LMFDB - Embedded newform 23.13.b.c.22.4

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			22.4
Root			$726.027 - 22106.6i$ of defining polynomial
Character	$\chi$	$=$	23.22
Dual form			23.13.b.c.22.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-108.578 q^{2} -822.605 q^{3} +7693.26 q^{4} +22106.6i q^{5} +89317.1 q^{6} +37537.4i q^{7} -390585. q^{8} +145238. q^{9} +O(q^{10})$	\(q-108.578 q^{2} -822.605 q^{3} +7693.26 q^{4} +22106.6i q^{5} +89317.1 q^{6} +37537.4i q^{7} -390585. q^{8} +145238. q^{9} -2.40030e6i q^{10} -1.71261e6i q^{11} -6.32852e6 q^{12} +2.82673e6 q^{13} -4.07575e6i q^{14} -1.81850e7i q^{15} +1.08975e7 q^{16} -3.09363e7i q^{17} -1.57697e7 q^{18} +6.04431e7i q^{19} +1.70072e8i q^{20} -3.08785e7i q^{21} +1.85952e8i q^{22} +(4.59207e7 - 1.40733e8i) q^{23} +3.21297e8 q^{24} -2.44563e8 q^{25} -3.06922e8 q^{26} +3.17693e8 q^{27} +2.88785e8i q^{28} +8.75516e8 q^{29} +1.97450e9i q^{30} -3.83695e8 q^{31} +4.16605e8 q^{32} +1.40880e9i q^{33} +3.35902e9i q^{34} -8.29826e8 q^{35} +1.11735e9 q^{36} -1.22482e9i q^{37} -6.56281e9i q^{38} -2.32528e9 q^{39} -8.63452e9i q^{40} +2.42141e9 q^{41} +3.35273e9i q^{42} -2.41300e9i q^{43} -1.31756e10i q^{44} +3.21072e9i q^{45} +(-4.98599e9 + 1.52806e10i) q^{46} -1.92249e10 q^{47} -8.96433e9 q^{48} +1.24322e10 q^{49} +2.65542e10 q^{50} +2.54484e10i q^{51} +2.17468e10 q^{52} +5.55632e9i q^{53} -3.44945e10 q^{54} +3.78600e10 q^{55} -1.46616e10i q^{56} -4.97208e10i q^{57} -9.50621e10 q^{58} +2.95376e10 q^{59} -1.39902e11i q^{60} +1.00189e11i q^{61} +4.16609e10 q^{62} +5.45186e9i q^{63} -8.98704e10 q^{64} +6.24895e10i q^{65} -1.52965e11i q^{66} +6.18938e10i q^{67} -2.38001e11i q^{68} +(-3.77746e10 + 1.15768e11i) q^{69} +9.01012e10 q^{70} -7.65896e10 q^{71} -5.67278e10 q^{72} +1.34703e11 q^{73} +1.32989e11i q^{74} +2.01179e11 q^{75} +4.65005e11i q^{76} +6.42869e10 q^{77} +2.52475e11 q^{78} +4.58260e11i q^{79} +2.40907e11i q^{80} -3.38521e11 q^{81} -2.62913e11 q^{82} -2.07317e11i q^{83} -2.37556e11i q^{84} +6.83899e11 q^{85} +2.62000e11i q^{86} -7.20204e11 q^{87} +6.68920e11i q^{88} -1.66099e11i q^{89} -3.48615e11i q^{90} +1.06108e11i q^{91} +(3.53280e11 - 1.08270e12i) q^{92} +3.15629e11 q^{93} +2.08741e12 q^{94} -1.33619e12 q^{95} -3.42702e11 q^{96} -5.52447e11i q^{97} -1.34987e12 q^{98} -2.48736e11i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−108.578			−1.69654			−0.848269	−	0.529566i	$-0.822355\pi$
$2$	−108.578			−1.69654			−0.848269	+	0.529566i	$0.822355\pi$
$3$	−822.605			−1.12840			−0.564201	−	0.825637i	$-0.690816\pi$
$3$	−822.605			−1.12840			−0.564201	+	0.825637i	$0.690816\pi$
$4$	7693.26			1.87824
$5$			22106.6i			1.41482i	0.706801	+	0.707412i	$0.250138\pi$
$5$			22106.6i			1.41482i	−0.706801	+	0.707412i	$0.749862\pi$
$6$	89317.1			1.91438
$7$			37537.4i			0.319063i	0.987193	+	0.159531i	$0.0509983\pi$
$7$			37537.4i			0.319063i	−0.987193	+	0.159531i	$0.949002\pi$
$8$	−390585.			−1.48996
$9$	145238.			0.273291
$10$		−	2.40030e6i		−	2.40030i
$11$		−	1.71261e6i		−	0.966723i	−0.875421	−	0.483362i	$-0.839416\pi$
$11$		−	1.71261e6i		−	0.966723i	0.875421	−	0.483362i	$-0.160584\pi$
$12$	−6.32852e6			−2.11941
$13$	2.82673e6			0.585631			0.292816	−	0.956169i	$-0.405408\pi$
$13$	2.82673e6			0.585631			0.292816	+	0.956169i	$0.405408\pi$
$14$		−	4.07575e6i		−	0.541302i
$15$		−	1.81850e7i		−	1.59649i
$16$	1.08975e7			0.649541
$17$		−	3.09363e7i		−	1.28167i	−0.767680	−	0.640834i	$-0.778589\pi$
$17$		−	3.09363e7i		−	1.28167i	0.767680	−	0.640834i	$-0.221411\pi$
$18$	−1.57697e7			−0.463648
$19$			6.04431e7i			1.28477i	0.766382	+	0.642385i	$0.222055\pi$
$19$			6.04431e7i			1.28477i	−0.766382	+	0.642385i	$0.777945\pi$
$20$			1.70072e8i			2.65738i
$21$		−	3.08785e7i		−	0.360031i
$22$			1.85952e8i			1.64008i
$23$	4.59207e7	−	1.40733e8i	0.310200	−	0.950671i
$23$	4.59207e7	−	1.40733e8i	0.310200	−	0.950671i
$24$	3.21297e8			1.68128
$25$	−2.44563e8			−1.00173
$26$	−3.06922e8			−0.993545
$27$	3.17693e8			0.820020
$28$			2.88785e8i			0.599276i
$29$	8.75516e8			1.47189			0.735946	−	0.677040i	$-0.236738\pi$
$29$	8.75516e8			1.47189			0.735946	+	0.677040i	$0.236738\pi$
$30$			1.97450e9i			2.70851i
$31$	−3.83695e8			−0.432330			−0.216165	−	0.976357i	$-0.569355\pi$
$31$	−3.83695e8			−0.432330			−0.216165	+	0.976357i	$0.569355\pi$
$32$	4.16605e8			0.387994
$33$			1.40880e9i			1.09085i
$34$			3.35902e9i			2.17440i
$35$	−8.29826e8			−0.451418
$36$	1.11735e9			0.513306
$37$		−	1.22482e9i		−	0.477377i	−0.971096	−	0.238688i	$-0.923283\pi$
$37$		−	1.22482e9i		−	0.477377i	0.971096	−	0.238688i	$-0.0767174\pi$
$38$		−	6.56281e9i		−	2.17966i
$39$	−2.32528e9			−0.660827
$40$		−	8.63452e9i		−	2.10804i
$41$	2.42141e9			0.509759			0.254880	−	0.966973i	$-0.417964\pi$
$41$	2.42141e9			0.509759			0.254880	+	0.966973i	$0.417964\pi$
$42$			3.35273e9i			0.610806i
$43$		−	2.41300e9i		−	0.381722i	−0.981617	−	0.190861i	$-0.938872\pi$
$43$		−	2.41300e9i		−	0.381722i	0.981617	−	0.190861i	$-0.0611280\pi$
$44$		−	1.31756e10i		−	1.81574i
$45$			3.21072e9i			0.386659i
$46$	−4.98599e9	+	1.52806e10i	−0.526265	+	1.61285i
$47$	−1.92249e10			−1.78352			−0.891758	−	0.452512i	$-0.850528\pi$
$47$	−1.92249e10			−1.78352			−0.891758	+	0.452512i	$0.850528\pi$
$48$	−8.96433e9			−0.732943
$49$	1.24322e10			0.898199
$50$	2.65542e10			1.69947
$51$			2.54484e10i			1.44624i
$52$	2.17468e10			1.09996
$53$			5.55632e9i			0.250687i	0.992113	+	0.125344i	$0.0400033\pi$
$53$			5.55632e9i			0.250687i	−0.992113	+	0.125344i	$0.959997\pi$
$54$	−3.44945e10			−1.39119
$55$	3.78600e10			1.36774
$56$		−	1.46616e10i		−	0.475392i
$57$		−	4.97208e10i		−	1.44974i
$58$	−9.50621e10			−2.49712
$59$	2.95376e10			0.700266			0.350133	−	0.936700i	$-0.386136\pi$
$59$	2.95376e10			0.700266			0.350133	+	0.936700i	$0.386136\pi$
$60$		−	1.39902e11i		−	2.99859i
$61$			1.00189e11i			1.94465i	0.233629	+	0.972326i	$0.424940\pi$
$61$			1.00189e11i			1.94465i	−0.233629	+	0.972326i	$0.575060\pi$
$62$	4.16609e10			0.733464
$63$			5.45186e9i			0.0871970i
$64$	−8.98704e10			−1.30779
$65$			6.24895e10i			0.828566i
$66$		−	1.52965e11i		−	1.85067i
$67$			6.18938e10i			0.684225i	0.939659	+	0.342112i	$0.111142\pi$
$67$			6.18938e10i			0.684225i	−0.939659	+	0.342112i	$0.888858\pi$
$68$		−	2.38001e11i		−	2.40728i
$69$	−3.77746e10	+	1.15768e11i	−0.350030	+	1.07274i
$70$	9.01012e10			0.765847
$71$	−7.65896e10			−0.597888			−0.298944	−	0.954271i	$-0.596634\pi$
$71$	−7.65896e10			−0.597888			−0.298944	+	0.954271i	$0.596634\pi$
$72$	−5.67278e10			−0.407194
$73$	1.34703e11			0.890106			0.445053	−	0.895504i	$-0.353185\pi$
$73$	1.34703e11			0.890106			0.445053	+	0.895504i	$0.353185\pi$
$74$			1.32989e11i			0.809887i
$75$	2.01179e11			1.13035
$76$			4.65005e11i			2.41310i
$77$	6.42869e10			0.308445
$78$	2.52475e11			1.12112
