LMFDB - Embedded newform 416.2.ba.b.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(17,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.17"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.ba (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.49
Dual form			416.2.ba.b.17.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.633975 - 0.366025i) q^{3} -3.73205 q^{5} +(3.00000 + 1.73205i) q^{7} +(-1.23205 + 2.13397i) q^{9} +(1.00000 + 1.73205i) q^{11} +(2.59808 + 2.50000i) q^{13} +(-2.36603 + 1.36603i) q^{15} +(0.232051 - 0.401924i) q^{17} +(0.633975 - 1.09808i) q^{19} +2.53590 q^{21} +(4.09808 + 7.09808i) q^{23} +8.92820 q^{25} +4.00000i q^{27} +(-2.59808 + 1.50000i) q^{29} -4.73205i q^{31} +(1.26795 + 0.732051i) q^{33} +(-11.1962 - 6.46410i) q^{35} +(-2.13397 - 3.69615i) q^{37} +(2.56218 + 0.633975i) q^{39} +(-7.96410 + 4.59808i) q^{41} +(2.19615 + 1.26795i) q^{43} +(4.59808 - 7.96410i) q^{45} -6.73205i q^{47} +(2.50000 + 4.33013i) q^{49} -0.339746i q^{51} +3.92820i q^{53} +(-3.73205 - 6.46410i) q^{55} -0.928203i q^{57} +(0.267949 - 0.464102i) q^{59} +(-0.866025 - 0.500000i) q^{61} +(-7.39230 + 4.26795i) q^{63} +(-9.69615 - 9.33013i) q^{65} +(3.63397 + 6.29423i) q^{67} +(5.19615 + 3.00000i) q^{69} +(-8.02628 - 4.63397i) q^{71} -1.73205i q^{73} +(5.66025 - 3.26795i) q^{75} +6.92820i q^{77} +10.3923 q^{79} +(-2.23205 - 3.86603i) q^{81} +1.46410 q^{83} +(-0.866025 + 1.50000i) q^{85} +(-1.09808 + 1.90192i) q^{87} +(6.46410 - 3.73205i) q^{89} +(3.46410 + 12.0000i) q^{91} +(-1.73205 - 3.00000i) q^{93} +(-2.36603 + 4.09808i) q^{95} +(-5.19615 - 3.00000i) q^{97} -4.92820 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{3} - 8 q^{5} + 12 q^{7} + 2 q^{9} + 4 q^{11} - 6 q^{15} - 6 q^{17} + 6 q^{19} + 24 q^{21} + 6 q^{23} + 8 q^{25} + 12 q^{33} - 24 q^{35} - 12 q^{37} - 14 q^{39} - 18 q^{41} - 12 q^{43} + 8 q^{45}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + 6 * q^3 - 8 * q^5 + 12 * q^7 + 2 * q^9 + 4 * q^11 - 6 * q^15 - 6 * q^17 + 6 * q^19 + 24 * q^21 + 6 * q^23 + 8 * q^25 + 12 * q^33 - 24 * q^35 - 12 * q^37 - 14 * q^39 - 18 * q^41 - 12 * q^43 + 8 * q^45 + 10 * q^49 - 8 * q^55 + 8 * q^59 + 12 * q^63 - 18 * q^65 + 18 * q^67 + 6 * q^71 - 12 * q^75 - 2 * q^81 - 8 * q^83 + 6 * q^87 + 12 * q^89 - 6 * q^95 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.633975	−	0.366025i	0.366025	−	0.211325i	−0.305695	−	0.952129i	$-0.598889\pi$
$3$	0.633975	−	0.366025i	0.366025	−	0.211325i	0.671721	+	0.740805i	$0.265556\pi$
$4$		0			0
$5$	−3.73205			−1.66902			−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−3.73205			−1.66902			−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$	3.00000	+	1.73205i	1.13389	+	0.654654i	0.944911	−	0.327327i	$-0.106148\pi$
$7$	3.00000	+	1.73205i	1.13389	+	0.654654i	0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$	−1.23205	+	2.13397i	−0.410684	+	0.711325i
$10$		0			0
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$	2.59808	+	2.50000i	0.720577	+	0.693375i
$13$	2.59808	+	2.50000i	0.720577	+	0.693375i
$14$		0			0
$15$	−2.36603	+	1.36603i	−0.610905	+	0.352706i
$16$		0			0
$17$	0.232051	−	0.401924i	0.0562806	−	0.0974808i	−0.836512	−	0.547948i	$-0.815409\pi$
$17$	0.232051	−	0.401924i	0.0562806	−	0.0974808i	0.892793	+	0.450467i	$0.148743\pi$
$18$		0			0
$19$	0.633975	−	1.09808i	0.145444	−	0.251916i	−0.784095	−	0.620641i	$-0.786872\pi$
$19$	0.633975	−	1.09808i	0.145444	−	0.251916i	0.929538	+	0.368725i	$0.120206\pi$
$20$		0			0
$21$	2.53590			0.553378
$22$		0			0
$23$	4.09808	+	7.09808i	0.854508	+	1.48005i	0.877101	+	0.480306i	$0.159475\pi$
$23$	4.09808	+	7.09808i	0.854508	+	1.48005i	−0.0225928	+	0.999745i	$0.507192\pi$
$24$		0			0
$25$	8.92820			1.78564
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	−2.59808	+	1.50000i	−0.482451	+	0.278543i	−0.721437	−	0.692480i	$-0.756518\pi$
$29$	−2.59808	+	1.50000i	−0.482451	+	0.278543i	0.238987	+	0.971023i	$0.423185\pi$
$30$		0			0
$31$		−	4.73205i		−	0.849901i	−0.905216	−	0.424951i	$-0.860291\pi$
$31$		−	4.73205i		−	0.849901i	0.905216	−	0.424951i	$-0.139709\pi$
$32$		0			0
$33$	1.26795	+	0.732051i	0.220722	+	0.127434i
$34$		0			0
$35$	−11.1962	−	6.46410i	−1.89250	−	1.09263i
$36$		0			0
$37$	−2.13397	−	3.69615i	−0.350823	−	0.607644i	0.635571	−	0.772043i	$-0.280765\pi$
$37$	−2.13397	−	3.69615i	−0.350823	−	0.607644i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$	2.56218	+	0.633975i	0.410277	+	0.101517i
$40$		0			0
$41$	−7.96410	+	4.59808i	−1.24378	+	0.718099i	−0.969862	−	0.243653i	$-0.921654\pi$
$41$	−7.96410	+	4.59808i	−1.24378	+	0.718099i	−0.273921	+	0.961752i	$0.588321\pi$
$42$		0			0
$43$	2.19615	+	1.26795i	0.334910	+	0.193360i	0.658019	−	0.753001i	$-0.271395\pi$
$43$	2.19615	+	1.26795i	0.334910	+	0.193360i	−0.323109	+	0.946362i	$0.604728\pi$
$44$		0			0
$45$	4.59808	−	7.96410i	0.685441	−	1.18722i
$46$		0			0
$47$		−	6.73205i		−	0.981971i	−0.871168	−	0.490985i	$-0.836637\pi$
$47$		−	6.73205i		−	0.981971i	0.871168	−	0.490985i	$-0.163363\pi$
$48$		0			0
$49$	2.50000	+	4.33013i	0.357143	+	0.618590i
$50$		0			0
$51$		−	0.339746i		−	0.0475740i
$52$		0			0
$53$			3.92820i			0.539580i	0.962919	+	0.269790i	$0.0869543\pi$
$53$			3.92820i			0.539580i	−0.962919	+	0.269790i	$0.913046\pi$
$54$		0			0
$55$	−3.73205	−	6.46410i	−0.503230	−	0.871619i
$56$		0			0
$57$		−	0.928203i		−	0.122944i
$58$		0			0
$59$	0.267949	−	0.464102i	0.0348840	−	0.0604209i	−0.848056	−	0.529906i	$-0.822227\pi$
$59$	0.267949	−	0.464102i	0.0348840	−	0.0604209i	0.882940	+	0.469485i	$0.155560\pi$
$60$		0			0
$61$	−0.866025	−	0.500000i	−0.110883	−	0.0640184i	0.443533	−	0.896258i	$-0.353725\pi$
$61$	−0.866025	−	0.500000i	−0.110883	−	0.0640184i	−0.554416	+	0.832240i	$0.687058\pi$
$62$		0			0
$63$	−7.39230	+	4.26795i	−0.931343	+	0.537711i
$64$		0			0
$65$	−9.69615	−	9.33013i	−1.20266	−	1.15726i
$66$		0			0
$67$	3.63397	+	6.29423i	0.443961	+	0.768962i	0.997979	−	0.0635419i	$-0.0202397\pi$
$67$	3.63397	+	6.29423i	0.443961	+	0.768962i	−0.554019	+	0.832504i	$0.686906\pi$
$68$		0			0
$69$	5.19615	+	3.00000i	0.625543	+	0.361158i
$70$		0			0
$71$	−8.02628	−	4.63397i	−0.952544	−	0.549952i	−0.0586738	−	0.998277i	$-0.518687\pi$
$71$	−8.02628	−	4.63397i	−0.952544	−	0.549952i	−0.893870	+	0.448326i	$0.852021\pi$
$72$		0			0
$73$		−	1.73205i		−	0.202721i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$		−	1.73205i		−	0.202721i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$	5.66025	−	3.26795i	0.653590	−	0.377350i
