LMFDB - Embedded newform 23.13.b.c.22.15

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			22.15
Root			$-962.760 + 10393.2i$ of defining polynomial
Character	$\chi$	$=$	23.22
Dual form			23.13.b.c.22.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+68.2708 q^{2} +1043.03 q^{3} +564.897 q^{4} -10393.2i q^{5} +71208.5 q^{6} -148862. i q^{7} -241071. q^{8} +556472. q^{9} +O(q^{10})$	\(q+68.2708 q^{2} +1043.03 q^{3} +564.897 q^{4} -10393.2i q^{5} +71208.5 q^{6} -148862. i q^{7} -241071. q^{8} +556472. q^{9} -709550. i q^{10} -2.82321e6i q^{11} +589205. q^{12} +7.25191e6 q^{13} -1.01629e7i q^{14} -1.08404e7i q^{15} -1.87719e7 q^{16} +4.15858e7i q^{17} +3.79907e7 q^{18} +1.46230e7i q^{19} -5.87108e6i q^{20} -1.55268e8i q^{21} -1.92743e8i q^{22} +(6.58008e7 + 1.32608e8i) q^{23} -2.51444e8 q^{24} +1.36122e8 q^{25} +4.95093e8 q^{26} +2.61077e7 q^{27} -8.40919e7i q^{28} -5.60391e8 q^{29} -7.40083e8i q^{30} +1.23606e9 q^{31} -2.94147e8 q^{32} -2.94469e9i q^{33} +2.83909e9i q^{34} -1.54715e9 q^{35} +3.14349e8 q^{36} -4.16625e9i q^{37} +9.98322e8i q^{38} +7.56396e9 q^{39} +2.50550e9i q^{40} -2.24738e9 q^{41} -1.06003e10i q^{42} +3.28047e9i q^{43} -1.59482e9i q^{44} -5.78351e9i q^{45} +(4.49227e9 + 9.05325e9i) q^{46} -3.04044e9 q^{47} -1.95797e10 q^{48} -8.31868e9 q^{49} +9.29318e9 q^{50} +4.33752e10i q^{51} +4.09658e9 q^{52} +3.72205e10i q^{53} +1.78239e9 q^{54} -2.93421e10 q^{55} +3.58864e10i q^{56} +1.52522e10i q^{57} -3.82583e10 q^{58} -2.34320e10 q^{59} -6.12372e9i q^{60} -2.00987e10i q^{61} +8.43869e10 q^{62} -8.28376e10i q^{63} +5.68082e10 q^{64} -7.53704e10i q^{65} -2.01037e11i q^{66} +5.90588e10i q^{67} +2.34917e10i q^{68} +(6.86322e10 + 1.38314e11i) q^{69} -1.05625e11 q^{70} +1.87511e11 q^{71} -1.34149e11 q^{72} -8.56695e9 q^{73} -2.84433e11i q^{74} +1.41980e11 q^{75} +8.26049e9i q^{76} -4.20269e11 q^{77} +5.16398e11 q^{78} -3.28520e10i q^{79} +1.95100e11i q^{80} -2.68501e11 q^{81} -1.53430e11 q^{82} +2.20312e11i q^{83} -8.77104e10i q^{84} +4.32209e11 q^{85} +2.23960e11i q^{86} -5.84505e11 q^{87} +6.80594e11i q^{88} -3.96343e11i q^{89} -3.94845e11i q^{90} -1.07954e12i q^{91} +(3.71707e10 + 7.49099e10i) q^{92} +1.28925e12 q^{93} -2.07573e11 q^{94} +1.51979e11 q^{95} -3.06804e11 q^{96} +1.87982e11i q^{97} -5.67923e11 q^{98} -1.57104e12i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	68.2708			1.06673			0.533365	−	0.845885i	$-0.320927\pi$
$2$	68.2708			1.06673			0.533365	+	0.845885i	$0.320927\pi$
$3$	1043.03			1.43077			0.715384	−	0.698731i	$-0.246252\pi$
$3$	1043.03			1.43077			0.715384	+	0.698731i	$0.246252\pi$
$4$	564.897			0.137914
$5$		−	10393.2i		−	0.665164i	−0.943074	−	0.332582i	$-0.892080\pi$
$5$		−	10393.2i		−	0.665164i	0.943074	−	0.332582i	$-0.107920\pi$
$6$	71208.5			1.52625
$7$		−	148862.i		−	1.26531i	−0.774434	−	0.632654i	$-0.781965\pi$
$7$		−	148862.i		−	1.26531i	0.774434	−	0.632654i	$-0.218035\pi$
$8$	−241071.			−0.919613
$9$	556472.			1.04710
$10$		−	709550.i		−	0.709550i
$11$		−	2.82321e6i		−	1.59363i	−0.604224	−	0.796814i	$-0.706517\pi$
$11$		−	2.82321e6i		−	1.59363i	0.604224	−	0.796814i	$-0.293483\pi$
$12$	589205.			0.197324
$13$	7.25191e6			1.50242			0.751212	−	0.660061i	$-0.229470\pi$
$13$	7.25191e6			1.50242			0.751212	+	0.660061i	$0.229470\pi$
$14$		−	1.01629e7i		−	1.34974i
$15$		−	1.08404e7i		−	0.951695i
$16$	−1.87719e7			−1.11889
$17$			4.15858e7i			1.72287i	0.507871	+	0.861433i	$0.330432\pi$
$17$			4.15858e7i			1.72287i	−0.507871	+	0.861433i	$0.669568\pi$
$18$	3.79907e7			1.11697
$19$			1.46230e7i			0.310824i	0.987850	+	0.155412i	$0.0496705\pi$
$19$			1.46230e7i			0.310824i	−0.987850	+	0.155412i	$0.950329\pi$
$20$		−	5.87108e6i		−	0.0917356i
$21$		−	1.55268e8i		−	1.81036i
$22$		−	1.92743e8i		−	1.69997i
$23$	6.58008e7	+	1.32608e8i	0.444492	+	0.895783i
$23$	6.58008e7	+	1.32608e8i	0.444492	+	0.895783i
$24$	−2.51444e8			−1.31575
$25$	1.36122e8			0.557557
$26$	4.95093e8			1.60268
$27$	2.61077e7			0.0673884
$28$		−	8.40919e7i		−	0.174504i
$29$	−5.60391e8			−0.942113			−0.471056	−	0.882103i	$-0.656127\pi$
$29$	−5.60391e8			−0.942113			−0.471056	+	0.882103i	$0.656127\pi$
$30$		−	7.40083e8i		−	1.01520i
$31$	1.23606e9			1.39274			0.696370	−	0.717683i	$-0.254797\pi$
$31$	1.23606e9			1.39274			0.696370	+	0.717683i	$0.254797\pi$
$32$	−2.94147e8			−0.273945
$33$		−	2.94469e9i		−	2.28011i
$34$			2.83909e9i			1.83783i
$35$	−1.54715e9			−0.841637
$36$	3.14349e8			0.144410
$37$		−	4.16625e9i		−	1.62381i	−0.583789	−	0.811905i	$-0.698430\pi$
$37$		−	4.16625e9i		−	1.62381i	0.583789	−	0.811905i	$-0.301570\pi$
$38$			9.98322e8i			0.331565i
$39$	7.56396e9			2.14962
$40$			2.50550e9i			0.611693i
$41$	−2.24738e9			−0.473121			−0.236561	−	0.971617i	$-0.576020\pi$
$41$	−2.24738e9			−0.473121			−0.236561	+	0.971617i	$0.576020\pi$
$42$		−	1.06003e10i		−	1.93117i
$43$			3.28047e9i			0.518949i	0.965750	+	0.259475i	$0.0835494\pi$
$43$			3.28047e9i			0.518949i	−0.965750	+	0.259475i	$0.916451\pi$
$44$		−	1.59482e9i		−	0.219784i
$45$		−	5.78351e9i		−	0.696492i
$46$	4.49227e9	+	9.05325e9i	0.474154	+	0.955559i
$47$	−3.04044e9			−0.282065			−0.141033	−	0.990005i	$-0.545042\pi$
$47$	−3.04044e9			−0.282065			−0.141033	+	0.990005i	$0.545042\pi$
$48$	−1.95797e10			−1.60088
$49$	−8.31868e9			−0.601005
$50$	9.29318e9			0.594764
$51$			4.33752e10i			2.46502i
$52$	4.09658e9			0.207206
$53$			3.72205e10i			1.67930i	0.543130	+	0.839649i	$0.317239\pi$
$53$			3.72205e10i			1.67930i	−0.543130	+	0.839649i	$0.682761\pi$
$54$	1.78239e9			0.0718853
$55$	−2.93421e10			−1.06002
$56$			3.58864e10i			1.16359i
$57$			1.52522e10i			0.444717i
$58$	−3.82583e10			−1.00498
$59$	−2.34320e10			−0.555517			−0.277759	−	0.960651i	$-0.589592\pi$
$59$	−2.34320e10			−0.555517			−0.277759	+	0.960651i	$0.589592\pi$
$60$		−	6.12372e9i		−	0.131252i
$61$		−	2.00987e10i		−	0.390113i	−0.980792	−	0.195056i	$-0.937511\pi$
$61$		−	2.00987e10i		−	0.390113i	0.980792	−	0.195056i	$-0.0624890\pi$
$62$	8.43869e10			1.48568
$63$		−	8.28376e10i		−	1.32490i
$64$	5.68082e10			0.826668
$65$		−	7.53704e10i		−	0.999357i
$66$		−	2.01037e11i		−	2.43227i
$67$			5.90588e10i			0.652883i	0.945217	+	0.326442i	$0.105850\pi$
$67$			5.90588e10i			0.652883i	−0.945217	+	0.326442i	$0.894150\pi$
$68$			2.34917e10i			0.237608i
$69$	6.86322e10	+	1.38314e11i	0.635966	+	1.28166i
$70$	−1.05625e11			−0.897800
$71$	1.87511e11			1.46378			0.731891	−	0.681422i	$-0.238638\pi$
$71$	1.87511e11			1.46378			0.731891	+	0.681422i	$0.238638\pi$
$72$	−1.34149e11			−0.962926
$73$	−8.56695e9			−0.0566095			−0.0283047	−	0.999599i	$-0.509011\pi$
$73$	−8.56695e9			−0.0566095			−0.0283047	+	0.999599i	$0.509011\pi$
$74$		−	2.84433e11i		−	1.73217i
$75$	1.41980e11			0.797736
$76$			8.26049e9i			0.0428671i
$77$	−4.20269e11			−2.01643
$78$	5.16398e11			2.29307
