LMFDB - Embedded newform 23.13.b.c.22.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,13,Mod(22,23)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 13, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	23.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.0218577974$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 3347471347 x^{18} + 50192028136 x^{17} + \cdots + 27\!\cdots\!00 $ x^20 - 10x^19 + 3347471347x^18 + 50192028136x^17 + 4614029959265316185x^16 + 103983573203959814986x^15 + 3379311419427059060814588431x^14 - 50075600880043830127484728724x^13 + 1418248117882107931183860815497994519x^12 - 186283301865915032190511169614543430870x^11 + 344909219364621128810886345868491344790294425x^10 - 125068945548217342185390941774282032763358152624x^9 + 47658267810007314724833617859245348674600894767076739x^8 - 34284587309212637136431349575614673086269699858260744042x^7 + 3657295051317463159335448490390059763836829894401828863741077x^6 - 4192488412085385429284373948564248993756540022283451136234513764x^5 + 145239704441603674542910095981883943467863635482546791782438264269772x^4 - 233898167710123853879870828485730417522747257713714973387426138036534400x^3 + 2377190267573030483458894159952368719674728986340398105981694428231002967376x^2 - 4888930139228117053934659002514858864803852057089485939941207548974713342600640*x + 2708707787738124609500305902915298085507964042734440369081759603145685551777502400
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	multiple of $ 2^{35}\cdot 3^{11}\cdot 67 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.14
Root			$1203.28 - 26094.4i$ of defining polynomial
Character	$\chi$	$=$	23.22
Dual form			23.13.b.c.22.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+52.2711 q^{2} -1139.01 q^{3} -1363.73 q^{4} +26094.4i q^{5} -59537.5 q^{6} +15165.4i q^{7} -285386. q^{8} +765911. q^{9} +O(q^{10})$	\(q+52.2711 q^{2} -1139.01 q^{3} -1363.73 q^{4} +26094.4i q^{5} -59537.5 q^{6} +15165.4i q^{7} -285386. q^{8} +765911. q^{9} +1.36399e6i q^{10} -55543.2i q^{11} +1.55331e6 q^{12} -2.41684e6 q^{13} +792711. i q^{14} -2.97219e7i q^{15} -9.33161e6 q^{16} -1.88550e7i q^{17} +4.00350e7 q^{18} -4.59212e7i q^{19} -3.55858e7i q^{20} -1.72736e7i q^{21} -2.90330e6i q^{22} +(2.86889e7 + 1.45229e8i) q^{23} +3.25059e8 q^{24} -4.36779e8 q^{25} -1.26331e8 q^{26} -2.67064e8 q^{27} -2.06815e7i q^{28} -1.19005e6 q^{29} -1.55360e9i q^{30} +1.39827e9 q^{31} +6.81168e8 q^{32} +6.32644e7i q^{33} -9.85571e8i q^{34} -3.95732e8 q^{35} -1.04450e9 q^{36} -3.52367e9i q^{37} -2.40035e9i q^{38} +2.75281e9 q^{39} -7.44699e9i q^{40} -4.39683e9 q^{41} -9.02908e8i q^{42} +1.17964e10i q^{43} +7.57460e7i q^{44} +1.99860e10i q^{45} +(1.49960e9 + 7.59130e9i) q^{46} -6.12026e9 q^{47} +1.06288e10 q^{48} +1.36113e10 q^{49} -2.28309e10 q^{50} +2.14761e10i q^{51} +3.29592e9 q^{52} -3.46235e10i q^{53} -1.39597e10 q^{54} +1.44937e9 q^{55} -4.32799e9i q^{56} +5.23049e10i q^{57} -6.22052e7 q^{58} -5.57560e10 q^{59} +4.05327e10i q^{60} -2.03921e10i q^{61} +7.30894e10 q^{62} +1.16153e10i q^{63} +7.38277e10 q^{64} -6.30661e10i q^{65} +3.30690e9i q^{66} +6.67992e10i q^{67} +2.57131e10i q^{68} +(-3.26771e10 - 1.65418e11i) q^{69} -2.06853e10 q^{70} +3.36649e8 q^{71} -2.18580e11 q^{72} -1.87523e11 q^{73} -1.84186e11i q^{74} +4.97498e11 q^{75} +6.26242e10i q^{76} +8.42333e8 q^{77} +1.43893e11 q^{78} -4.42058e11i q^{79} -2.43503e11i q^{80} -1.02847e11 q^{81} -2.29827e11 q^{82} +7.73175e10i q^{83} +2.35565e10i q^{84} +4.92010e11 q^{85} +6.16611e11i q^{86} +1.35548e9 q^{87} +1.58513e10i q^{88} -4.81209e11i q^{89} +1.04469e12i q^{90} -3.66523e10i q^{91} +(-3.91240e10 - 1.98054e11i) q^{92} -1.59265e12 q^{93} -3.19913e11 q^{94} +1.19829e12 q^{95} -7.75860e11 q^{96} +5.07361e11i q^{97} +7.11478e11 q^{98} -4.25411e10i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) $ 20 * q + 88 * q^2 + 318 * q^3 + 36072 * q^4 + 264240 * q^6 - 767728 * q^8 + 1971426 * q^9	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9} + 19833480 q^{12} + 8049118 q^{13} + 81743632 q^{16} + 36238080 q^{18} - 367189708 q^{23} + 530348736 q^{24} - 1824317212 q^{25} - 2465696728 q^{26} - 1765163178 q^{27} - 417897362 q^{29} - 1574125298 q^{31} + 1240884960 q^{32} + 4723363152 q^{35} + 3607269840 q^{36} + 1926187110 q^{39} - 132511250 q^{41} + 62599086544 q^{46} - 45052376402 q^{47} + 77275023312 q^{48} - 65534183260 q^{49} - 57446928200 q^{50} - 20407259400 q^{52} - 15082009344 q^{54} - 12572534832 q^{55} + 97529544392 q^{58} - 110336269112 q^{59} + 553127354432 q^{62} + 13799515808 q^{64} + 7914588558 q^{69} + 590935322064 q^{70} - 40523102210 q^{71} - 539385055680 q^{72} - 449748966242 q^{73} + 2570431278 q^{75} - 345974135760 q^{77} - 48894418992 q^{78} - 2694979782144 q^{81} - 1210586085208 q^{82} + 1051198266576 q^{85} - 2237359632618 q^{87} + 1936236755640 q^{92} - 116415989178 q^{93} + 3420883097024 q^{94} + 3777482201184 q^{95} + 3567911213376 q^{96} + 3590387133208 q^{98}+O(q^{100}) \) 20 * q + 88 * q^2 + 318 * q^3 + 36072 * q^4 + 264240 * q^6 - 767728 * q^8 + 1971426 * q^9 + 19833480 * q^12 + 8049118 * q^13 + 81743632 * q^16 + 36238080 * q^18 - 367189708 * q^23 + 530348736 * q^24 - 1824317212 * q^25 - 2465696728 * q^26 - 1765163178 * q^27 - 417897362 * q^29 - 1574125298 * q^31 + 1240884960 * q^32 + 4723363152 * q^35 + 3607269840 * q^36 + 1926187110 * q^39 - 132511250 * q^41 + 62599086544 * q^46 - 45052376402 * q^47 + 77275023312 * q^48 - 65534183260 * q^49 - 57446928200 * q^50 - 20407259400 * q^52 - 15082009344 * q^54 - 12572534832 * q^55 + 97529544392 * q^58 - 110336269112 * q^59 + 553127354432 * q^62 + 13799515808 * q^64 + 7914588558 * q^69 + 590935322064 * q^70 - 40523102210 * q^71 - 539385055680 * q^72 - 449748966242 * q^73 + 2570431278 * q^75 - 345974135760 * q^77 - 48894418992 * q^78 - 2694979782144 * q^81 - 1210586085208 * q^82 + 1051198266576 * q^85 - 2237359632618 * q^87 + 1936236755640 * q^92 - 116415989178 * q^93 + 3420883097024 * q^94 + 3777482201184 * q^95 + 3567911213376 * q^96 + 3590387133208 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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 $

$2$	52.2711			0.816736			0.408368	−	0.912817i	$-0.366098\pi$
$2$	52.2711			0.816736			0.408368	+	0.912817i	$0.366098\pi$
$3$	−1139.01			−1.56243			−0.781216	−	0.624260i	$-0.785400\pi$
$3$	−1139.01			−1.56243			−0.781216	+	0.624260i	$0.785400\pi$
$4$	−1363.73			−0.332942
$5$			26094.4i			1.67004i	0.550216	+	0.835022i	$0.314545\pi$
$5$			26094.4i			1.67004i	−0.550216	+	0.835022i	$0.685455\pi$
$6$	−59537.5			−1.27610
$7$			15165.4i			0.128904i	0.997921	+	0.0644518i	$0.0205299\pi$
$7$			15165.4i			0.128904i	−0.997921	+	0.0644518i	$0.979470\pi$
$8$	−285386.			−1.08866
$9$	765911.			1.44120
$10$			1.36399e6i			1.36399i
$11$		−	55543.2i		−	0.0313527i	−0.999877	−	0.0156763i	$-0.995010\pi$
$11$		−	55543.2i		−	0.0313527i	0.999877	−	0.0156763i	$-0.00499014\pi$
$12$	1.55331e6			0.520200
$13$	−2.41684e6			−0.500711			−0.250356	−	0.968154i	$-0.580548\pi$
$13$	−2.41684e6			−0.500711			−0.250356	+	0.968154i	$0.580548\pi$
$14$			792711.i			0.105280i
$15$		−	2.97219e7i		−	2.60933i
$16$	−9.33161e6			−0.556207
$17$		−	1.88550e7i		−	0.781147i	−0.920572	−	0.390573i	$-0.872277\pi$
$17$		−	1.88550e7i		−	0.781147i	0.920572	−	0.390573i	$-0.127723\pi$
$18$	4.00350e7			1.17708
$19$		−	4.59212e7i		−	0.976095i	−0.872817	−	0.488047i	$-0.837709\pi$
$19$		−	4.59212e7i		−	0.976095i	0.872817	−	0.488047i	$-0.162291\pi$
$20$		−	3.55858e7i		−	0.556028i
$21$		−	1.72736e7i		−	0.201403i
$22$		−	2.90330e6i		−	0.0256069i
