LMFDB - Embedded newform 108.3.f.a.91.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,3,Mod(19,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("108.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	108.f (of order $6$, degree $2$, not 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:	$2.94278685509$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			91.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	108.91
Dual form			108.3.f.a.19.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +(2.00000 - 3.46410i) q^{5} +(-3.00000 + 1.73205i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} +(2.00000 - 3.46410i) q^{5} +(-3.00000 + 1.73205i) q^{7} -8.00000 q^{8} +(-4.00000 + 6.92820i) q^{10} +(10.5000 - 6.06218i) q^{11} +(11.0000 - 19.0526i) q^{13} +(6.00000 - 3.46410i) q^{14} +16.0000 q^{16} +11.0000 q^{17} -15.5885i q^{19} +(8.00000 - 13.8564i) q^{20} +(-21.0000 + 12.1244i) q^{22} +(-21.0000 - 12.1244i) q^{23} +(4.50000 + 7.79423i) q^{25} +(-22.0000 + 38.1051i) q^{26} +(-12.0000 + 6.92820i) q^{28} +(17.0000 + 29.4449i) q^{29} +(6.00000 + 3.46410i) q^{31} -32.0000 q^{32} -22.0000 q^{34} +13.8564i q^{35} -16.0000 q^{37} +31.1769i q^{38} +(-16.0000 + 27.7128i) q^{40} +(6.50000 - 11.2583i) q^{41} +(-43.5000 + 25.1147i) q^{43} +(42.0000 - 24.2487i) q^{44} +(42.0000 + 24.2487i) q^{46} +(-3.00000 + 1.73205i) q^{47} +(-18.5000 + 32.0429i) q^{49} +(-9.00000 - 15.5885i) q^{50} +(44.0000 - 76.2102i) q^{52} -52.0000 q^{53} -48.4974i q^{55} +(24.0000 - 13.8564i) q^{56} +(-34.0000 - 58.8897i) q^{58} +(46.5000 + 26.8468i) q^{59} +(8.00000 + 13.8564i) q^{61} +(-12.0000 - 6.92820i) q^{62} +64.0000 q^{64} +(-44.0000 - 76.2102i) q^{65} +(100.500 + 58.0237i) q^{67} +44.0000 q^{68} -27.7128i q^{70} -25.0000 q^{73} +32.0000 q^{74} -62.3538i q^{76} +(-21.0000 + 36.3731i) q^{77} +(24.0000 - 13.8564i) q^{79} +(32.0000 - 55.4256i) q^{80} +(-13.0000 + 22.5167i) q^{82} +(-30.0000 + 17.3205i) q^{83} +(22.0000 - 38.1051i) q^{85} +(87.0000 - 50.2295i) q^{86} +(-84.0000 + 48.4974i) q^{88} +2.00000 q^{89} +76.2102i q^{91} +(-84.0000 - 48.4974i) q^{92} +(6.00000 - 3.46410i) q^{94} +(-54.0000 - 31.1769i) q^{95} +(21.5000 + 37.2391i) q^{97} +(37.0000 - 64.0859i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + 4 q^{5} - 6 q^{7} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 + 4 * q^5 - 6 * q^7 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} + 4 q^{5} - 6 q^{7} - 16 q^{8} - 8 q^{10} + 21 q^{11} + 22 q^{13} + 12 q^{14} + 32 q^{16} + 22 q^{17} + 16 q^{20} - 42 q^{22} - 42 q^{23} + 9 q^{25} - 44 q^{26} - 24 q^{28} + 34 q^{29} + 12 q^{31} - 64 q^{32} - 44 q^{34} - 32 q^{37} - 32 q^{40} + 13 q^{41} - 87 q^{43} + 84 q^{44} + 84 q^{46} - 6 q^{47} - 37 q^{49} - 18 q^{50} + 88 q^{52} - 104 q^{53} + 48 q^{56} - 68 q^{58} + 93 q^{59} + 16 q^{61} - 24 q^{62} + 128 q^{64} - 88 q^{65} + 201 q^{67} + 88 q^{68} - 50 q^{73} + 64 q^{74} - 42 q^{77} + 48 q^{79} + 64 q^{80} - 26 q^{82} - 60 q^{83} + 44 q^{85} + 174 q^{86} - 168 q^{88} + 4 q^{89} - 168 q^{92} + 12 q^{94} - 108 q^{95} + 43 q^{97} + 74 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + 4 * q^5 - 6 * q^7 - 16 * q^8 - 8 * q^10 + 21 * q^11 + 22 * q^13 + 12 * q^14 + 32 * q^16 + 22 * q^17 + 16 * q^20 - 42 * q^22 - 42 * q^23 + 9 * q^25 - 44 * q^26 - 24 * q^28 + 34 * q^29 + 12 * q^31 - 64 * q^32 - 44 * q^34 - 32 * q^37 - 32 * q^40 + 13 * q^41 - 87 * q^43 + 84 * q^44 + 84 * q^46 - 6 * q^47 - 37 * q^49 - 18 * q^50 + 88 * q^52 - 104 * q^53 + 48 * q^56 - 68 * q^58 + 93 * q^59 + 16 * q^61 - 24 * q^62 + 128 * q^64 - 88 * q^65 + 201 * q^67 + 88 * q^68 - 50 * q^73 + 64 * q^74 - 42 * q^77 + 48 * q^79 + 64 * q^80 - 26 * q^82 - 60 * q^83 + 44 * q^85 + 174 * q^86 - 168 * q^88 + 4 * q^89 - 168 * q^92 + 12 * q^94 - 108 * q^95 + 43 * q^97 + 74 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$

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 $

$2$	−2.00000			−1.00000
$2$	−2.00000			−1.00000
$3$		0			0
$3$		0			0
$4$	4.00000			1.00000
$5$	2.00000	−	3.46410i	0.400000	−	0.692820i	−0.593725	−	0.804668i	$-0.702343\pi$
$5$	2.00000	−	3.46410i	0.400000	−	0.692820i	0.993725	+	0.111847i	$0.0356768\pi$
$6$		0			0
$7$	−3.00000	+	1.73205i	−0.428571	+	0.247436i	−0.698738	−	0.715378i	$-0.746255\pi$
$7$	−3.00000	+	1.73205i	−0.428571	+	0.247436i	0.270166	+	0.962814i	$0.412921\pi$
$8$	−8.00000			−1.00000
$9$		0			0
$10$	−4.00000	+	6.92820i	−0.400000	+	0.692820i
$11$	10.5000	−	6.06218i	0.954545	−	0.551107i	0.0600555	−	0.998195i	$-0.480872\pi$
$11$	10.5000	−	6.06218i	0.954545	−	0.551107i	0.894490	+	0.447088i	$0.147539\pi$
$12$		0			0
$13$	11.0000	−	19.0526i	0.846154	−	1.46558i	−0.0384615	−	0.999260i	$-0.512246\pi$
$13$	11.0000	−	19.0526i	0.846154	−	1.46558i	0.884615	−	0.466321i	$-0.154421\pi$
$14$	6.00000	−	3.46410i	0.428571	−	0.247436i
$15$		0			0
$16$	16.0000			1.00000
$17$	11.0000			0.647059			0.323529	−	0.946218i	$-0.395131\pi$
$17$	11.0000			0.647059			0.323529	+	0.946218i	$0.395131\pi$
$18$		0			0
$19$		−	15.5885i		−	0.820445i	−0.911985	−	0.410223i	$-0.865451\pi$
$19$		−	15.5885i		−	0.820445i	0.911985	−	0.410223i	$-0.134549\pi$
$20$	8.00000	−	13.8564i	0.400000	−	0.692820i
$21$		0			0
$22$	−21.0000	+	12.1244i	−0.954545	+	0.551107i
$23$	−21.0000	−	12.1244i	−0.913043	−	0.527146i	−0.0316343	−	0.999500i	$-0.510071\pi$
$23$	−21.0000	−	12.1244i	−0.913043	−	0.527146i	−0.881409	+	0.472354i	$0.843405\pi$
$24$		0			0
$25$	4.50000	+	7.79423i	0.180000	+	0.311769i
$26$	−22.0000	+	38.1051i	−0.846154	+	1.46558i
$27$		0			0
$28$	−12.0000	+	6.92820i	−0.428571	+	0.247436i
$29$	17.0000	+	29.4449i	0.586207	+	1.01534i	0.994724	+	0.102589i	$0.0327128\pi$
$29$	17.0000	+	29.4449i	0.586207	+	1.01534i	−0.408517	+	0.912751i	$0.633954\pi$
$30$		0			0
$31$	6.00000	+	3.46410i	0.193548	+	0.111745i	0.593643	−	0.804729i	$-0.297689\pi$
$31$	6.00000	+	3.46410i	0.193548	+	0.111745i	−0.400094	+	0.916474i	$0.631023\pi$
$32$	−32.0000			−1.00000
$33$		0			0
$34$	−22.0000			−0.647059
$35$			13.8564i			0.395897i
$36$		0			0
$37$	−16.0000			−0.432432			−0.216216	−	0.976346i	$-0.569372\pi$
$37$	−16.0000			−0.432432			−0.216216	+	0.976346i	$0.569372\pi$
$38$			31.1769i			0.820445i
$39$		0			0
$40$	−16.0000	+	27.7128i	−0.400000	+	0.692820i
$41$	6.50000	−	11.2583i	0.158537	−	0.274593i	−0.775805	−	0.630973i	$-0.782656\pi$
$41$	6.50000	−	11.2583i	0.158537	−	0.274593i	0.934341	+	0.356380i	$0.115989\pi$
$42$		0			0
$43$	−43.5000	+	25.1147i	−1.01163	+	0.584064i	−0.911668	−	0.410928i	$-0.865205\pi$
$43$	−43.5000	+	25.1147i	−1.01163	+	0.584064i	−0.0999600	+	0.994991i	$0.531871\pi$
$44$	42.0000	−	24.2487i	0.954545	−	0.551107i
$45$		0			0
$46$	42.0000	+	24.2487i	0.913043	+	0.527146i
$47$	−3.00000	+	1.73205i	−0.0638298	+	0.0368521i	−0.531575	−	0.847011i	$-0.678400\pi$
$47$	−3.00000	+	1.73205i	−0.0638298	+	0.0368521i	0.467745	+	0.883863i	$0.345066\pi$
$48$		0			0
$49$	−18.5000	+	32.0429i	−0.377551	+	0.653938i
$50$	−9.00000	−	15.5885i	−0.180000	−	0.311769i
$51$		0			0
$52$	44.0000	−	76.2102i	0.846154	−	1.46558i
$53$	−52.0000			−0.981132			−0.490566	−	0.871404i	$-0.663210\pi$
$53$	−52.0000			−0.981132			−0.490566	+	0.871404i	$0.663210\pi$
$54$		0			0
$55$		−	48.4974i		−	0.881771i
$56$	24.0000	−	13.8564i	0.428571	−	0.247436i
$57$		0			0
$58$	−34.0000	−	58.8897i	−0.586207	−	1.01534i
$59$	46.5000	+	26.8468i	0.788136	+	0.455030i	0.839306	−	0.543660i	$-0.182962\pi$
