LMFDB - Embedded newform 147.8.c.b.146.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [147,8,Mod(146,147)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("147.146"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	147.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$45.9205987462$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			146.20
Character	$\chi$	$=$	147.146
Dual form			147.8.c.b.146.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+7.39832i q^{2} +(-33.2308 + 32.9046i) q^{3} +73.2648 q^{4} +203.095 q^{5} +(-243.439 - 245.852i) q^{6} +1489.02i q^{8} +(21.5784 - 2186.89i) q^{9} +1502.56i q^{10} +1946.33i q^{11} +(-2434.65 + 2410.75i) q^{12} -7873.88i q^{13} +(-6749.00 + 6682.74i) q^{15} -1638.36 q^{16} -9230.23 q^{17} +(16179.3 + 159.644i) q^{18} +10406.6i q^{19} +14879.7 q^{20} -14399.6 q^{22} +81873.3i q^{23} +(-48995.6 - 49481.5i) q^{24} -36877.6 q^{25} +58253.5 q^{26} +(71241.7 + 73382.4i) q^{27} +195123. i q^{29} +(-49441.0 - 49931.3i) q^{30} +97496.3i q^{31} +178474. i q^{32} +(-64043.1 - 64678.2i) q^{33} -68288.2i q^{34} +(1580.94 - 160222. i) q^{36} +570632. q^{37} -76991.4 q^{38} +(259087. + 261656. i) q^{39} +302412. i q^{40} -339491. q^{41} -356328. q^{43} +142598. i q^{44} +(4382.45 - 444146. i) q^{45} -605725. q^{46} -766151. q^{47} +(54444.1 - 53909.6i) q^{48} -272832. i q^{50} +(306728. - 303717. i) q^{51} -576879. i q^{52} +158435. i q^{53} +(-542906. + 527069. i) q^{54} +395289. i q^{55} +(-342425. - 345820. i) q^{57} -1.44358e6 q^{58} -2.56230e6 q^{59} +(-494465. + 489610. i) q^{60} -1.72901e6i q^{61} -721309. q^{62} -1.53012e6 q^{64} -1.59914e6i q^{65} +(478510. - 473812. i) q^{66} -2.20518e6 q^{67} -676251. q^{68} +(-2.69401e6 - 2.72072e6i) q^{69} +5.29805e6i q^{71} +(3.25633e6 + 32130.7i) q^{72} -2.54319e6i q^{73} +4.22172e6i q^{74} +(1.22547e6 - 1.21344e6i) q^{75} +762439. i q^{76} +(-1.93581e6 + 1.91681e6i) q^{78} +744581. q^{79} -332742. q^{80} +(-4.78204e6 - 94379.3i) q^{81} -2.51166e6i q^{82} +5.20177e6 q^{83} -1.87461e6 q^{85} -2.63623e6i q^{86} +(-6.42044e6 - 6.48411e6i) q^{87} -2.89813e6 q^{88} +1.01641e7 q^{89} +(3.28594e6 + 32422.8i) q^{90} +5.99844e6i q^{92} +(-3.20807e6 - 3.23989e6i) q^{93} -5.66823e6i q^{94} +2.11353e6i q^{95} +(-5.87260e6 - 5.93083e6i) q^{96} -3.36084e6i q^{97} +(4.25641e6 + 41998.7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 2300 q^{4} + 7032 q^{9} + 27576 q^{15} + 39188 q^{16} - 66996 q^{18} + 105132 q^{22} + 662504 q^{25} + 81324 q^{30} - 227244 q^{36} + 1237496 q^{37} - 3992208 q^{39} - 1416064 q^{43} + 6985680 q^{46}+ \cdots + 35642232 q^{99}+O(q^{100}) $ 32 * q - 2300 * q^4 + 7032 * q^9 + 27576 * q^15 + 39188 * q^16 - 66996 * q^18 + 105132 * q^22 + 662504 * q^25 + 81324 * q^30 - 227244 * q^36 + 1237496 * q^37 - 3992208 * q^39 - 1416064 * q^43 + 6985680 * q^46 - 1354680 * q^51 - 6279408 * q^57 + 4128684 * q^58 - 15042564 * q^60 - 589508 * q^64 + 3889456 * q^67 + 3088944 * q^72 + 1917432 * q^78 - 15252656 * q^79 - 17406072 * q^81 + 3418848 * q^85 - 24626796 * q^88 + 49664208 * q^93 + 35642232 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$-1$	$-1$

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 $

$2$			7.39832i			0.653925i	0.945037	+	0.326963i	$0.106025\pi$
$2$			7.39832i			0.653925i	−0.945037	+	0.326963i	$0.893975\pi$
$3$	−33.2308	+	32.9046i	−0.710587	+	0.703610i
$3$	−33.2308	+	32.9046i	−0.710587	+	0.703610i
$4$	73.2648			0.572382
$5$	203.095			0.726613			0.363306	−	0.931670i	$-0.381648\pi$
$5$	203.095			0.726613			0.363306	+	0.931670i	$0.381648\pi$
$6$	−243.439	−	245.852i	−0.460108	−	0.464671i
$7$		0			0
$7$		0			0
$8$			1489.02i			1.02822i
$9$	21.5784	−	2186.89i	0.00986667	−	0.999951i
$10$			1502.56i			0.475151i
$11$			1946.33i			0.440902i	0.975398	+	0.220451i	$0.0707529\pi$
$11$			1946.33i			0.440902i	−0.975398	+	0.220451i	$0.929247\pi$
$12$	−2434.65	+	2410.75i	−0.406727	+	0.402733i
$13$		−	7873.88i		−	0.994002i	−0.867750	−	0.497001i	$-0.834435\pi$
$13$		−	7873.88i		−	0.994002i	0.867750	−	0.497001i	$-0.165565\pi$
$14$		0			0
$15$	−6749.00	+	6682.74i	−0.516321	+	0.511252i
$16$	−1638.36			−0.0999977
$17$	−9230.23			−0.455660			−0.227830	−	0.973701i	$-0.573163\pi$
$17$	−9230.23			−0.455660			−0.227830	+	0.973701i	$0.573163\pi$
$18$	16179.3	+	159.644i	0.653894	+	0.00645206i
$19$			10406.6i			0.348074i	0.984739	+	0.174037i	$0.0556812\pi$
$19$			10406.6i			0.348074i	−0.984739	+	0.174037i	$0.944319\pi$
$20$	14879.7			0.415900
$21$		0			0
$22$	−14399.6			−0.288317
$23$			81873.3i			1.40312i	0.712610	+	0.701560i	$0.247513\pi$
$23$			81873.3i			1.40312i	−0.712610	+	0.701560i	$0.752487\pi$
$24$	−48995.6	−	49481.5i	−0.723466	−	0.730640i
$25$	−36877.6			−0.472034
$26$	58253.5			0.650003
$27$	71241.7	+	73382.4i	0.696564	+	0.717494i
$28$		0			0
$29$			195123.i			1.48565i	0.669487	+	0.742824i	$0.266514\pi$
$29$			195123.i			1.48565i	−0.669487	+	0.742824i	$0.733486\pi$
$30$	−49441.0	−	49931.3i	−0.334321	−	0.337636i
$31$			97496.3i			0.587790i	0.955838	+	0.293895i	$0.0949516\pi$
$31$			97496.3i			0.587790i	−0.955838	+	0.293895i	$0.905048\pi$
$32$			178474.i			0.962829i
$33$	−64043.1	−	64678.2i	−0.310223	−	0.313299i
$34$		−	68288.2i		−	0.297968i
$35$		0			0
$36$	1580.94	−	160222.i	0.00564750	−	0.572354i
$37$	570632.			1.85204			0.926019	−	0.377478i	$-0.123208\pi$
$37$	570632.			1.85204			0.926019	+	0.377478i	$0.123208\pi$
$38$	−76991.4			−0.227614
$39$	259087.	+	261656.i	0.699389	+	0.706324i
$40$			302412.i			0.747118i
$41$	−339491.			−0.769280			−0.384640	−	0.923067i	$-0.625674\pi$
$41$	−339491.			−0.769280			−0.384640	+	0.923067i	$0.625674\pi$
$42$		0			0
$43$	−356328.			−0.683455			−0.341727	−	0.939799i	$-0.611012\pi$
$43$	−356328.			−0.683455			−0.341727	+	0.939799i	$0.611012\pi$
$44$			142598.i			0.252364i
$45$	4382.45	−	444146.i	0.00716925	−	0.726578i
$46$	−605725.			−0.917536
$47$	−766151.			−1.07640			−0.538198	−	0.842819i	$-0.680895\pi$
$47$	−766151.			−1.07640			−0.538198	+	0.842819i	$0.680895\pi$
$48$	54444.1	−	53909.6i	0.0710570	−	0.0703593i
$49$		0			0
$50$		−	272832.i		−	0.308675i
$51$	306728.	−	303717.i	0.323786	−	0.320607i
$52$		−	576879.i		−	0.568948i
$53$			158435.i			0.146179i	0.997325	+	0.0730896i	$0.0232859\pi$
$53$			158435.i			0.146179i	−0.997325	+	0.0730896i	$0.976714\pi$
$54$	−542906.	+	527069.i	−0.469188	+	0.455501i
$55$			395289.i			0.320365i
$56$		0			0
$57$	−342425.	−	345820.i	−0.244908	−	0.247337i
$58$	−1.44358e6			−0.971503
$59$	−2.56230e6			−1.62423			−0.812117	−	0.583495i	$-0.801685\pi$
$59$	−2.56230e6			−1.62423			−0.812117	+	0.583495i	$0.801685\pi$
$60$	−494465.	+	489610.i	−0.295533	+	0.292631i
$61$		−	1.72901e6i		−	0.975310i	−0.873036	−	0.487655i	$-0.837852\pi$
$61$		−	1.72901e6i		−	0.975310i	0.873036	−	0.487655i	$-0.162148\pi$
$62$	−721309.			−0.384371
$63$		0			0
$64$	−1.53012e6			−0.729616
$65$		−	1.59914e6i		−	0.722255i
$66$	478510.	−	473812.i	0.204874	−	0.202863i
$67$	−2.20518e6			−0.895740			−0.447870	−	0.894099i	$-0.647817\pi$
$67$	−2.20518e6			−0.895740			−0.447870	+	0.894099i	$0.647817\pi$
$68$	−676251.			−0.260812
$69$	−2.69401e6	−	2.72072e6i	−0.987249	−	0.997039i
$70$		0			0
$71$			5.29805e6i			1.75676i	0.477965	+	0.878379i	$0.341375\pi$
$71$			5.29805e6i			1.75676i	−0.477965	+	0.878379i	$0.658625\pi$