$79$			4.58260e11i			1.88517i	0.333973	+	0.942583i	$0.391611\pi$
$79$			4.58260e11i			1.88517i	−0.333973	+	0.942583i	$0.608389\pi$
$80$			2.40907e11i			0.918986i
$81$	−3.38521e11			−1.19860
$82$	−2.62913e11			−0.864825
$83$		−	2.07317e11i		−	0.634113i	−0.948407	−	0.317056i	$-0.897306\pi$
$83$		−	2.07317e11i		−	0.634113i	0.948407	−	0.317056i	$-0.102694\pi$
$84$		−	2.37556e11i		−	0.676224i
$85$	6.83899e11			1.81334
$86$			2.62000e11i			0.647606i
$87$	−7.20204e11			−1.66089
$88$			6.68920e11i			1.44038i
$89$		−	1.66099e11i		−	0.334216i	−0.985939	−	0.167108i	$-0.946557\pi$
$89$		−	1.66099e11i		−	0.334216i	0.985939	−	0.167108i	$-0.0534429\pi$
$90$		−	3.48615e11i		−	0.655981i
$91$			1.06108e11i			0.186853i
$92$	3.53280e11	−	1.08270e12i	0.582629	−	1.78559i
$93$	3.15629e11			0.487842
$94$	2.08741e12			3.02580
$95$	−1.33619e12			−1.81772
$96$	−3.42702e11			−0.437813
$97$		−	5.52447e11i		−	0.663225i	−0.943416	−	0.331612i	$-0.892407\pi$
$97$		−	5.52447e11i		−	0.663225i	0.943416	−	0.331612i	$-0.107593\pi$
$98$	−1.34987e12			−1.52383
$99$		−	2.48736e11i		−	0.264197i
$100$	−1.88149e12			−1.88149
$101$	7.11301e11			0.670078			0.335039	−	0.942204i	$-0.391251\pi$
$101$	7.11301e11			0.670078			0.335039	+	0.942204i	$0.391251\pi$
$102$		−	2.76315e12i		−	2.45359i
$103$		−	1.10820e12i		−	0.928098i	−0.885810	−	0.464049i	$-0.846396\pi$
$103$		−	1.10820e12i		−	0.928098i	0.885810	−	0.464049i	$-0.153604\pi$
$104$	−1.10408e12			−0.872569
$105$	6.82619e11			0.509381
$106$		−	6.03297e11i		−	0.425300i
$107$		−	1.30514e12i		−	0.869672i	−0.900510	−	0.434836i	$-0.856806\pi$
$107$		−	1.30514e12i		−	0.869672i	0.900510	−	0.434836i	$-0.143194\pi$
$108$	2.44409e12			1.54019
$109$		−	7.45583e11i		−	0.444566i	−0.974982	−	0.222283i	$-0.928649\pi$
$109$		−	7.45583e11i		−	0.444566i	0.974982	−	0.222283i	$-0.0713510\pi$
$110$	−4.11078e12			−2.32043
$111$			1.00754e12i			0.538673i
$112$			4.09063e11i			0.207244i
$113$			3.83328e12i			1.84120i	0.390512	+	0.920598i	$0.372298\pi$
$113$			3.83328e12i			1.84120i	−0.390512	+	0.920598i	$0.627702\pi$
$114$			5.39860e12i			2.45953i
$115$	3.11114e12	+	1.01515e12i	1.34503	+	0.438878i
$116$	6.73557e12			2.76456
$117$	4.10549e11			0.160048
$118$	−3.20714e12			−1.18803
$119$	1.16127e12			0.408932
$120$			7.10280e12i			2.37871i
$121$	2.05397e11			0.0654458
$122$		−	1.08784e13i		−	3.29917i
$123$	−1.99186e12			−0.575213
$124$	−2.95186e12			−0.812019
$125$		−	9.33516e9i		−	0.00244716i
$126$		−	5.91954e11i		−	0.147933i
$127$	−6.82753e12			−1.62720			−0.813600	−	0.581425i	$-0.802495\pi$
$127$	−6.82753e12			−1.62720			−0.813600	+	0.581425i	$0.802495\pi$
$128$	8.05157e12			1.83071
$129$			1.98495e12i			0.430736i
$130$		−	6.78501e12i		−	1.40569i
$131$	−4.54636e12			−0.899573			−0.449786	−	0.893136i	$-0.648500\pi$
$131$	−4.54636e12			−0.899573			−0.449786	+	0.893136i	$0.648500\pi$
$132$			1.08383e13i			2.04888i
$133$	−2.26888e12			−0.409922
$134$		−	6.72033e12i		−	1.16081i
$135$			7.02311e12i			1.16018i
$136$			1.20833e13i			1.90964i
$137$		−	2.77077e12i		−	0.419060i	−0.977802	−	0.209530i	$-0.932807\pi$
$137$		−	2.77077e12i		−	0.419060i	0.977802	−	0.209530i	$-0.0671934\pi$
$138$	4.10150e12	−	1.25699e13i	0.593839	−	1.81994i
$139$	7.81592e12			1.08366			0.541828	−	0.840490i	$-0.317732\pi$
$139$	7.81592e12			1.08366			0.541828	+	0.840490i	$0.317732\pi$
$140$	−6.38407e12			−0.847870
$141$	1.58145e13			2.01252
$142$	8.31598e12			1.01434
$143$		−	4.84109e12i		−	0.566143i
$144$	1.58273e12			0.177514
$145$			1.93547e13i			2.08247i
$146$	−1.46259e13			−1.51010
$147$	−1.02268e13			−1.01353
$148$		−	9.42285e12i		−	0.896627i
$149$			1.73092e13i			1.58183i	0.611929	+	0.790913i	$0.290394\pi$
$149$			1.73092e13i			1.58183i	−0.611929	+	0.790913i	$0.709606\pi$
$150$	−2.18437e13			−1.91769
$151$	1.94038e13			1.63691			0.818457	−	0.574567i	$-0.194830\pi$
$151$	1.94038e13			1.63691			0.818457	+	0.574567i	$0.194830\pi$
$152$		−	2.36082e13i		−	1.91426i
$153$		−	4.49313e12i		−	0.350268i
$154$	−6.98017e12			−0.523289
$155$		−	8.48220e12i		−	0.611672i
$156$	−1.78890e13			−1.24119
$157$			2.82234e13i			1.88457i	0.334814	+	0.942284i	$0.391326\pi$
$157$			2.82234e13i			1.88457i	−0.334814	+	0.942284i	$0.608674\pi$
$158$		−	4.97571e13i		−	3.19825i
$159$		−	4.57066e12i		−	0.282876i
$160$			9.20974e12i			0.548943i
$161$	5.28277e12	+	1.72374e12i	0.303324	+	0.0989732i
$162$	3.67560e13			2.03347
$163$	1.11660e13			0.595351			0.297675	−	0.954667i	$-0.403789\pi$
$163$	1.11660e13			0.595351			0.297675	+	0.954667i	$0.403789\pi$
$164$	1.86285e13			0.957449
$165$	−3.11439e13			−1.54337
$166$			2.25102e13i			1.07580i
$167$	2.47671e13			1.14176			0.570882	−	0.821032i	$-0.306601\pi$
$167$	2.47671e13			1.14176			0.570882	+	0.821032i	$0.306601\pi$
$168$			1.20607e13i			0.536433i
$169$	−1.53077e13			−0.657036
$170$	−7.42566e13			−3.07639
$171$			8.77864e12i			0.351116i
$172$		−	1.85639e13i		−	0.716965i
$173$	7.71921e11			0.0287936			0.0143968	−	0.999896i	$-0.495417\pi$
$173$	7.71921e11			0.0287936			0.0143968	+	0.999896i	$0.495417\pi$
$174$	7.81985e13			2.81775
$175$		−	9.18026e12i		−	0.319615i
$176$		−	1.86631e13i		−	0.627926i
$177$	−2.42978e13			−0.790181
$178$			1.80348e13i			0.567010i
$179$	4.51782e12			0.137344			0.0686722	−	0.997639i	$-0.478124\pi$
$179$	4.51782e12			0.137344			0.0686722	+	0.997639i	$0.478124\pi$
$180$			2.47010e13i			0.726238i
$181$		−	3.93163e13i		−	1.11815i	−0.829116	−	0.559077i	$-0.811156\pi$
$181$		−	3.93163e13i		−	1.11815i	0.829116	−	0.559077i	$-0.188844\pi$
$182$		−	1.15210e13i		−	0.317003i
$183$		−	8.24161e13i		−	2.19435i
$184$	−1.79359e13	+	5.49684e13i	−0.462186	+	1.41647i
$185$	2.70766e13			0.675404
$186$	−3.42705e13			−0.827643
$187$	−5.29819e13			−1.23902
$188$	−1.47902e14			−3.34987
$189$			1.19254e13i			0.261638i
$190$	1.45082e14			3.08384
$191$		−	6.66728e13i		−	1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$191$		−	6.66728e13i		−	1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$192$	7.39279e13			1.47571
$193$	−5.08431e13			−0.983758			−0.491879	−	0.870663i	$-0.663690\pi$
$193$	−5.08431e13			−0.983758			−0.491879	+	0.870663i	$0.663690\pi$
$194$			5.99839e13i			1.12519i
$195$		−	5.14042e13i		−	0.934955i
$196$	9.56444e13			1.68703
$197$	4.30945e10			0.000737266			0.000368633	−	1.00000i	$-0.499883\pi$
$197$	4.30945e10			0.000737266			0.000368633		1.00000i	$0.499883\pi$
$198$			2.70074e13i			0.448220i
$199$			7.02576e13i			1.13129i	0.824648	+	0.565646i	$0.191373\pi$
$199$			7.02576e13i			1.13129i	−0.824648	+	0.565646i	$0.808627\pi$
$200$	9.55226e13			1.49254
$201$		−	5.09142e13i		−	0.772080i
$202$	−7.72319e13			−1.13681
$203$			3.28646e13i			0.469626i
$204$			1.95781e14i			2.71638i
$205$			5.35292e13i			0.721220i
$206$			1.20326e14i			1.57455i
$207$	6.66943e12	−	2.04399e13i	0.0847748	−	0.259810i
$208$	3.08043e13			0.380391
