LMFDB - Embedded newform 23.13.b.c.22.2

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			22.2
Root			$-891.853 - 7568.36i$ of defining polynomial
Character	$\chi$	$=$	23.22
Dual form			23.13.b.c.22.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-111.723 q^{2} +792.131 q^{3} +8385.93 q^{4} +7568.36i q^{5} -88498.9 q^{6} -231962. i q^{7} -479282. q^{8} +96030.0 q^{9} +O(q^{10})$	\(q-111.723 q^{2} +792.131 q^{3} +8385.93 q^{4} +7568.36i q^{5} -88498.9 q^{6} -231962. i q^{7} -479282. q^{8} +96030.0 q^{9} -845556. i q^{10} +956612. i q^{11} +6.64275e6 q^{12} +1.33056e6 q^{13} +2.59154e7i q^{14} +5.99513e6i q^{15} +1.91979e7 q^{16} -2.04742e7i q^{17} -1.07287e7 q^{18} -1.74722e6i q^{19} +6.34677e7i q^{20} -1.83744e8i q^{21} -1.06875e8i q^{22} +(-9.91035e7 + 1.09969e8i) q^{23} -3.79654e8 q^{24} +1.86861e8 q^{25} -1.48653e8 q^{26} -3.44902e8 q^{27} -1.94522e9i q^{28} -5.12386e8 q^{29} -6.69791e8i q^{30} -1.11066e9 q^{31} -1.81694e8 q^{32} +7.57762e8i q^{33} +2.28743e9i q^{34} +1.75557e9 q^{35} +8.05301e8 q^{36} -2.11420e9i q^{37} +1.95204e8i q^{38} +1.05398e9 q^{39} -3.62738e9i q^{40} +5.34076e8 q^{41} +2.05284e10i q^{42} -1.05964e10i q^{43} +8.02208e9i q^{44} +7.26789e8i q^{45} +(1.10721e10 - 1.22860e10i) q^{46} +2.06440e8 q^{47} +1.52072e10 q^{48} -3.99652e10 q^{49} -2.08765e10 q^{50} -1.62183e10i q^{51} +1.11580e10 q^{52} -3.93108e10i q^{53} +3.85334e10 q^{54} -7.23998e9 q^{55} +1.11175e11i q^{56} -1.38403e9i q^{57} +5.72451e10 q^{58} -2.74230e10 q^{59} +5.02747e10i q^{60} +7.12673e10i q^{61} +1.24085e11 q^{62} -2.22753e10i q^{63} -5.83351e10 q^{64} +1.00701e10i q^{65} -8.46591e10i q^{66} -1.12785e11i q^{67} -1.71695e11i q^{68} +(-7.85029e10 + 8.71096e10i) q^{69} -1.96137e11 q^{70} +1.71752e11 q^{71} -4.60255e10 q^{72} -2.22447e11 q^{73} +2.36204e11i q^{74} +1.48018e11 q^{75} -1.46521e10i q^{76} +2.21898e11 q^{77} -1.17753e11 q^{78} -2.90410e11i q^{79} +1.45296e11i q^{80} -3.24242e11 q^{81} -5.96684e10 q^{82} -1.27559e11i q^{83} -1.54087e12i q^{84} +1.54956e11 q^{85} +1.18386e12i q^{86} -4.05877e11 q^{87} -4.58487e11i q^{88} -1.91239e11i q^{89} -8.11988e10i q^{90} -3.08639e11i q^{91} +(-8.31075e11 + 9.22190e11i) q^{92} -8.79784e11 q^{93} -2.30640e10 q^{94} +1.32236e10 q^{95} -1.43925e11 q^{96} +9.67574e11i q^{97} +4.46502e12 q^{98} +9.18634e10i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−111.723			−1.74567			−0.872833	−	0.488020i	$-0.837719\pi$
$2$	−111.723			−1.74567			−0.872833	+	0.488020i	$0.837719\pi$
$3$	792.131			1.08660			0.543299	−	0.839539i	$-0.317175\pi$
$3$	792.131			1.08660			0.543299	+	0.839539i	$0.317175\pi$
$4$	8385.93			2.04735
$5$			7568.36i			0.484375i	0.970229	+	0.242187i	$0.0778649\pi$
$5$			7568.36i			0.484375i	−0.970229	+	0.242187i	$0.922135\pi$
$6$	−88498.9			−1.89684
$7$		−	231962.i		−	1.97165i	−0.167786	−	0.985823i	$-0.553662\pi$
$7$		−	231962.i		−	1.97165i	0.167786	−	0.985823i	$-0.446338\pi$
$8$	−479282.			−1.82832
$9$	96030.0			0.180697
$10$		−	845556.i		−	0.845556i
$11$			956612.i			0.539983i	0.962863	+	0.269991i	$0.0870208\pi$
$11$			956612.i			0.539983i	−0.962863	+	0.269991i	$0.912979\pi$
$12$	6.64275e6			2.22464
$13$	1.33056e6			0.275660			0.137830	−	0.990456i	$-0.455987\pi$
$13$	1.33056e6			0.275660			0.137830	+	0.990456i	$0.455987\pi$
$14$			2.59154e7i			3.44184i
$15$			5.99513e6i			0.526321i
$16$	1.91979e7			1.14428
$17$		−	2.04742e7i		−	0.848230i	−0.905608	−	0.424115i	$-0.860585\pi$
$17$		−	2.04742e7i		−	0.848230i	0.905608	−	0.424115i	$-0.139415\pi$
$18$	−1.07287e7			−0.315437
$19$		−	1.74722e6i		−	0.0371387i	−0.999828	−	0.0185694i	$-0.994089\pi$
$19$		−	1.74722e6i		−	0.0371387i	0.999828	−	0.0185694i	$-0.00591115\pi$
$20$			6.34677e7i			0.991683i
$21$		−	1.83744e8i		−	2.14239i
$22$		−	1.06875e8i		−	0.942629i
$23$	−9.91035e7	+	1.09969e8i	−0.669456	+	0.742852i
$23$	−9.91035e7	+	1.09969e8i	−0.669456	+	0.742852i
$24$	−3.79654e8			−1.98665
$25$	1.86861e8			0.765381
$26$	−1.48653e8			−0.481210
$27$	−3.44902e8			−0.890253
$28$		−	1.94522e9i		−	4.03664i
$29$	−5.12386e8			−0.861409			−0.430704	−	0.902493i	$-0.641735\pi$
$29$	−5.12386e8			−0.861409			−0.430704	+	0.902493i	$0.641735\pi$
$30$		−	6.69791e8i		−	0.918781i
$31$	−1.11066e9			−1.25144			−0.625719	−	0.780049i	$-0.715194\pi$
$31$	−1.11066e9			−1.25144			−0.625719	+	0.780049i	$0.715194\pi$
$32$	−1.81694e8			−0.169216
$33$			7.57762e8i			0.586744i
$34$			2.28743e9i			1.48073i
$35$	1.75557e9			0.955016
$36$	8.05301e8			0.369950
$37$		−	2.11420e9i		−	0.824016i	−0.911180	−	0.412008i	$-0.864828\pi$
$37$		−	2.11420e9i		−	0.824016i	0.911180	−	0.412008i	$-0.135172\pi$
$38$			1.95204e8i			0.0648318i
$39$	1.05398e9			0.299532
$40$		−	3.62738e9i		−	0.885590i
$41$	5.34076e8			0.112435			0.0562173	−	0.998419i	$-0.482096\pi$
$41$	5.34076e8			0.112435			0.0562173	+	0.998419i	$0.482096\pi$
$42$			2.05284e10i			3.73989i
$43$		−	1.05964e10i		−	1.67629i	−0.545451	−	0.838143i	$-0.683642\pi$
$43$		−	1.05964e10i		−	1.67629i	0.545451	−	0.838143i	$-0.316358\pi$
$44$			8.02208e9i			1.10553i
$45$			7.26789e8i			0.0875253i
$46$	1.10721e10	−	1.22860e10i	1.16865	−	1.29677i
$47$	2.06440e8			0.0191517			0.00957584	−	0.999954i	$-0.496952\pi$
$47$	2.06440e8			0.0191517			0.00957584	+	0.999954i	$0.496952\pi$
$48$	1.52072e10			1.24337
$49$	−3.99652e10			−2.88739
$50$	−2.08765e10			−1.33610
$51$		−	1.62183e10i		−	0.921686i
$52$	1.11580e10			0.564371
$53$		−	3.93108e10i		−	1.77360i	−0.462151	−	0.886801i	$-0.652922\pi$
$53$		−	3.93108e10i		−	1.77360i	0.462151	−	0.886801i	$-0.347078\pi$
$54$	3.85334e10			1.55408
$55$	−7.23998e9			−0.261554
$56$			1.11175e11i			3.60479i
$57$		−	1.38403e9i		−	0.0403549i
$58$	5.72451e10			1.50373
$59$	−2.74230e10			−0.650135			−0.325068	−	0.945691i	$-0.605387\pi$
$59$	−2.74230e10			−0.650135			−0.325068	+	0.945691i	$0.605387\pi$
$60$			5.02747e10i			1.07756i
$61$			7.12673e10i			1.38328i	0.722240	+	0.691642i	$0.243112\pi$
$61$			7.12673e10i			1.38328i	−0.722240	+	0.691642i	$0.756888\pi$
$62$	1.24085e11			2.18459
$63$		−	2.22753e10i		−	0.356271i
$64$	−5.83351e10			−0.848887
$65$			1.00701e10i			0.133523i
$66$		−	8.46591e10i		−	1.02426i
$67$		−	1.12785e11i		−	1.24681i	−0.781898	−	0.623406i	$-0.785748\pi$
$67$		−	1.12785e11i		−	1.24681i	0.781898	−	0.623406i	$-0.214252\pi$
$68$		−	1.71695e11i		−	1.73662i
$69$	−7.85029e10	+	8.71096e10i	−0.727430	+	0.807182i
$70$	−1.96137e11			−1.66714
$71$	1.71752e11			1.34076			0.670381	−	0.742017i	$-0.266131\pi$
$71$	1.71752e11			1.34076			0.670381	+	0.742017i	$0.266131\pi$
$72$	−4.60255e10			−0.330372
$73$	−2.22447e11			−1.46990			−0.734951	−	0.678120i	$-0.762795\pi$
$73$	−2.22447e11			−1.46990			−0.734951	+	0.678120i	$0.762795\pi$
$74$			2.36204e11i			1.43846i
$75$	1.48018e11			0.831662
$76$		−	1.46521e10i		−	0.0760359i
$77$	2.21898e11			1.06465
$78$	−1.17753e11			−0.522882
