LMFDB - Embedded newform 13.8.b.a.12.3

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			12.3
Root			$-2.43912i$ of defining polynomial
Character	$\chi$	$=$	13.12
Dual form			13.8.b.a.12.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.43912i q^{2} -51.4496 q^{3} +122.051 q^{4} +488.312i q^{5} +125.492i q^{6} +616.243i q^{7} -609.904i q^{8} +460.056 q^{9} +1191.05 q^{10} +4625.43i q^{11} -6279.45 q^{12} +(1257.79 - 7820.90i) q^{13} +1503.09 q^{14} -25123.4i q^{15} +14134.9 q^{16} -19536.2 q^{17} -1122.13i q^{18} +20077.3i q^{19} +59598.8i q^{20} -31705.4i q^{21} +11282.0 q^{22} +71368.5 q^{23} +31379.3i q^{24} -160323. q^{25} +(-19076.1 - 3067.89i) q^{26} +88850.5 q^{27} +75212.8i q^{28} +138095. q^{29} -61279.1 q^{30} -18285.7i q^{31} -112544. i q^{32} -237976. i q^{33} +47651.3i q^{34} -300918. q^{35} +56150.2 q^{36} -446035. i q^{37} +48970.9 q^{38} +(-64712.5 + 402382. i) q^{39} +297823. q^{40} +101639. i q^{41} -77333.4 q^{42} +103690. q^{43} +564537. i q^{44} +224651. i q^{45} -174076. i q^{46} +445129. i q^{47} -727232. q^{48} +443788. q^{49} +391048. i q^{50} +1.00513e6 q^{51} +(153514. - 954546. i) q^{52} +203578. q^{53} -216717. i q^{54} -2.25865e6 q^{55} +375849. q^{56} -1.03297e6i q^{57} -336832. i q^{58} +2.50176e6i q^{59} -3.06633e6i q^{60} -792320. q^{61} -44601.1 q^{62} +283506. i q^{63} +1.53475e6 q^{64} +(3.81904e6 + 614191. i) q^{65} -580454. q^{66} -3.17309e6i q^{67} -2.38441e6 q^{68} -3.67188e6 q^{69} +733977. i q^{70} +1.85875e6i q^{71} -280590. i q^{72} +4.25734e6i q^{73} -1.08793e6 q^{74} +8.24856e6 q^{75} +2.45045e6i q^{76} -2.85039e6 q^{77} +(981459. + 157842. i) q^{78} +2.38970e6 q^{79} +6.90221e6i q^{80} -5.57746e6 q^{81} +247910. q^{82} -1.12536e6i q^{83} -3.86967e6i q^{84} -9.53976e6i q^{85} -252913. i q^{86} -7.10495e6 q^{87} +2.82107e6 q^{88} -8.00227e6i q^{89} +547951. q^{90} +(4.81957e6 + 775101. i) q^{91} +8.71057e6 q^{92} +940791. i q^{93} +1.08572e6 q^{94} -9.80397e6 q^{95} +5.79036e6i q^{96} -276982. i q^{97} -1.08245e6i q^{98} +2.12796e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.43912i		−	0.215590i	−0.994173	−	0.107795i	$-0.965621\pi$
$2$		−	2.43912i		−	0.215590i	0.994173	−	0.107795i	$-0.0343790\pi$
$3$	−51.4496			−1.10016			−0.550082	−	0.835111i	$-0.685403\pi$
$3$	−51.4496			−1.10016			−0.550082	+	0.835111i	$0.685403\pi$
$4$	122.051			0.953521
$5$			488.312i			1.74704i	0.486791	+	0.873518i	$0.338167\pi$
$5$			488.312i			1.74704i	−0.486791	+	0.873518i	$0.661833\pi$
$6$			125.492i			0.237184i
$7$			616.243i			0.679061i	0.940595	+	0.339530i	$0.110268\pi$
$7$			616.243i			0.679061i	−0.940595	+	0.339530i	$0.889732\pi$
$8$		−	609.904i		−	0.421160i
$9$	460.056			0.210360
$10$	1191.05			0.376644
$11$			4625.43i			1.04780i	0.851780	+	0.523899i	$0.175523\pi$
$11$			4625.43i			1.04780i	−0.851780	+	0.523899i	$0.824477\pi$
$12$	−6279.45			−1.04903
$13$	1257.79	−	7820.90i	0.158783	−	0.987313i
$13$	1257.79	−	7820.90i	0.158783	−	0.987313i
$14$	1503.09			0.146399
$15$		−	25123.4i		−	1.92203i
$16$	14134.9			0.862723
$17$	−19536.2			−0.964427			−0.482214	−	0.876054i	$-0.660167\pi$
$17$	−19536.2			−0.964427			−0.482214	+	0.876054i	$0.660167\pi$
$18$		−	1122.13i		−	0.0453514i
$19$			20077.3i			0.671533i	0.941945	+	0.335766i	$0.108995\pi$
$19$			20077.3i			0.671533i	−0.941945	+	0.335766i	$0.891005\pi$
$20$			59598.8i			1.66584i
$21$		−	31705.4i		−	0.747078i
$22$	11282.0			0.225895
$23$	71368.5			1.22309			0.611546	−	0.791209i	$-0.290548\pi$
$23$	71368.5			1.22309			0.611546	+	0.791209i	$0.290548\pi$
$24$			31379.3i			0.463344i
$25$	−160323.			−2.05214
$26$	−19076.1	−	3067.89i	−0.212855	−	0.0342321i
$27$	88850.5			0.868734
$28$			75212.8i			0.647498i
$29$	138095.			1.05144			0.525722	−	0.850656i	$-0.323795\pi$
$29$	138095.			1.05144			0.525722	+	0.850656i	$0.323795\pi$
$30$	−61279.1			−0.414370
$31$		−	18285.7i		−	0.110242i	−0.998480	−	0.0551208i	$-0.982446\pi$
$31$		−	18285.7i		−	0.110242i	0.998480	−	0.0551208i	$-0.0175544\pi$
$32$		−	112544.i		−	0.607154i
$33$		−	237976.i		−	1.15275i
$34$			47651.3i			0.207921i
$35$	−300918.			−1.18634
$36$	56150.2			0.200582
$37$		−	446035.i		−	1.44765i	−0.689985	−	0.723824i	$-0.742383\pi$
$37$		−	446035.i		−	1.44765i	0.689985	−	0.723824i	$-0.257617\pi$
$38$	48970.9			0.144776
$39$	−64712.5	+	402382.i	−0.174688	+	1.08621i
$40$	297823.			0.735781
$41$			101639.i			0.230312i	0.993347	+	0.115156i	$0.0367368\pi$
$41$			101639.i			0.230312i	−0.993347	+	0.115156i	$0.963263\pi$
$42$	−77333.4			−0.161062
$43$	103690.			0.198883			0.0994416	−	0.995043i	$-0.468294\pi$
$43$	103690.			0.198883			0.0994416	+	0.995043i	$0.468294\pi$
$44$			564537.i			0.999098i
$45$			224651.i			0.367506i
$46$		−	174076.i		−	0.263686i
$47$			445129.i			0.625379i	0.949855	+	0.312690i	$0.101230\pi$
$47$			445129.i			0.625379i	−0.949855	+	0.312690i	$0.898770\pi$
$48$	−727232.			−0.949136
$49$	443788.			0.538877
$50$			391048.i			0.442420i
$51$	1.00513e6			1.06103
$52$	153514.	−	954546.i	0.151403	−	0.941424i
$53$	203578.			0.187830			0.0939149	−	0.995580i	$-0.470062\pi$
$53$	203578.			0.187830			0.0939149	+	0.995580i	$0.470062\pi$
$54$		−	216717.i		−	0.187290i
$55$	−2.25865e6			−1.83054
$56$	375849.			0.285993
$57$		−	1.03297e6i		−	0.738796i
$58$		−	336832.i		−	0.226681i
$59$			2.50176e6i			1.58585i	0.609317	+	0.792927i	$0.291444\pi$
$59$			2.50176e6i			1.58585i	−0.609317	+	0.792927i	$0.708556\pi$
$60$		−	3.06633e6i		−	1.83269i
$61$	−792320.			−0.446937			−0.223468	−	0.974711i	$-0.571738\pi$
$61$	−792320.			−0.446937			−0.223468	+	0.974711i	$0.571738\pi$
$62$	−44601.1			−0.0237670
$63$			283506.i			0.142847i
$64$	1.53475e6			0.731827
$65$	3.81904e6	+	614191.i	1.72487	+	0.277400i
$66$	−580454.			−0.248521
$67$		−	3.17309e6i		−	1.28890i	−0.764645	−	0.644452i	$-0.777085\pi$
$67$		−	3.17309e6i		−	1.28890i	0.764645	−	0.644452i	$-0.222915\pi$
$68$	−2.38441e6			−0.919601
$69$	−3.67188e6			−1.34560
$70$			733977.i			0.255764i
$71$			1.85875e6i			0.616333i	0.951332	+	0.308167i	$0.0997154\pi$
$71$			1.85875e6i			0.616333i	−0.951332	+	0.308167i	$0.900285\pi$
$72$		−	280590.i		−	0.0885950i
$73$			4.25734e6i			1.28088i	0.768008	+	0.640440i	$0.221248\pi$
$73$			4.25734e6i			1.28088i	−0.768008	+	0.640440i	$0.778752\pi$
$74$	−1.08793e6			−0.312098
$75$	8.24856e6			2.25769
$76$			2.45045e6i			0.640320i
$77$	−2.85039e6			−0.711519
$78$	981459.	+	157842.i	0.234175	+	0.0376609i
$79$	2.38970e6			0.545317			0.272658	−	0.962111i	$-0.412097\pi$
$79$	2.38970e6			0.545317			0.272658	+	0.962111i	$0.412097\pi$
$80$			6.90221e6i			1.50721i
$81$	−5.57746e6			−1.16611
$82$	247910.			0.0496530
$83$		−	1.12536e6i		−	0.216032i	−0.994149	−	0.108016i	$-0.965550\pi$
$83$		−	1.12536e6i		−	0.216032i	0.994149	−	0.108016i	$-0.0344498\pi$
$84$		−	3.86967e6i		−	0.712354i
