LMFDB - Embedded newform 69.6.c.b.68.18

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			68.18
Character	$\chi$	$=$	69.68
Dual form			69.6.c.b.68.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.27619i q^{2} +(11.5059 - 10.5173i) q^{3} +30.3713 q^{4} +80.0597 q^{5} +(13.4221 + 14.6838i) q^{6} +196.848i q^{7} +79.5978i q^{8} +(21.7727 - 242.023i) q^{9} +O(q^{10})$	\(q+1.27619i q^{2} +(11.5059 - 10.5173i) q^{3} +30.3713 q^{4} +80.0597 q^{5} +(13.4221 + 14.6838i) q^{6} +196.848i q^{7} +79.5978i q^{8} +(21.7727 - 242.023i) q^{9} +102.171i q^{10} -20.8886 q^{11} +(349.450 - 319.425i) q^{12} -791.678 q^{13} -251.216 q^{14} +(921.160 - 842.012i) q^{15} +870.301 q^{16} -174.605 q^{17} +(308.867 + 27.7861i) q^{18} +892.972i q^{19} +2431.52 q^{20} +(2070.32 + 2264.92i) q^{21} -26.6578i q^{22} +(-2039.46 - 1508.96i) q^{23} +(837.154 + 915.846i) q^{24} +3284.55 q^{25} -1010.33i q^{26} +(-2294.91 - 3013.68i) q^{27} +5978.55i q^{28} -5427.01i q^{29} +(1074.57 + 1175.58i) q^{30} +3748.89 q^{31} +3657.80i q^{32} +(-240.343 + 219.692i) q^{33} -222.829i q^{34} +15759.6i q^{35} +(661.265 - 7350.55i) q^{36} -10793.1i q^{37} -1139.60 q^{38} +(-9108.99 + 8326.32i) q^{39} +6372.57i q^{40} -5102.57i q^{41} +(-2890.48 + 2642.12i) q^{42} -10175.5i q^{43} -634.414 q^{44} +(1743.11 - 19376.2i) q^{45} +(1925.72 - 2602.74i) q^{46} -2207.39i q^{47} +(10013.6 - 9153.22i) q^{48} -21942.3 q^{49} +4191.71i q^{50} +(-2008.99 + 1836.37i) q^{51} -24044.3 q^{52} -38439.7 q^{53} +(3846.04 - 2928.75i) q^{54} -1672.33 q^{55} -15668.7 q^{56} +(9391.65 + 10274.5i) q^{57} +6925.91 q^{58} +49618.2i q^{59} +(27976.9 - 25573.0i) q^{60} +28860.1i q^{61} +4784.31i q^{62} +(47641.8 + 4285.92i) q^{63} +23181.6 q^{64} -63381.5 q^{65} +(-280.369 - 306.723i) q^{66} -61124.2i q^{67} -5302.99 q^{68} +(-39336.0 + 4087.65i) q^{69} -20112.3 q^{70} +11772.5i q^{71} +(19264.5 + 1733.06i) q^{72} +48746.9 q^{73} +13774.0 q^{74} +(37791.8 - 34544.6i) q^{75} +27120.7i q^{76} -4111.89i q^{77} +(-10626.0 - 11624.8i) q^{78} +65855.0i q^{79} +69676.0 q^{80} +(-58100.9 - 10539.0i) q^{81} +6511.86 q^{82} -53578.7 q^{83} +(62878.2 + 68788.8i) q^{84} -13978.8 q^{85} +12985.9 q^{86} +(-57077.5 - 62442.8i) q^{87} -1662.69i q^{88} +106893. q^{89} +(24727.8 + 2224.55i) q^{90} -155841. i q^{91} +(-61941.1 - 45829.1i) q^{92} +(43134.5 - 39428.3i) q^{93} +2817.05 q^{94} +71491.0i q^{95} +(38470.2 + 42086.4i) q^{96} +59613.7i q^{97} -28002.6i q^{98} +(-454.800 + 5055.51i) 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 $

$2$			1.27619i			0.225601i	0.993618	+	0.112800i	$0.0359821\pi$
$2$			1.27619i			0.225601i	−0.993618	+	0.112800i	$0.964018\pi$
$3$	11.5059	−	10.5173i	0.738106	−	0.674685i
$3$	11.5059	−	10.5173i	0.738106	−	0.674685i
$4$	30.3713			0.949104
$5$	80.0597			1.43215			0.716075	−	0.698023i	$-0.245937\pi$
$5$	80.0597			1.43215			0.716075	+	0.698023i	$0.245937\pi$
$6$	13.4221	+	14.6838i	0.152210	+	0.166517i
$7$			196.848i			1.51840i	0.650856	+	0.759201i	$0.274410\pi$
$7$			196.848i			1.51840i	−0.650856	+	0.759201i	$0.725590\pi$
$8$			79.5978i			0.439720i
$9$	21.7727	−	242.023i	0.0895995	−	0.995978i
$10$			102.171i			0.323095i
$11$	−20.8886			−0.0520508			−0.0260254	−	0.999661i	$-0.508285\pi$
$11$	−20.8886			−0.0520508			−0.0260254	+	0.999661i	$0.508285\pi$
$12$	349.450	−	319.425i	0.700539	−	0.640347i
$13$	−791.678			−1.29924			−0.649621	−	0.760258i	$-0.725073\pi$
$13$	−791.678			−1.29924			−0.649621	+	0.760258i	$0.725073\pi$
$14$	−251.216			−0.342553
$15$	921.160	−	842.012i	1.05708	−	0.966251i
$16$	870.301			0.849903
$17$	−174.605			−0.146533			−0.0732663	−	0.997312i	$-0.523342\pi$
$17$	−174.605			−0.146533			−0.0732663	+	0.997312i	$0.523342\pi$
$18$	308.867	+	27.7861i	0.224694	+	0.0202137i
$19$			892.972i			0.567484i	0.958901	+	0.283742i	$0.0915759\pi$
$19$			892.972i			0.567484i	−0.958901	+	0.283742i	$0.908424\pi$
$20$	2431.52			1.35926
$21$	2070.32	+	2264.92i	1.02444	+	1.12074i
$22$		−	26.6578i		−	0.0117427i
$23$	−2039.46	−	1508.96i	−0.803887	−	0.594781i
$23$	−2039.46	−	1508.96i	−0.803887	−	0.594781i
$24$	837.154	+	915.846i	0.296672	+	0.324560i
$25$	3284.55			1.05106
$26$		−	1010.33i		−	0.293110i
$27$	−2294.91	−	3013.68i	−0.605838	−	0.795588i
$28$			5978.55i			1.44112i
$29$		−	5427.01i		−	1.19830i	−0.800637	−	0.599150i	$-0.795505\pi$
$29$		−	5427.01i		−	1.19830i	0.800637	−	0.599150i	$-0.204495\pi$
$30$	1074.57	+	1175.58i	0.217987	+	0.238478i
$31$	3748.89			0.700647			0.350323	−	0.936629i	$-0.386072\pi$
$31$	3748.89			0.700647			0.350323	+	0.936629i	$0.386072\pi$
$32$			3657.80i			0.631459i
$33$	−240.343	+	219.692i	−0.0384190	+	0.0351179i
$34$		−	222.829i		−	0.0330579i
$35$			15759.6i			2.17458i
$36$	661.265	−	7350.55i	0.0850393	−	0.945287i
$37$		−	10793.1i		−	1.29611i	−0.761596	−	0.648053i	$-0.775584\pi$
$37$		−	10793.1i		−	1.29611i	0.761596	−	0.648053i	$-0.224416\pi$
$38$	−1139.60			−0.128025
$39$	−9108.99	+	8326.32i	−0.958978	+	0.876580i
$40$			6372.57i			0.629745i
$41$		−	5102.57i		−	0.474056i	−0.971503	−	0.237028i	$-0.923827\pi$
$41$		−	5102.57i		−	0.474056i	0.971503	−	0.237028i	$-0.0761733\pi$
$42$	−2890.48	+	2642.12i	−0.252840	+	0.231116i
$43$		−	10175.5i		−	0.839240i	−0.907700	−	0.419620i	$-0.862163\pi$
$43$		−	10175.5i		−	0.839240i	0.907700	−	0.419620i	$-0.137837\pi$
$44$	−634.414			−0.0494016
$45$	1743.11	−	19376.2i	0.128320	−	1.42639i
$46$	1925.72	−	2602.74i	0.134183	−	0.181358i
$47$		−	2207.39i		−	0.145759i	−0.997341	−	0.0728794i	$-0.976781\pi$
$47$		−	2207.39i		−	0.145759i	0.997341	−	0.0728794i	$-0.0232188\pi$
$48$	10013.6	−	9153.22i	0.627318	−	0.573417i
$49$	−21942.3			−1.30555
$50$			4191.71i			0.237119i
$51$	−2008.99	+	1836.37i	−0.108157	+	0.0988634i
$52$	−24044.3			−1.23312
$53$	−38439.7			−1.87971			−0.939854	−	0.341576i	$-0.889039\pi$
$53$	−38439.7			−1.87971			−0.939854	+	0.341576i	$0.889039\pi$
$54$	3846.04	−	2928.75i	0.179485	−	0.136678i
$55$	−1672.33			−0.0745446
$56$	−15668.7			−0.667672
$57$	9391.65	+	10274.5i	0.382873	+	0.418863i
$58$	6925.91			0.270338
$59$			49618.2i			1.85571i	0.372937	+	0.927856i	$0.378351\pi$
$59$			49618.2i			1.85571i	−0.372937	+	0.927856i	$0.621649\pi$
$60$	27976.9	−	25573.0i	1.00328	−	0.917073i
$61$			28860.1i			0.993053i	0.868021	+	0.496527i	$0.165392\pi$
$61$			28860.1i			0.993053i	−0.868021	+	0.496527i	$0.834608\pi$
$62$			4784.31i			0.158067i
$63$	47641.8	+	4285.92i	1.51230	+	0.136048i
$64$	23181.6			0.707445
$65$	−63381.5			−1.86071
$66$	−280.369	−	306.723i	−0.00792263	−	0.00866736i
$67$		−	61124.2i		−	1.66351i	−0.555140	−	0.831757i	$-0.687335\pi$
$67$		−	61124.2i		−	1.66351i	0.555140	−	0.831757i	$-0.312665\pi$
$68$	−5302.99			−0.139075
$69$	−39336.0	+	4087.65i	−0.994644	+	0.103360i
$70$	−20112.3			−0.490588
$71$			11772.5i			0.277154i	0.990352	+	0.138577i	$0.0442529\pi$
$71$			11772.5i			0.277154i	−0.990352	+	0.138577i	$0.955747\pi$
$72$	19264.5	+	1733.06i	0.437951	+	0.0393987i
$73$	48746.9			1.07063			0.535316	−	0.844652i	$-0.320193\pi$
$73$	48746.9			1.07063			0.535316	+	0.844652i	$0.320193\pi$
$74$	13774.0			0.292403
$75$	37791.8	−	34544.6i	0.775790	−	0.709132i
$76$			27120.7i			0.538602i
$77$		−	4111.89i		−	0.0790341i
$78$	−10626.0	−	11624.8i	−0.197757	−	0.216346i
$79$			65855.0i			1.18719i	0.804763	+	0.593596i	$0.202292\pi$
$79$			65855.0i			1.18719i	−0.804763	+	0.593596i	$0.797708\pi$
$80$	69676.0			1.21719
$81$	−58100.9	−	10539.0i	−0.983944	−	0.178478i
$82$	6511.86			0.106947