$79$		−	2.90410e11i		−	1.19467i	−0.801991	−	0.597336i	$-0.796226\pi$
$79$		−	2.90410e11i		−	1.19467i	0.801991	−	0.597336i	$-0.203774\pi$
$80$			1.45296e11i			0.554261i
$81$	−3.24242e11			−1.14805
$82$	−5.96684e10			−0.196273
$83$		−	1.27559e11i		−	0.390161i	−0.980787	−	0.195080i	$-0.937503\pi$
$83$		−	1.27559e11i		−	0.390161i	0.980787	−	0.195080i	$-0.0624968\pi$
$84$		−	1.54087e12i		−	4.38621i
$85$	1.54956e11			0.410861
$86$			1.18386e12i			2.92623i
$87$	−4.05877e11			−0.936006
$88$		−	4.58487e11i		−	0.987259i
$89$		−	1.91239e11i		−	0.384801i	−0.981316	−	0.192400i	$-0.938373\pi$
$89$		−	1.91239e11i		−	0.384801i	0.981316	−	0.192400i	$-0.0616273\pi$
$90$		−	8.11988e10i		−	0.152790i
$91$		−	3.08639e11i		−	0.543504i
$92$	−8.31075e11	+	9.22190e11i	−1.37061	+	1.52088i
$93$	−8.79784e11			−1.35981
$94$	−2.30640e10			−0.0334324
$95$	1.32236e10			0.0179891
$96$	−1.43925e11			−0.183870
$97$			9.67574e11i			1.16159i	0.814049	+	0.580796i	$0.197259\pi$
$97$			9.67574e11i			1.16159i	−0.814049	+	0.580796i	$0.802741\pi$
$98$	4.46502e12			5.04042
$99$			9.18634e10i			0.0975734i
$100$	1.56700e12			1.56700
$101$	6.96867e11			0.656480			0.328240	−	0.944594i	$-0.393545\pi$
$101$	6.96867e11			0.656480			0.328240	+	0.944594i	$0.393545\pi$
$102$			1.81195e12i			1.60896i
$103$			1.06863e12i			0.894959i	0.894294	+	0.447479i	$0.147678\pi$
$103$			1.06863e12i			0.894959i	−0.894294	+	0.447479i	$0.852322\pi$
$104$	−6.37713e11			−0.503994
$105$	1.39064e12			1.03772
$106$			4.39190e12i			3.09612i
$107$			1.74182e11i			0.116065i	0.998315	+	0.0580324i	$0.0184827\pi$
$107$			1.74182e11i			0.116065i	−0.998315	+	0.0580324i	$0.981517\pi$
$108$	−2.89233e12			−1.82266
$109$		−	2.46633e12i		−	1.47059i	−0.677745	−	0.735297i	$-0.737043\pi$
$109$		−	2.46633e12i		−	1.47059i	0.677745	−	0.735297i	$-0.262957\pi$
$110$	8.08869e11			0.456586
$111$		−	1.67472e12i		−	0.895375i
$112$		−	4.45318e12i		−	2.25612i
$113$			1.82230e12i			0.875283i	0.899149	+	0.437642i	$0.144186\pi$
$113$			1.82230e12i			0.875283i	−0.899149	+	0.437642i	$0.855814\pi$
$114$			1.54627e11i			0.0704462i
$115$	−8.32283e11	−	7.50051e11i	−0.359819	−	0.324268i
$116$	−4.29684e12			−1.76360
$117$	1.27773e11			0.0498110
$118$	3.06377e12			1.13492
$119$	−4.74925e12			−1.67241
$120$		−	2.87336e12i		−	0.962282i
$121$	2.22332e12			0.708419
$122$		−	7.96217e12i		−	2.41475i
$123$	4.23058e11			0.122171
$124$	−9.31388e12			−2.56213
$125$			3.26197e12i			0.855106i
$126$			2.48866e12i			0.621930i
$127$	8.60955e11			0.205191			0.102595	−	0.994723i	$-0.467285\pi$
$127$	8.60955e11			0.205191			0.102595	+	0.994723i	$0.467285\pi$
$128$	7.26157e12			1.65109
$129$		−	8.39374e12i		−	1.82145i
$130$		−	1.12506e12i		−	0.233086i
$131$	−3.85505e12			−0.762786			−0.381393	−	0.924413i	$-0.624556\pi$
$131$	−3.85505e12			−0.762786			−0.381393	+	0.924413i	$0.624556\pi$
$132$			6.35454e12i			1.20127i
$133$	−4.05290e11			−0.0732245
$134$			1.26006e13i			2.17652i
$135$		−	2.61034e12i		−	0.431216i
$136$			9.81293e12i			1.55083i
$137$			1.83704e12i			0.277841i	0.990304	+	0.138920i	$0.0443632\pi$
$137$			1.83704e12i			0.277841i	−0.990304	+	0.138920i	$0.955637\pi$
$138$	8.77055e12	−	9.73211e12i	1.26985	−	1.40907i
$139$	4.97123e12			0.689247			0.344624	−	0.938741i	$-0.388007\pi$
$139$	4.97123e12			0.689247			0.344624	+	0.938741i	$0.388007\pi$
$140$	1.47221e13			1.95525
$141$	1.63528e11			0.0208102
$142$	−1.91886e13			−2.34052
$143$			1.27283e12i			0.148852i
$144$	1.84357e12			0.206769
$145$		−	3.87792e12i		−	0.417245i
$146$	2.48523e13			2.56596
$147$	−3.16577e13			−3.13744
$148$		−	1.77295e13i		−	1.68705i
$149$			9.29618e12i			0.849546i	0.905300	+	0.424773i	$0.139646\pi$
$149$			9.29618e12i			0.849546i	−0.905300	+	0.424773i	$0.860354\pi$
$150$	−1.65370e13			−1.45180
$151$	5.81772e12			0.490784			0.245392	−	0.969424i	$-0.421083\pi$
$151$	5.81772e12			0.490784			0.245392	+	0.969424i	$0.421083\pi$
$152$			8.37413e11i			0.0679014i
$153$		−	1.96614e12i		−	0.153273i
$154$	−2.47910e13			−1.85853
$155$		−	8.40584e12i		−	0.606165i
$156$	8.83857e12			0.613245
$157$		−	1.58296e13i		−	1.05699i	−0.848936	−	0.528495i	$-0.822757\pi$
$157$		−	1.58296e13i		−	1.05699i	0.848936	−	0.528495i	$-0.177243\pi$
$158$			3.24453e13i			2.08550i
$159$		−	3.11393e13i		−	1.92719i
$160$		−	1.37513e12i		−	0.0819638i
$161$	2.55086e13	+	2.29883e13i	1.46464	+	1.31993i
$162$	3.62252e13			2.00410
$163$	7.35237e12			0.392014			0.196007	−	0.980602i	$-0.437202\pi$
$163$	7.35237e12			0.392014			0.196007	+	0.980602i	$0.437202\pi$
$164$	4.47873e12			0.230193
$165$	−5.73501e12			−0.284204
$166$			1.42513e13i			0.681090i
$167$	8.07303e12			0.372167			0.186084	−	0.982534i	$-0.440420\pi$
$167$	8.07303e12			0.372167			0.186084	+	0.982534i	$0.440420\pi$
$168$			8.80654e13i			3.91697i
$169$	−2.15277e13			−0.924012
$170$	−1.73121e13			−0.717227
$171$		−	1.67786e11i		−	0.00671087i
$172$		−	8.88608e13i		−	3.43194i
$173$	−3.30970e13			−1.23456			−0.617281	−	0.786743i	$-0.711766\pi$
$173$	−3.30970e13			−1.23456			−0.617281	+	0.786743i	$0.711766\pi$
$174$	4.53456e13			1.63395
$175$		−	4.33446e13i		−	1.50906i
$176$			1.83649e13i			0.617892i
$177$	−2.17226e13			−0.706436
$178$			2.13657e13i			0.671734i
$179$	−1.98742e13			−0.604188			−0.302094	−	0.953278i	$-0.597686\pi$
$179$	−1.98742e13			−0.604188			−0.302094	+	0.953278i	$0.597686\pi$
$180$			6.09480e12i			0.179195i
$181$			2.26848e13i			0.645154i	0.946543	+	0.322577i	$0.104549\pi$
$181$			2.26848e13i			0.645154i	−0.946543	+	0.322577i	$0.895451\pi$
$182$			3.44820e13i			0.948776i
$183$			5.64530e13i			1.50308i
$184$	4.74985e13	−	5.27060e13i	1.22398	−	1.35817i
$185$	1.60010e13			0.399133
$186$	9.82918e13			2.37377
$187$	1.95859e13			0.458030
$188$	1.73119e12			0.0392101
$189$			8.00044e13i			1.75527i
$190$	−1.47738e12			−0.0314029
$191$		−	6.25463e13i		−	1.28825i	−0.764919	−	0.644127i	$-0.777221\pi$
$191$		−	6.25463e13i		−	1.28825i	0.764919	−	0.644127i	$-0.222779\pi$
$192$	−4.62090e13			−0.922400
$193$	5.78720e13			1.11976			0.559880	−	0.828574i	$-0.310847\pi$
$193$	5.78720e13			1.11976			0.559880	+	0.828574i	$0.310847\pi$
$194$		−	1.08100e14i		−	2.02775i
$195$			7.97687e12i			0.145086i
$196$	−3.35146e14			−5.91149
$197$	9.63702e13			1.64871			0.824357	−	0.566071i	$-0.191537\pi$
$197$	9.63702e13			1.64871			0.824357	+	0.566071i	$0.191537\pi$
$198$		−	1.02632e13i		−	0.170330i
$199$			1.54021e13i			0.248006i	0.992282	+	0.124003i	$0.0395732\pi$
$199$			1.54021e13i			0.248006i	−0.992282	+	0.124003i	$0.960427\pi$
$200$	−8.95589e13			−1.39936
$201$		−	8.93401e13i		−	1.35478i
$202$	−7.78557e13			−1.14599
$203$			1.18854e14i			1.69839i
$204$		−	1.36005e14i		−	1.88701i
$205$			4.04208e12i			0.0544605i
$206$		−	1.19390e14i		−	1.56230i
$207$	−9.51691e12	+	1.05603e13i	−0.120969	+	0.134231i
$208$	2.55439e13			0.315433
$209$	1.67142e12			0.0200543
