LMFDB - Embedded newform 23.13.b.c.22.17

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			22.17
Root			$345.826 + 9266.28i$ of defining polynomial
Character	$\chi$	$=$	23.22
Dual form			23.13.b.c.22.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+91.3205 q^{2} -242.505 q^{3} +4243.43 q^{4} -9266.28i q^{5} -22145.7 q^{6} -93897.2i q^{7} +13463.2 q^{8} -472632. q^{9} +O(q^{10})$	\(q+91.3205 q^{2} -242.505 q^{3} +4243.43 q^{4} -9266.28i q^{5} -22145.7 q^{6} -93897.2i q^{7} +13463.2 q^{8} -472632. q^{9} -846201. i q^{10} +272509. i q^{11} -1.02905e6 q^{12} -344835. q^{13} -8.57474e6i q^{14} +2.24712e6i q^{15} -1.61516e7 q^{16} -4.46231e7i q^{17} -4.31610e7 q^{18} -5.25061e7i q^{19} -3.93208e7i q^{20} +2.27706e7i q^{21} +2.48857e7i q^{22} +(-8.59701e7 + 1.20515e8i) q^{23} -3.26489e6 q^{24} +1.58277e8 q^{25} -3.14905e7 q^{26} +2.43493e8 q^{27} -3.98446e8i q^{28} +5.91320e7 q^{29} +2.05208e8i q^{30} -4.72590e8 q^{31} -1.53012e9 q^{32} -6.60849e7i q^{33} -4.07500e9i q^{34} -8.70078e8 q^{35} -2.00558e9 q^{36} +1.00652e9i q^{37} -4.79488e9i q^{38} +8.36243e7 q^{39} -1.24753e8i q^{40} +2.53266e9 q^{41} +2.07942e9i q^{42} -1.64621e9i q^{43} +1.15637e9i q^{44} +4.37954e9i q^{45} +(-7.85083e9 + 1.10054e10i) q^{46} +6.65977e9 q^{47} +3.91685e9 q^{48} +5.02460e9 q^{49} +1.44539e10 q^{50} +1.08213e10i q^{51} -1.46328e9 q^{52} +3.43262e10i q^{53} +2.22359e10 q^{54} +2.52515e9 q^{55} -1.26415e9i q^{56} +1.27330e10i q^{57} +5.39996e9 q^{58} +4.10502e10 q^{59} +9.53550e9i q^{60} +3.36671e8i q^{61} -4.31572e10 q^{62} +4.43788e10i q^{63} -7.35741e10 q^{64} +3.19534e9i q^{65} -6.03490e9i q^{66} -1.47156e11i q^{67} -1.89355e11i q^{68} +(2.08482e10 - 2.92254e10i) q^{69} -7.94559e10 q^{70} -6.35591e10 q^{71} -6.36313e9 q^{72} -3.59793e10 q^{73} +9.19154e10i q^{74} -3.83829e10 q^{75} -2.22806e11i q^{76} +2.55878e10 q^{77} +7.63661e9 q^{78} +3.10679e11i q^{79} +1.49665e11i q^{80} +1.92128e11 q^{81} +2.31284e11 q^{82} -1.78215e11i q^{83} +9.66252e10i q^{84} -4.13490e11 q^{85} -1.50333e11i q^{86} -1.43398e10 q^{87} +3.66884e9i q^{88} -5.78171e11i q^{89} +3.99942e11i q^{90} +3.23790e10i q^{91} +(-3.64808e11 + 5.11395e11i) q^{92} +1.14606e11 q^{93} +6.08173e11 q^{94} -4.86536e11 q^{95} +3.71062e11 q^{96} -9.04646e11i q^{97} +4.58849e11 q^{98} -1.28797e11i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	91.3205			1.42688			0.713441	−	0.700715i	$-0.247136\pi$
$2$	91.3205			1.42688			0.713441	+	0.700715i	$0.247136\pi$
$3$	−242.505			−0.332655			−0.166327	−	0.986071i	$-0.553191\pi$
$3$	−242.505			−0.332655			−0.166327	+	0.986071i	$0.553191\pi$
$4$	4243.43			1.03599
$5$		−	9266.28i		−	0.593042i	−0.955026	−	0.296521i	$-0.904174\pi$
$5$		−	9266.28i		−	0.593042i	0.955026	−	0.296521i	$-0.0958264\pi$
$6$	−22145.7			−0.474659
$7$		−	93897.2i		−	0.798113i	−0.916926	−	0.399056i	$-0.869338\pi$
$7$		−	93897.2i		−	0.798113i	0.916926	−	0.399056i	$-0.130662\pi$
$8$	13463.2			0.0513579
$9$	−472632.			−0.889341
$10$		−	846201.i		−	0.846201i
$11$			272509.i			0.153824i	0.997038	+	0.0769122i	$0.0245061\pi$
$11$			272509.i			0.153824i	−0.997038	+	0.0769122i	$0.975494\pi$
$12$	−1.02905e6			−0.344628
$13$	−344835.			−0.0714416			−0.0357208	−	0.999362i	$-0.511373\pi$
$13$	−344835.			−0.0714416			−0.0357208	+	0.999362i	$0.511373\pi$
$14$		−	8.57474e6i		−	1.13881i
$15$			2.24712e6i			0.197278i
$16$	−1.61516e7			−0.962711
$17$		−	4.46231e7i		−	1.84870i	−0.381547	−	0.924350i	$-0.624609\pi$
$17$		−	4.46231e7i		−	1.84870i	0.381547	−	0.924350i	$-0.375391\pi$
$18$	−4.31610e7			−1.26898
$19$		−	5.25061e7i		−	1.11606i	−0.829820	−	0.558031i	$-0.811557\pi$
$19$		−	5.25061e7i		−	1.11606i	0.829820	−	0.558031i	$-0.188443\pi$
$20$		−	3.93208e7i		−	0.614387i
$21$			2.27706e7i			0.265496i
$22$			2.48857e7i			0.219489i
$23$	−8.59701e7	+	1.20515e8i	−0.580738	+	0.814090i
$23$	−8.59701e7	+	1.20515e8i	−0.580738	+	0.814090i
$24$	−3.26489e6			−0.0170844
$25$	1.58277e8			0.648301
$26$	−3.14905e7			−0.101939
$27$	2.43493e8			0.628498
$28$		−	3.98446e8i		−	0.826840i
$29$	5.91320e7			0.0994110			0.0497055	−	0.998764i	$-0.484172\pi$
$29$	5.91320e7			0.0994110			0.0497055	+	0.998764i	$0.484172\pi$
$30$			2.05208e8i			0.281493i
$31$	−4.72590e8			−0.532494			−0.266247	−	0.963905i	$-0.585784\pi$
$31$	−4.72590e8			−0.532494			−0.266247	+	0.963905i	$0.585784\pi$
$32$	−1.53012e9			−1.42503
$33$		−	6.60849e7i		−	0.0511704i
$34$		−	4.07500e9i		−	2.63788i
$35$	−8.70078e8			−0.473314
$36$	−2.00558e9			−0.921351
$37$			1.00652e9i			0.392292i	0.980575	+	0.196146i	$0.0628427\pi$
$37$			1.00652e9i			0.392292i	−0.980575	+	0.196146i	$0.937157\pi$
$38$		−	4.79488e9i		−	1.59249i
$39$	8.36243e7			0.0237654
$40$		−	1.24753e8i		−	0.0304574i
$41$	2.53266e9			0.533180			0.266590	−	0.963810i	$-0.414103\pi$
$41$	2.53266e9			0.533180			0.266590	+	0.963810i	$0.414103\pi$
$42$			2.07942e9i			0.378832i
$43$		−	1.64621e9i		−	0.260421i	−0.991486	−	0.130210i	$-0.958435\pi$
$43$		−	1.64621e9i		−	0.260421i	0.991486	−	0.130210i	$-0.0415653\pi$
$44$			1.15637e9i			0.159361i
$45$			4.37954e9i			0.527416i
$46$	−7.85083e9	+	1.10054e10i	−0.828645	+	1.16161i
$47$	6.65977e9			0.617834			0.308917	−	0.951089i	$-0.400033\pi$
$47$	6.65977e9			0.617834			0.308917	+	0.951089i	$0.400033\pi$
$48$	3.91685e9			0.320250
$49$	5.02460e9			0.363016
$50$	1.44539e10			0.925050
$51$			1.08213e10i			0.614978i
$52$	−1.46328e9			−0.0740130
$53$			3.43262e10i			1.54871i	0.632751	+	0.774355i	$0.281926\pi$
$53$			3.43262e10i			1.54871i	−0.632751	+	0.774355i	$0.718074\pi$
$54$	2.22359e10			0.896793
$55$	2.52515e9			0.0912242
$56$		−	1.26415e9i		−	0.0409894i
$57$			1.27330e10i			0.371263i
$58$	5.39996e9			0.141848
$59$	4.10502e10			0.973203			0.486602	−	0.873624i	$-0.338236\pi$
$59$	4.10502e10			0.973203			0.486602	+	0.873624i	$0.338236\pi$
$60$			9.53550e9i			0.204379i
$61$			3.36671e8i			0.00653471i	0.999995	+	0.00326735i	$0.00104003\pi$
$61$			3.36671e8i			0.00653471i	−0.999995	+	0.00326735i	$0.998960\pi$
$62$	−4.31572e10			−0.759806
$63$			4.43788e10i			0.709794i
$64$	−7.35741e10			−1.07064
$65$			3.19534e9i			0.0423678i
$66$		−	6.03490e9i		−	0.0730141i
$67$		−	1.47156e11i		−	1.62678i	−0.581718	−	0.813391i	$-0.697619\pi$
$67$		−	1.47156e11i		−	1.62678i	0.581718	−	0.813391i	$-0.302381\pi$
$68$		−	1.89355e11i		−	1.91524i
$69$	2.08482e10	−	2.92254e10i	0.193185	−	0.270811i
$70$	−7.94559e10			−0.675364
$71$	−6.35591e10			−0.496167			−0.248083	−	0.968739i	$-0.579801\pi$
$71$	−6.35591e10			−0.496167			−0.248083	+	0.968739i	$0.579801\pi$
$72$	−6.36313e9			−0.0456747
$73$	−3.59793e10			−0.237747			−0.118874	−	0.992909i	$-0.537928\pi$
$73$	−3.59793e10			−0.237747			−0.118874	+	0.992909i	$0.537928\pi$
$74$			9.19154e10i			0.559755i
$75$	−3.83829e10			−0.215660
$76$		−	2.22806e11i		−	1.15623i
$77$	2.55878e10			0.122769
$78$	7.63661e9			0.0339104
$79$			3.10679e11i			1.27806i	0.769184	+	0.639028i	$0.220663\pi$
$79$			3.10679e11i			1.27806i	−0.769184	+	0.639028i	$0.779337\pi$
$80$			1.49665e11i			0.570928i
$81$	1.92128e11			0.680268
$82$	2.31284e11			0.760785
$83$		−	1.78215e11i		−	0.545100i	−0.962142	−	0.272550i	$-0.912133\pi$
