LMFDB - Embedded newform 208.8.i.b.113.4

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			113.4
Root			$45.7672i$ of defining polynomial
Character	$\chi$	$=$	208.113
Dual form			208.8.i.b.81.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(39.6356 - 68.6509i) q^{3} +132.638 q^{5} +(754.459 + 1306.76i) q^{7} +(-2048.46 - 3548.04i) q^{9} +(-1463.11 + 2534.19i) q^{11} +(-7917.74 - 240.503i) q^{13} +(5257.18 - 9105.70i) q^{15} +(10869.5 + 18826.5i) q^{17} +(26739.1 + 46313.4i) q^{19} +119614. q^{21} +(-14224.0 + 24636.6i) q^{23} -60532.2 q^{25} -151402. q^{27} +(-76324.1 + 132197. i) q^{29} -83326.6 q^{31} +(115983. + 200888. i) q^{33} +(100070. + 173326. i) q^{35} +(-56229.4 + 97392.2i) q^{37} +(-330335. + 534028. i) q^{39} +(-55065.1 + 95375.6i) q^{41} +(52967.4 + 91742.2i) q^{43} +(-271703. - 470604. i) q^{45} -413100. q^{47} +(-726645. + 1.25859e6i) q^{49} +1.72327e6 q^{51} -889607. q^{53} +(-194064. + 336129. i) q^{55} +4.23928e6 q^{57} +(1.20535e6 + 2.08773e6i) q^{59} +(-1.19229e6 - 2.06512e6i) q^{61} +(3.09096e6 - 5.35370e6i) q^{63} +(-1.05019e6 - 31899.8i) q^{65} +(-184882. + 320224. i) q^{67} +(1.12755e6 + 1.95298e6i) q^{69} +(-1.59359e6 - 2.76018e6i) q^{71} +5.84239e6 q^{73} +(-2.39923e6 + 4.15559e6i) q^{75} -4.41544e6 q^{77} -1.07694e6 q^{79} +(-1.52092e6 + 2.63431e6i) q^{81} -647944. q^{83} +(1.44170e6 + 2.49710e6i) q^{85} +(6.05031e6 + 1.04794e7i) q^{87} +(5.81291e6 - 1.00682e7i) q^{89} +(-5.65933e6 - 1.05280e7i) q^{91} +(-3.30270e6 + 5.72045e6i) q^{93} +(3.54661e6 + 6.14291e6i) q^{95} +(-31183.4 - 54011.2i) q^{97} +1.19885e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	39.6356	−	68.6509i	0.847542	−	1.46799i	−0.0358539	−	0.999357i	$-0.511415\pi$
$3$	39.6356	−	68.6509i	0.847542	−	1.46799i	0.883396	−	0.468628i	$-0.155252\pi$
$4$		0			0
$5$	132.638			0.474539			0.237270	−	0.971444i	$-0.423748\pi$
$5$	132.638			0.474539			0.237270	+	0.971444i	$0.423748\pi$
$6$		0			0
$7$	754.459	+	1306.76i	0.831366	+	1.43997i	0.896955	+	0.442122i	$0.145774\pi$
$7$	754.459	+	1306.76i	0.831366	+	1.43997i	−0.0655888	+	0.997847i	$0.520893\pi$
$8$		0			0
$9$	−2048.46	−	3548.04i	−0.936654	−	1.62233i
$10$		0			0
$11$	−1463.11	+	2534.19i	−0.331439	+	0.574070i	−0.982794	−	0.184703i	$-0.940868\pi$
$11$	−1463.11	+	2534.19i	−0.331439	+	0.574070i	0.651355	+	0.758773i	$0.274201\pi$
$12$		0			0
$13$	−7917.74	−	240.503i	−0.999539	−	0.0303612i
$13$	−7917.74	−	240.503i	−0.999539	−	0.0303612i
$14$		0			0
$15$	5257.18	−	9105.70i	0.402192	−	0.696617i
$16$		0			0
$17$	10869.5	+	18826.5i	0.536583	+	0.929389i	0.999085	+	0.0427706i	$0.0136185\pi$
$17$	10869.5	+	18826.5i	0.536583	+	0.929389i	−0.462502	+	0.886618i	$0.653048\pi$
$18$		0			0
$19$	26739.1	+	46313.4i	0.894352	+	1.54906i	0.834604	+	0.550850i	$0.185697\pi$
$19$	26739.1	+	46313.4i	0.894352	+	1.54906i	0.0597485	+	0.998213i	$0.480970\pi$
$20$		0			0
$21$	119614.			2.81847
$22$		0			0
$23$	−14224.0	+	24636.6i	−0.243766	+	0.422215i	−0.961784	−	0.273809i	$-0.911716\pi$
$23$	−14224.0	+	24636.6i	−0.243766	+	0.422215i	0.718018	+	0.696025i	$0.245050\pi$
$24$		0			0
$25$	−60532.2			−0.774813
$26$		0			0
$27$	−151402.			−1.48033
$28$		0			0
$29$	−76324.1	+	132197.i	−0.581124	+	1.00654i	0.414222	+	0.910176i	$0.364054\pi$
$29$	−76324.1	+	132197.i	−0.581124	+	1.00654i	−0.995346	+	0.0963608i	$0.969280\pi$
$30$		0			0
$31$	−83326.6			−0.502363			−0.251182	−	0.967940i	$-0.580819\pi$
$31$	−83326.6			−0.502363			−0.251182	+	0.967940i	$0.580819\pi$
$32$		0			0
$33$	115983.	+	200888.i	0.561817	+	0.973096i
$34$		0			0
$35$	100070.	+	173326.i	0.394516	+	0.683322i
$36$		0			0
$37$	−56229.4	+	97392.2i	−0.182498	+	0.316095i	−0.942730	−	0.333555i	$-0.891752\pi$
$37$	−56229.4	+	97392.2i	−0.182498	+	0.316095i	0.760233	+	0.649651i	$0.225085\pi$
$38$		0			0
$39$	−330335.	+	534028.i	−0.891721	+	1.44158i
$40$		0			0
$41$	−55065.1	+	95375.6i	−0.124777	+	0.216119i	−0.921646	−	0.388033i	$-0.873155\pi$
$41$	−55065.1	+	95375.6i	−0.124777	+	0.216119i	0.796869	+	0.604152i	$0.206488\pi$
$42$		0			0
$43$	52967.4	+	91742.2i	0.101594	+	0.175966i	0.912342	−	0.409430i	$-0.134272\pi$
$43$	52967.4	+	91742.2i	0.101594	+	0.175966i	−0.810747	+	0.585396i	$0.800939\pi$
$44$		0			0
$45$	−271703.	−	470604.i	−0.444479	−	0.769860i
$46$		0			0
$47$	−413100.			−0.580381			−0.290190	−	0.956969i	$-0.593719\pi$
$47$	−413100.			−0.580381			−0.290190	+	0.956969i	$0.593719\pi$
$48$		0			0
$49$	−726645.	+	1.25859e6i	−0.882340	+	1.52826i
$50$		0			0
$51$	1.72327e6			1.81911
$52$		0			0
$53$	−889607.			−0.820791			−0.410395	−	0.911908i	$-0.634609\pi$
$53$	−889607.			−0.820791			−0.410395	+	0.911908i	$0.634609\pi$
$54$		0			0
$55$	−194064.	+	336129.i	−0.157281	+	0.272419i
$56$		0			0
$57$	4.23928e6			3.03200
$58$		0			0
$59$	1.20535e6	+	2.08773e6i	0.764066	+	1.32340i	0.940739	+	0.339132i	$0.110133\pi$
$59$	1.20535e6	+	2.08773e6i	0.764066	+	1.32340i	−0.176673	+	0.984270i	$0.556533\pi$
$60$		0			0
$61$	−1.19229e6	−	2.06512e6i	−0.672557	−	1.16490i	−0.977176	−	0.212429i	$-0.931863\pi$
$61$	−1.19229e6	−	2.06512e6i	−0.672557	−	1.16490i	0.304619	−	0.952474i	$-0.401471\pi$
$62$		0			0
$63$	3.09096e6	−	5.35370e6i	1.55740	−	2.69750i
$64$		0			0
$65$	−1.05019e6	−	31899.8i	−0.474320	−	0.0144076i
$66$		0			0
$67$	−184882.	+	320224.i	−0.0750986	+	0.130075i	−0.901129	−	0.433551i	$-0.857260\pi$
$67$	−184882.	+	320224.i	−0.0750986	+	0.130075i	0.826031	+	0.563625i	$0.190594\pi$
$68$		0			0
$69$	1.12755e6	+	1.95298e6i	0.413204	+	0.715690i
$70$		0			0
$71$	−1.59359e6	−	2.76018e6i	−0.528412	−	0.915236i	−0.999451	−	0.0331239i	$-0.989454\pi$
$71$	−1.59359e6	−	2.76018e6i	−0.528412	−	0.915236i	0.471039	−	0.882112i	$-0.343879\pi$
$72$		0			0
$73$	5.84239e6			1.75776			0.878882	−	0.477038i	$-0.158290\pi$
$73$	5.84239e6			1.75776			0.878882	+	0.477038i	$0.158290\pi$
$74$		0			0
$75$	−2.39923e6	+	4.15559e6i	−0.656686	+	1.13741i
$76$		0			0
$77$	−4.41544e6			−1.10219
$78$		0			0
$79$	−1.07694e6			−0.245751			−0.122876	−	0.992422i	$-0.539212\pi$
$79$	−1.07694e6			−0.245751			−0.122876	+	0.992422i	$0.539212\pi$
$80$		0			0
$81$	−1.52092e6	+	2.63431e6i	−0.317987	+	0.550769i
$82$		0			0
$83$	−647944.			−0.124384			−0.0621920	−	0.998064i	$-0.519809\pi$
$83$	−647944.			−0.124384			−0.0621920	+	0.998064i	$0.519809\pi$
$84$		0			0
$85$	1.44170e6	+	2.49710e6i	0.254630	+	0.441031i
$86$		0			0
$87$	6.05031e6	+	1.04794e7i	0.985054	+	1.70616i
$88$		0			0
$89$	5.81291e6	−	1.00682e7i	0.874034	−	1.51387i	0.0162455	−	0.999868i	$-0.494829\pi$
$89$	5.81291e6	−	1.00682e7i	0.874034	−	1.51387i	0.857788	−	0.514003i	$-0.171838\pi$
