LMFDB - Embedded newform 13.8.b.a.12.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			12.6
Root			$18.4900i$ of defining polynomial
Character	$\chi$	$=$	13.12
Dual form			13.8.b.a.12.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+18.4900i q^{2} -27.4160 q^{3} -213.880 q^{4} +70.5606i q^{5} -506.922i q^{6} -454.852i q^{7} -1587.93i q^{8} -1435.36 q^{9} -1304.67 q^{10} +6204.10i q^{11} +5863.75 q^{12} +(-6539.11 + 4470.85i) q^{13} +8410.21 q^{14} -1934.49i q^{15} +1984.12 q^{16} +27903.8 q^{17} -26539.9i q^{18} +19972.9i q^{19} -15091.5i q^{20} +12470.2i q^{21} -114714. q^{22} -45987.5 q^{23} +43534.7i q^{24} +73146.2 q^{25} +(-82666.1 - 120908. i) q^{26} +99310.7 q^{27} +97283.8i q^{28} +66421.0 q^{29} +35768.8 q^{30} -128237. i q^{31} -166568. i q^{32} -170092. i q^{33} +515941. i q^{34} +32094.6 q^{35} +306996. q^{36} +608049. i q^{37} -369299. q^{38} +(179276. - 122573. i) q^{39} +112045. q^{40} +32198.3i q^{41} -230574. q^{42} -623960. q^{43} -1.32693e6i q^{44} -101280. i q^{45} -850309. i q^{46} +342526. i q^{47} -54396.6 q^{48} +616653. q^{49} +1.35247e6i q^{50} -765010. q^{51} +(1.39859e6 - 956227. i) q^{52} -676681. q^{53} +1.83626e6i q^{54} -437765. q^{55} -722272. q^{56} -547577. i q^{57} +1.22812e6i q^{58} -600543. i q^{59} +413750. i q^{60} -39916.7 q^{61} +2.37110e6 q^{62} +652877. i q^{63} +3.33382e6 q^{64} +(-315466. - 461404. i) q^{65} +3.14500e6 q^{66} -3.50690e6i q^{67} -5.96807e6 q^{68} +1.26079e6 q^{69} +593430. i q^{70} -2.53886e6i q^{71} +2.27925e6i q^{72} +2.35690e6i q^{73} -1.12428e7 q^{74} -2.00538e6 q^{75} -4.27181e6i q^{76} +2.82194e6 q^{77} +(2.26637e6 + 3.31482e6i) q^{78} +1.39003e6 q^{79} +140001. i q^{80} +416432. q^{81} -595347. q^{82} +7.87325e6i q^{83} -2.66714e6i q^{84} +1.96891e6i q^{85} -1.15370e7i q^{86} -1.82100e6 q^{87} +9.85166e6 q^{88} -6.81445e6i q^{89} +1.87267e6 q^{90} +(2.03357e6 + 2.97433e6i) q^{91} +9.83582e6 q^{92} +3.51575e6i q^{93} -6.33331e6 q^{94} -1.40930e6 q^{95} +4.56664e6i q^{96} +1.63544e7i q^{97} +1.14019e7i q^{98} -8.90513e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9} - 406 q^{10} + 1898 q^{12} - 5018 q^{13} + 9558 q^{14} + 7778 q^{16} + 13152 q^{17} - 125080 q^{22} + 27264 q^{23} - 18262 q^{25} - 54210 q^{26} + 194560 q^{27} + 42924 q^{29}+ \cdots - 22075632 q^{95}+O(q^{100}) $ 6 * q - 56 * q^3 - 130 * q^4 - 1150 * q^9 - 406 * q^10 + 1898 * q^12 - 5018 * q^13 + 9558 * q^14 + 7778 * q^16 + 13152 * q^17 - 125080 * q^22 + 27264 * q^23 - 18262 * q^25 - 54210 * q^26 + 194560 * q^27 + 42924 * q^29 - 60114 * q^30 - 546720 * q^35 + 747772 * q^36 - 243492 * q^38 + 511160 * q^39 + 792058 * q^40 - 1138134 * q^42 - 1005576 * q^43 - 2807426 * q^48 + 3246846 * q^49 + 297984 * q^51 + 3252964 * q^52 + 1705524 * q^53 - 5320576 * q^55 - 2282730 * q^56 - 8237572 * q^61 + 9625800 * q^62 + 5070862 * q^64 + 7139028 * q^65 + 9289056 * q^66 - 16801086 * q^68 - 6017400 * q^69 - 28760826 * q^74 + 20426072 * q^75 + 4093560 * q^77 + 14088594 * q^78 + 15147200 * q^79 - 21318442 * q^81 - 9130240 * q^82 - 36474240 * q^87 + 38008000 * q^88 + 4769676 * q^90 + 21281832 * q^91 + 36462348 * q^92 - 34229770 * q^94 - 22075632 * q^95

Character values

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

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

Coefficient data

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

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

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

$2$			18.4900i			1.63430i	0.576424	+	0.817151i	$0.304448\pi$
$2$			18.4900i			1.63430i	−0.576424	+	0.817151i	$0.695552\pi$
$3$	−27.4160			−0.586246			−0.293123	−	0.956075i	$-0.594695\pi$
$3$	−27.4160			−0.586246			−0.293123	+	0.956075i	$0.594695\pi$
$4$	−213.880			−1.67094
$5$			70.5606i			0.252445i	0.992002	+	0.126223i	$0.0402854\pi$
$5$			70.5606i			0.252445i	−0.992002	+	0.126223i	$0.959715\pi$
$6$		−	506.922i		−	0.958103i
$7$		−	454.852i		−	0.501218i	−0.968088	−	0.250609i	$-0.919369\pi$
$7$		−	454.852i		−	0.501218i	0.968088	−	0.250609i	$-0.0806308\pi$
$8$		−	1587.93i		−	1.09652i
$9$	−1435.36			−0.656316
$10$	−1304.67			−0.412572
$11$			6204.10i			1.40541i	0.711479	+	0.702707i	$0.248026\pi$
$11$			6204.10i			1.40541i	−0.711479	+	0.702707i	$0.751974\pi$
$12$	5863.75			0.979582
$13$	−6539.11	+	4470.85i	−0.825500	+	0.564402i
$13$	−6539.11	+	4470.85i	−0.825500	+	0.564402i
$14$	8410.21			0.819141
$15$		−	1934.49i		−	0.147995i
$16$	1984.12			0.121101
$17$	27903.8			1.37750			0.688750	−	0.724999i	$-0.258160\pi$
$17$	27903.8			1.37750			0.688750	+	0.724999i	$0.258160\pi$
$18$		−	26539.9i		−	1.07262i
$19$			19972.9i			0.668042i	0.942566	+	0.334021i	$0.108406\pi$
$19$			19972.9i			0.668042i	−0.942566	+	0.334021i	$0.891594\pi$
$20$		−	15091.5i		−	0.421821i
$21$			12470.2i			0.293837i
$22$	−114714.			−2.29687
$23$	−45987.5			−0.788120			−0.394060	−	0.919085i	$-0.628930\pi$
$23$	−45987.5			−0.788120			−0.394060	+	0.919085i	$0.628930\pi$
$24$			43534.7i			0.642830i
$25$	73146.2			0.936271
$26$	−82666.1	−	120908.i	−0.922403	−	1.34912i
$27$	99310.7			0.971008
$28$			97283.8i			0.837505i
$29$	66421.0			0.505723			0.252861	−	0.967503i	$-0.418628\pi$
$29$	66421.0			0.505723			0.252861	+	0.967503i	$0.418628\pi$
$30$	35768.8			0.241869
$31$		−	128237.i		−	0.773121i	−0.922264	−	0.386560i	$-0.873663\pi$
$31$		−	128237.i		−	0.773121i	0.922264	−	0.386560i	$-0.126337\pi$
$32$		−	166568.i		−	0.898603i
$33$		−	170092.i		−	0.823918i
$34$			515941.i			2.25125i
$35$	32094.6			0.126530
$36$	306996.			1.09666
$37$			608049.i			1.97348i	0.162308	+	0.986740i	$0.448106\pi$
$37$			608049.i			1.97348i	−0.162308	+	0.986740i	$0.551894\pi$
$38$	−369299.			−1.09178
$39$	179276.	−	122573.i	0.483946	−	0.330878i
$40$	112045.			0.276811
$41$			32198.3i			0.0729608i	0.999334	+	0.0364804i	$0.0116147\pi$
$41$			32198.3i			0.0729608i	−0.999334	+	0.0364804i	$0.988385\pi$
$42$	−230574.			−0.480218
$43$	−623960.			−1.19679			−0.598394	−	0.801202i	$-0.704194\pi$
$43$	−623960.			−1.19679			−0.598394	+	0.801202i	$0.704194\pi$
$44$		−	1.32693e6i		−	2.34836i
$45$		−	101280.i		−	0.165684i
$46$		−	850309.i		−	1.28803i
$47$			342526.i			0.481229i	0.970621	+	0.240614i	$0.0773489\pi$
$47$			342526.i			0.481229i	−0.970621	+	0.240614i	$0.922651\pi$
$48$	−54396.6			−0.0709949
$49$	616653.			0.748781
$50$			1.35247e6i			1.53015i
$51$	−765010.			−0.807554
$52$	1.39859e6	−	956227.i	1.37936	−	0.943082i
$53$	−676681.			−0.624336			−0.312168	−	0.950027i	$-0.601055\pi$
$53$	−676681.			−0.624336			−0.312168	+	0.950027i	$0.601055\pi$
$54$			1.83626e6i			1.58692i
$55$	−437765.			−0.354790
$56$	−722272.			−0.549595
$57$		−	547577.i		−	0.391637i
$58$			1.22812e6i			0.826503i
$59$		−	600543.i		−	0.380682i	−0.981718	−	0.190341i	$-0.939041\pi$
$59$		−	600543.i		−	0.380682i	0.981718	−	0.190341i	$-0.0609593\pi$
$60$			413750.i			0.247291i
$61$	−39916.7			−0.0225165			−0.0112582	−	0.999937i	$-0.503584\pi$
$61$	−39916.7			−0.0225165			−0.0112582	+	0.999937i	$0.503584\pi$
$62$	2.37110e6			1.26351
$63$			652877.i			0.328957i
$64$	3.33382e6			1.58969
