LMFDB - Embedded newform 69.7.d.a.22.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,7,Mod(22,69)]

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([0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("69.22");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	69.d (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:	$15.8737317698$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.11
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.69366 q^{2} +15.5885 q^{3} -50.3569 q^{4} -218.943i q^{5} -57.5785 q^{6} -261.444i q^{7} +422.396 q^{8} +243.000 q^{9} +O(q^{10})$	\(q-3.69366 q^{2} +15.5885 q^{3} -50.3569 q^{4} -218.943i q^{5} -57.5785 q^{6} -261.444i q^{7} +422.396 q^{8} +243.000 q^{9} +808.702i q^{10} +1771.61i q^{11} -784.986 q^{12} -2889.11 q^{13} +965.685i q^{14} -3412.99i q^{15} +1662.65 q^{16} -559.041i q^{17} -897.560 q^{18} -1593.98i q^{19} +11025.3i q^{20} -4075.50i q^{21} -6543.74i q^{22} +(-11698.6 - 3343.37i) q^{23} +6584.50 q^{24} -32311.1 q^{25} +10671.4 q^{26} +3788.00 q^{27} +13165.5i q^{28} -19917.0 q^{29} +12606.4i q^{30} +14468.1 q^{31} -33174.6 q^{32} +27616.7i q^{33} +2064.91i q^{34} -57241.3 q^{35} -12236.7 q^{36} +63378.4i q^{37} +5887.61i q^{38} -45036.8 q^{39} -92480.7i q^{40} +4348.78 q^{41} +15053.5i q^{42} +26412.8i q^{43} -89212.9i q^{44} -53203.2i q^{45} +(43210.8 + 12349.3i) q^{46} -155401. q^{47} +25918.2 q^{48} +49296.2 q^{49} +119346. q^{50} -8714.59i q^{51} +145487. q^{52} +119055. i q^{53} -13991.6 q^{54} +387883. q^{55} -110433. i q^{56} -24847.6i q^{57} +73566.8 q^{58} +235457. q^{59} +171867. i q^{60} -210647. i q^{61} -53440.1 q^{62} -63530.8i q^{63} +16126.0 q^{64} +632551. i q^{65} -102007. i q^{66} -494703. i q^{67} +28151.5i q^{68} +(-182363. - 52118.0i) q^{69} +211430. q^{70} -508838. q^{71} +102642. q^{72} -39370.7 q^{73} -234098. i q^{74} -503681. q^{75} +80267.6i q^{76} +463177. q^{77} +166351. q^{78} -150541. i q^{79} -364027. i q^{80} +59049.0 q^{81} -16062.9 q^{82} +226820. i q^{83} +205230. i q^{84} -122398. q^{85} -97560.0i q^{86} -310476. q^{87} +748322. i q^{88} -576409. i q^{89} +196515. i q^{90} +755340. i q^{91} +(589106. + 168362. i) q^{92} +225535. q^{93} +573999. q^{94} -348990. q^{95} -517141. q^{96} +973550. i q^{97} -182083. q^{98} +430502. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

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 $

$2$	−3.69366			−0.461708			−0.230854	−	0.972988i	$-0.574152\pi$
$2$	−3.69366			−0.461708			−0.230854	+	0.972988i	$0.574152\pi$
$3$	15.5885			0.577350
$3$	15.5885			0.577350
$4$	−50.3569			−0.786826
$5$		−	218.943i		−	1.75155i	−0.482723	−	0.875773i	$-0.660352\pi$
$5$		−	218.943i		−	1.75155i	0.482723	−	0.875773i	$-0.339648\pi$
$6$	−57.5785			−0.266567
$7$		−	261.444i		−	0.762227i	−0.924528	−	0.381113i	$-0.875541\pi$
$7$		−	261.444i		−	0.762227i	0.924528	−	0.381113i	$-0.124459\pi$
$8$	422.396			0.824991
$9$	243.000			0.333333
$10$			808.702i			0.808702i
$11$			1771.61i			1.33104i	0.746380	+	0.665520i	$0.231790\pi$
$11$			1771.61i			1.33104i	−0.746380	+	0.665520i	$0.768210\pi$
$12$	−784.986			−0.454274
$13$	−2889.11			−1.31503			−0.657513	−	0.753444i	$-0.728391\pi$
$13$	−2889.11			−1.31503			−0.657513	+	0.753444i	$0.728391\pi$
$14$			965.685i			0.351926i
$15$		−	3412.99i		−	1.01126i
$16$	1662.65			0.405921
$17$		−	559.041i		−	0.113788i	−0.998380	−	0.0568940i	$-0.981880\pi$
$17$		−	559.041i		−	0.113788i	0.998380	−	0.0568940i	$-0.0181197\pi$
$18$	−897.560			−0.153903
$19$		−	1593.98i		−	0.232392i	−0.993226	−	0.116196i	$-0.962930\pi$
$19$		−	1593.98i		−	0.232392i	0.993226	−	0.116196i	$-0.0370701\pi$
$20$			11025.3i			1.37816i
$21$		−	4075.50i		−	0.440072i
$22$		−	6543.74i		−	0.614552i
$23$	−11698.6	−	3343.37i	−0.961504	−	0.274790i
$23$	−11698.6	−	3343.37i	−0.961504	−	0.274790i
$24$	6584.50			0.476309
$25$	−32311.1			−2.06791
$26$	10671.4			0.607157
$27$	3788.00			0.192450
$28$			13165.5i			0.599740i
$29$	−19917.0			−0.816640			−0.408320	−	0.912839i	$-0.633885\pi$
$29$	−19917.0			−0.816640			−0.408320	+	0.912839i	$0.633885\pi$
$30$			12606.4i			0.466904i
$31$	14468.1			0.485652			0.242826	−	0.970070i	$-0.421926\pi$
$31$	14468.1			0.485652			0.242826	+	0.970070i	$0.421926\pi$
$32$	−33174.6			−1.01241
$33$			27616.7i			0.768476i
$34$			2064.91i			0.0525368i
$35$	−57241.3			−1.33507
$36$	−12236.7			−0.262275
$37$			63378.4i			1.25123i	0.780133	+	0.625613i	$0.215151\pi$
$37$			63378.4i			1.25123i	−0.780133	+	0.625613i	$0.784849\pi$
$38$			5887.61i			0.107297i
$39$	−45036.8			−0.759230
$40$		−	92480.7i		−	1.44501i
$41$	4348.78			0.0630981			0.0315491	−	0.999502i	$-0.489956\pi$
$41$	4348.78			0.0630981			0.0315491	+	0.999502i	$0.489956\pi$
$42$			15053.5i			0.203185i
$43$			26412.8i			0.332207i	0.986108	+	0.166104i	$0.0531186\pi$
$43$			26412.8i			0.332207i	−0.986108	+	0.166104i	$0.946881\pi$
$44$		−	89212.9i		−	1.04730i
$45$		−	53203.2i		−	0.583849i
$46$	43210.8	+	12349.3i	0.443934	+	0.126873i
$47$	−155401.			−1.49679			−0.748394	−	0.663254i	$-0.769175\pi$
$47$	−155401.			−1.49679			−0.748394	+	0.663254i	$0.769175\pi$
$48$	25918.2			0.234359
$49$	49296.2			0.419011
$50$	119346.			0.954772
$51$		−	8714.59i		−	0.0656956i
$52$	145487.			1.03470
$53$			119055.i			0.799690i	0.916583	+	0.399845i	$0.130936\pi$
$53$			119055.i			0.799690i	−0.916583	+	0.399845i	$0.869064\pi$
$54$	−13991.6			−0.0888557
$55$	387883.			2.33138
$56$		−	110433.i		−	0.628830i
$57$		−	24847.6i		−	0.134172i
$58$	73566.8			0.377049
$59$	235457.			1.14645			0.573226	−	0.819398i	$-0.305692\pi$
$59$	235457.			1.14645			0.573226	+	0.819398i	$0.305692\pi$
$60$			171867.i			0.795682i
$61$		−	210647.i		−	0.928036i	−0.885826	−	0.464018i	$-0.846407\pi$
$61$		−	210647.i		−	0.928036i	0.885826	−	0.464018i	$-0.153593\pi$
$62$	−53440.1			−0.224229
$63$		−	63530.8i		−	0.254076i
$64$	16126.0			0.0615157
$65$			632551.i			2.30333i
$66$		−	102007.i		−	0.354811i
$67$		−	494703.i		−	1.64483i	−0.568891	−	0.822413i	$-0.692627\pi$
$67$		−	494703.i		−	1.64483i	0.568891	−	0.822413i	$-0.307373\pi$
$68$			28151.5i			0.0895314i
$69$	−182363.	−	52118.0i	−0.555125	−	0.158650i
$70$	211430.			0.616414
$71$	−508838.			−1.42169			−0.710844	−	0.703350i	$-0.751687\pi$
$71$	−508838.			−1.42169			−0.710844	+	0.703350i	$0.751687\pi$
$72$	102642.			0.274997
$73$	−39370.7			−0.101206			−0.0506028	−	0.998719i	$-0.516114\pi$
$73$	−39370.7			−0.101206			−0.0506028	+	0.998719i	$0.516114\pi$
$74$		−	234098.i		−	0.577701i
$75$	−503681.			−1.19391
$76$			80267.6i			0.182852i
$77$	463177.			1.01455
$78$	166351.			0.350542
$79$		−	150541.i		−	0.305333i	−0.988278	−	0.152667i	$-0.951214\pi$