$210$	−1.55366e14			−1.81151
$211$	4.65398e13			0.527388			0.263694	−	0.964606i	$-0.415059\pi$
$211$	4.65398e13			0.527388			0.263694	+	0.964606i	$0.415059\pi$
$212$		−	3.29657e14i		−	3.63118i
$213$	1.36050e14			1.45687
$214$		−	1.94601e13i		−	0.202610i
$215$	8.01974e13			0.811951
$216$	1.65306e14			1.62766
$217$			2.57630e14i			2.46739i
$218$			2.75545e14i			2.56716i
$219$	−1.76207e14			−1.59719
$220$	−6.07140e13			−0.535492
$221$		−	2.72421e13i		−	0.233823i
$222$			1.87104e14i			1.56303i
$223$	1.67886e14			1.36517			0.682585	−	0.730806i	$-0.260856\pi$
$223$	1.67886e14			1.36517			0.682585	+	0.730806i	$0.260856\pi$
$224$			4.21462e13i			0.333634i
$225$	1.79442e13			0.138302
$226$		−	2.03592e14i		−	1.52795i
$227$			2.44863e14i			1.78965i	0.446415	+	0.894826i	$0.352700\pi$
$227$			2.44863e14i			1.78965i	−0.446415	+	0.894826i	$0.647300\pi$
$228$		−	1.16064e13i		−	0.0826205i
$229$			1.15757e14i			0.802668i	0.915932	+	0.401334i	$0.131453\pi$
$229$			1.15757e14i			0.802668i	−0.915932	+	0.401334i	$0.868547\pi$
$230$	9.29848e13	+	8.37976e13i	0.628123	+	0.566063i
$231$	1.75772e14			1.15685
$232$	2.45578e14			1.57493
$233$	−2.01419e14			−1.25882			−0.629412	−	0.777072i	$-0.716704\pi$
$233$	−2.01419e14			−1.25882			−0.629412	+	0.777072i	$0.716704\pi$
$234$	−1.42752e13			−0.0869534
$235$			1.56241e12i			0.00927660i
$236$	−2.29968e14			−1.33105
$237$		−	2.30042e14i		−	1.29813i
$238$	5.30598e14			2.91947
$239$	−7.11653e13			−0.381840			−0.190920	−	0.981606i	$-0.561147\pi$
$239$	−7.11653e13			−0.381840			−0.190920	+	0.981606i	$0.561147\pi$
$240$			1.15094e14i			0.602260i
$241$		−	1.69668e13i		−	0.0865961i	−0.999062	−	0.0432981i	$-0.986213\pi$
$241$		−	1.69668e13i		−	0.0865961i	0.999062	−	0.0432981i	$-0.0137865\pi$
$242$	−2.48395e14			−1.23666
$243$	−7.35468e13			−0.357212
$244$			5.97643e14i			2.83206i
$245$		−	3.02471e14i		−	1.39858i
$246$	−4.72651e13			−0.213270
$247$		−	2.32478e12i		−	0.0102377i
$248$	5.32317e14			2.28802
$249$		−	1.01044e14i		−	0.423948i
$250$		−	3.64436e14i		−	1.49273i
$251$		−	4.79307e13i		−	0.191678i	−0.995397	−	0.0958388i	$-0.969447\pi$
$251$		−	4.79307e13i		−	0.191678i	0.995397	−	0.0958388i	$-0.0305533\pi$
$252$		−	1.86799e14i		−	0.729411i
$253$	−1.05197e14	−	9.48036e13i	−0.401127	−	0.361494i
$254$	−9.61881e13			−0.358194
$255$	1.22746e14			0.446442
$256$	−5.72340e14			−2.03336
$257$	−3.35579e14			−1.16465			−0.582325	−	0.812956i	$-0.697857\pi$
$257$	−3.35579e14			−1.16465			−0.582325	+	0.812956i	$0.697857\pi$
$258$			9.37770e14i			3.17964i
$259$	−4.90415e14			−1.62467
$260$			8.44475e13i			0.273367i
$261$	−4.92044e13			−0.155654
$262$	4.30696e14			1.33157
$263$		−	2.25829e14i		−	0.682410i	−0.939989	−	0.341205i	$-0.889165\pi$
$263$		−	2.25829e14i		−	0.682410i	0.939989	−	0.341205i	$-0.110835\pi$
$264$		−	3.63182e14i		−	1.07275i
$265$	2.97518e14			0.859088
$266$	4.52801e13			0.127825
$267$		−	1.51486e14i		−	0.418124i
$268$		−	9.45804e14i		−	2.55266i
$269$	−7.03869e13			−0.185771			−0.0928855	−	0.995677i	$-0.529609\pi$
$269$	−7.03869e13			−0.185771			−0.0928855	+	0.995677i	$0.529609\pi$
$270$			2.91634e14i			0.752759i
$271$	5.91195e14			1.49250			0.746251	−	0.665665i	$-0.231852\pi$
$271$	5.91195e14			1.49250			0.746251	+	0.665665i	$0.231852\pi$
$272$		−	3.93061e14i		−	0.970614i
$273$		−	2.44483e14i		−	0.590571i
$274$		−	2.05239e14i		−	0.485017i
$275$			1.78753e14i			0.413292i
$276$	−6.58320e14	+	7.30495e14i	−1.48930	+	1.65258i
$277$	−4.77761e14			−1.05763			−0.528813	−	0.848738i	$-0.677363\pi$
$277$	−4.77761e14			−1.05763			−0.528813	+	0.848738i	$0.677363\pi$
$278$	−5.55399e14			−1.20320
$279$	−1.06656e14			−0.226132
$280$	−8.41415e14			−1.74607
$281$		−	8.18776e14i		−	1.66313i	−0.555425	−	0.831567i	$-0.687444\pi$
$281$		−	8.18776e14i		−	1.66313i	0.555425	−	0.831567i	$-0.312556\pi$
$282$	−1.82697e13			−0.0363276
$283$			1.88967e14i			0.367848i	0.982940	+	0.183924i	$0.0588800\pi$
$283$			1.88967e14i			0.367848i	−0.982940	+	0.183924i	$0.941120\pi$
$284$	1.44030e15			2.74500
$285$	1.04748e13			0.0195469
$286$		−	1.42204e14i		−	0.259845i
$287$		−	1.23886e14i		−	0.221681i
$288$	−1.74481e13			−0.0305768
$289$	1.63429e14			0.280505
$290$			4.33251e14i			0.728370i
$291$			7.66445e14i			1.26219i
$292$	−1.86542e15			−3.00940
$293$			6.81718e14i			1.07745i	0.842480	+	0.538727i	$0.181095\pi$
$293$			6.81718e14i			1.07745i	−0.842480	+	0.538727i	$0.818905\pi$
$294$	3.53688e15			5.47691
$295$		−	2.07547e14i		−	0.314909i
$296$			1.01330e15i			1.50656i
$297$		−	3.29938e14i		−	0.480721i
$298$		−	1.03859e15i		−	1.48302i
$299$	−1.31863e14	+	1.46320e14i	−0.184542	+	0.204775i
$300$	1.24127e15			1.70270
$301$	−2.45797e15			−3.30504
$302$	−6.49970e14			−0.856745
$303$	5.52010e14			0.713330
$304$		−	3.35430e13i		−	0.0424972i
$305$	−5.39377e14			−0.670028
$306$			2.19662e14i			0.267563i
$307$	7.45905e14			0.890950			0.445475	−	0.895294i	$-0.353035\pi$
$307$	7.45905e14			0.890950			0.445475	+	0.895294i	$0.353035\pi$
$308$	1.86082e15			2.17972
$309$			8.46493e14i			0.972461i
$310$			9.39122e14i			1.05816i
$311$	4.61018e14			0.509514			0.254757	−	0.967005i	$-0.418005\pi$
$311$	4.61018e14			0.509514			0.254757	+	0.967005i	$0.418005\pi$
$312$	−5.05152e14			−0.547639
$313$		−	9.87907e14i		−	1.05063i	−0.850908	−	0.525315i	$-0.823947\pi$
$313$		−	9.87907e14i		−	1.05063i	0.850908	−	0.525315i	$-0.176053\pi$
$314$			1.76852e15i			1.84515i
$315$	1.68588e14			0.172569
$316$		−	2.43536e15i		−	2.44591i
$317$	1.46800e15			1.44667			0.723337	−	0.690495i	$-0.242607\pi$
$317$	1.46800e15			1.44667			0.723337	+	0.690495i	$0.242607\pi$
$318$			3.47896e15i			3.36424i
$319$		−	4.90155e14i		−	0.465146i
$320$		−	4.41501e14i		−	0.411180i
$321$			1.37975e14i			0.126116i
$322$	−2.84989e15	−	2.56831e15i	−2.55677	−	2.30416i
$323$	−3.57731e13			−0.0315022
$324$	−2.71907e15			−2.35045
$325$	2.48629e14			0.210985
$326$	−8.21426e14			−0.684326
$327$		−	1.95366e15i		−	1.59795i
$328$	−2.55973e14			−0.205566
$329$		−	4.78863e13i		−	0.0377604i
$330$	6.40730e14			0.496125
$331$	1.58575e12			0.00120578			0.000602888	−	1.00000i	$-0.499808\pi$
$331$	1.58575e12			0.00120578			0.000602888		1.00000i	$0.499808\pi$
$332$		−	1.06970e15i		−	0.798794i
$333$		−	2.03027e14i		−	0.148898i
$334$	−9.01940e14			−0.649679
$335$	8.53594e14			0.603924
$336$		−	3.52750e15i		−	2.45150i
$337$			7.60014e13i			0.0518850i	0.999663	+	0.0259425i	$0.00825869\pi$
$337$			7.60014e13i			0.0518850i	−0.999663	+	0.0259425i	$0.991741\pi$
$338$	2.40513e15			1.61301
$339$			1.44350e15i			0.951082i
$340$	1.29945e15			0.841176
$341$		−	1.06247e15i		−	0.675755i
$342$			1.87455e13i			0.0117149i
$343$			6.05977e15i			3.72127i
$344$			5.07867e15i			3.06478i
$345$	−6.59277e14	−	5.94138e14i	−0.390979	−	0.352349i
$346$	3.69769e15			2.15513
$347$	1.15982e15			0.664377			0.332189	−	0.943213i	$-0.392213\pi$
