LMFDB - Embedded newform 87.2.c.a.28.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [87,2,Mod(28,87)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(87, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("87.28");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	87.c (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:	$0.694698497585$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			28.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	87.28
Dual form			87.2.c.a.28.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} -1.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} +2.23607i q^{8} -1.00000 q^{9} -0.763932i q^{10} -2.23607i q^{11} +1.61803i q^{12} -1.00000 q^{13} +0.145898i q^{14} -1.23607i q^{15} +1.85410 q^{16} -3.00000i q^{17} -0.618034i q^{18} -5.70820i q^{19} -2.00000 q^{20} +0.236068i q^{21} +1.38197 q^{22} -3.23607 q^{23} -2.23607 q^{24} -3.47214 q^{25} -0.618034i q^{26} -1.00000i q^{27} +0.381966 q^{28} +(4.47214 + 3.00000i) q^{29} +0.763932 q^{30} -2.76393i q^{31} +5.61803i q^{32} +2.23607 q^{33} +1.85410 q^{34} -0.291796 q^{35} -1.61803 q^{36} +9.23607i q^{37} +3.52786 q^{38} -1.00000i q^{39} -2.76393i q^{40} +4.47214i q^{41} -0.145898 q^{42} +8.47214i q^{43} -3.61803i q^{44} +1.23607 q^{45} -2.00000i q^{46} -5.76393i q^{47} +1.85410i q^{48} -6.94427 q^{49} -2.14590i q^{50} +3.00000 q^{51} -1.61803 q^{52} +11.2361 q^{53} +0.618034 q^{54} +2.76393i q^{55} +0.527864i q^{56} +5.70820 q^{57} +(-1.85410 + 2.76393i) q^{58} -8.94427 q^{59} -2.00000i q^{60} +7.23607i q^{61} +1.70820 q^{62} -0.236068 q^{63} +0.236068 q^{64} +1.23607 q^{65} +1.38197i q^{66} -8.70820 q^{67} -4.85410i q^{68} -3.23607i q^{69} -0.180340i q^{70} +9.23607 q^{71} -2.23607i q^{72} +5.70820i q^{73} -5.70820 q^{74} -3.47214i q^{75} -9.23607i q^{76} -0.527864i q^{77} +0.618034 q^{78} -15.7082i q^{79} -2.29180 q^{80} +1.00000 q^{81} -2.76393 q^{82} -6.00000 q^{83} +0.381966i q^{84} +3.70820i q^{85} -5.23607 q^{86} +(-3.00000 + 4.47214i) q^{87} +5.00000 q^{88} -17.9443i q^{89} +0.763932i q^{90} -0.236068 q^{91} -5.23607 q^{92} +2.76393 q^{93} +3.56231 q^{94} +7.05573i q^{95} -5.61803 q^{96} -4.18034i q^{97} -4.29180i q^{98} +2.23607i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 4 q^{5} + 2 q^{6} - 8 q^{7} - 4 q^{9} - 4 q^{13} - 6 q^{16} - 8 q^{20} + 10 q^{22} - 4 q^{23} + 4 q^{25} + 6 q^{28} + 12 q^{30} - 6 q^{34} - 28 q^{35} - 2 q^{36} + 32 q^{38} - 14 q^{42}+ \cdots - 18 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 + 4 * q^5 + 2 * q^6 - 8 * q^7 - 4 * q^9 - 4 * q^13 - 6 * q^16 - 8 * q^20 + 10 * q^22 - 4 * q^23 + 4 * q^25 + 6 * q^28 + 12 * q^30 - 6 * q^34 - 28 * q^35 - 2 * q^36 + 32 * q^38 - 14 * q^42 - 4 * q^45 + 8 * q^49 + 12 * q^51 - 2 * q^52 + 36 * q^53 - 2 * q^54 - 4 * q^57 + 6 * q^58 - 20 * q^62 + 8 * q^63 - 8 * q^64 - 4 * q^65 - 8 * q^67 + 28 * q^71 + 4 * q^74 - 2 * q^78 - 36 * q^80 + 4 * q^81 - 20 * q^82 - 24 * q^83 - 12 * q^86 - 12 * q^87 + 20 * q^88 + 8 * q^91 - 12 * q^92 + 20 * q^93 - 26 * q^94 - 18 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$-1$	$1$

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 $

$2$			0.618034i			0.437016i	0.975835	+	0.218508i	$0.0701190\pi$
$2$			0.618034i			0.437016i	−0.975835	+	0.218508i	$0.929881\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	1.61803			0.809017
$5$	−1.23607			−0.552786			−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607			−0.552786			−0.276393	+	0.961045i	$0.589139\pi$
$6$	−0.618034			−0.252311
$7$	0.236068			0.0892253			0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068			0.0892253			0.0446127	+	0.999004i	$0.485795\pi$
$8$			2.23607i			0.790569i
$9$	−1.00000			−0.333333
$10$		−	0.763932i		−	0.241577i
$11$		−	2.23607i		−	0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$11$		−	2.23607i		−	0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$12$			1.61803i			0.467086i
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$			0.145898i			0.0389929i
$15$		−	1.23607i		−	0.319151i
$16$	1.85410			0.463525
$17$		−	3.00000i		−	0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$		−	3.00000i		−	0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$		−	0.618034i		−	0.145672i
$19$		−	5.70820i		−	1.30955i	−0.755823	−	0.654776i	$-0.772763\pi$
$19$		−	5.70820i		−	1.30955i	0.755823	−	0.654776i	$-0.227237\pi$
$20$	−2.00000			−0.447214
$21$			0.236068i			0.0515143i
$22$	1.38197			0.294636
$23$	−3.23607			−0.674767			−0.337383	−	0.941367i	$-0.609542\pi$
$23$	−3.23607			−0.674767			−0.337383	+	0.941367i	$0.609542\pi$
$24$	−2.23607			−0.456435
$25$	−3.47214			−0.694427
$26$		−	0.618034i		−	0.121206i
$27$		−	1.00000i		−	0.192450i
$28$	0.381966			0.0721848
$29$	4.47214	+	3.00000i	0.830455	+	0.557086i
$29$	4.47214	+	3.00000i	0.830455	+	0.557086i
$30$	0.763932			0.139474
$31$		−	2.76393i		−	0.496417i	−0.968707	−	0.248208i	$-0.920158\pi$
$31$		−	2.76393i		−	0.496417i	0.968707	−	0.248208i	$-0.0798418\pi$
$32$			5.61803i			0.993137i
$33$	2.23607			0.389249
$34$	1.85410			0.317976
$35$	−0.291796			−0.0493225
$36$	−1.61803			−0.269672
$37$			9.23607i			1.51840i	0.650857	+	0.759200i	$0.274410\pi$
$37$			9.23607i			1.51840i	−0.650857	+	0.759200i	$0.725590\pi$
$38$	3.52786			0.572295
$39$		−	1.00000i		−	0.160128i
$40$		−	2.76393i		−	0.437016i
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$	−0.145898			−0.0225126
$43$			8.47214i			1.29199i	0.763342	+	0.645994i	$0.223557\pi$
$43$			8.47214i			1.29199i	−0.763342	+	0.645994i	$0.776443\pi$
$44$		−	3.61803i		−	0.545439i
$45$	1.23607			0.184262
$46$		−	2.00000i		−	0.294884i
$47$		−	5.76393i		−	0.840756i	−0.907349	−	0.420378i	$-0.861897\pi$
$47$		−	5.76393i		−	0.840756i	0.907349	−	0.420378i	$-0.138103\pi$
$48$			1.85410i			0.267617i
$49$	−6.94427			−0.992039
$50$		−	2.14590i		−	0.303476i
$51$	3.00000			0.420084
$52$	−1.61803			−0.224381
$53$	11.2361			1.54339			0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361			1.54339			0.771696	+	0.635991i	$0.219409\pi$
$54$	0.618034			0.0841038
$55$			2.76393i			0.372689i
$56$			0.527864i			0.0705388i
$57$	5.70820			0.756070
$58$	−1.85410	+	2.76393i	−0.243456	+	0.362922i
$59$	−8.94427			−1.16445			−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427			−1.16445			−0.582223	+	0.813029i	$0.697817\pi$
$60$		−	2.00000i		−	0.258199i
$61$			7.23607i			0.926484i	0.886232	+	0.463242i	$0.153314\pi$
$61$			7.23607i			0.926484i	−0.886232	+	0.463242i	$0.846686\pi$
$62$	1.70820			0.216942
$63$	−0.236068			−0.0297418
$64$	0.236068			0.0295085
$65$	1.23607			0.153315
$66$			1.38197i			0.170108i
