LMFDB - Embedded newform 87.5.d.c.86.31

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [87,5,Mod(86,87)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87.86"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	87.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$8.99318678829$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			86.31
Character	$\chi$	$=$	87.86
Dual form			87.5.d.c.86.32

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+7.23588 q^{2} +(-2.40914 - 8.67157i) q^{3} +36.3580 q^{4} -19.6231i q^{5} +(-17.4322 - 62.7464i) q^{6} -0.405357 q^{7} +147.308 q^{8} +(-69.3921 + 41.7820i) q^{9} -141.990i q^{10} -13.1297 q^{11} +(-87.5913 - 315.280i) q^{12} +2.15048 q^{13} -2.93311 q^{14} +(-170.163 + 47.2747i) q^{15} +484.174 q^{16} +467.262 q^{17} +(-502.113 + 302.330i) q^{18} +421.813i q^{19} -713.455i q^{20} +(0.976560 + 3.51508i) q^{21} -95.0052 q^{22} -399.056i q^{23} +(-354.885 - 1277.39i) q^{24} +239.935 q^{25} +15.5606 q^{26} +(529.490 + 501.080i) q^{27} -14.7379 q^{28} +(-833.249 + 113.916i) q^{29} +(-1231.28 + 342.074i) q^{30} +1442.16i q^{31} +1146.50 q^{32} +(31.6313 + 113.855i) q^{33} +3381.05 q^{34} +7.95435i q^{35} +(-2522.96 + 1519.11i) q^{36} +1634.55i q^{37} +3052.19i q^{38} +(-5.18081 - 18.6480i) q^{39} -2890.63i q^{40} -1270.74 q^{41} +(7.06627 + 25.4347i) q^{42} -1809.33i q^{43} -477.370 q^{44} +(819.891 + 1361.69i) q^{45} -2887.52i q^{46} +1525.76 q^{47} +(-1166.44 - 4198.55i) q^{48} -2400.84 q^{49} +1736.14 q^{50} +(-1125.70 - 4051.89i) q^{51} +78.1871 q^{52} -4753.17i q^{53} +(3831.33 + 3625.75i) q^{54} +257.646i q^{55} -59.7122 q^{56} +(3657.78 - 1016.21i) q^{57} +(-6029.29 + 824.284i) q^{58} +2413.41i q^{59} +(-6186.77 + 1718.81i) q^{60} +5164.78i q^{61} +10435.3i q^{62} +(28.1286 - 16.9366i) q^{63} +549.159 q^{64} -42.1991i q^{65} +(228.881 + 823.844i) q^{66} -1685.57 q^{67} +16988.7 q^{68} +(-3460.44 + 961.380i) q^{69} +57.5567i q^{70} -3640.41i q^{71} +(-10222.0 + 6154.81i) q^{72} -8322.21i q^{73} +11827.4i q^{74} +(-578.036 - 2080.61i) q^{75} +15336.3i q^{76} +5.32223 q^{77} +(-37.4877 - 134.935i) q^{78} +5808.49i q^{79} -9500.99i q^{80} +(3069.53 - 5798.68i) q^{81} -9194.93 q^{82} -2307.90i q^{83} +(35.5057 + 127.801i) q^{84} -9169.12i q^{85} -13092.1i q^{86} +(2995.24 + 6951.13i) q^{87} -1934.11 q^{88} -4008.01 q^{89} +(5932.64 + 9853.00i) q^{90} -0.871712 q^{91} -14508.8i q^{92} +(12505.8 - 3474.36i) q^{93} +11040.2 q^{94} +8277.28 q^{95} +(-2762.08 - 9941.96i) q^{96} -11015.3i q^{97} -17372.2 q^{98} +(911.100 - 548.586i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 188 q^{4} - 36 q^{6} - 84 q^{7} - 452 q^{9} - 224 q^{13} - 52 q^{16} - 4216 q^{22} - 832 q^{24} - 7684 q^{25} - 396 q^{28} + 3384 q^{30} - 3308 q^{33} + 9124 q^{34} - 3680 q^{36} + 19764 q^{42} + 44 q^{45}+ \cdots + 13884 q^{96}+O(q^{100}) $ 32 * q + 188 * q^4 - 36 * q^6 - 84 * q^7 - 452 * q^9 - 224 * q^13 - 52 * q^16 - 4216 * q^22 - 832 * q^24 - 7684 * q^25 - 396 * q^28 + 3384 * q^30 - 3308 * q^33 + 9124 * q^34 - 3680 * q^36 + 19764 * q^42 + 44 * q^45 + 4796 * q^49 - 7052 * q^51 + 34048 * q^52 + 3008 * q^54 + 7976 * q^57 + 5936 * q^58 - 22848 * q^63 - 14668 * q^64 - 1872 * q^67 + 15192 * q^78 - 41088 * q^81 - 27028 * q^82 - 35572 * q^87 - 60032 * q^88 - 34272 * q^91 + 38000 * q^93 + 59636 * q^94 + 13884 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$-1$	$-1$

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 $

$2$	7.23588			1.80897			0.904485	−	0.426505i	$-0.140255\pi$
$2$	7.23588			1.80897			0.904485	+	0.426505i	$0.140255\pi$
$3$	−2.40914	−	8.67157i	−0.267682	−	0.963507i
$3$	−2.40914	−	8.67157i	−0.267682	−	0.963507i
$4$	36.3580			2.27237
$5$		−	19.6231i		−	0.784923i	−0.919768	−	0.392462i	$-0.871624\pi$
$5$		−	19.6231i		−	0.784923i	0.919768	−	0.392462i	$-0.128376\pi$
$6$	−17.4322	−	62.7464i	−0.484229	−	1.74296i
$7$	−0.405357			−0.00827259			−0.00413629	−	0.999991i	$-0.501317\pi$
$7$	−0.405357			−0.00827259			−0.00413629	+	0.999991i	$0.501317\pi$
$8$	147.308			2.30168
$9$	−69.3921	+	41.7820i	−0.856693	+	0.515827i
$10$		−	141.990i		−	1.41990i
$11$	−13.1297			−0.108510			−0.0542551	−	0.998527i	$-0.517278\pi$
$11$	−13.1297			−0.108510			−0.0542551	+	0.998527i	$0.517278\pi$
$12$	−87.5913	−	315.280i	−0.608273	−	2.18945i
$13$	2.15048			0.0127247			0.00636237	−	0.999980i	$-0.497975\pi$
$13$	2.15048			0.0127247			0.00636237	+	0.999980i	$0.497975\pi$
$14$	−2.93311			−0.0149649
$15$	−170.163	+	47.2747i	−0.756279	+	0.210110i
$16$	484.174			1.89130
$17$	467.262			1.61682			0.808411	−	0.588618i	$-0.200328\pi$
$17$	467.262			1.61682			0.808411	+	0.588618i	$0.200328\pi$
$18$	−502.113	+	302.330i	−1.54973	+	0.933116i
$19$			421.813i			1.16846i	0.811589	+	0.584229i	$0.198603\pi$
$19$			421.813i			1.16846i	−0.811589	+	0.584229i	$0.801397\pi$
$20$		−	713.455i		−	1.78364i
$21$	0.976560	+	3.51508i	0.00221442	+	0.00797070i
$22$	−95.0052			−0.196292
$23$		−	399.056i		−	0.754358i	−0.926140	−	0.377179i	$-0.876894\pi$
$23$		−	399.056i		−	0.754358i	0.926140	−	0.377179i	$-0.123106\pi$
$24$	−354.885	−	1277.39i	−0.616119	−	2.21769i
$25$	239.935			0.383895
$26$	15.5606			0.0230187
$27$	529.490	+	501.080i	0.726324	+	0.687352i
$28$	−14.7379			−0.0187984
$29$	−833.249	+	113.916i	−0.990784	+	0.135453i
$29$	−833.249	+	113.916i	−0.990784	+	0.135453i
$30$	−1231.28	+	342.074i	−1.36809	+	0.380082i
$31$			1442.16i			1.50069i	0.661048	+	0.750343i	$0.270112\pi$
$31$			1442.16i			1.50069i	−0.661048	+	0.750343i	$0.729888\pi$
$32$	1146.50			1.11963
$33$	31.6313	+	113.855i	0.0290462	+	0.104550i
$34$	3381.05			2.92478
$35$			7.95435i			0.00649335i
$36$	−2522.96	+	1519.11i	−1.94673	+	1.17215i
$37$			1634.55i			1.19398i	0.802250	+	0.596988i	$0.203636\pi$
$37$			1634.55i			1.19398i	−0.802250	+	0.596988i	$0.796364\pi$
$38$			3052.19i			2.11371i
$39$	−5.18081	−	18.6480i	−0.00340618	−	0.0122604i
$40$		−	2890.63i		−	1.80665i
$41$	−1270.74			−0.755944			−0.377972	−	0.925817i	$-0.623378\pi$
$41$	−1270.74			−0.755944			−0.377972	+	0.925817i	$0.623378\pi$
$42$	7.06627	+	25.4347i	0.00400582	+	0.0144188i
$43$		−	1809.33i		−	0.978547i	−0.872130	−	0.489274i	$-0.837262\pi$
$43$		−	1809.33i		−	0.978547i	0.872130	−	0.489274i	$-0.162738\pi$
$44$	−477.370			−0.246576
$45$	819.891	+	1361.69i	0.404885	+	0.672438i
$46$		−	2887.52i		−	1.36461i
$47$	1525.76			0.690700			0.345350	−	0.938474i	$-0.387760\pi$
$47$	1525.76			0.690700			0.345350	+	0.938474i	$0.387760\pi$
$48$	−1166.44	−	4198.55i	−0.506268	−	1.82229i
$49$	−2400.84			−0.999932
$50$	1736.14			0.694455
$51$	−1125.70	−	4051.89i	−0.432794	−	1.55782i
$52$	78.1871			0.0289154
$53$		−	4753.17i		−	1.69212i	−0.533086	−	0.846061i	$-0.678968\pi$
$53$		−	4753.17i		−	1.69212i	0.533086	−	0.846061i	$-0.321032\pi$
$54$	3831.33	+	3625.75i	1.31390	+	1.24340i
$55$			257.646i			0.0851722i
$56$	−59.7122			−0.0190409
$57$	3657.78	−	1016.21i	1.12582	−	0.312775i
$58$	−6029.29	+	824.284i	−1.79230	+	0.245031i
$59$			2413.41i			0.693309i	0.937993	+	0.346654i	$0.112682\pi$
$59$			2413.41i			0.693309i	−0.937993	+	0.346654i	$0.887318\pi$
$60$	−6186.77	+	1718.81i	−1.71855	+	0.477448i
$61$			5164.78i			1.38801i	0.719970	+	0.694005i	$0.244155\pi$
$61$			5164.78i			1.38801i	−0.719970	+	0.694005i	$0.755845\pi$
$62$			10435.3i			2.71470i
$63$	28.1286	−	16.9366i	0.00708706	−	0.00426722i
$64$	549.159			0.134072
$65$		−	42.1991i		−	0.00998795i
$66$	228.881	+	823.844i	0.0525437	+	0.189128i
