LMFDB - Embedded newform 162.6.c.d.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,6,Mod(55,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("162.55");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not 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:	$25.9821788097$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	162.109
Dual form			162.6.c.d.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-10.5000 + 18.1865i) q^{5} +(-37.0000 - 64.0859i) q^{7} +64.0000 q^{8} +84.0000 q^{10} +(-135.000 - 233.827i) q^{11} +(57.5000 - 99.5929i) q^{13} +(-148.000 + 256.344i) q^{14} +(-128.000 - 221.703i) q^{16} -861.000 q^{17} +1850.00 q^{19} +(-168.000 - 290.985i) q^{20} +(-540.000 + 935.307i) q^{22} +(-1809.00 + 3133.28i) q^{23} +(1342.00 + 2324.41i) q^{25} -460.000 q^{26} +1184.00 q^{28} +(-562.500 - 974.279i) q^{29} +(-2614.00 + 4527.58i) q^{31} +(-512.000 + 886.810i) q^{32} +(1722.00 + 2982.59i) q^{34} +1554.00 q^{35} +9917.00 q^{37} +(-3700.00 - 6408.59i) q^{38} +(-672.000 + 1163.94i) q^{40} +(-5379.00 + 9316.70i) q^{41} +(9857.00 + 17072.8i) q^{43} +4320.00 q^{44} +14472.0 q^{46} +(-4992.00 - 8646.40i) q^{47} +(5665.50 - 9812.93i) q^{49} +(5368.00 - 9297.65i) q^{50} +(920.000 + 1593.49i) q^{52} +36726.0 q^{53} +5670.00 q^{55} +(-2368.00 - 4101.50i) q^{56} +(-2250.00 + 3897.11i) q^{58} +(-13230.0 + 22915.0i) q^{59} +(26889.5 + 46574.0i) q^{61} +20912.0 q^{62} +4096.00 q^{64} +(1207.50 + 2091.45i) q^{65} +(6467.00 - 11201.2i) q^{67} +(6888.00 - 11930.4i) q^{68} +(-3108.00 - 5383.21i) q^{70} -4254.00 q^{71} -17521.0 q^{73} +(-19834.0 - 34353.5i) q^{74} +(-14800.0 + 25634.4i) q^{76} +(-9990.00 + 17303.2i) q^{77} +(18473.0 + 31996.2i) q^{79} +5376.00 q^{80} +43032.0 q^{82} +(38208.0 + 66178.2i) q^{83} +(9040.50 - 15658.6i) q^{85} +(39428.0 - 68291.3i) q^{86} +(-8640.00 - 14964.9i) q^{88} -45357.0 q^{89} -8510.00 q^{91} +(-28944.0 - 50132.5i) q^{92} +(-19968.0 + 34585.6i) q^{94} +(-19425.0 + 33645.1i) q^{95} +(-63787.0 - 110482. i) q^{97} -45324.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 16 q^{4} - 21 q^{5} - 74 q^{7} + 128 q^{8} + 168 q^{10} - 270 q^{11} + 115 q^{13} - 296 q^{14} - 256 q^{16} - 1722 q^{17} + 3700 q^{19} - 336 q^{20} - 1080 q^{22} - 3618 q^{23} + 2684 q^{25}+ \cdots - 90648 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 - 16 * q^4 - 21 * q^5 - 74 * q^7 + 128 * q^8 + 168 * q^10 - 270 * q^11 + 115 * q^13 - 296 * q^14 - 256 * q^16 - 1722 * q^17 + 3700 * q^19 - 336 * q^20 - 1080 * q^22 - 3618 * q^23 + 2684 * q^25 - 920 * q^26 + 2368 * q^28 - 1125 * q^29 - 5228 * q^31 - 1024 * q^32 + 3444 * q^34 + 3108 * q^35 + 19834 * q^37 - 7400 * q^38 - 1344 * q^40 - 10758 * q^41 + 19714 * q^43 + 8640 * q^44 + 28944 * q^46 - 9984 * q^47 + 11331 * q^49 + 10736 * q^50 + 1840 * q^52 + 73452 * q^53 + 11340 * q^55 - 4736 * q^56 - 4500 * q^58 - 26460 * q^59 + 53779 * q^61 + 41824 * q^62 + 8192 * q^64 + 2415 * q^65 + 12934 * q^67 + 13776 * q^68 - 6216 * q^70 - 8508 * q^71 - 35042 * q^73 - 39668 * q^74 - 29600 * q^76 - 19980 * q^77 + 36946 * q^79 + 10752 * q^80 + 86064 * q^82 + 76416 * q^83 + 18081 * q^85 + 78856 * q^86 - 17280 * q^88 - 90714 * q^89 - 17020 * q^91 - 57888 * q^92 - 39936 * q^94 - 38850 * q^95 - 127574 * q^97 - 90648 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−2.00000	−	3.46410i	−0.353553	−	0.612372i
$2$	−2.00000	−	3.46410i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−8.00000	+	13.8564i	−0.250000	+	0.433013i
$5$	−10.5000	+	18.1865i	−0.187830	+	0.325331i	−0.944526	−	0.328436i	$-0.893479\pi$
$5$	−10.5000	+	18.1865i	−0.187830	+	0.325331i	0.756697	+	0.653766i	$0.226812\pi$
$6$		0			0
$7$	−37.0000	−	64.0859i	−0.285402	−	0.494330i	0.687305	−	0.726369i	$-0.258794\pi$
$7$	−37.0000	−	64.0859i	−0.285402	−	0.494330i	−0.972707	+	0.232039i	$0.925460\pi$
$8$	64.0000			0.353553
$9$		0			0
$10$	84.0000			0.265631
$11$	−135.000	−	233.827i	−0.336397	−	0.582657i	0.647355	−	0.762188i	$-0.275875\pi$
$11$	−135.000	−	233.827i	−0.336397	−	0.582657i	−0.983752	+	0.179532i	$0.942542\pi$
$12$		0			0
$13$	57.5000	−	99.5929i	0.0943647	−	0.163444i	−0.814979	−	0.579491i	$-0.803251\pi$
$13$	57.5000	−	99.5929i	0.0943647	−	0.163444i	0.909343	+	0.416047i	$0.136585\pi$
$14$	−148.000	+	256.344i	−0.201810	+	0.349544i
$15$		0			0
$16$	−128.000	−	221.703i	−0.125000	−	0.216506i
$17$	−861.000			−0.722572			−0.361286	−	0.932455i	$-0.617662\pi$
$17$	−861.000			−0.722572			−0.361286	+	0.932455i	$0.617662\pi$
$18$		0			0
$19$	1850.00			1.17568			0.587838	−	0.808979i	$-0.299979\pi$
$19$	1850.00			1.17568			0.587838	+	0.808979i	$0.299979\pi$
$20$	−168.000	−	290.985i	−0.0939149	−	0.162665i
$21$		0			0
$22$	−540.000	+	935.307i	−0.237869	+	0.412000i
$23$	−1809.00	+	3133.28i	−0.713048	+	1.23504i	0.250659	+	0.968075i	$0.419353\pi$
$23$	−1809.00	+	3133.28i	−0.713048	+	1.23504i	−0.963707	+	0.266961i	$0.913981\pi$
$24$		0			0
$25$	1342.00	+	2324.41i	0.429440	+	0.743812i
$26$	−460.000			−0.133452
$27$		0			0
$28$	1184.00			0.285402
$29$	−562.500	−	974.279i	−0.124202	−	0.215124i	0.797219	−	0.603690i	$-0.206304\pi$
$29$	−562.500	−	974.279i	−0.124202	−	0.215124i	−0.921421	+	0.388567i	$0.872970\pi$
$30$		0			0
$31$	−2614.00	+	4527.58i	−0.488541	+	0.846178i	−0.999913	−	0.0131811i	$-0.995804\pi$
$31$	−2614.00	+	4527.58i	−0.488541	+	0.846178i	0.511372	+	0.859360i	$0.329138\pi$
$32$	−512.000	+	886.810i	−0.0883883	+	0.153093i
$33$		0			0
$34$	1722.00	+	2982.59i	0.255468	+	0.442483i
$35$	1554.00			0.214428
$36$		0			0
$37$	9917.00			1.19090			0.595451	−	0.803392i	$-0.296973\pi$
$37$	9917.00			1.19090			0.595451	+	0.803392i	$0.296973\pi$
$38$	−3700.00	−	6408.59i	−0.415664	−	0.719952i
$39$		0			0
$40$	−672.000	+	1163.94i	−0.0664078	+	0.115022i
$41$	−5379.00	+	9316.70i	−0.499737	+	0.865571i	−1.00000		0.000303132i	$-0.999904\pi$
$41$	−5379.00	+	9316.70i	−0.499737	+	0.865571i	0.500262	+	0.865874i	$0.333237\pi$
$42$		0			0
$43$	9857.00	+	17072.8i	0.812968	+	1.40810i	0.910778	+	0.412897i	$0.135483\pi$
$43$	9857.00	+	17072.8i	0.812968	+	1.40810i	−0.0978093	+	0.995205i	$0.531184\pi$
$44$	4320.00			0.336397
$45$		0			0
$46$	14472.0			1.00840
$47$	−4992.00	−	8646.40i	−0.329632	−	0.570940i	0.652806	−	0.757525i	$-0.273591\pi$
$47$	−4992.00	−	8646.40i	−0.329632	−	0.570940i	−0.982439	+	0.186585i	$0.940258\pi$
$48$		0			0
$49$	5665.50	−	9812.93i	0.337092	−	0.583860i
$50$	5368.00	−	9297.65i	0.303660	−	0.525954i
$51$		0			0
$52$	920.000	+	1593.49i	0.0471823	+	0.0817222i
$53$	36726.0			1.79591			0.897954	−	0.440090i	$-0.145053\pi$
$53$	36726.0			1.79591			0.897954	+	0.440090i	$0.145053\pi$
$54$		0			0
$55$	5670.00			0.252741
$56$	−2368.00	−	4101.50i	−0.100905	−	0.174772i
$57$		0			0
$58$	−2250.00	+	3897.11i	−0.0878239	+	0.152115i
$59$	−13230.0	+	22915.0i	−0.494800	+	0.857019i	−0.999982	−	0.00599391i	$-0.998092\pi$
$59$	−13230.0	+	22915.0i	−0.494800	+	0.857019i	0.505182	+	0.863013i	$0.331425\pi$
$60$		0			0
$61$	26889.5	+	46574.0i	0.925248	+	1.60258i	0.791162	+	0.611607i	$0.209477\pi$
$61$	26889.5	+	46574.0i	0.925248	+	1.60258i	0.134086	+	0.990970i	$0.457190\pi$
$62$	20912.0			0.690902
$63$		0			0
$64$	4096.00			0.125000
$65$	1207.50	+	2091.45i	0.0354490	+	0.0613994i
$66$		0			0
$67$	6467.00	−	11201.2i	0.176001	−	0.304843i	−0.764506	−	0.644617i	$-0.777017\pi$
$67$	6467.00	−	11201.2i	0.176001	−	0.304843i	0.940507	+	0.339773i	$0.110350\pi$