$23$	2.86889e7	+	1.45229e8i	0.193797	+	0.981042i
$23$	2.86889e7	+	1.45229e8i	0.193797	+	0.981042i
$24$	3.25059e8			1.70096
$25$	−4.36779e8			−1.78905
$26$	−1.26331e8			−0.408949
$27$	−2.67064e8			−0.689339
$28$		−	2.06815e7i		−	0.0429174i
$29$	−1.19005e6			−0.00200068			−0.00100034	−	0.999999i	$-0.500318\pi$
$29$	−1.19005e6			−0.00200068			−0.00100034	+	0.999999i	$0.500318\pi$
$30$		−	1.55360e9i		−	2.13114i
$31$	1.39827e9			1.57551			0.787757	−	0.615986i	$-0.211242\pi$
$31$	1.39827e9			1.57551			0.787757	+	0.615986i	$0.211242\pi$
$32$	6.81168e8			0.634387
$33$			6.32644e7i			0.0489865i
$34$		−	9.85571e8i		−	0.637991i
$35$	−3.95732e8			−0.215275
$36$	−1.04450e9			−0.479835
$37$		−	3.52367e9i		−	1.37336i	−0.726958	−	0.686682i	$-0.759067\pi$
$37$		−	3.52367e9i		−	1.37336i	0.726958	−	0.686682i	$-0.240933\pi$
$38$		−	2.40035e9i		−	0.797212i
$39$	2.75281e9			0.782328
$40$		−	7.44699e9i		−	1.81811i
$41$	−4.39683e9			−0.925628			−0.462814	−	0.886455i	$-0.653160\pi$
$41$	−4.39683e9			−0.925628			−0.462814	+	0.886455i	$0.653160\pi$
$42$		−	9.02908e8i		−	0.164493i
$43$			1.17964e10i			1.86612i	0.359726	+	0.933058i	$0.382870\pi$
$43$			1.17964e10i			1.86612i	−0.359726	+	0.933058i	$0.617130\pi$
$44$			7.57460e7i			0.0104386i
$45$			1.99860e10i			2.40686i
$46$	1.49960e9	+	7.59130e9i	0.158281	+	0.801252i
$47$	−6.12026e9			−0.567784			−0.283892	−	0.958856i	$-0.591626\pi$
$47$	−6.12026e9			−0.567784			−0.283892	+	0.958856i	$0.591626\pi$
$48$	1.06288e10			0.869037
$49$	1.36113e10			0.983384
$50$	−2.28309e10			−1.46118
$51$			2.14761e10i			1.22049i
$52$	3.29592e9			0.166708
$53$		−	3.46235e10i		−	1.56213i	−0.624452	−	0.781063i	$-0.714678\pi$
$53$		−	3.46235e10i		−	1.56213i	0.624452	−	0.781063i	$-0.285322\pi$
$54$	−1.39597e10			−0.563008
$55$	1.44937e9			0.0523604
$56$		−	4.32799e9i		−	0.140332i
$57$			5.23049e10i			1.52508i
$58$	−6.22052e7			−0.00163403
$59$	−5.57560e10			−1.32184			−0.660921	−	0.750455i	$-0.729834\pi$
$59$	−5.57560e10			−1.32184			−0.660921	+	0.750455i	$0.729834\pi$
$60$			4.05327e10i			0.868756i
$61$		−	2.03921e10i		−	0.395806i	−0.980222	−	0.197903i	$-0.936587\pi$
$61$		−	2.03921e10i		−	0.395806i	0.980222	−	0.197903i	$-0.0634131\pi$
$62$	7.30894e10			1.28678
$63$			1.16153e10i			0.185775i
$64$	7.38277e10			1.07433
$65$		−	6.30661e10i		−	0.836210i
$66$			3.30690e9i			0.0400090i
$67$			6.67992e10i			0.738452i	0.929340	+	0.369226i	$0.120377\pi$
$67$			6.67992e10i			0.738452i	−0.929340	+	0.369226i	$0.879623\pi$
$68$			2.57131e10i			0.260077i
$69$	−3.26771e10	−	1.65418e11i	−0.302795	−	1.53281i
$70$	−2.06853e10			−0.175823
$71$	3.36649e8			0.00262801			0.00131401	−	0.999999i	$-0.499582\pi$
$71$	3.36649e8			0.00262801			0.00131401	+	0.999999i	$0.499582\pi$
$72$	−2.18580e11			−1.56898
$73$	−1.87523e11			−1.23913			−0.619564	−	0.784946i	$-0.712691\pi$
$73$	−1.87523e11			−1.23913			−0.619564	+	0.784946i	$0.712691\pi$
$74$		−	1.84186e11i		−	1.12168i
$75$	4.97498e11			2.79527
$76$			6.26242e10i			0.324983i
$77$	8.42333e8			0.00404147
$78$	1.43893e11			0.638955
$79$		−	4.42058e11i		−	1.81851i	−0.416236	−	0.909257i	$-0.636651\pi$
$79$		−	4.42058e11i		−	1.81851i	0.416236	−	0.909257i	$-0.363349\pi$
$80$		−	2.43503e11i		−	0.928891i
$81$	−1.02847e11			−0.364151
$82$	−2.29827e11			−0.755994
$83$			7.73175e10i			0.236488i	0.992985	+	0.118244i	$0.0377265\pi$
$83$			7.73175e10i			0.236488i	−0.992985	+	0.118244i	$0.962273\pi$
$84$			2.35565e10i			0.0670556i
$85$	4.92010e11			1.30455
$86$			6.16611e11i			1.52412i
$87$	1.35548e9			0.00312592
$88$			1.58513e10i			0.0341325i
$89$		−	4.81209e11i		−	0.968264i	−0.874995	−	0.484132i	$-0.839135\pi$
$89$		−	4.81209e11i		−	0.968264i	0.874995	−	0.484132i	$-0.160865\pi$
$90$			1.04469e12i			1.96577i
$91$		−	3.66523e10i		−	0.0645435i
$92$	−3.91240e10	−	1.98054e11i	−0.0645232	−	0.326630i
$93$	−1.59265e12			−2.46164
$94$	−3.19913e11			−0.463730
$95$	1.19829e12			1.63012
$96$	−7.75860e11			−0.991187
$97$			5.07361e11i			0.609097i	0.952497	+	0.304549i	$0.0985056\pi$
$97$			5.07361e11i			0.609097i	−0.952497	+	0.304549i	$0.901494\pi$
$98$	7.11478e11			0.803165
$99$		−	4.25411e10i		−	0.0451853i
$100$	5.95650e11			0.595650
$101$	−4.32134e11			−0.407090			−0.203545	−	0.979066i	$-0.565246\pi$
$101$	−4.32134e11			−0.407090			−0.203545	+	0.979066i	$0.565246\pi$
$102$			1.12258e12i			0.996817i
$103$			3.50518e11i			0.293553i	0.989170	+	0.146777i	$0.0468898\pi$
$103$			3.50518e11i			0.293553i	−0.989170	+	0.146777i	$0.953110\pi$
$104$	6.89732e11			0.545106
$105$	4.50744e11			0.336352
$106$		−	1.80981e12i		−	1.27584i
$107$		−	2.42705e12i		−	1.61724i	−0.588328	−	0.808622i	$-0.700214\pi$
$107$		−	2.42705e12i		−	1.61724i	0.588328	−	0.808622i	$-0.299786\pi$
$108$	3.64203e11			0.229510
$109$		−	7.43557e11i		−	0.443359i	−0.975120	−	0.221679i	$-0.928846\pi$
$109$		−	7.43557e11i		−	0.443359i	0.975120	−	0.221679i	$-0.0711538\pi$
$110$	7.57601e10			0.0427646
$111$			4.01351e12i			2.14579i
$112$		−	1.41517e11i		−	0.0716971i
$113$		−	1.06174e12i		−	0.509971i	−0.966945	−	0.254986i	$-0.917929\pi$
$113$		−	1.06174e12i		−	0.509971i	0.966945	−	0.254986i	$-0.0820707\pi$
$114$			2.73404e12i			1.24559i
$115$	−3.78968e12	+	7.48622e11i	−1.63838	+	0.323650i
$116$	1.62291e9			0.000666110
$117$	−1.85108e12			−0.721623
$118$	−2.91443e12			−1.07960
$119$	2.85943e11			0.100693
$120$			8.48223e12i			2.84068i
$121$	3.13534e12			0.999017
$122$		−	1.06592e12i		−	0.323269i
$123$	5.00805e12			1.44623
$124$	−1.90687e12			−0.524555
$125$		−	5.02680e12i		−	1.31775i
$126$			6.07146e11i			0.151729i
$127$	7.37307e12			1.75722			0.878610	−	0.477540i	$-0.158471\pi$
$127$	7.37307e12			1.75722			0.878610	+	0.477540i	$0.158471\pi$
$128$	1.06899e12			0.243061
$129$		−	1.34363e13i		−	2.91568i
$130$		−	3.29653e12i		−	0.682963i
$131$	−7.36654e12			−1.45759			−0.728795	−	0.684732i	$-0.759919\pi$
$131$	−7.36654e12			−1.45759			−0.728795	+	0.684732i	$0.759919\pi$
$132$		−	8.62757e10i		−	0.0163097i
$133$	6.96413e11			0.125822
$134$			3.49167e12i			0.603121i
$135$		−	6.96888e12i		−	1.15123i
$136$			5.38095e12i			0.850405i
$137$		−	8.46093e12i		−	1.27966i	−0.768516	−	0.639830i	$-0.779005\pi$
$137$		−	8.46093e12i		−	1.27966i	0.768516	−	0.639830i	$-0.220995\pi$
$138$	−1.70807e12	−	8.64659e12i	−0.247304	−	1.25190i
$139$	3.98243e12			0.552153			0.276077	−	0.961136i	$-0.410966\pi$
$139$	3.98243e12			0.552153			0.276077	+	0.961136i	$0.410966\pi$
$140$	5.39672e11			0.0716740
$141$	6.97106e12			0.887124
$142$	1.75970e10			0.00214639
$143$			1.34239e11i			0.0156986i
$144$	−7.14718e12			−0.801604
$145$		−	3.10537e10i		−	0.00334122i
$146$	−9.80201e12			−1.01204
$147$	−1.55035e13			−1.53647
$148$			4.80534e12i			0.457250i
$149$			4.02536e11i			0.0367864i	0.999831	+	0.0183932i	$0.00585507\pi$
$149$			4.02536e11i			0.0367864i	−0.999831	+	0.0183932i	$0.994145\pi$
$150$	2.60048e13			2.28300
$151$	1.01082e13			0.852728			0.426364	−	0.904552i	$-0.359794\pi$
$151$	1.01082e13			0.852728			0.426364	+	0.904552i	$0.359794\pi$
$152$			1.31053e13i			1.06264i
$153$		−	1.44412e13i		−	1.12579i
$154$	4.40297e10			0.00330082
$155$			3.64872e13i			2.63118i
$156$	−3.75409e12			−0.260470
$157$			1.33865e13i			0.893859i	0.894569	+	0.446929i	$0.147482\pi$
$157$			1.33865e13i			0.893859i	−0.894569	+	0.446929i	$0.852518\pi$
$158$		−	2.31069e13i		−	1.48525i
$159$			3.94367e13i			2.44072i
$160$			1.77747e13i			1.05945i
$161$	−2.20246e12	+	4.35078e11i	−0.126460	+	0.0249811i
$162$	−5.37592e12			−0.297415
$163$	−2.64423e13			−1.40985			−0.704925	−	0.709282i	$-0.749019\pi$
$163$	−2.64423e13			−1.40985			−0.704925	+	0.709282i	$0.749019\pi$