$90$		0			0
$91$	−5.65933e6	−	1.05280e7i	−0.787264	−	1.46455i
$92$		0			0
$93$	−3.30270e6	+	5.72045e6i	−0.425774	+	0.737462i
$94$		0			0
$95$	3.54661e6	+	6.14291e6i	0.424405	+	0.735092i
$96$		0			0
$97$	−31183.4	−	54011.2i	−0.00346914	−	0.00600873i	0.864286	−	0.503001i	$-0.167771\pi$
$97$	−31183.4	−	54011.2i	−0.00346914	−	0.00600873i	−0.867755	+	0.496993i	$0.834438\pi$
$98$		0			0
$99$	1.19885e7			1.24178
$100$		0			0
$101$	7.40293e6	−	1.28222e7i	0.714955	−	1.23834i	−0.248022	−	0.968754i	$-0.579781\pi$
$101$	7.40293e6	−	1.28222e7i	0.714955	−	1.23834i	0.962977	−	0.269584i	$-0.0868862\pi$
$102$		0			0
$103$	1.24059e7			1.11866			0.559332	−	0.828944i	$-0.311058\pi$
$103$	1.24059e7			1.11866			0.559332	+	0.828944i	$0.311058\pi$
$104$		0			0
$105$	1.58653e7			1.33747
$106$		0			0
$107$	284651.	−	493030.i	0.0224631	−	0.0389072i	−0.854575	−	0.519327i	$-0.826183\pi$
$107$	284651.	−	493030.i	0.0224631	−	0.0389072i	0.877038	+	0.480420i	$0.159516\pi$
$108$		0			0
$109$	−3.87174e6			−0.286361			−0.143181	−	0.989697i	$-0.545733\pi$
$109$	−3.87174e6			−0.286361			−0.143181	+	0.989697i	$0.545733\pi$
$110$		0			0
$111$	4.45737e6	+	7.72040e6i	0.309349	+	0.535808i
$112$		0			0
$113$	1.39031e7	+	2.40808e7i	0.906433	+	1.56999i	0.818981	+	0.573820i	$0.194539\pi$
$113$	1.39031e7	+	2.40808e7i	0.906433	+	1.56999i	0.0874519	+	0.996169i	$0.472128\pi$
$114$		0			0
$115$	−1.88664e6	+	3.26775e6i	−0.115677	+	0.200358i
$116$		0			0
$117$	1.53659e7	+	2.85851e7i	0.886966	+	1.65002i
$118$		0			0
$119$	−1.64011e7	+	2.84076e7i	−0.892194	+	1.54533i
$120$		0			0
$121$	5.46218e6	+	9.46077e6i	0.280296	+	0.485487i
$122$		0			0
$123$	4.36508e6	+	7.56054e6i	0.211507	+	0.366341i
$124$		0			0
$125$	−1.83912e7			−0.842218
$126$		0			0
$127$	8.01861e6	−	1.38886e7i	0.347365	−	0.601654i	−0.638415	−	0.769692i	$-0.720410\pi$
$127$	8.01861e6	−	1.38886e7i	0.347365	−	0.601654i	0.985781	+	0.168038i	$0.0537431\pi$
$128$		0			0
$129$	8.39757e6			0.344421
$130$		0			0
$131$	−2.10431e7			−0.817824			−0.408912	−	0.912574i	$-0.634092\pi$
$131$	−2.10431e7			−0.817824			−0.408912	+	0.912574i	$0.634092\pi$
$132$		0			0
$133$	−4.03471e7	+	6.98831e7i	−1.48707	+	2.57568i
$134$		0			0
$135$	−2.00816e7			−0.702474
$136$		0			0
$137$	−2.19368e7	−	3.79957e7i	−0.728873	−	1.26244i	−0.957360	−	0.288897i	$-0.906711\pi$
$137$	−2.19368e7	−	3.79957e7i	−0.728873	−	1.26244i	0.228488	−	0.973547i	$-0.426622\pi$
$138$		0			0
$139$	6.95070e6	+	1.20390e7i	0.219521	+	0.380222i	0.954662	−	0.297693i	$-0.0962171\pi$
$139$	6.95070e6	+	1.20390e7i	0.219521	+	0.380222i	−0.735140	+	0.677915i	$0.762884\pi$
$140$		0			0
$141$	−1.63735e7	+	2.83597e7i	−0.491897	+	0.851991i
$142$		0			0
$143$	1.21940e7	−	1.97132e7i	0.348716	−	0.563742i
$144$		0			0
$145$	−1.01235e7	+	1.75343e7i	−0.275766	+	0.477641i
$146$		0			0
$147$	5.76020e7	+	9.97696e7i	1.49564	+	2.59052i
$148$		0			0
$149$	−2.03877e7	−	3.53125e7i	−0.504912	−	0.874533i	−0.999984	−	0.00568086i	$-0.998192\pi$
$149$	−2.03877e7	−	3.53125e7i	−0.504912	−	0.874533i	0.495072	−	0.868852i	$-0.335142\pi$
$150$		0			0
$151$	6.68855e7			1.58093			0.790465	−	0.612507i	$-0.209839\pi$
$151$	6.68855e7			1.58093			0.790465	+	0.612507i	$0.209839\pi$
$152$		0			0
$153$	4.45314e7	−	7.71306e7i	1.00518	−	1.74103i
$154$		0			0
$155$	−1.10523e7			−0.238391
$156$		0			0
$157$	7.74485e7			1.59722			0.798609	−	0.601850i	$-0.205569\pi$
$157$	7.74485e7			1.59722			0.798609	+	0.601850i	$0.205569\pi$
$158$		0			0
$159$	−3.52601e7	+	6.10723e7i	−0.695654	+	1.20491i
$160$		0			0
$161$	−4.29256e7			−0.810636
$162$		0			0
$163$	−1.53988e7	−	2.66715e7i	−0.278503	−	0.482381i	0.692510	−	0.721408i	$-0.256505\pi$
$163$	−1.53988e7	−	2.66715e7i	−0.278503	−	0.482381i	−0.971013	+	0.239027i	$0.923172\pi$
$164$		0			0
$165$	1.53837e7	+	2.66454e7i	0.266604	+	0.461772i
$166$		0			0
$167$	2.03626e7	−	3.52691e7i	0.338318	−	0.585985i	−0.645798	−	0.763508i	$-0.723475\pi$
$167$	2.03626e7	−	3.52691e7i	0.338318	−	0.585985i	0.984117	+	0.177524i	$0.0568086\pi$
$168$		0			0
$169$	6.26328e7	+	3.80849e6i	0.998156	+	0.0606945i
$170$		0			0
$171$	1.09548e8	−	1.89743e8i	1.67540	−	2.90187i
$172$		0			0
$173$	−1.47009e7	−	2.54627e7i	−0.215865	−	0.373889i	0.737675	−	0.675156i	$-0.235924\pi$
$173$	−1.47009e7	−	2.54627e7i	−0.215865	−	0.373889i	−0.953540	+	0.301267i	$0.902590\pi$
$174$		0			0
$175$	−4.56691e7	−	7.91012e7i	−0.644153	−	1.11571i
$176$		0			0
$177$	1.91099e8			2.59031
$178$		0			0
$179$	2.25181e7	−	3.90025e7i	0.293458	−	0.508285i	−0.681167	−	0.732128i	$-0.738527\pi$
$179$	2.25181e7	−	3.90025e7i	0.293458	−	0.508285i	0.974625	+	0.223844i	$0.0718605\pi$
$180$		0			0
$181$	−7.76931e7			−0.973885			−0.486942	−	0.873434i	$-0.661888\pi$
$181$	−7.76931e7			−0.973885			−0.486942	+	0.873434i	$0.661888\pi$
$182$		0			0
$183$	−1.89029e8			−2.28008
$184$		0			0
$185$	−7.45814e6	+	1.29179e7i	−0.0866024	+	0.150000i
$186$		0			0
$187$	−6.36131e7			−0.711379
$188$		0			0
$189$	−1.14226e8	−	1.97846e8i	−1.23070	−	2.13163i
$190$		0			0
$191$	8.27657e6	+	1.43354e7i	0.0859476	+	0.148866i	0.905795	−	0.423717i	$-0.139275\pi$
$191$	8.27657e6	+	1.43354e7i	0.0859476	+	0.148866i	−0.819847	+	0.572583i	$0.805942\pi$
$192$		0			0
$193$	2.90207e7	−	5.02653e7i	0.290574	−	0.503290i	−0.683371	−	0.730071i	$-0.739487\pi$
$193$	2.90207e7	−	5.02653e7i	0.290574	−	0.503290i	0.973946	+	0.226781i	$0.0728203\pi$
$194$		0			0
$195$	−4.38149e7	+	7.08322e7i	−0.423156	+	0.684084i
$196$		0			0
$197$	−1.93955e7	+	3.35940e7i	−0.180746	+	0.313062i	−0.942135	−	0.335234i	$-0.891185\pi$
$197$	−1.93955e7	+	3.35940e7i	−0.180746	+	0.313062i	0.761389	+	0.648296i	$0.224518\pi$
$198$		0			0
$199$	1.41836e7	+	2.45667e7i	0.127585	+	0.220984i	0.922740	−	0.385422i	$-0.125944\pi$
$199$	1.41836e7	+	2.45667e7i	0.127585	+	0.220984i	−0.795155	+	0.606406i	$0.792611\pi$
$200$		0			0
$201$	1.46558e7	+	2.53846e7i	0.127298	+	0.220487i
$202$		0			0
$203$	−2.30334e8			−1.93251
$204$		0			0
$205$	−7.30372e6	+	1.26504e7i	−0.0592114	+	0.102557i
$206$		0			0
$207$	1.16549e8			0.913298
$208$		0			0
$209$	−1.56489e8			−1.18569
$210$		0			0
$211$	−3.26467e7	+	5.65457e7i	−0.239249	+	0.414392i	−0.960499	−	0.278283i	$-0.910235\pi$
$211$	−3.26467e7	+	5.65457e7i	−0.239249	+	0.414392i	0.721250	+	0.692675i	$0.243568\pi$
$212$		0			0
$213$	−2.52652e8			−1.79140
$214$		0			0
$215$	7.02547e6	+	1.21685e7i	0.0482104	+	0.0835029i
$216$		0			0
$217$	−6.28665e7	−	1.08888e8i	−0.417648	−	0.723387i
$218$		0			0
$219$	2.31567e8	−	4.01085e8i	1.48978	−	2.58037i
$220$		0			0
$221$	−8.15338e7	−	1.51677e8i	−0.508118	−	0.945252i
$222$		0			0
