LMFDB - Embedded newform 2970.2.i.b.991.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2970,2,Mod(991,2970)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2970, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2970.991"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2970 = 2 \cdot 3^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2970.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,1,0,-1,1,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$23.7155694003$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 990)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			991.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2970.991
Dual form			2970.2.i.b.1981.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +1.00000 q^{10} +(-0.500000 + 0.866025i) q^{11} +(2.00000 + 3.46410i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +2.00000 q^{19} +(0.500000 - 0.866025i) q^{20} +(0.500000 + 0.866025i) q^{22} +(1.50000 + 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} +4.00000 q^{26} -1.00000 q^{28} +(-4.50000 + 7.79423i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} +1.00000 q^{35} -4.00000 q^{37} +(1.00000 - 1.73205i) q^{38} +(-0.500000 - 0.866025i) q^{40} +(1.50000 + 2.59808i) q^{41} +(2.00000 - 3.46410i) q^{43} +1.00000 q^{44} +3.00000 q^{46} +(-4.50000 + 7.79423i) q^{47} +(3.00000 + 5.19615i) q^{49} +(0.500000 + 0.866025i) q^{50} +(2.00000 - 3.46410i) q^{52} -1.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(4.50000 + 7.79423i) q^{58} +(0.500000 - 0.866025i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(0.500000 + 0.866025i) q^{67} +(0.500000 - 0.866025i) q^{70} +12.0000 q^{71} +8.00000 q^{73} +(-2.00000 + 3.46410i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(0.500000 + 0.866025i) q^{77} +(-1.00000 + 1.73205i) q^{79} -1.00000 q^{80} +3.00000 q^{82} +(4.50000 - 7.79423i) q^{83} +(-2.00000 - 3.46410i) q^{86} +(0.500000 - 0.866025i) q^{88} +3.00000 q^{89} +4.00000 q^{91} +(1.50000 - 2.59808i) q^{92} +(4.50000 + 7.79423i) q^{94} +(1.00000 + 1.73205i) q^{95} +(-4.00000 + 6.92820i) q^{97} +6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} - q^{16} + 4 q^{19} + q^{20} + q^{22} + 3 q^{23} - q^{25} + 8 q^{26} - 2 q^{28} - 9 q^{29} - 2 q^{31} + q^{32}+ \cdots + 12 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 + 4 * q^13 - q^14 - q^16 + 4 * q^19 + q^20 + q^22 + 3 * q^23 - q^25 + 8 * q^26 - 2 * q^28 - 9 * q^29 - 2 * q^31 + q^32 + 2 * q^35 - 8 * q^37 + 2 * q^38 - q^40 + 3 * q^41 + 4 * q^43 + 2 * q^44 + 6 * q^46 - 9 * q^47 + 6 * q^49 + q^50 + 4 * q^52 - 2 * q^55 - q^56 + 9 * q^58 + q^61 - 4 * q^62 + 2 * q^64 - 4 * q^65 + q^67 + q^70 + 24 * q^71 + 16 * q^73 - 4 * q^74 - 2 * q^76 + q^77 - 2 * q^79 - 2 * q^80 + 6 * q^82 + 9 * q^83 - 4 * q^86 + q^88 + 6 * q^89 + 8 * q^91 + 3 * q^92 + 9 * q^94 + 2 * q^95 - 8 * q^97 + 12 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2970\mathbb{Z}\right)^\times$.

$n$	$541$	$1541$	$2377$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	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
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	0.944911	+	0.327327i	$0.106148\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$12$		0			0
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	0.997927	+	0.0643593i	$0.0205004\pi$
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	−0.443227	+	0.896410i	$0.646166\pi$
$14$	−0.500000	−	0.866025i	−0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$	0.500000	−	0.866025i	0.111803	−	0.193649i
$21$		0			0
$22$	0.500000	+	0.866025i	0.106600	+	0.184637i
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	0.978961	−	0.204046i	$-0.0654092\pi$
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	−0.666190	+	0.745782i	$0.732076\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	4.00000			0.784465
$27$		0			0
$28$	−1.00000			−0.188982
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	0.0578882	+	0.998323i	$0.481563\pi$
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$	1.00000	−	1.73205i	0.162221	−	0.280976i
$39$		0			0
$40$	−0.500000	−	0.866025i	−0.0790569	−	0.136931i
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	0.959058	−	0.283211i	$-0.0913998\pi$
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	−0.724797	+	0.688963i	$0.758066\pi$
$42$		0			0
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	−0.672264	−	0.740312i	$-0.734678\pi$
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	0.977261	+	0.212041i	$0.0680112\pi$
$44$	1.00000			0.150756
$45$		0			0
$46$	3.00000			0.442326
$47$	−4.50000	+	7.79423i	−0.656392	+	1.13691i	0.325150	+	0.945662i	$0.394585\pi$
$47$	−4.50000	+	7.79423i	−0.656392	+	1.13691i	−0.981543	+	0.191243i	$0.938748\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$	0.500000	+	0.866025i	0.0707107	+	0.122474i
$51$		0			0
$52$	2.00000	−	3.46410i	0.277350	−	0.480384i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−1.00000			−0.134840
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	4.50000	+	7.79423i	0.590879	+	1.02343i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	0.896258	+	0.443533i	$0.146275\pi$
$62$	−2.00000			−0.254000
$63$		0			0
$64$	1.00000			0.125000
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	0.500000	+	0.866025i	0.0610847	+	0.105802i	0.894951	−	0.446165i	$-0.147211\pi$
$67$	0.500000	+	0.866025i	0.0610847	+	0.105802i	−0.833866	+	0.551967i	$0.813877\pi$
$68$		0			0
$69$		0			0
$70$	0.500000	−	0.866025i	0.0597614	−	0.103510i
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	8.00000			0.936329			0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000			0.936329			0.468165	+	0.883641i	$0.344915\pi$
$74$	−2.00000	+	3.46410i	−0.232495	+	0.402694i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	0.500000	+	0.866025i	0.0569803	+	0.0986928i
$78$		0			0
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	−0.916781	−	0.399390i	$-0.869222\pi$
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	0.804272	+	0.594261i	$0.202555\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	3.00000			0.331295
