LMFDB - Embedded newform 208.8.i.b.113.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,8,Mod(81,208)] 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("208.81"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	208.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0] 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:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400 $ x^8 + 4654x^6 + 7012369x^4 + 3763719168*x^2 + 637953638400
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			113.1
Root			$-41.0833i$ of defining polynomial
Character	$\chi$	$=$	208.113
Dual form			208.8.i.b.81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-35.5792 + 61.6250i) q^{3} +523.489 q^{5} +(-208.401 - 360.960i) q^{7} +(-1438.26 - 2491.14i) q^{9} +(1711.39 - 2964.21i) q^{11} +(-6741.24 + 4159.83i) q^{13} +(-18625.3 + 32260.0i) q^{15} +(3269.02 + 5662.11i) q^{17} +(13990.2 + 24231.8i) q^{19} +29658.9 q^{21} +(53089.0 - 91952.9i) q^{23} +195916. q^{25} +49065.2 q^{27} +(-34273.0 + 59362.6i) q^{29} -51572.8 q^{31} +(121780. + 210929. i) q^{33} +(-109095. - 188959. i) q^{35} +(-24067.7 + 41686.5i) q^{37} +(-16501.7 - 563432. i) q^{39} +(301049. - 521432. i) q^{41} +(458126. + 793497. i) q^{43} +(-752914. - 1.30409e6i) q^{45} +326201. q^{47} +(324910. - 562760. i) q^{49} -465237. q^{51} +934068. q^{53} +(895894. - 1.55173e6i) q^{55} -1.99105e6 q^{57} +(587217. + 1.01709e6i) q^{59} +(1.44545e6 + 2.50359e6i) q^{61} +(-599469. + 1.03831e6i) q^{63} +(-3.52897e6 + 2.17763e6i) q^{65} +(159079. - 275533. i) q^{67} +(3.77773e6 + 6.54322e6i) q^{69} +(641869. + 1.11175e6i) q^{71} -1.67209e6 q^{73} +(-6.97054e6 + 1.20733e7i) q^{75} -1.42662e6 q^{77} +8.09160e6 q^{79} +(1.39977e6 - 2.42448e6i) q^{81} -5.57141e6 q^{83} +(1.71130e6 + 2.96405e6i) q^{85} +(-2.43881e6 - 4.22415e6i) q^{87} +(-3.93786e6 + 6.82057e6i) q^{89} +(2.90641e6 + 1.56641e6i) q^{91} +(1.83492e6 - 3.17817e6i) q^{93} +(7.32374e6 + 1.26851e7i) q^{95} +(3.40424e6 + 5.89631e6i) q^{97} -9.84569e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 556 q^{5} + 548 q^{7} - 5214 q^{9} + 7392 q^{11} - 25818 q^{13} - 15528 q^{15} + 28316 q^{17} + 99888 q^{19} + 182148 q^{21} + 33388 q^{23} + 173756 q^{25} - 212544 q^{27} + 93140 q^{29} - 622320 q^{31}+ \cdots + 20715312 q^{99}+O(q^{100}) $ 8 * q + 556 * q^5 + 548 * q^7 - 5214 * q^9 + 7392 * q^11 - 25818 * q^13 - 15528 * q^15 + 28316 * q^17 + 99888 * q^19 + 182148 * q^21 + 33388 * q^23 + 173756 * q^25 - 212544 * q^27 + 93140 * q^29 - 622320 * q^31 + 238638 * q^33 - 141544 * q^35 - 9636 * q^37 + 22932 * q^39 + 82892 * q^41 + 569264 * q^43 - 2303394 * q^45 + 1148400 * q^47 - 717798 * q^49 + 5459856 * q^51 + 2470700 * q^53 + 1092512 * q^55 + 7056924 * q^57 - 231504 * q^59 + 685684 * q^61 + 5951712 * q^63 - 6216678 * q^65 - 3271056 * q^67 + 5600034 * q^69 + 175012 * q^71 + 14275780 * q^73 - 22200960 * q^75 - 27830412 * q^77 + 14107904 * q^79 + 3758004 * q^81 - 1314576 * q^83 + 11814998 * q^85 + 7182900 * q^87 - 11452234 * q^89 - 16457168 * q^91 + 2984688 * q^93 + 23334088 * q^95 - 428002 * q^97 + 20715312 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−35.5792	+	61.6250i	−0.760802	+	1.31775i	0.181635	+	0.983366i	$0.441861\pi$
$3$	−35.5792	+	61.6250i	−0.760802	+	1.31775i	−0.942437	+	0.334383i	$0.891472\pi$
$4$		0			0
$5$	523.489			1.87289			0.936446	−	0.350812i	$-0.114094\pi$
$5$	523.489			1.87289			0.936446	+	0.350812i	$0.114094\pi$
$6$		0			0
$7$	−208.401	−	360.960i	−0.229644	−	0.397756i	0.728058	−	0.685515i	$-0.240423\pi$
$7$	−208.401	−	360.960i	−0.229644	−	0.397756i	−0.957703	+	0.287759i	$0.907090\pi$
$8$		0			0
$9$	−1438.26	−	2491.14i	−0.657641	−	1.13907i
$10$		0			0
$11$	1711.39	−	2964.21i	0.387681	−	0.671483i	−0.604456	−	0.796638i	$-0.706610\pi$
$11$	1711.39	−	2964.21i	0.387681	−	0.671483i	0.992137	+	0.125155i	$0.0399429\pi$
$12$		0			0
$13$	−6741.24	+	4159.83i	−0.851017	+	0.525139i
$13$	−6741.24	+	4159.83i	−0.851017	+	0.525139i
$14$		0			0
$15$	−18625.3	+	32260.0i	−1.42490	+	2.46800i
$16$		0			0
$17$	3269.02	+	5662.11i	0.161379	+	0.279516i	0.935363	−	0.353688i	$-0.115073\pi$
$17$	3269.02	+	5662.11i	0.161379	+	0.279516i	−0.773985	+	0.633204i	$0.781739\pi$
$18$		0			0
$19$	13990.2	+	24231.8i	0.467937	+	0.810491i	0.999329	−	0.0366351i	$-0.0116639\pi$
$19$	13990.2	+	24231.8i	0.467937	+	0.810491i	−0.531391	+	0.847127i	$0.678331\pi$
$20$		0			0
$21$	29658.9			0.698856
$22$		0			0
$23$	53089.0	−	91952.9i	0.909824	−	1.57586i	0.0955160	−	0.995428i	$-0.469550\pi$
$23$	53089.0	−	91952.9i	0.909824	−	1.57586i	0.814308	−	0.580433i	$-0.197117\pi$
$24$		0			0
$25$	195916.			2.50772
$26$		0			0
$27$	49065.2			0.479734
$28$		0			0
$29$	−34273.0	+	59362.6i	−0.260951	+	0.451980i	−0.966495	−	0.256686i	$-0.917369\pi$
$29$	−34273.0	+	59362.6i	−0.260951	+	0.451980i	0.705544	+	0.708666i	$0.250703\pi$
$30$		0			0
$31$	−51572.8			−0.310924			−0.155462	−	0.987842i	$-0.549687\pi$
$31$	−51572.8			−0.310924			−0.155462	+	0.987842i	$0.549687\pi$
$32$		0			0
$33$	121780.	+	210929.i	0.589897	+	1.02173i
$34$		0			0
$35$	−109095.	−	188959.i	−0.430099	−	0.744954i
$36$		0			0
$37$	−24067.7	+	41686.5i	−0.0781140	+	0.135297i	−0.902436	−	0.430824i	$-0.858223\pi$
$37$	−24067.7	+	41686.5i	−0.0781140	+	0.135297i	0.824322	+	0.566121i	$0.191556\pi$
$38$		0			0
$39$	−16501.7	−	563432.i	−0.0445452	−	1.52095i
$40$		0			0
$41$	301049.	−	521432.i	0.682172	−	1.18156i	−0.292145	−	0.956374i	$-0.594369\pi$
$41$	301049.	−	521432.i	0.682172	−	1.18156i	0.974317	−	0.225182i	$-0.0722976\pi$
$42$		0			0
$43$	458126.	+	793497.i	0.878709	+	1.52197i	0.852758	+	0.522306i	$0.174928\pi$
$43$	458126.	+	793497.i	0.878709	+	1.52197i	0.0259511	+	0.999663i	$0.491739\pi$
$44$		0			0
$45$	−752914.	−	1.30409e6i	−1.23169	−	2.13335i
$46$		0			0
$47$	326201.			0.458293			0.229146	−	0.973392i	$-0.426407\pi$
$47$	326201.			0.458293			0.229146	+	0.973392i	$0.426407\pi$
$48$		0			0
$49$	324910.	−	562760.i	0.394527	−	0.683341i
$50$		0			0
$51$	−465237.			−0.491110
$52$		0			0
$53$	934068.			0.861813			0.430907	−	0.902397i	$-0.358194\pi$
$53$	934068.			0.861813			0.430907	+	0.902397i	$0.358194\pi$
$54$		0			0
$55$	895894.	−	1.55173e6i	0.726085	−	1.25762i
$56$		0			0
$57$	−1.99105e6			−1.42403
$58$		0			0
$59$	587217.	+	1.01709e6i	0.372235	+	0.644729i	0.989909	−	0.141705i	$-0.0452584\pi$
$59$	587217.	+	1.01709e6i	0.372235	+	0.644729i	−0.617674	+	0.786434i	$0.711925\pi$
$60$		0			0
$61$	1.44545e6	+	2.50359e6i	0.815358	+	1.41224i	0.909071	+	0.416642i	$0.136793\pi$
$61$	1.44545e6	+	2.50359e6i	0.815358	+	1.41224i	−0.0937130	+	0.995599i	$0.529874\pi$
$62$		0			0
$63$	−599469.	+	1.03831e6i	−0.302047	+	0.523161i
$64$		0			0
$65$	−3.52897e6	+	2.17763e6i	−1.59386	+	0.983528i
$66$		0			0
$67$	159079.	−	275533.i	0.0646176	−	0.111921i	−0.831907	−	0.554916i	$-0.812751\pi$
$67$	159079.	−	275533.i	0.0646176	−	0.111921i	0.896524	+	0.442995i	$0.146084\pi$
