LMFDB - Embedded newform 342.2.g.a.163.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,2,Mod(163,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("342.163");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	342.g (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.73088374913$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	342.163
Dual form			342.2.g.a.235.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +3.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +3.00000 q^{7} +1.00000 q^{8} +(-1.00000 - 1.73205i) q^{10} +2.00000 q^{11} +(0.500000 + 0.866025i) q^{13} +(-1.50000 + 2.59808i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.00000 + 5.19615i) q^{17} +(-4.00000 + 1.73205i) q^{19} +2.00000 q^{20} +(-1.00000 + 1.73205i) q^{22} +(2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} -1.00000 q^{26} +(-1.50000 - 2.59808i) q^{28} +(1.00000 + 1.73205i) q^{29} +7.00000 q^{31} +(-0.500000 - 0.866025i) q^{32} +(-3.00000 - 5.19615i) q^{34} +(-3.00000 + 5.19615i) q^{35} +1.00000 q^{37} +(0.500000 - 4.33013i) q^{38} +(-1.00000 + 1.73205i) q^{40} +(4.00000 - 6.92820i) q^{41} +(-3.50000 + 6.06218i) q^{43} +(-1.00000 - 1.73205i) q^{44} -4.00000 q^{46} +(-4.00000 - 6.92820i) q^{47} +2.00000 q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{52} +(-4.00000 - 6.92820i) q^{53} +(-2.00000 + 3.46410i) q^{55} +3.00000 q^{56} -2.00000 q^{58} +(6.00000 - 10.3923i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(-3.50000 + 6.06218i) q^{62} +1.00000 q^{64} -2.00000 q^{65} +(4.50000 + 7.79423i) q^{67} +6.00000 q^{68} +(-3.00000 - 5.19615i) q^{70} +(-1.00000 + 1.73205i) q^{71} +(7.50000 - 12.9904i) q^{73} +(-0.500000 + 0.866025i) q^{74} +(3.50000 + 2.59808i) q^{76} +6.00000 q^{77} +(-5.50000 + 9.52628i) q^{79} +(-1.00000 - 1.73205i) q^{80} +(4.00000 + 6.92820i) q^{82} +6.00000 q^{83} +(-6.00000 - 10.3923i) q^{85} +(-3.50000 - 6.06218i) q^{86} +2.00000 q^{88} +(1.50000 + 2.59808i) q^{91} +(2.00000 - 3.46410i) q^{92} +8.00000 q^{94} +(1.00000 - 8.66025i) q^{95} +(7.00000 - 12.1244i) q^{97} +(-1.00000 + 1.73205i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 2 q^{5} + 6 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - 2 * q^5 + 6 * q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - 2 q^{5} + 6 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} + q^{13} - 3 q^{14} - q^{16} - 6 q^{17} - 8 q^{19} + 4 q^{20} - 2 q^{22} + 4 q^{23} + q^{25} - 2 q^{26} - 3 q^{28} + 2 q^{29} + 14 q^{31} - q^{32} - 6 q^{34} - 6 q^{35} + 2 q^{37} + q^{38} - 2 q^{40} + 8 q^{41} - 7 q^{43} - 2 q^{44} - 8 q^{46} - 8 q^{47} + 4 q^{49} - 2 q^{50} + q^{52} - 8 q^{53} - 4 q^{55} + 6 q^{56} - 4 q^{58} + 12 q^{59} - 5 q^{61} - 7 q^{62} + 2 q^{64} - 4 q^{65} + 9 q^{67} + 12 q^{68} - 6 q^{70} - 2 q^{71} + 15 q^{73} - q^{74} + 7 q^{76} + 12 q^{77} - 11 q^{79} - 2 q^{80} + 8 q^{82} + 12 q^{83} - 12 q^{85} - 7 q^{86} + 4 q^{88} + 3 q^{91} + 4 q^{92} + 16 q^{94} + 2 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 2 * q^5 + 6 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 + q^13 - 3 * q^14 - q^16 - 6 * q^17 - 8 * q^19 + 4 * q^20 - 2 * q^22 + 4 * q^23 + q^25 - 2 * q^26 - 3 * q^28 + 2 * q^29 + 14 * q^31 - q^32 - 6 * q^34 - 6 * q^35 + 2 * q^37 + q^38 - 2 * q^40 + 8 * q^41 - 7 * q^43 - 2 * q^44 - 8 * q^46 - 8 * q^47 + 4 * q^49 - 2 * q^50 + q^52 - 8 * q^53 - 4 * q^55 + 6 * q^56 - 4 * q^58 + 12 * q^59 - 5 * q^61 - 7 * q^62 + 2 * q^64 - 4 * q^65 + 9 * q^67 + 12 * q^68 - 6 * q^70 - 2 * q^71 + 15 * q^73 - q^74 + 7 * q^76 + 12 * q^77 - 11 * q^79 - 2 * q^80 + 8 * q^82 + 12 * q^83 - 12 * q^85 - 7 * q^86 + 4 * q^88 + 3 * q^91 + 4 * q^92 + 16 * q^94 + 2 * q^95 + 14 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	−1.00000	+	1.73205i	−0.447214	+	0.774597i	−0.998203	−	0.0599153i	$-0.980917\pi$
$5$	−1.00000	+	1.73205i	−0.447214	+	0.774597i	0.550990	+	0.834512i	$0.314250\pi$
$6$		0			0
$7$	3.00000			1.13389			0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000			1.13389			0.566947	+	0.823754i	$0.308125\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000	−	1.73205i	−0.316228	−	0.547723i
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$	−1.50000	+	2.59808i	−0.400892	+	0.694365i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−3.00000	+	5.19615i	−0.727607	+	1.26025i	0.230285	+	0.973123i	$0.426034\pi$
$17$	−3.00000	+	5.19615i	−0.727607	+	1.26025i	−0.957892	+	0.287129i	$0.907299\pi$
$18$		0			0
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$20$	2.00000			0.447214
$21$		0			0
$22$	−1.00000	+	1.73205i	−0.213201	+	0.369274i
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$	−1.00000			−0.196116
$27$		0			0
$28$	−1.50000	−	2.59808i	−0.283473	−	0.490990i
$29$	1.00000	+	1.73205i	0.185695	+	0.321634i	0.943811	−	0.330487i	$-0.107213\pi$
$29$	1.00000	+	1.73205i	0.185695	+	0.321634i	−0.758115	+	0.652121i	$0.773880\pi$
$30$		0			0
$31$	7.00000			1.25724			0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000			1.25724			0.628619	+	0.777714i	$0.283621\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−3.00000	−	5.19615i	−0.514496	−	0.891133i
$35$	−3.00000	+	5.19615i	−0.507093	+	0.878310i
$36$		0			0
$37$	1.00000			0.164399			0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000			0.164399			0.0821995	+	0.996616i	$0.473806\pi$
$38$	0.500000	−	4.33013i	0.0811107	−	0.702439i
$39$		0			0
$40$	−1.00000	+	1.73205i	−0.158114	+	0.273861i
$41$	4.00000	−	6.92820i	0.624695	−	1.08200i	−0.363905	−	0.931436i	$-0.618557\pi$
$41$	4.00000	−	6.92820i	0.624695	−	1.08200i	0.988600	−	0.150567i	$-0.0481100\pi$
$42$		0			0
$43$	−3.50000	+	6.06218i	−0.533745	+	0.924473i	0.465478	+	0.885059i	$0.345882\pi$
$43$	−3.50000	+	6.06218i	−0.533745	+	0.924473i	−0.999223	+	0.0394140i	$0.987451\pi$
$44$	−1.00000	−	1.73205i	−0.150756	−	0.261116i
$45$		0			0
$46$	−4.00000			−0.589768
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	−0.995066	−	0.0992202i	$-0.968365\pi$
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	0.411606	−	0.911362i	$-0.364968\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$	−1.00000			−0.141421
$51$		0			0
$52$	0.500000	−	0.866025i	0.0693375	−	0.120096i
$53$	−4.00000	−	6.92820i	−0.549442	−	0.951662i	−0.998313	−	0.0580651i	$-0.981507\pi$
$53$	−4.00000	−	6.92820i	−0.549442	−	0.951662i	0.448871	−	0.893597i	$-0.351826\pi$
$54$		0			0
$55$	−2.00000	+	3.46410i	−0.269680	+	0.467099i
$56$	3.00000			0.400892
$57$		0			0
$58$	−2.00000			−0.262613
$59$	6.00000	−	10.3923i	0.781133	−	1.35296i	−0.150148	−	0.988663i	$-0.547975\pi$
$59$	6.00000	−	10.3923i	0.781133	−	1.35296i	0.931282	−	0.364299i	$-0.118692\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$	−3.50000	+	6.06218i	−0.444500	+	0.769897i
$63$		0			0
$64$	1.00000			0.125000
$65$	−2.00000			−0.248069
$66$		0			0
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	0.998290	+	0.0584478i	$0.0186151\pi$
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	−0.448528	+	0.893769i	$0.648052\pi$