$65$	−315466.	−	461404.i	−0.142481	−	0.208394i
$66$	3.14500e6			1.34653
$67$		−	3.50690e6i		−	1.42450i	−0.701926	−	0.712249i	$-0.747677\pi$
$67$		−	3.50690e6i		−	1.42450i	0.701926	−	0.712249i	$-0.252323\pi$
$68$	−5.96807e6			−2.30172
$69$	1.26079e6			0.462032
$70$			593430.i			0.206788i
$71$		−	2.53886e6i		−	0.841850i	−0.907095	−	0.420925i	$-0.861706\pi$
$71$		−	2.53886e6i		−	0.841850i	0.907095	−	0.420925i	$-0.138294\pi$
$72$			2.27925e6i			0.719662i
$73$			2.35690e6i			0.709105i	0.935036	+	0.354552i	$0.115367\pi$
$73$			2.35690e6i			0.709105i	−0.935036	+	0.354552i	$0.884633\pi$
$74$	−1.12428e7			−3.22526
$75$	−2.00538e6			−0.548885
$76$		−	4.27181e6i		−	1.11626i
$77$	2.82194e6			0.704419
$78$	2.26637e6	+	3.31482e6i	0.540755	+	0.790914i
$79$	1.39003e6			0.317196			0.158598	−	0.987343i	$-0.449303\pi$
$79$	1.39003e6			0.317196			0.158598	+	0.987343i	$0.449303\pi$
$80$			140001.i			0.0305714i
$81$	416432.			0.0870657
$82$	−595347.			−0.119240
$83$			7.87325e6i			1.51140i	0.654916	+	0.755702i	$0.272704\pi$
$83$			7.87325e6i			1.51140i	−0.654916	+	0.755702i	$0.727296\pi$
$84$		−	2.66714e6i		−	0.490984i
$85$			1.96891e6i			0.347743i
$86$		−	1.15370e7i		−	1.95591i
$87$	−1.82100e6			−0.296478
$88$	9.85166e6			1.54106
$89$		−	6.81445e6i		−	1.02463i	−0.858799	−	0.512313i	$-0.828789\pi$
$89$		−	6.81445e6i		−	1.02463i	0.858799	−	0.512313i	$-0.171211\pi$
$90$	1.87267e6			0.270777
$91$	2.03357e6	+	2.97433e6i	0.282888	+	0.413756i
$92$	9.83582e6			1.31690
$93$			3.51575e6i			0.453239i
$94$	−6.33331e6			−0.786473
$95$	−1.40930e6			−0.168644
$96$			4.56664e6i			0.526802i
$97$			1.63544e7i			1.81942i	0.415247	+	0.909709i	$0.363695\pi$
$97$			1.63544e7i			1.81942i	−0.415247	+	0.909709i	$0.636305\pi$
$98$			1.14019e7i			1.22373i
$99$		−	8.90513e6i		−	0.922395i
$100$	−1.56445e7			−1.56445
$101$	1.67008e7			1.61292			0.806460	−	0.591289i	$-0.201381\pi$
$101$	1.67008e7			1.61292			0.806460	+	0.591289i	$0.201381\pi$
$102$		−	1.41450e7i		−	1.31979i
$103$	−1.60361e7			−1.44600			−0.723001	−	0.690847i	$-0.757238\pi$
$103$	−1.60361e7			−1.44600			−0.723001	+	0.690847i	$0.757238\pi$
$104$	7.09939e6	+	1.03836e7i	0.618877	+	0.905176i
$105$	−879906.			−0.0741778
$106$		−	1.25118e7i		−	1.02035i
$107$	−3.53217e6			−0.278739			−0.139370	−	0.990240i	$-0.544508\pi$
$107$	−3.53217e6			−0.278739			−0.139370	+	0.990240i	$0.544508\pi$
$108$	−2.12406e7			−1.62250
$109$			7.71109e6i			0.570326i	0.958479	+	0.285163i	$0.0920477\pi$
$109$			7.71109e6i			0.570326i	−0.958479	+	0.285163i	$0.907952\pi$
$110$		−	8.09428e6i		−	0.579834i
$111$		−	1.66703e7i		−	1.15695i
$112$		−	902479.i		−	0.0606979i
$113$	1.82843e7			1.19207			0.596037	−	0.802957i	$-0.296741\pi$
$113$	1.82843e7			1.19207			0.596037	+	0.802957i	$0.296741\pi$
$114$	1.01247e7			0.640053
$115$		−	3.24491e6i		−	0.198957i
$116$	−1.42061e7			−0.845032
$117$	9.38600e6	−	6.41729e6i	0.541789	−	0.370426i
$118$	1.11040e7			0.622149
$119$		−	1.26921e7i		−	0.690428i
$120$	−3.07183e6			−0.162279
$121$	−1.90037e7			−0.975189
$122$		−	738060.i		−	0.0367987i
$123$		−	882750.i		−	0.0427730i
$124$			2.74274e7i			1.29184i
$125$			1.06738e7i			0.488803i
$126$	−1.20717e7			−0.537615
$127$	−4.26324e6			−0.184683			−0.0923414	−	0.995727i	$-0.529435\pi$
$127$	−4.26324e6			−0.184683			−0.0923414	+	0.995727i	$0.529435\pi$
$128$			4.03216e7i			1.69943i
$129$	1.71065e7			0.701612
$130$	8.53136e6	−	5.83297e6i	0.340578	−	0.232856i
$131$	−1.37009e7			−0.532476			−0.266238	−	0.963907i	$-0.585781\pi$
$131$	−1.37009e7			−0.532476			−0.266238	+	0.963907i	$0.585781\pi$
$132$			3.63793e7i			1.37672i
$133$	9.08471e6			0.334835
$134$	6.48427e7			2.32806
$135$			7.00743e6i			0.245127i
$136$		−	4.43092e7i		−	1.51045i
$137$		−	2.21666e7i		−	0.736508i	−0.929725	−	0.368254i	$-0.879956\pi$
$137$		−	2.21666e7i		−	0.736508i	0.929725	−	0.368254i	$-0.120044\pi$
$138$			2.33121e7i			0.755100i
$139$	−2.56615e7			−0.810459			−0.405229	−	0.914215i	$-0.632808\pi$
$139$	−2.56615e7			−0.810459			−0.405229	+	0.914215i	$0.632808\pi$
$140$	−6.86441e6			−0.211424
$141$		−	9.39071e6i		−	0.282118i
$142$	4.69436e7			1.37584
$143$	−2.77376e7	−	4.05693e7i	−0.793218	−	1.16017i
$144$	−2.84793e6			−0.0794804
$145$			4.68671e6i			0.127667i
$146$	−4.35790e7			−1.15889
$147$	−1.69062e7			−0.438970
$148$		−	1.30050e8i		−	3.29757i
$149$			6.01080e7i			1.48861i	0.667841	+	0.744304i	$0.267219\pi$
$149$			6.01080e7i			1.48861i	−0.667841	+	0.744304i	$0.732781\pi$
$150$		−	3.70794e7i		−	0.897044i
$151$			4.71692e7i			1.11491i	0.830208	+	0.557454i	$0.188222\pi$
$151$			4.71692e7i			1.11491i	−0.830208	+	0.557454i	$0.811778\pi$
$152$	3.17155e7			0.732520
$153$	−4.00520e7			−0.904075
$154$			5.21778e7i			1.15123i
$155$	9.04848e6			0.195171
$156$	−3.83437e7	+	2.62159e7i	−0.808645	+	0.552878i
$157$	−8.73361e6			−0.180113			−0.0900565	−	0.995937i	$-0.528705\pi$
$157$	−8.73361e6			−0.180113			−0.0900565	+	0.995937i	$0.528705\pi$
$158$			2.57016e7i			0.518394i
$159$	1.85519e7			0.366015
$160$	1.17532e7			0.226848
$161$			2.09175e7i			0.395020i
$162$			7.69984e6i			0.142292i
$163$		−	5.43357e7i		−	0.982718i	−0.870957	−	0.491359i	$-0.836500\pi$
$163$		−	5.43357e7i		−	0.982718i	0.870957	−	0.491359i	$-0.163500\pi$
$164$		−	6.88659e6i		−	0.121913i
$165$	1.20018e7			0.207994
$166$	−1.45576e8			−2.47009
$167$			6.71453e7i			1.11560i	0.829976	+	0.557799i	$0.188354\pi$
$167$			6.71453e7i			1.11560i	−0.829976	+	0.557799i	$0.811646\pi$
$168$	1.98018e7			0.322198
$169$	2.27715e7	−	5.84708e7i	0.362901	−	0.931828i
$170$	−3.64051e7			−0.568318
$171$		−	2.86684e7i		−	0.438446i
$172$	1.33453e8			1.99976
$173$	7.49724e7			1.10088			0.550440	−	0.834875i	$-0.314460\pi$
$173$	7.49724e7			1.10088			0.550440	+	0.834875i	$0.314460\pi$
$174$		−	3.36703e7i		−	0.484534i
$175$		−	3.32707e7i		−	0.469276i
$176$			1.23097e7i			0.170197i
$177$			1.64645e7i			0.223173i
$178$	1.25999e8			1.67455
$179$	1.40205e8			1.82717			0.913584	−	0.406650i	$-0.133303\pi$
$179$	1.40205e8			1.82717			0.913584	+	0.406650i	$0.133303\pi$
$180$			2.16618e7i			0.276848i
$181$	−3.70364e7			−0.464251			−0.232126	−	0.972686i	$-0.574568\pi$
$181$	−3.70364e7			−0.464251			−0.232126	+	0.972686i	$0.574568\pi$
$182$	−5.49953e7	+	3.76008e7i	−0.676201	+	0.462325i
$183$	1.09436e6			0.0132002
$184$			7.30249e7i			0.864188i
$185$	−4.29043e7			−0.498196
$186$	−6.50062e7			−0.740729
$187$			1.73118e8i			1.93596i
$188$		−	7.32597e7i		−	0.804104i
$189$		−	4.51717e7i		−	0.486687i
$190$		−	2.60580e7i		−	0.275615i
$191$	7.66608e7			0.796079			0.398040	−	0.917368i	$-0.369691\pi$
$191$	7.66608e7			0.796079			0.398040	+	0.917368i	$0.369691\pi$
$192$	−9.14000e7			−0.931949
$193$		−	4.92053e7i		−	0.492676i	−0.969184	−	0.246338i	$-0.920773\pi$
$193$		−	4.92053e7i		−	0.492676i	0.969184	−	0.246338i	$-0.0792273\pi$
$194$	−3.02392e8			−2.97348
$195$	8.64882e6	+	1.26499e7i	0.0835287	+	0.122170i
$196$	−1.31890e8			−1.25117
$197$		−	1.51075e8i		−	1.40786i	−0.710267	−	0.703932i	$-0.751426\pi$
$197$		−	1.51075e8i		−	1.40786i	0.710267	−	0.703932i	$-0.248574\pi$
$198$	1.64656e8			1.50747
