LMFDB - Embedded newform 189.2.i.a.152.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(143,189)]

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([1, 1]))

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

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

chi := DirichletCharacter("189.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	189.i (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:	$1.50917259820$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			152.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	189.152
Dual form			189.2.i.a.143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} -1.00000 q^{4} +(1.50000 + 2.59808i) q^{5} +(2.00000 - 1.73205i) q^{7} -1.73205i q^{8} +O(q^{10})$	\(q-1.73205i q^{2} -1.00000 q^{4} +(1.50000 + 2.59808i) q^{5} +(2.00000 - 1.73205i) q^{7} -1.73205i q^{8} +(4.50000 - 2.59808i) q^{10} +(-1.50000 - 0.866025i) q^{11} +(1.50000 + 0.866025i) q^{13} +(-3.00000 - 3.46410i) q^{14} -5.00000 q^{16} +(1.50000 + 2.59808i) q^{17} +(-4.50000 - 2.59808i) q^{19} +(-1.50000 - 2.59808i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(-4.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(1.50000 - 2.59808i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(4.50000 - 2.59808i) q^{29} +3.46410i q^{31} +5.19615i q^{32} +(4.50000 - 2.59808i) q^{34} +(7.50000 + 2.59808i) q^{35} +(-3.50000 + 6.06218i) q^{37} +(-4.50000 + 7.79423i) q^{38} +(4.50000 - 2.59808i) q^{40} +(1.50000 - 2.59808i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(1.50000 + 0.866025i) q^{44} +(4.50000 + 7.79423i) q^{46} +(1.00000 - 6.92820i) q^{49} +(6.00000 + 3.46410i) q^{50} +(-1.50000 - 0.866025i) q^{52} +(-7.50000 + 4.33013i) q^{53} -5.19615i q^{55} +(-3.00000 - 3.46410i) q^{56} +(-4.50000 - 7.79423i) q^{58} +13.8564i q^{61} +6.00000 q^{62} -1.00000 q^{64} +5.19615i q^{65} -4.00000 q^{67} +(-1.50000 - 2.59808i) q^{68} +(4.50000 - 12.9904i) q^{70} +3.46410i q^{71} +(-4.50000 + 2.59808i) q^{73} +(10.5000 + 6.06218i) q^{74} +(4.50000 + 2.59808i) q^{76} +(-4.50000 + 0.866025i) q^{77} +8.00000 q^{79} +(-7.50000 - 12.9904i) q^{80} +(-4.50000 - 2.59808i) q^{82} +(-7.50000 - 12.9904i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(-1.50000 + 0.866025i) q^{86} +(-1.50000 + 2.59808i) q^{88} +(1.50000 - 2.59808i) q^{89} +(4.50000 - 0.866025i) q^{91} +(4.50000 - 2.59808i) q^{92} -15.5885i q^{95} +(1.50000 - 0.866025i) q^{97} +(-12.0000 - 1.73205i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 3 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^4 + 3 * q^5 + 4 * q^7	$ 2 q - 2 q^{4} + 3 q^{5} + 4 q^{7} + 9 q^{10} - 3 q^{11} + 3 q^{13} - 6 q^{14} - 10 q^{16} + 3 q^{17} - 9 q^{19} - 3 q^{20} - 3 q^{22} - 9 q^{23} - 4 q^{25} + 3 q^{26} - 4 q^{28} + 9 q^{29} + 9 q^{34} + 15 q^{35} - 7 q^{37} - 9 q^{38} + 9 q^{40} + 3 q^{41} - q^{43} + 3 q^{44} + 9 q^{46} + 2 q^{49} + 12 q^{50} - 3 q^{52} - 15 q^{53} - 6 q^{56} - 9 q^{58} + 12 q^{62} - 2 q^{64} - 8 q^{67} - 3 q^{68} + 9 q^{70} - 9 q^{73} + 21 q^{74} + 9 q^{76} - 9 q^{77} + 16 q^{79} - 15 q^{80} - 9 q^{82} - 15 q^{83} - 9 q^{85} - 3 q^{86} - 3 q^{88} + 3 q^{89} + 9 q^{91} + 9 q^{92} + 3 q^{97} - 24 q^{98}+O(q^{100}) $ 2 * q - 2 * q^4 + 3 * q^5 + 4 * q^7 + 9 * q^10 - 3 * q^11 + 3 * q^13 - 6 * q^14 - 10 * q^16 + 3 * q^17 - 9 * q^19 - 3 * q^20 - 3 * q^22 - 9 * q^23 - 4 * q^25 + 3 * q^26 - 4 * q^28 + 9 * q^29 + 9 * q^34 + 15 * q^35 - 7 * q^37 - 9 * q^38 + 9 * q^40 + 3 * q^41 - q^43 + 3 * q^44 + 9 * q^46 + 2 * q^49 + 12 * q^50 - 3 * q^52 - 15 * q^53 - 6 * q^56 - 9 * q^58 + 12 * q^62 - 2 * q^64 - 8 * q^67 - 3 * q^68 + 9 * q^70 - 9 * q^73 + 21 * q^74 + 9 * q^76 - 9 * q^77 + 16 * q^79 - 15 * q^80 - 9 * q^82 - 15 * q^83 - 9 * q^85 - 3 * q^86 - 3 * q^88 + 3 * q^89 + 9 * q^91 + 9 * q^92 + 3 * q^97 - 24 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$136$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$

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 $

$2$		−	1.73205i		−	1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$		−	1.73205i		−	1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		−	1.73205i		−	0.612372i
$9$		0			0
$10$	4.50000	−	2.59808i	1.42302	−	0.821584i
$11$	−1.50000	−	0.866025i	−0.452267	−	0.261116i	0.256520	−	0.966539i	$-0.417424\pi$
$11$	−1.50000	−	0.866025i	−0.452267	−	0.261116i	−0.708787	+	0.705422i	$0.750757\pi$
$12$		0			0
$13$	1.50000	+	0.866025i	0.416025	+	0.240192i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	1.50000	+	0.866025i	0.416025	+	0.240192i	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−3.00000	−	3.46410i	−0.801784	−	0.925820i
$15$		0			0
$16$	−5.00000			−1.25000
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	−4.50000	−	2.59808i	−1.03237	−	0.596040i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−4.50000	−	2.59808i	−1.03237	−	0.596040i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.50000	−	2.59808i	−0.335410	−	0.580948i
$21$		0			0
$22$	−1.50000	+	2.59808i	−0.319801	+	0.553912i
$23$	−4.50000	+	2.59808i	−0.938315	+	0.541736i	−0.889432	−	0.457068i	$-0.848900\pi$
$23$	−4.50000	+	2.59808i	−0.938315	+	0.541736i	−0.0488832	+	0.998805i	$0.515566\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$	1.50000	−	2.59808i	0.294174	−	0.509525i
$27$		0			0
$28$	−2.00000	+	1.73205i	−0.377964	+	0.327327i
$29$	4.50000	−	2.59808i	0.835629	−	0.482451i	−0.0201471	−	0.999797i	$-0.506413\pi$
$29$	4.50000	−	2.59808i	0.835629	−	0.482451i	0.855776	+	0.517346i	$0.173080\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$			5.19615i			0.918559i
$33$		0			0
$34$	4.50000	−	2.59808i	0.771744	−	0.445566i
$35$	7.50000	+	2.59808i	1.26773	+	0.439155i
$36$		0			0
$37$	−3.50000	+	6.06218i	−0.575396	+	0.996616i	0.420602	+	0.907245i	$0.361819\pi$
$37$	−3.50000	+	6.06218i	−0.575396	+	0.996616i	−0.995998	+	0.0893706i	$0.971514\pi$
$38$	−4.50000	+	7.79423i	−0.729996	+	1.26439i
$39$		0			0
$40$	4.50000	−	2.59808i	0.711512	−	0.410792i
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	−0.724797	−	0.688963i	$-0.758066\pi$
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	0.959058	+	0.283211i	$0.0913998\pi$
$42$		0			0
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	$-0.190961\pi$
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	−0.901629	+	0.432511i	$0.857628\pi$
$44$	1.50000	+	0.866025i	0.226134	+	0.130558i
$45$		0			0
$46$	4.50000	+	7.79423i	0.663489	+	1.14920i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$	6.00000	+	3.46410i	0.848528	+	0.489898i
$51$		0			0
$52$	−1.50000	−	0.866025i	−0.208013	−	0.120096i
$53$	−7.50000	+	4.33013i	−1.03020	+	0.594789i	−0.917043	−	0.398788i	$-0.869431\pi$
$53$	−7.50000	+	4.33013i	−1.03020	+	0.594789i	−0.113161	+	0.993577i	$0.536098\pi$
$54$		0			0
$55$		−	5.19615i		−	0.700649i
$56$	−3.00000	−	3.46410i	−0.400892	−	0.462910i
$57$		0			0
$58$	−4.50000	−	7.79423i	−0.590879	−	1.02343i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			13.8564i			1.77413i	0.461644	+	0.887066i	$0.347260\pi$
$61$			13.8564i			1.77413i	−0.461644	+	0.887066i	$0.652740\pi$
$62$	6.00000			0.762001
$63$		0			0
$64$	−1.00000			−0.125000
$65$			5.19615i			0.644503i