$347$	1.15982e15			0.664377			0.332189	+	0.943213i	$0.392213\pi$
$348$	−3.40365e15			−1.91633
$349$	8.12282e14			0.449525			0.224763	−	0.974414i	$-0.427839\pi$
$349$	8.12282e14			0.449525			0.224763	+	0.974414i	$0.427839\pi$
$350$			4.84257e15i			2.63432i
$351$	−4.58913e14			−0.245407
$352$		−	1.73811e14i		−	0.0913735i
$353$	1.28013e15			0.661615			0.330807	−	0.943698i	$-0.392679\pi$
$353$	1.28013e15			0.661615			0.330807	+	0.943698i	$0.392679\pi$
$354$	2.42691e15			1.23320
$355$			1.29988e15i			0.649431i
$356$		−	1.60372e15i		−	0.787821i
$357$	−3.76202e15			−1.81724
$358$	2.22040e15			1.05471
$359$		−	1.55094e14i		−	0.0724485i	−0.999344	−	0.0362243i	$-0.988467\pi$
$359$		−	1.55094e14i		−	0.0724485i	0.999344	−	0.0362243i	$-0.0115331\pi$
$360$		−	3.48337e14i		−	0.160024i
$361$	2.21026e15			0.998621
$362$		−	2.53440e15i		−	1.12622i
$363$	1.76116e15			0.769767
$364$		−	2.58823e15i		−	1.11274i
$365$		−	1.68355e15i		−	0.711984i
$366$		−	6.30708e15i		−	2.62387i
$367$		−	3.46051e15i		−	1.41626i	−0.706080	−	0.708132i	$-0.749538\pi$
$367$		−	3.46051e15i		−	1.41626i	0.706080	−	0.708132i	$-0.250462\pi$
$368$	−1.90257e15	+	2.11116e15i	−0.766046	+	0.850031i
$369$	5.12873e13			0.0203166
$370$	−1.78768e15			−0.696752
$371$	−9.11861e15			−3.49692
$372$	−7.37781e15			−2.78400
$373$		−	2.87796e15i		−	1.06864i	−0.845282	−	0.534320i	$-0.820568\pi$
$373$		−	2.87796e15i		−	1.06864i	0.845282	−	0.534320i	$-0.179432\pi$
$374$	−2.18819e15			−0.799566
$375$			2.58391e15i			0.929158i
$376$	−9.89431e13			−0.0350153
$377$	−6.81760e14			−0.237456
$378$		−	8.93829e15i		−	3.06411i
$379$		−	4.91863e15i		−	1.65962i	−0.558046	−	0.829810i	$-0.688449\pi$
$379$		−	4.91863e15i		−	1.65962i	0.558046	−	0.829810i	$-0.311551\pi$
$380$	1.10892e14			0.0368299
$381$	6.81989e14			0.222960
$382$			6.98783e15i			2.24886i
$383$			1.34592e15i			0.426411i	0.977007	+	0.213205i	$0.0683903\pi$
$383$			1.34592e15i			0.426411i	−0.977007	+	0.213205i	$0.931610\pi$
$384$	5.75211e15			1.79407
$385$			1.67940e15i			0.515692i
$386$	−6.46561e15			−1.95472
$387$		−	1.01757e15i		−	0.302900i
$388$			8.11401e15i			2.37818i
$389$		−	1.76711e15i		−	0.509995i	−0.966942	−	0.254998i	$-0.917925\pi$
$389$		−	1.76711e15i		−	0.509995i	0.966942	−	0.254998i	$-0.0820747\pi$
$390$		−	8.91196e14i		−	0.253271i
$391$	2.25152e15	+	2.02907e15i	0.630109	+	0.567853i
$392$	1.91546e16			5.27907
$393$	−3.05371e15			−0.828842
$394$	−1.07667e16			−2.87810
$395$	2.19792e15			0.578669
$396$			7.70360e14i			0.199767i
$397$	2.47007e15			0.630908			0.315454	−	0.948941i	$-0.397843\pi$
$397$	2.47007e15			0.630908			0.315454	+	0.948941i	$0.397843\pi$
$398$		−	1.72076e15i		−	0.432935i
$399$	−3.21043e14			−0.0795656
$400$	3.58732e15			0.875811
$401$		−	9.25290e14i		−	0.222542i	−0.993790	−	0.111271i	$-0.964508\pi$
$401$		−	9.25290e14i		−	0.222542i	0.993790	−	0.111271i	$-0.0354921\pi$
$402$			9.98131e15i			2.36500i
$403$	−1.47779e15			−0.344971
$404$	5.84388e15			1.34404
$405$		−	2.45398e15i		−	0.556085i
$406$		−	1.32787e16i		−	2.96483i
$407$	2.02247e15			0.444954
$408$			7.77312e15i			1.68513i
$409$	−1.94278e15			−0.415034			−0.207517	−	0.978231i	$-0.566538\pi$
$409$	−1.94278e15			−0.415034			−0.207517	+	0.978231i	$0.566538\pi$
$410$		−	4.51591e14i		−	0.0950698i
$411$			1.45518e15i			0.301901i
$412$			8.96144e15i			1.83229i
$413$			6.36111e15i			1.28184i
$414$	1.06325e15	−	1.17982e15i	0.211171	−	0.234323i
$415$	9.65415e14			0.188984
$416$	−2.41754e14			−0.0466460
$417$	3.93786e15			0.748936
$418$	−1.86735e14			−0.0350080
$419$			1.47609e15i			0.272791i	0.990654	+	0.136396i	$0.0435518\pi$
$419$			1.47609e15i			0.272791i	−0.990654	+	0.136396i	$0.956448\pi$
$420$	1.16618e16			2.12457
$421$			2.20439e15i			0.395909i	0.980211	+	0.197955i	$0.0634299\pi$
$421$			2.20439e15i			0.395909i	−0.980211	+	0.197955i	$0.936570\pi$
$422$	−5.19955e15			−0.920642
$423$	1.98244e13			0.00346066
$424$			1.88409e16i			3.24271i
$425$		−	3.82582e15i		−	0.649219i
$426$	−1.51999e16			−2.54321
$427$	1.65313e16			2.72735
$428$			1.46068e15i			0.237625i
$429$			1.00825e15i			0.161742i
$430$	−8.95986e15			−1.41739
$431$			7.55030e15i			1.17788i	0.808177	+	0.588939i	$0.200454\pi$
$431$			7.55030e15i			1.17788i	−0.808177	+	0.588939i	$0.799546\pi$
$432$	−6.62139e15			−1.01870
$433$			4.52905e15i			0.687195i	0.939117	+	0.343597i	$0.111646\pi$
$433$			4.52905e15i			0.687195i	−0.939117	+	0.343597i	$0.888354\pi$
$434$		−	2.87831e16i		−	4.30724i
$435$		−	3.07182e15i		−	0.453378i
$436$		−	2.06825e16i		−	3.01081i
$437$	1.92140e14	+	1.73156e14i	0.0275886	+	0.0248627i
$438$	1.96863e16			2.78817
$439$	−1.51533e15			−0.211700			−0.105850	−	0.994382i	$-0.533756\pi$
$439$	−1.51533e15			−0.211700			−0.105850	+	0.994382i	$0.533756\pi$
$440$	3.46999e15			0.478203
$441$	−3.83786e15			−0.521744
$442$			3.04356e15i			0.408177i
$443$	−4.08853e15			−0.540934			−0.270467	−	0.962729i	$-0.587178\pi$
$443$	−4.08853e15			−0.540934			−0.270467	+	0.962729i	$0.587178\pi$
$444$		−	1.40441e16i		−	1.83314i
$445$	1.44736e15			0.186388
$446$	−1.87567e16			−2.38313
$447$			7.36379e15i			0.923116i
$448$			1.35315e16i			1.67371i
$449$	−9.15793e15			−1.11768			−0.558842	−	0.829274i	$-0.688754\pi$
$449$	−9.15793e15			−1.11768			−0.558842	+	0.829274i	$0.688754\pi$
$450$	−2.00477e15			−0.241430
$451$			5.10904e14i			0.0607127i
$452$			1.52817e16i			1.79201i
$453$	4.60839e15			0.533286
$454$		−	2.73568e16i		−	3.12413i
$455$	2.33589e15			0.263260
$456$			6.63341e14i			0.0737815i
$457$			1.01341e16i			1.11247i	0.831026	+	0.556234i	$0.187754\pi$
$457$			1.01341e16i			1.11247i	−0.831026	+	0.556234i	$0.812246\pi$
$458$		−	1.29327e16i		−	1.40119i
$459$			7.06161e15i			0.755140i
$460$	−6.97947e15	−	6.28987e15i	−0.736674	−	0.663888i
$461$	−4.10442e15			−0.427608			−0.213804	−	0.976877i	$-0.568585\pi$
$461$	−4.10442e15			−0.427608			−0.213804	+	0.976877i	$0.568585\pi$
$462$	−1.96377e16			−2.01948
$463$	1.11295e16			1.12977			0.564886	−	0.825169i	$-0.308920\pi$
$463$	1.11295e16			1.12977			0.564886	+	0.825169i	$0.308920\pi$
$464$	−9.83672e15			−0.985694
$465$		−	6.65852e15i		−	0.658658i
$466$	2.25030e16			2.19748
$467$		−	1.62066e16i		−	1.56239i	−0.624284	−	0.781197i	$-0.714609\pi$
$467$		−	1.62066e16i		−	1.56239i	0.624284	−	0.781197i	$-0.285391\pi$
$468$	1.07150e15			0.101980
$469$	−2.61618e16			−2.45827
$470$		−	1.74557e14i		−	0.0161938i
$471$		−	1.25391e16i		−	1.14852i
$472$	1.31434e16			1.18865
$473$	1.01367e16			0.905165
$474$			2.57009e16i			2.26610i
$475$		−	3.26487e14i		−	0.0284253i
$476$	−3.98269e16			−3.42400
$477$		−	3.77501e15i		−	0.320485i
$478$	7.95078e15			0.666564
$479$		−	2.24795e15i		−	0.186111i	−0.995661	−	0.0930557i	$-0.970337\pi$
$479$		−	2.24795e15i		−	0.186111i	0.995661	−	0.0930557i	$-0.0296635\pi$
$480$		−	1.08928e15i		−	0.0890618i
$481$		−	2.81307e15i		−	0.227148i
