LMFDB - Embedded newform 23.25.b.c.22.12

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			22.12
Character	$\chi$	$=$	23.22
Dual form			23.25.b.c.22.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4488.28 q^{2} +914024. q^{3} +3.36747e6 q^{4} +5.44878e7i q^{5} -4.10240e9 q^{6} +2.39870e10i q^{7} +6.01867e10 q^{8} +5.53011e11 q^{9} -2.44557e11i q^{10} +4.51091e10i q^{11} +3.07795e12 q^{12} +2.35222e13 q^{13} -1.07660e14i q^{14} +4.98032e13i q^{15} -3.26632e14 q^{16} -8.76299e14i q^{17} -2.48207e15 q^{18} -3.58837e15i q^{19} +1.83486e14i q^{20} +2.19247e16i q^{21} -2.02462e14i q^{22} +(2.07992e16 - 6.90252e15i) q^{23} +5.50121e16 q^{24} +5.66357e16 q^{25} -1.05574e17 q^{26} +2.47318e17 q^{27} +8.07754e16i q^{28} -1.57372e17 q^{29} -2.23531e17i q^{30} +1.17659e18 q^{31} +4.56250e17 q^{32} +4.12308e16i q^{33} +3.93308e18i q^{34} -1.30700e18 q^{35} +1.86225e18 q^{36} -8.68571e18i q^{37} +1.61056e19i q^{38} +2.14998e19 q^{39} +3.27944e18i q^{40} -4.27655e18 q^{41} -9.84042e19i q^{42} +4.84430e19i q^{43} +1.51903e17i q^{44} +3.01323e19i q^{45} +(-9.33526e19 + 3.09804e19i) q^{46} +7.36926e18 q^{47} -2.98549e20 q^{48} -3.83795e20 q^{49} -2.54197e20 q^{50} -8.00959e20i q^{51} +7.92101e19 q^{52} +5.63544e20i q^{53} -1.11003e21 q^{54} -2.45789e18 q^{55} +1.44370e21i q^{56} -3.27985e21i q^{57} +7.06329e20 q^{58} +1.95703e21 q^{59} +1.67711e20i q^{60} -4.34389e21i q^{61} -5.28085e21 q^{62} +1.32651e22i q^{63} +3.43219e21 q^{64} +1.28167e21i q^{65} -1.85055e20i q^{66} -4.19854e20i q^{67} -2.95091e21i q^{68} +(1.90110e22 - 6.30907e21i) q^{69} +5.86618e21 q^{70} -1.46861e20 q^{71} +3.32839e22 q^{72} +9.12685e21 q^{73} +3.89839e22i q^{74} +5.17664e22 q^{75} -1.20837e22i q^{76} -1.08203e21 q^{77} -9.64973e22 q^{78} -9.80031e21i q^{79} -1.77974e22i q^{80} +6.98678e22 q^{81} +1.91944e22 q^{82} +1.41915e23i q^{83} +7.38307e22i q^{84} +4.77476e22 q^{85} -2.17426e23i q^{86} -1.43842e23 q^{87} +2.71497e21i q^{88} -2.15040e23i q^{89} -1.35242e23i q^{90} +5.64226e23i q^{91} +(7.00406e22 - 2.32440e22i) q^{92} +1.07543e24 q^{93} -3.30753e22 q^{94} +1.95522e23 q^{95} +4.17024e23 q^{96} +6.51340e23i q^{97} +1.72258e24 q^{98} +2.49458e22i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4488.28			−1.09577			−0.547886	−	0.836553i	$-0.684567\pi$
$2$	−4488.28			−1.09577			−0.547886	+	0.836553i	$0.684567\pi$
$3$	914024.			1.71990			0.859949	−	0.510380i	$-0.170495\pi$
$3$	914024.			1.71990			0.859949	+	0.510380i	$0.170495\pi$
$4$	3.36747e6			0.200717
$5$			5.44878e7i			0.223182i	0.993754	+	0.111591i	$0.0355947\pi$
$5$			5.44878e7i			0.223182i	−0.993754	+	0.111591i	$0.964405\pi$
$6$	−4.10240e9			−1.88462
$7$			2.39870e10i			1.73300i	0.499174	+	0.866502i	$0.333637\pi$
$7$			2.39870e10i			1.73300i	−0.499174	+	0.866502i	$0.666363\pi$
$8$	6.01867e10			0.875832
$9$	5.53011e11			1.95805
$10$		−	2.44557e11i		−	0.244557i
$11$			4.51091e10i			0.0143731i	0.999974	+	0.00718657i	$0.00228758\pi$
$11$			4.51091e10i			0.0143731i	−0.999974	+	0.00718657i	$0.997712\pi$
$12$	3.07795e12			0.345212
$13$	2.35222e13			1.00962			0.504809	−	0.863231i	$-0.331563\pi$
$13$	2.35222e13			1.00962			0.504809	+	0.863231i	$0.331563\pi$
$14$		−	1.07660e14i		−	1.89898i
$15$			4.98032e13i			0.383850i
$16$	−3.26632e14			−1.16043
$17$		−	8.76299e14i		−	1.50406i	−0.659129	−	0.752030i	$-0.729075\pi$
$17$		−	8.76299e14i		−	1.50406i	0.659129	−	0.752030i	$-0.270925\pi$
$18$	−2.48207e15			−2.14557
$19$		−	3.58837e15i		−	1.62126i	−0.585556	−	0.810632i	$-0.699124\pi$
$19$		−	3.58837e15i		−	1.62126i	0.585556	−	0.810632i	$-0.300876\pi$
$20$			1.83486e14i			0.0447964i
$21$			2.19247e16i			2.98059i
$22$		−	2.02462e14i		−	0.0157497i
$23$	2.07992e16	−	6.90252e15i	0.949101	−	0.314973i
$23$	2.07992e16	−	6.90252e15i	0.949101	−	0.314973i
$24$	5.50121e16			1.50634
$25$	5.66357e16			0.950190
$26$	−1.05574e17			−1.10631
$27$	2.47318e17			1.64775
$28$			8.07754e16i			0.347843i
$29$	−1.57372e17			−0.444786			−0.222393	−	0.974957i	$-0.571387\pi$
$29$	−1.57372e17			−0.444786			−0.222393	+	0.974957i	$0.571387\pi$
$30$		−	2.23531e17i		−	0.420612i
$31$	1.17659e18			1.49377			0.746884	−	0.664954i	$-0.231549\pi$
$31$	1.17659e18			1.49377			0.746884	+	0.664954i	$0.231549\pi$
$32$	4.56250e17			0.395734
$33$			4.12308e16i			0.0247203i
$34$			3.93308e18i			1.64811i
$35$	−1.30700e18			−0.386775
$36$	1.86225e18			0.393013
$37$		−	8.68571e18i		−	1.31943i	−0.751518	−	0.659713i	$-0.770678\pi$
$37$		−	8.68571e18i		−	1.31943i	0.751518	−	0.659713i	$-0.229322\pi$
$38$			1.61056e19i			1.77654i
$39$	2.14998e19			1.73644
$40$			3.27944e18i			0.195470i
$41$	−4.27655e18			−0.189534			−0.0947671	−	0.995499i	$-0.530211\pi$
$41$	−4.27655e18			−0.189534			−0.0947671	+	0.995499i	$0.530211\pi$
$42$		−	9.84042e19i		−	3.26605i
$43$			4.84430e19i			1.21230i	0.795351	+	0.606149i	$0.207286\pi$
$43$			4.84430e19i			1.21230i	−0.795351	+	0.606149i	$0.792714\pi$
$44$			1.51903e17i			0.00288493i
$45$			3.01323e19i			0.437001i
$46$	−9.33526e19	+	3.09804e19i	−1.04000	+	0.345139i
$47$	7.36926e18			0.0634234			0.0317117	−	0.999497i	$-0.489904\pi$
$47$	7.36926e18			0.0634234			0.0317117	+	0.999497i	$0.489904\pi$
$48$	−2.98549e20			−1.99582
$49$	−3.83795e20			−2.00330
$50$	−2.54197e20			−1.04119
$51$		−	8.00959e20i		−	2.58683i
$52$	7.92101e19			0.202647
$53$			5.63544e20i			1.14714i	0.819155	+	0.573572i	$0.194443\pi$
$53$			5.63544e20i			1.14714i	−0.819155	+	0.573572i	$0.805557\pi$
$54$	−1.11003e21			−1.80555
$55$	−2.45789e18			−0.00320783
$56$			1.44370e21i			1.51782i
$57$		−	3.27985e21i		−	2.78841i
$58$	7.06329e20			0.487384
$59$	1.95703e21			1.09995			0.549976	−	0.835181i	$-0.314637\pi$
$59$	1.95703e21			1.09995			0.549976	+	0.835181i	$0.314637\pi$
$60$			1.67711e20i			0.0770451i
$61$		−	4.34389e21i		−	1.63652i	−0.574849	−	0.818259i	$-0.694939\pi$
$61$		−	4.34389e21i		−	1.63652i	0.574849	−	0.818259i	$-0.305061\pi$
$62$	−5.28085e21			−1.63683
$63$			1.32651e22i			3.39330i
$64$	3.43219e21			0.726795
$65$			1.28167e21i			0.225329i
$66$		−	1.85055e20i		−	0.0270879i
$67$		−	4.19854e20i		−	0.0513098i	−0.999671	−	0.0256549i	$-0.991833\pi$
$67$		−	4.19854e20i		−	0.0513098i	0.999671	−	0.0256549i	$-0.00816711\pi$
$68$		−	2.95091e21i		−	0.301890i
$69$	1.90110e22	−	6.30907e21i	1.63236	−	0.541722i
$70$	5.86618e21			0.423818
$71$	−1.46861e20			−0.00894964			−0.00447482	−	0.999990i	$-0.501424\pi$
$71$	−1.46861e20			−0.00894964			−0.00447482	+	0.999990i	$0.501424\pi$
$72$	3.32839e22			1.71492
$73$	9.12685e21			0.398517			0.199258	−	0.979947i	$-0.436147\pi$
$73$	9.12685e21			0.398517			0.199258	+	0.979947i	$0.436147\pi$
$74$			3.89839e22i			1.44579i
$75$	5.17664e22			1.63423
$76$		−	1.20837e22i		−	0.325415i
$77$	−1.08203e21			−0.0249087
$78$	−9.64973e22			−1.90274
$79$		−	9.80031e21i		−	0.165850i	−0.996556	−	0.0829248i	$-0.973574\pi$
$79$		−	9.80031e21i		−	0.165850i	0.996556	−	0.0829248i	$-0.0264261\pi$
