LMFDB - Embedded newform 69.6.c.b.68.7

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			68.7
Character	$\chi$	$=$	69.68
Dual form			69.6.c.b.68.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.06404i q^{2} +(-15.4896 + 1.75248i) q^{3} -17.9007 q^{4} -82.3579 q^{5} +(12.3796 + 109.419i) q^{6} +96.3127i q^{7} -99.5980i q^{8} +(236.858 - 54.2905i) q^{9} +O(q^{10})$	\(q-7.06404i q^{2} +(-15.4896 + 1.75248i) q^{3} -17.9007 q^{4} -82.3579 q^{5} +(12.3796 + 109.419i) q^{6} +96.3127i q^{7} -99.5980i q^{8} +(236.858 - 54.2905i) q^{9} +581.779i q^{10} +87.8590 q^{11} +(277.275 - 31.3706i) q^{12} +303.810 q^{13} +680.357 q^{14} +(1275.69 - 144.330i) q^{15} -1276.39 q^{16} +460.998 q^{17} +(-383.510 - 1673.17i) q^{18} +1560.29i q^{19} +1474.26 q^{20} +(-168.786 - 1491.85i) q^{21} -620.640i q^{22} +(2440.41 + 693.361i) q^{23} +(174.543 + 1542.74i) q^{24} +3657.82 q^{25} -2146.13i q^{26} +(-3573.70 + 1256.03i) q^{27} -1724.07i q^{28} -4091.99i q^{29} +(-1019.56 - 9011.55i) q^{30} +1861.74 q^{31} +5829.32i q^{32} +(-1360.90 + 153.971i) q^{33} -3256.51i q^{34} -7932.11i q^{35} +(-4239.92 + 971.838i) q^{36} +11817.5i q^{37} +11021.9 q^{38} +(-4705.91 + 532.421i) q^{39} +8202.68i q^{40} +12179.1i q^{41} +(-10538.5 + 1192.31i) q^{42} +22461.2i q^{43} -1572.74 q^{44} +(-19507.1 + 4471.25i) q^{45} +(4897.93 - 17239.2i) q^{46} +2743.55i q^{47} +(19770.8 - 2236.84i) q^{48} +7530.86 q^{49} -25839.0i q^{50} +(-7140.70 + 807.889i) q^{51} -5438.42 q^{52} -11551.9 q^{53} +(8872.63 + 25244.7i) q^{54} -7235.88 q^{55} +9592.56 q^{56} +(-2734.37 - 24168.3i) q^{57} -28906.0 q^{58} +39947.0i q^{59} +(-22835.8 + 2583.61i) q^{60} -24443.4i q^{61} -13151.4i q^{62} +(5228.87 + 22812.4i) q^{63} +334.155 q^{64} -25021.2 q^{65} +(1087.66 + 9613.48i) q^{66} -41740.7i q^{67} -8252.20 q^{68} +(-39016.1 - 6463.15i) q^{69} -56032.8 q^{70} -33816.0i q^{71} +(-5407.23 - 23590.6i) q^{72} +63699.0 q^{73} +83479.5 q^{74} +(-56658.2 + 6410.24i) q^{75} -27930.2i q^{76} +8461.94i q^{77} +(3761.04 + 33242.8i) q^{78} -41670.6i q^{79} +105121. q^{80} +(53154.1 - 25718.2i) q^{81} +86034.0 q^{82} -72364.0 q^{83} +(3021.39 + 26705.2i) q^{84} -37966.8 q^{85} +158667. q^{86} +(7171.13 + 63383.5i) q^{87} -8750.58i q^{88} +9268.15 q^{89} +(31585.1 + 137799. i) q^{90} +29260.8i q^{91} +(-43685.0 - 12411.6i) q^{92} +(-28837.6 + 3262.65i) q^{93} +19380.5 q^{94} -128502. i q^{95} +(-10215.8 - 90294.0i) q^{96} +28465.2i q^{97} -53198.3i q^{98} +(20810.1 - 4769.91i) 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 $

$2$		−	7.06404i		−	1.24876i	−0.781121	−	0.624379i	$-0.785352\pi$
$2$		−	7.06404i		−	1.24876i	0.781121	−	0.624379i	$-0.214648\pi$
$3$	−15.4896	+	1.75248i	−0.993661	+	0.112422i
$3$	−15.4896	+	1.75248i	−0.993661	+	0.112422i
$4$	−17.9007			−0.559397
$5$	−82.3579			−1.47326			−0.736631	−	0.676295i	$-0.763585\pi$
$5$	−82.3579			−1.47326			−0.736631	+	0.676295i	$0.763585\pi$
$6$	12.3796	+	109.419i	0.140387	+	1.24084i
$7$			96.3127i			0.742914i	0.928450	+	0.371457i	$0.121142\pi$
$7$			96.3127i			0.742914i	−0.928450	+	0.371457i	$0.878858\pi$
$8$		−	99.5980i		−	0.550207i
$9$	236.858	−	54.2905i	0.974723	−	0.223418i
$10$			581.779i			1.83975i
$11$	87.8590			0.218930			0.109465	−	0.993991i	$-0.465086\pi$
$11$	87.8590			0.218930			0.109465	+	0.993991i	$0.465086\pi$
$12$	277.275	−	31.3706i	0.555851	−	0.0628883i
$13$	303.810			0.498591			0.249295	−	0.968428i	$-0.419801\pi$
$13$	303.810			0.498591			0.249295	+	0.968428i	$0.419801\pi$
$14$	680.357			0.927720
$15$	1275.69	−	144.330i	1.46392	−	0.165626i
$16$	−1276.39			−1.24647
$17$	460.998			0.386881			0.193440	−	0.981112i	$-0.438035\pi$
$17$	460.998			0.386881			0.193440	+	0.981112i	$0.438035\pi$
$18$	−383.510	−	1673.17i	−0.278995	−	1.21719i
$19$			1560.29i			0.991563i	0.868447	+	0.495781i	$0.165118\pi$
$19$			1560.29i			0.991563i	−0.868447	+	0.495781i	$0.834882\pi$
$20$	1474.26			0.824138
$21$	−168.786	−	1491.85i	−0.0835195	−	0.738205i
$22$		−	620.640i		−	0.273390i
$23$	2440.41	+	693.361i	0.961929	+	0.273300i
$23$	2440.41	+	693.361i	0.961929	+	0.273300i
$24$	174.543	+	1542.74i	0.0618551	+	0.546719i
$25$	3657.82			1.17050
$26$		−	2146.13i		−	0.622619i
$27$	−3573.70	+	1256.03i	−0.943427	+	0.331581i
$28$		−	1724.07i		−	0.415584i
$29$		−	4091.99i		−	0.903525i	−0.892138	−	0.451762i	$-0.850796\pi$
$29$		−	4091.99i		−	0.903525i	0.892138	−	0.451762i	$-0.149204\pi$
$30$	−1019.56	−	9011.55i	−0.206827	−	1.82809i
$31$	1861.74			0.347947			0.173974	−	0.984750i	$-0.444339\pi$
$31$	1861.74			0.347947			0.173974	+	0.984750i	$0.444339\pi$
$32$			5829.32i			1.00634i
$33$	−1360.90	+	153.971i	−0.217542	+	0.0246124i
$34$		−	3256.51i		−	0.483120i
$35$		−	7932.11i		−	1.09451i
$36$	−4239.92	+	971.838i	−0.545257	+	0.124979i
$37$			11817.5i			1.41913i	0.704640	+	0.709565i	$0.251109\pi$
$37$			11817.5i			1.41913i	−0.704640	+	0.709565i	$0.748891\pi$
$38$	11021.9			1.23822
$39$	−4705.91	+	532.421i	−0.495430	+	0.0560523i
$40$			8202.68i			0.810599i
$41$			12179.1i			1.13151i	0.824574	+	0.565754i	$0.191415\pi$
$41$			12179.1i			1.13151i	−0.824574	+	0.565754i	$0.808585\pi$
$42$	−10538.5	+	1192.31i	−0.921839	+	0.104296i
$43$			22461.2i			1.85252i	0.376888	+	0.926259i	$0.376994\pi$
$43$			22461.2i			1.85252i	−0.376888	+	0.926259i	$0.623006\pi$
$44$	−1572.74			−0.122469
$45$	−19507.1	+	4471.25i	−1.43602	+	0.329153i
$46$	4897.93	−	17239.2i	0.341286	−	1.20122i
$47$			2743.55i			0.181162i	0.995889	+	0.0905811i	$0.0288724\pi$
$47$			2743.55i			0.181162i	−0.995889	+	0.0905811i	$0.971128\pi$
$48$	19770.8	−	2236.84i	1.23857	−	0.140130i
$49$	7530.86			0.448079
$50$		−	25839.0i		−	1.46167i
$51$	−7140.70	+	807.889i	−0.384428	+	0.0434937i
$52$	−5438.42			−0.278910
$53$	−11551.9			−0.564891			−0.282445	−	0.959283i	$-0.591146\pi$
$53$	−11551.9			−0.564891			−0.282445	+	0.959283i	$0.591146\pi$
$54$	8872.63	+	25244.7i	0.414065	+	1.17811i
$55$	−7235.88			−0.322541
$56$	9592.56			0.408756
$57$	−2734.37	−	24168.3i	−0.111473	−	0.985277i
$58$	−28906.0			−1.12828
$59$			39947.0i			1.49401i	0.664817	+	0.747006i	$0.268509\pi$
$59$			39947.0i			1.49401i	−0.664817	+	0.747006i	$0.731491\pi$
$60$	−22835.8	+	2583.61i	−0.818914	+	0.0926509i
$61$		−	24443.4i		−	0.841079i	−0.907274	−	0.420540i	$-0.861841\pi$
$61$		−	24443.4i		−	0.841079i	0.907274	−	0.420540i	$-0.138159\pi$
$62$		−	13151.4i		−	0.434502i
$63$	5228.87	+	22812.4i	0.165980	+	0.724135i
$64$	334.155			0.0101976
$65$	−25021.2			−0.734555
$66$	1087.66	+	9613.48i	0.0307349	+	0.271657i
$67$		−	41740.7i		−	1.13598i	−0.823034	−	0.567992i	$-0.807720\pi$
$67$		−	41740.7i		−	1.13598i	0.823034	−	0.567992i	$-0.192280\pi$
$68$	−8252.20			−0.216420
$69$	−39016.1	−	6463.15i	−0.986556	−	0.163426i
$70$	−56032.8			−1.36677
$71$		−	33816.0i		−	0.796115i	−0.917360	−	0.398058i	$-0.869684\pi$
$71$		−	33816.0i		−	0.796115i	0.917360	−	0.398058i	$-0.130316\pi$
$72$	−5407.23	−	23590.6i	−0.122926	−	0.536299i
$73$	63699.0			1.39902			0.699512	−	0.714621i	$-0.253401\pi$
$73$	63699.0			1.39902			0.699512	+	0.714621i	$0.253401\pi$
$74$	83479.5			1.77215
$75$	−56658.2	+	6410.24i	−1.16308	+	0.131589i
$76$		−	27930.2i		−	0.554677i
$77$			8461.94i			0.162646i
$78$	3761.04	+	33242.8i	0.0699958	+	0.618672i
$79$		−	41670.6i		−	0.751210i	−0.926780	−	0.375605i	$-0.877435\pi$
$79$		−	41670.6i		−	0.751210i	0.926780	−	0.375605i	$-0.122565\pi$
$80$	105121.			1.83638
$81$	53154.1	−	25718.2i	0.900169	−	0.435541i
$82$	86034.0			1.41298
