LMFDB - Embedded newform 189.2.g.a.172.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(100,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 2]))

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

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

chi := DirichletCharacter("189.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	189.g (of order $3$, 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:	$1.50917259820$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			172.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	189.172
Dual form			189.2.g.a.100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 2.59808i) q^{7} +3.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 2.59808i) q^{7} +3.00000 q^{8} +(0.500000 - 0.866025i) q^{10} -5.00000 q^{11} +(2.50000 - 4.33013i) q^{13} +(2.50000 + 0.866025i) q^{14} +(0.500000 - 0.866025i) q^{16} +(1.50000 - 2.59808i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(0.500000 + 0.866025i) q^{20} +(-2.50000 + 4.33013i) q^{22} -3.00000 q^{23} -4.00000 q^{25} +(-2.50000 - 4.33013i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(-0.500000 - 0.866025i) q^{29} +(2.50000 + 4.33013i) q^{32} +(-1.50000 - 2.59808i) q^{34} +(0.500000 + 2.59808i) q^{35} +(-1.50000 - 2.59808i) q^{37} -1.00000 q^{38} +3.00000 q^{40} +(-2.50000 + 4.33013i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-2.50000 - 4.33013i) q^{44} +(-1.50000 + 2.59808i) q^{46} +(-6.50000 + 2.59808i) q^{49} +(-2.00000 + 3.46410i) q^{50} +5.00000 q^{52} +(-4.50000 + 7.79423i) q^{53} -5.00000 q^{55} +(1.50000 + 7.79423i) q^{56} -1.00000 q^{58} +(7.00000 - 12.1244i) q^{61} +7.00000 q^{64} +(2.50000 - 4.33013i) q^{65} +(-2.00000 - 3.46410i) q^{67} +3.00000 q^{68} +(2.50000 + 0.866025i) q^{70} +12.0000 q^{71} +(-1.50000 + 2.59808i) q^{73} -3.00000 q^{74} +(0.500000 - 0.866025i) q^{76} +(-2.50000 - 12.9904i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(0.500000 - 0.866025i) q^{80} +(2.50000 + 4.33013i) q^{82} +(-4.50000 - 7.79423i) q^{83} +(1.50000 - 2.59808i) q^{85} +1.00000 q^{86} -15.0000 q^{88} +(-6.50000 - 11.2583i) q^{89} +(12.5000 + 4.33013i) q^{91} +(-1.50000 - 2.59808i) q^{92} +(-0.500000 - 0.866025i) q^{95} +(4.50000 + 7.79423i) q^{97} +(-1.00000 + 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + q^{4} + 2 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + q^4 + 2 * q^5 + q^7 + 6 * q^8	$ 2 q + q^{2} + q^{4} + 2 q^{5} + q^{7} + 6 q^{8} + q^{10} - 10 q^{11} + 5 q^{13} + 5 q^{14} + q^{16} + 3 q^{17} - q^{19} + q^{20} - 5 q^{22} - 6 q^{23} - 8 q^{25} - 5 q^{26} - 4 q^{28} - q^{29} + 5 q^{32} - 3 q^{34} + q^{35} - 3 q^{37} - 2 q^{38} + 6 q^{40} - 5 q^{41} + q^{43} - 5 q^{44} - 3 q^{46} - 13 q^{49} - 4 q^{50} + 10 q^{52} - 9 q^{53} - 10 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{61} + 14 q^{64} + 5 q^{65} - 4 q^{67} + 6 q^{68} + 5 q^{70} + 24 q^{71} - 3 q^{73} - 6 q^{74} + q^{76} - 5 q^{77} - 8 q^{79} + q^{80} + 5 q^{82} - 9 q^{83} + 3 q^{85} + 2 q^{86} - 30 q^{88} - 13 q^{89} + 25 q^{91} - 3 q^{92} - q^{95} + 9 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q + q^2 + q^4 + 2 * q^5 + q^7 + 6 * q^8 + q^10 - 10 * q^11 + 5 * q^13 + 5 * q^14 + q^16 + 3 * q^17 - q^19 + q^20 - 5 * q^22 - 6 * q^23 - 8 * q^25 - 5 * q^26 - 4 * q^28 - q^29 + 5 * q^32 - 3 * q^34 + q^35 - 3 * q^37 - 2 * q^38 + 6 * q^40 - 5 * q^41 + q^43 - 5 * q^44 - 3 * q^46 - 13 * q^49 - 4 * q^50 + 10 * q^52 - 9 * q^53 - 10 * q^55 + 3 * q^56 - 2 * q^58 + 14 * q^61 + 14 * q^64 + 5 * q^65 - 4 * q^67 + 6 * q^68 + 5 * q^70 + 24 * q^71 - 3 * q^73 - 6 * q^74 + q^76 - 5 * q^77 - 8 * q^79 + q^80 + 5 * q^82 - 9 * q^83 + 3 * q^85 + 2 * q^86 - 30 * q^88 - 13 * q^89 + 25 * q^91 - 3 * q^92 - q^95 + 9 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.500000	−	0.866025i	0.353553	−	0.612372i	−0.633316	−	0.773893i	$-0.718307\pi$
$2$	0.500000	−	0.866025i	0.353553	−	0.612372i	0.986869	+	0.161521i	$0.0516399\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	1.00000			0.447214			0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000			0.447214			0.223607	+	0.974679i	$0.428217\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$	3.00000			1.06066
$9$		0			0
$10$	0.500000	−	0.866025i	0.158114	−	0.273861i
$11$	−5.00000			−1.50756			−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000			−1.50756			−0.753778	+	0.657129i	$0.771771\pi$
$12$		0			0
$13$	2.50000	−	4.33013i	0.693375	−	1.20096i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	2.50000	−	4.33013i	0.693375	−	1.20096i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$	2.50000	+	0.866025i	0.668153	+	0.231455i
$15$		0			0
$16$	0.500000	−	0.866025i	0.125000	−	0.216506i
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	−0.624780	−	0.780801i	$-0.714811\pi$
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	0.988583	+	0.150675i	$0.0481447\pi$
$18$		0			0
$19$	−0.500000	−	0.866025i	−0.114708	−	0.198680i	0.802955	−	0.596040i	$-0.203260\pi$
$19$	−0.500000	−	0.866025i	−0.114708	−	0.198680i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0.500000	+	0.866025i	0.111803	+	0.193649i
$21$		0			0
$22$	−2.50000	+	4.33013i	−0.533002	+	0.923186i
$23$	−3.00000			−0.625543			−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000			−0.625543			−0.312772	+	0.949828i	$0.601257\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$	−2.50000	−	4.33013i	−0.490290	−	0.849208i
$27$		0			0
$28$	−2.00000	+	1.73205i	−0.377964	+	0.327327i
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	0.815861	−	0.578249i	$-0.196264\pi$
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	−0.908708	+	0.417432i	$0.862930\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	2.50000	+	4.33013i	0.441942	+	0.765466i
$33$		0			0
$34$	−1.50000	−	2.59808i	−0.257248	−	0.445566i
$35$	0.500000	+	2.59808i	0.0845154	+	0.439155i
$36$		0			0
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	0.715981	−	0.698119i	$-0.245980\pi$
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	−0.962580	+	0.270998i	$0.912646\pi$
$38$	−1.00000			−0.162221
$39$		0			0
$40$	3.00000			0.474342
$41$	−2.50000	+	4.33013i	−0.390434	+	0.676252i	−0.992507	−	0.122189i	$-0.961009\pi$
$41$	−2.50000	+	4.33013i	−0.390434	+	0.676252i	0.602072	+	0.798441i	$0.294342\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$	−2.50000	−	4.33013i	−0.376889	−	0.652791i
$45$		0			0
$46$	−1.50000	+	2.59808i	−0.221163	+	0.383065i
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$	−2.00000	+	3.46410i	−0.282843	+	0.489898i
$51$		0			0
$52$	5.00000			0.693375
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	0.371706	+	0.928351i	$0.378773\pi$
$53$	−4.50000	+	7.79423i	−0.618123	+	1.07062i	−0.989828	+	0.142269i	$0.954560\pi$
$54$		0			0
$55$	−5.00000			−0.674200
$56$	1.50000	+	7.79423i	0.200446	+	1.04155i
$57$		0			0
$58$	−1.00000			−0.131306
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	7.00000	−	12.1244i	0.896258	−	1.55236i	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	7.00000	−	12.1244i	0.896258	−	1.55236i	0.832240	−	0.554416i	$-0.187058\pi$
$62$		0			0
$63$		0			0
$64$	7.00000			0.875000
