LMFDB - Embedded newform 1045.2.b.c.419.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(419,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.419");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + \cdots + 4 $ x^20 + 26x^18 + 281x^16 + 1640x^14 + 5623x^12 + 11551x^10 + 13894x^8 + 9095x^6 + 2753x^4 + 276*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.2
Root			$-2.33380i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.c.419.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33380i q^{2} -1.97214i q^{3} -3.44661 q^{4} +(-1.75107 - 1.39059i) q^{5} -4.60257 q^{6} +4.79282i q^{7} +3.37609i q^{8} -0.889329 q^{9} +O(q^{10})$	\(q-2.33380i q^{2} -1.97214i q^{3} -3.44661 q^{4} +(-1.75107 - 1.39059i) q^{5} -4.60257 q^{6} +4.79282i q^{7} +3.37609i q^{8} -0.889329 q^{9} +(-3.24536 + 4.08665i) q^{10} +1.00000 q^{11} +6.79719i q^{12} +6.97588i q^{13} +11.1855 q^{14} +(-2.74244 + 3.45336i) q^{15} +0.985888 q^{16} -1.90231i q^{17} +2.07551i q^{18} +1.00000 q^{19} +(6.03526 + 4.79282i) q^{20} +9.45211 q^{21} -2.33380i q^{22} +4.07925i q^{23} +6.65811 q^{24} +(1.13251 + 4.87005i) q^{25} +16.2803 q^{26} -4.16254i q^{27} -16.5190i q^{28} -2.15857 q^{29} +(8.05943 + 6.40030i) q^{30} -10.6067 q^{31} +4.45131i q^{32} -1.97214i q^{33} -4.43961 q^{34} +(6.66486 - 8.39258i) q^{35} +3.06517 q^{36} +5.25363i q^{37} -2.33380i q^{38} +13.7574 q^{39} +(4.69476 - 5.91177i) q^{40} -4.11322 q^{41} -22.0593i q^{42} +2.83644i q^{43} -3.44661 q^{44} +(1.55728 + 1.23669i) q^{45} +9.52013 q^{46} +9.58734i q^{47} -1.94431i q^{48} -15.9712 q^{49} +(11.3657 - 2.64304i) q^{50} -3.75162 q^{51} -24.0431i q^{52} -4.41619i q^{53} -9.71451 q^{54} +(-1.75107 - 1.39059i) q^{55} -16.1810 q^{56} -1.97214i q^{57} +5.03767i q^{58} +3.08053 q^{59} +(9.45211 - 11.9024i) q^{60} -5.73895 q^{61} +24.7539i q^{62} -4.26240i q^{63} +12.3602 q^{64} +(9.70061 - 12.2153i) q^{65} -4.60257 q^{66} -5.17167i q^{67} +6.55652i q^{68} +8.04484 q^{69} +(-19.5866 - 15.5544i) q^{70} -8.60081 q^{71} -3.00245i q^{72} +2.32205i q^{73} +12.2609 q^{74} +(9.60442 - 2.23346i) q^{75} -3.44661 q^{76} +4.79282i q^{77} -32.1070i q^{78} +15.8136 q^{79} +(-1.72636 - 1.37097i) q^{80} -10.8771 q^{81} +9.59942i q^{82} +2.49359i q^{83} -32.5777 q^{84} +(-2.64534 + 3.33108i) q^{85} +6.61968 q^{86} +4.25700i q^{87} +3.37609i q^{88} -8.26610 q^{89} +(2.88619 - 3.63437i) q^{90} -33.4342 q^{91} -14.0596i q^{92} +20.9179i q^{93} +22.3749 q^{94} +(-1.75107 - 1.39059i) q^{95} +8.77861 q^{96} -0.339216i q^{97} +37.2735i q^{98} -0.889329 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) $ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9	$ 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) $ 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$.

$n$	$496$	$761$	$837$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	2.33380i		−	1.65024i	−0.564955	−	0.825122i	$-0.691106\pi$
$2$		−	2.33380i		−	1.65024i	0.564955	−	0.825122i	$-0.308894\pi$
$3$		−	1.97214i		−	1.13861i	−0.822125	−	0.569307i	$-0.807211\pi$
$3$		−	1.97214i		−	1.13861i	0.822125	−	0.569307i	$-0.192789\pi$
$4$	−3.44661			−1.72330
$5$	−1.75107	−	1.39059i	−0.783103	−	0.621892i
$5$	−1.75107	−	1.39059i	−0.783103	−	0.621892i
$6$	−4.60257			−1.87899
$7$			4.79282i			1.81152i	0.423794	+	0.905759i	$0.360698\pi$
$7$			4.79282i			1.81152i	−0.423794	+	0.905759i	$0.639302\pi$
$8$			3.37609i			1.19363i
$9$	−0.889329			−0.296443
$10$	−3.24536	+	4.08665i	−1.02627	+	1.29231i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$			6.79719i			1.96218i
$13$			6.97588i			1.93476i	0.253325	+	0.967381i	$0.418476\pi$
$13$			6.97588i			1.93476i	−0.253325	+	0.967381i	$0.581524\pi$
$14$	11.1855			2.98944
$15$	−2.74244	+	3.45336i	−0.708095	+	0.891653i
$16$	0.985888			0.246472
$17$		−	1.90231i		−	0.461378i	−0.973028	−	0.230689i	$-0.925902\pi$
$17$		−	1.90231i		−	0.461378i	0.973028	−	0.230689i	$-0.0740980\pi$
$18$			2.07551i			0.489203i
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	6.03526	+	4.79282i	1.34952	+	1.07171i
$21$	9.45211			2.06262
$22$		−	2.33380i		−	0.497567i
$23$			4.07925i			0.850582i	0.905057	+	0.425291i	$0.139828\pi$
$23$			4.07925i			0.850582i	−0.905057	+	0.425291i	$0.860172\pi$
$24$	6.65811			1.35908
$25$	1.13251	+	4.87005i	0.226502	+	0.974011i
$26$	16.2803			3.19283
$27$		−	4.16254i		−	0.801080i
$28$		−	16.5190i		−	3.12179i
$29$	−2.15857			−0.400837			−0.200418	−	0.979710i	$-0.564230\pi$
$29$	−2.15857			−0.400837			−0.200418	+	0.979710i	$0.564230\pi$
$30$	8.05943	+	6.40030i	1.47144	+	1.16853i
$31$	−10.6067			−1.90502			−0.952511	−	0.304505i	$-0.901509\pi$
$31$	−10.6067			−1.90502			−0.952511	+	0.304505i	$0.901509\pi$
$32$			4.45131i			0.786889i
$33$		−	1.97214i		−	0.343305i
$34$	−4.43961			−0.761386
$35$	6.66486	−	8.39258i	1.12657	−	1.41861i
$36$	3.06517			0.510861
$37$			5.25363i			0.863691i	0.901948	+	0.431845i	$0.142137\pi$
$37$			5.25363i			0.863691i	−0.901948	+	0.431845i	$0.857863\pi$
$38$		−	2.33380i		−	0.378592i
$39$	13.7574			2.20295
$40$	4.69476	−	5.91177i	0.742307	−	0.934734i
$41$	−4.11322			−0.642377			−0.321188	−	0.947015i	$-0.604082\pi$
$41$	−4.11322			−0.642377			−0.321188	+	0.947015i	$0.604082\pi$
$42$		−	22.0593i		−	3.40382i
$43$			2.83644i			0.432553i	0.976332	+	0.216277i	$0.0693913\pi$
$43$			2.83644i			0.432553i	−0.976332	+	0.216277i	$0.930609\pi$
$44$	−3.44661			−0.519596
$45$	1.55728	+	1.23669i	0.232145	+	0.184355i
$46$	9.52013			1.40367
$47$			9.58734i			1.39846i	0.714898	+	0.699229i	$0.246473\pi$
$47$			9.58734i			1.39846i	−0.714898	+	0.699229i	$0.753527\pi$
$48$		−	1.94431i		−	0.280637i
$49$	−15.9712			−2.28159
$50$	11.3657	−	2.64304i	1.60735	−	0.373783i
$51$	−3.75162			−0.525332
$52$		−	24.0431i		−	3.33418i
$53$		−	4.41619i		−	0.606610i	−0.952894	−	0.303305i	$-0.901910\pi$
$53$		−	4.41619i		−	0.606610i	0.952894	−	0.303305i	$-0.0980901\pi$
$54$	−9.71451			−1.32198
$55$	−1.75107	−	1.39059i	−0.236115	−	0.187507i
$56$	−16.1810			−2.16228
$57$		−	1.97214i		−	0.261216i
$58$			5.03767i			0.661478i
$59$	3.08053			0.401050			0.200525	−	0.979689i	$-0.435735\pi$
$59$	3.08053			0.401050			0.200525	+	0.979689i	$0.435735\pi$
$60$	9.45211	−	11.9024i	1.22026	−	1.53659i
$61$	−5.73895			−0.734797			−0.367398	−	0.930064i	$-0.619751\pi$
$61$	−5.73895			−0.734797			−0.367398	+	0.930064i	$0.619751\pi$
$62$			24.7539i			3.14375i
$63$		−	4.26240i		−	0.537011i
$64$	12.3602			1.54503
$65$	9.70061	−	12.2153i	1.20321	−	1.51512i
$66$	−4.60257			−0.566537
$67$		−	5.17167i		−	0.631820i	−0.948789	−	0.315910i	$-0.897690\pi$
$67$		−	5.17167i		−	0.631820i	0.948789	−	0.315910i	$-0.102310\pi$
$68$			6.55652i			0.795095i
$69$	8.04484			0.968485
$70$	−19.5866	−	15.5544i	−2.34104	−	1.85911i
$71$	−8.60081			−1.02073			−0.510364	−	0.859958i	$-0.670489\pi$
$71$	−8.60081			−1.02073			−0.510364	+	0.859958i	$0.670489\pi$
$72$		−	3.00245i		−	0.353842i
$73$			2.32205i			0.271776i	0.990724	+	0.135888i	$0.0433887\pi$
