LMFDB - Embedded newform 85.2.b.a.69.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [85,2,Mod(69,85)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("85.69");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	85.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:	$0.678728417181$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.619810816.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			69.8
Root			$1.18254 - 1.18254i$ of defining polynomial
Character	$\chi$	$=$	85.69
Dual form			85.2.b.a.69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.31627i q^{2} -0.203185i q^{3} -3.36509 q^{4} +(-1.55654 + 1.60536i) q^{5} +0.470630 q^{6} +0.683735i q^{7} -3.16190i q^{8} +2.95872 q^{9} +(-3.71844 - 3.60536i) q^{10} +3.68135 q^{11} +0.683735i q^{12} -4.43927i q^{13} -1.58371 q^{14} +(0.326185 + 0.316265i) q^{15} +0.593630 q^{16} -1.00000i q^{17} +6.85317i q^{18} +1.03890 q^{19} +(5.23789 - 5.40218i) q^{20} +0.138925 q^{21} +8.52699i q^{22} +4.52699i q^{23} -0.642450 q^{24} +(-0.154365 - 4.99762i) q^{25} +10.2825 q^{26} -1.21072i q^{27} -2.30083i q^{28} +3.69127 q^{29} +(-0.732555 + 0.755531i) q^{30} -10.8921 q^{31} -4.94880i q^{32} -0.747995i q^{33} +2.31627 q^{34} +(-1.09764 - 1.06426i) q^{35} -9.95633 q^{36} +0.308729i q^{37} +2.40637i q^{38} -0.901992 q^{39} +(5.07599 + 4.92163i) q^{40} -6.15198 q^{41} +0.321786i q^{42} -7.88454i q^{43} -12.3881 q^{44} +(-4.60536 + 4.74981i) q^{45} -10.4857 q^{46} +4.43927i q^{47} -0.120617i q^{48} +6.53251 q^{49} +(11.5758 - 0.357550i) q^{50} -0.203185 q^{51} +14.9385i q^{52} -11.4603i q^{53} +2.80435 q^{54} +(-5.73017 + 5.90990i) q^{55} +2.16190 q^{56} -0.211089i q^{57} +8.54996i q^{58} +2.00000 q^{59} +(-1.09764 - 1.06426i) q^{60} +9.94089 q^{61} -25.2289i q^{62} +2.02298i q^{63} +12.6500 q^{64} +(7.12662 + 6.90990i) q^{65} +1.73255 q^{66} +9.16944i q^{67} +3.36509i q^{68} +0.919815 q^{69} +(2.46511 - 2.54243i) q^{70} -9.37262 q^{71} -9.35517i q^{72} +2.26946i q^{73} -0.715099 q^{74} +(-1.01544 + 0.0313646i) q^{75} -3.49599 q^{76} +2.51707i q^{77} -2.08925i q^{78} -7.42696 q^{79} +(-0.924009 + 0.952991i) q^{80} +8.63015 q^{81} -14.2496i q^{82} +8.92344i q^{83} -0.467493 q^{84} +(1.60536 + 1.55654i) q^{85} +18.2627 q^{86} -0.750010i q^{87} -11.6401i q^{88} -11.5523 q^{89} +(-11.0018 - 10.6672i) q^{90} +3.03528 q^{91} -15.2337i q^{92} +2.21310i q^{93} -10.2825 q^{94} +(-1.61709 + 1.66781i) q^{95} -1.00552 q^{96} -12.5500i q^{97} +15.1310i q^{98} +10.8921 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 2 q^{5} - 8 q^{9} + 6 q^{10} - 4 q^{11} + 12 q^{14} - 8 q^{16} - 8 q^{19} - 2 q^{20} + 24 q^{21} + 12 q^{24} - 12 q^{25} + 8 q^{29} - 16 q^{30} - 24 q^{31} + 4 q^{34} + 44 q^{39} + 22 q^{40}+ \cdots + 24 q^{99}+O(q^{100}) $ 8 * q - 8 * q^4 - 2 * q^5 - 8 * q^9 + 6 * q^10 - 4 * q^11 + 12 * q^14 - 8 * q^16 - 8 * q^19 - 2 * q^20 + 24 * q^21 + 12 * q^24 - 12 * q^25 + 8 * q^29 - 16 * q^30 - 24 * q^31 + 4 * q^34 + 44 * q^39 + 22 * q^40 - 12 * q^41 - 32 * q^44 - 22 * q^45 - 8 * q^46 - 16 * q^49 + 44 * q^50 - 8 * q^51 - 20 * q^54 - 8 * q^55 - 8 * q^56 + 16 * q^59 + 12 * q^61 + 48 * q^64 + 20 * q^65 + 24 * q^66 - 8 * q^69 + 40 * q^70 - 20 * q^71 - 40 * q^74 + 4 * q^75 - 24 * q^76 + 24 * q^79 - 26 * q^80 - 8 * q^81 - 72 * q^84 - 2 * q^85 + 40 * q^86 - 48 * q^89 - 74 * q^90 + 36 * q^91 + 4 * q^95 + 16 * q^96 + 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/85\mathbb{Z}\right)^\times$.

$n$	$52$	$71$
$\chi(n)$	$-1$	$1$

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 $

$2$			2.31627i			1.63785i	0.573903	+	0.818924i	$0.305429\pi$
$2$			2.31627i			1.63785i	−0.573903	+	0.818924i	$0.694571\pi$
$3$		−	0.203185i		−	0.117309i	−0.998278	−	0.0586544i	$-0.981319\pi$
$3$		−	0.203185i		−	0.117309i	0.998278	−	0.0586544i	$-0.0186810\pi$
$4$	−3.36509			−1.68254
$5$	−1.55654	+	1.60536i	−0.696106	+	0.717939i
$5$	−1.55654	+	1.60536i	−0.696106	+	0.717939i
$6$	0.470630			0.192134
$7$			0.683735i			0.258427i	0.991617	+	0.129214i	$0.0412453\pi$
$7$			0.683735i			0.258427i	−0.991617	+	0.129214i	$0.958755\pi$
$8$		−	3.16190i		−	1.11790i
$9$	2.95872			0.986239
$10$	−3.71844	−	3.60536i	−1.17587	−	1.14012i
$11$	3.68135			1.10997			0.554985	−	0.831861i	$-0.312724\pi$
$11$	3.68135			1.10997			0.554985	+	0.831861i	$0.312724\pi$
$12$			0.683735i			0.197377i
$13$		−	4.43927i		−	1.23123i	−0.788047	−	0.615615i	$-0.788908\pi$
$13$		−	4.43927i		−	1.23123i	0.788047	−	0.615615i	$-0.211092\pi$
$14$	−1.58371			−0.423264
$15$	0.326185	+	0.316265i	0.0842206	+	0.0816594i
$16$	0.593630			0.148408
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$			6.85317i			1.61531i
$19$	1.03890			0.238340			0.119170	−	0.992874i	$-0.461977\pi$
$19$	1.03890			0.238340			0.119170	+	0.992874i	$0.461977\pi$
$20$	5.23789	−	5.40218i	1.17123	−	1.20796i
$21$	0.138925			0.0303158
$22$			8.52699i			1.81796i
$23$			4.52699i			0.943942i	0.881614	+	0.471971i	$0.156457\pi$
$23$			4.52699i			0.943942i	−0.881614	+	0.471971i	$0.843543\pi$
$24$	−0.642450			−0.131140
$25$	−0.154365	−	4.99762i	−0.0308729	−	0.999523i
$26$	10.2825			2.01657
$27$		−	1.21072i		−	0.233003i
$28$		−	2.30083i		−	0.434815i
$29$	3.69127			0.685452			0.342726	−	0.939435i	$-0.388650\pi$
$29$	3.69127			0.685452			0.342726	+	0.939435i	$0.388650\pi$
$30$	−0.732555	+	0.755531i	−0.133746	+	0.137940i
$31$	−10.8921			−1.95627			−0.978137	−	0.207962i	$-0.933317\pi$
$31$	−10.8921			−1.95627			−0.978137	+	0.207962i	$0.933317\pi$
$32$		−	4.94880i		−	0.874832i
$33$		−	0.747995i		−	0.130209i
$34$	2.31627			0.397236
$35$	−1.09764	−	1.06426i	−0.185535	−	0.179893i
$36$	−9.95633			−1.65939
$37$			0.308729i			0.0507548i	0.999678	+	0.0253774i	$0.00807874\pi$
$37$			0.308729i			0.0507548i	−0.999678	+	0.0253774i	$0.991921\pi$
$38$			2.40637i			0.390365i
$39$	−0.901992			−0.144434
$40$	5.07599	+	4.92163i	0.802585	+	0.778177i
$41$	−6.15198			−0.960778			−0.480389	−	0.877056i	$-0.659505\pi$
$41$	−6.15198			−0.960778			−0.480389	+	0.877056i	$0.659505\pi$
$42$			0.321786i			0.0496527i
$43$		−	7.88454i		−	1.20238i	−0.799106	−	0.601190i	$-0.794693\pi$
$43$		−	7.88454i		−	1.20238i	0.799106	−	0.601190i	$-0.205307\pi$
$44$	−12.3881			−1.86757
$45$	−4.60536	+	4.74981i	−0.686527	+	0.708059i
$46$	−10.4857			−1.54603
$47$			4.43927i			0.647533i	0.946137	+	0.323767i	$0.104949\pi$
$47$			4.43927i			0.647533i	−0.946137	+	0.323767i	$0.895051\pi$
$48$		−	0.120617i		−	0.0174095i
$49$	6.53251			0.933215
$50$	11.5758	−	0.357550i	1.63707	−	0.0505651i
$51$	−0.203185			−0.0284516
$52$			14.9385i			2.07160i
$53$		−	11.4603i		−	1.57420i	−0.616826	−	0.787100i	$-0.711582\pi$
$53$		−	11.4603i		−	1.57420i	0.616826	−	0.787100i	$-0.288418\pi$
$54$	2.80435			0.381624
$55$	−5.73017	+	5.90990i	−0.772656	+	0.796890i
$56$	2.16190			0.288896
$57$		−	0.211089i		−	0.0279594i
$58$			8.54996i			1.12267i
$59$	2.00000			0.260378			0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000			0.260378			0.130189	+	0.991489i	$0.458442\pi$
$60$	−1.09764	−	1.06426i	−0.141705	−	0.137395i
$61$	9.94089			1.27280			0.636400	−	0.771359i	$-0.280423\pi$
$61$	9.94089			1.27280			0.636400	+	0.771359i	$0.280423\pi$
