LMFDB - Embedded newform 23.13.b.c.22.7

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			22.7
Root			$1052.17 + 197.039i$ of defining polynomial
Character	$\chi$	$=$	23.22
Dual form			23.13.b.c.22.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-43.1697 q^{2} -1083.34 q^{3} -2232.38 q^{4} -197.039i q^{5} +46767.4 q^{6} -138687. i q^{7} +273194. q^{8} +642183. q^{9} +O(q^{10})$	\(q-43.1697 q^{2} -1083.34 q^{3} -2232.38 q^{4} -197.039i q^{5} +46767.4 q^{6} -138687. i q^{7} +273194. q^{8} +642183. q^{9} +8506.13i q^{10} -3.09310e6i q^{11} +2.41842e6 q^{12} -782061. q^{13} +5.98707e6i q^{14} +213460. i q^{15} -2.64989e6 q^{16} +1.68131e7i q^{17} -2.77228e7 q^{18} -8.31755e7i q^{19} +439866. i q^{20} +1.50245e8i q^{21} +1.33528e8i q^{22} +(-1.21849e8 - 8.40675e7i) q^{23} -2.95962e8 q^{24} +2.44102e8 q^{25} +3.37613e7 q^{26} -1.19971e8 q^{27} +3.09601e8i q^{28} -4.26269e8 q^{29} -9.21502e6i q^{30} -1.02830e9 q^{31} -1.00461e9 q^{32} +3.35087e9i q^{33} -7.25819e8i q^{34} -2.73268e7 q^{35} -1.43359e9 q^{36} -3.34404e9i q^{37} +3.59066e9i q^{38} +8.47237e8 q^{39} -5.38300e7i q^{40} +3.61362e9 q^{41} -6.48603e9i q^{42} -7.08764e9i q^{43} +6.90496e9i q^{44} -1.26535e8i q^{45} +(5.26020e9 + 3.62917e9i) q^{46} +1.41856e10 q^{47} +2.87073e9 q^{48} -5.39277e9 q^{49} -1.05378e10 q^{50} -1.82143e10i q^{51} +1.74586e9 q^{52} -7.07215e9i q^{53} +5.17910e9 q^{54} -6.09461e8 q^{55} -3.78884e10i q^{56} +9.01073e10i q^{57} +1.84019e10 q^{58} -3.08437e10 q^{59} -4.76524e8i q^{60} -1.27252e9i q^{61} +4.43914e10 q^{62} -8.90623e10i q^{63} +5.42226e10 q^{64} +1.54097e8i q^{65} -1.44656e11i q^{66} +5.88741e10i q^{67} -3.75333e10i q^{68} +(1.32004e11 + 9.10737e10i) q^{69} +1.17969e9 q^{70} -7.96554e10 q^{71} +1.75441e11 q^{72} -2.01818e11 q^{73} +1.44361e11i q^{74} -2.64445e11 q^{75} +1.85679e11i q^{76} -4.28972e11 q^{77} -3.65750e10 q^{78} +2.84935e11i q^{79} +5.22133e8i q^{80} -2.11313e11 q^{81} -1.55999e11 q^{82} +3.50338e11i q^{83} -3.35403e11i q^{84} +3.31285e9 q^{85} +3.05971e11i q^{86} +4.61794e11 q^{87} -8.45016e11i q^{88} -1.59579e11i q^{89} +5.46249e9i q^{90} +1.08462e11i q^{91} +(2.72014e11 + 1.87670e11i) q^{92} +1.11400e12 q^{93} -6.12390e11 q^{94} -1.63888e10 q^{95} +1.08833e12 q^{96} -1.07543e11i q^{97} +2.32804e11 q^{98} -1.98633e12i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−43.1697			−0.674527			−0.337263	−	0.941410i	$-0.609501\pi$
$2$	−43.1697			−0.674527			−0.337263	+	0.941410i	$0.609501\pi$
$3$	−1083.34			−1.48606			−0.743031	−	0.669257i	$-0.766612\pi$
$3$	−1083.34			−1.48606			−0.743031	+	0.669257i	$0.766612\pi$
$4$	−2232.38			−0.545014
$5$		−	197.039i		−	0.0126105i	−0.999980	−	0.00630526i	$-0.997993\pi$
$5$		−	197.039i		−	0.0126105i	0.999980	−	0.00630526i	$-0.00200704\pi$
$6$	46767.4			1.00239
$7$		−	138687.i		−	1.17882i	−0.807834	−	0.589410i	$-0.799360\pi$
$7$		−	138687.i		−	1.17882i	0.807834	−	0.589410i	$-0.200640\pi$
$8$	273194.			1.04215
$9$	642183.			1.20838
$10$			8506.13i			0.00850613i
$11$		−	3.09310e6i		−	1.74597i	−0.487745	−	0.872986i	$-0.662180\pi$
$11$		−	3.09310e6i		−	1.74597i	0.487745	−	0.872986i	$-0.337820\pi$
$12$	2.41842e6			0.809924
$13$	−782061.			−0.162024			−0.0810122	−	0.996713i	$-0.525815\pi$
$13$	−782061.			−0.162024			−0.0810122	+	0.996713i	$0.525815\pi$
$14$			5.98707e6i			0.795145i
$15$			213460.i			0.0187400i
$16$	−2.64989e6			−0.157946
$17$			1.68131e7i			0.696555i	0.937391	+	0.348278i	$0.113233\pi$
$17$			1.68131e7i			0.696555i	−0.937391	+	0.348278i	$0.886767\pi$
$18$	−2.77228e7			−0.815084
$19$		−	8.31755e7i		−	1.76797i	−0.467519	−	0.883983i	$-0.654852\pi$
$19$		−	8.31755e7i		−	1.76797i	0.467519	−	0.883983i	$-0.345148\pi$
$20$			439866.i			0.00687291i
$21$			1.50245e8i			1.75180i
$22$			1.33528e8i			1.17770i
$23$	−1.21849e8	−	8.40675e7i	−0.823107	−	0.567886i
$23$	−1.21849e8	−	8.40675e7i	−0.823107	−	0.567886i
$24$	−2.95962e8			−1.54870
$25$	2.44102e8			0.999841
$26$	3.37613e7			0.109290
$27$	−1.19971e8			−0.309666
$28$			3.09601e8i			0.642473i
$29$	−4.26269e8			−0.716631			−0.358315	−	0.933601i	$-0.616649\pi$
$29$	−4.26269e8			−0.716631			−0.358315	+	0.933601i	$0.616649\pi$
$30$		−	9.21502e6i		−	0.0126406i
$31$	−1.02830e9			−1.15864			−0.579321	−	0.815099i	$-0.696682\pi$
$31$	−1.02830e9			−1.15864			−0.579321	+	0.815099i	$0.696682\pi$
$32$	−1.00461e9			−0.935614
$33$			3.35087e9i			2.59462i
$34$		−	7.25819e8i		−	0.469845i
$35$	−2.73268e7			−0.0148655
$36$	−1.43359e9			−0.658584
$37$		−	3.34404e9i		−	1.30335i	−0.758499	−	0.651674i	$-0.774067\pi$
$37$		−	3.34404e9i		−	1.30335i	0.758499	−	0.651674i	$-0.225933\pi$
$38$			3.59066e9i			1.19254i
$39$	8.47237e8			0.240778
$40$		−	5.38300e7i		−	0.0131421i
$41$	3.61362e9			0.760746			0.380373	−	0.924833i	$-0.375796\pi$
$41$	3.61362e9			0.760746			0.380373	+	0.924833i	$0.375796\pi$
$42$		−	6.48603e9i		−	1.18163i
$43$		−	7.08764e9i		−	1.12122i	−0.828080	−	0.560610i	$-0.810567\pi$
$43$		−	7.08764e9i		−	1.12122i	0.828080	−	0.560610i	$-0.189433\pi$
$44$			6.90496e9i			0.951579i
$45$		−	1.26535e8i		−	0.0152383i
$46$	5.26020e9	+	3.62917e9i	0.555208	+	0.383054i
$47$	1.41856e10			1.31602			0.658009	−	0.753010i	$-0.271399\pi$
$47$	1.41856e10			1.31602			0.658009	+	0.753010i	$0.271399\pi$
$48$	2.87073e9			0.234717
$49$	−5.39277e9			−0.389615
$50$	−1.05378e10			−0.674419
$51$		−	1.82143e10i		−	1.03512i
$52$	1.74586e9			0.0883056
$53$		−	7.07215e9i		−	0.319077i	−0.987192	−	0.159539i	$-0.948999\pi$
$53$		−	7.07215e9i		−	0.319077i	0.987192	−	0.159539i	$-0.0510007\pi$
$54$	5.17910e9			0.208878
$55$	−6.09461e8			−0.0220176
$56$		−	3.78884e10i		−	1.22851i
$57$			9.01073e10i			2.62731i
$58$	1.84019e10			0.483387
$59$	−3.08437e10			−0.731231			−0.365615	−	0.930766i	$-0.619141\pi$
$59$	−3.08437e10			−0.731231			−0.365615	+	0.930766i	$0.619141\pi$
$60$		−	4.76524e8i		−	0.0102136i
$61$		−	1.27252e9i		−	0.0246994i	−0.999924	−	0.0123497i	$-0.996069\pi$
$61$		−	1.27252e9i		−	0.0246994i	0.999924	−	0.0123497i	$-0.00393114\pi$
$62$	4.43914e10			0.781535
$63$		−	8.90623e10i		−	1.42446i
$64$	5.42226e10			0.789042
$65$			1.54097e8i			0.00204321i
$66$		−	1.44656e11i		−	1.75014i
$67$			5.88741e10i			0.650842i	0.945569	+	0.325421i	$0.105506\pi$
$67$			5.88741e10i			0.650842i	−0.945569	+	0.325421i	$0.894494\pi$
$68$		−	3.75333e10i		−	0.379632i
$69$	1.32004e11	+	9.10737e10i	1.22319	+	0.843914i
$70$	1.17969e9			0.0100272
$71$	−7.96554e10			−0.621821			−0.310910	−	0.950439i	$-0.600634\pi$
$71$	−7.96554e10			−0.621821			−0.310910	+	0.950439i	$0.600634\pi$
$72$	1.75441e11			1.25932
$73$	−2.01818e11			−1.33359			−0.666796	−	0.745240i	$-0.732335\pi$
$73$	−2.01818e11			−1.33359			−0.666796	+	0.745240i	$0.732335\pi$
$74$			1.44361e11i			0.879143i
$75$	−2.64445e11			−1.48583
$76$			1.85679e11i			0.963566i
$77$	−4.28972e11			−2.05819
$78$	−3.65750e10			−0.162411