$68$	6.00000			0.727607
$69$		0			0
$70$	−3.00000	−	5.19615i	−0.358569	−	0.621059i
$71$	−1.00000	+	1.73205i	−0.118678	+	0.205557i	−0.919244	−	0.393688i	$-0.871199\pi$
$71$	−1.00000	+	1.73205i	−0.118678	+	0.205557i	0.800566	+	0.599245i	$0.204532\pi$
$72$		0			0
$73$	7.50000	−	12.9904i	0.877809	−	1.52041i	0.0240681	−	0.999710i	$-0.492338\pi$
$73$	7.50000	−	12.9904i	0.877809	−	1.52041i	0.853740	−	0.520699i	$-0.174329\pi$
$74$	−0.500000	+	0.866025i	−0.0581238	+	0.100673i
$75$		0			0
$76$	3.50000	+	2.59808i	0.401478	+	0.298020i
$77$	6.00000			0.683763
$78$		0			0
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	0.370907	+	0.928670i	$0.379047\pi$
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	−0.989705	+	0.143120i	$0.954286\pi$
$80$	−1.00000	−	1.73205i	−0.111803	−	0.193649i
$81$		0			0
$82$	4.00000	+	6.92820i	0.441726	+	0.765092i
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	−6.00000	−	10.3923i	−0.650791	−	1.12720i
$86$	−3.50000	−	6.06218i	−0.377415	−	0.653701i
$87$		0			0
$88$	2.00000			0.213201
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	1.50000	+	2.59808i	0.157243	+	0.272352i
$92$	2.00000	−	3.46410i	0.208514	−	0.361158i
$93$		0			0
$94$	8.00000			0.825137
$95$	1.00000	−	8.66025i	0.102598	−	0.888523i
$96$		0			0
$97$	7.00000	−	12.1244i	0.710742	−	1.23104i	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	7.00000	−	12.1244i	0.710742	−	1.23104i	0.964579	−	0.263795i	$-0.0849741\pi$
$98$	−1.00000	+	1.73205i	−0.101015	+	0.174964i
$99$		0			0
$100$	0.500000	−	0.866025i	0.0500000	−	0.0866025i
$101$	−7.00000	−	12.1244i	−0.696526	−	1.20642i	−0.969664	−	0.244443i	$-0.921395\pi$
$101$	−7.00000	−	12.1244i	−0.696526	−	1.20642i	0.273138	−	0.961975i	$-0.411939\pi$
$102$		0			0
$103$	3.00000			0.295599			0.147799	−	0.989017i	$-0.452781\pi$
$103$	3.00000			0.295599			0.147799	+	0.989017i	$0.452781\pi$
$104$	0.500000	+	0.866025i	0.0490290	+	0.0849208i
$105$		0			0
$106$	8.00000			0.777029
$107$	−10.0000			−0.966736			−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000			−0.966736			−0.483368	+	0.875417i	$0.660587\pi$
$108$		0			0
$109$	3.00000	−	5.19615i	0.287348	−	0.497701i	−0.685828	−	0.727764i	$-0.740560\pi$
$109$	3.00000	−	5.19615i	0.287348	−	0.497701i	0.973176	+	0.230063i	$0.0738931\pi$
$110$	−2.00000	−	3.46410i	−0.190693	−	0.330289i
$111$		0			0
$112$	−1.50000	+	2.59808i	−0.141737	+	0.245495i
$113$	20.0000			1.88144			0.940721	−	0.339182i	$-0.110150\pi$
$113$	20.0000			1.88144			0.940721	+	0.339182i	$0.110150\pi$
$114$		0			0
$115$	−8.00000			−0.746004
$116$	1.00000	−	1.73205i	0.0928477	−	0.160817i
$117$		0			0
$118$	6.00000	+	10.3923i	0.552345	+	0.956689i
$119$	−9.00000	+	15.5885i	−0.825029	+	1.42899i
$120$		0			0
$121$	−7.00000			−0.636364
$122$	5.00000			0.452679
$123$		0			0
$124$	−3.50000	−	6.06218i	−0.314309	−	0.544400i
$125$	−12.0000			−1.07331
$126$		0			0
$127$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$127$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	1.00000	−	1.73205i	0.0877058	−	0.151911i
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	−0.704692	−	0.709514i	$-0.748915\pi$
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	0.966803	+	0.255524i	$0.0822479\pi$
$132$		0			0
$133$	−12.0000	+	5.19615i	−1.04053	+	0.450564i
$134$	−9.00000			−0.777482
$135$		0			0
$136$	−3.00000	+	5.19615i	−0.257248	+	0.445566i
$137$	7.00000	+	12.1244i	0.598050	+	1.03585i	0.993109	+	0.117198i	$0.0373911\pi$
$137$	7.00000	+	12.1244i	0.598050	+	1.03585i	−0.395058	+	0.918656i	$0.629276\pi$
$138$		0			0
$139$	1.50000	+	2.59808i	0.127228	+	0.220366i	0.922602	−	0.385754i	$-0.126059\pi$
$139$	1.50000	+	2.59808i	0.127228	+	0.220366i	−0.795373	+	0.606120i	$0.792725\pi$
$140$	6.00000			0.507093
$141$		0			0
$142$	−1.00000	−	1.73205i	−0.0839181	−	0.145350i
$143$	1.00000	+	1.73205i	0.0836242	+	0.144841i
$144$		0			0
$145$	−4.00000			−0.332182
$146$	7.50000	+	12.9904i	0.620704	+	1.07509i
$147$		0			0
$148$	−0.500000	−	0.866025i	−0.0410997	−	0.0711868i
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	0.216394	+	0.976306i	$0.430570\pi$
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	−0.953703	+	0.300750i	$0.902763\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$	−4.00000	+	1.73205i	−0.324443	+	0.140488i
$153$		0			0
$154$	−3.00000	+	5.19615i	−0.241747	+	0.418718i
$155$	−7.00000	+	12.1244i	−0.562254	+	0.973852i
$156$		0			0
$157$	−6.50000	+	11.2583i	−0.518756	+	0.898513i	0.481006	+	0.876717i	$0.340272\pi$
$157$	−6.50000	+	11.2583i	−0.518756	+	0.898513i	−0.999762	+	0.0217953i	$0.993062\pi$
$158$	−5.50000	−	9.52628i	−0.437557	−	0.757870i
$159$		0			0
$160$	2.00000			0.158114
$161$	6.00000	+	10.3923i	0.472866	+	0.819028i
$162$		0			0
$163$	−13.0000			−1.01824			−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000			−1.01824			−0.509119	+	0.860696i	$0.670029\pi$
$164$	−8.00000			−0.624695
$165$		0			0
$166$	−3.00000	+	5.19615i	−0.232845	+	0.403300i
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	0.999169	−	0.0407502i	$-0.0129748\pi$
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	−0.534875	+	0.844931i	$0.679641\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$	12.0000			0.920358
$171$		0			0
$172$	7.00000			0.533745
$173$	9.00000	−	15.5885i	0.684257	−	1.18517i	−0.289412	−	0.957205i	$-0.593460\pi$
$173$	9.00000	−	15.5885i	0.684257	−	1.18517i	0.973670	−	0.227964i	$-0.0732068\pi$
$174$		0			0
$175$	1.50000	+	2.59808i	0.113389	+	0.196396i
$176$	−1.00000	+	1.73205i	−0.0753778	+	0.130558i
$177$		0			0
$178$		0			0
$179$	−24.0000			−1.79384			−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$	−11.0000	−	19.0526i	−0.817624	−	1.41617i	−0.907429	−	0.420206i	$-0.861958\pi$
$181$	−11.0000	−	19.0526i	−0.817624	−	1.41617i	0.0898051	−	0.995959i	$-0.471376\pi$
$182$	−3.00000			−0.222375
$183$		0			0
$184$	2.00000	+	3.46410i	0.147442	+	0.255377i
$185$	−1.00000	+	1.73205i	−0.0735215	+	0.127343i
$186$		0			0
$187$	−6.00000	+	10.3923i	−0.438763	+	0.759961i
$188$	−4.00000	+	6.92820i	−0.291730	+	0.505291i
$189$		0			0
$190$	7.00000	+	5.19615i	0.507833	+	0.376969i
$191$	22.0000			1.59186			0.795932	−	0.605386i	$-0.206981\pi$
$191$	22.0000			1.59186			0.795932	+	0.605386i	$0.206981\pi$
$192$		0			0
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	−0.657376	−	0.753563i	$-0.728333\pi$
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	0.981293	+	0.192522i	$0.0616668\pi$
$194$	7.00000	+	12.1244i	0.502571	+	0.870478i
$195$		0			0
$196$	−1.00000	−	1.73205i	−0.0714286	−	0.123718i
$197$	8.00000			0.569976			0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000			0.569976			0.284988	+	0.958531i	$0.408010\pi$
$198$		0			0
$199$	−5.50000	−	9.52628i	−0.389885	−	0.675300i	0.602549	−	0.798082i	$-0.294152\pi$
$199$	−5.50000	−	9.52628i	−0.389885	−	0.675300i	−0.992434	+	0.122782i	$0.960818\pi$
$200$	0.500000	+	0.866025i	0.0353553	+	0.0612372i
$201$		0			0
$202$	14.0000			0.985037
