−3.94793e93	−0.368660	−0.184330	+	0.982864i	$0.559011\pi$
$642$	2.17916e92	0.0193137
$643$	1.44715e94	1.21744	0.608719	−	0.793386i	$-0.291684\pi$
$643$	1.44715e94	1.21744	0.608719	+	0.793386i	$0.291684\pi$
$644$	−1.69921e93	−0.135697
$645$	1.37210e94	1.04024
$646$	3.40628e94	2.45184
$647$	2.75744e94	1.88457	0.942287	−	0.334806i	$-0.108671\pi$
$647$	2.75744e94	1.88457	0.942287	+	0.334806i	$0.108671\pi$
$648$	−8.29045e92	−0.0538041
$649$	−1.76667e94	−1.08882
$650$	−6.33697e94	−3.70923
$651$	5.38536e93	0.299400
$652$	−1.33002e94	−0.702365
$653$	1.26087e93	0.0632527	0.0316263	−	0.999500i	$-0.489931\pi$
$653$	1.26087e93	0.0632527	0.0316263	+	0.999500i	$0.489931\pi$
$654$	5.25226e93	0.250318
$655$	−1.17310e94	−0.531191
$656$	−2.43149e93	−0.104615
$657$	−9.93420e93	−0.406156
$658$	−4.27216e94	−1.65989
$659$	2.40277e94	0.887261	0.443630	−	0.896210i	$-0.353690\pi$
$659$	2.40277e94	0.887261	0.443630	+	0.896210i	$0.353690\pi$
$660$	1.62854e94	0.571580
$661$	2.08789e94	0.696561	0.348281	−	0.937390i	$-0.386766\pi$
$661$	2.08789e94	0.696561	0.348281	+	0.937390i	$0.386766\pi$
$662$	7.72973e94	2.45144
$663$	−3.70961e94	−1.11847
$664$	1.33075e93	0.0381474
$665$	−1.09939e95	−2.99659
$666$	1.89282e94	0.490594
$667$	−9.67977e93	−0.238588
$668$	2.80441e94	0.657401
$669$	4.75186e94	1.05947
$670$	−5.23868e94	−1.11101
$671$	−3.76253e94	−0.759058
$672$	−3.90672e94	−0.749793
$673$	1.04200e95	1.90267	0.951336	−	0.308155i	$-0.0997117\pi$
$673$	1.04200e95	1.90267	0.951336	+	0.308155i	$0.0997117\pi$
$674$	5.97615e94	1.03828
$675$	−2.32707e94	−0.384710
$676$	4.43385e94	0.697536
$677$	5.62652e94	0.842402	0.421201	−	0.906967i	$-0.361609\pi$
$677$	5.62652e94	0.842402	0.421201	+	0.906967i	$0.361609\pi$
$678$	4.12114e94	0.587249
$679$	8.34366e94	1.13167
$680$	−8.62979e94	−1.11417
$681$	−6.69996e94	−0.823463
$682$	4.36167e94	0.510360
$683$	7.48003e94	0.833316	0.416658	−	0.909063i	$-0.363201\pi$
$683$	7.48003e94	0.833316	0.416658	+	0.909063i	$0.363201\pi$
$684$	−2.82285e94	−0.299439
$685$	2.70715e95	2.73451
$686$	9.09454e94	0.874828
$687$	−1.18536e95	−1.08593
$688$	−1.47347e95	−1.28566
$689$	6.48774e94	0.539199
$690$	2.95024e94	0.233568
$691$	−1.71759e95	−1.29540	−0.647701	−	0.761895i	$-0.724269\pi$
$691$	−1.71759e95	−1.29540	−0.647701	+	0.761895i	$0.724269\pi$
$692$	−8.50883e94	−0.611386
$693$	−5.36166e94	−0.367059
$694$	−2.80237e95	−1.82803
$695$	−1.29673e95	−0.806043
$696$	−6.13423e94	−0.363371
$697$	−1.99239e94	−0.112480
$698$	1.67377e95	0.900614
$699$	−4.82872e94	−0.247655