$65$	2.50000	−	4.33013i	0.310087	−	0.537086i
$66$		0			0
$67$	−2.00000	−	3.46410i	−0.244339	−	0.423207i	0.717607	−	0.696449i	$-0.245238\pi$
$67$	−2.00000	−	3.46410i	−0.244339	−	0.423207i	−0.961946	+	0.273241i	$0.911904\pi$
$68$	3.00000			0.363803
$69$		0			0
$70$	2.50000	+	0.866025i	0.298807	+	0.103510i
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−1.50000	+	2.59808i	−0.175562	+	0.304082i	−0.940356	−	0.340193i	$-0.889507\pi$
$73$	−1.50000	+	2.59808i	−0.175562	+	0.304082i	0.764794	+	0.644275i	$0.222841\pi$
$74$	−3.00000			−0.348743
$75$		0			0
$76$	0.500000	−	0.866025i	0.0573539	−	0.0993399i
$77$	−2.50000	−	12.9904i	−0.284901	−	1.48039i
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$	0.500000	−	0.866025i	0.0559017	−	0.0968246i
$81$		0			0
$82$	2.50000	+	4.33013i	0.276079	+	0.478183i
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	0.506036	−	0.862512i	$-0.331110\pi$
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	−0.999976	+	0.00698436i	$0.997777\pi$
$84$		0			0
$85$	1.50000	−	2.59808i	0.162698	−	0.281801i
$86$	1.00000			0.107833
$87$		0			0
$88$	−15.0000			−1.59901
$89$	−6.50000	−	11.2583i	−0.688999	−	1.19338i	−0.972162	−	0.234309i	$-0.924717\pi$
$89$	−6.50000	−	11.2583i	−0.688999	−	1.19338i	0.283164	−	0.959072i	$-0.408616\pi$
$90$		0			0
$91$	12.5000	+	4.33013i	1.31036	+	0.453921i
$92$	−1.50000	−	2.59808i	−0.156386	−	0.270868i
$93$		0			0
$94$		0			0
$95$	−0.500000	−	0.866025i	−0.0512989	−	0.0888523i
$96$		0			0
$97$	4.50000	+	7.79423i	0.456906	+	0.791384i	0.998796	−	0.0490655i	$-0.0156243\pi$
$97$	4.50000	+	7.79423i	0.456906	+	0.791384i	−0.541890	+	0.840450i	$0.682291\pi$
$98$	−1.00000	+	6.92820i	−0.101015	+	0.699854i
$99$		0			0
$100$	−2.00000	−	3.46410i	−0.200000	−	0.346410i
$101$	17.0000			1.69156			0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000			1.69156			0.845782	+	0.533529i	$0.179135\pi$
$102$		0			0
$103$	−1.00000			−0.0985329			−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000			−0.0985329			−0.0492665	+	0.998786i	$0.515688\pi$
$104$	7.50000	−	12.9904i	0.735436	−	1.27381i
$105$		0			0
$106$	4.50000	+	7.79423i	0.437079	+	0.757042i
$107$	8.50000	+	14.7224i	0.821726	+	1.42327i	0.904396	+	0.426694i	$0.140322\pi$
$107$	8.50000	+	14.7224i	0.821726	+	1.42327i	−0.0826699	+	0.996577i	$0.526345\pi$
$108$		0			0
$109$	4.50000	−	7.79423i	0.431022	−	0.746552i	−0.565940	−	0.824447i	$-0.691487\pi$
$109$	4.50000	−	7.79423i	0.431022	−	0.746552i	0.996962	+	0.0778949i	$0.0248199\pi$
$110$	−2.50000	+	4.33013i	−0.238366	+	0.412861i
$111$		0			0
$112$	2.50000	+	0.866025i	0.236228	+	0.0818317i
$113$	−0.500000	+	0.866025i	−0.0470360	+	0.0814688i	−0.888585	−	0.458712i	$-0.848311\pi$
$113$	−0.500000	+	0.866025i	−0.0470360	+	0.0814688i	0.841549	+	0.540181i	$0.181644\pi$
$114$		0			0
$115$	−3.00000			−0.279751
$116$	0.500000	−	0.866025i	0.0464238	−	0.0804084i
$117$		0			0
$118$		0			0
$119$	7.50000	+	2.59808i	0.687524	+	0.238165i
$120$		0			0
$121$	14.0000			1.27273
$122$	−7.00000	−	12.1244i	−0.633750	−	1.09769i
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.50000	+	2.59808i	−0.132583	+	0.229640i
$129$		0			0
$130$	−2.50000	−	4.33013i	−0.219265	−	0.379777i
$131$	1.00000			0.0873704			0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000			0.0873704			0.0436852	+	0.999045i	$0.486090\pi$
$132$		0			0
$133$	2.00000	−	1.73205i	0.173422	−	0.150188i
$134$	−4.00000			−0.345547
$135$		0			0
$136$	4.50000	−	7.79423i	0.385872	−	0.668350i
$137$	9.00000			0.768922			0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000			0.768922			0.384461	+	0.923141i	$0.374387\pi$
$138$		0			0
$139$	−4.50000	+	7.79423i	−0.381685	+	0.661098i	−0.991303	−	0.131597i	$-0.957989\pi$
$139$	−4.50000	+	7.79423i	−0.381685	+	0.661098i	0.609618	+	0.792695i	$0.291323\pi$
$140$	−2.00000	+	1.73205i	−0.169031	+	0.146385i
$141$		0			0
$142$	6.00000	−	10.3923i	0.503509	−	0.872103i
$143$	−12.5000	+	21.6506i	−1.04530	+	1.81052i
$144$		0			0
$145$	−0.500000	−	0.866025i	−0.0415227	−	0.0719195i
$146$	1.50000	+	2.59808i	0.124141	+	0.215018i
$147$		0			0
$148$	1.50000	−	2.59808i	0.123299	−	0.213561i
$149$	−3.00000			−0.245770			−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000			−0.245770			−0.122885	+	0.992421i	$0.539215\pi$
$150$		0			0
$151$	5.00000			0.406894			0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000			0.406894			0.203447	+	0.979086i	$0.434786\pi$
$152$	−1.50000	−	2.59808i	−0.121666	−	0.210732i
$153$		0			0
$154$	−12.5000	−	4.33013i	−1.00728	−	0.348932i
$155$		0			0
$156$		0			0
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	−0.438948	+	0.898513i	$0.644649\pi$
$158$	4.00000	+	6.92820i	0.318223	+	0.551178i
$159$		0			0
$160$	2.50000	+	4.33013i	0.197642	+	0.342327i
$161$	−1.50000	−	7.79423i	−0.118217	−	0.614271i
$162$		0			0
$163$	5.50000	+	9.52628i	0.430793	+	0.746156i	0.996942	−	0.0781474i	$-0.0249005\pi$
$163$	5.50000	+	9.52628i	0.430793	+	0.746156i	−0.566149	+	0.824303i	$0.691567\pi$
$164$	−5.00000			−0.390434
$165$		0			0
$166$	−9.00000			−0.698535
$167$	−9.50000	+	16.4545i	−0.735132	+	1.27329i	0.219533	+	0.975605i	$0.429547\pi$
$167$	−9.50000	+	16.4545i	−0.735132	+	1.27329i	−0.954665	+	0.297681i	$0.903787\pi$
$168$		0			0
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$	−1.50000	−	2.59808i	−0.115045	−	0.199263i
$171$		0			0
$172$	−0.500000	+	0.866025i	−0.0381246	+	0.0660338i
$173$	−7.00000	+	12.1244i	−0.532200	+	0.921798i	0.467093	+	0.884208i	$0.345301\pi$
$173$	−7.00000	+	12.1244i	−0.532200	+	0.921798i	−0.999293	+	0.0375896i	$0.988032\pi$
$174$		0			0
$175$	−2.00000	−	10.3923i	−0.151186	−	0.785584i
$176$	−2.50000	+	4.33013i	−0.188445	+	0.326396i
$177$		0			0
$178$	−13.0000			−0.974391
$179$	9.50000	−	16.4545i	0.710063	−	1.22987i	−0.254770	−	0.967002i	$-0.582000\pi$
$179$	9.50000	−	16.4545i	0.710063	−	1.22987i	0.964833	−	0.262864i	$-0.0846670\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$	10.0000	−	8.66025i	0.741249	−	0.641941i
$183$		0			0
$184$	−9.00000			−0.663489
$185$	−1.50000	−	2.59808i	−0.110282	−	0.191014i
$186$		0			0
$187$	−7.50000	+	12.9904i	−0.548454	+	0.949951i
$188$		0			0
$189$		0			0
$190$	−1.00000			−0.0725476
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	−0.684244	−	0.729253i	$-0.739868\pi$
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	0.973674	+	0.227946i	$0.0732010\pi$
$192$		0			0
$193$	5.00000	+	8.66025i	0.359908	+	0.623379i	0.987945	−	0.154805i	$-0.0494748\pi$
$193$	5.00000	+	8.66025i	0.359908	+	0.623379i	−0.628037	+	0.778183i	$0.716141\pi$
$194$	9.00000			0.646162
$195$		0			0
$196$	−5.50000	−	4.33013i	−0.392857	−	0.309295i
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	−1.50000	+	2.59808i	−0.106332	+	0.184173i	−0.914282	−	0.405079i	$-0.867244\pi$
$199$	−1.50000	+	2.59808i	−0.106332	+	0.184173i	0.807950	+	0.589252i	$0.200577\pi$
$200$	−12.0000			−0.848528
$201$		0			0
$202$	8.50000	−	14.7224i	0.598058	−	1.03587i
$203$	2.00000	−	1.73205i	0.140372	−	0.121566i
$204$		0			0
$205$	−2.50000	+	4.33013i	−0.174608	+	0.302429i
$206$	−0.500000	+	0.866025i	−0.0348367	+	0.0603388i
$207$		0			0
$208$	−2.50000	−	4.33013i	−0.173344	−	0.300240i