$62$		−	25.2289i		−	3.20408i
$63$			2.02298i			0.254871i
$64$	12.6500			1.58125
$65$	7.12662	+	6.90990i	0.883949	+	0.857067i
$66$	1.73255			0.213263
$67$			9.16944i			1.12023i	0.828417	+	0.560113i	$0.189242\pi$
$67$			9.16944i			1.12023i	−0.828417	+	0.560113i	$0.810758\pi$
$68$			3.36509i			0.408077i
$69$	0.919815			0.110733
$70$	2.46511	−	2.54243i	0.294637	−	0.303878i
$71$	−9.37262			−1.11233			−0.556163	−	0.831073i	$-0.687727\pi$
$71$	−9.37262			−1.11233			−0.556163	+	0.831073i	$0.687727\pi$
$72$		−	9.35517i		−	1.10252i
$73$			2.26946i			0.265620i	0.991141	+	0.132810i	$0.0424001\pi$
$73$			2.26946i			0.265620i	−0.991141	+	0.132810i	$0.957600\pi$
$74$	−0.715099			−0.0831286
$75$	−1.01544	+	0.0313646i	−0.117253	+	0.00362167i
$76$	−3.49599			−0.401018
$77$			2.51707i			0.286846i
$78$		−	2.08925i		−	0.236561i
$79$	−7.42696			−0.835599			−0.417799	−	0.908539i	$-0.637198\pi$
$79$	−7.42696			−0.835599			−0.417799	+	0.908539i	$0.637198\pi$
$80$	−0.924009	+	0.952991i	−0.103307	+	0.106548i
$81$	8.63015			0.958905
$82$		−	14.2496i		−	1.57361i
$83$			8.92344i			0.979474i	0.871870	+	0.489737i	$0.162907\pi$
$83$			8.92344i			0.979474i	−0.871870	+	0.489737i	$0.837093\pi$
$84$	−0.467493			−0.0510077
$85$	1.60536	+	1.55654i	0.174126	+	0.168830i
$86$	18.2627			1.96932
$87$		−	0.750010i		−	0.0804096i
$88$		−	11.6401i		−	1.24084i
$89$	−11.5523			−1.22455			−0.612273	−	0.790646i	$-0.709745\pi$
$89$	−11.5523			−1.22455			−0.612273	+	0.790646i	$0.709745\pi$
$90$	−11.0018	−	10.6672i	−1.15969	−	1.12443i
$91$	3.03528			0.318184
$92$		−	15.2337i		−	1.58822i
$93$			2.21310i			0.229488i
$94$	−10.2825			−1.06056
$95$	−1.61709	+	1.66781i	−0.165910	+	0.171114i
$96$	−1.00552			−0.102626
$97$		−	12.5500i		−	1.27426i	−0.770758	−	0.637128i	$-0.780122\pi$
$97$		−	12.5500i		−	1.27426i	0.770758	−	0.637128i	$-0.219878\pi$
$98$			15.1310i			1.52846i
$99$	10.8921			1.09469
$100$	0.519450	+	16.8174i	0.0519450	+	1.68174i
$101$	−6.78653			−0.675285			−0.337642	−	0.941274i	$-0.609629\pi$
$101$	−6.78653			−0.675285			−0.337642	+	0.941274i	$0.609629\pi$
$102$		−	0.470630i		−	0.0465993i
$103$		−	8.74799i		−	0.861966i	−0.902360	−	0.430983i	$-0.858167\pi$
$103$		−	8.74799i		−	0.861966i	0.902360	−	0.430983i	$-0.141833\pi$
$104$	−14.0365			−1.37639
$105$	−0.216242	+	0.223024i	−0.0211030	+	0.0217649i
$106$	26.5452			2.57830
$107$			8.10078i			0.783132i	0.920150	+	0.391566i	$0.128066\pi$
$107$			8.10078i			0.783132i	−0.920150	+	0.391566i	$0.871934\pi$
$108$			4.07418i			0.392038i
$109$	3.11308			0.298179			0.149090	−	0.988824i	$-0.452366\pi$
$109$	3.11308			0.298179			0.149090	+	0.988824i	$0.452366\pi$
$110$	−13.6889	−	13.2726i	−1.30518	−	1.26549i
$111$	0.0627291			0.00595399
$112$			0.405885i			0.0383526i
$113$			6.93287i			0.652190i	0.945337	+	0.326095i	$0.105733\pi$
$113$			6.93287i			0.652190i	−0.945337	+	0.326095i	$0.894267\pi$
$114$	0.488938			0.0457932
$115$	−7.26745	−	7.04644i	−0.677693	−	0.657084i
$116$	−12.4214			−1.15330
$117$		−	13.1345i		−	1.21429i
$118$			4.63253i			0.426459i
$119$	0.683735			0.0626778
$120$	1.00000	−	1.03136i	0.0912871	−	0.0941503i
$121$	2.55235			0.232031
$122$			23.0257i			2.08465i
$123$			1.24999i			0.112708i
$124$	36.6528			3.29151
$125$	8.26325	+	7.53118i	0.739088	+	0.673609i
$126$	−4.68575			−0.417440
$127$			21.1496i			1.87672i	0.345655	+	0.938362i	$0.387657\pi$
$127$			21.1496i			1.87672i	−0.345655	+	0.938362i	$0.612343\pi$
$128$			19.4031i			1.71501i
$129$	−1.60202			−0.141050
$130$	−16.0052	+	16.5071i	−1.40374	+	1.44777i
$131$	8.71186			0.761159			0.380580	−	0.924748i	$-0.375725\pi$
$131$	8.71186			0.761159			0.380580	+	0.924748i	$0.375725\pi$
$132$			2.51707i			0.219083i
$133$			0.710332i			0.0615936i
$134$	−21.2388			−1.83476
$135$	1.94364	+	1.88454i	0.167282	+	0.162195i
$136$	−3.16190			−0.271131
$137$			0.243985i			0.0208450i	0.999946	+	0.0104225i	$0.00331765\pi$
$137$			0.243985i			0.0208450i	−0.999946	+	0.0104225i	$0.996682\pi$
$138$			2.13054i			0.181363i
$139$	−12.6884			−1.07622			−0.538108	−	0.842876i	$-0.680861\pi$
$139$	−12.6884			−1.07622			−0.538108	+	0.842876i	$0.680861\pi$
$140$	3.69365	+	3.58133i	0.312171	+	0.302677i
$141$	0.901992			0.0759614
$142$		−	21.7095i		−	1.82182i
$143$		−	16.3425i		−	1.36663i
$144$	1.75638			0.146365
$145$	−5.74561	+	5.92582i	−0.477147	+	0.492113i
$146$	−5.25667			−0.435045
$147$		−	1.32731i		−	0.109474i
$148$		−	1.03890i		−	0.0853971i
$149$	−12.0103			−0.983923			−0.491961	−	0.870617i	$-0.663720\pi$
$149$	−12.0103			−0.983923			−0.491961	+	0.870617i	$0.663720\pi$
$150$	−0.0726487	−	2.35203i	−0.00593174	−	0.192042i
$151$	8.95633			0.728856			0.364428	−	0.931232i	$-0.381265\pi$
$151$	8.95633			0.728856			0.364428	+	0.931232i	$0.381265\pi$
$152$		−	3.28490i		−	0.266441i
$153$		−	2.95872i		−	0.239198i
$154$	−5.83020			−0.469811
$155$	16.9539	−	17.4857i	1.36177	−	1.40449i
$156$	3.03528			0.243017
$157$		−	19.9127i		−	1.58920i	−0.607131	−	0.794602i	$-0.707680\pi$
$157$		−	19.9127i		−	1.58920i	0.607131	−	0.794602i	$-0.292320\pi$
$158$		−	17.2028i		−	1.36858i
$159$	−2.32857			−0.184667
$160$	7.94460	+	7.70300i	0.628076	+	0.608976i
$161$	−3.09526			−0.243940
$162$			19.9897i			1.57054i
$163$		−	22.4011i		−	1.75459i	−0.479951	−	0.877296i	$-0.659345\pi$
$163$		−	22.4011i		−	1.75459i	0.479951	−	0.877296i	$-0.340655\pi$
$164$	20.7019			1.61655
$165$	1.20080	+	1.16428i	0.0934823	+	0.0906394i
$166$	−20.6690			−1.60423
$167$		−	4.58133i		−	0.354514i	−0.984165	−	0.177257i	$-0.943278\pi$
$167$		−	4.58133i		−	0.354514i	0.984165	−	0.177257i	$-0.0567223\pi$
$168$		−	0.439266i		−	0.0338901i
$169$	−6.70708			−0.515929
$170$	−3.60536	+	3.71844i	−0.276519	+	0.285191i
$171$	3.07381			0.235060
$172$			26.5321i			2.02306i
$173$			8.09764i			0.615652i	0.951443	+	0.307826i	$0.0996015\pi$
$173$			8.09764i			0.615652i	−0.951443	+	0.307826i	$0.900399\pi$
$174$	1.73722			0.131699
$175$	3.41704	−	0.105544i	0.258304	−	0.00797841i
$176$	2.18536			0.164728
$177$		−	0.406370i		−	0.0305446i
$178$		−	26.7583i		−	2.00562i
$179$	−4.40637			−0.329348			−0.164674	−	0.986348i	$-0.552657\pi$
$179$	−4.40637			−0.329348			−0.164674	+	0.986348i	$0.552657\pi$
$180$	15.4974	−	15.9835i	1.15511	−	1.19134i
$181$	−5.93727			−0.441314			−0.220657	−	0.975351i	$-0.570820\pi$
$181$	−5.93727			−0.441314			−0.220657	+	0.975351i	$0.570820\pi$
$182$			7.03051i			0.521136i
$183$		−	2.01984i		−	0.149311i
$184$	14.3139			1.05523
$185$	−0.495622	−	0.480550i	−0.0364388	−	0.0353307i
$186$	−5.12614			−0.375867
$187$		−	3.68135i		−	0.269207i
$188$		−	14.9385i		−	1.08950i
$189$	0.827812			0.0602144
$190$	−3.86309	−	3.74561i	−0.280258	−	0.271735i
$191$	−9.89759			−0.716165			−0.358082	−	0.933690i	$-0.616569\pi$
$191$	−9.89759			−0.716165			−0.358082	+	0.933690i	$0.616569\pi$
$192$		−	2.57029i		−	0.185494i
$193$			8.47578i			0.610100i	0.952336	+	0.305050i	$0.0986732\pi$
$193$			8.47578i			0.610100i	−0.952336	+	0.305050i	$0.901327\pi$
$194$	29.0690			2.08704
$195$	1.40399	−	1.44802i	0.100542	−	0.103695i
$196$	−21.9824			−1.57017
$197$			13.5381i			0.964553i	0.876019	+	0.482276i	$0.160190\pi$
$197$			13.5381i			0.964553i	−0.876019	+	0.482276i	$0.839810\pi$
