LMFDB - Embedded newform 69.7.d.a.22.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("69.22");

S:= CuspForms(chi, 7);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.8737317698$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.19
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.8996 q^{2} -15.5885 q^{3} +54.8005 q^{4} -162.476i q^{5} -169.907 q^{6} +219.782i q^{7} -100.271 q^{8} +243.000 q^{9} +O(q^{10})$	\(q+10.8996 q^{2} -15.5885 q^{3} +54.8005 q^{4} -162.476i q^{5} -169.907 q^{6} +219.782i q^{7} -100.271 q^{8} +243.000 q^{9} -1770.91i q^{10} -1606.67i q^{11} -854.255 q^{12} -3928.95 q^{13} +2395.53i q^{14} +2532.75i q^{15} -4600.14 q^{16} -6864.44i q^{17} +2648.59 q^{18} +1474.89i q^{19} -8903.75i q^{20} -3426.06i q^{21} -17512.0i q^{22} +(7766.67 - 9365.61i) q^{23} +1563.06 q^{24} -10773.4 q^{25} -42823.9 q^{26} -3788.00 q^{27} +12044.2i q^{28} +18555.6 q^{29} +27605.8i q^{30} +27090.1 q^{31} -43722.2 q^{32} +25045.6i q^{33} -74819.4i q^{34} +35709.2 q^{35} +13316.5 q^{36} +74301.9i q^{37} +16075.6i q^{38} +61246.3 q^{39} +16291.5i q^{40} -53361.4 q^{41} -37342.6i q^{42} +68697.2i q^{43} -88046.5i q^{44} -39481.6i q^{45} +(84653.4 - 102081. i) q^{46} -105757. q^{47} +71709.0 q^{48} +69344.9 q^{49} -117425. q^{50} +107006. i q^{51} -215308. q^{52} -257751. i q^{53} -41287.5 q^{54} -261045. q^{55} -22037.7i q^{56} -22991.2i q^{57} +202248. q^{58} +158761. q^{59} +138796. i q^{60} +224111. i q^{61} +295270. q^{62} +53407.0i q^{63} -182144. q^{64} +638359. i q^{65} +272986. i q^{66} -530212. i q^{67} -376175. i q^{68} +(-121070. + 145995. i) q^{69} +389215. q^{70} +458854. q^{71} -24365.8 q^{72} +281567. q^{73} +809859. i q^{74} +167940. q^{75} +80824.5i q^{76} +353118. q^{77} +667558. q^{78} -253619. i q^{79} +747411. i q^{80} +59049.0 q^{81} -581616. q^{82} -147594. i q^{83} -187750. i q^{84} -1.11530e6 q^{85} +748770. i q^{86} -289253. q^{87} +161102. i q^{88} +260389. i q^{89} -430332. i q^{90} -863513. i q^{91} +(425618. - 513240. i) q^{92} -422293. q^{93} -1.15271e6 q^{94} +239633. q^{95} +681561. q^{96} -1.39769e6i q^{97} +755829. q^{98} -390422. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	10.8996			1.36245			0.681223	−	0.732076i	$-0.261449\pi$
$2$	10.8996			1.36245			0.681223	+	0.732076i	$0.261449\pi$
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	54.8005			0.856258
$5$		−	162.476i		−	1.29981i	−0.760017	−	0.649903i	$-0.774810\pi$
$5$		−	162.476i		−	1.29981i	0.760017	−	0.649903i	$-0.225190\pi$
$6$	−169.907			−0.786608
$7$			219.782i			0.640764i	0.947288	+	0.320382i	$0.103811\pi$
$7$			219.782i			0.640764i	−0.947288	+	0.320382i	$0.896189\pi$
$8$	−100.271			−0.195841
$9$	243.000			0.333333
$10$		−	1770.91i		−	1.77091i
$11$		−	1606.67i		−	1.20712i	−0.797318	−	0.603559i	$-0.793749\pi$
$11$		−	1606.67i		−	1.20712i	0.797318	−	0.603559i	$-0.206251\pi$
$12$	−854.255			−0.494361
$13$	−3928.95			−1.78833			−0.894163	−	0.447742i	$-0.852228\pi$
$13$	−3928.95			−1.78833			−0.894163	+	0.447742i	$0.852228\pi$
$14$			2395.53i			0.873006i
$15$			2532.75i			0.750443i
$16$	−4600.14			−1.12308
$17$		−	6864.44i		−	1.39720i	−0.715513	−	0.698599i	$-0.753807\pi$
$17$		−	6864.44i		−	1.39720i	0.715513	−	0.698599i	$-0.246193\pi$
$18$	2648.59			0.454149
$19$			1474.89i			0.215029i	0.994203	+	0.107515i	$0.0342893\pi$
$19$			1474.89i			0.215029i	−0.994203	+	0.107515i	$0.965711\pi$
$20$		−	8903.75i		−	1.11297i
$21$		−	3426.06i		−	0.369945i
$22$		−	17512.0i		−	1.64463i
$23$	7766.67	−	9365.61i	0.638339	−	0.769755i
$23$	7766.67	−	9365.61i	0.638339	−	0.769755i
$24$	1563.06			0.113069
$25$	−10773.4			−0.689496
$26$	−42823.9			−2.43650
$27$	−3788.00			−0.192450
$28$			12044.2i			0.548659i
$29$	18555.6			0.760819			0.380409	−	0.924818i	$-0.375783\pi$
$29$	18555.6			0.760819			0.380409	+	0.924818i	$0.375783\pi$
$30$			27605.8i			1.02244i
$31$	27090.1			0.909339			0.454670	−	0.890660i	$-0.349757\pi$
$31$	27090.1			0.909339			0.454670	+	0.890660i	$0.349757\pi$
$32$	−43722.2			−1.33429
$33$			25045.6i			0.696930i
$34$		−	74819.4i		−	1.90361i
$35$	35709.2			0.832869
$36$	13316.5			0.285419
$37$			74301.9i			1.46688i	0.679754	+	0.733441i	$0.262087\pi$
$37$			74301.9i			1.46688i	−0.679754	+	0.733441i	$0.737913\pi$
$38$			16075.6i			0.292966i
$39$	61246.3			1.03249
$40$			16291.5i			0.254555i
$41$	−53361.4			−0.774241			−0.387120	−	0.922029i	$-0.626530\pi$
$41$	−53361.4			−0.774241			−0.387120	+	0.922029i	$0.626530\pi$
$42$		−	37342.6i		−	0.504030i
$43$			68697.2i			0.864040i	0.901864	+	0.432020i	$0.142199\pi$
$43$			68697.2i			0.864040i	−0.901864	+	0.432020i	$0.857801\pi$
$44$		−	88046.5i		−	1.03360i
$45$		−	39481.6i		−	0.433269i
$46$	84653.4	−	102081.i	0.869702	−	1.04875i
$47$	−105757.			−1.01863			−0.509316	−	0.860580i	$-0.670101\pi$
$47$	−105757.			−1.01863			−0.509316	+	0.860580i	$0.670101\pi$
$48$	71709.0			0.648411
$49$	69344.9			0.589422
$50$	−117425.			−0.939400
$51$			107006.i			0.806673i
$52$	−215308.			−1.53127
$53$		−	257751.i		−	1.73130i	−0.500650	−	0.865650i	$-0.666906\pi$
$53$		−	257751.i		−	1.73130i	0.500650	−	0.865650i	$-0.333094\pi$
$54$	−41287.5			−0.262203
$55$	−261045.			−1.56902
$56$		−	22037.7i		−	0.125488i
$57$		−	22991.2i		−	0.124147i
$58$	202248.			1.03657
$59$	158761.			0.773013			0.386507	−	0.922287i	$-0.373682\pi$
$59$	158761.			0.773013			0.386507	+	0.922287i	$0.373682\pi$
$60$			138796.i			0.642573i
$61$			224111.i			0.987357i	0.869644	+	0.493679i	$0.164348\pi$
$61$			224111.i			0.987357i	−0.869644	+	0.493679i	$0.835652\pi$
$62$	295270.			1.23892
$63$			53407.0i			0.213588i
$64$	−182144.			−0.694824
$65$			638359.i			2.32448i
$66$			272986.i			0.949529i
$67$		−	530212.i		−	1.76289i	−0.472286	−	0.881445i	$-0.656571\pi$
$67$		−	530212.i		−	1.76289i	0.472286	−	0.881445i	$-0.343429\pi$
$68$		−	376175.i		−	1.19636i
$69$	−121070.	+	145995.i	−0.368545	+	0.444418i
$70$	389215.			1.13474
$71$	458854.			1.28203			0.641017	−	0.767526i	$-0.278513\pi$
$71$	458854.			1.28203			0.641017	+	0.767526i	$0.278513\pi$
$72$	−24365.8			−0.0652803
$73$	281567.			0.723791			0.361896	−	0.932219i	$-0.382130\pi$
$73$	281567.			0.723791			0.361896	+	0.932219i	$0.382130\pi$
$74$			809859.i			1.99855i
$75$	167940.			0.398080
$76$			80824.5i			0.184121i
$77$	353118.			0.773477
$78$	667558.			1.40671
$79$		−	253619.i		−	0.514399i	−0.966358	−	0.257200i	$-0.917200\pi$
$79$		−	253619.i		−	0.514399i	0.966358	−	0.257200i	$-0.0827998\pi$
$80$			747411.i			1.45979i
$81$	59049.0			0.111111
$82$	−581616.			−1.05486
$83$		−	147594.i		−	0.258127i	−0.991636	−	0.129063i	$-0.958803\pi$