$79$		−	3.28520e10i		−	0.135145i	−0.997714	−	0.0675724i	$-0.978475\pi$
$79$		−	3.28520e10i		−	0.135145i	0.997714	−	0.0675724i	$-0.0215254\pi$
$80$			1.95100e11i			0.744248i
$81$	−2.68501e11			−0.950682
$82$	−1.53430e11			−0.504693
$83$			2.20312e11i			0.673859i	0.941530	+	0.336929i	$0.109388\pi$
$83$			2.20312e11i			0.673859i	−0.941530	+	0.336929i	$0.890612\pi$
$84$		−	8.77104e10i		−	0.249675i
$85$	4.32209e11			1.14599
$86$			2.23960e11i			0.553579i
$87$	−5.84505e11			−1.34795
$88$			6.80594e11i			1.46552i
$89$		−	3.96343e11i		−	0.797502i	−0.917059	−	0.398751i	$-0.869444\pi$
$89$		−	3.96343e11i		−	0.797502i	0.917059	−	0.398751i	$-0.130556\pi$
$90$		−	3.94845e11i		−	0.742970i
$91$		−	1.07954e12i		−	1.90103i
$92$	3.71707e10	+	7.49099e10i	0.0613019	+	0.123541i
$93$	1.28925e12			1.99269
$94$	−2.07573e11			−0.300888
$95$	1.51979e11			0.206749
$96$	−3.06804e11			−0.391953
$97$			1.87982e11i			0.225676i	0.993613	+	0.112838i	$0.0359941\pi$
$97$			1.87982e11i			0.225676i	−0.993613	+	0.112838i	$0.964006\pi$
$98$	−5.67923e11			−0.641110
$99$		−	1.57104e12i		−	1.66869i
$100$	7.68952e10			0.0768952
$101$	1.23707e12			1.16538			0.582689	−	0.812695i	$-0.302000\pi$
$101$	1.23707e12			1.16538			0.582689	+	0.812695i	$0.302000\pi$
$102$			2.96126e12i			2.62952i
$103$			9.83820e11i			0.823933i	0.911199	+	0.411967i	$0.135158\pi$
$103$			9.83820e11i			0.823933i	−0.911199	+	0.411967i	$0.864842\pi$
$104$	−1.74823e12			−1.38165
$105$	−1.61373e12			−1.20419
$106$			2.54108e12i			1.79136i
$107$		−	2.65145e11i		−	0.176677i	−0.996090	−	0.0883387i	$-0.971844\pi$
$107$		−	2.65145e11i		−	0.176677i	0.996090	−	0.0883387i	$-0.0281558\pi$
$108$	1.47481e10			0.00929383
$109$			1.27223e12i			0.758587i	0.925276	+	0.379293i	$0.123833\pi$
$109$			1.27223e12i			0.758587i	−0.925276	+	0.379293i	$0.876167\pi$
$110$	−2.00321e12			−1.13076
$111$		−	4.34553e12i		−	2.32330i
$112$			2.79443e12i			1.41575i
$113$		−	1.05062e12i		−	0.504633i	−0.967645	−	0.252317i	$-0.918808\pi$
$113$		−	1.05062e12i		−	0.504633i	0.967645	−	0.252317i	$-0.0811924\pi$
$114$			1.04128e12i			0.474394i
$115$	1.37822e12	−	6.83880e11i	0.595842	−	0.295660i
$116$	−3.16563e11			−0.129931
$117$	4.03548e12			1.57319
$118$	−1.59972e12			−0.592587
$119$	6.19055e12			2.17996
$120$			2.61331e12i			0.875192i
$121$	−4.83209e12			−1.53965
$122$		−	1.37216e12i		−	0.416145i
$123$	−2.34408e12			−0.676927
$124$	6.98248e11			0.192079
$125$		−	3.95214e12i		−	1.03603i
$126$		−	5.65539e12i		−	1.41332i
$127$	−5.36235e12			−1.27801			−0.639003	−	0.769204i	$-0.720653\pi$
$127$	−5.36235e12			−1.27801			−0.639003	+	0.769204i	$0.720653\pi$
$128$	5.08316e12			1.15578
$129$			3.42163e12i			0.742497i
$130$		−	5.14560e12i		−	1.06605i
$131$	1.30374e12			0.257966			0.128983	−	0.991647i	$-0.458829\pi$
$131$	1.30374e12			0.257966			0.128983	+	0.991647i	$0.458829\pi$
$132$		−	1.66345e12i		−	0.314461i
$133$	2.17681e12			0.393288
$134$			4.03199e12i			0.696451i
$135$		−	2.71342e11i		−	0.0448243i
$136$		−	1.00251e13i		−	1.58437i
$137$		−	3.05378e11i		−	0.0461865i	−0.999733	−	0.0230932i	$-0.992649\pi$
$137$		−	3.05378e11i		−	0.0461865i	0.999733	−	0.0230932i	$-0.00735146\pi$
$138$	4.68558e12	+	9.44281e12i	0.678404	+	1.36718i
$139$	4.96100e12			0.687830			0.343915	−	0.939001i	$-0.388247\pi$
$139$	4.96100e12			0.687830			0.343915	+	0.939001i	$0.388247\pi$
$140$	−8.73982e11			−0.116074
$141$	−3.17127e12			−0.403570
$142$	1.28015e13			1.56146
$143$		−	2.04737e13i		−	2.39430i
$144$	−1.04460e13			−1.17159
$145$			5.82424e12i			0.626659i
$146$	−5.84872e11			−0.0603870
$147$	−8.67664e12			−0.859899
$148$		−	2.35351e12i		−	0.223947i
$149$			1.14195e13i			1.04359i	0.853072	+	0.521793i	$0.174737\pi$
$149$			1.14195e13i			1.04359i	−0.853072	+	0.521793i	$0.825263\pi$
$150$	9.69307e12			0.850969
$151$	2.62501e12			0.221447			0.110724	−	0.993851i	$-0.464683\pi$
$151$	2.62501e12			0.221447			0.110724	+	0.993851i	$0.464683\pi$
$152$		−	3.52518e12i		−	0.285838i
$153$			2.31413e13i			1.80401i
$154$	−2.86921e13			−2.15099
$155$		−	1.28466e13i		−	0.926401i
$156$	4.27286e12			0.296464
$157$		−	1.06529e13i		−	0.711328i	−0.934614	−	0.355664i	$-0.884255\pi$
$157$		−	1.06529e13i		−	0.711328i	0.934614	−	0.355664i	$-0.115745\pi$
$158$		−	2.24283e12i		−	0.144163i
$159$			3.88222e13i			2.40269i
$160$			3.05712e12i			0.182218i
$161$	1.97403e13	−	9.79526e12i	1.13344	−	0.562420i
$162$	−1.83308e13			−1.01412
$163$	5.13694e12			0.273892			0.136946	−	0.990579i	$-0.456271\pi$
$163$	5.13694e12			0.273892			0.136946	+	0.990579i	$0.456271\pi$
$164$	−1.26954e12			−0.0652503
$165$	−3.06047e13			−1.51665
$166$			1.50408e13i			0.718826i
$167$	7.47554e12			0.344623			0.172311	−	0.985043i	$-0.444876\pi$
$167$	7.47554e12			0.344623			0.172311	+	0.985043i	$0.444876\pi$
$168$			3.74306e13i			1.66483i
$169$	2.92921e13			1.25728
$170$	2.95072e13			1.22246
$171$			8.13728e12i			0.325464i
$172$			1.85313e12i			0.0715706i
$173$	−4.25146e13			−1.58585			−0.792925	−	0.609320i	$-0.791443\pi$
$173$	−4.25146e13			−1.58585			−0.792925	+	0.609320i	$0.791443\pi$
$174$	−3.99046e13			−1.43790
$175$		−	2.02635e13i		−	0.705482i
$176$			5.29971e13i			1.78310i
$177$	−2.44403e13			−0.794817
$178$		−	2.70587e13i		−	0.850719i
$179$	1.61669e13			0.491483			0.245741	−	0.969335i	$-0.420969\pi$
$179$	1.61669e13			0.491483			0.245741	+	0.969335i	$0.420969\pi$
$180$		−	3.26709e12i		−	0.0960563i
$181$			1.62096e13i			0.461001i	0.973072	+	0.230500i	$0.0740363\pi$
$181$			1.62096e13i			0.461001i	−0.973072	+	0.230500i	$0.925964\pi$
$182$		−	7.37007e13i		−	2.02789i
$183$		−	2.09636e13i		−	0.558161i
$184$	−1.58627e13	−	3.19680e13i	−0.408761	−	0.823774i
$185$	−4.33006e13			−1.08010
$186$	8.80182e13			2.12566
$187$	1.17405e14			2.74561
$188$	−1.71754e12			−0.0389008
$189$		−	3.88644e12i		−	0.0852671i
$190$	1.03757e13			0.220545
$191$		−	4.18020e13i		−	0.860987i	−0.902594	−	0.430493i	$-0.858340\pi$
$191$		−	4.18020e13i		−	0.860987i	0.902594	−	0.430493i	$-0.141660\pi$
$192$	5.92527e13			1.18277
$193$	−3.46212e13			−0.669882			−0.334941	−	0.942239i	$-0.608716\pi$
$193$	−3.46212e13			−0.669882			−0.334941	+	0.942239i	$0.608716\pi$
$194$			1.28337e13i			0.240735i
$195$		−	7.86136e13i		−	1.42985i
$196$	−4.69920e12			−0.0828872
$197$	−8.87746e13			−1.51877			−0.759384	−	0.650643i	$-0.774499\pi$
$197$	−8.87746e13			−1.51877			−0.759384	+	0.650643i	$0.774499\pi$
$198$		−	1.07256e14i		−	1.78004i
$199$		−	2.84774e13i		−	0.458545i	−0.973362	−	0.229272i	$-0.926365\pi$
$199$		−	2.84774e13i		−	0.458545i	0.973362	−	0.229272i	$-0.0736346\pi$
$200$	−3.28152e13			−0.512737
$201$			6.16001e13i			0.934125i
$202$	8.44559e13			1.24315
$203$			8.34210e13i			1.19206i
$204$			2.45026e13i			0.339962i
$205$			2.33574e13i			0.314703i
$206$			6.71661e13i			0.878915i
$207$	3.66163e13	+	7.37926e13i	0.465428	+	0.937974i
$208$	−1.36132e14			−1.68105
