LMFDB - Embedded newform 1365.2.f.c.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1365,2,Mod(274,1365)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1365, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1365.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1365 = 3 \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1365.f (of order $2$, degree $1$, 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:	$10.8995798759$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1365.274
Dual form			1365.2.f.c.274.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} +(1.90211 + 1.17557i) q^{5} -1.61803 q^{6} +1.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +(1.90211 - 3.07768i) q^{10} +2.80423 q^{11} +0.618034i q^{12} -1.00000i q^{13} +1.61803 q^{14} +(1.17557 - 1.90211i) q^{15} -4.85410 q^{16} -2.01062i q^{17} +1.61803i q^{18} +2.34458 q^{19} +(-1.17557 - 0.726543i) q^{20} +1.00000 q^{21} -4.53733i q^{22} +0.939503i q^{23} -2.23607 q^{24} +(2.23607 + 4.47214i) q^{25} -1.61803 q^{26} +1.00000i q^{27} -0.618034i q^{28} +5.69572 q^{29} +(-3.07768 - 1.90211i) q^{30} +0.895549 q^{31} +3.38197i q^{32} -2.80423i q^{33} -3.25325 q^{34} +(-1.17557 + 1.90211i) q^{35} +0.618034 q^{36} -5.52015i q^{37} -3.79360i q^{38} -1.00000 q^{39} +(2.62866 - 4.25325i) q^{40} +3.07768 q^{41} -1.61803i q^{42} -5.12454i q^{43} -1.73311 q^{44} +(-1.90211 - 1.17557i) q^{45} +1.52015 q^{46} -5.96261i q^{47} +4.85410i q^{48} -1.00000 q^{49} +(7.23607 - 3.61803i) q^{50} -2.01062 q^{51} +0.618034i q^{52} +1.30313i q^{53} +1.61803 q^{54} +(5.33395 + 3.29657i) q^{55} +2.23607 q^{56} -2.34458i q^{57} -9.21586i q^{58} +2.47027 q^{59} +(-0.726543 + 1.17557i) q^{60} -4.15838 q^{61} -1.44903i q^{62} -1.00000i q^{63} -4.23607 q^{64} +(1.17557 - 1.90211i) q^{65} -4.53733 q^{66} -15.4797i q^{67} +1.24263i q^{68} +0.939503 q^{69} +(3.07768 + 1.90211i) q^{70} -12.3416 q^{71} +2.23607i q^{72} +2.77455i q^{73} -8.93179 q^{74} +(4.47214 - 2.23607i) q^{75} -1.44903 q^{76} +2.80423i q^{77} +1.61803i q^{78} +6.16495 q^{79} +(-9.23305 - 5.70634i) q^{80} +1.00000 q^{81} -4.97980i q^{82} +15.5861i q^{83} -0.618034 q^{84} +(2.36363 - 3.82443i) q^{85} -8.29168 q^{86} -5.69572i q^{87} -6.27044i q^{88} +0.821412 q^{89} +(-1.90211 + 3.07768i) q^{90} +1.00000 q^{91} -0.580644i q^{92} -0.895549i q^{93} -9.64771 q^{94} +(4.45965 + 2.75621i) q^{95} +3.38197 q^{96} -1.98281i q^{97} +1.61803i q^{98} -2.80423 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 4 q^{6} - 8 q^{9} - 8 q^{11} + 4 q^{14} - 12 q^{16} + 4 q^{19} + 8 q^{21} - 4 q^{26} + 12 q^{29} - 4 q^{31} + 8 q^{34} - 4 q^{36} - 8 q^{39} - 4 q^{44} - 12 q^{46} - 8 q^{49} + 40 q^{50}+ \cdots + 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^4 - 4 * q^6 - 8 * q^9 - 8 * q^11 + 4 * q^14 - 12 * q^16 + 4 * q^19 + 8 * q^21 - 4 * q^26 + 12 * q^29 - 4 * q^31 + 8 * q^34 - 4 * q^36 - 8 * q^39 - 4 * q^44 - 12 * q^46 - 8 * q^49 + 40 * q^50 - 4 * q^51 + 4 * q^54 + 40 * q^55 - 8 * q^59 - 40 * q^61 - 16 * q^64 + 4 * q^66 + 16 * q^69 - 28 * q^71 - 20 * q^74 - 8 * q^76 + 52 * q^79 + 8 * q^81 + 4 * q^84 - 20 * q^85 + 8 * q^86 - 40 * q^89 + 8 * q^91 - 32 * q^94 + 20 * q^95 + 36 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1365\mathbb{Z}\right)^\times$.

$n$	$106$	$547$	$911$	$976$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		−	1.61803i		−	1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$2$		−	1.61803i		−	1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−0.618034			−0.309017
$5$	1.90211	+	1.17557i	0.850651	+	0.525731i
$5$	1.90211	+	1.17557i	0.850651	+	0.525731i
$6$	−1.61803			−0.660560
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		−	2.23607i		−	0.790569i
$9$	−1.00000			−0.333333
$10$	1.90211	−	3.07768i	0.601501	−	0.973249i
$11$	2.80423			0.845506			0.422753	−	0.906245i	$-0.361064\pi$
$11$	2.80423			0.845506			0.422753	+	0.906245i	$0.361064\pi$
$12$			0.618034i			0.178411i
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$	1.61803			0.432438
$15$	1.17557	−	1.90211i	0.303531	−	0.491123i
$16$	−4.85410			−1.21353
$17$		−	2.01062i		−	0.487647i	−0.969820	−	0.243824i	$-0.921598\pi$
$17$		−	2.01062i		−	0.487647i	0.969820	−	0.243824i	$-0.0784018\pi$
$18$			1.61803i			0.381374i
$19$	2.34458			0.537883			0.268941	−	0.963157i	$-0.413326\pi$
$19$	2.34458			0.537883			0.268941	+	0.963157i	$0.413326\pi$
$20$	−1.17557	−	0.726543i	−0.262866	−	0.162460i
$21$	1.00000			0.218218
$22$		−	4.53733i		−	0.967363i
$23$			0.939503i			0.195900i	0.995191	+	0.0979499i	$0.0312285\pi$
$23$			0.939503i			0.195900i	−0.995191	+	0.0979499i	$0.968772\pi$
$24$	−2.23607			−0.456435
$25$	2.23607	+	4.47214i	0.447214	+	0.894427i
$26$	−1.61803			−0.317323
$27$			1.00000i			0.192450i
$28$		−	0.618034i		−	0.116797i
$29$	5.69572			1.05767			0.528834	−	0.848725i	$-0.322629\pi$
$29$	5.69572			1.05767			0.528834	+	0.848725i	$0.322629\pi$
$30$	−3.07768	−	1.90211i	−0.561906	−	0.347277i
$31$	0.895549			0.160845			0.0804226	−	0.996761i	$-0.474373\pi$
$31$	0.895549			0.160845			0.0804226	+	0.996761i	$0.474373\pi$
$32$			3.38197i			0.597853i
$33$		−	2.80423i		−	0.488153i
$34$	−3.25325			−0.557928
$35$	−1.17557	+	1.90211i	−0.198708	+	0.321516i
$36$	0.618034			0.103006
$37$		−	5.52015i		−	0.907507i	−0.891127	−	0.453753i	$-0.850085\pi$
$37$		−	5.52015i		−	0.907507i	0.891127	−	0.453753i	$-0.149915\pi$
$38$		−	3.79360i		−	0.615404i
$39$	−1.00000			−0.160128
$40$	2.62866	−	4.25325i	0.415627	−	0.672499i
$41$	3.07768			0.480653			0.240327	−	0.970692i	$-0.422745\pi$
$41$	3.07768			0.480653			0.240327	+	0.970692i	$0.422745\pi$
$42$		−	1.61803i		−	0.249668i
$43$		−	5.12454i		−	0.781485i	−0.920500	−	0.390743i	$-0.872218\pi$
$43$		−	5.12454i		−	0.781485i	0.920500	−	0.390743i	$-0.127782\pi$
$44$	−1.73311			−0.261276
$45$	−1.90211	−	1.17557i	−0.283550	−	0.175244i
$46$	1.52015			0.224133
$47$		−	5.96261i		−	0.869736i	−0.900494	−	0.434868i	$-0.856795\pi$
$47$		−	5.96261i		−	0.869736i	0.900494	−	0.434868i	$-0.143205\pi$
$48$			4.85410i			0.700629i
$49$	−1.00000			−0.142857
$50$	7.23607	−	3.61803i	1.02333	−	0.511667i
$51$	−2.01062			−0.281543
$52$			0.618034i			0.0857059i
$53$			1.30313i			0.178999i	0.995987	+	0.0894993i	$0.0285267\pi$
$53$			1.30313i			0.178999i	−0.995987	+	0.0894993i	$0.971473\pi$
$54$	1.61803			0.220187
$55$	5.33395	+	3.29657i	0.719230	+	0.444509i
$56$	2.23607			0.298807
$57$		−	2.34458i		−	0.310547i
$58$		−	9.21586i		−	1.21010i
$59$	2.47027			0.321602			0.160801	−	0.986987i	$-0.448592\pi$
$59$	2.47027			0.321602			0.160801	+	0.986987i	$0.448592\pi$
$60$	−0.726543	+	1.17557i	−0.0937962	+	0.151765i
$61$	−4.15838			−0.532427			−0.266213	−	0.963914i	$-0.585773\pi$
$61$	−4.15838			−0.532427			−0.266213	+	0.963914i	$0.585773\pi$
$62$		−	1.44903i		−	0.184027i
$63$		−	1.00000i		−	0.125988i
$64$	−4.23607			−0.529508
$65$	1.17557	−	1.90211i	0.145812	−	0.235928i
$66$	−4.53733			−0.558507
$67$		−	15.4797i		−	1.89115i	−0.325402	−	0.945576i	$-0.605500\pi$