$79$		−	150541.i		−	0.305333i	0.988278	−	0.152667i	$-0.0487861\pi$
$80$		−	364027.i		−	0.710989i
$81$	59049.0			0.111111
$82$	−16062.9			−0.0291329
$83$			226820.i			0.396686i	0.980133	+	0.198343i	$0.0635560\pi$
$83$			226820.i			0.396686i	−0.980133	+	0.198343i	$0.936444\pi$
$84$			205230.i			0.346260i
$85$	−122398.			−0.199305
$86$		−	97560.0i		−	0.153383i
$87$	−310476.			−0.471488
$88$			748322.i			1.09810i
$89$		−	576409.i		−	0.817637i	−0.912616	−	0.408819i	$-0.865941\pi$
$89$		−	576409.i		−	0.817637i	0.912616	−	0.408819i	$-0.134059\pi$
$90$			196515.i			0.269567i
$91$			755340.i			1.00235i
$92$	589106.	+	168362.i	0.756536	+	0.216212i
$93$	225535.			0.280391
$94$	573999.			0.691079
$95$	−348990.			−0.407045
$96$	−517141.			−0.584514
$97$			973550.i			1.06670i	0.845894	+	0.533351i	$0.179068\pi$
$97$			973550.i			1.06670i	−0.845894	+	0.533351i	$0.820932\pi$
$98$	−182083.			−0.193460
$99$			430502.i			0.443680i
$100$	1.62709e6			1.62709
$101$	−1.22771e6			−1.19160			−0.595801	−	0.803132i	$-0.703165\pi$
$101$	−1.22771e6			−1.19160			−0.595801	+	0.803132i	$0.703165\pi$
$102$			32188.7i			0.0303322i
$103$		−	1.71152e6i		−	1.56628i	−0.621842	−	0.783142i	$-0.713616\pi$
$103$		−	1.71152e6i		−	1.56628i	0.621842	−	0.783142i	$-0.286384\pi$
$104$	−1.22035e6			−1.08488
$105$	−892304.			−0.770806
$106$		−	439751.i		−	0.369223i
$107$		−	2.25275e6i		−	1.83891i	−0.393190	−	0.919457i	$-0.628629\pi$
$107$		−	2.25275e6i		−	1.83891i	0.393190	−	0.919457i	$-0.371371\pi$
$108$	−190752.			−0.151425
$109$		−	1.94287e6i		−	1.50025i	−0.661293	−	0.750127i	$-0.729992\pi$
$109$		−	1.94287e6i		−	1.50025i	0.661293	−	0.750127i	$-0.270008\pi$
$110$	−1.43271e6			−1.07642
$111$			987971.i			0.722396i
$112$		−	434690.i		−	0.309404i
$113$			2.07914e6i			1.44095i	0.693481	+	0.720475i	$0.256076\pi$
$113$			2.07914e6i			1.44095i	−0.693481	+	0.720475i	$0.743924\pi$
$114$			91778.7i			0.0619480i
$115$	−732009.	+	2.56133e6i	−0.481308	+	1.68412i
$116$	1.00296e6			0.642554
$117$	−702054.			−0.438342
$118$	−869699.			−0.529325
$119$	−146158.			−0.0867323
$120$		−	1.44163e6i		−	0.834277i
$121$	−1.36706e6			−0.771668
$122$			778057.i			0.428481i
$123$	67790.8			0.0364297
$124$	−728566.			−0.382124
$125$			3.65332e6i			1.87050i
$126$			234661.i			0.117309i
$127$	3.21735e6			1.57068			0.785339	−	0.619066i	$-0.212489\pi$
$127$	3.21735e6			1.57068			0.785339	+	0.619066i	$0.212489\pi$
$128$	2.06361e6			0.984006
$129$			411735.i			0.191800i
$130$		−	2.33643e6i		−	1.06346i
$131$	−1.76389e6			−0.784618			−0.392309	−	0.919834i	$-0.628324\pi$
$131$	−1.76389e6			−0.784618			−0.392309	+	0.919834i	$0.628324\pi$
$132$		−	1.39069e6i		−	0.604657i
$133$	−416735.			−0.177135
$134$			1.82727e6i			0.759429i
$135$		−	829356.i		−	0.337085i
$136$		−	236136.i		−	0.0938742i
$137$			2.12948e6i			0.828154i	0.910242	+	0.414077i	$0.135896\pi$
$137$			2.12948e6i			0.828154i	−0.910242	+	0.414077i	$0.864104\pi$
$138$	673589.	+	192506.i	0.256305	+	0.0732500i
$139$	132686.			0.0494060			0.0247030	−	0.999695i	$-0.492136\pi$
$139$	132686.			0.0494060			0.0247030	+	0.999695i	$0.492136\pi$
$140$	2.88249e6			1.05047
$141$	−2.42246e6			−0.864171
$142$	1.87947e6			0.656404
$143$		−	5.11839e6i		−	1.75035i
$144$	404025.			0.135307
$145$			4.36070e6i			1.43038i
$146$	145422.			0.0467274
$147$	768451.			0.241916
$148$		−	3.19154e6i		−	0.984498i
$149$		−	5.17537e6i		−	1.56453i	−0.622949	−	0.782263i	$-0.714065\pi$
$149$		−	5.17537e6i		−	1.56453i	0.622949	−	0.782263i	$-0.285935\pi$
$150$	1.86043e6			0.551238
$151$	1.40318e6			0.407550			0.203775	−	0.979018i	$-0.434679\pi$
$151$	1.40318e6			0.407550			0.203775	+	0.979018i	$0.434679\pi$
$152$		−	673288.i		−	0.191721i
$153$		−	135847.i		−	0.0379294i
$154$	−1.71082e6			−0.468428
$155$		−	3.16768e6i		−	0.850642i
$156$	2.26791e6			0.597382
$157$		−	4.35430e6i		−	1.12517i	−0.826739	−	0.562586i	$-0.809806\pi$
$157$		−	4.35430e6i		−	1.12517i	0.826739	−	0.562586i	$-0.190194\pi$
$158$			556048.i			0.140975i
$159$			1.85589e6i			0.461701i
$160$			7.26335e6i			1.77328i
$161$	−874104.	+	3.05853e6i	−0.209452	+	0.732884i
$162$	−218107.			−0.0513009
$163$	−6.04372e6			−1.39554			−0.697769	−	0.716323i	$-0.745824\pi$
$163$	−6.04372e6			−1.39554			−0.697769	+	0.716323i	$0.745824\pi$
$164$	−218991.			−0.0496472
$165$	6.04650e6			1.34602
$166$		−	837796.i		−	0.183153i
$167$	5.03976e6			1.08208			0.541041	−	0.840996i	$-0.318030\pi$
$167$	5.03976e6			1.08208			0.541041	+	0.840996i	$0.318030\pi$
$168$		−	1.72147e6i		−	0.363055i
$169$	3.52015e6			0.729291
$170$	452098.			0.0920207
$171$		−	387336.i		−	0.0774640i
$172$		−	1.33007e6i		−	0.261389i
$173$	−2.13405e6			−0.412161			−0.206080	−	0.978535i	$-0.566071\pi$
$173$	−2.13405e6			−0.412161			−0.206080	+	0.978535i	$0.566071\pi$
$174$	1.14679e6			0.217689
$175$			8.44755e6i			1.57622i
$176$			2.94558e6i			0.540297i
$177$	3.67041e6			0.661904
$178$			2.12906e6i			0.377509i
$179$	−5.58887e6			−0.974463			−0.487231	−	0.873273i	$-0.661993\pi$
$179$	−5.58887e6			−0.974463			−0.487231	+	0.873273i	$0.661993\pi$
$180$			2.67915e6i			0.459387i
$181$		−	4.82682e6i		−	0.814001i	−0.913428	−	0.407001i	$-0.866575\pi$
$181$		−	4.82682e6i		−	0.814001i	0.913428	−	0.407001i	$-0.133425\pi$
$182$		−	2.78997e6i		−	0.462791i
$183$		−	3.28365e6i		−	0.535802i
$184$	−4.94145e6	−	1.41223e6i	−0.793233	−	0.226699i
$185$	1.38763e7			2.19158
$186$	−833049.			−0.129459
$187$	990405.			0.151457
$188$	7.82551e6			1.17771
$189$		−	990348.i		−	0.146691i
$190$	1.28905e6			0.187936
$191$		−	7.07156e6i		−	1.01488i	−0.861687	−	0.507440i	$-0.830592\pi$
$191$		−	7.07156e6i		−	1.01488i	0.861687	−	0.507440i	$-0.169408\pi$
$192$	251379.			0.0355161
$193$	−1.35096e7			−1.87919			−0.939597	−	0.342284i	$-0.888800\pi$
$193$	−1.35096e7			−1.87919			−0.939597	+	0.342284i	$0.888800\pi$
$194$		−	3.59597e6i		−	0.492505i
$195$			9.86050e6i			1.32983i
$196$	−2.48240e6			−0.329688
$197$	−6.44400e6			−0.842862			−0.421431	−	0.906860i	$-0.638472\pi$
$197$	−6.44400e6			−0.842862			−0.421431	+	0.906860i	$0.638472\pi$
$198$		−	1.59013e6i		−	0.204851i
$199$		−	1.20114e6i		−	0.152417i	−0.997092	−	0.0762084i	$-0.975719\pi$
$199$		−	1.20114e6i		−	0.152417i	0.997092	−	0.0762084i	$-0.0242814\pi$
$200$	−1.36481e7			−1.70601
$201$		−	7.71166e6i		−	0.949641i
$202$	4.53474e6			0.550172
$203$			5.20719e6i			0.622465i
$204$			438839.i			0.0516910i
$205$		−	952137.i		−	0.110519i
$206$			6.32178e6i			0.723166i
$207$	−2.84277e6	−	812439.i	−0.320501	−	0.0915967i
$208$	−4.80359e6			−0.533796
$209$	2.82391e6			0.309323
$210$	3.29587e6			0.355887
$211$	5.47137e6			0.582437			0.291218	−	0.956657i	$-0.405939\pi$