$76$		0			0
$77$			6.92820i			0.789542i
$78$		0			0
$79$	10.3923			1.16923			0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923			1.16923			0.584613	+	0.811312i	$0.301246\pi$
$80$		0			0
$81$	−2.23205	−	3.86603i	−0.248006	−	0.429558i
$82$		0			0
$83$	1.46410			0.160706			0.0803530	−	0.996766i	$-0.474395\pi$
$83$	1.46410			0.160706			0.0803530	+	0.996766i	$0.474395\pi$
$84$		0			0
$85$	−0.866025	+	1.50000i	−0.0939336	+	0.162698i
$86$		0			0
$87$	−1.09808	+	1.90192i	−0.117726	+	0.203908i
$88$		0			0
$89$	6.46410	−	3.73205i	0.685193	−	0.395597i	−0.116615	−	0.993177i	$-0.537205\pi$
$89$	6.46410	−	3.73205i	0.685193	−	0.395597i	0.801809	+	0.597581i	$0.203871\pi$
$90$		0			0
$91$	3.46410	+	12.0000i	0.363137	+	1.25794i
$92$		0			0
$93$	−1.73205	−	3.00000i	−0.179605	−	0.311086i
$94$		0			0
$95$	−2.36603	+	4.09808i	−0.242749	+	0.420454i
$96$		0			0
$97$	−5.19615	−	3.00000i	−0.527589	−	0.304604i	0.212445	−	0.977173i	$-0.431857\pi$
$97$	−5.19615	−	3.00000i	−0.527589	−	0.304604i	−0.740034	+	0.672569i	$0.765191\pi$
$98$		0			0
$99$	−4.92820			−0.495303
$100$		0			0
$101$	9.99038	−	5.76795i	0.994080	−	0.573932i	0.0875887	−	0.996157i	$-0.472084\pi$
$101$	9.99038	−	5.76795i	0.994080	−	0.573932i	0.906491	+	0.422224i	$0.138751\pi$
$102$		0			0
$103$	−6.19615			−0.610525			−0.305263	−	0.952268i	$-0.598744\pi$
$103$	−6.19615			−0.610525			−0.305263	+	0.952268i	$0.598744\pi$
$104$		0			0
$105$	−9.46410			−0.923602
$106$		0			0
$107$	15.2942	−	8.83013i	1.47855	−	0.853641i	0.478843	−	0.877900i	$-0.341056\pi$
$107$	15.2942	−	8.83013i	1.47855	−	0.853641i	0.999706	+	0.0242598i	$0.00772291\pi$
$108$		0			0
$109$	−6.00000			−0.574696			−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000			−0.574696			−0.287348	+	0.957826i	$0.592774\pi$
$110$		0			0
$111$	−2.70577	−	1.56218i	−0.256820	−	0.148275i
$112$		0			0
$113$	1.50000	−	2.59808i	0.141108	−	0.244406i	−0.786806	−	0.617200i	$-0.788267\pi$
$113$	1.50000	−	2.59808i	0.141108	−	0.244406i	0.927914	+	0.372794i	$0.121600\pi$
$114$		0			0
$115$	−15.2942	−	26.4904i	−1.42619	−	2.47024i
$116$		0			0
$117$	−8.53590	+	2.46410i	−0.789144	+	0.227806i
$118$		0			0
$119$	1.39230	−	0.803848i	0.127632	−	0.0736886i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$	−3.36603	+	5.83013i	−0.303504	+	0.525685i
$124$		0			0
$125$	−14.6603			−1.31125
$126$		0			0
$127$	0.169873	+	0.294229i	0.0150738	+	0.0261086i	0.873464	−	0.486889i	$-0.161868\pi$
$127$	0.169873	+	0.294229i	0.0150738	+	0.0261086i	−0.858390	+	0.512997i	$0.828535\pi$
$128$		0			0
$129$	1.85641			0.163447
$130$		0			0
$131$		−	10.7321i		−	0.937664i	−0.883287	−	0.468832i	$-0.844675\pi$
$131$		−	10.7321i		−	0.937664i	0.883287	−	0.468832i	$-0.155325\pi$
$132$		0			0
$133$	3.80385	−	2.19615i	0.329835	−	0.190431i
$134$		0			0
$135$		−	14.9282i		−	1.28482i
$136$		0			0
$137$	−11.7679	−	6.79423i	−1.00540	−	0.580470i	−0.0955611	−	0.995424i	$-0.530465\pi$
$137$	−11.7679	−	6.79423i	−1.00540	−	0.580470i	−0.909843	+	0.414953i	$0.863798\pi$
$138$		0			0
$139$	8.19615	+	4.73205i	0.695189	+	0.401367i	0.805553	−	0.592524i	$-0.201868\pi$
$139$	8.19615	+	4.73205i	0.695189	+	0.401367i	−0.110364	+	0.993891i	$0.535202\pi$
$140$		0			0
$141$	−2.46410	−	4.26795i	−0.207515	−	0.359426i
$142$		0			0
$143$	−1.73205	+	7.00000i	−0.144841	+	0.585369i
$144$		0			0
$145$	9.69615	−	5.59808i	0.805222	−	0.464895i
$146$		0			0
$147$	3.16987	+	1.83013i	0.261447	+	0.150946i
$148$		0			0
$149$	5.86603	−	10.1603i	0.480564	−	0.832360i	−0.519188	−	0.854660i	$-0.673765\pi$
$149$	5.86603	−	10.1603i	0.480564	−	0.832360i	0.999751	+	0.0222997i	$0.00709882\pi$
$150$		0			0
$151$		−	2.19615i		−	0.178720i	−0.995999	−	0.0893602i	$-0.971518\pi$
$151$		−	2.19615i		−	0.178720i	0.995999	−	0.0893602i	$-0.0284822\pi$
$152$		0			0
$153$	0.571797	+	0.990381i	0.0462270	+	0.0800676i
$154$		0			0
$155$			17.6603i			1.41851i
$156$		0			0
$157$		−	3.92820i		−	0.313505i	−0.987638	−	0.156752i	$-0.949898\pi$
$157$		−	3.92820i		−	0.313505i	0.987638	−	0.156752i	$-0.0501025\pi$
$158$		0			0
$159$	1.43782	+	2.49038i	0.114027	+	0.197500i
$160$		0			0
$161$			28.3923i			2.23763i
$162$		0			0
$163$	−8.19615	+	14.1962i	−0.641972	+	1.11193i	0.343020	+	0.939328i	$0.388550\pi$
$163$	−8.19615	+	14.1962i	−0.641972	+	1.11193i	−0.984992	+	0.172600i	$0.944783\pi$
$164$		0			0
$165$	−4.73205	−	2.73205i	−0.368390	−	0.212690i
$166$		0			0
$167$	4.26795	−	2.46410i	0.330264	−	0.190678i	−0.325694	−	0.945475i	$-0.605598\pi$
$167$	4.26795	−	2.46410i	0.330264	−	0.190678i	0.655958	+	0.754797i	$0.272265\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	1.56218	+	2.70577i	0.119463	+	0.206916i
$172$		0			0
$173$	16.3923	+	9.46410i	1.24628	+	0.719542i	0.970366	−	0.241639i	$-0.0776850\pi$
$173$	16.3923	+	9.46410i	1.24628	+	0.719542i	0.275918	+	0.961181i	$0.411018\pi$
$174$		0			0
$175$	26.7846	+	15.4641i	2.02473	+	1.16898i
$176$		0			0
$177$		−	0.392305i		−	0.0294874i
$178$		0			0
$179$	12.0000	−	6.92820i	0.896922	−	0.517838i	0.0207218	−	0.999785i	$-0.493404\pi$
$179$	12.0000	−	6.92820i	0.896922	−	0.517838i	0.876200	+	0.481947i	$0.160070\pi$
$180$		0			0
$181$			18.4641i			1.37243i	0.727401	+	0.686213i	$0.240728\pi$
$181$			18.4641i			1.37243i	−0.727401	+	0.686213i	$0.759272\pi$
$182$		0			0
$183$	−0.732051			−0.0541148
$184$		0			0
$185$	7.96410	+	13.7942i	0.585532	+	1.01417i
$186$		0			0
$187$	0.928203			0.0678769
$188$		0			0
$189$	−6.92820	+	12.0000i	−0.503953	+	0.872872i
$190$		0			0
$191$	1.90192	−	3.29423i	0.137618	−	0.238362i	−0.788976	−	0.614424i	$-0.789389\pi$
$191$	1.90192	−	3.29423i	0.137618	−	0.238362i	0.926595	+	0.376062i	$0.122722\pi$
$192$		0			0
$193$	11.3038	−	6.52628i	0.813669	−	0.469772i	−0.0345595	−	0.999403i	$-0.511003\pi$
$193$	11.3038	−	6.52628i	0.813669	−	0.469772i	0.848228	+	0.529631i	$0.177670\pi$
$194$		0			0
$195$	−9.56218	−	2.36603i	−0.684762	−	0.169435i
$196$		0			0
$197$	6.26795	+	10.8564i	0.446573	+	0.773487i	0.998160	−	0.0606302i	$-0.0193110\pi$
$197$	6.26795	+	10.8564i	0.446573	+	0.773487i	−0.551587	+	0.834117i	$0.685978\pi$
$198$		0			0
$199$	−7.56218	+	13.0981i	−0.536069	+	0.928498i	0.463042	+	0.886336i	$0.346758\pi$
$199$	−7.56218	+	13.0981i	−0.536069	+	0.928498i	−0.999111	+	0.0421618i	$0.986576\pi$
$200$		0			0
$201$	4.60770	+	2.66025i	0.325002	+	0.187640i
$202$		0			0
$203$	−10.3923			−0.729397
$204$		0			0
$205$	29.7224	−	17.1603i	2.07590	−	1.19852i
$206$		0			0
