LMFDB - Embedded newform 138.7.b.a.91.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,7,Mod(91,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, 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("138.91");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	138.b (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:	$31.7474635395$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.18
Character	$\chi$	$=$	138.91
Dual form			138.7.b.a.91.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685 q^{2} -15.5885 q^{3} +32.0000 q^{4} +223.235i q^{5} -88.1816 q^{6} +652.987i q^{7} +181.019 q^{8} +243.000 q^{9} +O(q^{10})$	\(q+5.65685 q^{2} -15.5885 q^{3} +32.0000 q^{4} +223.235i q^{5} -88.1816 q^{6} +652.987i q^{7} +181.019 q^{8} +243.000 q^{9} +1262.81i q^{10} -524.845i q^{11} -498.831 q^{12} -1117.97 q^{13} +3693.85i q^{14} -3479.89i q^{15} +1024.00 q^{16} +2162.11i q^{17} +1374.62 q^{18} -10390.2i q^{19} +7143.51i q^{20} -10179.1i q^{21} -2968.97i q^{22} +(-12063.0 - 1587.73i) q^{23} -2821.81 q^{24} -34208.8 q^{25} -6324.19 q^{26} -3788.00 q^{27} +20895.6i q^{28} +34508.2 q^{29} -19685.2i q^{30} +26785.9 q^{31} +5792.62 q^{32} +8181.52i q^{33} +12230.7i q^{34} -145769. q^{35} +7776.00 q^{36} +63171.7i q^{37} -58775.8i q^{38} +17427.4 q^{39} +40409.8i q^{40} -33696.9 q^{41} -57581.4i q^{42} +99077.8i q^{43} -16795.0i q^{44} +54246.0i q^{45} +(-68238.4 - 8981.54i) q^{46} +99139.2 q^{47} -15962.6 q^{48} -308743. q^{49} -193514. q^{50} -33703.9i q^{51} -35775.0 q^{52} -98979.9i q^{53} -21428.1 q^{54} +117164. q^{55} +118203. i q^{56} +161967. i q^{57} +195208. q^{58} -255470. q^{59} -111356. i q^{60} -324932. i q^{61} +151524. q^{62} +158676. i q^{63} +32768.0 q^{64} -249570. i q^{65} +46281.7i q^{66} -5997.63i q^{67} +69187.4i q^{68} +(188043. + 24750.2i) q^{69} -824596. q^{70} -62644.5 q^{71} +43987.7 q^{72} +353223. q^{73} +357353. i q^{74} +533262. q^{75} -332486. i q^{76} +342717. q^{77} +98584.3 q^{78} -216426. i q^{79} +228592. i q^{80} +59049.0 q^{81} -190619. q^{82} +161058. i q^{83} -325730. i q^{84} -482657. q^{85} +560468. i q^{86} -537929. q^{87} -95007.0i q^{88} +190490. i q^{89} +306862. i q^{90} -730019. i q^{91} +(-386015. - 50807.2i) q^{92} -417550. q^{93} +560816. q^{94} +2.31945e6 q^{95} -90298.0 q^{96} +1.16024e6i q^{97} -1.74651e6 q^{98} -127537. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 768 q^{4} + 5832 q^{9}+O(q^{10}) $ 24 * q + 768 * q^4 + 5832 * q^9	$ 24 q + 768 q^{4} + 5832 q^{9} - 768 q^{13} + 24576 q^{16} - 44104 q^{23} - 119448 q^{25} - 53888 q^{26} + 3456 q^{29} + 50976 q^{31} + 149008 q^{35} + 186624 q^{36} + 11664 q^{39} - 3920 q^{41} - 150720 q^{46} + 441088 q^{47} - 32472 q^{49} + 8320 q^{50} - 24576 q^{52} + 826176 q^{55} - 307200 q^{58} - 1210160 q^{59} + 783744 q^{62} + 786432 q^{64} + 361584 q^{69} - 2480064 q^{70} + 1531264 q^{71} + 593472 q^{73} + 23328 q^{75} + 1068784 q^{77} + 171072 q^{78} + 1417176 q^{81} + 1454592 q^{82} - 1318272 q^{85} + 697248 q^{87} - 1411328 q^{92} - 983664 q^{93} + 1115712 q^{94} + 4047632 q^{95} - 2409344 q^{98}+O(q^{100}) $ 24 * q + 768 * q^4 + 5832 * q^9 - 768 * q^13 + 24576 * q^16 - 44104 * q^23 - 119448 * q^25 - 53888 * q^26 + 3456 * q^29 + 50976 * q^31 + 149008 * q^35 + 186624 * q^36 + 11664 * q^39 - 3920 * q^41 - 150720 * q^46 + 441088 * q^47 - 32472 * q^49 + 8320 * q^50 - 24576 * q^52 + 826176 * q^55 - 307200 * q^58 - 1210160 * q^59 + 783744 * q^62 + 786432 * q^64 + 361584 * q^69 - 2480064 * q^70 + 1531264 * q^71 + 593472 * q^73 + 23328 * q^75 + 1068784 * q^77 + 171072 * q^78 + 1417176 * q^81 + 1454592 * q^82 - 1318272 * q^85 + 697248 * q^87 - 1411328 * q^92 - 983664 * q^93 + 1115712 * q^94 + 4047632 * q^95 - 2409344 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

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 $

$2$	5.65685			0.707107
$2$	5.65685			0.707107
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	32.0000			0.500000
$5$			223.235i			1.78588i	0.450178	+	0.892939i	$0.351361\pi$
$5$			223.235i			1.78588i	−0.450178	+	0.892939i	$0.648639\pi$
$6$	−88.1816			−0.408248
$7$			652.987i			1.90375i	0.306485	+	0.951876i	$0.400847\pi$
$7$			652.987i			1.90375i	−0.306485	+	0.951876i	$0.599153\pi$
$8$	181.019			0.353553
$9$	243.000			0.333333
$10$			1262.81i			1.26281i
$11$		−	524.845i		−	0.394324i	−0.980371	−	0.197162i	$-0.936828\pi$
$11$		−	524.845i		−	0.394324i	0.980371	−	0.197162i	$-0.0631724\pi$
$12$	−498.831			−0.288675
$13$	−1117.97			−0.508862			−0.254431	−	0.967091i	$-0.581888\pi$
$13$	−1117.97			−0.508862			−0.254431	+	0.967091i	$0.581888\pi$
$14$			3693.85i			1.34616i
$15$		−	3479.89i		−	1.03108i
$16$	1024.00			0.250000
$17$			2162.11i			0.440079i	0.975491	+	0.220039i	$0.0706186\pi$
$17$			2162.11i			0.440079i	−0.975491	+	0.220039i	$0.929381\pi$
$18$	1374.62			0.235702
$19$		−	10390.2i		−	1.51483i	−0.652936	−	0.757413i	$-0.726463\pi$
$19$		−	10390.2i		−	1.51483i	0.652936	−	0.757413i	$-0.273537\pi$
$20$			7143.51i			0.892939i
$21$		−	10179.1i		−	1.09913i
$22$		−	2968.97i		−	0.278829i
$23$	−12063.0	−	1587.73i	−0.991449	−	0.130494i
$23$	−12063.0	−	1587.73i	−0.991449	−	0.130494i
$24$	−2821.81			−0.204124
$25$	−34208.8			−2.18936
$26$	−6324.19			−0.359820
$27$	−3788.00			−0.192450
$28$			20895.6i			0.951876i
$29$	34508.2			1.41491			0.707453	−	0.706760i	$-0.249844\pi$
$29$	34508.2			1.41491			0.707453	+	0.706760i	$0.249844\pi$
$30$		−	19685.2i		−	0.729082i
$31$	26785.9			0.899126			0.449563	−	0.893249i	$-0.351580\pi$
$31$	26785.9			0.899126			0.449563	+	0.893249i	$0.351580\pi$
$32$	5792.62			0.176777
$33$			8181.52i			0.227663i
$34$			12230.7i			0.311183i
$35$	−145769.			−3.39987
$36$	7776.00			0.166667
$37$			63171.7i			1.24715i	0.781765	+	0.623573i	$0.214320\pi$
$37$			63171.7i			1.24715i	−0.781765	+	0.623573i	$0.785680\pi$
$38$		−	58775.8i		−	1.07114i
$39$	17427.4			0.293791
$40$			40409.8i			0.631403i
$41$	−33696.9			−0.488921			−0.244461	−	0.969659i	$-0.578611\pi$
$41$	−33696.9			−0.488921			−0.244461	+	0.969659i	$0.578611\pi$
$42$		−	57581.4i		−	0.777203i
$43$			99077.8i			1.24615i	0.782161	+	0.623076i	$0.214117\pi$
$43$			99077.8i			1.24615i	−0.782161	+	0.623076i	$0.785883\pi$
$44$		−	16795.0i		−	0.197162i
$45$			54246.0i			0.595293i
$46$	−68238.4	−	8981.54i	−0.701060	−	0.0922735i
$47$	99139.2			0.954887			0.477443	−	0.878663i	$-0.341564\pi$
$47$	99139.2			0.954887			0.477443	+	0.878663i	$0.341564\pi$
$48$	−15962.6			−0.144338
$49$	−308743.			−2.62427
$50$	−193514.			−1.54811
$51$		−	33703.9i		−	0.254080i
$52$	−35775.0			−0.254431
$53$		−	98979.9i		−	0.664843i	−0.943131	−	0.332422i	$-0.892134\pi$
$53$		−	98979.9i		−	0.664843i	0.943131	−	0.332422i	$-0.107866\pi$
$54$	−21428.1			−0.136083
$55$	117164.			0.704214
$56$			118203.i			0.673078i
$57$			161967.i			0.874585i
$58$	195208.			1.00049
$59$	−255470.			−1.24390			−0.621948	−	0.783059i	$-0.713658\pi$
$59$	−255470.			−1.24390			−0.621948	+	0.783059i	$0.713658\pi$
$60$		−	111356.i		−	0.515539i
