LMFDB - Embedded newform 52.3.k.a.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,3,Mod(33,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 11]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("52.33");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 52 = 2^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	52.k (of order $12$, degree $4$, 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:	$1.41689737467$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.44991500544.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 38x^{6} + 555x^{4} - 3674x^{2} + 9409 $ x^8 - 38x^6 + 555x^4 - 3674*x^2 + 9409
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.1
Root			$-3.38852 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	52.37
Dual form			52.3.k.a.45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.12727 - 3.68454i) q^{3} +(-0.923296 - 0.923296i) q^{5} +(-2.49330 - 9.30511i) q^{7} +(-4.55057 + 7.88181i) q^{9} +O(q^{10})$	\(q+(-2.12727 - 3.68454i) q^{3} +(-0.923296 - 0.923296i) q^{5} +(-2.49330 - 9.30511i) q^{7} +(-4.55057 + 7.88181i) q^{9} +(13.5944 + 3.64262i) q^{11} +(9.39716 + 8.98295i) q^{13} +(-1.43782 + 5.36603i) q^{15} +(1.66667 + 0.962254i) q^{17} +(-21.0402 + 5.63771i) q^{19} +(-28.9812 + 28.9812i) q^{21} +(38.2759 - 22.0986i) q^{23} -23.2950i q^{25} +0.430284 q^{27} +(10.8263 + 18.7518i) q^{29} +(6.79773 + 6.79773i) q^{31} +(-15.4977 - 57.8381i) q^{33} +(-6.28932 + 10.8934i) q^{35} +(8.26562 + 2.21477i) q^{37} +(13.1077 - 53.7334i) q^{39} +(-1.93917 + 7.23710i) q^{41} +(-49.6204 - 28.6483i) q^{43} +(11.4788 - 3.07573i) q^{45} +(-43.2442 + 43.2442i) q^{47} +(-37.9333 + 21.9008i) q^{49} -8.18790i q^{51} +92.7556 q^{53} +(-9.18847 - 15.9149i) q^{55} +(65.5306 + 65.5306i) q^{57} +(20.7683 + 77.5082i) q^{59} +(-5.30128 + 9.18209i) q^{61} +(84.6871 + 22.6918i) q^{63} +(-0.382439 - 16.9703i) q^{65} +(-29.6096 + 110.505i) q^{67} +(-162.847 - 94.0195i) q^{69} +(16.6551 - 4.46272i) q^{71} +(41.3034 - 41.3034i) q^{73} +(-85.8316 + 49.5549i) q^{75} -135.580i q^{77} -10.3418 q^{79} +(40.0398 + 69.3509i) q^{81} +(24.1449 + 24.1449i) q^{83} +(-0.650387 - 2.42728i) q^{85} +(46.0611 - 79.7802i) q^{87} +(-91.0982 - 24.4097i) q^{89} +(60.1574 - 109.839i) q^{91} +(10.5859 - 39.5071i) q^{93} +(24.6316 + 14.2211i) q^{95} +(22.9090 - 6.13844i) q^{97} +(-90.5728 + 90.5728i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{5} + 4 q^{7} - 6 q^{9}+O(q^{10}) $ 8 * q + 6 * q^5 + 4 * q^7 - 6 * q^9	$ 8 q + 6 q^{5} + 4 q^{7} - 6 q^{9} + 24 q^{11} + 18 q^{13} - 60 q^{15} - 54 q^{17} - 50 q^{19} - 54 q^{21} - 24 q^{23} + 36 q^{27} + 108 q^{29} + 176 q^{31} + 114 q^{33} - 30 q^{35} + 104 q^{37} + 120 q^{39} - 168 q^{41} - 198 q^{43} + 6 q^{45} - 96 q^{47} - 162 q^{49} - 72 q^{53} + 126 q^{55} + 318 q^{57} - 18 q^{61} + 180 q^{63} - 72 q^{65} - 326 q^{67} - 498 q^{69} - 162 q^{71} - 98 q^{73} - 240 q^{75} + 192 q^{79} + 240 q^{81} + 408 q^{83} + 318 q^{85} + 36 q^{87} + 54 q^{89} + 280 q^{91} - 66 q^{93} + 312 q^{95} + 182 q^{97}+O(q^{100}) $ 8 * q + 6 * q^5 + 4 * q^7 - 6 * q^9 + 24 * q^11 + 18 * q^13 - 60 * q^15 - 54 * q^17 - 50 * q^19 - 54 * q^21 - 24 * q^23 + 36 * q^27 + 108 * q^29 + 176 * q^31 + 114 * q^33 - 30 * q^35 + 104 * q^37 + 120 * q^39 - 168 * q^41 - 198 * q^43 + 6 * q^45 - 96 * q^47 - 162 * q^49 - 72 * q^53 + 126 * q^55 + 318 * q^57 - 18 * q^61 + 180 * q^63 - 72 * q^65 - 326 * q^67 - 498 * q^69 - 162 * q^71 - 98 * q^73 - 240 * q^75 + 192 * q^79 + 240 * q^81 + 408 * q^83 + 318 * q^85 + 36 * q^87 + 54 * q^89 + 280 * q^91 - 66 * q^93 + 312 * q^95 + 182 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/52\mathbb{Z}\right)^\times$.

$n$	$27$	$41$
$\chi(n)$	$1$	$e\left(\frac{7}{12}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−2.12727	−	3.68454i	−0.709090	−	1.22818i	−0.965195	−	0.261532i	$-0.915772\pi$
$3$	−2.12727	−	3.68454i	−0.709090	−	1.22818i	0.256104	−	0.966649i	$-0.417561\pi$
$4$		0			0
$5$	−0.923296	−	0.923296i	−0.184659	−	0.184659i	0.608723	−	0.793383i	$-0.291682\pi$
$5$	−0.923296	−	0.923296i	−0.184659	−	0.184659i	−0.793383	+	0.608723i	$0.791682\pi$
$6$		0			0
$7$	−2.49330	−	9.30511i	−0.356185	−	1.32930i	−0.878986	−	0.476847i	$-0.841780\pi$
$7$	−2.49330	−	9.30511i	−0.356185	−	1.32930i	0.522801	−	0.852455i	$-0.324887\pi$
$8$		0			0
$9$	−4.55057	+	7.88181i	−0.505619	+	0.875757i
$10$		0			0
$11$	13.5944	+	3.64262i	1.23586	+	0.331147i	0.816858	−	0.576839i	$-0.195714\pi$
$11$	13.5944	+	3.64262i	1.23586	+	0.331147i	0.419000	+	0.907986i	$0.362381\pi$
$12$		0			0
$13$	9.39716	+	8.98295i	0.722858	+	0.690996i
$13$	9.39716	+	8.98295i	0.722858	+	0.690996i
$14$		0			0
$15$	−1.43782	+	5.36603i	−0.0958548	+	0.357735i
$16$		0			0
$17$	1.66667	+	0.962254i	0.0980395	+	0.0566032i	0.548218	−	0.836335i	$-0.315306\pi$
$17$	1.66667	+	0.962254i	0.0980395	+	0.0566032i	−0.450179	+	0.892939i	$0.648640\pi$
$18$		0			0
$19$	−21.0402	+	5.63771i	−1.10738	+	0.296722i	−0.765766	−	0.643119i	$-0.777640\pi$
$19$	−21.0402	+	5.63771i	−1.10738	+	0.296722i	−0.341614	+	0.939840i	$0.610973\pi$
$20$		0			0
$21$	−28.9812	+	28.9812i	−1.38005	+	1.38005i
$22$		0			0
$23$	38.2759	−	22.0986i	1.66417	−	0.960809i	0.693480	−	0.720476i	$-0.256077\pi$
$23$	38.2759	−	22.0986i	1.66417	−	0.960809i	0.970690	−	0.240333i	$-0.0772567\pi$
$24$		0			0
$25$		−	23.2950i		−	0.931802i
$26$		0			0
$27$	0.430284			0.0159365
$28$		0			0
$29$	10.8263	+	18.7518i	0.373322	+	0.646613i	0.990074	−	0.140545i	$-0.0448853\pi$
$29$	10.8263	+	18.7518i	0.373322	+	0.646613i	−0.616752	+	0.787157i	$0.711552\pi$
$30$		0			0
$31$	6.79773	+	6.79773i	0.219282	+	0.219282i	0.808196	−	0.588914i	$-0.200444\pi$
$31$	6.79773	+	6.79773i	0.219282	+	0.219282i	−0.588914	+	0.808196i	$0.700444\pi$
$32$		0			0
$33$	−15.4977	−	57.8381i	−0.469626	−	1.75267i
$34$		0			0
$35$	−6.28932	+	10.8934i	−0.179695	+	0.311241i
$36$		0			0
$37$	8.26562	+	2.21477i	0.223395	+	0.0598585i	0.368780	−	0.929517i	$-0.379775\pi$
$37$	8.26562	+	2.21477i	0.223395	+	0.0598585i	−0.145385	+	0.989375i	$0.546442\pi$
$38$		0			0
$39$	13.1077	−	53.7334i	0.336096	−	1.37778i
$40$		0			0
$41$	−1.93917	+	7.23710i	−0.0472970	+	0.176515i	−0.985534	−	0.169479i	$-0.945791\pi$
$41$	−1.93917	+	7.23710i	−0.0472970	+	0.176515i	0.938237	+	0.345994i	$0.112458\pi$
$42$		0			0
$43$	−49.6204	−	28.6483i	−1.15396	−	0.666241i	−0.204113	−	0.978947i	$-0.565431\pi$
$43$	−49.6204	−	28.6483i	−1.15396	−	0.666241i	−0.949850	+	0.312707i	$0.898764\pi$
$44$		0			0
$45$	11.4788	−	3.07573i	0.255084	−	0.0683495i
$46$		0			0
$47$	−43.2442	+	43.2442i	−0.920090	+	0.920090i	−0.997035	−	0.0769453i	$-0.975483\pi$
$47$	−43.2442	+	43.2442i	−0.920090	+	0.920090i	0.0769453	+	0.997035i	$0.475483\pi$
$48$		0			0
$49$	−37.9333	+	21.9008i	−0.774149	+	0.446955i
$50$		0			0
$51$		−	8.18790i		−	0.160547i
$52$		0			0
$53$	92.7556			1.75010			0.875052	−	0.484028i	$-0.160827\pi$
$53$	92.7556			1.75010			0.875052	+	0.484028i	$0.160827\pi$
$54$		0			0
$55$	−9.18847	−	15.9149i	−0.167063	−	0.289362i
$56$		0			0