$83$		−	147594.i		−	0.258127i	0.991636	−	0.129063i	$-0.0411971\pi$
$84$		−	187750.i		−	0.316768i
$85$	−1.11530e6			−1.81609
$86$			748770.i			1.17721i
$87$	−289253.			−0.439259
$88$			161102.i			0.236403i
$89$			260389.i			0.369362i	0.982799	+	0.184681i	$0.0591252\pi$
$89$			260389.i			0.369362i	−0.982799	+	0.184681i	$0.940875\pi$
$90$		−	430332.i		−	0.590305i
$91$		−	863513.i		−	1.14589i
$92$	425618.	−	513240.i	0.546583	−	0.659109i
$93$	−422293.			−0.525007
$94$	−1.15271e6			−1.38783
$95$	239633.			0.279497
$96$	681561.			0.770356
$97$		−	1.39769e6i		−	1.53142i	−0.643186	−	0.765710i	$-0.722388\pi$
$97$		−	1.39769e6i		−	1.53142i	0.643186	−	0.765710i	$-0.277612\pi$
$98$	755829.			0.803055
$99$		−	390422.i		−	0.402373i
$100$	−590386.			−0.590386
$101$	−1.18689e6			−1.15199			−0.575993	−	0.817454i	$-0.695385\pi$
$101$	−1.18689e6			−1.15199			−0.575993	+	0.817454i	$0.695385\pi$
$102$			1.16632e6i			1.09905i
$103$		−	607038.i		−	0.555526i	−0.960650	−	0.277763i	$-0.910407\pi$
$103$		−	607038.i		−	0.555526i	0.960650	−	0.277763i	$-0.0895930\pi$
$104$	393958.			0.350227
$105$	−556652.			−0.480857
$106$		−	2.80937e6i		−	2.35880i
$107$		−	1.19097e6i		−	0.972184i	−0.873908	−	0.486092i	$-0.838422\pi$
$107$		−	1.19097e6i		−	0.972184i	0.873908	−	0.486092i	$-0.161578\pi$
$108$	−207584.			−0.164787
$109$			1.52787e6i			1.17979i	0.807478	+	0.589897i	$0.200832\pi$
$109$			1.52787e6i			1.17979i	−0.807478	+	0.589897i	$0.799168\pi$
$110$	−2.84528e6			−2.13770
$111$		−	1.15825e6i		−	0.846904i
$112$		−	1.01103e6i		−	0.719629i
$113$			1.05493e6i			0.731117i	0.930788	+	0.365559i	$0.119122\pi$
$113$			1.05493e6i			0.731117i	−0.930788	+	0.365559i	$0.880878\pi$
$114$		−	250594.i		−	0.169144i
$115$	−1.52168e6	−	1.26190e6i	−1.00053	−	0.829717i
$116$	1.01686e6			0.651457
$117$	−954735.			−0.596109
$118$	1.73042e6			1.05319
$119$	1.50868e6			0.895274
$120$		−	253960.i		−	0.146968i
$121$	−809839.			−0.457133
$122$			2.44272e6i			1.34522i
$123$	831822.			0.447008
$124$	1.48455e6			0.778629
$125$		−	788273.i		−	0.403596i
$126$			582113.i			0.291002i
$127$	−228919.			−0.111756			−0.0558779	−	0.998438i	$-0.517796\pi$
$127$	−228919.			−0.111756			−0.0558779	+	0.998438i	$0.517796\pi$
$128$	812931.			0.387636
$129$		−	1.07088e6i		−	0.498854i
$130$			6.95784e6i			3.16697i
$131$	−2.04939e6			−0.911614			−0.455807	−	0.890079i	$-0.650649\pi$
$131$	−2.04939e6			−0.911614			−0.455807	+	0.890079i	$0.650649\pi$
$132$			1.37251e6i			0.596751i
$133$	−324154.			−0.137783
$134$		−	5.77908e6i		−	2.40184i
$135$			615457.i			0.250148i
$136$			688301.i			0.273629i
$137$		−	3.85440e6i		−	1.49898i	−0.662018	−	0.749488i	$-0.730300\pi$
$137$		−	3.85440e6i		−	1.49898i	0.662018	−	0.749488i	$-0.269700\pi$
$138$	−1.31962e6	+	1.59129e6i	−0.502123	+	0.605496i
$139$	−4.05743e6			−1.51080			−0.755400	−	0.655264i	$-0.772557\pi$
$139$	−4.05743e6			−1.51080			−0.755400	+	0.655264i	$0.772557\pi$
$140$	1.95688e6			0.713150
$141$	1.64859e6			0.588107
$142$	5.00131e6			1.74670
$143$			6.31254e6i			2.15872i
$144$	−1.11783e6			−0.374360
$145$		−	3.01484e6i		−	0.988917i
$146$	3.06896e6			0.986126
$147$	−1.08098e6			−0.340303
$148$			4.07178e6i			1.25603i
$149$			4.37188e6i			1.32163i	0.750550	+	0.660814i	$0.229789\pi$
$149$			4.37188e6i			1.32163i	−0.750550	+	0.660814i	$0.770211\pi$
$150$	1.83047e6			0.542363
$151$	−1.27442e6			−0.370153			−0.185076	−	0.982724i	$-0.559253\pi$
$151$	−1.27442e6			−0.370153			−0.185076	+	0.982724i	$0.559253\pi$
$152$		−	147888.i		−	0.0421116i
$153$		−	1.66806e6i		−	0.465733i
$154$	3.84883e6			1.05382
$155$		−	4.40149e6i		−	1.18196i
$156$	3.35633e6			0.884078
$157$		−	1.84958e6i		−	0.477941i	−0.971027	−	0.238970i	$-0.923190\pi$
$157$		−	1.84958e6i		−	0.477941i	0.971027	−	0.238970i	$-0.0768099\pi$
$158$		−	2.76434e6i		−	0.700841i
$159$			4.01793e6i			0.999566i
$160$			7.10379e6i			1.73432i
$161$	2.05839e6	+	1.70697e6i	0.493231	+	0.409025i
$162$	643608.			0.151383
$163$	3.90418e6			0.901502			0.450751	−	0.892650i	$-0.351156\pi$
$163$	3.90418e6			0.901502			0.450751	+	0.892650i	$0.351156\pi$
$164$	−2.92423e6			−0.662949
$165$	4.06930e6			0.905873
$166$		−	1.60871e6i		−	0.351684i
$167$	−1.57936e6			−0.339103			−0.169552	−	0.985521i	$-0.554232\pi$
$167$	−1.57936e6			−0.339103			−0.169552	+	0.985521i	$0.554232\pi$
$168$			343533.i			0.0724504i
$169$	1.06099e7			2.19811
$170$	−1.21563e7			−2.47432
$171$			358398.i			0.0716765i
$172$			3.76464e6i			0.739841i
$173$	−3.44019e6			−0.664422			−0.332211	−	0.943205i	$-0.607795\pi$
$173$	−3.44019e6			−0.664422			−0.332211	+	0.943205i	$0.607795\pi$
$174$	−3.15273e6			−0.598466
$175$		−	2.36779e6i		−	0.441804i
$176$			7.39092e6i			1.35569i
$177$	−2.47483e6			−0.446299
$178$			2.83812e6i			0.503236i
$179$	−6.13454e6			−1.06960			−0.534802	−	0.844977i	$-0.679614\pi$
$179$	−6.13454e6			−1.06960			−0.534802	+	0.844977i	$0.679614\pi$
$180$		−	2.16361e6i		−	0.370990i
$181$			294540.i			0.0496717i	0.999692	+	0.0248358i	$0.00790631\pi$
$181$			294540.i			0.0496717i	−0.999692	+	0.0248358i	$0.992094\pi$
$182$		−	9.41191e6i		−	1.56122i
$183$		−	3.49355e6i		−	0.570051i
$184$	−778769.	+	939095.i	−0.125013	+	0.150750i
$185$	1.20723e7			1.90666
$186$	−4.60281e6			−0.715294
$187$	−1.10289e7			−1.68658
$188$	−5.79556e6			−0.872211
$189$		−	832533.i		−	0.123315i
$190$	2.61190e6			0.380799
$191$		−	2.24501e6i		−	0.322194i	−0.986939	−	0.161097i	$-0.948497\pi$
$191$		−	2.24501e6i		−	0.322194i	0.986939	−	0.161097i	$-0.0515032\pi$
$192$	2.83934e6			0.401157
$193$	709667.			0.0987148			0.0493574	−	0.998781i	$-0.484283\pi$
$193$	709667.			0.0987148			0.0493574	+	0.998781i	$0.484283\pi$
$194$		−	1.52342e7i		−	2.08648i
$195$		−	9.95104e6i		−	1.34204i
$196$	3.80013e6			0.504697
$197$	−2.48782e6			−0.325403			−0.162701	−	0.986675i	$-0.552021\pi$
$197$	−2.48782e6			−0.325403			−0.162701	+	0.986675i	$0.552021\pi$
$198$		−	4.25543e6i		−	0.548211i
$199$		−	1.47724e6i		−	0.187453i	−0.995598	−	0.0937266i	$-0.970122\pi$
$199$		−	1.47724e6i		−	0.187453i	0.995598	−	0.0937266i	$-0.0298780\pi$
$200$	1.08025e6			0.135031
$201$			8.26519e6i			1.01781i
$202$	−1.29366e7			−1.56952
$203$			4.07819e6i			0.487505i
$204$			5.86398e6i			0.690720i
$205$			8.66994e6i			1.00636i
$206$		−	6.61645e6i		−	0.756874i
$207$	1.88730e6	−	2.27584e6i	0.212780	−	0.256585i
$208$	1.80737e7			2.00843
$209$	2.36966e6			0.259566
$210$	−6.06726e6			−0.655141
$211$	7.31935e6			0.779157			0.389579	−	0.920993i	$-0.372621\pi$
$211$	7.31935e6			0.779157			0.389579	+	0.920993i	$0.372621\pi$
$212$		−	1.41249e7i		−	1.48244i
$213$	−7.15283e6			−0.740183
$214$		−	1.29810e7i		−	1.32455i