$79$			2.84935e11i			1.17215i	0.810256	+	0.586076i	$0.199328\pi$
$79$			2.84935e11i			1.17215i	−0.810256	+	0.586076i	$0.800672\pi$
$80$			5.22133e8i			0.00199178i
$81$	−2.11313e11			−0.748198
$82$	−1.55999e11			−0.513143
$83$			3.50338e11i			1.07156i	0.844356	+	0.535782i	$0.179983\pi$
$83$			3.50338e11i			1.07156i	−0.844356	+	0.535782i	$0.820017\pi$
$84$		−	3.35403e11i		−	0.954755i
$85$	3.31285e9			0.00878392
$86$			3.05971e11i			0.756292i
$87$	4.61794e11			1.06496
$88$		−	8.45016e11i		−	1.81957i
$89$		−	1.59579e11i		−	0.321097i	−0.987028	−	0.160549i	$-0.948674\pi$
$89$		−	1.59579e11i		−	0.321097i	0.987028	−	0.160549i	$-0.0513263\pi$
$90$			5.46249e9i			0.0102786i
$91$			1.08462e11i			0.190998i
$92$	2.72014e11	+	1.87670e11i	0.448605	+	0.309506i
$93$	1.11400e12			1.72182
$94$	−6.12390e11			−0.887689
$95$	−1.63888e10			−0.0222950
$96$	1.08833e12			1.39038
$97$		−	1.07543e11i		−	0.129108i	−0.997914	−	0.0645540i	$-0.979438\pi$
$97$		−	1.07543e11i		−	0.129108i	0.997914	−	0.0645540i	$-0.0205625\pi$
$98$	2.32804e11			0.262806
$99$		−	1.98633e12i		−	2.10980i
$100$	−5.44927e11			−0.544927
$101$	4.11180e11			0.387350			0.193675	−	0.981066i	$-0.437959\pi$
$101$	4.11180e11			0.387350			0.193675	+	0.981066i	$0.437959\pi$
$102$			7.86308e11i			0.698219i
$103$		−	3.73778e11i		−	0.313033i	−0.987675	−	0.156516i	$-0.949974\pi$
$103$		−	3.73778e11i		−	0.313033i	0.987675	−	0.156516i	$-0.0500264\pi$
$104$	−2.13654e11			−0.168854
$105$	2.96042e10			0.0220911
$106$			3.05302e11i			0.215226i
$107$		−	1.61213e12i		−	1.07423i	−0.843510	−	0.537114i	$-0.819515\pi$
$107$		−	1.61213e12i		−	1.07423i	0.843510	−	0.537114i	$-0.180485\pi$
$108$	2.67820e11			0.168772
$109$			2.83913e12i			1.69288i	0.532482	+	0.846441i	$0.321259\pi$
$109$			2.83913e12i			1.69288i	−0.532482	+	0.846441i	$0.678741\pi$
$110$	2.63103e10			0.0148515
$111$			3.62273e12i			1.93686i
$112$			3.67505e11i			0.186190i
$113$		−	2.73172e12i		−	1.31210i	−0.754718	−	0.656049i	$-0.772227\pi$
$113$		−	2.73172e12i		−	1.31210i	0.754718	−	0.656049i	$-0.227773\pi$
$114$		−	3.88990e12i		−	1.77219i
$115$	−1.65646e10	+	2.40091e10i	−0.00716134	+	0.0103798i
$116$	9.51593e11			0.390574
$117$	−5.02226e11			−0.195787
$118$	1.33151e12			0.493235
$119$	2.33176e12			0.821113
$120$			5.83161e10i			0.0195299i
$121$	−6.42882e12			−2.04842
$122$			5.49345e10i			0.0166604i
$123$	−3.91478e12			−1.13052
$124$	2.29555e12			0.631477
$125$		−	9.62029e10i		−	0.0252190i
$126$			3.84479e12i			0.960837i
$127$	1.65478e12			0.394383			0.197192	−	0.980365i	$-0.436818\pi$
$127$	1.65478e12			0.394383			0.197192	+	0.980365i	$0.436818\pi$
$128$	1.77410e12			0.403384
$129$			7.67831e12i			1.66620i
$130$		−	6.65231e9i		−	0.00137820i
$131$	2.38314e12			0.471543			0.235772	−	0.971808i	$-0.424238\pi$
$131$	2.38314e12			0.471543			0.235772	+	0.971808i	$0.424238\pi$
$132$		−	7.48041e12i		−	1.41411i
$133$	−1.15354e13			−2.08411
$134$		−	2.54158e12i		−	0.439010i
$135$			2.36390e10i			0.00390504i
$136$			4.59325e12i			0.725917i
$137$			8.82608e12i			1.33489i	0.744661	+	0.667443i	$0.232611\pi$
$137$			8.82608e12i			1.33489i	−0.744661	+	0.667443i	$0.767389\pi$
$138$	−5.69858e12	−	3.93162e12i	−0.825073	−	0.569242i
$139$	3.10954e12			0.431129			0.215564	−	0.976490i	$-0.430841\pi$
$139$	3.10954e12			0.431129			0.215564	+	0.976490i	$0.430841\pi$
$140$	6.10037e10			0.00810191
$141$	−1.53679e13			−1.95568
$142$	3.43870e12			0.419435
$143$			2.41899e12i			0.282890i
$144$	−1.70171e12			−0.190859
$145$			8.39917e10i			0.00903708i
$146$	8.71243e12			0.899543
$147$	5.84220e12			0.578992
$148$			7.46515e12i			0.710343i
$149$		−	1.17065e13i		−	1.06982i	−0.844910	−	0.534908i	$-0.820346\pi$
$149$		−	1.17065e13i		−	1.06982i	0.844910	−	0.534908i	$-0.179654\pi$
$150$	1.14160e13			1.00223
$151$	4.11075e12			0.346785			0.173392	−	0.984853i	$-0.444527\pi$
$151$	4.11075e12			0.346785			0.173392	+	0.984853i	$0.444527\pi$
$152$		−	2.27231e13i		−	1.84249i
$153$			1.07971e13i			0.841703i
$154$	1.85186e13			1.38830
$155$			2.02615e11i			0.0146111i
$156$	−1.89135e12			−0.131228
$157$		−	1.21477e13i		−	0.811141i	−0.914064	−	0.405570i	$-0.867073\pi$
$157$		−	1.21477e13i		−	0.811141i	0.914064	−	0.405570i	$-0.132927\pi$
$158$		−	1.23006e13i		−	0.790648i
$159$			7.66153e12i			0.474169i
$160$			1.97947e11i			0.0117986i
$161$	−1.16591e13	+	1.68989e13i	−0.669435	+	0.970294i
$162$	9.12233e12			0.504679
$163$	−1.82565e13			−0.973399			−0.486699	−	0.873569i	$-0.661799\pi$
$163$	−1.82565e13			−0.973399			−0.486699	+	0.873569i	$0.661799\pi$
$164$	−8.06697e12			−0.414617
$165$	6.60253e11			0.0327195
$166$		−	1.51240e13i		−	0.722798i
$167$	2.51714e13			1.16040			0.580200	−	0.814474i	$-0.302974\pi$
$167$	2.51714e13			1.16040			0.580200	+	0.814474i	$0.302974\pi$
$168$			4.10460e13i			1.82564i
$169$	−2.26865e13			−0.973748
$170$	−1.43015e11			−0.00592499
$171$		−	5.34139e13i		−	2.13637i
$172$			1.58223e13i			0.611080i
$173$	4.22256e13			1.57507			0.787533	−	0.616272i	$-0.211358\pi$
$173$	4.22256e13			1.57507			0.787533	+	0.616272i	$0.211358\pi$
$174$	−1.99355e13			−0.718342
$175$		−	3.38537e13i		−	1.17863i
$176$			8.19637e12i			0.275769i
$177$	3.34142e13			1.08665
$178$			6.88899e12i			0.216589i
$179$	2.19570e13			0.667507			0.333753	−	0.942660i	$-0.391685\pi$
$179$	2.19570e13			0.667507			0.333753	+	0.942660i	$0.391685\pi$
$180$			2.82474e11i			0.00830508i
$181$			4.45881e13i			1.26808i	0.773300	+	0.634041i	$0.218605\pi$
$181$			4.45881e13i			1.26808i	−0.773300	+	0.634041i	$0.781395\pi$
$182$		−	4.68226e12i		−	0.128833i
$183$			1.37858e12i			0.0367049i
$184$	−3.32885e13	−	2.29668e13i	−0.857803	−	0.591824i
$185$	−6.58907e11			−0.0164359
$186$	−4.80909e13			−1.16141
$187$	5.20047e13			1.21617
$188$	−3.16677e13			−0.717248
$189$			1.66384e13i			0.365040i
$190$	7.07501e11			0.0150385
$191$		−	5.08450e13i		−	1.04724i	−0.851951	−	0.523622i	$-0.824580\pi$
$191$		−	5.08450e13i		−	1.04724i	0.851951	−	0.523622i	$-0.175420\pi$
$192$	−5.87414e13			−1.17257
$193$	−3.49289e13			−0.675836			−0.337918	−	0.941176i	$-0.609723\pi$
$193$	−3.49289e13			−0.675836			−0.337918	+	0.941176i	$0.609723\pi$
$194$			4.64261e12i			0.0870868i
$195$		−	1.66939e11i		−	0.00303634i
$196$	1.20387e13			0.212346
$197$	6.41407e13			1.09733			0.548664	−	0.836043i	$-0.315137\pi$
$197$	6.41407e13			1.09733			0.548664	+	0.836043i	$0.315137\pi$
$198$			8.57494e13i			1.42311i
$199$			4.57253e13i			0.736272i	0.929772	+	0.368136i	$0.120004\pi$
$199$			4.57253e13i			0.736272i	−0.929772	+	0.368136i	$0.879996\pi$
$200$	6.66872e13			1.04199
$201$		−	6.37807e13i		−	0.967192i
$202$	−1.77505e13			−0.261278
$203$			5.91179e13i			0.844778i
$204$			4.06613e13i			0.564157i
$205$		−	7.12025e11i		−	0.00959340i
$206$			1.61359e13i			0.211149i
$207$	−7.82496e13	−	5.39867e13i	−0.994626	−	0.686222i
$208$	2.07238e12			0.0255911
$209$	−2.57270e14			−3.08682
$210$	−1.27800e12			−0.0149010