$198$			25.2289i			1.79294i
$199$	9.15388			0.648901			0.324451	−	0.945903i	$-0.394821\pi$
$199$	9.15388			0.648901			0.324451	+	0.945903i	$0.394821\pi$
$200$	−15.8020	+	0.488086i	−1.11737	+	0.0345129i
$201$	1.86309			0.131412
$202$		−	15.7194i		−	1.10601i
$203$			2.52385i			0.177139i
$204$	0.683735			0.0478710
$205$	9.57581	−	9.87615i	0.668803	−	0.689780i
$206$	20.2627			1.41177
$207$			13.3941i			0.930952i
$208$		−	2.63528i		−	0.182724i
$209$	3.82456			0.264550
$210$	−0.516583	−	0.500873i	−0.0356476	−	0.0345635i
$211$	7.72465			0.531787			0.265893	−	0.964002i	$-0.414333\pi$
$211$	7.72465			0.531787			0.265893	+	0.964002i	$0.414333\pi$
$212$			38.5650i			2.64866i
$213$			1.90438i			0.130486i
$214$	−18.7636			−1.28265
$215$	12.6575	+	12.2726i	0.863236	+	0.836984i
$216$	−3.82818			−0.260475
$217$		−	7.44729i		−	0.505555i
$218$			7.21072i			0.488372i
$219$	0.461120			0.0311596
$220$	19.2825	−	19.8873i	1.30003	−	1.34080i
$221$	−4.43927			−0.298617
$222$			0.145297i			0.00975172i
$223$			13.7194i			0.918719i	0.888250	+	0.459359i	$0.151921\pi$
$223$			13.7194i			0.918719i	−0.888250	+	0.459359i	$0.848079\pi$
$224$	3.38366			0.226080
$225$	−0.456721	−	14.7865i	−0.0304481	−	0.985769i
$226$	−16.0584			−1.06819
$227$		−	1.80044i		−	0.119499i	−0.998213	−	0.0597496i	$-0.980970\pi$
$227$		−	1.80044i		−	0.119499i	0.998213	−	0.0597496i	$-0.0190302\pi$
$228$			0.710332i			0.0470429i
$229$	3.24838			0.214659			0.107330	−	0.994223i	$-0.465770\pi$
$229$	3.24838			0.214659			0.107330	+	0.994223i	$0.465770\pi$
$230$	16.3214	−	16.8333i	1.07620	−	1.10996i
$231$	0.511430			0.0336496
$232$		−	11.6714i		−	0.766267i
$233$		−	5.38254i		−	0.352622i	−0.984335	−	0.176311i	$-0.943584\pi$
$233$		−	5.38254i		−	0.352622i	0.984335	−	0.176311i	$-0.0564164\pi$
$234$	30.4230			1.98882
$235$	−7.12662	−	6.90990i	−0.464890	−	0.450752i
$236$	−6.73017			−0.438097
$237$			1.50905i			0.0980231i
$238$			1.58371i			0.102657i
$239$	3.62071			0.234204			0.117102	−	0.993120i	$-0.462639\pi$
$239$	3.62071			0.234204			0.117102	+	0.993120i	$0.462639\pi$
$240$	0.193633	+	0.187745i	0.0124990	+	0.0121189i
$241$	−7.66781			−0.493927			−0.246964	−	0.969025i	$-0.579433\pi$
$241$	−7.66781			−0.493927			−0.246964	+	0.969025i	$0.579433\pi$
$242$			5.91191i			0.380032i
$243$		−	5.38568i		−	0.345491i
$244$	−33.4520			−2.14154
$245$	−10.1681	+	10.4870i	−0.649617	+	0.669992i
$246$	−2.89531			−0.184598
$247$		−	4.61196i		−	0.293452i
$248$			34.4397i			2.18692i
$249$	1.81311			0.114901
$250$	−17.4442	+	19.1399i	−1.10327	+	1.21051i
$251$	−6.04595			−0.381617			−0.190809	−	0.981627i	$-0.561111\pi$
$251$	−6.04595			−0.381617			−0.190809	+	0.981627i	$0.561111\pi$
$252$		−	6.80749i		−	0.428831i
$253$			16.6654i			1.04775i
$254$	−48.9881			−3.07379
$255$	0.316265	−	0.326185i	0.0198053	−	0.0204265i
$256$	−19.6428			−1.22768
$257$		−	6.13054i		−	0.382412i	−0.981550	−	0.191206i	$-0.938760\pi$
$257$		−	6.13054i		−	0.382412i	0.981550	−	0.191206i	$-0.0612399\pi$
$258$		−	3.71070i		−	0.231018i
$259$	−0.211089			−0.0131164
$260$	−23.9817	−	23.2524i	−1.48728	−	1.44205i
$261$	10.9214			0.676019
$262$			20.1790i			1.24666i
$263$			12.3012i			0.758525i	0.925289	+	0.379263i	$0.123822\pi$
$263$			12.3012i			0.758525i	−0.925289	+	0.379263i	$0.876178\pi$
$264$	−2.36509			−0.145561
$265$	18.3980	+	17.8385i	1.13018	+	1.09581i
$266$	−1.64532			−0.100881
$267$			2.34726i			0.143650i
$268$		−	30.8559i		−	1.88483i
$269$	−9.92508			−0.605143			−0.302572	−	0.953127i	$-0.597845\pi$
$269$	−9.92508			−0.605143			−0.302572	+	0.953127i	$0.597845\pi$
$270$	−4.36509	+	4.50199i	−0.265651	+	0.273983i
$271$	20.3040			1.23338			0.616689	−	0.787207i	$-0.288474\pi$
$271$	20.3040			1.23338			0.616689	+	0.787207i	$0.288474\pi$
$272$		−	0.593630i		−	0.0359941i
$273$		−	0.616723i		−	0.0373258i
$274$	−0.565133			−0.0341410
$275$	−0.568271	−	18.3980i	−0.0342680	−	1.10944i
$276$	−3.09526			−0.186313
$277$			18.1671i			1.09155i	0.837931	+	0.545776i	$0.183765\pi$
$277$			18.1671i			1.09155i	−0.837931	+	0.545776i	$0.816235\pi$
$278$		−	29.3897i		−	1.76268i
$279$	−32.2265			−1.92935
$280$	−3.36509	+	3.47063i	−0.201102	+	0.207410i
$281$	10.6983			0.638208			0.319104	−	0.947720i	$-0.396618\pi$
$281$	10.6983			0.638208			0.319104	+	0.947720i	$0.396618\pi$
$282$			2.08925i			0.124413i
$283$		−	10.0773i		−	0.599034i	−0.954091	−	0.299517i	$-0.903174\pi$
$283$		−	10.0773i		−	0.599034i	0.954091	−	0.299517i	$-0.0968256\pi$
$284$	31.5397			1.87154
$285$	0.338874	+	0.328568i	0.0200732	+	0.0194627i
$286$	37.8536			2.23833
$287$		−	4.20632i		−	0.248291i
$288$		−	14.6421i		−	0.862793i
$289$	−1.00000			−0.0588235
$290$	−13.7258	−	13.3084i	−0.806005	−	0.781494i
$291$	−2.54996			−0.149481
$292$		−	7.63693i		−	0.446918i
$293$			20.5349i			1.19966i	0.800127	+	0.599831i	$0.204765\pi$
$293$			20.5349i			1.19966i	−0.800127	+	0.599831i	$0.795235\pi$
$294$	3.07439			0.179302
$295$	−3.11308	+	3.21072i	−0.181251	+	0.186935i
$296$	0.976172			0.0567388
$297$		−	4.45709i		−	0.258627i
$298$		−	27.8191i		−	1.61151i
$299$	20.0965			1.16221
$300$	3.41704	−	0.105544i	0.197283	−	0.00609361i
$301$	5.39093			0.310728
$302$			20.7452i			1.19375i
$303$			1.37892i			0.0792169i
$304$	0.616723			0.0353715
$305$	−15.4734	+	15.9587i	−0.886004	+	0.913793i
$306$	6.85317			0.391770
$307$			13.1177i			0.748669i	0.927294	+	0.374335i	$0.122129\pi$
$307$			13.1177i			0.748669i	−0.927294	+	0.374335i	$0.877871\pi$
$308$		−	8.47015i		−	0.482631i
$309$	−1.77746			−0.101116
$310$	40.5015	+	39.2698i	2.30033	+	2.23038i
$311$	−12.3646			−0.701132			−0.350566	−	0.936538i	$-0.614011\pi$
$311$	−12.3646			−0.701132			−0.350566	+	0.936538i	$0.614011\pi$
$312$			2.85201i			0.161463i
$313$			5.16000i			0.291661i	0.989310	+	0.145830i	$0.0465853\pi$
$313$			5.16000i			0.291661i	−0.989310	+	0.145830i	$0.953415\pi$
$314$	46.1230			2.60287
$315$	−3.24761	−	3.14884i	−0.182982	−	0.177417i
$316$	24.9924			1.40593
$317$		−	23.7655i		−	1.33480i	−0.744699	−	0.667400i	$-0.767407\pi$
$317$		−	23.7655i		−	1.33480i	0.744699	−	0.667400i	$-0.232593\pi$
$318$		−	5.39358i		−	0.302457i
$319$	13.5889			0.760830
$320$	−19.6902	+	20.3078i	−1.10072	+	1.13524i
$321$	1.64596			0.0918683
$322$		−	7.16944i		−	0.399537i
$323$		−	1.03890i		−	0.0578060i
$324$	−29.0412			−1.61340
$325$	−22.1857	+	0.685266i	−1.23064	+	0.0380117i
$326$	51.8869			2.87375
$327$		−	0.632531i		−	0.0349790i
$328$			19.4520i			1.07405i
$329$	−3.03528			−0.167340
$330$	−2.69679	+	2.78137i	−0.148453	+	0.153110i
$331$	20.8341			1.14514			0.572572	−	0.819854i	$-0.305946\pi$
$331$	20.8341			1.14514			0.572572	+	0.819854i	$0.305946\pi$
$332$		−	30.0281i		−	1.64801i
$333$			0.913442i			0.0500563i
$334$	10.6116			0.580639
$335$	−14.7203	−	14.2726i	−0.804253	−	0.779795i
$336$	0.0824698			0.00449910
$337$			14.3080i			0.779406i	0.920941	+	0.389703i	$0.127422\pi$
$337$			14.3080i			0.779406i	−0.920941	+	0.389703i	$0.872578\pi$
$338$		−	15.5354i		−	0.845013i
$339$	1.40865			0.0765076
$340$	−5.40218	−	5.23789i	−0.292974	−	0.284065i
$341$	−40.0975			−2.17140
$342$			7.11976i			0.384993i
$343$			9.25264i			0.499596i
$344$	−24.9301			−1.34414