$83$		−	1.78215e11i		−	0.545100i	0.962142	−	0.272550i	$-0.0878669\pi$
$84$			9.66252e10i			0.275052i
$85$	−4.13490e11			−1.09636
$86$		−	1.50333e11i		−	0.371590i
$87$	−1.43398e10			−0.0330695
$88$			3.66884e9i			0.00790009i
$89$		−	5.78171e11i		−	1.16336i	−0.813416	−	0.581682i	$-0.802395\pi$
$89$		−	5.78171e11i		−	1.16336i	0.813416	−	0.581682i	$-0.197605\pi$
$90$			3.99942e11i			0.752561i
$91$			3.23790e10i			0.0570184i
$92$	−3.64808e11	+	5.11395e11i	−0.601641	+	0.843392i
$93$	1.14606e11			0.177137
$94$	6.08173e11			0.881577
$95$	−4.86536e11			−0.661872
$96$	3.71062e11			0.474044
$97$		−	9.04646e11i		−	1.08605i	−0.839718	−	0.543023i	$-0.817280\pi$
$97$		−	9.04646e11i		−	1.08605i	0.839718	−	0.543023i	$-0.182720\pi$
$98$	4.58849e11			0.517981
$99$		−	1.28797e11i		−	0.136802i
$100$	6.71636e11			0.671636
$101$	−1.46799e12			−1.38291			−0.691454	−	0.722420i	$-0.743030\pi$
$101$	−1.46799e12			−1.38291			−0.691454	+	0.722420i	$0.743030\pi$
$102$			9.88210e11i			0.877502i
$103$		−	3.29120e11i		−	0.275633i	−0.990458	−	0.137817i	$-0.955992\pi$
$103$		−	3.29120e11i		−	0.275633i	0.990458	−	0.137817i	$-0.0440084\pi$
$104$	−4.64257e9			−0.00366909
$105$	2.10998e11			0.157450
$106$			3.13468e12i			2.20983i
$107$		−	1.09043e12i		−	0.726602i	−0.931672	−	0.363301i	$-0.881650\pi$
$107$		−	1.09043e12i		−	0.726602i	0.931672	−	0.363301i	$-0.118350\pi$
$108$	1.03325e12			0.651120
$109$		−	7.53265e10i		−	0.0449147i	−0.999748	−	0.0224574i	$-0.992851\pi$
$109$		−	7.53265e10i		−	0.0449147i	0.999748	−	0.0224574i	$-0.00714900\pi$
$110$	2.30597e11			0.130166
$111$		−	2.44085e11i		−	0.130498i
$112$			1.51659e12i			0.768352i
$113$		−	1.70015e12i		−	0.816614i	−0.912845	−	0.408307i	$-0.866119\pi$
$113$		−	1.70015e12i		−	0.816614i	0.912845	−	0.408307i	$-0.133881\pi$
$114$			1.16278e12i			0.529749i
$115$	1.11672e12	+	7.96623e11i	0.482790	+	0.344402i
$116$	2.50922e11			0.102989
$117$	1.62980e11			0.0635359
$118$	3.74873e12			1.38865
$119$	−4.18998e12			−1.47547
$120$			3.02534e10i			0.0101318i
$121$	3.06417e12			0.976338
$122$			3.07449e10i			0.00932426i
$123$	−6.14183e11			−0.177365
$124$	−2.00540e12			−0.551660
$125$		−	3.72891e12i		−	0.977512i
$126$			4.05270e12i			1.01279i
$127$	−2.55365e12			−0.608610			−0.304305	−	0.952575i	$-0.598424\pi$
$127$	−2.55365e12			−0.608610			−0.304305	+	0.952575i	$0.598424\pi$
$128$	−4.51456e11			−0.102649
$129$			3.99215e11i			0.0866302i
$130$			2.91800e11i			0.0604539i
$131$	−2.07746e12			−0.411060			−0.205530	−	0.978651i	$-0.565892\pi$
$131$	−2.07746e12			−0.411060			−0.205530	+	0.978651i	$0.565892\pi$
$132$		−	2.80426e11i		−	0.0530122i
$133$	−4.93018e12			−0.890744
$134$		−	1.34384e13i		−	2.32123i
$135$		−	2.25627e12i		−	0.372726i
$136$		−	6.00768e11i		−	0.0949453i
$137$		−	7.97456e12i		−	1.20610i	−0.797704	−	0.603050i	$-0.793952\pi$
$137$		−	7.97456e12i		−	1.20610i	0.797704	−	0.603050i	$-0.206048\pi$
$138$	1.90387e12	−	2.66888e12i	0.275653	−	0.386415i
$139$	−4.12524e12			−0.571953			−0.285976	−	0.958237i	$-0.592318\pi$
$139$	−4.12524e12			−0.571953			−0.285976	+	0.958237i	$0.592318\pi$
$140$	−3.69211e12			−0.490350
$141$	−1.61503e12			−0.205525
$142$	−5.80425e12			−0.707972
$143$		−	9.39706e10i		−	0.0109894i
$144$	7.63377e12			0.856179
$145$		−	5.47933e11i		−	0.0589549i
$146$	−3.28565e12			−0.339238
$147$	−1.21849e12			−0.120759
$148$			4.27107e12i			0.406412i
$149$		−	9.14512e12i		−	0.835741i	−0.908507	−	0.417871i	$-0.862777\pi$
$149$		−	9.14512e12i		−	0.835741i	0.908507	−	0.417871i	$-0.137223\pi$
$150$	−3.50515e12			−0.307722
$151$	1.60655e13			1.35529			0.677644	−	0.735391i	$-0.263001\pi$
$151$	1.60655e13			1.35529			0.677644	+	0.735391i	$0.263001\pi$
$152$		−	7.06899e11i		−	0.0573186i
$153$			2.10903e13i			1.64412i
$154$	2.33669e12			0.175177
$155$			4.37915e12i			0.315791i
$156$	3.54854e11			0.0246208
$157$			2.16633e13i			1.44653i	0.690573	+	0.723263i	$0.257359\pi$
$157$			2.16633e13i			1.44653i	−0.690573	+	0.723263i	$0.742641\pi$
$158$			2.83714e13i			1.82364i
$159$		−	8.32428e12i		−	0.515186i
$160$			1.41785e13i			0.845105i
$161$	1.13160e13	+	8.07235e12i	0.649736	+	0.463495i
$162$	1.75452e13			0.970662
$163$	2.58844e13			1.38011			0.690054	−	0.723758i	$-0.257587\pi$
$163$	2.58844e13			1.38011			0.690054	+	0.723758i	$0.257587\pi$
$164$	1.07472e13			0.552371
$165$	−6.12361e11			−0.0303462
$166$		−	1.62747e13i		−	0.777793i
$167$	3.44527e13			1.58827			0.794136	−	0.607740i	$-0.207924\pi$
$167$	3.44527e13			1.58827			0.794136	+	0.607740i	$0.207924\pi$
$168$			3.06564e11i			0.0136353i
$169$	−2.31792e13			−0.994896
$170$	−3.77601e13			−1.56437
$171$			2.48161e13i			0.992560i
$172$		−	6.98559e12i		−	0.269794i
$173$	−1.32221e13			−0.493199			−0.246600	−	0.969117i	$-0.579313\pi$
$173$	−1.32221e13			−0.493199			−0.246600	+	0.969117i	$0.579313\pi$
$174$	−1.30952e12			−0.0471863
$175$		−	1.48617e13i		−	0.517418i
$176$		−	4.40146e12i		−	0.148088i
$177$	−9.95490e12			−0.323741
$178$		−	5.27988e13i		−	1.65998i
$179$	−5.21209e13			−1.58450			−0.792252	−	0.610194i	$-0.791092\pi$
$179$	−5.21209e13			−1.58450			−0.792252	+	0.610194i	$0.791092\pi$
$180$			1.85843e13i			0.546400i
$181$		−	4.95505e13i		−	1.40921i	−0.709598	−	0.704607i	$-0.751123\pi$
$181$		−	4.95505e13i		−	1.40921i	0.709598	−	0.704607i	$-0.248877\pi$
$182$			2.95687e12i			0.0813586i
$183$		−	8.16444e10i		−	0.00217380i
$184$	−1.15743e12	+	1.62251e12i	−0.0298255	+	0.0418100i
$185$	9.32665e12			0.232646
$186$	1.04658e13			0.252753
$187$	1.21602e13			0.284375
$188$	2.82602e13			0.640072
$189$		−	2.28633e13i		−	0.501612i
$190$	−4.44307e13			−0.944413
$191$			7.36959e13i			1.51790i	0.651149	+	0.758950i	$0.274287\pi$
$191$			7.36959e13i			1.51790i	−0.651149	+	0.758950i	$0.725713\pi$
$192$	1.78421e13			0.356155
$193$	−4.79301e13			−0.927395			−0.463698	−	0.885994i	$-0.653478\pi$
$193$	−4.79301e13			−0.927395			−0.463698	+	0.885994i	$0.653478\pi$
$194$		−	8.26127e13i		−	1.54966i
$195$		−	7.74886e11i		−	0.0140939i
$196$	2.13215e13			0.376082
$197$	5.30783e13			0.908071			0.454035	−	0.890984i	$-0.349984\pi$
$197$	5.30783e13			0.908071			0.454035	+	0.890984i	$0.349984\pi$
$198$		−	1.17618e13i		−	0.195201i
$199$		−	5.55668e12i		−	0.0894741i	−0.998999	−	0.0447370i	$-0.985755\pi$
$199$		−	5.55668e12i		−	0.0894741i	0.998999	−	0.0447370i	$-0.0142450\pi$
$200$	2.13091e12			0.0332954
$201$			3.56861e13i			0.541156i
$202$	−1.34057e14			−1.97325
$203$		−	5.55233e12i		−	0.0793412i
$204$			4.59196e13i			0.637113i
$205$		−	2.34683e13i		−	0.316198i
$206$		−	3.00554e13i		−	0.393296i
$207$	4.06323e13	−	5.69591e13i	0.516474	−	0.724004i
$208$	5.56964e12			0.0687776
$209$	1.43084e13			0.171677
$210$	1.92685e13			0.224663
$211$	1.99945e12			0.0226577			0.0113289	−	0.999936i	$-0.496394\pi$
$211$	1.99945e12			0.0226577			0.0113289	+	0.999936i	$0.496394\pi$
$212$			1.45661e14i			1.60445i
$213$	1.54134e13			0.165052
$214$		−	9.95789e13i		−	1.03678i
$215$	−1.52543e13			−0.154440
$216$	3.27819e12			0.0322783
$217$			4.43749e13i			0.424990i
$218$		−	6.87885e12i		−	0.0640880i
$219$	8.72518e12			0.0790878
$220$	1.07153e13			0.0945077
$221$			1.53876e13i			0.132074i
$222$		−	2.22900e13i		−	0.186205i