$83$	−53578.7			−0.853683			−0.426842	−	0.904326i	$-0.640374\pi$
$83$	−53578.7			−0.853683			−0.426842	+	0.904326i	$0.640374\pi$
$84$	62878.2	+	68788.8i	0.972304	+	1.06370i
$85$	−13978.8			−0.209857
$86$	12985.9			0.189333
$87$	−57077.5	−	62442.8i	−0.808476	−	0.884472i
$88$		−	1662.69i		−	0.0228878i
$89$	106893.			1.43045			0.715227	−	0.698892i	$-0.246323\pi$
$89$	106893.			1.43045			0.715227	+	0.698892i	$0.246323\pi$
$90$	24727.8	+	2224.55i	0.321795	+	0.0289491i
$91$		−	155841.i		−	1.97277i
$92$	−61941.1	−	45829.1i	−0.762973	−	0.564510i
$93$	43134.5	−	39428.3i	0.517151	−	0.472716i
$94$	2817.05			0.0328833
$95$			71491.0i			0.812723i
$96$	38470.2	+	42086.4i	0.426036	+	0.466083i
$97$			59613.7i			0.643304i	0.946858	+	0.321652i	$0.104238\pi$
$97$			59613.7i			0.643304i	−0.946858	+	0.321652i	$0.895762\pi$
$98$		−	28002.6i		−	0.294533i
$99$	−454.800	+	5055.51i	−0.00466372	+	0.0518414i
$100$	99756.1			0.997561
$101$		−	8155.58i		−	0.0795521i	−0.999209	−	0.0397760i	$-0.987336\pi$
$101$		−	8155.58i		−	0.0795521i	0.999209	−	0.0397760i	$-0.0126644\pi$
$102$	−2343.57	−	2563.86i	−0.0223037	−	0.0244002i
$103$			54139.3i			0.502828i	0.967880	+	0.251414i	$0.0808956\pi$
$103$			54139.3i			0.502828i	−0.967880	+	0.251414i	$0.919104\pi$
$104$		−	63015.8i		−	0.571303i
$105$	165749.	+	181329.i	1.46716	+	1.60507i
$106$		−	49056.4i		−	0.424064i
$107$	60053.1			0.507079			0.253540	−	0.967325i	$-0.418405\pi$
$107$	60053.1			0.507079			0.253540	+	0.967325i	$0.418405\pi$
$108$	−69699.5	−	91529.6i	−0.575003	−	0.755096i
$109$			152525.i			1.22963i	0.788670	+	0.614816i	$0.210770\pi$
$109$			152525.i			1.22963i	−0.788670	+	0.614816i	$0.789230\pi$
$110$		−	2134.22i		−	0.0168173i
$111$	−113514.	−	124184.i	−0.874463	−	0.956662i
$112$			171317.i			1.29049i
$113$	25828.9			0.190287			0.0951436	−	0.995464i	$-0.469669\pi$
$113$	25828.9			0.190287			0.0951436	+	0.995464i	$0.469669\pi$
$114$	−13112.2	+	11985.6i	−0.0944959	+	0.0863766i
$115$	−163278.	−	120807.i	−1.15129	−	0.851817i
$116$		−	164826.i		−	1.13731i
$117$	−17237.0	+	191604.i	−0.116412	+	1.29402i
$118$	−63322.3			−0.418651
$119$		−	34370.7i		−	0.222496i
$120$	67022.3	+	73322.3i	0.424880	+	0.464818i
$121$	−160615.			−0.997291
$122$	−36831.0			−0.224034
$123$	−53665.3	−	58709.8i	−0.319838	−	0.349903i
$124$	113859.			0.664987
$125$	12773.4			0.0731192
$126$	−5469.65	+	60800.1i	−0.0306926	+	0.341175i
$127$	234394.			1.28955			0.644775	−	0.764373i	$-0.276951\pi$
$127$	234394.			1.28955			0.644775	+	0.764373i	$0.276951\pi$
$128$			146634.i			0.791059i
$129$	−107019.	−	117079.i	−0.566223	−	0.619447i
$130$		−	80886.9i		−	0.419778i
$131$		−	50125.1i		−	0.255198i	−0.991826	−	0.127599i	$-0.959273\pi$
$131$		−	50125.1i		−	0.255198i	0.991826	−	0.127599i	$-0.0407270\pi$
$132$	−7299.52	+	6672.33i	−0.0364636	+	0.0333305i
$133$	−175780.			−0.861669
$134$	78006.3			0.375290
$135$	−183730.	−	241275.i	−0.867651	−	1.13940i
$136$		−	13898.2i		−	0.0644333i
$137$	−346120.			−1.57553			−0.787763	−	0.615979i	$-0.788761\pi$
$137$	−346120.			−1.57553			−0.787763	+	0.615979i	$0.788761\pi$
$138$	−5216.62	−	50200.3i	−0.0233180	−	0.224393i
$139$	5881.35			0.0258191			0.0129095	−	0.999917i	$-0.495891\pi$
$139$	5881.35			0.0258191			0.0129095	+	0.999917i	$0.495891\pi$
$140$			478641.i			2.06390i
$141$	−23215.8	−	25398.1i	−0.0983413	−	0.107585i
$142$	−15023.9			−0.0625262
$143$	16537.0			0.0676266
$144$	18948.8	−	210632.i	0.0761509	−	0.846485i
$145$		−	434485.i		−	1.71615i
$146$			62210.5i			0.241536i
$147$	−252467.	+	230774.i	−0.963631	+	0.880833i
$148$		−	327800.i		−	1.23014i
$149$	153793.			0.567505			0.283753	−	0.958898i	$-0.408421\pi$
$149$	153793.			0.567505			0.283753	+	0.958898i	$0.408421\pi$
$150$	44085.5	+	48229.6i	0.159981	+	0.175019i
$151$	−90252.7			−0.322120			−0.161060	−	0.986945i	$-0.551491\pi$
$151$	−90252.7			−0.322120			−0.161060	+	0.986945i	$0.551491\pi$
$152$	−71078.6			−0.249534
$153$	−3801.62	+	42258.4i	−0.0131293	+	0.145943i
$154$	5247.56			0.0178302
$155$	300135.			1.00343
$156$	−276652.	+	252881.i	−0.910170	+	0.831966i
$157$			259521.i			0.840280i	0.907459	+	0.420140i	$0.138019\pi$
$157$			259521.i			0.840280i	−0.907459	+	0.420140i	$0.861981\pi$
$158$	−84043.6			−0.267832
$159$	−442285.	+	404282.i	−1.38742	+	1.26821i
$160$			292842.i			0.904344i
$161$	297036.	−	401464.i	0.903118	−	1.22062i
$162$	13449.7	−	74147.9i	0.0402649	−	0.221979i
$163$	223631.			0.659271			0.329635	−	0.944108i	$-0.393074\pi$
$163$	223631.			0.659271			0.329635	+	0.944108i	$0.393074\pi$
$164$		−	154972.i		−	0.449928i
$165$	−19241.7	+	17588.4i	−0.0550218	+	0.0502941i
$166$		−	68376.7i		−	0.192592i
$167$		−	635080.i		−	1.76213i	−0.472998	−	0.881064i	$-0.656828\pi$
$167$		−	635080.i		−	1.76213i	0.472998	−	0.881064i	$-0.343172\pi$
$168$	−180283.	+	164793.i	−0.492812	+	0.450468i
$169$	255461.			0.688032
$170$		−	17839.7i		−	0.0473439i
$171$	216119.	+	19442.4i	0.565202	+	0.0508463i
$172$		−	309044.i		−	0.796526i
$173$			273947.i			0.695907i	0.937512	+	0.347953i	$0.113123\pi$
$173$			273947.i			0.695907i	−0.937512	+	0.347953i	$0.886877\pi$
$174$	79689.0	−	72841.9i	0.199538	−	0.182393i
$175$			646558.i			1.59593i
$176$	−18179.3			−0.0442381
$177$	521849.	+	570903.i	1.25202	+	1.36971i
$178$			136416.i			0.322712i
$179$		−	100929.i		−	0.235442i	−0.993047	−	0.117721i	$-0.962441\pi$
$179$		−	100929.i		−	0.235442i	0.993047	−	0.117721i	$-0.0375589\pi$
$180$	52940.7	−	588483.i	0.121789	−	1.35379i
$181$			347500.i			0.788421i	0.919020	+	0.394211i	$0.128982\pi$
$181$			347500.i			0.788421i	−0.919020	+	0.394211i	$0.871018\pi$
$182$	198883.			0.445060
$183$	303530.	+	332062.i	0.669999	+	0.732978i
$184$	120110.	−	162336.i	0.261537	−	0.353485i
$185$		−	864089.i		−	1.85622i
$186$	50318.0	+	55047.9i	0.106645	+	0.116670i
$187$	3647.25			0.00762714
$188$		−	67041.4i		−	0.138340i
$189$	593239.	−	451750.i	1.20802	−	0.919906i
$190$	−91236.2			−0.183351
$191$	869385.			1.72436			0.862182	−	0.506599i	$-0.169097\pi$
$191$	869385.			1.72436			0.862182	+	0.506599i	$0.169097\pi$
$192$	266725.	−	243808.i	0.522169	−	0.477303i
$193$	202279.			0.390893			0.195447	−	0.980714i	$-0.437384\pi$
$193$	202279.			0.390893			0.195447	+	0.980714i	$0.437384\pi$
$194$	−76078.5			−0.145130
$195$	−729263.	+	666602.i	−1.37340	+	1.25539i
$196$	−666418.			−1.23910
$197$			812892.i			1.49234i	0.665757	+	0.746169i	$0.268109\pi$
$197$			812892.i			1.49234i	−0.665757	+	0.746169i	$0.731891\pi$
$198$	−6451.80	−	580.413i	−0.0116955	−	0.00105214i
$199$		−	40521.5i		−	0.0725358i	−0.999342	−	0.0362679i	$-0.988453\pi$
$199$		−	40521.5i		−	0.0725358i	0.999342	−	0.0362679i	$-0.0115470\pi$
$200$			261443.i			0.462170i
$201$	−642862.	−	703291.i	−1.12235	−	1.22785i
$202$	10408.1			0.0179470
$203$	1.06830e6			1.81950
$204$	−61015.8	+	55773.1i	−0.102652	+	0.0938317i
$205$		−	408510.i		−	0.678919i
$206$	−69092.1			−0.113438
$207$	−409606.	+	460741.i	−0.664417	+	0.747362i
$208$	−688998.			−1.10423
$209$		−	18652.9i		−	0.0295380i
$210$	−231411.	+	211527.i	−0.362105	+	0.330992i
$211$	716481.			1.10789			0.553947	−	0.832552i	$-0.313121\pi$
$211$	716481.			1.10789			0.553947	+	0.832552i	$0.313121\pi$
$212$	−1.16747e6			−1.78404
$213$	123815.	+	135453.i	0.186992	+	0.204569i
$214$			76639.2i			0.114398i
$215$		−	814650.i		−	1.20192i
$216$	239883.	−	182670.i	0.349836	−	0.266399i
$217$			737964.i			1.06386i
$218$	−194651.			−0.277406
$219$	560879.	−	512686.i	0.790239	−	0.722340i
$220$	−50791.0			−0.0707506
$221$	138231.			0.190381