$85$		−	9.53976e6i		−	1.68489i
$86$		−	252913.i		−	0.0428773i
$87$	−7.10495e6			−1.15676
$88$	2.82107e6			0.441291
$89$		−	8.00227e6i		−	1.20323i	−0.798787	−	0.601614i	$-0.794524\pi$
$89$		−	8.00227e6i		−	1.20323i	0.798787	−	0.601614i	$-0.205476\pi$
$90$	547951.			0.0792306
$91$	4.81957e6	+	775101.i	0.670446	+	0.107823i
$92$	8.71057e6			1.16624
$93$			940791.i			0.121284i
$94$	1.08572e6			0.134826
$95$	−9.80397e6			−1.17319
$96$			5.79036e6i			0.667969i
$97$		−	276982.i		−	0.0308141i	−0.999881	−	0.0154071i	$-0.995096\pi$
$97$		−	276982.i		−	0.0308141i	0.999881	−	0.0154071i	$-0.00490442\pi$
$98$		−	1.08245e6i		−	0.116176i
$99$			2.12796e6i			0.220414i
$100$	−1.95676e7			−1.95676
$101$	1.32796e7			1.28251			0.641256	−	0.767327i	$-0.278414\pi$
$101$	1.32796e7			1.28251			0.641256	+	0.767327i	$0.278414\pi$
$102$		−	2.45164e6i		−	0.228747i
$103$	−7.07119e6			−0.637621			−0.318810	−	0.947819i	$-0.603283\pi$
$103$	−7.07119e6			−0.637621			−0.318810	+	0.947819i	$0.603283\pi$
$104$	−4.77000e6	−	767129.i	−0.415817	−	0.0668731i
$105$	1.54821e7			1.30517
$106$		−	496551.i		−	0.0404942i
$107$	2.81264e6			0.221958			0.110979	−	0.993823i	$-0.464601\pi$
$107$	2.81264e6			0.221958			0.110979	+	0.993823i	$0.464601\pi$
$108$	1.08443e7			0.828356
$109$		−	1.77688e7i		−	1.31421i	−0.753798	−	0.657107i	$-0.771780\pi$
$109$		−	1.77688e7i		−	1.31421i	0.753798	−	0.657107i	$-0.228220\pi$
$110$			5.50913e6i			0.394647i
$111$			2.29483e7i			1.59265i
$112$			8.71050e6i			0.585841i
$113$	−1.79731e7			−1.17179			−0.585894	−	0.810388i	$-0.699257\pi$
$113$	−1.79731e7			−1.17179			−0.585894	+	0.810388i	$0.699257\pi$
$114$	−2.51953e6			−0.159277
$115$			3.48501e7i			2.13679i
$116$	1.68546e7			1.00257
$117$	578652.	−	3.59805e6i	0.0334016	−	0.207691i
$118$	6.10209e6			0.341894
$119$		−	1.20391e7i		−	0.654904i
$120$	−1.53229e7			−0.809480
$121$	−1.90744e6			−0.0978819
$122$			1.93256e6i			0.0963551i
$123$		−	5.22928e6i		−	0.253381i
$124$		−	2.23178e6i		−	0.105118i
$125$		−	4.01383e7i		−	1.83812i
$126$	691507.			0.0307964
$127$	2.01354e7			0.872263			0.436131	−	0.899883i	$-0.356348\pi$
$127$	2.01354e7			0.872263			0.436131	+	0.899883i	$0.356348\pi$
$128$		−	1.81491e7i		−	0.764929i
$129$	−5.33482e6			−0.218804
$130$	1.49809e6	−	9.31510e6i	0.0598047	−	0.371865i
$131$	−2.41394e7			−0.938159			−0.469079	−	0.883156i	$-0.655414\pi$
$131$	−2.41394e7			−0.938159			−0.469079	+	0.883156i	$0.655414\pi$
$132$		−	2.90452e7i		−	1.09917i
$133$	−1.23725e7			−0.456011
$134$	−7.73956e6			−0.277875
$135$			4.33867e7i			1.51771i
$136$			1.19152e7i			0.406178i
$137$		−	8.35854e6i		−	0.277721i	−0.990312	−	0.138860i	$-0.955656\pi$
$137$		−	8.35854e6i		−	0.277721i	0.990312	−	0.138860i	$-0.0443439\pi$
$138$			8.95616e6i			0.290098i
$139$	4.59389e7			1.45087			0.725435	−	0.688290i	$-0.241639\pi$
$139$	4.59389e7			1.45087			0.725435	+	0.688290i	$0.241639\pi$
$140$	−3.67273e7			−1.13120
$141$		−	2.29017e7i		−	0.688020i
$142$	4.53371e6			0.132875
$143$	3.61750e7	+	5.81780e6i	1.03451	+	0.166373i
$144$	6.50283e6			0.181482
$145$			6.74336e7i			1.83691i
$146$	1.03842e7			0.276145
$147$	−2.28327e7			−0.592852
$148$		−	5.44389e7i		−	1.38036i
$149$		−	7.52366e6i		−	0.186328i	−0.995651	−	0.0931638i	$-0.970302\pi$
$149$		−	7.52366e6i		−	0.186328i	0.995651	−	0.0931638i	$-0.0296980\pi$
$150$		−	2.01192e7i		−	0.486735i
$151$			5.79965e7i			1.37083i	0.728155	+	0.685413i	$0.240378\pi$
$151$			5.79965e7i			1.37083i	−0.728155	+	0.685413i	$0.759622\pi$
$152$	1.22452e7			0.282822
$153$	−8.98776e6			−0.202876
$154$			6.95244e6i			0.153396i
$155$	8.92912e6			0.192596
$156$	−7.89820e6	+	4.91110e7i	−0.166568	+	1.03572i
$157$	−8.29958e7			−1.71162			−0.855810	−	0.517291i	$-0.826941\pi$
$157$	−8.29958e7			−1.71162			−0.855810	+	0.517291i	$0.826941\pi$
$158$		−	5.82878e6i		−	0.117565i
$159$	−1.04740e7			−0.206643
$160$	5.49567e7			1.06072
$161$			4.39803e7i			0.830553i
$162$			1.36041e7i			0.251401i
$163$		−	3.36561e7i		−	0.608705i	−0.952560	−	0.304352i	$-0.901560\pi$
$163$		−	3.36561e7i		−	0.608705i	0.952560	−	0.304352i	$-0.0984400\pi$
$164$			1.24051e7i			0.219608i
$165$	1.16207e8			2.01390
$166$	−2.74489e6			−0.0465743
$167$		−	7.76344e7i		−	1.28987i	−0.764237	−	0.644936i	$-0.776884\pi$
$167$		−	7.76344e7i		−	1.28987i	0.764237	−	0.644936i	$-0.223116\pi$
$168$	−1.93373e7			−0.314639
$169$	−5.95845e7	−	1.96740e7i	−0.949576	−	0.313538i
$170$	−2.32687e7			−0.363245
$171$			9.23668e6i			0.141263i
$172$	1.26555e7			0.189639
$173$	−668606.			−0.00981768			−0.00490884	−	0.999988i	$-0.501563\pi$
$173$	−668606.			−0.00981768			−0.00490884	+	0.999988i	$0.501563\pi$
$174$			1.73298e7i			0.249386i
$175$		−	9.87980e7i		−	1.39353i
$176$			6.53798e7i			0.903960i
$177$		−	1.28714e8i		−	1.74470i
$178$	−1.95185e7			−0.259404
$179$	−1.05607e8			−1.37628			−0.688142	−	0.725576i	$-0.741573\pi$
$179$	−1.05607e8			−1.37628			−0.688142	+	0.725576i	$0.741573\pi$
$180$			2.74188e7i			0.350425i
$181$	−1.42306e7			−0.178381			−0.0891904	−	0.996015i	$-0.528428\pi$
$181$	−1.42306e7			−0.178381			−0.0891904	+	0.996015i	$0.528428\pi$
$182$	1.89057e6	−	1.17555e7i	0.0232457	−	0.144541i
$183$	4.07645e7			0.491703
$184$		−	4.35279e7i		−	0.515117i
$185$	2.17804e8			2.52909
$186$	2.29470e6			0.0261476
$187$		−	9.03635e7i		−	1.01053i
$188$			5.43283e7i			0.596312i
$189$			5.47534e7i			0.589923i
$190$			2.39131e7i			0.252929i
$191$	−6.06460e7			−0.629775			−0.314887	−	0.949129i	$-0.601967\pi$
$191$	−6.06460e7			−0.629775			−0.314887	+	0.949129i	$0.601967\pi$
$192$	−7.89623e7			−0.805129
$193$			1.96802e8i			1.97051i	0.171102	+	0.985253i	$0.445267\pi$
$193$			1.96802e8i			1.97051i	−0.171102	+	0.985253i	$0.554733\pi$
$194$	−675593.			−0.00664322
$195$	−1.96488e8	−	3.15999e7i	−1.89764	−	0.305186i
$196$	5.41646e7			0.513830
$197$		−	7.08567e7i		−	0.660312i	−0.943926	−	0.330156i	$-0.892899\pi$
$197$		−	7.08567e7i		−	0.660312i	0.943926	−	0.330156i	$-0.107101\pi$
$198$	5.19035e6			0.0475192
$199$	1.26370e8			1.13674			0.568368	−	0.822774i	$-0.307575\pi$
$199$	1.26370e8			1.13674			0.568368	+	0.822774i	$0.307575\pi$
$200$			9.77818e7i			0.864277i
$201$			1.63254e8i			1.41801i
$202$		−	3.23907e7i		−	0.276497i
$203$			8.51003e7i			0.713995i
$204$	1.22677e8			1.01171
$205$	−4.96315e7			−0.402364
$206$			1.72475e7i			0.137465i
$207$	3.28335e7			0.257289
$208$	1.77786e7	−	1.10547e8i	0.136986	−	0.851778i
$209$	−9.28660e7			−0.703631
$210$		−	3.77628e7i		−	0.281382i
$211$	1.08405e8			0.794440			0.397220	−	0.917723i	$-0.369975\pi$
$211$	1.08405e8			0.794440			0.397220	+	0.917723i	$0.369975\pi$
$212$	2.48468e7			0.179100
$213$		−	9.56316e7i		−	0.678067i
$214$		−	6.86038e6i		−	0.0478520i
$215$			5.06332e7i			0.347456i
$216$		−	5.41903e7i		−	0.365876i
$217$	1.12684e7			0.0748607
$218$	−4.33403e7			−0.283331