$83$	4.50000	−	7.79423i	0.493939	−	0.855528i	−0.506036	−	0.862512i	$-0.668890\pi$
$83$	4.50000	−	7.79423i	0.493939	−	0.855528i	0.999976	+	0.00698436i	$0.00222321\pi$
$84$		0			0
$85$		0			0
$86$	−2.00000	−	3.46410i	−0.215666	−	0.373544i
$87$		0			0
$88$	0.500000	−	0.866025i	0.0533002	−	0.0923186i
$89$	3.00000			0.317999			0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$	1.50000	−	2.59808i	0.156386	−	0.270868i
$93$		0			0
$94$	4.50000	+	7.79423i	0.464140	+	0.803913i
$95$	1.00000	+	1.73205i	0.102598	+	0.177705i
$96$		0			0
$97$	−4.00000	+	6.92820i	−0.406138	+	0.703452i	−0.994453	−	0.105180i	$-0.966458\pi$
$97$	−4.00000	+	6.92820i	−0.406138	+	0.703452i	0.588315	+	0.808632i	$0.299792\pi$
$98$	6.00000			0.606092
$99$		0			0
$100$	1.00000			0.100000
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	0.927030	+	0.374987i	$0.122353\pi$
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	−0.138767	+	0.990325i	$0.544314\pi$
$104$	−2.00000	−	3.46410i	−0.196116	−	0.339683i
$105$		0			0
$106$		0			0
$107$	−3.00000			−0.290021			−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000			−0.290021			−0.145010	+	0.989430i	$0.546322\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$	−0.500000	+	0.866025i	−0.0476731	+	0.0825723i
$111$		0			0
$112$	0.500000	+	0.866025i	0.0472456	+	0.0818317i
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		0			0
$115$	−1.50000	+	2.59808i	−0.139876	+	0.242272i
$116$	9.00000			0.835629
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$	−0.500000	−	0.866025i	−0.0452679	−	0.0784063i
$123$		0			0
$124$	−1.00000	+	1.73205i	−0.0898027	+	0.155543i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$	2.00000	+	3.46410i	0.175412	+	0.303822i
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	1.00000	−	1.73205i	0.0867110	−	0.150188i
$134$	1.00000			0.0863868
$135$		0			0
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	−0.296866	+	0.954919i	$0.595942\pi$
$140$	−0.500000	−	0.866025i	−0.0422577	−	0.0731925i
$141$		0			0
$142$	6.00000	−	10.3923i	0.503509	−	0.872103i
$143$	−4.00000			−0.334497
$144$		0			0
$145$	−9.00000			−0.747409
$146$	4.00000	−	6.92820i	0.331042	−	0.573382i
$147$		0			0
$148$	2.00000	+	3.46410i	0.164399	+	0.284747i
$149$	−7.50000	−	12.9904i	−0.614424	−	1.06421i	−0.990485	−	0.137619i	$-0.956055\pi$
$149$	−7.50000	−	12.9904i	−0.614424	−	1.06421i	0.376061	−	0.926595i	$-0.377278\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$	−2.00000			−0.162221
$153$		0			0
$154$	1.00000			0.0805823
$155$	1.00000	−	1.73205i	0.0803219	−	0.139122i
$156$		0			0
$157$	11.0000	+	19.0526i	0.877896	+	1.52056i	0.853646	+	0.520854i	$0.174386\pi$
$157$	11.0000	+	19.0526i	0.877896	+	1.52056i	0.0242497	+	0.999706i	$0.492280\pi$
$158$	1.00000	+	1.73205i	0.0795557	+	0.137795i
$159$		0			0
$160$	−0.500000	+	0.866025i	−0.0395285	+	0.0684653i
$161$	3.00000			0.236433
$162$		0			0
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$	1.50000	−	2.59808i	0.117130	−	0.202876i
$165$		0			0
$166$	−4.50000	−	7.79423i	−0.349268	−	0.604949i
$167$	−1.50000	−	2.59808i	−0.116073	−	0.201045i	0.802135	−	0.597143i	$-0.203697\pi$
$167$	−1.50000	−	2.59808i	−0.116073	−	0.201045i	−0.918208	+	0.396098i	$0.870364\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$	−4.00000			−0.304997
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$	0.500000	+	0.866025i	0.0377964	+	0.0654654i
$176$	−0.500000	−	0.866025i	−0.0376889	−	0.0652791i
$177$		0			0
$178$	1.50000	−	2.59808i	0.112430	−	0.194734i
$179$	6.00000			0.448461			0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000			0.448461			0.224231	+	0.974536i	$0.428013\pi$
$180$		0			0
$181$	−25.0000			−1.85824			−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000			−1.85824			−0.929118	+	0.369784i	$0.879432\pi$
$182$	2.00000	−	3.46410i	0.148250	−	0.256776i
$183$		0			0
$184$	−1.50000	−	2.59808i	−0.110581	−	0.191533i
$185$	−2.00000	−	3.46410i	−0.147043	−	0.254686i
$186$		0			0
$187$		0			0
$188$	9.00000			0.656392
$189$		0			0
$190$	2.00000			0.145095
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	11.0000	+	19.0526i	0.791797	+	1.37143i	0.924853	+	0.380325i	$0.124188\pi$
$193$	11.0000	+	19.0526i	0.791797	+	1.37143i	−0.133056	+	0.991109i	$0.542479\pi$
$194$	4.00000	+	6.92820i	0.287183	+	0.497416i
$195$		0			0
$196$	3.00000	−	5.19615i	0.214286	−	0.371154i
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$	0.500000	−	0.866025i	0.0353553	−	0.0612372i
$201$		0			0
$202$	3.00000	+	5.19615i	0.211079	+	0.365600i
$203$	4.50000	+	7.79423i	0.315838	+	0.547048i
$204$		0			0
$205$	−1.50000	+	2.59808i	−0.104765	+	0.181458i
$206$	16.0000			1.11477
$207$		0			0
$208$	−4.00000			−0.277350
$209$	−1.00000	+	1.73205i	−0.0691714	+	0.119808i
$210$		0			0
$211$	−4.00000	−	6.92820i	−0.275371	−	0.476957i	0.694857	−	0.719148i	$-0.255467\pi$
$211$	−4.00000	−	6.92820i	−0.275371	−	0.476957i	−0.970229	+	0.242190i	$0.922134\pi$
$212$		0			0
$213$		0			0
$214$	−1.50000	+	2.59808i	−0.102538	+	0.177601i
$215$	4.00000			0.272798
$216$		0			0
$217$	−2.00000			−0.135769
$218$	2.50000	−	4.33013i	0.169321	−	0.293273i
$219$		0			0
$220$	0.500000	+	0.866025i	0.0337100	+	0.0583874i
$221$		0			0
$222$		0			0
$223$	−2.50000	+	4.33013i	−0.167412	+	0.289967i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−2.50000	+	4.33013i	−0.167412	+	0.289967i	0.770097	+	0.637927i	$0.220208\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	−6.00000			−0.399114
$227$	6.00000	−	10.3923i	0.398234	−	0.689761i	−0.595274	−	0.803523i	$-0.702957\pi$
$227$	6.00000	−	10.3923i	0.398234	−	0.689761i	0.993508	+	0.113761i	$0.0362899\pi$
$228$		0			0
$229$	−8.50000	−	14.7224i	−0.561696	−	0.972886i	−0.997349	−	0.0727709i	$-0.976816\pi$
$229$	−8.50000	−	14.7224i	−0.561696	−	0.972886i	0.435653	−	0.900115i	$-0.356518\pi$
$230$	1.50000	+	2.59808i	0.0989071	+	0.171312i
$231$		0			0
$232$	4.50000	−	7.79423i	0.295439	−	0.511716i
$233$	−24.0000			−1.57229			−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000			−1.57229			−0.786146	+	0.618041i	$0.787927\pi$
