LMFDB - Embedded newform 108.4.i.a.49.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(13,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 8]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, degree $6$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			49.9
Character	$\chi$	$=$	108.49
Dual form			108.4.i.a.97.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.17716 - 0.443910i) q^{3} +(-4.80495 + 1.74886i) q^{5} +(5.74341 + 32.5725i) q^{7} +(26.6059 - 4.59638i) q^{9} +O(q^{10})$	\(q+(5.17716 - 0.443910i) q^{3} +(-4.80495 + 1.74886i) q^{5} +(5.74341 + 32.5725i) q^{7} +(26.6059 - 4.59638i) q^{9} +(28.2317 + 10.2755i) q^{11} +(7.59849 + 6.37589i) q^{13} +(-24.0996 + 11.1871i) q^{15} +(44.7319 - 77.4779i) q^{17} +(-29.6509 - 51.3568i) q^{19} +(44.1938 + 166.083i) q^{21} +(-17.0562 + 96.7304i) q^{23} +(-75.7266 + 63.5421i) q^{25} +(135.702 - 35.6068i) q^{27} +(108.085 - 90.6938i) q^{29} +(4.10141 - 23.2603i) q^{31} +(150.721 + 40.6655i) q^{33} +(-84.5615 - 146.465i) q^{35} +(-114.087 + 197.604i) q^{37} +(42.1689 + 29.6359i) q^{39} +(-357.814 - 300.241i) q^{41} +(-10.1135 - 3.68102i) q^{43} +(-119.801 + 68.6153i) q^{45} +(-66.0561 - 374.623i) q^{47} +(-705.667 + 256.842i) q^{49} +(197.191 - 420.972i) q^{51} -202.984 q^{53} -153.622 q^{55} +(-176.305 - 252.720i) q^{57} +(766.955 - 279.149i) q^{59} +(-109.702 - 622.151i) q^{61} +(302.524 + 840.222i) q^{63} +(-47.6609 - 17.3471i) q^{65} +(-466.269 - 391.246i) q^{67} +(-45.3629 + 508.360i) q^{69} +(140.376 - 243.138i) q^{71} +(608.806 + 1054.48i) q^{73} +(-363.841 + 362.583i) q^{75} +(-172.552 + 978.592i) q^{77} +(278.680 - 233.841i) q^{79} +(686.746 - 244.582i) q^{81} +(453.825 - 380.804i) q^{83} +(-79.4366 + 450.507i) q^{85} +(519.311 - 517.516i) q^{87} +(-359.138 - 622.046i) q^{89} +(-164.037 + 284.121i) q^{91} +(10.9082 - 122.243i) q^{93} +(232.286 + 194.911i) q^{95} +(776.386 + 282.581i) q^{97} +(798.358 + 143.625i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) $ 54 * q + 12 * q^5 - 48 * q^9	$ 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) $ 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{7}{9}\right)$	$1$

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			0
$2$		0			0
$3$	5.17716	−	0.443910i	0.996344	−	0.0854306i
$3$	5.17716	−	0.443910i	0.996344	−	0.0854306i
$4$		0			0
$5$	−4.80495	+	1.74886i	−0.429767	+	0.156423i	−0.547841	−	0.836582i	$-0.684550\pi$
$5$	−4.80495	+	1.74886i	−0.429767	+	0.156423i	0.118074	+	0.993005i	$0.462328\pi$
$6$		0			0
$7$	5.74341	+	32.5725i	0.310115	+	1.75875i	0.598392	+	0.801203i	$0.295806\pi$
$7$	5.74341	+	32.5725i	0.310115	+	1.75875i	−0.288277	+	0.957547i	$0.593082\pi$
$8$		0			0
$9$	26.6059	−	4.59638i	0.985403	−	0.170236i
$10$		0			0