$215$	1.11616e7			1.12308
$216$	379824.			0.0376896
$217$			5.95392e6i			0.582671i
$218$			1.66531e7i			1.60741i
$219$	−4.38920e6			−0.417881
$220$	−1.43054e7			−1.34348
$221$			2.69700e7i			2.49865i
$222$		−	1.26244e7i		−	1.15386i
$223$	−703409.			−0.0634298			−0.0317149	−	0.999497i	$-0.510097\pi$
$223$	−703409.			−0.0634298			−0.0317149	+	0.999497i	$0.510097\pi$
$224$		−	9.60935e6i		−	0.854968i
$225$	−2.61793e6			−0.229832
$226$			1.14982e7i			0.996108i
$227$			9.48496e6i			0.810882i	0.914121	+	0.405441i	$0.132882\pi$
$227$			9.48496e6i			0.810882i	−0.914121	+	0.405441i	$0.867118\pi$
$228$		−	1.25993e6i		−	0.106302i
$229$			311688.i			0.0259546i	0.999916	+	0.0129773i	$0.00413091\pi$
$229$			311688.i			0.0259546i	−0.999916	+	0.0129773i	$0.995869\pi$
$230$	−1.65857e7	−	1.37541e7i	−1.36317	−	1.13044i
$231$	−5.50456e6			−0.446567
$232$	−1.86058e6			−0.148999
$233$	1.56282e7			1.23550			0.617749	−	0.786375i	$-0.288045\pi$
$233$	1.56282e7			1.23550			0.617749	+	0.786375i	$0.288045\pi$
$234$	−1.04062e7			−0.812165
$235$			1.71830e7i			1.32402i
$236$	8.70016e6			0.661899
$237$			3.95353e6i			0.296989i
$238$	1.64439e7			1.21976
$239$	−2.63176e6			−0.192776			−0.0963879	−	0.995344i	$-0.530729\pi$
$239$	−2.63176e6			−0.192776			−0.0963879	+	0.995344i	$0.530729\pi$
$240$		−	1.16510e7i		−	0.842808i
$241$		−	1.89669e7i		−	1.35502i	−0.735513	−	0.677510i	$-0.763059\pi$
$241$		−	1.89669e7i		−	1.35502i	0.735513	−	0.677510i	$-0.236941\pi$
$242$	−8.82689e6			−0.622819
$243$	−920483.			−0.0641500
$244$			1.22814e7i			0.845432i
$245$		−	1.12669e7i		−	0.766134i
$246$	9.06650e6			0.609024
$247$		−	5.79476e6i		−	0.384543i
$248$	−2.71634e6			−0.178086
$249$			2.30076e6i			0.149030i
$250$		−	8.59183e6i		−	0.549877i
$251$			6.64309e6i			0.420097i	0.977691	+	0.210048i	$0.0673621\pi$
$251$			6.64309e6i			0.420097i	−0.977691	+	0.210048i	$0.932638\pi$
$252$			2.92673e6i			0.182886i
$253$	−1.50475e7	−	1.24785e7i	−0.929185	−	0.770551i
$254$	−2.49511e6			−0.152261
$255$	1.73859e7			1.04852
$256$	2.05178e7			1.22296
$257$	2.36926e7			1.39577			0.697886	−	0.716209i	$-0.254124\pi$
$257$	2.36926e7			1.39577			0.697886	+	0.716209i	$0.254124\pi$
$258$		−	1.16722e7i		−	0.679661i
$259$	−1.63302e7			−0.939924
$260$			3.49824e7i			1.99035i
$261$	4.50901e6			0.253606
$262$	−2.23375e7			−1.24202
$263$		−	1.41309e7i		−	0.776788i	−0.921493	−	0.388394i	$-0.873030\pi$
$263$		−	1.41309e7i		−	0.776788i	0.921493	−	0.388394i	$-0.126970\pi$
$264$		−	2.51133e6i		−	0.136487i
$265$	−4.18782e7			−2.25035
$266$	−3.53313e6			−0.187722
$267$		−	4.05906e6i		−	0.213251i
$268$		−	2.90559e7i		−	1.50949i
$269$	2.51521e7			1.29216			0.646082	−	0.763268i	$-0.276407\pi$
$269$	2.51521e7			1.29216			0.646082	+	0.763268i	$0.276407\pi$
$270$			6.70822e6i			0.340813i
$271$	2.44564e7			1.22881			0.614405	−	0.788991i	$-0.289396\pi$
$271$	2.44564e7			1.22881			0.614405	+	0.788991i	$0.289396\pi$
$272$			3.15774e7i			1.56917i
$273$			1.34608e7i			0.661582i
$274$		−	4.20112e7i		−	2.04227i
$275$			1.73093e7i			0.832302i
$276$	−6.63472e6	+	8.00062e6i	−0.315570	+	0.380537i
$277$	3.06729e7			1.44316			0.721581	−	0.692330i	$-0.243416\pi$
$277$	3.06729e7			1.44316			0.721581	+	0.692330i	$0.243416\pi$
$278$	−4.42242e7			−2.05838
$279$	6.58290e6			0.303113
$280$	−3.58059e6			−0.163110
$281$			1.12468e7i			0.506885i	0.967350	+	0.253443i	$0.0815629\pi$
$281$			1.12468e7i			0.506885i	−0.967350	+	0.253443i	$0.918437\pi$
$282$	1.79690e7			0.801264
$283$		−	9.50856e6i		−	0.419523i	−0.977753	−	0.209761i	$-0.932731\pi$
$283$		−	9.50856e6i		−	0.419523i	0.977753	−	0.209761i	$-0.0672687\pi$
$284$	2.51454e7			1.09775
$285$	−3.73551e6			−0.161367
$286$			6.88040e7i			2.94114i
$287$		−	1.17279e7i		−	0.496105i
$288$	−1.06245e7			−0.444765
$289$	−2.29829e7			−0.952164
$290$		−	3.28604e7i		−	1.34735i
$291$			2.17878e7i			0.884166i
$292$	1.54300e7			0.619752
$293$		−	1.40459e7i		−	0.558400i	−0.960233	−	0.279200i	$-0.909931\pi$
$293$		−	1.40459e7i		−	0.558400i	0.960233	−	0.279200i	$-0.0900693\pi$
$294$	−1.17822e7			−0.463644
$295$		−	2.57948e7i		−	1.00477i
$296$		−	7.45030e6i		−	0.287275i
$297$			6.08607e6i			0.232310i
$298$			4.76515e7i			1.80065i
$299$	−3.05149e7	+	3.67970e7i	−1.14156	+	1.37657i
$300$	9.20321e6			0.340859
$301$	−1.50984e7			−0.553646
$302$	−1.38906e7			−0.504313
$303$	1.85018e7			0.665100
$304$		−	6.78468e6i		−	0.241495i
$305$	3.64127e7			1.28337
$306$		−	1.81811e7i		−	0.634536i
$307$	2.65536e7			0.917717			0.458859	−	0.888509i	$-0.348258\pi$
$307$	2.65536e7			0.917717			0.458859	+	0.888509i	$0.348258\pi$
$308$	1.93510e7			0.662296
$309$			9.46279e6i			0.320733i
$310$		−	4.79743e7i		−	1.61036i
$311$	−1.35991e7			−0.452093			−0.226047	−	0.974116i	$-0.572580\pi$
$311$	−1.35991e7			−0.452093			−0.226047	+	0.974116i	$0.572580\pi$
$312$	−6.14120e6			−0.202204
$313$			3.51071e7i			1.14489i	0.819945	+	0.572443i	$0.194004\pi$
$313$			3.51071e7i			1.14489i	−0.819945	+	0.572443i	$0.805996\pi$
$314$		−	2.01596e7i		−	0.651168i
$315$	8.67734e6			0.277623
$316$		−	1.38984e7i		−	0.440458i
$317$	−1.10945e7			−0.348282			−0.174141	−	0.984721i	$-0.555715\pi$
$317$	−1.10945e7			−0.348282			−0.174141	+	0.984721i	$0.555715\pi$
$318$			4.37937e7i			1.36185i
$319$		−	2.98128e7i		−	0.918398i
$320$			2.95940e7i			0.903136i
$321$			1.85653e7i			0.561290i
$322$	2.24356e7	+	1.86053e7i	0.672001	+	0.557274i
$323$	1.01243e7			0.300439
$324$	3.23591e6			0.0951397
$325$	4.23280e7			1.23304
$326$	4.25538e7			1.22825
$327$		−	2.38171e7i		−	0.681155i
$328$	5.35058e6			0.151628
$329$		−	2.32436e7i		−	0.652702i
$330$	4.43536e7			1.23420
$331$	−2.25040e7			−0.620549			−0.310274	−	0.950647i	$-0.600421\pi$
$331$	−2.25040e7			−0.620549			−0.310274	+	0.950647i	$0.600421\pi$
$332$		−	8.08821e6i		−	0.221023i
$333$			1.80554e7i			0.488960i
$334$	−1.72143e7			−0.462010
$335$	−8.61466e7			−2.29142
$336$			1.57604e7i			0.415478i
$337$			3.28613e7i			0.858608i	0.903160	+	0.429304i	$0.141241\pi$
$337$			3.28613e7i			0.858608i	−0.903160	+	0.429304i	$0.858759\pi$
$338$	1.15643e8			2.99480
$339$		−	1.64447e7i		−	0.422111i
$340$	−6.11192e7			−1.55504
$341$		−	4.35250e7i		−	1.09768i
$342$			3.90638e6i			0.0976553i
$343$			4.10979e7i			1.01844i
$344$		−	6.88831e6i		−	0.169214i
$345$	2.37207e7	+	1.96710e7i	0.577658	+	0.479037i
$346$	−3.74966e7			−0.905239
$347$	−4.39404e7			−1.05166			−0.525830	−	0.850590i	$-0.676245\pi$
$347$	−4.39404e7			−1.05166			−0.525830	+	0.850590i	$0.676245\pi$
$348$	−1.58512e7			−0.376119
$349$	−6.13364e7			−1.44292			−0.721460	−	0.692456i	$-0.756529\pi$
$349$	−6.13364e7			−1.44292			−0.721460	+	0.692456i	$0.756529\pi$