$73$			2.32205i			0.271776i	−0.990724	+	0.135888i	$0.956611\pi$
$74$	12.2609			1.42530
$75$	9.60442	−	2.23346i	1.10902	−	0.257898i
$76$	−3.44661			−0.395353
$77$			4.79282i			0.546193i
$78$		−	32.1070i		−	3.63540i
$79$	15.8136			1.77917			0.889584	−	0.456772i	$-0.150994\pi$
$79$	15.8136			1.77917			0.889584	+	0.456772i	$0.150994\pi$
$80$	−1.72636	−	1.37097i	−0.193013	−	0.153279i
$81$	−10.8771			−1.20856
$82$			9.59942i			1.06008i
$83$			2.49359i			0.273707i	0.990591	+	0.136854i	$0.0436990\pi$
$83$			2.49359i			0.273707i	−0.990591	+	0.136854i	$0.956301\pi$
$84$	−32.5777			−3.55452
$85$	−2.64534	+	3.33108i	−0.286927	+	0.361307i
$86$	6.61968			0.713818
$87$			4.25700i			0.456398i
$88$			3.37609i			0.359892i
$89$	−8.26610			−0.876205			−0.438102	−	0.898925i	$-0.644349\pi$
$89$	−8.26610			−0.876205			−0.438102	+	0.898925i	$0.644349\pi$
$90$	2.88619	−	3.63437i	0.304231	−	0.383096i
$91$	−33.4342			−3.50486
$92$		−	14.0596i		−	1.46581i
$93$			20.9179i			2.16908i
$94$	22.3749			2.30780
$95$	−1.75107	−	1.39059i	−0.179656	−	0.142672i
$96$	8.77861			0.895963
$97$		−	0.339216i		−	0.0344422i	−0.999852	−	0.0172211i	$-0.994518\pi$
$97$		−	0.339216i		−	0.0344422i	0.999852	−	0.0172211i	$-0.00548192\pi$
$98$			37.2735i			3.76519i
$99$	−0.889329			−0.0893809
$100$	−3.90331	−	16.7852i	−0.390331	−	1.67852i
$101$	−1.65580			−0.164758			−0.0823789	−	0.996601i	$-0.526252\pi$
$101$	−1.65580			−0.164758			−0.0823789	+	0.996601i	$0.526252\pi$
$102$			8.75552i			0.866925i
$103$		−	4.79043i		−	0.472015i	−0.971751	−	0.236008i	$-0.924161\pi$
$103$		−	4.79043i		−	0.472015i	0.971751	−	0.236008i	$-0.0758390\pi$
$104$	−23.5512			−2.30938
$105$	−16.5513	−	13.1440i	−1.61524	−	1.28273i
$106$	−10.3065			−1.00105
$107$			3.17144i			0.306595i	0.988180	+	0.153298i	$0.0489893\pi$
$107$			3.17144i			0.306595i	−0.988180	+	0.153298i	$0.951011\pi$
$108$			14.3466i			1.38050i
$109$	13.3407			1.27781			0.638905	−	0.769285i	$-0.279388\pi$
$109$	13.3407			1.27781			0.638905	+	0.769285i	$0.279388\pi$
$110$	−3.24536	+	4.08665i	−0.309433	+	0.389646i
$111$	10.3609			0.983411
$112$			4.72519i			0.446488i
$113$		−	5.03001i		−	0.473184i	−0.971609	−	0.236592i	$-0.923970\pi$
$113$		−	5.03001i		−	0.473184i	0.971609	−	0.236592i	$-0.0760304\pi$
$114$	−4.60257			−0.431070
$115$	5.67257	−	7.14306i	0.528970	−	0.666093i
$116$	7.43975			0.690763
$117$		−	6.20385i		−	0.573546i
$118$		−	7.18932i		−	0.661831i
$119$	9.11744			0.835794
$120$	−11.6588	−	9.25872i	−1.06430	−	0.845201i
$121$	1.00000			0.0909091
$122$			13.3935i			1.21259i
$123$			8.11184i			0.731420i
$124$	36.5572			3.28293
$125$	4.78915	−	10.1027i	0.428355	−	0.903611i
$126$	−9.94757			−0.886200
$127$		−	2.31203i		−	0.205159i	−0.994725	−	0.102580i	$-0.967290\pi$
$127$		−	2.31203i		−	0.205159i	0.994725	−	0.102580i	$-0.0327096\pi$
$128$		−	19.9437i		−	1.76279i
$129$	5.59385			0.492511
$130$	−28.5080	−	22.6392i	−2.50031	−	1.98559i
$131$	−6.53445			−0.570917			−0.285459	−	0.958391i	$-0.592146\pi$
$131$	−6.53445			−0.570917			−0.285459	+	0.958391i	$0.592146\pi$
$132$			6.79719i			0.591619i
$133$			4.79282i			0.415591i
$134$	−12.0696			−1.04266
$135$	−5.78839	+	7.28890i	−0.498185	+	0.627329i
$136$	6.42237			0.550713
$137$		−	1.80366i		−	0.154097i	−0.997027	−	0.0770487i	$-0.975450\pi$
$137$		−	1.80366i		−	0.154097i	0.997027	−	0.0770487i	$-0.0245497\pi$
$138$		−	18.7750i		−	1.59824i
$139$	5.61633			0.476371			0.238185	−	0.971220i	$-0.423447\pi$
$139$	5.61633			0.476371			0.238185	+	0.971220i	$0.423447\pi$
$140$	−22.9712	+	28.9259i	−1.94142	+	2.44469i
$141$	18.9076			1.59230
$142$			20.0725i			1.68445i
$143$			6.97588i			0.583353i
$144$	−0.876778			−0.0730649
$145$	3.77982	+	3.00169i	0.313897	+	0.249277i
$146$	5.41920			0.448496
$147$			31.4973i			2.59786i
$148$		−	18.1072i		−	1.48840i
$149$	12.4592			1.02070			0.510349	−	0.859968i	$-0.329516\pi$
$149$	12.4592			1.02070			0.510349	+	0.859968i	$0.329516\pi$
$150$	−5.21245	−	22.4148i	−0.425595	−	1.83016i
$151$	−15.1218			−1.23060			−0.615298	−	0.788294i	$-0.710964\pi$
$151$	−15.1218			−1.23060			−0.615298	+	0.788294i	$0.710964\pi$
$152$			3.37609i			0.273837i
$153$			1.69178i			0.136772i
$154$	11.1855			0.901351
$155$	18.5731	+	14.7496i	1.49183	+	1.18472i
$156$	−47.4164			−3.79635
$157$			11.8420i			0.945096i	0.881305	+	0.472548i	$0.156666\pi$
$157$			11.8420i			0.945096i	−0.881305	+	0.472548i	$0.843334\pi$
$158$		−	36.9057i		−	2.93606i
$159$	−8.70933			−0.690695
$160$	6.18996	−	7.79457i	0.489359	−	0.616215i
$161$	−19.5511			−1.54084
$162$			25.3849i			1.99443i
$163$			1.12465i			0.0880897i	0.999030	+	0.0440448i	$0.0140244\pi$
$163$			1.12465i			0.0880897i	−0.999030	+	0.0440448i	$0.985976\pi$
$164$	14.1767			1.10701
$165$	−2.74244	+	3.45336i	−0.213499	+	0.268843i
$166$	5.81954			0.451684
$167$		−	8.92311i		−	0.690491i	−0.938512	−	0.345246i	$-0.887796\pi$
$167$		−	8.92311i		−	0.690491i	0.938512	−	0.345246i	$-0.112204\pi$
$168$			31.9112i			2.46200i
$169$	−35.6630			−2.74330
$170$	7.77407	+	6.17368i	0.596244	+	0.473500i
$171$	−0.889329			−0.0680087
$172$		−	9.77610i		−	0.745421i
$173$		−	20.4108i		−	1.55180i	−0.630853	−	0.775902i	$-0.717295\pi$
$173$		−	20.4108i		−	1.55180i	0.630853	−	0.775902i	$-0.282705\pi$
$174$	9.93498			0.753169
$175$	−23.3413	+	5.42791i	−1.76444	+	0.410312i
$176$	0.985888			0.0743141
$177$		−	6.07522i		−	0.456642i
$178$			19.2914i			1.44595i
$179$	9.19132			0.686991			0.343496	−	0.939154i	$-0.388389\pi$
$179$	9.19132			0.686991			0.343496	+	0.939154i	$0.388389\pi$
$180$	−5.36733	−	4.26240i	−0.400057	−	0.317700i
$181$	4.87696			0.362501			0.181251	−	0.983437i	$-0.441985\pi$
$181$	4.87696			0.362501			0.181251	+	0.983437i	$0.441985\pi$
$182$			78.0286i			5.78386i
$183$			11.3180i			0.836650i
$184$	−13.7719			−1.01528
$185$	7.30565	−	9.19948i	0.537122	−	0.676359i
$186$	48.8181			3.57952
$187$		−	1.90231i		−	0.139111i
$188$		−	33.0438i		−	2.40997i
$189$	19.9503			1.45117
$190$	−3.24536	+	4.08665i	−0.235443	+	0.296477i
$191$	6.07137			0.439309			0.219655	−	0.975578i	$-0.429507\pi$
$191$	6.07137			0.439309			0.219655	+	0.975578i	$0.429507\pi$
$192$		−	24.3761i		−	1.75919i
$193$			22.7808i			1.63980i	0.572508	+	0.819899i	$0.305971\pi$
$193$			22.7808i			1.63980i	−0.572508	+	0.819899i	$0.694029\pi$
$194$	−0.791662			−0.0568380
$195$	−24.0902	−	19.1309i	−1.72514	−	1.36999i
$196$	55.0463			3.93188
$197$		−	7.06623i		−	0.503448i	−0.967799	−	0.251724i	$-0.919002\pi$
$197$		−	7.06623i		−	0.503448i	0.967799	−	0.251724i	$-0.0809976\pi$
$198$			2.07551i			0.147500i
$199$	23.6586			1.67711			0.838555	−	0.544817i	$-0.183401\pi$
$199$	23.6586			1.67711			0.838555	+	0.544817i	$0.183401\pi$
$200$	−16.4417	+	3.82345i	−1.16261	+	0.270359i
$201$	−10.1992			−0.719399
$202$			3.86429i			0.271891i
$203$		−	10.3457i		−	0.726123i
$204$	12.9304			0.905306
$205$	7.20255	+	5.71981i	0.503048	+	0.399489i
$206$	−11.1799			−0.778940