$68$	6888.00	−	11930.4i	0.180643	−	0.312883i
$69$		0			0
$70$	−3108.00	−	5383.21i	−0.0758116	−	0.131310i
$71$	−4254.00			−0.100150			−0.0500751	−	0.998745i	$-0.515946\pi$
$71$	−4254.00			−0.100150			−0.0500751	+	0.998745i	$0.515946\pi$
$72$		0			0
$73$	−17521.0			−0.384815			−0.192407	−	0.981315i	$-0.561630\pi$
$73$	−17521.0			−0.384815			−0.192407	+	0.981315i	$0.561630\pi$
$74$	−19834.0	−	34353.5i	−0.421047	−	0.729276i
$75$		0			0
$76$	−14800.0	+	25634.4i	−0.293919	+	0.509083i
$77$	−9990.00	+	17303.2i	−0.192017	+	0.332582i
$78$		0			0
$79$	18473.0	+	31996.2i	0.333020	+	0.576807i	0.983102	−	0.183057i	$-0.0585992\pi$
$79$	18473.0	+	31996.2i	0.333020	+	0.576807i	−0.650083	+	0.759863i	$0.725266\pi$
$80$	5376.00			0.0939149
$81$		0			0
$82$	43032.0			0.706735
$83$	38208.0	+	66178.2i	0.608778	+	1.05443i	0.991442	+	0.130547i	$0.0416734\pi$
$83$	38208.0	+	66178.2i	0.608778	+	1.05443i	−0.382664	+	0.923888i	$0.624993\pi$
$84$		0			0
$85$	9040.50	−	15658.6i	0.135720	−	0.235075i
$86$	39428.0	−	68291.3i	0.574855	−	0.995679i
$87$		0			0
$88$	−8640.00	−	14964.9i	−0.118934	−	0.206000i
$89$	−45357.0			−0.606973			−0.303486	−	0.952836i	$-0.598151\pi$
$89$	−45357.0			−0.606973			−0.303486	+	0.952836i	$0.598151\pi$
$90$		0			0
$91$	−8510.00			−0.107727
$92$	−28944.0	−	50132.5i	−0.356524	−	0.617518i
$93$		0			0
$94$	−19968.0	+	34585.6i	−0.233085	+	0.403716i
$95$	−19425.0	+	33645.1i	−0.220827	+	0.382483i
$96$		0			0
$97$	−63787.0	−	110482.i	−0.688340	−	1.19224i	−0.972375	−	0.233425i	$-0.925007\pi$
$97$	−63787.0	−	110482.i	−0.688340	−	1.19224i	0.284035	−	0.958814i	$-0.408327\pi$
$98$	−45324.0			−0.476720
$99$		0			0
$100$	−42944.0			−0.429440
$101$	39435.0	+	68303.4i	0.384661	+	0.666253i	0.991722	−	0.128403i	$-0.0409849\pi$
$101$	39435.0	+	68303.4i	0.384661	+	0.666253i	−0.607061	+	0.794655i	$0.707652\pi$
$102$		0			0
$103$	8744.00	−	15145.1i	0.0812114	−	0.140662i	−0.822559	−	0.568680i	$-0.807454\pi$
$103$	8744.00	−	15145.1i	0.0812114	−	0.140662i	0.903771	+	0.428017i	$0.140788\pi$
$104$	3680.00	−	6373.95i	0.0333630	−	0.0577863i
$105$		0			0
$106$	−73452.0	−	127223.i	−0.634949	−	1.09976i
$107$	−134364.			−1.13455			−0.567275	−	0.823529i	$-0.692002\pi$
$107$	−134364.			−1.13455			−0.567275	+	0.823529i	$0.692002\pi$
$108$		0			0
$109$	−123487.			−0.995531			−0.497766	−	0.867312i	$-0.665846\pi$
$109$	−123487.			−0.995531			−0.497766	+	0.867312i	$0.665846\pi$
$110$	−11340.0	−	19641.5i	−0.0893576	−	0.154772i
$111$		0			0
$112$	−9472.00	+	16406.0i	−0.0713504	+	0.123583i
$113$	−87019.5	+	150722.i	−0.641092	+	1.11040i	0.344097	+	0.938934i	$0.388185\pi$
$113$	−87019.5	+	150722.i	−0.641092	+	1.11040i	−0.985189	+	0.171470i	$0.945148\pi$
$114$		0			0
$115$	−37989.0	−	65798.9i	−0.267863	−	0.463953i
$116$	18000.0			0.124202
$117$		0			0
$118$	105840.			0.699753
$119$	31857.0	+	55177.9i	0.206223	+	0.357189i
$120$		0			0
$121$	44075.5	−	76341.0i	0.273674	−	0.474018i
$122$	107558.	−	186296.i	0.654249	−	1.13319i
$123$		0			0
$124$	−41824.0	−	72441.3i	−0.244271	−	0.423089i
$125$	−121989.			−0.698306
$126$		0			0
$127$	−312982.			−1.72191			−0.860954	−	0.508682i	$-0.830133\pi$
$127$	−312982.			−1.72191			−0.860954	+	0.508682i	$0.830133\pi$
$128$	−8192.00	−	14189.0i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	4830.00	−	8365.81i	0.0250662	−	0.0434160i
$131$	52905.0	−	91634.1i	0.269351	−	0.466529i	−0.699343	−	0.714786i	$-0.746524\pi$
$131$	52905.0	−	91634.1i	0.269351	−	0.466529i	0.968694	+	0.248256i	$0.0798575\pi$
$132$		0			0
$133$	−68450.0	−	118559.i	−0.335540	−	0.581172i
$134$	−51736.0			−0.248903
$135$		0			0
$136$	−55104.0			−0.255468
$137$	142240.	+	246368.i	0.647473	+	1.12146i	0.983724	+	0.179684i	$0.0575076\pi$
$137$	142240.	+	246368.i	0.647473	+	1.12146i	−0.336251	+	0.941772i	$0.609159\pi$
$138$		0			0
$139$	81164.0	−	140580.i	0.356309	−	0.617145i	−0.631032	−	0.775757i	$-0.717368\pi$
$139$	81164.0	−	140580.i	0.356309	−	0.617145i	0.987341	+	0.158612i	$0.0507018\pi$
$140$	−12432.0	+	21532.9i	−0.0536069	+	0.0928499i
$141$		0			0
$142$	8508.00	+	14736.3i	0.0354084	+	0.0613292i
$143$	−31050.0			−0.126976
$144$		0			0
$145$	23625.0			0.0933151
$146$	35042.0	+	60694.5i	0.136053	+	0.235650i
$147$		0			0
$148$	−79336.0	+	137414.i	−0.297725	+	0.515676i
$149$	135356.	−	234443.i	0.499471	−	0.865109i	−0.500529	−	0.865720i	$-0.666861\pi$
$149$	135356.	−	234443.i	0.499471	−	0.865109i	1.00000		0.000610783i	$0.000194418\pi$
$150$		0			0
$151$	8426.00	+	14594.3i	0.0300732	+	0.0520882i	0.880670	−	0.473730i	$-0.157093\pi$
$151$	8426.00	+	14594.3i	0.0300732	+	0.0520882i	−0.850597	+	0.525818i	$0.823759\pi$
$152$	118400.			0.415664
$153$		0			0
$154$	79920.0			0.271552
$155$	−54894.0	−	95079.2i	−0.183525	−	0.317875i
$156$		0			0
$157$	−123686.	+	214231.i	−0.400473	+	0.693639i	−0.993783	−	0.111335i	$-0.964487\pi$
$157$	−123686.	+	214231.i	−0.400473	+	0.693639i	0.593310	+	0.804974i	$0.297821\pi$
$158$	73892.0	−	127985.i	0.235480	−	0.407864i
$159$		0			0
$160$	−10752.0	−	18623.0i	−0.0332039	−	0.0575109i
$161$	267732.			0.814021
$162$		0			0
$163$	−200116.			−0.589947			−0.294973	−	0.955505i	$-0.595311\pi$
$163$	−200116.			−0.589947			−0.294973	+	0.955505i	$0.595311\pi$
$164$	−86064.0	−	149067.i	−0.249869	−	0.432785i
$165$		0			0
$166$	152832.	−	264713.i	0.430471	−	0.745598i
$167$	−7059.00	+	12226.5i	−0.0195863	+	0.0339244i	−0.875652	−	0.482942i	$-0.839568\pi$
$167$	−7059.00	+	12226.5i	−0.0195863	+	0.0339244i	0.856066	+	0.516866i	$0.172902\pi$
$168$		0			0
$169$	179034.	+	310096.i	0.482191	+	0.835179i
$170$	−72324.0			−0.191938
$171$		0			0
$172$	−315424.			−0.812968
$173$	24199.5	+	41914.8i	0.0614740	+	0.106476i	0.895124	−	0.445816i	$-0.147087\pi$
$173$	24199.5	+	41914.8i	0.0614740	+	0.106476i	−0.833651	+	0.552292i	$0.813753\pi$
$174$		0			0
$175$	99308.0	−	172007.i	0.245126	−	0.424570i
$176$	−34560.0	+	59859.7i	−0.0840992	+	0.145664i
$177$		0			0
$178$	90714.0	+	157121.i	0.214597	+	0.371693i
$179$	375396.			0.875703			0.437852	−	0.899047i	$-0.355739\pi$
$179$	375396.			0.875703			0.437852	+	0.899047i	$0.355739\pi$
$180$		0			0
$181$	440702.			0.999882			0.499941	−	0.866060i	$-0.333355\pi$
$181$	440702.			0.999882			0.499941	+	0.866060i	$0.333355\pi$
$182$	17020.0	+	29479.5i	0.0380874	+	0.0659693i
$183$		0			0
$184$	−115776.	+	200530.i	−0.252101	+	0.436651i
$185$	−104128.	+	180356.i	−0.223687	+	0.387437i
$186$		0			0
$187$	116235.	+	201325.i	0.243071	+	0.421011i
$188$	159744.			0.329632
$189$		0			0
$190$	155400.			0.312296
$191$	−243489.	−	421735.i	−0.482943	−	0.836482i	0.516865	−	0.856067i	$-0.327099\pi$
$191$	−243489.	−	421735.i	−0.482943	−	0.836482i	−0.999808	+	0.0195849i	$0.993766\pi$
$192$		0			0
$193$	52092.5	−	90226.9i	0.100666	−	0.174358i	−0.811293	−	0.584639i	$-0.801236\pi$
$193$	52092.5	−	90226.9i	0.100666	−	0.174358i	0.911959	+	0.410281i	$0.134569\pi$
$194$	−255148.	+	441929.i	−0.486730	+	0.843041i
$195$		0			0
$196$	90648.0	+	157007.i	0.168546	+	0.291930i
$197$	39369.0			0.0722751			0.0361376	−	0.999347i	$-0.488495\pi$
$197$	39369.0			0.0722751			0.0361376	+	0.999347i	$0.488495\pi$
$198$		0			0
$199$	952484.			1.70500			0.852501	−	0.522725i	$-0.175085\pi$
$199$	952484.			1.70500			0.852501	+	0.522725i	$0.175085\pi$
$200$	85888.0	+	148762.i	0.151830	+	0.262977i
$201$		0			0
$202$	157740.	−	273214.i	0.271997	−	0.471112i
$203$	−41625.0	+	72096.6i	−0.0708948	+	0.122793i
