LMFDB - Embedded newform 2.68.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2,68,Mod(1,2)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2.1"); S:= CuspForms(chi, 68); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 68, names="a")

Level:	$ N $	$=$	$ 2 $
Weight:	$ k $	$=$	$ 68 $
Character orbit:	$[\chi]$	$=$	2.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,25769803776] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.8580703860$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 195874426031875504236526x - 11619230000023993068059089554543524 $ x^3 - x^2 - 195874426031875504236526*x - 11619230000023993068059089554543524
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{7}\cdot 5^{2}\cdot 7\cdot 11 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.09246e11$ of defining polynomial
Character	$\chi$	$=$	2.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.58993e9 q^{2} +1.33494e16 q^{3} +7.37870e19 q^{4} -2.84057e23 q^{5} +1.14671e26 q^{6} -2.26934e28 q^{7} +6.33825e29 q^{8} +8.54976e31 q^{9} -2.44003e33 q^{10} -3.52937e34 q^{11} +9.85014e35 q^{12} +3.76926e37 q^{13} -1.94935e38 q^{14} -3.79199e39 q^{15} +5.44452e39 q^{16} -1.54564e41 q^{17} +7.34419e41 q^{18} -7.97842e42 q^{19} -2.09597e43 q^{20} -3.02944e44 q^{21} -3.03171e44 q^{22} -3.75304e45 q^{23} +8.46120e45 q^{24} +1.29255e46 q^{25} +3.23777e47 q^{26} -9.62738e46 q^{27} -1.67448e48 q^{28} -1.35341e49 q^{29} -3.25730e49 q^{30} -1.26339e49 q^{31} +4.67681e49 q^{32} -4.71150e50 q^{33} -1.32769e51 q^{34} +6.44620e51 q^{35} +6.30861e51 q^{36} -5.35615e52 q^{37} -6.85341e52 q^{38} +5.03174e53 q^{39} -1.80042e53 q^{40} +1.77137e54 q^{41} -2.60227e54 q^{42} -3.22312e54 q^{43} -2.60422e54 q^{44} -2.42862e55 q^{45} -3.22383e55 q^{46} +1.27712e56 q^{47} +7.26812e55 q^{48} +9.66117e55 q^{49} +1.11029e56 q^{50} -2.06333e57 q^{51} +2.78122e57 q^{52} -8.62187e57 q^{53} -8.26986e56 q^{54} +1.00254e58 q^{55} -1.43836e58 q^{56} -1.06507e59 q^{57} -1.16257e59 q^{58} -3.09921e58 q^{59} -2.79800e59 q^{60} -1.04651e59 q^{61} -1.08525e59 q^{62} -1.94023e60 q^{63} +4.01735e59 q^{64} -1.07068e61 q^{65} -4.04715e60 q^{66} -1.57101e61 q^{67} -1.14048e61 q^{68} -5.01009e61 q^{69} +5.53725e61 q^{70} +1.41255e62 q^{71} +5.41906e61 q^{72} +2.77405e62 q^{73} -4.60090e62 q^{74} +1.72548e62 q^{75} -5.88703e62 q^{76} +8.00933e62 q^{77} +4.32223e63 q^{78} +5.79087e63 q^{79} -1.54655e63 q^{80} -9.21164e63 q^{81} +1.52160e64 q^{82} +6.91367e62 q^{83} -2.23533e64 q^{84} +4.39048e64 q^{85} -2.76864e64 q^{86} -1.80672e65 q^{87} -2.23700e64 q^{88} +1.78683e65 q^{89} -2.08617e65 q^{90} -8.55372e65 q^{91} -2.76925e65 q^{92} -1.68656e65 q^{93} +1.09703e66 q^{94} +2.26632e66 q^{95} +6.24327e65 q^{96} +1.29164e66 q^{97} +8.29888e65 q^{98} -3.01753e66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 25769803776 q^{2} - 23\!\cdots\!44 q^{3} + 22\!\cdots\!92 q^{4} + 69\!\cdots\!90 q^{5} - 20\!\cdots\!48 q^{6} - 46\!\cdots\!92 q^{7} + 19\!\cdots\!64 q^{8} + 19\!\cdots\!51 q^{9} + 60\!\cdots\!80 q^{10}+ \cdots + 28\!\cdots\!32 q^{99}+O(q^{100}) $ 3 * q + 25769803776 * q^2 - 2382403442320044 * q^3 + 221360928884514619392 * q^4 + 69995731683175484205690 * q^5 - 20464689741284822690562048 * q^6 - 46930041031821809354401779192 * q^7 + 1901475900342344102245054808064 * q^8 + 191665880989256950329302542685151 * q^9 + 601258756877659476184806174228480 * q^10 - 94192717562712302465603901160680804 * q^11 - 175790346323210028954245448037564416 * q^12 + 9293162032083115751156028464346376386 * q^13 - 403125982863225532953405030547706609664 * q^14 - 2834416443023883562385768010875712279720 * q^15 + 16333553612205046246241981156724874149888 * q^16 - 456291990499287276743121870841967331662442 * q^17 + 1646397381215773458010101702644735139643392 * q^18 + 7094914417641504988662519373648786153621860 * q^19 + 5164773394446325046476466740820399263580160 * q^20 + 90866348206886606668322229671316231529787616 * q^21 - 809109282906428336293257840750280988549971968 * q^22 - 1211030869288683518792500686990840249736656744 * q^23 - 1510027576821401840195394559356413092426678272 * q^24 + 23527039678683417635278279831311248936865016725 * q^25 + 79827654008451769809995232895225577187953344512 * q^26 - 2113356067189354866826552899200490188176129267320 * q^27 - 3462825825130820210214089796088561712619835097088 * q^28 - 2991938816995158477528675338206092994290982287150 * q^29 - 24347451852064454494717698685108313124206008074240 * q^30 - 258768758082699807798920757172611699585086030318304 * q^31 + 140304157183766680147553743940763169886969524125696 * q^32 - 2181728345127367781475611482453366631578858604794608 * q^33 - 3919518353242363129841219656417171347581147402993664 * q^34 + 6685861313241550931298636663340184400338218304813040 * q^35 + 14142445817083583442996432100986108262700723485016064 * q^36 + 6329661235650435153849271583702324985112571136357018 * q^37 + 60944850783378298757052742981575881439807041301381120 * q^38 + 326230234453650802234252471202256216010008198357250872 * q^39 + 44365065640795748204004209370910646093478542148894720 * q^40 + 670194901684473649850829882936279861991784836038573486 * q^41 + 780535987711052435241718991256208087388803721091612672 * q^42 - 6590490969061452320864717359976138421991174495600681284 * q^43 - 6950195817946243065094464582636065897265360128553517056 * q^44 - 64368287466104027746747885379428542717642851620670372670 * q^45 - 10402675956082692992215983721366383048358826655715688448 * q^46 + 142496759244986398051581936313458913857843662485546058928 * q^47 - 12971038117012097073026875764504250079677616912307585024 * q^48 - 200510880128140115686545081532783362338669745164185519941 * q^49 + 202095731983279254162059735469236421961500031200736051200 * q^50 + 1384853383099710830200705854190773108901917972620490983016 * q^51 + 685714326565407317854499318401794497129966519705722159104 * q^52 - 7859044654180946725295267389956714422201615664275787862614 * q^53 - 18153590386762915584716960012960179810770722182015683133440 * q^54 - 48874001022465194932335655798148256348206330106827800478520 * q^55 - 29745447341362175449050721665215362550759884445811889668096 * q^56 - 312938298350217271758997292082690465354471302368807715396880 * q^57 - 25700558741254269302645622959593817436828967262049062092800 * q^58 - 55301513334625281723261507583433559952613870091854753243700 * q^59 - 209143018891102924277785441209844814172384781291145480110080 * q^60 - 1578188533688954960536201064862481464726063214655898847033614 * q^61 - 2222806706383462675763700792303849353249842539127160300371968 * q^62 - 6422990905330033092057765004674396147069390082610736922789464 * q^63 + 1205203533194242706656471569255871951891652245337094626476032 * q^64 - 24660802950023978954037674203067272398035309335453482039372420 * q^65 - 18740903782156491152003651877478575055457757105202060314279936 * q^66 - 26407626812894473557175218263553791675539117425031533908749932 * q^67 - 33668406286495450408848480194128214941378553604026238750425088 * q^68 - 76707923717026372170850229812154221436712384864328794865051488 * q^69 + 57431111371928145996491974556865308244124037916161132388679680 * q^70 + 133722587683066442512737537292042177005944363562526060398772296 * q^71 + 121482684539651977972313512236839808677229968007366281829285888 * q^72 + 599594953405705348016644397964150250082961686563655227588295886 * q^73 + 54371376003755136547902699930847224820444359818253897780166656 * q^74 + 1414205704990897986282727708114357439350697511376733150750330700 * q^75 + 523512281948419547115317960905923882652689269879906380063703040 * q^76 - 1116699559292699246432108790111828590176973601792672760678491744 * q^77 + 2802296375889685246800556189641744498351393641272546839414964224 * q^78 + 15040616041273693784225331917559164800461831418668702451543715280 * q^79 + 381093012024222043904097630988795917419440994814520769894154240 * q^80 + 10930595424114990446638976321107279614269803708106312987766962523 * q^81 + 5756930369361499274166139251341660918316218582908790633725427712 * q^82 + 12547243148793841808170951364938913025058218490201111704595731876 * q^83 + 6704753081140056214208681844535047384611244037303163692278349824 * q^84 + 14457146614409169395963615528990729489113749389068467459410461140 * q^85 - 56611886353404571064754519002761947725642283318468443980198576128 * q^86 - 454772454385715019021323102969817006950999802246232262453483445000 * q^87 - 59701727477750167701295049066104384997711235171599423082596401152 * q^88 - 397456754755248408754478323982148795788241603945122066031068971170 * q^89 - 552919379132887015412317476123604284282430019837919696427564400640 * q^90 - 995804409425932416113481886137438482063280754796390120190765025104 * q^91 - 89358306044521397346352065303473983253019893918463866716569993216 * q^92 + 1173987115649565041691486525937593995182689817613536077256661380992 * q^93 + 1224037841486404463192765075061321679318239446912564791594897637376 * q^94 + 2975480524015582606640572511450516916081877850724245393610141977400 * q^95 - 111420369015472756330055510275201503490441517723155155141638750208 * q^96 - 2195128909500657761105797202787371985980393874830510881776680235162 * q^97 - 1722375345285076172358735424825916186195009263249661917246701699072 * q^98 + 28054853016203263572503445694368856176570112697698383290929190617932 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.58993e9	0.707107
$2$	8.58993e9	0.707107
$3$	1.33494e16	1.38644	0.693219	−	0.720727i	$-0.256192\pi$
$3$	1.33494e16	1.38644	0.693219	+	0.720727i	$0.256192\pi$
$4$	7.37870e19	0.500000
$5$	−2.84057e23	−1.09121	−0.545607	−	0.838041i	$-0.683701\pi$
$5$	−2.84057e23	−1.09121	−0.545607	+	0.838041i	$0.683701\pi$
$6$	1.14671e26	0.980360
$7$	−2.26934e28	−1.10947	−0.554734	−	0.832028i	$-0.687180\pi$
$7$	−2.26934e28	−1.10947	−0.554734	+	0.832028i	$0.687180\pi$
$8$	6.33825e29	0.353553
$9$	8.54976e31	0.922210
$10$	−2.44003e33	−0.771605
$11$	−3.52937e34	−0.458186	−0.229093	−	0.973404i	$-0.573576\pi$
$11$	−3.52937e34	−0.458186	−0.229093	+	0.973404i	$0.573576\pi$
$12$	9.85014e35	0.693219
$13$	3.76926e37	1.81616	0.908079	−	0.418799i	$-0.137549\pi$
$13$	3.76926e37	1.81616	0.908079	+	0.418799i	$0.137549\pi$
$14$	−1.94935e38	−0.784512
$15$	−3.79199e39	−1.51290
$16$	5.44452e39	0.250000
$17$	−1.54564e41	−0.931254	−0.465627	−	0.884981i	$-0.654171\pi$
$17$	−1.54564e41	−0.931254	−0.465627	+	0.884981i	$0.654171\pi$
$18$	7.34419e41	0.652101
$19$	−7.97842e42	−1.15790	−0.578950	−	0.815363i	$-0.696537\pi$
$19$	−7.97842e42	−1.15790	−0.578950	+	0.815363i	$0.696537\pi$
$20$	−2.09597e43	−0.545607
$21$	−3.02944e44	−1.53821
$22$	−3.03171e44	−0.323987
$23$	−3.75304e45	−0.904691	−0.452346	−	0.891843i	$-0.649413\pi$
$23$	−3.75304e45	−0.904691	−0.452346	+	0.891843i	$0.649413\pi$
$24$	8.46120e45	0.490180
$25$	1.29255e46	0.190747
$26$	3.23777e47	1.28422
$27$	−9.62738e46	−0.107850
$28$	−1.67448e48	−0.554734
$29$	−1.35341e49	−1.38387	−0.691937	−	0.721958i	$-0.743242\pi$
$29$	−1.35341e49	−1.38387	−0.691937	+	0.721958i	$0.743242\pi$
$30$	−3.25730e49	−1.06978
$31$	−1.26339e49	−0.138332	−0.0691658	−	0.997605i	$-0.522034\pi$
$31$	−1.26339e49	−0.138332	−0.0691658	+	0.997605i	$0.522034\pi$
$32$	4.67681e49	0.176777
$33$	−4.71150e50	−0.635247
$34$	−1.32769e51	−0.658496
$35$	6.44620e51	1.21067
$36$	6.30861e51	0.461105
$37$	−5.35615e52	−1.56349	−0.781745	−	0.623599i	$-0.785670\pi$
$37$	−5.35615e52	−1.56349	−0.781745	+	0.623599i	$0.785670\pi$
$38$	−6.85341e52	−0.818759
$39$	5.03174e53	2.51799
$40$	−1.80042e53	−0.385802
$41$	1.77137e54	1.65978	0.829890	−	0.557926i	$-0.188403\pi$
$41$	1.77137e54	1.65978	0.829890	+	0.557926i	$0.188403\pi$
$42$	−2.60227e54	−1.08768
$43$	−3.22312e54	−0.612469	−0.306234	−	0.951956i	$-0.599069\pi$
$43$	−3.22312e54	−0.612469	−0.306234	+	0.951956i	$0.599069\pi$
$44$	−2.60422e54	−0.229093
$45$	−2.42862e55	−1.00633
$46$	−3.22383e55	−0.639713
$47$	1.27712e56	1.23297	0.616484	−	0.787367i	$-0.288556\pi$
$47$	1.27712e56	1.23297	0.616484	+	0.787367i	$0.288556\pi$
$48$	7.26812e55	0.346610
$49$	9.66117e55	0.230920
$50$	1.11029e56	0.134879
$51$	−2.06333e57	−1.29113
$52$	2.78122e57	0.908079
$53$	−8.62187e57	−1.48717	−0.743583	−	0.668643i	$-0.766875\pi$
$53$	−8.62187e57	−1.48717	−0.743583	+	0.668643i	$0.766875\pi$
$54$	−8.26986e56	−0.0762618
$55$	1.00254e58	0.499979
$56$	−1.43836e58	−0.392256
$57$	−1.06507e59	−1.60536
$58$	−1.16257e59	−0.978546
$59$	−3.09921e58	−0.147133	−0.0735663	−	0.997290i	$-0.523438\pi$
$59$	−3.09921e58	−0.147133	−0.0735663	+	0.997290i	$0.523438\pi$
$60$	−2.79800e59	−0.756450
$61$	−1.04651e59	−0.162628	−0.0813138	−	0.996689i	$-0.525912\pi$
$61$	−1.04651e59	−0.162628	−0.0813138	+	0.996689i	$0.525912\pi$
$62$	−1.08525e59	−0.0978152
$63$	−1.94023e60	−1.02316
$64$	4.01735e59	0.125000
$65$	−1.07068e61	−1.98182
$66$	−4.04715e60	−0.449188
$67$	−1.57101e61	−1.05359	−0.526797	−	0.849991i	$-0.676607\pi$
$67$	−1.57101e61	−1.05359	−0.526797	+	0.849991i	$0.676607\pi$
$68$	−1.14048e61	−0.465627
$69$	−5.01009e61	−1.25430
$70$	5.53725e61	0.856071
$71$	1.41255e62	1.35785	0.678923	−	0.734210i	$-0.262447\pi$
$71$	1.41255e62	1.35785	0.678923	+	0.734210i	$0.262447\pi$
$72$	5.41906e61	0.326051
$73$	2.77405e62	1.05147	0.525737	−	0.850647i	$-0.323790\pi$
$73$	2.77405e62	1.05147	0.525737	+	0.850647i	$0.323790\pi$
$74$	−4.60090e62	−1.10555
$75$	1.72548e62	0.264459
$76$	−5.88703e62	−0.578950
$77$	8.00933e62	0.508343
$78$	4.32223e63	1.78049
$79$	5.79087e63	1.55681	0.778406	−	0.627761i	$-0.216029\pi$
$79$	5.79087e63	1.55681	0.778406	+	0.627761i	$0.216029\pi$
$80$	−1.54655e63	−0.272803
$81$	−9.21164e63	−1.07174
$82$	1.52160e64	1.17364
$83$	6.91367e62	0.0355298	0.0177649	−	0.999842i	$-0.494345\pi$
$83$	6.91367e62	0.0355298	0.0177649	+	0.999842i	$0.494345\pi$
$84$	−2.23533e64	−0.769104
$85$	4.39048e64	1.01620
$86$	−2.76864e64	−0.433081
$87$	−1.80672e65	−1.91866
$88$	−2.23700e64	−0.161993
$89$	1.78683e65	0.886172	0.443086	−	0.896479i	$-0.353884\pi$
$89$	1.78683e65	0.886172	0.443086	+	0.896479i	$0.353884\pi$
$90$	−2.08617e65	−0.711582
$91$	−8.55372e65	−2.01497
$92$	−2.76925e65	−0.452346
$93$	−1.68656e65	−0.191788
$94$	1.09703e66	0.871841
$95$	2.26632e66	1.26352
$96$	6.24327e65	0.245090
$97$	1.29164e66	0.358334	0.179167	−	0.983819i	$-0.442660\pi$
$97$	1.29164e66	0.358334	0.179167	+	0.983819i	$0.442660\pi$
$98$	8.29888e65	0.163285
$99$	−3.01753e66	−0.422544
$100$	9.53735e65	0.0953735
$101$	1.38640e67	0.993397	0.496698	−	0.867923i	$-0.334545\pi$
$101$	1.38640e67	0.993397	0.496698	+	0.867923i	$0.334545\pi$
$102$	−1.77239e67	−0.912964
$103$	1.17940e67	0.438140	0.219070	−	0.975709i	$-0.429698\pi$
$103$	1.17940e67	0.438140	0.219070	+	0.975709i	$0.429698\pi$
$104$	2.38905e67	0.642109
$105$	8.60531e67	1.67851
$106$	−7.40613e67	−1.05159
$107$	1.61389e67	0.167308	0.0836539	−	0.996495i	$-0.473341\pi$
$107$	1.61389e67	0.167308	0.0836539	+	0.996495i	$0.473341\pi$
$108$	−7.10375e66	−0.0539252
$109$	−2.53505e68	−1.41318	−0.706592	−	0.707621i	$-0.749768\pi$
$109$	−2.53505e68	−1.41318	−0.706592	+	0.707621i	$0.749768\pi$
$110$	8.61176e67	0.353539
$111$	−7.15015e68	−2.16768
$112$	−1.23555e68	−0.277367
$113$	3.63995e68	0.606691	0.303346	−	0.952881i	$-0.401896\pi$
$113$	3.63995e68	0.606691	0.303346	+	0.952881i	$0.401896\pi$
$114$	−9.14890e68	−1.13516
$115$	1.06608e69	0.987211
$116$	−9.98640e68	−0.691937
$117$	3.22263e69	1.67488
$118$	−2.66220e68	−0.104038
$119$	3.50757e69	1.03320
$120$	−2.40346e69	−0.534891
$121$	−4.68784e69	−0.790065
$122$	−8.98946e68	−0.114995
