LMFDB - Embedded newform 23.13.b.c.22.20

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			22.20
Root			$-701.372 - 26263.6i$ of defining polynomial
Character	$\chi$	$=$	23.22
Dual form			23.13.b.c.22.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+117.945 q^{2} +831.317 q^{3} +9814.93 q^{4} +26263.6i q^{5} +98049.3 q^{6} -167269. i q^{7} +674517. q^{8} +159646. q^{9} +O(q^{10})$	\(q+117.945 q^{2} +831.317 q^{3} +9814.93 q^{4} +26263.6i q^{5} +98049.3 q^{6} -167269. i q^{7} +674517. q^{8} +159646. q^{9} +3.09765e6i q^{10} +728242. i q^{11} +8.15931e6 q^{12} -5.83177e6 q^{13} -1.97285e7i q^{14} +2.18333e7i q^{15} +3.93536e7 q^{16} +1.39261e7i q^{17} +1.88294e7 q^{18} -8.30706e7i q^{19} +2.57775e8i q^{20} -1.39054e8i q^{21} +8.58922e7i q^{22} +(1.37054e8 - 5.59538e7i) q^{23} +5.60737e8 q^{24} -4.45635e8 q^{25} -6.87825e8 q^{26} -3.09079e8 q^{27} -1.64174e9i q^{28} +6.50000e8 q^{29} +2.57513e9i q^{30} +4.47244e7 q^{31} +1.87873e9 q^{32} +6.05399e8i q^{33} +1.64251e9i q^{34} +4.39309e9 q^{35} +1.56692e9 q^{36} +1.55826e8i q^{37} -9.79773e9i q^{38} -4.84804e9 q^{39} +1.77152e10i q^{40} -9.18670e7 q^{41} -1.64006e10i q^{42} +1.58410e9i q^{43} +7.14764e9i q^{44} +4.19288e9i q^{45} +(1.61648e10 - 6.59944e9i) q^{46} -4.29921e9 q^{47} +3.27153e10 q^{48} -1.41378e10 q^{49} -5.25602e10 q^{50} +1.15770e10i q^{51} -5.72384e10 q^{52} -1.15006e10i q^{53} -3.64542e10 q^{54} -1.91262e10 q^{55} -1.12826e11i q^{56} -6.90580e10i q^{57} +7.66640e10 q^{58} -4.07090e10 q^{59} +2.14293e11i q^{60} +2.44060e10i q^{61} +5.27500e9 q^{62} -2.67039e10i q^{63} +6.03934e10 q^{64} -1.53163e11i q^{65} +7.14036e10i q^{66} +1.58761e11i q^{67} +1.36684e11i q^{68} +(1.13935e11 - 4.65153e10i) q^{69} +5.18142e11 q^{70} -9.47198e10 q^{71} +1.07684e11 q^{72} +9.19442e10 q^{73} +1.83788e10i q^{74} -3.70464e11 q^{75} -8.15332e11i q^{76} +1.21813e11 q^{77} -5.71801e11 q^{78} +3.66644e11i q^{79} +1.03357e12i q^{80} -3.41785e11 q^{81} -1.08352e10 q^{82} -1.44546e11i q^{83} -1.36480e12i q^{84} -3.65750e11 q^{85} +1.86836e11i q^{86} +5.40356e11 q^{87} +4.91211e11i q^{88} -3.73758e11i q^{89} +4.94527e11i q^{90} +9.75476e11i q^{91} +(1.34518e12 - 5.49182e11i) q^{92} +3.71801e10 q^{93} -5.07069e11 q^{94} +2.18173e12 q^{95} +1.56182e12 q^{96} -9.37374e11i q^{97} -1.66747e12 q^{98} +1.16261e11i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	117.945			1.84288			0.921442	−	0.388516i	$-0.127012\pi$
$2$	117.945			1.84288			0.921442	+	0.388516i	$0.127012\pi$
$3$	831.317			1.14035			0.570176	−	0.821523i	$-0.306875\pi$
$3$	831.317			1.14035			0.570176	+	0.821523i	$0.306875\pi$
$4$	9814.93			2.39622
$5$			26263.6i			1.68087i	0.541913	+	0.840435i	$0.317700\pi$
$5$			26263.6i			1.68087i	−0.541913	+	0.840435i	$0.682300\pi$
$6$	98049.3			2.10154
$7$		−	167269.i		−	1.42177i	−0.703310	−	0.710883i	$-0.748295\pi$
$7$		−	167269.i		−	1.42177i	0.703310	−	0.710883i	$-0.251705\pi$
$8$	674517.			2.57308
$9$	159646.			0.300402
$10$			3.09765e6i			3.09765i
$11$			728242.i			0.411074i	0.978649	+	0.205537i	$0.0658940\pi$
$11$			728242.i			0.411074i	−0.978649	+	0.205537i	$0.934106\pi$
$12$	8.15931e6			2.73254
$13$	−5.83177e6			−1.20820			−0.604102	−	0.796907i	$-0.706468\pi$
$13$	−5.83177e6			−1.20820			−0.604102	+	0.796907i	$0.706468\pi$
$14$		−	1.97285e7i		−	2.62015i
$15$			2.18333e7i			1.91678i
$16$	3.93536e7			2.34566
$17$			1.39261e7i			0.576949i	0.957488	+	0.288474i	$0.0931480\pi$
$17$			1.39261e7i			0.576949i	−0.957488	+	0.288474i	$0.906852\pi$
$18$	1.88294e7			0.553607
$19$		−	8.30706e7i		−	1.76574i	−0.469621	−	0.882868i	$-0.655609\pi$
$19$		−	8.30706e7i		−	1.76574i	0.469621	−	0.882868i	$-0.344391\pi$
$20$			2.57775e8i			4.02774i
$21$		−	1.39054e8i		−	1.62131i
$22$			8.58922e7i			0.757561i
$23$	1.37054e8	−	5.59538e7i	0.925816	−	0.377974i
$23$	1.37054e8	−	5.59538e7i	0.925816	−	0.377974i
$24$	5.60737e8			2.93421
$25$	−4.45635e8			−1.82532
$26$	−6.87825e8			−2.22658
$27$	−3.09079e8			−0.797787
$28$		−	1.64174e9i		−	3.40687i
$29$	6.50000e8			1.09276			0.546381	−	0.837537i	$-0.316005\pi$
$29$	6.50000e8			1.09276			0.546381	+	0.837537i	$0.316005\pi$
$30$			2.57513e9i			3.53241i
$31$	4.47244e7			0.0503935			0.0251967	−	0.999683i	$-0.491979\pi$
$31$	4.47244e7			0.0503935			0.0251967	+	0.999683i	$0.491979\pi$
$32$	1.87873e9			1.74970
$33$			6.05399e8i			0.468768i
$34$			1.64251e9i			1.06325i
$35$	4.39309e9			2.38980
$36$	1.56692e9			0.719831
$37$			1.55826e8i			0.0607336i	0.999539	+	0.0303668i	$0.00966753\pi$
$37$			1.55826e8i			0.0607336i	−0.999539	+	0.0303668i	$0.990332\pi$
$38$		−	9.79773e9i		−	3.25405i
$39$	−4.84804e9			−1.37778
$40$			1.77152e10i			4.32500i
$41$	−9.18670e7			−0.0193400			−0.00967000	−	0.999953i	$-0.503078\pi$
$41$	−9.18670e7			−0.0193400			−0.00967000	+	0.999953i	$0.503078\pi$
$42$		−	1.64006e10i		−	2.98789i
$43$			1.58410e9i			0.250594i	0.992119	+	0.125297i	$0.0399884\pi$
$43$			1.58410e9i			0.250594i	−0.992119	+	0.125297i	$0.960012\pi$
$44$			7.14764e9i			0.985024i
$45$			4.19288e9i			0.504937i
$46$	1.61648e10	−	6.59944e9i	1.70617	−	0.696563i
$47$	−4.29921e9			−0.398843			−0.199421	−	0.979914i	$-0.563906\pi$
$47$	−4.29921e9			−0.398843			−0.199421	+	0.979914i	$0.563906\pi$
$48$	3.27153e10			2.67488
$49$	−1.41378e10			−1.02142
$50$	−5.25602e10			−3.36386
$51$			1.15770e10i			0.657925i
$52$	−5.72384e10			−2.89512
$53$		−	1.15006e10i		−	0.518879i	−0.965759	−	0.259439i	$-0.916462\pi$
$53$		−	1.15006e10i		−	0.518879i	0.965759	−	0.259439i	$-0.0835378\pi$
$54$	−3.64542e10			−1.47023
$55$	−1.91262e10			−0.690961
$56$		−	1.12826e11i		−	3.65831i
$57$		−	6.90580e10i		−	2.01356i
$58$	7.66640e10			2.01383
$59$	−4.07090e10			−0.965114			−0.482557	−	0.875865i	$-0.660292\pi$
$59$	−4.07090e10			−0.965114			−0.482557	+	0.875865i	$0.660292\pi$
$60$			2.14293e11i			4.59304i
$61$			2.44060e10i			0.473715i	0.971544	+	0.236857i	$0.0761174\pi$
$61$			2.44060e10i			0.473715i	−0.971544	+	0.236857i	$0.923883\pi$
$62$	5.27500e9			0.0928694
$63$		−	2.67039e10i		−	0.427102i
$64$	6.03934e10			0.878840
$65$		−	1.53163e11i		−	2.03083i
$66$			7.14036e10i			0.863886i
$67$			1.58761e11i			1.75507i	0.479514	+	0.877534i	$0.340813\pi$
$67$			1.58761e11i			1.75507i	−0.479514	+	0.877534i	$0.659187\pi$
$68$			1.36684e11i			1.38250i
$69$	1.13935e11	−	4.65153e10i	1.05576	−	0.431024i
$70$	5.18142e11			4.40413
$71$	−9.47198e10			−0.739419			−0.369709	−	0.929147i	$-0.620543\pi$
$71$	−9.47198e10			−0.739419			−0.369709	+	0.929147i	$0.620543\pi$
$72$	1.07684e11			0.772958
$73$	9.19442e10			0.607557			0.303779	−	0.952743i	$-0.401752\pi$
$73$	9.19442e10			0.607557			0.303779	+	0.952743i	$0.401752\pi$
$74$			1.83788e10i			0.111925i
$75$	−3.70464e11			−2.08151
$76$		−	8.15332e11i		−	4.23110i
$77$	1.21813e11			0.584451
$78$	−5.71801e11			−2.53908
