LMFDB - Embedded newform 138.5.b.a.91.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,5,Mod(91,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("138.91");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	138.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$14.2650549056$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} + \cdots + 274129967370817 $ x^16 + 1428x^14 - 600x^13 + 788282x^12 - 529464x^11 + 213396724x^10 - 175079484x^9 + 29130946113x^8 - 26553654912x^7 + 1790717926120x^6 - 1672125302460x^5 + 32715955056348x^4 - 11255739266544x^3 + 140972261417692x^2 - 70961983286748x + 274129967370817
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.6
Root			$-0.707107 - 16.9956i$ of defining polynomial
Character	$\chi$	$=$	138.91
Dual form			138.5.b.a.91.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -7.78845i q^{5} -14.6969 q^{6} +25.8071i q^{7} -22.6274 q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -7.78845i q^{5} -14.6969 q^{6} +25.8071i q^{7} -22.6274 q^{8} +27.0000 q^{9} +22.0291i q^{10} +131.303i q^{11} +41.5692 q^{12} +98.7308 q^{13} -72.9935i q^{14} -40.4700i q^{15} +64.0000 q^{16} -271.085i q^{17} -76.3675 q^{18} +342.704i q^{19} -62.3076i q^{20} +134.098i q^{21} -371.382i q^{22} +(-359.824 + 387.773i) q^{23} -117.576 q^{24} +564.340 q^{25} -279.253 q^{26} +140.296 q^{27} +206.457i q^{28} +560.437 q^{29} +114.466i q^{30} +1081.60 q^{31} -181.019 q^{32} +682.273i q^{33} +766.745i q^{34} +200.997 q^{35} +216.000 q^{36} +2355.56i q^{37} -969.313i q^{38} +513.020 q^{39} +176.232i q^{40} +1281.30 q^{41} -379.285i q^{42} +3250.33i q^{43} +1050.43i q^{44} -210.288i q^{45} +(1017.74 - 1096.79i) q^{46} +1163.18 q^{47} +332.554 q^{48} +1734.99 q^{49} -1596.19 q^{50} -1408.60i q^{51} +789.846 q^{52} -1599.79i q^{53} -396.817 q^{54} +1022.65 q^{55} -583.948i q^{56} +1780.74i q^{57} -1585.16 q^{58} -3664.84 q^{59} -323.760i q^{60} -1882.61i q^{61} -3059.23 q^{62} +696.791i q^{63} +512.000 q^{64} -768.959i q^{65} -1929.76i q^{66} -1313.92i q^{67} -2168.68i q^{68} +(-1869.70 + 2014.93i) q^{69} -568.506 q^{70} -7245.58 q^{71} -610.940 q^{72} -753.067 q^{73} -6662.54i q^{74} +2932.40 q^{75} +2741.63i q^{76} -3388.56 q^{77} -1451.04 q^{78} -3742.68i q^{79} -498.461i q^{80} +729.000 q^{81} -3624.06 q^{82} -10443.1i q^{83} +1072.78i q^{84} -2111.33 q^{85} -9193.31i q^{86} +2912.12 q^{87} -2971.06i q^{88} +2622.70i q^{89} +594.784i q^{90} +2547.95i q^{91} +(-2878.59 + 3102.18i) q^{92} +5620.16 q^{93} -3289.96 q^{94} +2669.13 q^{95} -940.604 q^{96} -8542.41i q^{97} -4907.31 q^{98} +3545.19i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) $ 16 * q + 128 * q^4 + 432 * q^9	$ 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) $ 16 * q + 128 * q^4 + 432 * q^9 - 208 * q^13 + 1024 * q^16 + 840 * q^23 + 1056 * q^25 + 1920 * q^26 + 3600 * q^29 + 224 * q^31 - 3264 * q^35 + 3456 * q^36 - 2016 * q^39 - 6144 * q^41 + 1280 * q^46 + 8880 * q^47 - 13888 * q^49 + 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 - 18240 * q^59 + 8192 * q^64 + 10584 * q^69 + 19584 * q^70 - 30048 * q^71 + 9536 * q^73 - 4176 * q^75 + 14160 * q^77 + 6912 * q^78 + 11664 * q^81 - 19584 * q^82 - 32496 * q^85 - 8064 * q^87 + 6720 * q^92 - 11952 * q^93 - 21248 * q^94 - 20064 * q^95 + 21504 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−2.82843			−0.707107
$2$	−2.82843			−0.707107
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	8.00000			0.500000
$5$		−	7.78845i		−	0.311538i	−0.987794	−	0.155769i	$-0.950214\pi$
$5$		−	7.78845i		−	0.311538i	0.987794	−	0.155769i	$-0.0497855\pi$
$6$	−14.6969			−0.408248
$7$			25.8071i			0.526675i	0.964704	+	0.263338i	$0.0848233\pi$
$7$			25.8071i			0.526675i	−0.964704	+	0.263338i	$0.915177\pi$
$8$	−22.6274			−0.353553
$9$	27.0000			0.333333
$10$			22.0291i			0.220291i
$11$			131.303i			1.08515i	0.840007	+	0.542576i	$0.182551\pi$
$11$			131.303i			1.08515i	−0.840007	+	0.542576i	$0.817449\pi$
$12$	41.5692			0.288675
$13$	98.7308			0.584206			0.292103	−	0.956387i	$-0.405645\pi$
$13$	98.7308			0.584206			0.292103	+	0.956387i	$0.405645\pi$
$14$		−	72.9935i		−	0.372416i
$15$		−	40.4700i		−	0.179866i
$16$	64.0000			0.250000
$17$		−	271.085i		−	0.938011i	−0.883195	−	0.469006i	$-0.844612\pi$
$17$		−	271.085i		−	0.938011i	0.883195	−	0.469006i	$-0.155388\pi$
$18$	−76.3675			−0.235702
$19$			342.704i			0.949319i	0.880170	+	0.474659i	$0.157429\pi$
$19$			342.704i			0.949319i	−0.880170	+	0.474659i	$0.842571\pi$
$20$		−	62.3076i		−	0.155769i
$21$			134.098i			0.304076i
$22$		−	371.382i		−	0.767319i
$23$	−359.824	+	387.773i	−0.680196	+	0.733030i
$23$	−359.824	+	387.773i	−0.680196	+	0.733030i
$24$	−117.576			−0.204124
$25$	564.340			0.902944
$26$	−279.253			−0.413096
$27$	140.296			0.192450
$28$			206.457i			0.263338i
$29$	560.437			0.666394			0.333197	−	0.942857i	$-0.391873\pi$
$29$	560.437			0.666394			0.333197	+	0.942857i	$0.391873\pi$
$30$			114.466i			0.127185i
$31$	1081.60			1.12550			0.562748	−	0.826629i	$-0.309744\pi$
$31$	1081.60			1.12550			0.562748	+	0.826629i	$0.309744\pi$
$32$	−181.019			−0.176777
$33$			682.273i			0.626513i
$34$			766.745i			0.663274i
$35$	200.997			0.164079
$36$	216.000			0.166667
$37$			2355.56i			1.72065i	0.509750	+	0.860323i	$0.329738\pi$
$37$			2355.56i			1.72065i	−0.509750	+	0.860323i	$0.670262\pi$
$38$		−	969.313i		−	0.671270i
$39$	513.020			0.337291
$40$			176.232i			0.110145i
$41$	1281.30			0.762224			0.381112	−	0.924529i	$-0.375541\pi$
$41$	1281.30			0.762224			0.381112	+	0.924529i	$0.375541\pi$
$42$		−	379.285i		−	0.215014i
$43$			3250.33i			1.75788i	0.476929	+	0.878942i	$0.341750\pi$
$43$			3250.33i			1.75788i	−0.476929	+	0.878942i	$0.658250\pi$
$44$			1050.43i			0.542576i
$45$		−	210.288i		−	0.103846i
$46$	1017.74	−	1096.79i	0.480971	−	0.518331i
$47$	1163.18			0.526562			0.263281	−	0.964719i	$-0.415195\pi$
$47$	1163.18			0.526562			0.263281	+	0.964719i	$0.415195\pi$
$48$	332.554			0.144338
$49$	1734.99			0.722613
$50$	−1596.19			−0.638478
$51$		−	1408.60i		−	0.541561i
$52$	789.846			0.292103
$53$		−	1599.79i		−	0.569525i	−0.958598	−	0.284762i	$-0.908085\pi$
$53$		−	1599.79i		−	0.569525i	0.958598	−	0.284762i	$-0.0919147\pi$
$54$	−396.817			−0.136083
$55$	1022.65			0.338066
$56$		−	583.948i		−	0.186208i
$57$			1780.74i			0.548089i
$58$	−1585.16			−0.471212
$59$	−3664.84			−1.05281			−0.526406	−	0.850233i	$-0.676461\pi$
$59$	−3664.84			−1.05281			−0.526406	+	0.850233i	$0.676461\pi$
$60$		−	323.760i		−	0.0899332i
$61$		−	1882.61i		−	0.505941i	−0.967474	−	0.252971i	$-0.918592\pi$
$61$		−	1882.61i		−	0.505941i	0.967474	−	0.252971i	$-0.0814076\pi$