$68$		0			0
$69$	3.77773e6	+	6.54322e6i	1.38439	+	2.39784i
$70$		0			0
$71$	641869.	+	1.11175e6i	0.212834	+	0.368640i	0.952600	−	0.304224i	$-0.0983972\pi$
$71$	641869.	+	1.11175e6i	0.212834	+	0.368640i	−0.739766	+	0.672864i	$0.765064\pi$
$72$		0			0
$73$	−1.67209e6			−0.503070			−0.251535	−	0.967848i	$-0.580935\pi$
$73$	−1.67209e6			−0.503070			−0.251535	+	0.967848i	$0.580935\pi$
$74$		0			0
$75$	−6.97054e6	+	1.20733e7i	−1.90788	+	3.30455i
$76$		0			0
$77$	−1.42662e6			−0.356115
$78$		0			0
$79$	8.09160e6			1.84646			0.923229	−	0.384250i	$-0.125540\pi$
$79$	8.09160e6			1.84646			0.923229	+	0.384250i	$0.125540\pi$
$80$		0			0
$81$	1.39977e6	−	2.42448e6i	0.292658	−	0.506899i
$82$		0			0
$83$	−5.57141e6			−1.06953			−0.534764	−	0.845002i	$-0.679599\pi$
$83$	−5.57141e6			−1.06953			−0.534764	+	0.845002i	$0.679599\pi$
$84$		0			0
$85$	1.71130e6	+	2.96405e6i	0.302245	+	0.523504i
$86$		0			0
$87$	−2.43881e6	−	4.22415e6i	−0.397064	−	0.687736i
$88$		0			0
$89$	−3.93786e6	+	6.82057e6i	−0.592100	+	1.02555i	0.401849	+	0.915706i	$0.368368\pi$
$89$	−3.93786e6	+	6.82057e6i	−0.592100	+	1.02555i	−0.993949	+	0.109842i	$0.964966\pi$
$90$		0			0
$91$	2.90641e6	+	1.56641e6i	0.404308	+	0.217902i
$92$		0			0
$93$	1.83492e6	−	3.17817e6i	0.236552	−	0.409720i
$94$		0			0
$95$	7.32374e6	+	1.26851e7i	0.876396	+	1.51796i
$96$		0			0
$97$	3.40424e6	+	5.89631e6i	0.378720	+	0.655963i	0.990876	−	0.134774i	$-0.0430309\pi$
$97$	3.40424e6	+	5.89631e6i	0.378720	+	0.655963i	−0.612156	+	0.790737i	$0.709698\pi$
$98$		0			0
$99$	−9.84569e6			−1.01982
$100$		0			0
$101$	−805060.	+	1.39441e6i	−0.0777506	+	0.134668i	−0.902279	−	0.431152i	$-0.858107\pi$
$101$	−805060.	+	1.39441e6i	−0.0777506	+	0.134668i	0.824529	+	0.565820i	$0.191440\pi$
$102$		0			0
$103$	−4.93664e6			−0.445145			−0.222572	−	0.974916i	$-0.571445\pi$
$103$	−4.93664e6			−0.445145			−0.222572	+	0.974916i	$0.571445\pi$
$104$		0			0
$105$	1.55261e7			1.30888
$106$		0			0
$107$	−6.13515e6	+	1.06264e7i	−0.484152	+	0.838576i	−0.999834	−	0.0182040i	$-0.994205\pi$
$107$	−6.13515e6	+	1.06264e7i	−0.484152	+	0.838576i	0.515682	+	0.856780i	$0.327539\pi$
$108$		0			0
$109$	−1.65278e7			−1.22243			−0.611213	−	0.791467i	$-0.709318\pi$
$109$	−1.65278e7			−1.22243			−0.611213	+	0.791467i	$0.709318\pi$
$110$		0			0
$111$	−1.71262e6	−	2.96634e6i	−0.118859	−	0.205869i
$112$		0			0
$113$	−6.64380e6	−	1.15074e7i	−0.433154	−	0.750244i	0.563989	−	0.825782i	$-0.309266\pi$
$113$	−6.64380e6	−	1.15074e7i	−0.433154	−	0.750244i	−0.997143	+	0.0755379i	$0.975933\pi$
$114$		0			0
$115$	2.77915e7	−	4.81363e7i	1.70400	−	2.95142i
$116$		0			0
$117$	2.00584e7	+	1.08105e7i	1.15783	+	0.624013i
$118$		0			0
$119$	1.36253e6	−	2.35997e6i	0.0741195	−	0.128379i
$120$		0			0
$121$	3.88588e6	+	6.73054e6i	0.199407	+	0.345383i
$122$		0			0
$123$	2.14222e7	+	3.71043e7i	1.03800	+	1.79786i
$124$		0			0
$125$	6.16623e7			2.82381
$126$		0			0
$127$	1.36436e6	−	2.36314e6i	0.0591040	−	0.102371i	−0.834959	−	0.550311i	$-0.814509\pi$
$127$	1.36436e6	−	2.36314e6i	0.0591040	−	0.102371i	0.894063	+	0.447940i	$0.147842\pi$
$128$		0			0
$129$	−6.51990e7			−2.67410
$130$		0			0
$131$	−1.96041e7			−0.761898			−0.380949	−	0.924596i	$-0.624403\pi$
$131$	−1.96041e7			−0.761898			−0.380949	+	0.924596i	$0.624403\pi$
$132$		0			0
$133$	5.83115e6	−	1.00999e7i	0.214918	−	0.372250i
$134$		0			0
$135$	2.56851e7			0.898490
$136$		0			0
$137$	−326199.	−	564993.i	−0.0108383	−	0.0187724i	0.860555	−	0.509357i	$-0.170117\pi$
$137$	−326199.	−	564993.i	−0.0108383	−	0.0187724i	−0.871394	+	0.490584i	$0.836783\pi$
$138$		0			0
$139$	4.60831e6	+	7.98182e6i	0.145542	+	0.252087i	0.929575	−	0.368633i	$-0.120174\pi$
$139$	4.60831e6	+	7.98182e6i	0.145542	+	0.252087i	−0.784033	+	0.620719i	$0.786841\pi$
$140$		0			0
$141$	−1.16060e7	+	2.01021e7i	−0.348670	+	0.603915i
$142$		0			0
$143$	793743.	+	2.71016e7i	0.0226988	+	0.775030i
$144$		0			0
$145$	−1.79415e7	+	3.10757e7i	−0.488733	+	0.846510i
$146$		0			0
$147$	2.31201e7	+	4.00451e7i	0.600314	+	1.03977i
$148$		0			0
$149$	1.22231e6	+	2.11710e6i	0.0302712	+	0.0524312i	0.880764	−	0.473555i	$-0.157030\pi$
$149$	1.22231e6	+	2.11710e6i	0.0302712	+	0.0524312i	−0.850493	+	0.525986i	$0.823696\pi$
$150$		0			0
$151$	2.73196e7			0.645736			0.322868	−	0.946444i	$-0.395353\pi$
$151$	2.73196e7			0.645736			0.322868	+	0.946444i	$0.395353\pi$
$152$		0			0
$153$	9.40341e6	−	1.62872e7i	0.212259	−	0.367643i
$154$		0			0
$155$	−2.69978e7			−0.582328
$156$		0			0
$157$	5.54490e7			1.14352			0.571761	−	0.820420i	$-0.306260\pi$
$157$	5.54490e7			1.14352			0.571761	+	0.820420i	$0.306260\pi$
$158$		0			0
$159$	−3.32334e7	+	5.75620e7i	−0.655670	+	1.13565i
$160$		0			0
$161$	−4.42551e7			−0.835744
$162$		0			0
$163$	1.60007e7	+	2.77139e7i	0.289388	+	0.501235i	0.973664	−	0.227988i	$-0.0732148\pi$
$163$	1.60007e7	+	2.77139e7i	0.289388	+	0.501235i	−0.684275	+	0.729224i	$0.739881\pi$
$164$		0			0
$165$	6.37504e7	+	1.10419e8i	1.10481	+	1.91359i
$166$		0			0
$167$	−9.67304e6	+	1.67542e7i	−0.160715	+	0.278366i	−0.935125	−	0.354318i	$-0.884713\pi$
$167$	−9.67304e6	+	1.67542e7i	−0.160715	+	0.278366i	0.774411	+	0.632683i	$0.218047\pi$
$168$		0			0
$169$	2.81401e7	−	5.60848e7i	0.448459	−	0.893804i
$170$		0			0
$171$	4.02432e7	−	6.97033e7i	0.615469	−	1.06602i
$172$		0			0
$173$	−2.06874e7	−	3.58317e7i	−0.303770	−	0.526145i	0.673217	−	0.739445i	$-0.264912\pi$
$173$	−2.06874e7	−	3.58317e7i	−0.303770	−	0.526145i	−0.976987	+	0.213300i	$0.931579\pi$
$174$		0			0
$175$	−4.08290e7	−	7.07179e7i	−0.575885	−	0.997462i
$176$		0			0
$177$	−8.35709e7			−1.13279
$178$		0			0
$179$	−1.61438e7	+	2.79619e7i	−0.210388	+	0.364403i	−0.951836	−	0.306608i	$-0.900806\pi$
$179$	−1.61438e7	+	2.79619e7i	−0.210388	+	0.364403i	0.741448	+	0.671010i	$0.234139\pi$
$180$		0			0
$181$	−8.08975e7			−1.01405			−0.507026	−	0.861931i	$-0.669255\pi$
$181$	−8.08975e7			−1.01405			−0.507026	+	0.861931i	$0.669255\pi$
$182$		0			0
$183$	−2.05712e8			−2.48130
$184$		0			0
$185$	−1.25992e7	+	2.18224e7i	−0.146299	+	0.253397i
$186$		0			0
$187$	2.23783e7			0.250254
$188$		0			0
$189$	−1.02252e7	−	1.77106e7i	−0.110168	−	0.190817i
$190$		0			0
$191$	−6.81711e6	−	1.18076e7i	−0.0707919	−	0.122615i	0.828457	−	0.560053i	$-0.189219\pi$
$191$	−6.81711e6	−	1.18076e7i	−0.0707919	−	0.122615i	−0.899249	+	0.437438i	$0.855886\pi$
$192$		0			0
$193$	−8.66220e6	+	1.50034e7i	−0.0867316	+	0.150224i	−0.906128	−	0.423004i	$-0.860976\pi$
$193$	−8.66220e6	+	1.50034e7i	−0.0867316	+	0.150224i	0.819396	+	0.573228i	$0.194309\pi$
$194$		0			0
$195$	−8.63844e6	−	2.94951e8i	−0.0834284	−	2.84858i
$196$		0			0
$197$	−3.11874e7	+	5.40181e7i	−0.290634	+	0.503393i	−0.973960	−	0.226720i	$-0.927200\pi$
$197$	−3.11874e7	+	5.40181e7i	−0.290634	+	0.503393i	0.683326	+	0.730114i	$0.260533\pi$
$198$		0			0
