LMFDB - Embedded newform 23.13.b.c.22.9

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			22.9
Root			$273.019 + 7850.48i$ of defining polynomial
Character	$\chi$	$=$	23.22
Dual form			23.13.b.c.22.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.17467 q^{2} -251.844 q^{3} -4011.83 q^{4} -7850.48i q^{5} -2310.59 q^{6} +186545. i q^{7} -74386.7 q^{8} -468016. q^{9} +O(q^{10})$	\(q+9.17467 q^{2} -251.844 q^{3} -4011.83 q^{4} -7850.48i q^{5} -2310.59 q^{6} +186545. i q^{7} -74386.7 q^{8} -468016. q^{9} -72025.6i q^{10} -1.55970e6i q^{11} +1.01035e6 q^{12} +2.91464e6 q^{13} +1.71149e6i q^{14} +1.97710e6i q^{15} +1.57500e7 q^{16} -3.03835e6i q^{17} -4.29389e6 q^{18} -4.66485e7i q^{19} +3.14947e7i q^{20} -4.69803e7i q^{21} -1.43098e7i q^{22} +(1.33334e8 + 6.43178e7i) q^{23} +1.87338e7 q^{24} +1.82511e8 q^{25} +2.67409e7 q^{26} +2.51707e8 q^{27} -7.48386e8i q^{28} +3.96458e8 q^{29} +1.81392e7i q^{30} -1.56637e8 q^{31} +4.49189e8 q^{32} +3.92802e8i q^{33} -2.78759e7i q^{34} +1.46447e9 q^{35} +1.87760e9 q^{36} -2.63711e9i q^{37} -4.27985e8i q^{38} -7.34035e8 q^{39} +5.83971e8i q^{40} +1.40562e9 q^{41} -4.31029e8i q^{42} -2.75599e9i q^{43} +6.25725e9i q^{44} +3.67414e9i q^{45} +(1.22329e9 + 5.90095e8i) q^{46} -8.81837e9 q^{47} -3.96654e9 q^{48} -2.09578e10 q^{49} +1.67448e9 q^{50} +7.65191e8i q^{51} -1.16930e10 q^{52} +1.74988e10i q^{53} +2.30933e9 q^{54} -1.22444e10 q^{55} -1.38765e10i q^{56} +1.17481e10i q^{57} +3.63738e9 q^{58} +2.10425e10 q^{59} -7.93176e9i q^{60} +8.13588e10i q^{61} -1.43709e9 q^{62} -8.73060e10i q^{63} -6.03907e10 q^{64} -2.28813e10i q^{65} +3.60383e9i q^{66} +1.16096e8i q^{67} +1.21893e10i q^{68} +(-3.35793e10 - 1.61981e10i) q^{69} +1.34360e10 q^{70} +7.66140e10 q^{71} +3.48141e10 q^{72} +1.64447e11 q^{73} -2.41946e10i q^{74} -4.59642e10 q^{75} +1.87146e11i q^{76} +2.90955e11 q^{77} -6.73453e9 q^{78} -3.14639e11i q^{79} -1.23645e11i q^{80} +1.85332e11 q^{81} +1.28961e10 q^{82} -5.71832e11i q^{83} +1.88477e11i q^{84} -2.38525e10 q^{85} -2.52853e10i q^{86} -9.98457e10 q^{87} +1.16021e11i q^{88} +7.59123e11i q^{89} +3.37091e10i q^{90} +5.43711e11i q^{91} +(-5.34911e11 - 2.58032e11i) q^{92} +3.94480e10 q^{93} -8.09057e10 q^{94} -3.66213e11 q^{95} -1.13125e11 q^{96} -6.47739e11i q^{97} -1.92281e11 q^{98} +7.29965e11i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	9.17467			0.143354			0.0716771	−	0.997428i	$-0.477165\pi$
$2$	9.17467			0.143354			0.0716771	+	0.997428i	$0.477165\pi$
$3$	−251.844			−0.345465			−0.172733	−	0.984969i	$-0.555260\pi$
$3$	−251.844			−0.345465			−0.172733	+	0.984969i	$0.555260\pi$
$4$	−4011.83			−0.979450
$5$		−	7850.48i		−	0.502430i	−0.967931	−	0.251215i	$-0.919170\pi$
$5$		−	7850.48i		−	0.502430i	0.967931	−	0.251215i	$-0.0808302\pi$
$6$	−2310.59			−0.0495239
$7$			186545.i			1.58561i	0.609478	+	0.792803i	$0.291379\pi$
$7$			186545.i			1.58561i	−0.609478	+	0.792803i	$0.708621\pi$
$8$	−74386.7			−0.283763
$9$	−468016.			−0.880654
$10$		−	72025.6i		−	0.0720256i
$11$		−	1.55970e6i		−	0.880411i	−0.897897	−	0.440205i	$-0.854906\pi$
$11$		−	1.55970e6i		−	0.880411i	0.897897	−	0.440205i	$-0.145094\pi$
$12$	1.01035e6			0.338366
$13$	2.91464e6			0.603844			0.301922	−	0.953333i	$-0.402372\pi$
$13$	2.91464e6			0.603844			0.301922	+	0.953333i	$0.402372\pi$
$14$			1.71149e6i			0.227304i
$15$			1.97710e6i			0.173572i
$16$	1.57500e7			0.938771
$17$		−	3.03835e6i		−	0.125877i	−0.998017	−	0.0629383i	$-0.979953\pi$
$17$		−	3.03835e6i		−	0.125877i	0.998017	−	0.0629383i	$-0.0200471\pi$
$18$	−4.29389e6			−0.126246
$19$		−	4.66485e7i		−	0.991553i	−0.868450	−	0.495777i	$-0.834884\pi$
$19$		−	4.66485e7i		−	0.991553i	0.868450	−	0.495777i	$-0.165116\pi$
$20$			3.14947e7i			0.492105i
$21$		−	4.69803e7i		−	0.547772i
$22$		−	1.43098e7i		−	0.126211i
$23$	1.33334e8	+	6.43178e7i	0.900684	+	0.434475i
$23$	1.33334e8	+	6.43178e7i	0.900684	+	0.434475i
$24$	1.87338e7			0.0980301
$25$	1.82511e8			0.747564
$26$	2.67409e7			0.0865636
$27$	2.51707e8			0.649700
$28$		−	7.48386e8i		−	1.55302i
$29$	3.96458e8			0.666514			0.333257	−	0.942836i	$-0.391852\pi$
$29$	3.96458e8			0.666514			0.333257	+	0.942836i	$0.391852\pi$
$30$			1.81392e7i			0.0248823i
$31$	−1.56637e8			−0.176491			−0.0882457	−	0.996099i	$-0.528126\pi$
$31$	−1.56637e8			−0.176491			−0.0882457	+	0.996099i	$0.528126\pi$
$32$	4.49189e8			0.418339
$33$			3.92802e8i			0.304151i
$34$		−	2.78759e7i		−	0.0180449i
$35$	1.46447e9			0.796657
$36$	1.87760e9			0.862556
$37$		−	2.63711e9i		−	1.02782i	−0.857844	−	0.513910i	$-0.828196\pi$
$37$		−	2.63711e9i		−	1.02782i	0.857844	−	0.513910i	$-0.171804\pi$
$38$		−	4.27985e8i		−	0.142143i
$39$	−7.34035e8			−0.208607
$40$			5.83971e8i			0.142571i
$41$	1.40562e9			0.295913			0.147956	−	0.988994i	$-0.452731\pi$
$41$	1.40562e9			0.295913			0.147956	+	0.988994i	$0.452731\pi$
$42$		−	4.31029e8i		−	0.0785254i
$43$		−	2.75599e9i		−	0.435981i	−0.975951	−	0.217990i	$-0.930050\pi$
$43$		−	2.75599e9i		−	0.435981i	0.975951	−	0.217990i	$-0.0699501\pi$
$44$			6.25725e9i			0.862318i
$45$			3.67414e9i			0.442467i
$46$	1.22329e9	+	5.90095e8i	0.129117	+	0.0622838i
$47$	−8.81837e9			−0.818091			−0.409045	−	0.912514i	$-0.634138\pi$
$47$	−8.81837e9			−0.818091			−0.409045	+	0.912514i	$0.634138\pi$
$48$	−3.96654e9			−0.324313
$49$	−2.09578e10			−1.51415
$50$	1.67448e9			0.107166
$51$			7.65191e8i			0.0434860i
$52$	−1.16930e10			−0.591435
$53$			1.74988e10i			0.789503i	0.918788	+	0.394751i	$0.129169\pi$
$53$			1.74988e10i			0.789503i	−0.918788	+	0.394751i	$0.870831\pi$
$54$	2.30933e9			0.0931373
$55$	−1.22444e10			−0.442345
$56$		−	1.38765e10i		−	0.449936i
$57$			1.17481e10i			0.342547i
$58$	3.63738e9			0.0955477
$59$	2.10425e10			0.498867			0.249434	−	0.968392i	$-0.419756\pi$
$59$	2.10425e10			0.498867			0.249434	+	0.968392i	$0.419756\pi$
$60$		−	7.93176e9i		−	0.170005i
$61$			8.13588e10i			1.57916i	0.613649	+	0.789579i	$0.289701\pi$
$61$			8.13588e10i			1.57916i	−0.613649	+	0.789579i	$0.710299\pi$
$62$	−1.43709e9			−0.0253008
$63$		−	8.73060e10i		−	1.39637i
$64$	−6.03907e10			−0.878800
$65$		−	2.28813e10i		−	0.303390i
$66$			3.60383e9i			0.0436014i
$67$			1.16096e8i			0.00128342i	1.00000		0.000641711i	$0.000204263\pi$
$67$			1.16096e8i			0.00128342i	−1.00000		0.000641711i	$0.999796\pi$
$68$			1.21893e10i			0.123290i
$69$	−3.35793e10	−	1.61981e10i	−0.311155	−	0.150096i
$70$	1.34360e10			0.114204
$71$	7.66140e10			0.598078			0.299039	−	0.954241i	$-0.403334\pi$
$71$	7.66140e10			0.598078			0.299039	+	0.954241i	$0.403334\pi$
$72$	3.48141e10			0.249897
$73$	1.64447e11			1.08665			0.543324	−	0.839523i	$-0.317165\pi$
$73$	1.64447e11			1.08665			0.543324	+	0.839523i	$0.317165\pi$
$74$		−	2.41946e10i		−	0.147343i
$75$	−4.59642e10			−0.258257
$76$			1.87146e11i			0.971176i
$77$	2.90955e11			1.39599
$78$	−6.73453e9			−0.0299047
$79$		−	3.14639e11i		−	1.29435i	−0.762343	−	0.647173i	$-0.775951\pi$
