LMFDB - Embedded newform 69.6.c.b.68.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,6,Mod(68,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	69.c (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:	$11.0664835671$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			68.20
Character	$\chi$	$=$	69.68
Dual form			69.6.c.b.68.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.57119i q^{2} +(-12.8969 + 8.75610i) q^{3} +29.5314 q^{4} +37.1969 q^{5} +(-13.7575 - 20.2635i) q^{6} -122.242i q^{7} +96.6776i q^{8} +(89.6613 - 225.854i) q^{9} +O(q^{10})$	\(q+1.57119i q^{2} +(-12.8969 + 8.75610i) q^{3} +29.5314 q^{4} +37.1969 q^{5} +(-13.7575 - 20.2635i) q^{6} -122.242i q^{7} +96.6776i q^{8} +(89.6613 - 225.854i) q^{9} +58.4435i q^{10} +321.586 q^{11} +(-380.864 + 258.580i) q^{12} -207.773 q^{13} +192.066 q^{14} +(-479.726 + 325.700i) q^{15} +793.104 q^{16} +1644.37 q^{17} +(354.859 + 140.875i) q^{18} +2885.27i q^{19} +1098.48 q^{20} +(1070.37 + 1576.55i) q^{21} +505.273i q^{22} +(2534.43 - 114.122i) q^{23} +(-846.519 - 1246.84i) q^{24} -1741.39 q^{25} -326.451i q^{26} +(821.243 + 3697.90i) q^{27} -3609.98i q^{28} -2444.83i q^{29} +(-511.737 - 753.742i) q^{30} -244.199 q^{31} +4339.80i q^{32} +(-4147.47 + 2815.84i) q^{33} +2583.63i q^{34} -4547.03i q^{35} +(2647.82 - 6669.76i) q^{36} -9937.02i q^{37} -4533.31 q^{38} +(2679.63 - 1819.28i) q^{39} +3596.11i q^{40} +17465.4i q^{41} +(-2477.06 + 1681.75i) q^{42} -3358.44i q^{43} +9496.87 q^{44} +(3335.12 - 8401.06i) q^{45} +(179.307 + 3982.07i) q^{46} -15047.0i q^{47} +(-10228.6 + 6944.50i) q^{48} +1863.84 q^{49} -2736.06i q^{50} +(-21207.4 + 14398.3i) q^{51} -6135.81 q^{52} -7910.60 q^{53} +(-5810.11 + 1290.33i) q^{54} +11962.0 q^{55} +11818.1 q^{56} +(-25263.7 - 37211.1i) q^{57} +3841.31 q^{58} -11354.1i q^{59} +(-14167.0 + 9618.37i) q^{60} -25900.1i q^{61} -383.684i q^{62} +(-27608.8 - 10960.4i) q^{63} +18560.7 q^{64} -7728.51 q^{65} +(-4424.22 - 6516.47i) q^{66} -36948.4i q^{67} +48560.6 q^{68} +(-31687.0 + 23663.5i) q^{69} +7144.27 q^{70} +52830.9i q^{71} +(21835.0 + 8668.24i) q^{72} -71803.2 q^{73} +15613.0 q^{74} +(22458.6 - 15247.8i) q^{75} +85205.9i q^{76} -39311.4i q^{77} +(2858.44 + 4210.21i) q^{78} +43369.5i q^{79} +29501.0 q^{80} +(-42970.7 - 40500.7i) q^{81} -27441.5 q^{82} +107131. q^{83} +(31609.3 + 46557.6i) q^{84} +61165.6 q^{85} +5276.76 q^{86} +(21407.2 + 31530.8i) q^{87} +31090.1i q^{88} -1120.11 q^{89} +(13199.7 + 5240.12i) q^{90} +25398.6i q^{91} +(74845.1 - 3370.17i) q^{92} +(3149.42 - 2138.23i) q^{93} +23641.7 q^{94} +107323. i q^{95} +(-37999.8 - 55970.1i) q^{96} -26716.2i q^{97} +2928.45i q^{98} +(28833.8 - 72631.3i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) $ 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9	$ 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) $ 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9 - 2484 * q^12 + 520 * q^13 + 4936 * q^16 + 7188 * q^18 + 18660 * q^24 + 36032 * q^25 - 22032 * q^27 + 6544 * q^31 - 33912 * q^36 - 63912 * q^39 + 54328 * q^46 + 88284 * q^48 - 207664 * q^49 + 46296 * q^52 - 38628 * q^54 - 139296 * q^55 - 184144 * q^58 + 486584 * q^64 - 113580 * q^69 + 37176 * q^70 - 15504 * q^72 - 93896 * q^73 + 249840 * q^75 + 368028 * q^78 - 339372 * q^81 - 23512 * q^82 + 259584 * q^85 + 509928 * q^87 + 82740 * q^93 - 562000 * q^94 + 1404 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.57119i			0.277750i	0.990310	+	0.138875i	$0.0443487\pi$
$2$			1.57119i			0.277750i	−0.990310	+	0.138875i	$0.955651\pi$
$3$	−12.8969	+	8.75610i	−0.827338	+	0.561704i
$3$	−12.8969	+	8.75610i	−0.827338	+	0.561704i
$4$	29.5314			0.922855
$5$	37.1969			0.665399			0.332699	−	0.943033i	$-0.392041\pi$
$5$	37.1969			0.665399			0.332699	+	0.943033i	$0.392041\pi$
$6$	−13.7575	−	20.2635i	−0.156013	−	0.229793i
$7$		−	122.242i		−	0.942923i	−0.881887	−	0.471461i	$-0.843727\pi$
$7$		−	122.242i		−	0.942923i	0.881887	−	0.471461i	$-0.156273\pi$
$8$			96.6776i			0.534073i
$9$	89.6613	−	225.854i	0.368976	−	0.929439i
$10$			58.4435i			0.184815i
$11$	321.586			0.801337			0.400669	−	0.916223i	$-0.368778\pi$
$11$	321.586			0.801337			0.400669	+	0.916223i	$0.368778\pi$
$12$	−380.864	+	258.580i	−0.763513	+	0.518372i
$13$	−207.773			−0.340981			−0.170491	−	0.985359i	$-0.554535\pi$
$13$	−207.773			−0.340981			−0.170491	+	0.985359i	$0.554535\pi$
$14$	192.066			0.261897
$15$	−479.726	+	325.700i	−0.550510	+	0.373757i
$16$	793.104			0.774516
$17$	1644.37			1.38000			0.689999	−	0.723811i	$-0.257611\pi$
$17$	1644.37			1.38000			0.689999	+	0.723811i	$0.257611\pi$
$18$	354.859	+	140.875i	0.258152	+	0.102483i
$19$			2885.27i			1.83359i	0.399357	+	0.916795i	$0.369233\pi$
$19$			2885.27i			1.83359i	−0.399357	+	0.916795i	$0.630767\pi$
$20$	1098.48			0.614066
$21$	1070.37	+	1576.55i	0.529644	+	0.780116i
$22$			505.273i			0.222572i
$23$	2534.43	−	114.122i	0.998988	−	0.0449831i
$23$	2534.43	−	114.122i	0.998988	−	0.0449831i
$24$	−846.519	−	1246.84i	−0.299991	−	0.441859i
$25$	−1741.39			−0.557245
$26$		−	326.451i		−	0.0947076i
$27$	821.243	+	3697.90i	0.216801	+	0.976216i
$28$		−	3609.98i		−	0.870181i
$29$		−	2444.83i		−	0.539827i	−0.962885	−	0.269913i	$-0.913005\pi$
$29$		−	2444.83i		−	0.539827i	0.962885	−	0.269913i	$-0.0869951\pi$
$30$	−511.737	−	753.742i	−0.103811	−	0.152904i
$31$	−244.199			−0.0456394			−0.0228197	−	0.999740i	$-0.507264\pi$
$31$	−244.199			−0.0456394			−0.0228197	+	0.999740i	$0.507264\pi$
$32$			4339.80i			0.749195i
$33$	−4147.47	+	2815.84i	−0.662977	+	0.450114i
$34$			2583.63i			0.383294i
$35$		−	4547.03i		−	0.627420i
$36$	2647.82	−	6669.76i	0.340512	−	0.857737i
$37$		−	9937.02i		−	1.19331i	−0.802499	−	0.596653i	$-0.796497\pi$
$37$		−	9937.02i		−	1.19331i	0.802499	−	0.596653i	$-0.203503\pi$
$38$	−4533.31			−0.509280
$39$	2679.63	−	1819.28i	0.282107	−	0.191531i
$40$			3596.11i			0.355372i
$41$			17465.4i			1.62263i	0.584611	+	0.811313i	$0.301247\pi$
$41$			17465.4i			1.62263i	−0.584611	+	0.811313i	$0.698753\pi$
$42$	−2477.06	+	1681.75i	−0.216677	+	0.147109i
$43$		−	3358.44i		−	0.276992i	−0.990363	−	0.138496i	$-0.955773\pi$
$43$		−	3358.44i		−	0.276992i	0.990363	−	0.138496i	$-0.0442268\pi$
$44$	9496.87			0.739518
$45$	3335.12	−	8401.06i	0.245516	−	0.618447i
$46$	179.307	+	3982.07i	0.0124941	+	0.277469i
$47$		−	15047.0i		−	0.993583i	−0.867870	−	0.496791i	$-0.834511\pi$
$47$		−	15047.0i		−	0.993583i	0.867870	−	0.496791i	$-0.165489\pi$
$48$	−10228.6	+	6944.50i	−0.640786	+	0.435049i
$49$	1863.84			0.110896
$50$		−	2736.06i		−	0.154775i
$51$	−21207.4	+	14398.3i	−1.14172	+	0.775150i
$52$	−6135.81			−0.314676
$53$	−7910.60			−0.386830			−0.193415	−	0.981117i	$-0.561956\pi$
$53$	−7910.60			−0.386830			−0.193415	+	0.981117i	$0.561956\pi$
$54$	−5810.11	+	1290.33i	−0.271144	+	0.0602166i
$55$	11962.0			0.533209
$56$	11818.1			0.503590
$57$	−25263.7	−	37211.1i	−1.02994	−	1.51700i
$58$	3841.31			0.149937
$59$		−	11354.1i		−	0.424640i	−0.977200	−	0.212320i	$-0.931898\pi$
$59$		−	11354.1i		−	0.424640i	0.977200	−	0.212320i	$-0.0681019\pi$
$60$	−14167.0	+	9618.37i	−0.508041	+	0.344924i
$61$		−	25900.1i		−	0.891204i	−0.895231	−	0.445602i	$-0.852990\pi$
$61$		−	25900.1i		−	0.891204i	0.895231	−	0.445602i	$-0.147010\pi$
$62$		−	383.684i		−	0.0126764i
$63$	−27608.8	−	10960.4i	−0.876389	−	0.347916i
$64$	18560.7			0.566427
$65$	−7728.51			−0.226888
$66$	−4424.22	−	6516.47i	−0.125019	−	0.184142i
$67$		−	36948.4i		−	1.00556i	−0.864414	−	0.502781i	$-0.832310\pi$
$67$		−	36948.4i		−	1.00556i	0.864414	−	0.502781i	$-0.167690\pi$
$68$	48560.6			1.27354
$69$	−31687.0	+	23663.5i	−0.801233	+	0.598352i
$70$	7144.27			0.174266
