LMFDB - Embedded newform 45.9.g.b.37.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,9,Mod(28,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.3320374528$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3006 x^{14} + 3660359 x^{12} + 2360769624 x^{10} + 888292333775 x^{8} + 201214811046486 x^{6} + \cdots + 60\!\cdots\!84 $ x^16 + 3006x^14 + 3660359x^12 + 2360769624x^10 + 888292333775x^8 + 201214811046486x^6 + 26986219156502761x^4 + 1971352965206494164*x^2 + 60370178541549816384
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{12}\cdot 5^{19} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.5
Root			$27.6766i$ of defining polynomial
Character	$\chi$	$=$	45.37
Dual form			45.9.g.b.28.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.79807 + 3.79807i) q^{2} -227.149i q^{4} +(563.744 + 269.849i) q^{5} +(-1263.51 - 1263.51i) q^{7} +(1835.03 - 1835.03i) q^{8} +O(q^{10})$	\(q+(3.79807 + 3.79807i) q^{2} -227.149i q^{4} +(563.744 + 269.849i) q^{5} +(-1263.51 - 1263.51i) q^{7} +(1835.03 - 1835.03i) q^{8} +(1116.23 + 3166.04i) q^{10} -2393.22 q^{11} +(4601.53 - 4601.53i) q^{13} -9597.80i q^{14} -44211.1 q^{16} +(-62989.7 - 62989.7i) q^{17} -114324. i q^{19} +(61295.9 - 128054. i) q^{20} +(-9089.61 - 9089.61i) q^{22} +(264286. - 264286. i) q^{23} +(244988. + 304251. i) q^{25} +34953.9 q^{26} +(-287006. + 287006. i) q^{28} -1.26150e6i q^{29} +462025. q^{31} +(-637685. - 637685. i) q^{32} -478478. i q^{34} +(-371340. - 1.05325e6i) q^{35} +(-785681. - 785681. i) q^{37} +(434210. - 434210. i) q^{38} +(1.52967e6 - 539307. i) q^{40} +1.91676e6 q^{41} +(-3.06087e6 + 3.06087e6i) q^{43} +543619. i q^{44} +2.00755e6 q^{46} +(1.27822e6 + 1.27822e6i) q^{47} -2.57188e6i q^{49} +(-225082. + 2.08605e6i) q^{50} +(-1.04524e6 - 1.04524e6i) q^{52} +(-5.22439e6 + 5.22439e6i) q^{53} +(-1.34916e6 - 645807. i) q^{55} -4.63717e6 q^{56} +(4.79125e6 - 4.79125e6i) q^{58} +2.36986e7i q^{59} -1.07351e7 q^{61} +(1.75480e6 + 1.75480e6i) q^{62} +6.47410e6i q^{64} +(3.83580e6 - 1.35237e6i) q^{65} +(2.21877e7 + 2.21877e7i) q^{67} +(-1.43081e7 + 1.43081e7i) q^{68} +(2.58995e6 - 5.41070e6i) q^{70} +2.73960e7 q^{71} +(4.75412e6 - 4.75412e6i) q^{73} -5.96814e6i q^{74} -2.59686e7 q^{76} +(3.02386e6 + 3.02386e6i) q^{77} -8.87427e6i q^{79} +(-2.49237e7 - 1.19303e7i) q^{80} +(7.28000e6 + 7.28000e6i) q^{82} +(2.53098e7 - 2.53098e7i) q^{83} +(-1.85123e7 - 5.25077e7i) q^{85} -2.32508e7 q^{86} +(-4.39164e6 + 4.39164e6i) q^{88} -6.89392e7i q^{89} -1.16282e7 q^{91} +(-6.00324e7 - 6.00324e7i) q^{92} +9.70949e6i q^{94} +(3.08501e7 - 6.44493e7i) q^{95} +(5.46963e7 + 5.46963e7i) q^{97} +(9.76815e6 - 9.76815e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4220 q^{7}+O(q^{10}) $ 16 * q - 4220 * q^7	$ 16 q - 4220 q^{7} - 47000 q^{10} - 37940 q^{13} - 508916 q^{16} + 844700 q^{22} - 1664300 q^{25} + 6009380 q^{28} - 944752 q^{31} + 10616140 q^{37} - 17493600 q^{40} + 4050760 q^{43} - 34233160 q^{46} + 7180240 q^{52} + 17430500 q^{55} - 27842100 q^{58} + 32032232 q^{61} + 75463480 q^{67} + 272773500 q^{70} - 198258320 q^{73} - 196046088 q^{76} + 172139600 q^{82} + 183614500 q^{85} - 624395100 q^{88} - 477271600 q^{91} + 662476480 q^{97}+O(q^{100}) $ 16 * q - 4220 * q^7 - 47000 * q^10 - 37940 * q^13 - 508916 * q^16 + 844700 * q^22 - 1664300 * q^25 + 6009380 * q^28 - 944752 * q^31 + 10616140 * q^37 - 17493600 * q^40 + 4050760 * q^43 - 34233160 * q^46 + 7180240 * q^52 + 17430500 * q^55 - 27842100 * q^58 + 32032232 * q^61 + 75463480 * q^67 + 272773500 * q^70 - 198258320 * q^73 - 196046088 * q^76 + 172139600 * q^82 + 183614500 * q^85 - 624395100 * q^88 - 477271600 * q^91 + 662476480 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	3.79807	+	3.79807i	0.237379	+	0.237379i	0.815764	−	0.578385i	$-0.196317\pi$
$2$	3.79807	+	3.79807i	0.237379	+	0.237379i	−0.578385	+	0.815764i	$0.696317\pi$
$3$		0			0
$3$		0			0
$4$		−	227.149i		−	0.887302i
$5$	563.744	+	269.849i	0.901990	+	0.431758i
$5$	563.744	+	269.849i	0.901990	+	0.431758i
$6$		0			0
$7$	−1263.51	−	1263.51i	−0.526244	−	0.526244i	0.393206	−	0.919450i	$-0.371366\pi$
$7$	−1263.51	−	1263.51i	−0.526244	−	0.526244i	−0.919450	+	0.393206i	$0.871366\pi$
$8$	1835.03	−	1835.03i	0.448006	−	0.448006i
$9$		0			0
$10$	1116.23	+	3166.04i	0.111623	+	0.316604i
$11$	−2393.22			−0.163460			−0.0817301	−	0.996654i	$-0.526045\pi$
$11$	−2393.22			−0.163460			−0.0817301	+	0.996654i	$0.526045\pi$
$12$		0			0
$13$	4601.53	−	4601.53i	0.161113	−	0.161113i	−0.621947	−	0.783059i	$-0.713658\pi$
$13$	4601.53	−	4601.53i	0.161113	−	0.161113i	0.783059	+	0.621947i	$0.213658\pi$
$14$		−	9597.80i		−	0.249839i
$15$		0			0
$16$	−44211.1			−0.674608
$17$	−62989.7	−	62989.7i	−0.754177	−	0.754177i	0.221079	−	0.975256i	$-0.429042\pi$
$17$	−62989.7	−	62989.7i	−0.754177	−	0.754177i	−0.975256	+	0.221079i	$0.929042\pi$
$18$		0			0
$19$		−	114324.i		−	0.877248i	−0.898671	−	0.438624i	$-0.855466\pi$
$19$		−	114324.i		−	0.877248i	0.898671	−	0.438624i	$-0.144534\pi$
$20$	61295.9	−	128054.i	0.383100	−	0.800337i
$21$		0			0
$22$	−9089.61	−	9089.61i	−0.0388020	−	0.0388020i
$23$	264286.	−	264286.i	0.944415	−	0.944415i	−0.0541198	−	0.998534i	$-0.517235\pi$
$23$	264286.	−	264286.i	0.944415	−	0.944415i	0.998534	+	0.0541198i	$0.0172353\pi$
$24$		0			0
$25$	244988.	+	304251.i	0.627171	+	0.778882i
$26$	34953.9			0.0764895
$27$		0			0
$28$	−287006.	+	287006.i	−0.466938	+	0.466938i
$29$		−	1.26150e6i		−	1.78359i	−0.452443	−	0.891793i	$-0.649447\pi$
$29$		−	1.26150e6i		−	1.78359i	0.452443	−	0.891793i	$-0.350553\pi$
$30$		0			0
$31$	462025.			0.500287			0.250143	−	0.968209i	$-0.419522\pi$
$31$	462025.			0.500287			0.250143	+	0.968209i	$0.419522\pi$
$32$	−637685.	−	637685.i	−0.608144	−	0.608144i
$33$		0			0
$34$		−	478478.i		−	0.358052i
$35$	−371340.	−	1.05325e6i	−0.247457	−	0.701877i
$36$		0			0
$37$	−785681.	−	785681.i	−0.419218	−	0.419218i	0.465716	−	0.884934i	$-0.345797\pi$
$37$	−785681.	−	785681.i	−0.419218	−	0.419218i	−0.884934	+	0.465716i	$0.845797\pi$
$38$	434210.	−	434210.i	0.208240	−	0.208240i
$39$		0			0
$40$	1.52967e6	−	539307.i	0.597527	−	0.210667i
$41$	1.91676e6			0.678318			0.339159	−	0.940729i	$-0.389858\pi$
$41$	1.91676e6			0.678318			0.339159	+	0.940729i	$0.389858\pi$
$42$		0			0
$43$	−3.06087e6	+	3.06087e6i	−0.895306	+	0.895306i	−0.995017	−	0.0997103i	$-0.968208\pi$
$43$	−3.06087e6	+	3.06087e6i	−0.895306	+	0.895306i	0.0997103	+	0.995017i	$0.468208\pi$
$44$			543619.i			0.145039i
$45$		0			0
$46$	2.00755e6			0.448369
$47$	1.27822e6	+	1.27822e6i	0.261947	+	0.261947i	0.825844	−	0.563898i	$-0.190699\pi$
$47$	1.27822e6	+	1.27822e6i	0.261947	+	0.261947i	−0.563898	+	0.825844i	$0.690699\pi$
$48$		0			0
$49$		−	2.57188e6i		−	0.446134i
$50$	−225082.	+	2.08605e6i	−0.0360131	+	0.333767i
$51$		0			0
$52$	−1.04524e6	−	1.04524e6i	−0.142956	−	0.142956i
$53$	−5.22439e6	+	5.22439e6i	−0.662113	+	0.662113i	−0.955878	−	0.293765i	$-0.905092\pi$
$53$	−5.22439e6	+	5.22439e6i	−0.662113	+	0.662113i	0.293765	+	0.955878i	$0.405092\pi$
$54$		0			0
$55$	−1.34916e6	−	645807.i	−0.147439	−	0.0705752i
$56$	−4.63717e6			−0.471521
$57$		0			0
$58$	4.79125e6	−	4.79125e6i	0.423386	−	0.423386i
$59$			2.36986e7i			1.95576i	0.209169	+	0.977880i	$0.432924\pi$
$59$			2.36986e7i			1.95576i	−0.209169	+	0.977880i	$0.567076\pi$
$60$		0			0
$61$	−1.07351e7			−0.775329			−0.387665	−	0.921801i	$-0.626718\pi$
$61$	−1.07351e7			−0.775329			−0.387665	+	0.921801i	$0.626718\pi$
$62$	1.75480e6	+	1.75480e6i	0.118758	+	0.118758i
$63$		0			0