$57$	65.5306	+	65.5306i	1.14966	+	1.14966i
$58$		0			0
$59$	20.7683	+	77.5082i	0.352004	+	1.31370i	0.884213	+	0.467084i	$0.154696\pi$
$59$	20.7683	+	77.5082i	0.352004	+	1.31370i	−0.532208	+	0.846613i	$0.678638\pi$
$60$		0			0
$61$	−5.30128	+	9.18209i	−0.0869063	+	0.150526i	−0.906202	−	0.422845i	$-0.861031\pi$
$61$	−5.30128	+	9.18209i	−0.0869063	+	0.150526i	0.819296	+	0.573371i	$0.194365\pi$
$62$		0			0
$63$	84.6871	+	22.6918i	1.34424	+	0.360188i
$64$		0			0
$65$	−0.382439	−	16.9703i	−0.00588367	−	0.261081i
$66$		0			0
$67$	−29.6096	+	110.505i	−0.441935	+	1.64932i	0.281969	+	0.959424i	$0.409012\pi$
$67$	−29.6096	+	110.505i	−0.441935	+	1.64932i	−0.723904	+	0.689901i	$0.757654\pi$
$68$		0			0
$69$	−162.847	−	94.0195i	−2.36009	−	1.36260i
$70$		0			0
$71$	16.6551	−	4.46272i	0.234579	−	0.0628552i	−0.139614	−	0.990206i	$-0.544586\pi$
$71$	16.6551	−	4.46272i	0.234579	−	0.0628552i	0.374193	+	0.927351i	$0.377920\pi$
$72$		0			0
$73$	41.3034	−	41.3034i	0.565801	−	0.565801i	−0.365149	−	0.930949i	$-0.618982\pi$
$73$	41.3034	−	41.3034i	0.565801	−	0.565801i	0.930949	+	0.365149i	$0.118982\pi$
$74$		0			0
$75$	−85.8316	+	49.5549i	−1.14442	+	0.660732i
$76$		0			0
$77$		−	135.580i		−	1.76078i
$78$		0			0
$79$	−10.3418			−0.130908			−0.0654541	−	0.997856i	$-0.520850\pi$
$79$	−10.3418			−0.130908			−0.0654541	+	0.997856i	$0.520850\pi$
$80$		0			0
$81$	40.0398	+	69.3509i	0.494318	+	0.856184i
$82$		0			0
$83$	24.1449	+	24.1449i	0.290903	+	0.290903i	0.837437	−	0.546534i	$-0.184053\pi$
$83$	24.1449	+	24.1449i	0.290903	+	0.290903i	−0.546534	+	0.837437i	$0.684053\pi$
$84$		0			0
$85$	−0.650387	−	2.42728i	−0.00765161	−	0.0285562i
$86$		0			0
$87$	46.0611	−	79.7802i	0.529438	−	0.917014i
$88$		0			0
$89$	−91.0982	−	24.4097i	−1.02358	−	0.274266i	−0.292284	−	0.956331i	$-0.594415\pi$
$89$	−91.0982	−	24.4097i	−1.02358	−	0.274266i	−0.731291	+	0.682065i	$0.761082\pi$
$90$		0			0
$91$	60.1574	−	109.839i	0.661071	−	1.20702i
$92$		0			0
$93$	10.5859	−	39.5071i	0.113827	−	0.424808i
$94$		0			0
$95$	24.6316	+	14.2211i	0.259280	+	0.149696i
$96$		0			0
$97$	22.9090	−	6.13844i	0.236175	−	0.0632829i	−0.138790	−	0.990322i	$-0.544321\pi$
$97$	22.9090	−	6.13844i	0.236175	−	0.0632829i	0.374965	+	0.927039i	$0.377655\pi$
$98$		0			0
$99$	−90.5728	+	90.5728i	−0.914877	+	0.914877i
$100$		0			0
$101$	65.6452	−	37.9003i	0.649952	−	0.375250i	−0.138486	−	0.990364i	$-0.544223\pi$
$101$	65.6452	−	37.9003i	0.649952	−	0.375250i	0.788438	+	0.615114i	$0.210890\pi$
$102$		0			0
$103$		−	33.0643i		−	0.321013i	−0.987035	−	0.160506i	$-0.948687\pi$
$103$		−	33.0643i		−	0.321013i	0.987035	−	0.160506i	$-0.0513127\pi$
$104$		0			0
$105$	53.5164			0.509680
$106$		0			0
$107$	−40.0541	−	69.3758i	−0.374338	−	0.648372i	0.615890	−	0.787832i	$-0.288797\pi$
$107$	−40.0541	−	69.3758i	−0.374338	−	0.648372i	−0.990228	+	0.139460i	$0.955463\pi$
$108$		0			0
$109$	20.7093	+	20.7093i	0.189993	+	0.189993i	0.795693	−	0.605700i	$-0.207107\pi$
$109$	20.7093	+	20.7093i	0.189993	+	0.189993i	−0.605700	+	0.795693i	$0.707107\pi$
$110$		0			0
$111$	−9.42282	−	35.1664i	−0.0848902	−	0.316815i
$112$		0			0
$113$	−67.4382	+	116.806i	−0.596798	+	1.03368i	0.396493	+	0.918038i	$0.370227\pi$
$113$	−67.4382	+	116.806i	−0.596798	+	1.03368i	−0.993290	+	0.115646i	$0.963106\pi$
$114$		0			0
$115$	−55.7436	−	14.9364i	−0.484727	−	0.129882i
$116$		0			0
$117$	−113.564	+	33.1892i	−0.970635	+	0.283668i
$118$		0			0
$119$	4.79837	−	17.9078i	0.0403224	−	0.150485i
$120$		0			0
$121$	66.7509	+	38.5386i	0.551660	+	0.318501i
$122$		0			0
$123$	30.7906	−	8.25030i	0.250330	−	0.0670756i
$124$		0			0
$125$	−44.5906	+	44.5906i	−0.356725	+	0.356725i
$126$		0			0
$127$	91.2328	−	52.6733i	0.718368	−	0.414750i	−0.0957835	−	0.995402i	$-0.530536\pi$
$127$	91.2328	−	52.6733i	0.718368	−	0.414750i	0.814152	+	0.580652i	$0.197202\pi$
$128$		0			0
$129$			243.771i			1.88970i
$130$		0			0
$131$	−113.855			−0.869124			−0.434562	−	0.900642i	$-0.643097\pi$
$131$	−113.855			−0.869124			−0.434562	+	0.900642i	$0.643097\pi$
$132$		0			0
$133$	104.919	+	181.725i	0.788865	+	1.36635i
$134$		0			0
$135$	−0.397280	−	0.397280i	−0.00294281	−	0.00294281i
$136$		0			0
$137$	−43.7644	−	163.331i	−0.319448	−	1.19220i	−0.919776	−	0.392443i	$-0.871630\pi$
$137$	−43.7644	−	163.331i	−0.319448	−	1.19220i	0.600328	−	0.799754i	$-0.295037\pi$
$138$		0			0
$139$	34.0919	−	59.0490i	0.245266	−	0.424813i	−0.716941	−	0.697134i	$-0.754458\pi$
$139$	34.0919	−	59.0490i	0.245266	−	0.424813i	0.962206	+	0.272322i	$0.0877915\pi$
$140$		0			0
$141$	251.327	+	67.3430i	1.78246	+	0.477610i
$142$		0			0
$143$	95.0276	+	156.348i	0.664529	+	1.09335i
$144$		0			0
$145$	7.31752	−	27.3094i	0.0504657	−	0.188340i
$146$		0			0
$147$	161.389	+	93.1779i	1.09788	+	0.633863i
$148$		0			0
$149$	−177.839	+	47.6517i	−1.19355	+	0.319810i	−0.800287	−	0.599616i	$-0.795320\pi$
$149$	−177.839	+	47.6517i	−1.19355	+	0.319810i	−0.393261	+	0.919427i	$0.628653\pi$
$150$		0			0
$151$	28.0405	−	28.0405i	0.185699	−	0.185699i	−0.608135	−	0.793834i	$-0.708082\pi$
$151$	28.0405	−	28.0405i	0.185699	−	0.185699i	0.793834	+	0.608135i	$0.208082\pi$
$152$		0			0
$153$	−15.1686	+	8.75760i	−0.0991412	+	0.0572392i
$154$		0			0
$155$		−	12.5526i		−	0.0809847i
$156$		0			0
$157$	−50.3259			−0.320547			−0.160274	−	0.987073i	$-0.551238\pi$
$157$	−50.3259			−0.320547			−0.160274	+	0.987073i	$0.551238\pi$
$158$		0			0
$159$	−197.316	−	341.762i	−1.24098	−	2.14945i
$160$		0			0
$161$	−301.063	−	301.063i	−1.86996	−	1.86996i
$162$		0			0
$163$	21.9331	+	81.8556i	0.134559	+	0.502181i	0.999999	+	0.00116923i	$0.000372178\pi$
$163$	21.9331	+	81.8556i	0.134559	+	0.502181i	−0.865440	+	0.501012i	$0.832961\pi$
$164$		0			0
$165$	−39.0928	+	67.7106i	−0.236926	+	0.410367i
$166$		0			0
$167$	−11.4628	−	3.07144i	−0.0686395	−	0.0183919i	0.224336	−	0.974512i	$-0.427979\pi$
$167$	−11.4628	−	3.07144i	−0.0686395	−	0.0183919i	−0.292975	+	0.956120i	$0.594645\pi$
$168$		0			0
$169$	7.61323	+	168.828i	0.0450487	+	0.998985i
$170$		0			0
$171$	51.3096	−	191.490i	0.300056	−	1.11982i
$172$		0			0
$173$	26.8268	+	15.4885i	0.155068	+	0.0895286i	0.575526	−	0.817783i	$-0.304797\pi$
$173$	26.8268	+	15.4885i	0.155068	+	0.0895286i	−0.420458	+	0.907312i	$0.638131\pi$
$174$		0			0
$175$	−216.763	+	58.0815i	−1.23865	+	0.331894i
$176$		0			0
$177$	241.402	−	241.402i	1.36386	−	1.36386i
$178$		0			0
$179$	155.258	−	89.6380i	0.867361	−	0.500771i	0.000890680	−	1.00000i	$-0.499716\pi$
$179$	155.258	−	89.6380i	0.867361	−	0.500771i	0.866470	+	0.499228i	$0.166383\pi$
$180$		0			0
$181$		−	126.588i		−	0.699378i	−0.936866	−	0.349689i	$-0.886287\pi$
$181$		−	126.588i		−	0.699378i	0.936866	−	0.349689i	$-0.113713\pi$
$182$		0			0
$183$	45.1091			0.246498
$184$		0			0
$185$	−5.58673	−	9.67650i	−0.0301985	−	0.0523054i
$186$		0			0
$187$	19.1523	+	19.1523i	0.102419	+	0.102419i
$188$		0			0
$189$	−1.07283	−	4.00384i	−0.00567633	−	0.0211844i
$190$		0			0
$191$	−45.4036	+	78.6413i	−0.237715	+	0.411735i	−0.960058	−	0.279800i	$-0.909732\pi$
