LMFDB - Embedded newform 13.8.b.a.12.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [13,8,Mod(12,13)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13.12"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	13.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$4.06100533129$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 449x^{4} + 37224x^{2} + 205776 $ x^6 + 449x^4 + 37224x^2 + 205776
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			12.2
Root			$-10.0583i$ of defining polynomial
Character	$\chi$	$=$	13.12
Dual form			13.8.b.a.12.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-10.0583i q^{2} +50.8656 q^{3} +26.8297 q^{4} -8.88672i q^{5} -511.623i q^{6} -510.452i q^{7} -1557.33i q^{8} +400.306 q^{9} -89.3858 q^{10} +4065.46i q^{11} +1364.71 q^{12} +(2772.33 + 7420.43i) q^{13} -5134.30 q^{14} -452.028i q^{15} -12230.0 q^{16} -1791.54 q^{17} -4026.41i q^{18} +19743.0i q^{19} -238.428i q^{20} -25964.4i q^{21} +40891.8 q^{22} -11749.0 q^{23} -79214.5i q^{24} +78046.0 q^{25} +(74637.2 - 27885.0i) q^{26} -90881.2 q^{27} -13695.3i q^{28} -183054. q^{29} -4546.66 q^{30} +247198. i q^{31} -76325.0i q^{32} +206792. i q^{33} +18020.0i q^{34} -4536.25 q^{35} +10740.1 q^{36} -203775. i q^{37} +198582. q^{38} +(141016. + 377444. i) q^{39} -13839.6 q^{40} -419322. i q^{41} -261159. q^{42} +17481.7 q^{43} +109075. i q^{44} -3557.41i q^{45} +118175. i q^{46} -1.17985e6i q^{47} -622084. q^{48} +562982. q^{49} -785014. i q^{50} -91127.9 q^{51} +(74380.6 + 199088. i) q^{52} +1.32587e6 q^{53} +914115. i q^{54} +36128.6 q^{55} -794942. q^{56} +1.00424e6i q^{57} +1.84123e6i q^{58} +401485. i q^{59} -12127.8i q^{60} -3.28655e6 q^{61} +2.48640e6 q^{62} -204337. i q^{63} -2.33314e6 q^{64} +(65943.3 - 24636.9i) q^{65} +2.07999e6 q^{66} -2.10629e6i q^{67} -48066.5 q^{68} -597618. q^{69} +45627.1i q^{70} -4.22986e6i q^{71} -623408. i q^{72} +3.29726e6i q^{73} -2.04964e6 q^{74} +3.96986e6 q^{75} +529699. i q^{76} +2.07522e6 q^{77} +(3.79646e6 - 1.41839e6i) q^{78} +3.79387e6 q^{79} +108684. i q^{80} -5.49819e6 q^{81} -4.21768e6 q^{82} -4.36639e6i q^{83} -696617. i q^{84} +15920.9i q^{85} -175837. i q^{86} -9.31117e6 q^{87} +6.33127e6 q^{88} +8.72168e6i q^{89} -35781.6 q^{90} +(3.78777e6 - 1.41514e6i) q^{91} -315221. q^{92} +1.25738e7i q^{93} -1.18673e7 q^{94} +175451. q^{95} -3.88231e6i q^{96} +9.21482e6i q^{97} -5.66267e6i q^{98} +1.62743e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9} - 406 q^{10} + 1898 q^{12} - 5018 q^{13} + 9558 q^{14} + 7778 q^{16} + 13152 q^{17} - 125080 q^{22} + 27264 q^{23} - 18262 q^{25} - 54210 q^{26} + 194560 q^{27} + 42924 q^{29}+ \cdots - 22075632 q^{95}+O(q^{100}) $ 6 * q - 56 * q^3 - 130 * q^4 - 1150 * q^9 - 406 * q^10 + 1898 * q^12 - 5018 * q^13 + 9558 * q^14 + 7778 * q^16 + 13152 * q^17 - 125080 * q^22 + 27264 * q^23 - 18262 * q^25 - 54210 * q^26 + 194560 * q^27 + 42924 * q^29 - 60114 * q^30 - 546720 * q^35 + 747772 * q^36 - 243492 * q^38 + 511160 * q^39 + 792058 * q^40 - 1138134 * q^42 - 1005576 * q^43 - 2807426 * q^48 + 3246846 * q^49 + 297984 * q^51 + 3252964 * q^52 + 1705524 * q^53 - 5320576 * q^55 - 2282730 * q^56 - 8237572 * q^61 + 9625800 * q^62 + 5070862 * q^64 + 7139028 * q^65 + 9289056 * q^66 - 16801086 * q^68 - 6017400 * q^69 - 28760826 * q^74 + 20426072 * q^75 + 4093560 * q^77 + 14088594 * q^78 + 15147200 * q^79 - 21318442 * q^81 - 9130240 * q^82 - 36474240 * q^87 + 38008000 * q^88 + 4769676 * q^90 + 21281832 * q^91 + 36462348 * q^92 - 34229770 * q^94 - 22075632 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	10.0583i		−	0.889041i	−0.895769	−	0.444520i	$-0.853374\pi$
$2$		−	10.0583i		−	0.889041i	0.895769	−	0.444520i	$-0.146626\pi$
$3$	50.8656			1.08768			0.543838	−	0.839190i	$-0.316971\pi$
$3$	50.8656			1.08768			0.543838	+	0.839190i	$0.316971\pi$
$4$	26.8297			0.209607
$5$		−	8.88672i		−	0.0317941i	−0.999874	−	0.0158971i	$-0.994940\pi$
$5$		−	8.88672i		−	0.0317941i	0.999874	−	0.0158971i	$-0.00506040\pi$
$6$		−	511.623i		−	0.966988i
$7$		−	510.452i		−	0.562486i	−0.959637	−	0.281243i	$-0.909253\pi$
$7$		−	510.452i		−	0.562486i	0.959637	−	0.281243i	$-0.0907467\pi$
$8$		−	1557.33i		−	1.07539i
$9$	400.306			0.183039
$10$	−89.3858			−0.0282663
$11$			4065.46i			0.920949i	0.887673	+	0.460474i	$0.152321\pi$
$11$			4065.46i			0.920949i	−0.887673	+	0.460474i	$0.847679\pi$
$12$	1364.71			0.227984
$13$	2772.33	+	7420.43i	0.349980	+	0.936757i
$13$	2772.33	+	7420.43i	0.349980	+	0.936757i
$14$	−5134.30			−0.500073
$15$		−	452.028i		−	0.0345817i
$16$	−12230.0			−0.746458
$17$	−1791.54			−0.0884415			−0.0442207	−	0.999022i	$-0.514080\pi$
$17$	−1791.54			−0.0884415			−0.0442207	+	0.999022i	$0.514080\pi$
$18$		−	4026.41i		−	0.162729i
$19$			19743.0i			0.660353i	0.943919	+	0.330176i	$0.107108\pi$
$19$			19743.0i			0.660353i	−0.943919	+	0.330176i	$0.892892\pi$
$20$		−	238.428i		−	0.00666426i
$21$		−	25964.4i		−	0.611802i
$22$	40891.8			0.818761
$23$	−11749.0			−0.201350			−0.100675	−	0.994919i	$-0.532100\pi$
$23$	−11749.0			−0.201350			−0.100675	+	0.994919i	$0.532100\pi$
$24$		−	79214.5i		−	1.16968i
$25$	78046.0			0.998989
$26$	74637.2	−	27885.0i	0.832815	−	0.311146i
$27$	−90881.2			−0.888589
$28$		−	13695.3i		−	0.117901i
$29$	−183054.			−1.39376			−0.696879	−	0.717189i	$-0.745429\pi$
$29$	−183054.			−1.39376			−0.696879	+	0.717189i	$0.745429\pi$
$30$	−4546.66			−0.0307445
$31$			247198.i			1.49032i	0.666888	+	0.745158i	$0.267626\pi$
$31$			247198.i			1.49032i	−0.666888	+	0.745158i	$0.732374\pi$
$32$		−	76325.0i		−	0.411758i
$33$			206792.i			1.00169i
$34$			18020.0i			0.0786281i
$35$	−4536.25			−0.0178837
$36$	10740.1			0.0383662
$37$		−	203775.i		−	0.661372i	−0.943741	−	0.330686i	$-0.892720\pi$
$37$		−	203775.i		−	0.661372i	0.943741	−	0.330686i	$-0.107280\pi$
$38$	198582.			0.587081
$39$	141016.	+	377444.i	0.380665	+	1.01889i
$40$	−13839.6			−0.0341911
$41$		−	419322.i		−	0.950175i	−0.879938	−	0.475088i	$-0.842416\pi$
$41$		−	419322.i		−	0.950175i	0.879938	−	0.475088i	$-0.157584\pi$
$42$	−261159.			−0.543917
$43$	17481.7			0.0335309			0.0167654	−	0.999859i	$-0.494663\pi$
$43$	17481.7			0.0335309			0.0167654	+	0.999859i	$0.494663\pi$
$44$			109075.i			0.193037i
$45$		−	3557.41i		−	0.00581956i
$46$			118175.i			0.179009i
$47$		−	1.17985e6i		−	1.65761i	−0.559537	−	0.828806i	$-0.689021\pi$
$47$		−	1.17985e6i		−	1.65761i	0.559537	−	0.828806i	$-0.310979\pi$
$48$	−622084.			−0.811905
$49$	562982.			0.683610
$50$		−	785014.i		−	0.888142i
$51$	−91127.9			−0.0961957
$52$	74380.6	+	199088.i	0.0733581	+	0.196351i
$53$	1.32587e6			1.22330			0.611651	−	0.791127i	$-0.290506\pi$
$53$	1.32587e6			1.22330			0.611651	+	0.791127i	$0.290506\pi$
$54$			914115.i			0.789992i
$55$	36128.6			0.0292807
$56$	−794942.			−0.604891
$57$			1.00424e6i			0.718250i
$58$			1.84123e6i			1.23911i
$59$			401485.i			0.254499i	0.991871	+	0.127250i	$0.0406150\pi$
$59$			401485.i			0.254499i	−0.991871	+	0.127250i	$0.959385\pi$
$60$		−	12127.8i		−	0.00724855i
$61$	−3.28655e6			−1.85390			−0.926949	−	0.375188i	$-0.877578\pi$
$61$	−3.28655e6			−1.85390			−0.926949	+	0.375188i	$0.877578\pi$