$199$	−1.20854e6	−	2.09325e6i	−0.0108711	−	0.0188293i	0.860539	−	0.509385i	$-0.170127\pi$
$199$	−1.20854e6	−	2.09325e6i	−0.0108711	−	0.0188293i	−0.871410	+	0.490556i	$0.836794\pi$
$200$		0			0
$201$	1.13198e7	+	1.96065e7i	0.0983225	+	0.170300i
$202$		0			0
$203$	2.85700e7			0.239704
$204$		0			0
$205$	1.57596e8	−	2.72964e8i	1.27763	−	2.21293i
$206$		0			0
$207$	−3.05423e8			−2.39335
$208$		0			0
$209$	9.57711e7			0.725642
$210$		0			0
$211$	4.86339e7	−	8.42364e7i	0.356411	−	0.617321i	−0.630948	−	0.775825i	$-0.717334\pi$
$211$	4.86339e7	−	8.42364e7i	0.356411	−	0.617321i	0.987358	+	0.158504i	$0.0506671\pi$
$212$		0			0
$213$	−9.13487e7			−0.647700
$214$		0			0
$215$	2.39824e8	+	4.15387e8i	1.64573	+	2.85048i
$216$		0			0
$217$	1.07478e7	+	1.86157e7i	0.0714020	+	0.123672i
$218$		0			0
$219$	5.94915e7	−	1.03042e8i	0.382737	−	0.662920i
$220$		0			0
$221$	−4.55907e7	−	2.45711e7i	−0.284121	−	0.153127i
$222$		0			0
$223$	8.35602e7	−	1.44730e8i	0.504583	−	0.873963i	−0.495403	−	0.868663i	$-0.664980\pi$
$223$	8.35602e7	−	1.44730e8i	0.504583	−	0.873963i	0.999986	−	0.00529982i	$-0.00168699\pi$
$224$		0			0
$225$	−2.81778e8	−	4.88054e8i	−1.64918	−	2.85647i
$226$		0			0
$227$	2.85340e7	+	4.94223e7i	0.161909	+	0.280435i	0.935553	−	0.353185i	$-0.114902\pi$
$227$	2.85340e7	+	4.94223e7i	0.161909	+	0.280435i	−0.773644	+	0.633620i	$0.781568\pi$
$228$		0			0
$229$	2.68753e8			1.47886			0.739432	−	0.673231i	$-0.235094\pi$
$229$	2.68753e8			1.47886			0.739432	+	0.673231i	$0.235094\pi$
$230$		0			0
$231$	5.07580e7	−	8.79154e7i	0.270933	−	0.469270i
$232$		0			0
$233$	−6.53750e7			−0.338584			−0.169292	−	0.985566i	$-0.554148\pi$
$233$	−6.53750e7			−0.338584			−0.169292	+	0.985566i	$0.554148\pi$
$234$		0			0
$235$	1.70763e8			0.858333
$236$		0			0
$237$	−2.87893e8	+	4.98645e8i	−1.40479	+	2.43317i
$238$		0			0
$239$	2.06797e8			0.979832			0.489916	−	0.871770i	$-0.337027\pi$
$239$	2.06797e8			0.979832			0.489916	+	0.871770i	$0.337027\pi$
$240$		0			0
$241$	1.40238e8	+	2.42899e8i	0.645365	+	1.11780i	0.984217	+	0.176965i	$0.0566280\pi$
$241$	1.40238e8	+	2.42899e8i	0.645365	+	1.11780i	−0.338852	+	0.940840i	$0.610039\pi$
$242$		0			0
$243$	1.53259e8	+	2.65452e8i	0.685177	+	1.18676i
$244$		0			0
$245$	1.70087e8	−	2.94599e8i	0.738906	−	1.27982i
$246$		0			0
$247$	−1.95112e8	−	1.05155e8i	−0.823843	−	0.444010i
$248$		0			0
$249$	1.98226e8	−	3.43338e8i	0.813699	−	1.40937i
$250$		0			0
$251$	−1.55974e8	−	2.70156e8i	−0.622580	−	1.07834i	−0.989004	−	0.147892i	$-0.952751\pi$
$251$	−1.55974e8	−	2.70156e8i	−0.622580	−	1.07834i	0.366423	−	0.930448i	$-0.380582\pi$
$252$		0			0
$253$	−1.81712e8	−	3.14734e8i	−0.705443	−	1.22186i
$254$		0			0
$255$	−2.43546e8			−0.919795
$256$		0			0
$257$	−1.07145e8	+	1.85580e8i	−0.393736	+	0.681970i	−0.992939	−	0.118627i	$-0.962151\pi$
$257$	−1.07145e8	+	1.85580e8i	−0.393736	+	0.681970i	0.599203	+	0.800597i	$0.295484\pi$
$258$		0			0
$259$	2.00629e7			0.0717537
$260$		0			0
$261$	1.97174e8			0.686448
$262$		0			0
$263$	8.74186e7	−	1.51413e8i	0.296318	−	0.513239i	−0.678972	−	0.734164i	$-0.737574\pi$
$263$	8.74186e7	−	1.51413e8i	0.296318	−	0.513239i	0.975291	+	0.220925i	$0.0709077\pi$
$264$		0			0
$265$	4.88975e8			1.61408
$266$		0			0
$267$	−2.80212e8	−	4.85341e8i	−0.900943	−	1.56048i
$268$		0			0
$269$	−1.41548e8	−	2.45168e8i	−0.443374	−	0.767947i	0.554563	−	0.832142i	$-0.312885\pi$
$269$	−1.41548e8	−	2.45168e8i	−0.443374	−	0.767947i	−0.997937	+	0.0641947i	$0.979552\pi$
$270$		0			0
$271$	1.08878e8	−	1.88582e8i	0.332312	−	0.575581i	−0.650653	−	0.759375i	$-0.725505\pi$
$271$	1.08878e8	−	1.88582e8i	0.332312	−	0.575581i	0.982965	+	0.183794i	$0.0588380\pi$
$272$		0			0
$273$	−1.99938e8	+	1.23376e8i	−0.594738	+	0.366996i
$274$		0			0
$275$	3.35289e8	−	5.80737e8i	0.972197	−	1.68389i
$276$		0			0
$277$	2.75129e8	+	4.76538e8i	0.777782	+	1.34716i	0.933218	+	0.359312i	$0.116988\pi$
$277$	2.75129e8	+	4.76538e8i	0.777782	+	1.34716i	−0.155436	+	0.987846i	$0.549678\pi$
$278$		0			0
$279$	7.41751e7	+	1.28475e8i	0.204476	+	0.354164i
$280$		0			0
$281$	−1.42673e8			−0.383591			−0.191796	−	0.981435i	$-0.561431\pi$
$281$	−1.42673e8			−0.383591			−0.191796	+	0.981435i	$0.561431\pi$
$282$		0			0
$283$	−1.29624e7	+	2.24515e7i	−0.0339963	+	0.0588833i	−0.882523	−	0.470269i	$-0.844157\pi$
$283$	−1.29624e7	+	2.24515e7i	−0.0339963	+	0.0588833i	0.848527	+	0.529153i	$0.177490\pi$
$284$		0			0
$285$	−1.04229e9			−2.66706
$286$		0			0
$287$	−2.50955e8			−0.626627
$288$		0			0
$289$	1.83796e8	−	3.18345e8i	0.447914	−	0.775809i
$290$		0			0
$291$	−4.84480e8			−1.15253
$292$		0			0
$293$	−3.71932e8	−	6.44205e8i	−0.863827	−	1.49619i	−0.868207	−	0.496202i	$-0.834727\pi$
$293$	−3.71932e8	−	6.44205e8i	−0.863827	−	1.49619i	0.00437957	−	0.999990i	$-0.498606\pi$
$294$		0			0
$295$	3.07402e8	+	5.32436e8i	0.697155	+	1.20751i
$296$		0			0
$297$	8.39697e7	−	1.45440e8i	0.185984	−	0.322133i
$298$		0			0
$299$	2.46227e7	+	8.40718e8i	0.0532705	+	1.81887i
$300$		0			0
$301$	1.90947e8	−	3.30731e8i	0.403581	−	0.699023i
$302$		0			0
$303$	−5.72868e7	−	9.92237e7i	−0.118306	−	0.204911i
$304$		0			0
$305$	7.56677e8	+	1.31060e9i	1.52708	+	2.64497i
$306$		0			0
$307$	−2.86830e8			−0.565771			−0.282886	−	0.959154i	$-0.591292\pi$
$307$	−2.86830e8			−0.565771			−0.282886	+	0.959154i	$0.591292\pi$
$308$		0			0
$309$	1.75642e8	−	3.04221e8i	0.338667	−	0.586589i
$310$		0			0
$311$	−7.18890e8			−1.35519			−0.677596	−	0.735434i	$-0.736978\pi$
$311$	−7.18890e8			−1.35519			−0.677596	+	0.735434i	$0.736978\pi$
$312$		0			0
$313$	6.20382e8			1.14355			0.571774	−	0.820411i	$-0.306255\pi$
$313$	6.20382e8			1.14355			0.571774	+	0.820411i	$0.306255\pi$
$314$		0			0
$315$	−3.13815e8	+	5.43544e8i	−0.565701	+	0.979824i
$316$		0			0
$317$	1.67254e8			0.294897			0.147448	−	0.989070i	$-0.452894\pi$
$317$	1.67254e8			0.294897			0.147448	+	0.989070i	$0.452894\pi$
$318$		0			0
$319$	1.17309e8	+	2.03185e8i	0.202331	+	0.350448i
$320$		0			0
$321$	−4.36567e8	−	7.56157e8i	−0.736688	−	1.27598i
$322$		0			0
$323$	−9.14688e7	+	1.58429e8i	−0.151030	+	0.261592i
$324$		0			0
$325$	−1.32072e9	+	8.14978e8i	−2.13412	+	1.31690i
$326$		0			0
$327$	5.88046e8	−	1.01853e9i	0.930024	−	1.61085i
$328$		0			0
$329$	−6.79805e7	−	1.17746e8i	−0.105244	−	0.182289i
$330$		0			0
$331$	3.94643e8	+	6.83541e8i	0.598144	+	1.03602i	0.993095	+	0.117314i	$0.0374284\pi$
$331$	3.94643e8	+	6.83541e8i	0.598144	+	1.03602i	−0.394950	+	0.918702i	$0.629238\pi$
$332$		0			0
$333$	1.38463e8			0.205484
$334$		0			0
$335$	8.32761e7	−	1.44238e8i	0.121022	−	0.209616i
$336$		0			0
$337$	−6.94195e8			−0.988046			−0.494023	−	0.869449i	$-0.664474\pi$
$337$	−6.94195e8			−0.988046			−0.494023	+	0.869449i	$0.664474\pi$
$338$		0			0
$339$	9.45525e8			1.31818
$340$		0			0
$341$	−8.82611e7	+	1.52873e8i	−0.120539	+	0.208780i
$342$		0			0
$343$	−6.14099e8			−0.821692
$344$		0			0
$345$	1.97760e9	+	3.42531e9i	2.59282	+	4.49089i