$83$	−72364.0			−1.15299			−0.576497	−	0.817099i	$-0.695581\pi$
$83$	−72364.0			−1.15299			−0.576497	+	0.817099i	$0.695581\pi$
$84$	3021.39	+	26705.2i	0.0467206	+	0.412949i
$85$	−37966.8			−0.569977
$86$	158667.			2.31335
$87$	7171.13	+	63383.5i	0.101576	+	0.897797i
$88$		−	8750.58i		−	0.120457i
$89$	9268.15			0.124027			0.0620137	−	0.998075i	$-0.480248\pi$
$89$	9268.15			0.124027			0.0620137	+	0.998075i	$0.480248\pi$
$90$	31585.1	+	137799.i	0.411032	+	1.79324i
$91$			29260.8i			0.370410i
$92$	−43685.0	−	12411.6i	−0.538100	−	0.152883i
$93$	−28837.6	+	3262.65i	−0.345742	+	0.0391168i
$94$	19380.5			0.226228
$95$		−	128502.i		−	1.46083i
$96$	−10215.8	−	90294.0i	−0.113134	−	0.999956i
$97$			28465.2i			0.307174i	0.988135	+	0.153587i	$0.0490825\pi$
$97$			28465.2i			0.307174i	−0.988135	+	0.153587i	$0.950918\pi$
$98$		−	53198.3i		−	0.559542i
$99$	20810.1	−	4769.91i	0.213396	−	0.0489127i
$100$	−65477.5			−0.654775
$101$			105391.i			1.02802i	0.857785	+	0.514008i	$0.171840\pi$
$101$			105391.i			1.02802i	−0.857785	+	0.514008i	$0.828160\pi$
$102$	5706.97	+	50442.2i	0.0543131	+	0.480058i
$103$			208125.i			1.93300i	0.256673	+	0.966498i	$0.417374\pi$
$103$			208125.i			1.93300i	−0.256673	+	0.966498i	$0.582626\pi$
$104$		−	30258.9i		−	0.274328i
$105$	13900.8	+	122866.i	0.123046	+	1.08757i
$106$			81603.3i			0.705412i
$107$	183616.			1.55043			0.775215	−	0.631698i	$-0.217642\pi$
$107$	183616.			1.55043			0.775215	+	0.631698i	$0.217642\pi$
$108$	63971.7	−	22483.8i	0.527750	−	0.185485i
$109$		−	114970.i		−	0.926868i	−0.886131	−	0.463434i	$-0.846617\pi$
$109$		−	114970.i		−	0.926868i	0.886131	−	0.463434i	$-0.153383\pi$
$110$			51114.6i			0.402775i
$111$	−20710.0	−	183049.i	−0.159541	−	1.41013i
$112$		−	122932.i		−	0.926022i
$113$	−186197.			−1.37175			−0.685876	−	0.727719i	$-0.740581\pi$
$113$	−186197.			−1.37175			−0.685876	+	0.727719i	$0.740581\pi$
$114$	−170726.	+	19315.7i	−1.23037	+	0.139203i
$115$	−200987.	−	57103.7i	−1.41717	−	0.402643i
$116$			73249.6i			0.505429i
$117$	71959.8	−	16494.0i	0.485988	−	0.111394i
$118$	282187.			1.86566
$119$			44400.0i			0.287419i
$120$	−14375.0	−	127057.i	−0.0911287	−	0.805460i
$121$	−153332.			−0.952070
$122$	−172669.			−1.05030
$123$	−21343.7	−	188651.i	−0.127206	−	1.12433i
$124$	−33326.4			−0.194641
$125$	−43881.6			−0.251193
$126$	161148.	−	36936.9i	0.904270	−	0.207269i
$127$	130830.			0.719777			0.359888	−	0.932995i	$-0.382815\pi$
$127$	130830.			0.719777			0.359888	+	0.932995i	$0.382815\pi$
$128$			184178.i			0.993601i
$129$	−39362.8	−	347916.i	−0.208263	−	1.84077i
$130$			176751.i			0.917281i
$131$		−	54027.3i		−	0.275065i	−0.990497	−	0.137532i	$-0.956083\pi$
$131$		−	54027.3i		−	0.275065i	0.990497	−	0.137532i	$-0.0439171\pi$
$132$	24361.1	−	2756.19i	0.121692	−	0.0137681i
$133$	−150275.			−0.736646
$134$	−294858.			−1.41857
$135$	294322.	−	103444.i	1.38991	−	0.488506i
$136$		−	45914.5i		−	0.212864i
$137$	−162442.			−0.739428			−0.369714	−	0.929146i	$-0.620544\pi$
$137$	−162442.			−0.739428			−0.369714	+	0.929146i	$0.620544\pi$
$138$	−45655.9	+	275612.i	−0.204080	+	1.23197i
$139$	142959.			0.627587			0.313794	−	0.949491i	$-0.398400\pi$
$139$	142959.			0.627587			0.313794	+	0.949491i	$0.398400\pi$
$140$			141990.i			0.612264i
$141$	−4808.00	−	42496.5i	−0.0203665	−	0.180014i
$142$	−238877.			−0.994156
$143$	26692.5			0.109156
$144$	−302322.	+	69295.7i	−1.21496	+	0.278484i
$145$			337008.i			1.33113i
$146$		−	449972.i		−	1.74704i
$147$	−116650.	+	13197.7i	−0.445238	+	0.0503737i
$148$		−	211542.i		−	0.793857i
$149$	−333252.			−1.22972			−0.614862	−	0.788635i	$-0.710788\pi$
$149$	−333252.			−1.22972			−0.614862	+	0.788635i	$0.710788\pi$
$150$	45282.2	+	400236.i	0.164323	+	1.45241i
$151$	510659.			1.82259			0.911295	−	0.411755i	$-0.135084\pi$
$151$	510659.			1.82259			0.911295	+	0.411755i	$0.135084\pi$
$152$	155401.			0.545564
$153$	109191.	−	25027.8i	0.377101	−	0.0864360i
$154$	59775.5			0.203105
$155$	−153329.			−0.512618
$156$	84239.1	−	9530.71i	0.277142	−	0.0313555i
$157$			335833.i			1.08736i	0.839292	+	0.543680i	$0.182970\pi$
$157$			335833.i			1.08736i	−0.839292	+	0.543680i	$0.817030\pi$
$158$	−294363.			−0.938080
$159$	178935.	−	20244.5i	0.561310	−	0.0635059i
$160$		−	480090.i		−	1.48260i
$161$	−66779.5	+	235042.i	−0.203039	+	0.714631i
$162$	−181675.	−	375483.i	−0.543885	−	1.12409i
$163$	371286.			1.09456			0.547279	−	0.836950i	$-0.315664\pi$
$163$	371286.			1.09456			0.547279	+	0.836950i	$0.315664\pi$
$164$		−	218015.i		−	0.632962i
$165$	112081.	−	12680.7i	0.320496	−	0.0362605i
$166$			511182.i			1.43981i
$167$			352931.i			0.979262i	0.871930	+	0.489631i	$0.162869\pi$
$167$			352931.i			0.979262i	−0.871930	+	0.489631i	$0.837131\pi$
$168$	−148585.	+	16810.7i	−0.406165	+	0.0459530i
$169$	−278992.			−0.751407
$170$			268199.i			0.711763i
$171$	84708.7	+	369566.i	0.221533	+	0.966499i
$172$		−	402072.i		−	1.03629i
$173$			522540.i			1.32741i	0.747995	+	0.663704i	$0.231017\pi$
$173$			522540.i			1.32741i	−0.747995	+	0.663704i	$0.768983\pi$
$174$	447744.	−	50657.2i	1.12113	−	0.126843i
$175$			352294.i			0.869582i
$176$	−112142.			−0.272890
$177$	−70006.2	−	618765.i	−0.167959	−	1.48454i
$178$		−	65470.6i		−	0.154880i
$179$		−	45906.4i		−	0.107088i	−0.998565	−	0.0535440i	$-0.982948\pi$
$179$		−	45906.4i		−	0.107088i	0.998565	−	0.0535440i	$-0.0170517\pi$
$180$	349191.	−	80038.5i	0.803306	−	0.184127i
$181$			401676.i			0.911337i	0.890149	+	0.455669i	$0.150600\pi$
$181$			401676.i			0.911337i	−0.890149	+	0.455669i	$0.849400\pi$
$182$	206700.			0.462553
$183$	42836.5	+	378619.i	0.0945554	+	0.835747i
$184$	69057.4	−	243060.i	0.150372	−	0.529260i
$185$		−	973266.i		−	2.09075i
$186$	23047.5	+	203710.i	0.0488474	+	0.431748i
$187$	40502.9			0.0846997
$188$		−	49111.4i		−	0.101342i
$189$	−120971.	−	344192.i	−0.246336	−	0.700885i
$190$	−907742.			−1.82423
$191$	598178.			1.18644			0.593222	−	0.805039i	$-0.297856\pi$
$191$	598178.			1.18644			0.593222	+	0.805039i	$0.297856\pi$
$192$	−5175.95	+	585.600i	−0.0101330	+	0.00114643i
$193$	−535462.			−1.03475			−0.517375	−	0.855759i	$-0.673091\pi$
$193$	−535462.			−1.03475			−0.517375	+	0.855759i	$0.673091\pi$
$194$	201079.			0.383586
$195$	387569.	−	43849.0i	0.729898	−	0.0825797i
$196$	−134808.			−0.250654
$197$			620923.i			1.13991i	0.821675	+	0.569957i	$0.193040\pi$
$197$			620923.i			1.13991i	−0.821675	+	0.569957i	$0.806960\pi$
$198$	−33694.8	−	147003.i	−0.0610802	−	0.266480i
$199$		−	679618.i		−	1.21656i	−0.793724	−	0.608278i	$-0.791861\pi$
$199$		−	679618.i		−	1.21656i	0.793724	−	0.608278i	$-0.208139\pi$
$200$		−	364311.i		−	0.644017i
$201$	73149.6	+	646548.i	0.127709	+	1.12878i
$202$	744486.			1.28374
$203$	394111.			0.671241
$204$	127823.	−	14461.8i	0.215048	−	0.0243303i
$205$		−	1.00305e6i		−	1.66701i
$206$	1.47020e6			2.41385
$207$	615672.	+	31736.9i	0.998674	+	0.0514800i
$208$	−387780.			−0.621479
$209$			137085.i			0.217083i
$210$	867927.	−	98196.2i	1.35811	−	0.153655i
$211$	−99148.1			−0.153313			−0.0766564	−	0.997058i	$-0.524424\pi$
$211$	−99148.1			−0.153313			−0.0766564	+	0.997058i	$0.524424\pi$
$212$	206788.			0.315998
$213$	59261.7	+	523797.i	0.0895005	+	0.791069i
$214$		−	1.29707e6i		−	1.93611i
$215$		−	1.84986e6i		−	2.72924i
$216$	125098.	+	355933.i	0.182438	+	0.519080i
$217$			179309.i			0.258495i
$218$	−812152.			−1.15743
$219$	−986674.	+	111631.i	−1.39016	+	0.157280i
$220$	129527.			0.180428
$221$	140056.			0.192895
