LMFDB - Embedded newform 23.25.b.c.22.2

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			22.2
Character	$\chi$	$=$	23.22
Dual form			23.25.b.c.22.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-7585.13 q^{2} +776377. q^{3} +4.07570e7 q^{4} +3.88802e8i q^{5} -5.88892e9 q^{6} -7.00512e9i q^{7} -1.81889e11 q^{8} +3.20332e11 q^{9} -2.94912e12i q^{10} +3.53280e12i q^{11} +3.16428e13 q^{12} +7.20698e12 q^{13} +5.31348e13i q^{14} +3.01857e14i q^{15} +6.95866e14 q^{16} -3.92673e13i q^{17} -2.42976e15 q^{18} -2.86866e15i q^{19} +1.58464e16i q^{20} -5.43862e15i q^{21} -2.67967e16i q^{22} +(-1.94816e16 + 1.00359e16i) q^{23} -1.41215e17 q^{24} -9.15626e16 q^{25} -5.46659e16 q^{26} +2.94264e16 q^{27} -2.85508e17i q^{28} -1.47160e17 q^{29} -2.28963e18i q^{30} -4.74187e17 q^{31} -2.22664e18 q^{32} +2.74279e18i q^{33} +2.97847e17i q^{34} +2.72361e18 q^{35} +1.30557e19 q^{36} +1.14369e19i q^{37} +2.17591e19i q^{38} +5.59533e18 q^{39} -7.07190e19i q^{40} -3.61739e19 q^{41} +4.12526e19i q^{42} -7.05725e19i q^{43} +1.43986e20i q^{44} +1.24546e20i q^{45} +(1.47770e20 - 7.61235e19i) q^{46} -1.72045e20 q^{47} +5.40255e20 q^{48} +1.42509e20 q^{49} +6.94514e20 q^{50} -3.04862e19i q^{51} +2.93735e20 q^{52} +5.62272e20i q^{53} -2.23203e20 q^{54} -1.37356e21 q^{55} +1.27416e21i q^{56} -2.22716e21i q^{57} +1.11622e21 q^{58} +3.17850e21 q^{59} +1.23028e22i q^{60} -4.66747e21i q^{61} +3.59677e21 q^{62} -2.24396e21i q^{63} +5.21463e21 q^{64} +2.80209e21i q^{65} -2.08044e22i q^{66} +3.10815e21i q^{67} -1.60042e21i q^{68} +(-1.51250e22 + 7.79163e21i) q^{69} -2.06589e22 q^{70} -1.62037e22 q^{71} -5.82650e22 q^{72} -2.51668e21 q^{73} -8.67506e22i q^{74} -7.10871e22 q^{75} -1.16918e23i q^{76} +2.47477e22 q^{77} -4.24413e22 q^{78} +7.84739e22i q^{79} +2.70554e23i q^{80} -6.76252e22 q^{81} +2.74383e23 q^{82} +3.55880e22i q^{83} -2.21662e23i q^{84} +1.52672e22 q^{85} +5.35301e23i q^{86} -1.14251e23 q^{87} -6.42579e23i q^{88} +1.67467e22i q^{89} -9.44696e23i q^{90} -5.04858e22i q^{91} +(-7.94010e23 + 4.09032e23i) q^{92} -3.68148e23 q^{93} +1.30498e24 q^{94} +1.11534e24 q^{95} -1.72871e24 q^{96} -2.72120e23i q^{97} -1.08095e24 q^{98} +1.13167e24i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−7585.13			−1.85184			−0.925919	−	0.377722i	$-0.876707\pi$
$2$	−7585.13			−1.85184			−0.925919	+	0.377722i	$0.876707\pi$
$3$	776377.			1.46089			0.730445	−	0.682971i	$-0.239313\pi$
$3$	776377.			1.46089			0.730445	+	0.682971i	$0.239313\pi$
$4$	4.07570e7			2.42930
$5$			3.88802e8i			1.59253i	0.604945	+	0.796267i	$0.293195\pi$
$5$			3.88802e8i			1.59253i	−0.604945	+	0.796267i	$0.706805\pi$
$6$	−5.88892e9			−2.70533
$7$		−	7.00512e9i		−	0.506104i	−0.967453	−	0.253052i	$-0.918566\pi$
$7$		−	7.00512e9i		−	0.506104i	0.967453	−	0.253052i	$-0.0814343\pi$
$8$	−1.81889e11			−2.64684
$9$	3.20332e11			1.13420
$10$		−	2.94912e12i		−	2.94912i
$11$			3.53280e12i			1.12566i	0.826573	+	0.562830i	$0.190287\pi$
$11$			3.53280e12i			1.12566i	−0.826573	+	0.562830i	$0.809713\pi$
$12$	3.16428e13			3.54895
$13$	7.20698e12			0.309338			0.154669	−	0.987966i	$-0.450569\pi$
$13$	7.20698e12			0.309338			0.154669	+	0.987966i	$0.450569\pi$
$14$			5.31348e13i			0.937222i
$15$			3.01857e14i			2.32652i
$16$	6.95866e14			2.47221
$17$		−	3.92673e13i		−	0.0673975i	−0.999432	−	0.0336988i	$-0.989271\pi$
$17$		−	3.92673e13i		−	0.0673975i	0.999432	−	0.0336988i	$-0.0107287\pi$
$18$	−2.42976e15			−2.10036
$19$		−	2.86866e15i		−	1.29609i	−0.761602	−	0.648045i	$-0.775587\pi$
$19$		−	2.86866e15i		−	1.29609i	0.761602	−	0.648045i	$-0.224413\pi$
$20$			1.58464e16i			3.86875i
$21$		−	5.43862e15i		−	0.739362i
$22$		−	2.67967e16i		−	2.08454i
$23$	−1.94816e16	+	1.00359e16i	−0.888976	+	0.457954i
$23$	−1.94816e16	+	1.00359e16i	−0.888976	+	0.457954i
$24$	−1.41215e17			−3.86674
$25$	−9.15626e16			−1.53617
$26$	−5.46659e16			−0.572844
$27$	2.94264e16			0.196052
$28$		−	2.85508e17i		−	1.22948i
$29$	−1.47160e17			−0.415923			−0.207961	−	0.978137i	$-0.566683\pi$
$29$	−1.47160e17			−0.415923			−0.207961	+	0.978137i	$0.566683\pi$
$30$		−	2.28963e18i		−	4.30834i
$31$	−4.74187e17			−0.602018			−0.301009	−	0.953621i	$-0.597323\pi$
$31$	−4.74187e17			−0.602018			−0.301009	+	0.953621i	$0.597323\pi$
$32$	−2.22664e18			−1.93130
$33$			2.74279e18i			1.64446i
$34$			2.97847e17i			0.124809i
$35$	2.72361e18			0.805987
$36$	1.30557e19			2.75532
$37$			1.14369e19i			1.73736i	0.495377	+	0.868678i	$0.335030\pi$
$37$			1.14369e19i			1.73736i	−0.495377	+	0.868678i	$0.664970\pi$
$38$			2.17591e19i			2.40015i
$39$	5.59533e18			0.451909
$40$		−	7.07190e19i		−	4.21518i
$41$	−3.61739e19			−1.60320			−0.801602	−	0.597858i	$-0.796019\pi$
$41$	−3.61739e19			−1.60320			−0.801602	+	0.597858i	$0.796019\pi$
$42$			4.12526e19i			1.36918i
$43$		−	7.05725e19i		−	1.76609i	−0.469285	−	0.883047i	$-0.655488\pi$
$43$		−	7.05725e19i		−	1.76609i	0.469285	−	0.883047i	$-0.344512\pi$
$44$			1.43986e20i			2.73457i
$45$			1.24546e20i			1.80625i
$46$	1.47770e20	−	7.61235e19i	1.64624	−	0.848056i
$47$	−1.72045e20			−1.48070			−0.740351	−	0.672221i	$-0.765341\pi$
$47$	−1.72045e20			−1.48070			−0.740351	+	0.672221i	$0.765341\pi$
$48$	5.40255e20			3.61163
$49$	1.42509e20			0.743859
$50$	6.94514e20			2.84473
$51$		−	3.04862e19i		−	0.0984604i
$52$	2.93735e20			0.751476
$53$			5.62272e20i			1.14455i	0.820061	+	0.572277i	$0.193940\pi$
$53$			5.62272e20i			1.14455i	−0.820061	+	0.572277i	$0.806060\pi$
$54$	−2.23203e20			−0.363057
$55$	−1.37356e21			−1.79265
$56$			1.27416e21i			1.33957i
$57$		−	2.22716e21i		−	1.89345i
$58$	1.11622e21			0.770221
$59$	3.17850e21			1.78648			0.893239	−	0.449583i	$-0.148427\pi$
$59$	3.17850e21			1.78648			0.893239	+	0.449583i	$0.148427\pi$
$60$			1.23028e22i			5.65182i
$61$		−	4.66747e21i		−	1.75842i	−0.476432	−	0.879211i	$-0.658070\pi$
$61$		−	4.66747e21i		−	1.75842i	0.476432	−	0.879211i	$-0.341930\pi$
$62$	3.59677e21			1.11484
$63$		−	2.24396e21i		−	0.574023i
$64$	5.21463e21			1.10424
$65$			2.80209e21i			0.492631i
$66$		−	2.08044e22i		−	3.04528i
$67$			3.10815e21i			0.379843i	0.981799	+	0.189922i	$0.0608234\pi$
$67$			3.10815e21i			0.379843i	−0.981799	+	0.189922i	$0.939177\pi$
$68$		−	1.60042e21i		−	0.163729i
$69$	−1.51250e22	+	7.79163e21i	−1.29870	+	0.669020i
$70$	−2.06589e22			−1.49256
$71$	−1.62037e22			−0.987448			−0.493724	−	0.869619i	$-0.664365\pi$
$71$	−1.62037e22			−0.987448			−0.493724	+	0.869619i	$0.664365\pi$
$72$	−5.82650e22			−3.00205
$73$	−2.51668e21			−0.109889			−0.0549443	−	0.998489i	$-0.517498\pi$
$73$	−2.51668e21			−0.109889			−0.0549443	+	0.998489i	$0.517498\pi$
$74$		−	8.67506e22i		−	3.21730i
$75$	−7.10871e22			−2.24417
$76$		−	1.16918e23i		−	3.14860i
$77$	2.47477e22			0.569700
$78$	−4.24413e22			−0.836862
$79$			7.84739e22i			1.32801i	0.747730	+	0.664003i	$0.231144\pi$