$211$	5.47137e6			0.582437			0.291218	+	0.956657i	$0.405939\pi$
$212$		−	5.99526e6i		−	0.629217i
$213$	−7.93199e6			−0.820812
$214$			8.32089e6i			0.849041i
$215$	5.78290e6			0.581876
$216$	1.60003e6			0.158770
$217$		−	3.78258e6i		−	0.370177i
$218$			7.17632e6i			0.692679i
$219$	−613728.			−0.0584311
$220$	−1.95326e7			−1.83439
$221$			1.61513e6i			0.149634i
$222$		−	3.64923e6i		−	0.333536i
$223$	−1.00870e7			−0.909590			−0.454795	−	0.890596i	$-0.650287\pi$
$223$	−1.00870e7			−0.909590			−0.454795	+	0.890596i	$0.650287\pi$
$224$			8.67329e6i			0.771684i
$225$	−7.85161e6			−0.689304
$226$		−	7.67965e6i		−	0.665298i
$227$			1.98153e7i			1.69404i	0.531560	+	0.847021i	$0.321606\pi$
$227$			1.98153e7i			1.69404i	−0.531560	+	0.847021i	$0.678394\pi$
$228$			1.25125e6i			0.105570i
$229$		−	7.97432e6i		−	0.664029i	−0.943274	−	0.332015i	$-0.892272\pi$
$229$		−	7.97432e6i		−	0.664029i	0.943274	−	0.332015i	$-0.107728\pi$
$230$	2.70379e6	−	9.46070e6i	0.222223	−	0.777571i
$231$	7.22022e6			0.585753
$232$	−8.41287e6			−0.673721
$233$	5.30042e6			0.419028			0.209514	−	0.977806i	$-0.432812\pi$
$233$	5.30042e6			0.419028			0.209514	+	0.977806i	$0.432812\pi$
$234$	2.59315e6			0.202386
$235$			3.40240e7i			2.62169i
$236$	−1.18569e7			−0.902058
$237$		−	2.34670e6i		−	0.176284i
$238$	539857.			0.0400450
$239$	−6.35315e6			−0.465366			−0.232683	−	0.972553i	$-0.574751\pi$
$239$	−6.35315e6			−0.465366			−0.232683	+	0.972553i	$0.574751\pi$
$240$		−	5.67461e6i		−	0.410490i
$241$		−	9.34983e6i		−	0.667964i	−0.942580	−	0.333982i	$-0.891608\pi$
$241$		−	9.34983e6i		−	0.667964i	0.942580	−	0.333982i	$-0.108392\pi$
$242$	5.04945e6			0.356285
$243$	920483.			0.0641500
$244$			1.06075e7i			0.730203i
$245$		−	1.07931e7i		−	0.733916i
$246$	−250396.			−0.0168199
$247$			4.60517e6i			0.305601i
$248$	6.11125e6			0.400659
$249$			3.53577e6i			0.229027i
$250$		−	1.34941e7i		−	0.863624i
$251$			2.94710e6i			0.186369i	0.995649	+	0.0931847i	$0.0297047\pi$
$251$			2.94710e6i			0.186369i	−0.995649	+	0.0931847i	$0.970295\pi$
$252$			3.19921e6i			0.199913i
$253$	5.92317e6	−	2.07254e7i	0.365757	−	1.27980i
$254$	−1.18838e7			−0.725194
$255$	−1.90800e6			−0.115069
$256$	−8.65434e6			−0.515839
$257$	593358.			0.0349556			0.0174778	−	0.999847i	$-0.494436\pi$
$257$	593358.			0.0349556			0.0174778	+	0.999847i	$0.494436\pi$
$258$		−	1.52081e6i		−	0.0885555i
$259$	1.65699e7			0.953718
$260$		−	3.18533e7i		−	1.81232i
$261$	−4.83984e6			−0.272213
$262$	6.51522e6			0.362264
$263$			8.55513e6i			0.470283i	0.971961	+	0.235142i	$0.0755554\pi$
$263$			8.55513e6i			0.470283i	−0.971961	+	0.235142i	$0.924445\pi$
$264$			1.16652e7i			0.633986i
$265$	2.60664e7			1.40069
$266$	1.53928e6			0.0817847
$267$		−	8.98532e6i		−	0.472063i
$268$			2.49117e7i			1.29419i
$269$	2.39465e7			1.23023			0.615114	−	0.788439i	$-0.289110\pi$
$269$	2.39465e7			1.23023			0.615114	+	0.788439i	$0.289110\pi$
$270$			3.06336e6i			0.155635i
$271$	−8.20740e6			−0.412380			−0.206190	−	0.978512i	$-0.566106\pi$
$271$	−8.20740e6			−0.412380			−0.206190	+	0.978512i	$0.566106\pi$
$272$		−	929491.i		−	0.0461890i
$273$			1.17746e7i			0.578705i
$274$		−	7.86557e6i		−	0.382365i
$275$		−	5.72429e7i		−	2.75248i
$276$	9.18325e6	+	2.62450e6i	0.436787	+	0.124830i
$277$	2.22467e7			1.04671			0.523356	−	0.852114i	$-0.324680\pi$
$277$	2.22467e7			1.04671			0.523356	+	0.852114i	$0.324680\pi$
$278$	−490096.			−0.0228111
$279$	3.51574e6			0.161884
$280$	−2.41785e7			−1.10143
$281$			7.91487e6i			0.356718i	0.983965	+	0.178359i	$0.0570788\pi$
$281$			7.91487e6i			0.356718i	−0.983965	+	0.178359i	$0.942921\pi$
$282$	8.94776e6			0.398994
$283$			1.78585e6i			0.0787928i	0.999224	+	0.0393964i	$0.0125435\pi$
$283$			1.78585e6i			0.0787928i	−0.999224	+	0.0393964i	$0.987456\pi$
$284$	2.56235e7			1.11862
$285$	−5.44022e6			−0.235008
$286$			1.89056e7i			0.808151i
$287$		−	1.13696e6i		−	0.0480951i
$288$	−8.06143e6			−0.337469
$289$	2.38250e7			0.987052
$290$		−	1.61070e7i		−	0.660419i
$291$			1.51761e7i			0.615861i
$292$	1.98258e6			0.0796312
$293$			4.56796e7i			1.81602i	0.418953	+	0.908008i	$0.362397\pi$
$293$			4.56796e7i			1.81602i	−0.418953	+	0.908008i	$0.637603\pi$
$294$	−2.83840e6			−0.111694
$295$		−	5.15517e7i		−	2.00806i
$296$			2.67708e7i			1.03225i
$297$			6.71087e6i			0.256159i
$298$			1.91161e7i			0.722354i
$299$	3.37986e7	+	9.65937e6i	1.26440	+	0.361356i
$300$	2.53638e7			0.939400
$301$	6.90546e6			0.253217
$302$	−5.18286e6			−0.188169
$303$	−1.91381e7			−0.687972
$304$		−	2.65023e6i		−	0.0943328i
$305$	−4.61196e7			−1.62550
$306$			501773.i			0.0175123i
$307$	5.19187e7			1.79436			0.897178	−	0.441670i	$-0.145614\pi$
$307$	5.19187e7			1.79436			0.897178	+	0.441670i	$0.145614\pi$
$308$	−2.33242e7			−0.798278
$309$		−	2.66800e7i		−	0.904295i
$310$			1.17004e7i			0.392748i
$311$	−3.85711e7			−1.28227			−0.641137	−	0.767426i	$-0.721537\pi$
$311$	−3.85711e7			−1.28227			−0.641137	+	0.767426i	$0.721537\pi$
$312$	−1.90233e7			−0.626358
$313$		−	5.22230e7i		−	1.70306i	−0.524309	−	0.851528i	$-0.675676\pi$
$313$		−	5.22230e7i		−	1.70306i	0.524309	−	0.851528i	$-0.324324\pi$
$314$			1.60833e7i			0.519501i
$315$	−1.39096e7			−0.445025
$316$			7.58078e6i			0.240244i
$317$	4.17049e7			1.30921			0.654605	−	0.755972i	$-0.272835\pi$
$317$	4.17049e7			1.30921			0.654605	+	0.755972i	$0.272835\pi$
$318$		−	6.85504e6i		−	0.213171i
$319$		−	3.52853e7i		−	1.08698i
$320$		−	3.53067e6i		−	0.107747i
$321$		−	3.51169e7i		−	1.06170i
$322$	3.22864e6	−	1.12972e7i	0.0967058	−	0.338378i
$323$	−891098.			−0.0264434
$324$	−2.97352e6			−0.0874251
$325$	9.33505e7			2.71936
$326$	2.23235e7			0.644330
$327$		−	3.02864e7i		−	0.866173i
$328$	1.83691e6			0.0520554
$329$			4.06286e7i			1.14089i
$330$	−2.23337e7			−0.621469
$331$	−4.94227e7			−1.36283			−0.681417	−	0.731896i	$-0.738636\pi$
$331$	−4.94227e7			−1.36283			−0.681417	+	0.731896i	$0.738636\pi$
$332$		−	1.14219e7i		−	0.312123i
$333$			1.54010e7i			0.417076i
$334$	−1.86152e7			−0.499606
$335$	−1.08312e8			−2.88099
$336$		−	6.77615e6i		−	0.178634i
$337$			7.33390e7i			1.91622i	0.286402	+	0.958110i	$0.407541\pi$
$337$			7.33390e7i			1.91622i	−0.286402	+	0.958110i	$0.592459\pi$
$338$	−1.30022e7			−0.336719
$339$			3.24106e7i			0.831933i
$340$	6.16359e6			0.156818
$341$			2.56318e7i			0.646423i
$342$			1.43069e6i			0.0357657i
$343$		−	4.36468e7i		−	1.08161i
$344$			1.11567e7i			0.274068i
$345$	−1.14109e7	+	3.99272e7i	−0.277883	+	0.972326i
$346$	7.88246e6			0.190298
$347$	−7.00163e7			−1.67576			−0.837878	−	0.545857i	$-0.816204\pi$
$347$	−7.00163e7			−1.67576			−0.837878	+	0.545857i	$0.816204\pi$
$348$	1.56346e7			0.370979
$349$	−4.25755e6			−0.100157			−0.0500787	−	0.998745i	$-0.515947\pi$
