LMFDB - Embedded newform 138.7.b.a.91.20

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			91.20
Character	$\chi$	$=$	138.91
Dual form			138.7.b.a.91.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685 q^{2} +15.5885 q^{3} +32.0000 q^{4} -56.1927i q^{5} +88.1816 q^{6} +504.037i q^{7} +181.019 q^{8} +243.000 q^{9} +O(q^{10})$	\(q+5.65685 q^{2} +15.5885 q^{3} +32.0000 q^{4} -56.1927i q^{5} +88.1816 q^{6} +504.037i q^{7} +181.019 q^{8} +243.000 q^{9} -317.874i q^{10} +2009.79i q^{11} +498.831 q^{12} -3176.98 q^{13} +2851.26i q^{14} -875.957i q^{15} +1024.00 q^{16} -557.599i q^{17} +1374.62 q^{18} +872.926i q^{19} -1798.17i q^{20} +7857.16i q^{21} +11369.1i q^{22} +(-1589.36 + 12062.7i) q^{23} +2821.81 q^{24} +12467.4 q^{25} -17971.7 q^{26} +3788.00 q^{27} +16129.2i q^{28} -17563.3 q^{29} -4955.16i q^{30} +59384.5 q^{31} +5792.62 q^{32} +31329.5i q^{33} -3154.26i q^{34} +28323.2 q^{35} +7776.00 q^{36} -17225.3i q^{37} +4938.01i q^{38} -49524.2 q^{39} -10172.0i q^{40} -91342.0 q^{41} +44446.8i q^{42} +79561.5i q^{43} +64313.3i q^{44} -13654.8i q^{45} +(-8990.77 + 68237.2i) q^{46} +71324.4 q^{47} +15962.6 q^{48} -136404. q^{49} +70526.2 q^{50} -8692.11i q^{51} -101663. q^{52} -27688.8i q^{53} +21428.1 q^{54} +112935. q^{55} +91240.5i q^{56} +13607.6i q^{57} -99353.1 q^{58} +128610. q^{59} -28030.6i q^{60} +56533.0i q^{61} +335929. q^{62} +122481. i q^{63} +32768.0 q^{64} +178523. i q^{65} +177227. i q^{66} -145702. i q^{67} -17843.2i q^{68} +(-24775.6 + 188040. i) q^{69} +160220. q^{70} +105054. q^{71} +43987.7 q^{72} +156095. q^{73} -97440.9i q^{74} +194347. q^{75} +27933.6i q^{76} -1.01301e6 q^{77} -280151. q^{78} +377563. i q^{79} -57541.3i q^{80} +59049.0 q^{81} -516708. q^{82} -646548. i q^{83} +251429. i q^{84} -31333.0 q^{85} +450068. i q^{86} -273785. q^{87} +363811. i q^{88} +633856. i q^{89} -77243.3i q^{90} -1.60132e6i q^{91} +(-50859.4 + 386008. i) q^{92} +925713. q^{93} +403472. q^{94} +49052.0 q^{95} +90298.0 q^{96} -1.13938e6i q^{97} -771620. q^{98} +488379. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	56.1927i		−	0.449541i	−0.974412	−	0.224771i	$-0.927837\pi$
$5$		−	56.1927i		−	0.449541i	0.974412	−	0.224771i	$-0.0721633\pi$
$6$	88.1816			0.408248
$7$			504.037i			1.46950i	0.678340	+	0.734748i	$0.262700\pi$
$7$			504.037i			1.46950i	−0.678340	+	0.734748i	$0.737300\pi$
$8$	181.019			0.353553
$9$	243.000			0.333333
$10$		−	317.874i		−	0.317874i
$11$			2009.79i			1.50998i	0.655734	+	0.754992i	$0.272359\pi$
$11$			2009.79i			1.50998i	−0.655734	+	0.754992i	$0.727641\pi$
$12$	498.831			0.288675
$13$	−3176.98			−1.44605			−0.723027	−	0.690820i	$-0.757250\pi$
$13$	−3176.98			−1.44605			−0.723027	+	0.690820i	$0.757250\pi$
$14$			2851.26i			1.03909i
$15$		−	875.957i		−	0.259543i
$16$	1024.00			0.250000
$17$		−	557.599i		−	0.113495i	−0.998389	−	0.0567473i	$-0.981927\pi$
$17$		−	557.599i		−	0.113495i	0.998389	−	0.0567473i	$-0.0180729\pi$
$18$	1374.62			0.235702
$19$			872.926i			0.127267i	0.997973	+	0.0636336i	$0.0202689\pi$
$19$			872.926i			0.127267i	−0.997973	+	0.0636336i	$0.979731\pi$
$20$		−	1798.17i		−	0.224771i
$21$			7857.16i			0.848414i
$22$			11369.1i			1.06772i
$23$	−1589.36	+	12062.7i	−0.130629	+	0.991431i
$23$	−1589.36	+	12062.7i	−0.130629	+	0.991431i
$24$	2821.81			0.204124
$25$	12467.4			0.797913
$26$	−17971.7			−1.02251
$27$	3788.00			0.192450
$28$			16129.2i			0.734748i
$29$	−17563.3			−0.720132			−0.360066	−	0.932927i	$-0.617246\pi$
$29$	−17563.3			−0.720132			−0.360066	+	0.932927i	$0.617246\pi$
$30$		−	4955.16i		−	0.183524i
$31$	59384.5			1.99337			0.996685	−	0.0813560i	$-0.0259251\pi$
$31$	59384.5			1.99337			0.996685	+	0.0813560i	$0.0259251\pi$
$32$	5792.62			0.176777
$33$			31329.5i			0.871790i
$34$		−	3154.26i		−	0.0802528i
$35$	28323.2			0.660599
$36$	7776.00			0.166667
$37$		−	17225.3i		−	0.340064i	−0.985439	−	0.170032i	$-0.945613\pi$
$37$		−	17225.3i		−	0.340064i	0.985439	−	0.170032i	$-0.0543872\pi$
$38$			4938.01i			0.0899915i
$39$	−49524.2			−0.834880
$40$		−	10172.0i		−	0.158937i
$41$	−91342.0			−1.32531			−0.662657	−	0.748923i	$-0.730571\pi$
$41$	−91342.0			−1.32531			−0.662657	+	0.748923i	$0.730571\pi$
$42$			44446.8i			0.599919i
$43$			79561.5i			1.00069i	0.865827	+	0.500343i	$0.166793\pi$
$43$			79561.5i			1.00069i	−0.865827	+	0.500343i	$0.833207\pi$
$44$			64313.3i			0.754992i
$45$		−	13654.8i		−	0.149847i
$46$	−8990.77	+	68237.2i	−0.0923683	+	0.701048i
$47$	71324.4			0.686981			0.343491	−	0.939156i	$-0.388391\pi$
$47$	71324.4			0.686981			0.343491	+	0.939156i	$0.388391\pi$
$48$	15962.6			0.144338
$49$	−136404.			−1.15942
$50$	70526.2			0.564209
$51$		−	8692.11i		−	0.0655262i
$52$	−101663.			−0.723027
$53$		−	27688.8i		−	0.185984i	−0.995667	−	0.0929920i	$-0.970357\pi$
$53$		−	27688.8i		−	0.185984i	0.995667	−	0.0929920i	$-0.0296431\pi$
$54$	21428.1			0.136083
$55$	112935.			0.678800
$56$			91240.5i			0.519545i
$57$			13607.6i			0.0734778i
$58$	−99353.1			−0.509210
$59$	128610.			0.626206			0.313103	−	0.949719i	$-0.398631\pi$
$59$	128610.			0.626206			0.313103	+	0.949719i	$0.398631\pi$
$60$		−	28030.6i		−	0.129771i
$61$			56533.0i			0.249065i	0.992216	+	0.124532i	$0.0397431\pi$
$61$			56533.0i			0.249065i	−0.992216	+	0.124532i	$0.960257\pi$
$62$	335929.			1.40953
$63$			122481.i			0.489832i
$64$	32768.0			0.125000
$65$			178523.i			0.650061i
$66$			177227.i			0.616449i
$67$		−	145702.i		−	0.484441i	−0.970221	−	0.242220i	$-0.922124\pi$
$67$		−	145702.i		−	0.484441i	0.970221	−	0.242220i	$-0.0778757\pi$
$68$		−	17843.2i		−	0.0567473i
$69$	−24775.6	+	188040.i	−0.0754184	+	0.572403i
$70$	160220.			0.467114
$71$	105054.			0.293519			0.146760	−	0.989172i	$-0.453116\pi$
$71$	105054.			0.293519			0.146760	+	0.989172i	$0.453116\pi$
$72$	43987.7			0.117851
$73$	156095.			0.401254			0.200627	−	0.979668i	$-0.435702\pi$
$73$	156095.			0.401254			0.200627	+	0.979668i	$0.435702\pi$
$74$		−	97440.9i		−	0.240462i
$75$	194347.			0.460675
$76$			27933.6i			0.0636336i
$77$	−1.01301e6			−2.21892
$78$	−280151.			−0.590349
$79$			377563.i			0.765787i	0.923792	+	0.382893i	$0.125072\pi$
$79$			377563.i			0.765787i	−0.923792	+	0.382893i	$0.874928\pi$
$80$		−	57541.3i		−	0.112385i
$81$	59049.0			0.111111
$82$	−516708.			−0.937139
$83$		−	646548.i		−	1.13075i	−0.824834	−	0.565375i	$-0.808732\pi$
$83$		−	646548.i		−	1.13075i	0.824834	−	0.565375i	$-0.191268\pi$
$84$			251429.i			0.424207i
$85$	−31333.0			−0.0510205
$86$			450068.i			0.707592i
$87$	−273785.			−0.415769
$88$			363811.i			0.533860i
$89$			633856.i			0.899126i	0.893249	+	0.449563i	$0.148420\pi$