$211$	−1.42466e14			−1.61442			−0.807211	−	0.590263i	$-0.799024\pi$
$211$	−1.42466e14			−1.61442			−0.807211	+	0.590263i	$0.799024\pi$
$212$			1.57877e13i			0.173902i
$213$	8.62938e13			0.924064
$214$			6.95950e13i			0.724595i
$215$	−1.39654e12			−0.0141392
$216$	−3.27753e13			−0.322719
$217$			1.42612e14i			1.36583i
$218$		−	1.22565e14i		−	1.14189i
$219$	2.18637e14			1.98180
$220$	1.36055e12			0.0119999
$221$		−	1.31489e13i		−	0.112859i
$222$		−	1.56392e14i		−	1.30646i
$223$	8.84136e12			0.0718936			0.0359468	−	0.999354i	$-0.488555\pi$
$223$	8.84136e12			0.0718936			0.0359468	+	0.999354i	$0.488555\pi$
$224$			1.39326e14i			1.10292i
$225$	1.56758e14			1.20819
$226$			1.17928e14i			0.885045i
$227$		−	1.27980e14i		−	0.935378i	−0.883893	−	0.467689i	$-0.845087\pi$
$227$		−	1.27980e14i		−	0.935378i	0.883893	−	0.467689i	$-0.154913\pi$
$228$		−	2.01153e14i		−	1.43192i
$229$		−	8.38302e13i		−	0.581283i	−0.956832	−	0.290641i	$-0.906131\pi$
$229$		−	8.38302e13i		−	0.581283i	0.956832	−	0.290641i	$-0.0938687\pi$
$230$	7.15089e11	−	1.03647e12i	0.00483051	−	0.00700145i
$231$	4.64722e14			3.05859
$232$	−1.16454e14			−0.746839
$233$	4.36679e13			0.272915			0.136457	−	0.990646i	$-0.456428\pi$
$233$	4.36679e13			0.272915			0.136457	+	0.990646i	$0.456428\pi$
$234$	2.16809e13			0.132064
$235$		−	2.79513e12i		−	0.0165957i
$236$	6.88548e13			0.398531
$237$		−	3.08682e14i		−	1.74189i
$238$	−1.00662e14			−0.553862
$239$	−2.02270e14			−1.08528			−0.542642	−	0.839964i	$-0.682576\pi$
$239$	−2.02270e14			−1.08528			−0.542642	+	0.839964i	$0.682576\pi$
$240$		−	5.65647e11i		−	0.00295991i
$241$		−	5.60211e13i		−	0.285923i	−0.989728	−	0.142961i	$-0.954337\pi$
$241$		−	5.60211e13i		−	0.285923i	0.989728	−	0.142961i	$-0.0456625\pi$
$242$	2.77530e14			1.38171
$243$	2.92681e14			1.42153
$244$			2.84075e12i			0.0134615i
$245$			1.06259e12i			0.00491324i
$246$	1.69000e14			0.762563
$247$			6.50483e13i			0.286454i
$248$	−2.80925e14			−1.20748
$249$		−	3.79534e14i		−	1.59241i
$250$			4.15305e12i			0.0170109i
$251$			4.73764e14i			1.89461i	0.320333	+	0.947305i	$0.396205\pi$
$251$			4.73764e14i			1.89461i	−0.320333	+	0.947305i	$0.603795\pi$
$252$			1.98821e14i			0.776351i
$253$	−2.60029e14	+	3.76892e14i	−0.991514	+	1.43712i
$254$	−7.14364e13			−0.266022
$255$	−3.58894e12			−0.0130534
$256$	−2.98683e14			−1.06114
$257$	−3.96953e12			−0.0137765			−0.00688827	−	0.999976i	$-0.502193\pi$
$257$	−3.96953e12			−0.0137765			−0.00688827	+	0.999976i	$0.502193\pi$
$258$		−	3.31471e14i		−	1.12390i
$259$	−4.63774e14			−1.53641
$260$		−	3.44002e11i		−	0.00111358i
$261$	−2.73742e14			−0.865962
$262$	−1.02879e14			−0.318068
$263$		−	8.48743e13i		−	0.256473i	−0.991744	−	0.128237i	$-0.959068\pi$
$263$		−	8.48743e13i		−	0.256473i	0.991744	−	0.128237i	$-0.0409317\pi$
$264$			9.15439e14i			2.70399i
$265$	−1.39349e12			−0.00402373
$266$	4.97978e14			1.40579
$267$			1.72878e14i			0.477170i
$268$		−	1.31429e14i		−	0.354718i
$269$	2.77036e14			0.731178			0.365589	−	0.930776i	$-0.380868\pi$
$269$	2.77036e14			0.731178			0.365589	+	0.930776i	$0.380868\pi$
$270$		−	1.02049e12i		−	0.00263405i
$271$	−5.60984e14			−1.41623			−0.708117	−	0.706095i	$-0.750455\pi$
$271$	−5.60984e14			−1.41623			−0.708117	+	0.706095i	$0.750455\pi$
$272$		−	4.45530e13i		−	0.110018i
$273$		−	1.17501e14i		−	0.283834i
$274$		−	3.81019e14i		−	0.900417i
$275$		−	7.55030e14i		−	1.74569i
$276$	−2.94683e14	−	2.03311e14i	−0.666655	−	0.459945i
$277$	8.52398e14			1.88697			0.943483	−	0.331421i	$-0.107528\pi$
$277$	8.52398e14			1.88697			0.943483	+	0.331421i	$0.107528\pi$
$278$	−1.34238e14			−0.290808
$279$	−6.60356e14			−1.40008
$280$	−7.46551e12			−0.0154921
$281$			9.27542e14i			1.88406i	0.335524	+	0.942032i	$0.391087\pi$
$281$			9.27542e14i			1.88406i	−0.335524	+	0.942032i	$0.608913\pi$
$282$	6.63426e14			1.31916
$283$			9.58382e14i			1.86561i	0.360386	+	0.932803i	$0.382645\pi$
$283$			9.58382e14i			1.86561i	−0.360386	+	0.932803i	$0.617355\pi$
$284$	1.77821e14			0.338901
$285$	1.77547e13			0.0331317
$286$		−	1.04427e14i		−	0.190817i
$287$		−	5.01162e14i		−	0.896782i
$288$	−6.45142e14			−1.13058
$289$	2.99940e14			0.514811
$290$		−	3.62590e12i		−	0.00609575i
$291$			1.16506e14i			0.191862i
$292$	4.50534e14			0.726826
$293$		−	5.49275e14i		−	0.868129i	−0.900882	−	0.434065i	$-0.857079\pi$
$293$		−	5.49275e14i		−	0.868129i	0.900882	−	0.434065i	$-0.142921\pi$
$294$	−2.52206e14			−0.390545
$295$			6.07742e12i			0.00922120i
$296$		−	9.13571e14i		−	1.35829i
$297$			3.71081e14i			0.540667i
$298$			5.05366e14i			0.721619i
$299$	9.52937e13	+	6.57459e13i	0.133363	+	0.0920114i
$300$	5.90341e14			0.809796
$301$	−9.82962e14			−1.32172
$302$	−1.77460e14			−0.233915
$303$	−4.45447e14			−0.575626
$304$			2.20406e14i			0.279243i
$305$	−2.50737e11			−0.000311473
$306$		−	4.66108e14i		−	0.567751i
$307$	−1.73028e13			−0.0206674			−0.0103337	−	0.999947i	$-0.503289\pi$
$307$	−1.73028e13			−0.0206674			−0.0103337	+	0.999947i	$0.503289\pi$
$308$	9.57627e14			1.12174
$309$			4.04928e14i			0.465186i
$310$		−	8.74685e12i		−	0.00985556i
$311$	−1.19719e15			−1.32313			−0.661564	−	0.749889i	$-0.730107\pi$
$311$	−1.19719e15			−1.32313			−0.661564	+	0.749889i	$0.730107\pi$
$312$	2.31460e14			0.250928
$313$		−	4.06703e14i		−	0.432525i	−0.976335	−	0.216262i	$-0.930613\pi$
$313$		−	4.06703e14i		−	0.432525i	0.976335	−	0.216262i	$-0.0693867\pi$
$314$			5.24413e14i			0.547136i
$315$	−1.75488e13			−0.0179632
$316$		−	6.36083e14i		−	0.638839i
$317$	7.39236e14			0.728496			0.364248	−	0.931302i	$-0.381326\pi$
$317$	7.39236e14			0.728496			0.364248	+	0.931302i	$0.381326\pi$
$318$		−	3.30746e14i		−	0.319839i
$319$			1.31849e15i			1.25122i
$320$		−	1.06840e13i		−	0.00995023i
$321$			1.74648e15i			1.59637i
$322$	5.03318e14	−	7.29521e14i	0.451552	−	0.654489i
$323$	1.39844e15			1.23149
$324$	4.71731e14			0.407778
$325$	−1.90903e14			−0.161999
$326$	7.88126e14			0.656583
$327$		−	3.07575e15i		−	2.51573i
$328$	9.87220e14			0.792813
$329$		−	1.96736e15i		−	1.55135i
$330$	−2.85029e13			−0.0220702
$331$	4.84433e14			0.368354			0.184177	−	0.982893i	$-0.441038\pi$
$331$	4.84433e14			0.368354			0.184177	+	0.982893i	$0.441038\pi$
$332$		−	7.82086e14i		−	0.584017i
$333$		−	2.14748e15i		−	1.57494i
$334$	−1.08664e15			−0.782721
$335$	1.16005e13			0.00820746
$336$		−	3.98133e14i		−	0.276689i
$337$			1.35339e14i			0.0923937i	0.998932	+	0.0461968i	$0.0147101\pi$
$337$			1.35339e14i			0.0923937i	−0.998932	+	0.0461968i	$0.985290\pi$
$338$	9.79368e14			0.656819
$339$			2.95938e15i			1.94986i
$340$	−7.39553e12			−0.00478736
$341$			3.18063e15i			2.02296i
$342$			2.30586e15i			1.44104i
$343$		−	1.17170e15i		−	0.719534i
$344$		−	1.93630e15i		−	1.16848i
$345$	1.79451e13	−	2.60100e13i	0.0106422	−	0.0154250i
$346$	−1.82287e15			−1.06242
$347$	9.68253e14			0.554641			0.277321	−	0.960777i	$-0.410554\pi$