$345$	−1.43173	+	1.47664i	−0.0770817	+	0.0794994i
$346$	−18.7563			−1.00834
$347$		−	21.9988i		−	1.18096i	−0.807054	−	0.590478i	$-0.798939\pi$
$347$		−	21.9988i		−	1.18096i	0.807054	−	0.590478i	$-0.201061\pi$
$348$			2.52385i			0.135293i
$349$	6.06199			0.324491			0.162246	−	0.986750i	$-0.448126\pi$
$349$	6.06199			0.324491			0.162246	+	0.986750i	$0.448126\pi$
$350$	0.244469	+	7.91478i	0.0130674	+	0.423063i
$351$	−5.37471			−0.286881
$352$		−	18.2183i		−	0.971036i
$353$			20.8785i			1.11125i	0.831432	+	0.555626i	$0.187521\pi$
$353$			20.8785i			1.11125i	−0.831432	+	0.555626i	$0.812479\pi$
$354$	0.941260			0.0500274
$355$	14.5889	−	15.0464i	0.774296	−	0.798582i
$356$	38.8746			2.06035
$357$		−	0.138925i		−	0.00735267i
$358$		−	10.2063i		−	0.539421i
$359$	−18.4842			−0.975557			−0.487779	−	0.872967i	$-0.662193\pi$
$359$	−18.4842			−0.975557			−0.487779	+	0.872967i	$0.662193\pi$
$360$	15.0184	+	14.5617i	0.791540	+	0.767469i
$361$	−17.9207			−0.943194
$362$		−	13.7523i		−	0.722805i
$363$		−	0.518598i		−	0.0272193i
$364$	−10.2140			−0.535358
$365$	−3.64330	−	3.53251i	−0.190699	−	0.184900i
$366$	4.67848			0.244548
$367$			23.6044i			1.23214i	0.787691	+	0.616070i	$0.211276\pi$
$367$			23.6044i			1.23214i	−0.787691	+	0.616070i	$0.788724\pi$
$368$			2.68736i			0.140088i
$369$	−18.2020			−0.947556
$370$	1.11308	−	1.14799i	0.0578663	−	0.0596813i
$371$	7.83583			0.406816
$372$		−	7.44729i		−	0.386124i
$373$		−	18.0123i		−	0.932643i	−0.884615	−	0.466321i	$-0.845579\pi$
$373$		−	18.0123i		−	0.932643i	0.884615	−	0.466321i	$-0.154421\pi$
$374$	8.52699			0.440920
$375$	1.53022	−	1.67897i	0.0790203	−	0.0867015i
$376$	14.0365			0.723878
$377$		−	16.3865i		−	0.843949i
$378$			1.91743i			0.0986221i
$379$	5.74523			0.295112			0.147556	−	0.989054i	$-0.452859\pi$
$379$	5.74523			0.295112			0.147556	+	0.989054i	$0.452859\pi$
$380$	5.44165	−	5.61232i	0.279151	−	0.287906i
$381$	4.29728			0.220156
$382$		−	22.9255i		−	1.17297i
$383$		−	35.3281i		−	1.80518i	−0.430499	−	0.902591i	$-0.641663\pi$
$383$		−	35.3281i		−	1.80518i	0.430499	−	0.902591i	$-0.358337\pi$
$384$	3.94242			0.201186
$385$	−4.04080	−	3.91792i	−0.205938	−	0.199675i
$386$	−19.6322			−0.999251
$387$		−	23.3281i		−	1.18583i
$388$			42.2317i			2.14399i
$389$	18.2627			0.925955			0.462977	−	0.886370i	$-0.346781\pi$
$389$	18.2627			0.925955			0.462977	+	0.886370i	$0.346781\pi$
$390$	3.35400	+	3.25201i	0.169837	+	0.164672i
$391$	4.52699			0.228940
$392$		−	20.6551i		−	1.04324i
$393$		−	1.77012i		−	0.0892907i
$394$	−31.3579			−1.57979
$395$	11.5604	−	11.9230i	0.581665	−	0.599909i
$396$	−36.6528			−1.84187
$397$		−	4.64092i		−	0.232921i	−0.993195	−	0.116461i	$-0.962845\pi$
$397$		−	4.64092i		−	0.232921i	0.993195	−	0.116461i	$-0.0371549\pi$
$398$			21.2028i			1.06280i
$399$	0.144329			0.00722548
$400$	−0.0916355	−	2.96674i	−0.00458178	−	0.148337i
$401$	−27.4049			−1.36853			−0.684267	−	0.729232i	$-0.739878\pi$
$401$	−27.4049			−1.36853			−0.684267	+	0.729232i	$0.739878\pi$
$402$			4.31541i			0.215233i
$403$			48.3528i			2.40862i
$404$	22.8372			1.13620
$405$	−13.4332	+	13.8545i	−0.667500	+	0.688436i
$406$	−5.84590			−0.290127
$407$			1.13654i			0.0563363i
$408$			0.642450i			0.0318060i
$409$	−1.38254			−0.0683623			−0.0341811	−	0.999416i	$-0.510882\pi$
$409$	−1.38254			−0.0683623			−0.0341811	+	0.999416i	$0.510882\pi$
$410$	22.8758	+	22.1801i	1.12975	+	1.09540i
$411$	0.0495740			0.00244531
$412$			29.4378i			1.45029i
$413$			1.36747i			0.0672888i
$414$	−31.0242			−1.52476
$415$	−14.3253	−	13.8897i	−0.703203	−	0.681818i
$416$	−21.9690			−1.07712
$417$			2.57809i			0.126250i
$418$			8.85869i			0.433293i
$419$	33.2300			1.62339			0.811695	−	0.584081i	$-0.198545\pi$
$419$	33.2300			1.62339			0.811695	+	0.584081i	$0.198545\pi$
$420$	0.727672	−	0.750495i	0.0355067	−	0.0366204i
$421$	4.61109			0.224731			0.112365	−	0.993667i	$-0.464157\pi$
$421$	4.61109			0.224731			0.112365	+	0.993667i	$0.464157\pi$
$422$			17.8923i			0.870986i
$423$			13.1345i			0.638622i
$424$	−36.2365			−1.75980
$425$	−4.99762	+	0.154365i	−0.242420	+	0.00748779i
$426$	−4.41104			−0.213715
$427$			6.79693i			0.328927i
$428$		−	27.2598i		−	1.31765i
$429$	−3.32055			−0.160318
$430$	−28.4266	+	29.3182i	−1.37085	+	1.41385i
$431$	7.33812			0.353465			0.176732	−	0.984259i	$-0.443447\pi$
$431$	7.33812			0.353465			0.176732	+	0.984259i	$0.443447\pi$
$432$		−	0.718721i		−	0.0345795i
$433$			15.3487i			0.737610i	0.929507	+	0.368805i	$0.120233\pi$
$433$			15.3487i			0.737610i	−0.929507	+	0.368805i	$0.879767\pi$
$434$	17.2499			0.828021
$435$	1.20404	+	1.16742i	0.0577292	+	0.0559736i
$436$	−10.4758			−0.501699
$437$			4.70309i			0.224979i
$438$			1.06808i			0.0510347i
$439$	5.60355			0.267443			0.133721	−	0.991019i	$-0.457307\pi$
$439$	5.60355			0.267443			0.133721	+	0.991019i	$0.457307\pi$
$440$	18.6865	+	18.1182i	0.890844	+	0.863753i
$441$	19.3278			0.920373
$442$		−	10.2825i		−	0.489089i
$443$			29.6338i			1.40794i	0.710228	+	0.703971i	$0.248592\pi$
$443$			29.6338i			1.40794i	−0.710228	+	0.703971i	$0.751408\pi$
$444$	−0.211089			−0.0100178
$445$	17.9817	−	18.5457i	0.852414	−	0.879150i
$446$	−31.7778			−1.50472
$447$			2.44031i			0.115423i
$448$			8.64923i			0.408638i
$449$	34.4203			1.62439			0.812197	−	0.583383i	$-0.198271\pi$
$449$	34.4203			1.62439			0.812197	+	0.583383i	$0.198271\pi$
$450$	34.2495	−	1.05789i	1.61454	−	0.0498693i
$451$	−22.6476			−1.06643
$452$		−	23.3297i		−	1.09734i
$453$		−	1.81979i		−	0.0855013i
$454$	4.17029			0.195721
$455$	−4.72453	+	4.87272i	−0.221490	+	0.228437i
$456$	−0.667442			−0.0312558
$457$			9.32504i			0.436207i	0.975926	+	0.218103i	$0.0699870\pi$
$457$			9.32504i			0.436207i	−0.975926	+	0.218103i	$0.930013\pi$
$458$			7.52412i			0.351579i
$459$	−1.21072			−0.0565116
$460$	24.4556	+	23.7119i	1.14025	+	1.10557i
$461$	19.1715			0.892904			0.446452	−	0.894808i	$-0.352687\pi$
$461$	19.1715			0.892904			0.446452	+	0.894808i	$0.352687\pi$
$462$			1.18461i			0.0551129i
$463$		−	19.6385i		−	0.912680i	−0.889805	−	0.456340i	$-0.849160\pi$
$463$		−	19.6385i		−	0.912680i	0.889805	−	0.456340i	$-0.150840\pi$
$464$	2.19125			0.101726
$465$	−3.55283	−	3.44479i	−0.164759	−	0.159748i
$466$	12.4674			0.577541
$467$		−	21.1027i		−	0.976515i	−0.872699	−	0.488258i	$-0.837633\pi$
$467$		−	21.1027i		−	0.976515i	0.872699	−	0.488258i	$-0.162367\pi$
$468$			44.1988i			2.04309i
$469$	−6.26946			−0.289497
$470$	16.0052	−	16.5071i	0.738263	−	0.761418i
$471$	−4.04595			−0.186428
$472$		−	6.32380i		−	0.291077i
$473$		−	29.0257i		−	1.33461i
$474$	−3.49535			−0.160547
$475$	−0.160370	−	5.19203i	−0.00735826	−	0.238227i
$476$	−2.30083			−0.105458
$477$		−	33.9079i		−	1.55254i
$478$			8.38653i			0.383591i
$479$	28.6762			1.31025			0.655125	−	0.755521i	$-0.272616\pi$
$479$	28.6762			1.31025			0.655125	+	0.755521i	$0.272616\pi$
$480$	1.56513	−	1.61422i	0.0714382	−	0.0736789i
$481$	1.37053			0.0624909
$482$		−	17.7607i		−	0.808977i
$483$			0.628909i			0.0286164i
$484$	−8.58886			−0.390403
$485$	20.1472	+	19.5345i	0.914838	+	0.887017i
$486$	12.4747			0.565862
$487$			39.2139i			1.77695i	0.458925	+	0.888475i	$0.348234\pi$