$71$			52830.9i			1.24378i	0.783106	+	0.621888i	$0.213634\pi$
$71$			52830.9i			1.24378i	−0.783106	+	0.621888i	$0.786366\pi$
$72$	21835.0	+	8668.24i	0.496388	+	0.197060i
$73$	−71803.2			−1.57702			−0.788510	−	0.615023i	$-0.789147\pi$
$73$	−71803.2			−1.57702			−0.788510	+	0.615023i	$0.789147\pi$
$74$	15613.0			0.331441
$75$	22458.6	−	15247.8i	0.461030	−	0.313007i
$76$			85205.9i			1.69214i
$77$		−	39311.4i		−	0.755599i
$78$	2858.44	+	4210.21i	0.0531976	+	0.0783552i
$79$			43369.5i			0.781839i	0.920425	+	0.390919i	$0.127843\pi$
$79$			43369.5i			0.781839i	−0.920425	+	0.390919i	$0.872157\pi$
$80$	29501.0			0.515362
$81$	−42970.7	−	40500.7i	−0.727713	−	0.685882i
$82$	−27441.5			−0.450685
$83$	107131.			1.70694			0.853471	−	0.521140i	$-0.174493\pi$
$83$	107131.			1.70694			0.853471	+	0.521140i	$0.174493\pi$
$84$	31609.3	+	46557.6i	0.488784	+	0.719934i
$85$	61165.6			0.918248
$86$	5276.76			0.0769345
$87$	21407.2	+	31530.8i	0.303223	+	0.446619i
$88$			31090.1i			0.427973i
$89$	−1120.11			−0.0149895			−0.00749474	−	0.999972i	$-0.502386\pi$
$89$	−1120.11			−0.0149895			−0.00749474	+	0.999972i	$0.502386\pi$
$90$	13199.7	+	5240.12i	0.171774	+	0.0681923i
$91$			25398.6i			0.321519i
$92$	74845.1	−	3370.17i	0.921921	−	0.0415129i
$93$	3149.42	−	2138.23i	0.0377592	−	0.0256359i
$94$	23641.7			0.275968
$95$			107323.i			1.22007i
$96$	−37999.8	−	55970.1i	−0.420826	−	0.619838i
$97$		−	26716.2i		−	0.288301i	−0.989556	−	0.144150i	$-0.953955\pi$
$97$		−	26716.2i		−	0.288301i	0.989556	−	0.144150i	$-0.0460449\pi$
$98$			2928.45i			0.0308015i
$99$	28833.8	−	72631.3i	0.295675	−	0.744794i
$100$	−51425.6			−0.514256
$101$		−	79301.9i		−	0.773536i	−0.922177	−	0.386768i	$-0.873591\pi$
$101$		−	79301.9i		−	0.773536i	0.922177	−	0.386768i	$-0.126409\pi$
$102$	−22622.5	−	33320.8i	−0.215298	−	0.317114i
$103$			79210.1i			0.735678i	0.929890	+	0.367839i	$0.119902\pi$
$103$			79210.1i			0.735678i	−0.929890	+	0.367839i	$0.880098\pi$
$104$		−	20087.0i		−	0.182109i
$105$	39814.3	+	58642.8i	0.352424	+	0.519088i
$106$		−	12429.1i		−	0.107442i
$107$	−191353.			−1.61576			−0.807878	−	0.589350i	$-0.799384\pi$
$107$	−191353.			−1.61576			−0.807878	+	0.589350i	$0.799384\pi$
$108$	24252.4	+	109204.i	0.200076	+	0.900905i
$109$		−	123427.i		−	0.995047i	−0.867451	−	0.497523i	$-0.834243\pi$
$109$		−	123427.i		−	0.995047i	0.867451	−	0.497523i	$-0.165757\pi$
$110$			18794.6i			0.148099i
$111$	87009.5	+	128157.i	0.670285	+	0.987267i
$112$		−	96950.8i		−	0.730309i
$113$	−52135.8			−0.384096			−0.192048	−	0.981386i	$-0.561513\pi$
$113$	−52135.8			−0.384096			−0.192048	+	0.981386i	$0.561513\pi$
$114$	58465.8	−	39694.2i	0.421347	−	0.286065i
$115$	94272.9	−	4244.98i	0.664725	−	0.0299317i
$116$		−	72199.3i		−	0.498182i
$117$	−18629.2	+	46926.2i	−0.125814	+	0.316921i
$118$	17839.4			0.117944
$119$		−	201012.i		−	1.30123i
$120$	−31487.9	−	46378.7i	−0.199614	−	0.294013i
$121$	−57633.5			−0.357859
$122$	40694.1			0.247532
$123$	−152929.	−	225250.i	−0.911437	−	1.34246i
$124$	−7211.53			−0.0421185
$125$	−181015.			−1.03619
$126$	17220.9	−	43378.8i	0.0966338	−	0.243417i
$127$	−104033.			−0.572349			−0.286174	−	0.958178i	$-0.592384\pi$
$127$	−104033.			−0.572349			−0.286174	+	0.958178i	$0.592384\pi$
$128$			168036.i			0.906520i
$129$	29406.9	+	43313.6i	0.155588	+	0.229166i
$130$		−	12143.0i		−	0.0630183i
$131$			319871.i			1.62853i	0.580491	+	0.814267i	$0.302861\pi$
$131$			319871.i			1.62853i	−0.580491	+	0.814267i	$0.697139\pi$
$132$	−122480.	+	83155.5i	−0.611831	+	0.415390i
$133$	352702.			1.72893
$134$	58053.1			0.279295
$135$	30547.7	+	137551.i	0.144259	+	0.649573i
$136$			158974.i			0.737020i
$137$	−202011.			−0.919546			−0.459773	−	0.888037i	$-0.652069\pi$
$137$	−202011.			−0.919546			−0.459773	+	0.888037i	$0.652069\pi$
$138$	−37179.9	−	49786.4i	−0.166192	−	0.222543i
$139$	−197570.			−0.867330			−0.433665	−	0.901074i	$-0.642780\pi$
$139$	−197570.			−0.867330			−0.433665	+	0.901074i	$0.642780\pi$
$140$		−	134280.i		−	0.579017i
$141$	131753.	+	194059.i	0.558100	+	0.822029i
$142$	−83007.5			−0.345459
$143$	−66816.8			−0.273241
$144$	71110.8	−	179125.i	0.285778	−	0.719865i
$145$		−	90940.3i		−	0.359200i
$146$		−	112817.i		−	0.438017i
$147$	−24037.8	+	16320.0i	−0.0917489	+	0.0622910i
$148$		−	293454.i		−	1.10125i
$149$	−206334.			−0.761385			−0.380692	−	0.924702i	$-0.624314\pi$
$149$	−206334.			−0.761385			−0.380692	+	0.924702i	$0.624314\pi$
$150$	23957.2	+	35286.7i	0.0869377	+	0.128051i
$151$	−72590.0			−0.259080			−0.129540	−	0.991574i	$-0.541350\pi$
$151$	−72590.0			−0.259080			−0.129540	+	0.991574i	$0.541350\pi$
$152$	−278941.			−0.979272
$153$	147437.	−	371388.i	0.509186	−	1.28262i
$154$	61765.7			0.209868
$155$	−9083.46			−0.0303684
$156$	79133.1	−	53725.8i	0.260343	−	0.176755i
$157$		−	344412.i		−	1.11514i	−0.830131	−	0.557569i	$-0.811734\pi$
$157$		−	344412.i		−	1.11514i	0.830131	−	0.557569i	$-0.188266\pi$
$158$	−68141.9			−0.217156
$159$	102022.	−	69266.0i	0.320039	−	0.217284i
$160$			161427.i			0.498514i
$161$	−13950.5	−	309814.i	−0.0424156	−	0.941968i
$162$	63634.3	−	67515.2i	0.190504	−	0.202122i
$163$	52358.0			0.154353			0.0771764	−	0.997017i	$-0.475410\pi$
$163$	52358.0			0.154353			0.0771764	+	0.997017i	$0.475410\pi$
$164$			515777.i			1.49745i
$165$	−154273.	+	104741.i	−0.441144	+	0.299506i
$166$			168323.i			0.474103i
$167$		−	464676.i		−	1.28931i	−0.764472	−	0.644657i	$-0.777000\pi$
$167$		−	464676.i		−	1.28931i	0.764472	−	0.644657i	$-0.223000\pi$
$168$	−152417.	+	103480.i	−0.416639	+	0.282869i
$169$	−328123.			−0.883732
$170$			96103.0i			0.255044i
$171$	651649.	+	258697.i	1.70421	+	0.676552i
$172$		−	99179.4i		−	0.255623i
$173$		−	7055.13i		−	0.0179221i	−0.999960	−	0.00896107i	$-0.997148\pi$
$173$		−	7055.13i		−	0.0179221i	0.999960	−	0.00896107i	$-0.00285243\pi$
$174$	−49541.0	+	33634.9i	−0.124049	+	0.0842203i
$175$			212871.i			0.525439i
$176$	255051.			0.620648
$177$	99417.3	+	146432.i	0.238522	+	0.351321i
$178$		−	1759.91i		−	0.00416333i
$179$			610113.i			1.42324i	0.702565	+	0.711619i	$0.252038\pi$
$179$			610113.i			1.42324i	−0.702565	+	0.711619i	$0.747962\pi$
$180$	98490.7	−	248095.i	0.226576	−	0.570737i
$181$		−	58467.9i		−	0.132654i	−0.997798	−	0.0663271i	$-0.978872\pi$
$181$		−	58467.9i		−	0.132654i	0.997798	−	0.0663271i	$-0.0211281\pi$
$182$	−39906.1			−0.0893019
$183$	226784.	+	334032.i	0.500593	+	0.737327i
$184$	11033.0	+	245022.i	0.0240243	+	0.533533i
$185$		−	369626.i		−	0.794024i
$186$	3359.58	+	4948.34i	0.00712036	+	0.0104876i
$187$	528807.			1.10584
$188$		−	444357.i		−	0.916932i
$189$	452040.	−	100391.i	0.920496	−	0.204427i
$190$	−168625.			−0.338874
$191$	326167.			0.646929			0.323464	−	0.946240i	$-0.395152\pi$
$191$	326167.			0.646929			0.323464	+	0.946240i	$0.395152\pi$
$192$	−239376.	+	162519.i	−0.468626	+	0.318164i
$193$	196437.			0.379603			0.189802	−	0.981822i	$-0.439216\pi$
$193$	196437.			0.379603			0.189802	+	0.981822i	$0.439216\pi$
$194$	41976.4			0.0800757
$195$	99674.0	−	67671.6i	0.187713	−	0.127444i
$196$	55041.6			0.102341
$197$			66505.9i			0.122094i	0.998135	+	0.0610471i	$0.0194440\pi$
$197$			66505.9i			0.122094i	−0.998135	+	0.0610471i	$0.980556\pi$
$198$	114118.	+	45303.4i	0.206867	+	0.0821237i
$199$			826467.i			1.47942i	0.672923	+	0.739712i	$0.265038\pi$
$199$			826467.i			1.47942i	−0.672923	+	0.739712i	$0.734962\pi$
$200$		−	168353.i		−	0.297609i
$201$	323524.	+	476521.i	0.564829	+	0.831940i
$202$	124599.			0.214850
$203$	−298862.			−0.509015
$204$	−626282.	+	425201.i	−1.05365	+	0.715351i
$205$			649659.i			1.07969i
$206$	−124454.			−0.204335
$207$	201465.	−	582642.i	0.326794	−	0.945096i
$208$	−164785.			−0.264095