$67$		−	15.4797i		−	1.89115i	0.325402	−	0.945576i	$-0.394500\pi$
$68$			1.24263i			0.150691i
$69$	0.939503			0.113103
$70$	3.07768	+	1.90211i	0.367854	+	0.227346i
$71$	−12.3416			−1.46467			−0.732337	−	0.680943i	$-0.761570\pi$
$71$	−12.3416			−1.46467			−0.732337	+	0.680943i	$0.761570\pi$
$72$			2.23607i			0.263523i
$73$			2.77455i			0.324737i	0.986730	+	0.162368i	$0.0519133\pi$
$73$			2.77455i			0.324737i	−0.986730	+	0.162368i	$0.948087\pi$
$74$	−8.93179			−1.03830
$75$	4.47214	−	2.23607i	0.516398	−	0.258199i
$76$	−1.44903			−0.166215
$77$			2.80423i			0.319571i
$78$			1.61803i			0.183206i
$79$	6.16495			0.693611			0.346805	−	0.937937i	$-0.387266\pi$
$79$	6.16495			0.693611			0.346805	+	0.937937i	$0.387266\pi$
$80$	−9.23305	−	5.70634i	−1.03229	−	0.637988i
$81$	1.00000			0.111111
$82$		−	4.97980i		−	0.549927i
$83$			15.5861i			1.71079i	0.517975	+	0.855396i	$0.326686\pi$
$83$			15.5861i			1.71079i	−0.517975	+	0.855396i	$0.673314\pi$
$84$	−0.618034			−0.0674330
$85$	2.36363	−	3.82443i	0.256371	−	0.414818i
$86$	−8.29168			−0.894115
$87$		−	5.69572i		−	0.610645i
$88$		−	6.27044i		−	0.668431i
$89$	0.821412			0.0870695			0.0435348	−	0.999052i	$-0.486138\pi$
$89$	0.821412			0.0870695			0.0435348	+	0.999052i	$0.486138\pi$
$90$	−1.90211	+	3.07768i	−0.200500	+	0.324416i
$91$	1.00000			0.104828
$92$		−	0.580644i		−	0.0605364i
$93$		−	0.895549i		−	0.0928641i
$94$	−9.64771			−0.995085
$95$	4.45965	+	2.75621i	0.457550	+	0.282782i
$96$	3.38197			0.345170
$97$		−	1.98281i		−	0.201324i	−0.994921	−	0.100662i	$-0.967904\pi$
$97$		−	1.98281i		−	0.201324i	0.994921	−	0.100662i	$-0.0320961\pi$
$98$			1.61803i			0.163446i
$99$	−2.80423			−0.281835
$100$	−1.38197	−	2.76393i	−0.138197	−	0.276393i
$101$	7.71883			0.768052			0.384026	−	0.923322i	$-0.374537\pi$
$101$	7.71883			0.768052			0.384026	+	0.923322i	$0.374537\pi$
$102$			3.25325i			0.322120i
$103$			19.3309i			1.90473i	0.304954	+	0.952367i	$0.401359\pi$
$103$			19.3309i			1.90473i	−0.304954	+	0.952367i	$0.598641\pi$
$104$	−2.23607			−0.219265
$105$	1.90211	+	1.17557i	0.185627	+	0.114724i
$106$	2.10851			0.204796
$107$			7.81736i			0.755732i	0.925860	+	0.377866i	$0.123342\pi$
$107$			7.81736i			0.755732i	−0.925860	+	0.377866i	$0.876658\pi$
$108$		−	0.618034i		−	0.0594703i
$109$	−1.33686			−0.128048			−0.0640240	−	0.997948i	$-0.520393\pi$
$109$	−1.33686			−0.128048			−0.0640240	+	0.997948i	$0.520393\pi$
$110$	5.33395	−	8.63052i	0.508573	−	0.822888i
$111$	−5.52015			−0.523949
$112$		−	4.85410i		−	0.458670i
$113$			6.79766i			0.639470i	0.947507	+	0.319735i	$0.103594\pi$
$113$			6.79766i			0.639470i	−0.947507	+	0.319735i	$0.896406\pi$
$114$	−3.79360			−0.355304
$115$	−1.10445	+	1.78704i	−0.102991	+	0.166642i
$116$	−3.52015			−0.326837
$117$			1.00000i			0.0924500i
$118$		−	3.99698i		−	0.367952i
$119$	2.01062			0.184313
$120$	−4.25325	−	2.62866i	−0.388267	−	0.239962i
$121$	−3.13632			−0.285120
$122$			6.72841i			0.609161i
$123$		−	3.07768i		−	0.277505i
$124$	−0.553479			−0.0497039
$125$	−1.00406	+	11.1352i	−0.0898056	+	0.995959i
$126$	−1.61803			−0.144146
$127$		−	10.4912i		−	0.930942i	−0.885063	−	0.465471i	$-0.845885\pi$
$127$		−	10.4912i		−	0.930942i	0.885063	−	0.465471i	$-0.154115\pi$
$128$			13.6180i			1.20368i
$129$	−5.12454			−0.451191
$130$	−3.07768	−	1.90211i	−0.269931	−	0.166826i
$131$	−2.09319			−0.182883			−0.0914413	−	0.995810i	$-0.529147\pi$
$131$	−2.09319			−0.182883			−0.0914413	+	0.995810i	$0.529147\pi$
$132$			1.73311i			0.150848i
$133$			2.34458i			0.203301i
$134$	−25.0467			−2.16371
$135$	−1.17557	+	1.90211i	−0.101177	+	0.163708i
$136$	−4.49589			−0.385519
$137$			12.2704i			1.04833i	0.851615	+	0.524167i	$0.175623\pi$
$137$			12.2704i			1.04833i	−0.851615	+	0.524167i	$0.824377\pi$
$138$		−	1.52015i		−	0.129404i
$139$	−13.4538			−1.14114			−0.570568	−	0.821250i	$-0.693277\pi$
$139$	−13.4538			−1.14114			−0.570568	+	0.821250i	$0.693277\pi$
$140$	0.726543	−	1.17557i	0.0614041	−	0.0993538i
$141$	−5.96261			−0.502142
$142$			19.9691i			1.67577i
$143$		−	2.80423i		−	0.234501i
$144$	4.85410			0.404508
$145$	10.8339	+	6.69572i	0.899706	+	0.556049i
$146$	4.48932			0.371539
$147$			1.00000i			0.0824786i
$148$			3.41164i			0.280435i
$149$	−12.6216			−1.03400			−0.517000	−	0.855985i	$-0.672951\pi$
$149$	−12.6216			−1.03400			−0.517000	+	0.855985i	$0.672951\pi$
$150$	−3.61803	−	7.23607i	−0.295411	−	0.590822i
$151$	0.222429			0.0181010			0.00905051	−	0.999959i	$-0.497119\pi$
$151$	0.222429			0.0181010			0.00905051	+	0.999959i	$0.497119\pi$
$152$		−	5.24263i		−	0.425234i
$153$			2.01062i			0.162549i
$154$	4.53733			0.365629
$155$	1.70343	+	1.05278i	0.136823	+	0.0845614i
$156$	0.618034			0.0494823
$157$		−	19.5013i		−	1.55637i	−0.628033	−	0.778187i	$-0.716140\pi$
$157$		−	19.5013i		−	1.55637i	0.628033	−	0.778187i	$-0.283860\pi$
$158$		−	9.97510i		−	0.793576i
$159$	1.30313			0.103345
$160$	−3.97574	+	6.43288i	−0.314310	+	0.508564i
$161$	−0.939503			−0.0740432
$162$		−	1.61803i		−	0.127125i
$163$		−	1.14712i		−	0.0898494i	−0.998990	−	0.0449247i	$-0.985695\pi$
$163$		−	1.14712i		−	0.0898494i	0.998990	−	0.0449247i	$-0.0143048\pi$
$164$	−1.90211			−0.148530
$165$	3.29657	−	5.33395i	0.256637	−	0.415248i
$166$	25.2188			1.95736
$167$			17.8232i			1.37920i	0.724191	+	0.689599i	$0.242213\pi$
$167$			17.8232i			1.37920i	−0.724191	+	0.689599i	$0.757787\pi$
$168$		−	2.23607i		−	0.172516i
$169$	−1.00000			−0.0769231
$170$	−6.18806	−	3.82443i	−0.474602	−	0.293320i
$171$	−2.34458			−0.179294
$172$			3.16714i			0.241492i
$173$			2.11443i			0.160757i	0.996764	+	0.0803786i	$0.0256129\pi$
$173$			2.11443i			0.160757i	−0.996764	+	0.0803786i	$0.974387\pi$
$174$	−9.21586			−0.698653
$175$	−4.47214	+	2.23607i	−0.338062	+	0.169031i
$176$	−13.6120			−1.02604
$177$		−	2.47027i		−	0.185677i
$178$		−	1.32907i		−	0.0996182i
$179$	−18.6737			−1.39574			−0.697870	−	0.716225i	$-0.745869\pi$
$179$	−18.6737			−1.39574			−0.697870	+	0.716225i	$0.745869\pi$
$180$	1.17557	+	0.726543i	0.0876219	+	0.0541533i
$181$	6.07872			0.451828			0.225914	−	0.974147i	$-0.427463\pi$
$181$	6.07872			0.451828			0.225914	+	0.974147i	$0.427463\pi$
$182$		−	1.61803i		−	0.119937i
$183$			4.15838i			0.307397i
$184$	2.10079			0.154872
$185$	6.48932	−	10.4999i	0.477104	−	0.771971i
$186$	−1.44903			−0.106248
$187$		−	5.63824i		−	0.412309i
$188$			3.68510i			0.268763i
$189$	−1.00000			−0.0727393
$190$	4.45965	−	7.21586i	0.323537	−	0.523494i
$191$	13.9329			1.00815			0.504076	−	0.863659i	$-0.331833\pi$
$191$	13.9329			1.00815			0.504076	+	0.863659i	$0.331833\pi$
$192$			4.23607i			0.305712i
$193$			3.14339i			0.226266i	0.993580	+	0.113133i	$0.0360887\pi$
$193$			3.14339i			0.226266i	−0.993580	+	0.113133i	$0.963911\pi$
$194$	−3.20826			−0.230340
$195$	−1.90211	−	1.17557i	−0.136213	−	0.0841844i
$196$	0.618034			0.0441453
$197$			5.11141i			0.364173i	0.983283	+	0.182087i	$0.0582851\pi$
$197$			5.11141i			0.364173i	−0.983283	+	0.182087i	$0.941715\pi$
$198$			4.53733i			0.322454i
$199$	−16.5135			−1.17061			−0.585304	−	0.810814i	$-0.699025\pi$