$164$	5.99609e12			0.308181
$165$	−1.65085e12			−0.0818095
$166$			4.04147e12i			0.193148i
$167$	−2.34497e13			−1.08103			−0.540516	−	0.841334i	$-0.681771\pi$
$167$	−2.34497e13			−1.08103			−0.540516	+	0.841334i	$0.681771\pi$
$168$			4.92964e12i			0.219260i
$169$	−1.74570e13			−0.749288
$170$	2.57179e13			1.06547
$171$		−	3.51716e13i		−	1.40674i
$172$		−	1.60871e13i		−	0.621309i
$173$	1.12922e13			0.421214			0.210607	−	0.977571i	$-0.432456\pi$
$173$	1.12922e13			0.421214			0.210607	+	0.977571i	$0.432456\pi$
$174$	7.08526e10			0.00255305
$175$		−	6.62392e12i		−	0.230615i
$176$			5.18307e11i			0.0174386i
$177$	6.35069e13			2.06529
$178$		−	2.51533e13i		−	0.790816i
$179$	−2.54744e13			−0.774437			−0.387218	−	0.921988i	$-0.626564\pi$
$179$	−2.54744e13			−0.774437			−0.387218	+	0.921988i	$0.626564\pi$
$180$		−	2.72555e13i		−	0.801345i
$181$			4.34417e13i			1.23548i	0.786383	+	0.617739i	$0.211951\pi$
$181$			4.34417e13i			1.23548i	−0.786383	+	0.617739i	$0.788049\pi$
$182$		−	1.91585e12i		−	0.0527150i
$183$			2.32268e13i			0.618420i
$184$	−8.18743e12	−	4.14465e13i	−0.210980	−	1.06802i
$185$	9.19483e13			2.29358
$186$	−8.32498e13			−2.01051
$187$	−1.04727e12			−0.0244910
$188$	8.34639e12			0.189039
$189$		−	4.05012e12i		−	0.0888582i
$190$	6.26359e13			1.33138
$191$		−	3.34306e13i		−	0.688564i	−0.938866	−	0.344282i	$-0.888122\pi$
$191$		−	3.34306e13i		−	0.688564i	0.938866	−	0.344282i	$-0.111878\pi$
$192$	−8.40907e13			−1.67858
$193$	−4.37125e13			−0.845788			−0.422894	−	0.906179i	$-0.638986\pi$
$193$	−4.37125e13			−0.845788			−0.422894	+	0.906179i	$0.638986\pi$
$194$			2.65203e13i			0.497472i
$195$			7.18331e13i			1.30652i
$196$	−1.85621e13			−0.327410
$197$	−4.32949e13			−0.740695			−0.370348	−	0.928893i	$-0.620761\pi$
$197$	−4.32949e13			−0.740695			−0.370348	+	0.928893i	$0.620761\pi$
$198$		−	2.22367e12i		−	0.0369045i
$199$		−	6.07458e13i		−	0.978133i	−0.872246	−	0.489067i	$-0.837338\pi$
$199$		−	6.07458e13i		−	0.978133i	0.872246	−	0.489067i	$-0.162662\pi$
$200$	1.24651e14			1.94767
$201$		−	7.60852e13i		−	1.15378i
$202$	−2.25881e13			−0.332485
$203$		−	1.80475e10i		−	0.000257894i
$204$		−	2.92876e13i		−	0.406352i
$205$		−	1.14733e14i		−	1.54584i
$206$			1.83219e13i			0.239755i
$207$	2.19732e13	+	1.11233e14i	0.279300	+	1.41387i
$208$	2.25530e13			0.278499
$209$	−2.55061e12			−0.0306032
$210$	2.35609e13			0.274711
$211$	9.17364e13			1.03955			0.519777	−	0.854302i	$-0.326015\pi$
$211$	9.17364e13			1.03955			0.519777	+	0.854302i	$0.326015\pi$
$212$			4.72172e13i			0.520097i
$213$	−3.83448e11			−0.00410609
$214$		−	1.26865e14i		−	1.32086i
$215$	−3.07820e14			−3.11650
$216$	7.62164e13			0.750457
$217$			2.12054e13i			0.203089i
$218$		−	3.88665e13i		−	0.362107i
$219$	2.13591e14			1.93605
$220$	−1.97655e12			−0.0174330
$221$			4.55694e13i			0.391129i
$222$			2.09791e14i			1.75254i
$223$	1.78831e14			1.45416			0.727082	−	0.686551i	$-0.240876\pi$
$223$	1.78831e14			1.45416			0.727082	+	0.686551i	$0.240876\pi$
$224$			1.03302e13i			0.0817747i
$225$	−3.34534e14			−2.57837
$226$		−	5.54981e13i		−	0.416512i
$227$			2.23661e14i			1.63469i	0.576151	+	0.817343i	$0.304554\pi$
$227$			2.23661e14i			1.63469i	−0.576151	+	0.817343i	$0.695446\pi$
$228$		−	7.13298e13i		−	0.507764i
$229$		−	2.57413e14i		−	1.78491i	−0.451133	−	0.892456i	$-0.648980\pi$
$229$		−	2.57413e14i		−	1.78491i	0.451133	−	0.892456i	$-0.351020\pi$
$230$	−1.98091e14	+	3.91313e13i	−1.33813	+	0.264337i
$231$	−9.59429e11			−0.00631453
$232$	3.39624e11			0.00217806
$233$	2.23041e13			0.139396			0.0696978	−	0.997568i	$-0.477796\pi$
$233$	2.23041e13			0.139396			0.0696978	+	0.997568i	$0.477796\pi$
$234$	−9.67581e13			−0.589376
$235$		−	1.59705e14i		−	0.948224i
$236$	7.60362e13			0.440097
$237$			5.03510e14i			2.84130i
$238$	1.49465e13			0.0822392
$239$	−1.96517e13			−0.105442			−0.0527209	−	0.998609i	$-0.516789\pi$
$239$	−1.96517e13			−0.105442			−0.0527209	+	0.998609i	$0.516789\pi$
$240$			2.77353e14i			1.45133i
$241$		−	1.70339e14i		−	0.869384i	−0.900579	−	0.434692i	$-0.856857\pi$
$241$		−	1.70339e14i		−	0.869384i	0.900579	−	0.434692i	$-0.143143\pi$
$242$	1.63888e14			0.815933
$243$	2.59073e14			1.25830
$244$			2.78093e13i			0.131780i
$245$			3.55179e14i			1.64229i
$246$	2.61776e14			1.18119
$247$			1.10984e14i			0.488742i
$248$	−3.99048e14			−1.71520
$249$		−	8.80657e13i		−	0.369497i
$250$		−	2.62757e14i		−	1.07625i
$251$		−	7.38336e13i		−	0.295265i	−0.989042	−	0.147632i	$-0.952835\pi$
$251$		−	7.38336e13i		−	0.295265i	0.989042	−	0.147632i	$-0.0471653\pi$
$252$		−	1.58402e13i		−	0.0618524i
$253$	8.06650e12	−	1.59347e12i	0.0307583	−	0.00607606i
$254$	3.85399e14			1.43518
$255$	−5.60406e14			−2.03827
$256$	−2.46521e14			−0.875818
$257$	9.76832e13			0.339017			0.169508	−	0.985529i	$-0.445782\pi$
$257$	9.76832e13			0.339017			0.169508	+	0.985529i	$0.445782\pi$
$258$		−	7.02328e14i		−	2.38134i
$259$	5.34378e13			0.177031
$260$			8.60051e13i			0.278410i
$261$	−9.11472e11			−0.00288337
$262$	−3.85057e14			−1.19047
$263$		−	1.92657e14i		−	0.582170i	−0.956697	−	0.291085i	$-0.905984\pi$
$263$		−	1.92657e14i		−	0.582170i	0.956697	−	0.291085i	$-0.0940162\pi$
$264$		−	1.80548e13i		−	0.0533297i
$265$	9.03482e14			2.60882
$266$	3.64023e13			0.102763
$267$			5.48104e14i			1.51285i
$268$		−	9.10961e13i		−	0.245862i
$269$	−5.73155e14			−1.51272			−0.756360	−	0.654156i	$-0.773024\pi$
$269$	−5.73155e14			−1.51272			−0.756360	+	0.654156i	$0.773024\pi$
$270$		−	3.64271e14i		−	0.940248i
$271$	−8.83168e13			−0.222960			−0.111480	−	0.993767i	$-0.535559\pi$
$271$	−8.83168e13			−0.222960			−0.111480	+	0.993767i	$0.535559\pi$
$272$			1.75947e14i			0.434480i
$273$			4.17474e13i			0.100845i
$274$		−	4.42262e14i		−	1.04514i
$275$			2.42601e13i			0.0560915i
$276$	4.45627e13	+	2.25586e14i	0.100813	+	0.510337i
$277$	−3.08044e14			−0.681920			−0.340960	−	0.940078i	$-0.610752\pi$
$277$	−3.08044e14			−0.681920			−0.340960	+	0.940078i	$0.610752\pi$
$278$	2.08166e14			0.450963
$279$	1.07095e15			2.27062
$280$	1.12936e14			0.234361
$281$			5.45211e14i			1.10746i	0.832697	+	0.553728i	$0.186795\pi$
$281$			5.45211e14i			1.10746i	−0.832697	+	0.553728i	$0.813205\pi$
$282$	3.64385e14			0.724546
$283$		−	5.44216e14i		−	1.05938i	−0.848191	−	0.529691i	$-0.822308\pi$
$283$		−	5.44216e14i		−	1.05938i	0.848191	−	0.529691i	$-0.177692\pi$
$284$	−4.59099e11			−0.000874976
$285$	−1.36487e15			−2.54696
$286$			7.01682e12i			0.0128217i
$287$		−	6.66795e13i		−	0.119317i
$288$	5.21714e14			0.914276
$289$	2.27112e14			0.389810
$290$		−	1.62321e12i		−	0.00272890i
$291$		−	5.77891e14i		−	0.951673i
$292$	2.55730e14			0.412558
$293$		−	4.19965e14i		−	0.663755i	−0.943322	−	0.331878i	$-0.892318\pi$
$293$		−	4.19965e14i		−	0.663755i	0.943322	−	0.331878i	$-0.107682\pi$
$294$	−8.10383e14			−1.25489
$295$		−	1.45492e15i		−	2.20754i
$296$			1.00561e15i			1.49513i
$297$			1.48336e13i			0.0216126i
$298$			2.10410e13i			0.0300448i
$299$	−6.93365e13	−	3.50996e14i	−0.0970365	−	0.491219i
$300$	−6.78453e14			−0.930662
$301$	−1.78897e14			−0.240549
$302$	5.28365e14			0.696454
$303$	4.92207e14			0.636051
$304$			4.28519e14i			0.542911i
$305$	5.32119e14			0.661013
$306$		−	7.54859e14i		−	0.919469i
$307$	−1.01845e15			−1.21649			−0.608245	−	0.793749i	$-0.708126\pi$
$307$	−1.01845e15			−1.21649			−0.608245	+	0.793749i	$0.708126\pi$
$308$	−1.14872e12			−0.00134558
$309$		−	3.99244e14i		−	0.458657i
$310$			1.90723e15i			2.14898i
$311$	8.58928e14			0.949280			0.474640	−	0.880180i	$-0.342578\pi$
$311$	8.58928e14			0.949280			0.474640	+	0.880180i	$0.342578\pi$
$312$	−7.85614e14			−0.851691