$209$			927862.i			1.46932i
$210$	−92139.0	+	62555.9i	−0.144177	+	0.0978859i
$211$	69431.8			0.107362			0.0536812	−	0.998558i	$-0.482905\pi$
$211$	69431.8			0.107362			0.0536812	+	0.998558i	$0.482905\pi$
$212$	−233611.			−0.356988
$213$	−462593.	−	681356.i	−0.698635	−	1.02902i
$214$		−	300652.i		−	0.448776i
$215$		−	124924.i		−	0.184310i
$216$	−357504.	+	79395.8i	−0.521371	+	0.115788i
$217$			29851.5i			0.0430344i
$218$	193927.			0.276374
$219$	926041.	−	628717.i	1.30473	−	0.885818i
$220$	353254.			0.492074
$221$	−341656.			−0.470553
$222$	−201359.	+	136709.i	−0.274214	+	0.186172i
$223$	−1.36980e6			−1.84456			−0.922282	−	0.386517i	$-0.873678\pi$
$223$	−1.36980e6			−1.84456			−0.922282	+	0.386517i	$0.873678\pi$
$224$	530507.			0.706433
$225$	−156135.	+	393299.i	−0.205610	+	0.517925i
$226$		−	81915.3i		−	0.106683i
$227$	999585.			1.28752			0.643762	−	0.765226i	$-0.277373\pi$
$227$	999585.			1.28752			0.643762	+	0.765226i	$0.277373\pi$
$228$	−746072.	−	1.09889e6i	−0.950481	−	1.39997i
$229$		−	675351.i		−	0.851022i	−0.904953	−	0.425511i	$-0.860094\pi$
$229$		−	675351.i		−	0.851022i	0.904953	−	0.425511i	$-0.139906\pi$
$230$	6669.69	+	148121.i	0.00831354	+	0.184628i
$231$	344214.	+	506996.i	0.424423	+	0.625136i
$232$	236361.			0.288307
$233$			537073.i			0.648102i	0.946040	+	0.324051i	$0.105045\pi$
$233$			537073.i			0.648102i	−0.946040	+	0.324051i	$0.894955\pi$
$234$	−73730.1	−	29270.0i	−0.0880249	−	0.0349449i
$235$		−	559700.i		−	0.661129i
$236$		−	335301.i		−	0.391881i
$237$	−379748.	−	559334.i	−0.439162	−	0.646845i
$238$	315828.			0.361417
$239$		−	1.04144e6i		−	1.17934i	−0.807644	−	0.589670i	$-0.799258\pi$
$239$		−	1.04144e6i		−	1.17934i	0.807644	−	0.589670i	$-0.200742\pi$
$240$	−380473.	+	258314.i	−0.426379	+	0.289481i
$241$			993176.i			1.10150i	0.834671	+	0.550749i	$0.185658\pi$
$241$			993176.i			1.10150i	−0.834671	+	0.550749i	$0.814342\pi$
$242$		−	90553.4i		−	0.0993954i
$243$	908818.	+	146078.i	0.987327	+	0.158697i
$244$		−	764866.i		−	0.822452i
$245$	69329.0			0.0737904
$246$	353911.	−	240281.i	0.372869	−	0.253152i
$247$		−	599481.i		−	0.625220i
$248$		−	23608.6i		−	0.0243748i
$249$	−1.38166e6	+	938048.i	−1.41222	+	0.958797i
$250$		−	284409.i		−	0.287802i
$251$	648746.			0.649966			0.324983	−	0.945720i	$-0.394641\pi$
$251$	648746.			0.649966			0.324983	+	0.945720i	$0.394641\pi$
$252$	−815327.	−	323675.i	−0.808780	−	0.321076i
$253$	815036.	−	36700.0i	0.800526	−	0.0360466i
$254$		−	163455.i		−	0.158970i
$255$	−788848.	+	535573.i	−0.759702	+	0.515784i
$256$	329925.			0.314641
$257$			950768.i			0.897929i	0.893550	+	0.448964i	$0.148207\pi$
$257$			950768.i			0.897929i	−0.893550	+	0.448964i	$0.851793\pi$
$258$	−68054.0	+	46203.9i	−0.0636509	+	0.0432145i
$259$	−1.21472e6			−1.12520
$260$	−228233.			−0.209385
$261$	−552175.	−	219207.i	−0.501736	−	0.199183i
$262$	−502579.			−0.452326
$263$	−1.18331e6			−1.05490			−0.527450	−	0.849586i	$-0.676852\pi$
$263$	−1.18331e6			−1.05490			−0.527450	+	0.849586i	$0.676852\pi$
$264$	−272229.	−	400967.i	−0.240394	−	0.354078i
$265$	−294250.			−0.257396
$266$			554162.i			0.480212i
$267$	14446.0	−	9807.82i	0.0124014	−	0.00841966i
$268$		−	1.09114e6i		−	0.927988i
$269$		−	1.60834e6i		−	1.35518i	−0.735441	−	0.677588i	$-0.763025\pi$
$269$		−	1.60834e6i		−	1.35518i	0.735441	−	0.677588i	$-0.236975\pi$
$270$	−216118.	+	47996.3i	−0.180419	+	0.0400681i
$271$	453181.			0.374842			0.187421	−	0.982280i	$-0.439987\pi$
$271$	453181.			0.374842			0.187421	+	0.982280i	$0.439987\pi$
$272$	1.30416e6			1.06883
$273$	−222393.	−	327564.i	−0.180599	−	0.266005i
$274$		−	317398.i		−	0.255404i
$275$	−560006.			−0.446541
$276$	−935761.	+	698816.i	−0.739422	+	0.552192i
$277$	530675.			0.415555			0.207778	−	0.978176i	$-0.433377\pi$
$277$	530675.			0.415555			0.207778	+	0.978176i	$0.433377\pi$
$278$		−	310421.i		−	0.240901i
$279$	−21895.2	+	55153.3i	−0.0168399	+	0.0424190i
$280$	439596.			0.335088
$281$	−141938.			−0.107234			−0.0536171	−	0.998562i	$-0.517075\pi$
$281$	−141938.			−0.107234			−0.0536171	+	0.998562i	$0.517075\pi$
$282$	−304905.	+	207009.i	−0.228319	+	0.155012i
$283$			1.59400e6i			1.18310i	0.806268	+	0.591550i	$0.201484\pi$
$283$			1.59400e6i			1.18310i	−0.806268	+	0.591550i	$0.798516\pi$
$284$			1.56017e6i			1.14783i
$285$	−939733.	−	1.38414e6i	−0.685318	−	1.00941i
$286$		−	104982.i		−	0.0758927i
$287$	2.13501e6			1.53001
$288$	980160.	+	389112.i	0.696331	+	0.276435i
$289$	1.28411e6			0.904392
$290$	142885.			0.0997679
$291$	233930.	+	344557.i	0.161940	+	0.238522i
$292$	−2.12045e6			−1.45536
$293$	−831087.			−0.565558			−0.282779	−	0.959185i	$-0.591256\pi$
$293$	−831087.			−0.565558			−0.282779	+	0.959185i	$0.591256\pi$
$294$	−25641.8	−	37767.9i	−0.0173013	−	0.0254833i
$295$		−	422336.i		−	0.282555i
$296$	960687.			0.637313
$297$	264100.	+	1.18919e6i	0.173731	+	0.782278i
$298$		−	324190.i		−	0.211475i
$299$	−526585.	+	23711.4i	−0.340636	+	0.0153384i
$300$	663232.	−	450288.i	0.425463	−	0.288860i
$301$	−410544.			−0.261182
$302$		−	114053.i		−	0.0719596i
$303$	694376.	+	1.02275e6i	0.434498	+	0.639975i
$304$			2.28832e6i			1.42015i
$305$		−	963405.i		−	0.593006i
$306$	583521.	+	231651.i	0.356249	+	0.141427i
$307$	1.41110e6			0.854499			0.427250	−	0.904134i	$-0.359483\pi$
$307$	1.41110e6			0.854499			0.427250	+	0.904134i	$0.359483\pi$
$308$		−	1.16092e6i		−	0.697308i
$309$	−693572.	−	1.02157e6i	−0.413233	−	0.608654i
$310$		−	14271.9i		−	0.00843483i
$311$			885448.i			0.519113i	0.965728	+	0.259557i	$0.0835764\pi$
$311$			885448.i			0.519113i	−0.965728	+	0.259557i	$0.916424\pi$
$312$	175884.	+	259060.i	0.102291	+	0.150666i
$313$		−	1.01876e6i		−	0.587776i	−0.955840	−	0.293888i	$-0.905051\pi$
$313$		−	1.01876e6i		−	0.587776i	0.955840	−	0.293888i	$-0.0949492\pi$
$314$	541137.			0.309730
$315$	−1.02696e6	−	407693.i	−0.583148	−	0.231503i
$316$			1.28076e6i			0.721524i
$317$		−	75162.2i		−	0.0420099i	−0.999779	−	0.0210049i	$-0.993313\pi$
$317$		−	75162.2i		−	0.0420099i	0.999779	−	0.0210049i	$-0.00668657\pi$
$318$	108830.	+	160297.i	0.0603506	+	0.0888909i
$319$		−	786224.i		−	0.432583i
$320$	690400.			0.376900
$321$	2.46786e6	−	1.67551e6i	1.33678	−	0.907577i
$322$	486777.	−	21918.9i	0.261632	−	0.0117809i
$323$			4.74446e6i			2.53035i
$324$	−1.26898e6	−	1.19604e6i	−0.671573	−	0.632970i
$325$	361813.			0.190010
$326$			82264.6i			0.0428715i
$327$	1.08074e6	+	1.59183e6i	0.558922	+	0.823240i
$328$	−1.68851e6			−0.866602
$329$	−1.83937e6			−0.936872
$330$	−164568.	−	242393.i	−0.0831877	−	0.122528i
$331$	−574036.			−0.287985			−0.143992	−	0.989579i	$-0.545994\pi$
$331$	−574036.			−0.287985			−0.143992	+	0.989579i	$0.545994\pi$
$332$	3.16371e6			1.57526
$333$	−2.24431e6	−	890966.i	−1.10910	−	0.440302i
$334$	730095.			0.358107
$335$		−	1.37437e6i		−	0.669100i
$336$	848912.	+	1.25037e6i	0.410218	+	0.604212i
$337$			751869.i			0.360635i	0.983608	+	0.180317i	$0.0577125\pi$
$337$			751869.i			0.360635i	−0.983608	+	0.180317i	$0.942288\pi$
$338$		−	515545.i		−	0.245457i
$339$	672391.	−	456506.i	0.317777	−	0.215748i
$340$	1.80630e6			0.847410
$341$	−78531.0			−0.0365725
$342$	−406463.	+	1.02387e6i	−0.187912	+	0.473345i
$343$		−	2.28236e6i		−	1.04749i
$344$	324686.			0.147934
$345$	−1.17866e6	+	880210.i	−0.533140	+	0.398143i
$346$	11085.0			0.00497788
$347$		−	4.25948e6i		−	1.89903i	−0.313715	−	0.949517i	$-0.601574\pi$
$347$		−	4.25948e6i		−	1.89903i	0.313715	−	0.949517i	$-0.398426\pi$
$348$	632184.	+	931149.i	0.279831	+	0.412165i
$349$	918884.			0.403829			0.201914	−	0.979403i	$-0.435284\pi$
$349$	918884.			0.403829			0.201914	+	0.979403i	$0.435284\pi$
$350$	−334462.			−0.145941