$79$			7.84739e22i			1.32801i	−0.747730	+	0.664003i	$0.768856\pi$
$80$			2.70554e23i			3.93709i
$81$	−6.76252e22			−0.847790
$82$	2.74383e23			2.96887
$83$			3.55880e22i			0.332940i	0.986046	+	0.166470i	$0.0532369\pi$
$83$			3.55880e22i			0.332940i	−0.986046	+	0.166470i	$0.946763\pi$
$84$		−	2.21662e23i		−	1.79613i
$85$	1.52672e22			0.107333
$86$			5.35301e23i			3.27052i
$87$	−1.14251e23			−0.607617
$88$		−	6.42579e23i		−	2.97944i
$89$			1.67467e22i			0.0678029i	0.999425	+	0.0339014i	$0.0107932\pi$
$89$			1.67467e22i			0.0678029i	−0.999425	+	0.0339014i	$0.989207\pi$
$90$		−	9.44696e23i		−	3.34489i
$91$		−	5.04858e22i		−	0.156557i
$92$	−7.94010e23	+	4.09032e23i	−2.15959	+	1.11251i
$93$	−3.68148e23			−0.879482
$94$	1.30498e24			2.74202
$95$	1.11534e24			2.06407
$96$	−1.72871e24			−2.82142
$97$		−	2.72120e23i		−	0.392193i	−0.980585	−	0.196096i	$-0.937173\pi$
$97$		−	2.72120e23i		−	0.392193i	0.980585	−	0.196096i	$-0.0628265\pi$
$98$	−1.08095e24			−1.37751
$99$			1.13167e24i			1.27672i
$100$	−3.73181e24			−3.73181
$101$	6.45835e23			0.573145			0.286573	−	0.958059i	$-0.407484\pi$
$101$	6.45835e23			0.573145			0.286573	+	0.958059i	$0.407484\pi$
$102$			2.31242e23i			0.182333i
$103$		−	3.04489e22i		−	0.0213563i	−0.999943	−	0.0106781i	$-0.996601\pi$
$103$		−	3.04489e22i		−	0.0213563i	0.999943	−	0.0106781i	$-0.00339902\pi$
$104$	−1.31087e24			−0.818768
$105$	2.11455e24			1.17746
$106$		−	4.26490e24i		−	2.11953i
$107$		−	5.18109e23i		−	0.230046i	−0.993363	−	0.115023i	$-0.963306\pi$
$107$		−	5.18109e23i		−	0.230046i	0.993363	−	0.115023i	$-0.0366942\pi$
$108$	1.19933e24			0.476271
$109$			1.06840e24i			0.379854i	0.981798	+	0.189927i	$0.0608252\pi$
$109$			1.06840e24i			0.379854i	−0.981798	+	0.189927i	$0.939175\pi$
$110$	1.04186e25			3.31970
$111$			8.87937e24i			2.53809i
$112$		−	4.87463e24i		−	1.25120i
$113$		−	7.52777e24i		−	1.73670i	−0.495951	−	0.868350i	$-0.665181\pi$
$113$		−	7.52777e24i		−	1.73670i	0.495951	−	0.868350i	$-0.334819\pi$
$114$			1.68933e25i			3.50635i
$115$	−3.90198e24	−	7.57448e24i	−0.729307	−	1.41572i
$116$	−5.99777e24			−1.01040
$117$	2.30863e24			0.350851
$118$	−2.41093e25			−3.30827
$119$	−2.75072e23			−0.0341101
$120$		−	5.49046e25i		−	6.15792i
$121$	−2.63095e24			−0.267108
$122$			3.54033e25i			3.25631i
$123$	−2.80846e25			−2.34211
$124$	−1.93264e25			−1.46248
$125$		−	1.24253e25i		−	0.853863i
$126$			1.70208e25i			1.06300i
$127$	1.65652e25			0.940921			0.470460	−	0.882421i	$-0.344088\pi$
$127$	1.65652e25			0.940921			0.470460	+	0.882421i	$0.344088\pi$
$128$	−2.19687e24			−0.113576
$129$		−	5.47908e25i		−	2.58007i
$130$		−	2.12542e25i		−	0.912273i
$131$	−3.23922e25			−1.26819			−0.634095	−	0.773255i	$-0.718627\pi$
$131$	−3.23922e25			−1.26819			−0.634095	+	0.773255i	$0.718627\pi$
$132$			1.11788e26i			3.99490i
$133$	−2.00953e25			−0.655956
$134$		−	2.35757e25i		−	0.703408i
$135$			1.14411e25i			0.312220i
$136$			7.14230e24i			0.178390i
$137$		−	1.39722e25i		−	0.319609i	−0.987149	−	0.159804i	$-0.948914\pi$
$137$		−	1.39722e25i		−	0.319609i	0.987149	−	0.159804i	$-0.0510863\pi$
$138$	1.14725e26	−	5.91005e25i	2.40498	−	1.23892i
$139$	−1.86441e25			−0.358396			−0.179198	−	0.983813i	$-0.557350\pi$
$139$	−1.86441e25			−0.358396			−0.179198	+	0.983813i	$0.557350\pi$
$140$	1.11006e26			1.95799
$141$	−1.33572e26			−2.16314
$142$	1.22907e26			1.82859
$143$			2.54608e25i			0.348209i
$144$	2.22908e26			2.80399
$145$		−	5.72160e25i		−	0.662371i
$146$	1.90893e25			0.203496
$147$	1.10641e26			1.08670
$148$			4.66134e26i			4.22057i
$149$			4.43102e25i			0.370057i	0.982733	+	0.185028i	$0.0592377\pi$
$149$			4.43102e25i			0.370057i	−0.982733	+	0.185028i	$0.940762\pi$
$150$	5.39205e26			4.15584
$151$	−1.42601e26			−1.01484			−0.507422	−	0.861698i	$-0.669402\pi$
$151$	−1.42601e26			−1.01484			−0.507422	+	0.861698i	$0.669402\pi$
$152$			5.21778e26i			3.43054i
$153$		−	1.25786e25i		−	0.0764423i
$154$	−1.87715e26			−1.05499
$155$		−	1.84365e26i		−	0.958734i
$156$	2.28049e26			1.09782
$157$			4.56975e25i			0.203750i	0.994797	+	0.101875i	$0.0324841\pi$
$157$			4.56975e25i			0.203750i	−0.994797	+	0.101875i	$0.967516\pi$
$158$		−	5.95234e26i		−	2.45925i
$159$			4.36535e26i			1.67207i
$160$		−	8.65722e26i		−	3.07566i
$161$	7.03026e25	+	1.36471e26i	0.231772	+	0.449914i
$162$	5.12945e26			1.56997
$163$	1.25314e26			0.356244			0.178122	−	0.984008i	$-0.442998\pi$
$163$	1.25314e26			0.356244			0.178122	+	0.984008i	$0.442998\pi$
$164$	−1.47434e27			−3.89467
$165$	−1.06640e27			−2.61887
$166$		−	2.69939e26i		−	0.616551i
$167$	−1.55595e26			−0.330673			−0.165336	−	0.986237i	$-0.552871\pi$
$167$	−1.55595e26			−0.330673			−0.165336	+	0.986237i	$0.552871\pi$
$168$			9.89227e26i			1.95697i
$169$	−4.90860e26			−0.904310
$170$	−1.15804e26			−0.198763
$171$		−	9.18922e26i		−	1.47003i
$172$		−	2.87632e27i		−	4.29038i
$173$	3.74892e25			0.0521618			0.0260809	−	0.999660i	$-0.491697\pi$
$173$	3.74892e25			0.0521618			0.0260809	+	0.999660i	$0.491697\pi$
$174$	8.66611e26			1.12521
$175$			6.41408e26i			0.777459i
$176$			2.45836e27i			2.78287i
$177$	2.46771e27			2.60985
$178$		−	1.27026e26i		−	0.125560i
$179$	3.79011e25			0.0350280			0.0175140	−	0.999847i	$-0.494425\pi$
$179$	3.79011e25			0.0350280			0.0175140	+	0.999847i	$0.494425\pi$
$180$			5.07611e27i			4.38794i
$181$		−	3.03173e26i		−	0.245215i	−0.992455	−	0.122608i	$-0.960874\pi$
$181$		−	3.03173e26i		−	0.245215i	0.992455	−	0.122608i	$-0.0391257\pi$
$182$			3.82941e26i			0.289918i
$183$		−	3.62371e27i		−	2.56886i
$184$	3.54349e27	−	1.82542e27i	2.35298	−	1.21213i
$185$	−4.44671e27			−2.76680
$186$	2.79245e27			1.62866
$187$	1.38723e26			0.0758666
$188$	−7.01203e27			−3.59707
$189$		−	2.06136e26i		−	0.0992228i
$190$	−8.46000e27			−3.82232
$191$		−	2.52559e27i		−	1.07142i	−0.844400	−	0.535712i	$-0.820043\pi$
$191$		−	2.52559e27i		−	1.07142i	0.844400	−	0.535712i	$-0.179957\pi$
$192$	4.04852e27			1.61318
$193$	−3.32813e27			−1.24598			−0.622992	−	0.782229i	$-0.714083\pi$
$193$	−3.32813e27			−1.24598			−0.622992	+	0.782229i	$0.714083\pi$
$194$			2.06406e27i			0.726277i
$195$			2.17548e27i			0.719680i
$196$	5.80825e27			1.80706
$197$	1.60425e27			0.469545			0.234773	−	0.972050i	$-0.424565\pi$
$197$	1.60425e27			0.469545			0.234773	+	0.972050i	$0.424565\pi$
$198$		−	8.58385e27i		−	2.36428i
$199$			6.07370e27i			1.57477i	0.616463	+	0.787384i	$0.288565\pi$
$199$			6.07370e27i			1.57477i	−0.616463	+	0.787384i	$0.711435\pi$
$200$	1.66543e28			4.06599
$201$			2.41310e27i			0.554909i
$202$	−4.89874e27			−1.06137
$203$			1.03087e27i			0.210500i
$204$		−	1.24253e27i		−	0.239190i
$205$		−	1.40645e28i		−	2.55316i
$206$			2.30959e26i			0.0395483i
$207$	−6.24057e27	+	3.21481e27i	−1.00828	+	0.519412i
$208$	5.01510e27			0.764750
$209$	1.01344e28			1.45896
$210$	−1.60391e28			−2.18046
