LMFDB - Embedded newform 23.25.b.c.22.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [23,25,Mod(22,23)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(23, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 25, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("23.22"); S:= CuspForms(chi, 25); N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 25 $
Character orbit:	$[\chi]$	$=$	23.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$83.9424450193$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.18
Character	$\chi$	$=$	23.22
Dual form			23.25.b.c.22.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2739.52 q^{2} -10954.6 q^{3} -9.27226e6 q^{4} -2.86680e8i q^{5} +3.00102e7 q^{6} +2.46062e10i q^{7} +7.13630e10 q^{8} -2.82310e11 q^{9} +7.85365e11i q^{10} -4.90962e12i q^{11} +1.01573e11 q^{12} -1.92690e13 q^{13} -6.74091e13i q^{14} +3.14045e12i q^{15} -3.99374e13 q^{16} -1.36095e14i q^{17} +7.73392e14 q^{18} +2.47427e15i q^{19} +2.65817e15i q^{20} -2.69550e14i q^{21} +1.34500e16i q^{22} +(1.62284e16 + 1.47271e16i) q^{23} -7.81750e14 q^{24} -2.25809e16 q^{25} +5.27877e16 q^{26} +6.18646e15 q^{27} -2.28155e17i q^{28} -6.01690e17 q^{29} -8.60333e15i q^{30} -5.44628e17 q^{31} -1.08786e18 q^{32} +5.37826e16i q^{33} +3.72836e17i q^{34} +7.05411e18 q^{35} +2.61765e18 q^{36} +1.13458e19i q^{37} -6.77830e18i q^{38} +2.11083e17 q^{39} -2.04584e19i q^{40} -3.48372e19 q^{41} +7.38437e17i q^{42} +8.67301e18i q^{43} +4.55232e19i q^{44} +8.09326e19i q^{45} +(-4.44581e19 - 4.03452e19i) q^{46} +1.32257e20 q^{47} +4.37496e17 q^{48} -4.13884e20 q^{49} +6.18608e19 q^{50} +1.49086e18i q^{51} +1.78667e20 q^{52} -1.82135e20i q^{53} -1.69479e19 q^{54} -1.40749e21 q^{55} +1.75597e21i q^{56} -2.71045e19i q^{57} +1.64834e21 q^{58} -7.47014e20 q^{59} -2.91191e19i q^{60} +7.14079e20i q^{61} +1.49202e21 q^{62} -6.94657e21i q^{63} +3.65026e21 q^{64} +5.52404e21i q^{65} -1.47338e20i q^{66} -4.33668e21i q^{67} +1.26191e21i q^{68} +(-1.77775e20 - 1.61329e20i) q^{69} -1.93249e22 q^{70} +2.99449e22 q^{71} -2.01465e22 q^{72} -1.89373e22 q^{73} -3.10819e22i q^{74} +2.47364e20 q^{75} -2.29421e22i q^{76} +1.20807e23 q^{77} -5.78266e20 q^{78} -1.64671e22i q^{79} +1.14493e22i q^{80} +7.96648e22 q^{81} +9.54371e22 q^{82} +3.52773e22i q^{83} +2.49934e21i q^{84} -3.90159e22 q^{85} -2.37599e22i q^{86} +6.59125e21 q^{87} -3.50365e23i q^{88} +1.37936e23i q^{89} -2.21716e23i q^{90} -4.74136e23i q^{91} +(-1.50474e23 - 1.36554e23i) q^{92} +5.96616e21 q^{93} -3.62320e23 q^{94} +7.09323e23 q^{95} +1.19171e22 q^{96} +2.08536e23i q^{97} +1.13384e24 q^{98} +1.38603e24i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4232 q^{2} - 434562 q^{3} + 317760360 q^{4} - 8460029520 q^{6} - 198307023760 q^{8} + 4220041988298 q^{9} - 67439597688792 q^{12} + 5771152551358 q^{13} + 18\!\cdots\!92 q^{16} + 18\!\cdots\!68 q^{18}+ \cdots - 20\!\cdots\!92 q^{98}+O(q^{100}) $ 44 * q - 4232 * q^2 - 434562 * q^3 + 317760360 * q^4 - 8460029520 * q^6 - 198307023760 * q^8 + 4220041988298 * q^9 - 67439597688792 * q^12 + 5771152551358 * q^13 + 1801550792530192 * q^16 + 1842830558165568 * q^18 - 117204213263210644 * q^23 - 186493005437450784 * q^24 - 689533084157602612 * q^25 - 548650296251047576 * q^26 + 824037505793640630 * q^27 - 1583078054945709602 * q^29 - 77249813729122274 * q^31 - 2838716832733708512 * q^32 + 3418569899175948192 * q^35 + 71181245116739335248 * q^36 + 47106572467817156310 * q^39 - 9150679195666184738 * q^41 - 153049487518536134480 * q^46 + 22735751396127769438 * q^47 - 1317482931964495330032 * q^48 - 1842110772006482246836 * q^49 + 284936617115007894520 * q^50 + 207129175371045152568 * q^52 - 381774025126375304544 * q^54 - 5430040696094013021792 * q^55 - 2168901871328366220760 * q^58 + 16048018791948427185304 * q^59 + 14054450375064171442400 * q^62 + 20310379126091866412192 * q^64 + 24476879859710435470878 * q^69 - 118028399895318845638416 * q^70 - 68204988047209908888578 * q^71 - 106777935582637627887744 * q^72 - 7319375899831086678722 * q^73 + 12913295383070041819998 * q^75 + 111116199573084838704480 * q^77 - 422583593213386075404144 * q^78 - 331350288185757013618680 * q^81 + 1000764218021118649437320 * q^82 - 113385735487939722862944 * q^85 - 248828610584854760185290 * q^87 - 4238455111972177116774696 * q^92 + 4492289396741153284965270 * q^93 - 1602622056539190351258592 * q^94 - 1668692394291208047471936 * q^95 - 12092663288748189077973504 * q^96 - 2067304853876486831513192 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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 $

$2$	−2739.52			−0.668827			−0.334414	−	0.942426i	$-0.608538\pi$
$2$	−2739.52			−0.668827			−0.334414	+	0.942426i	$0.608538\pi$
$3$	−10954.6			−0.0206129			−0.0103065	−	0.999947i	$-0.503281\pi$
$3$	−10954.6			−0.0206129			−0.0103065	+	0.999947i	$0.503281\pi$
$4$	−9.27226e6			−0.552670
$5$		−	2.86680e8i		−	1.17424i	−0.809499	−	0.587121i	$-0.800261\pi$
$5$		−	2.86680e8i		−	1.17424i	0.809499	−	0.587121i	$-0.199739\pi$
$6$	3.00102e7			0.0137865
$7$			2.46062e10i			1.77774i	0.458160	+	0.888870i	$0.348509\pi$
$7$			2.46062e10i			1.77774i	−0.458160	+	0.888870i	$0.651491\pi$
$8$	7.13630e10			1.03847
$9$	−2.82310e11			−0.999575
$10$			7.85365e11i			0.785365i
$11$		−	4.90962e12i		−	1.56435i	−0.623056	−	0.782177i	$-0.714109\pi$
$11$		−	4.90962e12i		−	1.56435i	0.623056	−	0.782177i	$-0.285891\pi$
$12$	1.01573e11			0.0113921
$13$	−1.92690e13			−0.827063			−0.413531	−	0.910490i	$-0.635705\pi$
$13$	−1.92690e13			−0.827063			−0.413531	+	0.910490i	$0.635705\pi$
$14$		−	6.74091e13i		−	1.18900i
$15$			3.14045e12i			0.0242046i
$16$	−3.99374e13			−0.141886
$17$		−	1.36095e14i		−	0.233591i	−0.993156	−	0.116796i	$-0.962738\pi$
$17$		−	1.36095e14i		−	0.233591i	0.993156	−	0.116796i	$-0.0372622\pi$
$18$	7.73392e14			0.668543
$19$			2.47427e15i			1.11790i	0.829201	+	0.558950i	$0.188796\pi$
$19$			2.47427e15i			1.11790i	−0.829201	+	0.558950i	$0.811204\pi$
$20$			2.65817e15i			0.648968i
$21$		−	2.69550e14i		−	0.0366444i
$22$			1.34500e16i			1.04628i
$23$	1.62284e16	+	1.47271e16i	0.740530	+	0.672023i
$23$	1.62284e16	+	1.47271e16i	0.740530	+	0.672023i
$24$	−7.81750e14			−0.0214059
$25$	−2.25809e16			−0.378845
$26$	5.27877e16			0.553162
$27$	6.18646e15			0.0412171
$28$		−	2.28155e17i		−	0.982503i
$29$	−6.01690e17			−1.70058			−0.850290	−	0.526315i	$-0.823573\pi$
$29$	−6.01690e17			−1.70058			−0.850290	+	0.526315i	$0.823573\pi$
$30$		−	8.60333e15i		−	0.0161887i
$31$	−5.44628e17			−0.691448			−0.345724	−	0.938336i	$-0.612367\pi$
$31$	−5.44628e17			−0.691448			−0.345724	+	0.938336i	$0.612367\pi$
$32$	−1.08786e18			−0.943571
$33$			5.37826e16i			0.0322459i
$34$			3.72836e17i			0.156232i
$35$	7.05411e18			2.08750
$36$	2.61765e18			0.552435
$37$			1.13458e19i			1.72350i	0.507329	+	0.861752i	$0.330633\pi$
$37$			1.13458e19i			1.72350i	−0.507329	+	0.861752i	$0.669367\pi$
$38$		−	6.77830e18i		−	0.747683i
$39$	2.11083e17			0.0170482
$40$		−	2.04584e19i		−	1.21941i
$41$	−3.48372e19			−1.54396			−0.771981	−	0.635645i	$-0.780734\pi$
$41$	−3.48372e19			−1.54396			−0.771981	+	0.635645i	$0.780734\pi$
$42$			7.38437e17i			0.0245088i
$43$			8.67301e18i			0.217044i	0.994094	+	0.108522i	$0.0346119\pi$
$43$			8.67301e18i			0.217044i	−0.994094	+	0.108522i	$0.965388\pi$
$44$			4.55232e19i			0.864572i
$45$			8.09326e19i			1.17374i
$46$	−4.44581e19	−	4.03452e19i	−0.495287	−	0.449467i