$59$	46.5000	+	26.8468i	0.788136	+	0.455030i	−0.0511702	+	0.998690i	$0.516295\pi$
$60$		0			0
$61$	8.00000	+	13.8564i	0.131148	+	0.227154i	0.924119	−	0.382104i	$-0.124801\pi$
$61$	8.00000	+	13.8564i	0.131148	+	0.227154i	−0.792972	+	0.609259i	$0.791467\pi$
$62$	−12.0000	−	6.92820i	−0.193548	−	0.111745i
$63$		0			0
$64$	64.0000			1.00000
$65$	−44.0000	−	76.2102i	−0.676923	−	1.17247i
$66$		0			0
$67$	100.500	+	58.0237i	1.50000	+	0.866025i	1.00000			$0$
$67$	100.500	+	58.0237i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$68$	44.0000			0.647059
$69$		0			0
$70$		−	27.7128i		−	0.395897i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−25.0000			−0.342466			−0.171233	−	0.985231i	$-0.554775\pi$
$73$	−25.0000			−0.342466			−0.171233	+	0.985231i	$0.554775\pi$
$74$	32.0000			0.432432
$75$		0			0
$76$		−	62.3538i		−	0.820445i
$77$	−21.0000	+	36.3731i	−0.272727	+	0.472377i
$78$		0			0
$79$	24.0000	−	13.8564i	0.303797	−	0.175398i	−0.340350	−	0.940299i	$-0.610546\pi$
$79$	24.0000	−	13.8564i	0.303797	−	0.175398i	0.644148	+	0.764901i	$0.277212\pi$
$80$	32.0000	−	55.4256i	0.400000	−	0.692820i
$81$		0			0
$82$	−13.0000	+	22.5167i	−0.158537	+	0.274593i
$83$	−30.0000	+	17.3205i	−0.361446	+	0.208681i	−0.669715	−	0.742618i	$-0.733584\pi$
$83$	−30.0000	+	17.3205i	−0.361446	+	0.208681i	0.308269	+	0.951299i	$0.400250\pi$
$84$		0			0
$85$	22.0000	−	38.1051i	0.258824	−	0.448296i
$86$	87.0000	−	50.2295i	1.01163	−	0.584064i
$87$		0			0
$88$	−84.0000	+	48.4974i	−0.954545	+	0.551107i
$89$	2.00000			0.0224719			0.0112360	−	0.999937i	$-0.496423\pi$
$89$	2.00000			0.0224719			0.0112360	+	0.999937i	$0.496423\pi$
$90$		0			0
$91$			76.2102i			0.837475i
$92$	−84.0000	−	48.4974i	−0.913043	−	0.527146i
$93$		0			0
$94$	6.00000	−	3.46410i	0.0638298	−	0.0368521i
$95$	−54.0000	−	31.1769i	−0.568421	−	0.328178i
$96$		0			0
$97$	21.5000	+	37.2391i	0.221649	+	0.383908i	0.955309	−	0.295609i	$-0.0955226\pi$
$97$	21.5000	+	37.2391i	0.221649	+	0.383908i	−0.733659	+	0.679517i	$0.762189\pi$
$98$	37.0000	−	64.0859i	0.377551	−	0.653938i
$99$		0			0
$100$	18.0000	+	31.1769i	0.180000	+	0.311769i
$101$	−10.0000	−	17.3205i	−0.0990099	−	0.171490i	0.812265	−	0.583288i	$-0.198234\pi$
$101$	−10.0000	−	17.3205i	−0.0990099	−	0.171490i	−0.911275	+	0.411798i	$0.864901\pi$
$102$		0			0
$103$	−21.0000	−	12.1244i	−0.203883	−	0.117712i	0.394582	−	0.918861i	$-0.370889\pi$
$103$	−21.0000	−	12.1244i	−0.203883	−	0.117712i	−0.598466	+	0.801148i	$0.704223\pi$
$104$	−88.0000	+	152.420i	−0.846154	+	1.46558i
$105$		0			0
$106$	104.000			0.981132
$107$			15.5885i			0.145687i	0.997343	+	0.0728433i	$0.0232073\pi$
$107$			15.5885i			0.145687i	−0.997343	+	0.0728433i	$0.976793\pi$
$108$		0			0
$109$	−88.0000			−0.807339			−0.403670	−	0.914905i	$-0.632266\pi$
$109$	−88.0000			−0.807339			−0.403670	+	0.914905i	$0.632266\pi$
$110$			96.9948i			0.881771i
$111$		0			0
$112$	−48.0000	+	27.7128i	−0.428571	+	0.247436i
$113$	−25.0000	+	43.3013i	−0.221239	+	0.383197i	−0.955184	−	0.296011i	$-0.904343\pi$
$113$	−25.0000	+	43.3013i	−0.221239	+	0.383197i	0.733946	+	0.679208i	$0.237677\pi$
$114$		0			0
$115$	−84.0000	+	48.4974i	−0.730435	+	0.421717i
$116$	68.0000	+	117.779i	0.586207	+	1.01534i
$117$		0			0
$118$	−93.0000	−	53.6936i	−0.788136	−	0.455030i
$119$	−33.0000	+	19.0526i	−0.277311	+	0.160106i
$120$		0			0
$121$	13.0000	−	22.5167i	0.107438	−	0.186088i
$122$	−16.0000	−	27.7128i	−0.131148	−	0.227154i
$123$		0			0
$124$	24.0000	+	13.8564i	0.193548	+	0.111745i
$125$	136.000			1.08800
$126$		0			0
$127$		−	218.238i		−	1.71841i	−0.511629	−	0.859206i	$-0.670958\pi$
$127$		−	218.238i		−	1.71841i	0.511629	−	0.859206i	$-0.329042\pi$
$128$	−128.000			−1.00000
$129$		0			0
$130$	88.0000	+	152.420i	0.676923	+	1.17247i
$131$	168.000	+	96.9948i	1.28244	+	0.740419i	0.977294	−	0.211886i	$-0.0679606\pi$
$131$	168.000	+	96.9948i	1.28244	+	0.740419i	0.305148	+	0.952305i	$0.401294\pi$
$132$		0			0
$133$	27.0000	+	46.7654i	0.203008	+	0.351619i
$134$	−201.000	−	116.047i	−1.50000	−	0.866025i
$135$		0			0
$136$	−88.0000			−0.647059
$137$	84.5000	+	146.358i	0.616788	+	1.06831i	0.990068	+	0.140590i	$0.0448999\pi$
$137$	84.5000	+	146.358i	0.616788	+	1.06831i	−0.373280	+	0.927719i	$0.621767\pi$
$138$		0			0
$139$	−169.500	−	97.8609i	−1.21942	−	0.704035i	−0.254630	−	0.967039i	$-0.581954\pi$
$139$	−169.500	−	97.8609i	−1.21942	−	0.704035i	−0.964795	+	0.263004i	$0.915287\pi$
$140$			55.4256i			0.395897i
$141$		0			0
$142$		0			0
$143$		−	266.736i		−	1.86529i
$144$		0			0
$145$	136.000			0.937931
$146$	50.0000			0.342466
$147$		0			0
$148$	−64.0000			−0.432432
$149$	65.0000	−	112.583i	0.436242	−	0.755593i	−0.561154	−	0.827711i	$-0.689643\pi$
$149$	65.0000	−	112.583i	0.436242	−	0.755593i	0.997396	+	0.0721185i	$0.0229760\pi$
$150$		0			0
$151$	105.000	−	60.6218i	0.695364	−	0.401469i	−0.110254	−	0.993903i	$-0.535167\pi$
$151$	105.000	−	60.6218i	0.695364	−	0.401469i	0.805618	+	0.592435i	$0.201833\pi$
$152$			124.708i			0.820445i
$153$		0			0
$154$	42.0000	−	72.7461i	0.272727	−	0.472377i
$155$	24.0000	−	13.8564i	0.154839	−	0.0893962i
$156$		0			0
$157$	2.00000	−	3.46410i	0.0127389	−	0.0220643i	−0.859586	−	0.510992i	$-0.829278\pi$
$157$	2.00000	−	3.46410i	0.0127389	−	0.0220643i	0.872325	+	0.488927i	$0.162612\pi$
$158$	−48.0000	+	27.7128i	−0.303797	+	0.175398i
$159$		0			0
$160$	−64.0000	+	110.851i	−0.400000	+	0.692820i
$161$	84.0000			0.521739
$162$		0			0
$163$			311.769i			1.91269i	0.292233	+	0.956347i	$0.405602\pi$
$163$			311.769i			1.91269i	−0.292233	+	0.956347i	$0.594398\pi$
$164$	26.0000	−	45.0333i	0.158537	−	0.274593i
$165$		0			0
$166$	60.0000	−	34.6410i	0.361446	−	0.208681i
$167$	−156.000	−	90.0666i	−0.934132	−	0.539321i	−0.0460158	−	0.998941i	$-0.514652\pi$
$167$	−156.000	−	90.0666i	−0.934132	−	0.539321i	−0.888116	+	0.459620i	$0.847986\pi$
$168$		0			0
$169$	−157.500	−	272.798i	−0.931953	−	1.61419i
$170$	−44.0000	+	76.2102i	−0.258824	+	0.448296i
$171$		0			0
$172$	−174.000	+	100.459i	−1.01163	+	0.584064i
$173$	−1.00000	−	1.73205i	−0.00578035	−	0.0100119i	0.863121	−	0.504998i	$-0.168507\pi$
$173$	−1.00000	−	1.73205i	−0.00578035	−	0.0100119i	−0.868901	+	0.494986i	$0.835173\pi$
$174$		0			0
$175$	−27.0000	−	15.5885i	−0.154286	−	0.0890769i
$176$	168.000	−	96.9948i	0.954545	−	0.551107i
$177$		0			0
$178$	−4.00000			−0.0224719
$179$			187.061i			1.04504i	0.852628	+	0.522518i	$0.175007\pi$
$179$			187.061i			1.04504i	−0.852628	+	0.522518i	$0.824993\pi$
$180$		0			0
$181$	254.000			1.40331			0.701657	−	0.712514i	$-0.252444\pi$
$181$	254.000			1.40331			0.701657	+	0.712514i	$0.252444\pi$
$182$		−	152.420i		−	0.837475i
$183$		0			0
$184$	168.000	+	96.9948i	0.913043	+	0.527146i
$185$	−32.0000	+	55.4256i	−0.172973	+	0.299598i
$186$		0			0
$187$	115.500	−	66.6840i	0.617647	−	0.356599i
$188$	−12.0000	+	6.92820i	−0.0638298	+	0.0368521i
$189$		0			0
$190$	108.000	+	62.3538i	0.568421	+	0.328178i
$191$	−3.00000	+	1.73205i	−0.0157068	+	0.00906833i	−0.507833	−	0.861456i	$-0.669553\pi$
$191$	−3.00000	+	1.73205i	−0.0157068	+	0.00906833i	0.492126	+	0.870524i	$0.336220\pi$
$192$		0			0
$193$	33.5000	−	58.0237i	0.173575	−	0.300641i	−0.766092	−	0.642731i	$-0.777801\pi$
$193$	33.5000	−	58.0237i	0.173575	−	0.300641i	0.939667	+	0.342090i	$0.111135\pi$
$194$	−43.0000	−	74.4782i	−0.221649	−	0.383908i
$195$		0			0
$196$	−74.0000	+	128.172i	−0.377551	+	0.653938i
$197$	−268.000			−1.36041			−0.680203	−	0.733024i	$-0.738108\pi$