$204$		0			0
$205$	−112959.	−	195651.i	−0.187731	−	0.325160i
$206$	−69952.0			−0.114850
$207$		0			0
$208$	−29440.0			−0.0471823
$209$	−249750.	−	432580.i	−0.395494	−	0.685016i
$210$		0			0
$211$	−220555.	+	382012.i	−0.341044	+	0.590706i	−0.984627	−	0.174671i	$-0.944114\pi$
$211$	−220555.	+	382012.i	−0.341044	+	0.590706i	0.643583	+	0.765377i	$0.277447\pi$
$212$	−293808.	+	508890.i	−0.448977	+	0.777651i
$213$		0			0
$214$	268728.	+	465451.i	0.401124	+	0.694767i
$215$	−413994.			−0.610798
$216$		0			0
$217$	386872.			0.557722
$218$	246974.	+	427772.i	0.351974	+	0.609636i
$219$		0			0
$220$	−45360.0	+	78565.8i	−0.0631853	+	0.109440i
$221$	−49507.5	+	85749.5i	−0.0681852	+	0.118100i
$222$		0			0
$223$	412553.	+	714563.i	0.555543	+	0.962229i	0.997861	+	0.0653703i	$0.0208229\pi$
$223$	412553.	+	714563.i	0.555543	+	0.962229i	−0.442318	+	0.896858i	$0.645844\pi$
$224$	75776.0			0.100905
$225$		0			0
$226$	696156.			0.906641
$227$	−577851.	−	1.00087e6i	−0.744305	−	1.28917i	−0.950519	−	0.310668i	$-0.899447\pi$
$227$	−577851.	−	1.00087e6i	−0.744305	−	1.28917i	0.206213	−	0.978507i	$-0.433886\pi$
$228$		0			0
$229$	−671678.	+	1.16338e6i	−0.846394	+	1.46600i	0.0380105	+	0.999277i	$0.487898\pi$
$229$	−671678.	+	1.16338e6i	−0.846394	+	1.46600i	−0.884405	+	0.466721i	$0.845435\pi$
$230$	−151956.	+	263196.i	−0.189408	+	0.328064i
$231$		0			0
$232$	−36000.0	−	62353.8i	−0.0439119	−	0.0760577i
$233$	1.07555e6			1.29790			0.648950	−	0.760831i	$-0.275208\pi$
$233$	1.07555e6			1.29790			0.648950	+	0.760831i	$0.275208\pi$
$234$		0			0
$235$	209664.			0.247659
$236$	−211680.	−	366641.i	−0.247400	−	0.428509i
$237$		0			0
$238$	127428.	−	220712.i	0.145822	−	0.252571i
$239$	−18048.0	+	31260.1i	−0.0204378	+	0.0353993i	−0.876063	−	0.482196i	$-0.839839\pi$
$239$	−18048.0	+	31260.1i	−0.0204378	+	0.0353993i	0.855626	+	0.517595i	$0.173173\pi$
$240$		0			0
$241$	−36437.5	−	63111.6i	−0.0404116	−	0.0699949i	0.845112	−	0.534589i	$-0.179534\pi$
$241$	−36437.5	−	63111.6i	−0.0404116	−	0.0699949i	−0.885524	+	0.464594i	$0.846200\pi$
$242$	−352604.			−0.387034
$243$		0			0
$244$	−860464.			−0.925248
$245$	118976.	+	206072.i	0.126632	+	0.219332i
$246$		0			0
$247$	106375.	−	184247.i	0.110942	−	0.192158i
$248$	−167296.	+	289765.i	−0.172725	+	0.299169i
$249$		0			0
$250$	243978.	+	422582.i	0.246888	+	0.427623i
$251$	−1.65869e6			−1.66181			−0.830906	−	0.556413i	$-0.812177\pi$
$251$	−1.65869e6			−1.66181			−0.830906	+	0.556413i	$0.812177\pi$
$252$		0			0
$253$	976860.			0.959469
$254$	625964.	+	1.08420e6i	0.608787	+	1.05445i
$255$		0			0
$256$	−32768.0	+	56755.8i	−0.0312500	+	0.0541266i
$257$	397090.	−	687781.i	0.375022	−	0.649557i	−0.615308	−	0.788287i	$-0.710969\pi$
$257$	397090.	−	687781.i	0.375022	−	0.649557i	0.990330	+	0.138729i	$0.0443018\pi$
$258$		0			0
$259$	−366929.	−	635540.i	−0.339885	−	0.588699i
$260$	−38640.0			−0.0354490
$261$		0			0
$262$	−423240.			−0.380920
$263$	−795396.	−	1.37767e6i	−0.709078	−	1.22816i	−0.965200	−	0.261515i	$-0.915778\pi$
$263$	−795396.	−	1.37767e6i	−0.709078	−	1.22816i	0.256121	−	0.966645i	$-0.417555\pi$
$264$		0			0
$265$	−385623.	+	667919.i	−0.337325	+	0.584264i
$266$	−273800.	+	474236.i	−0.237263	+	0.410951i
$267$		0			0
$268$	103472.	+	179219.i	0.0880006	+	0.152422i
$269$	1.57975e6			1.33109			0.665545	−	0.746358i	$-0.268199\pi$
$269$	1.57975e6			1.33109			0.665545	+	0.746358i	$0.268199\pi$
$270$		0			0
$271$	−415762.			−0.343892			−0.171946	−	0.985106i	$-0.555005\pi$
$271$	−415762.			−0.343892			−0.171946	+	0.985106i	$0.555005\pi$
$272$	110208.	+	190886.i	0.0903214	+	0.156441i
$273$		0			0
$274$	568962.	−	985471.i	0.457833	−	0.792990i
$275$	362340.	−	627591.i	0.288925	−	0.500432i
$276$		0			0
$277$	−281203.	−	487058.i	−0.220202	−	0.381400i	0.734667	−	0.678427i	$-0.237338\pi$
$277$	−281203.	−	487058.i	−0.220202	−	0.381400i	−0.954869	+	0.297027i	$0.904005\pi$
$278$	−649312.			−0.503897
$279$		0			0
$280$	99456.0			0.0758116
$281$	1.21331e6	+	2.10152e6i	0.916657	+	1.58770i	0.804457	+	0.594011i	$0.202456\pi$
$281$	1.21331e6	+	2.10152e6i	0.916657	+	1.58770i	0.112200	+	0.993686i	$0.464210\pi$
$282$		0			0
$283$	601466.	−	1.04177e6i	0.446421	−	0.773225i	−0.551729	−	0.834024i	$-0.686032\pi$
$283$	601466.	−	1.04177e6i	0.446421	−	0.773225i	0.998150	+	0.0607992i	$0.0193649\pi$
$284$	34032.0	−	58945.2i	0.0250375	−	0.0433663i
$285$		0			0
$286$	62100.0	+	107560.i	0.0448928	+	0.0777566i
$287$	796092.			0.570504
$288$		0			0
$289$	−678536.			−0.477890
$290$	−47250.0	−	81839.4i	−0.0329919	−	0.0571436i
$291$		0			0
$292$	140168.	−	242778.i	0.0962037	−	0.166630i
$293$	336236.	−	582377.i	0.228810	−	0.396310i	−0.728646	−	0.684891i	$-0.759850\pi$
$293$	336236.	−	582377.i	0.228810	−	0.396310i	0.957456	+	0.288581i	$0.0931833\pi$
$294$		0			0
$295$	−277830.	−	481216.i	−0.185876	−	0.321947i
$296$	634688.			0.421047
$297$		0			0
$298$	−1.08284e6			−0.706359
$299$	208035.	+	360327.i	0.134573	+	0.233088i
$300$		0			0
$301$	729418.	−	1.26339e6i	0.464045	−	0.803750i
$302$	33704.0	−	58377.0i	0.0212649	−	0.0368320i
$303$		0			0
$304$	−236800.	−	410150.i	−0.146960	−	0.254541i
$305$	−1.12936e6			−0.695156
$306$		0			0
$307$	−2.79488e6			−1.69245			−0.846226	−	0.532823i	$-0.821131\pi$
$307$	−2.79488e6			−1.69245			−0.846226	+	0.532823i	$0.821131\pi$
$308$	−159840.	−	276851.i	−0.0960083	−	0.166291i
$309$		0			0
$310$	−219576.	+	380317.i	−0.129772	+	0.224772i
$311$	1.52980e6	−	2.64968e6i	0.896876	−	1.55344i	0.0654112	−	0.997858i	$-0.479164\pi$
$311$	1.52980e6	−	2.64968e6i	0.896876	−	1.55344i	0.831465	−	0.555577i	$-0.187503\pi$
$312$		0			0
$313$	−1.31737e6	−	2.28175e6i	−0.760059	−	1.31646i	−0.942820	−	0.333304i	$-0.891837\pi$
$313$	−1.31737e6	−	2.28175e6i	−0.760059	−	1.31646i	0.182760	−	0.983157i	$-0.441497\pi$
$314$	989492.			0.566354
$315$		0			0
$316$	−591136.			−0.333020
$317$	−404188.	−	700075.i	−0.225910	−	0.391288i	0.730682	−	0.682718i	$-0.239202\pi$
$317$	−404188.	−	700075.i	−0.225910	−	0.391288i	−0.956592	+	0.291430i	$0.905869\pi$
$318$		0			0
$319$	−151875.	+	263055.i	−0.0835621	+	0.144734i
$320$	−43008.0	+	74492.0i	−0.0234787	+	0.0406663i
$321$		0			0
$322$	−535464.	−	927451.i	−0.287800	−	0.498484i
$323$	−1.59285e6			−0.849510
$324$		0			0
$325$	308660.			0.162096
$326$	400232.	+	693222.i	0.208578	+	0.361267i
$327$		0			0
$328$	−344256.	+	596269.i	−0.176684	+	0.306025i
$329$	−369408.	+	639833.i	−0.188155	+	0.325895i
$330$		0			0
$331$	426839.	+	739307.i	0.214138	+	0.370898i	0.953006	−	0.302953i	$-0.0979724\pi$
$331$	426839.	+	739307.i	0.214138	+	0.370898i	−0.738867	+	0.673851i	$0.764639\pi$
$332$	−1.22266e6			−0.608778
$333$		0			0
$334$	56472.0			0.0276992
$335$	135807.	+	235225.i	0.0661165	+	0.114517i
$336$		0			0
$337$	−148939.	+	257970.i	−0.0714387	+	0.123735i	−0.899532	−	0.436855i	$-0.856092\pi$
$337$	−148939.	+	257970.i	−0.0714387	+	0.123735i	0.828093	+	0.560590i	$0.189426\pi$
$338$	716136.	−	1.24038e6i	0.340960	−	0.590560i
$339$		0			0
$340$	144648.	+	250538.i	0.0678602	+	0.117537i
$341$	1.41156e6			0.657375
$342$		0			0
$343$	−2.08221e6			−0.955630
$344$	630848.	+	1.09266e6i	0.287428	+	0.497839i
$345$		0			0
$346$	96798.0	−	167659.i	0.0434686	−	0.0752899i
$347$	−44373.0	+	76856.3i	−0.0197831	+	0.0342654i	−0.875747	−	0.482769i	$-0.839631\pi$
$347$	−44373.0	+	76856.3i	−0.0197831	+	0.0342654i	0.855964	+	0.517035i	$0.172964\pi$