$207$		−	3.62779i		−	0.252149i
$208$			6.87744i			0.476865i
$209$	1.00000			0.0691714
$210$	−30.6755	+	38.6274i	−2.11681	+	2.66555i
$211$	−0.374072			−0.0257522			−0.0128761	−	0.999917i	$-0.504099\pi$
$211$	−0.374072			−0.0257522			−0.0128761	+	0.999917i	$0.504099\pi$
$212$			15.2209i			1.04537i
$213$			16.9620i			1.16222i
$214$	7.40151			0.505956
$215$	3.94433	−	4.96681i	0.269001	−	0.338734i
$216$	14.0531			0.956191
$217$		−	50.8361i		−	3.45098i
$218$		−	31.1346i		−	2.10870i
$219$	4.57941			0.309448
$220$	6.03526	+	4.79282i	0.406897	+	0.323132i
$221$	13.2703			0.892657
$222$		−	24.1802i		−	1.62287i
$223$			17.9582i			1.20257i	0.799035	+	0.601285i	$0.205344\pi$
$223$			17.9582i			1.20257i	−0.799035	+	0.601285i	$0.794656\pi$
$224$	−21.3344			−1.42546
$225$	−1.00717	−	4.33108i	−0.0671448	−	0.288739i
$226$	−11.7390			−0.780869
$227$		−	4.54401i		−	0.301596i	−0.988565	−	0.150798i	$-0.951816\pi$
$227$		−	4.54401i		−	0.301596i	0.988565	−	0.150798i	$-0.0481844\pi$
$228$			6.79719i			0.450155i
$229$	−17.7415			−1.17239			−0.586196	−	0.810170i	$-0.699375\pi$
$229$	−17.7415			−1.17239			−0.586196	+	0.810170i	$0.699375\pi$
$230$	−16.6704	−	13.2386i	−1.09922	−	0.872929i
$231$	9.45211			0.621903
$232$		−	7.28753i		−	0.478450i
$233$		−	4.25651i		−	0.278853i	−0.990232	−	0.139427i	$-0.955474\pi$
$233$		−	4.25651i		−	0.278853i	0.990232	−	0.139427i	$-0.0445259\pi$
$234$	−14.4785			−0.946491
$235$	13.3321	−	16.7881i	0.869689	−	1.09514i
$236$	−10.6174			−0.691131
$237$		−	31.1866i		−	2.02579i
$238$		−	21.2783i		−	1.37926i
$239$	−27.3004			−1.76592			−0.882958	−	0.469452i	$-0.844452\pi$
$239$	−27.3004			−1.76592			−0.882958	+	0.469452i	$0.844452\pi$
$240$	−2.70374	+	3.40462i	−0.174526	+	0.219767i
$241$	25.7828			1.66081			0.830407	−	0.557157i	$-0.188108\pi$
$241$	25.7828			1.66081			0.830407	+	0.557157i	$0.188108\pi$
$242$		−	2.33380i		−	0.150022i
$243$			8.96350i			0.575009i
$244$	19.7799			1.26628
$245$	27.9667	+	22.2094i	1.78672	+	1.41890i
$246$	18.9314			1.20702
$247$			6.97588i			0.443865i
$248$		−	35.8092i		−	2.27389i
$249$	4.91771			0.311647
$250$	−23.5776	−	11.1769i	−1.49118	−	0.706890i
$251$	−1.69221			−0.106811			−0.0534055	−	0.998573i	$-0.517008\pi$
$251$	−1.69221			−0.106811			−0.0534055	+	0.998573i	$0.517008\pi$
$252$			14.6908i			0.925434i
$253$			4.07925i			0.256460i
$254$	−5.39580			−0.338562
$255$	6.56936	+	5.21697i	0.411389	+	0.326699i
$256$	−21.8240			−1.36400
$257$			22.4357i			1.39950i	0.714388	+	0.699750i	$0.246705\pi$
$257$			22.4357i			1.39950i	−0.714388	+	0.699750i	$0.753295\pi$
$258$		−	13.0549i		−	0.812764i
$259$	−25.1797			−1.56459
$260$	−33.4342	+	42.1013i	−2.07350	+	2.61101i
$261$	1.91968			0.118825
$262$			15.2501i			0.942153i
$263$			6.15036i			0.379247i	0.981857	+	0.189624i	$0.0607268\pi$
$263$			6.15036i			0.379247i	−0.981857	+	0.189624i	$0.939273\pi$
$264$	6.65811			0.409778
$265$	−6.14111	+	7.73306i	−0.377246	+	0.475038i
$266$	11.1855			0.685826
$267$			16.3019i			0.997659i
$268$			17.8247i			1.08882i
$269$	6.79348			0.414206			0.207103	−	0.978319i	$-0.433597\pi$
$269$	6.79348			0.414206			0.207103	+	0.978319i	$0.433597\pi$
$270$	17.0108	+	13.5089i	1.03525	+	0.822127i
$271$	−7.20564			−0.437712			−0.218856	−	0.975757i	$-0.570232\pi$
$271$	−7.20564			−0.437712			−0.218856	+	0.975757i	$0.570232\pi$
$272$		−	1.87546i		−	0.113717i
$273$			65.9368i			3.99068i
$274$	−4.20938			−0.254298
$275$	1.13251	+	4.87005i	0.0682928	+	0.293675i
$276$	−27.7274			−1.66899
$277$			28.8256i			1.73197i	0.500074	+	0.865983i	$0.333306\pi$
$277$			28.8256i			1.73197i	−0.500074	+	0.865983i	$0.666694\pi$
$278$		−	13.1074i		−	0.786128i
$279$	9.43285			0.564730
$280$	28.3341	+	22.5012i	1.69329	+	1.34470i
$281$	−17.7854			−1.06099			−0.530493	−	0.847689i	$-0.677993\pi$
$281$	−17.7854			−1.06099			−0.530493	+	0.847689i	$0.677993\pi$
$282$		−	44.1264i		−	2.62769i
$283$			12.3079i			0.731628i	0.930688	+	0.365814i	$0.119209\pi$
$283$			12.3079i			0.731628i	−0.930688	+	0.365814i	$0.880791\pi$
$284$	29.6436			1.75903
$285$	−2.74244	+	3.45336i	−0.162448	+	0.204559i
$286$	16.2803			0.962674
$287$		−	19.7139i		−	1.16368i
$288$		−	3.95868i		−	0.233267i
$289$	13.3812			0.787130
$290$	7.00534	−	8.82132i	0.411368	−	0.518006i
$291$	−0.668981			−0.0392164
$292$		−	8.00321i		−	0.468352i
$293$			32.4497i			1.89573i	0.318668	+	0.947866i	$0.396765\pi$
$293$			32.4497i			1.89573i	−0.318668	+	0.947866i	$0.603235\pi$
$294$	73.5084			4.28710
$295$	−5.39422	−	4.28375i	−0.314064	−	0.249410i
$296$	−17.7367			−1.03092
$297$		−	4.16254i		−	0.241535i
$298$		−	29.0772i		−	1.68440i
$299$	−28.4564			−1.64567
$300$	−33.1027	+	7.69787i	−1.91118	+	0.444437i
$301$	−13.5946			−0.783578
$302$			35.2913i			2.03078i
$303$			3.26546i			0.187596i
$304$	0.985888			0.0565445
$305$	10.0493	+	7.98053i	0.575422	+	0.456964i
$306$	3.94827			0.225708
$307$		−	21.8459i		−	1.24681i	−0.781898	−	0.623406i	$-0.785748\pi$
$307$		−	21.8459i		−	1.24681i	0.781898	−	0.623406i	$-0.214252\pi$
$308$		−	16.5190i		−	0.941256i
$309$	−9.44739			−0.537443
$310$	34.4226	−	43.3459i	1.95507	−	2.46188i
$311$	−3.21297			−0.182191			−0.0910953	−	0.995842i	$-0.529037\pi$
$311$	−3.21297			−0.182191			−0.0910953	+	0.995842i	$0.529037\pi$
$312$			46.4462i			2.62950i
$313$			19.1387i			1.08178i	0.841092	+	0.540892i	$0.181913\pi$
$313$			19.1387i			1.08178i	−0.841092	+	0.540892i	$0.818087\pi$
$314$	27.6369			1.55964
$315$	−5.92725	+	7.46376i	−0.333963	+	0.420535i
$316$	−54.5032			−3.06605
$317$			29.0432i			1.63123i	0.578597	+	0.815613i	$0.303600\pi$
$317$			29.0432i			1.63123i	−0.578597	+	0.815613i	$0.696400\pi$
$318$			20.3258i			1.13981i
$319$	−2.15857			−0.120857
$320$	−21.6437	−	17.1880i	−1.20992	−	0.960841i
$321$	6.25453			0.349094
$322$			45.6283i			2.54277i
$323$		−	1.90231i		−	0.105847i
$324$	37.4890			2.08272
$325$	−33.9729	+	7.90025i	−1.88448	+	0.438227i
$326$	2.62471			0.145369
$327$		−	26.3098i		−	1.45493i
$328$		−	13.8866i		−	0.766759i
$329$	−45.9504			−2.53333
$330$	8.05943	+	6.40030i	0.443657	+	0.352325i
$331$	6.66834			0.366525			0.183263	−	0.983064i	$-0.441334\pi$
$331$	6.66834			0.366525			0.183263	+	0.983064i	$0.441334\pi$
$332$		−	8.59443i		−	0.471681i
$333$		−	4.67220i		−	0.256035i
$334$	−20.8247			−1.13948
$335$	−7.19168	+	9.05596i	−0.392923	+	0.494780i
$336$	9.31872			0.508378
$337$		−	13.4640i		−	0.733432i	−0.930333	−	0.366716i	$-0.880482\pi$
$337$		−	13.4640i		−	0.733432i	0.930333	−	0.366716i	$-0.119518\pi$
$338$			83.2301i			4.52712i
$339$	−9.91988			−0.538774
$340$	9.11744	−	11.4809i	0.494463	−	0.622641i
$341$	−10.6067			−0.574386
$342$			2.07551i			0.112231i
$343$		−	42.9972i		−	2.32163i
$344$	−9.57608			−0.516307
$345$	−14.0871	−	11.1871i	−0.758424	−	0.602292i
$346$	−47.6347			−2.56085
$347$			31.5430i			1.69332i	0.532135	+	0.846659i	$0.321390\pi$
$347$			31.5430i			1.69332i	−0.532135	+	0.846659i	$0.678610\pi$