$203$	3.00000	+	5.19615i	0.210559	+	0.364698i
$204$		0			0
$205$	8.00000	+	13.8564i	0.558744	+	0.967773i
$206$	−1.50000	+	2.59808i	−0.104510	+	0.181017i
$207$		0			0
$208$	−1.00000			−0.0693375
$209$	−8.00000	+	3.46410i	−0.553372	+	0.239617i
$210$		0			0
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	−0.483499	−	0.875345i	$-0.660634\pi$
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	0.999820	−	0.0189499i	$-0.00603229\pi$
$212$	−4.00000	+	6.92820i	−0.274721	+	0.475831i
$213$		0			0
$214$	5.00000	−	8.66025i	0.341793	−	0.592003i
$215$	−7.00000	−	12.1244i	−0.477396	−	0.826874i
$216$		0			0
$217$	21.0000			1.42557
$218$	3.00000	+	5.19615i	0.203186	+	0.351928i
$219$		0			0
$220$	4.00000			0.269680
$221$	−6.00000			−0.403604
$222$		0			0
$223$	−5.50000	+	9.52628i	−0.368307	+	0.637927i	−0.989301	−	0.145889i	$-0.953396\pi$
$223$	−5.50000	+	9.52628i	−0.368307	+	0.637927i	0.620994	+	0.783815i	$0.286729\pi$
$224$	−1.50000	−	2.59808i	−0.100223	−	0.173591i
$225$		0			0
$226$	−10.0000	+	17.3205i	−0.665190	+	1.15214i
$227$	22.0000			1.46019			0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000			1.46019			0.730096	+	0.683345i	$0.239475\pi$
$228$		0			0
$229$	−25.0000			−1.65205			−0.826023	−	0.563636i	$-0.809402\pi$
$229$	−25.0000			−1.65205			−0.826023	+	0.563636i	$0.809402\pi$
$230$	4.00000	−	6.92820i	0.263752	−	0.456832i
$231$		0			0
$232$	1.00000	+	1.73205i	0.0656532	+	0.113715i
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	0.404674	+	0.914461i	$0.367385\pi$
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	−0.994283	+	0.106773i	$0.965948\pi$
$234$		0			0
$235$	16.0000			1.04372
$236$	−12.0000			−0.781133
$237$		0			0
$238$	−9.00000	−	15.5885i	−0.583383	−	1.01045i
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	0.849472	−	0.527633i	$-0.176921\pi$
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	−0.881680	+	0.471848i	$0.843587\pi$
$242$	3.50000	−	6.06218i	0.224989	−	0.389692i
$243$		0			0
$244$	−2.50000	+	4.33013i	−0.160046	+	0.277208i
$245$	−2.00000	+	3.46410i	−0.127775	+	0.221313i
$246$		0			0
$247$	−3.50000	−	2.59808i	−0.222700	−	0.165312i
$248$	7.00000			0.444500
$249$		0			0
$250$	6.00000	−	10.3923i	0.379473	−	0.657267i
$251$	8.00000	+	13.8564i	0.504956	+	0.874609i	0.999984	+	0.00573163i	$0.00182444\pi$
$251$	8.00000	+	13.8564i	0.504956	+	0.874609i	−0.495028	+	0.868877i	$0.664842\pi$
$252$		0			0
$253$	4.00000	+	6.92820i	0.251478	+	0.435572i
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−16.0000	−	27.7128i	−0.998053	−	1.72868i	−0.553047	−	0.833150i	$-0.686535\pi$
$257$	−16.0000	−	27.7128i	−0.998053	−	1.72868i	−0.445005	−	0.895528i	$-0.646798\pi$
$258$		0			0
$259$	3.00000			0.186411
$260$	1.00000	+	1.73205i	0.0620174	+	0.107417i
$261$		0			0
$262$	3.00000	+	5.19615i	0.185341	+	0.321019i
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	−0.442943	−	0.896550i	$-0.646065\pi$
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	0.997906	−	0.0646755i	$-0.0206012\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$	1.50000	−	12.9904i	0.0919709	−	0.796491i
$267$		0			0
$268$	4.50000	−	7.79423i	0.274881	−	0.476108i
$269$	−6.00000	+	10.3923i	−0.365826	+	0.633630i	−0.988908	−	0.148527i	$-0.952547\pi$
$269$	−6.00000	+	10.3923i	−0.365826	+	0.633630i	0.623082	+	0.782157i	$0.285880\pi$
$270$		0			0
$271$	4.00000	−	6.92820i	0.242983	−	0.420858i	−0.718580	−	0.695444i	$-0.755208\pi$
$271$	4.00000	−	6.92820i	0.242983	−	0.420858i	0.961563	+	0.274586i	$0.0885408\pi$
$272$	−3.00000	−	5.19615i	−0.181902	−	0.315063i
$273$		0			0
$274$	−14.0000			−0.845771
$275$	1.00000	+	1.73205i	0.0603023	+	0.104447i
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$	−3.00000			−0.179928
$279$		0			0
$280$	−3.00000	+	5.19615i	−0.179284	+	0.310530i
$281$	−11.0000	−	19.0526i	−0.656205	−	1.13658i	−0.981590	−	0.190999i	$-0.938827\pi$
$281$	−11.0000	−	19.0526i	−0.656205	−	1.13658i	0.325385	−	0.945582i	$-0.394506\pi$
$282$		0			0
$283$	6.00000	−	10.3923i	0.356663	−	0.617758i	−0.630738	−	0.775996i	$-0.717248\pi$
$283$	6.00000	−	10.3923i	0.356663	−	0.617758i	0.987401	+	0.158237i	$0.0505811\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	−2.00000			−0.118262
$287$	12.0000	−	20.7846i	0.708338	−	1.22688i
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$	2.00000	−	3.46410i	0.117444	−	0.203419i
$291$		0			0
$292$	−15.0000			−0.877809
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$	12.0000	+	20.7846i	0.698667	+	1.21013i
$296$	1.00000			0.0581238
$297$		0			0
$298$	−9.00000	−	15.5885i	−0.521356	−	0.903015i
$299$	−2.00000	+	3.46410i	−0.115663	+	0.200334i
$300$		0			0
$301$	−10.5000	+	18.1865i	−0.605210	+	1.04825i
$302$	2.00000	−	3.46410i	0.115087	−	0.199337i
$303$		0			0
$304$	0.500000	−	4.33013i	0.0286770	−	0.248350i
$305$	10.0000			0.572598
$306$		0			0
$307$	−6.00000	+	10.3923i	−0.342438	+	0.593120i	−0.984885	−	0.173210i	$-0.944586\pi$
$307$	−6.00000	+	10.3923i	−0.342438	+	0.593120i	0.642447	+	0.766330i	$0.277919\pi$
$308$	−3.00000	−	5.19615i	−0.170941	−	0.296078i
$309$		0			0
$310$	−7.00000	−	12.1244i	−0.397573	−	0.688617i
$311$	16.0000			0.907277			0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000			0.907277			0.453638	+	0.891186i	$0.350126\pi$
$312$		0			0
$313$	17.0000	+	29.4449i	0.960897	+	1.66432i	0.720257	+	0.693708i	$0.244024\pi$
$313$	17.0000	+	29.4449i	0.960897	+	1.66432i	0.240640	+	0.970614i	$0.422643\pi$
$314$	−6.50000	−	11.2583i	−0.366816	−	0.635344i
$315$		0			0
$316$	11.0000			0.618798
$317$	3.00000	+	5.19615i	0.168497	+	0.291845i	0.937892	−	0.346929i	$-0.112775\pi$
$317$	3.00000	+	5.19615i	0.168497	+	0.291845i	−0.769395	+	0.638774i	$0.779442\pi$
$318$		0			0
$319$	2.00000	+	3.46410i	0.111979	+	0.193952i
$320$	−1.00000	+	1.73205i	−0.0559017	+	0.0968246i
$321$		0			0
$322$	−12.0000			−0.668734
$323$	3.00000	−	25.9808i	0.166924	−	1.44561i
$324$		0			0
$325$	−0.500000	+	0.866025i	−0.0277350	+	0.0480384i
$326$	6.50000	−	11.2583i	0.360002	−	0.623541i
$327$		0			0
$328$	4.00000	−	6.92820i	0.220863	−	0.382546i
$329$	−12.0000	−	20.7846i	−0.661581	−	1.14589i
$330$		0			0
$331$	23.0000			1.26419			0.632097	−	0.774889i	$-0.282194\pi$
$331$	23.0000			1.26419			0.632097	+	0.774889i	$0.282194\pi$
$332$	−3.00000	−	5.19615i	−0.164646	−	0.285176i
$333$		0			0
$334$	−12.0000			−0.656611
$335$	−18.0000			−0.983445
$336$		0			0
$337$	11.5000	−	19.9186i	0.626445	−	1.08503i	−0.361815	−	0.932250i	$-0.617843\pi$
$337$	11.5000	−	19.9186i	0.626445	−	1.08503i	0.988260	−	0.152784i	$-0.0488240\pi$
$338$	6.00000	+	10.3923i	0.326357	+	0.565267i
$339$		0			0
$340$	−6.00000	+	10.3923i	−0.325396	+	0.563602i
$341$	14.0000			0.758143
$342$		0			0
$343$	−15.0000			−0.809924
$344$	−3.50000	+	6.06218i	−0.188707	+	0.326851i
$345$		0			0
$346$	9.00000	+	15.5885i	0.483843	+	0.838041i
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	−0.837935	−	0.545770i	$-0.816237\pi$
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	0.891618	+	0.452788i	$0.149571\pi$