$199$	−9.71940e7			−0.874287			−0.437143	−	0.899392i	$-0.644010\pi$
$199$	−9.71940e7			−0.874287			−0.437143	+	0.899392i	$0.644010\pi$
$200$		−	1.16151e8i		−	1.02664i
$201$			9.61453e7i			0.835107i
$202$			3.08798e8i			2.63600i
$203$		−	3.02117e7i		−	0.253477i
$204$	1.63621e8			1.34937
$205$	−2.27193e6			−0.0184186
$206$		−	2.96508e8i		−	2.36320i
$207$	6.60087e7			0.517256
$208$	−1.29744e7	+	8.87069e6i	−0.0999688	+	0.0683496i
$209$	−1.23914e8			−0.938875
$210$		−	1.62695e7i		−	0.121229i
$211$	1.37648e8			1.00875			0.504373	−	0.863486i	$-0.331724\pi$
$211$	1.37648e8			1.00875			0.504373	+	0.863486i	$0.331724\pi$
$212$	1.44729e8			1.04323
$213$			6.96055e7i			0.493531i
$214$		−	6.53099e7i		−	0.455544i
$215$		−	4.40270e7i		−	0.302124i
$216$		−	1.57698e8i		−	1.06473i
$217$	−5.83288e7			−0.387502
$218$	−1.42578e8			−0.932084
$219$		−	6.46167e7i		−	0.415710i
$220$	9.36293e7			0.592833
$221$	−1.82466e8	+	1.24754e8i	−1.13713	+	0.777464i
$222$	3.08234e8			1.89080
$223$		−	945701.i		−	0.00571067i	−0.999996	−	0.00285533i	$-0.999091\pi$
$223$		−	945701.i		−	0.00571067i	0.999996	−	0.00285533i	$-0.000908882\pi$
$224$	−7.57639e7			−0.450396
$225$	−1.04991e8			−0.614489
$226$			3.38076e8i			1.94821i
$227$			2.04286e8i			1.15917i	0.814911	+	0.579586i	$0.196786\pi$
$227$			2.04286e8i			1.15917i	−0.814911	+	0.579586i	$0.803214\pi$
$228$			1.17116e8i			0.654402i
$229$		−	1.71578e7i		−	0.0944144i	−0.998885	−	0.0472072i	$-0.984968\pi$
$229$		−	1.71578e7i		−	0.0944144i	0.998885	−	0.0472072i	$-0.0150321\pi$
$230$	5.99984e7			0.325156
$231$	−7.73665e7			−0.412963
$232$		−	1.05472e8i		−	0.554534i
$233$	1.76889e8			0.916128			0.458064	−	0.888919i	$-0.348543\pi$
$233$	1.76889e8			0.916128			0.458064	+	0.888919i	$0.348543\pi$
$234$	1.18656e8	+	1.73547e8i	0.605387	+	0.885446i
$235$	−2.41689e7			−0.121484
$236$			1.28444e8i			0.636097i
$237$	−3.81090e7			−0.185955
$238$	2.34677e8			1.12837
$239$			1.27144e8i			0.602423i	0.953557	+	0.301211i	$0.0973910\pi$
$239$			1.27144e8i			0.602423i	−0.953557	+	0.301211i	$0.902609\pi$
$240$		−	3.83826e6i		−	0.0179223i
$241$		−	2.24098e8i		−	1.03129i	−0.856804	−	0.515643i	$-0.827553\pi$
$241$		−	2.24098e8i		−	1.03129i	0.856804	−	0.515643i	$-0.172447\pi$
$242$		−	3.51378e8i		−	1.59375i
$243$	−2.28609e8			−1.02205
$244$	8.53739e6			0.0376236
$245$			4.35114e7i			0.189026i
$246$	1.63221e7			0.0699040
$247$	−8.92959e7	−	1.30605e8i	−0.377044	−	0.551469i
$248$	−2.03631e8			−0.847741
$249$		−	2.15853e8i		−	0.886055i
$250$	−1.97358e8			−0.798851
$251$	3.09551e8			1.23559			0.617795	−	0.786339i	$-0.288026\pi$
$251$	3.09551e8			1.23559			0.617795	+	0.786339i	$0.288026\pi$
$252$		−	1.39638e8i		−	0.549668i
$253$		−	2.85311e8i		−	1.10764i
$254$		−	7.88273e7i		−	0.301827i
$255$		−	5.39796e7i		−	0.203863i
$256$	−3.18817e8			−1.18769
$257$	7.43073e7			0.273065			0.136532	−	0.990636i	$-0.456404\pi$
$257$	7.43073e7			0.273065			0.136532	+	0.990636i	$0.456404\pi$
$258$			3.16299e8i			1.14665i
$259$	2.76572e8			0.989144
$260$	6.74720e7	+	9.86852e7i	0.238077	+	0.348213i
$261$	−9.53382e7			−0.331914
$262$		−	2.53330e8i		−	0.870226i
$263$	−1.42398e8			−0.482680			−0.241340	−	0.970441i	$-0.577587\pi$
$263$	−1.42398e8			−0.482680			−0.241340	+	0.970441i	$0.577587\pi$
$264$	−2.70093e8			−0.903442
$265$		−	4.77470e7i		−	0.157611i
$266$			1.67976e8i			0.547221i
$267$			1.86825e8i			0.600683i
$268$			7.50058e8i			2.38025i
$269$	−1.63438e7			−0.0511941			−0.0255971	−	0.999672i	$-0.508149\pi$
$269$	−1.63438e7			−0.0511941			−0.0255971	+	0.999672i	$0.508149\pi$
$270$	−1.29567e8			−0.400611
$271$		−	4.43345e8i		−	1.35316i	−0.736369	−	0.676580i	$-0.763461\pi$
$271$		−	4.43345e8i		−	1.35316i	0.736369	−	0.676580i	$-0.236539\pi$
$272$	5.53643e7			0.166816
$273$	−5.57525e7	−	8.15442e7i	−0.165842	−	0.242563i
$274$	4.09861e8			1.20368
$275$			4.53806e8i			1.31585i
$276$	−2.69659e8			−0.772028
$277$	−3.85754e8			−1.09051			−0.545257	−	0.838269i	$-0.683568\pi$
$277$	−3.85754e8			−1.09051			−0.545257	+	0.838269i	$0.683568\pi$
$278$		−	4.74482e8i		−	1.32453i
$279$			1.84067e8i			0.507411i
$280$		−	5.09639e7i		−	0.138743i
$281$			2.66195e8i			0.715695i	0.933780	+	0.357847i	$0.116489\pi$
$281$			2.66195e8i			0.715695i	−0.933780	+	0.357847i	$0.883511\pi$
$282$	1.73634e8			0.461066
$283$	4.35005e8			1.14088			0.570442	−	0.821338i	$-0.306772\pi$
$283$	4.35005e8			1.14088			0.570442	+	0.821338i	$0.306772\pi$
$284$			5.43013e8i			1.40668i
$285$	3.86374e7			0.0988669
$286$	7.50127e8	−	5.12868e8i	1.89607	−	1.29636i
$287$	1.46455e7			0.0365693
$288$			2.39086e8i			0.589767i
$289$	3.68282e8			0.897506
$290$	−8.66573e7			−0.208647
$291$		−	4.48371e8i		−	1.06663i
$292$		−	5.04094e8i		−	1.18487i
$293$			8.52483e7i			0.197993i	0.995088	+	0.0989963i	$0.0315632\pi$
$293$			8.52483e7i			0.197993i	−0.995088	+	0.0989963i	$0.968437\pi$
$294$		−	3.12595e8i		−	0.717409i
$295$	4.23747e7			0.0961014
$296$	9.65539e8			2.16396
$297$			6.16134e8i			1.36467i
$298$	−1.11140e9			−2.43283
$299$	3.00718e8	−	2.05603e8i	0.650593	−	0.444817i
$300$	4.28911e8			0.917155
$301$			2.83809e8i			0.599852i
$302$	−8.72158e8			−1.82210
$303$	−4.57869e8			−0.945567
$304$			3.96286e7i			0.0809005i
$305$		−	2.81655e6i		−	0.00568417i
$306$		−	7.40562e8i		−	1.47753i
$307$			6.24815e8i			1.23244i	0.787573	+	0.616222i	$0.211338\pi$
$307$			6.24815e8i			1.23244i	−0.787573	+	0.616222i	$0.788662\pi$
$308$	−6.03558e8			−1.17704
$309$	4.39646e8			0.847713
$310$			1.67306e8i			0.318968i
$311$	7.53251e7			0.141997			0.0709984	−	0.997476i	$-0.477381\pi$
$311$	7.53251e7			0.141997			0.0709984	+	0.997476i	$0.477381\pi$
$312$	−1.94637e8	−	2.84678e8i	−0.362814	−	0.530656i
$313$	−5.16335e8			−0.951757			−0.475878	−	0.879511i	$-0.657870\pi$
$313$	−5.16335e8			−0.951757			−0.475878	+	0.879511i	$0.657870\pi$
$314$		−	1.61485e8i		−	0.294359i
$315$	−4.60674e7			−0.0830437
$316$	−2.97299e8			−0.530016
$317$		−	3.81492e8i		−	0.672633i	−0.941749	−	0.336317i	$-0.890819\pi$
$317$		−	3.81492e8i		−	0.672633i	0.941749	−	0.336317i	$-0.109181\pi$
$318$			3.43025e8i			0.598178i
$319$			4.12082e8i			0.710750i
$320$			2.35236e8i			0.401310i
$321$	9.68380e7			0.163410
$322$	−3.86765e8			−0.645582
$323$			5.57319e8i			0.920228i
$324$	−8.90667e7			−0.145482
$325$	−4.78311e8	+	3.27026e8i	−0.772892	+	0.528433i
$326$	1.00467e9			1.60606
$327$		−	2.11407e8i		−	0.334351i
$328$	5.11286e7			0.0800029
$329$	1.55799e8			0.241200
$330$			2.21913e8i			0.339925i
$331$		−	6.24552e8i		−	0.946609i	−0.880899	−	0.473305i	$-0.843061\pi$
$331$		−	6.24552e8i		−	0.946609i	0.880899	−	0.473305i	$-0.156939\pi$
$332$		−	1.68393e9i		−	2.52547i
$333$		−	8.72771e8i		−	1.29523i
$334$	−1.24152e9			−1.82322
$335$	2.47449e8			0.359608
$336$			2.47424e7i			0.0355839i
$337$	4.36103e8			0.620703			0.310352	−	0.950622i	$-0.399553\pi$
$337$	4.36103e8			0.620703			0.310352	+	0.950622i	$0.399553\pi$
$338$	1.08113e9	+	4.21045e8i	1.52289	+	0.593089i
$339$	−5.01282e8			−0.698848
$340$		−	4.21111e8i		−	0.581059i
$341$	7.95595e8			1.08655
$342$	5.30078e8			0.716553
$343$		−	6.55076e8i		−	0.876520i