$62$	−3059.23			−0.795845
$63$			696.791i			0.175558i
$64$	512.000			0.125000
$65$		−	768.959i		−	0.182002i
$66$		−	1929.76i		−	0.443012i
$67$		−	1313.92i		−	0.292698i	−0.989233	−	0.146349i	$-0.953248\pi$
$67$		−	1313.92i		−	0.292698i	0.989233	−	0.146349i	$-0.0467523\pi$
$68$		−	2168.68i		−	0.469006i
$69$	−1869.70	+	2014.93i	−0.392711	+	0.423215i
$70$	−568.506			−0.116022
$71$	−7245.58			−1.43733			−0.718665	−	0.695357i	$-0.755246\pi$
$71$	−7245.58			−1.43733			−0.718665	+	0.695357i	$0.755246\pi$
$72$	−610.940			−0.117851
$73$	−753.067			−0.141315			−0.0706574	−	0.997501i	$-0.522510\pi$
$73$	−753.067			−0.141315			−0.0706574	+	0.997501i	$0.522510\pi$
$74$		−	6662.54i		−	1.21668i
$75$	2932.40			0.521315
$76$			2741.63i			0.474659i
$77$	−3388.56			−0.571523
$78$	−1451.04			−0.238501
$79$		−	3742.68i		−	0.599692i	−0.953988	−	0.299846i	$-0.903065\pi$
$79$		−	3742.68i		−	0.599692i	0.953988	−	0.299846i	$-0.0969354\pi$
$80$		−	498.461i		−	0.0778845i
$81$	729.000			0.111111
$82$	−3624.06			−0.538974
$83$		−	10443.1i		−	1.51591i	−0.652308	−	0.757954i	$-0.726199\pi$
$83$		−	10443.1i		−	1.51591i	0.652308	−	0.757954i	$-0.273801\pi$
$84$			1072.78i			0.152038i
$85$	−2111.33			−0.292226
$86$		−	9193.31i		−	1.24301i
$87$	2912.12			0.384743
$88$		−	2971.06i		−	0.383659i
$89$			2622.70i			0.331107i	0.986201	+	0.165553i	$0.0529410\pi$
$89$			2622.70i			0.331107i	−0.986201	+	0.165553i	$0.947059\pi$
$90$			594.784i			0.0734302i
$91$			2547.95i			0.307687i
$92$	−2878.59	+	3102.18i	−0.340098	+	0.366515i
$93$	5620.16			0.649805
$94$	−3289.96			−0.372336
$95$	2669.13			0.295749
$96$	−940.604			−0.102062
$97$		−	8542.41i		−	0.907898i	−0.891028	−	0.453949i	$-0.850015\pi$
$97$		−	8542.41i		−	0.907898i	0.891028	−	0.453949i	$-0.149985\pi$
$98$	−4907.31			−0.510965
$99$			3545.19i			0.361717i
$100$	4514.72			0.451472
$101$	−7160.70			−0.701960			−0.350980	−	0.936383i	$-0.614152\pi$
$101$	−7160.70			−0.701960			−0.350980	+	0.936383i	$0.614152\pi$
$102$			3984.12i			0.382942i
$103$		−	2989.95i		−	0.281831i	−0.990022	−	0.140916i	$-0.954995\pi$
$103$		−	2989.95i		−	0.281831i	0.990022	−	0.140916i	$-0.0450046\pi$
$104$	−2234.02			−0.206548
$105$	1044.41			0.0947312
$106$			4524.90i			0.402715i
$107$			8544.34i			0.746295i	0.927772	+	0.373148i	$0.121722\pi$
$107$			8544.34i			0.746295i	−0.927772	+	0.373148i	$0.878278\pi$
$108$	1122.37			0.0962250
$109$			12624.6i			1.06259i	0.847188	+	0.531293i	$0.178294\pi$
$109$			12624.6i			1.06259i	−0.847188	+	0.531293i	$0.821706\pi$
$110$	−2892.49			−0.239049
$111$			12239.9i			0.993415i
$112$			1651.65i			0.131669i
$113$		−	18761.7i		−	1.46931i	−0.678438	−	0.734657i	$-0.737343\pi$
$113$		−	18761.7i		−	1.46931i	0.678438	−	0.734657i	$-0.262657\pi$
$114$		−	5036.70i		−	0.387558i
$115$	3020.15	+	2802.47i	0.228367	+	0.211907i
$116$	4483.50			0.333197
$117$	2665.73			0.194735
$118$	10365.7			0.744451
$119$	6995.92			0.494027
$120$			915.730i			0.0635924i
$121$	−2599.59			−0.177556
$122$			5324.82i			0.357755i
$123$	6657.82			0.440070
$124$	8652.81			0.562748
$125$		−	9263.11i		−	0.592839i
$126$		−	1970.82i		−	0.124139i
$127$	−2470.20			−0.153153			−0.0765763	−	0.997064i	$-0.524399\pi$
$127$	−2470.20			−0.153153			−0.0765763	+	0.997064i	$0.524399\pi$
$128$	−1448.15			−0.0883883
$129$			16889.2i			1.01491i
$130$			2174.95i			0.128695i
$131$	3212.82			0.187216			0.0936082	−	0.995609i	$-0.470160\pi$
$131$	3212.82			0.187216			0.0936082	+	0.995609i	$0.470160\pi$
$132$			5458.18i			0.313256i
$133$	−8844.19			−0.499983
$134$			3716.33i			0.206969i
$135$		−	1092.69i		−	0.0599555i
$136$			6133.96i			0.331637i
$137$			22399.6i			1.19344i	0.802450	+	0.596719i	$0.203529\pi$
$137$			22399.6i			1.19344i	−0.802450	+	0.596719i	$0.796471\pi$
$138$	5288.31	−	5699.08i	0.277689	−	0.299258i
$139$	5427.20			0.280896			0.140448	−	0.990088i	$-0.455146\pi$
$139$	5427.20			0.280896			0.140448	+	0.990088i	$0.455146\pi$
$140$	1607.98			0.0820396
$141$	6044.04			0.304011
$142$	20493.6			1.01635
$143$			12963.7i			0.633952i
$144$	1728.00			0.0833333
$145$		−	4364.94i		−	0.207607i
$146$	2129.99			0.0999247
$147$	9015.30			0.417201
$148$			18844.5i			0.860323i
$149$			3494.78i			0.157415i	0.996898	+	0.0787077i	$0.0250794\pi$
$149$			3494.78i			0.157415i	−0.996898	+	0.0787077i	$0.974921\pi$
$150$	−8294.07			−0.368625
$151$	−16646.4			−0.730074			−0.365037	−	0.930993i	$-0.618944\pi$
$151$	−16646.4			−0.730074			−0.365037	+	0.930993i	$0.618944\pi$
$152$		−	7754.51i		−	0.335635i
$153$		−	7319.30i		−	0.312670i
$154$	9584.29			0.404128
$155$		−	8423.99i		−	0.350634i
$156$	4104.16			0.168646
$157$		−	45961.1i		−	1.86462i	−0.361657	−	0.932311i	$-0.617789\pi$
$157$		−	45961.1i		−	1.86462i	0.361657	−	0.932311i	$-0.382211\pi$
$158$			10585.9i			0.424047i
$159$		−	8312.78i		−	0.328815i
$160$			1409.86i			0.0550726i
$161$	−10007.3	−	9286.00i	−0.386069	−	0.358242i
$162$	−2061.92			−0.0785674
$163$	−24194.8			−0.910642			−0.455321	−	0.890327i	$-0.650475\pi$
$163$	−24194.8			−0.910642			−0.455321	+	0.890327i	$0.650475\pi$
$164$	10250.4			0.381112
$165$	5313.84			0.195183
$166$			29537.5i			1.07191i
$167$	7980.80			0.286163			0.143081	−	0.989711i	$-0.454299\pi$
$167$	7980.80			0.286163			0.143081	+	0.989711i	$0.454299\pi$
$168$		−	3034.28i		−	0.107507i
$169$	−18813.2			−0.658704
$170$	5971.75			0.206635
$171$			9253.01i			0.316440i
$172$			26002.6i			0.878942i
$173$	−9533.82			−0.318548			−0.159274	−	0.987234i	$-0.550915\pi$
$173$	−9533.82			−0.318548			−0.159274	+	0.987234i	$0.550915\pi$
$174$	−8236.71			−0.272054
$175$			14564.0i			0.475558i
$176$			8403.42i			0.271288i
$177$	−19043.1			−0.607841
$178$		−	7418.11i		−	0.234128i
$179$	32123.8			1.00258			0.501292	−	0.865278i	$-0.332858\pi$
$179$	32123.8			1.00258			0.501292	+	0.865278i	$0.332858\pi$
$180$		−	1682.30i		−	0.0519230i
$181$		−	46234.6i		−	1.41127i	−0.708576	−	0.705635i	$-0.750662\pi$
$181$		−	46234.6i		−	1.41127i	0.708576	−	0.705635i	$-0.249338\pi$
$182$		−	7206.70i		−	0.217567i
$183$		−	9782.32i		−	0.292105i
$184$	8141.88	−	8774.30i	0.240486	−	0.259165i
$185$	18346.2			0.536046
$186$	−15896.2			−0.459482
$187$	35594.4			1.01789
$188$	9305.40			0.263281
$189$			3620.63i			0.101359i
$190$	−7549.44			−0.209126
$191$			30406.5i			0.833488i	0.909024	+	0.416744i	$0.136829\pi$
$191$			30406.5i			0.833488i	−0.909024	+	0.416744i	$0.863171\pi$
$192$	2660.43			0.0721688
$193$	−53428.8			−1.43437			−0.717185	−	0.696883i	$-0.754570\pi$
$193$	−53428.8			−1.43437			−0.717185	+	0.696883i	$0.754570\pi$
$194$			24161.6i			0.641981i
$195$		−	3995.63i		−	0.105079i