$67$	−1685.57			−0.375489			−0.187745	−	0.982218i	$-0.560118\pi$
$67$	−1685.57			−0.375489			−0.187745	+	0.982218i	$0.560118\pi$
$68$	16988.7			3.67402
$69$	−3460.44	+	961.380i	−0.726830	+	0.201928i
$70$			57.5567i			0.0117463i
$71$		−	3640.41i		−	0.722160i	−0.932535	−	0.361080i	$-0.882408\pi$
$71$		−	3640.41i		−	0.722160i	0.932535	−	0.361080i	$-0.117592\pi$
$72$	−10222.0	+	6154.81i	−1.97184	+	1.18727i
$73$		−	8322.21i		−	1.56168i	−0.624730	−	0.780841i	$-0.714791\pi$
$73$		−	8322.21i		−	1.56168i	0.624730	−	0.780841i	$-0.285209\pi$
$74$			11827.4i			2.15987i
$75$	−578.036	−	2080.61i	−0.102762	−	0.369886i
$76$			15336.3i			2.65517i
$77$	5.32223			0.000897660
$78$	−37.4877	−	134.935i	−0.00616168	−	0.0221787i
$79$			5808.49i			0.930698i	0.885127	+	0.465349i	$0.154071\pi$
$79$			5808.49i			0.930698i	−0.885127	+	0.465349i	$0.845929\pi$
$80$		−	9500.99i		−	1.48453i
$81$	3069.53	−	5798.68i	0.467845	−	0.883811i
$82$	−9194.93			−1.36748
$83$		−	2307.90i		−	0.335012i	−0.985871	−	0.167506i	$-0.946429\pi$
$83$		−	2307.90i		−	0.335012i	0.985871	−	0.167506i	$-0.0535714\pi$
$84$	35.5057	+	127.801i	0.00503199	+	0.0181124i
$85$		−	9169.12i		−	1.26908i
$86$		−	13092.1i		−	1.77016i
$87$	2995.24	+	6951.13i	0.395725	+	0.918369i
$88$	−1934.11			−0.249756
$89$	−4008.01			−0.505998			−0.252999	−	0.967467i	$-0.581417\pi$
$89$	−4008.01			−0.505998			−0.252999	+	0.967467i	$0.581417\pi$
$90$	5932.64	+	9853.00i	0.732424	+	1.21642i
$91$	−0.871712			−0.000105267
$92$		−	14508.8i		−	1.71418i
$93$	12505.8	−	3474.36i	1.44592	−	0.401707i
$94$	11040.2			1.24946
$95$	8277.28			0.917150
$96$	−2762.08	−	9941.96i	−0.299705	−	1.07877i
$97$		−	11015.3i		−	1.17072i	−0.810775	−	0.585358i	$-0.800954\pi$
$97$		−	11015.3i		−	1.17072i	0.810775	−	0.585358i	$-0.199046\pi$
$98$	−17372.2			−1.80885
$99$	911.100	−	548.586i	0.0929599	−	0.0559725i
$100$	8723.54			0.872354
$101$	1511.24			0.148147			0.0740733	−	0.997253i	$-0.476400\pi$
$101$	1511.24			0.148147			0.0740733	+	0.997253i	$0.476400\pi$
$102$	−8145.42	−	29319.0i	−0.782912	−	2.81805i
$103$	−16214.6			−1.52838			−0.764192	−	0.644989i	$-0.776862\pi$
$103$	−16214.6			−1.52838			−0.764192	+	0.644989i	$0.776862\pi$
$104$	316.783			0.0292883
$105$	68.9767	−	19.1631i	0.00625639	−	0.00173815i
$106$		−	34393.4i		−	3.06100i
$107$			3253.67i			0.284188i	0.989853	+	0.142094i	$0.0453836\pi$
$107$			3253.67i			0.284188i	−0.989853	+	0.142094i	$0.954616\pi$
$108$	19251.2	+	18218.2i	1.65048	+	1.56192i
$109$	10961.4			0.922602			0.461301	−	0.887244i	$-0.347383\pi$
$109$	10961.4			0.922602			0.461301	+	0.887244i	$0.347383\pi$
$110$			1864.29i			0.154074i
$111$	14174.1	−	3937.86i	1.15040	−	0.319606i
$112$	−196.263			−0.0156460
$113$	11943.8			0.935373			0.467687	−	0.883894i	$-0.345088\pi$
$113$	11943.8			0.935373			0.467687	+	0.883894i	$0.345088\pi$
$114$	26467.3	−	7353.15i	2.03657	−	0.565801i
$115$	−7830.70			−0.592113
$116$	−30295.2	+	4141.76i	−2.25143	+	0.307800i
$117$	−149.226	+	89.8514i	−0.0109012	+	0.00656377i
$118$			17463.1i			1.25417i
$119$	−189.408			−0.0133753
$120$	−25066.3	+	6963.93i	−1.74072	+	0.483606i
$121$	−14468.6			−0.988226
$122$			37371.8i			2.51087i
$123$	3061.39	+	11019.3i	0.202353	+	0.728357i
$124$			52434.0i			3.41012i
$125$		−	16972.7i		−	1.08625i
$126$	203.535	−	122.551i	0.0128203	−	0.00771928i
$127$		−	14183.9i		−	0.879404i	−0.898144	−	0.439702i	$-0.855084\pi$
$127$		−	14183.9i		−	0.879404i	0.898144	−	0.439702i	$-0.144916\pi$
$128$	−14370.4			−0.877098
$129$	−15689.8	+	4358.93i	−0.942837	+	0.261939i
$130$		−	305.347i		−	0.0180679i
$131$	18372.1			1.07057			0.535287	−	0.844670i	$-0.320203\pi$
$131$	18372.1			1.07057			0.535287	+	0.844670i	$0.320203\pi$
$132$	1150.05	+	4139.55i	0.0660038	+	0.237577i
$133$		−	170.985i		−	0.00966617i
$134$	−12196.6			−0.679249
$135$	9832.73	−	10390.2i	0.539519	−	0.570109i
$136$	68831.3			3.72142
$137$	−16750.4			−0.892452			−0.446226	−	0.894920i	$-0.647232\pi$
$137$	−16750.4			−0.892452			−0.446226	+	0.894920i	$0.647232\pi$
$138$	−25039.3	+	6956.43i	−1.31481	+	0.365282i
$139$	24202.6			1.25266			0.626330	−	0.779558i	$-0.284556\pi$
$139$	24202.6			1.25266			0.626330	+	0.779558i	$0.284556\pi$
$140$			289.204i			0.0147553i
$141$	−3675.76	−	13230.7i	−0.184888	−	0.665494i
$142$		−	26341.6i		−	1.30637i
$143$	−28.2352			−0.00138076
$144$	−33597.9	+	20229.8i	−1.62027	+	0.975586i
$145$	2235.39	+	16350.9i	0.106320	+	0.777689i
$146$		−	60218.5i		−	2.82504i
$147$	5783.94	+	20819.0i	0.267664	+	0.963441i
$148$			59429.0i			2.71316i
$149$		−	14763.7i		−	0.665000i	−0.943103	−	0.332500i	$-0.892108\pi$
$149$		−	14763.7i		−	0.665000i	0.943103	−	0.332500i	$-0.107892\pi$
$150$	−4182.60	−	15055.0i	−0.185893	−	0.669113i
$151$	−24424.7			−1.07121			−0.535606	−	0.844468i	$-0.679917\pi$
$151$	−24424.7			−1.07121			−0.535606	+	0.844468i	$0.679917\pi$
$152$			62136.4i			2.68942i
$153$	−32424.3	+	19523.1i	−1.38512	+	0.834001i
$154$	38.5110			0.00162384
$155$	28299.6			1.17792
$156$	−188.364	−	678.005i	−0.00774012	−	0.0278602i
$157$		−	18163.8i		−	0.736898i	−0.929648	−	0.368449i	$-0.879889\pi$
$157$		−	18163.8i		−	0.736898i	0.929648	−	0.368449i	$-0.120111\pi$
$158$			42029.5i			1.68361i
$159$	−41217.4	+	11451.0i	−1.63037	+	0.452951i
$160$		−	22497.9i		−	0.878824i
$161$			161.760i			0.00624049i
$162$	22210.8	−	41958.6i	0.846317	−	1.59879i
$163$			40685.3i			1.53131i	0.643254	+	0.765653i	$0.277584\pi$
$163$			40685.3i			1.53131i	−0.643254	+	0.765653i	$0.722416\pi$
$164$	−46201.6			−1.71779
$165$	2234.19	−	620.704i	0.0820640	−	0.0227991i
$166$		−	16699.7i		−	0.606027i
$167$			34775.4i			1.24692i	0.781855	+	0.623460i	$0.214274\pi$
$167$			34775.4i			1.24692i	−0.781855	+	0.623460i	$0.785726\pi$
$168$	143.855	+	517.798i	0.00509690	+	0.0183460i
$169$	−28556.4			−0.999838
$170$		−	66346.6i		−	2.29573i
$171$	−17624.2	−	29270.5i	−0.602722	−	1.00101i
$172$		−	65783.7i		−	2.22362i
$173$			44699.3i			1.49351i	0.665099	+	0.746755i	$0.268389\pi$
$173$			44699.3i			1.49351i	−0.665099	+	0.746755i	$0.731611\pi$
$174$	21673.2	+	50297.6i	0.715855	+	1.66130i
$175$	−97.2591			−0.00317581
$176$	−6357.08			−0.205226
$177$	20928.0	−	5814.23i	0.668008	−	0.185586i
$178$	−29001.5			−0.915335
$179$			37939.6i			1.18410i	0.805902	+	0.592048i	$0.201681\pi$
$179$			37939.6i			1.18410i	−0.805902	+	0.592048i	$0.798319\pi$
$180$	29809.6	+	49508.2i	0.920049	+	1.52803i
$181$	18564.0			0.566648			0.283324	−	0.959024i	$-0.408563\pi$
$181$	18564.0			0.566648			0.283324	+	0.959024i	$0.408563\pi$
$182$	−6.30760			−0.000190424
$183$	44786.8	−	12442.7i	1.33736	−	0.371545i
$184$		−	58784.0i		−	1.73629i
$185$	32075.0			0.937179
$186$	90490.3	−	25140.1i	2.61563	−	0.726675i
$187$	−6135.02			−0.175442
$188$	55473.4			1.56953
$189$	−214.633	−	203.116i	−0.00600858	−	0.00568618i
$190$	59893.4			1.65910
$191$	−46653.9			−1.27886			−0.639428	−	0.768851i	$-0.720829\pi$
$191$	−46653.9			−1.27886			−0.639428	+	0.768851i	$0.720829\pi$
$192$	−1323.00	−	4762.07i	−0.0358887	−	0.129179i
$193$		−	5202.56i		−	0.139670i	−0.997559	−	0.0698350i	$-0.977753\pi$
$193$		−	5202.56i		−	0.139670i	0.997559	−	0.0698350i	$-0.0222473\pi$
$194$		−	79705.1i		−	2.11779i
$195$	−365.932	+	101.663i	−0.00962346	+	0.00267359i
$196$	−87289.5			−2.27222
$197$		−	21630.3i		−	0.557352i	−0.960385	−	0.278676i	$-0.910104\pi$
$197$		−	21630.3i		−	0.557352i	0.960385	−	0.278676i	$-0.0898956\pi$
$198$	6592.61	−	3969.51i	0.168162	−	0.101253i
$199$	15271.3			0.385630			0.192815	−	0.981235i	$-0.438238\pi$