$199$	−16.5135			−1.17061			−0.585304	+	0.810814i	$0.699025\pi$
$200$	10.0000	−	5.00000i	0.707107	−	0.353553i
$201$	−15.4797			−1.09186
$202$		−	12.4893i		−	0.878746i
$203$			5.69572i			0.399761i
$204$	1.24263			0.0870017
$205$	5.85410	+	3.61803i	0.408868	+	0.252694i
$206$	31.2781			2.17925
$207$		−	0.939503i		−	0.0652999i
$208$			4.85410i			0.336571i
$209$	6.57472			0.454783
$210$	1.90211	−	3.07768i	0.131258	−	0.212380i
$211$	−2.47078			−0.170096			−0.0850478	−	0.996377i	$-0.527104\pi$
$211$	−2.47078			−0.170096			−0.0850478	+	0.996377i	$0.527104\pi$
$212$		−	0.805379i		−	0.0553136i
$213$			12.3416i			0.845630i
$214$	12.6487			0.864650
$215$	6.02426	−	9.74746i	0.410851	−	0.664771i
$216$	2.23607			0.152145
$217$			0.895549i			0.0607938i
$218$			2.16308i			0.146503i
$219$	2.77455			0.187487
$220$	−3.29657	−	2.03739i	−0.222254	−	0.137361i
$221$	−2.01062			−0.135249
$222$			8.93179i			0.599462i
$223$			9.10445i			0.609679i	0.952404	+	0.304840i	$0.0986029\pi$
$223$			9.10445i			0.609679i	−0.952404	+	0.304840i	$0.901397\pi$
$224$	−3.38197			−0.225967
$225$	−2.23607	−	4.47214i	−0.149071	−	0.298142i
$226$	10.9988			0.731632
$227$			7.00779i			0.465123i	0.972582	+	0.232562i	$0.0747107\pi$
$227$			7.00779i			0.465123i	−0.972582	+	0.232562i	$0.925289\pi$
$228$			1.44903i			0.0959642i
$229$	20.2385			1.33740			0.668698	−	0.743534i	$-0.266852\pi$
$229$	20.2385			1.33740			0.668698	+	0.743534i	$0.266852\pi$
$230$	2.89149	+	1.78704i	0.190659	+	0.117834i
$231$	2.80423			0.184505
$232$		−	12.7360i		−	0.836160i
$233$			14.6750i			0.961391i	0.876887	+	0.480696i	$0.159616\pi$
$233$			14.6750i			0.961391i	−0.876887	+	0.480696i	$0.840384\pi$
$234$	1.61803			0.105774
$235$	7.00947	−	11.3416i	0.457247	−	0.739842i
$236$	−1.52671			−0.0993805
$237$		−	6.16495i		−	0.400456i
$238$		−	3.25325i		−	0.210877i
$239$	−23.9963			−1.55219			−0.776097	−	0.630613i	$-0.782803\pi$
$239$	−23.9963			−1.55219			−0.776097	+	0.630613i	$0.782803\pi$
$240$	−5.70634	+	9.23305i	−0.368343	+	0.595991i
$241$	25.2300			1.62521			0.812605	−	0.582815i	$-0.198049\pi$
$241$	25.2300			1.62521			0.812605	+	0.582815i	$0.198049\pi$
$242$			5.07467i			0.326212i
$243$		−	1.00000i		−	0.0641500i
$244$	2.57002			0.164529
$245$	−1.90211	−	1.17557i	−0.121522	−	0.0751044i
$246$	−4.97980			−0.317500
$247$		−	2.34458i		−	0.149182i
$248$		−	2.00251i		−	0.127159i
$249$	15.5861			0.987726
$250$	18.0171	+	1.62460i	1.13950	+	0.102749i
$251$	11.8992			0.751069			0.375534	−	0.926808i	$-0.377459\pi$
$251$	11.8992			0.751069			0.375534	+	0.926808i	$0.377459\pi$
$252$			0.618034i			0.0389325i
$253$			2.63458i			0.165634i
$254$	−16.9751			−1.06511
$255$	−3.82443	−	2.36363i	−0.239495	−	0.148016i
$256$	13.5623			0.847644
$257$			5.49151i			0.342551i	0.985223	+	0.171276i	$0.0547889\pi$
$257$			5.49151i			0.342551i	−0.985223	+	0.171276i	$0.945211\pi$
$258$			8.29168i			0.516218i
$259$	5.52015			0.343005
$260$	−0.726543	+	1.17557i	−0.0450583	+	0.0729058i
$261$	−5.69572			−0.352556
$262$			3.38685i			0.209240i
$263$			2.38614i			0.147135i	0.997290	+	0.0735677i	$0.0234385\pi$
$263$			2.38614i			0.147135i	−0.997290	+	0.0735677i	$0.976561\pi$
$264$	−6.27044			−0.385919
$265$	−1.53192	+	2.47870i	−0.0941052	+	0.152265i
$266$	3.79360			0.232601
$267$		−	0.821412i		−	0.0502696i
$268$			9.56701i			0.584398i
$269$	18.6149			1.13497			0.567485	−	0.823383i	$-0.307916\pi$
$269$	18.6149			1.13497			0.567485	+	0.823383i	$0.307916\pi$
$270$	3.07768	+	1.90211i	0.187302	+	0.115759i
$271$	−2.89400			−0.175798			−0.0878990	−	0.996129i	$-0.528015\pi$
$271$	−2.89400			−0.175798			−0.0878990	+	0.996129i	$0.528015\pi$
$272$			9.75976i			0.591773i
$273$		−	1.00000i		−	0.0605228i
$274$	19.8540			1.19942
$275$	6.27044	+	12.5409i	0.378122	+	0.756244i
$276$	−0.580644			−0.0349507
$277$		−	14.3261i		−	0.860769i	−0.902646	−	0.430385i	$-0.858378\pi$
$277$		−	14.3261i		−	0.860769i	0.902646	−	0.430385i	$-0.141622\pi$
$278$			21.7687i			1.30560i
$279$	−0.895549			−0.0536151
$280$	4.25325	+	2.62866i	0.254181	+	0.157092i
$281$	5.57543			0.332603			0.166301	−	0.986075i	$-0.446818\pi$
$281$	5.57543			0.332603			0.166301	+	0.986075i	$0.446818\pi$
$282$			9.64771i			0.574513i
$283$		−	18.6838i		−	1.11064i	−0.831637	−	0.555319i	$-0.812596\pi$
$283$		−	18.6838i		−	1.11064i	0.831637	−	0.555319i	$-0.187404\pi$
$284$	7.62750			0.452609
$285$	2.75621	−	4.45965i	0.163264	−	0.264167i
$286$	−4.53733			−0.268298
$287$			3.07768i			0.181670i
$288$		−	3.38197i		−	0.199284i
$289$	12.9574			0.762200
$290$	10.8339	−	17.5296i	0.636188	−	1.02937i
$291$	−1.98281			−0.116235
$292$		−	1.71477i		−	0.100349i
$293$			31.0585i			1.81446i	0.420637	+	0.907229i	$0.361807\pi$
$293$			31.0585i			1.81446i	−0.420637	+	0.907229i	$0.638193\pi$
$294$	1.61803			0.0943657
$295$	4.69873	+	2.90398i	0.273571	+	0.169076i
$296$	−12.3434			−0.717447
$297$			2.80423i			0.162718i
$298$			20.4221i			1.18302i
$299$	0.939503			0.0543328
$300$	−2.76393	+	1.38197i	−0.159576	+	0.0797878i
$301$	5.12454			0.295374
$302$		−	0.359898i		−	0.0207098i
$303$		−	7.71883i		−	0.443435i
$304$	−11.3808			−0.652734
$305$	−7.90972	−	4.88847i	−0.452909	−	0.279913i
$306$	3.25325			0.185976
$307$		−	1.72892i		−	0.0986745i	−0.998782	−	0.0493373i	$-0.984289\pi$
$307$		−	1.72892i		−	0.0986745i	0.998782	−	0.0493373i	$-0.0157109\pi$
$308$		−	1.73311i		−	0.0987529i
$309$	19.3309			1.09970
$310$	1.70343	−	2.75621i	0.0967486	−	0.156542i
$311$	17.7385			1.00586			0.502930	−	0.864327i	$-0.332256\pi$
$311$	17.7385			1.00586			0.502930	+	0.864327i	$0.332256\pi$
$312$			2.23607i			0.126592i
$313$			24.8522i			1.40473i	0.711817	+	0.702365i	$0.247872\pi$
$313$			24.8522i			1.40473i	−0.711817	+	0.702365i	$0.752128\pi$
$314$	−31.5538			−1.78068
$315$	1.17557	−	1.90211i	0.0662359	−	0.107172i
$316$	−3.81015			−0.214338
$317$			2.60626i			0.146382i	0.997318	+	0.0731911i	$0.0233183\pi$
$317$			2.60626i			0.146382i	−0.997318	+	0.0731911i	$0.976682\pi$
$318$		−	2.10851i		−	0.118239i
$319$	15.9721			0.894265
$320$	−8.05748	−	4.97980i	−0.450427	−	0.278379i
$321$	7.81736			0.436322
$322$			1.52015i			0.0847145i
$323$		−	4.71406i		−	0.262297i
$324$	−0.618034			−0.0343352
$325$	4.47214	−	2.23607i	0.248069	−	0.124035i
$326$	−1.85608			−0.102799
$327$			1.33686i			0.0739285i
$328$		−	6.88191i		−	0.379990i
$329$	5.96261			0.328729
$330$	−8.63052	−	5.33395i	−0.475095	−	0.293625i
$331$	−30.7543			−1.69041			−0.845205	−	0.534443i	$-0.820522\pi$
$331$	−30.7543			−1.69041			−0.845205	+	0.534443i	$0.820522\pi$
$332$		−	9.63271i		−	0.528664i
$333$			5.52015i			0.302502i
$334$	28.8385			1.57797
$335$	18.1975	−	29.4442i	0.994237	−	1.60871i
$336$	−4.85410			−0.264813
$337$			14.3356i			0.780912i	0.920622	+	0.390456i	$0.127683\pi$
$337$			14.3356i			0.780912i	−0.920622	+	0.390456i	$0.872317\pi$
$338$			1.61803i			0.0880094i
$339$	6.79766			0.369198
$340$	−1.46080	+	2.36363i	−0.0792231	+	0.128186i
$341$	2.51132			0.135996
$342$			3.79360i			0.205135i