$47$	1.32257e20			1.13827			0.569133	−	0.822245i	$-0.307279\pi$
$47$	1.32257e20			1.13827			0.569133	+	0.822245i	$0.307279\pi$
$48$	4.37496e17			0.00292469
$49$	−4.13884e20			−2.16036
$50$	6.18608e19			0.253382
$51$			1.49086e18i			0.00481500i
$52$	1.78667e20			0.457093
$53$		−	1.82135e20i		−	0.370752i	−0.982668	−	0.185376i	$-0.940650\pi$
$53$		−	1.82135e20i		−	0.370752i	0.982668	−	0.185376i	$-0.0593503\pi$
$54$	−1.69479e19			−0.0275671
$55$	−1.40749e21			−1.83693
$56$			1.75597e21i			1.84613i
$57$		−	2.71045e19i		−	0.0230432i
$58$	1.64834e21			1.13739
$59$	−7.47014e20			−0.419860			−0.209930	−	0.977716i	$-0.567324\pi$
$59$	−7.47014e20			−0.419860			−0.209930	+	0.977716i	$0.567324\pi$
$60$		−	2.91191e19i		−	0.0133771i
$61$			7.14079e20i			0.269022i	0.990912	+	0.134511i	$0.0429464\pi$
$61$			7.14079e20i			0.269022i	−0.990912	+	0.134511i	$0.957054\pi$
$62$	1.49202e21			0.462460
$63$		−	6.94657e21i		−	1.77698i
$64$	3.65026e21			0.772972
$65$			5.52404e21i			0.971172i
$66$		−	1.47338e20i		−	0.0215670i
$67$		−	4.33668e21i		−	0.529981i	−0.964251	−	0.264990i	$-0.914631\pi$
$67$		−	4.33668e21i		−	0.529981i	0.964251	−	0.264990i	$-0.0853688\pi$
$68$			1.26191e21i			0.129099i
$69$	−1.77775e20	−	1.61329e20i	−0.0152645	−	0.0138524i
$70$	−1.93249e22			−1.39617
$71$	2.99449e22			1.82483			0.912415	−	0.409265i	$-0.134215\pi$
$71$	2.99449e22			1.82483			0.912415	+	0.409265i	$0.134215\pi$
$72$	−2.01465e22			−1.03803
$73$	−1.89373e22			−0.826881			−0.413440	−	0.910531i	$-0.635673\pi$
$73$	−1.89373e22			−0.826881			−0.413440	+	0.910531i	$0.635673\pi$
$74$		−	3.10819e22i		−	1.15273i
$75$	2.47364e20			0.00780910
$76$		−	2.29421e22i		−	0.617830i
$77$	1.20807e23			2.78101
$78$	−5.78266e20			−0.0114023
$79$		−	1.64671e22i		−	0.278671i	−0.990245	−	0.139335i	$-0.955503\pi$
$79$		−	1.64671e22i		−	0.278671i	0.990245	−	0.139335i	$-0.0444966\pi$
$80$			1.14493e22i			0.166609i
$81$	7.96648e22			0.998726
$82$	9.54371e22			1.03264
$83$			3.52773e22i			0.330033i	0.986291	+	0.165017i	$0.0527678\pi$
$83$			3.52773e22i			0.330033i	−0.986291	+	0.165017i	$0.947232\pi$
$84$			2.49934e21i			0.0202523i
$85$	−3.90159e22			−0.274293
$86$		−	2.37599e22i		−	0.145165i
$87$	6.59125e21			0.0350539
$88$		−	3.50365e23i		−	1.62453i
$89$			1.37936e23i			0.558469i	0.960223	+	0.279235i	$0.0900807\pi$
$89$			1.37936e23i			0.558469i	−0.960223	+	0.279235i	$0.909919\pi$
$90$		−	2.21716e23i		−	0.785032i
$91$		−	4.74136e23i		−	1.47030i
$92$	−1.50474e23	−	1.36554e23i	−0.409269	−	0.371407i
$93$	5.96616e21			0.0142528
$94$	−3.62320e23			−0.761304
$95$	7.09323e23			1.31269
$96$	1.19171e22			0.0194498
$97$			2.08536e23i			0.300553i	0.988644	+	0.150277i	$0.0480164\pi$
$97$			2.08536e23i			0.300553i	−0.988644	+	0.150277i	$0.951984\pi$
$98$	1.13384e24			1.44491
$99$			1.38603e24i			1.56369i
$100$	2.09376e23			0.209376
$101$	1.19453e24			1.06008			0.530040	−	0.847972i	$-0.322177\pi$
$101$	1.19453e24			1.06008			0.530040	+	0.847972i	$0.322177\pi$
$102$		−	4.08425e21i		−	0.00322040i
$103$		−	3.59869e23i		−	0.252405i	−0.992004	−	0.126203i	$-0.959721\pi$
$103$		−	3.59869e23i		−	0.252405i	0.992004	−	0.126203i	$-0.0402789\pi$
$104$	−1.37509e24			−0.858878
$105$	−7.72746e22			−0.0430294
$106$			4.98962e23i			0.247969i
$107$		−	3.54957e24i		−	1.57605i	−0.615643	−	0.788026i	$-0.711103\pi$
$107$		−	3.54957e24i		−	1.57605i	0.615643	−	0.788026i	$-0.288897\pi$
$108$	−5.73625e22			−0.0227794
$109$		−	5.17485e24i		−	1.83984i	−0.392105	−	0.919920i	$-0.628253\pi$
$109$		−	5.17485e24i		−	1.83984i	0.392105	−	0.919920i	$-0.371747\pi$
$110$	3.85584e24			1.22859
$111$		−	1.24288e23i		−	0.0355265i
$112$		−	9.82707e23i		−	0.252236i
$113$		−	1.15328e24i		−	0.266069i	−0.991111	−	0.133035i	$-0.957528\pi$
$113$		−	1.15328e24i		−	0.266069i	0.991111	−	0.133035i	$-0.0424722\pi$
$114$			7.42532e22i			0.0154119i
$115$	4.22198e24	−	4.65237e24i	0.789118	−	0.869562i
$116$	5.57903e24			0.939859
$117$	5.43982e24			0.826711
$118$	2.04646e24			0.280814
$119$	3.34879e24			0.415264
$120$			2.24112e23i			0.0251357i
$121$	−1.42546e25			−1.44721
$122$		−	1.95623e24i		−	0.179929i
$123$	3.81626e23			0.0318256
$124$	5.04994e24			0.382143
$125$		−	1.06140e25i		−	0.729387i
$126$			1.90302e25i			1.18850i
$127$	1.21224e25			0.688562			0.344281	−	0.938867i	$-0.388123\pi$
$127$	1.21224e25			0.688562			0.344281	+	0.938867i	$0.388123\pi$
$128$	8.25137e24			0.426586
$129$		−	9.50090e22i		−	0.00447392i
$130$		−	1.51332e25i		−	0.649547i
$131$	2.80894e25			1.09973			0.549867	−	0.835252i	$-0.314678\pi$
$131$	2.80894e25			1.09973			0.549867	+	0.835252i	$0.314678\pi$
$132$		−	4.98687e23i		−	0.0178214i
$133$	−6.08823e25			−1.98734
$134$			1.18804e25i			0.354466i
$135$		−	1.77354e24i		−	0.0483988i
$136$		−	9.71218e24i		−	0.242577i
$137$		−	7.52786e24i		−	0.172197i	−0.996287	−	0.0860983i	$-0.972560\pi$
$137$		−	7.52786e24i		−	0.172197i	0.996287	−	0.0860983i	$-0.0274399\pi$
$138$	4.87019e23	+	4.41964e23i	0.0102093	+	0.00926483i
$139$	2.16977e24			0.0417095			0.0208548	−	0.999783i	$-0.493361\pi$
$139$	2.16977e24			0.0417095			0.0208548	+	0.999783i	$0.493361\pi$
$140$	−6.54076e25			−1.15370
$141$	−1.44882e24			−0.0234630
$142$	−8.20345e25			−1.22050
$143$			9.46033e25i			1.29382i
$144$	1.12747e25			0.141826
$145$			1.72493e26i			1.99689i
$146$	5.18790e25			0.553041
$147$	4.53391e24			0.0445313
$148$		−	1.05201e26i		−	0.952529i
$149$		−	1.34910e25i		−	0.112671i	−0.998412	−	0.0563353i	$-0.982058\pi$
$149$		−	1.34910e25i		−	0.112671i	0.998412	−	0.0563353i	$-0.0179416\pi$
$150$	−6.77657e23			−0.00522294
$151$	−1.07201e26			−0.762914			−0.381457	−	0.924387i	$-0.624578\pi$
$151$	−1.07201e26			−0.762914			−0.381457	+	0.924387i	$0.624578\pi$
$152$			1.76571e26i			1.16090i
$153$			3.84210e25i			0.233492i
$154$	−3.30953e26			−1.86002
$155$			1.56134e26i			0.811928i
$156$	−1.95722e24			−0.00942202
$157$		−	3.67688e26i		−	1.63940i	−0.572796	−	0.819698i	$-0.694141\pi$
$157$		−	3.67688e26i		−	1.63940i	0.572796	−	0.819698i	$-0.305859\pi$
$158$			4.51118e25i			0.186383i
$159$			1.99521e24i			0.00764228i
$160$			3.11869e26i			1.10798i
$161$	−3.62379e26	+	3.99320e26i	−1.19468	+	1.31647i
$162$	−2.18243e26			−0.667975
$163$	3.20281e26			0.910499			0.455250	−	0.890364i	$-0.349550\pi$
$163$	3.20281e26			0.910499			0.455250	+	0.890364i	$0.349550\pi$
$164$	3.23020e26			0.853302
$165$	1.54184e25			0.0378645
$166$		−	9.66427e25i		−	0.220735i
$167$	1.29054e26			0.274268			0.137134	−	0.990553i	$-0.456211\pi$
$167$	1.29054e26			0.274268			0.137134	+	0.990553i	$0.456211\pi$
$168$		−	1.92359e25i		−	0.0380540i
$169$	−1.71507e26			−0.315967
$170$	1.06885e26			0.183454
$171$		−	6.98509e26i		−	1.11743i
$172$		−	8.04185e25i		−	0.119954i
$173$	−2.62009e26			−0.364555			−0.182278	−	0.983247i	$-0.558347\pi$
$173$	−2.62009e26			−0.364555			−0.182278	+	0.983247i	$0.558347\pi$
$174$	−1.80568e25			−0.0234450
$175$		−	5.55631e26i		−	0.673488i
$176$			1.96077e26i			0.221960i
$177$	8.18320e24			0.00865454
$178$		−	3.77879e26i		−	0.373519i
$179$	4.90735e26			0.453534			0.226767	−	0.973949i	$-0.427184\pi$
$179$	4.90735e26			0.453534			0.226767	+	0.973949i	$0.427184\pi$
$180$		−	7.50428e26i		−	0.648693i
$181$		−	7.46775e26i		−	0.604014i	−0.953306	−	0.302007i	$-0.902343\pi$
$181$		−	7.46775e26i		−	0.604014i	0.953306	−	0.302007i	$-0.0976566\pi$
$182$			1.29890e27i			0.983378i
$183$		−	7.82241e24i		−	0.00554533i