$482$			1.89558e15i			0.151168i
$483$	2.02061e16	+	1.82097e16i	1.59148	+	1.43424i
$484$	1.86446e16			1.45038
$485$	−7.32295e15			−0.562646
$486$	8.21683e15			0.623572
$487$	8.99200e15			0.674035			0.337018	−	0.941498i	$-0.390582\pi$
$487$	8.99200e15			0.674035			0.337018	+	0.941498i	$0.390582\pi$
$488$		−	3.41572e16i		−	2.52908i
$489$	5.82404e15			0.425962
$490$			3.37928e16i			2.44145i
$491$	1.31958e16			0.941772			0.470886	−	0.882194i	$-0.343934\pi$
$491$	1.31958e16			0.941772			0.470886	+	0.882194i	$0.343934\pi$
$492$	3.54774e15			0.250127
$493$			1.04907e16i			0.730673i
$494$			2.59731e14i			0.0178715i
$495$	−6.95255e14			−0.0472621
$496$	−2.13222e16			−1.43200
$497$		−	3.98400e16i		−	2.64351i
$498$			1.12889e16i			0.740072i
$499$	3.42307e13			0.00221724			0.00110862	−	0.999999i	$-0.499647\pi$
$499$	3.42307e13			0.00221724			0.00110862	+	0.999999i	$0.499647\pi$
$500$			2.73547e16i			1.75070i
$501$	6.39490e15			0.404396
$502$			5.35494e15i			0.334605i
$503$			1.90470e16i			1.17603i	0.808849	+	0.588016i	$0.200091\pi$
$503$			1.90470e16i			1.17603i	−0.808849	+	0.588016i	$0.799909\pi$
$504$			1.06762e16i			0.651377i
$505$			5.27414e15i			0.317982i
$506$	1.17529e16	+	1.05917e16i	0.700233	+	0.631048i
$507$	−1.70528e16			−1.00403
$508$	7.21991e15			0.420097
$509$	−3.15135e16			−1.81213			−0.906066	−	0.423137i	$-0.860929\pi$
$509$	−3.15135e16			−1.81213			−0.906066	+	0.423137i	$0.860929\pi$
$510$	−1.37134e16			−0.779338
$511$			5.15992e16i			2.89813i
$512$	3.41999e16			1.89848
$513$			6.02622e14i			0.0330629i
$514$	3.74917e16			2.03309
$515$	−8.08776e15			−0.433496
$516$		−	7.03893e16i		−	3.72914i
$517$			1.97483e14i			0.0103416i
$518$	5.47904e16			2.83613
$519$	−2.62172e16			−1.34147
$520$		−	4.82644e15i		−	0.244122i
$521$		−	3.67283e16i		−	1.83643i	−0.396082	−	0.918215i	$-0.629630\pi$
$521$		−	3.67283e16i		−	1.83643i	0.396082	−	0.918215i	$-0.370370\pi$
$522$	5.49725e15			0.271720
$523$		−	2.06476e16i		−	1.00892i	−0.863434	−	0.504462i	$-0.831691\pi$
$523$		−	2.06476e16i		−	1.00892i	0.863434	−	0.504462i	$-0.168309\pi$
$524$	−3.23282e16			−1.56169
$525$		−	3.43346e16i		−	1.63974i
$526$			2.52302e16i			1.19126i
$527$			2.27398e16i			1.06151i
$528$			1.45474e16i			0.671401i
$529$	−2.27162e15	−	2.17966e16i	−0.103658	−	0.994613i
$530$	−3.32395e16			−1.49968
$531$	−2.63343e15			−0.117478
$532$	−3.39874e15			−0.149916
$533$	7.10619e14			0.0309937
$534$			1.69244e16i			0.729905i
$535$	−1.31827e15			−0.0562189
$536$			5.40556e16i			2.27957i
$537$	−1.57430e16			−0.656510
$538$	7.86380e15			0.324294
$539$		−	3.82312e16i		−	1.55914i
$540$		−	2.18902e16i		−	0.882849i
$541$	4.00081e16			1.59575			0.797874	−	0.602824i	$-0.205958\pi$
$541$	4.00081e16			1.59575			0.797874	+	0.602824i	$0.205958\pi$
$542$	−6.60498e16			−2.60541
$543$			1.79693e16i			0.701024i
$544$			3.72004e15i			0.143534i
$545$	1.86661e16			0.712319
$546$			2.73142e16i			1.03094i
$547$	−8.35881e15			−0.312047			−0.156024	−	0.987753i	$-0.549868\pi$
$547$	−8.35881e15			−0.312047			−0.156024	+	0.987753i	$0.549868\pi$
$548$			1.54053e16i			0.568836i
$549$			6.84380e15i			0.249956i
$550$		−	1.99708e16i		−	0.721470i
$551$			8.95254e14i			0.0319916i
$552$	3.76250e16	−	4.17501e16i	1.32997	−	1.47578i
$553$	−6.73641e16			−2.35547
$554$	5.33767e16			1.84626
$555$	1.26749e16			0.433697
$556$	4.16884e16			1.41113
$557$		−	4.12739e16i		−	1.38212i	−0.722799	−	0.691058i	$-0.757145\pi$
$557$		−	4.12739e16i		−	1.38212i	0.722799	−	0.691058i	$-0.242855\pi$
$558$	1.19159e16			0.394750
$559$		−	1.40991e16i		−	0.462085i
$560$	3.37032e16			1.09281
$561$	1.55146e16			0.497694
$562$			9.14757e16i			2.90327i
$563$			6.30566e15i			0.198007i	0.995087	+	0.0990035i	$0.0315655\pi$
$563$			6.30566e15i			0.198007i	−0.995087	+	0.0990035i	$0.968435\pi$
$564$	1.37133e15			0.0426057
$565$	−1.37918e16			−0.423965
$566$		−	2.11119e16i		−	0.642139i
$567$			7.52119e16i			2.26354i
$568$	−8.23176e16			−2.45134
$569$		−	1.79769e16i		−	0.529713i	−0.964288	−	0.264857i	$-0.914675\pi$
$569$		−	1.79769e16i		−	0.529713i	0.964288	−	0.264857i	$-0.0853246\pi$
$570$	−1.17028e15			−0.0341223
$571$		−	1.05021e16i		−	0.303013i	−0.988456	−	0.151507i	$-0.951588\pi$
$571$		−	1.05021e16i		−	0.303013i	0.988456	−	0.151507i	$-0.0484125\pi$
$572$			1.06738e16i			0.304751i
$573$		−	4.95448e16i		−	1.39981i
$574$			1.38408e16i			0.386981i
$575$	−1.85185e16	+	2.05488e16i	−0.512389	+	0.568565i
$576$	−5.60192e15			−0.153392
$577$	−3.21060e16			−0.870025			−0.435013	−	0.900424i	$-0.643256\pi$
$577$	−3.21060e16			−0.870025			−0.435013	+	0.900424i	$0.643256\pi$
$578$	−1.82587e16			−0.489668
$579$	4.58422e16			1.21673
$580$		−	3.25200e16i		−	0.854245i
$581$	−2.95890e16			−0.769259
$582$		−	8.56292e16i		−	2.20335i
$583$	3.76051e16			0.957714
$584$	1.06615e17			2.68745
$585$			9.67035e14i			0.0241272i
$586$		−	7.61633e16i		−	1.88087i
$587$	−3.87006e16			−0.945996			−0.472998	−	0.881064i	$-0.656828\pi$
$587$	−3.87006e16			−0.945996			−0.472998	+	0.881064i	$0.656828\pi$
$588$	−2.65479e17			−6.42342
$589$			1.94056e15i			0.0464768i
$590$			2.31877e16i			0.549726i
$591$	7.63378e16			1.79149
$592$		−	4.05881e16i		−	0.942906i
$593$	−4.69375e16			−1.07942			−0.539711	−	0.841850i	$-0.681467\pi$
$593$	−4.69375e16			−1.07942			−0.539711	+	0.841850i	$0.681467\pi$
$594$			3.68615e16i			0.839178i
$595$		−	3.59440e16i		−	0.810074i
$596$			7.79571e16i			1.73931i
$597$			1.22005e16i			0.269483i
$598$	1.47321e16	−	1.63472e16i	0.322149	−	0.357468i
$599$	1.08593e16			0.235093			0.117546	−	0.993067i	$-0.462497\pi$
$599$	1.08593e16			0.235093			0.117546	+	0.993067i	$0.462497\pi$
$600$	−7.09424e16			−1.52054
$601$	−1.24364e15			−0.0263905			−0.0131953	−	0.999913i	$-0.504200\pi$
$601$	−1.24364e15			−0.0263905			−0.0131953	+	0.999913i	$0.504200\pi$
$602$	2.74610e17			5.76950
$603$		−	1.08307e16i		−	0.225296i
$604$	4.87870e16			1.00481
$605$			1.68269e16i			0.343140i
$606$	−6.16719e16			−1.24524
$607$	5.40600e16			1.08080			0.540398	−	0.841409i	$-0.318274\pi$
$607$	5.40600e16			1.08080			0.540398	+	0.841409i	$0.318274\pi$
$608$			3.17460e14i			0.00628446i
$609$			9.41481e16i			1.84547i
$610$	6.02605e16			1.16964
$611$	2.74681e14			0.00527935
$612$		−	1.64879e16i		−	0.313803i
$613$			4.01852e15i			0.0757361i	0.999283	+	0.0378680i	$0.0120567\pi$
$613$			4.01852e15i			0.0757361i	−0.999283	+	0.0378680i	$0.987943\pi$
$614$	−8.33344e16			−1.55530
$615$			3.20185e15i			0.0591767i
$616$	−1.06352e17			−1.94653
$617$			9.62972e16i			1.74543i	0.488230	+	0.872715i	$0.337643\pi$
$617$			9.62972e16i			1.74543i	−0.488230	+	0.872715i	$0.662357\pi$
$618$		−	9.45723e16i		−	1.69759i
$619$			3.34623e16i			0.594856i	0.954744	+	0.297428i	$0.0961288\pi$
$619$			3.34623e16i			0.594856i	−0.954744	+	0.297428i	$0.903871\pi$
$620$		−	7.04908e16i		−	1.24103i
$621$	3.41810e16	−	3.79285e16i	0.595985	−	0.661326i