$123$	2.36468e70	2.30118
$124$	−9.32219e68	−0.0691658
$125$	1.55768e70	0.883068
$126$	−1.66665e70	−0.723486
$127$	−1.01462e70	−0.337970	−0.168985	−	0.985619i	$-0.554049\pi$
$127$	−1.01462e70	−0.337970	−0.168985	+	0.985619i	$0.554049\pi$
$128$	3.45087e69	0.0883883
$129$	−4.30269e70	−0.849150
$130$	−9.19709e70	−1.40136
$131$	3.94733e70	0.465280	0.232640	−	0.972563i	$-0.425264\pi$
$131$	3.94733e70	0.465280	0.232640	+	0.972563i	$0.425264\pi$
$132$	−3.47648e70	−0.317624
$133$	1.81057e71	1.28465
$134$	−1.34949e71	−0.745004
$135$	2.73472e70	0.117688
$136$	−9.79663e70	−0.329248
$137$	−2.01679e71	−0.530300	−0.265150	−	0.964207i	$-0.585422\pi$
$137$	−2.01679e71	−0.530300	−0.265150	+	0.964207i	$0.585422\pi$
$138$	−4.30363e71	−0.886923
$139$	−4.49517e71	−0.727362	−0.363681	−	0.931524i	$-0.618480\pi$
$139$	−4.49517e71	−0.727362	−0.363681	+	0.931524i	$0.618480\pi$
$140$	4.75646e71	0.605333
$141$	1.70488e72	1.70943
$142$	1.21337e72	0.960142
$143$	−1.33031e72	−0.832139
$144$	4.65493e71	0.230553
$145$	3.84445e72	1.51010
$146$	2.38289e72	0.743505
$147$	1.28971e72	0.320156
$148$	−3.95214e72	−0.781745
$149$	−3.09159e72	−0.488024	−0.244012	−	0.969772i	$-0.578464\pi$
$149$	−3.09159e72	−0.488024	−0.244012	+	0.969772i	$0.578464\pi$
$150$	1.48218e72	0.187001
$151$	−1.84888e73	−1.86716	−0.933578	−	0.358373i	$-0.883332\pi$
$151$	−1.84888e73	−1.86716	−0.933578	+	0.358373i	$0.883332\pi$
$152$	−5.05692e72	−0.409379
$153$	−1.32148e73	−0.858812
$154$	6.87996e72	0.359453
$155$	3.58875e72	0.150949
$156$	3.71277e73	1.25899
$157$	−2.51962e73	−0.689759	−0.344879	−	0.938647i	$-0.612080\pi$
$157$	−2.51962e73	−0.689759	−0.344879	+	0.938647i	$0.612080\pi$
$158$	4.97432e73	1.10083
$159$	−1.15097e74	−2.06186
$160$	−1.32848e73	−0.192901
$161$	8.51691e73	1.00373
$162$	−7.91274e73	−0.757833
$163$	9.00189e73	0.701535	0.350767	−	0.936463i	$-0.385921\pi$
$163$	9.00189e73	0.701535	0.350767	+	0.936463i	$0.385921\pi$
$164$	1.30704e74	0.829890
$165$	1.33833e74	0.693190
$166$	5.93880e72	0.0251234
$167$	−2.19165e74	−0.758175	−0.379088	−	0.925361i	$-0.623762\pi$
$167$	−2.19165e74	−0.758175	−0.379088	+	0.925361i	$0.623762\pi$
$168$	−1.92013e74	−0.543839
$169$	9.90001e74	2.29843
$170$	3.77139e74	0.718560
$171$	−6.82136e74	−1.06783
$172$	−2.37825e74	−0.306234
$173$	6.15969e73	0.0653152	0.0326576	−	0.999467i	$-0.489603\pi$
$173$	6.15969e73	0.0653152	0.0326576	+	0.999467i	$0.489603\pi$
$174$	−1.55196e75	−1.35669
$175$	−2.93324e74	−0.211628
$176$	−1.92157e74	−0.114547
$177$	−4.13727e74	−0.203990
$178$	1.53488e75	0.626618
$179$	−2.34799e75	−0.794546	−0.397273	−	0.917701i	$-0.630043\pi$
$179$	−2.34799e75	−0.794546	−0.397273	+	0.917701i	$0.630043\pi$
$180$	−1.79200e75	−0.503164
$181$	4.63546e75	1.08109	0.540544	−	0.841316i	$-0.318219\pi$
$181$	4.63546e75	1.08109	0.540544	+	0.841316i	$0.318219\pi$
$182$	−7.34759e75	−1.42480
$183$	−1.39703e75	−0.225473
$184$	−2.37877e75	−0.319857
$185$	1.52145e76	1.70610
$186$	−1.44874e75	−0.135615
$187$	5.45512e75	0.426688
$188$	9.42345e75	0.616484
$189$	2.18478e75	0.119657
$190$	1.94676e76	0.893441
$191$	3.82283e76	1.47152	0.735762	−	0.677241i	$-0.236824\pi$
$191$	3.82283e76	1.47152	0.735762	+	0.677241i	$0.236824\pi$
$192$	5.36292e75	0.173305
$193$	−4.59230e76	−1.24698	−0.623492	−	0.781830i	$-0.714287\pi$
$193$	−4.59230e76	−1.24698	−0.623492	+	0.781830i	$0.714287\pi$
$194$	1.10951e76	0.253381
$195$	−1.42930e77	−2.74766
$196$	7.12868e75	0.115460
$197$	−5.61841e76	−0.767353	−0.383676	−	0.923468i	$-0.625342\pi$
$197$	−5.61841e76	−0.767353	−0.383676	+	0.923468i	$0.625342\pi$
$198$	−2.59204e76	−0.298784
$199$	1.89731e77	1.84740	0.923698	−	0.383122i	$-0.125151\pi$
$199$	1.89731e77	1.84740	0.923698	+	0.383122i	$0.125151\pi$
$200$	8.19252e75	0.0674393
$201$	−2.09721e77	−1.46074
$202$	1.19091e77	0.702438
$203$	3.07134e77	1.53536
$204$	−1.52247e77	−0.645563
$205$	−5.03170e77	−1.81118
$206$	1.01309e77	0.309812
$207$	−3.20876e77	−0.834316
$208$	2.05218e77	0.454039
$209$	2.81588e77	0.530534
$210$	7.39191e77	1.18689
$211$	−4.06194e77	−0.556252	−0.278126	−	0.960545i	$-0.589713\pi$
$211$	−4.06194e77	−0.556252	−0.278126	+	0.960545i	$0.589713\pi$
$212$	−6.36181e77	−0.743583
$213$	1.88568e78	1.88257
$214$	1.38632e77	0.118304
$215$	9.15550e77	0.668334
$216$	−6.10208e76	−0.0381309
$217$	2.86706e77	0.153475
$218$	−2.17759e78	−0.999272
$219$	3.70320e78	1.45780
$220$	7.39745e77	0.249990
$221$	−5.82590e78	−1.69130
$222$	−6.14193e78	−1.53278
$223$	−9.95452e77	−0.213701	−0.106851	−	0.994275i	$-0.534077\pi$
$223$	−9.95452e77	−0.213701	−0.106851	+	0.994275i	$0.534077\pi$
$224$	−1.06133e78	−0.196128
$225$	1.10510e78	0.175909
$226$	3.12670e78	0.428996
$227$	−6.03085e78	−0.713694	−0.356847	−	0.934163i	$-0.616148\pi$
$227$	−6.03085e78	−0.713694	−0.356847	+	0.934163i	$0.616148\pi$
$228$	−7.85885e78	−0.802678
$229$	8.28002e78	0.730369	0.365185	−	0.930935i	$-0.381006\pi$
$229$	8.28002e78	0.730369	0.365185	+	0.930935i	$0.381006\pi$
$230$	9.15752e78	0.698064
$231$	1.06920e79	0.704786
$232$	−8.57825e78	−0.489273
$233$	−1.21664e79	−0.600813	−0.300407	−	0.953811i	$-0.597122\pi$
$233$	−1.21664e79	−0.600813	−0.300407	+	0.953811i	$0.597122\pi$
$234$	2.76821e79	1.18432
$235$	−3.62773e79	−1.34543
$236$	−2.28681e78	−0.0735663
$237$	7.73048e79	2.15842
$238$	3.01298e79	0.730581
$239$	4.87552e79	1.02729	0.513643	−	0.858004i	$-0.328296\pi$
$239$	4.87552e79	1.02729	0.513643	+	0.858004i	$0.328296\pi$
$240$	−2.06456e79	−0.378225
$241$	−1.48849e79	−0.237233	−0.118616	−	0.992940i	$-0.537846\pi$
$241$	−1.48849e79	−0.237233	−0.118616	+	0.992940i	$0.537846\pi$
$242$	−4.02682e79	−0.558660
$243$	−1.14045e80	−1.37805
$244$	−7.72189e78	−0.0813138
$245$	−2.74432e79	−0.251983
$246$	2.03125e80	1.62718
$247$	−3.00727e80	−2.10293
$248$	−8.00770e78	−0.0489076
$249$	9.22936e78	0.0492599
$250$	1.33804e80	0.624423
$251$	−1.98903e80	−0.812030	−0.406015	−	0.913866i	$-0.633082\pi$
$251$	−1.98903e80	−0.812030	−0.406015	+	0.913866i	$0.633082\pi$
$252$	−1.43164e80	−0.511582
$253$	1.32459e80	0.414517
$254$	−8.71552e79	−0.238981
$255$	5.86104e80	1.40889
$256$	2.96428e79	0.0625000
$257$	−3.40406e80	−0.629851	−0.314925	−	0.949116i	$-0.601979\pi$
$257$	−3.40406e80	−0.629851	−0.314925	+	0.949116i	$0.601979\pi$
$258$	−3.69598e80	−0.600440
$259$	1.21549e81	1.73464
$260$	−7.90024e80	−0.990908
$261$	−1.15713e81	−1.27622
$262$	3.39073e80	0.329003
$263$	7.60787e80	0.649749	0.324875	−	0.945757i	$-0.394678\pi$
$263$	7.60787e80	0.649749	0.324875	+	0.945757i	$0.394678\pi$
$264$	−2.98627e80	−0.224594
$265$	2.44910e81	1.62282
$266$	1.55527e81	0.908387
$267$	2.38532e81	1.22862
$268$	−1.15920e81	−0.526797
$269$	−3.25217e78	−0.00130458	−0.000652291	−	1.00000i	$-0.500208\pi$
$269$	−3.25217e78	−0.00130458	−0.000652291		1.00000i	$0.500208\pi$
$270$	2.34911e80	0.0832179
$271$	−2.30737e81	−0.722181	−0.361090	−	0.932531i	$-0.617595\pi$
$271$	−2.30737e81	−0.722181	−0.361090	+	0.932531i	$0.617595\pi$
$272$	−8.41524e80	−0.232814
$273$	−1.14187e82	−2.79363
$274$	−1.73241e81	−0.374979
$275$	−4.56189e80	−0.0873977
$276$	−3.69679e81	−0.627149
$277$	9.34211e81	1.40402	0.702009	−	0.712168i	$-0.252287\pi$
$277$	9.34211e81	1.40402	0.702009	+	0.712168i	$0.252287\pi$
$278$	−3.86133e81	−0.514323
$279$	−1.08017e81	−0.127571
$280$	4.08577e81	0.428035
$281$	−1.08069e82	−1.00470	−0.502352	−	0.864663i	$-0.667532\pi$
$281$	−1.08069e82	−1.00470	−0.502352	+	0.864663i	$0.667532\pi$
$282$	1.46448e82	1.20875
$283$	4.60163e81	0.337339	0.168669	−	0.985673i	$-0.446053\pi$
$283$	4.60163e81	0.337339	0.168669	+	0.985673i	$0.446053\pi$
$284$	1.04228e82	0.678923
$285$	3.02541e82	1.75179
$286$	−1.14273e82	−0.588411
$287$	−4.01984e82	−1.84147
$288$	3.99856e81	0.163025
$289$	−3.65732e81	−0.132765
$290$	3.30236e82	1.06780
$291$	1.72426e82	0.496808
$292$	2.04689e82	0.525737