$79$			3.66644e11i			1.50828i	0.656714	+	0.754139i	$0.271946\pi$
$79$			3.66644e11i			1.50828i	−0.656714	+	0.754139i	$0.728054\pi$
$80$			1.03357e12i			3.94275i
$81$	−3.41785e11			−1.21016
$82$	−1.08352e10			−0.0356414
$83$		−	1.44546e11i		−	0.442116i	−0.975261	−	0.221058i	$-0.929049\pi$
$83$		−	1.44546e11i		−	0.442116i	0.975261	−	0.221058i	$-0.0709510\pi$
$84$		−	1.36480e12i		−	3.88503i
$85$	−3.65750e11			−0.969776
$86$			1.86836e11i			0.461816i
$87$	5.40356e11			1.24613
$88$			4.91211e11i			1.05772i
$89$		−	3.73758e11i		−	0.752057i	−0.926608	−	0.376028i	$-0.877290\pi$
$89$		−	3.73758e11i		−	0.752057i	0.926608	−	0.376028i	$-0.122710\pi$
$90$			4.94527e11i			0.930541i
$91$			9.75476e11i			1.71778i
$92$	1.34518e12	−	5.49182e11i	2.21846	−	0.905710i
$93$	3.71801e10			0.0574663
$94$	−5.07069e11			−0.735021
$95$	2.18173e12			2.96797
$96$	1.56182e12			1.99528
$97$		−	9.37374e11i		−	1.12534i	−0.826683	−	0.562668i	$-0.809775\pi$
$97$		−	9.37374e11i		−	1.12534i	0.826683	−	0.562668i	$-0.190225\pi$
$98$	−1.66747e12			−1.88236
$99$			1.16261e11i			0.123487i
$100$	−4.37388e12			−4.37388
$101$	1.07239e12			1.01024			0.505120	−	0.863049i	$-0.331448\pi$
$101$	1.07239e12			1.01024			0.505120	+	0.863049i	$0.331448\pi$
$102$			1.36545e12i			1.21248i
$103$		−	8.73385e11i		−	0.731447i	−0.930724	−	0.365723i	$-0.880822\pi$
$103$		−	8.73385e11i		−	0.731447i	0.930724	−	0.365723i	$-0.119178\pi$
$104$	−3.93362e12			−3.10880
$105$	3.65205e12			2.72522
$106$		−	1.35644e12i		−	0.956234i
$107$			1.97819e12i			1.31815i	0.752076	+	0.659076i	$0.229052\pi$
$107$			1.97819e12i			1.31815i	−0.752076	+	0.659076i	$0.770948\pi$
$108$	−3.03359e12			−1.91168
$109$			2.10180e12i			1.25323i	0.779327	+	0.626617i	$0.215561\pi$
$109$			2.10180e12i			1.25323i	−0.779327	+	0.626617i	$0.784439\pi$
$110$	−2.25584e12			−1.27336
$111$			1.29540e11i			0.0692576i
$112$		−	6.58266e12i		−	3.33498i
$113$		−	5.92515e10i		−	0.0284596i	−0.999899	−	0.0142298i	$-0.995470\pi$
$113$		−	5.92515e10i		−	0.0284596i	0.999899	−	0.0142298i	$-0.00452963\pi$
$114$		−	8.14502e12i		−	3.71076i
$115$	1.46955e12	+	3.59953e12i	0.635325	+	1.55618i
$116$	6.37971e12			2.61850
$117$	−9.31019e11			−0.362947
$118$	−4.80141e12			−1.77859
$119$	2.32942e12			0.820287
$120$			1.47270e13i			4.93203i
$121$	2.60809e12			0.831019
$122$			2.87855e12i			0.873001i
$123$	−7.63706e10			−0.0220544
$124$	4.38967e11			0.120754
$125$		−	5.29196e12i		−	1.38726i
$126$		−	3.14958e12i		−	0.787100i
$127$	−3.23159e12			−0.770184			−0.385092	−	0.922878i	$-0.625830\pi$
$127$	−3.23159e12			−0.770184			−0.385092	+	0.922878i	$0.625830\pi$
$128$	−5.72197e11			−0.130102
$129$			1.31689e12i			0.285766i
$130$		−	1.80648e13i		−	3.74259i
$131$	5.17578e12			1.02411			0.512057	−	0.858952i	$-0.328884\pi$
$131$	5.17578e12			1.02411			0.512057	+	0.858952i	$0.328884\pi$
$132$			5.94195e12i			1.12327i
$133$	−1.38952e13			−2.51047
$134$			1.87250e13i			3.23439i
$135$		−	8.11753e12i		−	1.34098i
$136$			9.39342e12i			1.48453i
$137$			3.23332e12i			0.489018i	0.969647	+	0.244509i	$0.0786268\pi$
$137$			3.23332e12i			0.489018i	−0.969647	+	0.244509i	$0.921373\pi$
$138$	1.34380e13	−	5.48623e12i	1.94564	−	0.794327i
$139$	9.45027e12			1.31025			0.655127	−	0.755519i	$-0.272615\pi$
$139$	9.45027e12			1.31025			0.655127	+	0.755519i	$0.272615\pi$
$140$	4.31179e13			5.72650
$141$	−3.57401e12			−0.454821
$142$	−1.11717e13			−1.36266
$143$		−	4.24694e12i		−	0.496660i
$144$	6.28266e12			0.704642
$145$			1.70713e13i			1.83679i
$146$	1.08443e13			1.11966
$147$	−1.17530e13			−1.16478
$148$			1.52942e12i			0.145531i
$149$			7.34558e12i			0.671287i	0.941989	+	0.335644i	$0.108954\pi$
$149$			7.34558e12i			0.671287i	−0.941989	+	0.335644i	$0.891046\pi$
$150$	−4.36942e13			−3.83598
$151$	−1.84964e12			−0.156036			−0.0780182	−	0.996952i	$-0.524859\pi$
$151$	−1.84964e12			−0.156036			−0.0780182	+	0.996952i	$0.524859\pi$
$152$		−	5.60325e13i		−	4.54337i
$153$			2.22326e12i			0.173317i
$154$	1.43671e13			1.07707
$155$			1.17462e12i			0.0847049i
$156$	−4.75832e13			−3.30146
$157$		−	1.90075e13i		−	1.26919i	−0.772845	−	0.634595i	$-0.781167\pi$
$157$		−	1.90075e13i		−	1.26919i	0.772845	−	0.634595i	$-0.218833\pi$
$158$			4.32436e13i			2.77958i
$159$		−	9.56066e12i		−	0.591705i
$160$			4.93421e13i			2.94102i
$161$	−9.35935e12	−	2.29249e13i	−0.537391	−	1.31629i
$162$	−4.03117e13			−2.23019
$163$	2.84006e13			1.51427			0.757133	−	0.653261i	$-0.226600\pi$
$163$	2.84006e13			1.51427			0.757133	+	0.653261i	$0.226600\pi$
$164$	−9.01668e11			−0.0463430
$165$	−1.59000e13			−0.787938
$166$		−	1.70484e13i		−	0.814768i
$167$	2.30852e13			1.06423			0.532114	−	0.846673i	$-0.321398\pi$
$167$	2.30852e13			1.06423			0.532114	+	0.846673i	$0.321398\pi$
$168$		−	9.37941e13i		−	4.17176i
$169$	1.07114e13			0.459756
$170$	−4.31383e13			−1.78718
$171$		−	1.32619e13i		−	0.530431i
$172$			1.55478e13i			0.600480i
$173$	−2.24891e12			−0.0838872			−0.0419436	−	0.999120i	$-0.513355\pi$
$173$	−2.24891e12			−0.0838872			−0.0419436	+	0.999120i	$0.513355\pi$
$174$	6.37321e13			2.29648
$175$			7.45411e13i			2.59518i
$176$			2.86590e13i			0.964238i
$177$	−3.38421e13			−1.10057
$178$		−	4.40827e13i		−	1.38595i
$179$	−8.61999e12			−0.262053			−0.131026	−	0.991379i	$-0.541827\pi$
$179$	−8.61999e12			−0.262053			−0.131026	+	0.991379i	$0.541827\pi$
$180$			4.11528e13i			1.20994i
$181$		−	6.02455e13i		−	1.71338i	−0.515833	−	0.856689i	$-0.672518\pi$
$181$		−	6.02455e13i		−	1.71338i	0.515833	−	0.856689i	$-0.327482\pi$
$182$			1.15052e14i			3.16568i
$183$			2.02891e13i			0.540201i
$184$	9.24452e13	−	3.77417e13i	2.38220	−	0.972557i
$185$	−4.09254e12			−0.102085
$186$	4.38520e12			0.105904
$187$	−1.01416e13			−0.237168
$188$	−4.21965e13			−0.955716
$189$			5.16995e13i			1.13427i
$190$	2.57324e14			5.46963
$191$			4.26799e13i			0.879070i	0.898225	+	0.439535i	$0.144857\pi$
$191$			4.26799e13i			0.879070i	−0.898225	+	0.439535i	$0.855143\pi$
$192$	5.02060e13			1.00219
$193$	−5.00461e13			−0.968337			−0.484169	−	0.874975i	$-0.660878\pi$
$193$	−5.00461e13			−0.968337			−0.484169	+	0.874975i	$0.660878\pi$
$194$		−	1.10558e14i		−	2.07386i
$195$		−	1.27327e14i		−	2.31586i
$196$	−1.38761e14			−2.44755
$197$	6.09349e13			1.04248			0.521241	−	0.853410i	$-0.325469\pi$
$197$	6.09349e13			1.04248			0.521241	+	0.853410i	$0.325469\pi$
$198$			1.37124e13i			0.227573i
$199$			3.07058e13i			0.494427i	0.968961	+	0.247213i	$0.0795149\pi$
$199$			3.07058e13i			0.494427i	−0.968961	+	0.247213i	$0.920485\pi$
$200$	−3.00588e14			−4.69669
$201$			1.31980e14i			2.00140i
$202$	1.26483e14			1.86175
$203$		−	1.08725e14i		−	1.55365i
$204$			1.13628e14i			1.57653i
$205$		−	2.41276e12i		−	0.0325080i
$206$		−	1.03011e14i		−	1.34797i
$207$	2.18801e13	−	8.93280e12i	0.278117	−	0.113544i
$208$	−2.29501e14			−2.83403
$209$	6.04955e13			0.725848