$67$	−8.70820			−1.06388			−0.531938	−	0.846783i	$-0.678536\pi$
$67$	−8.70820			−1.06388			−0.531938	+	0.846783i	$0.678536\pi$
$68$		−	4.85410i		−	0.588646i
$69$		−	3.23607i		−	0.389577i
$70$		−	0.180340i		−	0.0215547i
$71$	9.23607			1.09612			0.548060	−	0.836439i	$-0.315367\pi$
$71$	9.23607			1.09612			0.548060	+	0.836439i	$0.315367\pi$
$72$		−	2.23607i		−	0.263523i
$73$			5.70820i			0.668095i	0.942556	+	0.334047i	$0.108415\pi$
$73$			5.70820i			0.668095i	−0.942556	+	0.334047i	$0.891585\pi$
$74$	−5.70820			−0.663565
$75$		−	3.47214i		−	0.400928i
$76$		−	9.23607i		−	1.05945i
$77$		−	0.527864i		−	0.0601557i
$78$	0.618034			0.0699786
$79$		−	15.7082i		−	1.76731i	−0.468138	−	0.883656i	$-0.655075\pi$
$79$		−	15.7082i		−	1.76731i	0.468138	−	0.883656i	$-0.344925\pi$
$80$	−2.29180			−0.256231
$81$	1.00000			0.111111
$82$	−2.76393			−0.305225
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$			0.381966i			0.0416759i
$85$			3.70820i			0.402211i
$86$	−5.23607			−0.564620
$87$	−3.00000	+	4.47214i	−0.321634	+	0.479463i
$88$	5.00000			0.533002
$89$		−	17.9443i		−	1.90209i	−0.309054	−	0.951045i	$-0.600012\pi$
$89$		−	17.9443i		−	1.90209i	0.309054	−	0.951045i	$-0.399988\pi$
$90$			0.763932i			0.0805255i
$91$	−0.236068			−0.0247466
$92$	−5.23607			−0.545898
$93$	2.76393			0.286606
$94$	3.56231			0.367424
$95$			7.05573i			0.723902i
$96$	−5.61803			−0.573388
$97$		−	4.18034i		−	0.424449i	−0.977221	−	0.212225i	$-0.931929\pi$
$97$		−	4.18034i		−	0.424449i	0.977221	−	0.212225i	$-0.0680708\pi$
$98$		−	4.29180i		−	0.433537i
$99$			2.23607i			0.224733i
$100$	−5.61803			−0.561803
$101$			13.9443i			1.38751i	0.720213	+	0.693753i	$0.244044\pi$
$101$			13.9443i			1.38751i	−0.720213	+	0.693753i	$0.755956\pi$
$102$			1.85410i			0.183583i
$103$	7.41641			0.730760			0.365380	−	0.930858i	$-0.380939\pi$
$103$	7.41641			0.730760			0.365380	+	0.930858i	$0.380939\pi$
$104$		−	2.23607i		−	0.219265i
$105$		−	0.291796i		−	0.0284764i
$106$			6.94427i			0.674487i
$107$	5.23607			0.506190			0.253095	−	0.967441i	$-0.418552\pi$
$107$	5.23607			0.506190			0.253095	+	0.967441i	$0.418552\pi$
$108$		−	1.61803i		−	0.155695i
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$	−1.70820			−0.162871
$111$	−9.23607			−0.876649
$112$	0.437694			0.0413582
$113$			13.4721i			1.26735i	0.773599	+	0.633676i	$0.218455\pi$
$113$			13.4721i			1.26735i	−0.773599	+	0.633676i	$0.781545\pi$
$114$			3.52786i			0.330415i
$115$	4.00000			0.373002
$116$	7.23607	+	4.85410i	0.671852	+	0.450692i
$117$	1.00000			0.0924500
$118$		−	5.52786i		−	0.508881i
$119$		−	0.708204i		−	0.0649209i
$120$	2.76393			0.252311
$121$	6.00000			0.545455
$122$	−4.47214			−0.404888
$123$	−4.47214			−0.403239
$124$		−	4.47214i		−	0.401610i
$125$	10.4721			0.936656
$126$		−	0.145898i		−	0.0129976i
$127$		−	3.52786i		−	0.313047i	−0.987674	−	0.156524i	$-0.949971\pi$
$127$		−	3.52786i		−	0.313047i	0.987674	−	0.156524i	$-0.0500287\pi$
$128$			11.3820i			1.00603i
$129$	−8.47214			−0.745930
$130$			0.763932i			0.0670013i
$131$			7.76393i			0.678338i	0.940725	+	0.339169i	$0.110146\pi$
$131$			7.76393i			0.678338i	−0.940725	+	0.339169i	$0.889854\pi$
$132$	3.61803			0.314909
$133$		−	1.34752i		−	0.116845i
$134$		−	5.38197i		−	0.464931i
$135$			1.23607i			0.106384i
$136$	6.70820			0.575224
$137$		−	12.4721i		−	1.06557i	−0.846252	−	0.532783i	$-0.821146\pi$
$137$		−	12.4721i		−	1.06557i	0.846252	−	0.532783i	$-0.178854\pi$
$138$	2.00000			0.170251
$139$	6.70820			0.568982			0.284491	−	0.958679i	$-0.408175\pi$
$139$	6.70820			0.568982			0.284491	+	0.958679i	$0.408175\pi$
$140$	−0.472136			−0.0399028
$141$	5.76393			0.485411
$142$			5.70820i			0.479022i
$143$			2.23607i			0.186989i
$144$	−1.85410			−0.154508
$145$	−5.52786	−	3.70820i	−0.459064	−	0.307950i
$146$	−3.52786			−0.291968
$147$		−	6.94427i		−	0.572754i
$148$			14.9443i			1.22841i
$149$	−14.4721			−1.18560			−0.592802	−	0.805348i	$-0.701978\pi$
$149$	−14.4721			−1.18560			−0.592802	+	0.805348i	$0.701978\pi$
$150$	2.14590			0.175212
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$	12.7639			1.03529
$153$			3.00000i			0.242536i
$154$	0.326238			0.0262890
$155$			3.41641i			0.274412i
$156$		−	1.61803i		−	0.129546i
$157$			2.00000i			0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$			2.00000i			0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	9.70820			0.772343
$159$			11.2361i			0.891078i
$160$		−	6.94427i		−	0.548993i
$161$	−0.763932			−0.0602063
$162$			0.618034i			0.0485573i
$163$		−	19.4164i		−	1.52081i	−0.649449	−	0.760405i	$-0.725000\pi$
$163$		−	19.4164i		−	1.52081i	0.649449	−	0.760405i	$-0.275000\pi$
$164$			7.23607i			0.565042i
$165$	−2.76393			−0.215172
$166$		−	3.70820i		−	0.287812i
$167$	−19.8885			−1.53902			−0.769511	−	0.638634i	$-0.779500\pi$
$167$	−19.8885			−1.53902			−0.769511	+	0.638634i	$0.779500\pi$
$168$	−0.527864			−0.0407256
$169$	−12.0000			−0.923077
$170$	−2.29180			−0.175773
$171$			5.70820i			0.436517i
$172$			13.7082i			1.04524i
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$	−2.76393	−	1.85410i	−0.209533	−	0.140559i
$175$	−0.819660			−0.0619605
$176$		−	4.14590i		−	0.312509i
$177$		−	8.94427i		−	0.672293i
$178$	11.0902			0.831243
$179$	−12.7639			−0.954021			−0.477011	−	0.878898i	$-0.658280\pi$
$179$	−12.7639			−0.954021			−0.477011	+	0.878898i	$0.658280\pi$
$180$	2.00000			0.149071
$181$	−6.41641			−0.476928			−0.238464	−	0.971151i	$-0.576644\pi$
$181$	−6.41641			−0.476928			−0.238464	+	0.971151i	$0.576644\pi$
$182$		−	0.145898i		−	0.0108147i
$183$	−7.23607			−0.534906
$184$		−	7.23607i		−	0.533450i
$185$		−	11.4164i		−	0.839351i
$186$			1.70820i			0.125252i
$187$	−6.70820			−0.490552
$188$		−	9.32624i		−	0.680186i
$189$		−	0.236068i		−	0.0171714i
$190$	−4.36068			−0.316357
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$			0.236068i			0.0170367i
$193$		−	9.41641i		−	0.677808i	−0.940821	−	0.338904i	$-0.889944\pi$
$193$		−	9.41641i		−	0.677808i	0.940821	−	0.338904i	$-0.110056\pi$
$194$	2.58359			0.185491
$195$			1.23607i			0.0885167i
$196$	−11.2361			−0.802576
$197$	9.70820			0.691681			0.345840	−	0.938293i	$-0.387594\pi$
$197$	9.70820			0.691681			0.345840	+	0.938293i	$0.387594\pi$
$198$	−1.38197			−0.0982120
$199$	20.1246			1.42660			0.713298	−	0.700861i	$-0.247201\pi$
$199$	20.1246			1.42660			0.713298	+	0.700861i	$0.247201\pi$
$200$		−	7.76393i		−	0.548993i
$201$		−	8.70820i		−	0.614229i
$202$	−8.61803			−0.606363
$203$	1.05573	+	0.708204i	0.0740976	+	0.0497062i
$204$	4.85410			0.339855
$205$		−	5.52786i		−	0.386083i
$206$			4.58359i			0.319354i