$219$		−	2.19038e8i		−	1.40918i
$220$	−2.75670e8			−1.74546
$221$	−2.45724e7	+	1.52791e8i	−0.153135	+	0.952192i
$222$	5.59737e7			0.343359
$223$			4.34061e7i			0.262110i	0.991375	+	0.131055i	$0.0418364\pi$
$223$			4.34061e7i			0.262110i	−0.991375	+	0.131055i	$0.958164\pi$
$224$	6.93546e7			0.412294
$225$	−7.37577e7			−0.431687
$226$			4.38387e7i			0.252626i
$227$		−	1.30814e8i		−	0.742272i	−0.928579	−	0.371136i	$-0.878968\pi$
$227$		−	1.30814e8i		−	0.742272i	0.928579	−	0.371136i	$-0.121032\pi$
$228$		−	1.26074e8i		−	0.704457i
$229$			1.82131e7i			0.100221i	0.998744	+	0.0501105i	$0.0159573\pi$
$229$			1.82131e7i			0.100221i	−0.998744	+	0.0501105i	$0.984043\pi$
$230$	8.50036e7			0.460670
$231$	1.46651e8			0.782787
$232$		−	8.42250e7i		−	0.442826i
$233$	1.76958e8			0.916482			0.458241	−	0.888828i	$-0.348480\pi$
$233$	1.76958e8			0.916482			0.458241	+	0.888828i	$0.348480\pi$
$234$	−8.77610e6	−	1.41140e6i	−0.0447761	−	0.00720105i
$235$	−2.17362e8			−1.09256
$236$			3.05341e8i			1.51214i
$237$	−1.22949e8			−0.599937
$238$	−2.93647e7			−0.141191
$239$		−	7.32549e7i		−	0.347092i	−0.984826	−	0.173546i	$-0.944478\pi$
$239$		−	7.32549e7i		−	0.347092i	0.984826	−	0.173546i	$-0.0555225\pi$
$240$		−	3.55116e8i		−	1.65818i
$241$		−	2.20158e8i		−	1.01315i	−0.862195	−	0.506576i	$-0.830911\pi$
$241$		−	2.20158e8i		−	1.01315i	0.862195	−	0.506576i	$-0.169089\pi$
$242$			4.65248e6i			0.0211024i
$243$	9.26418e7			0.414176
$244$	−9.67031e7			−0.426163
$245$			2.16707e8i			0.941437i
$246$	−1.27549e7			−0.0546264
$247$	1.57022e8	+	2.52529e7i	0.663013	+	0.106628i
$248$	−1.11525e7			−0.0464293
$249$			5.78992e7i			0.237670i
$250$	−9.79024e7			−0.396281
$251$	−1.29959e8			−0.518738			−0.259369	−	0.965778i	$-0.583515\pi$
$251$	−1.29959e8			−0.518738			−0.259369	+	0.965778i	$0.583515\pi$
$252$			3.46021e7i			0.136207i
$253$			3.30110e8i			1.28155i
$254$		−	4.91127e7i		−	0.188051i
$255$			4.90817e8i			1.85365i
$256$	1.52180e8			0.566916
$257$	1.11340e6			0.00409153			0.00204577	−	0.999998i	$-0.499349\pi$
$257$	1.11340e6			0.00409153			0.00204577	+	0.999998i	$0.499349\pi$
$258$			1.30123e7i			0.0471720i
$259$	2.74866e8			0.983040
$260$	4.66116e8	+	7.49624e7i	1.64470	+	0.264507i
$261$	6.35317e7			0.221181
$262$			5.88789e7i			0.202258i
$263$	−3.02037e8			−1.02380			−0.511900	−	0.859045i	$-0.671058\pi$
$263$	−3.02037e8			−1.02380			−0.511900	+	0.859045i	$0.671058\pi$
$264$	−1.45143e8			−0.485492
$265$			9.94093e7i			0.328145i
$266$			3.01780e7i			0.0983115i
$267$			4.11713e8i			1.32375i
$268$		−	3.87278e8i		−	1.22900i
$269$	−2.42963e8			−0.761040			−0.380520	−	0.924773i	$-0.624255\pi$
$269$	−2.42963e8			−0.761040			−0.380520	+	0.924773i	$0.624255\pi$
$270$	1.05826e8			0.327203
$271$		−	3.31493e8i		−	1.01177i	−0.862601	−	0.505886i	$-0.831166\pi$
$271$		−	3.31493e8i		−	1.01177i	0.862601	−	0.505886i	$-0.168834\pi$
$272$	−2.76142e8			−0.832034
$273$	−2.47965e8	−	3.98786e7i	−0.737600	−	0.118623i
$274$	−2.03875e7			−0.0598738
$275$		−	7.41564e8i		−	2.15023i
$276$	−4.48155e8			−1.28306
$277$	2.92754e8			0.827607			0.413803	−	0.910366i	$-0.364200\pi$
$277$	2.92754e8			0.827607			0.413803	+	0.910366i	$0.364200\pi$
$278$		−	1.12051e8i		−	0.312793i
$279$		−	8.41245e6i		−	0.0231904i
$280$			1.83531e8i			0.499640i
$281$		−	4.81806e8i		−	1.29539i	−0.761901	−	0.647694i	$-0.775734\pi$
$281$		−	4.81806e8i		−	1.29539i	0.761901	−	0.647694i	$-0.224266\pi$
$282$	−5.58601e7			−0.148330
$283$	6.21029e8			1.62877			0.814385	−	0.580326i	$-0.197075\pi$
$283$	6.21029e8			1.62877			0.814385	+	0.580326i	$0.197075\pi$
$284$			2.26861e8i			0.587687i
$285$	5.04410e8			1.29070
$286$	1.41903e7	−	8.82354e7i	0.0358683	−	0.223029i
$287$	−6.26343e7			−0.156396
$288$		−	5.17768e7i		−	0.127721i
$289$	−2.86746e7			−0.0698804
$290$	1.64479e8			0.396020
$291$			1.42506e7i			0.0339006i
$292$			5.19612e8i			1.22135i
$293$		−	2.25288e8i		−	0.523240i	−0.965171	−	0.261620i	$-0.915743\pi$
$293$		−	2.25288e8i		−	0.523240i	0.965171	−	0.261620i	$-0.0842568\pi$
$294$			5.56918e7i			0.127813i
$295$	−1.22164e9			−2.77054
$296$	−2.72039e8			−0.609691
$297$			4.10972e8i			0.910258i
$298$	−1.83511e7			−0.0401704
$299$	8.97662e7	−	5.58166e8i	0.194206	−	1.20757i
$300$	1.00674e9			2.15275
$301$			6.38983e7i			0.135054i
$302$	1.41460e8			0.295536
$303$	−6.83232e8			−1.41097
$304$			2.83789e8i			0.579347i
$305$		−	3.86899e8i		−	0.780815i
$306$			2.19223e7i			0.0437381i
$307$		−	1.94968e8i		−	0.384572i	−0.981339	−	0.192286i	$-0.938410\pi$
$307$		−	1.94968e8i		−	0.384572i	0.981339	−	0.192286i	$-0.0615902\pi$
$308$	−3.47892e8			−0.678448
$309$	3.63810e8			0.701487
$310$		−	2.17792e7i		−	0.0415218i
$311$	3.23645e8			0.610110			0.305055	−	0.952335i	$-0.401325\pi$
$311$	3.23645e8			0.610110			0.305055	+	0.952335i	$0.401325\pi$
$312$	2.45414e8	+	3.94684e7i	0.457466	+	0.0735713i
$313$	1.52942e8			0.281917			0.140959	−	0.990016i	$-0.454982\pi$
$313$	1.52942e8			0.281917			0.140959	+	0.990016i	$0.454982\pi$
$314$			2.02437e8i			0.369008i
$315$	−1.38439e8			−0.249559
$316$	2.91665e8			0.519971
$317$		−	1.05459e8i		−	0.185941i	−0.995669	−	0.0929707i	$-0.970364\pi$
$317$		−	1.05459e8i		−	0.185941i	0.995669	−	0.0929707i	$-0.0296363\pi$
$318$			2.55473e7i			0.0445503i
$319$			6.38751e8i			1.10170i
$320$			7.49437e8i			1.27853i
$321$	−1.44709e8			−0.244190
$322$	1.07273e8			0.179059
$323$		−	3.92234e8i		−	0.647644i
$324$	−6.80733e8			−1.11191
$325$	−2.01652e8	+	1.25387e9i	−0.325845	+	2.02610i
$326$	−8.20913e7			−0.131231
$327$			9.14198e8i			1.44585i
$328$	6.19901e7			0.0969982
$329$	−2.74308e8			−0.424670
$330$		−	2.83442e8i		−	0.434176i
$331$		−	6.53937e8i		−	0.991146i	−0.868566	−	0.495573i	$-0.834958\pi$
$331$		−	6.53937e8i		−	0.991146i	0.868566	−	0.495573i	$-0.165042\pi$
$332$		−	1.37351e8i		−	0.205991i
$333$		−	2.05201e8i		−	0.304526i
$334$	−1.89360e8			−0.278083
$335$	1.54946e9			2.25176
$336$		−	4.48151e8i		−	0.644521i
$337$	−6.58782e8			−0.937642			−0.468821	−	0.883293i	$-0.655321\pi$
$337$	−6.58782e8			−0.937642			−0.468821	+	0.883293i	$0.655321\pi$
$338$	−4.79874e7	+	1.45334e8i	−0.0675956	+	0.204719i
$339$	9.24709e8			1.28916
$340$		−	1.16433e9i		−	1.60658i
$341$	8.45792e7			0.115511
$342$	2.25294e7			0.0304550
$343$			7.80983e8i			1.04499i
$344$		−	6.32411e7i		−	0.0837616i
$345$		−	1.79302e9i		−	2.35081i
$346$			1.63081e6i			0.00211659i
$347$	−3.76864e8			−0.484208			−0.242104	−	0.970250i	$-0.577837\pi$
$347$	−3.76864e8			−0.484208			−0.242104	+	0.970250i	$0.577837\pi$
$348$	−8.67164e8			−1.10300
$349$		−	7.32436e8i		−	0.922318i	−0.887317	−	0.461159i	$-0.847434\pi$
$349$		−	7.32436e8i		−	0.922318i	0.887317	−	0.461159i	$-0.152566\pi$
$350$	−2.40980e8			−0.300430
$351$	1.11755e8	−	6.94891e8i	0.137940	−	0.857712i
$352$	5.20566e8			0.636175
$353$			1.08614e9i			1.31424i	0.753788	+	0.657118i	$0.228225\pi$