$344$			9.90804e8i			1.31230i
$345$			8.89624e7i			0.116638i
$346$			1.38624e9i			1.79917i
$347$	−5.03534e8			−0.646957			−0.323479	−	0.946235i	$-0.604852\pi$
$347$	−5.03534e8			−0.646957			−0.323479	+	0.946235i	$0.604852\pi$
$348$	3.89476e8			0.495397
$349$		−	7.56494e8i		−	0.952613i	−0.879279	−	0.476307i	$-0.841975\pi$
$349$		−	7.56494e8i		−	0.952613i	0.879279	−	0.476307i	$-0.158025\pi$
$350$	6.15175e8			0.766938
$351$	−6.49404e8	+	4.44004e8i	−0.801568	+	0.548039i
$352$	1.03341e9			1.26291
$353$			5.77975e8i			0.699354i	0.936870	+	0.349677i	$0.113709\pi$
$353$			5.77975e8i			0.699354i	−0.936870	+	0.349677i	$0.886291\pi$
$354$	−3.04429e8			−0.364732
$355$	1.79144e8			0.212521
$356$			1.45748e9i			1.71209i
$357$			3.47966e8i			0.404761i
$358$			2.59239e9i			2.98614i
$359$		−	1.60092e9i		−	1.82617i	−0.407774	−	0.913083i	$-0.633695\pi$
$359$		−	1.60092e9i		−	1.82617i	0.407774	−	0.913083i	$-0.366305\pi$
$360$	−1.60825e8			−0.181675
$361$	4.94955e8			0.553720
$362$		−	6.84802e8i		−	0.758727i
$363$	5.21005e8			0.571700
$364$	−4.34942e8	−	6.36150e8i	−0.472690	−	0.691361i
$365$	−1.66304e8			−0.179010
$366$			2.02347e7i			0.0215731i
$367$	−1.45572e9			−1.53726			−0.768629	−	0.639695i	$-0.779061\pi$
$367$	−1.45572e9			−1.53726			−0.768629	+	0.639695i	$0.779061\pi$
$368$	−9.12446e7			−0.0954421
$369$		−	4.62163e7i		−	0.0478853i
$370$		−	7.93302e8i		−	0.814202i
$371$			3.07790e8i			0.312929i
$372$		−	7.51949e8i		−	0.757335i
$373$	1.90149e8			0.189720			0.0948598	−	0.995491i	$-0.469760\pi$
$373$	1.90149e8			0.189720			0.0948598	+	0.995491i	$0.469760\pi$
$374$	−3.20095e9			−3.16394
$375$		−	2.92633e8i		−	0.286559i
$376$	5.43907e8			0.527676
$377$	−4.34335e8	+	2.96958e8i	−0.417474	+	0.285431i
$378$	8.35224e8			0.795393
$379$			3.66406e7i			0.0345721i	0.999851	+	0.0172861i	$0.00550259\pi$
$379$			3.66406e7i			0.0345721i	−0.999851	+	0.0172861i	$0.994497\pi$
$380$	3.01422e8			0.281794
$381$	1.16881e8			0.108270
$382$			1.41746e9i			1.30103i
$383$		−	1.19790e9i		−	1.08949i	−0.838600	−	0.544747i	$-0.816626\pi$
$383$		−	1.19790e9i		−	1.08949i	0.838600	−	0.544747i	$-0.183374\pi$
$384$		−	1.10546e9i		−	0.996282i
$385$			1.99118e8i			0.177827i
$386$	9.09807e8			0.805181
$387$	8.95609e8			0.785470
$388$		−	3.49787e9i		−	3.04014i
$389$	2.49771e8			0.215139			0.107569	−	0.994198i	$-0.465693\pi$
$389$	2.49771e8			0.215139			0.107569	+	0.994198i	$0.465693\pi$
$390$	−2.33896e8	+	1.59917e8i	−0.199663	+	0.136511i
$391$	−1.28322e9			−1.08564
$392$		−	9.79201e8i		−	0.821052i
$393$	3.75624e8			0.312162
$394$	2.79338e9			2.30087
$395$			9.80811e7i			0.0800747i
$396$			1.90463e9i			1.54127i
$397$			2.40099e9i			1.92586i	0.269758	+	0.962928i	$0.413056\pi$
$397$			2.40099e9i			1.92586i	−0.269758	+	0.962928i	$0.586944\pi$
$398$		−	1.79712e9i		−	1.42885i
$399$	−2.49067e8			−0.196295
$400$	1.45131e8			0.113383
$401$			7.88393e8i			0.610573i	0.952261	+	0.305287i	$0.0987522\pi$
$401$			7.88393e8i			0.610573i	−0.952261	+	0.305287i	$0.901248\pi$
$402$	−1.77773e9			−1.36482
$403$	5.73328e8	+	8.38556e8i	0.436351	+	0.638211i
$404$	−3.57197e9			−2.69509
$405$			2.93837e7i			0.0219793i
$406$	5.58615e8			0.414258
$407$	−3.77240e9			−2.77356
$408$			1.21478e9i			0.885498i
$409$		−	1.75342e9i		−	1.26723i	−0.773650	−	0.633614i	$-0.781571\pi$
$409$		−	1.75342e9i		−	1.26723i	0.773650	−	0.633614i	$-0.218429\pi$
$410$		−	4.20081e7i		−	0.0301016i
$411$			6.07720e8i			0.431775i
$412$	3.42981e9			2.41618
$413$	−2.73158e8			−0.190805
$414$			1.22050e9i			0.845351i
$415$	−5.55541e8			−0.381547
$416$	7.44703e8	+	1.08921e9i	0.507173	+	0.741797i
$417$	7.03537e8			0.475128
$418$		−	2.29117e9i		−	1.53440i
$419$	2.06615e9			1.37219			0.686094	−	0.727513i	$-0.259324\pi$
$419$	2.06615e9			1.37219			0.686094	+	0.727513i	$0.259324\pi$
$420$	1.88195e8			0.123947
$421$			8.35553e8i			0.545741i	0.962051	+	0.272871i	$0.0879731\pi$
$421$			8.35553e8i			0.545741i	−0.962051	+	0.272871i	$0.912027\pi$
$422$			2.54511e9i			1.64859i
$423$		−	4.91649e8i		−	0.315838i
$424$			1.07452e9i			0.684596i
$425$	2.04105e9			1.28971
$426$	−1.28701e9			−0.806579
$427$			1.81562e7i			0.0112857i
$428$	7.55462e8			0.465757
$429$	7.60454e8	+	1.11225e9i	0.465021	+	0.680145i
$430$	8.14060e8			0.493761
$431$		−	8.24930e8i		−	0.496303i	−0.968721	−	0.248151i	$-0.920177\pi$
$431$		−	8.24930e8i		−	0.496303i	0.968721	−	0.248151i	$-0.0798231\pi$
$432$	1.97044e8			0.117590
$433$	2.49718e8			0.147823			0.0739115	−	0.997265i	$-0.476452\pi$
$433$	2.49718e8			0.147823			0.0739115	+	0.997265i	$0.476452\pi$
$434$		−	1.07850e9i		−	0.633295i
$435$		−	1.28491e8i		−	0.0748445i
$436$		−	1.64925e9i		−	0.952981i
$437$		−	9.18504e8i		−	0.526497i
$438$	1.19476e9			0.679395
$439$	1.96422e9			1.10806			0.554030	−	0.832497i	$-0.313089\pi$
$439$	1.96422e9			1.10806			0.554030	+	0.832497i	$0.313089\pi$
$440$			6.95139e8i			0.389034i
$441$	−8.85120e8			−0.491436
$442$	−2.30669e9	−	3.37380e9i	−1.27061	−	1.85841i
$443$	−1.28165e9			−0.700419			−0.350209	−	0.936671i	$-0.613890\pi$
$443$	−1.28165e9			−0.700419			−0.350209	+	0.936671i	$0.613890\pi$
$444$			3.56545e9i			1.93319i
$445$	4.80832e8			0.258662
$446$	1.74860e7			0.00933295
$447$		−	1.64792e9i		−	0.872691i
$448$		−	1.51639e9i		−	0.796781i
$449$		−	7.78272e8i		−	0.405760i	−0.979204	−	0.202880i	$-0.934970\pi$
$449$		−	7.78272e8i		−	0.405760i	0.979204	−	0.202880i	$-0.0650302\pi$
$450$		−	1.94129e9i		−	1.00426i
$451$	−1.99762e8			−0.102540
$452$	−3.91065e9			−1.99188
$453$		−	1.29319e9i		−	0.653610i
$454$	−3.77725e9			−1.89443
$455$	−2.09870e8	+	1.43490e8i	−0.104451	+	0.0714139i
$456$	−8.69514e8			−0.429437
$457$			2.72247e9i			1.33431i	0.744918	+	0.667156i	$0.232488\pi$
$457$			2.72247e9i			1.33431i	−0.744918	+	0.667156i	$0.767512\pi$
$458$	3.17249e8			0.154302
$459$	2.77114e9			1.33756
$460$			6.94022e8i			0.332446i
$461$			1.14171e9i			0.542752i	0.962473	+	0.271376i	$0.0874787\pi$
$461$			1.14171e9i			0.542752i	−0.962473	+	0.271376i	$0.912521\pi$
$462$		−	1.43051e9i		−	0.674905i
$463$			2.34835e9i			1.09958i	0.835301	+	0.549792i	$0.185293\pi$
$463$			2.34835e9i			1.09958i	−0.835301	+	0.549792i	$0.814707\pi$
$464$	1.31787e8			0.0612435
$465$	−2.48073e8			−0.114418
$466$			3.27069e9i			1.49723i
$467$	2.68110e9			1.21816			0.609079	−	0.793109i	$-0.291539\pi$
$467$	2.68110e9			1.21816			0.609079	+	0.793109i	$0.291539\pi$
$468$	−2.00748e9	+	1.37253e9i	−0.905296	+	0.618959i
$469$	−1.59512e9			−0.713984
$470$		−	4.46883e8i		−	0.198541i
$471$	2.39441e8			0.105591
$472$	−9.53619e8			−0.417425
$473$		−	3.87111e9i		−	1.68198i
$474$		−	7.04635e8i		−	0.303907i
$475$			1.46094e9i			0.625468i
$476$			2.71459e9i			1.15366i
$477$	9.71282e8			0.409762
$478$	−2.35088e9			−0.984541
$479$			1.49608e9i			0.621985i	0.950412	+	0.310992i	$0.100661\pi$
$479$			1.49608e9i			0.621985i	−0.950412	+	0.310992i	$0.899339\pi$
$480$	−3.22225e8			−0.132989
$481$	−2.71850e9	−	3.97610e9i	−1.11384	−	1.62911i
$482$	4.14358e9			1.68543
$483$		−	5.73474e8i		−	0.231579i
$484$	4.06451e9			1.62948
$485$	−1.15397e9			−0.459303
$486$		−	4.22699e9i		−	1.67034i