$700$	−3.02254e95	−1.47770
$701$	2.16122e95	1.00727	0.503634	−	0.863917i	$-0.331996\pi$
$701$	2.16122e95	1.00727	0.503634	+	0.863917i	$0.331996\pi$
$702$	8.03675e94	0.357098
$703$	−3.95873e95	−1.67708
$704$	−3.41034e94	−0.137757
$705$	2.83736e95	1.09290
$706$	5.12179e94	0.188133
$707$	−2.92408e95	−1.02433
$708$	1.26307e95	0.422000
$709$	−2.44648e95	−0.779637	−0.389819	−	0.920892i	$-0.627462\pi$
$709$	−2.44648e95	−0.779637	−0.389819	+	0.920892i	$0.627462\pi$
$710$	5.39543e95	1.64010
$711$	−3.38708e94	−0.0982187
$712$	2.74739e95	0.760047
$713$	3.02251e94	0.0797752
$714$	−4.62555e95	−1.16486
$715$	9.69702e95	2.33016
$716$	−7.35691e94	−0.168698
$717$	−2.84258e95	−0.622043
$718$	−3.88354e95	−0.811072
$719$	9.95768e95	1.98491	0.992454	−	0.122619i	$-0.0391292\pi$
$719$	9.95768e95	1.98491	0.992454	+	0.122619i	$0.0391292\pi$
$720$	3.74910e95	0.713327
$721$	−1.58802e95	−0.288420
$722$	8.09427e95	1.40340
$723$	4.42565e95	0.732562
$724$	−2.49831e95	−0.394825
$725$	−1.72183e96	−2.59817
$726$	7.56756e94	0.109038
$727$	−1.33593e95	−0.183815	−0.0919077	−	0.995768i	$-0.529296\pi$
$727$	−1.33593e95	−0.183815	−0.0919077	+	0.995768i	$0.529296\pi$
$728$	−6.41181e95	−0.842518
$729$	2.95127e94	0.0370370
$730$	2.24031e96	2.68530
$731$	−1.20738e96	−1.38233
$732$	2.69000e95	0.294192
$733$	1.97695e95	0.206544	0.103272	−	0.994653i	$-0.467069\pi$
$733$	1.97695e95	0.206544	0.103272	+	0.994653i	$0.467069\pi$
$734$	−5.77018e95	−0.575933
$735$	4.44451e95	0.423836
$736$	−2.19263e95	−0.199783
$737$	5.34338e95	0.465216
$738$	4.31646e94	0.0359120
$739$	−1.35893e96	−1.08046	−0.540230	−	0.841518i	$-0.681663\pi$
$739$	−1.35893e96	−1.08046	−0.540230	+	0.841518i	$0.681663\pi$
$740$	−1.63282e96	−1.24073
$741$	−1.68085e96	−1.22073
$742$	8.08964e95	0.561563
$743$	5.58924e95	0.370875	0.185438	−	0.982656i	$-0.440630\pi$
$743$	5.58924e95	0.370875	0.185438	+	0.982656i	$0.440630\pi$
$744$	1.91541e95	0.121498
$745$	1.49714e96	0.907879
$746$	3.19995e96	1.85520
$747$	−4.73724e94	−0.0262594
$748$	−1.43304e96	−0.759545
$749$	6.18899e94	0.0313674
$750$	2.62264e96	1.27112
$751$	−1.55727e96	−0.721815	−0.360907	−	0.932602i	$-0.617533\pi$
$751$	−1.55727e96	−0.721815	−0.360907	+	0.932602i	$0.617533\pi$
$752$	−3.04698e96	−1.35074
$753$	−2.35991e96	−1.00061
$754$	5.94651e96	2.41169
$755$	−5.29287e96	−2.05338
$756$	3.83329e95	0.142263
$757$	1.94156e96	0.689349	0.344675	−	0.938722i	$-0.387989\pi$
$757$	1.94156e96	0.689349	0.344675	+	0.938722i	$0.387989\pi$
$758$	−4.64704e96	−1.57855