$353$			1.08614e9i			1.31424i	−0.753788	+	0.657118i	$0.771775\pi$
$354$	−3.13950e8			−0.376140
$355$	−9.07647e8			−1.07676
$356$		−	9.76682e8i		−	1.14730i
$357$			6.19404e8i			0.720502i
$358$			2.57589e8i			0.296713i
$359$			6.86701e8i			0.783316i	0.920111	+	0.391658i	$0.128098\pi$
$359$			6.86701e8i			0.783316i	−0.920111	+	0.391658i	$0.871902\pi$
$360$	1.37016e8			0.154779
$361$	4.90775e8			0.549044
$362$			3.47102e7i			0.0384571i
$363$	9.81370e7			0.107686
$364$	5.88232e8	+	9.46016e7i	0.639284	+	0.102812i
$365$	−2.07891e9			−2.23775
$366$		−	9.94296e7i		−	0.106006i
$367$	4.74836e8			0.501432			0.250716	−	0.968061i	$-0.419334\pi$
$367$	4.74836e8			0.501432			0.250716	+	0.968061i	$0.419334\pi$
$368$	1.00878e9			1.05519
$369$			4.67597e7i			0.0484484i
$370$		−	5.31251e8i		−	0.545247i
$371$			1.25453e8i			0.127548i
$372$			1.14824e8i			0.115647i
$373$	1.55637e9			1.55286			0.776431	−	0.630202i	$-0.217028\pi$
$373$	1.55637e9			1.55286			0.776431	+	0.630202i	$0.217028\pi$
$374$	−2.20408e8			−0.217859
$375$			2.06510e9i			2.02224i
$376$	2.71486e8			0.263385
$377$	1.73694e8	−	1.08003e9i	0.166952	−	1.03811i
$378$	1.33550e8			0.127181
$379$			1.39184e8i			0.131326i	0.997842	+	0.0656631i	$0.0209163\pi$
$379$			1.39184e8i			0.131326i	−0.997842	+	0.0656631i	$0.979084\pi$
$380$	−1.19658e9			−1.11866
$381$	−1.03596e9			−0.959631
$382$			1.47923e8i			0.135773i
$383$			9.91073e8i			0.901385i	0.892679	+	0.450692i	$0.148823\pi$
$383$			9.91073e8i			0.901385i	−0.892679	+	0.450692i	$0.851177\pi$
$384$			9.33765e8i			0.841547i
$385$		−	1.39188e9i		−	1.24305i
$386$	4.80023e8			0.424822
$387$	4.77034e7			0.0418370
$388$		−	3.38058e7i		−	0.0293819i
$389$	9.01206e8			0.776248			0.388124	−	0.921607i	$-0.373123\pi$
$389$	9.01206e8			0.776248			0.388124	+	0.921607i	$0.373123\pi$
$390$	−7.70759e7	+	4.79258e8i	−0.0657950	+	0.409113i
$391$	−1.39427e9			−1.17958
$392$		−	2.70668e8i		−	0.226953i
$393$	1.24196e9			1.03213
$394$	−1.72828e8			−0.142357
$395$			1.16692e9i			0.952688i
$396$			2.59719e8i			0.210170i
$397$			9.85599e8i			0.790558i	0.918561	+	0.395279i	$0.129352\pi$
$397$			9.85599e8i			0.790558i	−0.918561	+	0.395279i	$0.870648\pi$
$398$		−	3.08233e8i		−	0.245069i
$399$	6.36558e8			0.501687
$400$	−2.26615e9			−1.77043
$401$			2.69423e8i			0.208655i	0.994543	+	0.104328i	$0.0332690\pi$
$401$			2.69423e8i			0.208655i	−0.994543	+	0.104328i	$0.966731\pi$
$402$	3.98197e8			0.305708
$403$	−1.43011e8	−	2.29995e7i	−0.108843	−	0.0175045i
$404$	1.62079e9			1.22290
$405$		−	2.72354e9i		−	2.03723i
$406$	2.07570e8			0.153930
$407$	2.06310e9			1.51684
$408$		−	6.13033e8i		−	0.446862i
$409$			3.40992e8i			0.246441i	0.992379	+	0.123220i	$0.0393222\pi$
$409$			3.40992e8i			0.246441i	−0.992379	+	0.123220i	$0.960678\pi$
$410$			1.21057e8i			0.0867457i
$411$			4.30043e8i			0.305538i
$412$	−8.63044e8			−0.607985
$413$	−1.54169e9			−1.07689
$414$		−	8.00850e7i		−	0.0554689i
$415$	5.49526e8			0.377416
$416$	−8.80199e8	−	1.41557e8i	−0.599452	−	0.0964059i
$417$	−2.36353e9			−1.59619
$418$			2.26512e8i			0.151696i
$419$	−2.32463e9			−1.54385			−0.771923	−	0.635716i	$-0.780705\pi$
$419$	−2.32463e9			−1.54385			−0.771923	+	0.635716i	$0.780705\pi$
$420$	1.88960e9			1.24451
$421$		−	2.46436e9i		−	1.60959i	−0.593551	−	0.804797i	$-0.702274\pi$
$421$		−	2.46436e9i		−	1.60959i	0.593551	−	0.804797i	$-0.297726\pi$
$422$		−	2.64413e8i		−	0.171273i
$423$			2.04784e8i			0.131555i
$424$		−	1.24163e8i		−	0.0791063i
$425$	3.13211e9			1.97914
$426$	−2.33257e8			−0.146185
$427$		−	4.88261e8i		−	0.303497i
$428$	3.43285e8			0.211642
$429$	−1.86119e9	−	2.99323e8i	−1.13813	−	0.183037i
$430$	1.23500e8			0.0749081
$431$			8.64638e8i			0.520192i	0.965583	+	0.260096i	$0.0837543\pi$
$431$			8.64638e8i			0.520192i	−0.965583	+	0.260096i	$0.916246\pi$
$432$	1.25589e9			0.749476
$433$	−3.99758e8			−0.236641			−0.118320	−	0.992975i	$-0.537751\pi$
$433$	−3.99758e8			−0.236641			−0.118320	+	0.992975i	$0.537751\pi$
$434$		−	2.74851e7i		−	0.0161392i
$435$		−	3.46943e9i		−	2.02090i
$436$		−	2.16870e9i		−	1.25313i
$437$			1.43288e9i			0.821346i
$438$	−5.34262e8			−0.303805
$439$	−1.30846e8			−0.0738130			−0.0369065	−	0.999319i	$-0.511750\pi$
$439$	−1.30846e8			−0.0738130			−0.0369065	+	0.999319i	$0.511750\pi$
$440$			1.37756e9i			0.770951i
$441$	2.04168e8			0.113358
$442$	3.72676e8	+	5.99350e7i	0.205283	+	0.0330144i
$443$	−6.35046e8			−0.347050			−0.173525	−	0.984829i	$-0.555516\pi$
$443$	−6.35046e8			−0.347050			−0.173525	+	0.984829i	$0.555516\pi$
$444$			2.80085e9i			1.51862i
$445$	3.90760e9			2.10208
$446$	1.05873e8			0.0565083
$447$			3.87089e8i			0.204991i
$448$			9.45779e8i			0.496955i
$449$			6.68424e8i			0.348490i	0.984702	+	0.174245i	$0.0557484\pi$
$449$			6.68424e8i			0.348490i	−0.984702	+	0.174245i	$0.944252\pi$
$450$			1.79904e8i			0.0930673i
$451$	−4.70124e8			−0.241321
$452$	−2.19363e9			−1.11732
$453$		−	2.98389e9i		−	1.50813i
$454$	−3.19071e8			−0.160026
$455$	−3.78491e8	+	2.35345e9i	−0.188372	+	1.17129i
$456$	−6.30011e8			−0.311151
$457$			1.09854e9i			0.538406i	0.963083	+	0.269203i	$0.0867602\pi$
$457$			1.09854e9i			0.538406i	−0.963083	+	0.269203i	$0.913240\pi$
$458$	4.44239e7			0.0216066
$459$	−1.73580e9			−0.837830
$460$			4.25347e9i			2.03747i
$461$			1.96364e9i			0.933489i	0.884392	+	0.466745i	$0.154573\pi$
$461$			1.96364e9i			0.933489i	−0.884392	+	0.466745i	$0.845427\pi$
$462$		−	3.57700e8i		−	0.168761i
$463$			4.36528e6i			0.00204399i	0.999999	+	0.00102199i	$0.000325311\pi$
$463$			4.36528e6i			0.00204399i	−0.999999	+	0.00102199i	$0.999675\pi$
$464$	1.95196e9			0.907106
$465$	−4.59399e8			−0.211887
$466$		−	4.31622e8i		−	0.197584i
$467$	−1.77366e9			−0.805861			−0.402931	−	0.915230i	$-0.632008\pi$
$467$	−1.77366e9			−0.805861			−0.402931	+	0.915230i	$0.632008\pi$
$468$	7.06249e7	−	4.39145e8i	0.0318491	−	0.198038i
$469$	1.95539e9			0.875244
$470$			5.30172e8i			0.235545i
$471$	4.27009e9			1.88306
$472$	1.52583e9			0.667898
$473$			4.79612e8i			0.208390i
$474$			2.99888e8i			0.129341i
$475$		−	3.21885e9i		−	1.37808i
$476$		−	1.46937e9i		−	0.624465i
$477$	9.36571e7			0.0395118
$478$	−1.78678e8			−0.0748295
$479$		−	2.31639e9i		−	0.963026i	−0.876439	−	0.481513i	$-0.840087\pi$
$479$		−	2.31639e9i		−	0.963026i	0.876439	−	0.481513i	$-0.159913\pi$
$480$	−2.82750e9			−1.16697
$481$	−3.48839e9	−	5.61016e8i	−1.42928	−	0.229862i
$482$	−5.36992e8			−0.218425
$483$		−	2.26277e9i		−	0.913744i
$484$	−2.32804e8			−0.0933324
$485$	1.35253e8			0.0538334
$486$		−	2.25965e8i		−	0.0892923i
$487$			2.04846e9i			0.803668i	0.915713	+	0.401834i	$0.131627\pi$
$487$			2.04846e9i			0.803668i	−0.915713	+	0.401834i	$0.868373\pi$
$488$			4.83239e8i			0.188232i
$489$			1.73159e9i			0.669675i
$490$	5.28575e8			0.202965
$491$	3.64385e9			1.38923			0.694616	−	0.719381i	$-0.255574\pi$