$222$	158483.	−	144865.i	0.215824	−	0.197280i
$223$	−145527.			−0.195966			−0.0979831	−	0.995188i	$-0.531239\pi$
$223$	−145527.			−0.195966			−0.0979831	+	0.995188i	$0.531239\pi$
$224$	−720032.			−0.958809
$225$	71513.4	−	794935.i	0.0941740	−	1.04683i
$226$			32962.6i			0.0429290i
$227$	544805.			0.701740			0.350870	−	0.936424i	$-0.385886\pi$
$227$	544805.			0.701740			0.350870	+	0.936424i	$0.385886\pi$
$228$	285237.	+	312049.i	0.363387	+	0.397545i
$229$		−	1734.50i		−	0.00218567i	−0.999999	−	0.00109284i	$-0.999652\pi$
$229$		−	1734.50i		−	0.00218567i	0.999999	−	0.00109284i	$-0.000347861\pi$
$230$	154172.	−	208374.i	0.192171	−	0.259732i
$231$	−43246.0	−	47311.1i	−0.0533231	−	0.0583355i
$232$	431978.			0.526917
$233$			309502.i			0.373485i	0.982409	+	0.186742i	$0.0597930\pi$
$233$			309502.i			0.373485i	−0.982409	+	0.186742i	$0.940207\pi$
$234$	−244524.	−	21997.7i	−0.291932	−	0.0262625i
$235$		−	176723.i		−	0.208749i
$236$			1.50697e6i			1.76126i
$237$	692617.	+	757723.i	0.800981	+	0.876273i
$238$	43863.6			0.0501952
$239$			1.15098e6i			1.30339i	0.758483	+	0.651693i	$0.225941\pi$
$239$			1.15098e6i			1.30339i	−0.758483	+	0.651693i	$0.774059\pi$
$240$	801687.	−	732803.i	0.898414	−	0.821220i
$241$		−	1.55779e6i		−	1.72769i	−0.503756	−	0.863846i	$-0.668049\pi$
$241$		−	1.55779e6i		−	1.72769i	0.503756	−	0.863846i	$-0.331951\pi$
$242$		−	204975.i		−	0.224990i
$243$	−779346.	+	489804.i	−0.846671	+	0.532117i
$244$			876518.i			0.942511i
$245$	−1.75670e6			−1.86974
$246$	74925.0	−	68487.2i	0.0789385	−	0.0721559i
$247$		−	706946.i		−	0.737300i
$248$			298404.i			0.308088i
$249$	−616472.	+	563503.i	−0.630108	+	0.575968i
$250$			16301.3i			0.0164958i
$251$	−22290.6			−0.0223325			−0.0111663	−	0.999938i	$-0.503554\pi$
$251$	−22290.6			−0.0223325			−0.0111663	+	0.999938i	$0.503554\pi$
$252$	1.44694e6	+	130169.i	1.43533	+	0.129124i
$253$	42601.4	+	31520.0i	0.0418430	+	0.0309588i
$254$			299132.i			0.290924i
$255$	−160839.	+	147019.i	−0.154896	+	0.141587i
$256$	554677.			0.528982
$257$		−	1.38634e6i		−	1.30929i	−0.755935	−	0.654647i	$-0.772817\pi$
$257$		−	1.38634e6i		−	1.30929i	0.755935	−	0.654647i	$-0.227183\pi$
$258$	149415.	−	136577.i	0.139748	−	0.127740i
$259$	2.12460e6			1.96801
$260$	−1.92498e6			−1.76601
$261$	−1.31346e6	−	118161.i	−1.19348	−	0.107367i
$262$	63969.2			0.0575729
$263$	−1.66301e6			−1.48254			−0.741269	−	0.671208i	$-0.765776\pi$
$263$	−1.66301e6			−1.48254			−0.741269	+	0.671208i	$0.765776\pi$
$264$	−17487.0	−	19130.7i	−0.0154420	−	0.0168936i
$265$	−3.07747e6			−2.69203
$266$		−	224329.i		−	0.194393i
$267$	1.22990e6	−	1.12422e6i	1.05583	−	0.965106i
$268$		−	1.85642e6i		−	1.57885i
$269$			1.60222e6i			1.35002i	0.737809	+	0.675010i	$0.235861\pi$
$269$			1.60222e6i			1.35002i	−0.737809	+	0.675010i	$0.764139\pi$
$270$	307913.	−	234474.i	0.257050	−	0.195743i
$271$	−255594.			−0.211411			−0.105705	−	0.994397i	$-0.533710\pi$
$271$	−255594.			−0.211411			−0.105705	+	0.994397i	$0.533710\pi$
$272$	−151959.			−0.124539
$273$	−1.63902e6	−	1.79309e6i	−1.33100	−	1.45612i
$274$		−	441716.i		−	0.355440i
$275$	−68609.6			−0.0547083
$276$	−1.19469e6	+	124147.i	−0.944021	+	0.0980990i
$277$	−922483.			−0.722369			−0.361184	−	0.932494i	$-0.617627\pi$
$277$	−922483.			−0.722369			−0.361184	+	0.932494i	$0.617627\pi$
$278$			7505.74i			0.00582480i
$279$	81623.5	−	907317.i	0.0627776	−	0.697828i
$280$	−1.25443e6			−0.956207
$281$	−992232.			−0.749631			−0.374815	−	0.927100i	$-0.622294\pi$
$281$	−992232.			−0.749631			−0.374815	+	0.927100i	$0.622294\pi$
$282$	32412.8	−	29627.8i	0.0242714	−	0.0221859i
$283$			292415.i			0.217037i	0.994094	+	0.108518i	$0.0346106\pi$
$283$			292415.i			0.217037i	−0.994094	+	0.108518i	$0.965389\pi$
$284$			357545.i			0.263048i
$285$	751893.	+	822570.i	0.548332	+	0.599875i
$286$			21104.4i			0.0152566i
$287$	1.00443e6			0.719807
$288$	885270.	+	79640.1i	0.628919	+	0.0565784i
$289$	−1.38937e6			−0.978528
$290$	554486.			0.387164
$291$	626975.	+	685910.i	0.434028	+	0.474827i
$292$	1.48051e6			1.01614
$293$	150873.			0.102670			0.0513349	−	0.998681i	$-0.483652\pi$
$293$	150873.			0.102670			0.0513349	+	0.998681i	$0.483652\pi$
$294$	−294512.	−	322196.i	−0.198717	−	0.217396i
$295$			3.97241e6i			2.65766i
$296$	859104.			0.569923
$297$	47937.4	+	62951.6i	0.0315343	+	0.0414110i
$298$			196269.i			0.128030i
$299$	1.61459e6	+	1.19461e6i	1.04444	+	0.772766i
$300$	1.14779e6	−	1.04917e6i	0.736305	−	0.673040i
$301$	2.00304e6			1.27430
$302$		−	115180.i		−	0.0726706i
$303$	−85774.7	−	93837.5i	−0.0536726	−	0.0587178i
$304$			777154.i			0.482306i
$305$			2.31053e6i			1.42220i
$306$	−53929.8	−	4851.59i	−0.0329249	−	0.00296197i
$307$	1.91204e6			1.15785			0.578924	−	0.815382i	$-0.303473\pi$
$307$	1.91204e6			1.15785			0.578924	+	0.815382i	$0.303473\pi$
$308$		−	124883.i		−	0.0750116i
$309$	569399.	+	622923.i	0.339251	+	0.371140i
$310$			383030.i			0.226375i
$311$		−	1.82521e6i		−	1.07007i	−0.844829	−	0.535036i	$-0.820298\pi$
$311$		−	1.82521e6i		−	1.07007i	0.844829	−	0.535036i	$-0.179702\pi$
$312$	−662757.	−	725056.i	−0.385450	−	0.421682i
$313$		−	2.85121e6i		−	1.64501i	−0.568759	−	0.822504i	$-0.692576\pi$
$313$		−	2.85121e6i		−	1.64501i	0.568759	−	0.822504i	$-0.307424\pi$
$314$	−331199.			−0.189568
$315$	3.81419e6	+	343129.i	2.16584	+	0.194841i
$316$			2.00010e6i			1.12677i
$317$		−	2.10359e6i		−	1.17575i	−0.808953	−	0.587873i	$-0.799965\pi$
$317$		−	2.10359e6i		−	1.17575i	0.808953	−	0.587873i	$-0.200035\pi$
$318$	−515942.	−	564440.i	−0.286110	−	0.313004i
$319$			113363.i			0.0623725i
$320$	1.85591e6			1.01317
$321$	690966.	−	631596.i	0.374278	−	0.342119i
$322$	512345.	+	379075.i	0.275374	+	0.203744i
$323$		−	155917.i		−	0.0831550i
$324$	−1.76460e6	−	320082.i	−0.933865	−	0.169394i
$325$	−2.60031e6			−1.36558
$326$			285397.i			0.148732i
$327$	1.60415e6	+	1.75494e6i	0.829615	+	0.907598i
$328$	406153.			0.208452
$329$	434522.			0.221320
$330$	−22446.2	−	24556.2i	−0.0113464	−	0.0124130i
$331$	1.56186e6			0.783559			0.391779	−	0.920059i	$-0.371860\pi$
$331$	1.56186e6			0.783559			0.391779	+	0.920059i	$0.371860\pi$
$332$	−1.62726e6			−0.810234
$333$	−2.61216e6	−	234994.i	−1.29089	−	0.116130i
$334$	810484.			0.397538
$335$		−	4.89359e6i		−	2.38240i
$336$	1.80180e6	+	1.97116e6i	0.870678	+	0.952521i
$337$			1.97846e6i			0.948970i	0.880264	+	0.474485i	$0.157366\pi$
$337$			1.97846e6i			0.948970i	−0.880264	+	0.474485i	$0.842634\pi$
$338$			326018.i			0.155221i
$339$	297185.	−	271650.i	0.140452	−	0.128384i
$340$	−424555.			−0.199176
$341$	−78309.1			−0.0364692
$342$	−24812.2	+	275810.i	−0.0114710	+	0.127510i
$343$		−	1.01088e6i		−	0.463943i
$344$	809950.			0.369030
$345$	−3.14923e6	+	327256.i	−1.42448	+	0.148026i
$346$	−349609.			−0.156997
$347$			1.37659e6i			0.613734i	0.951752	+	0.306867i	$0.0992806\pi$
$347$			1.37659e6i			0.613734i	−0.951752	+	0.306867i	$0.900719\pi$
$348$	−1.73352e6	−	1.89647e6i	−0.767328	−	0.839456i
$349$	−1.69269e6			−0.743897			−0.371948	−	0.928253i	$-0.621310\pi$
$349$	−1.69269e6			−0.743897			−0.371948	+	0.928253i	$0.621310\pi$
$350$	−825133.			−0.360042
$351$	1.81683e6	+	2.38587e6i	0.787130	+	1.03366i
$352$		−	76406.3i		−	0.0328679i
$353$			2.61540e6i			1.11712i	0.829463	+	0.558561i	$0.188647\pi$
$353$			2.61540e6i			1.11712i	−0.829463	+	0.558561i	$0.811353\pi$
$354$	−728582.	+	665980.i	−0.309008	+	0.282457i
$355$			942499.i			0.396927i
$356$	3.24648e6			1.35765
$357$	−361487.	−	395467.i	−0.150114	−	0.164225i