$191$	−45.4036	+	78.6413i	−0.237715	+	0.411735i	0.722343	+	0.691535i	$0.243065\pi$
$192$		0			0
$193$	−280.061	−	75.0422i	−1.45110	−	0.388820i	−0.554691	−	0.832057i	$-0.687163\pi$
$193$	−280.061	−	75.0422i	−1.45110	−	0.388820i	−0.896405	+	0.443237i	$0.853830\pi$
$194$		0			0
$195$	−61.7142	+	37.5095i	−0.316483	+	0.192356i
$196$		0			0
$197$	−52.2492	+	194.997i	−0.265224	+	0.989831i	0.696888	+	0.717180i	$0.254567\pi$
$197$	−52.2492	+	194.997i	−0.265224	+	0.989831i	−0.962113	+	0.272651i	$0.912099\pi$
$198$		0			0
$199$	203.538	+	117.512i	1.02280	+	0.590515i	0.914914	−	0.403648i	$-0.132258\pi$
$199$	203.538	+	117.512i	1.02280	+	0.590515i	0.107888	+	0.994163i	$0.465591\pi$
$200$		0			0
$201$	470.147	−	125.976i	2.33904	−	0.626744i
$202$		0			0
$203$	147.494	−	147.494i	0.726572	−	0.726572i
$204$		0			0
$205$	8.47242	−	4.89155i	0.0413289	−	0.0238612i
$206$		0			0
$207$			402.245i			1.94321i
$208$		0			0
$209$	−306.566			−1.46682
$210$		0			0
$211$	60.7454	+	105.214i	0.287893	+	0.498645i	0.973307	−	0.229509i	$-0.0737119\pi$
$211$	60.7454	+	105.214i	0.287893	+	0.498645i	−0.685414	+	0.728154i	$0.740379\pi$
$212$		0			0
$213$	−51.8730	−	51.8730i	−0.243535	−	0.243535i
$214$		0			0
$215$	19.3634	+	72.2652i	0.0900624	+	0.336117i
$216$		0			0
$217$	46.3049	−	80.2024i	0.213386	−	0.369596i
$218$		0			0
$219$	−240.048	−	64.3207i	−1.09611	−	0.293702i
$220$		0			0
$221$	7.01811	+	24.0141i	0.0317562	+	0.108661i
$222$		0			0
$223$	56.0175	−	209.060i	0.251200	−	0.937489i	−0.718966	−	0.695045i	$-0.755384\pi$
$223$	56.0175	−	209.060i	0.251200	−	0.937489i	0.970165	−	0.242444i	$-0.0779491\pi$
$224$		0			0
$225$	183.607	+	106.006i	0.816032	+	0.471136i
$226$		0			0
$227$	−283.475	+	75.9570i	−1.24879	+	0.334612i	−0.821869	−	0.569677i	$-0.807068\pi$
$227$	−283.475	+	75.9570i	−1.24879	+	0.334612i	−0.426921	+	0.904289i	$0.640402\pi$
$228$		0			0
$229$	−5.31544	+	5.31544i	−0.0232115	+	0.0232115i	−0.718617	−	0.695406i	$-0.755225\pi$
$229$	−5.31544	+	5.31544i	−0.0232115	+	0.0232115i	0.695406	+	0.718617i	$0.255225\pi$
$230$		0			0
$231$	−499.550	+	288.415i	−2.16255	+	1.24855i
$232$		0			0
$233$			129.938i			0.557673i	0.960339	+	0.278836i	$0.0899487\pi$
$233$			129.938i			0.557673i	−0.960339	+	0.278836i	$0.910051\pi$
$234$		0			0
$235$	79.8545			0.339806
$236$		0			0
$237$	21.9997	+	38.1046i	0.0928258	+	0.160779i
$238$		0			0
$239$	153.226	+	153.226i	0.641111	+	0.641111i	0.950829	−	0.309718i	$-0.100235\pi$
$239$	153.226	+	153.226i	0.641111	+	0.641111i	−0.309718	+	0.950829i	$0.600235\pi$
$240$		0			0
$241$	113.403	+	423.227i	0.470553	+	1.75613i	0.637789	+	0.770211i	$0.279849\pi$
$241$	113.403	+	423.227i	0.470553	+	1.75613i	−0.167236	+	0.985917i	$0.553484\pi$
$242$		0			0
$243$	172.287	−	298.410i	0.709001	−	1.22803i
$244$		0			0
$245$	55.2446	+	14.8027i	0.225488	+	0.0604194i
$246$		0			0
$247$	−248.362	−	136.025i	−1.00551	−	0.550708i
$248$		0			0
$249$	37.6002	−	140.326i	0.151005	−	0.563558i
$250$		0			0
$251$	176.476	+	101.888i	0.703092	+	0.405930i	0.808498	−	0.588499i	$-0.200281\pi$
$251$	176.476	+	101.888i	0.703092	+	0.405930i	−0.105406	+	0.994429i	$0.533614\pi$
$252$		0			0
$253$	600.836	−	160.994i	2.37485	−	0.636338i
$254$		0			0
$255$	−7.55986	+	7.55986i	−0.0296465	+	0.0296465i
$256$		0			0
$257$	138.004	−	79.6765i	0.536980	−	0.310025i	−0.206874	−	0.978368i	$-0.566329\pi$
$257$	138.004	−	79.6765i	0.536980	−	0.310025i	0.743854	+	0.668342i	$0.232996\pi$
$258$		0			0
$259$		−	82.4346i		−	0.318280i
$260$		0			0
$261$	−197.064			−0.755034
$262$		0			0
$263$	109.486	+	189.635i	0.416296	+	0.721046i	0.995564	−	0.0940914i	$-0.0299946\pi$
$263$	109.486	+	189.635i	0.416296	+	0.721046i	−0.579267	+	0.815138i	$0.696661\pi$
$264$		0			0
$265$	−85.6409	−	85.6409i	−0.323173	−	0.323173i
$266$		0			0
$267$	103.852	+	387.581i	0.388959	+	1.45162i
$268$		0			0
$269$	136.950	−	237.205i	0.509109	−	0.881802i	−0.490836	−	0.871252i	$-0.663309\pi$
$269$	136.950	−	237.205i	0.509109	−	0.881802i	0.999944	−	0.0105498i	$-0.00335816\pi$
$270$		0			0
$271$	−231.100	−	61.9230i	−0.852767	−	0.228498i	−0.194146	−	0.980973i	$-0.562193\pi$
$271$	−231.100	−	61.9230i	−0.852767	−	0.228498i	−0.658621	+	0.752474i	$0.728860\pi$
$272$		0			0
$273$	−532.677	+	12.0043i	−1.95120	+	0.0439718i
$274$		0			0
$275$	84.8549	−	316.683i	0.308563	−	1.15157i
$276$		0			0
$277$	−393.677	−	227.289i	−1.42121	−	0.820539i	−0.424812	−	0.905282i	$-0.639660\pi$
$277$	−393.677	−	227.289i	−1.42121	−	0.820539i	−0.996403	+	0.0847428i	$0.972993\pi$
$278$		0			0
$279$	−84.5120	+	22.6449i	−0.302910	+	0.0811646i
$280$		0			0
$281$	257.440	−	257.440i	0.916157	−	0.916157i	−0.0805905	−	0.996747i	$-0.525681\pi$
$281$	257.440	−	257.440i	0.916157	−	0.916157i	0.996747	+	0.0805905i	$0.0256806\pi$
$282$		0			0
$283$	−210.263	+	121.395i	−0.742977	+	0.428958i	−0.823151	−	0.567823i	$-0.807786\pi$
$283$	−210.263	+	121.395i	−0.742977	+	0.428958i	0.0801734	+	0.996781i	$0.474453\pi$
$284$		0			0
$285$		−	121.008i		−	0.424591i
$286$		0			0
$287$	72.1770			0.251488
$288$		0			0
$289$	−142.648	−	247.074i	−0.493592	−	0.854927i
$290$		0			0
$291$	−71.3509	−	71.3509i	−0.245192	−	0.245192i
$292$		0			0
$293$	−115.688	−	431.752i	−0.394838	−	1.47356i	−0.822056	−	0.569407i	$-0.807173\pi$
$293$	−115.688	−	431.752i	−0.394838	−	1.47356i	0.427217	−	0.904149i	$-0.359494\pi$
$294$		0			0
$295$	52.3877	−	90.7382i	0.177586	−	0.307587i
$296$		0			0
$297$	5.84947	+	1.56736i	0.0196952	+	0.00527731i
$298$		0			0
$299$	558.196	+	136.166i	1.86687	+	0.455406i
$300$		0			0
$301$	−142.858	+	533.152i	−0.474610	+	1.77127i
$302$		0			0
$303$	−279.290	−	161.248i	−0.921750	−	0.532173i
$304$		0			0
$305$	13.3724	−	3.58314i	0.0438441	−	0.0117480i
$306$		0			0
$307$	294.287	−	294.287i	0.958588	−	0.958588i	−0.0405875	−	0.999176i	$-0.512923\pi$
$307$	294.287	−	294.287i	0.958588	−	0.958588i	0.999176	+	0.0405875i	$0.0129230\pi$
$308$		0			0
$309$	−121.827	+	70.3368i	−0.394262	+	0.227627i
$310$		0			0
$311$			149.387i			0.480345i	0.970730	+	0.240173i	$0.0772041\pi$
$311$			149.387i			0.480345i	−0.970730	+	0.240173i	$0.922796\pi$
$312$		0			0
$313$	−128.282			−0.409846			−0.204923	−	0.978778i	$-0.565694\pi$
$313$	−128.282			−0.409846			−0.204923	+	0.978778i	$0.565694\pi$
$314$		0			0
$315$	−57.2400	−	99.1425i	−0.181714	−	0.314738i
$316$		0			0
$317$	103.246	+	103.246i	0.325698	+	0.325698i	0.850948	−	0.525250i	$-0.176028\pi$
$317$	103.246	+	103.246i	0.325698	+	0.325698i	−0.525250	+	0.850948i	$0.676028\pi$
$318$		0			0
$319$	78.8724	+	294.356i	0.247249	+	0.922746i
$320$		0			0
$321$	−170.412	+	295.162i	−0.530878	+	0.919508i
$322$		0			0
$323$	−40.4921	−	10.8498i	−0.125362	−	0.0335908i
$324$		0			0
$325$	209.258	−	218.907i	0.643872	−	0.673561i
$326$		0			0
$327$	32.2499	−	120.358i	0.0986236	−	0.368068i
$328$		0			0
$329$	510.213	+	294.572i	1.55080	+	0.895355i
$330$		0			0
$331$	−303.909	+	81.4321i	−0.918153	+	0.246018i	−0.686796	−	0.726850i	$-0.740983\pi$
$331$	−303.909	+	81.4321i	−0.918153	+	0.246018i	−0.231358	+	0.972869i	$0.574317\pi$
$332$		0			0
$333$	−55.0696	+	55.0696i	−0.165374	+	0.165374i