$211$	−1.09266e28			−1.40312			−0.701561	−	0.712609i	$-0.747513\pi$
$211$	−1.09266e28			−1.40312			−0.701561	+	0.712609i	$0.747513\pi$
$212$			2.29165e28i			2.78047i
$213$	−1.25802e28			−1.44255
$214$			3.92992e27i			0.426009i
$215$	2.74387e28			2.81257
$216$	−5.35235e27			−0.518919
$217$			3.32174e27i			0.304683i
$218$		−	8.10398e27i		−	0.703429i
$219$	−1.95389e27			−0.160535
$220$	−5.59822e28			−4.35489
$221$		−	2.82999e26i		−	0.0208486i
$222$		−	6.73512e28i		−	4.70012i
$223$	−9.79037e26			−0.0647353			−0.0323676	−	0.999476i	$-0.510305\pi$
$223$	−9.79037e26			−0.0647353			−0.0323676	+	0.999476i	$0.510305\pi$
$224$			1.55979e28i			0.977438i
$225$	−2.93304e28			−1.74232
$226$			5.70991e28i			3.21609i
$227$		−	1.12221e28i		−	0.599467i	−0.954023	−	0.299733i	$-0.903102\pi$
$227$		−	1.12221e28i		−	0.599467i	0.954023	−	0.299733i	$-0.0968977\pi$
$228$		−	9.07722e28i		−	4.59976i
$229$		−	2.03649e28i		−	0.979165i	−0.871957	−	0.489583i	$-0.837149\pi$
$229$		−	2.03649e28i		−	0.979165i	0.871957	−	0.489583i	$-0.162851\pi$
$230$	2.95970e28	+	5.74534e28i	1.35056	+	2.62169i
$231$	1.92136e28			0.832269
$232$	2.67668e28			1.10088
$233$	−1.09927e28			−0.429372			−0.214686	−	0.976683i	$-0.568873\pi$
$233$	−1.09927e28			−0.429372			−0.214686	+	0.976683i	$0.568873\pi$
$234$	−1.75112e28			−0.649720
$235$		−	6.68915e28i		−	2.35807i
$236$	1.29546e29			4.33990
$237$			6.09253e28i			1.94007i
$238$	2.08646e27			0.0631664
$239$	−4.98514e28			−1.43517			−0.717583	−	0.696473i	$-0.754752\pi$
$239$	−4.98514e28			−1.43517			−0.717583	+	0.696473i	$0.754752\pi$
$240$			2.10052e29i			5.75165i
$241$			5.80118e28i			1.51116i	0.655054	+	0.755582i	$0.272646\pi$
$241$			5.80118e28i			1.51116i	−0.655054	+	0.755582i	$0.727354\pi$
$242$	1.99561e28			0.494642
$243$	−6.08135e28			−1.43458
$244$		−	1.90232e29i		−	4.27174i
$245$			5.54080e28i			1.18462i
$246$	2.13025e29			4.33720
$247$		−	2.06743e28i		−	0.400930i
$248$	8.62496e28			1.59344
$249$			2.76297e28i			0.486389i
$250$			9.42478e28i			1.58122i
$251$		−	1.31725e28i		−	0.210661i	−0.994437	−	0.105331i	$-0.966410\pi$
$251$		−	1.31725e28i		−	0.210661i	0.994437	−	0.105331i	$-0.0335901\pi$
$252$		−	9.14571e28i		−	1.39448i
$253$	−3.54548e28	−	6.88245e28i	−0.515500	−	1.00068i
$254$	−1.25649e29			−1.74243
$255$	1.18531e28			0.156802
$256$	−7.08234e28			−0.893918
$257$	5.76443e28			0.694319			0.347159	−	0.937806i	$-0.387146\pi$
$257$	5.76443e28			0.694319			0.347159	+	0.937806i	$0.387146\pi$
$258$			4.15596e29i			4.77787i
$259$	8.01171e28			0.879282
$260$			1.14205e29i			1.19675i
$261$	−4.71399e28			−0.471740
$262$	2.45699e29			2.34848
$263$		−	2.13176e29i		−	1.94657i	−0.229607	−	0.973283i	$-0.573744\pi$
$263$		−	2.13176e29i		−	1.94657i	0.229607	−	0.973283i	$-0.426256\pi$
$264$		−	4.98884e29i		−	4.35263i
$265$	−2.18613e29			−1.82274
$266$	1.52425e29			1.21472
$267$			1.30017e28i			0.0990525i
$268$			1.26679e29i			0.922754i
$269$	−1.02657e29			−0.715094			−0.357547	−	0.933895i	$-0.616387\pi$
$269$	−1.02657e29			−0.715094			−0.357547	+	0.933895i	$0.616387\pi$
$270$		−	8.67819e28i		−	0.578181i
$271$	1.40090e28			0.0892844			0.0446422	−	0.999003i	$-0.485785\pi$
$271$	1.40090e28			0.0892844			0.0446422	+	0.999003i	$0.485785\pi$
$272$		−	2.73248e28i		−	0.166621i
$273$		−	3.91960e28i		−	0.228713i
$274$			1.05981e29i			0.591863i
$275$		−	3.23473e29i		−	1.72920i
$276$	−6.16451e29	+	3.17563e29i	−3.15493	+	1.62525i
$277$	−2.25808e29			−1.10658			−0.553290	−	0.832989i	$-0.686628\pi$
$277$	−2.25808e29			−1.10658			−0.553290	+	0.832989i	$0.686628\pi$
$278$	1.41418e29			0.663691
$279$	−1.51897e29			−0.682809
$280$	−4.95396e29			−2.13332
$281$			1.94644e29i			0.803091i	0.915839	+	0.401545i	$0.131527\pi$
$281$			1.94644e29i			0.803091i	−0.915839	+	0.401545i	$0.868473\pi$
$282$	1.01316e30			4.00579
$283$		−	2.30282e29i		−	0.872615i	−0.899798	−	0.436308i	$-0.856286\pi$
$283$		−	2.30282e29i		−	0.872615i	0.899798	−	0.436308i	$-0.143714\pi$
$284$	−6.60414e29			−2.39881
$285$	8.65925e29			3.01538
$286$		−	1.93124e29i		−	0.644827i
$287$			2.53403e29i			0.811387i
$288$	−7.13263e29			−2.19048
$289$	3.37907e29			0.995458
$290$			4.33991e29i			1.22660i
$291$		−	2.11268e29i		−	0.572950i
$292$	−1.02572e29			−0.266953
$293$		−	1.00109e29i		−	0.250070i	−0.992152	−	0.125035i	$-0.960096\pi$
$293$		−	1.00109e29i		−	0.250070i	0.992152	−	0.125035i	$-0.0399042\pi$
$294$	−8.39227e29			−2.01239
$295$			1.23581e30i			2.84503i
$296$		−	2.08026e30i		−	4.59850i
$297$			1.03958e29i			0.220688i
$298$		−	3.36098e29i		−	0.685285i
$299$	−1.40403e29	+	7.23285e28i	−0.274994	+	0.141663i
$300$	−2.89730e30			−5.45177
$301$	−4.94369e29			−0.893826
$302$	1.08165e30			1.87933
$303$	5.01411e29			0.837303
$304$		−	1.99620e30i		−	3.20421i
$305$	1.81472e30			2.80035
$306$			9.54100e28i			0.141559i
$307$	6.98940e29			0.997193			0.498597	−	0.866834i	$-0.333849\pi$
$307$	6.98940e29			0.997193			0.498597	+	0.866834i	$0.333849\pi$
$308$	1.00864e30			1.38397
$309$		−	2.36398e28i		−	0.0311992i
$310$			1.39843e30i			1.77542i
$311$	−3.34364e29			−0.408409			−0.204204	−	0.978928i	$-0.565461\pi$
$311$	−3.34364e29			−0.408409			−0.204204	+	0.978928i	$0.565461\pi$
$312$	−1.01773e30			−1.19613
$313$			1.58699e30i			1.79491i	0.441103	+	0.897456i	$0.354587\pi$
$313$			1.58699e30i			1.79491i	−0.441103	+	0.897456i	$0.645413\pi$
$314$		−	3.46622e29i		−	0.377311i
$315$	8.72459e29			0.914151
$316$			3.19836e30i			3.22613i
$317$	−1.03048e30			−1.00076			−0.500378	−	0.865807i	$-0.666806\pi$
$317$	−1.03048e30			−1.00076			−0.500378	+	0.865807i	$0.666806\pi$
$318$		−	3.31117e30i		−	3.09640i
$319$		−	5.19885e29i		−	0.468187i
$320$			2.02746e30i			1.75854i
$321$		−	4.02248e29i		−	0.336073i
$322$	−5.33255e29	−	1.03515e30i	−0.429204	−	0.833168i
$323$	−1.12644e29			−0.0873533
$324$	−2.75620e30			−2.05954
$325$	−6.59890e29			−0.475194
$326$	−9.50522e29			−0.659707
$327$			8.29484e29i			0.554926i
$328$	6.57965e30			4.24342
$329$			1.20520e30i			0.749389i
$330$	8.08879e30			4.84972
$331$	7.48192e29			0.432591			0.216295	−	0.976328i	$-0.430603\pi$
$331$	7.48192e29			0.432591			0.216295	+	0.976328i	$0.430603\pi$
$332$			1.45046e30i			0.808813i
$333$			3.66361e30i			1.97051i
$334$	1.18021e30			0.612352
$335$	−1.20846e30			−0.604913
$336$		−	3.78455e30i		−	1.82786i
$337$			2.61722e30i			1.21978i	0.792487	+	0.609889i	$0.208786\pi$
$337$			2.61722e30i			1.21978i	−0.792487	+	0.609889i	$0.791214\pi$
$338$	3.72324e30			1.67464
$339$		−	5.84439e30i		−	2.53713i
$340$	6.22245e29			0.260744
$341$		−	1.67521e30i		−	0.677667i
$342$			6.97014e30i			2.72225i
$343$		−	2.34035e30i		−	0.882573i
$344$			1.28364e31i			4.67457i
$345$	−3.02941e30	−	5.88065e30i	−1.06544	−	2.06822i
$346$	−2.84360e29			−0.0965952
$347$	3.03827e30			0.996949			0.498474	−	0.866904i	$-0.333894\pi$
$347$	3.03827e30			0.996949			0.498474	+	0.866904i	$0.333894\pi$