$759$	−3.00920e95	−0.0978030
$760$	−3.91021e96	−1.21603
$761$	−2.46377e96	−0.733187	−0.366593	−	0.930381i	$-0.619476\pi$
$761$	−2.46377e96	−0.733187	−0.366593	+	0.930381i	$0.619476\pi$
$762$	3.58783e95	0.102174
$763$	1.49168e96	0.406541
$764$	−1.33624e96	−0.348543
$765$	3.07206e96	0.766959
$766$	−2.84734e96	−0.680418
$767$	7.52084e96	1.72037
$768$	−3.40784e96	−0.746240
$769$	−4.39714e96	−0.921802	−0.460901	−	0.887452i	$-0.652474\pi$
$769$	−4.39714e96	−0.921802	−0.460901	+	0.887452i	$0.652474\pi$
$770$	1.20913e97	2.42680
$771$	4.16386e96	0.800156
$772$	−6.64873e94	−0.0122337
$773$	1.04403e96	0.183950	0.0919748	−	0.995761i	$-0.470682\pi$
$773$	1.04403e96	0.183950	0.0919748	+	0.995761i	$0.470682\pi$
$774$	2.61575e96	0.441341
$775$	5.37642e96	0.868733
$776$	2.96759e96	0.459237
$777$	5.37575e96	0.796773
$778$	−6.54210e96	−0.928753
$779$	−9.02766e95	−0.122764
$780$	−6.93282e96	−0.903111
$781$	−5.50325e96	−0.686768
$782$	−2.59607e96	−0.310377
$783$	2.18369e96	0.250133
$784$	−4.77286e96	−0.523830
$785$	1.51559e97	1.59385
$786$	−2.23638e96	−0.225367
$787$	−6.22401e96	−0.601058	−0.300529	−	0.953773i	$-0.597163\pi$
$787$	−6.22401e96	−0.601058	−0.300529	+	0.953773i	$0.597163\pi$
$788$	1.14315e97	1.05797
$789$	−3.80235e96	−0.337266
$790$	7.63837e96	0.649371
$791$	1.17043e97	0.953750
$792$	−1.90698e96	−0.148954
$793$	1.60174e97	1.19933
$794$	−1.57266e97	−1.12888
$795$	−5.37274e96	−0.369740
$796$	−2.27025e95	−0.0149791
$797$	1.32909e97	0.840816	0.420408	−	0.907335i	$-0.361887\pi$
$797$	1.32909e97	0.840816	0.420408	+	0.907335i	$0.361887\pi$
$798$	−2.09587e97	−1.27135
$799$	−2.49674e97	−1.45230
$800$	−3.90024e97	−2.17559
$801$	−9.78026e96	−0.523192
$802$	5.62159e96	0.288415
$803$	−2.28508e97	−1.12443
$804$	−3.82021e96	−0.180306
$805$	8.37892e96	0.379338
$806$	−1.85680e97	−0.806382
$807$	6.30615e96	0.262725
$808$	−1.04001e97	−0.415677
$809$	1.27401e96	0.0488538	0.0244269	−	0.999702i	$-0.492224\pi$
$809$	1.27401e96	0.0488538	0.0244269	+	0.999702i	$0.492224\pi$
$810$	−6.65554e96	−0.244870
$811$	3.31277e96	0.116948	0.0584739	−	0.998289i	$-0.481377\pi$
$811$	3.31277e96	0.116948	0.0584739	+	0.998289i	$0.481377\pi$
$812$	2.83630e97	0.960784
$813$	2.16856e97	0.704918
$814$	4.35389e97	1.35819
$815$	6.55843e97	1.96345
$816$	−3.29902e97	−0.947906
$817$	−5.47071e97	−1.50871
$818$	3.16045e96	0.0836590
$819$	2.28250e97	0.579962
$820$	−3.72355e96	−0.0908225
$821$	3.50715e97	0.821218	0.410609	−	0.911812i	$-0.365316\pi$
$821$	3.50715e97	0.821218	0.410609	+	0.911812i	$0.365316\pi$