$350$		−	2.58079e7i		−	0.601934i
$351$	1.48828e7			0.344163
$352$			7.02473e7i			1.61065i
$353$	−2.62144e7			−0.595957			−0.297979	−	0.954573i	$-0.596312\pi$
$353$	−2.62144e7			−0.595957			−0.297979	+	0.954573i	$0.596312\pi$
$354$	−2.69746e7			−0.608059
$355$		−	7.45527e7i		−	1.66640i
$356$			1.42694e7i			0.316269i
$357$	−2.35180e7			−0.516887
$358$	−6.68638e7			−1.45728
$359$		−	5.52959e7i		−	1.19512i	−0.801826	−	0.597558i	$-0.796138\pi$
$359$		−	5.52959e7i		−	1.19512i	0.801826	−	0.597558i	$-0.203862\pi$
$360$			3.95884e6i			0.0848518i
$361$	4.48706e7			0.953762
$362$			3.21036e6i			0.0676750i
$363$	1.26241e7			0.263926
$364$		−	4.73209e7i		−	0.981181i
$365$		−	4.57478e7i		−	0.940788i
$366$		−	3.80782e7i		−	0.776663i
$367$		−	6.39160e7i		−	1.29304i	−0.762898	−	0.646519i	$-0.776224\pi$
$367$		−	6.39160e7i		−	1.29304i	0.762898	−	0.646519i	$-0.223776\pi$
$368$	−3.57278e7	+	4.30831e7i	−0.716906	+	0.864497i
$369$	−1.29668e7			−0.258080
$370$	1.31582e8			2.59772
$371$	5.66489e7			1.10935
$372$	−2.31419e7			−0.449541
$373$			1.00157e7i			0.192998i	0.995333	+	0.0964992i	$0.0307645\pi$
$373$			1.00157e7i			0.192998i	−0.995333	+	0.0964992i	$0.969235\pi$
$374$	−1.20210e8			−2.29788
$375$			1.22880e7i			0.233016i
$376$	1.06044e7			0.199490
$377$	−7.29041e7			−1.36059
$378$		−	9.07425e6i		−	0.168010i
$379$		−	3.47214e7i		−	0.637793i	−0.947790	−	0.318897i	$-0.896688\pi$
$379$		−	3.47214e7i		−	0.637793i	0.947790	−	0.318897i	$-0.103312\pi$
$380$	1.31320e7			0.239321
$381$	3.56849e6			0.0645222
$382$		−	2.44696e7i		−	0.438972i
$383$			7.09421e6i			0.126272i	0.998005	+	0.0631361i	$0.0201102\pi$
$383$			7.09421e6i			0.126272i	−0.998005	+	0.0631361i	$0.979890\pi$
$384$	−1.26723e7			−0.223801
$385$		−	5.73731e7i		−	1.00537i
$386$	7.73506e6			0.134494
$387$			1.66934e7i			0.288013i
$388$		−	7.65939e7i		−	1.31129i
$389$			7.07860e7i			1.20254i	0.799047	+	0.601268i	$0.205338\pi$
$389$			7.07860e7i			1.20254i	−0.799047	+	0.601268i	$0.794662\pi$
$390$		−	1.08462e8i		−	1.82845i
$391$	−6.42897e7	−	5.33138e7i	−1.07550	−	0.891887i
$392$	−6.95325e6			−0.115433
$393$	3.19468e7			0.526321
$394$	−2.71162e7			−0.443343
$395$	−4.12069e7			−0.668619
$396$		−	2.13953e7i		−	0.344535i
$397$	4.69239e7			0.749933			0.374967	−	0.927038i	$-0.377654\pi$
$397$	4.69239e7			0.749933			0.374967	+	0.927038i	$0.377654\pi$
$398$		−	1.61013e7i		−	0.255395i
$399$	5.05305e6			0.0795491
$400$	4.95590e7			0.774359
$401$			3.02762e7i			0.469535i	0.972052	+	0.234767i	$0.0754328\pi$
$401$			3.02762e7i			0.469535i	−0.972052	+	0.234767i	$0.924567\pi$
$402$			9.00870e7i			1.38670i
$403$	−1.06436e8			−1.62619
$404$	−6.50423e7			−0.986397
$405$		−	9.59403e6i		−	0.144423i
$406$			4.44505e7i			0.664199i
$407$	1.19379e8			1.77070
$408$		−	1.07296e7i		−	0.157980i
$409$	−7.70025e7			−1.12547			−0.562736	−	0.826637i	$-0.690251\pi$
$409$	−7.70025e7			−1.12547			−0.562736	+	0.826637i	$0.690251\pi$
$410$			9.44985e7i			1.37111i
$411$			6.00841e7i			0.865434i
$412$		−	3.32660e7i		−	0.475674i
$413$			3.48927e7i			0.495319i
$414$	2.05708e7	−	2.48057e7i	0.289901	−	0.349583i
$415$	−2.39804e7			−0.335515
$416$	1.71782e8			2.38615
$417$	6.32491e7			0.872261
$418$	2.58283e7			0.353644
$419$		−	1.00584e8i		−	1.36738i	−0.729775	−	0.683688i	$-0.760375\pi$
$419$		−	1.00584e8i		−	1.36738i	0.729775	−	0.683688i	$-0.239625\pi$
$420$	−3.05048e7			−0.411737
$421$			1.31330e8i			1.76002i	0.474960	+	0.880008i	$0.342463\pi$
$421$			1.31330e8i			1.76002i	−0.474960	+	0.880008i	$0.657537\pi$
$422$	7.97777e7			1.06156
$423$	−2.56990e7			−0.339544
$424$			2.58448e7i			0.339059i
$425$			7.39531e7i			0.963362i
$426$	−7.79627e7			−1.00846
$427$	−4.92556e7			−0.632663
$428$		−	6.52656e7i		−	0.832440i
$429$		−	9.84028e7i		−	1.24634i
$430$	1.21657e8			1.53014
$431$			1.07013e8i			1.33661i	0.743889	+	0.668303i	$0.232979\pi$
$431$			1.07013e8i			1.33661i	−0.743889	+	0.668303i	$0.767021\pi$
$432$	1.74253e7			0.216137
$433$		−	4.94367e7i		−	0.608956i	−0.952519	−	0.304478i	$-0.901518\pi$
$433$		−	4.94367e7i		−	0.608956i	0.952519	−	0.304478i	$-0.0984820\pi$
$434$			6.48951e7i			0.793858i
$435$			4.69966e7i			0.570951i
$436$			8.37279e7i			1.01021i
$437$	1.38132e7	+	1.14550e7i	0.165520	+	0.137262i
$438$	−4.78403e7			−0.569340
$439$	9.85699e7			1.16507			0.582533	−	0.812807i	$-0.302062\pi$
$439$	9.85699e7			1.16507			0.582533	+	0.812807i	$0.302062\pi$
$440$	2.61752e7			0.307278
$441$	1.68508e7			0.196474
$442$			2.93962e8i			3.40427i
$443$	1.30585e8			1.50204			0.751022	−	0.660277i	$-0.229561\pi$
$443$	1.30585e8			1.50204			0.751022	+	0.660277i	$0.229561\pi$
$444$		−	6.34728e7i		−	0.725168i
$445$	4.23069e7			0.480099
$446$	−7.66685e6			−0.0864196
$447$		−	6.81508e7i		−	0.763042i
$448$		−	4.00319e7i		−	0.445218i
$449$	9.76580e7			1.07887			0.539435	−	0.842027i	$-0.318638\pi$
$449$	9.76580e7			1.07887			0.539435	+	0.842027i	$0.318638\pi$
$450$	−2.85343e7			−0.313133
$451$			8.57344e7i			0.934599i
$452$			5.78105e7i			0.626025i
$453$	1.98662e7			0.213708
$454$			1.03382e8i			1.10478i
$455$	−1.40300e8			−1.48944
$456$			2.30534e6i			0.0243131i
$457$			1.64688e8i			1.72549i	0.505639	+	0.862745i	$0.331257\pi$
$457$			1.64688e8i			1.72549i	−0.505639	+	0.862745i	$0.668743\pi$
$458$			3.39726e6i			0.0353617i
$459$			2.60025e7i			0.268891i
$460$	−8.33891e7	−	6.91525e7i	−0.856714	−	0.710452i
$461$	8.62709e7			0.880565			0.440283	−	0.897859i	$-0.354878\pi$
$461$	8.62709e7			0.880565			0.440283	+	0.897859i	$0.354878\pi$
$462$	−5.99973e7			−0.608424
$463$	7.86303e7			0.792222			0.396111	−	0.918203i	$-0.370360\pi$
$463$	7.86303e7			0.792222			0.396111	+	0.918203i	$0.370360\pi$
$464$	−8.53583e7			−0.854461
$465$			6.86124e7i			0.682407i
$466$	1.70341e8			1.68330
$467$		−	1.69346e8i		−	1.66274i	−0.555721	−	0.831369i	$-0.687558\pi$
$467$		−	1.69346e8i		−	1.66274i	0.555721	−	0.831369i	$-0.312442\pi$
$468$	−5.23200e7			−0.510423
$469$	1.16531e8			1.12960
$470$			1.87287e8i			1.80391i
$471$			2.88321e7i			0.275939i
$472$	−1.59190e7			−0.151388
$473$	1.10374e8			1.04300
$474$			4.30917e7i			0.404631i
$475$		−	1.58895e7i		−	0.148262i
$476$	8.26764e7			0.766586
$477$		−	6.26334e7i		−	0.577100i
$478$	−2.86850e7			−0.262646
$479$		−	1.31802e8i		−	1.19927i	−0.800275	−	0.599633i	$-0.795313\pi$
$479$		−	1.31802e8i		−	1.19927i	0.800275	−	0.599633i	$-0.204687\pi$
$480$		−	1.10737e8i		−	1.00131i
$481$		−	2.91929e8i		−	2.62326i
$482$		−	2.06731e8i		−	1.84614i
$483$	−3.20872e7	−	2.66091e7i	−0.284767	−	0.236150i
$484$	−4.43796e7			−0.391424
$485$	−2.27090e8			−1.99055
$486$	−1.00329e7			−0.0874009
