LMFDB - Embedded newform 112.8.i.b.81.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,8,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$34.9871228542$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{949})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 238x^{2} + 237x + 56169 $ x^4 - x^3 + 238x^2 + 237x + 56169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.2
Root			$7.95146 - 13.7723i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.8.i.b.65.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(29.4029 - 50.9274i) q^{3} +(-212.141 - 367.439i) q^{5} +(30.7182 - 906.973i) q^{7} +(-635.564 - 1100.83i) q^{9} +(3944.43 - 6831.96i) q^{11} +6717.46 q^{13} -24950.2 q^{15} +(3537.51 - 6127.14i) q^{17} +(-13124.6 - 22732.4i) q^{19} +(-45286.5 - 28232.0i) q^{21} +(6035.27 + 10453.4i) q^{23} +(-50945.0 + 88239.4i) q^{25} +53858.7 q^{27} +3061.25 q^{29} +(-59262.6 + 102646. i) q^{31} +(-231956. - 401759. i) q^{33} +(-339774. + 181119. i) q^{35} +(229876. + 398156. i) q^{37} +(197513. - 342102. i) q^{39} +316857. q^{41} +31624.2 q^{43} +(-269658. + 467062. i) q^{45} +(403277. + 698496. i) q^{47} +(-821656. - 55721.1i) q^{49} +(-208026. - 360312. i) q^{51} +(-239664. + 415110. i) q^{53} -3.34710e6 q^{55} -1.54360e6 q^{57} +(337431. - 584448. i) q^{59} +(283165. + 490455. i) q^{61} +(-1.01794e6 + 542623. i) q^{63} +(-1.42505e6 - 2.46825e6i) q^{65} +(592349. - 1.02598e6i) q^{67} +709818. q^{69} -4.61736e6 q^{71} +(-1.52400e6 + 2.63965e6i) q^{73} +(2.99587e6 + 5.18899e6i) q^{75} +(-6.07524e6 - 3.78736e6i) q^{77} +(-3.45080e6 - 5.97696e6i) q^{79} +(2.97358e6 - 5.15039e6i) q^{81} +9.01927e6 q^{83} -3.00180e6 q^{85} +(90009.8 - 155902. i) q^{87} +(3.50739e6 + 6.07498e6i) q^{89} +(206348. - 6.09255e6i) q^{91} +(3.48499e6 + 6.03617e6i) q^{93} +(-5.56852e6 + 9.64496e6i) q^{95} +8.60029e6 q^{97} -1.00278e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 56 q^{3} + 14 q^{5} + 1848 q^{7} + 908 q^{9} + 2408 q^{11} + 21448 q^{13} - 52360 q^{15} + 35098 q^{17} - 2408 q^{19} - 92302 q^{21} - 61684 q^{23} - 215856 q^{25} + 83216 q^{27} - 191320 q^{29} - 166012 q^{31}+ \cdots - 43942528 q^{99}+O(q^{100}) $ 4 * q + 56 * q^3 + 14 * q^5 + 1848 * q^7 + 908 * q^9 + 2408 * q^11 + 21448 * q^13 - 52360 * q^15 + 35098 * q^17 - 2408 * q^19 - 92302 * q^21 - 61684 * q^23 - 215856 * q^25 + 83216 * q^27 - 191320 * q^29 - 166012 * q^31 - 479290 * q^33 - 166600 * q^35 + 559814 * q^37 + 383784 * q^39 + 1611960 * q^41 - 535952 * q^43 - 1494388 * q^45 + 1769292 * q^47 - 98588 * q^49 - 337424 * q^51 - 2317194 * q^53 - 11498536 * q^55 - 3220996 * q^57 + 660352 * q^59 - 1463042 * q^61 + 514472 * q^63 - 1094100 * q^65 - 1784280 * q^67 + 1833524 * q^69 - 548800 * q^71 - 4549062 * q^73 + 5671960 * q^75 - 15527806 * q^77 - 8673964 * q^79 + 1215782 * q^81 + 21188720 * q^83 + 18560332 * q^85 + 457016 * q^87 - 1779750 * q^89 + 7570640 * q^91 + 6836730 * q^93 - 21586180 * q^95 - 3497928 * q^97 - 43942528 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	29.4029	−	50.9274i	0.628733	−	1.08900i	−0.359074	−	0.933309i	$-0.616907\pi$
$3$	29.4029	−	50.9274i	0.628733	−	1.08900i	0.987806	−	0.155688i	$-0.0497594\pi$
$4$		0			0
$5$	−212.141	−	367.439i	−0.758978	−	1.31459i	−0.943372	−	0.331737i	$-0.892365\pi$
$5$	−212.141	−	367.439i	−0.758978	−	1.31459i	0.184394	−	0.982852i	$-0.440968\pi$
$6$		0			0
$7$	30.7182	−	906.973i	0.0338495	−	0.999427i
$7$	30.7182	−	906.973i	0.0338495	−	0.999427i
$8$		0			0
$9$	−635.564	−	1100.83i	−0.290610	−	0.503351i
$10$		0			0
$11$	3944.43	−	6831.96i	0.893532	−	1.54764i	0.0579219	−	0.998321i	$-0.481553\pi$
$11$	3944.43	−	6831.96i	0.893532	−	1.54764i	0.835611	−	0.549322i	$-0.185114\pi$
$12$		0			0
$13$	6717.46			0.848014			0.424007	−	0.905659i	$-0.360623\pi$
$13$	6717.46			0.848014			0.424007	+	0.905659i	$0.360623\pi$
$14$		0			0
$15$	−24950.2			−1.90878
$16$		0			0
$17$	3537.51	−	6127.14i	0.174633	−	0.302473i	−0.765401	−	0.643553i	$-0.777459\pi$
$17$	3537.51	−	6127.14i	0.174633	−	0.302473i	0.940034	+	0.341080i	$0.110793\pi$
$18$		0			0
$19$	−13124.6	−	22732.4i	−0.438983	−	0.760341i	0.558628	−	0.829418i	$-0.311328\pi$
$19$	−13124.6	−	22732.4i	−0.438983	−	0.760341i	−0.997611	+	0.0690773i	$0.977994\pi$
$20$		0			0
$21$	−45286.5	−	28232.0i	−1.06709	−	0.665234i
$22$		0			0
$23$	6035.27	+	10453.4i	0.103431	+	0.179147i	0.913096	−	0.407745i	$-0.133685\pi$
$23$	6035.27	+	10453.4i	0.103431	+	0.179147i	−0.809665	+	0.586892i	$0.800351\pi$
$24$		0			0
$25$	−50945.0	+	88239.4i	−0.652096	+	1.12946i
$26$		0			0
$27$	53858.7			0.526602
$28$		0			0
$29$	3061.25			0.0233081			0.0116540	−	0.999932i	$-0.496290\pi$
$29$	3061.25			0.0233081			0.0116540	+	0.999932i	$0.496290\pi$
$30$		0			0
$31$	−59262.6	+	102646.i	−0.357285	+	0.618835i	−0.987506	−	0.157580i	$-0.949631\pi$
$31$	−59262.6	+	102646.i	−0.357285	+	0.618835i	0.630221	+	0.776415i	$0.282964\pi$
$32$		0			0
$33$	−231956.	−	401759.i	−1.12359	−	1.94611i
$34$		0			0
$35$	−339774.	+	181119.i	−1.33953	+	0.714045i
$36$		0			0
$37$	229876.	+	398156.i	0.746083	+	1.29225i	0.949687	+	0.313200i	$0.101401\pi$
$37$	229876.	+	398156.i	0.746083	+	1.29225i	−0.203604	+	0.979053i	$0.565266\pi$
$38$		0			0
$39$	197513.	−	342102.i	0.533174	−	0.923485i
$40$		0			0
$41$	316857.			0.717992			0.358996	−	0.933339i	$-0.383119\pi$
$41$	316857.			0.717992			0.358996	+	0.933339i	$0.383119\pi$
$42$		0			0
$43$	31624.2			0.0606569			0.0303284	−	0.999540i	$-0.490345\pi$
$43$	31624.2			0.0606569			0.0303284	+	0.999540i	$0.490345\pi$
$44$		0			0
$45$	−269658.	+	467062.i	−0.441133	+	0.764065i
$46$		0			0
$47$	403277.	+	698496.i	0.566579	+	0.981344i	0.996901	+	0.0786682i	$0.0250668\pi$
$47$	403277.	+	698496.i	0.566579	+	0.981344i	−0.430322	+	0.902676i	$0.641600\pi$
$48$		0			0
$49$	−821656.	−	55721.1i	−0.997708	−	0.0676602i
$50$		0			0
$51$	−208026.	−	360312.i	−0.219595	−	0.380349i
$52$		0			0
$53$	−239664.	+	415110.i	−0.221125	+	0.382999i	−0.955150	−	0.296123i	$-0.904306\pi$
$53$	−239664.	+	415110.i	−0.221125	+	0.382999i	0.734025	+	0.679122i	$0.237639\pi$
$54$		0			0
$55$	−3.34710e6			−2.71269
$56$		0			0
$57$	−1.54360e6			−1.10401
$58$		0			0
$59$	337431.	−	584448.i	0.213896	−	0.370479i	−0.739034	−	0.673668i	$-0.764718\pi$
$59$	337431.	−	584448.i	0.213896	−	0.370479i	0.952931	+	0.303189i	$0.0980512\pi$
$60$		0			0
$61$	283165.	+	490455.i	0.159729	+	0.276659i	0.934771	−	0.355251i	$-0.115605\pi$
$61$	283165.	+	490455.i	0.159729	+	0.276659i	−0.775042	+	0.631910i	$0.782271\pi$
$62$		0			0
$63$	−1.01794e6	+	542623.i	−0.512899	+	0.273405i
$64$		0			0
$65$	−1.42505e6	−	2.46825e6i	−0.643625	−	1.11479i
$66$		0			0
$67$	592349.	−	1.02598e6i	0.240611	−	0.416751i	−0.720277	−	0.693686i	$-0.755985\pi$
$67$	592349.	−	1.02598e6i	0.240611	−	0.416751i	0.960889	+	0.276935i	$0.0893188\pi$
$68$		0			0
$69$	709818.			0.260121
$70$		0			0