$79$		−	3.14639e11i		−	1.29435i	0.762343	−	0.647173i	$-0.224049\pi$
$80$		−	1.23645e11i		−	0.471667i
$81$	1.85332e11			0.656205
$82$	1.28961e10			0.0424203
$83$		−	5.71832e11i		−	1.74904i	−0.484988	−	0.874521i	$-0.661176\pi$
$83$		−	5.71832e11i		−	1.74904i	0.484988	−	0.874521i	$-0.338824\pi$
$84$			1.88477e11i			0.536515i
$85$	−2.38525e10			−0.0632442
$86$		−	2.52853e10i		−	0.0624997i
$87$	−9.98457e10			−0.230257
$88$			1.16021e11i			0.249828i
$89$			7.59123e11i			1.52747i	0.645531	+	0.763734i	$0.276636\pi$
$89$			7.59123e11i			1.52747i	−0.645531	+	0.763734i	$0.723364\pi$
$90$			3.37091e10i			0.0634296i
$91$			5.43711e11i			0.957459i
$92$	−5.34911e11	−	2.58032e11i	−0.882175	−	0.425546i
$93$	3.94480e10			0.0609716
$94$	−8.09057e10			−0.117277
$95$	−3.66213e11			−0.498186
$96$	−1.13125e11			−0.144522
$97$		−	6.47739e11i		−	0.777624i	−0.921317	−	0.388812i	$-0.872886\pi$
$97$		−	6.47739e11i		−	0.777624i	0.921317	−	0.388812i	$-0.127114\pi$
$98$	−1.92281e11			−0.217060
$99$			7.29965e11i			0.775337i
$100$	−7.32201e11			−0.732201
$101$	1.01010e12			0.951563			0.475781	−	0.879564i	$-0.342165\pi$
$101$	1.01010e12			0.951563			0.475781	+	0.879564i	$0.342165\pi$
$102$			7.02038e9i			0.00623390i
$103$		−	1.73435e12i		−	1.45249i	−0.687437	−	0.726244i	$-0.741264\pi$
$103$		−	1.73435e12i		−	1.45249i	0.687437	−	0.726244i	$-0.258736\pi$
$104$	−2.16810e11			−0.171348
$105$	−3.68817e11			−0.275217
$106$			1.60546e11i			0.113179i
$107$			2.71214e12i			1.80721i	0.428362	+	0.903607i	$0.359091\pi$
$107$			2.71214e12i			1.80721i	−0.428362	+	0.903607i	$0.640909\pi$
$108$	−1.00981e12			−0.636349
$109$		−	9.62939e11i		−	0.574169i	−0.957905	−	0.287084i	$-0.907314\pi$
$109$		−	9.62939e11i		−	0.574169i	0.957905	−	0.287084i	$-0.0926860\pi$
$110$	−1.12338e11			−0.0634121
$111$			6.64140e11i			0.355076i
$112$			2.93808e12i			1.48852i
$113$		−	1.95847e12i		−	0.940691i	−0.882482	−	0.470345i	$-0.844129\pi$
$113$		−	1.95847e12i		−	0.940691i	0.882482	−	0.470345i	$-0.155871\pi$
$114$			1.07785e11i			0.0491056i
$115$	5.04926e11	−	1.04673e12i	0.218293	−	0.452531i
$116$	−1.59052e12			−0.652817
$117$	−1.36410e12			−0.531777
$118$	1.93058e11			0.0715147
$119$	5.66790e11			0.199591
$120$		−	1.47070e11i		−	0.0492533i
$121$	7.05760e11			0.224877
$122$			7.46440e11i			0.226379i
$123$	−3.53996e11			−0.102227
$124$	6.28399e11			0.172864
$125$		−	3.34942e12i		−	0.878029i
$126$		−	8.01004e11i		−	0.200176i
$127$	4.83918e12			1.15332			0.576660	−	0.816984i	$-0.304356\pi$
$127$	4.83918e12			1.15332			0.576660	+	0.816984i	$0.304356\pi$
$128$	−2.39394e12			−0.544319
$129$			6.94081e11i			0.150616i
$130$		−	2.09928e11i		−	0.0434922i
$131$	8.51685e12			1.68520			0.842600	−	0.538540i	$-0.181024\pi$
$131$	8.51685e12			1.68520			0.842600	+	0.538540i	$0.181024\pi$
$132$		−	1.57585e12i		−	0.297901i
$133$	8.70204e12			1.57221
$134$			1.06515e9i			0.000183984i
$135$		−	1.97602e12i		−	0.326429i
$136$			2.26013e11i			0.0357191i
$137$		−	1.63849e12i		−	0.247811i	−0.992294	−	0.123906i	$-0.960458\pi$
$137$		−	1.63849e12i		−	0.247811i	0.992294	−	0.123906i	$-0.0395420\pi$
$138$	−3.08079e11	−	1.48612e11i	−0.0446054	−	0.0215169i
$139$	−2.48277e11			−0.0344229			−0.0172114	−	0.999852i	$-0.505479\pi$
$139$	−2.48277e11			−0.0344229			−0.0172114	+	0.999852i	$0.505479\pi$
$140$	−5.87519e12			−0.780285
$141$	2.22086e12			0.282622
$142$	7.02908e11			0.0857371
$143$		−	4.54597e12i		−	0.531631i
$144$	−7.37123e12			−0.826732
$145$		−	3.11239e12i		−	0.334877i
$146$	1.50875e12			0.155776
$147$	5.27809e12			0.523086
$148$			1.05796e13i			1.00670i
$149$			9.55388e12i			0.873097i	0.899681	+	0.436548i	$0.143799\pi$
$149$			9.55388e12i			0.873097i	−0.899681	+	0.436548i	$0.856201\pi$
$150$	−4.21707e11			−0.0370223
$151$	−1.60623e13			−1.35502			−0.677509	−	0.735514i	$-0.736941\pi$
$151$	−1.60623e13			−1.35502			−0.677509	+	0.735514i	$0.736941\pi$
$152$			3.47003e12i			0.281366i
$153$			1.42200e12i			0.110854i
$154$	2.66941e12			0.200120
$155$			1.22967e12i			0.0886747i
$156$	2.94482e12			0.204320
$157$		−	1.83890e13i		−	1.22789i	−0.789347	−	0.613947i	$-0.789581\pi$
$157$		−	1.83890e13i		−	1.22789i	0.789347	−	0.613947i	$-0.210419\pi$
$158$		−	2.88671e12i		−	0.185550i
$159$		−	4.40698e12i		−	0.272746i
$160$		−	3.52634e12i		−	0.210186i
$161$	−1.19982e13	+	2.48727e13i	−0.688906	+	1.42813i
$162$	1.70036e12			0.0940698
$163$	−1.60347e13			−0.854937			−0.427468	−	0.904030i	$-0.640594\pi$
$163$	−1.60347e13			−0.854937			−0.427468	+	0.904030i	$0.640594\pi$
$164$	−5.63908e12			−0.289831
$165$	3.08368e12			0.152815
$166$		−	5.24637e12i		−	0.250733i
$167$	−1.91586e13			−0.883214			−0.441607	−	0.897209i	$-0.645591\pi$
$167$	−1.91586e13			−0.883214			−0.441607	+	0.897209i	$0.645591\pi$
$168$			3.49471e12i			0.155437i
$169$	−1.48030e13			−0.635373
$170$	−2.18839e11			−0.00906633
$171$			2.18322e13i			0.873215i
$172$			1.10566e13i			0.427021i
$173$	3.64376e13			1.35917			0.679584	−	0.733597i	$-0.262160\pi$
$173$	3.64376e13			1.35917			0.679584	+	0.733597i	$0.262160\pi$
$174$	−9.16051e11			−0.0330084
$175$			3.40465e13i			1.18534i
$176$		−	2.45652e13i		−	0.826504i
$177$	−5.29942e12			−0.172341
$178$			6.96470e12i			0.218969i
$179$	6.09040e10			0.00185152			0.000925758	−	1.00000i	$-0.499705\pi$
$179$	6.09040e10			0.00185152			0.000925758		1.00000i	$0.499705\pi$
$180$		−	1.47400e13i		−	0.433374i
$181$		−	4.37064e13i		−	1.24301i	−0.783412	−	0.621503i	$-0.786522\pi$
$181$		−	4.37064e13i		−	1.24301i	0.783412	−	0.621503i	$-0.213478\pi$
$182$			4.98838e12i			0.137256i
$183$		−	2.04897e13i		−	0.545544i
$184$	−9.91824e12	−	4.78439e12i	−0.255580	−	0.123288i
$185$	−2.07025e13			−0.516408
$186$	3.61923e11			0.00874054
$187$	−4.73892e12			−0.110823
$188$	3.53778e13			0.801278
$189$			4.69547e13i			1.03017i
$190$	−3.35988e12			−0.0714172
$191$		−	5.40621e13i		−	1.11351i	−0.830678	−	0.556753i	$-0.812047\pi$
$191$		−	5.40621e13i		−	1.11351i	0.830678	−	0.556753i	$-0.187953\pi$
$192$	1.52090e13			0.303595
$193$	8.87390e13			1.71700			0.858500	−	0.512813i	$-0.171396\pi$
$193$	8.87390e13			1.71700			0.858500	+	0.512813i	$0.171396\pi$
$194$		−	5.94279e12i		−	0.111476i
$195$			5.76252e12i			0.104811i
$196$	8.40789e13			1.48303
$197$	−1.31393e13			−0.224788			−0.112394	−	0.993664i	$-0.535852\pi$
$197$	−1.31393e13			−0.224788			−0.112394	+	0.993664i	$0.535852\pi$
$198$			6.69719e12i			0.111148i
$199$		−	2.68520e13i		−	0.432372i	−0.976352	−	0.216186i	$-0.930638\pi$
$199$		−	2.68520e13i		−	0.432372i	0.976352	−	0.216186i	$-0.0693617\pi$
$200$	−1.35764e13			−0.212131
$201$		−	2.92381e10i		−	0.000443377i
$202$	9.26736e12			0.136411
$203$			7.39573e13i			1.05683i
$204$		−	3.06981e12i		−	0.0425923i
$205$		−	1.10347e13i		−	0.148675i
$206$		−	1.59121e13i		−	0.208220i
$207$	−6.24022e13	−	3.01017e13i	−0.793191	−	0.382622i
$208$	4.59055e13			0.566871
$209$	−7.27577e13			−0.872974