$184$	1.15811e27	+	1.05097e27i	0.769017	+	0.697874i
$185$	3.25260e27			2.02381
$186$	−1.63444e25			−0.00953265
$187$	−6.68176e26			−0.365420
$188$	−1.22632e27			−0.629086
$189$			1.52225e26i			0.0732732i
$190$	−1.94320e27			−0.877961
$191$			9.55778e26i			0.405468i	0.979234	+	0.202734i	$0.0649827\pi$
$191$			9.55778e26i			0.405468i	−0.979234	+	0.202734i	$0.935017\pi$
$192$	−3.99869e25			−0.0159332
$193$	9.47524e26			0.354734			0.177367	−	0.984145i	$-0.443242\pi$
$193$	9.47524e26			0.354734			0.177367	+	0.984145i	$0.443242\pi$
$194$		−	5.71289e26i		−	0.201018i
$195$		−	6.05133e25i		−	0.0200187i
$196$	3.83764e27			1.19396
$197$	−6.73351e26			−0.197082			−0.0985410	−	0.995133i	$-0.531418\pi$
$197$	−6.73351e26			−0.197082			−0.0985410	+	0.995133i	$0.531418\pi$
$198$		−	3.79706e27i		−	1.04584i
$199$		−	9.68492e26i		−	0.251107i	−0.992087	−	0.125554i	$-0.959929\pi$
$199$		−	9.68492e26i		−	0.251107i	0.992087	−	0.125554i	$-0.0400707\pi$
$200$	−1.61144e27			−0.393418
$201$			4.75064e25i			0.0109245i
$202$	−3.27242e27			−0.709011
$203$		−	1.48053e28i		−	3.02319i
$204$		−	1.38237e25i		−	0.00266110i
$205$			9.98713e27i			1.81299i
$206$			9.85868e26i			0.168815i
$207$	−4.58145e27	−	4.15761e27i	−0.740216	−	0.671737i
$208$	7.69552e26			0.117349
$209$	1.21477e28			1.74879
$210$	2.11695e26			0.0287792
$211$	1.13991e28			1.46380			0.731900	−	0.681412i	$-0.238634\pi$
$211$	1.13991e28			1.46380			0.731900	+	0.681412i	$0.238634\pi$
$212$			1.68880e27i			0.204903i
$213$	−3.28033e26			−0.0376151
$214$			9.72410e27i			1.05411i
$215$	2.48638e27			0.254863
$216$	4.41485e26			0.0428026
$217$		−	1.34012e28i		−	1.22922i
$218$			1.41766e28i			1.23054i
$219$	2.07449e26			0.0170444
$220$	1.30506e28			1.01522
$221$			2.62242e27i			0.193195i
$222$			3.40488e26i			0.0237611i
$223$	−2.58963e28			−1.71230			−0.856149	−	0.516728i	$-0.827150\pi$
$223$	−2.58963e28			−1.71230			−0.856149	+	0.516728i	$0.827150\pi$
$224$		−	2.67682e28i		−	1.67742i
$225$	6.37481e27			0.378684
$226$			3.15944e27i			0.177954i
$227$		−	1.45554e28i		−	0.777522i	−0.921339	−	0.388761i	$-0.872903\pi$
$227$		−	1.45554e28i		−	0.777522i	0.921339	−	0.388761i	$-0.127097\pi$
$228$			2.51320e26i			0.0127353i
$229$			2.46317e28i			1.18432i	0.805820	+	0.592160i	$0.201725\pi$
$229$			2.46317e28i			1.18432i	−0.805820	+	0.592160i	$0.798275\pi$
$230$	−1.15662e28	+	1.27453e28i	−0.527783	+	0.581587i
$231$	−1.32339e27			−0.0573248
$232$	−4.29384e28			−1.76600
$233$	2.20706e28			0.862069			0.431035	−	0.902335i	$-0.358149\pi$
$233$	2.20706e28			0.862069			0.431035	+	0.902335i	$0.358149\pi$
$234$	−1.49025e28			−0.552927
$235$		−	3.79154e28i		−	1.33660i
$236$	6.92651e27			0.232044
$237$			1.80389e26i			0.00574422i
$238$	−9.17407e27			−0.277740
$239$	−3.39023e28			−0.976011			−0.488006	−	0.872841i	$-0.662275\pi$
$239$	−3.39023e28			−0.976011			−0.488006	+	0.872841i	$0.662275\pi$
$240$		−	1.25421e26i		−	0.00343429i
$241$			7.00766e28i			1.82544i	0.408582	+	0.912722i	$0.366023\pi$
$241$			7.00766e28i			1.82544i	−0.408582	+	0.912722i	$0.633977\pi$
$242$	3.90507e28			0.967931
$243$	−2.61993e27			−0.0618037
$244$		−	6.62113e27i		−	0.148680i
$245$			1.18652e29i			2.53678i
$246$	−1.04547e27			−0.0212858
$247$		−	4.76766e28i		−	0.924574i
$248$	−3.88663e28			−0.718047
$249$		−	3.86447e26i		−	0.00680295i
$250$			2.90772e28i			0.487834i
$251$			6.94667e28i			1.11094i	0.831535	+	0.555472i	$0.187462\pi$
$251$			6.94667e28i			1.11094i	−0.831535	+	0.555472i	$0.812538\pi$
$252$			6.44104e28i			0.982086i
$253$	7.23045e28	−	7.96754e28i	1.05128	−	1.15845i
$254$	−3.32094e28			−0.460529
$255$	4.27401e26			0.00565397
$256$	−8.38459e28			−1.05828
$257$	2.52176e28			0.303743			0.151871	−	0.988400i	$-0.451470\pi$
$257$	2.52176e28			0.303743			0.151871	+	0.988400i	$0.451470\pi$
$258$			2.60279e26i			0.00299228i
$259$	−2.79176e29			−3.06394
$260$		−	5.12203e28i		−	0.536738i
$261$	1.69863e29			1.69986
$262$	−7.69515e28			−0.735532
$263$		−	1.00843e29i		−	0.920826i	−0.887705	−	0.460413i	$-0.847701\pi$
$263$		−	1.00843e29i		−	0.920826i	0.887705	−	0.460413i	$-0.152299\pi$
$264$			3.83809e27i			0.0334864i
$265$	−5.22145e28			−0.435352
$266$	1.66788e29			1.32918
$267$		−	1.51103e27i		−	0.0115117i
$268$			4.02109e28i			0.292904i
$269$	−2.42276e29			−1.68765			−0.843825	−	0.536618i	$-0.819702\pi$
$269$	−2.42276e29			−1.68765			−0.843825	+	0.536618i	$0.819702\pi$
$270$			4.85863e27i			0.0323705i
$271$	1.13062e29			0.720587			0.360294	−	0.932839i	$-0.382677\pi$
$271$	1.13062e29			0.720587			0.360294	+	0.932839i	$0.382677\pi$
$272$			5.43529e27i			0.0331433i
$273$			5.19395e27i			0.0303072i
$274$			2.06227e28i			0.115170i
$275$			1.10864e29i			0.592648i
$276$	1.64838e27	+	1.49589e27i	0.00843623	+	0.00765578i
$277$	2.45377e29			1.20247			0.601237	−	0.799071i	$-0.294675\pi$
$277$	2.45377e29			1.20247			0.601237	+	0.799071i	$0.294675\pi$
$278$	−5.94411e27			−0.0278965
$279$	1.53754e29			0.691155
$280$	5.03403e29			2.16780
$281$		−	1.82027e29i		−	0.751034i	−0.926815	−	0.375517i	$-0.877465\pi$
$281$		−	1.82027e29i		−	0.751034i	0.926815	−	0.375517i	$-0.122535\pi$
$282$	3.96905e27			0.0156927
$283$			1.73983e29i			0.659278i	0.944107	+	0.329639i	$0.106927\pi$
$283$			1.73983e29i			0.659278i	−0.944107	+	0.329639i	$0.893073\pi$
$284$	−2.77657e29			−1.00853
$285$	−7.77032e27			−0.0270583
$286$		−	2.59167e29i		−	0.865342i
$287$		−	8.57211e29i		−	2.74476i
$288$	3.07114e29			0.943170
$289$	3.20927e29			0.945435
$290$		−	4.72547e29i		−	1.33558i
$291$		−	2.28442e27i		−	0.00619528i
$292$	1.75591e29			0.456992
$293$		−	1.41977e29i		−	0.354655i	−0.984152	−	0.177327i	$-0.943255\pi$
$293$		−	1.41977e29i		−	0.354655i	0.984152	−	0.177327i	$-0.0567451\pi$
$294$	−1.24207e28			−0.0297837
$295$			2.14154e29i			0.493017i
$296$			8.09667e29i			1.78980i
$297$		−	3.03732e28i		−	0.0644781i
$298$			3.69590e28i			0.0753572i
$299$	−3.12706e29	−	2.83777e29i	−0.612465	−	0.555805i
$300$	−2.29362e27			−0.00431586
$301$	−2.13410e29			−0.385848
$302$	2.93679e29			0.510258
$303$	−1.30855e28			−0.0218514
$304$		−	9.88157e28i		−	0.158614i
$305$	2.04712e29			0.315897
$306$		−	1.05255e29i		−	0.156166i
$307$	4.80574e29			0.685646			0.342823	−	0.939400i	$-0.388617\pi$
$307$	4.80574e29			0.685646			0.342823	+	0.939400i	$0.388617\pi$
$308$	−1.12015e30			−1.53698
$309$			3.94221e27i			0.00520281i
$310$		−	4.27732e29i		−	0.543040i
$311$	7.32540e29			0.894760			0.447380	−	0.894344i	$-0.352357\pi$
$311$	7.32540e29			0.894760			0.447380	+	0.894344i	$0.352357\pi$
$312$	1.50635e28			0.0177040
$313$		−	4.58754e29i		−	0.518857i	−0.965762	−	0.259428i	$-0.916466\pi$
$313$		−	4.58754e29i		−	0.518857i	0.965762	−	0.259428i	$-0.0835341\pi$
$314$			1.00729e30i			1.09647i
$315$	−1.99144e30			−2.08661
$316$			1.52687e29i			0.154013i
$317$	9.85269e29			0.956849			0.478425	−	0.878129i	$-0.341208\pi$
$317$	9.85269e29			0.956849			0.478425	+	0.878129i	$0.341208\pi$
$318$		−	5.46591e27i		−	0.00511137i
$319$			2.95407e30i			2.66031i
$320$		−	1.04646e30i		−	0.907656i
$321$			3.88839e28i			0.0324870i
$322$	9.92743e29	−	1.09395e30i	0.799036	−	0.880491i
$323$	3.36736e29			0.261132
$324$	−7.38673e29			−0.551966
$325$	4.35111e29			0.313329
$326$	−8.77415e29			−0.608967
$327$			5.66882e28i			0.0379245i
$328$	−2.48609e30			−1.60336
$329$			3.25434e30i			2.02354i
$330$	−4.22390e28			−0.0253248
$331$	3.09442e29			0.178913			0.0894567	−	0.995991i	$-0.471487\pi$