$622$	−5.15062e16			−0.889440
$623$	−4.43602e16			−0.758692
$624$	2.02341e16			0.342749
$625$	2.09325e16			0.351189
$626$			1.10371e17i			1.83405i
$627$	1.32398e15			0.0217909
$628$		−	1.32746e17i		−	2.16402i
$629$	−4.32866e16			−0.698956
$630$	−1.88350e16			−0.301248
$631$		−	7.00985e16i		−	1.11054i	−0.831672	−	0.555268i	$-0.812616\pi$
$631$		−	7.00985e16i		−	1.11054i	0.831672	−	0.555268i	$-0.187384\pi$
$632$			1.39188e17i			2.18424i
$633$	3.68656e16			0.573059
$634$	−1.64009e17			−2.52541
$635$			6.51601e15i			0.0993893i
$636$		−	2.61132e17i		−	3.94563i
$637$	−5.31760e16			−0.795938
$638$			5.47613e16i			0.811989i
$639$	1.64933e16			0.242272
$640$			5.49581e16i			0.799746i
$641$		−	3.96040e16i		−	0.570941i	−0.958388	−	0.285470i	$-0.907850\pi$
$641$		−	3.96040e16i		−	0.570941i	0.958388	−	0.285470i	$-0.0921498\pi$
$642$		−	1.54149e16i		−	0.220156i
$643$		−	2.71327e16i		−	0.383908i	−0.981404	−	0.191954i	$-0.938518\pi$
$643$		−	2.71327e16i		−	0.383908i	0.981404	−	0.191954i	$-0.0614824\pi$
$644$	2.13913e17	+	1.92778e17i	2.99863	+	2.70236i
$645$	6.35268e16			0.882265
$646$	3.99666e15			0.0549923
$647$	1.10223e17			1.50262			0.751309	−	0.659951i	$-0.229423\pi$
$647$	1.10223e17			1.50262			0.751309	+	0.659951i	$0.229423\pi$
$648$	1.55403e17			2.09899
$649$		−	2.62332e16i		−	0.351062i
$650$	−2.77775e16			−0.368309
$651$			2.04077e17i			2.68107i
$652$	6.16565e16			0.802589
$653$	7.33584e16			0.946173			0.473087	−	0.881016i	$-0.343140\pi$
$653$	7.33584e16			0.946173			0.473087	+	0.881016i	$0.343140\pi$
$654$			2.18268e17i			2.78948i
$655$		−	2.91764e16i		−	0.369474i
$656$	1.02531e16			0.128657
$657$	−2.13615e16			−0.265607
$658$			5.34998e15i			0.0659169i
$659$		−	4.26852e16i		−	0.521153i	−0.965453	−	0.260576i	$-0.916087\pi$
$659$		−	4.26852e16i		−	0.521153i	0.965453	−	0.260576i	$-0.0839126\pi$
$660$	−4.80934e16			−0.581865
$661$			9.61409e16i			1.15265i	0.817219	+	0.576327i	$0.195515\pi$
$661$			9.61409e16i			1.15265i	−0.817219	+	0.576327i	$0.804485\pi$
$662$	−1.77164e14			−0.00210488
$663$		−	2.15793e16i		−	0.254072i
$664$			6.11369e16i			0.713337i
$665$		−	3.06738e15i		−	0.0354681i
$666$			2.26827e16i			0.259925i
$667$	5.07793e16	−	5.63465e16i	0.576675	−	0.639899i
$668$	6.76999e16			0.761955
$669$	1.32988e17			1.48339
$670$	−9.53657e16			−1.05425
$671$	−6.81752e16			−0.746949
$672$			3.33853e16i			0.362526i
$673$	−9.04138e16			−0.973070			−0.486535	−	0.873661i	$-0.661739\pi$
$673$	−9.04138e16			−0.973070			−0.486535	+	0.873661i	$0.661739\pi$
$674$		−	8.49107e15i		−	0.0905739i
$675$	−6.44487e16			−0.681383
$676$	−1.80530e17			−1.89177
$677$			3.70409e16i			0.384724i	0.981324	+	0.192362i	$0.0616148\pi$
$677$			3.70409e16i			0.384724i	−0.981324	+	0.192362i	$0.938385\pi$
$678$		−	1.61271e17i		−	1.66027i
$679$	2.24441e17			2.29025
$680$	−7.42677e16			−0.751185
$681$			1.93964e17i			1.94463i
$682$			1.18701e17i			1.17964i
$683$	−9.57659e16			−0.943380			−0.471690	−	0.881764i	$-0.656356\pi$
$683$	−9.57659e16			−0.943380			−0.471690	+	0.881764i	$0.656356\pi$
$684$		−	1.40704e15i		−	0.0137395i
$685$	−1.39034e16			−0.134579
$686$		−	6.77013e17i		−	6.49609i
$687$			9.16950e16i			0.872179i
$688$		−	2.03428e17i		−	1.91814i
$689$		−	5.23052e16i		−	0.488911i
$690$	7.36561e16	+	6.63786e16i	0.682518	+	0.615083i
$691$	−1.26258e17			−1.15982			−0.579908	−	0.814682i	$-0.696911\pi$
$691$	−1.26258e17			−1.15982			−0.579908	+	0.814682i	$0.696911\pi$
$692$	−2.77550e17			−2.52757
$693$	2.13089e16			0.192380
$694$	−1.29578e17			−1.15978
$695$			3.76241e16i			0.333854i
$696$	1.94529e17			1.71132
$697$		−	1.09348e16i		−	0.0953704i
$698$	−9.07502e16			−0.784721
$699$	−1.59550e17			−1.36784
$700$		−	3.63485e17i		−	3.08957i
$701$		−	5.24413e16i		−	0.441942i	−0.975280	−	0.220971i	$-0.929077\pi$
$701$		−	5.24413e16i		−	0.441942i	0.975280	−	0.220971i	$-0.0709226\pi$
$702$	5.12709e16			0.428399
$703$	−3.69398e15			−0.0306029
$704$		−	5.58041e16i		−	0.458384i
$705$			1.23764e15i			0.0100799i
$706$	−1.43019e17			−1.15496
$707$		−	1.61647e17i		−	1.29435i
$708$	−1.82165e17			−1.44632
$709$			2.09036e17i			1.64567i	0.568278	+	0.822836i	$0.307610\pi$
$709$			2.09036e17i			1.64567i	−0.568278	+	0.822836i	$0.692390\pi$
$710$		−	1.45226e17i		−	1.13369i
$711$		−	2.78880e16i		−	0.215874i
$712$			9.16574e16i			0.703538i
$713$	1.10070e17	−	1.22137e17i	0.837782	−	0.929633i
$714$	4.20303e17			3.17229
$715$	−9.63322e15			−0.0721000
$716$	−1.66664e17			−1.23698
$717$	−5.63723e16			−0.414907
$718$			1.73275e16i			0.126471i
$719$	−2.06929e16			−0.149778			−0.0748890	−	0.997192i	$-0.523860\pi$
$719$	−2.06929e16			−0.149778			−0.0748890	+	0.997192i	$0.523860\pi$
$720$			1.39528e16i			0.100154i
$721$	2.47881e17			1.76454
$722$	−2.46936e17			−1.74326
$723$		−	1.34399e16i		−	0.0940952i
$724$			1.90233e17i			1.32085i
$725$	−9.57448e16			−0.659306
$726$	−1.96761e17			−1.34376
$727$			4.60296e16i			0.311767i	0.987775	+	0.155884i	$0.0498225\pi$
$727$			4.60296e16i			0.311767i	−0.987775	+	0.155884i	$0.950178\pi$
$728$			1.47925e17i			0.993697i
$729$	1.14057e17			0.759900
$730$			1.88091e17i			1.24289i
$731$	−2.16953e17			−1.42188
$732$			4.73411e17i			3.07732i
$733$		−	2.01711e17i		−	1.30049i	−0.759726	−	0.650243i	$-0.774667\pi$
$733$		−	2.01711e17i		−	1.30049i	0.759726	−	0.650243i	$-0.225333\pi$
$734$			3.86618e17i			2.47232i
$735$		−	2.39597e17i		−	1.51970i
$736$	1.80065e16	−	1.99807e16i	0.113282	−	0.125702i
$737$	1.07891e17			0.673257
$738$	−5.72995e15			−0.0354660
$739$	1.04830e16			0.0643605			0.0321802	−	0.999482i	$-0.489755\pi$
$739$	1.04830e16			0.0643605			0.0321802	+	0.999482i	$0.489755\pi$
$740$	1.34183e17			0.817163
$741$		−	1.84153e15i		−	0.0111242i
$742$	1.01875e18			6.10445
$743$		−	1.08652e17i		−	0.645812i	−0.946431	−	0.322906i	$-0.895340\pi$
$743$		−	1.08652e17i		−	0.645812i	0.946431	−	0.322906i	$-0.104660\pi$
$744$	4.21665e17			2.48616
$745$	−7.03568e16			−0.411499
$746$			3.21533e17i			1.86549i
$747$		−	1.22495e16i		−	0.0705010i
$748$	1.64246e17			0.937745
$749$	4.04037e16			0.228839
$750$		−	2.88681e17i		−	1.62200i
$751$			5.37097e16i			0.299373i	0.988733	+	0.149687i	$0.0478265\pi$
$751$			5.37097e16i			0.299373i	−0.988733	+	0.149687i	$0.952174\pi$
$752$	3.96321e15			0.0219149
$753$		−	3.79674e16i		−	0.208277i
$754$	7.61679e16			0.414519
$755$			4.40305e16i			0.237724i
$756$			6.70911e17i			3.59364i
$757$		−	7.69491e16i		−	0.408911i	−0.978876	−	0.204455i	$-0.934458\pi$
$757$		−	7.69491e16i		−	0.408911i	0.978876	−	0.204455i	$-0.0655423\pi$
$758$			5.49521e17i			2.89714i
$759$	−8.33301e16	−	7.50968e16i	−0.435864	−	0.392800i
$760$	−6.33784e15			−0.0328897
$761$	1.18624e17			0.610753			0.305377	−	0.952232i	$-0.401218\pi$
$761$	1.18624e17			0.610753			0.305377	+	0.952232i	$0.401218\pi$
$762$	−7.61935e16			−0.389214
$763$	−5.72096e17			−2.89949