$72$	3.25633e6	+	32130.7i	1.02817	+	0.0101451i
$73$		−	2.54319e6i		−	0.765155i	−0.923923	−	0.382577i	$-0.875037\pi$
$73$		−	2.54319e6i		−	0.765155i	0.923923	−	0.382577i	$-0.124963\pi$
$74$			4.22172e6i			1.21109i
$75$	1.22547e6	−	1.21344e6i	0.335421	−	0.332127i
$76$			762439.i			0.199231i
$77$		0			0
$78$	−1.93581e6	+	1.91681e6i	−0.461883	+	0.457348i
$79$	744581.			0.169909			0.0849547	−	0.996385i	$-0.472925\pi$
$79$	744581.			0.169909			0.0849547	+	0.996385i	$0.472925\pi$
$80$	−332742.			−0.0726596
$81$	−4.78204e6	−	94379.3i	−0.999805	−	0.0197324i
$82$		−	2.51166e6i		−	0.503052i
$83$	5.20177e6			0.998569			0.499284	−	0.866438i	$-0.333596\pi$
$83$	5.20177e6			0.998569			0.499284	+	0.866438i	$0.333596\pi$
$84$		0			0
$85$	−1.87461e6			−0.331089
$86$		−	2.63623e6i		−	0.446929i
$87$	−6.42044e6	−	6.48411e6i	−1.04532	−	1.05568i
$88$	−2.89813e6			−0.453344
$89$	1.01641e7			1.52828			0.764142	−	0.645048i	$-0.223163\pi$
$89$	1.01641e7			1.52828			0.764142	+	0.645048i	$0.223163\pi$
$90$	3.28594e6	+	32422.8i	0.475128	+	0.00468815i
$91$		0			0
$92$			5.99844e6i			0.803120i
$93$	−3.20807e6	−	3.23989e6i	−0.413575	−	0.417676i
$94$		−	5.66823e6i		−	0.703882i
$95$			2.11353e6i			0.252915i
$96$	−5.87260e6	−	5.93083e6i	−0.677456	−	0.684174i
$97$		−	3.36084e6i		−	0.373893i	−0.982370	−	0.186946i	$-0.940141\pi$
$97$		−	3.36084e6i		−	0.373893i	0.982370	−	0.186946i	$-0.0598591\pi$
$98$		0			0
$99$	4.25641e6	+	41998.7i	0.440880	+	0.00435023i
$100$	−2.70183e6			−0.270183
$101$	−1.27094e7			−1.22744			−0.613718	−	0.789526i	$-0.710327\pi$
$101$	−1.27094e7			−1.22744			−0.613718	+	0.789526i	$0.710327\pi$
$102$	2.24699e6	+	2.26927e6i	0.209653	+	0.211732i
$103$			8.58260e6i			0.773907i	0.922099	+	0.386954i	$0.126473\pi$
$103$			8.58260e6i			0.773907i	−0.922099	+	0.386954i	$0.873527\pi$
$104$	1.17244e7			1.02205
$105$		0			0
$106$	−1.17215e6			−0.0955903
$107$		−	494253.i		−	0.0390037i	−0.999810	−	0.0195019i	$-0.993792\pi$
$107$		−	494253.i		−	0.0390037i	0.999810	−	0.0195019i	$-0.00620803\pi$
$108$	5.21951e6	+	5.37635e6i	0.398701	+	0.410681i
$109$	−1.04794e7			−0.775073			−0.387537	−	0.921854i	$-0.626674\pi$
$109$	−1.04794e7			−0.775073			−0.387537	+	0.921854i	$0.626674\pi$
$110$	−2.92447e6			−0.209495
$111$	−1.89626e7	+	1.87764e7i	−1.31603	+	1.30311i
$112$		0			0
$113$			63166.9i			0.00411827i	0.999998	+	0.00205914i	$0.000655444\pi$
$113$			63166.9i			0.00411827i	−0.999998	+	0.00205914i	$0.999345\pi$
$114$	2.55849e6	−	2.53337e6i	0.161740	−	0.160152i
$115$			1.66280e7i			1.01953i
$116$			1.42957e7i			0.850358i
$117$	−1.72193e7	−	169906.i	−0.993953	−	0.00980748i
$118$		−	1.89567e7i		−	1.06213i
$119$		0			0
$120$	−9.95074e6	−	1.00494e7i	−0.525680	−	0.530892i
$121$	1.56990e7			0.805606
$122$	1.27918e7			0.637780
$123$	1.12816e7	−	1.11708e7i	0.546640	−	0.541273i
$124$			7.14305e6i			0.336440i
$125$	−2.33564e7			−1.06960
$126$		0			0
$127$	−1.51366e7			−0.655716			−0.327858	−	0.944727i	$-0.606327\pi$
$127$	−1.51366e7			−0.655716			−0.327858	+	0.944727i	$0.606327\pi$
$128$			1.15243e7i			0.485715i
$129$	1.18411e7	−	1.17248e7i	0.485654	−	0.480886i
$130$	1.18310e7			0.472301
$131$	−8.27433e6			−0.321576			−0.160788	−	0.986989i	$-0.551404\pi$
$131$	−8.27433e6			−0.321576			−0.160788	+	0.986989i	$0.551404\pi$
$132$	−4.69211e6	−	4.73864e6i	−0.177566	−	0.179327i
$133$		0			0
$134$		−	1.63146e7i		−	0.585747i
$135$	1.44688e7	+	1.49036e7i	0.506133	+	0.521341i
$136$		−	1.37440e7i		−	0.468519i
$137$		−	3.53464e7i		−	1.17442i	−0.809435	−	0.587210i	$-0.800226\pi$
$137$		−	3.53464e7i		−	1.17442i	0.809435	−	0.587210i	$-0.199774\pi$
$138$	2.01288e7	−	1.99311e7i	0.651989	−	0.645587i
$139$			3.41607e6i			0.107888i	0.998544	+	0.0539442i	$0.0171793\pi$
$139$			3.41607e6i			0.107888i	−0.998544	+	0.0539442i	$0.982821\pi$
$140$		0			0
$141$	2.54598e7	−	2.52099e7i	0.764872	−	0.757362i
$142$	−3.91967e7			−1.14879
$143$	1.53252e7			0.438257
$144$	−35353.2	+	3.58292e6i	−0.000986644	+	0.0999928i
$145$			3.96284e7i			1.07949i
$146$	1.88154e7			0.500354
$147$		0			0
$148$	4.18072e7			1.06007
$149$			2.08728e7i			0.516926i	0.966021	+	0.258463i	$0.0832161\pi$
$149$			2.08728e7i			0.516926i	−0.966021	+	0.258463i	$0.916784\pi$
$150$	8.97744e6	+	9.06645e6i	0.217187	+	0.219340i
$151$	7.97413e7			1.88480			0.942398	−	0.334495i	$-0.108566\pi$
$151$	7.97413e7			1.88480			0.942398	+	0.334495i	$0.108566\pi$
$152$	−1.54957e7			−0.357897
$153$	−199174.	+	2.01855e7i	−0.00449585	+	0.455638i
$154$		0			0
$155$			1.98010e7i			0.427096i
$156$	1.89819e7	+	1.91702e7i	0.400318	+	0.404287i
$157$			8.25879e7i			1.70321i	0.524187	+	0.851603i	$0.324369\pi$
$157$			8.25879e7i			1.70321i	−0.524187	+	0.851603i	$0.675631\pi$
$158$			5.50865e6i			0.111108i
$159$	−5.21324e6	−	5.26493e6i	−0.102853	−	0.103873i
$160$			3.62470e7i			0.699604i
$161$		0			0
$162$	698248.	−	3.53791e7i	0.0129035	−	0.653798i
$163$	−4.86596e7			−0.880059			−0.440029	−	0.897983i	$-0.645032\pi$
$163$	−4.86596e7			−0.880059			−0.440029	+	0.897983i	$0.645032\pi$
$164$	−2.48727e7			−0.440322
$165$	−1.30068e7	−	1.31358e7i	−0.225412	−	0.227647i
$166$			3.84844e7i			0.652989i
$167$	5.93974e7			0.986869			0.493435	−	0.869783i	$-0.335741\pi$
$167$	5.93974e7			0.986869			0.493435	+	0.869783i	$0.335741\pi$
$168$		0			0
$169$	750490.			0.0119603
$170$		−	1.38690e7i		−	0.216507i
$171$	2.27581e7	+	224558.i	0.348057	+	0.00343433i
$172$	−2.61063e7			−0.391197
$173$	1.01802e8			1.49485			0.747423	−	0.664348i	$-0.231291\pi$
$173$	1.01802e8			1.49485			0.747423	+	0.664348i	$0.231291\pi$
$174$	4.79715e7	−	4.75005e7i	0.690337	−	0.683559i
$175$		0			0
$176$		−	3.18879e6i		−	0.0440892i
$177$	8.51475e7	−	8.43115e7i	1.15416	−	1.14283i
$178$			7.51973e7i			0.999383i
$179$		−	6.33422e7i		−	0.825481i	−0.910849	−	0.412741i	$-0.864572\pi$
$179$		−	6.33422e7i		−	0.825481i	0.910849	−	0.412741i	$-0.135428\pi$
$180$	321080.	−	3.25403e7i	0.00410355	−	0.415880i
$181$		−	2.79149e6i		−	0.0349913i	−0.999847	−	0.0174957i	$-0.994431\pi$
$181$		−	2.79149e6i		−	0.0349913i	0.999847	−	0.0174957i	$-0.00556933\pi$
$182$		0			0
$183$	5.68923e7	+	5.74564e7i	0.686238	+	0.693042i
$184$	−1.21911e8			−1.44272
$185$	1.15892e8			1.34571
$186$	2.39697e7	−	2.37344e7i	0.273129	−	0.270447i
$187$		−	1.79651e7i		−	0.200901i
$188$	−5.61319e7			−0.616109
$189$		0			0
$190$	−1.56365e7			−0.165388
$191$			4.18650e7i			0.434745i	0.976089	+	0.217373i	$0.0697486\pi$
$191$			4.18650e7i			0.434745i	−0.976089	+	0.217373i	$0.930251\pi$
$192$	5.08470e7	−	5.03478e7i	0.518455	−	0.513365i
$193$	−8.04162e7			−0.805180			−0.402590	−	0.915380i	$-0.631890\pi$
$193$	−8.04162e7			−0.805180			−0.402590	+	0.915380i	$0.631890\pi$
$194$	2.48646e7			0.244498
$195$	5.26191e7	+	5.31409e7i	0.508185	+	0.513225i
$196$		0			0
$197$		−	2.46045e7i		−	0.229289i	−0.993407	−	0.114645i	$-0.963427\pi$
$197$		−	2.46045e7i		−	0.229289i	0.993407	−	0.114645i	$-0.0365729\pi$
$198$	−310720.	+	3.14903e7i	−0.00284473	+	0.288303i
$199$			3.02879e7i			0.272448i	0.990678	+	0.136224i	$0.0434966\pi$
$199$			3.02879e7i			0.272448i	−0.990678	+	0.136224i	$0.956503\pi$
$200$		−	5.49116e7i		−	0.485354i
$201$	7.32799e7	−	7.25605e7i	0.636501	−	0.630251i
$202$		−	9.40279e7i		−	0.802651i
$203$		0			0
$204$	2.24724e7	−	2.22518e7i	0.185329	−	0.183510i