$351$	−170632.	−	768323.i	−0.0739252	−	0.332871i
$352$			1.39562e6i			0.600358i
$353$		−	1.12183e6i		−	0.479170i	−0.970875	−	0.239585i	$-0.922989\pi$
$353$		−	1.12183e6i		−	0.479170i	0.970875	−	0.239585i	$-0.0770114\pi$
$354$	−230073.	+	156204.i	−0.0975795	+	0.0662496i
$355$			1.96515e6i			0.827607i
$356$	−33078.4			−0.0138331
$357$	1.76008e6	+	2.59243e6i	0.730907	+	1.07656i
$358$	−958605.			−0.395305
$359$	2.22040e6			0.909273			0.454636	−	0.890677i	$-0.349769\pi$
$359$	2.22040e6			0.909273			0.454636	+	0.890677i	$0.349769\pi$
$360$	812194.	+	322432.i	0.330296	+	0.131124i
$361$	−5.84868e6			−2.36206
$362$	91864.4			0.0368447
$363$	743295.	−	504645.i	0.296070	−	0.201011i
$364$			750055.i			0.296715i
$365$	−2.67086e6			−1.04935
$366$	−524828.	+	356322.i	−0.204793	+	0.139040i
$367$			4.35641e6i			1.68835i	0.536064	+	0.844177i	$0.319910\pi$
$367$			4.35641e6i			1.68835i	−0.536064	+	0.844177i	$0.680090\pi$
$368$	2.01006e6	−	90510.6i	0.773732	−	0.0348401i
$369$	3.94462e6	+	1.56597e6i	1.50813	+	0.598711i
$370$	580754.			0.220540
$371$			967010.i			0.364751i
$372$	93006.6	−	63144.9i	0.0348463	−	0.0236582i
$373$		−	4.60335e6i		−	1.71318i	−0.516001	−	0.856588i	$-0.672580\pi$
$373$		−	4.60335e6i		−	1.71318i	0.516001	−	0.856588i	$-0.327420\pi$
$374$			830858.i			0.307148i
$375$	2.33453e6	−	1.58498e6i	0.857278	−	0.582032i
$376$	1.45470e6			0.530646
$377$			507970.i			0.184071i
$378$	157733.	+	710241.i	0.0567797	+	0.255668i
$379$		−	496941.i		−	0.177708i	−0.996045	−	0.0888541i	$-0.971680\pi$
$379$		−	496941.i		−	0.177708i	0.996045	−	0.0888541i	$-0.0283205\pi$
$380$			3.16940e6i			1.12595i
$381$	1.34170e6	−	910922.i	0.473526	−	0.321491i
$382$			512471.i			0.179685i
$383$	−3.80492e6			−1.32540			−0.662702	−	0.748883i	$-0.730590\pi$
$383$	−3.80492e6			−1.32540			−0.662702	+	0.748883i	$0.730590\pi$
$384$	−1.47134e6	−	2.16715e6i	−0.509196	−	0.749999i
$385$		−	1.46226e6i		−	0.502775i
$386$			308640.i			0.105435i
$387$	−758517.	−	301122.i	−0.257447	−	0.102204i
$388$		−	788967.i		−	0.266060i
$389$	−1.45736e6			−0.488306			−0.244153	−	0.969737i	$-0.578510\pi$
$389$	−1.45736e6			−0.488306			−0.244153	+	0.969737i	$0.578510\pi$
$390$	106325.	+	156607.i	0.0353976	+	0.0521374i
$391$	4.16754e6	−	187659.i	1.37860	−	0.0620766i
$392$			180191.i			0.0592268i
$393$	−2.80082e6	−	4.12535e6i	−0.914755	−	1.34735i
$394$	−104494.			−0.0339117
$395$			1.61321e6i			0.520234i
$396$	851501.	−	2.14490e6i	0.272865	−	0.687336i
$397$	−4.37853e6			−1.39429			−0.697144	−	0.716931i	$-0.745546\pi$
$397$	−4.37853e6			−1.39429			−0.697144	+	0.716931i	$0.745546\pi$
$398$	−1.29854e6			−0.410911
$399$	−4.54877e6	+	3.08829e6i	−1.43041	+	0.971150i
$400$	−1.38110e6			−0.431595
$401$	−3.59017e6			−1.11495			−0.557473	−	0.830195i	$-0.688229\pi$
$401$	−3.59017e6			−1.11495			−0.557473	+	0.830195i	$0.688229\pi$
$402$	−748706.	+	508319.i	−0.231071	+	0.156881i
$403$	50737.9			0.0155622
$404$		−	2.34189e6i		−	0.713861i
$405$	−1.59838e6	−	1.50650e6i	−0.484219	−	0.456385i
$406$		−	469570.i		−	0.141379i
$407$		−	3.19560e6i		−	0.956240i
$408$	−1.39199e6	−	2.05028e6i	−0.413987	−	0.609764i
$409$	6.32230e6			1.86882			0.934409	−	0.356202i	$-0.115929\pi$
$409$	6.32230e6			1.86882			0.934409	+	0.356202i	$0.115929\pi$
$410$	−1.02074e6			−0.299885
$411$	2.60532e6	−	1.76883e6i	0.760775	−	0.516513i
$412$			2.33918e6i			0.678924i
$413$	−1.38795e6			−0.400403
$414$	915442.	+	316540.i	0.262500	+	0.0907671i
$415$	3.98493e6			1.13580
$416$		−	901693.i		−	0.255461i
$417$	2.54805e6	−	1.72995e6i	0.717575	−	0.487183i
$418$	−1.45785e6			−0.408105
$419$	−5.31646e6			−1.47941			−0.739704	−	0.672933i	$-0.765034\pi$
$419$	−5.31646e6			−1.47941			−0.739704	+	0.672933i	$0.765034\pi$
$420$	1.17577e6	+	1.73180e6i	0.325237	+	0.479043i
$421$		−	5.94869e6i		−	1.63575i	−0.575399	−	0.817873i	$-0.695153\pi$
$421$		−	5.94869e6i		−	1.63575i	0.575399	−	0.817873i	$-0.304847\pi$
$422$			109091.i			0.0298199i
$423$	−3.39841e6	−	1.34913e6i	−0.923474	−	0.366609i
$424$		−	764778.i		−	0.206595i
$425$	−2.86349e6			−0.768996
$426$	1.07054e6	−	726823.i	0.285811	−	0.194046i
$427$	−3.16609e6			−0.840337
$428$	−5.65091e6			−1.49111
$429$	861731.	−	585055.i	0.226063	−	0.153481i
$430$	196279.			0.0511922
$431$	738404.			0.191470			0.0957351	−	0.995407i	$-0.469480\pi$
$431$	738404.			0.191470			0.0957351	+	0.995407i	$0.469480\pi$
$432$	651331.	+	2.93282e6i	0.167916	+	0.756095i
$433$			546533.i			0.140087i	0.997544	+	0.0700433i	$0.0223137\pi$
$433$			546533.i			0.140087i	−0.997544	+	0.0700433i	$0.977686\pi$
$434$	−46902.4			−0.0119528
$435$	796283.	+	1.17285e6i	0.201764	+	0.297180i
$436$		−	3.64496e6i		−	0.918284i
$437$	329273.	+	7.31251e6i	0.0824806	+	1.83173i
$438$	987835.	+	1.45499e6i	0.246036	+	0.362388i
$439$	−4.25006e6			−1.05253			−0.526264	−	0.850321i	$-0.676408\pi$
$439$	−4.25006e6			−1.05253			−0.526264	+	0.850321i	$0.676408\pi$
$440$			1.15646e6i			0.284773i
$441$	167114.	−	420954.i	0.0409182	−	0.103071i
$442$		−	536807.i		−	0.130696i
$443$		−	4.67645e6i		−	1.13216i	−0.824351	−	0.566079i	$-0.808460\pi$
$443$		−	4.67645e6i		−	1.13216i	0.824351	−	0.566079i	$-0.191540\pi$
$444$	2.56951e6	+	3.78465e6i	0.618576	+	0.911104i
$445$	−41664.7			−0.00997398
$446$		−	2.15221e6i		−	0.512328i
$447$	2.66107e6	−	1.80668e6i	0.629923	−	0.427673i
$448$		−	2.26890e6i		−	0.534097i
$449$		−	2.01589e6i		−	0.471900i	−0.971765	−	0.235950i	$-0.924180\pi$
$449$		−	2.01589e6i		−	0.471900i	0.971765	−	0.235950i	$-0.0758202\pi$
$450$	−617948.	−	245318.i	−0.143854	−	0.0571083i
$451$			5.61662e6i			1.30027i
$452$	−1.53964e6			−0.354465
$453$	936188.	−	635606.i	0.214347	−	0.145527i
$454$			1.57054e6i			0.357610i
$455$			944750.i			0.213938i
$456$	3.59748e6	−	2.44244e6i	0.810189	−	0.550061i
$457$			8.34776e6i			1.86973i	0.355001	+	0.934866i	$0.384481\pi$
$457$			8.34776e6i			1.86973i	−0.355001	+	0.934866i	$0.615519\pi$
$458$	1.06111e6			0.236371
$459$	1.35043e6	+	6.08073e6i	0.299185	+	1.34717i
$460$	2.78401e6	−	125360.i	0.613445	−	0.0276226i
$461$			6.19226e6i			1.35705i	0.734576	+	0.678526i	$0.237381\pi$
$461$			6.19226e6i			1.35705i	−0.734576	+	0.678526i	$0.762619\pi$
$462$	−796588.	+	540827.i	−0.173632	+	0.117884i
$463$	6.00846e6			1.30260			0.651300	−	0.758821i	$-0.274224\pi$
$463$	6.00846e6			1.30260			0.651300	+	0.758821i	$0.274224\pi$
$464$		−	1.93901e6i		−	0.418105i
$465$	117149.	−	79535.7i	0.0251249	−	0.0170581i
$466$	−843845.			−0.180010
$467$	5.16831e6			1.09662			0.548311	−	0.836275i	$-0.315271\pi$
$467$	5.16831e6			1.09662			0.548311	+	0.836275i	$0.315271\pi$
$468$	−550145.	+	1.38580e6i	−0.116108	+	0.292472i
$469$	−4.51666e6			−0.948168
$470$	879397.			0.183629
$471$	3.01570e6	+	4.44185e6i	0.626378	+	0.922596i
$472$	1.09768e6			0.226789
$473$		−	1.08003e6i		−	0.221964i
$474$	878821.	−	596657.i	0.179661	−	0.121977i
$475$		−	5.02438e6i		−	1.02176i
$476$		−	5.93615e6i		−	1.20085i
$477$	−709275.	+	1.78664e6i	−0.142731	+	0.359535i
$478$	1.63630e6			0.327562
$479$	7.21936e6			1.43767			0.718836	−	0.695180i	$-0.244675\pi$
$479$	7.21936e6			1.43767			0.718836	+	0.695180i	$0.244675\pi$
$480$	−1.41347e6	−	2.08192e6i	−0.280017	−	0.412439i
$481$			2.06464e6i			0.406895i
$482$	−1.56047e6			−0.305941
$483$	2.89268e6	+	3.87350e6i	0.564200	+	0.755501i
$484$	−1.70200e6			−0.330252
$485$		−	993762.i		−	0.191835i
$486$	−229516.	+	1.42793e6i	−0.0440781	+	0.274230i
$487$	−1.69573e6			−0.323991			−0.161996	−	0.986791i	$-0.551793\pi$
$487$	−1.69573e6			−0.323991			−0.161996	+	0.986791i	$0.551793\pi$
$488$	2.50396e6			0.475968
$489$	−675258.	+	458452.i	−0.127702	+	0.0867006i
$490$			108929.i			0.0204953i
$491$		−	1.97941e6i		−	0.370537i	−0.982688	−	0.185269i	$-0.940684\pi$