$223$	2.35255e13			0.191298			0.0956490	−	0.995415i	$-0.469507\pi$
$223$	2.35255e13			0.191298			0.0956490	+	0.995415i	$0.469507\pi$
$224$			1.43674e14i			1.13734i
$225$	−7.48067e13			−0.576561
$226$		−	1.55259e14i		−	1.16521i
$227$			5.13267e13i			0.375135i	0.982252	+	0.187568i	$0.0600604\pi$
$227$			5.13267e13i			0.375135i	−0.982252	+	0.187568i	$0.939940\pi$
$228$			5.40316e13i			0.384626i
$229$			2.12886e12i			0.0147616i	0.999973	+	0.00738080i	$0.00234940\pi$
$229$			2.12886e12i			0.0147616i	−0.999973	+	0.00738080i	$0.997651\pi$
$230$	1.01980e14	+	7.27480e13i	0.688884	+	0.491421i
$231$	−6.20519e12			−0.0408397
$232$	7.96104e11			0.00510554
$233$	1.27896e14			0.799321			0.399660	−	0.916663i	$-0.369128\pi$
$233$	1.27896e14			0.799321			0.399660	+	0.916663i	$0.369128\pi$
$234$	1.48834e13			0.0906583
$235$		−	6.17113e13i		−	0.366402i
$236$	1.74194e14			1.00823
$237$		−	7.53414e13i		−	0.425151i
$238$	−3.82631e14			−2.10532
$239$	−9.78555e12			−0.0525047			−0.0262523	−	0.999655i	$-0.508357\pi$
$239$	−9.78555e12			−0.0525047			−0.0262523	+	0.999655i	$0.508357\pi$
$240$		−	3.62946e13i		−	0.189922i
$241$			1.45233e14i			0.741248i	0.928783	+	0.370624i	$0.120856\pi$
$241$			1.45233e14i			0.741248i	−0.928783	+	0.370624i	$0.879144\pi$
$242$	2.79821e14			1.39312
$243$	−1.75994e14			−0.854792
$244$			1.42864e12i			0.00676991i
$245$		−	4.65594e13i		−	0.215283i
$246$	−5.60875e13			−0.253079
$247$			1.81059e13i			0.0797332i
$248$	−6.36256e12			−0.0273478
$249$			4.32181e13i			0.181330i
$250$		−	3.40526e14i		−	1.39479i
$251$			4.68947e14i			1.87535i	0.347519	+	0.937673i	$0.387024\pi$
$251$			4.68947e14i			1.87535i	−0.347519	+	0.937673i	$0.612976\pi$
$252$			1.88318e14i			0.735342i
$253$	−3.28413e13	−	2.34276e13i	−0.125227	−	0.0893317i
$254$	−2.33200e14			−0.868414
$255$	1.00274e14			0.364708
$256$	2.60132e14			0.924176
$257$	4.08953e14			1.41930			0.709651	−	0.704553i	$-0.248853\pi$
$257$	4.08953e14			1.41930			0.709651	+	0.704553i	$0.248853\pi$
$258$			3.64565e13i			0.123611i
$259$	9.45089e13			0.313094
$260$			1.35592e13i			0.0438928i
$261$	−2.79477e13			−0.0884103
$262$	−1.89715e14			−0.586535
$263$			1.67884e14i			0.507313i	0.967294	+	0.253656i	$0.0816332\pi$
$263$			1.67884e14i			0.507313i	−0.967294	+	0.253656i	$0.918367\pi$
$264$		−	8.89712e11i		−	0.00262800i
$265$	3.18076e14			0.918450
$266$	−4.50226e14			−1.27099
$267$			1.40209e14i			0.386999i
$268$		−	6.24446e14i		−	1.68533i
$269$	4.16212e14			1.09850			0.549251	−	0.835657i	$-0.314913\pi$
$269$	4.16212e14			1.09850			0.549251	+	0.835657i	$0.314913\pi$
$270$		−	2.06044e14i		−	0.531836i
$271$	−3.64723e14			−0.920763			−0.460381	−	0.887721i	$-0.652287\pi$
$271$	−3.64723e14			−0.920763			−0.460381	+	0.887721i	$0.652287\pi$
$272$			7.20735e14i			1.77976i
$273$		−	7.85208e12i		−	0.0189675i
$274$		−	7.28240e14i		−	1.72096i
$275$			4.31318e13i			0.0997245i
$276$	8.84679e13	−	1.24016e14i	0.200139	−	0.280558i
$277$	−7.53943e14			−1.66901			−0.834507	−	0.550997i	$-0.814248\pi$
$277$	−7.53943e14			−1.66901			−0.834507	+	0.550997i	$0.814248\pi$
$278$	−3.76719e14			−0.816109
$279$	2.23361e14			0.473569
$280$	−1.17140e13			−0.0243084
$281$		−	3.09594e14i		−	0.628862i	−0.949280	−	0.314431i	$-0.898186\pi$
$281$		−	3.09594e14i		−	0.628862i	0.949280	−	0.314431i	$-0.101814\pi$
$282$	−1.47485e14			−0.293261
$283$		−	6.84137e14i		−	1.33176i	−0.746061	−	0.665878i	$-0.768057\pi$
$283$		−	6.84137e14i		−	1.33176i	0.746061	−	0.665878i	$-0.231943\pi$
$284$	−2.69708e14			−0.514025
$285$	1.17988e14			0.220175
$286$		−	8.58144e12i		−	0.0156807i
$287$		−	2.37810e14i		−	0.425538i
$288$	7.23183e14			1.26734
$289$	−1.40860e15			−2.41769
$290$		−	5.00375e13i		−	0.0841217i
$291$			2.19381e14i			0.361278i
$292$	−1.52676e14			−0.246305
$293$		−	3.95044e14i		−	0.624367i	−0.950022	−	0.312183i	$-0.898940\pi$
$293$		−	3.95044e14i		−	0.624367i	0.950022	−	0.312183i	$-0.101060\pi$
$294$	−1.11273e14			−0.172309
$295$		−	3.80383e14i		−	0.577150i
$296$			1.35509e13i			0.0201473i
$297$			6.63541e13i			0.0966783i
$298$		−	8.35137e14i		−	1.19250i
$299$	2.96455e13	−	4.15576e13i	0.0414889	−	0.0581599i
$300$	−1.62875e14			−0.223423
$301$	−1.54575e14			−0.207845
$302$	1.46710e15			1.93384
$303$	3.55994e14			0.460031
$304$			8.48059e14i			1.07445i
$305$	3.11968e12			0.00387536
$306$			1.92598e15i			2.34597i
$307$	2.53613e14			0.302929			0.151465	−	0.988463i	$-0.451601\pi$
$307$	2.53613e14			0.302929			0.151465	+	0.988463i	$0.451601\pi$
$308$	1.08580e14			0.127188
$309$			7.98134e13i			0.0916906i
$310$			3.99906e14i			0.450597i
$311$	1.55301e15			1.71637			0.858185	−	0.513341i	$-0.171593\pi$
$311$	1.55301e15			1.71637			0.858185	+	0.513341i	$0.171593\pi$
$312$	1.12585e12			0.00122054
$313$		−	4.98362e14i		−	0.530003i	−0.964248	−	0.265002i	$-0.914627\pi$
$313$		−	4.98362e14i		−	0.530003i	0.964248	−	0.265002i	$-0.0853725\pi$
$314$			1.97830e15i			2.06402i
$315$	4.11227e14			0.420938
$316$			1.31835e15i			1.32406i
$317$	−1.28629e15			−1.26760			−0.633801	−	0.773496i	$-0.718506\pi$
$317$	−1.28629e15			−1.26760			−0.633801	+	0.773496i	$0.718506\pi$
$318$		−	7.60177e14i		−	0.735109i
$319$			1.61140e13i			0.0152918i
$320$			6.81758e14i			0.634937i
$321$			2.64436e14i			0.241708i
$322$	1.03338e15	+	7.37171e14i	0.927097	+	0.661353i
$323$	−2.34299e15			−2.06326
$324$	8.15280e14			0.704753
$325$	−5.45793e13			−0.0463157
$326$	2.36378e15			1.96925
$327$			1.82671e13i			0.0149411i
$328$	3.40976e13			0.0273830
$329$		−	6.25334e14i		−	0.493101i
$330$	−5.59211e13			−0.0433004
$331$	9.45104e13			0.0718641			0.0359320	−	0.999354i	$-0.488560\pi$
$331$	9.45104e13			0.0718641			0.0359320	+	0.999354i	$0.488560\pi$
$332$		−	7.56243e14i		−	0.564719i
$333$		−	4.75711e14i		−	0.348882i
$334$	3.14624e15			2.26628
$335$	−1.36359e15			−0.964749
$336$		−	3.67781e14i		−	0.255596i
$337$		−	1.10074e15i		−	0.751457i	−0.926730	−	0.375729i	$-0.877392\pi$
$337$		−	1.10074e15i		−	0.751457i	0.926730	−	0.375729i	$-0.122608\pi$
$338$	−2.11673e15			−1.41960
$339$			4.12296e14i			0.271650i
$340$	−1.75462e15			−1.13582
$341$		−	1.28785e14i		−	0.0819105i
$342$			2.26622e15i			1.41627i
$343$		−	1.77145e15i		−	1.08784i
$344$		−	2.21632e13i		−	0.0133747i
$345$	−2.70811e14	−	1.93185e14i	−0.160602	−	0.114567i
$346$	−1.20744e15			−0.703737
$347$	8.30983e14			0.476009			0.238004	−	0.971264i	$-0.423507\pi$
$347$	8.30983e14			0.476009			0.238004	+	0.971264i	$0.423507\pi$
$348$	−6.08500e13			−0.0342598
$349$	2.62968e15			1.45529			0.727645	−	0.685954i	$-0.240615\pi$
$349$	2.62968e15			1.45529			0.727645	+	0.685954i	$0.240615\pi$
$350$		−	1.35718e15i		−	0.738294i
$351$	−8.39649e13			−0.0449009
$352$		−	4.16971e14i		−	0.219205i
$353$	−3.08064e15			−1.59218			−0.796091	−	0.605176i	$-0.793103\pi$
$353$	−3.08064e15			−1.59218			−0.796091	+	0.605176i	$0.793103\pi$
$354$	−9.09086e14			−0.461940
$355$			5.88956e14i			0.294248i
$356$		−	2.45342e15i		−	1.20524i
$357$	1.01609e15			0.490822
$358$	−4.75970e15			−2.26090
$359$		−	8.32200e14i		−	0.388742i	−0.980928	−	0.194371i	$-0.937734\pi$
$359$		−	8.32200e14i		−	0.388742i	0.980928	−	0.194371i	$-0.0622665\pi$