$223$	−5.24401e7	+	9.08290e7i	−0.316663	+	0.548476i	−0.979790	−	0.200031i	$-0.935896\pi$
$223$	−5.24401e7	+	9.08290e7i	−0.316663	+	0.548476i	0.663127	+	0.748507i	$0.269229\pi$
$224$		0			0
$225$	1.23998e8	+	2.14771e8i	0.725731	+	1.25700i
$226$		0			0
$227$	2.49235e7	+	4.31688e7i	0.141423	+	0.244951i	0.928033	−	0.372499i	$-0.121499\pi$
$227$	2.49235e7	+	4.31688e7i	0.141423	+	0.244951i	−0.786610	+	0.617450i	$0.788166\pi$
$228$		0			0
$229$	1.15632e8			0.636287			0.318144	−	0.948043i	$-0.396941\pi$
$229$	1.15632e8			0.636287			0.318144	+	0.948043i	$0.396941\pi$
$230$		0			0
$231$	−1.75009e8	+	3.03124e8i	−0.934152	+	1.61800i
$232$		0			0
$233$	7.04962e7			0.365107			0.182553	−	0.983196i	$-0.441564\pi$
$233$	7.04962e7			0.365107			0.182553	+	0.983196i	$0.441564\pi$
$234$		0			0
$235$	−5.47927e7			−0.275413
$236$		0			0
$237$	−4.26850e7	+	7.39327e7i	−0.208284	+	0.360759i
$238$		0			0
$239$	3.08705e8			1.46269			0.731343	−	0.682010i	$-0.238894\pi$
$239$	3.08705e8			1.46269			0.731343	+	0.682010i	$0.238894\pi$
$240$		0			0
$241$	−6.18820e7	−	1.07183e8i	−0.284777	−	0.493248i	0.687778	−	0.725921i	$-0.258586\pi$
$241$	−6.18820e7	−	1.07183e8i	−0.284777	−	0.493248i	−0.972555	+	0.232673i	$0.925253\pi$
$242$		0			0
$243$	−4.49928e7	−	7.79299e7i	−0.201151	−	0.348403i
$244$		0			0
$245$	−9.63805e7	+	1.66936e8i	−0.418705	+	0.725218i
$246$		0			0
$247$	−2.00575e8	−	3.73129e8i	−0.846909	−	1.57550i
$248$		0			0
$249$	−2.56817e7	+	4.44819e7i	−0.105421	+	0.182594i
$250$		0			0
$251$	1.68238e8	+	2.91397e8i	0.671531	+	1.16313i	0.977470	+	0.211075i	$0.0676963\pi$
$251$	1.68238e8	+	2.91397e8i	0.671531	+	1.16313i	−0.305939	+	0.952051i	$0.598970\pi$
$252$		0			0
$253$	−4.16226e7	−	7.20925e7i	−0.161587	−	0.279878i
$254$		0			0
$255$	2.28571e8			0.863237
$256$		0			0
$257$	−1.35250e8	+	2.34259e8i	−0.497016	+	0.860857i	−0.999994	−	0.00344215i	$-0.998904\pi$
$257$	−1.35250e8	+	2.34259e8i	−0.497016	+	0.860857i	0.502978	+	0.864299i	$0.332238\pi$
$258$		0			0
$259$	−1.69691e8			−0.606890
$260$		0			0
$261$	6.25388e8			2.17725
$262$		0			0
$263$	−2.01493e8	+	3.48997e8i	−0.682992	+	1.18298i	0.291072	+	0.956701i	$0.405988\pi$
$263$	−2.01493e8	+	3.48997e8i	−0.682992	+	1.18298i	−0.974063	+	0.226275i	$0.927345\pi$
$264$		0			0
$265$	−1.17995e8			−0.389497
$266$		0			0
$267$	−4.60796e8	−	7.98122e8i	−1.48156	−	2.56614i
$268$		0			0
$269$	7.27637e7	+	1.26030e8i	0.227920	+	0.394768i	0.957191	−	0.289456i	$-0.0934743\pi$
$269$	7.27637e7	+	1.26030e8i	0.227920	+	0.394768i	−0.729272	+	0.684224i	$0.760141\pi$
$270$		0			0
$271$	−1.78039e8	+	3.08372e8i	−0.543403	+	0.941202i	0.455302	+	0.890337i	$0.349531\pi$
$271$	−1.78039e8	+	3.08372e8i	−0.543403	+	0.941202i	−0.998706	+	0.0508652i	$0.983802\pi$
$272$		0			0
$273$	−9.47071e8	−	2.87675e7i	−2.81717	−	0.0855722i
$274$		0			0
$275$	8.85656e7	−	1.53400e8i	0.256803	−	0.444796i
$276$		0			0
$277$	−2.43177e8	−	4.21195e8i	−0.687454	−	1.19071i	−0.972659	−	0.232239i	$-0.925395\pi$
$277$	−2.43177e8	−	4.21195e8i	−0.687454	−	1.19071i	0.285205	−	0.958467i	$-0.407938\pi$
$278$		0			0
$279$	1.70691e8	+	2.95646e8i	0.470540	+	0.815000i
$280$		0			0
$281$	−4.89344e7			−0.131566			−0.0657828	−	0.997834i	$-0.520954\pi$
$281$	−4.89344e7			−0.131566			−0.0657828	+	0.997834i	$0.520954\pi$
$282$		0			0
$283$	−1.37703e8	+	2.38509e8i	−0.361153	+	0.625535i	−0.988151	−	0.153486i	$-0.950950\pi$
$283$	−1.37703e8	+	2.38509e8i	−0.361153	+	0.625535i	0.626998	+	0.779021i	$0.284283\pi$
$284$		0			0
$285$	5.62288e8			1.43880
$286$		0			0
$287$	−1.66178e8			−0.414940
$288$		0			0
$289$	−3.11211e7	+	5.39033e7i	−0.0758425	+	0.131363i
$290$		0			0
$291$	−4.94389e6			−0.0117610
$292$		0			0
$293$	1.53439e8	+	2.65765e8i	0.356369	+	0.617249i	0.987351	−	0.158548i	$-0.0506813\pi$
$293$	1.53439e8	+	2.65765e8i	0.356369	+	0.617249i	−0.630982	+	0.775797i	$0.717348\pi$
$294$		0			0
$295$	1.59875e8	+	2.76911e8i	0.362579	+	0.628006i
$296$		0			0
$297$	2.21518e8	−	3.83681e8i	0.490639	−	0.849812i
$298$		0			0
$299$	1.18547e8	−	1.91646e8i	0.256473	−	0.414620i
$300$		0			0
$301$	−7.99234e7	+	1.38431e8i	−0.168924	+	0.292585i
$302$		0			0
$303$	−5.86839e8	−	1.01643e9i	−1.21191	−	2.09909i
$304$		0			0
$305$	−1.58143e8	−	2.73912e8i	−0.319155	−	0.552792i
$306$		0			0
$307$	1.55157e8			0.306046			0.153023	−	0.988223i	$-0.451099\pi$
$307$	1.55157e8			0.306046			0.153023	+	0.988223i	$0.451099\pi$
$308$		0			0
$309$	4.91717e8	−	8.51679e8i	0.948114	−	1.64218i
$310$		0			0
$311$	5.61165e8			1.05786			0.528931	−	0.848665i	$-0.322593\pi$
$311$	5.61165e8			1.05786			0.528931	+	0.848665i	$0.322593\pi$
$312$		0			0
$313$	−3.88240e8			−0.715640			−0.357820	−	0.933791i	$-0.616480\pi$
$313$	−3.88240e8			−0.715640			−0.357820	+	0.933791i	$0.616480\pi$
$314$		0			0
$315$	4.09978e8	−	7.10103e8i	0.739050	−	1.28007i
$316$		0			0
$317$	6.56778e7			0.115801			0.0579003	−	0.998322i	$-0.481559\pi$
$317$	6.56778e7			0.115801			0.0579003	+	0.998322i	$0.481559\pi$
$318$		0			0
$319$	−2.23342e8	−	3.86840e8i	−0.385215	−	0.667212i
$320$		0			0
$321$	−2.25646e7	−	3.90831e7i	−0.0380768	−	0.0659509i
$322$		0			0
$323$	−5.81279e8	+	1.00680e9i	−0.959789	+	1.66240i
$324$		0			0
$325$	4.79279e8	+	1.45582e7i	0.774455	+	0.0235243i
$326$		0			0
$327$	−1.53459e8	+	2.65799e8i	−0.242703	+	0.420374i
$328$		0			0
$329$	−3.11667e8	−	5.39823e8i	−0.482509	−	0.835730i
$330$		0			0
$331$	3.41087e8	+	5.90779e8i	0.516971	+	0.895421i	0.999806	+	0.0197091i	$0.00627402\pi$
$331$	3.41087e8	+	5.90779e8i	0.516971	+	0.895421i	−0.482834	+	0.875712i	$0.660393\pi$
$332$		0			0
$333$	4.60735e8			0.683749
$334$		0			0
$335$	−2.45223e7	+	4.24738e7i	−0.0356372	+	0.0617255i
$336$		0			0
$337$	5.06834e8			0.721375			0.360687	−	0.932687i	$-0.382542\pi$
$337$	5.06834e8			0.721375			0.360687	+	0.932687i	$0.382542\pi$
$338$		0			0
$339$	2.20422e9			3.07296
$340$		0			0
$341$	1.21916e8	−	2.11165e8i	0.166503	−	0.288392i
$342$		0			0
$343$	−9.50235e8			−1.27146
$344$		0			0
$345$	1.49556e8	+	2.59038e8i	0.196081	+	0.339623i
$346$		0			0
$347$	−2.42354e8	−	4.19770e8i	−0.311385	−	0.539334i	0.667278	−	0.744809i	$-0.267459\pi$
$347$	−2.42354e8	−	4.19770e8i	−0.311385	−	0.539334i	−0.978662	+	0.205475i	$0.934126\pi$
$348$		0			0
$349$	−1.46232e8	+	2.53281e8i	−0.184142	+	0.318943i	−0.943287	−	0.331978i	$-0.892284\pi$
$349$	−1.46232e8	+	2.53281e8i	−0.184142	+	0.318943i	0.759145	+	0.650921i	$0.225617\pi$
$350$		0			0
$351$	1.19876e9	+	3.64127e7i	1.47965	+	0.0449446i
$352$		0			0
$353$	−3.95188e8	+	6.84485e8i	−0.478181	+	0.828233i	−0.999687	−	0.0250141i	$-0.992037\pi$