$209$	1.03515e14			1.24202
$210$	−7.41177e13			−0.864184
$211$	−1.43330e14			−1.62422			−0.812108	−	0.583507i	$-0.801680\pi$
$211$	−1.43330e14			−1.62422			−0.812108	+	0.583507i	$0.801680\pi$
$212$			4.27463e13i			0.470851i
$213$	6.30030e13			0.674658
$214$			1.41710e14i			1.47543i
$215$	5.33434e13			0.540070
$216$	−1.24086e14			−1.22180
$217$		−	1.44029e13i		−	0.137940i
$218$			8.09541e13i			0.754224i
$219$	−1.10808e14			−1.00440
$220$	2.91267e14			2.56895
$221$		−	8.74487e13i		−	0.750585i
$222$		−	1.09397e14i		−	0.913878i
$223$	1.30515e14			1.06128			0.530640	−	0.847597i	$-0.321951\pi$
$223$	1.30515e14			1.06128			0.530640	+	0.847597i	$0.321951\pi$
$224$			1.56383e13i			0.123794i
$225$	−3.55198e13			−0.273764
$226$		−	4.16211e14i		−	3.12366i
$227$			5.73526e13i			0.419177i	0.977790	+	0.209589i	$0.0672125\pi$
$227$			5.73526e13i			0.419177i	−0.977790	+	0.209589i	$0.932788\pi$
$228$		−	3.82515e14i		−	2.72295i
$229$		−	5.16217e13i		−	0.357947i	−0.983854	−	0.178974i	$-0.942722\pi$
$229$		−	5.16217e13i		−	0.357947i	0.983854	−	0.178974i	$-0.0572777\pi$
$230$	−3.37803e14	−	1.10224e14i	−2.28190	−	0.744573i
$231$	−5.28828e13			−0.348050
$232$	−3.41963e14			−2.19307
$233$	2.67630e13			0.167263			0.0836314	−	0.996497i	$-0.473348\pi$
$233$	2.67630e13			0.167263			0.0836314	+	0.996497i	$0.473348\pi$
$234$	−4.45767e13			−0.271527
$235$		−	4.24998e14i		−	2.52336i
$236$	2.27240e14			1.31527
$237$		−	3.76967e14i		−	2.12722i
$238$	−1.26089e14			−0.693769
$239$	2.94824e12			0.0158189			0.00790943	−	0.999969i	$-0.497482\pi$
$239$	2.94824e12			0.0158189			0.00790943	+	0.999969i	$0.497482\pi$
$240$		−	1.98171e14i		−	1.03699i
$241$			1.74099e14i			0.888577i	0.895884	+	0.444288i	$0.146543\pi$
$241$			1.74099e14i			0.888577i	−0.895884	+	0.444288i	$0.853457\pi$
$242$	−2.23017e13			−0.111031
$243$	1.09634e14			0.532486
$244$			7.70782e14i			3.65252i
$245$			2.74835e14i			1.27079i
$246$	2.16273e14			0.975871
$247$			1.70856e14i			0.752401i
$248$	1.49865e14			0.644156
$249$			1.70540e14i			0.715534i
$250$			1.01360e12i			0.00415169i
$251$			2.91930e14i			1.16744i	0.811953	+	0.583722i	$0.198404\pi$
$251$			2.91930e14i			1.16744i	−0.811953	+	0.583722i	$0.801596\pi$
$252$			4.19426e13i			0.163777i
$253$	−2.41022e14	−	7.86442e13i	−0.919036	−	0.299877i
$254$	7.41322e14			2.76061
$255$	−5.62579e14			−2.04617
$256$	−5.06117e14			−1.79809
$257$	1.74700e14			0.606309			0.303154	−	0.952941i	$-0.401960\pi$
$257$	1.74700e14			0.606309			0.303154	+	0.952941i	$0.401960\pi$
$258$		−	2.15523e14i		−	0.730759i
$259$	4.59765e13			0.152313
$260$			4.80748e14i			1.55624i
$261$	1.27158e14			0.402255
$262$	4.93637e14			1.52616
$263$			3.14336e13i			0.0949861i	0.998872	+	0.0474930i	$0.0151232\pi$
$263$			3.14336e13i			0.0949861i	−0.998872	+	0.0474930i	$0.984877\pi$
$264$		−	5.50257e14i		−	1.62533i
$265$	−1.22832e14			−0.354679
$266$	2.46351e14			0.695448
$267$			1.36634e14i			0.377130i
$268$			4.76166e14i			1.28514i
$269$	−2.89400e14			−0.763808			−0.381904	−	0.924202i	$-0.624731\pi$
$269$	−2.89400e14			−0.763808			−0.381904	+	0.924202i	$0.624731\pi$
$270$		−	7.62558e14i		−	1.96830i
$271$	5.62359e14			1.41971			0.709853	−	0.704350i	$-0.248762\pi$
$271$	5.62359e14			1.41971			0.709853	+	0.704350i	$0.248762\pi$
$272$		−	3.37128e14i		−	0.832496i
$273$		−	8.72851e13i		−	0.210845i
$274$			3.00845e14i			0.710951i
$275$			4.18841e14i			0.968396i
$276$	−2.90610e14	+	8.90634e14i	−0.657440	+	2.01486i
$277$	7.34185e14			1.62528			0.812638	−	0.582769i	$-0.198031\pi$
$277$	7.34185e14			1.62528			0.812638	+	0.582769i	$0.198031\pi$
$278$	−8.48640e14			−1.83846
$279$	−5.57271e13			−0.118152
$280$	3.24118e14			0.672596
$281$			2.85838e14i			0.580607i	0.956935	+	0.290303i	$0.0937562\pi$
$281$			2.85838e14i			0.580607i	−0.956935	+	0.290303i	$0.906244\pi$
$282$	−1.71711e15			−3.41432
$283$			2.70692e13i			0.0526934i	0.999653	+	0.0263467i	$0.00838739\pi$
$283$			2.70692e13i			0.0526934i	−0.999653	+	0.0263467i	$0.991613\pi$
$284$	−5.89224e14			−1.12298
$285$	1.09916e15			2.05112
$286$			5.25637e14i			0.960483i
$287$			9.08934e13i			0.162645i
$288$	6.05069e13			0.106035
$289$	−3.74435e14			−0.642673
$290$		−	2.10150e15i		−	3.53299i
$291$			4.54446e14i			0.748384i
$292$	1.03631e15			1.67183
$293$		−	7.38041e14i		−	1.16647i	−0.812303	−	0.583236i	$-0.801786\pi$
$293$		−	7.38041e14i		−	1.16647i	0.812303	−	0.583236i	$-0.198214\pi$
$294$	1.11041e15			1.71949
$295$			6.52977e14i			0.990754i
$296$			4.78396e14i			0.711274i
$297$		−	5.44083e14i		−	0.792732i
$298$		−	1.87940e15i		−	2.68363i
$299$	1.29805e14	−	3.97816e14i	0.181663	−	0.556743i
$300$	1.54772e15			2.12307
$301$	9.05779e13			0.121793
$302$	−2.10684e15			−2.77709
$303$	−5.85120e14			−0.756117
$304$			6.58678e14i			0.834510i
$305$	−2.21485e15			−2.75134
$306$			4.87857e14i			0.594243i
$307$	−3.15247e14			−0.376548			−0.188274	−	0.982116i	$-0.560289\pi$
$307$	−3.15247e14			−0.376548			−0.188274	+	0.982116i	$0.560289\pi$
$308$	4.94576e14			0.579334
$309$			9.11609e14i			1.04727i
$310$			9.20984e14i			1.03772i
$311$	7.90606e14			0.873772			0.436886	−	0.899517i	$-0.356081\pi$
$311$	7.90606e14			0.873772			0.436886	+	0.899517i	$0.356081\pi$
$312$	9.08221e14			0.984609
$313$			6.99599e14i			0.744017i	0.928229	+	0.372009i	$0.121331\pi$
$313$			6.99599e14i			0.744017i	−0.928229	+	0.372009i	$0.878669\pi$
$314$		−	3.06445e15i		−	3.19724i
$315$	−1.20522e14			−0.123368
$316$			3.52552e15i			3.54079i
$317$	1.17180e15			1.15477			0.577387	−	0.816470i	$-0.304072\pi$
$317$	1.17180e15			1.15477			0.577387	+	0.816470i	$0.304072\pi$
$318$			4.96275e14i			0.479910i
$319$		−	1.49942e15i		−	1.42291i
$320$		−	1.98673e15i		−	1.85029i
$321$			1.07362e15i			0.981340i
$322$	−5.73595e14	−	1.87161e14i	−0.514600	−	0.167912i
$323$	1.86989e15			1.64665
$324$	−2.60433e15			−2.25126
$325$	−6.91313e14			−0.586644
$326$	−1.21239e15			−1.01004
$327$			6.13320e14i			0.501650i
$328$	−9.45766e14			−0.759523
$329$		−	7.21654e14i		−	0.569054i
$330$	3.38155e15			2.61838
$331$	7.99031e14			0.607569			0.303785	−	0.952741i	$-0.401750\pi$
$331$	7.99031e14			0.607569			0.303785	+	0.952741i	$0.401750\pi$
$332$		−	1.59494e15i		−	1.19101i
$333$		−	1.77890e14i		−	0.130463i
$334$	−2.68917e15			−1.93704
$335$	−1.36827e15			−0.968058
$336$		−	3.36498e14i		−	0.233855i
$337$			1.76077e15i			1.20205i	0.799230	+	0.601026i	$0.205241\pi$
$337$			1.76077e15i			1.20205i	−0.799230	+	0.601026i	$0.794759\pi$
$338$	1.66208e15			1.11469
$339$		−	3.15328e15i		−	2.07761i
$340$	5.26141e15			3.40588
$341$			6.57119e14i			0.417944i
$342$		−	9.53170e14i		−	0.595681i
$343$			9.86240e14i			0.605645i
$344$			9.42483e14i			0.568752i
$345$	−2.55924e15	−	8.35069e14i	−1.51774	−	0.495231i
$346$	−8.38139e13			−0.0488494
$347$	3.03437e14			0.173817			0.0869084	−	0.996216i	$-0.472301\pi$