$487$			39.2139i			1.77695i	−0.458925	+	0.888475i	$0.651766\pi$
$488$		−	31.4321i		−	1.42287i
$489$	−4.55157			−0.205829
$490$	−24.2907	−	23.5520i	−1.09734	−	1.06397i
$491$	−36.3499			−1.64045			−0.820224	−	0.572042i	$-0.806151\pi$
$491$	−36.3499			−1.64045			−0.820224	+	0.572042i	$0.806151\pi$
$492$		−	4.20632i		−	0.189636i
$493$		−	3.69127i		−	0.166246i
$494$	10.6825			0.480629
$495$	−16.9539	+	17.4857i	−0.762023	+	0.785924i
$496$	−6.46586			−0.290326
$497$		−	6.40839i		−	0.287455i
$498$			4.19964i			0.188190i
$499$	8.52061			0.381435			0.190718	−	0.981645i	$-0.438919\pi$
$499$	8.52061			0.381435			0.190718	+	0.981645i	$0.438919\pi$
$500$	−27.8065	−	25.3431i	−1.24355	−	1.13338i
$501$	−0.930856			−0.0415876
$502$		−	14.0040i		−	0.625030i
$503$		−	18.6790i		−	0.832854i	−0.909169	−	0.416427i	$-0.863282\pi$
$503$		−	18.6790i		−	0.832854i	0.909169	−	0.416427i	$-0.136718\pi$
$504$	6.39645			0.284921
$505$	10.5635	−	10.8948i	0.470070	−	0.484813i
$506$	−38.6016			−1.71605
$507$			1.36278i			0.0605231i
$508$		−	71.1702i		−	3.15767i
$509$	−3.53567			−0.156716			−0.0783579	−	0.996925i	$-0.524968\pi$
$509$	−3.53567			−0.156716			−0.0783579	+	0.996925i	$0.524968\pi$
$510$	0.755531	+	0.732555i	0.0334555	+	0.0324381i
$511$	−1.55171			−0.0686436
$512$		−	6.69175i		−	0.295737i
$513$		−	1.25782i		−	0.0555341i
$514$	14.1999			0.626333
$515$	14.0437	+	13.6166i	0.618839	+	0.600019i
$516$	5.39093			0.237322
$517$			16.3425i			0.718742i
$518$		−	0.488938i		−	0.0214827i
$519$	1.64532			0.0722215
$520$	21.8484	−	22.5337i	0.958116	−	0.988167i
$521$	−4.64206			−0.203373			−0.101686	−	0.994817i	$-0.532424\pi$
$521$	−4.64206			−0.203373			−0.101686	+	0.994817i	$0.532424\pi$
$522$			25.2969i			1.10722i
$523$		−	18.3331i		−	0.801649i	−0.916155	−	0.400824i	$-0.868724\pi$
$523$		−	18.3331i		−	0.801649i	0.916155	−	0.400824i	$-0.131276\pi$
$524$	−29.3162			−1.28068
$525$	−0.0214450	−	0.694292i	−0.000935938	−	0.0303014i
$526$	−28.4929			−1.24235
$527$			10.8921i			0.474466i
$528$		−	0.444032i		−	0.0193240i
$529$	2.50639			0.108974
$530$	−41.3187	+	42.6146i	−1.79477	+	1.85106i
$531$	5.91743			0.256795
$532$		−	2.39033i		−	0.103634i
$533$			27.3103i			1.18294i
$534$	−5.43688			−0.235277
$535$	−13.0047	−	12.6092i	−0.562241	−	0.545143i
$536$	28.9928			1.25230
$537$			0.895308i			0.0386354i
$538$		−	22.9891i		−	0.991132i
$539$	24.0485			1.03584
$540$	−6.54053	−	6.34163i	−0.281459	−	0.272900i
$541$	22.4428			0.964891			0.482445	−	0.875926i	$-0.339749\pi$
$541$	22.4428			0.964891			0.482445	+	0.875926i	$0.339749\pi$
$542$			47.0294i			2.02008i
$543$			1.20636i			0.0517700i
$544$	−4.94880			−0.212178
$545$	−4.84564	+	4.99762i	−0.207564	+	0.214074i
$546$	1.42849			0.0611339
$547$		−	25.9778i		−	1.11073i	−0.831607	−	0.555365i	$-0.812578\pi$
$547$		−	25.9778i		−	1.11073i	0.831607	−	0.555365i	$-0.187422\pi$
$548$		−	0.821029i		−	0.0350726i
$549$	29.4123			1.25529
$550$	42.6146	−	1.31627i	1.81709	−	0.0561257i
$551$	3.83486			0.163371
$552$		−	2.90836i		−	0.123788i
$553$		−	5.07807i		−	0.215942i
$554$	−42.0797			−1.78780
$555$	−0.0976404	+	0.100703i	−0.00414461	+	0.00427460i
$556$	42.6976			1.81078
$557$		−	4.86144i		−	0.205986i	−0.994682	−	0.102993i	$-0.967158\pi$
$557$		−	4.86144i		−	0.205986i	0.994682	−	0.102993i	$-0.0328419\pi$
$558$		−	74.6452i		−	3.15998i
$559$	−35.0015			−1.48041
$560$	−0.651593	−	0.631777i	−0.0275348	−	0.0266975i
$561$	−0.747995			−0.0315804
$562$			24.7802i			1.04529i
$563$			1.53365i			0.0646357i	0.999478	+	0.0323179i	$0.0102889\pi$
$563$			1.53365i			0.0646357i	−0.999478	+	0.0323179i	$0.989711\pi$
$564$	−3.03528			−0.127808
$565$	−11.1298	−	10.7913i	−0.468232	−	0.453993i
$566$	23.3417			0.981127
$567$			5.90073i			0.247807i
$568$			29.6353i			1.24347i
$569$	16.5770			0.694946			0.347473	−	0.937690i	$-0.387040\pi$
$569$	16.5770			0.694946			0.347473	+	0.937690i	$0.387040\pi$
$570$	−0.761052	+	0.784922i	−0.0318769	+	0.0328768i
$571$	−14.2904			−0.598036			−0.299018	−	0.954248i	$-0.596659\pi$
$571$	−14.2904			−0.598036			−0.299018	+	0.954248i	$0.596659\pi$
$572$			54.9939i			2.29941i
$573$			2.01104i			0.0840125i
$574$	9.74296			0.406663
$575$	22.6241	−	0.698807i	0.943492	−	0.0291423i
$576$	37.4277			1.55949
$577$		−	28.5424i		−	1.18824i	−0.804377	−	0.594119i	$-0.797501\pi$
$577$		−	28.5424i		−	1.18824i	0.804377	−	0.594119i	$-0.202499\pi$
$578$		−	2.31627i		−	0.0963439i
$579$	1.72215			0.0715702
$580$	19.3345	−	19.9409i	0.802820	−	0.828000i
$581$	−6.10126			−0.253123
$582$		−	5.90639i		−	0.244828i
$583$		−	42.1895i		−	1.74731i
$584$	7.17581			0.296937
$585$	21.0856	+	20.4444i	0.871784	+	0.845273i
$586$	−47.5643			−1.96486
$587$			33.5281i			1.38385i	0.721967	+	0.691927i	$0.243238\pi$
$587$			33.5281i			1.38385i	−0.721967	+	0.691927i	$0.756762\pi$
$588$			4.46650i			0.184195i
$589$	−11.3158			−0.466259
$590$	−7.43688	−	7.21072i	−0.306172	−	0.296861i
$591$	2.75075			0.113151
$592$			0.183271i			0.00753239i
$593$			42.4729i			1.74415i	0.489368	+	0.872077i	$0.337227\pi$
$593$			42.4729i			1.74415i	−0.489368	+	0.872077i	$0.662773\pi$
$594$	10.3238			0.423591
$595$	−1.06426	+	1.09764i	−0.0436304	+	0.0449989i
$596$	40.4157			1.65549
$597$		−	1.85993i		−	0.0761219i
$598$			46.5488i			1.90352i
$599$	7.00705			0.286300			0.143150	−	0.989701i	$-0.454277\pi$
$599$	7.00705			0.286300			0.143150	+	0.989701i	$0.454277\pi$
$600$	0.0991717	+	3.21072i	0.00404867	+	0.131077i
$601$	39.3146			1.60368			0.801839	−	0.597541i	$-0.203855\pi$
$601$	39.3146			1.60368			0.801839	+	0.597541i	$0.203855\pi$
$602$			12.4868i			0.508925i
$603$			27.1298i			1.10481i
$604$	−30.1388			−1.22633
$605$	−3.97283	+	4.09744i	−0.161518	+	0.166584i
$606$	−3.19394			−0.129745
$607$			18.7928i			0.762777i	0.924415	+	0.381389i	$0.124554\pi$
$607$			18.7928i			0.762777i	−0.924415	+	0.381389i	$0.875446\pi$
$608$		−	5.14131i		−	0.208508i
$609$	0.512808			0.0207800
$610$	−36.9646	−	35.8405i	−1.49665	−	1.45114i
$611$	19.7071			0.797263
$612$			9.95633i			0.402461i
$613$		−	1.83560i		−	0.0741392i	−0.999313	−	0.0370696i	$-0.988198\pi$
$613$		−	1.83560i		−	0.0741392i	0.999313	−	0.0370696i	$-0.0118023\pi$
$614$	−30.3842			−1.22621
$615$	−2.00668	−	1.94566i	−0.0809173	−	0.0784566i
$616$	7.95872			0.320666
$617$		−	29.0011i		−	1.16754i	−0.811918	−	0.583771i	$-0.801577\pi$
$617$		−	29.0011i		−	1.16754i	0.811918	−	0.583771i	$-0.198423\pi$
$618$		−	4.11707i		−	0.165613i
$619$	−27.7941			−1.11714			−0.558569	−	0.829458i	$-0.688649\pi$
$619$	−27.7941			−1.11714			−0.558569	+	0.829458i	$0.688649\pi$
$620$	−57.0515	+	58.8409i	−2.29124	+	2.36311i
$621$	5.48092			0.219942
$622$		−	28.6397i		−	1.14835i
$623$		−	7.89874i		−	0.316456i
$624$	−0.535450			−0.0214351
$625$	−24.9523	+	1.54291i	−0.998094	+	0.0617164i
$626$	−11.9519			−0.477695
$627$		−	0.777092i		−	0.0310341i
$628$			67.0078i			2.67390i
$629$	0.308729			0.0123098
$630$	7.29356	−	7.52232i	0.290582	−	0.299696i
$631$	−14.1785			−0.564437			−0.282219	−	0.959350i	$-0.591070\pi$
$631$	−14.1785			−0.564437			−0.282219	+	0.959350i	$0.591070\pi$
$632$			23.4833i			0.934116i
$633$		−	1.56953i		−	0.0623833i
$634$	55.0471			2.18620