$199$	15271.3			0.385630			0.192815	+	0.981235i	$0.438238\pi$
$200$	35344.2			0.883606
$201$	4060.77	+	14616.5i	0.100512	+	0.361787i
$202$	10935.2			0.267993
$203$	337.763	−	46.1767i	0.00819634	−	0.00112055i
$204$	−40928.1	−	147319.i	−0.983470	−	3.53995i
$205$			24935.9i			0.593358i
$206$	−117327.			−2.76480
$207$	16673.3	+	27691.3i	0.389118	+	0.646253i
$208$	1041.21			0.0240664
$209$		−	5538.30i		−	0.126790i
$210$	499.107	−	138.662i	0.0113176	−	0.00314426i
$211$			42066.3i			0.944865i	0.881367	+	0.472433i	$0.156624\pi$
$211$			42066.3i			0.944865i	−0.881367	+	0.472433i	$0.843376\pi$
$212$		−	172816.i		−	3.84513i
$213$	−31568.1	+	8770.25i	−0.695807	+	0.193309i
$214$			23543.2i			0.514088i
$215$	−35504.7			−0.768085
$216$	77998.1	+	73812.9i	1.67177	+	1.58207i
$217$		−	584.589i		−	0.0124146i
$218$	79315.6			1.66896
$219$	−72166.6	+	20049.3i	−1.50469	+	0.418034i
$220$			9367.48i			0.193543i
$221$	1004.84			0.0205736
$222$	102562.	−	28493.9i	2.08105	−	0.578157i
$223$	−53606.9			−1.07798			−0.538990	−	0.842312i	$-0.681194\pi$
$223$	−53606.9			−1.07798			−0.538990	+	0.842312i	$0.681194\pi$
$224$	−464.742			−0.00926223
$225$	−16649.6	+	10024.9i	−0.328880	+	0.198024i
$226$	86423.8			1.69206
$227$			33548.0i			0.651050i	0.945533	+	0.325525i	$0.105541\pi$
$227$			33548.0i			0.651050i	−0.945533	+	0.325525i	$0.894459\pi$
$228$	132990.	−	36947.2i	2.55828	−	0.710742i
$229$			3684.52i			0.0702602i	0.999383	+	0.0351301i	$0.0111846\pi$
$229$			3684.52i			0.0702602i	−0.999383	+	0.0351301i	$0.988815\pi$
$230$	−56662.0			−1.07112
$231$	−12.8220	−	46.1520i	−0.000240287	−	0.000864902i
$232$	−122744.	+	16780.7i	−2.28047	+	0.311771i
$233$		−	93638.9i		−	1.72482i	−0.506208	−	0.862411i	$-0.668953\pi$
$233$		−	93638.9i		−	1.72482i	0.506208	−	0.862411i	$-0.331047\pi$
$234$	−1079.78	+	650.154i	−0.0197199	+	0.0118737i
$235$		−	29940.0i		−	0.542146i
$236$			87746.6i			1.57546i
$237$	50368.7	−	13993.4i	0.896735	−	0.249131i
$238$	−1370.53			−0.0241955
$239$			72162.3i			1.26332i	0.775244	+	0.631662i	$0.217627\pi$
$239$			72162.3i			1.26332i	−0.775244	+	0.631662i	$0.782373\pi$
$240$	−82388.4	+	22889.2i	−1.43035	+	0.397382i
$241$	−3326.50			−0.0572735			−0.0286368	−	0.999590i	$-0.509117\pi$
$241$	−3326.50			−0.0572735			−0.0286368	+	0.999590i	$0.509117\pi$
$242$	−104693.			−1.78767
$243$	−57678.6	−	12647.8i	−0.976792	−	0.214192i
$244$			187781.i			3.15408i
$245$			47111.8i			0.784870i
$246$	22151.9	+	79734.5i	0.366050	+	1.31758i
$247$			907.102i			0.0148683i
$248$			212441.i			3.45411i
$249$	−20013.1	+	5560.05i	−0.322787	+	0.0896768i
$250$		−	122812.i		−	1.96500i
$251$	37131.0			0.589371			0.294685	−	0.955594i	$-0.404785\pi$
$251$	37131.0			0.589371			0.294685	+	0.955594i	$0.404785\pi$
$252$	1022.70	−	615.781i	0.0161045	−	0.00969672i
$253$			5239.49i			0.0818556i
$254$		−	102633.i		−	1.59081i
$255$	−79510.6	+	22089.7i	−1.22277	+	0.339710i
$256$	−112769.			−1.72072
$257$		−	30001.3i		−	0.454227i	−0.973868	−	0.227114i	$-0.927071\pi$
$257$		−	30001.3i		−	0.454227i	0.973868	−	0.227114i	$-0.0729289\pi$
$258$	−113529.	+	31540.7i	−1.70556	+	0.473841i
$259$		−	662.577i		−	0.00987726i
$260$		−	1534.27i		−	0.0226963i
$261$	53061.3	−	42719.7i	0.778927	−	0.627115i
$262$	132939.			1.93664
$263$	94673.9			1.36873			0.684367	−	0.729138i	$-0.260079\pi$
$263$	94673.9			1.36873			0.684367	+	0.729138i	$0.260079\pi$
$264$	4659.54	+	16771.8i	0.0668552	+	0.240642i
$265$	−93271.9			−1.32819
$266$		−	1237.23i		−	0.0174858i
$267$	9655.84	+	34755.7i	0.135446	+	0.487532i
$268$	−61284.0			−0.853252
$269$	109966.			1.51969			0.759846	−	0.650103i	$-0.225274\pi$
$269$	109966.			1.51969			0.759846	+	0.650103i	$0.225274\pi$
$270$	71148.4	−	75182.5i	0.975973	−	1.03131i
$271$		−	65762.0i		−	0.895440i	−0.894174	−	0.447720i	$-0.852236\pi$
$271$		−	65762.0i		−	0.895440i	0.894174	−	0.447720i	$-0.147764\pi$
$272$	226236.			3.05790
$273$	2.10007	+	7.55911i	2.81779e−5	+	0.000101425i
$274$	−121204.			−1.61442
$275$	−3150.28			−0.0416566
$276$	−125814.	+	34953.8i	−1.65163	+	0.458856i
$277$	103352.			1.34698			0.673488	−	0.739198i	$-0.264795\pi$
$277$	103352.			1.34698			0.673488	+	0.739198i	$0.264795\pi$
$278$	175127.			2.26602
$279$	−60256.3	−	100075.i	−0.774095	−	1.28563i
$280$			1171.74i			0.0149456i
$281$		−	51896.8i		−	0.657246i	−0.944461	−	0.328623i	$-0.893415\pi$
$281$		−	51896.8i		−	0.657246i	0.944461	−	0.328623i	$-0.106585\pi$
$282$	−26597.3	−	95735.7i	−0.334457	−	1.20386i
$283$	77430.5			0.966805			0.483403	−	0.875398i	$-0.339401\pi$
$283$	77430.5			0.966805			0.483403	+	0.875398i	$0.339401\pi$
$284$		−	132358.i		−	1.64102i
$285$	−19941.1	−	71777.0i	−0.245505	−	0.883681i
$286$	−204.307			−0.00249776
$287$	515.104			0.00625361
$288$	−79558.1	+	47903.1i	−0.959179	+	0.577535i
$289$	134813.			1.61412
$290$	16175.0	+	118313.i	0.192331	+	1.40682i
$291$	−95519.6	+	26537.3i	−1.12799	+	0.313379i
$292$		−	302578.i		−	3.54872i
$293$	−25397.6			−0.295840			−0.147920	−	0.988999i	$-0.547258\pi$
$293$	−25397.6			−0.295840			−0.147920	+	0.988999i	$0.547258\pi$
$294$	41851.9	+	150644.i	0.484196	+	1.74284i
$295$	47358.5			0.544194
$296$			240782.i			2.74815i
$297$	−6952.07	−	6579.04i	−0.0788136	−	0.0745847i
$298$		−	106828.i		−	1.20297i
$299$		−	858.161i		−	0.00959901i
$300$	−21016.2	−	75646.7i	−0.233513	−	0.840519i
$301$			733.426i			0.00809512i
$302$	−176734.			−1.93779
$303$	−3640.79	−	13104.8i	−0.0396562	−	0.142740i
$304$			204231.i			2.20991i
$305$	101349.			1.08948
$306$	−234618.	+	141267.i	−2.50564	+	1.50868i
$307$		−	32257.5i		−	0.342258i	−0.985249	−	0.171129i	$-0.945259\pi$
$307$		−	32257.5i		−	0.342258i	0.985249	−	0.171129i	$-0.0547415\pi$
$308$	193.505			0.00203982
$309$	39063.3	+	140606.i	0.409121	+	1.47261i
$310$	204773.			2.13083
$311$	−96109.2			−0.993674			−0.496837	−	0.867844i	$-0.665505\pi$
$311$	−96109.2			−0.993674			−0.496837	+	0.867844i	$0.665505\pi$
$312$	−763.173	−	2747.00i	−0.00783996	−	0.0282195i
$313$	142981.			1.45945			0.729727	−	0.683739i	$-0.239647\pi$
$313$	142981.			1.45945			0.729727	+	0.683739i	$0.239647\pi$
$314$		−	131431.i		−	1.33303i
$315$	−332.349	−	551.969i	−0.00334944	−	0.00556280i
$316$			211185.i			2.11489i
$317$	47312.6			0.470823			0.235412	−	0.971896i	$-0.424356\pi$
$317$	47312.6			0.470823			0.235412	+	0.971896i	$0.424356\pi$
$318$	−298244.	+	82858.4i	−2.94929	+	0.819374i
$319$	10940.3	−	1495.69i	0.107510	−	0.0146981i
$320$		−	10776.2i		−	0.105236i
$321$	28214.4	−	7838.54i	0.273817	−	0.0760721i
$322$			1170.47i			0.0112889i
$323$			197097.i			1.88919i
$324$	111602.	−	210828.i	1.06312	−	2.00835i
$325$	515.975			0.00488497
$326$			294394.i			2.77009i
$327$	−26407.6	−	95052.8i	−0.246964	−	0.888933i
$328$	−187190.			−1.73994
$329$	−618.475			−0.00571387
$330$	16166.4	−	4491.34i	0.148451	−	0.0412428i
$331$		−	76511.8i		−	0.698349i	−0.937058	−	0.349174i	$-0.886462\pi$
$331$		−	76511.8i		−	0.698349i	0.937058	−	0.349174i	$-0.113538\pi$
$332$		−	83910.6i		−	0.761273i
$333$	−68294.8	−	113425.i	−0.615885	−	1.02287i
$334$			251630.i			2.25564i
$335$			33076.1i			0.294730i
$336$	472.825	+	1701.91i	0.00418815	+	0.0150750i
$337$		−	67924.0i		−	0.598086i	−0.954240	−	0.299043i	$-0.903333\pi$
$337$		−	67924.0i		−	0.598086i	0.954240	−	0.299043i	$-0.0966673\pi$
$338$	−206631.			−1.80868
$339$	−28774.2	−	103571.i	−0.250382	−	0.901239i
$340$		−	333370.i		−	2.88383i
$341$		−	18935.2i		−	0.162840i
$342$	−127527.	−	211798.i	−1.09031	−	1.81080i
$343$	1946.46			0.0165446