$207$	−20.1962			−1.40373
$208$		0			0
$209$	2.53590			0.175412
$210$		0			0
$211$	−24.7583	+	14.2942i	−1.70443	+	0.984055i	−0.763285	+	0.646061i	$0.776415\pi$
$211$	−24.7583	+	14.2942i	−1.70443	+	0.984055i	−0.941148	+	0.337994i	$0.890252\pi$
$212$		0			0
$213$	−6.78461			−0.464874
$214$		0			0
$215$	−8.19615	−	4.73205i	−0.558973	−	0.322723i
$216$		0			0
$217$	8.19615	−	14.1962i	0.556391	−	0.963698i
$218$		0			0
$219$	−0.633975	−	1.09808i	−0.0428400	−	0.0742011i
$220$		0			0
$221$	1.60770	−	0.464102i	0.108145	−	0.0312189i
$222$		0			0
$223$	22.0981	−	12.7583i	1.47980	−	0.854361i	0.480058	−	0.877237i	$-0.340615\pi$
$223$	22.0981	−	12.7583i	1.47980	−	0.854361i	0.999738	+	0.0228756i	$0.00728216\pi$
$224$		0			0
$225$	−11.0000	+	19.0526i	−0.733333	+	1.27017i
$226$		0			0
$227$	13.0263	−	22.5622i	0.864585	−	1.49750i	−0.00287459	−	0.999996i	$-0.500915\pi$
$227$	13.0263	−	22.5622i	0.864585	−	1.49750i	0.867459	−	0.497508i	$-0.165752\pi$
$228$		0			0
$229$	−12.9282			−0.854320			−0.427160	−	0.904176i	$-0.640486\pi$
$229$	−12.9282			−0.854320			−0.427160	+	0.904176i	$0.640486\pi$
$230$		0			0
$231$	2.53590	+	4.39230i	0.166850	+	0.288992i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			25.1244i			1.63893i
$236$		0			0
$237$	6.58846	−	3.80385i	0.427966	−	0.247086i
$238$		0			0
$239$			18.3923i			1.18970i	0.803837	+	0.594850i	$0.202788\pi$
$239$			18.3923i			1.18970i	−0.803837	+	0.594850i	$0.797212\pi$
$240$		0			0
$241$	9.69615	+	5.59808i	0.624584	+	0.360604i	0.778652	−	0.627457i	$-0.215904\pi$
$241$	9.69615	+	5.59808i	0.624584	+	0.360604i	−0.154068	+	0.988060i	$0.549237\pi$
$242$		0			0
$243$	−13.2224	−	7.63397i	−0.848219	−	0.489720i
$244$		0			0
$245$	−9.33013	−	16.1603i	−0.596080	−	1.03244i
$246$		0			0
$247$	4.39230	−	1.26795i	0.279476	−	0.0806777i
$248$		0			0
$249$	0.928203	−	0.535898i	0.0588225	−	0.0339612i
$250$		0			0
$251$	−7.09808	−	4.09808i	−0.448027	−	0.258668i	0.258970	−	0.965885i	$-0.416617\pi$
$251$	−7.09808	−	4.09808i	−0.448027	−	0.258668i	−0.706996	+	0.707217i	$0.749950\pi$
$252$		0			0
$253$	−8.19615	+	14.1962i	−0.515288	+	0.892504i
$254$		0			0
$255$			1.26795i			0.0794021i
$256$		0			0
$257$	−11.4282	−	19.7942i	−0.712872	−	1.23473i	−0.963775	−	0.266718i	$-0.914061\pi$
$257$	−11.4282	−	19.7942i	−0.712872	−	1.23473i	0.250903	−	0.968012i	$-0.419273\pi$
$258$		0			0
$259$		−	14.7846i		−	0.918671i
$260$		0			0
$261$		−	7.39230i		−	0.457572i
$262$		0			0
$263$	8.19615	+	14.1962i	0.505396	+	0.875372i	0.999981	+	0.00624249i	$0.00198706\pi$
$263$	8.19615	+	14.1962i	0.505396	+	0.875372i	−0.494584	+	0.869130i	$0.664680\pi$
$264$		0			0
$265$		−	14.6603i		−	0.900572i
$266$		0			0
$267$	2.73205	−	4.73205i	0.167199	−	0.289597i
$268$		0			0
$269$	−8.19615	−	4.73205i	−0.499728	−	0.288518i	0.228873	−	0.973456i	$-0.426496\pi$
$269$	−8.19615	−	4.73205i	−0.499728	−	0.288518i	−0.728601	+	0.684938i	$0.759829\pi$
$270$		0			0
$271$	15.0000	−	8.66025i	0.911185	−	0.526073i	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	15.0000	−	8.66025i	0.911185	−	0.526073i	0.880812	+	0.473466i	$0.156997\pi$
$272$		0			0
$273$	6.58846	+	6.33975i	0.398752	+	0.383699i
$274$		0			0
$275$	8.92820	+	15.4641i	0.538391	+	0.932520i
$276$		0			0
$277$	−12.4019	−	7.16025i	−0.745159	−	0.430218i	0.0787828	−	0.996892i	$-0.474897\pi$
$277$	−12.4019	−	7.16025i	−0.745159	−	0.430218i	−0.823942	+	0.566674i	$0.808230\pi$
$278$		0			0
$279$	10.0981	+	5.83013i	0.604556	+	0.349041i
$280$		0			0
$281$			10.6603i			0.635937i	0.948101	+	0.317969i	$0.103001\pi$
$281$			10.6603i			0.635937i	−0.948101	+	0.317969i	$0.896999\pi$
$282$		0			0
$283$	26.6603	−	15.3923i	1.58479	−	0.914978i	0.590641	−	0.806934i	$-0.298875\pi$
$283$	26.6603	−	15.3923i	1.58479	−	0.914978i	0.994146	−	0.108043i	$-0.0344586\pi$
$284$		0			0
$285$			3.46410i			0.205196i
$286$		0			0
$287$	−31.8564			−1.88042
$288$		0			0
$289$	8.39230	+	14.5359i	0.493665	+	0.855053i
$290$		0			0
$291$	−4.39230			−0.257481
$292$		0			0
$293$	−1.40192	+	2.42820i	−0.0819013	+	0.141857i	−0.904067	−	0.427392i	$-0.859433\pi$
$293$	−1.40192	+	2.42820i	−0.0819013	+	0.141857i	0.822165	+	0.569249i	$0.192766\pi$
$294$		0			0
$295$	−1.00000	+	1.73205i	−0.0582223	+	0.100844i
$296$		0			0
$297$	−6.92820	+	4.00000i	−0.402015	+	0.232104i
$298$		0			0
$299$	−7.09808	+	28.6865i	−0.410492	+	1.65899i
$300$		0			0
$301$	4.39230	+	7.60770i	0.253168	+	0.438500i
$302$		0			0
$303$	4.22243	−	7.31347i	0.242572	−	0.420148i
$304$		0			0
$305$	3.23205	+	1.86603i	0.185067	+	0.106848i
$306$		0			0
$307$	2.19615			0.125341			0.0626705	−	0.998034i	$-0.480038\pi$
$307$	2.19615			0.125341			0.0626705	+	0.998034i	$0.480038\pi$
$308$		0			0
$309$	−3.92820	+	2.26795i	−0.223468	+	0.129019i
$310$		0			0
$311$	18.5885			1.05405			0.527027	−	0.849848i	$-0.323307\pi$
$311$	18.5885			1.05405			0.527027	+	0.849848i	$0.323307\pi$
$312$		0			0
$313$	−2.53590			−0.143337			−0.0716687	−	0.997428i	$-0.522832\pi$
$313$	−2.53590			−0.143337			−0.0716687	+	0.997428i	$0.522832\pi$
$314$		0			0
$315$	27.5885	−	15.9282i	1.55443	−	0.897453i
$316$		0			0
$317$	−25.1962			−1.41516			−0.707578	−	0.706635i	$-0.750212\pi$
$317$	−25.1962			−1.41516			−0.707578	+	0.706635i	$0.750212\pi$
$318$		0			0
$319$	−5.19615	−	3.00000i	−0.290929	−	0.167968i
$320$		0			0
$321$	6.46410	−	11.1962i	0.360791	−	0.624908i
$322$		0			0
$323$	−0.294229	−	0.509619i	−0.0163713	−	0.0283560i
$324$		0			0
$325$	23.1962	+	22.3205i	1.28669	+	1.23812i
$326$		0			0
$327$	−3.80385	+	2.19615i	−0.210353	+	0.121448i
$328$		0			0
$329$	11.6603	−	20.1962i	0.642851	−	1.11345i
$330$		0			0
$331$	6.00000	−	10.3923i	0.329790	−	0.571213i	−0.652680	−	0.757634i	$-0.726355\pi$
$331$	6.00000	−	10.3923i	0.329790	−	0.571213i	0.982470	+	0.186421i	$0.0596888\pi$
$332$		0			0
$333$	10.5167			0.576309
$334$		0			0
$335$	−13.5622	−	23.4904i	−0.740981	−	1.28342i
$336$		0			0
$337$	−33.2487			−1.81117			−0.905586	−	0.424162i	$-0.860569\pi$
$337$	−33.2487			−1.81117			−0.905586	+	0.424162i	$0.860569\pi$
$338$		0			0
$339$		−	2.19615i		−	0.119279i
$340$		0			0
$341$	8.19615	−	4.73205i	0.443847	−	0.256255i
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$		0			0
$345$	−19.3923	−	11.1962i	−1.04405	−	0.602781i
$346$		0			0
$347$	−1.09808	−	0.633975i	−0.0589478	−	0.0340335i	0.470236	−	0.882540i	$-0.344169\pi$
$347$	−1.09808	−	0.633975i	−0.0589478	−	0.0340335i	−0.529184	+	0.848507i	$0.677502\pi$
$348$		0			0