$62$	2.48640e6			1.32495
$63$		−	204337.i		−	0.102957i
$64$	−2.33314e6			−1.11253
$65$	65943.3	−	24636.9i	0.0297834	−	0.0111273i
$66$	2.07999e6			0.890546
$67$		−	2.10629e6i		−	0.855572i	−0.903880	−	0.427786i	$-0.859294\pi$
$67$		−	2.10629e6i		−	0.855572i	0.903880	−	0.427786i	$-0.140706\pi$
$68$	−48066.5			−0.0185379
$69$	−597618.			−0.219004
$70$			45627.1i			0.0158994i
$71$		−	4.22986e6i		−	1.40256i	−0.712886	−	0.701280i	$-0.752612\pi$
$71$		−	4.22986e6i		−	1.40256i	0.712886	−	0.701280i	$-0.247388\pi$
$72$		−	623408.i		−	0.196838i
$73$			3.29726e6i			0.992027i	0.868315	+	0.496014i	$0.165203\pi$
$73$			3.29726e6i			0.992027i	−0.868315	+	0.496014i	$0.834797\pi$
$74$	−2.04964e6			−0.587986
$75$	3.96986e6			1.08658
$76$			529699.i			0.138414i
$77$	2.07522e6			0.518021
$78$	3.79646e6	−	1.41839e6i	0.905833	−	0.338426i
$79$	3.79387e6			0.865741			0.432870	−	0.901456i	$-0.357501\pi$
$79$	3.79387e6			0.865741			0.432870	+	0.901456i	$0.357501\pi$
$80$			108684.i			0.0237330i
$81$	−5.49819e6			−1.14954
$82$	−4.21768e6			−0.844745
$83$		−	4.36639e6i		−	0.838203i	−0.907939	−	0.419102i	$-0.862345\pi$
$83$		−	4.36639e6i		−	0.838203i	0.907939	−	0.419102i	$-0.137655\pi$
$84$		−	696617.i		−	0.128238i
$85$			15920.9i			0.00281192i
$86$		−	175837.i		−	0.0298103i
$87$	−9.31117e6			−1.51596
$88$	6.33127e6			0.990379
$89$			8.72168e6i			1.31140i	0.755022	+	0.655700i	$0.227626\pi$
$89$			8.72168e6i			1.31140i	−0.755022	+	0.655700i	$0.772374\pi$
$90$	−35781.6			−0.00517382
$91$	3.78777e6	−	1.41514e6i	0.526913	−	0.196859i
$92$	−315221.			−0.0422044
$93$			1.25738e7i			1.62098i
$94$	−1.18673e7			−1.47368
$95$	175451.			0.0209953
$96$		−	3.88231e6i		−	0.447859i
$97$			9.21482e6i			1.02515i	0.858644	+	0.512573i	$0.171308\pi$
$97$			9.21482e6i			1.02515i	−0.858644	+	0.512573i	$0.828692\pi$
$98$		−	5.66267e6i		−	0.607757i
$99$			1.62743e6i			0.168569i
$100$	2.09395e6			0.209395
$101$	2.06993e6			0.199908			0.0999539	−	0.994992i	$-0.468130\pi$
$101$	2.06993e6			0.199908			0.0999539	+	0.994992i	$0.468130\pi$
$102$			916596.i			0.0855219i
$103$	1.74931e7			1.57738			0.788688	−	0.614793i	$-0.210761\pi$
$103$	1.74931e7			1.57738			0.788688	+	0.614793i	$0.210761\pi$
$104$	1.15561e7	−	4.31743e6i	1.00738	−	0.376365i
$105$	−230739.			−0.0194517
$106$		−	1.33360e7i		−	1.08757i
$107$	−1.55162e7			−1.22445			−0.612226	−	0.790683i	$-0.709726\pi$
$107$	−1.55162e7			−1.22445			−0.612226	+	0.790683i	$0.709726\pi$
$108$	−2.43831e6			−0.186254
$109$		−	1.59529e7i		−	1.17990i	−0.807438	−	0.589952i	$-0.799147\pi$
$109$		−	1.59529e7i		−	1.17990i	0.807438	−	0.589952i	$-0.200853\pi$
$110$		−	363394.i		−	0.0260318i
$111$		−	1.03651e7i		−	0.719358i
$112$			6.24281e6i			0.419872i
$113$	−1.65661e7			−1.08006			−0.540028	−	0.841647i	$-0.681586\pi$
$113$	−1.65661e7			−1.08006			−0.540028	+	0.841647i	$0.681586\pi$
$114$	1.01010e7			0.638553
$115$			104410.i			0.00640176i
$116$	−4.91129e6			−0.292141
$117$	1.10978e6	+	2.97044e6i	0.0640599	+	0.171463i
$118$	4.03827e6			0.226260
$119$			914497.i			0.0497471i
$120$	−703957.			−0.0371888
$121$	2.95919e6			0.151853
$122$			3.30573e7i			1.64819i
$123$		−	2.13290e7i		−	1.03348i
$124$			6.63223e6i			0.312380i
$125$		−	1.38785e6i		−	0.0635561i
$126$	−2.05529e6			−0.0915327
$127$	1.98781e6			0.0861118			0.0430559	−	0.999073i	$-0.486291\pi$
$127$	1.98781e6			0.0861118			0.0430559	+	0.999073i	$0.486291\pi$
$128$			1.36979e7i			0.577325i
$129$	889218.			0.0364707
$130$	−247807.	−	663280.i	−0.00989262	−	0.0264786i
$131$	1.58779e7			0.617081			0.308541	−	0.951211i	$-0.400159\pi$
$131$	1.58779e7			0.617081			0.308541	+	0.951211i	$0.400159\pi$
$132$			5.54816e6i			0.209962i
$133$	1.00779e7			0.371439
$134$	−2.11858e7			−0.760638
$135$			807636.i			0.0282519i
$136$			2.79002e6i			0.0951090i
$137$			4.95975e7i			1.64793i	0.566643	+	0.823963i	$0.308242\pi$
$137$			4.95975e7i			1.64793i	−0.566643	+	0.823963i	$0.691758\pi$
$138$			6.01105e6i			0.194703i
$139$	−5.71630e6			−0.180536			−0.0902678	−	0.995918i	$-0.528772\pi$
$139$	−5.71630e6			−0.180536			−0.0902678	+	0.995918i	$0.528772\pi$
$140$	−121706.			−0.00374855
$141$		−	6.00135e7i		−	1.80294i
$142$	−4.25454e7			−1.24693
$143$	−3.01675e7	+	1.12708e7i	−0.862705	+	0.322313i
$144$	−4.89573e6			−0.136631
$145$			1.62675e6i			0.0443133i
$146$	3.31650e7			0.881953
$147$	2.86364e7			0.743546
$148$		−	5.46722e6i		−	0.138628i
$149$			1.51998e7i			0.376431i	0.982128	+	0.188216i	$0.0602704\pi$
$149$			1.51998e7i			0.376431i	−0.982128	+	0.188216i	$0.939730\pi$
$150$		−	3.99302e7i		−	0.966011i
$151$		−	6.47181e7i		−	1.52970i	−0.644208	−	0.764850i	$-0.722813\pi$
$151$		−	6.47181e7i		−	1.52970i	0.644208	−	0.764850i	$-0.277187\pi$
$152$	3.07464e7			0.710137
$153$	−717165.			−0.0161882
$154$		−	2.08733e7i		−	0.460541i
$155$	2.19678e6			0.0473833
$156$	3.78341e6	+	1.01267e7i	0.0797899	+	0.213566i
$157$	2.86572e6			0.0590997			0.0295499	−	0.999563i	$-0.490593\pi$
$157$	2.86572e6			0.0590997			0.0295499	+	0.999563i	$0.490593\pi$
$158$		−	3.81601e7i		−	0.769679i
$159$	6.74409e7			1.33056
$160$	−678279.			−0.0130915
$161$			5.99728e6i			0.113257i
$162$			5.53027e7i			1.02198i
$163$			3.56517e7i			0.644797i	0.946604	+	0.322399i	$0.104489\pi$
$163$			3.56517e7i			0.644797i	−0.946604	+	0.322399i	$0.895511\pi$
$164$		−	1.12503e7i		−	0.199163i
$165$	1.83770e6			0.0318480
$166$	−4.39187e7			−0.745197
$167$			2.20447e7i			0.366266i	0.983088	+	0.183133i	$0.0586239\pi$
$167$			2.20447e7i			0.366266i	−0.983088	+	0.183133i	$0.941376\pi$
$168$	−4.04352e7			−0.657926
$169$	−4.73769e7	+	4.11437e7i	−0.755028	+	0.655692i
$170$	160138.			0.00249991
$171$			7.90325e6i			0.120870i
$172$	469029.			0.00702830
$173$	−1.96908e7			−0.289136			−0.144568	−	0.989495i	$-0.546179\pi$
$173$	−1.96908e7			−0.289136			−0.144568	+	0.989495i	$0.546179\pi$
$174$			9.36550e7i			1.34775i
$175$		−	3.98387e7i		−	0.561917i
$176$		−	4.97205e7i		−	0.687450i
$177$			2.04217e7i			0.276813i
$178$	8.77257e7			1.16589
$179$	−7.90229e7			−1.02983			−0.514917	−	0.857240i	$-0.672177\pi$
$179$	−7.90229e7			−1.02983			−0.514917	+	0.857240i	$0.672177\pi$
$180$		−	95444.0i		−	0.00121982i
$181$	4.94136e7			0.619400			0.309700	−	0.950834i	$-0.399771\pi$
$181$	4.94136e7			0.619400			0.309700	+	0.950834i	$0.399771\pi$
$182$	−1.42340e7	−	3.80987e7i	−0.175015	−	0.468447i
$183$	−1.67172e8			−2.01644
$184$			1.82970e7i			0.216530i
$185$	−1.81090e6			−0.0210277
$186$	1.26472e8			1.44112
$187$		−	7.28345e6i		−	0.0814501i
$188$		−	3.16549e7i		−	0.347446i
$189$			4.63905e7i			0.499819i
$190$		−	1.76475e6i		−	0.0186657i
$191$	1.64464e8			1.70787			0.853935	−	0.520379i	$-0.174209\pi$
$191$	1.64464e8			1.70787			0.853935	+	0.520379i	$0.174209\pi$
$192$	−1.18676e8			−1.21007
$193$			8.78627e7i			0.879740i	0.898062	+	0.439870i	$0.144975\pi$
$193$			8.78627e7i			0.879740i	−0.898062	+	0.439870i	$0.855025\pi$
$194$	9.26859e7			0.911397
$195$	3.35424e6	−	1.25317e6i	0.0323946	−	0.0121029i
$196$	1.51046e7			0.143289
$197$		−	1.98762e7i		−	0.185225i	−0.995702	−	0.0926127i	$-0.970478\pi$
$197$		−	1.98762e7i		−	0.185225i	0.995702	−	0.0926127i	$-0.0295218\pi$