$348$		0			0
$349$	−11.0000			−0.588817			−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000			−0.588817			−0.294408	+	0.955680i	$0.595123\pi$
$350$	−3.00000			−0.160357
$351$		0			0
$352$	−1.00000	−	1.73205i	−0.0533002	−	0.0923186i
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$	−2.00000	−	3.46410i	−0.106149	−	0.183855i
$356$		0			0
$357$		0			0
$358$	12.0000	−	20.7846i	0.634220	−	1.09850i
$359$	−18.0000	+	31.1769i	−0.950004	+	1.64545i	−0.204595	+	0.978847i	$0.565588\pi$
$359$	−18.0000	+	31.1769i	−0.950004	+	1.64545i	−0.745409	+	0.666608i	$0.767746\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$	22.0000			1.15629
$363$		0			0
$364$	1.50000	−	2.59808i	0.0786214	−	0.136176i
$365$	15.0000	+	25.9808i	0.785136	+	1.35990i
$366$		0			0
$367$	5.50000	+	9.52628i	0.287098	+	0.497268i	0.973116	−	0.230317i	$-0.0739762\pi$
$367$	5.50000	+	9.52628i	0.287098	+	0.497268i	−0.686018	+	0.727585i	$0.740643\pi$
$368$	−4.00000			−0.208514
$369$		0			0
$370$	−1.00000	−	1.73205i	−0.0519875	−	0.0900450i
$371$	−12.0000	−	20.7846i	−0.623009	−	1.07908i
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$	−6.00000	−	10.3923i	−0.310253	−	0.537373i
$375$		0			0
$376$	−4.00000	−	6.92820i	−0.206284	−	0.357295i
$377$	−1.00000	+	1.73205i	−0.0515026	+	0.0892052i
$378$		0			0
$379$	3.00000			0.154100			0.0770498	−	0.997027i	$-0.475450\pi$
$379$	3.00000			0.154100			0.0770498	+	0.997027i	$0.475450\pi$
$380$	−8.00000	+	3.46410i	−0.410391	+	0.177705i
$381$		0			0
$382$	−11.0000	+	19.0526i	−0.562809	+	0.974814i
$383$	7.00000	−	12.1244i	0.357683	−	0.619526i	−0.629890	−	0.776684i	$-0.716900\pi$
$383$	7.00000	−	12.1244i	0.357683	−	0.619526i	0.987573	+	0.157159i	$0.0502334\pi$
$384$		0			0
$385$	−6.00000	+	10.3923i	−0.305788	+	0.529641i
$386$	4.50000	+	7.79423i	0.229044	+	0.396716i
$387$		0			0
$388$	−14.0000			−0.710742
$389$	−6.00000	−	10.3923i	−0.304212	−	0.526911i	0.672874	−	0.739758i	$-0.265060\pi$
$389$	−6.00000	−	10.3923i	−0.304212	−	0.526911i	−0.977086	+	0.212847i	$0.931726\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$	2.00000			0.101015
$393$		0			0
$394$	−4.00000	+	6.92820i	−0.201517	+	0.349038i
$395$	−11.0000	−	19.0526i	−0.553470	−	0.958638i
$396$		0			0
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	−0.764730	−	0.644351i	$-0.777127\pi$
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	0.940389	+	0.340099i	$0.110461\pi$
$398$	11.0000			0.551380
$399$		0			0
$400$	−1.00000			−0.0500000
$401$	−2.00000	+	3.46410i	−0.0998752	+	0.172989i	−0.911633	−	0.411005i	$-0.865178\pi$
$401$	−2.00000	+	3.46410i	−0.0998752	+	0.172989i	0.811758	+	0.583994i	$0.198511\pi$
$402$		0			0
$403$	3.50000	+	6.06218i	0.174347	+	0.301979i
$404$	−7.00000	+	12.1244i	−0.348263	+	0.603209i
$405$		0			0
$406$	−6.00000			−0.297775
$407$	2.00000			0.0991363
$408$		0			0
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	0.985558	−	0.169338i	$-0.0541630\pi$
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	−0.639430	+	0.768849i	$0.720830\pi$
$410$	−16.0000			−0.790184
$411$		0			0
$412$	−1.50000	−	2.59808i	−0.0738997	−	0.127998i
$413$	18.0000	−	31.1769i	0.885722	−	1.53412i
$414$		0			0
$415$	−6.00000	+	10.3923i	−0.294528	+	0.510138i
$416$	0.500000	−	0.866025i	0.0245145	−	0.0424604i
$417$		0			0
$418$	1.00000	−	8.66025i	0.0489116	−	0.423587i
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$	−13.0000	+	22.5167i	−0.633581	+	1.09739i	0.353233	+	0.935536i	$0.385082\pi$
$421$	−13.0000	+	22.5167i	−0.633581	+	1.09739i	−0.986814	+	0.161859i	$0.948251\pi$
$422$	7.50000	+	12.9904i	0.365094	+	0.632362i
$423$		0			0
$424$	−4.00000	−	6.92820i	−0.194257	−	0.336463i
$425$	−6.00000			−0.291043
$426$		0			0
$427$	−7.50000	−	12.9904i	−0.362950	−	0.628649i
$428$	5.00000	+	8.66025i	0.241684	+	0.418609i
$429$		0			0
$430$	14.0000			0.675140
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	0.973574	−	0.228373i	$-0.0733406\pi$
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	−0.684564	+	0.728953i	$0.740007\pi$
$432$		0			0
$433$	6.50000	+	11.2583i	0.312370	+	0.541041i	0.978875	−	0.204460i	$-0.0655438\pi$
$433$	6.50000	+	11.2583i	0.312370	+	0.541041i	−0.666505	+	0.745501i	$0.732210\pi$
$434$	−10.5000	+	18.1865i	−0.504016	+	0.872982i
$435$		0			0
$436$	−6.00000			−0.287348
$437$	−14.0000	−	10.3923i	−0.669711	−	0.497131i
$438$		0			0
$439$	9.50000	−	16.4545i	0.453410	−	0.785330i	−0.545185	−	0.838316i	$-0.683541\pi$
$439$	9.50000	−	16.4545i	0.453410	−	0.785330i	0.998595	+	0.0529862i	$0.0168739\pi$
$440$	−2.00000	+	3.46410i	−0.0953463	+	0.165145i
$441$		0			0
$442$	3.00000	−	5.19615i	0.142695	−	0.247156i
$443$	13.0000	+	22.5167i	0.617649	+	1.06980i	0.989914	+	0.141672i	$0.0452479\pi$
$443$	13.0000	+	22.5167i	0.617649	+	1.06980i	−0.372265	+	0.928126i	$0.621419\pi$
$444$		0			0
$445$		0			0
$446$	−5.50000	−	9.52628i	−0.260433	−	0.451082i
$447$		0			0
$448$	3.00000			0.141737
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$	8.00000	−	13.8564i	0.376705	−	0.652473i
$452$	−10.0000	−	17.3205i	−0.470360	−	0.814688i
$453$		0			0
$454$	−11.0000	+	19.0526i	−0.516256	+	0.894181i
$455$	−6.00000			−0.281284
$456$		0			0
$457$	17.0000			0.795226			0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000			0.795226			0.397613	+	0.917553i	$0.369839\pi$
$458$	12.5000	−	21.6506i	0.584087	−	1.01167i
$459$		0			0
$460$	4.00000	+	6.92820i	0.186501	+	0.323029i
$461$	−16.0000	+	27.7128i	−0.745194	+	1.29071i	0.204910	+	0.978781i	$0.434310\pi$
$461$	−16.0000	+	27.7128i	−0.745194	+	1.29071i	−0.950104	+	0.311933i	$0.899023\pi$
$462$		0			0
$463$	25.0000			1.16185			0.580924	−	0.813958i	$-0.302691\pi$
$463$	25.0000			1.16185			0.580924	+	0.813958i	$0.302691\pi$
$464$	−2.00000			−0.0928477
$465$		0			0
$466$	−9.00000	−	15.5885i	−0.416917	−	0.722121i
$467$	16.0000			0.740392			0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000			0.740392			0.370196	+	0.928954i	$0.379291\pi$
$468$		0			0
$469$	13.5000	+	23.3827i	0.623372	+	1.07971i
$470$	−8.00000	+	13.8564i	−0.369012	+	0.639148i
$471$		0			0
$472$	6.00000	−	10.3923i	0.276172	−	0.478345i
$473$	−7.00000	+	12.1244i	−0.321860	+	0.557478i
$474$		0			0
$475$	−3.50000	−	2.59808i	−0.160591	−	0.119208i
$476$	18.0000			0.825029
$477$		0			0
$478$	9.00000	−	15.5885i	0.411650	−	0.712999i
$479$	1.00000	+	1.73205i	0.0456912	+	0.0791394i	0.887967	−	0.459908i	$-0.152118\pi$
$479$	1.00000	+	1.73205i	0.0456912	+	0.0791394i	−0.842275	+	0.539048i	$0.818784\pi$
$480$		0			0
$481$	0.500000	+	0.866025i	0.0227980	+	0.0394874i
$482$	1.00000			0.0455488
$483$		0			0
$484$	3.50000	+	6.06218i	0.159091	+	0.275554i
$485$	14.0000	+	24.2487i	0.635707	+	1.10108i
$486$		0			0
$487$	−32.0000			−1.45006			−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000			−1.45006			−0.725029	+	0.688718i	$0.758174\pi$
$488$	−2.50000	−	4.33013i	−0.113170	−	0.196016i
$489$		0			0
$490$	−2.00000	−	3.46410i	−0.0903508	−	0.156492i
$491$	−19.0000	+	32.9090i	−0.857458	+	1.48516i	0.0168878	+	0.999857i	$0.494624\pi$