$347$	3.03437e14			0.173817			0.0869084	+	0.996216i	$0.472301\pi$
$348$	−5.54072e15			−3.11954
$349$	−2.30986e14			−0.127830			−0.0639150	−	0.997955i	$-0.520359\pi$
$349$	−2.30986e14			−0.127830			−0.0639150	+	0.997955i	$0.520359\pi$
$350$			9.96778e14i			0.542238i
$351$	8.98031e14			0.480229
$352$		−	7.13482e14i		−	0.375083i
$353$	1.03208e15			0.533416			0.266708	−	0.963777i	$-0.414064\pi$
$353$	1.03208e15			0.533416			0.266708	+	0.963777i	$0.414064\pi$
$354$	2.63821e15			1.34057
$355$		−	1.69314e15i		−	0.845907i
$356$		−	1.27785e15i		−	0.627738i
$357$	−9.55267e14			−0.461440
$358$	−4.90538e14			−0.233010
$359$		−	3.77448e14i		−	0.176315i	−0.996107	−	0.0881577i	$-0.971902\pi$
$359$		−	3.77448e14i		−	0.176315i	0.996107	−	0.0881577i	$-0.0280979\pi$
$360$		−	1.25406e15i		−	0.576108i
$361$	−1.44005e15			−0.650632
$362$			4.26890e15i			1.89699i
$363$	−1.68960e14			−0.0738491
$364$			8.16318e14i			0.350955i
$365$			2.97784e15i			1.25934i
$366$			8.94861e15i			3.72279i
$367$		−	9.53258e14i		−	0.390134i	−0.980790	−	0.195067i	$-0.937508\pi$
$367$		−	9.53258e14i		−	0.390134i	0.980790	−	0.195067i	$-0.0624924\pi$
$368$	5.00420e14	−	1.53364e15i	0.201487	−	0.617500i
$369$	3.51681e14			0.139313
$370$	−2.93993e15			−1.14585
$371$	−2.08570e14			−0.0799850
$372$	2.42822e15			0.916284
$373$		−	3.89233e14i		−	0.144529i	−0.997385	−	0.0722647i	$-0.976977\pi$
$373$		−	3.89233e14i		−	0.144529i	0.997385	−	0.0722647i	$-0.0230226\pi$
$374$	5.75269e15			2.10204
$375$			7.67915e12i			0.00276138i
$376$	7.50896e15			2.65738
$377$	2.47485e15			0.861986
$378$		−	1.29484e15i		−	0.443878i
$379$		−	2.40855e15i		−	0.812681i	−0.913722	−	0.406341i	$-0.866805\pi$
$379$		−	2.40855e15i		−	0.812681i	0.913722	−	0.406341i	$-0.133195\pi$
$380$	−1.02797e16			−3.41412
$381$	5.61636e15			1.83614
$382$			7.23923e15i			2.32976i
$383$		−	2.38575e15i		−	0.755843i	−0.925838	−	0.377921i	$-0.876639\pi$
$383$		−	2.38575e15i		−	0.755843i	0.925838	−	0.377921i	$-0.123361\pi$
$384$	−6.62326e15			−2.06578
$385$			1.42117e15i			0.436396i
$386$	5.52047e15			1.66898
$387$		−	3.50460e14i		−	0.104321i
$388$		−	4.25012e15i		−	1.24569i
$389$			2.93460e15i			0.846938i	0.905911	+	0.423469i	$0.139188\pi$
$389$			2.93460e15i			0.846938i	−0.905911	+	0.423469i	$0.860812\pi$
$390$			5.58138e15i			1.58619i
$391$	−4.35378e15	−	1.42062e15i	−1.21845	−	0.397573i
$392$	−4.85584e15			−1.33828
$393$	3.73986e15			1.01508
$394$	−4.67913e12			−0.00125080
$395$	−1.01306e16			−2.66718
$396$		−	1.91359e15i		−	0.496225i
$397$	5.59948e15			1.43022			0.715112	−	0.699009i	$-0.246375\pi$
$397$	5.59948e15			1.43022			0.715112	+	0.699009i	$0.246375\pi$
$398$		−	7.62845e15i		−	1.91928i
$399$	1.86639e15			0.462557
$400$	−2.66512e15			−0.650664
$401$			4.51535e15i			1.08599i	0.839736	+	0.542995i	$0.182710\pi$
$401$			4.51535e15i			1.08599i	−0.839736	+	0.542995i	$0.817290\pi$
$402$			5.52818e15i			1.30986i
$403$	−1.08460e15			−0.253186
$404$	5.47223e15			1.25857
$405$		−	7.48356e15i		−	1.69581i
$406$		−	3.56838e15i		−	0.796738i
$407$	−2.09763e15			−0.461491
$408$		−	9.93976e15i		−	2.15484i
$409$	−1.63133e14			−0.0348499			−0.0174249	−	0.999848i	$-0.505547\pi$
$409$	−1.63133e14			−0.0348499			−0.0174249	+	0.999848i	$0.505547\pi$
$410$		−	5.81212e15i		−	1.22358i
$411$			2.27925e15i			0.472868i
$412$		−	8.52565e15i		−	1.74319i
$413$			1.10876e15i			0.223429i
$414$	−7.24156e14	+	2.21933e15i	−0.143824	+	0.440777i
$415$	4.58308e15			0.897159
$416$	1.17763e15			0.227221
$417$	−6.42941e15			−1.22280
$418$	−1.12395e16			−2.10713
$419$		−	3.06518e15i		−	0.566463i	−0.959052	−	0.283232i	$-0.908593\pi$
$419$		−	3.06518e15i		−	0.566463i	0.959052	−	0.283232i	$-0.0914065\pi$
$420$	5.25157e15			0.956739
$421$			8.35913e15i			1.50130i	0.660698	+	0.750652i	$0.270260\pi$
$421$			8.35913e15i			1.50130i	−0.660698	+	0.750652i	$0.729740\pi$
$422$	1.55626e16			2.75554
$423$	−2.79219e15			−0.487419
$424$		−	2.17022e15i		−	0.373515i
$425$			7.56588e15i			1.28388i
$426$	−6.84076e15			−1.14458
$427$	−3.76084e15			−0.620466
$428$		−	1.00408e16i		−	1.63345i
$429$			3.98230e15i			0.638837i
$430$	−5.79194e15			−0.916249
$431$			6.08858e15i			0.949844i	0.880028	+	0.474922i	$0.157524\pi$
$431$			6.08858e15i			0.949844i	−0.880028	+	0.474922i	$0.842476\pi$
$432$	3.46205e15			0.532636
$433$		−	4.19989e15i		−	0.637251i	−0.947881	−	0.318625i	$-0.896779\pi$
$433$		−	4.19989e15i		−	0.637251i	0.947881	−	0.318625i	$-0.103221\pi$
$434$			1.56384e15i			0.234021i
$435$		−	1.59213e16i		−	2.34986i
$436$		−	5.73596e15i		−	0.835002i
$437$	8.50637e15	+	2.77559e15i	1.22139	+	0.398535i
$438$	1.20313e16			1.70400
$439$	2.48771e15			0.347546			0.173773	−	0.984786i	$-0.444404\pi$
$439$	2.48771e15			0.347546			0.173773	+	0.984786i	$0.444404\pi$
$440$	−1.47876e16			−2.03789
$441$	1.80563e15			0.245470
$442$			9.49504e15i			1.27340i
$443$	−1.09991e15			−0.145525			−0.0727623	−	0.997349i	$-0.523181\pi$
$443$	−1.09991e15			−0.145525			−0.0727623	+	0.997349i	$0.523181\pi$
$444$			7.75128e15i			1.01176i
$445$	3.67190e15			0.472857
$446$	−1.41711e16			−1.80050
$447$		−	1.42386e16i		−	1.78493i
$448$		−	3.37350e15i		−	0.417266i
$449$	−5.25324e15			−0.641134			−0.320567	−	0.947226i	$-0.603873\pi$
$449$	−5.25324e15			−0.641134			−0.320567	+	0.947226i	$0.603873\pi$
$450$	3.85669e15			0.464450
$451$		−	4.14693e15i		−	0.492796i
$452$			2.94904e16i			3.45820i
$453$	−1.59617e16			−1.84710
$454$		−	6.22725e15i		−	0.711150i
$455$	−2.34569e15			−0.264364
$456$			1.94202e16i			2.16005i
$457$			3.62790e15i			0.398252i	0.979974	+	0.199126i	$0.0638104\pi$
$457$			3.62790e15i			0.398252i	−0.979974	+	0.199126i	$0.936190\pi$
$458$			5.60500e15i			0.607271i
$459$		−	9.82825e15i		−	1.05099i
$460$	2.39349e16	+	7.80983e15i	2.52629	+	0.824318i
$461$	1.84337e16			1.92046			0.960231	−	0.279206i	$-0.0900711\pi$
$461$	1.84337e16			1.92046			0.960231	+	0.279206i	$0.0900711\pi$
$462$	5.74192e15			0.590480
$463$	1.22977e16			1.24836			0.624179	−	0.781282i	$-0.285434\pi$
$463$	1.22977e16			1.24836			0.624179	+	0.781282i	$0.285434\pi$
$464$	9.54092e15			0.956054
$465$			6.97750e15i			0.690212i
$466$	−2.90588e15			−0.283767
$467$			1.44278e16i			1.39091i	0.718571	+	0.695453i	$0.244796\pi$
$467$			1.44278e16i			1.39091i	−0.718571	+	0.695453i	$0.755204\pi$
$468$	3.15846e15			0.300608
$469$	−2.32333e15			−0.218311
$470$			4.61456e16i			4.28098i
$471$		−	2.32167e16i		−	2.12655i
$472$	−1.15369e16			−1.04337
$473$	−4.13253e15			−0.369020
$474$			4.09305e16i			3.60891i
$475$		−	1.47821e16i		−	1.28699i
$476$	8.93396e15			0.768073
$477$			8.06990e14i			0.0685106i
$478$	−3.20115e14			−0.0268373
$479$			1.71909e15i			0.142327i	0.997465	+	0.0711633i	$0.0226711\pi$
$479$			1.71909e15i			0.142327i	−0.997465	+	0.0711633i	$0.977329\pi$
$480$		−	7.57598e15i		−	0.619429i
$481$		−	3.46223e15i		−	0.279567i