$198$	1.63692e7			0.149865
$199$	−2.80120e7			−0.251975			−0.125988	−	0.992032i	$-0.540210\pi$
$199$	−2.80120e7			−0.251975			−0.125988	+	0.992032i	$0.540210\pi$
$200$		−	1.21543e8i		−	1.07430i
$201$		−	1.07138e8i		−	0.930585i
$202$		−	2.08200e7i		−	0.177726i
$203$			9.34405e7i			0.783969i
$204$	−2.44493e6			−0.0201633
$205$	−3.72640e6			−0.0302100
$206$		−	1.75951e8i		−	1.40235i
$207$	−4.70318e6			−0.0368549
$208$	−3.39055e7	−	9.07516e7i	−0.261245	−	0.699250i
$209$	−8.02645e7			−0.608151
$210$			2.32085e6i			0.0172934i
$211$	2.39701e7			0.175663			0.0878315	−	0.996135i	$-0.472006\pi$
$211$	2.39701e7			0.175663			0.0878315	+	0.996135i	$0.472006\pi$
$212$	3.55725e7			0.256413
$213$		−	2.15154e8i		−	1.52553i
$214$			1.56067e8i			1.08859i
$215$		−	155355.i		−	0.00106608i
$216$			1.41532e8i			0.955579i
$217$	1.26182e8			0.838281
$218$	−1.60460e8			−1.04898
$219$			1.67717e8i			1.07900i
$220$	969319.			0.00613744
$221$	−4.96675e6	−	1.32940e7i	−0.0309527	−	0.0828482i
$222$	−1.04256e8			−0.639538
$223$			1.88776e8i			1.13993i	0.821668	+	0.569967i	$0.193044\pi$
$223$			1.88776e8i			1.13993i	−0.821668	+	0.569967i	$0.806956\pi$
$224$	−3.89602e7			−0.231608
$225$	3.12423e7			0.182854
$226$			1.66628e8i			0.960213i
$227$		−	766382.i		−	0.00434865i	−0.999998	−	0.00217433i	$-0.999308\pi$
$227$		−	766382.i		−	0.00434865i	0.999998	−	0.00217433i	$-0.000692110\pi$
$228$			2.69434e7i			0.150550i
$229$		−	1.06420e8i		−	0.585597i	−0.956174	−	0.292798i	$-0.905414\pi$
$229$		−	1.06420e8i		−	0.585597i	0.956174	−	0.292798i	$-0.0945865\pi$
$230$	1.05019e6			0.00569142
$231$	1.05557e8			0.563439
$232$			2.85076e8i			1.49883i
$233$	−2.05485e7			−0.106423			−0.0532113	−	0.998583i	$-0.516946\pi$
$233$	−2.05485e7			−0.106423			−0.0532113	+	0.998583i	$0.516946\pi$
$234$	2.98777e7	−	1.11625e7i	0.152437	−	0.0569518i
$235$	−1.04850e7			−0.0527023
$236$			1.07717e7i			0.0533448i
$237$	1.92977e8			0.941645
$238$	9.19832e6			0.0442272
$239$		−	6.47473e7i		−	0.306781i	−0.988166	−	0.153391i	$-0.950981\pi$
$239$		−	6.47473e7i		−	0.306781i	0.988166	−	0.153391i	$-0.0490193\pi$
$240$			5.52829e6i			0.0258138i
$241$		−	4.01781e8i		−	1.84897i	−0.381217	−	0.924486i	$-0.624495\pi$
$241$		−	4.01781e8i		−	1.84897i	0.381217	−	0.924486i	$-0.375505\pi$
$242$		−	2.97646e7i		−	0.135004i
$243$	−8.09115e7			−0.361733
$244$	−8.81770e7			−0.388589
$245$		−	5.00306e6i		−	0.0217348i
$246$	−2.14535e8			−0.918808
$247$	−1.46502e8	+	5.47342e7i	−0.618590	+	0.231110i
$248$	3.84968e8			1.60267
$249$		−	2.22099e8i		−	0.911694i
$250$	−1.39595e7			−0.0565039
$251$	−2.77091e8			−1.10602			−0.553011	−	0.833174i	$-0.686521\pi$
$251$	−2.77091e8			−1.10602			−0.553011	+	0.833174i	$0.686521\pi$
$252$		−	5.48229e6i		−	0.0215804i
$253$		−	4.77650e7i		−	0.185433i
$254$		−	1.99941e7i		−	0.0765569i
$255$			809828.i			0.00305846i
$256$	−1.60863e8			−0.599263
$257$	4.67270e8			1.71713			0.858563	−	0.512708i	$-0.171358\pi$
$257$	4.67270e8			1.71713			0.858563	+	0.512708i	$0.171358\pi$
$258$		−	8.94407e6i		−	0.0324240i
$259$	−1.04017e8			−0.372012
$260$	1.76924e6	−	661000.i	0.00624279	−	0.00233236i
$261$	−7.32778e7			−0.255112
$262$		−	1.59705e8i		−	0.548610i
$263$	7.58768e7			0.257196			0.128598	−	0.991697i	$-0.458952\pi$
$263$	7.58768e7			0.257196			0.128598	+	0.991697i	$0.458952\pi$
$264$	3.22044e8			1.07721
$265$		−	1.17826e7i		−	0.0388938i
$266$		−	1.01367e8i		−	0.330225i
$267$			4.43633e8i			1.42638i
$268$		−	5.65111e7i		−	0.179334i
$269$	−1.93468e8			−0.606005			−0.303002	−	0.952990i	$-0.597989\pi$
$269$	−1.93468e8			−0.606005			−0.303002	+	0.952990i	$0.597989\pi$
$270$	8.12349e6			0.0251171
$271$		−	4.04769e8i		−	1.23542i	−0.786406	−	0.617710i	$-0.788060\pi$
$271$		−	4.04769e8i		−	1.23542i	0.786406	−	0.617710i	$-0.211940\pi$
$272$	2.19105e7			0.0660179
$273$	1.92667e8	−	7.19819e7i	0.573110	−	0.214118i
$274$	4.98869e8			1.46507
$275$			3.17293e8i			0.920018i
$276$	−1.60339e7			−0.0459047
$277$	−4.66835e8			−1.31973			−0.659864	−	0.751385i	$-0.729386\pi$
$277$	−4.66835e8			−1.31973			−0.659864	+	0.751385i	$0.729386\pi$
$278$			5.74965e7i			0.160503i
$279$			9.89546e7i			0.272786i
$280$			7.06443e6i			0.0192320i
$281$			2.83566e8i			0.762397i	0.924493	+	0.381199i	$0.124489\pi$
$281$			2.83566e8i			0.762397i	−0.924493	+	0.381199i	$0.875511\pi$
$282$	−6.03637e8			−1.60289
$283$	3.54304e8			0.929229			0.464615	−	0.885513i	$-0.346193\pi$
$283$	3.54304e8			0.929229			0.464615	+	0.885513i	$0.346193\pi$
$284$		−	1.13486e8i		−	0.293986i
$285$	8.92441e6			0.0228361
$286$	1.13366e8	+	3.03435e8i	0.286550	+	0.766980i
$287$	−2.14044e8			−0.534460
$288$		−	3.05533e7i		−	0.0753676i
$289$	−4.07129e8			−0.992178
$290$	1.63625e7			0.0393963
$291$			4.68717e8i			1.11503i
$292$			8.84645e7i			0.207936i
$293$			5.68237e8i			1.31975i	0.751374	+	0.659877i	$0.229392\pi$
$293$			5.68237e8i			1.31975i	−0.751374	+	0.659877i	$0.770608\pi$
$294$		−	2.88035e8i		−	0.661042i
$295$	3.56788e6			0.00809158
$296$	−3.17346e8			−0.711232
$297$		−	3.69474e8i		−	0.818345i
$298$	1.52885e8			0.334663
$299$	−3.25720e7	−	8.71824e7i	−0.0704685	−	0.188616i
$300$	1.06510e8			0.227754
$301$		−	8.92359e6i		−	0.0188606i
$302$	−6.50957e8			−1.35997
$303$	1.05288e8			0.217435
$304$		−	2.41457e8i		−	0.492926i
$305$			2.92067e7i			0.0589430i
$306$			7.21350e6i			0.0143920i
$307$		−	6.06885e8i		−	1.19708i	−0.801094	−	0.598538i	$-0.795749\pi$
$307$		−	6.06885e8i		−	1.19708i	0.801094	−	0.598538i	$-0.204251\pi$
$308$	5.56775e7			0.108581
$309$	8.89794e8			1.71567
$310$		−	2.20959e7i		−	0.0421256i
$311$	5.70878e8			1.07617			0.538086	−	0.842890i	$-0.319147\pi$
$311$	5.70878e8			1.07617			0.538086	+	0.842890i	$0.319147\pi$
$312$	5.87805e8	−	2.19609e8i	1.09570	−	0.409363i
$313$	1.19008e8			0.219367			0.109684	−	0.993967i	$-0.465016\pi$
$313$	1.19008e8			0.219367			0.109684	+	0.993967i	$0.465016\pi$
$314$		−	2.88244e7i		−	0.0525421i
$315$	−1.81589e6			−0.00327342
$316$	1.01788e8			0.181465
$317$			7.11011e8i			1.25363i	0.779169	+	0.626814i	$0.215642\pi$
$317$			7.11011e8i			1.25363i	−0.779169	+	0.626814i	$0.784358\pi$
$318$		−	6.78344e8i		−	1.18292i
$319$		−	7.44201e8i		−	1.28358i
$320$			2.07340e7i			0.0353718i
$321$	−7.89240e8			−1.33181
$322$	6.03228e7			0.100690
$323$		−	3.53705e7i		−	0.0584026i
$324$	−1.47515e8			−0.240950
$325$	2.16369e8	+	5.79135e8i	0.349626	+	0.935810i
$326$	3.58597e8			0.573251
$327$		−	8.11453e8i		−	1.28335i
$328$	−6.53022e8			−1.02181
$329$	−6.02254e8			−0.932383
$330$		−	1.84843e7i		−	0.0283141i
$331$			1.03380e9i			1.56689i	0.621459	+	0.783447i	$0.286540\pi$
$331$			1.03380e9i			1.56689i	−0.621459	+	0.783447i	$0.713460\pi$
$332$		−	1.17149e8i		−	0.175693i
$333$		−	8.15724e7i		−	0.121057i
$334$	2.21733e8			0.325625
$335$	−1.87180e7			−0.0272021
$336$			3.17544e8i			0.456685i
$337$	8.03307e8			1.14334			0.571672	−	0.820482i	$-0.306295\pi$
$337$	8.03307e8			1.14334			0.571672	+	0.820482i	$0.306295\pi$
$338$	4.13838e8	+	4.76533e8i	0.582937	+	0.671251i
$339$	−8.42645e8			−1.17475
$340$			427154.i			0.000589397i
$341$	−1.00497e9			−1.37250
$342$	7.94936e7			0.107459
$343$		−	7.07754e8i		−	0.947007i