$209$	4.12838e13			0.495338
$210$	−1.10170e14			−1.28454
$211$	−8.26614e13			−0.936716			−0.468358	−	0.883539i	$-0.655154\pi$
$211$	−8.26614e13			−0.936716			−0.468358	+	0.883539i	$0.655154\pi$
$212$			2.10258e13i			0.231599i
$213$	1.95580e14			2.09433
$214$		−	1.81017e13i		−	0.188467i
$215$	3.40945e13			0.345186
$216$	−6.29380e12			−0.0619713
$217$		−	1.84003e14i		−	1.76225i
$218$			8.68558e13i			0.809208i
$219$	−8.93559e12			−0.0809950
$220$	−1.65753e13			−0.146193
$221$			3.01576e14i			2.58847i
$222$		−	2.96673e14i		−	2.47833i
$223$	−1.21766e14			−0.990138			−0.495069	−	0.868854i	$-0.664857\pi$
$223$	−1.21766e14			−0.990138			−0.495069	+	0.868854i	$0.664857\pi$
$224$			4.37873e13i			0.346625i
$225$	7.57482e13			0.583818
$226$		−	7.17268e13i		−	0.538308i
$227$			8.98798e13i			0.656912i	0.944519	+	0.328456i	$0.106528\pi$
$227$			8.98798e13i			0.656912i	−0.944519	+	0.328456i	$0.893472\pi$
$228$			8.61594e12i			0.0613329i
$229$		−	1.12193e13i		−	0.0777950i	−0.999243	−	0.0388975i	$-0.987615\pi$
$229$		−	1.12193e13i		−	0.0777950i	0.999243	−	0.0388975i	$-0.0123846\pi$
$230$	9.40921e13	−	4.66890e13i	0.635603	−	0.315390i
$231$	−4.38354e14			−2.88505
$232$	1.35094e14			0.866379
$233$	−1.82378e13			−0.113982			−0.0569912	−	0.998375i	$-0.518151\pi$
$233$	−1.82378e13			−0.113982			−0.0569912	+	0.998375i	$0.518151\pi$
$234$	2.75505e14			1.67817
$235$			3.15999e13i			0.187620i
$236$	−1.32367e13			−0.0766138
$237$		−	3.42657e13i		−	0.193361i
$238$	4.22634e14			2.32543
$239$	1.16580e14			0.625515			0.312757	−	0.949833i	$-0.398747\pi$
$239$	1.16580e14			0.625515			0.312757	+	0.949833i	$0.398747\pi$
$240$			2.03495e14i			1.06485i
$241$		−	2.11307e14i		−	1.07848i	−0.842152	−	0.539239i	$-0.818712\pi$
$241$		−	2.11307e14i		−	1.07848i	0.842152	−	0.539239i	$-0.181288\pi$
$242$	−3.29890e14			−1.64239
$243$	−2.93929e14			−1.42759
$244$		−	1.13537e13i		−	0.0538021i
$245$			8.64576e13i			0.399767i
$246$	−1.60032e14			−0.722099
$247$			1.06045e14i			0.466989i
$248$	−2.97979e14			−1.28078
$249$			2.29792e14i			0.964136i
$250$		−	2.69816e14i		−	1.10517i
$251$			3.26554e14i			1.30591i	0.757397	+	0.652954i	$0.226471\pi$
$251$			3.26554e14i			1.30591i	−0.757397	+	0.652954i	$0.773529\pi$
$252$		−	4.67947e13i		−	0.182723i
$253$	3.74380e14	−	1.85770e14i	1.42755	−	0.708356i
$254$	−3.66092e14			−1.36329
$255$	4.50807e14			1.63964
$256$	1.14345e14			0.406235
$257$	2.60324e14			0.903474			0.451737	−	0.892151i	$-0.350804\pi$
$257$	2.60324e14			0.903474			0.451737	+	0.892151i	$0.350804\pi$
$258$			2.33597e14i			0.792044i
$259$	−6.20198e14			−2.05462
$260$		−	4.25765e13i		−	0.137826i
$261$	−3.11842e14			−0.986486
$262$	8.90073e13			0.275181
$263$			6.05564e14i			1.82989i	0.403575	+	0.914946i	$0.367767\pi$
$263$			6.05564e14i			1.82989i	−0.403575	+	0.914946i	$0.632233\pi$
$264$			7.09881e14i			2.09682i
$265$	3.86840e14			1.11701
$266$	1.48613e14			0.419532
$267$		−	4.13398e14i		−	1.14104i
$268$			3.33621e13i			0.0900420i
$269$	−5.19806e14			−1.37192			−0.685958	−	0.727642i	$-0.740616\pi$
$269$	−5.19806e14			−1.37192			−0.685958	+	0.727642i	$0.740616\pi$
$270$		−	1.85247e13i		−	0.0478155i
$271$	−4.61862e12			−0.0116599			−0.00582997	−	0.999983i	$-0.501856\pi$
$271$	−4.61862e12			−0.0116599			−0.00582997	+	0.999983i	$0.501856\pi$
$272$		−	7.80645e14i		−	1.92770i
$273$		−	1.12599e15i		−	2.71993i
$274$		−	2.08484e13i		−	0.0492685i
$275$		−	3.84302e14i		−	0.888539i
$276$	3.87702e13	+	7.81333e13i	0.0877088	+	0.176759i
$277$	6.53677e14			1.44705			0.723527	−	0.690296i	$-0.242520\pi$
$277$	6.53677e14			1.44705			0.723527	+	0.690296i	$0.242520\pi$
$278$	3.38692e14			0.733729
$279$	6.87834e14			1.45834
$280$	3.72974e14			0.773980
$281$		−	2.82301e14i		−	0.573422i	−0.958017	−	0.286711i	$-0.907438\pi$
$281$		−	2.82301e14i		−	0.573422i	0.958017	−	0.286711i	$-0.0925619\pi$
$282$	−2.16505e14			−0.430501
$283$		−	7.44895e14i		−	1.45003i	−0.688734	−	0.725014i	$-0.741833\pi$
$283$		−	7.44895e14i		−	1.45003i	0.688734	−	0.725014i	$-0.258167\pi$
$284$	1.05924e14			0.201877
$285$	1.58519e14			0.295810
$286$		−	1.39775e15i		−	2.55408i
$287$			3.34549e14i			0.598644i
$288$	−1.63684e14			−0.286848
$289$	−1.14676e15			−1.96827
$290$			3.97625e14i			0.668477i
$291$			1.96071e14i			0.322890i
$292$	−4.83945e12			−0.00780726
$293$		−	2.59561e14i		−	0.410235i	−0.978737	−	0.205118i	$-0.934242\pi$
$293$		−	2.59561e14i		−	0.410235i	0.978737	−	0.205118i	$-0.0657577\pi$
$294$	−5.92361e14			−0.917281
$295$			2.43533e14i			0.369510i
$296$			1.00436e15i			1.49328i
$297$		−	7.37074e13i		−	0.107392i
$298$			7.79615e14i			1.11322i
$299$	4.77181e14	+	9.61661e14i	0.667815	+	1.34584i
$300$	8.02040e13			0.110019
$301$	4.88338e14			0.656631
$302$	1.79212e14			0.236224
$303$	1.29030e15			1.66739
$304$		−	2.74502e14i		−	0.347779i
$305$	−2.08890e14			−0.259489
$306$			1.57987e15i			1.92439i
$307$	−1.08519e15			−1.29621			−0.648104	−	0.761551i	$-0.724438\pi$
$307$	−1.08519e15			−1.29621			−0.648104	+	0.761551i	$0.724438\pi$
$308$	−2.37409e14			−0.278095
$309$			1.02615e15i			1.17886i
$310$		−	8.77049e14i		−	0.988220i
$311$	−2.98035e13			−0.0329386			−0.0164693	−	0.999864i	$-0.505243\pi$
$311$	−2.98035e13			−0.0329386			−0.0164693	+	0.999864i	$0.505243\pi$
$312$	−1.82345e15			−1.97682
$313$		−	1.71487e15i		−	1.82375i	−0.410473	−	0.911873i	$-0.634637\pi$
$313$		−	1.71487e15i		−	1.82375i	0.410473	−	0.911873i	$-0.365363\pi$
$314$		−	7.27281e14i		−	0.758795i
$315$	−8.60946e14			−0.881278
$316$		−	1.85580e13i		−	0.0186384i
$317$	−8.60728e14			−0.848224			−0.424112	−	0.905610i	$-0.639414\pi$
$317$	−8.60728e14			−0.848224			−0.424112	+	0.905610i	$0.639414\pi$
$318$			2.65042e15i			2.56302i
$319$			1.58210e15i			1.50138i
$320$		−	5.90418e14i		−	0.549870i
$321$		−	2.76554e14i		−	0.252784i
$322$	1.34769e15	−	6.68730e14i	1.20908	−	0.599950i
$323$	−6.08108e14			−0.535508
$324$	−1.51675e14			−0.131113
$325$	9.87147e14			0.837687
$326$	3.50703e14			0.292169
$327$			1.32697e15i			1.08536i
$328$	5.41777e14			0.435089
$329$			4.52607e14i			0.356899i
$330$	−2.08941e15			−1.61786
$331$	5.77668e14			0.439249			0.219624	−	0.975585i	$-0.429517\pi$
$331$	5.77668e14			0.439249			0.219624	+	0.975585i	$0.429517\pi$
$332$			1.24453e14i			0.0929348i
$333$		−	2.31840e15i		−	1.70029i
$334$	5.10361e14			0.367620
$335$	6.13808e14			0.434274
$336$			2.91468e15i			2.02560i
$337$			1.41805e15i			0.968083i	0.875045	+	0.484042i	$0.160832\pi$
$337$			1.41805e15i			0.968083i	−0.875045	+	0.484042i	$0.839168\pi$
$338$	1.99979e15			1.34117
$339$		−	1.09583e15i		−	0.722013i
$340$	2.44154e14			0.158048
$341$		−	3.48967e15i		−	2.21951i
$342$			5.55538e14i			0.347182i
$343$		−	8.22108e14i		−	0.504852i
$344$		−	7.90826e14i		−	0.477233i
$345$	1.43752e15	−	7.13307e14i	0.852512	−	0.423021i
$346$	−2.90251e15			−1.69167