$197$	−268.000			−1.36041			−0.680203	+	0.733024i	$0.738108\pi$
$198$		0			0
$199$			31.1769i			0.156668i	0.996927	+	0.0783340i	$0.0249600\pi$
$199$			31.1769i			0.156668i	−0.996927	+	0.0783340i	$0.975040\pi$
$200$	−36.0000	−	62.3538i	−0.180000	−	0.311769i
$201$		0			0
$202$	20.0000	+	34.6410i	0.0990099	+	0.171490i
$203$	−102.000	−	58.8897i	−0.502463	−	0.290097i
$204$		0			0
$205$	−26.0000	−	45.0333i	−0.126829	−	0.219675i
$206$	42.0000	+	24.2487i	0.203883	+	0.117712i
$207$		0			0
$208$	176.000	−	304.841i	0.846154	−	1.46558i
$209$	−94.5000	−	163.679i	−0.452153	−	0.783152i
$210$		0			0
$211$	114.000	+	65.8179i	0.540284	+	0.311933i	0.745194	−	0.666848i	$-0.232357\pi$
$211$	114.000	+	65.8179i	0.540284	+	0.311933i	−0.204910	+	0.978781i	$0.565690\pi$
$212$	−208.000			−0.981132
$213$		0			0
$214$		−	31.1769i		−	0.145687i
$215$			200.918i			0.934502i
$216$		0			0
$217$	−24.0000			−0.110599
$218$	176.000			0.807339
$219$		0			0
$220$		−	193.990i		−	0.881771i
$221$	121.000	−	209.578i	0.547511	−	0.948317i
$222$		0			0
$223$	51.0000	−	29.4449i	0.228700	−	0.132040i	−0.381272	−	0.924463i	$-0.624514\pi$
$223$	51.0000	−	29.4449i	0.228700	−	0.132040i	0.609972	+	0.792423i	$0.291181\pi$
$224$	96.0000	−	55.4256i	0.428571	−	0.247436i
$225$		0			0
$226$	50.0000	−	86.6025i	0.221239	−	0.383197i
$227$	388.500	−	224.301i	1.71145	−	0.988108i	0.778847	−	0.627214i	$-0.215805\pi$
$227$	388.500	−	224.301i	1.71145	−	0.988108i	0.932607	−	0.360894i	$-0.117529\pi$
$228$		0			0
$229$	−205.000	+	355.070i	−0.895197	+	1.55053i	−0.0616353	+	0.998099i	$0.519632\pi$
$229$	−205.000	+	355.070i	−0.895197	+	1.55053i	−0.833561	+	0.552427i	$0.813702\pi$
$230$	168.000	−	96.9948i	0.730435	−	0.421717i
$231$		0			0
$232$	−136.000	−	235.559i	−0.586207	−	1.01534i
$233$	65.0000			0.278970			0.139485	−	0.990224i	$-0.455455\pi$
$233$	65.0000			0.278970			0.139485	+	0.990224i	$0.455455\pi$
$234$		0			0
$235$			13.8564i			0.0589634i
$236$	186.000	+	107.387i	0.788136	+	0.455030i
$237$		0			0
$238$	66.0000	−	38.1051i	0.277311	−	0.160106i
$239$	33.0000	+	19.0526i	0.138075	+	0.0797178i	0.567446	−	0.823410i	$-0.307931\pi$
$239$	33.0000	+	19.0526i	0.138075	+	0.0797178i	−0.429371	+	0.903128i	$0.641265\pi$
$240$		0			0
$241$	111.500	+	193.124i	0.462656	+	0.801343i	0.999092	−	0.0425975i	$-0.0135633\pi$
$241$	111.500	+	193.124i	0.462656	+	0.801343i	−0.536437	+	0.843941i	$0.680230\pi$
$242$	−26.0000	+	45.0333i	−0.107438	+	0.186088i
$243$		0			0
$244$	32.0000	+	55.4256i	0.131148	+	0.227154i
$245$	74.0000	+	128.172i	0.302041	+	0.523150i
$246$		0			0
$247$	−297.000	−	171.473i	−1.20243	−	0.694223i
$248$	−48.0000	−	27.7128i	−0.193548	−	0.111745i
$249$		0			0
$250$	−272.000			−1.08800
$251$			109.119i			0.434738i	0.976090	+	0.217369i	$0.0697475\pi$
$251$			109.119i			0.434738i	−0.976090	+	0.217369i	$0.930253\pi$
$252$		0			0
$253$	−294.000			−1.16206
$254$			436.477i			1.71841i
$255$		0			0
$256$	256.000			1.00000
$257$	−218.500	+	378.453i	−0.850195	+	1.47258i	0.0308379	+	0.999524i	$0.490182\pi$
$257$	−218.500	+	378.453i	−0.850195	+	1.47258i	−0.881032	+	0.473056i	$0.843151\pi$
$258$		0			0
$259$	48.0000	−	27.7128i	0.185328	−	0.106999i
$260$	−176.000	−	304.841i	−0.676923	−	1.17247i
$261$		0			0
$262$	−336.000	−	193.990i	−1.28244	−	0.740419i
$263$	−273.000	+	157.617i	−1.03802	+	0.599303i	−0.919273	−	0.393621i	$-0.871222\pi$
$263$	−273.000	+	157.617i	−1.03802	+	0.599303i	−0.118750	+	0.992924i	$0.537889\pi$
$264$		0			0
$265$	−104.000	+	180.133i	−0.392453	+	0.679748i
$266$	−54.0000	−	93.5307i	−0.203008	−	0.351619i
$267$		0			0
$268$	402.000	+	232.095i	1.50000	+	0.866025i
$269$	−304.000			−1.13011			−0.565056	−	0.825053i	$-0.691145\pi$
$269$	−304.000			−1.13011			−0.565056	+	0.825053i	$0.691145\pi$
$270$		0			0
$271$			311.769i			1.15044i	0.817999	+	0.575220i	$0.195083\pi$
$271$			311.769i			1.15044i	−0.817999	+	0.575220i	$0.804917\pi$
$272$	176.000			0.647059
$273$		0			0
$274$	−169.000	−	292.717i	−0.616788	−	1.06831i
$275$	94.5000	+	54.5596i	0.343636	+	0.198399i
$276$		0			0
$277$	17.0000	+	29.4449i	0.0613718	+	0.106299i	0.895079	−	0.445908i	$-0.147119\pi$
$277$	17.0000	+	29.4449i	0.0613718	+	0.106299i	−0.833707	+	0.552207i	$0.813786\pi$
$278$	339.000	+	195.722i	1.21942	+	0.704035i
$279$		0			0
$280$		−	110.851i		−	0.395897i
$281$	−109.000	−	188.794i	−0.387900	−	0.671863i	0.604267	−	0.796782i	$-0.293466\pi$
$281$	−109.000	−	188.794i	−0.387900	−	0.671863i	−0.992167	+	0.124919i	$0.960133\pi$
$282$		0			0
$283$	6.00000	+	3.46410i	0.0212014	+	0.0122406i	0.510563	−	0.859840i	$-0.329437\pi$
$283$	6.00000	+	3.46410i	0.0212014	+	0.0122406i	−0.489362	+	0.872081i	$0.662770\pi$
$284$		0			0
$285$		0			0
$286$			533.472i			1.86529i
$287$			45.0333i			0.156911i
$288$		0			0
$289$	−168.000			−0.581315
$290$	−272.000			−0.937931
$291$		0			0
$292$	−100.000			−0.342466
$293$	101.000	−	174.937i	0.344710	−	0.597055i	−0.640591	−	0.767882i	$-0.721311\pi$
$293$	101.000	−	174.937i	0.344710	−	0.597055i	0.985301	+	0.170827i	$0.0546440\pi$
$294$		0			0
$295$	186.000	−	107.387i	0.630508	−	0.364024i
$296$	128.000			0.432432
$297$		0			0
$298$	−130.000	+	225.167i	−0.436242	+	0.755593i
$299$	−462.000	+	266.736i	−1.54515	+	0.892093i
$300$		0			0
$301$	87.0000	−	150.688i	0.289037	−	0.500626i
$302$	−210.000	+	121.244i	−0.695364	+	0.401469i
$303$		0			0
$304$		−	249.415i		−	0.820445i
$305$	64.0000			0.209836
$306$		0			0
$307$		−	109.119i		−	0.355437i	−0.984081	−	0.177719i	$-0.943128\pi$
$307$		−	109.119i		−	0.355437i	0.984081	−	0.177719i	$-0.0568717\pi$
$308$	−84.0000	+	145.492i	−0.272727	+	0.472377i
$309$		0			0
$310$	−48.0000	+	27.7128i	−0.154839	+	0.0893962i
$311$	−237.000	−	136.832i	−0.762058	−	0.439974i	0.0679762	−	0.997687i	$-0.478346\pi$
$311$	−237.000	−	136.832i	−0.762058	−	0.439974i	−0.830034	+	0.557713i	$0.811679\pi$
$312$		0			0
$313$	39.5000	+	68.4160i	0.126198	+	0.218581i	0.922201	−	0.386712i	$-0.126389\pi$
$313$	39.5000	+	68.4160i	0.126198	+	0.218581i	−0.796003	+	0.605293i	$0.793056\pi$
$314$	−4.00000	+	6.92820i	−0.0127389	+	0.0220643i
$315$		0			0
$316$	96.0000	−	55.4256i	0.303797	−	0.175398i
$317$	251.000	+	434.745i	0.791798	+	1.37143i	0.924853	+	0.380326i	$0.124188\pi$
$317$	251.000	+	434.745i	0.791798	+	1.37143i	−0.133054	+	0.991109i	$0.542479\pi$
$318$		0			0
$319$	357.000	+	206.114i	1.11912	+	0.646126i
$320$	128.000	−	221.703i	0.400000	−	0.692820i
$321$		0			0
$322$	−168.000			−0.521739
$323$		−	171.473i		−	0.530876i
$324$		0			0
$325$	198.000			0.609231
$326$		−	623.538i		−	1.91269i
$327$		0			0
$328$	−52.0000	+	90.0666i	−0.158537	+	0.274593i
$329$	6.00000	−	10.3923i	0.0182371	−	0.0315876i
$330$		0			0
$331$	−354.000	+	204.382i	−1.06949	+	0.617468i	−0.928041	−	0.372478i	$-0.878508\pi$
$331$	−354.000	+	204.382i	−1.06949	+	0.617468i	−0.141445	+	0.989946i	$0.545175\pi$
$332$	−120.000	+	69.2820i	−0.361446	+	0.208681i
$333$		0			0
$334$	312.000	+	180.133i	0.934132	+	0.539321i
$335$	402.000	−	232.095i	1.20000	−	0.692820i
$336$		0			0
$337$	168.500	−	291.851i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	168.500	−	291.851i	0.500000	−	0.866025i	1.00000			$0$
$338$	315.000	+	545.596i	0.931953	+	1.61419i
$339$		0			0
$340$	88.0000	−	152.420i	0.258824	−	0.448296i
$341$	84.0000			0.246334
$342$		0			0
$343$		−	297.913i		−	0.868550i
$344$	348.000	−	200.918i	1.01163	−	0.584064i
$345$		0			0
$346$	2.00000	+	3.46410i	0.00578035	+	0.0100119i