$207$	3.23607			0.224922
$208$	−1.85410			−0.128559
$209$	−12.7639			−0.882900
$210$	0.180340			0.0124446
$211$			10.0000i			0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$			10.0000i			0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	18.1803			1.24863
$213$			9.23607i			0.632845i
$214$			3.23607i			0.221213i
$215$		−	10.4721i		−	0.714194i
$216$	2.23607			0.152145
$217$		−	0.652476i		−	0.0442929i
$218$			3.09017i			0.209293i
$219$	−5.70820			−0.385725
$220$			4.47214i			0.301511i
$221$			3.00000i			0.201802i
$222$		−	5.70820i		−	0.383110i
$223$	0.708204			0.0474248			0.0237124	−	0.999719i	$-0.492451\pi$
$223$	0.708204			0.0474248			0.0237124	+	0.999719i	$0.492451\pi$
$224$			1.32624i			0.0886130i
$225$	3.47214			0.231476
$226$	−8.32624			−0.553853
$227$	26.9443			1.78835			0.894177	−	0.447713i	$-0.147762\pi$
$227$	26.9443			1.78835			0.894177	+	0.447713i	$0.147762\pi$
$228$	9.23607			0.611674
$229$		−	0.180340i		−	0.0119172i	−0.999982	−	0.00595860i	$-0.998103\pi$
$229$		−	0.180340i		−	0.0119172i	0.999982	−	0.00595860i	$-0.00189669\pi$
$230$			2.47214i			0.163008i
$231$	0.527864			0.0347309
$232$	−6.70820	+	10.0000i	−0.440415	+	0.656532i
$233$	−1.52786			−0.100094			−0.0500469	−	0.998747i	$-0.515937\pi$
$233$	−1.52786			−0.100094			−0.0500469	+	0.998747i	$0.515937\pi$
$234$			0.618034i			0.0404021i
$235$			7.12461i			0.464758i
$236$	−14.4721			−0.942056
$237$	15.7082			1.02036
$238$	0.437694			0.0283715
$239$	3.81966			0.247073			0.123537	−	0.992340i	$-0.460576\pi$
$239$	3.81966			0.247073			0.123537	+	0.992340i	$0.460576\pi$
$240$		−	2.29180i		−	0.147935i
$241$	−3.00000			−0.193247			−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000			−0.193247			−0.0966235	+	0.995321i	$0.530804\pi$
$242$			3.70820i			0.238372i
$243$			1.00000i			0.0641500i
$244$			11.7082i			0.749541i
$245$	8.58359			0.548386
$246$		−	2.76393i		−	0.176222i
$247$			5.70820i			0.363204i
$248$	6.18034			0.392452
$249$		−	6.00000i		−	0.380235i
$250$			6.47214i			0.409334i
$251$		−	7.76393i		−	0.490055i	−0.969516	−	0.245028i	$-0.921203\pi$
$251$		−	7.76393i		−	0.490055i	0.969516	−	0.245028i	$-0.0787970\pi$
$252$	−0.381966			−0.0240616
$253$			7.23607i			0.454928i
$254$	2.18034			0.136807
$255$	−3.70820			−0.232217
$256$	−6.56231			−0.410144
$257$	−17.5279			−1.09336			−0.546679	−	0.837342i	$-0.684108\pi$
$257$	−17.5279			−1.09336			−0.546679	+	0.837342i	$0.684108\pi$
$258$		−	5.23607i		−	0.325983i
$259$			2.18034i			0.135480i
$260$	2.00000			0.124035
$261$	−4.47214	−	3.00000i	−0.276818	−	0.185695i
$262$	−4.79837			−0.296445
$263$			12.9443i			0.798178i	0.916912	+	0.399089i	$0.130674\pi$
$263$			12.9443i			0.798178i	−0.916912	+	0.399089i	$0.869326\pi$
$264$			5.00000i			0.307729i
$265$	−13.8885			−0.853166
$266$	0.832816			0.0510632
$267$	17.9443			1.09817
$268$	−14.0902			−0.860694
$269$			23.3607i			1.42433i	0.702014	+	0.712163i	$0.252284\pi$
$269$			23.3607i			1.42433i	−0.702014	+	0.712163i	$0.747716\pi$
$270$	−0.763932			−0.0464914
$271$			3.41641i			0.207532i	0.994602	+	0.103766i	$0.0330893\pi$
$271$			3.41641i			0.207532i	−0.994602	+	0.103766i	$0.966911\pi$
$272$		−	5.56231i		−	0.337264i
$273$		−	0.236068i		−	0.0142875i
$274$	7.70820			0.465670
$275$			7.76393i			0.468183i
$276$		−	5.23607i		−	0.315174i
$277$	−20.4164			−1.22670			−0.613352	−	0.789810i	$-0.710179\pi$
$277$	−20.4164			−1.22670			−0.613352	+	0.789810i	$0.710179\pi$
$278$			4.14590i			0.248654i
$279$			2.76393i			0.165472i
$280$		−	0.652476i		−	0.0389929i
$281$	25.4164			1.51622			0.758108	−	0.652129i	$-0.226124\pi$
$281$	25.4164			1.51622			0.758108	+	0.652129i	$0.226124\pi$
$282$			3.56231i			0.212132i
$283$	21.8885			1.30114			0.650569	−	0.759447i	$-0.274530\pi$
$283$	21.8885			1.30114			0.650569	+	0.759447i	$0.274530\pi$
$284$	14.9443			0.886779
$285$	−7.05573			−0.417945
$286$	−1.38197			−0.0817174
$287$			1.05573i			0.0623177i
$288$		−	5.61803i		−	0.331046i
$289$	8.00000			0.470588
$290$	2.29180	−	3.41641i	0.134579	−	0.200618i
$291$	4.18034			0.245056
$292$			9.23607i			0.540500i
$293$		−	12.0557i		−	0.704303i	−0.935943	−	0.352152i	$-0.885450\pi$
$293$		−	12.0557i		−	0.704303i	0.935943	−	0.352152i	$-0.114550\pi$
$294$	4.29180			0.250303
$295$	11.0557			0.643689
$296$	−20.6525			−1.20040
$297$	−2.23607			−0.129750
$298$		−	8.94427i		−	0.518128i
$299$	3.23607			0.187147
$300$		−	5.61803i		−	0.324357i
$301$			2.00000i			0.115278i
$302$			7.41641i			0.426766i
$303$	−13.9443			−0.801077
$304$		−	10.5836i		−	0.607011i
$305$		−	8.94427i		−	0.512148i
$306$	−1.85410			−0.105992
$307$			5.81966i			0.332146i	0.986113	+	0.166073i	$0.0531087\pi$
$307$			5.81966i			0.332146i	−0.986113	+	0.166073i	$0.946891\pi$
$308$		−	0.854102i		−	0.0486670i
$309$			7.41641i			0.421905i
$310$	−2.11146			−0.119923
$311$			15.6525i			0.887570i	0.896133	+	0.443785i	$0.146365\pi$
$311$			15.6525i			0.887570i	−0.896133	+	0.443785i	$0.853635\pi$
$312$	2.23607			0.126592
$313$	3.47214			0.196257			0.0981284	−	0.995174i	$-0.468714\pi$
$313$	3.47214			0.196257			0.0981284	+	0.995174i	$0.468714\pi$
$314$	−1.23607			−0.0697554
$315$	0.291796			0.0164408
$316$		−	25.4164i		−	1.42978i
$317$		−	29.8328i		−	1.67558i	−0.545994	−	0.837789i	$-0.683848\pi$
$317$		−	29.8328i		−	1.67558i	0.545994	−	0.837789i	$-0.316152\pi$
$318$	−6.94427			−0.389415
$319$	6.70820	−	10.0000i	0.375587	−	0.559893i
$320$	−0.291796			−0.0163119
$321$			5.23607i			0.292249i
$322$		−	0.472136i		−	0.0263111i
$323$	−17.1246			−0.952839
$324$	1.61803			0.0898908
$325$	3.47214			0.192599
$326$	12.0000			0.664619
$327$			5.00000i			0.276501i
$328$	−10.0000			−0.552158
$329$		−	1.36068i		−	0.0750167i
$330$		−	1.70820i		−	0.0940335i
$331$		−	21.7082i		−	1.19319i	−0.802542	−	0.596595i	$-0.796520\pi$
$331$		−	21.7082i		−	1.19319i	0.802542	−	0.596595i	$-0.203480\pi$
$332$	−9.70820			−0.532807
$333$		−	9.23607i		−	0.506133i
$334$		−	12.2918i		−	0.672577i
$335$	10.7639			0.588096
$336$			0.437694i			0.0238782i
$337$			14.7639i			0.804243i	0.915586	+	0.402121i	$0.131727\pi$
$337$			14.7639i			0.804243i	−0.915586	+	0.402121i	$0.868273\pi$
$338$		−	7.41641i		−	0.403399i
$339$	−13.4721			−0.731706
$340$			6.00000i			0.325396i
$341$	−6.18034			−0.334684
$342$	−3.52786			−0.190765
$343$	−3.29180			−0.177740
$344$	−18.9443			−1.02141
$345$			4.00000i			0.215353i
$346$		−	3.70820i		−	0.199354i
$347$	8.65248			0.464489			0.232245	−	0.972657i	$-0.425393\pi$
$347$	8.65248			0.464489			0.232245	+	0.972657i	$0.425393\pi$
$348$	−4.85410	+	7.23607i	−0.260207	+	0.387894i