$487$		−	8.18490e8i		−	0.321116i	−0.987026	−	0.160558i	$-0.948671\pi$
$487$		−	8.18490e8i		−	0.321116i	0.987026	−	0.160558i	$-0.0513294\pi$
$488$			6.33848e7i			0.0246897i
$489$			1.48967e9i			0.576115i
$490$	−8.04526e8			−0.308926
$491$	6.89475e8			0.262866			0.131433	−	0.991325i	$-0.458042\pi$
$491$	6.89475e8			0.262866			0.131433	+	0.991325i	$0.458042\pi$
$492$			1.88803e8i			0.0714711i
$493$	1.85340e9			0.696633
$494$	2.41489e9	−	1.65108e9i	0.901266	−	0.616204i
$495$	6.28351e8			0.232854
$496$		−	2.54437e8i		−	0.0936256i
$497$	−1.15481e9			−0.421950
$498$	3.99113e9			1.44808
$499$			2.18288e9i			0.786463i	0.919439	+	0.393232i	$0.128643\pi$
$499$			2.18288e9i			0.786463i	−0.919439	+	0.393232i	$0.871357\pi$
$500$		−	2.28291e9i		−	0.816760i
$501$		−	1.84086e9i		−	0.654015i
$502$			5.72360e9i			2.01933i
$503$	−4.03880e9			−1.41503			−0.707513	−	0.706700i	$-0.750183\pi$
$503$	−4.03880e9			−1.41503			−0.707513	+	0.706700i	$0.750183\pi$
$504$	1.03672e9			0.360708
$505$			1.17842e9i			0.407174i
$506$	5.27540e9			1.81021
$507$	−6.24304e8	+	1.60304e9i	−0.212749	+	0.546280i
$508$	9.11822e8			0.308594
$509$			3.33842e8i			0.112209i	0.998425	+	0.0561046i	$0.0178680\pi$
$509$			3.33842e8i			0.112209i	−0.998425	+	0.0561046i	$0.982132\pi$
$510$	9.98083e8			0.333174
$511$	1.07204e9			0.355416
$512$		−	7.33773e8i		−	0.241611i
$513$			1.98352e9i			0.648674i
$514$			1.37394e9i			0.446270i
$515$		−	1.13152e9i		−	0.365036i
$516$	−3.65874e9			−1.17235
$517$	−2.12507e9			−0.676326
$518$			5.11382e9i			1.61656i
$519$	−2.05545e9			−0.645387
$520$	−7.32676e8	+	5.00937e8i	−0.228507	+	0.156233i
$521$	−3.01841e9			−0.935075			−0.467538	−	0.883973i	$-0.654859\pi$
$521$	−3.01841e9			−0.935075			−0.467538	+	0.883973i	$0.654859\pi$
$522$		−	1.76280e9i		−	0.542447i
$523$	−4.89574e8			−0.149645			−0.0748226	−	0.997197i	$-0.523839\pi$
$523$	−4.89574e8			−0.149645			−0.0748226	+	0.997197i	$0.523839\pi$
$524$	2.93035e9			0.889735
$525$			9.12149e8i			0.275111i
$526$		−	2.63294e9i		−	0.788844i
$527$		−	3.57829e9i		−	1.06497i
$528$		−	3.37482e8i		−	0.0997773i
$529$	−1.28997e9			−0.378867
$530$	8.82843e8			0.257583
$531$			8.61997e8i			0.249847i
$532$	−1.94304e9			−0.559489
$533$	−1.43954e8	−	2.10549e8i	−0.0411792	−	0.0602292i
$534$	−3.45440e9			−0.981698
$535$		−	2.49232e8i		−	0.0703665i
$536$	−5.56871e9			−1.56199
$537$	−3.84387e9			−1.07117
$538$		−	3.02197e8i		−	0.0836666i
$539$			3.82578e9i			1.05235i
$540$		−	1.49875e9i		−	0.409592i
$541$		−	3.77991e9i		−	1.02634i	−0.858287	−	0.513170i	$-0.828471\pi$
$541$		−	3.77991e9i		−	1.02634i	0.858287	−	0.513170i	$-0.171529\pi$
$542$	8.19745e9			2.21147
$543$	1.01539e9			0.272166
$544$		−	4.64789e9i		−	1.23783i
$545$	−5.44099e8			−0.143976
$546$	1.50775e9	−	1.03086e9i	0.396420	−	0.271036i
$547$	7.41006e9			1.93583			0.967913	−	0.251286i	$-0.0808535\pi$
$547$	7.41006e9			1.93583			0.967913	+	0.251286i	$0.0808535\pi$
$548$			4.74100e9i			1.23066i
$549$	5.72949e7			0.0147779
$550$	−8.39088e9			−2.15049
$551$			1.32662e9i			0.337844i
$552$		−	2.00205e9i		−	0.506627i
$553$		−	6.32256e8i		−	0.158985i
$554$		−	7.13260e9i		−	1.78223i
$555$	1.17627e9			0.292065
$556$	5.48850e9			1.35423
$557$		−	3.25695e9i		−	0.798579i	−0.916825	−	0.399290i	$-0.869257\pi$
$557$		−	3.25695e9i		−	0.798579i	0.916825	−	0.399290i	$-0.130743\pi$
$558$	−3.40339e9			−0.829263
$559$	4.08015e9	−	2.78963e9i	0.987948	−	0.675469i
$560$	6.36795e7			0.0153229
$561$		−	4.74620e9i		−	1.13495i
$562$	−4.92195e9			−1.16966
$563$	2.62878e9			0.620834			0.310417	−	0.950600i	$-0.399531\pi$
$563$	2.62878e9			0.620834			0.310417	+	0.950600i	$0.399531\pi$
$564$			2.00849e9i			0.471403i
$565$			1.29015e9i			0.300933i
$566$			8.04325e9i			1.86455i
$567$		−	1.89415e8i		−	0.0436389i
$568$	−4.03153e9			−0.923104
$569$	3.01765e8			0.0686713			0.0343357	−	0.999410i	$-0.489068\pi$
$569$	3.01765e8			0.0686713			0.0343357	+	0.999410i	$0.489068\pi$
$570$			7.14406e8i			0.161578i
$571$	−1.35003e9			−0.303472			−0.151736	−	0.988421i	$-0.548486\pi$
$571$	−1.35003e9			−0.303472			−0.151736	+	0.988421i	$0.548486\pi$
$572$	5.93253e9	+	8.67698e9i	1.32542	+	1.93857i
$573$	−2.10173e9			−0.466698
$574$			2.70795e8i			0.0597652i
$575$	−3.36381e9			−0.737894
$576$	−4.78524e9			−1.04334
$577$			2.37915e9i			0.515593i	0.966199	+	0.257797i	$0.0829964\pi$
$577$			2.37915e9i			0.515593i	−0.966199	+	0.257797i	$0.917004\pi$
$578$			6.80953e9i			1.46680i
$579$			1.34901e9i			0.288829i
$580$		−	1.00239e9i		−	0.213325i
$581$	3.58116e9			0.757543
$582$	8.29039e9			1.74319
$583$		−	4.19820e9i		−	0.877451i
$584$	3.74258e9			0.777546
$585$	4.52808e8	+	6.62282e8i	0.0935123	+	0.136772i
$586$	−1.57624e9			−0.323580
$587$			4.98051e8i			0.101634i	0.998708	+	0.0508172i	$0.0161826\pi$
$587$			4.98051e8i			0.101634i	−0.998708	+	0.0508172i	$0.983817\pi$
$588$	3.61590e9			0.733492
$589$	2.56127e9			0.516477
$590$			7.83509e8i			0.157059i
$591$			4.14187e9i			0.825355i
$592$			1.20644e9i			0.238990i
$593$			7.06212e9i			1.39073i	0.718656	+	0.695366i	$0.244758\pi$
$593$			7.06212e9i			1.39073i	−0.718656	+	0.695366i	$0.755242\pi$
$594$	−1.13923e10			−2.23028
$595$	8.95561e8			0.174295
$596$		−	1.28559e10i		−	2.48738i
$597$	2.66467e9			0.512547
$598$	3.80161e9	+	5.56027e9i	0.726964	+	1.06327i
$599$	−1.27565e9			−0.242515			−0.121258	−	0.992621i	$-0.538693\pi$
$599$	−1.27565e9			−0.242515			−0.121258	+	0.992621i	$0.538693\pi$
$600$			3.18439e9i			0.601863i
$601$	1.88787e9			0.354742			0.177371	−	0.984144i	$-0.443241\pi$
$601$	1.88787e9			0.354742			0.177371	+	0.984144i	$0.443241\pi$
$602$	−5.24763e9			−0.980338
$603$			5.03368e9i			0.934921i
$604$		−	1.00886e10i		−	1.86294i
$605$		−	1.34091e9i		−	0.246182i
$606$		−	8.46601e9i		−	1.54534i
$607$	1.17192e9			0.212686			0.106343	−	0.994330i	$-0.466086\pi$
$607$	1.17192e9			0.212686			0.106343	+	0.994330i	$0.466086\pi$
$608$	3.32686e9			0.600304
$609$			8.28285e8i			0.148600i
$610$	5.20779e7			0.00928965
$611$	−1.53138e9	−	2.23982e9i	−0.271606	−	0.397254i
$612$	8.56634e9			1.51065
$613$		−	7.97221e9i		−	1.39787i	−0.715185	−	0.698935i	$-0.753657\pi$
$613$		−	7.97221e9i		−	1.39787i	0.715185	−	0.698935i	$-0.246343\pi$
$614$	−1.15528e10			−2.01418
$615$	6.22874e7			0.0107978
$616$		−	4.48105e9i		−	0.772408i
$617$		−	7.21859e9i		−	1.23724i	−0.785690	−	0.618620i	$-0.787692\pi$
$617$		−	7.21859e9i		−	1.23724i	0.785690	−	0.618620i	$-0.212308\pi$
$618$			8.12906e9i			1.38542i
$619$		−	2.25010e9i		−	0.381316i	−0.981657	−	0.190658i	$-0.938938\pi$
$619$		−	2.25010e9i		−	0.381316i	0.981657	−	0.190658i	$-0.0610621\pi$
$620$	−1.93529e9			−0.326119
$621$	−4.56705e9			−0.765271
$622$			1.39276e9i			0.232065i
$623$	−3.09956e9			−0.513561
$624$	3.55705e8	−	2.43199e8i	0.0586063	−	0.0400697i
$625$	4.96140e9			0.812875
$626$		−	9.54703e9i		−	1.55546i
$627$	3.39722e9			0.550412
$628$	1.86795e9			0.300958
$629$			1.69669e10i			2.71847i
$630$		−	8.51786e8i		−	0.135718i
$631$			1.03220e9i			0.163554i	0.996651	+	0.0817770i	$0.0260595\pi$
$631$			1.03220e9i			0.163554i	−0.996651	+	0.0817770i	$0.973940\pi$