$331$	3.09442e29			0.178913			0.0894567	+	0.995991i	$0.471487\pi$
$332$		−	3.27100e29i		−	0.182399i
$333$		−	3.20301e30i		−	1.72277i
$334$	−3.53547e29			−0.183438
$335$	−1.24324e30			−0.622326
$336$			1.07651e28i			0.00519933i
$337$			2.17132e30i			1.01196i	0.862544	+	0.505982i	$0.168870\pi$
$337$			2.17132e30i			1.01196i	−0.862544	+	0.505982i	$0.831130\pi$
$338$	4.69847e29			0.211327
$339$			1.26337e28i			0.00548447i
$340$	3.61765e29			0.151593
$341$			2.67392e30i			1.08167i
$342$			1.91358e30i			0.747365i
$343$		−	5.47003e30i		−	2.06281i
$344$			6.18932e29i			0.225394i
$345$	−4.62499e28	+	5.09647e28i	−0.0162660	+	0.0179242i
$346$	7.17778e29			0.243824
$347$	3.45573e30			1.13393			0.566964	−	0.823742i	$-0.308118\pi$
$347$	3.45573e30			1.13393			0.566964	+	0.823742i	$0.308118\pi$
$348$	−6.11158e28			−0.0193732
$349$	−1.82620e30			−0.559297			−0.279649	−	0.960102i	$-0.590218\pi$
$349$	−1.82620e30			−0.559297			−0.279649	+	0.960102i	$0.590218\pi$
$350$			1.52216e30i			0.450447i
$351$	−1.19207e29			−0.0340891
$352$			5.34099e30i			1.47608i
$353$	−6.92669e30			−1.85025			−0.925123	−	0.379667i	$-0.876038\pi$
$353$	−6.92669e30			−1.85025			−0.925123	+	0.379667i	$0.876038\pi$
$354$	−2.24180e28			−0.00578840
$355$		−	8.58461e30i		−	2.14279i
$356$		−	1.27898e30i		−	0.308649i
$357$	−3.66845e28			−0.00855981
$358$	−1.34438e30			−0.303336
$359$		−	5.71296e30i		−	1.24660i	−0.781982	−	0.623301i	$-0.785791\pi$
$359$		−	5.71296e30i		−	1.24660i	0.781982	−	0.623301i	$-0.214209\pi$
$360$			5.77559e30i			1.21890i
$361$	−1.22323e30			−0.249702
$362$			2.04580e30i			0.403981i
$363$	1.56153e29			0.0298311
$364$			4.39632e30i			0.812592i
$365$			5.42894e30i			0.970958i
$366$			2.14296e28i			0.00370887i
$367$			3.15787e30i			0.528934i	0.964395	+	0.264467i	$0.0851960\pi$
$367$			3.15787e30i			0.528934i	−0.964395	+	0.264467i	$0.914804\pi$
$368$	−6.48121e29	−	5.88163e29i	−0.105071	−	0.0953506i
$369$	9.83487e30			1.54331
$370$	−8.91056e30			−1.35358
$371$	4.48165e30			0.659100
$372$	−5.53198e28			−0.00787708
$373$		−	6.48513e30i		−	0.894154i	−0.894495	−	0.447077i	$-0.852465\pi$
$373$		−	6.48513e30i		−	0.894154i	0.894495	−	0.447077i	$-0.147535\pi$
$374$	1.83048e30			0.244403
$375$			1.16271e29i			0.0150348i
$376$	9.43825e30			1.18205
$377$	1.15940e31			1.40649
$378$		−	4.17024e29i		−	0.0490072i
$379$		−	2.92287e30i		−	0.332766i	−0.986061	−	0.166383i	$-0.946791\pi$
$379$		−	2.92287e30i		−	0.332766i	0.986061	−	0.166383i	$-0.0532088\pi$
$380$	−6.57703e30			−0.725482
$381$	−1.32795e29			−0.0141933
$382$		−	2.61837e30i		−	0.271188i
$383$			1.39480e31i			1.40000i	0.714143	+	0.700000i	$0.246817\pi$
$383$			1.39480e31i			1.40000i	−0.714143	+	0.700000i	$0.753183\pi$
$384$	−9.03901e28			−0.00879319
$385$		−	3.46330e31i		−	3.26559i
$386$	−2.59576e30			−0.237256
$387$		−	2.44847e30i		−	0.216952i
$388$		−	1.93360e30i		−	0.166107i
$389$			5.19165e30i			0.432424i	0.976346	+	0.216212i	$0.0693703\pi$
$389$			5.19165e30i			0.432424i	−0.976346	+	0.216212i	$0.930630\pi$
$390$			1.65777e29i			0.0133891i
$391$	2.00429e30	−	2.20862e30i	0.156979	−	0.172981i
$392$	−2.95360e31			−2.24346
$393$	−3.07707e29			−0.0226687
$394$	1.84466e30			0.131814
$395$	−4.72078e30			−0.327227
$396$		−	1.28516e31i		−	0.864204i
$397$	1.80049e31			1.17464			0.587321	−	0.809354i	$-0.300183\pi$
$397$	1.80049e31			1.17464			0.587321	+	0.809354i	$0.300183\pi$
$398$			2.65320e30i			0.167947i
$399$	6.66938e29			0.0409648
$400$	9.01822e29			0.0537528
$401$			2.42839e31i			1.40471i	0.711828	+	0.702354i	$0.247867\pi$
$401$			2.42839e31i			1.40471i	−0.711828	+	0.702354i	$0.752133\pi$
$402$		−	1.30145e29i		−	0.00730657i
$403$	1.04944e31			0.571871
$404$	−1.10760e31			−0.585875
$405$		−	2.28383e31i		−	1.17275i
$406$			4.05594e31i			2.02199i
$407$	5.57033e31			2.69617
$408$			1.06393e29i			0.00500022i
$409$	2.58216e31			1.17843			0.589215	−	0.807976i	$-0.299437\pi$
$409$	2.58216e31			1.17843			0.589215	+	0.807976i	$0.299437\pi$
$410$		−	2.73599e31i		−	1.21257i
$411$			8.24643e28i			0.00354948i
$412$			3.33680e30i			0.139497i
$413$		−	1.83812e31i		−	0.746402i
$414$	1.25509e31	+	1.13898e31i	0.495077	+	0.449276i
$415$	1.01133e31			0.387539
$416$	2.09620e31			0.780392
$417$	−2.37688e28			−0.000859756
$418$	−3.32788e31			−1.16964
$419$		−	5.03109e31i		−	1.71828i	−0.511741	−	0.859139i	$-0.670999\pi$
$419$		−	5.03109e31i		−	1.71828i	0.511741	−	0.859139i	$-0.329001\pi$
$420$	7.16511e29			0.0237811
$421$			2.14517e31i			0.691953i	0.938243	+	0.345976i	$0.112452\pi$
$421$			2.14517e31i			0.691953i	−0.938243	+	0.345976i	$0.887548\pi$
$422$	−3.12281e31			−0.979029
$423$	−3.73374e31			−1.13778
$424$		−	1.29977e31i		−	0.385014i
$425$			3.07316e30i			0.0884948i
$426$	8.98652e29			0.0251580
$427$	−1.75708e31			−0.478251
$428$			3.29125e31i			0.871036i
$429$		−	1.03634e30i		−	0.0266694i
$430$	−6.81148e30			−0.170459
$431$		−	3.72287e31i		−	0.906046i	−0.891499	−	0.453023i	$-0.850346\pi$
$431$		−	3.72287e31i		−	0.906046i	0.891499	−	0.453023i	$-0.149654\pi$
$432$	−2.47071e29			−0.00584813
$433$		−	7.72335e31i		−	1.77808i	−0.457831	−	0.889039i	$-0.651373\pi$
$433$		−	7.72335e31i		−	1.77808i	0.457831	−	0.889039i	$-0.348627\pi$
$434$			3.67129e31i			0.822133i
$435$		−	1.88958e30i		−	0.0411618i
$436$			4.79826e31i			1.01682i
$437$	−3.64388e31	+	4.01535e31i	−0.751255	+	0.827840i
$438$	−5.68311e29			−0.0113998
$439$	−2.70823e31			−0.528580			−0.264290	−	0.964443i	$-0.585138\pi$
$439$	−2.70823e31			−0.528580			−0.264290	+	0.964443i	$0.585138\pi$
$440$	−1.00443e32			−1.90759
$441$	1.16843e32			2.15944
$442$		−	7.18416e30i		−	0.129214i
$443$	1.86220e31			0.325974			0.162987	−	0.986628i	$-0.447887\pi$
$443$	1.86220e31			0.325974			0.162987	+	0.986628i	$0.447887\pi$
$444$			1.15243e30i			0.0196344i
$445$	3.95437e31			0.655778
$446$	7.09434e31			1.14523
$447$			1.47788e29i			0.00232247i
$448$			8.98190e31i			1.37414i
$449$	−7.82638e31			−1.16575			−0.582874	−	0.812563i	$-0.698072\pi$
$449$	−7.82638e31			−1.16575			−0.582874	+	0.812563i	$0.698072\pi$
$450$	−1.74639e31			−0.253274
$451$			1.71037e32i			2.41530i
$452$			1.06936e31i			0.147049i
$453$	1.17434e30			0.0157259
$454$			3.98747e31i			0.520028i
$455$	−1.35926e32			−1.72649
$456$		−	1.93426e30i		−	0.0239296i
$457$		−	7.58384e30i		−	0.0913892i	−0.998955	−	0.0456946i	$-0.985450\pi$
$457$		−	7.58384e30i		−	0.0913892i	0.998955	−	0.0456946i	$-0.0145501\pi$
$458$		−	6.74791e31i		−	0.792106i
$459$		−	8.41950e29i		−	0.00962795i
$460$	−3.91473e31	+	4.31380e31i	−0.436122	+	0.480581i
$461$	−1.54501e32			−1.67695			−0.838473	−	0.544944i	$-0.816551\pi$
$461$	−1.54501e32			−1.67695			−0.838473	+	0.544944i	$0.816551\pi$
$462$	3.62544e30			0.0383404
$463$	1.86902e31			0.192593			0.0962966	−	0.995353i	$-0.469300\pi$
$463$	1.86902e31			0.192593			0.0962966	+	0.995353i	$0.469300\pi$
$464$	2.40299e31			0.241288
$465$		−	1.71038e30i		−	0.0167362i
$466$	−6.04627e31			−0.576576
$467$		−	1.41735e31i		−	0.131727i	−0.997829	−	0.0658635i	$-0.979020\pi$
$467$		−	1.41735e31i		−	0.131727i	0.997829	−	0.0658635i	$-0.0209802\pi$
$468$	−5.04394e31			−0.456899
$469$	1.06709e32			0.942168
$470$			1.03870e32i			0.893955i
$471$			4.02786e30i			0.0337927i
$472$	−5.33091e31			−0.436011
$473$	4.25812e31			0.339534
$474$		−	4.94180e29i		−	0.00384189i
$475$		−	5.58712e31i		−	0.423511i
$476$	−3.10509e31			−0.229504
$477$			5.14185e31i			0.370594i