$71$	−4.61736e6			−1.53105			−0.765524	−	0.643407i	$-0.777520\pi$
$71$	−4.61736e6			−1.53105			−0.765524	+	0.643407i	$0.777520\pi$
$72$		0			0
$73$	−1.52400e6	+	2.63965e6i	−0.458517	+	0.794175i	−0.998883	−	0.0472553i	$-0.984953\pi$
$73$	−1.52400e6	+	2.63965e6i	−0.458517	+	0.794175i	0.540366	+	0.841430i	$0.318286\pi$
$74$		0			0
$75$	2.99587e6	+	5.18899e6i	0.819989	+	1.42026i
$76$		0			0
$77$	−6.07524e6	−	3.78736e6i	−1.51651	−	0.945407i
$78$		0			0
$79$	−3.45080e6	−	5.97696e6i	−0.787454	−	1.36391i	−0.927522	−	0.373768i	$-0.878066\pi$
$79$	−3.45080e6	−	5.97696e6i	−0.787454	−	1.36391i	0.140069	−	0.990142i	$-0.455268\pi$
$80$		0			0
$81$	2.97358e6	−	5.15039e6i	0.621702	−	1.07682i
$82$		0			0
$83$	9.01927e6			1.73140			0.865701	−	0.500562i	$-0.166873\pi$
$83$	9.01927e6			1.73140			0.865701	+	0.500562i	$0.166873\pi$
$84$		0			0
$85$	−3.00180e6			−0.530170
$86$		0			0
$87$	90009.8	−	155902.i	0.0146545	−	0.0253824i
$88$		0			0
$89$	3.50739e6	+	6.07498e6i	0.527374	+	0.913439i	0.999491	+	0.0319032i	$0.0101568\pi$
$89$	3.50739e6	+	6.07498e6i	0.527374	+	0.913439i	−0.472116	+	0.881536i	$0.656510\pi$
$90$		0			0
$91$	206348.	−	6.09255e6i	0.0287049	−	0.847528i
$92$		0			0
$93$	3.48499e6	+	6.03617e6i	0.449273	+	0.778164i
$94$		0			0
$95$	−5.56852e6	+	9.64496e6i	−0.666357	+	1.15416i
$96$		0			0
$97$	8.60029e6			0.956779			0.478390	−	0.878148i	$-0.341221\pi$
$97$	8.60029e6			0.956779			0.478390	+	0.878148i	$0.341221\pi$
$98$		0			0
$99$	−1.00278e7			−1.03868
$100$		0			0
$101$	2.95178e6	−	5.11262e6i	0.285075	−	0.493764i	−0.687553	−	0.726134i	$-0.741315\pi$
$101$	2.95178e6	−	5.11262e6i	0.285075	−	0.493764i	0.972627	+	0.232371i	$0.0746483\pi$
$102$		0			0
$103$	3.68525e6	+	6.38303e6i	0.332305	+	0.575568i	0.982963	−	0.183802i	$-0.0588405\pi$
$103$	3.68525e6	+	6.38303e6i	0.332305	+	0.575568i	−0.650659	+	0.759370i	$0.725507\pi$
$104$		0			0
$105$	−766427.	+	2.26292e7i	−0.0646112	+	1.90768i
$106$		0			0
$107$	3.30235e6	+	5.71983e6i	0.260603	+	0.451378i	0.966402	−	0.257034i	$-0.0827453\pi$
$107$	3.30235e6	+	5.71983e6i	0.260603	+	0.451378i	−0.705799	+	0.708412i	$0.749412\pi$
$108$		0			0
$109$	−1.57280e6	+	2.72417e6i	−0.116327	+	0.201484i	−0.918309	−	0.395863i	$-0.870445\pi$
$109$	−1.57280e6	+	2.72417e6i	−0.116327	+	0.201484i	0.801982	+	0.597348i	$0.203779\pi$
$110$		0			0
$111$	2.70361e7			1.87635
$112$		0			0
$113$	2.26100e7			1.47409			0.737047	−	0.675841i	$-0.236220\pi$
$113$	2.26100e7			1.47409			0.737047	+	0.675841i	$0.236220\pi$
$114$		0			0
$115$	2.56066e6	−	4.43519e6i	0.157003	−	0.271938i
$116$		0			0
$117$	−4.26937e6	−	7.39477e6i	−0.246441	−	0.426849i
$118$		0			0
$119$	−5.44848e6	−	3.39664e6i	−0.296388	−	0.184771i
$120$		0			0
$121$	−2.13735e7	−	3.70200e7i	−1.09680	−	1.89971i
$122$		0			0
$123$	9.31652e6	−	1.61367e7i	0.451425	−	0.781891i
$124$		0			0
$125$	1.00831e7			0.461751
$126$		0			0
$127$	1.76143e7			0.763047			0.381524	−	0.924359i	$-0.375399\pi$
$127$	1.76143e7			0.763047			0.381524	+	0.924359i	$0.375399\pi$
$128$		0			0
$129$	929844.	−	1.61054e6i	0.0381370	−	0.0660552i
$130$		0			0
$131$	2.41997e6	+	4.19152e6i	0.0940505	+	0.162900i	0.909212	−	0.416334i	$-0.136685\pi$
$131$	2.41997e6	+	4.19152e6i	0.0940505	+	0.162900i	−0.815161	+	0.579234i	$0.803352\pi$
$132$		0			0
$133$	−2.10209e7	+	1.12053e7i	−0.774764	+	0.412994i
$134$		0			0
$135$	−1.14256e7	−	1.97898e7i	−0.399679	−	0.692265i
$136$		0			0
$137$	2.73018e7	−	4.72881e7i	0.907130	−	1.57120i	0.0890984	−	0.996023i	$-0.471601\pi$
$137$	2.73018e7	−	4.72881e7i	0.907130	−	1.57120i	0.818032	−	0.575173i	$-0.195065\pi$
$138$		0			0
$139$	−5.28540e6			−0.166927			−0.0834633	−	0.996511i	$-0.526598\pi$
$139$	−5.28540e6			−0.166927			−0.0834633	+	0.996511i	$0.526598\pi$
$140$		0			0
$141$	4.74300e7			1.42491
$142$		0			0
$143$	2.64966e7	−	4.58934e7i	0.757728	−	1.31242i
$144$		0			0
$145$	−649417.	−	1.12482e6i	−0.0176903	−	0.0306405i
$146$		0			0
$147$	−2.69968e7	+	4.02064e7i	−0.700974	+	1.04396i
$148$		0			0
$149$	1.29671e7	+	2.24596e7i	0.321136	+	0.556225i	0.980723	−	0.195404i	$-0.0626019\pi$
$149$	1.29671e7	+	2.24596e7i	0.321136	+	0.556225i	−0.659586	+	0.751629i	$0.729269\pi$
$150$		0			0
$151$	−2.29593e7	+	3.97666e7i	−0.542673	+	0.939938i	0.456076	+	0.889941i	$0.349255\pi$
$151$	−2.29593e7	+	3.97666e7i	−0.542673	+	0.939938i	−0.998749	+	0.0499970i	$0.984079\pi$
$152$		0			0
$153$	−8.99324e6			−0.203000
$154$		0			0
$155$	5.02881e7			1.08469
$156$		0			0
$157$	−3.90042e6	+	6.75572e6i	−0.0804381	+	0.139323i	−0.903438	−	0.428718i	$-0.858965\pi$
$157$	−3.90042e6	+	6.75572e6i	−0.0804381	+	0.139323i	0.823000	+	0.568041i	$0.192299\pi$
$158$		0			0
$159$	1.40936e7	+	2.44109e7i	0.278057	+	0.481608i
$160$		0			0
$161$	9.66634e6	−	5.15272e6i	0.182546	−	0.0973073i
$162$		0			0
$163$	−3.28712e7	−	5.69346e7i	−0.594510	−	1.02972i	−0.993616	−	0.112817i	$-0.964013\pi$
$163$	−3.28712e7	−	5.69346e7i	−0.594510	−	1.02972i	0.399106	−	0.916905i	$-0.369321\pi$
$164$		0			0
$165$	−9.84146e7	+	1.70459e8i	−1.70556	+	2.95411i
$166$		0			0
$167$	−2.42648e7			−0.403152			−0.201576	−	0.979473i	$-0.564606\pi$
$167$	−2.42648e7			−0.403152			−0.201576	+	0.979473i	$0.564606\pi$
$168$		0			0
$169$	−1.76243e7			−0.280872
$170$		0			0
$171$	−1.66830e7	+	2.88958e7i	−0.255145	+	0.441925i
$172$		0			0
$173$	1.73925e7	+	3.01246e7i	0.255388	+	0.442344i	0.965001	−	0.262247i	$-0.0844636\pi$
$173$	1.73925e7	+	3.01246e7i	0.255388	+	0.442344i	−0.709613	+	0.704592i	$0.751130\pi$
$174$		0			0
$175$	7.84658e7	+	4.89163e7i	1.10674	+	0.689954i
$176$		0			0
$177$	−1.98429e7	−	3.43690e7i	−0.268967	−	0.465865i
$178$		0			0
$179$	−3.19751e7	+	5.53825e7i	−0.416703	+	0.721751i	−0.995606	−	0.0936461i	$-0.970148\pi$
$179$	−3.19751e7	+	5.53825e7i	−0.416703	+	0.721751i	0.578903	+	0.815397i	$0.303481\pi$
$180$		0			0
$181$	1.03303e8			1.29491			0.647454	−	0.762104i	$-0.275834\pi$
$181$	1.03303e8			1.29491			0.647454	+	0.762104i	$0.275834\pi$
$182$		0			0
$183$	3.33035e7			0.401708
$184$		0			0
$185$	9.75321e7	−	1.68931e8i	1.13252	−	1.96159i
$186$		0			0
$187$	−2.79069e7	−	4.83362e7i	−0.312080	−	0.540539i
$188$		0			0
$189$	1.65444e6	−	4.88483e7i	0.0178252	−	0.526300i
$190$		0			0
$191$	−4.27228e7	−	7.39981e7i	−0.443652	−	0.768429i	0.554305	−	0.832314i	$-0.312984\pi$
$191$	−4.27228e7	−	7.39981e7i	−0.443652	−	0.768429i	−0.997957	+	0.0638852i	$0.979651\pi$
$192$		0			0
$193$	−2.24321e7	+	3.88535e7i	−0.224605	+	0.389027i	−0.956201	−	0.292712i	$-0.905442\pi$
$193$	−2.24321e7	+	3.88535e7i	−0.224605	+	0.389027i	0.731596	+	0.681738i	$0.238776\pi$
$194$		0			0
$195$	−1.67602e8			−1.61867
$196$		0			0
$197$	−1.97306e8			−1.83869			−0.919345	−	0.393452i	$-0.871281\pi$
$197$	−1.97306e8			−1.83869			−0.919345	+	0.393452i	$0.871281\pi$
$198$		0			0
$199$	2.70550e7	−	4.68606e7i	0.243367	−	0.421523i	−0.718304	−	0.695729i	$-0.755081\pi$
$199$	2.70550e7	−	4.68606e7i	0.243367	−	0.421523i	0.961671	+	0.274205i	$0.0884148\pi$