$349$	−2.66025	−	4.60770i	−0.142400	−	0.246644i	0.786000	−	0.618227i	$-0.212149\pi$
$349$	−2.66025	−	4.60770i	−0.142400	−	0.246644i	−0.928400	+	0.371582i	$0.878815\pi$
$350$		0			0
$351$	−10.0000	+	10.3923i	−0.533761	+	0.554700i
$352$		0			0
$353$	−15.8205	+	9.13397i	−0.842041	+	0.486152i	−0.857957	−	0.513721i	$-0.828267\pi$
$353$	−15.8205	+	9.13397i	−0.842041	+	0.486152i	0.0159167	+	0.999873i	$0.494933\pi$
$354$		0			0
$355$	29.9545	+	17.2942i	1.58982	+	0.917882i
$356$		0			0
$357$	0.588457	−	1.01924i	0.0311445	−	0.0539438i
$358$		0			0
$359$			13.0718i			0.689903i	0.938621	+	0.344952i	$0.112105\pi$
$359$			13.0718i			0.689903i	−0.938621	+	0.344952i	$0.887895\pi$
$360$		0			0
$361$	8.69615	+	15.0622i	0.457692	+	0.792746i
$362$		0			0
$363$		−	5.12436i		−	0.268959i
$364$		0			0
$365$			6.46410i			0.338347i
$366$		0			0
$367$	−14.0981	−	24.4186i	−0.735914	−	1.27464i	−0.954321	−	0.298782i	$-0.903420\pi$
$367$	−14.0981	−	24.4186i	−0.735914	−	1.27464i	0.218408	−	0.975858i	$-0.429914\pi$
$368$		0			0
$369$		−	22.6603i		−	1.17965i
$370$		0			0
$371$	−6.80385	+	11.7846i	−0.353238	+	0.611826i
$372$		0			0
$373$	−22.3301	−	12.8923i	−1.15621	−	0.667538i	−0.205817	−	0.978590i	$-0.565985\pi$
$373$	−22.3301	−	12.8923i	−1.15621	−	0.667538i	−0.950393	+	0.311052i	$0.899319\pi$
$374$		0			0
$375$	−9.29423	+	5.36603i	−0.479952	+	0.277100i
$376$		0			0
$377$	−10.5000	−	2.59808i	−0.540778	−	0.133808i
$378$		0			0
$379$	−11.0263	−	19.0981i	−0.566382	−	0.981002i	−0.996920	−	0.0784297i	$-0.975009\pi$
$379$	−11.0263	−	19.0981i	−0.566382	−	0.981002i	0.430538	−	0.902573i	$-0.358324\pi$
$380$		0			0
$381$	0.215390	+	0.124356i	0.0110348	+	0.00637093i
$382$		0			0
$383$	−17.3205	−	10.0000i	−0.885037	−	0.510976i	−0.0127209	−	0.999919i	$-0.504049\pi$
$383$	−17.3205	−	10.0000i	−0.885037	−	0.510976i	−0.872316	+	0.488943i	$0.837383\pi$
$384$		0			0
$385$		−	25.8564i		−	1.31776i
$386$		0			0
$387$	−5.41154	+	3.12436i	−0.275084	+	0.158820i
$388$		0			0
$389$		−	18.7128i		−	0.948777i	−0.880316	−	0.474389i	$-0.842669\pi$
$389$		−	18.7128i		−	0.948777i	0.880316	−	0.474389i	$-0.157331\pi$
$390$		0			0
$391$	3.80385			0.192369
$392$		0			0
$393$	−3.92820	−	6.80385i	−0.198152	−	0.343209i
$394$		0			0
$395$	−38.7846			−1.95147
$396$		0			0
$397$	12.1244	−	21.0000i	0.608504	−	1.05396i	−0.382983	−	0.923755i	$-0.625103\pi$
$397$	12.1244	−	21.0000i	0.608504	−	1.05396i	0.991487	−	0.130204i	$-0.0415634\pi$
$398$		0			0
$399$	1.60770	−	2.78461i	0.0804854	−	0.139405i
$400$		0			0
$401$	8.42820	−	4.86603i	0.420884	−	0.242998i	−0.274571	−	0.961567i	$-0.588536\pi$
$401$	8.42820	−	4.86603i	0.420884	−	0.242998i	0.695456	+	0.718569i	$0.255203\pi$
$402$		0			0
$403$	11.8301	−	12.2942i	0.589301	−	0.612419i
$404$		0			0
$405$	8.33013	+	14.4282i	0.413927	+	0.716943i
$406$		0			0
$407$	4.26795	−	7.39230i	0.211554	−	0.366423i
$408$		0			0
$409$	0.696152	+	0.401924i	0.0344225	+	0.0198739i	0.517113	−	0.855917i	$-0.327007\pi$
$409$	0.696152	+	0.401924i	0.0344225	+	0.0198739i	−0.482690	+	0.875791i	$0.660340\pi$
$410$		0			0
$411$	−9.94744			−0.490671
$412$		0			0
$413$	1.60770	−	0.928203i	0.0791095	−	0.0456739i
$414$		0			0
$415$	−5.46410			−0.268222
$416$		0			0
$417$	6.92820			0.339276
$418$		0			0
$419$	−16.9019	+	9.75833i	−0.825713	+	0.476726i	−0.852383	−	0.522919i	$-0.824843\pi$
$419$	−16.9019	+	9.75833i	−0.825713	+	0.476726i	0.0266696	+	0.999644i	$0.491510\pi$
$420$		0			0
$421$	30.1244			1.46817			0.734086	−	0.679057i	$-0.237611\pi$
$421$	30.1244			1.46817			0.734086	+	0.679057i	$0.237611\pi$
$422$		0			0
$423$	14.3660	+	8.29423i	0.698500	+	0.403279i
$424$		0			0
$425$	2.07180	−	3.58846i	0.100497	−	0.174066i
$426$		0			0
$427$	−1.73205	−	3.00000i	−0.0838198	−	0.145180i
$428$		0			0
$429$	1.46410	+	5.07180i	0.0706875	+	0.244869i
$430$		0			0
$431$	−15.4641	+	8.92820i	−0.744880	+	0.430056i	−0.823841	−	0.566821i	$-0.808173\pi$
$431$	−15.4641	+	8.92820i	−0.744880	+	0.430056i	0.0789612	+	0.996878i	$0.474840\pi$
$432$		0			0
$433$	−10.5000	+	18.1865i	−0.504598	+	0.873989i	0.495388	+	0.868672i	$0.335026\pi$
$433$	−10.5000	+	18.1865i	−0.504598	+	0.873989i	−0.999986	+	0.00531724i	$0.998307\pi$
$434$		0			0
$435$	4.09808	−	7.09808i	0.196488	−	0.340327i
$436$		0			0
$437$	10.3923			0.497131
$438$		0			0
$439$	7.80385	+	13.5167i	0.372457	+	0.645115i	0.989943	−	0.141467i	$-0.0451820\pi$
$439$	7.80385	+	13.5167i	0.372457	+	0.645115i	−0.617486	+	0.786582i	$0.711849\pi$
$440$		0			0
$441$	−12.3205			−0.586691
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−24.1244	+	13.9282i	−1.14360	+	0.660260i
$446$		0			0
$447$		−	8.58846i		−	0.406220i
$448$		0			0
$449$	18.4641	+	10.6603i	0.871375	+	0.503088i	0.867805	−	0.496905i	$-0.165530\pi$
$449$	18.4641	+	10.6603i	0.871375	+	0.503088i	0.00356996	+	0.999994i	$0.498864\pi$
$450$		0			0
$451$	−15.9282	−	9.19615i	−0.750030	−	0.433030i
$452$		0			0
$453$	−0.803848	−	1.39230i	−0.0377681	−	0.0654162i
$454$		0			0
$455$	−12.9282	−	44.7846i	−0.606084	−	2.09953i
$456$		0			0
$457$	6.69615	−	3.86603i	0.313233	−	0.180845i	−0.335139	−	0.942169i	$-0.608783\pi$
$457$	6.69615	−	3.86603i	0.313233	−	0.180845i	0.648372	+	0.761324i	$0.275450\pi$
$458$		0			0
$459$	1.60770	+	0.928203i	0.0750408	+	0.0433248i
$460$		0			0
$461$	14.3301	−	24.8205i	0.667421	−	1.15601i	−0.311202	−	0.950344i	$-0.600732\pi$
$461$	14.3301	−	24.8205i	0.667421	−	1.15601i	0.978623	−	0.205663i	$-0.0659350\pi$
$462$		0			0
$463$			16.3923i			0.761815i	0.924613	+	0.380908i	$0.124388\pi$
$463$			16.3923i			0.761815i	−0.924613	+	0.380908i	$0.875612\pi$
$464$		0			0
$465$	6.46410	+	11.1962i	0.299766	+	0.519209i
$466$		0			0
$467$			20.5359i			0.950288i	0.879908	+	0.475144i	$0.157604\pi$
$467$			20.5359i			0.950288i	−0.879908	+	0.475144i	$0.842396\pi$
$468$		0			0
$469$			25.1769i			1.16256i
$470$		0			0
$471$	−1.43782	−	2.49038i	−0.0662513	−	0.114751i
$472$		0			0
$473$			5.07180i			0.233201i
$474$		0			0
$475$	5.66025	−	9.80385i	0.259710	−	0.449831i
$476$		0			0
$477$	−8.38269	−	4.83975i	−0.383817	−	0.221597i
$478$		0			0
$479$	25.3468	−	14.6340i	1.15812	−	0.668643i	0.207271	−	0.978284i	$-0.433542\pi$
$479$	25.3468	−	14.6340i	1.15812	−	0.668643i	0.950854	+	0.309640i	$0.100209\pi$
$480$		0			0
$481$	3.69615	−	14.9378i	0.168530	−	0.681106i
$482$		0			0
$483$	10.3923	+	18.0000i	0.472866	+	0.819028i
$484$		0			0
$485$	19.3923	+	11.1962i	0.880559	+	0.508391i
$486$		0			0
$487$	−20.1962	−	11.6603i	−0.915175	−	0.528377i	−0.0330824	−	0.999453i	$-0.510532\pi$