$344$		−	266529.i		−	2.25231i
$345$	18865.2	+	67904.4i	0.158498	+	0.570506i
$346$			323439.i			2.70172i
$347$		−	154548.i		−	1.28352i	−0.766904	−	0.641761i	$-0.778204\pi$
$347$		−	154548.i		−	1.28352i	0.766904	−	0.641761i	$-0.221796\pi$
$348$	108901.	+	252729.i	0.899235	+	2.08688i
$349$	51287.0			0.421072			0.210536	−	0.977586i	$-0.432479\pi$
$349$	51287.0			0.421072			0.210536	+	0.977586i	$0.432479\pi$
$350$	−703.755			−0.00574494
$351$	1138.66	+	1077.56i	0.00924229	+	0.00874638i
$352$	−15053.3			−0.121491
$353$		−	236679.i		−	1.89937i	−0.313203	−	0.949686i	$-0.601402\pi$
$353$		−	236679.i		−	1.89937i	0.313203	−	0.949686i	$-0.398598\pi$
$354$	151433.	−	42071.1i	1.20841	−	0.335720i
$355$	−71436.1			−0.566841
$356$	−145723.			−1.14982
$357$	456.309	+	1642.46i	0.00358033	+	0.0128872i
$358$			274527.i			2.14200i
$359$	2889.06			0.0224165			0.0112083	−	0.999937i	$-0.496432\pi$
$359$	2889.06			0.0224165			0.0112083	+	0.999937i	$0.496432\pi$
$360$	120776.	+	200587.i	0.931917	+	1.54774i
$361$	−47605.6			−0.365295
$362$	134327.			1.02505
$363$	34856.9	+	125466.i	0.264530	+	0.952163i
$364$	−31.6937			−0.000239205
$365$	−163307.			−1.22580
$366$	324072.	−	90033.7i	2.41924	−	0.672114i
$367$			229863.i			1.70662i	0.521402	+	0.853311i	$0.325409\pi$
$367$			229863.i			1.70662i	−0.521402	+	0.853311i	$0.674591\pi$
$368$		−	193212.i		−	1.42672i
$369$	88179.4	−	53094.1i	0.647612	−	0.389936i
$370$	232090.			1.69533
$371$			1926.73i			0.0139982i
$372$	454685.	−	126321.i	3.28567	−	0.912827i
$373$	−140417.			−1.00925			−0.504627	−	0.863337i	$-0.668370\pi$
$373$	−140417.			−1.00925			−0.504627	+	0.863337i	$0.668370\pi$
$374$	−44392.3			−0.317369
$375$	−147180.	+	40889.5i	−1.04661	+	0.290770i
$376$	224756.			1.58977
$377$	−1791.89	+	244.975i	−0.0126075	+	0.00172361i
$378$	−1553.06	−	1469.72i	−0.0108693	−	0.0102861i
$379$			35197.1i			0.245035i	0.992466	+	0.122518i	$0.0390968\pi$
$379$			35197.1i			0.245035i	−0.992466	+	0.122518i	$0.960903\pi$
$380$	300945.			2.08411
$381$	−122997.	+	34171.0i	−0.847312	+	0.235400i
$382$	−337582.			−2.31341
$383$			48532.0i			0.330850i	0.986222	+	0.165425i	$0.0528995\pi$
$383$			48532.0i			0.330850i	−0.986222	+	0.165425i	$0.947100\pi$
$384$	34620.2	+	124614.i	0.234783	+	0.845090i
$385$		−	104.438i		−	0.000704594i
$386$		−	37645.1i		−	0.252659i
$387$	75597.6	+	125554.i	0.504761	+	0.838314i
$388$		−	400492.i		−	2.66030i
$389$	−120748.			−0.797961			−0.398980	−	0.916959i	$-0.630636\pi$
$389$	−120748.			−0.797961			−0.398980	+	0.916959i	$0.630636\pi$
$390$	−2647.84	+	735.624i	−0.0174085	+	0.00483645i
$391$		−	186463.i		−	1.21966i
$392$	−353662.			−2.30153
$393$	−44261.0	−	159315.i	−0.286573	−	1.03151i
$394$		−	156514.i		−	1.00823i
$395$	113980.			0.730527
$396$	33125.7	−	19945.5i	0.211240	−	0.127190i
$397$	197150.			1.25088			0.625440	−	0.780272i	$-0.284919\pi$
$397$	197150.			1.25088			0.625440	+	0.780272i	$0.284919\pi$
$398$	110502.			0.697594
$399$	−1482.71	+	411.926i	−0.00931343	+	0.00258746i
$400$	116170.			0.726063
$401$		−	91127.4i		−	0.566709i	−0.959015	−	0.283355i	$-0.908553\pi$
$401$		−	91127.4i		−	0.566709i	0.959015	−	0.283355i	$-0.0914473\pi$
$402$	29383.3	+	105764.i	0.181823	+	0.654461i
$403$			3101.34i			0.0190958i
$404$	54945.7			0.336644
$405$	−113788.	−	60233.6i	−0.693724	−	0.367222i
$406$	2444.01	−	334.129i	0.0148269	−	0.00202704i
$407$		−	21461.2i		−	0.129558i
$408$	−165824.	−	596875.i	−0.996156	−	3.58561i
$409$		−	48919.9i		−	0.292442i	−0.989252	−	0.146221i	$-0.953289\pi$
$409$		−	48919.9i		−	0.292442i	0.989252	−	0.146221i	$-0.0467110\pi$
$410$			180433.i			1.07337i
$411$	40354.1	+	145252.i	0.238893	+	0.859884i
$412$	−589531.			−3.47306
$413$		−	978.291i		−	0.00573545i
$414$	120646.	+	200371.i	0.703904	+	1.16905i
$415$	−45288.1			−0.262959
$416$	2465.53			0.0142470
$417$	−58307.5	−	209875.i	−0.335314	−	1.20695i
$418$		−	40074.5i		−	0.229359i
$419$			29800.6i			0.169745i	0.996392	+	0.0848724i	$0.0270483\pi$
$419$			29800.6i			0.169745i	−0.996392	+	0.0848724i	$0.972952\pi$
$420$	2507.85	−	696.732i	0.0142168	−	0.00394973i
$421$			100307.i			0.565937i	0.959129	+	0.282968i	$0.0913192\pi$
$421$			100307.i			0.565937i	−0.959129	+	0.282968i	$0.908681\pi$
$422$			304387.i			1.70923i
$423$	−105875.	+	63749.1i	−0.591717	+	0.356282i
$424$		−	700179.i		−	3.89473i
$425$	112112.			0.620691
$426$	−228423.	+	63460.5i	−1.25869	+	0.349691i
$427$		−	2093.58i		−	0.0114824i
$428$			118297.i			0.645782i
$429$	68.0226	+	244.844i	0.000369606	+	0.00133038i
$430$	−256908.			−1.38944
$431$		−	142153.i		−	0.765246i	−0.923905	−	0.382623i	$-0.875021\pi$
$431$		−	142153.i		−	0.765246i	0.923905	−	0.382623i	$-0.124979\pi$
$432$	256366.	+	242610.i	1.37370	+	1.29999i
$433$			313179.i			1.67039i	0.549956	+	0.835193i	$0.314644\pi$
$433$			313179.i			1.67039i	−0.549956	+	0.835193i	$0.685356\pi$
$434$		−	4230.02i		−	0.0224576i
$435$	136403.	−	58775.9i	0.720849	−	0.310614i
$436$	398535.			2.09649
$437$	168327.			0.881436
$438$	−522189.	+	145075.i	−2.72194	+	0.756211i
$439$	−42216.8			−0.219056			−0.109528	−	0.993984i	$-0.534934\pi$
$439$	−42216.8			−0.219056			−0.109528	+	0.993984i	$0.534934\pi$
$440$			37953.2i			0.196039i
$441$	166599.	−	100312.i	0.856634	−	0.515792i
$442$	7270.88			0.0372171
$443$	82389.3			0.419820			0.209910	−	0.977721i	$-0.432683\pi$
$443$	82389.3			0.419820			0.209910	+	0.977721i	$0.432683\pi$
$444$	515342.	−	143173.i	2.61415	−	0.726263i
$445$			78649.5i			0.397169i
$446$	−387893.			−1.95003
$447$	−128024.	+	35567.7i	−0.640733	+	0.178009i
$448$	−222.605			−0.00110912
$449$	−381152.			−1.89062			−0.945312	−	0.326167i	$-0.894243\pi$
$449$	−381152.			−1.89062			−0.945312	+	0.326167i	$0.894243\pi$
$450$	−120474.	+	72539.3i	−0.594935	+	0.358219i
$451$	16684.5			0.0820276
$452$	434251.			2.12552
$453$	58842.5	+	211801.i	0.286744	+	1.03212i
$454$			242749.i			1.17773i
$455$			17.1057i			8.26261e-5i
$456$	538820.	−	149695.i	2.59128	−	0.719910i
$457$	−70384.3			−0.337011			−0.168505	−	0.985701i	$-0.553894\pi$
$457$	−70384.3			−0.337011			−0.168505	+	0.985701i	$0.553894\pi$
$458$			26660.7i			0.127099i
$459$	247411.	+	234135.i	1.17434	+	1.11133i
$460$	−284708.			−1.34550
$461$	−101625.			−0.478187			−0.239094	−	0.970997i	$-0.576850\pi$
$461$	−101625.			−0.478187			−0.239094	+	0.970997i	$0.576850\pi$
$462$	−92.7783	−	333.951i	−0.000434673	−	0.00156458i
$463$	190931.			0.890665			0.445333	−	0.895365i	$-0.353085\pi$
$463$	190931.			0.890665			0.445333	+	0.895365i	$0.353085\pi$
$464$	−403438.	+	55155.3i	−1.87387	+	0.256184i
$465$	−68177.7	−	245402.i	−0.315309	−	1.13494i
$466$		−	677560.i		−	3.12015i
$467$	319911.			1.46689			0.733443	−	0.679751i	$-0.237912\pi$
$467$	319911.			1.46689			0.733443	+	0.679751i	$0.237912\pi$
$468$	−5425.57	+	3266.81i	−0.0247716	+	0.0149153i
$469$	683.258			0.00310627
$470$		−	216642.i		−	0.980726i
$471$	−157509.	+	43759.1i	−0.710006	+	0.197254i
$472$			355514.i			1.59578i
$473$			23756.1i			0.106182i
$474$	364462.	−	101255.i	1.62217	−	0.450671i
$475$			101208.i			0.448566i
$476$	−6886.48			−0.0303937
$477$	198597.	+	329833.i	0.872843	+	1.44963i
$478$			522158.i			2.28532i
$479$	−208972.			−0.910788			−0.455394	−	0.890290i	$-0.650502\pi$
$479$	−208972.			−0.910788			−0.455394	+	0.890290i	$0.650502\pi$
$480$	−195092.	+	54200.5i	−0.846753	+	0.235245i
$481$			3515.07i			0.0151930i
$482$	−24070.2			−0.103606
$483$	1402.71	−	389.702i	0.00601276	−	0.00167047i