$491$	3.64385e9			1.38923			0.694616	+	0.719381i	$0.255574\pi$
$492$		−	6.38237e8i		−	0.241604i
$493$	−2.69786e9			−1.01404
$494$	6.15949e7	−	3.82997e8i	0.0229880	−	0.142939i
$495$	−1.03911e9			−0.385072
$496$		−	2.58466e8i		−	0.0951080i
$497$	−1.14544e9			−0.418528
$498$	1.41223e8			0.0512394
$499$		−	4.34184e9i		−	1.56431i	−0.623086	−	0.782153i	$-0.714121\pi$
$499$		−	4.34184e9i		−	1.56431i	0.623086	−	0.782153i	$-0.285879\pi$
$500$		−	4.89891e9i		−	1.75269i
$501$			3.99425e9i			1.41907i
$502$			3.16985e8i			0.111835i
$503$	1.29100e9			0.452311			0.226155	−	0.974091i	$-0.427384\pi$
$503$	1.29100e9			0.452311			0.226155	+	0.974091i	$0.427384\pi$
$504$	1.72912e8			0.0601613
$505$			6.48460e9i			2.24060i
$506$	8.05179e8			0.276290
$507$	3.06559e9	+	1.01222e9i	1.04469	+	0.344943i
$508$	2.45754e9			0.831721
$509$		−	4.23593e9i		−	1.42376i	−0.702301	−	0.711880i	$-0.747844\pi$
$509$		−	4.23593e9i		−	1.42376i	0.702301	−	0.711880i	$-0.252156\pi$
$510$	1.19716e9			0.399629
$511$	−2.62356e9			−0.869795
$512$		−	2.69428e9i		−	0.887150i
$513$			1.78388e9i			0.583383i
$514$		−	2.71572e6i		−	0.000882094i
$515$		−	3.45294e9i		−	1.11395i
$516$	−6.51118e8			−0.208634
$517$	−2.05891e9			−0.655272
$518$		−	6.70431e8i		−	0.211934i
$519$	3.43995e7			0.0108011
$520$	3.74598e8	−	2.32925e9i	0.116830	−	0.726447i
$521$	−2.70853e9			−0.839076			−0.419538	−	0.907738i	$-0.637808\pi$
$521$	−2.70853e9			−0.839076			−0.419538	+	0.907738i	$0.637808\pi$
$522$		−	1.54962e8i		−	0.0476845i
$523$	−2.07895e9			−0.635461			−0.317731	−	0.948181i	$-0.602921\pi$
$523$	−2.07895e9			−0.635461			−0.317731	+	0.948181i	$0.602921\pi$
$524$	−2.94623e9			−0.894554
$525$			5.08311e9i			1.53311i
$526$			7.36705e8i			0.220721i
$527$			3.57233e8i			0.106320i
$528$		−	3.36376e9i		−	0.994504i
$529$	1.68863e9			0.495953
$530$	2.42471e8			0.0707449
$531$			1.15095e9i			0.333599i
$532$	−1.51007e9			−0.434816
$533$	7.94909e8	+	1.27840e8i	0.227390	+	0.0365697i
$534$	1.00422e9			0.285387
$535$			1.37345e9i			0.387769i
$536$	−1.93528e9			−0.542835
$537$	5.43344e9			1.51414
$538$			5.92617e8i			0.164073i
$539$			2.05271e9i			0.564634i
$540$			5.29538e9i			1.44717i
$541$			3.70324e9i			1.00552i	0.864426	+	0.502761i	$0.167682\pi$
$541$			3.70324e9i			1.00552i	−0.864426	+	0.502761i	$0.832318\pi$
$542$	−8.08553e8			−0.218128
$543$	7.32158e8			0.196248
$544$			2.19869e9i			0.585556i
$545$	8.67672e9			2.29598
$546$	−9.72688e7	+	6.04817e8i	−0.0255740	+	0.159019i
$547$	2.64179e9			0.690149			0.345074	−	0.938575i	$-0.387854\pi$
$547$	2.64179e9			0.690149			0.345074	+	0.938575i	$0.387854\pi$
$548$		−	1.02017e9i		−	0.264813i
$549$	−3.64512e8			−0.0940174
$550$	−1.80877e9			−0.463567
$551$			2.77258e9i			0.706080i
$552$			2.23949e9i			0.566713i
$553$			1.47264e9i			0.370303i
$554$		−	7.14064e8i		−	0.178424i
$555$	−1.12059e10			−2.78242
$556$	5.60687e9			1.38344
$557$		−	5.09926e9i		−	1.25030i	−0.780505	−	0.625149i	$-0.785038\pi$
$557$		−	5.09926e9i		−	1.25030i	0.780505	−	0.625149i	$-0.214962\pi$
$558$	−2.05190e7			−0.00499961
$559$	1.30420e8	−	8.10951e8i	0.0315793	−	0.196360i
$560$	−4.25344e9			−1.02349
$561$			4.64916e9i			1.11174i
$562$	−1.17518e9			−0.279273
$563$	−1.20149e8			−0.0283754			−0.0141877	−	0.999899i	$-0.504516\pi$
$563$	−1.20149e8			−0.0283754			−0.0141877	+	0.999899i	$0.504516\pi$
$564$		−	2.79517e9i		−	0.656041i
$565$		−	8.77649e9i		−	2.04716i
$566$		−	1.51477e9i		−	0.351146i
$567$		−	3.43707e9i		−	0.791858i
$568$	1.13366e9			0.259575
$569$	−4.82795e9			−1.09868			−0.549338	−	0.835600i	$-0.685120\pi$
$569$	−4.82795e9			−1.09868			−0.549338	+	0.835600i	$0.685120\pi$
$570$		−	1.23032e9i		−	0.278263i
$571$	6.31138e9			1.41872			0.709362	−	0.704845i	$-0.248983\pi$
$571$	6.31138e9			1.41872			0.709362	+	0.704845i	$0.248983\pi$
$572$	4.41519e9	+	7.10066e8i	0.986423	+	0.158640i
$573$	3.12021e9			0.692855
$574$			1.52773e8i			0.0337174i
$575$	−1.14420e10			−2.50995
$576$	7.06072e8			0.153947
$577$		−	6.12272e9i		−	1.32687i	−0.748233	−	0.663436i	$-0.769097\pi$
$577$		−	6.12272e9i		−	1.32687i	0.748233	−	0.663436i	$-0.230903\pi$
$578$			6.99410e7i			0.0150655i
$579$		−	1.01254e10i		−	2.16788i
$580$			8.23032e9i			1.75153i
$581$	6.93494e8			0.146699
$582$	3.47589e7			0.00730863
$583$			9.41634e8i			0.196808i
$584$	2.59657e9			0.539455
$585$	1.75697e9	+	2.82562e8i	0.362843	+	0.0583538i
$586$	−5.49505e8			−0.112805
$587$			1.48927e9i			0.303907i	0.988388	+	0.151953i	$0.0485563\pi$
$587$			1.48927e9i			0.303907i	−0.988388	+	0.151953i	$0.951444\pi$
$588$	−2.78675e9			−0.565297
$589$	3.67127e8			0.0740309
$590$			2.97972e9i			0.597302i
$591$			3.64555e9i			0.726451i
$592$		−	6.30464e9i		−	1.24892i
$593$			5.49424e9i			1.08197i	0.841031	+	0.540987i	$0.181949\pi$
$593$			5.49424e9i			1.08197i	−0.841031	+	0.540987i	$0.818051\pi$
$594$	1.00241e9			0.196243
$595$	5.87881e9			1.14414
$596$		−	9.18267e8i		−	0.177667i
$597$	−6.50170e9			−1.25060
$598$	−1.36144e9	−	2.18951e8i	−0.260341	−	0.0418690i
$599$	4.52708e9			0.860645			0.430322	−	0.902675i	$-0.358400\pi$
$599$	4.52708e9			0.860645			0.430322	+	0.902675i	$0.358400\pi$
$600$		−	5.03083e9i		−	0.950846i
$601$	−2.93232e9			−0.550998			−0.275499	−	0.961301i	$-0.588843\pi$
$601$	−2.93232e9			−0.550998			−0.275499	+	0.961301i	$0.588843\pi$
$602$	1.55856e8			0.0291163
$603$		−	1.45980e9i		−	0.271133i
$604$			7.07851e9i			1.30711i
$605$		−	9.31425e8i		−	0.171003i
$606$			1.66649e9i			0.304192i
$607$	−1.02999e10			−1.86927			−0.934637	−	0.355604i	$-0.884275\pi$
$607$	−1.02999e10			−1.86927			−0.934637	+	0.355604i	$0.884275\pi$
$608$	2.25958e9			0.407724
$609$		−	4.37837e9i		−	0.785511i
$610$	−9.43694e8			−0.168336
$611$	3.48131e9	+	5.59877e8i	0.617446	+	0.0992998i
$612$	−1.09696e9			−0.193447
$613$			7.93265e9i			1.39093i	0.718558	+	0.695467i	$0.244803\pi$
$613$			7.93265e9i			1.39093i	−0.718558	+	0.695467i	$0.755197\pi$
$614$	−4.75550e8			−0.0829099
$615$	2.55352e9			0.442666
$616$			1.73846e9i			0.299663i
$617$		−	8.87840e9i		−	1.52173i	−0.648912	−	0.760863i	$-0.724776\pi$
$617$		−	8.87840e9i		−	1.52173i	0.648912	−	0.760863i	$-0.275224\pi$
$618$		−	8.87376e8i		−	0.151234i
$619$		−	7.70576e9i		−	1.30586i	−0.757416	−	0.652932i	$-0.773539\pi$
$619$		−	7.70576e9i		−	1.30586i	0.757416	−	0.652932i	$-0.226461\pi$
$620$	1.08981e9			0.183644
$621$	6.34112e9			1.06254
$622$		−	7.89411e8i		−	0.131534i
$623$	4.93134e9			0.817065
$624$	−9.14702e8	+	5.68761e9i	−0.150707	+	0.937095i
$625$	7.07477e9			1.15913
$626$		−	3.73044e8i		−	0.0607785i
$627$	4.77792e9			0.774109
$628$	−1.01297e10			−1.63206
$629$			8.71384e9i			1.39615i
$630$			3.37671e8i			0.0538024i
$631$			1.12137e10i			1.77683i	0.459045	+	0.888413i	$0.348192\pi$
$631$			1.12137e10i			1.77683i	−0.459045	+	0.888413i	$0.651808\pi$