$346$		0			0
$347$	−1.32264e7	−	2.29088e7i	−0.0169937	−	0.0294340i	0.857403	−	0.514645i	$-0.172076\pi$
$347$	−1.32264e7	−	2.29088e7i	−0.0169937	−	0.0294340i	−0.874397	+	0.485211i	$0.838743\pi$
$348$		0			0
$349$	−3.89696e8	+	6.74973e8i	−0.490723	+	0.849957i	−0.999943	−	0.0106791i	$-0.996601\pi$
$349$	−3.89696e8	+	6.74973e8i	−0.490723	+	0.849957i	0.509220	+	0.860636i	$0.329934\pi$
$350$		0			0
$351$	−3.30760e8	+	2.04103e8i	−0.408262	+	0.251927i
$352$		0			0
$353$	3.64706e8	−	6.31689e8i	0.441297	−	0.764349i	−0.556489	−	0.830855i	$-0.687852\pi$
$353$	3.64706e8	−	6.31689e8i	0.441297	−	0.764349i	0.997786	+	0.0665058i	$0.0211851\pi$
$354$		0			0
$355$	3.36011e8	+	5.81989e8i	0.398616	+	0.690423i
$356$		0			0
$357$	9.69556e7	+	1.67932e8i	0.112781	+	0.195342i
$358$		0			0
$359$	5.70058e8			0.650262			0.325131	−	0.945669i	$-0.394592\pi$
$359$	5.70058e8			0.650262			0.325131	+	0.945669i	$0.394592\pi$
$360$		0			0
$361$	5.54818e7	−	9.60973e7i	0.0620691	−	0.107507i
$362$		0			0
$363$	−5.53026e8			−0.606837
$364$		0			0
$365$	−8.75319e8			−0.942196
$366$		0			0
$367$	−2.41067e8	+	4.17540e8i	−0.254569	+	0.440927i	−0.964778	−	0.263064i	$-0.915267\pi$
$367$	−2.41067e8	+	4.17540e8i	−0.254569	+	0.440927i	0.710209	+	0.703991i	$0.248600\pi$
$368$		0			0
$369$	−1.73195e9			−1.79450
$370$		0			0
$371$	−1.94660e8	−	3.37162e8i	−0.197911	−	0.342791i
$372$		0			0
$373$	1.49680e8	+	2.59253e8i	0.149342	+	0.258668i	0.930984	−	0.365059i	$-0.118951\pi$
$373$	1.49680e8	+	2.59253e8i	0.149342	+	0.258668i	−0.781642	+	0.623727i	$0.785618\pi$
$374$		0			0
$375$	−2.19390e9	+	3.79994e9i	−2.14836	+	3.72107i
$376$		0			0
$377$	−1.58958e7	−	5.42747e8i	−0.0152788	−	0.521678i
$378$		0			0
$379$	5.05321e8	−	8.75242e8i	0.476793	−	0.825830i	−0.522853	−	0.852423i	$-0.675132\pi$
$379$	5.05321e8	−	8.75242e8i	0.476793	−	0.825830i	0.999646	+	0.0265924i	$0.00846563\pi$
$380$		0			0
$381$	9.70858e7	+	1.68158e8i	0.0899329	+	0.155768i
$382$		0			0
$383$	4.42877e8	+	7.67085e8i	0.402798	+	0.697667i	0.994062	−	0.108811i	$-0.0347044\pi$
$383$	4.42877e8	+	7.67085e8i	0.402798	+	0.697667i	−0.591264	+	0.806478i	$0.701371\pi$
$384$		0			0
$385$	−7.46819e8			−0.666965
$386$		0			0
$387$	1.31781e9	−	2.28251e9i	1.15575	−	2.00182i
$388$		0			0
$389$	2.15563e9			1.85674			0.928370	−	0.371656i	$-0.121210\pi$
$389$	2.15563e9			1.85674			0.928370	+	0.371656i	$0.121210\pi$
$390$		0			0
$391$	6.94197e8			0.587305
$392$		0			0
$393$	6.97498e8	−	1.20810e9i	0.579654	−	1.00399i
$394$		0			0
$395$	4.23586e9			3.45822
$396$		0			0
$397$	4.73536e8	+	8.20188e8i	0.379827	+	0.657880i	0.991037	−	0.133589i	$-0.0426501\pi$
$397$	4.73536e8	+	8.20188e8i	0.379827	+	0.657880i	−0.611210	+	0.791469i	$0.709317\pi$
$398$		0			0
$399$	4.14936e8	+	7.18689e8i	0.327021	+	0.566417i
$400$		0			0
$401$	7.68784e8	−	1.33157e9i	0.595386	−	1.03124i	−0.398106	−	0.917339i	$-0.630332\pi$
$401$	7.68784e8	−	1.33157e9i	0.595386	−	1.03124i	0.993492	−	0.113900i	$-0.0363343\pi$
$402$		0			0
$403$	3.47664e8	−	2.14534e8i	0.264602	−	0.163278i
$404$		0			0
$405$	7.32767e8	−	1.26919e9i	0.548117	−	0.949366i
$406$		0			0
$407$	8.23784e7	+	1.42684e8i	0.0605666	+	0.104904i
$408$		0			0
$409$	−4.78055e8	−	8.28015e8i	−0.345498	−	0.598421i	0.639946	−	0.768420i	$-0.278957\pi$
$409$	−4.78055e8	−	8.28015e8i	−0.345498	−	0.598421i	−0.985444	+	0.169999i	$0.945623\pi$
$410$		0			0
$411$	4.64236e7			0.0329831
$412$		0			0
$413$	2.44753e8	−	4.23924e8i	0.170963	−	0.296117i
$414$		0			0
$415$	−2.91657e9			−2.00311
$416$		0			0
$417$	−6.55839e8			−0.442916
$418$		0			0
$419$	1.00065e9	−	1.73318e9i	0.664559	−	1.15105i	−0.314846	−	0.949143i	$-0.601953\pi$
$419$	1.00065e9	−	1.73318e9i	0.664559	−	1.15105i	0.979405	−	0.201907i	$-0.0647137\pi$
$420$		0			0
$421$	2.08228e9			1.36004			0.680020	−	0.733194i	$-0.261971\pi$
$421$	2.08228e9			1.36004			0.680020	+	0.733194i	$0.261971\pi$
$422$		0			0
$423$	−4.69162e8	−	8.12613e8i	−0.301392	−	0.522026i
$424$		0			0
$425$	6.40454e8	+	1.10930e9i	0.404694	+	0.700950i
$426$		0			0
$427$	6.02465e8	−	1.04350e9i	0.374485	−	0.648626i
$428$		0			0
$429$	−1.69837e9	−	9.15338e8i	−1.03856	−	0.559733i
$430$		0			0
$431$	−3.68644e8	+	6.38511e8i	−0.221788	+	0.384147i	−0.955351	−	0.295474i	$-0.904522\pi$
$431$	−3.68644e8	+	6.38511e8i	−0.221788	+	0.384147i	0.733563	+	0.679621i	$0.237856\pi$
$432$		0			0
$433$	2.20052e8	+	3.81141e8i	0.130262	+	0.225620i	0.923778	−	0.382930i	$-0.125085\pi$
$433$	2.20052e8	+	3.81141e8i	0.130262	+	0.225620i	−0.793516	+	0.608550i	$0.791751\pi$
$434$		0			0
$435$	−1.27669e9	−	2.21129e9i	−0.743659	−	1.28805i
$436$		0			0
$437$	2.97091e9			1.70296
$438$		0			0
$439$	−1.08879e9	+	1.88584e9i	−0.614212	+	1.06385i	0.376310	+	0.926494i	$0.377193\pi$
$439$	−1.08879e9	+	1.88584e9i	−0.614212	+	1.06385i	−0.990522	+	0.137353i	$0.956140\pi$
$440$		0			0
$441$	−1.86922e9			−1.03783
$442$		0			0
$443$	−1.78526e9			−0.975639			−0.487819	−	0.872945i	$-0.662207\pi$
$443$	−1.78526e9			−0.975639			−0.487819	+	0.872945i	$0.662207\pi$
$444$		0			0
$445$	−2.06143e9	+	3.57050e9i	−1.10894	+	1.92074i
$446$		0			0
$447$	−1.73955e8			−0.0921216
$448$		0			0
$449$	4.68771e8	+	8.11935e8i	0.244398	+	0.423310i	0.961962	−	0.273182i	$-0.0880762\pi$
$449$	4.68771e8	+	8.11935e8i	0.244398	+	0.423310i	−0.717564	+	0.696493i	$0.754743\pi$
$450$		0			0
$451$	−1.03042e9	−	1.78475e9i	−0.528930	−	0.916133i
$452$		0			0
$453$	−9.72010e8	+	1.68357e9i	−0.491278	+	0.850918i
$454$		0			0
$455$	1.52148e9	+	8.19998e8i	0.757225	+	0.408106i
$456$		0			0
$457$	−1.53717e9	+	2.66245e9i	−0.753381	+	1.30489i	0.192794	+	0.981239i	$0.438245\pi$
$457$	−1.53717e9	+	2.66245e9i	−0.753381	+	1.30489i	−0.946175	+	0.323655i	$0.895088\pi$
$458$		0			0
$459$	1.60395e8	+	2.77813e8i	0.0774189	+	0.134093i
$460$		0			0
$461$	−1.64378e9	−	2.84710e9i	−0.781428	−	1.35347i	−0.931110	−	0.364739i	$-0.881158\pi$
$461$	−1.64378e9	−	2.84710e9i	−0.781428	−	1.35347i	0.149681	−	0.988734i	$-0.452175\pi$
$462$		0			0
$463$	−8.40096e7			−0.0393365			−0.0196682	−	0.999807i	$-0.506261\pi$
$463$	−8.40096e7			−0.0393365			−0.0196682	+	0.999807i	$0.506261\pi$
$464$		0			0
$465$	9.60560e8	−	1.66374e9i	0.443036	−	0.767361i
$466$		0			0
$467$	−3.33082e9			−1.51336			−0.756681	−	0.653785i	$-0.773180\pi$
$467$	−3.33082e9			−1.51336			−0.756681	+	0.653785i	$0.773180\pi$
$468$		0			0
$469$	−1.32609e8			−0.0593563
$470$		0			0
$471$	−1.97283e9	+	3.41704e9i	−0.869995	+	1.50688i
$472$		0			0
$473$	3.13613e9			1.36264
$474$		0			0
$475$	2.74091e9	+	4.74740e9i	1.17346	+	2.03249i
$476$		0			0
$477$	−1.34343e9	−	2.32690e9i	−0.566764	−	0.981663i
$478$		0			0
$479$	−4.49485e8	+	7.78532e8i	−0.186871	+	0.323670i	−0.944205	−	0.329357i	$-0.893168\pi$
$479$	−4.49485e8	+	7.78532e8i	−0.186871	+	0.323670i	0.757335	+	0.653027i	$0.226501\pi$
$480$		0			0
$481$	−1.11626e7	−	3.81136e8i	−0.00457360	−	0.156161i
$482$		0			0