$347$	−1.59311e15			−0.912574			−0.456287	−	0.889833i	$-0.650821\pi$
$347$	−1.59311e15			−0.912574			−0.456287	+	0.889833i	$0.650821\pi$
$348$	−3.30185e14			−0.185901
$349$	−1.56288e14			−0.0864912			−0.0432456	−	0.999064i	$-0.513770\pi$
$349$	−1.56288e14			−0.0864912			−0.0432456	+	0.999064i	$0.513770\pi$
$350$		−	1.38340e15i		−	0.752559i
$351$	1.89330e14			0.101246
$352$			8.30438e14i			0.436567i
$353$	−4.58896e14			−0.237173			−0.118587	−	0.992944i	$-0.537836\pi$
$353$	−4.58896e14			−0.237173			−0.118587	+	0.992944i	$0.537836\pi$
$354$	−1.66856e15			−0.847855
$355$		−	1.94884e15i		−	0.973655i
$356$		−	2.23893e14i		−	0.109987i
$357$	6.45694e15			3.11901
$358$	1.10373e15			0.524280
$359$		−	1.21776e14i		−	0.0568845i	−0.999595	−	0.0284423i	$-0.990945\pi$
$359$		−	1.21776e14i		−	0.0568845i	0.999595	−	0.0284423i	$-0.00905468\pi$
$360$			1.39424e15i			0.640504i
$361$	1.99948e15			0.903388
$362$			1.10664e15i			0.491764i
$363$	−5.04002e15			−2.20289
$364$		−	6.09827e14i		−	0.262179i
$365$			8.90378e13i			0.0376545i
$366$		−	1.43120e15i		−	0.595407i
$367$			3.18012e15i			1.30151i	0.759288	+	0.650755i	$0.225547\pi$
$367$			3.18012e15i			1.30151i	−0.759288	+	0.650755i	$0.774453\pi$
$368$	−1.23521e15	−	2.48931e15i	−0.497340	−	1.00229i
$369$	−1.25060e15			−0.495405
$370$	−2.95617e15			−1.15218
$371$	5.54073e15			2.12483
$372$	7.28294e14			0.274821
$373$		−	2.90513e15i		−	1.07873i	−0.842073	−	0.539364i	$-0.818665\pi$
$373$		−	2.90513e15i		−	1.07873i	0.842073	−	0.539364i	$-0.181335\pi$
$374$	8.01536e15			2.92882
$375$		−	4.12221e15i		−	1.48232i
$376$	7.32963e14			0.259391
$377$	−4.06390e15			−1.41545
$378$		−	2.65331e14i		−	0.0909571i
$379$			2.12577e15i			0.717266i	0.933479	+	0.358633i	$0.116757\pi$
$379$			2.12577e15i			0.717266i	−0.933479	+	0.358633i	$0.883243\pi$
$380$	8.58527e13			0.0285136
$381$	−5.59310e15			−1.82853
$382$		−	2.85385e15i		−	0.918441i
$383$			1.14258e15i			0.361988i	0.983484	+	0.180994i	$0.0579315\pi$
$383$			1.14258e15i			0.361988i	−0.983484	+	0.180994i	$0.942069\pi$
$384$	5.30189e15			1.65365
$385$			4.36794e15i			1.34126i
$386$	−2.36362e15			−0.714584
$387$			1.82549e15i			0.543392i
$388$			1.06190e14i			0.0311240i
$389$			2.76617e15i			0.798327i	0.916880	+	0.399164i	$0.130699\pi$
$389$			2.76617e15i			0.798327i	−0.916880	+	0.399164i	$0.869301\pi$
$390$		−	5.36701e15i		−	1.52526i
$391$	−5.51461e15	+	2.73638e15i	−1.54331	+	0.765800i
$392$	2.00539e15			0.552692
$393$	1.35984e15			0.369090
$394$	−6.06071e15			−1.62012
$395$	−3.41437e14			−0.0898934
$396$		−	8.87474e14i		−	0.230136i
$397$	3.95613e15			1.01048			0.505240	−	0.862979i	$-0.331404\pi$
$397$	3.95613e15			1.01048			0.505240	+	0.862979i	$0.331404\pi$
$398$		−	1.94417e15i		−	0.489144i
$399$	2.27048e15			0.562704
$400$	−2.55528e15			−0.623848
$401$			6.73560e15i			1.61998i	0.586442	+	0.809991i	$0.300528\pi$
$401$			6.73560e15i			1.61998i	−0.586442	+	0.809991i	$0.699472\pi$
$402$			4.20548e15i			0.996460i
$403$	8.96381e15			2.09249
$404$	6.98819e14			0.160722
$405$			2.79058e15i			0.632359i
$406$			5.69522e15i			1.27161i
$407$	−1.17622e16			−2.58775
$408$		−	1.04565e16i		−	2.26687i
$409$	−8.99006e15			−1.92054			−0.960269	−	0.279077i	$-0.909972\pi$
$409$	−8.99006e15			−1.92054			−0.960269	+	0.279077i	$0.909972\pi$
$410$			1.59463e15i			0.335704i
$411$		−	3.18519e14i		−	0.0660821i
$412$			5.55757e14i			0.113632i
$413$			3.48814e15i			0.702900i
$414$	2.49982e15	+	5.03788e15i	0.496486	+	1.00057i
$415$	2.28974e15			0.448226
$416$	−2.13312e15			−0.411582
$417$	5.17448e15			0.984125
$418$	2.81847e15			0.528392
$419$		−	4.96434e14i		−	0.0917439i	−0.998947	−	0.0458720i	$-0.985393\pi$
$419$		−	4.96434e14i		−	0.0917439i	0.998947	−	0.0458720i	$-0.0146066\pi$
$420$	−9.11590e14			−0.166075
$421$			8.76732e15i			1.57462i	0.616560	+	0.787308i	$0.288526\pi$
$421$			8.76732e15i			1.57462i	−0.616560	+	0.787308i	$0.711474\pi$
$422$	−5.64336e15			−0.999224
$423$	−1.69192e15			−0.295350
$424$		−	8.97280e15i		−	1.54430i
$425$			5.66076e15i			0.960596i
$426$	1.33524e16			2.23409
$427$	−2.99194e15			−0.493613
$428$		−	1.49780e14i		−	0.0243663i
$429$		−	2.13547e16i		−	3.42570i
$430$	2.32766e15			0.368221
$431$		−	9.41681e14i		−	0.146906i	−0.997299	−	0.0734531i	$-0.976598\pi$
$431$		−	9.41681e14i		−	0.146906i	0.997299	−	0.0734531i	$-0.0234019\pi$
$432$	−4.90091e14			−0.0754005
$433$		−	8.10712e15i		−	1.23010i	−0.788490	−	0.615048i	$-0.789137\pi$
$433$		−	8.10712e15i		−	1.23010i	0.788490	−	0.615048i	$-0.210863\pi$
$434$		−	1.25620e16i		−	1.87984i
$435$			6.07486e15i			0.896604i
$436$			7.18677e14i			0.104620i
$437$	−1.93912e15	+	9.62204e14i	−0.278431	+	0.138159i
$438$	−6.10039e14			−0.0863999
$439$	−7.63662e13			−0.0106688			−0.00533438	−	0.999986i	$-0.501698\pi$
$439$	−7.63662e13			−0.0106688			−0.00533438	+	0.999986i	$0.501698\pi$
$440$	7.07354e15			0.974812
$441$	−4.62911e15			−0.629312
$442$			2.05889e16i			2.76120i
$443$	7.46141e15			0.987184			0.493592	−	0.869693i	$-0.335684\pi$
$443$	7.46141e15			0.987184			0.493592	+	0.869693i	$0.335684\pi$
$444$		−	2.45478e15i		−	0.320416i
$445$	−4.11927e15			−0.530469
$446$	−8.31304e15			−1.05621
$447$			1.19108e16i			1.49313i
$448$		−	8.45660e15i		−	1.04599i
$449$	−1.89098e15			−0.230786			−0.115393	−	0.993320i	$-0.536813\pi$
$449$	−1.89098e15			−0.230786			−0.115393	+	0.993320i	$0.536813\pi$
$450$	5.17139e15			0.622777
$451$			6.34482e15i			0.753980i
$452$		−	5.93493e14i		−	0.0695962i
$453$	2.73797e15			0.316840
$454$			6.13617e15i			0.700748i
$455$	−1.12198e16			−1.26449
$456$		−	3.67687e15i		−	0.408968i
$457$			6.04805e14i			0.0663924i	0.999449	+	0.0331962i	$0.0105686\pi$
$457$			6.04805e14i			0.0663924i	−0.999449	+	0.0331962i	$0.989431\pi$
$458$		−	7.65949e14i		−	0.0829864i
$459$			1.08571e15i			0.116101i
$460$	7.78552e14	−	3.86322e14i	0.0821752	−	0.0407758i
$461$	1.76476e16			1.83857			0.919285	−	0.393592i	$-0.128768\pi$
$461$	1.76476e16			1.83857			0.919285	+	0.393592i	$0.128768\pi$
$462$	−2.99268e16			−3.07757
$463$	−4.81359e15			−0.488634			−0.244317	−	0.969695i	$-0.578564\pi$
$463$	−4.81359e15			−0.488634			−0.244317	+	0.969695i	$0.578564\pi$
$464$	1.05196e16			1.05412
$465$		−	1.33994e16i		−	1.32547i
$466$	−1.24511e15			−0.121589
$467$		−	1.02116e16i		−	0.984444i	−0.870470	−	0.492222i	$-0.836185\pi$
$467$		−	1.02116e16i		−	0.984444i	0.870470	−	0.492222i	$-0.163815\pi$
$468$	2.27963e15			0.216965
$469$	8.79162e15			0.826098
$470$			2.15735e15i			0.200139i
$471$		−	1.11113e16i		−	1.01775i
$472$	5.64878e15			0.510861
$473$	9.26145e15			0.827013
$474$		−	2.33934e15i		−	0.206264i
$475$			1.99052e15i			0.173302i
$476$	3.49703e15			0.300647
$477$			2.07122e16i			1.75839i
$478$	7.95902e15			0.667256
$479$			1.85008e16i			1.53171i	0.643012	+	0.765856i	$0.277685\pi$
$479$			1.85008e16i			1.53171i	−0.643012	+	0.765856i	$0.722315\pi$