$210$	4.30740e14			5.02226
$211$	4.57845e13			0.518828			0.259414	−	0.965766i	$-0.416471\pi$
$211$	4.57845e13			0.518828			0.259414	+	0.965766i	$0.416471\pi$
$212$		−	1.12878e14i		−	1.24335i
$213$	−7.87421e13			−0.843198
$214$			2.33317e14i			2.42920i
$215$	−4.16041e13			−0.421216
$216$	−2.08479e14			−2.05277
$217$		−	7.48103e12i		−	0.0716478i
$218$			2.47896e14i			2.30957i
$219$	7.64347e13			0.692829
$220$	−1.87723e14			−1.65570
$221$		−	8.12140e13i		−	0.697072i
$222$			1.52786e13i			0.127634i
$223$	−1.48437e14			−1.20701			−0.603507	−	0.797358i	$-0.706230\pi$
$223$	−1.48437e14			−1.20701			−0.603507	+	0.797358i	$0.706230\pi$
$224$		−	3.14254e14i		−	2.48767i
$225$	−7.11439e13			−0.548331
$226$		−	6.98839e12i		−	0.0524477i
$227$			6.16323e13i			0.450457i	0.974306	+	0.225229i	$0.0723129\pi$
$227$			6.16323e13i			0.450457i	−0.974306	+	0.225229i	$0.927687\pi$
$228$		−	6.77799e14i		−	4.82494i
$229$		−	7.13694e13i		−	0.494879i	−0.968903	−	0.247440i	$-0.920411\pi$
$229$		−	7.13694e13i		−	0.494879i	0.968903	−	0.247440i	$-0.0795892\pi$
$230$	1.73325e14	+	4.24545e14i	1.17083	+	2.86785i
$231$	1.01265e14			0.666479
$232$	4.38436e14			2.81176
$233$	1.15696e14			0.723072			0.361536	−	0.932358i	$-0.382252\pi$
$233$	1.15696e14			0.723072			0.361536	+	0.932358i	$0.382252\pi$
$234$	−1.09809e14			−0.668870
$235$		−	1.12913e14i		−	0.670403i
$236$	−3.99556e14			−2.31263
$237$			3.04797e14i			1.71997i
$238$	2.74742e14			1.51169
$239$	−3.06454e14			−1.64429			−0.822144	−	0.569280i	$-0.807222\pi$
$239$	−3.06454e14			−1.64429			−0.822144	+	0.569280i	$0.807222\pi$
$240$			8.59222e14i			4.49612i
$241$		−	3.36106e14i		−	1.71543i	−0.514121	−	0.857717i	$-0.671882\pi$
$241$		−	3.36106e14i		−	1.71543i	0.514121	−	0.857717i	$-0.328118\pi$
$242$	3.07610e14			1.53147
$243$	−1.19874e14			−0.582222
$244$			2.39543e14i			1.13513i
$245$		−	3.71308e14i		−	1.71687i
$246$	−9.00750e12			−0.0406437
$247$			4.84449e14i			2.13337i
$248$	3.01674e13			0.129666
$249$		−	1.20163e14i		−	0.504168i
$250$		−	6.24159e14i		−	2.55655i
$251$			9.52759e13i			0.381014i	0.981686	+	0.190507i	$0.0610132\pi$
$251$			9.52759e13i			0.381014i	−0.981686	+	0.190507i	$0.938987\pi$
$252$		−	2.62097e14i		−	1.02343i
$253$	4.07479e13	+	9.98085e13i	0.155375	+	0.380578i
$254$	−3.81149e14			−1.41936
$255$	−3.04054e14			−1.10589
$256$	−3.14859e14			−1.11860
$257$	−1.66162e14			−0.576678			−0.288339	−	0.957528i	$-0.593103\pi$
$257$	−1.66162e14			−0.576678			−0.288339	+	0.957528i	$0.593103\pi$
$258$			1.55320e14i			0.526633i
$259$	2.60649e13			0.0863489
$260$		−	1.50328e15i		−	4.86632i
$261$	1.03770e14			0.328268
$262$	6.10455e14			1.88732
$263$		−	1.17102e14i		−	0.353858i	−0.984224	−	0.176929i	$-0.943384\pi$
$263$		−	1.17102e14i		−	0.353858i	0.984224	−	0.176929i	$-0.0566164\pi$
$264$			4.08352e14i			1.20618i
$265$	3.02047e14			0.872168
$266$	−1.63886e15			−4.62650
$267$		−	3.10711e14i		−	0.857609i
$268$			1.55822e15i			4.20553i
$269$	6.87577e13			0.181471			0.0907356	−	0.995875i	$-0.471078\pi$
$269$	6.87577e13			0.181471			0.0907356	+	0.995875i	$0.471078\pi$
$270$		−	9.57418e14i		−	2.47126i
$271$	−3.79066e14			−0.956973			−0.478486	−	0.878095i	$-0.658814\pi$
$271$	−3.79066e14			−0.956973			−0.478486	+	0.878095i	$0.658814\pi$
$272$			5.48044e14i			1.35333i
$273$			8.10930e14i			1.95888i
$274$			3.81352e14i			0.901204i
$275$		−	3.24530e14i		−	0.750341i
$276$	1.11827e15	−	4.56544e14i	2.52983	−	1.03283i
$277$	−5.11180e14			−1.13161			−0.565803	−	0.824541i	$-0.691434\pi$
$277$	−5.11180e14			−1.13161			−0.565803	+	0.824541i	$0.691434\pi$
$278$	1.11461e15			2.41465
$279$	7.14008e12			0.0151383
$280$	2.96321e15			6.14915
$281$			4.43195e14i			0.900237i	0.892969	+	0.450119i	$0.148618\pi$
$281$			4.43195e14i			0.900237i	−0.892969	+	0.450119i	$0.851382\pi$
$282$	−4.21535e14			−0.838183
$283$			7.32918e14i			1.42671i	0.700801	+	0.713357i	$0.252826\pi$
$283$			7.32918e14i			1.42671i	−0.700801	+	0.713357i	$0.747174\pi$
$284$	−9.29668e14			−1.77181
$285$	1.81371e15			3.38453
$286$		−	5.00903e14i		−	0.915288i
$287$			1.53665e13i			0.0274970i
$288$	2.99932e14			0.525615
$289$	3.88685e14			0.667130
$290$			2.01347e15i			3.38499i
$291$		−	7.79254e14i		−	1.28328i
$292$	9.02426e14			1.45584
$293$		−	4.29215e14i		−	0.678374i	−0.940719	−	0.339187i	$-0.889848\pi$
$293$		−	4.29215e14i		−	0.678374i	0.940719	−	0.339187i	$-0.110152\pi$
$294$	−1.38620e15			−2.14655
$295$		−	1.06916e15i		−	1.62223i
$296$			1.05107e14i			0.156272i
$297$		−	2.25084e14i		−	0.327949i
$298$			8.66371e14i			1.23711i
$299$	−7.99267e14	+	3.26309e14i	−1.11857	+	0.456670i
$300$	−3.63607e15			−4.98776
$301$	2.64971e14			0.356287
$302$	−2.18155e14			−0.287557
$303$	8.91495e14			1.15203
$304$		−	3.26913e15i		−	4.14182i
$305$	−6.40988e14			−0.796252
$306$			2.62221e14i			0.319403i
$307$	−1.56451e15			−1.86873			−0.934366	−	0.356314i	$-0.884033\pi$
$307$	−1.56451e15			−1.86873			−0.934366	+	0.356314i	$0.884033\pi$
$308$	1.19558e15			1.40047
$309$		−	7.26060e14i		−	0.834106i
$310$			1.38540e14i			0.156101i
$311$	5.88063e14			0.649923			0.324961	−	0.945727i	$-0.394649\pi$
$311$	5.88063e14			0.649923			0.324961	+	0.945727i	$0.394649\pi$
$312$	−3.27009e15			−3.54513
$313$			2.84420e14i			0.302478i	0.988497	+	0.151239i	$0.0483264\pi$
$313$			2.84420e14i			0.302478i	−0.988497	+	0.151239i	$0.951674\pi$
$314$		−	2.24183e15i		−	2.33897i
$315$	7.01340e14			0.717903
$316$			3.59858e15i			3.61417i
$317$	−1.80440e14			−0.177819			−0.0889093	−	0.996040i	$-0.528338\pi$
$317$	−1.80440e14			−0.177819			−0.0889093	+	0.996040i	$0.528338\pi$
$318$		−	1.12763e15i		−	1.09044i
$319$			4.73357e14i			0.449206i
$320$			1.58615e15i			1.47721i
$321$			1.64450e15i			1.50316i
$322$	−1.10388e15	−	2.70387e15i	−0.990350	−	2.42578i
$323$	1.15685e15			1.01874
$324$	−3.35460e15			−2.89981
$325$	2.59884e15			2.20536
$326$	3.34970e15			2.79062
$327$			1.74726e15i			1.42913i
$328$	−6.19658e13			−0.0497633
$329$			7.19127e14i			0.567061i
$330$	−1.87531e15			−1.45208
$331$	8.42438e14			0.640575			0.320288	−	0.947320i	$-0.396220\pi$
$331$	8.42438e14			0.640575			0.320288	+	0.947320i	$0.396220\pi$
$332$		−	1.41870e15i		−	1.05941i
$333$			2.48770e13i			0.0182445i
$334$	2.72277e15			1.96125
$335$	−4.16962e15			−2.95004
$336$		−	5.47227e15i		−	3.80305i
$337$		−	9.56889e13i		−	0.0653254i	−0.999466	−	0.0326627i	$-0.989601\pi$
$337$		−	9.56889e13i		−	0.0653254i	0.999466	−	0.0326627i	$-0.0103987\pi$
$338$	1.26335e15			0.847276
$339$		−	4.92567e13i		−	0.0324539i
$340$	−3.58981e15			−2.32380
$341$			3.25702e13i			0.0207154i
$342$		−	1.56417e15i		−	0.977524i
$343$			4.95921e13i			0.0304542i
$344$			1.06850e15i			0.644799i
$345$	1.22166e15	+	2.99235e15i	0.724494	+	1.77459i
$346$	−2.65247e14			−0.154594
$347$	1.84321e15			1.05584			0.527921	−	0.849294i	$-0.322972\pi$