$353$	−3.95188e8	+	6.84485e8i	−0.478181	+	0.828233i	0.521506	+	0.853247i	$0.325370\pi$
$354$		0			0
$355$	−2.11370e8	−	3.66104e8i	−0.250752	−	0.434315i
$356$		0			0
$357$	1.30014e9	+	2.25190e9i	1.51234	+	2.61945i
$358$		0			0
$359$	1.31770e8			0.150309			0.0751544	−	0.997172i	$-0.476055\pi$
$359$	1.31770e8			0.150309			0.0751544	+	0.997172i	$0.476055\pi$
$360$		0			0
$361$	−9.83020e8	+	1.70264e9i	−1.09973	+	1.90479i
$362$		0			0
$363$	8.65986e8			0.950250
$364$		0			0
$365$	7.74922e8			0.834128
$366$		0			0
$367$	−4.36649e8	+	7.56299e8i	−0.461107	+	0.798661i	−0.999016	−	0.0443420i	$-0.985881\pi$
$367$	−4.36649e8	+	7.56299e8i	−0.461107	+	0.798661i	0.537909	+	0.843003i	$0.319214\pi$
$368$		0			0
$369$	4.51195e8			0.467490
$370$		0			0
$371$	−6.71172e8	−	1.16250e9i	−0.682378	−	1.18191i
$372$		0			0
$373$	−6.30678e8	−	1.09237e9i	−0.629255	−	1.08990i	−0.987701	−	0.156352i	$-0.950027\pi$
$373$	−6.30678e8	−	1.09237e9i	−0.629255	−	1.08990i	0.358446	−	0.933550i	$-0.383307\pi$
$374$		0			0
$375$	−7.28946e8	+	1.26257e9i	−0.713815	+	1.23636i
$376$		0			0
$377$	6.36109e8	−	1.02835e9i	0.611416	−	0.988429i
$378$		0			0
$379$	2.35511e8	−	4.07916e8i	0.222215	−	0.384887i	−0.733265	−	0.679943i	$-0.762005\pi$
$379$	2.35511e8	−	4.07916e8i	0.222215	−	0.384887i	0.955480	+	0.295055i	$0.0953380\pi$
$380$		0			0
$381$	−6.35645e8	−	1.10097e9i	−0.588813	−	1.01985i
$382$		0			0
$383$	4.16532e8	+	7.21454e8i	0.378837	+	0.656165i	0.990893	−	0.134649i	$-0.0429908\pi$
$383$	4.16532e8	+	7.21454e8i	0.378837	+	0.656165i	−0.612056	+	0.790814i	$0.709657\pi$
$384$		0			0
$385$	−5.85654e8			−0.523032
$386$		0			0
$387$	2.17003e8	−	3.75861e8i	0.190317	−	0.329639i
$388$		0			0
$389$	−1.21290e8			−0.104472			−0.0522362	−	0.998635i	$-0.516635\pi$
$389$	−1.21290e8			−0.104472			−0.0522362	+	0.998635i	$0.516635\pi$
$390$		0			0
$391$	−6.18428e8			−0.523203
$392$		0			0
$393$	−8.34055e8	+	1.44463e9i	−0.693140	+	1.20055i
$394$		0			0
$395$	−1.42842e8			−0.116619
$396$		0			0
$397$	3.95732e8	+	6.85429e8i	0.317420	+	0.549788i	0.979949	−	0.199248i	$-0.0638500\pi$
$397$	3.95732e8	+	6.85429e8i	0.317420	+	0.549788i	−0.662529	+	0.749037i	$0.730517\pi$
$398$		0			0
$399$	3.19836e9	+	5.53972e9i	2.52071	+	4.36599i
$400$		0			0
$401$	7.55394e8	−	1.30838e9i	0.585017	−	1.01328i	−0.409857	−	0.912150i	$-0.634421\pi$
$401$	7.55394e8	−	1.30838e9i	0.585017	−	1.01328i	0.994873	−	0.101129i	$-0.0322453\pi$
$402$		0			0
$403$	6.59759e8	+	2.00403e7i	0.502132	+	0.0152524i
$404$		0			0
$405$	−2.01731e8	+	3.49409e8i	−0.150897	+	0.261361i
$406$		0			0
$407$	−1.64540e8	−	2.84992e8i	−0.120974	−	0.209533i
$408$		0			0
$409$	1.16664e9	+	2.02068e9i	0.843151	+	1.46038i	0.887217	+	0.461351i	$0.152635\pi$
$409$	1.16664e9	+	2.02068e9i	0.843151	+	1.46038i	−0.0440666	+	0.999029i	$0.514031\pi$
$410$		0			0
$411$	−3.47792e9			−2.47100
$412$		0			0
$413$	−1.81877e9	+	3.15021e9i	−1.27044	+	2.20046i
$414$		0			0
$415$	−8.59418e7			−0.0590250
$416$		0			0
$417$	1.10198e9			0.744214
$418$		0			0
$419$	5.09928e8	−	8.83220e8i	0.338656	−	0.586570i	−0.645524	−	0.763740i	$-0.723361\pi$
$419$	5.09928e8	−	8.83220e8i	0.338656	−	0.586570i	0.984180	+	0.177170i	$0.0566942\pi$
$420$		0			0
$421$	−1.73659e9			−1.13426			−0.567128	−	0.823630i	$-0.691945\pi$
$421$	−1.73659e9			−1.13426			−0.567128	+	0.823630i	$0.691945\pi$
$422$		0			0
$423$	8.46220e8	+	1.46570e9i	0.543616	+	0.941570i
$424$		0			0
$425$	−6.57953e8	−	1.13961e9i	−0.415751	−	0.720102i
$426$		0			0
$427$	1.79907e9	−	3.11609e9i	1.11828	−	1.93692i
$428$		0			0
$429$	−8.70008e8	−	1.61848e9i	−0.532014	−	0.989705i
$430$		0			0
$431$	−1.20435e8	+	2.08599e8i	−0.0724573	+	0.125500i	−0.899978	−	0.435936i	$-0.856417\pi$
$431$	−1.20435e8	+	2.08599e8i	−0.0724573	+	0.125500i	0.827520	+	0.561436i	$0.189751\pi$
$432$		0			0
$433$	1.10516e9	+	1.91420e9i	0.654212	+	1.13313i	0.982091	+	0.188409i	$0.0603330\pi$
$433$	1.10516e9	+	1.91420e9i	0.654212	+	1.13313i	−0.327878	+	0.944720i	$0.606334\pi$
$434$		0			0
$435$	8.02499e8	+	1.38997e9i	0.467447	+	0.809641i
$436$		0			0
$437$	−1.52134e9			−0.872052
$438$		0			0
$439$	−1.15306e9	+	1.99716e9i	−0.650468	+	1.12664i	0.332542	+	0.943088i	$0.392094\pi$
$439$	−1.15306e9	+	1.99716e9i	−0.650468	+	1.12664i	−0.983010	+	0.183554i	$0.941240\pi$
$440$		0			0
$441$	5.95401e9			3.30579
$442$		0			0
$443$	−1.10170e9			−0.602073			−0.301036	−	0.953613i	$-0.597333\pi$
$443$	−1.10170e9			−0.602073			−0.301036	+	0.953613i	$0.597333\pi$
$444$		0			0
$445$	7.71011e8	−	1.33543e9i	0.414763	−	0.718391i
$446$		0			0
$447$	−3.23231e9			−1.71173
$448$		0			0
$449$	−9.03527e8	−	1.56495e9i	−0.471063	−	0.815905i	0.528389	−	0.849002i	$-0.322796\pi$
$449$	−9.03527e8	−	1.56495e9i	−0.471063	−	0.815905i	−0.999452	+	0.0330974i	$0.989463\pi$
$450$		0			0
$451$	−1.61133e8	−	2.79091e8i	−0.0827118	−	0.143261i
$452$		0			0
$453$	2.65105e9	−	4.59175e9i	1.33990	−	2.32078i
$454$		0			0
$455$	−7.50641e8	−	1.39642e9i	−0.373588	−	0.694985i
$456$		0			0
$457$	−9.01035e8	+	1.56064e9i	−0.441606	+	0.764884i	−0.997809	−	0.0661624i	$-0.978924\pi$
$457$	−9.01035e8	+	1.56064e9i	−0.441606	+	0.764884i	0.556203	+	0.831047i	$0.312258\pi$
$458$		0			0
$459$	−1.64566e9	−	2.85036e9i	−0.794319	−	1.37580i
$460$		0			0
$461$	3.90785e8	+	6.76859e8i	0.185774	+	0.321770i	0.943837	−	0.330412i	$-0.107188\pi$
$461$	3.90785e8	+	6.76859e8i	0.185774	+	0.321770i	−0.758063	+	0.652181i	$0.773854\pi$
$462$		0			0
$463$	7.80510e8			0.365464			0.182732	−	0.983163i	$-0.441506\pi$
$463$	7.80510e8			0.365464			0.182732	+	0.983163i	$0.441506\pi$
$464$		0			0
$465$	−4.38063e8	+	7.58747e8i	−0.202046	+	0.349955i
$466$		0			0
$467$	2.48897e9			1.13087			0.565433	−	0.824794i	$-0.308709\pi$
$467$	2.48897e9			1.13087			0.565433	+	0.824794i	$0.308709\pi$
$468$		0			0
$469$	−5.57942e8			−0.249738
$470$		0			0
$471$	3.06972e9	−	5.31691e9i	1.35371	−	2.34469i
$472$		0			0
$473$	−3.09989e8			−0.134689
$474$		0			0
$475$	−1.61858e9	−	2.80346e9i	−0.692956	−	1.20023i
$476$		0			0
$477$	1.82233e9	+	3.15636e9i	0.768797	+	1.33160i
$478$		0			0
$479$	−1.73346e9	+	3.00243e9i	−0.720673	+	1.24824i	0.240057	+	0.970759i	$0.422834\pi$
$479$	−1.73346e9	+	3.00243e9i	−0.720673	+	1.24824i	−0.960730	+	0.277484i	$0.910499\pi$
$480$		0			0
$481$	4.68633e8	−	7.57603e8i	0.192011	−	0.310409i
$482$		0			0
$483$	−1.70138e9	+	2.94688e9i	−0.687048	+	1.19000i
$484$		0			0
$485$	−4.13609e6	−	7.16392e6i	−0.00164624	−	0.00285138i
$486$		0			0
$487$	−5.00506e8	−	8.66901e8i	−0.196362	−	0.340109i	0.750984	−	0.660320i	$-0.229579\pi$
$487$	−5.00506e8	−	8.66901e8i	−0.196362	−	0.340109i	−0.947346	+	0.320211i	$0.896246\pi$
$488$		0			0