$491$	−19.0000	+	32.9090i	−0.857458	+	1.48516i	−0.874346	+	0.485303i	$0.838709\pi$
$492$		0			0
$493$	−12.0000			−0.540453
$494$	4.00000	−	1.73205i	0.179969	−	0.0779287i
$495$		0			0
$496$	−3.50000	+	6.06218i	−0.157155	+	0.272200i
$497$	−3.00000	+	5.19615i	−0.134568	+	0.233079i
$498$		0			0
$499$	11.5000	−	19.9186i	0.514811	−	0.891678i	−0.485042	−	0.874491i	$-0.661196\pi$
$499$	11.5000	−	19.9186i	0.514811	−	0.891678i	0.999852	−	0.0171872i	$-0.00547113\pi$
$500$	6.00000	+	10.3923i	0.268328	+	0.464758i
$501$		0			0
$502$	−16.0000			−0.714115
$503$	9.00000	+	15.5885i	0.401290	+	0.695055i	0.993882	−	0.110448i	$-0.0352286\pi$
$503$	9.00000	+	15.5885i	0.401290	+	0.695055i	−0.592592	+	0.805503i	$0.701895\pi$
$504$		0			0
$505$	28.0000			1.24598
$506$	−8.00000			−0.355643
$507$		0			0
$508$		0			0
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	0.967811	−	0.251679i	$-0.0809826\pi$
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	−0.701866	+	0.712309i	$0.747649\pi$
$510$		0			0
$511$	22.5000	−	38.9711i	0.995341	−	1.72398i
$512$	1.00000			0.0441942
$513$		0			0
$514$	32.0000			1.41146
$515$	−3.00000	+	5.19615i	−0.132196	+	0.228970i
$516$		0			0
$517$	−8.00000	−	13.8564i	−0.351840	−	0.609404i
$518$	−1.50000	+	2.59808i	−0.0659062	+	0.114153i
$519$		0			0
$520$	−2.00000			−0.0877058
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$	15.5000	+	26.8468i	0.677768	+	1.17393i	0.975652	+	0.219326i	$0.0703858\pi$
$523$	15.5000	+	26.8468i	0.677768	+	1.17393i	−0.297884	+	0.954602i	$0.596281\pi$
$524$	−6.00000			−0.262111
$525$		0			0
$526$	9.00000	+	15.5885i	0.392419	+	0.679689i
$527$	−21.0000	+	36.3731i	−0.914774	+	1.58444i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$	−8.00000	+	13.8564i	−0.347498	+	0.601884i
$531$		0			0
$532$	10.5000	+	7.79423i	0.455233	+	0.337923i
$533$	8.00000			0.346518
$534$		0			0
$535$	10.0000	−	17.3205i	0.432338	−	0.748831i
$536$	4.50000	+	7.79423i	0.194370	+	0.336659i
$537$		0			0
$538$	−6.00000	−	10.3923i	−0.258678	−	0.448044i
$539$	4.00000			0.172292
$540$		0			0
$541$	0.500000	+	0.866025i	0.0214967	+	0.0372333i	0.876574	−	0.481268i	$-0.159824\pi$
$541$	0.500000	+	0.866025i	0.0214967	+	0.0372333i	−0.855077	+	0.518501i	$0.826490\pi$
$542$	4.00000	+	6.92820i	0.171815	+	0.297592i
$543$		0			0
$544$	6.00000			0.257248
$545$	6.00000	+	10.3923i	0.257012	+	0.445157i
$546$		0			0
$547$	−21.5000	−	37.2391i	−0.919274	−	1.59223i	−0.800521	−	0.599305i	$-0.795444\pi$
$547$	−21.5000	−	37.2391i	−0.919274	−	1.59223i	−0.118753	−	0.992924i	$-0.537890\pi$
$548$	7.00000	−	12.1244i	0.299025	−	0.517927i
$549$		0			0
$550$	−2.00000			−0.0852803
$551$	−7.00000	−	5.19615i	−0.298210	−	0.221364i
$552$		0			0
$553$	−16.5000	+	28.5788i	−0.701651	+	1.21530i
$554$	−5.00000	+	8.66025i	−0.212430	+	0.367939i
$555$		0			0
$556$	1.50000	−	2.59808i	0.0636142	−	0.110183i
$557$	−5.00000	−	8.66025i	−0.211857	−	0.366947i	0.740439	−	0.672124i	$-0.234618\pi$
$557$	−5.00000	−	8.66025i	−0.211857	−	0.366947i	−0.952296	+	0.305177i	$0.901284\pi$
$558$		0			0
$559$	−7.00000			−0.296068
$560$	−3.00000	−	5.19615i	−0.126773	−	0.219578i
$561$		0			0
$562$	22.0000			0.928014
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$	−20.0000	+	34.6410i	−0.841406	+	1.45736i
$566$	6.00000	+	10.3923i	0.252199	+	0.436821i
$567$		0			0
$568$	−1.00000	+	1.73205i	−0.0419591	+	0.0726752i
$569$	4.00000			0.167689			0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000			0.167689			0.0838444	+	0.996479i	$0.473280\pi$
$570$		0			0
$571$	29.0000			1.21361			0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000			1.21361			0.606806	+	0.794850i	$0.292450\pi$
$572$	1.00000	−	1.73205i	0.0418121	−	0.0724207i
$573$		0			0
$574$	12.0000	+	20.7846i	0.500870	+	0.867533i
$575$	−2.00000	+	3.46410i	−0.0834058	+	0.144463i
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$	19.0000			0.790296
$579$		0			0
$580$	2.00000	+	3.46410i	0.0830455	+	0.143839i
$581$	18.0000			0.746766
$582$		0			0
$583$	−8.00000	−	13.8564i	−0.331326	−	0.573874i
$584$	7.50000	−	12.9904i	0.310352	−	0.537546i
$585$		0			0
$586$	−6.00000	+	10.3923i	−0.247858	+	0.429302i
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$	−28.0000	+	12.1244i	−1.15372	+	0.499575i
$590$	−24.0000			−0.988064
$591$		0			0
$592$	−0.500000	+	0.866025i	−0.0205499	+	0.0355934i
$593$	−8.00000	−	13.8564i	−0.328521	−	0.569014i	0.653698	−	0.756756i	$-0.273217\pi$
$593$	−8.00000	−	13.8564i	−0.328521	−	0.569014i	−0.982219	+	0.187741i	$0.939883\pi$
$594$		0			0
$595$	−18.0000	−	31.1769i	−0.737928	−	1.27813i
$596$	18.0000			0.737309
$597$		0			0
$598$	−2.00000	−	3.46410i	−0.0817861	−	0.141658i
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	−0.990752	−	0.135686i	$-0.956676\pi$
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	0.377869	−	0.925859i	$-0.376657\pi$
$600$		0			0
$601$	−41.0000			−1.67242			−0.836212	−	0.548406i	$-0.815235\pi$
$601$	−41.0000			−1.67242			−0.836212	+	0.548406i	$0.815235\pi$
$602$	−10.5000	−	18.1865i	−0.427948	−	0.741228i
$603$		0			0
$604$	2.00000	+	3.46410i	0.0813788	+	0.140952i
$605$	7.00000	−	12.1244i	0.284590	−	0.492925i
$606$		0			0
$607$	−13.0000			−0.527654			−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000			−0.527654			−0.263827	+	0.964570i	$0.584985\pi$
$608$	3.50000	+	2.59808i	0.141944	+	0.105366i
$609$		0			0
$610$	−5.00000	+	8.66025i	−0.202444	+	0.350643i
$611$	4.00000	−	6.92820i	0.161823	−	0.280285i
$612$		0			0
$613$	−23.0000	+	39.8372i	−0.928961	+	1.60901i	−0.143898	+	0.989593i	$0.545964\pi$
$613$	−23.0000	+	39.8372i	−0.928961	+	1.60901i	−0.785063	+	0.619416i	$0.787370\pi$
$614$	−6.00000	−	10.3923i	−0.242140	−	0.419399i
$615$		0			0
$616$	6.00000			0.241747
$617$	−3.00000	−	5.19615i	−0.120775	−	0.209189i	0.799298	−	0.600935i	$-0.205205\pi$
$617$	−3.00000	−	5.19615i	−0.120775	−	0.209189i	−0.920074	+	0.391745i	$0.871871\pi$
$618$		0			0
$619$	−23.0000			−0.924448			−0.462224	−	0.886763i	$-0.652948\pi$
$619$	−23.0000			−0.924448			−0.462224	+	0.886763i	$0.652948\pi$
$620$	14.0000			0.562254
$621$		0			0
$622$	−8.00000	+	13.8564i	−0.320771	+	0.555591i
$623$		0			0
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$	−34.0000			−1.35891
$627$		0			0
$628$	13.0000			0.518756
$629$	−3.00000	+	5.19615i	−0.119618	+	0.207184i
$630$		0			0
$631$	−23.5000	−	40.7032i	−0.935520	−	1.62037i	−0.773704	−	0.633548i	$-0.781598\pi$
$631$	−23.5000	−	40.7032i	−0.935520	−	1.62037i	−0.161817	−	0.986821i	$-0.551735\pi$
$632$	−5.50000	+	9.52628i	−0.218778	+	0.378935i
$633$		0			0
$634$	−6.00000			−0.238290
$635$		0			0
$636$		0			0
$637$	1.00000	+	1.73205i	0.0396214	+	0.0686264i
$638$	−4.00000			−0.158362
$639$		0			0
$640$	−1.00000	−	1.73205i	−0.0395285	−	0.0684653i
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−6.50000	+	11.2583i	−0.256335	+	0.443985i	−0.965257	−	0.261301i	$-0.915848\pi$
$643$	−6.50000	+	11.2583i	−0.256335	+	0.443985i	0.708922	+	0.705287i	$0.249182\pi$