$348$		−	14.6722i		−	0.786513i
$349$	−26.6513			−1.42661			−0.713306	−	0.700853i	$-0.752803\pi$
$349$	−26.6513			−1.42661			−0.713306	+	0.700853i	$0.752803\pi$
$350$	12.6676	+	54.4739i	0.677114	+	2.91175i
$351$	29.0374			1.54990
$352$			4.45131i			0.237256i
$353$			9.84002i			0.523732i	0.965104	+	0.261866i	$0.0843378\pi$
$353$			9.84002i			0.523732i	−0.965104	+	0.261866i	$0.915662\pi$
$354$	−14.1783			−0.753570
$355$	15.0606	+	11.9602i	0.799336	+	0.634782i
$356$	28.4900			1.50997
$357$		−	17.9809i		−	0.951648i
$358$		−	21.4507i		−	1.13370i
$359$	33.9480			1.79171			0.895854	−	0.444348i	$-0.146565\pi$
$359$	33.9480			1.79171			0.895854	+	0.444348i	$0.146565\pi$
$360$	−4.17518	+	5.25751i	−0.220052	+	0.277095i
$361$	1.00000			0.0526316
$362$		−	11.3818i		−	0.598216i
$363$		−	1.97214i		−	0.103510i
$364$	115.235			6.03993
$365$	3.22903	−	4.06608i	0.169015	−	0.212829i
$366$	26.4139			1.38068
$367$			18.9282i			0.988046i	0.869449	+	0.494023i	$0.164474\pi$
$367$			18.9282i			0.988046i	−0.869449	+	0.494023i	$0.835526\pi$
$368$			4.02168i			0.209645i
$369$	3.65800			0.190428
$370$	−21.4697	−	17.0499i	−1.11616	−	0.886382i
$371$	21.1660			1.09888
$372$		−	72.0958i		−	3.73799i
$373$		−	1.93691i		−	0.100290i	−0.998742	−	0.0501448i	$-0.984032\pi$
$373$		−	1.93691i		−	0.100290i	0.998742	−	0.0501448i	$-0.0159683\pi$
$374$	−4.43961			−0.229567
$375$	−19.9239	−	9.44487i	−1.02886	−	0.487731i
$376$	−32.3677			−1.66924
$377$		−	15.0579i		−	0.775524i
$378$		−	46.5599i		−	2.39479i
$379$	14.8053			0.760498			0.380249	−	0.924884i	$-0.375838\pi$
$379$	14.8053			0.760498			0.380249	+	0.924884i	$0.375838\pi$
$380$	6.03526	+	4.79282i	0.309602	+	0.245867i
$381$	−4.55963			−0.233597
$382$		−	14.1694i		−	0.724967i
$383$		−	24.1747i		−	1.23527i	−0.786465	−	0.617635i	$-0.788091\pi$
$383$		−	24.1747i		−	1.23527i	0.786465	−	0.617635i	$-0.211909\pi$
$384$	−39.3316			−2.00713
$385$	6.66486	−	8.39258i	0.339673	−	0.427726i
$386$	53.1658			2.70607
$387$		−	2.52253i		−	0.128227i
$388$			1.16915i			0.0593543i
$389$	−8.39826			−0.425809			−0.212904	−	0.977073i	$-0.568292\pi$
$389$	−8.39826			−0.425809			−0.212904	+	0.977073i	$0.568292\pi$
$390$	−44.6477	+	56.2217i	−2.26083	+	2.84689i
$391$	7.75999			0.392440
$392$		−	53.9201i		−	2.72337i
$393$			12.8868i			0.650055i
$394$	−16.4912			−0.830812
$395$	−27.6907	−	21.9903i	−1.39327	−	1.10645i
$396$	3.06517			0.154030
$397$			11.7512i			0.589777i	0.955532	+	0.294889i	$0.0952826\pi$
$397$			11.7512i			0.589777i	−0.955532	+	0.294889i	$0.904717\pi$
$398$		−	55.2143i		−	2.76764i
$399$	9.45211			0.473197
$400$	1.11653	+	4.80133i	0.0558263	+	0.240066i
$401$	13.1216			0.655259			0.327630	−	0.944806i	$-0.393750\pi$
$401$	13.1216			0.655259			0.327630	+	0.944806i	$0.393750\pi$
$402$			23.8030i			1.18718i
$403$		−	73.9912i		−	3.68576i
$404$	5.70688			0.283928
$405$	19.0466	+	15.1256i	0.946431	+	0.751596i
$406$	−24.1447			−1.19828
$407$			5.25363i			0.260413i
$408$		−	12.6658i		−	0.627050i
$409$	−28.1968			−1.39424			−0.697121	−	0.716954i	$-0.745536\pi$
$409$	−28.1968			−1.39424			−0.697121	+	0.716954i	$0.745536\pi$
$410$	13.3489	−	16.8093i	0.659254	−	0.830151i
$411$	−3.55707			−0.175457
$412$			16.5107i			0.813426i
$413$			14.7644i			0.726509i
$414$	−8.46653			−0.416107
$415$	3.46757	−	4.36646i	0.170216	−	0.214341i
$416$	−31.0518			−1.52244
$417$		−	11.0762i		−	0.542403i
$418$		−	2.33380i		−	0.114150i
$419$	−25.4857			−1.24506			−0.622529	−	0.782597i	$-0.713895\pi$
$419$	−25.4857			−1.24506			−0.622529	+	0.782597i	$0.713895\pi$
$420$	57.0459	+	45.3023i	2.78356	+	2.21053i
$421$	−25.7350			−1.25425			−0.627123	−	0.778920i	$-0.715768\pi$
$421$	−25.7350			−1.25425			−0.627123	+	0.778920i	$0.715768\pi$
$422$			0.873008i			0.0424974i
$423$		−	8.52630i		−	0.414563i
$424$	14.9094			0.724066
$425$	9.26435	−	2.15438i	0.449387	−	0.104503i
$426$	39.5858			1.91794
$427$		−	27.5058i		−	1.33110i
$428$		−	10.9307i		−	0.528356i
$429$	13.7574			0.664214
$430$	−11.5915	−	9.20527i	−0.558994	−	0.443918i
$431$	30.2074			1.45504			0.727519	−	0.686088i	$-0.240673\pi$
$431$	30.2074			1.45504			0.727519	+	0.686088i	$0.240673\pi$
$432$		−	4.10379i		−	0.197444i
$433$		−	34.1223i		−	1.63981i	−0.572499	−	0.819905i	$-0.694026\pi$
$433$		−	34.1223i		−	1.63981i	0.572499	−	0.819905i	$-0.305974\pi$
$434$	−118.641			−5.69496
$435$	5.91975	−	7.45432i	0.283830	−	0.357407i
$436$	−45.9803			−2.20206
$437$			4.07925i			0.195137i
$438$		−	10.6874i		−	0.510664i
$439$	31.5629			1.50641			0.753207	−	0.657784i	$-0.228506\pi$
$439$	31.5629			1.50641			0.753207	+	0.657784i	$0.228506\pi$
$440$	4.69476	−	5.91177i	0.223814	−	0.281833i
$441$	14.2036			0.676363
$442$		−	30.9702i		−	1.47310i
$443$		−	24.4776i		−	1.16297i	−0.813559	−	0.581483i	$-0.802473\pi$
$443$		−	24.4776i		−	1.16297i	0.813559	−	0.581483i	$-0.197527\pi$
$444$	−35.7099			−1.69472
$445$	14.4745	+	11.4948i	0.686159	+	0.544904i
$446$	41.9108			1.98453
$447$		−	24.5713i		−	1.16218i
$448$			59.2404i			2.79885i
$449$	−26.1058			−1.23201			−0.616005	−	0.787742i	$-0.711250\pi$
$449$	−26.1058			−1.23201			−0.616005	+	0.787742i	$0.711250\pi$
$450$	−10.1079	+	2.35054i	−0.476489	+	0.110805i
$451$	−4.11322			−0.193684
$452$			17.3365i			0.815439i
$453$			29.8223i			1.40117i
$454$	−10.6048			−0.497708
$455$	58.5457	+	46.4933i	2.74466	+	2.17964i
$456$	6.65811			0.311795
$457$		−	10.1270i		−	0.473722i	−0.971544	−	0.236861i	$-0.923881\pi$
$457$		−	10.1270i		−	0.473722i	0.971544	−	0.236861i	$-0.0761186\pi$
$458$			41.4051i			1.93473i
$459$	−7.91844			−0.369601
$460$	−19.5511	+	24.6193i	−0.911575	+	1.14788i
$461$	−27.9693			−1.30266			−0.651330	−	0.758795i	$-0.725789\pi$
$461$	−27.9693			−1.30266			−0.651330	+	0.758795i	$0.725789\pi$
$462$		−	22.0593i		−	1.02629i
$463$		−	30.8194i		−	1.43230i	−0.697947	−	0.716149i	$-0.745903\pi$
$463$		−	30.8194i		−	1.43230i	0.697947	−	0.716149i	$-0.254097\pi$
$464$	−2.12811			−0.0987950
$465$	29.0883	−	36.6288i	1.34894	−	1.69862i
$466$	−9.93382			−0.460176
$467$		−	23.4293i		−	1.08418i	−0.840320	−	0.542090i	$-0.817633\pi$
$467$		−	23.4293i		−	1.08418i	0.840320	−	0.542090i	$-0.182367\pi$
$468$			21.3822i			0.988395i
$469$	24.7869			1.14455
$470$	−39.1801	−	31.1144i	−1.80724	−	1.43520i
$471$	23.3541			1.07610
$472$			10.4001i			0.478704i
$473$			2.83644i			0.130420i
$474$	−72.7832			−3.34304
$475$	1.13251	+	4.87005i	0.0519631	+	0.223453i
$476$	−31.4242			−1.44033
$477$			3.92744i			0.179825i
$478$			63.7136i			2.91419i
$479$	−25.7398			−1.17608			−0.588040	−	0.808832i	$-0.700100\pi$
$479$	−25.7398			−1.17608			−0.588040	+	0.808832i	$0.700100\pi$
$480$	−15.3720	−	12.2075i	−0.701631	−	0.557192i
$481$	−36.6487			−1.67104
$482$		−	60.1717i		−	2.74075i
$483$			38.5575i			1.75443i
$484$	−3.44661			−0.156664
$485$	−0.471711	+	0.593992i	−0.0214193	+	0.0269718i
$486$	20.9190			0.948904
$487$		−	12.0033i		−	0.543923i	−0.962308	−	0.271962i	$-0.912328\pi$