$347$	235.500	+	135.966i	0.678674	+	0.391833i	0.799355	−	0.600859i	$-0.205174\pi$
$347$	235.500	+	135.966i	0.678674	+	0.391833i	−0.120681	+	0.992691i	$0.538508\pi$
$348$		0			0
$349$	−136.000	−	235.559i	−0.389685	−	0.674954i	0.602722	−	0.797951i	$-0.294083\pi$
$349$	−136.000	−	235.559i	−0.389685	−	0.674954i	−0.992407	+	0.122997i	$0.960749\pi$
$350$	54.0000	+	31.1769i	0.154286	+	0.0890769i
$351$		0			0
$352$	−336.000	+	193.990i	−0.954545	+	0.551107i
$353$	−230.500	−	399.238i	−0.652975	−	1.13099i	−0.982397	−	0.186803i	$-0.940188\pi$
$353$	−230.500	−	399.238i	−0.652975	−	1.13099i	0.329423	−	0.944182i	$-0.393146\pi$
$354$		0			0
$355$		0			0
$356$	8.00000			0.0224719
$357$		0			0
$358$		−	374.123i		−	1.04504i
$359$		−	530.008i		−	1.47634i	−0.674612	−	0.738172i	$-0.735689\pi$
$359$		−	530.008i		−	1.47634i	0.674612	−	0.738172i	$-0.264311\pi$
$360$		0			0
$361$	118.000			0.326870
$362$	−508.000			−1.40331
$363$		0			0
$364$			304.841i			0.837475i
$365$	−50.0000	+	86.6025i	−0.136986	+	0.237267i
$366$		0			0
$367$	−84.0000	+	48.4974i	−0.228883	+	0.132146i	−0.610057	−	0.792358i	$-0.708853\pi$
$367$	−84.0000	+	48.4974i	−0.228883	+	0.132146i	0.381174	+	0.924503i	$0.375520\pi$
$368$	−336.000	−	193.990i	−0.913043	−	0.527146i
$369$		0			0
$370$	64.0000	−	110.851i	0.172973	−	0.299598i
$371$	156.000	−	90.0666i	0.420485	−	0.242767i
$372$		0			0
$373$	173.000	−	299.645i	0.463807	−	0.803337i	−0.535340	−	0.844637i	$-0.679816\pi$
$373$	173.000	−	299.645i	0.463807	−	0.803337i	0.999147	+	0.0412995i	$0.0131498\pi$
$374$	−231.000	+	133.368i	−0.617647	+	0.356599i
$375$		0			0
$376$	24.0000	−	13.8564i	0.0638298	−	0.0368521i
$377$	748.000			1.98408
$378$		0			0
$379$		−	327.358i		−	0.863740i	−0.901936	−	0.431870i	$-0.857854\pi$
$379$		−	327.358i		−	0.863740i	0.901936	−	0.431870i	$-0.142146\pi$
$380$	−216.000	−	124.708i	−0.568421	−	0.328178i
$381$		0			0
$382$	6.00000	−	3.46410i	0.0157068	−	0.00906833i
$383$	546.000	+	315.233i	1.42559	+	0.823063i	0.996769	−	0.0803272i	$-0.0255965\pi$
$383$	546.000	+	315.233i	1.42559	+	0.823063i	0.428819	+	0.903390i	$0.358930\pi$
$384$		0			0
$385$	84.0000	+	145.492i	0.218182	+	0.377902i
$386$	−67.0000	+	116.047i	−0.173575	+	0.300641i
$387$		0			0
$388$	86.0000	+	148.956i	0.221649	+	0.383908i
$389$	−73.0000	−	126.440i	−0.187661	−	0.325038i	0.756809	−	0.653636i	$-0.226757\pi$
$389$	−73.0000	−	126.440i	−0.187661	−	0.325038i	−0.944470	+	0.328598i	$0.893424\pi$
$390$		0			0
$391$	−231.000	−	133.368i	−0.590793	−	0.341094i
$392$	148.000	−	256.344i	0.377551	−	0.653938i
$393$		0			0
$394$	536.000			1.36041
$395$		−	110.851i		−	0.280636i
$396$		0			0
$397$	488.000			1.22922			0.614610	−	0.788831i	$-0.289314\pi$
$397$	488.000			1.22922			0.614610	+	0.788831i	$0.289314\pi$
$398$		−	62.3538i		−	0.156668i
$399$		0			0
$400$	72.0000	+	124.708i	0.180000	+	0.311769i
$401$	222.500	−	385.381i	0.554863	−	0.961051i	−0.443051	−	0.896496i	$-0.646104\pi$
$401$	222.500	−	385.381i	0.554863	−	0.961051i	0.997914	−	0.0645544i	$-0.0205626\pi$
$402$		0			0
$403$	132.000	−	76.2102i	0.327543	−	0.189107i
$404$	−40.0000	−	69.2820i	−0.0990099	−	0.171490i
$405$		0			0
$406$	204.000	+	117.779i	0.502463	+	0.290097i
$407$	−168.000	+	96.9948i	−0.412776	+	0.238317i
$408$		0			0
$409$	33.5000	−	58.0237i	0.0819071	−	0.141867i	−0.822162	−	0.569254i	$-0.807232\pi$
$409$	33.5000	−	58.0237i	0.0819071	−	0.141867i	0.904069	+	0.427386i	$0.140566\pi$
$410$	52.0000	+	90.0666i	0.126829	+	0.219675i
$411$		0			0
$412$	−84.0000	−	48.4974i	−0.203883	−	0.117712i
$413$	−186.000			−0.450363
$414$		0			0
$415$			138.564i			0.333889i
$416$	−352.000	+	609.682i	−0.846154	+	1.46558i
$417$		0			0
$418$	189.000	+	327.358i	0.452153	+	0.783152i
$419$	−534.000	−	308.305i	−1.27446	−	0.735812i	−0.298638	−	0.954366i	$-0.596532\pi$
$419$	−534.000	−	308.305i	−1.27446	−	0.735812i	−0.975825	+	0.218555i	$0.929866\pi$
$420$		0			0
$421$	−136.000	−	235.559i	−0.323040	−	0.559522i	0.658073	−	0.752954i	$-0.271372\pi$
$421$	−136.000	−	235.559i	−0.323040	−	0.559522i	−0.981114	+	0.193431i	$0.938038\pi$
$422$	−228.000	−	131.636i	−0.540284	−	0.311933i
$423$		0			0
$424$	416.000			0.981132
$425$	49.5000	+	85.7365i	0.116471	+	0.201733i
$426$		0			0
$427$	−48.0000	−	27.7128i	−0.112412	−	0.0649012i
$428$			62.3538i			0.145687i
$429$		0			0
$430$		−	401.836i		−	0.934502i
$431$			405.300i			0.940371i	0.882568	+	0.470185i	$0.155813\pi$
$431$			405.300i			0.940371i	−0.882568	+	0.470185i	$0.844187\pi$
$432$		0			0
$433$	−439.000			−1.01386			−0.506928	−	0.861988i	$-0.669219\pi$
$433$	−439.000			−1.01386			−0.506928	+	0.861988i	$0.669219\pi$
$434$	48.0000			0.110599
$435$		0			0
$436$	−352.000			−0.807339
$437$	−189.000	+	327.358i	−0.432494	+	0.749102i
$438$		0			0
$439$	−732.000	+	422.620i	−1.66743	+	0.962689i	−0.698408	+	0.715700i	$0.746108\pi$
$439$	−732.000	+	422.620i	−1.66743	+	0.962689i	−0.969018	+	0.246989i	$0.920559\pi$
$440$			387.979i			0.881771i
$441$		0			0
$442$	−242.000	+	419.156i	−0.547511	+	0.948317i
$443$	−286.500	+	165.411i	−0.646727	+	0.373388i	−0.787201	−	0.616696i	$-0.788471\pi$
$443$	−286.500	+	165.411i	−0.646727	+	0.373388i	0.140474	+	0.990084i	$0.455137\pi$
$444$		0			0
$445$	4.00000	−	6.92820i	0.00898876	−	0.0155690i
$446$	−102.000	+	58.8897i	−0.228700	+	0.132040i
$447$		0			0
$448$	−192.000	+	110.851i	−0.428571	+	0.247436i
$449$	47.0000			0.104677			0.0523385	−	0.998629i	$-0.483333\pi$
$449$	47.0000			0.104677			0.0523385	+	0.998629i	$0.483333\pi$
$450$		0			0
$451$		−	157.617i		−	0.349483i
$452$	−100.000	+	173.205i	−0.221239	+	0.383197i
$453$		0			0
$454$	−777.000	+	448.601i	−1.71145	+	0.988108i
$455$	264.000	+	152.420i	0.580220	+	0.334990i
$456$		0			0
$457$	165.500	+	286.654i	0.362144	+	0.627253i	0.988314	−	0.152435i	$-0.0487114\pi$
$457$	165.500	+	286.654i	0.362144	+	0.627253i	−0.626169	+	0.779687i	$0.715378\pi$
$458$	410.000	−	710.141i	0.895197	−	1.55053i
$459$		0			0
$460$	−336.000	+	193.990i	−0.730435	+	0.421717i
$461$	269.000	+	465.922i	0.583514	+	1.01068i	0.995059	+	0.0992865i	$0.0316560\pi$
$461$	269.000	+	465.922i	0.583514	+	1.01068i	−0.411545	+	0.911390i	$0.635011\pi$
$462$		0			0
$463$	492.000	+	284.056i	1.06263	+	0.613513i	0.926160	−	0.377131i	$-0.123089\pi$
$463$	492.000	+	284.056i	1.06263	+	0.613513i	0.136475	+	0.990644i	$0.456423\pi$
$464$	272.000	+	471.118i	0.586207	+	1.01534i
$465$		0			0
$466$	−130.000			−0.278970
$467$		−	639.127i		−	1.36858i	−0.729210	−	0.684290i	$-0.760112\pi$
$467$		−	639.127i		−	1.36858i	0.729210	−	0.684290i	$-0.239888\pi$
$468$		0			0
$469$	−402.000			−0.857143
$470$		−	27.7128i		−	0.0589634i
$471$		0			0
$472$	−372.000	−	214.774i	−0.788136	−	0.455030i
$473$	−304.500	+	527.409i	−0.643763	+	1.11503i
$474$		0			0
$475$	121.500	−	70.1481i	0.255789	−	0.147680i
$476$	−132.000	+	76.2102i	−0.277311	+	0.160106i
$477$		0			0
$478$	−66.0000	−	38.1051i	−0.138075	−	0.0797178i
$479$	105.000	−	60.6218i	0.219207	−	0.126559i	−0.386376	−	0.922341i	$-0.626273\pi$
$479$	105.000	−	60.6218i	0.219207	−	0.126559i	0.605583	+	0.795782i	$0.292940\pi$
$480$		0			0
$481$	−176.000	+	304.841i	−0.365904	+	0.633765i
$482$	−223.000	−	386.247i	−0.462656	−	0.801343i
$483$		0			0
$484$	52.0000	−	90.0666i	0.107438	−	0.186088i
$485$	172.000			0.354639
$486$		0			0
$487$		−	405.300i		−	0.832238i	−0.909310	−	0.416119i	$-0.863390\pi$
$487$		−	405.300i		−	0.832238i	0.909310	−	0.416119i	$-0.136610\pi$