$80$		−	1.77974e22i		−	0.258987i
$81$	6.98678e22			0.875905
$82$	1.91944e22			0.207686
$83$			1.41915e23i			1.32767i	0.747877	+	0.663837i	$0.231073\pi$
$83$			1.41915e23i			1.32767i	−0.747877	+	0.663837i	$0.768927\pi$
$84$			7.38307e22i			0.598254i
$85$	4.77476e22			0.335679
$86$		−	2.17426e23i		−	1.32840i
$87$	−1.43842e23			−0.764986
$88$			2.71497e21i			0.0125885i
$89$		−	2.15040e23i		−	0.870640i	−0.900276	−	0.435320i	$-0.856635\pi$
$89$		−	2.15040e23i		−	0.870640i	0.900276	−	0.435320i	$-0.143365\pi$
$90$		−	1.35242e23i		−	0.478854i
$91$			5.64226e23i			1.74967i
$92$	7.00406e22	−	2.32440e22i	0.190500	−	0.0632204i
$93$	1.07543e24			2.56913
$94$	−3.30753e22			−0.0694976
$95$	1.95522e23			0.361837
$96$	4.17024e23			0.680622
$97$			6.51340e23i			0.938743i	0.883001	+	0.469371i	$0.155519\pi$
$97$			6.51340e23i			0.938743i	−0.883001	+	0.469371i	$0.844481\pi$
$98$	1.72258e24			2.19516
$99$			2.49458e22i			0.0281433i
$100$	1.90719e23			0.190719
$101$	−1.56949e24			−1.39284			−0.696422	−	0.717633i	$-0.745226\pi$
$101$	−1.56949e24			−1.39284			−0.696422	+	0.717633i	$0.745226\pi$
$102$			3.59493e24i			2.83458i
$103$		−	2.63581e24i		−	1.84870i	−0.381542	−	0.924352i	$-0.624607\pi$
$103$		−	2.63581e24i		−	1.84870i	0.381542	−	0.924352i	$-0.375393\pi$
$104$	1.41572e24			0.884256
$105$	−1.19463e24			−0.665214
$106$		−	2.52935e24i		−	1.25701i
$107$			9.06131e23i			0.402333i	0.979557	+	0.201166i	$0.0644732\pi$
$107$			9.06131e23i			0.402333i	−0.979557	+	0.201166i	$0.935527\pi$
$108$	8.32834e23			0.330730
$109$			2.61146e24i			0.928466i	0.885713	+	0.464233i	$0.153670\pi$
$109$			2.61146e24i			0.928466i	−0.885713	+	0.464233i	$0.846330\pi$
$110$	1.10317e22			0.00351505
$111$		−	7.93895e24i		−	2.26928i
$112$		−	7.83492e24i		−	2.01103i
$113$			4.43787e24i			1.02384i	0.859032	+	0.511922i	$0.171066\pi$
$113$			4.43787e24i			1.02384i	−0.859032	+	0.511922i	$0.828934\pi$
$114$			1.47209e25i			3.05546i
$115$	3.76103e23	+	1.13330e24i	0.0702963	+	0.211822i
$116$	−5.29945e23			−0.0892760
$117$	1.30080e25			1.97688
$118$	−8.78371e24			−1.20530
$119$	2.10198e25			2.60654
$120$			2.99749e24i			0.336188i
$121$	9.84770e24			0.999793
$122$			1.94966e25i			1.79325i
$123$	−3.90887e24			−0.325979
$124$	3.96211e24			0.299824
$125$			6.33368e24i			0.435247i
$126$		−	5.95374e25i		−	3.71829i
$127$	−4.61032e24			−0.261870			−0.130935	−	0.991391i	$-0.541798\pi$
$127$	−4.61032e24			−0.261870			−0.130935	+	0.991391i	$0.541798\pi$
$128$	−2.30593e25			−1.19214
$129$			4.42780e25i			2.08503i
$130$		−	5.75250e24i		−	0.246909i
$131$	2.23033e25			0.873200			0.436600	−	0.899656i	$-0.356183\pi$
$131$	2.23033e25			0.873200			0.436600	+	0.899656i	$0.356183\pi$
$132$			1.38843e23i			0.00496179i
$133$	8.60742e25			2.80966
$134$			1.88442e24i			0.0562239i
$135$			1.34758e25i			0.367747i
$136$		−	5.27416e25i		−	1.31730i
$137$			2.08478e25i			0.476885i	0.971157	+	0.238443i	$0.0766369\pi$
$137$			2.08478e25i			0.476885i	−0.971157	+	0.238443i	$0.923363\pi$
$138$	−8.53266e25	+	2.83169e25i	−1.78869	+	0.593603i
$139$	2.74955e25			0.528547			0.264273	−	0.964448i	$-0.414868\pi$
$139$	2.74955e25			0.528547			0.264273	+	0.964448i	$0.414868\pi$
$140$	−4.40128e24			−0.0776322
$141$	6.73568e24			0.109082
$142$	6.59153e23			0.00980677
$143$			1.06106e24i			0.0145114i
$144$	−1.80631e26			−2.27218
$145$		−	8.57484e24i		−	0.0992682i
$146$	−4.09639e25			−0.436684
$147$	−3.50798e26			−3.44547
$148$		−	2.92489e25i		−	0.264831i
$149$		−	4.40689e25i		−	0.368042i	−0.982922	−	0.184021i	$-0.941089\pi$
$149$		−	4.40689e25i		−	0.368042i	0.982922	−	0.184021i	$-0.0589114\pi$
$150$	−2.32342e26			−1.79074
$151$	−1.05459e26			−0.750514			−0.375257	−	0.926921i	$-0.622446\pi$
$151$	−1.05459e26			−0.750514			−0.375257	+	0.926921i	$0.622446\pi$
$152$		−	2.15972e26i		−	1.41996i
$153$		−	4.84603e26i		−	2.94502i
$154$	4.85646e24			0.0272943
$155$			6.41096e25i			0.333382i
$156$	7.24000e25			0.348533
$157$		−	1.02101e26i		−	0.455231i	−0.973751	−	0.227616i	$-0.926907\pi$
$157$		−	1.02101e26i		−	0.455231i	0.973751	−	0.227616i	$-0.0730930\pi$
$158$			4.39866e25i			0.181733i
$159$			5.15093e26i			1.97297i
$160$			2.48601e25i			0.0883207i
$161$	1.65571e26	+	4.98910e26i	0.545849	+	1.64479i
$162$	−3.13587e26			−0.959792
$163$	3.54318e26			1.00726			0.503630	−	0.863919i	$-0.331997\pi$
$163$	3.54318e26			1.00726			0.503630	+	0.863919i	$0.331997\pi$
$164$	−1.44011e25			−0.0380427
$165$	−2.24658e24			−0.00551714
$166$		−	6.36955e26i		−	1.45483i
$167$	6.43948e26			1.36853			0.684263	−	0.729236i	$-0.260124\pi$
$167$	6.43948e26			1.36853			0.684263	+	0.729236i	$0.260124\pi$
$168$			1.31958e27i			2.61050i
$169$	1.04918e25			0.0193289
$170$	−2.14305e26			−0.367828
$171$		−	1.98441e27i		−	3.17451i
$172$			1.63130e26i			0.243328i
$173$	4.64342e26			0.646078			0.323039	−	0.946386i	$-0.395295\pi$
$173$	4.64342e26			0.646078			0.323039	+	0.946386i	$0.395295\pi$
$174$	6.45602e26			0.838251
$175$			1.35852e27i			1.64668i
$176$		−	1.47341e25i		−	0.0166790i
$177$	1.78877e27			1.89180
$178$			9.65159e26i			0.954023i
$179$	−1.61443e27			−1.49205			−0.746025	−	0.665917i	$-0.768040\pi$
$179$	−1.61443e27			−1.49205			−0.746025	+	0.665917i	$0.768040\pi$
$180$			1.01470e26i			0.0877134i
$181$			1.96430e27i			1.58878i	0.607405	+	0.794392i	$0.292210\pi$
$181$			1.96430e27i			1.58878i	−0.607405	+	0.794392i	$0.707790\pi$
$182$		−	2.53241e27i		−	1.91724i
$183$		−	3.97042e27i		−	2.81464i
$184$	1.25184e27	−	4.15440e26i	0.831253	−	0.275864i
$185$	4.73265e26			0.294472
$186$	−4.82682e27			−2.81518
$187$	3.95291e25			0.0216181
$188$	2.48157e25			0.0127301
$189$			5.93241e27i			2.85555i
$190$	−8.77559e26			−0.396491
$191$			1.23002e27i			0.521808i	0.965365	+	0.260904i	$0.0840206\pi$
$191$			1.23002e27i			0.521808i	−0.965365	+	0.260904i	$0.915979\pi$
$192$	3.13711e27			1.25001
$193$	4.30534e26			0.161183			0.0805916	−	0.996747i	$-0.474319\pi$
$193$	4.30534e26			0.161183			0.0805916	+	0.996747i	$0.474319\pi$
$194$		−	2.92340e27i		−	1.02865i
$195$			1.17148e27i			0.387542i
$196$	−1.29242e27			−0.402096
$197$	−2.24684e27			−0.657623			−0.328811	−	0.944396i	$-0.606648\pi$
$197$	−2.24684e27			−0.657623			−0.328811	+	0.944396i	$0.606648\pi$
$198$		−	1.11964e26i		−	0.0308387i
$199$		−	1.07864e27i		−	0.279667i	−0.990175	−	0.139833i	$-0.955343\pi$
$199$		−	1.07864e27i		−	0.279667i	0.990175	−	0.139833i	$-0.0446567\pi$
$200$	3.40872e27			0.832207
$201$		−	3.83757e26i		−	0.0882477i
$202$	7.04432e27			1.52624
$203$		−	3.77488e27i		−	0.770816i
$204$		−	2.69720e27i		−	0.519220i
$205$		−	2.33020e26i		−	0.0423006i
$206$			1.18303e28i			2.02576i
$207$	1.15022e28	−	3.81717e27i	1.85838	−	0.616733i
$208$	−7.68309e27			−1.17159
$209$	1.61868e26			0.0233027
$210$	5.36183e27			0.728923
$211$	−6.87732e27			−0.883140			−0.441570	−	0.897227i	$-0.645578\pi$