$478$	9.28760e31			0.652783
$479$		−	1.66052e32i		−	1.13820i	−0.822268	−	0.569101i	$-0.807291\pi$
$479$		−	1.66052e32i		−	1.13820i	0.822268	−	0.569101i	$-0.192709\pi$
$480$		−	3.41638e30i		−	0.0228387i
$481$		−	2.18621e32i		−	1.42545i
$482$		−	1.91976e32i		−	1.22091i
$483$	3.96970e30	−	4.37438e30i	0.0246259	−	0.0271363i
$484$	1.32172e32			0.799827
$485$	5.97833e31			0.352922
$486$	7.17735e30			0.0413360
$487$	−1.23348e32			−0.693081			−0.346541	−	0.938035i	$-0.612644\pi$
$487$	−1.23348e32			−0.693081			−0.346541	+	0.938035i	$0.612644\pi$
$488$			5.09588e31i			0.279371i
$489$	−3.50853e30			−0.0187681
$490$		−	3.25050e32i		−	1.69667i
$491$	3.00997e32			1.53315			0.766573	−	0.642157i	$-0.221960\pi$
$491$	3.00997e32			1.53315			0.766573	+	0.642157i	$0.221960\pi$
$492$	−3.53853e30			−0.0175890
$493$			8.18873e31i			0.397241i
$494$			1.30611e32i			0.618381i
$495$	3.97348e32			1.83615
$496$	2.17510e31			0.0981069
$497$			7.36830e32i			3.24407i
$498$			1.05868e30i			0.00455000i
$499$	2.27582e32			0.954840			0.477420	−	0.878675i	$-0.341572\pi$
$499$	2.27582e32			0.954840			0.477420	+	0.878675i	$0.341572\pi$
$500$			9.84155e31i			0.403110i
$501$	−1.41373e30			−0.00565347
$502$		−	1.90305e32i		−	0.743029i
$503$			8.02741e31i			0.306027i	0.988224	+	0.153013i	$0.0488978\pi$
$503$			8.02741e31i			0.306027i	−0.988224	+	0.153013i	$0.951102\pi$
$504$		−	4.95728e32i		−	1.84534i
$505$		−	3.42447e32i		−	1.24479i
$506$	−1.98079e32	+	2.18272e32i	−0.703126	+	0.774805i
$507$	1.87878e30			0.00651300
$508$	−1.12402e32			−0.380547
$509$	−1.41901e32			−0.469216			−0.234608	−	0.972090i	$-0.575381\pi$
$509$	−1.41901e32			−0.469216			−0.234608	+	0.972090i	$0.575381\pi$
$510$	−1.17087e30			−0.00378153
$511$		−	4.65974e32i		−	1.46998i
$512$	9.12623e31			0.281224
$513$			1.53070e31i			0.0460766i
$514$	−6.90839e31			−0.203151
$515$	−1.03167e32			−0.296385
$516$			8.80948e29i			0.00247260i
$517$		−	6.49330e32i		−	1.78065i
$518$	7.64807e32			2.04925
$519$	2.87019e30			0.00751454
$520$			3.94212e32i			1.00853i
$521$		−	4.99447e32i		−	1.24864i	−0.781169	−	0.624319i	$-0.785377\pi$
$521$		−	4.99447e32i		−	1.24864i	0.781169	−	0.624319i	$-0.214623\pi$
$522$	−4.65342e32			−1.13691
$523$		−	1.12022e32i		−	0.267475i	−0.991017	−	0.133737i	$-0.957302\pi$
$523$		−	1.12022e32i		−	0.267475i	0.991017	−	0.133737i	$-0.0426979\pi$
$524$	−2.60453e32			−0.607790
$525$			6.08668e30i			0.0138825i
$526$			2.76262e32i			0.615873i
$527$			7.41214e31i			0.161516i
$528$		−	2.14794e30i		−	0.00457524i
$529$	4.64742e31	+	4.77997e32i	0.0967707	+	0.995307i
$530$	1.43043e32			0.291176
$531$	2.10889e32			0.419682
$532$	5.64517e32			1.09834
$533$	6.71277e32			1.27695
$534$			4.13950e30i			0.00769933i
$535$	−1.01759e33			−1.85067
$536$		−	3.09479e32i		−	0.550368i
$537$	−5.37578e30			−0.00934867
$538$	6.63718e32			1.12875
$539$			2.03201e33i			3.37957i
$540$			1.64447e31i			0.0267486i
$541$	−1.08434e33			−1.72503			−0.862516	−	0.506030i	$-0.831112\pi$
$541$	−1.08434e33			−1.72503			−0.862516	+	0.506030i	$0.831112\pi$
$542$	−3.09736e32			−0.481948
$543$			8.18059e30i			0.0124505i
$544$			1.48053e32i			0.220410i
$545$	−1.48353e33			−2.16042
$546$		−	1.42289e31i		−	0.0202703i
$547$	8.54290e32			1.19058			0.595288	−	0.803512i	$-0.297038\pi$
$547$	8.54290e32			1.19058			0.595288	+	0.803512i	$0.297038\pi$
$548$			6.98003e31i			0.0951679i
$549$		−	2.01591e32i		−	0.268908i
$550$		−	3.03713e32i		−	0.396379i
$551$		−	1.48874e33i		−	1.90108i
$552$	−1.26866e31	−	1.15129e31i	−0.0158517	−	0.0143852i
$553$	4.05192e32			0.495404
$554$	−6.72213e32			−0.804248
$555$	−3.56308e31			−0.0417167
$556$	−2.01186e31			−0.0230516
$557$			5.12682e31i			0.0574891i	0.999587	+	0.0287446i	$0.00915094\pi$
$557$			5.12682e31i			0.0574891i	−0.999587	+	0.0287446i	$0.990849\pi$
$558$	−4.21211e32			−0.462263
$559$		−	1.67120e32i		−	0.179509i
$560$	−2.81723e32			−0.296187
$561$	7.31957e30			0.00753236
$562$			4.98666e32i			0.502312i
$563$			1.50656e33i			1.48554i	0.669546	+	0.742770i	$0.266489\pi$
$563$			1.50656e33i			1.48554i	−0.669546	+	0.742770i	$0.733511\pi$
$564$	1.34338e31			0.0129673
$565$	−3.30624e32			−0.312430
$566$		−	4.76628e32i		−	0.440943i
$567$			1.96025e33i			1.77547i
$568$	2.13696e33			1.89503
$569$		−	3.51772e32i		−	0.305432i	−0.988270	−	0.152716i	$-0.951198\pi$
$569$		−	3.51772e32i		−	0.305432i	0.988270	−	0.152716i	$-0.0488020\pi$
$570$	2.12869e31			0.0180973
$571$			7.07820e32i			0.589237i	0.955615	+	0.294618i	$0.0951925\pi$
$571$			7.07820e32i			0.589237i	−0.955615	+	0.294618i	$0.904807\pi$
$572$		−	8.77186e32i		−	0.715055i
$573$		−	1.04701e31i		−	0.00835789i
$574$			2.34834e33i			1.83577i
$575$	−3.66453e32	−	3.32552e32i	−0.280546	−	0.254592i
$576$	−1.03050e33			−0.772644
$577$	−1.11213e33			−0.816666			−0.408333	−	0.912833i	$-0.633890\pi$
$577$	−1.11213e33			−0.816666			−0.408333	+	0.912833i	$0.633890\pi$
$578$	−8.79184e32			−0.632333
$579$	−1.03797e31			−0.00731210
$580$		−	1.59940e33i		−	1.10362i
$581$	−8.68039e32			−0.586713
$582$			6.25822e30i			0.00414357i
$583$	−8.94213e32			−0.579987
$584$	−1.35142e33			−0.858689
$585$		−	1.55949e33i		−	0.970760i
$586$			3.88948e32i			0.237203i
$587$	2.84195e33			1.69809			0.849043	−	0.528323i	$-0.177179\pi$
$587$	2.84195e33			1.69809			0.849043	+	0.528323i	$0.177179\pi$
$588$	−4.20396e31			−0.0246111
$589$		−	1.34756e33i		−	0.772971i
$590$		−	5.86679e32i		−	0.329744i
$591$	7.37626e30			0.00406244
$592$		−	4.53119e32i		−	0.244541i
$593$	1.84643e33			0.976508			0.488254	−	0.872701i	$-0.337634\pi$
$593$	1.84643e33			0.976508			0.488254	+	0.872701i	$0.337634\pi$
$594$			8.32078e31i			0.0431247i
$595$		−	9.60032e32i		−	0.487621i
$596$			1.25093e32i			0.0622696i
$597$			1.06094e31i			0.00517605i
$598$	8.56662e32	+	7.77411e32i	0.409634	+	0.371738i
$599$	6.34885e32			0.297559			0.148780	−	0.988870i	$-0.452466\pi$
$599$	6.34885e32			0.297559			0.148780	+	0.988870i	$0.452466\pi$
$600$	1.76526e31			0.00810950
$601$	9.60133e32			0.432352			0.216176	−	0.976354i	$-0.430641\pi$
$601$	9.60133e32			0.432352			0.216176	+	0.976354i	$0.430641\pi$
$602$	5.84640e32			0.258066
$603$			1.22429e33i			0.529755i
$604$	9.93996e32			0.421640
$605$			4.08651e33i			1.69937i
$606$	3.58479e31			0.0146148
$607$	−2.00017e33			−0.799472			−0.399736	−	0.916630i	$-0.630898\pi$
$607$	−2.00017e33			−0.799472			−0.399736	+	0.916630i	$0.630898\pi$
$608$		−	2.69166e33i		−	1.05482i
$609$			1.62186e32i			0.0623167i
$610$	−5.60813e32			−0.211281
$611$	−2.54846e33			−0.941418
$612$		−	3.56250e32i		−	0.129044i
$613$			2.49179e33i			0.885087i	0.896747	+	0.442544i	$0.145924\pi$
$613$			2.49179e33i			0.885087i	−0.896747	+	0.442544i	$0.854076\pi$
$614$	−1.31654e33			−0.458579
$615$		−	1.09405e32i		−	0.0373709i
$616$	8.62115e33			2.88800
$617$		−	2.10525e33i		−	0.691642i	−0.938300	−	0.345821i	$-0.887600\pi$
$617$		−	2.10525e33i		−	0.691642i	0.938300	−	0.345821i	$-0.112400\pi$
$618$		−	1.07997e31i		−	0.00347978i
$619$			8.35614e32i			0.264069i	0.991245	+	0.132035i	$0.0421510\pi$
$619$			8.35614e32i			0.264069i	−0.991245	+	0.132035i	$0.957849\pi$
$620$		−	1.44772e33i		−	0.448728i
$621$	1.00397e32	+	9.11088e31i	0.0305225	+	0.0276988i
$622$	−2.00680e33			−0.598440
$623$	−3.39409e33			−0.992812
$624$	−8.43010e30			−0.00241890
$625$	−4.38874e33			−1.23532
$626$			1.25676e33i			0.347026i
$627$	−1.33073e32			−0.0360477
$628$			3.40930e33i			0.906045i