$491$		−	1.97941e6i		−	0.370537i	0.982688	−	0.185269i	$-0.0593155\pi$
$492$	−4.51619e6	−	6.65193e6i	−0.841124	−	1.23890i
$493$		−	4.02022e6i		−	0.744960i
$494$	941899.			0.173655
$495$	1.07253e6	−	2.70166e6i	0.196741	−	0.495585i
$496$	−193675.			−0.0353484
$497$	6.45817e6			1.17279
$498$	−1.47385e6	−	2.17085e6i	−0.266306	−	0.392244i
$499$	−5.04988e6			−0.907882			−0.453941	−	0.891032i	$-0.649982\pi$
$499$	−5.04988e6			−0.907882			−0.453941	+	0.891032i	$0.649982\pi$
$500$	−5.34561e6			−0.956252
$501$	4.06875e6	+	5.99289e6i	0.724213	+	1.06670i
$502$			1.01930e6i			0.180528i
$503$	1.44925e6			0.255402			0.127701	−	0.991813i	$-0.459240\pi$
$503$	1.44925e6			0.255402			0.127701	+	0.991813i	$0.459240\pi$
$504$	1.05962e6	−	2.66916e6i	0.185813	−	0.468056i
$505$		−	2.94979e6i		−	0.514710i
$506$	57662.7	+	1.28058e6i	0.0100120	+	0.222346i
$507$	4.23178e6	−	2.87308e6i	0.731145	−	0.496396i
$508$	−3.07223e6			−0.528195
$509$			1.18111e6i			0.202068i	0.994883	+	0.101034i	$0.0322150\pi$
$509$			1.18111e6i			0.202068i	−0.994883	+	0.101034i	$0.967785\pi$
$510$	−841488.	−	1.23943e6i	−0.143259	−	0.211007i
$511$			8.77739e6i			1.48701i
$512$			5.89553e6i			0.993912i
$513$	−1.06694e7	+	2.36951e6i	−1.78998	+	0.397525i
$514$	−1.49384e6			−0.249400
$515$			2.94637e6i			0.489519i
$516$	868425.	+	1.27911e6i	0.143585	+	0.211487i
$517$		−	4.83889e6i		−	0.796195i
$518$		−	1.90856e6i		−	0.312523i
$519$	61775.4	+	90989.5i	0.0100669	+	0.0148277i
$520$		−	747174.i		−	0.121175i
$521$	8.05639e6			1.30031			0.650154	−	0.759802i	$-0.274704\pi$
$521$	8.05639e6			1.30031			0.650154	+	0.759802i	$0.274704\pi$
$522$	344416.	−	867573.i	0.0553232	−	0.139357i
$523$			2.65776e6i			0.424875i	0.977175	+	0.212437i	$0.0681401\pi$
$523$			2.65776e6i			0.424875i	−0.977175	+	0.212437i	$0.931860\pi$
$524$			9.44623e6i			1.50290i
$525$	−1.86392e6	−	2.74538e6i	−0.295141	−	0.434715i
$526$		−	1.85922e6i		−	0.292998i
$527$	−401555.			−0.0629822
$528$	−3.28937e6	+	2.23325e6i	−0.513486	+	0.348621i
$529$	6.41030e6	−	578467.i	0.995953	−	0.0898752i
$530$		−	462323.i		−	0.0714918i
$531$	−2.56435e6	−	1.01802e6i	−0.394677	−	0.156682i
$532$	1.04158e7			1.59556
$533$		−	3.62883e6i		−	0.553285i
$534$	15410.0	+	22697.5i	0.00233856	+	0.00344448i
$535$	−7.11774e6			−1.07512
$536$	3.57209e6			0.537044
$537$	−5.34221e6	−	7.86858e6i	−0.799439	−	1.17750i
$538$	2.52700e6			0.376401
$539$	599384.			0.0888655
$540$	902115.	+	4.06205e6i	0.133130	+	0.599461i
$541$	−2.23163e6			−0.327816			−0.163908	−	0.986476i	$-0.552410\pi$
$541$	−2.23163e6			−0.327816			−0.163908	+	0.986476i	$0.552410\pi$
$542$			712035.i			0.104113i
$543$	511951.	+	754056.i	0.0745125	+	0.109750i
$544$			7.13626e6i			1.03389i
$545$		−	4.59110e6i		−	0.662103i
$546$	514666.	−	349422.i	0.0738829	−	0.0501613i
$547$	7.42757e6			1.06140			0.530699	−	0.847560i	$-0.321929\pi$
$547$	7.42757e6			1.06140			0.530699	+	0.847560i	$0.321929\pi$
$548$	−5.96565e6			−0.848607
$549$	−5.84964e6	−	2.32224e6i	−0.828320	−	0.328833i
$550$		−	879877.i		−	0.124027i
$551$	7.05401e6			0.989822
$552$	−2.28773e6	−	3.06343e6i	−0.319564	−	0.427917i
$553$	5.30159e6			0.737213
$554$			833792.i			0.115421i
$555$	3.23649e6	+	4.76704e6i	0.446007	+	0.656926i
$556$	−5.83452e6			−0.800420
$557$	1.07965e7			1.47450			0.737252	−	0.675618i	$-0.236123\pi$
$557$	1.07965e7			1.47450			0.737252	+	0.675618i	$0.236123\pi$
$558$	−86656.4	−	34401.6i	−0.0117819	−	0.00467728i
$559$			697793.i			0.0944490i
$560$		−	3.60627e6i		−	0.485947i
$561$	−6.81999e6	+	4.63029e6i	−0.914906	+	0.621157i
$562$		−	223012.i		−	0.0297843i
$563$	−995065.			−0.132306			−0.0661531	−	0.997809i	$-0.521073\pi$
$563$	−995065.			−0.132306			−0.0661531	+	0.997809i	$0.521073\pi$
$564$	3.89084e6	+	5.73084e6i	0.515045	+	0.758613i
$565$	−1.93929e6			−0.255577
$566$	−2.50448e6			−0.328606
$567$	−4.95089e6	+	5.25284e6i	−0.646734	+	0.686177i
$568$	−5.10757e6			−0.664268
$569$	1.08672e7			1.40714			0.703568	−	0.710628i	$-0.251589\pi$
$569$	1.08672e7			1.40714			0.703568	+	0.710628i	$0.251589\pi$
$570$	2.17475e6	−	1.47650e6i	0.280364	−	0.190347i
$571$			3.65300e6i			0.468878i	0.972131	+	0.234439i	$0.0753253\pi$
$571$			3.65300e6i			0.468878i	−0.972131	+	0.234439i	$0.924675\pi$
$572$	−1.97319e6			−0.252162
$573$	−4.20655e6	+	2.85595e6i	−0.535229	+	0.363383i
$574$			3.35451e6i			0.424961i
$575$	−4.41342e6	+	198731.i	−0.556680	+	0.0250666i
$576$	1.66417e6	−	4.19199e6i	0.208998	−	0.526459i
$577$	−6.62252e6			−0.828102			−0.414051	−	0.910254i	$-0.635886\pi$
$577$	−6.62252e6			−0.828102			−0.414051	+	0.910254i	$0.635886\pi$
$578$			2.01758e6i			0.251195i
$579$	−2.53343e6	+	1.72002e6i	−0.314060	+	0.213225i
$580$		−	2.68559e6i		−	0.331490i
$581$		−	1.30959e7i		−	1.60951i
$582$	−541366.	+	367549.i	−0.0662496	+	0.0449788i
$583$	−2.54394e6			−0.309981
$584$		−	6.94176e6i		−	0.842244i
$585$	−692948.	+	1.74551e6i	−0.0837165	+	0.210879i
$586$		−	1.30580e6i		−	0.157084i
$587$			2.94644e6i			0.352941i	0.984306	+	0.176471i	$0.0564681\pi$
$587$			2.94644e6i			0.352941i	−0.984306	+	0.176471i	$0.943532\pi$
$588$	−709868.	+	481950.i	−0.0846709	+	0.0574856i
$589$		−	704581.i		−	0.0836840i
$590$	663571.			0.0784797
$591$	−582333.	−	857722.i	−0.0685808	−	0.101013i
$592$		−	7.88109e6i		−	0.924234i
$593$			6.52249e6i			0.761687i	0.924639	+	0.380844i	$0.124366\pi$
$593$			6.52249e6i			0.761687i	−0.924639	+	0.380844i	$0.875634\pi$
$594$	−1.86845e6	+	414952.i	−0.217278	+	0.0482538i
$595$		−	7.47702e6i		−	0.865837i
$596$	−6.09331e6			−0.702648
$597$	−7.23663e6	−	1.06589e7i	−0.830999	−	1.22398i
$598$	−37255.2	−	827366.i	−0.00426024	−	0.0946117i
$599$			1.00980e7i			1.14993i	0.818179	+	0.574964i	$0.194984\pi$
$599$			1.00980e7i			1.14993i	−0.818179	+	0.574964i	$0.805016\pi$
$600$	1.47412e6	+	2.17124e6i	0.167168	+	0.246224i
$601$	1.44237e7			1.62889			0.814445	−	0.580240i	$-0.197041\pi$
$601$	1.44237e7			1.62889			0.814445	+	0.580240i	$0.197041\pi$
$602$		−	645043.i		−	0.0725433i
$603$	−8.34494e6	−	3.31284e6i	−0.934609	−	0.371029i
$604$	−2.14368e6			−0.239094
$605$	−2.14379e6			−0.238119
$606$	−1.60694e6	+	1.09100e6i	−0.177753	+	0.120682i
$607$	1.46050e6			0.160890			0.0804451	−	0.996759i	$-0.474366\pi$
$607$	1.46050e6			0.160890			0.0804451	+	0.996759i	$0.474366\pi$
$608$	−1.25215e7			−1.37372
$609$	3.85440e6	−	2.61687e6i	0.421128	−	0.285916i
$610$	1.51369e6			0.164708
$611$			3.12635e6i			0.338793i
$612$	4.35400e6	−	1.09676e7i	0.469905	−	1.18367i
$613$			1.39592e6i			0.150040i	0.997182	+	0.0750202i	$0.0239021\pi$
$613$			1.39592e6i			0.150040i	−0.997182	+	0.0750202i	$0.976098\pi$
$614$			2.21711e6i			0.237337i
$615$	−5.68848e6	−	8.37860e6i	−0.606469	−	0.893272i
$616$	3.80053e6			0.403545
$617$	−1.33277e7			−1.40943			−0.704714	−	0.709492i	$-0.748925\pi$
$617$	−1.33277e7			−1.40943			−0.704714	+	0.709492i	$0.748925\pi$
$618$	1.60508e6	−	1.08974e6i	0.169054	−	0.114776i
$619$		−	1.40755e7i		−	1.47651i	−0.674519	−	0.738257i	$-0.735649\pi$
$619$		−	1.40755e7i		−	1.47651i	0.674519	−	0.738257i	$-0.264351\pi$
$620$	−268247.			−0.0280256
$621$	2.50339e6	+	9.27834e6i	0.260495	+	0.965475i
$622$	−1.39121e6			−0.144184
$623$			136925.i			0.0141339i
$624$	2.12523e6	−	1.44288e6i	0.218496	−	0.148343i
$625$	−1.29135e6			−0.132234
$626$	1.60067e6			0.163255
$627$	−8.12446e6	−	1.19666e7i	−0.825326	−	1.21563i
$628$		−	1.01709e7i		−	1.02911i
$629$		−	1.63402e7i		−	1.64676i
$630$	640564.	−	1.61356e6i	0.0643000	−	0.161970i
$631$			9.75276e6i			0.975112i	0.873092	+	0.487556i	$0.162111\pi$
$631$			9.75276e6i			0.975112i	−0.873092	+	0.487556i	$0.837889\pi$
$632$	−4.19286e6			−0.417559
$633$	−895457.	+	607952.i	−0.0888250	+	0.0603060i