$632$		−	2.20726e9i		−	0.347812i
$633$	−3.77376e9			−0.591373
$634$	7.05379e9			1.09928
$635$		−	3.00817e8i		−	0.0466223i
$636$	−3.96789e9			−0.611589
$637$	−4.03236e9	+	2.75696e9i	−0.618118	+	0.422613i
$638$	−7.61941e9			−1.16158
$639$			3.64419e9i			0.552519i
$640$	−2.84511e9			−0.429012
$641$	6.24903e9			0.937151			0.468576	−	0.883423i	$-0.344767\pi$
$641$	6.24903e9			0.937151			0.468576	+	0.883423i	$0.344767\pi$
$642$			1.79054e9i			0.267061i
$643$			5.34446e9i			0.792803i	0.918077	+	0.396401i	$0.129741\pi$
$643$			5.34446e9i			0.792803i	−0.918077	+	0.396401i	$0.870259\pi$
$644$		−	4.47384e9i		−	0.660055i
$645$			1.20704e9i			0.177119i
$646$	−1.03048e10			−1.50393
$647$	−4.01392e9			−0.582644			−0.291322	−	0.956625i	$-0.594095\pi$
$647$	−4.01392e9			−0.582644			−0.291322	+	0.956625i	$0.594095\pi$
$648$		−	6.61265e8i		−	0.0954691i
$649$	3.72583e9			0.535016
$650$	−6.04671e9	−	8.84398e9i	−0.863619	−	1.26314i
$651$	1.59914e9			0.227172
$652$			1.16213e10i			1.64206i
$653$	−8.30224e9			−1.16681			−0.583403	−	0.812183i	$-0.698279\pi$
$653$	−8.30224e9			−1.16681			−0.583403	+	0.812183i	$0.698279\pi$
$654$	3.90892e9			0.546431
$655$		−	9.66744e8i		−	0.134421i
$656$			6.38853e7i			0.00883562i
$657$		−	3.38300e9i		−	0.465397i
$658$			2.88072e9i			0.394194i
$659$	−9.76503e9			−1.32915			−0.664576	−	0.747220i	$-0.731388\pi$
$659$	−9.76503e9			−1.32915			−0.664576	+	0.747220i	$0.731388\pi$
$660$	−2.56694e9			−0.347546
$661$			2.64496e9i			0.356217i	0.984011	+	0.178108i	$0.0569978\pi$
$661$			2.64496e9i			0.356217i	−0.984011	+	0.178108i	$0.943002\pi$
$662$	1.15480e10			1.54704
$663$	5.00249e9	−	3.42025e9i	0.666636	−	0.455785i
$664$	1.25022e10			1.65728
$665$			6.41023e8i			0.0845274i
$666$	1.61375e10			2.11679
$667$	−3.05454e9			−0.398570
$668$		−	1.43611e10i		−	1.86410i
$669$			2.59273e7i			0.00334786i
$670$			4.57534e9i			0.587708i
$671$		−	2.47647e8i		−	0.0316449i
$672$	2.07715e9			0.264043
$673$	−5.84243e9			−0.738824			−0.369412	−	0.929266i	$-0.620441\pi$
$673$	−5.84243e9			−0.738824			−0.369412	+	0.929266i	$0.620441\pi$
$674$			8.06354e9i			1.01442i
$675$	7.26420e9			0.909127
$676$	−4.87037e9	+	1.25058e10i	−0.606386	+	1.55703i
$677$	1.28860e10			1.59609			0.798044	−	0.602600i	$-0.205868\pi$
$677$	1.28860e10			1.59609			0.798044	+	0.602600i	$0.205868\pi$
$678$		−	9.26870e9i		−	1.14213i
$679$	7.43880e9			0.911925
$680$	3.12648e9			0.381307
$681$		−	5.60070e9i		−	0.679560i
$682$			1.47106e10i			1.77576i
$683$		−	1.30209e10i		−	1.56376i	−0.623432	−	0.781878i	$-0.714262\pi$
$683$		−	1.30209e10i		−	1.56376i	0.623432	−	0.781878i	$-0.285738\pi$
$684$			6.13160e9i			0.732617i
$685$	1.56409e9			0.185928
$686$	1.21124e10			1.43250
$687$			4.70400e8i			0.0553501i
$688$	−1.23801e9			−0.144932
$689$	4.42489e9	−	3.02534e9i	0.515390	−	0.352377i
$690$	−1.64492e9			−0.190621
$691$		−	9.51655e8i		−	0.109725i	−0.998494	−	0.0548626i	$-0.982528\pi$
$691$		−	9.51655e8i		−	0.109725i	0.998494	−	0.0548626i	$-0.0174721\pi$
$692$	−1.60351e10			−1.83951
$693$	−4.05051e9			−0.462321
$694$		−	9.31035e9i		−	1.05732i
$695$		−	1.81069e9i		−	0.204597i
$696$			2.89162e9i			0.325094i
$697$			8.98455e8i			0.100504i
$698$	1.39876e10			1.55686
$699$	−4.84960e9			−0.537076
$700$			7.11594e9i			0.784132i
$701$	−1.23615e10			−1.35537			−0.677683	−	0.735354i	$-0.737016\pi$
$701$	−1.23615e10			−1.35537			−0.677683	+	0.735354i	$0.737016\pi$
$702$	−8.20963e9	−	1.20075e10i	−0.895661	−	1.31000i
$703$	−1.21445e10			−1.31837
$704$			2.06833e10i			2.23417i
$705$	6.62614e8			0.0712195
$706$	−1.06868e10			−1.14296
$707$		−	7.59639e9i		−	0.808424i
$708$		−	3.52143e9i		−	0.372909i
$709$		−	8.63273e9i		−	0.909675i	−0.890574	−	0.454838i	$-0.849697\pi$
$709$		−	8.63273e9i		−	0.909675i	0.890574	−	0.454838i	$-0.150303\pi$
$710$			3.31237e9i			0.347324i
$711$	−1.99519e9			−0.208181
$712$	−1.08209e10			−1.12352
$713$			5.89730e9i			0.609312i
$714$	−6.43390e9			−0.661501
$715$	2.86260e9	−	1.95718e9i	0.292879	−	0.200244i
$716$	−2.99871e10			−3.05309
$717$		−	3.48577e9i		−	0.353168i
$718$	2.96011e10			2.98450
$719$	1.27845e10			1.28272			0.641362	−	0.767238i	$-0.278370\pi$
$719$	1.27845e10			1.28272			0.641362	+	0.767238i	$0.278370\pi$
$720$		−	2.00951e8i		−	0.0200645i
$721$			7.29405e9i			0.724762i
$722$			9.15172e9i			0.904945i
$723$			6.14388e9i			0.604587i
$724$	7.92135e9			0.775736
$725$	4.85844e9			0.473494
$726$			9.63338e9i			0.934331i
$727$	−1.87595e10			−1.81072			−0.905359	−	0.424648i	$-0.860398\pi$
$727$	−1.87595e10			−1.81072			−0.905359	+	0.424648i	$0.860398\pi$
$728$	4.72302e9	−	3.22917e9i	0.453690	−	0.310192i
$729$	5.35682e9			0.512107
$730$		−	3.07496e9i		−	0.292557i
$731$	−1.74108e10			−1.64858
$732$	−2.34061e8			−0.0220567
$733$		−	1.15410e10i		−	1.08238i	−0.840899	−	0.541192i	$-0.817973\pi$
$733$		−	1.15410e10i		−	1.08238i	0.840899	−	0.541192i	$-0.182027\pi$
$734$		−	2.69163e10i		−	2.51234i
$735$		−	1.19291e9i		−	0.110816i
$736$			7.66007e9i			0.708207i
$737$	2.17572e10			2.00201
$738$	8.54539e8			0.0782591
$739$			6.88857e9i			0.627876i	0.949444	+	0.313938i	$0.101648\pi$
$739$			6.88857e9i			0.627876i	−0.949444	+	0.313938i	$0.898352\pi$
$740$	9.17639e9			0.832456
$741$	2.44814e9	+	3.58067e9i	0.221041	+	0.323296i
$742$	−5.69103e9			−0.511419
$743$		−	3.59487e9i		−	0.321530i	−0.986993	−	0.160765i	$-0.948604\pi$
$743$		−	3.59487e9i		−	0.321530i	0.986993	−	0.160765i	$-0.0513962\pi$
$744$	5.58275e9			0.496985
$745$	−4.24126e9			−0.375792
$746$			3.51585e9i			0.310059i
$747$		−	1.13010e10i		−	0.991958i
$748$		−	3.70265e10i		−	3.23487i
$749$			1.60661e9i			0.139709i
$750$	5.41078e9			0.468323
$751$	2.37589e8			0.0204685			0.0102343	−	0.999948i	$-0.496742\pi$
$751$	2.37589e8			0.0204685			0.0102343	+	0.999948i	$0.496742\pi$
$752$			6.79612e8i			0.0582772i
$753$	−8.48666e9			−0.724360
$754$	−5.49076e9	−	8.03085e9i	−0.466480	−	0.682279i
$755$	−3.32829e9			−0.281453
$756$			9.66133e9i			0.813225i
$757$	−6.54492e8			−0.0548364			−0.0274182	−	0.999624i	$-0.508729\pi$
$757$	−6.54492e8			−0.0548364			−0.0274182	+	0.999624i	$0.508729\pi$
$758$	−6.77486e8			−0.0565012
$759$			7.82209e9i			0.649347i
$760$			2.23787e9i			0.184921i
$761$			1.63474e10i			1.34463i	0.740267	+	0.672313i	$0.234699\pi$
$761$			1.63474e10i			1.34463i	−0.740267	+	0.672313i	$0.765301\pi$
$762$			2.16113e9i			0.176945i
$763$	3.50740e9			0.285858
$764$	−1.63962e10			−1.33020
$765$		−	2.82609e9i		−	0.228229i
$766$	2.21492e10			1.78056
$767$	2.68494e9	+	3.92702e9i	0.214858	+	0.314253i
$768$	8.74070e9			0.696277
$769$			1.66222e10i			1.31810i	0.752101	+	0.659048i	$0.229041\pi$
$769$			1.66222e10i			1.31810i	−0.752101	+	0.659048i	$0.770959\pi$
$770$	−3.68170e9			−0.290623
$771$	−2.03721e9			−0.160083
$772$			1.05241e10i			0.823232i
$773$			1.02282e10i			0.796474i	0.917283	+	0.398237i	$0.130378\pi$
$773$			1.02282e10i			0.796474i	−0.917283	+	0.398237i	$0.869622\pi$
$774$			1.65598e10i			1.28370i
$775$		−	9.38005e9i		−	0.723851i
$776$	2.59695e10			1.99502
$777$	−7.58251e9			−0.579882
$778$			4.61827e9i			0.351602i
$779$	−6.43094e8			−0.0487409