$629$	1.54410e33			0.402596
$630$	5.45559e33			1.39558
$631$		−	7.39626e33i		−	1.85635i	−0.372145	−	0.928174i	$-0.621378\pi$
$631$		−	7.39626e33i		−	1.85635i	0.372145	−	0.928174i	$-0.378622\pi$
$632$		−	1.17514e33i		−	0.289391i
$633$	−1.24872e32			−0.0301732
$634$	−2.69916e33			−0.639967
$635$		−	3.47524e33i		−	0.808538i
$636$		−	1.85001e31i		−	0.00422366i
$637$	7.97512e33			1.78675
$638$		−	8.09272e33i		−	1.77929i
$639$	−8.45373e33			−1.82406
$640$		−	2.36551e33i		−	0.500915i
$641$			6.20470e33i			1.28951i	0.764390	+	0.644754i	$0.223040\pi$
$641$			6.20470e33i			1.28951i	−0.764390	+	0.644754i	$0.776960\pi$
$642$		−	1.06523e32i		−	0.0217282i
$643$			3.78048e33i			0.756860i	0.925630	+	0.378430i	$0.123536\pi$
$643$			3.78048e33i			0.756860i	−0.925630	+	0.378430i	$0.876464\pi$
$644$	3.36007e33	−	3.70260e33i	0.660265	−	0.727573i
$645$	−2.72372e31			−0.00525347
$646$	−9.22495e32			−0.174652
$647$	3.91850e33			0.728230			0.364115	−	0.931354i	$-0.381372\pi$
$647$	3.91850e33			0.728230			0.364115	+	0.931354i	$0.381372\pi$
$648$	5.68512e33			1.03714
$649$			3.66755e33i			0.656810i
$650$	−1.19199e33			−0.209563
$651$			1.46805e32i			0.0253377i
$652$	−2.96973e33			−0.503206
$653$	8.74974e33			1.45558			0.727791	−	0.685799i	$-0.240547\pi$
$653$	8.74974e33			1.45558			0.727791	+	0.685799i	$0.240547\pi$
$654$		−	1.55298e32i		−	0.0253649i
$655$		−	8.05269e33i		−	1.29135i
$656$	1.39130e33			0.219067
$657$	5.34617e33			0.826530
$658$		−	8.91532e33i		−	1.35340i
$659$			1.05771e33i			0.157667i	0.996888	+	0.0788334i	$0.0251195\pi$
$659$			1.05771e33i			0.157667i	−0.996888	+	0.0788334i	$0.974880\pi$
$660$	−1.42964e32			−0.0209266
$661$			5.81304e33i			0.835576i	0.908545	+	0.417788i	$0.137194\pi$
$661$			5.81304e33i			0.835576i	−0.908545	+	0.417788i	$0.862806\pi$
$662$	−8.47721e32			−0.119662
$663$		−	2.87274e31i		−	0.00398231i
$664$			2.51749e33i			0.342729i
$665$			1.74538e34i			2.33361i
$666$			8.77471e33i			1.15224i
$667$	−9.76450e33	−	8.86117e33i	−1.25933	−	1.14283i
$668$	−1.19663e33			−0.151580
$669$	2.83683e32			0.0352955
$670$	3.40588e33			0.416228
$671$	3.50585e33			0.420846
$672$			2.93233e32i			0.0345766i
$673$	−3.28535e32			−0.0380541			−0.0190270	−	0.999819i	$-0.506057\pi$
$673$	−3.28535e32			−0.0380541			−0.0190270	+	0.999819i	$0.506057\pi$
$674$		−	5.94837e33i		−	0.676829i
$675$	−1.39696e32			−0.0156149
$676$	1.59026e33			0.174625
$677$		−	1.04976e34i		−	1.13247i	−0.824244	−	0.566234i	$-0.808400\pi$
$677$		−	1.04976e34i		−	1.13247i	0.824244	−	0.566234i	$-0.191600\pi$
$678$		−	3.46103e31i		−	0.00366816i
$679$	−5.13129e33			−0.534305
$680$	−2.78429e33			−0.284844
$681$			1.59448e32i			0.0160270i
$682$		−	7.32524e33i		−	0.723451i
$683$	−6.94793e33			−0.674229			−0.337114	−	0.941464i	$-0.609451\pi$
$683$	−6.94793e33			−0.674229			−0.337114	+	0.941464i	$0.609451\pi$
$684$			6.47676e33i			0.617568i
$685$	−2.15809e33			−0.202201
$686$			1.49852e34i			1.37967i
$687$		−	2.69830e32i		−	0.0244123i
$688$		−	3.46377e32i		−	0.0307956i
$689$			3.50956e33i			0.306635i
$690$	1.26702e32	−	1.39619e32i	0.0108792	−	0.0119882i
$691$	−3.88128e32			−0.0327521			−0.0163760	−	0.999866i	$-0.505213\pi$
$691$	−3.88128e32			−0.0327521			−0.0163760	+	0.999866i	$0.505213\pi$
$692$	2.42942e33			0.201479
$693$	−3.41050e34			−2.77983
$694$	−9.46702e33			−0.758402
$695$		−	6.22029e32i		−	0.0489771i
$696$	4.70371e32			0.0364024
$697$			4.74118e33i			0.360656i
$698$	5.00290e33			0.374073
$699$	−2.41773e32			−0.0177698
$700$			5.15195e33i			0.372216i
$701$		−	1.29242e34i		−	0.917884i	−0.888466	−	0.458942i	$-0.848229\pi$
$701$		−	1.29242e34i		−	0.917884i	0.888466	−	0.458942i	$-0.151771\pi$
$702$	3.26569e32			0.0227997
$703$	−2.80724e34			−1.92671
$704$		−	1.79214e34i		−	1.20920i
$705$			4.15347e32i			0.0275513i
$706$	1.89758e34			1.23750
$707$			2.93927e34i			1.88455i
$708$	−7.58768e31			−0.00478311
$709$			2.52294e34i			1.56369i	0.623471	+	0.781846i	$0.285722\pi$
$709$			2.52294e34i			1.56369i	−0.623471	+	0.781846i	$0.714278\pi$
$710$			2.35177e34i			1.43316i
$711$			4.64881e33i			0.278552i
$712$			9.84356e33i			0.579952i
$713$	−8.83847e33	−	8.02081e33i	−0.512039	−	0.464669i
$714$	1.00498e32			0.00572504
$715$	2.71209e34			1.51926
$716$	−4.55022e33			−0.250655
$717$	3.71385e32			0.0201184
$718$			1.56508e34i			0.833762i
$719$	8.59057e33			0.450065			0.225033	−	0.974351i	$-0.427751\pi$
$719$	8.59057e33			0.450065			0.225033	+	0.974351i	$0.427751\pi$
$720$		−	3.23223e33i		−	0.166538i
$721$	8.85502e33			0.448710
$722$	3.35107e33			0.167008
$723$		−	7.67657e32i		−	0.0376277i
$724$			6.92430e33i			0.333820i
$725$	1.35867e34			0.644256
$726$	−4.27783e32			−0.0199519
$727$			2.06739e34i			0.948441i	0.880406	+	0.474220i	$0.157270\pi$
$727$			2.06739e34i			0.948441i	−0.880406	+	0.474220i	$0.842730\pi$
$728$		−	3.38358e34i		−	1.52686i
$729$	−2.24710e34			−0.997452
$730$		−	1.48727e34i		−	0.649404i
$731$	1.18036e33			0.0506997
$732$			7.25315e31i			0.00306474i
$733$		−	4.39338e34i		−	1.82621i	−0.407721	−	0.913106i	$-0.633677\pi$
$733$		−	4.39338e34i		−	1.82621i	0.407721	−	0.913106i	$-0.366323\pi$
$734$		−	8.65103e33i		−	0.353765i
$735$		−	1.29978e33i		−	0.0522905i
$736$	−1.76543e34	−	1.60211e34i	−0.698743	−	0.634101i
$737$	−2.12914e34			−0.829078
$738$	−2.69428e34			−1.03221
$739$	3.93866e34			1.48462			0.742310	−	0.670056i	$-0.233730\pi$
$739$	3.93866e34			1.48462			0.742310	+	0.670056i	$0.233730\pi$
$740$	−3.01590e34			−1.11850
$741$			5.22276e32i			0.0190582i
$742$	−1.22776e34			−0.440824
$743$		−	3.60130e34i		−	1.27231i	−0.771561	−	0.636155i	$-0.780524\pi$
$743$		−	3.60130e34i		−	1.27231i	0.771561	−	0.636155i	$-0.219476\pi$
$744$	4.25763e32			0.0148011
$745$	−3.86762e33			−0.132303
$746$			1.77661e34i			0.598035i
$747$		−	9.95911e33i		−	0.329893i
$748$	6.19551e33			0.201956
$749$	8.73414e34			2.80181
$750$		−	3.18527e32i		−	0.0100557i
$751$			5.44722e33i			0.169237i	0.996413	+	0.0846186i	$0.0269672\pi$
$751$			5.44722e33i			0.169237i	−0.996413	+	0.0846186i	$0.973033\pi$
$752$	−5.28199e33			−0.161504
$753$		−	7.60977e32i		−	0.0228998i
$754$	−3.17618e34			−0.940697
$755$			3.07324e34i			0.895846i
$756$		−	1.41147e33i		−	0.0404959i
$757$		−	2.43566e34i		−	0.687805i	−0.939005	−	0.343903i	$-0.888251\pi$
$757$		−	2.43566e34i		−	0.687805i	0.939005	−	0.343903i	$-0.111749\pi$
$758$			8.00726e33i			0.222563i
$759$	−7.92064e32	+	8.72809e32i	−0.0216700	+	0.0238791i
$760$	5.06194e34			1.36318
$761$	−5.64836e33			−0.149729			−0.0748646	−	0.997194i	$-0.523852\pi$
$761$	−5.64836e33			−0.149729			−0.0748646	+	0.997194i	$0.523852\pi$
$762$	3.63794e32			0.00949285
$763$	1.27334e35			3.27076
$764$		−	8.86223e33i		−	0.224090i
$765$	1.10146e34			0.274176
$766$		−	3.82109e34i		−	0.936358i
$767$	1.43942e34			0.347251
$768$	9.18495e32			0.0218143
$769$		−	8.17177e33i		−	0.191073i	−0.995426	−	0.0955366i	$-0.969543\pi$
$769$		−	8.17177e33i		−	0.191073i	0.995426	−	0.0955366i	$-0.0304567\pi$
$770$			9.48776e34i			2.18411i
$771$	−2.76247e32			−0.00626102
$772$	−8.78570e33			−0.196051
$773$		−	2.00912e34i		−	0.441421i	−0.975339	−	0.220710i	$-0.929162\pi$
$773$		−	2.00912e34i		−	0.441421i	0.975339	−	0.220710i	$-0.0708376\pi$
$774$			6.70764e33i			0.145104i
$775$	1.22982e34			0.261952
$776$			1.48818e34i			0.312115i
$777$	3.05825e33			0.0631568
$778$		−	1.42226e34i		−	0.289217i
$779$		−	8.61965e34i		−	1.72600i
$780$			5.61096e32i			0.0110637i