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		−	0.506578i		−	0.0270777i
$351$			1.00000i			0.0533761i
$352$	12.5623			0.669573
$353$	−26.6525			−1.41857			−0.709284	−	0.704923i	$-0.750982\pi$
$353$	−26.6525			−1.41857			−0.709284	+	0.704923i	$0.750982\pi$
$354$	5.52786			0.293803
$355$	−11.4164			−0.605920
$356$		−	29.0344i		−	1.53882i
$357$	0.708204			0.0374821
$358$		−	7.88854i		−	0.416922i
$359$			37.3050i			1.96888i	0.175721	+	0.984440i	$0.443774\pi$
$359$			37.3050i			1.96888i	−0.175721	+	0.984440i	$0.556226\pi$
$360$			2.76393i			0.145672i
$361$	−13.5836			−0.714926
$362$		−	3.96556i		−	0.208425i
$363$			6.00000i			0.314918i
$364$	−0.381966			−0.0200205
$365$		−	7.05573i		−	0.369314i
$366$		−	4.47214i		−	0.233762i
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	−6.00000			−0.312772
$369$		−	4.47214i		−	0.232810i
$370$	7.05573			0.366810
$371$	2.65248			0.137710
$372$	4.47214			0.231869
$373$	31.8885			1.65113			0.825563	−	0.564310i	$-0.190858\pi$
$373$	31.8885			1.65113			0.825563	+	0.564310i	$0.190858\pi$
$374$		−	4.14590i		−	0.214379i
$375$			10.4721i			0.540779i
$376$	12.8885			0.664676
$377$	−4.47214	−	3.00000i	−0.230327	−	0.154508i
$378$	0.145898			0.00750419
$379$			27.7082i			1.42327i	0.702547	+	0.711637i	$0.252046\pi$
$379$			27.7082i			1.42327i	−0.702547	+	0.711637i	$0.747954\pi$
$380$			11.4164i			0.585649i
$381$	3.52786			0.180738
$382$	5.52786			0.282830
$383$	−36.6525			−1.87285			−0.936427	−	0.350862i	$-0.885888\pi$
$383$	−36.6525			−1.87285			−0.936427	+	0.350862i	$0.885888\pi$
$384$	−11.3820			−0.580834
$385$			0.652476i			0.0332532i
$386$	5.81966			0.296213
$387$		−	8.47214i		−	0.430663i
$388$		−	6.76393i		−	0.343387i
$389$		−	4.52786i		−	0.229572i	−0.993390	−	0.114786i	$-0.963382\pi$
$389$		−	4.52786i		−	0.229572i	0.993390	−	0.114786i	$-0.0366182\pi$
$390$	−0.763932			−0.0386832
$391$			9.70820i			0.490965i
$392$		−	15.5279i		−	0.784276i
$393$	−7.76393			−0.391639
$394$			6.00000i			0.302276i
$395$			19.4164i			0.976946i
$396$			3.61803i			0.181813i
$397$	−10.9443			−0.549277			−0.274639	−	0.961548i	$-0.588558\pi$
$397$	−10.9443			−0.549277			−0.274639	+	0.961548i	$0.588558\pi$
$398$			12.4377i			0.623445i
$399$	1.34752			0.0674606
$400$	−6.43769			−0.321885
$401$	20.2918			1.01332			0.506662	−	0.862145i	$-0.330879\pi$
$401$	20.2918			1.01332			0.506662	+	0.862145i	$0.330879\pi$
$402$	5.38197			0.268428
$403$			2.76393i			0.137681i
$404$			22.5623i			1.12252i
$405$	−1.23607			−0.0614207
$406$	−0.437694	+	0.652476i	−0.0217224	+	0.0323818i
$407$	20.6525			1.02371
$408$			6.70820i			0.332106i
$409$		−	37.4164i		−	1.85012i	−0.379818	−	0.925061i	$-0.624013\pi$
$409$		−	37.4164i		−	1.85012i	0.379818	−	0.925061i	$-0.375987\pi$
$410$	3.41641			0.168724
$411$	12.4721			0.615205
$412$	12.0000			0.591198
$413$	−2.11146			−0.103898
$414$			2.00000i			0.0982946i
$415$	7.41641			0.364057
$416$		−	5.61803i		−	0.275447i
$417$			6.70820i			0.328502i
$418$		−	7.88854i		−	0.385841i
$419$	26.1803			1.27899			0.639497	−	0.768794i	$-0.279143\pi$
$419$	26.1803			1.27899			0.639497	+	0.768794i	$0.279143\pi$
$420$		−	0.472136i		−	0.0230379i
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$	−6.18034			−0.300854
$423$			5.76393i			0.280252i
$424$			25.1246i			1.22016i
$425$			10.4164i			0.505270i
$426$	−5.70820			−0.276563
$427$			1.70820i			0.0826658i
$428$	8.47214			0.409516
$429$	−2.23607			−0.107958
$430$	6.47214			0.312114
$431$	−30.3607			−1.46242			−0.731211	−	0.682151i	$-0.761045\pi$
$431$	−30.3607			−1.46242			−0.731211	+	0.682151i	$0.761045\pi$
$432$		−	1.85410i		−	0.0892055i
$433$		−	27.7082i		−	1.33157i	−0.746143	−	0.665786i	$-0.768097\pi$
$433$		−	27.7082i		−	1.33157i	0.746143	−	0.665786i	$-0.231903\pi$
$434$	0.403252			0.0193567
$435$	3.70820	−	5.52786i	0.177795	−	0.265041i
$436$	8.09017			0.387449
$437$			18.4721i			0.883642i
$438$		−	3.52786i		−	0.168568i
$439$	26.7082			1.27471			0.637357	−	0.770569i	$-0.280028\pi$
$439$	26.7082			1.27471			0.637357	+	0.770569i	$0.280028\pi$
$440$	−6.18034			−0.294636
$441$	6.94427			0.330680
$442$	−1.85410			−0.0881906
$443$		−	21.6525i		−	1.02874i	−0.857568	−	0.514370i	$-0.828026\pi$
$443$		−	21.6525i		−	1.02874i	0.857568	−	0.514370i	$-0.171974\pi$
$444$	−14.9443			−0.709224
$445$			22.1803i			1.05145i
$446$			0.437694i			0.0207254i
$447$		−	14.4721i		−	0.684509i
$448$	0.0557281			0.00263290
$449$		−	16.8885i		−	0.797020i	−0.917164	−	0.398510i	$-0.869527\pi$
$449$		−	16.8885i		−	0.797020i	0.917164	−	0.398510i	$-0.130473\pi$
$450$			2.14590i			0.101159i
$451$	10.0000			0.470882
$452$			21.7984i			1.02531i
$453$			12.0000i			0.563809i
$454$			16.6525i			0.781539i
$455$	0.291796			0.0136796
$456$			12.7639i			0.597726i
$457$	−20.4164			−0.955039			−0.477520	−	0.878621i	$-0.658464\pi$
$457$	−20.4164			−0.955039			−0.477520	+	0.878621i	$0.658464\pi$
$458$	0.111456			0.00520801
$459$	−3.00000			−0.140028
$460$	6.47214			0.301765
$461$		−	1.05573i		−	0.0491702i	−0.999698	−	0.0245851i	$-0.992174\pi$
$461$		−	1.05573i		−	0.0491702i	0.999698	−	0.0245851i	$-0.00782646\pi$
$462$			0.326238i			0.0151780i
$463$	−12.7082			−0.590600			−0.295300	−	0.955405i	$-0.595420\pi$
$463$	−12.7082			−0.590600			−0.295300	+	0.955405i	$0.595420\pi$
$464$	8.29180	+	5.56231i	0.384937	+	0.258224i
$465$	−3.41641			−0.158432
$466$		−	0.944272i		−	0.0437426i
$467$		−	22.4721i		−	1.03989i	−0.854201	−	0.519943i	$-0.825953\pi$
$467$		−	22.4721i		−	1.03989i	0.854201	−	0.519943i	$-0.174047\pi$
$468$	1.61803			0.0747936
$469$	−2.05573			−0.0949247
$470$	−4.40325			−0.203107
$471$	−2.00000			−0.0921551
$472$		−	20.0000i		−	0.920575i
$473$	18.9443			0.871059
$474$			9.70820i			0.445913i
$475$			19.8197i			0.909388i
$476$		−	1.14590i		−	0.0525222i
$477$	−11.2361			−0.514464
$478$			2.36068i			0.107975i
$479$			1.52786i			0.0698099i	0.999391	+	0.0349049i	$0.0111128\pi$
$479$			1.52786i			0.0698099i	−0.999391	+	0.0349049i	$0.988887\pi$
$480$	6.94427			0.316961
$481$		−	9.23607i		−	0.421128i
$482$		−	1.85410i		−	0.0844520i
$483$		−	0.763932i		−	0.0347601i
$484$	9.70820			0.441282
$485$			5.16718i			0.234630i
$486$	−0.618034			−0.0280346
$487$	−24.3607			−1.10389			−0.551944	−	0.833881i	$-0.686114\pi$
$487$	−24.3607			−1.10389			−0.551944	+	0.833881i	$0.686114\pi$
$488$	−16.1803			−0.732450
$489$	19.4164			0.878040
$490$			5.30495i			0.239653i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	−7.23607			−0.326227
$493$	9.00000	−	13.4164i	0.405340	−	0.604245i
$494$	−3.52786			−0.158726
$495$		−	2.76393i		−	0.124230i
$496$		−	5.12461i		−	0.230102i
$497$	2.18034			0.0978016