$489$	−2.44136e9			−0.944172
$490$		0			0
$491$	−7.39363e7	+	1.28061e8i	−0.0281885	+	0.0488240i	−0.879776	−	0.475389i	$-0.842307\pi$
$491$	−7.39363e7	+	1.28061e8i	−0.0281885	+	0.0488240i	0.851587	+	0.524213i	$0.175641\pi$
$492$		0			0
$493$	−3.31841e9			−1.24729
$494$		0			0
$495$	1.59013e9			0.589271
$496$		0			0
$497$	2.40460e9	−	4.16488e9i	0.878607	−	1.52179i
$498$		0			0
$499$	3.44602e9			1.24155			0.620777	−	0.783988i	$-0.286817\pi$
$499$	3.44602e9			1.24155			0.620777	+	0.783988i	$0.286817\pi$
$500$		0			0
$501$	−1.61417e9	−	2.79582e9i	−0.573478	−	0.993293i
$502$		0			0
$503$	−2.83908e9	−	4.91744e9i	−0.994696	−	1.72286i	−0.586428	−	0.810001i	$-0.699467\pi$
$503$	−2.83908e9	−	4.91744e9i	−0.994696	−	1.72286i	−0.408267	−	0.912862i	$-0.633867\pi$
$504$		0			0
$505$	9.81908e8	−	1.70071e9i	0.339274	−	0.587640i
$506$		0			0
$507$	2.74395e9	−	4.14885e9i	0.935078	−	1.41384i
$508$		0			0
$509$	−1.82542e9	+	3.16171e9i	−0.613549	+	1.06270i	0.377088	+	0.926177i	$0.376925\pi$
$509$	−1.82542e9	+	3.16171e9i	−0.613549	+	1.06270i	−0.990637	+	0.136521i	$0.956408\pi$
$510$		0			0
$511$	4.40785e9	+	7.63461e9i	1.46135	+	2.53113i
$512$		0			0
$513$	−4.04835e9	−	7.01194e9i	−1.32394	−	2.29312i
$514$		0			0
$515$	1.64550e9			0.530850
$516$		0			0
$517$	6.04413e8	−	1.04687e9i	0.192361	−	0.333179i
$518$		0			0
$519$	−2.33071e9			−0.731819
$520$		0			0
$521$	3.14626e9			0.974680			0.487340	−	0.873212i	$-0.337967\pi$
$521$	3.14626e9			0.974680			0.487340	+	0.873212i	$0.337967\pi$
$522$		0			0
$523$	−3.21632e9	+	5.57083e9i	−0.983113	+	1.70280i	−0.333074	+	0.942901i	$0.608086\pi$
$523$	−3.21632e9	+	5.57083e9i	−0.983113	+	1.70280i	−0.650039	+	0.759901i	$0.725248\pi$
$524$		0			0
$525$	−7.24048e9			−2.18379
$526$		0			0
$527$	−9.05716e8	−	1.56875e9i	−0.269560	−	0.466891i
$528$		0			0
$529$	1.29777e9	+	2.24780e9i	0.381156	+	0.660182i
$530$		0			0
$531$	4.93822e9	−	8.55326e9i	1.43133	−	2.47914i
$532$		0			0
$533$	4.58930e8	−	7.41917e8i	0.131281	−	0.212231i
$534$		0			0
$535$	3.77555e7	−	6.53944e7i	0.0106596	−	0.0184630i
$536$		0			0
$537$	−1.78504e9	−	3.09178e9i	−0.497436	−	0.861585i
$538$		0			0
$539$	−2.12633e9	−	3.68291e9i	−0.584884	−	1.01305i
$540$		0			0
$541$	1.42344e9			0.386500			0.193250	−	0.981150i	$-0.438097\pi$
$541$	1.42344e9			0.386500			0.193250	+	0.981150i	$0.438097\pi$
$542$		0			0
$543$	−3.07941e9	+	5.33370e9i	−0.825408	+	1.42965i
$544$		0			0
$545$	−5.13539e8			−0.135890
$546$		0			0
$547$	1.52035e9			0.397181			0.198591	−	0.980083i	$-0.436364\pi$
$547$	1.52035e9			0.397181			0.198591	+	0.980083i	$0.436364\pi$
$548$		0			0
$549$	−4.88474e9	+	8.46062e9i	−1.25991	+	2.18222i
$550$		0			0
$551$	−8.16335e9			−2.07892
$552$		0			0
$553$	−8.12505e8	−	1.40730e9i	−0.204309	−	0.353874i
$554$		0			0
$555$	5.91216e8	+	1.02402e9i	0.146798	+	0.254262i
$556$		0			0
$557$	4.15106e8	−	7.18985e8i	0.101781	−	0.176290i	−0.810638	−	0.585548i	$-0.800879\pi$
$557$	4.15106e8	−	7.18985e8i	0.101781	−	0.176290i	0.912418	+	0.409259i	$0.134213\pi$
$558$		0			0
$559$	−3.97318e8	−	7.39130e8i	−0.0962048	−	0.178970i
$560$		0			0
$561$	−2.52134e9	+	4.36709e9i	−0.602923	+	1.04429i
$562$		0			0
$563$	−1.61860e9	−	2.80351e9i	−0.382262	−	0.662098i	0.609123	−	0.793076i	$-0.291522\pi$
$563$	−1.61860e9	−	2.80351e9i	−0.382262	−	0.662098i	−0.991385	+	0.130978i	$0.958188\pi$
$564$		0			0
$565$	1.84407e9	+	3.19402e9i	0.430138	+	0.745021i
$566$		0			0
$567$	−4.58989e9			−1.05745
$568$		0			0
$569$	2.35987e9	−	4.08742e9i	0.537026	−	0.930157i	−0.462036	−	0.886861i	$-0.652881\pi$
$569$	2.35987e9	−	4.08742e9i	0.537026	−	0.930157i	0.999062	−	0.0432957i	$-0.0137857\pi$
$570$		0			0
$571$	−3.95218e9			−0.888403			−0.444201	−	0.895927i	$-0.646513\pi$
$571$	−3.95218e9			−0.888403			−0.444201	+	0.895927i	$0.646513\pi$
$572$		0			0
$573$	1.31219e9			0.291377
$574$		0			0
$575$	8.61009e8	−	1.49131e9i	0.188873	−	0.327138i
$576$		0			0
$577$	−2.12731e9			−0.461016			−0.230508	−	0.973070i	$-0.574039\pi$
$577$	−2.12731e9			−0.461016			−0.230508	+	0.973070i	$0.574039\pi$
$578$		0			0
$579$	−2.30051e9	−	3.98459e9i	−0.492548	−	0.853118i
$580$		0			0
$581$	−4.88847e8	−	8.46708e8i	−0.103409	−	0.179109i
$582$		0			0
$583$	1.30160e9	−	2.25443e9i	0.272042	−	0.471191i
$584$		0			0
$585$	2.03810e9	+	3.79147e9i	0.420900	+	0.783000i
$586$		0			0
$587$	−2.65059e9	+	4.59096e9i	−0.540890	+	0.936850i	0.457963	+	0.888971i	$0.348579\pi$
$587$	−2.65059e9	+	4.59096e9i	−0.540890	+	0.936850i	−0.998853	+	0.0478782i	$0.984754\pi$
$588$		0			0
$589$	−2.22808e9	−	3.85914e9i	−0.449290	−	0.778193i
$590$		0			0
$591$	1.53750e9	+	2.66304e9i	0.306380	+	0.530666i
$592$		0			0
$593$	7.20568e9			1.41900			0.709501	−	0.704704i	$-0.248920\pi$
$593$	7.20568e9			1.41900			0.709501	+	0.704704i	$0.248920\pi$
$594$		0			0
$595$	−2.17541e9	+	3.76792e9i	−0.423381	+	0.733317i
$596$		0			0
$597$	2.24870e9			0.432535
$598$		0			0
$599$	1.75875e9			0.334357			0.167179	−	0.985927i	$-0.446534\pi$
$599$	1.75875e9			0.334357			0.167179	+	0.985927i	$0.446534\pi$
$600$		0			0
$601$	3.50633e9	−	6.07314e9i	0.658858	−	1.14118i	−0.322054	−	0.946721i	$-0.604373\pi$
$601$	3.50633e9	−	6.07314e9i	0.658858	−	1.14118i	0.980912	−	0.194454i	$-0.0622936\pi$
$602$		0			0
$603$	1.51489e9			0.281366
$604$		0			0
$605$	7.24491e8	+	1.25485e9i	0.133011	+	0.230383i
$606$		0			0
$607$	7.96698e8	+	1.37992e9i	0.144588	+	0.250435i	0.929219	−	0.369529i	$-0.120481\pi$
$607$	7.96698e8	+	1.37992e9i	0.144588	+	0.250435i	−0.784631	+	0.619963i	$0.787148\pi$
$608$		0			0
$609$	−9.12941e9	+	1.58126e10i	−1.63788	+	2.83689i
$610$		0			0
$611$	3.27082e9	+	9.93520e7i	0.580113	+	0.0176211i
$612$		0			0
$613$	5.40201e8	−	9.35656e8i	0.0947205	−	0.164061i	−0.814771	−	0.579782i	$-0.803138\pi$
$613$	5.40201e8	−	9.35656e8i	0.0947205	−	0.164061i	0.909492	+	0.415722i	$0.136471\pi$
$614$		0			0
$615$	5.78974e8	+	1.00281e9i	0.100368	+	0.173843i
$616$		0			0
$617$	−3.25522e9	−	5.63820e9i	−0.557933	−	0.966368i	−0.997669	−	0.0682409i	$-0.978261\pi$
$617$	−3.25522e9	−	5.63820e9i	−0.557933	−	0.966368i	0.439736	−	0.898127i	$-0.355072\pi$
$618$		0			0
$619$	2.46380e9			0.417531			0.208766	−	0.977966i	$-0.433055\pi$
$619$	2.46380e9			0.417531			0.208766	+	0.977966i	$0.433055\pi$
$620$		0			0
$621$	2.15354e9	−	3.73003e9i	0.360854	−	0.625018i
$622$		0			0
$623$	1.75424e10			2.90657
$624$		0			0
$625$	2.28972e9			0.375147
$626$		0			0
$627$	−6.20255e9	+	1.07431e10i	−1.00493	+	1.74058i
$628$		0			0
$629$	−2.44473e9			−0.391701
$630$		0			0
$631$	−1.56214e9	−	2.70571e9i	−0.247524	−	0.428725i	0.715314	−	0.698803i	$-0.246284\pi$