$480$			3.18867e15i			0.260713i
$481$		−	3.02133e16i		−	2.43965i
$482$		−	1.44261e16i		−	1.15045i
$483$	2.05898e16	−	1.02168e16i	1.62169	−	0.804693i
$484$	−2.72963e15			−0.212340
$485$	1.95373e15			0.150111
$486$	−2.00668e16			−1.52286
$487$	−4.25652e14			−0.0319067			−0.0159533	−	0.999873i	$-0.505078\pi$
$487$	−4.25652e14			−0.0319067			−0.0159533	+	0.999873i	$0.505078\pi$
$488$			4.84523e15i			0.358753i
$489$	5.35799e15			0.391876
$490$			5.90252e15i			0.426443i
$491$	−1.10029e16			−0.785268			−0.392634	−	0.919695i	$-0.628436\pi$
$491$	−1.10029e16			−0.785268			−0.392634	+	0.919695i	$0.628436\pi$
$492$	−1.32417e15			−0.0933580
$493$		−	2.33043e16i		−	1.62313i
$494$			7.23974e15i			0.498152i
$495$	−1.63281e16			−1.10995
$496$	−2.32033e16			−1.55833
$497$		−	2.79133e16i		−	1.85214i
$498$			1.56881e16i			1.02847i
$499$	2.69464e15			0.174541			0.0872703	−	0.996185i	$-0.472186\pi$
$499$	2.69464e15			0.174541			0.0872703	+	0.996185i	$0.472186\pi$
$500$		−	2.23255e15i		−	0.142883i
$501$	7.79722e15			0.493075
$502$			2.22941e16i			1.39305i
$503$		−	6.41773e14i		−	0.0396254i	−0.999804	−	0.0198127i	$-0.993693\pi$
$503$		−	6.41773e14i		−	0.0396254i	0.999804	−	0.0198127i	$-0.00630699\pi$
$504$			1.99698e16i			1.21840i
$505$		−	1.28571e16i		−	0.775167i
$506$	2.55592e16	−	1.26826e16i	1.52281	−	0.755625i
$507$	3.05526e16			1.79887
$508$	−3.02918e15			−0.176255
$509$	−1.40574e16			−0.808347			−0.404174	−	0.914682i	$-0.632441\pi$
$509$	−1.40574e16			−0.808347			−0.404174	+	0.914682i	$0.632441\pi$
$510$	3.07769e16			1.74906
$511$			1.27530e15i			0.0716284i
$512$	−1.30142e16			−0.722434
$513$			3.81772e14i			0.0209459i
$514$	1.77725e16			0.963764
$515$	1.02250e16			0.548051
$516$			1.93287e15i			0.102401i
$517$			8.58381e15i			0.449507i
$518$	−4.23414e16			−2.19173
$519$	−4.43441e16			−2.26898
$520$			1.81696e16i			0.919022i
$521$		−	2.61349e15i		−	0.130676i	−0.997863	−	0.0653379i	$-0.979187\pi$
$521$		−	2.61349e15i		−	0.130676i	0.997863	−	0.0653379i	$-0.0208125\pi$
$522$	−2.12897e16			−1.05231
$523$			2.09357e16i			1.02300i	0.859282	+	0.511501i	$0.170911\pi$
$523$			2.09357e16i			1.02300i	−0.859282	+	0.511501i	$0.829089\pi$
$524$	7.36479e14			0.0355773
$525$		−	2.11354e16i		−	1.00938i
$526$			4.13423e16i			1.95200i
$527$			5.14026e16i			2.39951i
$528$			5.52776e16i			2.55121i
$529$	−1.32551e16	+	1.74514e16i	−0.604853	+	0.796337i
$530$	2.64099e16			1.19155
$531$	−1.30392e16			−0.581682
$532$	1.22967e15			0.0542401
$533$	−1.62978e16			−0.710829
$534$		−	2.82230e16i		−	1.21718i
$535$	−2.75570e15			−0.117519
$536$		−	1.42374e16i		−	0.600400i
$537$	1.68626e16			0.703199
$538$	−3.54875e16			−1.46346
$539$			2.34854e16i			0.957779i
$540$		−	1.53280e14i		−	0.00618192i
$541$	−1.29978e16			−0.518427			−0.259213	−	0.965820i	$-0.583463\pi$
$541$	−1.29978e16			−0.518427			−0.259213	+	0.965820i	$0.583463\pi$
$542$	−3.15317e14			−0.0124380
$543$			1.69071e16i			0.659586i
$544$		−	1.22323e16i		−	0.471971i
$545$	1.32225e16			0.504584
$546$		−	7.68721e16i		−	2.90144i
$547$	1.65930e16			0.619443			0.309722	−	0.950827i	$-0.399764\pi$
$547$	1.65930e16			0.619443			0.309722	+	0.950827i	$0.399764\pi$
$548$		−	1.72507e14i		−	0.00636978i
$549$		−	1.11844e16i		−	0.408487i
$550$		−	2.62366e16i		−	0.947832i
$551$		−	8.19459e15i		−	0.292831i
$552$	−1.65452e16	−	3.33435e16i	−0.584842	−	1.17863i
$553$	−4.89043e15			−0.171000
$554$	4.46271e16			1.54362
$555$	−4.51639e16			−1.54537
$556$	2.80246e15			0.0948616
$557$			1.57559e16i			0.527608i	0.964576	+	0.263804i	$0.0849772\pi$
$557$			1.57559e16i			0.527608i	−0.964576	+	0.263804i	$0.915023\pi$
$558$	4.69589e16			1.55565
$559$			2.37897e16i			0.779682i
$560$	2.90430e16			0.941703
$561$	1.22457e17			3.92833
$562$		−	1.92729e16i		−	0.611686i
$563$		−	3.07741e16i		−	0.966352i	−0.875523	−	0.483176i	$-0.839483\pi$
$563$		−	3.07741e16i		−	0.966352i	0.875523	−	0.483176i	$-0.160517\pi$
$564$	−1.79144e15			−0.0556581
$565$	−1.09193e16			−0.335664
$566$		−	5.08546e16i		−	1.54679i
$567$			3.99696e16i			1.20291i
$568$	−4.52035e16			−1.34611
$569$		−	5.68155e16i		−	1.67415i	−0.547091	−	0.837073i	$-0.684265\pi$
$569$		−	5.68155e16i		−	1.67415i	0.547091	−	0.837073i	$-0.315735\pi$
$570$	1.08222e16			0.315549
$571$		−	1.99809e16i		−	0.576498i	−0.957555	−	0.288249i	$-0.906927\pi$
$571$		−	1.99809e16i		−	0.576498i	0.957555	−	0.288249i	$-0.0930731\pi$
$572$		−	1.15655e16i		−	0.330209i
$573$		−	4.36007e16i		−	1.23187i
$574$			2.28399e16i			0.638592i
$575$	8.95696e15	+	1.80509e16i	0.247830	+	0.499450i
$576$	3.16121e16			0.865604
$577$	−3.15202e16			−0.854150			−0.427075	−	0.904216i	$-0.640456\pi$
$577$	−3.15202e16			−0.854150			−0.427075	+	0.904216i	$0.640456\pi$
$578$	−7.82899e16			−2.09961
$579$	−3.61110e16			−0.958446
$580$			3.29010e15i			0.0864253i
$581$	3.27961e16			0.852639
$582$			1.33859e16i			0.344437i
$583$	1.05081e17			2.67618
$584$	2.06524e15			0.0520588
$585$		−	4.19415e16i		−	1.04643i
$586$		−	1.77204e16i		−	0.437610i
$587$	−2.31996e16			−0.567091			−0.283545	−	0.958959i	$-0.591511\pi$
$587$	−2.31996e16			−0.567091			−0.283545	+	0.958959i	$0.591511\pi$
$588$	−4.90141e15			−0.118592
$589$			1.80749e16i			0.432897i
$590$			1.66262e16i			0.394167i
$591$	−9.25946e16			−2.17301
$592$			7.82086e16i			1.81687i
$593$	−2.09796e16			−0.482469			−0.241235	−	0.970467i	$-0.577552\pi$
$593$	−2.09796e16			−0.482469			−0.241235	+	0.970467i	$0.577552\pi$
$594$		−	5.03206e15i		−	0.114558i
$595$		−	6.43396e16i		−	1.45003i
$596$			6.45082e15i			0.143925i
$597$		−	2.97028e16i		−	0.656071i
$598$	3.25775e16	+	6.56533e16i	0.712379	+	1.43565i
$599$	−7.16137e15			−0.155037			−0.0775184	−	0.996991i	$-0.524700\pi$
$599$	−7.16137e15			−0.155037			−0.0775184	+	0.996991i	$0.524700\pi$
$600$	−3.42272e16			−0.733608
$601$	−1.11246e16			−0.236067			−0.118034	−	0.993010i	$-0.537659\pi$
$601$	−1.11246e16			−0.236067			−0.118034	+	0.993010i	$0.537659\pi$
$602$	3.33392e16			0.700448
$603$			3.28645e16i			0.683634i
$604$	1.48286e15			0.0305407
$605$			5.02208e16i			1.02412i
$606$	8.80901e16			1.77865
$607$	6.81381e15			0.136225			0.0681127	−	0.997678i	$-0.478302\pi$
$607$	6.81381e15			0.136225			0.0681127	+	0.997678i	$0.478302\pi$
$608$		−	4.30130e15i		−	0.0851488i
$609$			8.70107e16i			1.70557i
$610$	−1.42611e16			−0.276805
$611$	−2.20490e16			−0.423781
$612$			1.30725e16i			0.248799i
$613$		−	4.49195e16i		−	0.846588i	−0.905992	−	0.423294i	$-0.860874\pi$
$613$		−	4.49195e16i		−	0.846588i	0.905992	−	0.423294i	$-0.139126\pi$
$614$	−7.40867e16			−1.38271
$615$			2.43625e16i			0.450268i
$616$	1.01315e17			1.85434
$617$		−	7.06093e16i		−	1.27983i	−0.768448	−	0.639913i	$-0.778971\pi$
$617$		−	7.06093e16i		−	1.27983i	0.768448	−	0.639913i	$-0.221029\pi$
$618$			7.00563e16i			1.25752i
$619$			9.36746e15i			0.166525i	0.996528	+	0.0832623i	$0.0265339\pi$