$644$	6.00000	−	10.3923i	0.236433	−	0.409514i
$645$		0			0
$646$	21.0000	+	15.5885i	0.826234	+	0.613320i
$647$	−10.0000			−0.393141			−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000			−0.393141			−0.196570	+	0.980490i	$0.562980\pi$
$648$		0			0
$649$	12.0000	−	20.7846i	0.471041	−	0.815867i
$650$	−0.500000	−	0.866025i	−0.0196116	−	0.0339683i
$651$		0			0
$652$	6.50000	+	11.2583i	0.254560	+	0.440910i
$653$	−18.0000			−0.704394			−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000			−0.704394			−0.352197	+	0.935926i	$0.614565\pi$
$654$		0			0
$655$	6.00000	+	10.3923i	0.234439	+	0.406061i
$656$	4.00000	+	6.92820i	0.156174	+	0.270501i
$657$		0			0
$658$	24.0000			0.935617
$659$	11.0000	+	19.0526i	0.428499	+	0.742182i	0.996740	−	0.0806799i	$-0.0257092\pi$
$659$	11.0000	+	19.0526i	0.428499	+	0.742182i	−0.568241	+	0.822862i	$0.692376\pi$
$660$		0			0
$661$	1.00000	+	1.73205i	0.0388955	+	0.0673690i	0.884818	−	0.465937i	$-0.154283\pi$
$661$	1.00000	+	1.73205i	0.0388955	+	0.0673690i	−0.845922	+	0.533306i	$0.820949\pi$
$662$	−11.5000	+	19.9186i	−0.446960	+	0.774158i
$663$		0			0
$664$	6.00000			0.232845
$665$	3.00000	−	25.9808i	0.116335	−	1.00749i
$666$		0			0
$667$	−4.00000	+	6.92820i	−0.154881	+	0.268261i
$668$	6.00000	−	10.3923i	0.232147	−	0.402090i
$669$		0			0
$670$	9.00000	−	15.5885i	0.347700	−	0.602235i
$671$	−5.00000	−	8.66025i	−0.193023	−	0.334325i
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$	11.5000	+	19.9186i	0.442963	+	0.767235i
$675$		0			0
$676$	−12.0000			−0.461538
$677$	−8.00000			−0.307465			−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000			−0.307465			−0.153732	+	0.988113i	$0.549129\pi$
$678$		0			0
$679$	21.0000	−	36.3731i	0.805906	−	1.39587i
$680$	−6.00000	−	10.3923i	−0.230089	−	0.398527i
$681$		0			0
$682$	−7.00000	+	12.1244i	−0.268044	+	0.464266i
$683$	46.0000			1.76014			0.880071	−	0.474843i	$-0.157495\pi$
$683$	46.0000			1.76014			0.880071	+	0.474843i	$0.157495\pi$
$684$		0			0
$685$	−28.0000			−1.06983
$686$	7.50000	−	12.9904i	0.286351	−	0.495975i
$687$		0			0
$688$	−3.50000	−	6.06218i	−0.133436	−	0.231118i
$689$	4.00000	−	6.92820i	0.152388	−	0.263944i
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$	−18.0000			−0.684257
$693$		0			0
$694$	1.00000	+	1.73205i	0.0379595	+	0.0657477i
$695$	−6.00000			−0.227593
$696$		0			0
$697$	24.0000	+	41.5692i	0.909065	+	1.57455i
$698$	5.50000	−	9.52628i	0.208178	−	0.360575i
$699$		0			0
$700$	1.50000	−	2.59808i	0.0566947	−	0.0981981i
$701$	17.0000	−	29.4449i	0.642081	−	1.11212i	−0.342886	−	0.939377i	$-0.611405\pi$
$701$	17.0000	−	29.4449i	0.642081	−	1.11212i	0.984967	−	0.172740i	$-0.0552621\pi$
$702$		0			0
$703$	−4.00000	+	1.73205i	−0.150863	+	0.0653255i
$704$	2.00000			0.0753778
$705$		0			0
$706$	−3.00000	+	5.19615i	−0.112906	+	0.195560i
$707$	−21.0000	−	36.3731i	−0.789786	−	1.36795i
$708$		0			0
$709$	4.50000	+	7.79423i	0.169001	+	0.292718i	0.938069	−	0.346449i	$-0.112613\pi$
$709$	4.50000	+	7.79423i	0.169001	+	0.292718i	−0.769068	+	0.639167i	$0.779279\pi$
$710$	4.00000			0.150117
$711$		0			0
$712$		0			0
$713$	14.0000	+	24.2487i	0.524304	+	0.908121i
$714$		0			0
$715$	−4.00000			−0.149592
$716$	12.0000	+	20.7846i	0.448461	+	0.776757i
$717$		0			0
$718$	−18.0000	−	31.1769i	−0.671754	−	1.16351i
$719$	22.0000	−	38.1051i	0.820462	−	1.42108i	−0.0848774	−	0.996391i	$-0.527050\pi$
$719$	22.0000	−	38.1051i	0.820462	−	1.42108i	0.905339	−	0.424690i	$-0.139617\pi$
$720$		0			0
$721$	9.00000			0.335178
$722$	5.50000	+	18.1865i	0.204689	+	0.676833i
$723$		0			0
$724$	−11.0000	+	19.0526i	−0.408812	+	0.708083i
$725$	−1.00000	+	1.73205i	−0.0371391	+	0.0643268i
$726$		0			0
$727$	−3.50000	+	6.06218i	−0.129808	+	0.224834i	−0.923602	−	0.383353i	$-0.874769\pi$
$727$	−3.50000	+	6.06218i	−0.129808	+	0.224834i	0.793794	+	0.608186i	$0.208103\pi$
$728$	1.50000	+	2.59808i	0.0555937	+	0.0962911i
$729$		0			0
$730$	−30.0000			−1.11035
$731$	−21.0000	−	36.3731i	−0.776713	−	1.34531i
$732$		0			0
$733$	−6.00000			−0.221615			−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000			−0.221615			−0.110808	+	0.993842i	$0.535344\pi$
$734$	−11.0000			−0.406017
$735$		0			0
$736$	2.00000	−	3.46410i	0.0737210	−	0.127688i
$737$	9.00000	+	15.5885i	0.331519	+	0.574208i
$738$		0			0
$739$	0.500000	−	0.866025i	0.0183928	−	0.0318573i	−0.856683	−	0.515844i	$-0.827478\pi$
$739$	0.500000	−	0.866025i	0.0183928	−	0.0318573i	0.875075	+	0.483987i	$0.160812\pi$
$740$	2.00000			0.0735215
$741$		0			0
$742$	24.0000			0.881068
$743$	5.00000	−	8.66025i	0.183432	−	0.317714i	−0.759615	−	0.650373i	$-0.774613\pi$
$743$	5.00000	−	8.66025i	0.183432	−	0.317714i	0.943047	+	0.332659i	$0.107946\pi$
$744$		0			0
$745$	−18.0000	−	31.1769i	−0.659469	−	1.14223i
$746$	11.0000	−	19.0526i	0.402739	−	0.697564i
$747$		0			0
$748$	12.0000			0.438763
$749$	−30.0000			−1.09618
$750$		0			0
$751$	18.5000	+	32.0429i	0.675075	+	1.16926i	0.976447	+	0.215757i	$0.0692219\pi$
$751$	18.5000	+	32.0429i	0.675075	+	1.16926i	−0.301373	+	0.953506i	$0.597445\pi$
$752$	8.00000			0.291730
$753$		0			0
$754$	−1.00000	−	1.73205i	−0.0364179	−	0.0630776i
$755$	4.00000	−	6.92820i	0.145575	−	0.252143i
$756$		0			0
$757$	20.5000	−	35.5070i	0.745085	−	1.29053i	−0.205070	−	0.978747i	$-0.565742\pi$
$757$	20.5000	−	35.5070i	0.745085	−	1.29053i	0.950155	−	0.311778i	$-0.100925\pi$
$758$	−1.50000	+	2.59808i	−0.0544825	+	0.0943664i
$759$		0			0
$760$	1.00000	−	8.66025i	0.0362738	−	0.314140i
$761$	−12.0000			−0.435000			−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000			−0.435000			−0.217500	+	0.976060i	$0.569790\pi$
$762$		0			0
$763$	9.00000	−	15.5885i	0.325822	−	0.564340i
$764$	−11.0000	−	19.0526i	−0.397966	−	0.689297i
$765$		0			0
$766$	7.00000	+	12.1244i	0.252920	+	0.438071i
$767$	12.0000			0.433295
$768$		0			0
$769$	14.5000	+	25.1147i	0.522883	+	0.905661i	0.999645	+	0.0266282i	$0.00847701\pi$
$769$	14.5000	+	25.1147i	0.522883	+	0.905661i	−0.476762	+	0.879032i	$0.658190\pi$
$770$	−6.00000	−	10.3923i	−0.216225	−	0.374513i
$771$		0			0
$772$	−9.00000			−0.323917
$773$	−24.0000	−	41.5692i	−0.863220	−	1.49514i	−0.868804	−	0.495156i	$-0.835111\pi$
$773$	−24.0000	−	41.5692i	−0.863220	−	1.49514i	0.00558380	−	0.999984i	$-0.498223\pi$
$774$		0			0
$775$	3.50000	+	6.06218i	0.125724	+	0.217760i
$776$	7.00000	−	12.1244i	0.251285	−	0.435239i
$777$		0			0
$778$	12.0000			0.430221
$779$	−4.00000	+	34.6410i	−0.143315	+	1.24114i
$780$		0			0
$781$	−2.00000	+	3.46410i	−0.0715656	+	0.123955i
$782$	12.0000	−	20.7846i	0.429119	−	0.743256i
$783$		0			0
$784$	−1.00000	+	1.73205i	−0.0357143	+	0.0618590i
$785$	−13.0000	−	22.5167i	−0.463990	−	0.803654i
$786$		0			0
$787$	−35.0000			−1.24762			−0.623808	−	0.781578i	$-0.714415\pi$
$787$	−35.0000			−1.24762			−0.623808	+	0.781578i	$0.714415\pi$
$788$	−4.00000	−	6.92820i	−0.142494	−	0.246807i
$789$		0			0
$790$	22.0000			0.782725
$791$	60.0000			2.13335
$792$		0			0
$793$	2.50000	−	4.33013i	0.0887776	−	0.153767i