$209$	2.50000	+	4.33013i	0.172929	+	0.299521i
$210$		0			0
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	0.550743	+	0.834675i	$0.314345\pi$
$212$	−9.00000			−0.618123
$213$		0			0
$214$	17.0000			1.16210
$215$	0.500000	+	0.866025i	0.0340997	+	0.0590624i
$216$		0			0
$217$		0			0
$218$	−4.50000	−	7.79423i	−0.304778	−	0.527892i
$219$		0			0
$220$	−2.50000	−	4.33013i	−0.168550	−	0.291937i
$221$	−7.50000	−	12.9904i	−0.504505	−	0.873828i
$222$		0			0
$223$	−9.50000	−	16.4545i	−0.636167	−	1.10187i	−0.986267	−	0.165161i	$-0.947186\pi$
$223$	−9.50000	−	16.4545i	−0.636167	−	1.10187i	0.350100	−	0.936713i	$-0.386148\pi$
$224$	−10.0000	+	8.66025i	−0.668153	+	0.578638i
$225$		0			0
$226$	0.500000	+	0.866025i	0.0332595	+	0.0576072i
$227$	3.00000			0.199117			0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000			0.199117			0.0995585	+	0.995032i	$0.468257\pi$
$228$		0			0
$229$	−1.00000			−0.0660819			−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000			−0.0660819			−0.0330409	+	0.999454i	$0.510519\pi$
$230$	−1.50000	+	2.59808i	−0.0989071	+	0.171312i
$231$		0			0
$232$	−1.50000	−	2.59808i	−0.0984798	−	0.170572i
$233$	1.50000	+	2.59808i	0.0982683	+	0.170206i	0.910968	−	0.412477i	$-0.135336\pi$
$233$	1.50000	+	2.59808i	0.0982683	+	0.170206i	−0.812700	+	0.582683i	$0.802003\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	6.00000	−	5.19615i	0.388922	−	0.336817i
$239$	−7.50000	+	12.9904i	−0.485135	+	0.840278i	−0.999854	−	0.0170808i	$-0.994563\pi$
$239$	−7.50000	+	12.9904i	−0.485135	+	0.840278i	0.514719	+	0.857359i	$0.327896\pi$
$240$		0			0
$241$	11.0000			0.708572			0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000			0.708572			0.354286	+	0.935137i	$0.384724\pi$
$242$	7.00000	−	12.1244i	0.449977	−	0.779383i
$243$		0			0
$244$	14.0000			0.896258
$245$	−6.50000	+	2.59808i	−0.415270	+	0.165985i
$246$		0			0
$247$	−5.00000			−0.318142
$248$		0			0
$249$		0			0
$250$	−4.50000	+	7.79423i	−0.284605	+	0.492950i
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$		0			0
$253$	15.0000			0.943042
$254$	−6.00000	+	10.3923i	−0.376473	+	0.652071i
$255$		0			0
$256$	8.50000	+	14.7224i	0.531250	+	0.920152i
$257$	29.0000			1.80897			0.904485	−	0.426505i	$-0.140255\pi$
$257$	29.0000			1.80897			0.904485	+	0.426505i	$0.140255\pi$
$258$		0			0
$259$	6.00000	−	5.19615i	0.372822	−	0.322873i
$260$	5.00000			0.310087
$261$		0			0
$262$	0.500000	−	0.866025i	0.0308901	−	0.0535032i
$263$	−5.00000			−0.308313			−0.154157	−	0.988046i	$-0.549266\pi$
$263$	−5.00000			−0.308313			−0.154157	+	0.988046i	$0.549266\pi$
$264$		0			0
$265$	−4.50000	+	7.79423i	−0.276433	+	0.478796i
$266$	−0.500000	−	2.59808i	−0.0306570	−	0.159298i
$267$		0			0
$268$	2.00000	−	3.46410i	0.122169	−	0.211604i
$269$	1.50000	−	2.59808i	0.0914566	−	0.158408i	−0.816668	−	0.577108i	$-0.804181\pi$
$269$	1.50000	−	2.59808i	0.0914566	−	0.158408i	0.908124	+	0.418701i	$0.137514\pi$
$270$		0			0
$271$	−0.500000	−	0.866025i	−0.0303728	−	0.0526073i	0.850439	−	0.526073i	$-0.176336\pi$
$271$	−0.500000	−	0.866025i	−0.0303728	−	0.0526073i	−0.880812	+	0.473466i	$0.843003\pi$
$272$	−1.50000	−	2.59808i	−0.0909509	−	0.157532i
$273$		0			0
$274$	4.50000	−	7.79423i	0.271855	−	0.470867i
$275$	20.0000			1.20605
$276$		0			0
$277$	19.0000			1.14160			0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000			1.14160			0.570800	+	0.821089i	$0.306633\pi$
$278$	4.50000	+	7.79423i	0.269892	+	0.467467i
$279$		0			0
$280$	1.50000	+	7.79423i	0.0896421	+	0.465794i
$281$	−14.5000	−	25.1147i	−0.864997	−	1.49822i	−0.867050	−	0.498222i	$-0.833987\pi$
$281$	−14.5000	−	25.1147i	−0.864997	−	1.49822i	0.00205220	−	0.999998i	$-0.499347\pi$
$282$		0			0
$283$	−14.0000	−	24.2487i	−0.832214	−	1.44144i	−0.896279	−	0.443491i	$-0.853740\pi$
$283$	−14.0000	−	24.2487i	−0.832214	−	1.44144i	0.0640654	−	0.997946i	$-0.479593\pi$
$284$	6.00000	+	10.3923i	0.356034	+	0.616670i
$285$		0			0
$286$	12.5000	+	21.6506i	0.739140	+	1.28023i
$287$	−12.5000	−	4.33013i	−0.737852	−	0.255599i
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$	−1.00000			−0.0587220
$291$		0			0
$292$	−3.00000			−0.175562
$293$	−2.50000	+	4.33013i	−0.146052	+	0.252969i	−0.929765	−	0.368154i	$-0.879990\pi$
$293$	−2.50000	+	4.33013i	−0.146052	+	0.252969i	0.783713	+	0.621123i	$0.213323\pi$
$294$		0			0
$295$		0			0
$296$	−4.50000	−	7.79423i	−0.261557	−	0.453030i
$297$		0			0
$298$	−1.50000	+	2.59808i	−0.0868927	+	0.150503i
$299$	−7.50000	+	12.9904i	−0.433736	+	0.751253i
$300$		0			0
$301$	−2.00000	+	1.73205i	−0.115278	+	0.0998337i
$302$	2.50000	−	4.33013i	0.143859	−	0.249171i
$303$		0			0
$304$	−1.00000			−0.0573539
$305$	7.00000	−	12.1244i	0.400819	−	0.694239i
$306$		0			0
$307$	28.0000			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$308$	10.0000	−	8.66025i	0.569803	−	0.493464i
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−7.00000	+	12.1244i	−0.395663	+	0.685309i	−0.993186	−	0.116543i	$-0.962819\pi$
$313$	−7.00000	+	12.1244i	−0.395663	+	0.685309i	0.597522	+	0.801852i	$0.296152\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$	−8.00000			−0.450035
$317$	−3.00000	+	5.19615i	−0.168497	+	0.291845i	−0.937892	−	0.346929i	$-0.887225\pi$
$317$	−3.00000	+	5.19615i	−0.168497	+	0.291845i	0.769395	+	0.638774i	$0.220558\pi$
$318$		0			0
$319$	2.50000	+	4.33013i	0.139973	+	0.242441i
$320$	7.00000			0.391312
$321$		0			0
$322$	−7.50000	−	2.59808i	−0.417959	−	0.144785i
$323$	−3.00000			−0.166924
$324$		0			0
$325$	−10.0000	+	17.3205i	−0.554700	+	0.960769i
$326$	11.0000			0.609234
$327$		0			0
$328$	−7.50000	+	12.9904i	−0.414118	+	0.717274i
$329$		0			0
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$	4.50000	−	7.79423i	0.246970	−	0.427764i
$333$		0			0
$334$	9.50000	+	16.4545i	0.519817	+	0.900349i
$335$	−2.00000	−	3.46410i	−0.109272	−	0.189264i
$336$		0			0
$337$	14.5000	−	25.1147i	0.789865	−	1.36809i	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	14.5000	−	25.1147i	0.789865	−	1.36809i	0.926049	−	0.377403i	$-0.123183\pi$
$338$	−12.0000			−0.652714
$339$		0			0
$340$	3.00000			0.162698
$341$		0			0
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$	1.50000	+	2.59808i	0.0808746	+	0.140079i
$345$		0			0
$346$	7.00000	+	12.1244i	0.376322	+	0.651809i
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	0.914702	−	0.404128i	$-0.132425\pi$
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	−0.807337	+	0.590091i	$0.799092\pi$
$348$		0			0
$349$	−9.50000	−	16.4545i	−0.508523	−	0.880788i	−0.999951	−	0.00987003i	$-0.996858\pi$
$349$	−9.50000	−	16.4545i	−0.508523	−	0.880788i	0.491428	−	0.870918i	$-0.336475\pi$
$350$	−10.0000	−	3.46410i	−0.534522	−	0.185164i
$351$		0			0
$352$	−12.5000	−	21.6506i	−0.666252	−	1.15398i
$353$	−11.0000			−0.585471			−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000			−0.585471			−0.292735	+	0.956193i	$0.594566\pi$
$354$		0			0
$355$	12.0000			0.636894
$356$	6.50000	−	11.2583i	0.344499	−	0.596690i
$357$		0			0
$358$	−9.50000	−	16.4545i	−0.502091	−	0.869646i
$359$	−5.50000	−	9.52628i	−0.290279	−	0.502778i	0.683597	−	0.729860i	$-0.260415\pi$