$348$		0			0
$349$	−1.18087e6	−	2.04533e6i	−0.518967	−	0.898877i	−0.999757	−	0.0220413i	$-0.992983\pi$
$349$	−1.18087e6	−	2.04533e6i	−0.518967	−	0.898877i	0.480790	−	0.876836i	$-0.340350\pi$
$350$	−794464.			−0.346660
$351$		0			0
$352$	276480.			0.118934
$353$	1.93038e6	+	3.34352e6i	0.824530	+	1.42813i	0.902278	+	0.431156i	$0.141894\pi$
$353$	1.93038e6	+	3.34352e6i	0.824530	+	1.42813i	−0.0777471	+	0.996973i	$0.524773\pi$
$354$		0			0
$355$	44667.0	−	77365.5i	0.0188112	−	0.0325819i
$356$	362856.	−	628485.i	0.151743	−	0.262827i
$357$		0			0
$358$	−750792.	−	1.30041e6i	−0.309608	−	0.536257i
$359$	−3.74852e6			−1.53505			−0.767527	−	0.641017i	$-0.778513\pi$
$359$	−3.74852e6			−1.53505			−0.767527	+	0.641017i	$0.778513\pi$
$360$		0			0
$361$	946401.			0.382215
$362$	−881404.	−	1.52664e6i	−0.353512	−	0.612300i
$363$		0			0
$364$	68080.0	−	117918.i	0.0269318	−	0.0466473i
$365$	183970.	−	318646.i	0.0722796	−	0.125192i
$366$		0			0
$367$	1.33170e6	+	2.30657e6i	0.516107	+	0.893924i	0.999825	+	0.0187000i	$0.00595273\pi$
$367$	1.33170e6	+	2.30657e6i	0.516107	+	0.893924i	−0.483718	+	0.875224i	$0.660714\pi$
$368$	926208.			0.356524
$369$		0			0
$370$	833028.			0.316341
$371$	−1.35886e6	−	2.35362e6i	−0.512555	−	0.887772i
$372$		0			0
$373$	−479935.	+	831272.i	−0.178612	+	0.309365i	−0.941405	−	0.337277i	$-0.890494\pi$
$373$	−479935.	+	831272.i	−0.178612	+	0.309365i	0.762793	+	0.646642i	$0.223827\pi$
$374$	464940.	−	805300.i	0.171877	−	0.297700i
$375$		0			0
$376$	−319488.	−	553369.i	−0.116543	−	0.201858i
$377$	−129375.			−0.0468810
$378$		0			0
$379$	−193780.			−0.0692964			−0.0346482	−	0.999400i	$-0.511031\pi$
$379$	−193780.			−0.0692964			−0.0346482	+	0.999400i	$0.511031\pi$
$380$	−310800.	−	538321.i	−0.110413	−	0.191242i
$381$		0			0
$382$	−973956.	+	1.68694e6i	−0.341492	+	0.591482i
$383$	−2.77696e6	+	4.80984e6i	−0.967327	+	1.67546i	−0.264100	+	0.964495i	$0.585075\pi$
$383$	−2.77696e6	+	4.80984e6i	−0.967327	+	1.67546i	−0.703228	+	0.710965i	$0.748259\pi$
$384$		0			0
$385$	−209790.	−	363367.i	−0.0721328	−	0.124938i
$386$	−416740.			−0.142363
$387$		0			0
$388$	2.04118e6			0.688340
$389$	2.41398e6	+	4.18113e6i	0.808833	+	1.40094i	0.913672	+	0.406451i	$0.133234\pi$
$389$	2.41398e6	+	4.18113e6i	0.808833	+	1.40094i	−0.104839	+	0.994489i	$0.533433\pi$
$390$		0			0
$391$	1.55755e6	−	2.69775e6i	0.515228	−	0.892402i
$392$	362592.	−	628028.i	0.119180	−	0.206426i
$393$		0			0
$394$	−78738.0	−	136378.i	−0.0255531	−	0.0442593i
$395$	−775866.			−0.250204
$396$		0			0
$397$	313409.			0.0998011			0.0499005	−	0.998754i	$-0.484110\pi$
$397$	313409.			0.0998011			0.0499005	+	0.998754i	$0.484110\pi$
$398$	−1.90497e6	−	3.29950e6i	−0.602809	−	1.04410i
$399$		0			0
$400$	343552.	−	595050.i	0.107360	−	0.185953i
$401$	−2.64145e6	+	4.57512e6i	−0.820316	+	1.42083i	0.0851315	+	0.996370i	$0.472869\pi$
$401$	−2.64145e6	+	4.57512e6i	−0.820316	+	1.42083i	−0.905447	+	0.424459i	$0.860464\pi$
$402$		0			0
$403$	300610.	+	520672.i	0.0922021	+	0.159699i
$404$	−1.26192e6			−0.384661
$405$		0			0
$406$	333000.			0.100260
$407$	−1.33880e6	−	2.31886e6i	−0.400616	−	0.693887i
$408$		0			0
$409$	−2.29406e6	+	3.97342e6i	−0.678104	+	1.17451i	0.297448	+	0.954738i	$0.403865\pi$
$409$	−2.29406e6	+	3.97342e6i	−0.678104	+	1.17451i	−0.975551	+	0.219772i	$0.929469\pi$
$410$	−451836.	+	782603.i	−0.132746	+	0.229923i
$411$		0			0
$412$	139904.	+	242321.i	0.0406057	+	0.0703312i
$413$	1.95804e6			0.564867
$414$		0			0
$415$	−1.60474e6			−0.457387
$416$	58880.0	+	101983.i	0.0166815	+	0.0288932i
$417$		0			0
$418$	−999000.	+	1.73032e6i	−0.279656	+	0.484379i
$419$	1.18349e6	−	2.04986e6i	0.329328	−	0.570413i	−0.653051	−	0.757314i	$-0.726511\pi$
$419$	1.18349e6	−	2.04986e6i	0.329328	−	0.570413i	0.982379	+	0.186901i	$0.0598444\pi$
$420$		0			0
$421$	227398.	+	393864.i	0.0625289	+	0.108303i	0.895595	−	0.444870i	$-0.146750\pi$
$421$	227398.	+	393864.i	0.0625289	+	0.108303i	−0.833066	+	0.553173i	$0.813417\pi$
$422$	1.76444e6			0.482309
$423$		0			0
$424$	2.35046e6			0.634949
$425$	−1.15546e6	−	2.00132e6i	−0.310301	−	0.537457i
$426$		0			0
$427$	1.98982e6	−	3.44647e6i	0.528135	−	0.914756i
$428$	1.07491e6	−	1.86180e6i	0.283637	−	0.491274i
$429$		0			0
$430$	827988.	+	1.43412e6i	0.215950	+	0.374036i
$431$	1.26286e6			0.327462			0.163731	−	0.986505i	$-0.447647\pi$
$431$	1.26286e6			0.327462			0.163731	+	0.986505i	$0.447647\pi$
$432$		0			0
$433$	5.48900e6			1.40693			0.703467	−	0.710728i	$-0.251634\pi$
$433$	5.48900e6			1.40693			0.703467	+	0.710728i	$0.251634\pi$
$434$	−773744.	−	1.34016e6i	−0.197185	−	0.341534i
$435$		0			0
$436$	987896.	−	1.71109e6i	0.248883	−	0.431078i
$437$	−3.34665e6	+	5.79657e6i	−0.838314	+	1.45200i
$438$		0			0
$439$	3.05822e6	+	5.29699e6i	0.757369	+	1.31180i	0.944188	+	0.329407i	$0.106849\pi$
$439$	3.05822e6	+	5.29699e6i	0.757369	+	1.31180i	−0.186819	+	0.982394i	$0.559818\pi$
$440$	362880.			0.0893576
$441$		0			0
$442$	396060.			0.0964285
$443$	−369600.	−	640166.i	−0.0894793	−	0.154983i	0.817812	−	0.575486i	$-0.195187\pi$
$443$	−369600.	−	640166.i	−0.0894793	−	0.154983i	−0.907291	+	0.420503i	$0.861854\pi$
$444$		0			0
$445$	476249.	−	824887.i	0.114008	−	0.197467i
$446$	1.65021e6	−	2.85825e6i	0.392828	−	0.680398i
$447$		0			0
$448$	−151552.	−	262496.i	−0.0356752	−	0.0617913i
$449$	7.31432e6			1.71221			0.856107	−	0.516799i	$-0.172876\pi$
$449$	7.31432e6			1.71221			0.856107	+	0.516799i	$0.172876\pi$
$450$		0			0
$451$	2.90466e6			0.672441
$452$	−1.39231e6	−	2.41156e6i	−0.320546	−	0.555202i
$453$		0			0
$454$	−2.31140e6	+	4.00347e6i	−0.526303	+	0.911584i
$455$	89355.0	−	154767.i	0.0202344	−	0.0350470i
$456$		0			0
$457$	3.84452e6	+	6.65890e6i	0.861096	+	1.49146i	0.870872	+	0.491509i	$0.163555\pi$
$457$	3.84452e6	+	6.65890e6i	0.861096	+	1.49146i	−0.00977649	+	0.999952i	$0.503112\pi$
$458$	5.37343e6			1.19698
$459$		0			0
$460$	1.21565e6			0.267863
$461$	2.96968e6	+	5.14365e6i	0.650816	+	1.12725i	0.982925	+	0.184005i	$0.0589063\pi$
$461$	2.96968e6	+	5.14365e6i	0.650816	+	1.12725i	−0.332110	+	0.943241i	$0.607760\pi$
$462$		0			0
$463$	227978.	−	394869.i	0.0494243	−	0.0856054i	−0.840255	−	0.542192i	$-0.817595\pi$
$463$	227978.	−	394869.i	0.0494243	−	0.0856054i	0.889679	+	0.456586i	$0.150928\pi$
$464$	−144000.	+	249415.i	−0.0310504	+	0.0537809i
$465$		0			0
$466$	−2.15110e6	−	3.72582e6i	−0.458877	−	0.794798i
$467$	−5.97097e6			−1.26693			−0.633465	−	0.773771i	$-0.718368\pi$
$467$	−5.97097e6			−1.26693			−0.633465	+	0.773771i	$0.718368\pi$
$468$		0			0
$469$	−957116.			−0.200924
$470$	−419328.	−	726297.i	−0.0875607	−	0.151660i
$471$		0			0
$472$	−846720.	+	1.46656e6i	−0.174938	+	0.303002i
$473$	2.66139e6	−	4.60966e6i	0.546960	−	0.947363i
$474$		0			0
$475$	2.48270e6	+	4.30016e6i	0.504882	+	0.874482i
$476$	−1.01942e6			−0.206223
$477$		0			0
$478$	144384.			0.0289034
$479$	−2.12420e6	−	3.67921e6i	−0.423015	−	0.732683i	0.573218	−	0.819403i	$-0.305695\pi$
$479$	−2.12420e6	−	3.67921e6i	−0.423015	−	0.732683i	−0.996233	+	0.0867198i	$0.972362\pi$
$480$		0			0
$481$	570228.	−	987663.i	0.112379	−	0.194646i
$482$	−145750.	+	252446.i	−0.0285753	+	0.0494939i
$483$		0			0
$484$	705208.	+	1.22146e6i	0.136837	+	0.237009i
$485$	2.67905e6			0.517163
$486$		0			0
$487$	−1.04048e6			−0.198797			−0.0993985	−	0.995048i	$-0.531692\pi$
$487$	−1.04048e6			−0.198797			−0.0993985	+	0.995048i	$0.531692\pi$