$632$		−	1.45749e9i		−	0.229665i
$633$	−5.57739e9			−0.874014
$634$	−2.57227e8			−0.0400871
$635$			9.83235e9i			1.52387i
$636$	−1.27836e9			−0.197039
$637$	5.58190e8	−	3.47082e9i	0.0855646	−	0.532040i
$638$	1.55799e9			0.237516
$639$			8.55127e8i			0.129652i
$640$	8.86243e9			1.33636
$641$	−9.37779e9			−1.40636			−0.703181	−	0.711011i	$-0.748238\pi$
$641$	−9.37779e9			−1.40636			−0.703181	+	0.711011i	$0.748238\pi$
$642$			3.52963e8i			0.0526450i
$643$		−	6.09443e8i		−	0.0904054i	−0.998978	−	0.0452027i	$-0.985607\pi$
$643$		−	6.09443e8i		−	0.0904054i	0.998978	−	0.0452027i	$-0.0143934\pi$
$644$			5.36782e9i			0.791950i
$645$		−	2.60505e9i		−	0.382259i
$646$	−9.56707e8			−0.139626
$647$	−3.63441e9			−0.527556			−0.263778	−	0.964583i	$-0.584969\pi$
$647$	−3.63441e9			−0.527556			−0.263778	+	0.964583i	$0.584969\pi$
$648$			3.40172e9i			0.491118i
$649$	−1.15717e10			−1.66166
$650$	3.05835e9	+	4.91854e8i	0.436808	+	0.0702490i
$651$	−5.79755e8			−0.0823591
$652$		−	4.10774e9i		−	0.580413i
$653$	−3.71780e9			−0.522505			−0.261252	−	0.965271i	$-0.584135\pi$
$653$	−3.71780e9			−0.522505			−0.261252	+	0.965271i	$0.584135\pi$
$654$	2.22984e9			0.311711
$655$		−	1.17875e10i		−	1.63900i
$656$			1.43665e9i			0.198696i
$657$			1.95862e9i			0.269445i
$658$			6.69070e8i			0.0915547i
$659$	6.01769e9			0.819088			0.409544	−	0.912290i	$-0.365688\pi$
$659$	6.01769e9			0.819088			0.409544	+	0.912290i	$0.365688\pi$
$660$	1.41831e10			1.92029
$661$			1.36622e9i			0.183999i	0.995759	+	0.0919993i	$0.0293257\pi$
$661$			1.36622e9i			0.183999i	−0.995759	+	0.0919993i	$0.970674\pi$
$662$	−1.59503e9			−0.213681
$663$	1.26424e9	−	7.86102e9i	0.168473	−	1.04757i
$664$	−6.86361e8			−0.0909839
$665$		−	6.04162e9i		−	0.796669i
$666$	−5.00511e8			−0.0656529
$667$	9.85566e9			1.28601
$668$		−	9.47533e9i		−	1.22992i
$669$		−	2.23322e9i		−	0.288364i
$670$		−	3.77932e9i		−	0.485458i
$671$		−	3.66482e9i		−	0.468300i
$672$	−3.56827e9			−0.453591
$673$	1.13200e9			0.143150			0.0715751	−	0.997435i	$-0.477197\pi$
$673$	1.13200e9			0.143150			0.0715751	+	0.997435i	$0.477197\pi$
$674$			1.60685e9i			0.202146i
$675$	−1.42448e10			−1.78276
$676$	−7.27233e9	−	2.40123e9i	−0.905440	−	0.298965i
$677$	2.50487e9			0.310259			0.155130	−	0.987894i	$-0.450420\pi$
$677$	2.50487e9			0.310259			0.155130	+	0.987894i	$0.450420\pi$
$678$		−	2.25548e9i		−	0.277930i
$679$	1.70688e8			0.0209247
$680$	−5.81834e9			−0.709608
$681$			6.73031e9i			0.816620i
$682$		−	2.06299e8i		−	0.0249030i
$683$			1.16246e9i			0.139606i	0.997561	+	0.0698032i	$0.0222371\pi$
$683$			1.16246e9i			0.139606i	−0.997561	+	0.0698032i	$0.977763\pi$
$684$			1.12734e9i			0.134698i
$685$	4.08157e9			0.485188
$686$	1.90491e9			0.225290
$687$		−	9.37053e8i		−	0.110259i
$688$	1.46565e9			0.171581
$689$	2.56057e8	−	1.59216e9i	0.0298242	−	0.185447i
$690$	−4.37340e9			−0.506812
$691$		−	1.00279e10i		−	1.15621i	−0.815963	−	0.578104i	$-0.803793\pi$
$691$		−	1.00279e10i		−	1.15621i	0.815963	−	0.578104i	$-0.196207\pi$
$692$	−8.16038e7			−0.00936136
$693$	−1.31134e9			−0.149675
$694$			9.19218e8i			0.104390i
$695$			2.24325e10i			2.53472i
$696$			4.33334e9i			0.487181i
$697$		−	1.98564e9i		−	0.222119i
$698$	−1.78650e9			−0.198843
$699$	−9.10440e9			−1.00828
$700$		−	1.20584e10i		−	1.32876i
$701$	1.58959e10			1.74290			0.871449	−	0.490486i	$-0.163181\pi$
$701$	1.58959e10			1.74290			0.871449	+	0.490486i	$0.163181\pi$
$702$	−1.69492e9	−	2.72584e8i	−0.184914	−	0.0297386i
$703$	8.95516e9			0.972142
$704$			7.09889e9i			0.766807i
$705$	1.11832e10			1.20200
$706$	2.64922e9			0.283336
$707$			8.18348e9i			0.870904i
$708$		−	1.57097e10i		−	1.66361i
$709$			3.06676e9i			0.323160i	0.986860	+	0.161580i	$0.0516590\pi$
$709$			3.06676e9i			0.323160i	−0.986860	+	0.161580i	$0.948341\pi$
$710$			2.21386e9i			0.232138i
$711$	1.09940e9			0.114713
$712$	−4.88062e9			−0.506751
$713$		−	1.30502e9i		−	0.134836i
$714$	1.51080e9			0.155333
$715$	−2.84090e9	+	1.76647e10i	−0.290660	+	1.80732i
$716$	−1.28894e10			−1.31232
$717$			3.76893e9i			0.381857i
$718$	1.67495e9			0.168875
$719$	7.17747e9			0.720145			0.360073	−	0.932924i	$-0.382752\pi$
$719$	7.17747e9			0.720145			0.360073	+	0.932924i	$0.382752\pi$
$720$			3.17541e9i			0.317056i
$721$		−	4.35757e9i		−	0.432983i
$722$		−	1.19706e9i		−	0.118368i
$723$			1.13270e10i			1.11463i
$724$	−1.73685e9			−0.170090
$725$	−2.21399e10			−2.15771
$726$		−	2.39368e8i		−	0.0232160i
$727$	−1.33854e10			−1.29200			−0.645998	−	0.763339i	$-0.723558\pi$
$727$	−1.33854e10			−1.29200			−0.645998	+	0.763339i	$0.723558\pi$
$728$	4.72737e8	−	2.93948e9i	0.0454109	−	0.282365i
$729$	7.43152e9			0.710447
$730$			5.07072e9i			0.482436i
$731$	−2.02572e9			−0.191808
$732$	4.97533e9			0.468849
$733$			1.23188e10i			1.15532i	0.816277	+	0.577661i	$0.196035\pi$
$733$			1.23188e10i			1.15532i	−0.816277	+	0.577661i	$0.803965\pi$
$734$		−	1.15818e9i		−	0.108104i
$735$		−	1.11495e10i		−	1.03574i
$736$		−	8.03212e9i		−	0.742605i
$737$	1.46769e10			1.35051
$738$	1.14053e8			0.0104450
$739$		−	3.87017e9i		−	0.352756i	−0.984323	−	0.176378i	$-0.943562\pi$
$739$		−	3.87017e9i		−	0.352756i	0.984323	−	0.176378i	$-0.0564381\pi$
$740$	2.65831e10			2.41154
$741$	−8.07873e9	−	1.29925e9i	−0.729423	−	0.117308i
$742$	3.05996e8			0.0274980
$743$			1.30810e10i			1.16998i	0.811040	+	0.584990i	$0.198902\pi$
$743$			1.30810e10i			1.16998i	−0.811040	+	0.584990i	$0.801098\pi$
$744$	5.73793e8			0.0510798
$745$	3.67389e9			0.325521
$746$		−	3.79619e9i		−	0.334782i
$747$		−	5.17729e8i		−	0.0454444i
$748$		−	1.10289e10i		−	0.963557i
$749$			1.73327e9i			0.150723i
$750$	5.03703e9			0.435974
$751$	2.33046e8			0.0200772			0.0100386	−	0.999950i	$-0.496805\pi$
$751$	2.33046e8			0.0200772			0.0100386	+	0.999950i	$0.496805\pi$
$752$			6.29184e9i			0.539529i
$753$	6.68632e9			0.570696
$754$	−2.63433e9	−	4.23662e8i	−0.223805	−	0.0359932i
$755$	−2.83203e10			−2.39488
$756$			6.68269e9i			0.562504i
$757$	−8.89085e9			−0.744917			−0.372458	−	0.928049i	$-0.621485\pi$
$757$	−8.89085e9			−0.744917			−0.372458	+	0.928049i	$0.621485\pi$
$758$	3.39486e8			0.0283126
$759$		−	1.69840e10i		−	1.40992i
$760$			5.97948e9i			0.494101i
$761$		−	1.69293e10i		−	1.39249i	−0.717803	−	0.696247i	$-0.754852\pi$
$761$		−	1.69293e10i		−	1.39249i	0.717803	−	0.696247i	$-0.245148\pi$
$762$			2.52683e9i			0.206887i
$763$	1.09499e10			0.892430
$764$	−7.40188e9			−0.600503
$765$		−	4.38883e9i		−	0.354433i
$766$	2.41735e9			0.194330
$767$	1.95660e10	+	3.14667e9i	1.56573	+	0.251807i
$768$	−7.82961e9			−0.623700
$769$		−	1.35031e10i		−	1.07076i	−0.844611	−	0.535380i	$-0.820169\pi$
$769$		−	1.35031e10i		−	1.07076i	0.844611	−	0.535380i	$-0.179831\pi$
$770$	−3.39496e9			−0.267989
$771$	−5.72840e7			−0.00450135
$772$			2.40198e10i			1.87892i