$234$		0			0
$235$	−9.00000			−0.587095
$236$		0			0
$237$		0			0
$238$		0			0
$239$	15.0000	+	25.9808i	0.970269	+	1.68056i	0.694737	+	0.719264i	$0.255521\pi$
$239$	15.0000	+	25.9808i	0.970269	+	1.68056i	0.275533	+	0.961292i	$0.411146\pi$
$240$		0			0
$241$	−5.50000	+	9.52628i	−0.354286	+	0.613642i	−0.986996	−	0.160748i	$-0.948609\pi$
$241$	−5.50000	+	9.52628i	−0.354286	+	0.613642i	0.632709	+	0.774389i	$0.281943\pi$
$242$	−1.00000			−0.0642824
$243$		0			0
$244$	−1.00000			−0.0640184
$245$	−3.00000	+	5.19615i	−0.191663	+	0.331970i
$246$		0			0
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$	1.00000	+	1.73205i	0.0635001	+	0.109985i
$249$		0			0
$250$	−0.500000	+	0.866025i	−0.0316228	+	0.0547723i
$251$	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
$253$	−3.00000			−0.188608
$254$	−3.50000	+	6.06218i	−0.219610	+	0.380375i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−12.0000	−	20.7846i	−0.748539	−	1.29651i	−0.948523	−	0.316709i	$-0.897422\pi$
$257$	−12.0000	−	20.7846i	−0.748539	−	1.29651i	0.199983	−	0.979799i	$-0.435911\pi$
$258$		0			0
$259$	−2.00000	+	3.46410i	−0.124274	+	0.215249i
$260$	4.00000			0.248069
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$	−1.00000	−	1.73205i	−0.0613139	−	0.106199i
$267$		0			0
$268$	0.500000	−	0.866025i	0.0305424	−	0.0529009i
$269$	9.00000			0.548740			0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000			0.548740			0.274370	+	0.961624i	$0.411531\pi$
$270$		0			0
$271$	−22.0000			−1.33640			−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000			−1.33640			−0.668202	+	0.743980i	$0.732936\pi$
$272$		0			0
$273$		0			0
$274$	−6.00000	−	10.3923i	−0.362473	−	0.627822i
$275$	−0.500000	−	0.866025i	−0.0301511	−	0.0522233i
$276$		0			0
$277$	2.00000	−	3.46410i	0.120168	−	0.208138i	−0.799666	−	0.600446i	$-0.794990\pi$
$277$	2.00000	−	3.46410i	0.120168	−	0.208138i	0.919834	+	0.392308i	$0.128323\pi$
$278$	16.0000			0.959616
$279$		0			0
$280$	−1.00000			−0.0597614
$281$	13.5000	−	23.3827i	0.805342	−	1.39489i	−0.110717	−	0.993852i	$-0.535315\pi$
$281$	13.5000	−	23.3827i	0.805342	−	1.39489i	0.916060	−	0.401042i	$-0.131352\pi$
$282$		0			0
$283$	3.50000	+	6.06218i	0.208053	+	0.360359i	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	3.50000	+	6.06218i	0.208053	+	0.360359i	−0.743048	+	0.669238i	$0.766621\pi$
$284$	−6.00000	−	10.3923i	−0.356034	−	0.616670i
$285$		0			0
$286$	−2.00000	+	3.46410i	−0.118262	+	0.204837i
$287$	3.00000			0.177084
$288$		0			0
$289$	−17.0000			−1.00000
$290$	−4.50000	+	7.79423i	−0.264249	+	0.457693i
$291$		0			0
$292$	−4.00000	−	6.92820i	−0.234082	−	0.405442i
$293$	6.00000	+	10.3923i	0.350524	+	0.607125i	0.986341	−	0.164714i	$-0.0526703\pi$
$293$	6.00000	+	10.3923i	0.350524	+	0.607125i	−0.635818	+	0.771839i	$0.719337\pi$
$294$		0			0
$295$		0			0
$296$	4.00000			0.232495
$297$		0			0
$298$	−15.0000			−0.868927
$299$	−6.00000	+	10.3923i	−0.346989	+	0.601003i
$300$		0			0
$301$	−2.00000	−	3.46410i	−0.115278	−	0.199667i
$302$	−5.00000	−	8.66025i	−0.287718	−	0.498342i
$303$		0			0
$304$	−1.00000	+	1.73205i	−0.0573539	+	0.0993399i
$305$	1.00000			0.0572598
$306$		0			0
$307$	23.0000			1.31268			0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000			1.31268			0.656340	+	0.754466i	$0.272104\pi$
$308$	0.500000	−	0.866025i	0.0284901	−	0.0493464i
$309$		0			0
$310$	−1.00000	−	1.73205i	−0.0567962	−	0.0983739i
$311$	−15.0000	−	25.9808i	−0.850572	−	1.47323i	−0.880693	−	0.473688i	$-0.842923\pi$
$311$	−15.0000	−	25.9808i	−0.850572	−	1.47323i	0.0301210	−	0.999546i	$-0.490411\pi$
$312$		0			0
$313$	−1.00000	+	1.73205i	−0.0565233	+	0.0979013i	−0.892903	−	0.450250i	$-0.851335\pi$
$313$	−1.00000	+	1.73205i	−0.0565233	+	0.0979013i	0.836379	+	0.548151i	$0.184668\pi$
$314$	22.0000			1.24153
$315$		0			0
$316$	2.00000			0.112509
$317$	15.0000	−	25.9808i	0.842484	−	1.45922i	−0.0453045	−	0.998973i	$-0.514426\pi$
$317$	15.0000	−	25.9808i	0.842484	−	1.45922i	0.887788	−	0.460252i	$-0.152241\pi$
$318$		0			0
$319$	−4.50000	−	7.79423i	−0.251952	−	0.436393i
$320$	0.500000	+	0.866025i	0.0279508	+	0.0484123i
$321$		0			0
$322$	1.50000	−	2.59808i	0.0835917	−	0.144785i
$323$		0			0
$324$		0			0
$325$	−4.00000			−0.221880
$326$	10.0000	−	17.3205i	0.553849	−	0.959294i
$327$		0			0
$328$	−1.50000	−	2.59808i	−0.0828236	−	0.143455i
$329$	4.50000	+	7.79423i	0.248093	+	0.429710i
$330$		0			0
$331$	17.0000	−	29.4449i	0.934405	−	1.61844i	0.158712	−	0.987325i	$-0.449266\pi$
$331$	17.0000	−	29.4449i	0.934405	−	1.61844i	0.775692	−	0.631111i	$-0.217401\pi$
$332$	−9.00000			−0.493939
$333$		0			0
$334$	−3.00000			−0.164153
$335$	−0.500000	+	0.866025i	−0.0273179	+	0.0473160i
$336$		0			0
$337$	−1.00000	−	1.73205i	−0.0544735	−	0.0943508i	0.837503	−	0.546433i	$-0.184015\pi$
$337$	−1.00000	−	1.73205i	−0.0544735	−	0.0943508i	−0.891976	+	0.452082i	$0.850681\pi$
$338$	1.50000	+	2.59808i	0.0815892	+	0.141317i
$339$		0			0
$340$		0			0
$341$	2.00000			0.108306
$342$		0			0
$343$	13.0000			0.701934
$344$	−2.00000	+	3.46410i	−0.107833	+	0.186772i
$345$		0			0
$346$	3.00000	+	5.19615i	0.161281	+	0.279347i
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	−5.50000	+	9.52628i	−0.294408	+	0.509930i	−0.974847	−	0.222875i	$-0.928456\pi$
$349$	−5.50000	+	9.52628i	−0.294408	+	0.509930i	0.680439	+	0.732805i	$0.261789\pi$
$350$	1.00000			0.0534522
$351$		0			0
$352$	−1.00000			−0.0533002
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$	6.00000	+	10.3923i	0.318447	+	0.551566i
$356$	−1.50000	−	2.59808i	−0.0794998	−	0.137698i
$357$		0			0
$358$	3.00000	−	5.19615i	0.158555	−	0.274625i
$359$	−30.0000			−1.58334			−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000			−1.58334			−0.791670	+	0.610949i	$0.790788\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$	−12.5000	+	21.6506i	−0.656985	+	1.13793i
$363$		0			0
$364$	−2.00000	−	3.46410i	−0.104828	−	0.181568i
$365$	4.00000	+	6.92820i	0.209370	+	0.362639i
$366$		0			0
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	0.913493	+	0.406855i	$0.133375\pi$
$368$	−3.00000			−0.156386
$369$		0			0