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$	−1.50000	−	2.59808i	−0.181902	−	0.315063i
$69$		0			0
$70$	4.50000	−	12.9904i	0.537853	−	1.55265i
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$	−4.50000	+	2.59808i	−0.526685	+	0.304082i	−0.739666	−	0.672975i	$-0.765016\pi$
$73$	−4.50000	+	2.59808i	−0.526685	+	0.304082i	0.212980	+	0.977056i	$0.431683\pi$
$74$	10.5000	+	6.06218i	1.22060	+	0.704714i
$75$		0			0
$76$	4.50000	+	2.59808i	0.516185	+	0.298020i
$77$	−4.50000	+	0.866025i	−0.512823	+	0.0986928i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	−7.50000	−	12.9904i	−0.838525	−	1.45237i
$81$		0			0
$82$	−4.50000	−	2.59808i	−0.496942	−	0.286910i
$83$	−7.50000	−	12.9904i	−0.823232	−	1.42588i	−0.903263	−	0.429087i	$-0.858835\pi$
$83$	−7.50000	−	12.9904i	−0.823232	−	1.42588i	0.0800311	−	0.996792i	$-0.474498\pi$
$84$		0			0
$85$	−4.50000	+	7.79423i	−0.488094	+	0.845403i
$86$	−1.50000	+	0.866025i	−0.161749	+	0.0933859i
$87$		0			0
$88$	−1.50000	+	2.59808i	−0.159901	+	0.276956i
$89$	1.50000	−	2.59808i	0.159000	−	0.275396i	−0.775509	−	0.631337i	$-0.782506\pi$
$89$	1.50000	−	2.59808i	0.159000	−	0.275396i	0.934508	+	0.355942i	$0.115840\pi$
$90$		0			0
$91$	4.50000	−	0.866025i	0.471728	−	0.0907841i
$92$	4.50000	−	2.59808i	0.469157	−	0.270868i
$93$		0			0
$94$		0			0
$95$		−	15.5885i		−	1.59934i
$96$		0			0
$97$	1.50000	−	0.866025i	0.152302	−	0.0879316i	−0.421912	−	0.906637i	$-0.638641\pi$
$97$	1.50000	−	0.866025i	0.152302	−	0.0879316i	0.574214	+	0.818705i	$0.305308\pi$
$98$	−12.0000	−	1.73205i	−1.21218	−	0.174964i
$99$		0			0
$100$	2.00000	−	3.46410i	0.200000	−	0.346410i
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	10.5000	−	6.06218i	1.03460	−	0.597324i	0.116298	−	0.993214i	$-0.462897\pi$
$103$	10.5000	−	6.06218i	1.03460	−	0.597324i	0.918298	+	0.395890i	$0.129564\pi$
$104$	1.50000	−	2.59808i	0.147087	−	0.254762i
$105$		0			0
$106$	7.50000	+	12.9904i	0.728464	+	1.26174i
$107$	−7.50000	−	4.33013i	−0.725052	−	0.418609i	0.0915571	−	0.995800i	$-0.470816\pi$
$107$	−7.50000	−	4.33013i	−0.725052	−	0.418609i	−0.816609	+	0.577191i	$0.804149\pi$
$108$		0			0
$109$	−9.50000	−	16.4545i	−0.909935	−	1.57605i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−9.50000	−	16.4545i	−0.909935	−	1.57605i	−0.0957826	−	0.995402i	$-0.530535\pi$
$110$	−9.00000			−0.858116
$111$		0			0
$112$	−10.0000	+	8.66025i	−0.944911	+	0.818317i
$113$	−1.50000	−	0.866025i	−0.141108	−	0.0814688i	0.427784	−	0.903881i	$-0.359294\pi$
$113$	−1.50000	−	0.866025i	−0.141108	−	0.0814688i	−0.568892	+	0.822412i	$0.692628\pi$
$114$		0			0
$115$	−13.5000	−	7.79423i	−1.25888	−	0.726816i
$116$	−4.50000	+	2.59808i	−0.417815	+	0.241225i
$117$		0			0
$118$		0			0
$119$	7.50000	+	2.59808i	0.687524	+	0.238165i
$120$		0			0
$121$	−4.00000	−	6.92820i	−0.363636	−	0.629837i
$122$	24.0000			2.17286
$123$		0			0
$124$		−	3.46410i		−	0.311086i
$125$	3.00000			0.268328
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$			12.1244i			1.07165i
$129$		0			0
$130$	9.00000			0.789352
$131$	4.50000	+	7.79423i	0.393167	+	0.680985i	0.992865	−	0.119241i	$-0.0380462\pi$
$131$	4.50000	+	7.79423i	0.393167	+	0.680985i	−0.599699	+	0.800226i	$0.704713\pi$
$132$		0			0
$133$	−13.5000	+	2.59808i	−1.17060	+	0.225282i
$134$			6.92820i			0.598506i
$135$		0			0
$136$	4.50000	−	2.59808i	0.385872	−	0.222783i
$137$	10.5000	+	6.06218i	0.897076	+	0.517927i	0.876250	−	0.481856i	$-0.160037\pi$
$137$	10.5000	+	6.06218i	0.897076	+	0.517927i	0.0208253	+	0.999783i	$0.493371\pi$
$138$		0			0
$139$	7.50000	+	4.33013i	0.636142	+	0.367277i	0.783127	−	0.621862i	$-0.213624\pi$
$139$	7.50000	+	4.33013i	0.636142	+	0.367277i	−0.146985	+	0.989139i	$0.546957\pi$
$140$	−7.50000	−	2.59808i	−0.633866	−	0.219578i
$141$		0			0
$142$	6.00000			0.503509
$143$	−1.50000	−	2.59808i	−0.125436	−	0.217262i
$144$		0			0
$145$	13.5000	+	7.79423i	1.12111	+	0.647275i
$146$	4.50000	+	7.79423i	0.372423	+	0.645055i
$147$		0			0
$148$	3.50000	−	6.06218i	0.287698	−	0.498308i
$149$	−1.50000	+	0.866025i	−0.122885	+	0.0709476i	−0.560182	−	0.828369i	$-0.689269\pi$
$149$	−1.50000	+	0.866025i	−0.122885	+	0.0709476i	0.437298	+	0.899317i	$0.355936\pi$
$150$		0			0
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	−0.279554	−	0.960130i	$-0.590186\pi$
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	0.971274	−	0.237964i	$-0.0764802\pi$
$152$	−4.50000	+	7.79423i	−0.364998	+	0.632195i
$153$		0			0
$154$	1.50000	+	7.79423i	0.120873	+	0.628077i
$155$	−9.00000	+	5.19615i	−0.722897	+	0.417365i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		−	13.8564i		−	1.10236i
$159$		0			0
$160$	−13.5000	+	7.79423i	−1.06727	+	0.616188i
$161$	−4.50000	+	12.9904i	−0.354650	+	1.02379i
$162$		0			0
$163$	−5.50000	+	9.52628i	−0.430793	+	0.746156i	−0.996942	−	0.0781474i	$-0.975100\pi$
$163$	−5.50000	+	9.52628i	−0.430793	+	0.746156i	0.566149	+	0.824303i	$0.308433\pi$
$164$	−1.50000	+	2.59808i	−0.117130	+	0.202876i
$165$		0			0
$166$	−22.5000	+	12.9904i	−1.74634	+	1.00825i
$167$	−4.50000	+	7.79423i	−0.348220	+	0.603136i	−0.985933	−	0.167139i	$-0.946547\pi$
$167$	−4.50000	+	7.79423i	−0.348220	+	0.603136i	0.637713	+	0.770274i	$0.279881\pi$
$168$		0			0
$169$	−5.00000	−	8.66025i	−0.384615	−	0.666173i
$170$	13.5000	+	7.79423i	1.03540	+	0.597790i
$171$		0			0
$172$	0.500000	+	0.866025i	0.0381246	+	0.0660338i
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$	2.00000	+	10.3923i	0.151186	+	0.785584i
$176$	7.50000	+	4.33013i	0.565334	+	0.326396i
$177$		0			0
$178$	−4.50000	−	2.59808i	−0.337289	−	0.194734i
$179$	13.5000	−	7.79423i	1.00904	−	0.582568i	0.0981277	−	0.995174i	$-0.468715\pi$
$179$	13.5000	−	7.79423i	1.00904	−	0.582568i	0.910910	+	0.412606i	$0.135381\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	−1.50000	−	7.79423i	−0.111187	−	0.577747i
$183$		0			0
$184$	4.50000	+	7.79423i	0.331744	+	0.574598i
$185$	−21.0000			−1.54395
$186$		0			0
$187$		−	5.19615i		−	0.379980i
$188$		0			0
$189$		0			0
$190$	−27.0000			−1.95879
$191$		−	17.3205i		−	1.25327i	−0.779314	−	0.626634i	$-0.784432\pi$
$191$		−	17.3205i		−	1.25327i	0.779314	−	0.626634i	$-0.215568\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−1.50000	−	2.59808i	−0.107694	−	0.186531i
$195$		0			0
$196$	−1.00000	+	6.92820i	−0.0714286	+	0.494872i
$197$		−	13.8564i		−	0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$		−	13.8564i		−	0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$		0			0
$199$	−7.50000	+	4.33013i	−0.531661	+	0.306955i	−0.741693	−	0.670740i	$-0.765977\pi$
$199$	−7.50000	+	4.33013i	−0.531661	+	0.306955i	0.210032	+	0.977695i	$0.432643\pi$
$200$	6.00000	+	3.46410i	0.424264	+	0.244949i
$201$		0			0
$202$	−4.50000	−	2.59808i	−0.316619	−	0.182800i
$203$	4.50000	−	12.9904i	0.315838	−	0.911746i
$204$		0			0
$205$	9.00000			0.628587
$206$	−10.5000	−	18.1865i	−0.731570	−	1.26712i
$207$		0			0
$208$	−7.50000	−	4.33013i	−0.520031	−	0.300240i
$209$	4.50000	+	7.79423i	0.311272	+	0.539138i
$210$		0			0
$211$	2.50000	−	4.33013i	0.172107	−	0.298098i	−0.767049	−	0.641588i	$-0.778276\pi$
$211$	2.50000	−	4.33013i	0.172107	−	0.298098i	0.939156	+	0.343490i	$0.111609\pi$
$212$	7.50000	−	4.33013i	0.515102	−	0.297394i
$213$		0			0