$200$		0			0
$201$	−3.48336e7	−	6.03335e7i	−0.302560	−	0.524050i
$202$		0			0
$203$	94036.2	−	2.77647e6i	0.000788967	−	0.0232947i
$204$		0			0
$205$	−6.72183e7	−	1.16426e8i	−0.544940	−	0.943864i
$206$		0			0
$207$	7.67160e6	−	1.32876e7i	0.0601159	−	0.104124i
$208$		0			0
$209$	−2.07076e8			−1.56898
$210$		0			0
$211$	−9.24251e7			−0.677331			−0.338666	−	0.940907i	$-0.609976\pi$
$211$	−9.24251e7			−0.677331			−0.338666	+	0.940907i	$0.609976\pi$
$212$		0			0
$213$	−1.35764e8	+	2.35150e8i	−0.962621	+	1.66731i
$214$		0			0
$215$	−6.70879e6	−	1.16200e7i	−0.0460373	−	0.0797389i
$216$		0			0
$217$	9.12765e7	+	5.69026e7i	0.606387	+	0.378027i
$218$		0			0
$219$	8.96202e7	+	1.55227e8i	0.576569	+	0.998648i
$220$		0			0
$221$	2.37630e7	−	4.11588e7i	0.148091	−	0.256501i
$222$		0			0
$223$	3.72959e7			0.225213			0.112607	−	0.993640i	$-0.464080\pi$
$223$	3.72959e7			0.225213			0.112607	+	0.993640i	$0.464080\pi$
$224$		0			0
$225$	1.29515e8			0.758022
$226$		0			0
$227$	−5.71956e7	+	9.90658e7i	−0.324543	+	0.562125i	−0.981420	−	0.191873i	$-0.938544\pi$
$227$	−5.71956e7	+	9.90658e7i	−0.324543	+	0.562125i	0.656877	+	0.753998i	$0.271877\pi$
$228$		0			0
$229$	4.17127e7	+	7.22486e7i	0.229533	+	0.397562i	0.957670	−	0.287869i	$-0.0929469\pi$
$229$	4.17127e7	+	7.22486e7i	0.229533	+	0.397562i	−0.728137	+	0.685432i	$0.759614\pi$
$230$		0			0
$231$	−3.71510e8	+	1.98036e8i	−1.98303	+	1.05707i
$232$		0			0
$233$	1.52762e7	+	2.64592e7i	0.0791170	+	0.137035i	0.902869	−	0.429915i	$-0.141457\pi$
$233$	1.52762e7	+	2.64592e7i	0.0791170	+	0.137035i	−0.823752	+	0.566950i	$0.808123\pi$
$234$		0			0
$235$	1.71103e8	−	2.96359e8i	0.860042	−	1.48964i
$236$		0			0
$237$	−4.05854e8			−1.98039
$238$		0			0
$239$	−5.35396e6			−0.0253678			−0.0126839	−	0.999920i	$-0.504038\pi$
$239$	−5.35396e6			−0.0253678			−0.0126839	+	0.999920i	$0.504038\pi$
$240$		0			0
$241$	9.27620e7	−	1.60668e8i	0.426884	−	0.739385i	−0.569710	−	0.821846i	$-0.692944\pi$
$241$	9.27620e7	−	1.60668e8i	0.426884	−	0.739385i	0.996594	+	0.0824604i	$0.0262778\pi$
$242$		0			0
$243$	−1.15969e8	−	2.00865e8i	−0.518467	−	0.898012i
$244$		0			0
$245$	1.53833e8	+	3.13729e8i	0.668294	+	1.36293i
$246$		0			0
$247$	−8.81638e7	−	1.52704e8i	−0.372264	−	0.644780i
$248$		0			0
$249$	2.65193e8	−	4.59327e8i	1.08859	−	1.88549i
$250$		0			0
$251$	−1.50737e6			−0.00601673			−0.00300837	−	0.999995i	$-0.500958\pi$
$251$	−1.50737e6			−0.00601673			−0.00300837	+	0.999995i	$0.500958\pi$
$252$		0			0
$253$	9.52229e7			0.369675
$254$		0			0
$255$	−8.82617e7	+	1.52874e8i	−0.333335	+	0.577354i
$256$		0			0
$257$	1.58688e8	+	2.74855e8i	0.583145	+	1.01004i	0.995104	+	0.0988349i	$0.0315116\pi$
$257$	1.58688e8	+	2.74855e8i	0.583145	+	1.01004i	−0.411958	+	0.911203i	$0.635155\pi$
$258$		0			0
$259$	3.68178e8	−	1.96260e8i	1.31677	−	0.701913i
$260$		0			0
$261$	−1.94562e6	−	3.36992e6i	−0.00677355	−	0.0117321i
$262$		0			0
$263$	6.83571e7	−	1.18398e8i	0.231707	−	0.401328i	−0.726604	−	0.687057i	$-0.758902\pi$
$263$	6.83571e7	−	1.18398e8i	0.231707	−	0.401328i	0.958310	+	0.285729i	$0.0922357\pi$
$264$		0			0
$265$	2.03370e8			0.671316
$266$		0			0
$267$	4.12510e8			1.32631
$268$		0			0
$269$	−9.36123e7	+	1.62141e8i	−0.293224	+	0.507879i	−0.974570	−	0.224082i	$-0.928062\pi$
$269$	−9.36123e7	+	1.62141e8i	−0.293224	+	0.507879i	0.681346	+	0.731961i	$0.261395\pi$
$270$		0			0
$271$	2.27535e8	+	3.94103e8i	0.694475	+	1.20287i	0.970357	+	0.241674i	$0.0776966\pi$
$271$	2.27535e8	+	3.94103e8i	0.694475	+	1.20287i	−0.275882	+	0.961191i	$0.588970\pi$
$272$		0			0
$273$	−3.04210e8	−	1.89648e8i	−0.904908	−	0.564128i
$274$		0			0
$275$	4.01899e8	+	6.96109e8i	1.16534	+	2.01843i
$276$		0			0
$277$	2.48231e8	−	4.29950e8i	0.701742	−	1.21545i	−0.266112	−	0.963942i	$-0.585739\pi$
$277$	2.48231e8	−	4.29950e8i	0.701742	−	1.21545i	0.967854	−	0.251511i	$-0.0809275\pi$
$278$		0			0
$279$	1.50661e8			0.415322
$280$		0			0
$281$	−2.68528e7			−0.0721968			−0.0360984	−	0.999348i	$-0.511493\pi$
$281$	−2.68528e7			−0.0721968			−0.0360984	+	0.999348i	$0.511493\pi$
$282$		0			0
$283$	−1.68539e8	+	2.91918e8i	−0.442026	+	0.765612i	−0.997840	−	0.0656950i	$-0.979074\pi$
$283$	−1.68539e8	+	2.91918e8i	−0.442026	+	0.765612i	0.555813	+	0.831307i	$0.312407\pi$
$284$		0			0
$285$	3.27461e8	+	5.67180e8i	0.837921	+	1.45132i
$286$		0			0
$287$	9.73327e6	−	2.87381e8i	0.0243037	−	0.717581i
$288$		0			0
$289$	1.80141e8	+	3.12014e8i	0.439007	+	0.760382i
$290$		0			0
$291$	2.52874e8	−	4.37990e8i	0.601559	−	1.04193i
$292$		0			0
$293$	−5.65397e8			−1.31316			−0.656579	−	0.754258i	$-0.727997\pi$
$293$	−5.65397e8			−1.31316			−0.656579	+	0.754258i	$0.727997\pi$
$294$		0			0
$295$	−2.86332e8			−0.649371
$296$		0			0
$297$	2.12442e8	−	3.67960e8i	0.470536	−	0.814992i
$298$		0			0
$299$	4.05417e7	+	7.02202e7i	0.0877107	+	0.151919i
$300$		0			0
$301$	971439.	−	2.86823e7i	0.00205321	−	0.0606221i
$302$		0			0
$303$	−1.73582e8	−	3.00652e8i	−0.358471	−	0.620891i
$304$		0			0
$305$	1.20142e8	−	2.08091e8i	0.242462	−	0.419957i
$306$		0			0
$307$	−7.13941e8			−1.40824			−0.704122	−	0.710079i	$-0.748659\pi$
$307$	−7.13941e8			−1.40824			−0.704122	+	0.710079i	$0.748659\pi$
$308$		0			0
$309$	4.33428e8			0.835723
$310$		0			0
$311$	2.28426e8	−	3.95646e8i	0.430611	−	0.745840i	−0.566315	−	0.824189i	$-0.691632\pi$
$311$	2.28426e8	−	3.95646e8i	0.430611	−	0.745840i	0.996926	+	0.0783491i	$0.0249649\pi$
$312$		0			0
$313$	−4.80069e8	−	8.31504e8i	−0.884909	−	1.53271i	−0.845818	−	0.533472i	$-0.820887\pi$
$313$	−4.80069e8	−	8.31504e8i	−0.884909	−	1.53271i	−0.0390914	−	0.999236i	$-0.512446\pi$
$314$		0			0
$315$	4.15329e8	+	2.58920e8i	0.748695	+	0.466744i
$316$		0			0
$317$	2.87942e8	+	4.98730e8i	0.507688	+	0.879342i	0.999960	+	0.00890063i	$0.00283320\pi$
$317$	2.87942e8	+	4.98730e8i	0.507688	+	0.879342i	−0.492272	+	0.870441i	$0.663833\pi$
$318$		0			0
$319$	1.20749e7	−	2.09144e7i	0.0208265	−	0.0360726i
$320$		0			0
$321$	3.88394e8			0.655398
$322$		0			0
$323$	−1.85713e8			−0.306643
$324$		0			0
$325$	−3.42221e8	+	5.92744e8i	−0.552987	+	0.957802i
$326$		0			0
$327$	9.24898e7	+	1.60197e8i	0.146277	+	0.253359i
$328$		0			0
$329$	6.45904e8	−	3.44304e8i	0.999960	−	0.533036i
$330$		0			0
$331$	−2.23337e8	−	3.86831e8i	−0.338503	−	0.586304i	0.645648	−	0.763635i	$-0.276587\pi$
$331$	−2.23337e8	−	3.86831e8i	−0.338503	−	0.586304i	−0.984151	+	0.177330i	$0.943254\pi$
$332$		0			0
$333$	2.92201e8	−	5.06108e8i	0.433638	−	0.751083i
$334$		0			0
$335$	−5.02646e8			−0.730475
$336$		0			0
$337$	−1.08980e9			−1.55111			−0.775555	−	0.631280i	$-0.782530\pi$
$337$	−1.08980e9			−1.55111			−0.775555	+	0.631280i	$0.782530\pi$
$338$		0			0
$339$	6.64799e8	−	1.15147e9i	0.926811	−	1.60528i
$340$		0			0
$341$	4.67515e8	+	8.09759e8i	0.638491	+	1.10590i
$342$		0			0
$343$	−7.57773e7	+	7.43508e8i	−0.101393	+	0.994846i
$344$		0			0