$222$	−1.29307e6	+	146296.i	−1.76092	+	0.199228i
$223$	434908.			0.585646			0.292823	−	0.956167i	$-0.405405\pi$
$223$	434908.			0.585646			0.292823	+	0.956167i	$0.405405\pi$
$224$	−561438.			−0.747621
$225$	866382.	−	198585.i	1.14091	−	0.261511i
$226$			1.31530e6i			1.71299i
$227$	−947182.			−1.22003			−0.610013	−	0.792392i	$-0.708836\pi$
$227$	−947182.			−1.22003			−0.610013	+	0.792392i	$0.708836\pi$
$228$	48947.1	+	432629.i	0.0623577	+	0.551161i
$229$		−	47648.3i		−	0.0600425i	−0.999549	−	0.0300213i	$-0.990443\pi$
$229$		−	47648.3i		−	0.0600425i	0.999549	−	0.0300213i	$-0.00955750\pi$
$230$	−403383.	+	1.41978e6i	−0.502803	+	1.76971i
$231$	−14829.4	−	131072.i	−0.0182849	−	0.161615i
$232$	−407555.			−0.497125
$233$		−	292242.i		−	0.352657i	−0.984331	−	0.176328i	$-0.943578\pi$
$233$		−	292242.i		−	0.352657i	0.984331	−	0.176328i	$-0.0564221\pi$
$234$	−116514.	−	508327.i	−0.139104	−	0.606881i
$235$		−	225953.i		−	0.266899i
$236$		−	715079.i		−	0.835746i
$237$	73026.7	+	645462.i	0.0844522	+	0.746448i
$238$	313644.			0.358917
$239$			1.53813e6i			1.74180i	0.491459	+	0.870901i	$0.336464\pi$
$239$			1.53813e6i			1.74180i	−0.491459	+	0.870901i	$0.663536\pi$
$240$	−1.62828e6	+	184221.i	−1.82474	+	0.206449i
$241$		−	370725.i		−	0.411158i	−0.978641	−	0.205579i	$-0.934092\pi$
$241$		−	370725.i		−	0.411158i	0.978641	−	0.205579i	$-0.0659078\pi$
$242$			1.08314e6i			1.18890i
$243$	−778267.	+	491517.i	−0.845498	+	0.533978i
$244$			437554.i			0.470497i
$245$	−620225.			−0.660137
$246$	−1.33264e6	+	150773.i	−1.40402	+	0.158849i
$247$			474031.i			0.494384i
$248$		−	185425.i		−	0.191443i
$249$	1.12089e6	−	126816.i	1.14569	−	0.129621i
$250$			309981.i			0.313679i
$251$	−209024.			−0.209417			−0.104709	−	0.994503i	$-0.533391\pi$
$251$	−209024.			−0.209417			−0.104709	+	0.994503i	$0.533391\pi$
$252$	−93600.4	−	408358.i	−0.0928488	−	0.405079i
$253$	214412.	+	60918.0i	0.210595	+	0.0598335i
$254$		−	924189.i		−	0.898827i
$255$	588092.	−	66536.0i	0.566363	−	0.0640776i
$256$	1.31173e6			1.25097
$257$			580419.i			0.548162i	0.961707	+	0.274081i	$0.0883736\pi$
$257$			580419.i			0.548162i	−0.961707	+	0.274081i	$0.911626\pi$
$258$	−2.45770e6	+	278061.i	−2.29868	+	0.260070i
$259$	−1.13818e6			−1.05429
$260$	447896.			0.410908
$261$	−222156.	−	969220.i	−0.201863	−	0.880686i
$262$	−381651.			−0.343490
$263$	1.58407e6			1.41216			0.706080	−	0.708132i	$-0.250462\pi$
$263$	1.58407e6			1.41216			0.706080	+	0.708132i	$0.250462\pi$
$264$	15335.2	+	135543.i	0.0135419	+	0.119693i
$265$	951391.			0.832232
$266$			1.06155e6i			0.919893i
$267$	−143560.	+	16242.2i	−0.123241	+	0.0139434i
$268$			747188.i			0.635466i
$269$		−	1.99855e6i		−	1.68397i	−0.539500	−	0.841986i	$-0.681387\pi$
$269$		−	1.99855e6i		−	1.68397i	0.539500	−	0.841986i	$-0.318613\pi$
$270$	−730731.	−	2.07910e6i	−0.610026	−	1.73567i
$271$	1.37973e6			1.14122			0.570612	−	0.821220i	$-0.306706\pi$
$271$	1.37973e6			1.14122			0.570612	+	0.821220i	$0.306706\pi$
$272$	−588412.			−0.482236
$273$	−51278.9	−	453239.i	−0.0416421	−	0.368062i
$274$			1.14749e6i			0.923367i
$275$	321372.			0.256257
$276$	698416.	+	115695.i	0.551876	+	0.0914200i
$277$	1.43497e6			1.12369			0.561843	−	0.827244i	$-0.310093\pi$
$277$	1.43497e6			1.12369			0.561843	+	0.827244i	$0.310093\pi$
$278$		−	1.00987e6i		−	0.783705i
$279$	440966.	−	101075.i	0.339152	−	0.0777376i
$280$	−790023.			−0.602205
$281$	1.17003e6			0.883954			0.441977	−	0.897026i	$-0.354277\pi$
$281$	1.17003e6			0.883954			0.441977	+	0.897026i	$0.354277\pi$
$282$	−300197.	+	33963.9i	−0.224794	+	0.0254329i
$283$			1.19800e6i			0.889179i	0.895734	+	0.444589i	$0.146650\pi$
$283$			1.19800e6i			0.889179i	−0.895734	+	0.444589i	$0.853350\pi$
$284$			605330.i			0.445345i
$285$	225197.	+	1.99045e6i	0.164229	+	1.45157i
$286$		−	188557.i		−	0.136310i
$287$	−1.17301e6			−0.840613
$288$	316477.	+	1.38072e6i	0.224833	+	0.980898i
$289$	−1.20734e6			−0.850323
$290$	2.38064e6			1.66226
$291$	−49884.5	−	440915.i	−0.0345329	−	0.305226i
$292$	−1.14026e6			−0.782610
$293$	−2.23825e6			−1.52314			−0.761568	−	0.648085i	$-0.775570\pi$
$293$	−2.23825e6			−1.52314			−0.761568	+	0.648085i	$0.775570\pi$
$294$	93228.8	+	824022.i	0.0629045	+	0.555995i
$295$		−	3.28995e6i		−	2.20107i
$296$	1.17700e6			0.780815
$297$	−313981.	+	110353.i	−0.206544	+	0.0725929i
$298$			2.35411e6i			1.53563i
$299$	741421.	+	210650.i	0.479609	+	0.136265i
$300$	1.01422e6	−	114748.i	0.650624	−	0.0736108i
$301$	−2.16330e6			−1.37626
$302$		−	3.60732e6i		−	2.27597i
$303$	−184695.	−	1.63247e6i	−0.115571	−	1.02150i
$304$		−	1.99153e6i		−	1.23596i
$305$			2.01310e6i			1.23913i
$306$	−176798.	−	771330.i	−0.107938	−	0.470909i
$307$	−2.25941e6			−1.36820			−0.684100	−	0.729388i	$-0.739805\pi$
$307$	−2.25941e6			−1.36820			−0.684100	+	0.729388i	$0.739805\pi$
$308$		−	151475.i		−	0.0909836i
$309$	−364734.	−	3.22378e6i	−0.217310	−	1.92074i
$310$			1.08312e6i			0.640136i
$311$		−	1.35140e6i		−	0.792285i	−0.918189	−	0.396142i	$-0.870349\pi$
$311$		−	1.35140e6i		−	0.792285i	0.918189	−	0.396142i	$-0.129651\pi$
$312$	53028.1	+	468700.i	0.0308404	+	0.272589i
$313$		−	913176.i		−	0.526858i	−0.964679	−	0.263429i	$-0.915147\pi$
$313$		−	913176.i		−	0.526858i	0.964679	−	0.263429i	$-0.0848535\pi$
$314$	2.37234e6			1.35785
$315$	−430638.	−	1.87878e6i	−0.244532	−	1.06684i
$316$			745932.i			0.420225i
$317$			2.07585e6i			1.16024i	0.814532	+	0.580119i	$0.196994\pi$
$317$			2.07585e6i			1.16024i	−0.814532	+	0.580119i	$0.803006\pi$
$318$	−143008.	−	1.26400e6i	−0.0793035	−	0.700940i
$319$		−	359518.i		−	0.197808i
$320$	−27520.3			−0.0150238
$321$	−2.84415e6	+	321784.i	−1.54060	+	0.174302i
$322$	1.66035e6	+	471733.i	0.892401	+	0.253546i
$323$			719289.i			0.383617i
$324$	−951495.	+	460374.i	−0.503552	+	0.243640i
$325$	1.11128e6			0.583601
$326$		−	2.62278e6i		−	1.36684i
$327$	201482.	+	1.78084e6i	0.104200	+	0.920992i
$328$	1.21302e6			0.622563
$329$	−264238.			−0.134588
$330$	−89577.1	−	791746.i	−0.0452806	−	0.400222i
$331$	−1.93557e6			−0.971043			−0.485522	−	0.874225i	$-0.661370\pi$
$331$	−1.93557e6			−0.971043			−0.485522	+	0.874225i	$0.661370\pi$
$332$	1.29537e6			0.644982
$333$	641579.	+	2.79907e6i	0.317059	+	1.38326i
$334$	2.49312e6			1.22286
$335$			3.43767e6i			1.67360i
$336$	215436.	+	1.90418e6i	0.104105	+	0.920151i
$337$		−	294207.i		−	0.141117i	−0.997508	−	0.0705583i	$-0.977522\pi$
$337$		−	294207.i		−	0.141117i	0.997508	−	0.0705583i	$-0.0224781\pi$
$338$			1.97081e6i			0.938326i
$339$	2.88412e6	−	326305.i	1.36306	−	0.154214i
$340$	679633.			0.318843
$341$	163570.			0.0761760
$342$	2.61063e6	−	598386.i	1.20692	−	0.276641i
$343$			2.34405e6i			1.07580i
$344$	2.23709e6			1.01927
$345$	3.21329e6	+	532291.i	1.45345	+	0.240769i
$346$	3.69125e6			1.65761
$347$		−	3.21410e6i		−	1.43296i	−0.697606	−	0.716482i	$-0.745751\pi$
$347$		−	3.21410e6i		−	1.43296i	0.697606	−	0.716482i	$-0.254249\pi$
$348$	−128368.	−	1.13461e6i	−0.0568211	−	0.502225i
$349$	464451.			0.204116			0.102058	−	0.994778i	$-0.467457\pi$
$349$	464451.			0.204116			0.102058	+	0.994778i	$0.467457\pi$
$350$	2.48862e6			1.08590
$351$	−1.08573e6	+	381594.i	−0.470384	+	0.165323i
$352$			512158.i			0.220317i
$353$		−	923698.i		−	0.394542i	−0.980349	−	0.197271i	$-0.936792\pi$
$353$		−	923698.i		−	0.394542i	0.980349	−	0.197271i	$-0.0632079\pi$
$354$	−4.37098e6	+	494527.i	−1.85383	+	0.209740i
$355$			2.78501e6i			1.17289i
$356$	−165906.			−0.0693806