$822$	5.16088e97	1.16016
$823$	3.83701e97	0.828130	0.414065	−	0.910247i	$-0.364109\pi$
$823$	3.83701e97	0.828130	0.414065	+	0.910247i	$0.364109\pi$
$824$	−5.64812e96	−0.117042
$825$	−5.35276e97	−1.06505
$826$	9.37783e97	1.79172
$827$	−1.64175e97	−0.301212	−0.150606	−	0.988594i	$-0.548122\pi$
$827$	−1.64175e97	−0.301212	−0.150606	+	0.988594i	$0.548122\pi$
$828$	2.15141e96	0.0379060
$829$	−6.42309e97	−1.08684	−0.543422	−	0.839460i	$-0.682871\pi$
$829$	−6.42309e97	−1.08684	−0.543422	+	0.839460i	$0.682871\pi$
$830$	1.06832e97	0.173614
$831$	7.09140e97	1.10687
$832$	1.45181e97	0.217660
$833$	−3.91095e97	−0.563215
$834$	−2.47207e97	−0.341977
$835$	−1.38288e98	−1.83775
$836$	−6.49317e97	−0.828985
$837$	−6.81856e96	−0.0836353
$838$	1.33247e98	1.57030
$839$	−7.70728e97	−0.872721	−0.436360	−	0.899772i	$-0.643733\pi$
$839$	−7.70728e97	−0.872721	−0.436360	+	0.899772i	$0.643733\pi$
$840$	5.30986e97	0.577733
$841$	6.59283e97	0.689297
$842$	7.73252e97	0.776901
$843$	2.90707e97	0.280693
$844$	−6.50966e97	−0.604069
$845$	−2.18637e98	−1.94995
$846$	5.40911e97	0.463680
$847$	2.14924e97	0.177089
$848$	5.76967e97	0.456972
$849$	−1.23378e98	−0.939353
$850$	−4.61788e98	−3.37993
$851$	3.01711e97	0.212300
$852$	3.93452e97	0.266174
$853$	2.10647e98	1.37013	0.685067	−	0.728480i	$-0.259773\pi$
$853$	2.10647e98	1.37013	0.685067	+	0.728480i	$0.259773\pi$
$854$	1.99722e98	1.24907
$855$	1.39197e98	0.837077
$856$	2.20124e96	0.0127291
$857$	−8.37787e97	−0.465882	−0.232941	−	0.972491i	$-0.574835\pi$
$857$	−8.37787e97	−0.465882	−0.232941	+	0.972491i	$0.574835\pi$
$858$	1.84863e98	0.988610
$859$	−8.68084e97	−0.446468	−0.223234	−	0.974765i	$-0.571661\pi$
$859$	−8.68084e97	−0.446468	−0.223234	+	0.974765i	$0.571661\pi$
$860$	−2.25645e98	−1.11616
$861$	1.22591e97	0.0583247
$862$	3.29293e98	1.50691
$863$	−1.17447e98	−0.516990	−0.258495	−	0.966013i	$-0.583227\pi$
$863$	−1.17447e98	−0.516990	−0.258495	+	0.966013i	$0.583227\pi$
$864$	4.94641e97	0.209450
$865$	4.19577e98	1.70912
$866$	−5.40921e98	−2.11975
$867$	−1.17190e98	−0.441825
$868$	−8.85636e97	−0.321251
$869$	−7.79102e97	−0.271914
$870$	−4.92453e98	−1.65375
$871$	−2.27472e98	−0.735054
$872$	5.30548e97	0.164977
$873$	−1.05641e98	−0.316124
$874$	−1.17629e98	−0.338753
$875$	7.44850e98	2.06442
$876$	1.63370e98	0.435799
$877$	4.45262e98	1.14322	0.571610	−	0.820526i	$-0.306319\pi$
$877$	4.45262e98	1.14322	0.571610	+	0.820526i	$0.306319\pi$
$878$	−4.22216e98	−1.04344
$879$	−1.52680e98	−0.363206
$880$	8.62374e98	1.97481