$211$	−6.87732e27			−0.883140			−0.441570	+	0.897227i	$0.645578\pi$
$212$			1.89772e27i			0.230251i
$213$	−1.34234e26			−0.0153925
$214$		−	4.06697e27i		−	0.440865i
$215$	−2.63955e27			−0.270563
$216$	1.48852e28			1.44315
$217$			2.82228e28i			2.58871i
$218$		−	1.17210e28i		−	1.01739i
$219$	8.34216e27			0.685408
$220$	−8.27688e24			−0.000643865
$221$		−	2.06125e28i		−	1.51853i
$222$			3.56323e28i			2.48661i
$223$	−7.68909e26			−0.0508413			−0.0254206	−	0.999677i	$-0.508093\pi$
$223$	−7.68909e26			−0.0508413			−0.0254206	+	0.999677i	$0.508093\pi$
$224$			1.09441e28i			0.685808i
$225$	3.13202e28			1.86052
$226$		−	1.99184e28i		−	1.12190i
$227$		−	6.33723e27i		−	0.338524i	−0.985571	−	0.169262i	$-0.945862\pi$
$227$		−	6.33723e27i		−	0.338524i	0.985571	−	0.169262i	$-0.0541384\pi$
$228$		−	1.10448e28i		−	0.559680i
$229$		−	1.11306e28i		−	0.535173i	−0.963534	−	0.267587i	$-0.913774\pi$
$229$		−	1.11306e28i		−	0.535173i	0.963534	−	0.267587i	$-0.0862262\pi$
$230$	−1.68806e27	−	5.08658e27i	−0.0770288	−	0.232109i
$231$	−9.89003e26			−0.0428404
$232$	−9.47170e27			−0.389558
$233$	−1.01191e28			−0.395250			−0.197625	−	0.980278i	$-0.563323\pi$
$233$	−1.01191e28			−0.395250			−0.197625	+	0.980278i	$0.563323\pi$
$234$	−5.83836e28			−2.16621
$235$			4.01535e26i			0.0141550i
$236$	6.59024e27			0.220779
$237$		−	8.95772e27i		−	0.285245i
$238$	−9.43427e28			−2.85618
$239$	−5.05563e28			−1.45546			−0.727730	−	0.685864i	$-0.759425\pi$
$239$	−5.05563e28			−1.45546			−0.727730	+	0.685864i	$0.759425\pi$
$240$		−	1.62673e28i		−	0.445431i
$241$		−	1.20702e27i		−	0.0314421i	−0.999876	−	0.0157211i	$-0.994996\pi$
$241$		−	1.20702e27i		−	0.0314421i	0.999876	−	0.0157211i	$-0.00500437\pi$
$242$	−4.41993e28			−1.09555
$243$	−5.98895e27			−0.141278
$244$		−	1.46279e28i		−	0.328477i
$245$		−	2.09121e28i		−	0.447101i
$246$	1.75441e28			0.357199
$247$		−	8.44062e28i		−	1.63686i
$248$	7.08149e28			1.30829
$249$			1.29714e29i			2.28346i
$250$		−	2.84274e28i		−	0.476932i
$251$			9.73478e28i			1.55683i	0.627749	+	0.778416i	$0.283976\pi$
$251$			9.73478e28i			1.55683i	−0.627749	+	0.778416i	$0.716024\pi$
$252$			4.46697e28i			0.681093i
$253$	3.11366e26	+	9.38232e26i	0.00452715	+	0.0136416i
$254$	2.06924e28			0.286950
$255$	4.36425e28			0.577334
$256$	4.59138e28			0.579514
$257$	−8.86594e28			−1.06789			−0.533946	−	0.845519i	$-0.679291\pi$
$257$	−8.86594e28			−1.06789			−0.533946	+	0.845519i	$0.679291\pi$
$258$		−	1.98732e29i		−	2.28471i
$259$	2.08344e29			2.28657
$260$			4.31599e27i			0.0452272i
$261$	−8.70283e28			−0.870912
$262$	−1.00104e29			−0.956828
$263$		−	4.26633e28i		−	0.389570i	−0.980846	−	0.194785i	$-0.937599\pi$
$263$		−	4.26633e28i		−	0.389570i	0.980846	−	0.194785i	$-0.0624010\pi$
$264$			2.48155e27i			0.0216509i
$265$	−3.07063e28			−0.256022
$266$	−3.86325e29			−3.07874
$267$		−	1.96551e29i		−	1.49741i
$268$		−	1.41384e27i		−	0.0102987i
$269$	5.54492e28			0.386249			0.193125	−	0.981174i	$-0.438138\pi$
$269$	5.54492e28			0.386249			0.193125	+	0.981174i	$0.438138\pi$
$270$		−	6.04832e28i		−	0.402967i
$271$	3.33534e28			0.212573			0.106287	−	0.994336i	$-0.466104\pi$
$271$	3.33534e28			0.212573			0.106287	+	0.994336i	$0.466104\pi$
$272$			2.86227e29i			1.74536i
$273$			5.15716e29i			3.00926i
$274$		−	9.35709e28i		−	0.522557i
$275$			2.55479e27i			0.0136572i
$276$	6.40188e28	−	2.12456e28i	0.327641	−	0.108733i
$277$	2.25531e29			1.10522			0.552611	−	0.833439i	$-0.313632\pi$
$277$	2.25531e29			1.10522			0.552611	+	0.833439i	$0.313632\pi$
$278$	−1.23407e29			−0.579167
$279$	6.50665e29			2.92487
$280$	−7.86640e28			−0.338750
$281$		−	2.34400e29i		−	0.967122i	−0.875311	−	0.483561i	$-0.839343\pi$
$281$		−	2.34400e29i		−	0.967122i	0.875311	−	0.483561i	$-0.160657\pi$
$282$	−3.02316e28			−0.119529
$283$		−	2.44527e28i		−	0.0926595i	−0.998926	−	0.0463297i	$-0.985248\pi$
$283$		−	2.44527e28i		−	0.0926595i	0.998926	−	0.0463297i	$-0.0147525\pi$
$284$	−4.94549e26			−0.00179634
$285$	1.78712e29			0.622322
$286$		−	4.76235e27i		−	0.0159012i
$287$		−	1.02582e29i		−	0.328463i
$288$	2.52311e29			0.774866
$289$	−4.28451e29			−1.26220
$290$			3.84863e28i			0.108775i
$291$			5.95340e29i			1.61454i
$292$	3.07344e28			0.0799890
$293$		−	3.45639e29i		−	0.863399i	−0.902018	−	0.431699i	$-0.857914\pi$
$293$		−	3.45639e29i		−	0.863399i	0.902018	−	0.431699i	$-0.142086\pi$
$294$	1.57448e30			3.77545
$295$			1.06634e29i			0.245489i
$296$		−	5.22765e29i		−	1.15560i
$297$			1.11563e28i			0.0236833i
$298$			1.97794e29i			0.403290i
$299$	4.89242e29	−	1.62362e29i	0.958229	−	0.318003i
$300$	1.74322e29			0.328017
$301$	−1.16200e30			−2.10092
$302$	4.73328e29			0.822392
$303$	−1.43455e30			−2.39555
$304$			1.17208e30i			1.88136i
$305$	2.36689e29			0.365242
$306$			2.17503e30i			3.22707i
$307$	1.01644e30			1.45018			0.725092	−	0.688652i	$-0.241797\pi$
$307$	1.01644e30			1.45018			0.725092	+	0.688652i	$0.241797\pi$
$308$	−3.64371e27			−0.00499959
$309$		−	2.40919e30i		−	3.17958i
$310$		−	2.87742e29i		−	0.365311i
$311$	2.26309e29			0.276425			0.138212	−	0.990403i	$-0.455864\pi$
$311$	2.26309e29			0.276425			0.138212	+	0.990403i	$0.455864\pi$
$312$	1.29400e30			1.52083
$313$			6.74907e29i			0.763329i	0.924301	+	0.381665i	$0.124649\pi$
$313$			6.74907e29i			0.763329i	−0.924301	+	0.381665i	$0.875351\pi$
$314$			4.58256e29i			0.498830i
$315$	−7.22784e29			−0.757325
$316$		−	3.30022e28i		−	0.0332888i
$317$	−7.54917e29			−0.733142			−0.366571	−	0.930390i	$-0.619468\pi$
$317$	−7.54917e29			−0.733142			−0.366571	+	0.930390i	$0.619468\pi$
$318$		−	2.31188e30i		−	2.16192i
$319$		−	7.09890e27i		−	0.00639297i
$320$			1.87013e29i			0.162208i
$321$			8.28225e29i			0.691971i
$322$	−7.43128e29	−	2.23925e30i	−0.598127	−	1.80232i
$323$	−3.14448e30			−2.43848
$324$	2.35278e29			0.175809
$325$	1.33220e30			0.959329
$326$	−1.59028e30			−1.10373
$327$			2.38694e30i			1.59687i
$328$	−2.57392e29			−0.166000
$329$			1.76766e29i			0.109913i
$330$	1.00833e28			0.00604552
$331$	−2.17616e30			−1.25821			−0.629107	−	0.777318i	$-0.716579\pi$
$331$	−2.17616e30			−1.25821			−0.629107	+	0.777318i	$0.716579\pi$
$332$			4.77894e29i			0.266486i
$333$		−	4.80329e30i		−	2.58350i
$334$	−2.89022e30			−1.49959
$335$	2.28769e28			0.0114514
$336$		−	7.16130e30i		−	3.45876i
$337$			2.84839e30i			1.32752i	0.747947	+	0.663758i	$0.231040\pi$
$337$			2.84839e30i			1.32752i	−0.747947	+	0.663758i	$0.768960\pi$
$338$	−4.70900e28			−0.0211801
$339$			4.05632e30i			1.76091i
$340$	1.60788e29			0.0673764
$341$			5.30747e28i			0.0214702i
$342$			8.90657e30i			3.47854i
$343$		−	4.61063e30i		−	1.73872i
$344$			2.91562e30i			1.06177i
$345$	3.43767e29	+	1.03587e30i	0.120902	+	0.364312i
$346$	−2.08410e30			−0.707954
$347$	−1.50747e30			−0.494646			−0.247323	−	0.968933i	$-0.579551\pi$
$347$	−1.50747e30			−0.494646			−0.247323	+	0.968933i	$0.579551\pi$
$348$	−4.84382e29			−0.153546