$348$	−4.65653e30			−1.47609
$349$	−3.45731e30			−1.05885			−0.529424	−	0.848358i	$-0.677592\pi$
$349$	−3.45731e30			−1.05885			−0.529424	+	0.848358i	$0.677592\pi$
$350$		−	4.86516e30i		−	1.43973i
$351$	2.12076e29			0.0606465
$352$		−	7.86627e30i		−	2.17399i
$353$	−4.13194e29			−0.110372			−0.0551858	−	0.998476i	$-0.517575\pi$
$353$	−4.13194e29			−0.110372			−0.0551858	+	0.998476i	$0.517575\pi$
$354$	−1.87179e31			−4.83301
$355$		−	6.30004e30i		−	1.57254i
$356$			6.82543e29i			0.164714i
$357$	−2.13560e29			−0.0498311
$358$	−2.87485e29			−0.0648662
$359$			2.07923e30i			0.453700i	0.973930	+	0.226850i	$0.0728428\pi$
$359$			2.07923e30i			0.453700i	−0.973930	+	0.226850i	$0.927157\pi$
$360$		−	2.26536e31i		−	4.78086i
$361$	−3.33042e30			−0.679850
$362$			2.29960e30i			0.454099i
$363$	−2.04261e30			−0.390216
$364$		−	2.05765e30i		−	0.380325i
$365$		−	9.78490e29i		−	0.175002i
$366$			2.74863e31i			4.75712i
$367$			4.10100e30i			0.686906i	0.939170	+	0.343453i	$0.111597\pi$
$367$			4.10100e30i			0.686906i	−0.939170	+	0.343453i	$0.888403\pi$
$368$	−1.35566e31	+	6.98364e30i	−2.19774	+	1.13216i
$369$	−1.15876e31			−1.81835
$370$	3.37288e31			5.12366
$371$	3.93879e30			0.579262
$372$	−1.50046e31			−2.13653
$373$			1.29773e31i			1.78928i	0.446788	+	0.894640i	$0.352568\pi$
$373$			1.29773e31i			1.78928i	−0.446788	+	0.894640i	$0.647432\pi$
$374$	−1.05224e30			−0.140493
$375$		−	9.64675e30i		−	1.24740i
$376$	3.12932e31			3.91918
$377$	−1.06058e30			−0.128661
$378$			1.56357e30i			0.183745i
$379$		−	1.48340e30i		−	0.168883i	−0.996428	−	0.0844417i	$-0.973089\pi$
$379$		−	1.48340e30i		−	0.168883i	0.996428	−	0.0844417i	$-0.0269107\pi$
$380$	4.54579e31			5.01425
$381$	1.28609e31			1.37458
$382$			1.91569e31i			1.98411i
$383$			6.43681e30i			0.646079i	0.946386	+	0.323039i	$0.104705\pi$
$383$			6.43681e30i			0.646079i	−0.946386	+	0.323039i	$0.895295\pi$
$384$	−1.70560e30			−0.165922
$385$			9.62197e30i			0.907267i
$386$	2.52443e31			2.30736
$387$		−	2.26066e31i		−	2.00310i
$388$		−	1.10908e31i		−	0.952755i
$389$		−	1.48758e31i		−	1.23904i	−0.784980	−	0.619521i	$-0.787327\pi$
$389$		−	1.48758e31i		−	1.23904i	0.784980	−	0.619521i	$-0.212673\pi$
$390$		−	1.65013e31i		−	1.33273i
$391$	3.94082e29	+	7.64989e29i	0.0308649	+	0.0599148i
$392$	−2.59210e31			−1.96888
$393$	−2.51485e31			−1.85269
$394$	−1.21684e31			−0.869522
$395$	−3.05108e31			−2.11489
$396$			4.61234e31i			3.10155i
$397$	−7.33844e30			−0.478760			−0.239380	−	0.970926i	$-0.576944\pi$
$397$	−7.33844e30			−0.478760			−0.239380	+	0.970926i	$0.576944\pi$
$398$		−	4.60698e31i		−	2.91621i
$399$	−1.56015e31			−0.958280
$400$	−6.37154e31			−3.79773
$401$			3.24739e30i			0.187846i	0.995579	+	0.0939231i	$0.0299408\pi$
$401$			3.24739e30i			0.187846i	−0.995579	+	0.0939231i	$0.970059\pi$
$402$		−	1.83036e31i		−	1.02760i
$403$	−3.41746e30			−0.186227
$404$	2.63223e31			1.39234
$405$		−	2.62928e31i		−	1.35013i
$406$		−	7.81929e30i		−	0.389812i
$407$	−4.04044e31			−1.95567
$408$			5.54512e30i			0.260609i
$409$	9.57044e30			0.436769			0.218385	−	0.975863i	$-0.429921\pi$
$409$	9.57044e30			0.436769			0.218385	+	0.975863i	$0.429921\pi$
$410$			1.06681e32i			4.72803i
$411$		−	1.08477e31i		−	0.466913i
$412$		−	1.24101e30i		−	0.0518809i
$413$		−	2.22658e31i		−	0.904142i
$414$	4.73355e31	−	2.43848e31i	1.86717	−	0.961866i
$415$	−1.38367e31			−0.530219
$416$	−1.60473e31			−0.597424
$417$	−1.44748e31			−0.523577
$418$	−7.68706e31			−2.70175
$419$		−	4.62348e30i		−	0.157907i	−0.996878	−	0.0789534i	$-0.974842\pi$
$419$		−	4.62348e30i		−	0.157907i	0.996878	−	0.0789534i	$-0.0251578\pi$
$420$	8.61825e31			2.86041
$421$			2.86447e30i			0.0923972i	0.998932	+	0.0461986i	$0.0147107\pi$
$421$			2.86447e30i			0.0923972i	−0.998932	+	0.0461986i	$0.985289\pi$
$422$	8.28796e31			2.59835
$423$	−5.51115e31			−1.67941
$424$		−	1.02271e32i		−	3.02945i
$425$			3.59542e30i			0.103534i
$426$	9.54223e31			2.67137
$427$	−3.26962e31			−0.889944
$428$		−	2.11165e31i		−	0.558853i
$429$			1.97672e31i			0.508695i
$430$	−2.08126e32			−5.20842
$431$			5.45776e31i			1.32827i	0.747611	+	0.664137i	$0.231201\pi$
$431$			5.45776e31i			1.32827i	−0.747611	+	0.664137i	$0.768799\pi$
$432$	2.04769e31			0.484684
$433$		−	4.90968e31i		−	1.13031i	−0.824984	−	0.565156i	$-0.808816\pi$
$433$		−	4.90968e31i		−	1.13031i	0.824984	−	0.565156i	$-0.191184\pi$
$434$		−	2.51958e31i		−	0.564224i
$435$		−	4.44212e31i		−	0.967651i
$436$			4.35449e31i			0.922782i
$437$	2.87895e31	+	5.58859e31i	0.593550	+	1.15219i
$438$	1.48205e31			0.297285
$439$	−7.31449e31			−1.42761			−0.713805	−	0.700345i	$-0.753030\pi$
$439$	−7.31449e31			−1.42761			−0.713805	+	0.700345i	$0.753030\pi$
$440$	2.49836e32			4.74486
$441$	4.56503e31			0.843686
$442$			2.14658e30i			0.0386082i
$443$	7.90449e31			1.38366			0.691830	−	0.722061i	$-0.256805\pi$
$443$	7.90449e31			1.38366			0.691830	+	0.722061i	$0.256805\pi$
$444$			3.61896e32i			6.16578i
$445$	−6.51114e30			−0.107978
$446$	7.42612e30			0.119879
$447$			3.44014e31i			0.540612i
$448$		−	3.65291e31i		−	0.558860i
$449$	7.38820e31			1.10048			0.550241	−	0.835006i	$-0.314536\pi$
$449$	7.38820e31			1.10048			0.550241	+	0.835006i	$0.314536\pi$
$450$	2.22475e32			3.22650
$451$		−	1.27795e32i		−	1.80466i
$452$		−	3.06809e32i		−	4.21897i
$453$	−1.10712e32			−1.48258
$454$			8.51213e31i			1.11012i
$455$	1.96290e31			0.249322
$456$			4.05097e32i			5.01165i
$457$			3.67034e30i			0.0442295i	0.999755	+	0.0221147i	$0.00703991\pi$
$457$			3.67034e30i			0.0442295i	−0.999755	+	0.0221147i	$0.992960\pi$
$458$			1.54470e32i			1.81325i
$459$		−	1.15550e30i		−	0.0132134i
$460$	−1.59033e32	−	3.08713e32i	−1.77171	−	3.43923i
$461$	1.04432e32			1.13350			0.566750	−	0.823890i	$-0.308200\pi$
$461$	1.04432e32			1.13350			0.566750	+	0.823890i	$0.308200\pi$
$462$	−1.45737e32			−1.54123
$463$	1.24125e32			1.27905			0.639523	−	0.768772i	$-0.279132\pi$
$463$	1.24125e32			1.27905			0.639523	+	0.768772i	$0.279132\pi$
$464$	−1.02403e32			−1.02825
$465$		−	1.43137e32i		−	1.40061i
$466$	8.33812e31			0.795128
$467$		−	4.52826e31i		−	0.420852i	−0.977610	−	0.210426i	$-0.932515\pi$
$467$		−	4.52826e31i		−	0.420852i	0.977610	−	0.210426i	$-0.0674850\pi$
$468$	9.40925e31			0.852324
$469$	2.17730e31			0.192240
$470$			5.07380e32i			4.36676i
$471$			3.54785e31i			0.297656i
$472$	−5.78135e32			−4.72852
$473$	2.49318e32			1.98802
$474$		−	4.62126e32i		−	3.59270i
$475$			2.62662e32i			1.99101i
$476$	−1.12111e31			−0.0828639
$477$			1.80114e32i			1.29815i
$478$	3.78129e32			2.65770
$479$		−	1.75664e32i		−	1.20408i	−0.798464	−	0.602042i	$-0.794354\pi$
$479$		−	1.75664e32i		−	1.20408i	0.798464	−	0.602042i	$-0.205646\pi$
$480$		−	6.72127e32i		−	4.49321i
$481$			8.24257e31i			0.537430i
$482$		−	4.40027e32i		−	2.79843i