$881$	3.62267e98	0.798613	0.399307	−	0.916817i	$-0.369251\pi$
$881$	3.62267e98	0.798613	0.399307	+	0.916817i	$0.369251\pi$
$882$	8.47295e97	0.179820
$883$	−9.21096e98	−1.88201	−0.941004	−	0.338396i	$-0.890116\pi$
$883$	−9.21096e98	−1.88201	−0.941004	+	0.338396i	$0.890116\pi$
$884$	6.10054e98	1.20010
$885$	−6.22830e98	−1.17969
$886$	7.20599e98	1.31421
$887$	−3.14448e97	−0.0552214	−0.0276107	−	0.999619i	$-0.508790\pi$
$887$	−3.14448e97	−0.0552214	−0.0276107	+	0.999619i	$0.508790\pi$
$888$	1.91199e98	0.323335
$889$	1.01897e98	0.165940
$890$	2.20559e99	3.45908
$891$	6.78855e97	0.102535
$892$	−7.81455e98	−1.13679
$893$	−1.13129e99	−1.58507
$894$	2.85413e98	0.385183
$895$	3.62775e98	0.471591
$896$	−8.56072e98	−1.07199
$897$	1.28104e98	0.154531
$898$	−1.77905e99	−2.06743
$899$	−5.04515e98	−0.564839
$900$	3.82692e98	0.412787
$901$	4.72774e98	0.491330
$902$	9.92879e97	0.0994208
$903$	7.42893e98	0.716781
$904$	4.16289e98	0.387037
$905$	1.23194e99	1.10373
$906$	−1.00903e99	−0.871180
$907$	−1.91163e99	−1.59059	−0.795297	−	0.606221i	$-0.792685\pi$
$907$	−1.91163e99	−1.59059	−0.795297	+	0.606221i	$0.792685\pi$
$908$	1.10183e99	0.883561
$909$	3.70225e98	0.286138
$910$	−5.14737e99	−3.83441
$911$	1.45855e98	0.104726	0.0523631	−	0.998628i	$-0.483325\pi$
$911$	1.45855e98	0.104726	0.0523631	+	0.998628i	$0.483325\pi$
$912$	−1.49481e99	−1.03457
$913$	−1.08967e98	−0.0726980
$914$	−1.57711e98	−0.101430
$915$	−1.32646e99	−0.822408
$916$	1.94936e99	1.16518
$917$	−6.35149e98	−0.366018
$918$	5.85654e98	0.325395
$919$	−4.29631e98	−0.230158	−0.115079	−	0.993356i	$-0.536712\pi$
$919$	−4.29631e98	−0.230158	−0.115079	+	0.993356i	$0.536712\pi$
$920$	2.98013e98	0.153937
$921$	−2.90054e98	−0.144471
$922$	1.54518e99	0.742153
$923$	2.34278e99	1.08511
$924$	8.81738e98	0.393848
$925$	5.36682e99	2.31190
$926$	−3.18367e99	−1.32270
$927$	2.01064e98	0.0805681
$928$	3.65992e99	1.41454
$929$	−1.14094e99	−0.425339	−0.212669	−	0.977124i	$-0.568216\pi$
$929$	−1.14094e99	−0.425339	−0.212669	+	0.977124i	$0.568216\pi$
$930$	1.53768e99	0.552953
$931$	−1.77207e99	−0.614706
$932$	7.94095e98	0.265729
$933$	−1.11058e99	−0.358520
$934$	−4.19520e99	−1.30657
$935$	7.06641e99	2.12329
$936$	8.11817e98	0.235352
$937$	5.31805e99	1.48757	0.743783	−	0.668421i	$-0.233030\pi$
$937$	5.31805e99	1.48757	0.743783	+	0.668421i	$0.233030\pi$
$938$	−2.83637e99	−0.765540
$939$	−2.63589e98	−0.0686483
$940$	−4.66611e99	−1.17266
$941$	−4.23002e99	−1.02586	−0.512932	−	0.858429i	$-0.671441\pi$