$349$	6.54957e29			0.200589			0.100295	−	0.994958i	$-0.468021\pi$
$349$	6.54957e29			0.200589			0.100295	+	0.994958i	$0.468021\pi$
$350$		−	6.09743e30i		−	1.80439i
$351$	5.81745e30			1.66359
$352$			2.05810e28i			0.00568794i
$353$	2.09776e30			0.560349			0.280175	−	0.959949i	$-0.409608\pi$
$353$	2.09776e30			0.560349			0.280175	+	0.959949i	$0.409608\pi$
$354$	−8.02852e30			−2.07299
$355$		−	8.00212e27i		−	0.00199740i
$356$		−	7.24139e29i		−	0.174752i
$357$	1.92126e31			4.48299
$358$	7.24603e30			1.63495
$359$		−	2.44473e30i		−	0.533454i	−0.963772	−	0.266727i	$-0.914058\pi$
$359$		−	2.44473e30i		−	0.533454i	0.963772	−	0.266727i	$-0.0859422\pi$
$360$			1.81357e30i			0.382740i
$361$	−7.97762e30			−1.62850
$362$		−	8.81633e30i		−	1.74095i
$363$	9.00103e30			1.71954
$364$			1.90001e30i			0.351188i
$365$			4.97302e29i			0.0889418i
$366$			1.78204e31i			3.08421i
$367$			1.04037e30i			0.174259i	0.996197	+	0.0871293i	$0.0277693\pi$
$367$			1.04037e30i			0.174259i	−0.996197	+	0.0871293i	$0.972231\pi$
$368$	−6.79368e30	+	2.25458e30i	−1.10136	+	0.365504i
$369$	−2.36498e30			−0.371117
$370$	−2.12415e30			−0.322674
$371$	−1.35177e31			−1.98800
$372$	3.62147e30			0.515667
$373$		−	7.10267e30i		−	0.979300i	−0.871919	−	0.489650i	$-0.837125\pi$
$373$		−	7.10267e30i		−	0.979300i	0.871919	−	0.489650i	$-0.162875\pi$
$374$	−1.77418e29			−0.0236885
$375$			5.78914e30i			0.748581i
$376$	4.43532e29			0.0555483
$377$	−3.70173e30			−0.449064
$378$		−	2.66263e31i		−	3.12903i
$379$			7.01346e30i			0.798476i	0.916847	+	0.399238i	$0.130725\pi$
$379$			7.01346e30i			0.798476i	−0.916847	+	0.399238i	$0.869275\pi$
$380$	6.58415e29			0.0726267
$381$	−4.21394e30			−0.450390
$382$		−	5.52066e30i		−	0.571782i
$383$			1.08152e30i			0.108555i	0.998526	+	0.0542775i	$0.0172856\pi$
$383$			1.08152e30i			0.108555i	−0.998526	+	0.0542775i	$0.982714\pi$
$384$	−2.10767e31			−2.05035
$385$		−	5.89575e28i		−	0.00555918i
$386$	−1.93236e30			−0.176620
$387$			2.67895e31i			2.37374i
$388$			2.19336e30i			0.188421i
$389$		−	7.24407e30i		−	0.603375i	−0.953407	−	0.301688i	$-0.902450\pi$
$389$		−	7.24407e30i		−	0.603375i	0.953407	−	0.301688i	$-0.0975500\pi$
$390$		−	5.25793e30i		−	0.424658i
$391$	−6.04867e30	−	1.82263e31i	−0.473739	−	1.42750i
$392$	−2.30994e31			−1.75456
$393$	2.03858e31			1.50181
$394$	1.00844e31			0.720605
$395$	5.33997e29			0.0370147
$396$			8.40042e28i			0.00564883i
$397$	2.46774e31			1.60996			0.804978	−	0.593305i	$-0.202177\pi$
$397$	2.46774e31			1.60996			0.804978	+	0.593305i	$0.202177\pi$
$398$			4.84126e30i			0.306451i
$399$	7.86739e31			4.83232
$400$	−1.84990e31			−1.10263
$401$			2.43418e31i			1.40806i	0.710172	+	0.704028i	$0.248617\pi$
$401$			2.43418e31i			1.40806i	−0.710172	+	0.704028i	$0.751383\pi$
$402$			1.72241e30i			0.0966993i
$403$	2.76759e31			1.50814
$404$	−5.28521e30			−0.279567
$405$			3.80694e30i			0.195486i
$406$			1.69427e31i			0.844638i
$407$	3.91805e29			0.0189643
$408$		−	4.82071e31i		−	2.26563i
$409$	4.65554e30			0.212466			0.106233	−	0.994341i	$-0.466121\pi$
$409$	4.65554e30			0.212466			0.106233	+	0.994341i	$0.466121\pi$
$410$			1.04586e30i			0.0463518i
$411$			1.90554e31i			0.820194i
$412$		−	8.87600e30i		−	0.371066i
$413$			4.69433e31i			1.90622i
$414$	−5.16250e31	+	1.71325e31i	−2.03637	+	0.675798i
$415$	−7.73264e30			−0.296313
$416$	1.07320e31			0.399540
$417$	2.51315e31			0.909047
$418$	−7.26509e29			−0.0255344
$419$		−	5.03570e30i		−	0.171986i	−0.996296	−	0.0859928i	$-0.972594\pi$
$419$		−	5.03570e30i		−	0.171986i	0.996296	−	0.0859928i	$-0.0274062\pi$
$420$	−4.02287e30			−0.133520
$421$		−	1.91386e31i		−	0.617341i	−0.951169	−	0.308670i	$-0.900116\pi$
$421$		−	1.91386e31i		−	0.617341i	0.951169	−	0.308670i	$-0.0998840\pi$
$422$	3.08673e31			0.967720
$423$	4.07528e30			0.124186
$424$			3.39179e31i			1.00471i
$425$		−	4.96298e31i		−	1.42914i
$426$	6.02481e29			0.0168666
$427$	1.04197e32			2.83609
$428$			3.05137e30i			0.0807549i
$429$			9.69838e29i			0.0249581i
$430$	1.18470e31			0.296475
$431$			4.84913e31i			1.18015i	0.807349	+	0.590074i	$0.200902\pi$
$431$			4.84913e31i			1.18015i	−0.807349	+	0.590074i	$0.799098\pi$
$432$	−8.07818e31			−1.91209
$433$			3.20456e31i			0.737758i	0.929477	+	0.368879i	$0.120258\pi$
$433$			3.20456e31i			0.737758i	−0.929477	+	0.368879i	$0.879742\pi$
$434$		−	1.26672e32i		−	2.83663i
$435$		−	7.83762e30i		−	0.170731i
$436$			8.79402e30i			0.186359i
$437$	−2.47688e31	−	7.46351e31i	−0.510654	−	1.53874i
$438$	−3.74420e31			−0.751051
$439$	−3.48481e31			−0.680151			−0.340076	−	0.940398i	$-0.610453\pi$
$439$	−3.48481e31			−0.680151			−0.340076	+	0.940398i	$0.610453\pi$
$440$	−1.47933e29			−0.00280952
$441$	−2.12243e32			−3.92256
$442$			9.25145e31i			1.66396i
$443$	8.18777e31			1.43325			0.716624	−	0.697460i	$-0.245687\pi$
$443$	8.18777e31			1.43325			0.716624	+	0.697460i	$0.245687\pi$
$444$		−	2.67342e31i		−	0.455482i
$445$	1.17170e31			0.194311
$446$	3.45108e30			0.0557105
$447$		−	4.02801e31i		−	0.632994i
$448$			8.23280e31i			1.25954i
$449$	−4.50242e31			−0.670641			−0.335320	−	0.942104i	$-0.608845\pi$
$449$	−4.50242e31			−0.670641			−0.335320	+	0.942104i	$0.608845\pi$
$450$	−1.40574e32			−2.03870
$451$		−	1.92911e29i		−	0.00272420i
$452$			1.49444e31i			0.205502i
$453$	−9.63917e31			−1.29081
$454$			2.84433e31i			0.370945i
$455$	−3.07434e31			−0.390495
$456$		−	1.97404e32i		−	2.44218i
$457$			5.52507e31i			0.665799i	0.942962	+	0.332900i	$0.108027\pi$
$457$			5.52507e31i			0.665799i	−0.942962	+	0.332900i	$0.891973\pi$
$458$			4.99575e31i			0.586428i
$459$		−	2.16724e32i		−	2.47831i
$460$	1.26651e30	+	3.81636e30i	0.0141096	+	0.0425162i
$461$	−1.77661e32			−1.92833			−0.964166	−	0.265301i	$-0.914529\pi$
$461$	−1.77661e32			−1.92833			−0.964166	+	0.265301i	$0.914529\pi$
$462$	4.43893e30			0.0469434
$463$	3.27152e31			0.337114			0.168557	−	0.985692i	$-0.446089\pi$
$463$	3.27152e31			0.337114			0.168557	+	0.985692i	$0.446089\pi$
$464$	5.14027e31			0.516143
$465$			5.85977e31i			0.573383i
$466$	4.54174e31			0.433104
$467$			6.90477e31i			0.641722i	0.947126	+	0.320861i	$0.103972\pi$
$467$			6.90477e31i			0.641722i	−0.947126	+	0.320861i	$0.896028\pi$
$468$	4.38041e31			0.396793
$469$	1.00710e31			0.0889201
$470$		−	1.80220e30i		−	0.0155106i
$471$		−	9.33223e31i		−	0.782951i
$472$	1.17787e32			0.963373
$473$	−2.18522e30			−0.0174245
$474$			4.02048e31i			0.312563i
$475$		−	2.03230e32i		−	1.54051i
$476$	7.07834e31			0.523177
$477$			3.11646e32i			2.24616i
$478$	2.26911e32			1.59485
$479$			7.34179e31i			0.503241i	0.967826	+	0.251620i	$0.0809635\pi$
$479$			7.34179e31i			0.503241i	−0.967826	+	0.251620i	$0.919037\pi$
$480$			2.27227e31i			0.151903i
$481$		−	2.04307e32i		−	1.33212i
$482$			5.41747e30i			0.0344534i
$483$	1.51336e32	+	4.56016e32i	0.938805	+	2.82888i
$484$	3.31618e31			0.200675
$485$	−3.54901e31			−0.209511