$210$	−3.38378e12			−0.0394536
$211$	5.49173e13			0.622321			0.311161	−	0.950357i	$-0.399282\pi$
$211$	5.49173e13			0.622321			0.311161	+	0.950357i	$0.399282\pi$
$212$		−	7.02022e13i		−	0.773278i
$213$	−1.92948e13			−0.206615
$214$			2.48830e13i			0.259072i
$215$	−2.16359e13			−0.219050
$216$	−1.87237e13			−0.184361
$217$		−	2.92198e13i		−	0.279846i
$218$		−	8.83465e12i		−	0.0823096i
$219$	−4.14150e13			−0.375399
$220$	4.91224e13			0.433255
$221$		−	8.85570e12i		−	0.0760098i
$222$			6.09327e12i			0.0509017i
$223$	1.44603e14			1.17584			0.587921	−	0.808919i	$-0.299947\pi$
$223$	1.44603e14			1.17584			0.587921	+	0.808919i	$0.299947\pi$
$224$			8.37939e13i			0.663322i
$225$	−8.54178e13			−0.658345
$226$		−	1.79684e13i		−	0.134852i
$227$			1.78521e14i			1.30477i	0.757887	+	0.652386i	$0.226232\pi$
$227$			1.78521e14i			1.30477i	−0.757887	+	0.652386i	$0.773768\pi$
$228$		−	4.71315e13i		−	0.335508i
$229$			4.74666e13i			0.329136i	0.986366	+	0.164568i	$0.0526230\pi$
$229$			4.74666e13i			0.329136i	−0.986366	+	0.164568i	$0.947377\pi$
$230$	4.63253e12	−	9.60343e12i	0.0312933	−	0.0648723i
$231$	−7.32752e13			−0.482264
$232$	−2.94912e13			−0.189132
$233$	−8.23059e13			−0.514394			−0.257197	−	0.966359i	$-0.582799\pi$
$233$	−8.23059e13			−0.514394			−0.257197	+	0.966359i	$0.582799\pi$
$234$	−1.25151e13			−0.0762326
$235$			6.92284e13i			0.411034i
$236$	−8.44187e13			−0.488615
$237$			7.92401e13i			0.447152i
$238$	5.20011e12			0.0286122
$239$	−2.40951e14			−1.29283			−0.646415	−	0.762986i	$-0.723733\pi$
$239$	−2.40951e14			−1.29283			−0.646415	+	0.762986i	$0.723733\pi$
$240$			3.11392e13i			0.162945i
$241$		−	1.81866e14i		−	0.928217i	−0.885778	−	0.464109i	$-0.846375\pi$
$241$		−	1.81866e14i		−	0.928217i	0.885778	−	0.464109i	$-0.153625\pi$
$242$	6.47512e12			0.0322371
$243$	−1.80442e14			−0.876396
$244$		−	3.26397e14i		−	1.54670i
$245$			1.64528e14i			0.760754i
$246$	−3.24780e12			−0.0146547
$247$		−	1.35964e14i		−	0.598743i
$248$	1.16517e13			0.0500817
$249$			1.44013e14i			0.604233i
$250$		−	3.07298e13i		−	0.125869i
$251$			3.32738e14i			1.33064i	0.746560	+	0.665318i	$0.231704\pi$
$251$			3.32738e14i			1.33064i	−0.746560	+	0.665318i	$0.768296\pi$
$252$			3.50256e14i			1.36767i
$253$	1.00317e14	−	2.07961e14i	0.382516	−	0.792972i
$254$	4.43979e13			0.165333
$255$	6.00712e12			0.0218487
$256$	2.25397e14			0.800770
$257$	−2.20367e14			−0.764799			−0.382400	−	0.923997i	$-0.624902\pi$
$257$	−2.20367e14			−0.764799			−0.382400	+	0.923997i	$0.624902\pi$
$258$			6.36796e12i			0.0215915i
$259$	4.91939e14			1.62972
$260$			9.17958e13i			0.297155i
$261$	−1.85549e14			−0.586968
$262$	7.81393e13			0.241581
$263$		−	5.30387e14i		−	1.60272i	−0.598180	−	0.801362i	$-0.704109\pi$
$263$		−	5.30387e14i		−	1.60272i	0.598180	−	0.801362i	$-0.295891\pi$
$264$		−	2.92192e13i		−	0.0863067i
$265$	1.37374e14			0.396670
$266$	7.98384e13			0.225384
$267$		−	1.91181e14i		−	0.527687i
$268$		−	4.65758e11i		−	0.00125705i
$269$	2.90294e14			0.766167			0.383084	−	0.923714i	$-0.374862\pi$
$269$	2.90294e14			0.766167			0.383084	+	0.923714i	$0.374862\pi$
$270$		−	1.81294e13i		−	0.0467950i
$271$	−1.28018e14			−0.323189			−0.161594	−	0.986857i	$-0.551664\pi$
$271$	−1.28018e14			−0.323189			−0.161594	+	0.986857i	$0.551664\pi$
$272$		−	4.78540e13i		−	0.118169i
$273$		−	1.36931e14i		−	0.330769i
$274$		−	1.50326e13i		−	0.0355248i
$275$		−	2.84662e14i		−	0.658163i
$276$	1.34714e14	+	6.49838e13i	0.304761	+	0.147011i
$277$	3.87507e14			0.857828			0.428914	−	0.903345i	$-0.358896\pi$
$277$	3.87507e14			0.857828			0.428914	+	0.903345i	$0.358896\pi$
$278$	−2.27786e12			−0.00493467
$279$	7.33084e13			0.155428
$280$	−1.08937e14			−0.226061
$281$			3.13265e14i			0.636317i	0.948037	+	0.318159i	$0.103064\pi$
$281$			3.13265e14i			0.636317i	−0.948037	+	0.318159i	$0.896936\pi$
$282$	2.03756e13			0.0405150
$283$			2.66833e14i			0.519422i	0.965686	+	0.259711i	$0.0836273\pi$
$283$			2.66833e14i			0.519422i	−0.965686	+	0.259711i	$0.916373\pi$
$284$	−3.07362e14			−0.585788
$285$	9.22285e13			0.172106
$286$		−	4.17078e13i		−	0.0762115i
$287$			2.62211e14i			0.469201i
$288$	−2.10227e14			−0.368412
$289$	5.73391e14			0.984155
$290$		−	2.85551e13i		−	0.0480061i
$291$			1.63129e14i			0.268642i
$292$	−6.59733e14			−1.06432
$293$		−	5.85575e14i		−	0.925500i	−0.886489	−	0.462750i	$-0.846863\pi$
$293$		−	5.85575e14i		−	0.925500i	0.886489	−	0.462750i	$-0.153137\pi$
$294$	4.84248e13			0.0749866
$295$		−	1.65193e14i		−	0.250646i
$296$			1.96166e14i			0.291657i
$297$		−	3.92588e14i		−	0.572003i
$298$			8.76538e13i			0.125162i
$299$	3.88619e14	+	1.87463e14i	0.543873	+	0.262355i
$300$	1.84400e14			0.252950
$301$	5.14117e14			0.691294
$302$	−1.47366e14			−0.194248
$303$	−2.54388e14			−0.328732
$304$		−	7.34712e14i		−	0.930841i
$305$	6.38705e14			0.793417
$306$			1.30464e13i			0.0158913i
$307$	2.80726e14			0.335315			0.167658	−	0.985845i	$-0.446380\pi$
$307$	2.80726e14			0.335315			0.167658	+	0.985845i	$0.446380\pi$
$308$	−1.16726e15			−1.36730
$309$			4.36785e14i			0.501784i
$310$			1.12819e13i			0.0127119i
$311$	−4.43093e14			−0.489703			−0.244852	−	0.969561i	$-0.578739\pi$
$311$	−4.43093e14			−0.489703			−0.244852	+	0.969561i	$0.578739\pi$
$312$	5.46024e13			0.0591949
$313$			1.33364e15i			1.41832i	0.705050	+	0.709158i	$0.250925\pi$
$313$			1.33364e15i			1.41832i	−0.705050	+	0.709158i	$0.749075\pi$
$314$		−	1.68713e14i		−	0.176024i
$315$	−6.85393e14			−0.701579
$316$			1.26228e15i			1.26775i
$317$	−1.79907e15			−1.77293			−0.886465	−	0.462796i	$-0.846846\pi$
$317$	−1.79907e15			−1.77293			−0.886465	+	0.462796i	$0.846846\pi$
$318$		−	4.04326e13i		−	0.0390993i
$319$		−	6.18356e14i		−	0.586806i
$320$			4.74096e14i			0.441536i
$321$		−	6.83037e14i		−	0.624330i
$322$	−1.10079e14	+	2.28199e14i	−0.0987576	+	0.204729i
$323$	−1.41735e14			−0.124813
$324$	−7.43518e14			−0.642720
$325$	5.31953e14			0.451412
$326$	−1.47113e14			−0.122559
$327$			2.42510e14i			0.198355i
$328$	−1.04559e14			−0.0839689
$329$		−	1.64502e15i		−	1.29717i
$330$	2.82918e13			0.0219067
$331$	−8.27243e14			−0.629021			−0.314510	−	0.949254i	$-0.601840\pi$
$331$	−8.27243e14			−0.629021			−0.314510	+	0.949254i	$0.601840\pi$
$332$			2.29409e15i			1.71310i
$333$			1.23421e15i			0.905154i
$334$	−1.75774e14			−0.126613
$335$	9.11411e11			0.000644830
$336$		−	7.39937e14i		−	0.514232i
$337$		−	1.66354e15i		−	1.13567i	−0.823141	−	0.567837i	$-0.807780\pi$
$337$		−	1.66354e15i		−	1.13567i	0.823141	−	0.567837i	$-0.192220\pi$
$338$	−1.35812e14			−0.0910834
$339$			4.93230e14i			0.324976i
$340$	9.56922e13			0.0619445
$341$			2.44307e14i			0.155385i
$342$			2.00303e14i			0.125179i
$343$		−	1.32754e15i		−	0.815237i
$344$			2.05009e14i			0.123715i
$345$	−1.27163e14	+	2.63613e14i	−0.0754127	+	0.156334i
$346$	3.34303e14			0.194843
$347$	1.57272e15			0.900896			0.450448	−	0.892803i	$-0.351264\pi$