$483$	5.45814e31	+	1.05953e32i	0.338594	+	0.657275i
$484$	−1.07229e32			−0.648888
$485$	1.05801e32			0.624580
$486$	4.61278e32			2.65661
$487$	−7.58562e31			−0.426230			−0.213115	−	0.977027i	$-0.568361\pi$
$487$	−7.58562e31			−0.426230			−0.213115	+	0.977027i	$0.568361\pi$
$488$			8.48963e32i			4.65426i
$489$	9.72909e31			0.520434
$490$		−	4.20277e32i		−	2.19373i
$491$	−2.75345e32			−1.40249			−0.701244	−	0.712922i	$-0.747371\pi$
$491$	−2.75345e32			−1.40249			−0.701244	+	0.712922i	$0.747371\pi$
$492$	−1.14464e33			−5.68969
$493$			5.77856e30i			0.0280321i
$494$			1.56818e32i			0.742457i
$495$	−4.39995e32			−2.03323
$496$	−3.29971e32			−1.48832
$497$			1.13509e32i			0.499751i
$498$		−	2.09575e32i		−	0.900714i
$499$	6.61065e31			0.277356			0.138678	−	0.990338i	$-0.455715\pi$
$499$	6.61065e31			0.277356			0.138678	+	0.990338i	$0.455715\pi$
$500$		−	5.06419e32i		−	2.07429i
$501$	−1.20800e32			−0.483076
$502$			9.99154e31i			0.390110i
$503$		−	3.92283e31i		−	0.149549i	−0.997200	−	0.0747746i	$-0.976176\pi$
$503$		−	3.92283e31i		−	0.149549i	0.997200	−	0.0747746i	$-0.0238237\pi$
$504$			4.08153e32i			1.51935i
$505$			2.51102e32i			0.912754i
$506$	2.68929e32	+	5.22043e32i	0.954622	+	1.85310i
$507$	−3.81093e32			−1.32110
$508$	6.75149e32			2.28578
$509$	−2.01147e31			−0.0665119			−0.0332560	−	0.999447i	$-0.510588\pi$
$509$	−2.01147e31			−0.0665119			−0.0332560	+	0.999447i	$0.510588\pi$
$510$	−8.99074e31			−0.290371
$511$			1.76296e31i			0.0556151i
$512$	5.74062e32			1.76897
$513$		−	8.44143e31i		−	0.254102i
$514$	−4.37240e32			−1.28577
$515$	1.18386e31			0.0340106
$516$		−	2.23311e33i		−	6.26777i
$517$		−	6.07800e32i		−	1.66677i
$518$	−6.07699e32			−1.62829
$519$	2.91057e31			0.0762027
$520$		−	5.09671e32i		−	1.30392i
$521$		−	1.11815e32i		−	0.279541i	−0.990184	−	0.139771i	$-0.955364\pi$
$521$		−	1.11815e32i		−	0.279541i	0.990184	−	0.139771i	$-0.0446365\pi$
$522$	3.57562e32			0.873585
$523$		−	2.13324e32i		−	0.509353i	−0.967026	−	0.254676i	$-0.918031\pi$
$523$		−	2.13324e32i		−	0.509353i	0.967026	−	0.254676i	$-0.0819690\pi$
$524$	−1.32021e33			−3.08082
$525$			4.97974e32i			1.13578i
$526$			1.61697e33i			3.60473i
$527$			1.86200e31i			0.0405745i
$528$			1.90861e33i			4.06547i
$529$	2.78813e32	−	3.91030e32i	0.580557	−	0.814220i
$530$	1.65821e33			3.37542
$531$	1.01817e33			2.02622
$532$	−8.19023e32			−1.59352
$533$	−2.60704e32			−0.495932
$534$		−	9.86197e31i		−	0.183429i
$535$	2.01442e32			0.366357
$536$		−	5.65340e32i		−	1.00538i
$537$	2.94255e31			0.0511720
$538$	7.78670e32			1.32424
$539$			5.03457e32i			0.837332i
$540$			4.66303e32i			0.758478i
$541$	7.25303e32			1.15386			0.576929	−	0.816795i	$-0.304251\pi$
$541$	7.25303e32			1.15386			0.576929	+	0.816795i	$0.304251\pi$
$542$	−1.06260e32			−0.165340
$543$		−	2.35376e32i		−	0.358232i
$544$			8.74340e31i			0.130165i
$545$	−4.15398e32			−0.604931
$546$			2.97307e32i			0.423539i
$547$	−6.58711e32			−0.918009			−0.459004	−	0.888434i	$-0.651794\pi$
$547$	−6.58711e32			−0.918009			−0.459004	+	0.888434i	$0.651794\pi$
$548$		−	5.69465e32i		−	0.776426i
$549$		−	1.49514e33i		−	1.99440i
$550$			2.45358e33i			3.20220i
$551$			4.22150e32i			0.539073i
$552$	2.75109e33	−	1.41722e33i	3.43744	−	1.77079i
$553$	5.49719e32			0.672108
$554$	1.71279e33			2.04921
$555$	−3.45232e33			−4.04199
$556$	−7.59875e32			−0.870652
$557$			1.26835e33i			1.42226i	0.703062	+	0.711129i	$0.251816\pi$
$557$			1.26835e33i			1.42226i	−0.703062	+	0.711129i	$0.748184\pi$
$558$	1.15216e33			1.26445
$559$		−	5.08614e32i		−	0.546320i
$560$	1.89527e33			1.99257
$561$	1.07702e32			0.110833
$562$		−	1.47640e33i		−	1.48719i
$563$			1.59371e33i			1.57148i	0.618560	+	0.785738i	$0.287717\pi$
$563$			1.59371e33i			1.57148i	−0.618560	+	0.785738i	$0.712283\pi$
$564$	−5.44398e33			−5.25493
$565$	2.92681e33			2.76576
$566$			1.74672e33i			1.61594i
$567$			4.73723e32i			0.429069i
$568$	2.94728e33			2.61362
$569$			4.21895e32i			0.366317i	0.983083	+	0.183159i	$0.0586322\pi$
$569$			4.21895e32i			0.366317i	−0.983083	+	0.183159i	$0.941368\pi$
$570$	−6.56815e33			−5.58399
$571$		−	1.92981e33i		−	1.60650i	−0.595640	−	0.803251i	$-0.703102\pi$
$571$		−	1.92981e33i		−	1.60650i	0.595640	−	0.803251i	$-0.296898\pi$
$572$			1.03771e33i			0.845906i
$573$		−	1.96081e33i		−	1.56523i
$574$		−	1.92209e33i		−	1.50256i
$575$	1.78378e33	−	9.18912e32i	1.36561	−	0.703493i
$576$	1.67041e33			1.25243
$577$	7.67282e32			0.563437			0.281719	−	0.959497i	$-0.409096\pi$
$577$	7.67282e32			0.563437			0.281719	+	0.959497i	$0.409096\pi$
$578$	−2.56307e33			−1.84343
$579$	−2.58388e33			−1.82024
$580$		−	2.33195e33i		−	1.60910i
$581$	2.49298e32			0.168502
$582$			1.60249e33i			1.06101i
$583$	−1.98639e33			−1.28838
$584$	4.57757e32			0.290858
$585$			8.97599e32i			0.558743i
$586$			7.59338e32i			0.463088i
$587$	9.72059e32			0.580812			0.290406	−	0.956904i	$-0.406210\pi$
$587$	9.72059e32			0.580812			0.290406	+	0.956904i	$0.406210\pi$
$588$	4.50939e33			2.63992
$589$			1.36028e33i			0.780269i
$590$		−	9.37375e33i		−	5.26853i
$591$	1.24550e33			0.685954
$592$			7.95857e33i			4.29512i
$593$	1.03826e31			0.00549100			0.00274550	−	0.999996i	$-0.499126\pi$
$593$	1.03826e31			0.00549100			0.00274550	+	0.999996i	$0.499126\pi$
$594$		−	7.88532e32i		−	0.408679i
$595$		−	1.06949e32i		−	0.0543215i
$596$			1.80595e33i			0.898980i
$597$			4.71548e33i			2.30056i
$598$	1.06498e33	−	5.48621e32i	0.509244	−	0.262336i
$599$	−2.49672e33			−1.17017			−0.585084	−	0.810972i	$-0.698939\pi$
$599$	−2.49672e33			−1.17017			−0.585084	+	0.810972i	$0.698939\pi$
$600$	1.29300e34			5.93996
$601$	2.06676e33			0.930672			0.465336	−	0.885134i	$-0.345933\pi$
$601$	2.06676e33			0.930672			0.465336	+	0.885134i	$0.345933\pi$
$602$	3.74985e33			1.65522
$603$			9.95639e32i			0.430818i
$604$	−5.81199e33			−2.46537
$605$		−	1.02292e33i		−	0.425379i
$606$	−3.80327e33			−1.55055
$607$	1.37636e33			0.550132			0.275066	−	0.961425i	$-0.411300\pi$
$607$	1.37636e33			0.550132			0.275066	+	0.961425i	$0.411300\pi$
$608$			6.38746e33i			2.50314i
$609$			8.00345e32i			0.307517i
$610$	−1.37649e34			−5.18579
$611$	−1.23992e33			−0.458037
$612$		−	5.12664e32i		−	0.185702i
$613$			2.20185e33i			0.782099i	0.920370	+	0.391049i	$0.127888\pi$
$613$			2.20185e33i			0.782099i	−0.920370	+	0.391049i	$0.872112\pi$
$614$	−5.30155e33			−1.84664
$615$		−	1.09193e34i		−	3.72988i
$616$	−4.50135e33			−1.50790
$617$			4.25946e33i			1.39937i	0.714451	+	0.699685i	$0.246676\pi$
$617$			4.25946e33i			1.39937i	−0.714451	+	0.699685i	$0.753324\pi$
$618$			1.79311e32i			0.0577758i
$619$		−	3.90011e33i		−	1.23250i	−0.787549	−	0.616252i	$-0.788650\pi$
$619$		−	3.90011e33i		−	1.23250i	0.787549	−	0.616252i	$-0.211350\pi$
$620$		−	7.51416e33i		−	2.32906i
$621$	−5.73273e32	+	2.95320e32i	−0.174286	+	0.0897830i
$622$	2.53620e33			0.756307