$780$	−1.84981e9	−	2.70556e9i	−0.139571	−	0.204139i
$781$	1.57513e10			1.18315
$782$		−	2.37268e10i		−	1.77426i
$783$	6.59632e9			0.491061
$784$	1.22351e9			0.0906780
$785$		−	6.16249e8i		−	0.0454687i
$786$			6.94529e9i			0.510166i
$787$			8.32343e9i			0.608682i	0.952563	+	0.304341i	$0.0984363\pi$
$787$			8.32343e9i			0.608682i	−0.952563	+	0.304341i	$0.901564\pi$
$788$			3.23120e10i			2.35246i
$789$	3.90399e9			0.282969
$790$	−1.81352e9			−0.130866
$791$		−	8.31663e9i		−	0.597489i
$792$	−1.41407e10			−1.01142
$793$	2.61020e8	−	1.78462e8i	0.0185873	−	0.0127083i
$794$	−4.43944e10			−3.14743
$795$			1.30903e9i			0.0923987i
$796$	2.07879e10			1.46088
$797$	9.22896e9			0.645727			0.322863	−	0.946446i	$-0.395355\pi$
$797$	9.22896e9			0.645727			0.322863	+	0.946446i	$0.395355\pi$
$798$		−	4.60524e9i		−	0.320806i
$799$			9.55778e9i			0.662892i
$800$		−	1.21838e10i		−	0.841336i
$801$			9.78120e9i			0.672479i
$802$	−1.45774e10			−0.997860
$803$	−1.46224e10			−0.996586
$804$		−	2.05636e10i		−	1.39541i
$805$	−1.47595e9			−0.0997210
$806$	−1.55049e10	+	1.06008e10i	−1.04303	+	0.713129i
$807$	4.48082e8			0.0300124
$808$		−	2.65197e10i		−	1.76860i
$809$	7.48827e9			0.497235			0.248618	−	0.968602i	$-0.420024\pi$
$809$	7.48827e9			0.497235			0.248618	+	0.968602i	$0.420024\pi$
$810$	−5.43305e8			−0.0359208
$811$		−	9.83512e9i		−	0.647451i	−0.946151	−	0.323725i	$-0.895065\pi$
$811$		−	9.83512e9i		−	0.647451i	0.946151	−	0.323725i	$-0.104935\pi$
$812$			6.46169e9i			0.423545i
$813$			1.21547e10i			0.793285i
$814$		−	6.97517e10i		−	4.53283i
$815$	3.83396e9			0.248083
$816$	−1.51787e9			−0.0977955
$817$		−	1.24623e10i		−	0.799504i
$818$	3.24208e10			2.07103
$819$	−2.91892e9	−	4.26924e9i	−0.185664	−	0.271554i
$820$	4.85922e8			0.0307764
$821$			5.31017e9i			0.334894i	0.985881	+	0.167447i	$0.0535524\pi$
$821$			5.31017e9i			0.334894i	−0.985881	+	0.167447i	$0.946448\pi$
$822$	−1.12368e10			−0.705650
$823$	1.19280e10			0.745880			0.372940	−	0.927855i	$-0.378350\pi$
$823$	1.19280e10			0.745880			0.372940	+	0.927855i	$0.378350\pi$
$824$			2.54642e10i			1.58557i
$825$		−	1.24416e10i		−	0.771411i
$826$		−	5.05070e9i		−	0.311832i
$827$		−	9.79288e9i		−	0.602062i	−0.953614	−	0.301031i	$-0.902669\pi$
$827$		−	9.79288e9i		−	0.602062i	0.953614	−	0.301031i	$-0.0973307\pi$
$828$	−1.41180e10			−0.864303
$829$	−7.77135e9			−0.473757			−0.236878	−	0.971539i	$-0.576124\pi$
$829$	−7.77135e9			−0.473757			−0.236878	+	0.971539i	$0.576124\pi$
$830$		−	1.02720e10i		−	0.623563i
$831$	1.05758e10			0.639310
$832$	−2.18002e10	+	1.49050e10i	−1.31229	+	0.897223i
$833$	1.72069e10			1.03145
$834$			1.30084e10i			0.776502i
$835$	−4.73781e9			−0.281628
$836$	2.65027e10			1.56880
$837$		−	1.27353e10i		−	0.750707i
$838$			3.82032e10i			2.24257i
$839$			2.10774e10i			1.23211i	0.787702	+	0.616056i	$0.211270\pi$
$839$			2.10774e10i			1.23211i	−0.787702	+	0.616056i	$0.788730\pi$
$840$			1.39723e9i			0.0813373i
$841$	−1.28381e10			−0.744245
$842$	−1.54494e10			−0.891906
$843$		−	7.29801e9i		−	0.419573i
$844$	−2.94402e10			−1.68555
$845$	4.12574e9	+	1.60677e9i	0.235236	+	0.0916126i
$846$	9.09060e9			0.516174
$847$			8.64385e9i			0.488782i
$848$	−1.34261e9			−0.0756077
$849$	−1.19261e10			−0.668839
$850$			3.77391e10i			2.10778i
$851$		−	2.79627e10i		−	1.55534i
$852$		−	1.48872e10i		−	0.824661i
$853$		−	2.23122e10i		−	1.23089i	−0.788178	−	0.615447i	$-0.788975\pi$
$853$		−	2.23122e10i		−	1.23089i	0.788178	−	0.615447i	$-0.211025\pi$
$854$	−3.35708e8			−0.0184442
$855$	2.02286e9			0.110684
$856$			5.60883e9i			0.305643i
$857$	1.81712e10			0.986166			0.493083	−	0.869982i	$-0.335870\pi$
$857$	1.81712e10			0.986166			0.493083	+	0.869982i	$0.335870\pi$
$858$	−2.05655e10	+	1.40608e10i	−1.11156	+	0.759985i
$859$	−6.97998e9			−0.375732			−0.187866	−	0.982195i	$-0.560157\pi$
$859$	−6.97998e9			−0.375732			−0.187866	+	0.982195i	$0.560157\pi$
$860$			9.41651e9i			0.504830i
$861$	−4.01520e8			−0.0214386
$862$	1.52530e10			0.811109
$863$			6.83713e9i			0.362106i	0.983473	+	0.181053i	$0.0579506\pi$
$863$			6.83713e9i			0.362106i	−0.983473	+	0.181053i	$0.942049\pi$
$864$		−	1.65420e10i		−	0.872551i
$865$			5.29010e9i			0.277912i
$866$			4.61729e9i			0.241587i
$867$	−1.00968e10			−0.526159
$868$	1.24754e10			0.647493
$869$			8.62386e9i			0.445792i
$870$	2.37580e9			0.122318
$871$	1.56788e10	+	2.29320e10i	0.803990	+	1.17592i
$872$	1.22447e10			0.625373
$873$		−	2.34744e10i		−	1.19411i
$874$	1.69831e10			0.860455
$875$	4.85499e9			0.244997
$876$			1.38202e10i			0.694626i
$877$			2.21947e9i			0.111109i	0.998456	+	0.0555547i	$0.0176927\pi$
$877$			2.21947e9i			0.111109i	−0.998456	+	0.0555547i	$0.982307\pi$
$878$			3.63184e10i			1.81090i
$879$		−	2.33717e9i		−	0.116072i
$880$	−8.68577e8			−0.0429654
$881$	1.17122e10			0.577063			0.288531	−	0.957470i	$-0.406833\pi$
$881$	1.17122e10			0.577063			0.288531	+	0.957470i	$0.406833\pi$
$882$		−	1.63659e10i		−	0.803155i
$883$	−2.78376e10			−1.36072			−0.680362	−	0.732876i	$-0.738177\pi$
$883$	−2.78376e10			−1.36072			−0.680362	+	0.732876i	$0.738177\pi$
$884$	3.90259e10	−	2.66823e10i	1.90007	−	1.29910i
$885$	−1.16175e9			−0.0563391
$886$		−	2.36978e10i		−	1.14470i
$887$	1.58706e10			0.763592			0.381796	−	0.924247i	$-0.375306\pi$
$887$	1.58706e10			0.763592			0.381796	+	0.924247i	$0.375306\pi$
$888$	−2.64712e10			−1.26861
$889$			1.93914e9i			0.0925663i
$890$			8.89058e9i			0.422732i
$891$			2.58359e9i			0.122363i
$892$			2.02267e8i			0.00954218i
$893$	−6.84125e9			−0.321481
$894$	3.04701e10			1.42624
$895$			9.89297e9i			0.461260i
$896$	1.83403e10			0.851783
$897$	−8.24448e9	+	5.63682e9i	−0.381408	+	0.260772i
$898$	1.43903e10			0.663134
$899$		−	8.51763e9i		−	0.390985i
$900$	2.24556e10			1.02678
$901$	−1.88820e10			−0.860023
$902$		−	3.69359e9i		−	0.167582i
$903$		−	7.78092e9i		−	0.351661i
$904$		−	2.90341e10i		−	1.30713i
$905$		−	2.61331e9i		−	0.117198i
$906$	2.39111e10			1.06820
$907$	−6.81086e9			−0.303094			−0.151547	−	0.988450i	$-0.548425\pi$
$907$	−6.81086e9			−0.303094			−0.151547	+	0.988450i	$0.548425\pi$
$908$		−	4.36927e10i		−	1.93691i
$909$	−2.39717e10			−1.05858
$910$	−2.65314e9	−	3.88050e9i	−0.116712	−	0.170704i
$911$	−6.32068e9			−0.276981			−0.138490	−	0.990364i	$-0.544225\pi$
$911$	−6.32068e9			−0.276981			−0.138490	+	0.990364i	$0.544225\pi$
$912$		−	1.08646e9i		−	0.0474276i
$913$	−4.88464e10			−2.12415
$914$	−5.03386e10			−2.18067
$915$			7.72185e7i			0.00333232i
$916$			3.66972e9i			0.157761i
$917$			6.23188e9i			0.266886i
$918$			5.12385e10i			2.18598i
$919$	2.78393e10			1.18319			0.591595	−	0.806236i	$-0.298499\pi$
$919$	2.78393e10			1.18319			0.591595	+	0.806236i	$0.298499\pi$
$920$	−5.15268e9			−0.218160
$921$		−	1.71299e10i		−	0.722515i
$922$	−2.11102e10			−0.887020
$923$	1.13509e10	+	1.66019e10i	0.475142	+	0.694947i
$924$	1.65472e10			0.690036
$925$			4.44765e10i			1.84771i
$926$	−4.34209e10			−1.79705
$927$	2.30176e10			0.949033
$928$		−	1.10636e10i		−	0.454444i
$929$		−	1.20484e10i		−	0.493030i	−0.969139	−	0.246515i	$-0.920714\pi$