$631$	−1.56214e9	−	2.70571e9i	−0.247524	−	0.428725i	−0.962838	+	0.270078i	$0.912950\pi$
$632$		0			0
$633$	2.58794e9	+	4.48245e9i	0.405547	+	0.702429i
$634$		0			0
$635$	1.06357e9	−	1.84216e9i	0.164838	−	0.285509i
$636$		0			0
$637$	6.05608e9	−	9.79040e9i	0.928333	−	1.50076i
$638$		0			0
$639$	−6.52882e9	+	1.13082e10i	−0.989878	+	1.71452i
$640$		0			0
$641$	4.15064e9	+	7.18911e9i	0.622460	+	1.07813i	0.989026	+	0.147741i	$0.0472001\pi$
$641$	4.15064e9	+	7.18911e9i	0.622460	+	1.07813i	−0.366566	+	0.930392i	$0.619467\pi$
$642$		0			0
$643$	3.87700e9	+	6.71515e9i	0.575118	+	0.996133i	0.996029	+	0.0890314i	$0.0283771\pi$
$643$	3.87700e9	+	6.71515e9i	0.575118	+	0.996133i	−0.420911	+	0.907102i	$0.638290\pi$
$644$		0			0
$645$	1.11384e9			0.163441
$646$		0			0
$647$	4.87730e9	−	8.44773e9i	0.707969	−	1.22624i	−0.257640	−	0.966241i	$-0.582945\pi$
$647$	4.87730e9	−	8.44773e9i	0.707969	−	1.22624i	0.965609	−	0.259998i	$-0.0837218\pi$
$648$		0			0
$649$	−7.05426e9			−1.01297
$650$		0			0
$651$	−9.96701e9			−1.41590
$652$		0			0
$653$	1.40039e9	−	2.42555e9i	0.196813	−	0.340890i	−0.750680	−	0.660666i	$-0.770274\pi$
$653$	1.40039e9	−	2.42555e9i	0.196813	−	0.340890i	0.947493	+	0.319775i	$0.103607\pi$
$654$		0			0
$655$	−2.79111e9			−0.388090
$656$		0			0
$657$	−1.19679e10	−	2.07290e10i	−1.64642	−	2.85168i
$658$		0			0
$659$	−4.64679e9	−	8.04848e9i	−0.632491	−	1.09551i	−0.987041	−	0.160469i	$-0.948699\pi$
$659$	−4.64679e9	−	8.04848e9i	−0.632491	−	1.09551i	0.354550	−	0.935037i	$-0.384634\pi$
$660$		0			0
$661$	−2.38956e9	+	4.13885e9i	−0.321820	+	0.557409i	−0.980864	−	0.194696i	$-0.937628\pi$
$661$	−2.38956e9	+	4.13885e9i	−0.321820	+	0.557409i	0.659043	+	0.752105i	$0.270961\pi$
$662$		0			0
$663$	−1.36444e10	−	4.14452e8i	−1.81827	−	0.0552303i
$664$		0			0
$665$	−5.35154e9	+	9.26914e9i	−0.705672	+	1.22226i
$666$		0			0
$667$	−2.17126e9	−	3.76074e9i	−0.283317	−	0.490719i
$668$		0			0
$669$	4.15699e9	+	7.20012e9i	0.536770	+	0.929712i
$670$		0			0
$671$	6.97786e9			0.891648
$672$		0			0
$673$	−1.48349e9	+	2.56948e9i	−0.187599	+	0.324932i	−0.944449	−	0.328657i	$-0.893404\pi$
$673$	−1.48349e9	+	2.56948e9i	−0.187599	+	0.324932i	0.756850	+	0.653589i	$0.226737\pi$
$674$		0			0
$675$	9.16469e9			1.14698
$676$		0			0
$677$	−7.59327e8			−0.0940521			−0.0470260	−	0.998894i	$-0.514974\pi$
$677$	−7.59327e8			−0.0940521			−0.0470260	+	0.998894i	$0.514974\pi$
$678$		0			0
$679$	4.70531e7	−	8.14984e7i	0.00576825	−	0.00999091i
$680$		0			0
$681$	3.95143e9			0.479446
$682$		0			0
$683$	−3.30422e9	−	5.72307e9i	−0.396822	−	0.687316i	0.596510	−	0.802606i	$-0.296554\pi$
$683$	−3.30422e9	−	5.72307e9i	−0.396822	−	0.687316i	−0.993332	+	0.115290i	$0.963220\pi$
$684$		0			0
$685$	−2.90965e9	−	5.03966e9i	−0.345879	−	0.599079i
$686$		0			0
$687$	4.58314e9	−	7.93823e9i	0.539280	−	0.934060i
$688$		0			0
$689$	7.04368e9	+	2.13953e8i	0.820412	+	0.0249202i
$690$		0			0
$691$	4.34206e9	−	7.52066e9i	0.500636	−	0.867128i	−0.499363	−	0.866393i	$-0.666433\pi$
$691$	4.34206e9	−	7.52066e9i	0.500636	−	0.867128i	1.00000		0.000734904i	$-0.000233927\pi$
$692$		0			0
$693$	9.04486e9	+	1.56662e10i	1.03237	+	1.78812i
$694$		0			0
$695$	9.21925e8	+	1.59682e9i	0.104171	+	0.180430i
$696$		0			0
$697$	−2.39411e9			−0.267812
$698$		0			0
$699$	2.79416e9	−	4.83962e9i	0.309443	−	0.535971i
$700$		0			0
$701$	7.50161e9			0.822510			0.411255	−	0.911520i	$-0.365091\pi$
$701$	7.50161e9			0.822510			0.411255	+	0.911520i	$0.365091\pi$
$702$		0			0
$703$	−6.01409e9			−0.652869
$704$		0			0
$705$	−2.17174e9	+	3.76157e9i	−0.233424	+	0.404303i
$706$		0			0
$707$	2.23408e10			2.37756
$708$		0			0
$709$	−3.60933e9	−	6.25154e9i	−0.380334	−	0.658757i	0.610776	−	0.791803i	$-0.290858\pi$
$709$	−3.60933e9	−	6.25154e9i	−0.380334	−	0.658757i	−0.991110	+	0.133046i	$0.957524\pi$
$710$		0			0
$711$	2.20606e9	+	3.82102e9i	0.230184	+	0.398690i
$712$		0			0
$713$	1.18524e9	−	2.05289e9i	0.122459	−	0.212106i
$714$		0			0
$715$	1.61739e9	−	2.61471e9i	0.165479	−	0.267518i
$716$		0			0
$717$	1.22357e10	−	2.11929e10i	1.23969	−	2.14720i
$718$		0			0
$719$	7.39310e9	+	1.28052e10i	0.741781	+	1.28480i	0.951684	+	0.307080i	$0.0993518\pi$
$719$	7.39310e9	+	1.28052e10i	0.741781	+	1.28480i	−0.209903	+	0.977722i	$0.567315\pi$
$720$		0			0
$721$	9.35978e9	+	1.62116e10i	0.930019	+	1.61084i
$722$		0			0
$723$	−9.81092e9			−0.965441
$724$		0			0
$725$	4.62007e9	−	8.00220e9i	0.450262	−	0.779877i
$726$		0			0
$727$	5.23529e9			0.505325			0.252662	−	0.967554i	$-0.418694\pi$
$727$	5.23529e9			0.505325			0.252662	+	0.967554i	$0.418694\pi$
$728$		0			0
$729$	−1.37858e10			−1.31791
$730$		0			0
$731$	−1.15145e9	+	1.99438e9i	−0.109027	+	0.188841i
$732$		0			0
$733$	2.06458e10			1.93628			0.968138	−	0.250417i	$-0.0805676\pi$
$733$	2.06458e10			1.93628			0.968138	+	0.250417i	$0.0805676\pi$
$734$		0			0
$735$	7.64020e9	+	1.32332e10i	0.709739	+	1.22930i
$736$		0			0
$737$	−5.41006e8	−	9.37050e8i	−0.0497813	−	0.0862237i
$738$		0			0
$739$	−4.34202e9	+	7.52060e9i	−0.395764	+	0.685483i	−0.993198	−	0.116435i	$-0.962853\pi$
$739$	−4.34202e9	+	7.52060e9i	−0.395764	+	0.685483i	0.597434	+	0.801918i	$0.296187\pi$
$740$		0			0
$741$	−3.35655e10	−	1.01956e9i	−3.03061	−	0.0920554i
$742$		0			0
$743$	5.13915e9	−	8.90127e9i	0.459653	−	0.796143i	−0.539289	−	0.842121i	$-0.681307\pi$
$743$	5.13915e9	−	8.90127e9i	0.459653	−	0.796143i	0.998942	+	0.0459776i	$0.0146403\pi$
$744$		0			0
$745$	−2.70417e9	−	4.68377e9i	−0.239600	−	0.415000i
$746$		0			0
$747$	1.32729e9	+	2.29893e9i	0.116505	+	0.201792i
$748$		0			0
$749$	8.59030e8			0.0747002
$750$		0			0
$751$	4.41272e9	−	7.64306e9i	0.380161	−	0.658457i	−0.610924	−	0.791689i	$-0.709202\pi$
$751$	4.41272e9	−	7.64306e9i	0.380161	−	0.658457i	0.991085	+	0.133232i	$0.0425354\pi$
$752$		0			0
$753$	2.66729e10			2.27660
$754$		0			0
$755$	8.87154e9			0.750213
$756$		0			0
$757$	−8.12765e9	+	1.40775e10i	−0.680972	+	1.17948i	0.293712	+	0.955894i	$0.405109\pi$
$757$	−8.12765e9	+	1.40775e10i	−0.680972	+	1.17948i	−0.974684	+	0.223585i	$0.928224\pi$
$758$		0			0
$759$	−6.59895e9			−0.547808
$760$		0			0
$761$	1.45722e8	+	2.52398e8i	0.0119862	+	0.0207606i	0.871956	−	0.489584i	$-0.162851\pi$
$761$	1.45722e8	+	2.52398e8i	0.0119862	+	0.0207606i	−0.859970	+	0.510344i	$0.829518\pi$
$762$		0			0
$763$	−2.92107e9	−	5.05944e9i	−0.238071	−	0.412351i
$764$		0			0
$765$	5.90654e9	−	1.02304e10i	0.477000	−	0.826188i
$766$		0			0
$767$	−9.04154e9	−	1.68200e10i	−0.723534	−	1.34599i
$768$		0			0
$769$	−2.22330e9	+	3.85087e9i	−0.176302	+	0.305363i	−0.940611	−	0.339486i	$-0.889747\pi$