$61$		−	324932.i		−	1.43154i	−0.698338	−	0.715769i	$-0.746077\pi$
$61$		−	324932.i		−	1.43154i	0.698338	−	0.715769i	$-0.253923\pi$
$62$	151524.			0.635778
$63$			158676.i			0.634584i
$64$	32768.0			0.125000
$65$		−	249570.i		−	0.908765i
$66$			46281.7i			0.160982i
$67$		−	5997.63i		−	0.0199414i	−0.999950	−	0.00997069i	$-0.996826\pi$
$67$		−	5997.63i		−	0.0199414i	0.999950	−	0.00997069i	$-0.00317382\pi$
$68$			69187.4i			0.220039i
$69$	188043.	+	24750.2i	0.572413	+	0.0753410i
$70$	−824596.			−2.40407
$71$	−62644.5			−0.175028			−0.0875140	−	0.996163i	$-0.527892\pi$
$71$	−62644.5			−0.175028			−0.0875140	+	0.996163i	$0.527892\pi$
$72$	43987.7			0.117851
$73$	353223.			0.907989			0.453994	−	0.891005i	$-0.349999\pi$
$73$	353223.			0.907989			0.453994	+	0.891005i	$0.349999\pi$
$74$			357353.i			0.881865i
$75$	533262.			1.26403
$76$		−	332486.i		−	0.757413i
$77$	342717.			0.750694
$78$	98584.3			0.207742
$79$		−	216426.i		−	0.438963i	−0.975617	−	0.219481i	$-0.929563\pi$
$79$		−	216426.i		−	0.438963i	0.975617	−	0.219481i	$-0.0704365\pi$
$80$			228592.i			0.446470i
$81$	59049.0			0.111111
$82$	−190619.			−0.345719
$83$			161058.i			0.281675i	0.990033	+	0.140838i	$0.0449795\pi$
$83$			161058.i			0.281675i	−0.990033	+	0.140838i	$0.955020\pi$
$84$		−	325730.i		−	0.549566i
$85$	−482657.			−0.785927
$86$			560468.i			0.881162i
$87$	−537929.			−0.816897
$88$		−	95007.0i		−	0.139414i
$89$			190490.i			0.270210i	0.990831	+	0.135105i	$0.0431372\pi$
$89$			190490.i			0.270210i	−0.990831	+	0.135105i	$0.956863\pi$
$90$			306862.i			0.420936i
$91$		−	730019.i		−	0.968746i
$92$	−386015.	−	50807.2i	−0.495725	−	0.0652472i
$93$	−417550.			−0.519111
$94$	560816.			0.675207
$95$	2.31945e6			2.70529
$96$	−90298.0			−0.102062
$97$			1.16024e6i			1.27125i	0.771997	+	0.635626i	$0.219258\pi$
$97$			1.16024e6i			1.27125i	−0.771997	+	0.635626i	$0.780742\pi$
$98$	−1.74651e6			−1.85564
$99$		−	127537.i		−	0.131441i
$100$	−1.09468e6			−1.09468
$101$	−1.87711e6			−1.82190			−0.910951	−	0.412515i	$-0.864651\pi$
$101$	−1.87711e6			−1.82190			−0.910951	+	0.412515i	$0.864651\pi$
$102$		−	190658.i		−	0.179661i
$103$			651291.i			0.596023i	0.954562	+	0.298012i	$0.0963234\pi$
$103$			651291.i			0.596023i	−0.954562	+	0.298012i	$0.903677\pi$
$104$	−202374.			−0.179910
$105$	2.27232e6			1.96291
$106$		−	559915.i		−	0.470115i
$107$		−	1.10278e6i		−	0.900199i	−0.892978	−	0.450099i	$-0.851389\pi$
$107$		−	1.10278e6i		−	0.900199i	0.892978	−	0.450099i	$-0.148611\pi$
$108$	−121216.			−0.0962250
$109$			1.99521e6i			1.54067i	0.637640	+	0.770334i	$0.279911\pi$
$109$			1.99521e6i			1.54067i	−0.637640	+	0.770334i	$0.720089\pi$
$110$	662777.			0.497954
$111$		−	984749.i		−	0.720040i
$112$			668658.i			0.475938i
$113$			1.83040e6i			1.26856i	0.773105	+	0.634278i	$0.218703\pi$
$113$			1.83040e6i			1.26856i	−0.773105	+	0.634278i	$0.781297\pi$
$114$			916224.i			0.618425i
$115$	354436.	−	2.69287e6i	0.233047	−	1.77061i
$116$	1.10426e6			0.707453
$117$	−271667.			−0.169621
$118$	−1.44516e6			−0.879567
$119$	−1.41183e6			−0.837800
$120$		−	629927.i		−	0.364541i
$121$	1.49610e6			0.844509
$122$		−	1.83809e6i		−	1.01225i
$123$	525283.			0.282279
$124$	857148.			0.449563
$125$		−	4.14854e6i		−	2.12405i
$126$			897606.i			0.448718i
$127$	−573188.			−0.279825			−0.139912	−	0.990164i	$-0.544682\pi$
$127$	−573188.			−0.279825			−0.139912	+	0.990164i	$0.544682\pi$
$128$	185364.			0.0883883
$129$		−	1.54447e6i		−	0.719466i
$130$		−	1.41178e6i		−	0.642594i
$131$	686788.			0.305498			0.152749	−	0.988265i	$-0.451187\pi$
$131$	686788.			0.305498			0.152749	+	0.988265i	$0.451187\pi$
$132$			261809.i			0.113831i
$133$	6.78465e6			2.88385
$134$		−	33927.7i		−	0.0141007i
$135$		−	845612.i		−	0.343692i
$136$			391383.i			0.155591i
$137$			420337.i			0.163469i	0.996654	+	0.0817347i	$0.0260460\pi$
$137$			420337.i			0.163469i	−0.996654	+	0.0817347i	$0.973954\pi$
$138$	1.06373e6	+	140008.i	0.404757	+	0.0532742i
$139$	2.98677e6			1.11213			0.556067	−	0.831137i	$-0.312310\pi$
$139$	2.98677e6			1.11213			0.556067	+	0.831137i	$0.312310\pi$
$140$	−4.66462e6			−1.69993
$141$	−1.54543e6			−0.551304
$142$	−354371.			−0.123764
$143$			586760.i			0.200656i
$144$	248832.			0.0833333
$145$			7.70342e6i			2.52685i
$146$	1.99813e6			0.642045
$147$	4.81282e6			1.51512
$148$			2.02149e6i			0.623573i
$149$			4.97553e6i			1.50411i	0.659099	+	0.752056i	$0.270938\pi$
$149$			4.97553e6i			1.50411i	−0.659099	+	0.752056i	$0.729062\pi$
$150$	3.01658e6			0.893803
$151$	2.79784e6			0.812628			0.406314	−	0.913733i	$-0.366814\pi$
$151$	2.79784e6			0.812628			0.406314	+	0.913733i	$0.366814\pi$
$152$		−	1.88082e6i		−	0.535572i
$153$			525392.i			0.146693i
$154$	1.93870e6			0.530821
$155$			5.97954e6i			1.60573i
$156$	557677.			0.146896
$157$		−	48304.7i		−	0.0124822i	−0.999981	−	0.00624109i	$-0.998013\pi$
$157$		−	48304.7i		−	0.0124822i	0.999981	−	0.00624109i	$-0.00198661\pi$
$158$		−	1.22429e6i		−	0.310394i
$159$			1.54294e6i			0.383847i
$160$			1.29311e6i			0.315702i
$161$	1.03676e6	−	7.87695e6i	0.248429	−	1.88747i
$162$	334032.			0.0785674
$163$	6.33436e6			1.46265			0.731324	−	0.682030i	$-0.238903\pi$
$163$	6.33436e6			1.46265			0.731324	+	0.682030i	$0.238903\pi$
$164$	−1.07830e6			−0.244461
$165$	−1.82640e6			−0.406578
$166$			911082.i			0.199174i
$167$	−2.95635e6			−0.634755			−0.317377	−	0.948299i	$-0.602802\pi$
$167$	−2.95635e6			−0.634755			−0.317377	+	0.948299i	$0.602802\pi$
$168$		−	1.84261e6i		−	0.388602i
$169$	−3.57695e6			−0.741060
$170$	−2.73032e6			−0.555734
$171$		−	2.52482e6i		−	0.504942i
$172$			3.17049e6i			0.623076i
$173$	−1.66884e6			−0.322313			−0.161156	−	0.986929i	$-0.551522\pi$
$173$	−1.66884e6			−0.322313			−0.161156	+	0.986929i	$0.551522\pi$
$174$	−3.04299e6			−0.577633
$175$		−	2.23379e7i		−	4.16800i
$176$		−	537441.i		−	0.0985809i
$177$	3.98238e6			0.718163
$178$			1.07757e6i			0.191067i
$179$	−4.01493e6			−0.700034			−0.350017	−	0.936743i	$-0.613824\pi$
$179$	−4.01493e6			−0.700034			−0.350017	+	0.936743i	$0.613824\pi$
$180$			1.73587e6i			0.297646i
$181$		−	9.09953e6i		−	1.53456i	−0.641314	−	0.767279i	$-0.721610\pi$
$181$		−	9.09953e6i		−	1.53456i	0.641314	−	0.767279i	$-0.278390\pi$
$182$		−	4.12961e6i		−	0.685007i
$183$			5.06518e6i			0.826498i
$184$	−2.18363e6	−	287409.i	−0.350530	−	0.0461368i
$185$	−1.41021e7			−2.22725
$186$	−2.36202e6			−0.367067
$187$	1.13477e6			0.173533
$188$	3.17245e6			0.477443
$189$		−	2.47351e6i		−	0.366377i
$190$	1.31208e7			1.91293
$191$			9.73798e6i			1.39755i	0.715339	+	0.698777i	$0.246272\pi$
$191$			9.73798e6i			1.39755i	−0.715339	+	0.698777i	$0.753728\pi$
$192$	−510803.			−0.0721688
$193$	−5.39492e6			−0.750435			−0.375218	−	0.926937i	$-0.622432\pi$
$193$	−5.39492e6			−0.750435			−0.375218	+	0.926937i	$0.622432\pi$
$194$			6.56330e6i			0.898911i
$195$			3.89040e6i			0.524676i
$196$	−9.87976e6			−1.31213
$197$	−6.49454e6			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$197$	−6.49454e6			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$198$		−	721460.i		−	0.0929430i