$498$	3.70820			0.166169
$499$	−1.18034			−0.0528393			−0.0264196	−	0.999651i	$-0.508411\pi$
$499$	−1.18034			−0.0528393			−0.0264196	+	0.999651i	$0.508411\pi$
$500$	16.9443			0.757771
$501$		−	19.8885i		−	0.888555i
$502$	4.79837			0.214162
$503$			2.81966i			0.125722i	0.998022	+	0.0628612i	$0.0200225\pi$
$503$			2.81966i			0.125722i	−0.998022	+	0.0628612i	$0.979977\pi$
$504$		−	0.527864i		−	0.0235129i
$505$		−	17.2361i		−	0.766995i
$506$	−4.47214			−0.198811
$507$		−	12.0000i		−	0.532939i
$508$		−	5.70820i		−	0.253261i
$509$	26.8328			1.18934			0.594672	−	0.803969i	$-0.297282\pi$
$509$	26.8328			1.18934			0.594672	+	0.803969i	$0.297282\pi$
$510$		−	2.29180i		−	0.101482i
$511$			1.34752i			0.0596110i
$512$			18.7082i			0.826794i
$513$	−5.70820			−0.252023
$514$		−	10.8328i		−	0.477815i
$515$	−9.16718			−0.403954
$516$	−13.7082			−0.603470
$517$	−12.8885			−0.566838
$518$	−1.34752			−0.0592068
$519$		−	6.00000i		−	0.263371i
$520$			2.76393i			0.121206i
$521$	−12.4721			−0.546414			−0.273207	−	0.961955i	$-0.588084\pi$
$521$	−12.4721			−0.546414			−0.273207	+	0.961955i	$0.588084\pi$
$522$	1.85410	−	2.76393i	0.0811518	−	0.120974i
$523$	−25.0689			−1.09619			−0.548093	−	0.836417i	$-0.684646\pi$
$523$	−25.0689			−1.09619			−0.548093	+	0.836417i	$0.684646\pi$
$524$			12.5623i			0.548787i
$525$		−	0.819660i		−	0.0357729i
$526$	−8.00000			−0.348817
$527$	−8.29180			−0.361196
$528$	4.14590			0.180427
$529$	−12.5279			−0.544690
$530$		−	8.58359i		−	0.372847i
$531$	8.94427			0.388148
$532$		−	2.18034i		−	0.0945297i
$533$		−	4.47214i		−	0.193710i
$534$			11.0902i			0.479919i
$535$	−6.47214			−0.279815
$536$		−	19.4721i		−	0.841068i
$537$		−	12.7639i		−	0.550804i
$538$	−14.4377			−0.622453
$539$			15.5279i			0.668832i
$540$			2.00000i			0.0860663i
$541$			20.6525i			0.887919i	0.896047	+	0.443960i	$0.146427\pi$
$541$			20.6525i			0.887919i	−0.896047	+	0.443960i	$0.853573\pi$
$542$	−2.11146			−0.0906948
$543$		−	6.41641i		−	0.275354i
$544$	16.8541			0.722614
$545$	−6.18034			−0.264737
$546$	0.145898			0.00624386
$547$	−37.6525			−1.60990			−0.804952	−	0.593340i	$-0.797809\pi$
$547$	−37.6525			−1.60990			−0.804952	+	0.593340i	$0.797809\pi$
$548$		−	20.1803i		−	0.862061i
$549$		−	7.23607i		−	0.308828i
$550$	−4.79837			−0.204603
$551$	17.1246	−	25.5279i	0.729533	−	1.08752i
$552$	7.23607			0.307988
$553$		−	3.70820i		−	0.157689i
$554$		−	12.6180i		−	0.536089i
$555$	11.4164			0.484600
$556$	10.8541			0.460316
$557$	−7.52786			−0.318966			−0.159483	−	0.987201i	$-0.550983\pi$
$557$	−7.52786			−0.318966			−0.159483	+	0.987201i	$0.550983\pi$
$558$	−1.70820			−0.0723140
$559$		−	8.47214i		−	0.358333i
$560$	−0.541020			−0.0228623
$561$		−	6.70820i		−	0.283221i
$562$			15.7082i			0.662611i
$563$		−	45.0689i		−	1.89943i	−0.313120	−	0.949713i	$-0.601374\pi$
$563$		−	45.0689i		−	1.89943i	0.313120	−	0.949713i	$-0.398626\pi$
$564$	9.32624			0.392705
$565$		−	16.6525i		−	0.700575i
$566$			13.5279i			0.568619i
$567$	0.236068			0.00991392
$568$			20.6525i			0.866559i
$569$			21.0000i			0.880366i	0.897908	+	0.440183i	$0.145086\pi$
$569$			21.0000i			0.880366i	−0.897908	+	0.440183i	$0.854914\pi$
$570$		−	4.36068i		−	0.182649i
$571$	−34.8328			−1.45771			−0.728854	−	0.684669i	$-0.759947\pi$
$571$	−34.8328			−1.45771			−0.728854	+	0.684669i	$0.759947\pi$
$572$			3.61803i			0.151278i
$573$	8.94427			0.373652
$574$	−0.652476			−0.0272338
$575$	11.2361			0.468576
$576$	−0.236068			−0.00983617
$577$			28.8328i			1.20033i	0.799878	+	0.600163i	$0.204898\pi$
$577$			28.8328i			1.20033i	−0.799878	+	0.600163i	$0.795102\pi$
$578$			4.94427i			0.205655i
$579$	9.41641			0.391333
$580$	−8.94427	−	6.00000i	−0.371391	−	0.249136i
$581$	−1.41641			−0.0587625
$582$			2.58359i			0.107093i
$583$		−	25.1246i		−	1.04056i
$584$	−12.7639			−0.528175
$585$	−1.23607			−0.0511051
$586$	7.45085			0.307792
$587$	16.9443			0.699365			0.349682	−	0.936868i	$-0.386289\pi$
$587$	16.9443			0.699365			0.349682	+	0.936868i	$0.386289\pi$
$588$		−	11.2361i		−	0.463368i
$589$	−15.7771			−0.650084
$590$			6.83282i			0.281303i
$591$			9.70820i			0.399342i
$592$			17.1246i			0.703817i
$593$	17.8197			0.731766			0.365883	−	0.930661i	$-0.380767\pi$
$593$	17.8197			0.731766			0.365883	+	0.930661i	$0.380767\pi$
$594$		−	1.38197i		−	0.0567028i
$595$			0.875388i			0.0358874i
$596$	−23.4164			−0.959173
$597$			20.1246i			0.823646i
$598$			2.00000i			0.0817861i
$599$		−	6.23607i		−	0.254799i	−0.991851	−	0.127399i	$-0.959337\pi$
$599$		−	6.23607i		−	0.254799i	0.991851	−	0.127399i	$-0.0406630\pi$
$600$	7.76393			0.316961
$601$			48.5410i			1.98003i	0.140964	+	0.990015i	$0.454980\pi$
$601$			48.5410i			1.98003i	−0.140964	+	0.990015i	$0.545020\pi$
$602$	−1.23607			−0.0503784
$603$	8.70820			0.354625
$604$	19.4164			0.790042
$605$	−7.41641			−0.301520
$606$		−	8.61803i		−	0.350084i
$607$		−	1.41641i		−	0.0574902i	−0.999587	−	0.0287451i	$-0.990849\pi$
$607$		−	1.41641i		−	0.0574902i	0.999587	−	0.0287451i	$-0.00915111\pi$
$608$	32.0689			1.30056
$609$	−0.708204	+	1.05573i	−0.0286979	+	0.0427803i
$610$	5.52786			0.223817
$611$			5.76393i			0.233184i
$612$			4.85410i			0.196215i
$613$	2.41641			0.0975978			0.0487989	−	0.998809i	$-0.484461\pi$
$613$	2.41641			0.0975978			0.0487989	+	0.998809i	$0.484461\pi$
$614$	−3.59675			−0.145153
$615$	5.52786			0.222905
$616$	1.18034			0.0475572
$617$		−	15.8885i		−	0.639649i	−0.947477	−	0.319824i	$-0.896376\pi$
$617$		−	15.8885i		−	0.639649i	0.947477	−	0.319824i	$-0.103624\pi$
$618$	−4.58359			−0.184379
$619$			34.5410i			1.38832i	0.719820	+	0.694160i	$0.244224\pi$
$619$			34.5410i			1.38832i	−0.719820	+	0.694160i	$0.755776\pi$
$620$			5.52786i			0.222004i
$621$			3.23607i			0.129859i
$622$	−9.67376			−0.387883
$623$		−	4.23607i		−	0.169714i
$624$		−	1.85410i		−	0.0742235i
$625$	4.41641			0.176656
$626$			2.14590i			0.0857673i
$627$		−	12.7639i		−	0.509742i
$628$			3.23607i			0.129133i
$629$	27.7082			1.10480
$630$			0.180340i			0.00718491i
$631$	15.2918			0.608757			0.304378	−	0.952551i	$-0.401551\pi$
$631$	15.2918			0.608757			0.304378	+	0.952551i	$0.401551\pi$
$632$	35.1246			1.39718
$633$	−10.0000			−0.397464
$634$	18.4377			0.732254
$635$			4.36068i			0.173048i
$636$			18.1803i			0.720897i
$637$	6.94427			0.275142
$638$	6.18034	+	4.14590i	0.244682	+	0.164138i
$639$	−9.23607			−0.365373
$640$		−	14.0689i		−	0.556121i
$641$		−	10.5279i		−	0.415826i	−0.978147	−	0.207913i	$-0.933333\pi$
$641$		−	10.5279i		−	0.415826i	0.978147	−	0.207913i	$-0.0666670\pi$
$642$	−3.23607			−0.127717
$643$	−12.7082			−0.501163			−0.250581	−	0.968096i	$-0.580622\pi$