$205$	−6.89487e7			−0.558969
$206$	−6.34969e7			−0.506077
$207$	1.79048e8	+	1.76670e6i	1.40305	+	0.0138441i
$208$			1.29003e7i			0.0993979i
$209$	−2.02547e7			−0.153466
$210$		0			0
$211$	6.47500e7			0.474517			0.237258	−	0.971447i	$-0.423751\pi$
$211$	6.47500e7			0.474517			0.237258	+	0.971447i	$0.423751\pi$
$212$			1.16077e7i			0.0836703i
$213$	−1.74330e8	−	1.76059e8i	−1.23607	−	1.24833i
$214$	3.65664e6			0.0255055
$215$	−7.23682e7			−0.496607
$216$	−1.09268e8	+	1.06080e8i	−0.737742	+	0.716222i
$217$		0			0
$218$		−	7.75298e7i		−	0.506840i
$219$	8.36827e7	+	8.45125e7i	0.538370	+	0.543709i
$220$			2.89608e7i			0.183371i
$221$			7.26777e7i			0.452927i
$222$	−1.38914e8	−	1.40291e8i	−0.852138	−	0.860587i
$223$		−	6.67311e7i		−	0.402959i	−0.979493	−	0.201480i	$-0.935425\pi$
$223$		−	6.67311e7i		−	0.402959i	0.979493	−	0.201480i	$-0.0645750\pi$
$224$		0			0
$225$	−795760.	+	8.06474e7i	−0.00465740	+	0.472011i
$226$	−467329.			−0.00269304
$227$	1.43456e8			0.814008			0.407004	−	0.913426i	$-0.366574\pi$
$227$	1.43456e8			0.814008			0.407004	+	0.913426i	$0.366574\pi$
$228$	−2.50877e7	−	2.53365e7i	−0.140181	−	0.141571i
$229$			3.62088e8i			1.99246i	0.0867559	+	0.996230i	$0.472350\pi$
$229$			3.62088e8i			1.99246i	−0.0867559	+	0.996230i	$0.527650\pi$
$230$	−1.23019e8			−0.666694
$231$		0			0
$232$	−2.90543e8			−1.52757
$233$		−	1.97126e8i		−	1.02094i	−0.859896	−	0.510469i	$-0.829472\pi$
$233$		−	1.97126e8i		−	1.02094i	0.859896	−	0.510469i	$-0.170528\pi$
$234$	1.25702e6	−	1.27394e8i	0.00641336	−	0.649971i
$235$	−1.55601e8			−0.782123
$236$	−1.87727e8			−0.929681
$237$	−2.47431e7	+	2.45001e7i	−0.120735	+	0.119550i
$238$		0			0
$239$		−	8.32422e7i		−	0.394413i	−0.980362	−	0.197206i	$-0.936813\pi$
$239$		−	8.32422e7i		−	0.394413i	0.980362	−	0.197206i	$-0.0631869\pi$
$240$	1.10573e7	−	1.09487e7i	0.0516309	−	0.0511240i
$241$			2.19094e8i			1.00826i	0.863629	+	0.504128i	$0.168186\pi$
$241$			2.19094e8i			1.00826i	−0.863629	+	0.504128i	$0.831814\pi$
$242$			1.16146e8i			0.526806i
$243$	1.62017e8	−	1.54215e8i	0.724332	−	0.689451i
$244$		−	1.26676e8i		−	0.558250i
$245$		0			0
$246$	8.26451e7	+	8.34646e7i	0.353952	+	0.357462i
$247$	8.19404e7			0.345986
$248$	−1.45174e8			−0.604378
$249$	−1.72859e8	+	1.71162e8i	−0.709569	+	0.702603i
$250$		−	1.72798e8i		−	0.699438i
$251$	6.98425e6			0.0278780			0.0139390	−	0.999903i	$-0.495563\pi$
$251$	6.98425e6			0.0278780			0.0139390	+	0.999903i	$0.495563\pi$
$252$		0			0
$253$	−1.59352e8			−0.618638
$254$		−	1.11986e8i		−	0.428789i
$255$	6.22948e7	−	6.16832e7i	0.235267	−	0.232957i
$256$	−2.81116e8			−1.04724
$257$	−2.58103e8			−0.948478			−0.474239	−	0.880396i	$-0.657277\pi$
$257$	−2.58103e8			−0.948478			−0.474239	+	0.880396i	$0.657277\pi$
$258$	8.67439e7	+	8.76040e7i	0.314463	+	0.317581i
$259$		0			0
$260$		−	1.17161e8i		−	0.413405i
$261$	4.26714e8	+	4.21045e6i	1.48558	+	0.0146584i
$262$		−	6.12162e7i		−	0.210287i
$263$			1.32458e8i			0.448986i	0.974476	+	0.224493i	$0.0720726\pi$
$263$			1.32458e8i			0.448986i	−0.974476	+	0.224493i	$0.927927\pi$
$264$	9.63072e7	−	9.53616e7i	0.322140	−	0.318977i
$265$			3.21773e7i			0.106216i
$266$		0			0
$267$	−3.37762e8	+	3.34445e8i	−1.08598	+	1.07532i
$268$	−1.61562e8			−0.512705
$269$	−4.55229e8			−1.42593			−0.712964	−	0.701201i	$-0.752648\pi$
$269$	−4.55229e8			−1.42593			−0.712964	+	0.701201i	$0.752648\pi$
$270$	−1.10261e8	+	1.07045e8i	−0.340918	+	0.330973i
$271$			5.52448e8i			1.68616i	0.537788	+	0.843080i	$0.319260\pi$
$271$			5.52448e8i			1.68616i	−0.537788	+	0.843080i	$0.680740\pi$
$272$	1.51225e7			0.0455650
$273$		0			0
$274$	2.61504e8			0.767983
$275$		−	7.17760e7i		−	0.208120i
$276$	−1.97376e8	−	1.99333e8i	−0.565083	−	0.570687i
$277$	5.25656e8			1.48601			0.743006	−	0.669285i	$-0.233399\pi$
$277$	5.25656e8			1.48601			0.743006	+	0.669285i	$0.233399\pi$
$278$	−2.52732e7			−0.0705510
$279$	2.13214e8	+	2.10381e6i	0.587762	+	0.00579953i
$280$		0			0
$281$			3.06215e8i			0.823292i	0.911344	+	0.411646i	$0.135046\pi$
$281$			3.06215e8i			0.823292i	−0.911344	+	0.411646i	$0.864954\pi$
$282$	1.86511e8	+	1.88360e8i	0.495258	+	0.500169i
$283$		−	5.56622e8i		−	1.45985i	−0.683528	−	0.729924i	$-0.739555\pi$
$283$		−	5.56622e8i		−	1.45985i	0.683528	−	0.729924i	$-0.260445\pi$
$284$			3.88161e8i			1.00554i
$285$	−6.95446e7	−	7.02342e7i	−0.177954	−	0.179718i
$286$			1.13381e8i			0.286588i
$287$		0			0
$288$	3.90303e8	+	3.85118e6i	0.962782	+	0.00949991i
$289$	−3.25142e8			−0.792374
$290$	−2.93184e8			−0.705907
$291$	1.10587e8	+	1.11684e8i	0.263075	+	0.265683i
$292$		−	1.86327e8i		−	0.437961i
$293$	1.60840e8			0.373557			0.186779	−	0.982402i	$-0.440195\pi$
$293$	1.60840e8			0.373557			0.186779	+	0.982402i	$0.440195\pi$
$294$		0			0
$295$	−5.20390e8			−1.18019
$296$			8.49683e8i			1.90430i
$297$	−1.42826e8	+	1.38660e8i	−0.316345	+	0.307117i
$298$	−1.54424e8			−0.338031
$299$	6.44661e8			1.39470
$300$	8.97842e7	−	8.89027e7i	0.191989	−	0.190104i
$301$		0			0
$302$			5.89952e8i			1.23252i
$303$	4.22343e8	−	4.18196e8i	0.872199	−	0.863635i
$304$		−	1.70498e7i		−	0.0348066i
$305$		−	3.51152e8i		−	0.708673i
$306$	−1.49339e8	−	1.47355e6i	−0.297953	−	0.00293995i
$307$		−	1.76458e8i		−	0.348063i	−0.984740	−	0.174032i	$-0.944321\pi$
$307$		−	1.76458e8i		−	0.348063i	0.984740	−	0.174032i	$-0.0556795\pi$
$308$		0			0
$309$	−2.82407e8	−	2.85207e8i	−0.544529	−	0.549928i
$310$	−1.46494e8			−0.279289
$311$	7.07166e8			1.33309			0.666546	−	0.745464i	$-0.267772\pi$
$311$	7.07166e8			1.33309			0.666546	+	0.745464i	$0.267772\pi$
$312$	−3.89611e8	+	3.85786e8i	−0.726257	+	0.719126i
$313$		−	5.14674e8i		−	0.948696i	−0.880337	−	0.474348i	$-0.842684\pi$
$313$		−	5.14674e8i		−	0.948696i	0.880337	−	0.474348i	$-0.157316\pi$
$314$	−6.11011e8			−1.11377
$315$		0			0
$316$	5.45516e7			0.0972530
$317$		−	8.11502e8i		−	1.43081i	−0.698710	−	0.715405i	$-0.746242\pi$
$317$		−	8.11502e8i		−	1.43081i	0.698710	−	0.715405i	$-0.253758\pi$
$318$	3.89517e7	−	3.85692e7i	0.0679252	−	0.0672583i
$319$	−3.79774e8			−0.655025
$320$	−3.10758e8			−0.530149
$321$	1.62632e7	+	1.64244e7i	0.0274434	+	0.0277155i
$322$		0			0
$323$		−	9.60554e7i		−	0.158604i
$324$	−3.50355e8	−	6.91469e6i	−0.572270	−	0.0112944i
$325$			2.90370e8i			0.469202i
$326$		−	3.59999e8i		−	0.575493i
$327$	3.48239e8	−	3.44819e8i	0.550757	−	0.545349i
$328$		−	5.05509e8i		−	0.790989i
$329$		0			0
$330$	9.71827e7	−	9.62285e7i	0.148864	−	0.147403i
$331$	7.17416e7			0.108736			0.0543680	−	0.998521i	$-0.482686\pi$
$331$	7.17416e7			0.108736			0.0543680	+	0.998521i	$0.482686\pi$
$332$	3.81107e8			0.571562
$333$	1.23133e7	−	1.24791e9i	0.0182734	−	1.85195i
$334$			4.39441e8i			0.645339i
$335$	−4.47860e8			−0.650856
$336$		0			0
$337$	8.22484e7			0.117064			0.0585320	−	0.998286i	$-0.481358\pi$
$337$	8.22484e7			0.117064			0.0585320	+	0.998286i	$0.481358\pi$
$338$			5.55236e6i			0.00782113i
$339$	−2.07848e6	−	2.09909e6i	−0.00289766	−	0.00292639i
$340$	−1.37343e8			−0.189509
$341$	−1.89760e8			−0.259158
$342$	−1.66135e6	+	1.68372e8i	−0.00224580	+	0.227603i
$343$		0			0
$344$		−	5.30580e8i		−	0.702742i
$345$	−5.47138e8	−	5.52563e8i	−0.717348	−	0.724461i
$346$			7.53167e8i			0.977518i
$347$		−	8.88130e7i		−	0.114110i	−0.998371	−	0.0570550i	$-0.981829\pi$
$347$		−	8.88130e7i		−	0.114110i	0.998371	−	0.0570550i	$-0.0181710\pi$