$487$		−	12.0033i		−	0.543923i	0.962308	−	0.271962i	$-0.0876724\pi$
$488$		−	19.3752i		−	0.877073i
$489$	2.21797			0.100300
$490$	51.8322	−	65.2685i	2.34154	−	2.94853i
$491$	−1.70788			−0.0770754			−0.0385377	−	0.999257i	$-0.512270\pi$
$491$	−1.70788			−0.0770754			−0.0385377	+	0.999257i	$0.512270\pi$
$492$		−	27.9583i		−	1.26046i
$493$			4.10627i			0.184937i
$494$	16.2803			0.732485
$495$	1.55728	+	1.23669i	0.0699945	+	0.0555852i
$496$	−10.4570			−0.469534
$497$		−	41.2222i		−	1.84907i
$498$		−	11.4769i		−	0.514294i
$499$	23.8364			1.06706			0.533532	−	0.845780i	$-0.320864\pi$
$499$	23.8364			1.06706			0.533532	+	0.845780i	$0.320864\pi$
$500$	−16.5063	+	34.8200i	−0.738185	+	1.55720i
$501$	−17.5976			−0.786203
$502$			3.94926i			0.176264i
$503$		−	26.3579i		−	1.17524i	−0.809137	−	0.587619i	$-0.800065\pi$
$503$		−	26.3579i		−	1.17524i	0.809137	−	0.587619i	$-0.199935\pi$
$504$	14.3902			0.640991
$505$	2.89942	+	2.30254i	0.129022	+	0.102461i
$506$	9.52013			0.423222
$507$			70.3323i			3.12357i
$508$			7.96864i			0.353551i
$509$	37.0820			1.64363			0.821816	−	0.569753i	$-0.192961\pi$
$509$	37.0820			1.64363			0.821816	+	0.569753i	$0.192961\pi$
$510$	12.1754	−	15.3315i	0.539134	−	0.678892i
$511$	−11.1292			−0.492327
$512$			11.0454i			0.488142i
$513$		−	4.16254i		−	0.183780i
$514$	52.3604			2.30952
$515$	−6.66154	+	8.38839i	−0.293542	+	0.369637i
$516$	−19.2798			−0.848747
$517$			9.58734i			0.421651i
$518$			58.7643i			2.58196i
$519$	−40.2529			−1.76691
$520$	41.2399	+	32.7501i	1.80849	+	1.43619i
$521$	−17.5841			−0.770374			−0.385187	−	0.922839i	$-0.625863\pi$
$521$	−17.5841			−0.770374			−0.385187	+	0.922839i	$0.625863\pi$
$522$		−	4.48014i		−	0.196090i
$523$		−	3.18765i		−	0.139386i	−0.997568	−	0.0696931i	$-0.977798\pi$
$523$		−	3.18765i		−	0.139386i	0.997568	−	0.0696931i	$-0.0222020\pi$
$524$	22.5217			0.983864
$525$	10.7046	+	46.0323i	0.467187	+	2.00901i
$526$	14.3537			0.625851
$527$			20.1773i			0.878935i
$528$		−	1.94431i		−	0.0846151i
$529$	6.35975			0.276511
$530$	18.0474	+	14.3321i	0.783929	+	0.622547i
$531$	−2.73960			−0.118888
$532$		−	16.5190i		−	0.716189i
$533$		−	28.6933i		−	1.24285i
$534$	38.0453			1.64638
$535$	4.41018	−	5.55343i	0.190669	−	0.240096i
$536$	17.4600			0.754157
$537$		−	18.1265i		−	0.782218i
$538$		−	15.8546i		−	0.683540i
$539$	−15.9712			−0.687927
$540$	19.9503	−	25.1220i	0.858524	−	1.08108i
$541$	−13.4906			−0.580005			−0.290002	−	0.957026i	$-0.593656\pi$
$541$	−13.4906			−0.580005			−0.290002	+	0.957026i	$0.593656\pi$
$542$			16.8165i			0.722331i
$543$		−	9.61803i		−	0.412749i
$544$	8.46778			0.363053
$545$	−23.3606	−	18.5515i	−1.00066	−	0.794660i
$546$	153.883			6.58559
$547$		−	34.6641i		−	1.48213i	−0.671433	−	0.741065i	$-0.734321\pi$
$547$		−	34.6641i		−	1.48213i	0.671433	−	0.741065i	$-0.265679\pi$
$548$			6.21652i			0.265557i
$549$	5.10381			0.217825
$550$	11.3657	−	2.64304i	0.484636	−	0.112700i
$551$	−2.15857			−0.0919582
$552$			27.1601i			1.15601i
$553$			75.7918i			3.22299i
$554$	67.2732			2.85816
$555$	−18.1426	−	14.4077i	−0.770112	−	0.611575i
$556$	−19.3573			−0.820931
$557$		−	6.73433i		−	0.285343i	−0.989770	−	0.142671i	$-0.954431\pi$
$557$		−	6.73433i		−	0.285343i	0.989770	−	0.142671i	$-0.0455692\pi$
$558$		−	22.0144i		−	0.931942i
$559$	−19.7867			−0.836888
$560$	6.57081	−	8.27414i	0.277667	−	0.349646i
$561$	−3.75162			−0.158393
$562$			41.5074i			1.75089i
$563$			0.220710i			0.00930180i	0.999989	+	0.00465090i	$0.00148043\pi$
$563$			0.220710i			0.00930180i	−0.999989	+	0.00465090i	$0.998520\pi$
$564$	−65.1669			−2.74402
$565$	−6.99470	+	8.80792i	−0.294269	+	0.370552i
$566$	28.7241			1.20736
$567$		−	52.1319i		−	2.18934i
$568$		−	29.0371i		−	1.21837i
$569$	0.0465835			0.00195288			0.000976440	−	1.00000i	$-0.499689\pi$
$569$	0.0465835			0.00195288			0.000976440		1.00000i	$0.499689\pi$
$570$	8.05943	+	6.40030i	0.337572	+	0.268079i
$571$	29.5536			1.23678			0.618390	−	0.785871i	$-0.287785\pi$
$571$	29.5536			1.23678			0.618390	+	0.785871i	$0.287785\pi$
$572$		−	24.0431i		−	1.00529i
$573$		−	11.9736i		−	0.500204i
$574$	−46.0083			−1.92035
$575$	−19.8662	+	4.61978i	−0.828476	+	0.192658i
$576$	−10.9923			−0.458013
$577$		−	6.64097i		−	0.276467i	−0.990400	−	0.138234i	$-0.955858\pi$
$577$		−	6.64097i		−	0.276467i	0.990400	−	0.138234i	$-0.0441425\pi$
$578$		−	31.2290i		−	1.29896i
$579$	44.9269			1.86710
$580$	−13.0275	−	10.3457i	−0.540939	−	0.429580i
$581$	−11.9514			−0.495826
$582$			1.56127i			0.0647166i
$583$		−	4.41619i		−	0.182900i
$584$	−7.83946			−0.324399
$585$	−8.62703	+	10.8634i	−0.356684	+	0.449146i
$586$	75.7311			3.12842
$587$			17.0513i			0.703783i	0.936041	+	0.351892i	$0.114461\pi$
$587$			17.0513i			0.703783i	−0.936041	+	0.351892i	$0.885539\pi$
$588$		−	108.559i		−	4.47690i
$589$	−10.6067			−0.437042
$590$	−9.99741	+	12.5890i	−0.411587	+	0.518282i
$591$	−13.9356			−0.573234
$592$			5.17949i			0.212876i
$593$			37.6513i			1.54615i	0.634313	+	0.773076i	$0.281283\pi$
$593$			37.6513i			1.54615i	−0.634313	+	0.773076i	$0.718717\pi$
$594$	−9.71451			−0.398591
$595$	−15.9653	−	12.6786i	−0.654513	−	0.519773i
$596$	−42.9420			−1.75897
$597$		−	46.6579i		−	1.90958i
$598$			66.4113i			2.71576i
$599$	5.66658			0.231530			0.115765	−	0.993277i	$-0.463068\pi$
$599$	5.66658			0.231530			0.115765	+	0.993277i	$0.463068\pi$
$600$	7.54037	+	32.4254i	0.307834	+	1.32376i
$601$	29.1095			1.18740			0.593701	−	0.804686i	$-0.297666\pi$
$601$	29.1095			1.18740			0.593701	+	0.804686i	$0.297666\pi$
$602$			31.7270i			1.29309i
$603$			4.59931i			0.187298i
$604$	52.1190			2.12069
$605$	−1.75107	−	1.39059i	−0.0711912	−	0.0565356i
$606$	7.62091			0.309578
$607$		−	27.0839i		−	1.09930i	−0.835394	−	0.549651i	$-0.814761\pi$
$607$		−	27.0839i		−	1.09930i	0.835394	−	0.549651i	$-0.185239\pi$
$608$			4.45131i			0.180525i
$609$	−20.4031			−0.826774
$610$	18.6249	−	23.4530i	0.754102	−	0.949586i
$611$	−66.8802			−2.70568
$612$		−	5.83090i		−	0.235700i
$613$			16.5050i			0.666632i	0.942815	+	0.333316i	$0.108168\pi$
$613$			16.5050i			0.666632i	−0.942815	+	0.333316i	$0.891832\pi$
$614$	−50.9839			−2.05754
$615$	11.2803	−	14.2044i	0.454864	−	0.572777i
$616$	−16.1810			−0.651951
$617$			9.17145i			0.369229i	0.982811	+	0.184614i	$0.0591036\pi$
$617$			9.17145i			0.369229i	−0.982811	+	0.184614i	$0.940896\pi$
$618$			22.0483i			0.886913i
$619$	−2.94996			−0.118569			−0.0592844	−	0.998241i	$-0.518882\pi$
$619$	−2.94996			−0.118569			−0.0592844	+	0.998241i	$0.518882\pi$
$620$	−64.0142	−	50.8361i	−2.57087	−	2.04163i
$621$	16.9800			0.681384
$622$			7.49841i			0.300659i
$623$		−	39.6180i		−	1.58726i
$624$	13.5633			0.542965
$625$	−22.4348	+	11.0308i	−0.897394	+	0.441230i
$626$	44.6658			1.78521
$627$		−	1.97214i		−	0.0787596i
$628$		−	40.8148i		−	1.62869i
$629$	9.99403			0.398488
$630$	17.4189	+	13.8330i	0.693986	+	0.551120i
$631$	−13.1349			−0.522894			−0.261447	−	0.965218i	$-0.584200\pi$