$370$	−4.00000			−0.207950
$371$		0			0
$372$		0			0
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	0.913147	−	0.407630i	$-0.133645\pi$
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	−0.809591	+	0.586994i	$0.800311\pi$
$374$		0			0
$375$		0			0
$376$	4.50000	−	7.79423i	0.232070	−	0.401957i
$377$	−36.0000			−1.85409
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$	1.00000	−	1.73205i	0.0512989	−	0.0888523i
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	−0.500000	+	0.866025i	−0.0254824	+	0.0441367i
$386$	22.0000			1.11977
$387$		0			0
$388$	8.00000			0.406138
$389$	7.50000	−	12.9904i	0.380265	−	0.658638i	−0.610835	−	0.791758i	$-0.709166\pi$
$389$	7.50000	−	12.9904i	0.380265	−	0.658638i	0.991100	+	0.133120i	$0.0424994\pi$
$390$		0			0
$391$		0			0
$392$	−3.00000	−	5.19615i	−0.151523	−	0.262445i
$393$		0			0
$394$	9.00000	−	15.5885i	0.453413	−	0.785335i
$395$	−2.00000			−0.100631
$396$		0			0
$397$	−16.0000			−0.803017			−0.401508	−	0.915855i	$-0.631514\pi$
$397$	−16.0000			−0.803017			−0.401508	+	0.915855i	$0.631514\pi$
$398$	−2.00000	+	3.46410i	−0.100251	+	0.173640i
$399$		0			0
$400$	−0.500000	−	0.866025i	−0.0250000	−	0.0433013i
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		0			0
$403$	4.00000	−	6.92820i	0.199254	−	0.345118i
$404$	6.00000			0.298511
$405$		0			0
$406$	9.00000			0.446663
$407$	2.00000	−	3.46410i	0.0991363	−	0.171709i
$408$		0			0
$409$	−1.00000	−	1.73205i	−0.0494468	−	0.0856444i	0.840243	−	0.542211i	$-0.182412\pi$
$409$	−1.00000	−	1.73205i	−0.0494468	−	0.0856444i	−0.889689	+	0.456566i	$0.849079\pi$
$410$	1.50000	+	2.59808i	0.0740797	+	0.128310i
$411$		0			0
$412$	8.00000	−	13.8564i	0.394132	−	0.682656i
$413$		0			0
$414$		0			0
$415$	9.00000			0.441793
$416$	−2.00000	+	3.46410i	−0.0980581	+	0.169842i
$417$		0			0
$418$	1.00000	+	1.73205i	0.0489116	+	0.0847174i
$419$	−6.00000	−	10.3923i	−0.293119	−	0.507697i	0.681426	−	0.731887i	$-0.261360\pi$
$419$	−6.00000	−	10.3923i	−0.293119	−	0.507697i	−0.974546	+	0.224189i	$0.928027\pi$
$420$		0			0
$421$	−7.00000	+	12.1244i	−0.341159	+	0.590905i	−0.984648	−	0.174550i	$-0.944153\pi$
$421$	−7.00000	+	12.1244i	−0.341159	+	0.590905i	0.643489	+	0.765455i	$0.277486\pi$
$422$	−8.00000			−0.389434
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.500000	−	0.866025i	−0.0241967	−	0.0419099i
$428$	1.50000	+	2.59808i	0.0725052	+	0.125583i
$429$		0			0
$430$	2.00000	−	3.46410i	0.0964486	−	0.167054i
$431$	−36.0000			−1.73406			−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000			−1.73406			−0.867029	+	0.498257i	$0.833974\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$	−1.00000	+	1.73205i	−0.0480015	+	0.0831411i
$435$		0			0
$436$	−2.50000	−	4.33013i	−0.119728	−	0.207375i
$437$	3.00000	+	5.19615i	0.143509	+	0.248566i
$438$		0			0
$439$	11.0000	−	19.0526i	0.525001	−	0.909329i	−0.474575	−	0.880215i	$-0.657398\pi$
$439$	11.0000	−	19.0526i	0.525001	−	0.909329i	0.999576	−	0.0291138i	$-0.00926853\pi$
$440$	1.00000			0.0476731
$441$		0			0
$442$		0			0
$443$	−13.5000	+	23.3827i	−0.641404	+	1.11094i	0.343715	+	0.939074i	$0.388315\pi$
$443$	−13.5000	+	23.3827i	−0.641404	+	1.11094i	−0.985119	+	0.171871i	$0.945019\pi$
$444$		0			0
$445$	1.50000	+	2.59808i	0.0711068	+	0.123161i
$446$	2.50000	+	4.33013i	0.118378	+	0.205037i
$447$		0			0
$448$	0.500000	−	0.866025i	0.0236228	−	0.0409159i
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−3.00000			−0.141264
$452$	−3.00000	+	5.19615i	−0.141108	+	0.244406i
$453$		0			0
$454$	−6.00000	−	10.3923i	−0.281594	−	0.487735i
$455$	2.00000	+	3.46410i	0.0937614	+	0.162400i
$456$		0			0
$457$	−13.0000	+	22.5167i	−0.608114	+	1.05328i	0.383437	+	0.923567i	$0.374740\pi$
$457$	−13.0000	+	22.5167i	−0.608114	+	1.05328i	−0.991551	+	0.129718i	$0.958593\pi$
$458$	−17.0000			−0.794358
$459$		0			0
$460$	3.00000			0.139876
$461$	16.5000	−	28.5788i	0.768482	−	1.33105i	−0.169904	−	0.985461i	$-0.554346\pi$
$461$	16.5000	−	28.5788i	0.768482	−	1.33105i	0.938386	−	0.345589i	$-0.112321\pi$
$462$		0			0
$463$	−16.0000	−	27.7128i	−0.743583	−	1.28792i	−0.950854	−	0.309640i	$-0.899791\pi$
$463$	−16.0000	−	27.7128i	−0.743583	−	1.28792i	0.207271	−	0.978284i	$-0.433542\pi$
$464$	−4.50000	−	7.79423i	−0.208907	−	0.361838i
$465$		0			0
$466$	−12.0000	+	20.7846i	−0.555889	+	0.962828i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	1.00000			0.0461757
$470$	−4.50000	+	7.79423i	−0.207570	+	0.359521i
$471$		0			0
$472$		0			0
$473$	2.00000	+	3.46410i	0.0919601	+	0.159280i
$474$		0			0
$475$	−1.00000	+	1.73205i	−0.0458831	+	0.0794719i
$476$		0			0
$477$		0			0
$478$	30.0000			1.37217
$479$	−15.0000	+	25.9808i	−0.685367	+	1.18709i	0.287954	+	0.957644i	$0.407025\pi$
$479$	−15.0000	+	25.9808i	−0.685367	+	1.18709i	−0.973321	+	0.229447i	$0.926308\pi$
$480$		0			0
$481$	−8.00000	−	13.8564i	−0.364769	−	0.631798i
$482$	5.50000	+	9.52628i	0.250518	+	0.433910i
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.0227273	+	0.0393648i
$485$	−8.00000			−0.363261
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	−0.500000	+	0.866025i	−0.0226339	+	0.0392031i
$489$		0			0
$490$	3.00000	+	5.19615i	0.135526	+	0.234738i
$491$	−9.00000	−	15.5885i	−0.406164	−	0.703497i	0.588292	−	0.808649i	$-0.299801\pi$
$491$	−9.00000	−	15.5885i	−0.406164	−	0.703497i	−0.994456	+	0.105151i	$0.966467\pi$
$492$		0			0
$493$		0			0
$494$	8.00000			0.359937
$495$		0			0
$496$	2.00000			0.0898027
$497$	6.00000	−	10.3923i	0.269137	−	0.466159i
$498$		0			0
$499$	8.00000	+	13.8564i	0.358129	+	0.620298i	0.987648	−	0.156687i	$-0.0500814\pi$
$499$	8.00000	+	13.8564i	0.358129	+	0.620298i	−0.629519	+	0.776985i	$0.716748\pi$
$500$	0.500000	+	0.866025i	0.0223607	+	0.0387298i
$501$		0			0
$502$	6.00000	−	10.3923i	0.267793	−	0.463831i
$503$	−27.0000			−1.20387			−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000			−1.20387			−0.601935	+	0.798545i	$0.705603\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$	−1.50000	+	2.59808i	−0.0666831	+	0.115499i
$507$		0			0
$508$	3.50000	+	6.06218i	0.155287	+	0.268966i