$348$	−4.70393e8	−	4.75057e8i	−0.598320	−	0.604253i
$349$			1.32408e8i			0.166735i	0.996519	+	0.0833673i	$0.0265675\pi$
$349$			1.32408e8i			0.166735i	−0.996519	+	0.0833673i	$0.973433\pi$
$350$		0			0
$351$	5.77804e8	−	5.60949e8i	0.713191	−	0.692386i
$352$	−3.47369e8			−0.424513
$353$	−2.87459e8			−0.347828			−0.173914	−	0.984761i	$-0.555641\pi$
$353$	−2.87459e8			−0.347828			−0.173914	+	0.984761i	$0.555641\pi$
$354$	6.23763e8	+	6.29949e8i	0.747323	+	0.754734i
$355$			1.07600e9i			1.27648i
$356$	7.44671e8			0.874761
$357$		0			0
$358$	4.68626e8			0.539803
$359$			4.47980e8i			0.511008i	0.966808	+	0.255504i	$0.0822414\pi$
$359$			4.47980e8i			0.511008i	−0.966808	+	0.255504i	$0.917759\pi$
$360$	6.61343e8	+	6.52557e6i	0.747082	+	0.00737157i
$361$	7.85574e8			0.878844
$362$	2.06523e7			0.0228817
$363$	−5.21690e8	+	5.16568e8i	−0.572453	+	0.566832i
$364$		0			0
$365$		−	5.16509e8i		−	0.555972i
$366$	−4.25081e8	+	4.20907e8i	−0.453198	+	0.448748i
$367$			6.90736e8i			0.729425i	0.931120	+	0.364713i	$0.118833\pi$
$367$			6.90736e8i			0.729425i	−0.931120	+	0.364713i	$0.881167\pi$
$368$		−	1.34138e8i		−	0.140309i
$369$	−7.32567e6	+	7.42430e8i	−0.00759023	+	0.769243i
$370$			8.57407e8i			0.879997i
$371$		0			0
$372$	−2.35039e8	−	2.37370e8i	−0.236723	−	0.239070i
$373$	3.63994e8			0.363172			0.181586	−	0.983375i	$-0.441877\pi$
$373$	3.63994e8			0.363172			0.181586	+	0.983375i	$0.441877\pi$
$374$	1.32911e8			0.131375
$375$	7.76153e8	−	7.68532e8i	0.760043	−	0.752580i
$376$		−	1.14082e9i		−	1.10677i
$377$	1.53638e9			1.47674
$378$		0			0
$379$	1.08976e9			1.02823			0.514117	−	0.857720i	$-0.328120\pi$
$379$	1.08976e9			1.02823			0.514117	+	0.857720i	$0.328120\pi$
$380$			1.54847e8i			0.144764i
$381$	5.03003e8	−	4.98064e8i	0.465943	−	0.461368i
$382$	−3.09731e8			−0.284291
$383$	1.76221e9			1.60273			0.801366	−	0.598174i	$-0.204107\pi$
$383$	1.76221e9			1.60273			0.801366	+	0.598174i	$0.204107\pi$
$384$	−3.79204e8	−	3.82964e8i	−0.341754	−	0.345142i
$385$		0			0
$386$		−	5.94945e8i		−	0.526528i
$387$	−7.68898e6	+	7.79251e8i	−0.00674342	+	0.683422i
$388$		−	2.46232e8i		−	0.214009i
$389$		−	7.00976e8i		−	0.603781i	−0.953343	−	0.301890i	$-0.902382\pi$
$389$		−	7.00976e8i		−	0.603781i	0.953343	−	0.301890i	$-0.0976177\pi$
$390$	−3.93153e8	+	3.89293e8i	−0.335611	+	0.332315i
$391$		−	7.55709e8i		−	0.639346i
$392$		0			0
$393$	2.74963e8	−	2.72263e8i	0.228508	−	0.226264i
$394$	1.82032e8			0.149938
$395$	1.51220e8			0.123458
$396$	3.11846e8	+	3.07703e6i	0.252352	+	0.00248999i
$397$		−	4.14313e8i		−	0.332324i	−0.986098	−	0.166162i	$-0.946863\pi$
$397$		−	4.14313e8i		−	0.332324i	0.986098	−	0.166162i	$-0.0531374\pi$
$398$	−2.24079e8			−0.178160
$399$		0			0
$400$	6.04189e7			0.0472023
$401$		−	6.24189e8i		−	0.483404i	−0.970350	−	0.241702i	$-0.922294\pi$
$401$		−	6.24189e8i		−	0.483404i	0.970350	−	0.241702i	$-0.0777057\pi$
$402$	5.36826e8	+	5.42149e8i	0.412137	+	0.416224i
$403$	7.67675e8			0.584265
$404$	−9.31149e8			−0.702561
$405$	−9.71206e8	−	1.91679e7i	−0.726472	−	0.0143378i
$406$		0			0
$407$			1.11064e9i			0.816567i
$408$	4.52241e8	+	4.56725e8i	0.329655	+	0.332923i
$409$			6.01550e8i			0.434751i	0.976088	+	0.217375i	$0.0697496\pi$
$409$			6.01550e8i			0.434751i	−0.976088	+	0.217375i	$0.930250\pi$
$410$		−	5.10105e8i		−	0.365524i
$411$	1.16306e9	+	1.17459e9i	0.826333	+	0.834527i
$412$			6.28803e8i			0.442970i
$413$		0			0
$414$	−1.30706e7	+	1.32466e9i	−0.00905302	+	0.917491i
$415$	1.05645e9			0.725573
$416$	1.40528e9			0.957054
$417$	−1.12404e8	−	1.13519e8i	−0.0759114	−	0.0766641i
$418$		−	1.49851e8i		−	0.100356i
$419$	8.13509e8			0.540273			0.270136	−	0.962822i	$-0.412931\pi$
$419$	8.13509e8			0.540273			0.270136	+	0.962822i	$0.412931\pi$
$420$		0			0
$421$	−5.73076e8			−0.374304			−0.187152	−	0.982331i	$-0.559926\pi$
$421$	−5.73076e8			−0.374304			−0.187152	+	0.982331i	$0.559926\pi$
$422$			4.79042e8i			0.310298i
$423$	−1.65323e7	+	1.67549e9i	−0.0106204	+	1.07634i
$424$	−2.35913e8			−0.150304
$425$	3.40389e8			0.215087
$426$	1.30254e9	−	1.28975e9i	0.816314	−	0.808299i
$427$		0			0
$428$		−	3.62114e7i		−	0.0223250i
$429$	−5.09268e8	+	5.04268e8i	−0.311420	+	0.308362i
$430$		−	5.35403e8i		−	0.324744i
$431$			2.11789e9i			1.27419i	0.770786	+	0.637094i	$0.219864\pi$
$431$			2.11789e9i			1.27419i	−0.770786	+	0.637094i	$0.780136\pi$
$432$	−1.16720e8	−	1.20227e8i	−0.0696548	−	0.0717478i
$433$			1.49662e9i			0.885938i	0.896537	+	0.442969i	$0.146075\pi$
$433$			1.49662e9i			0.885938i	−0.896537	+	0.442969i	$0.853925\pi$
$434$		0			0
$435$	−1.30396e9	−	1.31689e9i	−0.759540	−	0.767072i
$436$	−7.67770e8			−0.443638
$437$	−8.52024e8			−0.488390
$438$	−6.25250e8	+	6.19111e8i	−0.355545	+	0.352054i
$439$		−	6.71449e8i		−	0.378780i	−0.981902	−	0.189390i	$-0.939349\pi$
$439$		−	6.71449e8i		−	0.378780i	0.981902	−	0.189390i	$-0.0606511\pi$
$440$	−5.88594e8			−0.329406
$441$		0			0
$442$	−5.37693e8			−0.296181
$443$			1.63631e9i			0.894239i	0.894474	+	0.447119i	$0.147550\pi$
$443$			1.63631e9i			0.894239i	−0.894474	+	0.447119i	$0.852450\pi$
$444$	−1.38929e9	+	1.37565e9i	−0.753273	+	0.745877i
$445$	2.06427e9			1.11047
$446$	4.93698e8			0.263505
$447$	−6.86810e8	−	6.93621e8i	−0.363714	−	0.367321i
$448$		0			0
$449$			1.40788e9i			0.734011i	0.930219	+	0.367005i	$0.119617\pi$
$449$			1.40788e9i			0.734011i	−0.930219	+	0.367005i	$0.880383\pi$
$450$	−5.96656e8	−	5.88729e6i	−0.308660	−	0.00304559i
$451$		−	6.60761e8i		−	0.339177i
$452$			4.62791e6i			0.00235722i
$453$	−2.64987e9	+	2.62385e9i	−1.33931	+	1.32616i
$454$			1.06133e9i			0.532300i
$455$		0			0
$456$	5.14934e8	−	5.09878e8i	0.254317	−	0.251820i
$457$	1.13325e9			0.555419			0.277710	−	0.960665i	$-0.410425\pi$
$457$	1.13325e9			0.555419			0.277710	+	0.960665i	$0.410425\pi$
$458$	−2.67884e9			−1.30292
$459$	−6.57577e8	−	6.77336e8i	−0.317397	−	0.326934i
$460$			1.21825e9i			0.583558i
$461$	1.36884e9			0.650726			0.325363	−	0.945589i	$-0.394514\pi$
$461$	1.36884e9			0.650726			0.325363	+	0.945589i	$0.394514\pi$
$462$		0			0
$463$	1.03418e9			0.484241			0.242121	−	0.970246i	$-0.422157\pi$
$463$	1.03418e9			0.484241			0.242121	+	0.970246i	$0.422157\pi$
$464$		−	3.19682e8i		−	0.148561i
$465$	−6.51542e8	−	6.58003e8i	−0.300509	−	0.303489i
$466$	1.45840e9			0.667617
$467$	−1.77585e9			−0.806860			−0.403430	−	0.915010i	$-0.632182\pi$
$467$	−1.77585e9			−0.806860			−0.403430	+	0.915010i	$0.632182\pi$
$468$	−1.26157e9	−	1.24481e7i	−0.568921	−	0.00561362i
$469$		0			0
$470$		−	1.15119e9i		−	0.511450i
$471$	−2.71752e9	−	2.74446e9i	−1.19839	−	1.21028i
$472$		−	3.81533e9i		−	1.67007i
$473$		−	6.93531e8i		−	0.301337i
$474$	−1.81260e8	−	1.83057e8i	−0.0781767	−	0.0789519i
$475$		−	3.83771e8i		−	0.164303i
$476$		0			0
$477$	3.46481e8	+	3.41878e6i	0.146172	+	0.00144230i
$478$	6.15852e8			0.257916
$479$	2.60207e9			1.08179			0.540896	−	0.841089i	$-0.318085\pi$
$479$	2.60207e9			1.08179			0.540896	+	0.841089i	$0.318085\pi$
$480$	−1.19269e9	−	1.20452e9i	−0.492248	−	0.497129i
$481$		−	4.49309e9i		−	1.84093i
$482$	−1.62093e9			−0.659325
$483$		0			0
$484$	1.15018e9			0.461114
$485$		−	6.82569e8i		−	0.271675i
$486$	1.14093e9	+	1.19865e9i	0.450850	+	0.473659i
$487$	−3.25977e9			−1.27890			−0.639449	−	0.768834i	$-0.720837\pi$