$634$	118094.			0.0116682
$635$	−3.86970e6			−0.380840
$636$	3.01286e6	−	2.04552e6i	0.295349	−	0.200522i
$637$	−387255.			−0.0378136
$638$	1.23531e6			0.120150
$639$	1.19321e7	+	4.73689e6i	1.15601	+	0.458924i
$640$			6.25042e6i			0.603197i
$641$	1.64249e7			1.57891			0.789454	−	0.613810i	$-0.210364\pi$
$641$	1.64249e7			1.57891			0.789454	+	0.613810i	$0.210364\pi$
$642$	2.63254e6	+	3.87749e6i	0.252080	+	0.371290i
$643$		−	6.63786e6i		−	0.633141i	−0.948569	−	0.316571i	$-0.897469\pi$
$643$		−	6.63786e6i		−	0.633141i	0.948569	−	0.316571i	$-0.102531\pi$
$644$	−411978.	−	9.14923e6i	−0.0391434	−	0.869300i
$645$	1.09385e6	+	1.61113e6i	0.103528	+	0.152487i
$646$	−7.45446e6			−0.702805
$647$			1.25537e7i			1.17899i	0.807772	+	0.589495i	$0.200673\pi$
$647$			1.25537e7i			1.17899i	−0.807772	+	0.589495i	$0.799327\pi$
$648$	3.91551e6	−	4.15430e6i	0.366311	−	0.388652i
$649$		−	3.65130e6i		−	0.340280i
$650$			568478.i			0.0527753i
$651$	−261382.	−	384992.i	−0.0241726	−	0.0356040i
$652$	1.54620e6			0.142445
$653$			9.03858e6i			0.829502i	0.909935	+	0.414751i	$0.136131\pi$
$653$			9.03858e6i			0.829502i	−0.909935	+	0.414751i	$0.863869\pi$
$654$	−2.50107e6	+	1.69805e6i	−0.228655	+	0.155241i
$655$			1.18982e7i			1.08362i
$656$			1.38519e7i			1.25675i
$657$	−6.43797e6	+	1.62170e7i	−0.581883	+	1.46574i
$658$		−	2.89001e6i		−	0.260216i
$659$	658736.			0.0590878			0.0295439	−	0.999563i	$-0.490595\pi$
$659$	658736.			0.0590878			0.0295439	+	0.999563i	$0.490595\pi$
$660$	−4.55589e6	+	3.09313e6i	−0.407112	+	0.276400i
$661$			2.00503e7i			1.78491i	0.451132	+	0.892457i	$0.351020\pi$
$661$			2.00503e7i			1.78491i	−0.451132	+	0.892457i	$0.648980\pi$
$662$		−	901921.i		−	0.0799878i
$663$	4.40631e6	−	2.99158e6i	0.389306	−	0.264312i
$664$			1.03571e7i			0.911632i
$665$	1.31194e7			1.15043
$666$	1.39988e6	−	3.52624e6i	0.122294	−	0.308054i
$667$	−279009.	−	6.19625e6i	−0.0242831	−	0.539280i
$668$		−	1.37225e7i		−	1.18985i
$669$	1.76662e7	−	1.19941e7i	1.52608	−	1.03610i
$670$	2.15940e6			0.185843
$671$		−	8.32911e6i		−	0.714155i
$672$	−6.84191e6	+	4.64518e6i	−0.584459	+	0.396807i
$673$	−1.79212e7			−1.52521			−0.762604	−	0.646866i	$-0.776079\pi$
$673$	−1.79212e7			−1.52521			−0.762604	+	0.646866i	$0.776079\pi$
$674$	−1.18133e6			−0.100166
$675$	−1.43010e6	−	6.43948e6i	−0.120811	−	0.543991i
$676$	−9.68993e6			−0.815556
$677$	1.91825e7			1.60854			0.804271	−	0.594262i	$-0.202556\pi$
$677$	1.91825e7			1.60854			0.804271	+	0.594262i	$0.202556\pi$
$678$	717259.	+	1.05646e6i	0.0599241	+	0.0882626i
$679$	−3.26585e6			−0.271846
$680$			5.91335e6i			0.490412i
$681$	−1.28916e7	+	8.75247e6i	−1.06522	+	0.723207i
$682$		−	123387.i		−	0.0101580i
$683$			5.30024e6i			0.434754i	0.976088	+	0.217377i	$0.0697502\pi$
$683$			5.30024e6i			0.434754i	−0.976088	+	0.217377i	$0.930250\pi$
$684$	1.92441e7	+	7.63967e6i	1.57274	+	0.624359i
$685$	−7.51418e6			−0.611865
$686$	3.58603e6			0.290940
$687$	5.91344e6	+	8.70995e6i	0.478023	+	0.704083i
$688$		−	2.66360e6i		−	0.214535i
$689$	1.64361e6			0.131902
$690$	−1.38298e6	−	1.85190e6i	−0.110584	−	0.148080i
$691$	7.36822e6			0.587040			0.293520	−	0.955953i	$-0.405173\pi$
$691$	7.36822e6			0.587040			0.293520	+	0.955953i	$0.405173\pi$
$692$		−	208348.i		−	0.0165395i
$693$	−8.87862e6	−	3.52471e6i	−0.702283	−	0.278798i
$694$	6.69246e6			0.527457
$695$	−7.34900e6			−0.577120
$696$	−3.04833e6	+	2.06960e6i	−0.238527	+	0.161943i
$697$			2.87196e7i			2.23922i
$698$			1.44374e6i			0.112163i
$699$	−4.70267e6	−	6.92659e6i	−0.364042	−	0.536199i
$700$			6.28638e6i			0.484904i
$701$	6.12013e6			0.470398			0.235199	−	0.971947i	$-0.424426\pi$
$701$	6.12013e6			0.470398			0.235199	+	0.971947i	$0.424426\pi$
$702$	1.20718e6	−	268096.i	0.0924550	−	0.0205327i
$703$	2.86710e7			2.18803
$704$	5.96885e6			0.453899
$705$	4.90079e6	+	7.21841e6i	0.371359	+	0.546977i
$706$	1.76261e6			0.133090
$707$	−9.69404e6			−0.729384
$708$	2.93593e6	+	4.32435e6i	0.220121	+	0.324218i
$709$		−	1.54351e7i		−	1.15317i	−0.817036	−	0.576587i	$-0.804384\pi$
$709$		−	1.54351e7i		−	1.15317i	0.817036	−	0.576587i	$-0.195616\pi$
$710$	−3.08762e6			−0.229868
$711$	9.79517e6	+	3.88857e6i	0.726671	+	0.288480i
$712$		−	108290.i		−	0.00800548i
$713$	−618905.	+	27868.5i	−0.0455932	+	0.00205300i
$714$	−4.07321e6	+	2.76543e6i	−0.299014	+	0.203010i
$715$	−2.48538e6			−0.181814
$716$			1.80175e7i			1.31344i
$717$	9.11894e6	+	1.34314e7i	0.662440	+	0.975713i
$718$			3.48867e6i			0.252551i
$719$		−	1.94171e7i		−	1.40076i	−0.713772	−	0.700378i	$-0.753015\pi$
$719$		−	1.94171e7i		−	1.40076i	0.713772	−	0.700378i	$-0.246985\pi$
$720$	2.64510e6	−	6.66292e6i	0.190156	−	0.478997i
$721$	9.68282e6			0.693688
$722$		−	9.18941e6i		−	0.656061i
$723$	−8.69635e6	−	1.28089e7i	−0.618716	−	0.911311i
$724$		−	1.72664e6i		−	0.122421i
$725$			4.25741e6i			0.300816i
$726$	792895.	+	1.16786e6i	0.0558308	+	0.0822336i
$727$		−	1.77689e7i		−	1.24688i	−0.781872	−	0.623439i	$-0.785735\pi$
$727$		−	1.77689e7i		−	1.24688i	0.781872	−	0.623439i	$-0.214265\pi$
$728$	−2.45548e6			−0.171715
$729$	−1.30000e7	+	6.07375e6i	−0.905994	+	0.423290i
$730$		−	4.19643e6i		−	0.291456i
$731$		−	5.52254e6i		−	0.382248i
$732$	6.69724e6	+	9.86441e6i	0.461975	+	0.680446i
$733$		−	5.47786e6i		−	0.376575i	−0.982114	−	0.188287i	$-0.939706\pi$
$733$		−	5.47786e6i		−	0.376575i	0.982114	−	0.188287i	$-0.0602936\pi$
$734$	−6.84476e6			−0.468941
$735$	−894131.	+	607052.i	−0.0610496	+	0.0414484i
$736$	495267.	+	1.09989e7i	0.0337011	+	0.748437i
$737$		−	1.18821e7i		−	0.805794i
$738$	−2.46044e6	+	6.19776e6i	−0.166292	+	0.418884i
$739$	−5.18027e6			−0.348933			−0.174466	−	0.984663i	$-0.555820\pi$
$739$	−5.18027e6			−0.348933			−0.174466	+	0.984663i	$0.555820\pi$
$740$		−	1.09156e7i		−	0.732769i
$741$	5.24911e6	+	7.73146e6i	0.351189	+	0.517268i
$742$	−1.51936e6			−0.101310
$743$	−1.14016e7			−0.757697			−0.378848	−	0.925459i	$-0.623680\pi$
$743$	−1.14016e7			−0.757697			−0.378848	+	0.925459i	$0.623680\pi$
$744$	206719.	+	304478.i	0.0136914	+	0.0201662i
$745$	−7.67497e6			−0.506625
$746$	7.23275e6			0.475835
$747$	9.60548e6	−	2.41959e7i	0.629821	−	1.58650i
$748$	1.56164e7			1.02053
$749$			2.33914e7i			1.52353i
$750$	2.49031e6	+	3.66800e6i	0.161659	+	0.238109i
$751$		−	2.39349e7i		−	1.54857i	−0.632836	−	0.774286i	$-0.718109\pi$
$751$		−	2.39349e7i		−	1.54857i	0.632836	−	0.774286i	$-0.281891\pi$
$752$		−	1.19338e7i		−	0.769545i
$753$	−8.36683e6	+	5.68049e6i	−0.537741	+	0.365089i
$754$	−798119.			−0.0511257
$755$	−2.70013e6			−0.172392
$756$	1.33493e7	−	2.96467e6i	0.849484	−	0.188656i
$757$			1.62210e7i			1.02882i	0.857546	+	0.514408i	$0.171988\pi$
$757$			1.62210e7i			1.02882i	−0.857546	+	0.514408i	$0.828012\pi$
$758$	780791.			0.0493585
$759$	−1.01901e7	+	7.60986e6i	−0.642058	+	0.479482i
$760$	−1.03757e7			−0.651606
$761$			1.90740e7i			1.19393i	0.802265	+	0.596967i	$0.203628\pi$
$761$			1.90740e7i			1.19393i	−0.802265	+	0.596967i	$0.796372\pi$
$762$	1.43123e6	+	2.10807e6i	0.0892941	+	0.131522i
$763$	−1.50880e7			−0.938252
$764$	9.63215e6			0.597021
$765$	5.48419e6	−	1.38145e7i	0.338812	−	0.853456i
$766$		−	5.97825e6i		−	0.368131i
$767$			2.35906e6i			0.144794i
$768$	−4.25501e6	+	2.88885e6i	−0.260314	+	0.176735i
$769$		−	1.31501e7i		−	0.801885i	−0.916103	−	0.400943i	$-0.868683\pi$
$769$		−	1.31501e7i		−	0.801885i	0.916103	−	0.400943i	$-0.131317\pi$
$770$	2.29749e6			0.139646
$771$	−8.32502e6	−	1.22620e7i	−0.504370	−	0.742891i
$772$	5.80105e6			0.350318
$773$	−4.24089e6			−0.255275			−0.127638	−	0.991821i	$-0.540739\pi$
$773$	−4.24089e6			−0.255275			−0.127638	+	0.991821i	$0.540739\pi$
$774$	473121.	−	1.19178e6i	0.0283870	−	0.0715060i