$643$	−12.7082			−0.501163			−0.250581	+	0.968096i	$0.580622\pi$
$644$	−1.23607			−0.0487079
$645$	10.4721			0.412340
$646$		−	10.5836i		−	0.416406i
$647$	−31.5967			−1.24220			−0.621098	−	0.783733i	$-0.713313\pi$
$647$	−31.5967			−1.24220			−0.621098	+	0.783733i	$0.713313\pi$
$648$			2.23607i			0.0878410i
$649$			20.0000i			0.785069i
$650$			2.14590i			0.0841690i
$651$	0.652476			0.0255725
$652$		−	31.4164i		−	1.23036i
$653$			30.3050i			1.18592i	0.805230	+	0.592962i	$0.202042\pi$
$653$			30.3050i			1.18592i	−0.805230	+	0.592962i	$0.797958\pi$
$654$	−3.09017			−0.120835
$655$		−	9.59675i		−	0.374976i
$656$			8.29180i			0.323740i
$657$		−	5.70820i		−	0.222698i
$658$	0.840946			0.0327835
$659$		−	21.7639i		−	0.847802i	−0.905708	−	0.423901i	$-0.860660\pi$
$659$		−	21.7639i		−	0.847802i	0.905708	−	0.423901i	$-0.139340\pi$
$660$	−4.47214			−0.174078
$661$	23.8328			0.926989			0.463495	−	0.886100i	$-0.346595\pi$
$661$	23.8328			0.926989			0.463495	+	0.886100i	$0.346595\pi$
$662$	13.4164			0.521443
$663$	−3.00000			−0.116510
$664$		−	13.4164i		−	0.520658i
$665$			1.66563i			0.0645904i
$666$	5.70820			0.221188
$667$	−14.4721	−	9.70820i	−0.560363	−	0.375903i
$668$	−32.1803			−1.24509
$669$			0.708204i			0.0273807i
$670$			6.65248i			0.257008i
$671$	16.1803			0.624635
$672$	−1.32624			−0.0511607
$673$	22.4164			0.864089			0.432045	−	0.901852i	$-0.357792\pi$
$673$	22.4164			0.864089			0.432045	+	0.901852i	$0.357792\pi$
$674$	−9.12461			−0.351467
$675$			3.47214i			0.133643i
$676$	−19.4164			−0.746785
$677$			35.9443i			1.38145i	0.723117	+	0.690725i	$0.242709\pi$
$677$			35.9443i			1.38145i	−0.723117	+	0.690725i	$0.757291\pi$
$678$		−	8.32624i		−	0.319767i
$679$		−	0.986844i		−	0.0378716i
$680$	−8.29180			−0.317976
$681$			26.9443i			1.03251i
$682$		−	3.81966i		−	0.146262i
$683$	15.0557			0.576091			0.288046	−	0.957617i	$-0.406994\pi$
$683$	15.0557			0.576091			0.288046	+	0.957617i	$0.406994\pi$
$684$			9.23607i			0.353150i
$685$			15.4164i			0.589031i
$686$		−	2.03444i		−	0.0776754i
$687$	0.180340			0.00688040
$688$			15.7082i			0.598870i
$689$	−11.2361			−0.428060
$690$	−2.47214			−0.0941126
$691$	−25.7639			−0.980106			−0.490053	−	0.871693i	$-0.663023\pi$
$691$	−25.7639			−0.980106			−0.490053	+	0.871693i	$0.663023\pi$
$692$	−9.70820			−0.369051
$693$			0.527864i			0.0200519i
$694$			5.34752i			0.202989i
$695$	−8.29180			−0.314526
$696$	−10.0000	−	6.70820i	−0.379049	−	0.254274i
$697$	13.4164			0.508183
$698$			6.18034i			0.233929i
$699$		−	1.52786i		−	0.0577891i
$700$	−1.32624			−0.0501271
$701$	−14.1803			−0.535584			−0.267792	−	0.963477i	$-0.586294\pi$
$701$	−14.1803			−0.535584			−0.267792	+	0.963477i	$0.586294\pi$
$702$	−0.618034			−0.0233262
$703$	52.7214			1.98842
$704$		−	0.527864i		−	0.0198946i
$705$	−7.12461			−0.268328
$706$		−	16.4721i		−	0.619937i
$707$			3.29180i			0.123801i
$708$		−	14.4721i		−	0.543896i
$709$	13.4164			0.503864			0.251932	−	0.967745i	$-0.418934\pi$
$709$	13.4164			0.503864			0.251932	+	0.967745i	$0.418934\pi$
$710$		−	7.05573i		−	0.264797i
$711$			15.7082i			0.589104i
$712$	40.1246			1.50373
$713$			8.94427i			0.334966i
$714$			0.437694i			0.0163803i
$715$		−	2.76393i		−	0.103365i
$716$	−20.6525			−0.771819
$717$			3.81966i			0.142648i
$718$	−23.0557			−0.860432
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$	2.29180			0.0854102
$721$	1.75078			0.0652023
$722$		−	8.39512i		−	0.312434i
$723$		−	3.00000i		−	0.111571i
$724$	−10.3820			−0.385843
$725$	−15.5279	−	10.4164i	−0.576690	−	0.386856i
$726$	−3.70820			−0.137624
$727$			28.1803i			1.04515i	0.852593	+	0.522575i	$0.175029\pi$
$727$			28.1803i			1.04515i	−0.852593	+	0.522575i	$0.824971\pi$
$728$		−	0.527864i		−	0.0195639i
$729$	−1.00000			−0.0370370
$730$	4.36068			0.161396
$731$	25.4164			0.940060
$732$	−11.7082			−0.432748
$733$		−	0.472136i		−	0.0174387i	−0.999962	−	0.00871937i	$-0.997225\pi$
$733$		−	0.472136i		−	0.0174387i	0.999962	−	0.00871937i	$-0.00277550\pi$
$734$	11.1246			0.410617
$735$			8.58359i			0.316611i
$736$		−	18.1803i		−	0.670136i
$737$			19.4721i			0.717265i
$738$	2.76393			0.101742
$739$		−	10.1803i		−	0.374490i	−0.982313	−	0.187245i	$-0.940044\pi$
$739$		−	10.1803i		−	0.374490i	0.982313	−	0.187245i	$-0.0599558\pi$
$740$		−	18.4721i		−	0.679049i
$741$	−5.70820			−0.209696
$742$			1.63932i			0.0601813i
$743$		−	15.0689i		−	0.552824i	−0.961039	−	0.276412i	$-0.910855\pi$
$743$		−	15.0689i		−	0.552824i	0.961039	−	0.276412i	$-0.0891454\pi$
$744$			6.18034i			0.226582i
$745$	17.8885			0.655386
$746$			19.7082i			0.721569i
$747$	6.00000			0.219529
$748$	−10.8541			−0.396865
$749$	1.23607			0.0451649
$750$	−6.47214			−0.236329
$751$		−	26.8328i		−	0.979143i	−0.871963	−	0.489572i	$-0.837153\pi$
$751$		−	26.8328i		−	0.979143i	0.871963	−	0.489572i	$-0.162847\pi$
$752$		−	10.6869i		−	0.389712i
$753$	7.76393			0.282933
$754$	1.85410	−	2.76393i	0.0675224	−	0.100656i
$755$	−14.8328			−0.539821
$756$		−	0.381966i		−	0.0138920i
$757$		−	38.0000i		−	1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$		−	38.0000i		−	1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	−17.1246			−0.621994
$759$	−7.23607			−0.262653
$760$	−15.7771			−0.572295
$761$	−28.6525			−1.03865			−0.519326	−	0.854576i	$-0.673817\pi$
$761$	−28.6525			−1.03865			−0.519326	+	0.854576i	$0.673817\pi$
$762$			2.18034i			0.0789854i
$763$	1.18034			0.0427312
$764$		−	14.4721i		−	0.523584i
$765$		−	3.70820i		−	0.134070i
$766$		−	22.6525i		−	0.818467i
$767$	8.94427			0.322959
$768$		−	6.56231i		−	0.236797i
$769$		−	17.8197i		−	0.642593i	−0.946979	−	0.321297i	$-0.895881\pi$
$769$		−	17.8197i		−	0.642593i	0.946979	−	0.321297i	$-0.104119\pi$
$770$	−0.403252			−0.0145322
$771$		−	17.5279i		−	0.631251i
$772$		−	15.2361i		−	0.548358i
$773$		−	3.88854i		−	0.139861i	−0.997552	−	0.0699306i	$-0.977722\pi$
$773$		−	3.88854i		−	0.139861i	0.997552	−	0.0699306i	$-0.0222778\pi$
$774$	5.23607			0.188207
$775$			9.59675i			0.344725i
$776$	9.34752			0.335557
$777$	−2.18034			−0.0782193
$778$	2.79837			0.100327
$779$	25.5279			0.914631
$780$			2.00000i			0.0716115i
$781$		−	20.6525i		−	0.739004i
$782$	−6.00000			−0.214560
$783$	3.00000	−	4.47214i	0.107211	−	0.159821i
$784$	−12.8754			−0.459835
$785$		−	2.47214i		−	0.0882343i
$786$		−	4.79837i		−	0.171152i
$787$	−12.0000			−0.427754			−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000			−0.427754			−0.213877	+	0.976861i	$0.568609\pi$
$788$	15.7082			0.559582
$789$	−12.9443			−0.460828
$790$	−12.0000			−0.426941
$791$			3.18034i			0.113080i