$357$	−77810.0	−	687740.i	−0.0323121	−	0.285597i
$358$	−324285.			−0.133727
$359$	−2.76429e6			−1.13200			−0.566002	−	0.824404i	$-0.691511\pi$
$359$	−2.76429e6			−1.13200			−0.566002	+	0.824404i	$0.691511\pi$
$360$	445328.	+	1.94287e6i	0.181102	+	0.790109i
$361$	41606.1			0.0168031
$362$	2.83746e6			1.13804
$363$	2.37505e6	−	268711.i	0.946034	−	0.107033i
$364$		−	523789.i		−	0.207206i
$365$	−5.24611e6			−2.06113
$366$	2.67458e6	−	302599.i	1.04365	−	0.118077i
$367$		−	1.67359e6i		−	0.648610i	−0.945953	−	0.324305i	$-0.894870\pi$
$367$		−	1.67359e6i		−	0.648610i	0.945953	−	0.324305i	$-0.105130\pi$
$368$	−3.11491e6	−	884997.i	−1.19902	−	0.340661i
$369$	661212.	+	2.88472e6i	0.252799	+	1.10291i
$370$	−6.87520e6			−2.61084
$371$		−	1.11260e6i		−	0.419665i
$372$	516213.	−	58403.7i	0.193407	−	0.0218818i
$373$			496559.i			0.184799i	0.995722	+	0.0923993i	$0.0294536\pi$
$373$			496559.i			0.184799i	−0.995722	+	0.0923993i	$0.970546\pi$
$374$		−	286114.i		−	0.105769i
$375$	679709.	−	76901.5i	0.249600	−	0.0282395i
$376$	273252.			0.0996766
$377$		−	1.24319e6i		−	0.450489i
$378$	−2.43139e6	+	854548.i	−0.875236	+	0.307614i
$379$		−	3.34566e6i		−	1.19642i	−0.801340	−	0.598209i	$-0.795879\pi$
$379$		−	3.34566e6i		−	1.19642i	0.801340	−	0.598209i	$-0.204121\pi$
$380$			2.30027e6i			0.817185i
$381$	−2.02651e6	+	229277.i	−0.715214	+	0.0809184i
$382$		−	4.22556e6i		−	1.48158i
$383$	−338695.			−0.117981			−0.0589904	−	0.998259i	$-0.518788\pi$
$383$	−338695.			−0.117981			−0.0589904	+	0.998259i	$0.518788\pi$
$384$	−322767.	−	2.85285e6i	−0.111702	−	0.987302i
$385$		−	696907.i		−	0.239620i
$386$			3.78252e6i			1.29215i
$387$	1.21943e6	+	5.32011e6i	0.413885	+	1.80569i
$388$		−	509546.i		−	0.171832i
$389$	−3.12606e6			−1.04743			−0.523714	−	0.851894i	$-0.675454\pi$
$389$	−3.12606e6			−1.04743			−0.523714	+	0.851894i	$0.675454\pi$
$390$	−309751.	−	2.73780e6i	−0.103122	−	0.911466i
$391$	1.12502e6	+	319638.i	0.372152	+	0.105735i
$392$		−	750059.i		−	0.246536i
$393$	94681.7	+	836864.i	0.0309232	+	0.273321i
$394$	4.38622e6			1.42348
$395$			3.43190e6i			1.10673i
$396$	−372515.	+	85384.7i	−0.119373	+	0.0273616i
$397$	−1.16181e6			−0.369964			−0.184982	−	0.982742i	$-0.559223\pi$
$397$	−1.16181e6			−0.369964			−0.184982	+	0.982742i	$0.559223\pi$
$398$	−4.80085e6			−1.51918
$399$	2.32771e6	−	263354.i	0.731976	−	0.0828149i
$400$	−4.66879e6			−1.45900
$401$	−821803.			−0.255215			−0.127608	−	0.991825i	$-0.540730\pi$
$401$	−821803.			−0.255215			−0.127608	+	0.991825i	$0.540730\pi$
$402$	4.56724e6	−	516732.i	1.40958	−	0.159478i
$403$	565614.			0.173483
$404$		−	1.88657e6i		−	0.575069i
$405$	−4.37766e6	+	2.11810e6i	−1.32618	+	0.641665i
$406$		−	2.78402e6i		−	0.838218i
$407$			1.03828e6i			0.310690i
$408$	80464.2	+	711199.i	0.0239305	+	0.211515i
$409$	−2.29307e6			−0.677810			−0.338905	−	0.940821i	$-0.610057\pi$
$409$	−2.29307e6			−0.677810			−0.338905	+	0.940821i	$0.610057\pi$
$410$	−7.08558e6			−2.08169
$411$	2.51616e6	−	284675.i	0.734741	−	0.0831276i
$412$		−	3.72558e6i		−	1.08131i
$413$	−3.84741e6			−1.10992
$414$	224191.	−	4.34914e6i	0.0642860	−	1.24710i
$415$	5.95974e6			1.69866
$416$			1.77101e6i			0.501749i
$417$	−2.21438e6	+	250532.i	−0.623609	+	0.0705543i
$418$	968376.			0.271084
$419$	449527.			0.125090			0.0625448	−	0.998042i	$-0.480078\pi$
$419$	449527.			0.125090			0.0625448	+	0.998042i	$0.480078\pi$
$420$	−248835.	−	2.19938e6i	−0.0688316	−	0.608383i
$421$		−	4.57385e6i		−	1.25770i	−0.777527	−	0.628850i	$-0.783526\pi$
$421$		−	4.57385e6i		−	1.25770i	0.777527	−	0.628850i	$-0.216474\pi$
$422$			700387.i			0.191451i
$423$	148948.	+	649830.i	0.0404748	+	0.176583i
$424$			1.15055e6i			0.310807i
$425$	1.68625e6			0.452844
$426$	3.70013e6	−	418628.i	0.987853	−	0.111764i
$427$	2.35421e6			0.624849
$428$	−3.28686e6			−0.867306
$429$	−413457.	+	46778.0i	−0.108464	+	0.0122715i
$430$	−1.30675e7			−3.40817
$431$	4.58849e6			1.18981			0.594904	−	0.803797i	$-0.297190\pi$
$431$	4.58849e6			1.18981			0.594904	+	0.803797i	$0.297190\pi$
$432$	4.56142e6	−	1.60318e6i	1.17595	−	0.413307i
$433$			2.65527e6i			0.680596i	0.940318	+	0.340298i	$0.110528\pi$
$433$			2.65527e6i			0.680596i	−0.940318	+	0.340298i	$0.889472\pi$
$434$	1.26665e6			0.322798
$435$	−590599.	−	5.22013e6i	−0.149647	−	1.32269i
$436$			2.05804e6i			0.518487i
$437$	−1.08184e6	+	3.80774e6i	−0.270994	+	0.953813i
$438$	788566.	+	6.96991e6i	0.196405	+	1.73597i
$439$	−3.66951e6			−0.908754			−0.454377	−	0.890810i	$-0.650138\pi$
$439$	−3.66951e6			−0.908754			−0.454377	+	0.890810i	$0.650138\pi$
$440$			720679.i			0.177464i
$441$	1.78374e6	−	408854.i	0.436752	−	0.100109i
$442$		−	989362.i		−	0.240879i
$443$			670900.i			0.162423i	0.996697	+	0.0812117i	$0.0258790\pi$
$443$			670900.i			0.162423i	−0.996697	+	0.0812117i	$0.974121\pi$
$444$	370723.	+	3.27671e6i	0.0892466	+	0.788825i
$445$	−763305.			−0.182725
$446$		−	3.07221e6i		−	0.731330i
$447$	5.16196e6	−	584017.i	1.22193	−	0.138247i
$448$			32183.4i			0.00757595i
$449$		−	5.53849e6i		−	1.29651i	−0.761423	−	0.648255i	$-0.775499\pi$
$449$		−	5.53849e6i		−	1.29651i	0.761423	−	0.648255i	$-0.224501\pi$
$450$	−1.40281e6	−	6.12016e6i	−0.326563	−	1.42473i
$451$			1.07005e6i			0.247720i
$452$	3.33305e6			0.767354
$453$	−7.90993e6	+	894919.i	−1.81104	+	0.204898i
$454$			6.69094e6i			1.52352i
$455$		−	2.40986e6i		−	0.545711i
$456$	−2.40711e6	+	272338.i	−0.542106	+	0.0613332i
$457$			4.58966e6i			1.02799i	0.857792	+	0.513996i	$0.171836\pi$
$457$			4.58966e6i			1.02799i	−0.857792	+	0.513996i	$0.828164\pi$
$458$	−336590.			−0.0749786
$459$	−1.64747e6	+	579027.i	−0.364994	+	0.128282i
$460$	3.59781e6	+	1.02220e6i	0.792762	+	0.225237i
$461$		−	4.47210e6i		−	0.980074i	−0.871702	−	0.490037i	$-0.836983\pi$
$461$		−	4.47210e6i		−	0.980074i	0.871702	−	0.490037i	$-0.163017\pi$
$462$	−925901.	+	104755.i	−0.201818	+	0.0228334i
$463$	1.82230e6			0.395065			0.197532	−	0.980296i	$-0.436707\pi$
$463$	1.82230e6			0.395065			0.197532	+	0.980296i	$0.436707\pi$
$464$			5.22297e6i			1.12622i
$465$	2.37500e6	−	268705.i	0.509368	−	0.0576293i
$466$	−2.06441e6			−0.440383
$467$	4.99097e6			1.05899			0.529496	−	0.848312i	$-0.322381\pi$
$467$	4.99097e6			1.05899			0.529496	+	0.848312i	$0.322381\pi$
$468$	−1.28813e6	+	295254.i	−0.271860	+	0.0623134i
$469$	4.02016e6			0.843939
$470$	−1.59614e6			−0.333293
$471$	−588539.	−	5.20192e6i	−0.122243	−	1.08047i
$472$	3.97864e6			0.822015
$473$			1.97342e6i			0.405571i
$474$	4.55957e6	−	515864.i	0.932133	−	0.105460i
$475$			5.70724e6i			1.16063i
$476$		−	794792.i		−	0.160781i
$477$	−2.73616e6	+	627159.i	−0.550612	+	0.126207i
$478$	1.08654e7			2.17509
$479$	2.26574e6			0.451202			0.225601	−	0.974220i	$-0.427565\pi$
$479$	2.26574e6			0.451202			0.225601	+	0.974220i	$0.427565\pi$
$480$	841347.	+	7.43642e6i	0.166676	+	1.47320i
$481$			3.59029e6i			0.707565i
$482$	−2.61881e6			−0.513437
$483$	622483.	−	3.75775e6i	0.121412	−	0.732926i
$484$	2.74475e6			0.532585
$485$		−	2.34433e6i		−	0.452547i
$486$	3.47210e6	+	5.49771e6i	0.666809	+	1.05582i
$487$	919704.			0.175722			0.0878609	−	0.996133i	$-0.471997\pi$
$487$	919704.			0.175722			0.0878609	+	0.996133i	$0.471997\pi$
$488$	−2.43451e6			−0.462767
$489$	−5.75108e6	+	650670.i	−1.08762	+	0.123052i
$490$			4.38130e6i			0.824352i
$491$			6.77346e6i			1.26796i	0.773348	+	0.633982i	$0.218580\pi$
$491$			6.77346e6i			1.26796i	−0.773348	+	0.633982i	$0.781420\pi$
$492$	382067.	+	3.37698e6i	0.0711585	+	0.628949i