$347$	9.68253e14			0.554641			0.277321	+	0.960777i	$0.410554\pi$
$348$	−1.03090e15			−0.580417
$349$	−2.67043e15			−1.47785			−0.738923	−	0.673790i	$-0.764665\pi$
$349$	−2.67043e15			−1.47785			−0.738923	+	0.673790i	$0.764665\pi$
$350$			1.46146e15i			0.795018i
$351$	9.38245e13			0.0501734
$352$			3.10735e15i			1.63356i
$353$	6.07180e14			0.313812			0.156906	−	0.987614i	$-0.449848\pi$
$353$	6.07180e14			0.313812			0.156906	+	0.987614i	$0.449848\pi$
$354$	−1.44248e15			−0.732977
$355$			1.56952e13i			0.00784148i
$356$			3.56241e14i			0.175002i
$357$	−2.52609e15			−1.22022
$358$	−9.47879e14			−0.450251
$359$		−	1.09407e15i		−	0.511067i	−0.966800	−	0.255534i	$-0.917749\pi$
$359$		−	1.09407e15i		−	0.511067i	0.966800	−	0.255534i	$-0.0822512\pi$
$360$		−	3.45687e13i		−	0.0158806i
$361$	−4.70485e15			−2.12570
$362$		−	1.92485e15i		−	0.855354i
$363$	6.96459e15			3.04408
$364$		−	2.42127e14i		−	0.104096i
$365$			3.97661e13i			0.0168173i
$366$		−	5.95127e13i		−	0.0247584i
$367$		−	2.80459e14i		−	0.114782i	−0.998352	−	0.0573910i	$-0.981722\pi$
$367$		−	2.80459e14i		−	0.114782i	0.998352	−	0.0573910i	$-0.0182782\pi$
$368$	3.22888e14	+	2.22770e14i	0.130006	+	0.0896953i
$369$	2.32061e15			0.919270
$370$	2.84448e13			0.0110864
$371$	−9.80814e14			−0.376135
$372$	−2.48686e15			−0.938413
$373$			1.71276e15i			0.635981i	0.948094	+	0.317991i	$0.103008\pi$
$373$			1.71276e15i			0.635981i	−0.948094	+	0.317991i	$0.896992\pi$
$374$	−2.24503e15			−0.820336
$375$			1.04220e14i			0.0374770i
$376$	3.87543e15			1.37149
$377$	3.33368e14			0.116112
$378$		−	7.18274e14i		−	0.246229i
$379$			6.47468e14i			0.218465i	0.994016	+	0.109233i	$0.0348394\pi$
$379$			6.47468e14i			0.218465i	−0.994016	+	0.109233i	$0.965161\pi$
$380$	3.65861e13			0.0121511
$381$	−1.79269e15			−0.586078
$382$			2.19496e15i			0.706394i
$383$			1.26707e15i			0.401428i	0.979650	+	0.200714i	$0.0643262\pi$
$383$			1.26707e15i			0.401428i	−0.979650	+	0.200714i	$0.935674\pi$
$384$	−1.92195e15			−0.599454
$385$			8.45243e13i			0.0259548i
$386$	1.50787e15			0.455869
$387$		−	4.55156e15i		−	1.35486i
$388$			2.40077e14i			0.0703656i
$389$			1.30007e15i			0.375204i	0.982245	+	0.187602i	$0.0600716\pi$
$389$			1.30007e15i			0.375204i	−0.982245	+	0.187602i	$0.939928\pi$
$390$			7.20671e12i			0.00204809i
$391$	1.41344e15	−	2.04867e15i	0.395564	−	0.573339i
$392$	−1.47327e15			−0.406038
$393$	−2.58175e15			−0.700742
$394$	−2.76893e15			−0.740177
$395$	5.61435e13			0.0147814
$396$			4.43424e15i			1.14987i
$397$	−4.74502e14			−0.121198			−0.0605990	−	0.998162i	$-0.519301\pi$
$397$	−4.74502e14			−0.121198			−0.0605990	+	0.998162i	$0.519301\pi$
$398$		−	1.97395e15i		−	0.496635i
$399$	1.24967e16			3.09712
$400$	−6.46843e14			−0.157921
$401$			3.07327e14i			0.0739152i	0.999317	+	0.0369576i	$0.0117667\pi$
$401$			3.07327e14i			0.0739152i	−0.999317	+	0.0369576i	$0.988233\pi$
$402$			2.75339e15i			0.652397i
$403$	8.04193e14			0.187728
$404$	−9.17908e14			−0.211111
$405$			4.16370e13i			0.00943516i
$406$		−	2.55210e15i		−	0.569825i
$407$	−1.03434e16			−2.27561
$408$		−	4.97605e15i		−	1.07876i
$409$	3.89538e15			0.832165			0.416083	−	0.909327i	$-0.363403\pi$
$409$	3.89538e15			0.832165			0.416083	+	0.909327i	$0.363403\pi$
$410$			3.07379e13i			0.00647100i
$411$		−	9.56164e15i		−	1.98372i
$412$			8.34413e14i			0.170607i
$413$			4.27762e15i			0.861989i
$414$	3.37801e15	+	2.33059e15i	0.670902	+	0.462875i
$415$	6.90303e13			0.0135130
$416$	7.85665e14			0.151592
$417$	−3.36868e15			−0.640684
$418$	1.11063e16			2.08214
$419$		−	8.83024e15i		−	1.63188i	−0.578136	−	0.815940i	$-0.696220\pi$
$419$		−	8.83024e15i		−	1.63188i	0.578136	−	0.815940i	$-0.303780\pi$
$420$	−6.60876e13			−0.0120399
$421$			4.47391e15i			0.803516i	0.915746	+	0.401758i	$0.131601\pi$
$421$			4.47391e15i			0.803516i	−0.915746	+	0.401758i	$0.868399\pi$
$422$	6.15022e15			1.08897
$423$	9.10977e15			1.59025
$424$		−	1.93207e15i		−	0.332527i
$425$			4.10412e15i			0.696444i
$426$	−3.72528e15			−0.623306
$427$	−1.76482e14			−0.0291162
$428$			3.59887e15i			0.585469i
$429$		−	2.62059e15i		−	0.420392i
$430$	6.02883e13			0.00953724
$431$			5.48668e15i			0.855945i	0.903792	+	0.427973i	$0.140772\pi$
$431$			5.48668e15i			0.855945i	−0.903792	+	0.427973i	$0.859228\pi$
$432$	3.17909e14			0.0489104
$433$		−	1.04265e16i		−	1.58202i	−0.611806	−	0.791008i	$-0.709557\pi$
$433$		−	1.04265e16i		−	1.58202i	0.611806	−	0.791008i	$-0.290443\pi$
$434$		−	6.15651e15i		−	0.921289i
$435$		−	9.09915e13i		−	0.0134297i
$436$		−	6.33802e15i		−	0.922645i
$437$	−6.99236e15	+	1.01349e16i	−1.00400	+	1.45523i
$438$	−9.43851e15			−1.33678
$439$	1.26503e16			1.76732			0.883658	−	0.468133i	$-0.155073\pi$
$439$	1.26503e16			1.76732			0.883658	+	0.468133i	$0.155073\pi$
$440$	−1.66501e14			−0.0229457
$441$	−3.46314e15			−0.470803
$442$			5.67634e14i			0.0761264i
$443$	3.32307e15			0.439660			0.219830	−	0.975538i	$-0.429450\pi$
$443$	3.32307e15			0.439660			0.219830	+	0.975538i	$0.429450\pi$
$444$		−	8.08729e15i		−	1.05561i
$445$	−3.14434e13			−0.00404920
$446$	−3.81679e14			−0.0484941
$447$			1.26821e16i			1.58981i
$448$		−	7.51996e15i		−	0.930138i
$449$	−5.72605e15			−0.698839			−0.349420	−	0.936966i	$-0.613621\pi$
$449$	−5.72605e15			−0.698839			−0.349420	+	0.936966i	$0.613621\pi$
$450$	−6.76719e15			−0.814955
$451$		−	1.11773e16i		−	1.32824i
$452$			6.09824e15i			0.715112i
$453$	−4.45334e15			−0.515343
$454$			5.52486e15i			0.630937i
$455$	2.13712e13			0.00240858
$456$			2.46168e16i			2.73806i
$457$			1.38287e16i			1.51804i	0.651067	+	0.759020i	$0.274322\pi$
$457$			1.38287e16i			1.51804i	−0.651067	+	0.759020i	$0.725678\pi$
$458$			3.61892e15i			0.392091i
$459$		−	2.01709e15i		−	0.215699i
$460$	3.69785e13	−	5.35974e13i	0.00390303	−	0.00565714i
$461$	1.68026e15			0.175053			0.0875267	−	0.996162i	$-0.472104\pi$
$461$	1.68026e15			0.175053			0.0875267	+	0.996162i	$0.472104\pi$
$462$	−2.00619e16			−2.06310
$463$	−1.04060e16			−1.05633			−0.528163	−	0.849143i	$-0.677119\pi$
$463$	−1.04060e16			−1.05633			−0.528163	+	0.849143i	$0.677119\pi$
$464$	1.12957e15			0.113189
$465$		−	2.19501e14i		−	0.0217130i
$466$	−1.88513e15			−0.184088
$467$		−	5.47474e15i		−	0.527791i	−0.964551	−	0.263895i	$-0.914993\pi$
$467$		−	5.47474e15i		−	0.527791i	0.964551	−	0.263895i	$-0.0850074\pi$
$468$	1.12116e15			0.106707
$469$	8.16507e15			0.767226
$470$			1.20665e14i			0.0111942i
$471$			1.31601e16i			1.20541i
$472$	−8.42632e15			−0.762054
$473$	−2.19227e16			−1.95762
$474$			1.33257e16i			1.17495i
$475$		−	2.03033e16i		−	1.76768i
$476$	−5.20538e15			−0.447518
$477$		−	4.54161e15i		−	0.385567i
$478$	8.73192e15			0.732053
$479$		−	5.52874e15i		−	0.457734i	−0.973458	−	0.228867i	$-0.926498\pi$
$479$		−	5.52874e15i		−	0.457734i	0.973458	−	0.228867i	$-0.0735021\pi$
$480$		−	2.14444e14i		−	0.0175334i
$481$			2.61524e15i			0.211174i