$360$			5.89625e13i			0.0270870i
$361$	−5.43578e14			−0.245595
$362$		−	4.52498e15i		−	2.01078i
$363$	−7.43077e14			−0.324783
$364$			1.37398e14i			0.0590707i
$365$			3.33395e14i			0.140994i
$366$		−	7.45581e12i		−	0.00310176i
$367$			2.60714e15i			1.06701i	0.845797	+	0.533505i	$0.179125\pi$
$367$			2.60714e15i			1.06701i	−0.845797	+	0.533505i	$0.820875\pi$
$368$	1.38856e15	−	1.94651e15i	0.559084	−	0.783734i
$369$	−1.19702e15			−0.474179
$370$	8.51714e14			0.331958
$371$	3.22313e15			1.23605
$372$	4.86321e14			0.183512
$373$		−	2.81709e15i		−	1.04604i	−0.852321	−	0.523019i	$-0.824806\pi$
$373$		−	2.81709e15i		−	1.04604i	0.852321	−	0.523019i	$-0.175194\pi$
$374$	1.11048e15			0.405769
$375$			9.04281e14i			0.325174i
$376$	8.96616e13			0.0317307
$377$	−2.03908e13			−0.00710208
$378$		−	2.08789e15i		−	0.715742i
$379$			3.39965e15i			1.14709i	0.819173	+	0.573547i	$0.194433\pi$
$379$			3.39965e15i			1.14709i	−0.819173	+	0.573547i	$0.805567\pi$
$380$	−2.06458e15			−0.685694
$381$	6.19273e14			0.202457
$382$			6.72994e15i			2.16586i
$383$			1.53171e15i			0.485270i	0.970118	+	0.242635i	$0.0780118\pi$
$383$			1.53171e15i			0.485270i	−0.970118	+	0.242635i	$0.921988\pi$
$384$	1.09481e14			0.0341468
$385$		−	2.37104e14i		−	0.0728073i
$386$	−4.37700e15			−1.32328
$387$			7.78054e14i			0.231603i
$388$		−	3.83880e15i		−	1.12514i
$389$			2.15025e15i			0.620570i	0.950644	+	0.310285i	$0.100425\pi$
$389$			2.15025e15i			0.620570i	−0.950644	+	0.310285i	$0.899575\pi$
$390$		−	7.07629e13i		−	0.0201103i
$391$	5.37773e15	+	3.83625e15i	1.50501	+	1.07361i
$392$	6.76471e13			0.0186437
$393$	5.03796e14			0.136741
$394$	4.84714e15			1.29571
$395$	2.87884e15			0.757941
$396$		−	5.46539e14i		−	0.141726i
$397$	3.08353e15			0.787598			0.393799	−	0.919197i	$-0.371161\pi$
$397$	3.08353e15			0.787598			0.393799	+	0.919197i	$0.371161\pi$
$398$		−	5.07439e14i		−	0.127669i
$399$	1.19559e15			0.296310
$400$	−2.55642e15			−0.624127
$401$			3.97289e15i			0.955521i	0.878490	+	0.477761i	$0.158551\pi$
$401$			3.97289e15i			0.955521i	−0.878490	+	0.477761i	$0.841449\pi$
$402$			3.25887e15i			0.772167i
$403$	1.62966e14			0.0380422
$404$	−6.22929e15			−1.43268
$405$		−	1.78031e15i		−	0.403427i
$406$		−	5.07041e14i		−	0.113211i
$407$	−2.74285e14			−0.0603441
$408$			1.45689e14i			0.0315840i
$409$	−5.82134e15			−1.24361			−0.621804	−	0.783173i	$-0.713600\pi$
$409$	−5.82134e15			−1.24361			−0.621804	+	0.783173i	$0.713600\pi$
$410$		−	2.14314e15i		−	0.451177i
$411$			1.93387e15i			0.401215i
$412$		−	1.39660e15i		−	0.285554i
$413$		−	3.85450e15i		−	0.776726i
$414$	3.71056e15	−	5.20153e15i	0.736948	−	1.03307i
$415$	−1.65139e15			−0.323267
$416$	5.27638e14			0.101807
$417$	1.00039e15			0.190263
$418$	1.30665e15			0.244964
$419$			7.36581e15i			1.36125i	0.732634	+	0.680623i	$0.238291\pi$
$419$			7.36581e15i			1.36125i	−0.732634	+	0.680623i	$0.761709\pi$
$420$	8.95356e14			0.163117
$421$		−	5.81409e15i		−	1.04421i	−0.852880	−	0.522106i	$-0.825146\pi$
$421$		−	5.81409e15i		−	1.04421i	0.852880	−	0.522106i	$-0.174854\pi$
$422$	1.82591e14			0.0323299
$423$	−3.14762e15			−0.549465
$424$			4.62139e14i			0.0795385i
$425$		−	7.06280e15i		−	1.19851i
$426$	1.40756e15			0.235510
$427$	3.16124e13			0.00521544
$428$		−	4.62718e15i		−	0.752755i
$429$			2.27884e13i			0.00365569i
$430$	−1.39303e15			−0.220368
$431$			1.22956e16i			1.91816i	0.283139	+	0.959079i	$0.408624\pi$
$431$			1.22956e16i			1.91816i	−0.283139	+	0.959079i	$0.591376\pi$
$432$	−3.93281e15			−0.605062
$433$			2.08920e15i			0.316995i	0.987359	+	0.158497i	$0.0506649\pi$
$433$			2.08920e15i			0.316995i	−0.987359	+	0.158497i	$0.949335\pi$
$434$			4.05234e15i			0.606411i
$435$			1.32877e14i			0.0196116i
$436$		−	3.19642e14i		−	0.0465313i
$437$	6.32775e15	+	4.51396e15i	0.908575	+	0.648140i
$438$	7.96787e14			0.112849
$439$	5.81230e15			0.812009			0.406005	−	0.913871i	$-0.366922\pi$
$439$	5.81230e15			0.812009			0.406005	+	0.913871i	$0.366922\pi$
$440$	3.39965e13			0.00468509
$441$	−2.37479e15			−0.322845
$442$			1.40520e15i			0.188454i
$443$	1.18329e16			1.56556			0.782779	−	0.622300i	$-0.213801\pi$
$443$	1.18329e16			1.56556			0.782779	+	0.622300i	$0.213801\pi$
$444$		−	1.03576e15i		−	0.135195i
$445$	−5.35749e15			−0.689924
$446$	2.14836e15			0.272960
$447$			2.21774e15i			0.278013i
$448$			6.90840e15i			0.854495i
$449$	−3.63799e15			−0.444000			−0.222000	−	0.975047i	$-0.571258\pi$
$449$	−3.63799e15			−0.444000			−0.222000	+	0.975047i	$0.571258\pi$
$450$	−6.83138e15			−0.822685
$451$			6.90173e14i			0.0820160i
$452$		−	7.21447e15i		−	0.846006i
$453$	−3.89596e15			−0.450843
$454$			4.68717e15i			0.535274i
$455$	3.00033e14			0.0338143
$456$			1.71427e14i			0.0190673i
$457$		−	1.43341e16i		−	1.57352i	−0.617258	−	0.786761i	$-0.711756\pi$
$457$		−	1.43341e16i		−	1.57352i	0.617258	−	0.786761i	$-0.288244\pi$
$458$			1.94408e14i			0.0210631i
$459$		−	1.08654e16i		−	1.16190i
$460$	4.73873e15	+	3.38041e15i	0.500167	+	0.356798i
$461$	1.78359e15			0.185819			0.0929096	−	0.995675i	$-0.470383\pi$
$461$	1.78359e15			0.185819			0.0929096	+	0.995675i	$0.470383\pi$
$462$	−5.66661e14			−0.0582735
$463$	−7.35459e15			−0.746573			−0.373286	−	0.927716i	$-0.621769\pi$
$463$	−7.35459e15			−0.746573			−0.373286	+	0.927716i	$0.621769\pi$
$464$	−9.55077e14			−0.0957041
$465$		−	1.06197e15i		−	0.105049i
$466$	1.16795e16			1.14054
$467$			1.67998e16i			1.61959i	0.586716	+	0.809793i	$0.300421\pi$
$467$			1.67998e16i			1.61959i	−0.586716	+	0.809793i	$0.699579\pi$
$468$	6.91594e14			0.0658228
$469$	−1.38175e16			−1.29836
$470$		−	5.63550e15i		−	0.522812i
$471$		−	5.25346e15i		−	0.481194i
$472$	5.52666e14			0.0499817
$473$	4.48608e14			0.0400590
$474$		−	6.88021e15i		−	0.606641i
$475$		−	8.31050e15i		−	0.723545i
$476$	−1.77799e16			−1.52858
$477$		−	1.62237e16i		−	1.37733i
$478$	−8.93621e14			−0.0749180
$479$		−	1.44366e16i		−	1.19523i	−0.801784	−	0.597614i	$-0.796116\pi$
$479$		−	1.44366e16i		−	1.19523i	0.801784	−	0.597614i	$-0.203884\pi$
$480$		−	3.43836e15i		−	0.281128i
$481$		−	3.47081e14i		−	0.0280260i
$482$			1.32628e16i			1.05767i
$483$	−2.74418e15	−	1.95759e15i	−0.216138	−	0.154184i
$484$	1.30026e16			1.01148
$485$	−8.38270e15			−0.644071
$486$	−1.60719e16			−1.21969
$487$	3.08429e15			0.231197			0.115598	−	0.993296i	$-0.463121\pi$
$487$	3.08429e15			0.231197			0.115598	+	0.993296i	$0.463121\pi$
$488$			4.53265e12i			0.000335609i
$489$	−6.27711e15			−0.459099
$490$		−	4.25182e15i		−	0.307184i
$491$	1.65229e16			1.17922			0.589612	−	0.807686i	$-0.299281\pi$
$491$	1.65229e16			1.17922			0.589612	+	0.807686i	$0.299281\pi$
$492$	−2.60624e15			−0.183749
$493$		−	2.63865e15i		−	0.183781i
$494$			1.65344e15i			0.113770i
$495$	−1.19347e15			−0.0811294
$496$	7.63310e15			0.512638
$497$			5.96802e15i			0.395997i
$498$			3.94670e15i			0.258736i
$499$	−1.16249e15			−0.0752981			−0.0376491	−	0.999291i	$-0.511987\pi$
$499$	−1.16249e15			−0.0752981			−0.0376491	+	0.999291i	$0.511987\pi$
$500$		−	1.58234e16i		−	1.01270i
$501$	−8.35497e15			−0.528346
$502$			4.28244e16i			2.67590i
$503$			6.33015e15i			0.390846i	0.980719	+	0.195423i	$0.0626080\pi$