$11$	28.2317	+	10.2755i	0.773833	+	0.281652i	0.698599	−	0.715514i	$-0.253807\pi$
$11$	28.2317	+	10.2755i	0.773833	+	0.281652i	0.0752344	+	0.997166i	$0.476029\pi$
$12$		0			0
$13$	7.59849	+	6.37589i	0.162111	+	0.136027i	0.720235	−	0.693731i	$-0.244034\pi$
$13$	7.59849	+	6.37589i	0.162111	+	0.136027i	−0.558124	+	0.829758i	$0.688479\pi$
$14$		0			0
$15$	−24.0996	+	11.1871i	−0.414833	+	0.192566i
$16$		0			0
$17$	44.7319	−	77.4779i	0.638181	−	1.10536i	−0.347650	−	0.937624i	$-0.613020\pi$
$17$	44.7319	−	77.4779i	0.638181	−	1.10536i	0.985832	−	0.167738i	$-0.0536463\pi$
$18$		0			0
$19$	−29.6509	−	51.3568i	−0.358020	−	0.620108i	0.629610	−	0.776911i	$-0.283215\pi$
$19$	−29.6509	−	51.3568i	−0.358020	−	0.620108i	−0.987630	+	0.156803i	$0.949881\pi$
$20$		0			0
$21$	44.1938	+	166.083i	0.459232	+	1.72583i
$22$		0			0
$23$	−17.0562	+	96.7304i	−0.154629	+	0.876943i	0.804496	+	0.593958i	$0.202436\pi$
$23$	−17.0562	+	96.7304i	−0.154629	+	0.876943i	−0.959124	+	0.282984i	$0.908676\pi$
$24$		0			0
$25$	−75.7266	+	63.5421i	−0.605812	+	0.508337i
$26$		0			0
$27$	135.702	−	35.6068i	0.967257	−	0.253798i
$28$		0			0
$29$	108.085	−	90.6938i	0.692097	−	0.580738i	−0.227416	−	0.973798i	$-0.573028\pi$
$29$	108.085	−	90.6938i	0.692097	−	0.580738i	0.919513	+	0.393059i	$0.128583\pi$
$30$		0			0
$31$	4.10141	−	23.2603i	0.0237624	−	0.134764i	−0.970619	−	0.240623i	$-0.922648\pi$
$31$	4.10141	−	23.2603i	0.0237624	−	0.134764i	0.994381	+	0.105859i	$0.0337594\pi$
$32$		0			0
$33$	150.721	+	40.6655i	0.795066	+	0.214514i
$34$		0			0
$35$	−84.5615	−	146.465i	−0.408386	−	0.707345i
$36$		0			0
$37$	−114.087	+	197.604i	−0.506912	+	0.877997i	0.493056	+	0.869998i	$0.335880\pi$
$37$	−114.087	+	197.604i	−0.506912	+	0.877997i	−0.999968	+	0.00799976i	$0.997454\pi$
$38$		0			0
$39$	42.1689	+	29.6359i	0.173139	+	0.121681i
$40$		0			0
$41$	−357.814	−	300.241i	−1.36295	−	1.14365i	−0.975058	−	0.221951i	$-0.928758\pi$
$41$	−357.814	−	300.241i	−1.36295	−	1.14365i	−0.387896	−	0.921703i	$-0.626798\pi$
$42$		0			0
$43$	−10.1135	−	3.68102i	−0.0358673	−	0.0130546i	0.324024	−	0.946049i	$-0.394964\pi$
$43$	−10.1135	−	3.68102i	−0.0358673	−	0.0130546i	−0.359892	+	0.932994i	$0.617186\pi$
$44$		0			0
$45$	−119.801	+	68.6153i	−0.396865	+	0.227301i
$46$		0			0
$47$	−66.0561	−	374.623i	−0.205006	−	1.16265i	−0.897430	−	0.441156i	$-0.854569\pi$
$47$	−66.0561	−	374.623i	−0.205006	−	1.16265i	0.692424	−	0.721490i	$-0.256543\pi$
$48$		0			0
$49$	−705.667	+	256.842i	−2.05734	+	0.748810i
$50$		0			0
$51$	197.191	−	420.972i	0.541417	−	1.15584i
$52$		0			0
$53$	−202.984			−0.526076			−0.263038	−	0.964786i	$-0.584724\pi$
$53$	−202.984			−0.526076			−0.263038	+	0.964786i	$0.584724\pi$
$54$		0			0
$55$	−153.622			−0.376625
$56$		0			0
$57$	−176.305	−	252.720i	−0.409687	−	0.587255i