$488$	−64.0000	−	110.851i	−0.131148	−	0.227154i
$489$		0			0
$490$	−148.000	−	256.344i	−0.302041	−	0.523150i
$491$	−628.500	−	362.865i	−1.28004	−	0.739032i	−0.303185	−	0.952932i	$-0.598050\pi$
$491$	−628.500	−	362.865i	−1.28004	−	0.739032i	−0.976856	+	0.213900i	$0.931383\pi$
$492$		0			0
$493$	187.000	+	323.894i	0.379310	+	0.656985i
$494$	594.000	+	342.946i	1.20243	+	0.694223i
$495$		0			0
$496$	96.0000	+	55.4256i	0.193548	+	0.111745i
$497$		0			0
$498$		0			0
$499$	451.500	+	260.674i	0.904810	+	0.522392i	0.878758	−	0.477269i	$-0.158373\pi$
$499$	451.500	+	260.674i	0.904810	+	0.522392i	0.0260521	+	0.999661i	$0.491706\pi$
$500$	544.000			1.08800
$501$		0			0
$502$		−	218.238i		−	0.434738i
$503$			872.954i			1.73549i	0.497006	+	0.867747i	$0.334433\pi$
$503$			872.954i			1.73549i	−0.497006	+	0.867747i	$0.665567\pi$
$504$		0			0
$505$	−80.0000			−0.158416
$506$	588.000			1.16206
$507$		0			0
$508$		−	872.954i		−	1.71841i
$509$	380.000	−	658.179i	0.746562	−	1.29308i	−0.202900	−	0.979200i	$-0.565037\pi$
$509$	380.000	−	658.179i	0.746562	−	1.29308i	0.949461	−	0.313884i	$-0.101630\pi$
$510$		0			0
$511$	75.0000	−	43.3013i	0.146771	−	0.0847383i
$512$	−512.000			−1.00000
$513$		0			0
$514$	437.000	−	756.906i	0.850195	−	1.47258i
$515$	−84.0000	+	48.4974i	−0.163107	+	0.0941698i
$516$		0			0
$517$	−21.0000	+	36.3731i	−0.0406190	+	0.0703541i
$518$	−96.0000	+	55.4256i	−0.185328	+	0.106999i
$519$		0			0
$520$	352.000	+	609.682i	0.676923	+	1.17247i
$521$	−745.000			−1.42994			−0.714971	−	0.699154i	$-0.753560\pi$
$521$	−745.000			−1.42994			−0.714971	+	0.699154i	$0.753560\pi$
$522$		0			0
$523$		−	561.184i		−	1.07301i	−0.843897	−	0.536505i	$-0.819744\pi$
$523$		−	561.184i		−	1.07301i	0.843897	−	0.536505i	$-0.180256\pi$
$524$	672.000	+	387.979i	1.28244	+	0.740419i
$525$		0			0
$526$	546.000	−	315.233i	1.03802	−	0.599303i
$527$	66.0000	+	38.1051i	0.125237	+	0.0723057i
$528$		0			0
$529$	29.5000	+	51.0955i	0.0557656	+	0.0965888i
$530$	208.000	−	360.267i	0.392453	−	0.679748i
$531$		0			0
$532$	108.000	+	187.061i	0.203008	+	0.351619i
$533$	−143.000	−	247.683i	−0.268293	−	0.464697i
$534$		0			0
$535$	54.0000	+	31.1769i	0.100935	+	0.0582746i
$536$	−804.000	−	464.190i	−1.50000	−	0.866025i
$537$		0			0
$538$	608.000			1.13011
$539$			448.601i			0.832284i
$540$		0			0
$541$	−520.000			−0.961183			−0.480591	−	0.876945i	$-0.659578\pi$
$541$	−520.000			−0.961183			−0.480591	+	0.876945i	$0.659578\pi$
$542$		−	623.538i		−	1.15044i
$543$		0			0
$544$	−352.000			−0.647059
$545$	−176.000	+	304.841i	−0.322936	+	0.559341i
$546$		0			0
$547$	334.500	−	193.124i	0.611517	−	0.353060i	−0.162042	−	0.986784i	$-0.551808\pi$
$547$	334.500	−	193.124i	0.611517	−	0.353060i	0.773559	+	0.633724i	$0.218475\pi$
$548$	338.000	+	585.433i	0.616788	+	1.06831i
$549$		0			0
$550$	−189.000	−	109.119i	−0.343636	−	0.198399i
$551$	459.000	−	265.004i	0.833031	−	0.480951i
$552$		0			0
$553$	−48.0000	+	83.1384i	−0.0867993	+	0.150341i
$554$	−34.0000	−	58.8897i	−0.0613718	−	0.106299i
$555$		0			0
$556$	−678.000	−	391.443i	−1.21942	−	0.704035i
$557$	−934.000			−1.67684			−0.838420	−	0.545025i	$-0.816520\pi$
$557$	−934.000			−1.67684			−0.838420	+	0.545025i	$0.816520\pi$
$558$		0			0
$559$			1105.05i			1.97683i
$560$			221.703i			0.395897i
$561$		0			0
$562$	218.000	+	377.587i	0.387900	+	0.671863i
$563$	613.500	+	354.204i	1.08970	+	0.629137i	0.933496	−	0.358588i	$-0.116742\pi$
$563$	613.500	+	354.204i	1.08970	+	0.629137i	0.156202	+	0.987725i	$0.450075\pi$
$564$		0			0
$565$	100.000	+	173.205i	0.176991	+	0.306558i
$566$	−12.0000	−	6.92820i	−0.0212014	−	0.0122406i
$567$		0			0
$568$		0			0
$569$	−347.500	−	601.888i	−0.610721	−	1.05780i	−0.991119	−	0.132976i	$-0.957547\pi$
$569$	−347.500	−	601.888i	−0.610721	−	1.05780i	0.380399	−	0.924823i	$-0.375787\pi$
$570$		0			0
$571$	−466.500	−	269.334i	−0.816988	−	0.471688i	0.0323889	−	0.999475i	$-0.489689\pi$
$571$	−466.500	−	269.334i	−0.816988	−	0.471688i	−0.849377	+	0.527787i	$0.823022\pi$
$572$		−	1066.94i		−	1.86529i
$573$		0			0
$574$		−	90.0666i		−	0.156911i
$575$		−	218.238i		−	0.379545i
$576$		0			0
$577$	227.000			0.393414			0.196707	−	0.980462i	$-0.436975\pi$
$577$	227.000			0.393414			0.196707	+	0.980462i	$0.436975\pi$
$578$	336.000			0.581315
$579$		0			0
$580$	544.000			0.937931
$581$	60.0000	−	103.923i	0.103270	−	0.178869i
$582$		0			0
$583$	−546.000	+	315.233i	−0.936535	+	0.540709i
$584$	200.000			0.342466
$585$		0			0
$586$	−202.000	+	349.874i	−0.344710	+	0.597055i
$587$	−124.500	+	71.8801i	−0.212095	+	0.122453i	−0.602285	−	0.798281i	$-0.705743\pi$
$587$	−124.500	+	71.8801i	−0.212095	+	0.122453i	0.390189	+	0.920735i	$0.372410\pi$
$588$		0			0
$589$	54.0000	−	93.5307i	0.0916808	−	0.158796i
$590$	−372.000	+	214.774i	−0.630508	+	0.364024i
$591$		0			0
$592$	−256.000			−0.432432
$593$	506.000			0.853288			0.426644	−	0.904420i	$-0.359696\pi$
$593$	506.000			0.853288			0.426644	+	0.904420i	$0.359696\pi$
$594$		0			0
$595$			152.420i			0.256169i
$596$	260.000	−	450.333i	0.436242	−	0.755593i
$597$		0			0
$598$	924.000	−	533.472i	1.54515	−	0.892093i
$599$	−48.0000	−	27.7128i	−0.0801336	−	0.0462651i	0.459398	−	0.888231i	$-0.348065\pi$
$599$	−48.0000	−	27.7128i	−0.0801336	−	0.0462651i	−0.539531	+	0.841965i	$0.681399\pi$
$600$		0			0
$601$	−167.500	−	290.119i	−0.278702	−	0.482726i	0.692360	−	0.721552i	$-0.256571\pi$
$601$	−167.500	−	290.119i	−0.278702	−	0.482726i	−0.971062	+	0.238826i	$0.923238\pi$
$602$	−174.000	+	301.377i	−0.289037	+	0.500626i
$603$		0			0
$604$	420.000	−	242.487i	0.695364	−	0.401469i
$605$	−52.0000	−	90.0666i	−0.0859504	−	0.148870i
$606$		0			0
$607$	546.000	+	315.233i	0.899506	+	0.519330i	0.877040	−	0.480418i	$-0.159515\pi$
$607$	546.000	+	315.233i	0.899506	+	0.519330i	0.0224660	+	0.999748i	$0.492848\pi$
$608$			498.831i			0.820445i
$609$		0			0
$610$	−128.000			−0.209836
$611$			76.2102i			0.124730i
$612$		0			0
$613$	−340.000			−0.554649			−0.277325	−	0.960776i	$-0.589448\pi$
$613$	−340.000			−0.554649			−0.277325	+	0.960776i	$0.589448\pi$
$614$			218.238i			0.355437i
$615$		0			0
$616$	168.000	−	290.985i	0.272727	−	0.472377i
$617$	195.500	−	338.616i	0.316856	−	0.548810i	−0.662974	−	0.748642i	$-0.730706\pi$
$617$	195.500	−	338.616i	0.316856	−	0.548810i	0.979830	+	0.199832i	$0.0640396\pi$
$618$		0			0
$619$	10.5000	−	6.06218i	0.0169628	−	0.00979350i	−0.491495	−	0.870881i	$-0.663549\pi$
$619$	10.5000	−	6.06218i	0.0169628	−	0.00979350i	0.508457	+	0.861087i	$0.330216\pi$
$620$	96.0000	−	55.4256i	0.154839	−	0.0893962i
$621$		0			0
$622$	474.000	+	273.664i	0.762058	+	0.439974i
$623$	−6.00000	+	3.46410i	−0.00963082	+	0.00556036i
$624$		0			0
$625$	159.500	−	276.262i	0.255200	−	0.442019i
$626$	−79.0000	−	136.832i	−0.126198	−	0.218581i
$627$		0			0
$628$	8.00000	−	13.8564i	0.0127389	−	0.0220643i
$629$	−176.000			−0.279809
$630$		0			0
$631$			436.477i			0.691722i	0.938286	+	0.345861i	$0.112413\pi$
$631$			436.477i			0.691722i	−0.938286	+	0.345861i	$0.887587\pi$
$632$	−192.000	+	110.851i	−0.303797	+	0.175398i
$633$		0			0
$634$	−502.000	−	869.490i	−0.791798	−	1.37143i
$635$	−756.000	−	436.477i	−1.19055	−	0.687365i
$636$		0			0
$637$	407.000	+	704.945i	0.638932	+	1.10666i
$638$	−714.000	−	412.228i	−1.11912	−	0.646126i
$639$		0			0
$640$	−256.000	+	443.405i	−0.400000	+	0.692820i
$641$	210.500	+	364.597i	0.328393	+	0.568794i	0.982193	−	0.187874i	$-0.0601596\pi$