$349$	−4.25755e6			−0.100157			−0.0500787	+	0.998745i	$0.515947\pi$
$350$		−	3.12024e7i		−	0.727752i
$351$	−1.09439e7			−0.253077
$352$		−	5.87726e7i		−	1.34756i
$353$	−2.44696e7			−0.556292			−0.278146	−	0.960539i	$-0.589720\pi$
$353$	−2.44696e7			−0.556292			−0.278146	+	0.960539i	$0.589720\pi$
$354$	−1.35573e7			−0.305606
$355$			1.11407e8i			2.49015i
$356$			2.90261e7i			0.643338i
$357$	−2.27837e6			−0.0500749
$358$	2.06434e7			0.449917
$359$		−	4.27662e7i		−	0.924310i	−0.886799	−	0.462155i	$-0.847076\pi$
$359$		−	4.27662e7i		−	0.924310i	0.886799	−	0.462155i	$-0.152924\pi$
$360$		−	2.24728e7i		−	0.481670i
$361$	4.45051e7			0.945994
$362$			1.78286e7i			0.375831i
$363$	−2.13103e7			−0.445523
$364$		−	3.80365e7i		−	0.788673i
$365$			8.61995e6i			0.177266i
$366$			1.21287e7i			0.247384i
$367$			3.81457e7i			0.771699i	0.922562	+	0.385849i	$0.126092\pi$
$367$			3.81457e7i			0.771699i	−0.922562	+	0.385849i	$0.873908\pi$
$368$	−1.94507e7	−	5.55887e6i	−0.390295	−	0.111543i
$369$	1.05675e6			0.0210327
$370$	−5.12543e7			−1.01187
$371$	3.11263e7			0.609545
$372$	−1.13572e7			−0.220619
$373$			8.90738e7i			1.71642i	0.513299	+	0.858210i	$0.328423\pi$
$373$			8.90738e7i			1.71642i	−0.513299	+	0.858210i	$0.671577\pi$
$374$	−3.65822e6			−0.0699286
$375$			5.69496e7i			1.07993i
$376$	−6.56407e7			−1.23484
$377$	5.75425e7			1.07390
$378$			3.65801e6i			0.0677282i
$379$		−	6.77299e7i		−	1.24412i	−0.782969	−	0.622061i	$-0.786296\pi$
$379$		−	6.77299e7i		−	1.24412i	0.782969	−	0.622061i	$-0.213704\pi$
$380$	1.75741e7			0.320274
$381$	5.01535e7			0.906831
$382$			2.61199e7i			0.468578i
$383$		−	2.59319e7i		−	0.461570i	−0.973005	−	0.230785i	$-0.925871\pi$
$383$		−	2.59319e7i		−	0.461570i	0.973005	−	0.230785i	$-0.0741294\pi$
$384$	3.21685e7			0.568116
$385$		−	1.01410e8i		−	1.77704i
$386$	4.99000e7			0.867638
$387$			6.41831e6i			0.110736i
$388$		−	4.90249e7i		−	0.839309i
$389$		−	3.18918e7i		−	0.541788i	−0.962609	−	0.270894i	$-0.912681\pi$
$389$		−	3.18918e7i		−	0.541788i	0.962609	−	0.270894i	$-0.0873193\pi$
$390$		−	3.64213e7i		−	0.613991i
$391$	−1.86908e6	+	6.54001e6i	−0.0312678	+	0.109408i
$392$	2.08225e7			0.345680
$393$	−2.74964e7			−0.452999
$394$	2.38019e7			0.389156
$395$	−3.29600e7			−0.534805
$396$		−	2.16787e7i		−	0.349099i
$397$	−6.02801e7			−0.963391			−0.481696	−	0.876339i	$-0.659979\pi$
$397$	−6.02801e7			−0.963391			−0.481696	+	0.876339i	$0.659979\pi$
$398$			4.43659e6i			0.0703720i
$399$	−6.49626e6			−0.102269
$400$	−5.37222e7			−0.839410
$401$		−	8.25668e7i		−	1.28048i	−0.768176	−	0.640239i	$-0.778835\pi$
$401$		−	8.25668e7i		−	1.28048i	0.768176	−	0.640239i	$-0.221165\pi$
$402$			2.84842e7i			0.438457i
$403$	−4.17998e7			−0.638645
$404$	6.18236e7			0.937583
$405$		−	1.29284e7i		−	0.194616i
$406$		−	1.92336e7i		−	0.287397i
$407$	−1.12282e8			−1.66543
$408$		−	3.68100e6i		−	0.0541983i
$409$	1.28874e8			1.88363			0.941816	−	0.336129i	$-0.109118\pi$
$409$	1.28874e8			1.88363			0.941816	+	0.336129i	$0.109118\pi$
$410$			3.51687e6i			0.0510276i
$411$			3.31953e7i			0.478135i
$412$			8.61869e7i			1.23239i
$413$		−	6.15588e7i		−	0.873856i
$414$	1.05002e7	+	3.00088e6i	0.147978	+	0.0422909i
$415$	4.96607e7			0.694814
$416$	9.58451e7			1.33134
$417$	2.06837e6			0.0285246
$418$	−1.04306e7			−0.142817
$419$			3.80324e7i			0.517025i	0.966008	+	0.258513i	$0.0832323\pi$
$419$			3.80324e7i			0.517025i	−0.966008	+	0.258513i	$0.916768\pi$
$420$	4.49336e7			0.606490
$421$			2.33866e7i			0.313416i	0.987645	+	0.156708i	$0.0500882\pi$
$421$			2.33866e7i			0.313416i	−0.987645	+	0.156708i	$0.949912\pi$
$422$	−2.02094e7			−0.268916
$423$	−3.77624e7			−0.498929
$424$			5.02885e7i			0.659738i
$425$			1.80633e7i			0.235304i
$426$	2.92981e7			0.378975
$427$	−5.50722e7			−0.707374
$428$			1.13441e8i			1.44691i
$429$		−	7.97878e7i		−	1.01057i
$430$	−2.13601e7			−0.268657
$431$		−	3.70034e7i		−	0.462179i	−0.972932	−	0.231090i	$-0.925771\pi$
$431$		−	3.70034e7i		−	0.462179i	0.972932	−	0.231090i	$-0.0742290\pi$
$432$	6.29812e6			0.0781195
$433$			1.05066e8i			1.29419i	0.762408	+	0.647096i	$0.224017\pi$
$433$			1.05066e8i			1.29419i	−0.762408	+	0.647096i	$0.775983\pi$
$434$			1.39716e7i			0.170914i
$435$			6.79766e7i			0.825832i
$436$			9.78370e7i			1.18044i
$437$	−5.32925e6	+	1.86473e7i	−0.0638590	+	0.223446i
$438$	2.26691e6			0.0269781
$439$	−4.23482e7			−0.500543			−0.250271	−	0.968176i	$-0.580520\pi$
$439$	−4.23482e7			−0.500543			−0.250271	+	0.968176i	$0.580520\pi$
$440$	1.63840e8			1.92337
$441$	1.19790e7			0.139670
$442$		−	5.96575e6i		−	0.0690873i
$443$	−7.62775e7			−0.877375			−0.438688	−	0.898640i	$-0.644557\pi$
$443$	−7.62775e7			−0.877375			−0.438688	+	0.898640i	$0.644557\pi$
$444$		−	4.97511e7i		−	0.568400i
$445$	−1.26201e8			−1.43213
$446$	3.72578e7			0.419965
$447$		−	8.06760e7i		−	0.903279i
$448$		−	4.21603e6i		−	0.0468889i
$449$	−2.09418e7			−0.231353			−0.115676	−	0.993287i	$-0.536904\pi$
$449$	−2.09418e7			−0.231353			−0.115676	+	0.993287i	$0.536904\pi$
$450$	2.90012e7			0.318257
$451$			7.70437e6i			0.0839861i
$452$		−	1.04699e8i		−	1.13378i
$453$	2.18734e7			0.235299
$454$		−	7.31912e7i		−	0.782152i
$455$	1.65377e8			1.75566
$456$		−	1.04955e7i		−	0.110690i
$457$		−	1.44360e8i		−	1.51251i	−0.654274	−	0.756257i	$-0.727026\pi$
$457$		−	1.44360e8i		−	1.51251i	0.654274	−	0.756257i	$-0.272974\pi$
$458$			2.94544e7i			0.306587i
$459$		−	2.11764e6i		−	0.0218985i
$460$	3.68617e7	−	1.28981e8i	0.378705	−	1.32511i
$461$	5.85463e7			0.597581			0.298790	−	0.954319i	$-0.403417\pi$
$461$	5.85463e7			0.597581			0.298790	+	0.954319i	$0.403417\pi$
$462$	−2.66691e7			−0.270447
$463$	−3.56162e7			−0.358843			−0.179422	−	0.983772i	$-0.557423\pi$
$463$	−3.56162e7			−0.358843			−0.179422	+	0.983772i	$0.557423\pi$
$464$	−3.31151e7			−0.331492
$465$		−	4.93793e7i		−	0.491118i
$466$	−1.95780e7			−0.193468
$467$			9.94961e7i			0.976912i	0.872589	+	0.488456i	$0.162440\pi$
$467$			9.94961e7i			0.976912i	−0.872589	+	0.488456i	$0.837560\pi$
$468$	3.53532e7			0.344899
$469$	−1.29337e8			−1.25373
$470$		−	1.25673e8i		−	1.21046i
$471$		−	6.78768e7i		−	0.649619i
$472$	9.94560e7			0.945813
$473$	−4.67933e7			−0.442181
$474$			8.66793e6i			0.0813918i
$475$			5.15032e7i			0.480566i
$476$	7.36004e6			0.0682432
$477$			2.89305e7i			0.266563i
$478$	2.34664e7			0.214863
$479$			1.73205e8i			1.57599i	0.615679	+	0.787997i	$0.288882\pi$
$479$			1.73205e8i			1.57599i	−0.615679	+	0.787997i	$0.711118\pi$
$480$			1.13224e8i			1.02380i
$481$		−	1.83107e8i		−	1.64539i
$482$			3.45351e7i			0.308404i
$483$	−1.36259e7	+	4.76778e7i	−0.120927	+	0.423131i
$484$	6.88407e7			0.607168