$794$	3.50000	+	6.06218i	0.124210	+	0.215139i
$795$		0			0
$796$	−5.50000	+	9.52628i	−0.194942	+	0.337650i
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	48.0000			1.69812
$800$	0.500000	−	0.866025i	0.0176777	−	0.0306186i
$801$		0			0
$802$	−2.00000	−	3.46410i	−0.0706225	−	0.122322i
$803$	15.0000	−	25.9808i	0.529339	−	0.916841i
$804$		0			0
$805$	−24.0000			−0.845889
$806$	−7.00000			−0.246564
$807$		0			0
$808$	−7.00000	−	12.1244i	−0.246259	−	0.426533i
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	−6.00000	−	10.3923i	−0.210688	−	0.364923i	0.741242	−	0.671238i	$-0.234237\pi$
$811$	−6.00000	−	10.3923i	−0.210688	−	0.364923i	−0.951930	+	0.306315i	$0.900904\pi$
$812$	3.00000	−	5.19615i	0.105279	−	0.182349i
$813$		0			0
$814$	−1.00000	+	1.73205i	−0.0350500	+	0.0607083i
$815$	13.0000	−	22.5167i	0.455370	−	0.788724i
$816$		0			0
$817$	3.50000	−	30.3109i	0.122449	−	1.06044i
$818$	−14.0000			−0.489499
$819$		0			0
$820$	8.00000	−	13.8564i	0.279372	−	0.483887i
$821$	15.0000	+	25.9808i	0.523504	+	0.906735i	0.999626	+	0.0273557i	$0.00870868\pi$
$821$	15.0000	+	25.9808i	0.523504	+	0.906735i	−0.476122	+	0.879379i	$0.657958\pi$
$822$		0			0
$823$	8.00000	+	13.8564i	0.278862	+	0.483004i	0.971102	−	0.238664i	$-0.0767093\pi$
$823$	8.00000	+	13.8564i	0.278862	+	0.483004i	−0.692240	+	0.721668i	$0.743376\pi$
$824$	3.00000			0.104510
$825$		0			0
$826$	18.0000	+	31.1769i	0.626300	+	1.08478i
$827$	11.0000	+	19.0526i	0.382507	+	0.662522i	0.991420	−	0.130715i	$-0.0417273\pi$
$827$	11.0000	+	19.0526i	0.382507	+	0.662522i	−0.608913	+	0.793237i	$0.708394\pi$
$828$		0			0
$829$	−9.00000			−0.312583			−0.156291	−	0.987711i	$-0.549954\pi$
$829$	−9.00000			−0.312583			−0.156291	+	0.987711i	$0.549954\pi$
$830$	−6.00000	−	10.3923i	−0.208263	−	0.360722i
$831$		0			0
$832$	0.500000	+	0.866025i	0.0173344	+	0.0300240i
$833$	−6.00000	+	10.3923i	−0.207888	+	0.360072i
$834$		0			0
$835$	−24.0000			−0.830554
$836$	7.00000	+	5.19615i	0.242100	+	0.179713i
$837$		0			0
$838$	−3.00000	+	5.19615i	−0.103633	+	0.179498i
$839$	−7.00000	+	12.1244i	−0.241667	+	0.418579i	−0.961189	−	0.275890i	$-0.911027\pi$
$839$	−7.00000	+	12.1244i	−0.241667	+	0.418579i	0.719522	+	0.694469i	$0.244361\pi$
$840$		0			0
$841$	12.5000	−	21.6506i	0.431034	−	0.746574i
$842$	−13.0000	−	22.5167i	−0.448010	−	0.775975i
$843$		0			0
$844$	−15.0000			−0.516321
$845$	12.0000	+	20.7846i	0.412813	+	0.715012i
$846$		0			0
$847$	−21.0000			−0.721569
$848$	8.00000			0.274721
$849$		0			0
$850$	3.00000	−	5.19615i	0.102899	−	0.178227i
$851$	2.00000	+	3.46410i	0.0685591	+	0.118748i
$852$		0			0
$853$	−4.50000	+	7.79423i	−0.154077	+	0.266869i	−0.932723	−	0.360595i	$-0.882574\pi$
$853$	−4.50000	+	7.79423i	−0.154077	+	0.266869i	0.778646	+	0.627464i	$0.215907\pi$
$854$	15.0000			0.513289
$855$		0			0
$856$	−10.0000			−0.341793
$857$	−12.0000	+	20.7846i	−0.409912	+	0.709989i	−0.994880	−	0.101068i	$-0.967774\pi$
$857$	−12.0000	+	20.7846i	−0.409912	+	0.709989i	0.584967	+	0.811057i	$0.301107\pi$
$858$		0			0
$859$	2.50000	+	4.33013i	0.0852989	+	0.147742i	0.905519	−	0.424307i	$-0.139482\pi$
$859$	2.50000	+	4.33013i	0.0852989	+	0.147742i	−0.820220	+	0.572049i	$0.806149\pi$
$860$	−7.00000	+	12.1244i	−0.238698	+	0.413437i
$861$		0			0
$862$	−12.0000			−0.408722
$863$	−12.0000			−0.408485			−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000			−0.408485			−0.204242	+	0.978920i	$0.565473\pi$
$864$		0			0
$865$	18.0000	+	31.1769i	0.612018	+	1.06005i
$866$	−13.0000			−0.441758
$867$		0			0
$868$	−10.5000	−	18.1865i	−0.356393	−	0.617291i
$869$	−11.0000	+	19.0526i	−0.373149	+	0.646314i
$870$		0			0
$871$	−4.50000	+	7.79423i	−0.152477	+	0.264097i
$872$	3.00000	−	5.19615i	0.101593	−	0.175964i
$873$		0			0
$874$	16.0000	−	6.92820i	0.541208	−	0.234350i
$875$	−36.0000			−1.21702
$876$		0			0
$877$	−12.5000	+	21.6506i	−0.422095	+	0.731090i	−0.996144	−	0.0877308i	$-0.972038\pi$
$877$	−12.5000	+	21.6506i	−0.422095	+	0.731090i	0.574049	+	0.818821i	$0.305372\pi$
$878$	9.50000	+	16.4545i	0.320609	+	0.555312i
$879$		0			0
$880$	−2.00000	−	3.46410i	−0.0674200	−	0.116775i
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	14.5000	+	25.1147i	0.487964	+	0.845178i	0.999904	−	0.0138428i	$-0.00440645\pi$
$883$	14.5000	+	25.1147i	0.487964	+	0.845178i	−0.511940	+	0.859021i	$0.671073\pi$
$884$	3.00000	+	5.19615i	0.100901	+	0.174766i
$885$		0			0
$886$	−26.0000			−0.873487
$887$	4.00000	+	6.92820i	0.134307	+	0.232626i	0.925332	−	0.379157i	$-0.123786\pi$
$887$	4.00000	+	6.92820i	0.134307	+	0.232626i	−0.791026	+	0.611783i	$0.790453\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	11.0000			0.368307
$893$	28.0000	+	20.7846i	0.936984	+	0.695530i
$894$		0			0
$895$	24.0000	−	41.5692i	0.802232	−	1.38951i
$896$	−1.50000	+	2.59808i	−0.0501115	+	0.0867956i
$897$		0			0
$898$	7.00000	−	12.1244i	0.233593	−	0.404595i
$899$	7.00000	+	12.1244i	0.233463	+	0.404370i
$900$		0			0
$901$	48.0000			1.59911
$902$	8.00000	+	13.8564i	0.266371	+	0.461368i
$903$		0			0
$904$	20.0000			0.665190
$905$	44.0000			1.46261
$906$		0			0
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	−0.534333	−	0.845274i	$-0.679437\pi$
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	0.999195	+	0.0401089i	$0.0127705\pi$
$908$	−11.0000	−	19.0526i	−0.365048	−	0.632281i
$909$		0			0
$910$	3.00000	−	5.19615i	0.0994490	−	0.172251i
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$	12.0000			0.397142
$914$	−8.50000	+	14.7224i	−0.281155	+	0.486975i
$915$		0			0
$916$	12.5000	+	21.6506i	0.413012	+	0.715357i
$917$	9.00000	−	15.5885i	0.297206	−	0.514776i
$918$		0			0
$919$	17.0000			0.560778			0.280389	−	0.959886i	$-0.409536\pi$
$919$	17.0000			0.560778			0.280389	+	0.959886i	$0.409536\pi$
$920$	−8.00000			−0.263752
$921$		0			0
$922$	−16.0000	−	27.7128i	−0.526932	−	0.912673i
$923$	−2.00000			−0.0658308
$924$		0			0
$925$	0.500000	+	0.866025i	0.0164399	+	0.0284747i
$926$	−12.5000	+	21.6506i	−0.410775	+	0.711484i
$927$		0			0
$928$	1.00000	−	1.73205i	0.0328266	−	0.0568574i
$929$	17.0000	−	29.4449i	0.557752	−	0.966055i	−0.439932	−	0.898031i	$-0.644997\pi$
$929$	17.0000	−	29.4449i	0.557752	−	0.966055i	0.997684	−	0.0680235i	$-0.0216693\pi$
$930$		0			0
$931$	−8.00000	+	3.46410i	−0.262189	+	0.113531i
$932$	18.0000			0.589610
$933$		0			0
$934$	−8.00000	+	13.8564i	−0.261768	+	0.453395i
$935$	−12.0000	−	20.7846i	−0.392442	−	0.679729i
$936$		0			0
$937$	−23.5000	−	40.7032i	−0.767712	−	1.32972i	−0.938801	−	0.344460i	$-0.888062\pi$
$937$	−23.5000	−	40.7032i	−0.767712	−	1.32972i	0.171089	−	0.985255i	$-0.445271\pi$
$938$	−27.0000			−0.881581
$939$		0			0
$940$	−8.00000	−	13.8564i	−0.260931	−	0.451946i
$941$	−20.0000	−	34.6410i	−0.651981	−	1.12926i	−0.982641	−	0.185515i	$-0.940605\pi$
$941$	−20.0000	−	34.6410i	−0.651981	−	1.12926i	0.330660	−	0.943750i	$-0.392729\pi$
$942$		0			0
$943$	32.0000			1.04206
$944$	6.00000	+	10.3923i	0.195283	+	0.338241i
$945$		0			0
$946$	−7.00000	−	12.1244i	−0.227590	−	0.394197i