$196$	13880.0			0.361307
$197$	−5008.73			−0.129061			−0.0645305	−	0.997916i	$-0.520555\pi$
$197$	−5008.73			−0.129061			−0.0645305	+	0.997916i	$0.520555\pi$
$198$		−	10027.3i		−	0.255773i
$199$		−	530.184i		−	0.0133882i	−0.999978	−	0.00669408i	$-0.997869\pi$
$199$		−	530.184i		−	0.0133882i	0.999978	−	0.00669408i	$-0.00213081\pi$
$200$	−12769.6			−0.319239
$201$		−	6827.34i		−	0.168989i
$202$	20253.5			0.496361
$203$			14463.3i			0.350973i
$204$		−	11268.8i		−	0.270781i
$205$		−	9979.33i		−	0.237462i
$206$			8456.85i			0.199285i
$207$	−9715.24	+	10469.9i	−0.226732	+	0.244343i
$208$	6318.77			0.146051
$209$	−44998.2			−1.03016
$210$	−2954.04			−0.0669851
$211$	57164.1			1.28398			0.641991	−	0.766712i	$-0.278109\pi$
$211$	57164.1			1.28398			0.641991	+	0.766712i	$0.278109\pi$
$212$		−	12798.4i		−	0.284762i
$213$	−37649.1			−0.829842
$214$		−	24167.0i		−	0.527710i
$215$	25315.0			0.547647
$216$	−3174.54			−0.0680414
$217$			27913.0i			0.592770i
$218$		−	35707.7i		−	0.751362i
$219$	−3913.05			−0.0815881
$220$	8181.20			0.169033
$221$		−	26764.5i		−	0.547992i
$222$		−	34619.6i		−	0.702450i
$223$	−46659.2			−0.938269			−0.469134	−	0.883127i	$-0.655434\pi$
$223$	−46659.2			−0.938269			−0.469134	+	0.883127i	$0.655434\pi$
$224$		−	4671.58i		−	0.0931039i
$225$	15237.2			0.300981
$226$			53066.0i			1.03896i
$227$			27732.3i			0.538189i	0.963114	+	0.269094i	$0.0867243\pi$
$227$			27732.3i			0.538189i	−0.963114	+	0.269094i	$0.913276\pi$
$228$			14245.9i			0.274045i
$229$		−	45658.6i		−	0.870666i	−0.900269	−	0.435333i	$-0.856631\pi$
$229$		−	45658.6i		−	0.870666i	0.900269	−	0.435333i	$-0.143369\pi$
$230$	−8542.27	−	7926.57i	−0.161480	−	0.149841i
$231$	−17607.5			−0.329969
$232$	−12681.3			−0.235606
$233$	52406.3			0.965321			0.482661	−	0.875808i	$-0.339671\pi$
$233$	52406.3			0.965321			0.482661	+	0.875808i	$0.339671\pi$
$234$	−7539.83			−0.137699
$235$		−	9059.33i		−	0.164044i
$236$	−29318.7			−0.526406
$237$		−	19447.5i		−	0.346233i
$238$	−19787.5			−0.349330
$239$	10056.2			0.176051			0.0880253	−	0.996118i	$-0.471944\pi$
$239$	10056.2			0.176051			0.0880253	+	0.996118i	$0.471944\pi$
$240$		−	2590.08i		−	0.0449666i
$241$		−	9571.80i		−	0.164801i	−0.996599	−	0.0824004i	$-0.973741\pi$
$241$		−	9571.80i		−	0.164801i	0.996599	−	0.0824004i	$-0.0262586\pi$
$242$	7352.75			0.125551
$243$	3788.00			0.0641500
$244$		−	15060.9i		−	0.252971i
$245$		−	13512.9i		−	0.225121i
$246$	−18831.2			−0.311177
$247$			33835.4i			0.554597i
$248$	−24473.8			−0.397923
$249$		−	54263.9i		−	0.875210i
$250$			26200.0i			0.419201i
$251$			108627.i			1.72422i	0.506723	+	0.862109i	$0.330857\pi$
$251$			108627.i			1.72422i	−0.506723	+	0.862109i	$0.669143\pi$
$252$			5574.33i			0.0877792i
$253$	−50915.9	−	47246.1i	−0.795449	−	0.738116i
$254$	6986.78			0.108295
$255$	−10970.8			−0.168717
$256$	4096.00			0.0625000
$257$	35878.8			0.543215			0.271608	−	0.962408i	$-0.412445\pi$
$257$	35878.8			0.543215			0.271608	+	0.962408i	$0.412445\pi$
$258$		−	47769.8i		−	0.717653i
$259$	−60790.2			−0.906221
$260$		−	6151.67i		−	0.0910011i
$261$	15131.8			0.222131
$262$	−9087.23			−0.132382
$263$		−	111615.i		−	1.61366i	−0.590784	−	0.806829i	$-0.701182\pi$
$263$		−	111615.i		−	1.61366i	0.590784	−	0.806829i	$-0.298818\pi$
$264$		−	15438.1i		−	0.221506i
$265$	−12459.9			−0.177428
$266$	25015.1			0.353541
$267$			13627.9i			0.191165i
$268$		−	10511.4i		−	0.146349i
$269$	−32468.2			−0.448697			−0.224349	−	0.974509i	$-0.572025\pi$
$269$	−32468.2			−0.448697			−0.224349	+	0.974509i	$0.572025\pi$
$270$			3090.59i			0.0423949i
$271$	128445.			1.74896			0.874478	−	0.485066i	$-0.161204\pi$
$271$	128445.			1.74896			0.874478	+	0.485066i	$0.161204\pi$
$272$		−	17349.5i		−	0.234503i
$273$			13239.6i			0.177643i
$274$		−	63355.7i		−	0.843888i
$275$			74099.8i			0.979832i
$276$	−14957.6	+	16119.4i	−0.196356	+	0.211608i
$277$	−99093.6			−1.29148			−0.645738	−	0.763559i	$-0.723450\pi$
$277$	−99093.6			−1.29148			−0.645738	+	0.763559i	$0.723450\pi$
$278$	−15350.4			−0.198624
$279$	29203.2			0.375165
$280$	−4548.04			−0.0580108
$281$		−	6265.89i		−	0.0793542i	−0.999213	−	0.0396771i	$-0.987367\pi$
$281$		−	6265.89i		−	0.0793542i	0.999213	−	0.0396771i	$-0.0126329\pi$
$282$	−17095.1			−0.214968
$283$		−	13632.8i		−	0.170221i	−0.996372	−	0.0851103i	$-0.972876\pi$
$283$		−	13632.8i		−	0.170221i	0.996372	−	0.0851103i	$-0.0271243\pi$
$284$	−57964.6			−0.718665
$285$	13869.2			0.170751
$286$		−	36666.9i		−	0.448272i
$287$			33066.6i			0.401444i
$288$	−4887.52			−0.0589256
$289$	10033.8			0.120135
$290$			12345.9i			0.146800i
$291$		−	44387.7i		−	0.524175i
$292$	−6024.53			−0.0706574
$293$			46401.8i			0.540505i	0.962789	+	0.270253i	$0.0871072\pi$
$293$			46401.8i			0.540505i	−0.962789	+	0.270253i	$0.912893\pi$
$294$	−25499.1			−0.295006
$295$			28543.4i			0.327991i
$296$		−	53300.3i		−	0.608340i
$297$			18421.4i			0.208838i
$298$		−	9884.73i		−	0.111309i
$299$	−35525.7	+	38285.1i	−0.397374	+	0.428240i
$300$	23459.2			0.260658
$301$	−83881.4			−0.925834
$302$	47083.2			0.516240
$303$	−37208.1			−0.405277
$304$			21933.1i			0.237330i
$305$	−14662.6			−0.157620
$306$			20702.1i			0.221091i
$307$	62000.8			0.657841			0.328920	−	0.944358i	$-0.393315\pi$
$307$	62000.8			0.657841			0.328920	+	0.944358i	$0.393315\pi$
$308$	−27108.5			−0.285761
$309$		−	15536.2i		−	0.162715i
$310$			23826.6i			0.247936i
$311$	155947.			1.61234			0.806168	−	0.591687i	$-0.201538\pi$
$311$	155947.			1.61234			0.806168	+	0.591687i	$0.201538\pi$
$312$	−11608.3			−0.119251
$313$		−	104376.i		−	1.06540i	−0.846305	−	0.532699i	$-0.821178\pi$
$313$		−	104376.i		−	1.06540i	0.846305	−	0.532699i	$-0.178822\pi$
$314$			129998.i			1.31849i
$315$	5426.92			0.0546931
$316$		−	29941.4i		−	0.299846i
$317$	145401.			1.44693			0.723466	−	0.690360i	$-0.242548\pi$
$317$	145401.			1.44693			0.723466	+	0.690360i	$0.242548\pi$
$318$			23512.1i			0.232507i
$319$			73587.4i			0.723139i
$320$		−	3987.68i		−	0.0389422i
$321$			44397.7i			0.430874i
$322$	28304.9	+	26264.8i	0.272992	+	0.253316i
$323$	92902.0			0.890472
$324$	5832.00			0.0555556
$325$	55717.7			0.527505
$326$	68433.3			0.643921
$327$			65599.3i			0.613485i
$328$	−28992.5			−0.269487
$329$			30018.2i			0.277327i
$330$	−15029.8			−0.138015
$331$	130358.			1.18982			0.594911	−	0.803792i	$-0.297187\pi$
$331$	130358.			1.18982			0.594911	+	0.803792i	$0.297187\pi$
$332$		−	83544.7i		−	0.757954i
$333$			63600.2i			0.573548i
$334$	−22573.1			−0.202348
$335$	−10233.4			−0.0911866
$336$			8582.24i			0.0760190i
$337$			52770.8i			0.464658i	0.972637	+	0.232329i	$0.0746347\pi$