$89$			633856.i			0.899126i	−0.893249	+	0.449563i	$0.851580\pi$
$90$		−	77243.3i		−	0.105958i
$91$		−	1.60132e6i		−	2.12497i
$92$	−50859.4	+	386008.i	−0.0653143	+	0.495716i
$93$	925713.			1.15087
$94$	403472.			0.485769
$95$	49052.0			0.0572119
$96$	90298.0			0.102062
$97$		−	1.13938e6i		−	1.24839i	−0.781267	−	0.624197i	$-0.785426\pi$
$97$		−	1.13938e6i		−	1.24839i	0.781267	−	0.624197i	$-0.214574\pi$
$98$	−771620.			−0.819832
$99$			488379.i			0.503328i
$100$	398956.			0.398956
$101$	951449.			0.923467			0.461733	−	0.887019i	$-0.347228\pi$
$101$	951449.			0.923467			0.461733	+	0.887019i	$0.347228\pi$
$102$		−	49170.0i		−	0.0463340i
$103$			2.07362e6i			1.89765i	0.315796	+	0.948827i	$0.397729\pi$
$103$			2.07362e6i			1.89765i	−0.315796	+	0.948827i	$0.602271\pi$
$104$	−575095.			−0.511257
$105$	441515.			0.381397
$106$		−	156631.i		−	0.131511i
$107$		−	471816.i		−	0.385142i	−0.981283	−	0.192571i	$-0.938317\pi$
$107$		−	471816.i		−	0.385142i	0.981283	−	0.192571i	$-0.0616826\pi$
$108$	121216.			0.0962250
$109$		−	1.80335e6i		−	1.39252i	−0.717790	−	0.696260i	$-0.754846\pi$
$109$		−	1.80335e6i		−	1.39252i	0.717790	−	0.696260i	$-0.245154\pi$
$110$	638859.			0.479984
$111$		−	268515.i		−	0.196336i
$112$			516134.i			0.367374i
$113$		−	2.19136e6i		−	1.51872i	−0.650668	−	0.759362i	$-0.725511\pi$
$113$		−	2.19136e6i		−	1.51872i	0.650668	−	0.759362i	$-0.274489\pi$
$114$			76976.0i			0.0519566i
$115$	677838.	+	89310.3i	0.445689	+	0.0587229i
$116$	−562026.			−0.360066
$117$	−772006.			−0.482018
$118$	727526.			0.442794
$119$	281051.			0.166780
$120$		−	158565.i		−	0.0917622i
$121$	−2.26769e6			−1.28005
$122$			319799.i			0.176116i
$123$	−1.42388e6			−0.765171
$124$	1.90030e6			0.996685
$125$		−	1.57859e6i		−	0.808236i
$126$			692857.i			0.346363i
$127$	−3.72111e6			−1.81661			−0.908304	−	0.418312i	$-0.862622\pi$
$127$	−3.72111e6			−1.81661			−0.908304	+	0.418312i	$0.862622\pi$
$128$	185364.			0.0883883
$129$			1.24024e6i			0.577746i
$130$			1.00988e6i			0.459663i
$131$	231888.			0.103149			0.0515745	−	0.998669i	$-0.483576\pi$
$131$	231888.			0.103149			0.0515745	+	0.998669i	$0.483576\pi$
$132$			1.00254e6i			0.435895i
$133$	−439987.			−0.187019
$134$		−	824214.i		−	0.342551i
$135$		−	212858.i		−	0.0865143i
$136$		−	100936.i		−	0.0401264i
$137$		−	4.81585e6i		−	1.87289i	−0.350820	−	0.936443i	$-0.614097\pi$
$137$		−	4.81585e6i		−	1.87289i	0.350820	−	0.936443i	$-0.385903\pi$
$138$	−140152.	+	1.06371e6i	−0.0533289	+	0.404750i
$139$	1.72019e6			0.640518			0.320259	−	0.947330i	$-0.396230\pi$
$139$	1.72019e6			0.640518			0.320259	+	0.947330i	$0.396230\pi$
$140$	906342.			0.330300
$141$	1.11184e6			0.396629
$142$	594273.			0.207549
$143$		−	6.38506e6i		−	2.18352i
$144$	248832.			0.0833333
$145$			986929.i			0.323729i
$146$	883006.			0.283730
$147$	−2.12633e6			−0.669390
$148$		−	551209.i		−	0.170032i
$149$		−	1.97622e6i		−	0.597415i	−0.954345	−	0.298707i	$-0.903445\pi$
$149$		−	1.97622e6i		−	0.597415i	0.954345	−	0.298707i	$-0.0965554\pi$
$150$	1.09939e6			0.325746
$151$	4.52879e6			1.31538			0.657691	−	0.753288i	$-0.271533\pi$
$151$	4.52879e6			1.31538			0.657691	+	0.753288i	$0.271533\pi$
$152$			158016.i			0.0449958i
$153$		−	135497.i		−	0.0378315i
$154$	−5.73044e6			−1.56901
$155$		−	3.33697e6i		−	0.896102i
$156$	−1.58478e6			−0.417440
$157$		−	1.27088e6i		−	0.328402i	−0.986427	−	0.164201i	$-0.947495\pi$
$157$		−	1.27088e6i		−	0.328402i	0.986427	−	0.164201i	$-0.0525046\pi$
$158$			2.13582e6i			0.541493i
$159$		−	431625.i		−	0.107378i
$160$		−	325503.i		−	0.0794684i
$161$	−6.08007e6	−	801095.i	−1.45690	−	0.191958i
$162$	334032.			0.0785674
$163$	947145.			0.218702			0.109351	−	0.994003i	$-0.465123\pi$
$163$	947145.			0.218702			0.109351	+	0.994003i	$0.465123\pi$
$164$	−2.92294e6			−0.662657
$165$	1.76049e6			0.391906
$166$		−	3.65743e6i		−	0.799561i
$167$	149858.			0.0321759			0.0160879	−	0.999871i	$-0.494879\pi$
$167$	149858.			0.0321759			0.0160879	+	0.999871i	$0.494879\pi$
$168$			1.42230e6i			0.299960i
$169$	5.26639e6			1.09107
$170$	−177246.			−0.0360770
$171$			212121.i			0.0424224i
$172$			2.54597e6i			0.500343i
$173$	8.62215e6			1.66524			0.832620	−	0.553844i	$-0.186840\pi$
$173$	8.62215e6			1.66524			0.832620	+	0.553844i	$0.186840\pi$
$174$	−1.54876e6			−0.293993
$175$			6.28402e6i			1.17253i
$176$			2.05802e6i			0.377496i
$177$	2.00482e6			0.361540
$178$			3.58563e6i			0.635778i
$179$	−7.15675e6			−1.24783			−0.623917	−	0.781491i	$-0.714460\pi$
$179$	−7.15675e6			−1.24783			−0.623917	+	0.781491i	$0.714460\pi$
$180$		−	436954.i		−	0.0749236i
$181$			6.31047e6i			1.06421i	0.846680	+	0.532103i	$0.178598\pi$
$181$			6.31047e6i			1.06421i	−0.846680	+	0.532103i	$0.821402\pi$
$182$		−	9.05841e6i		−	1.50258i
$183$			881262.i			0.143798i
$184$	−287704.	+	2.18359e6i	−0.0461842	+	0.350524i
$185$	−967934.			−0.152873
$186$	5.23662e6			0.813790
$187$	1.12066e6			0.171375
$188$	2.28238e6			0.343491
$189$			1.90929e6i			0.282805i
$190$	277480.			0.0404549
$191$		−	555339.i		−	0.0797000i	−0.999206	−	0.0398500i	$-0.987312\pi$
$191$		−	555339.i		−	0.0797000i	0.999206	−	0.0398500i	$-0.0126880\pi$
$192$	510803.			0.0721688
$193$	−2.32926e6			−0.324001			−0.162001	−	0.986791i	$-0.551795\pi$
$193$	−2.32926e6			−0.324001			−0.162001	+	0.986791i	$0.551795\pi$
$194$		−	6.44528e6i		−	0.882748i
$195$			2.78290e6i			0.375313i
$196$	−4.36494e6			−0.579709
$197$	−8.92404e6			−1.16725			−0.583624	−	0.812024i	$-0.698366\pi$
$197$	−8.92404e6			−1.16725			−0.583624	+	0.812024i	$0.698366\pi$
$198$			2.76269e6i			0.355907i
$199$			1.03051e7i			1.30765i	0.756644	+	0.653827i	$0.226837\pi$
$199$			1.03051e7i			1.30765i	−0.756644	+	0.653827i	$0.773163\pi$
$200$	2.25684e6			0.282105
$201$		−	2.27127e6i		−	0.279692i
$202$	5.38221e6			0.652990
$203$		−	8.85256e6i		−	1.05823i
$204$		−	278147.i		−	0.0327631i
$205$			5.13275e6i			0.595784i
$206$			1.17302e7i			1.34184i
$207$	−386214.	+	2.93125e6i	−0.0435429	+	0.330477i
$208$	−3.25323e6			−0.361513
$209$	−1.75440e6			−0.192172
$210$	2.49758e6			0.269688
$211$	−1.20565e7			−1.28344			−0.641718	−	0.766941i	$-0.721778\pi$
$211$	−1.20565e7			−1.28344			−0.641718	+	0.766941i	$0.721778\pi$
$212$		−	886040.i		−	0.0929920i
$213$	1.63762e6			0.169463
$214$		−	2.66899e6i		−	0.272337i
$215$	4.47078e6			0.449850
$216$	685700.			0.0680414
$217$			2.99320e7i			2.92925i
$218$		−	1.02013e7i		−	0.984660i
$219$	2.43328e6			0.231664
$220$	3.61393e6			0.339400
$221$			1.77148e6i			0.164119i
$222$		−	1.51895e6i		−	0.138831i
$223$	2.06251e7			1.85987			0.929933	−	0.367729i	$-0.119864\pi$