$623$	1.17312e32			0.0343153
$624$	3.89361e33			1.11722
$625$	−6.26556e32			−0.176360
$626$		−	1.20376e34i		−	3.32389i
$627$	7.86811e33			2.13137
$628$			1.86249e33i			0.494970i
$629$	4.49097e32			0.117093
$630$	−6.61771e33			−1.69286
$631$		−	8.65919e32i		−	0.217333i	−0.994078	−	0.108666i	$-0.965342\pi$
$631$		−	8.65919e32i		−	0.217333i	0.994078	−	0.108666i	$-0.0346580\pi$
$632$		−	1.42736e34i		−	3.51502i
$633$	−8.48316e33			−2.04981
$634$	7.81632e33			1.85324
$635$			6.44060e33i			1.49845i
$636$			1.77918e34i			4.06196i
$637$	1.02706e33			0.230104
$638$			3.94340e33i			0.867006i
$639$	−5.19056e33			−1.11996
$640$		−	8.54149e32i		−	0.180873i
$641$			1.23355e33i			0.256366i	0.991751	+	0.128183i	$0.0409144\pi$
$641$			1.23355e33i			0.256366i	−0.991751	+	0.128183i	$0.959086\pi$
$642$			3.05110e33i			0.622352i
$643$			3.37076e33i			0.674833i	0.941355	+	0.337417i	$0.109553\pi$
$643$			3.37076e33i			0.674833i	−0.941355	+	0.337417i	$0.890447\pi$
$644$	2.86532e33	+	5.56214e33i	0.563045	+	1.09298i
$645$	2.13028e34			4.10885
$646$	8.54422e32			0.161764
$647$	5.56976e33			1.03511			0.517554	−	0.855651i	$-0.326843\pi$
$647$	5.56976e33			1.03511			0.517554	+	0.855651i	$0.326843\pi$
$648$	1.23003e34			2.24396
$649$			1.12290e34i			2.01096i
$650$	5.00535e33			0.879983
$651$			2.57892e33i			0.445109i
$652$	5.10741e33			0.865426
$653$	−7.17884e33			−1.19425			−0.597127	−	0.802147i	$-0.703691\pi$
$653$	−7.17884e33			−1.19425			−0.597127	+	0.802147i	$0.703691\pi$
$654$		−	6.29174e33i		−	1.02763i
$655$		−	1.25942e34i		−	2.01964i
$656$	−2.51722e34			−3.96346
$657$	−8.06171e32			−0.124636
$658$		−	9.14157e33i		−	1.38775i
$659$			8.04363e33i			1.19902i	0.800367	+	0.599511i	$0.204638\pi$
$659$			8.04363e33i			1.19902i	−0.800367	+	0.599511i	$0.795362\pi$
$660$	−4.34633e34			−6.36202
$661$			6.89259e33i			0.990752i	0.868679	+	0.495376i	$0.164970\pi$
$661$			6.89259e33i			0.990752i	−0.868679	+	0.495376i	$0.835030\pi$
$662$	−5.67513e33			−0.801088
$663$		−	2.19714e32i		−	0.0304575i
$664$		−	6.47308e33i		−	0.881239i
$665$		−	7.81310e33i		−	1.04463i
$666$		−	2.77890e34i		−	3.64907i
$667$	2.86690e33	−	1.47688e33i	0.369745	−	0.190473i
$668$	−6.34158e33			−0.803304
$669$	−7.60102e32			−0.0945711
$670$	9.16629e33			1.12020
$671$	1.64892e34			1.97938
$672$			1.21098e34i			1.42793i
$673$	3.55514e33			0.411790			0.205895	−	0.978574i	$-0.433989\pi$
$673$	3.55514e33			0.411790			0.205895	+	0.978574i	$0.433989\pi$
$674$		−	1.98519e34i		−	2.25883i
$675$	−2.69436e33			−0.301169
$676$	−2.00060e34			−2.19684
$677$			8.91448e33i			0.961683i	0.876808	+	0.480841i	$0.159669\pi$
$677$			8.91448e33i			0.961683i	−0.876808	+	0.480841i	$0.840331\pi$
$678$			4.43304e34i			4.69835i
$679$	−1.90623e33			−0.198490
$680$	−2.77694e33			−0.284093
$681$		−	8.71261e33i		−	0.875755i
$682$			1.27067e34i			1.25493i
$683$	−2.12262e33			−0.205979			−0.102989	−	0.994682i	$-0.532841\pi$
$683$	−2.12262e33			−0.205979			−0.102989	+	0.994682i	$0.532841\pi$
$684$		−	3.74524e34i		−	3.57114i
$685$	5.43243e33			0.508988
$686$			1.77518e34i			1.63438i
$687$		−	1.58108e34i		−	1.43045i
$688$		−	4.91090e34i		−	4.36616i
$689$			4.05228e33i			0.354054i
$690$	2.29784e34	+	4.46055e34i	1.97302	+	3.83001i
$691$	−1.26826e34			−1.07022			−0.535108	−	0.844784i	$-0.679729\pi$
$691$	−1.26826e34			−1.07022			−0.535108	+	0.844784i	$0.679729\pi$
$692$	1.52794e33			0.126717
$693$	7.92748e33			0.646154
$694$	−2.30457e34			−1.84619
$695$		−	7.24885e33i		−	0.570757i
$696$	2.07811e34			1.60827
$697$			1.42045e33i			0.108052i
$698$	2.62242e34			1.96081
$699$	−8.53450e33			−0.627266
$700$			2.61418e34i			1.88868i
$701$			5.71529e33i			0.405903i	0.979189	+	0.202951i	$0.0650533\pi$
$701$			5.71529e33i			0.405903i	−0.979189	+	0.202951i	$0.934947\pi$
$702$	−1.60862e33			−0.112307
$703$	3.28086e34			2.25177
$704$			1.84223e34i			1.24300i
$705$		−	5.19330e34i		−	3.44488i
$706$	3.13413e33			0.204390
$707$		−	4.52415e33i		−	0.290071i
$708$	1.00576e35			6.34011
$709$			2.19102e34i			1.35798i	0.734149	+	0.678988i	$0.237581\pi$
$709$			2.19102e34i			1.35798i	−0.734149	+	0.678988i	$0.762419\pi$
$710$			4.77866e34i			2.91210i
$711$			2.51377e34i			1.50622i
$712$		−	3.04604e33i		−	0.179463i
$713$	9.23791e33	−	4.75889e33i	0.535179	−	0.275696i
$714$	1.61988e33			0.0922792
$715$	−9.89923e33			−0.554535
$716$	1.54473e33			0.0850936
$717$	−3.87035e34			−2.09662
$718$		−	1.57712e34i		−	0.840180i
$719$	−2.93570e34			−1.53803			−0.769014	−	0.639232i	$-0.779252\pi$
$719$	−2.93570e34			−1.53803			−0.769014	+	0.639232i	$0.779252\pi$
$720$			8.66672e34i			4.46545i
$721$	−2.13299e32			−0.0108085
$722$	2.52617e34			1.25897
$723$			4.50390e34i			2.20764i
$724$		−	1.23564e34i		−	0.595702i
$725$	1.34743e34			0.638926
$726$	1.54934e34			0.722617
$727$		−	9.74007e33i		−	0.446837i	−0.974723	−	0.223418i	$-0.928278\pi$
$727$		−	9.74007e33i		−	0.446837i	0.974723	−	0.223418i	$-0.0717216\pi$
$728$			9.18283e33i			0.414381i
$729$	−2.81149e34			−1.24797
$730$			7.42197e33i			0.324074i
$731$	−2.77119e33			−0.119030
$732$		−	1.47692e35i		−	6.24055i
$733$		−	1.25245e33i		−	0.0520611i	−0.999661	−	0.0260305i	$-0.991713\pi$
$733$		−	1.25245e33i		−	0.0520611i	0.999661	−	0.0260305i	$-0.00828671\pi$
$734$		−	3.11066e34i		−	1.27204i
$735$			4.30175e34i			1.73060i
$736$	4.33784e34	−	2.23463e34i	1.71688	−	0.884446i
$737$	−1.09805e34			−0.427574
$738$	8.78937e34			3.36730
$739$	4.15383e34			1.56573			0.782863	−	0.622195i	$-0.213759\pi$
$739$	4.15383e34			1.56573			0.782863	+	0.622195i	$0.213759\pi$
$740$	−1.81234e35			−6.72140
$741$		−	1.60511e34i		−	0.585715i
$742$	−2.98762e34			−1.07270
$743$			1.00639e34i			0.355551i	0.984071	+	0.177776i	$0.0568901\pi$
$743$			1.00639e34i			0.355551i	−0.984071	+	0.177776i	$0.943110\pi$
$744$	6.69622e34			2.32785
$745$	−1.72279e34			−0.589328
$746$		−	9.84345e34i		−	3.31346i
$747$			1.14000e34i			0.377621i
$748$	5.65395e33			0.184303
$749$	−3.62942e33			−0.116427
$750$			7.31718e34i			2.30998i
$751$			2.24098e34i			0.696239i	0.937450	+	0.348119i	$0.113180\pi$
$751$			2.24098e34i			0.696239i	−0.937450	+	0.348119i	$0.886820\pi$
$752$	−1.19720e35			−3.66061
$753$		−	1.02269e34i		−	0.307753i
$754$	8.04461e33			0.238259
$755$		−	5.54436e34i		−	1.61617i
$756$		−	8.40147e33i		−	0.241042i
$757$			1.93694e34i			0.546974i	0.961876	+	0.273487i	$0.0881770\pi$
$757$			1.93694e34i			0.546974i	−0.961876	+	0.273487i	$0.911823\pi$
$758$			1.12518e34i			0.312745i
$759$	−2.75263e34	−	5.34338e34i	−0.753089	−	1.46189i
$760$	−2.02869e35			−5.46326
$761$	−1.63435e34			−0.433240			−0.216620	−	0.976256i	$-0.569503\pi$
$761$	−1.63435e34			−0.433240			−0.216620	+	0.976256i	$0.569503\pi$
$762$	−9.75514e34			−2.54550
$763$	7.48430e33			0.192246
$764$		−	1.02935e35i		−	2.60282i
$765$	4.89057e33			0.121737
$766$		−	4.88240e34i		−	1.19643i