$313$		−	1.20366e15i		−	1.28008i	−0.768340	−	0.640042i	$-0.778917\pi$
$313$		−	1.20366e15i		−	1.28008i	0.768340	−	0.640042i	$-0.221083\pi$
$314$			6.99727e14i			0.730047i
$315$	−3.03095e14			−0.310253
$316$			6.02848e14i			0.605460i
$317$	1.35177e15			1.33213			0.666065	−	0.745893i	$-0.267977\pi$
$317$	1.35177e15			1.33213			0.666065	+	0.745893i	$0.267977\pi$
$318$			2.06140e15i			1.99342i
$319$			6.60991e10i			6.27266e-5i
$320$			1.92649e15i			1.79419i
$321$			2.76444e15i			2.52684i
$322$	−1.15125e14	+	2.27420e13i	−0.103284	+	0.0204030i
$323$	−8.65844e14			−0.762473
$324$	1.40255e14			0.121241
$325$	1.05563e15			0.895797
$326$	−1.38217e15			−1.15148
$327$			8.46921e14i			0.692718i
$328$	1.25479e15			1.00770
$329$		−	9.28161e13i		−	0.0731893i
$330$	−8.62918e13			−0.0668168
$331$	−6.93332e14			−0.527198			−0.263599	−	0.964632i	$-0.584909\pi$
$331$	−6.93332e14			−0.527198			−0.263599	+	0.964632i	$0.584909\pi$
$332$		−	1.05440e14i		−	0.0787368i
$333$		−	2.69882e15i		−	1.97929i
$334$	−1.22574e15			−0.882917
$335$	−1.74309e15			−1.23325
$336$			1.61190e14i			0.112022i
$337$		−	6.94329e14i		−	0.474009i	−0.971509	−	0.237004i	$-0.923834\pi$
$337$		−	6.94329e14i		−	0.474009i	0.971509	−	0.237004i	$-0.0761655\pi$
$338$	−9.12496e14			−0.611971
$339$			1.20933e15i			0.796796i
$340$	−6.70970e14			−0.434339
$341$		−	7.76646e13i		−	0.0493966i
$342$		−	1.83846e15i		−	1.14894i
$343$			4.16329e14i			0.255665i
$344$		−	3.36653e15i		−	2.03157i
$345$	4.31650e15	−	8.52690e14i	2.55986	−	0.505681i
$346$	5.90258e14			0.344021
$347$	−5.45758e14			−0.312624			−0.156312	−	0.987708i	$-0.549961\pi$
$347$	−5.45758e14			−0.312624			−0.156312	+	0.987708i	$0.549961\pi$
$348$	−1.84851e12			−0.00104075
$349$	4.17304e14			0.230941			0.115470	−	0.993311i	$-0.463162\pi$
$349$	4.17304e14			0.230941			0.115470	+	0.993311i	$0.463162\pi$
$350$		−	3.46240e14i		−	0.188351i
$351$	6.45450e14			0.345160
$352$		−	3.78342e13i		−	0.0198897i
$353$	8.09282e14			0.418265			0.209133	−	0.977887i	$-0.432936\pi$
$353$	8.09282e14			0.418265			0.209133	+	0.977887i	$0.432936\pi$
$354$	3.31958e15			1.68680
$355$			8.78467e12i			0.00438890i
$356$			6.56240e14i			0.322376i
$357$	−3.25693e14			−0.157325
$358$	−1.33158e15			−0.632511
$359$		−	9.66526e14i		−	0.451489i	−0.974187	−	0.225744i	$-0.927519\pi$
$359$		−	9.66526e14i		−	0.451489i	0.974187	−	0.225744i	$-0.0724814\pi$
$360$		−	5.70373e15i		−	2.62026i
$361$	1.04555e14			0.0472390
$362$			2.27075e15i			1.00906i
$363$	−3.57120e15			−1.56090
$364$			4.99838e13i			0.0214892i
$365$		−	4.89330e15i		−	2.06940i
$366$			1.21409e15i			0.505086i
$367$		−	1.57782e15i		−	0.645746i	−0.946442	−	0.322873i	$-0.895351\pi$
$367$		−	1.57782e15i		−	0.645746i	0.946442	−	0.322873i	$-0.104649\pi$
$368$	−2.67714e14	−	1.35522e15i	−0.107791	−	0.545663i
$369$	−3.36758e15			−1.33401
$370$	4.80624e15			1.87325
$371$	5.25079e14			0.201364
$372$	2.17195e15			0.819582
$373$			1.50574e15i			0.559110i	0.960130	+	0.279555i	$0.0901869\pi$
$373$			1.50574e15i			0.559110i	−0.960130	+	0.279555i	$0.909813\pi$
$374$	−5.47417e13			−0.0200027
$375$			5.72559e15i			2.05889i
$376$	1.74664e15			0.618125
$377$	2.87616e12			0.00100176
$378$		−	2.11704e14i		−	0.0725737i
$379$			9.35406e14i			0.315620i	0.987469	+	0.157810i	$0.0504434\pi$
$379$			9.35406e14i			0.315620i	−0.987469	+	0.157810i	$0.949557\pi$
$380$	−1.63414e15			−0.542736
$381$	−8.39803e15			−2.74554
$382$		−	1.74746e15i		−	0.562375i
$383$			4.42391e15i			1.40157i	0.713375	+	0.700783i	$0.247166\pi$
$383$			4.42391e15i			1.40157i	−0.713375	+	0.700783i	$0.752834\pi$
$384$	−1.21760e15			−0.379766
$385$			2.19802e13i			0.00674944i
$386$	−2.28490e15			−0.690786
$387$			9.03498e15i			2.68944i
$388$		−	6.91904e14i		−	0.202794i
$389$		−	8.57904e14i		−	0.247594i	−0.992308	−	0.123797i	$-0.960493\pi$
$389$		−	8.57904e14i		−	0.247594i	0.992308	−	0.123797i	$-0.0395072\pi$
$390$			3.75480e15i			1.06708i
$391$	2.73830e15	−	5.40929e14i	0.766337	−	0.151384i
$392$	−3.88448e15			−1.07057
$393$	8.39058e15			2.27739
$394$	−2.26307e15			−0.604952
$395$	1.15353e16			3.03700
$396$			5.80146e13i			0.0150441i
$397$	−5.61516e15			−1.43423			−0.717115	−	0.696954i	$-0.754538\pi$
$397$	−5.61516e15			−1.43423			−0.717115	+	0.696954i	$0.754538\pi$
$398$		−	3.17525e15i		−	0.798877i
$399$	−7.93223e14			−0.196588
$400$	4.07586e15			0.995082
$401$		−	4.59305e15i		−	1.10468i	−0.833620	−	0.552338i	$-0.813736\pi$
$401$		−	4.59305e15i		−	1.10468i	0.833620	−	0.552338i	$-0.186264\pi$
$402$		−	3.97706e15i		−	0.942335i
$403$	−3.37940e15			−0.788878
$404$	5.89315e14			0.135537
$405$		−	2.68373e15i		−	0.608148i
$406$		−	9.43365e11i		−	0.000210632i
$407$	−1.95716e14			−0.0430586
$408$		−	6.12898e15i		−	1.32870i
$409$	7.85596e14			0.167826			0.0839130	−	0.996473i	$-0.473258\pi$
$409$	7.85596e14			0.167826			0.0839130	+	0.996473i	$0.473258\pi$
$410$		−	5.99721e15i		−	1.26254i
$411$			9.63711e15i			1.99938i
$412$		−	4.78012e14i		−	0.0977362i
$413$		−	8.45561e14i		−	0.170390i
$414$	1.14856e15	+	5.81426e15i	0.228114	+	1.15476i
$415$	−2.01756e15			−0.394946
$416$	−1.64627e15			−0.317645
$417$	−4.53604e15			−0.862702
$418$	−1.33323e14			−0.0249947
$419$			6.10925e15i			1.12903i	0.825424	+	0.564513i	$0.190936\pi$
$419$			6.10925e15i			1.12903i	−0.825424	+	0.564513i	$0.809064\pi$
$420$	−6.14693e14			−0.111986
$421$		−	5.51985e15i		−	0.991367i	−0.868503	−	0.495684i	$-0.834918\pi$
$421$		−	5.51985e15i		−	0.991367i	0.868503	−	0.495684i	$-0.165082\pi$
$422$	4.79517e15			0.849041
$423$	−4.68757e15			−0.818288
$424$			9.88108e15i			1.70063i
$425$			8.23547e15i			1.39751i
$426$	−2.00433e13			−0.00335359
$427$	3.09253e14			0.0510207
$428$			3.30984e15i			0.538449i
$429$		−	1.52900e14i		−	0.0245281i
$430$	−1.60901e16			−2.54536
$431$		−	9.05481e14i		−	0.141259i	−0.997503	−	0.0706294i	$-0.977499\pi$
$431$		−	9.05481e14i		−	0.141259i	0.997503	−	0.0706294i	$-0.0225008\pi$
$432$	2.49214e15			0.383415
$433$		−	6.19067e15i		−	0.939313i	−0.882849	−	0.469656i	$-0.844378\pi$
$433$		−	6.19067e15i		−	0.939313i	0.882849	−	0.469656i	$-0.155622\pi$
$434$			1.10843e15i			0.165870i
$435$			3.53706e13i			0.00522043i
$436$			1.01401e15i			0.147613i
$437$	6.66911e15	−	1.31743e15i	0.957590	−	0.189164i
$438$	1.11646e16			1.58125
$439$	−4.75973e15			−0.664959			−0.332480	−	0.943110i	$-0.607885\pi$
$439$	−4.75973e15			−0.664959			−0.332480	+	0.943110i	$0.607885\pi$
$440$	−4.13630e14			−0.0570027
$441$	1.04250e16			1.41725
$442$			2.38197e15i			0.319449i
$443$	3.76025e14			0.0497502			0.0248751	−	0.999691i	$-0.492081\pi$
$443$	3.76025e14			0.0497502			0.0248751	+	0.999691i	$0.492081\pi$
$444$		−	5.47335e15i		−	0.714423i
$445$	1.25569e16			1.61704
$446$	9.34769e15			1.18767
$447$		−	4.58494e14i		−	0.0574763i
$448$			1.11962e15i			0.138485i
$449$	−5.23958e15			−0.639467			−0.319734	−	0.947508i	$-0.603593\pi$
$449$	−5.23958e15			−0.639467			−0.319734	+	0.947508i	$0.603593\pi$
$450$	−1.74865e16			−2.10585
$451$			2.44214e14i			0.0290209i
$452$			1.44792e15i			0.169791i
$453$	−1.15133e16			−1.33233
$454$			1.16910e16i			1.33511i
$455$	9.56420e14			0.107790
$456$		−	1.49271e16i		−	1.66030i
$457$			9.13502e15i			1.00280i	0.865217	+	0.501398i	$0.167181\pi$
$457$			9.13502e15i			1.00280i	−0.865217	+	0.501398i	$0.832819\pi$
$458$		−	1.34552e16i		−	1.45780i
$459$			5.03549e15i			0.538475i
$460$	5.16810e15	−	1.02092e15i	0.545487	−	0.107757i
$461$	−1.41417e16			−1.47332			−0.736660	−	0.676263i	$-0.763598\pi$
$461$	−1.41417e16			−1.47332			−0.736660	+	0.676263i	$0.763598\pi$
$462$	−5.01504e13			−0.00515730