$483$	1.57456e9	−	2.72722e9i	0.635836	−	1.10130i
$484$		0			0
$485$	1.78208e9	+	3.08666e9i	0.709302	+	1.22855i
$486$		0			0
$487$	−1.33482e9	−	2.31197e9i	−0.523685	−	0.907048i	−0.999620	−	0.0275680i	$-0.991224\pi$
$487$	−1.33482e9	−	2.31197e9i	−0.523685	−	0.907048i	0.475935	−	0.879480i	$-0.342110\pi$
$488$		0			0
$489$	−2.27716e9			−0.880670
$490$		0			0
$491$	1.70549e9	−	2.95399e9i	0.650225	−	1.12622i	−0.332843	−	0.942982i	$-0.608008\pi$
$491$	1.70549e9	−	2.95399e9i	0.650225	−	1.12622i	0.983068	−	0.183241i	$-0.0586588\pi$
$492$		0			0
$493$	−4.48157e8			−0.168448
$494$		0			0
$495$	−5.15412e9			−1.91001
$496$		0			0
$497$	2.67532e8	−	4.63378e8i	0.0977525	−	0.169312i
$498$		0			0
$499$	5.43259e9			1.95729			0.978645	−	0.205559i	$-0.0659014\pi$
$499$	5.43259e9			1.95729			0.978645	+	0.205559i	$0.0659014\pi$
$500$		0			0
$501$	−6.88318e8	−	1.19220e9i	−0.244544	−	0.423563i
$502$		0			0
$503$	5.84567e8	+	1.01250e9i	0.204808	+	0.354737i	0.950071	−	0.312032i	$-0.101010\pi$
$503$	5.84567e8	+	1.01250e9i	0.204808	+	0.354737i	−0.745264	+	0.666770i	$0.767676\pi$
$504$		0			0
$505$	−4.21440e8	+	7.29956e8i	−0.145618	+	0.252219i
$506$		0			0
$507$	2.45503e9	+	3.72959e9i	0.836620	+	1.27096i
$508$		0			0
$509$	−1.08501e9	+	1.87929e9i	−0.364688	+	0.631658i	−0.988726	−	0.149736i	$-0.952158\pi$
$509$	−1.08501e9	+	1.87929e9i	−0.364688	+	0.631658i	0.624038	+	0.781394i	$0.285491\pi$
$510$		0			0
$511$	3.48464e8	+	6.03557e8i	0.115527	+	0.200099i
$512$		0			0
$513$	6.86434e8	+	1.18894e9i	0.224485	+	0.388820i
$514$		0			0
$515$	−2.58428e9			−0.833708
$516$		0			0
$517$	5.58257e8	−	9.66930e8i	0.177671	−	0.307736i
$518$		0			0
$519$	2.94417e9			0.924436
$520$		0			0
$521$	−5.57218e9			−1.72621			−0.863103	−	0.505027i	$-0.831482\pi$
$521$	−5.57218e9			−1.72621			−0.863103	+	0.505027i	$0.831482\pi$
$522$		0			0
$523$	−3.33759e8	+	5.78087e8i	−0.102018	+	0.176700i	−0.912516	−	0.409041i	$-0.865863\pi$
$523$	−3.33759e8	+	5.78087e8i	−0.102018	+	0.176700i	0.810498	+	0.585741i	$0.199197\pi$
$524$		0			0
$525$	5.81066e9			1.75254
$526$		0			0
$527$	−1.68593e8	−	2.92011e8i	−0.0501766	−	0.0869084i
$528$		0			0
$529$	−3.93448e9	−	6.81471e9i	−1.15556	−	2.00149i
$530$		0			0
$531$	1.68914e9	−	2.92568e9i	0.489593	−	0.848000i
$532$		0			0
$533$	1.39627e8	+	4.76741e9i	0.0399414	+	1.36376i
$534$		0			0
$535$	−3.21168e9	+	5.56280e9i	−0.906764	+	1.57056i
$536$		0			0
$537$	−1.14877e9	−	1.98973e9i	−0.320127	−	0.554477i
$538$		0			0
$539$	−1.11209e9	−	1.92620e9i	−0.305901	−	0.529836i
$540$		0			0
$541$	4.87524e9			1.32375			0.661875	−	0.749614i	$-0.269761\pi$
$541$	4.87524e9			1.32375			0.661875	+	0.749614i	$0.269761\pi$
$542$		0			0
$543$	2.87827e9	−	4.98531e9i	0.771493	−	1.33626i
$544$		0			0
$545$	−8.65212e9			−2.28947
$546$		0			0
$547$	−6.55899e8			−0.171349			−0.0856744	−	0.996323i	$-0.527304\pi$
$547$	−6.55899e8			−0.171349			−0.0856744	+	0.996323i	$0.527304\pi$
$548$		0			0
$549$	4.15786e9	−	7.20163e9i	1.07242	−	1.85749i
$550$		0			0
$551$	−1.91795e9			−0.488435
$552$		0			0
$553$	−1.68629e9	−	2.92075e9i	−0.424029	−	0.734439i
$554$		0			0
$555$	−8.96538e8	−	1.55285e9i	−0.222609	−	0.385571i
$556$		0			0
$557$	7.33190e8	−	1.26992e9i	0.179773	−	0.311375i	−0.762030	−	0.647542i	$-0.775797\pi$
$557$	7.33190e8	−	1.26992e9i	0.179773	−	0.311375i	0.941803	+	0.336167i	$0.109130\pi$
$558$		0			0
$559$	−6.38915e9	−	3.44343e9i	−1.54704	−	0.833777i
$560$		0			0
$561$	−7.96201e8	+	1.37906e9i	−0.190394	+	0.329772i
$562$		0			0
$563$	−1.50686e9	−	2.60996e9i	−0.355872	−	0.616388i	0.631395	−	0.775461i	$-0.282483\pi$
$563$	−1.50686e9	−	2.60996e9i	−0.355872	−	0.616388i	−0.987267	+	0.159074i	$0.949149\pi$
$564$		0			0
$565$	−3.47796e9	−	6.02400e9i	−0.811250	−	1.40513i
$566$		0			0
$567$	−1.16686e9			−0.268829
$568$		0			0
$569$	3.75097e9	−	6.49687e9i	0.853592	−	1.47847i	−0.0243527	−	0.999703i	$-0.507752\pi$
$569$	3.75097e9	−	6.49687e9i	0.853592	−	1.47847i	0.877945	−	0.478762i	$-0.158914\pi$
$570$		0			0
$571$	1.19452e9			0.268514			0.134257	−	0.990947i	$-0.457135\pi$
$571$	1.19452e9			0.268514			0.134257	+	0.990947i	$0.457135\pi$
$572$		0			0
$573$	9.70190e8			0.215435
$574$		0			0
$575$	1.04010e10	−	1.80150e10i	2.28159	−	3.95183i
$576$		0			0
$577$	−4.85637e9			−1.05244			−0.526219	−	0.850349i	$-0.676391\pi$
$577$	−4.85637e9			−1.05244			−0.526219	+	0.850349i	$0.676391\pi$
$578$		0			0
$579$	−6.16388e8	−	1.06762e9i	−0.131971	−	0.228581i
$580$		0			0
$581$	1.16109e9	+	2.01106e9i	0.245611	+	0.425411i
$582$		0			0
$583$	1.59856e9	−	2.76878e9i	0.334109	−	0.578693i
$584$		0			0
$585$	1.05003e10	+	5.65915e9i	2.16849	+	1.16871i
$586$		0			0
$587$	−1.01242e9	+	1.75356e9i	−0.206598	+	0.357839i	−0.950641	−	0.310294i	$-0.899573\pi$
$587$	−1.01242e9	+	1.75356e9i	−0.206598	+	0.357839i	0.744043	+	0.668132i	$0.232906\pi$
$588$		0			0
$589$	−7.21516e8	−	1.24970e9i	−0.145493	−	0.252001i
$590$		0			0
$591$	−2.21924e9	−	3.84384e9i	−0.442231	−	0.765966i
$592$		0			0
$593$	9.30040e8			0.183151			0.0915757	−	0.995798i	$-0.470810\pi$
$593$	9.30040e8			0.183151			0.0915757	+	0.995798i	$0.470810\pi$
$594$		0			0
$595$	7.13271e8	−	1.23542e9i	0.138818	−	0.240439i
$596$		0			0
$597$	1.71995e8			0.0330831
$598$		0			0
$599$	−2.25978e9			−0.429608			−0.214804	−	0.976657i	$-0.568911\pi$
$599$	−2.25978e9			−0.429608			−0.214804	+	0.976657i	$0.568911\pi$
$600$		0			0
$601$	−7.28350e8	+	1.26154e9i	−0.136861	+	0.237050i	−0.926307	−	0.376770i	$-0.877035\pi$
$601$	−7.28350e8	+	1.26154e9i	−0.136861	+	0.237050i	0.789446	+	0.613820i	$0.210368\pi$
$602$		0			0
$603$	−9.15188e8			−0.169981
$604$		0			0
$605$	2.03422e9	+	3.52336e9i	0.373468	+	0.646865i
$606$		0			0
$607$	−2.64143e9	−	4.57510e9i	−0.479379	−	0.830309i	0.520341	−	0.853958i	$-0.325805\pi$
$607$	−2.64143e9	−	4.57510e9i	−0.479379	−	0.830309i	−0.999720	+	0.0236495i	$0.992471\pi$
$608$		0			0
$609$	−1.01650e9	+	1.76063e9i	−0.182367	+	0.315869i
$610$		0			0
$611$	−2.19900e9	+	1.35694e9i	−0.390015	+	0.240667i
$612$		0			0
$613$	1.20643e9	−	2.08960e9i	0.211540	−	0.366397i	−0.740657	−	0.671883i	$-0.765486\pi$
$613$	1.20643e9	−	2.08960e9i	0.211540	−	0.366397i	0.952197	+	0.305486i	$0.0988190\pi$
$614$		0			0
$615$	1.12143e10	+	1.94237e10i	1.94405	+	3.36720i
$616$		0			0
$617$	−4.68931e9	−	8.12213e9i	−0.803732	−	1.39210i	−0.917144	−	0.398556i	$-0.869511\pi$
$617$	−4.68931e9	−	8.12213e9i	−0.803732	−	1.39210i	0.113412	−	0.993548i	$-0.463822\pi$
$618$		0			0
$619$	−2.73222e9			−0.463019			−0.231510	−	0.972833i	$-0.574366\pi$
$619$	−2.73222e9			−0.463019			−0.231510	+	0.972833i	$0.574366\pi$
$620$		0			0
$621$	2.60482e9	−	4.51169e9i	0.436473	−	0.755994i
$622$		0			0
$623$	3.28261e9			0.543890
$624$		0			0
$625$	1.69736e10			2.78096
$626$		0			0
$627$	−3.40746e9	+	5.90189e9i	−0.552070	+	0.956213i
$628$		0			0
$629$	−3.14711e8			−0.0504238
$630$		0			0
$631$	4.52633e9	+	7.83984e9i	0.717205	+	1.24224i	0.962103	+	0.272687i	$0.0879124\pi$