$775$	425246.			0.0254323
$776$	2.58286e6			0.153974
$777$	1.56662e7	−	1.06362e7i	0.930917	−	0.632027i
$778$		−	2.28979e6i		−	0.135627i
$779$	−5.03924e7			−2.97523
$780$	2.94351e6	−	1.99843e6i	0.173232	−	0.117612i
$781$			1.69897e7i			0.996684i
$782$	294848.	+	6.54801e6i	0.0172418	+	0.382906i
$783$	9.04076e6	−	2.00780e6i	0.526987	−	0.117035i
$784$	1.47822e6			0.0858911
$785$		−	1.28111e7i		−	0.742011i
$786$	6.48172e6	−	4.40063e6i	0.374226	−	0.254073i
$787$		−	2.27264e6i		−	0.130796i	−0.997859	−	0.0653980i	$-0.979168\pi$
$787$		−	2.27264e6i		−	0.130796i	0.997859	−	0.0653980i	$-0.0208317\pi$
$788$			1.96401e6i			0.112675i
$789$	1.52611e7	−	1.03612e7i	0.872758	−	0.592541i
$790$	−2.53467e6			−0.144495
$791$			6.37319e6i			0.362173i
$792$	7.02182e6	+	2.78758e6i	0.397774	+	0.157912i
$793$			5.38134e6i			0.303884i
$794$		−	6.87952e6i		−	0.387264i
$795$	3.79492e6	−	2.57648e6i	0.212954	−	0.144580i
$796$			2.44067e7i			1.36529i
$797$	−7.90668e6			−0.440908			−0.220454	−	0.975397i	$-0.570754\pi$
$797$	−7.90668e6			−0.440908			−0.220454	+	0.975397i	$0.570754\pi$
$798$	−4.85230e6	−	7.14699e6i	−0.269737	−	0.397298i
$799$		−	2.47428e7i		−	1.37114i
$800$		−	7.55728e6i		−	0.417485i
$801$	−100431.	+	252981.i	−0.00553077	+	0.0139318i
$802$		−	5.64084e6i		−	0.309676i
$803$	−2.30909e7			−1.26372
$804$	9.55411e6	+	1.40723e7i	0.521255	+	0.767760i
$805$	−518916.	−	1.15241e7i	−0.0282233	−	0.626785i
$806$			79719.1i			0.00432240i
$807$	1.40827e7	+	2.07426e7i	0.761209	+	1.12119i
$808$	7.66672e6			0.413125
$809$			8.57150e6i			0.460453i	0.973137	+	0.230227i	$0.0739467\pi$
$809$			8.57150e6i			0.460453i	−0.973137	+	0.230227i	$0.926053\pi$
$810$	2.36700e6	−	2.51136e6i	0.126761	−	0.134492i
$811$	−1.32411e7			−0.706921			−0.353461	−	0.935449i	$-0.614995\pi$
$811$	−1.32411e7			−0.706921			−0.353461	+	0.935449i	$0.614995\pi$
$812$	−8.82580e6			−0.469747
$813$	−5.84464e6	+	3.96810e6i	−0.310121	+	0.210551i
$814$	5.02091e6			0.265596
$815$	1.94756e6			0.102706
$816$	−1.68196e7	+	1.14194e7i	−0.884283	+	0.600366i
$817$	9.69002e6			0.507890
$818$			9.93355e6i			0.519064i
$819$	5.73637e6	+	2.27727e6i	0.298832	+	0.118633i
$820$			1.91853e7i			0.996401i
$821$		−	2.36284e7i		−	1.22342i	−0.791080	−	0.611712i	$-0.790481\pi$
$821$		−	2.36284e7i		−	1.22342i	0.791080	−	0.611712i	$-0.209519\pi$
$822$	2.77917e6	+	4.09346e6i	0.143462	+	0.211305i
$823$	−2.96088e7			−1.52378			−0.761888	−	0.647709i	$-0.775727\pi$
$823$	−2.96088e7			−1.52378			−0.761888	+	0.647709i	$0.775727\pi$
$824$	−7.65785e6			−0.392906
$825$	7.22236e6	−	4.90347e6i	0.369440	−	0.250824i
$826$		−	2.18073e6i		−	0.111212i
$827$	−1.58750e7			−0.807144			−0.403572	−	0.914948i	$-0.632232\pi$
$827$	−1.58750e7			−0.807144			−0.403572	+	0.914948i	$0.632232\pi$
$828$	5.94954e6	−	1.72062e7i	0.301583	−	0.872186i
$829$	1.94808e7			0.984509			0.492254	−	0.870451i	$-0.336173\pi$
$829$	1.94808e7			0.984509			0.492254	+	0.870451i	$0.336173\pi$
$830$			6.26110e6i			0.315468i
$831$	−6.84407e6	+	4.64664e6i	−0.343805	+	0.233419i
$832$	−3.85640e6			−0.193141
$833$	3.06484e6			0.153037
$834$	2.71808e6	+	4.00347e6i	0.135315	+	0.199307i
$835$		−	1.72845e7i		−	0.857908i
$836$			2.74010e7i			1.35597i
$837$	−200547.	−	903024.i	−0.00989469	−	0.0445539i
$838$		−	8.35319e6i		−	0.410906i
$839$	−3.57449e7			−1.75311			−0.876556	−	0.481299i	$-0.840165\pi$
$839$	−3.57449e7			−1.75311			−0.876556	+	0.481299i	$0.840165\pi$
$840$	−5.66944e6	+	3.84915e6i	−0.277231	+	0.188220i
$841$	1.45339e7			0.708587
$842$	9.34653e6			0.454329
$843$	1.83057e6	−	1.24283e6i	0.0887190	−	0.0602339i
$844$	2.05042e6			0.0990800
$845$	−1.22052e7			−0.588034
$846$	2.11974e6	−	5.33955e6i	0.101826	−	0.256495i
$847$			7.04525e6i			0.337433i
$848$	−6.27393e6			−0.299606
$849$	−1.39572e7	−	2.05577e7i	−0.664553	−	0.978824i
$850$		−	4.49910e6i		−	0.213589i
$851$	−1.13403e6	−	2.51846e7i	−0.0536786	−	1.19210i
$852$	−1.36610e7	−	2.01214e7i	−0.644738	−	0.949639i
$853$	−2.25616e7			−1.06169			−0.530845	−	0.847469i	$-0.678125\pi$
$853$	−2.25616e7			−1.06169			−0.530845	+	0.847469i	$0.678125\pi$
$854$		−	4.97453e6i		−	0.233404i
$855$	2.42393e7	+	9.62273e6i	1.13398	+	0.450177i
$856$		−	1.84995e7i		−	0.862932i
$857$			1.17266e7i			0.545408i	0.962098	+	0.272704i	$0.0879180\pi$
$857$			1.17266e7i			0.545408i	−0.962098	+	0.272704i	$0.912082\pi$
$858$	919234.	+	1.35395e6i	0.0426292	+	0.0627889i
$859$	1.30115e7			0.601649			0.300824	−	0.953680i	$-0.402738\pi$
$859$	1.30115e7			0.601649			0.300824	+	0.953680i	$0.402738\pi$
$860$		−	3.68917e6i		−	0.170091i
$861$	−2.75350e7	+	1.86944e7i	−1.26584	+	0.859414i
$862$			1.16017e6i			0.0531809i
$863$		−	1.62440e7i		−	0.742447i	−0.928544	−	0.371223i	$-0.878938\pi$
$863$		−	1.62440e7i		−	0.742447i	0.928544	−	0.371223i	$-0.121062\pi$
$864$	−1.60482e7	+	3.56403e6i	−0.731376	+	0.162427i
$865$		−	262429.i		−	0.0119254i
$866$	−858708.			−0.0389091
$867$	−1.65610e7	+	1.12438e7i	−0.748238	+	0.508001i
$868$			881554.i			0.0397145i
$869$			1.39470e7i			0.626516i
$870$	−1.84277e6	+	1.25111e6i	−0.0825418	+	0.0560401i
$871$			7.67688e6i			0.342878i
$872$	1.19326e7			0.531428
$873$	−6.03396e6	−	2.39541e6i	−0.267958	−	0.106376i
$874$	−1.14894e7	+	517350.i	−0.508765	+	0.0229090i
$875$			2.21276e7i			0.977046i
$876$	2.73472e7	−	1.85669e7i	1.20407	−	0.817482i
$877$	−5.44428e6			−0.239024			−0.119512	−	0.992833i	$-0.538133\pi$
$877$	−5.44428e6			−0.239024			−0.119512	+	0.992833i	$0.538133\pi$
$878$		−	6.67767e6i		−	0.292340i
$879$	1.07185e7	−	7.27708e6i	0.467908	−	0.317677i
$880$	9.48712e6			0.412979
$881$	2.73818e7			1.18856			0.594281	−	0.804258i	$-0.297437\pi$
$881$	2.73818e7			1.18856			0.594281	+	0.804258i	$0.297437\pi$
$882$	661400.	+	262568.i	0.0286281	+	0.0113650i
$883$	2.58182e7			1.11436			0.557179	−	0.830392i	$-0.311884\pi$
$883$	2.58182e7			1.11436			0.557179	+	0.830392i	$0.311884\pi$
$884$	−1.00896e7			−0.434252
$885$	3.69802e6	+	5.44684e6i	0.158712	+	0.233769i
$886$	7.34761e6			0.314457
$887$		−	4.06230e7i		−	1.73366i	−0.498606	−	0.866828i	$-0.666155\pi$
$887$		−	4.06230e7i		−	1.73366i	0.498606	−	0.866828i	$-0.333845\pi$
$888$	−1.23899e7	+	8.41187e6i	−0.527273	+	0.357981i
$889$			1.27172e7i			0.539681i
$890$		−	65463.3i		−	0.00277028i
$891$	−1.38188e7	−	1.30244e7i	−0.583143	−	0.549623i
$892$	−4.04519e7			−1.70227
$893$	4.34145e7			1.82182
$894$	2.83864e6	+	4.18105e6i	0.118786	+	0.174961i
$895$			2.26943e7i			0.947021i
$896$	2.05411e7			0.854779
$897$	6.58371e6	−	4.91664e6i	0.273205	−	0.204027i
$898$	3.16735e6			0.131070
$899$			597027.i			0.0246374i
$900$	−4.61088e6	+	1.16147e7i	−0.189748	+	0.477969i
$901$	−1.30080e7			−0.533824
$902$	−8.82479e6			−0.361151
$903$	5.29475e6	−	3.59476e6i	0.216086	−	0.146707i
$904$		−	5.04036e6i		−	0.205135i
$905$		−	2.17483e6i		−	0.0882680i
$906$	998659.	+	1.47093e6i	0.0404200	+	0.0595349i
$907$		−	3.44711e6i		−	0.139135i	−0.997577	−	0.0695675i	$-0.977838\pi$
$907$		−	3.44711e6i		−	0.139135i	0.997577	−	0.0695675i	$-0.0221619\pi$
$908$	2.95191e7			1.18820
$909$	−1.79106e7	−	7.11031e6i	−0.718954	−	0.285416i
$910$	−1.48438e6			−0.0594214
$911$	1.14733e7			0.458030			0.229015	−	0.973423i	$-0.426450\pi$
$911$	1.14733e7			0.458030			0.229015	+	0.973423i	$0.426450\pi$
$912$	−2.00368e7	−	2.95123e7i	−0.797702	−	1.17494i
$913$	3.44517e7			1.36784
$914$	−1.31159e7			−0.519318
$915$	8.43567e6	+	1.24250e7i	0.333094	+	0.490617i
$916$		−	1.99440e7i		−	0.785370i
$917$	3.91018e7			1.53558
$918$	−9.55400e6	+	2.12179e6i	−0.374178	+	0.0830988i
$919$			7.12254e6i			0.278193i	0.990279	+	0.139096i	$0.0444198\pi$
$919$			7.12254e6i			0.278193i	−0.990279	+	0.139096i	$0.955580\pi$
$920$	410395.	+	9.11407e6i	0.0159857	+	0.355012i