$486$	2.68801e31			0.154809
$487$	1.62681e32			0.914091			0.457046	−	0.889443i	$-0.348908\pi$
$487$	1.62681e32			0.914091			0.457046	+	0.889443i	$0.348908\pi$
$488$		−	2.61445e32i		−	1.43332i
$489$	3.23855e32			1.73239
$490$			9.38596e31i			0.489921i
$491$	−3.34438e32			−1.70348			−0.851742	−	0.523961i	$-0.824454\pi$
$491$	−3.34438e32			−1.70348			−0.851742	+	0.523961i	$0.824454\pi$
$492$	−1.31630e31			−0.0654295
$493$			1.37905e32i			0.668985i
$494$			3.78839e32i			1.79362i
$495$	−1.35924e30			−0.00628108
$496$	−3.84310e32			−1.73341
$497$		−	3.52275e30i		−	0.0155098i
$498$		−	5.82192e32i		−	2.50216i
$499$	1.45075e32			0.608674			0.304337	−	0.952564i	$-0.401565\pi$
$499$	1.45075e32			0.608674			0.304337	+	0.952564i	$0.401565\pi$
$500$			2.13285e31i			0.0873614i
$501$	5.88584e32			2.35372
$502$		−	4.36925e32i		−	1.70593i
$503$		−	1.57906e32i		−	0.601980i	−0.953627	−	0.300990i	$-0.902683\pi$
$503$		−	1.57906e32i		−	0.601980i	0.953627	−	0.300990i	$-0.0973170\pi$
$504$			7.98381e32i			2.97197i
$505$		−	8.55181e31i		−	0.310858i
$506$	−1.39750e30	−	4.21105e30i	−0.00496073	−	0.0149480i
$507$	9.58972e30			0.0332438
$508$	−1.55251e31			−0.0525618
$509$	−4.25625e32			−1.40739			−0.703694	−	0.710503i	$-0.748467\pi$
$509$	−4.25625e32			−1.40739			−0.703694	+	0.710503i	$0.748467\pi$
$510$	−1.95880e32			−0.632627
$511$			2.18926e32i			0.690631i
$512$	1.80796e32			0.557120
$513$		−	8.87467e32i		−	2.67143i
$514$	3.97929e32			1.17017
$515$	1.43619e32			0.412597
$516$			1.49105e32i			0.418500i
$517$			3.32420e29i			0.000911594i
$518$	−9.35108e32			−2.50556
$519$	4.24420e32			1.11119
$520$			7.71396e31i			0.197350i
$521$			5.38048e31i			0.134514i	0.997736	+	0.0672571i	$0.0214248\pi$
$521$			5.38048e31i			0.134514i	−0.997736	+	0.0672571i	$0.978575\pi$
$522$	3.90608e32			0.954322
$523$			3.16648e32i			0.756061i	0.925793	+	0.378030i	$0.123399\pi$
$523$			3.16648e32i			0.756061i	−0.925793	+	0.378030i	$0.876601\pi$
$524$	7.51057e31			0.175266
$525$			1.24172e33i			2.83213i
$526$			1.91485e32i			0.426880i
$527$		−	1.03104e33i		−	2.24672i
$528$		−	1.34673e31i		−	0.0286862i
$529$	3.84961e32	−	2.87133e32i	0.801584	−	0.597882i
$530$	1.37818e32			0.280541
$531$	1.08226e33			2.15376
$532$	2.89852e32			0.563945
$533$	−1.00594e32			−0.191357
$534$			8.82178e32i			1.64082i
$535$	−4.93731e31			−0.0897935
$536$		−	2.52696e31i		−	0.0449388i
$537$	−1.47563e33			−2.56618
$538$	−2.48872e32			−0.423241
$539$		−	1.73126e31i		−	0.0287937i
$540$			4.53793e31i			0.0738130i
$541$	−7.34223e32			−1.16805			−0.584024	−	0.811736i	$-0.698523\pi$
$541$	−7.34223e32			−1.16805			−0.584024	+	0.811736i	$0.698523\pi$
$542$	−1.49700e32			−0.232932
$543$			1.79542e33i			2.73255i
$544$		−	3.99812e32i		−	0.595208i
$545$	−1.42293e32			−0.207217
$546$		−	2.31468e33i		−	3.29746i
$547$	−3.98638e32			−0.555560			−0.277780	−	0.960645i	$-0.589599\pi$
$547$	−3.98638e32			−0.555560			−0.277780	+	0.960645i	$0.589599\pi$
$548$			7.02043e31i			0.0957188i
$549$		−	2.40222e33i		−	3.20438i
$550$		−	1.14666e31i		−	0.0149652i
$551$			5.64708e32i			0.721115i
$552$	1.14421e33	−	3.79722e32i	1.42967	−	0.474457i
$553$	2.35080e32			0.287418
$554$	−1.01225e33			−1.21107
$555$	4.32576e32			0.506462
$556$	9.25901e31			0.106088
$557$		−	1.10999e33i		−	1.24468i	−0.782748	−	0.622339i	$-0.786183\pi$
$557$		−	1.10999e33i		−	1.24468i	0.782748	−	0.622339i	$-0.213817\pi$
$558$	−2.92037e33			−3.20499
$559$			1.13948e33i			1.22396i
$560$	4.26907e32			0.448825
$561$	3.61305e31			0.0371809
$562$			1.05205e33i			1.05975i
$563$			9.96240e30i			0.00982343i	0.999988	+	0.00491172i	$0.00156345\pi$
$563$			9.96240e30i			0.00982343i	−0.999988	+	0.00491172i	$0.998437\pi$
$564$	2.26822e31			0.0218945
$565$	−2.41810e32			−0.228503
$566$			1.09751e32i			0.101534i
$567$			1.67592e33i			1.51795i
$568$	−8.83907e30			−0.00783838
$569$			7.96816e32i			0.691848i	0.938263	+	0.345924i	$0.112434\pi$
$569$			7.96816e32i			0.691848i	−0.938263	+	0.345924i	$0.887566\pi$
$570$	−8.02110e32			−0.681924
$571$		−	2.09995e33i		−	1.74814i	−0.485799	−	0.874071i	$-0.661471\pi$
$571$		−	2.09995e33i		−	1.74814i	0.485799	−	0.874071i	$-0.338529\pi$
$572$			3.57310e30i			0.00291268i
$573$			1.12426e33i			0.897456i
$574$			4.60415e32i			0.359921i
$575$	1.17798e33	−	3.90929e32i	0.901826	−	0.299284i
$576$	1.89804e33			1.42310
$577$	−6.71130e32			−0.492829			−0.246415	−	0.969164i	$-0.579253\pi$
$577$	−6.71130e32			−0.492829			−0.246415	+	0.969164i	$0.579253\pi$
$578$	1.92301e33			1.38308
$579$	3.93518e32			0.277219
$580$		−	2.88755e31i		−	0.0199248i
$581$	−3.40412e33			−2.30086
$582$		−	2.67206e33i		−	1.76917i
$583$	−2.54210e31			−0.0164881
$584$	5.49315e32			0.349034
$585$			7.08778e32i			0.441204i
$586$			1.55133e33i			0.946088i
$587$	−1.72724e33			−1.03204			−0.516018	−	0.856578i	$-0.672586\pi$
$587$	−1.72724e33			−1.03204			−0.516018	+	0.856578i	$0.672586\pi$
$588$	−1.18130e33			−0.691564
$589$		−	4.22202e33i		−	2.42179i
$590$		−	4.78605e32i		−	0.269000i
$591$	−2.05366e33			−1.13104
$592$			2.83703e33i			1.53110i
$593$	−2.04322e33			−1.08058			−0.540292	−	0.841478i	$-0.681686\pi$
$593$	−2.04322e33			−1.08058			−0.540292	+	0.841478i	$0.681686\pi$
$594$		−	5.00725e31i		−	0.0259515i
$595$			1.14532e33i			0.581733i
$596$		−	1.48401e32i		−	0.0738721i
$597$		−	9.85906e32i		−	0.480998i
$598$	−2.19586e33	+	7.28727e32i	−1.05000	+	0.348458i
$599$	2.52015e33			1.18115			0.590574	−	0.806983i	$-0.298901\pi$
$599$	2.52015e33			1.18115			0.590574	+	0.806983i	$0.298901\pi$
$600$	3.11565e33			1.43131
$601$	−2.12713e33			−0.957854			−0.478927	−	0.877855i	$-0.658974\pi$
$601$	−2.12713e33			−0.957854			−0.478927	+	0.877855i	$0.658974\pi$
$602$	5.21539e33			2.30212
$603$		−	2.32184e32i		−	0.100467i
$604$	−3.55129e32			−0.150641
$605$			5.36579e32i			0.223136i
$606$	6.43868e33			2.62498
$607$	4.49525e33			1.79676			0.898378	−	0.439222i	$-0.144746\pi$
$607$	4.49525e33			1.79676			0.898378	+	0.439222i	$0.144746\pi$
$608$		−	1.63719e33i		−	0.641589i
$609$		−	3.45033e33i		−	1.32572i
$610$	−1.06233e33			−0.400222
$611$	1.73341e32			0.0640334
$612$		−	1.63188e33i		−	0.591115i
$613$			1.24189e33i			0.441121i	0.975373	+	0.220560i	$0.0707886\pi$
$613$			1.24189e33i			0.441121i	−0.975373	+	0.220560i	$0.929211\pi$
$614$	−4.56209e33			−1.58907
$615$		−	2.12986e32i		−	0.0727527i
$616$	−6.51240e31			−0.0218159
$617$			5.73254e33i			1.88332i	0.336560	+	0.941662i	$0.390737\pi$
$617$			5.73254e33i			1.88332i	−0.336560	+	0.941662i	$0.609263\pi$
$618$			1.08131e34i			3.48410i
$619$			1.86040e33i			0.587919i	0.955818	+	0.293960i	$0.0949731\pi$
$619$			1.86040e33i			0.587919i	−0.955818	+	0.293960i	$0.905027\pi$
$620$			2.15887e32i			0.0669154i
$621$	5.14401e33	−	1.70711e33i	1.56388	−	0.518995i
$622$	−1.01574e33			−0.302899
$623$	5.15816e33			1.50882
$624$	−7.02253e33			−2.01502
$625$	3.03064e33			0.853050