$773$		−	1.01595e10i		−	0.791126i	−0.918439	−	0.395563i	$-0.870549\pi$
$773$		−	1.01595e10i		−	0.791126i	0.918439	−	0.395563i	$-0.129451\pi$
$774$		−	1.16354e8i		−	0.00901964i
$775$			2.93162e9i			0.226231i
$776$	−1.68932e8			−0.0129777
$777$	−1.41417e10			−1.08150
$778$		−	2.19815e9i		−	0.167351i
$779$	−2.04063e9			−0.154662
$780$	−2.39815e10	−	3.85678e9i	−1.80944	−	0.291001i
$781$	−8.59750e9			−0.645793
$782$			3.40080e9i			0.254306i
$783$	1.22698e10			0.913425
$784$	6.27288e9			0.464901
$785$		−	4.05278e10i		−	2.99026i
$786$		−	3.02929e9i		−	0.222516i
$787$			2.11128e10i			1.54396i	0.635650	+	0.771978i	$0.280732\pi$
$787$			2.11128e10i			1.54396i	−0.635650	+	0.771978i	$0.719268\pi$
$788$		−	8.64811e9i		−	0.629622i
$789$	1.55397e10			1.12635
$790$	2.84626e9			0.205390
$791$		−	1.10758e10i		−	0.795715i
$792$	1.29785e9			0.0928297
$793$	−9.96568e8	+	6.19665e9i	−0.0709661	+	0.441267i
$794$	2.40400e9			0.170436
$795$		−	5.11456e9i		−	0.361014i
$796$	1.54236e10			1.08390
$797$	1.50164e10			1.05066			0.525329	−	0.850899i	$-0.323942\pi$
$797$	1.50164e10			1.05066			0.525329	+	0.850899i	$0.323942\pi$
$798$		−	1.55264e9i		−	0.108159i
$799$		−	8.69614e9i		−	0.603133i
$800$			1.80435e10i			1.24596i
$801$		−	3.68149e9i		−	0.253111i
$802$	6.57155e8			0.0449840
$803$	−1.96920e10			−1.34210
$804$			1.99253e10i			1.35210i
$805$	−2.14761e10			−1.45101
$806$	−5.60986e7	+	3.48821e8i	−0.00377380	+	0.0234655i
$807$	1.25004e10			0.837269
$808$		−	8.09931e9i		−	0.540143i
$809$	−1.73621e10			−1.15288			−0.576439	−	0.817140i	$-0.695558\pi$
$809$	−1.73621e10			−1.15288			−0.576439	+	0.817140i	$0.695558\pi$
$810$	−6.64305e9			−0.439207
$811$			2.29338e10i			1.50974i	0.655872	+	0.754872i	$0.272301\pi$
$811$			2.29338e10i			1.50974i	−0.655872	+	0.754872i	$0.727699\pi$
$812$			1.03865e10i			0.680809i
$813$			1.70552e10i			1.11311i
$814$		−	5.03216e9i		−	0.327016i
$815$	1.64346e10			1.06343
$816$	1.42074e10			0.915373
$817$			2.08182e9i			0.133557i
$818$	8.31720e8			0.0531301
$819$	2.21727e9	+	3.56590e8i	0.141035	+	0.0226817i
$820$	−6.05756e9			−0.383662
$821$		−	1.35927e10i		−	0.857242i	−0.903484	−	0.428621i	$-0.859000\pi$
$821$		−	1.35927e10i		−	0.857242i	0.903484	−	0.428621i	$-0.141000\pi$
$822$	1.04893e9			0.0658710
$823$	−1.74580e9			−0.109168			−0.0545841	−	0.998509i	$-0.517383\pi$
$823$	−1.74580e9			−0.109168			−0.0545841	+	0.998509i	$0.517383\pi$
$824$			4.31275e9i			0.268540i
$825$			3.81531e10i			2.36560i
$826$			3.76037e9i			0.232167i
$827$			1.56621e10i			0.962900i	0.876473	+	0.481450i	$0.159890\pi$
$827$			1.56621e10i			0.962900i	−0.876473	+	0.481450i	$0.840110\pi$
$828$	4.00735e9			0.245330
$829$	6.04053e9			0.368243			0.184122	−	0.982903i	$-0.441056\pi$
$829$	6.04053e9			0.368243			0.184122	+	0.982903i	$0.441056\pi$
$830$		−	1.34036e9i		−	0.0813671i
$831$	−1.50621e10			−0.910503
$832$	1.93039e9	−	1.20031e10i	0.116202	−	0.722542i
$833$	−8.66994e9			−0.519707
$834$			5.76495e9i			0.344124i
$835$	3.79098e10			2.25345
$836$	−1.13344e10			−0.670927
$837$		−	1.62469e9i		−	0.0957706i
$838$			5.67005e9i			0.332838i
$839$		−	1.52652e10i		−	0.892348i	−0.894946	−	0.446174i	$-0.852786\pi$
$839$		−	1.52652e10i		−	0.892348i	0.894946	−	0.446174i	$-0.147214\pi$
$840$		−	9.44261e9i		−	0.549686i
$841$	1.82048e9			0.105536
$842$	−6.01087e9			−0.347012
$843$			2.47887e10i			1.42514i
$844$	1.32309e10			0.757515
$845$	9.60706e9	−	2.90958e10i	0.547762	−	1.65894i
$846$	4.99495e8			0.0283618
$847$		−	1.17545e9i		−	0.0664677i
$848$	2.87754e9			0.162045
$849$	−3.19517e10			−1.79191
$850$		−	7.63960e9i		−	0.426682i
$851$		−	3.18328e10i		−	1.77061i
$852$		−	1.16719e10i		−	0.646551i
$853$		−	4.49630e9i		−	0.248047i	−0.992279	−	0.124023i	$-0.960420\pi$
$853$		−	4.49630e9i		−	0.248047i	0.992279	−	0.124023i	$-0.0395798\pi$
$854$	−1.19093e9			−0.0654309
$855$	−4.51038e9			−0.246792
$856$		−	1.71544e9i		−	0.0934798i
$857$	−1.87369e10			−1.01687			−0.508435	−	0.861101i	$-0.669776\pi$
$857$	−1.87369e10			−1.01687			−0.508435	+	0.861101i	$0.669776\pi$
$858$	−7.30086e8	+	4.53967e9i	−0.0394610	+	0.245368i
$859$	5.28210e9			0.284335			0.142168	−	0.989843i	$-0.454593\pi$
$859$	5.28210e9			0.284335			0.142168	+	0.989843i	$0.454593\pi$
$860$			6.17981e9i			0.331307i
$861$	3.22251e9			0.172061
$862$	2.10896e9			0.112148
$863$			1.11634e10i			0.591231i	0.955307	+	0.295616i	$0.0955248\pi$
$863$			1.11634e10i			0.591231i	−0.955307	+	0.295616i	$0.904475\pi$
$864$		−	9.99962e9i		−	0.527455i
$865$		−	3.26488e8i		−	0.0171518i
$866$			9.75058e8i			0.0510174i
$867$	1.47530e9			0.0768799
$868$	1.37532e9			0.0713813
$869$			1.10534e10i			0.571382i
$870$	−8.46237e9			−0.435687
$871$	−2.48164e10	−	3.99107e9i	−1.27255	−	0.204656i
$872$	−1.08373e10			−0.553494
$873$		−	1.27427e8i		−	0.00648205i
$874$	3.49498e9			0.177074
$875$	2.47350e10			1.24820
$876$		−	2.67338e10i		−	1.34368i
$877$			1.55652e9i			0.0779210i	0.999241	+	0.0389605i	$0.0124047\pi$
$877$			1.55652e9i			0.0779210i	−0.999241	+	0.0389605i	$0.987595\pi$
$878$			3.19148e8i			0.0159134i
$879$			1.15910e10i			0.575650i
$880$	−3.19257e10			−1.57925
$881$	3.77180e10			1.85837			0.929187	−	0.369611i	$-0.120509\pi$
$881$	3.77180e10			1.85837			0.929187	+	0.369611i	$0.120509\pi$
$882$		−	4.97990e8i		−	0.0244388i
$883$	−1.53832e10			−0.751943			−0.375972	−	0.926631i	$-0.622691\pi$
$883$	−1.53832e10			−0.751943			−0.375972	+	0.926631i	$0.622691\pi$
$884$	−2.99907e9	+	1.86482e10i	−0.146017	+	0.907935i
$885$	6.28527e10			3.04805
$886$			1.54896e9i			0.0748205i
$887$	−1.58874e10			−0.764401			−0.382201	−	0.924079i	$-0.624834\pi$
$887$	−1.58874e10			−0.764401			−0.382201	+	0.924079i	$0.624834\pi$
$888$	1.39963e10			0.670759
$889$			1.24083e10i			0.592319i
$890$		−	9.53112e9i		−	0.453188i
$891$		−	2.57982e10i		−	1.22185i
$892$			5.29774e9i			0.249927i
$893$	−8.93698e9			−0.419963
$894$	9.44157e8			0.0441940
$895$		−	5.15692e10i		−	2.40442i
$896$	1.11843e10			0.519433
$897$	−4.61843e9	+	2.87174e10i	−0.213659	+	1.32853i
$898$	1.63037e9			0.0751309
$899$		−	2.52517e9i		−	0.115913i
$900$	−9.00218e9			−0.411622
$901$	−3.97714e9			−0.181148
$902$			1.14669e9i			0.0520264i
$903$		−	3.28754e9i		−	0.148581i
$904$			1.09619e10i			0.493510i
$905$		−	6.94896e9i		−	0.311638i
$906$	−7.27808e9			−0.325138
$907$	−1.04545e10			−0.465240			−0.232620	−	0.972568i	$-0.574730\pi$
$907$	−1.04545e10			−0.465240			−0.232620	+	0.972568i	$0.574730\pi$
$908$		−	1.59659e10i		−	0.707772i
$909$	6.10938e9			0.269789
$910$	5.74036e9	+	9.23185e8i	0.252519	+	0.0406110i
$911$	−4.40778e10			−1.93155			−0.965774	−	0.259386i	$-0.916480\pi$
$911$	−4.40778e10			−1.93155			−0.965774	+	0.259386i	$0.916480\pi$
$912$		−	1.46008e10i		−	0.637376i
$913$	5.20527e9			0.226358
$914$	2.67948e9			0.116075
$915$			1.99058e10i			0.859024i
$916$			2.22292e9i			0.0955628i