$345$	−1.50581e8	−	2.60815e8i	−0.197426	−	0.341952i
$346$		0			0
$347$	5.50402e8	−	9.53325e8i	0.707175	−	1.22486i	−0.258725	−	0.965951i	$-0.583302\pi$
$347$	5.50402e8	−	9.53325e8i	0.707175	−	1.22486i	0.965901	−	0.258913i	$-0.0833642\pi$
$348$		0			0
$349$	−9.17714e8			−1.15563			−0.577814	−	0.816168i	$-0.696094\pi$
$349$	−9.17714e8			−1.15563			−0.577814	+	0.816168i	$0.696094\pi$
$350$		0			0
$351$	3.61793e8			0.446566
$352$		0			0
$353$	−1.06680e8	+	1.84776e8i	−0.129084	+	0.223581i	−0.923322	−	0.384027i	$-0.874537\pi$
$353$	−1.06680e8	+	1.84776e8i	−0.129084	+	0.223581i	0.794238	+	0.607607i	$0.207870\pi$
$354$		0			0
$355$	9.79530e8	+	1.69660e9i	1.16203	+	2.01270i
$356$		0			0
$357$	−3.33183e8	+	1.77606e8i	−0.387565	+	0.206594i
$358$		0			0
$359$	5.94524e7	+	1.02975e8i	0.0678171	+	0.117463i	0.897940	−	0.440118i	$-0.145063\pi$
$359$	5.94524e7	+	1.02975e8i	0.0678171	+	0.117463i	−0.830123	+	0.557580i	$0.811730\pi$
$360$		0			0
$361$	1.02427e8	−	1.77409e8i	0.114588	−	0.198472i
$362$		0			0
$363$	−2.51378e9			−2.75838
$364$		0			0
$365$	1.29321e9			1.39202
$366$		0			0
$367$	1.37498e8	−	2.38153e8i	0.145199	−	0.251492i	−0.784248	−	0.620447i	$-0.786951\pi$
$367$	1.37498e8	−	2.38153e8i	0.145199	−	0.251492i	0.929447	+	0.368955i	$0.120284\pi$
$368$		0			0
$369$	−2.01383e8	−	3.48805e8i	−0.208656	−	0.361402i
$370$		0			0
$371$	3.69132e8	+	2.30120e8i	0.375295	+	0.233962i
$372$		0			0
$373$	−4.47876e8	−	7.75745e8i	−0.446866	−	0.773995i	0.551314	−	0.834298i	$-0.314127\pi$
$373$	−4.47876e8	−	7.75745e8i	−0.446866	−	0.773995i	−0.998180	+	0.0603032i	$0.980793\pi$
$374$		0			0
$375$	2.96472e8	−	5.13505e8i	0.290318	−	0.502846i
$376$		0			0
$377$	2.05638e7			0.0197656
$378$		0			0
$379$	−1.69750e9			−1.60167			−0.800833	−	0.598887i	$-0.795610\pi$
$379$	−1.69750e9			−1.60167			−0.800833	+	0.598887i	$0.795610\pi$
$380$		0			0
$381$	5.17911e8	−	8.97048e8i	0.479753	−	0.830956i
$382$		0			0
$383$	6.24533e8	+	1.08172e9i	0.568014	+	0.983830i	0.996762	+	0.0804057i	$0.0256216\pi$
$383$	6.24533e8	+	1.08172e9i	0.568014	+	0.983830i	−0.428748	+	0.903424i	$0.641045\pi$
$384$		0			0
$385$	−1.02817e8	+	3.03573e9i	−0.0918231	+	2.71113i
$386$		0			0
$387$	−2.00992e7	−	3.48128e7i	−0.0176275	−	0.0305317i
$388$		0			0
$389$	−6.80349e8	+	1.17840e9i	−0.586014	+	1.01501i	0.408734	+	0.912653i	$0.365970\pi$
$389$	−6.80349e8	+	1.17840e9i	−0.586014	+	1.01501i	−0.994748	+	0.102352i	$0.967363\pi$
$390$		0			0
$391$	8.53992e7			0.0722496
$392$		0			0
$393$	2.84617e8			0.236531
$394$		0			0
$395$	−1.46411e9	+	2.53592e9i	−1.19532	+	2.07036i
$396$		0			0
$397$	−1.03304e9	−	1.78928e9i	−0.828610	−	1.43519i	−0.899129	−	0.437684i	$-0.855799\pi$
$397$	−1.03304e9	−	1.78928e9i	−0.828610	−	1.43519i	0.0705188	−	0.997510i	$-0.477535\pi$
$398$		0			0
$399$	−4.74167e7	+	1.40001e9i	−0.0373703	+	1.10338i
$400$		0			0
$401$	7.29599e7	+	1.26370e8i	0.0565040	+	0.0978677i	0.892894	−	0.450267i	$-0.148671\pi$
$401$	7.29599e7	+	1.26370e8i	0.0565040	+	0.0978677i	−0.836390	+	0.548135i	$0.815338\pi$
$402$		0			0
$403$	−3.98094e8	+	6.89519e8i	−0.302983	+	0.524781i
$404$		0			0
$405$	−2.52327e9			−1.88743
$406$		0			0
$407$	3.62692e9			2.66660
$408$		0			0
$409$	−5.95951e8	+	1.03222e9i	−0.430704	+	0.746001i	−0.996934	−	0.0782460i	$-0.975068\pi$
$409$	−5.95951e8	+	1.03222e9i	−0.430704	+	0.746001i	0.566230	+	0.824247i	$0.308401\pi$
$410$		0			0
$411$	−1.60551e9	−	2.78082e9i	−1.14069	−	1.97572i
$412$		0			0
$413$	−5.19713e8	−	3.23994e8i	−0.363027	−	0.226314i
$414$		0			0
$415$	−1.91336e9	−	3.31403e9i	−1.31410	−	2.27608i
$416$		0			0
$417$	−1.55406e8	+	2.69171e8i	−0.104952	+	0.181783i
$418$		0			0
$419$	−5.89725e8			−0.391652			−0.195826	−	0.980639i	$-0.562739\pi$
$419$	−5.89725e8			−0.391652			−0.195826	+	0.980639i	$0.562739\pi$
$420$		0			0
$421$	2.35971e9			1.54124			0.770622	−	0.637292i	$-0.219946\pi$
$421$	2.35971e9			1.54124			0.770622	+	0.637292i	$0.219946\pi$
$422$		0			0
$423$	5.12616e8	−	8.87877e8i	0.329307	−	0.570376i
$424$		0			0
$425$	3.60437e8	+	6.24295e8i	0.227755	+	0.394483i
$426$		0			0
$427$	4.53528e8	−	2.41757e8i	0.281907	−	0.150273i
$428$		0			0
$429$	−1.55815e9	−	2.69880e9i	−0.952817	−	1.65033i
$430$		0			0
$431$	−1.29323e9	+	2.23994e9i	−0.778045	+	1.34761i	0.155022	+	0.987911i	$0.450455\pi$
$431$	−1.29323e9	+	2.23994e9i	−0.778045	+	1.34761i	−0.933067	+	0.359703i	$0.882878\pi$
$432$		0			0
$433$	−1.48937e9			−0.881649			−0.440825	−	0.897593i	$-0.645314\pi$
$433$	−1.48937e9			−0.881649			−0.440825	+	0.897593i	$0.645314\pi$
$434$		0			0
$435$	−7.63790e7			−0.0444899
$436$		0			0
$437$	1.58421e8	−	2.74393e8i	0.0908086	−	0.157285i
$438$		0			0
$439$	−1.33593e9	−	2.31390e9i	−0.753629	−	1.30532i	−0.946053	−	0.324012i	$-0.894968\pi$
$439$	−1.33593e9	−	2.31390e9i	−0.753629	−	1.30532i	0.192424	−	0.981312i	$-0.438365\pi$
$440$		0			0
$441$	4.60875e8	+	9.39916e8i	0.255887	+	0.521860i
$442$		0			0
$443$	6.94903e7	+	1.20361e8i	0.0379762	+	0.0657767i	0.884389	−	0.466751i	$-0.154576\pi$
$443$	6.94903e7	+	1.20361e8i	0.0379762	+	0.0657767i	−0.846413	+	0.532528i	$0.821242\pi$
$444$		0			0
$445$	1.48812e9	−	2.57750e9i	0.800532	−	1.38656i
$446$		0			0
$447$	1.52508e9			0.807636
$448$		0			0
$449$	−9.13167e8			−0.476089			−0.238044	−	0.971254i	$-0.576506\pi$
$449$	−9.13167e8			−0.476089			−0.238044	+	0.971254i	$0.576506\pi$
$450$		0			0
$451$	1.24982e9	−	2.16475e9i	0.641549	−	1.11120i
$452$		0			0
$453$	1.35014e9	+	2.33851e9i	0.682393	+	1.18194i
$454$		0			0
$455$	−2.28241e9	+	1.21666e9i	−1.13594	+	0.605521i
$456$		0			0
$457$	4.12098e8	+	7.13774e8i	0.201973	+	0.349828i	0.949164	−	0.314782i	$-0.101931\pi$
$457$	4.12098e8	+	7.13774e8i	0.201973	+	0.349828i	−0.747191	+	0.664609i	$0.768598\pi$
$458$		0			0
$459$	1.90525e8	−	3.30000e8i	0.0919620	−	0.159283i
$460$		0			0
$461$	1.64108e9			0.780147			0.390073	−	0.920784i	$-0.372450\pi$
$461$	1.64108e9			0.780147			0.390073	+	0.920784i	$0.372450\pi$
$462$		0			0
$463$	−2.87188e9			−1.34472			−0.672361	−	0.740224i	$-0.734720\pi$
$463$	−2.87188e9			−1.34472			−0.672361	+	0.740224i	$0.734720\pi$
$464$		0			0
$465$	1.47862e9	−	2.56104e9i	0.681977	−	1.18122i
$466$		0			0
$467$	1.47195e9	+	2.54949e9i	0.668782	+	1.15836i	0.978245	+	0.207453i	$0.0665173\pi$
$467$	1.47195e9	+	2.54949e9i	0.668782	+	1.15836i	−0.309463	+	0.950911i	$0.600149\pi$
$468$		0			0
$469$	−9.12338e8	−	5.68760e8i	−0.408367	−	0.254580i
$470$		0			0
$471$	2.29367e8	+	3.97276e8i	0.101148	+	0.175194i
$472$		0			0
$473$	1.24740e8	−	2.16055e8i	0.0541989	−	0.0938752i
$474$		0			0
$475$	2.67453e9			1.14504
$476$		0			0
$477$	6.09287e8			0.257044
$478$		0			0
$479$	4.19445e8	−	7.26500e8i	0.174382	−	0.302038i	−0.765566	−	0.643358i	$-0.777541\pi$
$479$	4.19445e8	−	7.26500e8i	0.174382	−	0.302038i	0.939947	+	0.341320i	$0.110874\pi$
$480$		0			0
$481$	1.54418e9	+	2.67460e9i	0.632689	+	1.09585i
$482$		0			0
$483$	2.18043e7	−	6.43786e8i	0.00880497	−	0.259972i