$293$	−1.33934e82	−0.306779	−0.153390	−	0.988166i	$-0.549019\pi$
$293$	−1.33934e82	−0.306779	−0.153390	+	0.988166i	$0.549019\pi$
$294$	1.10785e82	0.226384
$295$	8.80352e81	0.160553
$296$	−3.39486e82	−0.552777
$297$	3.39786e81	0.0494156
$298$	−2.65565e82	−0.345085
$299$	−1.41462e83	−1.64306
$300$	1.27318e82	0.132229
$301$	7.31436e82	0.679514
$302$	−1.58818e83	−1.32028
$303$	1.85077e83	1.37728
$304$	−4.34386e82	−0.289475
$305$	2.97268e82	0.177461
$306$	−1.13514e83	−0.607272
$307$	−2.96493e83	−1.42193	−0.710966	−	0.703227i	$-0.751742\pi$
$307$	−2.96493e83	−1.42193	−0.710966	+	0.703227i	$0.751742\pi$
$308$	5.90984e82	0.254172
$309$	1.57443e83	0.607454
$310$	3.08271e82	0.106737
$311$	2.70719e83	0.841484	0.420742	−	0.907180i	$-0.361770\pi$
$311$	2.70719e83	0.841484	0.420742	+	0.907180i	$0.361770\pi$
$312$	3.18925e83	0.890244
$313$	−2.54292e83	−0.637672	−0.318836	−	0.947810i	$-0.603292\pi$
$313$	−2.54292e83	−0.637672	−0.318836	+	0.947810i	$0.603292\pi$
$314$	−2.16434e83	−0.487733
$315$	5.51135e83	1.11649
$316$	4.27291e83	0.778406
$317$	−1.66696e83	−0.273174	−0.136587	−	0.990628i	$-0.543613\pi$
$317$	−1.66696e83	−0.273174	−0.136587	+	0.990628i	$0.543613\pi$
$318$	−9.88675e83	−1.45796
$319$	4.77668e83	0.634072
$320$	−1.14115e83	−0.136402
$321$	2.15444e83	0.231962
$322$	7.31597e83	0.709742
$323$	1.23317e84	1.07830
$324$	−6.79699e83	−0.535869
$325$	4.87196e83	0.346427
$326$	7.73257e83	0.496060
$327$	−3.38414e84	−1.95929
$328$	1.12274e84	0.586821
$329$	−2.89821e84	−1.36794
$330$	1.14962e84	0.490159
$331$	−3.94958e84	−1.52164	−0.760822	−	0.648960i	$-0.775204\pi$
$331$	−3.94958e84	−1.52164	−0.760822	+	0.648960i	$0.775204\pi$
$332$	5.10139e82	0.0177649
$333$	−4.57938e84	−1.44187
$334$	−1.88261e84	−0.536111
$335$	4.46256e84	1.14970
$336$	−1.64938e84	−0.384552
$337$	1.55199e84	0.327558	0.163779	−	0.986497i	$-0.447632\pi$
$337$	1.55199e84	0.327558	0.163779	+	0.986497i	$0.447632\pi$
$338$	8.50404e84	1.62523
$339$	4.85913e84	0.841140
$340$	3.23960e84	0.508099
$341$	4.45898e83	0.0633817
$342$	−5.85950e84	−0.755068
$343$	7.30196e84	0.853270
$344$	−2.04290e84	−0.216540
$345$	1.42315e85	1.36871
$346$	5.29113e83	0.0461848
$347$	−2.39872e84	−0.190082	−0.0950412	−	0.995473i	$-0.530298\pi$
$347$	−2.39872e84	−0.190082	−0.0950412	+	0.995473i	$0.530298\pi$
$348$	−1.33313e85	−0.959328
$349$	−8.55230e84	−0.559025	−0.279513	−	0.960142i	$-0.590173\pi$
$349$	−8.55230e84	−0.559025	−0.279513	+	0.960142i	$0.590173\pi$
$350$	−2.51963e84	−0.149643
$351$	−3.62881e84	−0.195873
$352$	−1.65062e84	−0.0809967
$353$	1.26346e85	0.563777	0.281889	−	0.959447i	$-0.409039\pi$
$353$	1.26346e85	0.563777	0.281889	+	0.959447i	$0.409039\pi$
$354$	−3.55389e84	−0.144243
$355$	−4.01245e85	−1.48170
$356$	1.31845e85	0.443086
$357$	4.68240e85	1.43246
$358$	−2.01690e85	−0.561829
$359$	−1.57541e84	−0.0399696	−0.0199848	−	0.999800i	$-0.506362\pi$
$359$	−1.57541e84	−0.0399696	−0.0199848	+	0.999800i	$0.506362\pi$
$360$	−1.53932e85	−0.355791
$361$	1.61772e85	0.340732
$362$	3.98183e85	0.764444
$363$	−6.25800e85	−1.09538
$364$	−6.31153e85	−1.00748
$365$	−7.87988e85	−1.14738
$366$	−1.20004e85	−0.159434
$367$	−2.25002e85	−0.272819	−0.136409	−	0.990653i	$-0.543556\pi$
$367$	−2.25002e85	−0.272819	−0.136409	+	0.990653i	$0.543556\pi$
$368$	−2.04335e85	−0.226173
$369$	1.51448e86	1.53067
$370$	1.30692e86	1.20640
$371$	1.95659e86	1.64996
$372$	−1.24446e85	−0.0958941
$373$	−9.34358e85	−0.658063	−0.329032	−	0.944319i	$-0.606722\pi$
$373$	−9.34358e85	−0.658063	−0.329032	+	0.944319i	$0.606722\pi$
$374$	4.68591e85	0.301714
$375$	2.07942e86	1.22432
$376$	8.09468e85	0.435920
$377$	−5.10135e86	−2.51333
$378$	1.87671e85	0.0846101
$379$	−1.49483e85	−0.0616846	−0.0308423	−	0.999524i	$-0.509819\pi$
$379$	−1.49483e85	−0.0616846	−0.0308423	+	0.999524i	$0.509819\pi$
$380$	1.67225e86	0.631758
$381$	−1.35446e86	−0.468575
$382$	3.28379e86	1.04052
$383$	4.96835e85	0.144229	0.0721147	−	0.997396i	$-0.477025\pi$
$383$	4.96835e85	0.144229	0.0721147	+	0.997396i	$0.477025\pi$
$384$	4.60672e85	0.122545
$385$	−2.27510e86	−0.554711
$386$	−3.94476e86	−0.881751
$387$	−2.75569e86	−0.564825
$388$	9.53062e85	0.179167
$389$	1.18330e86	0.204071	0.102036	−	0.994781i	$-0.467464\pi$
$389$	1.18330e86	0.204071	0.102036	+	0.994781i	$0.467464\pi$
$390$	−1.22776e87	−1.94289
$391$	5.80083e86	0.842498
$392$	6.12349e85	0.0816424
$393$	5.26945e86	0.645082
$394$	−4.82618e86	−0.542600
$395$	−1.64493e87	−1.69881
$396$	−2.22654e86	−0.211272
$397$	4.87241e85	0.0424876	0.0212438	−	0.999774i	$-0.493237\pi$
$397$	4.87241e85	0.0424876	0.0212438	+	0.999774i	$0.493237\pi$
$398$	1.62978e87	1.30631
$399$	2.41701e87	1.78109
$400$	7.03732e85	0.0476868
$401$	1.14071e87	0.710947	0.355473	−	0.934686i	$-0.384320\pi$
$401$	1.14071e87	0.710947	0.355473	+	0.934686i	$0.384320\pi$
$402$	−1.80149e87	−1.03290
$403$	−4.76205e86	−0.251232
$404$	1.02298e87	0.496698
$405$	2.61663e87	1.16950
$406$	2.63827e87	1.08567
$407$	1.89038e87	0.716369
$408$	−1.30779e87	−0.456482
$409$	−2.28658e87	−0.735285	−0.367643	−	0.929967i	$-0.619835\pi$
$409$	−2.28658e87	−0.735285	−0.367643	+	0.929967i	$0.619835\pi$
$410$	−4.32220e87	−1.28069
$411$	−2.69230e87	−0.735229
$412$	8.70241e86	0.219070
$413$	7.03316e86	0.163239
$414$	−2.75630e87	−0.589950
$415$	−1.96388e86	−0.0387706
$416$	1.76281e87	0.321054
$417$	−6.00080e87	−1.00844
$418$	2.41882e87	0.375144
$419$	−1.00340e88	−1.43649	−0.718247	−	0.695788i	$-0.755055\pi$
$419$	−1.00340e88	−1.43649	−0.718247	+	0.695788i	$0.755055\pi$
$420$	6.34960e87	0.839257
$421$	2.08698e87	0.254723	0.127361	−	0.991856i	$-0.459349\pi$
$421$	2.08698e87	0.254723	0.127361	+	0.991856i	$0.459349\pi$
$422$	−3.48918e87	−0.393330
$423$	1.09190e88	1.13706
$424$	−5.46476e87	−0.525793
$425$	−1.99781e87	−0.177634
$426$	1.61979e88	1.33118
$427$	2.37489e87	0.180430
$428$	1.19084e87	0.0836539
$429$	−1.77589e88	−1.15371
$430$	7.86451e87	0.472584
$431$	−2.38656e88	−1.32673	−0.663366	−	0.748295i	$-0.730873\pi$
$431$	−2.38656e88	−1.32673	−0.663366	+	0.748295i	$0.730873\pi$
$432$	−5.24165e86	−0.0269626
$433$	−2.22104e88	−1.05733	−0.528667	−	0.848830i	$-0.677308\pi$
$433$	−2.22104e88	−1.05733	−0.528667	+	0.848830i	$0.677308\pi$
$434$	2.46279e87	0.108523
$435$	5.13212e88	2.09366
$436$	−1.87054e88	−0.706592
$437$	2.99433e88	1.04754
$438$	3.18102e88	1.03082
$439$	4.90254e88	1.47184	0.735918	−	0.677070i	$-0.236751\pi$
$439$	4.90254e88	1.47184	0.735918	+	0.677070i	$0.236751\pi$
$440$	6.35436e87	0.176769
$441$	8.26007e87	0.212957
$442$	−5.00441e88	−1.19593
$443$	3.26102e88	0.722484	0.361242	−	0.932472i	$-0.382353\pi$
$443$	3.26102e88	0.722484	0.361242	+	0.932472i	$0.382353\pi$
$444$	−5.27588e88	−1.08384
$445$	−5.07561e88	−0.967003
$446$	−8.55087e87	−0.151110
$447$	−4.12709e88	−0.676615
$448$	−9.11671e87	−0.138684
$449$	3.35516e88	0.473653	0.236827	−	0.971552i	$-0.423893\pi$
$449$	3.35516e88	0.473653	0.236827	+	0.971552i	$0.423893\pi$
$450$	9.49275e87	0.124386
$451$	−6.25183e88	−0.760489
$452$	2.68581e88	0.303346
$453$	−2.46815e89	−2.58870
$454$	−5.18046e88	−0.504658
$455$	2.42974e89	2.19876
$456$	−6.75070e88	−0.567579
$457$	6.03090e87	0.0471183	0.0235592	−	0.999722i	$-0.492500\pi$
$457$	6.03090e87	0.0471183	0.0235592	+	0.999722i	$0.492500\pi$
$458$	7.11248e88	0.516449
$459$	1.48804e88	0.100436
$460$	7.86625e88	0.493606
$461$	−4.47552e88	−0.261133	−0.130567	−	0.991440i	$-0.541680\pi$
$461$	−4.47552e88	−0.261133	−0.130567	+	0.991440i	$0.541680\pi$
$462$	9.18436e88	0.498359
$463$	1.96658e89	0.992539	0.496269	−	0.868169i	$-0.334703\pi$
$463$	1.96658e89	0.992539	0.496269	+	0.868169i	$0.334703\pi$
$464$	−7.36866e88	−0.345968
$465$	4.79077e88	0.209282
$466$	−1.04509e89	−0.424839
$467$	1.94015e87	0.00734039	0.00367019	−	0.999993i	$-0.498832\pi$
$467$	1.94015e87	0.00734039	0.00367019	+	0.999993i	$0.498832\pi$