$334$		0			0
$335$	129.367	−	74.6901i	0.386170	−	0.222956i
$336$		0			0
$337$		−	8.00621i		−	0.0237573i	−0.999929	−	0.0118786i	$-0.996219\pi$
$337$		−	8.00621i		−	0.0237573i	0.999929	−	0.0118786i	$-0.00378118\pi$
$338$		0			0
$339$	573.837			1.69273
$340$		0			0
$341$	67.6497	+	117.173i	0.198386	+	0.343615i
$342$		0			0
$343$	−35.4105	−	35.4105i	−0.103237	−	0.103237i
$344$		0			0
$345$	63.5477	+	237.163i	0.184196	+	0.687430i
$346$		0			0
$347$	−238.098	+	412.397i	−0.686160	+	1.18846i	0.286911	+	0.957957i	$0.407372\pi$
$347$	−238.098	+	412.397i	−0.686160	+	1.18846i	−0.973071	+	0.230507i	$0.925962\pi$
$348$		0			0
$349$	175.198	+	46.9441i	0.501999	+	0.134510i	0.500928	−	0.865489i	$-0.332992\pi$
$349$	175.198	+	46.9441i	0.501999	+	0.134510i	0.00107165	+	0.999999i	$0.499659\pi$
$350$		0			0
$351$	4.04345	+	3.86522i	0.0115198	+	0.0110120i
$352$		0			0
$353$	−98.5790	+	367.902i	−0.279261	+	1.04221i	0.673672	+	0.739031i	$0.264716\pi$
$353$	−98.5790	+	367.902i	−0.279261	+	1.04221i	−0.952932	+	0.303184i	$0.901950\pi$
$354$		0			0
$355$	−19.4980	−	11.2572i	−0.0549240	−	0.0317104i
$356$		0			0
$357$	−76.1893	+	20.4149i	−0.213415	+	0.0571845i
$358$		0			0
$359$	−456.460	+	456.460i	−1.27148	+	1.27148i	−0.326164	+	0.945313i	$0.605756\pi$
$359$	−456.460	+	456.460i	−1.27148	+	1.27148i	−0.945313	+	0.326164i	$0.894244\pi$
$360$		0			0
$361$	98.2719	−	56.7373i	0.272221	−	0.157167i
$362$		0			0
$363$		−	327.928i		−	0.903384i
$364$		0			0
$365$	−76.2706			−0.208961
$366$		0			0
$367$	−117.212	−	203.017i	−0.319379	−	0.553181i	0.660980	−	0.750404i	$-0.270141\pi$
$367$	−117.212	−	203.017i	−0.319379	−	0.553181i	−0.980359	+	0.197223i	$0.936808\pi$
$368$		0			0
$369$	−48.2171	−	48.2171i	−0.130670	−	0.130670i
$370$		0			0
$371$	−231.267	−	863.101i	−0.623362	−	2.32642i
$372$		0			0
$373$	−275.822	+	477.738i	−0.739469	+	1.28080i	0.213266	+	0.976994i	$0.431590\pi$
$373$	−275.822	+	477.738i	−0.739469	+	1.28080i	−0.952735	+	0.303804i	$0.901743\pi$
$374$		0			0
$375$	259.152	+	69.4397i	0.691073	+	0.185173i
$376$		0			0
$377$	−66.7094	+	273.466i	−0.176948	+	0.725374i
$378$		0			0
$379$	15.1389	−	56.4991i	0.0399443	−	0.149074i	−0.943074	−	0.332584i	$-0.892079\pi$
$379$	15.1389	−	56.4991i	0.0399443	−	0.149074i	0.983018	+	0.183510i	$0.0587461\pi$
$380$		0			0
$381$	−388.154	−	224.101i	−1.01878	−	0.588191i
$382$		0			0
$383$	−91.7644	+	24.5882i	−0.239594	+	0.0641990i	−0.376618	−	0.926369i	$-0.622913\pi$
$383$	−91.7644	+	24.5882i	−0.239594	+	0.0641990i	0.137024	+	0.990568i	$0.456246\pi$
$384$		0			0
$385$	−125.180	+	125.180i	−0.325144	+	0.325144i
$386$		0			0
$387$	451.602	−	260.732i	1.16693	−	0.673727i
$388$		0			0
$389$			179.334i			0.461013i	0.973071	+	0.230506i	$0.0740383\pi$
$389$			179.334i			0.461013i	−0.973071	+	0.230506i	$0.925962\pi$
$390$		0			0
$391$	85.0579			0.217539
$392$		0			0
$393$	242.201	+	419.504i	0.616287	+	1.06744i
$394$		0			0
$395$	9.54850	+	9.54850i	0.0241734	+	0.0241734i
$396$		0			0
$397$	−120.789	−	450.789i	−0.304254	−	1.13549i	−0.933586	−	0.358354i	$-0.883338\pi$
$397$	−120.789	−	450.789i	−0.304254	−	1.13549i	0.629332	−	0.777136i	$-0.283328\pi$
$398$		0			0
$399$	446.383	−	773.157i	1.11875	−	1.93774i
$400$		0			0
$401$	599.716	+	160.693i	1.49555	+	0.400732i	0.911607	−	0.411062i	$-0.134842\pi$
$401$	599.716	+	160.693i	1.49555	+	0.400732i	0.583944	+	0.811794i	$0.301509\pi$
$402$		0			0
$403$	2.81569	+	124.943i	0.00698682	+	0.310032i
$404$		0			0
$405$	27.0629	−	101.000i	0.0668219	−	0.249383i
$406$		0			0
$407$	104.299	+	60.2170i	0.256263	+	0.147953i
$408$		0			0
$409$	−250.361	+	67.0840i	−0.612129	+	0.164019i	−0.551547	−	0.834144i	$-0.685962\pi$
$409$	−250.361	+	67.0840i	−0.612129	+	0.164019i	−0.0605819	+	0.998163i	$0.519296\pi$
$410$		0			0
$411$	−508.701	+	508.701i	−1.23772	+	1.23772i
$412$		0			0
$413$	669.441	−	386.502i	1.62092	−	0.935840i
$414$		0			0
$415$		−	44.5859i		−	0.107436i
$416$		0			0
$417$	−290.091			−0.695662
$418$		0			0
$419$	−123.190	−	213.372i	−0.294011	−	0.509241i	0.680744	−	0.732522i	$-0.261657\pi$
$419$	−123.190	−	213.372i	−0.294011	−	0.509241i	−0.974754	+	0.223280i	$0.928323\pi$
$420$		0			0
$421$	68.2337	+	68.2337i	0.162075	+	0.162075i	0.783485	−	0.621410i	$-0.213440\pi$
$421$	68.2337	+	68.2337i	0.162075	+	0.162075i	−0.621410	+	0.783485i	$0.713440\pi$
$422$		0			0
$423$	−144.057	−	537.629i	−0.340561	−	1.27099i
$424$		0			0
$425$	22.4157	−	38.8252i	0.0527429	−	0.0913534i
$426$		0			0
$427$	98.6581	+	26.4353i	0.231049	+	0.0619095i
$428$		0			0
$429$	373.923	−	682.729i	0.871614	−	1.59144i
$430$		0			0
$431$	96.4525	−	359.966i	0.223788	−	0.835187i	−0.759099	−	0.650975i	$-0.774360\pi$
$431$	96.4525	−	359.966i	0.223788	−	0.835187i	0.982887	−	0.184212i	$-0.0589733\pi$
$432$		0			0
$433$	−351.782	−	203.101i	−0.812430	−	0.469057i	0.0353692	−	0.999374i	$-0.488739\pi$
$433$	−351.782	−	203.101i	−0.812430	−	0.469057i	−0.847799	+	0.530318i	$0.822073\pi$
$434$		0			0
$435$	−116.189	+	31.1327i	−0.267101	+	0.0715694i
$436$		0			0
$437$	−680.748	+	680.748i	−1.55778	+	1.55778i
$438$		0			0
$439$	−314.845	+	181.776i	−0.717186	+	0.414068i	−0.813716	−	0.581262i	$-0.802559\pi$
$439$	−314.845	+	181.776i	−0.717186	+	0.414068i	0.0965300	+	0.995330i	$0.469226\pi$
$440$		0			0
$441$		−	398.644i		−	0.903956i
$442$		0			0
$443$	−443.990			−1.00223			−0.501117	−	0.865379i	$-0.667078\pi$
$443$	−443.990			−1.00223			−0.501117	+	0.865379i	$0.667078\pi$
$444$		0			0
$445$	61.5733	+	106.648i	0.138367	+	0.239658i
$446$		0			0
$447$	553.886	+	553.886i	1.23912	+	1.23912i
$448$		0			0
$449$	−71.0846	−	265.291i	−0.158318	−	0.590849i	−0.998798	−	0.0490078i	$-0.984394\pi$
$449$	−71.0846	−	265.291i	−0.158318	−	0.590849i	0.840481	−	0.541841i	$-0.182273\pi$
$450$		0			0
$451$	−52.7240	+	91.3206i	−0.116905	+	0.202485i
$452$		0			0
$453$	−162.966	−	43.6667i	−0.359749	−	0.0963944i
$454$		0			0
$455$	−156.957	+	45.8706i	−0.344960	+	0.100814i
$456$		0			0
$457$	124.096	−	463.131i	0.271544	−	1.01342i	−0.686578	−	0.727056i	$-0.740888\pi$
$457$	124.096	−	463.131i	0.271544	−	1.01342i	0.958122	−	0.286360i	$-0.0924452\pi$
$458$		0			0
$459$	0.717143	+	0.414043i	0.00156240	+	0.000902054i
$460$		0			0
$461$	−221.281	+	59.2921i	−0.480002	+	0.128616i	−0.490704	−	0.871327i	$-0.663260\pi$
$461$	−221.281	+	59.2921i	−0.480002	+	0.128616i	0.0107014	+	0.999943i	$0.496594\pi$
$462$		0			0
$463$	−136.651	+	136.651i	−0.295142	+	0.295142i	−0.839108	−	0.543965i	$-0.816922\pi$
$463$	−136.651	+	136.651i	−0.295142	+	0.295142i	0.543965	+	0.839108i	$0.316922\pi$
$464$		0			0
$465$	−46.2507	+	26.7029i	−0.0994639	+	0.0574255i
$466$		0			0
$467$		−	578.439i		−	1.23863i	−0.785144	−	0.619314i	$-0.787411\pi$
$467$		−	578.439i		−	1.23863i	0.785144	−	0.619314i	$-0.212589\pi$
$468$		0			0
$469$	1102.08			2.34986
$470$		0			0
$471$	107.057	+	185.428i	0.227297	+	0.393690i
$472$		0			0
$473$	−570.206	−	570.206i	−1.20551	−	1.20551i
$474$		0			0
$475$	131.331	+	490.133i	0.276486	+	1.03186i
$476$		0			0
$477$	−422.090	+	731.082i	−0.884886	+	1.53267i