$359$	−5.50000	−	9.52628i	−0.290279	−	0.502778i	−0.973876	+	0.227082i	$0.927081\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$	−7.00000	+	12.1244i	−0.367912	+	0.637242i
$363$		0			0
$364$	2.50000	+	12.9904i	0.131036	+	0.680881i
$365$	−1.50000	+	2.59808i	−0.0785136	+	0.135990i
$366$		0			0
$367$	−3.00000			−0.156599			−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000			−0.156599			−0.0782994	+	0.996930i	$0.524949\pi$
$368$	−1.50000	+	2.59808i	−0.0781929	+	0.135434i
$369$		0			0
$370$	−3.00000			−0.155963
$371$	−22.5000	−	7.79423i	−1.16814	−	0.404656i
$372$		0			0
$373$	−25.0000			−1.29445			−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000			−1.29445			−0.647225	+	0.762299i	$0.724071\pi$
$374$	7.50000	+	12.9904i	0.387816	+	0.671717i
$375$		0			0
$376$		0			0
$377$	−5.00000			−0.257513
$378$		0			0
$379$	−12.0000			−0.616399			−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000			−0.616399			−0.308199	+	0.951322i	$0.599726\pi$
$380$	0.500000	−	0.866025i	0.0256495	−	0.0444262i
$381$		0			0
$382$	−4.00000	−	6.92820i	−0.204658	−	0.354478i
$383$	−27.0000			−1.37964			−0.689818	−	0.723983i	$-0.742309\pi$
$383$	−27.0000			−1.37964			−0.689818	+	0.723983i	$0.742309\pi$
$384$		0			0
$385$	−2.50000	−	12.9904i	−0.127412	−	0.662051i
$386$	10.0000			0.508987
$387$		0			0
$388$	−4.50000	+	7.79423i	−0.228453	+	0.395692i
$389$	9.00000			0.456318			0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000			0.456318			0.228159	+	0.973624i	$0.426729\pi$
$390$		0			0
$391$	−4.50000	+	7.79423i	−0.227575	+	0.394171i
$392$	−19.5000	+	7.79423i	−0.984899	+	0.393668i
$393$		0			0
$394$	−1.00000	+	1.73205i	−0.0503793	+	0.0872595i
$395$	−4.00000	+	6.92820i	−0.201262	+	0.348596i
$396$		0			0
$397$	−7.50000	−	12.9904i	−0.376414	−	0.651969i	0.614123	−	0.789210i	$-0.289510\pi$
$397$	−7.50000	−	12.9904i	−0.376414	−	0.651969i	−0.990538	+	0.137241i	$0.956176\pi$
$398$	1.50000	+	2.59808i	0.0751882	+	0.130230i
$399$		0			0
$400$	−2.00000	+	3.46410i	−0.100000	+	0.173205i
$401$	−3.00000			−0.149813			−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000			−0.149813			−0.0749064	+	0.997191i	$0.523866\pi$
$402$		0			0
$403$		0			0
$404$	8.50000	+	14.7224i	0.422891	+	0.732468i
$405$		0			0
$406$	−0.500000	−	2.59808i	−0.0248146	−	0.128940i
$407$	7.50000	+	12.9904i	0.371761	+	0.643909i
$408$		0			0
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	0.639430	−	0.768849i	$-0.279170\pi$
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	−0.985558	+	0.169338i	$0.945837\pi$
$410$	2.50000	+	4.33013i	0.123466	+	0.213850i
$411$		0			0
$412$	−0.500000	−	0.866025i	−0.0246332	−	0.0426660i
$413$		0			0
$414$		0			0
$415$	−4.50000	−	7.79423i	−0.220896	−	0.382604i
$416$	25.0000			1.22573
$417$		0			0
$418$	5.00000			0.244558
$419$	4.50000	−	7.79423i	0.219839	−	0.380773i	−0.734919	−	0.678155i	$-0.762780\pi$
$419$	4.50000	−	7.79423i	0.219839	−	0.380773i	0.954759	+	0.297382i	$0.0961133\pi$
$420$		0			0
$421$	0.500000	+	0.866025i	0.0243685	+	0.0422075i	0.877952	−	0.478748i	$-0.158909\pi$
$421$	0.500000	+	0.866025i	0.0243685	+	0.0422075i	−0.853584	+	0.520955i	$0.825576\pi$
$422$	6.50000	+	11.2583i	0.316415	+	0.548047i
$423$		0			0
$424$	−13.5000	+	23.3827i	−0.655618	+	1.13556i
$425$	−6.00000	+	10.3923i	−0.291043	+	0.504101i
$426$		0			0
$427$	35.0000	+	12.1244i	1.69377	+	0.586739i
$428$	−8.50000	+	14.7224i	−0.410863	+	0.711636i
$429$		0			0
$430$	1.00000			0.0482243
$431$	−4.50000	+	7.79423i	−0.216757	+	0.375435i	−0.953815	−	0.300395i	$-0.902881\pi$
$431$	−4.50000	+	7.79423i	−0.216757	+	0.375435i	0.737057	+	0.675830i	$0.236215\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$	9.00000			0.431022
$437$	1.50000	+	2.59808i	0.0717547	+	0.124283i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$	−15.0000			−0.715097
$441$		0			0
$442$	−15.0000			−0.713477
$443$	18.0000	−	31.1769i	0.855206	−	1.48126i	−0.0212481	−	0.999774i	$-0.506764\pi$
$443$	18.0000	−	31.1769i	0.855206	−	1.48126i	0.876454	−	0.481486i	$-0.159903\pi$
$444$		0			0
$445$	−6.50000	−	11.2583i	−0.308130	−	0.533696i
$446$	−19.0000			−0.899676
$447$		0			0
$448$	3.50000	+	18.1865i	0.165359	+	0.859233i
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	12.5000	−	21.6506i	0.588602	−	1.01949i
$452$	−1.00000			−0.0470360
$453$		0			0
$454$	1.50000	−	2.59808i	0.0703985	−	0.121934i
$455$	12.5000	+	4.33013i	0.586009	+	0.202999i
$456$		0			0
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	0.485299	+	0.874348i	$0.338711\pi$
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	−0.999857	+	0.0168929i	$0.994623\pi$
$458$	−0.500000	+	0.866025i	−0.0233635	+	0.0404667i
$459$		0			0
$460$	−1.50000	−	2.59808i	−0.0699379	−	0.121136i
$461$	9.50000	+	16.4545i	0.442459	+	0.766362i	0.997871	−	0.0652135i	$-0.0207728\pi$
$461$	9.50000	+	16.4545i	0.442459	+	0.766362i	−0.555412	+	0.831575i	$0.687440\pi$
$462$		0			0
$463$	−6.50000	+	11.2583i	−0.302081	+	0.523219i	−0.976607	−	0.215032i	$-0.931015\pi$
$463$	−6.50000	+	11.2583i	−0.302081	+	0.523219i	0.674526	+	0.738251i	$0.264348\pi$
$464$	−1.00000			−0.0464238
$465$		0			0
$466$	3.00000			0.138972
$467$	−13.5000	−	23.3827i	−0.624705	−	1.08202i	−0.988598	−	0.150581i	$-0.951886\pi$
$467$	−13.5000	−	23.3827i	−0.624705	−	1.08202i	0.363892	−	0.931441i	$-0.381448\pi$
$468$		0			0
$469$	8.00000	−	6.92820i	0.369406	−	0.319915i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.50000	−	4.33013i	−0.114950	−	0.199099i
$474$		0			0
$475$	2.00000	+	3.46410i	0.0917663	+	0.158944i
$476$	1.50000	+	7.79423i	0.0687524	+	0.357248i
$477$		0			0
$478$	7.50000	+	12.9904i	0.343042	+	0.594166i
$479$	−25.0000			−1.14228			−0.571140	−	0.820853i	$-0.693499\pi$
$479$	−25.0000			−1.14228			−0.571140	+	0.820853i	$0.693499\pi$
$480$		0			0
$481$	−15.0000			−0.683941
$482$	5.50000	−	9.52628i	0.250518	−	0.433910i
$483$		0			0
$484$	7.00000	+	12.1244i	0.318182	+	0.551107i
$485$	4.50000	+	7.79423i	0.204334	+	0.353918i
$486$		0			0
$487$	−9.50000	+	16.4545i	−0.430486	+	0.745624i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−9.50000	+	16.4545i	−0.430486	+	0.745624i	0.566429	+	0.824110i	$0.308325\pi$
$488$	21.0000	−	36.3731i	0.950625	−	1.64653i
$489$		0			0
$490$	−1.00000	+	6.92820i	−0.0451754	+	0.312984i
$491$	6.50000	−	11.2583i	0.293341	−	0.508081i	−0.681257	−	0.732045i	$-0.738566\pi$
$491$	6.50000	−	11.2583i	0.293341	−	0.508081i	0.974598	+	0.223963i	$0.0718996\pi$
$492$		0			0
$493$	−3.00000			−0.135113
$494$	−2.50000	+	4.33013i	−0.112480	+	0.194822i
$495$		0			0
$496$		0			0
$497$	6.00000	+	31.1769i	0.269137	+	1.39848i
$498$		0			0
$499$	31.0000			1.38775			0.693875	−	0.720095i	$-0.255902\pi$
$499$	31.0000			1.38775			0.693875	+	0.720095i	$0.255902\pi$
$500$	−4.50000	−	7.79423i	−0.201246	−	0.348569i
$501$		0			0
$502$	14.0000	−	24.2487i	0.624851	−	1.08227i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	17.0000			0.756490
$506$	7.50000	−	12.9904i	0.333416	−	0.577493i
$507$		0			0
$508$	−6.00000	−	10.3923i	−0.266207	−	0.461084i
$509$	29.0000			1.28540			0.642701	−	0.766117i	$-0.277814\pi$
$509$	29.0000			1.28540			0.642701	+	0.766117i	$0.277814\pi$
$510$		0			0
$511$	−7.50000	−	2.59808i	−0.331780	−	0.114932i