$769$	−2.22330e9	+	3.85087e9i	−0.176302	+	0.305363i	0.764309	+	0.644850i	$0.223080\pi$
$770$		0			0
$771$	1.07214e10	+	1.85700e10i	0.842484	+	1.45922i
$772$		0			0
$773$	−2.56726e9	−	4.44663e9i	−0.199913	−	0.346260i	0.748587	−	0.663037i	$-0.230733\pi$
$773$	−2.56726e9	−	4.44663e9i	−0.199913	−	0.346260i	−0.948500	+	0.316777i	$0.897399\pi$
$774$		0			0
$775$	5.04395e9			0.389237
$776$		0			0
$777$	−6.72581e9	+	1.16494e10i	−0.514365	+	0.890905i
$778$		0			0
$779$	−5.88956e9			−0.446377
$780$		0			0
$781$	9.32642e9			0.700546
$782$		0			0
$783$	1.15556e10	−	2.00149e10i	0.860255	−	1.49000i
$784$		0			0
$785$	1.02726e10			0.757943
$786$		0			0
$787$	1.07057e10	+	1.85428e10i	0.782895	+	1.35601i	0.930248	+	0.366930i	$0.119591\pi$
$787$	1.07057e10	+	1.85428e10i	0.782895	+	1.35601i	−0.147353	+	0.989084i	$0.547075\pi$
$788$		0			0
$789$	1.59726e10	+	2.76654e10i	1.15773	+	2.00524i
$790$		0			0
$791$	−2.09786e10	+	3.63360e10i	−1.50716	+	2.61047i
$792$		0			0
$793$	8.94362e9	+	1.66378e10i	0.636879	+	1.18479i
$794$		0			0
$795$	−4.67682e9	+	8.10049e9i	−0.330115	+	0.571776i
$796$		0			0
$797$	−2.24946e9	−	3.89618e9i	−0.157389	−	0.272605i	0.776537	−	0.630071i	$-0.216974\pi$
$797$	−2.24946e9	−	3.89618e9i	−0.157389	−	0.272605i	−0.933926	+	0.357466i	$0.883641\pi$
$798$		0			0
$799$	−4.49018e9	−	7.77722e9i	−0.311422	−	0.539400i
$800$		0			0
$801$	−4.76301e10			−3.27467
$802$		0			0
$803$	−8.54809e9	+	1.48057e10i	−0.582592	+	1.00908i
$804$		0			0
$805$	−5.69356e9			−0.384679
$806$		0			0
$807$	1.15361e10			0.772685
$808$		0			0
$809$	1.23533e10	−	2.13966e10i	0.820282	−	1.42077i	−0.0851894	−	0.996365i	$-0.527150\pi$
$809$	1.23533e10	−	2.13966e10i	0.820282	−	1.42077i	0.905472	−	0.424406i	$-0.139517\pi$
$810$		0			0
$811$	−1.66270e9			−0.109456			−0.0547282	−	0.998501i	$-0.517429\pi$
$811$	−1.66270e9			−0.109456			−0.0547282	+	0.998501i	$0.517429\pi$
$812$		0			0
$813$	1.41134e10	+	2.44450e10i	0.921114	+	1.59542i
$814$		0			0
$815$	−2.04246e9	−	3.53765e9i	−0.132161	−	0.228909i
$816$		0			0
$817$	−2.83260e9	+	4.90620e9i	−0.181722	+	0.314752i
$818$		0			0
$819$	−2.57610e10	+	4.16458e10i	−1.63859	+	2.64898i
$820$		0			0
$821$	7.44317e9	−	1.28920e10i	0.469415	−	0.813051i	−0.529974	−	0.848014i	$-0.677798\pi$
$821$	7.44317e9	−	1.28920e10i	0.469415	−	0.813051i	0.999389	+	0.0349635i	$0.0111315\pi$
$822$		0			0
$823$	−4.69118e9	−	8.12536e9i	−0.293347	−	0.508093i	0.681252	−	0.732049i	$-0.261436\pi$
$823$	−4.69118e9	−	8.12536e9i	−0.293347	−	0.508093i	−0.974599	+	0.223957i	$0.928103\pi$
$824$		0			0
$825$	−7.02070e9	−	1.21602e10i	−0.435303	−	0.753967i
$826$		0			0
$827$	5.95599e9			0.366172			0.183086	−	0.983097i	$-0.441391\pi$
$827$	5.95599e9			0.366172			0.183086	+	0.983097i	$0.441391\pi$
$828$		0			0
$829$	−1.02687e10	+	1.77859e10i	−0.626002	+	1.08427i	0.362345	+	0.932044i	$0.381976\pi$
$829$	−1.02687e10	+	1.77859e10i	−0.626002	+	1.08427i	−0.988346	+	0.152223i	$0.951357\pi$
$830$		0			0
$831$	−3.85539e10			−2.33058
$832$		0			0
$833$	−3.15929e10			−1.89379
$834$		0			0
$835$	2.70085e9	−	4.67801e9i	0.160545	−	0.278073i
$836$		0			0
$837$	1.26158e10			0.743663
$838$		0			0
$839$	−6.87493e9	−	1.19077e10i	−0.401884	−	0.696084i	0.592069	−	0.805887i	$-0.298311\pi$
$839$	−6.87493e9	−	1.19077e10i	−0.401884	−	0.696084i	−0.993953	+	0.109803i	$0.964978\pi$
$840$		0			0
$841$	−3.02581e9	−	5.24086e9i	−0.175411	−	0.303820i
$842$		0			0
$843$	−1.93954e9	+	3.35939e9i	−0.111507	+	0.193136i
$844$		0			0
$845$	8.30748e9	+	5.05149e8i	0.473664	+	0.0288019i
$846$		0			0
$847$	−8.24197e9	+	1.42755e10i	−0.466057	+	0.807235i
$848$		0			0
$849$	1.09159e10	+	1.89069e10i	0.612184	+	1.06033i
$850$		0			0
$851$	−1.59961e9	−	2.77061e9i	−0.0889736	−	0.154107i
$852$		0			0
$853$	9.49367e9			0.523736			0.261868	−	0.965104i	$-0.415661\pi$
$853$	9.49367e9			0.523736			0.261868	+	0.965104i	$0.415661\pi$
$854$		0			0
$855$	1.45302e10	−	2.51670e10i	0.795042	−	1.37705i
$856$		0			0
$857$	−1.03739e9			−0.0562998			−0.0281499	−	0.999604i	$-0.508962\pi$
$857$	−1.03739e9			−0.0562998			−0.0281499	+	0.999604i	$0.508962\pi$
$858$		0			0
$859$	3.21586e10			1.73110			0.865549	−	0.500824i	$-0.166970\pi$
$859$	3.21586e10			1.73110			0.865549	+	0.500824i	$0.166970\pi$
$860$		0			0
$861$	−6.58655e9	+	1.14082e10i	−0.351679	+	0.609126i
$862$		0			0
$863$	1.58573e10			0.839828			0.419914	−	0.907564i	$-0.362060\pi$
$863$	1.58573e10			0.839828			0.419914	+	0.907564i	$0.362060\pi$
$864$		0			0
$865$	−1.94989e9	−	3.37731e9i	−0.102436	−	0.177425i
$866$		0			0
$867$	2.46701e9	+	4.27298e9i	0.128559	+	0.222671i
$868$		0			0
$869$	1.57568e9	−	2.72916e9i	0.0814516	−	0.141078i
$870$		0			0
$871$	1.54086e9	−	2.49099e9i	0.0790132	−	0.127735i
$872$		0			0
$873$	−1.27756e8	+	2.21280e8i	−0.00649877	+	0.0112562i
$874$		0			0
$875$	−1.38754e10	−	2.40329e10i	−0.700192	−	1.21277i
$876$		0			0
$877$	1.44053e10	+	2.49507e10i	0.721145	+	1.24906i	0.960541	+	0.278138i	$0.0897174\pi$
$877$	1.44053e10	+	2.49507e10i	0.721145	+	1.24906i	−0.239395	+	0.970922i	$0.576949\pi$
$878$		0			0
$879$	2.43266e10			1.20815
$880$		0			0
$881$	1.65245e10	−	2.86213e10i	0.814167	−	1.41018i	−0.0957580	−	0.995405i	$-0.530527\pi$
$881$	1.65245e10	−	2.86213e10i	0.814167	−	1.41018i	0.909925	−	0.414773i	$-0.136139\pi$
$882$		0			0
$883$	1.91851e9			0.0937783			0.0468891	−	0.998900i	$-0.485069\pi$
$883$	1.91851e9			0.0937783			0.0468891	+	0.998900i	$0.485069\pi$
$884$		0			0
$885$	2.53469e10			1.22920
$886$		0			0
$887$	1.12577e10	−	1.94989e10i	0.541646	−	0.938159i	−0.457163	−	0.889383i	$-0.651135\pi$
$887$	1.12577e10	−	1.94989e10i	0.541646	−	0.938159i	0.998810	−	0.0487763i	$-0.0155321\pi$
$888$		0			0
$889$	2.41989e10			1.15515
$890$		0			0
$891$	−4.45056e9	−	7.70860e9i	−0.210786	−	0.365093i
$892$		0			0
$893$	−1.10459e10	−	1.91321e10i	−0.519065	−	0.899047i
$894$		0			0
$895$	2.98675e9	−	5.17321e9i	0.139257	−	0.241201i
$896$		0			0
$897$	−8.45797e9	−	1.57343e10i	−0.391284	−	0.727906i
$898$		0			0
$899$	6.35983e9	−	1.10156e10i	0.291935	−	0.505647i
$900$		0			0
$901$	−9.66955e9	−	1.67481e10i	−0.440422	−	0.762834i
$902$		0			0
$903$	6.33562e9	+	1.09736e10i	0.286340	+	0.495956i
$904$		0			0
$905$	−1.03050e10			−0.462146
$906$		0			0
$907$	1.77982e10	−	3.08274e10i	0.792047	−	1.37187i	−0.132650	−	0.991163i	$-0.542349\pi$
$907$	1.77982e10	−	3.08274e10i	0.792047	−	1.37187i	0.924697	−	0.380703i	$-0.124318\pi$
$908$		0			0
$909$	−6.06585e10			−2.67866
$910$		0			0
$911$	2.01254e10			0.881923			0.440962	−	0.897526i	$-0.354637\pi$
$911$	2.01254e10			0.881923			0.440962	+	0.897526i	$0.354637\pi$