$214$	−7.50000	+	12.9904i	−0.512689	+	0.888004i
$215$	1.50000	−	2.59808i	0.102299	−	0.177187i
$216$		0			0
$217$	6.00000	+	6.92820i	0.407307	+	0.470317i
$218$	−28.5000	+	16.4545i	−1.93026	+	1.11444i
$219$		0			0
$220$			5.19615i			0.350325i
$221$			5.19615i			0.349531i
$222$		0			0
$223$	4.50000	−	2.59808i	0.301342	−	0.173980i	−0.341703	−	0.939808i	$-0.611004\pi$
$223$	4.50000	−	2.59808i	0.301342	−	0.173980i	0.643046	+	0.765828i	$0.277671\pi$
$224$	9.00000	+	10.3923i	0.601338	+	0.694365i
$225$		0			0
$226$	−1.50000	+	2.59808i	−0.0997785	+	0.172821i
$227$	−10.5000	+	18.1865i	−0.696909	+	1.20708i	0.272623	+	0.962121i	$0.412109\pi$
$227$	−10.5000	+	18.1865i	−0.696909	+	1.20708i	−0.969533	+	0.244962i	$0.921225\pi$
$228$		0			0
$229$	7.50000	−	4.33013i	0.495614	−	0.286143i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	7.50000	−	4.33013i	0.495614	−	0.286143i	0.726900	+	0.686743i	$0.240960\pi$
$230$	−13.5000	+	23.3827i	−0.890164	+	1.54181i
$231$		0			0
$232$	−4.50000	−	7.79423i	−0.295439	−	0.511716i
$233$	4.50000	+	2.59808i	0.294805	+	0.170206i	0.640107	−	0.768286i	$-0.278890\pi$
$233$	4.50000	+	2.59808i	0.294805	+	0.170206i	−0.345302	+	0.938492i	$0.612223\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	4.50000	−	12.9904i	0.291692	−	0.842041i
$239$	−1.50000	−	0.866025i	−0.0970269	−	0.0560185i	0.450701	−	0.892675i	$-0.351174\pi$
$239$	−1.50000	−	0.866025i	−0.0970269	−	0.0560185i	−0.547728	+	0.836656i	$0.684507\pi$
$240$		0			0
$241$	19.5000	+	11.2583i	1.25611	+	0.725213i	0.972315	−	0.233674i	$-0.0750747\pi$
$241$	19.5000	+	11.2583i	1.25611	+	0.725213i	0.283790	+	0.958886i	$0.408408\pi$
$242$	−12.0000	+	6.92820i	−0.771389	+	0.445362i
$243$		0			0
$244$		−	13.8564i		−	0.887066i
$245$	19.5000	−	7.79423i	1.24581	−	0.497955i
$246$		0			0
$247$	−4.50000	−	7.79423i	−0.286328	−	0.495935i
$248$	6.00000			0.381000
$249$		0			0
$250$		−	5.19615i		−	0.328634i
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	9.00000			0.565825
$254$		−	34.6410i		−	2.17357i
$255$		0			0
$256$	19.0000			1.18750
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	0.909010	−	0.416775i	$-0.136840\pi$
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	−0.815442	+	0.578838i	$0.803506\pi$
$258$		0			0
$259$	3.50000	+	18.1865i	0.217479	+	1.13006i
$260$		−	5.19615i		−	0.322252i
$261$		0			0
$262$	13.5000	−	7.79423i	0.834033	−	0.481529i
$263$	−19.5000	−	11.2583i	−1.20242	−	0.694218i	−0.241329	−	0.970443i	$-0.577583\pi$
$263$	−19.5000	−	11.2583i	−1.20242	−	0.694218i	−0.961093	+	0.276225i	$0.910916\pi$
$264$		0			0
$265$	−22.5000	−	12.9904i	−1.38216	−	0.797993i
$266$	4.50000	+	23.3827i	0.275913	+	1.43368i
$267$		0			0
$268$	4.00000			0.244339
$269$	7.50000	+	12.9904i	0.457283	+	0.792038i	0.998816	−	0.0486418i	$-0.0154893\pi$
$269$	7.50000	+	12.9904i	0.457283	+	0.792038i	−0.541533	+	0.840679i	$0.682156\pi$
$270$		0			0
$271$	−10.5000	−	6.06218i	−0.637830	−	0.368251i	0.145948	−	0.989292i	$-0.453377\pi$
$271$	−10.5000	−	6.06218i	−0.637830	−	0.368251i	−0.783778	+	0.621041i	$0.786710\pi$
$272$	−7.50000	−	12.9904i	−0.454754	−	0.787658i
$273$		0			0
$274$	10.5000	−	18.1865i	0.634328	−	1.09869i
$275$	6.00000	−	3.46410i	0.361814	−	0.208893i
$276$		0			0
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	−0.850613	−	0.525792i	$-0.823769\pi$
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	0.880656	+	0.473757i	$0.157103\pi$
$278$	7.50000	−	12.9904i	0.449820	−	0.779111i
$279$		0			0
$280$	4.50000	−	12.9904i	0.268926	−	0.776324i
$281$	16.5000	−	9.52628i	0.984307	−	0.568290i	0.0807396	−	0.996735i	$-0.474272\pi$
$281$	16.5000	−	9.52628i	0.984307	−	0.568290i	0.903568	+	0.428445i	$0.140938\pi$
$282$		0			0
$283$		−	3.46410i		−	0.205919i	−0.994686	−	0.102960i	$-0.967169\pi$
$283$		−	3.46410i		−	0.205919i	0.994686	−	0.102960i	$-0.0328313\pi$
$284$		−	3.46410i		−	0.205557i
$285$		0			0
$286$	−4.50000	+	2.59808i	−0.266091	+	0.153627i
$287$	−1.50000	−	7.79423i	−0.0885422	−	0.460079i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	13.5000	−	23.3827i	0.792747	−	1.37308i
$291$		0			0
$292$	4.50000	−	2.59808i	0.263343	−	0.152041i
$293$	−4.50000	+	7.79423i	−0.262893	+	0.455344i	−0.967009	−	0.254741i	$-0.918010\pi$
$293$	−4.50000	+	7.79423i	−0.262893	+	0.455344i	0.704117	+	0.710084i	$0.251343\pi$
$294$		0			0
$295$		0			0
$296$	10.5000	+	6.06218i	0.610300	+	0.352357i
$297$		0			0
$298$	1.50000	+	2.59808i	0.0868927	+	0.150503i
$299$	−9.00000			−0.520483
$300$		0			0
$301$	−2.50000	−	0.866025i	−0.144098	−	0.0499169i
$302$	−25.5000	−	14.7224i	−1.46736	−	0.847181i
$303$		0			0
$304$	22.5000	+	12.9904i	1.29046	+	0.745049i
$305$	−36.0000	+	20.7846i	−2.06135	+	1.19012i
$306$		0			0
$307$			24.2487i			1.38395i	0.721923	+	0.691974i	$0.243259\pi$
$307$			24.2487i			1.38395i	−0.721923	+	0.691974i	$0.756741\pi$
$308$	4.50000	−	0.866025i	0.256411	−	0.0493464i
$309$		0			0
$310$	9.00000	+	15.5885i	0.511166	+	0.885365i
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$		−	20.7846i		−	1.17482i	−0.809291	−	0.587408i	$-0.800148\pi$
$313$		−	20.7846i		−	1.17482i	0.809291	−	0.587408i	$-0.199852\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000			−0.450035
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$	−9.00000			−0.503903
$320$	−1.50000	−	2.59808i	−0.0838525	−	0.145237i
$321$		0			0
$322$	22.5000	+	7.79423i	1.25388	+	0.434355i
$323$		−	15.5885i		−	0.867365i
$324$		0			0
$325$	−6.00000	+	3.46410i	−0.332820	+	0.192154i
$326$	16.5000	+	9.52628i	0.913850	+	0.527612i
$327$		0			0
$328$	−4.50000	−	2.59808i	−0.248471	−	0.143455i
$329$		0			0
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$	7.50000	+	12.9904i	0.411616	+	0.712940i
$333$		0			0
$334$	13.5000	+	7.79423i	0.738687	+	0.426481i
$335$	−6.00000	−	10.3923i	−0.327815	−	0.567792i
$336$		0			0
$337$	−9.50000	+	16.4545i	−0.517498	+	0.896333i	0.482295	+	0.876009i	$0.339803\pi$
$337$	−9.50000	+	16.4545i	−0.517498	+	0.896333i	−0.999793	+	0.0203242i	$0.993530\pi$
$338$	−15.0000	+	8.66025i	−0.815892	+	0.471056i
$339$		0			0
$340$	4.50000	−	7.79423i	0.244047	−	0.422701i
$341$	3.00000	−	5.19615i	0.162459	−	0.281387i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$	−1.50000	+	0.866025i	−0.0808746	+	0.0466930i
$345$		0			0
$346$		−	10.3923i		−	0.558694i
$347$			3.46410i			0.185963i	0.995668	+	0.0929814i	$0.0296397\pi$
$347$			3.46410i			0.185963i	−0.995668	+	0.0929814i	$0.970360\pi$
$348$		0			0
$349$	−10.5000	+	6.06218i	−0.562052	+	0.324501i	−0.753969	−	0.656910i	$-0.771863\pi$
$349$	−10.5000	+	6.06218i	−0.562052	+	0.324501i	0.191917	+	0.981411i	$0.438530\pi$
$350$	18.0000	−	3.46410i	0.962140	−	0.185164i
$351$		0			0
$352$	4.50000	−	7.79423i	0.239851	−	0.415434i
$353$	−10.5000	+	18.1865i	−0.558859	+	0.967972i	0.438733	+	0.898617i	$0.355427\pi$
$353$	−10.5000	+	18.1865i	−0.558859	+	0.967972i	−0.997592	+	0.0693543i	$0.977906\pi$
$354$		0			0
$355$	−9.00000	+	5.19615i	−0.477670	+	0.275783i
$356$	−1.50000	+	2.59808i	−0.0794998	+	0.137698i
$357$		0			0
$358$	−13.5000	−	23.3827i	−0.713497	−	1.23581i
$359$	−19.5000	−	11.2583i	−1.02917	−	0.594192i	−0.112424	−	0.993660i	$-0.535861\pi$
$359$	−19.5000	−	11.2583i	−1.02917	−	0.594192i	−0.916747	+	0.399468i	$0.869195\pi$