$463$	3.91829e15			0.397751			0.198875	−	0.980025i	$-0.436271\pi$
$463$	3.91829e15			0.397751			0.198875	+	0.980025i	$0.436271\pi$
$464$	1.11051e13			0.00111279
$465$		−	4.15594e16i		−	4.11104i
$466$	1.16586e15			0.113849
$467$			5.24033e15i			0.505193i	0.967572	+	0.252596i	$0.0812845\pi$
$467$			5.24033e15i			0.505193i	−0.967572	+	0.252596i	$0.918715\pi$
$468$	2.52438e15			0.240259
$469$	−1.01303e15			−0.0951891
$470$		−	8.34795e15i		−	0.774449i
$471$		−	1.52474e16i		−	1.39659i
$472$	1.59120e16			1.43904
$473$	6.55209e14			0.0585077
$474$			2.63190e16i			2.32060i
$475$			2.00575e16i			1.74628i
$476$	−3.89949e14			−0.0335248
$477$		−	2.65185e16i		−	2.25133i
$478$	−1.02722e15			−0.0861182
$479$			6.97514e15i			0.577484i	0.957407	+	0.288742i	$0.0932370\pi$
$479$			6.97514e15i			0.577484i	−0.957407	+	0.288742i	$0.906763\pi$
$480$		−	2.02456e16i		−	1.65533i
$481$			8.51615e15i			0.687659i
$482$		−	8.90380e15i		−	0.710057i
$483$	2.50863e15	−	4.95560e14i	0.197585	−	0.0390313i
$484$	−4.27576e15			−0.332615
$485$	−1.32393e16			−1.01722
$486$	1.35420e16			1.02770
$487$	−1.44081e16			−1.08002			−0.540011	−	0.841658i	$-0.681580\pi$
$487$	−1.44081e16			−1.08002			−0.540011	+	0.841658i	$0.681580\pi$
$488$			5.81961e15i			0.430899i
$489$	3.01181e16			2.20280
$490$			1.85656e16i			1.34132i
$491$	−4.18942e15			−0.298996			−0.149498	−	0.988762i	$-0.547766\pi$
$491$	−4.18942e15			−0.298996			−0.149498	+	0.988762i	$0.547766\pi$
$492$	−6.82963e15			−0.481511
$493$			2.24384e13i			0.00156282i
$494$			5.80127e15i			0.399173i
$495$	1.11009e15			0.0754615
$496$	−1.30482e16			−0.876313
$497$			5.10541e12i			0.000338760i
$498$		−	4.60329e15i		−	0.301781i
$499$	5.14399e15			0.333194			0.166597	−	0.986025i	$-0.446722\pi$
$499$	5.14399e15			0.333194			0.166597	+	0.986025i	$0.446722\pi$
$500$			6.85520e15i			0.438733i
$501$	2.67095e16			1.68904
$502$		−	3.85937e15i		−	0.241154i
$503$			2.75223e16i			1.69933i	0.527326	+	0.849663i	$0.323195\pi$
$503$			2.75223e16i			1.69933i	−0.527326	+	0.849663i	$0.676805\pi$
$504$		−	3.31485e15i		−	0.202246i
$505$		−	1.12763e16i		−	0.679858i
$506$	4.21645e14	−	8.32927e13i	0.0251214	−	0.00496254i
$507$	1.98837e16			1.17071
$508$	−1.00549e16			−0.585052
$509$	−2.18254e16			−1.25503			−0.627517	−	0.778603i	$-0.715929\pi$
$509$	−2.18254e16			−1.25503			−0.627517	+	0.778603i	$0.715929\pi$
$510$	−2.92931e16			−1.66473
$511$		−	2.84385e15i		−	0.159728i
$512$	−1.72645e16			−0.958373
$513$			1.22639e16i			0.672860i
$514$	5.10601e15			0.276887
$515$	−9.14656e15			−0.490247
$516$			1.83234e16i			0.970753i
$517$			3.39939e14i			0.0178015i
$518$	2.79325e15			0.144588
$519$	−1.28620e16			−0.658119
$520$			1.79982e16i			0.910350i
$521$			8.70619e15i			0.435313i	0.976025	+	0.217657i	$0.0698413\pi$
$521$			8.70619e15i			0.435313i	−0.976025	+	0.217657i	$0.930159\pi$
$522$	−4.76436e13			−0.00235495
$523$			1.88483e16i			0.921007i	0.887658	+	0.460504i	$0.152331\pi$
$523$			1.88483e16i			0.921007i	−0.887658	+	0.460504i	$0.847669\pi$
$524$	1.00460e16			0.485293
$525$			7.54473e15i			0.360320i
$526$		−	1.00704e16i		−	0.475479i
$527$		−	2.63644e16i		−	1.23071i
$528$		−	5.90359e14i		−	0.0272466i
$529$	−2.02685e16	+	8.33295e15i	−0.924885	+	0.380246i
$530$	4.72260e16			2.13072
$531$	−4.27041e16			−1.90503
$532$	−9.49719e14			−0.0418915
$533$	1.06264e16			0.463473
$534$			2.86500e16i			1.23560i
$535$	6.33325e16			2.70087
$536$		−	1.90636e16i		−	0.803925i
$537$	2.90157e16			1.21001
$538$	−2.99594e16			−1.23549
$539$		−	7.56015e14i		−	0.0308317i
$540$			9.50368e15i			0.383292i
$541$	1.07007e16			0.426803			0.213402	−	0.976965i	$-0.431546\pi$
$541$	1.07007e16			0.426803			0.213402	+	0.976965i	$0.431546\pi$
$542$	−4.61642e15			−0.182100
$543$		−	4.94807e16i		−	1.93035i
$544$		−	1.28434e16i		−	0.495549i
$545$	1.94027e16			0.740429
$546$			2.18218e15i			0.0823636i
$547$	−2.33885e16			−0.873128			−0.436564	−	0.899673i	$-0.643805\pi$
$547$	−2.33885e16			−0.873128			−0.436564	+	0.899673i	$0.643805\pi$
$548$			1.15384e16i			0.426053i
$549$		−	1.56185e16i		−	0.570434i
$550$			1.26810e15i			0.0458119i
$551$			5.46486e13i			0.00195285i
$552$	9.32559e15	+	4.72081e16i	0.329641	+	1.66871i
$553$	6.70397e15			0.234413
$554$	−1.61018e16			−0.556949
$555$	−1.04730e17			−3.58356
$556$	−5.43096e15			−0.183835
$557$			2.33538e16i			0.782034i	0.920383	+	0.391017i	$0.127877\pi$
$557$			2.33538e16i			0.782034i	−0.920383	+	0.391017i	$0.872123\pi$
$558$	5.59799e16			1.85450
$559$		−	2.85100e16i		−	0.934386i
$560$	3.69282e15			0.119737
$561$	1.19285e15			0.0382656
$562$			2.84988e16i			0.904500i
$563$		−	2.03292e14i		−	0.00638368i	−0.999995	−	0.00319184i	$-0.998984\pi$
$563$		−	2.03292e14i		−	0.00638368i	0.999995	−	0.00319184i	$-0.00101600\pi$
$564$	−9.50665e15			−0.295361
$565$	2.77054e16			0.851675
$566$		−	2.84468e16i		−	0.865236i
$567$		−	1.55971e15i		−	0.0469403i
$568$	−9.60750e13			−0.00286102
$569$		−	6.35528e16i		−	1.87267i	−0.351110	−	0.936334i	$-0.614196\pi$
$569$		−	6.35528e16i		−	1.87267i	0.351110	−	0.936334i	$-0.385804\pi$
$570$	−7.13431e16			−2.08019
$571$			3.37742e15i			0.0974468i	0.998812	+	0.0487234i	$0.0155153\pi$
$571$			3.37742e15i			0.0974468i	−0.998812	+	0.0487234i	$0.984485\pi$
$572$		−	1.83066e14i		−	0.00522674i
$573$			3.80780e16i			1.07584i
$574$		−	3.48541e15i		−	0.0974503i
$575$	−1.25307e16	−	6.34332e16i	−0.346713	−	1.75513i
$576$	5.65454e16			1.54833
$577$	−2.86457e16			−0.776255			−0.388127	−	0.921606i	$-0.626878\pi$
$577$	−2.86457e16			−0.776255			−0.388127	+	0.921606i	$0.626878\pi$
$578$	1.18714e16			0.318372
$579$	4.97891e16			1.32149
$580$			4.23489e13i			0.00111243i
$581$	−1.17255e15			−0.0304841
$582$		−	3.02070e16i		−	0.777266i
$583$	−1.92310e15			−0.0489768
$584$	5.35163e16			1.34899
$585$		−	4.83030e16i		−	1.20514i
$586$		−	2.19521e16i		−	0.542113i
$587$	4.37743e16			1.07002			0.535009	−	0.844846i	$-0.320308\pi$
$587$	4.37743e16			1.07002			0.535009	+	0.844846i	$0.320308\pi$
$588$	2.11425e16			0.511556
$589$		−	6.42105e16i		−	1.53785i
$590$		−	7.60504e16i		−	1.80297i
$591$	4.93135e16			1.15729
$592$			3.28816e16i			0.763875i
$593$	−5.65644e16			−1.30081			−0.650407	−	0.759586i	$-0.725402\pi$
$593$	−5.65644e16			−1.30081			−0.650407	+	0.759586i	$0.725402\pi$
$594$			7.75368e14i			0.0176518i
$595$			7.46152e15i			0.168161i
$596$		−	5.48951e14i		−	0.0122477i
$597$			6.91903e16i			1.52827i
$598$	−3.62430e15	−	1.83469e16i	−0.0792532	−	0.401196i
$599$	8.05243e16			1.74327			0.871637	−	0.490151i	$-0.163058\pi$
$599$	8.05243e16			1.74327			0.871637	+	0.490151i	$0.163058\pi$
$600$	−1.41979e17			−3.04310
$601$	−5.43921e16			−1.15422			−0.577111	−	0.816666i	$-0.695820\pi$
$601$	−5.43921e16			−1.15422			−0.577111	+	0.816666i	$0.695820\pi$
$602$	−9.35113e15			−0.196465
$603$			5.11622e16i			1.06425i
$604$	−1.37848e16			−0.283909
$605$			8.18150e16i			1.66840i
$606$	2.57282e16			0.519486
$607$	−1.83635e16			−0.367132			−0.183566	−	0.983007i	$-0.558764\pi$
$607$	−1.83635e16			−0.367132			−0.183566	+	0.983007i	$0.558764\pi$
$608$		−	3.12801e16i		−	0.619222i
$609$			2.05564e13i			0.000402943i
$610$	2.78145e16			0.539873
$611$	1.47917e16			0.284296
$612$			1.96939e16i			0.374821i
$613$			2.99220e16i			0.563933i	0.959424	+	0.281967i	$0.0909867\pi$
$613$			2.99220e16i			0.563933i	−0.959424	+	0.281967i	$0.909013\pi$
$614$	−5.32354e16			−0.993552
$615$			1.30682e17i			2.41527i
$616$	−2.40390e14			−0.00439980
$617$			4.78430e16i			0.867177i	0.901111	+	0.433589i	$0.142753\pi$
$617$			4.78430e16i			0.867177i	−0.901111	+	0.433589i	$0.857247\pi$
$618$		−	2.08689e16i		−	0.374602i