$478$		0			0
$479$	410.305	+	109.941i	0.856587	+	0.229522i	0.660279	−	0.751020i	$-0.270438\pi$
$479$	410.305	+	109.941i	0.856587	+	0.229522i	0.196308	+	0.980542i	$0.437105\pi$
$480$		0			0
$481$	57.7782	+	95.0621i	0.120121	+	0.197634i
$482$		0			0
$483$	−468.837	+	1749.72i	−0.970677	+	3.62262i
$484$		0			0
$485$	−26.8194	−	15.4842i	−0.0552976	−	0.0319261i
$486$		0			0
$487$	729.744	−	195.534i	1.49845	−	0.401508i	0.585868	−	0.810406i	$-0.300754\pi$
$487$	729.744	−	195.534i	1.49845	−	0.401508i	0.912580	+	0.408898i	$0.134087\pi$
$488$		0			0
$489$	254.943	−	254.943i	0.521355	−	0.521355i
$490$		0			0
$491$	379.546	−	219.131i	0.773007	−	0.446296i	−0.0609393	−	0.998141i	$-0.519410\pi$
$491$	379.546	−	219.131i	0.773007	−	0.446296i	0.833946	+	0.551846i	$0.186076\pi$
$492$		0			0
$493$			41.6707i			0.0845248i
$494$		0			0
$495$	167.251			0.337881
$496$		0			0
$497$	−83.0522	−	143.851i	−0.167107	−	0.289438i
$498$		0			0
$499$	500.722	+	500.722i	1.00345	+	1.00345i	0.999994	+	0.00345740i	$0.00110053\pi$
$499$	500.722	+	500.722i	1.00345	+	1.00345i	0.00345740	+	0.999994i	$0.498899\pi$
$500$		0			0
$501$	13.0676	+	48.7689i	0.0260830	+	0.0973432i
$502$		0			0
$503$	−428.181	+	741.632i	−0.851255	+	1.47442i	0.0288208	+	0.999585i	$0.490825\pi$
$503$	−428.181	+	741.632i	−0.851255	+	1.47442i	−0.880076	+	0.474833i	$0.842509\pi$
$504$		0			0
$505$	−95.6031	−	25.6168i	−0.189313	−	0.0507263i
$506$		0			0
$507$	605.860	−	387.195i	1.19499	−	0.763699i
$508$		0			0
$509$	−65.0313	+	242.700i	−0.127763	+	0.476817i	−0.999923	−	0.0124030i	$-0.996052\pi$
$509$	−65.0313	+	242.700i	−0.127763	+	0.476817i	0.872160	+	0.489220i	$0.162719\pi$
$510$		0			0
$511$	−487.315	−	281.351i	−0.953649	−	0.550590i
$512$		0			0
$513$	−9.05328	+	2.42582i	−0.0176477	+	0.00472869i
$514$		0			0
$515$	−30.5282	+	30.5282i	−0.0592780	+	0.0592780i
$516$		0			0
$517$	−745.403	+	430.359i	−1.44179	+	0.832415i
$518$		0			0
$519$		−	131.793i		−	0.253936i
$520$		0			0
$521$	687.675			1.31991			0.659957	−	0.751303i	$-0.270574\pi$
$521$	687.675			1.31991			0.659957	+	0.751303i	$0.270574\pi$
$522$		0			0
$523$	171.524	+	297.089i	0.327962	+	0.568047i	0.982107	−	0.188322i	$-0.0603049\pi$
$523$	171.524	+	297.089i	0.327962	+	0.568047i	−0.654145	+	0.756369i	$0.726972\pi$
$524$		0			0
$525$	675.117	+	675.117i	1.28594	+	1.28594i
$526$		0			0
$527$	4.78845	+	17.8707i	0.00908624	+	0.0339103i
$528$		0			0
$529$	712.197	−	1233.56i	1.34631	−	2.33187i
$530$		0			0
$531$	−705.412	−	189.015i	−1.32846	−	0.355960i
$532$		0			0
$533$	−83.2332	+	50.5887i	−0.156160	+	0.0949131i
$534$		0			0
$535$	−27.0726	+	101.036i	−0.0506029	+	0.188853i
$536$		0			0
$537$	−660.550	−	381.369i	−1.23007	−	0.710184i
$538$		0			0
$539$	−595.458	+	159.552i	−1.10475	+	0.296016i
$540$		0			0
$541$	311.783	−	311.783i	0.576308	−	0.576308i	−0.357576	−	0.933884i	$-0.616397\pi$
$541$	311.783	−	311.783i	0.576308	−	0.576308i	0.933884	+	0.357576i	$0.116397\pi$
$542$		0			0
$543$	−466.417	+	269.286i	−0.858963	+	0.495923i
$544$		0			0
$545$		−	38.2415i		−	0.0701680i
$546$		0			0
$547$	−835.886			−1.52813			−0.764064	−	0.645140i	$-0.776799\pi$
$547$	−835.886			−1.52813			−0.764064	+	0.645140i	$0.776799\pi$
$548$		0			0
$549$	−48.2477	−	83.5675i	−0.0878829	−	0.152218i
$550$		0			0
$551$	−333.506	−	333.506i	−0.605273	−	0.605273i
$552$		0			0
$553$	25.7851	+	96.2311i	0.0466276	+	0.174017i
$554$		0			0
$555$	−23.7690	+	41.1691i	−0.0428270	+	0.0741785i
$556$		0			0
$557$	238.419	+	63.8841i	0.428041	+	0.114693i	0.466406	−	0.884571i	$-0.345548\pi$
$557$	238.419	+	63.8841i	0.428041	+	0.114693i	−0.0383655	+	0.999264i	$0.512215\pi$
$558$		0			0
$559$	−208.944	−	714.951i	−0.373782	−	1.27898i
$560$		0			0
$561$	29.8254	−	111.310i	0.0531647	−	0.198413i
$562$		0			0
$563$	−518.673	−	299.456i	−0.921267	−	0.531894i	−0.0372279	−	0.999307i	$-0.511853\pi$
$563$	−518.673	−	299.456i	−0.921267	−	0.531894i	−0.884039	+	0.467413i	$0.845186\pi$
$564$		0			0
$565$	170.112	−	45.5814i	0.301084	−	0.0806751i
$566$		0			0
$567$	545.487	−	545.487i	0.962058	−	0.962058i
$568$		0			0
$569$	−322.009	+	185.912i	−0.565921	+	0.326735i	−0.755519	−	0.655127i	$-0.772615\pi$
$569$	−322.009	+	185.912i	−0.565921	+	0.326735i	0.189597	+	0.981862i	$0.439282\pi$
$570$		0			0
$571$		−	26.0544i		−	0.0456294i	−0.999740	−	0.0228147i	$-0.992737\pi$
$571$		−	26.0544i		−	0.0456294i	0.999740	−	0.0228147i	$-0.00726277\pi$
$572$		0			0
$573$	386.343			0.674246
$574$		0			0
$575$	−514.788	−	891.639i	−0.895284	−	1.55068i
$576$		0			0
$577$	418.760	+	418.760i	0.725754	+	0.725754i	0.969771	−	0.244017i	$-0.0784652\pi$
$577$	418.760	+	418.760i	0.725754	+	0.725754i	−0.244017	+	0.969771i	$0.578465\pi$
$578$		0			0
$579$	319.270	+	1191.53i	0.551417	+	2.05792i
$580$		0			0
$581$	164.471	−	284.872i	0.283082	−	0.490313i
$582$		0			0
$583$	1260.96	+	337.873i	2.16288	+	0.579542i
$584$		0			0
$585$	135.497	+	74.2101i	0.231619	+	0.126855i
$586$		0			0
$587$	185.349	−	691.731i	0.315756	−	1.17842i	−0.607528	−	0.794298i	$-0.707839\pi$
$587$	185.349	−	691.731i	0.315756	−	1.17842i	0.923284	−	0.384118i	$-0.125495\pi$
$588$		0			0
$589$	−181.349	−	104.702i	−0.307894	−	0.177762i
$590$		0			0
$591$	829.622	−	222.297i	1.40376	−	0.376136i
$592$		0			0
$593$	−387.924	+	387.924i	−0.654172	+	0.654172i	−0.953995	−	0.299823i	$-0.903072\pi$
$593$	−387.924	+	387.924i	−0.654172	+	0.654172i	0.299823	+	0.953995i	$0.403072\pi$
$594$		0			0
$595$	−20.9645	+	12.1038i	−0.0352344	+	0.0203426i
$596$		0			0
$597$		−	999.924i		−	1.67491i
$598$		0			0
$599$	502.437			0.838793			0.419396	−	0.907803i	$-0.362242\pi$
$599$	502.437			0.838793			0.419396	+	0.907803i	$0.362242\pi$
$600$		0			0
$601$	331.194	+	573.645i	0.551071	+	0.954484i	0.998198	+	0.0600127i	$0.0191141\pi$
$601$	331.194	+	573.645i	0.551071	+	0.954484i	−0.447126	+	0.894471i	$0.647553\pi$
$602$		0			0
$603$	−736.237	−	736.237i	−1.22096	−	1.22096i
$604$		0			0
$605$	−26.0482	−	97.2134i	−0.0430549	−	0.160683i
$606$		0			0
$607$	55.8571	−	96.7473i	0.0920215	−	0.159386i	−0.816340	−	0.577571i	$-0.804000\pi$
$607$	55.8571	−	96.7473i	0.0920215	−	0.159386i	0.908362	+	0.418186i	$0.137334\pi$
$608$		0			0
$609$	−857.208	−	229.688i	−1.40757	−	0.377156i
$610$		0			0
$611$	−794.834	+	17.9122i	−1.30087	+	0.0293162i
$612$		0			0
$613$	79.0782	−	295.124i	0.129002	−	0.481442i	−0.870949	−	0.491374i	$-0.836495\pi$
$613$	79.0782	−	295.124i	0.129002	−	0.481442i	0.999951	+	0.00993192i	$0.00316148\pi$
$614$		0			0
$615$	−36.0463	−	20.8113i	−0.0586118	−	0.0338396i
$616$		0			0
$617$	−237.311	+	63.5872i	−0.384620	+	0.103059i	−0.445948	−	0.895059i	$-0.647133\pi$
$617$	−237.311	+	63.5872i	−0.384620	+	0.103059i	0.0613275	+	0.998118i	$0.480467\pi$
$618$		0			0
$619$	−302.816	+	302.816i	−0.489201	+	0.489201i	−0.908054	−	0.418853i	$-0.862432\pi$
$619$	−302.816	+	302.816i	−0.489201	+	0.489201i	0.418853	+	0.908054i	$0.362432\pi$
$620$		0			0
$621$	16.4695	−	9.50869i	0.0265210	−	0.0153119i
$622$		0			0
$623$			908.540i			1.45833i
$624$		0			0
$625$	−500.035			−0.800057