$337$			52770.8i			0.464658i	−0.972637	+	0.232329i	$0.925365\pi$
$338$	53211.9			0.465774
$339$		−	97488.5i		−	0.848309i
$340$	−16890.7			−0.146113
$341$			142018.i			1.22133i
$342$		−	26171.5i		−	0.223757i
$343$			106738.i			0.907258i
$344$		−	73546.5i		−	0.621506i
$345$	15693.2	+	14562.0i	0.131848	+	0.122344i
$346$	26965.7			0.225247
$347$	−62012.9			−0.515019			−0.257509	−	0.966276i	$-0.582902\pi$
$347$	−62012.9			−0.515019			−0.257509	+	0.966276i	$0.582902\pi$
$348$	23296.9			0.192371
$349$	−38439.9			−0.315596			−0.157798	−	0.987471i	$-0.550439\pi$
$349$	−38439.9			−0.315596			−0.157798	+	0.987471i	$0.550439\pi$
$350$		−	41193.1i		−	0.336270i
$351$	13851.5			0.112430
$352$		−	23768.5i		−	0.191830i
$353$	−107559.			−0.863175			−0.431588	−	0.902071i	$-0.642046\pi$
$353$	−107559.			−0.863175			−0.431588	+	0.902071i	$0.642046\pi$
$354$	53861.9			0.429809
$355$			56431.8i			0.447782i
$356$			20981.6i			0.165553i
$357$	36351.9			0.285227
$358$	−90859.9			−0.708935
$359$			162229.i			1.25875i	0.777102	+	0.629375i	$0.216689\pi$
$359$			162229.i			1.25875i	−0.777102	+	0.629375i	$0.783311\pi$
$360$			4758.28i			0.0367151i
$361$	12875.0			0.0987943
$362$			130771.i			0.997919i
$363$	−13507.9			−0.102512
$364$			20383.6i			0.153843i
$365$			5865.22i			0.0440249i
$366$			27668.6i			0.206550i
$367$		−	115685.i		−	0.858908i	−0.903089	−	0.429454i	$-0.858706\pi$
$367$		−	115685.i		−	0.858908i	0.903089	−	0.429454i	$-0.141294\pi$
$368$	−23028.7	+	24817.5i	−0.170049	+	0.183258i
$369$	34595.1			0.254075
$370$	−51890.8			−0.379042
$371$	41286.0			0.299954
$372$	44961.3			0.324903
$373$			46264.2i			0.332527i	0.986081	+	0.166264i	$0.0531703\pi$
$373$			46264.2i			0.332527i	−0.986081	+	0.166264i	$0.946830\pi$
$374$	−100676.			−0.719754
$375$		−	48132.5i		−	0.342276i
$376$	−26319.7			−0.186168
$377$	55332.4			0.389311
$378$		−	10240.7i		−	0.0716714i
$379$		−	45500.3i		−	0.316764i	−0.987378	−	0.158382i	$-0.949372\pi$
$379$		−	45500.3i		−	0.316764i	0.987378	−	0.158382i	$-0.0506278\pi$
$380$	21353.1			0.147874
$381$	−12835.5			−0.0884227
$382$		−	86002.5i		−	0.589365i
$383$		−	198234.i		−	1.35139i	−0.737180	−	0.675696i	$-0.763843\pi$
$383$		−	198234.i		−	1.35139i	0.737180	−	0.675696i	$-0.236157\pi$
$384$	−7524.83			−0.0510310
$385$			26391.6i			0.178051i
$386$	151120.			1.01425
$387$			87758.8i			0.585961i
$388$		−	68339.3i		−	0.453949i
$389$		−	230824.i		−	1.52539i	−0.646756	−	0.762697i	$-0.723875\pi$
$389$		−	230824.i		−	1.52539i	0.646756	−	0.762697i	$-0.276125\pi$
$390$			11301.3i			0.0743021i
$391$	105120.	+	97542.9i	0.687591	+	0.638032i
$392$	−39258.4			−0.255482
$393$	16694.3			0.108089
$394$	14166.8			0.0912599
$395$	−29149.7			−0.186827
$396$			28361.5i			0.180859i
$397$	−172503.			−1.09450			−0.547251	−	0.836968i	$-0.684326\pi$
$397$	−172503.			−1.09450			−0.547251	+	0.836968i	$0.684326\pi$
$398$			1499.59i			0.00946685i
$399$	−45955.8			−0.288665
$400$	36117.8			0.225736
$401$			161378.i			1.00359i	0.864987	+	0.501795i	$0.167327\pi$
$401$			161378.i			1.00359i	−0.864987	+	0.501795i	$0.832673\pi$
$402$			19310.6i			0.119494i
$403$	106787.			0.657521
$404$	−57285.6			−0.350980
$405$		−	5677.78i		−	0.0346153i
$406$		−	40908.3i		−	0.248176i
$407$	−309294.			−1.86716
$408$			31873.0i			0.191471i
$409$	317644.			1.89887			0.949434	−	0.313967i	$-0.101658\pi$
$409$	317644.			1.89887			0.949434	+	0.313967i	$0.101658\pi$
$410$			28225.8i			0.167911i
$411$			116392.i			0.689032i
$412$		−	23919.6i		−	0.140916i
$413$		−	94578.8i		−	0.554490i
$414$	27478.8	−	29613.3i	0.160324	−	0.172777i
$415$	−81335.4			−0.472263
$416$	−17872.2			−0.103274
$417$	28200.6			0.162176
$418$	127274.			0.728430
$419$		−	97451.5i		−	0.555086i	−0.960713	−	0.277543i	$-0.910480\pi$
$419$		−	97451.5i		−	0.555086i	0.960713	−	0.277543i	$-0.0895201\pi$
$420$	8355.29			0.0473656
$421$			72442.0i			0.408721i	0.978896	+	0.204360i	$0.0655114\pi$
$421$			72442.0i			0.408721i	−0.978896	+	0.204360i	$0.934489\pi$
$422$	−161685.			−0.907912
$423$	31405.7			0.175521
$424$			36199.2i			0.201357i
$425$		−	152984.i		−	0.846972i
$426$	106488.			0.586787
$427$	48584.6			0.266467
$428$			68354.7i			0.373148i
$429$			67361.3i			0.366012i
$430$	−71601.6			−0.387245
$431$		−	187900.i		−	1.01151i	−0.862676	−	0.505757i	$-0.831213\pi$
$431$		−	187900.i		−	1.01151i	0.862676	−	0.505757i	$-0.168787\pi$
$432$	8978.95			0.0481125
$433$		−	258865.i		−	1.38069i	−0.723479	−	0.690346i	$-0.757458\pi$
$433$		−	258865.i		−	1.38069i	0.723479	−	0.690346i	$-0.242542\pi$
$434$		−	78949.8i		−	0.419152i
$435$		−	22680.9i		−	0.119862i
$436$			100997.i			0.531293i
$437$	−132891.	−	123313.i	−0.695879	−	0.645723i
$438$	11067.8			0.0576915
$439$	−330835.			−1.71665			−0.858326	−	0.513105i	$-0.828495\pi$
$439$	−330835.			−1.71665			−0.858326	+	0.513105i	$0.828495\pi$
$440$	−23139.9			−0.119524
$441$	46844.9			0.240871
$442$			75701.3i			0.387489i
$443$	−330611.			−1.68465			−0.842324	−	0.538971i	$-0.818813\pi$
$443$	−330611.			−1.68465			−0.842324	+	0.538971i	$0.818813\pi$
$444$			97918.9i			0.496707i
$445$	20426.7			0.103152
$446$	131972.			0.663456
$447$			18159.4i			0.0908838i
$448$			13213.2i			0.0658344i
$449$	180789.			0.896767			0.448383	−	0.893841i	$-0.352000\pi$
$449$	180789.			0.896767			0.448383	+	0.893841i	$0.352000\pi$
$450$	−43097.3			−0.212826
$451$			168239.i			0.827129i
$452$		−	150093.i		−	0.734657i
$453$	−86497.3			−0.421508
$454$		−	78438.8i		−	0.380557i
$455$	19844.6			0.0958560
$456$		−	40293.6i		−	0.193779i
$457$		−	8836.98i		−	0.0423128i	−0.999776	−	0.0211564i	$-0.993265\pi$
$457$		−	8836.98i		−	0.0423128i	0.999776	−	0.0211564i	$-0.00673479\pi$
$458$			129142.i			0.615654i
$459$		−	38032.2i		−	0.180520i
$460$	24161.2	+	22419.7i	0.114183	+	0.105953i
$461$	−96996.3			−0.456408			−0.228204	−	0.973613i	$-0.573285\pi$
$461$	−96996.3			−0.456408			−0.228204	+	0.973613i	$0.573285\pi$
$462$	49801.4			0.233323
$463$	−72759.5			−0.339413			−0.169706	−	0.985495i	$-0.554282\pi$
$463$	−72759.5			−0.339413			−0.169706	+	0.985495i	$0.554282\pi$
$464$	35868.0			0.166599
$465$		−	43772.3i		−	0.202439i
$466$	−148227.			−0.682585
$467$		−	368036.i		−	1.68755i	−0.536697	−	0.843775i	$-0.680328\pi$
$467$		−	368036.i		−	1.68755i	0.536697	−	0.843775i	$-0.319672\pi$
$468$	21325.8			0.0973676
$469$	33908.5			0.154157
$470$			25623.7i			0.115997i
$471$		−	238821.i		−	1.07654i
$472$	82925.9			0.372225
$473$	−426779.			−1.90757
$474$			55005.9i			0.244823i
$475$			193402.i			0.857182i
$476$	55967.4			0.247014
$477$		−	43194.5i		−	0.189842i
$478$	−28443.2			−0.124487
$479$		−	356541.i		−	1.55395i	−0.629529	−	0.776977i	$-0.716752\pi$