$781$		−	1.47018e35i		−	2.85468i
$782$	−5.49080e33	+	6.05055e33i	−0.104992	+	0.115695i
$783$	−3.72234e33			−0.0700930
$784$	1.65294e34			0.306524
$785$	−1.05409e35			−1.92505
$786$	8.42969e32			0.0151615
$787$			4.59658e34i			0.814212i	0.913381	+	0.407106i	$0.133462\pi$
$787$			4.59658e34i			0.814212i	−0.913381	+	0.407106i	$0.866538\pi$
$788$	6.24349e33			0.108921
$789$			1.10469e33i			0.0189809i
$790$	1.29327e34			0.218858
$791$	2.83779e34			0.473002
$792$			9.89113e34i			1.62384i
$793$		−	1.37596e34i		−	0.222498i
$794$	−4.93249e34			−0.785633
$795$	5.71987e32			0.00897389
$796$			8.98011e33i			0.138779i
$797$		−	8.12717e34i		−	1.23720i	−0.785707	−	0.618599i	$-0.787700\pi$
$797$		−	8.12717e34i		−	1.23720i	0.785707	−	0.618599i	$-0.212300\pi$
$798$	−1.82709e33			−0.0273984
$799$		−	1.79996e34i		−	0.265889i
$800$	2.45649e34			0.357467
$801$		−	3.89408e34i		−	0.558232i
$802$		−	6.65261e34i		−	0.939507i
$803$			9.29747e34i			1.29353i
$804$		−	4.40492e32i		−	0.00603762i
$805$	1.14477e35	+	1.03887e35i	1.54585	+	1.40285i
$806$	−2.87497e34			−0.382483
$807$	2.65402e33			0.0347874
$808$	8.52449e34			1.10086
$809$	−1.28489e35			−1.63488			−0.817439	−	0.576015i	$-0.804607\pi$
$809$	−1.28489e35			−1.63488			−0.817439	+	0.576015i	$0.804607\pi$
$810$			6.25660e34i			0.784364i
$811$	−1.11678e35			−1.37949			−0.689744	−	0.724053i	$-0.742277\pi$
$811$	−1.11678e35			−1.37949			−0.689744	+	0.724053i	$0.742277\pi$
$812$			1.37279e35i			1.67083i
$813$	−1.23855e33			−0.0148534
$814$	−1.52600e35			−1.80327
$815$		−	9.18182e34i		−	1.06915i
$816$		−	5.95412e31i		−	0.000683181i
$817$	−2.14593e34			−0.242634
$818$	−7.07388e34			−0.788166
$819$			1.33853e35i			1.46968i
$820$		−	9.26033e34i		−	1.00198i
$821$	4.46275e33			0.0475866			0.0237933	−	0.999717i	$-0.492426\pi$
$821$	4.46275e33			0.0475866			0.0237933	+	0.999717i	$0.492426\pi$
$822$		−	2.25912e32i		−	0.00237399i
$823$	6.13916e34			0.635786			0.317893	−	0.948127i	$-0.397025\pi$
$823$	6.13916e34			0.635786			0.317893	+	0.948127i	$0.397025\pi$
$824$		−	2.56813e34i		−	0.262115i
$825$		−	1.21446e33i		−	0.0122162i
$826$			5.03555e34i			0.499214i
$827$			9.05474e34i			0.884729i	0.896835	+	0.442364i	$0.145860\pi$
$827$			9.05474e34i			0.884729i	−0.896835	+	0.442364i	$0.854140\pi$
$828$	4.24804e34	+	3.85504e34i	0.409095	+	0.371249i
$829$	4.87698e34			0.462910			0.231455	−	0.972846i	$-0.425651\pi$
$829$	4.87698e34			0.462910			0.231455	+	0.972846i	$0.425651\pi$
$830$	−2.77055e34			−0.259197
$831$	−2.68799e33			−0.0247865
$832$	−7.03367e34			−0.639296
$833$			5.63277e34i			0.504641i
$834$	6.51151e31			0.000575028
$835$		−	3.69973e34i		−	0.322057i
$836$	−1.12637e35			−0.966506
$837$	−3.36932e33			−0.0284995
$838$			1.37828e35i			1.14923i
$839$			1.36779e34i			0.112429i	0.998419	+	0.0562144i	$0.0179030\pi$
$839$			1.36779e34i			0.112429i	−0.998419	+	0.0562144i	$0.982097\pi$
$840$	−5.51455e33			−0.0446847
$841$	2.36846e35			1.89197
$842$		−	5.87673e34i		−	0.462797i
$843$			1.99402e33i			0.0154810i
$844$	−1.05696e35			−0.808998
$845$			4.91677e34i			0.371022i
$846$	1.02286e35			0.760980
$847$		−	3.50751e35i		−	2.57275i
$848$			7.27400e33i			0.0526045i
$849$		−	1.90590e33i		−	0.0135896i
$850$		−	8.41897e33i		−	0.0591878i
$851$	−1.67090e35	+	1.84124e35i	−1.15823	+	1.27631i
$852$	3.04161e33			0.0207887
$853$	−2.02411e35			−1.36410			−0.682049	−	0.731306i	$-0.738911\pi$
$853$	−2.02411e35			−1.36410			−0.682049	+	0.731306i	$0.738911\pi$
$854$	4.81354e34			0.319868
$855$	−2.00249e35			−1.31213
$856$		−	2.53308e35i		−	1.63668i
$857$	2.08398e35			1.32777			0.663887	−	0.747833i	$-0.268905\pi$
$857$	2.08398e35			1.32777			0.663887	+	0.747833i	$0.268905\pi$
$858$			2.83906e33i			0.0178372i
$859$	1.82146e35			1.12850			0.564251	−	0.825604i	$-0.309165\pi$
$859$	1.82146e35			1.12850			0.564251	+	0.825604i	$0.309165\pi$
$860$	−2.30544e34			−0.140855
$861$			9.39036e33i			0.0565776i
$862$			1.01989e35i			0.605988i
$863$	−4.74578e34			−0.278086			−0.139043	−	0.990286i	$-0.544403\pi$
$863$	−4.74578e34			−0.278086			−0.139043	+	0.990286i	$0.544403\pi$
$864$	−6.73003e33			−0.0388912
$865$			7.51128e34i			0.428076i
$866$			2.11582e35i			1.18923i
$867$	−3.51561e33			−0.0194882
$868$			1.24260e35i			0.679350i
$869$	−8.08470e34			−0.435940
$870$			5.17654e33i			0.0275301i
$871$			8.35635e34i			0.438327i
$872$		−	3.69293e35i		−	1.91062i
$873$		−	5.88718e34i		−	0.300425i
$874$	9.98248e34	−	1.10001e35i	0.502460	−	0.553682i
$875$	2.61170e35			1.29666
$876$	−1.92352e33			−0.00941995
$877$	2.37984e35			1.14961			0.574807	−	0.818289i	$-0.305077\pi$
$877$	2.37984e35			1.14961			0.574807	+	0.818289i	$0.305077\pi$
$878$	7.41923e34			0.353529
$879$			1.55529e33i			0.00731047i
$880$	5.62114e34			0.260635
$881$		−	9.63897e34i		−	0.440879i	−0.975401	−	0.220440i	$-0.929251\pi$
$881$		−	9.63897e34i		−	0.440879i	0.975401	−	0.220440i	$-0.0707492\pi$
$882$	−3.20094e35			−1.44429
$883$	−4.82464e34			−0.214752			−0.107376	−	0.994219i	$-0.534245\pi$
$883$	−4.82464e34			−0.214752			−0.107376	+	0.994219i	$0.534245\pi$
$884$		−	2.43158e34i		−	0.106773i
$885$		−	2.34596e33i		−	0.0101625i
$886$	−5.10154e34			−0.218020
$887$	−3.30209e35			−1.39221			−0.696105	−	0.717940i	$-0.745085\pi$
$887$	−3.30209e35			−1.39221			−0.696105	+	0.717940i	$0.745085\pi$
$888$		−	8.86954e33i		−	0.0368931i
$889$			2.98285e35i			1.22408i
$890$	−1.08331e35			−0.438602
$891$		−	3.91123e35i		−	1.56236i
$892$	2.40117e35			0.946336
$893$			3.27239e35i			1.27247i
$894$		−	4.04869e32i		−	0.00155333i
$895$		−	1.40684e35i		−	0.532559i
$896$			2.03035e35i			0.758359i
$897$	3.42555e33	+	3.10865e33i	0.0126247	+	0.0114568i
$898$	2.14405e35			0.779684
$899$	3.27698e35			1.17586
$900$	−5.91089e34			−0.209287
$901$	−2.47878e34			−0.0866044
$902$		−	4.68559e35i		−	1.61542i
$903$	2.33781e33			0.00795346
$904$		−	8.23018e34i		−	0.276305i
$905$	−2.14086e35			−0.709259
$906$	−3.21712e33			−0.0105179
$907$		−	1.55779e35i		−	0.502600i	−0.967909	−	0.251300i	$-0.919142\pi$
$907$		−	1.55779e35i		−	0.502600i	0.967909	−	0.251300i	$-0.0808581\pi$
$908$			1.34961e35i			0.429713i
$909$	−3.37226e35			−1.05963
$910$	3.72370e35			1.15472
$911$			4.16565e35i			1.27486i	0.770508	+	0.637430i	$0.220003\pi$
$911$			4.16565e35i			1.27486i	−0.770508	+	0.637430i	$0.779997\pi$
$912$			1.08248e33i			0.00326951i
$913$	1.73198e35			0.516289
$914$			2.07761e34i			0.0611236i
$915$	−2.24253e33			−0.00651157
$916$		−	2.28392e35i		−	0.654539i
$917$			6.91175e35i			1.95504i
$918$			2.30653e33i			0.00643944i
$919$		−	4.82434e34i		−	0.132939i	−0.997788	−	0.0664694i	$-0.978827\pi$
$919$		−	4.82434e34i		−	0.132939i	0.997788	−	0.0664694i	$-0.0211735\pi$
$920$	3.01293e35	−	3.32007e35i	0.819473	−	0.903013i
$921$	−5.26448e33			−0.0141332
$922$	4.23257e35			1.12159
$923$	−5.77008e35			−1.50925
$924$	1.22708e34			0.0316817
$925$		−	2.56197e35i		−	0.652941i
$926$	−5.12020e34			−0.128812
$927$			1.01595e35i			0.252298i
$928$	6.54557e35			1.60462
$929$	1.54125e35			0.372978			0.186489	−	0.982457i	$-0.440289\pi$
$929$	1.54125e35			0.372978			0.186489	+	0.982457i	$0.440289\pi$
$930$			4.68561e33i			0.0111936i
$931$		−	1.02406e36i		−	2.41507i
$932$	−2.04644e35			−0.476440
$933$	−8.02464e33			−0.0184436
$934$			3.88286e34i			0.0881026i
$935$			1.91553e35i			0.429091i
$936$	3.88202e35			0.858514
$937$		−	5.55643e35i		−	1.21317i	−0.795020	−	0.606584i	$-0.792539\pi$