$468$	2.37788e89	0.837440
$469$	3.56516e89	1.16893
$470$	−3.11620e89	−0.951364
$471$	−3.36355e89	−0.956308
$472$	−1.96436e88	−0.0520192
$473$	1.13756e89	0.280625
$474$	6.64043e89	1.52624
$475$	−1.03125e89	−0.220866
$476$	2.58813e89	0.516599
$477$	−7.37149e89	−1.37148
$478$	4.18804e89	0.726400
$479$	−9.36805e88	−0.151499	−0.0757495	−	0.997127i	$-0.524135\pi$
$479$	−9.36805e88	−0.151499	−0.0757495	+	0.997127i	$0.524135\pi$
$480$	−1.77344e89	−0.267445
$481$	−2.01887e90	−2.83954
$482$	−1.27860e89	−0.167749
$483$	1.13696e90	1.39160
$484$	−3.45902e89	−0.395033
$485$	−3.66899e89	−0.391019
$486$	−9.79635e89	−0.974427
$487$	−7.41719e89	−0.688683	−0.344341	−	0.938845i	$-0.611898\pi$
$487$	−7.41719e89	−0.688683	−0.344341	+	0.938845i	$0.611898\pi$
$488$	−6.63305e88	−0.0574976
$489$	1.20170e90	0.972634
$490$	−2.35735e89	−0.178179
$491$	2.47488e90	1.74713	0.873563	−	0.486712i	$-0.161804\pi$
$491$	2.47488e90	1.74713	0.873563	+	0.486712i	$0.161804\pi$
$492$	1.74483e90	1.15059
$493$	2.09188e90	1.28874
$494$	−2.58323e90	−1.48700
$495$	8.57149e89	0.461086
$496$	−6.87856e88	−0.0345829
$497$	−3.20556e90	−1.50649
$498$	7.92796e88	0.0348320
$499$	1.17805e90	0.483945	0.241972	−	0.970283i	$-0.422206\pi$
$499$	1.17805e90	0.483945	0.241972	+	0.970283i	$0.422206\pi$
$500$	1.14937e90	0.441534
$501$	−2.92573e90	−1.05116
$502$	−1.70856e90	−0.574192
$503$	6.22741e90	1.95785	0.978927	−	0.204211i	$-0.0654629\pi$
$503$	6.22741e90	1.95785	0.978927	+	0.204211i	$0.0654629\pi$
$504$	−1.22977e90	−0.361743
$505$	−3.93816e90	−1.08401
$506$	1.13781e90	0.293108
$507$	1.32159e91	3.18663
$508$	−7.48657e89	−0.168985
$509$	−7.44558e90	−1.57345	−0.786725	−	0.617304i	$-0.788225\pi$
$509$	−7.44558e90	−1.57345	−0.786725	+	0.617304i	$0.788225\pi$
$510$	5.03459e90	0.996239
$511$	−6.29526e90	−1.16658
$512$	2.54629e89	0.0441942
$513$	7.68113e89	0.124880
$514$	−2.92407e90	−0.445372
$515$	−3.35016e90	−0.478104
$516$	−3.17482e90	−0.424575
$517$	−4.50741e90	−0.564930
$518$	1.04410e91	1.22658
$519$	8.22283e89	0.0905555
$520$	−6.78626e90	−0.700678
$521$	−1.54016e91	−1.49108	−0.745542	−	0.666458i	$-0.767809\pi$
$521$	−1.54016e91	−1.49108	−0.745542	+	0.666458i	$0.767809\pi$
$522$	−9.93970e90	−0.902426
$523$	3.42226e90	0.291411	0.145706	−	0.989328i	$-0.453455\pi$
$523$	3.42226e90	0.291411	0.145706	+	0.989328i	$0.453455\pi$
$524$	2.91261e90	0.232640
$525$	−3.91570e90	−0.293409
$526$	6.53511e90	0.459442
$527$	1.95274e90	0.128822
$528$	−2.56519e90	−0.158812
$529$	−3.12408e90	−0.181534
$530$	2.10376e91	1.14750
$531$	−2.64975e90	−0.135687
$532$	1.33597e91	0.642327
$533$	6.67676e91	3.01442
$534$	2.04897e91	0.868767
$535$	−4.58435e90	−0.182568
$536$	−9.95747e90	−0.372502
$537$	−3.13443e91	−1.10159
$538$	−2.79359e88	−0.000922479 0
$539$	−3.40978e90	−0.105804
$540$	2.01787e90	0.0588440
$541$	−6.34854e91	−1.74006	−0.870032	−	0.492995i	$-0.835902\pi$
$541$	−6.34854e91	−1.74006	−0.870032	+	0.492995i	$0.835902\pi$
$542$	−1.98202e91	−0.510659
$543$	6.18807e91	1.49886
$544$	−7.22864e90	−0.164624
$545$	7.20097e91	1.54208
$546$	−9.80861e91	−1.97539
$547$	−7.02150e91	−1.33001	−0.665004	−	0.746840i	$-0.731570\pi$
$547$	−7.02150e91	−1.33001	−0.665004	+	0.746840i	$0.731570\pi$
$548$	−1.48813e91	−0.265150
$549$	−8.94742e90	−0.149977
$550$	−3.91864e90	−0.0617995
$551$	1.07981e92	1.60239
$552$	−3.17552e91	−0.443461
$553$	−1.31414e92	−1.72723
$554$	8.02481e91	0.992790
$555$	2.03105e92	2.36540
$556$	−3.31685e91	−0.363681
$557$	−1.63262e92	−1.68553	−0.842767	−	0.538279i	$-0.819075\pi$
$557$	−1.63262e92	−1.68553	−0.842767	+	0.538279i	$0.819075\pi$
$558$	−9.27860e90	−0.0902062
$559$	−1.21488e92	−1.11234
$560$	3.50965e91	0.302667
$561$	7.28227e91	0.591577
$562$	−9.28304e91	−0.710434
$563$	2.45056e91	0.176699	0.0883494	−	0.996090i	$-0.471841\pi$
$563$	2.45056e91	0.176699	0.0883494	+	0.996090i	$0.471841\pi$
$564$	1.25798e92	0.854717
$565$	−1.03395e92	−0.662030
$566$	3.95277e91	0.238535
$567$	2.09043e92	1.18906
$568$	8.95313e91	0.480071
$569$	1.31507e92	0.664797	0.332398	−	0.943139i	$-0.392142\pi$
$569$	1.31507e92	0.664797	0.332398	+	0.943139i	$0.392142\pi$
$570$	2.59881e92	1.23870
$571$	−2.70090e92	−1.21395	−0.606973	−	0.794723i	$-0.707616\pi$
$571$	−2.70090e92	−1.21395	−0.606973	+	0.794723i	$0.707616\pi$
$572$	−9.81596e91	−0.416069
$573$	5.10326e92	2.04018
$574$	−3.45302e92	−1.30212
$575$	−4.85100e91	−0.172567
$576$	3.43473e91	0.115276
$577$	−1.79781e92	−0.569318	−0.284659	−	0.958629i	$-0.591880\pi$
$577$	−1.79781e92	−0.569318	−0.284659	+	0.958629i	$0.591880\pi$
$578$	−3.14161e91	−0.0938794
$579$	−6.13046e92	−1.72887
$580$	2.83670e92	0.755051
$581$	−1.56895e91	−0.0394192
$582$	1.48113e92	0.351296
$583$	3.04298e92	0.681400
$584$	1.75826e92	0.371752
$585$	−9.15408e92	−1.82765
$586$	−1.15049e92	−0.216926
$587$	3.71694e92	0.661926	0.330963	−	0.943644i	$-0.392626\pi$
$587$	3.71694e92	0.661926	0.330963	+	0.943644i	$0.392626\pi$
$588$	9.51638e91	0.160078
$589$	1.00799e92	0.160174
$590$	7.56216e91	0.113528
$591$	−7.50026e92	−1.06389
$592$	−2.91616e92	−0.390872
$593$	−4.17953e90	−0.00529414	−0.00264707	−	0.999996i	$-0.500843\pi$
$593$	−4.17953e90	−0.00529414	−0.00264707	+	0.999996i	$0.500843\pi$
$594$	2.91874e91	0.0349421
$595$	−9.96348e92	−1.12744
$596$	−2.28119e92	−0.244012
$597$	2.53280e93	2.56130
$598$	−1.21515e93	−1.16182
$599$	3.30837e92	0.299099	0.149550	−	0.988754i	$-0.452218\pi$
$599$	3.30837e92	0.299099	0.149550	+	0.988754i	$0.452218\pi$
$600$	1.09365e92	0.0935004
$601$	2.10095e92	0.169872	0.0849359	−	0.996386i	$-0.472931\pi$
$601$	2.10095e92	0.169872	0.0849359	+	0.996386i	$0.472931\pi$
$602$	6.28299e92	0.480489
$603$	−1.34318e93	−0.971636
$604$	−1.36423e93	−0.933578
$605$	1.33161e93	0.862130
$606$	1.58980e93	0.973886
$607$	3.30489e92	0.191574	0.0957868	−	0.995402i	$-0.469463\pi$
$607$	3.30489e92	0.191574	0.0957868	+	0.995402i	$0.469463\pi$
$608$	−3.73135e92	−0.204690
$609$	4.10007e93	2.12869
$610$	2.55352e92	0.125484
$611$	4.81378e93	2.23927
$612$	−9.75082e92	−0.429406
$613$	2.95128e93	1.23051	0.615254	−	0.788329i	$-0.289053\pi$
$613$	2.95128e93	1.23051	0.615254	+	0.788329i	$0.289053\pi$
$614$	−2.54685e93	−1.00546
$615$	−6.71703e93	−2.51108
$616$	5.07652e92	0.179726
$617$	−5.74654e93	−1.92687	−0.963437	−	0.267935i	$-0.913659\pi$
$617$	−5.74654e93	−1.92687	−0.963437	+	0.267935i	$0.913659\pi$
$618$	1.35242e93	0.429535
$619$	−6.22167e93	−1.87185	−0.935923	−	0.352206i	$-0.885432\pi$
$619$	−6.22167e93	−1.87185	−0.935923	+	0.352206i	$0.885432\pi$
$620$	2.64803e92	0.0754747
$621$	3.61319e92	0.0975714
$622$	2.32546e93	0.595019
$623$	−4.05492e93	−0.983180
$624$	2.73954e93	0.629497
$625$	−5.30057e93	−1.15436
$626$	−2.18435e93	−0.450902
$627$	3.75903e93	0.735553
$628$	−1.85915e93	−0.344879
$629$	8.27865e93	1.45601
$630$	4.73422e93	0.789477
$631$	7.09912e92	0.112259	0.0561295	−	0.998423i	$-0.482124\pi$
$631$	7.09912e92	0.112259	0.0561295	+	0.998423i	$0.482124\pi$
$632$	3.67040e93	0.550416
$633$	−5.42246e93	−0.771209
$634$	−1.43191e93	−0.193163
$635$	2.88209e93	0.368798
$636$	−8.49266e93	−1.03093
$637$	3.64154e93	0.419386
$638$	4.10314e93	0.448357
$639$	1.20770e94	1.25222
$640$	−9.80243e92	−0.0964506
$641$	−1.58485e93	−0.147994	−0.0739970	−	0.997258i	$-0.523576\pi$
$641$	−1.58485e93	−0.147994	−0.0739970	+	0.997258i	$0.523576\pi$
$642$	1.85065e93	0.164022
$643$	−2.21457e94	−1.86304	−0.931519	−	0.363693i	$-0.881516\pi$
$643$	−2.21457e94	−1.86304	−0.931519	+	0.363693i	$0.881516\pi$
$644$	6.28437e93	0.501863
$645$	1.22221e94	0.926604
$646$	1.05929e94	0.762473
$647$	−1.50119e93	−0.102599	−0.0512994	−	0.998683i	$-0.516336\pi$
$647$	−1.50119e93	−0.102599	−0.0512994	+	0.998683i	$0.516336\pi$
$648$	−5.83857e93	−0.378917
$649$	1.09383e93	0.0674141
$650$	4.18498e93	0.244961
$651$	3.82737e93	0.212783
$652$	6.64223e93	0.350767