$223$	2.06251e7			1.85987			0.929933	+	0.367729i	$0.119864\pi$
$224$			2.91969e6i			0.259773i
$225$	3.02957e6			0.265971
$226$		−	1.23962e7i		−	1.07390i
$227$			2.02503e7i			1.73123i	0.500714	+	0.865613i	$0.333071\pi$
$227$			2.02503e7i			1.73123i	−0.500714	+	0.865613i	$0.666929\pi$
$228$			435442.i			0.0367389i
$229$			7.28654e6i			0.606757i	0.952870	+	0.303379i	$0.0981147\pi$
$229$			7.28654e6i			0.606757i	−0.952870	+	0.303379i	$0.901885\pi$
$230$	3.83443e6	+	505215.i	0.315150	+	0.0415234i
$231$	−1.57912e7			−1.28109
$232$	−3.17930e6			−0.254605
$233$	2.10101e7			1.66096			0.830480	−	0.557048i	$-0.188066\pi$
$233$	2.10101e7			1.66096			0.830480	+	0.557048i	$0.188066\pi$
$234$	−4.36713e6			−0.340838
$235$		−	4.00791e6i		−	0.308826i
$236$	4.11551e6			0.313103
$237$			5.88562e6i			0.442127i
$238$	1.58986e6			0.117931
$239$	−306482.			−0.0224497			−0.0112249	−	0.999937i	$-0.503573\pi$
$239$	−306482.			−0.0224497			−0.0112249	+	0.999937i	$0.503573\pi$
$240$		−	896980.i		−	0.0648857i
$241$			2.17886e7i			1.55660i	0.627891	+	0.778301i	$0.283918\pi$
$241$			2.17886e7i			1.55660i	−0.627891	+	0.778301i	$0.716082\pi$
$242$	−1.28280e7			−0.905134
$243$	920483.			0.0641500
$244$			1.80906e6i			0.124532i
$245$			7.66492e6i			0.521206i
$246$	−8.05469e6			−0.541057
$247$		−	2.77327e6i		−	0.184035i
$248$	1.07497e7			0.704763
$249$		−	1.00787e7i		−	0.652839i
$250$		−	8.92983e6i		−	0.571509i
$251$		−	1.75152e7i		−	1.10763i	−0.832641	−	0.553814i	$-0.813172\pi$
$251$		−	1.75152e7i		−	1.10763i	0.832641	−	0.553814i	$-0.186828\pi$
$252$			3.91939e6i			0.244916i
$253$	−2.42436e7	−	3.19427e6i	−1.49705	−	0.197247i
$254$	−2.10498e7			−1.28454
$255$	−488433.			−0.0294567
$256$	1.04858e6			0.0625000
$257$	1.23640e7			0.728382			0.364191	−	0.931324i	$-0.381346\pi$
$257$	1.23640e7			0.728382			0.364191	+	0.931324i	$0.381346\pi$
$258$			7.01587e6i			0.408528i
$259$	8.68218e6			0.499723
$260$			5.71274e6i			0.325030i
$261$	−4.26788e6			−0.240044
$262$	1.31176e6			0.0729373
$263$			2.20267e7i			1.21083i	0.795911	+	0.605413i	$0.206992\pi$
$263$			2.20267e7i			1.21083i	−0.795911	+	0.605413i	$0.793008\pi$
$264$			5.67125e6i			0.308224i
$265$	−1.55590e6			−0.0836075
$266$	−2.48894e6			−0.132242
$267$			9.88083e6i			0.519110i
$268$		−	4.66246e6i		−	0.242220i
$269$	−2.04037e7			−1.04822			−0.524111	−	0.851650i	$-0.675602\pi$
$269$	−2.04037e7			−1.04822			−0.524111	+	0.851650i	$0.675602\pi$
$270$		−	1.20410e6i		−	0.0611748i
$271$	2.88433e6			0.144923			0.0724613	−	0.997371i	$-0.476915\pi$
$271$	2.88433e6			0.144923			0.0724613	+	0.997371i	$0.476915\pi$
$272$		−	570981.i		−	0.0283737i
$273$		−	2.49620e7i		−	1.22685i
$274$		−	2.72426e7i		−	1.32433i
$275$			2.50568e7i			1.20484i
$276$	−792820.	+	6.01727e6i	−0.0377092	+	0.286202i
$277$	1.56862e7			0.738036			0.369018	−	0.929422i	$-0.379694\pi$
$277$	1.56862e7			0.738036			0.369018	+	0.929422i	$0.379694\pi$
$278$	9.73085e6			0.452915
$279$	1.44304e7			0.664457
$280$	5.12704e6			0.233557
$281$		−	4.92923e6i		−	0.222157i	−0.993812	−	0.111078i	$-0.964570\pi$
$281$		−	4.92923e6i		−	0.222157i	0.993812	−	0.111078i	$-0.0354305\pi$
$282$	6.28951e6			0.280459
$283$		−	6.47321e6i		−	0.285602i	−0.989751	−	0.142801i	$-0.954389\pi$
$283$		−	6.47321e6i		−	0.285602i	0.989751	−	0.142801i	$-0.0456108\pi$
$284$	3.36172e6			0.146760
$285$	764646.			0.0330313
$286$		−	3.61194e7i		−	1.54398i
$287$		−	4.60398e7i		−	1.94754i
$288$	1.40761e6			0.0589256
$289$	2.38267e7			0.987119
$290$			5.58291e6i			0.228911i
$291$		−	1.77611e7i		−	0.720761i
$292$	4.99503e6			0.200627
$293$		−	2.79002e7i		−	1.10918i	−0.832122	−	0.554592i	$-0.812874\pi$
$293$		−	2.79002e7i		−	1.10918i	0.832122	−	0.554592i	$-0.187126\pi$
$294$	−1.20284e7			−0.473330
$295$		−	7.22691e6i		−	0.281505i
$296$		−	3.11811e6i		−	0.120231i
$297$			7.61307e6i			0.290597i
$298$		−	1.11792e7i		−	0.422436i
$299$	5.04936e6	−	3.83231e7i	0.188896	−	1.43366i
$300$	6.21911e6			0.230338
$301$	−4.01020e7			−1.47050
$302$	2.56187e7			0.930115
$303$	1.48316e7			0.533164
$304$			893876.i			0.0318168i
$305$	3.17674e6			0.111965
$306$		−	766484.i		−	0.0267509i
$307$	−1.56431e7			−0.540639			−0.270319	−	0.962771i	$-0.587129\pi$
$307$	−1.56431e7			−0.540639			−0.270319	+	0.962771i	$0.587129\pi$
$308$	−3.24163e7			−1.10946
$309$			3.23245e7i			1.09561i
$310$		−	1.88768e7i		−	0.633640i
$311$	−3.92717e7			−1.30557			−0.652783	−	0.757545i	$-0.726399\pi$
$311$	−3.92717e7			−1.30557			−0.652783	+	0.757545i	$0.726399\pi$
$312$	−8.96484e6			−0.295175
$313$		−	1.90400e7i		−	0.620916i	−0.950587	−	0.310458i	$-0.899518\pi$
$313$		−	1.90400e7i		−	0.620916i	0.950587	−	0.310458i	$-0.100482\pi$
$314$		−	7.18919e6i		−	0.232215i
$315$	6.88253e6			0.220200
$316$			1.20820e7i			0.382893i
$317$	−1.66129e7			−0.521516			−0.260758	−	0.965404i	$-0.583972\pi$
$317$	−1.66129e7			−0.521516			−0.260758	+	0.965404i	$0.583972\pi$
$318$		−	2.44164e6i		−	0.0759277i
$319$		−	3.52985e7i		−	1.08739i
$320$		−	1.84132e6i		−	0.0561927i
$321$		−	7.35488e6i		−	0.222362i
$322$	−3.43941e7	−	4.53168e6i	−1.03019	−	0.135735i
$323$	486743.			0.0144441
$324$	1.88957e6			0.0555556
$325$	−3.96086e7			−1.15382
$326$	5.35786e6			0.154646
$327$		−	2.81115e7i		−	0.803972i
$328$	−1.65347e7			−0.468569
$329$			3.59502e7i			1.00952i
$330$	9.95883e6			0.277119
$331$	1.87042e7			0.515768			0.257884	−	0.966176i	$-0.416975\pi$
$331$	1.87042e7			0.515768			0.257884	+	0.966176i	$0.416975\pi$
$332$		−	2.06895e7i		−	0.565375i
$333$		−	4.18574e6i		−	0.113355i
$334$	847724.			0.0227518
$335$	−8.18737e6			−0.217776
$336$			8.04573e6i			0.212103i
$337$		−	1.39573e7i		−	0.364680i	−0.983236	−	0.182340i	$-0.941633\pi$
$337$		−	1.39573e7i		−	0.364680i	0.983236	−	0.182340i	$-0.0583671\pi$
$338$	2.97912e7			0.771504
$339$		−	3.41600e7i		−	0.876836i
$340$	−1.00266e6			−0.0255103
$341$			1.19350e8i			3.00996i
$342$			1.19994e6i			0.0299972i
$343$		−	9.45340e6i		−	0.234264i
$344$			1.44022e7i			0.353796i
$345$	1.05664e7	+	1.39221e6i	0.257319	+	0.0339037i
$346$	4.87742e7			1.17750
$347$	1.95300e7			0.467426			0.233713	−	0.972306i	$-0.424912\pi$
$347$	1.95300e7			0.467426			0.233713	+	0.972306i	$0.424912\pi$
$348$	−8.76112e6			−0.207884
$349$	5.81284e7			1.36745			0.683726	−	0.729739i	$-0.260359\pi$
$349$	5.81284e7			1.36745			0.683726	+	0.729739i	$0.260359\pi$
$350$			3.55478e7i			0.829103i
$351$	−1.20344e7			−0.278293
$352$			1.16419e7i			0.266930i
$353$	9.47223e6			0.215342			0.107671	−	0.994187i	$-0.465661\pi$
$353$	9.47223e6			0.215342			0.107671	+	0.994187i	$0.465661\pi$
$354$	1.13410e7			0.255648