$641$	210.500	+	364.597i	0.328393	+	0.568794i	−0.653800	+	0.756667i	$0.726826\pi$
$642$		0			0
$643$	−358.500	−	206.980i	−0.557543	−	0.321897i	0.194616	−	0.980880i	$-0.437654\pi$
$643$	−358.500	−	206.980i	−0.557543	−	0.321897i	−0.752159	+	0.658982i	$0.770987\pi$
$644$	336.000			0.521739
$645$		0			0
$646$			342.946i			0.530876i
$647$			405.300i			0.626430i	0.949682	+	0.313215i	$0.101406\pi$
$647$			405.300i			0.626430i	−0.949682	+	0.313215i	$0.898594\pi$
$648$		0			0
$649$	651.000			1.00308
$650$	−396.000			−0.609231
$651$		0			0
$652$			1247.08i			1.91269i
$653$	443.000	−	767.299i	0.678407	−	1.17504i	−0.297053	−	0.954861i	$-0.596004\pi$
$653$	443.000	−	767.299i	0.678407	−	1.17504i	0.975460	−	0.220175i	$-0.0706628\pi$
$654$		0			0
$655$	672.000	−	387.979i	1.02595	−	0.592335i
$656$	104.000	−	180.133i	0.158537	−	0.274593i
$657$		0			0
$658$	−12.0000	+	20.7846i	−0.0182371	+	0.0315876i
$659$	726.000	−	419.156i	1.10167	−	0.636049i	0.165010	−	0.986292i	$-0.447234\pi$
$659$	726.000	−	419.156i	1.10167	−	0.636049i	0.936659	+	0.350243i	$0.113901\pi$
$660$		0			0
$661$	−124.000	+	214.774i	−0.187595	+	0.324923i	−0.944448	−	0.328662i	$-0.893402\pi$
$661$	−124.000	+	214.774i	−0.187595	+	0.324923i	0.756853	+	0.653585i	$0.226736\pi$
$662$	708.000	−	408.764i	1.06949	−	0.617468i
$663$		0			0
$664$	240.000	−	138.564i	0.361446	−	0.208681i
$665$	216.000			0.324812
$666$		0			0
$667$		−	824.456i		−	1.23607i
$668$	−624.000	−	360.267i	−0.934132	−	0.539321i
$669$		0			0
$670$	−804.000	+	464.190i	−1.20000	+	0.692820i
$671$	168.000	+	96.9948i	0.250373	+	0.144553i
$672$		0			0
$673$	−577.000	−	999.393i	−0.857355	−	1.48498i	−0.874443	−	0.485129i	$-0.838773\pi$
$673$	−577.000	−	999.393i	−0.857355	−	1.48498i	0.0170877	−	0.999854i	$-0.494561\pi$
$674$	−337.000	+	583.701i	−0.500000	+	0.866025i
$675$		0			0
$676$	−630.000	−	1091.19i	−0.931953	−	1.61419i
$677$	566.000	+	980.341i	0.836041	+	1.44807i	0.893180	+	0.449700i	$0.148469\pi$
$677$	566.000	+	980.341i	0.836041	+	1.44807i	−0.0571384	+	0.998366i	$0.518198\pi$
$678$		0			0
$679$	−129.000	−	74.4782i	−0.189985	−	0.109688i
$680$	−176.000	+	304.841i	−0.258824	+	0.448296i
$681$		0			0
$682$	−168.000			−0.246334
$683$		−	795.011i		−	1.16400i	−0.813189	−	0.582000i	$-0.802271\pi$
$683$		−	795.011i		−	1.16400i	0.813189	−	0.582000i	$-0.197729\pi$
$684$		0			0
$685$	676.000			0.986861
$686$			595.825i			0.868550i
$687$		0			0
$688$	−696.000	+	401.836i	−1.01163	+	0.584064i
$689$	−572.000	+	990.733i	−0.830189	+	1.43793i
$690$		0			0
$691$	780.000	−	450.333i	1.12880	−	0.651712i	0.185166	−	0.982707i	$-0.440718\pi$
$691$	780.000	−	450.333i	1.12880	−	0.651712i	0.943633	+	0.330995i	$0.107384\pi$
$692$	−4.00000	−	6.92820i	−0.00578035	−	0.0100119i
$693$		0			0
$694$	−471.000	−	271.932i	−0.678674	−	0.391833i
$695$	−678.000	+	391.443i	−0.975540	+	0.563228i
$696$		0			0
$697$	71.5000	−	123.842i	0.102582	−	0.177678i
$698$	272.000	+	471.118i	0.389685	+	0.674954i
$699$		0			0
$700$	−108.000	−	62.3538i	−0.154286	−	0.0890769i
$701$	−142.000			−0.202568			−0.101284	−	0.994858i	$-0.532295\pi$
$701$	−142.000			−0.202568			−0.101284	+	0.994858i	$0.532295\pi$
$702$		0			0
$703$			249.415i			0.354787i
$704$	672.000	−	387.979i	0.954545	−	0.551107i
$705$		0			0
$706$	461.000	+	798.475i	0.652975	+	1.13099i
$707$	60.0000	+	34.6410i	0.0848656	+	0.0489972i
$708$		0			0
$709$	−370.000	−	640.859i	−0.521862	−	0.903891i	−0.999677	−	0.0254305i	$-0.991904\pi$
$709$	−370.000	−	640.859i	−0.521862	−	0.903891i	0.477815	−	0.878461i	$-0.341429\pi$
$710$		0			0
$711$		0			0
$712$	−16.0000			−0.0224719
$713$	−84.0000	−	145.492i	−0.117812	−	0.204056i
$714$		0			0
$715$	−924.000	−	533.472i	−1.29231	−	0.746114i
$716$			748.246i			1.04504i
$717$		0			0
$718$			1060.02i			1.47634i
$719$			124.708i			0.173446i	0.996232	+	0.0867230i	$0.0276395\pi$
$719$			124.708i			0.173446i	−0.996232	+	0.0867230i	$0.972360\pi$
$720$		0			0
$721$	84.0000			0.116505
$722$	−236.000			−0.326870
$723$		0			0
$724$	1016.00			1.40331
$725$	−153.000	+	265.004i	−0.211034	+	0.365522i
$726$		0			0
$727$	−705.000	+	407.032i	−0.969739	+	0.559879i	−0.899157	−	0.437627i	$-0.855819\pi$
$727$	−705.000	+	407.032i	−0.969739	+	0.559879i	−0.0705821	+	0.997506i	$0.522486\pi$
$728$		−	609.682i		−	0.837475i
$729$		0			0
$730$	100.000	−	173.205i	0.136986	−	0.237267i
$731$	−478.500	+	276.262i	−0.654583	+	0.377924i
$732$		0			0
$733$	−457.000	+	791.547i	−0.623465	+	1.07987i	0.365370	+	0.930862i	$0.380942\pi$
$733$	−457.000	+	791.547i	−0.623465	+	1.07987i	−0.988836	+	0.149011i	$0.952391\pi$
$734$	168.000	−	96.9948i	0.228883	−	0.132146i
$735$		0			0
$736$	672.000	+	387.979i	0.913043	+	0.527146i
$737$	1407.00			1.90909
$738$		0			0
$739$			358.535i			0.485162i	0.970131	+	0.242581i	$0.0779940\pi$
$739$			358.535i			0.485162i	−0.970131	+	0.242581i	$0.922006\pi$
$740$	−128.000	+	221.703i	−0.172973	+	0.299598i
$741$		0			0
$742$	−312.000	+	180.133i	−0.420485	+	0.242767i
$743$	−345.000	−	199.186i	−0.464334	−	0.268083i	0.249531	−	0.968367i	$-0.419724\pi$
$743$	−345.000	−	199.186i	−0.464334	−	0.268083i	−0.713865	+	0.700284i	$0.753057\pi$
$744$		0			0
$745$	−260.000	−	450.333i	−0.348993	−	0.604474i
$746$	−346.000	+	599.290i	−0.463807	+	0.803337i
$747$		0			0
$748$	462.000	−	266.736i	0.617647	−	0.356599i
$749$	−27.0000	−	46.7654i	−0.0360481	−	0.0624371i
$750$		0			0
$751$	−966.000	−	557.720i	−1.28628	−	0.742637i	−0.308295	−	0.951291i	$-0.599759\pi$
$751$	−966.000	−	557.720i	−1.28628	−	0.742637i	−0.977990	+	0.208654i	$0.933092\pi$
$752$	−48.0000	+	27.7128i	−0.0638298	+	0.0368521i
$753$		0			0
$754$	−1496.00			−1.98408
$755$		−	484.974i		−	0.642350i
$756$		0			0
$757$	758.000			1.00132			0.500661	−	0.865644i	$-0.333091\pi$
$757$	758.000			1.00132			0.500661	+	0.865644i	$0.333091\pi$
$758$			654.715i			0.863740i
$759$		0			0
$760$	432.000	+	249.415i	0.568421	+	0.328178i
$761$	−187.000	+	323.894i	−0.245729	+	0.425616i	−0.962336	−	0.271861i	$-0.912361\pi$
$761$	−187.000	+	323.894i	−0.245729	+	0.425616i	0.716607	+	0.697477i	$0.245694\pi$
$762$		0			0
$763$	264.000	−	152.420i	0.346003	−	0.199765i
$764$	−12.0000	+	6.92820i	−0.0157068	+	0.00906833i
$765$		0			0
$766$	−1092.00	−	630.466i	−1.42559	−	0.823063i
$767$	1023.00	−	590.629i	1.33377	−	0.770051i
$768$		0			0
$769$	11.0000	−	19.0526i	0.0143043	−	0.0247758i	−0.858785	−	0.512337i	$-0.828780\pi$
$769$	11.0000	−	19.0526i	0.0143043	−	0.0247758i	0.873089	+	0.487561i	$0.162113\pi$
$770$	−168.000	−	290.985i	−0.218182	−	0.377902i
$771$		0			0
$772$	134.000	−	232.095i	0.173575	−	0.300641i
$773$	1334.00			1.72574			0.862872	−	0.505423i	$-0.168663\pi$
$773$	1334.00			1.72574			0.862872	+	0.505423i	$0.168663\pi$
$774$		0			0
$775$			62.3538i			0.0804566i
$776$	−172.000	−	297.913i	−0.221649	−	0.383908i
$777$		0			0
$778$	146.000	+	252.879i	0.187661	+	0.325038i
$779$	−175.500	−	101.325i	−0.225289	−	0.130071i
$780$		0			0
$781$		0			0
$782$	462.000	+	266.736i	0.590793	+	0.341094i
$783$		0			0
$784$	−296.000	+	512.687i	−0.377551	+	0.653938i
$785$	−8.00000	−	13.8564i	−0.0101911	−	0.0176515i
$786$		0			0
$787$	762.000	+	439.941i	0.968234	+	0.559010i	0.898697	−	0.438569i	$-0.144515\pi$
$787$	762.000	+	439.941i	0.968234	+	0.559010i	0.0695365	+	0.997579i	$0.477848\pi$
$788$	−1072.00			−1.36041
$789$		0			0
$790$			221.703i			0.280636i
$791$		−	173.205i		−	0.218970i
$792$		0			0
$793$	352.000			0.443884
$794$	−976.000			−1.22922