$635$	−33.9527	−	32.9202i	−1.34737	−	1.30640i
$636$	7.83583			0.310711
$637$		−	28.9995i		−	1.14900i
$638$			31.4754i			1.24612i
$639$	−27.7309			−1.09702
$640$	−31.1490	−	30.2018i	−1.23127	−	1.19383i
$641$	−27.8734			−1.10093			−0.550466	−	0.834857i	$-0.685550\pi$
$641$	−27.8734			−1.10093			−0.550466	+	0.834857i	$0.685550\pi$
$642$			3.81247i			0.150466i
$643$			25.8277i			1.01854i	0.860605	+	0.509272i	$0.170085\pi$
$643$			25.8277i			1.01854i	−0.860605	+	0.509272i	$0.829915\pi$
$644$	10.4158			0.410440
$645$	2.49361	−	2.57182i	0.0981857	−	0.101265i
$646$	2.40637			0.0946774
$647$		−	4.40912i		−	0.173340i	−0.996237	−	0.0866702i	$-0.972377\pi$
$647$		−	4.40912i		−	0.173340i	0.996237	−	0.0866702i	$-0.0276226\pi$
$648$		−	27.2877i		−	1.07196i
$649$	7.36270			0.289011
$650$	−1.58726	−	51.3881i	−0.0622574	−	2.01561i
$651$	−1.51318			−0.0593060
$652$			75.3817i			2.95217i
$653$		−	30.9245i		−	1.21017i	−0.796161	−	0.605084i	$-0.793139\pi$
$653$		−	30.9245i		−	1.21017i	0.796161	−	0.605084i	$-0.206861\pi$
$654$	1.46511			0.0572903
$655$	−13.5604	+	13.9857i	−0.529847	+	0.546466i
$656$	−3.65200			−0.142587
$657$			6.71469i			0.261965i
$658$		−	7.03051i		−	0.274078i
$659$	−39.7992			−1.55036			−0.775179	−	0.631742i	$-0.782340\pi$
$659$	−39.7992			−1.55036			−0.775179	+	0.631742i	$0.782340\pi$
$660$	−4.04080	−	3.91792i	−0.157288	−	0.152505i
$661$	−13.0969			−0.509409			−0.254704	−	0.967019i	$-0.581978\pi$
$661$	−13.0969			−0.509409			−0.254704	+	0.967019i	$0.581978\pi$
$662$			48.2573i			1.87557i
$663$			0.901992i			0.0350305i
$664$	28.2150			1.09496
$665$	−1.14034	−	1.10566i	−0.0442205	−	0.0428757i
$666$	−2.11578			−0.0819846
$667$			16.7103i			0.647027i
$668$			15.4166i			0.596485i
$669$	2.78757			0.107774
$670$	33.0591	−	34.0960i	1.27719	−	1.31724i
$671$	36.5959			1.41277
$672$		−	0.687509i		−	0.0265212i
$673$		−	22.2830i		−	0.858946i	−0.903080	−	0.429473i	$-0.858699\pi$
$673$		−	22.2830i		−	0.858946i	0.903080	−	0.429473i	$-0.141301\pi$
$674$	−33.1411			−1.27655
$675$	−6.05072	+	0.186893i	−0.232892	+	0.00719350i
$676$	22.5699			0.868073
$677$		−	10.5576i		−	0.405762i	−0.979203	−	0.202881i	$-0.934970\pi$
$677$		−	10.5576i		−	0.405762i	0.979203	−	0.202881i	$-0.0650305\pi$
$678$			3.26282i			0.125308i
$679$	8.58084			0.329303
$680$	4.92163	−	5.07599i	0.188736	−	0.194655i
$681$	−0.365822			−0.0140183
$682$		−	92.8766i		−	3.55643i
$683$		−	1.80882i		−	0.0692128i	−0.999401	−	0.0346064i	$-0.988982\pi$
$683$		−	1.80882i		−	0.0692128i	0.999401	−	0.0346064i	$-0.0110178\pi$
$684$	−10.3436			−0.395499
$685$	−0.391683	−	0.379772i	−0.0149655	−	0.0145103i
$686$	−21.4316			−0.818261
$687$		−	0.660022i		−	0.0251814i
$688$		−	4.68050i		−	0.178442i
$689$	−50.8755			−1.93820
$690$	−3.42028	−	3.31627i	−0.130208	−	0.126248i
$691$	3.84649			0.146327			0.0731636	−	0.997320i	$-0.476690\pi$
$691$	3.84649			0.146327			0.0731636	+	0.997320i	$0.476690\pi$
$692$		−	27.2493i		−	1.03586i
$693$			7.44729i			0.282899i
$694$	50.9550			1.93423
$695$	19.7500	−	20.3695i	0.749161	−	0.772658i
$696$	−2.37146			−0.0898899
$697$			6.15198i			0.233023i
$698$			14.0412i			0.531467i
$699$	−1.09365			−0.0413657
$700$	−11.4986	+	0.355166i	−0.434608	+	0.0134240i
$701$	43.9484			1.65991			0.829955	−	0.557830i	$-0.188366\pi$
$701$	43.9484			1.65991			0.829955	+	0.557830i	$0.188366\pi$
$702$		−	12.4493i		−	0.469867i
$703$			0.320739i			0.0120969i
$704$	46.5690			1.75514
$705$	−1.40399	+	1.44802i	−0.0528772	+	0.0545357i
$706$	−48.3602			−1.82006
$707$		−	4.64018i		−	0.174512i
$708$			1.36747i			0.0513926i
$709$	−45.9706			−1.72646			−0.863232	−	0.504808i	$-0.831563\pi$
$709$	−45.9706			−1.72646			−0.863232	+	0.504808i	$0.831563\pi$
$710$	34.8515	+	33.7917i	1.30796	+	1.26818i
$711$	−21.9743			−0.824100
$712$			36.5274i			1.36892i
$713$		−	49.3083i		−	1.84661i
$714$	0.321786			0.0120425
$715$	26.2356	+	25.4378i	0.981156	+	0.951318i
$716$	14.8278			0.554141
$717$		−	0.735674i		−	0.0274742i
$718$		−	42.8142i		−	1.59781i
$719$	−6.24809			−0.233014			−0.116507	−	0.993190i	$-0.537170\pi$
$719$	−6.24809			−0.233014			−0.116507	+	0.993190i	$0.537170\pi$
$720$	−2.73388	+	2.81963i	−0.101886	+	0.105081i
$721$	5.98131			0.222755
$722$		−	41.5091i		−	1.54481i
$723$			1.55798i			0.0579420i
$724$	19.9794			0.742529
$725$	−0.569802	−	18.4476i	−0.0211619	−	0.685125i
$726$	1.20121			0.0445811
$727$			49.9606i			1.85294i	0.376372	+	0.926469i	$0.377171\pi$
$727$			49.9606i			1.85294i	−0.376372	+	0.926469i	$0.622829\pi$
$728$		−	9.59725i		−	0.355698i
$729$	24.7962			0.918376
$730$	8.18222	−	8.43886i	0.302838	−	0.312336i
$731$	−7.88454			−0.291620
$732$			6.79693i			0.251222i
$733$			37.4111i			1.38181i	0.722945	+	0.690906i	$0.242788\pi$
$733$			37.4111i			1.38181i	−0.722945	+	0.690906i	$0.757212\pi$
$734$	−54.6741			−2.01806
$735$	2.13081	+	2.06601i	0.0785960	+	0.0762058i
$736$	22.4031			0.825790
$737$			33.7559i			1.24342i
$738$		−	42.1606i		−	1.55195i
$739$	−38.9348			−1.43224			−0.716120	−	0.697978i	$-0.754084\pi$
$739$	−38.9348			−1.43224			−0.716120	+	0.697978i	$0.754084\pi$
$740$	1.66781	+	1.61709i	0.0613099	+	0.0594454i
$741$	−0.937080			−0.0344245
$742$			18.1499i			0.666303i
$743$		−	13.0253i		−	0.477850i	−0.971038	−	0.238925i	$-0.923205\pi$
$743$		−	13.0253i		−	0.477850i	0.971038	−	0.238925i	$-0.0767951\pi$
$744$	6.99762			0.256545
$745$	18.6945	−	19.2809i	0.684914	−	0.706396i
$746$	41.7213			1.52753
$747$			26.4019i			0.965996i
$748$			12.3881i			0.452952i
$749$	−5.53878			−0.202383
$750$	3.88893	+	3.54440i	0.142004	+	0.129423i
$751$	−42.3558			−1.54559			−0.772793	−	0.634659i	$-0.781141\pi$
$751$	−42.3558			−1.54559			−0.772793	+	0.634659i	$0.781141\pi$
$752$			2.63528i			0.0960989i
$753$			1.22845i			0.0447671i
$754$	37.9556			1.38226
$755$	−13.9409	+	14.3781i	−0.507361	+	0.523274i
$756$	−2.78566			−0.101313
$757$			4.20834i			0.152955i	0.997071	+	0.0764773i	$0.0243673\pi$
$757$			4.20834i			0.152955i	−0.997071	+	0.0764773i	$0.975633\pi$
$758$			13.3075i			0.483349i
$759$	3.38616			0.122910
$760$	5.27345	+	5.11308i	0.191288	+	0.185471i
$761$	31.8563			1.15479			0.577395	−	0.816465i	$-0.304069\pi$
$761$	31.8563			1.15479			0.577395	+	0.816465i	$0.304069\pi$
$762$			9.95364i			0.360582i
$763$			2.12852i			0.0770576i
$764$	33.3062			1.20498
$765$	4.74981	+	4.60536i	0.171730	+	0.166507i
$766$	81.8293			2.95661
$767$		−	8.87853i		−	0.320585i
$768$			3.99113i			0.144017i
$769$	7.43010			0.267936			0.133968	−	0.990986i	$-0.457228\pi$
$769$	7.43010			0.267936			0.133968	+	0.990986i	$0.457228\pi$
$770$	9.07493	−	9.35957i	0.327038	−	0.337295i
$771$	−1.24563			−0.0448604
$772$		−	28.5217i		−	1.02652i
$773$		−	7.33384i		−	0.263780i	−0.991264	−	0.131890i	$-0.957895\pi$
$773$		−	7.33384i		−	0.263780i	0.991264	−	0.131890i	$-0.0421045\pi$
$774$	54.0341			1.94221
$775$	1.68135	+	54.4344i	0.0603959	+	1.95534i
$776$	−39.6817			−1.42449
$777$			0.0428901i			0.00153867i
$778$			42.3012i			1.51657i
$779$	−6.39130			−0.228992
$780$	−4.72453	+	4.87272i	−0.169165	+	0.174471i
$781$	−34.5039			−1.23465
$782$			10.4857i			0.374968i
$783$		−	4.46910i		−	0.159713i
$784$	3.87789			0.138496