$941$	−4.23002e99	−1.02586	−0.512932	+	0.858429i	$0.671441\pi$
$942$	2.88929e99	0.676218
$943$	6.88035e97	0.0155406
$944$	6.68843e99	1.45802
$945$	−1.89022e99	−0.397693
$946$	6.01679e99	1.22183
$947$	−3.02686e99	−0.593291	−0.296645	−	0.954988i	$-0.595868\pi$
$947$	−3.02686e99	−0.593291	−0.296645	+	0.954988i	$0.595868\pi$
$948$	5.57014e98	0.105387
$949$	9.72776e99	1.77662
$950$	−2.09239e100	−3.68894
$951$	−9.70491e98	−0.165175
$952$	−4.67241e99	−0.767721
$953$	−1.12836e100	−1.78993	−0.894967	−	0.446132i	$-0.852801\pi$
$953$	−1.12836e100	−1.78993	−0.894967	+	0.446132i	$0.852801\pi$
$954$	−1.02425e99	−0.156869
$955$	6.58910e99	0.974347
$956$	4.67469e99	0.667442
$957$	5.02295e99	0.692482
$958$	−8.13180e98	−0.108254
$959$	1.46573e100	1.88422
$960$	−1.20230e99	−0.149254
$961$	−6.76595e99	−0.811139
$962$	−1.85348e100	−2.14597
$963$	−7.83606e97	−0.00876228
$964$	−7.27810e99	−0.786027
$965$	3.27854e98	0.0341992
$966$	1.59735e99	0.160940
$967$	−1.26244e100	−1.22864	−0.614318	−	0.789059i	$-0.710569\pi$
$967$	−1.26244e100	−1.22864	−0.614318	+	0.789059i	$0.710569\pi$
$968$	7.64422e98	0.0718636
$969$	−1.22487e100	−1.11235
$970$	2.38237e100	2.09005
$971$	−2.90768e99	−0.246435	−0.123217	−	0.992380i	$-0.539321\pi$
$971$	−2.90768e99	−0.246435	−0.123217	+	0.992380i	$0.539321\pi$
$972$	−4.85343e98	−0.0397401
$973$	−7.02086e99	−0.555405
$974$	1.58585e100	1.21210
$975$	2.27871e100	1.68281
$976$	1.42446e100	1.01644
$977$	1.09692e100	0.756324	0.378162	−	0.925739i	$-0.376556\pi$
$977$	1.09692e100	0.756324	0.378162	+	0.925739i	$0.376556\pi$
$978$	1.25029e100	0.833027
$979$	−2.24967e100	−1.44843
$980$	−7.30911e99	−0.454769
$981$	−1.88866e99	−0.113565
$982$	−7.19043e99	−0.417850
$983$	−8.14594e99	−0.457508	−0.228754	−	0.973484i	$-0.573465\pi$
$983$	−8.14594e99	−0.457508	−0.228754	+	0.973484i	$0.573465\pi$
$984$	4.36019e98	0.0236685
$985$	−5.63695e100	−2.95754
$986$	4.33334e100	2.19759
$987$	1.53623e100	0.753062
$988$	2.76419e100	1.30982
$989$	4.16945e99	0.190987
$990$	−1.53092e100	−0.677911
$991$	−1.51919e100	−0.650347	−0.325174	−	0.945654i	$-0.605423\pi$
$991$	−1.51919e100	−0.650347	−0.325174	+	0.945654i	$0.605423\pi$
$992$	−1.14281e100	−0.472969
$993$	−2.77954e100	−1.11217
$994$	2.92124e100	1.13012
$995$	1.11948e99	0.0418738
$996$	7.79051e98	0.0281759
$997$	−2.34865e100	−0.821354	−0.410677	−	0.911781i	$-0.634708\pi$
$997$	−2.34865e100	−0.821354	−0.410677	+	0.911781i	$0.634708\pi$
$998$	−3.17213e100	−1.07270
$999$	−6.80639e99	−0.222573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3.68.a.b.1.5	✓	6