$355$		−	5.90325e6i		−	0.131949i
$356$			2.02834e7i			0.449563i
$357$	4.38115e6			0.0962904
$358$	−4.04847e7			−0.882351
$359$		−	6.25196e7i		−	1.35124i	−0.737250	−	0.675620i	$-0.763876\pi$
$359$		−	6.25196e7i		−	1.35124i	0.737250	−	0.675620i	$-0.236124\pi$
$360$		−	2.47179e6i		−	0.0529790i
$361$	4.62839e7			0.983803
$362$			3.56974e7i			0.752507i
$363$	−3.53498e7			−0.739039
$364$		−	5.12421e7i		−	1.06249i
$365$		−	8.77138e6i		−	0.180380i
$366$			4.98517e6i			0.101680i
$367$		−	8.76436e7i		−	1.77305i	−0.462677	−	0.886527i	$-0.653111\pi$
$367$		−	8.76436e7i		−	1.77305i	0.462677	−	0.886527i	$-0.346889\pi$
$368$	−1.62750e6	+	1.23523e7i	−0.0326571	+	0.247858i
$369$	−2.21961e7			−0.441771
$370$	−5.47546e6			−0.108098
$371$	1.39562e7			0.273303
$372$	2.96228e7			0.575436
$373$			6.96613e7i			1.34235i	0.741300	+	0.671174i	$0.234210\pi$
$373$			6.96613e7i			1.34235i	−0.741300	+	0.671174i	$0.765790\pi$
$374$	6.33939e6			0.121181
$375$		−	2.46077e7i		−	0.466635i
$376$	1.29111e7			0.242885
$377$	5.57983e7			1.04135
$378$			1.08006e7i			0.199973i
$379$			1.03340e8i			1.89824i	0.314915	+	0.949120i	$0.398024\pi$
$379$			1.03340e8i			1.89824i	−0.314915	+	0.949120i	$0.601976\pi$
$380$	1.56967e6			0.0286059
$381$	−5.80063e7			−1.04882
$382$		−	3.14147e6i		−	0.0563564i
$383$			5.37054e7i			0.955920i	0.878382	+	0.477960i	$0.158624\pi$
$383$			5.37054e7i			0.955920i	−0.878382	+	0.477960i	$0.841376\pi$
$384$	2.88954e6			0.0510310
$385$			5.69236e7i			0.997494i
$386$	−1.31763e7			−0.229104
$387$			1.93335e7i			0.333562i
$388$		−	3.64600e7i		−	0.624197i
$389$			7.33206e7i			1.24560i	0.782382	+	0.622798i	$0.214004\pi$
$389$			7.33206e7i			1.24560i	−0.782382	+	0.622798i	$0.785996\pi$
$390$			1.57424e7i			0.265386i
$391$	6.72618e6	+	886224.i	0.112522	+	0.0148256i
$392$	−2.46918e7			−0.409916
$393$	3.61478e6			0.0595531
$394$	−5.04820e7			−0.825368
$395$	2.12163e7			0.344253
$396$			1.56281e7i			0.251664i
$397$	1.80323e7			0.288190			0.144095	−	0.989564i	$-0.453973\pi$
$397$	1.80323e7			0.288190			0.144095	+	0.989564i	$0.453973\pi$
$398$			5.82944e7i			0.924651i
$399$	−6.85872e6			−0.107975
$400$	1.27666e7			0.199478
$401$		−	4.84643e7i		−	0.751604i	−0.926700	−	0.375802i	$-0.877367\pi$
$401$		−	4.84643e7i		−	0.751604i	0.926700	−	0.375802i	$-0.122633\pi$
$402$		−	1.28482e7i		−	0.197772i
$403$	−1.88663e8			−2.88252
$404$	3.04464e7			0.461733
$405$		−	3.31812e6i		−	0.0499490i
$406$		−	5.00776e7i		−	0.748283i
$407$	3.46192e7			0.513492
$408$		−	1.57344e6i		−	0.0231670i
$409$	6.74072e7			0.985227			0.492613	−	0.870248i	$-0.336042\pi$
$409$	6.74072e7			0.985227			0.492613	+	0.870248i	$0.336042\pi$
$410$			2.90352e7i			0.421283i
$411$		−	7.50717e7i		−	1.08131i
$412$			6.63558e7i			0.948827i
$413$			6.48240e7i			0.920207i
$414$	−2.18476e6	+	1.65816e7i	−0.0307894	+	0.233683i
$415$	−3.63313e7			−0.508319
$416$	−1.84030e7			−0.255629
$417$	2.68151e7			0.369803
$418$	−9.92437e6			−0.135886
$419$		−	5.18747e7i		−	0.705202i	−0.935774	−	0.352601i	$-0.885297\pi$
$419$		−	5.18747e7i		−	0.705202i	0.935774	−	0.352601i	$-0.114703\pi$
$420$	1.41285e7			0.190699
$421$		−	7.41300e7i		−	0.993454i	−0.867907	−	0.496727i	$-0.834535\pi$
$421$		−	7.41300e7i		−	0.993454i	0.867907	−	0.496727i	$-0.165465\pi$
$422$	−6.82019e7			−0.907526
$423$	1.73318e7			0.228994
$424$		−	5.01220e6i		−	0.0657553i
$425$		−	6.95180e6i		−	0.0905588i
$426$	9.26381e6			0.119829
$427$	−2.84947e7			−0.366000
$428$		−	1.50981e7i		−	0.192571i
$429$		−	9.95333e7i		−	1.26066i
$430$	2.52905e7			0.318092
$431$			2.83083e7i			0.353576i	0.984249	+	0.176788i	$0.0565707\pi$
$431$			2.83083e7i			0.353576i	−0.984249	+	0.176788i	$0.943429\pi$
$432$	3.87891e6			0.0481125
$433$			3.57864e7i			0.440813i	0.975408	+	0.220407i	$0.0707384\pi$
$433$			3.57864e7i			0.440813i	−0.975408	+	0.220407i	$0.929262\pi$
$434$			1.69321e8i			2.07129i
$435$			1.53847e7i			0.186905i
$436$		−	5.77073e7i		−	0.696260i
$437$	−1.05299e7	−	1.38739e6i	−0.126177	−	0.0166247i
$438$	1.37647e7			0.163811
$439$	2.86339e7			0.338444			0.169222	−	0.985578i	$-0.445875\pi$
$439$	2.86339e7			0.338444			0.169222	+	0.985578i	$0.445875\pi$
$440$	2.04435e7			0.239992
$441$	−3.31463e7			−0.386473
$442$			1.00210e7i			0.116050i
$443$	3.92830e7			0.451850			0.225925	−	0.974145i	$-0.427460\pi$
$443$	3.92830e7			0.451850			0.225925	+	0.974145i	$0.427460\pi$
$444$		−	8.59250e6i		−	0.0981681i
$445$	3.56180e7			0.404194
$446$	1.16673e8			1.31512
$447$		−	3.08062e7i		−	0.344918i
$448$			1.65163e7i			0.183687i
$449$	−5.84331e7			−0.645535			−0.322768	−	0.946478i	$-0.604613\pi$
$449$	−5.84331e7			−0.645535			−0.322768	+	0.946478i	$0.604613\pi$
$450$	1.71379e7			0.188070
$451$		−	1.83578e8i		−	2.00120i
$452$		−	7.01236e7i		−	0.759362i
$453$	7.05969e7			0.759436
$454$			1.14553e8i			1.22416i
$455$	−8.99822e7			−0.955262
$456$			2.46323e6i			0.0259783i
$457$			1.46617e8i			1.53616i	0.640356	+	0.768078i	$0.278787\pi$
$457$			1.46617e8i			1.53616i	−0.640356	+	0.768078i	$0.721213\pi$
$458$			4.12189e7i			0.429042i
$459$		−	2.11218e6i		−	0.0218421i
$460$	2.16908e7	+	2.85793e6i	0.222845	+	0.0293615i
$461$	1.69676e7			0.173188			0.0865939	−	0.996244i	$-0.472402\pi$
$461$	1.69676e7			0.173188			0.0865939	+	0.996244i	$0.472402\pi$
$462$	−8.93287e7			−0.905869
$463$	1.19581e8			1.20481			0.602407	−	0.798189i	$-0.294208\pi$
$463$	1.19581e8			1.20481			0.602407	+	0.798189i	$0.294208\pi$
$464$	−1.79848e7			−0.180033
$465$		−	5.20183e7i		−	0.517365i
$466$	1.18851e8			1.17448
$467$		−	7.93380e7i		−	0.778988i	−0.921029	−	0.389494i	$-0.872650\pi$
$467$		−	7.93380e7i		−	0.778988i	0.921029	−	0.389494i	$-0.127350\pi$
$468$	−2.47042e7			−0.241009
$469$	7.34391e7			0.711883
$470$		−	2.26722e7i		−	0.218373i
$471$		−	1.98111e7i		−	0.189603i
$472$	2.32808e7			0.221397
$473$	−1.59902e8			−1.51102
$474$			3.32941e7i			0.312631i
$475$			1.08831e7i			0.101548i
$476$	8.99362e6			0.0833899
$477$		−	6.72837e6i		−	0.0619947i
$478$	−1.73372e6			−0.0158744
$479$			1.17677e8i			1.07074i	0.844617	+	0.535371i	$0.179828\pi$
$479$			1.17677e8i			1.07074i	−0.844617	+	0.535371i	$0.820172\pi$
$480$		−	5.07408e6i		−	0.0458811i
$481$			5.47244e7i			0.491751i
$482$			1.23255e8i			1.10068i
$483$	−9.47789e7	−	1.24878e7i	−0.841144	−	0.110827i
$484$	−7.25662e7			−0.640027
$485$	−6.40246e7			−0.561205
$486$	5.20704e6			0.0453609
$487$	3.67280e7			0.317988			0.158994	−	0.987280i	$-0.449175\pi$
$487$	3.67280e7			0.317988			0.158994	+	0.987280i	$0.449175\pi$
$488$			1.02336e7i			0.0880578i
$489$	1.47645e7			0.126268
$490$			4.33594e7i			0.368548i
$491$	1.68575e8			1.42413			0.712064	−	0.702115i	$-0.247761\pi$