$917$		−	1.48757e10i		−	0.637067i
$918$			4.23384e9i			0.180628i
$919$	−4.22709e10			−1.79654			−0.898270	−	0.439445i	$-0.855175\pi$
$919$	−4.22709e10			−1.79654			−0.898270	+	0.439445i	$0.855175\pi$
$920$	2.12552e10			0.899928
$921$			1.00310e10i			0.423092i
$922$	4.78957e9			0.201251
$923$	1.45371e10	+	2.33790e9i	0.608514	+	0.0978634i
$924$	1.78989e10			0.746404
$925$			7.15097e10i			2.97077i
$926$	1.06475e7			0.000440664
$927$	−3.25315e9			−0.134130
$928$		−	1.55419e10i		−	0.638389i
$929$			4.13059e10i			1.69027i	0.534551	+	0.845136i	$0.320481\pi$
$929$			4.13059e10i			1.69027i	−0.534551	+	0.845136i	$0.679519\pi$
$930$			1.12053e9i			0.0456808i
$931$			8.91006e9i			0.361873i
$932$	2.15978e10			0.873885
$933$	−1.66514e10			−0.671221
$934$			4.32617e9i			0.173736i
$935$	4.41255e10			1.76542
$936$	−2.19447e9	−	3.52922e8i	−0.0874710	−	0.0140674i
$937$	3.77754e10			1.50010			0.750050	−	0.661382i	$-0.230030\pi$
$937$	3.77754e10			1.50010			0.750050	+	0.661382i	$0.230030\pi$
$938$		−	4.76944e9i		−	0.188694i
$939$	−7.86879e9			−0.310155
$940$	−2.65291e10			−1.04178
$941$		−	3.17351e9i		−	0.124159i	−0.998071	−	0.0620793i	$-0.980227\pi$
$941$		−	3.17351e9i		−	0.124159i	0.998071	−	0.0620793i	$-0.0197732\pi$
$942$		−	1.04153e10i		−	0.405969i
$943$			7.25382e9i			0.281693i
$944$			3.53620e10i			1.36815i
$945$	−2.67367e10			−1.03062
$946$	1.16983e9			0.0449267
$947$			3.54610e10i			1.35683i	0.734677	+	0.678417i	$0.237334\pi$
$947$			3.54610e10i			1.35683i	−0.734677	+	0.678417i	$0.762666\pi$
$948$	−1.50060e10			−0.572053
$949$	3.32963e10	+	5.35482e9i	1.26463	+	0.203382i
$950$	−7.85118e9			−0.297100
$951$			5.42582e9i			0.204566i
$952$	−7.34267e9			−0.275819
$953$	5.38620e9			0.201585			0.100792	−	0.994907i	$-0.467862\pi$
$953$	5.38620e9			0.201585			0.100792	+	0.994907i	$0.467862\pi$
$954$		−	2.28441e8i		−	0.00851834i
$955$		−	2.96141e10i		−	1.10024i
$956$		−	8.94081e9i		−	0.330959i
$957$		−	3.28635e10i		−	1.21205i
$958$	−5.64997e9			−0.207619
$959$	5.15089e9			0.188589
$960$		−	3.85582e10i		−	1.40659i
$961$	2.71782e10			0.987847
$962$	−1.36839e9	+	8.50862e9i	−0.0495560	+	0.308139i
$963$	1.29397e9			0.0466910
$964$		−	2.68704e10i		−	0.966061i
$965$	−9.61005e10			−3.44255
$966$	−5.51916e9			−0.196994
$967$		−	3.69599e10i		−	1.31443i	−0.753703	−	0.657216i	$-0.771734\pi$
$967$		−	3.69599e10i		−	1.31443i	0.753703	−	0.657216i	$-0.228266\pi$
$968$			1.16336e9i			0.0412239i
$969$			2.01803e10i			0.712515i
$970$		−	3.29900e8i		−	0.0116060i
$971$	−9.10466e9			−0.319151			−0.159576	−	0.987186i	$-0.551013\pi$
$971$	−9.10466e9			−0.319151			−0.159576	+	0.987186i	$0.551013\pi$
$972$	1.13070e10			0.394926
$973$			2.83095e10i			0.985229i
$974$	4.99645e9			0.173263
$975$	1.03749e10	−	6.45112e10i	0.358483	−	2.22904i
$976$	−1.11993e10			−0.385583
$977$		−	1.66549e10i		−	0.571363i	−0.958325	−	0.285681i	$-0.907780\pi$
$977$		−	1.66549e10i		−	0.571363i	0.958325	−	0.285681i	$-0.0922199\pi$
$978$	4.22356e9			0.144375
$979$	3.70139e10			1.26074
$980$			2.64492e10i			0.897680i
$981$		−	8.17466e9i		−	0.276457i
$982$		−	8.88779e9i		−	0.299505i
$983$		−	5.63122e10i		−	1.89088i	−0.325790	−	0.945442i	$-0.605630\pi$
$983$		−	5.63122e10i		−	1.89088i	0.325790	−	0.945442i	$-0.394370\pi$
$984$	−3.18936e9			−0.106714
$985$	3.46002e10			1.15359
$986$			6.58042e9i			0.218617i
$987$	1.41130e10			0.467207
$988$	1.91647e10	+	3.08213e9i	0.632197	+	0.101672i
$989$	7.40022e9			0.243253
$990$			2.53451e9i			0.0830177i
$991$	3.57927e9			0.116825			0.0584127	−	0.998293i	$-0.481396\pi$
$991$	3.57927e9			0.116825			0.0584127	+	0.998293i	$0.481396\pi$
$992$	−2.05795e9			−0.0669337
$993$			3.36447e10i			1.09042i
$994$			2.79386e9i			0.0902304i
$995$			6.17082e10i			1.98592i
$996$			7.06664e9i			0.226624i
$997$	−1.48467e10			−0.474456			−0.237228	−	0.971454i	$-0.576239\pi$
$997$	−1.48467e10			−0.474456			−0.237228	+	0.971454i	$0.576239\pi$
$998$	−1.05903e10			−0.337249
$999$		−	3.96304e10i		−	1.25762i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.8.b.a.12.3	✓	6
3.2	odd	2		117.8.b.b.64.4		6
4.3	odd	2		208.8.f.a.129.6		6
13.5	odd	4		169.8.a.d.1.3		6
13.8	odd	4		169.8.a.d.1.4		6
13.12	even	2	inner	13.8.b.a.12.4	yes	6
39.38	odd	2		117.8.b.b.64.3		6
52.51	odd	2		208.8.f.a.129.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.b.a.12.3	✓	6	1.1	even	1	trivial
13.8.b.a.12.4	yes	6	13.12	even	2	inner
117.8.b.b.64.3		6	39.38	odd	2
117.8.b.b.64.4		6	3.2	odd	2
169.8.a.d.1.3		6	13.5	odd	4
169.8.a.d.1.4		6	13.8	odd	4
208.8.f.a.129.5		6	52.51	odd	2
208.8.f.a.129.6		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)

\(f(q)\)	\(=\)	\(q-2.43912i q^{2} -51.4496 q^{3} +122.051 q^{4} +488.312i q^{5} +125.492i q^{6} +616.243i q^{7} -609.904i q^{8} +460.056 q^{9} +1191.05 q^{10} +4625.43i q^{11} -6279.45 q^{12} +(1257.79 - 7820.90i) q^{13} +1503.09 q^{14} -25123.4i q^{15} +14134.9 q^{16} -19536.2 q^{17} -1122.13i q^{18} +20077.3i q^{19} +59598.8i q^{20} -31705.4i q^{21} +11282.0 q^{22} +71368.5 q^{23} +31379.3i q^{24} -160323. q^{25} +(-19076.1 - 3067.89i) q^{26} +88850.5 q^{27} +75212.8i q^{28} +138095. q^{29} -61279.1 q^{30} -18285.7i q^{31} -112544. i q^{32} -237976. i q^{33} +47651.3i q^{34} -300918. q^{35} +56150.2 q^{36} -446035. i q^{37} +48970.9 q^{38} +(-64712.5 + 402382. i) q^{39} +297823. q^{40} +101639. i q^{41} -77333.4 q^{42} +103690. q^{43} +564537. i q^{44} +224651. i q^{45} -174076. i q^{46} +445129. i q^{47} -727232. q^{48} +443788. q^{49} +391048. i q^{50} +1.00513e6 q^{51} +(153514. - 954546. i) q^{52} +203578. q^{53} -216717. i q^{54} -2.25865e6 q^{55} +375849. q^{56} -1.03297e6i q^{57} -336832. i q^{58} +2.50176e6i q^{59} -3.06633e6i q^{60} -792320. q^{61} -44601.1 q^{62} +283506. i q^{63} +1.53475e6 q^{64} +(3.81904e6 + 614191. i) q^{65} -580454. q^{66} -3.17309e6i q^{67} -2.38441e6 q^{68} -3.67188e6 q^{69} +733977. i q^{70} +1.85875e6i q^{71} -280590. i q^{72} +4.25734e6i q^{73} -1.08793e6 q^{74} +8.24856e6 q^{75} +2.45045e6i q^{76} -2.85039e6 q^{77} +(981459. + 157842. i) q^{78} +2.38970e6 q^{79} +6.90221e6i q^{80} -5.57746e6 q^{81} +247910. q^{82} -1.12536e6i q^{83} -3.86967e6i q^{84} -9.53976e6i q^{85} -252913. i q^{86} -7.10495e6 q^{87} +2.82107e6 q^{88} -8.00227e6i q^{89} +547951. q^{90} +(4.81957e6 + 775101. i) q^{91} +8.71057e6 q^{92} +940791. i q^{93} +1.08572e6 q^{94} -9.80397e6 q^{95} +5.79036e6i q^{96} -276982. i q^{97} -1.08245e6i q^{98} +2.12796e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9} - 406 q^{10} + 1898 q^{12} - 5018 q^{13} + 9558 q^{14} + 7778 q^{16} + 13152 q^{17} - 125080 q^{22} + 27264 q^{23} - 18262 q^{25} - 54210 q^{26} + 194560 q^{27} + 42924 q^{29}+ \cdots - 22075632 q^{95}+O(q^{100}) \) 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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