$487$	−20.1962	−	11.6603i	−0.915175	−	0.528377i	−0.882093	+	0.471076i	$0.843866\pi$
$488$		0			0
$489$			12.0000i			0.542659i
$490$		0			0
$491$	−6.58846	+	3.80385i	−0.297333	+	0.171665i	−0.641244	−	0.767337i	$-0.721581\pi$
$491$	−6.58846	+	3.80385i	−0.297333	+	0.171665i	0.343911	+	0.939002i	$0.388248\pi$
$492$		0			0
$493$			1.39230i			0.0627063i
$494$		0			0
$495$	18.3923			0.826673
$496$		0			0
$497$	−16.0526	−	27.8038i	−0.720056	−	1.24717i
$498$		0			0
$499$	8.87564			0.397328			0.198664	−	0.980068i	$-0.436340\pi$
$499$	8.87564			0.397328			0.198664	+	0.980068i	$0.436340\pi$
$500$		0			0
$501$	1.80385	−	3.12436i	0.0805900	−	0.139586i
$502$		0			0
$503$	4.39230	−	7.60770i	0.195843	−	0.339210i	−0.751333	−	0.659923i	$-0.770589\pi$
$503$	4.39230	−	7.60770i	0.195843	−	0.339210i	0.947177	+	0.320712i	$0.103922\pi$
$504$		0			0
$505$	−37.2846	+	21.5263i	−1.65914	+	0.957907i
$506$		0			0
$507$	5.07180	+	8.05256i	0.225246	+	0.357627i
$508$		0			0
$509$	−3.93782	−	6.82051i	−0.174541	−	0.302314i	0.765461	−	0.643482i	$-0.222511\pi$
$509$	−3.93782	−	6.82051i	−0.174541	−	0.302314i	−0.940002	+	0.341168i	$0.889177\pi$
$510$		0			0
$511$	3.00000	−	5.19615i	0.132712	−	0.229864i
$512$		0			0
$513$	4.39230	+	2.53590i	0.193925	+	0.111963i
$514$		0			0
$515$	23.1244			1.01898
$516$		0			0
$517$	11.6603	−	6.73205i	0.512817	−	0.296075i
$518$		0			0
$519$	13.8564			0.608229
$520$		0			0
$521$	9.24871			0.405193			0.202597	−	0.979262i	$-0.435062\pi$
$521$	9.24871			0.405193			0.202597	+	0.979262i	$0.435062\pi$
$522$		0			0
$523$	−0.803848	+	0.464102i	−0.0351498	+	0.0202937i	−0.517472	−	0.855700i	$-0.673127\pi$
$523$	−0.803848	+	0.464102i	−0.0351498	+	0.0202937i	0.482322	+	0.875994i	$0.339793\pi$
$524$		0			0
$525$	22.6410			0.988135
$526$		0			0
$527$	−1.90192	−	1.09808i	−0.0828491	−	0.0478330i
$528$		0			0
$529$	−22.0885	+	38.2583i	−0.960368	+	1.66341i
$530$		0			0
$531$	0.660254	+	1.14359i	0.0286526	+	0.0496277i
$532$		0			0
$533$	−32.1865	−	7.96410i	−1.39415	−	0.344964i
$534$		0			0
$535$	−57.0788	+	32.9545i	−2.46773	+	1.42475i
$536$		0			0
$537$	5.07180	−	8.78461i	0.218864	−	0.379084i
$538$		0			0
$539$	−5.00000	+	8.66025i	−0.215365	+	0.373024i
$540$		0			0
$541$	−35.4449			−1.52389			−0.761947	−	0.647640i	$-0.775756\pi$
$541$	−35.4449			−1.52389			−0.761947	+	0.647640i	$0.775756\pi$
$542$		0			0
$543$	6.75833	+	11.7058i	0.290028	+	0.502343i
$544$		0			0
$545$	22.3923			0.959181
$546$		0			0
$547$		−	18.5885i		−	0.794785i	−0.917649	−	0.397393i	$-0.869915\pi$
$547$		−	18.5885i		−	0.794785i	0.917649	−	0.397393i	$-0.130085\pi$
$548$		0			0
$549$	2.13397	−	1.23205i	0.0910758	−	0.0525826i
$550$		0			0
$551$			3.80385i			0.162049i
$552$		0			0
$553$	31.1769	+	18.0000i	1.32578	+	0.765438i
$554$		0			0
$555$	10.0981	+	5.83013i	0.428639	+	0.247475i
$556$		0			0
$557$	2.40192	+	4.16025i	0.101773	+	0.176276i	0.912415	−	0.409266i	$-0.134215\pi$
$557$	2.40192	+	4.16025i	0.101773	+	0.176276i	−0.810642	+	0.585542i	$0.800882\pi$
$558$		0			0
$559$	2.53590	+	8.78461i	0.107257	+	0.371549i
$560$		0			0
$561$	0.588457	−	0.339746i	0.0248447	−	0.0143441i
$562$		0			0
$563$	13.9808	+	8.07180i	0.589219	+	0.340186i	0.764789	−	0.644281i	$-0.222843\pi$
$563$	13.9808	+	8.07180i	0.589219	+	0.340186i	−0.175570	+	0.984467i	$0.556177\pi$
$564$		0			0
$565$	−5.59808	+	9.69615i	−0.235513	+	0.407920i
$566$		0			0
$567$		−	15.4641i		−	0.649431i
$568$		0			0
$569$	12.0000	+	20.7846i	0.503066	+	0.871336i	0.999994	+	0.00354413i	$0.00112814\pi$
$569$	12.0000	+	20.7846i	0.503066	+	0.871336i	−0.496928	+	0.867792i	$0.665539\pi$
$570$		0			0
$571$			2.39230i			0.100115i	0.998746	+	0.0500574i	$0.0159404\pi$
$571$			2.39230i			0.100115i	−0.998746	+	0.0500574i	$0.984060\pi$
$572$		0			0
$573$		−	2.78461i		−	0.116329i
$574$		0			0
$575$	36.5885	+	63.3731i	1.52584	+	2.64284i
$576$		0			0
$577$		−	34.2679i		−	1.42659i	−0.700862	−	0.713297i	$-0.747201\pi$
$577$		−	34.2679i		−	1.42659i	0.700862	−	0.713297i	$-0.252799\pi$
$578$		0			0
$579$	4.77757	−	8.27499i	0.198549	−	0.343897i
$580$		0			0
$581$	4.39230	+	2.53590i	0.182224	+	0.105207i
$582$		0			0
$583$	−6.80385	+	3.92820i	−0.281787	+	0.162690i
$584$		0			0
$585$	31.8564	−	9.19615i	1.31710	−	0.380214i
$586$		0			0
$587$	−22.5885	−	39.1244i	−0.932325	−	1.61483i	−0.779335	−	0.626607i	$-0.784443\pi$
$587$	−22.5885	−	39.1244i	−0.932325	−	1.61483i	−0.152990	−	0.988228i	$-0.548890\pi$
$588$		0			0
$589$	−5.19615	−	3.00000i	−0.214104	−	0.123613i
$590$		0			0
$591$	7.94744	+	4.58846i	0.326914	+	0.188744i
$592$		0			0
$593$			1.58846i			0.0652301i	0.999468	+	0.0326151i	$0.0103835\pi$
$593$			1.58846i			0.0652301i	−0.999468	+	0.0326151i	$0.989616\pi$
$594$		0			0
$595$	−5.19615	+	3.00000i	−0.213021	+	0.122988i
$596$		0			0
$597$			11.0718i			0.453138i
$598$		0			0
$599$	13.2679			0.542114			0.271057	−	0.962563i	$-0.412627\pi$
$599$	13.2679			0.542114			0.271057	+	0.962563i	$0.412627\pi$
$600$		0			0
$601$	−13.0359	−	22.5788i	−0.531745	−	0.921010i	−0.999313	−	0.0370529i	$-0.988203\pi$
$601$	−13.0359	−	22.5788i	−0.531745	−	0.921010i	0.467568	−	0.883957i	$-0.345130\pi$
$602$		0			0
$603$	−17.9090			−0.729309
$604$		0			0
$605$	−13.0622	+	22.6244i	−0.531053	+	0.919811i
$606$		0			0
$607$	1.29423	−	2.24167i	0.0525311	−	0.0909866i	−0.838564	−	0.544803i	$-0.816604\pi$
$607$	1.29423	−	2.24167i	0.0525311	−	0.0909866i	0.891095	+	0.453816i	$0.149938\pi$
$608$		0			0
$609$	−6.58846	+	3.80385i	−0.266978	+	0.154140i
$610$		0			0
$611$	16.8301	−	17.4904i	0.680874	−	0.707585i
$612$		0			0
$613$	−20.2583	−	35.0885i	−0.818226	−	1.41721i	−0.906988	−	0.421157i	$-0.861624\pi$
$613$	−20.2583	−	35.0885i	−0.818226	−	1.41721i	0.0887617	−	0.996053i	$-0.471709\pi$
$614$		0			0
$615$	12.5622	−	21.7583i	0.506556	−	0.877381i
$616$		0			0
$617$	15.3564	+	8.86603i	0.618226	+	0.356933i	0.776178	−	0.630514i	$-0.217156\pi$
$617$	15.3564	+	8.86603i	0.618226	+	0.356933i	−0.157952	+	0.987447i	$0.550489\pi$
$618$		0			0
$619$	−24.2487			−0.974638			−0.487319	−	0.873224i	$-0.662025\pi$
$619$	−24.2487			−0.974638			−0.487319	+	0.873224i	$0.662025\pi$
$620$		0			0
$621$	−28.3923	+	16.3923i	−1.13934	+	0.657801i
$622$		0			0
$623$	25.8564			1.03592
$624$		0			0
$625$	10.0718			0.402872
$626$		0			0
$627$	1.60770	−	0.928203i	0.0642052	−	0.0370689i
$628$		0			0
$629$	−1.98076			−0.0789782
$630$		0			0
$631$	14.7846	+	8.53590i	0.588566	+	0.339809i	0.764530	−	0.644588i	$-0.222971\pi$