$491$	1.68575e8			1.42413			0.712064	+	0.702115i	$0.247761\pi$
$492$	−4.55642e7			−0.382585
$493$			9.79328e6i			0.0817311i
$494$		−	1.56880e7i		−	0.130133i
$495$	2.74433e7			0.226267
$496$	6.08097e7			0.498343
$497$			5.29510e7i			0.431325i
$498$		−	5.70136e7i		−	0.461627i
$499$	−1.21210e8			−0.975521			−0.487761	−	0.872977i	$-0.662186\pi$
$499$	−1.21210e8			−0.975521			−0.487761	+	0.872977i	$0.662186\pi$
$500$		−	5.05148e7i		−	0.404118i
$501$	2.33605e6			0.0185767
$502$		−	9.90809e7i		−	0.783211i
$503$		−	1.60471e8i		−	1.26094i	−0.776215	−	0.630469i	$-0.782863\pi$
$503$		−	1.60471e8i		−	1.26094i	0.776215	−	0.630469i	$-0.217137\pi$
$504$			2.21714e7i			0.173182i
$505$		−	5.34644e7i		−	0.415136i
$506$	−1.37142e8	−	1.80695e7i	−1.05857	−	0.139475i
$507$	8.20950e7			0.629931
$508$	−1.19075e8			−0.908304
$509$	−2.17030e8			−1.64576			−0.822880	−	0.568215i	$-0.807634\pi$
$509$	−2.17030e8			−1.64576			−0.822880	+	0.568215i	$0.807634\pi$
$510$	−2.76299e6			−0.0208290
$511$			7.86776e7i			0.589642i
$512$	5.93164e6			0.0441942
$513$			3.30664e6i			0.0244926i
$514$	6.99412e7			0.515044
$515$	1.16522e8			0.853074
$516$			3.96877e7i			0.288873i
$517$			1.43347e8i			1.03733i
$518$	4.91138e7			0.353358
$519$	1.34406e8			0.961427
$520$			3.23161e7i			0.229831i
$521$			5.49011e7i			0.388211i	0.980981	+	0.194105i	$0.0621804\pi$
$521$			5.49011e7i			0.388211i	−0.980981	+	0.194105i	$0.937820\pi$
$522$	−2.41428e7			−0.169737
$523$			1.76307e8i			1.23244i	0.787575	+	0.616218i	$0.211336\pi$
$523$			1.76307e8i			1.23244i	−0.787575	+	0.616218i	$0.788664\pi$
$524$	7.42042e6			0.0515745
$525$			9.79582e7i			0.676960i
$526$			1.24602e8i			0.856183i
$527$		−	3.31127e7i		−	0.226237i
$528$			3.20814e7i			0.217947i
$529$	−1.42984e8	−	3.83440e7i	−0.965872	−	0.259019i
$530$	−8.80153e6			−0.0591195
$531$	3.12521e7			0.208735
$532$	−1.40796e7			−0.0935093
$533$	2.90192e8			1.91648
$534$			5.58944e7i			0.367066i
$535$	−2.65126e7			−0.173137
$536$		−	2.63748e7i		−	0.171276i
$537$	−1.11563e8			−0.720437
$538$	−1.15421e8			−0.741205
$539$		−	2.74144e8i		−	1.75070i
$540$		−	6.81144e6i		−	0.0432571i
$541$	1.59463e8			1.00709			0.503545	−	0.863969i	$-0.332029\pi$
$541$	1.59463e8			1.00709			0.503545	+	0.863969i	$0.332029\pi$
$542$	1.63162e7			0.102476
$543$			9.83704e7i			0.614420i
$544$		−	3.22996e6i		−	0.0200632i
$545$	−1.01335e8			−0.625995
$546$		−	1.41207e8i		−	0.867515i
$547$	−1.34272e8			−0.820393			−0.410197	−	0.911997i	$-0.634540\pi$
$547$	−1.34272e8			−0.820393			−0.410197	+	0.911997i	$0.634540\pi$
$548$		−	1.54107e8i		−	0.936443i
$549$			1.37375e7i			0.0830217i
$550$			1.41743e8i			0.851947i
$551$		−	1.53315e7i		−	0.0916492i
$552$	−4.48487e6	+	3.40388e7i	−0.0266644	+	0.202375i
$553$	−1.90306e8			−1.12532
$554$	8.87343e7			0.521870
$555$	−1.50886e7			−0.0882612
$556$	5.50460e7			0.320259
$557$		−	4.58035e7i		−	0.265053i	−0.991179	−	0.132527i	$-0.957691\pi$
$557$		−	4.58035e7i		−	0.265053i	0.991179	−	0.132527i	$-0.0423090\pi$
$558$	8.16308e7			0.469842
$559$		−	2.52765e8i		−	1.44705i
$560$	2.90029e7			0.165150
$561$	1.74693e7			0.0989435
$562$		−	2.78839e7i		−	0.157089i
$563$		−	9.45196e7i		−	0.529659i	−0.964295	−	0.264830i	$-0.914684\pi$
$563$		−	9.45196e7i		−	0.529659i	0.964295	−	0.264830i	$-0.0853157\pi$
$564$	3.55788e7			0.198314
$565$	−1.23139e8			−0.682729
$566$		−	3.66180e7i		−	0.201951i
$567$			2.97629e7i			0.163277i
$568$	1.90167e7			0.103775
$569$		−	2.45619e8i		−	1.33329i	−0.745375	−	0.666646i	$-0.767729\pi$
$569$		−	2.45619e8i		−	1.33329i	0.745375	−	0.666646i	$-0.232271\pi$
$570$	4.32549e6			0.0233567
$571$		−	1.64102e8i		−	0.881464i	−0.897639	−	0.440732i	$-0.854719\pi$
$571$		−	1.64102e8i		−	0.881464i	0.897639	−	0.440732i	$-0.145281\pi$
$572$		−	2.04322e8i		−	1.09176i
$573$		−	8.65688e6i		−	0.0460148i
$574$		−	2.60440e8i		−	1.37712i
$575$	−1.98151e7	+	1.50391e8i	−0.104230	+	0.791076i
$576$	7.96262e6			0.0416667
$577$	9.79856e7			0.510076			0.255038	−	0.966931i	$-0.417912\pi$
$577$	9.79856e7			0.510076			0.255038	+	0.966931i	$0.417912\pi$
$578$	1.34784e8			0.697999
$579$	−3.63096e7			−0.187062
$580$			3.15817e7i			0.161865i
$581$	3.25884e8			1.66163
$582$		−	1.00472e8i		−	0.509655i
$583$	5.56486e7			0.280833
$584$	2.82562e7			0.141865
$585$			4.33811e7i			0.216687i
$586$		−	1.57827e8i		−	0.784312i
$587$	7.35873e7			0.363822			0.181911	−	0.983315i	$-0.441772\pi$
$587$	7.35873e7			0.363822			0.181911	+	0.983315i	$0.441772\pi$
$588$	−6.80427e7			−0.334695
$589$			5.18383e7i			0.253691i
$590$		−	4.08816e7i		−	0.199054i
$591$	−1.39112e8			−0.673911
$592$		−	1.76387e7i		−	0.0850161i
$593$	7.53049e7			0.361126			0.180563	−	0.983563i	$-0.442208\pi$
$593$	7.53049e7			0.361126			0.180563	+	0.983563i	$0.442208\pi$
$594$			4.30660e7i			0.205483i
$595$		−	1.57930e7i		−	0.0749744i
$596$		−	6.32390e7i		−	0.298707i
$597$			1.60641e8i			0.754974i
$598$	2.85635e7	−	2.16788e8i	0.133570	−	1.01375i
$599$	−2.91653e8			−1.35702			−0.678509	−	0.734592i	$-0.737374\pi$
$599$	−2.91653e8			−1.35702			−0.678509	+	0.734592i	$0.737374\pi$
$600$	3.51806e7			0.162873
$601$	−9.30759e7			−0.428760			−0.214380	−	0.976750i	$-0.568773\pi$
$601$	−9.30759e7			−0.428760			−0.214380	+	0.976750i	$0.568773\pi$
$602$	−2.26851e8			−1.03980
$603$		−	3.54055e7i		−	0.161480i
$604$	1.44921e8			0.657691
$605$			1.27428e8i			0.575437i
$606$	8.39003e7			0.377004
$607$	2.94208e8			1.31549			0.657745	−	0.753240i	$-0.271510\pi$
$607$	2.94208e8			1.31549			0.657745	+	0.753240i	$0.271510\pi$
$608$			5.05653e6i			0.0224979i
$609$		−	1.37998e8i		−	0.610970i
$610$	1.79704e7			0.0791712
$611$	−2.26596e8			−0.993412
$612$		−	4.33589e6i		−	0.0189158i
$613$			3.36000e8i			1.45867i	0.684154	+	0.729337i	$0.260171\pi$
$613$			3.36000e8i			1.45867i	−0.684154	+	0.729337i	$0.739829\pi$
$614$	−8.84906e7			−0.382289
$615$			8.00117e7i			0.343976i
$616$	−1.83374e8			−0.784505
$617$		−	1.71784e8i		−	0.731352i	−0.930742	−	0.365676i	$-0.880838\pi$
$617$		−	1.71784e8i		−	0.731352i	0.930742	−	0.365676i	$-0.119162\pi$
$618$			1.82855e8i			0.774714i
$619$		−	3.72664e8i		−	1.57125i	−0.618704	−	0.785625i	$-0.712342\pi$
$619$		−	3.72664e8i		−	1.57125i	0.618704	−	0.785625i	$-0.287658\pi$
$620$		−	1.06783e8i		−	0.448051i
$621$	−6.02048e6	+	4.56936e7i	−0.0251395	+	0.190801i
$622$	−2.22154e8			−0.923174
$623$	−3.19487e8			−1.32126
$624$	−5.07128e7			−0.208720
$625$	1.06098e8			0.434577
$626$		−	1.07706e8i		−	0.439054i
$627$	−2.73483e7			−0.110950
$628$		−	4.06682e7i		−	0.164201i
$629$	−9.60480e6			−0.0385955
$630$	3.89335e7			0.155705