$785$	31.9670	+	30.9949i	1.14095	+	1.10625i
$786$	4.10007			0.146244
$787$		−	35.7257i		−	1.27348i	−0.771077	−	0.636742i	$-0.780282\pi$
$787$		−	35.7257i		−	1.27348i	0.771077	−	0.636742i	$-0.219718\pi$
$788$		−	45.5570i		−	1.62290i
$789$	2.49942			0.0889817
$790$	27.6167	+	26.7769i	0.982559	+	0.952679i
$791$	−4.74024			−0.168544
$792$		−	34.4397i		−	1.22376i
$793$		−	44.1303i		−	1.56711i
$794$	10.7496			0.381489
$795$	3.62451	−	3.73819i	0.128548	−	0.132580i
$796$	−30.8036			−1.09180
$797$			28.2025i			0.998985i	0.866318	+	0.499492i	$0.166480\pi$
$797$			28.2025i			0.998985i	−0.866318	+	0.499492i	$0.833520\pi$
$798$			0.334304i			0.0118342i
$799$	4.43927			0.157050
$800$	−24.7322	+	0.763919i	−0.874415	+	0.0270086i
$801$	−34.1801			−1.20769
$802$		−	63.4769i		−	2.24145i
$803$			8.35468i			0.294830i
$804$	−6.26946			−0.221107
$805$	4.81789	−	4.96900i	0.169808	−	0.175134i
$806$	−111.998			−3.94496
$807$			2.01663i			0.0709886i
$808$			21.4583i			0.754901i
$809$	41.9318			1.47424			0.737121	−	0.675761i	$-0.236185\pi$
$809$	41.9318			1.47424			0.737121	+	0.675761i	$0.236185\pi$
$810$	−32.0907	−	31.1148i	−1.12755	−	1.09326i
$811$	4.46604			0.156824			0.0784119	−	0.996921i	$-0.475015\pi$
$811$	4.46604			0.156824			0.0784119	+	0.996921i	$0.475015\pi$
$812$		−	8.49297i		−	0.298045i
$813$		−	4.12546i		−	0.144686i
$814$	−2.63253			−0.0922702
$815$	35.9619	+	34.8682i	1.25969	+	1.22138i
$816$	−0.120617			−0.00422243
$817$		−	8.19125i		−	0.286576i
$818$		−	3.20233i		−	0.111967i
$819$	8.98053			0.313805
$820$	−32.2234	+	33.2341i	−1.12529	+	1.16058i
$821$	19.6083			0.684334			0.342167	−	0.939639i	$-0.388839\pi$
$821$	19.6083			0.684334			0.342167	+	0.939639i	$0.388839\pi$
$822$			0.114827i			0.00400504i
$823$		−	46.9128i		−	1.63528i	−0.575732	−	0.817638i	$-0.695283\pi$
$823$		−	46.9128i		−	1.63528i	0.575732	−	0.817638i	$-0.304717\pi$
$824$	−27.6603			−0.963592
$825$	−3.73819	+	0.115464i	−0.130147	+	0.00401994i
$826$	−3.16742			−0.110209
$827$			14.0134i			0.487295i	0.969864	+	0.243648i	$0.0783440\pi$
$827$			14.0134i			0.487295i	−0.969864	+	0.243648i	$0.921656\pi$
$828$		−	45.0722i		−	1.56637i
$829$	40.5206			1.40734			0.703669	−	0.710528i	$-0.251544\pi$
$829$	40.5206			1.40734			0.703669	+	0.710528i	$0.251544\pi$
$830$	32.1722	−	33.1813i	1.11671	−	1.15174i
$831$	3.69127			0.128049
$832$		−	56.1566i		−	1.94688i
$833$		−	6.53251i		−	0.226338i
$834$	−5.97154			−0.206778
$835$	7.35468	+	7.13102i	0.254519	+	0.246779i
$836$	−12.8700			−0.445117
$837$			13.1873i			0.455818i
$838$			76.9694i			2.65887i
$839$	−8.68840			−0.299957			−0.149978	−	0.988689i	$-0.547920\pi$
$839$	−8.68840			−0.299957			−0.149978	+	0.988689i	$0.547920\pi$
$840$	0.705180	+	0.683735i	0.0243310	+	0.0235911i
$841$	−15.3745			−0.530156
$842$			10.6805i			0.368074i
$843$		−	2.17374i		−	0.0748675i
$844$	−25.9941			−0.894754
$845$	10.4398	−	10.7673i	0.359141	−	0.370406i
$846$	−30.4230			−1.04597
$847$			1.74513i			0.0599633i
$848$		−	6.80321i		−	0.233623i
$849$	−2.04756			−0.0702720
$850$	−0.357550	−	11.5758i	−0.0122638	−	0.397047i
$851$	−1.39761			−0.0479096
$852$		−	6.40839i		−	0.219548i
$853$			23.1523i			0.792721i	0.918095	+	0.396361i	$0.129727\pi$
$853$			23.1523i			0.792721i	−0.918095	+	0.396361i	$0.870273\pi$
$854$	−15.7435			−0.538731
$855$	−4.78451	+	4.93458i	−0.163627	+	0.168759i
$856$	25.6139			0.875464
$857$			43.6043i			1.48950i	0.667346	+	0.744748i	$0.267430\pi$
$857$			43.6043i			1.48950i	−0.667346	+	0.744748i	$0.732570\pi$
$858$		−	7.69127i		−	0.262576i
$859$	14.2532			0.486314			0.243157	−	0.969987i	$-0.421817\pi$
$859$	14.2532			0.486314			0.243157	+	0.969987i	$0.421817\pi$
$860$	−42.5936	−	41.2983i	−1.45243	−	1.40826i
$861$	−0.854661			−0.0291268
$862$			16.9970i			0.578921i
$863$			24.1162i			0.820926i	0.911877	+	0.410463i	$0.134633\pi$
$863$			24.1162i			0.820926i	−0.911877	+	0.410463i	$0.865367\pi$
$864$	−5.99161			−0.203839
$865$	−12.9996	−	12.6043i	−0.442001	−	0.428559i
$866$	−35.5516			−1.20809
$867$			0.203185i			0.00690052i
$868$			25.0608i			0.850617i
$869$	−27.3413			−0.927489
$870$	−2.70406	+	2.78887i	−0.0916761	+	0.0945515i
$871$	40.7056			1.37926
$872$		−	9.84325i		−	0.333335i
$873$		−	37.1318i		−	1.25672i
$874$	−10.8936			−0.368482
$875$	−5.14933	+	5.64987i	−0.174079	+	0.191000i
$876$	−1.55171			−0.0524274
$877$		−	1.56540i		−	0.0528599i	−0.999651	−	0.0264299i	$-0.991586\pi$
$877$		−	1.56540i		−	0.0528599i	0.999651	−	0.0264299i	$-0.00841389\pi$
$878$			12.9793i			0.438030i
$879$	4.17238			0.140731
$880$	−3.40160	+	3.50829i	−0.114668	+	0.118265i
$881$	−39.0973			−1.31722			−0.658610	−	0.752484i	$-0.728855\pi$
$881$	−39.0973			−1.31722			−0.658610	+	0.752484i	$0.728855\pi$
$882$			44.7684i			1.50743i
$883$			30.9249i			1.04071i	0.853951	+	0.520354i	$0.174200\pi$
$883$			30.9249i			1.04071i	−0.853951	+	0.520354i	$0.825800\pi$
$884$	14.9385			0.502436
$885$	0.652370	+	0.632531i	0.0219292	+	0.0212623i
$886$	−68.6397			−2.30599
$887$		−	45.1839i		−	1.51713i	−0.651599	−	0.758564i	$-0.725901\pi$
$887$		−	45.1839i		−	1.51713i	0.651599	−	0.758564i	$-0.274099\pi$
$888$		−	0.198343i		−	0.00665597i
$889$	−14.4607			−0.484997
$890$	42.9567	+	41.6504i	1.43991	+	1.39612i
$891$	31.7706			1.06436
$892$		−	46.1670i		−	1.54578i
$893$			4.61196i			0.154333i
$894$	−5.65241			−0.189045
$895$	6.85869	−	7.07381i	0.229261	−	0.236451i
$896$	−13.2666			−0.443206
$897$		−	4.08330i		−	0.136338i
$898$			79.7265i			2.66051i
$899$	−40.2056			−1.34093
$900$	1.53691	+	49.7579i	0.0512302	+	1.65860i
$901$	−11.4603			−0.381799
$902$		−	52.4579i		−	1.74666i
$903$		−	1.09536i		−	0.0364511i
$904$	21.9211			0.729083
$905$	9.24160	−	9.53146i	0.307201	−	0.316836i
$906$	4.21512			0.140038
$907$		−	33.9862i		−	1.12849i	−0.825606	−	0.564246i	$-0.809167\pi$
$907$		−	33.9862i		−	1.12849i	0.825606	−	0.564246i	$-0.190833\pi$
$908$			6.05862i			0.201062i
$909$	−20.0794			−0.665992
$910$	−11.2865	−	10.9433i	−0.374144	−	0.362766i
$911$	22.1997			0.735509			0.367754	−	0.929923i	$-0.380127\pi$
$911$	22.1997			0.735509			0.367754	+	0.929923i	$0.380127\pi$
$912$		−	0.125309i		−	0.00414939i
$913$			32.8503i			1.08719i
$914$	−21.5993			−0.714440
$915$	3.24257	+	3.14396i	0.107196	+	0.103936i
$916$	−10.9311			−0.361173
$917$			5.95660i			0.196704i
$918$		−	2.80435i		−	0.0925574i
$919$	8.84668			0.291825			0.145913	−	0.989297i	$-0.453388\pi$
$919$	8.84668			0.291825			0.145913	+	0.989297i	$0.453388\pi$
$920$	−22.2801	+	22.9789i	−0.734554	+	0.757593i
$921$	2.66533			0.0878256
$922$			44.4062i			1.46244i
$923$			41.6076i			1.36953i
$924$	−1.72101			−0.0566169
$925$	1.54291	−	0.0476569i	0.0507306	−	0.00156695i
$926$	45.4881			1.49483
$927$		−	25.8828i		−	0.850104i
$928$		−	18.2673i		−	0.599655i
$929$	−13.9941			−0.459131			−0.229566	−	0.973293i	$-0.573731\pi$
$929$	−13.9941			−0.459131			−0.229566	+	0.973293i	$0.573731\pi$
$930$	7.97904	−	8.22930i	0.261643	−	0.269849i
$931$	6.78663			0.222423
$932$			18.1127i			0.593302i
$933$			2.51230i			0.0822490i
$934$	48.8794			1.59938
$935$	5.90990	+	5.73017i	0.193274	+	0.187397i
$936$	−41.5301			−1.35745
$937$			5.19129i			0.169592i	0.996398	+	0.0847960i	$0.0270239\pi$