$619$			9.36746e15i			0.166525i	−0.996528	+	0.0832623i	$0.973466\pi$
$620$		−	7.25702e15i		−	0.127764i
$621$	1.71790e15	+	3.46208e15i	0.0299536	+	0.0603654i
$622$	−2.03471e15			−0.0351366
$623$	−5.90006e16			−1.00909
$624$	−1.41990e17			−2.40520
$625$	−7.84233e15			−0.131572
$626$		−	1.17075e17i		−	1.94544i
$627$	4.30602e16			0.708714
$628$		−	6.01779e15i		−	0.0981023i
$629$	1.73257e17			2.79761
$630$	−5.87775e16			−0.940086
$631$			6.73533e16i			1.06705i	0.845786	+	0.533523i	$0.179132\pi$
$631$			6.73533e16i			1.06705i	−0.845786	+	0.533523i	$0.820868\pi$
$632$			7.91967e15i			0.124281i
$633$	−8.62183e16			−1.34022
$634$	−5.87626e16			−0.904826
$635$			5.57319e16i			0.850083i
$636$			2.19305e16i			0.331365i
$637$	−6.03263e16			−0.902964
$638$			1.08011e17i			1.60157i
$639$	1.04345e17			1.53273
$640$		−	5.28302e16i		−	0.768781i
$641$			3.52998e16i			0.508890i	0.967087	+	0.254445i	$0.0818928\pi$
$641$			3.52998e16i			0.508890i	−0.967087	+	0.254445i	$0.918107\pi$
$642$		−	1.88806e16i		−	0.269653i
$643$		−	1.09544e17i		−	1.54997i	−0.631981	−	0.774984i	$-0.717758\pi$
$643$		−	1.09544e17i		−	1.54997i	0.631981	−	0.774984i	$-0.282242\pi$
$644$	1.11513e16	−	5.53331e15i	0.156318	−	0.0775658i
$645$	3.55616e16			0.493882
$646$	−4.15160e16			−0.571243
$647$	1.05835e15			0.0144279			0.00721395	−	0.999974i	$-0.497704\pi$
$647$	1.05835e15			0.0144279			0.00721395	+	0.999974i	$0.497704\pi$
$648$	6.47278e16			0.874260
$649$			6.61535e16i			0.885288i
$650$	6.73933e16			0.893587
$651$		−	1.91921e17i		−	2.52137i
$652$	2.90184e15			0.0377736
$653$	−9.90505e16			−1.27755			−0.638775	−	0.769394i	$-0.720558\pi$
$653$	−9.90505e16			−1.27755			−0.638775	+	0.769394i	$0.720558\pi$
$654$			9.05933e16i			1.15779i
$655$		−	1.35500e16i		−	0.171590i
$656$	4.21876e16			0.529373
$657$	−4.76726e15			−0.0592757
$658$			3.08998e16i			0.380716i
$659$			4.65255e16i			0.568040i	0.958818	+	0.284020i	$0.0916682\pi$
$659$			4.65255e16i			0.568040i	−0.958818	+	0.284020i	$0.908332\pi$
$660$	−1.72885e16			−0.209168
$661$		−	8.43548e16i		−	1.01135i	−0.862724	−	0.505674i	$-0.831244\pi$
$661$		−	8.43548e16i		−	1.01135i	0.862724	−	0.505674i	$-0.168756\pi$
$662$	3.94378e16			0.468560
$663$			3.14553e17i			3.70351i
$664$		−	5.31108e16i		−	0.619689i
$665$		−	2.26240e16i		−	0.261601i
$666$		−	1.58279e17i		−	1.81375i
$667$	−3.68742e16	−	7.43123e16i	−0.418762	−	0.843928i
$668$	4.22291e15			0.0475284
$669$	−1.27005e17			−1.41666
$670$	4.19052e16			0.463254
$671$	−5.67430e16			−0.621695
$672$			4.56715e16i			0.495941i
$673$	8.29178e16			0.892396			0.446198	−	0.894934i	$-0.352778\pi$
$673$	8.29178e16			0.892396			0.446198	+	0.894934i	$0.352778\pi$
$674$			9.68115e16i			1.03268i
$675$	3.55384e15			0.0375729
$676$	1.65470e16			0.173396
$677$			9.46984e16i			0.983582i	0.870713	+	0.491791i	$0.163658\pi$
$677$			9.46984e16i			0.983582i	−0.870713	+	0.491791i	$0.836342\pi$
$678$		−	7.48132e16i		−	0.770194i
$679$	2.79834e16			0.285550
$680$	−1.04193e17			−1.05387
$681$			9.37474e16i			0.939889i
$682$		−	2.38242e17i		−	2.36762i
$683$	8.88743e16			0.875492			0.437746	−	0.899099i	$-0.355777\pi$
$683$	8.88743e16			0.875492			0.437746	+	0.899099i	$0.355777\pi$
$684$			4.59673e15i			0.0448861i
$685$	−3.17385e15			−0.0307216
$686$		−	5.61259e16i		−	0.538541i
$687$		−	1.17020e16i		−	0.111307i
$688$		−	6.15807e16i		−	0.580649i
$689$			2.69920e17i			2.52301i
$690$	9.81409e16	−	4.86980e16i	0.909401	−	0.451250i
$691$	1.96285e17			1.80309			0.901546	−	0.432684i	$-0.142433\pi$
$691$	1.96285e17			1.80309			0.901546	+	0.432684i	$0.142433\pi$
$692$	−2.40164e16			−0.218711
$693$	−2.33868e17			−2.11140
$694$	−1.08763e17			−0.973471
$695$		−	5.15606e16i		−	0.457519i
$696$	1.40907e17			1.23959
$697$		−	9.34589e16i		−	0.815125i
$698$	−1.06699e16			−0.0922628
$699$	−1.90226e16			−0.163083
$700$		−	1.14468e16i		−	0.0972961i
$701$		−	4.43313e16i		−	0.373596i	−0.982398	−	0.186798i	$-0.940189\pi$
$701$		−	4.43313e16i		−	0.373596i	0.982398	−	0.186798i	$-0.0598110\pi$
$702$	1.29257e16			0.108002
$703$	6.09231e16			0.504719
$704$		−	1.60381e17i		−	1.31740i
$705$			3.29596e16i			0.268440i
$706$	−3.13292e16			−0.253000
$707$		−	1.84153e17i		−	1.47456i
$708$	−1.38063e16			−0.109617
$709$		−	1.54769e17i		−	1.21845i	−0.792999	−	0.609224i	$-0.791481\pi$
$709$		−	1.54769e17i		−	1.21845i	0.792999	−	0.609224i	$-0.208519\pi$
$710$		−	1.33048e17i		−	1.03863i
$711$		−	1.82812e16i		−	0.141510i
$712$			9.55469e16i			0.733393i
$713$	8.13339e16	+	1.63912e17i	0.619063	+	1.24759i
$714$	4.40820e17			3.32715
$715$	−2.12787e17			−1.59260
$716$	9.13264e15			0.0677826
$717$	1.21597e17			0.894967
$718$		−	8.31372e15i		−	0.0606805i
$719$	−2.58319e17			−1.86975			−0.934875	−	0.354977i	$-0.884489\pi$
$719$	−2.58319e17			−1.86975			−0.934875	+	0.354977i	$0.884489\pi$
$720$			1.08568e17i			0.779301i
$721$	1.46454e17			1.04253
$722$	1.36506e17			0.963672
$723$		−	2.20400e17i		−	1.54305i
$724$			9.15678e15i			0.0635786i
$725$	−7.62817e16			−0.525282
$726$	−3.44086e17			−2.34989
$727$			9.68333e16i			0.655871i	0.944700	+	0.327936i	$0.106353\pi$
$727$			9.68333e16i			0.655871i	−0.944700	+	0.327936i	$0.893647\pi$
$728$			2.60245e17i			1.74821i
$729$	−1.63885e17			−1.09188
$730$			6.07868e15i			0.0401673i
$731$	−1.36421e17			−0.894080
$732$		−	1.18423e16i		−	0.0769784i
$733$			1.48795e17i			0.959320i	0.877454	+	0.479660i	$0.159240\pi$
$733$			1.48795e17i			0.959320i	−0.877454	+	0.479660i	$0.840760\pi$
$734$			2.17109e17i			1.38836i
$735$			9.01779e16i			0.571974i
$736$	−1.93551e16	−	3.90062e16i	−0.121767	−	0.245396i
$737$	1.66735e17			1.04045
$738$	−8.53795e16			−0.528464
$739$	1.15348e17			0.708179			0.354089	−	0.935212i	$-0.384791\pi$
$739$	1.15348e17			0.708179			0.354089	+	0.935212i	$0.384791\pi$
$740$	−2.44604e16			−0.148961
$741$			1.10608e17i			0.668153i
$742$	3.78270e17			2.26662
$743$		−	1.53935e17i		−	0.914965i	−0.889219	−	0.457482i	$-0.848751\pi$
$743$		−	1.53935e17i		−	0.914965i	0.889219	−	0.457482i	$-0.151249\pi$
$744$	−3.10801e17			−1.83250
$745$	1.18685e17			0.694155
$746$		−	1.98335e17i		−	1.15071i
$747$			1.22597e17i			0.705597i
$748$	6.63220e16			0.378659
$749$	−3.94701e16			−0.223551
$750$		−	2.81426e17i		−	1.58124i
$751$		−	3.11798e17i		−	1.73793i	−0.494870	−	0.868967i	$-0.664784\pi$
$751$		−	3.11798e17i		−	1.73793i	0.494870	−	0.868967i	$-0.335216\pi$
$752$	5.70749e16			0.315601
$753$			3.40606e17i			1.86845i
$754$	−2.77446e17			−1.50991
$755$		−	2.72823e16i		−	0.147299i
$756$		−	2.19544e15i		−	0.0117596i
$757$			1.83666e17i			0.976010i	0.872841	+	0.488005i	$0.162275\pi$
$757$			1.83666e17i			0.976010i	−0.872841	+	0.488005i	$0.837725\pi$
$758$			1.45128e17i			0.765130i
$759$	3.90490e17	−	1.93763e17i	2.04249	−	1.01349i
$760$	−3.66378e16			−0.190129
$761$	−2.77886e17			−1.43073			−0.715366	−	0.698750i	$-0.753740\pi$