$479$		−	356541.i		−	1.55395i	0.629529	−	0.776977i	$-0.283248\pi$
$480$			7325.84i			0.0317962i
$481$			232567.i			1.00521i
$482$			27073.1i			0.116532i
$483$	−51999.4	−	48251.5i	−0.222897	−	0.206831i
$484$	−20796.7			−0.0887778
$485$	−66532.1			−0.282845
$486$	−10714.1			−0.0453609
$487$	−141982.			−0.598653			−0.299326	−	0.954151i	$-0.596762\pi$
$487$	−141982.			−0.598653			−0.299326	+	0.954151i	$0.596762\pi$
$488$			42598.6i			0.178877i
$489$	−125720.			−0.525759
$490$			38220.3i			0.159185i
$491$	−121954.			−0.505864			−0.252932	−	0.967484i	$-0.581395\pi$
$491$	−121954.			−0.505864			−0.252932	+	0.967484i	$0.581395\pi$
$492$	53262.6			0.220035
$493$		−	151926.i		−	0.625085i
$494$		−	95701.0i		−	0.392160i
$495$	27611.5			0.112689
$496$	69222.5			0.281374
$497$		−	186987.i		−	0.757006i
$498$			153481.i			0.618867i
$499$	370036.			1.48608			0.743040	−	0.669247i	$-0.233383\pi$
$499$	370036.			1.48608			0.743040	+	0.669247i	$0.233383\pi$
$500$		−	74104.9i		−	0.296420i
$501$	41469.4			0.165216
$502$		−	307245.i		−	1.21921i
$503$		−	302069.i		−	1.19391i	−0.802277	−	0.596953i	$-0.796378\pi$
$503$		−	302069.i		−	1.19391i	0.802277	−	0.596953i	$-0.203622\pi$
$504$		−	15766.6i		−	0.0620693i
$505$			55770.7i			0.218687i
$506$	144012.	+	133632.i	0.562468	+	0.521927i
$507$	−97756.4			−0.380303
$508$	−19761.6			−0.0765763
$509$	−351023.			−1.35488			−0.677439	−	0.735579i	$-0.736910\pi$
$509$	−351023.			−1.35488			−0.677439	+	0.735579i	$0.736910\pi$
$510$	31030.1			0.119301
$511$		−	19434.5i		−	0.0744270i
$512$	−11585.2			−0.0441942
$513$			48080.0i			0.182696i
$514$	−101481.			−0.384111
$515$	−23287.0			−0.0878011
$516$			135114.i			0.507457i
$517$			152729.i			0.571400i
$518$	171941.			0.640795
$519$	−49539.2			−0.183914
$520$			17399.6i			0.0643475i
$521$			357021.i			1.31528i	0.753332	+	0.657641i	$0.228446\pi$
$521$			357021.i			1.31528i	−0.753332	+	0.657641i	$0.771554\pi$
$522$	−42799.2			−0.157071
$523$		−	44504.4i		−	0.162704i	−0.996685	−	0.0813522i	$-0.974076\pi$
$523$		−	44504.4i		−	0.162704i	0.996685	−	0.0813522i	$-0.0259239\pi$
$524$	25702.6			0.0936082
$525$			75676.6i			0.274564i
$526$			315695.i			1.14103i
$527$		−	293206.i		−	1.05573i
$528$			43665.4i			0.156628i
$529$	−20894.8	−	279060.i	−0.0746667	−	0.997209i
$530$	35242.0			0.125461
$531$	−98950.7			−0.350937
$532$	−70753.5			−0.249991
$533$	126504.			0.445296
$534$		−	38545.6i		−	0.135174i
$535$	66547.1			0.232499
$536$			29730.7i			0.103484i
$537$	166920.			0.578843
$538$	91833.9			0.317277
$539$			227811.i			0.784145i
$540$		−	8741.51i		−	0.0299777i
$541$	482897.			1.64991			0.824954	−	0.565200i	$-0.191201\pi$
$541$	482897.			1.64991			0.824954	+	0.565200i	$0.191201\pi$
$542$	−363297.			−1.23670
$543$		−	240242.i		−	0.814797i
$544$			49071.7i			0.165819i
$545$	98326.0			0.331036
$546$		−	37447.1i		−	0.125613i
$547$	418376.			1.39827			0.699137	−	0.714988i	$-0.253568\pi$
$547$	418376.			1.39827			0.699137	+	0.714988i	$0.253568\pi$
$548$			179197.i			0.596719i
$549$		−	50830.4i		−	0.168647i
$550$		−	209586.i		−	0.692846i
$551$			192064.i			0.632620i
$552$	42306.5	−	45592.6i	0.138844	−	0.149629i
$553$	96587.7			0.315843
$554$	280279.			0.913211
$555$	95329.5			0.309486
$556$	43417.6			0.140448
$557$			89511.3i			0.288514i	0.989540	+	0.144257i	$0.0460793\pi$
$557$			89511.3i			0.288514i	−0.989540	+	0.144257i	$0.953921\pi$
$558$	−82599.2			−0.265282
$559$			320907.i			1.02697i
$560$	12863.8			0.0410198
$561$	184954.			0.587676
$562$			17722.6i			0.0561119i
$563$			256215.i			0.808328i	0.914686	+	0.404164i	$0.132438\pi$
$563$			256215.i			0.808328i	−0.914686	+	0.404164i	$0.867562\pi$
$564$	48352.3			0.152005
$565$	−146124.			−0.457747
$566$			38559.4i			0.120364i
$567$			18813.4i			0.0585195i
$568$	163949.			0.508173
$569$			463862.i			1.43273i	0.697725	+	0.716366i	$0.254196\pi$
$569$			463862.i			1.43273i	−0.697725	+	0.716366i	$0.745804\pi$
$570$	−39228.1			−0.120739
$571$			615484.i			1.88775i	0.330301	+	0.943876i	$0.392850\pi$
$571$			615484.i			1.88775i	−0.330301	+	0.943876i	$0.607150\pi$
$572$			103710.i			0.316976i
$573$			157997.i			0.481214i
$574$		−	93526.4i		−	0.283864i
$575$	−203063.	+	218836.i	−0.614179	+	0.661885i
$576$	13824.0			0.0416667
$577$	−477717.			−1.43489			−0.717445	−	0.696615i	$-0.754689\pi$
$577$	−477717.			−1.43489			−0.717445	+	0.696615i	$0.754689\pi$
$578$	−28379.8			−0.0849481
$579$	−277624.			−0.828134
$580$		−	34919.5i		−	0.103803i
$581$	269506.			0.798391
$582$			125547.i			0.370648i
$583$	210059.			0.618021
$584$	17040.0			0.0499623
$585$		−	20761.9i		−	0.0606674i
$586$		−	131244.i		−	0.382195i
$587$	459886.			1.33467			0.667334	−	0.744758i	$-0.267435\pi$
$587$	459886.			1.33467			0.667334	+	0.744758i	$0.267435\pi$
$588$	72122.4			0.208600
$589$			370669.i			1.06845i
$590$		−	80732.9i		−	0.231925i
$591$	−26026.1			−0.0745134
$592$			150756.i			0.430161i
$593$	673907.			1.91642			0.958210	−	0.286064i	$-0.0923471\pi$
$593$	673907.			1.91642			0.958210	+	0.286064i	$0.0923471\pi$
$594$		−	52103.5i		−	0.147671i
$595$		−	54487.3i		−	0.153908i
$596$			27958.2i			0.0787077i
$597$		−	2754.92i		−	0.00772965i
$598$	100482.	−	108287.i	0.280986	−	0.302812i
$599$	−135272.			−0.377011			−0.188506	−	0.982072i	$-0.560364\pi$
$599$	−135272.			−0.377011			−0.188506	+	0.982072i	$0.560364\pi$
$600$	−66352.6			−0.184313
$601$	−334468.			−0.925988			−0.462994	−	0.886361i	$-0.653225\pi$
$601$	−334468.			−0.925988			−0.462994	+	0.886361i	$0.653225\pi$
$602$	237253.			0.654663
$603$		−	35475.9i		−	0.0975661i
$604$	−133171.			−0.365037
$605$			20246.8i			0.0553153i
$606$	105240.			0.286574
$607$	365940.			0.993190			0.496595	−	0.867982i	$-0.334583\pi$
$607$	365940.			0.993190			0.496595	+	0.867982i	$0.334583\pi$
$608$		−	62036.0i		−	0.167817i
$609$			75153.3i			0.202634i
$610$	41472.1			0.111454
$611$	114841.			0.307621
$612$		−	58554.4i		−	0.156335i
$613$		−	59835.4i		−	0.159235i	−0.996826	−	0.0796173i	$-0.974630\pi$
$613$		−	59835.4i		−	0.159235i	0.996826	−	0.0796173i	$-0.0253698\pi$
$614$	−175365.			−0.465163
$615$		−	51854.1i		−	0.137099i
$616$	76674.3			0.202064
$617$			126109.i			0.331264i	0.986188	+	0.165632i	$0.0529664\pi$
$617$			126109.i			0.331264i	−0.986188	+	0.165632i	$0.947034\pi$
$618$			43943.1i			0.115057i
$619$		−	242271.i		−	0.632295i	−0.948710	−	0.316147i	$-0.897611\pi$
$619$		−	242271.i		−	0.632295i	0.948710	−	0.316147i	$-0.102389\pi$
$620$		−	67391.9i		−	0.175317i
$621$	−50481.9	+	54403.0i	−0.130904	+	0.141072i
$622$	−441084.			−1.14009
$623$	−67684.2			−0.174386