$58$		0			0
$59$	766.955	−	279.149i	1.69236	−	0.615967i	0.697437	−	0.716646i	$-0.254324\pi$
$59$	766.955	−	279.149i	1.69236	−	0.615967i	0.994919	+	0.100678i	$0.0321013\pi$
$60$		0			0
$61$	−109.702	−	622.151i	−0.230261	−	1.30587i	−0.852368	−	0.522942i	$-0.824835\pi$
$61$	−109.702	−	622.151i	−0.230261	−	1.30587i	0.622108	−	0.782932i	$-0.286277\pi$
$62$		0			0
$63$	302.524	+	840.222i	0.604992	+	1.68029i
$64$		0			0
$65$	−47.6609	−	17.3471i	−0.0909477	−	0.0331023i
$66$		0			0
$67$	−466.269	−	391.246i	−0.850206	−	0.713408i	0.109629	−	0.993973i	$-0.465034\pi$
$67$	−466.269	−	391.246i	−0.850206	−	0.713408i	−0.959835	+	0.280565i	$0.909478\pi$
$68$		0			0
$69$	−45.3629	+	508.360i	−0.0791457	+	0.886947i
$70$		0			0
$71$	140.376	−	243.138i	0.234642	−	0.406411i	−0.724527	−	0.689247i	$-0.757942\pi$
$71$	140.376	−	243.138i	0.234642	−	0.406411i	0.959168	+	0.282835i	$0.0912750\pi$
$72$		0			0
$73$	608.806	+	1054.48i	0.976101	+	1.69066i	0.676252	+	0.736671i	$0.263603\pi$
$73$	608.806	+	1054.48i	0.976101	+	1.69066i	0.299849	+	0.953987i	$0.403064\pi$
$74$		0			0
$75$	−363.841	+	362.583i	−0.560170	+	0.558233i
$76$		0			0
$77$	−172.552	+	978.592i	−0.255379	+	1.44832i
$78$		0			0
$79$	278.680	−	233.841i	0.396886	−	0.333027i	−0.422402	−	0.906408i	$-0.638813\pi$
$79$	278.680	−	233.841i	0.396886	−	0.333027i	0.819289	+	0.573381i	$0.194369\pi$
$80$		0			0
$81$	686.746	−	244.582i	0.942039	−	0.335503i
$82$		0			0
$83$	453.825	−	380.804i	0.600166	−	0.503599i	−0.291333	−	0.956622i	$-0.594099\pi$
$83$	453.825	−	380.804i	0.600166	−	0.503599i	0.891499	+	0.453023i	$0.149654\pi$
$84$		0			0
$85$	−79.4366	+	450.507i	−0.101366	+	0.574875i
$86$		0			0
$87$	519.311	−	517.516i	0.639954	−	0.637741i
$88$		0			0
$89$	−359.138	−	622.046i	−0.427737	−	0.740862i	0.568935	−	0.822383i	$-0.307356\pi$
$89$	−359.138	−	622.046i	−0.427737	−	0.740862i	−0.996672	+	0.0815204i	$0.974022\pi$
$90$		0			0
$91$	−164.037	+	284.121i	−0.188965	+	0.327297i
$92$		0			0
$93$	10.9082	−	122.243i	0.0121626	−	0.136301i
$94$		0			0
$95$	232.286	+	194.911i	0.250864	+	0.210500i
$96$		0			0
$97$	776.386	+	282.581i	0.812681	+	0.295792i	0.714731	−	0.699400i	$-0.246549\pi$
$97$	776.386	+	282.581i	0.812681	+	0.295792i	0.0979501	+	0.995191i	$0.468771\pi$
$98$		0			0
$99$	798.358	+	143.625i	0.810485	+	0.145806i
$100$		0			0
$101$	171.883	+	974.796i	0.169336	+	0.960355i	0.944480	+	0.328569i	$0.106566\pi$
$101$	171.883	+	974.796i	0.169336	+	0.960355i	−0.775143	+	0.631785i	$0.782322\pi$
$102$		0			0
$103$	645.975	−	235.116i	0.617959	−	0.224919i	−0.0140232	−	0.999902i	$-0.504464\pi$
$103$	645.975	−	235.116i	0.617959	−	0.224919i	0.631982	+	0.774983i	$0.282242\pi$
$104$		0			0
$105$	−502.805	−	720.733i	−0.467321	−	0.669870i
$106$		0			0