$358$	128805.			0.0531159
$359$	−3.58025e6			−1.46615			−0.733073	−	0.680150i	$-0.761915\pi$
$359$	−3.58025e6			−1.46615			−0.733073	+	0.680150i	$0.761915\pi$
$360$	1.54231e6	+	138748.i	0.627212	+	0.0564248i
$361$	1.67870e6			0.677962
$362$	−443476.			−0.177869
$363$	−1.84802e6	+	1.68923e6i	−0.736106	+	0.672857i
$364$		−	4.73309e6i		−	1.87237i
$365$	3.90266e6			1.53331
$366$	−423774.	+	387362.i	−0.165361	+	0.151152i
$367$		−	3.14970e6i		−	1.22068i	−0.792138	−	0.610342i	$-0.791032\pi$
$367$		−	3.14970e6i		−	1.22068i	0.792138	−	0.610342i	$-0.208968\pi$
$368$	−1.77494e6	−	1.31325e6i	−0.683226	−	0.505507i
$369$	−1.23494e6	−	111097.i	−0.472149	−	0.0424752i
$370$	1.10274e6			0.418765
$371$		−	7.56680e6i		−	2.85415i
$372$	1.31005e6	−	1.19749e6i	0.490830	−	0.448657i
$373$		−	2.01867e6i		−	0.751266i	−0.926769	−	0.375633i	$-0.877425\pi$
$373$		−	2.01867e6i		−	0.751266i	0.926769	−	0.375633i	$-0.122575\pi$
$374$			4654.59i			0.00172069i
$375$	146970.	−	134342.i	0.0539697	−	0.0493325i
$376$	175703.			0.0640930
$377$			4.29645e6i			1.55688i
$378$	576519.	+	757087.i	0.207532	+	0.272531i
$379$			1.69715e6i			0.606908i	0.952846	+	0.303454i	$0.0981399\pi$
$379$			1.69715e6i			0.606908i	−0.952846	+	0.303454i	$0.901860\pi$
$380$			2.17128e6i			0.771359i
$381$	2.69693e6	−	2.46520e6i	0.951824	−	0.870040i
$382$			1.10950e6i			0.389018i
$383$	1.88013e6			0.654924			0.327462	−	0.944864i	$-0.393807\pi$
$383$	1.88013e6			0.654924			0.327462	+	0.944864i	$0.393807\pi$
$384$	1.54219e6	+	1.68716e6i	0.533716	+	0.583885i
$385$		−	329196.i		−	0.113189i
$386$			258147.i			0.0881859i
$387$	−2.46271e6	−	221549.i	−0.835864	−	0.0751954i
$388$			1.81055e6i			0.610563i
$389$	−1.23969e6			−0.415373			−0.207687	−	0.978195i	$-0.566593\pi$
$389$	−1.23969e6			−0.415373			−0.207687	+	0.978195i	$0.566593\pi$
$390$	−850712.	−	930679.i	−0.283218	−	0.309841i
$391$	356100.	+	263472.i	0.117796	+	0.0871549i
$392$		−	1.74656e6i		−	0.574075i
$393$	−527181.	−	576736.i	−0.172178	−	0.188363i
$394$	−1.03741e6			−0.336673
$395$			5.27233e6i			1.70024i
$396$	−13812.9	+	153543.i	−0.00442636	+	0.0492029i
$397$	556498.			0.177210			0.0886049	−	0.996067i	$-0.471759\pi$
$397$	556498.			0.177210			0.0886049	+	0.996067i	$0.471759\pi$
$398$	51713.2			0.0163641
$399$	−2.02251e6	+	1.84873e6i	−0.636003	+	0.581356i
$400$	2.85854e6			0.893295
$401$	2.64380e6			0.821047			0.410523	−	0.911850i	$-0.365346\pi$
$401$	2.64380e6			0.821047			0.410523	+	0.911850i	$0.365346\pi$
$402$	897534.	−	820415.i	0.277004	−	0.253203i
$403$	−2.96792e6			−0.910310
$404$		−	247696.i		−	0.0755032i
$405$	−4.65154e6	−	843746.i	−1.40916	−	0.255608i
$406$			1.36335e6i			0.410482i
$407$			225452.i			0.0674633i
$408$	−146171.	−	159911.i	−0.0434722	−	0.0475586i
$409$	−3.94099e6			−1.16492			−0.582461	−	0.812859i	$-0.697910\pi$
$409$	−3.94099e6			−1.16492			−0.582461	+	0.812859i	$0.697910\pi$
$410$	521337.			0.153165
$411$	−3.98243e6	+	3.64025e6i	−1.16290	+	1.06298i
$412$			1.64428e6i			0.477236i
$413$	−9.76726e6			−2.81772
$414$	−587994.	−	522736.i	−0.168606	−	0.149893i
$415$	−4.28949e6			−1.22260
$416$		−	2.89580e6i		−	0.820418i
$417$	67670.4	−	61856.0i	0.0190572	−	0.0174197i
$418$	23804.7			0.00666380
$419$	2.63743e6			0.733917			0.366958	−	0.930237i	$-0.380399\pi$
$419$	2.63743e6			0.733917			0.366958	+	0.930237i	$0.380399\pi$
$420$	5.03401e6	+	5.50721e6i	1.39249	+	1.52338i
$421$		−	599700.i		−	0.164903i	−0.996595	−	0.0824515i	$-0.973725\pi$
$421$		−	599700.i		−	0.164903i	0.996595	−	0.0824515i	$-0.0262750\pi$
$422$			914367.i			0.249942i
$423$	−534239.	−	48060.8i	−0.145172	−	0.0130599i
$424$		−	3.05972e6i		−	0.826545i
$425$	−573499.			−0.154014
$426$	−172864.	+	158011.i	−0.0461510	+	0.0421855i
$427$	−5.68106e6			−1.50786
$428$	1.82389e6			0.481271
$429$	190274.	−	173925.i	0.0499156	−	0.0456267i
$430$	1.03965e6			0.271154
$431$	6.87699e6			1.78322			0.891610	−	0.452804i	$-0.149576\pi$
$431$	6.87699e6			1.78322			0.891610	+	0.452804i	$0.149576\pi$
$432$	−1.99726e6	−	2.62281e6i	−0.514903	−	0.676173i
$433$		−	6.73591e6i		−	1.72654i	−0.504743	−	0.863270i	$-0.668413\pi$
$433$		−	6.73591e6i		−	1.72654i	0.504743	−	0.863270i	$-0.331587\pi$
$434$	−941784.			−0.240009
$435$	−4.56961e6	−	4.99915e6i	−1.15786	−	1.26670i
$436$			4.63239e6i			1.16705i
$437$	1.34746e6	−	1.82118e6i	0.337529	−	0.456193i
$438$	654286.	+	715789.i	0.162961	+	0.178279i
$439$	1.75564e6			0.434784			0.217392	−	0.976084i	$-0.430245\pi$
$439$	1.75564e6			0.434784			0.217392	+	0.976084i	$0.430245\pi$
$440$		−	133114.i		−	0.0327787i
$441$	−477743.	+	5.31054e6i	−0.116976	+	1.30030i
$442$			176409.i			0.0429503i
$443$			7.53505e6i			1.82422i	0.409946	+	0.912110i	$0.365547\pi$
$443$			7.53505e6i			1.82422i	−0.409946	+	0.912110i	$0.634453\pi$
$444$	−3.44757e6	−	3.77164e6i	−0.829957	−	0.907972i
$445$	8.55781e6			2.04863
$446$		−	185720.i		−	0.0442102i
$447$	1.76953e6	−	1.61748e6i	0.418879	−	0.382887i
$448$			4.56326e6i			1.07419i
$449$		−	5.86864e6i		−	1.37379i	−0.726755	−	0.686897i	$-0.758972\pi$
$449$		−	5.86864e6i		−	1.37379i	0.726755	−	0.686897i	$-0.241028\pi$
$450$	1.01449e6	+	91264.8i	0.236165	+	0.0212458i
$451$			106585.i			0.0246750i
$452$	784458.			0.180602
$453$	−1.03844e6	+	949215.i	−0.237759	+	0.217330i
$454$			695275.i			0.158313i
$455$		−	1.24765e7i		−	2.82531i
$456$	−817825.	+	747555.i	−0.184182	+	0.168357i
$457$		−	5.34684e6i		−	1.19759i	−0.800904	−	0.598793i	$-0.795647\pi$
$457$		−	5.34684e6i		−	1.19759i	0.800904	−	0.598793i	$-0.204353\pi$
$458$	2213.55			0.000493090
$459$	400703.	+	526204.i	0.0887750	+	0.116580i
$460$	−4.95898e6	−	3.66906e6i	−1.09269	−	0.808463i
$461$			5.80691e6i			1.27260i	0.771441	+	0.636301i	$0.219536\pi$
$461$			5.80691e6i			1.27260i	−0.771441	+	0.636301i	$0.780464\pi$
$462$	60378.0	−	55190.1i	0.0131605	−	0.0120297i
$463$	−4.53602e6			−0.983383			−0.491691	−	0.870769i	$-0.663621\pi$
$463$	−4.53602e6			−0.983383			−0.491691	+	0.870769i	$0.663621\pi$
$464$		−	4.72313e6i		−	1.01844i
$465$	3.45333e6	−	3.15661e6i	0.740638	−	0.677000i
$466$	−394983.			−0.0842586
$467$	2.16503e6			0.459379			0.229689	−	0.973264i	$-0.426229\pi$
$467$	2.16503e6			0.459379			0.229689	+	0.973264i	$0.426229\pi$
$468$	−523509.	+	5.81927e6i	−0.110487	+	1.22816i
$469$	1.20322e7			2.52588
$470$	225532.			0.0470939
$471$	2.72947e6	+	2.98603e6i	0.566925	+	0.620215i
$472$	−3.94950e6			−0.815994
$473$			212552.i			0.0436831i
$474$	−967000.	+	883912.i	−0.197688	+	0.180702i
$475$			2.93301e6i			0.596457i
$476$		−	1.04389e6i		−	0.211171i
$477$	−836935.	+	9.30328e6i	−0.168421	+	1.87215i
$478$	−1.46887e6			−0.294045
$479$	4.63191e6			0.922403			0.461202	−	0.887295i	$-0.347418\pi$
$479$	4.63191e6			0.922403			0.461202	+	0.887295i	$0.347418\pi$
$480$	3.07991e6	+	3.36942e6i	0.610148	+	0.667501i
$481$			8.54463e6i			1.68396i
$482$	1.98804e6			0.389769
$483$	−804647.	−	7.74324e6i	−0.156941	−	1.51027i
$484$	−4.87808e6			−0.946533
$485$			4.77265e6i			0.921309i
$486$	−625084.	−	994595.i	−0.120046	−	0.191010i
$487$	9.50630e6			1.81631			0.908153	−	0.418638i	$-0.137492\pi$
$487$	9.50630e6			1.81631			0.908153	+	0.418638i	$0.137492\pi$
$488$	−2.29720e6			−0.436665
$489$	2.57309e6	−	2.35200e6i	0.486611	−	0.444800i
$490$		−	2.24188e6i		−	0.421815i
$491$		−	4.09756e6i		−	0.767047i	−0.923531	−	0.383523i	$-0.874711\pi$
$491$		−	4.09756e6i		−	0.767047i	0.923531	−	0.383523i	$-0.125289\pi$
$492$	−1.62989e6	−	1.78309e6i	−0.303560	−	0.332095i
$493$			947583.i			0.175590i