$493$		−	1.88640e6i		−	0.349556i
$494$	3.34858e6			0.617366
$495$	−1.71387e6	+	392839.i	−0.314388	+	0.0720613i
$496$	−2.37630e6			−0.433707
$497$	3.25691e6			0.591445
$498$	−895836.	−	7.91803e6i	−0.161866	−	1.43068i
$499$	−2.57927e6			−0.463709			−0.231855	−	0.972750i	$-0.574479\pi$
$499$	−2.57927e6			−0.463709			−0.231855	+	0.972750i	$0.574479\pi$
$500$	785511.			0.140516
$501$	−618504.	−	5.46678e6i	−0.110090	−	0.973054i
$502$			1.47656e6i			0.261512i
$503$	2.59926e6			0.458068			0.229034	−	0.973418i	$-0.426443\pi$
$503$	2.59926e6			0.458068			0.229034	+	0.973418i	$0.426443\pi$
$504$	2.27207e6	−	520785.i	0.398424	−	0.0913234i
$505$		−	8.67977e6i		−	1.51454i
$506$	430327.	−	1.51461e6i	0.0747176	−	0.262982i
$507$	4.32149e6	−	488928.i	0.746644	−	0.0844744i
$508$	−2.34195e6			−0.402641
$509$			7.89088e6i			1.34999i	0.737822	+	0.674996i	$0.235854\pi$
$509$			7.89088e6i			1.34999i	−0.737822	+	0.674996i	$0.764146\pi$
$510$	−470013.	−	4.15431e6i	−0.0800175	−	0.707251i
$511$			6.13502e6i			1.03935i
$512$		−	3.37245e6i		−	0.568552i
$513$	−1.95976e6	−	5.57599e6i	−0.328784	−	0.935467i
$514$	4.10011e6			0.684522
$515$		−	1.71407e7i		−	2.84781i
$516$	704622.	+	6.22795e6i	0.116502	+	1.02972i
$517$			241045.i			0.0396618i
$518$			8.04014e6i			1.31656i
$519$	−915740.	−	8.09396e6i	−0.149229	−	1.31899i
$520$			2.49206e6i			0.404157i
$521$	9.26811e6			1.49588			0.747940	−	0.663766i	$-0.231043\pi$
$521$	9.26811e6			1.49588			0.747940	+	0.663766i	$0.231043\pi$
$522$	−6.84661e6	+	1.56932e6i	−1.09976	+	0.252078i
$523$		−	7.44892e6i		−	1.19080i	−0.803429	−	0.595401i	$-0.796993\pi$
$523$		−	7.44892e6i		−	1.19080i	0.803429	−	0.595401i	$-0.203007\pi$
$524$			967127.i			0.153871i
$525$	−617388.	−	5.45691e6i	−0.0977597	−	0.864069i
$526$		−	1.11899e7i		−	1.76345i
$527$	858257.			0.134614
$528$	1.73704e6	−	196527.i	0.271160	−	0.0306787i
$529$	5.47484e6	+	3.38417e6i	0.850614	+	0.525790i
$530$		−	6.72067e6i		−	1.03926i
$531$	2.16874e6	+	9.46175e6i	0.333789	+	1.45625i
$532$	2.69004e6			0.412078
$533$			3.70015e6i			0.564159i
$534$	114736.	+	1.01412e6i	0.0174119	+	0.153898i
$535$	−1.51223e7			−2.28419
$536$	−4.15729e6			−0.625026
$537$	80450.0	+	711074.i	0.0120390	+	0.106409i
$538$	−1.41179e7			−2.10287
$539$	661654.			0.0980977
$540$	−5.26857e6	+	1.85172e6i	−0.777514	+	0.273269i
$541$	1.12804e6			0.165703			0.0828517	−	0.996562i	$-0.473597\pi$
$541$	1.12804e6			0.165703			0.0828517	+	0.996562i	$0.473597\pi$
$542$		−	9.74648e6i		−	1.42511i
$543$	−703928.	−	6.22181e6i	−0.102454	−	0.905560i
$544$			2.68731e6i			0.389332i
$545$			9.46867e6i			1.36552i
$546$	−3.20170e6	+	362236.i	−0.459620	+	0.0520009i
$547$	−816790.			−0.116719			−0.0583596	−	0.998296i	$-0.518587\pi$
$547$	−816790.			−0.116719			−0.0583596	+	0.998296i	$0.518587\pi$
$548$	2.90782e6			0.413634
$549$	−1.32704e6	−	5.78960e6i	−0.187912	−	0.819819i
$550$		−	2.27019e6i		−	0.320003i
$551$	6.38468e6			0.895901
$552$	−643717.	+	3.88593e6i	−0.0899181	+	0.542809i
$553$	4.01341e6			0.558085
$554$		−	1.01367e7i		−	1.40321i
$555$	1.70563e6	+	1.50755e7i	0.235045	+	2.07750i
$556$	−2.55906e6			−0.351070
$557$	62846.8			0.00858311			0.00429156	−	0.999991i	$-0.498634\pi$
$557$	62846.8			0.00858311			0.00429156	+	0.999991i	$0.498634\pi$
$558$	−713995.	−	3.11500e6i	−0.0970755	−	0.423519i
$559$			6.82395e6i			0.923648i
$560$			1.01244e7i			1.36427i
$561$	−627375.	+	70980.4i	−0.0841627	+	0.00952206i
$562$		−	8.26512e6i		−	1.10385i
$563$	−6.57966e6			−0.874848			−0.437424	−	0.899255i	$-0.644109\pi$
$563$	−6.57966e6			−0.874848			−0.437424	+	0.899255i	$0.644109\pi$
$564$	86066.6	+	760718.i	0.0113930	+	0.100699i
$565$	1.53347e7			2.02095
$566$	8.46269e6			1.11037
$567$	2.47699e6	+	5.11942e6i	0.323569	+	0.668748i
$568$	−3.36800e6			−0.438028
$569$	−8.19458e6			−1.06108			−0.530538	−	0.847661i	$-0.678010\pi$
$569$	−8.19458e6			−1.06108			−0.530538	+	0.847661i	$0.678010\pi$
$570$	1.40606e7	−	1.59080e6i	1.81266	−	0.205082i
$571$		−	3.95838e6i		−	0.508074i	−0.967195	−	0.254037i	$-0.918242\pi$
$571$		−	3.95838e6i		−	0.508074i	0.967195	−	0.254037i	$-0.0817585\pi$
$572$	−477814.			−0.0610617
$573$	−9.26556e6	+	1.04829e6i	−1.17892	+	0.133382i
$574$			8.28617e6i			1.04972i
$575$	8.92656e6	+	2.53619e6i	1.12594	+	0.319898i
$576$	79147.3	−	18141.5i	0.00993985	−	0.00227833i
$577$	−5.93437e6			−0.742053			−0.371027	−	0.928622i	$-0.620994\pi$
$577$	−5.93437e6			−0.742053			−0.371027	+	0.928622i	$0.620994\pi$
$578$			8.52868e6i			1.06185i
$579$	8.29411e6	−	938385.i	1.02819	−	0.116328i
$580$		−	6.03268e6i		−	0.744629i
$581$		−	6.96958e6i		−	0.856576i
$582$	−3.11464e6	+	352387.i	−0.381154	+	0.0431233i
$583$	−1.01494e6			−0.123671
$584$		−	6.34429e6i		−	0.769752i
$585$	−5.92645e6	+	1.35841e6i	−0.715987	+	0.164112i
$586$			1.58111e7i			1.90203i
$587$		−	1.32192e7i		−	1.58347i	−0.610866	−	0.791734i	$-0.709179\pi$
$587$		−	1.32192e7i		−	1.58347i	0.610866	−	0.791734i	$-0.290821\pi$
$588$	2.08812e6	−	236247.i	0.249065	−	0.0281789i
$589$			2.90484e6i			0.345012i
$590$	−2.32403e7			−2.74861
$591$	−1.08815e6	−	9.61787e6i	−0.128151	−	1.13269i
$592$		−	1.50837e7i		−	1.76891i
$593$		−	7.51285e6i		−	0.877340i	−0.898648	−	0.438670i	$-0.855450\pi$
$593$		−	7.51285e6i		−	0.877340i	0.898648	−	0.438670i	$-0.144550\pi$
$594$	779541.	+	2.21798e6i	0.0906510	+	0.257924i
$595$		−	3.65669e6i		−	0.423444i
$596$	5.96545e6			0.687904
$597$	1.19101e6	+	1.05270e7i	0.136767	+	1.20884i
$598$	1.48804e6	−	5.23743e6i	0.170162	−	0.598915i
$599$			7.65322e6i			0.871519i	0.900063	+	0.435760i	$0.143520\pi$
$599$			7.65322e6i			0.871519i	−0.900063	+	0.435760i	$0.856480\pi$
$600$	638447.	+	5.64305e6i	0.0724014	+	0.639935i
$601$	−462278.			−0.0522056			−0.0261028	−	0.999659i	$-0.508310\pi$
$601$	−462278.			−0.0522056			−0.0261028	+	0.999659i	$0.508310\pi$
$602$			1.52817e7i			1.71862i
$603$	−2.26612e6	−	9.88660e6i	−0.253799	−	1.10727i
$604$	−9.14116e6			−1.01955
$605$	1.26281e7			1.40265
$606$	−1.15318e7	+	1.30470e6i	−1.27561	+	0.144320i
$607$	−4.48728e6			−0.494323			−0.247162	−	0.968974i	$-0.579498\pi$
$607$	−4.48728e6			−0.494323			−0.247162	+	0.968974i	$0.579498\pi$
$608$	−9.09540e6			−0.997845
$609$	−6.10464e6	+	690671.i	−0.666986	+	0.0754619i
$610$	1.42207e7			1.54737
$611$			833517.i			0.0903258i
$612$	−1.95460e6	+	448016.i	−0.210949	+	0.0483520i
$613$		−	3.41404e6i		−	0.366959i	−0.983024	−	0.183479i	$-0.941264\pi$
$613$		−	3.41404e6i		−	0.366959i	0.983024	−	0.183479i	$-0.0587361\pi$
$614$			1.59606e7i			1.70855i
$615$	1.75782e6	+	1.55369e7i	0.187407	+	1.65644i
$616$	842793.			0.0894889
$617$	1.60232e7			1.69448			0.847238	−	0.531214i	$-0.178264\pi$
$617$	1.60232e7			1.69448			0.847238	+	0.531214i	$0.178264\pi$
$618$	−2.27729e7	+	2.57650e6i	−2.39854	+	0.271368i
$619$			1.72859e7i			1.81328i	0.421901	+	0.906642i	$0.361363\pi$
$619$			1.72859e7i			1.81328i	−0.421901	+	0.906642i	$0.638637\pi$
$620$	2.74469e6			0.286757
$621$	−9.59216e6	+	587360.i	−0.998130	+	0.0611188i
$622$	−9.54631e6			−0.989372
$623$			892640.i			0.0921418i
$624$	6.00657e6	−	679575.i	0.617539	−	0.0698676i
$625$	−7.81668e6			−0.800428
$626$	−6.45071e6			−0.657918
$627$	−240239.	−	2.12340e6i	−0.0244047	−	0.215706i
$628$		−	6.01164e6i		−	0.608266i
$629$			5.44786e6i			0.549034i
$630$	−1.32718e7	+	3.04205e6i	−1.33223	+	0.305362i
$631$		−	4.13451e6i		−	0.413382i	−0.978406	−	0.206691i	$-0.933731\pi$
$631$		−	4.13451e6i		−	0.413382i	0.978406	−	0.206691i	$-0.0662694\pi$