$482$			2.41841e15i			0.192863i
$483$	1.26307e16	−	1.83073e16i	0.994822	−	1.44192i
$484$	1.43515e16			1.11642
$485$	−2.11903e13			−0.00162812
$486$	−1.26350e16			−0.958862
$487$	1.69381e16			1.26967			0.634833	−	0.772649i	$-0.281069\pi$
$487$	1.69381e16			1.26967			0.634833	+	0.772649i	$0.281069\pi$
$488$		−	3.47646e14i		−	0.0257406i
$489$	1.97779e16			1.44653
$490$		−	4.58716e13i		−	0.00331411i
$491$	−4.59682e14			−0.0328071			−0.0164036	−	0.999865i	$-0.505222\pi$
$491$	−4.59682e14			−0.0328071			−0.0164036	+	0.999865i	$0.505222\pi$
$492$	8.73926e15			0.616147
$493$		−	7.16692e15i		−	0.499173i
$494$		−	2.80812e15i		−	0.193221i
$495$	−3.91386e14			−0.0266056
$496$	2.72488e15			0.183003
$497$			1.10472e16i			0.733014i
$498$			1.63844e16i			1.07412i
$499$	2.02798e16			1.31359			0.656794	−	0.754070i	$-0.271912\pi$
$499$	2.02798e16			1.31359			0.656794	+	0.754070i	$0.271912\pi$
$500$			2.14761e14i			0.0137447i
$501$	−2.72691e16			−1.72443
$502$		−	2.04523e16i		−	1.27796i
$503$			8.92318e15i			0.550950i	0.961308	+	0.275475i	$0.0888351\pi$
$503$			8.92318e15i			0.550950i	−0.961308	+	0.275475i	$0.911165\pi$
$504$		−	2.43313e16i		−	1.48451i
$505$		−	8.10185e13i		−	0.00488468i
$506$	1.12254e16	−	1.62703e16i	0.668802	−	0.969377i
$507$	2.45771e16			1.44705
$508$	−3.69410e15			−0.214944
$509$	−1.11310e16			−0.640073			−0.320036	−	0.947405i	$-0.603695\pi$
$509$	−1.11310e16			−0.640073			−0.320036	+	0.947405i	$0.603695\pi$
$510$	1.54933e14			0.00880490
$511$			2.79895e16i			1.57206i
$512$	5.62734e15			0.312380
$513$			9.97863e15i			0.547478i
$514$	1.71363e14			0.00929264
$515$	−7.36489e13			−0.00394751
$516$		−	1.71409e16i		−	0.908103i
$517$		−	4.38776e16i		−	2.29773i
$518$	2.00210e16			1.03635
$519$	−4.57446e16			−2.34065
$520$			4.20983e13i			0.00212934i
$521$			1.60810e16i			0.804057i	0.915627	+	0.402029i	$0.131695\pi$
$521$			1.60810e16i			0.804057i	−0.915627	+	0.402029i	$0.868305\pi$
$522$	1.18174e16			0.584115
$523$			2.43390e15i			0.118930i	0.998230	+	0.0594652i	$0.0189395\pi$
$523$			2.43390e15i			0.118930i	−0.998230	+	0.0594652i	$0.981060\pi$
$524$	−5.32006e15			−0.256998
$525$			3.66751e16i			1.75152i
$526$			3.66400e15i			0.172998i
$527$		−	1.72890e16i		−	0.807059i
$528$		−	8.87945e15i		−	0.409810i
$529$	7.77992e15	+	2.04872e16i	0.355010	+	0.934862i
$530$	6.01566e13			0.00271411
$531$	−1.98073e16			−0.883605
$532$	2.57513e16			1.13587
$533$	−2.82607e15			−0.123259
$534$		−	7.46311e15i		−	0.321864i
$535$	−3.17652e14			−0.0135466
$536$			1.60841e16i			0.678277i
$537$	−2.37869e16			−0.991957
$538$	−1.19596e16			−0.493199
$539$			1.66804e16i			0.680257i
$540$		−	5.27711e13i		−	0.00212830i
$541$	1.80900e16			0.721532			0.360766	−	0.932656i	$-0.382515\pi$
$541$	1.80900e16			0.721532			0.360766	+	0.932656i	$0.382515\pi$
$542$	2.42175e16			0.955287
$543$		−	4.83040e16i		−	1.88445i
$544$		−	1.68906e16i		−	0.651707i
$545$	5.59421e14			0.0213481
$546$			5.07247e15i			0.191454i
$547$	−4.12296e16			−1.53917			−0.769583	−	0.638547i	$-0.779536\pi$
$547$	−4.12296e16			−1.53917			−0.769583	+	0.638547i	$0.779536\pi$
$548$		−	1.97031e16i		−	0.727532i
$549$		−	8.17193e14i		−	0.0298463i
$550$			3.25944e16i			1.17752i
$551$			3.54551e16i			1.26698i
$552$	3.60628e16	+	2.48808e16i	1.27475	+	0.879487i
$553$	3.95168e16			1.38176
$554$	−3.67978e16			−1.27281
$555$	7.13819e14			0.0244248
$556$	−6.94166e15			−0.234971
$557$		−	5.55098e16i		−	1.85882i	−0.369043	−	0.929412i	$-0.620315\pi$
$557$		−	5.55098e16i		−	1.85882i	0.369043	−	0.929412i	$-0.379685\pi$
$558$	2.85074e16			0.944392
$559$			5.54296e15i			0.181665i
$560$	7.24130e13			0.00234795
$561$	−5.63387e16			−1.80730
$562$		−	4.00417e16i		−	1.27085i
$563$			5.10025e16i			1.60155i	0.598964	+	0.800776i	$0.295579\pi$
$563$			5.10025e16i			1.60155i	−0.598964	+	0.800776i	$0.704421\pi$
$564$	3.43069e16			1.06588
$565$	−5.38257e14			−0.0165462
$566$		−	4.13731e16i		−	1.25840i
$567$			2.93064e16i			0.881990i
$568$	−2.17614e16			−0.648032
$569$		−	3.79289e16i		−	1.11763i	−0.829293	−	0.558814i	$-0.811257\pi$
$569$		−	3.79289e16i		−	1.11763i	0.829293	−	0.558814i	$-0.188743\pi$
$570$	−7.66464e14			−0.0223482
$571$		−	7.91017e15i		−	0.228228i	−0.993468	−	0.114114i	$-0.963597\pi$
$571$		−	7.91017e15i		−	0.228228i	0.993468	−	0.114114i	$-0.0364029\pi$
$572$		−	5.40010e15i		−	0.154179i
$573$			5.50823e16i			1.55627i
$574$			2.16350e16i			0.604903i
$575$	−2.97437e16	−	2.05210e16i	−0.822976	−	0.567796i
$576$	3.48208e16			0.953463
$577$	−4.17872e16			−1.13237			−0.566186	−	0.824278i	$-0.691581\pi$
$577$	−4.17872e16			−1.13237			−0.566186	+	0.824278i	$0.691581\pi$
$578$	−1.29483e16			−0.347254
$579$	3.78399e16			1.00433
$580$		−	1.87501e14i		−	0.00492534i
$581$	4.85872e16			1.26318
$582$		−	5.02952e15i		−	0.129416i
$583$	−2.18748e16			−0.557100
$584$	−5.51355e16			−1.38981
$585$			9.89583e13i			0.00246898i
$586$			2.37121e16i			0.585576i
$587$	7.23158e16			1.76768			0.883842	−	0.467785i	$-0.154948\pi$
$587$	7.23158e16			1.76768			0.883842	+	0.467785i	$0.154948\pi$
$588$	−1.30420e16			−0.315559
$589$			8.55294e16i			2.04844i
$590$		−	2.62360e14i		−	0.00621994i
$591$	−6.94861e16			−1.63070
$592$			8.86133e15i			0.205859i
$593$	−4.60776e16			−1.05965			−0.529824	−	0.848107i	$-0.677742\pi$
$593$	−4.60776e16			−1.05965			−0.529824	+	0.848107i	$0.677742\pi$
$594$		−	1.60195e16i		−	0.364694i
$595$		−	4.59449e14i		−	0.0103547i
$596$			2.61333e16i			0.583065i
$597$		−	4.95360e16i		−	1.09415i
$598$	−4.11380e15	−	2.83823e15i	−0.0899572	−	0.0620642i
$599$	2.79755e16			0.605643			0.302822	−	0.953047i	$-0.402071\pi$
$599$	2.79755e16			0.605643			0.302822	+	0.953047i	$0.402071\pi$
$600$	−7.22448e16			−1.54846
$601$	−3.14835e16			−0.668092			−0.334046	−	0.942557i	$-0.608414\pi$
$601$	−3.14835e16			−0.668092			−0.334046	+	0.942557i	$0.608414\pi$
$602$	4.24342e16			0.891532
$603$			3.78080e16i			0.786465i
$604$	−9.17675e15			−0.189002
$605$			1.26673e15i			0.0258316i
$606$	1.92298e16			0.388275
$607$	−1.94987e16			−0.389828			−0.194914	−	0.980820i	$-0.562443\pi$
$607$	−1.94987e16			−0.389828			−0.194914	+	0.980820i	$0.562443\pi$
$608$			8.35588e16i			1.65413i
$609$		−	6.40447e16i		−	1.25539i
$610$	1.08243e13			0.000210097
$611$	−1.10940e16			−0.213227
$612$		−	2.41032e16i		−	0.458740i
$613$		−	1.00393e16i		−	0.189209i	−0.995515	−	0.0946043i	$-0.969841\pi$
$613$		−	1.00393e16i		−	0.189209i	0.995515	−	0.0946043i	$-0.0301586\pi$
$614$	7.46957e14			0.0139407
$615$			7.71365e14i			0.0142564i
$616$	−1.17193e17			−2.14494
$617$			1.08743e16i			0.197101i	0.995132	+	0.0985507i	$0.0314207\pi$
$617$			1.08743e16i			0.197101i	−0.995132	+	0.0985507i	$0.968579\pi$
$618$		−	1.74806e16i		−	0.313780i
$619$		−	9.07446e16i		−	1.61316i	−0.591126	−	0.806579i	$-0.701316\pi$
$619$		−	9.07446e16i		−	1.61316i	0.591126	−	0.806579i	$-0.298684\pi$
$620$		−	4.52314e14i		−	0.00796324i