$64$			6.47410e6i			0.385887i
$65$	3.83580e6	−	1.35237e6i	0.214883	−	0.0757602i
$66$		0			0
$67$	2.21877e7	+	2.21877e7i	1.10107	+	1.10107i	0.994282	+	0.106783i	$0.0340551\pi$
$67$	2.21877e7	+	2.21877e7i	1.10107	+	1.10107i	0.106783	+	0.994282i	$0.465945\pi$
$68$	−1.43081e7	+	1.43081e7i	−0.669183	+	0.669183i
$69$		0			0
$70$	2.58995e6	−	5.41070e6i	0.107870	−	0.225352i
$71$	2.73960e7			1.07809			0.539043	−	0.842278i	$-0.318786\pi$
$71$	2.73960e7			1.07809			0.539043	+	0.842278i	$0.318786\pi$
$72$		0			0
$73$	4.75412e6	−	4.75412e6i	0.167409	−	0.167409i	−0.618430	−	0.785840i	$-0.712231\pi$
$73$	4.75412e6	−	4.75412e6i	0.167409	−	0.167409i	0.785840	+	0.618430i	$0.212231\pi$
$74$		−	5.96814e6i		−	0.199027i
$75$		0			0
$76$	−2.59686e7			−0.778384
$77$	3.02386e6	+	3.02386e6i	0.0860200	+	0.0860200i
$78$		0			0
$79$		−	8.87427e6i		−	0.227837i	−0.993490	−	0.113919i	$-0.963660\pi$
$79$		−	8.87427e6i		−	0.227837i	0.993490	−	0.113919i	$-0.0363403\pi$
$80$	−2.49237e7	−	1.19303e7i	−0.608489	−	0.291267i
$81$		0			0
$82$	7.28000e6	+	7.28000e6i	0.161019	+	0.161019i
$83$	2.53098e7	−	2.53098e7i	0.533305	−	0.533305i	−0.388249	−	0.921554i	$-0.626920\pi$
$83$	2.53098e7	−	2.53098e7i	0.533305	−	0.533305i	0.921554	+	0.388249i	$0.126920\pi$
$84$		0			0
$85$	−1.85123e7	−	5.25077e7i	−0.354638	−	1.00588i
$86$	−2.32508e7			−0.425054
$87$		0			0
$88$	−4.39164e6	+	4.39164e6i	−0.0732312	+	0.0732312i
$89$		−	6.89392e7i		−	1.09877i	−0.835570	−	0.549384i	$-0.814862\pi$
$89$		−	6.89392e7i		−	1.09877i	0.835570	−	0.549384i	$-0.185138\pi$
$90$		0			0
$91$	−1.16282e7			−0.169569
$92$	−6.00324e7	−	6.00324e7i	−0.837981	−	0.837981i
$93$		0			0
$94$			9.70949e6i			0.124361i
$95$	3.08501e7	−	6.44493e7i	0.378759	−	0.791269i
$96$		0			0
$97$	5.46963e7	+	5.46963e7i	0.617833	+	0.617833i	0.944975	−	0.327142i	$-0.106086\pi$
$97$	5.46963e7	+	5.46963e7i	0.617833	+	0.617833i	−0.327142	+	0.944975i	$0.606086\pi$
$98$	9.76815e6	−	9.76815e6i	0.105903	−	0.105903i
$99$		0			0
$100$	6.91104e7	−	5.56490e7i	0.691104	−	0.556490i
$101$	1.29927e8			1.24857			0.624287	−	0.781195i	$-0.285389\pi$
$101$	1.29927e8			1.24857			0.624287	+	0.781195i	$0.285389\pi$
$102$		0			0
$103$	−1.27844e8	+	1.27844e8i	−1.13588	+	1.13588i	−0.146696	+	0.989182i	$0.546864\pi$
$103$	−1.27844e8	+	1.27844e8i	−1.13588	+	1.13588i	−0.989182	+	0.146696i	$0.953136\pi$
$104$		−	1.68879e7i		−	0.144359i
$105$		0			0
$106$	−3.96852e7			−0.314344
$107$	3.20088e7	+	3.20088e7i	0.244194	+	0.244194i	0.818583	−	0.574389i	$-0.194760\pi$
$107$	3.20088e7	+	3.20088e7i	0.244194	+	0.244194i	−0.574389	+	0.818583i	$0.694760\pi$
$108$		0			0
$109$		−	1.68791e8i		−	1.19576i	−0.801585	−	0.597880i	$-0.796010\pi$
$109$		−	1.68791e8i		−	1.19576i	0.801585	−	0.597880i	$-0.203990\pi$
$110$	−2.67139e6	−	7.57703e6i	−0.0182460	−	0.0517521i
$111$		0			0
$112$	5.58612e7	+	5.58612e7i	0.355008	+	0.355008i
$113$	4.34518e7	−	4.34518e7i	0.266498	−	0.266498i	−0.561189	−	0.827687i	$-0.689656\pi$
$113$	4.34518e7	−	4.34518e7i	0.266498	−	0.266498i	0.827687	+	0.561189i	$0.189656\pi$
$114$		0			0
$115$	2.20307e8	−	7.76723e7i	1.25961	−	0.444094i
$116$	−2.86548e8			−1.58258
$117$		0			0
$118$	−9.00090e7	+	9.00090e7i	−0.464256	+	0.464256i
$119$			1.59176e8i			0.793763i
$120$		0			0
$121$	−2.08631e8			−0.973281
$122$	−4.07726e7	−	4.07726e7i	−0.184047	−	0.184047i
$123$		0			0
$124$		−	1.04949e8i		−	0.443906i
$125$	5.60090e7	+	2.37629e8i	0.229413	+	0.973329i
$126$		0			0
$127$	2.83349e8	+	2.83349e8i	1.08920	+	1.08920i	0.995611	+	0.0935869i	$0.0298333\pi$
$127$	2.83349e8	+	2.83349e8i	1.08920	+	1.08920i	0.0935869	+	0.995611i	$0.470167\pi$
$128$	−1.87836e8	+	1.87836e8i	−0.699745	+	0.699745i
$129$		0			0
$130$	1.97050e7	+	9.43225e6i	0.0689927	+	0.0330249i
$131$	1.56673e8			0.531997			0.265998	−	0.963973i	$-0.414298\pi$
$131$	1.56673e8			0.531997			0.265998	+	0.963973i	$0.414298\pi$
$132$		0			0
$133$	−1.44450e8	+	1.44450e8i	−0.461647	+	0.461647i
$134$			1.68541e8i			0.522740i
$135$		0			0
$136$	−2.31176e8			−0.675752
$137$	4.29110e8	+	4.29110e8i	1.21811	+	1.21811i	0.968293	+	0.249817i	$0.0803704\pi$
$137$	4.29110e8	+	4.29110e8i	1.21811	+	1.21811i	0.249817	+	0.968293i	$0.419630\pi$
$138$		0			0
$139$		−	3.83809e8i		−	1.02815i	−0.857746	−	0.514074i	$-0.828136\pi$
$139$		−	3.83809e8i		−	1.02815i	0.857746	−	0.514074i	$-0.171864\pi$
$140$	−2.39246e8	+	8.43496e7i	−0.622777	+	0.219569i
$141$		0			0
$142$	1.04052e8	+	1.04052e8i	0.255915	+	0.255915i
$143$	−1.10125e7	+	1.10125e7i	−0.0263355	+	0.0263355i
$144$		0			0
$145$	3.40413e8	−	7.11161e8i	0.770077	−	1.60878i
$146$	3.61129e7			0.0794788
$147$		0			0
$148$	−1.78467e8	+	1.78467e8i	−0.371973	+	0.371973i
$149$		−	1.64089e8i		−	0.332916i	−0.986049	−	0.166458i	$-0.946767\pi$
$149$		−	1.64089e8i		−	0.332916i	0.986049	−	0.166458i	$-0.0532330\pi$
$150$		0			0
$151$	−6.72571e8			−1.29369			−0.646845	−	0.762622i	$-0.723912\pi$
$151$	−6.72571e8			−1.29369			−0.646845	+	0.762622i	$0.723912\pi$
$152$	−2.09788e8	−	2.09788e8i	−0.393013	−	0.393013i
$153$		0			0
$154$			2.29697e7i			0.0408387i
$155$	2.60464e8	+	1.24677e8i	0.451253	+	0.216003i
$156$		0			0
$157$	7.08820e8	+	7.08820e8i	1.16664	+	1.16664i	0.982992	+	0.183649i	$0.0587911\pi$
$157$	7.08820e8	+	7.08820e8i	1.16664	+	1.16664i	0.183649	+	0.982992i	$0.441209\pi$
$158$	3.37051e7	−	3.37051e7i	0.0540838	−	0.0540838i
$159$		0			0
$160$	−1.87412e8	−	5.31569e8i	−0.285969	−	0.811110i
$161$	−6.67857e8			−0.993985
$162$		0			0
$163$	−1.86978e8	+	1.86978e8i	−0.264874	+	0.264874i	−0.827031	−	0.562157i	$-0.809972\pi$
$163$	−1.86978e8	+	1.86978e8i	−0.264874	+	0.264874i	0.562157	+	0.827031i	$0.309972\pi$
$164$		−	4.35392e8i		−	0.601873i
$165$		0			0
$166$	1.92256e8			0.253191
$167$	−2.92261e8	−	2.92261e8i	−0.375756	−	0.375756i	0.493813	−	0.869568i	$-0.335603\pi$
$167$	−2.92261e8	−	2.92261e8i	−0.375756	−	0.375756i	−0.869568	+	0.493813i	$0.835603\pi$
$168$		0			0
$169$			7.73382e8i			0.948086i
$170$	1.29117e8	−	2.69739e8i	0.154592	−	0.322959i
$171$		0			0
$172$	6.95276e8	+	6.95276e8i	0.794407	+	0.794407i
$173$	9.48183e8	−	9.48183e8i	1.05854	−	1.05854i	0.0603642	−	0.998176i	$-0.480774\pi$
$173$	9.48183e8	−	9.48183e8i	1.05854	−	1.05854i	0.998176	−	0.0603642i	$-0.0192262\pi$
$174$		0			0
$175$	7.48786e7	−	6.93970e8i	0.0798373	−	0.739927i
$176$	1.05807e8			0.110272
$177$		0			0
$178$	2.61836e8	−	2.61836e8i	0.260825	−	0.260825i
$179$		−	8.32961e8i		−	0.811358i	−0.914016	−	0.405679i	$-0.867035\pi$
$179$		−	8.32961e8i		−	0.811358i	0.914016	−	0.405679i	$-0.132965\pi$
$180$		0			0
$181$	−1.03144e9			−0.961011			−0.480505	−	0.876992i	$-0.659547\pi$
$181$	−1.03144e9			−0.961011			−0.480505	+	0.876992i	$0.659547\pi$
$182$	−4.41646e7	−	4.41646e7i	−0.0402521	−	0.0402521i
$183$		0			0
$184$		−	9.69947e8i		−	0.846207i
$185$	−2.30908e8	−	6.54938e8i	−0.197129	−	0.559130i
$186$		0			0
$187$	1.50748e8	+	1.50748e8i	0.123278	+	0.123278i
$188$	2.90346e8	−	2.90346e8i	0.232426	−	0.232426i
$189$		0			0
$190$	3.61954e8	−	1.27612e8i	0.277740	−	0.0979213i
$191$	1.15496e9			0.867827			0.433913	−	0.900955i	$-0.357132\pi$
$191$	1.15496e9			0.867827			0.433913	+	0.900955i	$0.357132\pi$
$192$		0			0
$193$	−7.80665e8	+	7.80665e8i	−0.562646	+	0.562646i	−0.930058	−	0.367412i	$-0.880244\pi$
$193$	−7.80665e8	+	7.80665e8i	−0.562646	+	0.562646i	0.367412	+	0.930058i	$0.380244\pi$
$194$			4.15480e8i			0.293321i
$195$		0			0
$196$	−5.84200e8			−0.395856
$197$	−8.77672e8	−	8.77672e8i	−0.582730	−	0.582730i	0.352923	−	0.935653i	$-0.385188\pi$
$197$	−8.77672e8	−	8.77672e8i	−0.582730	−	0.582730i	−0.935653	+	0.352923i	$0.885188\pi$