$360$		0			0
$361$	4.00000	+	6.92820i	0.210526	+	0.364642i
$362$		0			0
$363$		0			0
$364$	−4.50000	+	0.866025i	−0.235864	+	0.0453921i
$365$	−13.5000	−	7.79423i	−0.706622	−	0.407969i
$366$		0			0
$367$	−4.50000	−	2.59808i	−0.234898	−	0.135618i	0.377932	−	0.925834i	$-0.376635\pi$
$367$	−4.50000	−	2.59808i	−0.234898	−	0.135618i	−0.612830	+	0.790215i	$0.709969\pi$
$368$	22.5000	−	12.9904i	1.17289	−	0.677170i
$369$		0			0
$370$			36.3731i			1.89095i
$371$	−7.50000	+	21.6506i	−0.389381	+	1.12404i
$372$		0			0
$373$	18.5000	+	32.0429i	0.957894	+	1.65912i	0.727603	+	0.685999i	$0.240634\pi$
$373$	18.5000	+	32.0429i	0.957894	+	1.65912i	0.230291	+	0.973122i	$0.426032\pi$
$374$	−9.00000			−0.465379
$375$		0			0
$376$		0			0
$377$	9.00000			0.463524
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$			15.5885i			0.799671i
$381$		0			0
$382$	−30.0000			−1.53493
$383$	4.50000	+	7.79423i	0.229939	+	0.398266i	0.957790	−	0.287469i	$-0.0928139\pi$
$383$	4.50000	+	7.79423i	0.229939	+	0.398266i	−0.727851	+	0.685736i	$0.759481\pi$
$384$		0			0
$385$	−9.00000	−	10.3923i	−0.458682	−	0.529641i
$386$			3.46410i			0.176318i
$387$		0			0
$388$	−1.50000	+	0.866025i	−0.0761510	+	0.0439658i
$389$	−31.5000	−	18.1865i	−1.59711	−	0.922094i	−0.992040	−	0.125924i	$-0.959810\pi$
$389$	−31.5000	−	18.1865i	−1.59711	−	0.922094i	−0.605074	−	0.796170i	$-0.706856\pi$
$390$		0			0
$391$	−13.5000	−	7.79423i	−0.682724	−	0.394171i
$392$	−12.0000	−	1.73205i	−0.606092	−	0.0874818i
$393$		0			0
$394$	−24.0000			−1.20910
$395$	12.0000	+	20.7846i	0.603786	+	1.04579i
$396$		0			0
$397$	7.50000	+	4.33013i	0.376414	+	0.217323i	0.676257	−	0.736666i	$-0.263601\pi$
$397$	7.50000	+	4.33013i	0.376414	+	0.217323i	−0.299843	+	0.953989i	$0.596934\pi$
$398$	7.50000	+	12.9904i	0.375941	+	0.651149i
$399$		0			0
$400$	10.0000	−	17.3205i	0.500000	−	0.866025i
$401$	28.5000	−	16.4545i	1.42322	−	0.821698i	0.426649	−	0.904417i	$-0.359694\pi$
$401$	28.5000	−	16.4545i	1.42322	−	0.821698i	0.996573	+	0.0827195i	$0.0263606\pi$
$402$		0			0
$403$	−3.00000	+	5.19615i	−0.149441	+	0.258839i
$404$	−1.50000	+	2.59808i	−0.0746278	+	0.129259i
$405$		0			0
$406$	−22.5000	−	7.79423i	−1.11666	−	0.386821i
$407$	10.5000	−	6.06218i	0.520466	−	0.300491i
$408$		0			0
$409$			6.92820i			0.342578i	0.985221	+	0.171289i	$0.0547931\pi$
$409$			6.92820i			0.342578i	−0.985221	+	0.171289i	$0.945207\pi$
$410$		−	15.5885i		−	0.769859i
$411$		0			0
$412$	−10.5000	+	6.06218i	−0.517298	+	0.298662i
$413$		0			0
$414$		0			0
$415$	22.5000	−	38.9711i	1.10448	−	1.91302i
$416$	−4.50000	+	7.79423i	−0.220631	+	0.382143i
$417$		0			0
$418$	13.5000	−	7.79423i	0.660307	−	0.381228i
$419$	−16.5000	+	28.5788i	−0.806078	+	1.39617i	0.109483	+	0.993989i	$0.465080\pi$
$419$	−16.5000	+	28.5788i	−0.806078	+	1.39617i	−0.915561	+	0.402179i	$0.868253\pi$
$420$		0			0
$421$	−5.50000	−	9.52628i	−0.268054	−	0.464282i	0.700306	−	0.713843i	$-0.253047\pi$
$421$	−5.50000	−	9.52628i	−0.268054	−	0.464282i	−0.968359	+	0.249561i	$0.919714\pi$
$422$	−7.50000	−	4.33013i	−0.365094	−	0.210787i
$423$		0			0
$424$	7.50000	+	12.9904i	0.364232	+	0.630869i
$425$	−12.0000			−0.582086
$426$		0			0
$427$	24.0000	+	27.7128i	1.16144	+	1.34112i
$428$	7.50000	+	4.33013i	0.362526	+	0.209305i
$429$		0			0
$430$	−4.50000	−	2.59808i	−0.217009	−	0.125290i
$431$	13.5000	−	7.79423i	0.650272	−	0.375435i	−0.138288	−	0.990392i	$-0.544160\pi$
$431$	13.5000	−	7.79423i	0.650272	−	0.375435i	0.788560	+	0.614957i	$0.210827\pi$
$432$		0			0
$433$		−	13.8564i		−	0.665896i	−0.942945	−	0.332948i	$-0.891957\pi$
$433$		−	13.8564i		−	0.665896i	0.942945	−	0.332948i	$-0.108043\pi$
$434$	12.0000	−	10.3923i	0.576018	−	0.498847i
$435$		0			0
$436$	9.50000	+	16.4545i	0.454967	+	0.788027i
$437$	27.0000			1.29159
$438$		0			0
$439$			31.1769i			1.48799i	0.668184	+	0.743996i	$0.267072\pi$
$439$			31.1769i			1.48799i	−0.668184	+	0.743996i	$0.732928\pi$
$440$	−9.00000			−0.429058
$441$		0			0
$442$	9.00000			0.428086
$443$			31.1769i			1.48126i	0.671913	+	0.740630i	$0.265473\pi$
$443$			31.1769i			1.48126i	−0.671913	+	0.740630i	$0.734527\pi$
$444$		0			0
$445$	9.00000			0.426641
$446$	−4.50000	−	7.79423i	−0.213081	−	0.369067i
$447$		0			0
$448$	−2.00000	+	1.73205i	−0.0944911	+	0.0818317i
$449$			34.6410i			1.63481i	0.576063	+	0.817405i	$0.304588\pi$
$449$			34.6410i			1.63481i	−0.576063	+	0.817405i	$0.695412\pi$
$450$		0			0
$451$	−4.50000	+	2.59808i	−0.211897	+	0.122339i
$452$	1.50000	+	0.866025i	0.0705541	+	0.0407344i
$453$		0			0
$454$	31.5000	+	18.1865i	1.47837	+	0.853536i
$455$	9.00000	+	10.3923i	0.421927	+	0.487199i
$456$		0			0
$457$	−26.0000			−1.21623			−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000			−1.21623			−0.608114	+	0.793849i	$0.708074\pi$
$458$	−7.50000	−	12.9904i	−0.350452	−	0.607001i
$459$		0			0
$460$	13.5000	+	7.79423i	0.629441	+	0.363408i
$461$	7.50000	+	12.9904i	0.349310	+	0.605022i	0.986127	−	0.165992i	$-0.0530827\pi$
$461$	7.50000	+	12.9904i	0.349310	+	0.605022i	−0.636817	+	0.771015i	$0.719749\pi$
$462$		0			0
$463$	0.500000	−	0.866025i	0.0232370	−	0.0402476i	−0.854173	−	0.519989i	$-0.825936\pi$
$463$	0.500000	−	0.866025i	0.0232370	−	0.0402476i	0.877410	+	0.479741i	$0.159269\pi$
$464$	−22.5000	+	12.9904i	−1.04454	+	0.603063i
$465$		0			0
$466$	4.50000	−	7.79423i	0.208458	−	0.361061i
$467$	1.50000	−	2.59808i	0.0694117	−	0.120225i	−0.829231	−	0.558906i	$-0.811221\pi$
$467$	1.50000	−	2.59808i	0.0694117	−	0.120225i	0.898642	+	0.438682i	$0.144554\pi$
$468$		0			0
$469$	−8.00000	+	6.92820i	−0.369406	+	0.319915i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.73205i			0.0796398i
$474$		0			0
$475$	18.0000	−	10.3923i	0.825897	−	0.476832i
$476$	−7.50000	−	2.59808i	−0.343762	−	0.119083i
$477$		0			0
$478$	−1.50000	+	2.59808i	−0.0686084	+	0.118833i
$479$	13.5000	−	23.3827i	0.616831	−	1.06838i	−0.373230	−	0.927739i	$-0.621750\pi$
$479$	13.5000	−	23.3827i	0.616831	−	1.06838i	0.990060	−	0.140643i	$-0.0449170\pi$
$480$		0			0
$481$	−10.5000	+	6.06218i	−0.478759	+	0.276412i
$482$	19.5000	−	33.7750i	0.888201	−	1.53841i
$483$		0			0
$484$	4.00000	+	6.92820i	0.181818	+	0.314918i
$485$	4.50000	+	2.59808i	0.204334	+	0.117973i
$486$		0			0
$487$	11.5000	+	19.9186i	0.521115	+	0.902597i	0.999698	+	0.0245553i	$0.00781698\pi$
$487$	11.5000	+	19.9186i	0.521115	+	0.902597i	−0.478584	+	0.878042i	$0.658850\pi$
$488$	24.0000			1.08643
$489$		0			0
$490$	−13.5000	−	33.7750i	−0.609868	−	1.52580i
$491$	22.5000	+	12.9904i	1.01541	+	0.586248i	0.912771	−	0.408471i	$-0.133938\pi$
$491$	22.5000	+	12.9904i	1.01541	+	0.586248i	0.102639	+	0.994719i	$0.467271\pi$
$492$		0			0
$493$	13.5000	+	7.79423i	0.608009	+	0.351034i
$494$	−13.5000	+	7.79423i	−0.607394	+	0.350679i
$495$		0			0
$496$		−	17.3205i		−	0.777714i
$497$	6.00000	+	6.92820i	0.269137	+	0.310772i
$498$		0			0
$499$	−12.5000	−	21.6506i	−0.559577	−	0.969216i	−0.997532	−	0.0702185i	$-0.977630\pi$
$499$	−12.5000	−	21.6506i	−0.559577	−	0.969216i	0.437955	−	0.898997i	$-0.355703\pi$
$500$	−3.00000			−0.134164
$501$		0			0
$502$		−	20.7846i		−	0.927663i
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	9.00000			0.400495
$506$		−	15.5885i		−	0.692991i