$512$	11.0000			0.486136
$513$		0			0
$514$	14.5000	−	25.1147i	0.639568	−	1.10776i
$515$	−1.00000			−0.0440653
$516$		0			0
$517$		0			0
$518$	−1.50000	−	7.79423i	−0.0659062	−	0.342459i
$519$		0			0
$520$	7.50000	−	12.9904i	0.328897	−	0.569666i
$521$	1.50000	−	2.59808i	0.0657162	−	0.113824i	−0.831295	−	0.555831i	$-0.812400\pi$
$521$	1.50000	−	2.59808i	0.0657162	−	0.113824i	0.897011	+	0.442007i	$0.145733\pi$
$522$		0			0
$523$	−0.500000	−	0.866025i	−0.0218635	−	0.0378686i	0.854887	−	0.518815i	$-0.173627\pi$
$523$	−0.500000	−	0.866025i	−0.0218635	−	0.0378686i	−0.876750	+	0.480946i	$0.840293\pi$
$524$	0.500000	+	0.866025i	0.0218426	+	0.0378325i
$525$		0			0
$526$	−2.50000	+	4.33013i	−0.109005	+	0.188803i
$527$		0			0
$528$		0			0
$529$	−14.0000			−0.608696
$530$	4.50000	+	7.79423i	0.195468	+	0.338560i
$531$		0			0
$532$	2.50000	+	0.866025i	0.108389	+	0.0375470i
$533$	12.5000	+	21.6506i	0.541435	+	0.937793i
$534$		0			0
$535$	8.50000	+	14.7224i	0.367487	+	0.636506i
$536$	−6.00000	−	10.3923i	−0.259161	−	0.448879i
$537$		0			0
$538$	−1.50000	−	2.59808i	−0.0646696	−	0.112011i
$539$	32.5000	−	12.9904i	1.39987	−	0.559535i
$540$		0			0
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	0.999042	+	0.0437584i	$0.0139332\pi$
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	−0.461625	+	0.887075i	$0.652733\pi$
$542$	−1.00000			−0.0429537
$543$		0			0
$544$	15.0000			0.643120
$545$	4.50000	−	7.79423i	0.192759	−	0.333868i
$546$		0			0
$547$	14.5000	+	25.1147i	0.619975	+	1.07383i	0.989490	+	0.144604i	$0.0461907\pi$
$547$	14.5000	+	25.1147i	0.619975	+	1.07383i	−0.369514	+	0.929225i	$0.620476\pi$
$548$	4.50000	+	7.79423i	0.192230	+	0.332953i
$549$		0			0
$550$	10.0000	−	17.3205i	0.426401	−	0.738549i
$551$	−0.500000	+	0.866025i	−0.0213007	+	0.0368939i
$552$		0			0
$553$	−20.0000	−	6.92820i	−0.850487	−	0.294617i
$554$	9.50000	−	16.4545i	0.403616	−	0.699084i
$555$		0			0
$556$	−9.00000			−0.381685
$557$	−18.5000	+	32.0429i	−0.783870	+	1.35770i	0.145802	+	0.989314i	$0.453424\pi$
$557$	−18.5000	+	32.0429i	−0.783870	+	1.35770i	−0.929672	+	0.368389i	$0.879909\pi$
$558$		0			0
$559$	5.00000			0.211477
$560$	2.50000	+	0.866025i	0.105644	+	0.0365963i
$561$		0			0
$562$	−29.0000			−1.22329
$563$	−14.0000	−	24.2487i	−0.590030	−	1.02196i	−0.994228	−	0.107290i	$-0.965783\pi$
$563$	−14.0000	−	24.2487i	−0.590030	−	1.02196i	0.404198	−	0.914671i	$-0.367551\pi$
$564$		0			0
$565$	−0.500000	+	0.866025i	−0.0210352	+	0.0364340i
$566$	−28.0000			−1.17693
$567$		0			0
$568$	36.0000			1.51053
$569$	−17.0000	+	29.4449i	−0.712677	+	1.23439i	0.251172	+	0.967943i	$0.419184\pi$
$569$	−17.0000	+	29.4449i	−0.712677	+	1.23439i	−0.963849	+	0.266450i	$0.914149\pi$
$570$		0			0
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	−0.978022	−	0.208502i	$-0.933141\pi$
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	0.308443	−	0.951243i	$-0.400192\pi$
$572$	−25.0000			−1.04530
$573$		0			0
$574$	−10.0000	+	8.66025i	−0.417392	+	0.361472i
$575$	12.0000			0.500435
$576$		0			0
$577$	−15.5000	+	26.8468i	−0.645273	+	1.11765i	0.338965	+	0.940799i	$0.389923\pi$
$577$	−15.5000	+	26.8468i	−0.645273	+	1.11765i	−0.984238	+	0.176847i	$0.943410\pi$
$578$	8.00000			0.332756
$579$		0			0
$580$	0.500000	−	0.866025i	0.0207614	−	0.0359597i
$581$	18.0000	−	15.5885i	0.746766	−	0.646718i
$582$		0			0
$583$	22.5000	−	38.9711i	0.931855	−	1.61402i
$584$	−4.50000	+	7.79423i	−0.186211	+	0.322527i
$585$		0			0
$586$	2.50000	+	4.33013i	0.103274	+	0.178876i
$587$	−18.5000	−	32.0429i	−0.763577	−	1.32255i	−0.940996	−	0.338418i	$-0.890108\pi$
$587$	−18.5000	−	32.0429i	−0.763577	−	1.32255i	0.177419	−	0.984135i	$-0.443225\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−3.00000			−0.123299
$593$	7.50000	+	12.9904i	0.307988	+	0.533451i	0.977922	−	0.208970i	$-0.0670110\pi$
$593$	7.50000	+	12.9904i	0.307988	+	0.533451i	−0.669934	+	0.742421i	$0.733678\pi$
$594$		0			0
$595$	7.50000	+	2.59808i	0.307470	+	0.106511i
$596$	−1.50000	−	2.59808i	−0.0614424	−	0.106421i
$597$		0			0
$598$	7.50000	+	12.9904i	0.306698	+	0.531216i
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	0.509631	−	0.860393i	$-0.329782\pi$
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	−0.999938	+	0.0111569i	$0.996449\pi$
$600$		0			0
$601$	4.50000	+	7.79423i	0.183559	+	0.317933i	0.943090	−	0.332538i	$-0.107905\pi$
$601$	4.50000	+	7.79423i	0.183559	+	0.317933i	−0.759531	+	0.650471i	$0.774572\pi$
$602$	0.500000	+	2.59808i	0.0203785	+	0.105890i
$603$		0			0
$604$	2.50000	+	4.33013i	0.101724	+	0.176190i
$605$	14.0000			0.569181
$606$		0			0
$607$	−1.00000			−0.0405887			−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000			−0.0405887			−0.0202944	+	0.999794i	$0.506460\pi$
$608$	2.50000	−	4.33013i	0.101388	−	0.175610i
$609$		0			0
$610$	−7.00000	−	12.1244i	−0.283422	−	0.490901i
$611$		0			0
$612$		0			0
$613$	−9.50000	+	16.4545i	−0.383701	+	0.664590i	−0.991588	−	0.129433i	$-0.958684\pi$
$613$	−9.50000	+	16.4545i	−0.383701	+	0.664590i	0.607887	+	0.794024i	$0.292017\pi$
$614$	14.0000	−	24.2487i	0.564994	−	0.978598i
$615$		0			0
$616$	−7.50000	−	38.9711i	−0.302184	−	1.57019i
$617$	13.5000	−	23.3827i	0.543490	−	0.941351i	−0.455211	−	0.890384i	$-0.650436\pi$
$617$	13.5000	−	23.3827i	0.543490	−	0.941351i	0.998700	−	0.0509678i	$-0.0162306\pi$
$618$		0			0
$619$	25.0000			1.00483			0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000			1.00483			0.502417	+	0.864625i	$0.332444\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	26.0000	−	22.5167i	1.04167	−	0.902111i
$624$		0			0
$625$	11.0000			0.440000
$626$	7.00000	+	12.1244i	0.279776	+	0.484587i
$627$		0			0
$628$	−7.00000	+	12.1244i	−0.279330	+	0.483814i
$629$	−9.00000			−0.358854
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$	−12.0000	+	20.7846i	−0.477334	+	0.826767i
$633$		0			0
$634$	3.00000	+	5.19615i	0.119145	+	0.206366i
$635$	−12.0000			−0.476205
$636$		0			0
$637$	−5.00000	+	34.6410i	−0.198107	+	1.37253i
$638$	5.00000			0.197952
$639$		0			0
$640$	−1.50000	+	2.59808i	−0.0592927	+	0.102698i
$641$	9.00000			0.355479			0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000			0.355479			0.177739	+	0.984078i	$0.443122\pi$
$642$		0			0
$643$	9.50000	−	16.4545i	0.374643	−	0.648901i	−0.615630	−	0.788035i	$-0.711098\pi$
$643$	9.50000	−	16.4545i	0.374643	−	0.648901i	0.990274	+	0.139134i	$0.0444318\pi$
$644$	6.00000	−	5.19615i	0.236433	−	0.204757i
$645$		0			0
$646$	−1.50000	+	2.59808i	−0.0590167	+	0.102220i
$647$	15.5000	−	26.8468i	0.609368	−	1.05546i	−0.381977	−	0.924172i	$-0.624757\pi$
$647$	15.5000	−	26.8468i	0.609368	−	1.05546i	0.991345	−	0.131284i	$-0.0419101\pi$
$648$		0			0
$649$		0			0
$650$	10.0000	+	17.3205i	0.392232	+	0.679366i
$651$		0			0
$652$	−5.50000	+	9.52628i	−0.215397	+	0.373078i
$653$	−3.00000			−0.117399			−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000			−0.117399			−0.0586995	+	0.998276i	$0.518695\pi$
$654$		0			0
$655$	1.00000			0.0390732
$656$	2.50000	+	4.33013i	0.0976086	+	0.169063i
$657$		0			0
$658$		0			0
$659$	13.5000	+	23.3827i	0.525885	+	0.910860i	0.999545	+	0.0301523i	$0.00959924\pi$