$484$	−526049.			−2.24562
$485$	−216153.			−0.918922
$486$	−417355.	−	91518.0i	−1.76699	−	0.387467i
$487$	−39411.8			−0.166176			−0.0830879	−	0.996542i	$-0.526478\pi$
$487$	−39411.8			−0.166176			−0.0830879	+	0.996542i	$0.526478\pi$
$488$			760813.i			3.19476i
$489$	352805.	−	98016.4i	1.47542	−	0.409903i
$490$			340895.i			1.41981i
$491$	272351.			1.12971			0.564854	−	0.825191i	$-0.308932\pi$
$491$	272351.			1.12971			0.564854	+	0.825191i	$0.308932\pi$
$492$	111306.	+	400640.i	0.459820	+	1.65510i
$493$	−389345.	+	53228.7i	−1.60192	+	0.219004i
$494$			6563.68i			0.0268964i
$495$	−10765.0	−	17878.6i	−0.0439341	−	0.0729664i
$496$			698256.i			2.83826i
$497$			1475.67i			0.00597413i
$498$	−144812.	+	40231.9i	−0.583912	+	0.162223i
$499$	−292173.			−1.17338			−0.586690	−	0.809812i	$-0.699569\pi$
$499$	−292173.			−1.17338			−0.586690	+	0.809812i	$0.699569\pi$
$500$		−	617092.i		−	2.46837i
$501$	301557.	−	83778.6i	1.20142	−	0.333778i
$502$	268675.			1.06615
$503$	−116211.			−0.459315			−0.229658	−	0.973271i	$-0.573761\pi$
$503$	−116211.			−0.459315			−0.229658	+	0.973271i	$0.573761\pi$
$504$	4143.56	−	2494.89i	0.0163122	−	0.00982180i
$505$		−	29655.3i		−	0.116284i
$506$			37912.3i			0.148074i
$507$	68796.2	+	247628.i	0.267639	+	0.963351i
$508$		−	515698.i		−	1.99833i
$509$		−	304519.i		−	1.17538i	−0.809086	−	0.587690i	$-0.800037\pi$
$509$		−	304519.i		−	1.17538i	0.809086	−	0.587690i	$-0.199963\pi$
$510$	−575329.	+	159838.i	−2.21195	+	0.614526i
$511$			3373.46i			0.0129192i
$512$	−586056.			−2.23562
$513$	−211362.	+	223346.i	−0.803142	+	0.848680i
$514$		−	217086.i		−	0.821684i
$515$			318181.i			1.19966i
$516$	−570448.	+	158482.i	−2.14248	+	0.595224i
$517$	−20032.8			−0.0749480
$518$		−	4794.33i		−	0.0178677i
$519$	387613.	−	107687.i	1.43901	−	0.399786i
$520$		−	6216.25i		−	0.0229891i
$521$			174738.i			0.643742i	0.946783	+	0.321871i	$0.104312\pi$
$521$			174738.i			0.643742i	−0.946783	+	0.321871i	$0.895688\pi$
$522$	383945.	−	309115.i	1.40906	−	1.13443i
$523$	34433.4			0.125886			0.0629429	−	0.998017i	$-0.479951\pi$
$523$	34433.4			0.125886			0.0629429	+	0.998017i	$0.479951\pi$
$524$	667973.			2.43274
$525$	234.311	+	843.389i	0.000850107	+	0.00305991i
$526$	685049.			2.47600
$527$			673866.i			2.42634i
$528$	15315.1	+	55125.8i	0.0549353	+	0.197737i
$529$	120596.			0.430944
$530$	−674904.			−2.40265
$531$	−100837.	−	167471.i	−0.357627	−	0.593952i
$532$		−	6216.66i		−	0.0219651i
$533$	−2732.71			−0.00961919
$534$	69868.5	+	251488.i	0.245019	+	0.881932i
$535$	63847.1			0.223066
$536$	−248298.			−0.864258
$537$	328996.	−	91401.8i	1.14089	−	0.316961i
$538$	795704.			2.74908
$539$	31522.3			0.108503
$540$	357498.	−	377768.i	1.22599	−	1.29550i
$541$		−	563577.i		−	1.92557i	−0.270271	−	0.962784i	$-0.587113\pi$
$541$		−	563577.i		−	1.92557i	0.270271	−	0.962784i	$-0.412887\pi$
$542$		−	475846.i		−	1.61982i
$543$	−44723.2	−	160979.i	−0.151682	−	0.545970i
$544$	535716.			1.81024
$545$		−	215097.i		−	0.724172i
$546$	15.1959	+	54.6968i	5.09731e−5	+	0.000183475i
$547$	−308082.			−1.02965			−0.514827	−	0.857294i	$-0.672144\pi$
$547$	−308082.			−1.02965			−0.514827	+	0.857294i	$0.672144\pi$
$548$	−609012.			−2.02798
$549$	−215795.	−	358395.i	−0.715973	−	1.18910i
$550$	−22795.0			−0.0753555
$551$	−48051.4	−	351476.i	−0.158272	−	1.15769i
$552$	−509749.	+	141619.i	−1.67293	+	0.464775i
$553$		−	2354.51i		−	0.00769928i
$554$	747844.			2.43664
$555$	−77273.0	−	278140.i	−0.250866	−	0.902979i
$556$	879958.			2.84651
$557$		−	252223.i		−	0.812968i	−0.913658	−	0.406484i	$-0.866755\pi$
$557$		−	252223.i		−	0.812968i	0.913658	−	0.406484i	$-0.133245\pi$
$558$	−436007.	−	724127.i	−1.40031	−	2.32566i
$559$		−	3890.94i		−	0.0124518i
$560$			3851.29i			0.0122809i
$561$	14780.1	+	53200.2i	0.0469626	+	0.169039i
$562$		−	375519.i		−	1.18894i
$563$	−100456.			−0.316926			−0.158463	−	0.987365i	$-0.550654\pi$
$563$	−100456.			−0.316926			−0.158463	+	0.987365i	$0.550654\pi$
$564$	−133643.	−	481041.i	−0.420134	−	1.51225i
$565$		−	234374.i		−	0.734196i
$566$	560278.			1.74892
$567$	−1244.25	+	2350.53i	−0.00387029	+	0.00731140i
$568$		−	536261.i		−	1.66219i
$569$	−387102.			−1.19564			−0.597820	−	0.801630i	$-0.703966\pi$
$569$	−387102.			−1.19564			−0.597820	+	0.801630i	$0.703966\pi$
$570$	−144291.	−	519370.i	−0.444110	−	1.59855i
$571$	230611.			0.707305			0.353653	−	0.935377i	$-0.384939\pi$
$571$	230611.			0.707305			0.353653	+	0.935377i	$0.384939\pi$
$572$	−1026.58			−0.00313761
$573$	112396.	+	404563.i	0.342327	+	1.23219i
$574$	3727.23			0.0113126
$575$		−	95747.3i		−	0.289595i
$576$	−38107.3	+	22945.0i	−0.114859	+	0.0691580i
$577$		−	199189.i		−	0.598292i	−0.954207	−	0.299146i	$-0.903298\pi$
$577$		−	199189.i		−	0.598292i	0.954207	−	0.299146i	$-0.0967017\pi$
$578$	975487.			2.91989
$579$	−45114.4	+	12533.7i	−0.134573	+	0.0373871i
$580$	81274.1	+	594486.i	0.241600	+	1.76720i
$581$			935.523i			0.00277142i
$582$	−691168.	+	192021.i	−2.04051	+	0.566894i
$583$			62407.9i			0.183613i
$584$		−	1.22593e6i		−	3.59450i
$585$	1763.16	+	2928.28i	0.00515205	+	0.00855660i
$586$	−183774.			−0.535166
$587$			268148.i			0.778214i	0.921193	+	0.389107i	$0.127216\pi$
$587$			268148.i			0.778214i	−0.921193	+	0.389107i	$0.872784\pi$
$588$	210292.	+	756937.i	0.608232	+	2.18930i
$589$	−608322.			−1.75349
$590$	342680.			0.984431
$591$	−187568.	+	52110.3i	−0.537013	+	0.149193i
$592$			791408.i			2.25817i
$593$			326728.i			0.929131i	0.885539	+	0.464565i	$0.153789\pi$
$593$			326728.i			0.929131i	−0.885539	+	0.464565i	$0.846211\pi$
$594$	−50304.3	−	47605.2i	−0.142571	−	0.134921i
$595$			3716.76i			0.0104986i
$596$		−	536777.i		−	1.51113i
$597$	−36790.8	−	132426.i	−0.103226	−	0.371558i
$598$		−	6209.55i		−	0.0173643i
$599$	393139.			1.09570			0.547852	−	0.836575i	$-0.315446\pi$
$599$	393139.			1.09570			0.547852	+	0.836575i	$0.315446\pi$
$600$	−85149.1	−	306490.i	−0.236525	−	0.851361i
$601$			342216.i			0.947438i	0.880676	+	0.473719i	$0.157089\pi$
$601$			342216.i			0.947438i	−0.880676	+	0.473719i	$0.842911\pi$
$602$			5306.98i			0.0146438i
$603$	116965.	−	70426.6i	0.321679	−	0.193688i
$604$	−888033.			−2.43419
$605$			283919.i			0.775681i
$606$	−26344.4	−	94825.1i	−0.0717368	−	0.258213i
$607$		−	251048.i		−	0.681365i	−0.940178	−	0.340682i	$-0.889342\pi$
$607$		−	251048.i		−	0.681365i	0.940178	−	0.340682i	$-0.110658\pi$
$608$			483610.i			1.30824i
$609$	−1214.14	−	2817.69i	−0.00327367	−	0.00759729i
$610$	733349.			1.97084
$611$	3281.11			0.00878897
$612$	−1.17888e6	+	709821.i	−3.14751	+	1.89516i
$613$	268670.			0.714987			0.357494	−	0.933916i	$-0.383631\pi$
$613$	268670.			0.714987			0.357494	+	0.933916i	$0.383631\pi$
$614$		−	233411.i		−	0.619134i
$615$	216233.	−	60073.9i	0.571705	−	0.158831i
$616$	784.005			0.00206613
$617$	−251765.			−0.661340			−0.330670	−	0.943746i	$-0.607275\pi$
$617$	−251765.			−0.661340			−0.330670	+	0.943746i	$0.607275\pi$
$618$	282657.	+	1.01741e6i	0.740088	+	2.66391i
$619$		−	378681.i		−	0.988308i	−0.869374	−	0.494154i	$-0.835478\pi$
$619$		−	378681.i		−	0.988308i	0.869374	−	0.494154i	$-0.164522\pi$
$620$	1.02892e6			2.67668
$621$	199959.	−	211296.i	0.518510	−	0.547909i
$622$	−695435.			−1.79753
$623$	1624.67			0.00418591
$624$	−2508.41	−	9028.90i	−0.00644213	−	0.0231881i
$625$	−183097.			−0.468729
$626$	1.03459e6			2.64011
$627$	−48025.7	+	13342.5i	−0.122163	+	0.0339393i
$628$		−	660398.i		−	1.67451i
$629$			763764.i			1.93045i