$624$	32833.3			0.0843228
$625$	280567.			0.718252
$626$			295220.i			0.753350i
$627$	−233818.			−0.594760
$628$		−	367689.i		−	0.932311i
$629$	638558.			1.61398
$630$	−15349.7			−0.0386738
$631$		−	459389.i		−	1.15378i	−0.816823	−	0.576888i	$-0.804267\pi$
$631$		−	459389.i		−	1.15378i	0.816823	−	0.576888i	$-0.195733\pi$
$632$			84687.2i			0.212023i
$633$	297034.			0.741307
$634$	−411255.			−1.02314
$635$			19239.0i			0.0477128i
$636$		−	66502.2i		−	0.164408i
$637$	171297.			0.422155
$638$		−	208136.i		−	0.511337i
$639$	−195631.			−0.479110
$640$			11278.9i			0.0275363i
$641$			198280.i			0.482573i	0.970454	+	0.241286i	$0.0775693\pi$
$641$			198280.i			0.482573i	−0.970454	+	0.241286i	$0.922431\pi$
$642$		−	125576.i		−	0.304674i
$643$			417638.i			1.01013i	0.863081	+	0.505066i	$0.168532\pi$
$643$			417638.i			1.01013i	−0.863081	+	0.505066i	$0.831468\pi$
$644$	−80058.3	−	74288.0i	−0.193034	−	0.179121i
$645$	131541.			0.316184
$646$	−262767.			−0.629658
$647$	−595986.			−1.42373			−0.711865	−	0.702316i	$-0.752149\pi$
$647$	−595986.			−1.42373			−0.711865	+	0.702316i	$0.752149\pi$
$648$	−16495.4			−0.0392837
$649$		−	481206.i		−	1.14246i
$650$	−157594.			−0.373002
$651$			145040.i			0.342236i
$652$	−193559.			−0.455321
$653$	406716.			0.953818			0.476909	−	0.878953i	$-0.341757\pi$
$653$	406716.			0.953818			0.476909	+	0.878953i	$0.341757\pi$
$654$		−	185543.i		−	0.433799i
$655$		−	25022.9i		−	0.0583250i
$656$	82003.1			0.190556
$657$	−20332.8			−0.0471049
$658$		−	84904.2i		−	0.196100i
$659$		−	253852.i		−	0.584533i	−0.956337	−	0.292267i	$-0.905591\pi$
$659$		−	253852.i		−	0.584533i	0.956337	−	0.292267i	$-0.0944095\pi$
$660$	42510.7			0.0975913
$661$		−	9196.14i		−	0.0210476i	−0.999945	−	0.0105238i	$-0.996650\pi$
$661$		−	9196.14i		−	0.0210476i	0.999945	−	0.0105238i	$-0.00334989\pi$
$662$	−368708.			−0.841331
$663$		−	139072.i		−	0.316383i
$664$			236300.i			0.535954i
$665$			68882.5i			0.155763i
$666$		−	179889.i		−	0.405560i
$667$	−201659.	+	217322.i	−0.453279	+	0.488487i
$668$	63846.4			0.143081
$669$	−242448.			−0.541710
$670$	28944.5			0.0644786
$671$	247193.			0.549024
$672$		−	24274.2i		−	0.0537536i
$673$	−795973.			−1.75739			−0.878695	−	0.477383i	$-0.841585\pi$
$673$	−795973.			−1.75739			−0.878695	+	0.477383i	$0.841585\pi$
$674$		−	149258.i		−	0.328563i
$675$	79174.7			0.173772
$676$	−150506.			−0.329352
$677$		−	95036.2i		−	0.207354i	−0.994611	−	0.103677i	$-0.966939\pi$
$677$		−	95036.2i		−	0.207354i	0.994611	−	0.103677i	$-0.0330608\pi$
$678$			275739.i			0.599845i
$679$	220455.			0.478167
$680$	47774.0			0.103317
$681$			144101.i			0.310723i
$682$		−	401687.i		−	0.863613i
$683$	53573.6			0.114844			0.0574221	−	0.998350i	$-0.481712\pi$
$683$	53573.6			0.114844			0.0574221	+	0.998350i	$0.481712\pi$
$684$			74024.1i			0.158220i
$685$	174458.			0.371801
$686$		−	301901.i		−	0.641528i
$687$		−	237249.i		−	0.502679i
$688$			208021.i			0.439471i
$689$		−	157949.i		−	0.332720i
$690$	−44386.9	−	41187.7i	−0.0932303	−	0.0865106i
$691$	−369867.			−0.774620			−0.387310	−	0.921949i	$-0.626596\pi$
$691$	−369867.			−0.774620			−0.387310	+	0.921949i	$0.626596\pi$
$692$	−76270.5			−0.159274
$693$	−91491.1			−0.190508
$694$	175399.			0.364173
$695$		−	42269.5i		−	0.0875099i
$696$	−65893.7			−0.136027
$697$		−	347341.i		−	0.714975i
$698$	108724.			0.223160
$699$	272311.			0.557328
$700$			116512.i			0.237779i
$701$			875236.i			1.78110i	0.454882	+	0.890552i	$0.349681\pi$
$701$			875236.i			1.78110i	−0.454882	+	0.890552i	$0.650319\pi$
$702$	−39178.1			−0.0795003
$703$	−807261.			−1.63344
$704$			67227.4i			0.135644i
$705$		−	47073.7i		−	0.0947108i
$706$	304224.			0.610357
$707$		−	184797.i		−	0.369705i
$708$	−152345.			−0.303921
$709$		−	87459.7i		−	0.173987i	−0.996209	−	0.0869933i	$-0.972274\pi$
$709$		−	87459.7i		−	0.173987i	0.996209	−	0.0869933i	$-0.0277259\pi$
$710$		−	159613.i		−	0.316630i
$711$		−	101052.i		−	0.199897i
$712$		−	59344.9i		−	0.117064i
$713$	−389186.	+	419416.i	−0.765557	+	0.825022i
$714$	−102819.			−0.201686
$715$	100967.			0.197500
$716$	256991.			0.501292
$717$	52253.5			0.101643
$718$		−	458852.i		−	0.890070i
$719$	−823238.			−1.59246			−0.796228	−	0.604997i	$-0.793174\pi$
$719$	−823238.			−1.59246			−0.796228	+	0.604997i	$0.793174\pi$
$720$		−	13458.4i		−	0.0259615i
$721$	77161.8			0.148433
$722$	−36415.9			−0.0698581
$723$		−	49736.5i		−	0.0951478i
$724$		−	369877.i		−	0.705635i
$725$	316277.			0.601717
$726$	38206.0			0.0724868
$727$		−	873047.i		−	1.65184i	−0.563785	−	0.825922i	$-0.690655\pi$
$727$		−	873047.i		−	1.65184i	0.563785	−	0.825922i	$-0.309345\pi$
$728$		−	57653.6i		−	0.108784i
$729$	19683.0			0.0370370
$730$		−	16589.3i		−	0.0311303i
$731$	881116.			1.64891
$732$		−	78258.6i		−	0.146053i
$733$		−	358681.i		−	0.667576i	−0.942648	−	0.333788i	$-0.891673\pi$
$733$		−	358681.i		−	0.667576i	0.942648	−	0.333788i	$-0.108327\pi$
$734$			327208.i			0.607340i
$735$		−	70215.1i		−	0.129974i
$736$	65135.0	−	70194.4i	0.120243	−	0.129583i
$737$	172522.			0.317622
$738$	−97849.6			−0.179658
$739$	−288631.			−0.528511			−0.264255	−	0.964453i	$-0.585126\pi$
$739$	−288631.			−0.528511			−0.264255	+	0.964453i	$0.585126\pi$
$740$	146769.			0.268023
$741$			175814.i			0.320197i
$742$	−116775.			−0.212100
$743$			424532.i			0.769012i	0.923123	+	0.384506i	$0.125628\pi$
$743$			424532.i			0.769012i	−0.923123	+	0.384506i	$0.874372\pi$
$744$	−127170.			−0.229741
$745$	27218.9			0.0490408
$746$		−	130855.i		−	0.235132i
$747$		−	281963.i		−	0.505303i
$748$	284755.			0.508943
$749$	−220504.			−0.393055
$750$			136139.i			0.242026i
$751$		−	355475.i		−	0.630274i	−0.949046	−	0.315137i	$-0.897950\pi$
$751$		−	355475.i		−	0.630274i	0.949046	−	0.315137i	$-0.102050\pi$
$752$	74443.2			0.131641
$753$			564445.i			0.995478i
$754$	−156504.			−0.275285
$755$			129650.i			0.227446i
$756$			28965.1i			0.0506793i
$757$			243048.i			0.424132i	0.977255	+	0.212066i	$0.0680192\pi$
$757$			243048.i			0.424132i	−0.977255	+	0.212066i	$0.931981\pi$
$758$			128694.i			0.223986i
$759$	−264567.	−	245498.i	−0.459253	−	0.426152i
$760$	−60395.6			−0.104563
$761$	621196.			1.07265			0.536326	−	0.844011i	$-0.319812\pi$
$761$	621196.			1.07265			0.536326	+	0.844011i	$0.319812\pi$
$762$	36304.3			0.0625243
$763$	−325804.			−0.559638
$764$			243252.i			0.416744i
$765$	−57006.0			−0.0974087
$766$			560692.i			0.955579i
$767$	−361832.			−0.615059
$768$	21283.4			0.0360844
$769$			784252.i			1.32618i	0.748539	+	0.663091i	$0.230756\pi$
$769$			784252.i			1.32618i	−0.748539	+	0.663091i	$0.769244\pi$
$770$		−	74646.7i		−	0.125901i