$199$		−	5.12255e6i		−	0.650021i	−0.945710	−	0.325010i	$-0.894632\pi$
$199$		−	5.12255e6i		−	0.650021i	0.945710	−	0.325010i	$-0.105368\pi$
$200$	−6.19245e6			−0.774056
$201$			93493.8i			0.0115132i
$202$	−1.06185e7			−1.28828
$203$			2.25334e7i			2.69363i
$204$		−	1.07853e6i		−	0.127040i
$205$		−	7.52233e6i		−	0.873154i
$206$			3.68426e6i			0.421452i
$207$	−2.93130e6	−	385818.i	−0.330483	−	0.0434982i
$208$	−1.14480e6			−0.127215
$209$	−5.45324e6			−0.597331
$210$	1.28542e7			1.38799
$211$	1.32326e7			1.40864			0.704318	−	0.709885i	$-0.251253\pi$
$211$	1.32326e7			1.40864			0.704318	+	0.709885i	$0.251253\pi$
$212$		−	3.16736e6i		−	0.332422i
$213$	976530.			0.101052
$214$		−	6.23828e6i		−	0.636537i
$215$	−2.21176e7			−2.22547
$216$	−685700.			−0.0680414
$217$			1.74908e7i			1.71171i
$218$			1.12866e7i			1.08942i
$219$	−5.50620e6			−0.524227
$220$	3.74924e6			0.352107
$221$		−	2.41717e6i		−	0.223939i
$222$		−	5.57058e6i		−	0.509145i
$223$	−2.57972e6			−0.232626			−0.116313	−	0.993213i	$-0.537108\pi$
$223$	−2.57972e6			−0.232626			−0.116313	+	0.993213i	$0.537108\pi$
$224$			3.78250e6i			0.336539i
$225$	−8.31273e6			−0.729787
$226$			1.03543e7i			0.897005i
$227$			2.87927e6i			0.246153i	0.992397	+	0.123076i	$0.0392760\pi$
$227$			2.87927e6i			0.246153i	−0.992397	+	0.123076i	$0.960724\pi$
$228$			5.18294e6i			0.437292i
$229$			8.06087e6i			0.671236i	0.941998	+	0.335618i	$0.108945\pi$
$229$			8.06087e6i			0.671236i	−0.941998	+	0.335618i	$0.891055\pi$
$230$	2.00499e6	−	1.52332e7i	0.164789	−	1.25201i
$231$	−5.34242e6			−0.433413
$232$	6.24664e6			0.500245
$233$	−1.20730e7			−0.954441			−0.477220	−	0.878784i	$-0.658356\pi$
$233$	−1.20730e7			−0.954441			−0.477220	+	0.878784i	$0.658356\pi$
$234$	−1.53678e6			−0.119940
$235$			2.21313e7i			1.70531i
$236$	−8.17504e6			−0.621948
$237$			3.37374e6i			0.253435i
$238$	−7.98650e6			−0.592414
$239$	2.67659e7			1.96060			0.980300	−	0.197516i	$-0.0632874\pi$
$239$	2.67659e7			1.96060			0.980300	+	0.197516i	$0.0632874\pi$
$240$		−	3.56340e6i		−	0.257769i
$241$			4.86048e6i			0.347239i	0.984813	+	0.173619i	$0.0555462\pi$
$241$			4.86048e6i			0.347239i	−0.984813	+	0.173619i	$0.944454\pi$
$242$	8.46321e6			0.597158
$243$	−920483.			−0.0641500
$244$		−	1.03978e7i		−	0.715769i
$245$		−	6.89221e7i		−	4.68662i
$246$	2.97145e6			0.199601
$247$			1.16159e7i			0.770837i
$248$	4.84876e6			0.317889
$249$		−	2.51065e6i		−	0.162625i
$250$		−	2.34677e7i		−	1.50193i
$251$		−	1.98006e7i		−	1.25216i	−0.779761	−	0.626078i	$-0.784659\pi$
$251$		−	1.98006e7i		−	1.25216i	0.779761	−	0.626078i	$-0.215341\pi$
$252$			5.07762e6i			0.317292i
$253$	−833310.	+	6.33118e6i	−0.0514571	+	0.390952i
$254$	−3.24244e6			−0.197866
$255$	7.52388e6			0.453755
$256$	1.04858e6			0.0625000
$257$	−2.13611e7			−1.25842			−0.629208	−	0.777237i	$-0.716621\pi$
$257$	−2.13611e7			−1.25842			−0.629208	+	0.777237i	$0.716621\pi$
$258$		−	8.73684e6i		−	0.508739i
$259$	−4.12503e7			−2.37426
$260$		−	7.98623e6i		−	0.454383i
$261$	8.38548e6			0.471635
$262$	3.88506e6			0.216020
$263$			1.81176e7i			0.995942i	0.867194	+	0.497971i	$0.165921\pi$
$263$			1.81176e7i			0.995942i	−0.867194	+	0.497971i	$0.834079\pi$
$264$			1.48101e6i			0.0804910i
$265$	2.20958e7			1.18733
$266$	3.83798e7			2.03919
$267$		−	2.96944e6i		−	0.156006i
$268$		−	191924.i		−	0.00997069i
$269$	−2.74813e7			−1.41182			−0.705912	−	0.708300i	$-0.749462\pi$
$269$	−2.74813e7			−1.41182			−0.705912	+	0.708300i	$0.749462\pi$
$270$		−	4.78350e6i		−	0.243027i
$271$	3.15899e7			1.58723			0.793616	−	0.608418i	$-0.208196\pi$
$271$	3.15899e7			1.58723			0.793616	+	0.608418i	$0.208196\pi$
$272$			2.21400e6i			0.110020i
$273$			1.13799e7i			0.559306i
$274$			2.37779e6i			0.115590i
$275$			1.79543e7i			0.863317i
$276$	6.01737e6	+	792007.i	0.286207	+	0.0376705i
$277$	1.54669e7			0.727721			0.363861	−	0.931453i	$-0.381458\pi$
$277$	1.54669e7			0.727721			0.363861	+	0.931453i	$0.381458\pi$
$278$	1.68957e7			0.786398
$279$	6.50897e6			0.299709
$280$	−2.63871e7			−1.20203
$281$		−	397242.i		−	0.0179034i	−0.999960	−	0.00895172i	$-0.997151\pi$
$281$		−	397242.i		−	0.0179034i	0.999960	−	0.00895172i	$-0.00284946\pi$
$282$	−8.74226e6			−0.389831
$283$			4.25425e7i			1.87700i	0.345286	+	0.938498i	$0.387782\pi$
$283$			4.25425e7i			1.87700i	−0.345286	+	0.938498i	$0.612218\pi$
$284$	−2.00462e6			−0.0875140
$285$	−3.61567e7			−1.56190
$286$			3.31922e6i			0.141885i
$287$		−	2.20037e7i		−	0.930784i
$288$	1.40761e6			0.0589256
$289$	1.94629e7			0.806331
$290$			4.35771e7i			1.78675i
$291$		−	1.80863e7i		−	0.733958i
$292$	1.13031e7			0.453994
$293$			5.69284e6i			0.226322i	0.993577	+	0.113161i	$0.0360976\pi$
$293$			5.69284e6i			0.226322i	−0.993577	+	0.113161i	$0.963902\pi$
$294$	2.72254e7			1.07135
$295$		−	5.70298e7i		−	2.22145i
$296$			1.14353e7i			0.440933i
$297$			1.98811e6i			0.0758876i
$298$			2.81458e7i			1.06357i
$299$	1.34860e7	+	1.77503e6i	0.504510	+	0.0664036i
$300$	1.70644e7			0.632014
$301$	−6.46965e7			−2.37236
$302$	1.58270e7			0.574615
$303$	2.92612e7			1.05188
$304$		−	1.06396e7i		−	0.378706i
$305$	7.25361e7			2.55655
$306$			2.97207e6i			0.103728i
$307$	−1.01118e6			−0.0349473			−0.0174737	−	0.999847i	$-0.505562\pi$
$307$	−1.01118e6			−0.0349473			−0.0174737	+	0.999847i	$0.505562\pi$
$308$	1.09669e7			0.375347
$309$		−	1.01526e7i		−	0.344114i
$310$			3.38254e7i			1.13542i
$311$	−3.54306e7			−1.17787			−0.588935	−	0.808181i	$-0.700452\pi$
$311$	−3.54306e7			−1.17787			−0.588935	+	0.808181i	$0.700452\pi$
$312$	3.15470e6			0.103871
$313$		−	7.69318e6i		−	0.250884i	−0.992101	−	0.125442i	$-0.959965\pi$
$313$		−	7.69318e6i		−	0.250884i	0.992101	−	0.125442i	$-0.0400349\pi$
$314$		−	273253.i		−	0.00882624i
$315$	−3.54219e7			−1.13329
$316$		−	6.92563e6i		−	0.219481i
$317$	1.49285e7			0.468638			0.234319	−	0.972160i	$-0.424714\pi$
$317$	1.49285e7			0.468638			0.234319	+	0.972160i	$0.424714\pi$
$318$			8.72821e6i			0.271421i
$319$		−	1.81114e7i		−	0.557931i
$320$			7.31496e6i			0.223235i
$321$			1.71907e7i			0.519730i
$322$	5.86482e6	−	4.45588e7i	0.175666	−	1.33464i
$323$	2.24647e7			0.666642
$324$	1.88957e6			0.0555556
$325$	3.82443e7			1.11408
$326$	3.58326e7			1.03425
$327$		−	3.11023e7i		−	0.889505i
$328$	−6.09980e6			−0.172860
$329$			6.47366e7i			1.81787i
$330$	−1.03317e7			−0.287494
$331$	4.50695e7			1.24279			0.621397	−	0.783496i	$-0.286566\pi$
$331$	4.50695e7			1.24279			0.621397	+	0.783496i	$0.286566\pi$
$332$			5.15386e6i			0.140838i
$333$			1.53507e7i			0.415715i
$334$	−1.67236e7			−0.448839
$335$	1.33888e6			0.0356129
$336$		−	1.04234e7i		−	0.274783i
$337$			3.25819e7i			0.851308i	0.904886	+	0.425654i	$0.139956\pi$
$337$			3.25819e7i			0.851308i	−0.904886	+	0.425654i	$0.860044\pi$
$338$	−2.02343e7			−0.524008
$339$		−	2.85330e7i		−	0.732401i
$340$	−1.54450e7			−0.392963
$341$		−	1.40584e7i		−	0.354547i
$342$		−	1.42825e7i		−	0.357048i
$343$		−	1.24782e8i		−	3.09220i