$198$		0			0
$199$		−	1.28132e9i		−	0.817041i	−0.912749	−	0.408521i	$-0.866045\pi$
$199$		−	1.28132e9i		−	0.817041i	0.912749	−	0.408521i	$-0.133955\pi$
$200$	1.00787e9	+	1.08748e8i	0.629920	+	0.0679677i
$201$		0			0
$202$	4.93472e8	+	4.93472e8i	0.296386	+	0.296386i
$203$	−1.59392e9	+	1.59392e9i	−0.938602	+	0.938602i
$204$		0			0
$205$	1.08056e9	+	5.17236e8i	0.611836	+	0.292869i
$206$	−9.71120e8			−0.539267
$207$		0			0
$208$	−2.03439e8	+	2.03439e8i	−0.108688	+	0.108688i
$209$			2.73602e8i			0.143395i
$210$		0			0
$211$	−2.42670e8			−0.122429			−0.0612147	−	0.998125i	$-0.519497\pi$
$211$	−2.42670e8			−0.122429			−0.0612147	+	0.998125i	$0.519497\pi$
$212$	1.18672e9	+	1.18672e9i	0.587495	+	0.587495i
$213$		0			0
$214$			2.43143e8i			0.115933i
$215$	−2.55152e9	+	8.99575e8i	−1.19411	+	0.421002i
$216$		0			0
$217$	−5.83775e8	−	5.83775e8i	−0.263273	−	0.263273i
$218$	6.41081e8	−	6.41081e8i	0.283849	−	0.283849i
$219$		0			0
$220$	−1.46695e8	+	3.06461e8i	−0.0626215	+	0.130823i
$221$	−5.79698e8			−0.243015
$222$		0			0
$223$	−8.08119e7	+	8.08119e7i	−0.0326780	+	0.0326780i	−0.723257	−	0.690579i	$-0.757356\pi$
$223$	−8.08119e7	+	8.08119e7i	−0.0326780	+	0.0326780i	0.690579	+	0.723257i	$0.257356\pi$
$224$			1.61145e9i			0.640064i
$225$		0			0
$226$	3.30065e8			0.126522
$227$	2.04069e9	+	2.04069e9i	0.768552	+	0.768552i	0.977851	−	0.209300i	$-0.0671185\pi$
$227$	2.04069e9	+	2.04069e9i	0.768552	+	0.768552i	−0.209300	+	0.977851i	$0.567118\pi$
$228$		0			0
$229$		−	9.69115e8i		−	0.352398i	−0.984355	−	0.176199i	$-0.943620\pi$
$229$		−	9.69115e8i		−	0.352398i	0.984355	−	0.176199i	$-0.0563802\pi$
$230$	1.13174e9	+	5.41735e8i	0.404424	+	0.193587i
$231$		0			0
$232$	−2.31489e9	−	2.31489e9i	−0.799058	−	0.799058i
$233$	−3.48900e9	+	3.48900e9i	−1.18380	+	1.18380i	−0.205046	+	0.978752i	$0.565735\pi$
$233$	−3.48900e9	+	3.48900e9i	−1.18380	+	1.18380i	−0.978752	+	0.205046i	$0.934265\pi$
$234$		0			0
$235$	3.75661e8	+	1.06551e9i	0.123176	+	0.349370i
$236$	5.38313e9			1.73535
$237$		0			0
$238$	−6.04562e8	+	6.04562e8i	−0.188423	+	0.188423i
$239$			3.23778e9i			0.992330i	0.868228	+	0.496165i	$0.165259\pi$
$239$			3.23778e9i			0.992330i	−0.868228	+	0.496165i	$0.834741\pi$
$240$		0			0
$241$	5.80806e9			1.72172			0.860861	−	0.508840i	$-0.169925\pi$
$241$	5.80806e9			1.72172			0.860861	+	0.508840i	$0.169925\pi$
$242$	−7.92396e8	−	7.92396e8i	−0.231036	−	0.231036i
$243$		0			0
$244$			2.43847e9i			0.687951i
$245$	6.94017e8	−	1.44988e9i	0.192622	−	0.402409i
$246$		0			0
$247$	−5.26065e8	−	5.26065e8i	−0.141336	−	0.141336i
$248$	8.47832e8	−	8.47832e8i	0.224132	−	0.224132i
$249$		0			0
$250$	−6.89805e8	+	1.11526e9i	−0.176590	+	0.285506i
$251$	−4.71877e9			−1.18887			−0.594434	−	0.804144i	$-0.702624\pi$
$251$	−4.71877e9			−1.18887			−0.594434	+	0.804144i	$0.702624\pi$
$252$		0			0
$253$	−6.32495e8	+	6.32495e8i	−0.154374	+	0.154374i
$254$			2.15236e9i			0.517106i
$255$		0			0
$256$	2.30540e8			0.0536767
$257$	−4.32801e9	−	4.32801e9i	−0.992100	−	0.992100i	0.00786898	−	0.999969i	$-0.497495\pi$
$257$	−4.32801e9	−	4.32801e9i	−0.992100	−	0.992100i	−0.999969	+	0.00786898i	$0.997495\pi$
$258$		0			0
$259$			1.98544e9i			0.441222i
$260$	−3.07189e8	−	8.71300e8i	−0.0672222	−	0.190667i
$261$		0			0
$262$	5.95054e8	+	5.95054e8i	0.126285	+	0.126285i
$263$	2.02736e9	−	2.02736e9i	0.423748	−	0.423748i	−0.462744	−	0.886492i	$-0.653135\pi$
$263$	2.02736e9	−	2.02736e9i	0.423748	−	0.423748i	0.886492	+	0.462744i	$0.153135\pi$
$264$		0			0
$265$	−4.35501e9	+	1.53542e9i	−0.883092	+	0.311347i
$266$	−1.09726e9			−0.219171
$267$		0			0
$268$	5.03992e9	−	5.03992e9i	0.976978	−	0.976978i
$269$		−	6.96389e9i		−	1.32997i	−0.746855	−	0.664987i	$-0.768437\pi$
$269$		−	6.96389e9i		−	1.32997i	0.746855	−	0.664987i	$-0.231563\pi$
$270$		0			0
$271$	−4.60908e8			−0.0854548			−0.0427274	−	0.999087i	$-0.513605\pi$
$271$	−4.60908e8			−0.0854548			−0.0427274	+	0.999087i	$0.513605\pi$
$272$	2.78484e9	+	2.78484e9i	0.508774	+	0.508774i
$273$		0			0
$274$			3.25958e9i			0.578308i
$275$	−5.86312e8	−	7.28139e8i	−0.102517	−	0.127316i
$276$		0			0
$277$	−7.03729e9	−	7.03729e9i	−1.19533	−	1.19533i	−0.975550	−	0.219776i	$-0.929467\pi$
$277$	−7.03729e9	−	7.03729e9i	−1.19533	−	1.19533i	−0.219776	−	0.975550i	$-0.570533\pi$
$278$	1.45773e9	−	1.45773e9i	0.244061	−	0.244061i
$279$		0			0
$280$	−2.61418e9	−	1.25133e9i	−0.425307	−	0.203583i
$281$	6.38779e9			1.02453			0.512266	−	0.858827i	$-0.328806\pi$
$281$	6.38779e9			1.02453			0.512266	+	0.858827i	$0.328806\pi$
$282$		0			0
$283$	7.73852e9	−	7.73852e9i	1.20646	−	1.20646i	0.234292	−	0.972166i	$-0.424723\pi$
$283$	7.73852e9	−	7.73852e9i	1.20646	−	1.20646i	0.972166	−	0.234292i	$-0.0752772\pi$
$284$		−	6.22298e9i		−	0.956589i
$285$		0			0
$286$	−8.36523e7			−0.0125030
$287$	−2.42186e9	−	2.42186e9i	−0.356961	−	0.356961i
$288$		0			0
$289$			9.59636e8i			0.137567i
$290$	3.99395e9	−	1.40812e9i	0.564690	−	0.199090i
$291$		0			0
$292$	−1.07990e9	−	1.07990e9i	−0.148542	−	0.148542i
$293$	3.65403e9	−	3.65403e9i	0.495794	−	0.495794i	−0.414332	−	0.910126i	$-0.635985\pi$
$293$	3.65403e9	−	3.65403e9i	0.495794	−	0.495794i	0.910126	+	0.414332i	$0.135985\pi$
$294$		0			0
$295$	−6.39504e9	+	1.33600e10i	−0.844414	+	1.76407i
$296$	−2.88350e9			−0.375624
$297$		0			0
$298$	6.23221e8	−	6.23221e8i	0.0790273	−	0.0790273i
$299$		−	2.43224e9i		−	0.304314i
$300$		0			0
$301$	7.73490e9			0.942299
$302$	−2.55447e9	−	2.55447e9i	−0.307095	−	0.307095i
$303$		0			0
$304$			5.05438e9i			0.591798i
$305$	−6.05183e9	−	2.89685e9i	−0.699339	−	0.334754i
$306$		0			0
$307$	−1.90099e9	−	1.90099e9i	−0.214006	−	0.214006i	0.591961	−	0.805967i	$-0.298354\pi$
$307$	−1.90099e9	−	1.90099e9i	−0.214006	−	0.214006i	−0.805967	+	0.591961i	$0.798354\pi$
$308$	6.86869e8	−	6.86869e8i	0.0763257	−	0.0763257i
$309$		0			0
$310$	5.15728e8	+	1.46279e9i	0.0558436	+	0.158393i
$311$	−3.14139e9			−0.335799			−0.167900	−	0.985804i	$-0.553698\pi$
$311$	−3.14139e9			−0.335799			−0.167900	+	0.985804i	$0.553698\pi$
$312$		0			0
$313$	2.51498e9	−	2.51498e9i	0.262034	−	0.262034i	−0.563846	−	0.825880i	$-0.690679\pi$
$313$	2.51498e9	−	2.51498e9i	0.262034	−	0.262034i	0.825880	+	0.563846i	$0.190679\pi$
$314$			5.38429e9i			0.553872i
$315$		0			0
$316$	−2.01579e9			−0.202160
$317$	8.48738e9	+	8.48738e9i	0.840498	+	0.840498i	0.988924	−	0.148426i	$-0.0474206\pi$
$317$	8.48738e9	+	8.48738e9i	0.840498	+	0.840498i	−0.148426	+	0.988924i	$0.547421\pi$
$318$		0			0
$319$			3.01904e9i			0.291545i
$320$	−1.74703e9	+	3.64973e9i	−0.166609	+	0.348066i
$321$		0			0
$322$	−2.53656e9	−	2.53656e9i	−0.235951	−	0.235951i
$323$	−7.20122e9	+	7.20122e9i	−0.661601	+	0.661601i
$324$		0			0
$325$	2.52734e9	+	2.72697e8i	0.226533	+	0.0244426i
$326$	−1.42031e9			−0.125751
$327$		0			0
$328$	3.51733e9	−	3.51733e9i	0.303891	−	0.303891i
$329$		−	3.23008e9i		−	0.275696i
$330$		0			0
$331$	1.47303e10			1.22716			0.613579	−	0.789634i	$-0.289729\pi$
$331$	1.47303e10			1.22716			0.613579	+	0.789634i	$0.289729\pi$
$332$	−5.74909e9	−	5.74909e9i	−0.473203	−	0.473203i
$333$		0			0
$334$		−	2.22006e9i		−	0.178393i
$335$	6.52085e9	+	1.84955e10i	0.517756	+	1.46854i
$336$		0			0
$337$	−1.23187e10	−	1.23187e10i	−0.955093	−	0.955093i	0.0439408	−	0.999034i	$-0.486009\pi$
$337$	−1.23187e10	−	1.23187e10i	−0.955093	−	0.955093i	−0.999034	+	0.0439408i	$0.986009\pi$
$338$	−2.93736e9	+	2.93736e9i	−0.225056	+	0.225056i
$339$		0			0
$340$	−1.19271e10	+	4.20507e9i	−0.892522	+	0.314671i
$341$	−1.10573e9			−0.0817770