$631$	4.52633e9	+	7.83984e9i	0.717205	+	1.24224i	−0.244897	+	0.969549i	$0.578754\pi$
$632$		0			0
$633$	3.46071e9	+	5.99413e9i	0.542316	+	0.939319i
$634$		0			0
$635$	7.14229e8	−	1.23708e9i	0.110695	−	0.191730i
$636$		0			0
$637$	1.50693e8	+	5.14527e9i	0.0230997	+	0.788716i
$638$		0			0
$639$	1.84635e9	−	3.19797e9i	0.279937	−	0.484866i
$640$		0			0
$641$	5.18052e9	+	8.97293e9i	0.776910	+	1.34565i	0.933715	+	0.358018i	$0.116547\pi$
$641$	5.18052e9	+	8.97293e9i	0.776910	+	1.34565i	−0.156805	+	0.987630i	$0.550119\pi$
$642$		0			0
$643$	−1.96400e9	−	3.40175e9i	−0.291342	−	0.504619i	0.682785	−	0.730619i	$-0.260768\pi$
$643$	−1.96400e9	−	3.40175e9i	−0.291342	−	0.504619i	−0.974127	+	0.226000i	$0.927435\pi$
$644$		0			0
$645$	−3.41310e10			−5.00829
$646$		0			0
$647$	−5.63673e9	+	9.76311e9i	−0.818206	+	1.41717i	0.0887973	+	0.996050i	$0.471698\pi$
$647$	−5.63673e9	+	9.76311e9i	−0.818206	+	1.41717i	−0.907003	+	0.421124i	$0.861636\pi$
$648$		0			0
$649$	4.01983e9			0.577233
$650$		0			0
$651$	−1.52959e9			−0.217291
$652$		0			0
$653$	6.93321e9	−	1.20087e10i	0.974401	−	1.68771i	0.292505	−	0.956264i	$-0.405511\pi$
$653$	6.93321e9	−	1.20087e10i	0.974401	−	1.68771i	0.681896	−	0.731449i	$-0.261156\pi$
$654$		0			0
$655$	−1.02625e10			−1.42695
$656$		0			0
$657$	2.40490e9	+	4.16540e9i	0.330840	+	0.573031i
$658$		0			0
$659$	−7.94245e7	−	1.37567e8i	−0.0108107	−	0.0187248i	0.860569	−	0.509333i	$-0.170108\pi$
$659$	−7.94245e7	−	1.37567e8i	−0.0108107	−	0.0187248i	−0.871380	+	0.490608i	$0.836775\pi$
$660$		0			0
$661$	−5.58771e9	+	9.67820e9i	−0.752539	+	1.30344i	0.194050	+	0.980992i	$0.437838\pi$
$661$	−5.58771e9	+	9.67820e9i	−0.752539	+	1.30344i	−0.946589	+	0.322443i	$0.895496\pi$
$662$		0			0
$663$	3.13627e9	−	1.93531e9i	0.417942	−	0.257901i
$664$		0			0
$665$	3.05255e9	−	5.28716e9i	0.402519	−	0.697183i
$666$		0			0
$667$	3.63904e9	+	6.30300e9i	0.474839	+	0.822445i
$668$		0			0
$669$	5.94601e9	+	1.02988e10i	0.767776	+	1.32983i
$670$		0			0
$671$	9.89490e9			1.26439
$672$		0			0
$673$	2.69956e9	−	4.67577e9i	0.341381	−	0.591289i	−0.643308	−	0.765607i	$-0.722439\pi$
$673$	2.69956e9	−	4.67577e9i	0.341381	−	0.591289i	0.984689	+	0.174318i	$0.0557720\pi$
$674$		0			0
$675$	9.61266e9			1.20304
$676$		0			0
$677$	−7.69486e9			−0.953104			−0.476552	−	0.879146i	$-0.658114\pi$
$677$	−7.69486e9			−0.953104			−0.476552	+	0.879146i	$0.658114\pi$
$678$		0			0
$679$	1.41889e9	−	2.45759e9i	0.173942	−	0.301276i
$680$		0			0
$681$	−4.06086e9			−0.492724
$682$		0			0
$683$	−2.54926e9	−	4.41544e9i	−0.306155	−	0.530275i	0.671363	−	0.741129i	$-0.265709\pi$
$683$	−2.54926e9	−	4.41544e9i	−0.306155	−	0.530275i	−0.977518	+	0.210853i	$0.932376\pi$
$684$		0			0
$685$	−1.70761e8	−	2.95768e8i	−0.0202989	−	0.0351588i
$686$		0			0
$687$	−9.56200e9	+	1.65619e10i	−1.12512	+	1.94877i
$688$		0			0
$689$	−6.29678e9	+	3.88557e9i	−0.733417	+	0.452572i
$690$		0			0
$691$	−1.53611e8	+	2.66062e8i	−0.0177112	+	0.0306768i	−0.874745	−	0.484583i	$-0.838971\pi$
$691$	−1.53611e8	+	2.66062e8i	−0.0177112	+	0.0306768i	0.857034	+	0.515260i	$0.172305\pi$
$692$		0			0
$693$	2.05185e9	+	3.55391e9i	0.234196	+	0.405639i
$694$		0			0
$695$	2.41240e9	+	4.17840e9i	0.272585	+	0.472131i
$696$		0			0
$697$	3.93654e9			0.440352
$698$		0			0
$699$	2.32599e9	−	4.02874e9i	0.257595	−	0.446168i
$700$		0			0
$701$	1.35330e9			0.148382			0.0741910	−	0.997244i	$-0.476363\pi$
$701$	1.35330e9			0.148382			0.0741910	+	0.997244i	$0.476363\pi$
$702$		0			0
$703$	−1.34685e9			−0.146210
$704$		0			0
$705$	−6.07561e9	+	1.05233e10i	−0.653022	+	1.13107i
$706$		0			0
$707$	6.71100e8			0.0714199
$708$		0			0
$709$	−5.45562e9	−	9.44942e9i	−0.574887	−	0.995734i	−0.996054	−	0.0887504i	$-0.971713\pi$
$709$	−5.45562e9	−	9.44942e9i	−0.574887	−	0.995734i	0.421167	−	0.906983i	$-0.361621\pi$
$710$		0			0
$711$	−1.16378e10	−	2.01573e10i	−1.21431	−	2.10324i
$712$		0			0
$713$	−2.73795e9	+	4.74227e9i	−0.282886	+	0.489973i
$714$		0			0
$715$	4.15516e8	+	1.41874e10i	0.0425125	+	1.45155i
$716$		0			0
$717$	−7.35768e9	+	1.27439e10i	−0.745459	+	1.29117i
$718$		0			0
$719$	−5.14659e9	−	8.91415e9i	−0.516379	−	0.894394i	−0.999819	−	0.0190170i	$-0.993946\pi$
$719$	−5.14659e9	−	8.91415e9i	−0.516379	−	0.894394i	0.483440	−	0.875377i	$-0.339387\pi$
$720$		0			0
$721$	1.02880e9	+	1.78193e9i	0.102225	+	0.177059i
$722$		0			0
$723$	−1.99582e10			−1.96398
$724$		0			0
$725$	−6.71463e9	+	1.16301e10i	−0.654393	+	1.13344i
$726$		0			0
$727$	−7.61641e9			−0.735156			−0.367578	−	0.929993i	$-0.619813\pi$
$727$	−7.61641e9			−0.735156			−0.367578	+	0.929993i	$0.619813\pi$
$728$		0			0
$729$	−1.56887e10			−1.49982
$730$		0			0
$731$	−2.99525e9	+	5.18792e9i	−0.283610	+	0.491227i
$732$		0			0
$733$	5.63920e9			0.528875			0.264438	−	0.964403i	$-0.414814\pi$
$733$	5.63920e9			0.528875			0.264438	+	0.964403i	$0.414814\pi$
$734$		0			0
$735$	1.21031e10	+	2.09632e10i	1.12432	+	1.94739i
$736$		0			0
$737$	−5.44492e8	−	9.43088e8i	−0.0501020	−	0.0867793i
$738$		0			0
$739$	1.09173e10	−	1.89093e10i	0.995083	−	1.72353i	0.411766	−	0.911289i	$-0.364912\pi$
$739$	1.09173e10	−	1.89093e10i	0.995083	−	1.72353i	0.583317	−	0.812245i	$-0.301755\pi$
$740$		0			0
$741$	1.34221e10	−	8.28242e9i	1.21187	−	0.747814i
$742$		0			0
$743$	−5.74324e9	+	9.94759e9i	−0.513684	+	0.889728i	0.486190	+	0.873853i	$0.338386\pi$
$743$	−5.74324e9	+	9.94759e9i	−0.513684	+	0.889728i	−0.999874	+	0.0158742i	$0.994947\pi$
$744$		0			0
$745$	6.39866e8	+	1.10828e9i	0.0566947	+	0.0981980i
$746$		0			0
$747$	8.01314e9	+	1.38792e10i	0.703365	+	1.21826i
$748$		0			0
$749$	5.11427e9			0.444731
$750$		0			0
$751$	3.48748e9	−	6.04050e9i	0.300450	−	0.520395i	−0.675788	−	0.737096i	$-0.736196\pi$
$751$	3.48748e9	−	6.04050e9i	0.300450	−	0.520395i	0.976238	+	0.216701i	$0.0695298\pi$
$752$		0			0
$753$	2.21978e10			1.89464
$754$		0			0
$755$	1.43015e10			1.20939
$756$		0			0
$757$	1.58758e9	−	2.74977e9i	0.133015	−	0.230389i	−0.791822	−	0.610751i	$-0.790867\pi$
$757$	1.58758e9	−	2.74977e9i	0.133015	−	0.230389i	0.924837	+	0.380363i	$0.124201\pi$
$758$		0			0
$759$	2.58607e10			2.14681
$760$		0			0
$761$	−7.65367e9	−	1.32565e10i	−0.629540	−	1.09040i	−0.987644	−	0.156714i	$-0.949910\pi$
$761$	−7.65367e9	−	1.32565e10i	−0.629540	−	1.09040i	0.358104	−	0.933682i	$-0.383423\pi$
$762$		0			0
$763$	3.44440e9	+	5.96588e9i	0.280723	+	0.486227i
$764$		0			0
$765$	4.92258e9	−	8.52616e9i	0.397537	−	0.688555i
$766$		0			0
$767$	−8.18950e9	−	4.41372e9i	−0.655350	−	0.353200i
$768$		0			0
$769$	−1.47679e9	+	2.55788e9i	−0.117105	+	0.202833i	−0.918619	−	0.395143i	$-0.870695\pi$
$769$	−1.47679e9	+	2.55788e9i	−0.117105	+	0.202833i	0.801514	+	0.597976i	$0.204028\pi$
$770$		0			0
$771$	−7.62425e9	−	1.32056e10i	−0.599110	−	1.03769i
$772$		0			0
$773$	−4.92294e9	−	8.52678e9i	−0.383351	−	0.663983i	0.608188	−	0.793793i	$-0.291897\pi$