$344$			1.79350e7i			0.440581i
$345$	−5.52511e6	+	4.19777e7i	−0.134550	+	1.02226i
$346$	−9.44040e6			−0.227909
$347$	5.63856e7			1.34952			0.674760	−	0.738037i	$-0.264247\pi$
$347$	5.63856e7			1.34952			0.674760	+	0.738037i	$0.264247\pi$
$348$	−1.72137e7			−0.408448
$349$	2.68794e6			0.0632328			0.0316164	−	0.999500i	$-0.489935\pi$
$349$	2.68794e6			0.0632328			0.0316164	+	0.999500i	$0.489935\pi$
$350$		−	1.26362e8i		−	2.94722i
$351$	4.23486e6			0.0979305
$352$		−	3.04023e6i		−	0.0697072i
$353$	−7.65579e6			−0.174047			−0.0870234	−	0.996206i	$-0.527735\pi$
$353$	−7.65579e6			−0.174047			−0.0870234	+	0.996206i	$0.527735\pi$
$354$	2.25278e7			0.507818
$355$		−	1.39844e7i		−	0.312579i
$356$			6.09567e6i			0.135105i
$357$	2.20082e7			0.483704
$358$	−2.27119e7			−0.494998
$359$			2.10444e7i			0.454834i	0.973798	+	0.227417i	$0.0730280\pi$
$359$			2.10444e7i			0.454834i	−0.973798	+	0.227417i	$0.926972\pi$
$360$			9.81958e6i			0.210468i
$361$	−6.09101e7			−1.29470
$362$		−	5.14747e7i		−	1.08510i
$363$	−2.33219e7			−0.487577
$364$		−	2.33606e7i		−	0.484373i
$365$			7.88516e7i			1.62156i
$366$			2.86530e7i			0.584423i
$367$			3.12750e7i			0.632701i	0.948642	+	0.316351i	$0.102458\pi$
$367$			3.12750e7i			0.632701i	−0.948642	+	0.316351i	$0.897542\pi$
$368$	−1.23525e7	−	1.62583e6i	−0.247862	−	0.0326236i
$369$	−8.18836e6			−0.162974
$370$	−7.97736e7			−1.57490
$371$	6.46325e7			1.26570
$372$	−1.33616e7			−0.259555
$373$		−	1.02928e7i		−	0.198339i	−0.995071	−	0.0991696i	$-0.968381\pi$
$373$		−	1.02928e7i		−	0.198339i	0.995071	−	0.0991696i	$-0.0316186\pi$
$374$	6.41923e6			0.122707
$375$			6.46694e7i			1.22632i
$376$	1.79461e7			0.337603
$377$	−3.85790e7			−0.719992
$378$		−	1.39923e7i		−	0.259068i
$379$			2.58783e7i			0.475356i	0.971344	+	0.237678i	$0.0763863\pi$
$379$			2.58783e7i			0.475356i	−0.971344	+	0.237678i	$0.923614\pi$
$380$	7.42224e7			1.35265
$381$	8.93512e6			0.161557
$382$			5.50863e7i			0.988220i
$383$		−	6.24623e7i		−	1.11179i	−0.831254	−	0.555893i	$-0.812376\pi$
$383$		−	6.24623e7i		−	1.11179i	0.831254	−	0.555893i	$-0.187624\pi$
$384$	−2.88954e6			−0.0510310
$385$			7.65063e7i			1.34065i
$386$	−3.05183e7			−0.530638
$387$			2.40759e7i			0.415384i
$388$			3.71276e7i			0.635626i
$389$			1.31761e7i			0.223840i	0.993717	+	0.111920i	$0.0357000\pi$
$389$			1.31761e7i			0.223840i	−0.993717	+	0.111920i	$0.964300\pi$
$390$			2.20075e7i			0.371002i
$391$	3.43283e6	−	2.60814e7i	0.0574279	−	0.436316i
$392$	−5.58884e7			−0.927819
$393$	−1.07060e7			−0.176379
$394$	−3.67387e7			−0.600668
$395$	4.83138e7			0.783934
$396$		−	4.08119e6i		−	0.0657206i
$397$	7.26229e7			1.16065			0.580326	−	0.814384i	$-0.302925\pi$
$397$	7.26229e7			1.16065			0.580326	+	0.814384i	$0.302925\pi$
$398$		−	2.89775e7i		−	0.459634i
$399$	−1.05762e8			−1.66499
$400$	−3.50298e7			−0.547340
$401$			3.83883e7i			0.595341i	0.954669	+	0.297671i	$0.0962098\pi$
$401$			3.83883e7i			0.595341i	−0.954669	+	0.297671i	$0.903790\pi$
$402$			528881.i			0.00814103i
$403$	−2.99458e7			−0.457531
$404$	−6.00674e7			−0.910951
$405$			1.31818e7i			0.198431i
$406$			1.27468e8i			1.90468i
$407$	3.31553e7			0.491779
$408$		−	6.10106e6i		−	0.0898307i
$409$	−1.00182e8			−1.46426			−0.732131	−	0.681164i	$-0.761474\pi$
$409$	−1.00182e8			−1.46426			−0.732131	+	0.681164i	$0.761474\pi$
$410$		−	4.25527e7i		−	0.617413i
$411$		−	6.55241e6i		−	0.0943790i
$412$			2.08413e7i			0.298012i
$413$		−	1.66819e8i		−	2.36807i
$414$	−1.65819e7	−	2.18251e6i	−0.233687	−	0.0307578i
$415$	−3.59538e7			−0.503037
$416$	−6.47597e6			−0.0899549
$417$	−4.65591e7			−0.642091
$418$	−3.08482e7			−0.422377
$419$			6.88046e7i			0.935352i	0.883900	+	0.467676i	$0.154909\pi$
$419$			6.88046e7i			0.935352i	−0.883900	+	0.467676i	$0.845091\pi$
$420$	7.27142e7			0.981457
$421$		−	1.05045e7i		−	0.140776i	−0.997520	−	0.0703880i	$-0.977576\pi$
$421$		−	1.05045e7i		−	0.140776i	0.997520	−	0.0703880i	$-0.0224237\pi$
$422$	7.48550e7			0.996056
$423$	2.40908e7			0.318296
$424$		−	1.79173e7i		−	0.235058i
$425$		−	7.39630e7i		−	0.963491i
$426$	5.52409e6			0.0714549
$427$	2.12176e8			2.72529
$428$		−	3.52890e7i		−	0.450099i
$429$		−	9.14669e6i		−	0.115849i
$430$	−1.25116e8			−1.57365
$431$		−	5.92009e7i		−	0.739429i	−0.929145	−	0.369715i	$-0.879455\pi$
$431$		−	5.92009e7i		−	0.739429i	0.929145	−	0.369715i	$-0.120545\pi$
$432$	−3.87891e6			−0.0481125
$433$		−	1.49652e8i		−	1.84340i	−0.387905	−	0.921699i	$-0.626801\pi$
$433$		−	1.49652e8i		−	1.84340i	0.387905	−	0.921699i	$-0.373199\pi$
$434$			9.89430e7i			1.21036i
$435$		−	1.20084e8i		−	1.45888i
$436$			6.38467e7i			0.770334i
$437$	−1.64968e7	+	1.25336e8i	−0.197676	+	1.50187i
$438$	−3.11478e7			−0.370685
$439$	6.52686e7			0.771455			0.385728	−	0.922613i	$-0.373950\pi$
$439$	6.52686e7			0.771455			0.385728	+	0.922613i	$0.373950\pi$
$440$	2.12089e7			0.248977
$441$	−7.50245e7			−0.874756
$442$		−	1.36736e7i		−	0.158349i
$443$	−1.15808e8			−1.33207			−0.666033	−	0.745922i	$-0.732009\pi$
$443$	−1.15808e8			−1.33207			−0.666033	+	0.745922i	$0.732009\pi$
$444$		−	3.15120e7i		−	0.360020i
$445$	−4.25239e7			−0.482562
$446$	−1.45931e7			−0.164491
$447$		−	7.75608e7i		−	0.868400i
$448$			2.13971e7i			0.237969i
$449$	−7.63250e7			−0.843195			−0.421597	−	0.906783i	$-0.638530\pi$
$449$	−7.63250e7			−0.843195			−0.421597	+	0.906783i	$0.638530\pi$
$450$	−4.70239e7			−0.516037
$451$			1.76857e7i			0.192793i
$452$			5.85727e7i			0.634278i
$453$	−4.36140e7			−0.469171
$454$			1.62876e7i			0.174056i
$455$	1.62966e8			1.73006
$456$			2.93192e7i			0.309212i
$457$			4.60920e7i			0.482922i	0.970410	+	0.241461i	$0.0776267\pi$
$457$			4.60920e7i			0.482922i	−0.970410	+	0.241461i	$0.922373\pi$
$458$			4.55991e7i			0.474636i
$459$		−	8.19005e6i		−	0.0846932i
$460$	1.13419e7	−	8.61719e7i	0.116524	−	0.885304i
$461$	1.68932e8			1.72429			0.862143	−	0.506665i	$-0.169122\pi$
$461$	1.68932e8			1.72429			0.862143	+	0.506665i	$0.169122\pi$
$462$	−3.02213e7			−0.306470
$463$	−4.01206e7			−0.404226			−0.202113	−	0.979362i	$-0.564781\pi$
$463$	−4.01206e7			−0.404226			−0.202113	+	0.979362i	$0.564781\pi$
$464$	3.53363e7			0.353727
$465$		−	9.32118e7i		−	0.927069i
$466$	−6.82954e7			−0.674891
$467$		−	1.71891e7i		−	0.168773i	−0.996433	−	0.0843865i	$-0.973107\pi$
$467$		−	1.71891e7i		−	0.168773i	0.996433	−	0.0843865i	$-0.0268930\pi$
$468$	−8.69333e6			−0.0848103
$469$	3.91637e6			0.0379634
$470$			1.25194e8i			1.20584i
$471$			752996.i			0.00720659i
$472$	−4.62450e7			−0.439784
$473$	5.20004e7			0.491387
$474$			1.90848e7i			0.179206i
$475$			3.55435e8i			3.31650i
$476$	−4.51785e7			−0.418900
$477$		−	2.40521e7i		−	0.221614i
$478$	1.51411e8			1.38635
$479$			7.17632e7i			0.652973i	0.945202	+	0.326486i	$0.105865\pi$
$479$			7.17632e7i			0.652973i	−0.945202	+	0.326486i	$0.894135\pi$
$480$		−	2.01577e7i		−	0.182270i
$481$		−	7.06240e7i		−	0.634625i
$482$			2.74950e7i			0.245535i
$483$	−1.61616e7	+	1.22790e8i	−0.143431	+	1.08973i
$484$	4.78752e7			0.422254
$485$	−2.59005e8			−2.27030