$342$		0			0
$343$	−1.05335e10	+	1.05335e10i	−0.761020	+	0.761020i
$344$			1.12336e10i			0.802205i
$345$		0			0
$346$	7.20252e9			0.502551
$347$	1.53260e10	+	1.53260e10i	1.05709	+	1.05709i	0.998269	+	0.0588205i	$0.0187340\pi$
$347$	1.53260e10	+	1.53260e10i	1.05709	+	1.05709i	0.0588205	+	0.998269i	$0.481266\pi$
$348$		0			0
$349$			2.75699e9i			0.185837i	0.995674	+	0.0929187i	$0.0296197\pi$
$349$			2.75699e9i			0.185837i	−0.995674	+	0.0929187i	$0.970380\pi$
$350$	2.92014e9	−	2.35135e9i	0.194595	−	0.156691i
$351$		0			0
$352$	1.52612e9	+	1.52612e9i	0.0994073	+	0.0994073i
$353$	1.01786e10	−	1.01786e10i	0.655525	−	0.655525i	−0.298793	−	0.954318i	$-0.596584\pi$
$353$	1.01786e10	−	1.01786e10i	0.655525	−	0.655525i	0.954318	+	0.298793i	$0.0965841\pi$
$354$		0			0
$355$	1.54443e10	+	7.39277e9i	0.972423	+	0.465472i
$356$	−1.56595e10			−0.974940
$357$		0			0
$358$	3.16364e9	−	3.16364e9i	0.192599	−	0.192599i
$359$			2.18455e10i			1.31517i	0.753378	+	0.657587i	$0.228423\pi$
$359$			2.18455e10i			1.31517i	−0.753378	+	0.657587i	$0.771577\pi$
$360$		0			0
$361$	3.91362e9			0.230436
$362$	−3.91746e9	−	3.91746e9i	−0.228124	−	0.228124i
$363$		0			0
$364$			2.64134e9i			0.150459i
$365$	3.96300e9	−	1.39721e9i	0.223281	−	0.0787211i
$366$		0			0
$367$	−1.57287e10	−	1.57287e10i	−0.867020	−	0.867020i	0.125121	−	0.992141i	$-0.460068\pi$
$367$	−1.57287e10	−	1.57287e10i	−0.867020	−	0.867020i	−0.992141	+	0.125121i	$0.960068\pi$
$368$	−1.16844e10	+	1.16844e10i	−0.637109	+	0.637109i
$369$		0			0
$370$	1.61049e9	−	3.36450e9i	0.0859314	−	0.179520i
$371$	1.32022e10			0.696866
$372$		0			0
$373$	−1.34880e10	+	1.34880e10i	−0.696809	+	0.696809i	−0.963721	−	0.266912i	$-0.913997\pi$
$373$	−1.34880e10	+	1.34880e10i	−0.696809	+	0.696809i	0.266912	+	0.963721i	$0.413997\pi$
$374$			1.14510e9i			0.0585272i
$375$		0			0
$376$	4.69114e9			0.234707
$377$	−5.80482e9	−	5.80482e9i	−0.287358	−	0.287358i
$378$		0			0
$379$		−	1.99838e10i		−	0.968546i	−0.874917	−	0.484273i	$-0.839084\pi$
$379$		−	1.99838e10i		−	0.968546i	0.874917	−	0.484273i	$-0.160916\pi$
$380$	−1.46396e10	−	7.00759e9i	−0.702095	−	0.336073i
$381$		0			0
$382$	4.38661e9	+	4.38661e9i	0.206004	+	0.206004i
$383$	−1.80028e9	+	1.80028e9i	−0.0836653	+	0.0836653i	−0.747701	−	0.664036i	$-0.768842\pi$
$383$	−1.80028e9	+	1.80028e9i	−0.0836653	+	0.0836653i	0.664036	+	0.747701i	$0.268842\pi$
$384$		0			0
$385$	8.88698e8	+	2.52067e9i	0.0404493	+	0.114729i
$386$	−5.93003e9			−0.267121
$387$		0			0
$388$	1.24242e10	−	1.24242e10i	0.548204	−	0.548204i
$389$		−	3.97690e10i		−	1.73679i	−0.495876	−	0.868393i	$-0.665153\pi$
$389$		−	3.97690e10i		−	1.73679i	0.495876	−	0.868393i	$-0.334847\pi$
$390$		0			0
$391$	−3.32946e10			−1.42451
$392$	−4.71948e9	−	4.71948e9i	−0.199871	−	0.199871i
$393$		0			0
$394$		−	6.66691e9i		−	0.276656i
$395$	2.39471e9	−	5.00281e9i	0.0983704	−	0.205507i
$396$		0			0
$397$	2.14166e10	+	2.14166e10i	0.862162	+	0.862162i	0.991589	−	0.129427i	$-0.0413137\pi$
$397$	2.14166e10	+	2.14166e10i	0.862162	+	0.862162i	−0.129427	+	0.991589i	$0.541314\pi$
$398$	4.86652e9	−	4.86652e9i	0.193949	−	0.193949i
$399$		0			0
$400$	−1.08312e10	−	1.34513e10i	−0.423094	−	0.525440i
$401$	1.76006e10			0.680690			0.340345	−	0.940301i	$-0.389456\pi$
$401$	1.76006e10			0.680690			0.340345	+	0.940301i	$0.389456\pi$
$402$		0			0
$403$	2.12603e9	−	2.12603e9i	0.0806025	−	0.0806025i
$404$		−	2.95129e10i		−	1.10786i
$405$		0			0
$406$	−1.21076e10			−0.445609
$407$	1.88031e9	+	1.88031e9i	0.0685254	+	0.0685254i
$408$		0			0
$409$			1.01202e10i			0.361655i	0.983515	+	0.180827i	$0.0578776\pi$
$409$			1.01202e10i			0.361655i	−0.983515	+	0.180827i	$0.942122\pi$
$410$	2.13955e9	+	6.06855e9i	0.0757160	+	0.214758i
$411$		0			0
$412$	2.90397e10	+	2.90397e10i	1.00787	+	1.00787i
$413$	2.99435e10	−	2.99435e10i	1.02921	−	1.02921i
$414$		0			0
$415$	2.10980e10	−	7.43841e9i	0.711294	−	0.250777i
$416$	−5.86866e9			−0.195959
$417$		0			0
$418$	−1.03916e9	+	1.03916e9i	−0.0340390	+	0.0340390i
$419$			1.92609e10i			0.624913i	0.949932	+	0.312457i	$0.101152\pi$
$419$			1.92609e10i			0.624913i	−0.949932	+	0.312457i	$0.898848\pi$
$420$		0			0
$421$	5.32612e9			0.169544			0.0847720	−	0.996400i	$-0.472984\pi$
$421$	5.32612e9			0.169544			0.0847720	+	0.996400i	$0.472984\pi$
$422$	−9.21676e8	−	9.21676e8i	−0.0290622	−	0.0290622i
$423$		0			0
$424$			1.91739e10i			0.593262i
$425$	3.73291e9	−	3.45964e10i	0.114417	−	1.06041i
$426$		0			0
$427$	1.35639e10	+	1.35639e10i	0.408012	+	0.408012i
$428$	7.27078e9	−	7.27078e9i	0.216674	−	0.216674i
$429$		0			0
$430$	−1.31075e10	−	6.27420e9i	−0.383394	−	0.183520i
$431$	2.68556e10			0.778262			0.389131	−	0.921182i	$-0.372775\pi$
$431$	2.68556e10			0.778262			0.389131	+	0.921182i	$0.372775\pi$
$432$		0			0
$433$	−1.08434e10	+	1.08434e10i	−0.308472	+	0.308472i	−0.844317	−	0.535845i	$-0.819993\pi$
$433$	−1.08434e10	+	1.08434e10i	−0.308472	+	0.308472i	0.535845	+	0.844317i	$0.319993\pi$
$434$		−	4.43443e9i		−	0.124991i
$435$		0			0
$436$	−3.83409e10			−1.06100
$437$	−3.02142e10	−	3.02142e10i	−0.828486	−	0.828486i
$438$		0			0
$439$		−	2.77833e10i		−	0.748041i	−0.927420	−	0.374021i	$-0.877979\pi$
$439$		−	2.77833e10i		−	0.748041i	0.927420	−	0.374021i	$-0.122021\pi$
$440$	−3.66084e9	+	1.29068e9i	−0.0976719	+	0.0344356i
$441$		0			0
$442$	−2.20173e9	−	2.20173e9i	−0.0576866	−	0.0576866i
$443$	−4.95264e10	+	4.95264e10i	−1.28594	+	1.28594i	−0.348714	+	0.937229i	$0.613382\pi$
$443$	−4.95264e10	+	4.95264e10i	−1.28594	+	1.28594i	−0.937229	+	0.348714i	$0.886618\pi$
$444$		0			0
$445$	1.86031e10	−	3.88640e10i	0.474402	−	0.991078i
$446$	−6.13858e8			−0.0155142
$447$		0			0
$448$	8.18010e9	−	8.18010e9i	0.203070	−	0.203070i
$449$		−	5.42287e9i		−	0.133427i	−0.997772	−	0.0667136i	$-0.978749\pi$
$449$		−	5.42287e9i		−	0.133427i	0.997772	−	0.0667136i	$-0.0212514\pi$
$450$		0			0
$451$	−4.58724e9			−0.110878
$452$	−9.87005e9	−	9.87005e9i	−0.236464	−	0.236464i
$453$		0			0
$454$			1.55013e10i			0.364876i
$455$	−6.55532e9	−	3.13785e9i	−0.152949	−	0.0732127i
$456$		0			0
$457$	1.55109e10	+	1.55109e10i	0.355609	+	0.355609i	0.862191	−	0.506583i	$-0.169091\pi$
$457$	1.55109e10	+	1.55109e10i	0.355609	+	0.355609i	−0.506583	+	0.862191i	$0.669091\pi$
$458$	3.68076e9	−	3.68076e9i	0.0836519	−	0.0836519i
$459$		0			0
$460$	−1.76432e10	−	5.00425e10i	−0.394046	−	1.11766i
$461$	−3.26888e10			−0.723761			−0.361881	−	0.932224i	$-0.617865\pi$
$461$	−3.26888e10			−0.723761			−0.361881	+	0.932224i	$0.617865\pi$
$462$		0			0
$463$	−2.92199e10	+	2.92199e10i	−0.635849	+	0.635849i	−0.949529	−	0.313680i	$-0.898438\pi$
$463$	−2.92199e10	+	2.92199e10i	−0.635849	+	0.635849i	0.313680	+	0.949529i	$0.398438\pi$
$464$			5.57722e10i			1.20322i
$465$		0			0
$466$	−2.65029e10			−0.562018
$467$	4.90240e10	+	4.90240e10i	1.03072	+	1.03072i	0.999513	+	0.0312094i	$0.00993587\pi$
$467$	4.90240e10	+	4.90240e10i	1.03072	+	1.03072i	0.0312094	+	0.999513i	$0.490064\pi$
$468$		0			0
$469$		−	5.60689e10i		−	1.15886i
$470$	−2.62009e9	+	5.47366e9i	−0.0536939	+	0.112173i
$471$		0			0
$472$	4.34878e10	+	4.34878e10i	0.876192	+	0.876192i
$473$	7.32535e9	−	7.32535e9i	0.146347	−	0.146347i
$474$		0			0
$475$	3.47831e10	−	2.80080e10i	0.683273	−	0.550184i
$476$	3.61568e10			0.704308
$477$		0			0
$478$	−1.22973e10	+	1.22973e10i	−0.235558	+	0.235558i
$479$			7.35533e10i			1.39721i	0.715510	+	0.698603i	$0.246195\pi$
$479$			7.35533e10i			1.39721i	−0.715510	+	0.698603i	$0.753805\pi$
$480$		0			0
$481$	−7.23068e9			−0.135082
$482$	2.20594e10	+	2.20594e10i	0.408701	+	0.408701i