$929$		−	1.20484e10i		−	0.493030i	0.969139	−	0.246515i	$-0.0792855\pi$
$930$		−	4.58688e9i		−	0.186994i
$931$			1.23164e10i			0.500217i
$932$	−3.78332e10			−1.53080
$933$	−2.06511e9			−0.0832450
$934$			4.95736e10i			1.99084i
$935$	−1.22153e10			−0.488724
$936$	−1.01902e10	−	1.49043e10i	−0.406179	−	0.594081i
$937$	−8.29223e9			−0.329293			−0.164647	−	0.986353i	$-0.552648\pi$
$937$	−8.29223e9			−0.329293			−0.164647	+	0.986353i	$0.552648\pi$
$938$		−	2.94938e10i		−	1.16687i
$939$	1.41558e10			0.557964
$940$	5.16925e9			0.202992
$941$		−	3.30723e10i		−	1.29390i	−0.762532	−	0.646950i	$-0.776044\pi$
$941$		−	3.30723e10i		−	1.29390i	0.762532	−	0.646950i	$-0.223956\pi$
$942$			4.42726e9i			0.172567i
$943$		−	1.48072e9i		−	0.0575019i
$944$		−	1.19155e9i		−	0.0461009i
$945$	3.18734e9			0.122862
$946$	7.15768e10			2.74887
$947$			1.65097e10i			0.631706i	0.948808	+	0.315853i	$0.102291\pi$
$947$			1.65097e10i			0.631706i	−0.948808	+	0.315853i	$0.897709\pi$
$948$	8.15076e9			0.310720
$949$	−1.05373e10	−	1.54120e10i	−0.400220	−	0.585366i
$950$	−2.70128e10			−1.02220
$951$			1.04590e10i			0.394328i
$952$	−2.01541e10			−0.757067
$953$	−3.77080e10			−1.41126			−0.705632	−	0.708579i	$-0.749337\pi$
$953$	−3.77080e10			−1.41126			−0.705632	+	0.708579i	$0.749337\pi$
$954$			1.79590e10i			0.669674i
$955$			5.40923e9i			0.200967i
$956$		−	2.71935e10i		−	1.00661i
$957$		−	1.12977e10i		−	0.416674i
$958$	−2.76625e10			−1.01651
$959$	−1.00825e10			−0.369151
$960$		−	6.44924e9i		−	0.235266i
$961$	1.10679e10			0.402284
$962$	7.35182e10	−	5.02651e10i	2.66245	−	1.82034i
$963$	5.06994e9			0.182941
$964$			4.79302e10i			1.72322i
$965$	3.47196e9			0.124374
$966$	1.06035e10			0.378470
$967$		−	4.52479e10i		−	1.60919i	−0.593827	−	0.804593i	$-0.702384\pi$
$967$		−	4.52479e10i		−	1.60919i	0.593827	−	0.804593i	$-0.297616\pi$
$968$			3.01765e10i			1.06931i
$969$		−	1.52795e10i		−	0.539480i
$970$		−	2.13370e10i		−	0.750640i
$971$	−1.62549e10			−0.569795			−0.284897	−	0.958558i	$-0.591959\pi$
$971$	−1.62549e10			−0.569795			−0.284897	+	0.958558i	$0.591959\pi$
$972$	4.88951e10			1.70779
$973$			1.16722e10i			0.406216i
$974$	1.51339e10			0.524800
$975$	1.31134e10	−	8.96574e9i	0.453105	−	0.309792i
$976$	−7.91994e7			−0.00272676
$977$		−	3.76446e9i		−	0.129143i	−0.997913	−	0.0645717i	$-0.979432\pi$
$977$		−	3.76446e9i		−	0.129143i	0.997913	−	0.0645717i	$-0.0205681\pi$
$978$	−2.75440e10			−0.941545
$979$	4.22775e10			1.44002
$980$		−	9.30624e9i		−	0.315851i
$981$		−	1.10682e10i		−	0.374314i
$982$			1.27484e10i			0.429601i
$983$			2.70334e10i			0.907743i	0.891067	+	0.453871i	$0.149957\pi$
$983$			2.70334e10i			0.907743i	−0.891067	+	0.453871i	$0.850043\pi$
$984$	−1.40174e9			−0.0469014
$985$	1.06599e10			0.355409
$986$			3.42693e10i			1.13851i
$987$	−4.27138e9			−0.141403
$988$	1.90986e10	+	2.79339e10i	0.630018	+	0.921471i
$989$	2.86944e10			0.943213
$990$			1.16182e10i			0.380554i
$991$	3.34211e10			1.09085			0.545423	−	0.838161i	$-0.316369\pi$
$991$	3.34211e10			1.09085			0.545423	+	0.838161i	$0.316369\pi$
$992$	−2.13602e10			−0.694729
$993$			1.71227e10i			0.554946i
$994$		−	2.13524e10i		−	0.689594i
$995$		−	6.85807e9i		−	0.220710i
$996$			4.61667e10i			1.48054i
$997$	−2.13341e9			−0.0681775			−0.0340888	−	0.999419i	$-0.510853\pi$
$997$	−2.13341e9			−0.0681775			−0.0340888	+	0.999419i	$0.510853\pi$
$998$	−4.03615e10			−1.28532
$999$			6.03858e10i			1.91627i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.8.b.a.12.6	yes	6
3.2	odd	2		117.8.b.b.64.1		6
4.3	odd	2		208.8.f.a.129.4		6
13.5	odd	4		169.8.a.d.1.6		6
13.8	odd	4		169.8.a.d.1.1		6
13.12	even	2	inner	13.8.b.a.12.1	✓	6
39.38	odd	2		117.8.b.b.64.6		6
52.51	odd	2		208.8.f.a.129.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.b.a.12.1	✓	6	13.12	even	2	inner
13.8.b.a.12.6	yes	6	1.1	even	1	trivial
117.8.b.b.64.1		6	3.2	odd	2
117.8.b.b.64.6		6	39.38	odd	2
169.8.a.d.1.1		6	13.8	odd	4
169.8.a.d.1.6		6	13.5	odd	4
208.8.f.a.129.3		6	52.51	odd	2
208.8.f.a.129.4		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+18.4900i q^{2} -27.4160 q^{3} -213.880 q^{4} +70.5606i q^{5} -506.922i q^{6} -454.852i q^{7} -1587.93i q^{8} -1435.36 q^{9} -1304.67 q^{10} +6204.10i q^{11} +5863.75 q^{12} +(-6539.11 + 4470.85i) q^{13} +8410.21 q^{14} -1934.49i q^{15} +1984.12 q^{16} +27903.8 q^{17} -26539.9i q^{18} +19972.9i q^{19} -15091.5i q^{20} +12470.2i q^{21} -114714. q^{22} -45987.5 q^{23} +43534.7i q^{24} +73146.2 q^{25} +(-82666.1 - 120908. i) q^{26} +99310.7 q^{27} +97283.8i q^{28} +66421.0 q^{29} +35768.8 q^{30} -128237. i q^{31} -166568. i q^{32} -170092. i q^{33} +515941. i q^{34} +32094.6 q^{35} +306996. q^{36} +608049. i q^{37} -369299. q^{38} +(179276. - 122573. i) q^{39} +112045. q^{40} +32198.3i q^{41} -230574. q^{42} -623960. q^{43} -1.32693e6i q^{44} -101280. i q^{45} -850309. i q^{46} +342526. i q^{47} -54396.6 q^{48} +616653. q^{49} +1.35247e6i q^{50} -765010. q^{51} +(1.39859e6 - 956227. i) q^{52} -676681. q^{53} +1.83626e6i q^{54} -437765. q^{55} -722272. q^{56} -547577. i q^{57} +1.22812e6i q^{58} -600543. i q^{59} +413750. i q^{60} -39916.7 q^{61} +2.37110e6 q^{62} +652877. i q^{63} +3.33382e6 q^{64} +(-315466. - 461404. i) q^{65} +3.14500e6 q^{66} -3.50690e6i q^{67} -5.96807e6 q^{68} +1.26079e6 q^{69} +593430. i q^{70} -2.53886e6i q^{71} +2.27925e6i q^{72} +2.35690e6i q^{73} -1.12428e7 q^{74} -2.00538e6 q^{75} -4.27181e6i q^{76} +2.82194e6 q^{77} +(2.26637e6 + 3.31482e6i) q^{78} +1.39003e6 q^{79} +140001. i q^{80} +416432. q^{81} -595347. q^{82} +7.87325e6i q^{83} -2.66714e6i q^{84} +1.96891e6i q^{85} -1.15370e7i q^{86} -1.82100e6 q^{87} +9.85166e6 q^{88} -6.81445e6i q^{89} +1.87267e6 q^{90} +(2.03357e6 + 2.97433e6i) q^{91} +9.83582e6 q^{92} +3.51575e6i q^{93} -6.33331e6 q^{94} -1.40930e6 q^{95} +4.56664e6i q^{96} +1.63544e7i q^{97} +1.14019e7i q^{98} -8.90513e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9} - 406 q^{10} + 1898 q^{12} - 5018 q^{13} + 9558 q^{14} + 7778 q^{16} + 13152 q^{17} - 125080 q^{22} + 27264 q^{23} - 18262 q^{25} - 54210 q^{26} + 194560 q^{27} + 42924 q^{29}+ \cdots - 22075632 q^{95}+O(q^{100}) \) 6 * q - 56 * q^3 - 130 * q^4 - 1150 * q^9 - 406 * q^10 + 1898 * q^12 - 5018 * q^13 + 9558 * q^14 + 7778 * q^16 + 13152 * q^17 - 125080 * q^22 + 27264 * q^23 - 18262 * q^25 - 54210 * q^26 + 194560 * q^27 + 42924 * q^29 - 60114 * q^30 - 546720 * q^35 + 747772 * q^36 - 243492 * q^38 + 511160 * q^39 + 792058 * q^40 - 1138134 * q^42 - 1005576 * q^43 - 2807426 * q^48 + 3246846 * q^49 + 297984 * q^51 + 3252964 * q^52 + 1705524 * q^53 - 5320576 * q^55 - 2282730 * q^56 - 8237572 * q^61 + 9625800 * q^62 + 5070862 * q^64 + 7139028 * q^65 + 9289056 * q^66 - 16801086 * q^68 - 6017400 * q^69 - 28760826 * q^74 + 20426072 * q^75 + 4093560 * q^77 + 14088594 * q^78 + 15147200 * q^79 - 21318442 * q^81 - 9130240 * q^82 - 36474240 * q^87 + 38008000 * q^88 + 4769676 * q^90 + 21281832 * q^91 + 36462348 * q^92 - 34229770 * q^94 - 22075632 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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