3.2	odd	2		9.68.a.c.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.68.a.b.1.5	✓	6	1.1	even	1	trivial
9.68.a.c.1.2		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)

\(f(q)\)	\(=\)	\(q+1.54594e10 q^{2} -5.55906e15 q^{3} +9.14202e19 q^{4} -4.50800e23 q^{5} -8.59399e25 q^{6} -2.44076e28 q^{7} -8.68106e29 q^{8} +3.09032e31 q^{9} +O(q^{10})\)	\(q+1.54594e10 q^{2} -5.55906e15 q^{3} +9.14202e19 q^{4} -4.50800e23 q^{5} -8.59399e25 q^{6} -2.44076e28 q^{7} -8.68106e29 q^{8} +3.09032e31 q^{9} -6.96912e33 q^{10} +7.10839e34 q^{11} -5.08210e35 q^{12} -3.02610e37 q^{13} -3.77328e38 q^{14} +2.50603e39 q^{15} -2.69117e40 q^{16} -2.20518e41 q^{17} +4.77745e41 q^{18} -9.99180e42 q^{19} -4.12122e43 q^{20} +1.35683e44 q^{21} +1.09892e45 q^{22} +7.61516e44 q^{23} +4.82585e45 q^{24} +1.35458e47 q^{25} -4.67817e47 q^{26} -1.71793e47 q^{27} -2.23135e48 q^{28} -1.27112e49 q^{29} +3.87417e49 q^{30} +3.96906e49 q^{31} -2.87929e50 q^{32} -3.95160e50 q^{33} -3.40908e51 q^{34} +1.10029e52 q^{35} +2.82517e51 q^{36} +3.96198e52 q^{37} -1.54468e53 q^{38} +1.68223e53 q^{39} +3.91342e53 q^{40} +9.03507e52 q^{41} +2.09759e54 q^{42} +5.47520e54 q^{43} +6.49850e54 q^{44} -1.39311e55 q^{45} +1.17726e55 q^{46} +1.13222e56 q^{47} +1.49604e56 q^{48} +1.77353e56 q^{49} +2.09411e57 q^{50} +1.22587e57 q^{51} -2.76646e57 q^{52} -2.14393e57 q^{53} -2.65581e57 q^{54} -3.20446e58 q^{55} +2.11884e58 q^{56} +5.55450e58 q^{57} -1.96508e59 q^{58} -2.48533e59 q^{59} +2.29101e59 q^{60} -5.29308e59 q^{61} +6.13595e59 q^{62} -7.54272e59 q^{63} -4.79763e59 q^{64} +1.36416e61 q^{65} -6.10895e60 q^{66} +7.51700e60 q^{67} -2.01598e61 q^{68} -4.23331e60 q^{69} +1.70099e62 q^{70} -7.74191e61 q^{71} -2.68272e61 q^{72} -3.21462e62 q^{73} +6.12500e62 q^{74} -7.53020e62 q^{75} -9.13452e62 q^{76} -1.73499e63 q^{77} +2.60063e63 q^{78} -1.09603e63 q^{79} +1.21318e64 q^{80} +9.55005e62 q^{81} +1.39677e63 q^{82} -1.53293e63 q^{83} +1.24042e64 q^{84} +9.94094e64 q^{85} +8.46435e64 q^{86} +7.06622e64 q^{87} -6.17084e64 q^{88} -3.16481e65 q^{89} -2.15368e65 q^{90} +7.38597e65 q^{91} +6.96179e64 q^{92} -2.20643e65 q^{93} +1.75034e66 q^{94} +4.50430e66 q^{95} +1.60062e66 q^{96} -3.41847e66 q^{97} +2.74178e66 q^{98} +2.19672e66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 13735355166 q^{2} - 33\!\cdots\!38 q^{3}+ \cdots + 18\!\cdots\!74 q^{9}+O(q^{10}) \) 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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