$626$		0			0
$627$	652.149	+	1129.55i	1.04011	+	1.80152i
$628$		0			0
$629$	11.6449	+	11.6449i	0.0185134	+	0.0185134i
$630$		0			0
$631$	−134.079	−	500.391i	−0.212487	−	0.793012i	−0.987036	−	0.160498i	$-0.948690\pi$
$631$	−134.079	−	500.391i	−0.212487	−	0.793012i	0.774549	−	0.632514i	$-0.217977\pi$
$632$		0			0
$633$	258.444	−	447.638i	0.408284	−	0.707169i
$634$		0			0
$635$	−132.868	−	35.6019i	−0.209241	−	0.0560659i
$636$		0			0
$637$	−553.199	−	134.948i	−0.868445	−	0.211849i
$638$		0			0
$639$	−40.6158	+	151.580i	−0.0635616	+	0.237215i
$640$		0			0
$641$	418.401	+	241.564i	0.652732	+	0.376855i	0.789502	−	0.613748i	$-0.210339\pi$
$641$	418.401	+	241.564i	0.652732	+	0.376855i	−0.136770	+	0.990603i	$0.543672\pi$
$642$		0			0
$643$	885.346	−	237.228i	1.37690	−	0.368939i	0.506906	−	0.862002i	$-0.330789\pi$
$643$	885.346	−	237.228i	1.37690	−	0.368939i	0.869994	+	0.493063i	$0.164123\pi$
$644$		0			0
$645$	225.073	−	225.073i	0.348950	−	0.348950i
$646$		0			0
$647$	275.550	−	159.089i	0.425889	−	0.245887i	−0.271705	−	0.962381i	$-0.587587\pi$
$647$	275.550	−	159.089i	0.425889	−	0.245887i	0.697594	+	0.716493i	$0.254254\pi$
$648$		0			0
$649$			1129.33i			1.74011i
$650$		0			0
$651$	−394.012			−0.605241
$652$		0			0
$653$	−154.046	−	266.816i	−0.235905	−	0.408600i	0.723630	−	0.690188i	$-0.242472\pi$
$653$	−154.046	−	266.816i	−0.235905	−	0.408600i	−0.959535	+	0.281588i	$0.909139\pi$
$654$		0			0
$655$	105.122	+	105.122i	0.160492	+	0.160492i
$656$		0			0
$657$	137.592	+	513.500i	0.209425	+	0.781583i
$658$		0			0
$659$	−100.403	+	173.902i	−0.152356	+	0.263888i	−0.932093	−	0.362219i	$-0.882019\pi$
$659$	−100.403	+	173.902i	−0.152356	+	0.263888i	0.779737	+	0.626107i	$0.215353\pi$
$660$		0			0
$661$	−1124.54	−	301.319i	−1.70127	−	0.455853i	−0.728009	−	0.685567i	$-0.759554\pi$
$661$	−1124.54	−	301.319i	−1.70127	−	0.455853i	−0.973258	+	0.229714i	$0.926221\pi$
$662$		0			0
$663$	73.5515	−	76.9430i	0.110937	−	0.116053i
$664$		0			0
$665$	70.9147	−	264.657i	0.106639	−	0.397981i
$666$		0			0
$667$	828.776	+	478.494i	1.24254	+	0.717383i
$668$		0			0
$669$	−889.455	+	238.329i	−1.32953	+	0.356246i
$670$		0			0
$671$	−105.515	+	105.515i	−0.157250	+	0.157250i
$672$		0			0
$673$	−186.498	+	107.675i	−0.277114	+	0.159992i	−0.632116	−	0.774874i	$-0.717813\pi$
$673$	−186.498	+	107.675i	−0.277114	+	0.159992i	0.355002	+	0.934866i	$0.384480\pi$
$674$		0			0
$675$		−	10.0235i		−	0.0148496i
$676$		0			0
$677$	−152.374			−0.225072			−0.112536	−	0.993648i	$-0.535897\pi$
$677$	−152.374			−0.225072			−0.112536	+	0.993648i	$0.535897\pi$
$678$		0			0
$679$	−114.238	−	197.865i	−0.168244	−	0.291407i
$680$		0			0
$681$	882.896	+	882.896i	1.29647	+	1.29647i
$682$		0			0
$683$	−138.666	−	517.507i	−0.203024	−	0.757697i	−0.990043	−	0.140768i	$-0.955043\pi$
$683$	−138.666	−	517.507i	−0.203024	−	0.757697i	0.787018	−	0.616930i	$-0.211624\pi$
$684$		0			0
$685$	−110.395	+	191.210i	−0.161161	+	0.279139i
$686$		0			0
$687$	30.8923	+	8.27758i	0.0449670	+	0.0120489i
$688$		0			0
$689$	871.639	+	833.219i	1.26508	+	1.20932i
$690$		0			0
$691$	−243.448	+	908.562i	−0.352313	+	1.31485i	0.531519	+	0.847046i	$0.321621\pi$
$691$	−243.448	+	908.562i	−0.352313	+	1.31485i	−0.883832	+	0.467804i	$0.845045\pi$
$692$		0			0
$693$	1068.61	+	616.965i	1.54201	+	0.890282i
$694$		0			0
$695$	−85.9966	+	23.0427i	−0.123736	+	0.0331550i
$696$		0			0
$697$	−10.1959	+	10.1959i	−0.0146283	+	0.0146283i
$698$		0			0
$699$	478.761	−	276.413i	0.684923	−	0.395440i
$700$		0			0
$701$			316.887i			0.452051i	0.974121	+	0.226025i	$0.0725732\pi$
$701$			316.887i			0.452051i	−0.974121	+	0.226025i	$0.927427\pi$
$702$		0			0
$703$	−186.397			−0.265145
$704$		0			0
$705$	−169.872	−	294.227i	−0.240953	−	0.417343i
$706$		0			0
$707$	−516.339	−	516.339i	−0.730324	−	0.730324i
$708$		0			0
$709$	59.3214	+	221.391i	0.0836692	+	0.312258i	0.995059	−	0.0992869i	$-0.0316561\pi$
$709$	59.3214	+	221.391i	0.0836692	+	0.312258i	−0.911390	+	0.411544i	$0.864989\pi$
$710$		0			0
$711$	47.0608	−	81.5118i	0.0661896	−	0.114644i
$712$		0			0
$713$	410.410	+	109.969i	0.575610	+	0.154234i
$714$		0			0
$715$	56.6172	−	232.094i	0.0791849	−	0.324608i
$716$		0			0
$717$	238.614	−	890.518i	0.332794	−	1.24201i
$718$		0			0
$719$	762.823	+	440.416i	1.06095	+	0.612540i	0.925695	−	0.378271i	$-0.123481\pi$
$719$	762.823	+	440.416i	1.06095	+	0.612540i	0.135255	+	0.990811i	$0.456815\pi$
$720$		0			0
$721$	−307.667	+	82.4392i	−0.426723	+	0.114340i
$722$		0			0
$723$	1318.16	−	1318.16i	1.82318	−	1.82318i
$724$		0			0
$725$	436.823	−	252.200i	0.602515	−	0.347862i
$726$		0			0
$727$			153.689i			0.211402i	0.994398	+	0.105701i	$0.0337086\pi$
$727$			153.689i			0.211402i	−0.994398	+	0.105701i	$0.966291\pi$
$728$		0			0
$729$	−745.291			−1.02235
$730$		0			0
$731$	−55.1339	−	95.4948i	−0.0754226	−	0.130636i
$732$		0			0
$733$	−792.425	−	792.425i	−1.08107	−	1.08107i	−0.996410	−	0.0846613i	$-0.973019\pi$
$733$	−792.425	−	792.425i	−1.08107	−	1.08107i	−0.0846613	−	0.996410i	$-0.526981\pi$
$734$		0			0
$735$	−62.9789	−	235.041i	−0.0856856	−	0.319783i
$736$		0			0
$737$	−805.053	+	1394.39i	−1.09234	+	1.89198i
$738$		0			0
$739$	−845.089	−	226.441i	−1.14356	−	0.306415i	−0.363177	−	0.931720i	$-0.618308\pi$
$739$	−845.089	−	226.441i	−1.14356	−	0.306415i	−0.780381	+	0.625305i	$0.784975\pi$
$740$		0			0
$741$	27.1435	+	1204.46i	0.0366309	+	1.62545i
$742$		0			0
$743$	88.8305	−	331.520i	0.119557	−	0.446191i	−0.880031	−	0.474917i	$-0.842478\pi$
$743$	88.8305	−	331.520i	0.119557	−	0.446191i	0.999587	+	0.0287256i	$0.00914490\pi$
$744$		0			0
$745$	208.194	+	120.201i	0.279456	+	0.161344i
$746$		0			0
$747$	−300.179	+	80.4328i	−0.401846	+	0.107674i
$748$		0			0
$749$	−545.682	+	545.682i	−0.728548	+	0.728548i
$750$		0			0
$751$	468.567	−	270.527i	0.623924	−	0.360223i	−0.154471	−	0.987997i	$-0.549367\pi$
$751$	468.567	−	270.527i	0.623924	−	0.360223i	0.778395	+	0.627775i	$0.216034\pi$
$752$		0			0
$753$		−	866.978i		−	1.15136i
$754$		0			0
$755$	−51.7794			−0.0685820
$756$		0			0
$757$	−485.878	−	841.565i	−0.641846	−	1.11171i	−0.985020	−	0.172439i	$-0.944835\pi$
$757$	−485.878	−	841.565i	−0.641846	−	1.11171i	0.343174	−	0.939272i	$-0.388498\pi$
$758$		0			0
$759$	−1871.33	−	1871.33i	−2.46552	−	2.46552i
$760$		0			0
$761$	−156.736	−	584.947i	−0.205961	−	0.768655i	−0.989155	−	0.146878i	$-0.953078\pi$
$761$	−156.736	−	584.947i	−0.205961	−	0.768655i	0.783194	−	0.621777i	$-0.213589\pi$
$762$		0			0
$763$	141.068	−	244.336i	0.184885	−	0.320231i
$764$		0			0
$765$	22.0910	+	5.91926i	0.0288771	+	0.00773759i
$766$		0			0
$767$	−501.089	+	914.917i	−0.653311	+	1.19285i
$768$		0			0
$769$	216.110	−	806.535i	0.281028	−	1.04881i	−0.670665	−	0.741760i	$-0.733991\pi$
$769$	216.110	−	806.535i	0.281028	−	1.04881i	0.951693	−	0.307050i	$-0.0993420\pi$
$770$		0			0
$771$	−587.143	−	338.987i	−0.761534	−	0.439672i
$772$		0			0
$773$	657.949	−	176.297i	0.851163	−	0.228069i	0.193238	−	0.981152i	$-0.438101\pi$
$773$	657.949	−	176.297i	0.851163	−	0.228069i	0.657925	+	0.753083i	$0.271434\pi$