$764$		−	5.24509e17i		−	2.63750i
$765$	1.48804e16			0.0742416
$766$		−	1.50370e17i		−	0.744370i
$767$	−3.64880e16			−0.179216
$768$	−4.53368e17			−2.20945
$769$		−	1.49752e17i		−	0.724126i	−0.932154	−	0.362063i	$-0.882072\pi$
$769$		−	1.49752e17i		−	0.724126i	0.932154	−	0.362063i	$-0.117928\pi$
$770$		−	1.87627e17i		−	0.900226i
$771$	−2.65822e17			−1.26551
$772$	4.85311e17			2.29253
$773$		−	4.89890e16i		−	0.229626i	−0.993387	−	0.114813i	$-0.963373\pi$
$773$		−	4.89890e16i		−	0.229626i	0.993387	−	0.114813i	$-0.0366270\pi$
$774$			1.13686e17i			0.528763i
$775$	−2.07538e17			−0.957827
$776$		−	4.63741e17i		−	2.12376i
$777$	−3.88473e17			−1.76536
$778$			1.97426e17i			0.890281i
$779$		−	9.33151e14i		−	0.00417568i
$780$			6.68934e16i			0.297041i
$781$			1.64300e17i			0.723988i
$782$	−2.51546e17	−	2.26693e17i	−1.09996	−	0.991281i
$783$	1.76723e17			0.766872
$784$	−7.67246e17			−3.30399
$785$	1.19804e17			0.511979
$786$	3.41168e17			1.44688
$787$		−	2.81296e17i		−	1.18390i	−0.805974	−	0.591951i	$-0.798358\pi$
$787$		−	2.81296e17i		−	1.18390i	0.805974	−	0.591951i	$-0.201642\pi$
$788$	8.08154e17			3.37549
$789$		−	1.78886e17i		−	0.741506i
$790$	−2.45558e17			−1.01016
$791$	4.22704e17			1.72575
$792$		−	4.40285e16i		−	0.178395i
$793$			9.48253e16i			0.381316i
$794$	−2.75962e17			−1.10135
$795$	2.35673e17			0.933484
$796$			1.29161e17i			0.507753i
$797$			4.20209e17i			1.63952i	0.572709	+	0.819758i	$0.305892\pi$
$797$			4.20209e17i			1.63952i	−0.572709	+	0.819758i	$0.694108\pi$
$798$	3.58677e16			0.138895
$799$		−	4.22670e15i		−	0.0162450i
$800$	−3.39514e16			−0.129514
$801$		−	1.83647e16i		−	0.0695325i
$802$			1.03376e17i			0.388484i
$803$		−	2.12795e17i		−	0.793722i
$804$		−	7.49200e17i		−	2.77371i
$805$	−1.73983e17	+	1.93058e17i	−0.639341	+	0.709436i
$806$	1.65103e17			0.602204
$807$	−5.57556e16			−0.201859
$808$	−3.33996e17			−1.20025
$809$	−2.85070e17			−1.01686			−0.508429	−	0.861104i	$-0.669773\pi$
$809$	−2.85070e17			−1.01686			−0.508429	+	0.861104i	$0.669773\pi$
$810$			2.74165e17i			0.970737i
$811$	3.03956e17			1.06828			0.534141	−	0.845396i	$-0.320635\pi$
$811$	3.03956e17			1.06828			0.534141	+	0.845396i	$0.320635\pi$
$812$			9.96704e17i			3.47720i
$813$	4.68304e17			1.62175
$814$	−2.25955e17			−0.776741
$815$			5.56454e16i			0.189882i
$816$		−	3.11356e17i		−	1.05467i
$817$	−1.85143e16			−0.0622551
$818$	2.17052e17			0.724510
$819$		−	2.96386e16i		−	0.0982098i
$820$			3.38966e16i			0.111500i
$821$	−3.93145e16			−0.128379			−0.0641895	−	0.997938i	$-0.520446\pi$
$821$	−3.93145e16			−0.128379			−0.0641895	+	0.997938i	$0.520446\pi$
$822$		−	1.62576e17i		−	0.527019i
$823$	1.85166e17			0.595886			0.297943	−	0.954584i	$-0.403700\pi$
$823$	1.85166e17			0.595886			0.297943	+	0.954584i	$0.403700\pi$
$824$		−	5.12174e17i		−	1.63627i
$825$			1.41596e17i			0.449083i
$826$		−	7.10680e17i		−	2.23766i
$827$			2.07427e17i			0.648385i	0.945991	+	0.324192i	$0.105092\pi$
$827$			2.07427e17i			0.648385i	−0.945991	+	0.324192i	$0.894908\pi$
$828$	−7.98081e16	+	8.85579e16i	−0.247665	+	0.274818i
$829$	1.15761e17			0.356645			0.178323	−	0.983972i	$-0.442933\pi$
$829$	1.15761e17			0.356645			0.178323	+	0.983972i	$0.442933\pi$
$830$	−1.07859e17			−0.329903
$831$	−3.78449e17			−1.14922
$832$	−7.76182e16			−0.234004
$833$			8.18257e17i			2.44917i
$834$	−4.39948e17			−1.30739
$835$			6.10996e16i			0.180268i
$836$	1.40164e16			0.0410580
$837$	3.83068e17			1.11410
$838$		−	1.64913e17i		−	0.476202i
$839$			4.61990e17i			1.32453i	0.749271	+	0.662264i	$0.230404\pi$
$839$			4.61990e17i			1.32453i	−0.749271	+	0.662264i	$0.769596\pi$
$840$	−6.66511e17			−1.89728
$841$	−9.12752e16			−0.257975
$842$		−	2.46280e17i		−	0.691125i
$843$		−	6.48577e17i		−	1.80716i
$844$	3.90280e17			1.07975
$845$		−	1.62929e17i		−	0.447568i
$846$	−2.21484e15			−0.00604115
$847$		−	5.15727e17i		−	1.39675i
$848$		−	7.54682e17i		−	2.02950i
$849$			1.49687e17i			0.399703i
$850$			4.27431e17i			1.13332i
$851$	2.32496e17	+	2.09525e17i	0.612122	+	0.551643i
$852$	1.14091e18			2.98272
$853$	−1.88602e17			−0.489612			−0.244806	−	0.969572i	$-0.578724\pi$
$853$	−1.88602e17			−0.489612			−0.244806	+	0.969572i	$0.578724\pi$
$854$	−1.84692e18			−4.76104
$855$	1.26986e15			0.00325058
$856$		−	8.34823e16i		−	0.212203i
$857$	−2.33141e17			−0.588484			−0.294242	−	0.955731i	$-0.595067\pi$
$857$	−2.33141e17			−0.588484			−0.294242	+	0.955731i	$0.595067\pi$
$858$		−	1.12644e17i		−	0.282347i
$859$	−6.31515e16			−0.157190			−0.0785949	−	0.996907i	$-0.525043\pi$
$859$	−6.31515e16			−0.157190			−0.0785949	+	0.996907i	$0.525043\pi$
$860$	6.72530e17			1.66234
$861$		−	9.81335e16i		−	0.240879i
$862$		−	8.43539e17i		−	2.05618i
$863$	5.13360e17			1.24267			0.621337	−	0.783543i	$-0.286590\pi$
$863$	5.13360e17			1.24267			0.621337	+	0.783543i	$0.286590\pi$
$864$	6.26667e16			0.150645
$865$		−	2.50490e17i		−	0.597990i
$866$		−	5.05997e17i		−	1.19961i
$867$	1.29457e17			0.304797
$868$			2.16047e18i			5.05161i
$869$	2.77809e17			0.645102
$870$			3.43192e17i			0.791446i
$871$		−	1.50066e17i		−	0.343696i
$872$			1.18207e18i			2.68871i
$873$			9.29161e16i			0.209897i
$874$	−2.14664e16	−	1.93454e16i	−0.0481604	−	0.0434020i
$875$	7.56654e17			1.68597
$876$	−1.47766e18			−3.27001
$877$	9.02951e16			0.198457			0.0992286	−	0.995065i	$-0.468362\pi$
$877$	9.02951e16			0.198457			0.0992286	+	0.995065i	$0.468362\pi$
$878$	1.69297e17			0.369557
$879$			5.40010e17i			1.17076i
$880$	−1.38992e17			−0.299291
$881$		−	3.39144e17i		−	0.725319i	−0.931922	−	0.362660i	$-0.881869\pi$
$881$		−	3.39144e17i		−	0.725319i	0.931922	−	0.362660i	$-0.118131\pi$
$882$	4.28775e17			0.910790
$883$	−3.53331e17			−0.745449			−0.372724	−	0.927942i	$-0.621576\pi$
$883$	−3.53331e17			−0.745449			−0.372724	+	0.927942i	$0.621576\pi$
$884$		−	2.28451e17i		−	0.478717i
$885$		−	1.64405e17i		−	0.342180i
$886$	4.56781e17			0.944290
$887$	−6.67281e17			−1.37015			−0.685073	−	0.728474i	$-0.740230\pi$
$887$	−6.67281e17			−1.37015			−0.685073	+	0.728474i	$0.740230\pi$
$888$			8.02665e17i			1.63703i
$889$		−	1.99709e17i		−	0.404564i
$890$	−1.61703e17			−0.325371
$891$		−	3.10174e17i		−	0.619925i
$892$	1.40788e18			2.79498
$893$		−	3.60697e14i		−	0.000711269i
$894$		−	8.22701e17i		−	1.61145i
$895$		−	1.50415e17i		−	0.292653i
$896$		−	1.68441e18i		−	3.25536i
$897$	−1.04453e17	+	1.15904e17i	−0.200523	+	0.222508i
$898$	1.02315e18			1.95110
$899$	5.69085e17			1.07800
$900$	1.50479e17			0.283153
$901$	−8.04857e17			−1.50442
$902$		−	5.70795e16i		−	0.105984i
$903$	−1.94703e18			−3.59126
$904$		−	8.73395e17i		−	1.60030i
$905$	−1.71687e17			−0.312496
$906$	−5.14861e17			−0.930939
$907$			4.75850e17i			0.854724i	0.904081	+	0.427362i	$0.140557\pi$
$907$			4.75850e17i			0.854724i	−0.904081	+	0.427362i	$0.859443\pi$
$908$			2.05341e18i			3.66404i