$485$	2.13152e8			1.86838
$486$	−3.39995e6			−0.0296186
$487$	4.78345e7			0.414147			0.207073	−	0.978325i	$-0.433606\pi$
$487$	4.78345e7			0.414147			0.207073	+	0.978325i	$0.433606\pi$
$488$		−	8.89762e7i		−	0.765622i
$489$	−9.42123e7			−0.805714
$490$			3.98659e7i			0.338855i
$491$	1.10447e8			0.933059			0.466530	−	0.884506i	$-0.345504\pi$
$491$	1.10447e8			0.933059			0.466530	+	0.884506i	$0.345504\pi$
$492$	−3.41373e6			−0.0286638
$493$			1.11344e7i			0.0929240i
$494$		−	1.70099e7i		−	0.141098i
$495$	9.42556e7			0.777126
$496$	2.40554e7			0.197136
$497$			1.33032e8i			1.08365i
$498$		−	1.30600e7i		−	0.105743i
$499$	3.75153e7			0.301930			0.150965	−	0.988539i	$-0.451762\pi$
$499$	3.75153e7			0.301930			0.150965	+	0.988539i	$0.451762\pi$
$500$		−	1.83970e8i		−	1.47176i
$501$	7.85621e7			0.624741
$502$		−	1.08856e7i		−	0.0860482i
$503$		−	101609.i		−	0.000798411i	−1.00000		0.000399205i	$-0.999873\pi$
$503$		−	101609.i		−	0.000798411i	1.00000		0.000399205i	$-0.000127071\pi$
$504$		−	2.68351e7i		−	0.209610i
$505$			2.68799e8i			2.08715i
$506$	−2.18782e7	+	7.65528e7i	−0.168873	+	0.590894i
$507$	5.48737e7			0.421056
$508$	−1.62016e8			−1.23585
$509$	9.08093e7			0.688616			0.344308	−	0.938857i	$-0.388114\pi$
$509$	9.08093e7			0.688616			0.344308	+	0.938857i	$0.388114\pi$
$510$	7.04751e6			0.0531282
$511$			1.02932e7i			0.0771416i
$512$	−1.00105e8			−0.745839
$513$		−	6.03797e6i		−	0.0447238i
$514$	−2.19166e6			−0.0161393
$515$	−3.74726e8			−2.74342
$516$		−	2.07337e7i		−	0.150913i
$517$		−	2.75311e8i		−	1.99228i
$518$	−6.12036e7			−0.440339
$519$	−3.32666e7			−0.237961
$520$			2.67187e8i			1.90022i
$521$		−	1.52672e8i		−	1.07956i	−0.841806	−	0.539779i	$-0.818508\pi$
$521$		−	1.52672e8i		−	1.07956i	0.841806	−	0.539779i	$-0.181492\pi$
$522$	1.78767e7			0.125683
$523$			1.29320e8i			0.903982i	0.892023	+	0.451991i	$0.149286\pi$
$523$			1.29320e8i			0.903982i	−0.892023	+	0.451991i	$0.850714\pi$
$524$	8.88241e7			0.617358
$525$			1.31684e8i			0.910030i
$526$		−	3.15998e7i		−	0.217133i
$527$		−	8.08824e6i		−	0.0552614i
$528$			4.59170e7i			0.311941i
$529$	1.25680e8	+	7.82257e7i	0.848981	+	0.528424i
$530$	−9.62805e7			−0.646711
$531$	5.72161e7			0.382150
$532$	2.09855e7			0.139375
$533$	−1.25641e7			−0.0829756
$534$			3.31888e7i			0.217955i
$535$	−4.93224e8			−3.22094
$536$		−	2.08960e8i		−	1.35697i
$537$	−8.71219e7			−0.562606
$538$	−8.84503e7			−0.568005
$539$			8.73338e7i			0.557720i
$540$			4.17638e7i			0.265227i
$541$	3.69700e7			0.233484			0.116742	−	0.993162i	$-0.462755\pi$
$541$	3.69700e7			0.233484			0.116742	+	0.993162i	$0.462755\pi$
$542$	3.03153e7			0.190399
$543$		−	7.52426e7i		−	0.469964i
$544$			1.85460e7i			0.115200i
$545$	−4.25379e8			−2.62777
$546$		−	4.34913e7i		−	0.267193i
$547$	−1.45850e8			−0.891140			−0.445570	−	0.895247i	$-0.646999\pi$
$547$	−1.45850e8			−0.891140			−0.445570	+	0.895247i	$0.646999\pi$
$548$		−	1.07234e8i		−	0.651613i
$549$		−	5.11871e7i		−	0.309345i
$550$			2.11436e8i			1.27084i
$551$			3.17473e7i			0.189781i
$552$	−7.70295e7	−	2.20144e7i	−0.457973	−	0.130885i
$553$	−3.93580e7			−0.232733
$554$	−8.21719e7			−0.483275
$555$	2.16310e8			1.26531
$556$	−6.68164e6			−0.0388739
$557$		−	1.78399e8i		−	1.03235i	−0.856483	−	0.516175i	$-0.827355\pi$
$557$		−	1.78399e8i		−	1.03235i	0.856483	−	0.516175i	$-0.172645\pi$
$558$	−1.29860e7			−0.0747431
$559$		−	7.63095e7i		−	0.436861i
$560$	−9.51725e7			−0.541935
$561$	1.54389e7			0.0874435
$562$		−	2.92349e7i		−	0.164699i
$563$			1.68361e8i			0.943442i	0.881748	+	0.471721i	$0.156367\pi$
$563$			1.68361e8i			0.943442i	−0.881748	+	0.471721i	$0.843633\pi$
$564$	1.21988e8			0.679952
$565$	4.55214e8			2.52389
$566$		−	6.59634e6i		−	0.0363792i
$567$		−	1.54380e7i		−	0.0846918i
$568$	−2.14931e8			−1.17288
$569$		−	1.19163e8i		−	0.646852i	−0.946253	−	0.323426i	$-0.895165\pi$
$569$		−	1.19163e8i		−	0.646852i	0.946253	−	0.323426i	$-0.104835\pi$
$570$	2.00943e7			0.108505
$571$		−	2.59509e7i		−	0.139394i	−0.997568	−	0.0696970i	$-0.977797\pi$
$571$		−	2.59509e7i		−	0.139394i	0.997568	−	0.0696970i	$-0.0222032\pi$
$572$			2.57746e8i			1.37722i
$573$		−	1.10235e8i		−	0.585942i
$574$			4.19956e6i			0.0222059i
$575$	3.77996e8	+	1.08028e8i	1.98831	+	0.568242i
$576$	3.91861e6			0.0205052
$577$	8.32275e7			0.433251			0.216625	−	0.976255i	$-0.430495\pi$
$577$	8.32275e7			0.433251			0.216625	+	0.976255i	$0.430495\pi$
$578$	−8.80017e7			−0.455730
$579$	−2.10594e8			−1.08495
$580$		−	2.19591e8i		−	1.12546i
$581$	5.93007e7			0.302365
$582$		−	5.60556e7i		−	0.284348i
$583$	−2.10920e8			−1.06442
$584$	−1.66300e7			−0.0834937
$585$			1.53710e8i			0.767776i
$586$		−	1.68725e8i		−	0.838468i
$587$	−6.50290e6			−0.0321509			−0.0160754	−	0.999871i	$-0.505117\pi$
$587$	−6.50290e6			−0.0321509			−0.0160754	+	0.999871i	$0.505117\pi$
$588$	−3.86968e7			−0.190346
$589$		−	2.30617e7i		−	0.112862i
$590$			1.90415e8i			0.927138i
$591$	−1.00452e8			−0.486627
$592$			1.05376e8i			0.507899i
$593$	7.43776e7			0.356680			0.178340	−	0.983969i	$-0.442927\pi$
$593$	7.43776e7			0.356680			0.178340	+	0.983969i	$0.442927\pi$
$594$		−	2.47877e7i		−	0.118270i
$595$			3.20002e7i			0.151916i
$596$			2.60615e8i			1.23101i
$597$		−	1.87238e7i		−	0.0879978i
$598$	−1.24841e8	−	3.56784e7i	−0.583784	−	0.166841i
$599$	1.01274e8			0.471213			0.235606	−	0.971849i	$-0.424292\pi$
$599$	1.01274e8			0.471213			0.235606	+	0.971849i	$0.424292\pi$
$600$	−2.12753e8			−0.984966
$601$	2.15901e8			0.994561			0.497281	−	0.867590i	$-0.334332\pi$
$601$	2.15901e8			0.994561			0.497281	+	0.867590i	$0.334332\pi$
$602$	−2.55064e7			−0.116912
$603$		−	1.20213e8i		−	0.548275i
$604$	−7.06595e7			−0.320671
$605$			2.99308e8i			1.35161i
$606$	7.06896e7			0.317642
$607$	3.76162e8			1.68193			0.840966	−	0.541088i	$-0.181987\pi$
$607$	3.76162e8			1.68193			0.840966	+	0.541088i	$0.181987\pi$
$608$			5.28795e7i			0.235275i
$609$			8.11720e7i			0.359380i
$610$	1.70350e8			0.750505
$611$	4.48971e8			1.96831
$612$			6.84083e6i			0.0298438i
$613$		−	2.12207e8i		−	0.921250i	−0.887595	−	0.460625i	$-0.847625\pi$
$613$		−	2.12207e8i		−	0.921250i	0.887595	−	0.460625i	$-0.152375\pi$
$614$	−1.91770e8			−0.828468
$615$		−	1.48423e7i		−	0.0638083i
$616$	1.95644e8			0.836998
$617$		−	2.28882e8i		−	0.974443i	−0.873278	−	0.487222i	$-0.838010\pi$
$617$		−	2.28882e8i		−	0.974443i	0.873278	−	0.487222i	$-0.161990\pi$
$618$			9.85468e7i			0.417520i
$619$		−	2.46253e8i		−	1.03827i	−0.854693	−	0.519133i	$-0.826255\pi$
$619$		−	2.46253e8i		−	1.03827i	0.854693	−	0.519133i	$-0.173745\pi$
$620$			1.59515e8i			0.669307i
$621$	−4.43143e7	−	1.26647e7i	−0.185042	−	0.0528834i