$347$	1.84321e15			1.05584			0.527921	+	0.849294i	$0.322972\pi$
$348$	5.30356e15			2.98601
$349$	−5.67050e14			−0.313811			−0.156906	−	0.987614i	$-0.550152\pi$
$349$	−5.67050e14			−0.313811			−0.156906	+	0.987614i	$0.550152\pi$
$350$			8.79172e15i			4.78262i
$351$	1.80248e15			0.963890
$352$			1.36817e15i			0.719256i
$353$	3.55541e15			1.83756			0.918781	−	0.394768i	$-0.129175\pi$
$353$	3.55541e15			1.83756			0.918781	+	0.394768i	$0.129175\pi$
$354$	−3.99149e15			−2.02822
$355$		−	2.48768e15i		−	1.24287i
$356$		−	3.66841e15i		−	1.80210i
$357$	1.93648e15			0.935416
$358$	−1.01668e15			−0.482933
$359$		−	4.10709e15i		−	1.91853i	−0.282514	−	0.959263i	$-0.591168\pi$
$359$		−	4.10709e15i		−	1.91853i	0.282514	−	0.959263i	$-0.408832\pi$
$360$			2.82817e15i			1.29924i
$361$	−4.68742e15			−2.11783
$362$		−	7.10563e15i		−	3.15756i
$363$	2.16815e15			0.947654
$364$			9.57423e15i			4.11619i
$365$			2.41478e15i			1.02122i
$366$			2.39299e15i			0.995529i
$367$			2.02227e14i			0.0827641i	0.999143	+	0.0413820i	$0.0131761\pi$
$367$			2.02227e14i			0.0827641i	−0.999143	+	0.0413820i	$0.986824\pi$
$368$	5.39357e15	−	2.20198e15i	2.17165	−	0.886599i
$369$	−1.46662e13			−0.00580978
$370$	−4.82693e14			−0.188131
$371$	−1.92370e15			−0.737725
$372$	3.64920e14			0.137702
$373$		−	5.10595e15i		−	1.89593i	−0.318368	−	0.947967i	$-0.603135\pi$
$373$		−	5.10595e15i		−	1.89593i	0.318368	−	0.947967i	$-0.396865\pi$
$374$	−1.19615e15			−0.437074
$375$		−	4.39930e15i		−	1.58196i
$376$	−2.89989e15			−1.02625
$377$	−3.79065e15			−1.32028
$378$			6.09768e15i			2.09032i
$379$		−	1.68961e15i		−	0.570099i	−0.958513	−	0.285050i	$-0.907990\pi$
$379$		−	1.68961e15i		−	0.570099i	0.958513	−	0.285050i	$-0.0920101\pi$
$380$	2.14135e16			7.11192
$381$	−2.68648e15			−0.878281
$382$			5.03387e15i			1.62002i
$383$			4.33182e15i			1.37239i	0.727417	+	0.686196i	$0.240721\pi$
$383$			4.33182e15i			1.37239i	−0.727417	+	0.686196i	$0.759279\pi$
$384$	−4.75677e14			−0.148363
$385$			3.19923e15i			0.982385i
$386$	−5.90267e15			−1.78453
$387$			2.52895e14i			0.0752792i
$388$		−	9.20026e15i		−	2.69656i
$389$		−	1.99337e15i		−	0.575295i	−0.957736	−	0.287648i	$-0.907127\pi$
$389$		−	1.99337e15i		−	0.575295i	0.957736	−	0.287648i	$-0.0928732\pi$
$390$		−	1.50175e16i		−	4.26787i
$391$	7.79220e14	+	1.90863e15i	0.218072	+	0.534149i
$392$	−9.53616e15			−2.62819
$393$	4.30271e15			1.16785
$394$	7.18694e15			1.92117
$395$	−9.62938e15			−2.53522
$396$			1.14109e15i			0.295903i
$397$	−2.64755e15			−0.676241			−0.338120	−	0.941103i	$-0.609791\pi$
$397$	−2.64755e15			−0.676241			−0.338120	+	0.941103i	$0.609791\pi$
$398$			3.62158e15i			0.911172i
$399$	−1.15513e16			−2.86281
$400$	−1.75374e16			−4.28158
$401$			4.94776e15i			1.18999i	0.803730	+	0.594994i	$0.202846\pi$
$401$			4.94776e15i			1.18999i	−0.803730	+	0.594994i	$0.797154\pi$
$402$			1.55664e16i			3.68834i
$403$	−2.60822e14			−0.0608856
$404$	1.05254e16			2.42076
$405$		−	8.97650e15i		−	2.03412i
$406$		−	1.28235e16i		−	2.86320i
$407$	−1.13479e14			−0.0249660
$408$			7.80890e15i			1.69289i
$409$	−3.62223e15			−0.773813			−0.386907	−	0.922119i	$-0.626456\pi$
$409$	−3.62223e15			−0.773813			−0.386907	+	0.922119i	$0.626456\pi$
$410$		−	2.84572e14i		−	0.0599085i
$411$			2.68791e15i			0.557653i
$412$		−	8.57221e15i		−	1.75271i
$413$			6.80937e15i			1.37217i
$414$	2.58064e15	−	1.05358e15i	0.512538	−	0.209249i
$415$	3.79628e15			0.743139
$416$	−1.09563e16			−2.11400
$417$	7.85616e15			1.49415
$418$	7.13512e15			1.33765
$419$		−	4.01511e15i		−	0.742016i	−0.928630	−	0.371008i	$-0.879012\pi$
$419$		−	4.01511e15i		−	0.742016i	0.928630	−	0.371008i	$-0.120988\pi$
$420$	3.58446e16			6.53023
$421$		−	6.86836e14i		−	0.123356i	−0.998096	−	0.0616780i	$-0.980355\pi$
$421$		−	6.86836e14i		−	0.123356i	0.998096	−	0.0616780i	$-0.0196452\pi$
$422$	5.40003e15			0.956140
$423$	−6.86353e14			−0.119813
$424$		−	7.75736e15i		−	1.33512i
$425$		−	6.20598e15i		−	1.05312i
$426$	−9.28721e15			−1.55392
$427$	4.08237e15			0.673512
$428$			1.94158e16i			3.15859i
$429$		−	3.53055e15i		−	0.566368i
$430$	−4.90698e15			−0.776253
$431$		−	3.39399e15i		−	0.529476i	−0.964320	−	0.264738i	$-0.914714\pi$
$431$		−	3.39399e15i		−	0.529476i	0.964320	−	0.264738i	$-0.0852856\pi$
$432$	−1.21634e16			−1.87134
$433$		−	7.43618e14i		−	0.112830i	−0.998407	−	0.0564148i	$-0.982033\pi$
$433$		−	7.43618e14i		−	0.112830i	0.998407	−	0.0564148i	$-0.0179669\pi$
$434$		−	8.82346e14i		−	0.132039i
$435$			1.41917e16i			2.09459i
$436$			2.06290e16i			3.00303i
$437$	−4.64811e15	−	1.13852e16i	−0.667403	−	1.63475i
$438$	9.01506e15			1.27680
$439$	−3.35548e15			−0.468778			−0.234389	−	0.972143i	$-0.575309\pi$
$439$	−3.35548e15			−0.468778			−0.234389	+	0.972143i	$0.575309\pi$
$440$	−1.29010e16			−1.77789
$441$	−2.25704e15			−0.306837
$442$		−	9.57876e15i		−	1.28462i
$443$	2.13780e15			0.282843			0.141421	−	0.989949i	$-0.454833\pi$
$443$	2.13780e15			0.282843			0.141421	+	0.989949i	$0.454833\pi$
$444$			1.27143e15i			0.165957i
$445$	9.81623e15			1.26411
$446$	−1.75073e16			−2.22439
$447$			6.10650e15i			0.765504i
$448$		−	1.01020e16i		−	1.24950i
$449$	−6.35381e15			−0.775454			−0.387727	−	0.921774i	$-0.626740\pi$
$449$	−6.35381e15			−0.775454			−0.387727	+	0.921774i	$0.626740\pi$
$450$	−8.39104e15			−1.01051
$451$		−	6.69014e13i		−	0.00795016i
$452$		−	5.81549e14i		−	0.0681955i
$453$	−1.53764e15			−0.177936
$454$			7.26920e15i			0.830140i
$455$	−2.56195e16			−2.88737
$456$		−	4.65808e16i		−	5.18105i
$457$			3.19131e15i			0.350326i	0.984539	+	0.175163i	$0.0560452\pi$
$457$			3.19131e15i			0.350326i	−0.984539	+	0.175163i	$0.943955\pi$
$458$		−	8.41764e15i		−	0.912005i
$459$		−	4.30428e15i		−	0.460283i
$460$	1.44235e16	+	3.53291e16i	1.52238	+	3.72894i
$461$	−1.33828e14			−0.0139425			−0.00697126	−	0.999976i	$-0.502219\pi$
$461$	−1.33828e14			−0.0139425			−0.00697126	+	0.999976i	$0.502219\pi$
$462$	1.19436e16			1.22824
$463$	1.82297e16			1.85052			0.925259	−	0.379336i	$-0.123848\pi$
$463$	1.82297e16			1.85052			0.925259	+	0.379336i	$0.123848\pi$
$464$	2.55799e16			2.56325
$465$			9.76484e14i			0.0965933i
$466$	1.36457e16			1.33254
$467$		−	3.01774e15i		−	0.290925i	−0.989364	−	0.145462i	$-0.953533\pi$
$467$		−	3.01774e15i		−	0.290925i	0.989364	−	0.145462i	$-0.0464669\pi$
$468$	−9.13788e15			−0.869702
$469$	2.65558e16			2.49530
$470$		−	1.33174e16i		−	1.23547i
$471$		−	1.58012e16i		−	1.44732i
$472$	−2.74589e16			−2.48331
$473$	−1.15361e15			−0.103013
$474$			3.59492e16i			3.16970i
$475$			3.70192e16i			3.22304i
$476$	2.28631e16			1.96559
$477$		−	1.83603e15i		−	0.155872i
$478$	−3.61446e16			−3.03023
$479$			2.38919e15i			0.197805i	0.995097	+	0.0989024i	$0.0315332\pi$
$479$			2.38919e15i			0.197805i	−0.995097	+	0.0989024i	$0.968467\pi$
$480$			4.10189e16i			3.35380i