$937$		−	5.55643e35i		−	1.21317i	0.795020	−	0.606584i	$-0.207461\pi$
$938$	−2.92332e35			−0.630147
$939$			5.02544e33i			0.0106952i
$940$			3.51562e35i			0.738699i
$941$			2.53976e34i			0.0526888i	0.999653	+	0.0263444i	$0.00838665\pi$
$941$			2.53976e34i			0.0526888i	−0.999653	+	0.0263444i	$0.991613\pi$
$942$		−	1.10344e34i		−	0.0226015i
$943$	−5.65353e35	−	5.13052e35i	−1.14335	−	1.03758i
$944$	2.98338e34			0.0595723
$945$	4.36400e34			0.0860405
$946$	−1.16652e35			−0.227090
$947$	−5.49575e35			−1.05640			−0.528199	−	0.849121i	$-0.677132\pi$
$947$	−5.49575e35			−1.05640			−0.528199	+	0.849121i	$0.677132\pi$
$948$		−	1.67262e33i		−	0.00317466i
$949$	3.64902e35			0.683882
$950$			1.53060e35i			0.283256i
$951$	−1.07932e34			−0.0197235
$952$	2.38980e35			0.431239
$953$		−	1.99150e35i		−	0.354866i	−0.984133	−	0.177433i	$-0.943221\pi$
$953$		−	1.99150e35i		−	0.354866i	0.984133	−	0.177433i	$-0.0567793\pi$
$954$		−	1.40862e35i		−	0.247864i
$955$	2.74003e35			0.476118
$956$	3.14351e35			0.539412
$957$		−	3.23605e34i		−	0.0548368i
$958$			4.54904e35i			0.761260i
$959$	1.85232e35			0.306121
$960$			1.14635e34i			0.0187095i
$961$	−3.23793e35			−0.521899
$962$			5.98916e35i			0.953378i
$963$			1.00208e36i			1.57538i
$964$		−	6.49768e35i		−	1.00887i
$965$		−	2.71637e35i		−	0.416544i
$966$	−1.08751e34	+	1.19837e34i	−0.0164705	+	0.0181495i
$967$	3.14175e35			0.469953			0.234976	−	0.972001i	$-0.424499\pi$
$967$	3.14175e35			0.469953			0.234976	+	0.972001i	$0.424499\pi$
$968$	−1.01725e36			−1.50288
$969$	−3.68880e33			−0.00538269
$970$	−1.63777e35			−0.236044
$971$			1.25851e36i			1.79154i	0.444521	+	0.895768i	$0.353374\pi$
$971$			1.25851e36i			1.79154i	−0.444521	+	0.895768i	$0.646626\pi$
$972$	2.42927e34			0.0341571
$973$			5.33897e34i			0.0741487i
$974$	3.37913e35			0.463552
$975$	−4.76645e33			−0.00645862
$976$		−	2.85184e34i		−	0.0381705i
$977$			3.15353e35i			0.416929i	0.978030	+	0.208464i	$0.0668465\pi$
$977$			3.15353e35i			0.416929i	−0.978030	+	0.208464i	$0.933154\pi$
$978$	9.61169e33			0.0125526
$979$	6.77215e35			0.873644
$980$		−	1.10018e36i		−	1.40200i
$981$			1.46091e36i			1.83906i
$982$	−8.24585e35			−1.02541
$983$			3.36453e35i			0.413316i	0.978413	+	0.206658i	$0.0662588\pi$
$983$			3.36453e35i			0.413316i	−0.978413	+	0.206658i	$0.933741\pi$
$984$	2.72340e34			0.0330499
$985$			1.93036e35i			0.231422i
$986$		−	2.24332e35i		−	0.265685i
$987$		−	3.56498e34i		−	0.0417111i
$988$			4.42070e35i			0.510984i
$989$	−1.27729e35	+	1.40750e35i	−0.145859	+	0.160728i
$990$	−1.08854e36			−1.22807
$991$	−6.07030e34			−0.0676591			−0.0338296	−	0.999428i	$-0.510770\pi$
$991$	−6.07030e34			−0.0676591			−0.0338296	+	0.999428i	$0.510770\pi$
$992$	5.92481e35			0.652431
$993$	−3.38979e33			−0.00368793
$994$		−	2.01856e36i		−	2.16973i
$995$	−2.77647e35			−0.294861
$996$			3.58323e33i			0.00375979i
$997$	1.08803e36			1.12797			0.563986	−	0.825784i	$-0.309267\pi$
$997$	1.08803e36			1.12797			0.563986	+	0.825784i	$0.309267\pi$
$998$	−6.23464e35			−0.638623
$999$			7.01901e34i			0.0710379i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.25.b.c.22.18	yes	44
23.22	odd	2	inner	23.25.b.c.22.17	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.25.b.c.22.17	✓	44	23.22	odd	2	inner
23.25.b.c.22.18	yes	44	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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 25 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2739.52 q^{2} -10954.6 q^{3} -9.27226e6 q^{4} -2.86680e8i q^{5} +3.00102e7 q^{6} +2.46062e10i q^{7} +7.13630e10 q^{8} -2.82310e11 q^{9} +7.85365e11i q^{10} -4.90962e12i q^{11} +1.01573e11 q^{12} -1.92690e13 q^{13} -6.74091e13i q^{14} +3.14045e12i q^{15} -3.99374e13 q^{16} -1.36095e14i q^{17} +7.73392e14 q^{18} +2.47427e15i q^{19} +2.65817e15i q^{20} -2.69550e14i q^{21} +1.34500e16i q^{22} +(1.62284e16 + 1.47271e16i) q^{23} -7.81750e14 q^{24} -2.25809e16 q^{25} +5.27877e16 q^{26} +6.18646e15 q^{27} -2.28155e17i q^{28} -6.01690e17 q^{29} -8.60333e15i q^{30} -5.44628e17 q^{31} -1.08786e18 q^{32} +5.37826e16i q^{33} +3.72836e17i q^{34} +7.05411e18 q^{35} +2.61765e18 q^{36} +1.13458e19i q^{37} -6.77830e18i q^{38} +2.11083e17 q^{39} -2.04584e19i q^{40} -3.48372e19 q^{41} +7.38437e17i q^{42} +8.67301e18i q^{43} +4.55232e19i q^{44} +8.09326e19i q^{45} +(-4.44581e19 - 4.03452e19i) q^{46} +1.32257e20 q^{47} +4.37496e17 q^{48} -4.13884e20 q^{49} +6.18608e19 q^{50} +1.49086e18i q^{51} +1.78667e20 q^{52} -1.82135e20i q^{53} -1.69479e19 q^{54} -1.40749e21 q^{55} +1.75597e21i q^{56} -2.71045e19i q^{57} +1.64834e21 q^{58} -7.47014e20 q^{59} -2.91191e19i q^{60} +7.14079e20i q^{61} +1.49202e21 q^{62} -6.94657e21i q^{63} +3.65026e21 q^{64} +5.52404e21i q^{65} -1.47338e20i q^{66} -4.33668e21i q^{67} +1.26191e21i q^{68} +(-1.77775e20 - 1.61329e20i) q^{69} -1.93249e22 q^{70} +2.99449e22 q^{71} -2.01465e22 q^{72} -1.89373e22 q^{73} -3.10819e22i q^{74} +2.47364e20 q^{75} -2.29421e22i q^{76} +1.20807e23 q^{77} -5.78266e20 q^{78} -1.64671e22i q^{79} +1.14493e22i q^{80} +7.96648e22 q^{81} +9.54371e22 q^{82} +3.52773e22i q^{83} +2.49934e21i q^{84} -3.90159e22 q^{85} -2.37599e22i q^{86} +6.59125e21 q^{87} -3.50365e23i q^{88} +1.37936e23i q^{89} -2.21716e23i q^{90} -4.74136e23i q^{91} +(-1.50474e23 - 1.36554e23i) q^{92} +5.96616e21 q^{93} -3.62320e23 q^{94} +7.09323e23 q^{95} +1.19171e22 q^{96} +2.08536e23i q^{97} +1.13384e24 q^{98} +1.38603e24i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4232 q^{2} - 434562 q^{3} + 317760360 q^{4} - 8460029520 q^{6} - 198307023760 q^{8} + 4220041988298 q^{9} - 67439597688792 q^{12} + 5771152551358 q^{13} + 18\!\cdots\!92 q^{16} + 18\!\cdots\!68 q^{18}+ \cdots - 20\!\cdots\!92 q^{98}+O(q^{100}) \) 44 * q - 4232 * q^2 - 434562 * q^3 + 317760360 * q^4 - 8460029520 * q^6 - 198307023760 * q^8 + 4220041988298 * q^9 - 67439597688792 * q^12 + 5771152551358 * q^13 + 1801550792530192 * q^16 + 1842830558165568 * q^18 - 117204213263210644 * q^23 - 186493005437450784 * q^24 - 689533084157602612 * q^25 - 548650296251047576 * q^26 + 824037505793640630 * q^27 - 1583078054945709602 * q^29 - 77249813729122274 * q^31 - 2838716832733708512 * q^32 + 3418569899175948192 * q^35 + 71181245116739335248 * q^36 + 47106572467817156310 * q^39 - 9150679195666184738 * q^41 - 153049487518536134480 * q^46 + 22735751396127769438 * q^47 - 1317482931964495330032 * q^48 - 1842110772006482246836 * q^49 + 284936617115007894520 * q^50 + 207129175371045152568 * q^52 - 381774025126375304544 * q^54 - 5430040696094013021792 * q^55 - 2168901871328366220760 * q^58 + 16048018791948427185304 * q^59 + 14054450375064171442400 * q^62 + 20310379126091866412192 * q^64 + 24476879859710435470878 * q^69 - 118028399895318845638416 * q^70 - 68204988047209908888578 * q^71 - 106777935582637627887744 * q^72 - 7319375899831086678722 * q^73 + 12913295383070041819998 * q^75 + 111116199573084838704480 * q^77 - 422583593213386075404144 * q^78 - 331350288185757013618680 * q^81 + 1000764218021118649437320 * q^82 - 113385735487939722862944 * q^85 - 248828610584854760185290 * q^87 - 4238455111972177116774696 * q^92 + 4492289396741153284965270 * q^93 - 1602622056539190351258592 * q^94 - 1668692394291208047471936 * q^95 - 12092663288748189077973504 * q^96 - 2067304853876486831513192 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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