$509$	−13.5000	−	23.3827i	−0.598377	−	1.03642i	−0.993061	−	0.117602i	$-0.962479\pi$
$509$	−13.5000	−	23.3827i	−0.598377	−	1.03642i	0.394684	−	0.918817i	$-0.370854\pi$
$510$		0			0
$511$	4.00000	−	6.92820i	0.176950	−	0.306486i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−24.0000			−1.05859
$515$	−8.00000	+	13.8564i	−0.352522	+	0.610586i
$516$		0			0
$517$	−4.50000	−	7.79423i	−0.197910	−	0.342790i
$518$	2.00000	+	3.46410i	0.0878750	+	0.152204i
$519$		0			0
$520$	2.00000	−	3.46410i	0.0877058	−	0.151911i
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$	11.0000			0.480996			0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000			0.480996			0.240498	+	0.970650i	$0.422689\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$		0			0
$531$		0			0
$532$	−2.00000			−0.0867110
$533$	−6.00000	+	10.3923i	−0.259889	+	0.450141i
$534$		0			0
$535$	−1.50000	−	2.59808i	−0.0648507	−	0.112325i
$536$	−0.500000	−	0.866025i	−0.0215967	−	0.0374066i
$537$		0			0
$538$	4.50000	−	7.79423i	0.194009	−	0.336033i
$539$	−6.00000			−0.258438
$540$		0			0
$541$	−43.0000			−1.84871			−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000			−1.84871			−0.924357	+	0.381528i	$0.875398\pi$
$542$	−11.0000	+	19.0526i	−0.472490	+	0.818377i
$543$		0			0
$544$		0			0
$545$	2.50000	+	4.33013i	0.107088	+	0.185482i
$546$		0			0
$547$	12.5000	−	21.6506i	0.534461	−	0.925714i	−0.464728	−	0.885454i	$-0.653848\pi$
$547$	12.5000	−	21.6506i	0.534461	−	0.925714i	0.999189	−	0.0402607i	$-0.0128188\pi$
$548$	−12.0000			−0.512615
$549$		0			0
$550$	−1.00000			−0.0426401
$551$	−9.00000	+	15.5885i	−0.383413	+	0.664091i
$552$		0			0
$553$	1.00000	+	1.73205i	0.0425243	+	0.0736543i
$554$	−2.00000	−	3.46410i	−0.0849719	−	0.147176i
$555$		0			0
$556$	8.00000	−	13.8564i	0.339276	−	0.587643i
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$	−0.500000	+	0.866025i	−0.0211289	+	0.0365963i
$561$		0			0
$562$	−13.5000	−	23.3827i	−0.569463	−	0.986339i
$563$	19.5000	+	33.7750i	0.821827	+	1.42345i	0.904320	+	0.426855i	$0.140378\pi$
$563$	19.5000	+	33.7750i	0.821827	+	1.42345i	−0.0824933	+	0.996592i	$0.526288\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$	7.00000			0.294232
$567$		0			0
$568$	−12.0000			−0.503509
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	0.358954	+	0.933355i	$0.383134\pi$
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	−0.987786	+	0.155815i	$0.950200\pi$
$570$		0			0
$571$	11.0000	+	19.0526i	0.460336	+	0.797325i	0.998978	−	0.0452101i	$-0.0143957\pi$
$571$	11.0000	+	19.0526i	0.460336	+	0.797325i	−0.538642	+	0.842535i	$0.681062\pi$
$572$	2.00000	+	3.46410i	0.0836242	+	0.144841i
$573$		0			0
$574$	1.50000	−	2.59808i	0.0626088	−	0.108442i
$575$	−3.00000			−0.125109
$576$		0			0
$577$	20.0000			0.832611			0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000			0.832611			0.416305	+	0.909225i	$0.363325\pi$
$578$	−8.50000	+	14.7224i	−0.353553	+	0.612372i
$579$		0			0
$580$	4.50000	+	7.79423i	0.186852	+	0.323638i
$581$	−4.50000	−	7.79423i	−0.186691	−	0.323359i
$582$		0			0
$583$		0			0
$584$	−8.00000			−0.331042
$585$		0			0
$586$	12.0000			0.495715
$587$	10.5000	−	18.1865i	0.433381	−	0.750639i	−0.563781	−	0.825925i	$-0.690654\pi$
$587$	10.5000	−	18.1865i	0.433381	−	0.750639i	0.997162	+	0.0752860i	$0.0239870\pi$
$588$		0			0
$589$	−2.00000	−	3.46410i	−0.0824086	−	0.142736i
$590$		0			0
$591$		0			0
$592$	2.00000	−	3.46410i	0.0821995	−	0.142374i
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$		0			0
$596$	−7.50000	+	12.9904i	−0.307212	+	0.532107i
$597$		0			0
$598$	6.00000	+	10.3923i	0.245358	+	0.424973i
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	5.00000	−	8.66025i	0.203954	−	0.353259i	−0.745845	−	0.666120i	$-0.767954\pi$
$601$	5.00000	−	8.66025i	0.203954	−	0.353259i	0.949799	+	0.312861i	$0.101287\pi$
$602$	−4.00000			−0.163028
$603$		0			0
$604$	−10.0000			−0.406894
$605$	0.500000	−	0.866025i	0.0203279	−	0.0352089i
$606$		0			0
$607$	21.5000	+	37.2391i	0.872658	+	1.51149i	0.859237	+	0.511578i	$0.170939\pi$
$607$	21.5000	+	37.2391i	0.872658	+	1.51149i	0.0134214	+	0.999910i	$0.495728\pi$
$608$	1.00000	+	1.73205i	0.0405554	+	0.0702439i
$609$		0			0
$610$	0.500000	−	0.866025i	0.0202444	−	0.0350643i
$611$	−36.0000			−1.45640
$612$		0			0
$613$	8.00000			0.323117			0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000			0.323117			0.161558	+	0.986863i	$0.448348\pi$
$614$	11.5000	−	19.9186i	0.464102	−	0.803849i
$615$		0			0
$616$	−0.500000	−	0.866025i	−0.0201456	−	0.0348932i
$617$	−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
$619$	−13.0000	+	22.5167i	−0.522514	+	0.905021i	0.477143	+	0.878826i	$0.341672\pi$
$619$	−13.0000	+	22.5167i	−0.522514	+	0.905021i	−0.999657	+	0.0261952i	$0.991661\pi$
$620$	−2.00000			−0.0803219
$621$		0			0
$622$	−30.0000			−1.20289
$623$	1.50000	−	2.59808i	0.0600962	−	0.104090i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$	1.00000	+	1.73205i	0.0399680	+	0.0692267i
$627$		0			0
$628$	11.0000	−	19.0526i	0.438948	−	0.760280i
$629$		0			0
$630$		0			0
$631$	14.0000			0.557331			0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000			0.557331			0.278666	+	0.960388i	$0.410108\pi$
$632$	1.00000	−	1.73205i	0.0397779	−	0.0688973i
$633$		0			0
$634$	−15.0000	−	25.9808i	−0.595726	−	1.03183i
$635$	−3.50000	−	6.06218i	−0.138893	−	0.240570i
$636$		0			0
$637$	−12.0000	+	20.7846i	−0.475457	+	0.823516i
$638$	−9.00000			−0.356313
$639$		0			0
$640$	1.00000			0.0395285
$641$	−1.50000	+	2.59808i	−0.0592464	+	0.102618i	−0.894127	−	0.447813i	$-0.852203\pi$
$641$	−1.50000	+	2.59808i	−0.0592464	+	0.102618i	0.834881	+	0.550431i	$0.185536\pi$
$642$		0			0
$643$	15.5000	+	26.8468i	0.611260	+	1.05873i	0.991028	+	0.133652i	$0.0426705\pi$
$643$	15.5000	+	26.8468i	0.611260	+	1.05873i	−0.379768	+	0.925082i	$0.623996\pi$
$644$	−1.50000	−	2.59808i	−0.0591083	−	0.102379i
$645$		0			0
$646$		0			0
$647$	33.0000			1.29736			0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000			1.29736			0.648682	+	0.761060i	$0.275321\pi$
$648$		0			0
$649$		0			0
$650$	−2.00000	+	3.46410i	−0.0784465	+	0.135873i