$482$		−	1.89034e16i		−	1.50750i
$483$	−4.34563e15	−	1.41796e15i	−0.342271	−	0.111682i
$484$	1.58017e15			0.122923
$485$	1.22128e16			0.938347
$486$	−1.19039e16			−0.903382
$487$	−1.66403e16			−1.24735			−0.623675	−	0.781683i	$-0.714361\pi$
$487$	−1.66403e16			−1.24735			−0.623675	+	0.781683i	$0.714361\pi$
$488$		−	3.91324e16i		−	2.89746i
$489$	−9.18523e15			−0.671795
$490$		−	2.98411e16i		−	2.15595i
$491$	−2.47463e16			−1.76612			−0.883061	−	0.469258i	$-0.844522\pi$
$491$	−2.47463e16			−1.76612			−0.883061	+	0.469258i	$0.844522\pi$
$492$	−1.53239e16			−1.08039
$493$		−	2.70853e16i		−	1.88648i
$494$		−	1.85513e16i		−	1.27648i
$495$	5.49872e15			0.373792
$496$	−4.18131e15			−0.280816
$497$		−	2.87498e15i		−	0.190764i
$498$		−	1.85170e16i		−	1.21393i
$499$	−1.38850e15			−0.0899382			−0.0449691	−	0.998988i	$-0.514319\pi$
$499$	−1.38850e15			−0.0899382			−0.0449691	+	0.998988i	$0.514319\pi$
$500$		−	7.18179e13i		−	0.00459634i
$501$	−2.03735e16			−1.28837
$502$		−	3.16973e16i		−	1.98061i
$503$			1.94679e16i			1.20202i	0.799243	+	0.601008i	$0.205234\pi$
$503$			1.94679e16i			1.20202i	−0.799243	+	0.601008i	$0.794766\pi$
$504$		−	2.12942e15i		−	0.129920i
$505$			1.57245e16i			0.948043i
$506$	2.61697e16	+	8.53906e15i	1.55918	+	0.508753i
$507$	1.25922e16			0.741401
$508$	−5.25260e16			−3.05627
$509$	2.51862e16			1.44829			0.724147	−	0.689646i	$-0.242234\pi$
$509$	2.51862e16			1.44829			0.724147	+	0.689646i	$0.242234\pi$
$510$	6.10839e16			3.47141
$511$			5.05642e15i			0.284000i
$512$	2.19741e16			1.21981
$513$			1.92023e16i			1.05354i
$514$	−1.89686e16			−1.02863
$515$	2.44985e16			1.31310
$516$			1.52707e16i			0.809025i
$517$			3.29248e16i			1.72417i
$518$	−4.99205e15			−0.258405
$519$	−6.34986e14			−0.0324908
$520$		−	2.44075e16i		−	1.23453i
$521$			3.05158e16i			1.52580i	0.646515	+	0.762901i	$0.276226\pi$
$521$			3.05158e16i			1.52580i	−0.646515	+	0.762901i	$0.723774\pi$
$522$	−1.38066e16			−0.682440
$523$		−	4.86422e15i		−	0.237686i	−0.992913	−	0.118843i	$-0.962082\pi$
$523$		−	4.86422e15i		−	0.237686i	0.992913	−	0.118843i	$-0.0379185\pi$
$524$	−3.49764e16			−1.68961
$525$			7.55173e15i			0.360654i
$526$		−	3.41301e15i		−	0.161147i
$527$			1.18701e16i			0.554104i
$528$			1.53524e16i			0.708553i
$529$	−1.76972e16	−	1.29252e16i	−0.807552	−	0.589796i
$530$	1.33369e16			0.601726
$531$	4.28998e15			0.191376
$532$	−1.74551e16			−0.769931
$533$	6.84467e15			0.298531
$534$		−	1.48355e16i		−	0.639815i
$535$	2.88523e16			1.23043
$536$		−	2.41748e16i		−	1.01947i
$537$	−3.71638e15			−0.154980
$538$	3.14225e16			1.29583
$539$		−	2.12916e16i		−	0.868310i
$540$			5.40307e16i			2.17910i
$541$	5.92409e15			0.236286			0.118143	−	0.992997i	$-0.462306\pi$
$541$	5.92409e15			0.236286			0.118143	+	0.992997i	$0.462306\pi$
$542$	−6.10601e16			−2.40858
$543$			3.23418e16i			1.26173i
$544$		−	1.28882e16i		−	0.497279i
$545$	1.64823e16			0.628984
$546$			9.47727e15i			0.357707i
$547$	5.14072e15			0.191911			0.0959556	−	0.995386i	$-0.469409\pi$
$547$	5.14072e15			0.191911			0.0959556	+	0.995386i	$0.469409\pi$
$548$		−	2.13162e16i		−	0.787095i
$549$			1.45513e16i			0.531456i
$550$		−	4.54771e16i		−	1.64292i
$551$			5.29189e16i			1.89104i
$552$	1.47542e16	−	4.52173e16i	0.521532	−	1.59834i
$553$	−1.72019e16			−0.601486
$554$	−7.97166e16			−2.75734
$555$	−2.22734e16			−0.762128
$556$	6.01299e16			2.03536
$557$		−	1.73634e16i		−	0.581439i	−0.956808	−	0.290719i	$-0.906105\pi$
$557$		−	1.73634e16i		−	0.581439i	0.956808	−	0.290719i	$-0.0938946\pi$
$558$	6.05075e15			0.200449
$559$		−	6.82091e15i		−	0.223548i
$560$	−9.04302e15			−0.293214
$561$	4.35832e16			1.39811
$562$		−	3.10358e16i		−	0.985021i
$563$		−	5.88683e16i		−	1.84855i	−0.381729	−	0.924274i	$-0.624671\pi$
$563$		−	5.88683e16i		−	1.84855i	0.381729	−	0.924274i	$-0.375329\pi$
$564$	1.21665e17			3.78000
$565$	−8.47410e16			−2.60497
$566$		−	2.93913e15i		−	0.0893964i
$567$		−	1.27072e16i		−	0.382430i
$568$	2.99148e16			0.890832
$569$			1.23856e16i			0.364960i	0.983210	+	0.182480i	$0.0584125\pi$
$569$			1.23856e16i			0.364960i	−0.983210	+	0.182480i	$0.941588\pi$
$570$	−1.19345e17			−3.47981
$571$		−	1.08700e16i		−	0.313627i	−0.987628	−	0.156813i	$-0.949878\pi$
$571$		−	1.08700e16i		−	0.313627i	0.987628	−	0.156813i	$-0.0501221\pi$
$572$		−	3.72437e16i		−	1.06335i
$573$			5.48454e16i			1.54957i
$574$		−	9.86906e15i		−	0.275934i
$575$	−1.12305e16	+	3.44182e16i	−0.310736	+	0.952316i
$576$	−1.30526e16			−0.357406
$577$	5.70377e16			1.54564			0.772818	−	0.634628i	$-0.218847\pi$
$577$	5.70377e16			1.54564			0.772818	+	0.634628i	$0.218847\pi$
$578$	4.06556e16			1.09032
$579$	4.18238e16			1.11007
$580$			1.48901e17i			3.91137i
$581$	7.78215e15			0.202322
$582$		−	4.93430e16i		−	1.26966i
$583$	9.51581e15			0.242345
$584$	−5.26132e16			−1.32623
$585$			9.07585e15i			0.226440i
$586$			8.01353e16i			1.97896i
$587$	5.08401e16			1.24273			0.621366	−	0.783521i	$-0.286578\pi$
$587$	5.08401e16			1.24273			0.621366	+	0.783521i	$0.286578\pi$
$588$	−7.86776e16			−1.90365
$589$		−	2.31917e16i		−	0.555445i
$590$		−	7.08992e16i		−	1.68085i
$591$	−3.54497e13			−0.000831932
$592$		−	1.33474e16i		−	0.310076i
$593$	1.47881e16			0.340083			0.170041	−	0.985437i	$-0.445610\pi$
$593$	1.47881e16			0.340083			0.170041	+	0.985437i	$0.445610\pi$
$594$			5.90757e16i			1.34490i
$595$			2.56718e16i			0.578568i
$596$			1.33164e17i			2.97104i
$597$		−	5.77942e16i		−	1.27655i
$598$	−1.40941e16	+	4.31942e16i	−0.308197	+	0.944535i
$599$	−3.49420e16			−0.756462			−0.378231	−	0.925711i	$-0.623467\pi$
$599$	−3.49420e16			−0.756462			−0.378231	+	0.925711i	$0.623467\pi$
$600$	−7.85774e16			−1.68419
$601$	−1.31081e16			−0.278158			−0.139079	−	0.990281i	$-0.544414\pi$
$601$	−1.31081e16			−0.278158			−0.139079	+	0.990281i	$0.544414\pi$
$602$	−9.83480e15			−0.206627
$603$			8.98934e15i			0.186992i
$604$	1.49279e17			3.07452
$605$			4.54063e15i			0.0925943i
$606$	6.35314e16			1.28278
$607$	−9.31741e16			−1.86279			−0.931393	−	0.364015i	$-0.881406\pi$
$607$	−9.31741e16			−1.86279			−0.931393	+	0.364015i	$0.881406\pi$
$608$			2.51809e16i			0.498483i
$609$		−	2.70346e16i		−	0.529927i
$610$	2.40484e17			4.66775
$611$	−5.43436e16			−1.04448
$612$		−	3.45669e16i		−	0.657887i
$613$			8.06671e16i			1.52031i	0.649739	+	0.760157i	$0.274878\pi$
$613$			8.06671e16i			1.52031i	−0.649739	+	0.760157i	$0.725122\pi$
$614$	3.42290e16			0.638828
$615$		−	4.40334e16i		−	0.813826i
$616$	−2.51095e16			−0.459573
$617$		−	2.90426e16i		−	0.526411i	−0.964740	−	0.263206i	$-0.915220\pi$
$617$		−	2.90426e16i		−	0.526411i	0.964740	−	0.263206i	$-0.0847798\pi$
$618$		−	9.89810e16i		−	1.77673i
$619$		−	3.79265e16i		−	0.674216i	−0.941466	−	0.337108i	$-0.890551\pi$
$619$		−	3.79265e16i		−	0.674216i	0.941466	−	0.337108i	$-0.109449\pi$
$620$		−	6.52558e16i		−	1.14887i