$487$	1.02076e8			0.883761			0.441881	−	0.897074i	$-0.354311\pi$
$487$	1.02076e8			0.883761			0.441881	+	0.897074i	$0.354311\pi$
$488$		−	2.24718e7i		−	0.193365i
$489$	−6.08601e7			−0.520483
$490$		−	1.22804e8i		−	1.04382i
$491$	−1.70391e8			−1.43947			−0.719733	−	0.694251i	$-0.755736\pi$
$491$	−1.70391e8			−1.43947			−0.719733	+	0.694251i	$0.755736\pi$
$492$	4.55843e7			0.382754
$493$		−	1.27374e8i		−	1.06302i
$494$		−	6.31604e7i		−	0.523919i
$495$	−6.34341e7			−0.523006
$496$	−1.24618e8			−1.02126
$497$			1.00848e8i			0.821481i
$498$			2.50773e7i			0.203045i
$499$	−9.77836e7			−0.786981			−0.393491	−	0.919329i	$-0.628733\pi$
$499$	−9.77836e7			−0.786981			−0.393491	+	0.919329i	$0.628733\pi$
$500$		−	4.31977e7i		−	0.345582i
$501$	2.46198e7			0.195781
$502$			7.24068e7i			0.572359i
$503$			1.05813e8i			0.831446i	0.909491	+	0.415723i	$0.136471\pi$
$503$			1.05813e8i			0.831446i	−0.909491	+	0.415723i	$0.863529\pi$
$504$		−	5.35515e6i		−	0.0418293i
$505$			1.92841e8i			1.49736i
$506$	−1.64011e8	−	1.36010e8i	−1.26596	−	1.04983i
$507$	−1.65391e8			−1.26908
$508$	−1.25449e7			−0.0956918
$509$	−2.11018e8			−1.60017			−0.800085	−	0.599886i	$-0.795213\pi$
$509$	−2.11018e8			−1.60017			−0.800085	+	0.599886i	$0.795213\pi$
$510$	1.89498e8			1.42855
$511$			6.18834e7i			0.463779i
$512$	1.71607e8			1.27858
$513$		−	5.58687e6i		−	0.0413824i
$514$	2.58240e8			1.90166
$515$	−9.86290e7			−0.722076
$516$		−	5.86850e7i		−	0.427147i
$517$			1.69918e8i			1.22961i
$518$	−1.77992e8			−1.28060
$519$	5.36273e7			0.383604
$520$		−	6.40087e7i		−	0.455228i
$521$			6.93858e6i			0.0490634i	0.999699	+	0.0245317i	$0.00780947\pi$
$521$			6.93858e6i			0.0490634i	−0.999699	+	0.0245317i	$0.992191\pi$
$522$	4.91463e7			0.345525
$523$			1.77380e8i			1.23994i	0.784626	+	0.619970i	$0.212855\pi$
$523$			1.77380e8i			1.23994i	−0.784626	+	0.619970i	$0.787145\pi$
$524$	−1.12308e8			−0.780576
$525$			3.69102e7i			0.255076i
$526$		−	1.54021e8i		−	1.05833i
$527$		−	1.85958e8i		−	1.27053i
$528$		−	1.15213e8i		−	0.782708i
$529$	−2.73935e7	−	1.45479e8i	−0.185046	−	0.982730i
$530$	−4.56454e8			−3.06598
$531$	3.85788e7			0.257671
$532$	−1.77638e7			−0.117978
$533$	2.09654e8			1.38459
$534$		−	4.42420e7i		−	0.290543i
$535$	−1.93503e8			−1.26365
$536$			5.31647e7i			0.345246i
$537$	9.56280e7			0.617536
$538$	2.74147e8			1.76050
$539$		−	1.11415e8i		−	0.711501i
$540$			3.37274e7i			0.214191i
$541$	−1.01572e8			−0.641477			−0.320739	−	0.947168i	$-0.603931\pi$
$541$	−1.01572e8			−0.641477			−0.320739	+	0.947168i	$0.603931\pi$
$542$	2.66564e8			1.67419
$543$		−	4.59143e6i		−	0.0286780i
$544$			3.00128e8i			1.86428i
$545$	2.48241e8			1.53350
$546$			1.46717e8i			0.901370i
$547$	−1.68367e7			−0.102871			−0.0514356	−	0.998676i	$-0.516380\pi$
$547$	−1.68367e7			−0.102871			−0.0514356	+	0.998676i	$0.516380\pi$
$548$		−	2.11223e8i		−	1.28351i
$549$			5.44590e7i			0.329119i
$550$			1.88664e8i			1.13397i
$551$			2.73674e7i			0.163598i
$552$	1.21398e7	−	1.46390e7i	0.0721763	−	0.0870353i
$553$	5.57409e7			0.329608
$554$	3.34321e8			1.96623
$555$	−1.88188e8			−1.10081
$556$	−2.22349e8			−1.29363
$557$		−	2.43652e8i		−	1.40995i	−0.709232	−	0.704975i	$-0.750958\pi$
$557$		−	2.43652e8i		−	1.40995i	0.709232	−	0.704975i	$-0.249042\pi$
$558$	7.17507e7			0.412975
$559$		−	2.69908e8i		−	1.54519i
$560$	−1.64267e8			−0.935378
$561$	1.71924e8			0.973749
$562$			1.22585e8i			0.690603i
$563$		−	2.01599e8i		−	1.12970i	−0.825193	−	0.564851i	$-0.808934\pi$
$563$		−	2.01599e8i		−	1.12970i	0.825193	−	0.564851i	$-0.191066\pi$
$564$	9.03438e7			0.503571
$565$	1.71400e8			0.950311
$566$		−	1.03639e8i		−	0.571577i
$567$			1.29779e7i			0.0711960i
$568$	−4.60096e7			−0.251075
$569$			3.56375e8i			1.93451i	0.253810	+	0.967254i	$0.418316\pi$
$569$			3.56375e8i			1.93451i	−0.253810	+	0.967254i	$0.581684\pi$
$570$	−4.07155e7			−0.219854
$571$		−	1.34654e8i		−	0.723290i	−0.932316	−	0.361645i	$-0.882215\pi$
$571$		−	1.34654e8i		−	0.723290i	0.932316	−	0.361645i	$-0.117785\pi$
$572$			3.45930e8i			1.84842i
$573$			3.49962e7i			0.186019i
$574$		−	1.27829e8i		−	0.675916i
$575$	−8.36732e7	+	1.00899e8i	−0.440132	+	0.530743i
$576$	−4.42610e7			−0.231608
$577$	2.37855e8			1.23818			0.619092	−	0.785319i	$-0.287501\pi$
$577$	2.37855e8			1.23818			0.619092	+	0.785319i	$0.287501\pi$
$578$	−2.50504e8			−1.29727
$579$	−1.10626e7			−0.0569930
$580$		−	1.65215e8i		−	0.846768i
$581$	3.24384e7			0.165398
$582$			2.37477e8i			1.20463i
$583$	−4.14121e8			−2.08988
$584$	−2.82329e7			−0.141748
$585$			1.55121e8i			0.774825i
$586$		−	1.53094e8i		−	0.760790i
$587$	−1.33484e8			−0.659954			−0.329977	−	0.943989i	$-0.607041\pi$
$587$	−1.33484e8			−0.659954			−0.329977	+	0.943989i	$0.607041\pi$
$588$	−5.92382e7			−0.291387
$589$			3.99549e7i			0.195535i
$590$		−	2.81152e8i		−	1.36894i
$591$	3.87813e7			0.187871
$592$		−	3.41799e8i		−	1.64743i
$593$	−3.38405e8			−1.62283			−0.811414	−	0.584472i	$-0.801302\pi$
$593$	−3.38405e8			−1.62283			−0.811414	+	0.584472i	$0.801302\pi$
$594$			6.63355e7i			0.316510i
$595$		−	2.45124e8i		−	1.16368i
$596$			2.39581e8i			1.13165i
$597$			2.30279e7i			0.108226i
$598$	−3.32599e8	+	4.01072e8i	−1.55531	+	1.87551i
$599$	3.60443e8			1.67709			0.838544	−	0.544834i	$-0.183407\pi$
$599$	3.60443e8			1.67709			0.838544	+	0.544834i	$0.183407\pi$
$600$	−1.68395e7			−0.0779605
$601$	3.11612e7			0.143546			0.0717729	−	0.997421i	$-0.477134\pi$
$601$	3.11612e7			0.143546			0.0717729	+	0.997421i	$0.477134\pi$
$602$	−1.64566e8			−0.754312
$603$		−	1.28842e8i		−	0.587630i
$604$	−6.98387e7			−0.316946
$605$			1.31579e8i			0.594184i
$606$	2.01662e8			0.906162
$607$	3.01125e8			1.34642			0.673210	−	0.739451i	$-0.264915\pi$
$607$	3.01125e8			1.34642			0.673210	+	0.739451i	$0.264915\pi$
$608$		−	6.44853e7i		−	0.286913i
$609$		−	6.35727e7i		−	0.281461i
$610$	3.96882e8			1.74853
$611$	4.15516e8			1.82164
$612$		−	9.14104e7i		−	0.398787i
$613$		−	6.71266e7i		−	0.291416i	−0.989328	−	0.145708i	$-0.953454\pi$
$613$		−	6.71266e7i		−	0.291416i	0.989328	−	0.145708i	$-0.0465460\pi$
$614$	2.89423e8			1.25034
$615$		−	1.35151e8i		−	0.581024i
$616$	−3.54073e7			−0.151479
$617$			2.93868e8i			1.25111i	0.780178	+	0.625557i	$0.215128\pi$
$617$			2.93868e8i			1.25111i	−0.780178	+	0.625557i	$0.784872\pi$
$618$			1.03140e8i			0.436981i
$619$			3.23663e8i			1.36465i	0.731049	+	0.682325i	$0.239031\pi$
$619$			3.23663e8i			1.36465i	−0.731049	+	0.682325i	$0.760969\pi$
$620$		−	2.41204e8i		−	1.01207i
$621$	−2.94201e7	+	3.54769e7i	−0.122848	+	0.148139i
$622$	−1.48224e8			−0.615952
$623$	−5.72287e7			−0.236674
$624$	−2.81741e8			−1.15957