$632$	−4.15031e6			−0.413321
$633$	1.53577e6	−	173755.i	0.152341	−	0.0172357i
$634$	1.46639e7			1.44886
$635$	−1.07749e7			−1.06042
$636$	−3.20306e6	+	362391.i	−0.313995	+	0.0355250i
$637$	2.28795e6			0.223408
$638$	−2.53965e6			−0.247015
$639$	−1.83589e6	−	8.00957e6i	−0.177866	−	0.775992i
$640$		−	1.51685e7i		−	1.46383i
$641$	8.64292e6			0.830836			0.415418	−	0.909631i	$-0.363635\pi$
$641$	8.64292e6			0.830836			0.415418	+	0.909631i	$0.363635\pi$
$642$	2.27309e6	+	2.00912e7i	0.217661	+	1.92384i
$643$			1.25671e6i			0.119870i	0.998202	+	0.0599348i	$0.0190893\pi$
$643$			1.25671e6i			0.119870i	−0.998202	+	0.0599348i	$0.980911\pi$
$644$	1.19540e6	−	4.20742e6i	0.113579	−	0.399762i
$645$	3.24184e6	+	2.86536e7i	0.306826	+	2.71194i
$646$	5.08109e6			0.479044
$647$		−	3.40198e6i		−	0.319500i	−0.987157	−	0.159750i	$-0.948931\pi$
$647$		−	3.40198e6i		−	0.319500i	0.987157	−	0.159750i	$-0.0510689\pi$
$648$	−2.56149e6	−	5.29404e6i	−0.239637	−	0.495279i
$649$			3.50970e6i			0.327084i
$650$		−	7.85014e6i		−	0.728776i
$651$	−314235.	−	2.77743e6i	−0.0290604	−	0.256856i
$652$	−6.64627e6			−0.612293
$653$		−	2.05797e7i		−	1.88867i	−0.328990	−	0.944333i	$-0.606708\pi$
$653$		−	2.05797e7i		−	1.88867i	0.328990	−	0.944333i	$-0.393292\pi$
$654$	1.25799e7	−	1.42328e6i	1.15010	−	0.130120i
$655$			4.44958e6i			0.405243i
$656$		−	1.55453e7i		−	1.41039i
$657$	1.50876e7	−	3.45825e6i	1.36366	−	0.312567i
$658$			1.86659e6i			0.168068i
$659$	−1.51070e7			−1.35508			−0.677539	−	0.735486i	$-0.736954\pi$
$659$	−1.51070e7			−1.35508			−0.677539	+	0.735486i	$0.736954\pi$
$660$	−2.00633e6	+	226994.i	−0.179284	+	0.0202840i
$661$		−	1.12641e7i		−	1.00275i	−0.865229	−	0.501376i	$-0.832827\pi$
$661$		−	1.12641e7i		−	1.00275i	0.865229	−	0.501376i	$-0.167173\pi$
$662$			1.36729e7i			1.21260i
$663$	−2.16942e6	+	245445.i	−0.191672	+	0.0216856i
$664$			7.20731e6i			0.634385i
$665$	1.23764e7			1.08527
$666$	1.97728e7	−	4.53214e6i	1.72736	−	0.395930i
$667$	2.83723e6	−	9.98614e6i	0.246933	−	0.869126i
$668$		−	6.31772e6i		−	0.547796i
$669$	−6.73656e6	+	762166.i	−0.581933	+	0.0658392i
$670$	2.42839e7			2.08993
$671$		−	2.14757e6i		−	0.184137i
$672$	8.69646e6	−	983907.i	0.742881	−	0.0840487i
$673$	−1.33396e7			−1.13528			−0.567642	−	0.823276i	$-0.692144\pi$
$673$	−1.33396e7			−1.13528			−0.567642	+	0.823276i	$0.692144\pi$
$674$	−2.07829e6			−0.176221
$675$	−1.30719e7	+	4.59432e6i	−1.10428	+	0.388116i
$676$	4.99416e6			0.420335
$677$	5.60773e6			0.470236			0.235118	−	0.971967i	$-0.424452\pi$
$677$	5.60773e6			0.470236			0.235118	+	0.971967i	$0.424452\pi$
$678$	−2.30503e6	−	2.03735e7i	−0.192576	−	1.70213i
$679$	−2.74156e6			−0.228204
$680$			3.78142e6i			0.313605i
$681$	1.46715e7	−	1.65992e6i	1.21229	−	0.137157i
$682$		−	1.15547e6i		−	0.0951254i
$683$			2.32238e7i			1.90494i	0.304637	+	0.952469i	$0.401465\pi$
$683$			2.32238e7i			1.90494i	−0.304637	+	0.952469i	$0.598535\pi$
$684$	−1.51635e6	−	6.61549e6i	−0.123925	−	0.540657i
$685$	1.33783e7			1.08937
$686$	1.65584e7			1.34341
$687$	83502.6	+	738055.i	0.00675007	+	0.0596619i
$688$		−	2.86692e7i		−	2.30911i
$689$	−3.50959e6			−0.281649
$690$	3.76013e6	−	2.26988e7i	0.300663	−	1.81501i
$691$	1.09068e7			0.868964			0.434482	−	0.900681i	$-0.356932\pi$
$691$	1.09068e7			0.868964			0.434482	+	0.900681i	$0.356932\pi$
$692$		−	9.35384e6i		−	0.742548i
$693$	459403.	+	2.00428e6i	0.0363380	+	0.158535i
$694$	−2.27045e7			−1.78943
$695$	−1.17738e7			−0.924601
$696$	6.31287e6	−	714230.i	0.493974	−	0.0558876i
$697$			5.61457e6i			0.437758i
$698$		−	3.28090e6i		−	0.254891i
$699$	512147.	+	4.52672e6i	0.0396462	+	0.350421i
$700$		−	6.30631e6i		−	0.486441i
$701$	2.24487e7			1.72543			0.862713	−	0.505695i	$-0.168764\pi$
$701$	2.24487e7			1.72543			0.862713	+	0.505695i	$0.168764\pi$
$702$	2.69560e6	+	7.66961e6i	0.206449	+	0.587396i
$703$	−1.84387e7			−1.40716
$704$	29358.6			0.00223256
$705$	395977.	+	3.49992e6i	0.0300052	+	0.265207i
$706$	−6.52504e6			−0.492687
$707$	−1.01505e7			−0.763728
$708$	1.25316e6	+	1.10763e7i	0.0939558	+	0.830448i
$709$			2.21859e7i			1.65753i	0.559596	+	0.828766i	$0.310956\pi$
$709$			2.21859e7i			1.65753i	−0.559596	+	0.828766i	$0.689044\pi$
$710$	1.96734e7			1.46465
$711$	−2.26231e6	−	9.86999e6i	−0.167834	−	0.732222i
$712$		−	923089.i		−	0.0682407i
$713$	4.54339e6	+	1.29085e6i	0.334701	+	0.0950941i
$714$	−4.85823e6	+	549654.i	−0.356642	+	0.0403500i
$715$	−2.19833e6			−0.160816
$716$			821757.i			0.0599047i
$717$	−2.69554e6	−	2.38251e7i	−0.195816	−	1.73076i
$718$			1.95271e7i			1.41360i
$719$			471342.i			0.0340028i	0.999855	+	0.0170014i	$0.00541197\pi$
$719$			471342.i			0.0340028i	−0.999855	+	0.0170014i	$0.994588\pi$
$720$	2.48986e7	−	5.70704e6i	1.78996	−	0.410280i
$721$	−2.00451e7			−1.43605
$722$		−	293907.i		−	0.0209830i
$723$	649687.	+	5.74239e6i	0.0462230	+	0.408552i
$724$		−	7.19028e6i		−	0.509799i
$725$		−	1.49678e7i		−	1.05758i
$726$	−1.89818e6	−	1.67775e7i	−0.133658	−	1.18137i
$727$			1.10840e7i			0.777784i	0.921283	+	0.388892i	$0.127142\pi$
$727$			1.10840e7i			0.777784i	−0.921283	+	0.388892i	$0.872858\pi$
$728$	2.91432e6			0.203802
$729$	1.11937e7	−	8.97732e6i	0.780108	−	0.625645i
$730$			3.70587e7i			2.57385i
$731$			1.03546e7i			0.716703i
$732$	−766803.	−	6.77755e6i	−0.0528940	−	0.467514i
$733$			7.54820e6i			0.518900i	0.965757	+	0.259450i	$0.0835413\pi$
$733$			7.54820e6i			0.518900i	−0.965757	+	0.259450i	$0.916459\pi$
$734$	−1.18223e7			−0.809957
$735$	9.60706e6	−	1.08693e6i	0.655952	−	0.0742136i
$736$	−4.04182e6	+	1.42259e7i	−0.275032	+	0.968023i
$737$		−	3.66729e6i		−	0.248701i
$738$	2.03778e7	−	4.67083e6i	1.37726	−	0.315684i
$739$	2.33132e7			1.57033			0.785164	−	0.619287i	$-0.212578\pi$
$739$	2.33132e7			1.57033			0.785164	+	0.619287i	$0.212578\pi$
$740$			1.74222e7i			1.16956i
$741$	−830729.	−	7.34257e6i	−0.0555794	−	0.491250i
$742$	−7.85943e6			−0.524061
$743$	−5.07752e6			−0.337427			−0.168713	−	0.985665i	$-0.553961\pi$
$743$	−5.07752e6			−0.337427			−0.168713	+	0.985665i	$0.553961\pi$
$744$	324954.	+	2.87217e6i	0.0215223	+	0.190229i
$745$	2.74459e7			1.81171
$746$	3.50771e6			0.230769
$747$	−1.71400e7	+	3.92868e6i	−1.12385	+	0.257599i
$748$	−725030.			−0.0473807
$749$			1.76846e7i			1.15184i
$750$	−543235.	−	4.80150e6i	−0.0352643	−	0.311690i
$751$		−	1.75809e7i		−	1.13747i	−0.822519	−	0.568737i	$-0.807432\pi$
$751$		−	1.75809e7i		−	1.13747i	0.822519	−	0.568737i	$-0.192568\pi$
$752$		−	3.50183e6i		−	0.225814i
$753$	3.23771e6	−	366311.i	0.208090	−	0.0235430i
$754$	−8.78195e6			−0.562552
$755$	−4.20568e7			−2.68515
$756$	2.16547e6	+	6.16129e6i	0.137800	+	0.392073i
$757$		−	1.79381e7i		−	1.13772i	−0.822433	−	0.568862i	$-0.807384\pi$
$757$		−	1.79381e7i		−	1.13772i	0.822433	−	0.568862i	$-0.192616\pi$
$758$	−2.36339e7			−1.49404
$759$	−3.42792e6	−	567846.i	−0.215986	−	0.0357788i
$760$	−1.27985e7			−0.803759
$761$			1.50206e7i			0.940214i	0.882609	+	0.470107i	$0.155785\pi$
$761$			1.50206e7i			0.940214i	−0.882609	+	0.470107i	$0.844215\pi$
$762$	1.61962e6	+	1.43153e7i	0.101048	+	0.893129i
$763$	1.10731e7			0.688583
$764$	−1.07078e7			−0.663693
$765$	−8.99274e6	+	2.06124e6i	−0.555569	+	0.127343i
$766$			2.39255e6i			0.147329i
$767$			1.21363e7i			0.744901i
$768$	−2.03183e7	+	2.29878e6i	−1.24303	+	0.140635i
$769$			2.25008e7i			1.37209i	0.727559	+	0.686045i	$0.240655\pi$
$769$			2.25008e7i			1.37209i	−0.727559	+	0.686045i	$0.759345\pi$
$770$	−4.92298e6			−0.299228