$343$		−	1.00000i		−	0.0539949i
$344$	−11.4588			−0.617818
$345$	1.78704	+	1.10445i	0.0962110	+	0.0594617i
$346$	3.42122			0.183926
$347$		−	28.2949i		−	1.51895i	−0.650537	−	0.759475i	$-0.725456\pi$
$347$		−	28.2949i		−	1.51895i	0.650537	−	0.759475i	$-0.274544\pi$
$348$			3.52015i			0.188700i
$349$	7.82174			0.418688			0.209344	−	0.977842i	$-0.432867\pi$
$349$	7.82174			0.418688			0.209344	+	0.977842i	$0.432867\pi$
$350$	3.61803	+	7.23607i	0.193392	+	0.386784i
$351$	1.00000			0.0533761
$352$			9.48380i			0.505488i
$353$		−	12.6656i		−	0.674122i	−0.941483	−	0.337061i	$-0.890567\pi$
$353$		−	12.6656i		−	0.674122i	0.941483	−	0.337061i	$-0.109433\pi$
$354$	−3.99698			−0.212437
$355$	−23.4750	−	14.5084i	−1.24593	−	0.770024i
$356$	−0.507661			−0.0269060
$357$		−	2.01062i		−	0.106413i
$358$			30.2147i			1.59690i
$359$	−4.35926			−0.230073			−0.115036	−	0.993361i	$-0.536698\pi$
$359$	−4.35926			−0.230073			−0.115036	+	0.993361i	$0.536698\pi$
$360$	−2.62866	+	4.25325i	−0.138542	+	0.224166i
$361$	−13.5030			−0.710682
$362$		−	9.83558i		−	0.516947i
$363$			3.13632i			0.164614i
$364$	−0.618034			−0.0323938
$365$	−3.26168	+	5.27751i	−0.170724	+	0.276238i
$366$	6.72841			0.351699
$367$		−	23.7244i		−	1.23840i	−0.785233	−	0.619201i	$-0.787457\pi$
$367$		−	23.7244i		−	1.23840i	0.785233	−	0.619201i	$-0.212543\pi$
$368$		−	4.56044i		−	0.237729i
$369$	−3.07768			−0.160218
$370$	−16.9893	−	10.4999i	−0.883230	−	0.545866i
$371$	−1.30313			−0.0676551
$372$			0.553479i			0.0286966i
$373$			12.2595i			0.634773i	0.948296	+	0.317387i	$0.102805\pi$
$373$			12.2595i			0.634773i	−0.948296	+	0.317387i	$0.897195\pi$
$374$	−9.12286			−0.471732
$375$	11.1352	+	1.00406i	0.575017	+	0.0518493i
$376$	−13.3328			−0.687587
$377$		−	5.69572i		−	0.293344i
$378$			1.61803i			0.0832227i
$379$	9.44164			0.484984			0.242492	−	0.970153i	$-0.422035\pi$
$379$	9.44164			0.484984			0.242492	+	0.970153i	$0.422035\pi$
$380$	−2.75621	−	1.70343i	−0.141391	−	0.0873843i
$381$	−10.4912			−0.537480
$382$		−	22.5440i		−	1.15345i
$383$		−	1.81922i		−	0.0929578i	−0.998919	−	0.0464789i	$-0.985200\pi$
$383$		−	1.81922i		−	0.0929578i	0.998919	−	0.0464789i	$-0.0148000\pi$
$384$	13.6180			0.694942
$385$	−3.29657	+	5.33395i	−0.168009	+	0.271844i
$386$	5.08611			0.258876
$387$			5.12454i			0.260495i
$388$			1.22545i			0.0622126i
$389$	11.2846			0.572152			0.286076	−	0.958207i	$-0.407649\pi$
$389$	11.2846			0.572152			0.286076	+	0.958207i	$0.407649\pi$
$390$	−1.90211	+	3.07768i	−0.0963172	+	0.155845i
$391$	1.88898			0.0955300
$392$			2.23607i			0.112938i
$393$			2.09319i			0.105587i
$394$	8.27044			0.416659
$395$	11.7264	+	7.24733i	0.590021	+	0.364653i
$396$	1.73311			0.0870919
$397$			21.6658i			1.08738i	0.839287	+	0.543688i	$0.182973\pi$
$397$			21.6658i			1.08738i	−0.839287	+	0.543688i	$0.817027\pi$
$398$			26.7194i			1.33932i
$399$	2.34458			0.117376
$400$	−10.8541	−	21.7082i	−0.542705	−	1.08541i
$401$	15.1128			0.754696			0.377348	−	0.926072i	$-0.376836\pi$
$401$	15.1128			0.754696			0.377348	+	0.926072i	$0.376836\pi$
$402$			25.0467i			1.24922i
$403$		−	0.895549i		−	0.0446105i
$404$	−4.77050			−0.237341
$405$	1.90211	+	1.17557i	0.0945168	+	0.0584146i
$406$	9.21586			0.457376
$407$		−	15.4797i		−	0.767302i
$408$			4.49589i			0.222580i
$409$	−0.0919651			−0.00454738			−0.00227369	−	0.999997i	$-0.500724\pi$
$409$	−0.0919651			−0.00454738			−0.00227369	+	0.999997i	$0.500724\pi$
$410$	5.85410	−	9.47214i	0.289113	−	0.467795i
$411$	12.2704			0.605256
$412$		−	11.9472i		−	0.588595i
$413$			2.47027i			0.121554i
$414$	−1.52015			−0.0747112
$415$	−18.3225	+	29.6464i	−0.899417	+	1.45529i
$416$	3.38197			0.165815
$417$			13.4538i			0.658835i
$418$		−	10.6381i		−	0.520328i
$419$	−15.8705			−0.775323			−0.387661	−	0.921802i	$-0.626717\pi$
$419$	−15.8705			−0.775323			−0.387661	+	0.921802i	$0.626717\pi$
$420$	−1.17557	−	0.726543i	−0.0573620	−	0.0354516i
$421$	−13.6257			−0.664076			−0.332038	−	0.943266i	$-0.607736\pi$
$421$	−13.6257			−0.664076			−0.332038	+	0.943266i	$0.607736\pi$
$422$			3.99781i			0.194610i
$423$			5.96261i			0.289912i
$424$	2.91389			0.141511
$425$	8.99177	−	4.49589i	0.436165	−	0.218083i
$426$	19.9691			0.967504
$427$		−	4.15838i		−	0.201238i
$428$		−	4.83139i		−	0.233534i
$429$	−2.80423			−0.135389
$430$	−15.7717	−	9.74746i	−0.760580	−	0.470064i
$431$	−25.4756			−1.22712			−0.613559	−	0.789649i	$-0.710263\pi$
$431$	−25.4756			−1.22712			−0.613559	+	0.789649i	$0.710263\pi$
$432$		−	4.85410i		−	0.233543i
$433$			20.3956i			0.980151i	0.871680	+	0.490076i	$0.163031\pi$
$433$			20.3956i			0.980151i	−0.871680	+	0.490076i	$0.836969\pi$
$434$	1.44903			0.0695556
$435$	6.69572	−	10.8339i	0.321035	−	0.519446i
$436$	0.826225			0.0395690
$437$			2.20274i			0.105371i
$438$		−	4.48932i		−	0.214508i
$439$	−22.8539			−1.09076			−0.545379	−	0.838190i	$-0.683614\pi$
$439$	−22.8539			−1.09076			−0.545379	+	0.838190i	$0.683614\pi$
$440$	7.37134	−	11.9271i	0.351415	−	0.568602i
$441$	1.00000			0.0476190
$442$			3.25325i			0.154742i
$443$		−	15.9067i		−	0.755750i	−0.925857	−	0.377875i	$-0.876655\pi$
$443$		−	15.9067i		−	0.755750i	0.925857	−	0.377875i	$-0.123345\pi$
$444$	3.41164			0.161909
$445$	1.56242	+	0.965628i	0.0740658	+	0.0457752i
$446$	14.7313			0.697548
$447$			12.6216i			0.596980i
$448$		−	4.23607i		−	0.200135i
$449$	23.3073			1.09994			0.549969	−	0.835185i	$-0.314640\pi$
$449$	23.3073			1.09994			0.549969	+	0.835185i	$0.314640\pi$
$450$	−7.23607	+	3.61803i	−0.341112	+	0.170556i
$451$	8.63052			0.406395
$452$		−	4.20119i		−	0.197607i
$453$		−	0.222429i		−	0.0104506i
$454$	11.3388			0.532158
$455$	1.90211	+	1.17557i	0.0891724	+	0.0551116i
$456$	−5.24263			−0.245509
$457$			18.1531i			0.849164i	0.905389	+	0.424582i	$0.139579\pi$
$457$			18.1531i			0.849164i	−0.905389	+	0.424582i	$0.860421\pi$
$458$		−	32.7465i		−	1.53014i
$459$	2.01062			0.0938478
$460$	0.682589	−	1.10445i	0.0318259	−	0.0514953i
$461$	30.5416			1.42246			0.711232	−	0.702957i	$-0.248138\pi$
$461$	30.5416			1.42246			0.711232	+	0.702957i	$0.248138\pi$
$462$		−	4.53733i		−	0.211096i
$463$			18.8199i			0.874636i	0.899307	+	0.437318i	$0.144072\pi$
$463$			18.8199i			0.874636i	−0.899307	+	0.437318i	$0.855928\pi$
$464$	−27.6476			−1.28351
$465$	1.05278	−	1.70343i	0.0488215	−	0.0789949i
$466$	23.7447			1.09995
$467$		−	14.3948i		−	0.666114i	−0.942907	−	0.333057i	$-0.891920\pi$
$467$		−	14.3948i		−	0.666114i	0.942907	−	0.333057i	$-0.108080\pi$
$468$		−	0.618034i		−	0.0285686i
$469$	15.4797			0.714788
$470$	−18.3510	−	11.3416i	−0.846470	−	0.523147i
$471$	−19.5013			−0.898572
$472$		−	5.52369i		−	0.254249i
$473$		−	14.3704i		−	0.660751i
$474$	−9.97510			−0.458171
$475$	5.24263	+	10.4853i	0.240548	+	0.481097i
$476$	−1.24263			−0.0569560
$477$		−	1.30313i		−	0.0596662i
$478$			38.8269i			1.77590i
$479$	25.3568			1.15858			0.579292	−	0.815120i	$-0.303329\pi$
$479$	25.3568			1.15858			0.579292	+	0.815120i	$0.303329\pi$
$480$	6.43288	+	3.97574i	0.293620	+	0.181467i
$481$	−5.52015			−0.251697