$631$	14.7846	+	8.53590i	0.588566	+	0.339809i	−0.175964	+	0.984397i	$0.556304\pi$
$632$		0			0
$633$	−10.4641	+	18.1244i	−0.415911	+	0.720378i
$634$		0			0
$635$	−0.633975	−	1.09808i	−0.0251585	−	0.0435758i
$636$		0			0
$637$	−4.33013	+	17.5000i	−0.171566	+	0.693375i
$638$		0			0
$639$	19.7776	−	11.4186i	0.782389	−	0.451712i
$640$		0			0
$641$	7.03590	−	12.1865i	0.277901	−	0.481339i	−0.692962	−	0.720974i	$-0.743695\pi$
$641$	7.03590	−	12.1865i	0.277901	−	0.481339i	0.970863	+	0.239635i	$0.0770279\pi$
$642$		0			0
$643$	−19.3923	+	33.5885i	−0.764758	+	1.32460i	0.175617	+	0.984459i	$0.443808\pi$
$643$	−19.3923	+	33.5885i	−0.764758	+	1.32460i	−0.940375	+	0.340141i	$0.889525\pi$
$644$		0			0
$645$	−6.92820			−0.272798
$646$		0			0
$647$	14.8301	+	25.6865i	0.583032	+	1.00984i	0.995118	+	0.0986965i	$0.0314673\pi$
$647$	14.8301	+	25.6865i	0.583032	+	1.00984i	−0.412085	+	0.911145i	$0.635199\pi$
$648$		0			0
$649$	1.07180			0.0420717
$650$		0			0
$651$		−	12.0000i		−	0.470317i
$652$		0			0
$653$	31.1769	−	18.0000i	1.22005	−	0.704394i	0.255119	−	0.966910i	$-0.417885\pi$
$653$	31.1769	−	18.0000i	1.22005	−	0.704394i	0.964928	+	0.262515i	$0.0845520\pi$
$654$		0			0
$655$			40.0526i			1.56498i
$656$		0			0
$657$	3.69615	+	2.13397i	0.144201	+	0.0832543i
$658$		0			0
$659$	26.7846	+	15.4641i	1.04338	+	0.602396i	0.920789	−	0.390061i	$-0.127546\pi$
$659$	26.7846	+	15.4641i	1.04338	+	0.602396i	0.122591	+	0.992457i	$0.460880\pi$
$660$		0			0
$661$	−6.06218	−	10.5000i	−0.235791	−	0.408403i	0.723711	−	0.690103i	$-0.242435\pi$
$661$	−6.06218	−	10.5000i	−0.235791	−	0.408403i	−0.959502	+	0.281701i	$0.909102\pi$
$662$		0			0
$663$	0.849365	−	0.882686i	0.0329866	−	0.0342807i
$664$		0			0
$665$	−14.1962	+	8.19615i	−0.550503	+	0.317833i
$666$		0			0
$667$	−21.2942	−	12.2942i	−0.824516	−	0.476034i
$668$		0			0
$669$	9.33975	−	16.1769i	0.361095	−	0.625436i
$670$		0			0
$671$		−	2.00000i		−	0.0772091i
$672$		0			0
$673$	5.50000	+	9.52628i	0.212009	+	0.367211i	0.952343	−	0.305028i	$-0.0986659\pi$
$673$	5.50000	+	9.52628i	0.212009	+	0.367211i	−0.740334	+	0.672239i	$0.765333\pi$
$674$		0			0
$675$			35.7128i			1.37459i
$676$		0			0
$677$		−	7.85641i		−	0.301946i	−0.988538	−	0.150973i	$-0.951759\pi$
$677$		−	7.85641i		−	0.301946i	0.988538	−	0.150973i	$-0.0482407\pi$
$678$		0			0
$679$	−10.3923	−	18.0000i	−0.398820	−	0.690777i
$680$		0			0
$681$		−	19.0718i		−	0.730833i
$682$		0			0
$683$	5.63397	−	9.75833i	0.215578	−	0.373392i	−0.737873	−	0.674939i	$-0.764170\pi$
$683$	5.63397	−	9.75833i	0.215578	−	0.373392i	0.953451	+	0.301547i	$0.0975031\pi$
$684$		0			0
$685$	43.9186	+	25.3564i	1.67804	+	0.968818i
$686$		0			0
$687$	−8.19615	+	4.73205i	−0.312703	+	0.180539i
$688$		0			0
$689$	−9.82051	+	10.2058i	−0.374132	+	0.388809i
$690$		0			0
$691$	9.00000	+	15.5885i	0.342376	+	0.593013i	0.984873	−	0.173275i	$-0.0554350\pi$
$691$	9.00000	+	15.5885i	0.342376	+	0.593013i	−0.642497	+	0.766288i	$0.722102\pi$
$692$		0			0
$693$	−14.7846	−	8.53590i	−0.561621	−	0.324252i
$694$		0			0
$695$	−30.5885	−	17.6603i	−1.16029	−	0.669892i
$696$		0			0
$697$			4.26795i			0.161660i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			8.78461i			0.331790i	0.986143	+	0.165895i	$0.0530513\pi$
$701$			8.78461i			0.331790i	−0.986143	+	0.165895i	$0.946949\pi$
$702$		0			0
$703$	−5.41154			−0.204100
$704$		0			0
$705$	9.19615	+	15.9282i	0.346347	+	0.599891i
$706$		0			0
$707$	39.9615			1.50291
$708$		0			0
$709$	−20.5981	+	35.6769i	−0.773577	+	1.33987i	0.162014	+	0.986788i	$0.448201\pi$
$709$	−20.5981	+	35.6769i	−0.773577	+	1.33987i	−0.935591	+	0.353086i	$0.885132\pi$
$710$		0			0
$711$	−12.8038	+	22.1769i	−0.480182	+	0.831699i
$712$		0			0
$713$	33.5885	−	19.3923i	1.25790	−	0.726248i
$714$		0			0
$715$	6.46410	−	26.1244i	0.241744	−	0.976996i
$716$		0			0
$717$	6.73205	+	11.6603i	0.251413	+	0.435460i
$718$		0			0
$719$	−16.8564	+	29.1962i	−0.628638	+	1.08883i	0.359187	+	0.933265i	$0.383054\pi$
$719$	−16.8564	+	29.1962i	−0.628638	+	1.08883i	−0.987825	+	0.155567i	$0.950279\pi$
$720$		0			0
$721$	−18.5885	−	10.7321i	−0.692270	−	0.399682i
$722$		0			0
$723$	8.19615			0.304818
$724$		0			0
$725$	−23.1962	+	13.3923i	−0.861483	+	0.497378i
$726$		0			0
$727$	13.6077			0.504681			0.252341	−	0.967638i	$-0.418800\pi$
$727$	13.6077			0.504681			0.252341	+	0.967638i	$0.418800\pi$
$728$		0			0
$729$	2.21539			0.0820515
$730$		0			0
$731$	1.01924	−	0.588457i	0.0376979	−	0.0217649i
$732$		0			0
$733$	−4.94744			−0.182738			−0.0913690	−	0.995817i	$-0.529124\pi$
$733$	−4.94744			−0.182738			−0.0913690	+	0.995817i	$0.529124\pi$
$734$		0			0
$735$	−11.8301	−	6.83013i	−0.436361	−	0.251933i
$736$		0			0
$737$	−7.26795	+	12.5885i	−0.267718	+	0.463702i
$738$		0			0
$739$	9.46410	+	16.3923i	0.348143	+	0.603001i	0.985920	−	0.167220i	$-0.0534792\pi$
$739$	9.46410	+	16.3923i	0.348143	+	0.603001i	−0.637777	+	0.770221i	$0.720146\pi$
$740$		0			0
$741$	2.32051	−	2.41154i	0.0852460	−	0.0885902i
$742$		0			0
$743$	−4.73205	+	2.73205i	−0.173602	+	0.100229i	−0.584283	−	0.811550i	$-0.698624\pi$
$743$	−4.73205	+	2.73205i	−0.173602	+	0.100229i	0.410681	+	0.911779i	$0.365291\pi$
$744$		0			0
$745$	−21.8923	+	37.9186i	−0.802072	+	1.38923i
$746$		0			0
$747$	−1.80385	+	3.12436i	−0.0659993	+	0.114314i
$748$		0			0
$749$	61.1769			2.23536
$750$		0			0
$751$	−18.1962	−	31.5167i	−0.663987	−	1.15006i	−0.979559	−	0.201158i	$-0.935530\pi$
$751$	−18.1962	−	31.5167i	−0.663987	−	1.15006i	0.315572	−	0.948902i	$-0.397804\pi$
$752$		0			0
$753$	−6.00000			−0.218652
$754$		0			0
$755$			8.19615i			0.298289i
$756$		0			0
$757$	18.3397	−	10.5885i	0.666569	−	0.384844i	−0.128206	−	0.991748i	$-0.540922\pi$
$757$	18.3397	−	10.5885i	0.666569	−	0.384844i	0.794776	+	0.606904i	$0.207589\pi$
$758$		0			0
$759$			12.0000i			0.435572i
$760$		0			0
$761$	−10.8564	−	6.26795i	−0.393544	−	0.227213i	0.290150	−	0.956981i	$-0.406295\pi$
$761$	−10.8564	−	6.26795i	−0.393544	−	0.227213i	−0.683695	+	0.729768i	$0.739628\pi$
$762$		0			0
$763$	−18.0000	−	10.3923i	−0.651644	−	0.376227i
$764$		0			0
$765$	−2.13397	−	3.69615i	−0.0771540	−	0.133635i
$766$		0			0
$767$	1.85641	−	0.535898i	0.0670310	−	0.0193502i
$768$		0			0
$769$	19.3923	−	11.1962i	0.699304	−	0.403744i	−0.107784	−	0.994174i	$-0.534375\pi$
$769$	19.3923	−	11.1962i	0.699304	−	0.403744i	0.807088	+	0.590431i	$0.201042\pi$
$770$		0			0
$771$	−14.4904	−	8.36603i	−0.521858	−	0.301295i
$772$		0			0
$773$	−26.2487	+	45.4641i	−0.944101	+	1.63523i	−0.186558	+	0.982444i	$0.559733\pi$