$486$	−5.20704e6			−0.0453609
$487$	3.39708e7			0.294116			0.147058	−	0.989128i	$-0.453020\pi$
$487$	3.39708e7			0.294116			0.147058	+	0.989128i	$0.453020\pi$
$488$		−	5.88189e7i		−	0.506125i
$489$	−9.87429e7			−0.844460
$490$		−	3.89882e8i		−	3.31394i
$491$	1.66199e7			0.140405			0.0702027	−	0.997533i	$-0.477635\pi$
$491$	1.66199e7			0.140405			0.0702027	+	0.997533i	$0.477635\pi$
$492$	1.68091e7			0.141139
$493$			7.46103e7i			0.622670i
$494$			6.57095e7i			0.545064i
$495$	2.84708e7			0.234738
$496$	2.74287e7			0.224782
$497$		−	4.09060e7i		−	0.333210i
$498$		−	1.42024e7i		−	0.114993i
$499$	3.32441e6			0.0267555			0.0133777	−	0.999911i	$-0.495742\pi$
$499$	3.32441e6			0.0267555			0.0133777	+	0.999911i	$0.495742\pi$
$500$		−	1.32753e8i		−	1.06203i
$501$	4.60849e7			0.366476
$502$		−	1.12009e8i		−	0.885408i
$503$		−	1.18744e8i		−	0.933057i	−0.884506	−	0.466529i	$-0.845504\pi$
$503$		−	1.18744e8i		−	0.933057i	0.884506	−	0.466529i	$-0.154496\pi$
$504$			2.87234e7i			0.224359i
$505$		−	4.19036e8i		−	3.25369i
$506$	−4.71391e6	+	3.58146e7i	−0.0363856	+	0.276445i
$507$	5.57592e7			0.427851
$508$	−1.83420e7			−0.139912
$509$	9.45736e7			0.717161			0.358581	−	0.933499i	$-0.383261\pi$
$509$	9.45736e7			0.717161			0.358581	+	0.933499i	$0.383261\pi$
$510$	4.25615e7			0.320853
$511$			2.30650e8i			1.72858i
$512$	5.93164e6			0.0441942
$513$			3.93580e7i			0.291528i
$514$	−1.20837e8			−0.889834
$515$	−1.45391e8			−1.06442
$516$		−	4.94230e7i		−	0.359733i
$517$		−	5.20327e7i		−	0.376534i
$518$	−2.33347e8			−1.67885
$519$	2.60147e7			0.186087
$520$		−	4.51769e7i		−	0.321297i
$521$			3.07896e7i			0.217716i	0.994057	+	0.108858i	$0.0347194\pi$
$521$			3.07896e7i			0.217716i	−0.994057	+	0.108858i	$0.965281\pi$
$522$	4.74354e7			0.333497
$523$			2.12609e8i			1.48620i	0.669183	+	0.743098i	$0.266644\pi$
$523$			2.12609e8i			1.48620i	−0.669183	+	0.743098i	$0.733356\pi$
$524$	2.19772e7			0.152749
$525$			3.48213e8i			2.40639i
$526$			1.02489e8i			0.704237i
$527$			5.79139e7i			0.395686i
$528$			8.37788e6i			0.0569157i
$529$	1.42994e8	+	3.83054e7i	0.965942	+	0.258757i
$530$	1.24992e8			0.839569
$531$	−6.20792e7			−0.414632
$532$	2.17109e8			1.44193
$533$	3.76721e7			0.248793
$534$		−	1.67977e7i		−	0.110313i
$535$	2.46179e8			1.60765
$536$		−	1.08569e6i		−	0.00705034i
$537$	6.25866e7			0.404165
$538$	−1.55458e8			−0.998310
$539$			1.62042e8i			1.03481i
$540$		−	2.70596e7i		−	0.171846i
$541$	−6.80769e7			−0.429940			−0.214970	−	0.976621i	$-0.568965\pi$
$541$	−6.80769e7			−0.429940			−0.214970	+	0.976621i	$0.568965\pi$
$542$	1.78700e8			1.12234
$543$			1.41848e8i			0.885977i
$544$			1.25243e7i			0.0777957i
$545$	−4.45400e8			−2.75145
$546$			6.43743e7i			0.395489i
$547$	1.91663e8			1.17105			0.585526	−	0.810653i	$-0.300888\pi$
$547$	1.91663e8			1.17105			0.585526	+	0.810653i	$0.300888\pi$
$548$			1.34508e7i			0.0817347i
$549$		−	7.89584e7i		−	0.477179i
$550$			1.01565e8i			0.610457i
$551$		−	3.58546e8i		−	2.14334i
$552$	3.40394e7	+	4.48027e6i	0.202379	+	0.0266371i
$553$	1.41323e8			0.835676
$554$	8.74942e7			0.514577
$555$	2.19830e8			1.28590
$556$	9.55766e7			0.556067
$557$			1.70667e8i			0.987607i	0.869573	+	0.493804i	$0.164394\pi$
$557$			1.70667e8i			0.987607i	−0.869573	+	0.493804i	$0.835606\pi$
$558$	3.68203e7			0.211926
$559$		−	1.10766e8i		−	0.634119i
$560$	−1.49268e8			−0.849967
$561$	−1.76893e7			−0.100190
$562$		−	2.24714e6i		−	0.0126596i
$563$		−	1.86990e8i		−	1.04783i	−0.851769	−	0.523917i	$-0.824470\pi$
$563$		−	1.86990e8i		−	1.04783i	0.851769	−	0.523917i	$-0.175530\pi$
$564$	−4.94537e7			−0.275652
$565$	−4.08608e8			−2.26549
$566$			2.40656e8i			1.32724i
$567$			3.85582e7i			0.211528i
$568$	−1.13399e7			−0.0618818
$569$		−	3.07226e8i		−	1.66771i	−0.551981	−	0.833856i	$-0.686128\pi$
$569$		−	3.07226e8i		−	1.66771i	0.551981	−	0.833856i	$-0.313872\pi$
$570$	−2.04533e8			−1.10443
$571$		−	1.54595e8i		−	0.830401i	−0.909730	−	0.415201i	$-0.863711\pi$
$571$		−	1.54595e8i		−	0.830401i	0.909730	−	0.415201i	$-0.136289\pi$
$572$			1.87763e7i			0.100328i
$573$		−	1.51800e8i		−	0.806879i
$574$		−	1.24471e8i		−	0.658164i
$575$	4.12659e8	+	5.43142e7i	2.17064	+	0.285700i
$576$	7.96262e6			0.0416667
$577$	−4.29223e7			−0.223437			−0.111719	−	0.993740i	$-0.535636\pi$
$577$	−4.29223e7			−0.223437			−0.111719	+	0.993740i	$0.535636\pi$
$578$	1.10099e8			0.570162
$579$	8.40985e7			0.433264
$580$			2.46509e8i			1.26343i
$581$	−1.05169e8			−0.536239
$582$		−	1.02312e8i		−	0.518987i
$583$	−5.19491e7			−0.262163
$584$	6.39402e7			0.321022
$585$		−	6.06454e7i		−	0.302922i
$586$			3.22036e7i			0.160034i
$587$	3.41503e8			1.68842			0.844210	−	0.536012i	$-0.180070\pi$
$587$	3.41503e8			1.68842			0.844210	+	0.536012i	$0.180070\pi$
$588$	1.54010e8			0.757561
$589$		−	2.78310e8i		−	1.36202i
$590$		−	3.22609e8i		−	1.57080i
$591$	1.01240e8			0.490443
$592$			6.46878e7i			0.311786i
$593$	−3.07133e8			−1.47286			−0.736431	−	0.676513i	$-0.763490\pi$
$593$	−3.07133e8			−1.47286			−0.736431	+	0.676513i	$0.763490\pi$
$594$			1.12464e7i			0.0536606i
$595$		−	3.15169e8i		−	1.49621i
$596$			1.59217e8i			0.752056i
$597$			7.98527e7i			0.375290i
$598$	7.62884e7	+	1.00411e7i	0.356743	+	0.0469545i
$599$	−1.22069e8			−0.567969			−0.283984	−	0.958829i	$-0.591656\pi$
$599$	−1.22069e8			−0.567969			−0.283984	+	0.958829i	$0.591656\pi$
$600$	9.65307e7			0.446901
$601$	2.17918e7			0.100385			0.0501927	−	0.998740i	$-0.484016\pi$
$601$	2.17918e7			0.100385			0.0501927	+	0.998740i	$0.484016\pi$
$602$	−3.65978e8			−1.67751
$603$		−	1.45742e6i		−	0.00664713i
$604$	8.95309e7			0.406314
$605$			3.33981e8i			1.50819i
$606$	1.65526e8			0.743788
$607$	−4.03226e8			−1.80295			−0.901473	−	0.432836i	$-0.857513\pi$
$607$	−4.03226e8			−1.80295			−0.901473	+	0.432836i	$0.857513\pi$
$608$		−	6.01864e7i		−	0.267786i
$609$		−	3.51260e8i		−	1.55517i
$610$	4.10326e8			1.80775
$611$	−1.10835e8			−0.485905
$612$			1.68125e7i			0.0733465i
$613$			2.99288e8i			1.29930i	0.760235	+	0.649649i	$0.225084\pi$
$613$			2.99288e8i			1.29930i	−0.760235	+	0.649649i	$0.774916\pi$
$614$	−5.72011e6			−0.0247115
$615$			1.17261e8i			0.504116i
$616$	6.20383e7			0.265410
$617$			1.33500e8i			0.568365i	0.958770	+	0.284182i	$0.0917221\pi$
$617$			1.33500e8i			0.568365i	−0.958770	+	0.284182i	$0.908278\pi$
$618$		−	5.74319e7i		−	0.243325i
$619$			3.82388e7i			0.161225i	0.996746	+	0.0806124i	$0.0256876\pi$
$619$			3.82388e7i			0.161225i	−0.996746	+	0.0806124i	$0.974312\pi$
$620$			1.91345e8i			0.802865i
$621$	4.56944e7	+	6.01430e6i	0.190804	+	0.0251137i
$622$	−2.00426e8			−0.832879
$623$	−1.24387e8			−0.514413
$624$	1.78457e7			0.0734479
$625$	3.91587e8			1.60394
$626$		−	4.35192e7i		−	0.177402i
$627$	8.50075e7			0.344869
$628$		−	1.54575e6i		−	0.00624109i
$629$	−1.36584e8			−0.548842
$630$	−2.00377e8			−0.801356
$631$		−	3.12539e8i		−	1.24399i	−0.783022	−	0.621994i	$-0.786323\pi$
$631$		−	3.12539e8i		−	1.24399i	0.783022	−	0.621994i	$-0.213677\pi$
$632$		−	3.91773e7i		−	0.155197i