$773$	−4.92294e9	−	8.52678e9i	−0.383351	−	0.663983i	−0.991539	+	0.129810i	$0.958563\pi$
$774$		0			0
$775$	−1.01039e10			−0.779712
$776$		0			0
$777$	−7.13822e8	+	1.23638e9i	−0.0545904	+	0.0945534i
$778$		0			0
$779$	1.68470e10			1.27685
$780$		0			0
$781$	4.39395e9			0.330047
$782$		0			0
$783$	−1.68161e9	+	2.91264e9i	−0.125187	+	0.216830i
$784$		0			0
$785$	2.90269e10			2.14169
$786$		0			0
$787$	−2.60565e9	−	4.51312e9i	−0.190548	−	0.330039i	0.754884	−	0.655858i	$-0.227693\pi$
$787$	−2.60565e9	−	4.51312e9i	−0.190548	−	0.330039i	−0.945432	+	0.325820i	$0.894360\pi$
$788$		0			0
$789$	6.22057e9	+	1.07743e10i	0.450880	+	0.780946i
$790$		0			0
$791$	−2.76914e9	+	4.79630e9i	−0.198943	+	0.344579i
$792$		0			0
$793$	−2.01586e10	−	1.08645e10i	−1.43551	−	0.773665i
$794$		0			0
$795$	−1.73973e10	+	3.01331e10i	−1.22800	+	2.12696i
$796$		0			0
$797$	−3.92980e9	−	6.80662e9i	−0.274958	−	0.476241i	0.695166	−	0.718849i	$-0.255331\pi$
$797$	−3.92980e9	−	6.80662e9i	−0.274958	−	0.476241i	−0.970125	+	0.242607i	$0.921997\pi$
$798$		0			0
$799$	1.06636e9	+	1.84699e9i	0.0739588	+	0.128100i
$800$		0			0
$801$	2.26547e10			1.55756
$802$		0			0
$803$	−2.86159e9	+	4.95642e9i	−0.195031	+	0.337803i
$804$		0			0
$805$	−2.31671e10			−1.56526
$806$		0			0
$807$	2.01447e10			1.34928
$808$		0			0
$809$	3.22934e9	−	5.59338e9i	0.214434	−	0.371411i	−0.738663	−	0.674075i	$-0.764543\pi$
$809$	3.22934e9	−	5.59338e9i	0.214434	−	0.371411i	0.953097	+	0.302664i	$0.0978760\pi$
$810$		0			0
$811$	2.35320e10			1.54912			0.774562	−	0.632498i	$-0.217970\pi$
$811$	2.35320e10			1.54912			0.774562	+	0.632498i	$0.217970\pi$
$812$		0			0
$813$	7.74756e9	+	1.34192e10i	0.505648	+	0.875807i
$814$		0			0
$815$	8.37617e9	+	1.45080e10i	0.541993	+	0.938760i
$816$		0			0
$817$	−1.28186e10	+	2.22024e10i	−0.822362	+	1.42437i
$818$		0			0
$819$	−2.78034e8	−	9.49319e9i	−0.0176850	−	0.603835i
$820$		0			0
$821$	1.26199e9	−	2.18583e9i	0.0795894	−	0.137853i	−0.823483	−	0.567340i	$-0.807972\pi$
$821$	1.26199e9	−	2.18583e9i	0.0795894	−	0.137853i	0.903073	+	0.429487i	$0.141306\pi$
$822$		0			0
$823$	−8.68821e9	−	1.50484e10i	−0.543289	−	0.941004i	−0.998712	−	0.0507293i	$-0.983845\pi$
$823$	−8.68821e9	−	1.50484e10i	−0.543289	−	0.941004i	0.455423	−	0.890275i	$-0.349488\pi$
$824$		0			0
$825$	2.38586e10	+	4.13243e10i	1.47930	+	2.56222i
$826$		0			0
$827$	−1.96660e10			−1.20906			−0.604530	−	0.796583i	$-0.706639\pi$
$827$	−1.96660e10			−1.20906			−0.604530	+	0.796583i	$0.706639\pi$
$828$		0			0
$829$	−1.53344e10	+	2.65600e10i	−0.934817	+	1.61915i	−0.159857	+	0.987140i	$0.551103\pi$
$829$	−1.53344e10	+	2.65600e10i	−0.934817	+	1.61915i	−0.774960	+	0.632010i	$0.782230\pi$
$830$		0			0
$831$	−3.91555e10			−2.36695
$832$		0			0
$833$	4.24855e9			0.254673
$834$		0			0
$835$	−5.06373e9	+	8.77064e9i	−0.301001	+	0.521349i
$836$		0			0
$837$	−2.53043e9			−0.149161
$838$		0			0
$839$	−1.50419e10	−	2.60533e10i	−0.879297	−	1.52299i	−0.852114	−	0.523356i	$-0.824680\pi$
$839$	−1.50419e10	−	2.60533e10i	−0.879297	−	1.52299i	−0.0271823	−	0.999630i	$-0.508653\pi$
$840$		0			0
$841$	6.27566e9	+	1.08698e10i	0.363809	+	0.630136i
$842$		0			0
$843$	5.07618e9	−	8.79220e9i	0.291837	−	0.505477i
$844$		0			0
$845$	1.47310e10	−	2.93598e10i	0.839915	−	1.67400i
$846$		0			0
$847$	1.61964e9	−	2.80530e9i	0.0915854	−	0.158631i
$848$		0			0
$849$	−9.22381e8	−	1.59761e9i	−0.0517289	−	0.0895971i
$850$		0			0
$851$	2.55546e9	+	4.42619e9i	0.142140	+	0.246194i
$852$		0			0
$853$	−8.80922e9			−0.485977			−0.242989	−	0.970029i	$-0.578128\pi$
$853$	−8.80922e9			−0.485977			−0.242989	+	0.970029i	$0.578128\pi$
$854$		0			0
$855$	2.10669e10	−	3.64889e10i	1.15271	−	1.99655i
$856$		0			0
$857$	1.71306e10			0.929696			0.464848	−	0.885391i	$-0.346109\pi$
$857$	1.71306e10			0.929696			0.464848	+	0.885391i	$0.346109\pi$
$858$		0			0
$859$	−9.23130e9			−0.496921			−0.248460	−	0.968642i	$-0.579925\pi$
$859$	−9.23130e9			−0.496921			−0.248460	+	0.968642i	$0.579925\pi$
$860$		0			0
$861$	8.92879e9	−	1.54651e10i	0.476740	−	0.825737i
$862$		0			0
$863$	2.02167e10			1.07071			0.535356	−	0.844627i	$-0.320178\pi$
$863$	2.02167e10			1.07071			0.535356	+	0.844627i	$0.320178\pi$
$864$		0			0
$865$	−1.08296e10	−	1.87575e10i	−0.568929	−	0.985414i
$866$		0			0
$867$	1.30787e10	+	2.26529e10i	0.681548	+	1.18048i
$868$		0			0
$869$	1.38479e10	−	2.39852e10i	0.715837	−	1.23987i
$870$		0			0
$871$	7.37809e7	+	2.51918e9i	0.00378338	+	0.129180i
$872$		0			0
$873$	9.79236e9	−	1.69609e10i	0.498124	−	0.862776i
$874$		0			0
$875$	−1.28505e10	−	2.22577e10i	−0.648471	−	1.12318i
$876$		0			0
$877$	7.10449e9	+	1.23053e10i	0.355659	+	0.616020i	0.987231	−	0.159298i	$-0.0509230\pi$
$877$	7.10449e9	+	1.23053e10i	0.355659	+	0.616020i	−0.631571	+	0.775318i	$0.717590\pi$
$878$		0			0
$879$	5.29322e10			2.62881
$880$		0			0
$881$	7.59598e9	−	1.31566e10i	0.374256	−	0.648230i	−0.615960	−	0.787778i	$-0.711232\pi$
$881$	7.59598e9	−	1.31566e10i	0.374256	−	0.648230i	0.990215	+	0.139548i	$0.0445649\pi$
$882$		0			0
$883$	−5.15139e8			−0.0251803			−0.0125902	−	0.999921i	$-0.504008\pi$
$883$	−5.15139e8			−0.0251803			−0.0125902	+	0.999921i	$0.504008\pi$
$884$		0			0
$885$	−4.37485e10			−2.12159
$886$		0			0
$887$	1.37994e10	−	2.39012e10i	0.663936	−	1.14997i	−0.315636	−	0.948880i	$-0.602218\pi$
$887$	1.37994e10	−	2.39012e10i	0.663936	−	1.14997i	0.979573	−	0.201091i	$-0.0644487\pi$
$888$		0			0
$889$	−1.13734e9			−0.0542916
$890$		0			0
$891$	−4.79112e9	−	8.29846e9i	−0.226916	−	0.393030i
$892$		0			0
$893$	4.56364e9	+	7.90445e9i	0.214452	+	0.371442i
$894$		0			0
$895$	−8.45112e9	+	1.46378e10i	−0.394034	+	0.682487i
$896$		0			0
$897$	−5.26853e10	−	2.83947e10i	−2.43734	−	1.31360i
$898$		0			0
$899$	1.76755e9	−	3.06149e9i	0.0811360	−	0.140532i
$900$		0			0
$901$	3.05349e9	+	5.28880e9i	0.139078	+	0.240891i
$902$		0			0
$903$	1.35875e10	+	2.35343e10i	0.614091	+	1.06364i
$904$		0			0
$905$	−4.23489e10			−1.89921
$906$		0			0
$907$	−1.19671e10	+	2.07277e10i	−0.532555	+	0.922412i	0.466723	+	0.884404i	$0.345435\pi$
$907$	−1.19671e10	+	2.07277e10i	−0.532555	+	0.922412i	−0.999277	+	0.0380081i	$0.987899\pi$
$908$		0			0
$909$	4.63155e9			0.204528
$910$		0			0
$911$	2.98472e10			1.30794			0.653972	−	0.756519i	$-0.273102\pi$
$911$	2.98472e10			1.30794			0.653972	+	0.756519i	$0.273102\pi$
$912$		0			0
$913$	−9.53485e9	+	1.65148e10i	−0.414635	+	0.718169i
$914$		0			0
$915$	−1.07688e11			−4.64722
$916$		0			0
$917$	4.08550e9	+	7.07630e9i	0.174966	+	0.303049i
$918$		0			0
$919$	−4.11876e9	−	7.13390e9i	−0.175050	−	0.303196i	0.765129	−	0.643878i	$-0.222675\pi$
$919$	−4.11876e9	−	7.13390e9i	−0.175050	−	0.303196i	−0.940179	+	0.340682i	$0.889342\pi$
$920$		0			0
$921$	1.02052e10	−	1.76759e10i	0.430440	−	0.745544i
$922$		0			0
$923$	−8.95168e9	−	4.82450e9i	−0.374713	−	0.201951i
$924$		0			0
$925$	−4.71525e9	+	8.16705e9i	−0.195888	+	0.339289i