$487$	−3.25977e9			−1.27890			−0.639449	+	0.768834i	$0.720837\pi$
$488$	2.57453e9			1.00283
$489$	1.61700e9	−	1.60112e9i	0.625358	−	0.619218i
$490$		0			0
$491$			9.35676e7i			0.0356730i	0.999841	+	0.0178365i	$0.00567784\pi$
$491$			9.35676e7i			0.0356730i	−0.999841	+	0.0178365i	$0.994322\pi$
$492$	8.26542e8	−	8.18427e8i	0.312887	−	0.309815i
$493$		−	1.80103e9i		−	0.676951i
$494$			6.06221e8i			0.226249i
$495$	8.64454e8	+	8.52970e6i	0.320349	+	0.00316093i
$496$		−	1.59734e8i		−	0.0587776i
$497$		0			0
$498$	−1.26631e9	−	1.27887e9i	−0.459450	−	0.464005i
$499$	3.57650e9			1.28857			0.644283	−	0.764788i	$-0.277156\pi$
$499$	3.57650e9			1.28857			0.644283	+	0.764788i	$0.277156\pi$
$500$	−1.71120e9			−0.612219
$501$	−1.97382e9	+	1.95444e9i	−0.701256	+	0.694371i
$502$			5.16717e7i			0.0182301i
$503$	−3.25287e9			−1.13967			−0.569835	−	0.821759i	$-0.692993\pi$
$503$	−3.25287e9			−1.13967			−0.569835	+	0.821759i	$0.692993\pi$
$504$		0			0
$505$	−2.58120e9			−0.891870
$506$		−	1.17894e9i		−	0.404543i
$507$	−2.49394e7	+	2.46945e7i	−0.00849881	+	0.00841537i
$508$	−1.10898e9			−0.375320
$509$	−2.67600e9			−0.899443			−0.449722	−	0.893169i	$-0.648477\pi$
$509$	−2.67600e9			−0.899443			−0.449722	+	0.893169i	$0.648477\pi$
$510$	4.56352e8	+	4.60877e8i	0.152337	+	0.153847i
$511$		0			0
$512$		−	6.04668e8i		−	0.199100i
$513$	−7.63661e8	+	7.41385e8i	−0.249741	+	0.242456i
$514$		−	1.90953e9i		−	0.620234i
$515$			1.74308e9i			0.562331i
$516$	8.67534e8	−	8.59016e8i	0.277979	−	0.275250i
$517$		−	1.49118e9i		−	0.474585i
$518$		0			0
$519$	−3.38298e9	+	3.34976e9i	−1.06222	+	1.05179i
$520$	2.38116e9			0.742637
$521$	2.20727e9			0.683792			0.341896	−	0.939738i	$-0.388931\pi$
$521$	2.20727e9			0.683792			0.341896	+	0.939738i	$0.388931\pi$
$522$	−3.11502e7	+	3.15696e9i	−0.00958549	+	0.971456i
$523$		−	2.88210e9i		−	0.880955i	−0.897764	−	0.440477i	$-0.854809\pi$
$523$		−	2.88210e9i		−	0.880955i	0.897764	−	0.440477i	$-0.145191\pi$
$524$	−6.06218e8			−0.184064
$525$		0			0
$526$	−9.79966e8			−0.293603
$527$		−	8.99913e8i		−	0.267833i
$528$	1.04926e8	+	1.05966e8i	0.0310216	+	0.0313292i
$529$	−3.29842e9			−0.968747
$530$	−2.38058e8			−0.0694572
$531$	−5.52904e7	+	5.60348e9i	−0.0160258	+	1.62415i
$532$		0			0
$533$			2.67311e9i			0.764666i
$534$	−2.47433e9	−	2.49887e9i	−0.703176	−	0.710148i
$535$		−	1.00380e8i		−	0.0283406i
$536$		−	3.28356e9i		−	0.921018i
$537$	2.08425e9	+	2.10491e9i	0.580817	+	0.586576i
$538$		−	3.36793e9i		−	0.932450i
$539$		0			0
$540$	1.06005e9	+	1.09191e9i	0.289701	+	0.298406i
$541$	1.79049e9			0.486161			0.243081	−	0.970006i	$-0.421842\pi$
$541$	1.79049e9			0.486161			0.243081	+	0.970006i	$0.421842\pi$
$542$	−4.08719e9			−1.10262
$543$	9.18526e7	+	9.27634e7i	0.0246202	+	0.0248644i
$544$		−	1.64735e9i		−	0.438723i
$545$	−2.12830e9			−0.563178
$546$		0			0
$547$	6.81952e9			1.78155			0.890776	−	0.454443i	$-0.150162\pi$
$547$	6.81952e9			1.78155			0.890776	+	0.454443i	$0.150162\pi$
$548$		−	2.58965e9i		−	0.672216i
$549$	−3.78116e9	−	3.73092e7i	−0.975263	−	0.00962306i
$550$	5.31022e8			0.136095
$551$	−2.03057e9			−0.517115
$552$	4.05121e9	−	4.01144e9i	1.02518	−	1.01511i
$553$		0			0
$554$			3.88897e9i			0.971741i
$555$	−3.85119e9	+	3.81338e9i	−0.956247	+	0.946858i
$556$			2.50278e8i			0.0617534i
$557$			5.28664e9i			1.29624i	0.761537	+	0.648121i	$0.224445\pi$
$557$			5.28664e9i			1.29624i	−0.761537	+	0.648121i	$0.775555\pi$
$558$	−1.55647e7	+	1.57743e9i	−0.00379246	+	0.384352i
$559$			2.80568e9i			0.679355i
$560$		0			0
$561$	5.91133e8	+	5.96994e8i	0.141356	+	0.142758i
$562$	−2.26547e9			−0.538371
$563$	−6.69389e9			−1.58088			−0.790441	−	0.612538i	$-0.790148\pi$
$563$	−6.69389e9			−1.58088			−0.790441	+	0.612538i	$0.790148\pi$
$564$	1.86531e9	−	1.84700e9i	0.437799	−	0.433500i
$565$			1.28288e7i			0.00299239i
$566$	4.11807e9			0.954632
$567$		0			0
$568$	−7.88891e9			−1.80633
$569$			1.65712e9i			0.377103i	0.982063	+	0.188552i	$0.0603793\pi$
$569$			1.65712e9i			0.377103i	−0.982063	+	0.188552i	$0.939621\pi$
$570$	5.19615e8	−	5.14514e8i	0.117522	−	0.116368i
$571$	−3.51892e7			−0.00791012			−0.00395506	−	0.999992i	$-0.501259\pi$
$571$	−3.51892e7			−0.00791012			−0.00395506	+	0.999992i	$0.501259\pi$
$572$	1.12280e9			0.250850
$573$	−1.37755e9	−	1.39121e9i	−0.305891	−	0.308924i
$574$		0			0
$575$		−	3.01929e9i		−	0.662320i
$576$	−3.30174e7	+	3.34620e9i	−0.00719888	+	0.729581i
$577$		−	6.83968e9i		−	1.48225i	−0.671369	−	0.741123i	$-0.734293\pi$
$577$		−	6.83968e9i		−	1.48225i	0.671369	−	0.741123i	$-0.265707\pi$
$578$		−	2.40550e9i		−	0.518153i
$579$	2.67230e9	−	2.64606e9i	0.572150	−	0.566533i
$580$			2.90337e9i			0.617881i
$581$		0			0
$582$	−8.26272e8	+	8.18159e8i	−0.173737	+	0.172031i
$583$	−3.08367e8			−0.0644507
$584$	3.78687e9			0.786748
$585$	−3.49715e9	−	3.45069e7i	−0.722220	−	0.00712625i
$586$			1.18995e9i			0.244279i
$587$	3.21879e9			0.656839			0.328420	−	0.944532i	$-0.393484\pi$
$587$	3.21879e9			0.656839			0.328420	+	0.944532i	$0.393484\pi$
$588$		0			0
$589$	−1.01461e9			−0.204594
$590$		−	3.85001e9i		−	0.771756i
$591$	8.09602e8	+	8.17630e8i	0.161330	+	0.162930i
$592$	−9.34901e8			−0.185199
$593$	−1.53639e9			−0.302560			−0.151280	−	0.988491i	$-0.548339\pi$
$593$	−1.53639e9			−0.302560			−0.151280	+	0.988491i	$0.548339\pi$
$594$	−1.02585e9	−	1.05667e9i	−0.200831	−	0.206866i
$595$		0			0
$596$			1.52924e9i			0.295879i
$597$	−9.96610e8	−	1.00649e9i	−0.191697	−	0.193598i
$598$			4.76941e9i			0.912033i
$599$		−	1.13949e9i		−	0.216629i	−0.994117	−	0.108315i	$-0.965455\pi$
$599$		−	1.13949e9i		−	0.216629i	0.994117	−	0.108315i	$-0.0345454\pi$
$600$	1.80684e9	+	1.82476e9i	0.341500	+	0.344886i
$601$		−	4.72513e9i		−	0.887878i	−0.896057	−	0.443939i	$-0.853581\pi$
$601$		−	4.72513e9i		−	0.887878i	0.896057	−	0.443939i	$-0.146419\pi$
$602$		0			0
$603$	−4.75842e7	+	4.82249e9i	−0.00883797	+	0.895696i
$604$	5.84224e9			1.07882
$605$	3.18838e9			0.585363
$606$	3.09395e9	+	3.12463e9i	0.564753	+	0.570353i
$607$		−	6.37886e8i		−	0.115766i	−0.998323	−	0.0578832i	$-0.981565\pi$
$607$		−	6.37886e8i		−	0.115766i	0.998323	−	0.0578832i	$-0.0184351\pi$
$608$	−1.85731e9			−0.335136
$609$		0			0
$610$	2.59794e9			0.463419
$611$			6.03258e9i			1.06994i
$612$	−1.45924e7	+	1.47889e9i	−0.00257334	+	0.260799i
$613$	5.36259e9			0.940293			0.470147	−	0.882588i	$-0.344201\pi$
$613$	5.36259e9			0.940293			0.470147	+	0.882588i	$0.344201\pi$
$614$	1.30550e9			0.227607
$615$	2.29122e9	−	2.26873e9i	0.397196	−	0.393296i
$616$		0			0
$617$		−	9.68182e9i		−	1.65943i	−0.558187	−	0.829715i	$-0.688503\pi$
$617$		−	9.68182e9i		−	1.65943i	0.558187	−	0.829715i	$-0.311497\pi$
$618$	2.11005e9	−	2.08934e9i	0.359612	−	0.356081i
$619$			2.98416e8i			0.0505714i	0.999680	+	0.0252857i	$0.00804954\pi$
$619$			2.98416e8i			0.0505714i	−0.999680	+	0.0252857i	$0.991950\pi$
$620$			1.45071e9i			0.244462i
$621$	−6.00806e9	+	5.83280e9i	−1.00673	+	0.977364i
$622$			5.23184e9i			0.871743i
$623$		0			0
$624$	−4.24478e8	−	4.28687e8i	−0.0699373	−	0.0706308i
$625$	−1.86249e9			−0.305151
$626$	3.80772e9			0.620376
$627$	6.73080e8	−	6.66472e8i	0.109051	−	0.107981i
$628$			6.05079e9i			0.974884i
$629$	−5.26706e9			−0.843900
$630$		0			0
$631$	3.70007e9			0.586282			0.293141	−	0.956069i	$-0.405299\pi$