$494$	902199.			0.166336
$495$	−36411.2	+	404742.i	−0.00667916	+	0.0742447i
$496$	3.26267e6			0.595482
$497$	−2.31739e6			−0.420832
$498$	−719138.	−	786737.i	−0.129939	−	0.142153i
$499$	−1443.25			−0.000259472			−0.000129736	−	1.00000i	$-0.500041\pi$
$499$	−1443.25			−0.000259472			−0.000129736		1.00000i	$0.500041\pi$
$500$	387945.			0.0693978
$501$	−6.67933e6	−	7.30718e6i	−1.18888	−	1.30064i
$502$		−	28447.1i		−	0.00503824i
$503$	−5.18638e6			−0.913996			−0.456998	−	0.889468i	$-0.651075\pi$
$503$	−5.18638e6			−0.913996			−0.456998	+	0.889468i	$0.651075\pi$
$504$	−341150.	+	3.79218e6i	−0.0598231	+	0.664986i
$505$		−	652933.i		−	0.113931i
$506$	−40225.5	+	54367.6i	−0.00698435	+	0.00943982i
$507$	2.93932e6	−	2.68677e6i	0.507840	−	0.464205i
$508$	7.11887e6			1.22392
$509$		−	4.48149e6i		−	0.766705i	−0.923602	−	0.383353i	$-0.874769\pi$
$509$		−	4.48149e6i		−	0.766705i	0.923602	−	0.383353i	$-0.125231\pi$
$510$	−187625.	−	205262.i	−0.0319422	−	0.0349448i
$511$			9.59576e6i			1.62565i
$512$			5.40015e6i			0.910398i
$513$	2.69113e6	−	2.04929e6i	0.451484	−	0.343803i
$514$	1.76924e6			0.295378
$515$			4.33437e6i			0.720125i
$516$	−3.25031e6	−	3.55584e6i	−0.537404	−	0.587920i
$517$			46109.3i			0.00758686i
$518$			2.71139e6i			0.443985i
$519$	2.88118e6	+	3.15201e6i	0.469518	+	0.513653i
$520$		−	5.04503e6i		−	0.818192i
$521$	1.55720e6			0.251333			0.125667	−	0.992073i	$-0.459893\pi$
$521$	1.55720e6			0.251333			0.125667	+	0.992073i	$0.459893\pi$
$522$	150796.	−	1.67623e6i	0.0242221	−	0.269250i
$523$		−	5.42090e6i		−	0.866597i	−0.901251	−	0.433298i	$-0.857350\pi$
$523$		−	5.42090e6i		−	0.866597i	0.901251	−	0.433298i	$-0.142650\pi$
$524$		−	1.52237e6i		−	0.242209i
$525$	6.80005e6	+	7.43925e6i	1.07675	+	1.17796i
$526$		−	2.12232e6i		−	0.334462i
$527$	−654576.			−0.102668
$528$	−209170.	+	191198.i	−0.0326524	+	0.0298468i
$529$	1.88244e6	+	6.15491e6i	0.292470	+	0.956275i
$530$		−	3.92744e6i		−	0.607324i
$531$	1.20087e7	+	1.08032e6i	1.84825	+	0.166271i
$532$	−5.33868e6			−0.817814
$533$			4.03959e6i			0.615913i
$534$	1.43473e6	+	1.56959e6i	0.217729	+	0.238195i
$535$	4.80783e6			0.726214
$536$	4.86535e6			0.731480
$537$	−1.06150e6	−	1.16128e6i	−0.158849	−	0.173781i
$538$	−2.04473e6			−0.304566
$539$	458344.			0.0679547
$540$	−5.58012e6	−	7.32783e6i	−0.823491	−	1.08141i
$541$	5.88262e6			0.864127			0.432064	−	0.901843i	$-0.357786\pi$
$541$	5.88262e6			0.864127			0.432064	+	0.901843i	$0.357786\pi$
$542$		−	326187.i		−	0.0476945i
$543$	3.65476e6	+	3.99831e6i	0.531936	+	0.581938i
$544$		−	638670.i		−	0.0925293i
$545$			1.22111e7i			1.76102i
$546$	2.28833e6	−	2.09171e6i	0.328501	−	0.300275i
$547$	−8.94337e6			−1.27801			−0.639003	−	0.769204i	$-0.720653\pi$
$547$	−8.94337e6			−1.27801			−0.639003	+	0.769204i	$0.720653\pi$
$548$	−1.05121e7			−1.49534
$549$	6.98479e6	+	628361.i	0.989059	+	0.0889771i
$550$		−	87559.0i		−	0.0123422i
$551$	4.84617e6			0.680017
$552$	−325368.	−	3.13106e6i	−0.0454493	−	0.437365i
$553$	−1.29635e7			−1.80264
$554$		−	1.17726e6i		−	0.162967i
$555$	−9.08788e6	−	9.94214e6i	−1.25236	−	1.37008i
$556$	178625.			0.0245050
$557$	76523.7			0.0104510			0.00522550	−	0.999986i	$-0.498337\pi$
$557$	76523.7			0.0104510			0.00522550	+	0.999986i	$0.498337\pi$
$558$	1.15791e6	+	104167.i	0.157431	+	0.0141627i
$559$			8.05575e6i			1.09038i
$560$			1.37156e7i			1.84818i
$561$	41965.0	−	38359.3i	0.00562963	−	0.00514592i
$562$		−	1.26628e6i		−	0.169117i
$563$	−7.73965e6			−1.02908			−0.514542	−	0.857465i	$-0.672038\pi$
$563$	−7.73965e6			−1.02908			−0.514542	+	0.857465i	$0.672038\pi$
$564$	−705095.	−	771374.i	−0.0933361	−	0.102110i
$565$	2.06785e6			0.272520
$566$	−373177.			−0.0489637
$567$	2.07458e6	−	1.14371e7i	0.271002	−	1.49402i
$568$	−937062.			−0.121870
$569$	−1.52774e7			−1.97819			−0.989096	−	0.147271i	$-0.952951\pi$
$569$	−1.52774e7			−1.97819			−0.989096	+	0.147271i	$0.952951\pi$
$570$	−1.04976e6	+	959559.i	−0.135332	+	0.123704i
$571$			1.36736e6i			0.175506i	0.996142	+	0.0877532i	$0.0279687\pi$
$571$			1.36736e6i			0.175506i	−0.996142	+	0.0877532i	$0.972031\pi$
$572$	502252.			0.0641847
$573$	1.00031e7	−	9.14359e6i	1.27276	−	1.16340i
$574$			1.28185e6i			0.162389i
$575$	−6.69870e6	−	4.95624e6i	−0.844930	−	0.625148i
$576$	504725.	−	5.61046e6i	0.0633867	−	0.704600i
$577$	−7.52623e6			−0.941105			−0.470552	−	0.882372i	$-0.655945\pi$
$577$	−7.52623e6			−0.941105			−0.470552	+	0.882372i	$0.655945\pi$
$578$		−	1.77310e6i		−	0.220757i
$579$	2.32741e6	−	2.12743e6i	0.288520	−	0.263730i
$580$		−	1.31959e7i		−	1.62880i
$581$		−	1.05469e7i		−	1.29624i
$582$	−875353.	+	800140.i	−0.107121	+	0.0979171i
$583$	802951.			0.0978403
$584$			3.88015e6i			0.470778i
$585$	−1.37998e6	+	1.53398e7i	−0.166719	+	1.85323i
$586$			192543.i			0.0231624i
$587$			620332.i			0.0743069i	0.999310	+	0.0371534i	$0.0118290\pi$
$587$			620332.i			0.0743069i	−0.999310	+	0.0371534i	$0.988171\pi$
$588$	−7.66775e6	+	7.00892e6i	−0.914587	+	0.836003i
$589$			3.34766e6i			0.397606i
$590$	−5.06956e6			−0.599571
$591$	8.54943e6	+	9.35307e6i	1.00686	+	1.10150i
$592$		−	9.39321e6i		−	1.10156i
$593$		−	4.96024e6i		−	0.579250i	−0.957140	−	0.289625i	$-0.906469\pi$
$593$		−	4.96024e6i		−	0.579250i	0.957140	−	0.289625i	$-0.0935307\pi$
$594$	−80338.3	+	61177.4i	−0.00934236	+	0.00711418i
$595$		−	2.75171e6i		−	0.318647i
$596$	4.67089e6			0.538622
$597$	−426177.	−	466237.i	−0.0489388	−	0.0535391i
$598$	−1.52455e6	+	2.06053e6i	−0.174337	+	0.235628i
$599$		−	2.30566e6i		−	0.262560i	−0.991345	−	0.131280i	$-0.958091\pi$
$599$		−	2.30566e6i		−	0.262560i	0.991345	−	0.131280i	$-0.0419087\pi$
$600$	2.74967e6	+	3.00814e6i	0.311819	+	0.341130i
$601$	−3.65731e6			−0.413025			−0.206512	−	0.978444i	$-0.566211\pi$
$601$	−3.65731e6			−0.413025			−0.206512	+	0.978444i	$0.566211\pi$
$602$			2.55626e6i			0.287484i
$603$	−1.47934e7	−	1.33084e6i	−1.65682	−	0.149050i
$604$	−2.74110e6			−0.305726
$605$	−1.28588e7			−1.42827
$606$	119755.	−	109465.i	0.0132468	−	0.0121086i
$607$	5.25634e6			0.579044			0.289522	−	0.957171i	$-0.406504\pi$
$607$	5.25634e6			0.579044			0.289522	+	0.957171i	$0.406504\pi$
$608$	−3.26631e6			−0.358343
$609$	1.22918e7	−	1.12356e7i	1.34299	−	1.22759i
$610$	−2.94867e6			−0.320850
$611$			1.74754e6i			0.189376i
$612$	−115460.	+	1.28344e6i	−0.0124610	+	0.138515i
$613$		−	3.67092e6i		−	0.394570i	−0.980346	−	0.197285i	$-0.936788\pi$
$613$		−	3.67092e6i		−	0.394570i	0.980346	−	0.197285i	$-0.0632124\pi$
$614$			2.44013e6i			0.261212i
$615$	−4.29642e6	−	4.70029e6i	−0.458057	−	0.501114i
$616$	327297.			0.0347528
$617$	1.73511e6			0.183490			0.0917451	−	0.995783i	$-0.470755\pi$
$617$	1.73511e6			0.183490			0.0917451	+	0.995783i	$0.470755\pi$
$618$	−794969.	+	726663.i	−0.0837296	+	0.0765353i
$619$		−	7.42838e6i		−	0.779233i	−0.920977	−	0.389617i	$-0.872607\pi$
$619$		−	7.42838e6i		−	0.779233i	0.920977	−	0.389617i	$-0.127393\pi$
$620$	9.11551e6			0.952361
$621$	132853.	+	9.60921e6i	0.0138242	+	0.999904i
$622$	2.32932e6			0.241409
$623$			2.10417e7i			2.17200i
$624$	−7.92756e6	+	7.24640e6i	−0.815038	+	0.745008i
$625$	−9.24158e6			−0.946338
$626$	3.63869e6			0.371115
$627$	−196178.	−	214619.i	−0.0199289	−	0.0218022i
$628$			7.88201e6i			0.797513i
$629$			1.88452e6i			0.189922i
$630$	−437899.	+	4.86763e6i	−0.0439564	+	0.488615i
$631$		−	9.24226e6i		−	0.924070i	−0.886862	−	0.462035i	$-0.847120\pi$
$631$		−	9.24226e6i		−	0.924070i	0.886862	−	0.462035i	$-0.152880\pi$