$792$	−5.00000			−0.177667
$793$		−	7.23607i		−	0.256960i
$794$		−	6.76393i		−	0.240043i
$795$		−	13.8885i		−	0.492576i
$796$	32.5623			1.15414
$797$			8.83282i			0.312874i	0.987688	+	0.156437i	$0.0500009\pi$
$797$			8.83282i			0.312874i	−0.987688	+	0.156437i	$0.949999\pi$
$798$			0.832816i			0.0294814i
$799$	−17.2918			−0.611740
$800$		−	19.5066i		−	0.689662i
$801$			17.9443i			0.634030i
$802$			12.5410i			0.442839i
$803$	12.7639			0.450429
$804$		−	14.0902i		−	0.496922i
$805$	0.944272			0.0332812
$806$	−1.70820			−0.0601689
$807$	−23.3607			−0.822335
$808$	−31.1803			−1.09692
$809$		−	1.36068i		−	0.0478390i	−0.999714	−	0.0239195i	$-0.992385\pi$
$809$		−	1.36068i		−	0.0478390i	0.999714	−	0.0239195i	$-0.00761453\pi$
$810$		−	0.763932i		−	0.0268418i
$811$	−2.59675			−0.0911841			−0.0455921	−	0.998960i	$-0.514517\pi$
$811$	−2.59675			−0.0911841			−0.0455921	+	0.998960i	$0.514517\pi$
$812$	1.70820	+	1.14590i	0.0599462	+	0.0402131i
$813$	−3.41641			−0.119819
$814$			12.7639i			0.447376i
$815$			24.0000i			0.840683i
$816$	5.56231			0.194720
$817$	48.3607			1.69193
$818$	23.1246			0.808533
$819$	0.236068			0.00824888
$820$		−	8.94427i		−	0.312348i
$821$	−3.52786			−0.123123			−0.0615617	−	0.998103i	$-0.519608\pi$
$821$	−3.52786			−0.123123			−0.0615617	+	0.998103i	$0.519608\pi$
$822$			7.70820i			0.268854i
$823$		−	18.3607i		−	0.640013i	−0.947415	−	0.320007i	$-0.896315\pi$
$823$		−	18.3607i		−	0.640013i	0.947415	−	0.320007i	$-0.103685\pi$
$824$			16.5836i			0.577717i
$825$	−7.76393			−0.270305
$826$		−	1.30495i		−	0.0454051i
$827$		−	5.88854i		−	0.204765i	−0.994745	−	0.102382i	$-0.967353\pi$
$827$		−	5.88854i		−	0.204765i	0.994745	−	0.102382i	$-0.0326465\pi$
$828$	5.23607			0.181966
$829$		−	35.3050i		−	1.22619i	−0.790009	−	0.613096i	$-0.789924\pi$
$829$		−	35.3050i		−	1.22619i	0.790009	−	0.613096i	$-0.210076\pi$
$830$			4.58359i			0.159099i
$831$		−	20.4164i		−	0.708237i
$832$	−0.236068			−0.00818418
$833$			20.8328i			0.721814i
$834$	−4.14590			−0.143561
$835$	24.5836			0.850750
$836$	−20.6525			−0.714281
$837$	−2.76393			−0.0955355
$838$			16.1803i			0.558941i
$839$			12.7082i			0.438736i	0.975642	+	0.219368i	$0.0703995\pi$
$839$			12.7082i			0.438736i	−0.975642	+	0.219368i	$0.929600\pi$
$840$	0.652476			0.0225126
$841$	11.0000	+	26.8328i	0.379310	+	0.925270i
$842$	−12.3607			−0.425977
$843$			25.4164i			0.875388i
$844$			16.1803i			0.556950i
$845$	14.8328			0.510264
$846$	−3.56231			−0.122475
$847$	1.41641			0.0486684
$848$	20.8328			0.715402
$849$			21.8885i			0.751213i
$850$	−6.43769			−0.220811
$851$		−	29.8885i		−	1.02457i
$852$			14.9443i			0.511982i
$853$			53.1935i			1.82131i	0.413167	+	0.910655i	$0.364423\pi$
$853$			53.1935i			1.82131i	−0.413167	+	0.910655i	$0.635577\pi$
$854$	−1.05573			−0.0361263
$855$		−	7.05573i		−	0.241301i
$856$			11.7082i			0.400178i
$857$	−11.5967			−0.396137			−0.198069	−	0.980188i	$-0.563467\pi$
$857$	−11.5967			−0.396137			−0.198069	+	0.980188i	$0.563467\pi$
$858$		−	1.38197i		−	0.0471795i
$859$			8.36068i			0.285263i	0.989776	+	0.142631i	$0.0455563\pi$
$859$			8.36068i			0.285263i	−0.989776	+	0.142631i	$0.954444\pi$
$860$		−	16.9443i		−	0.577795i
$861$	−1.05573			−0.0359791
$862$		−	18.7639i		−	0.639102i
$863$	−15.3475			−0.522436			−0.261218	−	0.965280i	$-0.584124\pi$
$863$	−15.3475			−0.522436			−0.261218	+	0.965280i	$0.584124\pi$
$864$	5.61803			0.191129
$865$	7.41641			0.252165
$866$	17.1246			0.581918
$867$			8.00000i			0.271694i
$868$		−	1.05573i		−	0.0358337i
$869$	−35.1246			−1.19152
$870$	3.41641	+	2.29180i	0.115827	+	0.0776992i
$871$	8.70820			0.295066
$872$			11.1803i			0.378614i
$873$			4.18034i			0.141483i
$874$	−11.4164			−0.386166
$875$	2.47214			0.0835734
$876$	−9.23607			−0.312058
$877$	23.5279			0.794480			0.397240	−	0.917715i	$-0.369968\pi$
$877$	23.5279			0.794480			0.397240	+	0.917715i	$0.369968\pi$
$878$			16.5066i			0.557070i
$879$	12.0557			0.406630
$880$			5.12461i			0.172751i
$881$			25.2492i			0.850668i	0.905036	+	0.425334i	$0.139843\pi$
$881$			25.2492i			0.850668i	−0.905036	+	0.425334i	$0.860157\pi$
$882$			4.29180i			0.144512i
$883$	−46.2492			−1.55641			−0.778205	−	0.628010i	$-0.783870\pi$
$883$	−46.2492			−1.55641			−0.778205	+	0.628010i	$0.783870\pi$
$884$			4.85410i			0.163261i
$885$			11.0557i			0.371634i
$886$	13.3820			0.449576
$887$			41.0689i			1.37896i	0.724306	+	0.689479i	$0.242160\pi$
$887$			41.0689i			1.37896i	−0.724306	+	0.689479i	$0.757840\pi$
$888$		−	20.6525i		−	0.693052i
$889$		−	0.832816i		−	0.0279317i
$890$	−13.7082			−0.459500
$891$		−	2.23607i		−	0.0749111i
$892$	1.14590			0.0383675
$893$	−32.9017			−1.10101
$894$	8.94427			0.299141
$895$	15.7771			0.527370
$896$			2.68692i			0.0897636i
$897$			3.23607i			0.108049i
$898$	10.4377			0.348310
$899$	8.29180	−	12.3607i	0.276547	−	0.412252i
$900$	5.61803			0.187268
$901$		−	33.7082i		−	1.12298i
$902$			6.18034i			0.205783i
$903$	−2.00000			−0.0665558
$904$	−30.1246			−1.00193
$905$	7.93112			0.263639
$906$	−7.41641			−0.246394
$907$			17.5279i			0.582003i	0.956723	+	0.291002i	$0.0939885\pi$
$907$			17.5279i			0.582003i	−0.956723	+	0.291002i	$0.906012\pi$
$908$	43.5967			1.44681
$909$		−	13.9443i		−	0.462502i
$910$			0.180340i			0.00597821i
$911$		−	28.8197i		−	0.954838i	−0.878676	−	0.477419i	$-0.841572\pi$
$911$		−	28.8197i		−	0.954838i	0.878676	−	0.477419i	$-0.158428\pi$
$912$	10.5836			0.350458
$913$			13.4164i			0.444018i
$914$		−	12.6180i		−	0.417367i
$915$	8.94427			0.295689
$916$		−	0.291796i		−	0.00964121i
$917$			1.83282i			0.0605249i
$918$		−	1.85410i		−	0.0611945i
$919$	5.65248			0.186458			0.0932290	−	0.995645i	$-0.470281\pi$
$919$	5.65248			0.186458			0.0932290	+	0.995645i	$0.470281\pi$
$920$			8.94427i			0.294884i
$921$	−5.81966			−0.191764
$922$	0.652476			0.0214881
$923$	−9.23607			−0.304009
$924$	0.854102			0.0280979
$925$		−	32.0689i		−	1.05442i
$926$		−	7.85410i		−	0.258102i
$927$	−7.41641			−0.243587
$928$	−16.8541	+	25.1246i	−0.553263	+	0.824756i
$929$	12.3607			0.405541			0.202770	−	0.979226i	$-0.435006\pi$
$929$	12.3607			0.405541			0.202770	+	0.979226i	$0.435006\pi$
$930$		−	2.11146i		−	0.0692374i
$931$			39.6393i			1.29913i
$932$	−2.47214			−0.0809775
$933$	−15.6525			−0.512439
$934$	13.8885			0.454447
$935$	8.29180			0.271171
$936$			2.23607i			0.0730882i
$937$	33.0000			1.07806			0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000			1.07806			0.539032	+	0.842286i	$0.318790\pi$
$938$		−	1.27051i		−	0.0414836i
$939$			3.47214i			0.113309i
$940$			11.5279i			0.375997i
$941$	8.83282			0.287942			0.143971	−	0.989582i	$-0.454013\pi$