$481$		−	9.08739e14i		−	0.0733785i
$482$		−	3.96419e16i		−	3.16135i
$483$	−7.78058e15	−	1.90579e16i	−0.612815	−	1.50104i
$484$	2.55982e16			1.99131
$485$	2.46188e16			1.89154
$486$	−1.41385e16			−1.07297
$487$	−7.86229e15			−0.589353			−0.294676	−	0.955597i	$-0.595212\pi$
$487$	−7.86229e15			−0.589353			−0.294676	+	0.955597i	$0.595212\pi$
$488$			1.64622e16i			1.21890i
$489$	2.36099e16			1.72680
$490$		−	4.37938e16i		−	3.16400i
$491$	2.66234e15			0.190009			0.0950047	−	0.995477i	$-0.469713\pi$
$491$	2.66234e15			0.190009			0.0950047	+	0.995477i	$0.469713\pi$
$492$	−7.49572e14			−0.0528473
$493$			9.05200e15i			0.630468i
$494$			5.71381e16i			3.93155i
$495$	−3.05343e15			−0.207566
$496$	1.76007e15			0.118206
$497$			1.58437e16i			1.05128i
$498$		−	1.41726e16i		−	0.929123i
$499$	−7.76435e15			−0.502923			−0.251462	−	0.967867i	$-0.580911\pi$
$499$	−7.76435e15			−0.502923			−0.251462	+	0.967867i	$0.580911\pi$
$500$		−	5.19402e16i		−	3.32418i
$501$	1.91911e16			1.21359
$502$			1.12373e16i			0.702164i
$503$		−	2.41511e16i		−	1.49118i	−0.666407	−	0.745589i	$-0.732168\pi$
$503$		−	2.41511e16i		−	1.49118i	0.666407	−	0.745589i	$-0.267832\pi$
$504$		−	1.80122e16i		−	1.09897i
$505$			2.81648e16i			1.69808i
$506$	4.80599e15	+	1.17719e16i	0.286339	+	0.701362i
$507$	8.90458e15			0.524283
$508$	−3.17179e16			−1.84553
$509$	−1.77309e16			−1.01958			−0.509792	−	0.860298i	$-0.670278\pi$
$509$	−1.77309e16			−1.01958			−0.509792	+	0.860298i	$0.670278\pi$
$510$	−3.58616e16			−2.03802
$511$		−	1.53795e16i		−	0.863805i
$512$	−3.47922e16			−1.93135
$513$			2.56754e16i			1.40868i
$514$	−1.95979e16			−1.06275
$515$	2.29382e16			1.22947
$516$			1.29251e16i			0.684758i
$517$		−	3.13087e15i		−	0.163954i
$518$	3.07421e15			0.159131
$519$	−1.86956e15			−0.0956609
$520$		−	1.03311e17i		−	5.22549i
$521$			1.64611e15i			0.0823062i	0.999153	+	0.0411531i	$0.0131031\pi$
$521$			1.64611e15i			0.0823062i	−0.999153	+	0.0411531i	$0.986897\pi$
$522$	1.22391e16			0.604961
$523$		−	1.65674e16i		−	0.809552i	−0.914416	−	0.404776i	$-0.867350\pi$
$523$		−	1.65674e16i		−	0.809552i	0.914416	−	0.404776i	$-0.132650\pi$
$524$	5.07999e16			2.45400
$525$			6.19673e16i			2.95942i
$526$		−	1.38115e16i		−	0.652120i
$527$			6.22839e14i			0.0290745i
$528$			2.38247e16i			1.09957i
$529$	1.56530e16	−	1.53374e16i	0.714271	−	0.699869i
$530$	3.56249e16			1.60730
$531$	−6.49904e15			−0.289922
$532$	−1.36380e17			−6.01563
$533$	5.35747e14			0.0233667
$534$		−	3.66467e16i		−	1.58047i
$535$	−5.19544e16			−2.21564
$536$			1.07087e17i			4.51592i
$537$	−7.16594e15			−0.298832
$538$	8.10960e15			0.334430
$539$		−	1.02957e16i		−	0.419879i
$540$		−	7.96729e16i		−	3.21328i
$541$	1.70095e16			0.678436			0.339218	−	0.940708i	$-0.389838\pi$
$541$	1.70095e16			0.678436			0.339218	+	0.940708i	$0.389838\pi$
$542$	−4.47088e16			−1.76359
$543$		−	5.00831e16i		−	1.95385i
$544$			2.61635e16i			1.00949i
$545$	−5.52008e16			−2.10652
$546$			9.56448e16i			3.60998i
$547$	−2.93062e16			−1.09405			−0.547023	−	0.837117i	$-0.684239\pi$
$547$	−2.93062e16			−1.09405			−0.547023	+	0.837117i	$0.684239\pi$
$548$			3.17348e16i			1.17180i
$549$			3.89632e15i			0.142305i
$550$		−	3.82766e16i		−	1.38279i
$551$		−	5.39959e16i		−	1.92953i
$552$	7.68512e16	−	3.13753e16i	2.71654	−	1.10906i
$553$	6.13283e16			2.14442
$554$	−6.02909e16			−2.08542
$555$	−3.40220e15			−0.116413
$556$	9.27537e16			3.13966
$557$		−	4.44897e16i		−	1.48980i	−0.667175	−	0.744901i	$-0.732497\pi$
$557$		−	4.44897e16i		−	1.48980i	0.667175	−	0.744901i	$-0.267503\pi$
$558$	8.42134e14			0.0278982
$559$		−	9.23809e15i		−	0.302769i
$560$	1.72884e17			5.60566
$561$	−8.43088e15			−0.270455
$562$			5.22725e16i			1.65903i
$563$			3.93461e16i			1.23552i	0.786365	+	0.617762i	$0.211961\pi$
$563$			3.93461e16i			1.23552i	−0.786365	+	0.617762i	$0.788039\pi$
$564$	−3.50786e16			−1.08985
$565$	1.55616e15			0.0478368
$566$			8.64437e16i			2.62927i
$567$			5.71702e16i			1.72057i
$568$	−6.38901e16			−1.90258
$569$		−	2.45556e16i		−	0.723565i	−0.932263	−	0.361782i	$-0.882168\pi$
$569$		−	2.45556e16i		−	0.723565i	0.932263	−	0.361782i	$-0.117832\pi$
$570$	2.13917e17			6.23730
$571$		−	4.68809e16i		−	1.35263i	−0.736612	−	0.676316i	$-0.763576\pi$
$571$		−	4.68809e16i		−	1.35263i	0.736612	−	0.676316i	$-0.236424\pi$
$572$		−	4.16834e16i		−	1.19011i
$573$			3.54805e16i			1.00245i
$574$			1.81240e15i			0.0506737i
$575$	−6.10761e16	+	2.49350e16i	−1.68991	+	0.689924i
$576$	9.64157e15			0.264006
$577$	6.47404e15			0.175437			0.0877184	−	0.996145i	$-0.472042\pi$
$577$	6.47404e15			0.175437			0.0877184	+	0.996145i	$0.472042\pi$
$578$	4.58433e16			1.22944
$579$	−4.16042e16			−1.10425
$580$			1.67554e17i			4.40136i
$581$	−2.41780e16			−0.628585
$582$		−	9.19088e16i		−	2.36494i
$583$	8.37523e15			0.213297
$584$	6.20179e16			1.56329
$585$		−	2.44519e16i		−	0.610067i
$586$		−	5.06236e16i		−	1.25016i
$587$	2.45857e16			0.600971			0.300486	−	0.953786i	$-0.402851\pi$
$587$	2.45857e16			0.600971			0.300486	+	0.953786i	$0.402851\pi$
$588$	−1.15354e17			−2.79107
$589$		−	3.71528e15i		−	0.0889816i
$590$		−	1.26102e17i		−	2.98958i
$591$	5.06561e16			1.18880
$592$			6.13231e15i			0.142460i
$593$	6.77742e16			1.55860			0.779302	−	0.626648i	$-0.215574\pi$
$593$	6.77742e16			1.55860			0.779302	+	0.626648i	$0.215574\pi$
$594$		−	2.65475e16i		−	0.604373i
$595$			6.11789e16i			1.37879i
$596$			7.20963e16i			1.60855i
$597$			2.55262e16i			0.563821i
$598$	−9.42692e16	+	3.84864e16i	−2.06140	+	0.841590i
$599$	−5.71218e16			−1.23663			−0.618316	−	0.785929i	$-0.712185\pi$
$599$	−5.71218e16			−1.23663			−0.618316	+	0.785929i	$0.712185\pi$
$600$	−2.49884e17			−5.35588
$601$	6.42057e16			1.36247			0.681235	−	0.732065i	$-0.261443\pi$
$601$	6.42057e16			1.36247			0.681235	+	0.732065i	$0.261443\pi$
$602$	3.12519e16			0.656595
$603$			2.53455e16i			0.527227i
$604$	−1.81541e16			−0.373898
$605$			6.84978e16i			1.39683i
$606$	1.05147e17			2.12305
$607$	2.40842e16			0.481503			0.240752	−	0.970587i	$-0.422606\pi$
$607$	2.40842e16			0.481503			0.240752	+	0.970587i	$0.422606\pi$
$608$		−	1.56067e17i		−	3.08951i
$609$		−	9.03850e16i		−	1.77171i
$610$	−7.56011e16			−1.46740
$611$	2.50720e16			0.481883
$612$			2.18211e16i			0.415306i
$613$		−	7.56028e16i		−	1.42487i	−0.701739	−	0.712434i	$-0.747593\pi$
$613$		−	7.56028e16i		−	1.42487i	0.701739	−	0.712434i	$-0.252407\pi$
$614$	−1.84525e17			−3.44386
$615$		−	2.00576e15i		−	0.0370706i
$616$	8.21646e16			1.50384
$617$			2.86649e16i			0.519565i	0.965667	+	0.259783i	$0.0836509\pi$
$617$			2.86649e16i			0.519565i	−0.965667	+	0.259783i	$0.916349\pi$
$618$		−	8.56348e16i		−	1.53716i
$619$			4.67397e16i			0.830888i	0.909619	+	0.415444i	$0.136374\pi$
$619$			4.67397e16i			0.830888i	−0.909619	+	0.415444i	$0.863626\pi$
$620$			1.15288e16i			0.202972i