$507$		0			0
$508$	−20.0000			−0.887357
$509$	−16.5000	−	28.5788i	−0.731350	−	1.26673i	−0.956306	−	0.292366i	$-0.905557\pi$
$509$	−16.5000	−	28.5788i	−0.731350	−	1.26673i	0.224957	−	0.974369i	$-0.427776\pi$
$510$		0			0
$511$	−4.50000	+	12.9904i	−0.199068	+	0.574661i
$512$		−	8.66025i		−	0.382733i
$513$		0			0
$514$	4.50000	−	2.59808i	0.198486	−	0.114596i
$515$	31.5000	+	18.1865i	1.38806	+	0.801394i
$516$		0			0
$517$		0			0
$518$	31.5000	−	6.06218i	1.38403	−	0.266357i
$519$		0			0
$520$	9.00000			0.394676
$521$	−22.5000	−	38.9711i	−0.985743	−	1.70736i	−0.638588	−	0.769549i	$-0.720481\pi$
$521$	−22.5000	−	38.9711i	−0.985743	−	1.70736i	−0.347155	−	0.937808i	$-0.612852\pi$
$522$		0			0
$523$	−16.5000	−	9.52628i	−0.721495	−	0.416555i	0.0938079	−	0.995590i	$-0.470096\pi$
$523$	−16.5000	−	9.52628i	−0.721495	−	0.416555i	−0.815303	+	0.579035i	$0.803429\pi$
$524$	−4.50000	−	7.79423i	−0.196583	−	0.340492i
$525$		0			0
$526$	−19.5000	+	33.7750i	−0.850240	+	1.47266i
$527$	−9.00000	+	5.19615i	−0.392046	+	0.226348i
$528$		0			0
$529$	2.00000	−	3.46410i	0.0869565	−	0.150613i
$530$	−22.5000	+	38.9711i	−0.977338	+	1.69280i
$531$		0			0
$532$	13.5000	−	2.59808i	0.585299	−	0.112641i
$533$	4.50000	−	2.59808i	0.194917	−	0.112535i
$534$		0			0
$535$		−	25.9808i		−	1.12325i
$536$			6.92820i			0.299253i
$537$		0			0
$538$	22.5000	−	12.9904i	0.970044	−	0.560055i
$539$	−7.50000	+	9.52628i	−0.323048	+	0.410326i
$540$		0			0
$541$	6.50000	−	11.2583i	0.279457	−	0.484033i	−0.691793	−	0.722096i	$-0.743179\pi$
$541$	6.50000	−	11.2583i	0.279457	−	0.484033i	0.971250	+	0.238062i	$0.0765123\pi$
$542$	−10.5000	+	18.1865i	−0.451014	+	0.781179i
$543$		0			0
$544$	−13.5000	+	7.79423i	−0.578808	+	0.334175i
$545$	28.5000	−	49.3634i	1.22081	−	2.11450i
$546$		0			0
$547$	9.50000	+	16.4545i	0.406191	+	0.703543i	0.994459	−	0.105123i	$-0.0335235\pi$
$547$	9.50000	+	16.4545i	0.406191	+	0.703543i	−0.588269	+	0.808666i	$0.700190\pi$
$548$	−10.5000	−	6.06218i	−0.448538	−	0.258963i
$549$		0			0
$550$	−6.00000	−	10.3923i	−0.255841	−	0.443129i
$551$	−27.0000			−1.15024
$552$		0			0
$553$	16.0000	−	13.8564i	0.680389	−	0.589234i
$554$	−1.50000	−	0.866025i	−0.0637289	−	0.0367939i
$555$		0			0
$556$	−7.50000	−	4.33013i	−0.318071	−	0.183638i
$557$	10.5000	−	6.06218i	0.444899	−	0.256863i	−0.260774	−	0.965400i	$-0.583978\pi$
$557$	10.5000	−	6.06218i	0.444899	−	0.256863i	0.705674	+	0.708537i	$0.250645\pi$
$558$		0			0
$559$		−	1.73205i		−	0.0732579i
$560$	−37.5000	−	12.9904i	−1.58466	−	0.548944i
$561$		0			0
$562$	−16.5000	−	28.5788i	−0.696010	−	1.20553i
$563$	−36.0000			−1.51722			−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000			−1.51722			−0.758610	+	0.651546i	$0.774121\pi$
$564$		0			0
$565$		−	5.19615i		−	0.218604i
$566$	−6.00000			−0.252199
$567$		0			0
$568$	6.00000			0.251754
$569$			6.92820i			0.290445i	0.989399	+	0.145223i	$0.0463899\pi$
$569$			6.92820i			0.290445i	−0.989399	+	0.145223i	$0.953610\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$	1.50000	+	2.59808i	0.0627182	+	0.108631i
$573$		0			0
$574$	−13.5000	+	2.59808i	−0.563479	+	0.108442i
$575$		−	20.7846i		−	0.866778i
$576$		0			0
$577$	−34.5000	+	19.9186i	−1.43625	+	0.829222i	−0.997587	−	0.0694283i	$-0.977883\pi$
$577$	−34.5000	+	19.9186i	−1.43625	+	0.829222i	−0.438667	+	0.898650i	$0.644549\pi$
$578$	−12.0000	−	6.92820i	−0.499134	−	0.288175i
$579$		0			0
$580$	−13.5000	−	7.79423i	−0.560557	−	0.323638i
$581$	−37.5000	−	12.9904i	−1.55576	−	0.538932i
$582$		0			0
$583$	15.0000			0.621237
$584$	4.50000	+	7.79423i	0.186211	+	0.322527i
$585$		0			0
$586$	13.5000	+	7.79423i	0.557680	+	0.321977i
$587$	10.5000	+	18.1865i	0.433381	+	0.750639i	0.997162	−	0.0752860i	$-0.0239870\pi$
$587$	10.5000	+	18.1865i	0.433381	+	0.750639i	−0.563781	+	0.825925i	$0.690654\pi$
$588$		0			0
$589$	9.00000	−	15.5885i	0.370839	−	0.642311i
$590$		0			0
$591$		0			0
$592$	17.5000	−	30.3109i	0.719246	−	1.24577i
$593$	19.5000	−	33.7750i	0.800769	−	1.38697i	−0.118342	−	0.992973i	$-0.537758\pi$
$593$	19.5000	−	33.7750i	0.800769	−	1.38697i	0.919111	−	0.394000i	$-0.128909\pi$
$594$		0			0
$595$	4.50000	+	23.3827i	0.184482	+	0.958597i
$596$	1.50000	−	0.866025i	0.0614424	−	0.0354738i
$597$		0			0
$598$			15.5885i			0.637459i
$599$			24.2487i			0.990775i	0.868672	+	0.495388i	$0.164974\pi$
$599$			24.2487i			0.990775i	−0.868672	+	0.495388i	$0.835026\pi$
$600$		0			0
$601$	25.5000	−	14.7224i	1.04017	−	0.600541i	0.120286	−	0.992739i	$-0.461619\pi$
$601$	25.5000	−	14.7224i	1.04017	−	0.600541i	0.919881	+	0.392199i	$0.128285\pi$
$602$	−1.50000	+	4.33013i	−0.0611354	+	0.176483i
$603$		0			0
$604$	−8.50000	+	14.7224i	−0.345860	+	0.599047i
$605$	12.0000	−	20.7846i	0.487869	−	0.845015i
$606$		0			0
$607$	−13.5000	+	7.79423i	−0.547948	+	0.316358i	−0.748294	−	0.663367i	$-0.769127\pi$
$607$	−13.5000	+	7.79423i	−0.547948	+	0.316358i	0.200346	+	0.979725i	$0.435793\pi$
$608$	13.5000	−	23.3827i	0.547497	−	0.948293i
$609$		0			0
$610$	36.0000	+	62.3538i	1.45760	+	2.52463i
$611$		0			0
$612$		0			0
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$	42.0000			1.69498
$615$		0			0
$616$	1.50000	+	7.79423i	0.0604367	+	0.314038i
$617$	4.50000	+	2.59808i	0.181163	+	0.104595i	0.587839	−	0.808978i	$-0.299979\pi$
$617$	4.50000	+	2.59808i	0.181163	+	0.104595i	−0.406676	+	0.913573i	$0.633312\pi$
$618$		0			0
$619$	−16.5000	−	9.52628i	−0.663191	−	0.382893i	0.130301	−	0.991475i	$-0.458406\pi$
$619$	−16.5000	−	9.52628i	−0.663191	−	0.382893i	−0.793492	+	0.608581i	$0.791739\pi$
$620$	9.00000	−	5.19615i	0.361449	−	0.208683i
$621$		0			0
$622$		−	41.5692i		−	1.66677i
$623$	−1.50000	−	7.79423i	−0.0600962	−	0.312269i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$	−36.0000			−1.43885
$627$		0			0
$628$		0			0
$629$	−21.0000			−0.837325
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		−	13.8564i		−	0.551178i
$633$		0			0
$634$		0			0
$635$	30.0000	+	51.9615i	1.19051	+	2.06203i
$636$		0			0
$637$	7.50000	−	9.52628i	0.297161	−	0.377445i
$638$			15.5885i			0.617153i
$639$		0			0
$640$	−31.5000	+	18.1865i	−1.24515	+	0.718886i
$641$	10.5000	+	6.06218i	0.414725	+	0.239442i	0.692818	−	0.721113i	$-0.256369\pi$
$641$	10.5000	+	6.06218i	0.414725	+	0.239442i	−0.278093	+	0.960554i	$0.589702\pi$
$642$		0			0
$643$	−10.5000	−	6.06218i	−0.414080	−	0.239069i	0.278462	−	0.960447i	$-0.410176\pi$
$643$	−10.5000	−	6.06218i	−0.414080	−	0.239069i	−0.692541	+	0.721378i	$0.743509\pi$
$644$	4.50000	−	12.9904i	0.177325	−	0.511893i
$645$		0			0
$646$	−27.0000			−1.06230
$647$	−1.50000	−	2.59808i	−0.0589711	−	0.102141i	0.835033	−	0.550200i	$-0.185449\pi$
$647$	−1.50000	−	2.59808i	−0.0589711	−	0.102141i	−0.894004	+	0.448059i	$0.852115\pi$
$648$		0			0
$649$		0			0
$650$	6.00000	+	10.3923i	0.235339	+	0.407620i
$651$		0			0
$652$	5.50000	−	9.52628i	0.215397	−	0.373078i
$653$	34.5000	−	19.9186i	1.35009	−	0.779474i	0.361828	−	0.932245i	$-0.382153\pi$
$653$	34.5000	−	19.9186i	1.35009	−	0.779474i	0.988262	+	0.152771i	$0.0488196\pi$
$654$		0			0
$655$	−13.5000	+	23.3827i	−0.527489	+	0.913637i
$656$	−7.50000	+	12.9904i	−0.292826	+	0.507189i
$657$		0			0
$658$		0			0
$659$	−10.5000	+	6.06218i	−0.409022	+	0.236149i	−0.690369	−	0.723457i	$-0.742552\pi$