$347$	1.57272e15			0.900896			0.450448	+	0.892803i	$0.351264\pi$
$348$	4.00563e14			0.225526
$349$	2.92770e15			1.62022			0.810111	−	0.586276i	$-0.199407\pi$
$349$	2.92770e15			1.62022			0.810111	+	0.586276i	$0.199407\pi$
$350$			3.12365e14i			0.169924i
$351$	7.33636e14			0.392318
$352$		−	7.00600e14i		−	0.368311i
$353$	1.99586e15			1.03153			0.515765	−	0.856730i	$-0.327508\pi$
$353$	1.99586e15			1.03153			0.515765	+	0.856730i	$0.327508\pi$
$354$	−4.86205e13			−0.0247058
$355$		−	6.01456e14i		−	0.300493i
$356$		−	3.04547e15i		−	1.49608i
$357$	−1.42743e14			−0.0689516
$358$	5.58774e11			0.000265423
$359$			3.64076e15i			1.70069i	0.526227	+	0.850344i	$0.323606\pi$
$359$			3.64076e15i			1.70069i	−0.526227	+	0.850344i	$0.676394\pi$
$360$		−	2.73307e14i		−	0.125556i
$361$	3.72334e13			0.0168225
$362$		−	4.00992e14i		−	0.178190i
$363$	−1.77741e14			−0.0776871
$364$		−	2.18128e15i		−	0.937783i
$365$		−	1.29099e15i		−	0.545965i
$366$		−	1.87987e14i		−	0.0782060i
$367$		−	1.75363e15i		−	0.717698i	−0.933396	−	0.358849i	$-0.883169\pi$
$367$		−	1.75363e15i		−	0.717698i	0.933396	−	0.358849i	$-0.116831\pi$
$368$	2.10000e15	+	1.01300e15i	0.845536	+	0.407872i
$369$	−6.57850e14			−0.260597
$370$	−1.89939e14			−0.0740294
$371$	−3.26432e15			−1.25184
$372$	−1.58259e14			−0.0597186
$373$		−	5.61125e11i		−	0.000208356i	−1.00000		0.000104178i	$-0.999967\pi$
$373$		−	5.61125e11i		−	0.000208356i	1.00000		0.000104178i	$-3.31609e-5\pi$
$374$	−4.34781e13			−0.0158870
$375$			8.43531e14i			0.303328i
$376$	6.55969e14			0.232144
$377$	1.15553e15			0.402471
$378$			4.30794e14i			0.147679i
$379$			3.60849e15i			1.21756i	0.793339	+	0.608780i	$0.208341\pi$
$379$			3.60849e15i			1.21756i	−0.793339	+	0.608780i	$0.791659\pi$
$380$	1.46918e15			0.487949
$381$	−1.21872e15			−0.398432
$382$		−	4.96002e14i		−	0.159626i
$383$			2.89206e15i			0.916252i	0.888887	+	0.458126i	$0.151479\pi$
$383$			2.89206e15i			0.916252i	−0.888887	+	0.458126i	$0.848521\pi$
$384$	6.02900e14			0.188043
$385$		−	2.28413e15i		−	0.701385i
$386$	8.14151e14			0.246139
$387$			1.28985e15i			0.383948i
$388$			2.59862e15i			0.761643i
$389$		−	2.80591e15i		−	0.809798i	−0.914361	−	0.404899i	$-0.867307\pi$
$389$		−	2.80591e15i		−	0.809798i	0.914361	−	0.404899i	$-0.132693\pi$
$390$			5.28693e13i			0.0150250i
$391$	1.95420e14	−	4.05115e14i	0.0546902	−	0.113375i
$392$	1.55898e15			0.429659
$393$	−2.14492e15			−0.582178
$394$	−1.20548e14			−0.0322243
$395$	−2.47007e15			−0.650319
$396$		−	2.92849e15i		−	0.759404i
$397$	1.07586e15			0.274798			0.137399	−	0.990516i	$-0.456126\pi$
$397$	1.07586e15			0.274798			0.137399	+	0.990516i	$0.456126\pi$
$398$		−	2.46358e14i		−	0.0619824i
$399$	−2.19156e15			−0.543145
$400$	2.87454e15			0.701791
$401$		−	3.74352e15i		−	0.900355i	−0.892939	−	0.450177i	$-0.851361\pi$
$401$		−	3.74352e15i		−	0.900355i	0.892939	−	0.450177i	$-0.148639\pi$
$402$		−	2.68250e11i		−	6.35600e-5i
$403$	−4.56540e14			−0.106573
$404$	−4.05236e15			−0.932007
$405$		−	1.45494e15i		−	0.329697i
$406$			6.78534e14i			0.151501i
$407$	−4.11310e15			−0.904904
$408$		−	5.69200e13i		−	0.0123397i
$409$	2.05179e14			0.0438322			0.0219161	−	0.999760i	$-0.493023\pi$
$409$	2.05179e14			0.0438322			0.0219161	+	0.999760i	$0.493023\pi$
$410$		−	1.01240e14i		−	0.0213133i
$411$			4.12644e14i			0.0856101i
$412$			6.95789e15i			1.42264i
$413$			3.92537e15i			0.791007i
$414$	−5.72520e14	−	2.76174e14i	−0.113707	−	0.0548505i
$415$	−4.48915e15			−0.878772
$416$	1.30922e15			0.252612
$417$	6.25270e13			0.0118919
$418$	−6.67528e14			−0.125145
$419$			2.78781e15i			0.515203i	0.966251	+	0.257602i	$0.0829322\pi$
$419$			2.78781e15i			0.515203i	−0.966251	+	0.257602i	$0.917068\pi$
$420$	1.47963e15			0.269561
$421$			9.84555e15i			1.76827i	0.467236	+	0.884133i	$0.345250\pi$
$421$			9.84555e15i			1.76827i	−0.467236	+	0.884133i	$0.654750\pi$
$422$	5.03849e14			0.0892124
$423$	4.12714e15			0.720455
$424$		−	1.30168e15i		−	0.224031i
$425$		−	5.54532e14i		−	0.0941007i
$426$	−1.77023e14			−0.0296192
$427$	−1.51771e16			−2.50392
$428$		−	1.08806e16i		−	1.77008i
$429$			1.14487e15i			0.183660i
$430$	−1.98502e14			−0.0314018
$431$		−	1.42530e14i		−	0.0222353i	−0.999938	−	0.0111176i	$-0.996461\pi$
$431$		−	1.42530e14i		−	0.0222353i	0.999938	−	0.0111176i	$-0.00353893\pi$
$432$	3.96438e15			0.609920
$433$			1.19687e15i			0.181602i	0.995869	+	0.0908008i	$0.0289427\pi$
$433$			1.19687e15i			0.181602i	−0.995869	+	0.0908008i	$0.971057\pi$
$434$		−	2.68082e14i		−	0.0401171i
$435$			7.83836e14i			0.115688i
$436$			3.86314e15i			0.562369i
$437$	3.00033e15	−	6.21981e15i	0.430805	−	0.893076i
$438$	−3.79970e14			−0.0538151
$439$	−1.17513e16			−1.64173			−0.820863	−	0.571125i	$-0.806507\pi$
$439$	−1.17513e16			−1.64173			−0.820863	+	0.571125i	$0.806507\pi$
$440$	9.10820e14			0.125521
$441$	9.80856e15			1.33344
$442$		−	8.12482e13i		−	0.0108963i
$443$	−1.67326e15			−0.221382			−0.110691	−	0.993855i	$-0.535306\pi$
$443$	−1.67326e15			−0.221382			−0.110691	+	0.993855i	$0.535306\pi$
$444$		−	2.66441e15i		−	0.347779i
$445$	5.95947e15			0.767446
$446$	1.32669e15			0.168562
$447$		−	2.40609e15i		−	0.301625i
$448$		−	1.12656e16i		−	1.39343i
$449$	7.66604e15			0.935606			0.467803	−	0.883833i	$-0.345046\pi$
$449$	7.66604e15			0.935606			0.467803	+	0.883833i	$0.345046\pi$
$450$	−7.83681e14			−0.0943765
$451$		−	2.19234e15i		−	0.260525i
$452$			7.85705e15i			0.921359i
$453$	4.04519e15			0.468112
$454$			1.63787e15i			0.187045i
$455$	4.26839e15			0.481056
$456$		−	8.73905e14i		−	0.0972020i
$457$			3.53833e15i			0.388419i	0.980960	+	0.194210i	$0.0622142\pi$
$457$			3.53833e15i			0.388419i	−0.980960	+	0.194210i	$0.937786\pi$
$458$			4.35491e14i			0.0471831i
$459$		−	7.64776e14i		−	0.0817820i
$460$	−2.02567e15	+	4.19931e15i	−0.213807	+	0.443231i
$461$	1.42057e16			1.47998			0.739991	−	0.672617i	$-0.234830\pi$
$461$	1.42057e16			1.47998			0.739991	+	0.672617i	$0.234830\pi$
$462$	−6.72276e14			−0.0691347
$463$	−6.34766e15			−0.644358			−0.322179	−	0.946679i	$-0.604415\pi$
$463$	−6.34766e15			−0.644358			−0.322179	+	0.946679i	$0.604415\pi$
$464$	6.24420e15			0.625704
$465$		−	3.09686e14i		−	0.0306340i
$466$	−7.55130e14			−0.0737405
$467$		−	5.84366e15i		−	0.563357i	−0.959509	−	0.281678i	$-0.909109\pi$
$467$		−	5.84366e15i		−	0.563357i	0.959509	−	0.281678i	$-0.0908911\pi$
$468$	5.47252e15			0.520849
$469$	−2.16572e13			−0.00203500
$470$			6.35148e14i			0.0589234i
$471$			4.63117e15i			0.424195i
$472$	−1.56528e15			−0.141560
$473$	−4.29853e15			−0.383842
$474$			7.27002e14i			0.0641011i
$475$		−	8.51385e15i		−	0.741249i
$476$	−2.27386e15			−0.195489
$477$		−	8.18972e15i		−	0.695279i
$478$	−2.21065e15			−0.185333
$479$			1.06857e16i			0.884690i	0.896845	+	0.442345i	$0.145853\pi$
$479$			1.06857e16i			0.884690i	−0.896845	+	0.442345i	$0.854147\pi$
$480$			8.88089e14i			0.0726121i
$481$		−	7.68621e15i		−	0.620643i
$482$		−	1.66856e15i		−	0.133064i