$795$		0			0
$796$			124.708i			0.156668i
$797$	416.000	−	720.533i	0.521957	−	0.904057i	−0.477716	−	0.878514i	$-0.658535\pi$
$797$	416.000	−	720.533i	0.521957	−	0.904057i	0.999674	−	0.0255425i	$-0.00813132\pi$
$798$		0			0
$799$	−33.0000	+	19.0526i	−0.0413016	+	0.0238455i
$800$	−144.000	−	249.415i	−0.180000	−	0.311769i
$801$		0			0
$802$	−445.000	+	770.763i	−0.554863	+	0.961051i
$803$	−262.500	+	151.554i	−0.326899	+	0.188735i
$804$		0			0
$805$	168.000	−	290.985i	0.208696	−	0.361471i
$806$	−264.000	+	152.420i	−0.327543	+	0.189107i
$807$		0			0
$808$	80.0000	+	138.564i	0.0990099	+	0.171490i
$809$	−493.000			−0.609394			−0.304697	−	0.952449i	$-0.598555\pi$
$809$	−493.000			−0.609394			−0.304697	+	0.952449i	$0.598555\pi$
$810$		0			0
$811$			327.358i			0.403647i	0.979422	+	0.201823i	$0.0646867\pi$
$811$			327.358i			0.403647i	−0.979422	+	0.201823i	$0.935313\pi$
$812$	−408.000	−	235.559i	−0.502463	−	0.290097i
$813$		0			0
$814$	336.000	−	193.990i	0.412776	−	0.238317i
$815$	1080.00	+	623.538i	1.32515	+	0.765078i
$816$		0			0
$817$	391.500	+	678.098i	0.479192	+	0.829985i
$818$	−67.0000	+	116.047i	−0.0819071	+	0.141867i
$819$		0			0
$820$	−104.000	−	180.133i	−0.126829	−	0.219675i
$821$	−379.000	−	656.447i	−0.461632	−	0.799570i	0.537410	−	0.843321i	$-0.319403\pi$
$821$	−379.000	−	656.447i	−0.461632	−	0.799570i	−0.999042	+	0.0437505i	$0.986069\pi$
$822$		0			0
$823$	−750.000	−	433.013i	−0.911300	−	0.526139i	−0.0304509	−	0.999536i	$-0.509694\pi$
$823$	−750.000	−	433.013i	−0.911300	−	0.526139i	−0.880849	+	0.473397i	$0.843028\pi$
$824$	168.000	+	96.9948i	0.203883	+	0.117712i
$825$		0			0
$826$	372.000			0.450363
$827$			436.477i			0.527783i	0.964552	+	0.263892i	$0.0850061\pi$
$827$			436.477i			0.527783i	−0.964552	+	0.263892i	$0.914994\pi$
$828$		0			0
$829$	−718.000			−0.866104			−0.433052	−	0.901369i	$-0.642563\pi$
$829$	−718.000			−0.866104			−0.433052	+	0.901369i	$0.642563\pi$
$830$		−	277.128i		−	0.333889i
$831$		0			0
$832$	704.000	−	1219.36i	0.846154	−	1.46558i
$833$	−203.500	+	352.472i	−0.244298	+	0.423136i
$834$		0			0
$835$	−624.000	+	360.267i	−0.747305	+	0.431457i
$836$	−378.000	−	654.715i	−0.452153	−	0.783152i
$837$		0			0
$838$	1068.00	+	616.610i	1.27446	+	0.735812i
$839$	−786.000	+	453.797i	−0.936830	+	0.540879i	−0.888965	−	0.457975i	$-0.848575\pi$
$839$	−786.000	+	453.797i	−0.936830	+	0.540879i	−0.0478645	+	0.998854i	$0.515242\pi$
$840$		0			0
$841$	−157.500	+	272.798i	−0.187277	+	0.324373i
$842$	272.000	+	471.118i	0.323040	+	0.559522i
$843$		0			0
$844$	456.000	+	263.272i	0.540284	+	0.311933i
$845$	−1260.00			−1.49112
$846$		0			0
$847$			90.0666i			0.106336i
$848$	−832.000			−0.981132
$849$		0			0
$850$	−99.0000	−	171.473i	−0.116471	−	0.201733i
$851$	336.000	+	193.990i	0.394830	+	0.227955i
$852$		0			0
$853$	−73.0000	−	126.440i	−0.0855803	−	0.148229i	0.820058	−	0.572280i	$-0.193941\pi$
$853$	−73.0000	−	126.440i	−0.0855803	−	0.148229i	−0.905638	+	0.424051i	$0.860608\pi$
$854$	96.0000	+	55.4256i	0.112412	+	0.0649012i
$855$		0			0
$856$		−	124.708i		−	0.145687i
$857$	−73.0000	−	126.440i	−0.0851809	−	0.147538i	0.820287	−	0.571952i	$-0.193813\pi$
$857$	−73.0000	−	126.440i	−0.0851809	−	0.147538i	−0.905468	+	0.424414i	$0.860480\pi$
$858$		0			0
$859$	73.5000	+	42.4352i	0.0855646	+	0.0494008i	0.542172	−	0.840268i	$-0.317602\pi$
$859$	73.5000	+	42.4352i	0.0855646	+	0.0494008i	−0.456607	+	0.889668i	$0.650936\pi$
$860$			803.672i			0.934502i
$861$		0			0
$862$		−	810.600i		−	0.940371i
$863$		−	1184.72i		−	1.37280i	−0.727226	−	0.686398i	$-0.759191\pi$
$863$		−	1184.72i		−	1.37280i	0.727226	−	0.686398i	$-0.240809\pi$
$864$		0			0
$865$	−8.00000			−0.00924855
$866$	878.000			1.01386
$867$		0			0
$868$	−96.0000			−0.110599
$869$	168.000	−	290.985i	0.193326	−	0.334850i
$870$		0			0
$871$	2211.00	−	1276.52i	2.53846	−	1.46558i
$872$	704.000			0.807339
$873$		0			0
$874$	378.000	−	654.715i	0.432494	−	0.749102i
$875$	−408.000	+	235.559i	−0.466286	+	0.269210i
$876$		0			0
$877$	740.000	−	1281.72i	0.843786	−	1.46148i	−0.0428860	−	0.999080i	$-0.513655\pi$
$877$	740.000	−	1281.72i	0.843786	−	1.46148i	0.886672	−	0.462400i	$-0.153011\pi$
$878$	1464.00	−	845.241i	1.66743	−	0.962689i
$879$		0			0
$880$		−	775.959i		−	0.881771i
$881$	−142.000			−0.161180			−0.0805902	−	0.996747i	$-0.525681\pi$
$881$	−142.000			−0.161180			−0.0805902	+	0.996747i	$0.525681\pi$
$882$		0			0
$883$			1200.31i			1.35936i	0.733511	+	0.679678i	$0.237880\pi$
$883$			1200.31i			1.35936i	−0.733511	+	0.679678i	$0.762120\pi$
$884$	484.000	−	838.313i	0.547511	−	0.948317i
$885$		0			0
$886$	573.000	−	330.822i	0.646727	−	0.373388i
$887$	546.000	+	315.233i	0.615558	+	0.355393i	0.775138	−	0.631792i	$-0.217681\pi$
$887$	546.000	+	315.233i	0.615558	+	0.355393i	−0.159580	+	0.987185i	$0.551014\pi$
$888$		0			0
$889$	378.000	+	654.715i	0.425197	+	0.736463i
$890$	−8.00000	+	13.8564i	−0.00898876	+	0.0155690i
$891$		0			0
$892$	204.000	−	117.779i	0.228700	−	0.132040i
$893$	27.0000	+	46.7654i	0.0302352	+	0.0523688i
$894$		0			0
$895$	648.000	+	374.123i	0.724022	+	0.418014i
$896$	384.000	−	221.703i	0.428571	−	0.247436i
$897$		0			0
$898$	−94.0000			−0.104677
$899$			235.559i			0.262023i
$900$		0			0
$901$	−572.000			−0.634850
$902$			315.233i			0.349483i
$903$		0			0
$904$	200.000	−	346.410i	0.221239	−	0.383197i
$905$	508.000	−	879.882i	0.561326	−	0.972245i
$906$		0			0
$907$	−556.500	+	321.295i	−0.613561	+	0.354240i	−0.774358	−	0.632748i	$-0.781927\pi$
$907$	−556.500	+	321.295i	−0.613561	+	0.354240i	0.160797	+	0.986988i	$0.448594\pi$
$908$	1554.00	−	897.202i	1.71145	−	0.988108i
$909$		0			0
$910$	−528.000	−	304.841i	−0.580220	−	0.334990i
$911$	348.000	−	200.918i	0.381998	−	0.220547i	−0.296689	−	0.954974i	$-0.595883\pi$
$911$	348.000	−	200.918i	0.381998	−	0.220547i	0.678687	+	0.734428i	$0.262549\pi$
$912$		0			0
$913$	−210.000	+	363.731i	−0.230011	+	0.398391i
$914$	−331.000	−	573.309i	−0.362144	−	0.627253i
$915$		0			0
$916$	−820.000	+	1420.28i	−0.895197	+	1.55053i
$917$	−672.000			−0.732824
$918$		0			0
$919$		−	779.423i		−	0.848121i	−0.905634	−	0.424060i	$-0.860604\pi$
$919$		−	779.423i		−	0.848121i	0.905634	−	0.424060i	$-0.139396\pi$
$920$	672.000	−	387.979i	0.730435	−	0.421717i
$921$		0			0
$922$	−538.000	−	931.843i	−0.583514	−	1.01068i
$923$		0			0
$924$		0			0
$925$	−72.0000	−	124.708i	−0.0778378	−	0.134819i
$926$	−984.000	−	568.113i	−1.06263	−	0.613513i
$927$		0			0
$928$	−544.000	−	942.236i	−0.586207	−	1.01534i
$929$	−379.000	−	656.447i	−0.407966	−	0.706617i	0.586696	−	0.809807i	$-0.300428\pi$
$929$	−379.000	−	656.447i	−0.407966	−	0.706617i	−0.994662	+	0.103190i	$0.967095\pi$
$930$		0			0
$931$	499.500	+	288.386i	0.536520	+	0.309760i
$932$	260.000			0.278970
$933$		0			0
$934$			1278.25i			1.36858i
$935$		−	533.472i		−	0.570558i
$936$		0			0
$937$	−754.000			−0.804696			−0.402348	−	0.915487i	$-0.631806\pi$
$937$	−754.000			−0.804696			−0.402348	+	0.915487i	$0.631806\pi$
$938$	804.000			0.857143
$939$		0			0
$940$			55.4256i			0.0589634i
$941$	−898.000	+	1555.38i	−0.954304	+	1.65290i	−0.218351	+	0.975870i	$0.570068\pi$
$941$	−898.000	+	1555.38i	−0.954304	+	1.65290i	−0.735953	+	0.677033i	$0.763266\pi$
$942$		0			0
$943$	−273.000	+	157.617i	−0.289502	+	0.167144i
$944$	744.000	+	429.549i	0.788136	+	0.455030i
$945$		0			0
$946$	609.000	−	1054.82i	0.643763	−	1.11503i
$947$	91.5000	−	52.8275i	0.0966209	−	0.0557841i	−0.450911	−	0.892569i	$-0.648901\pi$