$651$		0			0
$652$	−10.0000	−	17.3205i	−0.391630	−	0.678323i
$653$	−15.0000	−	25.9808i	−0.586995	−	1.01671i	−0.994623	−	0.103558i	$-0.966977\pi$
$653$	−15.0000	−	25.9808i	−0.586995	−	1.01671i	0.407628	−	0.913148i	$-0.366356\pi$
$654$		0			0
$655$		0			0
$656$	−3.00000			−0.117130
$657$		0			0
$658$	9.00000			0.350857
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	−0.958902	−	0.283738i	$-0.908425\pi$
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	0.725175	+	0.688565i	$0.241759\pi$
$660$		0			0
$661$	−19.0000	−	32.9090i	−0.739014	−	1.28001i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−19.0000	−	32.9090i	−0.739014	−	1.28001i	0.213925	−	0.976850i	$-0.431375\pi$
$662$	−17.0000	−	29.4449i	−0.660724	−	1.14441i
$663$		0			0
$664$	−4.50000	+	7.79423i	−0.174634	+	0.302475i
$665$	2.00000			0.0775567
$666$		0			0
$667$	−27.0000			−1.04544
$668$	−1.50000	+	2.59808i	−0.0580367	+	0.100523i
$669$		0			0
$670$	0.500000	+	0.866025i	0.0193167	+	0.0334575i
$671$	0.500000	+	0.866025i	0.0193023	+	0.0334325i
$672$		0			0
$673$	−19.0000	+	32.9090i	−0.732396	+	1.26855i	0.223460	+	0.974713i	$0.428265\pi$
$673$	−19.0000	+	32.9090i	−0.732396	+	1.26855i	−0.955856	+	0.293834i	$0.905069\pi$
$674$	−2.00000			−0.0770371
$675$		0			0
$676$	3.00000			0.115385
$677$	6.00000	−	10.3923i	0.230599	−	0.399409i	−0.727386	−	0.686229i	$-0.759265\pi$
$677$	6.00000	−	10.3923i	0.230599	−	0.399409i	0.957984	+	0.286820i	$0.0925982\pi$
$678$		0			0
$679$	4.00000	+	6.92820i	0.153506	+	0.265880i
$680$		0			0
$681$		0			0
$682$	1.00000	−	1.73205i	0.0382920	−	0.0663237i
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$	6.50000	−	11.2583i	0.248171	−	0.429845i
$687$		0			0
$688$	2.00000	+	3.46410i	0.0762493	+	0.132068i
$689$		0			0
$690$		0			0
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	−0.466691	−	0.884420i	$-0.654554\pi$
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	0.999276	−	0.0380440i	$-0.0121127\pi$
$692$	6.00000			0.228086
$693$		0			0
$694$		0			0
$695$	−8.00000	+	13.8564i	−0.303457	+	0.525603i
$696$		0			0
$697$		0			0
$698$	5.50000	+	9.52628i	0.208178	+	0.360575i
$699$		0			0
$700$	0.500000	−	0.866025i	0.0188982	−	0.0327327i
$701$	33.0000			1.24639			0.623196	−	0.782065i	$-0.285834\pi$
$701$	33.0000			1.24639			0.623196	+	0.782065i	$0.285834\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$	−0.500000	+	0.866025i	−0.0188445	+	0.0326396i
$705$		0			0
$706$	3.00000	+	5.19615i	0.112906	+	0.195560i
$707$	3.00000	+	5.19615i	0.112827	+	0.195421i
$708$		0			0
$709$	9.50000	−	16.4545i	0.356780	−	0.617961i	−0.630641	−	0.776075i	$-0.717208\pi$
$709$	9.50000	−	16.4545i	0.356780	−	0.617961i	0.987421	+	0.158114i	$0.0505412\pi$
$710$	12.0000			0.450352
$711$		0			0
$712$	−3.00000			−0.112430
$713$	3.00000	−	5.19615i	0.112351	−	0.194597i
$714$		0			0
$715$	−2.00000	−	3.46410i	−0.0747958	−	0.129550i
$716$	−3.00000	−	5.19615i	−0.112115	−	0.194189i
$717$		0			0
$718$	−15.0000	+	25.9808i	−0.559795	+	0.969593i
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$	−7.50000	+	12.9904i	−0.279121	+	0.483452i
$723$		0			0
$724$	12.5000	+	21.6506i	0.464559	+	0.804640i
$725$	−4.50000	−	7.79423i	−0.167126	−	0.289470i
$726$		0			0
$727$	18.5000	−	32.0429i	0.686127	−	1.18841i	−0.286954	−	0.957944i	$-0.592643\pi$
$727$	18.5000	−	32.0429i	0.686127	−	1.18841i	0.973081	−	0.230463i	$-0.0740239\pi$
$728$	−4.00000			−0.148250
$729$		0			0
$730$	8.00000			0.296093
$731$		0			0
$732$		0			0
$733$	2.00000	+	3.46410i	0.0738717	+	0.127950i	0.900595	−	0.434659i	$-0.143131\pi$
$733$	2.00000	+	3.46410i	0.0738717	+	0.127950i	−0.826723	+	0.562609i	$0.809798\pi$
$734$	−2.00000	−	3.46410i	−0.0738213	−	0.127862i
$735$		0			0
$736$	−1.50000	+	2.59808i	−0.0552907	+	0.0957664i
$737$	−1.00000			−0.0368355
$738$		0			0
$739$	−34.0000			−1.25071			−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000			−1.25071			−0.625355	+	0.780340i	$0.715046\pi$
$740$	−2.00000	+	3.46410i	−0.0735215	+	0.127343i
$741$		0			0
$742$		0			0
$743$	10.5000	+	18.1865i	0.385208	+	0.667199i	0.991798	−	0.127815i	$-0.0407965\pi$
$743$	10.5000	+	18.1865i	0.385208	+	0.667199i	−0.606590	+	0.795015i	$0.707463\pi$
$744$		0			0
$745$	7.50000	−	12.9904i	0.274779	−	0.475931i
$746$	4.00000			0.146450
$747$		0			0
$748$		0			0
$749$	−1.50000	+	2.59808i	−0.0548088	+	0.0949316i
$750$		0			0
$751$	−7.00000	−	12.1244i	−0.255434	−	0.442424i	0.709580	−	0.704625i	$-0.248885\pi$
$751$	−7.00000	−	12.1244i	−0.255434	−	0.442424i	−0.965013	+	0.262201i	$0.915552\pi$
$752$	−4.50000	−	7.79423i	−0.164098	−	0.284226i
$753$		0			0
$754$	−18.0000	+	31.1769i	−0.655521	+	1.13540i
$755$	10.0000			0.363937
$756$		0			0
$757$	−52.0000			−1.88997			−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000			−1.88997			−0.944986	+	0.327111i	$0.893925\pi$
$758$	−2.00000	+	3.46410i	−0.0726433	+	0.125822i
$759$		0			0
$760$	−1.00000	−	1.73205i	−0.0362738	−	0.0628281i
$761$	10.5000	+	18.1865i	0.380625	+	0.659261i	0.991152	−	0.132734i	$-0.0423756\pi$
$761$	10.5000	+	18.1865i	0.380625	+	0.659261i	−0.610527	+	0.791995i	$0.709042\pi$
$762$		0			0
$763$	2.50000	−	4.33013i	0.0905061	−	0.156761i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−5.50000	−	9.52628i	−0.198335	−	0.343526i	0.749654	−	0.661830i	$-0.230220\pi$
$769$	−5.50000	−	9.52628i	−0.198335	−	0.343526i	−0.947989	+	0.318304i	$0.896887\pi$
$770$	0.500000	+	0.866025i	0.0180187	+	0.0312094i
$771$		0			0
$772$	11.0000	−	19.0526i	0.395899	−	0.685717i
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$	2.00000			0.0718421
$776$	4.00000	−	6.92820i	0.143592	−	0.248708i
$777$		0			0
$778$	−7.50000	−	12.9904i	−0.268888	−	0.465728i
$779$	3.00000	+	5.19615i	0.107486	+	0.186171i
$780$		0			0
$781$	−6.00000	+	10.3923i	−0.214697	+	0.371866i
$782$		0			0
$783$		0			0
$784$	−6.00000			−0.214286
$785$	−11.0000	+	19.0526i	−0.392607	+	0.680015i
$786$		0			0
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	0.899468	−	0.436987i	$-0.143954\pi$
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	−0.828176	+	0.560469i	$0.810621\pi$