$483$	3.02167e15	−	6.26405e15i	0.237993	−	0.493369i
$484$	−2.83139e15			−0.220256
$485$	−5.08506e15			−0.390702
$486$	−1.65550e15			−0.125635
$487$	6.89856e15			0.517112			0.258556	−	0.965996i	$-0.416753\pi$
$487$	6.89856e15			0.517112			0.258556	+	0.965996i	$0.416753\pi$
$488$		−	6.05201e15i		−	0.448106i
$489$	4.03823e15			0.295351
$490$			1.50949e15i			0.109057i
$491$	1.37586e16			0.981943			0.490971	−	0.871176i	$-0.336642\pi$
$491$	1.37586e16			0.981943			0.490971	+	0.871176i	$0.336642\pi$
$492$	1.42017e15			0.100127
$493$		−	1.20458e15i		−	0.0838985i
$494$		−	1.24742e15i		−	0.0858324i
$495$	5.73057e15			0.389553
$496$	−2.46702e15			−0.165685
$497$			1.42920e16i			0.948317i
$498$			1.32127e15i			0.0866194i
$499$	−2.38285e16			−1.54345			−0.771727	−	0.635954i	$-0.780607\pi$
$499$	−2.38285e16			−1.54345			−0.771727	+	0.635954i	$0.780607\pi$
$500$			1.34373e16i			0.859985i
$501$	4.82499e15			0.305120
$502$			3.05276e15i			0.190752i
$503$		−	3.79419e15i		−	0.234267i	−0.993116	−	0.117134i	$-0.962629\pi$
$503$		−	3.79419e15i		−	0.234267i	0.993116	−	0.117134i	$-0.0373706\pi$
$504$			6.49440e15i			0.396238i
$505$		−	7.92979e15i		−	0.478094i
$506$	9.20372e14	−	1.90797e15i	0.0548353	−	0.113676i
$507$	3.72804e15			0.219499
$508$	−1.94139e16			−1.12962
$509$	−1.66090e16			−0.955076			−0.477538	−	0.878611i	$-0.658471\pi$
$509$	−1.66090e16			−0.955076			−0.477538	+	0.878611i	$0.658471\pi$
$510$	5.51133e13			0.00313210
$511$			3.06768e16i			1.72300i
$512$	1.18735e16			0.659113
$513$		−	1.17418e16i		−	0.644212i
$514$	−2.02179e15			−0.109637
$515$	−1.36154e16			−0.729774
$516$		−	2.78453e15i		−	0.147521i
$517$			1.37540e16i			0.720256i
$518$	4.51338e15			0.233627
$519$	−9.17660e15			−0.469545
$520$			1.70206e15i			0.0860906i
$521$		−	3.96036e16i		−	1.98020i	−0.140368	−	0.990099i	$-0.544829\pi$
$521$		−	3.96036e16i		−	1.98020i	0.140368	−	0.990099i	$-0.455171\pi$
$522$	−1.70235e15			−0.0841444
$523$			1.42152e16i			0.694612i	0.937752	+	0.347306i	$0.112904\pi$
$523$			1.42152e16i			0.694612i	−0.937752	+	0.347306i	$0.887096\pi$
$524$	−3.41681e16			−1.65057
$525$		−	8.57440e15i		−	0.409494i
$526$		−	4.86613e15i		−	0.229757i
$527$			4.75918e14i			0.0222161i
$528$			6.18661e15i			0.285528i
$529$	1.36411e16	+	1.71515e16i	0.622464	+	0.782649i
$530$	1.26036e15			0.0568644
$531$	−9.84821e15			−0.439329
$532$	−3.49111e16			−1.53990
$533$	4.09686e15			0.178685
$534$		−	1.75402e15i		−	0.0756462i
$535$	2.12916e16			0.908000
$536$		−	8.63601e12i		−	0.000364187i
$537$	−1.53383e13			−0.000639635
$538$	2.66335e15			0.109833
$539$			3.26879e16i			1.33307i
$540$			7.92745e15i			0.319721i
$541$	1.68476e16			0.671978			0.335989	−	0.941866i	$-0.390930\pi$
$541$	1.68476e16			0.671978			0.335989	+	0.941866i	$0.390930\pi$
$542$	−1.17453e15			−0.0463305
$543$			1.10072e16i			0.429415i
$544$		−	1.36479e15i		−	0.0526591i
$545$	−7.55953e15			−0.288480
$546$		−	1.25629e15i		−	0.0474171i
$547$	−1.11358e16			−0.415717			−0.207858	−	0.978159i	$-0.566649\pi$
$547$	−1.11358e16			−0.415717			−0.207858	+	0.978159i	$0.566649\pi$
$548$			6.57334e15i			0.242718i
$549$		−	3.80772e16i		−	1.39069i
$550$		−	2.61168e15i		−	0.0943505i
$551$		−	1.84942e16i		−	0.660884i
$552$	2.49785e15	+	1.20492e15i	0.0882941	+	0.0425916i
$553$	5.86944e16			2.05233
$554$	3.55525e15			0.122973
$555$	5.21381e15			0.178401
$556$	9.96043e14			0.0337155
$557$		−	4.39402e16i		−	1.47140i	−0.677307	−	0.735701i	$-0.736853\pi$
$557$		−	4.39402e16i		−	1.47140i	0.677307	−	0.735701i	$-0.263147\pi$
$558$	6.72581e14			0.0222812
$559$		−	8.03272e15i		−	0.263264i
$560$	2.30653e16			0.747879
$561$	1.19347e15			0.0382855
$562$			2.87410e15i			0.0912188i
$563$			7.34378e15i			0.230605i	0.993330	+	0.115303i	$0.0367838\pi$
$563$			7.34378e15i			0.230605i	−0.993330	+	0.115303i	$0.963216\pi$
$564$	−8.90969e15			−0.276814
$565$	−1.53749e16			−0.472632
$566$			2.44810e15i			0.0744613i
$567$			3.45727e16i			1.04048i
$568$	−5.69906e15			−0.169712
$569$		−	2.64256e15i		−	0.0778665i	−0.999242	−	0.0389332i	$-0.987604\pi$
$569$		−	2.64256e15i		−	0.0778665i	0.999242	−	0.0389332i	$-0.0123960\pi$
$570$	8.46167e14			0.0246721
$571$		−	6.56969e16i		−	1.89552i	−0.318984	−	0.947760i	$-0.603342\pi$
$571$		−	6.56969e16i		−	1.89552i	0.318984	−	0.947760i	$-0.396658\pi$
$572$			1.82376e16i			0.520705i
$573$			1.36152e16i			0.384677i
$574$			2.40570e15i			0.0672620i
$575$	2.43348e16	+	1.17387e16i	0.673319	+	0.324797i
$576$	2.82638e16			0.773919
$577$	−2.93905e16			−0.796439			−0.398219	−	0.917290i	$-0.630372\pi$
$577$	−2.93905e16			−0.796439			−0.398219	+	0.917290i	$0.630372\pi$
$578$	5.26067e15			0.141083
$579$	−2.23484e16			−0.593164
$580$			1.24863e16i			0.327995i
$581$	1.06672e17			2.77329
$582$			1.49666e15i			0.0385110i
$583$	2.72929e16			0.695087
$584$	−1.22327e16			−0.308350
$585$			1.07088e16i			0.267181i
$586$		−	5.37246e15i		−	0.132674i
$587$	−5.59779e16			−1.36832			−0.684160	−	0.729332i	$-0.739831\pi$
$587$	−5.59779e16			−1.36832			−0.684160	+	0.729332i	$0.739831\pi$
$588$	−2.11748e16			−0.512336
$589$			7.30687e15i			0.175001i
$590$		−	1.51560e15i		−	0.0359312i
$591$	3.30904e15			0.0776564
$592$		−	4.15343e16i		−	0.964888i
$593$	−2.18544e16			−0.502586			−0.251293	−	0.967911i	$-0.580856\pi$
$593$	−2.18544e16			−0.502586			−0.251293	+	0.967911i	$0.580856\pi$
$594$		−	3.60187e15i		−	0.0819991i
$595$		−	4.44957e15i		−	0.100280i
$596$		−	3.83285e16i		−	0.855154i
$597$			6.76251e15i			0.149369i
$598$	3.56545e15	+	1.71991e15i	0.0779665	+	0.0376097i
$599$	8.34170e16			1.80590			0.902950	−	0.429745i	$-0.141397\pi$
$599$	8.34170e16			1.80590			0.902950	+	0.429745i	$0.141397\pi$
$600$	3.41913e15			0.0732837
$601$	−2.82324e16			−0.599103			−0.299551	−	0.954080i	$-0.596837\pi$
$601$	−2.82324e16			−0.599103			−0.299551	+	0.954080i	$0.596837\pi$
$602$	4.71685e15			0.0991000
$603$		−	5.43348e13i		−	0.00113025i
$604$	6.44390e16			1.32717
$605$		−	5.54055e15i		−	0.112985i
$606$	−2.33393e15			−0.0471251
$607$	−6.29603e16			−1.25874			−0.629368	−	0.777108i	$-0.716686\pi$
$607$	−6.29603e16			−1.25874			−0.629368	+	0.777108i	$0.716686\pi$
$608$		−	2.09540e16i		−	0.414806i
$609$		−	1.86257e16i		−	0.365098i
$610$	5.85991e15			0.113740
$611$	−2.57024e16			−0.493999
$612$		−	5.70480e15i		−	0.108576i
$613$			3.01470e16i			0.568173i	0.958799	+	0.284087i	$0.0916904\pi$
$613$			3.01470e16i			0.568173i	−0.958799	+	0.284087i	$0.908310\pi$
$614$	2.57557e15			0.0480688
$615$			2.77904e15i			0.0513622i
$616$	−2.16431e16			−0.396128
$617$			7.67733e16i			1.39155i	0.718259	+	0.695776i	$0.244939\pi$
$617$			7.67733e16i			1.39155i	−0.718259	+	0.695776i	$0.755061\pi$
$618$			4.00736e15i			0.0719329i
$619$			2.94639e16i			0.523776i	0.965098	+	0.261888i	$0.0843451\pi$
$619$			2.94639e16i			0.523776i	−0.965098	+	0.261888i	$0.915655\pi$
$620$		−	4.93323e15i		−	0.0868523i
$621$	3.35610e16	+	1.61893e16i	0.585175	+	0.282278i