$482$		−	40.8231i		−	1.85944i
$483$			0.939503i			0.0427488i
$484$	1.93835			0.0881068
$485$	2.33094	−	3.77154i	0.105842	−	0.171257i
$486$	−1.61803			−0.0733955
$487$		−	6.55671i		−	0.297113i	−0.988904	−	0.148556i	$-0.952537\pi$
$487$		−	6.55671i		−	0.297113i	0.988904	−	0.148556i	$-0.0474627\pi$
$488$			9.29843i			0.420920i
$489$	−1.14712			−0.0518746
$490$	−1.90211	+	3.07768i	−0.0859287	+	0.139036i
$491$	−36.7728			−1.65954			−0.829768	−	0.558109i	$-0.811527\pi$
$491$	−36.7728			−1.65954			−0.829768	+	0.558109i	$0.811527\pi$
$492$			1.90211i			0.0857539i
$493$		−	11.4519i		−	0.515769i
$494$	−3.79360			−0.170682
$495$	−5.33395	−	3.29657i	−0.239743	−	0.148170i
$496$	−4.34708			−0.195190
$497$		−	12.3416i		−	0.553595i
$498$		−	25.2188i		−	1.13008i
$499$	−33.8073			−1.51342			−0.756712	−	0.653748i	$-0.773196\pi$
$499$	−33.8073			−1.51342			−0.756712	+	0.653748i	$0.773196\pi$
$500$	0.620541	−	6.88191i	0.0277515	−	0.307768i
$501$	17.8232			0.796280
$502$		−	19.2533i		−	0.859315i
$503$		−	2.57397i		−	0.114768i	−0.998352	−	0.0573838i	$-0.981724\pi$
$503$		−	2.57397i		−	0.114768i	0.998352	−	0.0573838i	$-0.0182759\pi$
$504$	−2.23607			−0.0996024
$505$	14.6821	+	9.07402i	0.653344	+	0.403789i
$506$	4.26284			0.189506
$507$			1.00000i			0.0444116i
$508$			6.48391i			0.287677i
$509$	−14.2906			−0.633422			−0.316711	−	0.948522i	$-0.602578\pi$
$509$	−14.2906			−0.633422			−0.316711	+	0.948522i	$0.602578\pi$
$510$	−3.82443	+	6.18806i	−0.169349	+	0.274012i
$511$	−2.77455			−0.122739
$512$			5.29180i			0.233867i
$513$			2.34458i			0.103516i
$514$	8.88546			0.391921
$515$	−22.7249	+	36.7696i	−1.00138	+	1.62026i
$516$	3.16714			0.139426
$517$		−	16.7205i		−	0.735367i
$518$		−	8.93179i		−	0.392440i
$519$	2.11443			0.0928132
$520$	−4.25325	−	2.62866i	−0.186518	−	0.115274i
$521$	15.9346			0.698108			0.349054	−	0.937103i	$-0.386503\pi$
$521$	15.9346			0.698108			0.349054	+	0.937103i	$0.386503\pi$
$522$			9.21586i			0.403367i
$523$			42.2459i			1.84729i	0.383254	+	0.923643i	$0.374803\pi$
$523$			42.2459i			1.84729i	−0.383254	+	0.923643i	$0.625197\pi$
$524$	1.29366			0.0565138
$525$	2.23607	+	4.47214i	0.0975900	+	0.195180i
$526$	3.86085			0.168341
$527$		−	1.80061i		−	0.0784358i
$528$			13.6120i			0.592386i
$529$	22.1173			0.961623
$530$	4.01062	+	2.47870i	0.174210	+	0.107668i
$531$	−2.47027			−0.107201
$532$		−	1.44903i		−	0.0628233i
$533$		−	3.07768i		−	0.133309i
$534$	−1.32907			−0.0575146
$535$	−9.18985	+	14.8695i	−0.397312	+	0.642864i
$536$	−34.6138			−1.49509
$537$			18.6737i			0.805830i
$538$		−	30.1195i		−	1.29855i
$539$	−2.80423			−0.120787
$540$	0.726543	−	1.17557i	0.0312654	−	0.0505885i
$541$	−32.0729			−1.37892			−0.689460	−	0.724324i	$-0.742152\pi$
$541$	−32.0729			−1.37892			−0.689460	+	0.724324i	$0.742152\pi$
$542$			4.68259i			0.201134i
$543$		−	6.07872i		−	0.260863i
$544$	6.79985			0.291541
$545$	−2.54286	−	1.57157i	−0.108924	−	0.0673188i
$546$	−1.61803			−0.0692455
$547$			19.6201i			0.838895i	0.907779	+	0.419448i	$0.137776\pi$
$547$			19.6201i			0.838895i	−0.907779	+	0.419448i	$0.862224\pi$
$548$		−	7.58355i		−	0.323953i
$549$	4.15838			0.177476
$550$	20.2916	−	10.1458i	0.865235	−	0.432618i
$551$	13.3540			0.568901
$552$		−	2.10079i		−	0.0894156i
$553$			6.16495i			0.262160i
$554$	−23.1800			−0.984826
$555$	−10.4999	−	6.48932i	−0.445698	−	0.275456i
$556$	8.31490			0.352630
$557$			0.768499i			0.0325624i	0.999867	+	0.0162812i	$0.00518269\pi$
$557$			0.768499i			0.0325624i	−0.999867	+	0.0162812i	$0.994817\pi$
$558$			1.44903i			0.0613422i
$559$	−5.12454			−0.216745
$560$	5.70634	−	9.23305i	0.241137	−	0.390168i
$561$	−5.63824			−0.238047
$562$		−	9.02124i		−	0.380538i
$563$		−	35.0266i		−	1.47620i	−0.674694	−	0.738098i	$-0.735724\pi$
$563$		−	35.0266i		−	1.47620i	0.674694	−	0.738098i	$-0.264276\pi$
$564$	3.68510			0.155171
$565$	−7.99113	+	12.9299i	−0.336189	+	0.543966i
$566$	−30.2311			−1.27071
$567$			1.00000i			0.0419961i
$568$			27.5966i			1.15793i
$569$	−26.3124			−1.10307			−0.551537	−	0.834150i	$-0.685958\pi$
$569$	−26.3124			−1.10307			−0.551537	+	0.834150i	$0.685958\pi$
$570$	−7.21586	−	4.45965i	−0.302239	−	0.186794i
$571$	−40.3772			−1.68973			−0.844866	−	0.534978i	$-0.820320\pi$
$571$	−40.3772			−1.68973			−0.844866	+	0.534978i	$0.820320\pi$
$572$			1.73311i			0.0724648i
$573$		−	13.9329i		−	0.582057i
$574$	4.97980			0.207853
$575$	−4.20158	+	2.10079i	−0.175218	+	0.0876091i
$576$	4.23607			0.176503
$577$			33.7497i			1.40502i	0.711675	+	0.702509i	$0.247937\pi$
$577$			33.7497i			1.40502i	−0.711675	+	0.702509i	$0.752063\pi$
$578$		−	20.9655i		−	0.872050i
$579$	3.14339			0.130635
$580$	−6.69572	−	4.13818i	−0.278025	−	0.171829i
$581$	−15.5861			−0.646619
$582$			3.20826i			0.132987i
$583$			3.65427i			0.151344i
$584$	6.20409			0.256727
$585$	−1.17557	+	1.90211i	−0.0486039	+	0.0786427i
$586$	50.2537			2.07596
$587$			35.8502i			1.47970i	0.672774	+	0.739848i	$0.265103\pi$
$587$			35.8502i			1.47970i	−0.672774	+	0.739848i	$0.734897\pi$
$588$		−	0.618034i		−	0.0254873i
$589$	2.09968			0.0865159
$590$	4.69873	−	7.60271i	0.193444	−	0.312999i
$591$	5.11141			0.210255
$592$			26.7954i			1.10128i
$593$			28.7642i			1.18120i	0.806963	+	0.590602i	$0.201110\pi$
$593$			28.7642i			1.18120i	−0.806963	+	0.590602i	$0.798890\pi$
$594$	4.53733			0.186169
$595$	3.82443	+	2.36363i	0.156786	+	0.0968993i
$596$	7.80057			0.319524
$597$			16.5135i			0.675851i
$598$		−	1.52015i		−	0.0621634i
$599$	16.1612			0.660327			0.330163	−	0.943924i	$-0.392896\pi$
$599$	16.1612			0.660327			0.330163	+	0.943924i	$0.392896\pi$
$600$	−5.00000	−	10.0000i	−0.204124	−	0.408248i
$601$	−40.3491			−1.64587			−0.822936	−	0.568134i	$-0.807666\pi$
$601$	−40.3491			−1.64587			−0.822936	+	0.568134i	$0.807666\pi$
$602$		−	8.29168i		−	0.337944i
$603$			15.4797i			0.630384i
$604$	−0.137469			−0.00559352
$605$	−5.96563	−	3.68696i	−0.242537	−	0.149896i
$606$	−12.4893			−0.507344
$607$		−	34.9319i		−	1.41784i	−0.705289	−	0.708920i	$-0.749183\pi$
$607$		−	34.9319i		−	1.41784i	0.705289	−	0.708920i	$-0.250817\pi$
$608$			7.92928i			0.321575i
$609$	5.69572			0.230802
$610$	−7.90972	+	12.7982i	−0.320255	+	0.518184i
$611$	−5.96261			−0.241221
$612$		−	1.24263i		−	0.0502304i
$613$		−	36.1928i		−	1.46181i	−0.682477	−	0.730907i	$-0.739097\pi$
$613$		−	36.1928i		−	1.46181i	0.682477	−	0.730907i	$-0.260903\pi$
$614$	−2.79745			−0.112896
$615$	3.61803	−	5.85410i	0.145893	−	0.236060i
$616$	6.27044			0.252643
$617$			27.8387i			1.12074i	0.828242	+	0.560371i	$0.189342\pi$
$617$			27.8387i			1.12074i	−0.828242	+	0.560371i	$0.810658\pi$
$618$		−	31.2781i		−	1.25819i
$619$	−14.9807			−0.602126			−0.301063	−	0.953604i	$-0.597341\pi$
$619$	−14.9807			−0.602126			−0.301063	+	0.953604i	$0.597341\pi$
$620$	−1.05278	−	0.650654i	−0.0422807	−	0.0261309i
$621$	−0.939503			−0.0377009
$622$		−	28.7015i		−	1.15083i
$623$			0.821412i			0.0329092i
$624$	4.85410			0.194320
$625$	−15.0000	+	20.0000i	−0.600000	+	0.800000i
$626$	40.2117			1.60718