$659$	−10.5000	+	6.06218i	−0.409022	+	0.236149i	0.281347	+	0.959606i	$0.409219\pi$
$660$		0			0
$661$		−	41.5692i		−	1.61686i	−0.588596	−	0.808428i	$-0.700319\pi$
$661$		−	41.5692i		−	1.61686i	0.588596	−	0.808428i	$-0.299681\pi$
$662$			13.8564i			0.538545i
$663$		0			0
$664$	−22.5000	+	12.9904i	−0.873169	+	0.504125i
$665$	−27.0000	−	31.1769i	−1.04702	−	1.20899i
$666$		0			0
$667$	−13.5000	+	23.3827i	−0.522722	+	0.905381i
$668$	4.50000	−	7.79423i	0.174110	−	0.301568i
$669$		0			0
$670$	−18.0000	+	10.3923i	−0.695401	+	0.401490i
$671$	12.0000	−	20.7846i	0.463255	−	0.802381i
$672$		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	$0.644544\pi$
$674$	28.5000	+	16.4545i	1.09778	+	0.633803i
$675$		0			0
$676$	5.00000	+	8.66025i	0.192308	+	0.333087i
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	1.50000	−	4.33013i	0.0575647	−	0.166175i
$680$	13.5000	+	7.79423i	0.517701	+	0.298895i
$681$		0			0
$682$	−9.00000	−	5.19615i	−0.344628	−	0.198971i
$683$	7.50000	−	4.33013i	0.286980	−	0.165688i	−0.349599	−	0.936899i	$-0.613682\pi$
$683$	7.50000	−	4.33013i	0.286980	−	0.165688i	0.636579	+	0.771212i	$0.280349\pi$
$684$		0			0
$685$			36.3731i			1.38974i
$686$	−27.0000	+	17.3205i	−1.03086	+	0.661300i
$687$		0			0
$688$	2.50000	+	4.33013i	0.0953116	+	0.165085i
$689$	−15.0000			−0.571454
$690$		0			0
$691$		−	3.46410i		−	0.131781i	−0.997827	−	0.0658903i	$-0.979011\pi$
$691$		−	3.46410i		−	0.131781i	0.997827	−	0.0658903i	$-0.0209887\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$	6.00000			0.227757
$695$			25.9808i			0.985506i
$696$		0			0
$697$	9.00000			0.340899
$698$	10.5000	+	18.1865i	0.397431	+	0.688370i
$699$		0			0
$700$	−2.00000	−	10.3923i	−0.0755929	−	0.392792i
$701$			34.6410i			1.30837i	0.756333	+	0.654187i	$0.226989\pi$
$701$			34.6410i			1.30837i	−0.756333	+	0.654187i	$0.773011\pi$
$702$		0			0
$703$	31.5000	−	18.1865i	1.18805	−	0.685918i
$704$	1.50000	+	0.866025i	0.0565334	+	0.0326396i
$705$		0			0
$706$	31.5000	+	18.1865i	1.18552	+	0.684459i
$707$	−1.50000	−	7.79423i	−0.0564133	−	0.293132i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$	9.00000	+	15.5885i	0.337764	+	0.585024i
$711$		0			0
$712$	−4.50000	−	2.59808i	−0.168645	−	0.0973670i
$713$	−9.00000	−	15.5885i	−0.337053	−	0.583792i
$714$		0			0
$715$	4.50000	−	7.79423i	0.168290	−	0.291488i
$716$	−13.5000	+	7.79423i	−0.504519	+	0.291284i
$717$		0			0
$718$	−19.5000	+	33.7750i	−0.727734	+	1.26047i
$719$	−4.50000	+	7.79423i	−0.167822	+	0.290676i	−0.937654	−	0.347571i	$-0.887007\pi$
$719$	−4.50000	+	7.79423i	−0.167822	+	0.290676i	0.769832	+	0.638247i	$0.220340\pi$
$720$		0			0
$721$	10.5000	−	30.3109i	0.391040	−	1.12884i
$722$	12.0000	−	6.92820i	0.446594	−	0.257841i
$723$		0			0
$724$		0			0
$725$			20.7846i			0.771921i
$726$		0			0
$727$	10.5000	−	6.06218i	0.389423	−	0.224834i	−0.292487	−	0.956270i	$-0.594483\pi$
$727$	10.5000	−	6.06218i	0.389423	−	0.224834i	0.681910	+	0.731436i	$0.261149\pi$
$728$	−1.50000	−	7.79423i	−0.0555937	−	0.288873i
$729$		0			0
$730$	−13.5000	+	23.3827i	−0.499657	+	0.865432i
$731$	1.50000	−	2.59808i	0.0554795	−	0.0960933i
$732$		0			0
$733$	37.5000	−	21.6506i	1.38509	−	0.799684i	0.392337	−	0.919822i	$-0.371667\pi$
$733$	37.5000	−	21.6506i	1.38509	−	0.799684i	0.992757	+	0.120137i	$0.0383334\pi$
$734$	−4.50000	+	7.79423i	−0.166098	+	0.287690i
$735$		0			0
$736$	−13.5000	−	23.3827i	−0.497617	−	0.861897i
$737$	6.00000	+	3.46410i	0.221013	+	0.127602i
$738$		0			0
$739$	3.50000	+	6.06218i	0.128750	+	0.223001i	0.923192	−	0.384338i	$-0.125570\pi$
$739$	3.50000	+	6.06218i	0.128750	+	0.223001i	−0.794443	+	0.607339i	$0.792237\pi$
$740$	21.0000			0.771975
$741$		0			0
$742$	37.5000	+	12.9904i	1.37667	+	0.476892i
$743$	10.5000	+	6.06218i	0.385208	+	0.222400i	0.680082	−	0.733136i	$-0.261944\pi$
$743$	10.5000	+	6.06218i	0.385208	+	0.222400i	−0.294874	+	0.955536i	$0.595278\pi$
$744$		0			0
$745$	−4.50000	−	2.59808i	−0.164867	−	0.0951861i
$746$	55.5000	−	32.0429i	2.03200	−	1.17318i
$747$		0			0
$748$			5.19615i			0.189990i
$749$	−22.5000	+	4.33013i	−0.822132	+	0.158219i
$750$		0			0
$751$	−18.5000	−	32.0429i	−0.675075	−	1.16926i	−0.976447	−	0.215757i	$-0.930778\pi$
$751$	−18.5000	−	32.0429i	−0.675075	−	1.16926i	0.301373	−	0.953506i	$-0.402555\pi$
$752$		0			0
$753$		0			0
$754$		−	15.5885i		−	0.567698i
$755$	51.0000			1.85608
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$			34.6410i			1.25822i
$759$		0			0
$760$	−27.0000			−0.979393
$761$	−22.5000	−	38.9711i	−0.815624	−	1.41270i	−0.908879	−	0.417061i	$-0.863060\pi$
$761$	−22.5000	−	38.9711i	−0.815624	−	1.41270i	0.0932544	−	0.995642i	$-0.470273\pi$
$762$		0			0
$763$	−47.5000	−	16.4545i	−1.71962	−	0.595692i
$764$			17.3205i			0.626634i
$765$		0			0
$766$	13.5000	−	7.79423i	0.487775	−	0.281617i
$767$		0			0
$768$		0			0
$769$	13.5000	+	7.79423i	0.486822	+	0.281067i	0.723255	−	0.690581i	$-0.242645\pi$
$769$	13.5000	+	7.79423i	0.486822	+	0.281067i	−0.236433	+	0.971648i	$0.575978\pi$
$770$	−18.0000	+	15.5885i	−0.648675	+	0.561769i
$771$		0			0
$772$	2.00000			0.0719816
$773$	25.5000	+	44.1673i	0.917171	+	1.58859i	0.803691	+	0.595047i	$0.202867\pi$
$773$	25.5000	+	44.1673i	0.917171	+	1.58859i	0.113480	+	0.993540i	$0.463800\pi$
$774$		0			0
$775$	−12.0000	−	6.92820i	−0.431053	−	0.248868i
$776$	−1.50000	−	2.59808i	−0.0538469	−	0.0932655i
$777$		0			0
$778$	−31.5000	+	54.5596i	−1.12933	+	1.95606i
$779$	−13.5000	+	7.79423i	−0.483688	+	0.279257i
$780$		0			0
$781$	3.00000	−	5.19615i	0.107348	−	0.185933i
$782$	−13.5000	+	23.3827i	−0.482759	+	0.836163i
$783$		0			0
$784$	−5.00000	+	34.6410i	−0.178571	+	1.23718i
$785$		0			0
$786$		0			0
$787$			38.1051i			1.35830i	0.733999	+	0.679150i	$0.237652\pi$
$787$			38.1051i			1.35830i	−0.733999	+	0.679150i	$0.762348\pi$
$788$			13.8564i			0.493614i
$789$		0			0
$790$	36.0000	−	20.7846i	1.28082	−	0.739483i
$791$	−4.50000	+	0.866025i	−0.160002	+	0.0307923i
$792$		0			0
$793$	−12.0000	+	20.7846i	−0.426132	+	0.738083i
$794$	7.50000	−	12.9904i	0.266165	−	0.461011i
$795$		0			0
$796$	7.50000	−	4.33013i	0.265830	−	0.153477i
$797$	−22.5000	+	38.9711i	−0.796991	+	1.38043i	0.124576	+	0.992210i	$0.460243\pi$
$797$	−22.5000	+	38.9711i	−0.796991	+	1.38043i	−0.921567	+	0.388219i	$0.873091\pi$
$798$		0			0
$799$		0			0
$800$	−18.0000	−	10.3923i	−0.636396	−	0.367423i
$801$		0			0
$802$	−28.5000	−	49.3634i	−1.00637	−	1.74308i
$803$	9.00000			0.317603
$804$		0			0
$805$	−40.5000	+	7.79423i	−1.42744	+	0.274710i
$806$	9.00000	+	5.19615i	0.317011	+	0.183027i
$807$		0			0
$808$	−4.50000	−	2.59808i	−0.158309	−	0.0914000i
$809$	−1.50000	+	0.866025i	−0.0527372	+	0.0304478i	−0.526137	−	0.850400i	$-0.676360\pi$
$809$	−1.50000	+	0.866025i	−0.0527372	+	0.0304478i	0.473400	+	0.880848i	$0.343027\pi$
$810$		0			0
$811$			10.3923i			0.364923i	0.983213	+	0.182462i	$0.0584065\pi$
$811$			10.3923i			0.364923i	−0.983213	+	0.182462i	$0.941593\pi$
$812$	−4.50000	+	12.9904i	−0.157919	+	0.455873i
$813$		0			0
$814$	−10.5000	−	18.1865i	−0.368025	−	0.637438i
$815$	−33.0000			−1.15594
$816$		0			0
$817$			5.19615i			0.181790i
$818$	12.0000			0.419570
$819$		0			0
$820$	−9.00000			−0.314294