$622$	−4.06524e15			−0.0702011
$623$	−1.41611e17			−2.42196
$624$	−1.15610e16			−0.195834
$625$	1.82638e16			0.306415
$626$			1.22357e16i			0.203322i
$627$	1.83236e16			0.301582
$628$			7.37736e16i			1.20266i
$629$	−8.01246e15			−0.129379
$630$	−6.28826e15			−0.100574
$631$		−	8.23161e16i		−	1.30409i	−0.758179	−	0.652046i	$-0.773911\pi$
$631$		−	8.23161e16i		−	1.30409i	0.758179	−	0.652046i	$-0.226089\pi$
$632$			2.34050e16i			0.367287i
$633$	−1.38306e16			−0.214990
$634$	−1.65058e16			−0.254157
$635$		−	3.79899e16i		−	0.579463i
$636$			1.76800e16i			0.267141i
$637$	−6.10843e16			−0.914309
$638$		−	5.67322e15i		−	0.0841212i
$639$	−3.58565e16			−0.526700
$640$			1.87936e16i			0.273483i
$641$			1.15769e17i			1.66895i	0.551048	+	0.834473i	$0.314228\pi$
$641$			1.15769e17i			1.66895i	−0.551048	+	0.834473i	$0.685772\pi$
$642$		−	6.26664e15i		−	0.0895003i
$643$			2.91827e16i			0.412914i	0.978456	+	0.206457i	$0.0661933\pi$
$643$			2.91827e16i			0.412914i	−0.978456	+	0.206457i	$0.933807\pi$
$644$	4.81346e16	−	9.97850e16i	0.674748	−	1.39878i
$645$	5.44886e15			0.0756742
$646$	−1.30037e15			−0.0178925
$647$	−1.31350e17			−1.79063			−0.895313	−	0.445437i	$-0.853048\pi$
$647$	−1.31350e17			−1.79063			−0.895313	+	0.445437i	$0.853048\pi$
$648$	−1.37862e16			−0.186206
$649$		−	3.28200e16i		−	0.439208i
$650$	4.88049e15			0.0647118
$651$			7.35884e15i			0.0966770i
$652$	6.43282e16			0.837367
$653$	−2.11441e16			−0.272716			−0.136358	−	0.990660i	$-0.543540\pi$
$653$	−2.11441e16			−0.272716			−0.136358	+	0.990660i	$0.543540\pi$
$654$			2.22495e15i			0.0284351i
$655$		−	6.68613e16i		−	0.846695i
$656$	2.21384e16			0.277794
$657$	−7.69638e16			−0.956961
$658$		−	1.50926e16i		−	0.185955i
$659$			4.73654e16i			0.578294i	0.957285	+	0.289147i	$0.0933717\pi$
$659$			4.73654e16i			0.578294i	−0.957285	+	0.289147i	$0.906628\pi$
$660$	−1.23712e16			−0.149674
$661$			9.89472e15i			0.118630i	0.998239	+	0.0593150i	$0.0188916\pi$
$661$			9.89472e15i			0.118630i	−0.998239	+	0.0593150i	$0.981108\pi$
$662$	−7.58969e15			−0.0901729
$663$			2.23026e15i			0.0262587i
$664$			4.25367e16i			0.496312i
$665$		−	6.83152e16i		−	0.789928i
$666$			1.13234e16i			0.129758i
$667$	5.28612e16	+	2.54993e16i	0.600319	+	0.289583i
$668$	7.68611e16			0.865064
$669$	−3.64175e16			−0.406212
$670$	8.36190e12			9.24391e−5
$671$	1.26895e17			1.39031
$672$		−	2.11030e16i		−	0.229155i
$673$	1.14542e17			1.23274			0.616372	−	0.787455i	$-0.288602\pi$
$673$	1.14542e17			1.23274			0.616372	+	0.787455i	$0.288602\pi$
$674$		−	1.52624e16i		−	0.162804i
$675$	4.59393e16			0.485692
$676$	5.93869e16			0.622315
$677$			1.11716e17i			1.16033i	0.814499	+	0.580165i	$0.197012\pi$
$677$			1.11716e17i			1.16033i	−0.814499	+	0.580165i	$0.802988\pi$
$678$			4.52522e15i			0.0465867i
$679$	1.20832e17			1.23301
$680$	1.77431e15			0.0179463
$681$		−	4.49595e16i		−	0.450753i
$682$			2.24143e15i			0.0222751i
$683$	−1.59995e17			−1.57610			−0.788048	−	0.615613i	$-0.788908\pi$
$683$	−1.59995e17			−1.57610			−0.788048	+	0.615613i	$0.788908\pi$
$684$		−	8.75870e16i		−	0.855270i
$685$	−1.28629e16			−0.124508
$686$		−	1.21798e16i		−	0.116868i
$687$		−	1.19542e16i		−	0.113705i
$688$		−	4.34068e16i		−	0.409286i
$689$			5.10028e16i			0.476736i
$690$	−1.16667e15	+	2.41857e15i	−0.0108107	+	0.0224111i
$691$	1.99962e17			1.83687			0.918435	−	0.395572i	$-0.129454\pi$
$691$	1.99962e17			1.83687			0.918435	+	0.395572i	$0.129454\pi$
$692$	−1.46181e17			−1.33124
$693$	−1.36171e17			−1.22938
$694$	1.44292e16			0.129147
$695$			1.94909e15i			0.0172951i
$696$	7.42719e15			0.0653384
$697$		−	4.27076e15i		−	0.0372484i
$698$	2.68607e16			0.232266
$699$	2.07283e16			0.177705
$700$		−	1.36588e17i		−	1.16098i
$701$		−	2.17146e17i		−	1.82997i	−0.403486	−	0.914986i	$-0.632202\pi$
$701$		−	2.17146e17i		−	1.82997i	0.403486	−	0.914986i	$-0.367798\pi$
$702$	6.73087e15			0.0562404
$703$	−1.23017e17			−1.01914
$704$			9.41914e16i			0.773705i
$705$		−	1.74348e16i		−	0.141998i
$706$	1.83114e16			0.147874
$707$			1.88430e17i			1.50880i
$708$	2.12604e16			0.168799
$709$		−	1.05419e17i		−	0.829931i	−0.909837	−	0.414965i	$-0.863794\pi$
$709$		−	1.05419e17i		−	0.829931i	0.909837	−	0.414965i	$-0.136206\pi$
$710$		−	5.51817e15i		−	0.0430769i
$711$			1.47256e17i			1.13987i
$712$		−	5.64686e16i		−	0.433438i
$713$	−2.08849e16	−	1.00745e16i	−0.158963	−	0.0766810i
$714$	−1.30962e15			−0.00988451
$715$	−3.56880e16			−0.267107
$716$	−2.44336e14			−0.00181347
$717$	6.06821e16			0.446628
$718$			3.34027e16i			0.243801i
$719$	7.44887e16			0.539159			0.269579	−	0.962978i	$-0.413115\pi$
$719$	7.44887e16			0.539159			0.269579	+	0.962978i	$0.413115\pi$
$720$			5.78676e16i			0.415375i
$721$	3.23534e17			2.30307
$722$	3.41604e14			0.00241157
$723$			4.58019e16i			0.320667i
$724$			1.75342e17i			1.21746i
$725$	7.23578e16			0.498262
$726$	−1.63072e15			−0.0111368
$727$		−	1.35202e17i		−	0.915751i	−0.889016	−	0.457876i	$-0.848610\pi$
$727$		−	1.35202e17i		−	0.915751i	0.889016	−	0.457876i	$-0.151390\pi$
$728$		−	4.04449e16i		−	0.271691i
$729$	−5.30495e16			−0.353441
$730$		−	1.18444e16i		−	0.0782665i
$731$	−8.37368e15			−0.0548798
$732$			8.22012e16i			0.534333i
$733$		−	1.66675e17i		−	1.07460i	−0.843392	−	0.537299i	$-0.819445\pi$
$733$		−	1.66675e17i		−	1.07460i	0.843392	−	0.537299i	$-0.180555\pi$
$734$		−	1.60890e16i		−	0.102885i
$735$		−	4.14355e16i		−	0.262814i
$736$	5.98919e16	+	2.88908e16i	0.376792	+	0.181758i
$737$	1.81075e14			0.00112994
$738$	−6.03556e15			−0.0373576
$739$	−1.60012e17			−0.982393			−0.491197	−	0.871049i	$-0.663440\pi$
$739$	−1.60012e17			−0.982393			−0.491197	+	0.871049i	$0.663440\pi$
$740$	8.30550e16			0.505796
$741$			3.42416e16i			0.206845i
$742$	−2.99491e16			−0.179457
$743$		−	2.39870e17i		−	1.42575i	−0.701293	−	0.712873i	$-0.747393\pi$
$743$		−	2.39870e17i		−	1.42575i	0.701293	−	0.712873i	$-0.252607\pi$
$744$	−2.93441e15			−0.0173015
$745$	7.50025e16			0.438670
$746$		−	5.14814e12i		−	2.98688e-5i
$747$			2.67626e17i			1.54030i
$748$	1.90117e16			0.108546
$749$	−5.05937e17			−2.86553
$750$			7.73912e15i			0.0434834i
$751$		−	2.21865e17i		−	1.23666i	−0.785920	−	0.618329i	$-0.787810\pi$
$751$		−	2.21865e17i		−	1.23666i	0.785920	−	0.618329i	$-0.212190\pi$
$752$	−1.38889e17			−0.768000
$753$		−	8.37980e16i		−	0.459689i
$754$	1.06016e16			0.0576959
$755$			1.26096e17i			0.680803i
$756$		−	1.88374e17i		−	1.00900i
$757$			2.13854e17i			1.13643i	0.822881	+	0.568214i	$0.192366\pi$
$757$			2.13854e17i			1.13643i	−0.822881	+	0.568214i	$0.807634\pi$
$758$			3.31067e16i			0.174542i
$759$	−2.52641e16	+	5.23736e16i	−0.132146	+	0.273944i
$760$	2.72413e16			0.141367
$761$	−1.83421e17			−0.944367			−0.472184	−	0.881500i	$-0.656534\pi$
$761$	−1.83421e17			−0.944367			−0.472184	+	0.881500i	$0.656534\pi$
$762$	−1.11814e16			−0.0571169
$763$	1.79631e17			0.910406
$764$			2.16888e17i			1.09062i