$632$	−5.24191e6			−0.522032
$633$	8.24378e6	−	7.53545e6i	0.817743	−	0.747480i
$634$	2.68459e6			0.265249
$635$	1.87655e7			1.84683
$636$	−1.34328e7	+	1.22786e7i	−1.31681	+	1.20366i
$637$	1.73713e7			1.69622
$638$	−144672.			−0.0140713
$639$	2.84920e6	+	256318.i	0.276039	+	0.0248329i
$640$			1.17394e7i			1.13292i
$641$	−8.97691e6			−0.862942			−0.431471	−	0.902127i	$-0.642005\pi$
$641$	−8.97691e6			−0.862942			−0.431471	+	0.902127i	$0.642005\pi$
$642$	806038.	+	881805.i	0.0771823	+	0.0844374i
$643$		−	5.63467e6i		−	0.537453i	−0.963216	−	0.268727i	$-0.913397\pi$
$643$		−	5.63467e6i		−	0.537453i	0.963216	−	0.268727i	$-0.0866029\pi$
$644$	9.02138e6	−	1.21930e7i	0.857153	−	1.15850i
$645$	−8.56792e6	−	9.37330e6i	−0.810916	−	0.887142i
$646$	198980.			0.0187598
$647$			1.01191e7i			0.950349i	0.879892	+	0.475174i	$0.157615\pi$
$647$			1.01191e7i			0.950349i	−0.879892	+	0.475174i	$0.842385\pi$
$648$	838878.	−	4.62470e6i	0.0784804	−	0.432660i
$649$		−	1.03645e6i		−	0.0965913i
$650$		−	3.31849e6i		−	0.308075i
$651$	7.76139e6	+	8.49096e6i	0.717773	+	0.785244i
$652$	6.79198e6			0.625717
$653$			4.41505e6i			0.405184i	0.979263	+	0.202592i	$0.0649365\pi$
$653$			4.41505e6i			0.405184i	−0.979263	+	0.202592i	$0.935063\pi$
$654$	−2.23964e6	+	2.04721e6i	−0.204755	+	0.187162i
$655$		−	4.01300e6i		−	0.365482i
$656$		−	4.44077e6i		−	0.402901i
$657$	1.06135e6	−	1.17979e7i	0.0959281	−	1.06633i
$658$			554533.i			0.0499301i
$659$	−7.07020e6			−0.634188			−0.317094	−	0.948394i	$-0.602707\pi$
$659$	−7.07020e6			−0.634188			−0.317094	+	0.948394i	$0.602707\pi$
$660$	−584397.	+	534184.i	−0.0522214	+	0.0477344i
$661$			1.36132e6i			0.121187i	0.998163	+	0.0605937i	$0.0192994\pi$
$661$			1.36132e6i			0.121187i	−0.998163	+	0.0605937i	$0.980701\pi$
$662$			1.99323e6i			0.176772i
$663$	1.59048e6	−	1.45382e6i	0.140522	−	0.128448i
$664$		−	4.26474e6i		−	0.375381i
$665$	−1.40729e7			−1.23404
$666$	299897.	−	3.33362e6i	0.0261991	−	0.291226i
$667$	−8.18913e6	+	1.10682e7i	−0.712727	+	0.963299i
$668$		−	1.92882e7i		−	1.67244i
$669$	−1.67442e6	+	1.53055e6i	−0.144644	+	0.132216i
$670$	6.24515e6			0.537472
$671$		−	602846.i		−	0.0516892i
$672$	−8.28464e6	+	7.57280e6i	−0.707702	+	0.646894i
$673$	1.60775e7			1.36830			0.684148	−	0.729344i	$-0.260174\pi$
$673$	1.60775e7			1.36830			0.684148	+	0.729344i	$0.260174\pi$
$674$	−2.52489e6			−0.214089
$675$	−7.53774e6	−	9.89859e6i	−0.636769	−	0.836207i
$676$	7.75870e6			0.653014
$677$	9.18709e6			0.770382			0.385191	−	0.922837i	$-0.374135\pi$
$677$	9.18709e6			0.770382			0.385191	+	0.922837i	$0.374135\pi$
$678$	346678.	+	379265.i	0.0289636	+	0.0316861i
$679$	−1.17349e7			−0.976795
$680$		−	1.11268e6i		−	0.0922782i
$681$	6.26848e6	−	5.72988e6i	0.517958	−	0.473454i
$682$		−	99937.4i		−	0.00822749i
$683$			3.89239e6i			0.319275i	0.987176	+	0.159637i	$0.0510325\pi$
$683$			3.89239e6i			0.319275i	−0.987176	+	0.159637i	$0.948967\pi$
$684$	6.56383e6	+	590491.i	0.536435	+	0.0482584i
$685$	−2.77103e7			−2.25639
$686$	1.29008e6			0.104666
$687$	−18242.3	−	19957.0i	−0.00147464	−	0.00161326i
$688$		−	8.85577e6i		−	0.713272i
$689$	3.04319e7			2.44220
$690$	−417641.	−	4.01902e6i	−0.0333949	−	0.321364i
$691$	1.11648e7			0.889518			0.444759	−	0.895650i	$-0.353289\pi$
$691$	1.11648e7			0.889518			0.444759	+	0.895650i	$0.353289\pi$
$692$			8.32013e6i			0.660488i
$693$	−995169.	−	89526.8i	−0.0787162	−	0.00708141i
$694$	−1.75679e6			−0.138459
$695$	470859.			0.0369768
$696$	4.97031e6	−	4.54324e6i	0.388920	−	0.355503i
$697$			890934.i			0.0694646i
$698$		−	2.16019e6i		−	0.167824i
$699$	3.25512e6	+	3.56110e6i	0.251985	+	0.275671i
$700$			1.96368e7i			1.51470i
$701$	−4.27810e6			−0.328818			−0.164409	−	0.986392i	$-0.552572\pi$
$701$	−4.27810e6			−0.328818			−0.164409	+	0.986392i	$0.552572\pi$
$702$	−3.04483e6	+	2.31862e6i	−0.233195	+	0.177577i
$703$	9.63790e6			0.735519
$704$	−484230.			−0.0368231
$705$	−1.85865e6	−	2.03336e6i	−0.140840	−	0.154078i
$706$	−3.33775e6			−0.252024
$707$	1.60541e6			0.120792
$708$	1.58493e7	+	1.73391e7i	1.18830	+	1.30000i
$709$			4.55553e6i			0.340348i	0.985414	+	0.170174i	$0.0544330\pi$
$709$			4.55553e6i			0.340348i	−0.985414	+	0.170174i	$0.945567\pi$
$710$	−1.20281e6			−0.0895470
$711$	1.59384e7	+	1.43384e6i	1.18242	+	0.106372i
$712$			8.50844e6i			0.628999i
$713$	−7.64571e6	−	5.65692e6i	−0.563241	−	0.416732i
$714$	504692.	−	461327.i	0.0370494	−	0.0338660i
$715$	1.32395e6			0.0968515
$716$		−	3.06535e6i		−	0.223459i
$717$	1.21052e7	+	1.32431e7i	0.879375	+	0.962036i
$718$		−	4.56909e6i		−	0.330764i
$719$			1.76274e6i			0.127164i	0.997977	+	0.0635821i	$0.0202525\pi$
$719$			1.76274e6i			0.127164i	−0.997977	+	0.0635821i	$0.979748\pi$
$720$	1.51703e6	−	1.68632e7i	0.109060	−	1.21229i
$721$	−1.06572e7			−0.763495
$722$			2.14234e6i			0.152949i
$723$	−1.63838e7	−	1.79238e7i	−1.16565	−	1.27522i
$724$			1.05540e7i			0.748294i
$725$		−	1.78253e7i		−	1.25948i
$726$	−2.15579e6	−	2.35843e6i	−0.151797	−	0.166066i
$727$		−	222075.i		−	0.0155834i	−0.999970	−	0.00779172i	$-0.997520\pi$
$727$		−	222075.i		−	0.0155834i	0.999970	−	0.00779172i	$-0.00248021\pi$
$728$	1.24046e7			0.867468
$729$	−3.81568e6	+	1.38323e7i	−0.265921	+	0.963995i
$730$			4.98055e6i			0.345915i
$731$			1.77670e6i			0.122976i
$732$	9.21861e6	+	1.00852e7i	0.635898	+	0.695673i
$733$			3.03919e6i			0.208929i	0.994529	+	0.104464i	$0.0333128\pi$
$733$			3.03919e6i			0.208929i	−0.994529	+	0.104464i	$0.966687\pi$
$734$	4.01962e6			0.275388
$735$	−2.02124e7	+	1.84757e7i	−1.38007	+	1.26149i
$736$	5.51946e6	−	7.45993e6i	0.375580	−	0.507622i
$737$			1.27680e6i			0.0865872i
$738$	141781.	−	1.57602e6i	0.00958244	−	0.106517i
$739$	1.00967e7			0.680096			0.340048	−	0.940408i	$-0.389557\pi$
$739$	1.00967e7			0.680096			0.340048	+	0.940408i	$0.389557\pi$
$740$		−	2.62435e7i		−	1.76174i
$741$	−7.43517e6	−	8.13407e6i	−0.497445	−	0.544205i
$742$	9.65669e6			0.643900
$743$	5.27625e6			0.350633			0.175317	−	0.984512i	$-0.443905\pi$
$743$	5.27625e6			0.350633			0.175317	+	0.984512i	$0.443905\pi$
$744$	3.13840e6	+	3.43341e6i	0.207863	+	0.227402i
$745$	1.23126e7			0.812753
$746$	2.57621e6			0.169486
$747$	−1.16655e6	+	1.29673e7i	−0.0764896	+	0.850250i
$748$	110772.			0.00723895
$749$			1.18214e7i			0.769950i
$750$	171446.	+	187562.i	0.0111295	+	0.0121756i
$751$		−	8.55220e6i		−	0.553322i	−0.960968	−	0.276661i	$-0.910772\pi$
$751$		−	8.55220e6i		−	0.553322i	0.960968	−	0.276661i	$-0.0892279\pi$
$752$		−	1.92109e6i		−	0.123881i
$753$	−256474.	+	234437.i	−0.0164838	+	0.0150674i
$754$	−5.48309e6			−0.351234
$755$	−7.22560e6			−0.461325
$756$	1.80175e7	−	1.37202e7i	1.14654	−	0.873086i
$757$			2.04752e7i			1.29864i	0.760516	+	0.649320i	$0.224946\pi$
$757$			2.04752e7i			1.29864i	−0.760516	+	0.649320i	$0.775054\pi$
$758$	−2.16589e6			−0.136919
$759$	821674.	−	85385.2i	0.0517720	−	0.00537995i
$760$	−5.69053e6			−0.357370
$761$		−	1.69707e6i		−	0.106228i	−0.998588	−	0.0531140i	$-0.983085\pi$
$761$		−	1.69707e6i		−	0.106228i	0.998588	−	0.0531140i	$-0.0169147\pi$
$762$	3.14607e6	+	3.44179e6i	0.196282	+	0.214732i
$763$	−3.00243e7			−1.86708
$764$	2.64044e7			1.63660
$765$	−304356.	+	3.38319e6i	−0.0188031	+	0.209013i
$766$			2.39941e6i			0.147751i
$767$		−	3.92816e7i		−	2.41102i
$768$	6.38208e6	−	5.83371e6i	0.390444	−	0.356896i
$769$		−	8.50809e6i		−	0.518819i	−0.965767	−	0.259410i	$-0.916472\pi$
$769$		−	8.50809e6i		−	0.518819i	0.965767	−	0.259410i	$-0.0835280\pi$
$770$	420117.			0.0255355