$344$		−	2.72248e7i		−	0.0360588i
$345$			5.31087e6i			0.00696303i
$346$			1.98057e8i			0.257054i
$347$	3.48273e8			0.447473			0.223737	−	0.974650i	$-0.428174\pi$
$347$	3.48273e8			0.447473			0.223737	+	0.974650i	$0.428174\pi$
$348$	−2.49816e8			−0.317755
$349$			6.90910e8i			0.870027i	0.900424	+	0.435013i	$0.143256\pi$
$349$			6.90910e8i			0.870027i	−0.900424	+	0.435013i	$0.856744\pi$
$350$	−4.00712e8			−0.499567
$351$	−2.51953e8	−	6.74377e8i	−0.310988	−	0.832392i
$352$	3.10296e8			0.379208
$353$			3.16345e8i			0.382780i	0.981514	+	0.191390i	$0.0612996\pi$
$353$			3.16345e8i			0.382780i	−0.981514	+	0.191390i	$0.938700\pi$
$354$	2.05409e8			0.246098
$355$	−3.75896e7			−0.0445932
$356$			2.34000e8i			0.274878i
$357$			4.65164e7i			0.0541087i
$358$			7.94840e8i			0.915565i
$359$			5.14441e8i			0.586820i	0.955987	+	0.293410i	$0.0947901\pi$
$359$			5.14441e8i			0.586820i	−0.955987	+	0.293410i	$0.905210\pi$
$360$	−5.54006e6			−0.00625829
$361$	5.04085e8			0.563934
$362$		−	4.97019e8i		−	0.550672i
$363$	1.50521e8			0.165167
$364$	1.01625e8	−	3.79677e7i	0.110444	−	0.0412629i
$365$	2.93019e7			0.0315406
$366$			1.68148e9i			1.79270i
$367$	1.47338e9			1.55591			0.777953	−	0.628323i	$-0.216258\pi$
$367$	1.47338e9			1.55591			0.777953	+	0.628323i	$0.216258\pi$
$368$	1.43690e8			0.150300
$369$		−	1.67857e8i		−	0.173919i
$370$			1.82146e7i			0.0186945i
$371$		−	6.76791e8i		−	0.688091i
$372$			3.37352e8i			0.339768i
$373$	−9.69375e8			−0.967188			−0.483594	−	0.875292i	$-0.660669\pi$
$373$	−9.69375e8			−0.967188			−0.483594	+	0.875292i	$0.660669\pi$
$374$	−7.32595e7			−0.0724124
$375$		−	7.05937e7i		−	0.0691284i
$376$	−1.83741e9			−1.78258
$377$	−5.07487e8	−	1.35834e9i	−0.487787	−	1.30561i
$378$	4.66612e8			0.444359
$379$		−	3.46921e8i		−	0.327335i	−0.986516	−	0.163668i	$-0.947668\pi$
$379$		−	3.46921e8i		−	0.327335i	0.986516	−	0.163668i	$-0.0523325\pi$
$380$	4.70729e6			0.00440076
$381$	1.01111e8			0.0936617
$382$		−	1.65424e9i		−	1.51837i
$383$		−	1.83888e9i		−	1.67247i	−0.548373	−	0.836234i	$-0.684753\pi$
$383$		−	1.83888e9i		−	1.67247i	0.548373	−	0.836234i	$-0.315247\pi$
$384$			6.96753e8i			0.627942i
$385$		−	1.84419e7i		−	0.0164700i
$386$	8.83754e8			0.782124
$387$	6.99804e6			0.00613745
$388$			2.47231e8i			0.214878i
$389$	1.00176e9			0.862861			0.431431	−	0.902146i	$-0.358009\pi$
$389$	1.00176e9			0.862861			0.431431	+	0.902146i	$0.358009\pi$
$390$	−1.26048e7	−	3.37381e7i	−0.0107600	−	0.0288002i
$391$	2.10488e7			0.0178077
$392$		−	8.76749e8i		−	0.735147i
$393$	8.07636e8			0.671184
$394$	−1.99921e8			−0.164673
$395$		−	3.37151e7i		−	0.0275255i
$396$			4.36633e7i			0.0353333i
$397$		−	1.39793e9i		−	1.12129i	−0.828055	−	0.560647i	$-0.810552\pi$
$397$		−	1.39793e9i		−	1.12129i	0.828055	−	0.560647i	$-0.189448\pi$
$398$			2.81754e8i			0.224016i
$399$	5.12616e8			0.404005
$400$	−9.54501e8			−0.745704
$401$			1.21136e9i			0.938144i	0.883160	+	0.469072i	$0.155411\pi$
$401$			1.21136e9i			0.938144i	−0.883160	+	0.469072i	$0.844589\pi$
$402$	−1.07763e9			−0.827328
$403$	−1.83431e9	+	6.85313e8i	−1.39606	+	0.521580i
$404$	5.55354e7			0.0419020
$405$			4.88609e7i			0.0365485i
$406$	9.39857e8			0.696981
$407$	8.28441e8			0.609089
$408$			1.41916e8i			0.103448i
$409$			1.45644e9i			1.05260i	0.850300	+	0.526299i	$0.176421\pi$
$409$			1.45644e9i			1.05260i	−0.850300	+	0.526299i	$0.823579\pi$
$410$			3.74814e7i			0.0268579i
$411$			2.52281e9i			1.79241i
$412$	4.69333e8			0.330629
$413$	2.04939e8			0.143152
$414$			4.73062e7i			0.0327655i
$415$	−3.88029e7			−0.0266499
$416$	5.66364e8	−	2.11598e8i	0.385717	−	0.144107i
$417$	−2.90763e8			−0.196364
$418$			8.07328e8i			0.540671i
$419$	1.35223e9			0.898051			0.449026	−	0.893519i	$-0.351771\pi$
$419$	1.35223e9			0.898051			0.449026	+	0.893519i	$0.351771\pi$
$420$	−6.19064e6			−0.00407721
$421$			1.35611e9i			0.885742i	0.896585	+	0.442871i	$0.146040\pi$
$421$			1.35611e9i			0.885742i	−0.896585	+	0.442871i	$0.853960\pi$
$422$		−	2.41099e8i		−	0.156172i
$423$		−	4.72299e8i		−	0.303407i
$424$		−	2.06481e9i		−	1.31553i
$425$	−1.39823e8			−0.0883521
$426$	−2.16409e9			−1.35626
$427$			1.67763e9i			1.04279i
$428$	−4.16294e8			−0.256654
$429$	−1.53448e9	+	5.73295e8i	−0.938344	+	0.350573i
$430$	−1.56262e6			−0.000947793
$431$		−	1.27281e9i		−	0.765764i	−0.923797	−	0.382882i	$-0.874932\pi$
$431$		−	1.27281e9i		−	0.765764i	0.923797	−	0.382882i	$-0.125068\pi$
$432$	1.11147e9			0.663295
$433$	−3.39003e8			−0.200676			−0.100338	−	0.994953i	$-0.531992\pi$
$433$	−3.39003e8			−0.200676			−0.100338	+	0.994953i	$0.531992\pi$
$434$		−	1.26919e9i		−	0.745266i
$435$			8.27458e7i			0.0481985i
$436$		−	4.28011e8i		−	0.247316i
$437$		−	2.31960e8i		−	0.132962i
$438$	1.68696e9			0.959279
$439$	−1.40195e9			−0.790870			−0.395435	−	0.918494i	$-0.629406\pi$
$439$	−1.40195e9			−0.790870			−0.395435	+	0.918494i	$0.629406\pi$
$440$		−	5.62642e7i		−	0.0314882i
$441$	2.25365e8			0.125127
$442$	−1.33716e8	+	4.99572e7i	−0.0736554	+	0.0275182i
$443$	−3.31716e9			−1.81282			−0.906408	−	0.422404i	$-0.861187\pi$
$443$	−3.31716e9			−1.81282			−0.906408	+	0.422404i	$0.861187\pi$
$444$		−	2.78093e8i		−	0.150782i
$445$	7.75071e7			0.0416948
$446$	1.89877e9			1.01345
$447$			7.73146e8i			0.409435i
$448$			1.19096e9i			0.625781i
$449$			8.90007e8i			0.464014i	0.972714	+	0.232007i	$0.0745292\pi$
$449$			8.90007e8i			0.464014i	−0.972714	+	0.232007i	$0.925471\pi$
$450$		−	3.14246e8i		−	0.162564i
$451$	1.70474e9			0.875063
$452$	−4.44463e8			−0.226387
$453$		−	3.29192e9i		−	1.66382i
$454$	−7.70853e6			−0.00386613
$455$	−1.25760e7	−	3.36609e7i	−0.00625895	−	0.0167527i
$456$	1.56393e9			0.772399
$457$			1.66381e9i			0.815448i	0.913105	+	0.407724i	$0.133677\pi$
$457$			1.66381e9i			0.815448i	−0.913105	+	0.407724i	$0.866323\pi$
$458$	−1.07041e9			−0.520619
$459$	1.62818e8			0.0785881
$460$			2.80128e6i			0.00134185i
$461$		−	2.68236e7i		−	0.0127516i	−0.999980	−	0.00637580i	$-0.997971\pi$
$461$		−	2.68236e7i		−	0.0127516i	0.999980	−	0.00637580i	$-0.00202949\pi$
$462$		−	1.06173e9i		−	0.500920i
$463$		−	3.55847e9i		−	1.66621i	−0.553115	−	0.833105i	$-0.686561\pi$
$463$		−	3.55847e9i		−	1.66621i	0.553115	−	0.833105i	$-0.313439\pi$
$464$	2.23875e9			1.04038
$465$	1.11740e8			0.0515376
$466$			2.06684e8i			0.0946141i
$467$	2.59791e9			1.18036			0.590180	−	0.807272i	$-0.299057\pi$
$467$	2.59791e9			1.18036			0.590180	+	0.807272i	$0.299057\pi$
$468$	2.97750e7	+	7.96959e7i	0.0134274	+	0.0359398i
$469$	−1.07516e9			−0.481247
$470$			1.05461e8i			0.0468545i
$471$	1.45767e8			0.0642813
$472$	6.25244e8			0.273686
$473$			7.10713e7i			0.0308802i
$474$		−	1.94103e9i		−	0.837161i
$475$			1.54086e9i			0.659685i
$476$			2.45356e7i			0.0104273i
$477$	5.30752e8			0.223912
$478$	−6.51250e8			−0.272741
$479$		−	2.90167e9i		−	1.20635i	−0.797608	−	0.603176i	$-0.793902\pi$
$479$		−	2.90167e9i		−	1.20635i	0.797608	−	0.603176i	$-0.206098\pi$
$480$	−3.45011e7			−0.0142393
$481$	1.51210e9	−	5.64932e8i	0.619545	−	0.231467i
$482$	−4.04126e9			−1.64381
$483$			3.05055e8i			0.123187i
$484$	7.93941e7			0.0318295
$485$	8.18896e7			0.0325936