$503$			6.33015e15i			0.390846i	−0.980719	+	0.195423i	$0.937392\pi$
$504$			5.97480e14i			0.0364536i
$505$			1.36028e16i			0.820123i
$506$	−2.99908e15	−	2.13942e15i	−0.178684	−	0.127466i
$507$	5.62107e15			0.330957
$508$	−1.08362e16			−0.630515
$509$	2.07562e15			0.119355			0.0596777	−	0.998218i	$-0.480993\pi$
$509$	2.07562e15			0.119355			0.0596777	+	0.998218i	$0.480993\pi$
$510$	9.15702e15			0.520395
$511$			3.37836e15i			0.189749i
$512$	2.56046e16			1.42134
$513$		−	1.27849e16i		−	0.701443i
$514$	3.73458e16			2.02518
$515$	−3.04972e15			−0.163462
$516$			1.69404e15i			0.0897482i
$517$			1.81485e15i			0.0950379i
$518$	8.63060e15			0.446748
$519$	3.20642e15			0.164065
$520$			4.30193e13i			0.00217592i
$521$		−	9.70512e15i		−	0.485260i	−0.970119	−	0.242630i	$-0.921990\pi$
$521$		−	9.70512e15i		−	0.485260i	0.970119	−	0.242630i	$-0.0780102\pi$
$522$	−2.55220e15			−0.126151
$523$			2.37204e16i			1.15907i	0.814946	+	0.579537i	$0.196767\pi$
$523$			2.37204e16i			1.15907i	−0.814946	+	0.579537i	$0.803233\pi$
$524$	−8.81557e15			−0.425856
$525$			3.60405e15i			0.172121i
$526$			1.53313e16i			0.723875i
$527$			2.10884e16i			0.984421i
$528$			1.06738e15i			0.0492623i
$529$	−7.13290e15	−	2.07213e16i	−0.325486	−	0.945547i
$530$	2.90468e16			1.31052
$531$	−1.94017e16			−0.865509
$532$	−2.09209e16			−0.922804
$533$	−8.73349e14			−0.0380912
$534$			1.28040e16i			0.552202i
$535$	−1.01043e16			−0.430905
$536$		−	1.98119e15i		−	0.0835481i
$537$	1.26396e16			0.527093
$538$	3.80087e16			1.56743
$539$			1.36925e15i			0.0558406i
$540$		−	9.57434e15i		−	0.386141i
$541$	−6.59482e15			−0.263039			−0.131519	−	0.991314i	$-0.541986\pi$
$541$	−6.59482e15			−0.263039			−0.131519	+	0.991314i	$0.541986\pi$
$542$	−3.33067e16			−1.31382
$543$			1.20163e16i			0.468782i
$544$			6.82786e16i			2.63446i
$545$	−6.97996e14			−0.0266363
$546$		−	7.17056e14i		−	0.0270643i
$547$	4.57303e16			1.70718			0.853592	−	0.520942i	$-0.174419\pi$
$547$	4.57303e16			1.70718			0.853592	+	0.520942i	$0.174419\pi$
$548$		−	3.38395e16i		−	1.24951i
$549$		−	1.59121e14i		−	0.00581158i
$550$			3.93882e15i			0.142295i
$551$		−	3.10479e15i		−	0.110949i
$552$	2.80683e14	−	3.93467e14i	0.00992159	−	0.0139083i
$553$	2.91719e16			1.02003
$554$	−6.88505e16			−2.38149
$555$	−2.26176e15			−0.0773907
$556$	−1.75052e16			−0.592539
$557$		−	5.32932e16i		−	1.78460i	−0.451444	−	0.892299i	$-0.649091\pi$
$557$		−	5.32932e16i		−	1.78460i	0.451444	−	0.892299i	$-0.350909\pi$
$558$	2.03975e16			0.675727
$559$			5.67672e14i			0.0186049i
$560$	1.40532e16			0.455665
$561$	−2.94891e15			−0.0945986
$562$		−	2.82723e16i		−	0.897312i
$563$			2.27760e16i			0.715198i	0.933875	+	0.357599i	$0.116405\pi$
$563$			2.27760e16i			0.715198i	−0.933875	+	0.357599i	$0.883595\pi$
$564$	−6.85326e15			−0.212923
$565$	−1.57541e16			−0.484286
$566$		−	6.24757e16i		−	1.90026i
$567$		−	1.80403e16i		−	0.542931i
$568$	−8.55707e14			−0.0254821
$569$			5.21115e15i			0.153553i	0.997048	+	0.0767767i	$0.0244629\pi$
$569$			5.21115e15i			0.153553i	−0.997048	+	0.0767767i	$0.975537\pi$
$570$	1.07747e16			0.314163
$571$		−	5.47345e16i		−	1.57923i	−0.613605	−	0.789613i	$-0.710281\pi$
$571$		−	5.47345e16i		−	1.57923i	0.613605	−	0.789613i	$-0.289719\pi$
$572$		−	3.98758e14i		−	0.0113850i
$573$		−	1.78716e16i		−	0.504936i
$574$		−	2.17169e16i		−	0.607192i
$575$	−1.36071e16	+	1.90746e16i	−0.376494	+	0.527776i
$576$	3.47735e16			0.952167
$577$	−6.15989e16			−1.66924			−0.834619	−	0.550827i	$-0.814312\pi$
$577$	−6.15989e16			−1.66924			−0.834619	+	0.550827i	$0.814312\pi$
$578$	−1.28634e17			−3.44976
$579$	1.16233e16			0.308502
$580$		−	2.32512e15i		−	0.0610769i
$581$	−1.67339e16			−0.435051
$582$			2.00340e16i			0.515502i
$583$	−9.35420e15			−0.238229
$584$	−4.84396e14			−0.0122102
$585$		−	1.51022e15i		−	0.0376795i
$586$		−	3.60756e16i		−	0.890898i
$587$	−4.26035e15			−0.104140			−0.0520699	−	0.998643i	$-0.516582\pi$
$587$	−4.26035e15			−0.104140			−0.0520699	+	0.998643i	$0.516582\pi$
$588$	−5.17059e15			−0.125105
$589$			2.48139e16i			0.594296i
$590$		−	3.47367e16i		−	0.823525i
$591$	−1.28718e16			−0.302074
$592$		−	1.62568e16i		−	0.377664i
$593$	8.35523e16			1.92145			0.960727	−	0.277494i	$-0.0895040\pi$
$593$	8.35523e16			1.92145			0.960727	+	0.277494i	$0.0895040\pi$
$594$			6.05949e15i			0.137949i
$595$			3.88256e16i			0.875016i
$596$		−	3.88067e16i		−	0.865822i
$597$			1.34753e15i			0.0297640i
$598$	2.70724e15	−	3.79506e15i	0.0591997	−	0.0829873i
$599$	6.53819e16			1.41546			0.707728	−	0.706485i	$-0.249720\pi$
$599$	6.53819e16			1.41546			0.707728	+	0.706485i	$0.249720\pi$
$600$	−5.16756e14			−0.0110759
$601$	−8.66340e16			−1.83841			−0.919204	−	0.393782i	$-0.871166\pi$
$601$	−8.66340e16			−1.83841			−0.919204	+	0.393782i	$0.871166\pi$
$602$	−1.41158e16			−0.296570
$603$			6.95507e16i			1.44676i
$604$	6.81726e16			1.40407
$605$		−	2.83934e16i		−	0.579009i
$606$	3.25096e16			0.656410
$607$	6.00782e16			1.20111			0.600557	−	0.799582i	$-0.294945\pi$
$607$	6.00782e16			1.20111			0.600557	+	0.799582i	$0.294945\pi$
$608$			8.03406e16i			1.59043i
$609$			1.34647e15i			0.0263932i
$610$	2.84891e14			0.00552968
$611$	−2.29652e15			−0.0441390
$612$			8.94952e16i			1.70330i
$613$			5.17495e14i			0.00975311i	0.999988	+	0.00487656i	$0.00155226\pi$
$613$			5.17495e14i			0.00975311i	−0.999988	+	0.00487656i	$0.998448\pi$
$614$	2.31600e16			0.432244
$615$			5.69119e15i			0.105185i
$616$	3.44493e14			0.00630517
$617$			8.48253e16i			1.53750i	0.639551	+	0.768749i	$0.279120\pi$
$617$			8.48253e16i			1.53750i	−0.639551	+	0.768749i	$0.720880\pi$
$618$			7.28860e15i			0.130832i
$619$		−	3.76467e16i		−	0.669242i	−0.942353	−	0.334621i	$-0.891392\pi$
$619$		−	3.76467e16i		−	0.669242i	0.942353	−	0.334621i	$-0.108608\pi$
$620$			1.85826e16i			0.327158i
$621$	−2.09331e16	+	2.93445e16i	−0.364993	+	0.511654i
$622$	1.41821e17			2.44906
$623$	−5.42886e16			−0.928497
$624$	−1.35067e15			−0.0228792
$625$	4.08864e15			0.0685961
$626$		−	4.55106e16i		−	0.756252i
$627$	−3.46986e15			−0.0571093
$628$			9.19265e16i			1.49859i
$629$	4.49138e16			0.725231
$630$	3.75534e16			0.600629
$631$		−	5.78447e16i		−	0.916405i	−0.888848	−	0.458203i	$-0.848493\pi$
$631$		−	5.78447e16i		−	0.916405i	0.888848	−	0.458203i	$-0.151507\pi$
$632$			4.18273e15i			0.0656383i
$633$	−4.84878e14			−0.00753720
$634$	−1.17465e17			−1.80872
$635$			2.36628e16i			0.360931i
$636$		−	3.53235e16i		−	0.533729i
$637$	−1.73266e15			−0.0259344
$638$			1.47154e15i			0.0218196i
$639$	3.00401e16			0.441261
$640$			4.18332e15i			0.0608753i
$641$		−	1.14940e17i		−	1.65700i	−0.559987	−	0.828501i	$-0.689194\pi$
$641$		−	1.14940e17i		−	1.65700i	0.559987	−	0.828501i	$-0.310806\pi$
$642$			2.41484e16i			0.344888i
$643$		−	2.02450e16i		−	0.286451i	−0.989690	−	0.143226i	$-0.954253\pi$
$643$		−	2.02450e16i		−	0.286451i	0.989690	−	0.143226i	$-0.0457475\pi$
$644$	4.80185e16	+	3.42544e16i	0.673122	+	0.480177i
$645$	3.69924e15			0.0513753
$646$	−2.13963e17			−2.94403
$647$	−7.16620e16			−0.976929			−0.488465	−	0.872584i	$-0.662443\pi$
$647$	−7.16620e16			−0.976929			−0.488465	+	0.872584i	$0.662443\pi$
$648$	2.58665e15			0.0349371