$631$	−13.1349			−0.522894			−0.261447	+	0.965218i	$0.584200\pi$
$632$			53.3881i			2.12366i
$633$			0.737722i			0.0293218i
$634$	67.7808			2.69192
$635$	−3.21508	+	4.04852i	−0.127587	+	0.160661i
$636$	30.0176			1.19028
$637$		−	111.413i		−	4.41434i
$638$			5.03767i			0.199443i
$639$	7.64895			0.302588
$640$	−27.7335	+	34.9228i	−1.09626	+	1.38044i
$641$	17.1097			0.675793			0.337897	−	0.941183i	$-0.390285\pi$
$641$	17.1097			0.675793			0.337897	+	0.941183i	$0.390285\pi$
$642$		−	14.5968i		−	0.576089i
$643$		−	14.7732i		−	0.582599i	−0.956632	−	0.291300i	$-0.905912\pi$
$643$		−	14.7732i		−	0.582599i	0.956632	−	0.291300i	$-0.0940876\pi$
$644$	67.3850			2.65534
$645$	−9.79524	−	7.77877i	−0.385687	−	0.306289i
$646$	−4.43961			−0.174674
$647$		−	33.9680i		−	1.33542i	−0.744422	−	0.667710i	$-0.767275\pi$
$647$		−	33.9680i		−	1.33542i	0.744422	−	0.667710i	$-0.232725\pi$
$648$		−	36.7220i		−	1.44258i
$649$	3.08053			0.120921
$650$	18.4376	+	79.2859i	0.723181	+	3.10985i
$651$	−100.256			−3.92933
$652$		−	3.87624i		−	0.151805i
$653$			25.3457i			0.991855i	0.868364	+	0.495927i	$0.165172\pi$
$653$			25.3457i			0.991855i	−0.868364	+	0.495927i	$0.834828\pi$
$654$	−61.4017			−2.40099
$655$	11.4423	+	9.08675i	0.447087	+	0.355049i
$656$	−4.05517			−0.158328
$657$		−	2.06507i		−	0.0805660i
$658$			107.239i			4.18061i
$659$	27.4327			1.06863			0.534313	−	0.845287i	$-0.320570\pi$
$659$	27.4327			1.06863			0.534313	+	0.845287i	$0.320570\pi$
$660$	9.45211	−	11.9024i	0.367923	−	0.463299i
$661$	−13.2280			−0.514509			−0.257255	−	0.966344i	$-0.582818\pi$
$661$	−13.2280			−0.514509			−0.257255	+	0.966344i	$0.582818\pi$
$662$		−	15.5626i		−	0.604856i
$663$		−	26.1709i		−	1.01639i
$664$	−8.41859			−0.326705
$665$	6.66486	−	8.39258i	0.258452	−	0.325450i
$666$	−10.9040			−0.422520
$667$		−	8.80535i		−	0.340944i
$668$			30.7545i			1.18993i
$669$	35.4160			1.36926
$670$	21.1348	+	16.7839i	0.816508	+	0.648419i
$671$	−5.73895			−0.221550
$672$			42.0743i			1.62305i
$673$			42.4318i			1.63562i	0.575486	+	0.817812i	$0.304813\pi$
$673$			42.4318i			1.63562i	−0.575486	+	0.817812i	$0.695187\pi$
$674$	−31.4223			−1.21034
$675$	20.2718	−	4.71411i	0.780261	−	0.181446i
$676$	122.916			4.72755
$677$		−	3.83814i		−	0.147512i	−0.997276	−	0.0737559i	$-0.976501\pi$
$677$		−	3.83814i		−	0.147512i	0.997276	−	0.0737559i	$-0.0234986\pi$
$678$			23.1510i			0.889108i
$679$	1.62580			0.0623926
$680$	−11.2460	−	8.93089i	−0.431266	−	0.342484i
$681$	−8.96141			−0.343402
$682$			24.7539i			0.947876i
$683$			16.2575i			0.622076i	0.950398	+	0.311038i	$0.100677\pi$
$683$			16.2575i			0.622076i	−0.950398	+	0.311038i	$0.899323\pi$
$684$	3.06517			0.117200
$685$	−2.50816	+	3.15835i	−0.0958319	+	0.120674i
$686$	−100.347			−3.83126
$687$			34.9887i			1.33490i
$688$			2.79641i			0.106612i
$689$	30.8068			1.17365
$690$	−26.1084	+	32.8764i	−0.993929	+	1.25158i
$691$	36.4802			1.38777			0.693887	−	0.720084i	$-0.255897\pi$
$691$	36.4802			1.38777			0.693887	+	0.720084i	$0.255897\pi$
$692$			70.3480i			2.67423i
$693$		−	4.26240i		−	0.161915i
$694$	73.6150			2.79439
$695$	−9.83460	−	7.81002i	−0.373047	−	0.296251i
$696$	−14.3720			−0.544770
$697$			7.82462i			0.296379i
$698$			62.1987i			2.35426i
$699$	−8.39442			−0.317506
$700$	80.4483	−	18.7079i	3.04066	−	0.707092i
$701$	−4.34336			−0.164046			−0.0820232	−	0.996630i	$-0.526138\pi$
$701$	−4.34336			−0.164046			−0.0820232	+	0.996630i	$0.526138\pi$
$702$		−	67.7673i		−	2.55771i
$703$			5.25363i			0.198144i
$704$	12.3602			0.465844
$705$	−33.1085	−	26.2927i	−1.24694	−	0.990240i
$706$	22.9646			0.864285
$707$		−	7.93594i		−	0.298462i
$708$			20.9389i			0.786932i
$709$	37.5895			1.41170			0.705852	−	0.708360i	$-0.250564\pi$
$709$	37.5895			1.41170			0.705852	+	0.708360i	$0.250564\pi$
$710$	27.9127	−	35.1485i	1.04755	−	1.31910i
$711$	−14.0635			−0.527422
$712$		−	27.9071i		−	1.04586i
$713$		−	43.2674i		−	1.62038i
$714$	−41.9637			−1.57045
$715$	9.70061	−	12.2153i	0.362782	−	0.456826i
$716$	−31.6789			−1.18389
$717$			53.8402i			2.01070i
$718$		−	79.2278i		−	2.95676i
$719$	−17.5352			−0.653952			−0.326976	−	0.945033i	$-0.606030\pi$
$719$	−17.5352			−0.653952			−0.326976	+	0.945033i	$0.606030\pi$
$720$	1.53530	+	1.21924i	0.0572173	+	0.0454384i
$721$	22.9597			0.855064
$722$		−	2.33380i		−	0.0868549i
$723$		−	50.8472i		−	1.89103i
$724$	−16.8090			−0.624700
$725$	−2.44460	−	10.5124i	−0.0907902	−	0.390419i
$726$	−4.60257			−0.170817
$727$		−	0.0700926i		−	0.00259959i	−0.999999	−	0.00129980i	$-0.999586\pi$
$727$		−	0.0700926i		−	0.00259959i	0.999999	−	0.00129980i	$-0.000413738\pi$
$728$		−	112.877i		−	4.18349i
$729$	−14.9540			−0.553851
$730$	−9.48941	−	7.53590i	−0.351219	−	0.278916i
$731$	5.39579			0.199571
$732$		−	39.0087i		−	1.44180i
$733$			5.41119i			0.199867i	0.994994	+	0.0999335i	$0.0318630\pi$
$733$			5.41119i			0.199867i	−0.994994	+	0.0999335i	$0.968137\pi$
$734$	44.1747			1.63052
$735$	43.7999	−	55.1541i	1.61559	−	2.03439i
$736$	−18.1580			−0.669313
$737$		−	5.17167i		−	0.190501i
$738$		−	8.53704i		−	0.314253i
$739$	−16.5265			−0.607939			−0.303969	−	0.952682i	$-0.598312\pi$
$739$	−16.5265			−0.607939			−0.303969	+	0.952682i	$0.598312\pi$
$740$	−25.1797	+	31.7070i	−0.925624	+	1.16557i
$741$	13.7574			0.505391
$742$		−	49.3972i		−	1.81343i
$743$		−	11.5689i		−	0.424423i	−0.977224	−	0.212211i	$-0.931933\pi$
$743$		−	11.5689i		−	0.424423i	0.977224	−	0.212211i	$-0.0680665\pi$
$744$	−70.6207			−2.58908
$745$	−21.8170	−	17.3257i	−0.799311	−	0.634763i
$746$	−4.52036			−0.165502
$747$		−	2.21762i		−	0.0811386i
$748$			6.55652i			0.239730i
$749$	−15.2002			−0.555402
$750$	−22.0424	+	46.4983i	−0.804875	+	1.69788i
$751$	−27.5998			−1.00713			−0.503566	−	0.863957i	$-0.667979\pi$
$751$	−27.5998			−1.00713			−0.503566	+	0.863957i	$0.667979\pi$
$752$			9.45204i			0.344681i
$753$			3.33726i			0.121617i
$754$	−35.1422			−1.27980
$755$	26.4794	+	21.0283i	0.963684	+	0.765298i
$756$	−68.7609			−2.50081
$757$			12.5688i			0.456821i	0.973565	+	0.228411i	$0.0733529\pi$
$757$			12.5688i			0.456821i	−0.973565	+	0.228411i	$0.926647\pi$
$758$		−	34.5526i		−	1.25501i
$759$	8.04484			0.292009
$760$	4.69476	−	5.91177i	0.170297	−	0.214443i
$761$	13.1005			0.474893			0.237446	−	0.971401i	$-0.423690\pi$
$761$	13.1005			0.474893			0.237446	+	0.971401i	$0.423690\pi$
$762$			10.6413i			0.385492i
$763$			63.9398i			2.31478i
$764$	−20.9256			−0.757063
$765$	2.35257	−	2.96243i	0.0850575	−	0.107107i
$766$	−56.4189			−2.03850
$767$			21.4894i			0.775937i
$768$			43.0399i			1.55307i
$769$	4.22081			0.152206			0.0761031	−	0.997100i	$-0.475752\pi$
$769$	4.22081			0.152206			0.0761031	+	0.997100i	$0.475752\pi$
$770$	−19.5866	−	15.5544i	−0.705851	−	0.560543i
$771$	44.2463			1.59349
$772$		−	78.5165i		−	2.82587i
$773$			52.3823i			1.88406i	0.335527	+	0.942031i	$0.391086\pi$
$773$			52.3823i			1.88406i	−0.335527	+	0.942031i	$0.608914\pi$