$621$	1.45887e16	−	4.47100e16i	0.254370	−	0.779569i
$622$	−8.58428e16			−1.48239
$623$	6.23493e15			0.106636
$624$	−2.53397e16			−0.429234
$625$	−5.95014e16			−0.998267
$626$		−	7.59613e16i		−	1.26225i
$627$	−8.51523e16			−1.40149
$628$			2.17130e17i			3.53967i
$629$	−3.78914e16			−0.611838
$630$	1.30861e16			0.209299
$631$		−	9.14680e15i		−	0.144908i	−0.997372	−	0.0724541i	$-0.976917\pi$
$631$		−	9.14680e15i		−	0.144908i	0.997372	−	0.0724541i	$-0.0230831\pi$
$632$		−	1.78990e17i		−	2.80883i
$633$	1.17904e17			1.83277
$634$	−1.27232e17			−1.95912
$635$		−	1.50934e17i		−	2.30220i
$636$		−	3.51633e16i		−	0.531309i
$637$	3.51426e16			0.526013
$638$			1.62804e17i			2.41402i
$639$	−1.11237e16			−0.163397
$640$			1.77993e17i			2.59014i
$641$		−	6.40790e16i		−	0.923778i	−0.886938	−	0.461889i	$-0.847172\pi$
$641$		−	6.40790e16i		−	0.923778i	0.886938	−	0.461889i	$-0.152828\pi$
$642$		−	1.16572e17i		−	1.66488i
$643$			6.53715e16i			0.924959i	0.886630	+	0.462480i	$0.153040\pi$
$643$			6.53715e16i			0.924959i	−0.886630	+	0.462480i	$0.846960\pi$
$644$	4.06418e16	+	1.32612e16i	0.569714	+	0.185895i
$645$	−4.38806e16			−0.609416
$646$	−2.03029e17			−2.79360
$647$	9.58061e14			0.0130607			0.00653037	−	0.999979i	$-0.497921\pi$
$647$	9.58061e14			0.0130607			0.00653037	+	0.999979i	$0.497921\pi$
$648$	1.32221e17			1.78588
$649$		−	5.05864e16i		−	0.676963i
$650$	7.50617e16			0.995264
$651$			1.18479e16i			0.155652i
$652$	8.59032e16			1.11821
$653$	−1.33744e16			−0.172502			−0.0862511	−	0.996273i	$-0.527489\pi$
$653$	−1.33744e16			−0.172502			−0.0862511	+	0.996273i	$0.527489\pi$
$654$		−	6.65933e16i		−	0.851067i
$655$		−	1.00505e17i		−	1.27274i
$656$	2.63873e16			0.331109
$657$	1.95641e16			0.243258
$658$			7.83560e16i			0.965421i
$659$		−	8.08322e16i		−	0.986897i	−0.869775	−	0.493449i	$-0.835736\pi$
$659$		−	8.08322e16i		−	0.986897i	0.869775	−	0.493449i	$-0.164264\pi$
$660$	−2.39598e17			−2.89881
$661$		−	8.49707e16i		−	1.01873i	−0.860550	−	0.509366i	$-0.829880\pi$
$661$		−	8.49707e16i		−	1.01873i	0.860550	−	0.509366i	$-0.170120\pi$
$662$	−8.67575e16			−1.03076
$663$			7.19358e16i			0.846961i
$664$			8.09750e16i			0.944805i
$665$		−	5.01573e16i		−	0.579968i
$666$			1.93150e16i			0.221335i
$667$	4.02043e16	−	1.23214e17i	0.456580	−	1.39929i
$668$	1.90540e17			2.14450
$669$	−1.07362e17			−1.19755
$670$	1.48564e17			1.64235
$671$	1.71585e17			1.87994
$672$		−	1.28641e16i		−	0.139690i
$673$	1.21371e17			1.30624			0.653120	−	0.757254i	$-0.273460\pi$
$673$	1.21371e17			1.30624			0.653120	+	0.757254i	$0.273460\pi$
$674$		−	1.91181e17i		−	2.03932i
$675$	−7.76958e16			−0.821438
$676$	−1.17766e17			−1.23407
$677$			1.89816e17i			1.97152i	0.168168	+	0.985758i	$0.446215\pi$
$677$			1.89816e17i			1.97152i	−0.168168	+	0.985758i	$0.553785\pi$
$678$			3.42378e17i			3.52474i
$679$	2.07374e16			0.211610
$680$	−2.67121e17			−2.70180
$681$		−	4.71785e16i		−	0.473000i
$682$		−	7.13489e16i		−	0.709057i
$683$	−1.56171e17			−1.53842			−0.769212	−	0.638993i	$-0.779351\pi$
$683$	−1.56171e17			−1.53842			−0.769212	+	0.638993i	$0.779351\pi$
$684$			6.75364e16i			0.659479i
$685$	6.12523e16			0.592897
$686$		−	1.07084e17i		−	1.02750i
$687$			4.24642e16i			0.403908i
$688$		−	2.62957e16i		−	0.247944i
$689$			1.57062e16i			0.146810i
$690$	2.77878e17	+	9.06705e16i	2.57490	+	0.840178i
$691$	−2.86766e16			−0.263426			−0.131713	−	0.991288i	$-0.542048\pi$
$691$	−2.86766e16			−0.263426			−0.131713	+	0.991288i	$0.542048\pi$
$692$	5.93859e15			0.0540813
$693$	9.33691e15			0.0842954
$694$	−3.29467e16			−0.294887
$695$			1.72784e17i			1.53318i
$696$	2.81301e17			2.47466
$697$		−	7.49096e16i		−	0.653342i
$698$	2.50801e16			0.216868
$699$	−2.20154e16			−0.188740
$700$		−	7.06262e16i		−	0.600312i
$701$			1.62122e17i			1.36626i	0.730296	+	0.683131i	$0.239382\pi$
$701$			1.62122e17i			1.36626i	−0.730296	+	0.683131i	$0.760618\pi$
$702$	−9.75068e16			−0.814727
$703$	7.40318e16			0.613319
$704$			1.53913e17i			1.26427i
$705$			3.49606e17i			2.84737i
$706$	−1.12062e17			−0.904960
$707$			2.67004e16i			0.213797i
$708$	−1.86929e17			−1.48415
$709$		−	5.04179e16i		−	0.396924i	−0.980109	−	0.198462i	$-0.936405\pi$
$709$		−	5.04179e16i		−	0.396924i	0.980109	−	0.198462i	$-0.0635947\pi$
$710$			1.83838e17i			1.43511i
$711$			6.65568e16i			0.515199i
$712$			6.48759e16i			0.497970i
$713$	−1.76195e16	+	5.39987e16i	−0.134109	+	0.411004i
$714$	1.03721e17			0.782850
$715$	1.07020e17			0.800994
$716$	3.47568e16			0.257965
$717$	−2.42523e15			−0.0178500
$718$			4.09827e16i			0.299126i
$719$	−1.99020e16			−0.144054			−0.0720268	−	0.997403i	$-0.522947\pi$
$719$	−1.99020e16			−0.144054			−0.0720268	+	0.997403i	$0.522947\pi$
$720$			3.49888e16i			0.251151i
$721$	4.15989e16			0.296121
$722$	1.56359e17			1.10382
$723$		−	1.43215e17i		−	1.00267i
$724$		−	3.02471e17i		−	2.10016i
$725$	−2.14119e17			−1.47444
$726$	1.83455e16			0.125288
$727$			2.77962e17i			1.88269i	0.337443	+	0.941346i	$0.390438\pi$
$727$			2.77962e17i			1.88269i	−0.337443	+	0.941346i	$0.609562\pi$
$728$		−	4.14443e16i		−	0.278404i
$729$	8.97183e16			0.597745
$730$		−	3.23329e17i		−	2.13652i
$731$	−7.46495e16			−0.489241
$732$		−	6.34049e17i		−	4.12151i
$733$		−	2.17514e16i		−	0.140237i	−0.997539	−	0.0701185i	$-0.977662\pi$
$733$		−	2.17514e16i		−	0.140237i	0.997539	−	0.0701185i	$-0.0223378\pi$
$734$			1.03503e17i			0.661877i
$735$		−	2.26081e17i		−	1.43397i
$736$	1.91308e16	−	5.86303e16i	0.120356	−	0.368855i
$737$	1.06000e17			0.661456
$738$	−3.81849e16			−0.236349
$739$	6.21321e16			0.381460			0.190730	−	0.981642i	$-0.438914\pi$
$739$	6.21321e16			0.381460			0.190730	+	0.981642i	$0.438914\pi$
$740$	2.08307e17			1.26857
$741$		−	1.40547e17i		−	0.849011i
$742$	2.26462e16			0.135698
$743$		−	2.02759e17i		−	1.20517i	−0.798055	−	0.602584i	$-0.794138\pi$
$743$		−	2.02759e17i		−	1.20517i	0.798055	−	0.602584i	$-0.205862\pi$
$744$	−1.23280e17			−0.726867
$745$	−3.82647e17			−2.23801
$746$			4.22623e16i			0.245200i
$747$		−	3.01103e16i		−	0.173297i
$748$	−4.07604e17			−2.32717
$749$	4.89917e16			0.277480
$750$		−	8.33790e14i		−	0.00468478i
$751$			1.79393e16i			0.0999921i	0.998749	+	0.0499961i	$0.0159209\pi$
$751$			1.79393e16i			0.0999921i	−0.998749	+	0.0499961i	$0.984079\pi$
$752$	−2.09503e17			−1.15847
$753$		−	2.40143e17i		−	1.31735i
$754$	−2.68715e17			−1.46239
$755$			4.28954e17i			2.31595i
$756$			9.17449e16i			0.491418i
$757$			1.82824e17i			0.971534i	0.874088	+	0.485767i	$0.161460\pi$
$757$			1.82824e17i			0.971534i	−0.874088	+	0.485767i	$0.838540\pi$
$758$			2.61516e17i			1.37874i
$759$	1.98266e17	+	6.46931e16i	1.03704	+	0.338382i
$760$	5.21897e17			2.70834
$761$	2.57894e17			1.32780			0.663900	−	0.747821i	$-0.268900\pi$
$761$	2.57894e17			1.32780			0.663900	+	0.747821i	$0.268900\pi$
$762$	−6.09815e17			−3.11507