$621$	1.46184e16	+	1.00856e16i	0.254888	+	0.175855i
$622$	5.16824e16			0.892485
$623$	−2.21316e16			−0.378516
$624$	−2.24509e15			−0.0380299
$625$	5.95762e16			0.999523
$626$			1.75572e16i			0.291750i
$627$	2.78710e17			4.58720
$628$			2.71183e16i			0.442083i
$629$	5.62238e16			0.907854
$630$	7.57575e14			0.0121166
$631$		−	6.54578e16i		−	1.03702i	−0.855073	−	0.518508i	$-0.826488\pi$
$631$		−	6.54578e16i		−	1.03702i	0.855073	−	0.518508i	$-0.173512\pi$
$632$			7.78427e16i			1.22156i
$633$	1.54339e17			2.39913
$634$	−3.19126e16			−0.491390
$635$		−	3.26057e14i		−	0.00497338i
$636$		−	1.71034e16i		−	0.258429i
$637$	4.21748e15			0.0631271
$638$		−	5.69188e16i		−	0.843980i
$639$	−5.11533e16			−0.751396
$640$		−	3.49568e14i		−	0.00508688i
$641$		−	1.02117e17i		−	1.47214i	−0.676905	−	0.736070i	$-0.736679\pi$
$641$		−	1.02117e17i		−	1.47214i	0.676905	−	0.736070i	$-0.263321\pi$
$642$		−	7.53950e16i		−	1.07679i
$643$		−	8.63624e16i		−	1.22197i	−0.791644	−	0.610983i	$-0.790774\pi$
$643$		−	8.63624e16i		−	1.22197i	0.791644	−	0.610983i	$-0.209226\pi$
$644$	2.60274e16	−	3.77248e16i	0.364852	−	0.528824i
$645$	1.51293e15			0.0210117
$646$	−6.03703e16			−0.830670
$647$	5.86706e16			0.799825			0.399912	−	0.916553i	$-0.369041\pi$
$647$	5.86706e16			0.799825			0.399912	+	0.916553i	$0.369041\pi$
$648$	−5.77295e16			−0.779737
$649$			9.54026e16i			1.27671i
$650$	8.24120e15			0.109272
$651$		−	1.54497e17i		−	2.02971i
$652$	4.07553e16			0.530516
$653$	−5.01181e16			−0.646422			−0.323211	−	0.946327i	$-0.604762\pi$
$653$	−5.01181e16			−0.646422			−0.323211	+	0.946327i	$0.604762\pi$
$654$			1.32779e17i			1.69693i
$655$		−	4.69572e14i		−	0.00594640i
$656$	−9.57570e15			−0.120157
$657$	−1.29604e17			−1.61149
$658$			8.49305e16i			1.04643i
$659$		−	1.84832e16i		−	0.225665i	−0.993614	−	0.112833i	$-0.964008\pi$
$659$		−	1.84832e16i		−	0.225665i	0.993614	−	0.112833i	$-0.0359924\pi$
$660$	−1.47393e15			−0.0178326
$661$			1.47351e17i			1.76663i	0.468784	+	0.883313i	$0.344692\pi$
$661$			1.47351e17i			1.76663i	−0.468784	+	0.883313i	$0.655308\pi$
$662$	−2.09128e16			−0.248465
$663$			1.42447e16i			0.167715i
$664$			9.57102e16i			1.11673i
$665$			2.27292e15i			0.0262817i
$666$			9.27062e16i			1.06234i
$667$	5.19406e16	+	3.58354e16i	0.589864	+	0.406965i
$668$	−5.61920e16			−0.632434
$669$	−9.57819e15			−0.106838
$670$	−5.00791e14			−0.00553615
$671$	−3.93604e15			−0.0431245
$672$		−	1.50937e17i		−	1.63901i
$673$	−3.84688e15			−0.0414018			−0.0207009	−	0.999786i	$-0.506590\pi$
$673$	−3.84688e15			−0.0414018			−0.0207009	+	0.999786i	$0.506590\pi$
$674$		−	5.84253e15i		−	0.0623220i
$675$	−2.92851e16			−0.309616
$676$	5.06447e16			0.530706
$677$		−	1.50179e17i		−	1.55983i	−0.625885	−	0.779916i	$-0.715262\pi$
$677$		−	1.50179e17i		−	1.55983i	0.625885	−	0.779916i	$-0.284738\pi$
$678$		−	1.27756e17i		−	1.31523i
$679$	−1.49149e16			−0.152195
$680$	9.05051e14			0.00915419
$681$			1.38646e17i			1.39003i
$682$		−	1.37307e17i		−	1.36454i
$683$	7.47720e16			0.736572			0.368286	−	0.929713i	$-0.379945\pi$
$683$	7.47720e16			0.736572			0.368286	+	0.929713i	$0.379945\pi$
$684$			1.19240e17i			1.16435i
$685$	1.73908e15			0.0168336
$686$			5.05819e16i			0.485345i
$687$			9.08165e16i			0.863823i
$688$			1.87815e16i			0.177092i
$689$			5.53085e15i			0.0516983i
$690$	−7.74684e14	+	1.12284e15i	−0.00717844	+	0.0104046i
$691$	2.57585e16			0.236620			0.118310	−	0.992977i	$-0.462252\pi$
$691$	2.57585e16			0.236620			0.118310	+	0.992977i	$0.462252\pi$
$692$	−9.42634e16			−0.858433
$693$	−2.75478e17			−2.48707
$694$	−4.17992e16			−0.374120
$695$		−	6.12701e14i		−	0.00543676i
$696$	1.26159e17			1.10985
$697$			6.07564e16i			0.529901i
$698$	1.15282e17			0.996846
$699$	−4.73072e16			−0.405568
$700$			7.55743e16i			0.642371i
$701$			5.23286e16i			0.440992i	0.975388	+	0.220496i	$0.0707676\pi$
$701$			5.23286e16i			0.440992i	−0.975388	+	0.220496i	$0.929232\pi$
$702$	−4.05037e15			−0.0338433
$703$	−2.78142e17			−2.30428
$704$		−	1.67716e17i		−	1.37765i
$705$			3.02807e15i			0.0246622i
$706$	−2.62118e16			−0.211674
$707$		−	5.70252e16i		−	0.456615i
$708$	−7.45931e16			−0.592242
$709$		−	5.12273e16i		−	0.403296i	−0.979458	−	0.201648i	$-0.935370\pi$
$709$		−	5.12273e16i		−	0.403296i	0.979458	−	0.201648i	$-0.0646298\pi$
$710$		−	6.77559e14i		−	0.00528929i
$711$			1.82981e17i			1.41640i
$712$		−	4.35961e16i		−	0.334632i
$713$	1.25298e17	+	8.64466e16i	0.953687	+	0.657977i
$714$	1.09051e17			0.823074
$715$	4.76636e14			0.00356739
$716$	−4.90164e16			−0.363801
$717$	2.19127e17			1.61280
$718$			4.72306e16i			0.344728i
$719$	3.98857e16			0.288698			0.144349	−	0.989527i	$-0.453891\pi$
$719$	3.98857e16			0.288698			0.144349	+	0.989527i	$0.453891\pi$
$720$			3.35305e14i			0.00240682i
$721$	−5.18381e16			−0.369009
$722$	2.03107e17			1.43384
$723$			6.06898e16i			0.424899i
$724$		−	9.95374e16i		−	0.691122i
$725$	−1.04053e17			−0.716517
$726$	−3.00659e17			−2.05331
$727$		−	1.33841e17i		−	0.906530i	−0.891376	−	0.453265i	$-0.850259\pi$
$727$		−	1.33841e17i		−	0.906530i	0.891376	−	0.453265i	$-0.149741\pi$
$728$			2.96311e16i			0.199049i
$729$	−2.04773e17			−1.36429
$730$		−	1.71669e15i		−	0.0113437i
$731$	1.19165e17			0.780991
$732$		−	3.07750e15i		−	0.0200047i
$733$		−	3.53101e16i		−	0.227654i	−0.993501	−	0.113827i	$-0.963689\pi$
$733$		−	3.53101e16i		−	0.227654i	0.993501	−	0.113827i	$-0.0363109\pi$
$734$			1.21073e16i			0.0774235i
$735$		−	1.15114e15i		−	0.00730138i
$736$	1.22411e17	+	8.44549e16i	0.770111	+	0.531322i
$737$	1.82103e17			1.13635
$738$	−1.00180e17			−0.620072
$739$	−2.12756e17			−1.30622			−0.653109	−	0.757264i	$-0.726536\pi$
$739$	−2.12756e17			−1.30622			−0.653109	+	0.757264i	$0.726536\pi$
$740$	1.47093e15			0.00895779
$741$		−	7.04694e16i		−	0.425688i
$742$	4.23414e16			0.253713
$743$			2.09986e17i			1.24812i	0.781375	+	0.624062i	$0.214519\pi$
$743$			2.09986e17i			1.24812i	−0.781375	+	0.624062i	$0.785481\pi$
$744$	3.04338e17			1.79439
$745$	−2.30664e15			−0.0134909
$746$		−	7.39395e16i		−	0.428986i
$747$			2.24981e17i			1.29486i
$748$	−1.16094e17			−0.662827
$749$	−2.23581e17			−1.26632
$750$		−	4.49916e15i		−	0.0252792i
$751$			2.57245e17i			1.43386i	0.697144	+	0.716931i	$0.254454\pi$
$751$			2.57245e17i			1.43386i	−0.697144	+	0.716931i	$0.745546\pi$
$752$	−3.75904e16			−0.207860
$753$		−	5.13247e17i		−	2.81551i
$754$	−1.43914e16			−0.0783204
$755$		−	8.09980e14i		−	0.00437313i
$756$		−	3.71431e16i		−	0.198952i
$757$		−	2.67529e17i		−	1.42166i	−0.703365	−	0.710829i	$-0.748320\pi$
$757$		−	2.67529e17i		−	1.42166i	0.703365	−	0.710829i	$-0.251680\pi$
$758$		−	2.79510e16i		−	0.147361i
$759$	2.81700e17	−	4.08302e17i	1.47345	−	2.13565i
$760$	−4.47734e15			−0.0232348
$761$	4.34486e16			0.223701			0.111850	−	0.993725i	$-0.464322\pi$
$761$	4.34486e16			0.223701			0.111850	+	0.993725i	$0.464322\pi$