$483$		0			0
$484$			4.73905e10i			0.863594i
$485$	1.60750e10	+	4.55944e10i	0.290525	+	0.824033i
$486$		0			0
$487$	2.47809e10	+	2.47809e10i	0.440556	+	0.440556i	0.892199	−	0.451643i	$-0.149162\pi$
$487$	2.47809e10	+	2.47809e10i	0.440556	+	0.440556i	−0.451643	+	0.892199i	$0.649162\pi$
$488$	−1.96992e10	+	1.96992e10i	−0.347352	+	0.347352i
$489$		0			0
$490$	8.14266e9	−	2.87081e9i	0.141248	−	0.0497990i
$491$	−2.36469e10			−0.406864			−0.203432	−	0.979089i	$-0.565210\pi$
$491$	−2.36469e10			−0.406864			−0.203432	+	0.979089i	$0.565210\pi$
$492$		0			0
$493$	−7.94613e10	+	7.94613e10i	−1.34514	+	1.34514i
$494$		−	3.99606e9i		−	0.0671003i
$495$		0			0
$496$	−2.04266e10			−0.337497
$497$	−3.46152e10	−	3.46152e10i	−0.567337	−	0.567337i
$498$		0			0
$499$		−	8.08962e10i		−	1.30475i	−0.757898	−	0.652373i	$-0.773773\pi$
$499$		−	8.08962e10i		−	1.30475i	0.757898	−	0.652373i	$-0.226227\pi$
$500$	5.39773e10	−	1.27224e10i	0.863637	−	0.203559i
$501$		0			0
$502$	−1.79222e10	−	1.79222e10i	−0.282213	−	0.282213i
$503$	−3.19123e10	+	3.19123e10i	−0.498525	+	0.498525i	−0.910978	−	0.412454i	$-0.864672\pi$
$503$	−3.19123e10	+	3.19123e10i	−0.498525	+	0.498525i	0.412454	+	0.910978i	$0.364672\pi$
$504$		0			0
$505$	7.32456e10	+	3.50607e10i	1.12620	+	0.539082i
$506$	−4.80451e9			−0.0732904
$507$		0			0
$508$	6.43626e10	−	6.43626e10i	0.966448	−	0.966448i
$509$		−	4.29973e10i		−	0.640576i	−0.947320	−	0.320288i	$-0.896220\pi$
$509$		−	4.29973e10i		−	0.640576i	0.947320	−	0.320288i	$-0.103780\pi$
$510$		0			0
$511$	−1.20138e10			−0.176196
$512$	4.89617e10	+	4.89617e10i	0.712487	+	0.712487i
$513$		0			0
$514$		−	3.28761e10i		−	0.471008i
$515$	−1.06570e11	+	3.75727e10i	−1.51497	+	0.534126i
$516$		0			0
$517$	−3.05905e9	−	3.05905e9i	−0.0428178	−	0.0428178i
$518$	−7.54081e9	+	7.54081e9i	−0.104737	+	0.104737i
$519$		0			0
$520$	4.55719e9	−	9.52046e9i	0.0623280	−	0.130210i
$521$	2.79302e9			0.0379073			0.0189536	−	0.999820i	$-0.493967\pi$
$521$	2.79302e9			0.0379073			0.0189536	+	0.999820i	$0.493967\pi$
$522$		0			0
$523$	−8.03667e10	+	8.03667e10i	−1.07416	+	1.07416i	−0.0771414	+	0.997020i	$0.524579\pi$
$523$	−8.03667e10	+	8.03667e10i	−1.07416	+	1.07416i	−0.997020	+	0.0771414i	$0.975421\pi$
$524$		−	3.55882e10i		−	0.472042i
$525$		0			0
$526$	1.54001e10			0.201178
$527$	−2.91028e10	−	2.91028e10i	−0.377305	−	0.377305i
$528$		0			0
$529$		−	6.13831e10i		−	0.783838i
$530$	−2.23723e10	−	1.07090e10i	−0.283535	−	0.135720i
$531$		0			0
$532$	3.28116e10	+	3.28116e10i	0.409620	+	0.409620i
$533$	8.82006e9	−	8.82006e9i	0.109286	−	0.109286i
$534$		0			0
$535$	9.40722e9	+	2.66823e10i	0.114828	+	0.325693i
$536$	8.14303e10			0.986568
$537$		0			0
$538$	2.64493e10	−	2.64493e10i	0.315708	−	0.315708i
$539$			6.15507e9i			0.0729252i
$540$		0			0
$541$	−2.13690e10			−0.249457			−0.124728	−	0.992191i	$-0.539806\pi$
$541$	−2.13690e10			−0.249457			−0.124728	+	0.992191i	$0.539806\pi$
$542$	−1.75056e9	−	1.75056e9i	−0.0202852	−	0.0202852i
$543$		0			0
$544$			8.03351e10i			0.917297i
$545$	4.55481e10	−	9.51550e10i	0.516279	−	1.07856i
$546$		0			0
$547$	1.23893e10	+	1.23893e10i	0.138387	+	0.138387i	0.772907	−	0.634520i	$-0.218802\pi$
$547$	1.23893e10	+	1.23893e10i	0.138387	+	0.138387i	−0.634520	+	0.772907i	$0.718802\pi$
$548$	9.74721e10	−	9.74721e10i	1.08083	−	1.08083i
$549$		0			0
$550$	5.38671e8	−	4.99237e9i	0.00588671	−	0.0545577i
$551$	−1.44219e11			−1.56465
$552$		0			0
$553$	−1.12128e10	+	1.12128e10i	−0.119898	+	0.119898i
$554$		−	5.34562e10i		−	0.567491i
$555$		0			0
$556$	−8.71820e10			−0.912278
$557$	−4.20197e10	−	4.20197e10i	−0.436549	−	0.436549i	0.454300	−	0.890849i	$-0.349889\pi$
$557$	−4.20197e10	−	4.20197e10i	−0.436549	−	0.436549i	−0.890849	+	0.454300i	$0.849889\pi$
$558$		0			0
$559$			2.81694e10i			0.288490i
$560$	1.64173e10	+	4.65655e10i	0.166936	+	0.473491i
$561$		0			0
$562$	2.42613e10	+	2.42613e10i	0.243202	+	0.243202i
$563$	5.40384e10	−	5.40384e10i	0.537860	−	0.537860i	−0.385040	−	0.922900i	$-0.625812\pi$
$563$	5.40384e10	−	5.40384e10i	0.537860	−	0.537860i	0.922900	+	0.385040i	$0.125812\pi$
$564$		0			0
$565$	3.62211e10	−	1.27703e10i	0.355441	−	0.125316i
$566$	5.87828e10			0.572776
$567$		0			0
$568$	5.02726e10	−	5.02726e10i	0.482989	−	0.482989i
$569$		−	1.43322e11i		−	1.36730i	−0.729812	−	0.683648i	$-0.760392\pi$
$569$		−	1.43322e11i		−	1.36730i	0.729812	−	0.683648i	$-0.239608\pi$
$570$		0			0
$571$	1.10340e11			1.03798			0.518990	−	0.854780i	$-0.326308\pi$
$571$	1.10340e11			1.03798			0.518990	+	0.854780i	$0.326308\pi$
$572$	2.50148e9	+	2.50148e9i	0.0233675	+	0.0233675i
$573$		0			0
$574$		−	1.83967e10i		−	0.169470i
$575$	1.45156e11	+	1.56622e10i	1.32790	+	0.143279i
$576$		0			0
$577$	−1.29057e11	−	1.29057e11i	−1.16434	−	1.16434i	−0.983515	−	0.180825i	$-0.942123\pi$
$577$	−1.29057e11	−	1.29057e11i	−1.16434	−	1.16434i	−0.180825	−	0.983515i	$-0.557877\pi$
$578$	−3.64476e9	+	3.64476e9i	−0.0326556	+	0.0326556i
$579$		0			0
$580$	−1.61540e11	−	7.73247e10i	−1.42747	−	0.683291i
$581$	−6.39583e10			−0.561297
$582$		0			0
$583$	1.25031e10	−	1.25031e10i	0.108229	−	0.108229i
$584$		−	1.74479e10i		−	0.150001i
$585$		0			0
$586$	2.77565e10			0.235382
$587$	−3.68418e10	−	3.68418e10i	−0.310304	−	0.310304i	0.534723	−	0.845027i	$-0.320416\pi$
$587$	−3.68418e10	−	3.68418e10i	−0.310304	−	0.310304i	−0.845027	+	0.534723i	$0.820416\pi$
$588$		0			0
$589$		−	5.28205e10i		−	0.438876i
$590$	−7.50308e10	+	2.64532e10i	−0.619201	+	0.218308i
$591$		0			0
$592$	3.47358e10	+	3.47358e10i	0.282807	+	0.282807i
$593$	4.73580e10	−	4.73580e10i	0.382979	−	0.382979i	−0.489195	−	0.872174i	$-0.662710\pi$
$593$	4.73580e10	−	4.73580e10i	0.382979	−	0.382979i	0.872174	+	0.489195i	$0.162710\pi$
$594$		0			0
$595$	−4.29535e10	+	8.97346e10i	−0.342713	+	0.715966i
$596$	−3.72727e10			−0.295397
$597$		0			0
$598$	9.23781e9	−	9.23781e9i	0.0722378	−	0.0722378i
$599$			1.11240e11i			0.864076i	0.901855	+	0.432038i	$0.142205\pi$
$599$			1.11240e11i			0.864076i	−0.901855	+	0.432038i	$0.857795\pi$
$600$		0			0
$601$	6.47420e10			0.496236			0.248118	−	0.968730i	$-0.420188\pi$
$601$	6.47420e10			0.496236			0.248118	+	0.968730i	$0.420188\pi$
$602$	2.93777e10	+	2.93777e10i	0.223682	+	0.223682i
$603$		0			0
$604$			1.52774e11i			1.14789i
$605$	−1.17615e11	−	5.62989e10i	−0.877889	−	0.420222i
$606$		0			0
$607$	−8.31206e10	−	8.31206e10i	−0.612285	−	0.612285i	0.331256	−	0.943541i	$-0.392528\pi$
$607$	−8.31206e10	−	8.31206e10i	−0.612285	−	0.612285i	−0.943541	+	0.331256i	$0.892528\pi$
$608$	−7.29026e10	+	7.29026e10i	−0.533493	+	0.533493i
$609$		0			0
$610$	−1.19828e10	−	3.39877e10i	−0.0865447	−	0.245472i
$611$	1.17635e10			0.0844057
$612$		0			0
$613$	5.95760e10	−	5.95760e10i	0.421919	−	0.421919i	−0.463945	−	0.885864i	$-0.653566\pi$
$613$	5.95760e10	−	5.95760e10i	0.421919	−	0.421919i	0.885864	+	0.463945i	$0.153566\pi$
$614$		−	1.44401e10i		−	0.101601i
$615$		0			0
$616$	1.10978e10			0.0770749
$617$	1.40069e10	+	1.40069e10i	0.0966502	+	0.0966502i	0.753779	−	0.657128i	$-0.228229\pi$
$617$	1.40069e10	+	1.40069e10i	0.0966502	+	0.0966502i	−0.657128	+	0.753779i	$0.728229\pi$
$618$		0			0
$619$			2.09904e11i			1.42974i	0.699255	+	0.714872i	$0.253515\pi$
$619$			2.09904e11i			1.42974i	−0.699255	+	0.714872i	$0.746485\pi$
$620$	2.83203e10	−	5.91642e10i	0.191660	−	0.400398i
$621$		0			0
$622$	−1.19312e10	−	1.19312e10i	−0.0797118	−	0.0797118i
$623$	−8.71055e10	+	8.71055e10i	−0.578220	+	0.578220i
$624$		0			0
$625$	−3.25492e10	+	1.49076e11i	−0.213314	+	0.976984i