$619$			4.44337e16i			0.789894i	0.918704	+	0.394947i	$0.129237\pi$
$619$			4.44337e16i			0.789894i	−0.918704	+	0.394947i	$0.870763\pi$
$620$		−	4.97587e16i		−	0.876030i
$621$	−7.66178e15	−	3.87855e16i	−0.133592	−	0.676270i
$622$	4.48971e16			0.775311
$623$	7.29771e15			0.124813
$624$	−2.56882e16			−0.435137
$625$	2.45360e16			0.411646
$626$		−	6.29167e16i		−	1.04549i
$627$	2.90518e15			0.0478154
$628$		−	1.82556e16i		−	0.297603i
$629$	−6.64388e16			−1.07280
$630$	−1.58431e16			−0.253395
$631$		−	5.28553e16i		−	0.837360i	−0.908134	−	0.418680i	$-0.862493\pi$
$631$		−	5.28553e16i		−	0.837360i	0.908134	−	0.418680i	$-0.137507\pi$
$632$			1.26157e17i			1.97975i
$633$	−1.04489e17			−1.62423
$634$	7.06585e16			1.08800
$635$			1.92396e17i			2.93464i
$636$		−	5.37810e16i		−	0.812617i
$637$	−3.28963e16			−0.492392
$638$			3.45508e12i			5.12311e-5i
$639$	2.57843e14			0.00378748
$640$			2.78947e16i			0.405922i
$641$		−	1.19489e17i		−	1.72258i	−0.508113	−	0.861290i	$-0.669657\pi$
$641$		−	1.19489e17i		−	1.72258i	0.508113	−	0.861290i	$-0.330343\pi$
$642$			1.44500e17i			2.06376i
$643$			1.00380e17i			1.42030i	0.704048	+	0.710152i	$0.251374\pi$
$643$			1.00380e17i			1.42030i	−0.704048	+	0.710152i	$0.748626\pi$
$644$	3.00356e15	−	5.93330e14i	0.0421038	−	0.00831727i
$645$	3.50612e17			4.86932
$646$	−4.52586e16			−0.622739
$647$	−1.22482e17			−1.66974			−0.834868	−	0.550450i	$-0.814456\pi$
$647$	−1.22482e17			−1.66974			−0.834868	+	0.550450i	$0.814456\pi$
$648$	2.93511e16			0.396437
$649$			3.09687e15i			0.0414433i
$650$	5.51787e16			0.731630
$651$		−	2.41532e16i		−	0.317313i
$652$	3.60601e16			0.469398
$653$	−3.76554e16			−0.485678			−0.242839	−	0.970067i	$-0.578079\pi$
$653$	−3.76554e16			−0.485678			−0.242839	+	0.970067i	$0.578079\pi$
$654$			4.42695e16i			0.565768i
$655$		−	1.92226e17i		−	2.43424i
$656$	4.10295e16			0.514841
$657$	−1.43625e17			−1.78583
$658$		−	4.85160e15i		−	0.0597764i
$659$			4.46581e15i			0.0545240i	0.999628	+	0.0272620i	$0.00867884\pi$
$659$			4.46581e15i			0.0545240i	−0.999628	+	0.0272620i	$0.991321\pi$
$660$	2.25132e15			0.0272378
$661$		−	1.05898e17i		−	1.26964i	−0.772661	−	0.634819i	$-0.781075\pi$
$661$		−	1.05898e17i		−	1.26964i	0.772661	−	0.634819i	$-0.218925\pi$
$662$	−3.62412e16			−0.430581
$663$		−	5.19042e16i		−	0.611113i
$664$		−	2.20653e16i		−	0.257456i
$665$			1.81725e16i			0.210128i
$666$		−	1.41070e17i		−	1.61655i
$667$	−3.41413e13	−	1.72830e14i	−0.000387726	−	0.00196275i
$668$	3.19791e16			0.359921
$669$	−2.03691e17			−2.27203
$670$	−9.11131e16			−1.00724
$671$	−1.13264e15			−0.0124096
$672$		−	1.17662e16i		−	0.127768i
$673$	1.29385e17			1.39249			0.696246	−	0.717803i	$-0.254852\pi$
$673$	1.29385e17			1.39249			0.696246	+	0.717803i	$0.254852\pi$
$674$		−	3.62934e16i		−	0.387140i
$675$	1.16648e17			1.23326
$676$	2.38066e16			0.249470
$677$		−	3.56085e16i		−	0.369847i	−0.982753	−	0.184924i	$-0.940796\pi$
$677$		−	3.56085e16i		−	0.369847i	0.982753	−	0.184924i	$-0.0592038\pi$
$678$			6.32131e16i			0.650772i
$679$	−7.69432e15			−0.0785148
$680$	−1.40413e17			−1.42021
$681$		−	2.54752e17i		−	2.55409i
$682$		−	4.05962e15i		−	0.0403440i
$683$	−6.52076e16			−0.642354			−0.321177	−	0.947019i	$-0.604078\pi$
$683$	−6.52076e16			−0.642354			−0.321177	+	0.947019i	$0.604078\pi$
$684$			4.79645e16i			0.468364i
$685$	2.20783e17			2.13709
$686$			2.17620e16i			0.208811i
$687$			2.93196e17i			2.78881i
$688$		−	1.10079e17i		−	1.03795i
$689$			8.36795e16i			0.782174i
$690$	2.25628e17	−	4.45711e16i	2.09073	−	0.413008i
$691$	−5.06181e16			−0.464984			−0.232492	−	0.972598i	$-0.574688\pi$
$691$	−5.06181e16			−0.464984			−0.232492	+	0.972598i	$0.574688\pi$
$692$	−1.53996e16			−0.140240
$693$	6.45152e14			0.00582455
$694$	−2.85274e16			−0.255332
$695$			1.03919e17i			0.922120i
$696$	−3.86836e14			−0.00340307
$697$			8.29021e16i			0.723051i
$698$	2.18130e16			0.188618
$699$	−2.54047e16			−0.217796
$700$			9.03325e15i			0.0767813i
$701$			4.80747e16i			0.405143i	0.979267	+	0.202572i	$0.0649299\pi$
$701$			4.80747e16i			0.405143i	−0.979267	+	0.202572i	$0.935070\pi$
$702$	3.37384e16			0.281904
$703$	−1.61811e17			−1.34053
$704$		−	4.10063e15i		−	0.0336833i
$705$			1.81906e17i			1.48154i
$706$	4.23021e16			0.341612
$707$		−	6.55348e15i		−	0.0524753i
$708$	−8.66063e16			−0.687622
$709$		−	1.72785e17i		−	1.36028i	−0.733083	−	0.680140i	$-0.761919\pi$
$709$		−	1.72785e17i		−	1.36028i	0.733083	−	0.680140i	$-0.238081\pi$
$710$			4.59185e14i			0.00358457i
$711$		−	3.38577e17i		−	2.62083i
$712$			1.37330e17i			1.05411i
$713$	4.01150e16	+	2.03071e17i	0.305330	+	1.54565i
$714$	−1.70243e16			−0.128493
$715$	−3.50289e15			−0.0262174
$716$	3.47402e16			0.257843
$717$	2.23836e16			0.164746
$718$		−	5.05214e16i		−	0.368747i
$719$	9.34648e16			0.676511			0.338255	−	0.941054i	$-0.390163\pi$
$719$	9.34648e16			0.676511			0.338255	+	0.941054i	$0.390163\pi$
$720$		−	1.86502e17i		−	1.33871i
$721$	−5.31573e15			−0.0378400
$722$	5.46520e15			0.0385818
$723$			1.94018e17i			1.35835i
$724$		−	5.92428e16i		−	0.411343i
$725$	5.19789e14			0.00357931
$726$	−1.86670e17			−1.27484
$727$		−	5.83130e16i		−	0.394965i	−0.980306	−	0.197483i	$-0.936723\pi$
$727$		−	5.83130e16i		−	0.394965i	0.980306	−	0.197483i	$-0.0632766\pi$
$728$			1.04600e16i			0.0702660i
$729$	−2.40430e17			−1.60186
$730$		−	2.55778e17i		−	1.69015i
$731$	2.22421e17			1.45771
$732$		−	3.16751e16i		−	0.205898i
$733$		−	1.09361e17i		−	0.705083i	−0.935796	−	0.352542i	$-0.885317\pi$
$733$		−	1.09361e17i		−	0.705083i	0.935796	−	0.352542i	$-0.114683\pi$
$734$		−	8.24746e16i		−	0.527404i
$735$		−	4.04554e17i		−	2.56597i
$736$	1.95420e16	+	9.89256e16i	0.122942	+	0.622360i
$737$	3.71024e15			0.0231525
$738$	−1.76027e17			−1.08954
$739$	1.02360e17			0.628439			0.314220	−	0.949350i	$-0.398257\pi$
$739$	1.02360e17			0.628439			0.314220	+	0.949350i	$0.398257\pi$
$740$	−1.25393e17			−0.763629
$741$		−	1.26413e17i		−	0.763626i
$742$	2.74464e16			0.164461
$743$			9.90196e16i			0.588557i	0.955720	+	0.294278i	$0.0950792\pi$
$743$			9.90196e16i			0.588557i	−0.955720	+	0.294278i	$0.904921\pi$
$744$	4.54521e17			2.67989
$745$	−1.05040e16			−0.0614349
$746$			7.87067e16i			0.456645i
$747$			5.92183e16i			0.340826i
$748$	1.42819e15			0.00815410
$749$	3.68071e16			0.208469
$750$			2.99283e17i			1.68157i
$751$		−	1.85388e17i		−	1.03334i	−0.856185	−	0.516669i	$-0.827172\pi$
$751$		−	1.85388e17i		−	1.03334i	0.856185	−	0.516669i	$-0.172828\pi$
$752$	5.71119e16			0.315806
$753$			8.40975e16i			0.461332i
$754$	1.50340e14			0.000818175
$755$			2.63767e17i			1.42409i
$756$			5.52328e15i			0.0295846i
$757$			1.41816e17i			0.753614i	0.926292	+	0.376807i	$0.122978\pi$
$757$			1.41816e17i			0.753614i	−0.926292	+	0.376807i	$0.877022\pi$
$758$			4.88947e16i			0.257778i
$759$	−9.18785e15	+	1.81499e15i	−0.0480577	+	0.00949344i
$760$	−3.41975e17			−1.77465
$761$	6.89641e16			0.355071			0.177536	−	0.984114i	$-0.443188\pi$
$761$	6.89641e16			0.355071			0.177536	+	0.984114i	$0.443188\pi$
$762$	−4.38974e17			−2.24238
$763$	1.12763e16			0.0571505
$764$			4.55904e16i			0.229252i
$765$	3.76836e17			1.88011
$766$			2.31243e17i			1.14471i
$767$	1.34753e17			0.661862
$768$	2.80791e17			1.36841
$769$			9.57379e16i			0.462941i	0.972842	+	0.231471i	$0.0743537\pi$
$769$			9.57379e16i			0.462941i	−0.972842	+	0.231471i	$0.925646\pi$
$770$			1.14893e15i			0.00551251i
$771$	−1.11262e17			−0.529691
$772$	5.96121e16			0.281599
$773$			3.07198e17i			1.43993i	0.694011	+	0.719964i	$0.255842\pi$
$773$			3.07198e17i			1.43993i	−0.694011	+	0.719964i	$0.744158\pi$
$774$			4.72269e17i			2.19656i
$775$	−6.10738e17			−2.81867