$659$	13.5000	+	23.3827i	0.525885	+	0.910860i	−0.473660	+	0.880708i	$0.657067\pi$
$660$		0			0
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	0.969442	−	0.245319i	$-0.0788928\pi$
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	−0.697174	+	0.716902i	$0.745559\pi$
$662$	4.00000	+	6.92820i	0.155464	+	0.269272i
$663$		0			0
$664$	−13.5000	−	23.3827i	−0.523902	−	0.907424i
$665$	2.00000	−	1.73205i	0.0775567	−	0.0671660i
$666$		0			0
$667$	1.50000	+	2.59808i	0.0580802	+	0.100598i
$668$	−19.0000			−0.735132
$669$		0			0
$670$	−4.00000			−0.154533
$671$	−35.0000	+	60.6218i	−1.35116	+	2.34028i
$672$		0			0
$673$	14.5000	+	25.1147i	0.558934	+	0.968102i	0.997586	+	0.0694449i	$0.0221228\pi$
$673$	14.5000	+	25.1147i	0.558934	+	0.968102i	−0.438652	+	0.898657i	$0.644544\pi$
$674$	−14.5000	−	25.1147i	−0.558519	−	0.967384i
$675$		0			0
$676$	6.00000	−	10.3923i	0.230769	−	0.399704i
$677$	21.0000	−	36.3731i	0.807096	−	1.39793i	−0.107772	−	0.994176i	$-0.534372\pi$
$677$	21.0000	−	36.3731i	0.807096	−	1.39793i	0.914867	−	0.403755i	$-0.132295\pi$
$678$		0			0
$679$	−18.0000	+	15.5885i	−0.690777	+	0.598230i
$680$	4.50000	−	7.79423i	0.172567	−	0.298895i
$681$		0			0
$682$		0			0
$683$	−4.50000	+	7.79423i	−0.172188	+	0.298238i	−0.939184	−	0.343413i	$-0.888417\pi$
$683$	−4.50000	+	7.79423i	−0.172188	+	0.298238i	0.766997	+	0.641651i	$0.221750\pi$
$684$		0			0
$685$	9.00000			0.343872
$686$	−18.5000	+	0.866025i	−0.706333	+	0.0330650i
$687$		0			0
$688$	1.00000			0.0381246
$689$	22.5000	+	38.9711i	0.857182	+	1.48468i
$690$		0			0
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	−0.466691	−	0.884420i	$-0.654554\pi$
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	0.999276	−	0.0380440i	$-0.0121127\pi$
$692$	−14.0000			−0.532200
$693$		0			0
$694$	4.00000			0.151838
$695$	−4.50000	+	7.79423i	−0.170695	+	0.295652i
$696$		0			0
$697$	7.50000	+	12.9904i	0.284083	+	0.492046i
$698$	−19.0000			−0.719161
$699$		0			0
$700$	8.00000	−	6.92820i	0.302372	−	0.261861i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	−1.50000	+	2.59808i	−0.0565736	+	0.0979883i
$704$	−35.0000			−1.31911
$705$		0			0
$706$	−5.50000	+	9.52628i	−0.206995	+	0.358526i
$707$	8.50000	+	44.1673i	0.319675	+	1.66108i
$708$		0			0
$709$	3.00000	−	5.19615i	0.112667	−	0.195146i	−0.804178	−	0.594389i	$-0.797394\pi$
$709$	3.00000	−	5.19615i	0.112667	−	0.195146i	0.916845	+	0.399244i	$0.130727\pi$
$710$	6.00000	−	10.3923i	0.225176	−	0.390016i
$711$		0			0
$712$	−19.5000	−	33.7750i	−0.730793	−	1.26577i
$713$		0			0
$714$		0			0
$715$	−12.5000	+	21.6506i	−0.467473	+	0.809688i
$716$	19.0000			0.710063
$717$		0			0
$718$	−11.0000			−0.410516
$719$	−13.5000	−	23.3827i	−0.503465	−	0.872027i	−0.999992	−	0.00400572i	$-0.998725\pi$
$719$	−13.5000	−	23.3827i	−0.503465	−	0.872027i	0.496527	−	0.868021i	$-0.334608\pi$
$720$		0			0
$721$	−0.500000	−	2.59808i	−0.0186210	−	0.0967574i
$722$	−9.00000	−	15.5885i	−0.334945	−	0.580142i
$723$		0			0
$724$	−7.00000	−	12.1244i	−0.260153	−	0.450598i
$725$	2.00000	+	3.46410i	0.0742781	+	0.128654i
$726$		0			0
$727$	−23.5000	−	40.7032i	−0.871567	−	1.50960i	−0.860376	−	0.509661i	$-0.829771\pi$
$727$	−23.5000	−	40.7032i	−0.871567	−	1.50960i	−0.0111912	−	0.999937i	$-0.503562\pi$
$728$	37.5000	+	12.9904i	1.38984	+	0.481456i
$729$		0			0
$730$	1.50000	+	2.59808i	0.0555175	+	0.0961591i
$731$	3.00000			0.110959
$732$		0			0
$733$	27.0000			0.997268			0.498634	−	0.866813i	$-0.333835\pi$
$733$	27.0000			0.997268			0.498634	+	0.866813i	$0.333835\pi$
$734$	−1.50000	+	2.59808i	−0.0553660	+	0.0958967i
$735$		0			0
$736$	−7.50000	−	12.9904i	−0.276454	−	0.478832i
$737$	10.0000	+	17.3205i	0.368355	+	0.638009i
$738$		0			0
$739$	4.50000	−	7.79423i	0.165535	−	0.286715i	−0.771310	−	0.636460i	$-0.780398\pi$
$739$	4.50000	−	7.79423i	0.165535	−	0.286715i	0.936845	+	0.349744i	$0.113732\pi$
$740$	1.50000	−	2.59808i	0.0551411	−	0.0955072i
$741$		0			0
$742$	−18.0000	+	15.5885i	−0.660801	+	0.572270i
$743$	−7.50000	+	12.9904i	−0.275148	+	0.476571i	−0.970173	−	0.242415i	$-0.922060\pi$
$743$	−7.50000	+	12.9904i	−0.275148	+	0.476571i	0.695024	+	0.718986i	$0.255394\pi$
$744$		0			0
$745$	−3.00000			−0.109911
$746$	−12.5000	+	21.6506i	−0.457658	+	0.792686i
$747$		0			0
$748$	−15.0000			−0.548454
$749$	−34.0000	+	29.4449i	−1.24233	+	1.07589i
$750$		0			0
$751$	31.0000			1.13121			0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000			1.13121			0.565603	+	0.824678i	$0.308643\pi$
$752$		0			0
$753$		0			0
$754$	−2.50000	+	4.33013i	−0.0910446	+	0.157694i
$755$	5.00000			0.181969
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	−6.00000	+	10.3923i	−0.217930	+	0.377466i
$759$		0			0
$760$	−1.50000	−	2.59808i	−0.0544107	−	0.0942421i
$761$	−27.0000			−0.978749			−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000			−0.978749			−0.489375	+	0.872074i	$0.662775\pi$
$762$		0			0
$763$	22.5000	+	7.79423i	0.814555	+	0.282170i
$764$	8.00000			0.289430
$765$		0			0
$766$	−13.5000	+	23.3827i	−0.487775	+	0.844851i
$767$		0			0
$768$		0			0
$769$	−11.5000	+	19.9186i	−0.414701	+	0.718283i	−0.995397	−	0.0958377i	$-0.969447\pi$
$769$	−11.5000	+	19.9186i	−0.414701	+	0.718283i	0.580696	+	0.814120i	$0.302780\pi$
$770$	−12.5000	−	4.33013i	−0.450469	−	0.156047i
$771$		0			0
$772$	−5.00000	+	8.66025i	−0.179954	+	0.311689i
$773$	15.5000	−	26.8468i	0.557496	−	0.965612i	−0.440208	−	0.897896i	$-0.645095\pi$
$773$	15.5000	−	26.8468i	0.557496	−	0.965612i	0.997705	−	0.0677162i	$-0.0215712\pi$
$774$		0			0
$775$		0			0
$776$	13.5000	+	23.3827i	0.484622	+	0.839390i
$777$		0			0
$778$	4.50000	−	7.79423i	0.161333	−	0.279437i
$779$	5.00000			0.179144
$780$		0			0
$781$	−60.0000			−2.14697
$782$	4.50000	+	7.79423i	0.160920	+	0.278721i
$783$		0			0
$784$	−1.00000	+	6.92820i	−0.0357143	+	0.247436i
$785$	7.00000	+	12.1244i	0.249841	+	0.432737i
$786$		0			0
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	0.500953	−	0.865474i	$-0.332983\pi$
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	−0.999999	+	0.00110111i	$0.999650\pi$
$788$	−1.00000	−	1.73205i	−0.0356235	−	0.0617018i
$789$		0			0
$790$	4.00000	+	6.92820i	0.142314	+	0.246494i
$791$	−2.50000	−	0.866025i	−0.0888898	−	0.0307923i
$792$		0			0
$793$	−35.0000	−	60.6218i	−1.24289	−	2.15274i
$794$	−15.0000			−0.532330
$795$		0			0
$796$	−3.00000			−0.106332
$797$	11.5000	−	19.9186i	0.407351	−	0.705552i	−0.587241	−	0.809412i	$-0.699786\pi$
$797$	11.5000	−	19.9186i	0.407351	−	0.705552i	0.994592	+	0.103860i	$0.0331193\pi$
$798$		0			0
$799$		0			0
$800$	−10.0000	−	17.3205i	−0.353553	−	0.612372i
$801$		0			0
$802$	−1.50000	+	2.59808i	−0.0529668	+	0.0917413i
$803$	7.50000	−	12.9904i	0.264669	−	0.458421i
$804$		0			0
$805$	−1.50000	−	7.79423i	−0.0528681	−	0.274710i
$806$		0			0
$807$		0			0
$808$	51.0000			1.79417
$809$	−4.50000	+	7.79423i	−0.158212	+	0.274030i	−0.934224	−	0.356687i	$-0.883906\pi$
$809$	−4.50000	+	7.79423i	−0.158212	+	0.274030i	0.776012	+	0.630718i	$0.217239\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$	2.50000	+	0.866025i	0.0877328	+	0.0303915i
$813$		0			0
$814$	15.0000			0.525750
$815$	5.50000	+	9.52628i	0.192657	+	0.333691i
$816$		0			0