$626$	1.91041e10			0.124403
$627$		0			0
$628$	1.61008e11	−	1.61008e11i	1.03516	−	1.03516i
$629$			9.89796e10i			0.632329i
$630$		0			0
$631$	1.62984e11			1.02808			0.514039	−	0.857767i	$-0.328148\pi$
$631$	1.62984e11			1.02808			0.514039	+	0.857767i	$0.328148\pi$
$632$	−1.62846e10	−	1.62846e10i	−0.102072	−	0.102072i
$633$		0			0
$634$			6.44713e10i			0.399033i
$635$	8.32748e10	+	2.36198e11i	0.512176	+	1.45272i
$636$		0			0
$637$	−1.18346e10	−	1.18346e10i	−0.0718778	−	0.0718778i
$638$	−1.14665e10	+	1.14665e10i	−0.0692068	+	0.0692068i
$639$		0			0
$640$	−1.56579e11	+	5.52042e10i	−0.933283	+	0.329043i
$641$	1.17439e11			0.695631			0.347815	−	0.937563i	$-0.386924\pi$
$641$	1.17439e11			0.695631			0.347815	+	0.937563i	$0.386924\pi$
$642$		0			0
$643$	2.23655e10	−	2.23655e10i	0.130838	−	0.130838i	−0.638655	−	0.769493i	$-0.720509\pi$
$643$	2.23655e10	−	2.23655e10i	0.130838	−	0.130838i	0.769493	+	0.638655i	$0.220509\pi$
$644$			1.51703e11i			0.881965i
$645$		0			0
$646$	−5.47014e10			−0.314100
$647$	1.67021e11	+	1.67021e11i	0.953136	+	0.953136i	0.998950	−	0.0458137i	$-0.0145880\pi$
$647$	1.67021e11	+	1.67021e11i	0.953136	+	0.953136i	−0.0458137	+	0.998950i	$0.514588\pi$
$648$		0			0
$649$		−	5.67161e10i		−	0.319689i
$650$	8.56329e9	+	1.06347e10i	0.0479719	+	0.0595763i
$651$		0			0
$652$	4.24719e10	+	4.24719e10i	0.235023	+	0.235023i
$653$	1.48677e9	−	1.48677e9i	0.00817696	−	0.00817696i	−0.703006	−	0.711183i	$-0.748160\pi$
$653$	1.48677e9	−	1.48677e9i	0.00817696	−	0.00817696i	0.711183	+	0.703006i	$0.248160\pi$
$654$		0			0
$655$	8.83234e10	+	4.22780e10i	0.479855	+	0.229694i
$656$	−8.47423e10			−0.457599
$657$		0			0
$658$	1.22681e10	−	1.22681e10i	0.0654444	−	0.0654444i
$659$			3.54544e10i			0.187987i	0.995573	+	0.0939937i	$0.0299633\pi$
$659$			3.54544e10i			0.187987i	−0.995573	+	0.0939937i	$0.970037\pi$
$660$		0			0
$661$	−5.32084e10			−0.278724			−0.139362	−	0.990241i	$-0.544505\pi$
$661$	−5.32084e10			−0.278724			−0.139362	+	0.990241i	$0.544505\pi$
$662$	5.59467e10	+	5.59467e10i	0.291302	+	0.291302i
$663$		0			0
$664$		−	9.28885e10i		−	0.477848i
$665$	−1.20412e11	+	4.24530e10i	−0.615720	+	0.217081i
$666$		0			0
$667$	−3.33396e11	−	3.33396e11i	−1.68445	−	1.68445i
$668$	−6.63870e10	+	6.63870e10i	−0.333409	+	0.333409i
$669$		0			0
$670$	−4.54805e10	+	9.50137e10i	−0.225697	+	0.471506i
$671$	2.56914e10			0.126735
$672$		0			0
$673$	−4.97227e10	+	4.97227e10i	−0.242379	+	0.242379i	−0.817834	−	0.575455i	$-0.804825\pi$
$673$	−4.97227e10	+	4.97227e10i	−0.242379	+	0.242379i	0.575455	+	0.817834i	$0.304825\pi$
$674$		−	9.35746e10i		−	0.453438i
$675$		0			0
$676$	1.75673e11			0.841238
$677$	5.21560e10	+	5.21560e10i	0.248284	+	0.248284i	0.820266	−	0.571982i	$-0.193825\pi$
$677$	5.21560e10	+	5.21560e10i	0.248284	+	0.248284i	−0.571982	+	0.820266i	$0.693825\pi$
$678$		0			0
$679$		−	1.38219e11i		−	0.650262i
$680$	−1.30324e11	−	6.23826e10i	−0.609521	−	0.291761i
$681$		0			0
$682$	−4.19963e9	−	4.19963e9i	−0.0194121	−	0.0194121i
$683$	−8.03643e10	+	8.03643e10i	−0.369301	+	0.369301i	−0.867222	−	0.497921i	$-0.834097\pi$
$683$	−8.03643e10	+	8.03643e10i	−0.369301	+	0.369301i	0.497921	+	0.867222i	$0.334097\pi$
$684$		0			0
$685$	1.26113e11	+	3.57703e11i	0.572794	+	1.62465i
$686$	−8.00138e10			−0.361300
$687$		0			0
$688$	1.35325e11	−	1.35325e11i	0.603981	−	0.603981i
$689$			4.80804e10i			0.213349i
$690$		0			0
$691$	1.30282e11			0.571443			0.285722	−	0.958313i	$-0.407767\pi$
$691$	1.30282e11			0.571443			0.285722	+	0.958313i	$0.407767\pi$
$692$	−2.15379e11	−	2.15379e11i	−0.939246	−	0.939246i
$693$		0			0
$694$			1.16418e11i			0.501862i
$695$	1.03570e11	−	2.16370e11i	0.443911	−	0.927379i
$696$		0			0
$697$	−1.20736e11	−	1.20736e11i	−0.511572	−	0.511572i
$698$	−1.04712e10	+	1.04712e10i	−0.0441139	+	0.0441139i
$699$		0			0
$700$	−1.57635e11	−	1.70086e10i	−0.656539	−	0.0708398i
$701$	2.33961e11			0.968882			0.484441	−	0.874824i	$-0.339023\pi$
$701$	2.33961e11			0.968882			0.484441	+	0.874824i	$0.339023\pi$
$702$		0			0
$703$	−8.98221e10	+	8.98221e10i	−0.367758	+	0.367758i
$704$		−	1.54940e10i		−	0.0630771i
$705$		0			0
$706$	7.73179e10			0.311216
$707$	−1.64165e11	−	1.64165e11i	−0.657055	−	0.657055i
$708$		0			0
$709$			2.64363e10i			0.104620i	0.998631	+	0.0523102i	$0.0166584\pi$
$709$			2.64363e10i			0.104620i	−0.998631	+	0.0523102i	$0.983342\pi$
$710$	3.05803e10	+	8.67367e10i	0.120340	+	0.341326i
$711$		0			0
$712$	−1.26506e11	−	1.26506e11i	−0.492255	−	0.492255i
$713$	1.22107e11	−	1.22107e11i	0.472478	−	0.472478i
$714$		0			0
$715$	−9.17992e9	+	3.23651e9i	−0.0351249	+	0.0123838i
$716$	−1.89207e11			−0.719920
$717$		0			0
$718$	−8.29705e10	+	8.29705e10i	−0.312195	+	0.312195i
$719$		−	4.16438e9i		−	0.0155824i	−0.999970	−	0.00779120i	$-0.997520\pi$
$719$		−	4.16438e9i		−	0.0155824i	0.999970	−	0.00779120i	$-0.00248004\pi$
$720$		0			0
$721$	3.23065e11			1.19550
$722$	1.48642e10	+	1.48642e10i	0.0547006	+	0.0547006i
$723$		0			0
$724$			2.34290e11i			0.852707i
$725$	3.83811e11	−	3.09052e11i	1.38920	−	1.11861i
$726$		0			0
$727$	−1.08982e11	−	1.08982e11i	−0.390137	−	0.390137i	0.484600	−	0.874736i	$-0.338965\pi$
$727$	−1.08982e11	−	1.08982e11i	−0.390137	−	0.390137i	−0.874736	+	0.484600i	$0.838965\pi$
$728$	−2.13381e10	+	2.13381e10i	−0.0759679	+	0.0759679i
$729$		0			0
$730$	2.03584e10	+	9.74503e9i	0.0716891	+	0.0343156i
$731$	3.85607e11			1.35044
$732$		0			0
$733$	7.49618e10	−	7.49618e10i	0.259672	−	0.259672i	−0.565249	−	0.824920i	$-0.691220\pi$
$733$	7.49618e10	−	7.49618e10i	0.259672	−	0.259672i	0.824920	+	0.565249i	$0.191220\pi$
$734$		−	1.19477e11i		−	0.411625i
$735$		0			0
$736$	−3.37062e11			−1.14868
$737$	−5.31001e10	−	5.31001e10i	−0.179980	−	0.179980i
$738$		0			0
$739$		−	2.78382e11i		−	0.933391i	−0.884418	−	0.466696i	$-0.845444\pi$
$739$		−	2.78382e11i		−	0.933391i	0.884418	−	0.466696i	$-0.154556\pi$
$740$	−1.48769e11	+	5.24506e10i	−0.496118	+	0.174913i
$741$		0			0
$742$	5.01427e10	+	5.01427e10i	0.165421	+	0.165421i
$743$	3.99433e11	−	3.99433e11i	1.31066	−	1.31066i	0.389723	−	0.920932i	$-0.372571\pi$
$743$	3.99433e11	−	3.99433e11i	1.31066	−	1.31066i	0.920932	−	0.389723i	$-0.127429\pi$
$744$		0			0
$745$	4.42792e10	−	9.25041e10i	0.143739	−	0.300287i
$746$	−1.02457e11			−0.330816
$747$		0			0
$748$	3.42424e10	−	3.42424e10i	0.109385	−	0.109385i
$749$		−	8.08870e10i		−	0.257011i
$750$		0			0
$751$	3.16261e11			0.994228			0.497114	−	0.867685i	$-0.334393\pi$
$751$	3.16261e11			0.994228			0.497114	+	0.867685i	$0.334393\pi$
$752$	−5.65113e10	−	5.65113e10i	−0.176711	−	0.176711i
$753$		0			0
$754$		−	4.40942e10i		−	0.136426i
$755$	−3.79157e11	−	1.81492e11i	−1.16689	−	0.558561i
$756$		0			0
$757$	2.03260e10	+	2.03260e10i	0.0618968	+	0.0618968i	0.737378	−	0.675481i	$-0.236064\pi$
$757$	2.03260e10	+	2.03260e10i	0.0618968	+	0.0618968i	−0.675481	+	0.737378i	$0.736064\pi$
$758$	7.58996e10	−	7.58996e10i	0.229913	−	0.229913i
$759$		0			0
$760$	−6.16556e10	−	1.74878e11i	−0.184807	−	0.524179i
$761$	−2.39323e11			−0.713586			−0.356793	−	0.934183i	$-0.616130\pi$
$761$	−2.39323e11			−0.713586			−0.356793	+	0.934183i	$0.616130\pi$
$762$		0			0
$763$	−2.13270e11	+	2.13270e11i	−0.629262	+	0.629262i
$764$		−	2.62348e11i		−	0.770025i
$765$		0			0
$766$	−1.36752e10			−0.0397208
$767$	1.09050e11	+	1.09050e11i	0.315097	+	0.315097i
$768$		0			0
$769$			3.89431e11i			1.11359i	0.830650	+	0.556795i	$0.187969\pi$
$769$			3.89431e11i			1.11359i	−0.830650	+	0.556795i	$0.812031\pi$
$770$	−6.19833e9	+	1.29490e10i	−0.0176324	+	0.0368361i