$630$	−2404.83	−	3993.98i	−0.00605904	−	0.0100629i
$631$	156349.			0.392677			0.196339	−	0.980536i	$-0.437095\pi$
$631$	156349.			0.392677			0.196339	+	0.980536i	$0.437095\pi$
$632$			855636.i			2.14217i
$633$	364781.	−	101344.i	0.910384	−	0.252923i
$634$	342348.			0.851705
$635$	−278332.			−0.690264
$636$	−1.49858e6	+	416337.i	−3.70481	+	1.02927i
$637$	−5162.95			−0.0127239
$638$	79163.0	−	10822.6i	0.194483	−	0.0265884i
$639$	152104.	+	252616.i	0.372510	+	0.618670i
$640$			281991.i			0.688454i
$641$	−313782.			−0.763680			−0.381840	−	0.924228i	$-0.624710\pi$
$641$	−313782.			−0.763680			−0.381840	+	0.924228i	$0.624710\pi$
$642$	204156.	−	56718.8i	0.495328	−	0.137612i
$643$	−520220.			−1.25825			−0.629123	−	0.777306i	$-0.716586\pi$
$643$	−520220.			−1.25825			−0.629123	+	0.777306i	$0.716586\pi$
$644$			5881.26i			0.0141807i
$645$	85535.7	+	307881.i	0.205602	+	0.740055i
$646$			1.42617e6i			3.41749i
$647$		−	71922.9i		−	0.171814i	−0.996303	−	0.0859070i	$-0.972621\pi$
$647$		−	71922.9i		−	0.171814i	0.996303	−	0.0859070i	$-0.0273788\pi$
$648$	452166.	−	854191.i	1.07683	−	2.03425i
$649$		−	31687.4i		−	0.0752310i
$650$	3733.53			0.00883676
$651$	−5069.30	+	1408.36i	−0.0119615	+	0.00332315i
$652$			1.47923e6i			3.47970i
$653$	−154050.			−0.361273			−0.180637	−	0.983550i	$-0.557816\pi$
$653$	−154050.			−0.361273			−0.180637	+	0.983550i	$0.557816\pi$
$654$	−191082.	−	687790.i	−0.446750	−	1.60805i
$655$		−	360518.i		−	0.840319i
$656$	−615260.			−1.42972
$657$	347718.	+	577495.i	0.805558	+	1.33788i
$658$	−4475.21			−0.0103362
$659$	262520.			0.604493			0.302247	−	0.953230i	$-0.402263\pi$
$659$	262520.			0.604493			0.302247	+	0.953230i	$0.402263\pi$
$660$	81230.7	−	22567.5i	0.186480	−	0.0518079i
$661$	−137844.			−0.315490			−0.157745	−	0.987480i	$-0.550422\pi$
$661$	−137844.			−0.315490			−0.157745	+	0.987480i	$0.550422\pi$
$662$		−	553630.i		−	1.26329i
$663$	−2420.79	−	8713.52i	−0.00550720	−	0.0198229i
$664$		−	339972.i		−	0.771093i
$665$	−3355.25			−0.00758720
$666$	−494173.	−	820730.i	−1.11412	−	1.85034i
$667$	45458.9	+	332513.i	0.102180	+	0.747406i
$668$			1.26436e6i			2.83347i
$669$	129146.	+	464856.i	0.288556	+	1.03864i
$670$			239335.i			0.533158i
$671$		−	67812.2i		−	0.150613i
$672$	1119.63	+	4030.04i	0.00247933	+	0.00892423i
$673$	710569.			1.56883			0.784415	−	0.620236i	$-0.212963\pi$
$673$	710569.			1.56883			0.784415	+	0.620236i	$0.212963\pi$
$674$		−	491490.i		−	1.08192i
$675$	127043.	+	120226.i	0.278833	+	0.263871i
$676$	−1.03825e6			−2.27200
$677$	−648634.			−1.41522			−0.707608	−	0.706605i	$-0.750226\pi$
$677$	−648634.			−1.41522			−0.707608	+	0.706605i	$0.750226\pi$
$678$	−208207.	−	749429.i	−0.452934	−	1.63031i
$679$			4465.11i			0.00968484i
$680$		−	1.35068e6i		−	2.92103i
$681$	290914.	−	80821.7i	0.627292	−	0.174274i
$682$		−	137013.i		−	0.294572i
$683$		−	658451.i		−	1.41150i	−0.708459	−	0.705752i	$-0.750609\pi$
$683$		−	658451.i		−	1.41150i	0.708459	−	0.705752i	$-0.249391\pi$
$684$	−640780.	−	1.06422e6i	−1.36961	−	2.27467i
$685$			328695.i			0.700506i
$686$	14084.3			0.0299287
$687$	31950.5	−	8876.51i	0.0676962	−	0.0188074i
$688$		−	876033.i		−	1.85073i
$689$		−	10221.6i		−	0.0215318i
$690$	136507.	+	491348.i	0.286718	+	1.03203i
$691$	728040.			1.52475			0.762376	−	0.647134i	$-0.224033\pi$
$691$	728040.			1.52475			0.762376	+	0.647134i	$0.224033\pi$
$692$			1.62517e6i			3.39381i
$693$	−369.320	+	222.373i	−0.000769019	+	0.000463037i
$694$		−	1.11829e6i		−	2.32185i
$695$		−	474930.i		−	0.983241i
$696$	441223.	+	1.02396e6i	0.910834	+	2.11380i
$697$	−593769.			−1.22223
$698$	371107.			0.761707
$699$	−811996.	+	225589.i	−1.66188	+	0.461704i
$700$	−3536.14			−0.00721662
$701$			310664.i			0.632200i	0.948726	+	0.316100i	$0.102374\pi$
$701$			310664.i			0.632200i	−0.948726	+	0.316100i	$0.897626\pi$
$702$	8239.20	+	7797.11i	0.0167190	+	0.0158219i
$703$	−689476.			−1.39511
$704$	−7210.31			−0.0145482
$705$	−259627.	+	72129.7i	−0.522362	+	0.145123i
$706$		−	1.71258e6i		−	3.43591i
$707$	−612.593			−0.00122556
$708$	760900.	−	211394.i	1.51796	−	0.421721i
$709$	34103.8			0.0678439			0.0339219	−	0.999424i	$-0.489200\pi$
$709$	34103.8			0.0678439			0.0339219	+	0.999424i	$0.489200\pi$
$710$	−516903.			−1.02540
$711$	−242690.	−	403063.i	−0.480079	−	0.797323i
$712$	−590411.			−1.16465
$713$	575502.			1.13206
$714$	3301.80	+	11884.7i	0.00647671	+	0.0233126i
$715$			554.062i			0.00108379i
$716$			1.37941e6i			2.69071i
$717$	625760.	−	173849.i	1.21722	−	0.338169i
$718$	20904.9			0.0405508
$719$		−	317802.i		−	0.614750i	−0.951588	−	0.307375i	$-0.900549\pi$
$719$		−	317802.i		−	0.614750i	0.951588	−	0.307375i	$-0.0994507\pi$
$720$	396970.	+	659294.i	0.765760	+	1.27179i
$721$	6572.71			0.0126437
$722$	−344468.			−0.660807
$723$	8014.01	+	28846.0i	0.0153311	+	0.0551835i
$724$	674948.			1.28764
$725$	−199925.	+	27332.5i	−0.380357	+	0.0519999i
$726$	252220.	+	907853.i	0.478527	+	1.72243i
$727$			590844.i			1.11790i	0.829200	+	0.558952i	$0.188796\pi$
$727$			590844.i			1.11790i	−0.829200	+	0.558952i	$0.811204\pi$
$728$	−128.410			−0.000242290
$729$	29279.3	+	530634.i	0.0550942	+	0.998481i
$730$	−1.18167e6			−2.21744
$731$		−	845433.i		−	1.58214i
$732$	1.62836e6	−	452390.i	3.03897	−	0.844289i
$733$			246513.i			0.458808i	0.973331	+	0.229404i	$0.0736777\pi$
$733$			246513.i			0.458808i	−0.973331	+	0.229404i	$0.926322\pi$
$734$			1.66326e6i			3.08723i
$735$	408533.	−	113499.i	0.756228	−	0.210095i
$736$		−	457518.i		−	0.844602i
$737$	22131.1			0.0407444
$738$	638056.	−	384183.i	1.17151	−	0.705383i
$739$		−	434180.i		−	0.795025i	−0.917597	−	0.397513i	$-0.869874\pi$
$739$		−	434180.i		−	0.795025i	0.917597	−	0.397513i	$-0.130126\pi$
$740$	1.16618e6			2.12962
$741$	7865.99	−	2185.33i	0.0143257	−	0.00397998i
$742$			13941.6i			0.0253224i
$743$	−383205.			−0.694150			−0.347075	−	0.937837i	$-0.612825\pi$
$743$	−383205.			−0.694150			−0.347075	+	0.937837i	$0.612825\pi$
$744$	1.84220e6	−	511800.i	3.32806	−	0.924602i
$745$	−289709.			−0.521974
$746$	−1.01604e6			−1.82571
$747$	96428.7	+	160150.i	0.172808	+	0.287003i
$748$	−223057.			−0.398669
$749$		−	1318.90i		−	0.00235097i
$750$	−1.06498e6	+	295872.i	−1.89329	+	0.525994i
$751$			155076.i			0.274957i	0.990505	+	0.137478i	$0.0438998\pi$
$751$			155076.i			0.274957i	−0.990505	+	0.137478i	$0.956100\pi$
$752$	738731.			1.30632
$753$	−89453.6	−	321983.i	−0.157764	−	0.567863i
$754$	−12965.9	+	1772.61i	−0.0228065	+	0.00311796i
$755$			479288.i			0.840820i
$756$	−7803.60	−	7384.88i	−0.0136537	−	0.0129211i
$757$			614348.i			1.07207i	0.844196	+	0.536034i	$0.180078\pi$
$757$			614348.i			1.07207i	−0.844196	+	0.536034i	$0.819922\pi$
$758$			254682.i			0.443261i
$759$	45434.6	−	12622.7i	0.0788684	−	0.0219113i
$760$	1.21931e6			2.11099
$761$			185648.i			0.320569i	0.987071	+	0.160284i	$0.0512412\pi$
$761$			185648.i			0.320569i	−0.987071	+	0.160284i	$0.948759\pi$
$762$	−889989.	+	247257.i	−1.53276	+	0.425832i
$763$	−4443.29			−0.00763230
$764$	−1.69624e6			−2.90604
$765$	383104.	+	636264.i	0.654627	+	1.08721i
$766$			351172.i			0.598497i
$767$			5189.99i			0.00882217i
$768$	271676.	+	977882.i	0.460604	+	1.65792i
$769$		−	529269.i		−	0.895002i	−0.894284	−	0.447501i	$-0.852314\pi$
$769$		−	529269.i		−	0.895002i	0.894284	−	0.447501i	$-0.147686\pi$
$770$		−	755.704i		−	0.00127459i
$771$	−260158.	+	72277.2i	−0.437651	+	0.121588i
$772$		−	189155.i		−	0.317382i
$773$	−711620.			−1.19094			−0.595469	−	0.803378i	$-0.703034\pi$