$947$	91.5000	−	52.8275i	0.0966209	−	0.0557841i	0.547532	+	0.836785i	$0.315567\pi$
$948$		0			0
$949$	−275.000	+	476.314i	−0.289779	+	0.501911i
$950$	−243.000	+	140.296i	−0.255789	+	0.147680i
$951$		0			0
$952$	264.000	−	152.420i	0.277311	−	0.160106i
$953$	−1213.00			−1.27282			−0.636411	−	0.771350i	$-0.719582\pi$
$953$	−1213.00			−1.27282			−0.636411	+	0.771350i	$0.719582\pi$
$954$		0			0
$955$			13.8564i			0.0145093i
$956$	132.000	+	76.2102i	0.138075	+	0.0797178i
$957$		0			0
$958$	−210.000	+	121.244i	−0.219207	+	0.126559i
$959$	−507.000	−	292.717i	−0.528676	−	0.305231i
$960$		0			0
$961$	−456.500	−	790.681i	−0.475026	−	0.822769i
$962$	352.000	−	609.682i	0.365904	−	0.633765i
$963$		0			0
$964$	446.000	+	772.495i	0.462656	+	0.801343i
$965$	−134.000	−	232.095i	−0.138860	−	0.240513i
$966$		0			0
$967$	303.000	+	174.937i	0.313340	+	0.180907i	0.648420	−	0.761283i	$-0.275430\pi$
$967$	303.000	+	174.937i	0.313340	+	0.180907i	−0.335080	+	0.942190i	$0.608763\pi$
$968$	−104.000	+	180.133i	−0.107438	+	0.186088i
$969$		0			0
$970$	−344.000			−0.354639
$971$			1434.14i			1.47697i	0.674270	+	0.738485i	$0.264458\pi$
$971$			1434.14i			1.47697i	−0.674270	+	0.738485i	$0.735542\pi$
$972$		0			0
$973$	678.000			0.696814
$974$			810.600i			0.832238i
$975$		0			0
$976$	128.000	+	221.703i	0.131148	+	0.227154i
$977$	78.5000	−	135.966i	0.0803480	−	0.139167i	−0.823051	−	0.567967i	$-0.807730\pi$
$977$	78.5000	−	135.966i	0.0803480	−	0.139167i	0.903399	+	0.428800i	$0.141064\pi$
$978$		0			0
$979$	21.0000	−	12.1244i	0.0214505	−	0.0123844i
$980$	296.000	+	512.687i	0.302041	+	0.523150i
$981$		0			0
$982$	1257.00	+	725.729i	1.28004	+	0.739032i
$983$	−1218.00	+	703.213i	−1.23906	+	0.715374i	−0.968903	−	0.247441i	$-0.920410\pi$
$983$	−1218.00	+	703.213i	−1.23906	+	0.715374i	−0.270161	+	0.962815i	$0.587077\pi$
$984$		0			0
$985$	−536.000	+	928.379i	−0.544162	+	0.942517i
$986$	−374.000	−	647.787i	−0.379310	−	0.656985i
$987$		0			0
$988$	−1188.00	−	685.892i	−1.20243	−	0.694223i
$989$	1218.00			1.23155
$990$		0			0
$991$			249.415i			0.251680i	0.992051	+	0.125840i	$0.0401627\pi$
$991$			249.415i			0.251680i	−0.992051	+	0.125840i	$0.959837\pi$
$992$	−192.000	−	110.851i	−0.193548	−	0.111745i
$993$		0			0
$994$		0			0
$995$	108.000	+	62.3538i	0.108543	+	0.0626672i
$996$		0			0
$997$	206.000	+	356.802i	0.206620	+	0.357876i	0.950648	−	0.310273i	$-0.100420\pi$
$997$	206.000	+	356.802i	0.206620	+	0.357876i	−0.744028	+	0.668149i	$0.767087\pi$
$998$	−903.000	−	521.347i	−0.904810	−	0.522392i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.3.f.a.91.1		2
3.2	odd	2		36.3.f.b.31.1	yes	2
4.3	odd	2		108.3.f.b.91.1		2
8.3	odd	2		1728.3.o.b.1279.1		2
8.5	even	2		1728.3.o.a.1279.1		2
9.2	odd	6		36.3.f.a.7.1	✓	2
9.4	even	3		324.3.d.c.163.1		2
9.5	odd	6		324.3.d.b.163.2		2
9.7	even	3		108.3.f.b.19.1		2
12.11	even	2		36.3.f.a.31.1	yes	2
24.5	odd	2		576.3.o.b.319.1		2
24.11	even	2		576.3.o.a.319.1		2
36.7	odd	6	inner	108.3.f.a.19.1		2
36.11	even	6		36.3.f.b.7.1	yes	2
36.23	even	6		324.3.d.b.163.1		2
36.31	odd	6		324.3.d.c.163.2		2
72.11	even	6		576.3.o.b.511.1		2
72.29	odd	6		576.3.o.a.511.1		2
72.43	odd	6		1728.3.o.a.127.1		2
72.61	even	6		1728.3.o.b.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.3.f.a.7.1	✓	2	9.2	odd	6
36.3.f.a.31.1	yes	2	12.11	even	2
36.3.f.b.7.1	yes	2	36.11	even	6
36.3.f.b.31.1	yes	2	3.2	odd	2
108.3.f.a.19.1		2	36.7	odd	6	inner
108.3.f.a.91.1		2	1.1	even	1	trivial
108.3.f.b.19.1		2	9.7	even	3
108.3.f.b.91.1		2	4.3	odd	2
324.3.d.b.163.1		2	36.23	even	6
324.3.d.b.163.2		2	9.5	odd	6
324.3.d.c.163.1		2	9.4	even	3
324.3.d.c.163.2		2	36.31	odd	6
576.3.o.a.319.1		2	24.11	even	2
576.3.o.a.511.1		2	72.29	odd	6
576.3.o.b.319.1		2	24.5	odd	2
576.3.o.b.511.1		2	72.11	even	6
1728.3.o.a.127.1		2	72.43	odd	6
1728.3.o.a.1279.1		2	8.5	even	2
1728.3.o.b.127.1		2	72.61	even	6
1728.3.o.b.1279.1		2	8.3	odd	2

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	108.f (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +(2.00000 - 3.46410i) q^{5} +(-3.00000 + 1.73205i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +(2.00000 - 3.46410i) q^{5} +(-3.00000 + 1.73205i) q^{7} -8.00000 q^{8} +(-4.00000 + 6.92820i) q^{10} +(10.5000 - 6.06218i) q^{11} +(11.0000 - 19.0526i) q^{13} +(6.00000 - 3.46410i) q^{14} +16.0000 q^{16} +11.0000 q^{17} -15.5885i q^{19} +(8.00000 - 13.8564i) q^{20} +(-21.0000 + 12.1244i) q^{22} +(-21.0000 - 12.1244i) q^{23} +(4.50000 + 7.79423i) q^{25} +(-22.0000 + 38.1051i) q^{26} +(-12.0000 + 6.92820i) q^{28} +(17.0000 + 29.4449i) q^{29} +(6.00000 + 3.46410i) q^{31} -32.0000 q^{32} -22.0000 q^{34} +13.8564i q^{35} -16.0000 q^{37} +31.1769i q^{38} +(-16.0000 + 27.7128i) q^{40} +(6.50000 - 11.2583i) q^{41} +(-43.5000 + 25.1147i) q^{43} +(42.0000 - 24.2487i) q^{44} +(42.0000 + 24.2487i) q^{46} +(-3.00000 + 1.73205i) q^{47} +(-18.5000 + 32.0429i) q^{49} +(-9.00000 - 15.5885i) q^{50} +(44.0000 - 76.2102i) q^{52} -52.0000 q^{53} -48.4974i q^{55} +(24.0000 - 13.8564i) q^{56} +(-34.0000 - 58.8897i) q^{58} +(46.5000 + 26.8468i) q^{59} +(8.00000 + 13.8564i) q^{61} +(-12.0000 - 6.92820i) q^{62} +64.0000 q^{64} +(-44.0000 - 76.2102i) q^{65} +(100.500 + 58.0237i) q^{67} +44.0000 q^{68} -27.7128i q^{70} -25.0000 q^{73} +32.0000 q^{74} -62.3538i q^{76} +(-21.0000 + 36.3731i) q^{77} +(24.0000 - 13.8564i) q^{79} +(32.0000 - 55.4256i) q^{80} +(-13.0000 + 22.5167i) q^{82} +(-30.0000 + 17.3205i) q^{83} +(22.0000 - 38.1051i) q^{85} +(87.0000 - 50.2295i) q^{86} +(-84.0000 + 48.4974i) q^{88} +2.00000 q^{89} +76.2102i q^{91} +(-84.0000 - 48.4974i) q^{92} +(6.00000 - 3.46410i) q^{94} +(-54.0000 - 31.1769i) q^{95} +(21.5000 + 37.2391i) q^{97} +(37.0000 - 64.0859i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 4 q^{5} - 6 q^{7} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 + 4 * q^5 - 6 * q^7 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} + 4 q^{5} - 6 q^{7} - 16 q^{8} - 8 q^{10} + 21 q^{11} + 22 q^{13} + 12 q^{14} + 32 q^{16} + 22 q^{17} + 16 q^{20} - 42 q^{22} - 42 q^{23} + 9 q^{25} - 44 q^{26} - 24 q^{28} + 34 q^{29} + 12 q^{31} - 64 q^{32} - 44 q^{34} - 32 q^{37} - 32 q^{40} + 13 q^{41} - 87 q^{43} + 84 q^{44} + 84 q^{46} - 6 q^{47} - 37 q^{49} - 18 q^{50} + 88 q^{52} - 104 q^{53} + 48 q^{56} - 68 q^{58} + 93 q^{59} + 16 q^{61} - 24 q^{62} + 128 q^{64} - 88 q^{65} + 201 q^{67} + 88 q^{68} - 50 q^{73} + 64 q^{74} - 42 q^{77} + 48 q^{79} + 64 q^{80} - 26 q^{82} - 60 q^{83} + 44 q^{85} + 174 q^{86} - 168 q^{88} + 4 q^{89} - 168 q^{92} + 12 q^{94} - 108 q^{95} + 43 q^{97} + 74 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + 4 * q^5 - 6 * q^7 - 16 * q^8 - 8 * q^10 + 21 * q^11 + 22 * q^13 + 12 * q^14 + 32 * q^16 + 22 * q^17 + 16 * q^20 - 42 * q^22 - 42 * q^23 + 9 * q^25 - 44 * q^26 - 24 * q^28 + 34 * q^29 + 12 * q^31 - 64 * q^32 - 44 * q^34 - 32 * q^37 - 32 * q^40 + 13 * q^41 - 87 * q^43 + 84 * q^44 + 84 * q^46 - 6 * q^47 - 37 * q^49 - 18 * q^50 + 88 * q^52 - 104 * q^53 + 48 * q^56 - 68 * q^58 + 93 * q^59 + 16 * q^61 - 24 * q^62 + 128 * q^64 - 88 * q^65 + 201 * q^67 + 88 * q^68 - 50 * q^73 + 64 * q^74 - 42 * q^77 + 48 * q^79 + 64 * q^80 - 26 * q^82 - 60 * q^83 + 44 * q^85 + 174 * q^86 - 168 * q^88 + 4 * q^89 - 168 * q^92 + 12 * q^94 - 108 * q^95 + 43 * q^97 + 74 * q^98

Introduction

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