$788$	−9.00000	−	15.5885i	−0.320612	−	0.555316i
$789$		0			0
$790$	−1.00000	+	1.73205i	−0.0355784	+	0.0616236i
$791$	−6.00000			−0.213335
$792$		0			0
$793$	4.00000			0.142044
$794$	−8.00000	+	13.8564i	−0.283909	+	0.491745i
$795$		0			0
$796$	2.00000	+	3.46410i	0.0708881	+	0.122782i
$797$	24.0000	+	41.5692i	0.850124	+	1.47246i	0.881096	+	0.472937i	$0.156806\pi$
$797$	24.0000	+	41.5692i	0.850124	+	1.47246i	−0.0309726	+	0.999520i	$0.509860\pi$
$798$		0			0
$799$		0			0
$800$	−1.00000			−0.0353553
$801$		0			0
$802$	18.0000			0.635602
$803$	−4.00000	+	6.92820i	−0.141157	+	0.244491i
$804$		0			0
$805$	1.50000	+	2.59808i	0.0528681	+	0.0915702i
$806$	−4.00000	−	6.92820i	−0.140894	−	0.244036i
$807$		0			0
$808$	3.00000	−	5.19615i	0.105540	−	0.182800i
$809$	−42.0000			−1.47664			−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$	4.50000	−	7.79423i	0.157919	−	0.273524i
$813$		0			0
$814$	−2.00000	−	3.46410i	−0.0701000	−	0.121417i
$815$	10.0000	+	17.3205i	0.350285	+	0.606711i
$816$		0			0
$817$	4.00000	−	6.92820i	0.139942	−	0.242387i
$818$	−2.00000			−0.0699284
$819$		0			0
$820$	3.00000			0.104765
$821$	25.5000	−	44.1673i	0.889956	−	1.54145i	0.0500305	−	0.998748i	$-0.484068\pi$
$821$	25.5000	−	44.1673i	0.889956	−	1.54145i	0.839926	−	0.542702i	$-0.182599\pi$
$822$		0			0
$823$	9.50000	+	16.4545i	0.331149	+	0.573567i	0.982737	−	0.185006i	$-0.0592303\pi$
$823$	9.50000	+	16.4545i	0.331149	+	0.573567i	−0.651588	+	0.758573i	$0.725897\pi$
$824$	−8.00000	−	13.8564i	−0.278693	−	0.482711i
$825$		0			0
$826$		0			0
$827$	39.0000			1.35616			0.678081	−	0.734987i	$-0.262812\pi$
$827$	39.0000			1.35616			0.678081	+	0.734987i	$0.262812\pi$
$828$		0			0
$829$	−1.00000			−0.0347314			−0.0173657	−	0.999849i	$-0.505528\pi$
$829$	−1.00000			−0.0347314			−0.0173657	+	0.999849i	$0.505528\pi$
$830$	4.50000	−	7.79423i	0.156197	−	0.270542i
$831$		0			0
$832$	2.00000	+	3.46410i	0.0693375	+	0.120096i
$833$		0			0
$834$		0			0
$835$	1.50000	−	2.59808i	0.0519096	−	0.0899101i
$836$	2.00000			0.0691714
$837$		0			0
$838$	−12.0000			−0.414533
$839$	−15.0000	+	25.9808i	−0.517858	+	0.896956i	0.481927	+	0.876211i	$0.339937\pi$
$839$	−15.0000	+	25.9808i	−0.517858	+	0.896956i	−0.999785	+	0.0207443i	$0.993396\pi$
$840$		0			0
$841$	−26.0000	−	45.0333i	−0.896552	−	1.55287i
$842$	7.00000	+	12.1244i	0.241236	+	0.417833i
$843$		0			0
$844$	−4.00000	+	6.92820i	−0.137686	+	0.238479i
$845$	−3.00000			−0.103203
$846$		0			0
$847$	−1.00000			−0.0343604
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000	−	10.3923i	−0.205677	−	0.356244i
$852$		0			0
$853$	−1.00000	+	1.73205i	−0.0342393	+	0.0593043i	−0.882637	−	0.470055i	$-0.844234\pi$
$853$	−1.00000	+	1.73205i	−0.0342393	+	0.0593043i	0.848398	+	0.529359i	$0.177568\pi$
$854$	−1.00000			−0.0342193
$855$		0			0
$856$	3.00000			0.102538
$857$	24.0000	−	41.5692i	0.819824	−	1.41998i	−0.0859870	−	0.996296i	$-0.527404\pi$
$857$	24.0000	−	41.5692i	0.819824	−	1.41998i	0.905811	−	0.423681i	$-0.139262\pi$
$858$		0			0
$859$	−22.0000	−	38.1051i	−0.750630	−	1.30013i	−0.947518	−	0.319704i	$-0.896417\pi$
$859$	−22.0000	−	38.1051i	−0.750630	−	1.30013i	0.196887	−	0.980426i	$-0.436917\pi$
$860$	−2.00000	−	3.46410i	−0.0681994	−	0.118125i
$861$		0			0
$862$	−18.0000	+	31.1769i	−0.613082	+	1.06189i
$863$	39.0000			1.32758			0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000			1.32758			0.663788	+	0.747921i	$0.268948\pi$
$864$		0			0
$865$	−6.00000			−0.204006
$866$	1.00000	−	1.73205i	0.0339814	−	0.0588575i
$867$		0			0
$868$	1.00000	+	1.73205i	0.0339422	+	0.0587896i
$869$	−1.00000	−	1.73205i	−0.0339227	−	0.0587558i
$870$		0			0
$871$	−2.00000	+	3.46410i	−0.0677674	+	0.117377i
$872$	−5.00000			−0.169321
$873$		0			0
$874$	6.00000			0.202953
$875$	−0.500000	+	0.866025i	−0.0169031	+	0.0292770i
$876$		0			0
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	0.989788	−	0.142548i	$-0.0455296\pi$
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	−0.618344	+	0.785907i	$0.712196\pi$
$878$	−11.0000	−	19.0526i	−0.371232	−	0.642993i
$879$		0			0
$880$	0.500000	−	0.866025i	0.0168550	−	0.0291937i
$881$	−57.0000			−1.92038			−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000			−1.92038			−0.960189	+	0.279350i	$0.909881\pi$
$882$		0			0
$883$	−7.00000			−0.235569			−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000			−0.235569			−0.117784	+	0.993039i	$0.537579\pi$
$884$		0			0
$885$		0			0
$886$	13.5000	+	23.3827i	0.453541	+	0.785557i
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	0.915722	+	0.401813i	$0.131620\pi$
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	−0.109881	+	0.993945i	$0.535047\pi$
$888$		0			0
$889$	−3.50000	+	6.06218i	−0.117386	+	0.203319i
$890$	3.00000			0.100560
$891$		0			0
$892$	5.00000			0.167412
$893$	−9.00000	+	15.5885i	−0.301174	+	0.521648i
$894$		0			0
$895$	3.00000	+	5.19615i	0.100279	+	0.173688i
$896$	−0.500000	−	0.866025i	−0.0167038	−	0.0289319i
$897$		0			0
$898$	−3.00000	+	5.19615i	−0.100111	+	0.173398i
$899$	18.0000			0.600334
$900$		0			0
$901$		0			0
$902$	−1.50000	+	2.59808i	−0.0499445	+	0.0865065i
$903$		0			0
$904$	3.00000	+	5.19615i	0.0997785	+	0.172821i
$905$	−12.5000	−	21.6506i	−0.415514	−	0.719691i
$906$		0			0
$907$	24.5000	−	42.4352i	0.813509	−	1.40904i	−0.0968843	−	0.995296i	$-0.530888\pi$
$907$	24.5000	−	42.4352i	0.813509	−	1.40904i	0.910393	−	0.413744i	$-0.135779\pi$
$908$	−12.0000			−0.398234
$909$		0			0
$910$	4.00000			0.132599
$911$	3.00000	−	5.19615i	0.0993944	−	0.172156i	−0.812040	−	0.583602i	$-0.801643\pi$
$911$	3.00000	−	5.19615i	0.0993944	−	0.172156i	0.911434	+	0.411446i	$0.134976\pi$
$912$		0			0
$913$	4.50000	+	7.79423i	0.148928	+	0.257951i
$914$	13.0000	+	22.5167i	0.430002	+	0.744785i
$915$		0			0
$916$	−8.50000	+	14.7224i	−0.280848	+	0.486443i
$917$		0			0
$918$		0			0
$919$	44.0000			1.45143			0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000			1.45143			0.725713	+	0.687998i	$0.241510\pi$
$920$	1.50000	−	2.59808i	0.0494535	−	0.0856560i
$921$		0			0
$922$	−16.5000	−	28.5788i	−0.543399	−	0.941194i
$923$	24.0000	+	41.5692i	0.789970	+	1.36827i