$627$		−	6.57472i		−	0.262569i
$628$			12.0525i			0.480946i
$629$	−11.0989			−0.442543
$630$	−3.07768	−	1.90211i	−0.122618	−	0.0757820i
$631$	10.7583			0.428281			0.214140	−	0.976803i	$-0.431305\pi$
$631$	10.7583			0.428281			0.214140	+	0.976803i	$0.431305\pi$
$632$		−	13.7852i		−	0.548348i
$633$			2.47078i			0.0982047i
$634$	4.21702			0.167479
$635$	12.3331	−	19.9554i	0.489425	−	0.791907i
$636$	−0.805379			−0.0319353
$637$			1.00000i			0.0396214i
$638$		−	25.8434i		−	1.02315i
$639$	12.3416			0.488225
$640$	−16.0090	+	25.9030i	−0.632810	+	1.02391i
$641$	−12.9112			−0.509961			−0.254981	−	0.966946i	$-0.582069\pi$
$641$	−12.9112			−0.509961			−0.254981	+	0.966946i	$0.582069\pi$
$642$		−	12.6487i		−	0.499206i
$643$		−	20.8057i		−	0.820495i	−0.911974	−	0.410248i	$-0.865442\pi$
$643$		−	20.8057i		−	0.820495i	0.911974	−	0.410248i	$-0.134558\pi$
$644$	0.580644			0.0228806
$645$	−9.74746	−	6.02426i	−0.383806	−	0.237205i
$646$	−7.62750			−0.300100
$647$			25.7610i			1.01277i	0.862308	+	0.506384i	$0.169018\pi$
$647$			25.7610i			1.01277i	−0.862308	+	0.506384i	$0.830982\pi$
$648$		−	2.23607i		−	0.0878410i
$649$	6.92720			0.271916
$650$	−3.61803	−	7.23607i	−0.141911	−	0.283822i
$651$	0.895549			0.0350993
$652$			0.708959i			0.0277650i
$653$		−	33.0209i		−	1.29221i	−0.763250	−	0.646103i	$-0.776398\pi$
$653$		−	33.0209i		−	1.29221i	0.763250	−	0.646103i	$-0.223602\pi$
$654$	2.16308			0.0845833
$655$	−3.98148	−	2.46069i	−0.155569	−	0.0961471i
$656$	−14.9394			−0.583285
$657$		−	2.77455i		−	0.108246i
$658$		−	9.64771i		−	0.376107i
$659$	36.4979			1.42176			0.710879	−	0.703315i	$-0.248298\pi$
$659$	36.4979			1.42176			0.710879	+	0.703315i	$0.248298\pi$
$660$	−2.03739	+	3.29657i	−0.0793053	+	0.128319i
$661$	35.1393			1.36676			0.683380	−	0.730063i	$-0.260509\pi$
$661$	35.1393			1.36676			0.683380	+	0.730063i	$0.260509\pi$
$662$			49.7615i			1.93404i
$663$			2.01062i			0.0780861i
$664$	34.8515			1.35250
$665$	−2.75621	+	4.45965i	−0.106881	+	0.172938i
$666$	8.93179			0.346100
$667$			5.35114i			0.207197i
$668$		−	11.0153i		−	0.426196i
$669$	9.10445			0.351998
$670$	−47.6417	−	29.4442i	−1.84056	−	1.13753i
$671$	−11.6611			−0.450170
$672$			3.38197i			0.130462i
$673$		−	10.0652i		−	0.387985i	−0.981003	−	0.193992i	$-0.937856\pi$
$673$		−	10.0652i		−	0.387985i	0.981003	−	0.193992i	$-0.0621437\pi$
$674$	23.1955			0.893459
$675$	−4.47214	+	2.23607i	−0.172133	+	0.0860663i
$676$	0.618034			0.0237705
$677$		−	34.4229i		−	1.32298i	−0.749955	−	0.661489i	$-0.769925\pi$
$677$		−	34.4229i		−	1.32298i	0.749955	−	0.661489i	$-0.230075\pi$
$678$		−	10.9988i		−	0.422408i
$679$	1.98281			0.0760934
$680$	−8.55168	−	5.28523i	−0.327942	−	0.202679i
$681$	7.00779			0.268539
$682$		−	4.06340i		−	0.155596i
$683$		−	7.19340i		−	0.275248i	−0.990485	−	0.137624i	$-0.956053\pi$
$683$		−	7.19340i		−	0.275248i	0.990485	−	0.137624i	$-0.0439465\pi$
$684$	1.44903			0.0554050
$685$	−14.4248	+	23.3398i	−0.551142	+	0.891767i
$686$	−1.61803			−0.0617768
$687$		−	20.2385i		−	0.772146i
$688$			24.8750i			0.948352i
$689$	1.30313			0.0496453
$690$	1.78704	−	2.89149i	0.0680315	−	0.110077i
$691$	2.19402			0.0834645			0.0417323	−	0.999129i	$-0.486712\pi$
$691$	2.19402			0.0834645			0.0417323	+	0.999129i	$0.486712\pi$
$692$		−	1.30679i		−	0.0496767i
$693$		−	2.80423i		−	0.106524i
$694$	−45.7821			−1.73787
$695$	−25.5906	−	15.8159i	−0.970708	−	0.599931i
$696$	−12.7360			−0.482757
$697$		−	6.18806i		−	0.234389i
$698$		−	12.6558i		−	0.479031i
$699$	14.6750			0.555060
$700$	2.76393	−	1.38197i	0.104467	−	0.0522334i
$701$	9.08651			0.343193			0.171596	−	0.985167i	$-0.445108\pi$
$701$	9.08651			0.343193			0.171596	+	0.985167i	$0.445108\pi$
$702$		−	1.61803i		−	0.0610688i
$703$		−	12.9424i		−	0.488132i
$704$	−11.8789			−0.447703
$705$	−11.3416	−	7.00947i	−0.427148	−	0.263992i
$706$	−20.4934			−0.771278
$707$			7.71883i			0.290296i
$708$			1.52671i			0.0573773i
$709$	17.0828			0.641557			0.320778	−	0.947154i	$-0.396056\pi$
$709$	17.0828			0.641557			0.320778	+	0.947154i	$0.396056\pi$
$710$	−23.4750	+	37.9834i	−0.881003	+	1.42549i
$711$	−6.16495			−0.231204
$712$		−	1.83673i		−	0.0688345i
$713$			0.841370i			0.0315096i
$714$	−3.25325			−0.121750
$715$	3.29657	−	5.33395i	0.123285	−	0.199479i
$716$	11.5410			0.431307
$717$			23.9963i			0.896160i
$718$			7.05342i			0.263231i
$719$	46.0179			1.71618			0.858088	−	0.513502i	$-0.171652\pi$
$719$	46.0179			1.71618			0.858088	+	0.513502i	$0.171652\pi$
$720$	9.23305	+	5.70634i	0.344095	+	0.212663i
$721$	−19.3309			−0.719922
$722$			21.8482i			0.813108i
$723$		−	25.2300i		−	0.938315i
$724$	−3.75686			−0.139623
$725$	12.7360	+	25.4720i	0.473004	+	0.946007i
$726$	5.07467			0.188339
$727$		−	41.4822i		−	1.53849i	−0.638954	−	0.769245i	$-0.720632\pi$
$727$		−	41.4822i		−	1.53849i	0.638954	−	0.769245i	$-0.279368\pi$
$728$		−	2.23607i		−	0.0828742i
$729$	−1.00000			−0.0370370
$730$	8.53920	+	5.27751i	0.316050	+	0.195330i
$731$	−10.3035			−0.381089
$732$		−	2.57002i		−	0.0949908i
$733$			8.95740i			0.330849i	0.986222	+	0.165425i	$0.0528994\pi$
$733$			8.95740i			0.330849i	−0.986222	+	0.165425i	$0.947101\pi$
$734$	−38.3868			−1.41688
$735$	−1.17557	+	1.90211i	−0.0433616	+	0.0701605i
$736$	−3.17737			−0.117119
$737$		−	43.4087i		−	1.59898i
$738$			4.97980i			0.183309i
$739$	−41.1448			−1.51354			−0.756769	−	0.653683i	$-0.773223\pi$
$739$	−41.1448			−1.51354			−0.756769	+	0.653683i	$0.773223\pi$
$740$	−4.01062	+	6.48932i	−0.147433	+	0.238552i
$741$	−2.34458			−0.0861302
$742$			2.10851i			0.0774058i
$743$			51.2608i			1.88058i	0.340379	+	0.940288i	$0.389445\pi$
$743$			51.2608i			1.88058i	−0.340379	+	0.940288i	$0.610555\pi$
$744$	−2.00251			−0.0734155
$745$	−24.0077	−	14.8376i	−0.879573	−	0.543606i
$746$	19.8363			0.726258
$747$		−	15.5861i		−	0.570264i
$748$			3.48462i			0.127410i
$749$	−7.81736			−0.285640
$750$	1.62460	−	18.0171i	0.0593219	−	0.657890i
$751$	−9.69851			−0.353904			−0.176952	−	0.984220i	$-0.556624\pi$
$751$	−9.69851			−0.353904			−0.176952	+	0.984220i	$0.556624\pi$
$752$			28.9431i			1.05545i
$753$		−	11.8992i		−	0.433630i
$754$	−9.21586			−0.335622
$755$	0.423085	+	0.261481i	0.0153976	+	0.00951627i
$756$	0.618034			0.0224777
$757$		−	42.0784i		−	1.52936i	−0.644408	−	0.764682i	$-0.722896\pi$
$757$		−	42.0784i		−	1.52936i	0.644408	−	0.764682i	$-0.277104\pi$
$758$		−	15.2769i		−	0.554882i
$759$	2.63458			0.0956291
$760$	6.16308	−	9.97208i	0.223559	−	0.361725i
$761$	−20.1288			−0.729668			−0.364834	−	0.931072i	$-0.618874\pi$
$761$	−20.1288			−0.729668			−0.364834	+	0.931072i	$0.618874\pi$
$762$			16.9751i			0.614943i
$763$		−	1.33686i		−	0.0483976i
$764$	−8.61103			−0.311536
$765$	−2.36363	+	3.82443i	−0.0854571	+	0.138273i
$766$	−2.94356			−0.106355
$767$		−	2.47027i		−	0.0891963i
$768$		−	13.5623i		−	0.489388i
$769$	−23.5408			−0.848902			−0.424451	−	0.905451i	$-0.639533\pi$
$769$	−23.5408			−0.848902			−0.424451	+	0.905451i	$0.639533\pi$
$770$	8.63052	+	5.33395i	0.311022	+	0.192222i