$941$	8.83282			0.287942			0.143971	+	0.989582i	$0.454013\pi$
$942$		−	1.23607i		−	0.0402733i
$943$		−	14.4721i		−	0.471278i
$944$	−16.5836			−0.539750
$945$			0.291796i			0.00949213i
$946$			11.7082i			0.380667i
$947$		−	39.1803i		−	1.27319i	−0.771198	−	0.636595i	$-0.780342\pi$
$947$		−	39.1803i		−	1.27319i	0.771198	−	0.636595i	$-0.219658\pi$
$948$	25.4164			0.825487
$949$		−	5.70820i		−	0.185296i
$950$	−12.2492			−0.397417
$951$	29.8328			0.967395
$952$	1.58359			0.0513245
$953$	47.6656			1.54404			0.772021	−	0.635597i	$-0.219246\pi$
$953$	47.6656			1.54404			0.772021	+	0.635597i	$0.219246\pi$
$954$		−	6.94427i		−	0.224829i
$955$			11.0557i			0.357755i
$956$	6.18034			0.199886
$957$	10.0000	+	6.70820i	0.323254	+	0.216845i
$958$	−0.944272			−0.0305080
$959$		−	2.94427i		−	0.0950755i
$960$		−	0.291796i		−	0.00941768i
$961$	23.3607			0.753570
$962$	5.70820			0.184040
$963$	−5.23607			−0.168730
$964$	−4.85410			−0.156340
$965$			11.6393i			0.374683i
$966$	0.472136			0.0151907
$967$		−	13.1246i		−	0.422059i	−0.977480	−	0.211030i	$-0.932318\pi$
$967$		−	13.1246i		−	0.422059i	0.977480	−	0.211030i	$-0.0676816\pi$
$968$			13.4164i			0.431220i
$969$		−	17.1246i		−	0.550122i
$970$	−3.19350			−0.102537
$971$			43.4164i			1.39330i	0.717412	+	0.696649i	$0.245327\pi$
$971$			43.4164i			1.39330i	−0.717412	+	0.696649i	$0.754673\pi$
$972$			1.61803i			0.0518985i
$973$	1.58359			0.0507676
$974$		−	15.0557i		−	0.482417i
$975$			3.47214i			0.111197i
$976$			13.4164i			0.429449i
$977$	−25.0132			−0.800242			−0.400121	−	0.916462i	$-0.631032\pi$
$977$	−25.0132			−0.800242			−0.400121	+	0.916462i	$0.631032\pi$
$978$			12.0000i			0.383718i
$979$	−40.1246			−1.28239
$980$	13.8885			0.443653
$981$	−5.00000			−0.159638
$982$		0			0
$983$			21.8885i			0.698136i	0.937097	+	0.349068i	$0.113502\pi$
$983$			21.8885i			0.698136i	−0.937097	+	0.349068i	$0.886498\pi$
$984$		−	10.0000i		−	0.318788i
$985$	−12.0000			−0.382352
$986$	8.29180	+	5.56231i	0.264065	+	0.177140i
$987$	1.36068			0.0433109
$988$			9.23607i			0.293838i
$989$		−	27.4164i		−	0.871791i
$990$	1.70820			0.0542903
$991$	−34.7082			−1.10254			−0.551271	−	0.834326i	$-0.685857\pi$
$991$	−34.7082			−1.10254			−0.551271	+	0.834326i	$0.685857\pi$
$992$	15.5279			0.493010
$993$	21.7082			0.688889
$994$			1.34752i			0.0427409i
$995$	−24.8754			−0.788603
$996$		−	9.70820i		−	0.307616i
$997$		−	51.4164i		−	1.62837i	−0.580603	−	0.814187i	$-0.697183\pi$
$997$		−	51.4164i		−	1.62837i	0.580603	−	0.814187i	$-0.302817\pi$
$998$		−	0.729490i		−	0.0230916i
$999$	9.23607			0.292216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.2.c.a.28.3	yes	4
3.2	odd	2		261.2.c.b.28.2		4
4.3	odd	2		1392.2.o.i.289.1		4
5.2	odd	4		2175.2.f.b.724.2		4
5.3	odd	4		2175.2.f.a.724.3		4
5.4	even	2		2175.2.d.e.376.2		4
12.11	even	2		4176.2.o.l.289.3		4
29.12	odd	4		2523.2.a.e.1.1		2
29.17	odd	4		2523.2.a.d.1.2		2
29.28	even	2	inner	87.2.c.a.28.2	✓	4
87.17	even	4		7569.2.a.n.1.1		2
87.41	even	4		7569.2.a.f.1.2		2
87.86	odd	2		261.2.c.b.28.3		4
116.115	odd	2		1392.2.o.i.289.3		4
145.28	odd	4		2175.2.f.b.724.1		4
145.57	odd	4		2175.2.f.a.724.4		4
145.144	even	2		2175.2.d.e.376.3		4
348.347	even	2		4176.2.o.l.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.c.a.28.2	✓	4	29.28	even	2	inner
87.2.c.a.28.3	yes	4	1.1	even	1	trivial
261.2.c.b.28.2		4	3.2	odd	2
261.2.c.b.28.3		4	87.86	odd	2
1392.2.o.i.289.1		4	4.3	odd	2
1392.2.o.i.289.3		4	116.115	odd	2
2175.2.d.e.376.2		4	5.4	even	2
2175.2.d.e.376.3		4	145.144	even	2
2175.2.f.a.724.3		4	5.3	odd	4
2175.2.f.a.724.4		4	145.57	odd	4
2175.2.f.b.724.1		4	145.28	odd	4
2175.2.f.b.724.2		4	5.2	odd	4
2523.2.a.d.1.2		2	29.17	odd	4
2523.2.a.e.1.1		2	29.12	odd	4
4176.2.o.l.289.3		4	12.11	even	2
4176.2.o.l.289.4		4	348.347	even	2
7569.2.a.f.1.2		2	87.41	even	4
7569.2.a.n.1.1		2	87.17	even	4

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:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	87.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} -1.23607 q^{5} -0.618034 q^{6} +0.236068 q^{7} +2.23607i q^{8} -1.00000 q^{9} -0.763932i q^{10} -2.23607i q^{11} +1.61803i q^{12} -1.00000 q^{13} +0.145898i q^{14} -1.23607i q^{15} +1.85410 q^{16} -3.00000i q^{17} -0.618034i q^{18} -5.70820i q^{19} -2.00000 q^{20} +0.236068i q^{21} +1.38197 q^{22} -3.23607 q^{23} -2.23607 q^{24} -3.47214 q^{25} -0.618034i q^{26} -1.00000i q^{27} +0.381966 q^{28} +(4.47214 + 3.00000i) q^{29} +0.763932 q^{30} -2.76393i q^{31} +5.61803i q^{32} +2.23607 q^{33} +1.85410 q^{34} -0.291796 q^{35} -1.61803 q^{36} +9.23607i q^{37} +3.52786 q^{38} -1.00000i q^{39} -2.76393i q^{40} +4.47214i q^{41} -0.145898 q^{42} +8.47214i q^{43} -3.61803i q^{44} +1.23607 q^{45} -2.00000i q^{46} -5.76393i q^{47} +1.85410i q^{48} -6.94427 q^{49} -2.14590i q^{50} +3.00000 q^{51} -1.61803 q^{52} +11.2361 q^{53} +0.618034 q^{54} +2.76393i q^{55} +0.527864i q^{56} +5.70820 q^{57} +(-1.85410 + 2.76393i) q^{58} -8.94427 q^{59} -2.00000i q^{60} +7.23607i q^{61} +1.70820 q^{62} -0.236068 q^{63} +0.236068 q^{64} +1.23607 q^{65} +1.38197i q^{66} -8.70820 q^{67} -4.85410i q^{68} -3.23607i q^{69} -0.180340i q^{70} +9.23607 q^{71} -2.23607i q^{72} +5.70820i q^{73} -5.70820 q^{74} -3.47214i q^{75} -9.23607i q^{76} -0.527864i q^{77} +0.618034 q^{78} -15.7082i q^{79} -2.29180 q^{80} +1.00000 q^{81} -2.76393 q^{82} -6.00000 q^{83} +0.381966i q^{84} +3.70820i q^{85} -5.23607 q^{86} +(-3.00000 + 4.47214i) q^{87} +5.00000 q^{88} -17.9443i q^{89} +0.763932i q^{90} -0.236068 q^{91} -5.23607 q^{92} +2.76393 q^{93} +3.56231 q^{94} +7.05573i q^{95} -5.61803 q^{96} -4.18034i q^{97} -4.29180i q^{98} +2.23607i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 4 q^{5} + 2 q^{6} - 8 q^{7} - 4 q^{9} - 4 q^{13} - 6 q^{16} - 8 q^{20} + 10 q^{22} - 4 q^{23} + 4 q^{25} + 6 q^{28} + 12 q^{30} - 6 q^{34} - 28 q^{35} - 2 q^{36} + 32 q^{38} - 14 q^{42}+ \cdots - 18 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 + 4 * q^5 + 2 * q^6 - 8 * q^7 - 4 * q^9 - 4 * q^13 - 6 * q^16 - 8 * q^20 + 10 * q^22 - 4 * q^23 + 4 * q^25 + 6 * q^28 + 12 * q^30 - 6 * q^34 - 28 * q^35 - 2 * q^36 + 32 * q^38 - 14 * q^42 - 4 * q^45 + 8 * q^49 + 12 * q^51 - 2 * q^52 + 36 * q^53 - 2 * q^54 - 4 * q^57 + 6 * q^58 - 20 * q^62 + 8 * q^63 - 8 * q^64 - 4 * q^65 - 8 * q^67 + 28 * q^71 + 4 * q^74 - 2 * q^78 - 36 * q^80 + 4 * q^81 - 20 * q^82 - 24 * q^83 - 12 * q^86 - 12 * q^87 + 20 * q^88 + 8 * q^91 - 12 * q^92 + 20 * q^93 - 26 * q^94 - 18 * q^96

Introduction

L-functions

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