$649$			1.11866e16i			0.149702i
$650$	−4.98421e15			−0.0660870
$651$		−	1.07611e16i		−	0.141375i
$652$	1.09839e17			1.42978
$653$	6.45055e16			0.831989			0.415995	−	0.909367i	$-0.363433\pi$
$653$	6.45055e16			0.831989			0.415995	+	0.909367i	$0.363433\pi$
$654$			1.66816e15i			0.0213192i
$655$			1.92504e16i			0.243776i
$656$	−4.09065e16			−0.513298
$657$	1.70050e16			0.211439
$658$		−	5.71057e16i		−	0.703598i
$659$			7.31754e16i			0.893414i	0.894680	+	0.446707i	$0.147403\pi$
$659$			7.31754e16i			0.893414i	−0.894680	+	0.446707i	$0.852597\pi$
$660$	−2.59851e15			−0.0314384
$661$		−	1.25581e17i		−	1.50562i	−0.658236	−	0.752812i	$-0.728697\pi$
$661$		−	1.25581e17i		−	1.50562i	0.658236	−	0.752812i	$-0.271303\pi$
$662$	8.63074e15			0.102542
$663$		−	3.73157e15i		−	0.0439350i
$664$		−	2.39934e15i		−	0.0279952i
$665$			4.56844e16i			0.528248i
$666$		−	4.34422e16i		−	0.497813i
$667$	−5.08358e15	+	7.12627e15i	−0.0577318	+	0.0809295i
$668$	1.46198e17			1.64544
$669$	−5.70507e15			−0.0636362
$670$	−1.24524e17			−1.37658
$671$	−9.17458e13			−0.00100520
$672$		−	3.48417e16i		−	0.378341i
$673$	8.30698e16			0.894031			0.447016	−	0.894526i	$-0.352487\pi$
$673$	8.30698e16			0.894031			0.447016	+	0.894526i	$0.352487\pi$
$674$		−	1.00520e17i		−	1.07224i
$675$	3.85393e16			0.407456
$676$	−9.83591e16			−1.03071
$677$			1.63261e17i			1.69571i	0.530228	+	0.847855i	$0.322106\pi$
$677$			1.63261e17i			1.69571i	−0.530228	+	0.847855i	$0.677894\pi$
$678$			3.76510e16i			0.387613i
$679$	−8.49437e16			−0.866787
$680$	−5.56689e15			−0.0563065
$681$		−	1.24470e16i		−	0.124790i
$682$		−	1.17607e16i		−	0.116877i
$683$	−4.46570e16			−0.439912			−0.219956	−	0.975510i	$-0.570591\pi$
$683$	−4.46570e16			−0.439912			−0.219956	+	0.975510i	$0.570591\pi$
$684$			1.05305e17i			1.02828i
$685$	−7.38945e16			−0.715268
$686$		−	1.61770e17i		−	1.55222i
$687$		−	5.16259e14i		−	0.00491052i
$688$			2.65890e16i			0.250710i
$689$		−	1.18369e16i		−	0.110642i
$690$	−2.47306e16	−	1.76418e16i	−0.229160	−	0.163474i
$691$	1.09324e17			1.00426			0.502130	−	0.864792i	$-0.332550\pi$
$691$	1.09324e17			1.00426			0.502130	+	0.864792i	$0.332550\pi$
$692$	−5.61068e16			−0.510951
$693$	−1.20936e16			−0.109184
$694$	7.58857e16			0.679209
$695$			3.82256e16i			0.339192i
$696$	−1.93059e14			−0.00169838
$697$		−	1.13015e17i		−	0.985689i
$698$	2.40143e17			2.07653
$699$	−3.10154e16			−0.265898
$700$		−	6.30647e16i		−	0.536041i
$701$			9.95060e16i			0.838573i	0.907854	+	0.419287i	$0.137720\pi$
$701$			9.95060e16i			0.838573i	−0.907854	+	0.419287i	$0.862280\pi$
$702$	−7.66771e15			−0.0640683
$703$	5.28482e16			0.437823
$704$		−	2.00496e16i		−	0.164691i
$705$			1.49653e16i			0.121885i
$706$	−2.81326e17			−2.27186
$707$			1.37840e17i			1.10372i
$708$	−4.22429e16			−0.335393
$709$			1.35540e17i			1.06706i	0.845781	+	0.533531i	$0.179135\pi$
$709$			1.35540e17i			1.06706i	−0.845781	+	0.533531i	$0.820865\pi$
$710$			5.37838e16i			0.419857i
$711$		−	1.46837e17i		−	1.13663i
$712$		−	7.78401e15i		−	0.0597480i
$713$	4.06287e16	−	5.69540e16i	0.309240	−	0.433498i
$714$	9.27901e16			0.700346
$715$	−8.70758e14			−0.00651720
$716$	−2.21171e17			−1.64154
$717$	2.37305e15			0.0174659
$718$		−	7.59969e16i		−	0.554688i
$719$	2.19269e17			1.58710			0.793548	−	0.608508i	$-0.208232\pi$
$719$	2.19269e17			1.58710			0.793548	+	0.608508i	$0.208232\pi$
$720$		−	7.07367e16i		−	0.507750i
$721$	−3.09035e16			−0.219986
$722$	−4.96398e16			−0.350435
$723$		−	3.52198e16i		−	0.246580i
$724$		−	2.10264e17i		−	1.45994i
$725$	9.35922e15			0.0644483
$726$	−6.78581e16			−0.463428
$727$		−	8.46583e16i		−	0.573407i	−0.958019	−	0.286703i	$-0.907441\pi$
$727$		−	8.46583e16i		−	0.573407i	0.958019	−	0.286703i	$-0.0925594\pi$
$728$			4.35924e14i			0.00292835i
$729$	−5.94251e16			−0.395917
$730$			3.04457e16i			0.201182i
$731$	−7.34591e16			−0.481439
$732$		−	3.46452e14i		−	0.00225204i
$733$			1.11211e17i			0.717010i	0.933528	+	0.358505i	$0.116713\pi$
$733$			1.11211e17i			0.717010i	−0.933528	+	0.358505i	$0.883287\pi$
$734$			2.38085e17i			1.52250i
$735$			1.12909e16i			0.0716151i
$736$	1.31544e17	−	1.84402e17i	0.827572	−	1.16011i
$737$	4.01014e16			0.250239
$738$	−1.09312e17			−0.676597
$739$	−1.01613e17			−0.623856			−0.311928	−	0.950106i	$-0.600975\pi$
$739$	−1.01613e17			−0.623856			−0.311928	+	0.950106i	$0.600975\pi$
$740$	3.95770e16			0.241019
$741$		−	4.39079e15i		−	0.0265236i
$742$	2.94338e17			1.76369
$743$			2.87742e17i			1.71029i	0.518386	+	0.855147i	$0.326533\pi$
$743$			2.87742e17i			1.71029i	−0.518386	+	0.855147i	$0.673467\pi$
$744$	1.54295e15			0.00909737
$745$	−8.47413e16			−0.495630
$746$		−	2.57258e17i		−	1.49257i
$747$			8.42302e16i			0.484779i
$748$	5.16009e16			0.294610
$749$	−1.02389e17			−0.579910
$750$			8.25793e16i			0.463985i
$751$		−	2.46803e17i		−	1.37566i	−0.725874	−	0.687828i	$-0.758564\pi$
$751$		−	2.46803e17i		−	1.37566i	0.725874	−	0.687828i	$-0.241436\pi$
$752$	−1.07566e17			−0.594796
$753$		−	1.13722e17i		−	0.623842i
$754$	−1.86209e15			−0.0101338
$755$		−	1.48867e17i		−	0.803742i
$756$		−	9.70188e16i		−	0.519667i
$757$			4.84778e16i			0.257613i	0.991670	+	0.128806i	$0.0411146\pi$
$757$			4.84778e16i			0.257613i	−0.991670	+	0.128806i	$0.958885\pi$
$758$			3.10458e17i			1.63677i
$759$	7.96419e15	+	5.68133e15i	0.0416573	+	0.0297166i
$760$	−6.55032e15			−0.0339923
$761$	1.66889e17			0.859251			0.429625	−	0.903007i	$-0.358646\pi$
$761$	1.66889e17			0.859251			0.429625	+	0.903007i	$0.358646\pi$
$762$	5.65523e16			0.288882
$763$	−7.07294e15			−0.0358470
$764$			3.12723e17i			1.57253i
$765$	1.95429e17			0.975034
$766$			1.39876e17i			0.692424i
$767$	−1.41555e16			−0.0695272
$768$	−6.30835e16			−0.307431
$769$		−	6.73923e15i		−	0.0325876i	−0.999867	−	0.0162938i	$-0.994813\pi$
$769$		−	6.73923e15i		−	0.0325876i	0.999867	−	0.0162938i	$-0.00518671\pi$
$770$		−	2.16525e16i		−	0.103887i
$771$	−9.91733e16			−0.472137
$772$	−2.03388e17			−0.960775
$773$			1.92675e16i			0.0903124i	0.998980	+	0.0451562i	$0.0143786\pi$
$773$			1.92675e16i			0.0903124i	−0.998980	+	0.0451562i	$0.985621\pi$
$774$			7.10522e16i			0.330470i
$775$	−7.48000e16			−0.345217
$776$		−	1.21794e16i		−	0.0557770i
$777$	−2.29189e16			−0.104152
$778$			1.96362e17i			0.885481i
$779$		−	1.32980e17i		−	0.595062i
$780$		−	3.28817e15i		−	0.0146011i
$781$		−	1.73204e16i		−	0.0763225i
$782$	4.91097e17	+	3.50328e17i	2.14747	+	1.53192i
$783$	1.43982e16			0.0624796
$784$	−8.11555e16			−0.349479
$785$	2.00738e17			0.857850
$786$	4.60069e16			0.195114
$787$			2.23263e16i			0.0939654i	0.998896	+	0.0469827i	$0.0149606\pi$
$787$			2.23263e16i			0.0939654i	−0.998896	+	0.0469827i	$0.985039\pi$
$788$	2.25234e17			0.940755
$789$		−	4.07128e16i		−	0.168760i
$790$	2.62897e17			1.08149
$791$	−1.59639e17			−0.651750
$792$		−	1.73401e15i		−	0.00702588i
$793$		−	1.16096e14i		−	0.000466850i
$794$	2.81589e17			1.12381
$795$	−7.71351e16			−0.305527
$796$		−	2.35794e16i		−	0.0926945i
$797$			1.74225e17i			0.679766i	0.940468	+	0.339883i	$0.110388\pi$
$797$			1.74225e17i			0.679766i	−0.940468	+	0.339883i	$0.889612\pi$
$798$	1.09182e17			0.422800