$774$		0			0
$775$	158.353	−	158.353i	0.204327	−	0.204327i
$776$		0			0
$777$	−303.734	+	175.361i	−0.390906	+	0.225689i
$778$		0			0
$779$		−	163.203i		−	0.209503i
$780$		0			0
$781$	242.673			0.310720
$782$		0			0
$783$	4.65840	+	8.06859i	0.00594943	+	0.0103047i
$784$		0			0
$785$	46.4657	+	46.4657i	0.0591920	+	0.0591920i
$786$		0			0
$787$	−1.29745	−	4.84215i	−0.00164860	−	0.00615267i	0.965097	−	0.261894i	$-0.0843471\pi$
$787$	−1.29745	−	4.84215i	−0.00164860	−	0.00615267i	−0.966745	+	0.255741i	$0.917680\pi$
$788$		0			0
$789$	465.813	−	806.811i	0.590383	−	1.02257i
$790$		0			0
$791$	1255.04	+	336.287i	1.58665	+	0.425141i
$792$		0			0
$793$	−132.299	+	38.6644i	−0.166834	+	0.0487572i
$794$		0			0
$795$	−133.366	+	497.729i	−0.167756	+	0.626074i
$796$		0			0
$797$	1190.55	+	687.363i	1.49379	+	0.862438i	0.999975	−	0.00713149i	$-0.00227004\pi$
$797$	1190.55	+	687.363i	1.49379	+	0.862438i	0.493811	+	0.869569i	$0.335603\pi$
$798$		0			0
$799$	−113.686	+	30.4620i	−0.142285	+	0.0381252i
$800$		0			0
$801$	606.941	−	606.941i	0.757730	−	0.757730i
$802$		0			0
$803$	711.950	−	411.044i	0.886612	−	0.511886i
$804$		0			0
$805$			555.941i			0.690610i
$806$		0			0
$807$	−1165.32			−1.44402
$808$		0			0
$809$	−524.805	−	908.990i	−0.648709	−	1.12360i	−0.983432	−	0.181280i	$-0.941976\pi$
$809$	−524.805	−	908.990i	−0.648709	−	1.12360i	0.334723	−	0.942317i	$-0.391357\pi$
$810$		0			0
$811$	405.873	+	405.873i	0.500460	+	0.500460i	0.911581	−	0.411121i	$-0.134863\pi$
$811$	405.873	+	405.873i	0.500460	+	0.500460i	−0.411121	+	0.911581i	$0.634863\pi$
$812$		0			0
$813$	263.454	+	983.224i	0.324052	+	1.20938i
$814$		0			0
$815$	55.3262	−	95.8277i	0.0678849	−	0.117580i
$816$		0			0
$817$	1205.53	+	323.022i	1.47556	+	0.395376i
$818$		0			0
$819$	591.978	+	973.978i	0.722806	+	1.18923i
$820$		0			0
$821$	59.1226	−	220.649i	0.0720129	−	0.268756i	−0.920526	−	0.390680i	$-0.872240\pi$
$821$	59.1226	−	220.649i	0.0720129	−	0.268756i	0.992539	+	0.121924i	$0.0389065\pi$
$822$		0			0
$823$	−280.482	−	161.936i	−0.340804	−	0.196763i	0.319824	−	0.947477i	$-0.396376\pi$
$823$	−280.482	−	161.936i	−0.340804	−	0.196763i	−0.660628	+	0.750714i	$0.729710\pi$
$824$		0			0
$825$	−1347.34	+	361.019i	−1.63314	+	0.437599i
$826$		0			0
$827$	887.535	−	887.535i	1.07320	−	1.07320i	0.0760981	−	0.997100i	$-0.475754\pi$
$827$	887.535	−	887.535i	1.07320	−	1.07320i	0.997100	−	0.0760981i	$-0.0242462\pi$
$828$		0			0
$829$	−1096.52	+	633.078i	−1.32271	+	0.763664i	−0.984159	−	0.177285i	$-0.943268\pi$
$829$	−1096.52	+	633.078i	−1.32271	+	0.763664i	−0.338546	+	0.940950i	$0.609935\pi$
$830$		0			0
$831$			1934.02i			2.32735i
$832$		0			0
$833$	−84.2965			−0.101196
$834$		0			0
$835$	7.74770	+	13.4194i	0.00927868	+	0.0160711i
$836$		0			0
$837$	2.92496	+	2.92496i	0.00349457	+	0.00349457i
$838$		0			0
$839$	−297.266	−	1109.41i	−0.354310	−	1.32230i	−0.881351	−	0.472463i	$-0.843365\pi$
$839$	−297.266	−	1109.41i	−0.354310	−	1.32230i	0.527040	−	0.849840i	$-0.323302\pi$
$840$		0			0
$841$	186.081	−	322.301i	0.221261	−	0.383236i
$842$		0			0
$843$	−1496.19	−	400.904i	−1.77484	−	0.475568i
$844$		0			0
$845$	148.849	−	162.908i	0.176153	−	0.192790i
$846$		0			0
$847$	192.176	−	717.212i	0.226891	−	0.846768i
$848$		0			0
$849$	894.571	+	516.481i	1.05368	+	0.608340i
$850$		0			0
$851$	365.317	−	97.8865i	0.429280	−	0.115025i
$852$		0			0
$853$	304.237	−	304.237i	0.356667	−	0.356667i	−0.505916	−	0.862583i	$-0.668845\pi$
$853$	304.237	−	304.237i	0.356667	−	0.356667i	0.862583	+	0.505916i	$0.168845\pi$
$854$		0			0
$855$	−224.176	+	129.428i	−0.262194	+	0.151378i
$856$		0			0
$857$			747.181i			0.871856i	0.899982	+	0.435928i	$0.143580\pi$
$857$			747.181i			0.871856i	−0.899982	+	0.435928i	$0.856420\pi$
$858$		0			0
$859$	425.406			0.495234			0.247617	−	0.968858i	$-0.420353\pi$
$859$	425.406			0.495234			0.247617	+	0.968858i	$0.420353\pi$
$860$		0			0
$861$	−153.540	−	265.939i	−0.178327	−	0.308872i
$862$		0			0
$863$	790.201	+	790.201i	0.915644	+	0.915644i	0.996709	−	0.0810650i	$-0.0258321\pi$
$863$	790.201	+	790.201i	0.915644	+	0.915644i	−0.0810650	+	0.996709i	$0.525832\pi$
$864$		0			0
$865$	−10.4686	−	39.0695i	−0.0121025	−	0.0451671i
$866$		0			0
$867$	−606.903	+	1051.19i	−0.700003	+	1.21244i
$868$		0			0
$869$	−140.590	−	37.6710i	−0.161784	−	0.0433499i
$870$		0			0
$871$	−1270.90	+	772.449i	−1.45913	+	0.886852i
$872$		0			0
$873$	−55.8668	+	208.498i	−0.0639940	+	0.238829i
$874$		0			0
$875$	526.098	+	303.743i	0.601255	+	0.347135i
$876$		0			0
$877$	−603.991	+	161.839i	−0.688701	+	0.184537i	−0.586164	−	0.810192i	$-0.699363\pi$
$877$	−603.991	+	161.839i	−0.688701	+	0.184537i	−0.102537	+	0.994729i	$0.532696\pi$
$878$		0			0
$879$	−1344.71	+	1344.71i	−1.52982	+	1.52982i
$880$		0			0
$881$	592.665	−	342.175i	0.672719	−	0.388394i	−0.124387	−	0.992234i	$-0.539696\pi$
$881$	592.665	−	342.175i	0.672719	−	0.388394i	0.797106	+	0.603839i	$0.206363\pi$
$882$		0			0
$883$		−	573.501i		−	0.649491i	−0.945801	−	0.324746i	$-0.894721\pi$
$883$		−	573.501i		−	0.649491i	0.945801	−	0.324746i	$-0.105279\pi$
$884$		0			0
$885$	−445.772			−0.503697
$886$		0			0
$887$	810.522	+	1403.87i	0.913779	+	1.58271i	0.808679	+	0.588251i	$0.200183\pi$
$887$	810.522	+	1403.87i	0.913779	+	1.58271i	0.105101	+	0.994462i	$0.466484\pi$
$888$		0			0
$889$	−717.601	−	717.601i	−0.807200	−	0.807200i
$890$		0			0
$891$	291.699	+	1088.64i	0.327384	+	1.22181i
$892$		0			0
$893$	666.070	−	1153.67i	0.745879	−	1.29190i
$894$		0			0
$895$	−226.111	−	60.5863i	−0.252638	−	0.0676942i
$896$		0			0
$897$	−685.723	−	2346.36i	−0.764462	−	2.61578i
$898$		0			0
$899$	−53.8749	+	201.064i	−0.0599276	+	0.223653i
$900$		0			0
$901$	154.593	+	89.2544i	0.171579	+	0.0990615i
$902$		0			0
$903$	2268.32	−	607.794i	2.51198	−	0.673083i
$904$		0			0
$905$	−116.878	+	116.878i	−0.129147	+	0.129147i
$906$		0			0
$907$	−485.987	+	280.585i	−0.535818	+	0.309355i	−0.743382	−	0.668867i	$-0.766780\pi$
$907$	−485.987	+	280.585i	−0.535818	+	0.309355i	0.207564	+	0.978221i	$0.433446\pi$
$908$		0			0
$909$			689.871i			0.758934i
$910$		0			0
$911$	−97.0104			−0.106488			−0.0532439	−	0.998582i	$-0.516956\pi$
$911$	−97.0104			−0.106488			−0.0532439	+	0.998582i	$0.516956\pi$
$912$		0			0
$913$	240.286	+	416.188i	0.263183	+	0.455846i
$914$		0			0
$915$	−41.6490	−	41.6490i	−0.0455181	−	0.0455181i
$916$		0			0
$917$	283.875	+	1059.44i	0.309569	+	1.15533i
$918$		0			0
$919$	−436.806	+	756.571i	−0.475306	+	0.823254i	−0.999600	−	0.0282832i	$-0.990996\pi$
$919$	−436.806	+	756.571i	−0.475306	+	0.823254i	0.524294	+	0.851537i	$0.324329\pi$
$920$		0			0
$921$	−1710.34	−	458.284i	−1.85705	−	0.497594i
$922$		0			0
$923$	196.599	+	107.675i	0.213000	+	0.116658i
$924$		0			0
$925$	51.5931	−	192.548i	0.0557763	−	0.208160i
$926$		0			0
$927$	260.607	+	150.461i	0.281129	+	0.162310i
$928$		0			0
$929$	844.364	−	226.247i	0.908896	−	0.243538i	0.226063	−	0.974113i	$-0.427414\pi$
$929$	844.364	−	226.247i	0.908896	−	0.243538i	0.682833	+	0.730575i	$0.260748\pi$