$773$	−711620.			−1.19094			−0.595469	+	0.803378i	$0.703034\pi$
$774$	547015.	+	908490.i	0.913098	+	1.51649i
$775$			346024.i			0.576107i
$776$		−	1.62263e6i		−	2.69462i
$777$	−5745.58	+	1596.24i	−0.00951682	+	0.00264397i
$778$	−873720.			−1.44349
$779$		−	536016.i		−	0.883289i
$780$	−13304.5	+	3696.27i	−0.0218681	+	0.00607540i
$781$			47797.6i			0.0783618i
$782$		−	1.34923e6i		−	2.20633i
$783$	−498279.	−	357207.i	−0.812734	−	0.582634i
$784$	−1.16242e6			−1.89118
$785$	−356430.			−0.578408
$786$	−320267.	−	1.15279e6i	−0.518403	−	1.86596i
$787$	−1.09692e6			−1.77103			−0.885516	−	0.464609i	$-0.846195\pi$
$787$	−1.09692e6			−1.77103			−0.885516	+	0.464609i	$0.846195\pi$
$788$		−	786433.i		−	1.26651i
$789$	−228082.	−	820971.i	−0.366385	−	1.31878i
$790$	824749.			1.32150
$791$	−4841.49			−0.00773795
$792$	134212.	−	80811.0i	0.213964	−	0.128831i
$793$			11106.8i			0.0176621i
$794$	1.42655e6			2.26281
$795$	224705.	+	808813.i	0.355532	+	1.27972i
$796$	555235.			0.876296
$797$	832139.			1.31002			0.655012	−	0.755618i	$-0.272664\pi$
$797$	832139.			1.31002			0.655012	+	0.755618i	$0.272664\pi$
$798$	−10728.7	+	2980.65i	−0.0168477	+	0.00468064i
$799$	712927.			1.11674
$800$	275085.			0.429821
$801$	278124.	−	167463.i	0.433485	−	0.261007i
$802$		−	659387.i		−	1.02516i
$803$			109268.i			0.169458i
$804$	147642.	+	531428.i	0.228400	+	0.822114i
$805$	3174.23			0.00489831
$806$			22440.9i			0.0345438i
$807$	−264924.	−	953582.i	−0.406794	−	1.46424i
$808$	222618.			0.340987
$809$	551020.			0.841919			0.420960	−	0.907079i	$-0.361693\pi$
$809$	551020.			0.841919			0.420960	+	0.907079i	$0.361693\pi$
$810$	−823356.	−	435843.i	−1.25493	−	0.664294i
$811$	922048.			1.40188			0.700942	−	0.713219i	$-0.252763\pi$
$811$	922048.			1.40188			0.700942	+	0.713219i	$0.252763\pi$
$812$	12280.4	−	1678.89i	0.0186251	−	0.00254631i
$813$	−570260.	+	158430.i	−0.862763	+	0.239693i
$814$		−	155291.i		−	0.234367i
$815$	798370.			1.20196
$816$	−545034.	−	1.96182e6i	−0.818546	−	2.94631i
$817$	763201.			1.14339
$818$		−	353979.i		−	0.529018i
$819$	60.4899	−	36.4219i	9.01811e−5	−	5.42993e-5i
$820$			906617.i			1.34833i
$821$			542818.i			0.805319i	0.915350	+	0.402660i	$0.131914\pi$
$821$			542818.i			0.805319i	−0.915350	+	0.402660i	$0.868086\pi$
$822$	291997.	+	1.05103e6i	0.432151	+	1.55550i
$823$			850262.i			1.25532i	0.778489	+	0.627658i	$0.215986\pi$
$823$			850262.i			1.25532i	−0.778489	+	0.627658i	$0.784014\pi$
$824$	−2.38854e6			−3.51786
$825$	7589.45	+	27317.8i	0.0111507	+	0.0401364i
$826$		−	7078.79i		−	0.0103753i
$827$	192231.			0.281069			0.140535	−	0.990076i	$-0.455118\pi$
$827$	192231.			0.281069			0.140535	+	0.990076i	$0.455118\pi$
$828$	606209.	+	1.00680e6i	0.884222	+	1.46853i
$829$			545493.i			0.793744i	0.917874	+	0.396872i	$0.129904\pi$
$829$			545493.i			0.793744i	−0.917874	+	0.396872i	$0.870096\pi$
$830$	−327699.			−0.475685
$831$	−248990.	−	896225.i	−0.360561	−	1.29782i
$832$	1180.96			0.00170603
$833$	−1.12182e6			−1.61671
$834$	−421906.	−	1.51863e6i	−0.606573	−	2.18333i
$835$	682400.			0.978737
$836$		−	201361.i		−	0.288113i
$837$	−722637.	+	763610.i	−1.03150	+	1.08999i
$838$			215633.i			0.307063i
$839$	241031.			0.342412			0.171206	−	0.985235i	$-0.445234\pi$
$839$	241031.			0.342412			0.171206	+	0.985235i	$0.445234\pi$
$840$	10160.8	−	2822.88i	0.0144002	−	0.00400068i
$841$	681327.	−	189841.i	0.963305	−	0.268410i
$842$			725811.i			1.02376i
$843$	−450027.	+	125027.i	−0.633261	+	0.175933i
$844$			1.52945e6i			2.14709i
$845$			560364.i			0.784796i
$846$	−766102.	+	461281.i	−1.07040	+	0.644503i
$847$	5864.95			0.00817518
$848$		−	2.30136e6i		−	3.20032i
$849$	−186541.	−	671443.i	−0.258796	−	0.931524i
$850$	811231.			1.12281
$851$	652277.			0.900685
$852$	−1.14775e6	+	318868.i	−1.58113	+	0.439271i
$853$			659310.i			0.906132i	0.891477	+	0.453066i	$0.149670\pi$
$853$			659310.i			0.906132i	−0.891477	+	0.453066i	$0.850330\pi$
$854$		−	15148.9i		−	0.0207714i
$855$	−574378.	+	345841.i	−0.785716	+	0.473091i
$856$			479291.i			0.654112i
$857$		−	1.07814e6i		−	1.46796i	−0.679174	−	0.733978i	$-0.737662\pi$
$857$		−	1.07814e6i		−	1.46796i	0.679174	−	0.733978i	$-0.262338\pi$
$858$	492.203	+	1771.66i	0.000668606	+	0.00240661i
$859$		−	348516.i		−	0.472320i	−0.971714	−	0.236160i	$-0.924111\pi$
$859$		−	348516.i		−	0.472320i	0.971714	−	0.236160i	$-0.0758889\pi$
$860$	−1.29088e6			−1.74537
$861$	−1240.96	−	4466.76i	−0.00167398	−	0.00602540i
$862$		−	1.02860e6i		−	1.38431i
$863$			289079.i			0.388146i	0.980987	+	0.194073i	$0.0621699\pi$
$863$			289079.i			0.388146i	−0.980987	+	0.194073i	$0.937830\pi$
$864$	607061.	+	574488.i	0.813214	+	0.769580i
$865$	877138.			1.17229
$866$			2.26613e6i			3.02168i
$867$	−324782.	−	1.16904e6i	−0.432070	−	1.55521i
$868$		−	21254.5i		−	0.0282105i
$869$		−	76263.9i		−	0.100990i
$870$	986994.	−	425296.i	1.30399	−	0.561891i
$871$	−3624.79			−0.00477801
$872$	1.61470e6			2.12354
$873$	460240.	+	764372.i	0.603887	+	1.00294i
$874$	1.21799e6			1.59449
$875$			6879.99i			0.00898611i
$876$	−2.62383e6	+	728953.i	−3.41922	+	0.949930i
$877$	−428834.			−0.557558			−0.278779	−	0.960355i	$-0.589930\pi$
$877$	−428834.			−0.557558			−0.278779	+	0.960355i	$0.589930\pi$
$878$	−305475.			−0.396266
$879$	61186.3	+	220237.i	0.0791911	+	0.285044i
$880$			124745.i			0.161087i
$881$	675379.			0.870153			0.435077	−	0.900393i	$-0.356721\pi$
$881$	675379.			0.870153			0.435077	+	0.900393i	$0.356721\pi$
$882$	1.20549e6	−	725843.i	1.54963	−	0.933052i
$883$	519933.			0.666847			0.333423	−	0.942777i	$-0.391796\pi$
$883$	519933.			0.666847			0.333423	+	0.942777i	$0.391796\pi$
$884$	36533.8			0.0467510
$885$	−114093.	−	410672.i	−0.145671	−	0.524335i
$886$	596159.			0.759443
$887$	10596.0			0.0134678			0.00673388	−	0.999977i	$-0.497857\pi$
$887$	10596.0			0.0134678			0.00673388	+	0.999977i	$0.497857\pi$
$888$	2.08796e6	−	580078.i	2.64787	−	0.735631i
$889$			5749.54i			0.00727494i
$890$			569098.i			0.718468i
$891$	−40302.1	+	76135.1i	−0.0507659	+	0.0959025i
$892$	−1.94904e6			−2.44957
$893$			643584.i			0.807054i
$894$	−926368.	+	257364.i	−1.15907	+	0.322012i
$895$	744493.			0.929425
$896$	5825.12			0.00725587
$897$	−7441.60	+	2067.43i	−0.00924872	+	0.00256948i
$898$	−2.75797e6			−3.42008
$899$	−164285.	−	1.20168e6i	−0.203273	−	1.48686i
$900$	−605345.	+	364487.i	−0.747339	+	0.449984i
$901$		−	2.22098e6i		−	2.73586i
$902$	120727.			0.148385
$903$	6359.95	−	1766.92i	0.00779970	−	0.00216692i
$904$	1.75941e6			2.15293
$905$		−	364282.i		−	0.444775i
$906$	425777.	+	1.53256e6i	0.518712	+	1.86708i
$907$		−	1.38317e6i		−	1.68136i	−0.541530	−	0.840681i	$-0.682155\pi$
$907$		−	1.38317e6i		−	1.68136i	0.541530	−	0.840681i	$-0.317845\pi$
$908$			1.21974e6i			1.47943i
$909$	−104868.	+	63142.8i	−0.126916	+	0.0764180i
$910$			123.775i			0.000149468i
$911$	−799842.			−0.963757			−0.481879	−	0.876238i	$-0.660045\pi$
$911$	−799842.			−0.963757			−0.481879	+	0.876238i	$0.660045\pi$
$912$	1.77100e6	−	492021.i	2.12927	−	0.591553i
$913$			30302.1i			0.0363523i
$914$	−509293.			−0.609642
$915$	−244164.	−	878854.i	−0.291634	−	1.04972i
$916$			133962.i			0.159657i
$917$	−7447.27			−0.00885642
$918$	1.79023e6	+	1.69418e6i	2.12434	+	2.01036i
$919$	−619130.			−0.733080			−0.366540	−	0.930402i	$-0.619458\pi$
$919$	−619130.			−0.733080			−0.366540	+	0.930402i	$0.619458\pi$
$920$	−1.15352e6			−1.36286
$921$	−279723.	+	77712.6i	−0.329768	+	0.0916162i
$922$	−735345.			−0.865026