$765$	1.11634e16			0.0556963
$766$			2.65337e16i			0.131349i
$767$	6.13312e16			0.301238
$768$	−5.67648e16			−0.276638
$769$			1.53717e17i			0.743300i	0.928373	+	0.371650i	$0.121208\pi$
$769$			1.53717e17i			0.743300i	−0.928373	+	0.371650i	$0.878792\pi$
$770$		−	2.09562e16i		−	0.100547i
$771$	5.54981e16			0.264212
$772$	−3.56005e17			−1.68172
$773$			2.93224e17i			1.37443i	0.726454	+	0.687215i	$0.241167\pi$
$773$			2.93224e17i			1.37443i	−0.726454	+	0.687215i	$0.758833\pi$
$774$			1.18339e16i			0.0550406i
$775$	−2.85879e16			−0.131939
$776$			4.81831e16i			0.220661i
$777$	−1.23892e17			−0.563011
$778$		−	2.57434e16i		−	0.116088i
$779$		−	6.55698e16i		−	0.293413i
$780$		−	2.31182e16i		−	0.102657i
$781$		−	1.19495e17i		−	0.526555i
$782$	1.79292e15	−	3.71679e15i	0.00784007	−	0.0162528i
$783$	9.97914e16			0.433035
$784$	−3.30084e17			−1.42144
$785$	−1.44363e17			−0.616932
$786$	−1.96789e16			−0.0834577
$787$		−	1.57526e17i		−	0.662986i	−0.943458	−	0.331493i	$-0.892448\pi$
$787$		−	1.57526e17i		−	0.662986i	0.943458	−	0.331493i	$-0.107552\pi$
$788$	5.27124e16			0.220169
$789$			1.33575e17i			0.553685i
$790$	−2.26621e16			−0.0932261
$791$	3.65343e17			1.49157
$792$		−	5.42996e16i		−	0.220012i
$793$			2.37131e17i			0.953564i
$794$	9.87067e15			0.0393934
$795$	−3.45969e16			−0.137036
$796$			1.07725e17i			0.423486i
$797$		−	1.26188e17i		−	0.492342i	−0.969226	−	0.246171i	$-0.920827\pi$
$797$		−	1.26188e17i		−	0.492342i	0.969226	−	0.246171i	$-0.0791726\pi$
$798$	−2.01068e16			−0.0778621
$799$			2.67933e16i			0.102978i
$800$	8.19817e16			0.312735
$801$		−	3.55281e17i		−	1.34517i
$802$		−	3.43456e16i		−	0.129070i
$803$		−	2.56488e17i		−	0.956697i
$804$			1.17298e14i			0.000434266i
$805$	1.95263e17	+	9.41914e16i	0.717536	+	0.346127i
$806$	−4.18860e15			−0.0152777
$807$	−7.31087e16			−0.264684
$808$	−7.51382e16			−0.270018
$809$	2.18223e17			0.778411			0.389205	−	0.921151i	$-0.372750\pi$
$809$	2.18223e17			0.778411			0.389205	+	0.921151i	$0.372750\pi$
$810$		−	1.33486e16i		−	0.0472635i
$811$	5.43014e17			1.90847			0.954236	−	0.299055i	$-0.0966714\pi$
$811$	5.43014e17			1.90847			0.954236	+	0.299055i	$0.0966714\pi$
$812$		−	2.96704e17i		−	1.03511i
$813$	3.22407e16			0.111651
$814$	−3.77363e16			−0.129722
$815$			1.25880e17i			0.429546i
$816$			1.20517e16i			0.0408234i
$817$	−1.28563e17			−0.432298
$818$	1.88245e15			0.00628353
$819$		−	2.54465e17i		−	0.843190i
$820$			4.42695e16i			0.145620i
$821$	1.57873e17			0.515525			0.257762	−	0.966208i	$-0.417015\pi$
$821$	1.57873e17			0.515525			0.257762	+	0.966208i	$0.417015\pi$
$822$			3.78588e15i			0.0122726i
$823$	3.83737e17			1.23491			0.617454	−	0.786607i	$-0.288164\pi$
$823$	3.83737e17			1.23491			0.617454	+	0.786607i	$0.288164\pi$
$824$			1.29012e17i			0.412162i
$825$			7.16905e16i			0.227372i
$826$			3.60140e16i			0.113394i
$827$			1.90457e16i			0.0595340i	0.999557	+	0.0297670i	$0.00947652\pi$
$827$			1.90457e16i			0.0595340i	−0.999557	+	0.0297670i	$0.990523\pi$
$828$	2.50347e17	+	1.20763e17i	0.776890	+	0.374759i
$829$	−1.91078e17			−0.588687			−0.294343	−	0.955700i	$-0.595101\pi$
$829$	−1.91078e17			−0.588687			−0.294343	+	0.955700i	$0.595101\pi$
$830$	−4.11865e16			−0.125976
$831$	−9.75912e16			−0.296350
$832$	−1.76017e17			−0.530658
$833$			6.36771e16i			0.190596i
$834$	5.73665e14			0.00170476
$835$			1.50404e17i			0.443754i
$836$	2.91891e17			0.855034
$837$	−3.94266e16			−0.114667
$838$			2.55772e16i			0.0738566i
$839$		−	6.54049e17i		−	1.87516i	−0.347769	−	0.937580i	$-0.613061\pi$
$839$		−	6.54049e17i		−	1.87516i	0.347769	−	0.937580i	$-0.386939\pi$
$840$	2.74351e16			0.0780964
$841$	−1.96636e17			−0.555759
$842$			9.03297e16i			0.253488i
$843$		−	7.88939e16i		−	0.219825i
$844$	−2.20319e17			−0.609532
$845$			1.16210e17i			0.319231i
$846$	3.78651e16			0.103280
$847$			1.31656e17i			0.356566i
$848$			2.75606e17i			0.741162i
$849$		−	6.72002e16i		−	0.179442i
$850$		−	5.08765e15i		−	0.0134897i
$851$	1.69613e17	−	3.51615e17i	0.446562	−	0.925742i
$852$	7.74073e16			0.202369
$853$	−3.97491e16			−0.103189			−0.0515945	−	0.998668i	$-0.516430\pi$
$853$	−3.97491e16			−0.103189			−0.0515945	+	0.998668i	$0.516430\pi$
$854$	−1.39245e17			−0.358948
$855$	1.71393e17			0.438730
$856$		−	2.01747e17i		−	0.512820i
$857$	3.41589e17			0.862222			0.431111	−	0.902299i	$-0.358122\pi$
$857$	3.41589e17			0.862222			0.431111	+	0.902299i	$0.358122\pi$
$858$			1.05039e16i			0.0263284i
$859$	3.27050e16			0.0814056			0.0407028	−	0.999171i	$-0.487040\pi$
$859$	3.27050e16			0.0814056			0.0407028	+	0.999171i	$0.487040\pi$
$860$	8.67993e16			0.214548
$861$		−	6.60362e16i		−	0.162093i
$862$		−	1.30767e15i		−	0.00318752i
$863$	2.31877e17			0.561298			0.280649	−	0.959810i	$-0.409450\pi$
$863$	2.31877e17			0.561298			0.280649	+	0.959810i	$0.409450\pi$
$864$	1.13064e17			0.271795
$865$		−	2.86053e17i		−	0.682888i
$866$			1.09809e16i			0.0260334i
$867$	−1.44405e17			−0.339991
$868$			1.17225e17i			0.274095i
$869$	−4.90744e17			−1.13956
$870$			7.19144e15i			0.0165844i
$871$			3.38379e14i			0.000774986i
$872$			7.16298e16i			0.162928i
$873$			3.03152e17i			0.684817i
$874$	2.75270e16	−	5.70647e16i	0.0617577	−	0.128026i
$875$	6.24817e17			1.39221
$876$	1.66150e17			0.367685
$877$	−1.89485e17			−0.416463			−0.208232	−	0.978080i	$-0.566771\pi$
$877$	−1.89485e17			−0.416463			−0.208232	+	0.978080i	$0.566771\pi$
$878$	−1.07815e17			−0.235348
$879$			1.47474e17i			0.319728i
$880$	−1.92849e17			−0.415261
$881$			3.28345e16i			0.0702223i	0.999383	+	0.0351111i	$0.0111785\pi$
$881$			3.28345e16i			0.0702223i	−0.999383	+	0.0351111i	$0.988821\pi$
$882$	8.99903e16			0.191154
$883$	−6.29031e17			−1.32711			−0.663556	−	0.748126i	$-0.730954\pi$
$883$	−6.29031e17			−1.32711			−0.663556	+	0.748126i	$0.730954\pi$
$884$			3.55275e16i			0.0744477i
$885$			4.16030e16i			0.0865895i
$886$	−1.53516e16			−0.0317360
$887$	−3.82030e17			−0.784432			−0.392216	−	0.919873i	$-0.628291\pi$
$887$	−3.82030e17			−0.784432			−0.392216	+	0.919873i	$0.628291\pi$
$888$		−	4.94031e16i		−	0.100757i
$889$			9.02725e17i			1.82871i
$890$	5.46762e16			0.110017
$891$		−	2.89062e17i		−	0.577730i
$892$	−5.80123e17			−1.15168
$893$			4.11364e17i			0.811180i
$894$		−	2.20751e16i		−	0.0432392i
$895$		−	4.78125e14i		−	0.000930259i
$896$		−	4.46578e17i		−	0.863076i
$897$	−9.78715e16	−	4.72115e16i	−0.187889	−	0.0906344i
$898$	7.03334e16			0.134123
$899$	−6.20999e16			−0.117634
$900$	3.42681e17			0.644816
$901$	5.31676e16			0.0993799
$902$		−	2.01140e16i		−	0.0373473i
$903$	−1.29477e17			−0.238818
$904$			1.45684e17i			0.266933i
$905$	−3.43116e17			−0.624524
$906$	3.71133e16			0.0671058
$907$		−	8.24798e17i		−	1.48151i	−0.671777	−	0.740754i	$-0.734469\pi$
$907$		−	8.24798e17i		−	1.48151i	0.671777	−	0.740754i	$-0.265531\pi$
$908$		−	7.16196e17i		−	1.27796i
$909$	−4.72744e17			−0.837997
$910$	3.91611e16			0.0689615