$626$		−	3.02917e33i		−	0.836435i
$627$	1.47951e32			0.0400782
$628$		−	3.43820e32i		−	0.0913725i
$629$	−7.61128e33			−1.98450
$630$	3.24406e33			0.829855
$631$		−	5.99707e33i		−	1.50517i	−0.658492	−	0.752587i	$-0.728805\pi$
$631$		−	5.99707e33i		−	1.50517i	0.658492	−	0.752587i	$-0.271195\pi$
$632$		−	5.89849e32i		−	0.145257i
$633$	−6.28604e33			−1.51891
$634$	3.38828e33			0.803357
$635$		−	2.51206e32i		−	0.0584448i
$636$			1.73456e33i			0.396008i
$637$	−9.02769e33			−2.02257
$638$			3.18619e31i			0.00700524i
$639$	−8.12156e31			−0.0175238
$640$		−	1.25645e33i		−	0.266063i
$641$		−	4.57985e33i		−	0.951820i	−0.879494	−	0.475910i	$-0.842119\pi$
$641$		−	4.57985e33i		−	0.951820i	0.879494	−	0.475910i	$-0.157881\pi$
$642$		−	3.71731e33i		−	0.758243i
$643$		−	9.19096e33i		−	1.84005i	−0.391862	−	0.920024i	$-0.628169\pi$
$643$		−	9.19096e33i		−	1.84005i	0.391862	−	0.920024i	$-0.371831\pi$
$644$	5.57554e32	+	1.68006e33i	0.109561	+	0.330138i
$645$	−2.41261e33			−0.465341
$646$	1.41133e34			2.67202
$647$	−4.09734e32			−0.0761467			−0.0380733	−	0.999275i	$-0.512122\pi$
$647$	−4.09734e32			−0.0761467			−0.0380733	+	0.999275i	$0.512122\pi$
$648$	4.20512e33			0.767146
$649$			8.82799e31i			0.0158098i
$650$	−5.97927e33			−1.05121
$651$			2.57963e34i			4.45231i
$652$	1.19315e33			0.202174
$653$	−2.44646e33			−0.406986			−0.203493	−	0.979076i	$-0.565229\pi$
$653$	−2.44646e33			−0.406986			−0.203493	+	0.979076i	$0.565229\pi$
$654$		−	1.07133e34i		−	1.74980i
$655$			1.21526e33i			0.194883i
$656$	1.39686e33			0.219941
$657$	5.04725e33			0.780315
$658$		−	7.93377e32i		−	0.120440i
$659$			4.00906e33i			0.597610i	0.954314	+	0.298805i	$0.0965881\pi$
$659$			4.00906e33i			0.597610i	−0.954314	+	0.298805i	$0.903412\pi$
$660$	−7.56527e30			−0.00110738
$661$		−	1.14497e33i		−	0.164580i	−0.996608	−	0.0822899i	$-0.973777\pi$
$661$		−	1.14497e33i		−	0.164580i	0.996608	−	0.0822899i	$-0.0262233\pi$
$662$	9.76722e33			1.37872
$663$		−	1.88403e34i		−	2.61171i
$664$			8.54141e33i			1.16282i
$665$			4.68999e33i			0.627065i
$666$			2.15585e34i			2.83093i
$667$	−3.27321e33	+	1.08626e33i	−0.422147	+	0.140096i
$668$	2.16847e33			0.274686
$669$	−7.02801e32			−0.0874418
$670$	−1.02678e32			−0.0125482
$671$	1.95949e32			0.0235219
$672$			1.00031e34i			1.17952i
$673$	9.63959e33			1.11655			0.558275	−	0.829656i	$-0.311464\pi$
$673$	9.63959e33			1.11655			0.558275	+	0.829656i	$0.311464\pi$
$674$		−	1.27844e34i		−	1.45466i
$675$	1.40070e34			1.56567
$676$	3.53307e31			0.00387964
$677$			7.19769e33i			0.776477i	0.921559	+	0.388239i	$0.126916\pi$
$677$			7.19769e33i			0.776477i	−0.921559	+	0.388239i	$0.873084\pi$
$678$		−	1.82059e34i		−	1.92955i
$679$	−1.56237e34			−1.62684
$680$	2.87377e33			0.293999
$681$		−	5.79238e33i		−	0.582226i
$682$		−	2.38214e32i		−	0.0235264i
$683$	5.56249e33			0.539785			0.269893	−	0.962890i	$-0.413012\pi$
$683$	5.56249e33			0.539785			0.269893	+	0.962890i	$0.413012\pi$
$684$		−	6.68242e33i		−	0.637178i
$685$	−1.13595e33			−0.106432
$686$			2.06938e34i			1.90525i
$687$		−	1.01737e34i		−	0.920443i
$688$		−	1.58230e34i		−	1.40679i
$689$			1.32558e34i			1.15818i
$690$	−1.54292e33	−	4.64926e33i	−0.132482	−	0.399203i
$691$	−1.35580e34			−1.14409			−0.572043	−	0.820223i	$-0.693849\pi$
$691$	−1.35580e34			−1.14409			−0.572043	+	0.820223i	$0.693849\pi$
$692$	1.56366e33			0.129679
$693$	−5.98375e32			−0.0487725
$694$	6.76595e33			0.542020
$695$			1.49817e33i			0.117962i
$696$	−8.65736e33			−0.670000
$697$			3.74754e33i			0.285071i
$698$	−2.93963e33			−0.219800
$699$	−9.24911e33			−0.679789
$700$			4.57478e33i			0.330517i
$701$		−	6.13799e33i		−	0.435923i	−0.975957	−	0.217962i	$-0.930059\pi$
$701$		−	6.13799e33i		−	0.435923i	0.975957	−	0.217962i	$-0.0699408\pi$
$702$	−2.61104e34			−1.82292
$703$	−3.11675e34			−2.13914
$704$			1.54823e32i			0.0104463i
$705$			3.67012e32i			0.0243451i
$706$	−9.41533e33			−0.614015
$707$		−	3.76474e34i		−	2.41380i
$708$	6.02364e33			0.379717
$709$			1.56670e34i			0.971028i	0.874229	+	0.485514i	$0.161368\pi$
$709$			1.56670e34i			0.971028i	−0.874229	+	0.485514i	$0.838632\pi$
$710$			3.59158e31i			0.00218869i
$711$		−	5.41968e33i		−	0.324742i
$712$		−	1.29425e34i		−	0.762534i
$713$	2.44720e34	−	8.12140e33i	1.41774	−	0.470497i
$714$	−8.62316e34			−4.91233
$715$	−5.78150e31			−0.00323868
$716$	−5.43655e33			−0.299480
$717$	−4.62096e34			−2.50324
$718$			1.09726e34i			0.584544i
$719$	1.75708e33			0.0920545			0.0460273	−	0.998940i	$-0.485344\pi$
$719$	1.75708e33			0.0920545			0.0460273	+	0.998940i	$0.485344\pi$
$720$		−	9.84218e33i		−	0.507109i
$721$	6.32251e34			3.20381
$722$	3.58058e34			1.78446
$723$		−	1.10325e33i		−	0.0540772i
$724$			6.61472e33i			0.318895i
$725$	−8.91287e33			−0.422631
$726$	−4.03992e34			−1.88423
$727$			3.74282e34i			1.71706i	0.512763	+	0.858530i	$0.328622\pi$
$727$			3.74282e34i			1.71706i	−0.512763	+	0.858530i	$0.671378\pi$
$728$			3.39589e34i			1.53242i
$729$	−2.52068e34			−1.11889
$730$		−	2.23203e33i		−	0.0974599i
$731$	4.24505e34			1.82337
$732$		−	1.33703e34i		−	0.564946i
$733$		−	9.35848e33i		−	0.389007i	−0.980902	−	0.194504i	$-0.937690\pi$
$733$		−	9.35848e33i		−	0.389007i	0.980902	−	0.194504i	$-0.0623096\pi$
$734$		−	4.66947e33i		−	0.190948i
$735$		−	1.91142e34i		−	0.768968i
$736$	9.48963e33	−	3.14927e33i	0.375591	−	0.124646i
$737$	1.89392e31			0.000737484
$738$	1.06147e34			0.406660
$739$	2.21957e34			0.836634			0.418317	−	0.908301i	$-0.362620\pi$
$739$	2.21957e34			0.836634			0.418317	+	0.908301i	$0.362620\pi$
$740$	1.59371e33			0.0591055
$741$		−	7.71493e34i		−	2.81523i
$742$	6.06714e34			2.17840
$743$		−	3.04555e34i		−	1.07597i	−0.842954	−	0.537985i	$-0.819186\pi$
$743$		−	3.04555e34i		−	1.07597i	0.842954	−	0.537985i	$-0.180814\pi$
$744$	6.47265e34			2.25013
$745$	2.40122e33			0.0821403
$746$			3.18788e34i			1.07309i
$747$			7.84806e34i			2.59965i
$748$	1.33113e32			0.00433911
$749$	−2.17354e34			−0.697244
$750$		−	2.59833e34i		−	0.820274i
$751$			1.47879e33i			0.0459440i	0.999736	+	0.0229720i	$0.00731285\pi$
$751$			1.47879e33i			0.0459440i	−0.999736	+	0.0229720i	$0.992687\pi$
$752$	−2.40703e33			−0.0735984
$753$			8.89783e34i			2.67759i
$754$	1.66144e34			0.492072
$755$		−	5.74621e33i		−	0.167501i
$756$			1.99772e34i			0.573156i
$757$			6.99037e34i			1.97401i	0.160683	+	0.987006i	$0.448630\pi$
$757$			6.99037e34i			1.97401i	−0.160683	+	0.987006i	$0.551370\pi$
$758$		−	3.14784e34i		−	0.874948i
$759$	2.84596e32	+	8.57567e32i	0.00778624	+	0.0234621i
$760$	1.17678e34			0.316908
$761$	3.55196e34			0.941570			0.470785	−	0.882248i	$-0.343971\pi$
$761$	3.55196e34			0.941570			0.470785	+	0.882248i	$0.343971\pi$
$762$	1.89134e34			0.493525
$763$	−6.26412e34			−1.60903
$764$			4.14204e33i			0.104736i
$765$	2.64049e34			0.657276
$766$		−	4.85417e33i		−	0.118952i
$767$	4.60336e34			1.11053