$486$			8.13836e8i			0.321595i
$487$			1.15313e9i			0.452406i	0.974080	+	0.226203i	$0.0726313\pi$
$487$			1.15313e9i			0.452406i	−0.974080	+	0.226203i	$0.927369\pi$
$488$			5.11824e9i			1.99366i
$489$			1.81344e9i			0.701331i
$490$	−5.03226e7			−0.0193231
$491$	−3.78661e9			−1.44366			−0.721831	−	0.692069i	$-0.756699\pi$
$491$	−3.78661e9			−1.44366			−0.721831	+	0.692069i	$0.756699\pi$
$492$		−	5.72251e8i		−	0.216625i
$493$	3.27950e8			0.123266
$494$	5.50535e8	+	1.47356e9i	0.205466	+	0.549952i
$495$	1.44625e7			0.00535951
$496$		−	3.02322e9i		−	1.11246i
$497$	−2.15914e9			−0.788920
$498$	−2.23395e9			−0.810533
$499$			1.98153e9i			0.713918i	0.934120	+	0.356959i	$0.116186\pi$
$499$			1.98153e9i			0.713918i	−0.934120	+	0.356959i	$0.883814\pi$
$500$		−	3.72355e7i		−	0.0133218i
$501$			1.12132e9i			0.398379i
$502$			2.78707e9i			0.983299i
$503$	2.32772e9			0.815536			0.407768	−	0.913086i	$-0.366307\pi$
$503$	2.32772e9			0.815536			0.407768	+	0.913086i	$0.366307\pi$
$504$	−3.18220e8			−0.110719
$505$		−	1.83949e7i		−	0.00635589i
$506$	−4.80437e8			−0.164858
$507$	−2.40985e9	+	2.09280e9i	−0.821226	+	0.713181i
$508$	5.33324e7			0.0180496
$509$		−	1.44356e9i		−	0.485203i	−0.970126	−	0.242601i	$-0.921999\pi$
$509$		−	1.44356e9i		−	0.485203i	0.970126	−	0.242601i	$-0.0780007\pi$
$510$	8.14553e6			0.00271909
$511$	1.68309e9			0.558001
$512$			3.37135e9i			1.11009i
$513$		−	1.79427e9i		−	0.586782i
$514$		−	4.69997e9i		−	1.52660i
$515$		−	1.55456e8i		−	0.0501513i
$516$	2.38574e7			0.00764451
$517$	4.79662e9			1.52657
$518$			1.04624e9i			0.330734i
$519$	−1.00158e9			−0.314486
$520$	−3.83678e7	−	1.02695e8i	−0.0119662	−	0.0320287i
$521$	2.02069e9			0.625990			0.312995	−	0.949755i	$-0.398668\pi$
$521$	2.02069e9			0.625990			0.312995	+	0.949755i	$0.398668\pi$
$522$			7.37053e8i			0.226805i
$523$	3.62640e9			1.10846			0.554230	−	0.832364i	$-0.313013\pi$
$523$	3.62640e9			1.10846			0.554230	+	0.832364i	$0.313013\pi$
$524$	4.25998e8			0.129344
$525$		−	2.02642e9i		−	0.611184i
$526$		−	7.63195e8i		−	0.228658i
$527$		−	4.42865e8i		−	0.131806i
$528$		−	2.52906e9i		−	0.747723i
$529$	−3.26679e9			−0.959458
$530$	−1.18513e8			−0.0345782
$531$			1.60717e8i			0.0465833i
$532$	2.70386e8			0.0778562
$533$	3.11155e9	−	1.16250e9i	0.890084	−	0.332542i
$534$	4.46222e9			1.26811
$535$			1.37888e8i			0.0389304i
$536$	−3.28019e9			−0.920073
$537$	−4.01955e9			−1.12013
$538$			1.94597e9i			0.538763i
$539$			2.28878e9i			0.629569i
$540$			2.16686e7i			0.00592179i
$541$		−	7.29172e9i		−	1.97988i	−0.141481	−	0.989941i	$-0.545186\pi$
$541$		−	7.29172e9i		−	1.97988i	0.141481	−	0.989941i	$-0.454814\pi$
$542$	−4.07131e9			−1.09834
$543$	2.51345e9			0.673706
$544$			1.36740e8i			0.0364165i
$545$	−1.41769e8			−0.0375140
$546$	−7.24019e8	−	1.93791e9i	−0.190360	−	0.509518i
$547$	−2.20432e9			−0.575862			−0.287931	−	0.957651i	$-0.592967\pi$
$547$	−2.20432e9			−0.575862			−0.287931	+	0.957651i	$0.592967\pi$
$548$			1.33068e9i			0.345417i
$549$	−1.31562e9			−0.339335
$550$	3.19144e9			0.817933
$551$		−	3.61405e9i		−	0.920372i
$552$			9.30689e8i			0.235515i
$553$		−	1.93659e9i		−	0.486967i
$554$			4.69559e9i			1.17329i
$555$	−9.21122e7			−0.0228713
$556$	−1.53366e8			−0.0378415
$557$			6.48577e8i			0.159026i	0.996834	+	0.0795131i	$0.0253365\pi$
$557$			6.48577e8i			0.159026i	−0.996834	+	0.0795131i	$0.974663\pi$
$558$	9.95320e8			0.242517
$559$	4.84651e7	+	1.29722e8i	0.0117351	+	0.0314103i
$560$	5.54782e7			0.0133495
$561$		−	3.70477e8i		−	0.0885913i
$562$	2.85220e9			0.677802
$563$	−7.79259e8			−0.184036			−0.0920180	−	0.995757i	$-0.529332\pi$
$563$	−7.79259e8			−0.184036			−0.0920180	+	0.995757i	$0.529332\pi$
$564$		−	1.61014e9i		−	0.377909i
$565$			1.47218e8i			0.0343394i
$566$		−	3.56371e9i		−	0.826123i
$567$			2.80656e9i			0.646598i
$568$	−6.58729e9			−1.50830
$569$	4.16166e9			0.947051			0.473525	−	0.880780i	$-0.342981\pi$
$569$	4.16166e9			0.947051			0.473525	+	0.880780i	$0.342981\pi$
$570$		−	8.97648e7i		−	0.0203022i
$571$	−8.48385e9			−1.90707			−0.953535	−	0.301282i	$-0.902585\pi$
$571$	−8.48385e9			−1.90707			−0.953535	+	0.301282i	$0.902585\pi$
$572$	−8.09383e8	+	3.02392e8i	−0.180829	+	0.0675591i
$573$	8.36557e9			1.85761
$574$			2.15292e9i			0.475157i
$575$	−9.16960e8			−0.201147
$576$	−9.33969e8			−0.203636
$577$		−	4.94714e9i		−	1.07211i	−0.844184	−	0.536054i	$-0.819914\pi$
$577$		−	4.94714e9i		−	1.07211i	0.844184	−	0.536054i	$-0.180086\pi$
$578$			4.09505e9i			0.882087i
$579$			4.46919e9i			0.956872i
$580$			4.36453e7i			0.00928837i
$581$	−2.22883e9			−0.471478
$582$	4.71452e9			0.991304
$583$			5.39026e9i			1.12660i
$584$	5.13493e9			1.06682
$585$	2.63975e7	−	9.86230e6i	0.00545151	−	0.00203673i
$586$	5.71553e9			1.17332
$587$			6.91524e9i			1.41115i	0.708634	+	0.705577i	$0.249312\pi$
$587$			6.91524e9i			1.41115i	−0.708634	+	0.705577i	$0.750688\pi$
$588$	7.68305e8			0.155852
$589$	−4.88043e9			−0.984134
$590$		−	3.58870e7i		−	0.00719375i
$591$		−	1.01101e9i		−	0.201465i
$592$			2.49217e9i			0.493686i
$593$			4.73029e9i			0.931529i	0.884909	+	0.465765i	$0.154221\pi$
$593$			4.73029e9i			0.931529i	−0.884909	+	0.465765i	$0.845779\pi$
$594$	−3.71630e9			−0.727542
$595$	8.12688e6			0.00158166
$596$			4.07805e8i			0.0789025i
$597$	−1.42484e9			−0.274067
$598$	−8.76910e8	+	3.27621e8i	−0.167688	+	0.0626494i
$599$	−5.05854e9			−0.961682			−0.480841	−	0.876808i	$-0.659669\pi$
$599$	−5.05854e9			−0.961682			−0.480841	+	0.876808i	$0.659669\pi$
$600$		−	6.18238e9i		−	1.16849i
$601$	−2.81440e9			−0.528840			−0.264420	−	0.964408i	$-0.585181\pi$
$601$	−2.81440e9			−0.528840			−0.264420	+	0.964408i	$0.585181\pi$
$602$	−8.97565e7			−0.0167679
$603$		−	8.43160e8i		−	0.156603i
$604$		−	1.73636e9i		−	0.320635i
$605$		−	2.62975e7i		−	0.00482804i
$606$		−	1.05902e9i		−	0.193308i
$607$	−3.54047e9			−0.642541			−0.321271	−	0.946987i	$-0.604110\pi$
$607$	−3.54047e9			−0.642541			−0.321271	+	0.946987i	$0.604110\pi$
$608$	1.50689e9			0.271905
$609$			4.75290e9i			0.852704i
$610$	2.93771e8			0.0524028
$611$	8.75496e9	−	3.27092e9i	1.55278	−	0.580130i
$612$	−1.92413e7			−0.00339316
$613$		−	1.90580e8i		−	0.0334169i	−0.999860	−	0.0167085i	$-0.994681\pi$
$613$		−	1.90580e8i		−	0.0334169i	0.999860	−	0.0167085i	$-0.00531872\pi$
$614$	−6.10426e9			−1.06425
$615$	−1.89545e8			−0.0328587
$616$		−	3.23181e9i		−	0.557074i
$617$		−	9.31051e9i		−	1.59579i	−0.602798	−	0.797894i	$-0.705947\pi$
$617$		−	9.31051e9i		−	1.59579i	0.602798	−	0.797894i	$-0.294053\pi$
$618$		−	8.94986e9i		−	1.52530i
$619$			8.75671e9i			1.48396i	0.670419	+	0.741982i	$0.266114\pi$
$619$			8.75671e9i			1.48396i	−0.670419	+	0.741982i	$0.733886\pi$
$620$	5.89388e7			0.00993185
$621$	1.06776e9			0.178918
$622$		−	5.74209e9i		−	0.956761i
$623$	4.45200e9			0.737644
$624$	−1.72462e9	−	4.61613e9i	−0.284150	−	0.760558i
$625$	6.08501e9			0.996968
$626$		−	1.19702e9i		−	0.195026i
$627$	−4.08270e9			−0.661471
$628$	7.68864e7			0.0123877
$629$			3.65072e8i			0.0584927i
$630$			1.82648e7i			0.00291020i
$631$			2.59032e9i			0.410440i	0.978716	+	0.205220i	$0.0657910\pi$
$631$			2.59032e9i			0.410440i	−0.978716	+	0.205220i	$0.934209\pi$