$923$		−	7828.64i		−	0.00918930i
$924$	−466.181	−	1677.99i	−0.000546022	−	0.00196538i
$925$			392186.i			0.458362i
$926$	1.38155e6			1.61119
$927$	1.12517e6	−	677480.i	1.30936	−	0.788382i
$928$	−955321.	+	130605.i	−1.10931	+	0.151658i
$929$		−	918103.i		−	1.06380i	−0.846807	−	0.531900i	$-0.821478\pi$
$929$		−	918103.i		−	1.06380i	0.846807	−	0.531900i	$-0.178522\pi$
$930$	−493326.	−	1.77570e6i	−0.570384	−	2.05307i
$931$		−	1.01270e6i		−	1.16838i
$932$		−	3.40452e6i		−	3.91944i
$933$	231540.	+	833417.i	0.265989	+	0.957413i
$934$	2.31484e6			2.65355
$935$			120388.i			0.137708i
$936$	−21982.2	+	13235.8i	−0.0250911	+	0.0151077i
$937$	564657.			0.643140			0.321570	−	0.946886i	$-0.395789\pi$
$937$	564657.			0.643140			0.321570	+	0.946886i	$0.395789\pi$
$938$	4943.97			0.00561915
$939$	−344461.	−	1.23987e6i	−0.390669	−	1.40619i
$940$		−	1.08856e6i		−	1.23196i
$941$			1.47710e6i			1.66813i	0.551665	+	0.834066i	$0.313993\pi$
$941$			1.47710e6i			1.66813i	−0.551665	+	0.834066i	$0.686007\pi$
$942$	−1.13971e6	+	316635.i	−1.28438	+	0.356827i
$943$			507096.i			0.570252i
$944$			1.16851e6i			1.31126i
$945$	−3985.76	+	4211.75i	−0.00446321	+	0.00471627i
$946$			171896.i			0.192081i
$947$	−1.09520e6			−1.22122			−0.610611	−	0.791931i	$-0.709076\pi$
$947$	−1.09520e6			−1.22122			−0.610611	+	0.791931i	$0.709076\pi$
$948$	1.83130e6	−	508773.i	2.03772	−	0.566119i
$949$		−	17896.7i		−	0.0198720i
$950$			732327.i			0.811442i
$951$	−113982.	−	410274.i	−0.126031	−	0.453642i
$952$	−27901.2			−0.0307857
$953$		−	1.64631e6i		−	1.81270i	−0.422532	−	0.906348i	$-0.638859\pi$
$953$		−	1.64631e6i		−	1.81270i	0.422532	−	0.906348i	$-0.361141\pi$
$954$	1.43702e6	+	2.38663e6i	1.57895	+	2.62234i
$955$			915494.i			1.00380i
$956$			2.62368e6i			2.87074i
$957$	−39326.8	−	91266.5i	−0.0429402	−	0.0996524i
$958$	−1.51210e6			−1.64759
$959$	6789.90			0.00738289
$960$	−93446.5	+	25961.3i	−0.101396	+	0.0281699i
$961$	−1.15630e6			−1.25206
$962$			25434.7i			0.0274837i
$963$	−135945.	−	225779.i	−0.146592	−	0.243462i
$964$	−120945.			−0.130147
$965$	−102090.			−0.109630
$966$	10149.8	−	2819.84i	0.0108769	−	0.00302183i
$967$		−	49950.4i		−	0.0534178i	−0.999643	−	0.0267089i	$-0.991497\pi$
$967$		−	49950.4i		−	0.0534178i	0.999643	−	0.0267089i	$-0.00850271\pi$
$968$	−2.13134e6			−2.27458
$969$	1.70914e6	−	474835.i	1.82025	−	0.505702i
$970$	−1.56406e6			−1.66230
$971$	1.31590e6			1.39567			0.697837	−	0.716257i	$-0.254146\pi$
$971$	1.31590e6			1.39567			0.697837	+	0.716257i	$0.254146\pi$
$972$	−2.09708e6	−	459849.i	−2.21963	−	0.486724i
$973$	−9810.70			−0.0103627
$974$	−285179.			−0.300607
$975$	−1243.05	−	4474.31i	−0.00130762	−	0.00470670i
$976$			2.50065e6i			2.62515i
$977$		−	328035.i		−	0.343662i	−0.985126	−	0.171831i	$-0.945032\pi$
$977$		−	328035.i		−	0.343662i	0.985126	−	0.171831i	$-0.0549683\pi$
$978$	2.55285e6	−	709235.i	2.66900	−	0.741502i
$979$	52624.1			0.0549059
$980$			1.71289e6i			1.78352i
$981$	−760637.	+	457990.i	−0.790386	+	0.475903i
$982$	1.97070e6			2.04361
$983$	1.27241e6			1.31680			0.658399	−	0.752669i	$-0.271234\pi$
$983$	1.27241e6			1.31680			0.658399	+	0.752669i	$0.271234\pi$
$984$	450967.	+	1.62323e6i	0.465752	+	1.67645i
$985$	−424453.			−0.437479
$986$	−2.81726e6	+	385156.i	−2.89783	+	0.396172i
$987$	1489.99	+	5363.15i	0.00152950	+	0.00550536i
$988$			32980.4i			0.0337864i
$989$	−722025.			−0.738175
$990$	−77893.9	−	129367.i	−0.0794755	−	0.131994i
$991$	1.32060e6			1.34470			0.672349	−	0.740234i	$-0.265285\pi$
$991$	1.32060e6			1.34470			0.672349	+	0.740234i	$0.265285\pi$
$992$			1.65344e6i			1.68021i
$993$	−663477.	+	184327.i	−0.672864	+	0.186935i
$994$			10677.7i			0.0108070i
$995$		−	299671.i		−	0.302690i
$996$	−727636.	+	202152.i	−0.733492	+	0.203779i
$997$		−	1.47306e6i		−	1.48194i	−0.671537	−	0.740971i	$-0.734365\pi$
$997$		−	1.47306e6i		−	1.48194i	0.671537	−	0.740971i	$-0.265635\pi$
$998$	−2.11413e6			−2.12261
$999$	−819041.	+	865480.i	−0.820681	+	0.867213i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.5.d.c.86.31	yes	32
3.2	odd	2	inner	87.5.d.c.86.1	✓	32
29.28	even	2	inner	87.5.d.c.86.2	yes	32
87.86	odd	2	inner	87.5.d.c.86.32	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.5.d.c.86.1	✓	32	3.2	odd	2	inner
87.5.d.c.86.2	yes	32	29.28	even	2	inner
87.5.d.c.86.31	yes	32	1.1	even	1	trivial
87.5.d.c.86.32	yes	32	87.86	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	87.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.23588 q^{2} +(-2.40914 - 8.67157i) q^{3} +36.3580 q^{4} -19.6231i q^{5} +(-17.4322 - 62.7464i) q^{6} -0.405357 q^{7} +147.308 q^{8} +(-69.3921 + 41.7820i) q^{9} -141.990i q^{10} -13.1297 q^{11} +(-87.5913 - 315.280i) q^{12} +2.15048 q^{13} -2.93311 q^{14} +(-170.163 + 47.2747i) q^{15} +484.174 q^{16} +467.262 q^{17} +(-502.113 + 302.330i) q^{18} +421.813i q^{19} -713.455i q^{20} +(0.976560 + 3.51508i) q^{21} -95.0052 q^{22} -399.056i q^{23} +(-354.885 - 1277.39i) q^{24} +239.935 q^{25} +15.5606 q^{26} +(529.490 + 501.080i) q^{27} -14.7379 q^{28} +(-833.249 + 113.916i) q^{29} +(-1231.28 + 342.074i) q^{30} +1442.16i q^{31} +1146.50 q^{32} +(31.6313 + 113.855i) q^{33} +3381.05 q^{34} +7.95435i q^{35} +(-2522.96 + 1519.11i) q^{36} +1634.55i q^{37} +3052.19i q^{38} +(-5.18081 - 18.6480i) q^{39} -2890.63i q^{40} -1270.74 q^{41} +(7.06627 + 25.4347i) q^{42} -1809.33i q^{43} -477.370 q^{44} +(819.891 + 1361.69i) q^{45} -2887.52i q^{46} +1525.76 q^{47} +(-1166.44 - 4198.55i) q^{48} -2400.84 q^{49} +1736.14 q^{50} +(-1125.70 - 4051.89i) q^{51} +78.1871 q^{52} -4753.17i q^{53} +(3831.33 + 3625.75i) q^{54} +257.646i q^{55} -59.7122 q^{56} +(3657.78 - 1016.21i) q^{57} +(-6029.29 + 824.284i) q^{58} +2413.41i q^{59} +(-6186.77 + 1718.81i) q^{60} +5164.78i q^{61} +10435.3i q^{62} +(28.1286 - 16.9366i) q^{63} +549.159 q^{64} -42.1991i q^{65} +(228.881 + 823.844i) q^{66} -1685.57 q^{67} +16988.7 q^{68} +(-3460.44 + 961.380i) q^{69} +57.5567i q^{70} -3640.41i q^{71} +(-10222.0 + 6154.81i) q^{72} -8322.21i q^{73} +11827.4i q^{74} +(-578.036 - 2080.61i) q^{75} +15336.3i q^{76} +5.32223 q^{77} +(-37.4877 - 134.935i) q^{78} +5808.49i q^{79} -9500.99i q^{80} +(3069.53 - 5798.68i) q^{81} -9194.93 q^{82} -2307.90i q^{83} +(35.5057 + 127.801i) q^{84} -9169.12i q^{85} -13092.1i q^{86} +(2995.24 + 6951.13i) q^{87} -1934.11 q^{88} -4008.01 q^{89} +(5932.64 + 9853.00i) q^{90} -0.871712 q^{91} -14508.8i q^{92} +(12505.8 - 3474.36i) q^{93} +11040.2 q^{94} +8277.28 q^{95} +(-2762.08 - 9941.96i) q^{96} -11015.3i q^{97} -17372.2 q^{98} +(911.100 - 548.586i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 188 q^{4} - 36 q^{6} - 84 q^{7} - 452 q^{9} - 224 q^{13} - 52 q^{16} - 4216 q^{22} - 832 q^{24} - 7684 q^{25} - 396 q^{28} + 3384 q^{30} - 3308 q^{33} + 9124 q^{34} - 3680 q^{36} + 19764 q^{42} + 44 q^{45}+ \cdots + 13884 q^{96}+O(q^{100}) \) 32 * q + 188 * q^4 - 36 * q^6 - 84 * q^7 - 452 * q^9 - 224 * q^13 - 52 * q^16 - 4216 * q^22 - 832 * q^24 - 7684 * q^25 - 396 * q^28 + 3384 * q^30 - 3308 * q^33 + 9124 * q^34 - 3680 * q^36 + 19764 * q^42 + 44 * q^45 + 4796 * q^49 - 7052 * q^51 + 34048 * q^52 + 3008 * q^54 + 7976 * q^57 + 5936 * q^58 - 22848 * q^63 - 14668 * q^64 - 1872 * q^67 + 15192 * q^78 - 41088 * q^81 - 27028 * q^82 - 35572 * q^87 - 60032 * q^88 - 34272 * q^91 + 38000 * q^93 + 59636 * q^94 + 13884 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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