$484$		0			0
$485$	−1.82447e9	−	3.16008e9i	−0.726175	−	1.25777i
$486$		0			0
$487$	1.22668e8	−	2.12468e8i	0.0481261	−	0.0833569i	−0.840959	−	0.541099i	$-0.818008\pi$
$487$	1.22668e8	−	2.12468e8i	0.0481261	−	0.0833569i	0.889085	+	0.457742i	$0.151342\pi$
$488$		0			0
$489$	−3.86604e9			−1.49515
$490$		0			0
$491$	4.86528e9			1.85491			0.927455	−	0.373936i	$-0.121992\pi$
$491$	4.86528e9			1.85491			0.927455	+	0.373936i	$0.121992\pi$
$492$		0			0
$493$	1.08292e7	−	1.87567e7i	0.00407036	−	0.00705006i
$494$		0			0
$495$	2.12730e9	+	3.68459e9i	0.788333	+	1.36543i
$496$		0			0
$497$	−1.41837e8	+	4.18782e9i	−0.0518253	+	1.53017i
$498$		0			0
$499$	9.41538e8	+	1.63079e9i	0.339223	+	0.587552i	0.984287	−	0.176577i	$-0.0565024\pi$
$499$	9.41538e8	+	1.63079e9i	0.339223	+	0.587552i	−0.645063	+	0.764129i	$0.723169\pi$
$500$		0			0
$501$	−7.13456e8	+	1.23574e9i	−0.253475	+	0.439031i
$502$		0			0
$503$	−4.05510e9			−1.42074			−0.710369	−	0.703830i	$-0.751472\pi$
$503$	−4.05510e9			−1.42074			−0.710369	+	0.703830i	$0.751472\pi$
$504$		0			0
$505$	−2.50477e9			−0.865462
$506$		0			0
$507$	−5.18206e8	+	8.97558e8i	−0.176593	+	0.305869i
$508$		0			0
$509$	2.94236e9	+	5.09632e9i	0.988972	+	1.71295i	0.622745	+	0.782425i	$0.286017\pi$
$509$	2.94236e9	+	5.09632e9i	0.988972	+	1.71295i	0.366227	+	0.930526i	$0.380649\pi$
$510$		0			0
$511$	2.34727e9	+	1.46331e9i	0.778199	+	0.485137i
$512$		0			0
$513$	−7.06872e8	−	1.22434e9i	−0.231169	−	0.400397i
$514$		0			0
$515$	1.56358e9	−	2.70821e9i	0.504424	−	0.873688i
$516$		0			0
$517$	6.36279e9			2.02503
$518$		0			0
$519$	2.04556e9			0.642282
$520$		0			0
$521$	1.82384e9	−	3.15899e9i	0.565009	−	0.978625i	−0.432039	−	0.901855i	$-0.642206\pi$
$521$	1.82384e9	−	3.15899e9i	0.565009	−	0.978625i	0.997049	−	0.0767702i	$-0.0244608\pi$
$522$		0			0
$523$	1.17775e9	+	2.03993e9i	0.359996	+	0.623532i	0.987960	−	0.154711i	$-0.0494446\pi$
$523$	1.17775e9	+	2.03993e9i	0.359996	+	0.623532i	−0.627964	+	0.778243i	$0.716111\pi$
$524$		0			0
$525$	4.79830e9	−	2.55777e9i	1.44720	−	0.771444i
$526$		0			0
$527$	4.19283e8	+	7.26220e8i	0.124787	+	0.216138i
$528$		0			0
$529$	1.62956e9	−	2.82249e9i	0.478604	−	0.828967i
$530$		0			0
$531$	−8.57836e8			−0.248641
$532$		0			0
$533$	2.12847e9			0.608867
$534$		0			0
$535$	1.40113e9	−	2.42682e9i	0.395584	−	0.685172i
$536$		0			0
$537$	1.88032e9	+	3.25682e9i	0.523990	+	0.907576i
$538$		0			0
$539$	−3.62165e9	+	5.39373e9i	−0.996199	+	1.48364i
$540$		0			0
$541$	2.74393e8	+	4.75263e8i	0.0745045	+	0.129046i	0.900871	−	0.434088i	$-0.142929\pi$
$541$	2.74393e8	+	4.75263e8i	0.0745045	+	0.129046i	−0.826366	+	0.563133i	$0.809596\pi$
$542$		0			0
$543$	3.03742e9	−	5.26096e9i	0.814151	−	1.41015i
$544$		0			0
$545$	1.33462e9			0.353159
$546$		0			0
$547$	−1.19987e8			−0.0313457			−0.0156729	−	0.999877i	$-0.504989\pi$
$547$	−1.19987e8			−0.0313457			−0.0156729	+	0.999877i	$0.504989\pi$
$548$		0			0
$549$	3.59938e8	−	6.23431e8i	0.0928378	−	0.160800i
$550$		0			0
$551$	−4.01777e7	−	6.95897e7i	−0.0102318	−	0.0177221i
$552$		0			0
$553$	−5.52694e9	+	2.94618e9i	−1.38978	+	0.740835i
$554$		0			0
$555$	−5.73546e9	−	9.93410e9i	−1.42411	−	2.46663i
$556$		0			0
$557$	1.53796e8	−	2.66382e8i	0.0377096	−	0.0653149i	−0.846555	−	0.532302i	$-0.821327\pi$
$557$	1.53796e8	−	2.66382e8i	0.0377096	−	0.0653149i	0.884264	+	0.466987i	$0.154660\pi$
$558$		0			0
$559$	2.12434e8			0.0514379
$560$		0			0
$561$	−3.28218e9			−0.784860
$562$		0			0
$563$	6.69100e8	−	1.15892e9i	0.158020	−	0.273699i	−0.776135	−	0.630567i	$-0.782822\pi$
$563$	6.69100e8	−	1.15892e9i	0.158020	−	0.273699i	0.934155	+	0.356869i	$0.116156\pi$
$564$		0			0
$565$	−4.79650e9	−	8.30778e9i	−1.11881	−	1.93783i
$566$		0			0
$567$	−4.57992e9	−	2.85517e9i	−1.05516	−	0.657795i
$568$		0			0
$569$	3.24443e9	+	5.61952e9i	0.738322	+	1.27881i	0.953250	+	0.302182i	$0.0977149\pi$
$569$	3.24443e9	+	5.61952e9i	0.738322	+	1.27881i	−0.214928	+	0.976630i	$0.568952\pi$
$570$		0			0
$571$	3.48596e9	−	6.03786e9i	0.783603	−	1.35724i	−0.146227	−	0.989251i	$-0.546713\pi$
$571$	3.48596e9	−	6.03786e9i	0.783603	−	1.35724i	0.929830	−	0.367989i	$-0.119954\pi$
$572$		0			0
$573$	−5.02470e9			−1.11576
$574$		0			0
$575$	−1.22987e9			−0.269787
$576$		0			0
$577$	2.26789e9	−	3.92810e9i	0.491481	−	0.851270i	−0.508471	−	0.861079i	$-0.669789\pi$
$577$	2.26789e9	−	3.92810e9i	0.491481	−	0.851270i	0.999952	+	0.00980933i	$0.00312246\pi$
$578$		0			0
$579$	1.31914e9	+	2.28481e9i	0.282433	+	0.489188i
$580$		0			0
$581$	2.77055e8	−	8.18023e9i	0.0586071	−	1.73041i
$582$		0			0
$583$	1.89068e9	+	3.27475e9i	0.395164	+	0.684445i
$584$		0			0
$585$	−1.81142e9	+	3.13747e9i	−0.374087	+	0.647938i
$586$		0			0
$587$	5.05375e9			1.03129			0.515644	−	0.856803i	$-0.327553\pi$
$587$	5.05375e9			1.03129			0.515644	+	0.856803i	$0.327553\pi$
$588$		0			0
$589$	3.11118e9			0.627368
$590$		0			0
$591$	−5.80137e9	+	1.00483e10i	−1.15604	+	2.00233i
$592$		0			0
$593$	−1.90968e8	−	3.30766e8i	−0.0376070	−	0.0651373i	0.846609	−	0.532215i	$-0.178640\pi$
$593$	−1.90968e8	−	3.30766e8i	−0.0376070	−	0.0651373i	−0.884216	+	0.467078i	$0.845307\pi$
$594$		0			0
$595$	−9.22098e7	+	2.72255e9i	−0.0179460	+	0.529867i
$596$		0			0
$597$	−1.59099e9	−	2.75568e9i	−0.306025	−	0.530051i
$598$		0			0
$599$	8.94729e8	−	1.54972e9i	0.170097	−	0.294617i	−0.768356	−	0.640022i	$-0.778925\pi$
$599$	8.94729e8	−	1.54972e9i	0.170097	−	0.294617i	0.938454	+	0.345405i	$0.112258\pi$
$600$		0			0
$601$	−3.92179e8			−0.0736925			−0.0368463	−	0.999321i	$-0.511731\pi$
$601$	−3.92179e8			−0.0736925			−0.0368463	+	0.999321i	$0.511731\pi$
$602$		0			0
$603$	−1.50590e9			−0.279696
$604$		0			0
$605$	−9.06840e9	+	1.57069e10i	−1.66490	+	2.88368i
$606$		0			0
$607$	−5.59514e8	−	9.69107e8i	−0.101543	−	0.175878i	0.810777	−	0.585354i	$-0.199045\pi$
$607$	−5.59514e8	−	9.69107e8i	−0.101543	−	0.175878i	−0.912321	+	0.409477i	$0.865711\pi$
$608$		0			0
$609$	−1.38634e8	−	8.64254e7i	−0.0248718	−	0.0155053i
$610$		0			0
$611$	2.70899e9	+	4.69211e9i	0.480467	+	0.832193i
$612$		0			0
$613$	2.22809e9	−	3.85916e9i	0.390680	−	0.676677i	−0.601860	−	0.798602i	$-0.705573\pi$
$613$	2.22809e9	−	3.85916e9i	0.390680	−	0.676677i	0.992539	+	0.121925i	$0.0389066\pi$
$614$		0			0
$615$	−7.90566e9			−1.37049
$616$		0			0
$617$	−2.50707e9			−0.429704			−0.214852	−	0.976647i	$-0.568927\pi$
$617$	−2.50707e9			−0.429704			−0.214852	+	0.976647i	$0.568927\pi$
$618$		0			0
$619$	−7.42231e7	+	1.28558e8i	−0.0125783	+	0.0217862i	−0.872246	−	0.489067i	$-0.837337\pi$
$619$	−7.42231e7	+	1.28558e8i	−0.0125783	+	0.0217862i	0.859668	+	0.510854i	$0.170671\pi$
$620$		0			0
$621$	3.25052e8	+	5.63006e8i	0.0544668	+	0.0943392i
$622$		0			0
$623$	5.61758e9	−	2.99450e9i	0.930767	−	0.496153i
$624$		0			0
$625$	1.84105e9	+	3.18879e9i	0.301637	+	0.522451i
$626$		0			0
$627$	−6.08864e9	+	1.05458e10i	−0.986470	+	1.70862i
$628$		0			0