$771$	−1.45806e7	−	1.59511e7i	−0.883361	−	0.966397i
$772$	6.14349e6			0.370998
$773$	−1.74244e7			−1.04884			−0.524420	−	0.851460i	$-0.675718\pi$
$773$	−1.74244e7			−1.04884			−0.524420	+	0.851460i	$0.675718\pi$
$774$	282738.	−	3.14289e6i	0.0169642	−	0.188572i
$775$	1.23134e7			0.736418
$776$	−4.74512e6			−0.282874
$777$	2.44455e7	−	2.23450e7i	1.45260	−	1.32779i
$778$		−	1.58208e6i		−	0.0937087i
$779$	4.55645e6			0.269019
$780$	−2.21487e7	+	2.02456e7i	−1.30350	+	1.19150i
$781$		−	245910.i		−	0.0144261i
$782$	−336240.	+	454451.i	−0.0196622	+	0.0265748i
$783$	−1.63553e7	+	1.24545e7i	−0.953354	+	0.725976i
$784$	−1.90964e7			−1.10959
$785$			2.07772e7i			1.20341i
$786$	736025.	−	672784.i	0.0424949	−	0.0388436i
$787$			2.50321e7i			1.44066i	0.693634	+	0.720328i	$0.256009\pi$
$787$			2.50321e7i			1.44066i	−0.693634	+	0.720328i	$0.743991\pi$
$788$			2.46886e7i			1.41638i
$789$	−1.91345e7	+	1.74904e7i	−1.09427	+	1.00025i
$790$	−6.72850e6			−0.383575
$791$			5.08438e6i			0.288933i
$792$	−402407.	−	36201.1i	−0.0227957	−	0.00205073i
$793$		−	2.28479e7i		−	1.29022i
$794$			710199.i			0.0399787i
$795$	−3.54091e7	+	3.23667e7i	−1.98700	+	1.81627i
$796$		−	1.23069e6i		−	0.0688440i
$797$	2.04001e7			1.13759			0.568797	−	0.822478i	$-0.307409\pi$
$797$	2.04001e7			1.13759			0.568797	+	0.822478i	$0.307409\pi$
$798$	−2.35934e6	−	2.58111e6i	−0.131154	−	0.143483i
$799$			385422.i			0.0213584i
$800$			1.20142e7i			0.663698i
$801$	2.32734e6	−	2.58705e7i	0.128168	−	1.42470i
$802$			3.37400e6i			0.185229i
$803$	−1.01825e6			−0.0557272
$804$	−1.95246e7	−	2.13599e7i	−1.06523	−	1.16536i
$805$	2.37806e7	−	3.21411e7i	1.29340	−	1.74812i
$806$		−	3.78763e6i		−	0.205367i
$807$	1.68510e7	+	1.84350e7i	0.910839	+	0.996457i
$808$	649166.			0.0349806
$809$		−	1.70605e7i		−	0.916475i	−0.888830	−	0.458238i	$-0.848481\pi$
$809$		−	1.70605e7i		−	0.916475i	0.888830	−	0.458238i	$-0.151519\pi$
$810$	1.07678e6	−	5.93626e6i	0.0576654	−	0.317907i
$811$	−2.15503e7			−1.15054			−0.575269	−	0.817964i	$-0.695103\pi$
$811$	−2.15503e7			−1.15054			−0.575269	+	0.817964i	$0.695103\pi$
$812$	3.24457e7			1.72690
$813$	−2.94084e6	+	2.68816e6i	−0.156043	+	0.142636i
$814$	−287720.			−0.0152198
$815$	1.79039e7			0.944175
$816$	−1.74843e6	+	1.59820e6i	−0.0919226	+	0.0840243i
$817$	9.08646e6			0.476255
$818$		−	5.02946e6i		−	0.262808i
$819$	−3.77170e7	−	3.39307e6i	−1.96484	−	0.176760i
$820$		−	1.24070e7i		−	0.644365i
$821$			1.63166e7i			0.844834i	0.906402	+	0.422417i	$0.138818\pi$
$821$			1.63166e7i			0.844834i	−0.906402	+	0.422417i	$0.861182\pi$
$822$	−4.64566e6	−	5.08235e6i	−0.239810	−	0.262352i
$823$	1.20206e7			0.618624			0.309312	−	0.950961i	$-0.399901\pi$
$823$	1.20206e7			0.618624			0.309312	+	0.950961i	$0.399901\pi$
$824$	−4.30937e6			−0.221103
$825$	−789417.	+	721588.i	−0.0403805	+	0.0369109i
$826$		−	1.24649e7i		−	0.635680i
$827$	−2.90002e7			−1.47447			−0.737236	−	0.675635i	$-0.763869\pi$
$827$	−2.90002e7			−1.47447			−0.737236	+	0.675635i	$0.763869\pi$
$828$	−1.24403e7	+	1.39933e7i	−0.630601	+	0.709324i
$829$	4.57153e6			0.231034			0.115517	−	0.993306i	$-0.463148\pi$
$829$	4.57153e6			0.231034			0.115517	+	0.993306i	$0.463148\pi$
$830$		−	5.47421e6i		−	0.275820i
$831$	−1.06140e7	+	9.70203e6i	−0.533184	+	0.487371i
$832$	−1.83523e7			−0.919143
$833$	3.83124e6			0.191305
$834$	78940.1	+	86360.5i	0.00392991	+	0.00429932i
$835$		−	5.08443e7i		−	2.52363i
$836$		−	566514.i		−	0.0280346i
$837$	−8.60338e6	−	1.12980e7i	−0.424478	−	0.557426i
$838$			3.36587e6i			0.165572i
$839$	−1.67247e7			−0.820263			−0.410132	−	0.912026i	$-0.634517\pi$
$839$	−1.67247e7			−0.820263			−0.410132	+	0.912026i	$0.634517\pi$
$840$	−1.44334e7	+	1.31932e7i	−0.705781	+	0.645139i
$841$	−8.94131e6			−0.435924
$842$	765332.			0.0372023
$843$	−1.14165e7	+	1.04356e7i	−0.553306	+	0.505765i
$844$	2.17605e7			1.05151
$845$	2.04522e7			0.985365
$846$	61334.8	−	681791.i	0.00294633	−	0.0327511i
$847$		−	3.16168e7i		−	1.51429i
$848$	−3.34541e7			−1.59757
$849$	3.07541e6	+	3.36450e6i	0.146431	+	0.160196i
$850$		−	731894.i		−	0.0347457i
$851$	−1.62863e7	+	2.20120e7i	−0.770899	+	1.04192i
$852$	3.76041e6	+	4.11389e6i	0.177475	+	0.194157i
$853$	−1.92985e7			−0.908137			−0.454068	−	0.890967i	$-0.650028\pi$
$853$	−1.92985e7			−0.908137			−0.454068	+	0.890967i	$0.650028\pi$
$854$		−	7.25012e6i		−	0.340174i
$855$	1.73024e7	+	1.55655e6i	0.809454	+	0.0728196i
$856$			4.78009e6i			0.222973i
$857$		−	1.36216e7i		−	0.633545i	−0.948502	−	0.316772i	$-0.897401\pi$
$857$		−	1.36216e7i		−	0.633545i	0.948502	−	0.316772i	$-0.102599\pi$
$858$	221962.	+	242826.i	0.0102934	+	0.0112610i
$859$	2.08230e7			0.962854			0.481427	−	0.876486i	$-0.340119\pi$
$859$	2.08230e7			0.962854			0.481427	+	0.876486i	$0.340119\pi$
$860$		−	2.47420e7i		−	1.14075i
$861$	1.15569e7	−	1.05639e7i	0.531294	−	0.485643i
$862$			8.77635e6i			0.402296i
$863$			3.02999e7i			1.38489i	0.721473	+	0.692443i	$0.243465\pi$
$863$			3.02999e7i			1.38489i	−0.721473	+	0.692443i	$0.756535\pi$
$864$	1.10235e7	−	8.39432e6i	0.502381	−	0.382562i
$865$			2.19321e7i			0.996643i
$866$	8.59631e6			0.389509
$867$	−1.59860e7	+	1.46124e7i	−0.722257	+	0.660199i
$868$			2.24130e7i			1.00972i
$869$		−	1.37562e6i		−	0.0617943i
$870$	6.37987e6	−	5.83170e6i	0.285768	−	0.261214i
$871$			4.83907e7i			2.16131i
$872$	−1.21407e7			−0.540694
$873$	1.44279e7	+	1.29795e6i	0.640717	+	0.0576398i
$874$	2.32417e6	+	1.71961e6i	0.102918	+	0.0761469i
$875$			2.51442e6i			0.111024i
$876$	1.70346e7	−	1.55710e7i	0.750020	−	0.685576i
$877$	6.18389e6			0.271496			0.135748	−	0.990743i	$-0.456656\pi$
$877$	6.18389e6			0.271496			0.135748	+	0.990743i	$0.456656\pi$
$878$			2.24053e6i			0.0980876i
$879$	1.73593e6	−	1.58678e6i	0.0757811	−	0.0692698i
$880$	−1.45543e6			−0.0633557
$881$	−2.91955e7			−1.26729			−0.633645	−	0.773624i	$-0.718442\pi$
$881$	−2.91955e7			−1.26729			−0.633645	+	0.773624i	$0.718442\pi$
$882$	−6.77727e6	−	609692.i	−0.293348	−	0.0263900i
$883$	−3.67569e7			−1.58649			−0.793244	−	0.608905i	$-0.791609\pi$
$883$	−3.67569e7			−1.58649			−0.793244	+	0.608905i	$0.791609\pi$
$884$	4.19826e6			0.180692
$885$	4.17791e7	+	4.57063e7i	1.79308	+	1.96163i
$886$	−9.61617e6			−0.411546
$887$		−	3.76607e7i		−	1.60724i	−0.595145	−	0.803618i	$-0.702906\pi$
$887$		−	3.76607e7i		−	1.60724i	0.595145	−	0.803618i	$-0.297094\pi$
$888$	9.88478e6	−	9.03545e6i	0.420663	−	0.384519i
$889$			4.61402e7i			1.95806i
$890$			1.09214e7i			0.462172i
$891$	1.21365e6	+	220144.i	0.0512151	+	0.00928993i
$892$	−4.41985e6			−0.185992
$893$	1.97114e6			0.0827158
$894$	2.06422e6	+	2.25826e6i	0.0863798	+	0.0944995i
$895$		−	8.08035e6i		−	0.337188i
$896$	−2.88646e7			−1.20115
$897$	3.11415e7	−	3.23610e6i	1.29228	−	0.134289i
$898$	7.48951e6			0.309929
$899$		−	2.03453e7i		−	0.839585i
$900$	2.17196e6	−	2.41432e7i	0.0893810	−	0.993549i
$901$	6.71177e6			0.275439
$902$	−136024.			−0.00556670
$903$	2.30468e7	−	2.10666e7i	0.940571	−	0.859754i
$904$			2.05592e6i			0.0836731i
$905$			2.78207e7i			1.12914i
$906$	−1.21138e6	−	1.32525e6i	−0.0490298	−	0.0536386i
$907$		−	1.30537e7i		−	0.526885i	−0.964675	−	0.263443i	$-0.915142\pi$
$907$		−	1.30537e7i		−	0.526885i	0.964675	−	0.263443i	$-0.0848579\pi$
$908$	1.65464e7			0.666024
$909$	−1.97384e6	−	177569.i	−0.0792321	−	0.00712783i
$910$	1.59225e7			0.637393
$911$	2.20067e7			0.878534			0.439267	−	0.898357i	$-0.355238\pi$
$911$	2.20067e7			0.878534			0.439267	+	0.898357i	$0.355238\pi$