$911$			1.71355e17i			0.299768i	0.988704	+	0.149884i	$0.0478901\pi$
$911$			1.71355e17i			0.299768i	−0.988704	+	0.149884i	$0.952110\pi$
$912$			1.85033e17i			0.321573i
$913$	−8.91887e17			−1.53987
$914$			3.24630e16i			0.0556816i
$915$	−1.60854e17			−0.274098
$916$		−	1.90428e17i		−	0.322372i
$917$			1.58878e18i			2.67206i
$918$		−	7.01657e15i		−	0.0117238i
$919$			6.13338e17i			1.01814i	0.860726	+	0.509069i	$0.170010\pi$
$919$			6.13338e17i			1.01814i	−0.860726	+	0.509069i	$0.829990\pi$
$920$	−3.75597e16	+	7.78629e16i	−0.0619435	+	0.128411i
$921$	−7.06993e16			−0.115840
$922$	1.30332e17			0.212162
$923$	2.23302e17			0.361146
$924$	2.93967e17			0.472354
$925$		−	4.81300e17i		−	0.768361i
$926$	−5.82377e16			−0.0923715
$927$			8.11701e17i			1.27914i
$928$	1.78084e17			0.278829
$929$	−6.79449e17			−1.05697			−0.528485	−	0.848943i	$-0.677240\pi$
$929$	−6.79449e17			−1.05697			−0.528485	+	0.848943i	$0.677240\pi$
$930$		−	2.84127e15i		−	0.00439152i
$931$			9.77648e17i			1.50136i
$932$	3.30197e17			0.503823
$933$	1.11590e17			0.169175
$934$		−	5.36137e16i		−	0.0807596i
$935$			3.72028e16i			0.0556809i
$936$	1.01471e17			0.150899
$937$		−	9.84332e17i		−	1.45447i	−0.686390	−	0.727234i	$-0.740805\pi$
$937$		−	9.84332e17i		−	1.45447i	0.686390	−	0.727234i	$-0.259195\pi$
$938$	−1.98698e14			−0.000291726
$939$		−	3.35870e17i		−	0.489979i
$940$		−	2.77732e17i		−	0.402587i
$941$		−	9.37248e17i		−	1.34995i	−0.737842	−	0.674974i	$-0.764155\pi$
$941$		−	9.37248e17i		−	1.34995i	0.737842	−	0.674974i	$-0.235845\pi$
$942$			4.24895e16i			0.0608101i
$943$	1.87416e17	+	9.04061e16i	0.266524	+	0.128566i
$944$	3.31418e17			0.468322
$945$	3.68617e17			0.517588
$946$	−3.94376e16			−0.0550254
$947$	−5.36903e17			−0.744383			−0.372191	−	0.928156i	$-0.621393\pi$
$947$	−5.36903e17			−0.744383			−0.372191	+	0.928156i	$0.621393\pi$
$948$		−	3.17897e17i		−	0.437963i
$949$	4.79304e17			0.656166
$950$		−	7.81118e16i		−	0.106261i
$951$	4.53084e17			0.612485
$952$	−4.21616e16			−0.0566364
$953$		−	1.32415e17i		−	0.176759i	−0.996087	−	0.0883794i	$-0.971831\pi$
$953$		−	1.32415e17i		−	0.176759i	0.996087	−	0.0883794i	$-0.0281688\pi$
$954$		−	7.51380e16i		−	0.0996712i
$955$	−4.24413e17			−0.559459
$956$	9.66654e17			1.26626
$957$			1.55729e17i			0.202721i
$958$			9.80381e16i			0.126824i
$959$	3.05652e17			0.392931
$960$		−	1.19398e17i		−	0.152535i
$961$	−7.63128e17			−0.968851
$962$		−	7.05185e16i		−	0.0889719i
$963$		−	1.26932e18i		−	1.59153i
$964$			7.29615e17i			0.909142i
$965$		−	6.96643e17i		−	0.862674i
$966$	2.77228e16	−	5.74706e16i	0.0341173	−	0.0707266i
$967$	3.05768e17			0.373967			0.186984	−	0.982363i	$-0.440129\pi$
$967$	3.05768e17			0.373967			0.186984	+	0.982363i	$0.440129\pi$
$968$	−5.24991e16			−0.0638116
$969$	3.56950e16			0.0431186
$970$	−4.66538e16			−0.0560088
$971$			2.11293e17i			0.252098i	0.992024	+	0.126049i	$0.0402296\pi$
$971$			2.11293e17i			0.252098i	−0.992024	+	0.126049i	$0.959770\pi$
$972$	7.23903e17			0.858386
$973$		−	4.63148e16i		−	0.0545811i
$974$	6.32920e16			0.0741302
$975$	−1.33969e17			−0.155947
$976$			1.28140e18i			1.48247i
$977$		−	8.07021e16i		−	0.0927934i	−0.998923	−	0.0463967i	$-0.985226\pi$
$977$		−	8.07021e16i		−	0.0927934i	0.998923	−	0.0463967i	$-0.0147738\pi$
$978$	3.70495e16			0.0423398
$979$	1.18400e18			1.34480
$980$		−	6.60059e17i		−	0.745120i
$981$			4.50670e17i			0.505644i
$982$	1.26231e17			0.140766
$983$			6.49313e17i			0.719669i	0.933016	+	0.359834i	$0.117167\pi$
$983$			6.49313e17i			0.719669i	−0.933016	+	0.359834i	$0.882833\pi$
$984$	2.63326e16			0.0290083
$985$			1.03149e17i			0.112940i
$986$		−	1.10516e16i		−	0.0120272i
$987$			4.14290e17i			0.448127i
$988$			5.45462e17i			0.586439i
$989$	1.77259e17	−	3.67466e17i	0.189423	−	0.392681i
$990$	5.25761e16			0.0558441
$991$	1.27432e17			0.134535			0.0672677	−	0.997735i	$-0.478572\pi$
$991$	1.27432e17			0.134535			0.0672677	+	0.997735i	$0.478572\pi$
$992$	−7.03594e16			−0.0738333
$993$	2.08336e17			0.217305
$994$			1.31124e17i			0.135945i
$995$	−2.10801e17			−0.217237
$996$		−	5.77753e17i		−	0.591816i
$997$	−1.28143e18			−1.30474			−0.652369	−	0.757902i	$-0.726225\pi$
$997$	−1.28143e18			−1.30474			−0.652369	+	0.757902i	$0.726225\pi$
$998$	−2.18619e17			−0.221261
$999$		−	6.63779e17i		−	0.667775i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.c.22.9	✓	20
23.22	odd	2	inner	23.13.b.c.22.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.c.22.9	✓	20	1.1	even	1	trivial
23.13.b.c.22.10	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)

\(f(q)\)	\(=\)	\(q+9.17467 q^{2} -251.844 q^{3} -4011.83 q^{4} -7850.48i q^{5} -2310.59 q^{6} +186545. i q^{7} -74386.7 q^{8} -468016. q^{9} +O(q^{10})\)	\(q+9.17467 q^{2} -251.844 q^{3} -4011.83 q^{4} -7850.48i q^{5} -2310.59 q^{6} +186545. i q^{7} -74386.7 q^{8} -468016. q^{9} -72025.6i q^{10} -1.55970e6i q^{11} +1.01035e6 q^{12} +2.91464e6 q^{13} +1.71149e6i q^{14} +1.97710e6i q^{15} +1.57500e7 q^{16} -3.03835e6i q^{17} -4.29389e6 q^{18} -4.66485e7i q^{19} +3.14947e7i q^{20} -4.69803e7i q^{21} -1.43098e7i q^{22} +(1.33334e8 + 6.43178e7i) q^{23} +1.87338e7 q^{24} +1.82511e8 q^{25} +2.67409e7 q^{26} +2.51707e8 q^{27} -7.48386e8i q^{28} +3.96458e8 q^{29} +1.81392e7i q^{30} -1.56637e8 q^{31} +4.49189e8 q^{32} +3.92802e8i q^{33} -2.78759e7i q^{34} +1.46447e9 q^{35} +1.87760e9 q^{36} -2.63711e9i q^{37} -4.27985e8i q^{38} -7.34035e8 q^{39} +5.83971e8i q^{40} +1.40562e9 q^{41} -4.31029e8i q^{42} -2.75599e9i q^{43} +6.25725e9i q^{44} +3.67414e9i q^{45} +(1.22329e9 + 5.90095e8i) q^{46} -8.81837e9 q^{47} -3.96654e9 q^{48} -2.09578e10 q^{49} +1.67448e9 q^{50} +7.65191e8i q^{51} -1.16930e10 q^{52} +1.74988e10i q^{53} +2.30933e9 q^{54} -1.22444e10 q^{55} -1.38765e10i q^{56} +1.17481e10i q^{57} +3.63738e9 q^{58} +2.10425e10 q^{59} -7.93176e9i q^{60} +8.13588e10i q^{61} -1.43709e9 q^{62} -8.73060e10i q^{63} -6.03907e10 q^{64} -2.28813e10i q^{65} +3.60383e9i q^{66} +1.16096e8i q^{67} +1.21893e10i q^{68} +(-3.35793e10 - 1.61981e10i) q^{69} +1.34360e10 q^{70} +7.66140e10 q^{71} +3.48141e10 q^{72} +1.64447e11 q^{73} -2.41946e10i q^{74} -4.59642e10 q^{75} +1.87146e11i q^{76} +2.90955e11 q^{77} -6.73453e9 q^{78} -3.14639e11i q^{79} -1.23645e11i q^{80} +1.85332e11 q^{81} +1.28961e10 q^{82} -5.71832e11i q^{83} +1.88477e11i q^{84} -2.38525e10 q^{85} -2.52853e10i q^{86} -9.98457e10 q^{87} +1.16021e11i q^{88} +7.59123e11i q^{89} +3.37091e10i q^{90} +5.43711e11i q^{91} +(-5.34911e11 - 2.58032e11i) q^{92} +3.94480e10 q^{93} -8.09057e10 q^{94} -3.66213e11 q^{95} -1.13125e11 q^{96} -6.47739e11i q^{97} -1.92281e11 q^{98} +7.29965e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 88 q^{2} + 318 q^{3} + 36072 q^{4} + 264240 q^{6} - 767728 q^{8} + 1971426 q^{9}+O(q^{10}) \) 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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