$488$	1.72093e6	+	2.98073e6i	0.327125	+	0.566596i
$489$		0			0
$490$	475902.	−	824286.i	0.0895421	−	0.155091i
$491$	−2.77849e6	+	4.81248e6i	−0.520122	+	0.900877i	0.479605	+	0.877485i	$0.340780\pi$
$491$	−2.77849e6	+	4.81248e6i	−0.520122	+	0.900877i	−0.999726	+	0.0233925i	$0.992553\pi$
$492$		0			0
$493$	484312.	+	838854.i	0.0897446	+	0.155442i
$494$	−851000.			−0.156896
$495$		0			0
$496$	1.33837e6			0.244271
$497$	157398.	+	272621.i	0.0285830	+	0.0495073i
$498$		0			0
$499$	176639.	−	305948.i	0.0317567	−	0.0550042i	−0.849710	−	0.527250i	$-0.823223\pi$
$499$	176639.	−	305948.i	0.0317567	−	0.0550042i	0.881467	+	0.472246i	$0.156556\pi$
$500$	975912.	−	1.69033e6i	0.174576	−	0.302375i
$501$		0			0
$502$	3.31739e6	+	5.74588e6i	0.587539	+	1.01765i
$503$	4.21978e6			0.743651			0.371826	−	0.928303i	$-0.378732\pi$
$503$	4.21978e6			0.743651			0.371826	+	0.928303i	$0.378732\pi$
$504$		0			0
$505$	−1.65627e6			−0.289003
$506$	−1.95372e6	−	3.38394e6i	−0.339224	−	0.587552i
$507$		0			0
$508$	2.50386e6	−	4.33681e6i	0.430477	−	0.745608i
$509$	−173067.	+	299761.i	−0.0296087	+	0.0512838i	−0.880450	−	0.474139i	$-0.842759\pi$
$509$	−173067.	+	299761.i	−0.0296087	+	0.0512838i	0.850841	+	0.525423i	$0.176093\pi$
$510$		0			0
$511$	648277.	+	1.12285e6i	0.109827	+	0.190226i
$512$	262144.			0.0441942
$513$		0			0
$514$	−3.17672e6			−0.530361
$515$	183624.	+	318046.i	0.0305078	+	0.0528411i
$516$		0			0
$517$	−1.34784e6	+	2.33453e6i	−0.221775	+	0.384125i
$518$	−1.46772e6	+	2.54216e6i	−0.240335	+	0.416273i
$519$		0			0
$520$	77280.0	+	133853.i	0.0125331	+	0.0217080i
$521$	1.90025e6			0.306703			0.153351	−	0.988172i	$-0.450993\pi$
$521$	1.90025e6			0.306703			0.153351	+	0.988172i	$0.450993\pi$
$522$		0			0
$523$	−5.16589e6			−0.825831			−0.412915	−	0.910769i	$-0.635489\pi$
$523$	−5.16589e6			−0.825831			−0.412915	+	0.910769i	$0.635489\pi$
$524$	846480.	+	1.46615e6i	0.134675	+	0.233265i
$525$		0			0
$526$	−3.18158e6	+	5.51067e6i	−0.501394	+	0.868440i
$527$	2.25065e6	−	3.89825e6i	0.353006	−	0.611424i
$528$		0			0
$529$	−3.32679e6	−	5.76217e6i	−0.516876	−	0.895255i
$530$	3.08498e6			0.477049
$531$		0			0
$532$	2.19040e6			0.335540
$533$	618585.	+	1.07142e6i	0.0943151	+	0.163359i
$534$		0			0
$535$	1.41082e6	−	2.44362e6i	0.213102	−	0.369104i
$536$	413888.	−	716875.i	0.0622259	−	0.107778i
$537$		0			0
$538$	−3.15950e6	−	5.47241e6i	−0.470611	−	0.815123i
$539$	−3.05937e6			−0.453586
$540$		0			0
$541$	4.15125e6			0.609797			0.304899	−	0.952385i	$-0.401377\pi$
$541$	4.15125e6			0.609797			0.304899	+	0.952385i	$0.401377\pi$
$542$	831524.	+	1.44024e6i	0.121584	+	0.210590i
$543$		0			0
$544$	440832.	−	763543.i	0.0638669	−	0.110621i
$545$	1.29661e6	−	2.24580e6i	0.186990	−	0.323877i
$546$		0			0
$547$	−4.56262e6	−	7.90269e6i	−0.651998	−	1.12929i	−0.982637	−	0.185536i	$-0.940598\pi$
$547$	−4.56262e6	−	7.90269e6i	−0.651998	−	1.12929i	0.330640	−	0.943757i	$-0.392735\pi$
$548$	−4.55170e6			−0.647473
$549$		0			0
$550$	−2.89872e6			−0.408601
$551$	−1.04062e6	−	1.80242e6i	−0.146021	−	0.252916i
$552$		0			0
$553$	1.36700e6	−	2.36772e6i	0.190089	−	0.329243i
$554$	−1.12481e6	+	1.94823e6i	−0.155706	+	0.269691i
$555$		0			0
$556$	1.29862e6	+	2.24928e6i	0.178154	+	0.308572i
$557$	−3.78858e6			−0.517415			−0.258707	−	0.965956i	$-0.583297\pi$
$557$	−3.78858e6			−0.517415			−0.258707	+	0.965956i	$0.583297\pi$
$558$		0			0
$559$	2.26711e6			0.306862
$560$	−198912.	−	344526.i	−0.0268035	−	0.0464250i
$561$		0			0
$562$	4.85325e6	−	8.40608e6i	0.648174	−	1.12267i
$563$	−2.84333e6	+	4.92479e6i	−0.378056	+	0.654812i	−0.990779	−	0.135485i	$-0.956741\pi$
$563$	−2.84333e6	+	4.92479e6i	−0.378056	+	0.654812i	0.612723	+	0.790297i	$0.290074\pi$
$564$		0			0
$565$	−1.82741e6	−	3.16517e6i	−0.240832	−	0.417134i
$566$	−4.81173e6			−0.631335
$567$		0			0
$568$	−272256.			−0.0354084
$569$	5.25194e6	+	9.09663e6i	0.680048	+	1.17788i	0.974966	+	0.222354i	$0.0713742\pi$
$569$	5.25194e6	+	9.09663e6i	0.680048	+	1.17788i	−0.294918	+	0.955522i	$0.595292\pi$
$570$		0			0
$571$	−3.50496e6	+	6.07077e6i	−0.449876	+	0.779208i	−0.998377	−	0.0569418i	$-0.981865\pi$
$571$	−3.50496e6	+	6.07077e6i	−0.449876	+	0.779208i	0.548502	+	0.836149i	$0.315198\pi$
$572$	248400.	−	430241.i	0.0317440	−	0.0549822i
$573$		0			0
$574$	−1.59218e6	−	2.75774e6i	−0.201704	−	0.349361i
$575$	−9.71071e6			−1.22485
$576$		0			0
$577$	−3.54196e6			−0.442898			−0.221449	−	0.975172i	$-0.571079\pi$
$577$	−3.54196e6			−0.442898			−0.221449	+	0.975172i	$0.571079\pi$
$578$	1.35707e6	+	2.35052e6i	0.168960	+	0.292647i
$579$		0			0
$580$	−189000.	+	327358.i	−0.0233288	+	0.0404066i
$581$	2.82739e6	−	4.89719e6i	0.347493	−	0.601875i
$582$		0			0
$583$	−4.95801e6	−	8.58753e6i	−0.604138	−	1.04640i
$584$	−1.12134e6			−0.136053
$585$		0			0
$586$	−2.68988e6			−0.323586
$587$	−7.63892e6	−	1.32310e7i	−0.915032	−	1.58488i	−0.806854	−	0.590751i	$-0.798832\pi$
$587$	−7.63892e6	−	1.32310e7i	−0.915032	−	1.58488i	−0.108178	−	0.994132i	$-0.534502\pi$
$588$		0			0
$589$	−4.83590e6	+	8.37602e6i	−0.574366	+	0.994832i
$590$	−1.11132e6	+	1.92486e6i	−0.131434	+	0.227651i
$591$		0			0
$592$	−1.26938e6	−	2.19862e6i	−0.148863	−	0.257838i
$593$	−278457.			−0.0325178			−0.0162589	−	0.999868i	$-0.505176\pi$
$593$	−278457.			−0.0325178			−0.0162589	+	0.999868i	$0.505176\pi$
$594$		0			0
$595$	−1.33799e6			−0.154939
$596$	2.16569e6	+	3.75108e6i	0.249735	+	0.432555i
$597$		0			0
$598$	832140.	−	1.44131e6i	0.0951576	−	0.164818i
$599$	5.03181e6	−	8.71535e6i	0.573003	−	0.992471i	−0.423252	−	0.906012i	$-0.639112\pi$
$599$	5.03181e6	−	8.71535e6i	0.573003	−	0.992471i	0.996255	−	0.0864590i	$-0.0275551\pi$
$600$		0			0
$601$	−6.82889e6	−	1.18280e7i	−0.771194	−	1.33575i	−0.936909	−	0.349574i	$-0.886326\pi$
$601$	−6.82889e6	−	1.18280e7i	−0.771194	−	1.33575i	0.165715	−	0.986174i	$-0.447007\pi$
$602$	−5.83534e6			−0.656259
$603$		0			0
$604$	−269632.			−0.0300732
$605$	925586.	+	1.60316e6i	0.102808	+	0.178069i
$606$		0			0
$607$	4.47969e6	−	7.75905e6i	0.493487	−	0.854745i	−0.506484	−	0.862249i	$-0.669055\pi$
$607$	4.47969e6	−	7.75905e6i	0.493487	−	0.854745i	0.999972	+	0.00750376i	$0.00238854\pi$
$608$	−947200.	+	1.64060e6i	−0.103916	+	0.179988i
$609$		0			0
$610$	2.25872e6	+	3.91221e6i	0.245775	+	0.425695i
$611$	−1.14816e6			−0.124423
$612$		0			0
$613$	529958.			0.0569627			0.0284813	−	0.999594i	$-0.490933\pi$
$613$	529958.			0.0569627			0.0284813	+	0.999594i	$0.490933\pi$
$614$	5.58975e6	+	9.68173e6i	0.598372	+	1.03641i
$615$		0			0
$616$	−639360.	+	1.10740e6i	−0.0678881	+	0.117586i
$617$	2.98514e6	−	5.17041e6i	0.315683	−	0.546779i	−0.663899	−	0.747822i	$-0.731100\pi$
$617$	2.98514e6	−	5.17041e6i	0.315683	−	0.546779i	0.979582	+	0.201043i	$0.0644330\pi$
$618$		0			0
$619$	585668.	+	1.01441e6i	0.0614363	+	0.106411i	0.895108	−	0.445850i	$-0.147099\pi$
$619$	585668.	+	1.01441e6i	0.0614363	+	0.106411i	−0.833671	+	0.552261i	$0.813765\pi$
$620$	1.75661e6			0.183525
$621$		0			0
$622$	−1.22384e7			−1.26837
$623$	1.67821e6	+	2.90674e6i	0.173231	+	0.300045i
$624$		0			0
$625$	−2.91287e6	+	5.04523e6i	−0.298277	+	0.516632i
$626$	−5.26949e6	+	9.12702e6i	−0.537443	+	0.930879i
$627$		0			0
$628$	−1.97898e6	−	3.42770e6i	−0.200236	−	0.346820i
$629$	−8.53854e6			−0.860512
$630$		0			0
$631$	−1.49126e7			−1.49101			−0.745504	−	0.666501i	$-0.767791\pi$