$768$	4.19664e34			0.996705
$769$			2.03369e34i			0.475520i	0.971324	+	0.237760i	$0.0764131\pi$
$769$			2.03369e34i			0.475520i	−0.971324	+	0.237760i	$0.923587\pi$
$770$			2.64618e32i			0.00609159i
$771$	−8.10368e34			−1.83667
$772$	1.44981e33			0.0323521
$773$		−	4.49080e34i		−	0.986665i	−0.869841	−	0.493332i	$-0.835779\pi$
$773$		−	4.49080e34i		−	0.986665i	0.869841	−	0.493332i	$-0.164221\pi$
$774$		−	1.20239e35i		−	2.60107i
$775$	6.66368e34			1.41936
$776$			3.92020e34i			0.822182i
$777$	1.90432e35			3.93266
$778$			3.25134e34i			0.661162i
$779$			1.53458e34i			0.307285i
$780$			3.94492e33i			0.0777862i
$781$		−	6.62475e30i		−	0.000128634i
$782$	2.71481e34	+	8.18048e34i	0.519110	+	1.56422i
$783$	−3.89208e34			−0.732894
$784$	1.25360e35			2.32469
$785$	5.56323e33			0.101599
$786$	−9.14971e34			−1.64565
$787$			1.28190e34i			0.227069i	0.993534	+	0.113534i	$0.0362172\pi$
$787$			1.28190e34i			0.227069i	−0.993534	+	0.113534i	$0.963783\pi$
$788$	−7.56615e33			−0.131996
$789$		−	3.89953e34i		−	0.670021i
$790$	−2.39673e33			−0.0405596
$791$	−1.06451e35			−1.77432
$792$			1.50141e33i			0.0246488i
$793$		−	1.02178e35i		−	1.65226i
$794$	−1.10759e35			−1.76414
$795$	−2.80663e34			−0.440331
$796$		−	3.63230e33i		−	0.0561338i
$797$			4.79997e34i			0.730699i	0.930870	+	0.365349i	$0.119050\pi$
$797$			4.79997e34i			0.730699i	−0.930870	+	0.365349i	$0.880950\pi$
$798$	−3.53111e35			−5.29512
$799$		−	6.45767e33i		−	0.0953926i
$800$	2.58401e34			0.376022
$801$		−	1.18919e35i		−	1.70475i
$802$		−	1.09253e35i		−	1.54291i
$803$			4.11704e32i			0.00572794i
$804$		−	1.29229e33i		−	0.0177128i
$805$	−2.71845e34	+	9.02158e33i	−0.367089	+	0.121824i
$806$	−1.24217e35			−1.65257
$807$	5.06819e34			0.664310
$808$	−9.44626e34			−1.21990
$809$	8.17358e34			1.03999			0.519996	−	0.854169i	$-0.325933\pi$
$809$	8.17358e34			1.03999			0.519996	+	0.854169i	$0.325933\pi$
$810$		−	1.70866e34i		−	0.214208i
$811$	−6.11715e34			−0.755613			−0.377806	−	0.925885i	$-0.623321\pi$
$811$	−6.11715e34			−0.755613			−0.377806	+	0.925885i	$0.623321\pi$
$812$		−	1.27118e34i		−	0.154716i
$813$	3.04858e34			0.365605
$814$	−1.75853e33			−0.0207805
$815$			1.93060e34i			0.224803i
$816$			2.61619e35i			3.00183i
$817$	1.73831e35			1.96545
$818$	−2.08954e34			−0.232815
$819$			3.12023e35i			3.42594i
$820$		−	7.84687e32i		−	0.00849044i
$821$	−1.26105e35			−1.34467			−0.672335	−	0.740247i	$-0.734708\pi$
$821$	−1.26105e35			−1.34467			−0.672335	+	0.740247i	$0.734708\pi$
$822$		−	8.55261e34i		−	0.898745i
$823$	1.64389e35			1.70246			0.851228	−	0.524796i	$-0.175858\pi$
$823$	1.64389e35			1.70246			0.851228	+	0.524796i	$0.175858\pi$
$824$		−	1.58641e35i		−	1.61915i
$825$			2.33514e33i			0.0234890i
$826$		−	2.10695e35i		−	2.08878i
$827$		−	1.52246e35i		−	1.48758i	−0.668413	−	0.743790i	$-0.733026\pi$
$827$		−	1.52246e35i		−	1.48758i	0.668413	−	0.743790i	$-0.266974\pi$
$828$	3.87332e34	−	1.28542e34i	0.373009	−	0.123789i
$829$	−7.20474e34			−0.683855			−0.341927	−	0.939726i	$-0.611080\pi$
$829$	−7.20474e34			−0.683855			−0.341927	+	0.939726i	$0.611080\pi$
$830$	3.47063e34			0.324692
$831$	2.06141e35			1.90087
$832$	8.07326e34			0.733786
$833$			3.36319e35i			3.01309i
$834$	−1.12797e35			−0.996108
$835$			3.50873e34i			0.305430i
$836$	5.45085e32			0.00467723
$837$	2.90991e35			2.46135
$838$			2.26017e34i			0.188457i
$839$		−	2.04809e35i		−	1.68347i	−0.539888	−	0.841737i	$-0.681533\pi$
$839$		−	2.04809e35i		−	1.68347i	0.539888	−	0.841737i	$-0.318467\pi$
$840$	−7.19008e34			−0.582616
$841$	−1.00419e35			−0.802165
$842$			8.58994e34i			0.676465i
$843$		−	2.14247e35i		−	1.66335i
$844$	−2.31591e34			−0.177261
$845$			5.71673e32i			0.00431387i
$846$	−1.82910e34			−0.136080
$847$			2.36217e35i			1.73265i
$848$		−	1.84072e35i		−	1.33118i
$849$		−	2.23504e34i		−	0.159365i
$850$			2.22753e35i			1.56602i
$851$	−5.99533e34	−	1.80656e35i	−0.415584	−	1.25227i
$852$	−4.52029e32			−0.00308952
$853$	8.80667e34			0.593504			0.296752	−	0.954955i	$-0.404097\pi$
$853$	8.80667e34			0.593504			0.296752	+	0.954955i	$0.404097\pi$
$854$	−4.67665e35			−3.10771
$855$	1.08126e35			0.708494
$856$			5.45371e34i			0.352376i
$857$	−1.97178e35			−1.25629			−0.628143	−	0.778098i	$-0.716185\pi$
$857$	−1.97178e35			−1.25629			−0.628143	+	0.778098i	$0.716185\pi$
$858$		−	4.35291e33i		−	0.0273484i
$859$	2.19132e35			1.35765			0.678826	−	0.734299i	$-0.262489\pi$
$859$	2.19132e35			1.35765			0.678826	+	0.734299i	$0.262489\pi$
$860$	−8.88860e33			−0.0543065
$861$		−	9.37621e34i		−	0.564923i
$862$		−	2.17643e35i		−	1.29317i
$863$	−2.91101e35			−1.70574			−0.852872	−	0.522120i	$-0.825141\pi$
$863$	−2.91101e35			−1.70574			−0.852872	+	0.522120i	$0.825141\pi$
$864$	1.12839e35			0.652069
$865$			2.53010e34i			0.144193i
$866$		−	1.43830e35i		−	0.808415i
$867$	−3.91615e35			−2.17085
$868$			9.50392e34i			0.519597i
$869$	4.42083e32			0.00238378
$870$			3.51774e34i			0.187082i
$871$		−	9.87588e33i		−	0.0518033i
$872$			1.57175e35i			0.813180i
$873$			3.60198e35i			1.83810i
$874$	1.11169e35	+	3.34983e35i	0.559561	+	1.68611i
$875$	−1.51926e35			−0.754285
$876$	2.80920e34			0.137573
$877$	−3.16698e34			−0.152985			−0.0764927	−	0.997070i	$-0.524372\pi$
$877$	−3.16698e34			−0.152985			−0.0764927	+	0.997070i	$0.524372\pi$
$878$	1.56408e35			0.745291
$879$		−	3.15922e35i		−	1.48496i
$880$	8.02827e32			0.00372246
$881$			2.28390e35i			1.04464i	0.852750	+	0.522320i	$0.174933\pi$
$881$			2.28390e35i			1.04464i	−0.852750	+	0.522320i	$0.825067\pi$
$882$	9.52605e35			4.29823
$883$	6.48209e34			0.288527			0.144264	−	0.989539i	$-0.453919\pi$
$883$	6.48209e34			0.288527			0.144264	+	0.989539i	$0.453919\pi$
$884$		−	6.94118e34i		−	0.304794i
$885$			9.74663e34i			0.422217i
$886$	−3.67490e35			−1.57051
$887$	3.25439e34			0.137210			0.0686050	−	0.997644i	$-0.478145\pi$
$887$	3.25439e34			0.137210			0.0686050	+	0.997644i	$0.478145\pi$
$888$		−	4.77820e35i		−	1.98751i
$889$		−	1.10588e35i		−	0.453822i
$890$	−5.25894e34			−0.212921
$891$			3.15167e33i			0.0125895i
$892$	−2.58928e33			−0.0102047
$893$		−	2.64436e34i		−	0.102826i
$894$			1.80788e35i			0.693617i
$895$		−	8.79669e34i		−	0.332999i
$896$		−	5.53123e35i		−	2.06598i
$897$	4.47179e35	−	1.48403e35i	1.64806	−	0.546932i
$898$	2.02081e35			0.734869
$899$	−1.85161e35			−0.664407
$900$	1.05470e35			0.373437
$901$	4.93833e35			1.72537
$902$			8.65841e32i			0.00298510i
$903$	−1.06210e36			−3.61336
$904$			2.67101e35i			0.896715i
$905$	−1.07030e35			−0.354588
$906$	4.32633e35			1.41443
$907$			1.97071e35i			0.635822i	0.948120	+	0.317911i	$0.102981\pi$
$907$			1.97071e35i			0.635822i	−0.948120	+	0.317911i	$0.897019\pi$
$908$		−	2.13404e34i		−	0.0679474i
$909$	−8.67945e35			−2.72726
$910$	1.37985e35			0.427894
$911$		−	4.14938e35i		−	1.26988i	−0.772561	−	0.634940i	$-0.781025\pi$
$911$		−	4.14938e35i		−	1.26988i	0.772561	−	0.634940i	$-0.218975\pi$