$912$	8.17356e6	+	8.94188e6i	0.325405	+	0.355993i
$913$	1.11918e6			0.0444349
$914$	6.82359e6			0.270176
$915$	2.43005e7	+	2.65847e7i	0.959539	+	1.04974i
$916$		−	52679.1i		−	0.00207443i
$917$	9.86705e6			0.387493
$918$	−671538.	+	511374.i	−0.0263005	+	0.0200277i
$919$		−	6.53274e6i		−	0.255156i	−0.991829	−	0.127578i	$-0.959280\pi$
$919$		−	6.53274e6i		−	0.255156i	0.991829	−	0.127578i	$-0.0407204\pi$
$920$	9.61594e6	−	1.29966e7i	0.374561	−	0.506244i
$921$	2.19998e7	−	2.01095e7i	0.854614	−	0.781183i
$922$	−7.41073e6			−0.287100
$923$		−	9.32000e6i		−	0.360091i
$924$	−1.31344e6	−	1.43690e6i	−0.0506092	−	0.0553664i
$925$		−	3.54503e7i		−	1.36228i
$926$		−	5.78883e6i		−	0.221852i
$927$	1.31029e7	+	1.17876e6i	0.500806	+	0.0450531i
$928$	1.98509e7			0.756677
$929$			1.39556e7i			0.530529i	0.964176	+	0.265265i	$0.0854593\pi$
$929$			1.39556e7i			0.530529i	−0.964176	+	0.265265i	$0.914541\pi$
$930$	4.02844e6	+	4.40712e6i	0.152732	+	0.167089i
$931$		−	1.95939e7i		−	0.740877i
$932$			9.39998e6i			0.354476i
$933$	−1.91963e7	−	2.10008e7i	−0.721962	−	0.789826i
$934$			2.76299e6i			0.103636i
$935$	291998.			0.0109232
$936$	−1.52513e7	−	1.37202e6i	−0.569005	−	0.0511884i
$937$			4.24419e7i			1.57923i	0.613602	+	0.789615i	$0.289720\pi$
$937$			4.24419e7i			1.57923i	−0.613602	+	0.789615i	$0.710280\pi$
$938$			1.53554e7i			0.569842i
$939$	−2.99870e7	−	3.28058e7i	−1.10986	−	1.21419i
$940$		−	5.36731e6i		−	0.198124i
$941$	−3.11592e6			−0.114713			−0.0573565	−	0.998354i	$-0.518267\pi$
$941$	−3.11592e6			−0.114713			−0.0573565	+	0.998354i	$0.518267\pi$
$942$	−3.81075e6	+	3.48332e6i	−0.139921	+	0.127899i
$943$	−7.69956e6	+	1.04065e7i	−0.281960	+	0.381087i
$944$			4.31827e7i			1.57718i
$945$	4.74945e7	−	3.61669e7i	1.73007	−	1.31744i
$946$	−271258.			−0.00985495
$947$			1.39901e7i			0.506929i	0.967345	+	0.253464i	$0.0815701\pi$
$947$			1.39901e7i			0.506929i	−0.967345	+	0.253464i	$0.918430\pi$
$948$	2.10357e7	+	2.30131e7i	0.760215	+	0.831675i
$949$	−3.85919e7			−1.39101
$950$	−3.74308e6			−0.134561
$951$	−2.21241e7	−	2.42038e7i	−0.793259	−	0.867825i
$952$	2.73583e6			0.0978357
$953$	−7.04777e6			−0.251373			−0.125687	−	0.992070i	$-0.540113\pi$
$953$	−7.04777e6			−0.251373			−0.125687	+	0.992070i	$0.540113\pi$
$954$	−1.18728e7	−	1.06809e6i	−0.422358	−	0.0379959i
$955$	6.96027e7			2.46955
$956$			3.49568e7i			1.23705i
$957$	1.19227e6	+	1.30434e6i	0.0420818	+	0.0460375i
$958$			5.91120e6i			0.208095i
$959$		−	6.81332e7i		−	2.39228i
$960$	2.13539e7	−	1.95191e7i	0.747825	−	0.683570i
$961$	−1.45749e7			−0.509094
$962$	−1.09046e7			−0.379902
$963$	1.30752e6	−	1.45342e7i	0.0454340	−	0.505040i
$964$		−	4.73122e7i		−	1.63976i
$965$	1.61944e7			0.559818
$966$	9.88185e6	−	1.02688e6i	0.340718	−	0.0354061i
$967$	−2.59611e7			−0.892804			−0.446402	−	0.894832i	$-0.647295\pi$
$967$	−2.59611e7			−0.892804			−0.446402	+	0.894832i	$0.647295\pi$
$968$		−	1.27846e7i		−	0.438528i
$969$	−1.63983e6	−	1.79397e6i	−0.0561034	−	0.0613771i
$970$	−6.09082e6			−0.207848
$971$	5.77704e6			0.196633			0.0983167	−	0.995155i	$-0.468654\pi$
$971$	5.77704e6			0.196633			0.0983167	+	0.995155i	$0.468654\pi$
$972$	−2.36698e7	+	1.48760e7i	−0.803579	+	0.505034i
$973$			1.15774e6i			0.0392037i
$974$			1.21319e7i			0.409760i
$975$	−2.99189e7	+	2.73482e7i	−1.00794	+	0.921334i
$976$			2.51169e7i			0.843999i
$977$	7.49681e6			0.251270			0.125635	−	0.992077i	$-0.459903\pi$
$977$	7.49681e6			0.251270			0.125635	+	0.992077i	$0.459903\pi$
$978$	3.00160e6	+	3.28375e6i	0.100347	+	0.109780i
$979$	−2.23284e6			−0.0744562
$980$	−5.33532e7			−1.77458
$981$	3.69145e7	+	3.32088e6i	1.22469	+	0.110174i
$982$	5.22928e6			0.173046
$983$	2.54694e7			0.840688			0.420344	−	0.907365i	$-0.361909\pi$
$983$	2.54694e7			0.840688			0.420344	+	0.907365i	$0.361909\pi$
$984$	4.67317e6	−	4.27164e6i	0.153859	−	0.140639i
$985$			6.50798e7i			2.13725i
$986$	−1.20930e6			−0.0396133
$987$	4.99957e6	−	4.57000e6i	0.163358	−	0.149322i
$988$		−	2.14709e7i		−	0.699774i
$989$	−1.53544e7	+	2.07526e7i	−0.499164	+	0.674654i
$990$	−516529.	−	46467.6i	−0.0167497	−	0.00150682i
$991$	−4.73562e7			−1.53177			−0.765884	−	0.642979i	$-0.777698\pi$
$991$	−4.73562e7			−1.53177			−0.765884	+	0.642979i	$0.777698\pi$
$992$			1.37127e7i			0.442429i
$993$	1.79706e7	−	1.64265e7i	0.578349	−	0.528656i
$994$		−	2.95744e6i		−	0.0949400i
$995$		−	3.24413e6i		−	0.103882i
$996$	−1.87231e7	+	1.71143e7i	−0.598038	+	0.546653i
$997$	−2.11118e7			−0.672648			−0.336324	−	0.941746i	$-0.609184\pi$
$997$	−2.11118e7			−0.672648			−0.336324	+	0.941746i	$0.609184\pi$
$998$		−	1841.87i		−	5.85372e-5i
$999$	−3.25269e7	+	2.47691e7i	−1.03117	+	0.785229i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.c.b.68.18	yes	32
3.2	odd	2	inner	69.6.c.b.68.15	✓	32
23.22	odd	2	inner	69.6.c.b.68.17	yes	32
69.68	even	2	inner	69.6.c.b.68.16	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.c.b.68.15	✓	32	3.2	odd	2	inner
69.6.c.b.68.16	yes	32	69.68	even	2	inner
69.6.c.b.68.17	yes	32	23.22	odd	2	inner
69.6.c.b.68.18	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)

\(f(q)\)	\(=\)	\(q+1.27619i q^{2} +(11.5059 - 10.5173i) q^{3} +30.3713 q^{4} +80.0597 q^{5} +(13.4221 + 14.6838i) q^{6} +196.848i q^{7} +79.5978i q^{8} +(21.7727 - 242.023i) q^{9} +O(q^{10})\)	\(q+1.27619i q^{2} +(11.5059 - 10.5173i) q^{3} +30.3713 q^{4} +80.0597 q^{5} +(13.4221 + 14.6838i) q^{6} +196.848i q^{7} +79.5978i q^{8} +(21.7727 - 242.023i) q^{9} +102.171i q^{10} -20.8886 q^{11} +(349.450 - 319.425i) q^{12} -791.678 q^{13} -251.216 q^{14} +(921.160 - 842.012i) q^{15} +870.301 q^{16} -174.605 q^{17} +(308.867 + 27.7861i) q^{18} +892.972i q^{19} +2431.52 q^{20} +(2070.32 + 2264.92i) q^{21} -26.6578i q^{22} +(-2039.46 - 1508.96i) q^{23} +(837.154 + 915.846i) q^{24} +3284.55 q^{25} -1010.33i q^{26} +(-2294.91 - 3013.68i) q^{27} +5978.55i q^{28} -5427.01i q^{29} +(1074.57 + 1175.58i) q^{30} +3748.89 q^{31} +3657.80i q^{32} +(-240.343 + 219.692i) q^{33} -222.829i q^{34} +15759.6i q^{35} +(661.265 - 7350.55i) q^{36} -10793.1i q^{37} -1139.60 q^{38} +(-9108.99 + 8326.32i) q^{39} +6372.57i q^{40} -5102.57i q^{41} +(-2890.48 + 2642.12i) q^{42} -10175.5i q^{43} -634.414 q^{44} +(1743.11 - 19376.2i) q^{45} +(1925.72 - 2602.74i) q^{46} -2207.39i q^{47} +(10013.6 - 9153.22i) q^{48} -21942.3 q^{49} +4191.71i q^{50} +(-2008.99 + 1836.37i) q^{51} -24044.3 q^{52} -38439.7 q^{53} +(3846.04 - 2928.75i) q^{54} -1672.33 q^{55} -15668.7 q^{56} +(9391.65 + 10274.5i) q^{57} +6925.91 q^{58} +49618.2i q^{59} +(27976.9 - 25573.0i) q^{60} +28860.1i q^{61} +4784.31i q^{62} +(47641.8 + 4285.92i) q^{63} +23181.6 q^{64} -63381.5 q^{65} +(-280.369 - 306.723i) q^{66} -61124.2i q^{67} -5302.99 q^{68} +(-39336.0 + 4087.65i) q^{69} -20112.3 q^{70} +11772.5i q^{71} +(19264.5 + 1733.06i) q^{72} +48746.9 q^{73} +13774.0 q^{74} +(37791.8 - 34544.6i) q^{75} +27120.7i q^{76} -4111.89i q^{77} +(-10626.0 - 11624.8i) q^{78} +65855.0i q^{79} +69676.0 q^{80} +(-58100.9 - 10539.0i) q^{81} +6511.86 q^{82} -53578.7 q^{83} +(62878.2 + 68788.8i) q^{84} -13978.8 q^{85} +12985.9 q^{86} +(-57077.5 - 62442.8i) q^{87} -1662.69i q^{88} +106893. q^{89} +(24727.8 + 2224.55i) q^{90} -155841. i q^{91} +(-61941.1 - 45829.1i) q^{92} +(43134.5 - 39428.3i) q^{93} +2817.05 q^{94} +71491.0i q^{95} +(38470.2 + 42086.4i) q^{96} +59613.7i q^{97} -28002.6i q^{98} +(-454.800 + 5055.51i) 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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