$631$	−1.49126e7			−1.49101			−0.745504	+	0.666501i	$0.767791\pi$
$632$	1.18227e6	+	2.04776e6i	0.117740	+	0.203932i
$633$		0			0
$634$	−1.61675e6	+	2.80030e6i	−0.159743	+	0.276682i
$635$	3.28631e6	−	5.69206e6i	0.323426	−	0.560190i
$636$		0			0
$637$	−651532.	−	1.12849e6i	−0.0636191	−	0.110192i
$638$	1.21500e6			0.118175
$639$		0			0
$640$	344064.			0.0332039
$641$	−3.94252e6	−	6.82864e6i	−0.378991	−	0.656431i	0.611925	−	0.790916i	$-0.290395\pi$
$641$	−3.94252e6	−	6.82864e6i	−0.378991	−	0.656431i	−0.990916	+	0.134485i	$0.957062\pi$
$642$		0			0
$643$	−929020.	+	1.60911e6i	−0.0886130	+	0.153482i	−0.906925	−	0.421292i	$-0.861577\pi$
$643$	−929020.	+	1.60911e6i	−0.0886130	+	0.153482i	0.818312	+	0.574774i	$0.194910\pi$
$644$	−2.14186e6	+	3.70980e6i	−0.203505	+	0.352481i
$645$		0			0
$646$	3.18570e6	+	5.51779e6i	0.300347	+	0.520217i
$647$	−1.54147e6			−0.144768			−0.0723841	−	0.997377i	$-0.523061\pi$
$647$	−1.54147e6			−0.144768			−0.0723841	+	0.997377i	$0.523061\pi$
$648$		0			0
$649$	7.14420e6			0.665797
$650$	−617320.	−	1.06923e6i	−0.0573095	−	0.0992630i
$651$		0			0
$652$	1.60093e6	−	2.77289e6i	0.147487	−	0.255454i
$653$	−2.06668e6	+	3.57959e6i	−0.189666	+	0.328512i	−0.945139	−	0.326669i	$-0.894074\pi$
$653$	−2.06668e6	+	3.57959e6i	−0.189666	+	0.328512i	0.755473	+	0.655180i	$0.227407\pi$
$654$		0			0
$655$	1.11100e6	+	1.92432e6i	0.101184	+	0.175256i
$656$	2.75405e6			0.249869
$657$		0			0
$658$	2.95526e6			0.266092
$659$	8.54102e6	+	1.47935e7i	0.766118	+	1.32696i	0.939653	+	0.342128i	$0.111148\pi$
$659$	8.54102e6	+	1.47935e7i	0.766118	+	1.32696i	−0.173535	+	0.984828i	$0.555519\pi$
$660$		0			0
$661$	3.07322e6	−	5.32298e6i	0.273584	−	0.473861i	−0.696193	−	0.717855i	$-0.745124\pi$
$661$	3.07322e6	−	5.32298e6i	0.273584	−	0.473861i	0.969777	+	0.243994i	$0.0784576\pi$
$662$	1.70736e6	−	2.95723e6i	0.151419	−	0.262265i
$663$		0			0
$664$	2.44531e6	+	4.23540e6i	0.215236	+	0.372799i
$665$	2.87490e6			0.252098
$666$		0			0
$667$	4.07025e6			0.354247
$668$	−112944.	−	195625.i	−0.00979314	−	0.0169622i
$669$		0			0
$670$	543228.	−	940898.i	0.0467515	−	0.0809759i
$671$	7.26016e6	−	1.25750e7i	0.622501	−	1.07820i
$672$		0			0
$673$	−4.36248e6	−	7.55604e6i	−0.371275	−	0.643067i	0.618487	−	0.785795i	$-0.287746\pi$
$673$	−4.36248e6	−	7.55604e6i	−0.371275	−	0.643067i	−0.989762	+	0.142728i	$0.954413\pi$
$674$	1.19151e6			0.101030
$675$		0			0
$676$	−5.72909e6			−0.482191
$677$	−5.92244e6	−	1.02580e7i	−0.496625	−	0.860180i	0.503367	−	0.864073i	$-0.332094\pi$
$677$	−5.92244e6	−	1.02580e7i	−0.496625	−	0.860180i	−0.999992	+	0.00389267i	$0.998761\pi$
$678$		0			0
$679$	−4.72024e6	+	8.17569e6i	−0.392907	+	0.680534i
$680$	578592.	−	1.00215e6i	0.0479844	−	0.0831114i
$681$		0			0
$682$	−2.82312e6	−	4.88979e6i	−0.232417	−	0.402559i
$683$	4.03085e6			0.330632			0.165316	−	0.986241i	$-0.447136\pi$
$683$	4.03085e6			0.330632			0.165316	+	0.986241i	$0.447136\pi$
$684$		0			0
$685$	−5.97410e6			−0.486459
$686$	4.16442e6	+	7.21299e6i	0.337866	+	0.585201i
$687$		0			0
$688$	2.52339e6	−	4.37064e6i	0.203242	−	0.352026i
$689$	2.11174e6	−	3.65765e6i	0.169470	−	0.293531i
$690$		0			0
$691$	−1.61461e6	−	2.79658e6i	−0.128639	−	0.222809i	0.794511	−	0.607250i	$-0.207727\pi$
$691$	−1.61461e6	−	2.79658e6i	−0.128639	−	0.222809i	−0.923149	+	0.384441i	$0.874394\pi$
$692$	−774384.			−0.0614740
$693$		0			0
$694$	354984.			0.0279776
$695$	1.70444e6	+	2.95218e6i	0.133851	+	0.231836i
$696$		0			0
$697$	4.63132e6	−	8.02168e6i	0.361096	−	0.625437i
$698$	−4.72349e6	+	8.18133e6i	−0.366965	+	0.635602i
$699$		0			0
$700$	1.58893e6	+	2.75210e6i	0.122563	+	0.212285i
$701$	1.34398e7			1.03300			0.516499	−	0.856288i	$-0.327235\pi$
$701$	1.34398e7			1.03300			0.516499	+	0.856288i	$0.327235\pi$
$702$		0			0
$703$	1.83464e7			1.40012
$704$	−552960.	−	957755.i	−0.0420496	−	0.0728321i
$705$		0			0
$706$	7.72153e6	−	1.33741e7i	0.583031	−	1.00984i
$707$	2.91819e6	−	5.05445e6i	0.219566	−	0.380299i
$708$		0			0
$709$	1.04150e7	+	1.80393e7i	0.778114	+	1.34773i	0.933028	+	0.359805i	$0.117157\pi$
$709$	1.04150e7	+	1.80393e7i	0.778114	+	1.34773i	−0.154913	+	0.987928i	$0.549510\pi$
$710$	−357336.			−0.0266030
$711$		0			0
$712$	−2.90285e6			−0.214597
$713$	−9.45745e6	−	1.63808e7i	−0.696707	−	1.20673i
$714$		0			0
$715$	326025.	−	564692.i	0.0238499	−	0.0413092i
$716$	−3.00317e6	+	5.20164e6i	−0.218926	+	0.379191i
$717$		0			0
$718$	7.49704e6	+	1.29852e7i	0.542723	+	0.940024i
$719$	2.57779e7			1.85963			0.929813	−	0.368032i	$-0.119968\pi$
$719$	2.57779e7			1.85963			0.929813	+	0.368032i	$0.119968\pi$
$720$		0			0
$721$	−1.29411e6			−0.0927115
$722$	−1.89280e6	−	3.27843e6i	−0.135133	−	0.234058i
$723$		0			0
$724$	−3.52562e6	+	6.10655e6i	−0.249970	+	0.432961i
$725$	1.50975e6	−	2.61496e6i	0.106674	−	0.184765i
$726$		0			0
$727$	3.62246e6	+	6.27428e6i	0.254195	+	0.440279i	0.964677	−	0.263437i	$-0.0848561\pi$
$727$	3.62246e6	+	6.27428e6i	0.254195	+	0.440279i	−0.710481	+	0.703716i	$0.751523\pi$
$728$	−544640.			−0.0380874
$729$		0			0
$730$	−1.47176e6			−0.102219
$731$	−8.48688e6	−	1.46997e7i	−0.587428	−	1.01745i
$732$		0			0
$733$	2.82136e6	−	4.88674e6i	0.193954	−	0.335938i	−0.752603	−	0.658474i	$-0.771202\pi$
$733$	2.82136e6	−	4.88674e6i	0.193954	−	0.335938i	0.946557	+	0.322536i	$0.104535\pi$
$734$	5.32678e6	−	9.22626e6i	0.364943	−	0.632100i
$735$		0			0
$736$	−1.85242e6	−	3.20848e6i	−0.126050	−	0.218326i
$737$	−3.49218e6			−0.236825
$738$		0			0
$739$	1.72501e7			1.16193			0.580967	−	0.813927i	$-0.302674\pi$
$739$	1.72501e7			1.16193			0.580967	+	0.813927i	$0.302674\pi$
$740$	−1.66606e6	−	2.88569e6i	−0.111843	−	0.193718i
$741$		0			0
$742$	−5.43545e6	+	9.41447e6i	−0.362431	+	0.627749i
$743$	−6.46732e6	+	1.12017e7i	−0.429786	+	0.744412i	−0.996854	−	0.0792596i	$-0.974744\pi$
$743$	−6.46732e6	+	1.12017e7i	−0.429786	+	0.744412i	0.567068	+	0.823671i	$0.308078\pi$
$744$		0			0
$745$	2.84247e6	+	4.92329e6i	0.187631	+	0.324986i
$746$	3.83948e6			0.252595
$747$		0			0
$748$	−3.71952e6			−0.243071
$749$	4.97147e6	+	8.61084e6i	0.323802	+	0.560842i
$750$		0			0
$751$	−2.84905e6	+	4.93469e6i	−0.184332	+	0.319272i	−0.943351	−	0.331796i	$-0.892345\pi$
$751$	−2.84905e6	+	4.93469e6i	−0.184332	+	0.319272i	0.759019	+	0.651068i	$0.225679\pi$
$752$	−1.27795e6	+	2.21348e6i	−0.0824081	+	0.142735i
$753$		0			0
$754$	258750.	+	448168.i	0.0165749	+	0.0287086i
$755$	−353892.			−0.0225945
$756$		0			0
$757$	3.72388e6			0.236187			0.118093	−	0.993002i	$-0.462322\pi$
$757$	3.72388e6			0.236187			0.118093	+	0.993002i	$0.462322\pi$
$758$	387560.	+	671274.i	0.0245000	+	0.0424352i
$759$		0			0
$760$	−1.24320e6	+	2.15329e6i	−0.0780741	+	0.135228i
$761$	−8.84733e6	+	1.53240e7i	−0.553797	+	0.959205i	0.444199	+	0.895928i	$0.353488\pi$
$761$	−8.84733e6	+	1.53240e7i	−0.553797	+	0.959205i	−0.997996	+	0.0632765i	$0.979845\pi$
$762$		0			0
$763$	4.56902e6	+	7.91377e6i	0.284126	+	0.492121i
$764$	7.79165e6			0.482943
$765$		0			0
$766$	2.22157e7			1.36801
$767$	1.52145e6	+	2.63523e6i	0.0933833	+	0.161745i
$768$		0			0
$769$	−8.32902e6	+	1.44263e7i	−0.507900	+	0.879708i	0.492058	+	0.870562i	$0.336245\pi$
$769$	−8.32902e6	+	1.44263e7i	−0.507900	+	0.879708i	−0.999958	+	0.00914603i	$0.997089\pi$
$770$	−839160.	+	1.45347e6i	−0.0510056	+	0.0883443i
$771$		0			0
$772$	833480.	+	1.44363e6i	0.0503329	+	0.0871791i
$773$	2.04852e6			0.123308			0.0616539	−	0.998098i	$-0.480362\pi$