$622$	1.42469e8			0.592036
$623$	−1.50698e8			−0.623225
$624$	−7.48805e7			−0.308187
$625$	2.95008e8			1.20835
$626$			1.92894e8i			0.786314i
$627$	4.40204e7			0.178588
$628$			2.19269e8i			0.885315i
$629$	3.54311e7			0.142375
$630$	5.13775e7			0.205471
$631$			7.39331e7i			0.294273i	0.989116	+	0.147137i	$0.0470057\pi$
$631$			7.39331e7i			0.294273i	−0.989116	+	0.147137i	$0.952994\pi$
$632$		−	6.35879e7i		−	0.251897i
$633$	8.52903e7			0.336270
$634$	−1.54044e8			−0.604472
$635$		−	7.04417e8i		−	2.75111i
$636$		−	9.34569e7i		−	0.363279i
$637$	−1.42422e8			−0.551010
$638$			1.30332e8i			0.501868i
$639$	−1.23648e8			−0.473896
$640$		−	4.51814e8i		−	1.72353i
$641$		−	4.71156e8i		−	1.78892i	−0.447149	−	0.894460i	$-0.647561\pi$
$641$		−	4.71156e8i		−	1.78892i	0.447149	−	0.894460i	$-0.352439\pi$
$642$			1.29710e8i			0.490194i
$643$		−	4.36101e8i		−	1.64042i	−0.572065	−	0.820208i	$-0.693857\pi$
$643$		−	4.36101e8i		−	1.64042i	0.572065	−	0.820208i	$-0.306143\pi$
$644$	4.40171e7	−	1.54018e8i	0.164803	−	0.576652i
$645$	9.01466e7			0.335946
$646$	3.29141e6			0.0122091
$647$	2.21024e8			0.816067			0.408034	−	0.912967i	$-0.366215\pi$
$647$	2.21024e8			0.816067			0.408034	+	0.912967i	$0.366215\pi$
$648$	2.49420e7			0.0916657
$649$			4.17139e8i			1.52597i
$650$	−3.44805e8			−1.25555
$651$		−	5.89647e7i		−	0.213722i
$652$	3.04343e8			1.09805
$653$	−1.19383e8			−0.428749			−0.214374	−	0.976752i	$-0.568771\pi$
$653$	−1.19383e8			−0.428749			−0.214374	+	0.976752i	$0.568771\pi$
$654$			1.11868e8i			0.399919i
$655$			3.86192e8i			1.37429i
$656$	7.23052e6			0.0256129
$657$	−9.56708e6			−0.0337352
$658$		−	1.50068e8i		−	0.526758i
$659$			3.11263e8i			1.08760i	0.839213	+	0.543802i	$0.183016\pi$
$659$			3.11263e8i			1.08760i	−0.839213	+	0.543802i	$0.816984\pi$
$660$	−3.04483e8			−1.05908
$661$			1.51166e8i			0.523421i	0.965146	+	0.261710i	$0.0842865\pi$
$661$			1.51166e8i			0.523421i	−0.965146	+	0.261710i	$0.915714\pi$
$662$	1.82551e8			0.629231
$663$			2.51774e7i			0.0863913i
$664$			9.58078e7i			0.327263i
$665$			9.12413e7i			0.310261i
$666$		−	5.68859e7i		−	0.192567i
$667$	2.33002e8	+	6.65901e7i	0.785203	+	0.224405i
$668$	−2.53787e8			−0.851411
$669$	−1.57240e8			−0.525152
$670$	4.00067e8			1.33017
$671$	3.73184e8			1.23525
$672$			1.35203e8i			0.445532i
$673$	4.67833e6			0.0153478			0.00767389	−	0.999971i	$-0.497557\pi$
$673$	4.67833e6			0.0153478			0.00767389	+	0.999971i	$0.497557\pi$
$674$		−	2.70889e8i		−	0.884733i
$675$	−1.22394e8			−0.397970
$676$	−1.77264e8			−0.573825
$677$			1.72661e8i			0.556454i	0.960515	+	0.278227i	$0.0897468\pi$
$677$			1.72661e8i			0.556454i	−0.960515	+	0.278227i	$0.910253\pi$
$678$		−	1.19714e8i		−	0.384110i
$679$	2.54529e8			0.813069
$680$	−5.17005e7			−0.164425
$681$			3.08891e8i			0.978056i
$682$		−	9.46753e7i		−	0.298458i
$683$	−4.96528e8			−1.55841			−0.779205	−	0.626769i	$-0.784377\pi$
$683$	−4.96528e8			−1.55841			−0.779205	+	0.626769i	$0.784377\pi$
$684$			1.95050e7i			0.0609507i
$685$	4.66235e8			1.45055
$686$			1.61216e8i			0.499387i
$687$		−	1.24307e8i		−	0.383377i
$688$			4.39153e7i			0.134850i
$689$		−	3.43964e8i		−	1.05161i
$690$	4.21480e7	−	1.47478e8i	0.128301	−	0.448931i
$691$	3.22850e8			0.978512			0.489256	−	0.872140i	$-0.337268\pi$
$691$	3.22850e8			0.978512			0.489256	+	0.872140i	$0.337268\pi$
$692$	1.07464e8			0.324299
$693$	1.12552e8			0.338185
$694$	2.58617e8			0.773710
$695$		−	2.90506e7i		−	0.0865369i
$696$	−1.31144e8			−0.388973
$697$		−	2.43115e6i		−	0.00717981i
$698$	1.57259e7			0.0462435
$699$	8.26254e7			0.241926
$700$		−	4.25392e8i		−	1.24021i
$701$			7.25912e7i			0.210732i	0.994434	+	0.105366i	$0.0336014\pi$
$701$			7.25912e7i			0.210732i	−0.994434	+	0.105366i	$0.966399\pi$
$702$	4.04232e7			0.116847
$703$	1.01024e8			0.290775
$704$			2.85690e7i			0.0818798i
$705$			5.30382e8i			1.51364i
$706$	9.03824e7			0.256844
$707$			3.20977e8i			0.908271i
$708$	−1.84830e8			−0.520803
$709$			3.61431e8i			1.01411i	0.861912	+	0.507057i	$0.169267\pi$
$709$			3.61431e8i			1.01411i	−0.861912	+	0.507057i	$0.830733\pi$
$710$		−	4.11498e8i		−	1.14972i
$711$		−	3.65815e7i		−	0.101778i
$712$		−	2.43473e8i		−	0.674544i
$713$	−1.69256e8	−	4.83721e7i	−0.466957	−	0.133452i
$714$	8.41554e6			0.0231200
$715$	−1.12064e9			−3.06582
$716$	2.81438e8			0.766733
$717$	−9.90357e7			−0.268679
$718$			1.57964e8i			0.426761i
$719$	−3.68121e8			−0.990384			−0.495192	−	0.868784i	$-0.664902\pi$
$719$	−3.68121e8			−0.990384			−0.495192	+	0.868784i	$0.664902\pi$
$720$		−	8.84585e7i		−	0.236996i
$721$	−4.47467e8			−1.19386
$722$	−1.64387e8			−0.436773
$723$		−	1.45749e8i		−	0.385649i
$724$			2.43063e8i			0.640478i
$725$	6.43542e8			1.68874
$726$	7.87131e7			0.205701
$727$		−	4.93322e8i		−	1.28389i	−0.766751	−	0.641944i	$-0.778128\pi$
$727$		−	4.93322e8i		−	1.28389i	0.766751	−	0.641944i	$-0.221872\pi$
$728$			3.19052e8i			0.826928i
$729$	1.43489e7			0.0370370
$730$		−	3.18392e7i		−	0.0818452i
$731$	1.47658e7			0.0378012
$732$			1.65355e8i			0.421583i
$733$			6.84962e8i			1.73922i	0.493738	+	0.869611i	$0.335630\pi$
$733$			6.84962e8i			1.73922i	−0.493738	+	0.869611i	$0.664370\pi$
$734$		−	1.40897e8i		−	0.356299i
$735$		−	1.68247e8i		−	0.423727i
$736$	3.88097e8	+	1.10915e8i	0.973435	+	0.278200i
$737$	8.76423e8			2.18933
$738$	−3.90329e6			−0.00971096
$739$	−2.60679e8			−0.645911			−0.322956	−	0.946414i	$-0.604676\pi$
$739$	−2.60679e8			−0.645911			−0.322956	+	0.946414i	$0.604676\pi$
$740$	−6.98766e8			−1.72439
$741$			7.17875e7i			0.176439i
$742$	−1.14970e8			−0.281432
$743$			6.77473e7i			0.165168i	0.996584	+	0.0825839i	$0.0263172\pi$
$743$			6.77473e7i			0.165168i	−0.996584	+	0.0825839i	$0.973683\pi$
$744$	9.52649e7			0.231320
$745$	−1.13311e9			−2.74034
$746$		−	3.29008e8i		−	0.792484i
$747$			5.51173e7i			0.132229i
$748$	−4.98737e7			−0.119170
$749$	−5.88967e8			−1.40167
$750$		−	2.10353e8i		−	0.498613i
$751$			1.58709e8i			0.374698i	0.982293	+	0.187349i	$0.0599896\pi$
$751$			1.58709e8i			0.374698i	−0.982293	+	0.187349i	$0.940010\pi$
$752$	−2.58378e8			−0.607578
$753$			4.59408e7i			0.107600i
$754$	−2.12543e8			−0.495829
$755$		−	3.07216e8i		−	0.713843i
$756$			4.98708e7i			0.115420i
$757$		−	4.53416e8i		−	1.04522i	−0.852571	−	0.522612i	$-0.824958\pi$
$757$		−	4.53416e8i		−	1.04522i	0.852571	−	0.522612i	$-0.175042\pi$
$758$			2.50171e8i			0.574421i
$759$	9.23330e7	−	3.23078e8i	0.211170	−	0.738893i
$760$	−1.47412e8			−0.335809
$761$	1.89469e8			0.429917			0.214958	−	0.976623i	$-0.431038\pi$
$761$	1.89469e8			0.429917			0.214958	+	0.976623i	$0.431038\pi$
$762$	−1.85250e8			−0.418691
$763$	−5.07952e8			−1.14353
$764$			3.56101e8i			0.798534i