$629$	3.25275e9			0.521163
$630$		0			0
$631$	−6.90977e9			−1.09487			−0.547433	−	0.836850i	$-0.684395\pi$
$631$	−6.90977e9			−1.09487			−0.547433	+	0.836850i	$0.684395\pi$
$632$		0			0
$633$	−2.71757e9	+	4.70696e9i	−0.425860	+	0.737612i
$634$		0			0
$635$	−3.73671e9	−	6.47217e9i	−0.579136	−	1.00309i
$636$		0			0
$637$	−5.51944e9	−	3.74304e8i	−0.846071	−	0.0573768i
$638$		0			0
$639$	2.93462e9	+	5.08292e9i	0.444938	+	0.770655i
$640$		0			0
$641$	7.83493e8	−	1.35705e9i	0.117498	−	0.203513i	−0.801277	−	0.598293i	$-0.795846\pi$
$641$	7.83493e8	−	1.35705e9i	0.117498	−	0.203513i	0.918776	+	0.394780i	$0.129179\pi$
$642$		0			0
$643$	−7.91354e9			−1.17390			−0.586952	−	0.809622i	$-0.699672\pi$
$643$	−7.91354e9			−1.17390			−0.586952	+	0.809622i	$0.699672\pi$
$644$		0			0
$645$	−7.89032e8			−0.115781
$646$		0			0
$647$	1.93426e9	−	3.35023e9i	0.280769	−	0.486307i	−0.690805	−	0.723041i	$-0.742744\pi$
$647$	1.93426e9	−	3.35023e9i	0.280769	−	0.486307i	0.971574	+	0.236734i	$0.0760771\pi$
$648$		0			0
$649$	−2.66195e9	−	4.61063e9i	−0.382247	−	0.662070i
$650$		0			0
$651$	5.58170e9	−	2.97537e9i	0.792926	−	0.422675i
$652$		0			0
$653$	−3.34714e9	−	5.79742e9i	−0.470412	−	0.814777i	0.529016	−	0.848612i	$-0.322561\pi$
$653$	−3.34714e9	−	5.79742e9i	−0.470412	−	0.814777i	−0.999427	+	0.0338348i	$0.989228\pi$
$654$		0			0
$655$	1.02675e9	−	1.77838e9i	0.142765	−	0.247276i
$656$		0			0
$657$	3.87440e9			0.532998
$658$		0			0
$659$	4.20015e9			0.571696			0.285848	−	0.958275i	$-0.407725\pi$
$659$	4.20015e9			0.571696			0.285848	+	0.958275i	$0.407725\pi$
$660$		0			0
$661$	4.50337e9	−	7.80006e9i	0.606502	−	1.05049i	−0.385310	−	0.922787i	$-0.625906\pi$
$661$	4.50337e9	−	7.80006e9i	0.606502	−	1.05049i	0.991812	−	0.127705i	$-0.0407611\pi$
$662$		0			0
$663$	−1.39741e9	−	2.42038e9i	−0.186220	−	0.322542i
$664$		0			0
$665$	8.57666e9	+	5.34677e9i	1.13095	+	0.705043i
$666$		0			0
$667$	1.84755e7	+	3.20005e7i	0.00241077	+	0.00417557i
$668$		0			0
$669$	1.09661e9	−	1.89938e9i	0.141599	−	0.245257i
$670$		0			0
$671$	4.46770e9			0.570893
$672$		0			0
$673$	−6.02502e9			−0.761913			−0.380957	−	0.924593i	$-0.624405\pi$
$673$	−6.02502e9			−0.761913			−0.380957	+	0.924593i	$0.624405\pi$
$674$		0			0
$675$	−2.74383e9	+	4.75246e9i	−0.343395	+	0.594778i
$676$		0			0
$677$	3.88772e9	+	6.73374e9i	0.481543	+	0.834057i	0.999776	−	0.0211827i	$-0.00674317\pi$
$677$	3.88772e9	+	6.73374e9i	0.481543	+	0.834057i	−0.518233	+	0.855240i	$0.673410\pi$
$678$		0			0
$679$	2.64185e8	−	7.80022e9i	0.0323865	−	0.956231i
$680$		0			0
$681$	3.36344e9	+	5.82565e9i	0.408102	+	0.706853i
$682$		0			0
$683$	−3.80036e9	+	6.58242e9i	−0.456407	+	0.790520i	−0.998768	−	0.0496253i	$-0.984197\pi$
$683$	−3.80036e9	+	6.58242e9i	−0.456407	+	0.790520i	0.542361	+	0.840146i	$0.317531\pi$
$684$		0			0
$685$	−2.31673e10			−2.75397
$686$		0			0
$687$	4.90590e9			0.577259
$688$		0			0
$689$	−1.60993e9	+	2.78849e9i	−0.187517	+	0.324789i
$690$		0			0
$691$	−2.51431e9	−	4.35491e9i	−0.289898	−	0.502118i	0.683887	−	0.729588i	$-0.260288\pi$
$691$	−2.51431e9	−	4.35491e9i	−0.289898	−	0.502118i	−0.973785	+	0.227470i	$0.926955\pi$
$692$		0			0
$693$	−3.08034e8	+	9.09490e9i	−0.0351587	+	1.03808i
$694$		0			0
$695$	1.12125e9	+	1.94206e9i	0.126694	+	0.219440i
$696$		0			0
$697$	1.12088e9	−	1.94143e9i	0.125385	−	0.217173i
$698$		0			0
$699$	1.79666e9			0.198974
$700$		0			0
$701$	1.07254e10			1.17598			0.587989	−	0.808869i	$-0.299920\pi$
$701$	1.07254e10			1.17598			0.587989	+	0.808869i	$0.299920\pi$
$702$		0			0
$703$	6.03404e9	−	1.04513e10i	0.655035	−	1.13455i
$704$		0			0
$705$	−1.00619e10	−	1.74276e10i	−1.08147	−	1.87317i
$706$		0			0
$707$	−4.54634e9	−	2.83423e9i	−0.483831	−	0.301625i
$708$		0			0
$709$	7.38414e9	+	1.27897e10i	0.778105	+	1.34772i	0.933033	+	0.359792i	$0.117152\pi$
$709$	7.38414e9	+	1.27897e10i	0.778105	+	1.34772i	−0.154927	+	0.987926i	$0.549514\pi$
$710$		0			0
$711$	−4.38641e9	+	7.59748e9i	−0.457683	+	0.792731i
$712$		0			0
$713$	−1.43066e9			−0.147817
$714$		0			0
$715$	−2.24840e10			−2.30040
$716$		0			0
$717$	−1.57422e8	+	2.72663e8i	−0.0159496	+	0.0276255i
$718$		0			0
$719$	6.19788e9	+	1.07350e10i	0.621859	+	1.07709i	0.989139	+	0.146981i	$0.0469557\pi$
$719$	6.19788e9	+	1.07350e10i	0.621859	+	1.07709i	−0.367280	+	0.930110i	$0.619711\pi$
$720$		0			0
$721$	5.90244e9	−	3.14634e9i	0.586487	−	0.312631i
$722$		0			0
$723$	−5.45494e9	−	9.44824e9i	−0.536792	−	0.929752i
$724$		0			0
$725$	−1.55956e8	+	2.70123e8i	−0.0151991	+	0.0263256i
$726$		0			0
$727$	−1.18455e10			−1.14336			−0.571681	−	0.820476i	$-0.693709\pi$
$727$	−1.18455e10			−1.14336			−0.571681	+	0.820476i	$0.693709\pi$
$728$		0			0
$729$	−6.32921e8			−0.0605066
$730$		0			0
$731$	1.11871e8	−	1.93766e8i	0.0105927	−	0.0183471i
$732$		0			0
$733$	3.69991e9	+	6.40844e9i	0.346998	+	0.601019i	0.985715	−	0.168423i	$-0.0538676\pi$
$733$	3.69991e9	+	6.40844e9i	0.346998	+	0.601019i	−0.638716	+	0.769442i	$0.720534\pi$
$734$		0			0
$735$	2.05005e10	+	1.39026e9i	1.90440	+	0.129148i
$736$		0			0
$737$	−4.67296e9	−	8.09381e9i	−0.429988	−	0.744760i
$738$		0			0
$739$	−5.92456e9	+	1.02616e10i	−0.540009	+	0.935323i	0.458894	+	0.888491i	$0.348246\pi$
$739$	−5.92456e9	+	1.02616e10i	−0.540009	+	0.935323i	−0.998903	+	0.0468318i	$0.985088\pi$
$740$		0			0
$741$	−1.03691e10			−0.936218
$742$		0			0
$743$	−4.53308e9			−0.405446			−0.202723	−	0.979236i	$-0.564979\pi$
$743$	−4.53308e9			−0.405446			−0.202723	+	0.979236i	$0.564979\pi$
$744$		0			0
$745$	5.50169e9	−	9.52920e9i	0.487471	−	0.844325i
$746$		0			0
$747$	−5.73232e9	−	9.92866e9i	−0.503162	−	0.871503i
$748$		0			0
$749$	5.28917e9	−	2.81943e9i	0.459940	−	0.245175i
$750$		0			0
$751$	1.08837e10	+	1.88510e10i	0.937638	+	1.62404i	0.769862	+	0.638211i	$0.220325\pi$
$751$	1.08837e10	+	1.88510e10i	0.937638	+	1.62404i	0.167776	+	0.985825i	$0.446342\pi$
$752$		0			0
$753$	−4.43210e7	+	7.67661e7i	−0.00378292	+	0.00655220i
$754$		0			0
$755$	1.94824e10			1.64751
$756$		0			0
$757$	2.03442e9			0.170453			0.0852266	−	0.996362i	$-0.472839\pi$
$757$	2.03442e9			0.170453			0.0852266	+	0.996362i	$0.472839\pi$
$758$		0			0
$759$	2.79983e9	−	4.84945e9i	0.232427	−	0.402575i
$760$		0			0
$761$	−6.30476e9	−	1.09202e10i	−0.518587	−	0.898220i	−0.999767	−	0.0215975i	$-0.993125\pi$
$761$	−6.30476e9	−	1.09202e10i	−0.518587	−	0.898220i	0.481179	−	0.876622i	$-0.340209\pi$
$762$		0			0
$763$	2.42243e9	+	1.51017e9i	0.197431	+	0.123080i
$764$		0			0
$765$	1.90783e9	+	3.30447e9i	0.154073	+	0.266862i
$766$		0			0
$767$	2.26668e9	−	3.92601e9i	0.181387	−	0.314172i
$768$		0			0
$769$	1.47113e10			1.16657			0.583284	−	0.812268i	$-0.301768\pi$
$769$	1.47113e10			1.16657			0.583284	+	0.812268i	$0.301768\pi$
$770$		0			0
$771$	1.86635e10			1.46657
$772$		0			0
$773$	−2.89303e8	+	5.01088e8i	−0.0225281	+	0.0390198i	−0.877070	−	0.480363i	$-0.840505\pi$