$761$	−2.77886e17			−1.43073			−0.715366	+	0.698750i	$0.753740\pi$
$762$	−3.81845e17			−1.95055
$763$	1.89386e17			0.959846
$764$		−	2.36138e16i		−	0.118742i
$765$	2.40512e17			1.19996
$766$			7.80049e16i			0.386144i
$767$	−1.69927e17			−0.834622
$768$	1.19265e17			0.581229
$769$		−	2.50946e17i		−	1.21345i	−0.794912	−	0.606724i	$-0.792483\pi$
$769$		−	2.50946e17i		−	1.21345i	0.794912	−	0.606724i	$-0.207517\pi$
$770$			2.98202e17i			1.43076i
$771$	2.71526e17			1.29266
$772$	−1.95574e16			−0.0923863
$773$		−	3.12437e16i		−	0.146449i	−0.997315	−	0.0732244i	$-0.976671\pi$
$773$		−	3.12437e16i		−	0.146449i	0.997315	−	0.0732244i	$-0.0233289\pi$
$774$			1.24627e17i			0.579653i
$775$	1.68256e17			0.776533
$776$		−	4.53170e16i		−	0.207535i
$777$	−6.46885e17			−2.93969
$778$			1.88848e17i			0.851600i
$779$		−	3.28634e16i		−	0.147057i
$780$		−	4.44086e16i		−	0.197197i
$781$		−	5.29383e17i		−	2.33273i
$782$	−3.76487e17	+	1.86815e17i	−1.64630	+	0.816903i
$783$	−1.46305e16			−0.0634875
$784$	1.56158e17			0.672461
$785$	−1.10717e17			−0.473149
$786$	9.28373e16			0.393720
$787$			2.82661e16i			0.118965i	0.998229	+	0.0594823i	$0.0189450\pi$
$787$			2.82661e16i			0.118965i	−0.998229	+	0.0594823i	$0.981055\pi$
$788$	−5.01485e16			−0.209460
$789$			6.31622e17i			2.61815i
$790$	−2.33102e16			−0.0958921
$791$	−1.56398e17			−0.638516
$792$			3.78731e17i			1.53455i
$793$		−	1.45754e17i		−	0.586114i
$794$	2.70088e17			1.07791
$795$	4.03486e17			1.59818
$796$		−	1.60868e16i		−	0.0632399i
$797$			3.74064e17i			1.45947i	0.683728	+	0.729737i	$0.260357\pi$
$797$			3.74064e17i			1.45947i	−0.683728	+	0.729737i	$0.739643\pi$
$798$	1.55007e17			0.600254
$799$		−	1.26439e17i		−	0.485960i
$800$	−4.00399e16			−0.152740
$801$		−	2.20554e17i		−	0.835063i
$802$			4.59845e17i			1.72809i
$803$			2.41863e16i			0.0902144i
$804$			3.47977e16i			0.128829i
$805$	−1.01804e17	−	2.05165e17i	−0.374101	−	0.753924i
$806$	6.11966e17			2.23212
$807$	−5.42173e17			−1.96289
$808$	−2.98222e17			−1.07170
$809$	5.32808e17			1.90055			0.950276	−	0.311409i	$-0.100801\pi$
$809$	5.32808e17			1.90055			0.950276	+	0.311409i	$0.100801\pi$
$810$			1.90515e17i			0.674557i
$811$	9.72580e16			0.341822			0.170911	−	0.985286i	$-0.445329\pi$
$811$	9.72580e16			0.341822			0.170911	+	0.985286i	$0.445329\pi$
$812$			4.71243e16i			0.164403i
$813$	−4.81736e15			−0.0166827
$814$	−8.03015e17			−2.76043
$815$		−	5.33892e16i		−	0.182183i
$816$		−	8.14237e17i		−	2.75810i
$817$	−4.79702e16			−0.161302
$818$	−6.13758e17			−2.04870
$819$		−	6.00731e17i		−	1.99057i
$820$			1.31945e16i			0.0434021i
$821$	5.12832e17			1.67462			0.837310	−	0.546729i	$-0.184127\pi$
$821$	5.12832e17			1.67462			0.837310	+	0.546729i	$0.184127\pi$
$822$		−	2.17455e16i		−	0.0704919i
$823$	−3.04844e17			−0.981021			−0.490510	−	0.871435i	$-0.663190\pi$
$823$	−3.04844e17			−0.981021			−0.490510	+	0.871435i	$0.663190\pi$
$824$		−	2.37170e17i		−	0.757700i
$825$		−	4.00839e17i		−	1.27129i
$826$			2.38138e17i			0.749805i
$827$			8.84378e16i			0.276443i	0.990401	+	0.138221i	$0.0441385\pi$
$827$			8.84378e16i			0.276443i	−0.990401	+	0.138221i	$0.955861\pi$
$828$	2.06844e16	+	4.16852e16i	0.0641892	+	0.129360i
$829$	−3.20991e17			−0.988932			−0.494466	−	0.869197i	$-0.664636\pi$
$829$	−3.20991e17			−0.988932			−0.494466	+	0.869197i	$0.664636\pi$
$830$	1.56322e17			0.478137
$831$	6.81805e17			2.07040
$832$	4.11968e17			1.24201
$833$		−	3.45939e17i		−	1.03545i
$834$	3.53266e17			1.04980
$835$		−	7.76947e16i		−	0.229230i
$836$	2.33211e16			0.0683142
$837$	3.22707e16			0.0938546
$838$		−	3.38919e16i		−	0.0978660i
$839$			4.26539e17i			1.22289i	0.791287	+	0.611445i	$0.209411\pi$
$839$			4.26539e17i			1.22289i	−0.791287	+	0.611445i	$0.790589\pi$
$840$	3.89023e17			1.10739
$841$	−3.97770e16			−0.112423
$842$			5.98552e17i			1.67969i
$843$		−	2.94448e17i		−	0.820434i
$844$	−4.66952e16			−0.129187
$845$		−	3.04438e17i		−	0.836294i
$846$	−1.15509e17			−0.315059
$847$			7.19316e17i			1.94813i
$848$		−	6.98701e17i		−	1.87896i
$849$		−	7.76948e17i		−	2.07466i
$850$			3.86464e17i			1.02470i
$851$	5.52479e17	−	2.74143e17i	1.45458	−	0.721771i
$852$	1.10482e17			0.288839
$853$	−3.38388e17			−0.878458			−0.439229	−	0.898375i	$-0.644748\pi$
$853$	−3.38388e17			−0.878458			−0.439229	+	0.898375i	$0.644748\pi$
$854$	−2.04262e17			−0.526552
$855$	8.45722e16			0.216487
$856$			6.39188e16i			0.162475i
$857$	9.27814e16			0.234194			0.117097	−	0.993120i	$-0.462641\pi$
$857$	9.27814e16			0.234194			0.117097	+	0.993120i	$0.462641\pi$
$858$		−	1.45790e18i		−	3.65430i
$859$	4.82803e17			1.20174			0.600870	−	0.799347i	$-0.294821\pi$
$859$	4.82803e17			1.20174			0.600870	+	0.799347i	$0.294821\pi$
$860$	1.92599e16			0.0476062
$861$			3.48945e17i			0.856522i
$862$		−	6.42893e16i		−	0.156709i
$863$	4.52375e17			1.09505			0.547525	−	0.836790i	$-0.315570\pi$
$863$	4.52375e17			1.09505			0.547525	+	0.836790i	$0.315570\pi$
$864$	−7.67948e15			−0.0184607
$865$			4.41862e17i			1.05485i
$866$		−	5.53479e17i		−	1.31218i
$867$	−1.19610e18			−2.81613
$868$		−	1.03943e17i		−	0.243039i
$869$	−9.27482e16			−0.215371
$870$			4.14736e17i			0.956436i
$871$			4.28289e17i			0.980907i
$872$		−	3.06697e17i		−	0.697606i
$873$			1.04607e17i			0.236305i
$874$	−1.32386e17	+	6.56904e16i	−0.297011	+	0.147378i
$875$	−5.88325e17			−1.31090
$876$	−5.04769e15			−0.0111704
$877$	2.48136e17			0.545370			0.272685	−	0.962103i	$-0.412088\pi$
$877$	2.48136e17			0.545370			0.272685	+	0.962103i	$0.412088\pi$
$878$	−5.21358e15			−0.0113807
$879$		−	2.70730e17i		−	0.586952i
$880$	5.50808e17			1.18605
$881$			2.86246e17i			0.612187i	0.952001	+	0.306094i	$0.0990221\pi$
$881$			2.86246e17i			0.612187i	−0.952001	+	0.306094i	$0.900978\pi$
$882$	−3.16033e17			−0.671306
$883$	−3.17828e17			−0.670546			−0.335273	−	0.942121i	$-0.608829\pi$
$883$	−3.17828e17			−0.670546			−0.335273	+	0.942121i	$0.608829\pi$
$884$			1.70360e17i			0.356988i
$885$			2.54012e17i			0.528683i
$886$	5.09396e17			1.05306
$887$	−7.03567e17			−1.44465			−0.722326	−	0.691552i	$-0.756927\pi$
$887$	−7.03567e17			−1.44465			−0.722326	+	0.691552i	$0.756927\pi$
$888$			1.04758e18i			2.13654i
$889$			7.98252e17i			1.61707i
$890$	−2.81226e17			−0.565868
$891$			7.58034e17i			1.51503i
$892$	−6.87851e16			−0.136554
$893$		−	4.44603e16i		−	0.0876726i
$894$			8.13163e17i			1.59277i
$895$		−	1.68025e17i		−	0.326917i
$896$		−	7.56691e17i		−	1.46241i
$897$	4.97715e17	+	1.00304e18i	0.955490	+	1.92559i
$898$	−1.29099e17			−0.246186
$899$	−6.92678e17			−1.31212
$900$	4.27900e16			0.0805169
$901$	−1.54785e18			−2.89320
$902$			4.33165e17i			0.804294i
$903$	5.09351e17			0.939487
$904$			2.53275e17i			0.464067i
$905$	1.68470e17			0.306641
$906$	1.86923e17			0.337983
$907$			9.60564e17i			1.72537i	0.505740	+	0.862686i	$0.331220\pi$