$631$	3.70007e9			0.586282			0.293141	+	0.956069i	$0.405299\pi$
$632$			1.10870e9i			0.174704i
$633$	−2.15170e9	+	2.13057e9i	−0.337185	+	0.333875i
$634$	6.00375e9			0.935643
$635$	−3.07416e9			−0.476452
$636$	−3.81947e8	−	3.85734e8i	−0.0588713	−	0.0594550i
$637$		0			0
$638$		−	2.80969e9i		−	0.428337i
$639$	1.15863e10	+	1.14323e8i	1.75667	+	0.0173333i
$640$			2.34053e9i			0.352927i
$641$			1.07467e9i			0.161165i	0.996748	+	0.0805826i	$0.0256781\pi$
$641$			1.07467e9i			0.161165i	−0.996748	+	0.0805826i	$0.974322\pi$
$642$	−1.21513e8	+	1.20320e8i	−0.0181239	+	0.0179459i
$643$			5.89391e9i			0.874309i	0.899386	+	0.437155i	$0.144014\pi$
$643$			5.89391e9i			0.874309i	−0.899386	+	0.437155i	$0.855986\pi$
$644$		0			0
$645$	2.40486e9	−	2.38124e9i	0.352882	−	0.349418i
$646$	7.10648e8			0.103715
$647$	6.85020e9			0.994348			0.497174	−	0.867651i	$-0.334371\pi$
$647$	6.85020e9			0.994348			0.497174	+	0.867651i	$0.334371\pi$
$648$	1.40533e8	−	7.12056e9i	0.0202892	−	1.02802i
$649$		−	4.98709e9i		−	0.716128i
$650$	−2.14825e9			−0.306823
$651$		0			0
$652$	−3.56504e9			−0.503730
$653$			2.68881e9i			0.377889i	0.981988	+	0.188944i	$0.0605066\pi$
$653$			2.68881e9i			0.377889i	−0.981988	+	0.188944i	$0.939493\pi$
$654$	2.55108e9	+	2.57638e9i	0.356618	+	0.360154i
$655$	−1.68047e9			−0.233661
$656$	5.56209e8			0.0769262
$657$	−5.56169e9	−	5.48780e7i	−0.765118	−	0.00754953i
$658$		0			0
$659$		−	7.57949e9i		−	1.03167i	−0.856688	−	0.515835i	$-0.827482\pi$
$659$		−	7.57949e9i		−	1.03167i	0.856688	−	0.515835i	$-0.172518\pi$
$660$	−9.52942e8	−	9.62391e8i	−0.129022	−	0.130301i
$661$			1.21582e10i			1.63744i	0.574194	+	0.818720i	$0.305316\pi$
$661$			1.21582e10i			1.63744i	−0.574194	+	0.818720i	$0.694684\pi$
$662$			5.30767e8i			0.0711052i
$663$	−2.39143e9	−	2.41514e9i	−0.318684	−	0.321844i
$664$			7.74555e9i			1.02675i
$665$		0			0
$666$	9.23244e9	+	9.10979e7i	1.21104	+	0.0119495i
$667$	−1.59754e10			−2.08454
$668$	4.35174e9			0.564866
$669$	2.19576e9	+	2.21753e9i	0.283526	+	0.286337i
$670$		−	3.31341e9i		−	0.425611i
$671$	3.36522e9			0.430016
$672$		0			0
$673$	1.01796e10			1.28729			0.643647	−	0.765323i	$-0.277421\pi$
$673$	1.01796e10			1.28729			0.643647	+	0.765323i	$0.277421\pi$
$674$			6.08500e8i			0.0765511i
$675$	−2.62723e9	−	2.70617e9i	−0.328802	−	0.338681i
$676$	5.49845e7			0.00684584
$677$	5.08427e9			0.629750			0.314875	−	0.949133i	$-0.398037\pi$
$677$	5.08427e9			0.629750			0.314875	+	0.949133i	$0.398037\pi$
$678$	1.55297e7	−	1.53773e7i	0.00191364	−	0.00189485i
$679$		0			0
$680$		−	2.79133e9i		−	0.340432i
$681$	−4.76717e9	+	4.72036e9i	−0.578423	+	0.572744i
$682$		−	1.40391e9i		−	0.169470i
$683$		−	1.31072e10i		−	1.57412i	−0.616875	−	0.787061i	$-0.711602\pi$
$683$		−	1.31072e10i		−	1.57412i	0.616875	−	0.787061i	$-0.288398\pi$
$684$	1.66737e9	+	1.64522e7i	0.199221	+	0.00196575i
$685$		−	7.17866e9i		−	0.853348i
$686$		0			0
$687$	−1.19143e10	−	1.20325e10i	−1.40191	−	1.41581i
$688$	5.83794e8			0.0683439
$689$	1.24750e9			0.145302
$690$	4.08804e9	−	4.04790e9i	0.473744	−	0.469092i
$691$			4.92137e9i			0.567430i	0.958909	+	0.283715i	$0.0915670\pi$
$691$			4.92137e9i			0.567430i	−0.958909	+	0.283715i	$0.908433\pi$
$692$	7.45854e9			0.855623
$693$		0			0
$694$	6.57067e8			0.0746194
$695$			6.93785e8i			0.0783932i
$696$	9.65498e9	−	9.56018e9i	1.08547	−	1.07482i
$697$	3.13358e9			0.350530
$698$	−9.79598e8			−0.109032
$699$	6.48636e9	+	6.55068e9i	0.718341	+	0.725464i
$700$		0			0
$701$			7.59763e9i			0.833039i	0.909127	+	0.416519i	$0.136750\pi$
$701$			7.59763e9i			0.833039i	−0.909127	+	0.416519i	$0.863250\pi$
$702$	4.15008e9	+	4.27478e9i	0.452769	+	0.466373i
$703$			5.93834e9i			0.644646i
$704$		−	2.97811e9i		−	0.321689i
$705$	5.17075e9	−	5.11998e9i	0.555766	−	0.550309i
$706$		−	2.12671e9i		−	0.227453i
$707$		0			0
$708$	6.23832e9	−	6.17707e9i	0.660619	−	0.654133i
$709$	−7.88873e9			−0.831276			−0.415638	−	0.909530i	$-0.636442\pi$
$709$	−7.88873e9			−0.831276			−0.415638	+	0.909530i	$0.636442\pi$
$710$	−7.96063e9			−0.834725
$711$	1.60669e7	−	1.62832e9i	0.00167644	−	0.169901i
$712$			1.51346e10i			1.57141i
$713$	−7.98235e9			−0.824740
$714$		0			0
$715$	3.11246e9			0.318443
$716$		−	4.64075e9i		−	0.472490i
$717$	2.73905e9	+	2.76621e9i	0.277513	+	0.280264i
$718$	−3.31430e9			−0.334161
$719$	−3.50177e9			−0.351347			−0.175674	−	0.984448i	$-0.556210\pi$
$719$	−3.50177e9			−0.351347			−0.175674	+	0.984448i	$0.556210\pi$
$720$	−7.18004e6	+	7.27672e8i	−0.000716908	+	0.0726561i
$721$		0			0
$722$			5.81193e9i			0.574699i
$723$	−7.20920e9	−	7.28068e9i	−0.709419	−	0.716454i
$724$		−	2.04518e8i		−	0.0200284i
$725$		−	7.19568e9i		−	0.701276i
$726$	−3.82174e9	−	3.85963e9i	−0.370666	−	0.374341i
$727$		−	2.03022e10i		−	1.95962i	−0.199923	−	0.979812i	$-0.564069\pi$
$727$		−	2.03022e10i		−	1.95962i	0.199923	−	0.979812i	$-0.435931\pi$
$728$		0			0
$729$	−3.09586e8	+	1.04558e10i	−0.0295962	+	0.999562i
$730$	3.82130e9			0.363564
$731$	3.28899e9			0.311423
$732$	4.16820e9	+	4.20954e9i	0.392790	+	0.396685i
$733$		−	3.53965e9i		−	0.331968i	−0.986128	−	0.165984i	$-0.946920\pi$
$733$		−	3.53965e9i		−	0.331968i	0.986128	−	0.165984i	$-0.0530800\pi$
$734$	−5.11028e9			−0.476990
$735$		0			0
$736$	−1.46122e10			−1.35097
$737$		−	4.29200e9i		−	0.394933i
$738$	−5.49274e9	−	5.41976e7i	−0.503027	−	0.00496344i
$739$	1.32681e9			0.120935			0.0604677	−	0.998170i	$-0.480741\pi$
$739$	1.32681e9			0.120935			0.0604677	+	0.998170i	$0.480741\pi$
$740$	8.49082e9			0.770262
$741$	−2.72295e9	+	2.69621e9i	−0.245853	+	0.243439i
$742$		0			0
$743$			5.32031e9i			0.475857i	0.971283	+	0.237928i	$0.0764683\pi$
$743$			5.32031e9i			0.475857i	−0.971283	+	0.237928i	$0.923532\pi$
$744$	4.82426e9	−	4.77689e9i	0.429463	−	0.425246i
$745$			4.23915e9i			0.375605i
$746$			2.69294e9i			0.237488i
$747$	1.12246e8	−	1.13757e10i	0.00985254	−	0.998520i
$748$		−	1.31621e9i		−	0.114992i
$749$		0			0
$750$	5.68585e9	+	5.74223e9i	0.492131	+	0.497011i
$751$	6.16154e9			0.530823			0.265411	−	0.964135i	$-0.414492\pi$
$751$	6.16154e9			0.530823			0.265411	+	0.964135i	$0.414492\pi$
$752$	1.25523e9			0.107637
$753$	−2.32092e8	+	2.29814e8i	−0.0198097	+	0.0196152i
$754$			1.13666e10i			0.965676i
$755$	1.61950e10			1.36952
$756$		0			0
$757$	−4.11040e9			−0.344388			−0.172194	−	0.985063i	$-0.555086\pi$
$757$	−4.11040e9			−0.344388			−0.172194	+	0.985063i	$0.555086\pi$
$758$			8.06236e9i			0.672388i
$759$	5.29542e9	−	5.24342e9i	0.439596	−	0.435280i
$760$	−3.14709e9			−0.260052
$761$	4.48555e9			0.368951			0.184476	−	0.982837i	$-0.440941\pi$
$761$	4.48555e9			0.368951			0.184476	+	0.982837i	$0.440941\pi$
$762$	3.68484e9	+	3.72137e9i	0.301700	+	0.304692i
$763$		0			0
$764$			3.06724e9i			0.248840i
$765$	−4.04510e7	+	4.09957e9i	−0.00326674	+	0.331073i
$766$			1.30374e10i			1.04807i
$767$			2.01753e10i			1.61449i
$768$	9.34171e9	−	9.24999e9i	0.744153	−	0.736846i
$769$		−	1.62822e9i		−	0.129114i	−0.997914	−	0.0645568i	$-0.979437\pi$
$769$		−	1.62822e9i		−	0.129114i	0.997914	−	0.0645568i	$-0.0205634\pi$
$770$		0			0
$771$	8.57698e9	−	8.49277e9i	0.673975	−	0.667358i
$772$	−5.89168e9			−0.460870
$773$	−1.90021e10			−1.47970			−0.739849	−	0.672773i	$-0.765103\pi$
$773$	−1.90021e10			−1.47970			−0.739849	+	0.672773i	$0.765103\pi$