$653$	−6.29126e93	−0.315607	−0.157803	−	0.987471i	$-0.550441\pi$
$653$	−6.29126e93	−0.315607	−0.157803	+	0.987471i	$0.550441\pi$
$654$	−2.90696e94	−1.38543
$655$	−1.12126e94	−0.507720
$656$	9.64427e93	0.414945
$657$	2.37175e94	0.969680
$658$	−2.48954e94	−0.967279
$659$	1.46135e94	0.539627	0.269813	−	0.962913i	$-0.413038\pi$
$659$	1.46135e94	0.539627	0.269813	+	0.962913i	$0.413038\pi$
$660$	9.87516e93	0.346595
$661$	5.76501e94	1.92332	0.961660	−	0.274245i	$-0.0884279\pi$
$661$	5.76501e94	1.92332	0.961660	+	0.274245i	$0.0884279\pi$
$662$	−3.39266e94	−1.07597
$663$	−7.77724e94	−2.34489
$664$	4.38206e92	0.0125617
$665$	−5.14305e94	−1.40183
$666$	−3.93366e94	−1.01955
$667$	5.07940e94	1.25198
$668$	−1.61715e94	−0.379088
$669$	−1.32887e94	−0.296284
$670$	3.83331e94	0.812958
$671$	3.69352e93	0.0745138
$672$	−1.41681e94	−0.271919
$673$	3.15852e94	0.576739	0.288369	−	0.957519i	$-0.406887\pi$
$673$	3.15852e94	0.576739	0.288369	+	0.957519i	$0.406887\pi$
$674$	1.33315e94	0.231619
$675$	−1.24439e93	−0.0205722
$676$	7.30492e94	1.14921
$677$	8.67061e94	1.29816	0.649081	−	0.760719i	$-0.275153\pi$
$677$	8.67061e94	1.29816	0.649081	+	0.760719i	$0.275153\pi$
$678$	4.17396e94	0.594776
$679$	−2.93117e94	−0.397560
$680$	2.78280e94	0.359280
$681$	−8.05084e94	−0.989493
$682$	3.83023e93	0.0448176
$683$	−5.43627e94	−0.605630	−0.302815	−	0.953049i	$-0.597926\pi$
$683$	−5.43627e94	−0.605630	−0.302815	+	0.953049i	$0.597926\pi$
$684$	−5.03327e94	−0.533914
$685$	5.72883e94	0.578671
$686$	6.27234e94	0.603353
$687$	1.10533e95	1.01261
$688$	−1.75484e94	−0.153117
$689$	−3.24980e95	−2.70093
$690$	1.22248e95	0.967822
$691$	1.85794e95	1.40126	0.700628	−	0.713527i	$-0.252903\pi$
$691$	1.85794e95	1.40126	0.700628	+	0.713527i	$0.252903\pi$
$692$	4.54505e93	0.0326576
$693$	6.84779e94	0.468799
$694$	−2.06048e94	−0.134409
$695$	1.27688e95	0.793708
$696$	−1.14515e95	−0.678347
$697$	−2.73790e95	−1.54568
$698$	−7.34637e94	−0.395290
$699$	−1.62415e95	−0.832991
$700$	−2.16435e94	−0.105814
$701$	−2.53419e95	−1.18110	−0.590548	−	0.807002i	$-0.701088\pi$
$701$	−2.53419e95	−1.18110	−0.590548	+	0.807002i	$0.701088\pi$
$702$	−3.11712e94	−0.138503
$703$	4.27336e95	1.81036
$704$	−1.41787e94	−0.0572733
$705$	−4.84281e95	−1.86536
$706$	1.08530e95	0.398651
$707$	−3.14621e95	−1.10214
$708$	−3.05277e94	−0.101995
$709$	3.83871e95	1.22331	0.611655	−	0.791125i	$-0.290504\pi$
$709$	3.83871e95	1.22331	0.611655	+	0.791125i	$0.290504\pi$
$710$	−3.44667e95	−1.04772
$711$	4.95106e95	1.43571
$712$	1.13254e95	0.313309
$713$	4.74156e94	0.125147
$714$	4.02215e95	1.01290
$715$	3.77883e95	0.908041
$716$	−1.73251e95	−0.397273
$717$	6.50853e95	1.42427
$718$	−1.35327e94	−0.0282627
$719$	3.51252e95	0.700166	0.350083	−	0.936719i	$-0.386153\pi$
$719$	3.51252e95	0.700166	0.350083	+	0.936719i	$0.386153\pi$
$720$	−1.32226e95	−0.251582
$721$	−2.67645e95	−0.486103
$722$	1.38961e95	0.240934
$723$	−1.98705e95	−0.328909
$724$	3.42037e95	0.540544
$725$	−1.74935e95	−0.263970
$726$	−5.37558e95	−0.774548
$727$	−2.50767e95	−0.345038	−0.172519	−	0.985006i	$-0.555191\pi$
$727$	−2.50767e95	−0.345038	−0.172519	+	0.985006i	$0.555191\pi$
$728$	−5.42156e95	−0.712399
$729$	−6.68423e95	−0.838840
$730$	−6.76876e95	−0.811322
$731$	4.98178e95	0.570364
$732$	−1.03083e95	−0.112737
$733$	1.47901e96	1.54521	0.772604	−	0.634888i	$-0.218954\pi$
$733$	1.47901e96	1.54521	0.772604	+	0.634888i	$0.218954\pi$
$734$	−1.93276e95	−0.192912
$735$	−3.66351e95	−0.349358
$736$	−1.75522e95	−0.159928
$737$	5.54468e95	0.482743
$738$	1.30093e96	1.08235
$739$	−1.30836e96	−1.04025	−0.520126	−	0.854090i	$-0.674115\pi$
$739$	−1.30836e96	−1.04025	−0.520126	+	0.854090i	$0.674115\pi$
$740$	1.12263e96	0.853050
$741$	−4.01453e96	−2.91558
$742$	1.68070e96	1.16670
$743$	−2.39596e95	−0.158984	−0.0794922	−	0.996835i	$-0.525330\pi$
$743$	−2.39596e95	−0.158984	−0.0794922	+	0.996835i	$0.525330\pi$
$744$	−1.06898e95	−0.0678074
$745$	8.78185e95	0.532538
$746$	−8.02608e95	−0.465321
$747$	5.91103e94	0.0327660
$748$	4.02517e95	0.213344
$749$	−3.66245e95	−0.185623
$750$	1.78621e96	0.865724
$751$	−2.64738e96	−1.22710	−0.613548	−	0.789657i	$-0.710258\pi$
$751$	−2.64738e96	−1.22710	−0.613548	+	0.789657i	$0.710258\pi$
$752$	6.95328e95	0.308242
$753$	−2.65524e96	−1.12583
$754$	−4.38203e96	−1.77719
$755$	5.25186e96	2.03747
$756$	1.61208e95	0.0598283
$757$	1.56772e96	0.556618	0.278309	−	0.960492i	$-0.410226\pi$
$757$	1.56772e96	0.556618	0.278309	+	0.960492i	$0.410226\pi$
$758$	−1.28405e95	−0.0436176
$759$	1.76825e96	0.574702
$760$	1.43645e96	0.446720
$761$	2.30382e96	0.685587	0.342793	−	0.939411i	$-0.388627\pi$
$761$	2.30382e96	0.685587	0.342793	+	0.939411i	$0.388627\pi$
$762$	−1.16347e96	−0.331332
$763$	5.75288e96	1.56788
$764$	2.82075e96	0.735762
$765$	3.75376e96	0.937148
$766$	4.26778e95	0.101986
$767$	−1.16817e96	−0.267216
$768$	3.95714e95	0.0866524
$769$	−6.53211e96	−1.36937	−0.684686	−	0.728838i	$-0.740061\pi$
$769$	−6.53211e96	−1.36937	−0.684686	+	0.728838i	$0.740061\pi$
$770$	−1.95430e96	−0.392240
$771$	−4.54422e96	−0.873249
$772$	−3.38852e96	−0.623492
$773$	−4.96093e96	−0.874078	−0.437039	−	0.899443i	$-0.643973\pi$
$773$	−4.96093e96	−0.874078	−0.437039	+	0.899443i	$0.643973\pi$
$774$	−2.36712e96	−0.399391
$775$	−1.63300e95	−0.0263864
$776$	8.18674e95	0.126690
$777$	1.62261e97	2.40497
$778$	1.01644e96	0.144300
$779$	−1.41327e97	−1.92186
$780$	−1.05464e97	−1.37383
$781$	−4.98543e96	−0.622147
$782$	4.98287e96	0.595736
$783$	1.30298e96	0.149251
$784$	5.26004e95	0.0577299
$785$	7.15716e96	0.752674
$786$	4.52643e96	0.456142
$787$	−6.47932e96	−0.625714	−0.312857	−	0.949800i	$-0.601286\pi$
$787$	−6.47932e96	−0.625714	−0.312857	+	0.949800i	$0.601286\pi$
$788$	−4.14566e96	−0.383676
$789$	1.01561e97	0.900837
$790$	−1.41299e97	−1.20124
$791$	−8.26028e96	−0.673105
$792$	−1.91259e96	−0.149392
$793$	−3.94457e96	−0.295357
$794$	4.18537e95	0.0300432
$795$	3.26940e97	2.24993
$796$	1.39997e97	0.923698
$797$	6.80262e96	0.430351	0.215175	−	0.976575i	$-0.430968\pi$
$797$	6.80262e96	0.430351	0.215175	+	0.976575i	$0.430968\pi$
$798$	2.07620e97	1.25942
$799$	−1.97396e97	−1.14821
$800$	6.04501e95	0.0337196
$801$	1.52770e97	0.817237
$802$	9.79858e96	0.502715
$803$	−9.79065e96	−0.481771
$804$	−1.54747e97	−0.730372
$805$	−2.41928e97	−1.09528
$806$	−4.09057e96	−0.177648
$807$	−4.34145e94	−0.00180872
$808$	8.78736e96	0.351219
$809$	4.28933e97	1.64480	0.822401	−	0.568908i	$-0.192634\pi$
$809$	4.28933e97	1.64480	0.822401	+	0.568908i	$0.192634\pi$
$810$	2.24767e97	0.826958
$811$	−3.83378e97	−1.35341	−0.676704	−	0.736256i	$-0.736592\pi$
$811$	−3.83378e97	−1.35341	−0.676704	+	0.736256i	$0.736592\pi$
$812$	2.26625e97	0.767682
$813$	−3.08021e97	−1.00126
$814$	1.62383e97	0.506550
$815$	−2.55705e97	−0.765524
$816$	−1.12339e97	−0.322782
$817$	2.57154e97	0.709177
$818$	−1.96416e97	−0.519925
$819$	−7.31323e97	−1.85823
$820$	−3.71274e97	−0.905588
$821$	6.54109e97	1.53163	0.765816	−	0.643060i	$-0.222335\pi$
$821$	6.54109e97	1.53163	0.765816	+	0.643060i	$0.222335\pi$
$822$	−2.31267e97	−0.519885
$823$	−7.78040e97	−1.67922	−0.839610	−	0.543190i	$-0.817216\pi$
$823$	−7.78040e97	−1.67922	−0.839610	+	0.543190i	$0.817216\pi$
$824$	7.47532e96	0.154906
$825$	−6.08987e96	−0.121171
$826$	6.04144e96	0.115427
$827$	6.63061e97	1.21652	0.608260	−	0.793738i	$-0.291868\pi$
$827$	6.63061e97	1.21652	0.608260	+	0.793738i	$0.291868\pi$
$828$	−2.36765e97	−0.417158
$829$	8.47922e96	0.143476	0.0717379	−	0.997424i	$-0.477145\pi$
$829$	8.47922e96	0.143476	0.0717379	+	0.997424i	$0.477145\pi$
$830$	−1.68696e96	−0.0274150
$831$	1.24712e98	1.94658
$832$	1.51424e97	0.227020
$833$	−1.49326e97	−0.215045
$834$	−5.15465e97	−0.713077
$835$	6.22553e97	0.827331
$836$	2.07775e97	0.265267
$837$	1.21632e96	0.0149191
$838$	−8.61913e97	−1.01575