$909$	6.69201e16			0.118624
$910$	−2.60972e17			−0.459563
$911$		−	2.13503e17i		−	0.373502i	−0.982407	−	0.186751i	$-0.940204\pi$
$911$		−	2.13503e17i		−	0.373502i	0.982407	−	0.186751i	$-0.0597958\pi$
$912$		−	2.65704e16i		−	0.0461774i
$913$	1.22025e17			0.210680
$914$		−	1.13221e18i		−	1.94200i
$915$	−4.27257e17			−0.728052
$916$			9.70734e17i			1.64334i
$917$			8.94227e17i			1.50394i
$918$		−	7.88941e17i		−	1.31822i
$919$			2.83161e16i			0.0470046i	0.999724	+	0.0235023i	$0.00748170\pi$
$919$			2.83161e16i			0.0470046i	−0.999724	+	0.0235023i	$0.992518\pi$
$920$	3.98898e17	+	3.59486e17i	0.657862	+	0.592864i
$921$	5.90854e17			0.968105
$922$	4.58556e17			0.746461
$923$	2.28526e17			0.369594
$924$	1.47401e18			2.36848
$925$		−	3.95061e17i		−	0.630686i
$926$	−1.24342e18			−1.97220
$927$			1.02620e17i			0.161717i
$928$	9.30975e16			0.145764
$929$	1.27279e18			1.97998			0.989991	−	0.141131i	$-0.0450737\pi$
$929$	1.27279e18			1.97998			0.989991	+	0.141131i	$0.0450737\pi$
$930$			7.43907e17i			1.14980i
$931$			6.98282e16i			0.107234i
$932$	−1.68909e18			−2.57725
$933$	3.65187e17			0.553637
$934$			1.81064e18i			2.72742i
$935$			1.48233e17i			0.221858i
$936$	−6.12395e16			−0.0910703
$937$		−	2.82721e17i		−	0.417754i	−0.977942	−	0.208877i	$-0.933019\pi$
$937$		−	2.82721e17i		−	0.417754i	0.977942	−	0.208877i	$-0.0669808\pi$
$938$	2.92286e18			4.29132
$939$		−	7.82551e17i		−	1.14161i
$940$			1.31023e16i			0.0189924i
$941$			1.06068e18i			1.52773i	0.645378	+	0.763864i	$0.276700\pi$
$941$			1.06068e18i			1.52773i	−0.645378	+	0.763864i	$0.723300\pi$
$942$			1.40090e18i			2.00494i
$943$	−5.29288e16	+	5.87317e16i	−0.0752700	+	0.0835223i
$944$	−5.26464e17			−0.743937
$945$	−6.05502e17			−0.850206
$946$	−1.13249e18			−1.58011
$947$	−7.04354e17			−0.976543			−0.488271	−	0.872692i	$-0.662372\pi$
$947$	−7.04354e17			−0.976543			−0.488271	+	0.872692i	$0.662372\pi$
$948$		−	1.92912e18i		−	2.65772i
$949$	−2.95978e17			−0.405193
$950$			3.64760e16i			0.0496210i
$951$	1.16285e18			1.57195
$952$	2.27623e18			3.05770
$953$		−	5.17590e16i		−	0.0690921i	−0.999403	−	0.0345460i	$-0.989001\pi$
$953$		−	5.17590e16i		−	0.0690921i	0.999403	−	0.0345460i	$-0.0109985\pi$
$954$			4.21754e17i			0.559460i
$955$	4.73373e17			0.623997
$956$	−5.96788e17			−0.781758
$957$		−	3.88267e17i		−	0.505427i
$958$			2.51146e17i			0.324888i
$959$	4.26124e17			0.547804
$960$		−	3.49726e17i		−	0.446787i
$961$	4.45893e17			0.566097
$962$			3.14283e17i			0.396525i
$963$			1.67267e16i			0.0209726i
$964$		−	1.42283e17i		−	0.177292i
$965$			4.37996e17i			0.542383i
$966$	−2.25748e18	−	2.03444e18i	−2.77819	−	2.50369i
$967$	−9.80540e17			−1.19924			−0.599620	−	0.800285i	$-0.704682\pi$
$967$	−9.80540e17			−1.19924			−0.599620	+	0.800285i	$0.704682\pi$
$968$	−1.06560e18			−1.29521
$969$	−2.83369e16			−0.0342303
$970$	8.18139e17			0.982192
$971$		−	1.25448e18i		−	1.49675i	−0.663274	−	0.748376i	$-0.730834\pi$
$971$		−	1.25448e18i		−	1.49675i	0.663274	−	0.748376i	$-0.269166\pi$
$972$	−6.16758e17			−0.731337
$973$		−	1.15314e18i		−	1.35895i
$974$	−1.00461e18			−1.17664
$975$	1.96947e17			0.229256
$976$			1.36818e18i			1.58287i
$977$		−	1.09837e18i		−	1.26294i	−0.775402	−	0.631468i	$-0.782453\pi$
$977$		−	1.09837e18i		−	1.26294i	0.775402	−	0.631468i	$-0.217547\pi$
$978$	−6.50677e17			−0.743588
$979$	1.82941e17			0.207786
$980$		−	2.53650e18i		−	2.86338i
$981$		−	2.36842e17i		−	0.265732i
$982$	−1.47427e18			−1.64402
$983$			8.99943e17i			0.997456i	0.866759	+	0.498728i	$0.166199\pi$
$983$			8.99943e17i			0.997456i	−0.866759	+	0.498728i	$0.833801\pi$
$984$	−2.02764e17			−0.223368
$985$			7.29364e17i			0.798595i
$986$		−	1.17205e18i		−	1.27551i
$987$		−	3.79322e16i		−	0.0410304i
$988$		−	1.94955e16i		−	0.0209600i
$989$	1.16527e18	+	1.05014e18i	1.24523	+	1.12220i
$990$	7.76757e16			0.0825038
$991$	−1.40137e18			−1.47949			−0.739743	−	0.672889i	$-0.765053\pi$
$991$	−1.40137e18			−1.47949			−0.739743	+	0.672889i	$0.765053\pi$
$992$	2.01799e17			0.211763
$993$	1.25612e15			0.00131019
$994$			4.45102e18i			4.61468i
$995$	−1.16569e17			−0.120128
$996$		−	8.47345e17i		−	0.867969i
$997$	2.18779e17			0.222759			0.111379	−	0.993778i	$-0.464473\pi$
$997$	2.18779e17			0.222759			0.111379	+	0.993778i	$0.464473\pi$
$998$	−3.82434e15			−0.00387056
$999$			7.29193e17i			0.733583i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.c.22.2	yes	20
23.22	odd	2	inner	23.13.b.c.22.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.c.22.1	✓	20	23.22	odd	2	inner
23.13.b.c.22.2	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)

\(f(q)\)	\(=\)	\(q-111.723 q^{2} +792.131 q^{3} +8385.93 q^{4} +7568.36i q^{5} -88498.9 q^{6} -231962. i q^{7} -479282. q^{8} +96030.0 q^{9} +O(q^{10})\)	\(q-111.723 q^{2} +792.131 q^{3} +8385.93 q^{4} +7568.36i q^{5} -88498.9 q^{6} -231962. i q^{7} -479282. q^{8} +96030.0 q^{9} -845556. i q^{10} +956612. i q^{11} +6.64275e6 q^{12} +1.33056e6 q^{13} +2.59154e7i q^{14} +5.99513e6i q^{15} +1.91979e7 q^{16} -2.04742e7i q^{17} -1.07287e7 q^{18} -1.74722e6i q^{19} +6.34677e7i q^{20} -1.83744e8i q^{21} -1.06875e8i q^{22} +(-9.91035e7 + 1.09969e8i) q^{23} -3.79654e8 q^{24} +1.86861e8 q^{25} -1.48653e8 q^{26} -3.44902e8 q^{27} -1.94522e9i q^{28} -5.12386e8 q^{29} -6.69791e8i q^{30} -1.11066e9 q^{31} -1.81694e8 q^{32} +7.57762e8i q^{33} +2.28743e9i q^{34} +1.75557e9 q^{35} +8.05301e8 q^{36} -2.11420e9i q^{37} +1.95204e8i q^{38} +1.05398e9 q^{39} -3.62738e9i q^{40} +5.34076e8 q^{41} +2.05284e10i q^{42} -1.05964e10i q^{43} +8.02208e9i q^{44} +7.26789e8i q^{45} +(1.10721e10 - 1.22860e10i) q^{46} +2.06440e8 q^{47} +1.52072e10 q^{48} -3.99652e10 q^{49} -2.08765e10 q^{50} -1.62183e10i q^{51} +1.11580e10 q^{52} -3.93108e10i q^{53} +3.85334e10 q^{54} -7.23998e9 q^{55} +1.11175e11i q^{56} -1.38403e9i q^{57} +5.72451e10 q^{58} -2.74230e10 q^{59} +5.02747e10i q^{60} +7.12673e10i q^{61} +1.24085e11 q^{62} -2.22753e10i q^{63} -5.83351e10 q^{64} +1.00701e10i q^{65} -8.46591e10i q^{66} -1.12785e11i q^{67} -1.71695e11i q^{68} +(-7.85029e10 + 8.71096e10i) q^{69} -1.96137e11 q^{70} +1.71752e11 q^{71} -4.60255e10 q^{72} -2.22447e11 q^{73} +2.36204e11i q^{74} +1.48018e11 q^{75} -1.46521e10i q^{76} +2.21898e11 q^{77} -1.17753e11 q^{78} -2.90410e11i q^{79} +1.45296e11i q^{80} -3.24242e11 q^{81} -5.96684e10 q^{82} -1.27559e11i q^{83} -1.54087e12i q^{84} +1.54956e11 q^{85} +1.18386e12i q^{86} -4.05877e11 q^{87} -4.58487e11i q^{88} -1.91239e11i q^{89} -8.11988e10i q^{90} -3.08639e11i q^{91} +(-8.31075e11 + 9.22190e11i) q^{92} -8.79784e11 q^{93} -2.30640e10 q^{94} +1.32236e10 q^{95} -1.43925e11 q^{96} +9.67574e11i q^{97} +4.46502e12 q^{98} +9.18634e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) 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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