$631$			2.96139e8i			1.17871i	0.807873	+	0.589357i	$0.200619\pi$
$631$			2.96139e8i			1.17871i	−0.807873	+	0.589357i	$0.799381\pi$
$632$			6.83462e7i			0.270747i
$633$	−1.87942e8			−0.740992
$634$	−9.39767e7			−0.368767
$635$			2.09099e8i			0.816640i
$636$		−	1.38120e7i		−	0.0536890i
$637$	4.33354e8			1.67658
$638$		−	1.99679e8i		−	0.768900i
$639$	2.55280e7			0.0978397
$640$		−	1.04161e7i		−	0.0397342i
$641$		−	4.07102e8i		−	1.54571i	−0.634581	−	0.772856i	$-0.718827\pi$
$641$		−	4.07102e8i		−	1.54571i	0.634581	−	0.772856i	$-0.281173\pi$
$642$		−	4.16055e7i		−	0.157234i
$643$		−	1.63152e8i		−	0.613704i	−0.951757	−	0.306852i	$-0.900724\pi$
$643$		−	1.63152e8i		−	0.613704i	0.951757	−	0.306852i	$-0.0992756\pi$
$644$	−1.94562e8	−	2.56350e7i	−0.728452	−	0.0959791i
$645$	6.96925e7			0.259721
$646$	2.75343e6			0.0102136
$647$	−2.26469e8			−0.836173			−0.418087	−	0.908407i	$-0.637299\pi$
$647$	−2.26469e8			−0.836173			−0.418087	+	0.908407i	$0.637299\pi$
$648$	1.06890e7			0.0392837
$649$			2.58478e8i			0.945561i
$650$	−2.24060e8			−0.815877
$651$			4.66593e8i			1.69120i
$652$	3.03086e7			0.109351
$653$	2.88209e8			1.03507			0.517534	−	0.855663i	$-0.326850\pi$
$653$	2.88209e8			1.03507			0.517534	+	0.855663i	$0.326850\pi$
$654$		−	1.59023e8i		−	0.568494i
$655$		−	1.30304e7i		−	0.0463697i
$656$	−9.35342e7			−0.331329
$657$	3.79310e7			0.133751
$658$			2.03365e8i			0.713836i
$659$			2.34397e8i			0.819021i	0.912305	+	0.409511i	$0.134301\pi$
$659$			2.34397e8i			0.819021i	−0.912305	+	0.409511i	$0.865699\pi$
$660$	5.63357e7			0.195953
$661$		−	2.37172e8i		−	0.821219i	−0.911811	−	0.410609i	$-0.865316\pi$
$661$		−	2.37172e8i		−	0.821219i	0.911811	−	0.410609i	$-0.134684\pi$
$662$	1.05807e8			0.364703
$663$			2.76147e7i			0.0947543i
$664$		−	1.17038e8i		−	0.399780i
$665$			2.47240e7i			0.0840726i
$666$		−	2.36781e7i		−	0.0801539i
$667$	2.79144e7	−	2.11862e8i	0.0940699	−	0.713962i
$668$	4.79545e6			0.0160879
$669$	3.21514e8			1.07379
$670$	−4.63148e7			−0.153991
$671$	−1.13619e8			−0.376084
$672$			4.55135e7i			0.149980i
$673$	1.30154e8			0.426985			0.213492	−	0.976945i	$-0.431516\pi$
$673$	1.30154e8			0.426985			0.213492	+	0.976945i	$0.431516\pi$
$674$		−	7.89544e7i		−	0.257867i
$675$	4.72264e7			0.153558
$676$	1.68525e8			0.545536
$677$			2.04801e8i			0.660034i	0.943975	+	0.330017i	$0.107055\pi$
$677$			2.04801e8i			0.660034i	−0.943975	+	0.330017i	$0.892945\pi$
$678$		−	1.93238e8i		−	0.620017i
$679$	5.74288e8			1.83451
$680$	−5.67188e6			−0.0180385
$681$			3.15671e8i			0.999524i
$682$			6.75147e8i			2.12836i
$683$	5.34411e8			1.67731			0.838655	−	0.544663i	$-0.183342\pi$
$683$	5.34411e8			1.67731			0.838655	+	0.544663i	$0.183342\pi$
$684$			6.78787e6i			0.0212112i
$685$	−2.70615e8			−0.841940
$686$		−	5.34765e7i		−	0.165650i
$687$			1.13586e8i			0.350311i
$688$			8.14710e7i			0.250172i
$689$			8.79666e7i			0.268943i
$690$	5.97728e7	+	7.87552e6i	0.181952	+	0.0239735i
$691$	−4.07967e8			−1.23649			−0.618246	−	0.785985i	$-0.712156\pi$
$691$	−4.07967e8			−1.23649			−0.618246	+	0.785985i	$0.712156\pi$
$692$	2.75909e8			0.832620
$693$	−2.46161e8			−0.739639
$694$	1.10478e8			0.330520
$695$		−	9.66620e7i		−	0.287939i
$696$	−4.95604e7			−0.146996
$697$			5.09322e7i			0.150416i
$698$	3.28824e8			0.966934
$699$	3.27514e8			0.958956
$700$			2.01089e8i			0.586265i
$701$		−	4.05310e8i		−	1.17661i	−0.808639	−	0.588306i	$-0.799795\pi$
$701$		−	4.05310e8i		−	1.17661i	0.808639	−	0.588306i	$-0.200205\pi$
$702$	−6.80768e7			−0.196783
$703$	1.50364e7			0.0432790
$704$			6.58568e7i			0.188748i
$705$		−	6.24771e7i		−	0.178301i
$706$	5.35830e7			0.152270
$707$			4.79565e8i			1.35703i
$708$	6.41544e7			0.180770
$709$		−	2.67784e7i		−	0.0751355i	−0.999294	−	0.0375678i	$-0.988039\pi$
$709$		−	2.67784e7i		−	0.0751355i	0.999294	−	0.0375678i	$-0.0119610\pi$
$710$		−	3.33938e7i		−	0.0933020i
$711$			9.17478e7i			0.255262i
$712$			1.14740e8i			0.317889i
$713$	−9.43832e7	+	7.16340e8i	−0.260391	+	1.97629i
$714$	2.47835e7			0.0680876
$715$	−3.58794e8			−0.981582
$716$	−2.29016e8			−0.623917
$717$	−4.77758e6			−0.0129614
$718$		−	3.53664e8i		−	0.955472i
$719$	−4.02786e8			−1.08365			−0.541823	−	0.840493i	$-0.682266\pi$
$719$	−4.02786e8			−1.08365			−0.541823	+	0.840493i	$0.682266\pi$
$720$		−	1.39825e7i		−	0.0374618i
$721$	−1.04518e9			−2.78860
$722$	2.61821e8			0.695654
$723$			3.39650e8i			0.898705i
$724$			2.01935e8i			0.532103i
$725$	−2.18969e8			−0.574603
$726$	−1.99969e8			−0.522580
$727$		−	4.34880e8i		−	1.13179i	−0.824477	−	0.565896i	$-0.808530\pi$
$727$		−	4.34880e8i		−	1.13179i	0.824477	−	0.565896i	$-0.191470\pi$
$728$		−	2.89869e8i		−	0.751290i
$729$	1.43489e7			0.0370370
$730$		−	4.96184e7i		−	0.127548i
$731$	4.43634e7			0.113572
$732$			2.82004e7i			0.0718989i
$733$			6.28739e8i			1.59646i	0.602352	+	0.798231i	$0.294231\pi$
$733$			6.28739e8i			1.59646i	−0.602352	+	0.798231i	$0.705769\pi$
$734$		−	4.95787e8i		−	1.25374i
$735$			1.19484e8i			0.300919i
$736$	−9.20654e6	+	6.98749e7i	−0.0230921	+	0.175262i
$737$	2.92830e8			0.731498
$738$	−1.25560e8			−0.312380
$739$	−3.04077e8			−0.753442			−0.376721	−	0.926327i	$-0.622948\pi$
$739$	−3.04077e8			−0.753442			−0.376721	+	0.926327i	$0.622948\pi$
$740$	−3.09739e7			−0.0764365
$741$		−	4.32310e7i		−	0.106253i
$742$	7.89479e7			0.193254
$743$		−	68594.9i		−	0.000167234i	−1.00000		8.36172e-5i	$-0.999973\pi$
$743$		−	68594.9i		−	0.000167234i	1.00000		8.36172e-5i	$-2.66162e-5\pi$
$744$	1.67572e8			0.406895
$745$	−1.11049e8			−0.268563
$746$			3.94064e8i			0.949184i
$747$		−	1.57111e8i		−	0.376917i
$748$	3.58610e7			0.0856876
$749$	2.37813e8			0.565965
$750$		−	1.39202e8i		−	0.329961i
$751$			4.14970e8i			0.979709i	0.871804	+	0.489855i	$0.162950\pi$
$751$			4.14970e8i			0.979709i	−0.871804	+	0.489855i	$0.837050\pi$
$752$	7.30362e7			0.171745
$753$		−	2.73035e8i		−	0.639489i
$754$	3.15643e8			0.736346
$755$		−	2.54485e8i		−	0.591318i
$756$			6.10973e7i			0.141402i
$757$		−	6.70503e8i		−	1.54566i	−0.634615	−	0.772828i	$-0.718841\pi$
$757$		−	6.70503e8i		−	1.54566i	0.634615	−	0.772828i	$-0.281159\pi$
$758$			5.84580e8i			1.34226i
$759$	−3.77920e8	−	4.97938e7i	−0.864320	−	0.113881i
$760$	8.87937e6			0.0202275
$761$	4.98998e8			1.13226			0.566129	−	0.824317i	$-0.308441\pi$
$761$	4.98998e8			1.13226			0.566129	+	0.824317i	$0.308441\pi$
$762$	−3.28133e8			−0.741627
$763$	9.08957e8			2.04630
$764$		−	1.77708e7i		−	0.0398500i
$765$	−7.61391e6			−0.0170068
$766$			3.03803e8i			0.675937i
$767$	−4.08590e8			−0.905528
$768$	1.63457e7			0.0360844
$769$			5.37875e7i			0.118278i	0.998250	+	0.0591388i	$0.0188354\pi$