$621$	−4.23605e16	+	1.72941e16i	−0.738604	+	0.301543i
$622$	6.93589e16			1.19773
$623$	−6.25183e16			−1.06925
$624$	−1.90788e17			−3.23180
$625$	3.01883e16			0.506476
$626$			3.35458e16i			0.557433i
$627$	5.02909e16			0.827722
$628$		−	1.86557e17i		−	3.04126i
$629$	−2.17005e15			−0.0350402
$630$	8.27193e16			1.32301
$631$		−	8.31113e16i		−	1.31669i	−0.752716	−	0.658346i	$-0.771256\pi$
$631$		−	8.31113e16i		−	1.31669i	0.752716	−	0.658346i	$-0.228744\pi$
$632$			2.47307e17i			3.88092i
$633$	3.80614e16			0.591647
$634$	−2.12819e16			−0.327699
$635$		−	8.48733e16i		−	1.29458i
$636$		−	9.38371e16i		−	1.41786i
$637$	8.24482e16			1.23408
$638$			5.58300e16i			0.827834i
$639$	−1.51216e16			−0.222123
$640$		−	1.50279e16i		−	0.218685i
$641$			1.24633e17i			1.79674i	0.439243	+	0.898368i	$0.355247\pi$
$641$			1.24633e17i			1.79674i	−0.439243	+	0.898368i	$0.644753\pi$
$642$			1.93960e17i			2.77014i
$643$			1.08518e17i			1.53545i	0.640782	+	0.767723i	$0.278610\pi$
$643$			1.08518e17i			1.53545i	−0.640782	+	0.767723i	$0.721390\pi$
$644$	−9.18614e16	−	2.25007e17i	−1.28771	−	3.15413i
$645$	−3.45862e16			−0.480335
$646$	1.36445e17			1.87742
$647$	−9.72207e16			−1.32536			−0.662679	−	0.748904i	$-0.730580\pi$
$647$	−9.72207e16			−1.32536			−0.662679	+	0.748904i	$0.730580\pi$
$648$	−2.30540e17			−3.11384
$649$		−	2.96460e16i		−	0.396733i
$650$	3.06519e17			4.06422
$651$		−	6.21910e15i		−	0.0817037i
$652$	2.78750e17			3.62852
$653$	4.27446e16			0.551318			0.275659	−	0.961256i	$-0.411104\pi$
$653$	4.27446e16			0.551318			0.275659	+	0.961256i	$0.411104\pi$
$654$			2.06080e17i			2.63372i
$655$			1.35935e17i			1.72140i
$656$	−3.61530e15			−0.0453651
$657$	1.46785e16			0.182512
$658$			8.48171e16i			1.04503i
$659$			7.60268e16i			0.928228i	0.885776	+	0.464114i	$0.153627\pi$
$659$			7.60268e16i			0.928228i	−0.885776	+	0.464114i	$0.846373\pi$
$660$	−1.56057e17			−1.88808
$661$			6.38690e16i			0.765740i	0.923802	+	0.382870i	$0.125064\pi$
$661$			6.38690e16i			0.765740i	−0.923802	+	0.382870i	$0.874936\pi$
$662$	9.93610e16			1.18051
$663$		−	6.75146e16i		−	0.794907i
$664$		−	9.74983e16i		−	1.13760i
$665$		−	3.64937e17i		−	4.21976i
$666$			2.93410e15i			0.0336225i
$667$	8.90852e16	−	3.63700e16i	1.01170	−	0.413036i
$668$	2.26579e17			2.55013
$669$	−1.23398e17			−1.37642
$670$	−4.91784e17			−5.43658
$671$	−1.77734e16			−0.194732
$672$		−	2.61244e17i		−	2.83682i
$673$	−1.14842e17			−1.23598			−0.617989	−	0.786187i	$-0.712052\pi$
$673$	−1.14842e17			−1.23598			−0.617989	+	0.786187i	$0.712052\pi$
$674$		−	1.12860e16i		−	0.120387i
$675$	1.37737e17			1.45622
$676$	1.05132e17			1.10168
$677$		−	1.35478e17i		−	1.40714i	−0.710625	−	0.703571i	$-0.751588\pi$
$677$		−	1.35478e17i		−	1.40714i	0.710625	−	0.703571i	$-0.248412\pi$
$678$		−	5.80956e15i		−	0.0598088i
$679$	−1.56794e17			−1.59997
$680$	−2.46705e17			−2.49531
$681$			5.12360e16i			0.513680i
$682$			3.84148e15i			0.0381761i
$683$	−4.08573e16			−0.402481			−0.201240	−	0.979542i	$-0.564497\pi$
$683$	−4.08573e16			−0.402481			−0.201240	+	0.979542i	$0.564497\pi$
$684$		−	1.30165e17i		−	1.27103i
$685$	−8.49185e16			−0.821976
$686$			5.84912e15i			0.0561236i
$687$		−	5.93306e16i		−	0.564337i
$688$			6.23400e16i			0.587809i
$689$			6.70689e16i			0.626911i
$690$	1.44088e17	+	3.52931e17i	1.33516	+	3.27036i
$691$	−3.76769e16			−0.346105			−0.173052	−	0.984913i	$-0.555363\pi$
$691$	−3.76769e16			−0.346105			−0.173052	+	0.984913i	$0.555363\pi$
$692$	−2.20729e16			−0.201012
$693$	1.94469e16			0.175570
$694$	2.17397e17			1.94579
$695$			2.48198e17i			2.20236i
$696$	3.64479e17			3.20640
$697$		−	1.27935e15i		−	0.0111582i
$698$	−6.68805e16			−0.578318
$699$	9.61797e16			0.824556
$700$			7.31616e17i			6.21863i
$701$		−	4.74910e16i		−	0.400224i	−0.979773	−	0.200112i	$-0.935869\pi$
$701$		−	4.74910e16i		−	0.400224i	0.979773	−	0.200112i	$-0.0641306\pi$
$702$	2.12593e17			1.77634
$703$	1.29445e16			0.107239
$704$			4.39810e16i			0.361268i
$705$		−	9.38662e16i		−	0.764495i
$706$	4.19342e17			3.38641
$707$		−	1.79378e17i		−	1.43632i
$708$	−3.32157e17			−2.63721
$709$			6.07289e16i			0.478099i	0.971007	+	0.239049i	$0.0768358\pi$
$709$			6.07289e16i			0.478099i	−0.971007	+	0.239049i	$0.923164\pi$
$710$		−	2.93408e17i		−	2.29046i
$711$			5.85332e16i			0.453091i
$712$		−	2.52106e17i		−	1.93510i
$713$	6.12966e15	−	2.50250e15i	0.0466551	−	0.0190474i
$714$	2.28398e17			1.72386
$715$	1.11540e17			0.834821
$716$	−8.46046e16			−0.627937
$717$	−2.54760e17			−1.87507
$718$		−	4.84409e17i		−	3.53562i
$719$	8.00394e16			0.579336			0.289668	−	0.957127i	$-0.406455\pi$
$719$	8.00394e16			0.579336			0.289668	+	0.957127i	$0.406455\pi$
$720$			1.65005e17i			1.18441i
$721$	−1.46091e17			−1.03995
$722$	−5.52855e17			−3.90291
$723$		−	2.79411e17i		−	1.95620i
$724$		−	5.91305e17i		−	4.10563i
$725$	−2.89663e17			−1.99464
$726$	2.55722e17			1.74642
$727$			8.83125e16i			0.598157i	0.954228	+	0.299079i	$0.0966793\pi$
$727$			8.83125e16i			0.598157i	−0.954228	+	0.299079i	$0.903321\pi$
$728$			6.57975e17i			4.41999i
$729$	8.19852e16			0.546223
$730$			2.84811e17i			1.88200i
$731$	−2.20604e16			−0.144580
$732$			1.99136e17i			1.29444i
$733$			2.07956e17i			1.34075i	0.742022	+	0.670376i	$0.233867\pi$
$733$			2.07956e17i			1.34075i	−0.742022	+	0.670376i	$0.766133\pi$
$734$			2.38515e16i			0.152525i
$735$		−	3.08675e17i		−	1.95784i
$736$	2.57487e17	−	1.05122e17i	1.61990	−	0.661343i
$737$	−1.15616e17			−0.721462
$738$	−1.72980e15			−0.0107068
$739$	2.67465e17			1.64210			0.821051	−	0.570856i	$-0.193388\pi$
$739$	2.67465e17			1.64210			0.821051	+	0.570856i	$0.193388\pi$
$740$	−4.01680e16			−0.244619
$741$			4.02730e17i			2.43279i
$742$	−2.26890e17			−1.35954
$743$			1.84339e17i			1.09568i	0.836582	+	0.547842i	$0.184551\pi$
$743$			1.84339e17i			1.09568i	−0.836582	+	0.547842i	$0.815449\pi$
$744$	2.50786e16			0.147865
$745$	−1.92921e17			−1.12835
$746$		−	6.02219e17i		−	3.49399i
$747$		−	2.30761e16i		−	0.132813i
$748$	−9.95391e16			−0.568308
$749$	3.30891e17			1.87410
$750$		−	5.18873e17i		−	2.91537i
$751$		−	9.86281e16i		−	0.549745i	−0.961481	−	0.274872i	$-0.911364\pi$
$751$		−	9.86281e16i		−	0.549745i	0.961481	−	0.274872i	$-0.0886356\pi$
$752$	−1.69190e17			−0.935550
$753$			7.92044e16i			0.434490i
$754$	−4.47087e17			−2.43312
$755$		−	4.85782e16i		−	0.262277i
$756$			5.07427e17i			2.71796i
$757$			4.19579e16i			0.222966i	0.993766	+	0.111483i	$0.0355601\pi$
$757$			4.19579e16i			0.222966i	−0.993766	+	0.111483i	$0.964440\pi$
$758$		−	1.99280e17i		−	1.05063i
$759$	3.38744e16	+	8.29724e16i	0.177182	+	0.433993i
$760$	1.47161e18			7.63682
$761$	−3.65254e17			−1.88056			−0.940279	−	0.340405i	$-0.889436\pi$
$761$	−3.65254e17			−1.88056			−0.940279	+	0.340405i	$0.889436\pi$
$762$	−3.16856e17			−1.61857