$799$		−	2.97179e17i		−	1.14219i
$800$	−2.42182e17			−0.923851
$801$			2.73262e17i			1.03463i
$802$			3.62806e17i			1.36342i
$803$		−	9.80470e15i		−	0.0365713i
$804$			1.51431e17i			0.560634i
$805$	7.48007e16	−	1.04857e17i	0.274872	−	0.385321i
$806$	1.48821e16			0.0542818
$807$	−1.00934e17			−0.365422
$808$	−1.97637e16			−0.0710233
$809$	3.80295e16			0.135653			0.0678266	−	0.997697i	$-0.478394\pi$
$809$	3.80295e16			0.135653			0.0678266	+	0.997697i	$0.478394\pi$
$810$		−	1.62579e17i		−	0.575643i
$811$	−4.18058e16			−0.146930			−0.0734651	−	0.997298i	$-0.523406\pi$
$811$	−4.18058e16			−0.146930			−0.0734651	+	0.997298i	$0.523406\pi$
$812$		−	2.35609e16i		−	0.0821970i
$813$	8.84473e16			0.306296
$814$	−2.50478e16			−0.0861039
$815$		−	2.39852e17i		−	0.818462i
$816$		−	1.74782e17i		−	0.592047i
$817$	−8.64363e16			−0.290646
$818$	−5.31608e17			−1.77448
$819$		−	1.53034e16i		−	0.0507088i
$820$		−	9.95861e16i		−	0.327579i
$821$	−3.74484e17			−1.22285			−0.611427	−	0.791301i	$-0.709404\pi$
$821$	−3.74484e17			−1.22285			−0.611427	+	0.791301i	$0.709404\pi$
$822$			1.76602e17i			0.572486i
$823$	−4.61014e17			−1.48359			−0.741797	−	0.670625i	$-0.766026\pi$
$823$	−4.61014e17			−1.48359			−0.741797	+	0.670625i	$0.766026\pi$
$824$		−	4.43100e15i		−	0.0141559i
$825$		−	1.04597e16i		−	0.0331738i
$826$		−	3.51995e17i		−	1.10830i
$827$		−	3.73880e17i		−	1.16869i	−0.811506	−	0.584344i	$-0.801352\pi$
$827$		−	3.73880e17i		−	1.16869i	0.811506	−	0.584344i	$-0.198648\pi$
$828$	1.72420e17	−	2.41702e17i	0.535064	−	0.750063i
$829$	−1.62093e17			−0.499386			−0.249693	−	0.968325i	$-0.580330\pi$
$829$	−1.62093e17			−0.499386			−0.249693	+	0.968325i	$0.580330\pi$
$830$	−1.50806e17			−0.461264
$831$	1.82835e17			0.555206
$832$	2.53709e16			0.0764885
$833$		−	2.24213e17i		−	0.671107i
$834$	9.13563e16			0.271483
$835$		−	3.19249e17i		−	0.941912i
$836$	6.07167e16			0.177857
$837$	−1.15072e17			−0.334671
$838$			6.72649e17i			1.94234i
$839$		−	5.58404e17i		−	1.60095i	−0.599368	−	0.800474i	$-0.704581\pi$
$839$		−	5.58404e17i		−	1.60095i	0.599368	−	0.800474i	$-0.295419\pi$
$840$	2.84071e15			0.00808631
$841$	−3.50318e17			−0.990117
$842$		−	5.30945e17i		−	1.48997i
$843$			7.50783e16i			0.209194i
$844$	8.48454e15			0.0234733
$845$			2.14785e17i			0.590015i
$846$	−2.87442e17			−0.784022
$847$		−	2.87717e17i		−	0.779228i
$848$		−	5.54423e17i		−	1.49096i
$849$			1.65907e17i			0.443015i
$850$		−	6.44978e17i		−	1.71014i
$851$	−1.21300e17	−	8.65302e16i	−0.319361	−	0.227819i
$852$	6.54057e16			0.170993
$853$	1.60645e17			0.417036			0.208518	−	0.978018i	$-0.433136\pi$
$853$	1.60645e17			0.417036			0.208518	+	0.978018i	$0.433136\pi$
$854$	2.88686e15			0.00744181
$855$	2.29953e17			0.588629
$856$		−	1.46807e16i		−	0.0373168i
$857$	4.07305e17			1.02810			0.514049	−	0.857761i	$-0.328145\pi$
$857$	4.07305e17			1.02810			0.514049	+	0.857761i	$0.328145\pi$
$858$			2.08105e15i			0.00521624i
$859$	−4.27676e17			−1.06453			−0.532263	−	0.846579i	$-0.678658\pi$
$859$	−4.27676e17			−1.06453			−0.532263	+	0.846579i	$0.678658\pi$
$860$	−6.47304e16			−0.159999
$861$			5.76701e16i			0.141557i
$862$			1.12284e18i			2.73699i
$863$	−8.27127e16			−0.200220			−0.100110	−	0.994976i	$-0.531919\pi$
$863$	−8.27127e16			−0.200220			−0.100110	+	0.994976i	$0.531919\pi$
$864$	−3.72573e17			−0.895631
$865$			1.22519e17i			0.292488i
$866$			1.90786e17i			0.452314i
$867$	3.41593e17			0.804255
$868$			1.88302e17i			0.440287i
$869$	−8.46630e16			−0.196596
$870$			1.21344e16i			0.0279835i
$871$			5.07445e16i			0.116220i
$872$		−	1.01413e15i		−	0.00230673i
$873$			4.27565e17i			0.965865i
$874$	5.77853e17	+	4.12217e17i	1.29643	+	0.924820i
$875$	−3.50134e17			−0.780165
$876$	3.70247e16			0.0819344
$877$	5.77616e17			1.26953			0.634763	−	0.772707i	$-0.281098\pi$
$877$	5.77616e17			1.26953			0.634763	+	0.772707i	$0.281098\pi$
$878$	5.30782e17			1.15864
$879$			9.58003e16i			0.207699i
$880$	−4.07852e16			−0.0878226
$881$			2.49640e17i			0.533899i	0.963711	+	0.266949i	$0.0860157\pi$
$881$			2.49640e17i			0.533899i	−0.963711	+	0.266949i	$0.913984\pi$
$882$	−2.16867e17			−0.460661
$883$	3.75519e17			0.792260			0.396130	−	0.918195i	$-0.370353\pi$
$883$	3.75519e17			0.792260			0.396130	+	0.918195i	$0.370353\pi$
$884$			6.52962e16i			0.136828i
$885$			9.22448e16i			0.191992i
$886$	1.08059e18			2.23387
$887$	−6.65872e17			−1.36725			−0.683627	−	0.729832i	$-0.739598\pi$
$887$	−6.65872e17			−1.36725			−0.683627	+	0.729832i	$0.739598\pi$
$888$		−	3.28616e15i		−	0.00670210i
$889$			2.39780e17i			0.485739i
$890$	−4.89248e17			−0.984440
$891$			5.23566e16i			0.104642i
$892$	9.98289e16			0.198183
$893$		−	3.49679e17i		−	0.689541i
$894$			2.02525e17i			0.396692i
$895$			4.82967e17i			0.939677i
$896$			4.23905e16i			0.0819257i
$897$	−7.18919e15	+	1.00779e16i	−0.0138015	+	0.0193472i
$898$	−3.32223e17			−0.633536
$899$	−2.79452e16			−0.0529358
$900$	−3.17437e17			−0.597313
$901$	1.53174e18			2.86310
$902$			6.30269e16i			0.117027i
$903$	3.74852e16			0.0691406
$904$		−	2.28894e16i		−	0.0419396i
$905$	−4.59149e17			−0.835723
$906$	−3.55781e17			−0.643299
$907$			1.88235e17i			0.338110i	0.985607	+	0.169055i	$0.0540715\pi$
$907$			1.88235e17i			0.338110i	−0.985607	+	0.169055i	$0.945929\pi$
$908$			2.17801e17i			0.388637i
$909$	6.93817e17			1.22988
$910$	2.73992e16			0.0482491
$911$			6.20001e17i			1.08463i	0.840175	+	0.542316i	$0.182452\pi$
$911$			6.20001e17i			1.08463i	−0.840175	+	0.542316i	$0.817548\pi$
$912$		−	2.05659e17i		−	0.357419i
$913$	4.85652e16			0.0838496
$914$		−	1.30900e18i		−	2.24523i
$915$	−7.56540e14			−0.00128916
$916$			9.03365e15i			0.0152929i
$917$			1.95068e17i			0.328073i
$918$		−	9.92235e17i		−	1.65790i
$919$			3.07432e17i			0.510336i	0.966897	+	0.255168i	$0.0821307\pi$
$919$			3.07432e17i			0.510336i	−0.966897	+	0.255168i	$0.917869\pi$
$920$	1.50346e16	+	1.07251e16i	0.0247951	+	0.0176878i
$921$	−6.15024e16			−0.100771
$922$	1.62879e17			0.265142
$923$	2.19174e16			0.0354469
$924$	−2.63313e16			−0.0423097
$925$			1.59308e17i			0.254324i
$926$	−6.71624e17			−1.06527
$927$			1.55553e17i			0.245132i
$928$	−9.04789e16			−0.141664
$929$	8.04027e17			1.25077			0.625383	−	0.780318i	$-0.284943\pi$
$929$	8.04027e17			1.25077			0.625383	+	0.780318i	$0.284943\pi$
$930$		−	9.69794e16i		−	0.149893i
$931$		−	2.63823e17i		−	0.405148i
$932$	5.42717e17			0.828091
$933$	−3.76612e17			−0.570958
$934$			1.53417e18i			2.31096i
$935$		−	1.12680e17i		−	0.168646i
$936$	2.19423e15			0.00326307
$937$			1.32782e18i			1.96202i	0.193961	+	0.981009i	$0.437866\pi$
$937$			1.32782e18i			1.96202i	−0.193961	+	0.981009i	$0.562134\pi$
$938$	−1.26182e18			−1.85260
$939$			1.20855e17i			0.176308i
$940$		−	2.61867e17i		−	0.379589i
$941$		−	3.78279e17i		−	0.544847i	−0.962177	−	0.272423i	$-0.912175\pi$
$941$		−	3.78279e17i		−	0.544847i	0.962177	−	0.272423i	$-0.0878252\pi$
$942$		−	4.79748e17i		−	0.686607i
$943$	−2.17733e17	+	3.05222e17i	−0.309638	+	0.434056i
$944$	−6.63028e17			−0.936914
$945$	−2.11858e17			−0.297477
$946$	4.09671e16			0.0571595
$947$	4.65783e17			0.645778			0.322889	−	0.946437i	$-0.395346\pi$