$912$		0			0
$913$	9.48016e8	−	1.64201e9i	0.0412257	−	0.0714050i
$914$		0			0
$915$	−2.50724e10			−1.08199
$916$		0			0
$917$	−1.58761e10	−	2.74983e10i	−0.679911	−	1.17764i
$918$		0			0
$919$	5.03019e9	+	8.71254e9i	0.213786	+	0.370289i	0.952896	−	0.303296i	$-0.0980870\pi$
$919$	5.03019e9	+	8.71254e9i	0.213786	+	0.370289i	−0.739110	+	0.673585i	$0.764754\pi$
$920$		0			0
$921$	6.14973e9	−	1.06517e10i	0.259387	−	0.449271i
$922$		0			0
$923$	1.19538e10	+	2.22377e10i	0.500381	+	0.930857i
$924$		0			0
$925$	3.40369e9	−	5.89537e9i	0.141402	−	0.244915i
$926$		0			0
$927$	−2.54131e10	−	4.40168e10i	−1.04780	−	1.81484i
$928$		0			0
$929$	−1.24643e10	−	2.15889e10i	−0.510052	−	0.883436i	−0.999932	−	0.0116460i	$-0.996293\pi$
$929$	−1.24643e10	−	2.15889e10i	−0.510052	−	0.883436i	0.489880	−	0.871790i	$-0.337040\pi$
$930$		0			0
$931$	−7.77192e10			−3.15649
$932$		0			0
$933$	2.22421e10	−	3.85245e10i	0.896582	−	1.55293i
$934$		0			0
$935$	−8.43750e9			−0.337577
$936$		0			0
$937$	3.64214e9			0.144633			0.0723166	−	0.997382i	$-0.476961\pi$
$937$	3.64214e9			0.144633			0.0723166	+	0.997382i	$0.476961\pi$
$938$		0			0
$939$	−1.53881e10	+	2.66530e10i	−0.606535	+	1.05055i
$940$		0			0
$941$	−3.92765e10			−1.53663			−0.768314	−	0.640073i	$-0.778904\pi$
$941$	−3.92765e10			−1.53663			−0.768314	+	0.640073i	$0.778904\pi$
$942$		0			0
$943$	−1.56649e9	−	2.71324e9i	−0.0608327	−	0.105365i
$944$		0			0
$945$	−1.51507e10	−	2.62419e10i	−0.584013	−	1.01154i
$946$		0			0
$947$	4.01982e9	−	6.96253e9i	0.153809	−	0.266405i	−0.778816	−	0.627253i	$-0.784179\pi$
$947$	4.01982e9	−	6.96253e9i	0.153809	−	0.266405i	0.932625	+	0.360848i	$0.117513\pi$
$948$		0			0
$949$	−4.62586e10	−	1.40512e9i	−1.75695	−	0.0533679i
$950$		0			0
$951$	2.60318e9	−	4.50884e9i	0.0981459	−	0.169994i
$952$		0			0
$953$	7.75981e9	+	1.34404e10i	0.290420	+	0.503022i	0.973909	−	0.226939i	$-0.0728718\pi$
$953$	7.75981e9	+	1.34404e10i	0.290420	+	0.503022i	−0.683489	+	0.729961i	$0.739538\pi$
$954$		0			0
$955$	1.09779e9	+	1.90142e9i	0.0407855	+	0.0706425i
$956$		0			0
$957$	−3.54092e10			−1.30594
$958$		0			0
$959$	3.31008e10	−	5.73323e10i	1.21192	−	2.09911i
$960$		0			0
$961$	−2.05693e10			−0.747631
$962$		0			0
$963$	−2.33239e9			−0.0841605
$964$		0			0
$965$	3.84924e9	−	6.66708e9i	0.137889	−	0.238831i
$966$		0			0
$967$	1.65901e10			0.590006			0.295003	−	0.955496i	$-0.404679\pi$
$967$	1.65901e10			0.590006			0.295003	+	0.955496i	$0.404679\pi$
$968$		0			0
$969$	4.60787e10	+	7.98106e10i	1.62692	+	2.81791i
$970$		0			0
$971$	2.75084e9	+	4.76459e9i	0.0964268	+	0.167016i	0.910203	−	0.414162i	$-0.135925\pi$
$971$	2.75084e9	+	4.76459e9i	0.0964268	+	0.167016i	−0.813776	+	0.581178i	$0.802592\pi$
$972$		0			0
$973$	−1.04880e10	+	1.81658e10i	−0.365005	+	0.632208i
$974$		0			0
$975$	1.99959e10	−	3.23259e10i	0.690916	−	1.11695i
$976$		0			0
$977$	1.36203e10	−	2.35910e10i	0.467256	−	0.809312i	−0.532044	−	0.846717i	$-0.678576\pi$
$977$	1.36203e10	−	2.35910e10i	0.467256	−	0.809312i	0.999300	+	0.0374051i	$0.0119092\pi$
$978$		0			0
$979$	1.70099e10	+	2.94620e10i	0.579378	+	1.00351i
$980$		0			0
$981$	7.93112e9	+	1.37371e10i	0.268221	+	0.464573i
$982$		0			0
$983$	−3.12589e10			−1.04963			−0.524816	−	0.851216i	$-0.675866\pi$
$983$	−3.12589e10			−1.04963			−0.524816	+	0.851216i	$0.675866\pi$
$984$		0			0
$985$	−2.57258e9	+	4.45583e9i	−0.0857712	+	0.148560i
$986$		0			0
$987$	−4.94125e10			−1.63579
$988$		0			0
$989$	−3.01363e9			−0.0990609
$990$		0			0
$991$	−2.70471e10	+	4.68470e10i	−0.882802	+	1.52906i	−0.0345889	+	0.999402i	$0.511012\pi$
$991$	−2.70471e10	+	4.68470e10i	−0.882802	+	1.52906i	−0.848213	+	0.529656i	$0.822321\pi$
$992$		0			0
$993$	5.40767e10			1.75262
$994$		0			0
$995$	1.88128e9	+	3.25847e9i	0.0605441	+	0.104866i
$996$		0			0
$997$	−1.77119e10	−	3.06779e10i	−0.566021	−	0.980377i	−0.996954	−	0.0779930i	$-0.975149\pi$
$997$	−1.77119e10	−	3.06779e10i	−0.566021	−	0.980377i	0.430933	−	0.902384i	$-0.358184\pi$
$998$		0			0
$999$	8.51324e9	−	1.47454e10i	0.270157	−	0.467925i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.i.b.113.4		8
4.3	odd	2		26.8.c.b.9.1	yes	8
12.11	even	2		234.8.h.b.217.2		8
13.3	even	3	inner	208.8.i.b.81.4		8
52.3	odd	6		26.8.c.b.3.1	✓	8
52.7	even	12		338.8.b.h.337.4		8
52.19	even	12		338.8.b.h.337.8		8
52.35	odd	6		338.8.a.i.1.4		4
52.43	odd	6		338.8.a.j.1.4		4
156.107	even	6		234.8.h.b.55.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.c.b.3.1	✓	8	52.3	odd	6
26.8.c.b.9.1	yes	8	4.3	odd	2
208.8.i.b.81.4		8	13.3	even	3	inner
208.8.i.b.113.4		8	1.1	even	1	trivial
234.8.h.b.55.2		8	156.107	even	6
234.8.h.b.217.2		8	12.11	even	2
338.8.a.i.1.4		4	52.35	odd	6
338.8.a.j.1.4		4	52.43	odd	6
338.8.b.h.337.4		8	52.7	even	12
338.8.b.h.337.8		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)

\(f(q)\)	\(=\)	\(q+(39.6356 - 68.6509i) q^{3} +132.638 q^{5} +(754.459 + 1306.76i) q^{7} +(-2048.46 - 3548.04i) q^{9} +(-1463.11 + 2534.19i) q^{11} +(-7917.74 - 240.503i) q^{13} +(5257.18 - 9105.70i) q^{15} +(10869.5 + 18826.5i) q^{17} +(26739.1 + 46313.4i) q^{19} +119614. q^{21} +(-14224.0 + 24636.6i) q^{23} -60532.2 q^{25} -151402. q^{27} +(-76324.1 + 132197. i) q^{29} -83326.6 q^{31} +(115983. + 200888. i) q^{33} +(100070. + 173326. i) q^{35} +(-56229.4 + 97392.2i) q^{37} +(-330335. + 534028. i) q^{39} +(-55065.1 + 95375.6i) q^{41} +(52967.4 + 91742.2i) q^{43} +(-271703. - 470604. i) q^{45} -413100. q^{47} +(-726645. + 1.25859e6i) q^{49} +1.72327e6 q^{51} -889607. q^{53} +(-194064. + 336129. i) q^{55} +4.23928e6 q^{57} +(1.20535e6 + 2.08773e6i) q^{59} +(-1.19229e6 - 2.06512e6i) q^{61} +(3.09096e6 - 5.35370e6i) q^{63} +(-1.05019e6 - 31899.8i) q^{65} +(-184882. + 320224. i) q^{67} +(1.12755e6 + 1.95298e6i) q^{69} +(-1.59359e6 - 2.76018e6i) q^{71} +5.84239e6 q^{73} +(-2.39923e6 + 4.15559e6i) q^{75} -4.41544e6 q^{77} -1.07694e6 q^{79} +(-1.52092e6 + 2.63431e6i) q^{81} -647944. q^{83} +(1.44170e6 + 2.49710e6i) q^{85} +(6.05031e6 + 1.04794e7i) q^{87} +(5.81291e6 - 1.00682e7i) q^{89} +(-5.65933e6 - 1.05280e7i) q^{91} +(-3.30270e6 + 5.72045e6i) q^{93} +(3.54661e6 + 6.14291e6i) q^{95} +(-31183.4 - 54011.2i) q^{97} +1.19885e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 556 q^{5} + 548 q^{7} - 5214 q^{9} + 7392 q^{11} - 25818 q^{13} - 15528 q^{15} + 28316 q^{17} + 99888 q^{19} + 182148 q^{21} + 33388 q^{23} + 173756 q^{25} - 212544 q^{27} + 93140 q^{29} - 622320 q^{31}+ \cdots + 20715312 q^{99}+O(q^{100}) \) 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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