$763$	3.51567e17			1.78181
$764$			4.18900e17i			2.10645i
$765$	−5.83906e16			−0.291323
$766$			5.10915e17i			2.52916i
$767$	2.37405e17			1.16605
$768$	−2.61747e17			−1.27560
$769$		−	1.00221e16i		−	0.0484617i	−0.999706	−	0.0242309i	$-0.992286\pi$
$769$		−	1.00221e16i		−	0.0484617i	0.999706	−	0.0242309i	$-0.00771368\pi$
$770$			3.77332e17i			1.81042i
$771$	−1.38133e17			−0.657616
$772$	−4.91199e17			−2.32035
$773$			1.58288e17i			0.741944i	0.928644	+	0.370972i	$0.120975\pi$
$773$			1.58288e17i			0.741944i	−0.928644	+	0.370972i	$0.879025\pi$
$774$			2.98276e16i			0.138731i
$775$	−1.99308e16			−0.0919843
$776$		−	6.32274e17i		−	2.89558i
$777$	2.16682e16			0.0984682
$778$		−	2.35107e17i		−	1.06020i
$779$			7.63145e15i			0.0341494i
$780$		−	1.24971e18i		−	5.54932i
$781$		−	6.89789e16i		−	0.303956i
$782$	9.19048e16	+	2.25113e17i	0.401881	+	0.984374i
$783$	−2.00902e17			−0.871792
$784$	−5.56373e17			−2.39590
$785$	4.99205e17			2.13334
$786$	5.07482e17			2.15221
$787$		−	2.38932e17i		−	1.00560i	−0.864402	−	0.502801i	$-0.832303\pi$
$787$		−	2.38932e17i		−	1.00560i	0.864402	−	0.502801i	$-0.167697\pi$
$788$	5.98071e17			2.49802
$789$		−	9.73488e16i		−	0.403523i
$790$	−1.13573e18			−4.67212
$791$	−9.91096e15			−0.0404629
$792$			7.84200e16i			0.317743i
$793$		−	1.42330e17i		−	0.572344i
$794$	−3.12264e17			−1.24623
$795$	2.51097e17			0.994578
$796$			3.01375e17i			1.18476i
$797$		−	2.76378e17i		−	1.07833i	−0.842199	−	0.539167i	$-0.818739\pi$
$797$		−	2.76378e17i		−	1.07833i	0.842199	−	0.539167i	$-0.181261\pi$
$798$	−1.36241e18			−5.27583
$799$		−	5.98715e16i		−	0.230112i
$800$	−8.37227e17			−3.19377
$801$		−	5.96690e16i		−	0.225920i
$802$			5.83562e17i			2.19301i
$803$			6.69576e16i			0.249751i
$804$			1.29538e18i			4.79579i
$805$	6.02091e17	−	2.45810e17i	2.21252	−	0.903284i
$806$	−3.07626e16			−0.112205
$807$	5.71594e16			0.206941
$808$	7.23344e17			2.59942
$809$	1.84376e17			0.657679			0.328840	−	0.944386i	$-0.393342\pi$
$809$	1.84376e17			0.657679			0.328840	+	0.944386i	$0.393342\pi$
$810$		−	1.05873e18i		−	3.74865i
$811$	2.89684e17			1.01812			0.509060	−	0.860731i	$-0.329993\pi$
$811$	2.89684e17			1.01812			0.509060	+	0.860731i	$0.329993\pi$
$812$		−	1.06713e18i		−	3.72290i
$813$	−3.15124e17			−1.09129
$814$	−1.33842e16			−0.0460094
$815$			7.45902e17i			2.54528i
$816$			4.55598e17i			1.54327i
$817$	1.31592e17			0.442484
$818$	−4.27222e17			−1.42605
$819$			1.55731e17i			0.516026i
$820$		−	2.36810e16i		−	0.0778964i
$821$	3.43366e15			0.0112124			0.00560619	−	0.999984i	$-0.498215\pi$
$821$	3.43366e15			0.0112124			0.00560619	+	0.999984i	$0.498215\pi$
$822$			3.17025e17i			1.02769i
$823$	2.23087e17			0.717920			0.358960	−	0.933353i	$-0.383131\pi$
$823$	2.23087e17			0.717920			0.358960	+	0.933353i	$0.383131\pi$
$824$		−	5.89113e17i		−	1.88207i
$825$		−	2.69787e17i		−	0.855653i
$826$			8.03129e17i			2.52874i
$827$			3.95464e17i			1.23616i	0.786116	+	0.618078i	$0.212089\pi$
$827$			3.95464e17i			1.23616i	−0.786116	+	0.618078i	$0.787911\pi$
$828$	2.14752e17	−	8.76748e16i	0.666431	−	0.272078i
$829$	2.66222e17			0.820194			0.410097	−	0.912042i	$-0.365495\pi$
$829$	2.66222e17			0.820194			0.410097	+	0.912042i	$0.365495\pi$
$830$	4.47751e17			1.36952
$831$	−4.24952e17			−1.29043
$832$	−3.52200e17			−1.06182
$833$		−	1.96885e17i		−	0.589307i
$834$	9.26592e17			2.75355
$835$			6.06300e17i			1.78883i
$836$	5.93759e17			1.73929
$837$	−1.38234e16			−0.0402033
$838$		−	4.73561e17i		−	1.36745i
$839$			2.44630e17i			0.701356i	0.936496	+	0.350678i	$0.114049\pi$
$839$			2.44630e17i			0.701356i	−0.936496	+	0.350678i	$0.885951\pi$
$840$	2.46337e18			7.01219
$841$	6.86857e16			0.194129
$842$		−	8.10086e16i		−	0.227331i
$843$			3.68435e17i			1.02659i
$844$	4.49371e17			1.24323
$845$			2.81320e17i			0.772789i
$846$	−8.09516e16			−0.220802
$847$		−	4.36254e17i		−	1.18151i
$848$		−	4.52591e17i		−	1.21711i
$849$			6.09287e17i			1.62696i
$850$		−	7.31962e17i		−	1.94077i
$851$	8.71903e15	+	2.13565e16i	0.0229557	+	0.0562281i
$852$	−7.72848e17			−2.02049
$853$	−6.38967e17			−1.65876			−0.829382	−	0.558682i	$-0.811307\pi$
$853$	−6.38967e17			−1.65876			−0.829382	+	0.558682i	$0.811307\pi$
$854$	4.81494e17			1.24120
$855$	3.48305e17			0.891586
$856$			1.33432e18i			3.39171i
$857$	−4.66601e17			−1.17777			−0.588885	−	0.808217i	$-0.700433\pi$
$857$	−4.66601e17			−1.17777			−0.588885	+	0.808217i	$0.700433\pi$
$858$		−	4.16409e17i		−	1.04375i
$859$	−2.23550e17			−0.556436			−0.278218	−	0.960518i	$-0.589744\pi$
$859$	−2.23550e17			−0.556436			−0.278218	+	0.960518i	$0.589744\pi$
$860$	−4.08341e17			−1.00933
$861$			1.27745e16i			0.0313562i
$862$		−	4.00303e17i		−	0.975764i
$863$	2.98104e17			0.721611			0.360805	−	0.932641i	$-0.382502\pi$
$863$	2.98104e17			0.721611			0.360805	+	0.932641i	$0.382502\pi$
$864$	−5.80676e17			−1.39589
$865$		−	5.90644e16i		−	0.141003i
$866$		−	8.77058e16i		−	0.207932i
$867$	3.23120e17			0.760763
$868$		−	7.34257e16i		−	0.171684i
$869$	−2.67005e17			−0.620014
$870$			1.67383e18i			3.86008i
$871$		−	9.25855e17i		−	2.12048i
$872$			1.41770e18i			3.22467i
$873$		−	1.49648e17i		−	0.338054i
$874$	−5.48220e17	−	1.34282e18i	−1.22995	−	3.01265i
$875$	−8.85184e17			−1.97236
$876$	7.50201e17			1.66017
$877$	−2.02431e17			−0.444918			−0.222459	−	0.974942i	$-0.571408\pi$
$877$	−2.02431e17			−0.444918			−0.222459	+	0.974942i	$0.571408\pi$
$878$	−3.95761e17			−0.863905
$879$		−	3.56814e17i		−	0.773585i
$880$	−7.52687e17			−1.62076
$881$		−	8.81666e17i		−	1.88560i	−0.333365	−	0.942798i	$-0.608184\pi$
$881$		−	8.81666e17i		−	1.88560i	0.333365	−	0.942798i	$-0.391816\pi$
$882$	−2.66206e17			−0.565465
$883$	−2.64938e16			−0.0558958			−0.0279479	−	0.999609i	$-0.508897\pi$
$883$	−2.64938e16			−0.0558958			−0.0279479	+	0.999609i	$0.508897\pi$
$884$		−	7.97110e17i		−	1.67034i
$885$		−	8.88814e17i		−	1.84991i
$886$	2.52142e17			0.521247
$887$	−2.31474e17			−0.475291			−0.237646	−	0.971352i	$-0.576376\pi$
$887$	−2.31474e17			−0.475291			−0.237646	+	0.971352i	$0.576376\pi$
$888$			8.73772e16i			0.178205i
$889$			5.40547e17i			1.09502i
$890$	1.15777e18			2.32961
$891$		−	2.48902e17i		−	0.497465i
$892$	−1.45690e18			−2.89227
$893$			3.57138e17i			0.704252i
$894$			7.20229e17i			1.41073i
$895$		−	2.26392e17i		−	0.440476i
$896$			9.57110e16i			0.184975i
$897$	−6.64444e17	+	2.71266e17i	−1.27557	+	0.520764i
$898$	−7.49397e17			−1.42907
$899$	2.90709e16			0.0550681
$900$	−6.98272e17			−1.31392
$901$	1.60159e17			0.299367
$902$		−	7.89066e15i		−	0.0146512i
$903$	2.20275e17			0.406292
$904$		−	3.99661e16i		−	0.0732286i
$905$	1.58226e18			2.87996
$906$	−1.81356e17			−0.327916
$907$			4.59338e17i			0.825067i	0.910943	+	0.412533i	$0.135356\pi$
$907$			4.59338e17i			0.825067i	−0.910943	+	0.412533i	$0.864644\pi$
$908$			6.04917e17i			1.07940i