$625$	−2.96409e8			−1.21409
$626$			3.82652e8i			1.55984i
$627$	−3.69394e7			−0.149860
$628$		−	1.01358e8i		−	0.409241i
$629$	5.10041e8			2.04952
$630$	9.45793e7			0.378246
$631$			3.09659e8i			1.23253i	0.787541	+	0.616263i	$0.211354\pi$
$631$			3.09659e8i			1.23253i	−0.787541	+	0.616263i	$0.788646\pi$
$632$			2.54305e7i			0.100740i
$633$	−1.14097e8			−0.449847
$634$	−1.20926e8			−0.474516
$635$			3.71937e7i			0.145261i
$636$			2.20185e8i			0.855886i
$637$	−2.72453e8			−1.05408
$638$		−	3.24947e8i		−	1.25127i
$639$	1.11502e8			0.427345
$640$		−	1.32082e8i		−	0.503851i
$641$		−	1.43431e8i		−	0.544590i	−0.962214	−	0.272295i	$-0.912217\pi$
$641$		−	1.43431e8i		−	0.544590i	0.962214	−	0.272295i	$-0.0877826\pi$
$642$			2.02354e8i			0.764728i
$643$		−	3.10636e7i		−	0.116847i	−0.998292	−	0.0584237i	$-0.981393\pi$
$643$		−	3.10636e7i		−	0.116847i	0.998292	−	0.0584237i	$-0.0186074\pi$
$644$	1.12801e8	+	9.35431e7i	0.422333	+	0.350230i
$645$	−1.73993e8			−0.648413
$646$	1.10350e8			0.409332
$647$	−5.28652e8			−1.95190			−0.975949	−	0.218000i	$-0.930047\pi$
$647$	−5.28652e8			−1.95190			−0.975949	+	0.218000i	$0.930047\pi$
$648$	−5.92088e6			−0.0217601
$649$		−	2.55077e8i		−	0.933118i
$650$	4.61357e8			1.67995
$651$		−	9.28124e7i		−	0.336406i
$652$	2.13951e8			0.771918
$653$	2.78700e8			1.00091			0.500457	−	0.865761i	$-0.333165\pi$
$653$	2.78700e8			1.00091			0.500457	+	0.865761i	$0.333165\pi$
$654$		−	2.59596e8i		−	0.928036i
$655$			3.32976e8i			1.18492i
$656$	2.45470e8			0.869534
$657$	6.84208e7			0.241264
$658$		−	2.53345e8i		−	0.889271i
$659$			1.58187e8i			0.552733i	0.961052	+	0.276367i	$0.0891304\pi$
$659$			1.58187e8i			0.552733i	−0.961052	+	0.276367i	$0.910870\pi$
$660$	2.22999e8			0.775661
$661$		−	3.68083e7i		−	0.127450i	−0.997967	−	0.0637252i	$-0.979702\pi$
$661$		−	3.68083e7i		−	0.127450i	0.997967	−	0.0637252i	$-0.0202981\pi$
$662$	−2.45284e8			−0.845464
$663$		−	4.20421e8i		−	1.44259i
$664$			1.47993e7i			0.0505518i
$665$			5.26671e7i			0.179091i
$666$			1.96796e8i			0.666182i
$667$	1.44115e8	−	1.73785e8i	0.485660	−	0.585644i
$668$	−8.65498e7			−0.290360
$669$	1.09651e7			0.0366212
$670$	−9.38961e8			−3.12193
$671$	3.60074e8			1.19186
$672$			1.49795e8i			0.493616i
$673$	3.37285e8			1.10650			0.553250	−	0.833015i	$-0.313387\pi$
$673$	3.37285e8			1.10650			0.553250	+	0.833015i	$0.313387\pi$
$674$			3.58174e8i			1.16981i
$675$	4.08095e7			0.132693
$676$	5.81425e8			1.88215
$677$		−	3.81974e8i		−	1.23103i	−0.788127	−	0.615513i	$-0.788949\pi$
$677$		−	3.81974e8i		−	1.23103i	0.788127	−	0.615513i	$-0.211051\pi$
$678$		−	1.79240e8i		−	0.575103i
$679$	3.07186e8			0.981278
$680$	1.11832e8			0.355664
$681$		−	1.47856e8i		−	0.468163i
$682$		−	4.74403e8i		−	1.49553i
$683$	−4.48113e8			−1.40645			−0.703226	−	0.710966i	$-0.748258\pi$
$683$	−4.48113e8			−1.40645			−0.703226	+	0.710966i	$0.748258\pi$
$684$			1.96404e7i			0.0613736i
$685$	−6.26246e8			−1.94838
$686$			4.47949e8i			1.38757i
$687$		−	4.85874e6i		−	0.0149849i
$688$		−	3.16017e8i		−	0.970387i
$689$			1.01269e9i			3.09613i
$690$	2.58545e8	+	2.14405e8i	0.787027	+	0.652662i
$691$	−7.90521e7			−0.239596			−0.119798	−	0.992798i	$-0.538225\pi$
$691$	−7.90521e7			−0.239596			−0.119798	+	0.992798i	$0.538225\pi$
$692$	−1.88524e8			−0.568917
$693$	8.58076e7			0.257826
$694$	−4.78931e8			−1.43283
$695$			6.59234e8i			1.96375i
$696$	2.90036e7			0.0860249
$697$			3.66296e8i			1.08177i
$698$	−6.68540e8			−1.96590
$699$	−2.43620e8			−0.713316
$700$		−	1.29756e8i		−	0.378298i
$701$			5.04427e8i			1.46435i	0.681118	+	0.732174i	$0.261494\pi$
$701$			5.04427e8i			1.46435i	−0.681118	+	0.732174i	$0.738506\pi$
$702$	1.62217e8			0.468904
$703$	−1.09587e8			−0.315423
$704$			2.92646e8i			0.838734i
$705$		−	2.67857e8i		−	0.764425i
$706$	−2.85725e8			−0.811959
$707$		−	2.60858e8i		−	0.738151i
$708$	−1.35622e8			−0.382147
$709$			2.86560e8i			0.804038i	0.915631	+	0.402019i	$0.131691\pi$
$709$			2.86560e8i			0.804038i	−0.915631	+	0.402019i	$0.868309\pi$
$710$		−	8.12592e8i		−	2.27037i
$711$		−	6.16294e7i		−	0.171466i
$712$		−	2.61093e7i		−	0.0723362i
$713$	2.10400e8	−	2.53716e8i	0.580467	−	0.699968i
$714$	−2.56336e8			−0.704230
$715$	1.02564e9			2.80592
$716$	−3.36176e8			−0.915857
$717$	4.10251e7			0.111299
$718$		−	6.02702e8i		−	1.62828i
$719$	6.86464e8			1.84685			0.923423	−	0.383783i	$-0.125379\pi$
$719$	6.86464e8			1.84685			0.923423	+	0.383783i	$0.125379\pi$
$720$			1.81621e8i			0.486596i
$721$	1.33416e8			0.355961
$722$	4.89070e8			1.29945
$723$			2.95665e8i			0.782322i
$724$			1.61410e7i			0.0425318i
$725$	−1.99906e8			−0.524581
$726$	1.37598e8			0.359585
$727$			1.53301e8i			0.398972i	0.979901	+	0.199486i	$0.0639272\pi$
$727$			1.53301e8i			0.398972i	−0.979901	+	0.199486i	$0.936073\pi$
$728$			8.65849e7i			0.224413i
$729$	1.43489e7			0.0370370
$730$		−	4.98631e8i		−	1.28177i
$731$	4.71568e8			1.20724
$732$		−	1.91448e8i		−	0.488111i
$733$		−	2.31274e8i		−	0.587240i	−0.955922	−	0.293620i	$-0.905140\pi$
$733$		−	2.31274e8i		−	0.587240i	0.955922	−	0.293620i	$-0.0948601\pi$
$734$		−	6.96657e8i		−	1.76169i
$735$			1.75633e8i			0.442328i
$736$	−3.39576e8	+	4.09485e8i	−0.851733	+	1.02708i
$737$	−8.51878e8			−2.12802
$738$	−1.41333e8			−0.351620
$739$	−2.62854e8			−0.651299			−0.325650	−	0.945491i	$-0.605583\pi$
$739$	−2.62854e8			−0.651299			−0.325650	+	0.945491i	$0.605583\pi$
$740$	6.61566e8			1.63259
$741$			9.03314e7i			0.222016i
$742$	6.17449e8			1.51143
$743$		−	3.46168e8i		−	0.843958i	−0.906606	−	0.421979i	$-0.861336\pi$
$743$		−	3.46168e8i		−	0.843958i	0.906606	−	0.421979i	$-0.138664\pi$
$744$	4.23436e7			0.102818
$745$	7.10324e8			1.71786
$746$			1.09166e8i			0.262950i
$747$		−	3.58653e7i		−	0.0860423i
$748$	−6.04390e8			−1.44415
$749$	2.61753e8			0.622940
$750$			1.33933e8i			0.317472i
$751$		−	2.30256e8i		−	0.543615i	−0.962352	−	0.271808i	$-0.912379\pi$
$751$		−	2.30256e8i		−	0.543615i	0.962352	−	0.271808i	$-0.0876215\pi$
$752$	4.86498e8			1.14400
$753$		−	1.03556e8i		−	0.242543i
$754$	−7.94623e8			−1.85373
$755$			2.07062e8i			0.481127i
$756$		−	4.56232e7i		−	0.105589i
$757$			4.07649e8i			0.939721i	0.882741	+	0.469860i	$0.155696\pi$
$757$			4.07649e8i			0.939721i	−0.882741	+	0.469860i	$0.844304\pi$
$758$		−	3.78448e8i		−	0.868958i
$759$	2.34567e8	+	1.94521e8i	0.536465	+	0.444878i
$760$	−2.40282e7			−0.0547369
$761$	−1.04973e8			−0.238190			−0.119095	−	0.992883i	$-0.537999\pi$
$761$	−1.04973e8			−0.238190			−0.119095	+	0.992883i	$0.537999\pi$
$762$	3.88950e7			0.0879080
$763$	−3.35798e8			−0.755969
$764$		−	1.23027e8i		−	0.275881i
$765$	−2.71019e8			−0.605362