$774$	−5.88707			−0.211606
$775$	−12.0122	−	51.6552i	−0.431491	−	1.85551i
$776$	1.14522			0.0411111
$777$			49.6578i			1.78147i
$778$			19.5998i			0.702688i
$779$	−4.11322			−0.147371
$780$	83.0295	+	65.9368i	2.97293	+	2.36092i
$781$	−8.60081			−0.307761
$782$		−	18.1102i		−	0.647621i
$783$			8.98513i			0.321102i
$784$	−15.7458			−0.562349
$785$	16.4674	−	20.7362i	0.587747	−	0.740108i
$786$	30.0753			1.07275
$787$			22.1383i			0.789146i	0.918865	+	0.394573i	$0.129108\pi$
$787$			22.1383i			0.789146i	−0.918865	+	0.394573i	$0.870892\pi$
$788$			24.3545i			0.867594i
$789$	12.1294			0.431817
$790$	−51.3208	+	64.6246i	−1.82591	+	2.29924i
$791$	24.1080			0.857181
$792$		−	3.00245i		−	0.106687i
$793$		−	40.0342i		−	1.42166i
$794$	27.4250			0.973276
$795$	15.2507	+	12.1111i	0.540885	+	0.429537i
$796$	−81.5417			−2.89017
$797$			27.9114i			0.988672i	0.869271	+	0.494336i	$0.164589\pi$
$797$			27.9114i			0.988672i	−0.869271	+	0.494336i	$0.835411\pi$
$798$		−	22.0593i		−	0.780891i
$799$	18.2381			0.645218
$800$	−21.6781	+	5.04115i	−0.766438	+	0.178232i
$801$	7.35128			0.259745
$802$		−	30.6231i		−	1.08134i
$803$			2.32205i			0.0819435i
$804$	35.1528			1.23974
$805$	34.2354	+	27.1876i	1.20664	+	0.958238i
$806$	−172.680			−6.08241
$807$		−	13.3977i		−	0.471621i
$808$		−	5.59011i		−	0.196659i
$809$	50.2065			1.76517			0.882583	−	0.470157i	$-0.155803\pi$
$809$	50.2065			1.76517			0.882583	+	0.470157i	$0.155803\pi$
$810$	35.3000	−	44.4508i	1.24032	−	1.56184i
$811$	−20.8048			−0.730556			−0.365278	−	0.930899i	$-0.619026\pi$
$811$	−20.8048			−0.730556			−0.365278	+	0.930899i	$0.619026\pi$
$812$			35.6574i			1.25133i
$813$			14.2105i			0.498385i
$814$	12.2609			0.429744
$815$	1.56393	−	1.96935i	0.0547822	−	0.0689833i
$816$	−3.69868			−0.129480
$817$			2.83644i			0.0992345i
$818$			65.8056i			2.30084i
$819$	29.7340			1.03899
$820$	−24.8243	−	19.7139i	−0.866904	−	0.688441i
$821$	−31.5126			−1.09980			−0.549899	−	0.835231i	$-0.685334\pi$
$821$	−31.5126			−1.09980			−0.549899	+	0.835231i	$0.685334\pi$
$822$			8.30149i			0.289548i
$823$			0.755361i			0.0263302i	0.999913	+	0.0131651i	$0.00419071\pi$
$823$			0.755361i			0.0263302i	−0.999913	+	0.0131651i	$0.995809\pi$
$824$	16.1729			0.563410
$825$	9.60442	−	2.23346i	0.334383	−	0.0777592i
$826$	34.4571			1.19892
$827$			48.2285i			1.67707i	0.544849	+	0.838534i	$0.316587\pi$
$827$			48.2285i			1.67707i	−0.544849	+	0.838534i	$0.683413\pi$
$828$			12.5036i			0.434529i
$829$	−33.2069			−1.15332			−0.576662	−	0.816983i	$-0.695645\pi$
$829$	−33.2069			−1.15332			−0.576662	+	0.816983i	$0.695645\pi$
$830$	−10.1904	−	8.09260i	−0.353715	−	0.280898i
$831$	56.8482			1.97204
$832$			86.2236i			2.98926i
$833$			30.3821i			1.05268i
$834$	−25.8495			−0.895096
$835$	−12.4084	+	15.6250i	−0.429411	+	0.540726i
$836$	−3.44661			−0.119203
$837$			44.1508i			1.52608i
$838$			59.4784i			2.05465i
$839$	−26.6232			−0.919134			−0.459567	−	0.888143i	$-0.651995\pi$
$839$	−26.6232			−0.919134			−0.459567	+	0.888143i	$0.651995\pi$
$840$	44.3754	−	55.8787i	1.53110	−	1.92800i
$841$	−24.3406			−0.839330
$842$			60.0602i			2.06981i
$843$			35.0752i			1.20805i
$844$	1.28928			0.0443788
$845$	62.4484	+	49.5926i	2.14829	+	1.70604i
$846$	−19.8986			−0.684129
$847$			4.79282i			0.164683i
$848$		−	4.35387i		−	0.149512i
$849$	24.2729			0.833042
$850$	−5.02789	−	21.6211i	−0.172455	−	0.741598i
$851$	−21.4308			−0.734639
$852$		−	58.4613i		−	2.00285i
$853$		−	20.7758i		−	0.711349i	−0.934610	−	0.355675i	$-0.884251\pi$
$853$		−	20.7758i		−	0.711349i	0.934610	−	0.355675i	$-0.115749\pi$
$854$	−64.1928			−2.19663
$855$	1.55728	+	1.23669i	0.0532578	+	0.0422940i
$856$	−10.7071			−0.365960
$857$		−	30.3639i		−	1.03721i	−0.855014	−	0.518605i	$-0.826452\pi$
$857$		−	30.3639i		−	1.03721i	0.855014	−	0.518605i	$-0.173548\pi$
$858$		−	32.1070i		−	1.09611i
$859$	−12.4982			−0.426434			−0.213217	−	0.977005i	$-0.568394\pi$
$859$	−12.4982			−0.426434			−0.213217	+	0.977005i	$0.568394\pi$
$860$	−13.5946	+	17.1187i	−0.463571	+	0.583741i
$861$	−38.8786			−1.32498
$862$		−	70.4979i		−	2.40117i
$863$		−	25.1151i		−	0.854928i	−0.904032	−	0.427464i	$-0.859407\pi$
$863$		−	25.1151i		−	0.854928i	0.904032	−	0.427464i	$-0.140593\pi$
$864$	18.5288			0.630361
$865$	−28.3831	+	35.7408i	−0.965054	+	1.21522i
$866$	−79.6344			−2.70609
$867$		−	26.3896i		−	0.896238i
$868$			175.212i			5.94709i
$869$	15.8136			0.536439
$870$	−17.3969	−	13.8155i	−0.589809	−	0.468389i
$871$	36.0769			1.22242
$872$			45.0395i			1.52523i
$873$			0.301675i			0.0102101i
$874$	9.52013			0.322023
$875$	48.4203	+	22.9536i	1.63691	+	0.775972i
$876$	−15.7834			−0.533273
$877$			39.6014i			1.33724i	0.743602	+	0.668622i	$0.233116\pi$
$877$			39.6014i			1.33724i	−0.743602	+	0.668622i	$0.766884\pi$
$878$		−	73.6613i		−	2.48595i
$879$	63.9953			2.15851
$880$	−1.72636	−	1.37097i	−0.0581956	−	0.0462153i
$881$	−15.1847			−0.511585			−0.255792	−	0.966732i	$-0.582336\pi$
$881$	−15.1847			−0.511585			−0.255792	+	0.966732i	$0.582336\pi$
$882$		−	33.1483i		−	1.11616i
$883$		−	1.91096i		−	0.0643088i	−0.999483	−	0.0321544i	$-0.989763\pi$
$883$		−	1.91096i		−	0.0643088i	0.999483	−	0.0321544i	$-0.0102368\pi$
$884$	−45.7375			−1.53832
$885$	−8.44815	+	10.6382i	−0.283982	+	0.357598i
$886$	−57.1257			−1.91918
$887$		−	48.2069i		−	1.61863i	−0.587375	−	0.809315i	$-0.699839\pi$
$887$		−	48.2069i		−	1.61863i	0.587375	−	0.809315i	$-0.300161\pi$
$888$			34.9792i			1.17383i
$889$	11.0811			0.371649
$890$	26.8265	−	33.7806i	0.899225	−	1.13233i
$891$	−10.8771			−0.364396
$892$		−	61.8948i		−	2.07239i
$893$			9.58734i			0.320828i
$894$	−57.3443			−1.91788
$895$	−16.0947	−	12.7814i	−0.537985	−	0.427234i
$896$	95.5864			3.19332
$897$			56.1199i			1.87379i
$898$			60.9257i			2.03312i
$899$	22.8953			0.763602
$900$	3.47133	+	14.9275i	0.115711	+	0.497584i
$901$	−8.40096			−0.279877
$902$			9.59942i			0.319626i
$903$			26.8104i			0.892193i
$904$	16.9818			0.564805
$905$	−8.53990	−	6.78186i	−0.283876	−	0.225437i
$906$	69.5992			2.31228
$907$		−	4.98348i		−	0.165474i	−0.996571	−	0.0827369i	$-0.973634\pi$
$907$		−	4.98348i		−	0.165474i	0.996571	−	0.0827369i	$-0.0263661\pi$
$908$			15.6614i			0.519742i
$909$	1.47255			0.0488413
$910$	108.506	−	136.634i	3.59694	−	4.52936i
$911$	−21.7087			−0.719240			−0.359620	−	0.933099i	$-0.617094\pi$
$911$	−21.7087			−0.719240			−0.359620	+	0.933099i	$0.617094\pi$
$912$		−	1.94431i		−	0.0643824i
$913$			2.49359i			0.0825259i
$914$	−23.6344			−0.781757
$915$	15.7387	−	19.8186i	0.520306	−	0.655184i
$916$	61.1480			2.02039
$917$		−	31.3185i		−	1.03423i
$918$			18.4800i			0.609931i
$919$	12.7377			0.420178			0.210089	−	0.977682i	$-0.432625\pi$
$919$	12.7377			0.420178			0.210089	+	0.977682i	$0.432625\pi$
$920$	24.1156	+	19.1511i	0.795067	+	0.631393i
$921$	−43.0832			−1.41964
$922$			65.2747i			2.14971i
$923$		−	59.9983i		−	1.97487i
$924$	−32.5777			−1.07173