$771$	186432.			0.313626
$772$	−427431.			−0.717185
$773$		−	140783.i		−	0.235608i	−0.993037	−	0.117804i	$-0.962414\pi$
$773$		−	140783.i		−	0.235608i	0.993037	−	0.117804i	$-0.0375855\pi$
$774$		−	248219.i		−	0.414337i
$775$	610391.			1.01626
$776$			193293.i			0.320991i
$777$	−315875.			−0.523207
$778$			652869.i			1.07862i
$779$			439106.i			0.723593i
$780$		−	31965.0i		−	0.0525395i
$781$		−	951369.i		−	1.55972i
$782$	−297323.	−	275893.i	−0.486200	−	0.451156i
$783$	78627.2			0.128248
$784$	111040.			0.180653
$785$	−357965.			−0.580900
$786$	−47218.6			−0.0764308
$787$			451236.i			0.728542i	0.931293	+	0.364271i	$0.118682\pi$
$787$			451236.i			0.728542i	−0.931293	+	0.364271i	$0.881318\pi$
$788$	−40069.8			−0.0645305
$789$		−	579969.i		−	0.931646i
$790$	82447.7			0.132107
$791$	484184.			0.773852
$792$		−	80218.6i		−	0.127886i
$793$		−	185871.i		−	0.295574i
$794$	487913.			0.773930
$795$	−64743.6			−0.102438
$796$		−	4241.47i		−	0.00669408i
$797$		−	43780.4i		−	0.0689228i	−0.999406	−	0.0344614i	$-0.989028\pi$
$797$		−	43780.4i		−	0.0689228i	0.999406	−	0.0344614i	$-0.0109716\pi$
$798$	129983.			0.204117
$799$		−	315320.i		−	0.493921i
$800$	−102156.			−0.159619
$801$			70812.8i			0.110369i
$802$		−	456447.i		−	0.709645i
$803$		−	98880.2i		−	0.153348i
$804$		−	54618.7i		−	0.0844947i
$805$	−72323.5	+	77941.2i	−0.111606	+	0.120275i
$806$	−302040.			−0.464937
$807$	−168710.			−0.259055
$808$	162028.			0.248180
$809$	388364.			0.593393			0.296696	−	0.954972i	$-0.404115\pi$
$809$	388364.			0.593393			0.296696	+	0.954972i	$0.404115\pi$
$810$			16059.2i			0.0244767i
$811$	313900.			0.477254			0.238627	−	0.971111i	$-0.423303\pi$
$811$	313900.			0.477254			0.238627	+	0.971111i	$0.423303\pi$
$812$			115706.i			0.175487i
$813$	667420.			1.00976
$814$	874814.			1.32028
$815$			188440.i			0.283699i
$816$		−	90150.4i		−	0.135390i
$817$	−1.11390e6			−1.66879
$818$	−898434.			−1.34270
$819$			68794.7i			0.102562i
$820$		−	79834.6i		−	0.118731i
$821$	848276.			1.25849			0.629247	−	0.777206i	$-0.283364\pi$
$821$	848276.			1.25849			0.629247	+	0.777206i	$0.283364\pi$
$822$		−	329206.i		−	0.487219i
$823$	48012.0			0.0708843			0.0354421	−	0.999372i	$-0.488716\pi$
$823$	48012.0			0.0708843			0.0354421	+	0.999372i	$0.488716\pi$
$824$			67654.8i			0.0996424i
$825$			385034.i			0.565706i
$826$			267509.i			0.392084i
$827$			243648.i			0.356248i	0.984008	+	0.178124i	$0.0570028\pi$
$827$			243648.i			0.356248i	−0.984008	+	0.178124i	$0.942997\pi$
$828$	−77721.9	+	83759.0i	−0.113366	+	0.122172i
$829$	−684629.			−0.996199			−0.498100	−	0.867120i	$-0.665969\pi$
$829$	−684629.			−0.996199			−0.498100	+	0.867120i	$0.665969\pi$
$830$	230051.			0.333940
$831$	−514906.			−0.745634
$832$	50550.2			0.0730257
$833$		−	470331.i		−	0.677819i
$834$	−79763.2			−0.114675
$835$		−	62158.0i		−	0.0891506i
$836$	−359986.			−0.515078
$837$	151744.			0.216602
$838$			275634.i			0.392505i
$839$		−	1.02529e6i		−	1.45654i	−0.685292	−	0.728268i	$-0.740326\pi$
$839$		−	1.02529e6i		−	1.45654i	0.685292	−	0.728268i	$-0.259674\pi$
$840$	−23632.3			−0.0334925
$841$	−393191.			−0.555919
$842$		−	204897.i		−	0.289009i
$843$		−	32558.5i		−	0.0458152i
$844$	457313.			0.641991
$845$			146526.i			0.205211i
$846$	−88828.8			−0.124112
$847$		−	67087.9i		−	0.0935141i
$848$		−	102387.i		−	0.142381i
$849$		−	70838.1i		−	0.0982769i
$850$			432705.i			0.598900i
$851$	−913424.	−	847587.i	−1.26128	−	1.17038i
$852$	−301193.			−0.414921
$853$	−691358.			−0.950178			−0.475089	−	0.879938i	$-0.657584\pi$
$853$	−691358.			−0.950178			−0.475089	+	0.879938i	$0.657584\pi$
$854$	−137418.			−0.188420
$855$	72066.6			0.0985829
$856$		−	193336.i		−	0.263855i
$857$	4730.13			0.00644038			0.00322019	−	0.999995i	$-0.498975\pi$
$857$	4730.13			0.00644038			0.00322019	+	0.999995i	$0.498975\pi$
$858$		−	190527.i		−	0.258810i
$859$	505051.			0.684461			0.342230	−	0.939616i	$-0.388818\pi$
$859$	505051.			0.684461			0.342230	+	0.939616i	$0.388818\pi$
$860$	202520.			0.273824
$861$			171819.i			0.231774i
$862$			531461.i			0.715248i
$863$	−435898.			−0.585280			−0.292640	−	0.956223i	$-0.594534\pi$
$863$	−435898.			−0.585280			−0.292640	+	0.956223i	$0.594534\pi$
$864$	−25396.3			−0.0340207
$865$			74253.6i			0.0992397i
$866$			732180.i			0.976297i
$867$	52137.0			0.0693598
$868$			223304.i			0.296385i
$869$	491427.			0.650758
$870$			64151.2i			0.0847552i
$871$		−	129725.i		−	0.170996i
$872$		−	285662.i		−	0.375681i
$873$		−	230645.i		−	0.302633i
$874$	375874.	+	348782.i	0.492061	+	0.456595i
$875$	239054.			0.312234
$876$	−31304.4			−0.0407941
$877$	412107.			0.535810			0.267905	−	0.963445i	$-0.413669\pi$
$877$	412107.			0.535810			0.267905	+	0.963445i	$0.413669\pi$
$878$	935742.			1.21386
$879$			241111.i			0.312061i
$880$	65449.6			0.0845165
$881$			775702.i			0.999409i	0.866196	+	0.499704i	$0.166558\pi$
$881$			775702.i			0.999409i	−0.866196	+	0.499704i	$0.833442\pi$
$882$	−132497.			−0.170322
$883$	1.21481e6			1.55807			0.779037	−	0.626979i	$-0.215709\pi$
$883$	1.21481e6			1.55807			0.779037	+	0.626979i	$0.215709\pi$
$884$		−	214116.i		−	0.273996i
$885$			148316.i			0.189366i
$886$	935108.			1.19123
$887$	−1.33334e6			−1.69470			−0.847350	−	0.531034i	$-0.821804\pi$
$887$	−1.33334e6			−1.69470			−0.847350	+	0.531034i	$0.821804\pi$
$888$		−	276957.i		−	0.351225i
$889$		−	63748.6i		−	0.0806617i
$890$	−57775.5			−0.0729397
$891$			95720.2i			0.120572i
$892$	−373273.			−0.469134
$893$			398625.i			0.499875i
$894$		−	51362.5i		−	0.0642645i
$895$		−	250195.i		−	0.312343i
$896$		−	37372.6i		−	0.0465519i
$897$	−184597.	+	198935.i	−0.229424	+	0.247245i
$898$	−511349.			−0.634110
$899$	606170.			0.750023
$900$	121897.			0.150491
$901$	−433681.			−0.534221
$902$		−	475852.i		−	0.584869i
$903$	−435861.			−0.534530
$904$			424528.i			0.519481i
$905$	−360096.			−0.439664
$906$	244651.			0.298051
$907$		−	381066.i		−	0.463219i	−0.972809	−	0.231609i	$-0.925601\pi$
$907$		−	381066.i		−	0.463219i	0.972809	−	0.231609i	$-0.0743991\pi$
$908$			221859.i			0.269094i
$909$	−193339.			−0.233987
$910$	−56129.0			−0.0677805
$911$		−	266311.i		−	0.320887i	−0.987045	−	0.160444i	$-0.948708\pi$
$911$		−	266311.i		−	0.320887i	0.987045	−	0.160444i	$-0.0512925\pi$
$912$			113967.i			0.137022i
$913$	1.37121e6			1.64499
$914$			24994.8i			0.0299197i
$915$	−76189.1			−0.0910019
$916$		−	365269.i		−	0.435333i
$917$			82913.5i			0.0986022i
$918$			107571.i			0.127647i
$919$			1.04927e6i			1.24238i	0.783660	+	0.621191i	$0.213351\pi$
$919$			1.04927e6i			1.24238i	−0.783660	+	0.621191i	$0.786649\pi$
$920$	−68338.2	−	63412.6i	−0.0807398	−	0.0749204i