$821$			6.92820i			0.241796i	0.992665	+	0.120898i	$0.0385774\pi$
$821$			6.92820i			0.241796i	−0.992665	+	0.120898i	$0.961423\pi$
$822$		0			0
$823$	−16.0000			−0.557725			−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000			−0.557725			−0.278862	+	0.960331i	$0.589957\pi$
$824$	−10.5000	−	18.1865i	−0.365785	−	0.633558i
$825$		0			0
$826$		0			0
$827$		−	24.2487i		−	0.843210i	−0.906780	−	0.421605i	$-0.861467\pi$
$827$		−	24.2487i		−	0.843210i	0.906780	−	0.421605i	$-0.138533\pi$
$828$		0			0
$829$	31.5000	−	18.1865i	1.09404	−	0.631644i	0.159391	−	0.987216i	$-0.449047\pi$
$829$	31.5000	−	18.1865i	1.09404	−	0.631644i	0.934649	+	0.355571i	$0.115714\pi$
$830$	−67.5000	−	38.9711i	−2.34296	−	1.35271i
$831$		0			0
$832$	−1.50000	−	0.866025i	−0.0520031	−	0.0300240i
$833$	19.5000	−	7.79423i	0.675635	−	0.270054i
$834$		0			0
$835$	−27.0000			−0.934374
$836$	−4.50000	−	7.79423i	−0.155636	−	0.269569i
$837$		0			0
$838$	49.5000	+	28.5788i	1.70995	+	0.987240i
$839$	−19.5000	−	33.7750i	−0.673215	−	1.16604i	−0.976987	−	0.213298i	$-0.931580\pi$
$839$	−19.5000	−	33.7750i	−0.673215	−	1.16604i	0.303773	−	0.952745i	$-0.401754\pi$
$840$		0			0
$841$	−1.00000	+	1.73205i	−0.0344828	+	0.0597259i
$842$	−16.5000	+	9.52628i	−0.568628	+	0.328297i
$843$		0			0
$844$	−2.50000	+	4.33013i	−0.0860535	+	0.149049i
$845$	15.0000	−	25.9808i	0.516016	−	0.893765i
$846$		0			0
$847$	−20.0000	−	6.92820i	−0.687208	−	0.238056i
$848$	37.5000	−	21.6506i	1.28776	−	0.743486i
$849$		0			0
$850$			20.7846i			0.712906i
$851$		−	36.3731i		−	1.24685i
$852$		0			0
$853$	−22.5000	+	12.9904i	−0.770385	+	0.444782i	−0.833012	−	0.553255i	$-0.813386\pi$
$853$	−22.5000	+	12.9904i	−0.770385	+	0.444782i	0.0626267	+	0.998037i	$0.480052\pi$
$854$	48.0000	−	41.5692i	1.64253	−	1.42247i
$855$		0			0
$856$	−7.50000	+	12.9904i	−0.256345	+	0.444002i
$857$	13.5000	−	23.3827i	0.461151	−	0.798737i	−0.537867	−	0.843029i	$-0.680770\pi$
$857$	13.5000	−	23.3827i	0.461151	−	0.798737i	0.999019	+	0.0442921i	$0.0141032\pi$
$858$		0			0
$859$	−43.5000	+	25.1147i	−1.48420	+	0.856904i	−0.999839	−	0.0179638i	$-0.994282\pi$
$859$	−43.5000	+	25.1147i	−1.48420	+	0.856904i	−0.484362	+	0.874868i	$0.660948\pi$
$860$	−1.50000	+	2.59808i	−0.0511496	+	0.0885937i
$861$		0			0
$862$	−13.5000	−	23.3827i	−0.459812	−	0.796417i
$863$	−37.5000	−	21.6506i	−1.27651	−	0.736996i	−0.300309	−	0.953842i	$-0.597090\pi$
$863$	−37.5000	−	21.6506i	−1.27651	−	0.736996i	−0.976206	+	0.216846i	$0.930423\pi$
$864$		0			0
$865$	9.00000	+	15.5885i	0.306009	+	0.530023i
$866$	−24.0000			−0.815553
$867$		0			0
$868$	−6.00000	−	6.92820i	−0.203653	−	0.235159i
$869$	−12.0000	−	6.92820i	−0.407072	−	0.235023i
$870$		0			0
$871$	−6.00000	−	3.46410i	−0.203302	−	0.117377i
$872$	−28.5000	+	16.4545i	−0.965132	+	0.557219i
$873$		0			0
$874$		−	46.7654i		−	1.58186i
$875$	6.00000	−	5.19615i	0.202837	−	0.175662i
$876$		0			0
$877$	−11.5000	−	19.9186i	−0.388327	−	0.672603i	0.603897	−	0.797062i	$-0.293614\pi$
$877$	−11.5000	−	19.9186i	−0.388327	−	0.672603i	−0.992225	+	0.124459i	$0.960280\pi$
$878$	54.0000			1.82241
$879$		0			0
$880$			25.9808i			0.875811i
$881$	−54.0000			−1.81931			−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000			−1.81931			−0.909653	+	0.415369i	$0.863653\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		−	5.19615i		−	0.174766i
$885$		0			0
$886$	54.0000			1.81417
$887$	−7.50000	−	12.9904i	−0.251825	−	0.436174i	0.712203	−	0.701974i	$-0.247698\pi$
$887$	−7.50000	−	12.9904i	−0.251825	−	0.436174i	−0.964028	+	0.265799i	$0.914364\pi$
$888$		0			0
$889$	40.0000	−	34.6410i	1.34156	−	1.16182i
$890$		−	15.5885i		−	0.522526i
$891$		0			0
$892$	−4.50000	+	2.59808i	−0.150671	+	0.0869900i
$893$		0			0
$894$		0			0
$895$	40.5000	+	23.3827i	1.35377	+	0.781597i
$896$	21.0000	+	24.2487i	0.701561	+	0.810093i
$897$		0			0
$898$	60.0000			2.00223
$899$	9.00000	+	15.5885i	0.300167	+	0.519904i
$900$		0			0
$901$	−22.5000	−	12.9904i	−0.749584	−	0.432772i
$902$	4.50000	+	7.79423i	0.149834	+	0.259519i
$903$		0			0
$904$	−1.50000	+	2.59808i	−0.0498893	+	0.0864107i
$905$		0			0
$906$		0			0
$907$	−9.50000	+	16.4545i	−0.315442	+	0.546362i	−0.979531	−	0.201291i	$-0.935486\pi$
$907$	−9.50000	+	16.4545i	−0.315442	+	0.546362i	0.664089	+	0.747653i	$0.268820\pi$
$908$	10.5000	−	18.1865i	0.348455	−	0.603541i
$909$		0			0
$910$	18.0000	−	15.5885i	0.596694	−	0.516752i
$911$	−4.50000	+	2.59808i	−0.149092	+	0.0860781i	−0.572690	−	0.819772i	$-0.694100\pi$
$911$	−4.50000	+	2.59808i	−0.149092	+	0.0860781i	0.423598	+	0.905850i	$0.360767\pi$
$912$		0			0
$913$			25.9808i			0.859838i
$914$			45.0333i			1.48957i
$915$		0			0
$916$	−7.50000	+	4.33013i	−0.247807	+	0.143071i
$917$	22.5000	+	7.79423i	0.743015	+	0.257388i
$918$		0			0
$919$	14.5000	−	25.1147i	0.478311	−	0.828459i	−0.521380	−	0.853325i	$-0.674583\pi$
$919$	14.5000	−	25.1147i	0.478311	−	0.828459i	0.999691	+	0.0248659i	$0.00791589\pi$
$920$	−13.5000	+	23.3827i	−0.445082	+	0.770904i
$921$		0			0
$922$	22.5000	−	12.9904i	0.740998	−	0.427815i
$923$	−3.00000	+	5.19615i	−0.0987462	+	0.171033i
$924$		0			0
$925$	−14.0000	−	24.2487i	−0.460317	−	0.797293i
$926$	−1.50000	−	0.866025i	−0.0492931	−	0.0284594i
$927$		0			0
$928$	13.5000	+	23.3827i	0.443159	+	0.767574i
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$	−22.5000	+	28.5788i	−0.737408	+	0.936634i
$932$	−4.50000	−	2.59808i	−0.147402	−	0.0851028i
$933$		0			0
$934$	−4.50000	−	2.59808i	−0.147244	−	0.0850117i
$935$	13.5000	−	7.79423i	0.441497	−	0.254899i
$936$		0			0
$937$		−	13.8564i		−	0.452669i	−0.974050	−	0.226335i	$-0.927326\pi$
$937$		−	13.8564i		−	0.452669i	0.974050	−	0.226335i	$-0.0726743\pi$
$938$	12.0000	+	13.8564i	0.391814	+	0.452428i
$939$		0			0
$940$		0			0
$941$	−18.0000			−0.586783			−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000			−0.586783			−0.293392	+	0.955992i	$0.594784\pi$
$942$		0			0
$943$			15.5885i			0.507630i
$944$		0			0
$945$		0			0
$946$	3.00000			0.0975384
$947$		−	51.9615i		−	1.68852i	−0.535932	−	0.844261i	$-0.680040\pi$
$947$		−	51.9615i		−	1.68852i	0.535932	−	0.844261i	$-0.319960\pi$
$948$		0			0
$949$	−9.00000			−0.292152
$950$	−18.0000	−	31.1769i	−0.583997	−	1.01151i
$951$		0			0
$952$	4.50000	−	12.9904i	0.145846	−	0.421021i
$953$			20.7846i			0.673280i	0.941634	+	0.336640i	$0.109290\pi$
$953$			20.7846i			0.673280i	−0.941634	+	0.336640i	$0.890710\pi$
$954$		0			0
$955$	45.0000	−	25.9808i	1.45617	−	0.840718i
$956$	1.50000	+	0.866025i	0.0485135	+	0.0280093i
$957$		0			0
$958$	−40.5000	−	23.3827i	−1.30850	−	0.755460i
$959$	31.5000	−	6.06218i	1.01719	−	0.195758i
$960$		0			0
$961$	19.0000			0.612903
$962$	10.5000	+	18.1865i	0.338534	+	0.586357i
$963$		0			0
$964$	−19.5000	−	11.2583i	−0.628053	−	0.362606i
$965$	−3.00000	−	5.19615i	−0.0965734	−	0.167270i
$966$		0			0
$967$	12.5000	−	21.6506i	0.401973	−	0.696237i	−0.591991	−	0.805945i	$-0.701658\pi$
$967$	12.5000	−	21.6506i	0.401973	−	0.696237i	0.993964	+	0.109707i	$0.0349913\pi$
$968$	−12.0000	+	6.92820i	−0.385695	+	0.222681i
$969$		0			0
$970$	4.50000	−	7.79423i	0.144486	−	0.250258i
$971$	−28.5000	+	49.3634i	−0.914609	+	1.58415i	−0.107135	+	0.994244i	$0.534168\pi$