$633$	−2.06276e8			−0.813276
$634$	8.44482e7			0.331377
$635$		−	1.27956e8i		−	0.499733i
$636$			4.93742e7i			0.191924i
$637$	3.45165e8			1.33539
$638$		−	1.02454e8i		−	0.394517i
$639$	−1.52226e7			−0.0583427
$640$			4.13796e7i			0.157851i
$641$			1.11030e8i			0.421568i	0.977533	+	0.210784i	$0.0676017\pi$
$641$			1.11030e8i			0.421568i	−0.977533	+	0.210784i	$0.932398\pi$
$642$			9.72451e7i			0.367505i
$643$			8.67489e7i			0.326310i	0.986600	+	0.163155i	$0.0521671\pi$
$643$			8.67489e7i			0.326310i	−0.986600	+	0.163155i	$0.947833\pi$
$644$	3.31765e7	−	2.52062e8i	0.124215	−	0.943736i
$645$	3.44779e8			1.28488
$646$	1.27080e8			0.471387
$647$	8.57067e7			0.316448			0.158224	−	0.987403i	$-0.449423\pi$
$647$	8.57067e7			0.316448			0.158224	+	0.987403i	$0.449423\pi$
$648$	1.06890e7			0.0392837
$649$			1.34082e8i			0.490497i
$650$	2.16343e8			0.787775
$651$		−	2.72655e8i		−	0.988258i
$652$	2.02700e8			0.731324
$653$	4.38775e8			1.57580			0.787902	−	0.615801i	$-0.211168\pi$
$653$	4.38775e8			1.57580			0.787902	+	0.615801i	$0.211168\pi$
$654$		−	1.75941e8i		−	0.628975i
$655$			1.53315e8i			0.545582i
$656$	−3.45057e7			−0.122230
$657$	8.58332e7			0.302663
$658$			3.66205e8i			1.28543i
$659$		−	1.43384e8i		−	0.501009i	−0.968115	−	0.250505i	$-0.919403\pi$
$659$		−	1.43384e8i		−	0.501009i	0.968115	−	0.250505i	$-0.0805965\pi$
$660$	−5.84448e7			−0.203289
$661$			5.55836e8i			1.92461i	0.271978	+	0.962303i	$0.412322\pi$
$661$			5.55836e8i			1.92461i	−0.271978	+	0.962303i	$0.587678\pi$
$662$	2.54952e8			0.878788
$663$			3.76799e7i			0.129291i
$664$			2.91546e7i			0.0995872i
$665$			1.51457e9i			5.15021i
$666$			8.68368e7i			0.293955i
$667$	−4.16270e8	−	5.47895e7i	−1.40281	−	0.184637i
$668$	−9.46031e7			−0.317377
$669$	4.02138e7			0.134307
$670$	7.57385e6			0.0251821
$671$	−1.70539e8			−0.564489
$672$		−	5.89634e7i		−	0.194301i
$673$	−5.68482e8			−1.86497			−0.932484	−	0.361211i	$-0.882363\pi$
$673$	−5.68482e8			−1.86497			−0.932484	+	0.361211i	$0.882363\pi$
$674$			1.84311e8i			0.601966i
$675$	1.29583e8			0.421343
$676$	−1.14463e8			−0.370530
$677$		−	3.52106e8i		−	1.13477i	−0.823454	−	0.567384i	$-0.807956\pi$
$677$		−	3.52106e8i		−	1.13477i	0.823454	−	0.567384i	$-0.192044\pi$
$678$		−	1.61407e8i		−	0.517886i
$679$	−7.57620e8			−2.42015
$680$	−8.73703e7			−0.277867
$681$		−	4.48833e7i		−	0.142116i
$682$		−	7.95264e7i		−	0.250702i
$683$	−2.46734e8			−0.774404			−0.387202	−	0.921995i	$-0.626558\pi$
$683$	−2.46734e8			−0.774404			−0.387202	+	0.921995i	$0.626558\pi$
$684$		−	8.07941e7i		−	0.252471i
$685$	−9.38339e7			−0.291936
$686$		−	7.05871e8i		−	2.18652i
$687$		−	1.25656e8i		−	0.387538i
$688$			1.01456e8i			0.311538i
$689$			1.10656e8i			0.338313i
$690$	−3.12547e7	+	2.37462e8i	−0.0951411	+	0.722847i
$691$	3.12068e8			0.945834			0.472917	−	0.881107i	$-0.343201\pi$
$691$	3.12068e8			0.945834			0.472917	+	0.881107i	$0.343201\pi$
$692$	−5.34030e7			−0.161156
$693$	8.32801e7			0.250231
$694$	3.18965e8			0.954255
$695$			6.66751e8i			1.98614i
$696$	−9.73755e7			−0.288817
$697$		−	7.28564e7i		−	0.215164i
$698$	1.52053e7			0.0447124
$699$	1.88200e8			0.551047
$700$		−	7.14812e8i		−	2.08400i
$701$		−	6.64072e8i		−	1.92780i	−0.266269	−	0.963899i	$-0.585791\pi$
$701$		−	6.64072e8i		−	1.92780i	0.266269	−	0.963899i	$-0.414209\pi$
$702$	2.39560e7			0.0692473
$703$	6.56366e8			1.88921
$704$		−	1.71981e7i		−	0.0492905i
$705$		−	3.44993e8i		−	0.984562i
$706$	−4.33077e7			−0.123070
$707$		−	1.22573e9i		−	3.46845i
$708$	1.27436e8			0.359082
$709$			5.94483e8i			1.66802i	0.551752	+	0.834009i	$0.313960\pi$
$709$			5.94483e8i			1.66802i	−0.551752	+	0.834009i	$0.686040\pi$
$710$		−	7.91078e7i		−	0.221027i
$711$		−	5.25915e7i		−	0.146321i
$712$			3.44823e7i			0.0955337i
$713$	−3.23117e8	−	4.25286e7i	−0.891438	−	0.117331i
$714$	1.24497e8			0.342031
$715$	−1.30985e8			−0.358348
$716$	−1.28478e8			−0.350017
$717$	−4.17240e8			−1.13195
$718$			1.19045e8i			0.321616i
$719$	−1.34441e8			−0.361698			−0.180849	−	0.983511i	$-0.557884\pi$
$719$	−1.34441e8			−0.361698			−0.180849	+	0.983511i	$0.557884\pi$
$720$			5.55480e7i			0.148823i
$721$	−4.25284e8			−1.13468
$722$	−3.44560e8			−0.915489
$723$		−	7.57674e7i		−	0.200478i
$724$		−	2.91185e8i		−	0.767279i
$725$	−1.18048e9			−3.09774
$726$	−1.31928e8			−0.344769
$727$		−	7.41139e8i		−	1.92884i	−0.264372	−	0.964421i	$-0.585165\pi$
$727$		−	7.41139e8i		−	1.92884i	0.264372	−	0.964421i	$-0.414835\pi$
$728$		−	1.32148e8i		−	0.342503i
$729$	1.43489e7			0.0370370
$730$			4.46052e8i			1.14661i
$731$	−2.14217e8			−0.548405
$732$			1.62086e8i			0.413249i
$733$		−	3.05172e8i		−	0.774878i	−0.921895	−	0.387439i	$-0.873360\pi$
$733$		−	3.05172e8i		−	0.774878i	0.921895	−	0.387439i	$-0.126640\pi$
$734$			1.76918e8i			0.447387i
$735$			1.07439e9i			2.70582i
$736$	−6.98761e7	−	9.19709e6i	−0.175265	−	0.0230684i
$737$	−3.14782e6			−0.00786336
$738$	−4.63203e7			−0.115240
$739$	6.64941e8			1.64759			0.823797	−	0.566885i	$-0.191852\pi$
$739$	6.64941e8			1.64759			0.823797	+	0.566885i	$0.191852\pi$
$740$	−4.51268e8			−1.11363
$741$		−	1.81074e8i		−	0.445043i
$742$	3.65617e8			0.894982
$743$		−	2.54814e8i		−	0.621237i	−0.950535	−	0.310619i	$-0.899464\pi$
$743$		−	2.54814e8i		−	0.621237i	0.950535	−	0.310619i	$-0.100536\pi$
$744$	−7.55847e7			−0.183533
$745$	−1.11071e9			−2.68616
$746$		−	5.82250e7i		−	0.140247i
$747$			3.91371e7i			0.0938917i
$748$	3.63127e7			0.0867667
$749$	7.20102e8			1.71375
$750$			3.65825e8i			0.867141i
$751$			3.83346e7i			0.0905047i	0.998976	+	0.0452524i	$0.0144092\pi$
$751$			3.83346e7i			0.0905047i	−0.998976	+	0.0452524i	$0.985591\pi$
$752$	1.01519e8			0.238722
$753$			3.08662e8i			0.722932i
$754$	−2.18236e8			−0.509111
$755$			6.24575e8i			1.45126i
$756$		−	7.91523e7i		−	0.183189i
$757$		−	7.35616e7i		−	0.169576i	−0.996399	−	0.0847878i	$-0.972979\pi$
$757$		−	7.35616e7i		−	0.169576i	0.996399	−	0.0847878i	$-0.0270212\pi$
$758$			1.46390e8i			0.336127i
$759$	1.29900e7	−	9.86934e7i	0.0297087	−	0.225716i
$760$	4.19866e8			0.956466
$761$	6.35953e8			1.44302			0.721508	−	0.692406i	$-0.243449\pi$
$761$	6.35953e8			1.44302			0.721508	+	0.692406i	$0.243449\pi$
$762$	5.05447e7			0.114238
$763$	−1.30285e9			−2.93305
$764$			3.11615e8i			0.698777i
$765$	−1.17286e8			−0.261976
$766$		−	3.53340e8i		−	0.786152i
$767$	2.85608e8			0.632971
$768$	−1.63457e7			−0.0360844
$769$		−	1.17356e8i		−	0.258063i	−0.991640	−	0.129032i	$-0.958813\pi$
$769$		−	1.17356e8i		−	0.258063i	0.991640	−	0.129032i	$-0.0411869\pi$
$770$			4.32785e8i			0.947981i
$771$	3.32987e8			0.726547
$772$	−1.72637e8			−0.375218
$773$			2.53070e8i			0.547901i	0.961744	+	0.273951i	$0.0883305\pi$
$773$			2.53070e8i			0.547901i	−0.961744	+	0.273951i	$0.911669\pi$
$774$			1.36194e8i			0.293721i
$775$	−9.16311e8			−1.96851
$776$			2.10026e8i			0.449456i
$777$	6.43028e8			1.37078
$778$			7.45352e7i			0.158279i
$779$			3.50118e8i			0.740630i
$780$			1.24493e8i			0.262338i
$781$			3.28786e7i			0.0690177i