$937$			5.19129i			0.169592i	−0.996398	+	0.0847960i	$0.972976\pi$
$938$		−	14.5217i		−	0.474151i
$939$	1.04843			0.0342144
$940$	23.9817	+	23.2524i	0.782197	+	0.758409i
$941$	−26.2746			−0.856527			−0.428264	−	0.903654i	$-0.640875\pi$
$941$	−26.2746			−0.856527			−0.428264	+	0.903654i	$0.640875\pi$
$942$		−	9.37150i		−	0.305340i
$943$		−	27.8499i		−	0.906919i
$944$	1.18726			0.0386420
$945$	−1.28852	+	1.32894i	−0.0419156	+	0.0432303i
$946$	67.2313			2.18588
$947$		−	13.9416i		−	0.453040i	−0.974007	−	0.226520i	$-0.927265\pi$
$947$		−	13.9416i		−	0.453040i	0.974007	−	0.226520i	$-0.0727348\pi$
$948$		−	5.07807i		−	0.164928i
$949$	10.0747			0.327040
$950$	12.0261	−	0.371458i	0.390179	−	0.0120517i
$951$	−4.82878			−0.156584
$952$		−	2.16190i		−	0.0700676i
$953$			19.3298i			0.626154i	0.949728	+	0.313077i	$0.101360\pi$
$953$			19.3298i			0.626154i	−0.949728	+	0.313077i	$0.898640\pi$
$954$	78.5397			2.54282
$955$	15.4060	−	15.8892i	0.498527	−	0.514163i
$956$	−12.1840			−0.394059
$957$		−	2.76105i		−	0.0892521i
$958$			66.4217i			2.14599i
$959$	−0.166821			−0.00538692
$960$	4.12624	+	4.00075i	0.133174	+	0.129124i
$961$	87.6372			2.82701
$962$			3.17451i			0.102350i
$963$			23.9679i			0.772355i
$964$	25.8028			0.831053
$965$	−13.6067	−	13.1929i	−0.438015	−	0.424694i
$966$	−1.45672			−0.0468692
$967$		−	27.0663i		−	0.870393i	−0.900335	−	0.435197i	$-0.856679\pi$
$967$		−	27.0663i		−	0.870393i	0.900335	−	0.435197i	$-0.143321\pi$
$968$		−	8.07027i		−	0.259388i
$969$	−0.211089			−0.00678115
$970$	−45.2471	+	46.6663i	−1.45280	+	1.49836i
$971$	9.36344			0.300487			0.150244	−	0.988649i	$-0.451994\pi$
$971$	9.36344			0.300487			0.150244	+	0.988649i	$0.451994\pi$
$972$			18.1233i			0.581304i
$973$		−	8.67550i		−	0.278124i
$974$	−90.8297			−2.91037
$975$	0.139236	+	4.50781i	0.00445911	+	0.144365i
$976$	5.90121			0.188893
$977$		−	14.1127i		−	0.451506i	−0.974185	−	0.225753i	$-0.927516\pi$
$977$		−	14.1127i		−	0.451506i	0.974185	−	0.225753i	$-0.0724842\pi$
$978$		−	10.5426i		−	0.337116i
$979$	−42.5282			−1.35921
$980$	34.2166	−	35.2898i	1.09301	−	1.12729i
$981$	9.21072			0.294076
$982$		−	84.1961i		−	2.68680i
$983$		−	48.9970i		−	1.56276i	−0.624054	−	0.781381i	$-0.714515\pi$
$983$		−	48.9970i		−	1.56276i	0.624054	−	0.781381i	$-0.285485\pi$
$984$	3.95234			0.125996
$985$	−21.7336	−	21.0727i	−0.692490	−	0.671431i
$986$	8.54996			0.272286
$987$			0.616723i			0.0196305i
$988$			15.5196i			0.493745i
$989$	35.6932			1.13498
$990$	−40.5015	−	39.2698i	−1.28722	−	1.24808i
$991$	−20.7542			−0.659279			−0.329639	−	0.944107i	$-0.606927\pi$
$991$	−20.7542			−0.659279			−0.329639	+	0.944107i	$0.606927\pi$
$992$			53.9026i			1.71141i
$993$		−	4.23317i		−	0.134336i
$994$	14.8435			0.470808
$995$	−14.2484	+	14.6953i	−0.451704	+	0.465872i
$996$	−6.10126			−0.193326
$997$			9.54268i			0.302220i	0.988517	+	0.151110i	$0.0482847\pi$
$997$			9.54268i			0.302220i	−0.988517	+	0.151110i	$0.951715\pi$
$998$			19.7360i			0.624732i
$999$	0.373785			0.0118260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.2.b.a.69.8	yes	8
3.2	odd	2		765.2.b.c.154.1		8
4.3	odd	2		1360.2.e.d.1089.5		8
5.2	odd	4		425.2.a.g.1.1		4
5.3	odd	4		425.2.a.h.1.4		4
5.4	even	2	inner	85.2.b.a.69.1	✓	8
15.2	even	4		3825.2.a.bj.1.4		4
15.8	even	4		3825.2.a.bh.1.1		4
15.14	odd	2		765.2.b.c.154.8		8
17.16	even	2		1445.2.b.e.579.8		8
20.3	even	4		6800.2.a.bt.1.3		4
20.7	even	4		6800.2.a.bw.1.2		4
20.19	odd	2		1360.2.e.d.1089.4		8
85.33	odd	4		7225.2.a.w.1.4		4
85.67	odd	4		7225.2.a.v.1.1		4
85.84	even	2		1445.2.b.e.579.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.b.a.69.1	✓	8	5.4	even	2	inner
85.2.b.a.69.8	yes	8	1.1	even	1	trivial
425.2.a.g.1.1		4	5.2	odd	4
425.2.a.h.1.4		4	5.3	odd	4
765.2.b.c.154.1		8	3.2	odd	2
765.2.b.c.154.8		8	15.14	odd	2
1360.2.e.d.1089.4		8	20.19	odd	2
1360.2.e.d.1089.5		8	4.3	odd	2
1445.2.b.e.579.1		8	85.84	even	2
1445.2.b.e.579.8		8	17.16	even	2
3825.2.a.bh.1.1		4	15.8	even	4
3825.2.a.bj.1.4		4	15.2	even	4
6800.2.a.bt.1.3		4	20.3	even	4
6800.2.a.bw.1.2		4	20.7	even	4
7225.2.a.v.1.1		4	85.67	odd	4
7225.2.a.w.1.4		4	85.33	odd	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:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.31627i q^{2} -0.203185i q^{3} -3.36509 q^{4} +(-1.55654 + 1.60536i) q^{5} +0.470630 q^{6} +0.683735i q^{7} -3.16190i q^{8} +2.95872 q^{9} +(-3.71844 - 3.60536i) q^{10} +3.68135 q^{11} +0.683735i q^{12} -4.43927i q^{13} -1.58371 q^{14} +(0.326185 + 0.316265i) q^{15} +0.593630 q^{16} -1.00000i q^{17} +6.85317i q^{18} +1.03890 q^{19} +(5.23789 - 5.40218i) q^{20} +0.138925 q^{21} +8.52699i q^{22} +4.52699i q^{23} -0.642450 q^{24} +(-0.154365 - 4.99762i) q^{25} +10.2825 q^{26} -1.21072i q^{27} -2.30083i q^{28} +3.69127 q^{29} +(-0.732555 + 0.755531i) q^{30} -10.8921 q^{31} -4.94880i q^{32} -0.747995i q^{33} +2.31627 q^{34} +(-1.09764 - 1.06426i) q^{35} -9.95633 q^{36} +0.308729i q^{37} +2.40637i q^{38} -0.901992 q^{39} +(5.07599 + 4.92163i) q^{40} -6.15198 q^{41} +0.321786i q^{42} -7.88454i q^{43} -12.3881 q^{44} +(-4.60536 + 4.74981i) q^{45} -10.4857 q^{46} +4.43927i q^{47} -0.120617i q^{48} +6.53251 q^{49} +(11.5758 - 0.357550i) q^{50} -0.203185 q^{51} +14.9385i q^{52} -11.4603i q^{53} +2.80435 q^{54} +(-5.73017 + 5.90990i) q^{55} +2.16190 q^{56} -0.211089i q^{57} +8.54996i q^{58} +2.00000 q^{59} +(-1.09764 - 1.06426i) q^{60} +9.94089 q^{61} -25.2289i q^{62} +2.02298i q^{63} +12.6500 q^{64} +(7.12662 + 6.90990i) q^{65} +1.73255 q^{66} +9.16944i q^{67} +3.36509i q^{68} +0.919815 q^{69} +(2.46511 - 2.54243i) q^{70} -9.37262 q^{71} -9.35517i q^{72} +2.26946i q^{73} -0.715099 q^{74} +(-1.01544 + 0.0313646i) q^{75} -3.49599 q^{76} +2.51707i q^{77} -2.08925i q^{78} -7.42696 q^{79} +(-0.924009 + 0.952991i) q^{80} +8.63015 q^{81} -14.2496i q^{82} +8.92344i q^{83} -0.467493 q^{84} +(1.60536 + 1.55654i) q^{85} +18.2627 q^{86} -0.750010i q^{87} -11.6401i q^{88} -11.5523 q^{89} +(-11.0018 - 10.6672i) q^{90} +3.03528 q^{91} -15.2337i q^{92} +2.21310i q^{93} -10.2825 q^{94} +(-1.61709 + 1.66781i) q^{95} -1.00552 q^{96} -12.5500i q^{97} +15.1310i q^{98} +10.8921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 2 q^{5} - 8 q^{9} + 6 q^{10} - 4 q^{11} + 12 q^{14} - 8 q^{16} - 8 q^{19} - 2 q^{20} + 24 q^{21} + 12 q^{24} - 12 q^{25} + 8 q^{29} - 16 q^{30} - 24 q^{31} + 4 q^{34} + 44 q^{39} + 22 q^{40}+ \cdots + 24 q^{99}+O(q^{100}) \) 8 * q - 8 * q^4 - 2 * q^5 - 8 * q^9 + 6 * q^10 - 4 * q^11 + 12 * q^14 - 8 * q^16 - 8 * q^19 - 2 * q^20 + 24 * q^21 + 12 * q^24 - 12 * q^25 + 8 * q^29 - 16 * q^30 - 24 * q^31 + 4 * q^34 + 44 * q^39 + 22 * q^40 - 12 * q^41 - 32 * q^44 - 22 * q^45 - 8 * q^46 - 16 * q^49 + 44 * q^50 - 8 * q^51 - 20 * q^54 - 8 * q^55 - 8 * q^56 + 16 * q^59 + 12 * q^61 + 48 * q^64 + 20 * q^65 + 24 * q^66 - 8 * q^69 + 40 * q^70 - 20 * q^71 - 40 * q^74 + 4 * q^75 - 24 * q^76 + 24 * q^79 - 26 * q^80 - 8 * q^81 - 72 * q^84 - 2 * q^85 + 40 * q^86 - 48 * q^89 - 74 * q^90 + 36 * q^91 + 4 * q^95 + 16 * q^96 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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