$771$	−1.01717e6	−	8.99048e6i	−0.0616252	−	0.544687i
$772$	9.58514e6			0.578836
$773$	8.67404e6			0.522123			0.261061	−	0.965322i	$-0.415927\pi$
$773$	8.67404e6			0.522123			0.261061	+	0.965322i	$0.415927\pi$
$774$	3.75815e7	−	8.61411e6i	2.25487	−	0.516842i
$775$	6.80988e6			0.407273
$776$	2.83507e6			0.169009
$777$	1.76300e7	−	1.99463e6i	1.04761	−	0.118525i
$778$			2.20827e7i			1.30798i
$779$	−1.90030e7			−1.12196
$780$	−6.93775e6	+	784929.i	−0.408303	+	0.0461949i
$781$		−	2.97104e6i		−	0.174293i
$782$	2.25794e6	−	7.94722e6i	0.132037	−	0.464728i
$783$	5.13966e6	+	1.46235e7i	0.299592	+	0.852409i
$784$	−9.61229e6			−0.558517
$785$		−	2.76584e7i		−	1.60197i
$786$	5.91164e6	−	668836.i	0.341312	−	0.0386156i
$787$		−	2.50677e7i		−	1.44271i	−0.692568	−	0.721353i	$-0.743521\pi$
$787$		−	2.50677e7i		−	1.44271i	0.692568	−	0.721353i	$-0.256479\pi$
$788$		−	1.11150e7i		−	0.637664i
$789$	−2.45366e7	+	2.77604e6i	−1.40321	+	0.158757i
$790$	2.42431e7			1.38204
$791$		−	1.79331e7i		−	1.01909i
$792$	−475074.	−	2.07264e6i	−0.0269121	−	0.117412i
$793$		−	7.42615e6i		−	0.419354i
$794$			8.20710e6i			0.461996i
$795$	−1.47367e7	+	1.66729e6i	−0.826956	+	0.0935608i
$796$			1.21656e7i			0.680538i
$797$	8.45754e6			0.471626			0.235813	−	0.971798i	$-0.424225\pi$
$797$	8.45754e6			0.471626			0.235813	+	0.971798i	$0.424225\pi$
$798$	−1.86035e6	−	1.64431e7i	−0.103416	−	0.914061i
$799$			1.26477e6i			0.0700882i
$800$			2.13226e7i			1.17792i
$801$	2.19523e6	−	503172.i	0.120892	−	0.0277099i
$802$			5.80525e6i			0.318702i
$803$	5.59653e6			0.306288
$804$	−1.30943e6	−	1.15737e7i	−0.0714401	−	0.631438i
$805$	5.49982e6	−	1.93576e7i	0.299129	−	1.05284i
$806$		−	3.99552e6i		−	0.216639i
$807$	3.50242e6	+	3.09568e7i	0.189315	+	1.67330i
$808$	1.04967e7			0.565621
$809$		−	792665.i		−	0.0425812i	−0.999773	−	0.0212906i	$-0.993222\pi$
$809$		−	792665.i		−	0.0425812i	0.999773	−	0.0212906i	$-0.00677753\pi$
$810$	1.49623e7	+	3.09240e7i	0.801285	+	1.65608i
$811$	−5.20675e6			−0.277981			−0.138990	−	0.990294i	$-0.544386\pi$
$811$	−5.20675e6			−0.277981			−0.138990	+	0.990294i	$0.544386\pi$
$812$	−7.05487e6			−0.375490
$813$	−2.13715e7	+	2.41795e6i	−1.13399	+	0.128298i
$814$	7.33443e6			0.387976
$815$	−3.05783e7			−1.61257
$816$	9.11430e6	−	1.03118e6i	0.479179	−	0.0542137i
$817$	−3.50459e7			−1.83689
$818$			1.61983e7i			0.846421i
$819$	1.58858e6	+	6.93064e6i	0.0827561	+	0.361047i
$820$			1.79553e7i			0.932518i
$821$			4.93224e6i			0.255380i	0.991814	+	0.127690i	$0.0407562\pi$
$821$			4.93224e6i			0.255380i	−0.991814	+	0.127690i	$0.959244\pi$
$822$	−2.01096e6	−	1.77743e7i	−0.103806	−	0.917513i
$823$	9.20715e6			0.473833			0.236917	−	0.971530i	$-0.423863\pi$
$823$	9.20715e6			0.473833			0.236917	+	0.971530i	$0.423863\pi$
$824$	2.07288e7			1.06355
$825$	−4.97794e6	+	563197.i	−0.254633	+	0.0288088i
$826$			2.71782e7i			1.38603i
$827$	−2.00455e7			−1.01919			−0.509593	−	0.860415i	$-0.670204\pi$
$827$	−2.00455e7			−1.01919			−0.509593	+	0.860415i	$0.670204\pi$
$828$	−1.10210e7	−	568112.i	−0.558655	−	0.0287977i
$829$	−2.99048e7			−1.51131			−0.755657	−	0.654968i	$-0.772682\pi$
$829$	−2.99048e7			−1.51131			−0.755657	+	0.654968i	$0.772682\pi$
$830$		−	4.20999e7i		−	2.12122i
$831$	−2.22272e7	+	2.51476e6i	−1.11656	+	0.126326i
$832$	101520.			0.00508443
$833$	3.47171e6			0.173353
$834$	1.76977e6	+	1.56425e7i	0.0881053	+	0.778737i
$835$		−	2.90667e7i		−	1.44271i
$836$		−	2.45392e6i		−	0.121435i
$837$	−6.65328e6	+	2.33839e6i	−0.328263	+	0.115373i
$838$		−	3.17548e6i		−	0.156207i
$839$	9.08698e6			0.445671			0.222836	−	0.974856i	$-0.428469\pi$
$839$	9.08698e6			0.445671			0.222836	+	0.974856i	$0.428469\pi$
$840$	1.22372e7	−	1.38450e6i	0.598388	−	0.0677008i
$841$	3.76674e6			0.183643
$842$	−3.23099e7			−1.57056
$843$	−1.81233e7	+	2.05045e6i	−0.878351	+	0.0993755i
$844$	1.77482e6			0.0857627
$845$	2.29772e7			1.10702
$846$	4.59043e6	−	1.05218e6i	0.220509	−	0.0505433i
$847$		−	1.47678e7i		−	0.707306i
$848$	1.47447e7			0.704121
$849$	−2.09946e6	−	1.85565e7i	−0.0999628	−	0.883542i
$850$		−	1.19117e7i		−	0.565493i
$851$	−8.19381e6	+	2.88396e7i	−0.387849	+	1.36510i
$852$	−1.06083e6	−	9.37634e6i	−0.0500663	−	0.442521i
$853$	6.90592e6			0.324974			0.162487	−	0.986711i	$-0.448048\pi$
$853$	6.90592e6			0.324974			0.162487	+	0.986711i	$0.448048\pi$
$854$		−	1.66302e7i		−	0.780286i
$855$	−6.97643e6	−	3.04366e7i	−0.326376	−	1.42391i
$856$		−	1.82878e7i		−	0.853057i
$857$			1.09221e7i			0.507989i	0.967206	+	0.253994i	$0.0817444\pi$
$857$			1.09221e7i			0.507989i	−0.967206	+	0.253994i	$0.918256\pi$
$858$	330442.	+	2.92068e6i	0.0153242	+	0.135446i
$859$	2.15152e7			0.994860			0.497430	−	0.867504i	$-0.334277\pi$
$859$	2.15152e7			0.994860			0.497430	+	0.867504i	$0.334277\pi$
$860$			3.31138e7i			1.52673i
$861$	1.81695e7	−	2.05567e6i	0.835284	−	0.0945029i
$862$		−	3.24133e7i		−	1.48578i
$863$			1.63382e6i			0.0746755i	0.999303	+	0.0373377i	$0.0118877\pi$
$863$			1.63382e6i			0.0746755i	−0.999303	+	0.0373377i	$0.988112\pi$
$864$	−7.32178e6	−	2.08322e7i	−0.333682	−	0.949404i
$865$		−	4.30353e7i		−	1.95562i
$866$	1.87570e7			0.849900
$867$	1.87012e7	−	2.11583e6i	0.844933	−	0.0955946i
$868$		−	3.20975e6i		−	0.144601i
$869$		−	3.66113e6i		−	0.164462i
$870$	−3.68752e7	+	4.17201e6i	−1.65172	+	0.186874i
$871$		−	1.26812e7i		−	0.566391i
$872$	−1.14508e7			−0.509969
$873$	1.54539e6	+	6.74219e6i	0.0686280	+	0.299409i
$874$	2.68980e7	+	7.64217e6i	1.19108	+	0.338406i
$875$		−	4.22635e6i		−	0.186615i
$876$	1.76622e7	−	1.99827e6i	0.777649	−	0.0879822i
$877$	−3.43998e7			−1.51028			−0.755139	−	0.655564i	$-0.772431\pi$
$877$	−3.43998e7			−1.51028			−0.755139	+	0.655564i	$0.772431\pi$
$878$			2.59215e7i			1.13481i
$879$	3.46696e7	−	3.92248e6i	1.51348	−	0.171233i
$880$	9.23578e6			0.402038
$881$	−3.08894e7			−1.34082			−0.670409	−	0.741992i	$-0.733881\pi$
$881$	−3.08894e7			−1.34082			−0.670409	+	0.741992i	$0.733881\pi$
$882$	−2.88816e6	−	1.26004e7i	−0.125012	−	0.545398i
$883$	9.24667e6			0.399102			0.199551	−	0.979887i	$-0.436052\pi$
$883$	9.24667e6			0.399102			0.199551	+	0.979887i	$0.436052\pi$
$884$	−2.50710e6			−0.107905
$885$	5.76556e6	+	5.09601e7i	0.247448	+	2.18712i
$886$	4.73926e6			0.202827
$887$			1.89393e7i			0.808266i	0.914700	+	0.404133i	$0.132427\pi$
$887$			1.89393e7i			0.808266i	−0.914700	+	0.404133i	$0.867573\pi$
$888$	−1.82313e7	+	2.06267e6i	−0.775865	+	0.0877804i
$889$			1.26006e7i			0.534732i
$890$			5.39202e6i			0.228179i
$891$	4.67007e6	−	2.25958e6i	0.197074	−	0.0953527i
$892$	−7.78515e6			−0.327608
$893$	−4.28072e6			−0.179634
$894$	−4.12552e6	−	3.64643e7i	−0.172638	−	1.52589i
$895$			3.78076e6i			0.157769i
$896$	−1.77387e7			−0.738160
$897$	−1.18535e7	−	1.96357e6i	−0.491887	−	0.0814827i
$898$	−3.91242e7			−1.61903
$899$		−	7.61821e6i		−	0.314379i
$900$	−1.55088e7	+	3.55480e6i	−0.638224	+	0.146288i
$901$	−5.32542e6			−0.218545
$902$	7.55886e6			0.309343
$903$	3.35088e7	−	3.79114e6i	1.36754	−	0.154721i
$904$			1.85448e7i			0.754747i
$905$		−	3.30812e7i		−	1.34264i
$906$	6.32175e6	+	5.58761e7i	0.255868	+	2.26154i
$907$			4.42182e6i			0.178477i	0.996010	+	0.0892387i	$0.0284434\pi$
$907$			4.42182e6i			0.178477i	−0.996010	+	0.0892387i	$0.971557\pi$
$908$	1.69552e7			0.682479
$909$	5.72173e6	+	2.49627e7i	0.229677	+	1.00203i
$910$	−1.70233e7			−0.681461
$911$	8.58070e6			0.342552			0.171276	−	0.985223i	$-0.445211\pi$
$911$	8.58070e6			0.342552			0.171276	+	0.985223i	$0.445211\pi$