$767$	2.29074e34			0.552625
$768$	−5.49857e34			−1.30592
$769$			5.56456e34i			1.30111i	0.759458	+	0.650556i	$0.225464\pi$
$769$			5.56456e34i			1.30111i	−0.759458	+	0.650556i	$0.774536\pi$
$770$		−	7.29839e34i		−	1.68011i
$771$	4.47537e34			1.01432
$772$	−1.35644e35			−3.02687
$773$		−	8.90328e34i		−	1.95612i	−0.208314	−	0.978062i	$-0.566798\pi$
$773$		−	8.90328e34i		−	1.95612i	0.208314	−	0.978062i	$-0.433202\pi$
$774$			1.71474e35i			3.70943i
$775$	4.34178e34			0.924799
$776$			4.94957e34i			1.03807i
$777$	6.22011e34			1.28453
$778$			1.12835e35i			2.29450i
$779$			1.03770e35i			2.07790i
$780$			8.86659e34i			1.74832i
$781$		−	5.72445e34i		−	1.11153i
$782$	−2.98916e33	−	5.80254e33i	−0.0571569	−	0.110952i
$783$	−4.33038e33			−0.0815426
$784$	9.91675e34			1.83898
$785$	−1.77673e34			−0.324478
$786$	1.90755e35			3.43088
$787$		−	7.06584e34i		−	1.25160i	−0.779982	−	0.625802i	$-0.784772\pi$
$787$		−	7.06584e34i		−	1.25160i	0.779982	−	0.625802i	$-0.215228\pi$
$788$	6.53844e34			1.14067
$789$		−	1.65505e35i		−	2.84372i
$790$	2.31429e35			3.91644
$791$	−5.27330e34			−0.878950
$792$		−	2.05838e35i		−	3.37928i
$793$		−	3.36384e34i		−	0.543947i
$794$	5.56630e34			0.886586
$795$	−1.69726e35			−2.66282
$796$			2.47546e35i			3.82559i
$797$			1.08128e35i			1.64603i	0.568019	+	0.823015i	$0.307710\pi$
$797$			1.08128e35i			1.64603i	−0.568019	+	0.823015i	$0.692290\pi$
$798$	1.18340e35			1.77458
$799$			6.75574e33i			0.0997956i
$800$	2.03877e35			2.96680
$801$			5.36449e33i			0.0769020i
$802$		−	2.46319e34i		−	0.347861i
$803$		−	8.89091e33i		−	0.123697i
$804$			9.83505e34i			1.34804i
$805$	−5.30602e34	+	2.73338e34i	−0.716503	+	0.369105i
$806$	2.59219e34			0.344862
$807$	−7.97009e34			−1.04467
$808$	−1.17470e35			−1.51702
$809$	−8.40942e34			−1.07000			−0.535000	−	0.844852i	$-0.679688\pi$
$809$	−8.40942e34			−1.07000			−0.535000	+	0.844852i	$0.679688\pi$
$810$			1.99434e35i			2.50023i
$811$	5.29013e34			0.653456			0.326728	−	0.945118i	$-0.394054\pi$
$811$	5.29013e34			0.653456			0.326728	+	0.945118i	$0.394054\pi$
$812$			4.20152e34i			0.511368i
$813$	1.08763e34			0.130435
$814$	3.06472e35			3.62158
$815$			4.87223e34i			0.567332i
$816$		−	2.12143e34i		−	0.243415i
$817$	−2.02448e35			−2.28902
$818$	−7.25930e34			−0.808826
$819$		−	1.61722e34i		−	0.177567i
$820$		−	5.73226e35i		−	6.20240i
$821$	5.15760e34			0.549958			0.274979	−	0.961450i	$-0.411329\pi$
$821$	5.15760e34			0.549958			0.274979	+	0.961450i	$0.411329\pi$
$822$			8.22812e34i			0.864647i
$823$	−4.44982e34			−0.460834			−0.230417	−	0.973092i	$-0.574009\pi$
$823$	−4.44982e34			−0.460834			−0.230417	+	0.973092i	$0.574009\pi$
$824$			5.53834e33i			0.0565266i
$825$		−	2.51137e35i		−	2.52617i
$826$			1.68889e35i			1.67433i
$827$			4.20884e34i			0.411241i	0.978632	+	0.205621i	$0.0659213\pi$
$827$			4.20884e34i			0.411241i	−0.978632	+	0.205621i	$0.934079\pi$
$828$	−2.54347e35	+	1.31026e35i	−2.44941	+	1.26181i
$829$	5.72020e34			0.542946			0.271473	−	0.962446i	$-0.412489\pi$
$829$	5.72020e34			0.542946			0.271473	+	0.962446i	$0.412489\pi$
$830$	1.04953e35			0.981879
$831$	−1.75312e35			−1.61659
$832$	3.75818e34			0.341584
$833$		−	5.59596e33i		−	0.0501343i
$834$	1.09793e35			0.969579
$835$		−	6.04957e34i		−	0.526607i
$836$	4.13047e35			3.54425
$837$	−1.39536e34			−0.118027
$838$			3.50697e34i			0.292418i
$839$			2.29875e35i			1.88951i	0.327778	+	0.944755i	$0.393700\pi$
$839$			2.29875e35i			1.88951i	−0.327778	+	0.944755i	$0.606300\pi$
$840$	−3.84614e35			−3.11655
$841$	−1.03529e35			−0.827008
$842$		−	2.17274e34i		−	0.171105i
$843$			1.51117e35i			1.17323i
$844$	−4.45335e35			−3.40861
$845$		−	1.90848e35i		−	1.44014i
$846$	4.18028e35			3.11000
$847$			1.84301e34i			0.135185i
$848$			3.91266e35i			2.82958i
$849$		−	1.78786e35i		−	1.27480i
$850$		−	2.72717e34i		−	0.191728i
$851$	−1.14780e35	−	2.22809e35i	−0.795629	−	1.54447i
$852$	−5.12730e35			−3.50440
$853$	−1.08005e35			−0.727874			−0.363937	−	0.931424i	$-0.618568\pi$
$853$	−1.08005e35			−0.727874			−0.363937	+	0.931424i	$0.618568\pi$
$854$	2.48005e35			1.64803
$855$	3.57279e35			2.34107
$856$			9.42385e34i			0.608896i
$857$	−1.45055e35			−0.924193			−0.462097	−	0.886830i	$-0.652903\pi$
$857$	−1.45055e35			−0.924193			−0.462097	+	0.886830i	$0.652903\pi$
$858$		−	1.49937e35i		−	0.942021i
$859$	−2.52052e34			−0.156161			−0.0780805	−	0.996947i	$-0.524879\pi$
$859$	−2.52052e34			−0.156161			−0.0780805	+	0.996947i	$0.524879\pi$
$860$	1.11832e36			6.83258
$861$			1.96736e35i			1.18535i
$862$		−	4.13978e35i		−	2.45975i
$863$	2.04820e35			1.20017			0.600085	−	0.799936i	$-0.295133\pi$
$863$	2.04820e35			1.20017			0.600085	+	0.799936i	$0.295133\pi$
$864$	−6.55220e34			−0.378636
$865$			1.45759e34i			0.0830694i
$866$			3.72405e35i			2.09315i
$867$	2.62343e35			1.45425
$868$			1.35384e35i			0.740169i
$869$	−2.77233e35			−1.49488
$870$			3.36940e35i			1.79193i
$871$			2.24004e34i			0.117500i
$872$		−	1.94331e35i		−	1.00541i
$873$		−	8.71686e34i		−	0.444825i
$874$	−2.18372e35	−	4.23902e35i	−1.09916	−	2.13367i
$875$	−8.70411e34			−0.432143
$876$	−7.96346e34			−0.389989
$877$	2.26575e35			1.09450			0.547252	−	0.836968i	$-0.315674\pi$
$877$	2.26575e35			1.09450			0.547252	+	0.836968i	$0.315674\pi$
$878$	5.54813e35			2.64370
$879$		−	7.77222e34i		−	0.365324i
$880$	−9.55815e35			−4.43182
$881$			9.51911e34i			0.435397i	0.976016	+	0.217698i	$0.0698549\pi$
$881$			9.51911e34i			0.435397i	−0.976016	+	0.217698i	$0.930145\pi$
$882$	−3.46263e35			−1.56237
$883$	2.37106e35			1.05539			0.527697	−	0.849433i	$-0.323056\pi$
$883$	2.37106e35			1.05539			0.527697	+	0.849433i	$0.323056\pi$
$884$		−	1.15342e34i		−	0.0506476i
$885$			9.59452e35i			4.15627i
$886$	−5.99566e35			−2.56231
$887$	4.28739e35			1.80763			0.903814	−	0.427925i	$-0.140755\pi$
$887$	4.28739e35			1.80763			0.903814	+	0.427925i	$0.140755\pi$
$888$		−	1.61506e36i		−	6.71791i
$889$		−	1.16042e35i		−	0.476203i
$890$	4.93878e34			0.199958
$891$		−	2.38906e35i		−	0.954322i
$892$	−3.99026e34			−0.157262
$893$			4.93538e35i			1.91912i
$894$		−	2.60939e35i		−	1.00113i
$895$			1.47360e34i			0.0557833i
$896$			1.53894e34i			0.0574810i
$897$	−1.09006e35	+	5.61541e34i	−0.401736	+	0.206953i
$898$	−5.60405e35			−2.03791
$899$	6.97811e34			0.250393
$900$	−1.19542e36			−4.23263
$901$	2.20789e34			0.0771400
$902$			9.69342e35i			3.34194i
$903$	−3.83817e35			−1.30578
$904$			1.36922e36i			4.59677i
$905$	1.17874e35			0.390514
$906$	8.39766e35			2.74549
$907$		−	2.84024e35i		−	0.916363i	−0.888859	−	0.458182i	$-0.848501\pi$
$907$		−	2.84024e35i		−	0.916363i	0.888859	−	0.458182i	$-0.151499\pi$
$908$		−	4.57380e35i		−	1.45629i
$909$	2.06881e35			0.650062
$910$	−1.48888e35			−0.461705
$911$			1.00037e35i			0.306155i	0.988214	+	0.153077i	$0.0489184\pi$