$763$	2.79872e16			0.141845
$764$		−	5.12932e17i		−	2.57929i
$765$	9.93281e16			0.495568
$766$			2.59040e17i			1.28232i
$767$	8.34948e16			0.410098
$768$	4.16334e17			2.02897
$769$		−	4.98115e16i		−	0.240864i	−0.992722	−	0.120432i	$-0.961572\pi$
$769$		−	4.98115e16i		−	0.240864i	0.992722	−	0.120432i	$-0.0384280\pi$
$770$		−	1.54308e17i		−	0.740363i
$771$	−1.43709e17			−0.684160
$772$	−3.91150e17			−1.84773
$773$		−	2.85993e17i		−	1.34053i	−0.742120	−	0.670267i	$-0.766180\pi$
$773$		−	2.85993e17i		−	1.34053i	0.742120	−	0.670267i	$-0.233820\pi$
$774$			3.80524e16i			0.176985i
$775$	9.38375e16			0.433078
$776$			2.15778e17i			0.988181i
$777$	−3.78205e16			−0.171870
$778$		−	3.18634e17i		−	1.43686i
$779$			1.46357e17i			0.654923i
$780$		−	3.95466e17i		−	1.75607i
$781$			1.31168e17i			0.577992i
$782$	4.72726e17	+	1.54248e17i	2.06714	+	0.674497i
$783$	2.78145e17			1.20698
$784$	1.35480e17			0.583417
$785$	−6.23925e17			−2.66633
$786$	−4.06068e17			−1.72212
$787$		−	3.26272e17i		−	1.37319i	−0.727039	−	0.686596i	$-0.759104\pi$
$787$		−	3.26272e17i		−	1.37319i	0.727039	−	0.686596i	$-0.240896\pi$
$788$	3.31537e14			0.00138476
$789$		−	2.58575e16i		−	0.107182i
$790$	1.09996e18			4.52497
$791$	−1.43891e17			−0.587457
$792$			9.71526e16i			0.393644i
$793$			2.83208e17i			1.13885i
$794$	−6.07982e17			−2.42643
$795$	1.01042e17			0.400220
$796$			5.40510e17i			2.12484i
$797$		−	3.11112e17i		−	1.21385i	−0.794758	−	0.606927i	$-0.792402\pi$
$797$		−	3.11112e17i		−	1.21385i	0.794758	−	0.606927i	$-0.207598\pi$
$798$	−2.02650e17			−0.784745
$799$			5.94749e17i			2.28588i
$800$	−1.01886e17			−0.388665
$801$		−	2.41239e16i		−	0.0913383i
$802$		−	4.90270e17i		−	1.84242i
$803$		−	2.30694e17i		−	0.860486i
$804$		−	3.91696e17i		−	1.45015i
$805$	−3.81062e16	+	1.16784e17i	−0.140030	+	0.429150i
$806$	1.17764e17			0.429540
$807$	2.38062e17			0.861882
$808$	−2.77824e17			−0.998392
$809$	2.82839e17			1.00890			0.504450	−	0.863441i	$-0.331695\pi$
$809$	2.82839e17			1.00890			0.504450	+	0.863441i	$0.331695\pi$
$810$			8.12553e17i			2.87701i
$811$	5.32574e16			0.187178			0.0935890	−	0.995611i	$-0.470166\pi$
$811$	5.32574e16			0.187178			0.0935890	+	0.995611i	$0.470166\pi$
$812$			2.52836e17i			0.882069i
$813$	−4.62600e17			−1.60200
$814$	2.27758e17			0.782937
$815$			2.46843e17i			0.842317i
$816$			2.77324e17i			0.939390i
$817$	1.45849e17			0.490425
$818$	1.77127e16			0.0591241
$819$			1.54109e16i			0.0510653i
$820$			4.11814e17i			1.35462i
$821$	4.17544e17			1.36346			0.681732	−	0.731602i	$-0.261227\pi$
$821$	4.17544e17			1.36346			0.681732	+	0.731602i	$0.261227\pi$
$822$		−	2.47477e17i		−	0.802239i
$823$	3.22325e17			1.03728			0.518639	−	0.854994i	$-0.326439\pi$
$823$	3.22325e17			1.03728			0.518639	+	0.854994i	$0.326439\pi$
$824$			4.32845e17i			1.38283i
$825$		−	3.44541e17i		−	1.09274i
$826$		−	1.20388e17i		−	0.379055i
$827$			2.39216e17i			0.747752i	0.927479	+	0.373876i	$0.121972\pi$
$827$			2.39216e17i			0.747752i	−0.927479	+	0.373876i	$0.878028\pi$
$828$	5.13097e16	−	1.57249e17i	0.159227	−	0.487985i
$829$	−4.08241e17			−1.25773			−0.628867	−	0.777512i	$-0.716481\pi$
$829$	−4.08241e17			−1.25773			−0.628867	+	0.777512i	$0.716481\pi$
$830$	−4.97624e17			−1.52206
$831$	−6.03944e17			−1.83396
$832$	−2.54039e17			−0.765881
$833$		−	3.84608e17i		−	1.15119i
$834$	6.98095e17			2.07452
$835$			5.47517e17i			1.61540i
$836$	7.96372e17			2.33280
$837$	−1.21897e17			−0.354519
$838$			3.32812e17i			0.961026i
$839$			2.29956e15i			0.00659285i	0.999995	+	0.00329643i	$0.00104929\pi$
$839$			2.29956e15i			0.00659285i	−0.999995	+	0.00329643i	$0.998951\pi$
$840$	−2.66621e17			−0.758959
$841$	4.12713e17			1.16647
$842$		−	9.07621e17i		−	2.54702i
$843$		−	2.35132e17i		−	0.655158i
$844$	−1.10268e18			−3.05066
$845$		−	3.38401e17i		−	0.929591i
$846$	3.03171e17			0.826925
$847$			7.71006e15i			0.0208813i
$848$			6.05500e16i			0.162832i
$849$		−	2.22672e16i		−	0.0594594i
$850$		−	8.21491e17i		−	2.17816i
$851$	−1.72373e17	−	5.62445e16i	−0.453828	−	0.148082i
$852$	4.84699e17			1.26717
$853$	−2.07068e17			−0.537549			−0.268775	−	0.963203i	$-0.586619\pi$
$853$	−2.07068e17			−0.537549			−0.268775	+	0.963203i	$0.586619\pi$
$854$	4.08346e17			1.05264
$855$	−1.94066e17			−0.496767
$856$			5.09769e17i			1.29578i
$857$	−3.91618e17			−0.988502			−0.494251	−	0.869319i	$-0.664558\pi$
$857$	−3.91618e17			−0.988502			−0.494251	+	0.869319i	$0.664558\pi$
$858$		−	4.32392e17i		−	1.08381i
$859$	3.71326e17			0.924266			0.462133	−	0.886811i	$-0.347084\pi$
$859$	3.71326e17			0.924266			0.462133	+	0.886811i	$0.347084\pi$
$860$	4.10385e17			1.01438
$861$		−	7.47694e16i		−	0.183529i
$862$		−	6.61088e17i		−	1.61145i
$863$	−1.84894e17			−0.447568			−0.223784	−	0.974639i	$-0.571841\pi$
$863$	−1.84894e17			−0.447568			−0.223784	+	0.974639i	$0.571841\pi$
$864$	1.32352e17			0.318163
$865$			1.70646e16i			0.0407379i
$866$			4.56017e17i			1.08112i
$867$	3.08012e17			0.725193
$868$		−	1.10805e17i		−	0.259085i
$869$	7.84821e17			1.82243
$870$			1.72871e18i			3.98663i
$871$			1.74957e17i			0.400703i
$872$			2.91213e17i			0.662388i
$873$		−	8.02364e16i		−	0.181253i
$874$	−9.23608e17	−	3.01369e17i	−2.07214	−	0.676130i
$875$	3.50418e14			0.000780797
$876$	−8.52473e17			−1.88650
$877$	−7.37894e16			−0.162180			−0.0810898	−	0.996707i	$-0.525840\pi$
$877$	−7.37894e16			−0.162180			−0.0810898	+	0.996707i	$0.525840\pi$
$878$	−2.70111e17			−0.589625
$879$			6.07116e17i			1.31625i
$880$	4.12579e17			0.888406
$881$		−	9.59554e16i		−	0.205217i	−0.994722	−	0.102609i	$-0.967281\pi$
$881$		−	9.59554e16i		−	0.205217i	0.994722	−	0.102609i	$-0.0327189\pi$
$882$	−1.96053e17			−0.416448
$883$	−7.27208e17			−1.53424			−0.767122	−	0.641501i	$-0.778312\pi$
$883$	−7.27208e17			−1.53424			−0.767122	+	0.641501i	$0.778312\pi$
$884$		−	6.72766e17i		−	1.40978i
$885$		−	5.37142e17i		−	1.11797i
$886$	1.19427e17			0.246888
$887$	−4.16193e17			−0.854580			−0.427290	−	0.904115i	$-0.640532\pi$
$887$	−4.16193e17			−0.854580			−0.427290	+	0.904115i	$0.640532\pi$
$888$		−	3.93531e17i		−	0.802603i
$889$		−	2.56288e17i		−	0.519179i
$890$	−3.98688e17			−0.802220
$891$			5.79754e17i			1.15872i
$892$	1.00408e18			1.99334
$893$		−	1.16201e18i		−	2.29141i
$894$			1.54600e18i			3.02821i
$895$			9.98738e16i			0.194318i
$896$			3.02235e17i			0.584113i
$897$	−1.06779e17	+	3.27245e17i	−0.204988	+	0.628230i
$898$	5.70388e17			1.08771
$899$	−3.35931e17			−0.636343
$900$	−2.73263e17			−0.514193
$901$	1.71892e17			0.321298
$902$			4.50267e17i			0.836047i
$903$	−7.45098e16			−0.137432
$904$		−	1.49722e18i		−	2.74332i
$905$	8.69152e17			1.58199
$906$	1.73310e18			3.13367
$907$			3.01361e17i			0.541306i	0.962677	+	0.270653i	$0.0872396\pi$
$907$			3.01361e17i			0.541306i	−0.962677	+	0.270653i	$0.912760\pi$
$908$			4.41228e17i			0.787315i