$632$		−	5.90831e9i		−	0.931009i
$633$	1.21925e9			0.191065
$634$	7.15159e9			1.11453
$635$		−	1.76652e7i		−	0.00273785i
$636$	1.80942e9			0.278894
$637$	1.56077e9	+	4.17757e9i	0.239250	+	0.640376i
$638$	−7.48543e9			−1.14115
$639$		−	1.69324e9i		−	0.256723i
$640$	1.21730e8			0.0183555
$641$	4.49376e9			0.673918			0.336959	−	0.941519i	$-0.390602\pi$
$641$	4.49376e9			0.673918			0.336959	+	0.941519i	$0.390602\pi$
$642$			7.93845e9i			1.18403i
$643$		−	5.61571e9i		−	0.833041i	−0.909126	−	0.416521i	$-0.863249\pi$
$643$		−	5.61571e9i		−	0.833041i	0.909126	−	0.416521i	$-0.136751\pi$
$644$			1.60905e8i			0.0237394i
$645$		−	7.90224e6i		−	0.00115955i
$646$	−3.55769e8			−0.0519223
$647$	−1.45087e9			−0.210602			−0.105301	−	0.994440i	$-0.533581\pi$
$647$	−1.45087e9			−0.210602			−0.105301	+	0.994440i	$0.533581\pi$
$648$			8.56250e9i			1.23620i
$649$	−1.63222e9			−0.234381
$650$	5.82514e9	−	2.17632e9i	0.831973	−	0.310832i
$651$	6.41834e9			0.911779
$652$			9.56522e8i			0.135154i
$653$	−9.22610e9			−1.29665			−0.648324	−	0.761365i	$-0.724530\pi$
$653$	−9.22610e9			−1.29665			−0.648324	+	0.761365i	$0.724530\pi$
$654$	−8.16187e9			−1.14095
$655$		−	1.41102e8i		−	0.0196195i
$656$			5.12829e9i			0.709266i
$657$			1.31991e9i			0.181579i
$658$			6.05768e9i			0.828926i
$659$	4.26023e9			0.579875			0.289937	−	0.957046i	$-0.406366\pi$
$659$	4.26023e9			0.579875			0.289937	+	0.957046i	$0.406366\pi$
$660$	4.93050e7			0.00667555
$661$		−	1.14507e9i		−	0.154216i	−0.997023	−	0.0771078i	$-0.975431\pi$
$661$		−	1.14507e9i		−	0.154216i	0.997023	−	0.0771078i	$-0.0245685\pi$
$662$	1.03983e10			1.39303
$663$	−2.52636e8	−	6.76207e8i	−0.0336665	−	0.0901120i
$664$	−6.79992e9			−0.901395
$665$		−	8.95592e7i		−	0.0118096i
$666$	−8.20484e8			−0.107624
$667$	2.15070e9			0.280634
$668$			5.91452e8i			0.0767718i
$669$			9.60219e9i			1.23988i
$670$			1.88272e8i			0.0241838i
$671$		−	1.33613e10i		−	1.70735i
$672$	−1.98173e9			−0.251914
$673$	−5.46085e9			−0.690570			−0.345285	−	0.938498i	$-0.612218\pi$
$673$	−5.46085e9			−0.690570			−0.345285	+	0.938498i	$0.612218\pi$
$674$		−	8.07994e9i		−	1.01648i
$675$	−7.09292e9			−0.887691
$676$	−1.27111e9	+	1.10387e9i	−0.158259	+	0.137438i
$677$	−8.92821e9			−1.10587			−0.552935	−	0.833225i	$-0.686492\pi$
$677$	−8.92821e9			−1.10587			−0.552935	+	0.833225i	$0.686492\pi$
$678$			8.47561e9i			1.04440i
$679$	4.70372e9			0.576630
$680$	2.47942e7			0.00302391
$681$		−	3.89824e7i		−	0.00472992i
$682$			1.01084e10i			1.22021i
$683$		−	8.50973e9i		−	1.02198i	−0.859586	−	0.510991i	$-0.829279\pi$
$683$		−	8.50973e9i		−	1.02198i	0.859586	−	0.510991i	$-0.170721\pi$
$684$			2.12041e8i			0.0253352i
$685$	4.40759e8			0.0523944
$686$	−7.11884e9			−0.841927
$687$		−	5.41311e9i		−	0.636940i
$688$	−2.13801e8			−0.0250294
$689$	3.67573e9	+	9.83849e9i	0.428131	+	1.14594i
$690$	5.34185e7			0.00619042
$691$		−	1.11787e9i		−	0.128889i	−0.997921	−	0.0644447i	$-0.979472\pi$
$691$		−	1.11787e9i		−	0.128889i	0.997921	−	0.0644447i	$-0.0205276\pi$
$692$	−5.28298e8			−0.0606049
$693$	8.30724e8			0.0948179
$694$		−	3.50306e9i		−	0.397822i
$695$			5.07991e7i			0.00573997i
$696$			1.45006e10i			1.63024i
$697$			7.51233e8i			0.0840349i
$698$	6.94942e9			0.773489
$699$	−1.04521e9			−0.115753
$700$		−	1.06886e9i		−	0.117782i
$701$	3.84858e9			0.421976			0.210988	−	0.977489i	$-0.432332\pi$
$701$	3.84858e9			0.421976			0.210988	+	0.977489i	$0.432332\pi$
$702$	−6.78312e9	+	2.53423e9i	−0.740030	+	0.276481i
$703$	4.02314e9			0.436739
$704$		−	9.48529e9i		−	1.02458i
$705$	−5.33324e8			−0.0573230
$706$	3.18191e9			0.340307
$707$		−	1.05660e9i		−	0.112445i
$708$			5.47908e8i			0.0580219i
$709$			5.36316e9i			0.565144i	0.959246	+	0.282572i	$0.0911875\pi$
$709$			5.36316e9i			0.565144i	−0.959246	+	0.282572i	$0.908812\pi$
$710$			3.78089e8i			0.0396451i
$711$	1.51871e9			0.158464
$712$	1.35825e10			1.41027
$713$		−	2.90432e9i		−	0.300076i
$714$	4.67878e8			0.0481048
$715$	1.00160e8	+	2.68090e8i	0.0102477	+	0.0274290i
$716$	−2.12016e9			−0.215860
$717$		−	3.29341e9i		−	0.333679i
$718$	5.17443e9			0.521707
$719$	6.62611e9			0.664825			0.332413	−	0.943134i	$-0.392137\pi$
$719$	6.62611e9			0.664825			0.332413	+	0.943134i	$0.392137\pi$
$720$			4.35070e7i			0.00434406i
$721$		−	8.92936e9i		−	0.887252i
$722$		−	5.07026e9i		−	0.501360i
$723$		−	2.04368e10i		−	2.01108i
$724$	1.32575e9			0.129830
$725$	−1.42867e10			−1.39235
$726$		−	1.51399e9i		−	0.146840i
$727$	−9.52174e8			−0.0919064			−0.0459532	−	0.998944i	$-0.514633\pi$
$727$	−9.52174e8			−0.0919064			−0.0459532	+	0.998944i	$0.514633\pi$
$728$	−2.20384e9	−	5.89881e9i	−0.211700	−	0.566636i
$729$	7.90894e9			0.756087
$730$		−	2.94728e8i		−	0.0280409i
$731$	−3.13193e7			−0.00296552
$732$	−4.48517e9			−0.422659
$733$			1.46510e10i			1.37405i	0.726632	+	0.687027i	$0.241084\pi$
$733$			1.46510e10i			1.37405i	−0.726632	+	0.687027i	$0.758916\pi$
$734$		−	1.48198e10i		−	1.38326i
$735$		−	2.54484e8i		−	0.0236404i
$736$			8.96740e8i			0.0829076i
$737$	8.56305e9			0.787938
$738$	−1.68836e9			−0.154621
$739$		−	1.21913e10i		−	1.11121i	−0.831447	−	0.555604i	$-0.812487\pi$
$739$		−	1.21913e10i		−	1.11121i	0.831447	−	0.555604i	$-0.187513\pi$
$740$	−4.85857e7			−0.00440755
$741$	−7.45189e9	+	2.78408e9i	−0.672826	+	0.251373i
$742$	−6.80739e9			−0.611740
$743$			3.89555e9i			0.348424i	0.984708	+	0.174212i	$0.0557378\pi$
$743$			3.89555e9i			0.348424i	−0.984708	+	0.174212i	$0.944262\pi$
$744$	1.95816e10			1.74319
$745$	1.35076e8			0.0119683
$746$			9.75031e9i			0.859870i
$747$		−	1.74789e9i		−	0.153424i
$748$		−	1.95413e8i		−	0.0170725i
$749$			7.92027e9i			0.688737i
$750$	−7.10056e8			−0.0614580
$751$	−1.53353e10			−1.32115			−0.660574	−	0.750761i	$-0.729687\pi$
$751$	−1.53353e10			−1.32115			−0.660574	+	0.750761i	$0.729687\pi$
$752$			1.44295e10i			1.23734i
$753$	−1.40944e10			−1.20299
$754$	−1.36627e10	+	5.10448e9i	−1.16074	+	0.433663i
$755$	−5.75132e8			−0.0486355
$756$			1.24464e9i			0.104765i
$757$	1.71312e10			1.43533			0.717665	−	0.696389i	$-0.245211\pi$
$757$	1.71312e10			1.43533			0.717665	+	0.696389i	$0.245211\pi$
$758$	−3.48945e9			−0.291014
$759$		−	2.42959e9i		−	0.201691i
$760$		−	2.73235e8i		−	0.0225782i
$761$			9.75251e9i			0.802177i	0.916039	+	0.401088i	$0.131368\pi$
$761$			9.75251e9i			0.802177i	−0.916039	+	0.401088i	$0.868632\pi$
$762$		−	1.01701e9i		−	0.0832691i
$763$	−8.14318e9			−0.663679
$764$	4.41252e9			0.357981
$765$			6.37325e6i			0.000514690i
$766$	−1.84961e10			−1.48689
$767$	−2.97919e9	+	1.11305e9i	−0.238404	+	0.0890697i
$768$	−8.18241e9			−0.651804
$769$		−	6.98230e9i		−	0.553677i	−0.960916	−	0.276839i	$-0.910713\pi$
$769$		−	6.98230e9i		−	0.553677i	0.960916	−	0.276839i	$-0.0892867\pi$
$770$	−1.85495e8			−0.0146425
$771$	2.37680e10			1.86768
$772$			2.35733e9i			0.184399i
$773$			1.86799e10i			1.45461i	0.686314	+	0.727305i	$0.259227\pi$
$773$			1.86799e10i			1.45461i	−0.686314	+	0.727305i	$0.740773\pi$
$774$		−	7.03887e7i		−	0.00545644i
$775$			1.92928e10i			1.48881i
$776$	1.43505e10			1.10243
$777$	−5.29091e9			−0.404629
$778$		−	1.00761e10i		−	0.767119i