$773$	−26.2487	+	45.4641i	−0.944101	+	1.63523i	−0.757542	+	0.652786i	$0.773600\pi$
$774$		0			0
$775$		−	42.2487i		−	1.51762i
$776$		0			0
$777$	−5.41154	−	9.37307i	−0.194138	−	0.336257i
$778$		0			0
$779$			11.6603i			0.417772i
$780$		0			0
$781$		−	18.5359i		−	0.663267i
$782$		0			0
$783$	−6.00000	−	10.3923i	−0.214423	−	0.371391i
$784$		0			0
$785$			14.6603i			0.523247i
$786$		0			0
$787$	−20.3660	+	35.2750i	−0.725970	+	1.25742i	0.232603	+	0.972572i	$0.425276\pi$
$787$	−20.3660	+	35.2750i	−0.725970	+	1.25742i	−0.958573	+	0.284846i	$0.908058\pi$
$788$		0			0
$789$	10.3923	+	6.00000i	0.369976	+	0.213606i
$790$		0			0
$791$	9.00000	−	5.19615i	0.320003	−	0.184754i
$792$		0			0
$793$	−1.00000	−	3.46410i	−0.0355110	−	0.123014i
$794$		0			0
$795$	−5.36603	−	9.29423i	−0.190313	−	0.329632i
$796$		0			0
$797$	−15.8038	−	9.12436i	−0.559801	−	0.323201i	0.193265	−	0.981147i	$-0.438092\pi$
$797$	−15.8038	−	9.12436i	−0.559801	−	0.323201i	−0.753066	+	0.657945i	$0.771426\pi$
$798$		0			0
$799$	−2.70577	−	1.56218i	−0.0957233	−	0.0552659i
$800$		0			0
$801$			18.3923i			0.649860i
$802$		0			0
$803$	3.00000	−	1.73205i	0.105868	−	0.0611227i
$804$		0			0
$805$		−	105.962i		−	3.73465i
$806$		0			0
$807$	−6.92820			−0.243884
$808$		0			0
$809$	−3.35641	−	5.81347i	−0.118005	−	0.204391i	0.800972	−	0.598702i	$-0.204317\pi$
$809$	−3.35641	−	5.81347i	−0.118005	−	0.204391i	−0.918977	+	0.394311i	$0.870983\pi$
$810$		0			0
$811$	32.4449			1.13929			0.569647	−	0.821890i	$-0.307080\pi$
$811$	32.4449			1.13929			0.569647	+	0.821890i	$0.307080\pi$
$812$		0			0
$813$	6.33975	−	10.9808i	0.222345	−	0.385112i
$814$		0			0
$815$	30.5885	−	52.9808i	1.07147	−	1.85584i
$816$		0			0
$817$	2.78461	−	1.60770i	0.0974212	−	0.0562461i
$818$		0			0
$819$	−29.8756	−	7.39230i	−1.04394	−	0.258308i
$820$		0			0
$821$	17.0000	+	29.4449i	0.593304	+	1.02763i	0.993784	+	0.111327i	$0.0355102\pi$
$821$	17.0000	+	29.4449i	0.593304	+	1.02763i	−0.400480	+	0.916306i	$0.631157\pi$
$822$		0			0
$823$	15.7846	−	27.3397i	0.550217	−	0.953004i	−0.448042	−	0.894013i	$-0.647878\pi$
$823$	15.7846	−	27.3397i	0.550217	−	0.953004i	0.998259	−	0.0589909i	$-0.0187883\pi$
$824$		0			0
$825$	11.3205	+	6.53590i	0.394130	+	0.227551i
$826$		0			0
$827$	−27.5692			−0.958676			−0.479338	−	0.877630i	$-0.659123\pi$
$827$	−27.5692			−0.958676			−0.479338	+	0.877630i	$0.659123\pi$
$828$		0			0
$829$	−16.4545	+	9.50000i	−0.571488	+	0.329949i	−0.757743	−	0.652553i	$-0.773698\pi$
$829$	−16.4545	+	9.50000i	−0.571488	+	0.329949i	0.186256	+	0.982501i	$0.440365\pi$
$830$		0			0
$831$	−10.4833			−0.363663
$832$		0			0
$833$	2.32051			0.0804008
$834$		0			0
$835$	−15.9282	+	9.19615i	−0.551218	+	0.318246i
$836$		0			0
$837$	18.9282			0.654254
$838$		0			0
$839$	29.8301	+	17.2224i	1.02985	+	0.594584i	0.916942	−	0.399021i	$-0.130650\pi$
$839$	29.8301	+	17.2224i	1.02985	+	0.594584i	0.112909	+	0.993605i	$0.463983\pi$
$840$		0			0
$841$	−10.0000	+	17.3205i	−0.344828	+	0.597259i
$842$		0			0
$843$	3.90192	+	6.75833i	0.134389	+	0.232769i
$844$		0			0
$845$	−1.86603	−	48.4808i	−0.0641932	−	1.66779i
$846$		0			0
$847$	21.0000	−	12.1244i	0.721569	−	0.416598i
$848$		0			0
$849$	11.2679	−	19.5167i	0.386715	−	0.669810i
$850$		0			0
$851$	17.4904	−	30.2942i	0.599563	−	1.03847i
$852$		0			0
$853$	23.1962			0.794221			0.397111	−	0.917771i	$-0.370013\pi$
$853$	23.1962			0.794221			0.397111	+	0.917771i	$0.370013\pi$
$854$		0			0
$855$	−5.83013	−	10.0981i	−0.199386	−	0.345347i
$856$		0			0
$857$	−9.67949			−0.330645			−0.165323	−	0.986240i	$-0.552867\pi$
$857$	−9.67949			−0.330645			−0.165323	+	0.986240i	$0.552867\pi$
$858$		0			0
$859$		−	19.8038i		−	0.675699i	−0.941200	−	0.337849i	$-0.890301\pi$
$859$		−	19.8038i		−	0.675699i	0.941200	−	0.337849i	$-0.109699\pi$
$860$		0			0
$861$	−20.1962	+	11.6603i	−0.688283	+	0.397380i
$862$		0			0
$863$			9.32051i			0.317274i	0.987337	+	0.158637i	$0.0507099\pi$
$863$			9.32051i			0.317274i	−0.987337	+	0.158637i	$0.949290\pi$
$864$		0			0
$865$	−61.1769	−	35.3205i	−2.08008	−	1.20093i
$866$		0			0
$867$	10.6410	+	6.14359i	0.361388	+	0.208647i
$868$		0			0
$869$	10.3923	+	18.0000i	0.352535	+	0.610608i
$870$		0			0
$871$	−6.29423	+	25.4378i	−0.213272	+	0.861928i
$872$		0			0
$873$	12.8038	−	7.39230i	0.433345	−	0.250192i
$874$		0			0
$875$	−43.9808	−	25.3923i	−1.48682	−	0.858417i
$876$		0			0
$877$	−2.59808	+	4.50000i	−0.0877308	+	0.151954i	−0.906552	−	0.422095i	$-0.861295\pi$
$877$	−2.59808	+	4.50000i	−0.0877308	+	0.151954i	0.818821	+	0.574049i	$0.194628\pi$
$878$		0			0
$879$			2.05256i			0.0692311i
$880$		0			0
$881$	−14.3038	−	24.7750i	−0.481909	−	0.834691i	0.517876	−	0.855456i	$-0.326723\pi$
$881$	−14.3038	−	24.7750i	−0.481909	−	0.834691i	−0.999784	+	0.0207653i	$0.993390\pi$
$882$		0			0
$883$		−	42.1962i		−	1.42001i	−0.704195	−	0.710007i	$-0.748692\pi$
$883$		−	42.1962i		−	1.42001i	0.704195	−	0.710007i	$-0.251308\pi$
$884$		0			0
$885$			1.46410i			0.0492152i
$886$		0			0
$887$	−17.1962	−	29.7846i	−0.577390	−	1.00007i	−0.995777	−	0.0918005i	$-0.970738\pi$
$887$	−17.1962	−	29.7846i	−0.577390	−	1.00007i	0.418387	−	0.908269i	$-0.362596\pi$
$888$		0			0
$889$			1.17691i			0.0394724i
$890$		0			0
$891$	4.46410	−	7.73205i	0.149553	−	0.259033i
$892$		0			0
$893$	−7.39230	−	4.26795i	−0.247374	−	0.142821i
$894$		0			0
$895$	−44.7846	+	25.8564i	−1.49698	+	0.864284i
$896$		0			0
$897$	6.00000	+	20.7846i	0.200334	+	0.693978i
$898$		0			0
$899$	7.09808	+	12.2942i	0.236734	+	0.410035i
$900$		0			0
$901$	1.57884	+	0.911543i	0.0525987	+	0.0303679i
$902$		0			0
$903$	5.56922	+	3.21539i	0.185332	+	0.107001i
$904$		0			0
$905$		−	68.9090i		−	2.29061i
$906$		0			0
$907$	−4.48334	+	2.58846i	−0.148867	+	0.0859483i	−0.572583	−	0.819847i	$-0.694059\pi$
$907$	−4.48334	+	2.58846i	−0.148867	+	0.0859483i	0.423716	+	0.905795i	$0.360725\pi$
$908$		0			0
$909$			28.4256i			0.942818i
$910$		0			0
$911$	−42.4974			−1.40800			−0.704001	−	0.710199i	$-0.748605\pi$
$911$	−42.4974			−1.40800			−0.704001	+	0.710199i	$0.748605\pi$
$912$		0			0
$913$	1.46410	+	2.53590i	0.0484547	+	0.0839260i
$914$		0			0
$915$	2.73205			0.0903188
$916$		0			0
$917$	18.5885	−	32.1962i	0.613845	−	1.06321i
$918$		0			0
$919$	−12.7583	+	22.0981i	−0.420858	+	0.728948i	−0.996024	−	0.0890890i	$-0.971604\pi$
$919$	−12.7583	+	22.0981i	−0.420858	+	0.728948i	0.575165	+	0.818037i	$0.304938\pi$
$920$		0			0
$921$	1.39230	−	0.803848i	0.0458780	−	0.0264877i