$766$			7.73238e7i			0.172039i
$767$	−6.23763e8			−1.38240
$768$	−3.19841e8			−0.706074
$769$			5.72554e8i			1.25903i	0.776986	+	0.629517i	$0.216747\pi$
$769$			5.72554e8i			1.25903i	−0.776986	+	0.629517i	$0.783253\pi$
$770$		−	6.25342e8i		−	1.36976i
$771$	−3.69332e8			−0.805849
$772$	3.88901e7			0.0845253
$773$			1.70666e8i			0.369496i	0.982786	+	0.184748i	$0.0591469\pi$
$773$			1.70666e8i			0.369496i	−0.982786	+	0.184748i	$0.940853\pi$
$774$			1.81951e8i			0.392403i
$775$	−2.91852e8			−0.626985
$776$			1.40147e8i			0.299915i
$777$	2.54563e8			0.542666
$778$			7.71536e8i			1.63839i
$779$		−	7.87021e7i		−	0.166485i
$780$		−	5.45322e8i		−	1.14913i
$781$		−	7.37229e8i		−	1.54757i
$782$	−7.00729e8	−	5.81098e8i	−1.46531	−	1.21515i
$783$	−7.02886e7			−0.146420
$784$	−3.18996e8			−0.661968
$785$	−3.00512e8			−0.621230
$786$	3.48207e8			0.717083
$787$		−	1.62065e8i		−	0.332480i	−0.986085	−	0.166240i	$-0.946837\pi$
$787$		−	1.62065e8i		−	0.332480i	0.986085	−	0.166240i	$-0.0531626\pi$
$788$	−1.36334e8			−0.278628
$789$			2.20279e8i			0.448479i
$790$	−4.49137e8			−0.910957
$791$	−2.31854e8			−0.468473
$792$			3.91478e7i			0.0788010i
$793$		−	8.80522e8i		−	1.76572i
$794$	5.11450e8			1.02174
$795$	6.52817e8			1.29924
$796$		−	8.09537e7i		−	0.160508i
$797$		−	3.21457e8i		−	0.634963i	−0.948264	−	0.317482i	$-0.897163\pi$
$797$		−	3.21457e8i		−	0.634963i	0.948264	−	0.317482i	$-0.102837\pi$
$798$	5.50761e7			0.108381
$799$			7.25965e8i			1.42323i
$800$	4.71035e8			0.919990
$801$			6.32745e7i			0.123121i
$802$			3.29997e8i			0.639715i
$803$		−	4.52386e8i		−	0.873701i
$804$			4.52936e8i			0.871504i
$805$	2.77342e8	−	3.34439e8i	0.531653	−	0.641105i
$806$	−1.16010e9			−2.21560
$807$	−3.92082e8			−0.746031
$808$	1.19010e8			0.225606
$809$	−4.47863e8			−0.845862			−0.422931	−	0.906162i	$-0.638999\pi$
$809$	−4.47863e8			−0.845862			−0.422931	+	0.906162i	$0.638999\pi$
$810$		−	1.04571e8i		−	0.196768i
$811$	9.94216e8			1.86388			0.931941	−	0.362610i	$-0.118114\pi$
$811$	9.94216e8			1.86388			0.931941	+	0.362610i	$0.118114\pi$
$812$			2.23487e8i			0.417430i
$813$	−3.81238e8			−0.709454
$814$	1.30118e9			2.41248
$815$		−	6.34334e8i		−	1.17178i
$816$		−	4.92242e8i		−	0.905959i
$817$	−1.01321e8			−0.185794
$818$	−8.39294e8			−1.53339
$819$		−	2.09834e8i		−	0.381965i
$820$			4.75117e8i			0.861706i
$821$	−1.70550e8			−0.308192			−0.154096	−	0.988056i	$-0.549247\pi$
$821$	−1.70550e8			−0.308192			−0.154096	+	0.988056i	$0.549247\pi$
$822$			6.54890e8i			1.17911i
$823$	−5.45105e8			−0.977869			−0.488934	−	0.872321i	$-0.662614\pi$
$823$	−5.45105e8			−0.977869			−0.488934	+	0.872321i	$0.662614\pi$
$824$			6.08681e7i			0.108795i
$825$		−	2.69825e8i		−	0.480530i
$826$			3.80316e8i			0.674845i
$827$		−	5.63227e8i		−	0.995788i	−0.867238	−	0.497894i	$-0.834107\pi$
$827$		−	5.63227e8i		−	0.995788i	0.867238	−	0.497894i	$-0.165893\pi$
$828$	1.03425e8	−	1.24717e8i	0.182194	−	0.219703i
$829$	4.00995e8			0.703843			0.351921	−	0.936030i	$-0.385528\pi$
$829$	4.00995e8			0.703843			0.351921	+	0.936030i	$0.385528\pi$
$830$	−2.61376e8			−0.457121
$831$	−4.78143e8			−0.833210
$832$	7.15634e8			1.24257
$833$		−	4.76014e8i		−	0.823539i
$834$	6.89388e8			1.18841
$835$			2.56608e8i			0.440768i
$836$	1.29859e8			0.222255
$837$	−1.02617e8			−0.175002
$838$		−	1.09632e9i		−	1.86297i
$839$		−	2.15528e8i		−	0.364937i	−0.983212	−	0.182468i	$-0.941591\pi$
$839$		−	2.15528e8i		−	0.364937i	0.983212	−	0.182468i	$-0.0584087\pi$
$840$	5.58158e7			0.0941715
$841$	−2.50513e8			−0.421155
$842$			1.43144e9i			2.39792i
$843$		−	1.75320e8i		−	0.292650i
$844$	4.01104e8			0.667160
$845$		−	1.72384e9i		−	2.85711i
$846$	−2.80108e8			−0.462610
$847$		−	1.77988e8i		−	0.292914i
$848$			1.18569e9i			1.94439i
$849$			1.48224e8i			0.242212i
$850$			8.06057e8i			1.31253i
$851$	6.95883e8	+	5.77079e8i	1.12914	+	0.936368i
$852$	−3.91979e8			−0.633787
$853$	−1.76071e8			−0.283688			−0.141844	−	0.989889i	$-0.545303\pi$
$853$	−1.76071e8			−0.283688			−0.141844	+	0.989889i	$0.545303\pi$
$854$	−5.36865e8			−0.861968
$855$	5.82309e7			0.0931655
$856$			1.19419e8i			0.190393i
$857$	3.30301e7			0.0524769			0.0262384	−	0.999656i	$-0.491647\pi$
$857$	3.30301e7			0.0524769			0.0262384	+	0.999656i	$0.491647\pi$
$858$		−	1.07255e9i		−	1.69807i
$859$	−1.49960e8			−0.236589			−0.118295	−	0.992979i	$-0.537743\pi$
$859$	−1.49960e8			−0.236589			−0.118295	+	0.992979i	$0.537743\pi$
$860$	6.11663e8			0.961650
$861$			1.82820e8i			0.286427i
$862$			1.16639e9i			1.82105i
$863$	−8.36556e7			−0.130156			−0.0650778	−	0.997880i	$-0.520730\pi$
$863$	−8.36556e7			−0.130156			−0.0650778	+	0.997880i	$0.520730\pi$
$864$	1.65619e8			0.256785
$865$			5.58948e8i			0.863620i
$866$		−	5.38839e8i		−	0.829669i
$867$	3.58269e8			0.549732
$868$			3.26278e8i			0.498917i
$869$	−4.07483e8			−0.620940
$870$			5.12243e8i			0.777890i
$871$			2.08318e9i			3.15262i
$872$		−	1.53200e8i		−	0.231052i
$873$		−	3.39638e8i		−	0.510473i
$874$	1.50558e8	+	1.24854e8i	0.225512	+	0.187012i
$875$	1.73248e8			0.258609
$876$	−2.40530e8			−0.357814
$877$	6.75754e8			1.00182			0.500910	−	0.865499i	$-0.332999\pi$
$877$	6.75754e8			1.00182			0.500910	+	0.865499i	$0.332999\pi$
$878$	1.07437e9			1.58734
$879$			2.18953e8i			0.322393i
$880$	1.20085e9			1.76213
$881$		−	4.95490e8i		−	0.724614i	−0.932059	−	0.362307i	$-0.881989\pi$
$881$		−	4.95490e8i		−	0.724614i	0.932059	−	0.362307i	$-0.118011\pi$
$882$	1.83666e8			0.267685
$883$	6.44084e8			0.935536			0.467768	−	0.883851i	$-0.345058\pi$
$883$	6.44084e8			0.935536			0.467768	+	0.883851i	$0.345058\pi$
$884$			1.47797e9i			2.13949i
$885$			4.02101e8i			0.580103i
$886$	1.42332e9			2.04645
$887$	−8.93336e8			−1.28010			−0.640050	−	0.768333i	$-0.721086\pi$
$887$	−8.93336e8			−1.28010			−0.640050	+	0.768333i	$0.721086\pi$
$888$			1.16139e8i			0.165859i
$889$		−	5.03122e7i		−	0.0716091i
$890$	4.61126e8			0.654109
$891$		−	9.48725e7i		−	0.134124i
$892$	−3.85472e7			−0.0543123
$893$		−	1.55980e8i		−	0.219036i
$894$		−	7.42814e8i		−	1.03960i
$895$			9.96714e8i			1.39028i
$896$			1.78667e8i			0.248383i
$897$	4.75680e8	−	5.73609e8i	0.659079	−	0.794765i
$898$	1.06443e9			1.46990
$899$	5.02674e8			0.691842
$900$	−1.43464e8			−0.196795
$901$	−1.76931e9			−2.41897
$902$			9.34468e8i			1.27334i
$903$	2.35361e8			0.319647
$904$		−	1.05778e8i		−	0.143183i
$905$	4.78556e7			0.0645636
$906$	2.16533e8			0.291165
$907$		−	1.50772e8i		−	0.202068i	−0.994883	−	0.101034i	$-0.967785\pi$
$907$		−	1.50772e8i		−	0.202068i	0.994883	−	0.101034i	$-0.0322151\pi$
$908$			5.19780e8i			0.694324i
$909$	−2.88415e8			−0.383995