$773$	−2.89303e8	+	5.01088e8i	−0.0225281	+	0.0390198i	0.854542	+	0.519383i	$0.173838\pi$
$774$		0			0
$775$	−6.03827e9	−	1.04586e10i	−0.465968	−	0.807081i
$776$		0			0
$777$	8.30499e8	−	2.45210e10i	0.0635135	−	1.87527i
$778$		0			0
$779$	−4.15861e9	−	7.20293e9i	−0.315186	−	0.545919i
$780$		0			0
$781$	−1.82129e10	+	3.15456e10i	−1.36804	+	2.36952i
$782$		0			0
$783$	1.64875e8			0.0122741
$784$		0			0
$785$	3.30975e9			0.244203
$786$		0			0
$787$	1.66639e9	−	2.88627e9i	0.121861	−	0.211069i	−0.798641	−	0.601808i	$-0.794447\pi$
$787$	1.66639e9	−	2.88627e9i	0.121861	−	0.211069i	0.920502	+	0.390739i	$0.127780\pi$
$788$		0			0
$789$	−4.01980e9	−	6.96249e9i	−0.291363	−	0.504656i
$790$		0			0
$791$	6.94537e8	−	2.05066e10i	0.0498974	−	1.47325i
$792$		0			0
$793$	1.90215e9	+	3.29461e9i	0.135453	+	0.234611i
$794$		0			0
$795$	5.97968e9	−	1.03571e10i	0.422078	−	0.731061i
$796$		0			0
$797$	1.88465e10			1.31864			0.659322	−	0.751861i	$-0.270843\pi$
$797$	1.88465e10			1.31864			0.659322	+	0.751861i	$0.270843\pi$
$798$		0			0
$799$	5.70637e9			0.395773
$800$		0			0
$801$	4.45834e9	−	7.72207e9i	0.306520	−	0.530909i
$802$		0			0
$803$	1.20227e10	+	2.08238e10i	0.819400	+	1.41924i
$804$		0			0
$805$	−3.94393e9	−	2.45869e9i	−0.266467	−	0.166118i
$806$		0			0
$807$	5.50495e9	+	9.53485e9i	0.368719	+	0.638641i
$808$		0			0
$809$	−4.69404e8	+	8.13032e8i	−0.0311693	+	0.0539868i	−0.881189	−	0.472764i	$-0.843256\pi$
$809$	−4.69404e8	+	8.13032e8i	−0.0311693	+	0.0539868i	0.850020	+	0.526750i	$0.176590\pi$
$810$		0			0
$811$	2.78193e10			1.83136			0.915679	−	0.401910i	$-0.131654\pi$
$811$	2.78193e10			1.83136			0.915679	+	0.401910i	$0.131654\pi$
$812$		0			0
$813$	2.67608e10			1.74656
$814$		0			0
$815$	−1.39467e10	+	2.41563e10i	−0.902441	+	1.56307i
$816$		0			0
$817$	−4.15054e8	−	7.18895e8i	−0.0266273	−	0.0461199i
$818$		0			0
$819$	−6.83800e9	+	3.64505e9i	−0.434946	+	0.231851i
$820$		0			0
$821$	4.27495e9	+	7.40443e9i	0.269606	+	0.466972i	0.968760	−	0.248000i	$-0.0797732\pi$
$821$	4.27495e9	+	7.40443e9i	0.269606	+	0.466972i	−0.699154	+	0.714971i	$0.746440\pi$
$822$		0			0
$823$	6.44756e9	−	1.11675e10i	0.403177	−	0.698324i	−0.590930	−	0.806723i	$-0.701239\pi$
$823$	6.44756e9	−	1.11675e10i	0.403177	−	0.698324i	0.994107	+	0.108399i	$0.0345724\pi$
$824$		0			0
$825$	4.72680e10			2.93075
$826$		0			0
$827$	−4.73751e9			−0.291260			−0.145630	−	0.989339i	$-0.546521\pi$
$827$	−4.73751e9			−0.291260			−0.145630	+	0.989339i	$0.546521\pi$
$828$		0			0
$829$	−7.03830e9	+	1.21907e10i	−0.429069	+	0.743169i	−0.996791	−	0.0800516i	$-0.974491\pi$
$829$	−7.03830e9	+	1.21907e10i	−0.429069	+	0.743169i	0.567722	+	0.823220i	$0.307825\pi$
$830$		0			0
$831$	−1.45975e10	−	2.52835e10i	−0.882417	−	1.52839i
$832$		0			0
$833$	−3.24802e9	+	4.83729e9i	−0.194698	+	0.289964i
$834$		0			0
$835$	5.14755e9	+	8.91582e9i	0.305984	+	0.529979i
$836$		0			0
$837$	−3.19180e9	+	5.52837e9i	−0.188147	+	0.325880i
$838$		0			0
$839$	−1.66762e10			−0.974833			−0.487416	−	0.873170i	$-0.662061\pi$
$839$	−1.66762e10			−0.974833			−0.487416	+	0.873170i	$0.662061\pi$
$840$		0			0
$841$	−1.72405e10			−0.999457
$842$		0			0
$843$	−7.89551e8	+	1.36754e9i	−0.0453925	+	0.0786221i
$844$		0			0
$845$	3.73883e9	+	6.47585e9i	0.213176	+	0.369231i
$846$		0			0
$847$	−3.42327e10	+	1.82480e10i	−1.93575	+	1.03187i
$848$		0			0
$849$	9.91108e9	+	1.71665e10i	0.555833	+	0.962731i
$850$		0			0
$851$	−2.77472e9	+	4.80596e9i	−0.154336	+	0.267317i
$852$		0			0
$853$	2.75175e10			1.51805			0.759027	−	0.651060i	$-0.225675\pi$
$853$	2.75175e10			1.51805			0.759027	+	0.651060i	$0.225675\pi$
$854$		0			0
$855$	1.41566e10			0.774600
$856$		0			0
$857$	−1.78022e10	+	3.08344e10i	−0.966144	+	1.67341i	−0.259634	+	0.965707i	$0.583602\pi$
$857$	−1.78022e10	+	3.08344e10i	−0.966144	+	1.67341i	−0.706510	+	0.707704i	$0.749731\pi$
$858$		0			0
$859$	7.28789e9	+	1.26230e10i	0.392307	+	0.679495i	0.992753	−	0.120169i	$-0.0383438\pi$
$859$	7.28789e9	+	1.26230e10i	0.392307	+	0.679495i	−0.600447	+	0.799665i	$0.705010\pi$
$860$		0			0
$861$	−1.43493e10	−	8.94552e9i	−0.766163	−	0.477633i
$862$		0			0
$863$	3.26274e9	+	5.65123e9i	0.172800	+	0.299299i	0.939398	−	0.342829i	$-0.111385\pi$
$863$	3.26274e9	+	5.65123e9i	0.172800	+	0.299299i	−0.766598	+	0.642128i	$0.778052\pi$
$864$		0			0
$865$	7.37931e9	−	1.27813e10i	0.387667	−	0.671459i
$866$		0			0
$867$	2.11867e10			1.10407
$868$		0			0
$869$	−5.44458e10			−2.81446
$870$		0			0
$871$	3.97908e9	−	6.89196e9i	0.204042	−	0.353411i
$872$		0			0
$873$	−5.46603e9	−	9.46744e9i	−0.278049	−	0.481596i
$874$		0			0
$875$	3.09734e8	−	9.14508e9i	0.0156301	−	0.461487i
$876$		0			0
$877$	−1.88472e9	−	3.26443e9i	−0.0943514	−	0.163421i	0.814986	−	0.579480i	$-0.196744\pi$
$877$	−1.88472e9	−	3.26443e9i	−0.0943514	−	0.163421i	−0.909338	+	0.416059i	$0.863411\pi$
$878$		0			0
$879$	−1.66243e10	+	2.87942e10i	−0.825625	+	1.43002i
$880$		0			0
$881$	−6.65654e9			−0.327969			−0.163985	−	0.986463i	$-0.552435\pi$
$881$	−6.65654e9			−0.327969			−0.163985	+	0.986463i	$0.552435\pi$
$882$		0			0
$883$	2.18423e10			1.06767			0.533834	−	0.845590i	$-0.320751\pi$
$883$	2.18423e10			1.06767			0.533834	+	0.845590i	$0.320751\pi$
$884$		0			0
$885$	−8.41900e9	+	1.45821e10i	−0.408281	+	0.707163i
$886$		0			0
$887$	−4.55208e9	−	7.88443e9i	−0.219017	−	0.379348i	0.735491	−	0.677534i	$-0.236952\pi$
$887$	−4.55208e9	−	7.88443e9i	−0.219017	−	0.379348i	−0.954508	+	0.298187i	$0.903618\pi$
$888$		0			0
$889$	5.41078e8	−	1.59757e10i	0.0258288	−	0.762610i
$890$		0			0
$891$	−2.34582e10	−	4.06308e10i	−1.11102	−	1.92435i
$892$		0			0
$893$	1.05857e10	−	1.83349e10i	0.497437	−	0.861586i
$894$		0			0
$895$	2.71329e10			1.26507
$896$		0			0
$897$	4.76817e9			0.220586
$898$		0			0
$899$	−1.81418e8	+	3.14225e8i	−0.00832762	+	0.0144239i
$900$		0			0
$901$	1.69563e9	+	2.93691e9i	0.0772313	+	0.133769i
$902$		0			0
$903$	−1.43215e9	−	8.92816e8i	−0.0647264	−	0.0403511i
$904$		0			0
$905$	−2.19148e10	−	3.79576e10i	−0.982807	−	1.70227i
$906$		0			0
$907$	7.58589e9	−	1.31391e10i	0.337583	−	0.584712i	−0.646394	−	0.763004i	$-0.723724\pi$
$907$	7.58589e9	−	1.31391e10i	0.337583	−	0.584712i	0.983978	+	0.178292i	$0.0570572\pi$
$908$		0			0
$909$	−7.50416e9			−0.331382
$910$		0			0
$911$	−1.48713e10			−0.651679			−0.325839	−	0.945425i	$-0.605647\pi$
$911$	−1.48713e10			−0.651679			−0.325839	+	0.945425i	$0.605647\pi$
$912$		0			0
$913$	3.55759e10	−	6.16193e10i	1.54706	−	2.67959i
$914$		0			0
$915$	−7.06503e9	−	1.22370e10i	−0.304888	−	0.528081i
$916$		0			0
$917$	3.87593e9	−	2.06609e9i	0.165990	−	0.0884825i
$918$		0			0
$919$	−1.55406e10	−	2.69171e10i	−0.660486	−	1.14400i	−0.980488	−	0.196579i	$-0.937017\pi$
$919$	−1.55406e10	−	2.69171e10i	−0.660486	−	1.14400i	0.320002	−	0.947417i	$-0.396316\pi$
$920$		0			0
$921$	−2.09920e10	+	3.63591e10i	−0.885410	+	1.53357i
$922$		0			0
$923$	−3.10169e10			−1.29835