$779$	8.27868e9			0.627451
$780$	8.99932e7	−	3.36221e7i	0.00679014	−	0.00253685i
$781$	1.71963e10			1.29169
$782$		−	2.11716e8i		−	0.0158318i
$783$	1.66362e10			1.23848
$784$	−6.88525e9			−0.510286
$785$		−	2.54669e7i		−	0.00187902i
$786$		−	8.12348e9i		−	0.596710i
$787$			5.17067e9i			0.378125i	0.981965	+	0.189062i	$0.0605448\pi$
$787$			5.17067e9i			0.378125i	−0.981965	+	0.189062i	$0.939455\pi$
$788$		−	5.33271e8i		−	0.0388245i
$789$	3.85952e9			0.279746
$790$	−3.39118e8			−0.0244713
$791$			8.45620e9i			0.607516i
$792$	2.53444e9			0.181278
$793$	−9.11140e9	−	2.43876e10i	−0.648827	−	1.73665i
$794$	−1.40609e10			−0.996877
$795$		−	5.99329e8i		−	0.0423039i
$796$	−7.51552e8			−0.0528157
$797$	−8.89261e9			−0.622193			−0.311097	−	0.950378i	$-0.600696\pi$
$797$	−8.89261e9			−0.622193			−0.311097	+	0.950378i	$0.600696\pi$
$798$		−	5.15607e9i		−	0.359177i
$799$			2.11374e9i			0.146602i
$800$		−	5.95686e9i		−	0.411342i
$801$			3.49134e9i			0.240037i
$802$	1.21843e10			0.834048
$803$	−1.34049e10			−0.913606
$804$		−	2.87447e9i		−	0.195057i
$805$	5.32962e7			0.00360090
$806$	6.89311e9	+	1.84501e10i	0.463706	+	1.24116i
$807$	−9.84086e9			−0.659137
$808$		−	3.22356e9i		−	0.214979i
$809$	−1.93280e10			−1.28342			−0.641709	−	0.766948i	$-0.721774\pi$
$809$	−1.93280e10			−1.28342			−0.641709	+	0.766948i	$0.721774\pi$
$810$	4.91460e8			0.0324931
$811$		−	1.58565e9i		−	0.104384i	−0.998637	−	0.0521922i	$-0.983379\pi$
$811$		−	1.58565e9i		−	0.104384i	0.998637	−	0.0521922i	$-0.0166208\pi$
$812$			2.50698e9i			0.164325i
$813$		−	2.05888e10i		−	1.34374i
$814$		−	8.33274e9i		−	0.541505i
$815$	3.16827e8			0.0205008
$816$	1.11449e9			0.0718061
$817$			3.45142e8i			0.0221422i
$818$	1.46494e10			0.935802
$819$	1.51627e9	−	5.66489e8i	0.0964455	−	0.0360328i
$820$	−9.99780e7			−0.00633222
$821$			1.92167e10i			1.21193i	0.795490	+	0.605967i	$0.207214\pi$
$821$			1.92167e10i			1.21193i	−0.795490	+	0.605967i	$0.792786\pi$
$822$	2.53753e10			1.59353
$823$	7.27465e9			0.454897			0.227448	−	0.973790i	$-0.426962\pi$
$823$	7.27465e9			0.454897			0.227448	+	0.973790i	$0.426962\pi$
$824$		−	2.72425e10i		−	1.69629i
$825$			1.61393e10i			1.00068i
$826$		−	2.06134e9i		−	0.127268i
$827$		−	7.39476e9i		−	0.454626i	−0.973822	−	0.227313i	$-0.927006\pi$
$827$		−	7.39476e9i		−	0.454626i	0.973822	−	0.227313i	$-0.0729941\pi$
$828$	−1.26185e8			−0.00772504
$829$	−1.20575e10			−0.735048			−0.367524	−	0.930014i	$-0.619794\pi$
$829$	−1.20575e10			−0.735048			−0.367524	+	0.930014i	$0.619794\pi$
$830$			3.90293e8i			0.0236929i
$831$	−2.37458e10			−1.43544
$832$	−6.46823e9	−	1.73129e10i	−0.389362	−	1.04217i
$833$	−1.00861e9			−0.0604594
$834$			2.92459e9i			0.174576i
$835$	1.95905e8			0.0116451
$836$	−2.15347e9			−0.127473
$837$		−	2.24656e10i		−	1.32428i
$838$		−	1.36012e10i		−	0.798404i
$839$		−	1.74685e9i		−	0.102115i	−0.998696	−	0.0510575i	$-0.983741\pi$
$839$		−	1.74685e9i		−	0.102115i	0.998696	−	0.0510575i	$-0.0162592\pi$
$840$			3.59336e8i			0.0209182i
$841$	1.62591e10			0.942562
$842$	1.36402e10			0.787461
$843$			1.44237e10i			0.829241i
$844$	6.43109e8			0.0368202
$845$	3.65633e8	+	4.21026e8i	0.0208471	+	0.0240055i
$846$	−4.75055e9			−0.269741
$847$		−	1.51052e9i		−	0.0854153i
$848$	−1.62153e10			−0.913144
$849$	1.80218e10			1.01070
$850$			1.40639e9i			0.0785486i
$851$			2.39415e9i			0.133167i
$852$		−	5.77251e9i		−	0.319762i
$853$		−	2.10759e9i		−	0.116269i	−0.998309	−	0.0581346i	$-0.981485\pi$
$853$		−	2.10759e9i		−	0.116269i	0.998309	−	0.0581346i	$-0.0185153\pi$
$854$	1.68741e10			0.927084
$855$	7.02340e7			0.00384296
$856$			2.41638e10i			1.31676i
$857$	2.74086e10			1.48749			0.743744	−	0.668464i	$-0.233048\pi$
$857$	2.74086e10			1.48749			0.743744	+	0.668464i	$0.233048\pi$
$858$	5.76640e9	+	1.54344e10i	0.311673	+	0.834226i
$859$	2.95561e9			0.159100			0.0795502	−	0.996831i	$-0.474652\pi$
$859$	2.95561e9			0.159100			0.0795502	+	0.996831i	$0.474652\pi$
$860$		−	4.16813e6i		−	0.000223459i
$861$	−1.08874e10			−0.581320
$862$	−1.28024e10			−0.680795
$863$			2.24829e10i			1.19073i	0.803454	+	0.595366i	$0.202993\pi$
$863$			2.24829e10i			1.19073i	−0.803454	+	0.595366i	$0.797007\pi$
$864$			6.93651e9i			0.365883i
$865$			1.74987e8i			0.00919283i
$866$			3.40981e9i			0.178409i
$867$	−2.07088e10			−1.07917
$868$	3.38543e9			0.175709
$869$			1.54238e10i			0.797303i
$870$	8.32286e8			0.0428504
$871$	1.56296e10	−	5.83933e9i	0.801463	−	0.299433i
$872$	−2.48439e10			−1.26886
$873$			3.68875e9i			0.187642i
$874$	−2.33314e9			−0.118209
$875$	−7.08430e8			−0.0357494
$876$			4.49980e9i			0.226167i
$877$		−	3.27881e10i		−	1.64141i	−0.571350	−	0.820707i	$-0.693580\pi$
$877$		−	3.27881e10i		−	1.64141i	0.571350	−	0.820707i	$-0.306420\pi$
$878$			1.41012e10i			0.703116i
$879$			2.89037e10i			1.43546i
$880$	−4.41852e8			−0.0218569
$881$	−3.78138e10			−1.86310			−0.931548	−	0.363619i	$-0.881541\pi$
$881$	−3.78138e10			−1.86310			−0.931548	+	0.363619i	$0.881541\pi$
$882$		−	2.26680e9i		−	0.111243i
$883$	8.05584e9			0.393775			0.196888	−	0.980426i	$-0.436917\pi$
$883$	8.05584e9			0.393775			0.196888	+	0.980426i	$0.436917\pi$
$884$	−1.33256e8	−	3.56674e8i	−0.00648790	−	0.0173655i
$885$	1.81482e8			0.00880102
$886$			3.33652e10i			1.61167i
$887$	−3.15643e10			−1.51867			−0.759335	−	0.650700i	$-0.774476\pi$
$887$	−3.15643e10			−1.51867			−0.759335	+	0.650700i	$0.774476\pi$
$888$	−1.61420e10			−0.773590
$889$		−	1.01468e9i		−	0.0484367i
$890$		−	7.79594e8i		−	0.0370684i
$891$		−	2.23527e10i		−	1.05866i
$892$			5.06479e9i			0.238938i
$893$	2.32937e10			1.09461
$894$	7.77657e9			0.364004
$895$			7.02255e8i			0.0327427i
$896$	6.99213e9			0.324737
$897$	−1.65679e9	−	4.43458e9i	−0.0766469	−	0.205154i
$898$	8.95200e9			0.412527
$899$		−	4.52506e10i		−	2.07714i
$900$	8.38220e8			0.0383274
$901$	−2.37535e9			−0.108191
$902$		−	1.71468e10i		−	0.777967i
$903$		−	4.53903e8i		−	0.0205143i
$904$			2.57989e10i			1.16148i
$905$		−	4.39125e8i		−	0.0196933i
$906$	−3.31113e10			−1.47920
$907$	3.74494e10			1.66656			0.833278	−	0.552854i	$-0.186461\pi$
$907$	3.74494e10			1.66656			0.833278	+	0.552854i	$0.186461\pi$
$908$		−	2.05618e7i		−	0.000911507i
$909$	8.28603e8			0.0365909
$910$	−3.38573e8	+	1.26493e8i	−0.0148939	+	0.00556446i
$911$	4.69150e9			0.205588			0.102794	−	0.994703i	$-0.467222\pi$
$911$	4.69150e9			0.205588			0.102794	+	0.994703i	$0.467222\pi$
$912$		−	1.22818e10i		−	0.536144i
$913$	1.77514e10			0.771942
$914$	1.67351e10			0.724966
$915$			1.48561e9i			0.0641109i
$916$		−	2.85521e9i		−	0.122745i
$917$		−	8.10488e9i		−	0.347099i
$918$		−	1.63768e9i		−	0.0698680i
$919$	−3.13816e10			−1.33374			−0.666870	−	0.745174i	$-0.732366\pi$
$919$	−3.13816e10			−1.33374			−0.666870	+	0.745174i	$0.732366\pi$
$920$	1.62601e8			0.00688438
$921$		−	3.08695e10i		−	1.30203i
$922$	−2.69801e8			−0.0113367
$923$	3.13873e10	−	1.17266e10i	1.31386	−	0.490868i
$924$	2.83207e9			0.118101
$925$		−	1.59039e10i		−	0.660703i
$926$	−3.57923e10			−1.48133
$927$	7.00257e9			0.288721
$928$			1.39716e10i			0.573891i
$929$			2.99605e10i			1.22601i	0.790079	+	0.613005i	$0.210039\pi$