$947$	4.65783e17			0.645778			0.322889	+	0.946437i	$0.395346\pi$
$948$		−	3.19706e17i		−	0.440454i
$949$	1.24069e16			0.0169851
$950$		−	7.58918e17i		−	1.03241i
$951$	3.11932e17			0.421674
$952$	−5.64104e16			−0.0757771
$953$		−	8.81097e17i		−	1.17616i	−0.808803	−	0.588080i	$-0.799884\pi$
$953$		−	8.81097e17i		−	1.17616i	0.808803	−	0.588080i	$-0.200116\pi$
$954$		−	1.48155e18i		−	1.96529i
$955$	6.82887e17			0.900178
$956$	−4.15243e16			−0.0543945
$957$		−	3.90773e15i		−	0.00508690i
$958$		−	1.31835e18i		−	1.70545i
$959$	−7.48789e17			−0.962604
$960$		−	1.65330e17i		−	0.211215i
$961$	−5.64321e17			−0.716450
$962$		−	3.16956e16i		−	0.0399898i
$963$			5.15374e17i			0.646197i
$964$			6.16286e17i			0.767928i
$965$			4.44134e17i			0.549984i
$966$	−2.50600e17	−	1.78768e17i	−0.308403	−	0.220002i
$967$	9.08423e15			0.0111104			0.00555519	−	0.999985i	$-0.498232\pi$
$967$	9.08423e15			0.0111104			0.00555519	+	0.999985i	$0.498232\pi$
$968$	4.12534e16			0.0501427
$969$	5.68186e17			0.686354
$970$	−7.65512e17			−0.919013
$971$		−	8.41350e17i		−	1.00383i	−0.864916	−	0.501916i	$-0.832629\pi$
$971$		−	8.41350e17i		−	1.00383i	0.864916	−	0.501916i	$-0.167371\pi$
$972$	−7.46819e17			−0.885559
$973$			3.87348e17i			0.456483i
$974$	2.81659e17			0.329891
$975$	1.32358e16			0.0154071
$976$		−	5.43778e15i		−	0.00629104i
$977$			8.17997e17i			0.940555i	0.882519	+	0.470278i	$0.155846\pi$
$977$			8.17997e17i			0.940555i	−0.882519	+	0.470278i	$0.844154\pi$
$978$	−5.73229e17			−0.655081
$979$	1.57557e17			0.178954
$980$		−	1.97571e17i		−	0.223032i
$981$			3.56017e16i			0.0399445i
$982$	1.50888e18			1.68261
$983$		−	1.06872e18i		−	1.18452i	−0.805748	−	0.592259i	$-0.798236\pi$
$983$		−	1.06872e18i		−	1.18452i	0.805748	−	0.592259i	$-0.201764\pi$
$984$	−8.26885e15			−0.00910908
$985$		−	4.91838e17i		−	0.538524i
$986$		−	2.40963e17i		−	0.262234i
$987$			1.51647e17i			0.164033i
$988$			7.68313e16i			0.0826031i
$989$	1.98393e17	+	1.41525e17i	0.212006	+	0.151236i
$990$	−1.08988e17			−0.115762
$991$	−8.46061e17			−0.893223			−0.446612	−	0.894728i	$-0.647369\pi$
$991$	−8.46061e17			−0.893223			−0.446612	+	0.894728i	$0.647369\pi$
$992$	7.23119e17			0.758822
$993$	−2.29193e16			−0.0239059
$994$			5.45003e17i			0.565041i
$995$	−5.14898e16			−0.0530619
$996$			1.83393e17i			0.187857i
$997$	−9.87076e17			−1.00503			−0.502516	−	0.864568i	$-0.667592\pi$
$997$	−9.87076e17			−1.00503			−0.502516	+	0.864568i	$0.667592\pi$
$998$	−1.06159e17			−0.107442
$999$			2.45079e17i			0.246555i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.c.22.17	✓	20
23.22	odd	2	inner	23.13.b.c.22.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.c.22.17	✓	20	1.1	even	1	trivial
23.13.b.c.22.18	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)

\(f(q)\)	\(=\)	\(q+91.3205 q^{2} -242.505 q^{3} +4243.43 q^{4} -9266.28i q^{5} -22145.7 q^{6} -93897.2i q^{7} +13463.2 q^{8} -472632. q^{9} +O(q^{10})\)	\(q+91.3205 q^{2} -242.505 q^{3} +4243.43 q^{4} -9266.28i q^{5} -22145.7 q^{6} -93897.2i q^{7} +13463.2 q^{8} -472632. q^{9} -846201. i q^{10} +272509. i q^{11} -1.02905e6 q^{12} -344835. q^{13} -8.57474e6i q^{14} +2.24712e6i q^{15} -1.61516e7 q^{16} -4.46231e7i q^{17} -4.31610e7 q^{18} -5.25061e7i q^{19} -3.93208e7i q^{20} +2.27706e7i q^{21} +2.48857e7i q^{22} +(-8.59701e7 + 1.20515e8i) q^{23} -3.26489e6 q^{24} +1.58277e8 q^{25} -3.14905e7 q^{26} +2.43493e8 q^{27} -3.98446e8i q^{28} +5.91320e7 q^{29} +2.05208e8i q^{30} -4.72590e8 q^{31} -1.53012e9 q^{32} -6.60849e7i q^{33} -4.07500e9i q^{34} -8.70078e8 q^{35} -2.00558e9 q^{36} +1.00652e9i q^{37} -4.79488e9i q^{38} +8.36243e7 q^{39} -1.24753e8i q^{40} +2.53266e9 q^{41} +2.07942e9i q^{42} -1.64621e9i q^{43} +1.15637e9i q^{44} +4.37954e9i q^{45} +(-7.85083e9 + 1.10054e10i) q^{46} +6.65977e9 q^{47} +3.91685e9 q^{48} +5.02460e9 q^{49} +1.44539e10 q^{50} +1.08213e10i q^{51} -1.46328e9 q^{52} +3.43262e10i q^{53} +2.22359e10 q^{54} +2.52515e9 q^{55} -1.26415e9i q^{56} +1.27330e10i q^{57} +5.39996e9 q^{58} +4.10502e10 q^{59} +9.53550e9i q^{60} +3.36671e8i q^{61} -4.31572e10 q^{62} +4.43788e10i q^{63} -7.35741e10 q^{64} +3.19534e9i q^{65} -6.03490e9i q^{66} -1.47156e11i q^{67} -1.89355e11i q^{68} +(2.08482e10 - 2.92254e10i) q^{69} -7.94559e10 q^{70} -6.35591e10 q^{71} -6.36313e9 q^{72} -3.59793e10 q^{73} +9.19154e10i q^{74} -3.83829e10 q^{75} -2.22806e11i q^{76} +2.55878e10 q^{77} +7.63661e9 q^{78} +3.10679e11i q^{79} +1.49665e11i q^{80} +1.92128e11 q^{81} +2.31284e11 q^{82} -1.78215e11i q^{83} +9.66252e10i q^{84} -4.13490e11 q^{85} -1.50333e11i q^{86} -1.43398e10 q^{87} +3.66884e9i q^{88} -5.78171e11i q^{89} +3.99942e11i q^{90} +3.23790e10i q^{91} +(-3.64808e11 + 5.11395e11i) q^{92} +1.14606e11 q^{93} +6.08173e11 q^{94} -4.86536e11 q^{95} +3.71062e11 q^{96} -9.04646e11i q^{97} +4.58849e11 q^{98} -1.28797e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) 20 * q + 88 * q^2 + 318 * q^3 + 36072 * q^4 + 264240 * q^6 - 767728 * q^8 + 1971426 * q^9	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9} + 19833480 q^{12} + 8049118 q^{13} + 81743632 q^{16} + 36238080 q^{18} - 367189708 q^{23} + 530348736 q^{24} - 1824317212 q^{25} - 2465696728 q^{26} - 1765163178 q^{27} - 417897362 q^{29} - 1574125298 q^{31} + 1240884960 q^{32} + 4723363152 q^{35} + 3607269840 q^{36} + 1926187110 q^{39} - 132511250 q^{41} + 62599086544 q^{46} - 45052376402 q^{47} + 77275023312 q^{48} - 65534183260 q^{49} - 57446928200 q^{50} - 20407259400 q^{52} - 15082009344 q^{54} - 12572534832 q^{55} + 97529544392 q^{58} - 110336269112 q^{59} + 553127354432 q^{62} + 13799515808 q^{64} + 7914588558 q^{69} + 590935322064 q^{70} - 40523102210 q^{71} - 539385055680 q^{72} - 449748966242 q^{73} + 2570431278 q^{75} - 345974135760 q^{77} - 48894418992 q^{78} - 2694979782144 q^{81} - 1210586085208 q^{82} + 1051198266576 q^{85} - 2237359632618 q^{87} + 1936236755640 q^{92} - 116415989178 q^{93} + 3420883097024 q^{94} + 3777482201184 q^{95} + 3567911213376 q^{96} + 3590387133208 q^{98}+O(q^{100}) \) 20 * q + 88 * q^2 + 318 * q^3 + 36072 * q^4 + 264240 * q^6 - 767728 * q^8 + 1971426 * q^9 + 19833480 * q^12 + 8049118 * q^13 + 81743632 * q^16 + 36238080 * q^18 - 367189708 * q^23 + 530348736 * q^24 - 1824317212 * q^25 - 2465696728 * q^26 - 1765163178 * q^27 - 417897362 * q^29 - 1574125298 * q^31 + 1240884960 * q^32 + 4723363152 * q^35 + 3607269840 * q^36 + 1926187110 * q^39 - 132511250 * q^41 + 62599086544 * q^46 - 45052376402 * q^47 + 77275023312 * q^48 - 65534183260 * q^49 - 57446928200 * q^50 - 20407259400 * q^52 - 15082009344 * q^54 - 12572534832 * q^55 + 97529544392 * q^58 - 110336269112 * q^59 + 553127354432 * q^62 + 13799515808 * q^64 + 7914588558 * q^69 + 590935322064 * q^70 - 40523102210 * q^71 - 539385055680 * q^72 - 449748966242 * q^73 + 2570431278 * q^75 - 345974135760 * q^77 - 48894418992 * q^78 - 2694979782144 * q^81 - 1210586085208 * q^82 + 1051198266576 * q^85 - 2237359632618 * q^87 + 1936236755640 * q^92 - 116415989178 * q^93 + 3420883097024 * q^94 + 3777482201184 * q^95 + 3567911213376 * q^96 + 3590387133208 * q^98

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