$911$			1.00037e35i			0.306155i	−0.988214	+	0.153077i	$0.951082\pi$
$912$		−	1.54980e36i		−	4.68100i
$913$	−1.25725e35			−0.374777
$914$		−	2.78400e34i		−	0.0819058i
$915$	1.40891e36			4.09100
$916$		−	8.30010e35i		−	2.37869i
$917$			2.26911e35i			0.641836i
$918$			8.76458e33i			0.0244692i
$919$			2.23591e35i			0.616124i	0.951366	+	0.308062i	$0.0996804\pi$
$919$			2.23591e35i			0.616124i	−0.951366	+	0.308062i	$0.900320\pi$
$920$	7.09728e35	+	1.37772e36i	1.93036	+	3.74720i
$921$	5.42641e35			1.45679
$922$	−7.92129e35			−2.09906
$923$	−1.16780e35			−0.305455
$924$	7.83086e35			2.02184
$925$		−	1.04720e36i		−	2.66887i
$926$	−9.41501e35			−2.36858
$927$		−	9.75376e33i		−	0.0242223i
$928$	3.27671e35			0.803271
$929$	3.21770e35			0.778675			0.389338	−	0.921095i	$-0.372704\pi$
$929$	3.21770e35			0.778675			0.389338	+	0.921095i	$0.372704\pi$
$930$			1.08571e36i			2.59369i
$931$		−	4.08811e35i		−	0.964109i
$932$	−4.48030e35			−1.04308
$933$	−2.59593e35			−0.596641
$934$			3.43474e35i			0.779349i
$935$			5.39360e34i			0.120820i
$936$	−4.19914e35			−0.928647
$937$			5.20194e35i			1.13577i	0.823108	+	0.567885i	$0.192238\pi$
$937$			5.20194e35i			1.13577i	−0.823108	+	0.567885i	$0.807762\pi$
$938$	−1.65151e35			−0.355997
$939$			1.23211e36i			2.62217i
$940$		−	2.72629e36i		−	5.72847i
$941$		−	6.58812e34i		−	0.136674i	−0.997662	−	0.0683371i	$-0.978231\pi$
$941$		−	6.58812e34i		−	0.136674i	0.997662	−	0.0683371i	$-0.0217693\pi$
$942$		−	2.69109e35i		−	0.551211i
$943$	7.04724e35	−	3.63037e35i	1.42521	−	0.734193i
$944$	2.21181e36			4.41655
$945$	8.01461e34			0.158016
$946$	−1.89111e36			−3.68149
$947$	−9.06536e35			−1.74255			−0.871275	−	0.490795i	$-0.836706\pi$
$947$	−9.06536e35			−1.74255			−0.871275	+	0.490795i	$0.836706\pi$
$948$			2.48313e36i			4.71302i
$949$	−1.81376e34			−0.0339927
$950$		−	1.99232e36i		−	3.68703i
$951$	−8.00041e35			−1.46199
$952$	5.00327e34			0.0902840
$953$		−	3.40744e34i		−	0.0607174i	−0.999539	−	0.0303587i	$-0.990335\pi$
$953$		−	3.40744e34i		−	0.0607174i	0.999539	−	0.0303587i	$-0.00966497\pi$
$954$		−	1.36618e36i		−	2.40397i
$955$	9.81954e35			1.70628
$956$	−2.03179e36			−3.48646
$957$		−	4.03627e35i		−	0.683970i
$958$			1.33243e36i			2.22977i
$959$	−9.78771e34			−0.161755
$960$			1.57407e36i			2.56904i
$961$	−3.95559e35			−0.637575
$962$		−	6.25210e35i		−	0.995233i
$963$		−	1.65967e35i		−	0.260919i
$964$			2.36438e36i			3.67108i
$965$		−	1.29398e36i		−	1.98427i
$966$	−4.14007e35	−	8.03666e35i	−0.627021	−	1.21717i
$967$	6.33786e35			0.948036			0.474018	−	0.880515i	$-0.342803\pi$
$967$	6.33786e35			0.948036			0.474018	+	0.880515i	$0.342803\pi$
$968$	4.78541e35			0.706993
$969$	−8.74545e34			−0.127614
$970$	−8.02513e35			−1.15662
$971$		−	1.07342e36i		−	1.52806i	−0.645180	−	0.764030i	$-0.723218\pi$
$971$		−	1.07342e36i		−	1.52806i	0.645180	−	0.764030i	$-0.276782\pi$
$972$	−2.47857e36			−3.48503
$973$			1.30604e35i			0.181385i
$974$	5.75379e35			0.789309
$975$	−5.12324e35			−0.694207
$976$		−	3.24793e36i		−	4.34720i
$977$			1.20936e36i			1.59890i	0.600736	+	0.799448i	$0.294874\pi$
$977$			1.20936e36i			1.59890i	−0.600736	+	0.799448i	$0.705126\pi$
$978$	−7.37964e35			−0.963760
$979$	−5.91626e34			−0.0763229
$980$			2.25826e36i			2.87781i
$981$			3.42244e35i			0.430831i
$982$	2.08853e36			2.59718
$983$		−	1.89469e35i		−	0.232754i	−0.993205	−	0.116377i	$-0.962872\pi$
$983$		−	1.89469e35i		−	0.232754i	0.993205	−	0.116377i	$-0.0371281\pi$
$984$	5.10829e36			6.19918
$985$			6.23736e35i			0.747767i
$986$		−	4.38311e34i		−	0.0519110i
$987$			9.35687e35i			1.09477i
$988$		−	8.42624e35i		−	0.973981i
$989$	7.08257e35	+	1.37486e36i	0.808789	+	1.57001i
$990$	3.33742e36			3.76521
$991$	−1.46997e36			−1.63841			−0.819206	−	0.573499i	$-0.805586\pi$
$991$	−1.46997e36			−1.63841			−0.819206	+	0.573499i	$0.805586\pi$
$992$	1.05584e36			1.16268
$993$	5.80879e35			0.631968
$994$		−	8.60980e35i		−	0.925458i
$995$	−2.36147e36			−2.50787
$996$			1.12610e36i			1.18159i
$997$	3.00297e35			0.311322			0.155661	−	0.987811i	$-0.450249\pi$
$997$	3.00297e35			0.311322			0.155661	+	0.987811i	$0.450249\pi$
$998$	−5.01426e35			−0.513618
$999$			3.36548e35i			0.340613i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.25.b.c.22.2	yes	44
23.22	odd	2	inner	23.25.b.c.22.1	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.25.b.c.22.1	✓	44	23.22	odd	2	inner
23.25.b.c.22.2	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)

\(f(q)\)	\(=\)	\(q-7585.13 q^{2} +776377. q^{3} +4.07570e7 q^{4} +3.88802e8i q^{5} -5.88892e9 q^{6} -7.00512e9i q^{7} -1.81889e11 q^{8} +3.20332e11 q^{9} -2.94912e12i q^{10} +3.53280e12i q^{11} +3.16428e13 q^{12} +7.20698e12 q^{13} +5.31348e13i q^{14} +3.01857e14i q^{15} +6.95866e14 q^{16} -3.92673e13i q^{17} -2.42976e15 q^{18} -2.86866e15i q^{19} +1.58464e16i q^{20} -5.43862e15i q^{21} -2.67967e16i q^{22} +(-1.94816e16 + 1.00359e16i) q^{23} -1.41215e17 q^{24} -9.15626e16 q^{25} -5.46659e16 q^{26} +2.94264e16 q^{27} -2.85508e17i q^{28} -1.47160e17 q^{29} -2.28963e18i q^{30} -4.74187e17 q^{31} -2.22664e18 q^{32} +2.74279e18i q^{33} +2.97847e17i q^{34} +2.72361e18 q^{35} +1.30557e19 q^{36} +1.14369e19i q^{37} +2.17591e19i q^{38} +5.59533e18 q^{39} -7.07190e19i q^{40} -3.61739e19 q^{41} +4.12526e19i q^{42} -7.05725e19i q^{43} +1.43986e20i q^{44} +1.24546e20i q^{45} +(1.47770e20 - 7.61235e19i) q^{46} -1.72045e20 q^{47} +5.40255e20 q^{48} +1.42509e20 q^{49} +6.94514e20 q^{50} -3.04862e19i q^{51} +2.93735e20 q^{52} +5.62272e20i q^{53} -2.23203e20 q^{54} -1.37356e21 q^{55} +1.27416e21i q^{56} -2.22716e21i q^{57} +1.11622e21 q^{58} +3.17850e21 q^{59} +1.23028e22i q^{60} -4.66747e21i q^{61} +3.59677e21 q^{62} -2.24396e21i q^{63} +5.21463e21 q^{64} +2.80209e21i q^{65} -2.08044e22i q^{66} +3.10815e21i q^{67} -1.60042e21i q^{68} +(-1.51250e22 + 7.79163e21i) q^{69} -2.06589e22 q^{70} -1.62037e22 q^{71} -5.82650e22 q^{72} -2.51668e21 q^{73} -8.67506e22i q^{74} -7.10871e22 q^{75} -1.16918e23i q^{76} +2.47477e22 q^{77} -4.24413e22 q^{78} +7.84739e22i q^{79} +2.70554e23i q^{80} -6.76252e22 q^{81} +2.74383e23 q^{82} +3.55880e22i q^{83} -2.21662e23i q^{84} +1.52672e22 q^{85} +5.35301e23i q^{86} -1.14251e23 q^{87} -6.42579e23i q^{88} +1.67467e22i q^{89} -9.44696e23i q^{90} -5.04858e22i q^{91} +(-7.94010e23 + 4.09032e23i) q^{92} -3.68148e23 q^{93} +1.30498e24 q^{94} +1.11534e24 q^{95} -1.72871e24 q^{96} -2.72120e23i q^{97} -1.08095e24 q^{98} +1.13167e24i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4232 q^{2} - 434562 q^{3} + 317760360 q^{4} - 8460029520 q^{6} - 198307023760 q^{8} + 4220041988298 q^{9} - 67439597688792 q^{12} + 5771152551358 q^{13} + 18\!\cdots\!92 q^{16} + 18\!\cdots\!68 q^{18}+ \cdots - 20\!\cdots\!92 q^{98}+O(q^{100}) \) 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

Related objects

Downloads

Learn more