$765$	−2.97428e7			−0.0664350
$766$			9.57836e7i			0.213110i
$767$	−6.80261e8			−1.50761
$768$	−1.34908e8			−0.297820
$769$		−	4.26727e8i		−	0.938363i	−0.883102	−	0.469182i	$-0.844549\pi$
$769$		−	4.26727e8i		−	0.938363i	0.883102	−	0.469182i	$-0.155451\pi$
$770$			3.74573e8i			0.820472i
$771$	9.24953e6			0.0201816
$772$	6.80303e8			1.47860
$773$			3.03849e8i			0.657839i	0.944358	+	0.328919i	$0.106684\pi$
$773$			3.03849e8i			0.657839i	−0.944358	+	0.328919i	$0.893316\pi$
$774$		−	2.37071e7i		−	0.0511276i
$775$	−4.67480e8			−1.00429
$776$			4.11223e8i			0.880020i
$777$	2.58299e8			0.550630
$778$			1.17797e8i			0.250148i
$779$		−	6.93186e6i		−	0.0146635i
$780$		−	4.96544e8i		−	1.04634i
$781$		−	9.01464e8i		−	1.89232i
$782$	6.90376e6	−	2.41566e7i	0.0144366	−	0.0505144i
$783$	−7.54457e7			−0.157163
$784$	8.19624e7			0.170085
$785$	−9.53344e8			−1.97079
$786$	1.01562e8			0.209153
$787$			2.54470e8i			0.522050i	0.965332	+	0.261025i	$0.0840604\pi$
$787$			2.54470e8i			0.522050i	−0.965332	+	0.261025i	$0.915940\pi$
$788$	3.24499e8			0.663186
$789$			1.33361e8i			0.271518i
$790$	1.21743e8			0.246924
$791$	5.43579e8			1.09833
$792$			1.81842e8i			0.366032i
$793$			6.08581e8i			1.22039i
$794$	2.22654e8			0.444805
$795$	4.06335e8			0.808691
$796$			6.04854e7i			0.119925i
$797$			1.67207e8i			0.330277i	0.986270	+	0.165139i	$0.0528071\pi$
$797$			1.67207e8i			0.330277i	−0.986270	+	0.165139i	$0.947193\pi$
$798$	2.39950e7			0.0472184
$799$			8.68755e7i			0.170317i
$800$	1.07191e9			2.09357
$801$		−	1.40067e8i		−	0.272546i
$802$			3.04974e8i			0.591207i
$803$		−	6.97497e7i		−	0.134709i
$804$			3.88335e8i			0.747202i
$805$	6.69645e8	+	1.91379e8i	1.28368	+	0.366865i
$806$	1.54394e8			0.294867
$807$	3.73289e8			0.710272
$808$	−5.18579e8			−0.983061
$809$	3.49295e8			0.659700			0.329850	−	0.944033i	$-0.393002\pi$
$809$	3.49295e8			0.659700			0.329850	+	0.944033i	$0.393002\pi$
$810$			4.77531e7i			0.0898558i
$811$	−1.94555e8			−0.364737			−0.182369	−	0.983230i	$-0.558376\pi$
$811$	−1.94555e8			−0.364737			−0.182369	+	0.983230i	$0.558376\pi$
$812$		−	2.62218e8i		−	0.489772i
$813$	−1.27941e8			−0.238088
$814$	4.14732e8			0.768943
$815$			1.32323e9i			2.44435i
$816$		−	1.44893e7i		−	0.0266672i
$817$	4.21014e7			0.0772023
$818$	−4.76018e8			−0.869687
$819$			1.83548e8i			0.334116i
$820$			4.79466e7i			0.0869594i
$821$	−5.79161e8			−1.04657			−0.523287	−	0.852157i	$-0.675294\pi$
$821$	−5.79161e8			−1.04657			−0.523287	+	0.852157i	$0.675294\pi$
$822$		−	1.22612e8i		−	0.220759i
$823$	7.10616e8			1.27478			0.637390	−	0.770541i	$-0.280014\pi$
$823$	7.10616e8			1.27478			0.637390	+	0.770541i	$0.280014\pi$
$824$		−	7.22939e8i		−	1.29217i
$825$		−	8.92328e8i		−	1.58914i
$826$			2.27377e8i			0.403466i
$827$		−	8.70562e7i		−	0.153916i	−0.997034	−	0.0769579i	$-0.975479\pi$
$827$		−	8.70562e7i		−	0.153916i	0.997034	−	0.0769579i	$-0.0245207\pi$
$828$	1.43153e8	+	4.09119e7i	0.252179	+	0.0720707i
$829$	−3.58917e8			−0.629985			−0.314993	−	0.949094i	$-0.602002\pi$
$829$	−3.58917e8			−0.629985			−0.314993	+	0.949094i	$0.602002\pi$
$830$	−1.83430e8			−0.320801
$831$	3.46792e8			0.604319
$832$	−4.65897e7			−0.0808946
$833$		−	2.75586e7i		−	0.0476784i
$834$	−7.63985e6			−0.0131700
$835$		−	1.10342e9i		−	1.89532i
$836$	−1.42203e8			−0.243383
$837$	5.48050e7			0.0934638
$838$		−	1.40479e8i		−	0.238714i
$839$		−	5.88198e8i		−	0.995950i	−0.867191	−	0.497975i	$-0.834077\pi$
$839$		−	5.88198e8i		−	0.995950i	0.867191	−	0.497975i	$-0.165923\pi$
$840$	−3.76905e8			−0.635908
$841$	−1.98135e8			−0.333098
$842$		−	8.63823e7i		−	0.144707i
$843$			1.23381e8i			0.205951i
$844$	−2.75521e8			−0.458277
$845$		−	7.70713e8i		−	1.27739i
$846$	1.39482e8			0.230360
$847$			3.57408e8i			0.588186i
$848$			1.97948e8i			0.324611i
$849$			2.78387e7i			0.0454910i
$850$		−	6.67195e7i		−	0.108642i
$851$	2.11898e8	−	7.41440e8i	0.343825	−	1.20306i
$852$	3.99430e8			0.645836
$853$	−4.28088e8			−0.689740			−0.344870	−	0.938650i	$-0.612077\pi$
$853$	−4.28088e8			−0.689740			−0.344870	+	0.938650i	$0.612077\pi$
$854$	2.03418e8			0.326600
$855$	−8.48046e7			−0.135682
$856$		−	9.51551e8i		−	1.51709i
$857$	2.10898e8			0.335066			0.167533	−	0.985867i	$-0.446420\pi$
$857$	2.10898e8			0.335066			0.167533	+	0.985867i	$0.446420\pi$
$858$			2.94709e8i			0.466586i
$859$	2.05527e7			0.0324257			0.0162128	−	0.999869i	$-0.494839\pi$
$859$	2.05527e7			0.0324257			0.0162128	+	0.999869i	$0.494839\pi$
$860$	−2.91209e8			−0.457835
$861$		−	1.77235e7i		−	0.0277677i
$862$			1.36678e8i			0.213392i
$863$	6.24282e8			0.971289			0.485644	−	0.874156i	$-0.338585\pi$
$863$	6.24282e8			0.971289			0.485644	+	0.874156i	$0.338585\pi$
$864$	−1.25665e8			−0.194838
$865$			4.67236e8i			0.721918i
$866$		−	3.88079e8i		−	0.597539i
$867$	3.71396e8			0.569875
$868$			1.90479e8i			0.291265i
$869$	2.66701e8			0.406411
$870$		−	2.51083e8i		−	0.381293i
$871$			1.42925e9i			2.16299i
$872$		−	8.20661e8i		−	1.23770i
$873$			2.36573e8i			0.355567i
$874$	1.96845e7	−	6.88769e7i	0.0294842	−	0.103167i
$875$	9.55137e8			1.42574
$876$	3.09054e7			0.0459751
$877$	1.30626e8			0.193656			0.0968281	−	0.995301i	$-0.469130\pi$
$877$	1.30626e8			0.193656			0.0968281	+	0.995301i	$0.469130\pi$
$878$	1.56420e8			0.231104
$879$			7.12075e8i			1.04848i
$880$	6.44915e8			0.946355
$881$			5.42231e8i			0.792970i	0.918041	+	0.396485i	$0.129770\pi$
$881$			5.42231e8i			0.792970i	−0.918041	+	0.396485i	$0.870230\pi$
$882$	−4.42463e7			−0.0644868
$883$	−9.62220e8			−1.39763			−0.698815	−	0.715302i	$-0.746289\pi$
$883$	−9.62220e8			−1.39763			−0.698815	+	0.715302i	$0.746289\pi$
$884$		−	8.13329e7i		−	0.117736i
$885$		−	8.03612e8i		−	1.15936i
$886$	2.81743e8			0.405091
$887$	−2.90513e8			−0.416288			−0.208144	−	0.978098i	$-0.566742\pi$
$887$	−2.90513e8			−0.416288			−0.208144	+	0.978098i	$0.566742\pi$
$888$			4.17315e8i			0.595971i
$889$		−	8.41156e8i		−	1.19721i
$890$	4.66143e8			0.661225
$891$			1.04612e8i			0.147893i
$892$	5.07947e8			0.715689
$893$			2.47705e8i			0.347841i
$894$			2.97990e8i			0.417051i
$895$			1.22365e9i			1.70682i
$896$		−	5.39518e8i		−	0.750036i
$897$	5.26868e8	+	1.50575e8i	0.730003	+	0.208629i
$898$	7.73518e7			0.106817
$899$	−2.88161e8			−0.396603
$900$	3.95382e8			0.542363
$901$	6.65569e7			0.0909952
$902$		−	2.84573e7i		−	0.0387770i
$903$	1.07645e8			0.146195
$904$			8.78220e8i			1.18877i
$905$	−1.05680e9			−1.42576
$906$	−8.07928e7			−0.108640
$907$			2.33369e8i			0.312768i	0.987696	+	0.156384i	$0.0499837\pi$
$907$			2.33369e8i			0.312768i	−0.987696	+	0.156384i	$0.950016\pi$
$908$		−	9.97839e8i		−	1.33292i
$909$	−2.98333e8			−0.397201
$910$	−6.10845e8			−0.810600