$912$			1.07130e36i			3.23575i
$913$	−6.40166e33			−0.0190829
$914$		−	2.47981e35i		−	0.729564i
$915$	2.16340e35			0.628178
$916$		−	3.74821e34i		−	0.107418i
$917$			5.34989e35i			1.51326i
$918$			9.72720e35i			2.71566i
$919$		−	1.59871e35i		−	0.440539i	−0.975439	−	0.220269i	$-0.929306\pi$
$919$		−	1.59871e35i		−	0.440539i	0.975439	−	0.220269i	$-0.0706936\pi$
$920$	2.26364e34	+	6.82097e34i	0.0615678	+	0.185521i
$921$	9.29054e35			2.49417
$922$	7.97394e35			2.11301
$923$	−3.45448e33			−0.00903572
$924$	−3.33044e33			−0.00859879
$925$		−	4.91922e35i		−	1.25370i
$926$	−1.46835e35			−0.369400
$927$		−	1.45763e36i		−	3.61985i
$928$	−7.18009e34			−0.176017
$929$	−4.84522e35			−1.17253			−0.586267	−	0.810118i	$-0.699403\pi$
$929$	−4.84522e35			−1.17253			−0.586267	+	0.810118i	$0.699403\pi$
$930$		−	2.63003e35i		−	0.628298i
$931$			1.37720e36i			3.24788i
$932$	−3.40758e34			−0.0793332
$933$	2.06852e35			0.475422
$934$		−	3.09906e35i		−	0.703181i
$935$			2.15385e33i			0.00482477i
$936$	7.82910e35			1.73142
$937$		−	5.02566e34i		−	0.109728i	−0.998494	−	0.0548640i	$-0.982527\pi$
$937$		−	5.02566e34i		−	0.109728i	0.998494	−	0.0548640i	$-0.0174725\pi$
$938$	−4.52017e34			−0.0974362
$939$			6.16881e35i			1.31285i
$940$			1.35215e33i			0.00284114i
$941$		−	7.86938e35i		−	1.63254i	−0.577667	−	0.816272i	$-0.696037\pi$
$941$		−	7.86938e35i		−	1.63254i	0.577667	−	0.816272i	$-0.303963\pi$
$942$			4.18857e35i			0.857936i
$943$	−8.89488e34	+	2.95190e34i	−0.179887	+	0.0596981i
$944$	−6.39228e35			−1.27642
$945$	−3.23244e35			−0.637307
$946$	9.80787e33			0.0190933
$947$	3.40488e34			0.0654488			0.0327244	−	0.999464i	$-0.489582\pi$
$947$	3.40488e34			0.0654488			0.0327244	+	0.999464i	$0.489582\pi$
$948$		−	3.01648e34i		−	0.0572533i
$949$	2.14683e35			0.402350
$950$			9.12153e35i			1.68805i
$951$	−6.90012e35			−1.26093
$952$	1.26511e36			2.28289
$953$		−	4.11597e35i		−	0.733428i	−0.930334	−	0.366714i	$-0.880483\pi$
$953$		−	4.11597e35i		−	0.733428i	0.930334	−	0.366714i	$-0.119517\pi$
$954$		−	1.39876e36i		−	2.46128i
$955$	−6.70209e34			−0.116458
$956$	−1.70247e35			−0.292135
$957$		−	6.48857e33i		−	0.0109953i
$958$		−	3.29520e35i		−	0.551437i
$959$	−5.00077e35			−0.826444
$960$			1.70934e35i			0.278981i
$961$	7.63942e35			1.23134
$962$			9.16987e35i			1.45970i
$963$			5.01100e35i			0.787787i
$964$		−	4.06462e33i		−	0.00631096i
$965$			2.34588e34i			0.0359732i
$966$	−6.79237e35	−	2.04673e36i	−1.02872	−	3.09981i
$967$	5.85891e35			0.876393			0.438197	−	0.898879i	$-0.355617\pi$
$967$	5.85891e35			0.876393			0.438197	+	0.898879i	$0.355617\pi$
$968$	5.92701e35			0.875651
$969$	−2.87413e36			−4.19393
$970$	1.59289e35			0.229576
$971$		−	2.39384e34i		−	0.0340773i	−0.999855	−	0.0170386i	$-0.994576\pi$
$971$		−	2.39384e34i		−	0.0340773i	0.999855	−	0.0170386i	$-0.00542383\pi$
$972$	−2.01676e34			−0.0283569
$973$			6.59534e35i			0.915974i
$974$	−7.30159e35			−1.00164
$975$	1.21766e36			1.64995
$976$			1.41885e36i			1.89906i
$977$		−	1.38440e36i		−	1.83032i	−0.403093	−	0.915159i	$-0.632065\pi$
$977$		−	1.38440e36i		−	1.83032i	0.403093	−	0.915159i	$-0.367935\pi$
$978$	−1.45355e36			−1.89830
$979$	9.70024e33			0.0125138
$980$		−	7.04209e34i		−	0.0897406i
$981$			1.44417e36i			1.81798i
$982$	1.50105e36			1.86663
$983$			2.87478e35i			0.353152i	0.984287	+	0.176576i	$0.0565022\pi$
$983$			2.87478e35i			0.353152i	−0.984287	+	0.176576i	$0.943498\pi$
$984$	−2.35262e35			−0.285503
$985$		−	1.22425e35i		−	0.146770i
$986$		−	6.18956e35i		−	0.733055i
$987$			1.61569e35i			0.189039i
$988$		−	2.84235e35i		−	0.328545i
$989$	3.34378e35	+	1.00757e36i	0.381841	+	1.15059i
$990$	6.10066e33			0.00688263
$991$	5.22824e35			0.582736			0.291368	−	0.956611i	$-0.405890\pi$
$991$	5.22824e35			0.582736			0.291368	+	0.956611i	$0.405890\pi$
$992$	5.36818e35			0.591135
$993$	−1.98906e36			−2.16400
$994$			1.58111e34i			0.0169952i
$995$	5.87729e34			0.0624166
$996$			4.36807e35i			0.458329i
$997$	−4.89819e35			−0.507801			−0.253901	−	0.967230i	$-0.581714\pi$
$997$	−4.89819e35			−0.507801			−0.253901	+	0.967230i	$0.581714\pi$
$998$	−6.51136e35			−0.666968
$999$		−	2.14813e36i		−	2.17408i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.25.b.c.22.12	yes	44
23.22	odd	2	inner	23.25.b.c.22.11	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.25.b.c.22.11	✓	44	23.22	odd	2	inner
23.25.b.c.22.12	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)

\(f(q)\)	\(=\)	\(q-4488.28 q^{2} +914024. q^{3} +3.36747e6 q^{4} +5.44878e7i q^{5} -4.10240e9 q^{6} +2.39870e10i q^{7} +6.01867e10 q^{8} +5.53011e11 q^{9} -2.44557e11i q^{10} +4.51091e10i q^{11} +3.07795e12 q^{12} +2.35222e13 q^{13} -1.07660e14i q^{14} +4.98032e13i q^{15} -3.26632e14 q^{16} -8.76299e14i q^{17} -2.48207e15 q^{18} -3.58837e15i q^{19} +1.83486e14i q^{20} +2.19247e16i q^{21} -2.02462e14i q^{22} +(2.07992e16 - 6.90252e15i) q^{23} +5.50121e16 q^{24} +5.66357e16 q^{25} -1.05574e17 q^{26} +2.47318e17 q^{27} +8.07754e16i q^{28} -1.57372e17 q^{29} -2.23531e17i q^{30} +1.17659e18 q^{31} +4.56250e17 q^{32} +4.12308e16i q^{33} +3.93308e18i q^{34} -1.30700e18 q^{35} +1.86225e18 q^{36} -8.68571e18i q^{37} +1.61056e19i q^{38} +2.14998e19 q^{39} +3.27944e18i q^{40} -4.27655e18 q^{41} -9.84042e19i q^{42} +4.84430e19i q^{43} +1.51903e17i q^{44} +3.01323e19i q^{45} +(-9.33526e19 + 3.09804e19i) q^{46} +7.36926e18 q^{47} -2.98549e20 q^{48} -3.83795e20 q^{49} -2.54197e20 q^{50} -8.00959e20i q^{51} +7.92101e19 q^{52} +5.63544e20i q^{53} -1.11003e21 q^{54} -2.45789e18 q^{55} +1.44370e21i q^{56} -3.27985e21i q^{57} +7.06329e20 q^{58} +1.95703e21 q^{59} +1.67711e20i q^{60} -4.34389e21i q^{61} -5.28085e21 q^{62} +1.32651e22i q^{63} +3.43219e21 q^{64} +1.28167e21i q^{65} -1.85055e20i q^{66} -4.19854e20i q^{67} -2.95091e21i q^{68} +(1.90110e22 - 6.30907e21i) q^{69} +5.86618e21 q^{70} -1.46861e20 q^{71} +3.32839e22 q^{72} +9.12685e21 q^{73} +3.89839e22i q^{74} +5.17664e22 q^{75} -1.20837e22i q^{76} -1.08203e21 q^{77} -9.64973e22 q^{78} -9.80031e21i q^{79} -1.77974e22i q^{80} +6.98678e22 q^{81} +1.91944e22 q^{82} +1.41915e23i q^{83} +7.38307e22i q^{84} +4.77476e22 q^{85} -2.17426e23i q^{86} -1.43842e23 q^{87} +2.71497e21i q^{88} -2.15040e23i q^{89} -1.35242e23i q^{90} +5.64226e23i q^{91} +(7.00406e22 - 2.32440e22i) q^{92} +1.07543e24 q^{93} -3.30753e22 q^{94} +1.95522e23 q^{95} +4.17024e23 q^{96} +6.51340e23i q^{97} +1.72258e24 q^{98} +2.49458e22i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4232 q^{2} - 434562 q^{3} + 317760360 q^{4} - 8460029520 q^{6} - 198307023760 q^{8} + 4220041988298 q^{9} - 67439597688792 q^{12} + 5771152551358 q^{13} + 18\!\cdots\!92 q^{16} + 18\!\cdots\!68 q^{18}+ \cdots - 20\!\cdots\!92 q^{98}+O(q^{100}) \) 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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