$839$	3.96426e97	0.448887	0.224443	−	0.974487i	$-0.427944\pi$
$839$	3.96426e97	0.448887	0.224443	+	0.974487i	$0.427944\pi$
$840$	5.45426e97	0.593444
$841$	8.75261e97	0.915106
$842$	1.79270e97	0.180116
$843$	−1.44266e98	−1.39296
$844$	−2.99719e97	−0.278126
$845$	−2.81216e98	−2.50808
$846$	9.37938e97	0.804020
$847$	1.06383e98	0.876552
$848$	−4.69419e97	−0.371792
$849$	6.14292e97	0.467700
$850$	−1.71611e97	−0.125606
$851$	2.01018e98	1.41447
$852$	1.39139e98	0.941285
$853$	1.90163e97	0.123690	0.0618449	−	0.998086i	$-0.480302\pi$
$853$	1.90163e97	0.123690	0.0618449	+	0.998086i	$0.480302\pi$
$854$	2.04001e97	0.127583
$855$	1.93765e98	1.16523
$856$	1.02292e97	0.0591522
$857$	1.17745e98	0.654767	0.327383	−	0.944892i	$-0.393833\pi$
$857$	1.17745e98	0.654767	0.327383	+	0.944892i	$0.393833\pi$
$858$	−1.52548e98	−0.815795
$859$	−1.78895e98	−0.920086	−0.460043	−	0.887897i	$-0.652166\pi$
$859$	−1.78895e98	−0.920086	−0.460043	+	0.887897i	$0.652166\pi$
$860$	6.75557e97	0.334167
$861$	−5.36626e98	−2.55309
$862$	−2.05004e98	−0.938141
$863$	9.31222e97	0.409913	0.204957	−	0.978771i	$-0.434295\pi$
$863$	9.31222e97	0.409913	0.204957	+	0.978771i	$0.434295\pi$
$864$	−4.50254e96	−0.0190655
$865$	−1.74970e97	−0.0712729
$866$	−1.90786e98	−0.747647
$867$	−4.88231e97	−0.184071
$868$	2.11552e97	0.0767373
$869$	−2.04381e98	−0.713310
$870$	4.40846e98	1.48044
$871$	−5.92155e98	−1.91349
$872$	−1.60678e98	−0.499636
$873$	1.10432e98	0.330460
$874$	2.57211e98	0.740724
$875$	−3.53491e98	−0.979736
$876$	2.73248e98	0.728902
$877$	4.13147e98	1.06076	0.530381	−	0.847759i	$-0.322049\pi$
$877$	4.13147e98	1.06076	0.530381	+	0.847759i	$0.322049\pi$
$878$	4.21125e98	1.04075
$879$	−1.78794e98	−0.425330
$880$	5.45835e97	0.124995
$881$	3.38780e98	0.746836	0.373418	−	0.927663i	$-0.378186\pi$
$881$	3.38780e98	0.746836	0.373418	+	0.927663i	$0.378186\pi$
$882$	7.09534e97	0.150583
$883$	7.60653e98	1.55419	0.777093	−	0.629386i	$-0.216694\pi$
$883$	7.60653e98	1.55419	0.777093	+	0.629386i	$0.216694\pi$
$884$	−4.29876e98	−0.845652
$885$	1.17522e98	0.222597
$886$	2.80119e98	0.510873
$887$	7.59244e98	1.33334	0.666669	−	0.745354i	$-0.267720\pi$
$887$	7.59244e98	1.33334	0.666669	+	0.745354i	$0.267720\pi$
$888$	−4.53195e98	−0.766391
$889$	2.30251e98	0.374967
$890$	−4.35992e98	−0.683774
$891$	3.25113e98	0.491056
$892$	−7.34514e97	−0.106851
$893$	−1.01894e99	−1.42765
$894$	−3.54514e98	−0.478439
$895$	6.66961e98	0.867019
$896$	−7.83120e97	−0.0980641
$897$	−1.88843e99	−2.27800
$898$	2.88206e98	0.334924
$899$	1.70989e98	0.191434
$900$	8.15421e97	0.0879544
$901$	1.33263e99	1.38493
$902$	−5.37028e98	−0.537747
$903$	9.76425e98	0.942105
$904$	2.30709e98	0.214498
$905$	−1.31673e99	−1.17970
$906$	−2.12012e99	−1.83049
$907$	2.98307e97	0.0248210	0.0124105	−	0.999923i	$-0.496050\pi$
$907$	2.98307e97	0.0248210	0.0124105	+	0.999923i	$0.496050\pi$
$908$	−4.44998e98	−0.356847
$909$	1.18534e99	0.916121
$910$	2.08713e99	1.55476
$911$	−2.36041e99	−1.69481	−0.847405	−	0.530946i	$-0.821837\pi$
$911$	−2.36041e99	−1.69481	−0.847405	+	0.530946i	$0.821837\pi$
$912$	−5.79881e98	−0.401339
$913$	−2.44009e97	−0.0162793
$914$	5.18051e97	0.0333177
$915$	3.96836e98	0.246039
$916$	6.10957e98	0.365185
$917$	−8.95782e98	−0.516214
$918$	1.27822e98	0.0710191
$919$	6.02104e98	0.322553	0.161277	−	0.986909i	$-0.448439\pi$
$919$	6.02104e98	0.322553	0.161277	+	0.986909i	$0.448439\pi$
$920$	6.75705e98	0.349032
$921$	−3.95801e99	−1.97142
$922$	−3.84444e98	−0.184649
$923$	5.32428e99	2.46606
$924$	7.88930e98	0.352393
$925$	−6.92310e98	−0.298231
$926$	1.68928e99	0.701831
$927$	1.00836e99	0.404057
$928$	−6.32963e98	−0.244637
$929$	2.90527e98	0.108308	0.0541540	−	0.998533i	$-0.482754\pi$
$929$	2.90527e98	0.108308	0.0541540	+	0.998533i	$0.482754\pi$
$930$	4.11524e98	0.147985
$931$	−7.70808e98	−0.267382
$932$	−8.97724e98	−0.300407
$933$	3.61394e99	1.16667
$934$	1.66657e97	0.00519044
$935$	−1.54956e99	−0.465608
$936$	2.04258e99	0.592159
$937$	2.86297e99	0.800830	0.400415	−	0.916334i	$-0.368866\pi$
$937$	2.86297e99	0.800830	0.400415	+	0.916334i	$0.368866\pi$
$938$	3.06245e99	0.826558
$939$	−3.39465e99	−0.884093
$940$	−2.67679e99	−0.672716
$941$	−2.68094e99	−0.650181	−0.325090	−	0.945683i	$-0.605395\pi$
$941$	−2.68094e99	−0.650181	−0.325090	+	0.945683i	$0.605395\pi$
$942$	−2.88927e99	−0.676212
$943$	−6.64803e99	−1.50159
$944$	−1.68737e98	−0.0367831
$945$	−6.20601e98	−0.130571
$946$	9.77156e98	0.198432
$947$	−2.19670e99	−0.430572	−0.215286	−	0.976551i	$-0.569068\pi$
$947$	−2.19670e99	−0.430572	−0.215286	+	0.976551i	$0.569068\pi$
$948$	5.70408e99	1.07921
$949$	1.04561e100	1.90964
$950$	−8.85839e98	−0.156176
$951$	−2.22529e99	−0.378739
$952$	2.22319e99	0.365290
$953$	−3.27444e99	−0.519427	−0.259714	−	0.965686i	$-0.583628\pi$
$953$	−3.27444e99	−0.519427	−0.259714	+	0.965686i	$0.583628\pi$
$954$	−6.33206e99	−0.969783
$955$	−1.08590e100	−1.60575
$956$	3.59750e99	0.513643
$957$	6.37660e99	0.879102
$958$	−8.04710e98	−0.107126
$959$	4.57678e99	0.588352
$960$	−1.52337e99	−0.189112
$961$	−8.18168e99	−0.980864
$962$	−1.73420e100	−2.00786
$963$	1.37983e99	0.154293
$964$	−1.09831e99	−0.118616
$965$	1.30447e100	1.36073
$966$	9.76640e99	0.984013
$967$	−1.22820e100	−1.19531	−0.597656	−	0.801752i	$-0.703901\pi$
$967$	−1.22820e100	−1.19531	−0.597656	+	0.801752i	$0.703901\pi$
$968$	−2.97127e99	−0.279330
$969$	1.64621e100	1.49500
$970$	−3.15164e99	−0.276492
$971$	1.81836e100	1.54112	0.770558	−	0.637370i	$-0.219978\pi$
$971$	1.81836e100	1.54112	0.770558	+	0.637370i	$0.219978\pi$
$972$	−8.41500e99	−0.689024
$973$	1.02011e100	0.806985
$974$	−6.37132e99	−0.486972
$975$	6.50379e99	0.480299
$976$	−5.69775e98	−0.0406569
$977$	2.59046e99	0.178612	0.0893058	−	0.996004i	$-0.471535\pi$
$977$	2.59046e99	0.178612	0.0893058	+	0.996004i	$0.471535\pi$
$978$	1.03225e100	0.687756
$979$	−6.30638e99	−0.406032
$980$	−2.02495e99	−0.125991
$981$	−2.16741e100	−1.30325
$982$	2.12591e100	1.23540
$983$	2.24498e100	1.26087	0.630433	−	0.776244i	$-0.282877\pi$
$983$	2.24498e100	1.26087	0.630433	+	0.776244i	$0.282877\pi$
$984$	1.49879e100	0.813591
$985$	1.59595e100	0.837346
$986$	1.79691e100	0.911276
$987$	−3.86894e100	−1.89656
$988$	−2.21897e100	−1.05146
$989$	1.20965e100	0.554095
$990$	7.36285e99	0.326037
$991$	−3.10559e99	−0.132946	−0.0664732	−	0.997788i	$-0.521175\pi$
$991$	−3.10559e99	−0.132946	−0.0664732	+	0.997788i	$0.521175\pi$
$992$	−5.90864e98	−0.0244538
$993$	−5.27246e100	−2.10967
$994$	−2.75356e100	−1.06525
$995$	−5.38943e100	−2.01590
$996$	6.81006e98	0.0246299
$997$	−4.04193e100	−1.41352	−0.706758	−	0.707455i	$-0.749843\pi$
$997$	−4.04193e100	−1.41352	−0.706758	+	0.707455i	$0.749843\pi$
$998$	1.01194e100	0.342201
$999$	5.15657e99	0.168623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2.68.a.b.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.68.a.b.1.3	✓	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2 \)
Weight:	\( k \)	\(=\)	\( 68 \)
Character orbit:	\([\chi]\)	\(=\)	2.a (trivial)