$924$		0			0
$925$	2.00000	−	3.46410i	0.0657596	−	0.113899i
$926$	−32.0000			−1.05159
$927$		0			0
$928$	−9.00000			−0.295439
$929$	27.0000	−	46.7654i	0.885841	−	1.53432i	0.0410949	−	0.999155i	$-0.486915\pi$
$929$	27.0000	−	46.7654i	0.885841	−	1.53432i	0.844746	−	0.535167i	$-0.179751\pi$
$930$		0			0
$931$	6.00000	+	10.3923i	0.196642	+	0.340594i
$932$	12.0000	+	20.7846i	0.393073	+	0.680823i
$933$		0			0
$934$	−6.00000	+	10.3923i	−0.196326	+	0.340047i
$935$		0			0
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$	0.500000	−	0.866025i	0.0163256	−	0.0282767i
$939$		0			0
$940$	4.50000	+	7.79423i	0.146774	+	0.254220i
$941$	−7.50000	−	12.9904i	−0.244493	−	0.423474i	0.717496	−	0.696563i	$-0.245288\pi$
$941$	−7.50000	−	12.9904i	−0.244493	−	0.423474i	−0.961989	+	0.273088i	$0.911955\pi$
$942$		0			0
$943$	−4.50000	+	7.79423i	−0.146540	+	0.253815i
$944$		0			0
$945$		0			0
$946$	4.00000			0.130051
$947$	7.50000	−	12.9904i	0.243717	−	0.422131i	−0.718053	−	0.695988i	$-0.754966\pi$
$947$	7.50000	−	12.9904i	0.243717	−	0.422131i	0.961770	+	0.273858i	$0.0882998\pi$
$948$		0			0
$949$	16.0000	+	27.7128i	0.519382	+	0.899596i
$950$	1.00000	+	1.73205i	0.0324443	+	0.0561951i
$951$		0			0
$952$		0			0
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$		0			0
$956$	15.0000	−	25.9808i	0.485135	−	0.840278i
$957$		0			0
$958$	15.0000	+	25.9808i	0.484628	+	0.839400i
$959$	−6.00000	−	10.3923i	−0.193750	−	0.335585i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$	−16.0000			−0.515861
$963$		0			0
$964$	11.0000			0.354286
$965$	−11.0000	+	19.0526i	−0.354103	+	0.613324i
$966$		0			0
$967$	−11.5000	−	19.9186i	−0.369815	−	0.640538i	0.619721	−	0.784822i	$-0.287246\pi$
$967$	−11.5000	−	19.9186i	−0.369815	−	0.640538i	−0.989536	+	0.144283i	$0.953912\pi$
$968$	0.500000	+	0.866025i	0.0160706	+	0.0278351i
$969$		0			0
$970$	−4.00000	+	6.92820i	−0.128432	+	0.222451i
$971$	−36.0000			−1.15529			−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000			−1.15529			−0.577647	+	0.816286i	$0.696029\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$	−8.00000	+	13.8564i	−0.256337	+	0.443988i
$975$		0			0
$976$	0.500000	+	0.866025i	0.0160046	+	0.0277208i
$977$	24.0000	+	41.5692i	0.767828	+	1.32992i	0.938738	+	0.344631i	$0.111996\pi$
$977$	24.0000	+	41.5692i	0.767828	+	1.32992i	−0.170910	+	0.985287i	$0.554671\pi$
$978$		0			0
$979$	−1.50000	+	2.59808i	−0.0479402	+	0.0830349i
$980$	6.00000			0.191663
$981$		0			0
$982$	−18.0000			−0.574403
$983$	−4.50000	+	7.79423i	−0.143528	+	0.248597i	−0.928823	−	0.370525i	$-0.879178\pi$
$983$	−4.50000	+	7.79423i	−0.143528	+	0.248597i	0.785295	+	0.619122i	$0.212511\pi$
$984$		0			0
$985$	9.00000	+	15.5885i	0.286764	+	0.496690i
$986$		0			0
$987$		0			0
$988$	4.00000	−	6.92820i	0.127257	−	0.220416i
$989$	12.0000			0.381578
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$	1.00000	−	1.73205i	0.0317500	−	0.0549927i
$993$		0			0
$994$	−6.00000	−	10.3923i	−0.190308	−	0.329624i
$995$	−2.00000	−	3.46410i	−0.0634043	−	0.109819i
$996$		0			0
$997$	23.0000	−	39.8372i	0.728417	−	1.26166i	−0.229135	−	0.973395i	$-0.573590\pi$
$997$	23.0000	−	39.8372i	0.728417	−	1.26166i	0.957552	−	0.288261i	$-0.0930771\pi$
$998$	16.0000			0.506471
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2970.2.i.b.991.1		2
3.2	odd	2		990.2.i.a.331.1	✓	2
9.2	odd	6		8910.2.a.m.1.1		1
9.4	even	3	inner	2970.2.i.b.1981.1		2
9.5	odd	6		990.2.i.a.661.1	yes	2
9.7	even	3		8910.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.i.a.331.1	✓	2	3.2	odd	2
990.2.i.a.661.1	yes	2	9.5	odd	6
2970.2.i.b.991.1		2	1.1	even	1	trivial
2970.2.i.b.1981.1		2	9.4	even	3	inner
8910.2.a.c.1.1		1	9.7	even	3
8910.2.a.m.1.1		1	9.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2970 = 2 \cdot 3^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2970.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +1.00000 q^{10} +(-0.500000 + 0.866025i) q^{11} +(2.00000 + 3.46410i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +2.00000 q^{19} +(0.500000 - 0.866025i) q^{20} +(0.500000 + 0.866025i) q^{22} +(1.50000 + 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} +4.00000 q^{26} -1.00000 q^{28} +(-4.50000 + 7.79423i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} +1.00000 q^{35} -4.00000 q^{37} +(1.00000 - 1.73205i) q^{38} +(-0.500000 - 0.866025i) q^{40} +(1.50000 + 2.59808i) q^{41} +(2.00000 - 3.46410i) q^{43} +1.00000 q^{44} +3.00000 q^{46} +(-4.50000 + 7.79423i) q^{47} +(3.00000 + 5.19615i) q^{49} +(0.500000 + 0.866025i) q^{50} +(2.00000 - 3.46410i) q^{52} -1.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(4.50000 + 7.79423i) q^{58} +(0.500000 - 0.866025i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(0.500000 + 0.866025i) q^{67} +(0.500000 - 0.866025i) q^{70} +12.0000 q^{71} +8.00000 q^{73} +(-2.00000 + 3.46410i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(0.500000 + 0.866025i) q^{77} +(-1.00000 + 1.73205i) q^{79} -1.00000 q^{80} +3.00000 q^{82} +(4.50000 - 7.79423i) q^{83} +(-2.00000 - 3.46410i) q^{86} +(0.500000 - 0.866025i) q^{88} +3.00000 q^{89} +4.00000 q^{91} +(1.50000 - 2.59808i) q^{92} +(4.50000 + 7.79423i) q^{94} +(1.00000 + 1.73205i) q^{95} +(-4.00000 + 6.92820i) q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} - q^{16} + 4 q^{19} + q^{20} + q^{22} + 3 q^{23} - q^{25} + 8 q^{26} - 2 q^{28} - 9 q^{29} - 2 q^{31} + q^{32}+ \cdots + 12 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 + 4 * q^13 - q^14 - q^16 + 4 * q^19 + q^20 + q^22 + 3 * q^23 - q^25 + 8 * q^26 - 2 * q^28 - 9 * q^29 - 2 * q^31 + q^32 + 2 * q^35 - 8 * q^37 + 2 * q^38 - q^40 + 3 * q^41 + 4 * q^43 + 2 * q^44 + 6 * q^46 - 9 * q^47 + 6 * q^49 + q^50 + 4 * q^52 - 2 * q^55 - q^56 + 9 * q^58 + q^61 - 4 * q^62 + 2 * q^64 - 4 * q^65 + q^67 + q^70 + 24 * q^71 + 16 * q^73 - 4 * q^74 - 2 * q^76 + q^77 - 2 * q^79 - 2 * q^80 + 6 * q^82 + 9 * q^83 - 4 * q^86 + q^88 + 6 * q^89 + 8 * q^91 + 3 * q^92 + 9 * q^94 + 2 * q^95 - 8 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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