$922$		0			0
$923$	−9.26795	−	32.1051i	−0.305058	−	1.05675i
$924$		0			0
$925$	−19.0526	−	33.0000i	−0.626444	−	1.08503i
$926$		0			0
$927$	7.63397	−	13.2224i	0.250733	−	0.434282i
$928$		0			0
$929$	23.2128	+	13.4019i	0.761588	+	0.439703i	0.829865	−	0.557963i	$-0.188417\pi$
$929$	23.2128	+	13.4019i	0.761588	+	0.439703i	−0.0682778	+	0.997666i	$0.521750\pi$
$930$		0			0
$931$	6.33975			0.207777
$932$		0			0
$933$	11.7846	−	6.80385i	0.385811	−	0.222748i
$934$		0			0
$935$	−3.46410			−0.113288
$936$		0			0
$937$	−43.6410			−1.42569			−0.712845	−	0.701322i	$-0.752594\pi$
$937$	−43.6410			−1.42569			−0.712845	+	0.701322i	$0.752594\pi$
$938$		0			0
$939$	−1.60770	+	0.928203i	−0.0524651	+	0.0302908i
$940$		0			0
$941$	−7.21539			−0.235215			−0.117608	−	0.993060i	$-0.537522\pi$
$941$	−7.21539			−0.235215			−0.117608	+	0.993060i	$0.537522\pi$
$942$		0			0
$943$	−65.2750	−	37.6865i	−2.12565	−	1.22724i
$944$		0			0
$945$	25.8564	−	44.7846i	0.841109	−	1.45684i
$946$		0			0
$947$	24.3468	+	42.1699i	0.791164	+	1.37034i	0.925246	+	0.379367i	$0.123858\pi$
$947$	24.3468	+	42.1699i	0.791164	+	1.37034i	−0.134082	+	0.990970i	$0.542809\pi$
$948$		0			0
$949$	4.33013	−	4.50000i	0.140562	−	0.146076i
$950$		0			0
$951$	−15.9737	+	9.22243i	−0.517983	+	0.299058i
$952$		0			0
$953$	−12.4641	+	21.5885i	−0.403752	+	0.699319i	−0.994175	−	0.107775i	$-0.965627\pi$
$953$	−12.4641	+	21.5885i	−0.403752	+	0.699319i	0.590423	+	0.807094i	$0.298961\pi$
$954$		0			0
$955$	−7.09808	+	12.2942i	−0.229688	+	0.397832i
$956$		0			0
$957$	−4.39230			−0.141983
$958$		0			0
$959$	−23.5359	−	40.7654i	−0.760014	−	1.31638i
$960$		0			0
$961$	8.60770			0.277668
$962$		0			0
$963$			43.5167i			1.40230i
$964$		0			0
$965$	−42.1865	+	24.3564i	−1.35803	+	0.784060i
$966$		0			0
$967$			8.87564i			0.285421i	0.989764	+	0.142711i	$0.0455819\pi$
$967$			8.87564i			0.285421i	−0.989764	+	0.142711i	$0.954418\pi$
$968$		0			0
$969$	−0.373067	−	0.215390i	−0.0119846	−	0.00691933i
$970$		0			0
$971$	5.70577	+	3.29423i	0.183107	+	0.105717i	0.588752	−	0.808314i	$-0.299620\pi$
$971$	5.70577	+	3.29423i	0.183107	+	0.105717i	−0.405645	+	0.914031i	$0.632953\pi$
$972$		0			0
$973$	16.3923	+	28.3923i	0.525513	+	0.910216i
$974$		0			0
$975$	22.8756	+	5.66025i	0.732607	+	0.181273i
$976$		0			0
$977$	28.1603	−	16.2583i	0.900926	−	0.520150i	0.0234257	−	0.999726i	$-0.492543\pi$
$977$	28.1603	−	16.2583i	0.900926	−	0.520150i	0.877501	+	0.479576i	$0.159209\pi$
$978$		0			0
$979$	12.9282	+	7.46410i	0.413187	+	0.238554i
$980$		0			0
$981$	7.39230	−	12.8038i	0.236018	−	0.408795i
$982$		0			0
$983$			59.7654i			1.90622i	0.302625	+	0.953110i	$0.402137\pi$
$983$			59.7654i			1.90622i	−0.302625	+	0.953110i	$0.597863\pi$
$984$		0			0
$985$	−23.3923	−	40.5167i	−0.745341	−	1.29097i
$986$		0			0
$987$		−	17.0718i		−	0.543401i
$988$		0			0
$989$			20.7846i			0.660912i
$990$		0			0
$991$	11.2942	+	19.5622i	0.358773	+	0.621413i	0.987756	−	0.156006i	$-0.0498619\pi$
$991$	11.2942	+	19.5622i	0.358773	+	0.621413i	−0.628983	+	0.777419i	$0.716529\pi$
$992$		0			0
$993$		−	8.78461i		−	0.278771i
$994$		0			0
$995$	28.2224	−	48.8827i	0.894711	−	1.54969i
$996$		0			0
$997$	45.9904	+	26.5526i	1.45653	+	0.840928i	0.998839	−	0.0481831i	$-0.0153431\pi$
$997$	45.9904	+	26.5526i	1.45653	+	0.840928i	0.457691	+	0.889111i	$0.348676\pi$
$998$		0			0
$999$	14.7846	−	8.53590i	0.467764	−	0.270064i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.ba.b.49.1		4
4.3	odd	2		104.2.s.a.101.1	yes	4
8.3	odd	2		104.2.s.b.101.2	yes	4
8.5	even	2		416.2.ba.a.49.2		4
12.11	even	2		936.2.dg.b.829.2		4
13.4	even	6		416.2.ba.a.17.2		4
24.11	even	2		936.2.dg.a.829.1		4
52.43	odd	6		104.2.s.b.69.2	yes	4
104.43	odd	6		104.2.s.a.69.2	✓	4
104.69	even	6	inner	416.2.ba.b.17.1		4
156.95	even	6		936.2.dg.a.901.1		4
312.251	even	6		936.2.dg.b.901.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.s.a.69.2	✓	4	104.43	odd	6
104.2.s.a.101.1	yes	4	4.3	odd	2
104.2.s.b.69.2	yes	4	52.43	odd	6
104.2.s.b.101.2	yes	4	8.3	odd	2
416.2.ba.a.17.2		4	13.4	even	6
416.2.ba.a.49.2		4	8.5	even	2
416.2.ba.b.17.1		4	104.69	even	6	inner
416.2.ba.b.49.1		4	1.1	even	1	trivial
936.2.dg.a.829.1		4	24.11	even	2
936.2.dg.a.901.1		4	156.95	even	6
936.2.dg.b.829.2		4	12.11	even	2
936.2.dg.b.901.1		4	312.251	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.ba (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.633975 - 0.366025i) q^{3} -3.73205 q^{5} +(3.00000 + 1.73205i) q^{7} +(-1.23205 + 2.13397i) q^{9} +(1.00000 + 1.73205i) q^{11} +(2.59808 + 2.50000i) q^{13} +(-2.36603 + 1.36603i) q^{15} +(0.232051 - 0.401924i) q^{17} +(0.633975 - 1.09808i) q^{19} +2.53590 q^{21} +(4.09808 + 7.09808i) q^{23} +8.92820 q^{25} +4.00000i q^{27} +(-2.59808 + 1.50000i) q^{29} -4.73205i q^{31} +(1.26795 + 0.732051i) q^{33} +(-11.1962 - 6.46410i) q^{35} +(-2.13397 - 3.69615i) q^{37} +(2.56218 + 0.633975i) q^{39} +(-7.96410 + 4.59808i) q^{41} +(2.19615 + 1.26795i) q^{43} +(4.59808 - 7.96410i) q^{45} -6.73205i q^{47} +(2.50000 + 4.33013i) q^{49} -0.339746i q^{51} +3.92820i q^{53} +(-3.73205 - 6.46410i) q^{55} -0.928203i q^{57} +(0.267949 - 0.464102i) q^{59} +(-0.866025 - 0.500000i) q^{61} +(-7.39230 + 4.26795i) q^{63} +(-9.69615 - 9.33013i) q^{65} +(3.63397 + 6.29423i) q^{67} +(5.19615 + 3.00000i) q^{69} +(-8.02628 - 4.63397i) q^{71} -1.73205i q^{73} +(5.66025 - 3.26795i) q^{75} +6.92820i q^{77} +10.3923 q^{79} +(-2.23205 - 3.86603i) q^{81} +1.46410 q^{83} +(-0.866025 + 1.50000i) q^{85} +(-1.09808 + 1.90192i) q^{87} +(6.46410 - 3.73205i) q^{89} +(3.46410 + 12.0000i) q^{91} +(-1.73205 - 3.00000i) q^{93} +(-2.36603 + 4.09808i) q^{95} +(-5.19615 - 3.00000i) q^{97} -4.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} - 8 q^{5} + 12 q^{7} + 2 q^{9} + 4 q^{11} - 6 q^{15} - 6 q^{17} + 6 q^{19} + 24 q^{21} + 6 q^{23} + 8 q^{25} + 12 q^{33} - 24 q^{35} - 12 q^{37} - 14 q^{39} - 18 q^{41} - 12 q^{43} + 8 q^{45}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + 6 * q^3 - 8 * q^5 + 12 * q^7 + 2 * q^9 + 4 * q^11 - 6 * q^15 - 6 * q^17 + 6 * q^19 + 24 * q^21 + 6 * q^23 + 8 * q^25 + 12 * q^33 - 24 * q^35 - 12 * q^37 - 14 * q^39 - 18 * q^41 - 12 * q^43 + 8 * q^45 + 10 * q^49 - 8 * q^55 + 8 * q^59 + 12 * q^63 - 18 * q^65 + 18 * q^67 + 6 * q^71 - 12 * q^75 - 2 * q^81 - 8 * q^83 + 6 * q^87 + 12 * q^89 - 6 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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