$921$	−1.81988e7	+	1.23557e7i	−0.706960	+	0.479976i
$922$	−9.72923e6			−0.376922
$923$		−	1.09768e7i		−	0.424104i
$924$	1.01651e7	+	1.49723e7i	0.391681	+	0.576910i
$925$			1.73042e7i			0.664963i
$926$			9.44045e6i			0.361797i
$927$	1.78899e7	+	7.10208e6i	0.683768	+	0.271448i
$928$	1.06101e7			0.404436
$929$			1.96846e7i			0.748319i	0.927364	+	0.374160i	$0.122069\pi$
$929$			1.96846e7i			0.748319i	−0.927364	+	0.374160i	$0.877931\pi$
$930$	124966.	+	184063.i	0.00473788	+	0.00697846i
$931$			5.37767e6i			0.203339i
$932$			1.58605e7i			0.598104i
$933$	−7.75307e6	−	1.14196e7i	−0.291588	−	0.429482i
$934$			8.12041e6i			0.304587i
$935$	1.96700e7			0.735826
$936$	−4.53672e6	−	1.80102e6i	−0.169259	−	0.0671939i
$937$		−	6.94080e6i		−	0.258262i	−0.991628	−	0.129131i	$-0.958781\pi$
$937$		−	6.94080e6i		−	0.258262i	0.991628	−	0.129131i	$-0.0412188\pi$
$938$		−	7.09654e6i		−	0.263354i
$939$	8.92038e6	+	1.31389e7i	0.330156	+	0.486289i
$940$		−	1.65287e7i		−	0.610126i
$941$	1.83924e7			0.677119			0.338559	−	0.940945i	$-0.390060\pi$
$941$	1.83924e7			0.677119			0.338559	+	0.940945i	$0.390060\pi$
$942$	−6.97900e6	+	4.73825e6i	−0.256251	+	0.173977i
$943$	1.99318e6	+	4.42648e7i	0.0729908	+	1.62098i
$944$		−	9.00495e6i		−	0.328891i
$945$	1.68145e7	−	3.73422e6i	0.612497	−	0.136026i
$946$	1.69693e6			0.0616505
$947$			2.87108e7i			1.04033i	0.854066	+	0.520164i	$0.174129\pi$
$947$			2.87108e7i			1.04033i	−0.854066	+	0.520164i	$0.825871\pi$
$948$	−1.12145e7	−	1.65179e7i	−0.405283	−	0.596944i
$949$	1.49188e7			0.537734
$950$	7.89426e6			0.283794
$951$	658128.	+	969361.i	0.0235971	+	0.0347564i
$952$	1.94333e7			0.694953
$953$	2.58238e7			0.921060			0.460530	−	0.887644i	$-0.347659\pi$
$953$	2.58238e7			0.921060			0.460530	+	0.887644i	$0.347659\pi$
$954$	−2.80715e6	−	1.11441e6i	−0.0998608	−	0.0396436i
$955$	1.21324e7			0.430466
$956$		−	3.07551e7i		−	1.08836i
$957$	6.88426e6	+	1.01399e7i	0.242984	+	0.357893i
$958$			1.13430e7i			0.399314i
$959$			2.46943e7i			0.867061i
$960$	−8.90403e6	+	6.04521e6i	−0.311823	+	0.211706i
$961$	−2.85695e7			−0.997917
$962$	−3.24395e6			−0.113015
$963$	−1.71570e7	+	4.32178e7i	−0.596176	+	1.50175i
$964$			2.93298e7i			1.01652i
$965$	7.30685e6			0.252587
$966$	−6.08601e6	+	4.54496e6i	−0.209841	+	0.156707i
$967$	−2.62029e7			−0.901121			−0.450560	−	0.892746i	$-0.648776\pi$
$967$	−2.62029e7			−0.901121			−0.450560	+	0.892746i	$0.648776\pi$
$968$		−	5.57187e6i		−	0.191123i
$969$	−4.15430e7	−	6.11890e7i	−1.42131	−	2.09345i
$970$	1.56139e6			0.0532822
$971$	−2.22674e7			−0.757915			−0.378958	−	0.925414i	$-0.623717\pi$
$971$	−2.22674e7			−0.757915			−0.378958	+	0.925414i	$0.623717\pi$
$972$	2.68386e7	+	4.31388e6i	0.911160	+	0.146454i
$973$			2.41514e7i			0.817825i
$974$		−	2.66431e6i		−	0.0899886i
$975$	−4.66628e6	+	3.16807e6i	−0.157202	+	0.106729i
$976$		−	2.05415e7i		−	0.690252i
$977$	1.24196e7			0.416267			0.208133	−	0.978100i	$-0.433261\pi$
$977$	1.24196e7			0.416267			0.208133	+	0.978100i	$0.433261\pi$
$978$	−720317.	−	1.06096e6i	−0.0240811	−	0.0354692i
$979$	−360212.			−0.0120116
$980$	2.04738e6			0.0680978
$981$	−2.78764e7	−	1.10666e7i	−0.924835	−	0.367149i
$982$	3.11003e6			0.102917
$983$	4.07722e7			1.34580			0.672900	−	0.739733i	$-0.265048\pi$
$983$	4.07722e7			1.34580			0.672900	+	0.739733i	$0.265048\pi$
$984$	2.17766e7	−	1.47848e7i	0.716973	−	0.486774i
$985$			2.47382e6i			0.0812413i
$986$	6.31654e6			0.206913
$987$	2.37223e7	−	1.61057e7i	0.775110	−	0.526245i
$988$		−	1.77035e7i		−	0.576987i
$989$	−383272.	−	8.51173e6i	−0.0124600	−	0.276712i
$990$	4.24483e6	+	1.68515e6i	0.137649	+	0.0546450i
$991$	−4.46092e7			−1.44291			−0.721457	−	0.692459i	$-0.756527\pi$
$991$	−4.46092e7			−1.44291			−0.721457	+	0.692459i	$0.756527\pi$
$992$		−	1.05978e6i		−	0.0341928i
$993$	7.40330e6	−	5.02632e6i	0.238261	−	0.161762i
$994$			1.01470e7i			0.325741i
$995$			3.07420e7i			0.984407i
$996$	−4.08022e7	+	2.77018e7i	−1.30327	+	0.884830i
$997$	8.22183e6			0.261957			0.130979	−	0.991385i	$-0.458188\pi$
$997$	8.22183e6			0.261957			0.130979	+	0.991385i	$0.458188\pi$
$998$		−	7.93433e6i		−	0.252164i
$999$	3.67461e7	−	8.16070e6i	1.16492	−	0.258710i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.c.b.68.20	yes	32
3.2	odd	2	inner	69.6.c.b.68.13	✓	32
23.22	odd	2	inner	69.6.c.b.68.19	yes	32
69.68	even	2	inner	69.6.c.b.68.14	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.c.b.68.13	✓	32	3.2	odd	2	inner
69.6.c.b.68.14	yes	32	69.68	even	2	inner
69.6.c.b.68.19	yes	32	23.22	odd	2	inner
69.6.c.b.68.20	yes	32	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.57119i q^{2} +(-12.8969 + 8.75610i) q^{3} +29.5314 q^{4} +37.1969 q^{5} +(-13.7575 - 20.2635i) q^{6} -122.242i q^{7} +96.6776i q^{8} +(89.6613 - 225.854i) q^{9} +O(q^{10})\)	\(q+1.57119i q^{2} +(-12.8969 + 8.75610i) q^{3} +29.5314 q^{4} +37.1969 q^{5} +(-13.7575 - 20.2635i) q^{6} -122.242i q^{7} +96.6776i q^{8} +(89.6613 - 225.854i) q^{9} +58.4435i q^{10} +321.586 q^{11} +(-380.864 + 258.580i) q^{12} -207.773 q^{13} +192.066 q^{14} +(-479.726 + 325.700i) q^{15} +793.104 q^{16} +1644.37 q^{17} +(354.859 + 140.875i) q^{18} +2885.27i q^{19} +1098.48 q^{20} +(1070.37 + 1576.55i) q^{21} +505.273i q^{22} +(2534.43 - 114.122i) q^{23} +(-846.519 - 1246.84i) q^{24} -1741.39 q^{25} -326.451i q^{26} +(821.243 + 3697.90i) q^{27} -3609.98i q^{28} -2444.83i q^{29} +(-511.737 - 753.742i) q^{30} -244.199 q^{31} +4339.80i q^{32} +(-4147.47 + 2815.84i) q^{33} +2583.63i q^{34} -4547.03i q^{35} +(2647.82 - 6669.76i) q^{36} -9937.02i q^{37} -4533.31 q^{38} +(2679.63 - 1819.28i) q^{39} +3596.11i q^{40} +17465.4i q^{41} +(-2477.06 + 1681.75i) q^{42} -3358.44i q^{43} +9496.87 q^{44} +(3335.12 - 8401.06i) q^{45} +(179.307 + 3982.07i) q^{46} -15047.0i q^{47} +(-10228.6 + 6944.50i) q^{48} +1863.84 q^{49} -2736.06i q^{50} +(-21207.4 + 14398.3i) q^{51} -6135.81 q^{52} -7910.60 q^{53} +(-5810.11 + 1290.33i) q^{54} +11962.0 q^{55} +11818.1 q^{56} +(-25263.7 - 37211.1i) q^{57} +3841.31 q^{58} -11354.1i q^{59} +(-14167.0 + 9618.37i) q^{60} -25900.1i q^{61} -383.684i q^{62} +(-27608.8 - 10960.4i) q^{63} +18560.7 q^{64} -7728.51 q^{65} +(-4424.22 - 6516.47i) q^{66} -36948.4i q^{67} +48560.6 q^{68} +(-31687.0 + 23663.5i) q^{69} +7144.27 q^{70} +52830.9i q^{71} +(21835.0 + 8668.24i) q^{72} -71803.2 q^{73} +15613.0 q^{74} +(22458.6 - 15247.8i) q^{75} +85205.9i q^{76} -39311.4i q^{77} +(2858.44 + 4210.21i) q^{78} +43369.5i q^{79} +29501.0 q^{80} +(-42970.7 - 40500.7i) q^{81} -27441.5 q^{82} +107131. q^{83} +(31609.3 + 46557.6i) q^{84} +61165.6 q^{85} +5276.76 q^{86} +(21407.2 + 31530.8i) q^{87} +31090.1i q^{88} -1120.11 q^{89} +(13199.7 + 5240.12i) q^{90} +25398.6i q^{91} +(74845.1 - 3370.17i) q^{92} +(3149.42 - 2138.23i) q^{93} +23641.7 q^{94} +107323. i q^{95} +(-37999.8 - 55970.1i) q^{96} -26716.2i q^{97} +2928.45i q^{98} +(28833.8 - 72631.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) \) 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9	\( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) \) 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9 - 2484 * q^12 + 520 * q^13 + 4936 * q^16 + 7188 * q^18 + 18660 * q^24 + 36032 * q^25 - 22032 * q^27 + 6544 * q^31 - 33912 * q^36 - 63912 * q^39 + 54328 * q^46 + 88284 * q^48 - 207664 * q^49 + 46296 * q^52 - 38628 * q^54 - 139296 * q^55 - 184144 * q^58 + 486584 * q^64 - 113580 * q^69 + 37176 * q^70 - 15504 * q^72 - 93896 * q^73 + 249840 * q^75 + 368028 * q^78 - 339372 * q^81 - 23512 * q^82 + 259584 * q^85 + 509928 * q^87 + 82740 * q^93 - 562000 * q^94 + 1404 * q^96

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