$911$			1.10739e8i			0.146469i	0.997315	+	0.0732347i	$0.0233322\pi$
$911$			1.10739e8i			0.146469i	−0.997315	+	0.0732347i	$0.976668\pi$
$912$		−	4.13130e7i		−	0.0544630i
$913$	−4.01838e8			−0.528005
$914$			5.33219e8i			0.698340i
$915$	−7.18934e8			−0.938481
$916$			4.01562e8i			0.522475i
$917$			4.61158e8i			0.598056i
$918$			7.82186e6i			0.0101107i
$919$			7.56808e8i			0.975077i	0.873101	+	0.487539i	$0.162105\pi$
$919$			7.56808e8i			0.975077i	−0.873101	+	0.487539i	$0.837895\pi$
$920$	−3.09197e8	+	1.08190e9i	−0.397075	+	1.38938i
$921$	8.09332e8			1.03597
$922$	−2.16250e8			−0.275908
$923$	1.47009e9			1.86955
$924$	−3.63588e8			−0.460886
$925$		−	2.04783e9i		−	2.58743i
$926$	1.31554e8			0.165681
$927$		−	4.15900e8i		−	0.522095i
$928$	6.60740e8			0.826773
$929$	−3.59892e7			−0.0448874			−0.0224437	−	0.999748i	$-0.507145\pi$
$929$	−3.59892e7			−0.0448874			−0.0224437	+	0.999748i	$0.507145\pi$
$930$			1.82391e8i			0.226753i
$931$		−	7.85769e7i		−	0.0973747i
$932$	−2.66913e8			−0.329702
$933$	−6.01264e8			−0.740322
$934$		−	3.67505e8i		−	0.451048i
$935$		−	2.16842e8i		−	0.265283i
$936$	−2.96544e8			−0.361628
$937$		−	4.13887e8i		−	0.503110i	−0.967843	−	0.251555i	$-0.919058\pi$
$937$		−	4.13887e8i		−	0.503110i	0.967843	−	0.251555i	$-0.0809419\pi$
$938$	4.77727e8			0.578857
$939$		−	8.14076e8i		−	0.983260i
$940$		−	1.71334e9i		−	2.06282i
$941$			1.03553e9i			1.24278i	0.783500	+	0.621391i	$0.213432\pi$
$941$			1.03553e9i			1.24278i	−0.783500	+	0.621391i	$0.786568\pi$
$942$			2.50714e8i			0.299934i
$943$	−5.08748e7	−	1.45396e7i	−0.0606691	−	0.0173387i
$944$	3.91483e8			0.465369
$945$	−2.16830e8			−0.256935
$946$	1.72839e8			0.204158
$947$	1.48320e8			0.174643			0.0873214	−	0.996180i	$-0.472169\pi$
$947$	1.48320e8			0.174643			0.0873214	+	0.996180i	$0.472169\pi$
$948$			1.18173e8i			0.138705i
$949$	1.13746e8			0.133088
$950$		−	1.90235e8i		−	0.221881i
$951$	6.50115e8			0.755872
$952$	−6.17364e7			−0.0715534
$953$		−	9.98605e8i		−	1.15376i	−0.816829	−	0.576879i	$-0.804270\pi$
$953$		−	9.98605e8i		−	1.15376i	0.816829	−	0.576879i	$-0.195730\pi$
$954$		−	1.06859e8i		−	0.123074i
$955$	−1.54827e9			−1.77761
$956$	3.19924e8			0.366162
$957$		−	5.50044e8i		−	0.627569i
$958$		−	6.39762e8i		−	0.727649i
$959$	5.56738e8			0.631241
$960$		−	5.50377e7i		−	0.0622080i
$961$	−6.78179e8			−0.764142
$962$			6.76336e8i			0.759692i
$963$		−	5.47418e8i		−	0.612971i
$964$			4.70828e8i			0.525571i
$965$			2.95784e9i			3.29149i
$966$	5.03296e7	−	1.76106e8i	0.0558331	−	0.195363i
$967$	−1.25410e9			−1.38692			−0.693462	−	0.720493i	$-0.743915\pi$
$967$	−1.25410e9			−1.38692			−0.693462	+	0.720493i	$0.743915\pi$
$968$	−5.77439e8			−0.636619
$969$	−1.38908e7			−0.0152671
$970$	−7.87312e8			−0.862644
$971$		−	2.21220e8i		−	0.241639i	−0.992675	−	0.120819i	$-0.961448\pi$
$971$		−	2.21220e8i		−	0.241639i	0.992675	−	0.120819i	$-0.0385522\pi$
$972$	−4.63526e7			−0.0504749
$973$		−	3.46899e7i		−	0.0376586i
$974$	−1.76684e8			−0.191215
$975$	1.45519e9			1.57002
$976$		−	3.50232e8i		−	0.376709i
$977$		−	8.32816e8i		−	0.893029i	−0.894776	−	0.446514i	$-0.852665\pi$
$977$		−	8.32816e8i		−	0.893029i	0.894776	−	0.446514i	$-0.147335\pi$
$978$	3.47988e8			0.372004
$979$	1.02117e9			1.08831
$980$			5.43505e8i			0.577464i
$981$		−	4.72118e8i		−	0.500085i
$982$	−4.07954e8			−0.430801
$983$		−	7.30746e8i		−	0.769319i	−0.923059	−	0.384659i	$-0.874319\pi$
$983$		−	7.30746e8i		−	0.769319i	0.923059	−	0.384659i	$-0.125681\pi$
$984$	2.86346e7			0.0300542
$985$			1.41087e9i			1.47631i
$986$		−	4.11269e7i		−	0.0429037i
$987$			6.33337e8i			0.658694i
$988$		−	2.31902e8i		−	0.240455i
$989$	8.83078e7	−	3.08993e8i	0.0912873	−	0.319419i
$990$	−3.48148e8			−0.358805
$991$	7.30256e7			0.0750333			0.0375167	−	0.999296i	$-0.488055\pi$
$991$	7.30256e7			0.0750333			0.0375167	+	0.999296i	$0.488055\pi$
$992$	−4.79972e8			−0.491678
$993$	−7.70424e8			−0.786832
$994$		−	4.91377e8i		−	0.500329i
$995$	−2.62980e8			−0.266965
$996$		−	1.78050e8i		−	0.180204i
$997$	1.36699e8			0.137937			0.0689685	−	0.997619i	$-0.478029\pi$
$997$	1.36699e8			0.137937			0.0689685	+	0.997619i	$0.478029\pi$
$998$	−1.38569e8			−0.139404
$999$			2.40077e8i			0.240799i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.11	✓	24
3.2	odd	2		207.7.d.e.91.14		24
23.22	odd	2	inner	69.7.d.a.22.12	yes	24
69.68	even	2		207.7.d.e.91.13		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.11	✓	24	1.1	even	1	trivial
69.7.d.a.22.12	yes	24	23.22	odd	2	inner
207.7.d.e.91.13		24	69.68	even	2
207.7.d.e.91.14		24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.69366 q^{2} +15.5885 q^{3} -50.3569 q^{4} -218.943i q^{5} -57.5785 q^{6} -261.444i q^{7} +422.396 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-3.69366 q^{2} +15.5885 q^{3} -50.3569 q^{4} -218.943i q^{5} -57.5785 q^{6} -261.444i q^{7} +422.396 q^{8} +243.000 q^{9} +808.702i q^{10} +1771.61i q^{11} -784.986 q^{12} -2889.11 q^{13} +965.685i q^{14} -3412.99i q^{15} +1662.65 q^{16} -559.041i q^{17} -897.560 q^{18} -1593.98i q^{19} +11025.3i q^{20} -4075.50i q^{21} -6543.74i q^{22} +(-11698.6 - 3343.37i) q^{23} +6584.50 q^{24} -32311.1 q^{25} +10671.4 q^{26} +3788.00 q^{27} +13165.5i q^{28} -19917.0 q^{29} +12606.4i q^{30} +14468.1 q^{31} -33174.6 q^{32} +27616.7i q^{33} +2064.91i q^{34} -57241.3 q^{35} -12236.7 q^{36} +63378.4i q^{37} +5887.61i q^{38} -45036.8 q^{39} -92480.7i q^{40} +4348.78 q^{41} +15053.5i q^{42} +26412.8i q^{43} -89212.9i q^{44} -53203.2i q^{45} +(43210.8 + 12349.3i) q^{46} -155401. q^{47} +25918.2 q^{48} +49296.2 q^{49} +119346. q^{50} -8714.59i q^{51} +145487. q^{52} +119055. i q^{53} -13991.6 q^{54} +387883. q^{55} -110433. i q^{56} -24847.6i q^{57} +73566.8 q^{58} +235457. q^{59} +171867. i q^{60} -210647. i q^{61} -53440.1 q^{62} -63530.8i q^{63} +16126.0 q^{64} +632551. i q^{65} -102007. i q^{66} -494703. i q^{67} +28151.5i q^{68} +(-182363. - 52118.0i) q^{69} +211430. q^{70} -508838. q^{71} +102642. q^{72} -39370.7 q^{73} -234098. i q^{74} -503681. q^{75} +80267.6i q^{76} +463177. q^{77} +166351. q^{78} -150541. i q^{79} -364027. i q^{80} +59049.0 q^{81} -16062.9 q^{82} +226820. i q^{83} +205230. i q^{84} -122398. q^{85} -97560.0i q^{86} -310476. q^{87} +748322. i q^{88} -576409. i q^{89} +196515. i q^{90} +755340. i q^{91} +(589106. + 168362. i) q^{92} +225535. q^{93} +573999. q^{94} -348990. q^{95} -517141. q^{96} +973550. i q^{97} -182083. q^{98} +430502. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Introduction

L-functions

Modular forms

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Database

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