$782$	1.94190e7	−	1.47539e8i	0.0406076	−	0.308522i
$783$	−1.30717e8			−0.272299
$784$	−3.16152e8			−0.656067
$785$	1.07833e7			0.0222917
$786$	−6.05620e7			−0.124719
$787$		−	7.03440e7i		−	0.144312i	−0.997393	−	0.0721561i	$-0.977012\pi$
$787$		−	7.03440e7i		−	0.144312i	0.997393	−	0.0721561i	$-0.0229880\pi$
$788$	−2.07825e8			−0.424736
$789$		−	2.82426e8i		−	0.575007i
$790$	2.73304e8			0.554325
$791$	−1.19522e9			−2.41502
$792$		−	2.30867e7i		−	0.0464715i
$793$			3.63264e8i			0.728454i
$794$	4.10817e8			0.820705
$795$	−3.44439e8			−0.685505
$796$		−	1.63922e8i		−	0.325010i
$797$		−	4.07957e8i		−	0.805823i	−0.915239	−	0.402911i	$-0.867998\pi$
$797$		−	4.07957e8i		−	0.805823i	0.915239	−	0.402911i	$-0.132002\pi$
$798$	−5.98282e8			−1.17733
$799$			2.14350e8i			0.420225i
$800$	−1.98158e8			−0.387028
$801$			4.62890e7i			0.0900701i
$802$			2.17157e8i			0.420970i
$803$		−	1.85387e8i		−	0.358041i
$804$			2.99180e6i			0.00575658i
$805$	1.75841e9	+	2.31442e8i	3.37080	+	0.443664i
$806$	−1.69399e8			−0.323523
$807$	4.28391e8			0.815117
$808$	−3.39793e8			−0.644140
$809$	2.44789e8			0.462324			0.231162	−	0.972915i	$-0.425747\pi$
$809$	2.44789e8			0.462324			0.231162	+	0.972915i	$0.425747\pi$
$810$			7.45675e7i			0.140312i
$811$	−1.89970e7			−0.0356142			−0.0178071	−	0.999841i	$-0.505668\pi$
$811$	−1.89970e7			−0.0356142			−0.0178071	+	0.999841i	$0.505668\pi$
$812$			7.21068e8i			1.34681i
$813$	−4.92438e8			−0.916389
$814$	1.87555e8			0.347740
$815$			1.41405e9i			2.61211i
$816$		−	3.45128e7i		−	0.0635199i
$817$	1.02944e9			1.88770
$818$	−5.66714e8			−1.03539
$819$		−	1.77395e8i		−	0.322915i
$820$		−	2.40714e8i		−	0.436577i
$821$	5.31510e8			0.960466			0.480233	−	0.877141i	$-0.340552\pi$
$821$	5.31510e8			0.960466			0.480233	+	0.877141i	$0.340552\pi$
$822$		−	3.70660e7i		−	0.0667361i
$823$	1.33570e8			0.239613			0.119807	−	0.992797i	$-0.461773\pi$
$823$	1.33570e8			0.239613			0.119807	+	0.992797i	$0.461773\pi$
$824$			1.17896e8i			0.210726i
$825$		−	2.79880e8i		−	0.498436i
$826$		−	9.43668e8i		−	1.67448i
$827$		−	5.36311e8i		−	0.948201i	−0.880471	−	0.474100i	$-0.842773\pi$
$827$		−	5.36311e8i		−	0.948201i	0.880471	−	0.474100i	$-0.157227\pi$
$828$	−9.38016e7	−	1.23462e7i	−0.165242	−	0.0217491i
$829$	−2.04963e8			−0.359759			−0.179879	−	0.983689i	$-0.557571\pi$
$829$	−2.04963e8			−0.359759			−0.179879	+	0.983689i	$0.557571\pi$
$830$	−2.03385e8			−0.355701
$831$	−2.41106e8			−0.420150
$832$	−3.66336e7			−0.0636077
$833$		−	6.67534e8i		−	1.15488i
$834$	−2.63378e8			−0.454027
$835$		−	6.59959e8i		−	1.13359i
$836$	−1.74504e8			−0.298666
$837$	−1.01465e8			−0.173037
$838$			3.89217e8i			0.661394i
$839$			1.33088e8i			0.225347i	0.993632	+	0.112673i	$0.0359414\pi$
$839$			1.33088e8i			0.225347i	−0.993632	+	0.112673i	$0.964059\pi$
$840$	4.11334e8			0.693995
$841$	5.95989e8			1.00196
$842$		−	5.94224e7i		−	0.0995437i
$843$			6.19240e6i			0.0103366i
$844$	4.23444e8			0.704318
$845$		−	7.98500e8i		−	1.32344i
$846$	1.36278e8			0.225069
$847$			9.76933e8i			1.60773i
$848$		−	1.01355e8i		−	0.166211i
$849$		−	6.63171e8i		−	1.08368i
$850$		−	4.18398e8i		−	0.681291i
$851$	1.00299e8	−	7.62037e8i	0.162746	−	1.23648i
$852$	3.12490e7			0.0505262
$853$	5.03907e8			0.811902			0.405951	−	0.913895i	$-0.366940\pi$
$853$	5.03907e8			0.811902			0.405951	+	0.913895i	$0.366940\pi$
$854$	1.20025e9			1.92707
$855$	5.63627e8			0.901765
$856$		−	1.99625e8i		−	0.318268i
$857$	−2.16449e8			−0.343885			−0.171942	−	0.985107i	$-0.555004\pi$
$857$	−2.16449e8			−0.343885			−0.171942	+	0.985107i	$0.555004\pi$
$858$		−	5.17415e7i		−	0.0819175i
$859$	−1.87634e8			−0.296028			−0.148014	−	0.988985i	$-0.547288\pi$
$859$	−1.87634e8			−0.296028			−0.148014	+	0.988985i	$0.547288\pi$
$860$	−7.07763e8			−1.11274
$861$			3.43003e8i			0.537389i
$862$		−	3.34891e8i		−	0.522855i
$863$	4.98471e8			0.775546			0.387773	−	0.921755i	$-0.373244\pi$
$863$	4.98471e8			0.775546			0.387773	+	0.921755i	$0.373244\pi$
$864$	−2.19424e7			−0.0340207
$865$		−	3.72544e8i		−	0.575611i
$866$		−	8.46560e8i		−	1.30348i
$867$	−3.03396e8			−0.465535
$868$			5.59706e8i			0.855856i
$869$	−1.13590e8			−0.173093
$870$		−	6.79300e8i		−	1.03158i
$871$			6.70517e6i			0.0101474i
$872$			3.61172e8i			0.544709i
$873$			2.81938e8i			0.423751i
$874$	−9.33199e7	+	7.09010e8i	−0.139778	+	1.06198i
$875$	2.70894e9			4.04367
$876$	−1.76198e8			−0.262114
$877$	9.56146e8			1.41751			0.708754	−	0.705456i	$-0.249258\pi$
$877$	9.56146e8			1.41751			0.708754	+	0.705456i	$0.249258\pi$
$878$	3.69215e8			0.545501
$879$		−	8.87427e7i		−	0.130667i
$880$	1.19976e8			0.176053
$881$		−	1.13220e8i		−	0.165575i	−0.996567	−	0.0827876i	$-0.973618\pi$
$881$		−	1.13220e8i		−	0.165575i	0.996567	−	0.0827876i	$-0.0263823\pi$
$882$	−4.24402e8			−0.618546
$883$	1.91751e7			0.0278519			0.0139260	−	0.999903i	$-0.495567\pi$
$883$	1.91751e7			0.0278519			0.0139260	+	0.999903i	$0.495567\pi$
$884$		−	7.73494e7i		−	0.111970i
$885$			8.89007e8i			1.28255i
$886$	−6.55107e8			−0.941913
$887$	2.63415e8			0.377459			0.188729	−	0.982029i	$-0.439563\pi$
$887$	2.63415e8			0.377459			0.188729	+	0.982029i	$0.439563\pi$
$888$		−	1.78259e8i		−	0.254573i
$889$		−	3.74284e8i		−	0.532717i
$890$	−2.40552e8			−0.341223
$891$		−	3.09916e7i		−	0.0438137i
$892$	−8.25510e7			−0.116313
$893$		−	1.03007e9i		−	1.44649i
$894$		−	4.38750e8i		−	0.614051i
$895$		−	8.96272e8i		−	1.25017i
$896$			1.21040e8i			0.168269i
$897$	−2.10226e8	−	2.76700e7i	−0.291279	−	0.0383382i
$898$	−4.31760e8			−0.596229
$899$	9.24331e8			1.27218
$900$	−2.66007e8			−0.364893
$901$	2.14005e8			0.292583
$902$			1.00045e8i			0.136325i
$903$	1.00852e9			1.36968
$904$			3.31337e8i			0.448502i
$905$	2.03133e9			2.74053
$906$	−2.46718e8			−0.331754
$907$			6.44372e8i			0.863605i	0.901968	+	0.431802i	$0.142122\pi$
$907$			6.44372e8i			0.863605i	−0.901968	+	0.431802i	$0.857878\pi$
$908$			9.21365e7i			0.123076i
$909$	−4.56137e8			−0.607301
$910$	9.21873e8			1.22334
$911$			1.19502e9i			1.58060i	0.612721	+	0.790299i	$0.290075\pi$
$911$			1.19502e9i			1.58060i	−0.612721	+	0.790299i	$0.709925\pi$
$912$			1.65854e8i			0.218646i
$913$	8.45305e7			0.111071
$914$			2.60736e8i			0.341478i
$915$	−1.13073e9			−1.47603
$916$			2.57948e8i			0.335618i
$917$			4.48463e8i			0.581592i
$918$		−	4.63299e7i		−	0.0598871i
$919$		−	1.26236e8i		−	0.162643i	−0.996688	−	0.0813217i	$-0.974086\pi$
$919$		−	1.26236e8i		−	0.162643i	0.996688	−	0.0813217i	$-0.0259141\pi$
$920$	6.41597e7	−	4.87462e8i	0.0823946	−	0.626004i
$921$	1.57628e7			0.0201768
$922$	9.55624e8			1.21925
$923$	7.00346e7			0.0890651
$924$	−1.70958e8			−0.216707
$925$		−	2.16102e9i		−	2.73045i
$926$	−2.26957e8			−0.285831
$927$			1.58264e8i			0.198674i
$928$	1.99893e8			0.250122
$929$	−3.22745e7			−0.0402543			−0.0201272	−	0.999797i	$-0.506407\pi$
$929$	−3.22745e7			−0.0402543			−0.0201272	+	0.999797i	$0.506407\pi$
$930$		−	5.27285e8i		−	0.655536i
$931$			3.20789e9i			3.97531i
$932$	−3.86337e8			−0.477220
$933$	5.52308e8			0.680043