$930$		0			0
$931$	674.655	−	674.655i	0.724656	−	0.724656i
$932$		0			0
$933$	550.424	−	317.788i	0.589951	−	0.340608i
$934$		0			0
$935$		−	35.3666i		−	0.0378252i
$936$		0			0
$937$	1108.22			1.18274			0.591368	−	0.806402i	$-0.298588\pi$
$937$	1108.22			1.18274			0.591368	+	0.806402i	$0.298588\pi$
$938$		0			0
$939$	272.890	+	472.660i	0.290618	+	0.503365i
$940$		0			0
$941$	−541.655	−	541.655i	−0.575617	−	0.575617i	0.358076	−	0.933693i	$-0.383433\pi$
$941$	−541.655	−	541.655i	−0.575617	−	0.575617i	−0.933693	+	0.358076i	$0.883433\pi$
$942$		0			0
$943$	85.7061	+	319.860i	0.0908867	+	0.339194i
$944$		0			0
$945$	−2.70620	+	4.68727i	−0.00286370	+	0.00496007i
$946$		0			0
$947$	−353.063	−	94.6031i	−0.372823	−	0.0998976i	0.0675420	−	0.997716i	$-0.478484\pi$
$947$	−353.063	−	94.6031i	−0.372823	−	0.0998976i	−0.440365	+	0.897819i	$0.645151\pi$
$948$		0			0
$949$	759.162	−	17.1083i	0.799960	−	0.0180277i
$950$		0			0
$951$	160.782	−	600.047i	0.169066	−	0.630965i
$952$		0			0
$953$	−847.716	−	489.429i	−0.889524	−	0.513567i	−0.0157371	−	0.999876i	$-0.505009\pi$
$953$	−847.716	−	489.429i	−0.889524	−	0.513567i	−0.873787	+	0.486309i	$0.838343\pi$
$954$		0			0
$955$	114.530	−	30.6883i	0.119927	−	0.0321343i
$956$		0			0
$957$	916.784	−	916.784i	0.957977	−	0.957977i
$958$		0			0
$959$	−1410.69	+	814.465i	−1.47101	+	0.849286i
$960$		0			0
$961$		−	868.582i		−	0.903831i
$962$		0			0
$963$	729.076			0.757088
$964$		0			0
$965$	189.293	+	327.866i	0.196159	+	0.339757i
$966$		0			0
$967$	602.892	+	602.892i	0.623466	+	0.623466i	0.946416	−	0.322950i	$-0.104675\pi$
$967$	602.892	+	602.892i	0.623466	+	0.623466i	−0.322950	+	0.946416i	$0.604675\pi$
$968$		0			0
$969$	46.1610	+	172.275i	0.0476378	+	0.177787i
$970$		0			0
$971$	599.798	−	1038.88i	0.617712	−	1.06991i	−0.372191	−	0.928156i	$-0.621393\pi$
$971$	599.798	−	1038.88i	0.617712	−	1.06991i	0.989902	−	0.141752i	$-0.0452734\pi$
$972$		0			0
$973$	−634.458	−	170.003i	−0.652064	−	0.174720i
$974$		0			0
$975$	−1251.72	−	305.346i	−1.28382	−	0.313175i
$976$		0			0
$977$	−134.792	+	503.052i	−0.137966	+	0.514894i	0.862003	+	0.506904i	$0.169210\pi$
$977$	−134.792	+	503.052i	−0.137966	+	0.514894i	−0.999968	+	0.00799044i	$0.997457\pi$
$978$		0			0
$979$	−1149.51	−	663.672i	−1.17417	−	0.677908i
$980$		0			0
$981$	−257.465	+	68.9876i	−0.262452	+	0.0703238i
$982$		0			0
$983$	118.087	−	118.087i	0.120129	−	0.120129i	−0.644487	−	0.764616i	$-0.722929\pi$
$983$	118.087	−	118.087i	0.120129	−	0.120129i	0.764616	+	0.644487i	$0.222929\pi$
$984$		0			0
$985$	228.281	−	131.798i	0.231758	−	0.133805i
$986$		0			0
$987$		−	2506.54i		−	2.53955i
$988$		0			0
$989$	−2532.35			−2.56052
$990$		0			0
$991$	−381.140	−	660.154i	−0.384602	−	0.666150i	0.607112	−	0.794616i	$-0.292328\pi$
$991$	−381.140	−	660.154i	−0.384602	−	0.666150i	−0.991714	+	0.128466i	$0.958995\pi$
$992$		0			0
$993$	946.537	+	946.537i	0.953209	+	0.953209i
$994$		0			0
$995$	−79.4266	−	296.424i	−0.0798258	−	0.297914i
$996$		0			0
$997$	617.798	−	1070.06i	0.619657	−	1.07328i	−0.369892	−	0.929075i	$-0.620605\pi$
$997$	617.798	−	1070.06i	0.619657	−	1.07328i	0.989548	−	0.144202i	$-0.0460615\pi$
$998$		0			0
$999$	3.55657	+	0.952979i	0.00356013	+	0.000953933i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.3.k.a.37.1	✓	8
3.2	odd	2		468.3.cd.b.37.2		8
4.3	odd	2		208.3.bd.e.193.2		8
13.2	odd	12		676.3.g.d.577.4		8
13.3	even	3		676.3.g.d.437.4		8
13.6	odd	12	inner	52.3.k.a.45.1	yes	8
13.10	even	6		676.3.g.c.437.4		8
13.11	odd	12		676.3.g.c.577.4		8
39.32	even	12		468.3.cd.b.253.2		8
52.19	even	12		208.3.bd.e.97.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.3.k.a.37.1	✓	8	1.1	even	1	trivial
52.3.k.a.45.1	yes	8	13.6	odd	12	inner
208.3.bd.e.97.2		8	52.19	even	12
208.3.bd.e.193.2		8	4.3	odd	2
468.3.cd.b.37.2		8	3.2	odd	2
468.3.cd.b.253.2		8	39.32	even	12
676.3.g.c.437.4		8	13.10	even	6
676.3.g.c.577.4		8	13.11	odd	12
676.3.g.d.437.4		8	13.3	even	3
676.3.g.d.577.4		8	13.2	odd	12

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:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	52.k (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.12727 - 3.68454i) q^{3} +(-0.923296 - 0.923296i) q^{5} +(-2.49330 - 9.30511i) q^{7} +(-4.55057 + 7.88181i) q^{9} +O(q^{10})\)	\(q+(-2.12727 - 3.68454i) q^{3} +(-0.923296 - 0.923296i) q^{5} +(-2.49330 - 9.30511i) q^{7} +(-4.55057 + 7.88181i) q^{9} +(13.5944 + 3.64262i) q^{11} +(9.39716 + 8.98295i) q^{13} +(-1.43782 + 5.36603i) q^{15} +(1.66667 + 0.962254i) q^{17} +(-21.0402 + 5.63771i) q^{19} +(-28.9812 + 28.9812i) q^{21} +(38.2759 - 22.0986i) q^{23} -23.2950i q^{25} +0.430284 q^{27} +(10.8263 + 18.7518i) q^{29} +(6.79773 + 6.79773i) q^{31} +(-15.4977 - 57.8381i) q^{33} +(-6.28932 + 10.8934i) q^{35} +(8.26562 + 2.21477i) q^{37} +(13.1077 - 53.7334i) q^{39} +(-1.93917 + 7.23710i) q^{41} +(-49.6204 - 28.6483i) q^{43} +(11.4788 - 3.07573i) q^{45} +(-43.2442 + 43.2442i) q^{47} +(-37.9333 + 21.9008i) q^{49} -8.18790i q^{51} +92.7556 q^{53} +(-9.18847 - 15.9149i) q^{55} +(65.5306 + 65.5306i) q^{57} +(20.7683 + 77.5082i) q^{59} +(-5.30128 + 9.18209i) q^{61} +(84.6871 + 22.6918i) q^{63} +(-0.382439 - 16.9703i) q^{65} +(-29.6096 + 110.505i) q^{67} +(-162.847 - 94.0195i) q^{69} +(16.6551 - 4.46272i) q^{71} +(41.3034 - 41.3034i) q^{73} +(-85.8316 + 49.5549i) q^{75} -135.580i q^{77} -10.3418 q^{79} +(40.0398 + 69.3509i) q^{81} +(24.1449 + 24.1449i) q^{83} +(-0.650387 - 2.42728i) q^{85} +(46.0611 - 79.7802i) q^{87} +(-91.0982 - 24.4097i) q^{89} +(60.1574 - 109.839i) q^{91} +(10.5859 - 39.5071i) q^{93} +(24.6316 + 14.2211i) q^{95} +(22.9090 - 6.13844i) q^{97} +(-90.5728 + 90.5728i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{5} + 4 q^{7} - 6 q^{9}+O(q^{10}) \) 8 * q + 6 * q^5 + 4 * q^7 - 6 * q^9	\( 8 q + 6 q^{5} + 4 q^{7} - 6 q^{9} + 24 q^{11} + 18 q^{13} - 60 q^{15} - 54 q^{17} - 50 q^{19} - 54 q^{21} - 24 q^{23} + 36 q^{27} + 108 q^{29} + 176 q^{31} + 114 q^{33} - 30 q^{35} + 104 q^{37} + 120 q^{39} - 168 q^{41} - 198 q^{43} + 6 q^{45} - 96 q^{47} - 162 q^{49} - 72 q^{53} + 126 q^{55} + 318 q^{57} - 18 q^{61} + 180 q^{63} - 72 q^{65} - 326 q^{67} - 498 q^{69} - 162 q^{71} - 98 q^{73} - 240 q^{75} + 192 q^{79} + 240 q^{81} + 408 q^{83} + 318 q^{85} + 36 q^{87} + 54 q^{89} + 280 q^{91} - 66 q^{93} + 312 q^{95} + 182 q^{97}+O(q^{100}) \) 8 * q + 6 * q^5 + 4 * q^7 - 6 * q^9 + 24 * q^11 + 18 * q^13 - 60 * q^15 - 54 * q^17 - 50 * q^19 - 54 * q^21 - 24 * q^23 + 36 * q^27 + 108 * q^29 + 176 * q^31 + 114 * q^33 - 30 * q^35 + 104 * q^37 + 120 * q^39 - 168 * q^41 - 198 * q^43 + 6 * q^45 - 96 * q^47 - 162 * q^49 - 72 * q^53 + 126 * q^55 + 318 * q^57 - 18 * q^61 + 180 * q^63 - 72 * q^65 - 326 * q^67 - 498 * q^69 - 162 * q^71 - 98 * q^73 - 240 * q^75 + 192 * q^79 + 240 * q^81 + 408 * q^83 + 318 * q^85 + 36 * q^87 + 54 * q^89 + 280 * q^91 - 66 * q^93 + 312 * q^95 + 182 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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