$773$	2.04852e6			0.123308			0.0616539	+	0.998098i	$0.480362\pi$
$774$		0			0
$775$	−1.40320e7			−0.839197
$776$	−4.08237e6	−	7.07087e6i	−0.243365	−	0.421520i
$777$		0			0
$778$	9.65591e6	−	1.67245e7i	0.571932	−	0.990615i
$779$	−9.95115e6	+	1.72359e7i	−0.587529	+	1.01763i
$780$		0			0
$781$	574290.	+	994699.i	0.0336902	+	0.0583531i
$782$	−1.24604e7			−0.728643
$783$		0			0
$784$	−2.90074e6			−0.168546
$785$	−2.59742e6	−	4.49886e6i	−0.150441	−	0.260572i
$786$		0			0
$787$	−3.78298e6	+	6.55232e6i	−0.217720	+	0.377101i	−0.954110	−	0.299455i	$-0.903195\pi$
$787$	−3.78298e6	+	6.55232e6i	−0.217720	+	0.377101i	0.736391	+	0.676556i	$0.236529\pi$
$788$	−314952.	+	545513.i	−0.0180688	+	0.0312960i
$789$		0			0
$790$	1.55173e6	+	2.68768e6i	0.0884604	+	0.153218i
$791$	1.28789e7			0.731875
$792$		0			0
$793$	6.18458e6			0.349243
$794$	−626818.	−	1.08568e6i	−0.0352850	−	0.0611154i
$795$		0			0
$796$	−7.61987e6	+	1.31980e7i	−0.426251	+	0.738288i
$797$	−503674.	+	872390.i	−0.0280869	+	0.0486480i	−0.879727	−	0.475479i	$-0.842275\pi$
$797$	−503674.	+	872390.i	−0.0280869	+	0.0486480i	0.851640	+	0.524127i	$0.175608\pi$
$798$		0			0
$799$	4.29811e6	+	7.44455e6i	0.238183	+	0.412545i
$800$	−2.74842e6			−0.151830
$801$		0			0
$802$	2.11316e7			1.16010
$803$	2.36534e6	+	4.09688e6i	0.129451	+	0.224215i
$804$		0			0
$805$	−2.81119e6	+	4.86912e6i	−0.152897	+	0.264826i
$806$	1.20244e6	−	2.08269e6i	0.0651967	−	0.112924i
$807$		0			0
$808$	2.52384e6	+	4.37142e6i	0.135998	+	0.235556i
$809$	−2.37253e7			−1.27450			−0.637251	−	0.770657i	$-0.719928\pi$
$809$	−2.37253e7			−1.27450			−0.637251	+	0.770657i	$0.719928\pi$
$810$		0			0
$811$	2.09278e7			1.11730			0.558652	−	0.829402i	$-0.311319\pi$
$811$	2.09278e7			1.11730			0.558652	+	0.829402i	$0.311319\pi$
$812$	−666000.	−	1.15355e6i	−0.0354474	−	0.0613967i
$813$		0			0
$814$	−5.35518e6	+	9.27544e6i	−0.283278	+	0.490652i
$815$	2.10122e6	−	3.63942e6i	0.110810	−	0.191928i
$816$		0			0
$817$	1.82354e7	+	3.15847e7i	0.955787	+	1.65547i
$818$	1.83525e7			0.958983
$819$		0			0
$820$	3.61469e6			0.187731
$821$	−6.96434e6	−	1.20626e7i	−0.360597	−	0.624572i	0.627462	−	0.778647i	$-0.284094\pi$
$821$	−6.96434e6	−	1.20626e7i	−0.360597	−	0.624572i	−0.988059	+	0.154075i	$0.950760\pi$
$822$		0			0
$823$	9.97764e6	−	1.72818e7i	0.513486	−	0.889383i	−0.486392	−	0.873741i	$-0.661687\pi$
$823$	9.97764e6	−	1.72818e7i	0.513486	−	0.889383i	0.999878	−	0.0156426i	$-0.00497938\pi$
$824$	559616.	−	969283.i	0.0287126	−	0.0497316i
$825$		0			0
$826$	−3.91608e6	−	6.78285e6i	−0.199711	−	0.345909i
$827$	−2.48097e7			−1.26141			−0.630706	−	0.776022i	$-0.717235\pi$
$827$	−2.48097e7			−1.26141			−0.630706	+	0.776022i	$0.717235\pi$
$828$		0			0
$829$	1.22092e6			0.0617021			0.0308511	−	0.999524i	$-0.490178\pi$
$829$	1.22092e6			0.0617021			0.0308511	+	0.999524i	$0.490178\pi$
$830$	3.20947e6	+	5.55897e6i	0.161711	+	0.280091i
$831$		0			0
$832$	235520.	−	407933.i	0.0117956	−	0.0204306i
$833$	−4.87800e6	+	8.44894e6i	−0.243573	+	0.421881i
$834$		0			0
$835$	−148239.	−	256757.i	−0.00735777	−	0.0127440i
$836$	7.99200e6			0.395494
$837$		0			0
$838$	−9.46790e6			−0.465740
$839$	8.80920e6	+	1.52580e7i	0.432048	+	0.748329i	0.997050	−	0.0767611i	$-0.0244579\pi$
$839$	8.80920e6	+	1.52580e7i	0.432048	+	0.748329i	−0.565002	+	0.825090i	$0.691125\pi$
$840$		0			0
$841$	9.62276e6	−	1.66671e7i	0.469148	−	0.812588i
$842$	909590.	−	1.57546e6i	0.0442146	−	0.0765819i
$843$		0			0
$844$	−3.52888e6	−	6.11220e6i	−0.170522	−	0.295353i
$845$	−7.51943e6			−0.362279
$846$		0			0
$847$	−6.52317e6			−0.312428
$848$	−4.70093e6	−	8.14225e6i	−0.224488	−	0.388825i
$849$		0			0
$850$	−4.62185e6	+	8.00528e6i	−0.219416	+	0.380040i
$851$	−1.79399e7	+	3.10727e7i	−0.849171	+	1.47081i
$852$		0			0
$853$	−1.71780e7	−	2.97531e7i	−0.808350	−	1.40010i	−0.914006	−	0.405701i	$-0.867027\pi$
$853$	−1.71780e7	−	2.97531e7i	−0.808350	−	1.40010i	0.105656	−	0.994403i	$-0.466306\pi$
$854$	−1.59186e7			−0.746895
$855$		0			0
$856$	−8.59930e6			−0.401124
$857$	493631.	+	854993.i	0.0229588	+	0.0397659i	0.877277	−	0.479985i	$-0.159358\pi$
$857$	493631.	+	854993.i	0.0229588	+	0.0397659i	−0.854318	+	0.519751i	$0.826025\pi$
$858$		0			0
$859$	−2.05926e7	+	3.56675e7i	−0.952202	+	1.64926i	−0.211555	+	0.977366i	$0.567853\pi$
$859$	−2.05926e7	+	3.56675e7i	−0.952202	+	1.64926i	−0.740646	+	0.671895i	$0.765480\pi$
$860$	3.31195e6	−	5.73647e6i	0.152700	−	0.264483i
$861$		0			0
$862$	−2.52571e6	−	4.37466e6i	−0.115775	−	0.200529i
$863$	2.07182e7			0.946946			0.473473	−	0.880808i	$-0.343000\pi$
$863$	2.07182e7			0.946946			0.473473	+	0.880808i	$0.343000\pi$
$864$		0			0
$865$	−1.01638e6			−0.0461865
$866$	−1.09780e7	−	1.90145e7i	−0.497426	−	0.861568i
$867$		0			0
$868$	−3.09498e6	+	5.36066e6i	−0.139431	+	0.241501i
$869$	4.98771e6	−	8.63897e6i	0.224054	−	0.388072i
$870$		0			0
$871$	−743705.	−	1.28813e6i	−0.0332166	−	0.0575329i
$872$	−7.90317e6			−0.351974
$873$		0			0
$874$	2.67732e7			1.18555
$875$	4.51359e6	+	7.81777e6i	0.199298	+	0.345194i
$876$		0			0
$877$	1.85173e7	−	3.20730e7i	0.812979	−	1.40812i	−0.0977904	−	0.995207i	$-0.531177\pi$
$877$	1.85173e7	−	3.20730e7i	0.812979	−	1.40812i	0.910770	−	0.412915i	$-0.135489\pi$
$878$	1.22329e7	−	2.11880e7i	0.535541	−	0.927584i
$879$		0			0
$880$	−725760.	−	1.25705e6i	−0.0315927	−	0.0547201i
$881$	−7.00502e6			−0.304067			−0.152034	−	0.988375i	$-0.548582\pi$
$881$	−7.00502e6			−0.304067			−0.152034	+	0.988375i	$0.548582\pi$
$882$		0			0
$883$	5.47102e6			0.236139			0.118069	−	0.993005i	$-0.462330\pi$
$883$	5.47102e6			0.236139			0.118069	+	0.993005i	$0.462330\pi$
$884$	−792120.	−	1.37199e6i	−0.0340926	−	0.0590501i
$885$		0			0
$886$	−1.47840e6	+	2.56066e6i	−0.0632714	+	0.109589i
$887$	−1.09490e6	+	1.89642e6i	−0.0467268	+	0.0809331i	−0.888443	−	0.458987i	$-0.848212\pi$
$887$	−1.09490e6	+	1.89642e6i	−0.0467268	+	0.0809331i	0.841716	+	0.539920i	$0.181546\pi$
$888$		0			0
$889$	1.15803e7	+	2.00577e7i	0.491436	+	0.851192i
$890$	−3.80999e6			−0.161231
$891$		0			0
$892$	−1.32017e7			−0.555543
$893$	−9.23520e6	−	1.59958e7i	−0.387541	−	0.671241i
$894$		0			0
$895$	−3.94166e6	+	6.82715e6i	−0.164483	+	0.284893i
$896$	−606208.	+	1.04998e6i	−0.0252262	+	0.0436930i
$897$		0			0
$898$	−1.46286e7	−	2.53375e7i	−0.605359	−	1.04851i
$899$	5.88150e6			0.242711
$900$		0			0
$901$	−3.16211e7			−1.29767
$902$	−5.80932e6	−	1.00620e7i	−0.237744	−	0.411784i
$903$		0			0
$904$	−5.56925e6	+	9.64622e6i	−0.226660	+	0.392587i
$905$	−4.62737e6	+	8.01484e6i	−0.187807	+	0.325292i
$906$		0			0
$907$	−2.43272e7	−	4.21360e7i	−0.981916	−	1.70073i	−0.654912	−	0.755705i	$-0.727294\pi$
$907$	−2.43272e7	−	4.21360e7i	−0.981916	−	1.70073i	−0.327004	−	0.945023i	$-0.606039\pi$
$908$	1.84912e7			0.744305
$909$		0			0
$910$	−714840.			−0.0286158
$911$	2.35396e7	+	4.07718e7i	0.939730	+	1.62766i	0.765975	+	0.642871i	$0.222257\pi$
$911$	2.35396e7	+	4.07718e7i	0.939730	+	1.62766i	0.173755	+	0.984789i	$0.444410\pi$
$912$		0			0
$913$	1.03162e7	−	1.78681e7i	0.409582	−	0.709417i
$914$	1.53781e7	−	2.66356e7i	0.608887	−	1.05462i
$915$		0			0
$916$	−1.07469e7	−	1.86141e7i	−0.423197	−	0.732999i
$917$	−7.82994e6			−0.307493
$918$		0			0
$919$	−2.23428e7			−0.872667			−0.436333	−	0.899785i	$-0.643723\pi$
$919$	−2.23428e7			−0.872667			−0.436333	+	0.899785i	$0.643723\pi$
$920$	−2.43130e6	−	4.21113e6i	−0.0947040	−	0.164032i