$929$			2.99605e10i			1.22601i	−0.790079	+	0.613005i	$0.789961\pi$
$930$		−	1.12392e9i		−	0.0458190i
$931$			1.11150e10i			0.451424i
$932$	−5.51309e8			−0.0223069
$933$	2.90380e10			1.17053
$934$		−	2.61307e10i		−	1.04939i
$935$	−6.47260e7			−0.00258963
$936$	4.62596e9	−	1.72829e9i	0.184389	−	0.0688893i
$937$	3.17700e10			1.26162			0.630810	−	0.775937i	$-0.282723\pi$
$937$	3.17700e10			1.26162			0.630810	+	0.775937i	$0.282723\pi$
$938$			1.08143e10i			0.427848i
$939$	6.05341e9			0.238600
$940$	−2.81308e8			−0.0110468
$941$			3.63923e8i			0.0142379i	0.999975	+	0.00711895i	$0.00226605\pi$
$941$			3.63923e8i			0.0142379i	−0.999975	+	0.00711895i	$0.997734\pi$
$942$		−	1.46617e9i		−	0.0571487i
$943$			4.92660e9i			0.191318i
$944$		−	4.91015e9i		−	0.189973i
$945$	4.12259e8			0.0158913
$946$	7.14860e8			0.0274538
$947$		−	8.10602e9i		−	0.310158i	−0.987902	−	0.155079i	$-0.950437\pi$
$947$		−	8.10602e9i		−	0.310158i	0.987902	−	0.155079i	$-0.0495632\pi$
$948$	5.17752e9			0.197375
$949$	−2.44671e10	+	9.14110e9i	−0.929289	+	0.347190i
$950$	1.54986e10			0.586487
$951$			3.61660e10i			1.36354i
$952$	1.42417e9			0.0534975
$953$	−3.07889e10			−1.15231			−0.576155	−	0.817340i	$-0.695448\pi$
$953$	−3.07889e10			−1.15231			−0.576155	+	0.817340i	$0.695448\pi$
$954$		−	5.33848e9i		−	0.199067i
$955$		−	1.46155e9i		−	0.0543002i
$956$		−	1.73715e9i		−	0.0643034i
$957$		−	3.78542e10i		−	1.39612i
$958$	−2.91860e10			−1.07250
$959$	2.53171e10			0.926936
$960$			1.05465e9i			0.0384731i
$961$	−3.35940e10			−1.22104
$962$	−5.68228e9	−	1.52092e10i	−0.205783	−	0.550800i
$963$	−6.21122e9			−0.224122
$964$		−	1.07797e10i		−	0.387557i
$965$	7.80812e8			0.0279705
$966$	3.06835e9			0.109518
$967$			2.71301e10i			0.964846i	0.875938	+	0.482423i	$0.160243\pi$
$967$			2.71301e10i			0.964846i	−0.875938	+	0.482423i	$0.839757\pi$
$968$		−	4.60844e9i		−	0.163301i
$969$		−	1.79914e9i		−	0.0635231i
$970$		−	8.23674e8i		−	0.0289770i
$971$	−2.35980e10			−0.827197			−0.413598	−	0.910459i	$-0.635728\pi$
$971$	−2.35980e10			−0.827197			−0.413598	+	0.910459i	$0.635728\pi$
$972$	−2.17083e9			−0.0758217
$973$			2.91789e9i			0.101549i
$974$	1.15986e10			0.402207
$975$	1.10057e10	+	2.94580e10i	0.380280	+	1.01786i
$976$	4.01944e10			1.38386
$977$			1.46911e9i			0.0503992i	0.999682	+	0.0251996i	$0.00802212\pi$
$977$			1.46911e9i			0.0503992i	−0.999682	+	0.0251996i	$0.991978\pi$
$978$	1.82402e10			0.623511
$979$	−3.54576e10			−1.20773
$980$		−	1.34231e8i		−	0.00455575i
$981$		−	6.38603e9i		−	0.215968i
$982$			3.80871e10i			1.28347i
$983$		−	3.36024e10i		−	1.12832i	−0.825665	−	0.564161i	$-0.809200\pi$
$983$		−	3.36024e10i		−	1.12832i	0.825665	−	0.564161i	$-0.190800\pi$
$984$	−3.32164e10			−1.11140
$985$	−1.76634e8			−0.00588908
$986$		−	3.29863e9i		−	0.109589i
$987$	−3.06340e10			−1.01413
$988$	−3.93059e9	+	1.46850e9i	−0.129661	+	0.0484422i
$989$	−2.05392e8			−0.00675146
$990$		−	1.45469e8i		−	0.00476482i
$991$	−2.03355e10			−0.663737			−0.331869	−	0.943326i	$-0.607679\pi$
$991$	−2.03355e10			−0.663737			−0.331869	+	0.943326i	$0.607679\pi$
$992$	1.88673e10			0.613649
$993$			5.25849e10i			1.70427i
$994$			2.17174e10i			0.701382i
$995$			2.48935e8i			0.00801133i
$996$		−	5.95884e9i		−	0.191097i
$997$	8.04557e9			0.257113			0.128556	−	0.991702i	$-0.458966\pi$
$997$	8.04557e9			0.257113			0.128556	+	0.991702i	$0.458966\pi$
$998$	1.99309e10			0.634702
$999$			1.85193e10i			0.587688i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.8.b.a.12.2	✓	6
3.2	odd	2		117.8.b.b.64.5		6
4.3	odd	2		208.8.f.a.129.1		6
13.5	odd	4		169.8.a.d.1.2		6
13.8	odd	4		169.8.a.d.1.5		6
13.12	even	2	inner	13.8.b.a.12.5	yes	6
39.38	odd	2		117.8.b.b.64.2		6
52.51	odd	2		208.8.f.a.129.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.b.a.12.2	✓	6	1.1	even	1	trivial
13.8.b.a.12.5	yes	6	13.12	even	2	inner
117.8.b.b.64.2		6	39.38	odd	2
117.8.b.b.64.5		6	3.2	odd	2
169.8.a.d.1.2		6	13.5	odd	4
169.8.a.d.1.5		6	13.8	odd	4
208.8.f.a.129.1		6	4.3	odd	2
208.8.f.a.129.2		6	52.51	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	13.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.0583i q^{2} +50.8656 q^{3} +26.8297 q^{4} -8.88672i q^{5} -511.623i q^{6} -510.452i q^{7} -1557.33i q^{8} +400.306 q^{9} -89.3858 q^{10} +4065.46i q^{11} +1364.71 q^{12} +(2772.33 + 7420.43i) q^{13} -5134.30 q^{14} -452.028i q^{15} -12230.0 q^{16} -1791.54 q^{17} -4026.41i q^{18} +19743.0i q^{19} -238.428i q^{20} -25964.4i q^{21} +40891.8 q^{22} -11749.0 q^{23} -79214.5i q^{24} +78046.0 q^{25} +(74637.2 - 27885.0i) q^{26} -90881.2 q^{27} -13695.3i q^{28} -183054. q^{29} -4546.66 q^{30} +247198. i q^{31} -76325.0i q^{32} +206792. i q^{33} +18020.0i q^{34} -4536.25 q^{35} +10740.1 q^{36} -203775. i q^{37} +198582. q^{38} +(141016. + 377444. i) q^{39} -13839.6 q^{40} -419322. i q^{41} -261159. q^{42} +17481.7 q^{43} +109075. i q^{44} -3557.41i q^{45} +118175. i q^{46} -1.17985e6i q^{47} -622084. q^{48} +562982. q^{49} -785014. i q^{50} -91127.9 q^{51} +(74380.6 + 199088. i) q^{52} +1.32587e6 q^{53} +914115. i q^{54} +36128.6 q^{55} -794942. q^{56} +1.00424e6i q^{57} +1.84123e6i q^{58} +401485. i q^{59} -12127.8i q^{60} -3.28655e6 q^{61} +2.48640e6 q^{62} -204337. i q^{63} -2.33314e6 q^{64} +(65943.3 - 24636.9i) q^{65} +2.07999e6 q^{66} -2.10629e6i q^{67} -48066.5 q^{68} -597618. q^{69} +45627.1i q^{70} -4.22986e6i q^{71} -623408. i q^{72} +3.29726e6i q^{73} -2.04964e6 q^{74} +3.96986e6 q^{75} +529699. i q^{76} +2.07522e6 q^{77} +(3.79646e6 - 1.41839e6i) q^{78} +3.79387e6 q^{79} +108684. i q^{80} -5.49819e6 q^{81} -4.21768e6 q^{82} -4.36639e6i q^{83} -696617. i q^{84} +15920.9i q^{85} -175837. i q^{86} -9.31117e6 q^{87} +6.33127e6 q^{88} +8.72168e6i q^{89} -35781.6 q^{90} +(3.78777e6 - 1.41514e6i) q^{91} -315221. q^{92} +1.25738e7i q^{93} -1.18673e7 q^{94} +175451. q^{95} -3.88231e6i q^{96} +9.21482e6i q^{97} -5.66267e6i q^{98} +1.62743e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9} - 406 q^{10} + 1898 q^{12} - 5018 q^{13} + 9558 q^{14} + 7778 q^{16} + 13152 q^{17} - 125080 q^{22} + 27264 q^{23} - 18262 q^{25} - 54210 q^{26} + 194560 q^{27} + 42924 q^{29}+ \cdots - 22075632 q^{95}+O(q^{100}) \) 6 * q - 56 * q^3 - 130 * q^4 - 1150 * q^9 - 406 * q^10 + 1898 * q^12 - 5018 * q^13 + 9558 * q^14 + 7778 * q^16 + 13152 * q^17 - 125080 * q^22 + 27264 * q^23 - 18262 * q^25 - 54210 * q^26 + 194560 * q^27 + 42924 * q^29 - 60114 * q^30 - 546720 * q^35 + 747772 * q^36 - 243492 * q^38 + 511160 * q^39 + 792058 * q^40 - 1138134 * q^42 - 1005576 * q^43 - 2807426 * q^48 + 3246846 * q^49 + 297984 * q^51 + 3252964 * q^52 + 1705524 * q^53 - 5320576 * q^55 - 2282730 * q^56 - 8237572 * q^61 + 9625800 * q^62 + 5070862 * q^64 + 7139028 * q^65 + 9289056 * q^66 - 16801086 * q^68 - 6017400 * q^69 - 28760826 * q^74 + 20426072 * q^75 + 4093560 * q^77 + 14088594 * q^78 + 15147200 * q^79 - 21318442 * q^81 - 9130240 * q^82 - 36474240 * q^87 + 38008000 * q^88 + 4769676 * q^90 + 21281832 * q^91 + 36462348 * q^92 - 34229770 * q^94 - 22075632 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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