$776$		−	1.44794e17i		−	0.663101i
$777$	−6.08664e16			−0.276600
$778$		−	4.48436e16i		−	0.202219i
$779$			2.01908e17i			0.903501i
$780$		−	9.79610e16i		−	0.434996i
$781$		−	1.86986e13i		−	8.23953e-5i
$782$	1.43134e17	−	2.82750e16i	0.625895	−	0.123641i
$783$	3.17819e14			0.00137914
$784$	−1.27015e17			−0.546965
$785$	−3.49313e17			−1.49278
$786$	4.38585e17			1.86002
$787$		−	3.38550e17i		−	1.42487i	−0.701739	−	0.712434i	$-0.747593\pi$
$787$		−	3.38550e17i		−	1.42487i	0.701739	−	0.712434i	$-0.252407\pi$
$788$	5.90426e16			0.246609
$789$			2.19439e17i			0.909601i
$790$	6.02960e17			2.48043
$791$	1.61016e16			0.0657371
$792$			1.21406e16i			0.0491916i
$793$			4.92843e16i			0.198184i
$794$	−2.93511e17			−1.17139
$795$	−1.02908e18			−4.07611
$796$			8.28409e16i			0.325662i
$797$			3.01967e16i			0.117818i	0.998263	+	0.0589088i	$0.0187621\pi$
$797$			3.01967e16i			0.117818i	−0.998263	+	0.0589088i	$0.981238\pi$
$798$	−4.14627e16			−0.160561
$799$			1.15397e17i			0.443522i
$800$	−2.97520e17			−1.13495
$801$		−	3.68563e17i		−	1.39546i
$802$		−	2.40084e17i		−	0.902229i
$803$			1.04156e16i			0.0388500i
$804$			1.03760e17i			0.384142i
$805$	−1.13531e16	−	5.74719e16i	−0.0417196	−	0.211193i
$806$	−1.76645e17			−0.644305
$807$	6.52831e17			2.36352
$808$	1.23325e17			0.443183
$809$	4.87868e17			1.74025			0.870123	−	0.492834i	$-0.164039\pi$
$809$	4.87868e17			1.74025			0.870123	+	0.492834i	$0.164039\pi$
$810$		−	1.40282e17i		−	0.496696i
$811$	4.95884e17			1.74283			0.871415	−	0.490547i	$-0.163203\pi$
$811$	4.95884e17			1.74283			0.871415	+	0.490547i	$0.163203\pi$
$812$			2.46120e13i			8.58639e-5i
$813$	1.00594e17			0.348360
$814$	−1.02303e16			−0.0351675
$815$		−	6.89996e17i		−	2.35451i
$816$		−	2.00406e17i		−	0.678845i
$817$	5.41705e17			1.82151
$818$	4.10640e16			0.137070
$819$		−	2.80723e16i		−	0.0930198i
$820$			1.56465e17i			0.514675i
$821$	−4.22650e17			−1.38014			−0.690069	−	0.723744i	$-0.742420\pi$
$821$	−4.22650e17			−1.38014			−0.690069	+	0.723744i	$0.742420\pi$
$822$			5.03743e17i			1.63297i
$823$	−3.81463e16			−0.122759			−0.0613796	−	0.998114i	$-0.519550\pi$
$823$	−3.81463e16			−0.122759			−0.0613796	+	0.998114i	$0.519550\pi$
$824$		−	1.00033e17i		−	0.319580i
$825$		−	2.76326e16i		−	0.0876391i
$826$		−	4.41984e16i		−	0.139164i
$827$		−	2.93983e17i		−	0.918944i	−0.888192	−	0.459472i	$-0.848039\pi$
$827$		−	2.93983e17i		−	0.918944i	0.888192	−	0.459472i	$-0.151961\pi$
$828$	−2.99655e16	−	1.51691e17i	−0.0929906	−	0.470738i
$829$	9.43405e16			0.290650			0.145325	−	0.989384i	$-0.453577\pi$
$829$	9.43405e16			0.290650			0.145325	+	0.989384i	$0.453577\pi$
$830$	−1.05460e17			−0.322566
$831$	3.50866e17			1.06545
$832$	−1.78430e17			−0.537932
$833$		−	2.56641e17i		−	0.768167i
$834$	−2.37104e17			−0.704600
$835$		−	6.11907e17i		−	1.80537i
$836$	3.47835e15			0.0101891
$837$	−3.73429e17			−1.08606
$838$			3.19337e17i			0.922116i
$839$		−	3.21541e17i		−	0.921860i	−0.887437	−	0.460930i	$-0.847516\pi$
$839$		−	3.21541e17i		−	0.921860i	0.887437	−	0.460930i	$-0.152484\pi$
$840$	−1.28636e17			−0.366174
$841$	−3.53813e17			−0.999996
$842$		−	2.88529e17i		−	0.809685i
$843$		−	6.21003e17i		−	1.73033i
$844$	−1.25104e17			−0.346111
$845$		−	4.55530e17i		−	1.25134i
$846$	−2.45025e17			−0.668325
$847$			4.75486e16i			0.128777i
$848$			3.23093e17i			0.868866i
$849$			6.19869e17i			1.65521i
$850$			4.30477e17i			1.14140i
$851$	5.11741e17	−	1.01090e17i	1.34733	−	0.266154i
$852$	5.22920e14			0.00136709
$853$	2.52473e17			0.655421			0.327711	−	0.944778i	$-0.393723\pi$
$853$	2.52473e17			0.655421			0.327711	+	0.944778i	$0.393723\pi$
$854$	1.61650e16			0.0416705
$855$	9.17782e17			2.34932
$856$			6.92646e17i			1.76063i
$857$	−5.92290e17			−1.49503			−0.747514	−	0.664246i	$-0.768753\pi$
$857$	−5.92290e17			−1.49503			−0.747514	+	0.664246i	$0.768753\pi$
$858$		−	7.99225e15i		−	0.0200330i
$859$	−4.79389e17			−1.19324			−0.596622	−	0.802522i	$-0.703491\pi$
$859$	−4.79389e17			−1.19324			−0.596622	+	0.802522i	$0.703491\pi$
$860$	4.19784e17			1.03761
$861$			7.59489e16i			0.186424i
$862$		−	4.73305e16i		−	0.115371i
$863$	3.85601e16			0.0933411			0.0466706	−	0.998910i	$-0.485139\pi$
$863$	3.85601e16			0.0933411			0.0466706	+	0.998910i	$0.485139\pi$
$864$	−1.81915e17			−0.437308
$865$			2.94665e17i			0.703447i
$866$		−	3.23593e17i		−	0.767171i
$867$	−2.58684e17			−0.609052
$868$		−	2.89184e16i		−	0.0676170i
$869$	−2.45533e16			−0.0570153
$870$			1.84886e15i			0.00426372i
$871$		−	1.61443e17i		−	0.369751i
$872$			2.12201e17i			0.482668i
$873$			3.88593e17i			0.877828i
$874$	3.48602e17	−	6.88636e16i	0.782098	−	0.154497i
$875$	7.62333e16			0.169862
$876$	−2.91280e17			−0.644594
$877$	−2.91114e17			−0.639830			−0.319915	−	0.947446i	$-0.603654\pi$
$877$	−2.91114e17			−0.639830			−0.319915	+	0.947446i	$0.603654\pi$
$878$	−2.48796e17			−0.543096
$879$			4.78346e17i			1.03707i
$880$	−1.35249e16			−0.0291232
$881$		−	6.01248e17i		−	1.28587i	−0.765919	−	0.642937i	$-0.777716\pi$
$881$		−	6.01248e17i		−	1.28587i	0.765919	−	0.642937i	$-0.222284\pi$
$882$	5.44928e17			1.15752
$883$	2.28013e17			0.481055			0.240528	−	0.970642i	$-0.422680\pi$
$883$	2.28013e17			0.481055			0.240528	+	0.970642i	$0.422680\pi$
$884$		−	6.21445e16i		−	0.130223i
$885$			1.65718e18i			3.44913i
$886$	1.96553e16			0.0406328
$887$	3.14219e17			0.645195			0.322597	−	0.946536i	$-0.395444\pi$
$887$	3.14219e17			0.645195			0.322597	+	0.946536i	$0.395444\pi$
$888$		−	1.14540e18i		−	2.33604i
$889$			1.11815e17i			0.226512i
$890$	6.56362e17			1.32070
$891$			5.71244e15i			0.0114171i
$892$	−2.43877e17			−0.484152
$893$			2.81050e17i			0.554211i
$894$		−	2.39660e16i		−	0.0469430i
$895$		−	6.64741e17i		−	1.29334i
$896$			1.62117e16i			0.0313314i
$897$	7.89752e16	+	3.99789e17i	0.151613	+	0.767496i
$898$	−2.73879e17			−0.522276
$899$	−1.66402e15			−0.00315210
$900$	4.56214e17			0.858448
$901$	−6.52826e17			−1.22025
$902$			1.27653e16i			0.0237024i
$903$	2.03766e17			0.375841
$904$			3.03005e17i			0.555186i
$905$	−1.13359e18			−2.06330
$906$	−6.01815e17			−1.08816
$907$			7.18418e17i			1.29043i	0.764002	+	0.645214i	$0.223232\pi$
$907$			7.18418e17i			1.29043i	−0.764002	+	0.645214i	$0.776768\pi$
$908$		−	3.05013e17i		−	0.544256i
$909$	−3.30976e17			−0.586696
$910$	4.99931e16			0.0880364
$911$		−	3.73230e17i		−	0.652930i	−0.945209	−	0.326465i	$-0.894142\pi$
$911$		−	3.73230e17i		−	0.652930i	0.945209	−	0.326465i	$-0.105858\pi$
$912$		−	4.88089e17i		−	0.848262i
$913$	4.29446e15			0.00741453
$914$			4.77498e17i			0.819020i
$915$	−6.06091e17			−1.03279
$916$			3.51042e17i			0.594273i
$917$		−	1.11716e17i		−	0.187889i
$918$			2.63210e17i			0.439792i
$919$			5.66738e17i			0.940782i	0.882458	+	0.470391i	$0.155887\pi$
$919$			5.66738e17i			0.940782i	−0.882458	+	0.470391i	$0.844113\pi$
$920$	1.08152e18	−	2.13646e17i	1.78365	−	0.352345i
$921$	1.16003e18			1.90068
$922$	−7.39204e17			−1.20331
$923$	−8.13627e14			−0.00131588
$924$	1.30840e15			0.00210237
$925$			1.53907e18i			2.45701i
$926$	2.04813e17			0.324857
$927$			2.68465e17i			0.423067i
$928$	−8.10624e14			−0.00126920
$929$	−1.23689e17			−0.192414			−0.0962071	−	0.995361i	$-0.530671\pi$
$929$	−1.23689e17			−0.192414			−0.0962071	+	0.995361i	$0.530671\pi$
$930$		−	2.17236e18i		−	3.35764i
$931$		−	6.25048e17i		−	0.959876i
$932$	−3.04168e16			−0.0464107
$933$	−9.78331e17			−1.48319
$934$			2.73918e17i			0.412609i
$935$		−	2.73278e16i		−	0.0409011i
$936$	5.28273e17			0.785604
$937$			7.95860e17i			1.17598i	0.808869	+	0.587989i	$0.200080\pi$
$937$			7.95860e17i			1.17598i	−0.808869	+	0.587989i	$0.799920\pi$