$947$	−3.00000	+	5.19615i	−0.0974869	+	0.168852i	−0.910644	−	0.413192i	$-0.864414\pi$
$947$	−3.00000	+	5.19615i	−0.0974869	+	0.168852i	0.813157	+	0.582045i	$0.197747\pi$
$948$		0			0
$949$	15.0000			0.486921
$950$	4.00000	−	1.73205i	0.129777	−	0.0561951i
$951$		0			0
$952$	−9.00000	+	15.5885i	−0.291692	+	0.505225i
$953$	−13.0000	+	22.5167i	−0.421111	+	0.729386i	−0.996048	−	0.0888114i	$-0.971693\pi$
$953$	−13.0000	+	22.5167i	−0.421111	+	0.729386i	0.574937	+	0.818198i	$0.305026\pi$
$954$		0			0
$955$	−22.0000	+	38.1051i	−0.711903	+	1.23305i
$956$	9.00000	+	15.5885i	0.291081	+	0.504167i
$957$		0			0
$958$	−2.00000			−0.0646171
$959$	21.0000	+	36.3731i	0.678125	+	1.17455i
$960$		0			0
$961$	18.0000			0.580645
$962$	−1.00000			−0.0322413
$963$		0			0
$964$	−0.500000	+	0.866025i	−0.0161039	+	0.0278928i
$965$	9.00000	+	15.5885i	0.289720	+	0.501810i
$966$		0			0
$967$	−4.50000	+	7.79423i	−0.144710	+	0.250645i	−0.929265	−	0.369414i	$-0.879558\pi$
$967$	−4.50000	+	7.79423i	−0.144710	+	0.250645i	0.784555	+	0.620060i	$0.212892\pi$
$968$	−7.00000			−0.224989
$969$		0			0
$970$	−28.0000			−0.899026
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	−0.980677	−	0.195633i	$-0.937324\pi$
$971$	−10.0000	+	17.3205i	−0.320915	+	0.555842i	0.659762	+	0.751475i	$0.270657\pi$
$972$		0			0
$973$	4.50000	+	7.79423i	0.144263	+	0.249871i
$974$	16.0000	−	27.7128i	0.512673	−	0.887976i
$975$		0			0
$976$	5.00000			0.160046
$977$	−4.00000			−0.127971			−0.0639857	−	0.997951i	$-0.520381\pi$
$977$	−4.00000			−0.127971			−0.0639857	+	0.997951i	$0.520381\pi$
$978$		0			0
$979$		0			0
$980$	4.00000			0.127775
$981$		0			0
$982$	−19.0000	−	32.9090i	−0.606314	−	1.05017i
$983$	−3.00000	+	5.19615i	−0.0956851	+	0.165732i	−0.909894	−	0.414840i	$-0.863838\pi$
$983$	−3.00000	+	5.19615i	−0.0956851	+	0.165732i	0.814209	+	0.580572i	$0.197171\pi$
$984$		0			0
$985$	−8.00000	+	13.8564i	−0.254901	+	0.441502i
$986$	6.00000	−	10.3923i	0.191079	−	0.330958i
$987$		0			0
$988$	−0.500000	+	4.33013i	−0.0159071	+	0.137760i
$989$	−28.0000			−0.890348
$990$		0			0
$991$	5.50000	−	9.52628i	0.174713	−	0.302612i	−0.765349	−	0.643616i	$-0.777433\pi$
$991$	5.50000	−	9.52628i	0.174713	−	0.302612i	0.940062	+	0.341004i	$0.110767\pi$
$992$	−3.50000	−	6.06218i	−0.111125	−	0.192474i
$993$		0			0
$994$	−3.00000	−	5.19615i	−0.0951542	−	0.164812i
$995$	22.0000			0.697447
$996$		0			0
$997$	−0.500000	−	0.866025i	−0.0158352	−	0.0274273i	0.857999	−	0.513651i	$-0.171707\pi$
$997$	−0.500000	−	0.866025i	−0.0158352	−	0.0274273i	−0.873834	+	0.486224i	$0.838374\pi$
$998$	11.5000	+	19.9186i	0.364026	+	0.630512i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.2.g.a.163.1	✓	2
3.2	odd	2		342.2.g.e.163.1	yes	2
4.3	odd	2		2736.2.s.d.1873.1		2
12.11	even	2		2736.2.s.o.1873.1		2
19.7	even	3	inner	342.2.g.a.235.1	yes	2
19.8	odd	6		6498.2.a.k.1.1		1
19.11	even	3		6498.2.a.w.1.1		1
57.8	even	6		6498.2.a.q.1.1		1
57.11	odd	6		6498.2.a.c.1.1		1
57.26	odd	6		342.2.g.e.235.1	yes	2
76.7	odd	6		2736.2.s.d.577.1		2
228.83	even	6		2736.2.s.o.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.2.g.a.163.1	✓	2	1.1	even	1	trivial
342.2.g.a.235.1	yes	2	19.7	even	3	inner
342.2.g.e.163.1	yes	2	3.2	odd	2
342.2.g.e.235.1	yes	2	57.26	odd	6
2736.2.s.d.577.1		2	76.7	odd	6
2736.2.s.d.1873.1		2	4.3	odd	2
2736.2.s.o.577.1		2	228.83	even	6
2736.2.s.o.1873.1		2	12.11	even	2
6498.2.a.c.1.1		1	57.11	odd	6
6498.2.a.k.1.1		1	19.8	odd	6
6498.2.a.q.1.1		1	57.8	even	6
6498.2.a.w.1.1		1	19.11	even	3

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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	342.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +3.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +3.00000 q^{7} +1.00000 q^{8} +(-1.00000 - 1.73205i) q^{10} +2.00000 q^{11} +(0.500000 + 0.866025i) q^{13} +(-1.50000 + 2.59808i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.00000 + 5.19615i) q^{17} +(-4.00000 + 1.73205i) q^{19} +2.00000 q^{20} +(-1.00000 + 1.73205i) q^{22} +(2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} -1.00000 q^{26} +(-1.50000 - 2.59808i) q^{28} +(1.00000 + 1.73205i) q^{29} +7.00000 q^{31} +(-0.500000 - 0.866025i) q^{32} +(-3.00000 - 5.19615i) q^{34} +(-3.00000 + 5.19615i) q^{35} +1.00000 q^{37} +(0.500000 - 4.33013i) q^{38} +(-1.00000 + 1.73205i) q^{40} +(4.00000 - 6.92820i) q^{41} +(-3.50000 + 6.06218i) q^{43} +(-1.00000 - 1.73205i) q^{44} -4.00000 q^{46} +(-4.00000 - 6.92820i) q^{47} +2.00000 q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{52} +(-4.00000 - 6.92820i) q^{53} +(-2.00000 + 3.46410i) q^{55} +3.00000 q^{56} -2.00000 q^{58} +(6.00000 - 10.3923i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(-3.50000 + 6.06218i) q^{62} +1.00000 q^{64} -2.00000 q^{65} +(4.50000 + 7.79423i) q^{67} +6.00000 q^{68} +(-3.00000 - 5.19615i) q^{70} +(-1.00000 + 1.73205i) q^{71} +(7.50000 - 12.9904i) q^{73} +(-0.500000 + 0.866025i) q^{74} +(3.50000 + 2.59808i) q^{76} +6.00000 q^{77} +(-5.50000 + 9.52628i) q^{79} +(-1.00000 - 1.73205i) q^{80} +(4.00000 + 6.92820i) q^{82} +6.00000 q^{83} +(-6.00000 - 10.3923i) q^{85} +(-3.50000 - 6.06218i) q^{86} +2.00000 q^{88} +(1.50000 + 2.59808i) q^{91} +(2.00000 - 3.46410i) q^{92} +8.00000 q^{94} +(1.00000 - 8.66025i) q^{95} +(7.00000 - 12.1244i) q^{97} +(-1.00000 + 1.73205i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 2 q^{5} + 6 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - 2 * q^5 + 6 * q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - 2 q^{5} + 6 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} + q^{13} - 3 q^{14} - q^{16} - 6 q^{17} - 8 q^{19} + 4 q^{20} - 2 q^{22} + 4 q^{23} + q^{25} - 2 q^{26} - 3 q^{28} + 2 q^{29} + 14 q^{31} - q^{32} - 6 q^{34} - 6 q^{35} + 2 q^{37} + q^{38} - 2 q^{40} + 8 q^{41} - 7 q^{43} - 2 q^{44} - 8 q^{46} - 8 q^{47} + 4 q^{49} - 2 q^{50} + q^{52} - 8 q^{53} - 4 q^{55} + 6 q^{56} - 4 q^{58} + 12 q^{59} - 5 q^{61} - 7 q^{62} + 2 q^{64} - 4 q^{65} + 9 q^{67} + 12 q^{68} - 6 q^{70} - 2 q^{71} + 15 q^{73} - q^{74} + 7 q^{76} + 12 q^{77} - 11 q^{79} - 2 q^{80} + 8 q^{82} + 12 q^{83} - 12 q^{85} - 7 q^{86} + 4 q^{88} + 3 q^{91} + 4 q^{92} + 16 q^{94} + 2 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 2 * q^5 + 6 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 + q^13 - 3 * q^14 - q^16 - 6 * q^17 - 8 * q^19 + 4 * q^20 - 2 * q^22 + 4 * q^23 + q^25 - 2 * q^26 - 3 * q^28 + 2 * q^29 + 14 * q^31 - q^32 - 6 * q^34 - 6 * q^35 + 2 * q^37 + q^38 - 2 * q^40 + 8 * q^41 - 7 * q^43 - 2 * q^44 - 8 * q^46 - 8 * q^47 + 4 * q^49 - 2 * q^50 + q^52 - 8 * q^53 - 4 * q^55 + 6 * q^56 - 4 * q^58 + 12 * q^59 - 5 * q^61 - 7 * q^62 + 2 * q^64 - 4 * q^65 + 9 * q^67 + 12 * q^68 - 6 * q^70 - 2 * q^71 + 15 * q^73 - q^74 + 7 * q^76 + 12 * q^77 - 11 * q^79 - 2 * q^80 + 8 * q^82 + 12 * q^83 - 12 * q^85 - 7 * q^86 + 4 * q^88 + 3 * q^91 + 4 * q^92 + 16 * q^94 + 2 * q^95 + 14 * q^97 - 2 * q^98

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