$771$		0			0
$772$	1.77328e11	+	1.77328e11i	0.499237	+	0.499237i
$773$	−2.96297e11	+	2.96297e11i	−0.829868	+	0.829868i	−0.987498	−	0.157630i	$-0.949615\pi$
$773$	−2.96297e11	+	2.96297e11i	−0.829868	+	0.829868i	0.157630	+	0.987498i	$0.449615\pi$
$774$		0			0
$775$	1.13191e11	+	1.40572e11i	0.313765	+	0.389664i
$776$	2.00739e11			0.553586
$777$		0			0
$778$	1.51045e11	−	1.51045e11i	0.412277	−	0.412277i
$779$		−	2.19132e11i		−	0.595053i
$780$		0			0
$781$	−6.55647e10			−0.176224
$782$	−1.26455e11	−	1.26455e11i	−0.338149	−	0.338149i
$783$		0			0
$784$			1.13705e11i			0.300966i
$785$	2.08319e11	+	5.90867e11i	0.548592	+	1.55600i
$786$		0			0
$787$	1.21354e11	+	1.21354e11i	0.316342	+	0.316342i	0.847360	−	0.531019i	$-0.178191\pi$
$787$	1.21354e11	+	1.21354e11i	0.316342	+	0.316342i	−0.531019	+	0.847360i	$0.678191\pi$
$788$	−1.99363e11	+	1.99363e11i	−0.517058	+	0.517058i
$789$		0			0
$790$	2.80963e10	−	9.90575e9i	0.0721341	−	0.0254319i
$791$	−1.09804e11			−0.280486
$792$		0			0
$793$	−4.93979e10	+	4.93979e10i	−0.124915	+	0.124915i
$794$			1.62683e11i			0.409319i
$795$		0			0
$796$	−2.91050e11			−0.724963
$797$	3.88575e10	+	3.88575e10i	0.0963033	+	0.0963033i	0.753617	−	0.657314i	$-0.228307\pi$
$797$	3.88575e10	+	3.88575e10i	0.0963033	+	0.0963033i	−0.657314	+	0.753617i	$0.728307\pi$
$798$		0			0
$799$		−	1.61029e11i		−	0.395108i
$800$	3.77907e10	−	3.50242e11i	0.0922624	−	0.855082i
$801$		0			0
$802$	6.68481e10	+	6.68481e10i	0.161581	+	0.161581i
$803$	−1.13777e10	+	1.13777e10i	−0.0273647	+	0.0273647i
$804$		0			0
$805$	−3.76500e11	−	1.80220e11i	−0.896564	−	0.429161i
$806$	1.61496e10			0.0382667
$807$		0			0
$808$	2.38421e11	−	2.38421e11i	0.559369	−	0.559369i
$809$			5.38642e11i			1.25749i	0.777610	+	0.628747i	$0.216432\pi$
$809$			5.38642e11i			1.25749i	−0.777610	+	0.628747i	$0.783568\pi$
$810$		0			0
$811$	−5.26130e11			−1.21621			−0.608107	−	0.793855i	$-0.708071\pi$
$811$	−5.26130e11			−1.21621			−0.608107	+	0.793855i	$0.708071\pi$
$812$	3.62057e11	+	3.62057e11i	0.832824	+	0.832824i
$813$		0			0
$814$			1.42831e10i			0.0325330i
$815$	−1.55863e11	+	5.49518e10i	−0.353275	+	0.124552i
$816$		0			0
$817$	3.49931e11	+	3.49931e11i	0.785406	+	0.785406i
$818$	−3.84370e10	+	3.84370e10i	−0.0858493	+	0.0858493i
$819$		0			0
$820$	1.17490e11	−	2.45449e11i	0.259863	−	0.542883i
$821$	8.36261e10			0.184064			0.0920320	−	0.995756i	$-0.470664\pi$
$821$	8.36261e10			0.184064			0.0920320	+	0.995756i	$0.470664\pi$
$822$		0			0
$823$	2.08310e11	−	2.08310e11i	0.454058	−	0.454058i	−0.442641	−	0.896699i	$-0.645958\pi$
$823$	2.08310e11	−	2.08310e11i	0.454058	−	0.454058i	0.896699	+	0.442641i	$0.145958\pi$
$824$			4.69196e11i			1.01776i
$825$		0			0
$826$	2.27455e11			0.488624
$827$	6.03123e10	+	6.03123e10i	0.128939	+	0.128939i	0.768631	−	0.639692i	$-0.220938\pi$
$827$	6.03123e10	+	6.03123e10i	0.128939	+	0.128939i	−0.639692	+	0.768631i	$0.720938\pi$
$828$		0			0
$829$		−	7.70491e11i		−	1.63136i	−0.578504	−	0.815680i	$-0.696363\pi$
$829$		−	7.70491e11i		−	1.63136i	0.578504	−	0.815680i	$-0.303637\pi$
$830$	1.08383e11	+	5.18801e10i	0.228375	+	0.109317i
$831$		0			0
$832$	2.97908e10	+	2.97908e10i	0.0621711	+	0.0621711i
$833$	−1.62002e11	+	1.62002e11i	−0.336465	+	0.336465i
$834$		0			0
$835$	−8.58942e10	−	2.43627e11i	−0.176692	−	0.501163i
$836$	6.21486e10			0.127235
$837$		0			0
$838$	−7.31540e10	+	7.31540e10i	−0.148341	+	0.148341i
$839$		−	2.81465e11i		−	0.568037i	−0.958819	−	0.284018i	$-0.908332\pi$
$839$		−	2.81465e11i		−	0.568037i	0.958819	−	0.284018i	$-0.0916677\pi$
$840$		0			0
$841$	−1.09113e12			−2.18118
$842$	2.02290e10	+	2.02290e10i	0.0402462	+	0.0402462i
$843$		0			0
$844$			5.51223e10i			0.108632i
$845$	−2.08696e11	+	4.35989e11i	−0.409343	+	0.855163i
$846$		0			0
$847$	2.63608e11	+	2.63608e11i	0.512183	+	0.512183i
$848$	2.30976e11	−	2.30976e11i	0.446667	−	0.446667i
$849$		0			0
$850$	1.45577e11	−	1.17222e11i	0.278880	−	0.224560i
$851$	−4.15289e11			−0.791830
$852$		0			0
$853$	−3.95360e11	+	3.95360e11i	−0.746787	+	0.746787i	−0.973874	−	0.227087i	$-0.927080\pi$
$853$	−3.95360e11	+	3.95360e11i	−0.746787	+	0.746787i	0.227087	+	0.973874i	$0.427080\pi$
$854$			1.03033e11i			0.193707i
$855$		0			0
$856$	1.17474e11			0.218800
$857$	5.29610e11	+	5.29610e11i	0.981822	+	0.981822i	0.999838	−	0.0180158i	$-0.00573493\pi$
$857$	5.29610e11	+	5.29610e11i	0.981822	+	0.981822i	−0.0180158	+	0.999838i	$0.505735\pi$
$858$		0			0
$859$			5.68396e11i			1.04395i	0.852962	+	0.521973i	$0.174804\pi$
$859$			5.68396e11i			1.04395i	−0.852962	+	0.521973i	$0.825196\pi$
$860$	2.04338e11	+	5.79576e11i	0.373556	+	1.05954i
$861$		0			0
$862$	1.01999e11	+	1.01999e11i	0.184743	+	0.184743i
$863$	−2.15388e11	+	2.15388e11i	−0.388310	+	0.388310i	−0.874084	−	0.485774i	$-0.838538\pi$
$863$	−2.15388e11	+	2.15388e11i	−0.388310	+	0.388310i	0.485774	+	0.874084i	$0.338538\pi$
$864$		0			0
$865$	7.90397e11	−	2.78666e11i	1.41183	−	0.497760i
$866$	−8.23682e10			−0.146449
$867$		0			0
$868$	−1.32604e11	+	1.32604e11i	−0.233603	+	0.233603i
$869$			2.12381e10i			0.0372423i
$870$		0			0
$871$	2.04195e11			0.354791
$872$	−3.09738e11	−	3.09738e11i	−0.535708	−	0.535708i
$873$		0			0
$874$		−	2.29511e11i		−	0.393331i
$875$	2.29479e11	−	3.71015e11i	0.391481	−	0.632936i
$876$		0			0
$877$	4.94884e11	+	4.94884e11i	0.836575	+	0.836575i	0.988406	−	0.151832i	$-0.0485171\pi$
$877$	4.94884e11	+	4.94884e11i	0.836575	+	0.836575i	−0.151832	+	0.988406i	$0.548517\pi$
$878$	1.05523e11	−	1.05523e11i	0.177569	−	0.177569i
$879$		0			0
$880$	5.96480e10	+	2.85518e10i	0.0994638	+	0.0476106i
$881$	−1.03433e12			−1.71694			−0.858469	−	0.512865i	$-0.828584\pi$
$881$	−1.03433e12			−1.71694			−0.858469	+	0.512865i	$0.828584\pi$
$882$		0			0
$883$	−7.47773e11	+	7.47773e11i	−1.23006	+	1.23006i	−0.266123	+	0.963939i	$0.585743\pi$
$883$	−7.47773e11	+	7.47773e11i	−1.23006	+	1.23006i	−0.963939	+	0.266123i	$0.914257\pi$
$884$			1.31678e11i			0.215628i
$885$		0			0
$886$	−3.76209e11			−0.610512
$887$	−6.17384e11	−	6.17384e11i	−0.997380	−	0.997380i	0.00261651	−	0.999997i	$-0.499167\pi$
$887$	−6.17384e11	−	6.17384e11i	−0.997380	−	0.997380i	−0.999997	+	0.00261651i	$0.999167\pi$
$888$		0			0
$889$		−	7.16030e11i		−	1.14637i
$890$	2.18264e11	−	7.69521e10i	0.347874	−	0.122648i
$891$		0			0
$892$	1.83564e10	+	1.83564e10i	0.0289953	+	0.0289953i
$893$	1.46131e11	−	1.46131e11i	0.229792	−	0.229792i
$894$		0			0
$895$	2.24773e11	−	4.69576e11i	0.350310	−	0.731836i
$896$	4.74667e11			0.736474
$897$		0			0
$898$	2.05964e10	−	2.05964e10i	0.0316728	−	0.0316728i
$899$		−	5.82844e11i		−	0.892305i
$900$		0			0
$901$	6.58165e11			0.998702
$902$	−1.74226e10	−	1.74226e10i	−0.0263201	−	0.0263201i
$903$		0			0
$904$		−	1.59471e11i		−	0.238785i
$905$	−5.81466e11	−	2.78332e11i	−0.866822	−	0.414924i
$906$		0			0
$907$	1.19603e11	+	1.19603e11i	0.176731	+	0.176731i	0.789929	−	0.613198i	$-0.210117\pi$
$907$	1.19603e11	+	1.19603e11i	0.176731	+	0.176731i	−0.613198	+	0.789929i	$0.710117\pi$
$908$	4.63541e11	−	4.63541e11i	0.681938	−	0.681938i
$909$		0			0
$910$	−1.29798e10	−	3.68153e10i	−0.0189278	−	0.0536862i
$911$	8.32730e11			1.20901			0.604506	−	0.796601i	$-0.293371\pi$
$911$	8.32730e11			1.20901			0.604506	+	0.796601i	$0.293371\pi$
$912$		0			0
$913$	−6.05718e10	+	6.05718e10i	−0.0871741	+	0.0871741i
$914$			1.17823e11i			0.168828i
$915$		0			0
$916$	−2.20134e11			−0.312683
$917$	−1.97958e11	−	1.97958e11i	−0.279960	−	0.279960i
$918$		0			0
$919$		−	1.64422e11i		−	0.230514i	−0.993336	−	0.115257i	$-0.963231\pi$