$762$	7.73899e16			0.395325
$763$	3.93751e17			1.99560
$764$			1.13505e17i			0.570762i
$765$	2.12746e15			0.0106143
$766$		−	5.46990e16i		−	0.270774i
$767$	2.41217e16			0.118477
$768$	3.23575e17			1.57691
$769$			4.49253e16i			0.217237i	0.994084	+	0.108618i	$0.0346427\pi$
$769$			4.49253e16i			0.217237i	−0.994084	+	0.108618i	$0.965357\pi$
$770$		−	3.64889e15i		−	0.0175072i
$771$	4.30035e15			0.0204728
$772$	7.79745e16			0.368340
$773$		−	2.47476e15i		−	0.0115999i	−0.999983	−	0.00579996i	$-0.998154\pi$
$773$		−	2.47476e15i		−	0.0115999i	0.999983	−	0.00579996i	$-0.00184620\pi$
$774$			1.96489e17i			0.913889i
$775$	−2.51010e17			−1.15846
$776$		−	2.93802e16i		−	0.134550i
$777$	5.02425e17			2.28320
$778$		−	5.61235e16i		−	0.253085i
$779$		−	3.00565e17i		−	1.34497i
$780$			3.72671e14i			0.00165485i
$781$			2.46382e17i			1.08568i
$782$	−6.10178e16	+	8.84405e16i	−0.266818	+	0.386733i
$783$	5.11398e16			0.221916
$784$	1.42903e16			0.0615380
$785$	−2.39357e15			−0.0102289
$786$	1.11453e17			0.472669
$787$		−	9.96355e15i		−	0.0419339i	−0.999780	−	0.0209670i	$-0.993326\pi$
$787$		−	9.96355e15i		−	0.0419339i	0.999780	−	0.0209670i	$-0.00667448\pi$
$788$	−1.43186e17			−0.598059
$789$			9.19477e16i			0.381135i
$790$	−2.42370e15			−0.00997047
$791$	−3.78854e17			−1.54673
$792$		−	5.42654e17i		−	2.19873i
$793$			9.95192e14i			0.00400191i
$794$	2.04841e16			0.0817512
$795$	1.50962e15			0.00597951
$796$		−	1.02076e17i		−	0.401278i
$797$			1.11543e17i			0.435203i	0.976038	+	0.217602i	$0.0698233\pi$
$797$			1.11543e17i			0.435203i	−0.976038	+	0.217602i	$0.930177\pi$
$798$	−5.39479e17			−2.08909
$799$			2.38505e17i			0.916679i
$800$	−2.45227e17			−0.935465
$801$		−	1.02479e17i		−	0.388007i
$802$		−	1.32672e16i		−	0.0498578i
$803$			6.24243e17i			2.32841i
$804$			1.42383e17i			0.527133i
$805$	3.32975e15	+	2.29729e15i	0.0122359	+	0.00844192i
$806$	−3.47168e16			−0.126628
$807$	−3.00124e17			−1.08658
$808$	1.12332e17			0.403678
$809$	7.67640e16			0.273821			0.136910	−	0.990583i	$-0.456283\pi$
$809$	7.67640e16			0.273821			0.136910	+	0.990583i	$0.456283\pi$
$810$		−	1.79746e15i		−	0.00636427i
$811$	−5.07809e17			−1.78474			−0.892371	−	0.451303i	$-0.850959\pi$
$811$	−5.07809e17			−1.78474			−0.892371	+	0.451303i	$0.850959\pi$
$812$		−	1.31973e17i		−	0.460416i
$813$	6.07736e17			2.10461
$814$	4.46523e17			1.53496
$815$			3.59724e15i			0.0122751i
$816$			4.82660e16i			0.163494i
$817$	−5.89518e17			−1.98228
$818$	−1.68162e17			−0.561318
$819$			6.96522e16i			0.230798i
$820$			1.58951e15i			0.00522853i
$821$	−5.34524e17			−1.74545			−0.872726	−	0.488210i	$-0.837650\pi$
$821$	−5.34524e17			−1.74545			−0.872726	+	0.488210i	$0.837650\pi$
$822$			4.12773e17i			1.33807i
$823$	1.84477e17			0.593669			0.296834	−	0.954929i	$-0.404069\pi$
$823$	1.84477e17			0.593669			0.296834	+	0.954929i	$0.404069\pi$
$824$		−	1.02114e17i		−	0.326228i
$825$			8.17954e17i			2.59421i
$826$		−	1.84664e17i		−	0.581435i
$827$		−	2.43107e17i		−	0.759915i	−0.925004	−	0.379957i	$-0.875939\pi$
$827$		−	2.43107e17i		−	0.759915i	0.925004	−	0.379957i	$-0.124061\pi$
$828$	1.74683e17	+	1.20519e17i	0.542085	+	0.374001i
$829$	5.28902e17			1.62948			0.814739	−	0.579829i	$-0.196880\pi$
$829$	5.28902e17			1.62948			0.814739	+	0.579829i	$0.196880\pi$
$830$	−2.98002e15			−0.00911486
$831$	−9.23437e17			−2.80415
$832$	−4.24054e16			−0.127844
$833$		−	9.06695e16i		−	0.271388i
$834$	1.45425e17			0.432158
$835$		−	4.95975e15i		−	0.0146332i
$836$	5.74323e17			1.68236
$837$	1.23366e17			0.358792
$838$			3.81199e17i			1.10075i
$839$		−	3.72458e17i		−	1.06784i	−0.845536	−	0.533919i	$-0.820719\pi$
$839$		−	3.72458e17i		−	1.06784i	0.845536	−	0.533919i	$-0.179281\pi$
$840$	8.08768e15			0.0230223
$841$	−1.72110e17			−0.486440
$842$		−	1.93137e17i		−	0.541993i
$843$		−	1.00484e18i		−	2.79983i
$844$	3.18038e17			0.879882
$845$			4.47012e15i			0.0122795i
$846$	−3.93266e17			−1.07267
$847$			8.91593e17i			2.41472i
$848$			1.87404e16i			0.0503969i
$849$		−	1.03825e18i		−	2.77241i
$850$		−	1.77174e17i		−	0.469770i
$851$	−2.81125e17	+	4.07469e17i	−0.740154	+	1.07280i
$852$	−1.92640e17			−0.503628
$853$	1.71441e17			0.445062			0.222531	−	0.974926i	$-0.428568\pi$
$853$	1.71441e17			0.445062			0.222531	+	0.974926i	$0.428568\pi$
$854$	7.61869e15			0.0196396
$855$	−1.05246e16			−0.0269408
$856$		−	4.40423e17i		−	1.11951i
$857$	7.39162e17			1.86576			0.932878	−	0.360194i	$-0.117289\pi$
$857$	7.39162e17			1.86576			0.932878	+	0.360194i	$0.117289\pi$
$858$			1.13130e17i			0.283566i
$859$	7.74483e16			0.192776			0.0963880	−	0.995344i	$-0.469271\pi$
$859$	7.74483e16			0.192776			0.0963880	+	0.995344i	$0.469271\pi$
$860$	3.11761e15			0.00770604
$861$			5.42928e17i			1.33267i
$862$		−	2.36858e17i		−	0.577358i
$863$	9.60821e16			0.232583			0.116291	−	0.993215i	$-0.462899\pi$
$863$	9.60821e16			0.232583			0.116291	+	0.993215i	$0.462899\pi$
$864$	1.20524e17			0.289727
$865$		−	8.32010e15i		−	0.0198624i
$866$			4.50108e17i			1.06711i
$867$	−3.24937e17			−0.765041
$868$		−	3.18363e17i		−	0.744397i
$869$	8.81333e17			2.04654
$870$			3.92808e15i			0.00905867i
$871$		−	4.60432e16i		−	0.105452i
$872$			7.75635e17i			1.76424i
$873$		−	6.90625e16i		−	0.156011i
$874$	3.01858e17	−	4.37520e17i	0.677227	−	0.981588i
$875$	−1.33421e16			−0.0297287
$876$	−4.88081e17			−1.08011
$877$	−3.35782e17			−0.738007			−0.369003	−	0.929428i	$-0.620301\pi$
$877$	−3.35782e17			−0.738007			−0.369003	+	0.929428i	$0.620301\pi$
$878$	−5.46110e17			−1.19210
$879$			5.95051e17i			1.29009i
$880$	1.61501e15			0.00347759
$881$		−	8.78827e16i		−	0.187952i	−0.995574	−	0.0939762i	$-0.970042\pi$
$881$		−	8.78827e16i		−	0.187952i	0.995574	−	0.0939762i	$-0.0299578\pi$
$882$	1.49503e17			0.317569
$883$	−7.17005e17			−1.51272			−0.756359	−	0.654156i	$-0.773024\pi$
$883$	−7.17005e17			−1.51272			−0.756359	+	0.654156i	$0.773024\pi$
$884$			2.93533e16i			0.0615097i
$885$		−	6.58391e15i		−	0.0137033i
$886$	−1.43456e17			−0.296562
$887$	−4.93950e17			−1.01424			−0.507121	−	0.861875i	$-0.669290\pi$
$887$	−4.93950e17			−1.01424			−0.507121	+	0.861875i	$0.669290\pi$
$888$			9.89707e17i			2.01850i
$889$		−	2.29497e17i		−	0.464907i
$890$	1.35740e15			0.00273129
$891$			6.53612e17i			1.30633i
$892$	−1.97373e16			−0.0391830
$893$		−	1.17990e18i		−	2.32668i
$894$		−	5.47483e17i		−	1.07237i
$895$		−	4.32640e15i		−	0.00841761i
$896$		−	2.46045e17i		−	0.475517i
$897$	−1.03235e17	−	7.12252e16i	−0.198186	−	0.136735i
$898$	2.47192e17			0.471385
$899$	4.38332e17			0.830319
$900$	−3.49943e17			−0.658479
$901$	1.18905e17			0.222255
$902$			4.82520e17i			0.895934i
$903$	1.06488e18			1.96415
$904$		−	7.46291e17i		−	1.36741i
$905$	8.78560e15			0.0159912
$906$	1.92249e17			0.347613
$907$		−	4.02229e17i		−	0.722487i	−0.932471	−	0.361244i	$-0.882352\pi$
$907$		−	4.02229e17i		−	0.722487i	0.932471	−	0.361244i	$-0.117648\pi$