$909$	1.71203e17			0.303478
$910$	−3.02168e18			−5.32109
$911$			1.48072e17i			0.259038i	0.991577	+	0.129519i	$0.0413433\pi$
$911$			1.48072e17i			0.259038i	−0.991577	+	0.129519i	$0.958657\pi$
$912$		−	2.71768e18i		−	4.72313i
$913$	1.05264e17			0.181742
$914$			3.76398e17i			0.645610i
$915$	−5.32864e17			−0.908008
$916$		−	7.00486e17i		−	1.18584i
$917$		−	8.65750e17i		−	1.45605i
$918$		−	5.07667e17i		−	0.848248i
$919$		−	1.28042e17i		−	0.212549i	−0.994337	−	0.106275i	$-0.966108\pi$
$919$		−	1.28042e17i		−	0.212549i	0.994337	−	0.106275i	$-0.0338923\pi$
$920$	9.91233e17	+	2.42794e18i	1.63474	+	4.00416i
$921$	−1.30060e18			−2.13101
$922$	−1.57843e16			−0.0256944
$923$	5.52384e17			0.893369
$924$	9.93907e17			1.59703
$925$		−	6.94414e16i		−	0.110858i
$926$	2.15009e18			3.41029
$927$		−	1.39433e17i		−	0.219728i
$928$	1.22117e18			1.91201
$929$	1.98810e16			0.0309274			0.0154637	−	0.999880i	$-0.495078\pi$
$929$	1.98810e16			0.0309274			0.0154637	+	0.999880i	$0.495078\pi$
$930$			1.15171e17i			0.178010i
$931$			1.17443e18i			1.80356i
$932$	1.13554e18			1.73264
$933$	4.88867e17			0.741141
$934$		−	3.55926e17i		−	0.536140i
$935$		−	2.66355e17i		−	0.398649i
$936$	−6.27988e17			−0.933891
$937$			7.49480e17i			1.10745i	0.832701	+	0.553723i	$0.186793\pi$
$937$			7.49480e17i			1.10745i	−0.832701	+	0.553723i	$0.813207\pi$
$938$	3.13211e18			4.59854
$939$			2.36443e17i			0.344932i
$940$		−	1.10823e18i		−	1.60643i
$941$		−	5.83078e17i		−	0.839825i	−0.907565	−	0.419913i	$-0.862061\pi$
$941$		−	5.83078e17i		−	0.839825i	0.907565	−	0.419913i	$-0.137939\pi$
$942$		−	1.86367e18i		−	2.66725i
$943$	−1.25907e16	+	5.14031e15i	−0.0179053	+	0.00731002i
$944$	−1.60205e18			−2.26383
$945$	−1.35781e18			−1.90656
$946$	−1.36062e17			−0.189841
$947$	−1.14754e17			−0.159099			−0.0795494	−	0.996831i	$-0.525348\pi$
$947$	−1.14754e17			−0.159099			−0.0795494	+	0.996831i	$0.525348\pi$
$948$			2.99156e18i			4.12143i
$949$	−5.36197e17			−0.734053
$950$			4.36621e18i			5.93968i
$951$	−1.50003e17			−0.202776
$952$	1.57123e18			2.11066
$953$			2.85930e17i			0.381682i	0.981621	+	0.190841i	$0.0611215\pi$
$953$			2.85930e17i			0.381682i	−0.981621	+	0.190841i	$0.938878\pi$
$954$		−	2.16550e17i		−	0.287255i
$955$	−1.12093e18			−1.47760
$956$	−3.00782e18			−3.94008
$957$			3.93510e17i			0.512252i
$958$			2.81792e17i			0.364531i
$959$	5.40835e17			0.695270
$960$			1.31859e18i			1.68454i
$961$	−7.85663e17			−0.997460
$962$		−	1.07181e17i		−	0.135228i
$963$			3.15811e17i			0.395976i
$964$		−	3.29886e18i		−	4.11056i
$965$		−	1.31439e18i		−	1.62765i
$966$	−9.17678e17	−	2.24777e18i	−1.12935	−	2.76624i
$967$	−8.48205e17			−1.03739			−0.518695	−	0.854959i	$-0.673582\pi$
$967$	−8.48205e17			−1.03739			−0.518695	+	0.854959i	$0.673582\pi$
$968$	1.75920e18			2.13827
$969$	9.61712e17			1.16172
$970$	2.90365e18			3.48590
$971$		−	1.64200e18i		−	1.95911i	−0.201189	−	0.979553i	$-0.564480\pi$
$971$		−	1.64200e18i		−	1.95911i	0.201189	−	0.979553i	$-0.435520\pi$
$972$	−1.17656e18			−1.39513
$973$		−	1.58074e18i		−	1.86287i
$974$	−9.27315e17			−1.08611
$975$	2.16046e18			2.51489
$976$			9.60463e17i			1.11117i
$977$			1.24577e18i			1.43242i	0.697887	+	0.716208i	$0.254124\pi$
$977$			1.24577e18i			1.43242i	−0.697887	+	0.716208i	$0.745876\pi$
$978$	2.78466e18			3.18228
$979$	2.72186e17			0.309151
$980$		−	3.64436e18i		−	4.11401i
$981$			3.35544e17i			0.376475i
$982$	3.14009e17			0.350165
$983$		−	2.00181e17i		−	0.221872i	−0.993828	−	0.110936i	$-0.964615\pi$
$983$		−	2.00181e17i		−	0.221872i	0.993828	−	0.110936i	$-0.0353849\pi$
$984$	−5.15132e16			−0.0567477
$985$			1.60037e18i			1.75228i
$986$			1.06763e18i			1.16188i
$987$			5.97822e17i			0.646650i
$988$			4.75483e18i			5.11203i
$989$	8.86363e16	+	2.17107e17i	0.0947182	+	0.232004i
$990$	−3.60136e17			−0.382521
$991$	−7.33300e17			−0.774176			−0.387088	−	0.922043i	$-0.626519\pi$
$991$	−7.33300e17			−0.774176			−0.387088	+	0.922043i	$0.626519\pi$
$992$	8.40250e16			0.0881736
$993$	7.00333e17			0.730481
$994$			1.86868e18i			1.93739i
$995$	−8.06445e17			−0.831067
$996$		−	1.17939e18i		−	1.20810i
$997$	−1.55600e18			−1.58430			−0.792150	−	0.610326i	$-0.791038\pi$
$997$	−1.55600e18			−1.58430			−0.792150	+	0.610326i	$0.791038\pi$
$998$	−9.15763e17			−0.926830
$999$		−	4.81625e16i		−	0.0484525i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.c.22.20	yes	20
23.22	odd	2	inner	23.13.b.c.22.19	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.c.22.19	✓	20	23.22	odd	2	inner
23.13.b.c.22.20	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+117.945 q^{2} +831.317 q^{3} +9814.93 q^{4} +26263.6i q^{5} +98049.3 q^{6} -167269. i q^{7} +674517. q^{8} +159646. q^{9} +O(q^{10})\)	\(q+117.945 q^{2} +831.317 q^{3} +9814.93 q^{4} +26263.6i q^{5} +98049.3 q^{6} -167269. i q^{7} +674517. q^{8} +159646. q^{9} +3.09765e6i q^{10} +728242. i q^{11} +8.15931e6 q^{12} -5.83177e6 q^{13} -1.97285e7i q^{14} +2.18333e7i q^{15} +3.93536e7 q^{16} +1.39261e7i q^{17} +1.88294e7 q^{18} -8.30706e7i q^{19} +2.57775e8i q^{20} -1.39054e8i q^{21} +8.58922e7i q^{22} +(1.37054e8 - 5.59538e7i) q^{23} +5.60737e8 q^{24} -4.45635e8 q^{25} -6.87825e8 q^{26} -3.09079e8 q^{27} -1.64174e9i q^{28} +6.50000e8 q^{29} +2.57513e9i q^{30} +4.47244e7 q^{31} +1.87873e9 q^{32} +6.05399e8i q^{33} +1.64251e9i q^{34} +4.39309e9 q^{35} +1.56692e9 q^{36} +1.55826e8i q^{37} -9.79773e9i q^{38} -4.84804e9 q^{39} +1.77152e10i q^{40} -9.18670e7 q^{41} -1.64006e10i q^{42} +1.58410e9i q^{43} +7.14764e9i q^{44} +4.19288e9i q^{45} +(1.61648e10 - 6.59944e9i) q^{46} -4.29921e9 q^{47} +3.27153e10 q^{48} -1.41378e10 q^{49} -5.25602e10 q^{50} +1.15770e10i q^{51} -5.72384e10 q^{52} -1.15006e10i q^{53} -3.64542e10 q^{54} -1.91262e10 q^{55} -1.12826e11i q^{56} -6.90580e10i q^{57} +7.66640e10 q^{58} -4.07090e10 q^{59} +2.14293e11i q^{60} +2.44060e10i q^{61} +5.27500e9 q^{62} -2.67039e10i q^{63} +6.03934e10 q^{64} -1.53163e11i q^{65} +7.14036e10i q^{66} +1.58761e11i q^{67} +1.36684e11i q^{68} +(1.13935e11 - 4.65153e10i) q^{69} +5.18142e11 q^{70} -9.47198e10 q^{71} +1.07684e11 q^{72} +9.19442e10 q^{73} +1.83788e10i q^{74} -3.70464e11 q^{75} -8.15332e11i q^{76} +1.21813e11 q^{77} -5.71801e11 q^{78} +3.66644e11i q^{79} +1.03357e12i q^{80} -3.41785e11 q^{81} -1.08352e10 q^{82} -1.44546e11i q^{83} -1.36480e12i q^{84} -3.65750e11 q^{85} +1.86836e11i q^{86} +5.40356e11 q^{87} +4.91211e11i q^{88} -3.73758e11i q^{89} +4.94527e11i q^{90} +9.75476e11i q^{91} +(1.34518e12 - 5.49182e11i) q^{92} +3.71801e10 q^{93} -5.07069e11 q^{94} +2.18173e12 q^{95} +1.56182e12 q^{96} -9.37374e11i q^{97} -1.66747e12 q^{98} +1.16261e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) 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

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