\(f(q)\)	\(=\)	\(q+8.58993e9 q^{2} +1.33494e16 q^{3} +7.37870e19 q^{4} -2.84057e23 q^{5} +1.14671e26 q^{6} -2.26934e28 q^{7} +6.33825e29 q^{8} +8.54976e31 q^{9} -2.44003e33 q^{10} -3.52937e34 q^{11} +9.85014e35 q^{12} +3.76926e37 q^{13} -1.94935e38 q^{14} -3.79199e39 q^{15} +5.44452e39 q^{16} -1.54564e41 q^{17} +7.34419e41 q^{18} -7.97842e42 q^{19} -2.09597e43 q^{20} -3.02944e44 q^{21} -3.03171e44 q^{22} -3.75304e45 q^{23} +8.46120e45 q^{24} +1.29255e46 q^{25} +3.23777e47 q^{26} -9.62738e46 q^{27} -1.67448e48 q^{28} -1.35341e49 q^{29} -3.25730e49 q^{30} -1.26339e49 q^{31} +4.67681e49 q^{32} -4.71150e50 q^{33} -1.32769e51 q^{34} +6.44620e51 q^{35} +6.30861e51 q^{36} -5.35615e52 q^{37} -6.85341e52 q^{38} +5.03174e53 q^{39} -1.80042e53 q^{40} +1.77137e54 q^{41} -2.60227e54 q^{42} -3.22312e54 q^{43} -2.60422e54 q^{44} -2.42862e55 q^{45} -3.22383e55 q^{46} +1.27712e56 q^{47} +7.26812e55 q^{48} +9.66117e55 q^{49} +1.11029e56 q^{50} -2.06333e57 q^{51} +2.78122e57 q^{52} -8.62187e57 q^{53} -8.26986e56 q^{54} +1.00254e58 q^{55} -1.43836e58 q^{56} -1.06507e59 q^{57} -1.16257e59 q^{58} -3.09921e58 q^{59} -2.79800e59 q^{60} -1.04651e59 q^{61} -1.08525e59 q^{62} -1.94023e60 q^{63} +4.01735e59 q^{64} -1.07068e61 q^{65} -4.04715e60 q^{66} -1.57101e61 q^{67} -1.14048e61 q^{68} -5.01009e61 q^{69} +5.53725e61 q^{70} +1.41255e62 q^{71} +5.41906e61 q^{72} +2.77405e62 q^{73} -4.60090e62 q^{74} +1.72548e62 q^{75} -5.88703e62 q^{76} +8.00933e62 q^{77} +4.32223e63 q^{78} +5.79087e63 q^{79} -1.54655e63 q^{80} -9.21164e63 q^{81} +1.52160e64 q^{82} +6.91367e62 q^{83} -2.23533e64 q^{84} +4.39048e64 q^{85} -2.76864e64 q^{86} -1.80672e65 q^{87} -2.23700e64 q^{88} +1.78683e65 q^{89} -2.08617e65 q^{90} -8.55372e65 q^{91} -2.76925e65 q^{92} -1.68656e65 q^{93} +1.09703e66 q^{94} +2.26632e66 q^{95} +6.24327e65 q^{96} +1.29164e66 q^{97} +8.29888e65 q^{98} -3.01753e66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 25769803776 q^{2} - 23\!\cdots\!44 q^{3} + 22\!\cdots\!92 q^{4} + 69\!\cdots\!90 q^{5} - 20\!\cdots\!48 q^{6} - 46\!\cdots\!92 q^{7} + 19\!\cdots\!64 q^{8} + 19\!\cdots\!51 q^{9} + 60\!\cdots\!80 q^{10}+ \cdots + 28\!\cdots\!32 q^{99}+O(q^{100}) \) 3 * q + 25769803776 * q^2 - 2382403442320044 * q^3 + 221360928884514619392 * q^4 + 69995731683175484205690 * q^5 - 20464689741284822690562048 * q^6 - 46930041031821809354401779192 * q^7 + 1901475900342344102245054808064 * q^8 + 191665880989256950329302542685151 * q^9 + 601258756877659476184806174228480 * q^10 - 94192717562712302465603901160680804 * q^11 - 175790346323210028954245448037564416 * q^12 + 9293162032083115751156028464346376386 * q^13 - 403125982863225532953405030547706609664 * q^14 - 2834416443023883562385768010875712279720 * q^15 + 16333553612205046246241981156724874149888 * q^16 - 456291990499287276743121870841967331662442 * q^17 + 1646397381215773458010101702644735139643392 * q^18 + 7094914417641504988662519373648786153621860 * q^19 + 5164773394446325046476466740820399263580160 * q^20 + 90866348206886606668322229671316231529787616 * q^21 - 809109282906428336293257840750280988549971968 * q^22 - 1211030869288683518792500686990840249736656744 * q^23 - 1510027576821401840195394559356413092426678272 * q^24 + 23527039678683417635278279831311248936865016725 * q^25 + 79827654008451769809995232895225577187953344512 * q^26 - 2113356067189354866826552899200490188176129267320 * q^27 - 3462825825130820210214089796088561712619835097088 * q^28 - 2991938816995158477528675338206092994290982287150 * q^29 - 24347451852064454494717698685108313124206008074240 * q^30 - 258768758082699807798920757172611699585086030318304 * q^31 + 140304157183766680147553743940763169886969524125696 * q^32 - 2181728345127367781475611482453366631578858604794608 * q^33 - 3919518353242363129841219656417171347581147402993664 * q^34 + 6685861313241550931298636663340184400338218304813040 * q^35 + 14142445817083583442996432100986108262700723485016064 * q^36 + 6329661235650435153849271583702324985112571136357018 * q^37 + 60944850783378298757052742981575881439807041301381120 * q^38 + 326230234453650802234252471202256216010008198357250872 * q^39 + 44365065640795748204004209370910646093478542148894720 * q^40 + 670194901684473649850829882936279861991784836038573486 * q^41 + 780535987711052435241718991256208087388803721091612672 * q^42 - 6590490969061452320864717359976138421991174495600681284 * q^43 - 6950195817946243065094464582636065897265360128553517056 * q^44 - 64368287466104027746747885379428542717642851620670372670 * q^45 - 10402675956082692992215983721366383048358826655715688448 * q^46 + 142496759244986398051581936313458913857843662485546058928 * q^47 - 12971038117012097073026875764504250079677616912307585024 * q^48 - 200510880128140115686545081532783362338669745164185519941 * q^49 + 202095731983279254162059735469236421961500031200736051200 * q^50 + 1384853383099710830200705854190773108901917972620490983016 * q^51 + 685714326565407317854499318401794497129966519705722159104 * q^52 - 7859044654180946725295267389956714422201615664275787862614 * q^53 - 18153590386762915584716960012960179810770722182015683133440 * q^54 - 48874001022465194932335655798148256348206330106827800478520 * q^55 - 29745447341362175449050721665215362550759884445811889668096 * q^56 - 312938298350217271758997292082690465354471302368807715396880 * q^57 - 25700558741254269302645622959593817436828967262049062092800 * q^58 - 55301513334625281723261507583433559952613870091854753243700 * q^59 - 209143018891102924277785441209844814172384781291145480110080 * q^60 - 1578188533688954960536201064862481464726063214655898847033614 * q^61 - 2222806706383462675763700792303849353249842539127160300371968 * q^62 - 6422990905330033092057765004674396147069390082610736922789464 * q^63 + 1205203533194242706656471569255871951891652245337094626476032 * q^64 - 24660802950023978954037674203067272398035309335453482039372420 * q^65 - 18740903782156491152003651877478575055457757105202060314279936 * q^66 - 26407626812894473557175218263553791675539117425031533908749932 * q^67 - 33668406286495450408848480194128214941378553604026238750425088 * q^68 - 76707923717026372170850229812154221436712384864328794865051488 * q^69 + 57431111371928145996491974556865308244124037916161132388679680 * q^70 + 133722587683066442512737537292042177005944363562526060398772296 * q^71 + 121482684539651977972313512236839808677229968007366281829285888 * q^72 + 599594953405705348016644397964150250082961686563655227588295886 * q^73 + 54371376003755136547902699930847224820444359818253897780166656 * q^74 + 1414205704990897986282727708114357439350697511376733150750330700 * q^75 + 523512281948419547115317960905923882652689269879906380063703040 * q^76 - 1116699559292699246432108790111828590176973601792672760678491744 * q^77 + 2802296375889685246800556189641744498351393641272546839414964224 * q^78 + 15040616041273693784225331917559164800461831418668702451543715280 * q^79 + 381093012024222043904097630988795917419440994814520769894154240 * q^80 + 10930595424114990446638976321107279614269803708106312987766962523 * q^81 + 5756930369361499274166139251341660918316218582908790633725427712 * q^82 + 12547243148793841808170951364938913025058218490201111704595731876 * q^83 + 6704753081140056214208681844535047384611244037303163692278349824 * q^84 + 14457146614409169395963615528990729489113749389068467459410461140 * q^85 - 56611886353404571064754519002761947725642283318468443980198576128 * q^86 - 454772454385715019021323102969817006950999802246232262453483445000 * q^87 - 59701727477750167701295049066104384997711235171599423082596401152 * q^88 - 397456754755248408754478323982148795788241603945122066031068971170 * q^89 - 552919379132887015412317476123604284282430019837919696427564400640 * q^90 - 995804409425932416113481886137438482063280754796390120190765025104 * q^91 - 89358306044521397346352065303473983253019893918463866716569993216 * q^92 + 1173987115649565041691486525937593995182689817613536077256661380992 * q^93 + 1224037841486404463192765075061321679318239446912564791594897637376 * q^94 + 2975480524015582606640572511450516916081877850724245393610141977400 * q^95 - 111420369015472756330055510275201503490441517723155155141638750208 * q^96 - 2195128909500657761105797202787371985980393874830510881776680235162 * q^97 - 1722375345285076172358735424825916186195009263249661917246701699072 * q^98 + 28054853016203263572503445694368856176570112697698383290929190617932 * q^99

Introduction

L-functions

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