$769$			5.37875e7i			0.118278i	−0.998250	+	0.0591388i	$0.981165\pi$
$770$			3.22009e8i			0.705335i
$771$	1.92735e8			0.420531
$772$	−7.45365e7			−0.162001
$773$			3.84173e8i			0.831742i	0.909424	+	0.415871i	$0.136523\pi$
$773$			3.84173e8i			0.831742i	−0.909424	+	0.415871i	$0.863477\pi$
$774$			1.09367e8i			0.235864i
$775$	7.40369e8			1.59054
$776$		−	2.06249e8i		−	0.441374i
$777$	1.35342e8			0.288515
$778$			4.14764e8i			0.880770i
$779$		−	7.97348e7i		−	0.168669i
$780$			8.90527e7i			0.187656i
$781$			2.11136e8i			0.443209i
$782$	3.80490e7	+	5.01324e6i	0.0795652	+	0.0104833i
$783$	−6.65297e7			−0.138590
$784$	−1.39678e8			−0.289854
$785$	−7.14142e7			−0.147630
$786$	2.04483e7			0.0421104
$787$			1.67986e8i			0.344628i	0.985042	+	0.172314i	$0.0551243\pi$
$787$			1.67986e8i			0.344628i	−0.985042	+	0.172314i	$0.944876\pi$
$788$	−2.85569e8			−0.583624
$789$			3.43362e8i			0.699071i
$790$	1.20017e8			0.243424
$791$	1.10453e9			2.23176
$792$			8.84060e7i			0.177953i
$793$		−	1.79604e8i		−	0.360161i
$794$	1.02006e8			0.203781
$795$	−2.42542e7			−0.0482708
$796$			3.29763e8i			0.653827i
$797$		−	5.80016e8i		−	1.14568i	−0.819666	−	0.572842i	$-0.805841\pi$
$797$		−	5.80016e8i		−	1.14568i	0.819666	−	0.572842i	$-0.194159\pi$
$798$	−3.87988e7			−0.0763500
$799$		−	3.97704e7i		−	0.0779687i
$800$	7.22188e7			0.141052
$801$			1.54027e8i			0.299709i
$802$		−	2.74156e8i		−	0.531464i
$803$			3.13718e8i			0.605888i
$804$		−	7.26805e7i		−	0.139846i
$805$	−4.50157e7	+	3.41655e8i	−0.0862931	+	0.654939i
$806$	−1.06724e9			−2.03825
$807$	−3.18063e8			−0.605191
$808$	1.72231e8			0.326495
$809$	2.31723e8			0.437646			0.218823	−	0.975765i	$-0.429778\pi$
$809$	2.31723e8			0.437646			0.218823	+	0.975765i	$0.429778\pi$
$810$		−	1.87701e7i		−	0.0353193i
$811$	−6.40203e8			−1.20020			−0.600102	−	0.799923i	$-0.704873\pi$
$811$	−6.40203e8			−1.20020			−0.600102	+	0.799923i	$0.704873\pi$
$812$		−	2.83282e8i		−	0.529116i
$813$	4.49622e7			0.0836711
$814$	1.95836e8			0.363094
$815$		−	5.32226e7i		−	0.0983158i
$816$		−	8.90072e6i		−	0.0163815i
$817$	−6.94513e7			−0.127355
$818$	3.81312e8			0.696660
$819$		−	3.89120e8i		−	0.708323i
$820$			1.64248e8i			0.297892i
$821$	−7.50652e8			−1.35647			−0.678233	−	0.734847i	$-0.737254\pi$
$821$	−7.50652e8			−1.35647			−0.678233	+	0.734847i	$0.737254\pi$
$822$		−	4.24670e8i		−	0.764602i
$823$	−3.86794e8			−0.693873			−0.346936	−	0.937889i	$-0.612778\pi$
$823$	−3.86794e8			−0.693873			−0.346936	+	0.937889i	$0.612778\pi$
$824$			3.75365e8i			0.670922i
$825$			3.90597e8i			0.695612i
$826$			3.66700e8i			0.650685i
$827$		−	1.06143e9i		−	1.87661i	−0.345808	−	0.938305i	$-0.612395\pi$
$827$		−	1.06143e9i		−	1.87661i	0.345808	−	0.938305i	$-0.387605\pi$
$828$	−1.23588e7	+	9.37999e7i	−0.0217714	+	0.165239i
$829$	−2.03269e8			−0.356787			−0.178393	−	0.983959i	$-0.557090\pi$
$829$	−2.03269e8			−0.356787			−0.178393	+	0.983959i	$0.557090\pi$
$830$	−2.05521e8			−0.359436
$831$	2.44523e8			0.426105
$832$	−1.04103e8			−0.180757
$833$			7.60589e7i			0.131588i
$834$	1.51689e8			0.261491
$835$		−	8.42091e6i		−	0.0144644i
$836$	−5.61407e7			−0.0960858
$837$	2.24948e8			0.383624
$838$		−	2.93448e8i		−	0.498653i
$839$			4.81937e8i			0.816027i	0.912976	+	0.408014i	$0.133778\pi$
$839$			4.81937e8i			0.816027i	−0.912976	+	0.408014i	$0.866222\pi$
$840$	7.99227e7			0.134844
$841$	−2.86354e8			−0.481409
$842$		−	4.19342e8i		−	0.702478i
$843$		−	7.68390e7i		−	0.128262i
$844$	−3.85808e8			−0.641718
$845$		−	2.95933e8i		−	0.490482i
$846$	9.80437e7			0.161923
$847$		−	1.14300e9i		−	1.88103i
$848$		−	2.83533e7i		−	0.0464960i
$849$		−	1.00907e8i		−	0.164892i
$850$		−	3.93253e7i		−	0.0640347i
$851$	2.07784e8	+	2.73771e7i	0.337150	+	0.0444221i
$852$	5.24040e7			0.0847316
$853$	5.26646e8			0.848539			0.424270	−	0.905536i	$-0.360531\pi$
$853$	5.26646e8			0.848539			0.424270	+	0.905536i	$0.360531\pi$
$854$	−1.61191e8			−0.258801
$855$	1.19196e7			0.0190706
$856$		−	8.54078e7i		−	0.136168i
$857$	−6.15604e8			−0.978045			−0.489022	−	0.872271i	$-0.662646\pi$
$857$	−6.15604e8			−0.978045			−0.489022	+	0.872271i	$0.662646\pi$
$858$		−	5.63045e8i		−	0.891418i
$859$	7.56648e8			1.19375			0.596877	−	0.802333i	$-0.296408\pi$
$859$	7.56648e8			1.19375			0.596877	+	0.802333i	$0.296408\pi$
$860$	1.43065e8			0.224925
$861$		−	7.17689e8i		−	1.12442i
$862$			1.60136e8i			0.250016i
$863$	3.33286e8			0.518543			0.259272	−	0.965804i	$-0.416518\pi$
$863$	3.33286e8			0.518543			0.259272	+	0.965804i	$0.416518\pi$
$864$	2.19424e7			0.0340207
$865$		−	4.84501e8i		−	0.748595i
$866$			2.02439e8i			0.311702i
$867$	3.71421e8			0.569913
$868$			9.57824e8i			1.46462i
$869$	−7.58822e8			−1.15633
$870$			8.70290e7i			0.132162i
$871$			4.62892e8i			0.700527i
$872$		−	3.26442e8i		−	0.492330i
$873$		−	2.76868e8i		−	0.416131i
$874$	−5.95660e7	−	7.84827e6i	−0.0892204	−	0.0117555i
$875$	7.95666e8			1.18770
$876$	7.78649e7			0.115832
$877$	−9.11109e8			−1.35074			−0.675369	−	0.737480i	$-0.736016\pi$
$877$	−9.11109e8			−1.35074			−0.675369	+	0.737480i	$0.736016\pi$
$878$	1.61978e8			0.239316
$879$		−	4.34921e8i		−	0.640388i
$880$	1.15646e8			0.169700
$881$		−	5.84068e8i		−	0.854153i	−0.904216	−	0.427076i	$-0.859544\pi$
$881$		−	5.84068e8i		−	0.854153i	0.904216	−	0.427076i	$-0.140456\pi$
$882$	−1.87504e8			−0.273277
$883$	9.35782e7			0.135923			0.0679615	−	0.997688i	$-0.478351\pi$
$883$	9.35782e7			0.135923			0.0679615	+	0.997688i	$0.478351\pi$
$884$			5.66874e7i			0.0820597i
$885$		−	1.12656e8i		−	0.162527i
$886$	2.22218e8			0.319506
$887$	−8.65394e8			−1.24006			−0.620030	−	0.784578i	$-0.712880\pi$
$887$	−8.65394e8			−1.24006			−0.620030	+	0.784578i	$0.712880\pi$
$888$		−	4.86065e7i		−	0.0694153i
$889$		−	1.87558e9i		−	2.66950i
$890$	2.01486e8			0.285808
$891$			1.18676e8i			0.167776i
$892$	6.60004e8			0.929933
$893$			6.22610e7i			0.0874302i
$894$		−	1.74266e8i		−	0.243894i
$895$			4.02157e8i			0.560953i
$896$			9.34302e7i			0.129886i
$897$	7.87117e7	−	5.97398e8i	0.109059	−	0.827726i
$898$	−3.30548e8			−0.456462
$899$	−1.04299e9			−1.43549
$900$	9.69464e7			0.132985
$901$	−1.54392e7			−0.0211082
$902$		−	1.03847e9i		−	1.41507i
$903$	−6.25128e8			−0.848996
$904$		−	3.96679e8i		−	0.536950i
$905$	3.54602e8			0.478405
$906$	3.99356e8			0.537002
$907$		−	9.68754e8i		−	1.29835i	−0.760639	−	0.649175i	$-0.775114\pi$
$907$		−	9.68754e8i		−	1.29835i	0.760639	−	0.649175i	$-0.224886\pi$
$908$			6.48009e8i			0.865613i
$909$	2.31202e8			0.307822
$910$	−5.09016e8			−0.675472
$911$			1.24422e9i			1.64567i	0.568281	+	0.822834i	$0.307608\pi$