$909$	1.03308e17			0.183126
$910$	2.54692e17			0.448504
$911$			6.16198e17i			1.07798i	0.842313	+	0.538989i	$0.181194\pi$
$911$			6.16198e17i			1.07798i	−0.842313	+	0.538989i	$0.818806\pi$
$912$		−	5.41832e17i		−	0.941663i
$913$	−3.55053e17			−0.613012
$914$		−	3.93912e17i		−	0.675650i
$915$	1.82194e18			3.10462
$916$		−	3.97139e17i		−	0.672310i
$917$		−	1.70659e17i		−	0.287020i
$918$			1.06714e18i			1.78305i
$919$			3.00736e17i			0.499220i	0.968346	+	0.249610i	$0.0803024\pi$
$919$			3.00736e17i			0.499220i	−0.968346	+	0.249610i	$0.919698\pi$
$920$	−1.21517e18	−	3.96503e17i	−2.00405	−	0.653913i
$921$	2.59324e17			0.424898
$922$	−2.00150e18			−3.25814
$923$	−2.16498e17			−0.350142
$924$	−4.06841e17			−0.653722
$925$			2.99545e17i			0.478202i
$926$	−1.33527e18			−2.11789
$927$		−	1.60952e17i		−	0.253641i
$928$	3.64744e17			0.571085
$929$	−5.06489e17			−0.787908			−0.393954	−	0.919130i	$-0.628893\pi$
$929$	−5.06489e17			−0.787908			−0.393954	+	0.919130i	$0.628893\pi$
$930$		−	7.57606e17i		−	1.17097i
$931$			7.51443e17i			1.15398i
$932$	2.05895e17			0.314159
$933$	−6.50357e17			−0.985966
$934$		−	1.56654e18i		−	2.35972i
$935$		−	1.17125e18i		−	1.75299i
$936$	−1.60354e17			−0.238465
$937$			6.56767e16i			0.0970452i	0.998822	+	0.0485226i	$0.0154513\pi$
$937$			6.56767e16i			0.0970452i	−0.998822	+	0.0485226i	$0.984549\pi$
$938$	2.52264e17			0.370372
$939$		−	5.75493e17i		−	0.839550i
$940$		−	3.26962e18i		−	4.73948i
$941$		−	4.45760e17i		−	0.642042i	−0.947072	−	0.321021i	$-0.895974\pi$
$941$		−	4.45760e17i		−	0.642042i	0.947072	−	0.321021i	$-0.104026\pi$
$942$			2.52084e18i			3.60777i
$943$	1.11193e17	−	3.40773e17i	0.158127	−	0.484614i
$944$	3.21885e17			0.454851
$945$	−2.63630e17			−0.370172
$946$	4.48704e17			0.626056
$947$	−9.93630e15			−0.0137761			−0.00688803	−	0.999976i	$-0.502193\pi$
$947$	−9.93630e15			−0.0137761			−0.00688803	+	0.999976i	$0.502193\pi$
$948$		−	2.90011e18i		−	3.99543i
$949$	3.80770e17			0.521274
$950$			1.60502e18i			2.18343i
$951$	−9.63927e17			−1.30305
$952$	−4.53575e17			−0.609295
$953$		−	9.80134e17i		−	1.30836i	−0.756338	−	0.654181i	$-0.773013\pi$
$953$		−	9.80134e17i		−	1.30836i	0.756338	−	0.654181i	$-0.226987\pi$
$954$		−	8.76216e16i		−	0.116231i
$955$	1.47391e18			1.94290
$956$	2.26816e16			0.0297116
$957$			1.23343e18i			1.60562i
$958$		−	1.86656e17i		−	0.241462i
$959$	1.04007e17			0.133706
$960$			1.63430e18i			2.08787i
$961$	−6.40441e17			−0.813091
$962$			3.75923e17i			0.474295i
$963$		−	1.89556e17i		−	0.237673i
$964$			1.33939e18i			1.66896i
$965$		−	1.12397e18i		−	1.39185i
$966$	4.71842e17	+	1.53960e17i	0.580676	+	0.189472i
$967$	9.68915e16			0.118502			0.0592512	−	0.998243i	$-0.481129\pi$
$967$	9.68915e16			0.118502			0.0592512	+	0.998243i	$0.481129\pi$
$968$	−8.02249e16			−0.0975118
$969$	−1.53818e18			−1.85808
$970$	−1.32604e18			−1.59194
$971$			9.16079e17i			1.09299i	0.837461	+	0.546497i	$0.184039\pi$
$971$			9.16079e17i			1.09299i	−0.837461	+	0.546497i	$0.815961\pi$
$972$	8.43445e17			1.00014
$973$			2.93389e17i			0.345754i
$974$	1.80678e18			2.11618
$975$	5.68678e17			0.661970
$976$			1.09181e18i			1.26313i
$977$			4.78743e17i			0.550471i	0.961377	+	0.275236i	$0.0887559\pi$
$977$			4.78743e17i			0.550471i	−0.961377	+	0.275236i	$0.911244\pi$
$978$	9.97317e17			1.13973
$979$	−2.84463e17			−0.323095
$980$			2.11438e18i			2.38685i
$981$		−	1.08287e17i		−	0.121496i
$982$	2.68691e18			2.99629
$983$			1.99085e17i			0.220657i	0.993895	+	0.110328i	$0.0351903\pi$
$983$			1.99085e17i			0.220657i	−0.993895	+	0.110328i	$0.964810\pi$
$984$	7.77992e17			0.857047
$985$			9.52674e14i			0.00104310i
$986$			2.94087e18i			3.20048i
$987$			5.93636e17i			0.642121i
$988$			1.31444e18i			1.41319i
$989$	−3.39590e17	−	1.10807e17i	−0.362892	−	0.118410i
$990$	−5.97042e17			−0.634152
$991$	8.46987e17			0.894200			0.447100	−	0.894484i	$-0.352457\pi$
$991$	8.46987e17			0.894200			0.447100	+	0.894484i	$0.352457\pi$
$992$	−1.59849e17			−0.167741
$993$	−6.57287e17			−0.685583
$994$			3.12160e17i			0.323638i
$995$	−1.55316e18			−1.60058
$996$			1.31201e18i			1.34394i
$997$	−4.70432e17			−0.478989			−0.239495	−	0.970898i	$-0.576982\pi$
$997$	−4.70432e17			−0.478989			−0.239495	+	0.970898i	$0.576982\pi$
$998$	1.50762e17			0.152583
$999$		−	3.89115e17i		−	0.391458i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.c.22.4	yes	20
23.22	odd	2	inner	23.13.b.c.22.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.c.22.3	✓	20	23.22	odd	2	inner
23.13.b.c.22.4	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)

\(f(q)\)	\(=\)	\(q-108.578 q^{2} -822.605 q^{3} +7693.26 q^{4} +22106.6i q^{5} +89317.1 q^{6} +37537.4i q^{7} -390585. q^{8} +145238. q^{9} +O(q^{10})\)	\(q-108.578 q^{2} -822.605 q^{3} +7693.26 q^{4} +22106.6i q^{5} +89317.1 q^{6} +37537.4i q^{7} -390585. q^{8} +145238. q^{9} -2.40030e6i q^{10} -1.71261e6i q^{11} -6.32852e6 q^{12} +2.82673e6 q^{13} -4.07575e6i q^{14} -1.81850e7i q^{15} +1.08975e7 q^{16} -3.09363e7i q^{17} -1.57697e7 q^{18} +6.04431e7i q^{19} +1.70072e8i q^{20} -3.08785e7i q^{21} +1.85952e8i q^{22} +(4.59207e7 - 1.40733e8i) q^{23} +3.21297e8 q^{24} -2.44563e8 q^{25} -3.06922e8 q^{26} +3.17693e8 q^{27} +2.88785e8i q^{28} +8.75516e8 q^{29} +1.97450e9i q^{30} -3.83695e8 q^{31} +4.16605e8 q^{32} +1.40880e9i q^{33} +3.35902e9i q^{34} -8.29826e8 q^{35} +1.11735e9 q^{36} -1.22482e9i q^{37} -6.56281e9i q^{38} -2.32528e9 q^{39} -8.63452e9i q^{40} +2.42141e9 q^{41} +3.35273e9i q^{42} -2.41300e9i q^{43} -1.31756e10i q^{44} +3.21072e9i q^{45} +(-4.98599e9 + 1.52806e10i) q^{46} -1.92249e10 q^{47} -8.96433e9 q^{48} +1.24322e10 q^{49} +2.65542e10 q^{50} +2.54484e10i q^{51} +2.17468e10 q^{52} +5.55632e9i q^{53} -3.44945e10 q^{54} +3.78600e10 q^{55} -1.46616e10i q^{56} -4.97208e10i q^{57} -9.50621e10 q^{58} +2.95376e10 q^{59} -1.39902e11i q^{60} +1.00189e11i q^{61} +4.16609e10 q^{62} +5.45186e9i q^{63} -8.98704e10 q^{64} +6.24895e10i q^{65} -1.52965e11i q^{66} +6.18938e10i q^{67} -2.38001e11i q^{68} +(-3.77746e10 + 1.15768e11i) q^{69} +9.01012e10 q^{70} -7.65896e10 q^{71} -5.67278e10 q^{72} +1.34703e11 q^{73} +1.32989e11i q^{74} +2.01179e11 q^{75} +4.65005e11i q^{76} +6.42869e10 q^{77} +2.52475e11 q^{78} +4.58260e11i q^{79} +2.40907e11i q^{80} -3.38521e11 q^{81} -2.62913e11 q^{82} -2.07317e11i q^{83} -2.37556e11i q^{84} +6.83899e11 q^{85} +2.62000e11i q^{86} -7.20204e11 q^{87} +6.68920e11i q^{88} -1.66099e11i q^{89} -3.48615e11i q^{90} +1.06108e11i q^{91} +(3.53280e11 - 1.08270e12i) q^{92} +3.15629e11 q^{93} +2.08741e12 q^{94} -1.33619e12 q^{95} -3.42702e11 q^{96} -5.52447e11i q^{97} -1.34987e12 q^{98} -2.48736e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) 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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