$912$	3.49011e6	+	3.08481e7i	0.138948	+	1.22812i
$913$	−6.35783e6			−0.252425
$914$	3.24216e7			1.28371
$915$	−3.52792e6	−	3.11823e7i	−0.139305	−	1.23127i
$916$			852938.i			0.0335876i
$917$	5.20352e6			0.204350
$918$	4.09027e6	+	1.16378e7i	0.160194	+	0.455789i
$919$			3.91773e6i			0.153019i	0.997069	+	0.0765095i	$0.0243776\pi$
$919$			3.91773e6i			0.153019i	−0.997069	+	0.0765095i	$0.975622\pi$
$920$	−5.68742e6	+	2.00179e7i	−0.221537	+	0.779738i
$921$	3.49975e7	−	3.95957e6i	1.35953	−	0.153815i
$922$	−3.15911e7			−1.22388
$923$		−	1.02736e7i		−	0.396936i
$924$	265456.	+	2.34629e6i	0.0102285	+	0.0904069i
$925$			4.32263e7i			1.66109i
$926$		−	1.28728e7i		−	0.493341i
$927$	1.12992e7	+	4.92960e7i	0.431866	+	1.88414i
$928$	2.38535e7			0.909249
$929$		−	2.27456e7i		−	0.864684i	−0.901710	−	0.432342i	$-0.857687\pi$
$929$		−	2.27456e7i		−	0.864684i	0.901710	−	0.432342i	$-0.142313\pi$
$930$	−1.89814e6	−	1.67771e7i	−0.0719650	−	0.636078i
$931$			1.17503e7i			0.444298i
$932$			5.23133e6i			0.197275i
$933$	2.36829e6	+	2.09326e7i	0.0890699	+	0.787262i
$934$		−	3.52564e7i		−	1.32243i
$935$	−3.33573e6			−0.124785
$936$	−1.64277e6	−	7.16705e6i	−0.0612897	−	0.267394i
$937$			1.14888e6i			0.0427489i	0.999772	+	0.0213744i	$0.00680421\pi$
$937$			1.14888e6i			0.0427489i	−0.999772	+	0.0213744i	$0.993196\pi$
$938$		−	2.83986e7i		−	1.05388i
$939$	1.60032e6	+	1.41448e7i	0.0592302	+	0.523518i
$940$			4.04471e6i			0.149303i
$941$	4.70391e7			1.73175			0.865874	−	0.500261i	$-0.166763\pi$
$941$	4.70391e7			1.73175			0.865874	+	0.500261i	$0.166763\pi$
$942$	−3.67466e7	+	4.15747e6i	−1.34924	+	0.152652i
$943$	−8.44455e6	+	2.97221e7i	−0.309241	+	1.08843i
$944$		−	5.09878e7i		−	1.86224i
$945$	9.96295e6	+	2.83469e7i	0.362918	+	1.03259i
$946$	1.39403e7			0.506460
$947$			8.69406e6i			0.315027i	0.987517	+	0.157513i	$0.0503478\pi$
$947$			8.69406e6i			0.315027i	−0.987517	+	0.157513i	$0.949652\pi$
$948$	−1.30723e6	−	1.15542e7i	−0.0472423	−	0.417561i
$949$	1.93524e7			0.697540
$950$	4.03162e7			1.44934
$951$	−3.63787e6	−	3.21541e7i	−0.130436	−	1.15288i
$952$	4.42215e6			0.158140
$953$	1.28078e7			0.456815			0.228408	−	0.973566i	$-0.426648\pi$
$953$	1.28078e7			0.456815			0.228408	+	0.973566i	$0.426648\pi$
$954$	4.43028e6	+	1.93284e7i	0.157602	+	0.687581i
$955$	−4.92647e7			−1.74794
$956$		−	2.75336e7i		−	0.974359i
$957$	630048.	+	5.56881e6i	0.0222379	+	0.196554i
$958$		−	1.60053e7i		−	0.563442i
$959$		−	1.56452e7i		−	0.549332i
$960$	426280.	−	48228.8i	0.0149285	−	0.00168899i
$961$	−2.51631e7			−0.878933
$962$	2.53619e7			0.883578
$963$	4.34909e7	−	9.96862e6i	1.51124	−	0.346393i
$964$			6.63623e6i			0.230001i
$965$	4.40995e7			1.52446
$966$	−2.65449e7	−	4.39725e6i	−0.915247	−	0.151614i
$967$	5.03888e7			1.73288			0.866439	−	0.499283i	$-0.166403\pi$
$967$	5.03888e7			1.73288			0.866439	+	0.499283i	$0.166403\pi$
$968$			1.52715e7i			0.523835i
$969$	−1.26054e6	−	1.11415e7i	−0.0431268	−	0.381185i
$970$	−1.65604e7			−0.565122
$971$	−670405.			−0.0228186			−0.0114093	−	0.999935i	$-0.503632\pi$
$971$	−670405.			−0.0228186			−0.0114093	+	0.999935i	$0.503632\pi$
$972$	1.39315e7	−	8.79851e6i	0.472969	−	0.298706i
$973$			1.37688e7i			0.466243i
$974$		−	6.49683e6i		−	0.219434i
$975$	−1.72134e7	+	1.94750e6i	−0.579901	+	0.0656093i
$976$			3.11992e7i			1.04838i
$977$	3.59434e7			1.20471			0.602355	−	0.798228i	$-0.294229\pi$
$977$	3.59434e7			1.20471			0.602355	+	0.798228i	$0.294229\pi$
$978$	4.59636e6	+	4.06259e7i	0.153662	+	1.35817i
$979$	814290.			0.0271533
$980$	1.11025e7			0.369279
$981$	−6.24177e6	−	2.72315e7i	−0.207079	−	0.903439i
$982$	4.78480e7			1.58338
$983$	1.84333e7			0.608443			0.304222	−	0.952601i	$-0.401604\pi$
$983$	1.84333e7			0.608443			0.304222	+	0.952601i	$0.401604\pi$
$984$	−1.87892e7	+	2.12579e6i	−0.618616	+	0.0699894i
$985$		−	5.11379e7i		−	1.67939i
$986$	−1.33256e7			−0.436511
$987$	4.09296e6	−	463072.i	0.133735	−	0.0151306i
$988$		−	8.48549e6i		−	0.276557i
$989$	−1.55737e7	+	5.48146e7i	−0.506293	+	1.78199i
$990$	2.77503e6	+	1.21069e7i	0.0899871	+	0.392594i
$991$	9.89527e6			0.320069			0.160034	−	0.987111i	$-0.448839\pi$
$991$	9.89527e6			0.320069			0.160034	+	0.987111i	$0.448839\pi$
$992$			1.08526e7i			0.350152i
$993$	2.99813e7	−	3.39204e6i	0.964888	−	0.109166i
$994$		−	2.30069e7i		−	0.738572i
$995$			5.59719e7i			1.79231i
$996$	−2.00648e7	+	2.27010e6i	−0.640893	+	0.0725098i
$997$	5.64735e7			1.79931			0.899657	−	0.436598i	$-0.143817\pi$
$997$	5.64735e7			1.79931			0.899657	+	0.436598i	$0.143817\pi$
$998$			1.82201e7i			0.579061i
$999$	−1.48431e7	−	4.22323e7i	−0.470557	−	1.33885i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.c.b.68.7	✓	32
3.2	odd	2	inner	69.6.c.b.68.26	yes	32
23.22	odd	2	inner	69.6.c.b.68.8	yes	32
69.68	even	2	inner	69.6.c.b.68.25	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.c.b.68.7	✓	32	1.1	even	1	trivial
69.6.c.b.68.8	yes	32	23.22	odd	2	inner
69.6.c.b.68.25	yes	32	69.68	even	2	inner
69.6.c.b.68.26	yes	32	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-7.06404i q^{2} +(-15.4896 + 1.75248i) q^{3} -17.9007 q^{4} -82.3579 q^{5} +(12.3796 + 109.419i) q^{6} +96.3127i q^{7} -99.5980i q^{8} +(236.858 - 54.2905i) q^{9} +O(q^{10})\)	\(q-7.06404i q^{2} +(-15.4896 + 1.75248i) q^{3} -17.9007 q^{4} -82.3579 q^{5} +(12.3796 + 109.419i) q^{6} +96.3127i q^{7} -99.5980i q^{8} +(236.858 - 54.2905i) q^{9} +581.779i q^{10} +87.8590 q^{11} +(277.275 - 31.3706i) q^{12} +303.810 q^{13} +680.357 q^{14} +(1275.69 - 144.330i) q^{15} -1276.39 q^{16} +460.998 q^{17} +(-383.510 - 1673.17i) q^{18} +1560.29i q^{19} +1474.26 q^{20} +(-168.786 - 1491.85i) q^{21} -620.640i q^{22} +(2440.41 + 693.361i) q^{23} +(174.543 + 1542.74i) q^{24} +3657.82 q^{25} -2146.13i q^{26} +(-3573.70 + 1256.03i) q^{27} -1724.07i q^{28} -4091.99i q^{29} +(-1019.56 - 9011.55i) q^{30} +1861.74 q^{31} +5829.32i q^{32} +(-1360.90 + 153.971i) q^{33} -3256.51i q^{34} -7932.11i q^{35} +(-4239.92 + 971.838i) q^{36} +11817.5i q^{37} +11021.9 q^{38} +(-4705.91 + 532.421i) q^{39} +8202.68i q^{40} +12179.1i q^{41} +(-10538.5 + 1192.31i) q^{42} +22461.2i q^{43} -1572.74 q^{44} +(-19507.1 + 4471.25i) q^{45} +(4897.93 - 17239.2i) q^{46} +2743.55i q^{47} +(19770.8 - 2236.84i) q^{48} +7530.86 q^{49} -25839.0i q^{50} +(-7140.70 + 807.889i) q^{51} -5438.42 q^{52} -11551.9 q^{53} +(8872.63 + 25244.7i) q^{54} -7235.88 q^{55} +9592.56 q^{56} +(-2734.37 - 24168.3i) q^{57} -28906.0 q^{58} +39947.0i q^{59} +(-22835.8 + 2583.61i) q^{60} -24443.4i q^{61} -13151.4i q^{62} +(5228.87 + 22812.4i) q^{63} +334.155 q^{64} -25021.2 q^{65} +(1087.66 + 9613.48i) q^{66} -41740.7i q^{67} -8252.20 q^{68} +(-39016.1 - 6463.15i) q^{69} -56032.8 q^{70} -33816.0i q^{71} +(-5407.23 - 23590.6i) q^{72} +63699.0 q^{73} +83479.5 q^{74} +(-56658.2 + 6410.24i) q^{75} -27930.2i q^{76} +8461.94i q^{77} +(3761.04 + 33242.8i) q^{78} -41670.6i q^{79} +105121. q^{80} +(53154.1 - 25718.2i) q^{81} +86034.0 q^{82} -72364.0 q^{83} +(3021.39 + 26705.2i) q^{84} -37966.8 q^{85} +158667. q^{86} +(7171.13 + 63383.5i) q^{87} -8750.58i q^{88} +9268.15 q^{89} +(31585.1 + 137799. i) q^{90} +29260.8i q^{91} +(-43685.0 - 12411.6i) q^{92} +(-28837.6 + 3262.65i) q^{93} +19380.5 q^{94} -128502. i q^{95} +(-10215.8 - 90294.0i) q^{96} +28465.2i q^{97} -53198.3i q^{98} +(20810.1 - 4769.91i) 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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