$908$			2.85700e17i			0.509794i
$909$	2.64052e17			0.468066
$910$	−9.22588e14			−0.00162465
$911$			2.71856e17i			0.475585i	0.971316	+	0.237793i	$0.0764238\pi$
$911$			2.71856e17i			0.475585i	−0.971316	+	0.237793i	$0.923576\pi$
$912$		−	2.38774e17i		−	0.414972i
$913$	1.08363e18			1.87092
$914$		−	5.96980e17i		−	1.02396i
$915$	2.71633e14			0.000462868
$916$			1.87141e17i			0.316807i
$917$		−	3.30510e17i		−	0.555864i
$918$			8.70770e16i			0.145495i
$919$			6.97302e17i			1.15752i	0.815499	+	0.578759i	$0.196463\pi$
$919$			6.97302e17i			1.15752i	−0.815499	+	0.578759i	$0.803537\pi$
$920$	−4.52535e15	+	6.55915e15i	−0.00746321	+	0.0108173i
$921$	1.87448e16			0.0307131
$922$	−7.25363e16			−0.118078
$923$	6.22954e16			0.100750
$924$	−1.03744e18			−1.66697
$925$		−	8.16285e17i		−	1.30314i
$926$	4.49224e17			0.712520
$927$		−	2.40034e17i		−	0.378263i
$928$	4.28233e17			0.670490
$929$	−4.99060e17			−0.776352			−0.388176	−	0.921585i	$-0.626895\pi$
$929$	−4.99060e17			−0.776352			−0.388176	+	0.921585i	$0.626895\pi$
$930$			9.47580e15i			0.0146460i
$931$			4.48546e17i			0.688826i
$932$	−9.74833e16			−0.148742
$933$	1.29697e18			1.96625
$934$			2.36343e17i			0.356009i
$935$		−	1.02470e16i		−	0.0153365i
$936$	−1.37205e17			−0.204040
$937$			2.02687e17i			0.299495i	0.988724	+	0.149747i	$0.0478460\pi$
$937$			2.02687e17i			0.299495i	−0.988724	+	0.149747i	$0.952154\pi$
$938$	−3.52484e17			−0.517514
$939$			4.40597e17i			0.642759i
$940$			6.23978e15i			0.00904487i
$941$		−	1.07140e18i		−	1.54318i	−0.636123	−	0.771588i	$-0.719463\pi$
$941$		−	1.07140e18i		−	1.54318i	0.636123	−	0.771588i	$-0.280537\pi$
$942$		−	5.68117e17i		−	0.813078i
$943$	−4.40318e17	−	3.03788e17i	−0.626175	−	0.432017i
$944$	8.17325e16			0.115495
$945$	3.27841e15			0.00460334
$946$	9.46398e17			1.32047
$947$	−4.67317e17			−0.647906			−0.323953	−	0.946073i	$-0.605012\pi$
$947$	−4.67317e17			−0.647906			−0.323953	+	0.946073i	$0.605012\pi$
$948$			6.89094e17i			0.949354i
$949$	1.57834e17			0.216075
$950$			8.76487e17i			1.19235i
$951$	−8.00843e17			−1.08259
$952$	6.37024e17			0.855725
$953$			1.45697e17i			0.194489i	0.995261	+	0.0972444i	$0.0310029\pi$
$953$			1.45697e17i			0.194489i	−0.995261	+	0.0972444i	$0.968997\pi$
$954$			1.96060e17i			0.260075i
$955$	−1.00185e16			−0.0132063
$956$	4.51542e17			0.591495
$957$		−	1.42837e18i		−	1.85939i
$958$			2.38674e17i			0.308754i
$959$	1.22406e18			1.57359
$960$			1.15744e16i			0.0147867i
$961$	2.69738e17			0.342453
$962$		−	1.12899e17i		−	0.142443i
$963$		−	1.03528e18i		−	1.29808i
$964$			1.25060e17i			0.155832i
$965$			6.88237e15i			0.00852264i
$966$	−5.45265e17	+	7.90319e17i	−0.671034	+	0.972612i
$967$	5.00948e17			0.612680			0.306340	−	0.951922i	$-0.400895\pi$
$967$	5.00948e17			0.612680			0.306340	+	0.951922i	$0.400895\pi$
$968$	−1.75631e18			−2.13477
$969$	−1.51499e18			−1.83006
$970$	9.14777e14			0.00109821
$971$			8.15663e16i			0.0973185i	0.998815	+	0.0486593i	$0.0154948\pi$
$971$			8.15663e16i			0.0973185i	−0.998815	+	0.0486593i	$0.984505\pi$
$972$	−6.53375e17			−0.774756
$973$		−	4.31252e17i		−	0.508223i
$974$	−7.31211e17			−0.856424
$975$	2.06812e17			0.240740
$976$			3.37205e15i			0.00390117i
$977$			1.61924e18i			1.86185i	0.365210	+	0.930925i	$0.380997\pi$
$977$			1.61924e18i			1.86185i	−0.365210	+	0.930925i	$0.619003\pi$
$978$	−8.53807e17			−0.975724
$979$	−4.93594e17			−0.560627
$980$		−	2.37210e15i		−	0.00267779i
$981$			1.82324e18i			2.04565i
$982$	1.98443e16			0.0221293
$983$		−	7.50585e17i		−	0.831915i	−0.909384	−	0.415957i	$-0.863447\pi$
$983$		−	7.50585e17i		−	0.831915i	0.909384	−	0.415957i	$-0.136553\pi$
$984$	−1.06949e18			−1.17817
$985$		−	1.26382e16i		−	0.0138379i
$986$			3.09394e17i			0.336705i
$987$			2.13132e18i			2.30540i
$988$		−	1.45212e17i		−	0.156121i
$989$	−5.95840e17	+	8.63624e17i	−0.636725	+	0.922884i
$990$	1.68960e16			0.0179462
$991$	9.73285e16			0.102754			0.0513769	−	0.998679i	$-0.483639\pi$
$991$	9.73285e16			0.102754			0.0513769	+	0.998679i	$0.483639\pi$
$992$	1.03304e18			1.08404
$993$	−5.24805e17			−0.547397
$994$		−	4.76903e17i		−	0.494438i
$995$	9.00968e15			0.00928476
$996$			8.47264e17i			0.867886i
$997$	−1.01590e18			−1.03438			−0.517191	−	0.855870i	$-0.673022\pi$
$997$	−1.01590e18			−1.03438			−0.517191	+	0.855870i	$0.673022\pi$
$998$	−8.75471e17			−0.886051
$999$			4.01187e17i			0.403602i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.c.22.7	✓	20
23.22	odd	2	inner	23.13.b.c.22.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.c.22.7	✓	20	1.1	even	1	trivial
23.13.b.c.22.8	yes	20	23.22	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-43.1697 q^{2} -1083.34 q^{3} -2232.38 q^{4} -197.039i q^{5} +46767.4 q^{6} -138687. i q^{7} +273194. q^{8} +642183. q^{9} +O(q^{10})\)	\(q-43.1697 q^{2} -1083.34 q^{3} -2232.38 q^{4} -197.039i q^{5} +46767.4 q^{6} -138687. i q^{7} +273194. q^{8} +642183. q^{9} +8506.13i q^{10} -3.09310e6i q^{11} +2.41842e6 q^{12} -782061. q^{13} +5.98707e6i q^{14} +213460. i q^{15} -2.64989e6 q^{16} +1.68131e7i q^{17} -2.77228e7 q^{18} -8.31755e7i q^{19} +439866. i q^{20} +1.50245e8i q^{21} +1.33528e8i q^{22} +(-1.21849e8 - 8.40675e7i) q^{23} -2.95962e8 q^{24} +2.44102e8 q^{25} +3.37613e7 q^{26} -1.19971e8 q^{27} +3.09601e8i q^{28} -4.26269e8 q^{29} -9.21502e6i q^{30} -1.02830e9 q^{31} -1.00461e9 q^{32} +3.35087e9i q^{33} -7.25819e8i q^{34} -2.73268e7 q^{35} -1.43359e9 q^{36} -3.34404e9i q^{37} +3.59066e9i q^{38} +8.47237e8 q^{39} -5.38300e7i q^{40} +3.61362e9 q^{41} -6.48603e9i q^{42} -7.08764e9i q^{43} +6.90496e9i q^{44} -1.26535e8i q^{45} +(5.26020e9 + 3.62917e9i) q^{46} +1.41856e10 q^{47} +2.87073e9 q^{48} -5.39277e9 q^{49} -1.05378e10 q^{50} -1.82143e10i q^{51} +1.74586e9 q^{52} -7.07215e9i q^{53} +5.17910e9 q^{54} -6.09461e8 q^{55} -3.78884e10i q^{56} +9.01073e10i q^{57} +1.84019e10 q^{58} -3.08437e10 q^{59} -4.76524e8i q^{60} -1.27252e9i q^{61} +4.43914e10 q^{62} -8.90623e10i q^{63} +5.42226e10 q^{64} +1.54097e8i q^{65} -1.44656e11i q^{66} +5.88741e10i q^{67} -3.75333e10i q^{68} +(1.32004e11 + 9.10737e10i) q^{69} +1.17969e9 q^{70} -7.96554e10 q^{71} +1.75441e11 q^{72} -2.01818e11 q^{73} +1.44361e11i q^{74} -2.64445e11 q^{75} +1.85679e11i q^{76} -4.28972e11 q^{77} -3.65750e10 q^{78} +2.84935e11i q^{79} +5.22133e8i q^{80} -2.11313e11 q^{81} -1.55999e11 q^{82} +3.50338e11i q^{83} -3.35403e11i q^{84} +3.31285e9 q^{85} +3.05971e11i q^{86} +4.61794e11 q^{87} -8.45016e11i q^{88} -1.59579e11i q^{89} +5.46249e9i q^{90} +1.08462e11i q^{91} +(2.72014e11 + 1.87670e11i) q^{92} +1.11400e12 q^{93} -6.12390e11 q^{94} -1.63888e10 q^{95} +1.08833e12 q^{96} -1.07543e11i q^{97} +2.32804e11 q^{98} -1.98633e12i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) 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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