$921$	322166.			0.379804
$922$	274347.			0.322729
$923$	−715361.			−0.839696
$924$	−140860.			−0.164984
$925$			1.32934e6i			1.55365i
$926$	205795.			0.240001
$927$		−	80728.6i		−	0.0939437i
$928$	−101450.			−0.117803
$929$	−112325.			−0.130150			−0.0650751	−	0.997880i	$-0.520729\pi$
$929$	−112325.			−0.130150			−0.0650751	+	0.997880i	$0.520729\pi$
$930$			123807.i			0.143146i
$931$			594590.i			0.685990i
$932$	419251.			0.482661
$933$	810323.			0.930883
$934$			1.04096e6i			1.19328i
$935$		−	277225.i		−	0.317110i
$936$	−60318.6			−0.0688493
$937$		−	1.67155e6i		−	1.90389i	−0.306273	−	0.951944i	$-0.599082\pi$
$937$		−	1.67155e6i		−	1.90389i	0.306273	−	0.951944i	$-0.400918\pi$
$938$	−95907.7			−0.109005
$939$		−	542354.i		−	0.615108i
$940$		−	72474.6i		−	0.0820220i
$941$			1.07993e6i			1.21960i	0.792556	+	0.609799i	$0.208750\pi$
$941$			1.07993e6i			1.21960i	−0.792556	+	0.609799i	$0.791250\pi$
$942$			675487.i			0.761229i
$943$	−461042.	+	496853.i	−0.518462	+	0.558733i
$944$	−234550.			−0.263203
$945$	28199.1			0.0315771
$946$	1.20711e6			1.34886
$947$	795156.			0.886650			0.443325	−	0.896361i	$-0.353799\pi$
$947$	795156.			0.886650			0.443325	+	0.896361i	$0.353799\pi$
$948$		−	155580.i		−	0.173116i
$949$	−74350.9			−0.0825569
$950$		−	547022.i		−	0.606119i
$951$	755524.			0.835386
$952$	−158300.			−0.174665
$953$		−	1.24724e6i		−	1.37330i	−0.726990	−	0.686649i	$-0.759081\pi$
$953$		−	1.24724e6i		−	1.37330i	0.726990	−	0.686649i	$-0.240919\pi$
$954$			122172.i			0.134238i
$955$	236819.			0.259663
$956$	80449.5			0.0880253
$957$			382371.i			0.417505i
$958$			1.00845e6i			1.09881i
$959$	−578069.			−0.628554
$960$		−	20720.6i		−	0.0224833i
$961$	246340.			0.266740
$962$		−	657798.i		−	0.710791i
$963$			230697.i			0.248765i
$964$		−	76574.4i		−	0.0824004i
$965$			416128.i			0.446860i
$966$	147077.	+	136476.i	0.157612	+	0.146252i
$967$	478617.			0.511841			0.255920	−	0.966698i	$-0.417622\pi$
$967$	478617.			0.511841			0.255920	+	0.966698i	$0.417622\pi$
$968$	58822.0			0.0627754
$969$	482733.			0.514114
$970$	188181.			0.200001
$971$			581067.i			0.616293i	0.951339	+	0.308147i	$0.0997088\pi$
$971$			581067.i			0.616293i	−0.951339	+	0.308147i	$0.900291\pi$
$972$	30304.0			0.0320750
$973$			140060.i			0.147941i
$974$	401585.			0.423311
$975$	289518.			0.304555
$976$		−	120487.i		−	0.126485i
$977$		−	1.27496e6i		−	1.33569i	−0.744299	−	0.667846i	$-0.767216\pi$
$977$		−	1.27496e6i		−	1.33569i	0.744299	−	0.667846i	$-0.232784\pi$
$978$	355590.			0.371768
$979$	−344369.			−0.359301
$980$		−	108103.i		−	0.112561i
$981$			340864.i			0.354196i
$982$	344939.			0.357700
$983$			293603.i			0.303846i	0.988392	+	0.151923i	$0.0485466\pi$
$983$			293603.i			0.303846i	−0.988392	+	0.151923i	$0.951453\pi$
$984$	−150649.			−0.155588
$985$			39010.2i			0.0402074i
$986$			429713.i			0.442002i
$987$			155979.i			0.160115i
$988$			270683.i			0.277299i
$989$	−1.26039e6	−	1.16954e6i	−1.28858	−	1.19571i
$990$	−78097.2			−0.0796829
$991$	−1.32846e6			−1.35269			−0.676347	−	0.736583i	$-0.736438\pi$
$991$	−1.32846e6			−1.35269			−0.676347	+	0.736583i	$0.736438\pi$
$992$	−195791.			−0.198961
$993$	677360.			0.686944
$994$			528880.i			0.535284i
$995$	−4129.31			−0.00417092
$996$		−	434111.i		−	0.437605i
$997$	−371136.			−0.373373			−0.186686	−	0.982420i	$-0.559775\pi$
$997$	−371136.			−0.373373			−0.186686	+	0.982420i	$0.559775\pi$
$998$	−1.04662e6			−1.05082
$999$			330476.i			0.331138i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.b.a.91.6	✓	16
3.2	odd	2		414.5.b.b.91.13		16
4.3	odd	2		1104.5.c.a.1057.3		16
23.22	odd	2	inner	138.5.b.a.91.7	yes	16
69.68	even	2		414.5.b.b.91.12		16
92.91	even	2		1104.5.c.a.1057.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.b.a.91.6	✓	16	1.1	even	1	trivial
138.5.b.a.91.7	yes	16	23.22	odd	2	inner
414.5.b.b.91.12		16	69.68	even	2
414.5.b.b.91.13		16	3.2	odd	2
1104.5.c.a.1057.3		16	4.3	odd	2
1104.5.c.a.1057.6		16	92.91	even	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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -7.78845i q^{5} -14.6969 q^{6} +25.8071i q^{7} -22.6274 q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -7.78845i q^{5} -14.6969 q^{6} +25.8071i q^{7} -22.6274 q^{8} +27.0000 q^{9} +22.0291i q^{10} +131.303i q^{11} +41.5692 q^{12} +98.7308 q^{13} -72.9935i q^{14} -40.4700i q^{15} +64.0000 q^{16} -271.085i q^{17} -76.3675 q^{18} +342.704i q^{19} -62.3076i q^{20} +134.098i q^{21} -371.382i q^{22} +(-359.824 + 387.773i) q^{23} -117.576 q^{24} +564.340 q^{25} -279.253 q^{26} +140.296 q^{27} +206.457i q^{28} +560.437 q^{29} +114.466i q^{30} +1081.60 q^{31} -181.019 q^{32} +682.273i q^{33} +766.745i q^{34} +200.997 q^{35} +216.000 q^{36} +2355.56i q^{37} -969.313i q^{38} +513.020 q^{39} +176.232i q^{40} +1281.30 q^{41} -379.285i q^{42} +3250.33i q^{43} +1050.43i q^{44} -210.288i q^{45} +(1017.74 - 1096.79i) q^{46} +1163.18 q^{47} +332.554 q^{48} +1734.99 q^{49} -1596.19 q^{50} -1408.60i q^{51} +789.846 q^{52} -1599.79i q^{53} -396.817 q^{54} +1022.65 q^{55} -583.948i q^{56} +1780.74i q^{57} -1585.16 q^{58} -3664.84 q^{59} -323.760i q^{60} -1882.61i q^{61} -3059.23 q^{62} +696.791i q^{63} +512.000 q^{64} -768.959i q^{65} -1929.76i q^{66} -1313.92i q^{67} -2168.68i q^{68} +(-1869.70 + 2014.93i) q^{69} -568.506 q^{70} -7245.58 q^{71} -610.940 q^{72} -753.067 q^{73} -6662.54i q^{74} +2932.40 q^{75} +2741.63i q^{76} -3388.56 q^{77} -1451.04 q^{78} -3742.68i q^{79} -498.461i q^{80} +729.000 q^{81} -3624.06 q^{82} -10443.1i q^{83} +1072.78i q^{84} -2111.33 q^{85} -9193.31i q^{86} +2912.12 q^{87} -2971.06i q^{88} +2622.70i q^{89} +594.784i q^{90} +2547.95i q^{91} +(-2878.59 + 3102.18i) q^{92} +5620.16 q^{93} -3289.96 q^{94} +2669.13 q^{95} -940.604 q^{96} -8542.41i q^{97} -4907.31 q^{98} +3545.19i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) \) 16 * q + 128 * q^4 + 432 * q^9	\( 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) \) 16 * q + 128 * q^4 + 432 * q^9 - 208 * q^13 + 1024 * q^16 + 840 * q^23 + 1056 * q^25 + 1920 * q^26 + 3600 * q^29 + 224 * q^31 - 3264 * q^35 + 3456 * q^36 - 2016 * q^39 - 6144 * q^41 + 1280 * q^46 + 8880 * q^47 - 13888 * q^49 + 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 - 18240 * q^59 + 8192 * q^64 + 10584 * q^69 + 19584 * q^70 - 30048 * q^71 + 9536 * q^73 - 4176 * q^75 + 14160 * q^77 + 6912 * q^78 + 11664 * q^81 - 19584 * q^82 - 32496 * q^85 - 8064 * q^87 + 6720 * q^92 - 11952 * q^93 - 21248 * q^94 - 20064 * q^95 + 21504 * q^98

Introduction

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