$771$	5.49151			0.197772
$772$		−	1.94272i		−	0.0699201i
$773$		−	9.93937i		−	0.357494i	−0.983895	−	0.178747i	$-0.942796\pi$
$773$		−	9.93937i		−	0.357494i	0.983895	−	0.178747i	$-0.0572044\pi$
$774$	8.29168			0.298038
$775$	2.00251	+	4.00501i	0.0719322	+	0.143864i
$776$	−4.43371			−0.159161
$777$		−	5.52015i		−	0.198034i
$778$		−	18.2589i		−	0.654613i
$779$	7.21586			0.258535
$780$	1.17557	+	0.726543i	0.0420922	+	0.0260144i
$781$	−34.6085			−1.23839
$782$		−	3.05644i		−	0.109298i
$783$			5.69572i			0.203548i
$784$	4.85410			0.173361
$785$	22.9252	−	37.0937i	0.818234	−	1.32393i
$786$	3.38685			0.120805
$787$			37.9659i			1.35334i	0.736287	+	0.676670i	$0.236577\pi$
$787$			37.9659i			1.35334i	−0.736287	+	0.676670i	$0.763423\pi$
$788$		−	3.15903i		−	0.112536i
$789$	2.38614			0.0849487
$790$	11.7264	−	18.9738i	0.417208	−	0.675056i
$791$	−6.79766			−0.241697
$792$			6.27044i			0.222810i
$793$			4.15838i			0.147669i
$794$	35.0560			1.24409
$795$	2.47870	+	1.53192i	0.0879104	+	0.0543316i
$796$	10.2059			0.361738
$797$			42.9372i			1.52091i	0.649389	+	0.760456i	$0.275025\pi$
$797$			42.9372i			1.52091i	−0.649389	+	0.760456i	$0.724975\pi$
$798$		−	3.79360i		−	0.134292i
$799$	−11.9886			−0.424125
$800$	−15.1246	+	7.56231i	−0.534736	+	0.267368i
$801$	−0.821412			−0.0290232
$802$		−	24.4530i		−	0.863464i
$803$			7.78048i			0.274567i
$804$	9.56701			0.337402
$805$	−1.78704	−	1.10445i	−0.0629849	−	0.0389268i
$806$	−1.44903			−0.0510398
$807$		−	18.6149i		−	0.655276i
$808$		−	17.2598i		−	0.607198i
$809$	−47.5910			−1.67321			−0.836605	−	0.547807i	$-0.815463\pi$
$809$	−47.5910			−1.67321			−0.836605	+	0.547807i	$0.815463\pi$
$810$	1.90211	−	3.07768i	0.0668334	−	0.108139i
$811$	−40.0579			−1.40662			−0.703312	−	0.710882i	$-0.748296\pi$
$811$	−40.0579			−1.40662			−0.703312	+	0.710882i	$0.748296\pi$
$812$		−	3.52015i		−	0.123533i
$813$			2.89400i			0.101497i
$814$	−25.0467			−0.877888
$815$	1.34852	−	2.18195i	0.0472366	−	0.0764305i
$816$	9.75976			0.341660
$817$		−	12.0149i		−	0.420347i
$818$			0.148803i			0.00520276i
$819$	−1.00000			−0.0349428
$820$	−3.61803	−	2.23607i	−0.126347	−	0.0780869i
$821$	−2.41652			−0.0843372			−0.0421686	−	0.999111i	$-0.513427\pi$
$821$	−2.41652			−0.0843372			−0.0421686	+	0.999111i	$0.513427\pi$
$822$		−	19.8540i		−	0.692487i
$823$		−	57.1896i		−	1.99350i	−0.0805350	−	0.996752i	$-0.525663\pi$
$823$		−	57.1896i		−	1.99350i	0.0805350	−	0.996752i	$-0.474337\pi$
$824$	43.2253			1.50582
$825$	12.5409	−	6.27044i	0.436617	−	0.218309i
$826$	3.99698			0.139073
$827$		−	5.71058i		−	0.198576i	−0.995059	−	0.0992881i	$-0.968343\pi$
$827$		−	5.71058i		−	0.198576i	0.995059	−	0.0992881i	$-0.0316566\pi$
$828$			0.580644i			0.0201788i
$829$	32.2179			1.11898			0.559488	−	0.828839i	$-0.310998\pi$
$829$	32.2179			1.11898			0.559488	+	0.828839i	$0.310998\pi$
$830$	47.9689	+	29.6464i	1.66503	+	1.02904i
$831$	−14.3261			−0.496965
$832$			4.23607i			0.146859i
$833$			2.01062i			0.0696639i
$834$	21.7687			0.753788
$835$	−20.9524	+	33.9017i	−0.725087	+	1.17322i
$836$	−4.06340			−0.140536
$837$			0.895549i			0.0309547i
$838$			25.6789i			0.887065i
$839$	−39.2649			−1.35557			−0.677787	−	0.735258i	$-0.737061\pi$
$839$	−39.2649			−1.35557			−0.677787	+	0.735258i	$0.737061\pi$
$840$	2.62866	−	4.25325i	0.0906972	−	0.146751i
$841$	3.44120			0.118662
$842$			22.0469i			0.759785i
$843$		−	5.57543i		−	0.192028i
$844$	1.52703			0.0525624
$845$	−1.90211	−	1.17557i	−0.0654347	−	0.0404409i
$846$	9.64771			0.331695
$847$		−	3.13632i		−	0.107765i
$848$		−	6.32553i		−	0.217219i
$849$	−18.6838			−0.641227
$850$	−7.27450	−	14.5490i	−0.249513	−	0.499026i
$851$	5.18619			0.177780
$852$		−	7.62750i		−	0.261314i
$853$			13.4267i			0.459722i	0.973224	+	0.229861i	$0.0738271\pi$
$853$			13.4267i			0.459722i	−0.973224	+	0.229861i	$0.926173\pi$
$854$	−6.72841			−0.230241
$855$	−4.45965	−	2.75621i	−0.152517	−	0.0942606i
$856$	17.4801			0.597459
$857$		−	48.4140i		−	1.65379i	−0.562356	−	0.826895i	$-0.690105\pi$
$857$		−	48.4140i		−	1.65379i	0.562356	−	0.826895i	$-0.309895\pi$
$858$			4.53733i			0.154902i
$859$	16.8574			0.575168			0.287584	−	0.957755i	$-0.407148\pi$
$859$	16.8574			0.575168			0.287584	+	0.957755i	$0.407148\pi$
$860$	−3.72320	+	6.02426i	−0.126960	+	0.205426i
$861$	3.07768			0.104887
$862$			41.2205i			1.40397i
$863$			47.6860i			1.62325i	0.584178	+	0.811625i	$0.301417\pi$
$863$			47.6860i			1.62325i	−0.584178	+	0.811625i	$0.698583\pi$
$864$	−3.38197			−0.115057
$865$	−2.48566	+	4.02189i	−0.0845150	+	0.136748i
$866$	33.0008			1.12141
$867$		−	12.9574i		−	0.440056i
$868$		−	0.553479i		−	0.0187863i
$869$	17.2879			0.586452
$870$	−17.5296	−	10.8339i	−0.594310	−	0.367304i
$871$	−15.4797			−0.524511
$872$			2.98931i			0.101231i
$873$			1.98281i			0.0671081i
$874$	3.56410			0.120558
$875$	−11.1352	−	1.00406i	−0.376437	−	0.0339433i
$876$	−1.71477			−0.0579367
$877$			5.74990i			0.194160i	0.995277	+	0.0970801i	$0.0309503\pi$
$877$			5.74990i			0.194160i	−0.995277	+	0.0970801i	$0.969050\pi$
$878$			36.9784i			1.24796i
$879$	31.0585			1.04758
$880$	−25.8916	−	16.0019i	−0.872804	−	0.539423i
$881$	−16.9157			−0.569905			−0.284952	−	0.958542i	$-0.591978\pi$
$881$	−16.9157			−0.569905			−0.284952	+	0.958542i	$0.591978\pi$
$882$		−	1.61803i		−	0.0544820i
$883$			8.09873i			0.272544i	0.990671	+	0.136272i	$0.0435121\pi$
$883$			8.09873i			0.272544i	−0.990671	+	0.136272i	$0.956488\pi$
$884$	1.24263			0.0417943
$885$	2.90398	−	4.69873i	0.0976161	−	0.157946i
$886$	−25.7376			−0.864671
$887$		−	44.5786i		−	1.49680i	−0.663247	−	0.748401i	$-0.730822\pi$
$887$		−	44.5786i		−	1.49680i	0.663247	−	0.748401i	$-0.269178\pi$
$888$			12.3434i			0.414218i
$889$	10.4912			0.351863
$890$	1.56242	−	2.52805i	0.0523724	−	0.0847403i
$891$	2.80423			0.0939451
$892$		−	5.62686i		−	0.188401i
$893$		−	13.9798i		−	0.467816i
$894$	20.4221			0.683019
$895$	−35.5195	−	21.9523i	−1.18729	−	0.733784i
$896$	−13.6180			−0.454947
$897$		−	0.939503i		−	0.0313691i
$898$		−	37.7119i		−	1.25846i
$899$	5.10079			0.170121
$900$	1.38197	+	2.76393i	0.0460655	+	0.0921311i
$901$	2.62010			0.0872882
$902$		−	13.9645i		−	0.464966i
$903$		−	5.12454i		−	0.170534i
$904$	15.2000			0.505546
$905$	11.5624	+	7.14597i	0.384348	+	0.237540i
$906$	−0.359898			−0.0119568
$907$		−	24.1868i		−	0.803110i	−0.915835	−	0.401555i	$-0.868470\pi$
$907$		−	24.1868i		−	0.803110i	0.915835	−	0.401555i	$-0.131530\pi$
$908$		−	4.33105i		−	0.143731i
$909$	−7.71883			−0.256017
$910$	1.90211	−	3.07768i	0.0630544	−	0.102024i
$911$	9.27522			0.307302			0.153651	−	0.988125i	$-0.450897\pi$
$911$	9.27522			0.307302			0.153651	+	0.988125i	$0.450897\pi$
$912$			11.3808i			0.376856i
$913$			43.7068i			1.44648i
$914$	29.3723			0.971548
$915$	−4.88847	+	7.90972i	−0.161608	+	0.261487i
$916$	−12.5081			−0.413278
$917$		−	2.09319i		−	0.0691231i
$918$		−	3.25325i		−	0.107373i
$919$	2.91209			0.0960611			0.0480305	−	0.998846i	$-0.484706\pi$
$919$	2.91209			0.0960611			0.0480305	+	0.998846i	$0.484706\pi$