$774$	−5.76515e9	−	5.68855e7i	−0.446907	−	0.00440969i
$775$		−	3.59543e9i		−	0.277457i
$776$	5.00437e9			0.384444
$777$		0			0
$778$	5.18604e9			0.394828
$779$		−	3.53295e9i		−	0.267766i
$780$	3.85513e9	+	3.89336e9i	0.290876	+	0.293760i
$781$	−1.03118e10			−0.774558
$782$	5.59098e9			0.418085
$783$	−1.43186e10	+	1.39009e10i	−1.06594	+	1.03485i
$784$		0			0
$785$			1.67731e10i			1.23757i
$786$	2.01429e9	+	2.03427e9i	0.147960	+	0.149427i
$787$		−	6.43494e9i		−	0.470580i	−0.971925	−	0.235290i	$-0.924396\pi$
$787$		−	6.43494e9i		−	0.470580i	0.971925	−	0.235290i	$-0.0756039\pi$
$788$		−	1.80265e9i		−	0.131241i
$789$	−4.35847e9	−	4.40169e9i	−0.315911	−	0.319043i
$790$			1.11878e9i			0.0807326i
$791$		0			0
$792$	−6.25369e7	+	6.33790e9i	−0.00447300	+	0.453322i
$793$	−1.36140e10			−0.969460
$794$	3.06522e9			0.217315
$795$	−1.05878e9	−	1.06928e9i	−0.0747344	−	0.0754755i
$796$			2.21904e9i			0.155944i
$797$	1.31128e10			0.917468			0.458734	−	0.888574i	$-0.348303\pi$
$797$	1.31128e10			0.917468			0.458734	+	0.888574i	$0.348303\pi$
$798$		0			0
$799$	7.07175e9			0.490470
$800$		−	6.58168e9i		−	0.454488i
$801$	2.19325e8	−	2.22278e10i	0.0150791	−	1.52821i
$802$	4.61795e9			0.316110
$803$	4.94989e9			0.337358
$804$	5.36884e9	−	5.31613e9i	0.364321	−	0.360744i
$805$		0			0
$806$			5.67950e9i			0.382065i
$807$	1.51277e10	−	1.49791e10i	1.01324	−	1.00330i
$808$		−	1.89245e10i		−	1.26207i
$809$		−	1.15552e10i		−	0.767287i	−0.923481	−	0.383643i	$-0.874669\pi$
$809$		−	1.15552e10i		−	0.767287i	0.923481	−	0.383643i	$-0.125331\pi$
$810$	1.41810e8	−	7.18529e9i	0.00937585	−	0.475058i
$811$			1.69436e9i			0.111541i	0.998444	+	0.0557703i	$0.0177615\pi$
$811$			1.69436e9i			0.111541i	−0.998444	+	0.0557703i	$0.982239\pi$
$812$		0			0
$813$	−1.81781e10	−	1.83583e10i	−1.18640	−	1.19816i
$814$	−8.21685e9			−0.533974
$815$	−9.88249e9			−0.639462
$816$	−5.02532e8	+	4.97598e8i	−0.0323779	+	0.0320600i
$817$		−	3.70816e9i		−	0.237893i
$818$	−4.45046e9			−0.284295
$819$		0			0
$820$	−5.05152e9			−0.319943
$821$			2.40857e10i			1.51900i	0.650508	+	0.759500i	$0.274556\pi$
$821$			2.40857e10i			1.51900i	−0.650508	+	0.759500i	$0.725444\pi$
$822$	−8.69000e9	+	8.60468e9i	−0.545718	+	0.540360i
$823$	−1.87468e10			−1.17227			−0.586136	−	0.810213i	$-0.699352\pi$
$823$	−1.87468e10			−1.17227			−0.586136	+	0.810213i	$0.699352\pi$
$824$	−1.27797e10			−0.795747
$825$	2.36176e9	+	2.38518e9i	0.146436	+	0.147888i
$826$		0			0
$827$		−	1.55760e10i		−	0.957605i	−0.877923	−	0.478803i	$-0.841071\pi$
$827$		−	1.55760e10i		−	0.957605i	0.877923	−	0.478803i	$-0.158929\pi$
$828$	1.31179e10	+	1.29437e8i	0.803081	+	0.00792412i
$829$		−	8.16621e9i		−	0.497828i	−0.968526	−	0.248914i	$-0.919926\pi$
$829$		−	8.16621e9i		−	0.497828i	0.968526	−	0.248914i	$-0.0800737\pi$
$830$			7.81596e9i			0.474471i
$831$	−1.74680e10	+	1.72965e10i	−1.05594	+	1.04557i
$832$			1.20480e10i			0.725240i
$833$		0			0
$834$	8.39849e8	−	8.31603e8i	0.0501326	−	0.0496404i
$835$	1.20633e10			0.717072
$836$	−1.48396e9			−0.0878414
$837$	−7.15451e9	+	6.94581e9i	−0.421736	+	0.409434i
$838$			6.01860e9i			0.353298i
$839$	−1.89484e10			−1.10766			−0.553830	−	0.832630i	$-0.686834\pi$
$839$	−1.89484e10			−1.10766			−0.553830	+	0.832630i	$0.686834\pi$
$840$		0			0
$841$	−2.08232e10			−1.20715
$842$		−	4.23980e9i		−	0.244767i
$843$	−1.00759e10	−	1.01758e10i	−0.579276	−	0.585020i
$844$	4.74390e9			0.271605
$845$	1.52420e8			0.00869049
$846$	−1.23958e10	−	1.22311e8i	−0.703848	−	0.00694497i
$847$		0			0
$848$		−	2.59574e8i		−	0.0146176i
$849$	1.83154e10	+	1.84970e10i	1.02716	+	1.03735i
$850$			2.51831e9i			0.140651i
$851$			4.67195e10i			2.59863i
$852$	−1.27723e10	−	1.28989e10i	−0.707505	−	0.714520i
$853$		−	9.10063e9i		−	0.502053i	−0.967980	−	0.251027i	$-0.919232\pi$
$853$		−	9.10063e9i		−	0.502053i	0.967980	−	0.251027i	$-0.0807682\pi$
$854$		0			0
$855$	4.62205e9	+	4.56065e7i	0.252903	+	0.00249543i
$856$	7.35954e8			0.0401044
$857$	1.30628e10			0.708928			0.354464	−	0.935070i	$-0.384663\pi$
$857$	1.30628e10			0.708928			0.354464	+	0.935070i	$0.384663\pi$
$858$	−3.73074e9	−	3.76773e9i	−0.201646	−	0.203645i
$859$			1.92933e10i			1.03856i	0.854604	+	0.519280i	$0.173800\pi$
$859$			1.92933e10i			1.03856i	−0.854604	+	0.519280i	$0.826200\pi$
$860$	−5.30204e9			−0.284249
$861$		0			0
$862$	−1.56688e10			−0.833224
$863$		−	3.14459e10i		−	1.66543i	−0.553701	−	0.832716i	$-0.686785\pi$
$863$		−	3.14459e10i		−	1.66543i	0.553701	−	0.832716i	$-0.313215\pi$
$864$	−1.30968e10	+	1.27148e10i	−0.690824	+	0.670673i
$865$	2.06755e10			1.08618
$866$	−1.10725e10			−0.579337
$867$	1.08047e10	−	1.06986e10i	0.563050	−	0.557522i
$868$		0			0
$869$			1.44920e9i			0.0749134i
$870$	9.74275e9	−	9.64709e9i	0.501608	−	0.496683i
$871$			1.73633e10i			0.890367i
$872$		−	1.56040e10i		−	0.796946i
$873$	−7.34981e9	−	7.25216e7i	−0.373875	−	0.00368908i
$874$		−	6.30354e9i		−	0.319370i
$875$		0			0
$876$	6.13100e9	+	6.19179e9i	0.308153	+	0.311209i
$877$	−1.84424e10			−0.923248			−0.461624	−	0.887076i	$-0.652733\pi$
$877$	−1.84424e10			−0.923248			−0.461624	+	0.887076i	$0.652733\pi$
$878$	4.96760e9			0.247694
$879$	−5.34485e9	+	5.29237e9i	−0.265445	+	0.262839i
$880$		−	6.47626e8i		−	0.0320358i
$881$	−2.28663e10			−1.12663			−0.563314	−	0.826243i	$-0.690474\pi$
$881$	−2.28663e10			−1.12663			−0.563314	+	0.826243i	$0.690474\pi$
$882$		0			0
$883$	9.87721e9			0.482805			0.241402	−	0.970425i	$-0.422393\pi$
$883$	9.87721e9			0.482805			0.241402	+	0.970425i	$0.422393\pi$
$884$			5.32472e9i			0.259247i
$885$	1.72930e10	−	1.71232e10i	0.838627	−	0.830393i
$886$	−1.21060e10			−0.584765
$887$	−2.76055e10			−1.32820			−0.664099	−	0.747644i	$-0.731185\pi$
$887$	−2.76055e10			−1.32820			−0.664099	+	0.747644i	$0.731185\pi$
$888$	−2.79585e10	−	2.82357e10i	−1.33989	−	1.35317i
$889$		0			0
$890$			1.52722e10i			0.726165i
$891$	1.83693e8	−	9.30742e9i	0.00870004	−	0.440816i
$892$		−	4.88904e9i		−	0.230646i
$893$		−	7.97303e9i		−	0.374665i
$894$	5.13163e9	−	5.08124e9i	0.240200	−	0.237842i
$895$		−	1.28644e10i		−	0.599805i
$896$		0			0
$897$	−2.14226e10	+	2.12123e10i	−0.991058	+	0.981328i
$898$	−1.04159e10			−0.479988
$899$	−1.90238e10			−0.873249
$900$	−5.83012e7	+	5.90862e9i	−0.00266581	+	0.270170i
$901$		−	1.46239e9i		−	0.0666081i
$902$	4.88852e9			0.221796
$903$		0			0
$904$	−9.40569e7			−0.00423449
$905$		−	5.66935e8i		−	0.0254252i
$906$	−1.94121e10	−	1.96046e10i	−0.867210	−	0.875809i
$907$	1.45634e10			0.648094			0.324047	−	0.946041i	$-0.394956\pi$
$907$	1.45634e10			0.648094			0.324047	+	0.946041i	$0.394956\pi$
$908$	1.05103e10			0.465923
$909$	−2.74247e8	+	2.77940e10i	−0.0121107	+	1.22738i
$910$		0			0
$911$		−	1.79017e10i		−	0.784478i	−0.919863	−	0.392239i	$-0.871701\pi$
$911$		−	1.79017e10i		−	0.784478i	0.919863	−	0.392239i	$-0.128299\pi$
$912$	5.61016e8	+	5.66579e8i	0.0244903	+	0.0247331i
$913$			1.01244e10i			0.440271i
$914$			8.38418e9i			0.363203i
$915$	1.15545e10	+	1.16691e10i	0.498629	+	0.503573i
$916$			2.65283e10i			1.14045i
$917$		0			0
$918$	5.01115e9	−	4.86497e9i	0.213790	−	0.207554i
$919$	1.97477e10			0.839292			0.419646	−	0.907688i	$-0.362154\pi$
$919$	1.97477e10			0.839292			0.419646	+	0.907688i	$0.362154\pi$
$920$	−2.47595e10			−1.04830
$921$	5.80629e9	+	5.86386e9i	0.244901	+	0.247329i