$817$	0.500000	−	0.866025i	0.0174928	−	0.0302984i
$818$	−14.0000			−0.489499
$819$		0			0
$820$	−5.00000			−0.174608
$821$	11.0000	−	19.0526i	0.383903	−	0.664939i	−0.607714	−	0.794156i	$-0.707913\pi$
$821$	11.0000	−	19.0526i	0.383903	−	0.664939i	0.991616	+	0.129217i	$0.0412465\pi$
$822$		0			0
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	0.995768	−	0.0919029i	$-0.0292950\pi$
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	−0.577474	+	0.816409i	$0.695962\pi$
$824$	−3.00000			−0.104510
$825$		0			0
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$		0			0
$829$	12.5000	−	21.6506i	0.434143	−	0.751958i	−0.563082	−	0.826401i	$-0.690385\pi$
$829$	12.5000	−	21.6506i	0.434143	−	0.751958i	0.997225	+	0.0744432i	$0.0237179\pi$
$830$	−9.00000			−0.312395
$831$		0			0
$832$	17.5000	−	30.3109i	0.606703	−	1.05084i
$833$	−3.00000	+	20.7846i	−0.103944	+	0.720144i
$834$		0			0
$835$	−9.50000	+	16.4545i	−0.328761	+	0.569431i
$836$	−2.50000	+	4.33013i	−0.0864643	+	0.149761i
$837$		0			0
$838$	−4.50000	−	7.79423i	−0.155450	−	0.269247i
$839$	−18.5000	−	32.0429i	−0.638691	−	1.10625i	−0.985720	−	0.168391i	$-0.946143\pi$
$839$	−18.5000	−	32.0429i	−0.638691	−	1.10625i	0.347029	−	0.937854i	$-0.387190\pi$
$840$		0			0
$841$	14.0000	−	24.2487i	0.482759	−	0.836162i
$842$	1.00000			0.0344623
$843$		0			0
$844$	−13.0000			−0.447478
$845$	−6.00000	−	10.3923i	−0.206406	−	0.357506i
$846$		0			0
$847$	7.00000	+	36.3731i	0.240523	+	1.24979i
$848$	4.50000	+	7.79423i	0.154531	+	0.267655i
$849$		0			0
$850$	6.00000	+	10.3923i	0.205798	+	0.356453i
$851$	4.50000	+	7.79423i	0.154258	+	0.267183i
$852$		0			0
$853$	18.5000	+	32.0429i	0.633428	+	1.09713i	0.986846	+	0.161664i	$0.0516860\pi$
$853$	18.5000	+	32.0429i	0.633428	+	1.09713i	−0.353418	+	0.935466i	$0.614981\pi$
$854$	28.0000	−	24.2487i	0.958140	−	0.829774i
$855$		0			0
$856$	25.5000	+	44.1673i	0.871572	+	1.50961i
$857$	−11.0000			−0.375753			−0.187876	−	0.982193i	$-0.560160\pi$
$857$	−11.0000			−0.375753			−0.187876	+	0.982193i	$0.560160\pi$
$858$		0			0
$859$	−1.00000			−0.0341196			−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000			−0.0341196			−0.0170598	+	0.999854i	$0.505431\pi$
$860$	−0.500000	+	0.866025i	−0.0170499	+	0.0295312i
$861$		0			0
$862$	4.50000	+	7.79423i	0.153271	+	0.265472i
$863$	−19.5000	−	33.7750i	−0.663788	−	1.14971i	−0.979612	−	0.200897i	$-0.935615\pi$
$863$	−19.5000	−	33.7750i	−0.663788	−	1.14971i	0.315825	−	0.948818i	$-0.397719\pi$
$864$		0			0
$865$	−7.00000	+	12.1244i	−0.238007	+	0.412240i
$866$	−7.00000	+	12.1244i	−0.237870	+	0.412002i
$867$		0			0
$868$		0			0
$869$	20.0000	−	34.6410i	0.678454	−	1.17512i
$870$		0			0
$871$	−20.0000			−0.677674
$872$	13.5000	−	23.3827i	0.457168	−	0.791838i
$873$		0			0
$874$	3.00000			0.101477
$875$	−4.50000	−	23.3827i	−0.152128	−	0.790479i
$876$		0			0
$877$	−53.0000			−1.78968			−0.894841	−	0.446384i	$-0.852711\pi$
$877$	−53.0000			−1.78968			−0.894841	+	0.446384i	$0.852711\pi$
$878$		0			0
$879$		0			0
$880$	−2.50000	+	4.33013i	−0.0842750	+	0.145969i
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$	7.50000	−	12.9904i	0.252252	−	0.436914i
$885$		0			0
$886$	−18.0000	−	31.1769i	−0.604722	−	1.04741i
$887$	29.0000			0.973725			0.486862	−	0.873479i	$-0.338141\pi$
$887$	29.0000			0.973725			0.486862	+	0.873479i	$0.338141\pi$
$888$		0			0
$889$	−6.00000	−	31.1769i	−0.201234	−	1.04564i
$890$	−13.0000			−0.435761
$891$		0			0
$892$	9.50000	−	16.4545i	0.318084	−	0.550937i
$893$		0			0
$894$		0			0
$895$	9.50000	−	16.4545i	0.317550	−	0.550013i
$896$	−7.50000	−	2.59808i	−0.250557	−	0.0867956i
$897$		0			0
$898$	−15.0000	+	25.9808i	−0.500556	+	0.866989i
$899$		0			0
$900$		0			0
$901$	13.5000	+	23.3827i	0.449750	+	0.778990i
$902$	−12.5000	−	21.6506i	−0.416204	−	0.720887i
$903$		0			0
$904$	−1.50000	+	2.59808i	−0.0498893	+	0.0864107i
$905$	−14.0000			−0.465376
$906$		0			0
$907$	5.00000			0.166022			0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000			0.166022			0.0830111	+	0.996549i	$0.473546\pi$
$908$	1.50000	+	2.59808i	0.0497792	+	0.0862202i
$909$		0			0
$910$	10.0000	−	8.66025i	0.331497	−	0.287085i
$911$	13.5000	+	23.3827i	0.447275	+	0.774703i	0.998208	−	0.0598468i	$-0.0190612\pi$
$911$	13.5000	+	23.3827i	0.447275	+	0.774703i	−0.550933	+	0.834550i	$0.685728\pi$
$912$		0			0
$913$	22.5000	+	38.9711i	0.744641	+	1.28976i
$914$	11.0000	+	19.0526i	0.363848	+	0.630203i
$915$		0			0
$916$	−0.500000	−	0.866025i	−0.0165205	−	0.0286143i
$917$	0.500000	+	2.59808i	0.0165115	+	0.0857960i
$918$		0			0
$919$	−8.50000	−	14.7224i	−0.280389	−	0.485648i	0.691091	−	0.722767i	$-0.257130\pi$
$919$	−8.50000	−	14.7224i	−0.280389	−	0.485648i	−0.971481	+	0.237119i	$0.923797\pi$
$920$	−9.00000			−0.296721
$921$		0			0
$922$	19.0000			0.625732
$923$	30.0000	−	51.9615i	0.987462	−	1.71033i
$924$		0			0
$925$	6.00000	+	10.3923i	0.197279	+	0.341697i
$926$	6.50000	+	11.2583i	0.213603	+	0.369972i
$927$		0			0
$928$	2.50000	−	4.33013i	0.0820665	−	0.142143i
$929$	−7.00000	+	12.1244i	−0.229663	+	0.397787i	−0.957708	−	0.287742i	$-0.907096\pi$
$929$	−7.00000	+	12.1244i	−0.229663	+	0.397787i	0.728046	+	0.685529i	$0.240429\pi$
$930$		0			0
$931$	5.50000	+	4.33013i	0.180255	+	0.141914i
$932$	−1.50000	+	2.59808i	−0.0491341	+	0.0851028i
$933$		0			0
$934$	−27.0000			−0.883467
$935$	−7.50000	+	12.9904i	−0.245276	+	0.424831i
$936$		0			0
$937$	42.0000			1.37208			0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000			1.37208			0.686040	+	0.727564i	$0.259347\pi$
$938$	−2.00000	−	10.3923i	−0.0653023	−	0.339321i
$939$		0			0
$940$		0			0
$941$	−7.00000	−	12.1244i	−0.228193	−	0.395243i	0.729079	−	0.684429i	$-0.239949\pi$
$941$	−7.00000	−	12.1244i	−0.228193	−	0.395243i	−0.957273	+	0.289187i	$0.906615\pi$
$942$		0			0
$943$	7.50000	−	12.9904i	0.244234	−	0.423025i
$944$		0			0
$945$		0			0
$946$	−5.00000			−0.162564
$947$	−10.0000	+	17.3205i	−0.324956	+	0.562841i	−0.981504	−	0.191444i	$-0.938683\pi$
$947$	−10.0000	+	17.3205i	−0.324956	+	0.562841i	0.656547	+	0.754285i	$0.272016\pi$
$948$		0			0
$949$	7.50000	+	12.9904i	0.243460	+	0.421686i
$950$	4.00000			0.129777
$951$		0			0
$952$	22.5000	+	7.79423i	0.729229	+	0.252612i
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$		0			0
$955$	4.00000	−	6.92820i	0.129437	−	0.224191i
$956$	−15.0000			−0.485135
$957$		0			0
$958$	−12.5000	+	21.6506i	−0.403857	+	0.699500i
$959$	4.50000	+	23.3827i	0.145313	+	0.755066i
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$	−7.50000	+	12.9904i	−0.241810	+	0.418827i
$963$		0			0
$964$	5.50000	+	9.52628i	0.177143	+	0.306821i
$965$	5.00000	+	8.66025i	0.160956	+	0.278783i
$966$		0			0
$967$	−6.50000	+	11.2583i	−0.209026	+	0.362043i	−0.951408	−	0.307933i	$-0.900363\pi$
$967$	−6.50000	+	11.2583i	−0.209026	+	0.362043i	0.742382	+	0.669977i	$0.233696\pi$
$968$	42.0000			1.34993
$969$		0			0
$970$	9.00000			0.288973
$971$	28.5000	+	49.3634i	0.914609	+	1.58415i	0.807473	+	0.589904i	$0.200834\pi$
$971$	28.5000	+	49.3634i	0.914609	+	1.58415i	0.107135	+	0.994244i	$0.465832\pi$
$972$		0			0
$973$	−22.5000	−	7.79423i	−0.721317	−	0.249871i
$974$	9.50000	+	16.4545i	0.304400	+	0.527236i
$975$		0			0