$920$	3.99594	+	2.46963i	0.131742	+	0.0814212i
$921$	−1.72892			−0.0569698
$922$		−	49.4173i		−	1.62747i
$923$			12.3416i			0.406227i
$924$	−1.73311			−0.0570150
$925$	24.6868	−	12.3434i	0.811699	−	0.405849i
$926$	30.4513			1.00069
$927$		−	19.3309i		−	0.634911i
$928$			19.2627i			0.632330i
$929$	−35.9359			−1.17902			−0.589510	−	0.807761i	$-0.700679\pi$
$929$	−35.9359			−1.17902			−0.589510	+	0.807761i	$0.700679\pi$
$930$	−2.75621	−	1.70343i	−0.0903799	−	0.0558578i
$931$	−2.34458			−0.0768404
$932$		−	9.06965i		−	0.297086i
$933$		−	17.7385i		−	0.580733i
$934$	−23.2914			−0.762116
$935$	6.62815	−	10.7246i	0.216764	−	0.350731i
$936$	2.23607			0.0730882
$937$			25.0249i			0.817527i	0.912640	+	0.408763i	$0.134040\pi$
$937$			25.0249i			0.817527i	−0.912640	+	0.408763i	$0.865960\pi$
$938$		−	25.0467i		−	0.817805i
$939$	24.8522			0.811021
$940$	−4.33209	+	7.00947i	−0.141297	+	0.228624i
$941$	12.7170			0.414562			0.207281	−	0.978281i	$-0.433538\pi$
$941$	12.7170			0.414562			0.207281	+	0.978281i	$0.433538\pi$
$942$			31.5538i			1.02808i
$943$			2.89149i			0.0941599i
$944$	−11.9909			−0.390272
$945$	−1.90211	−	1.17557i	−0.0618757	−	0.0382413i
$946$	−23.2518			−0.755980
$947$		−	42.4461i		−	1.37931i	−0.724137	−	0.689656i	$-0.757762\pi$
$947$		−	42.4461i		−	1.37931i	0.724137	−	0.689656i	$-0.242238\pi$
$948$			3.81015i			0.123748i
$949$	2.77455			0.0900658
$950$	16.9655	−	8.48276i	0.550434	−	0.275217i
$951$	2.60626			0.0845138
$952$		−	4.49589i		−	0.145713i
$953$			44.1291i			1.42948i	0.699389	+	0.714741i	$0.253456\pi$
$953$			44.1291i			1.42948i	−0.699389	+	0.714741i	$0.746544\pi$
$954$	−2.10851			−0.0682655
$955$	26.5020	+	16.3792i	0.857585	+	0.530017i
$956$	14.8306			0.479654
$957$		−	15.9721i		−	0.516304i
$958$		−	41.0282i		−	1.32556i
$959$	−12.2704			−0.396233
$960$	−4.97980	+	8.05748i	−0.160722	+	0.260054i
$961$	−30.1980			−0.974129
$962$			8.93179i			0.287972i
$963$		−	7.81736i		−	0.251911i
$964$	−15.5930			−0.502217
$965$	−3.69528	+	5.97908i	−0.118955	+	0.192474i
$966$	1.52015			0.0489099
$967$		−	5.88712i		−	0.189317i	−0.995510	−	0.0946585i	$-0.969824\pi$
$967$		−	5.88712i		−	0.189317i	0.995510	−	0.0946585i	$-0.0301759\pi$
$968$			7.01302i			0.225407i
$969$	−4.71406			−0.151437
$970$	−6.10247	−	3.77154i	−0.195939	−	0.121097i
$971$	44.7978			1.43763			0.718814	−	0.695202i	$-0.244685\pi$
$971$	44.7978			1.43763			0.718814	+	0.695202i	$0.244685\pi$
$972$			0.618034i			0.0198234i
$973$		−	13.4538i		−	0.431309i
$974$	−10.6090			−0.339934
$975$	−2.23607	−	4.47214i	−0.0716115	−	0.143223i
$976$	20.1852			0.646113
$977$			7.32375i			0.234308i	0.993114	+	0.117154i	$0.0373770\pi$
$977$			7.32375i			0.234308i	−0.993114	+	0.117154i	$0.962623\pi$
$978$			1.85608i			0.0593509i
$979$	2.30343			0.0736178
$980$	1.17557	+	0.726543i	0.0375522	+	0.0232085i
$981$	1.33686			0.0426826
$982$			59.4997i			1.89871i
$983$			15.2176i			0.485367i	0.970106	+	0.242683i	$0.0780276\pi$
$983$			15.2176i			0.485367i	−0.970106	+	0.242683i	$0.921972\pi$
$984$	−6.88191			−0.219387
$985$	−6.00883	+	9.72249i	−0.191457	+	0.309784i
$986$	−18.5296			−0.590103
$987$		−	5.96261i		−	0.189792i
$988$			1.44903i			0.0460997i
$989$	4.81452			0.153093
$990$	−5.33395	+	8.63052i	−0.169524	+	0.274296i
$991$	1.65833			0.0526785			0.0263393	−	0.999653i	$-0.491615\pi$
$991$	1.65833			0.0526785			0.0263393	+	0.999653i	$0.491615\pi$
$992$			3.02871i			0.0961618i
$993$			30.7543i			0.975958i
$994$	−19.9691			−0.633380
$995$	−31.4105	−	19.4127i	−0.995779	−	0.615425i
$996$	−9.63271			−0.305224
$997$		−	2.49409i		−	0.0789887i	−0.999220	−	0.0394943i	$-0.987425\pi$
$997$		−	2.49409i		−	0.0789887i	0.999220	−	0.0394943i	$-0.0125747\pi$
$998$			54.7014i			1.73154i
$999$	5.52015			0.174650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1365.2.f.c.274.2	✓	8
5.2	odd	4		6825.2.a.bm.1.4		4
5.3	odd	4		6825.2.a.be.1.2		4
5.4	even	2	inner	1365.2.f.c.274.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1365.2.f.c.274.2	✓	8	1.1	even	1	trivial
1365.2.f.c.274.8	yes	8	5.4	even	2	inner
6825.2.a.be.1.2		4	5.3	odd	4
6825.2.a.bm.1.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 1365 = 3 \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1365.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} +(1.90211 + 1.17557i) q^{5} -1.61803 q^{6} +1.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +(1.90211 - 3.07768i) q^{10} +2.80423 q^{11} +0.618034i q^{12} -1.00000i q^{13} +1.61803 q^{14} +(1.17557 - 1.90211i) q^{15} -4.85410 q^{16} -2.01062i q^{17} +1.61803i q^{18} +2.34458 q^{19} +(-1.17557 - 0.726543i) q^{20} +1.00000 q^{21} -4.53733i q^{22} +0.939503i q^{23} -2.23607 q^{24} +(2.23607 + 4.47214i) q^{25} -1.61803 q^{26} +1.00000i q^{27} -0.618034i q^{28} +5.69572 q^{29} +(-3.07768 - 1.90211i) q^{30} +0.895549 q^{31} +3.38197i q^{32} -2.80423i q^{33} -3.25325 q^{34} +(-1.17557 + 1.90211i) q^{35} +0.618034 q^{36} -5.52015i q^{37} -3.79360i q^{38} -1.00000 q^{39} +(2.62866 - 4.25325i) q^{40} +3.07768 q^{41} -1.61803i q^{42} -5.12454i q^{43} -1.73311 q^{44} +(-1.90211 - 1.17557i) q^{45} +1.52015 q^{46} -5.96261i q^{47} +4.85410i q^{48} -1.00000 q^{49} +(7.23607 - 3.61803i) q^{50} -2.01062 q^{51} +0.618034i q^{52} +1.30313i q^{53} +1.61803 q^{54} +(5.33395 + 3.29657i) q^{55} +2.23607 q^{56} -2.34458i q^{57} -9.21586i q^{58} +2.47027 q^{59} +(-0.726543 + 1.17557i) q^{60} -4.15838 q^{61} -1.44903i q^{62} -1.00000i q^{63} -4.23607 q^{64} +(1.17557 - 1.90211i) q^{65} -4.53733 q^{66} -15.4797i q^{67} +1.24263i q^{68} +0.939503 q^{69} +(3.07768 + 1.90211i) q^{70} -12.3416 q^{71} +2.23607i q^{72} +2.77455i q^{73} -8.93179 q^{74} +(4.47214 - 2.23607i) q^{75} -1.44903 q^{76} +2.80423i q^{77} +1.61803i q^{78} +6.16495 q^{79} +(-9.23305 - 5.70634i) q^{80} +1.00000 q^{81} -4.97980i q^{82} +15.5861i q^{83} -0.618034 q^{84} +(2.36363 - 3.82443i) q^{85} -8.29168 q^{86} -5.69572i q^{87} -6.27044i q^{88} +0.821412 q^{89} +(-1.90211 + 3.07768i) q^{90} +1.00000 q^{91} -0.580644i q^{92} -0.895549i q^{93} -9.64771 q^{94} +(4.45965 + 2.75621i) q^{95} +3.38197 q^{96} -1.98281i q^{97} +1.61803i q^{98} -2.80423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 4 q^{6} - 8 q^{9} - 8 q^{11} + 4 q^{14} - 12 q^{16} + 4 q^{19} + 8 q^{21} - 4 q^{26} + 12 q^{29} - 4 q^{31} + 8 q^{34} - 4 q^{36} - 8 q^{39} - 4 q^{44} - 12 q^{46} - 8 q^{49} + 40 q^{50}+ \cdots + 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^4 - 4 * q^6 - 8 * q^9 - 8 * q^11 + 4 * q^14 - 12 * q^16 + 4 * q^19 + 8 * q^21 - 4 * q^26 + 12 * q^29 - 4 * q^31 + 8 * q^34 - 4 * q^36 - 8 * q^39 - 4 * q^44 - 12 * q^46 - 8 * q^49 + 40 * q^50 - 4 * q^51 + 4 * q^54 + 40 * q^55 - 8 * q^59 - 40 * q^61 - 16 * q^64 + 4 * q^66 + 16 * q^69 - 28 * q^71 - 20 * q^74 - 8 * q^76 + 52 * q^79 + 8 * q^81 + 4 * q^84 - 20 * q^85 + 8 * q^86 - 40 * q^89 + 8 * q^91 - 32 * q^94 + 20 * q^95 + 36 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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