$919$		−	1.64422e11i		−	0.230514i	0.993336	−	0.115257i	$-0.0367691\pi$
$920$	2.61739e11	−	5.46801e11i	0.365356	−	0.763270i
$921$		0			0
$922$	−1.24154e11	−	1.24154e11i	−0.171806	−	0.171806i
$923$	1.26064e11	−	1.26064e11i	0.173693	−	0.173693i
$924$		0			0
$925$	4.65613e10	−	4.31527e11i	0.0636001	−	0.589442i
$926$	−2.21958e11			−0.301875
$927$		0			0
$928$	−8.04438e11	+	8.04438e11i	−1.08468	+	1.08468i
$929$		−	4.23919e11i		−	0.569142i	−0.958655	−	0.284571i	$-0.908149\pi$
$929$		−	4.23919e11i		−	0.569142i	0.958655	−	0.284571i	$-0.0918511\pi$
$930$		0			0
$931$	−2.94027e11			−0.391371
$932$	7.92525e11	+	7.92525e11i	1.05039	+	1.05039i
$933$		0			0
$934$			3.72393e11i			0.489344i
$935$	4.43041e10	+	1.25662e11i	0.0579692	+	0.164422i
$936$		0			0
$937$	−1.39851e11	−	1.39851e11i	−0.181430	−	0.181430i	0.610549	−	0.791979i	$-0.290949\pi$
$937$	−1.39851e11	−	1.39851e11i	−0.181430	−	0.181430i	−0.791979	+	0.610549i	$0.790949\pi$
$938$	2.12953e11	−	2.12953e11i	0.275089	−	0.275089i
$939$		0			0
$940$	2.42030e11	−	8.53312e10i	0.309997	−	0.109294i
$941$	3.60703e11			0.460035			0.230018	−	0.973186i	$-0.426122\pi$
$941$	3.60703e11			0.460035			0.230018	+	0.973186i	$0.426122\pi$
$942$		0			0
$943$	5.06574e11	−	5.06574e11i	0.640614	−	0.640614i
$944$		−	1.04774e12i		−	1.31937i
$945$		0			0
$946$	5.56443e10			0.0694794
$947$	−6.11835e11	−	6.11835e11i	−0.760737	−	0.760737i	0.215718	−	0.976456i	$-0.430791\pi$
$947$	−6.11835e11	−	6.11835e11i	−0.760737	−	0.760737i	−0.976456	+	0.215718i	$0.930791\pi$
$948$		0			0
$949$		−	4.37525e10i		−	0.0539434i
$950$	2.38485e11	+	2.57323e10i	0.292797	+	0.0315925i
$951$		0			0
$952$	2.92094e11	+	2.92094e11i	0.355611	+	0.355611i
$953$	−3.04156e11	+	3.04156e11i	−0.368744	+	0.368744i	−0.867019	−	0.498275i	$-0.833967\pi$
$953$	−3.04156e11	+	3.04156e11i	−0.368744	+	0.368744i	0.498275	+	0.867019i	$0.333967\pi$
$954$		0			0
$955$	6.51101e11	+	3.11664e11i	0.782771	+	0.374691i
$956$	7.35460e11			0.880497
$957$		0			0
$958$	−2.79360e11	+	2.79360e11i	−0.331667	+	0.331667i
$959$		−	1.08437e12i		−	1.28205i
$960$		0			0
$961$	−6.39424e11			−0.749713
$962$	−2.74626e10	−	2.74626e10i	−0.0320657	−	0.0320657i
$963$		0			0
$964$		−	1.31930e12i		−	1.52769i
$965$	−6.50756e11	+	2.29433e11i	−0.750428	+	0.264574i
$966$		0			0
$967$	6.65811e11	+	6.65811e11i	0.761457	+	0.761457i	0.976586	−	0.215129i	$-0.0690172\pi$
$967$	6.65811e11	+	6.65811e11i	0.761457	+	0.761457i	−0.215129	+	0.976586i	$0.569017\pi$
$968$	−3.82845e11	+	3.82845e11i	−0.436036	+	0.436036i
$969$		0			0
$970$	−1.12117e11	+	2.34224e11i	−0.126644	+	0.264573i
$971$	−1.11212e12			−1.25105			−0.625525	−	0.780205i	$-0.715115\pi$
$971$	−1.11212e12			−1.25105			−0.625525	+	0.780205i	$0.715115\pi$
$972$		0			0
$973$	−4.84947e11	+	4.84947e11i	−0.541057	+	0.541057i
$974$			1.88239e11i			0.209157i
$975$		0			0
$976$	4.74610e11			0.523043
$977$	−7.14891e10	−	7.14891e10i	−0.0784624	−	0.0784624i	0.666786	−	0.745249i	$-0.267669\pi$
$977$	−7.14891e10	−	7.14891e10i	−0.0784624	−	0.0784624i	−0.745249	+	0.666786i	$0.767669\pi$
$978$		0			0
$979$			1.64987e11i			0.179605i
$980$	−3.29339e11	−	1.57646e11i	−0.357058	−	0.170914i
$981$		0			0
$982$	−8.98126e10	−	8.98126e10i	−0.0965810	−	0.0965810i
$983$	5.26321e11	−	5.26321e11i	0.563685	−	0.563685i	−0.366667	−	0.930352i	$-0.619501\pi$
$983$	5.26321e11	−	5.26321e11i	0.563685	−	0.563685i	0.930352	+	0.366667i	$0.119501\pi$
$984$		0			0
$985$	−2.57943e11	−	7.31621e11i	−0.274018	−	0.777215i
$986$	−6.03598e11			−0.638617
$987$		0			0
$988$	−1.19495e11	+	1.19495e11i	−0.125407	+	0.125407i
$989$			1.61789e12i			1.69108i
$990$		0			0
$991$	−1.21186e12			−1.25649			−0.628245	−	0.778016i	$-0.716226\pi$
$991$	−1.21186e12			−1.25649			−0.628245	+	0.778016i	$0.716226\pi$
$992$	−2.94627e11	−	2.94627e11i	−0.304246	−	0.304246i
$993$		0			0
$994$		−	2.62941e11i		−	0.269348i
$995$	3.45761e11	−	7.22334e11i	0.352764	−	0.736963i
$996$		0			0
$997$	−5.69885e11	−	5.69885e11i	−0.576775	−	0.576775i	0.357238	−	0.934013i	$-0.383718\pi$
$997$	−5.69885e11	−	5.69885e11i	−0.576775	−	0.576775i	−0.934013	+	0.357238i	$0.883718\pi$
$998$	3.07249e11	−	3.07249e11i	0.309720	−	0.309720i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.9.g.b.37.5	yes	16
3.2	odd	2	inner	45.9.g.b.37.4	yes	16
5.3	odd	4	inner	45.9.g.b.28.5	yes	16
15.8	even	4	inner	45.9.g.b.28.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.g.b.28.4	✓	16	15.8	even	4	inner
45.9.g.b.28.5	yes	16	5.3	odd	4	inner
45.9.g.b.37.4	yes	16	3.2	odd	2	inner
45.9.g.b.37.5	yes	16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.79807 + 3.79807i) q^{2} -227.149i q^{4} +(563.744 + 269.849i) q^{5} +(-1263.51 - 1263.51i) q^{7} +(1835.03 - 1835.03i) q^{8} +O(q^{10})\)	\(q+(3.79807 + 3.79807i) q^{2} -227.149i q^{4} +(563.744 + 269.849i) q^{5} +(-1263.51 - 1263.51i) q^{7} +(1835.03 - 1835.03i) q^{8} +(1116.23 + 3166.04i) q^{10} -2393.22 q^{11} +(4601.53 - 4601.53i) q^{13} -9597.80i q^{14} -44211.1 q^{16} +(-62989.7 - 62989.7i) q^{17} -114324. i q^{19} +(61295.9 - 128054. i) q^{20} +(-9089.61 - 9089.61i) q^{22} +(264286. - 264286. i) q^{23} +(244988. + 304251. i) q^{25} +34953.9 q^{26} +(-287006. + 287006. i) q^{28} -1.26150e6i q^{29} +462025. q^{31} +(-637685. - 637685. i) q^{32} -478478. i q^{34} +(-371340. - 1.05325e6i) q^{35} +(-785681. - 785681. i) q^{37} +(434210. - 434210. i) q^{38} +(1.52967e6 - 539307. i) q^{40} +1.91676e6 q^{41} +(-3.06087e6 + 3.06087e6i) q^{43} +543619. i q^{44} +2.00755e6 q^{46} +(1.27822e6 + 1.27822e6i) q^{47} -2.57188e6i q^{49} +(-225082. + 2.08605e6i) q^{50} +(-1.04524e6 - 1.04524e6i) q^{52} +(-5.22439e6 + 5.22439e6i) q^{53} +(-1.34916e6 - 645807. i) q^{55} -4.63717e6 q^{56} +(4.79125e6 - 4.79125e6i) q^{58} +2.36986e7i q^{59} -1.07351e7 q^{61} +(1.75480e6 + 1.75480e6i) q^{62} +6.47410e6i q^{64} +(3.83580e6 - 1.35237e6i) q^{65} +(2.21877e7 + 2.21877e7i) q^{67} +(-1.43081e7 + 1.43081e7i) q^{68} +(2.58995e6 - 5.41070e6i) q^{70} +2.73960e7 q^{71} +(4.75412e6 - 4.75412e6i) q^{73} -5.96814e6i q^{74} -2.59686e7 q^{76} +(3.02386e6 + 3.02386e6i) q^{77} -8.87427e6i q^{79} +(-2.49237e7 - 1.19303e7i) q^{80} +(7.28000e6 + 7.28000e6i) q^{82} +(2.53098e7 - 2.53098e7i) q^{83} +(-1.85123e7 - 5.25077e7i) q^{85} -2.32508e7 q^{86} +(-4.39164e6 + 4.39164e6i) q^{88} -6.89392e7i q^{89} -1.16282e7 q^{91} +(-6.00324e7 - 6.00324e7i) q^{92} +9.70949e6i q^{94} +(3.08501e7 - 6.44493e7i) q^{95} +(5.46963e7 + 5.46963e7i) q^{97} +(9.76815e6 - 9.76815e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4220 q^{7}+O(q^{10}) \) 16 * q - 4220 * q^7	\( 16 q - 4220 q^{7} - 47000 q^{10} - 37940 q^{13} - 508916 q^{16} + 844700 q^{22} - 1664300 q^{25} + 6009380 q^{28} - 944752 q^{31} + 10616140 q^{37} - 17493600 q^{40} + 4050760 q^{43} - 34233160 q^{46} + 7180240 q^{52} + 17430500 q^{55} - 27842100 q^{58} + 32032232 q^{61} + 75463480 q^{67} + 272773500 q^{70} - 198258320 q^{73} - 196046088 q^{76} + 172139600 q^{82} + 183614500 q^{85} - 624395100 q^{88} - 477271600 q^{91} + 662476480 q^{97}+O(q^{100}) \) 16 * q - 4220 * q^7 - 47000 * q^10 - 37940 * q^13 - 508916 * q^16 + 844700 * q^22 - 1664300 * q^25 + 6009380 * q^28 - 944752 * q^31 + 10616140 * q^37 - 17493600 * q^40 + 4050760 * q^43 - 34233160 * q^46 + 7180240 * q^52 + 17430500 * q^55 - 27842100 * q^58 + 32032232 * q^61 + 75463480 * q^67 + 272773500 * q^70 - 198258320 * q^73 - 196046088 * q^76 + 172139600 * q^82 + 183614500 * q^85 - 624395100 * q^88 - 477271600 * q^91 + 662476480 * q^97

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