$926$		0			0
$927$	7.10018e9	+	1.22979e10i	0.292745	+	0.507050i
$928$		0			0
$929$	−1.65384e10	−	2.86454e10i	−0.676767	−	1.17219i	−0.975949	−	0.217999i	$-0.930047\pi$
$929$	−1.65384e10	−	2.86454e10i	−0.676767	−	1.17219i	0.299182	−	0.954196i	$-0.403286\pi$
$930$		0			0
$931$	1.81823e10			0.738456
$932$		0			0
$933$	2.55775e10	−	4.43016e10i	1.03103	−	1.78580i
$934$		0			0
$935$	1.17148e10			0.468699
$936$		0			0
$937$	6.86759e9			0.272719			0.136360	−	0.990659i	$-0.456460\pi$
$937$	6.86759e9			0.272719			0.136360	+	0.990659i	$0.456460\pi$
$938$		0			0
$939$	−2.20727e10	+	3.82311e10i	−0.870014	+	1.50691i
$940$		0			0
$941$	−3.10041e10			−1.21298			−0.606492	−	0.795090i	$-0.707424\pi$
$941$	−3.10041e10			−1.21298			−0.606492	+	0.795090i	$0.707424\pi$
$942$		0			0
$943$	−3.19648e10	−	5.53646e10i	−1.24131	−	2.15002i
$944$		0			0
$945$	−5.35279e9	−	9.27131e9i	−0.206333	−	0.357380i
$946$		0			0
$947$	−1.08273e10	+	1.87534e10i	−0.414280	+	0.717555i	−0.995353	−	0.0962973i	$-0.969300\pi$
$947$	−1.08273e10	+	1.87534e10i	−0.414280	+	0.717555i	0.581072	+	0.813852i	$0.302633\pi$
$948$		0			0
$949$	1.12719e10	−	6.95560e9i	0.428121	−	0.264182i
$950$		0			0
$951$	−5.95078e9	+	1.03070e10i	−0.224358	+	0.388600i
$952$		0			0
$953$	6.18367e9	+	1.07104e10i	0.231431	+	0.400850i	0.958229	−	0.286001i	$-0.0923260\pi$
$953$	6.18367e9	+	1.07104e10i	0.231431	+	0.400850i	−0.726799	+	0.686851i	$0.758993\pi$
$954$		0			0
$955$	−3.56868e9	−	6.18114e9i	−0.132586	−	0.229645i
$956$		0			0
$957$	−1.66950e10			−0.615737
$958$		0			0
$959$	−1.35960e8	+	2.35490e8i	−0.00497790	+	0.00862197i
$960$		0			0
$961$	−2.48529e10			−0.903326
$962$		0			0
$963$	3.52958e10			1.27359
$964$		0			0
$965$	−4.53457e9	+	7.85410e9i	−0.162439	+	0.281353i
$966$		0			0
$967$	−3.71093e10			−1.31974			−0.659872	−	0.751378i	$-0.729390\pi$
$967$	−3.71093e10			−1.31974			−0.659872	+	0.751378i	$0.729390\pi$
$968$		0			0
$969$	−6.50878e9	−	1.12735e10i	−0.229809	−	0.398040i
$970$		0			0
$971$	−1.35012e10	−	2.33848e10i	−0.473267	−	0.819723i	0.526265	−	0.850321i	$-0.323592\pi$
$971$	−1.35012e10	−	2.33848e10i	−0.473267	−	0.819723i	−0.999532	+	0.0305981i	$0.990259\pi$
$972$		0			0
$973$	1.92075e9	−	3.32683e9i	0.0668460	−	0.115781i
$974$		0			0
$975$	−3.23294e9	−	1.10385e11i	−0.111707	−	3.81413i
$976$		0			0
$977$	−1.50462e10	+	2.60608e10i	−0.516175	+	0.894041i	0.483649	+	0.875262i	$0.339311\pi$
$977$	−1.50462e10	+	2.60608e10i	−0.516175	+	0.894041i	−0.999824	+	0.0187788i	$0.994022\pi$
$978$		0			0
$979$	1.34784e10	+	2.33453e10i	0.459092	+	0.795171i
$980$		0			0
$981$	2.37713e10	+	4.11731e10i	0.803917	+	1.39242i
$982$		0			0
$983$	5.51343e10			1.85133			0.925667	−	0.378340i	$-0.123505\pi$
$983$	5.51343e10			1.85133			0.925667	+	0.378340i	$0.123505\pi$
$984$		0			0
$985$	−1.63262e10	+	2.82779e10i	−0.544327	+	0.942801i
$986$		0			0
$987$	9.67477e9			0.320281
$988$		0			0
$989$	9.72858e10			3.19788
$990$		0			0
$991$	−7.69281e9	+	1.33243e10i	−0.251089	+	0.434898i	−0.963826	−	0.266533i	$-0.914122\pi$
$991$	−7.69281e9	+	1.33243e10i	−0.251089	+	0.434898i	0.712737	+	0.701431i	$0.247455\pi$
$992$		0			0
$993$	−5.61643e10			−1.82028
$994$		0			0
$995$	−6.32656e8	−	1.09579e9i	−0.0203604	−	0.0352653i
$996$		0			0
$997$	1.39828e10	+	2.42190e10i	0.446850	+	0.773967i	0.998179	−	0.0603210i	$-0.0192124\pi$
$997$	1.39828e10	+	2.42190e10i	0.446850	+	0.773967i	−0.551329	+	0.834288i	$0.685879\pi$
$998$		0			0
$999$	−1.18089e9	+	2.04536e9i	−0.0374739	+	0.0649067i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.i.b.113.1		8
4.3	odd	2		26.8.c.b.9.4	yes	8
12.11	even	2		234.8.h.b.217.1		8
13.3	even	3	inner	208.8.i.b.81.1		8
52.3	odd	6		26.8.c.b.3.4	✓	8
52.7	even	12		338.8.b.h.337.1		8
52.19	even	12		338.8.b.h.337.5		8
52.35	odd	6		338.8.a.i.1.1		4
52.43	odd	6		338.8.a.j.1.1		4
156.107	even	6		234.8.h.b.55.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.c.b.3.4	✓	8	52.3	odd	6
26.8.c.b.9.4	yes	8	4.3	odd	2
208.8.i.b.81.1		8	13.3	even	3	inner
208.8.i.b.113.1		8	1.1	even	1	trivial
234.8.h.b.55.1		8	156.107	even	6
234.8.h.b.217.1		8	12.11	even	2
338.8.a.i.1.1		4	52.35	odd	6
338.8.a.j.1.1		4	52.43	odd	6
338.8.b.h.337.1		8	52.7	even	12
338.8.b.h.337.5		8	52.19	even	12

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	208.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-35.5792 + 61.6250i) q^{3} +523.489 q^{5} +(-208.401 - 360.960i) q^{7} +(-1438.26 - 2491.14i) q^{9} +(1711.39 - 2964.21i) q^{11} +(-6741.24 + 4159.83i) q^{13} +(-18625.3 + 32260.0i) q^{15} +(3269.02 + 5662.11i) q^{17} +(13990.2 + 24231.8i) q^{19} +29658.9 q^{21} +(53089.0 - 91952.9i) q^{23} +195916. q^{25} +49065.2 q^{27} +(-34273.0 + 59362.6i) q^{29} -51572.8 q^{31} +(121780. + 210929. i) q^{33} +(-109095. - 188959. i) q^{35} +(-24067.7 + 41686.5i) q^{37} +(-16501.7 - 563432. i) q^{39} +(301049. - 521432. i) q^{41} +(458126. + 793497. i) q^{43} +(-752914. - 1.30409e6i) q^{45} +326201. q^{47} +(324910. - 562760. i) q^{49} -465237. q^{51} +934068. q^{53} +(895894. - 1.55173e6i) q^{55} -1.99105e6 q^{57} +(587217. + 1.01709e6i) q^{59} +(1.44545e6 + 2.50359e6i) q^{61} +(-599469. + 1.03831e6i) q^{63} +(-3.52897e6 + 2.17763e6i) q^{65} +(159079. - 275533. i) q^{67} +(3.77773e6 + 6.54322e6i) q^{69} +(641869. + 1.11175e6i) q^{71} -1.67209e6 q^{73} +(-6.97054e6 + 1.20733e7i) q^{75} -1.42662e6 q^{77} +8.09160e6 q^{79} +(1.39977e6 - 2.42448e6i) q^{81} -5.57141e6 q^{83} +(1.71130e6 + 2.96405e6i) q^{85} +(-2.43881e6 - 4.22415e6i) q^{87} +(-3.93786e6 + 6.82057e6i) q^{89} +(2.90641e6 + 1.56641e6i) q^{91} +(1.83492e6 - 3.17817e6i) q^{93} +(7.32374e6 + 1.26851e7i) q^{95} +(3.40424e6 + 5.89631e6i) q^{97} -9.84569e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 556 q^{5} + 548 q^{7} - 5214 q^{9} + 7392 q^{11} - 25818 q^{13} - 15528 q^{15} + 28316 q^{17} + 99888 q^{19} + 182148 q^{21} + 33388 q^{23} + 173756 q^{25} - 212544 q^{27} + 93140 q^{29} - 622320 q^{31}+ \cdots + 20715312 q^{99}+O(q^{100}) \) 8 * q + 556 * q^5 + 548 * q^7 - 5214 * q^9 + 7392 * q^11 - 25818 * q^13 - 15528 * q^15 + 28316 * q^17 + 99888 * q^19 + 182148 * q^21 + 33388 * q^23 + 173756 * q^25 - 212544 * q^27 + 93140 * q^29 - 622320 * q^31 + 238638 * q^33 - 141544 * q^35 - 9636 * q^37 + 22932 * q^39 + 82892 * q^41 + 569264 * q^43 - 2303394 * q^45 + 1148400 * q^47 - 717798 * q^49 + 5459856 * q^51 + 2470700 * q^53 + 1092512 * q^55 + 7056924 * q^57 - 231504 * q^59 + 685684 * q^61 + 5951712 * q^63 - 6216678 * q^65 - 3271056 * q^67 + 5600034 * q^69 + 175012 * q^71 + 14275780 * q^73 - 22200960 * q^75 - 27830412 * q^77 + 14107904 * q^79 + 3758004 * q^81 - 1314576 * q^83 + 11814998 * q^85 + 7182900 * q^87 - 11452234 * q^89 - 16457168 * q^91 + 2984688 * q^93 + 23334088 * q^95 - 428002 * q^97 + 20715312 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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