$911$			1.24422e9i			1.64567i	−0.568281	+	0.822834i	$0.692392\pi$
$912$			1.39341e7i			0.0183694i
$913$	1.29943e9			1.70741
$914$			8.29390e8i			1.08623i
$915$	4.95205e7			0.0646430
$916$			2.33169e8i			0.303379i
$917$			1.16880e8i			0.151577i
$918$		−	1.19483e7i		−	0.0154447i
$919$		−	3.56122e8i		−	0.458831i	−0.973329	−	0.229415i	$-0.926319\pi$
$919$		−	3.56122e8i		−	0.458831i	0.973329	−	0.229415i	$-0.0736814\pi$
$920$	1.22702e8	+	1.61669e7i	0.157575	+	0.0207617i
$921$	−2.43851e8			−0.312138
$922$	9.59831e7			0.122462
$923$	−3.33753e8			−0.424444
$924$	−5.05320e8			−0.640546
$925$		−	2.14754e8i		−	0.271342i
$926$	6.76454e8			0.851933
$927$			5.03889e8i			0.632551i
$928$	−1.01738e8			−0.127303
$929$	−7.74495e7			−0.0965987			−0.0482994	−	0.998833i	$-0.515380\pi$
$929$	−7.74495e7			−0.0965987			−0.0482994	+	0.998833i	$0.515380\pi$
$930$		−	2.94260e8i		−	0.365832i
$931$		−	1.19071e8i		−	0.147556i
$932$	6.72322e8			0.830480
$933$	−6.12185e8			−0.753769
$934$		−	4.48803e8i		−	0.550827i
$935$		−	6.29727e7i		−	0.0770402i
$936$	−1.39748e8			−0.170419
$937$		−	7.97959e8i		−	0.969978i	−0.874520	−	0.484989i	$-0.838824\pi$
$937$		−	7.97959e8i		−	0.969978i	0.874520	−	0.484989i	$-0.161176\pi$
$938$	4.15434e8			0.503378
$939$		−	2.96804e8i		−	0.358486i
$940$		−	1.28253e8i		−	0.154413i
$941$		−	1.58099e7i		−	0.0189741i	−0.999955	−	0.00948705i	$-0.996980\pi$
$941$		−	1.58099e7i		−	0.0189741i	0.999955	−	0.00948705i	$-0.00301987\pi$
$942$		−	1.12068e8i		−	0.134070i
$943$	1.45175e8	−	1.10184e9i	0.173124	−	1.31396i
$944$	1.31696e8			0.156551
$945$	1.07288e8			0.127132
$946$	−9.04542e8			−1.06845
$947$	−1.60050e9			−1.88454			−0.942269	−	0.334858i	$-0.891312\pi$
$947$	−1.60050e9			−1.88454			−0.942269	+	0.334858i	$0.891312\pi$
$948$			1.88340e8i			0.221064i
$949$	−4.95910e8			−0.580236
$950$			6.15641e7i			0.0718054i
$951$	−2.58969e8			−0.301097
$952$	5.08756e7			0.0589656
$953$		−	8.71910e8i		−	1.00738i	−0.863885	−	0.503690i	$-0.831976\pi$
$953$		−	8.71910e8i		−	1.00738i	0.863885	−	0.503690i	$-0.168024\pi$
$954$		−	3.80614e7i		−	0.0438369i
$955$	−3.12060e7			−0.0358284
$956$	−9.80742e6			−0.0112249
$957$		−	5.50250e8i		−	0.627804i
$958$			6.65681e8i			0.757129i
$959$	2.42737e9			2.75220
$960$		−	2.87034e7i		−	0.0324429i
$961$	2.63901e9			2.97352
$962$			3.09568e8i			0.347721i
$963$		−	1.14651e8i		−	0.128381i
$964$			6.97234e8i			0.778301i
$965$			1.30888e8i			0.145652i
$966$	−5.36151e8	−	7.06419e7i	−0.594779	−	0.0783666i
$967$	−1.66083e9			−1.83673			−0.918366	−	0.395733i	$-0.870491\pi$
$967$	−1.66083e9			−1.83673			−0.918366	+	0.395733i	$0.870491\pi$
$968$	−4.10496e8			−0.452567
$969$	7.58757e6			0.00833933
$970$	−3.62178e8			−0.396832
$971$			1.14257e9i			1.24803i	0.781412	+	0.624016i	$0.214500\pi$
$971$			1.14257e9i			1.24803i	−0.781412	+	0.624016i	$0.785500\pi$
$972$	2.94555e7			0.0320750
$973$			8.67039e8i			0.941239i
$974$	2.07765e8			0.224851
$975$	−6.17437e8			−0.666161
$976$			5.78898e7i			0.0622662i
$977$			3.54149e8i			0.379753i	0.981808	+	0.189877i	$0.0608088\pi$
$977$			3.54149e8i			0.379753i	−0.981808	+	0.189877i	$0.939191\pi$
$978$	8.35208e7			0.0892849
$979$	−1.27392e9			−1.35767
$980$			2.45278e8i			0.260603i
$981$		−	4.38215e8i		−	0.464173i
$982$	9.53605e8			1.00701
$983$			1.66681e9i			1.75479i	0.479769	+	0.877395i	$0.340720\pi$
$983$			1.66681e9i			1.75479i	−0.479769	+	0.877395i	$0.659280\pi$
$984$	−2.57750e8			−0.270529
$985$			5.01466e8i			0.524726i
$986$			5.53992e7i			0.0577926i
$987$			5.60408e8i			0.582844i
$988$		−	8.87446e7i		−	0.0920176i
$989$	−9.59731e8	−	1.26452e8i	−0.992112	−	0.130718i
$990$	1.55243e8			0.159995
$991$	1.06907e8			0.109846			0.0549229	−	0.998491i	$-0.482509\pi$
$991$	1.06907e8			0.109846			0.0549229	+	0.998491i	$0.482509\pi$
$992$	3.43992e8			0.352381
$993$	2.91569e8			0.297779
$994$			2.99536e8i			0.304993i
$995$	5.79071e8			0.587844
$996$		−	3.22518e8i		−	0.326419i
$997$	−4.93070e8			−0.497535			−0.248767	−	0.968563i	$-0.580025\pi$
$997$	−4.93070e8			−0.497535			−0.248767	+	0.968563i	$0.580025\pi$
$998$	−6.85667e8			−0.689798
$999$		−	6.52493e7i		−	0.0654454i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.7.b.a.91.20	✓	24
3.2	odd	2		414.7.b.c.91.9		24
23.22	odd	2	inner	138.7.b.a.91.23	yes	24
69.68	even	2		414.7.b.c.91.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.7.b.a.91.20	✓	24	1.1	even	1	trivial
138.7.b.a.91.23	yes	24	23.22	odd	2	inner
414.7.b.c.91.4		24	69.68	even	2
414.7.b.c.91.9		24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.65685 q^{2} +15.5885 q^{3} +32.0000 q^{4} -56.1927i q^{5} +88.1816 q^{6} +504.037i q^{7} +181.019 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q+5.65685 q^{2} +15.5885 q^{3} +32.0000 q^{4} -56.1927i q^{5} +88.1816 q^{6} +504.037i q^{7} +181.019 q^{8} +243.000 q^{9} -317.874i q^{10} +2009.79i q^{11} +498.831 q^{12} -3176.98 q^{13} +2851.26i q^{14} -875.957i q^{15} +1024.00 q^{16} -557.599i q^{17} +1374.62 q^{18} +872.926i q^{19} -1798.17i q^{20} +7857.16i q^{21} +11369.1i q^{22} +(-1589.36 + 12062.7i) q^{23} +2821.81 q^{24} +12467.4 q^{25} -17971.7 q^{26} +3788.00 q^{27} +16129.2i q^{28} -17563.3 q^{29} -4955.16i q^{30} +59384.5 q^{31} +5792.62 q^{32} +31329.5i q^{33} -3154.26i q^{34} +28323.2 q^{35} +7776.00 q^{36} -17225.3i q^{37} +4938.01i q^{38} -49524.2 q^{39} -10172.0i q^{40} -91342.0 q^{41} +44446.8i q^{42} +79561.5i q^{43} +64313.3i q^{44} -13654.8i q^{45} +(-8990.77 + 68237.2i) q^{46} +71324.4 q^{47} +15962.6 q^{48} -136404. q^{49} +70526.2 q^{50} -8692.11i q^{51} -101663. q^{52} -27688.8i q^{53} +21428.1 q^{54} +112935. q^{55} +91240.5i q^{56} +13607.6i q^{57} -99353.1 q^{58} +128610. q^{59} -28030.6i q^{60} +56533.0i q^{61} +335929. q^{62} +122481. i q^{63} +32768.0 q^{64} +178523. i q^{65} +177227. i q^{66} -145702. i q^{67} -17843.2i q^{68} +(-24775.6 + 188040. i) q^{69} +160220. q^{70} +105054. q^{71} +43987.7 q^{72} +156095. q^{73} -97440.9i q^{74} +194347. q^{75} +27933.6i q^{76} -1.01301e6 q^{77} -280151. q^{78} +377563. i q^{79} -57541.3i q^{80} +59049.0 q^{81} -516708. q^{82} -646548. i q^{83} +251429. i q^{84} -31333.0 q^{85} +450068. i q^{86} -273785. q^{87} +363811. i q^{88} +633856. i q^{89} -77243.3i q^{90} -1.60132e6i q^{91} +(-50859.4 + 386008. i) q^{92} +925713. q^{93} +403472. q^{94} +49052.0 q^{95} +90298.0 q^{96} -1.13938e6i q^{97} -771620. q^{98} +488379. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 768 q^{4} + 5832 q^{9}+O(q^{10}) \) 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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