$907$			9.60564e17i			1.72537i	−0.505740	+	0.862686i	$0.668780\pi$
$908$			5.07729e16i			0.0905976i
$909$	6.88396e17			1.22027
$910$	−7.65985e17			−1.34888
$911$		−	9.91135e17i		−	1.73389i	−0.498400	−	0.866947i	$-0.666079\pi$
$911$		−	9.91135e17i		−	1.73389i	0.498400	−	0.866947i	$-0.333921\pi$
$912$		−	2.86314e17i		−	0.497591i
$913$	6.21986e17			1.07388
$914$			4.12905e16i			0.0708228i
$915$	−2.17879e17			−0.371268
$916$		−	6.33774e15i		−	0.0107291i
$917$		−	1.94078e17i		−	0.326407i
$918$			7.41221e16i			0.123849i
$919$			7.32888e16i			0.121659i	0.998148	+	0.0608296i	$0.0193746\pi$
$919$			7.32888e16i			0.121659i	−0.998148	+	0.0608296i	$0.980625\pi$
$920$	−3.32249e17	+	1.64864e17i	−0.547944	+	0.271893i
$921$	−1.13188e18			−1.85458
$922$	1.20482e18			1.96126
$923$	1.35981e18			2.19922
$924$	−2.47625e17			−0.397889
$925$		−	5.67121e17i		−	0.905368i
$926$	−3.28628e17			−0.521240
$927$			5.47468e17i			0.862740i
$928$	1.64837e17			0.258087
$929$	5.97529e17			0.929533			0.464767	−	0.885433i	$-0.346138\pi$
$929$	5.97529e17			0.929533			0.464767	+	0.885433i	$0.346138\pi$
$930$		−	9.14789e17i		−	1.41391i
$931$		−	1.21644e17i		−	0.186807i
$932$	−1.03025e16			−0.0157198
$933$	−3.10860e16			−0.0471275
$934$		−	6.97152e17i		−	1.05014i
$935$		−	1.22022e18i		−	1.82628i
$936$	−9.72838e17			−1.44672
$937$			2.98912e15i			0.00441678i	0.999998	+	0.00220839i	$0.000702953\pi$
$937$			2.98912e15i			0.00441678i	−0.999998	+	0.00220839i	$0.999297\pi$
$938$	6.00211e17			0.881225
$939$		−	1.78866e18i		−	2.60936i
$940$			1.78507e16i			0.0258754i
$941$			1.06570e18i			1.53497i	0.641069	+	0.767484i	$0.278491\pi$
$941$			1.06570e18i			1.53497i	−0.641069	+	0.767484i	$0.721509\pi$
$942$		−	7.58576e17i		−	1.08566i
$943$	−1.47879e17	−	2.98020e17i	−0.210299	−	0.423814i
$944$	4.39864e17			0.621565
$945$	−4.03925e16			−0.0567166
$946$	6.32286e17			0.882200
$947$	1.07304e18			1.48770			0.743852	−	0.668344i	$-0.232997\pi$
$947$	1.07304e18			1.48770			0.743852	+	0.668344i	$0.232997\pi$
$948$		−	1.93566e16i		−	0.0266673i
$949$	−6.21267e16			−0.0850513
$950$			1.35894e17i			0.184867i
$951$	−8.97766e17			−1.21361
$952$	−1.49236e18			−2.00472
$953$		−	7.79194e17i		−	1.04013i	−0.854126	−	0.520065i	$-0.825908\pi$
$953$		−	7.79194e17i		−	1.04013i	0.854126	−	0.520065i	$-0.174092\pi$
$954$			1.41404e18i			1.87573i
$955$	−4.34455e17			−0.572697
$956$	6.58559e16			0.0862675
$957$			1.65018e18i			2.14813i
$958$			1.26306e18i			1.63393i
$959$	−4.54593e16			−0.0584401
$960$		−	6.15824e17i		−	0.786736i
$961$	7.40188e17			0.939727
$962$		−	2.06269e18i		−	2.60245i
$963$		−	1.47546e17i		−	0.184999i
$964$		−	1.19367e17i		−	0.148738i
$965$			3.59824e17i			0.445581i
$966$	1.40568e18	−	6.97505e17i	1.72991	−	0.858390i
$967$	−1.09103e17			−0.133437			−0.0667186	−	0.997772i	$-0.521253\pi$
$967$	−1.09103e17			−0.133437			−0.0667186	+	0.997772i	$0.521253\pi$
$968$	1.16488e18			1.41588
$969$	−6.34276e17			−0.766188
$970$	1.33383e17			0.160128
$971$			1.59391e18i			1.90173i	0.309611	+	0.950863i	$0.399801\pi$
$971$			1.59391e18i			1.90173i	−0.309611	+	0.950863i	$0.600199\pi$
$972$	−1.66040e17			−0.196886
$973$		−	7.38506e17i		−	0.870316i
$974$	−2.90596e16			−0.0340358
$975$	1.02962e18			1.19854
$976$			3.77292e17i			0.436495i
$977$		−	8.34968e17i		−	0.960069i	−0.877250	−	0.480035i	$-0.840624\pi$
$977$		−	8.34968e17i		−	0.960069i	0.877250	−	0.480035i	$-0.159376\pi$
$978$	3.65794e17			0.418026
$979$	−1.11896e18			−1.27092
$980$			4.88396e16i			0.0551336i
$981$			7.07958e17i			0.794316i
$982$	−7.51176e17			−0.837669
$983$			1.31816e17i			0.146099i	0.997328	+	0.0730495i	$0.0232731\pi$
$983$			1.31816e17i			0.146099i	−0.997328	+	0.0730495i	$0.976727\pi$
$984$	5.65090e17			0.622511
$985$			9.22650e17i			1.01023i
$986$		−	1.59100e18i		−	1.73145i
$987$			4.72083e17i			0.510641i
$988$			5.99043e16i			0.0644045i
$989$	−4.35016e17	+	2.15857e17i	−0.464866	+	0.230669i
$990$	−1.11473e18			−1.18402
$991$	−5.22789e17			−0.551931			−0.275965	−	0.961168i	$-0.588997\pi$
$991$	−5.22789e17			−0.551931			−0.275965	+	0.961168i	$0.588997\pi$
$992$	−3.63584e17			−0.381535
$993$	6.02525e17			0.628463
$994$		−	1.90566e18i		−	1.97573i
$995$	−2.95971e17			−0.305007
$996$			1.29809e17i			0.132968i
$997$	1.47350e18			1.50031			0.750153	−	0.661265i	$-0.229980\pi$
$997$	1.47350e18			1.50031			0.750153	+	0.661265i	$0.229980\pi$
$998$	1.83965e17			0.186188
$999$		−	1.08771e17i		−	0.109426i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.c.22.15	✓	20
23.22	odd	2	inner	23.13.b.c.22.16	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.c.22.15	✓	20	1.1	even	1	trivial
23.13.b.c.22.16	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)

\(f(q)\)	\(=\)	\(q+68.2708 q^{2} +1043.03 q^{3} +564.897 q^{4} -10393.2i q^{5} +71208.5 q^{6} -148862. i q^{7} -241071. q^{8} +556472. q^{9} +O(q^{10})\)	\(q+68.2708 q^{2} +1043.03 q^{3} +564.897 q^{4} -10393.2i q^{5} +71208.5 q^{6} -148862. i q^{7} -241071. q^{8} +556472. q^{9} -709550. i q^{10} -2.82321e6i q^{11} +589205. q^{12} +7.25191e6 q^{13} -1.01629e7i q^{14} -1.08404e7i q^{15} -1.87719e7 q^{16} +4.15858e7i q^{17} +3.79907e7 q^{18} +1.46230e7i q^{19} -5.87108e6i q^{20} -1.55268e8i q^{21} -1.92743e8i q^{22} +(6.58008e7 + 1.32608e8i) q^{23} -2.51444e8 q^{24} +1.36122e8 q^{25} +4.95093e8 q^{26} +2.61077e7 q^{27} -8.40919e7i q^{28} -5.60391e8 q^{29} -7.40083e8i q^{30} +1.23606e9 q^{31} -2.94147e8 q^{32} -2.94469e9i q^{33} +2.83909e9i q^{34} -1.54715e9 q^{35} +3.14349e8 q^{36} -4.16625e9i q^{37} +9.98322e8i q^{38} +7.56396e9 q^{39} +2.50550e9i q^{40} -2.24738e9 q^{41} -1.06003e10i q^{42} +3.28047e9i q^{43} -1.59482e9i q^{44} -5.78351e9i q^{45} +(4.49227e9 + 9.05325e9i) q^{46} -3.04044e9 q^{47} -1.95797e10 q^{48} -8.31868e9 q^{49} +9.29318e9 q^{50} +4.33752e10i q^{51} +4.09658e9 q^{52} +3.72205e10i q^{53} +1.78239e9 q^{54} -2.93421e10 q^{55} +3.58864e10i q^{56} +1.52522e10i q^{57} -3.82583e10 q^{58} -2.34320e10 q^{59} -6.12372e9i q^{60} -2.00987e10i q^{61} +8.43869e10 q^{62} -8.28376e10i q^{63} +5.68082e10 q^{64} -7.53704e10i q^{65} -2.01037e11i q^{66} +5.90588e10i q^{67} +2.34917e10i q^{68} +(6.86322e10 + 1.38314e11i) q^{69} -1.05625e11 q^{70} +1.87511e11 q^{71} -1.34149e11 q^{72} -8.56695e9 q^{73} -2.84433e11i q^{74} +1.41980e11 q^{75} +8.26049e9i q^{76} -4.20269e11 q^{77} +5.16398e11 q^{78} -3.28520e10i q^{79} +1.95100e11i q^{80} -2.68501e11 q^{81} -1.53430e11 q^{82} +2.20312e11i q^{83} -8.77104e10i q^{84} +4.32209e11 q^{85} +2.23960e11i q^{86} -5.84505e11 q^{87} +6.80594e11i q^{88} -3.96343e11i q^{89} -3.94845e11i q^{90} -1.07954e12i q^{91} +(3.71707e10 + 7.49099e10i) q^{92} +1.28925e12 q^{93} -2.07573e11 q^{94} +1.51979e11 q^{95} -3.06804e11 q^{96} +1.87982e11i q^{97} -5.67923e11 q^{98} -1.57104e12i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more