$921$		0			0
$922$	1.18787e7	−	2.05746e7i	0.460196	−	0.797083i
$923$	−244605.	+	423668.i	−0.00945064	+	0.0163690i
$924$		0			0
$925$	1.33086e7	+	2.30512e7i	0.511421	+	0.885807i
$926$	−1.82382e6			−0.0698965
$927$		0			0
$928$	1.15200e6			0.0439119
$929$	1.01093e7	+	1.75098e7i	0.384309	+	0.665642i	0.991673	−	0.128781i	$-0.0411065\pi$
$929$	1.01093e7	+	1.75098e7i	0.384309	+	0.665642i	−0.607364	+	0.794424i	$0.707773\pi$
$930$		0			0
$931$	1.04812e7	−	1.81539e7i	0.396311	−	0.686430i
$932$	−8.60441e6	+	1.49033e7i	−0.324475	+	0.562007i
$933$		0			0
$934$	1.19419e7	+	2.06841e7i	0.447928	+	0.775833i
$935$	−4.88187e6			−0.182624
$936$		0			0
$937$	−3.88053e7			−1.44392			−0.721959	−	0.691936i	$-0.756758\pi$
$937$	−3.88053e7			−1.44392			−0.721959	+	0.691936i	$0.756758\pi$
$938$	1.91423e6	+	3.31555e6i	0.0710375	+	0.123041i
$939$		0			0
$940$	−1.67731e6	+	2.90519e6i	−0.0619148	+	0.107240i
$941$	1.39057e7	−	2.40854e7i	0.511940	−	0.886707i	−0.487964	−	0.872864i	$-0.662260\pi$
$941$	1.39057e7	−	2.40854e7i	0.511940	−	0.886707i	0.999904	−	0.0138427i	$-0.00440642\pi$
$942$		0			0
$943$	−1.94612e7	−	3.37078e7i	−0.712674	−	1.23439i
$944$	6.77376e6			0.247400
$945$		0			0
$946$	−2.12911e7			−0.773518
$947$	1.91196e7	+	3.31162e7i	0.692794	+	1.19995i	0.970919	+	0.239410i	$0.0769540\pi$
$947$	1.91196e7	+	3.31162e7i	0.692794	+	1.19995i	−0.278124	+	0.960545i	$0.589713\pi$
$948$		0			0
$949$	−1.00746e6	+	1.74497e6i	−0.0363129	+	0.0628958i
$950$	9.93080e6	−	1.72007e7i	0.357006	−	0.618352i
$951$		0			0
$952$	2.03885e6	+	3.53139e6i	0.0729109	+	0.126285i
$953$	1.01512e7			0.362065			0.181033	−	0.983477i	$-0.442056\pi$
$953$	1.01512e7			0.362065			0.181033	+	0.983477i	$0.442056\pi$
$954$		0			0
$955$	1.02265e7			0.362844
$956$	−288768.	−	500161.i	−0.0102189	−	0.0176997i
$957$		0			0
$958$	−8.49678e6	+	1.47169e7i	−0.299117	+	0.518085i
$959$	1.05258e7	−	1.82312e7i	0.369580	−	0.640131i
$960$		0			0
$961$	648584.	+	1.12338e6i	0.0226547	+	0.0392390i
$962$	−4.56182e6			−0.158928
$963$		0			0
$964$	1.16600e6			0.0404116
$965$	1.09394e6	+	1.89476e6i	0.0378161	+	0.0654993i
$966$		0			0
$967$	7.07872e6	−	1.22607e7i	0.243438	−	0.421647i	−0.718253	−	0.695782i	$-0.755058\pi$
$967$	7.07872e6	−	1.22607e7i	0.243438	−	0.421647i	0.961691	+	0.274135i	$0.0883914\pi$
$968$	2.82083e6	−	4.88582e6i	0.0967584	−	0.167591i
$969$		0			0
$970$	−5.35811e6	−	9.28052e6i	−0.182845	−	0.316696i
$971$	1.50291e6			0.0511546			0.0255773	−	0.999673i	$-0.491858\pi$
$971$	1.50291e6			0.0511546			0.0255773	+	0.999673i	$0.491858\pi$
$972$		0			0
$973$	−1.20123e7			−0.406765
$974$	2.08095e6	+	3.60431e6i	0.0702853	+	0.121738i
$975$		0			0
$976$	6.88371e6	−	1.19229e7i	0.231312	−	0.400644i
$977$	1.44460e7	−	2.50212e7i	0.484185	−	0.838634i	−0.515650	−	0.856800i	$-0.672449\pi$
$977$	1.44460e7	−	2.50212e7i	0.484185	−	0.838634i	0.999835	+	0.0181658i	$0.00578268\pi$
$978$		0			0
$979$	6.12320e6	+	1.06057e7i	0.204184	+	0.353657i
$980$	−3.80722e6			−0.126632
$981$		0			0
$982$	2.22279e7			0.735563
$983$	−3.85799e6	−	6.68223e6i	−0.127344	−	0.220566i	0.795303	−	0.606212i	$-0.207312\pi$
$983$	−3.85799e6	−	6.68223e6i	−0.127344	−	0.220566i	−0.922647	+	0.385647i	$0.873978\pi$
$984$		0			0
$985$	−413374.	+	715986.i	−0.0135754	+	0.0235133i
$986$	1.93725e6	−	3.35542e6i	0.0634590	−	0.109914i
$987$		0			0
$988$	1.70200e6	+	2.94795e6i	0.0554712	+	0.0960789i
$989$	−7.13253e7			−2.31874
$990$		0			0
$991$	−1.87209e7			−0.605538			−0.302769	−	0.953064i	$-0.597911\pi$
$991$	−1.87209e7			−0.605538			−0.302769	+	0.953064i	$0.597911\pi$
$992$	−2.67674e6	−	4.63624e6i	−0.0863627	−	0.149585i
$993$		0			0
$994$	629592.	−	1.09049e6i	0.0202113	−	0.0350069i
$995$	−1.00011e7	+	1.73224e7i	−0.320250	+	0.554689i
$996$		0			0
$997$	−2.68938e6	−	4.65815e6i	−0.0856869	−	0.148414i	0.819997	−	0.572368i	$-0.193975\pi$
$997$	−2.68938e6	−	4.65815e6i	−0.0856869	−	0.148414i	−0.905684	+	0.423954i	$0.860642\pi$
$998$	−1.41311e6			−0.0449107
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.6.c.d.109.1		2
3.2	odd	2		162.6.c.i.109.1		2
9.2	odd	6		162.6.c.i.55.1		2
9.4	even	3		162.6.a.b.1.1	yes	1
9.5	odd	6		162.6.a.a.1.1	✓	1
9.7	even	3	inner	162.6.c.d.55.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.6.a.a.1.1	✓	1	9.5	odd	6
162.6.a.b.1.1	yes	1	9.4	even	3
162.6.c.d.55.1		2	9.7	even	3	inner
162.6.c.d.109.1		2	1.1	even	1	trivial
162.6.c.i.55.1		2	9.2	odd	6
162.6.c.i.109.1		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-10.5000 + 18.1865i) q^{5} +(-37.0000 - 64.0859i) q^{7} +64.0000 q^{8} +84.0000 q^{10} +(-135.000 - 233.827i) q^{11} +(57.5000 - 99.5929i) q^{13} +(-148.000 + 256.344i) q^{14} +(-128.000 - 221.703i) q^{16} -861.000 q^{17} +1850.00 q^{19} +(-168.000 - 290.985i) q^{20} +(-540.000 + 935.307i) q^{22} +(-1809.00 + 3133.28i) q^{23} +(1342.00 + 2324.41i) q^{25} -460.000 q^{26} +1184.00 q^{28} +(-562.500 - 974.279i) q^{29} +(-2614.00 + 4527.58i) q^{31} +(-512.000 + 886.810i) q^{32} +(1722.00 + 2982.59i) q^{34} +1554.00 q^{35} +9917.00 q^{37} +(-3700.00 - 6408.59i) q^{38} +(-672.000 + 1163.94i) q^{40} +(-5379.00 + 9316.70i) q^{41} +(9857.00 + 17072.8i) q^{43} +4320.00 q^{44} +14472.0 q^{46} +(-4992.00 - 8646.40i) q^{47} +(5665.50 - 9812.93i) q^{49} +(5368.00 - 9297.65i) q^{50} +(920.000 + 1593.49i) q^{52} +36726.0 q^{53} +5670.00 q^{55} +(-2368.00 - 4101.50i) q^{56} +(-2250.00 + 3897.11i) q^{58} +(-13230.0 + 22915.0i) q^{59} +(26889.5 + 46574.0i) q^{61} +20912.0 q^{62} +4096.00 q^{64} +(1207.50 + 2091.45i) q^{65} +(6467.00 - 11201.2i) q^{67} +(6888.00 - 11930.4i) q^{68} +(-3108.00 - 5383.21i) q^{70} -4254.00 q^{71} -17521.0 q^{73} +(-19834.0 - 34353.5i) q^{74} +(-14800.0 + 25634.4i) q^{76} +(-9990.00 + 17303.2i) q^{77} +(18473.0 + 31996.2i) q^{79} +5376.00 q^{80} +43032.0 q^{82} +(38208.0 + 66178.2i) q^{83} +(9040.50 - 15658.6i) q^{85} +(39428.0 - 68291.3i) q^{86} +(-8640.00 - 14964.9i) q^{88} -45357.0 q^{89} -8510.00 q^{91} +(-28944.0 - 50132.5i) q^{92} +(-19968.0 + 34585.6i) q^{94} +(-19425.0 + 33645.1i) q^{95} +(-63787.0 - 110482. i) q^{97} -45324.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 16 q^{4} - 21 q^{5} - 74 q^{7} + 128 q^{8} + 168 q^{10} - 270 q^{11} + 115 q^{13} - 296 q^{14} - 256 q^{16} - 1722 q^{17} + 3700 q^{19} - 336 q^{20} - 1080 q^{22} - 3618 q^{23} + 2684 q^{25}+ \cdots - 90648 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 - 16 * q^4 - 21 * q^5 - 74 * q^7 + 128 * q^8 + 168 * q^10 - 270 * q^11 + 115 * q^13 - 296 * q^14 - 256 * q^16 - 1722 * q^17 + 3700 * q^19 - 336 * q^20 - 1080 * q^22 - 3618 * q^23 + 2684 * q^25 - 920 * q^26 + 2368 * q^28 - 1125 * q^29 - 5228 * q^31 - 1024 * q^32 + 3444 * q^34 + 3108 * q^35 + 19834 * q^37 - 7400 * q^38 - 1344 * q^40 - 10758 * q^41 + 19714 * q^43 + 8640 * q^44 + 28944 * q^46 - 9984 * q^47 + 11331 * q^49 + 10736 * q^50 + 1840 * q^52 + 73452 * q^53 + 11340 * q^55 - 4736 * q^56 - 4500 * q^58 - 26460 * q^59 + 53779 * q^61 + 41824 * q^62 + 8192 * q^64 + 2415 * q^65 + 12934 * q^67 + 13776 * q^68 - 6216 * q^70 - 8508 * q^71 - 35042 * q^73 - 39668 * q^74 - 29600 * q^76 - 19980 * q^77 + 36946 * q^79 + 10752 * q^80 + 86064 * q^82 + 76416 * q^83 + 18081 * q^85 + 78856 * q^86 - 17280 * q^88 - 90714 * q^89 - 17020 * q^91 - 57888 * q^92 - 39936 * q^94 - 38850 * q^95 - 127574 * q^97 - 90648 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more