$971$	−28.5000	+	49.3634i	−0.914609	+	1.58415i	−0.807473	+	0.589904i	$0.799166\pi$
$972$		0			0
$973$	22.5000	−	4.33013i	0.721317	−	0.138817i
$974$	34.5000	−	19.9186i	1.10545	−	0.638233i
$975$		0			0
$976$		−	69.2820i		−	2.21766i
$977$			41.5692i			1.32992i	0.746880	+	0.664959i	$0.231551\pi$
$977$			41.5692i			1.32992i	−0.746880	+	0.664959i	$0.768449\pi$
$978$		0			0
$979$	−4.50000	+	2.59808i	−0.143821	+	0.0830349i
$980$	−19.5000	+	7.79423i	−0.622905	+	0.248978i
$981$		0			0
$982$	22.5000	−	38.9711i	0.718004	−	1.24362i
$983$	19.5000	−	33.7750i	0.621953	−	1.07725i	−0.367168	−	0.930155i	$-0.619673\pi$
$983$	19.5000	−	33.7750i	0.621953	−	1.07725i	0.989122	−	0.147100i	$-0.0469940\pi$
$984$		0			0
$985$	36.0000	−	20.7846i	1.14706	−	0.662253i
$986$	13.5000	−	23.3827i	0.429928	−	0.744656i
$987$		0			0
$988$	4.50000	+	7.79423i	0.143164	+	0.247967i
$989$	4.50000	+	2.59808i	0.143092	+	0.0826140i
$990$		0			0
$991$	23.5000	+	40.7032i	0.746502	+	1.29298i	0.949490	+	0.313798i	$0.101602\pi$
$991$	23.5000	+	40.7032i	0.746502	+	1.29298i	−0.202988	+	0.979181i	$0.565065\pi$
$992$	−18.0000			−0.571501
$993$		0			0
$994$	12.0000	−	10.3923i	0.380617	−	0.329624i
$995$	−22.5000	−	12.9904i	−0.713298	−	0.411823i
$996$		0			0
$997$	7.50000	+	4.33013i	0.237527	+	0.137136i	0.614040	−	0.789275i	$-0.289543\pi$
$997$	7.50000	+	4.33013i	0.237527	+	0.137136i	−0.376512	+	0.926412i	$0.622877\pi$
$998$	−37.5000	+	21.6506i	−1.18704	+	0.685339i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.i.a.152.1		2
3.2	odd	2		63.2.i.a.5.1	✓	2
4.3	odd	2		3024.2.ca.a.2609.1		2
7.2	even	3		1323.2.o.b.881.1		2
7.3	odd	6		189.2.s.a.17.1		2
7.4	even	3		1323.2.s.a.962.1		2
7.5	odd	6		1323.2.o.a.881.1		2
7.6	odd	2		1323.2.i.a.1097.1		2
9.2	odd	6		189.2.s.a.89.1		2
9.4	even	3		567.2.p.b.404.1		2
9.5	odd	6		567.2.p.a.404.1		2
9.7	even	3		63.2.s.a.47.1	yes	2
12.11	even	2		1008.2.ca.a.257.1		2
21.2	odd	6		441.2.o.a.293.1		2
21.5	even	6		441.2.o.b.293.1		2
21.11	odd	6		441.2.s.a.374.1		2
21.17	even	6		63.2.s.a.59.1	yes	2
21.20	even	2		441.2.i.a.68.1		2
28.3	even	6		3024.2.df.a.17.1		2
36.7	odd	6		1008.2.df.a.929.1		2
36.11	even	6		3024.2.df.a.1601.1		2
63.2	odd	6		1323.2.o.a.440.1		2
63.11	odd	6		1323.2.i.a.521.1		2
63.16	even	3		441.2.o.b.146.1		2
63.20	even	6		1323.2.s.a.656.1		2
63.25	even	3		441.2.i.a.227.1		2
63.31	odd	6		567.2.p.a.80.1		2
63.34	odd	6		441.2.s.a.362.1		2
63.38	even	6	inner	189.2.i.a.143.1		2
63.47	even	6		1323.2.o.b.440.1		2
63.52	odd	6		63.2.i.a.38.1	yes	2
63.59	even	6		567.2.p.b.80.1		2
63.61	odd	6		441.2.o.a.146.1		2
84.59	odd	6		1008.2.df.a.689.1		2
252.115	even	6		1008.2.ca.a.353.1		2
252.227	odd	6		3024.2.ca.a.2033.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.i.a.5.1	✓	2	3.2	odd	2
63.2.i.a.38.1	yes	2	63.52	odd	6
63.2.s.a.47.1	yes	2	9.7	even	3
63.2.s.a.59.1	yes	2	21.17	even	6
189.2.i.a.143.1		2	63.38	even	6	inner
189.2.i.a.152.1		2	1.1	even	1	trivial
189.2.s.a.17.1		2	7.3	odd	6
189.2.s.a.89.1		2	9.2	odd	6
441.2.i.a.68.1		2	21.20	even	2
441.2.i.a.227.1		2	63.25	even	3
441.2.o.a.146.1		2	63.61	odd	6
441.2.o.a.293.1		2	21.2	odd	6
441.2.o.b.146.1		2	63.16	even	3
441.2.o.b.293.1		2	21.5	even	6
441.2.s.a.362.1		2	63.34	odd	6
441.2.s.a.374.1		2	21.11	odd	6
567.2.p.a.80.1		2	63.31	odd	6
567.2.p.a.404.1		2	9.5	odd	6
567.2.p.b.80.1		2	63.59	even	6
567.2.p.b.404.1		2	9.4	even	3
1008.2.ca.a.257.1		2	12.11	even	2
1008.2.ca.a.353.1		2	252.115	even	6
1008.2.df.a.689.1		2	84.59	odd	6
1008.2.df.a.929.1		2	36.7	odd	6
1323.2.i.a.521.1		2	63.11	odd	6
1323.2.i.a.1097.1		2	7.6	odd	2
1323.2.o.a.440.1		2	63.2	odd	6
1323.2.o.a.881.1		2	7.5	odd	6
1323.2.o.b.440.1		2	63.47	even	6
1323.2.o.b.881.1		2	7.2	even	3
1323.2.s.a.656.1		2	63.20	even	6
1323.2.s.a.962.1		2	7.4	even	3
3024.2.ca.a.2033.1		2	252.227	odd	6
3024.2.ca.a.2609.1		2	4.3	odd	2
3024.2.df.a.17.1		2	28.3	even	6
3024.2.df.a.1601.1		2	36.11	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.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -1.00000 q^{4} +(1.50000 + 2.59808i) q^{5} +(2.00000 - 1.73205i) q^{7} -1.73205i q^{8} +O(q^{10})\)	\(q-1.73205i q^{2} -1.00000 q^{4} +(1.50000 + 2.59808i) q^{5} +(2.00000 - 1.73205i) q^{7} -1.73205i q^{8} +(4.50000 - 2.59808i) q^{10} +(-1.50000 - 0.866025i) q^{11} +(1.50000 + 0.866025i) q^{13} +(-3.00000 - 3.46410i) q^{14} -5.00000 q^{16} +(1.50000 + 2.59808i) q^{17} +(-4.50000 - 2.59808i) q^{19} +(-1.50000 - 2.59808i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(-4.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(1.50000 - 2.59808i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(4.50000 - 2.59808i) q^{29} +3.46410i q^{31} +5.19615i q^{32} +(4.50000 - 2.59808i) q^{34} +(7.50000 + 2.59808i) q^{35} +(-3.50000 + 6.06218i) q^{37} +(-4.50000 + 7.79423i) q^{38} +(4.50000 - 2.59808i) q^{40} +(1.50000 - 2.59808i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(1.50000 + 0.866025i) q^{44} +(4.50000 + 7.79423i) q^{46} +(1.00000 - 6.92820i) q^{49} +(6.00000 + 3.46410i) q^{50} +(-1.50000 - 0.866025i) q^{52} +(-7.50000 + 4.33013i) q^{53} -5.19615i q^{55} +(-3.00000 - 3.46410i) q^{56} +(-4.50000 - 7.79423i) q^{58} +13.8564i q^{61} +6.00000 q^{62} -1.00000 q^{64} +5.19615i q^{65} -4.00000 q^{67} +(-1.50000 - 2.59808i) q^{68} +(4.50000 - 12.9904i) q^{70} +3.46410i q^{71} +(-4.50000 + 2.59808i) q^{73} +(10.5000 + 6.06218i) q^{74} +(4.50000 + 2.59808i) q^{76} +(-4.50000 + 0.866025i) q^{77} +8.00000 q^{79} +(-7.50000 - 12.9904i) q^{80} +(-4.50000 - 2.59808i) q^{82} +(-7.50000 - 12.9904i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(-1.50000 + 0.866025i) q^{86} +(-1.50000 + 2.59808i) q^{88} +(1.50000 - 2.59808i) q^{89} +(4.50000 - 0.866025i) q^{91} +(4.50000 - 2.59808i) q^{92} -15.5885i q^{95} +(1.50000 - 0.866025i) q^{97} +(-12.0000 - 1.73205i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 3 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^4 + 3 * q^5 + 4 * q^7	\( 2 q - 2 q^{4} + 3 q^{5} + 4 q^{7} + 9 q^{10} - 3 q^{11} + 3 q^{13} - 6 q^{14} - 10 q^{16} + 3 q^{17} - 9 q^{19} - 3 q^{20} - 3 q^{22} - 9 q^{23} - 4 q^{25} + 3 q^{26} - 4 q^{28} + 9 q^{29} + 9 q^{34} + 15 q^{35} - 7 q^{37} - 9 q^{38} + 9 q^{40} + 3 q^{41} - q^{43} + 3 q^{44} + 9 q^{46} + 2 q^{49} + 12 q^{50} - 3 q^{52} - 15 q^{53} - 6 q^{56} - 9 q^{58} + 12 q^{62} - 2 q^{64} - 8 q^{67} - 3 q^{68} + 9 q^{70} - 9 q^{73} + 21 q^{74} + 9 q^{76} - 9 q^{77} + 16 q^{79} - 15 q^{80} - 9 q^{82} - 15 q^{83} - 9 q^{85} - 3 q^{86} - 3 q^{88} + 3 q^{89} + 9 q^{91} + 9 q^{92} + 3 q^{97} - 24 q^{98}+O(q^{100}) \) 2 * q - 2 * q^4 + 3 * q^5 + 4 * q^7 + 9 * q^10 - 3 * q^11 + 3 * q^13 - 6 * q^14 - 10 * q^16 + 3 * q^17 - 9 * q^19 - 3 * q^20 - 3 * q^22 - 9 * q^23 - 4 * q^25 + 3 * q^26 - 4 * q^28 + 9 * q^29 + 9 * q^34 + 15 * q^35 - 7 * q^37 - 9 * q^38 + 9 * q^40 + 3 * q^41 - q^43 + 3 * q^44 + 9 * q^46 + 2 * q^49 + 12 * q^50 - 3 * q^52 - 15 * q^53 - 6 * q^56 - 9 * q^58 + 12 * q^62 - 2 * q^64 - 8 * q^67 - 3 * q^68 + 9 * q^70 - 9 * q^73 + 21 * q^74 + 9 * q^76 - 9 * q^77 + 16 * q^79 - 15 * q^80 - 9 * q^82 - 15 * q^83 - 9 * q^85 - 3 * q^86 - 3 * q^88 + 3 * q^89 + 9 * q^91 + 9 * q^92 + 3 * q^97 - 24 * q^98

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