$925$	−25.5854	+	5.94978i	−0.841244	+	0.195627i
$926$	−71.9262			−2.36364
$927$			4.26027i			0.139926i
$928$		−	9.60848i		−	0.315414i
$929$	−28.0934			−0.921713			−0.460857	−	0.887475i	$-0.652458\pi$
$929$	−28.0934			−0.921713			−0.460857	+	0.887475i	$0.652458\pi$
$930$	−85.4841	−	67.8861i	−2.80313	−	2.22607i
$931$	−15.9712			−0.523434
$932$			14.6705i			0.480549i
$933$			6.33642i			0.207445i
$934$	−54.6793			−1.78916
$935$	−2.64534	+	3.33108i	−0.0865118	+	0.108938i
$936$	20.9448			0.684601
$937$		−	20.4039i		−	0.666566i	−0.942827	−	0.333283i	$-0.891843\pi$
$937$		−	20.4039i		−	0.666566i	0.942827	−	0.333283i	$-0.108157\pi$
$938$		−	57.8476i		−	1.88879i
$939$	37.7442			1.23173
$940$	−45.9504	+	57.8621i	−1.49874	+	1.88725i
$941$	30.8314			1.00508			0.502538	−	0.864555i	$-0.332400\pi$
$941$	30.8314			1.00508			0.502538	+	0.864555i	$0.332400\pi$
$942$		−	54.5037i		−	1.77583i
$943$		−	16.7788i		−	0.546394i
$944$	3.03705			0.0988476
$945$	−34.9344	−	27.7427i	−1.13642	−	0.902471i
$946$	6.61968			0.215224
$947$		−	24.9048i		−	0.809297i	−0.914472	−	0.404648i	$-0.867394\pi$
$947$		−	24.9048i		−	0.809297i	0.914472	−	0.404648i	$-0.132606\pi$
$948$			107.488i			3.49105i
$949$	−16.1984			−0.525822
$950$	11.3657	−	2.64304i	0.368753	−	0.0857517i
$951$	57.2771			1.85734
$952$			30.7813i			0.997627i
$953$		−	50.7623i		−	1.64435i	−0.569233	−	0.822176i	$-0.692760\pi$
$953$		−	50.7623i		−	1.64435i	0.569233	−	0.822176i	$-0.307240\pi$
$954$	9.16585			0.296755
$955$	−10.6314	−	8.44280i	−0.344025	−	0.273203i
$956$	94.0938			3.04321
$957$			4.25700i			0.137609i
$958$			60.0714i			1.94082i
$959$	8.64464			0.279150
$960$	−33.8972	+	42.6843i	−1.09403	+	1.37763i
$961$	81.5023			2.62911
$962$			85.5306i			2.75762i
$963$		−	2.82046i		−	0.0908879i
$964$	−88.8631			−2.86209
$965$	31.6788	−	39.8908i	1.01978	−	1.28413i
$966$	89.9854			2.89523
$967$			41.8598i			1.34612i	0.739588	+	0.673060i	$0.235020\pi$
$967$			41.8598i			1.34612i	−0.739588	+	0.673060i	$0.764980\pi$
$968$			3.37609i			0.108512i
$969$	−3.75162			−0.120519
$970$	1.38626	+	1.10088i	0.0445100	+	0.0353471i
$971$	52.9614			1.69961			0.849806	−	0.527095i	$-0.176719\pi$
$971$	52.9614			1.69961			0.849806	+	0.527095i	$0.176719\pi$
$972$		−	30.8937i		−	0.990915i
$973$			26.9181i			0.862954i
$974$	−28.0134			−0.897606
$975$	15.5804	+	66.9993i	0.498972	+	2.14570i
$976$	−5.65796			−0.181107
$977$			32.4341i			1.03766i	0.854878	+	0.518829i	$0.173632\pi$
$977$			32.4341i			1.03766i	−0.854878	+	0.518829i	$0.826368\pi$
$978$		−	5.17630i		−	0.165520i
$979$	−8.26610			−0.264186
$980$	−96.3901	−	76.5470i	−3.07907	−	2.44520i
$981$	−11.8643			−0.378798
$982$			3.98583i			0.127193i
$983$		−	47.5546i		−	1.51675i	−0.651816	−	0.758377i	$-0.725992\pi$
$983$		−	47.5546i		−	1.51675i	0.651816	−	0.758377i	$-0.274008\pi$
$984$	−27.3863			−0.873043
$985$	−9.82625	+	12.3735i	−0.313090	+	0.394252i
$986$	9.58321			0.305192
$987$			90.6206i			2.88449i
$988$		−	24.0431i		−	0.764914i
$989$	−11.5705			−0.367922
$990$	2.88619	−	3.63437i	0.0917292	−	0.115508i
$991$	36.3103			1.15344			0.576718	−	0.816943i	$-0.304333\pi$
$991$	36.3103			1.15344			0.576718	+	0.816943i	$0.304333\pi$
$992$		−	47.2138i		−	1.49904i
$993$		−	13.1509i		−	0.417331i
$994$	−96.2042			−3.05141
$995$	−41.4278	−	32.8994i	−1.31335	−	1.04298i
$996$	−16.9494			−0.537063
$997$			2.22395i			0.0704333i	0.999380	+	0.0352167i	$0.0112121\pi$
$997$			2.22395i			0.0704333i	−0.999380	+	0.0352167i	$0.988788\pi$
$998$		−	55.6294i		−	1.76092i
$999$	21.8684			0.691886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.c.419.2	✓	20
5.2	odd	4		5225.2.a.ba.1.19		20
5.3	odd	4		5225.2.a.ba.1.2		20
5.4	even	2	inner	1045.2.b.c.419.19	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.c.419.2	✓	20	1.1	even	1	trivial
1045.2.b.c.419.19	yes	20	5.4	even	2	inner
5225.2.a.ba.1.2		20	5.3	odd	4
5225.2.a.ba.1.19		20	5.2	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.33380i q^{2} -1.97214i q^{3} -3.44661 q^{4} +(-1.75107 - 1.39059i) q^{5} -4.60257 q^{6} +4.79282i q^{7} +3.37609i q^{8} -0.889329 q^{9} +O(q^{10})\)	\(q-2.33380i q^{2} -1.97214i q^{3} -3.44661 q^{4} +(-1.75107 - 1.39059i) q^{5} -4.60257 q^{6} +4.79282i q^{7} +3.37609i q^{8} -0.889329 q^{9} +(-3.24536 + 4.08665i) q^{10} +1.00000 q^{11} +6.79719i q^{12} +6.97588i q^{13} +11.1855 q^{14} +(-2.74244 + 3.45336i) q^{15} +0.985888 q^{16} -1.90231i q^{17} +2.07551i q^{18} +1.00000 q^{19} +(6.03526 + 4.79282i) q^{20} +9.45211 q^{21} -2.33380i q^{22} +4.07925i q^{23} +6.65811 q^{24} +(1.13251 + 4.87005i) q^{25} +16.2803 q^{26} -4.16254i q^{27} -16.5190i q^{28} -2.15857 q^{29} +(8.05943 + 6.40030i) q^{30} -10.6067 q^{31} +4.45131i q^{32} -1.97214i q^{33} -4.43961 q^{34} +(6.66486 - 8.39258i) q^{35} +3.06517 q^{36} +5.25363i q^{37} -2.33380i q^{38} +13.7574 q^{39} +(4.69476 - 5.91177i) q^{40} -4.11322 q^{41} -22.0593i q^{42} +2.83644i q^{43} -3.44661 q^{44} +(1.55728 + 1.23669i) q^{45} +9.52013 q^{46} +9.58734i q^{47} -1.94431i q^{48} -15.9712 q^{49} +(11.3657 - 2.64304i) q^{50} -3.75162 q^{51} -24.0431i q^{52} -4.41619i q^{53} -9.71451 q^{54} +(-1.75107 - 1.39059i) q^{55} -16.1810 q^{56} -1.97214i q^{57} +5.03767i q^{58} +3.08053 q^{59} +(9.45211 - 11.9024i) q^{60} -5.73895 q^{61} +24.7539i q^{62} -4.26240i q^{63} +12.3602 q^{64} +(9.70061 - 12.2153i) q^{65} -4.60257 q^{66} -5.17167i q^{67} +6.55652i q^{68} +8.04484 q^{69} +(-19.5866 - 15.5544i) q^{70} -8.60081 q^{71} -3.00245i q^{72} +2.32205i q^{73} +12.2609 q^{74} +(9.60442 - 2.23346i) q^{75} -3.44661 q^{76} +4.79282i q^{77} -32.1070i q^{78} +15.8136 q^{79} +(-1.72636 - 1.37097i) q^{80} -10.8771 q^{81} +9.59942i q^{82} +2.49359i q^{83} -32.5777 q^{84} +(-2.64534 + 3.33108i) q^{85} +6.61968 q^{86} +4.25700i q^{87} +3.37609i q^{88} -8.26610 q^{89} +(2.88619 - 3.63437i) q^{90} -33.4342 q^{91} -14.0596i q^{92} +20.9179i q^{93} +22.3749 q^{94} +(-1.75107 - 1.39059i) q^{95} +8.77861 q^{96} -0.339216i q^{97} +37.2735i q^{98} -0.889329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) \) 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9	\( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) \) 20 * q - 12 * q^4 - 8 * q^6 - 10 * q^9 - 6 * q^10 + 20 * q^11 + 24 * q^14 - 6 * q^15 - 4 * q^16 + 20 * q^19 - 6 * q^20 - 30 * q^21 + 38 * q^24 + 2 * q^25 + 8 * q^26 + 50 * q^29 - 20 * q^30 - 50 * q^31 + 28 * q^34 + 6 * q^35 - 12 * q^36 + 48 * q^39 + 12 * q^40 - 34 * q^41 - 12 * q^44 - 18 * q^45 - 36 * q^46 - 6 * q^49 + 26 * q^50 - 40 * q^51 - 6 * q^54 - 40 * q^56 + 30 * q^59 - 30 * q^60 - 14 * q^61 + 36 * q^64 + 30 * q^65 - 8 * q^66 - 12 * q^69 - 54 * q^70 - 40 * q^71 + 50 * q^74 - 8 * q^75 - 12 * q^76 + 106 * q^79 + 8 * q^80 - 30 * q^84 - 22 * q^85 + 56 * q^86 + 36 * q^89 - 64 * q^90 - 56 * q^91 + 28 * q^94 + 66 * q^96 - 10 * q^99

Introduction

L-functions

Modular forms

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