$924$		0			0
$925$	−4.68441e10			−1.94607
$926$		0			0
$927$	4.68442e9	−	8.11365e9i	0.193142	−	0.334532i
$928$		0			0
$929$	1.49789e10	+	2.59442e10i	0.612950	+	1.06166i	0.990740	+	0.135770i	$0.0433508\pi$
$929$	1.49789e10	+	2.59442e10i	0.612950	+	1.06166i	−0.377790	+	0.925891i	$0.623316\pi$
$930$		0			0
$931$	9.51721e9	+	1.94095e10i	0.386532	+	0.788300i
$932$		0			0
$933$	−1.34328e10	−	2.32663e10i	−0.541478	−	0.937868i
$934$		0			0
$935$	−1.18404e10	+	2.05082e10i	−0.473724	+	0.820515i
$936$		0			0
$937$	−1.75637e10			−0.697473			−0.348736	−	0.937221i	$-0.613389\pi$
$937$	−1.75637e10			−0.697473			−0.348736	+	0.937221i	$0.613389\pi$
$938$		0			0
$939$	−5.64618e10			−2.22549
$940$		0			0
$941$	8.69121e9	−	1.50536e10i	0.340029	−	0.588948i	−0.644408	−	0.764681i	$-0.722896\pi$
$941$	8.69121e9	−	1.50536e10i	0.340029	−	0.588948i	0.984438	+	0.175733i	$0.0562296\pi$
$942$		0			0
$943$	1.91232e9	+	3.31223e9i	0.0742624	+	0.128626i
$944$		0			0
$945$	−1.82998e10	+	9.75483e9i	−0.705397	+	0.376018i
$946$		0			0
$947$	2.00818e10	+	3.47828e10i	0.768385	+	1.33088i	0.938438	+	0.345447i	$0.112273\pi$
$947$	2.00818e10	+	3.47828e10i	0.768385	+	1.33088i	−0.170054	+	0.985435i	$0.554394\pi$
$948$		0			0
$949$	−1.02374e10	+	1.77317e10i	−0.388829	+	0.673472i
$950$		0			0
$951$	3.38653e10			1.27680
$952$		0			0
$953$	3.96693e10			1.48467			0.742335	−	0.670029i	$-0.233718\pi$
$953$	3.96693e10			1.48467			0.742335	+	0.670029i	$0.233718\pi$
$954$		0			0
$955$	−1.81265e10	+	3.13960e10i	−0.673445	+	1.16644i
$956$		0			0
$957$	−7.10075e8	−	1.22989e9i	−0.0261886	−	0.0453600i
$958$		0			0
$959$	−4.20504e10	−	2.62146e10i	−1.53959	−	0.959795i
$960$		0			0
$961$	6.73220e9	+	1.16605e10i	0.244695	+	0.423824i
$962$		0			0
$963$	4.19770e9	−	7.27063e9i	0.151468	−	0.262349i
$964$		0			0
$965$	1.90350e10			0.681880
$966$		0			0
$967$	−3.37872e10			−1.20160			−0.600800	−	0.799400i	$-0.705151\pi$
$967$	−3.37872e10			−1.20160			−0.600800	+	0.799400i	$0.705151\pi$
$968$		0			0
$969$	−5.46051e9	+	9.45788e9i	−0.192797	+	0.333934i
$970$		0			0
$971$	−7.77704e9	−	1.34702e10i	−0.272613	−	0.472180i	0.696917	−	0.717152i	$-0.254555\pi$
$971$	−7.77704e9	−	1.34702e10i	−0.272613	−	0.472180i	−0.969530	+	0.244972i	$0.921221\pi$
$972$		0			0
$973$	−1.62358e8	+	4.79371e9i	−0.00565039	+	0.166831i
$974$		0			0
$975$	2.01246e10	+	3.48568e10i	0.695362	+	1.20440i
$976$		0			0
$977$	1.01465e8	−	1.75742e8i	0.00348083	−	0.00602898i	−0.864280	−	0.503011i	$-0.832225\pi$
$977$	1.01465e8	−	1.75742e8i	0.00348083	−	0.00602898i	0.867761	+	0.496982i	$0.165559\pi$
$978$		0			0
$979$	5.53387e10			1.88490
$980$		0			0
$981$	3.99846e9			0.135223
$982$		0			0
$983$	2.28447e10	−	3.95681e10i	0.767092	−	1.32864i	−0.172041	−	0.985090i	$-0.555036\pi$
$983$	2.28447e10	−	3.95681e10i	0.767092	−	1.32864i	0.939133	−	0.343553i	$-0.111631\pi$
$984$		0			0
$985$	4.18567e10	+	7.24979e10i	1.39553	+	2.41712i
$986$		0			0
$987$	1.45696e9	−	4.30177e10i	0.0482324	−	1.42409i
$988$		0			0
$989$	1.90861e8	+	3.30580e8i	0.00627378	+	0.0108665i
$990$		0			0
$991$	−5.59741e9	+	9.69501e9i	−0.182696	+	0.316439i	−0.942798	−	0.333365i	$-0.891816\pi$
$991$	−5.59741e9	+	9.69501e9i	−0.182696	+	0.316439i	0.760102	+	0.649804i	$0.225149\pi$
$992$		0			0
$993$	−2.62670e10			−0.851312
$994$		0			0
$995$	−2.29579e10			−0.738840
$996$		0			0
$997$	−9.08278e9	+	1.57318e10i	−0.290259	+	0.502744i	−0.973871	−	0.227102i	$-0.927075\pi$
$997$	−9.08278e9	+	1.57318e10i	−0.290259	+	0.502744i	0.683612	+	0.729846i	$0.260408\pi$
$998$		0			0
$999$	1.23808e10	+	2.14442e10i	0.392889	+	0.680503i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.8.i.b.81.2		4
4.3	odd	2		14.8.c.b.11.1	yes	4
7.2	even	3	inner	112.8.i.b.65.2		4
12.11	even	2		126.8.g.d.109.2		4
28.3	even	6		98.8.a.d.1.1		2
28.11	odd	6		98.8.a.f.1.2		2
28.19	even	6		98.8.c.m.79.2		4
28.23	odd	6		14.8.c.b.9.1	✓	4
28.27	even	2		98.8.c.m.67.2		4
84.23	even	6		126.8.g.d.37.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.c.b.9.1	✓	4	28.23	odd	6
14.8.c.b.11.1	yes	4	4.3	odd	2
98.8.a.d.1.1		2	28.3	even	6
98.8.a.f.1.2		2	28.11	odd	6
98.8.c.m.67.2		4	28.27	even	2
98.8.c.m.79.2		4	28.19	even	6
112.8.i.b.65.2		4	7.2	even	3	inner
112.8.i.b.81.2		4	1.1	even	1	trivial
126.8.g.d.37.2		4	84.23	even	6
126.8.g.d.109.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(29.4029 - 50.9274i) q^{3} +(-212.141 - 367.439i) q^{5} +(30.7182 - 906.973i) q^{7} +(-635.564 - 1100.83i) q^{9} +(3944.43 - 6831.96i) q^{11} +6717.46 q^{13} -24950.2 q^{15} +(3537.51 - 6127.14i) q^{17} +(-13124.6 - 22732.4i) q^{19} +(-45286.5 - 28232.0i) q^{21} +(6035.27 + 10453.4i) q^{23} +(-50945.0 + 88239.4i) q^{25} +53858.7 q^{27} +3061.25 q^{29} +(-59262.6 + 102646. i) q^{31} +(-231956. - 401759. i) q^{33} +(-339774. + 181119. i) q^{35} +(229876. + 398156. i) q^{37} +(197513. - 342102. i) q^{39} +316857. q^{41} +31624.2 q^{43} +(-269658. + 467062. i) q^{45} +(403277. + 698496. i) q^{47} +(-821656. - 55721.1i) q^{49} +(-208026. - 360312. i) q^{51} +(-239664. + 415110. i) q^{53} -3.34710e6 q^{55} -1.54360e6 q^{57} +(337431. - 584448. i) q^{59} +(283165. + 490455. i) q^{61} +(-1.01794e6 + 542623. i) q^{63} +(-1.42505e6 - 2.46825e6i) q^{65} +(592349. - 1.02598e6i) q^{67} +709818. q^{69} -4.61736e6 q^{71} +(-1.52400e6 + 2.63965e6i) q^{73} +(2.99587e6 + 5.18899e6i) q^{75} +(-6.07524e6 - 3.78736e6i) q^{77} +(-3.45080e6 - 5.97696e6i) q^{79} +(2.97358e6 - 5.15039e6i) q^{81} +9.01927e6 q^{83} -3.00180e6 q^{85} +(90009.8 - 155902. i) q^{87} +(3.50739e6 + 6.07498e6i) q^{89} +(206348. - 6.09255e6i) q^{91} +(3.48499e6 + 6.03617e6i) q^{93} +(-5.56852e6 + 9.64496e6i) q^{95} +8.60029e6 q^{97} -1.00278e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 56 q^{3} + 14 q^{5} + 1848 q^{7} + 908 q^{9} + 2408 q^{11} + 21448 q^{13} - 52360 q^{15} + 35098 q^{17} - 2408 q^{19} - 92302 q^{21} - 61684 q^{23} - 215856 q^{25} + 83216 q^{27} - 191320 q^{29} - 166012 q^{31}+ \cdots - 43942528 q^{99}+O(q^{100}) \) 4 * q + 56 * q^3 + 14 * q^5 + 1848 * q^7 + 908 * q^9 + 2408 * q^11 + 21448 * q^13 - 52360 * q^15 + 35098 * q^17 - 2408 * q^19 - 92302 * q^21 - 61684 * q^23 - 215856 * q^25 + 83216 * q^27 - 191320 * q^29 - 166012 * q^31 - 479290 * q^33 - 166600 * q^35 + 559814 * q^37 + 383784 * q^39 + 1611960 * q^41 - 535952 * q^43 - 1494388 * q^45 + 1769292 * q^47 - 98588 * q^49 - 337424 * q^51 - 2317194 * q^53 - 11498536 * q^55 - 3220996 * q^57 + 660352 * q^59 - 1463042 * q^61 + 514472 * q^63 - 1094100 * q^65 - 1784280 * q^67 + 1833524 * q^69 - 548800 * q^71 - 4549062 * q^73 + 5671960 * q^75 - 15527806 * q^77 - 8673964 * q^79 + 1215782 * q^81 + 21188720 * q^83 + 18560332 * q^85 + 457016 * q^87 - 1779750 * q^89 + 7570640 * q^91 + 6836730 * q^93 - 21586180 * q^95 - 3497928 * q^97 - 43942528 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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