$934$		−	9.72363e7i		−	0.119340i
$935$			2.53320e8i			0.309910i
$936$	−4.91769e7			−0.0599699
$937$			5.43533e8i			0.660704i	0.943858	+	0.330352i	$0.107167\pi$
$937$			5.43533e8i			0.660704i	−0.943858	+	0.330352i	$0.892833\pi$
$938$	2.21543e7			0.0268442
$939$			1.19925e8i			0.144848i
$940$			7.08202e8i			0.852656i
$941$		−	3.61272e8i		−	0.433577i	−0.976219	−	0.216788i	$-0.930442\pi$
$941$		−	3.61272e8i		−	0.433577i	0.976219	−	0.216788i	$-0.0695581\pi$
$942$			4.25959e6i			0.00509583i
$943$	4.06485e8	+	5.35015e7i	0.484740	+	0.0638015i
$944$	−2.61601e8			−0.310974
$945$	5.52173e8			0.654305
$946$	2.94159e8			0.347463
$947$	−7.21732e8			−0.849818			−0.424909	−	0.905236i	$-0.639694\pi$
$947$	−7.21732e8			−0.849818			−0.424909	+	0.905236i	$0.639694\pi$
$948$			1.07960e8i			0.126718i
$949$	−3.94892e8			−0.462041
$950$			2.01065e9i			2.34512i
$951$	−2.32712e8			−0.270568
$952$	−2.55568e8			−0.296207
$953$			1.52292e9i			1.75954i	0.475403	+	0.879768i	$0.342302\pi$
$953$			1.52292e9i			1.75954i	−0.475403	+	0.879768i	$0.657698\pi$
$954$		−	1.36059e8i		−	0.156705i
$955$	−2.17386e9			−2.49586
$956$	8.56510e8			0.980300
$957$			2.82329e8i			0.322122i
$958$			4.05954e8i			0.461722i
$959$	−2.74475e8			−0.311205
$960$		−	1.14029e8i		−	0.128885i
$961$	−1.70021e8			−0.191572
$962$		−	3.99510e8i		−	0.448747i
$963$		−	2.67976e8i		−	0.300066i
$964$			1.55535e8i			0.173619i
$965$		−	1.20433e9i		−	1.34019i
$966$	−9.14236e7	+	6.94603e8i	−0.101421	+	0.770557i
$967$	3.02166e8			0.334169			0.167084	−	0.985943i	$-0.446565\pi$
$967$	3.02166e8			0.334169			0.167084	+	0.985943i	$0.446565\pi$
$968$	2.70823e8			0.298579
$969$	−3.50190e8			−0.384886
$970$	−1.46516e9			−1.60535
$971$		−	7.11635e7i		−	0.0777319i	−0.999244	−	0.0388660i	$-0.987625\pi$
$971$		−	7.11635e7i		−	0.0777319i	0.999244	−	0.0388660i	$-0.0123745\pi$
$972$	−2.94555e7			−0.0320750
$973$			1.95032e9i			2.11723i
$974$	1.92168e8			0.207971
$975$	−5.96170e8			−0.643215
$976$		−	3.32730e8i		−	0.357884i
$977$		−	9.81298e8i		−	1.05225i	−0.850409	−	0.526123i	$-0.823645\pi$
$977$		−	9.81298e8i		−	1.05225i	0.850409	−	0.526123i	$-0.176355\pi$
$978$	−5.58574e8			−0.597124
$979$	9.99776e7			0.106550
$980$		−	2.20551e9i		−	2.34331i
$981$			4.84836e8i			0.513556i
$982$	9.40163e7			0.0992816
$983$		−	3.42557e7i		−	0.0360639i	−0.999837	−	0.0180320i	$-0.994260\pi$
$983$		−	3.42557e7i		−	0.0360639i	0.999837	−	0.0180320i	$-0.00574006\pi$
$984$	9.50864e7			0.0998006
$985$		−	1.44981e9i		−	1.51706i
$986$			4.22060e8i			0.440294i
$987$		−	1.00914e9i		−	1.04955i
$988$			3.71709e8i			0.385418i
$989$	1.57308e8	−	1.19517e9i	0.162616	−	1.23550i
$990$	1.61055e8			0.165985
$991$	−1.09607e9			−1.12621			−0.563104	−	0.826386i	$-0.690393\pi$
$991$	−1.09607e9			−1.12621			−0.563104	+	0.826386i	$0.690393\pi$
$992$	1.55160e8			0.158945
$993$	−7.02564e8			−0.717527
$994$		−	2.31399e8i		−	0.235615i
$995$	1.14353e9			1.16086
$996$		−	8.03407e7i		−	0.0813126i
$997$	5.87307e8			0.592625			0.296312	−	0.955091i	$-0.404243\pi$
$997$	5.87307e8			0.592625			0.296312	+	0.955091i	$0.404243\pi$
$998$	1.88057e7			0.0189190
$999$		−	2.39294e8i		−	0.240013i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.7.b.a.91.18	yes	24
3.2	odd	2		414.7.b.c.91.2		24
23.22	odd	2	inner	138.7.b.a.91.13	✓	24
69.68	even	2		414.7.b.c.91.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.7.b.a.91.13	✓	24	23.22	odd	2	inner
138.7.b.a.91.18	yes	24	1.1	even	1	trivial
414.7.b.c.91.2		24	3.2	odd	2
414.7.b.c.91.11		24	69.68	even	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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.65685 q^{2} -15.5885 q^{3} +32.0000 q^{4} +223.235i q^{5} -88.1816 q^{6} +652.987i q^{7} +181.019 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q+5.65685 q^{2} -15.5885 q^{3} +32.0000 q^{4} +223.235i q^{5} -88.1816 q^{6} +652.987i q^{7} +181.019 q^{8} +243.000 q^{9} +1262.81i q^{10} -524.845i q^{11} -498.831 q^{12} -1117.97 q^{13} +3693.85i q^{14} -3479.89i q^{15} +1024.00 q^{16} +2162.11i q^{17} +1374.62 q^{18} -10390.2i q^{19} +7143.51i q^{20} -10179.1i q^{21} -2968.97i q^{22} +(-12063.0 - 1587.73i) q^{23} -2821.81 q^{24} -34208.8 q^{25} -6324.19 q^{26} -3788.00 q^{27} +20895.6i q^{28} +34508.2 q^{29} -19685.2i q^{30} +26785.9 q^{31} +5792.62 q^{32} +8181.52i q^{33} +12230.7i q^{34} -145769. q^{35} +7776.00 q^{36} +63171.7i q^{37} -58775.8i q^{38} +17427.4 q^{39} +40409.8i q^{40} -33696.9 q^{41} -57581.4i q^{42} +99077.8i q^{43} -16795.0i q^{44} +54246.0i q^{45} +(-68238.4 - 8981.54i) q^{46} +99139.2 q^{47} -15962.6 q^{48} -308743. q^{49} -193514. q^{50} -33703.9i q^{51} -35775.0 q^{52} -98979.9i q^{53} -21428.1 q^{54} +117164. q^{55} +118203. i q^{56} +161967. i q^{57} +195208. q^{58} -255470. q^{59} -111356. i q^{60} -324932. i q^{61} +151524. q^{62} +158676. i q^{63} +32768.0 q^{64} -249570. i q^{65} +46281.7i q^{66} -5997.63i q^{67} +69187.4i q^{68} +(188043. + 24750.2i) q^{69} -824596. q^{70} -62644.5 q^{71} +43987.7 q^{72} +353223. q^{73} +357353. i q^{74} +533262. q^{75} -332486. i q^{76} +342717. q^{77} +98584.3 q^{78} -216426. i q^{79} +228592. i q^{80} +59049.0 q^{81} -190619. q^{82} +161058. i q^{83} -325730. i q^{84} -482657. q^{85} +560468. i q^{86} -537929. q^{87} -95007.0i q^{88} +190490. i q^{89} +306862. i q^{90} -730019. i q^{91} +(-386015. - 50807.2i) q^{92} -417550. q^{93} +560816. q^{94} +2.31945e6 q^{95} -90298.0 q^{96} +1.16024e6i q^{97} -1.74651e6 q^{98} -127537. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 768 q^{4} + 5832 q^{9}+O(q^{10}) \) 24 * q + 768 * q^4 + 5832 * q^9	\( 24 q + 768 q^{4} + 5832 q^{9} - 768 q^{13} + 24576 q^{16} - 44104 q^{23} - 119448 q^{25} - 53888 q^{26} + 3456 q^{29} + 50976 q^{31} + 149008 q^{35} + 186624 q^{36} + 11664 q^{39} - 3920 q^{41} - 150720 q^{46} + 441088 q^{47} - 32472 q^{49} + 8320 q^{50} - 24576 q^{52} + 826176 q^{55} - 307200 q^{58} - 1210160 q^{59} + 783744 q^{62} + 786432 q^{64} + 361584 q^{69} - 2480064 q^{70} + 1531264 q^{71} + 593472 q^{73} + 23328 q^{75} + 1068784 q^{77} + 171072 q^{78} + 1417176 q^{81} + 1454592 q^{82} - 1318272 q^{85} + 697248 q^{87} - 1411328 q^{92} - 983664 q^{93} + 1115712 q^{94} + 4047632 q^{95} - 2409344 q^{98}+O(q^{100}) \) 24 * q + 768 * q^4 + 5832 * q^9 - 768 * q^13 + 24576 * q^16 - 44104 * q^23 - 119448 * q^25 - 53888 * q^26 + 3456 * q^29 + 50976 * q^31 + 149008 * q^35 + 186624 * q^36 + 11664 * q^39 - 3920 * q^41 - 150720 * q^46 + 441088 * q^47 - 32472 * q^49 + 8320 * q^50 - 24576 * q^52 + 826176 * q^55 - 307200 * q^58 - 1210160 * q^59 + 783744 * q^62 + 786432 * q^64 + 361584 * q^69 - 2480064 * q^70 + 1531264 * q^71 + 593472 * q^73 + 23328 * q^75 + 1068784 * q^77 + 171072 * q^78 + 1417176 * q^81 + 1454592 * q^82 - 1318272 * q^85 + 697248 * q^87 - 1411328 * q^92 - 983664 * q^93 + 1115712 * q^94 + 4047632 * q^95 - 2409344 * q^98

Introduction

L-functions

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Database

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