$976$	−7.00000	−	12.1244i	−0.224065	−	0.388091i
$977$	9.00000	+	15.5885i	0.287936	+	0.498719i	0.973317	−	0.229465i	$-0.0736978\pi$
$977$	9.00000	+	15.5885i	0.287936	+	0.498719i	−0.685381	+	0.728184i	$0.740364\pi$
$978$		0			0
$979$	32.5000	+	56.2917i	1.03870	+	1.79909i
$980$	−5.50000	−	4.33013i	−0.175691	−	0.138321i
$981$		0			0
$982$	−6.50000	−	11.2583i	−0.207423	−	0.359268i
$983$	3.00000			0.0956851			0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000			0.0956851			0.0478426	+	0.998855i	$0.484765\pi$
$984$		0			0
$985$	−2.00000			−0.0637253
$986$	−1.50000	+	2.59808i	−0.0477697	+	0.0827396i
$987$		0			0
$988$	−2.50000	−	4.33013i	−0.0795356	−	0.137760i
$989$	−1.50000	−	2.59808i	−0.0476972	−	0.0826140i
$990$		0			0
$991$	18.5000	−	32.0429i	0.587672	−	1.01788i	−0.406865	−	0.913488i	$-0.633378\pi$
$991$	18.5000	−	32.0429i	0.587672	−	1.01788i	0.994537	−	0.104389i	$-0.0332887\pi$
$992$		0			0
$993$		0			0
$994$	30.0000	+	10.3923i	0.951542	+	0.329624i
$995$	−1.50000	+	2.59808i	−0.0475532	+	0.0823646i
$996$		0			0
$997$	−17.0000			−0.538395			−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000			−0.538395			−0.269198	+	0.963085i	$0.586759\pi$
$998$	15.5000	−	26.8468i	0.490644	−	0.849820i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.g.a.172.1		2
3.2	odd	2		63.2.g.a.4.1	✓	2
4.3	odd	2		3024.2.t.d.1873.1		2
7.2	even	3		189.2.h.a.37.1		2
7.3	odd	6		1323.2.f.b.442.1		2
7.4	even	3		1323.2.f.a.442.1		2
7.5	odd	6		1323.2.h.a.226.1		2
7.6	odd	2		1323.2.g.a.361.1		2
9.2	odd	6		63.2.h.a.25.1	yes	2
9.4	even	3		567.2.e.b.487.1		2
9.5	odd	6		567.2.e.a.487.1		2
9.7	even	3		189.2.h.a.46.1		2
12.11	even	2		1008.2.t.d.193.1		2
21.2	odd	6		63.2.h.a.58.1	yes	2
21.5	even	6		441.2.h.a.373.1		2
21.11	odd	6		441.2.f.b.148.1		2
21.17	even	6		441.2.f.a.148.1		2
21.20	even	2		441.2.g.a.67.1		2
28.23	odd	6		3024.2.q.b.2305.1		2
36.7	odd	6		3024.2.q.b.2881.1		2
36.11	even	6		1008.2.q.c.529.1		2
63.2	odd	6		63.2.g.a.16.1	yes	2
63.4	even	3		3969.2.a.c.1.1		1
63.11	odd	6		441.2.f.b.295.1		2
63.16	even	3	inner	189.2.g.a.100.1		2
63.20	even	6		441.2.h.a.214.1		2
63.23	odd	6		567.2.e.a.163.1		2
63.25	even	3		1323.2.f.a.883.1		2
63.31	odd	6		3969.2.a.a.1.1		1
63.32	odd	6		3969.2.a.d.1.1		1
63.34	odd	6		1323.2.h.a.802.1		2
63.38	even	6		441.2.f.a.295.1		2
63.47	even	6		441.2.g.a.79.1		2
63.52	odd	6		1323.2.f.b.883.1		2
63.58	even	3		567.2.e.b.163.1		2
63.59	even	6		3969.2.a.f.1.1		1
63.61	odd	6		1323.2.g.a.667.1		2
84.23	even	6		1008.2.q.c.625.1		2
252.79	odd	6		3024.2.t.d.289.1		2
252.191	even	6		1008.2.t.d.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.g.a.4.1	✓	2	3.2	odd	2
63.2.g.a.16.1	yes	2	63.2	odd	6
63.2.h.a.25.1	yes	2	9.2	odd	6
63.2.h.a.58.1	yes	2	21.2	odd	6
189.2.g.a.100.1		2	63.16	even	3	inner
189.2.g.a.172.1		2	1.1	even	1	trivial
189.2.h.a.37.1		2	7.2	even	3
189.2.h.a.46.1		2	9.7	even	3
441.2.f.a.148.1		2	21.17	even	6
441.2.f.a.295.1		2	63.38	even	6
441.2.f.b.148.1		2	21.11	odd	6
441.2.f.b.295.1		2	63.11	odd	6
441.2.g.a.67.1		2	21.20	even	2
441.2.g.a.79.1		2	63.47	even	6
441.2.h.a.214.1		2	63.20	even	6
441.2.h.a.373.1		2	21.5	even	6
567.2.e.a.163.1		2	63.23	odd	6
567.2.e.a.487.1		2	9.5	odd	6
567.2.e.b.163.1		2	63.58	even	3
567.2.e.b.487.1		2	9.4	even	3
1008.2.q.c.529.1		2	36.11	even	6
1008.2.q.c.625.1		2	84.23	even	6
1008.2.t.d.193.1		2	12.11	even	2
1008.2.t.d.961.1		2	252.191	even	6
1323.2.f.a.442.1		2	7.4	even	3
1323.2.f.a.883.1		2	63.25	even	3
1323.2.f.b.442.1		2	7.3	odd	6
1323.2.f.b.883.1		2	63.52	odd	6
1323.2.g.a.361.1		2	7.6	odd	2
1323.2.g.a.667.1		2	63.61	odd	6
1323.2.h.a.226.1		2	7.5	odd	6
1323.2.h.a.802.1		2	63.34	odd	6
3024.2.q.b.2305.1		2	28.23	odd	6
3024.2.q.b.2881.1		2	36.7	odd	6
3024.2.t.d.289.1		2	252.79	odd	6
3024.2.t.d.1873.1		2	4.3	odd	2
3969.2.a.a.1.1		1	63.31	odd	6
3969.2.a.c.1.1		1	63.4	even	3
3969.2.a.d.1.1		1	63.32	odd	6
3969.2.a.f.1.1		1	63.59	even	6

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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 2.59808i) q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 2.59808i) q^{7} +3.00000 q^{8} +(0.500000 - 0.866025i) q^{10} -5.00000 q^{11} +(2.50000 - 4.33013i) q^{13} +(2.50000 + 0.866025i) q^{14} +(0.500000 - 0.866025i) q^{16} +(1.50000 - 2.59808i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(0.500000 + 0.866025i) q^{20} +(-2.50000 + 4.33013i) q^{22} -3.00000 q^{23} -4.00000 q^{25} +(-2.50000 - 4.33013i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(-0.500000 - 0.866025i) q^{29} +(2.50000 + 4.33013i) q^{32} +(-1.50000 - 2.59808i) q^{34} +(0.500000 + 2.59808i) q^{35} +(-1.50000 - 2.59808i) q^{37} -1.00000 q^{38} +3.00000 q^{40} +(-2.50000 + 4.33013i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-2.50000 - 4.33013i) q^{44} +(-1.50000 + 2.59808i) q^{46} +(-6.50000 + 2.59808i) q^{49} +(-2.00000 + 3.46410i) q^{50} +5.00000 q^{52} +(-4.50000 + 7.79423i) q^{53} -5.00000 q^{55} +(1.50000 + 7.79423i) q^{56} -1.00000 q^{58} +(7.00000 - 12.1244i) q^{61} +7.00000 q^{64} +(2.50000 - 4.33013i) q^{65} +(-2.00000 - 3.46410i) q^{67} +3.00000 q^{68} +(2.50000 + 0.866025i) q^{70} +12.0000 q^{71} +(-1.50000 + 2.59808i) q^{73} -3.00000 q^{74} +(0.500000 - 0.866025i) q^{76} +(-2.50000 - 12.9904i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(0.500000 - 0.866025i) q^{80} +(2.50000 + 4.33013i) q^{82} +(-4.50000 - 7.79423i) q^{83} +(1.50000 - 2.59808i) q^{85} +1.00000 q^{86} -15.0000 q^{88} +(-6.50000 - 11.2583i) q^{89} +(12.5000 + 4.33013i) q^{91} +(-1.50000 - 2.59808i) q^{92} +(-0.500000 - 0.866025i) q^{95} +(4.50000 + 7.79423i) q^{97} +(-1.00000 + 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} + 2 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + q^4 + 2 * q^5 + q^7 + 6 * q^8	\( 2 q + q^{2} + q^{4} + 2 q^{5} + q^{7} + 6 q^{8} + q^{10} - 10 q^{11} + 5 q^{13} + 5 q^{14} + q^{16} + 3 q^{17} - q^{19} + q^{20} - 5 q^{22} - 6 q^{23} - 8 q^{25} - 5 q^{26} - 4 q^{28} - q^{29} + 5 q^{32} - 3 q^{34} + q^{35} - 3 q^{37} - 2 q^{38} + 6 q^{40} - 5 q^{41} + q^{43} - 5 q^{44} - 3 q^{46} - 13 q^{49} - 4 q^{50} + 10 q^{52} - 9 q^{53} - 10 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{61} + 14 q^{64} + 5 q^{65} - 4 q^{67} + 6 q^{68} + 5 q^{70} + 24 q^{71} - 3 q^{73} - 6 q^{74} + q^{76} - 5 q^{77} - 8 q^{79} + q^{80} + 5 q^{82} - 9 q^{83} + 3 q^{85} + 2 q^{86} - 30 q^{88} - 13 q^{89} + 25 q^{91} - 3 q^{92} - q^{95} + 9 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q + q^2 + q^4 + 2 * q^5 + q^7 + 6 * q^8 + q^10 - 10 * q^11 + 5 * q^13 + 5 * q^14 + q^16 + 3 * q^17 - q^19 + q^20 - 5 * q^22 - 6 * q^23 - 8 * q^25 - 5 * q^26 - 4 * q^28 - q^29 + 5 * q^32 - 3 * q^34 + q^35 - 3 * q^37 - 2 * q^38 + 6 * q^40 - 5 * q^41 + q^43 - 5 * q^44 - 3 * q^46 - 13 * q^49 - 4 * q^50 + 10 * q^52 - 9 * q^53 - 10 * q^55 + 3 * q^56 - 2 * q^58 + 14 * q^61 + 14 * q^64 + 5 * q^65 - 4 * q^67 + 6 * q^68 + 5 * q^70 + 24 * q^71 - 3 * q^73 - 6 * q^74 + q^76 - 5 * q^77 - 8 * q^79 + q^80 + 5 * q^82 - 9 * q^83 + 3 * q^85 + 2 * q^86 - 30 * q^88 - 13 * q^89 + 25 * q^91 - 3 * q^92 - q^95 + 9 * q^97 - 2 * q^98

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