$938$	−5.29524e16			−0.0777444
$939$			1.37099e18i			2.00004i
$940$			2.17794e17i			0.315704i
$941$			7.85797e17i			1.13181i	0.824471	+	0.565904i	$0.191473\pi$
$941$			7.85797e17i			1.13181i	−0.824471	+	0.565904i	$0.808527\pi$
$942$		−	7.96998e17i		−	1.14065i
$943$	−1.26140e17	−	6.38549e17i	−0.179384	−	0.908080i
$944$	5.20294e17			0.735219
$945$	1.05686e17			0.148397
$946$	3.42485e16			0.0477854
$947$	1.14043e18			1.58113			0.790567	−	0.612376i	$-0.209786\pi$
$947$	1.14043e18			1.58113			0.790567	+	0.612376i	$0.209786\pi$
$948$		−	6.86652e17i		−	0.945990i
$949$	4.53212e17			0.620446
$950$			1.04843e18i			1.42625i
$951$	−1.53968e18			−2.08136
$952$	−8.16041e16			−0.109620
$953$			9.13462e17i			1.21936i	0.792647	+	0.609681i	$0.208703\pi$
$953$			9.13462e17i			1.21936i	−0.792647	+	0.609681i	$0.791297\pi$
$954$		−	1.38615e18i		−	1.83874i
$955$	8.72354e17			1.14993
$956$	2.67997e16			0.0351060
$957$		−	7.52878e13i		−	9.80061e-5i
$958$			3.64599e17i			0.471652i
$959$	1.28313e17			0.164953
$960$		−	2.19430e18i		−	2.80330i
$961$	1.16751e18			1.48225
$962$			4.45149e17i			0.561636i
$963$		−	1.85890e18i		−	2.33077i
$964$			2.32296e17i			0.289455i
$965$		−	1.14065e18i		−	1.41250i
$966$	1.31129e17	−	2.59035e16i	0.161375	−	0.0318783i
$967$	−5.95186e17			−0.727937			−0.363968	−	0.931411i	$-0.618578\pi$
$967$	−5.95186e17			−0.727937			−0.363968	+	0.931411i	$0.618578\pi$
$968$	−8.94784e17			−1.08759
$969$	9.86208e17			1.19131
$970$	−6.92033e17			−0.830800
$971$		−	1.27194e18i		−	1.51758i	−0.651334	−	0.758791i	$-0.725791\pi$
$971$		−	1.27194e18i		−	1.51758i	0.651334	−	0.758791i	$-0.274209\pi$
$972$	−3.53305e17			−0.418941
$973$			6.03950e16i			0.0711745i
$974$	−7.53127e17			−0.882094
$975$	−1.20237e18			−1.39962
$976$			1.90291e17i			0.220150i
$977$		−	2.06774e17i		−	0.237755i	−0.992909	−	0.118877i	$-0.962070\pi$
$977$		−	2.06774e17i		−	0.237755i	0.992909	−	0.118877i	$-0.0379295\pi$
$978$	1.57431e18			1.79910
$979$	−2.67279e16			−0.0303577
$980$		−	4.84369e17i		−	0.546789i
$981$		−	5.69498e17i		−	0.638967i
$982$	−2.18986e17			−0.244201
$983$			1.48395e18i			1.64474i	0.568953	+	0.822370i	$0.307349\pi$
$983$			1.48395e18i			1.64474i	−0.568953	+	0.822370i	$0.692651\pi$
$984$	−1.42923e18			−1.57446
$985$		−	1.12976e18i		−	1.23699i
$986$			1.17288e15i			0.00127641i
$987$			1.05719e17i			0.114353i
$988$		−	1.51353e17i		−	0.162723i
$989$	−1.71318e18	+	3.38426e17i	−1.83074	+	0.361648i
$990$	5.80255e16			0.0616322
$991$	2.50527e17			0.264492			0.132246	−	0.991217i	$-0.457781\pi$
$991$	2.50527e17			0.264492			0.132246	+	0.991217i	$0.457781\pi$
$992$	9.52460e17			0.999486
$993$	7.89715e17			0.823711
$994$			2.66865e14i			0.000276678i
$995$	1.58513e18			1.63353
$996$			1.20098e17i			0.123021i
$997$	−3.83287e17			−0.390259			−0.195130	−	0.980777i	$-0.562513\pi$
$997$	−3.83287e17			−0.390259			−0.195130	+	0.980777i	$0.562513\pi$
$998$	2.68882e17			0.272131
$999$			9.41046e17i			0.946712i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.c.22.14	yes	20
23.22	odd	2	inner	23.13.b.c.22.13	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.c.22.13	✓	20	23.22	odd	2	inner
23.13.b.c.22.14	yes	20	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+52.2711 q^{2} -1139.01 q^{3} -1363.73 q^{4} +26094.4i q^{5} -59537.5 q^{6} +15165.4i q^{7} -285386. q^{8} +765911. q^{9} +O(q^{10})\)	\(q+52.2711 q^{2} -1139.01 q^{3} -1363.73 q^{4} +26094.4i q^{5} -59537.5 q^{6} +15165.4i q^{7} -285386. q^{8} +765911. q^{9} +1.36399e6i q^{10} -55543.2i q^{11} +1.55331e6 q^{12} -2.41684e6 q^{13} +792711. i q^{14} -2.97219e7i q^{15} -9.33161e6 q^{16} -1.88550e7i q^{17} +4.00350e7 q^{18} -4.59212e7i q^{19} -3.55858e7i q^{20} -1.72736e7i q^{21} -2.90330e6i q^{22} +(2.86889e7 + 1.45229e8i) q^{23} +3.25059e8 q^{24} -4.36779e8 q^{25} -1.26331e8 q^{26} -2.67064e8 q^{27} -2.06815e7i q^{28} -1.19005e6 q^{29} -1.55360e9i q^{30} +1.39827e9 q^{31} +6.81168e8 q^{32} +6.32644e7i q^{33} -9.85571e8i q^{34} -3.95732e8 q^{35} -1.04450e9 q^{36} -3.52367e9i q^{37} -2.40035e9i q^{38} +2.75281e9 q^{39} -7.44699e9i q^{40} -4.39683e9 q^{41} -9.02908e8i q^{42} +1.17964e10i q^{43} +7.57460e7i q^{44} +1.99860e10i q^{45} +(1.49960e9 + 7.59130e9i) q^{46} -6.12026e9 q^{47} +1.06288e10 q^{48} +1.36113e10 q^{49} -2.28309e10 q^{50} +2.14761e10i q^{51} +3.29592e9 q^{52} -3.46235e10i q^{53} -1.39597e10 q^{54} +1.44937e9 q^{55} -4.32799e9i q^{56} +5.23049e10i q^{57} -6.22052e7 q^{58} -5.57560e10 q^{59} +4.05327e10i q^{60} -2.03921e10i q^{61} +7.30894e10 q^{62} +1.16153e10i q^{63} +7.38277e10 q^{64} -6.30661e10i q^{65} +3.30690e9i q^{66} +6.67992e10i q^{67} +2.57131e10i q^{68} +(-3.26771e10 - 1.65418e11i) q^{69} -2.06853e10 q^{70} +3.36649e8 q^{71} -2.18580e11 q^{72} -1.87523e11 q^{73} -1.84186e11i q^{74} +4.97498e11 q^{75} +6.26242e10i q^{76} +8.42333e8 q^{77} +1.43893e11 q^{78} -4.42058e11i q^{79} -2.43503e11i q^{80} -1.02847e11 q^{81} -2.29827e11 q^{82} +7.73175e10i q^{83} +2.35565e10i q^{84} +4.92010e11 q^{85} +6.16611e11i q^{86} +1.35548e9 q^{87} +1.58513e10i q^{88} -4.81209e11i q^{89} +1.04469e12i q^{90} -3.66523e10i q^{91} +(-3.91240e10 - 1.98054e11i) q^{92} -1.59265e12 q^{93} -3.19913e11 q^{94} +1.19829e12 q^{95} -7.75860e11 q^{96} +5.07361e11i q^{97} +7.11478e11 q^{98} -4.25411e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) 20 * q + 88 * q^2 + 318 * q^3 + 36072 * q^4 + 264240 * q^6 - 767728 * q^8 + 1971426 * q^9	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9} + 19833480 q^{12} + 8049118 q^{13} + 81743632 q^{16} + 36238080 q^{18} - 367189708 q^{23} + 530348736 q^{24} - 1824317212 q^{25} - 2465696728 q^{26} - 1765163178 q^{27} - 417897362 q^{29} - 1574125298 q^{31} + 1240884960 q^{32} + 4723363152 q^{35} + 3607269840 q^{36} + 1926187110 q^{39} - 132511250 q^{41} + 62599086544 q^{46} - 45052376402 q^{47} + 77275023312 q^{48} - 65534183260 q^{49} - 57446928200 q^{50} - 20407259400 q^{52} - 15082009344 q^{54} - 12572534832 q^{55} + 97529544392 q^{58} - 110336269112 q^{59} + 553127354432 q^{62} + 13799515808 q^{64} + 7914588558 q^{69} + 590935322064 q^{70} - 40523102210 q^{71} - 539385055680 q^{72} - 449748966242 q^{73} + 2570431278 q^{75} - 345974135760 q^{77} - 48894418992 q^{78} - 2694979782144 q^{81} - 1210586085208 q^{82} + 1051198266576 q^{85} - 2237359632618 q^{87} + 1936236755640 q^{92} - 116415989178 q^{93} + 3420883097024 q^{94} + 3777482201184 q^{95} + 3567911213376 q^{96} + 3590387133208 q^{98}+O(q^{100}) \) 20 * q + 88 * q^2 + 318 * q^3 + 36072 * q^4 + 264240 * q^6 - 767728 * q^8 + 1971426 * q^9 + 19833480 * q^12 + 8049118 * q^13 + 81743632 * q^16 + 36238080 * q^18 - 367189708 * q^23 + 530348736 * q^24 - 1824317212 * q^25 - 2465696728 * q^26 - 1765163178 * q^27 - 417897362 * q^29 - 1574125298 * q^31 + 1240884960 * q^32 + 4723363152 * q^35 + 3607269840 * q^36 + 1926187110 * q^39 - 132511250 * q^41 + 62599086544 * q^46 - 45052376402 * q^47 + 77275023312 * q^48 - 65534183260 * q^49 - 57446928200 * q^50 - 20407259400 * q^52 - 15082009344 * q^54 - 12572534832 * q^55 + 97529544392 * q^58 - 110336269112 * q^59 + 553127354432 * q^62 + 13799515808 * q^64 + 7914588558 * q^69 + 590935322064 * q^70 - 40523102210 * q^71 - 539385055680 * q^72 - 449748966242 * q^73 + 2570431278 * q^75 - 345974135760 * q^77 - 48894418992 * q^78 - 2694979782144 * q^81 - 1210586085208 * q^82 + 1051198266576 * q^85 - 2237359632618 * q^87 + 1936236755640 * q^92 - 116415989178 * q^93 + 3420883097024 * q^94 + 3777482201184 * q^95 + 3567911213376 * q^96 + 3590387133208 * q^98

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