$922$			1.01271e10i			0.425526i
$923$	4.17162e10			1.74622
$924$		0			0
$925$	−2.10435e10			−0.874224
$926$			7.65118e9i			0.316657i
$927$	1.87692e10	+	1.85199e8i	0.773869	+	0.00763588i
$928$	−3.48244e10			−1.43043
$929$	2.77242e10			1.13450			0.567249	−	0.823546i	$-0.308008\pi$
$929$	2.77242e10			1.13450			0.567249	+	0.823546i	$0.308008\pi$
$930$	4.86812e9	−	4.82032e9i	0.198459	−	0.196510i
$931$		0			0
$932$		−	1.44424e10i		−	0.584366i
$933$	−2.34997e10	+	2.32690e10i	−0.947278	+	0.937977i
$934$		−	1.31383e10i		−	0.527626i
$935$		−	3.64861e9i		−	0.145978i
$936$	2.52993e8	−	2.56400e10i	0.0100843	−	1.02200i
$937$		−	2.09195e10i		−	0.830736i	−0.909654	−	0.415368i	$-0.863653\pi$
$937$		−	2.09195e10i		−	0.830736i	0.909654	−	0.415368i	$-0.136347\pi$
$938$		0			0
$939$	1.69351e10	+	1.71031e10i	0.667512	+	0.674131i
$940$	−1.14001e10			−0.447673
$941$	−3.59853e9			−0.140787			−0.0703934	−	0.997519i	$-0.522425\pi$
$941$	−3.59853e9			−0.140787			−0.0703934	+	0.997519i	$0.522425\pi$
$942$	2.03044e10	−	2.01051e10i	0.791430	−	0.783660i
$943$		−	2.77952e10i		−	1.07939i
$944$	4.19798e9			0.162420
$945$		0			0
$946$	5.13096e9			0.197052
$947$			6.32978e9i			0.242194i	0.992641	+	0.121097i	$0.0386413\pi$
$947$			6.32978e9i			0.242194i	−0.992641	+	0.121097i	$0.961359\pi$
$948$	−1.81280e9	+	1.79500e9i	−0.0691067	+	0.0684282i
$949$	−2.00248e10			−0.760565
$950$	2.83926e9			0.107442
$951$	2.67021e10	+	2.69669e10i	1.00673	+	1.01671i
$952$		0			0
$953$			2.82104e10i			1.05581i	0.849304	+	0.527904i	$0.177022\pi$
$953$			2.82104e10i			1.05581i	−0.849304	+	0.527904i	$0.822978\pi$
$954$	−2.52932e7	+	2.56338e9i	−0.000943158	+	0.0955857i
$955$			8.50256e9i			0.315891i
$956$		−	6.09872e9i		−	0.225754i
$957$	1.26202e10	−	1.24963e10i	0.465452	−	0.460882i
$958$			1.92509e10i			0.707412i
$959$		0			0
$960$	1.03268e10	−	1.02254e10i	0.376716	−	0.373018i
$961$	1.80071e10			0.654503
$962$	3.32413e10			1.20383
$963$	−1.08088e9	−	1.06652e7i	−0.0390018	−	0.000384837i
$964$			1.60519e10i			0.577108i
$965$	−1.63321e10			−0.585054
$966$		0			0
$967$	2.10247e10			0.747718			0.373859	−	0.927486i	$-0.378034\pi$
$967$	2.10247e10			0.747718			0.373859	+	0.927486i	$0.378034\pi$
$968$			2.33761e10i			0.828340i
$969$	3.16066e9	+	3.19200e9i	0.111595	+	0.112702i
$970$	5.04986e9			0.177655
$971$	1.08425e10			0.380068			0.190034	−	0.981778i	$-0.439140\pi$
$971$	1.08425e10			0.380068			0.190034	+	0.981778i	$0.439140\pi$
$972$	1.18701e10	−	1.12985e10i	0.414594	−	0.394629i
$973$		0			0
$974$		−	2.41168e10i		−	0.836304i
$975$	−9.55450e9	−	9.64924e9i	−0.330135	−	0.333409i
$976$			2.83274e9i			0.0975287i
$977$		−	2.45894e10i		−	0.843562i	−0.906698	−	0.421781i	$-0.861405\pi$
$977$		−	2.45894e10i		−	0.843562i	0.906698	−	0.421781i	$-0.138595\pi$
$978$	1.18456e10	+	1.19631e10i	0.404922	+	0.408938i
$979$			1.97827e10i			0.673823i
$980$		0			0
$981$	−2.26128e8	+	2.29173e10i	−0.00764739	+	0.775035i
$982$	−6.92243e8			−0.0233275
$983$	2.06353e10			0.692905			0.346452	−	0.938068i	$-0.387386\pi$
$983$	2.06353e10			0.692905			0.346452	+	0.938068i	$0.387386\pi$
$984$	1.66336e10	+	1.67985e10i	0.556548	+	0.562066i
$985$		−	4.99705e9i		−	0.166604i
$986$	1.33246e10			0.442675
$987$		0			0
$988$	6.00335e9			0.198036
$989$		−	2.91737e10i		−	0.958970i
$990$	−6.31054e7	+	6.39551e9i	−0.00206702	+	0.209485i
$991$	8.42393e9			0.274952			0.137476	−	0.990505i	$-0.456101\pi$
$991$	8.42393e9			0.274952			0.137476	+	0.990505i	$0.456101\pi$
$992$	−1.74005e10			−0.565942
$993$	−2.38403e9	+	2.36063e9i	−0.0772663	+	0.0765077i
$994$		0			0
$995$			6.15130e9i			0.197964i
$996$	−1.26645e10	+	1.25402e10i	−0.406145	+	0.402157i
$997$			3.50078e10i			1.11875i	0.828916	+	0.559373i	$0.188958\pi$
$997$			3.50078e10i			1.11875i	−0.828916	+	0.559373i	$0.811042\pi$
$998$			2.64601e10i			0.842625i
$999$	4.06528e10	+	4.18743e10i	1.29006	+	1.32883i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.8.c.b.146.20		32
3.2	odd	2	inner	147.8.c.b.146.13		32
7.2	even	3		21.8.g.b.17.10	yes	32
7.3	odd	6		21.8.g.b.5.7	✓	32
7.6	odd	2	inner	147.8.c.b.146.14		32
21.2	odd	6		21.8.g.b.17.7	yes	32
21.17	even	6		21.8.g.b.5.10	yes	32
21.20	even	2	inner	147.8.c.b.146.19		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.g.b.5.7	✓	32	7.3	odd	6
21.8.g.b.5.10	yes	32	21.17	even	6
21.8.g.b.17.7	yes	32	21.2	odd	6
21.8.g.b.17.10	yes	32	7.2	even	3
147.8.c.b.146.13		32	3.2	odd	2	inner
147.8.c.b.146.14		32	7.6	odd	2	inner
147.8.c.b.146.19		32	21.20	even	2	inner
147.8.c.b.146.20		32	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	147.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.39832i q^{2} +(-33.2308 + 32.9046i) q^{3} +73.2648 q^{4} +203.095 q^{5} +(-243.439 - 245.852i) q^{6} +1489.02i q^{8} +(21.5784 - 2186.89i) q^{9} +1502.56i q^{10} +1946.33i q^{11} +(-2434.65 + 2410.75i) q^{12} -7873.88i q^{13} +(-6749.00 + 6682.74i) q^{15} -1638.36 q^{16} -9230.23 q^{17} +(16179.3 + 159.644i) q^{18} +10406.6i q^{19} +14879.7 q^{20} -14399.6 q^{22} +81873.3i q^{23} +(-48995.6 - 49481.5i) q^{24} -36877.6 q^{25} +58253.5 q^{26} +(71241.7 + 73382.4i) q^{27} +195123. i q^{29} +(-49441.0 - 49931.3i) q^{30} +97496.3i q^{31} +178474. i q^{32} +(-64043.1 - 64678.2i) q^{33} -68288.2i q^{34} +(1580.94 - 160222. i) q^{36} +570632. q^{37} -76991.4 q^{38} +(259087. + 261656. i) q^{39} +302412. i q^{40} -339491. q^{41} -356328. q^{43} +142598. i q^{44} +(4382.45 - 444146. i) q^{45} -605725. q^{46} -766151. q^{47} +(54444.1 - 53909.6i) q^{48} -272832. i q^{50} +(306728. - 303717. i) q^{51} -576879. i q^{52} +158435. i q^{53} +(-542906. + 527069. i) q^{54} +395289. i q^{55} +(-342425. - 345820. i) q^{57} -1.44358e6 q^{58} -2.56230e6 q^{59} +(-494465. + 489610. i) q^{60} -1.72901e6i q^{61} -721309. q^{62} -1.53012e6 q^{64} -1.59914e6i q^{65} +(478510. - 473812. i) q^{66} -2.20518e6 q^{67} -676251. q^{68} +(-2.69401e6 - 2.72072e6i) q^{69} +5.29805e6i q^{71} +(3.25633e6 + 32130.7i) q^{72} -2.54319e6i q^{73} +4.22172e6i q^{74} +(1.22547e6 - 1.21344e6i) q^{75} +762439. i q^{76} +(-1.93581e6 + 1.91681e6i) q^{78} +744581. q^{79} -332742. q^{80} +(-4.78204e6 - 94379.3i) q^{81} -2.51166e6i q^{82} +5.20177e6 q^{83} -1.87461e6 q^{85} -2.63623e6i q^{86} +(-6.42044e6 - 6.48411e6i) q^{87} -2.89813e6 q^{88} +1.01641e7 q^{89} +(3.28594e6 + 32422.8i) q^{90} +5.99844e6i q^{92} +(-3.20807e6 - 3.23989e6i) q^{93} -5.66823e6i q^{94} +2.11353e6i q^{95} +(-5.87260e6 - 5.93083e6i) q^{96} -3.36084e6i q^{97} +(4.25641e6 + 41998.7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2300 q^{4} + 7032 q^{9} + 27576 q^{15} + 39188 q^{16} - 66996 q^{18} + 105132 q^{22} + 662504 q^{25} + 81324 q^{30} - 227244 q^{36} + 1237496 q^{37} - 3992208 q^{39} - 1416064 q^{43} + 6985680 q^{46}+ \cdots + 35642232 q^{99}+O(q^{100}) \) 32 * q - 2300 * q^4 + 7032 * q^9 + 27576 * q^15 + 39188 * q^16 - 66996 * q^18 + 105132 * q^22 + 662504 * q^25 + 81324 * q^30 - 227244 * q^36 + 1237496 * q^37 - 3992208 * q^39 - 1416064 * q^43 + 6985680 * q^46 - 1354680 * q^51 - 6279408 * q^57 + 4128684 * q^58 - 15042564 * q^60 - 589508 * q^64 + 3889456 * q^67 + 3088944 * q^72 + 1917432 * q^78 - 15252656 * q^79 - 17406072 * q^81 + 3418848 * q^85 - 24626796 * q^88 + 49664208 * q^93 + 35642232 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more