$910$	−1.52921e9			−2.02928
$911$		−	1.18502e9i		−	1.56737i	−0.621159	−	0.783685i	$-0.713338\pi$
$911$		−	1.18502e9i		−	1.56737i	0.621159	−	0.783685i	$-0.286662\pi$
$912$			1.05763e8i			0.139427i
$913$	−2.37135e8			−0.311590
$914$			1.79502e9i			2.35089i
$915$	−5.67617e8			−0.740956
$916$			1.70807e7i			0.0222238i
$917$		−	4.50419e8i		−	0.584129i
$918$			2.83415e8i			0.366349i
$919$		−	7.20023e8i		−	0.927684i	−0.885918	−	0.463842i	$-0.846471\pi$
$919$		−	7.20023e8i		−	0.927684i	0.885918	−	0.463842i	$-0.153529\pi$
$920$	1.52580e8	+	1.26531e8i	0.195945	+	0.162493i
$921$	−4.13930e8			−0.529844
$922$	9.40315e8			1.19972
$923$	−1.80282e9			−2.29270
$924$	−3.01653e8			−0.382377
$925$		−	8.00482e8i		−	1.01141i
$926$	8.57035e8			1.07936
$927$		−	1.47510e8i		−	0.185175i
$928$	−8.11292e8			−1.01516
$929$	−8.30705e7			−0.103609			−0.0518047	−	0.998657i	$-0.516497\pi$
$929$	−8.30705e7			−0.103609			−0.0518047	+	0.998657i	$0.516497\pi$
$930$			7.47845e8i			0.929743i
$931$			1.02276e8i			0.126743i
$932$	8.56435e8			1.05791
$933$	2.11989e8			0.261016
$934$		−	1.84580e9i		−	2.26539i
$935$			1.79193e9i			2.19223i
$936$	9.57319e7			0.116742
$937$		−	1.17181e8i		−	0.142442i	−0.997461	−	0.0712209i	$-0.977310\pi$
$937$		−	1.17181e8i		−	0.142442i	0.997461	−	0.0712209i	$-0.0226895\pi$
$938$	1.27014e9			1.53901
$939$		−	5.47266e8i		−	0.661000i
$940$			9.41637e8i			1.13370i
$941$			1.60300e8i			0.192382i	0.995363	+	0.0961908i	$0.0306659\pi$
$941$			1.60300e8i			0.192382i	−0.995363	+	0.0961908i	$0.969334\pi$
$942$			3.14257e8i			0.375952i
$943$	−4.14441e8	+	4.99762e8i	−0.494228	+	0.595976i
$944$	−7.30321e8			−0.868156
$945$	−1.35266e8			−0.160286
$946$	1.20303e9			1.42103
$947$	2.05644e7			0.0242139			0.0121070	−	0.999927i	$-0.496146\pi$
$947$	2.05644e7			0.0242139			0.0121070	+	0.999927i	$0.496146\pi$
$948$			2.16655e8i			0.254299i
$949$	−1.10626e9			−1.29437
$950$		−	1.73189e8i		−	0.201999i
$951$	1.72947e8			0.201081
$952$	−1.51276e8			−0.175331
$953$		−	3.96103e8i		−	0.457645i	−0.973468	−	0.228823i	$-0.926512\pi$
$953$		−	3.96103e8i		−	0.457645i	0.973468	−	0.228823i	$-0.0734876\pi$
$954$		−	6.82677e8i		−	0.786267i
$955$	−3.64759e8			−0.418789
$956$	−1.44222e8			−0.165066
$957$			4.64736e8i			0.530237i
$958$		−	1.43658e9i		−	1.63393i
$959$	8.47127e8			0.960489
$960$		−	4.61324e8i		−	0.521426i
$961$	−1.53629e8			−0.173102
$962$		−	3.18190e9i		−	3.57405i
$963$		−	2.89405e8i		−	0.324061i
$964$		−	1.03940e9i		−	1.16025i
$965$		−	1.15304e8i		−	0.128310i
$966$	−3.49736e8	−	2.90028e8i	−0.387980	−	0.321742i
$967$	1.37417e9			1.51971			0.759855	−	0.650092i	$-0.225270\pi$
$967$	1.37417e9			1.51971			0.759855	+	0.650092i	$0.225270\pi$
$968$	8.12030e7			0.0895254
$969$	−1.57822e8			−0.173458
$970$	−2.47518e9			−2.71201
$971$			8.46449e7i			0.0924577i	0.998931	+	0.0462289i	$0.0147203\pi$
$971$			8.46449e7i			0.0924577i	−0.998931	+	0.0462289i	$0.985280\pi$
$972$	−5.04429e7			−0.0549290
$973$		−	8.91750e8i		−	0.968066i
$974$	1.11258e9			1.20408
$975$	−6.59829e8			−0.711897
$976$		−	1.03094e9i		−	1.10888i
$977$		−	7.26228e8i		−	0.778734i	−0.921083	−	0.389367i	$-0.872694\pi$
$977$		−	7.26228e8i		−	0.778734i	0.921083	−	0.389367i	$-0.127306\pi$
$978$	−6.63349e8			−0.709129
$979$	4.18360e8			0.445863
$980$		−	6.17430e8i		−	0.656008i
$981$			3.71272e8i			0.393265i
$982$	−1.85718e9			−1.96119
$983$			1.23399e9i			1.29913i	0.760306	+	0.649565i	$0.225049\pi$
$983$			1.23399e9i			1.29913i	−0.760306	+	0.649565i	$0.774951\pi$
$984$	−8.34073e7			−0.0875425
$985$			4.04211e8i			0.422960i
$986$		−	1.38832e9i		−	1.44830i
$987$			3.62331e8i			0.376838i
$988$		−	3.17556e8i		−	0.329268i
$989$	6.43392e8	+	5.33549e8i	0.665099	+	0.551551i
$990$	−6.91404e8			−0.712567
$991$	1.54592e9			1.58842			0.794211	−	0.607642i	$-0.207884\pi$
$991$	1.54592e9			1.58842			0.794211	+	0.607642i	$0.207884\pi$
$992$	−1.18444e9			−1.21333
$993$	3.50803e8			0.358274
$994$			1.09920e9i			1.11922i
$995$	−2.40016e8			−0.243653
$996$			1.26083e8i			0.127608i
$997$	−1.06995e9			−1.07964			−0.539818	−	0.841782i	$-0.681507\pi$
$997$	−1.06995e9			−1.07964			−0.539818	+	0.841782i	$0.681507\pi$
$998$	−1.06580e9			−1.07222
$999$		−	2.81455e8i		−	0.282301i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.19	✓	24
3.2	odd	2		207.7.d.e.91.6		24
23.22	odd	2	inner	69.7.d.a.22.20	yes	24
69.68	even	2		207.7.d.e.91.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.19	✓	24	1.1	even	1	trivial
69.7.d.a.22.20	yes	24	23.22	odd	2	inner
207.7.d.e.91.5		24	69.68	even	2
207.7.d.e.91.6		24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+10.8996 q^{2} -15.5885 q^{3} +54.8005 q^{4} -162.476i q^{5} -169.907 q^{6} +219.782i q^{7} -100.271 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q+10.8996 q^{2} -15.5885 q^{3} +54.8005 q^{4} -162.476i q^{5} -169.907 q^{6} +219.782i q^{7} -100.271 q^{8} +243.000 q^{9} -1770.91i q^{10} -1606.67i q^{11} -854.255 q^{12} -3928.95 q^{13} +2395.53i q^{14} +2532.75i q^{15} -4600.14 q^{16} -6864.44i q^{17} +2648.59 q^{18} +1474.89i q^{19} -8903.75i q^{20} -3426.06i q^{21} -17512.0i q^{22} +(7766.67 - 9365.61i) q^{23} +1563.06 q^{24} -10773.4 q^{25} -42823.9 q^{26} -3788.00 q^{27} +12044.2i q^{28} +18555.6 q^{29} +27605.8i q^{30} +27090.1 q^{31} -43722.2 q^{32} +25045.6i q^{33} -74819.4i q^{34} +35709.2 q^{35} +13316.5 q^{36} +74301.9i q^{37} +16075.6i q^{38} +61246.3 q^{39} +16291.5i q^{40} -53361.4 q^{41} -37342.6i q^{42} +68697.2i q^{43} -88046.5i q^{44} -39481.6i q^{45} +(84653.4 - 102081. i) q^{46} -105757. q^{47} +71709.0 q^{48} +69344.9 q^{49} -117425. q^{50} +107006. i q^{51} -215308. q^{52} -257751. i q^{53} -41287.5 q^{54} -261045. q^{55} -22037.7i q^{56} -22991.2i q^{57} +202248. q^{58} +158761. q^{59} +138796. i q^{60} +224111. i q^{61} +295270. q^{62} +53407.0i q^{63} -182144. q^{64} +638359. i q^{65} +272986. i q^{66} -530212. i q^{67} -376175. i q^{68} +(-121070. + 145995. i) q^{69} +389215. q^{70} +458854. q^{71} -24365.8 q^{72} +281567. q^{73} +809859. i q^{74} +167940. q^{75} +80824.5i q^{76} +353118. q^{77} +667558. q^{78} -253619. i q^{79} +747411. i q^{80} +59049.0 q^{81} -581616. q^{82} -147594. i q^{83} -187750. i q^{84} -1.11530e6 q^{85} +748770. i q^{86} -289253. q^{87} +161102. i q^{88} +260389. i q^{89} -430332. i q^{90} -863513. i q^{91} +(425618. - 513240. i) q^{92} -422293. q^{93} -1.15271e6 q^{94} +239633. q^{95} +681561. q^{96} -1.39769e6i q^{97} +755829. q^{98} -390422. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Introduction

L-functions

Modular forms

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