$107$	−1298.75			−1.17341			−0.586706	−	0.809800i	$-0.699576\pi$
$107$	−1298.75			−1.17341			−0.586706	+	0.809800i	$0.699576\pi$
$108$		0			0
$109$	576.373			0.506482			0.253241	−	0.967403i	$-0.418503\pi$
$109$	576.373			0.506482			0.253241	+	0.967403i	$0.418503\pi$
$110$		0			0
$111$	−502.926	+	1073.67i	−0.430051	+	0.918093i
$112$		0			0
$113$	−1844.78	+	671.443i	−1.53577	+	0.558974i	−0.965026	−	0.262155i	$-0.915567\pi$
$113$	−1844.78	+	671.443i	−1.53577	+	0.558974i	−0.570743	+	0.821129i	$0.693345\pi$
$114$		0			0
$115$	−87.2136	−	494.613i	−0.0707193	−	0.401069i
$116$		0			0
$117$	231.471	+	134.711i	0.182901	+	0.106444i
$118$		0			0
$119$	2780.57	+	1012.04i	2.14197	+	0.779612i
$120$		0			0
$121$	−328.164	−	275.362i	−0.246555	−	0.206884i
$122$		0			0
$123$	−1985.74	−	1395.56i	−1.45567	−	1.02303i
$124$		0			0
$125$	572.318	−	991.284i	0.409518	−	0.709305i
$126$		0			0
$127$	−245.983	−	426.055i	−0.171870	−	0.297687i	0.767204	−	0.641403i	$-0.221647\pi$
$127$	−245.983	−	426.055i	−0.171870	−	0.297687i	−0.939073	+	0.343716i	$0.888314\pi$
$128$		0			0
$129$	−53.9933	−	14.5677i	−0.0368515	−	0.00994275i
$130$		0			0
$131$	−377.965	+	2143.55i	−0.252084	+	1.42964i	0.551365	+	0.834264i	$0.314107\pi$
$131$	−377.965	+	2143.55i	−0.252084	+	1.42964i	−0.803449	+	0.595374i	$0.797004\pi$
$132$		0			0
$133$	1502.52	−	1260.77i	0.979588	−	0.821972i
$134$		0			0
$135$	−589.772	+	408.413i	−0.375996	+	0.260375i
$136$		0			0
$137$	1065.12	−	893.740i	0.664228	−	0.557353i	−0.247123	−	0.968984i	$-0.579485\pi$
$137$	1065.12	−	893.740i	0.664228	−	0.557353i	0.911351	+	0.411631i	$0.135041\pi$
$138$		0			0
$139$	191.282	−	1084.82i	0.116722	−	0.661963i	−0.869161	−	0.494529i	$-0.835341\pi$
$139$	191.282	−	1084.82i	0.116722	−	0.661963i	0.985883	−	0.167435i	$-0.0535483\pi$
$140$		0			0
$141$	−508.282	−	1910.16i	−0.303582	−	1.14088i
$142$		0			0
$143$	149.003	+	258.080i	0.0871344	+	0.150921i
$144$		0			0
$145$	−360.730	+	624.803i	−0.206600	+	0.357842i
$146$		0			0
$147$	−3539.33	+	1642.96i	−1.98585	+	0.921832i
$148$		0			0
$149$	1499.38	+	1258.13i	0.824388	+	0.691744i	0.953995	−	0.299821i	$-0.0969271\pi$
$149$	1499.38	+	1258.13i	0.824388	+	0.691744i	−0.129607	+	0.991565i	$0.541372\pi$
$150$		0			0
$151$	617.466	+	224.739i	0.332773	+	0.121119i	0.503002	−	0.864285i	$-0.332229\pi$
$151$	617.466	+	224.739i	0.332773	+	0.121119i	−0.170230	+	0.985404i	$0.554451\pi$
$152$		0			0
$153$	834.014	−	2266.97i	0.440693	−	1.19787i
$154$		0			0
$155$	20.9718	+	118.937i	0.0108677	+	0.0616339i
$156$		0			0
$157$	−1247.25	+	453.962i	−0.634022	+	0.230765i	−0.638981	−	0.769222i	$-0.720644\pi$
$157$	−1247.25	+	453.962i	−0.634022	+	0.230765i	0.00495878	+	0.999988i	$0.498422\pi$
$158$		0			0
$159$	−1050.88	+	90.1067i	−0.524152	+	0.0449429i
$160$		0			0
$161$	−3248.71			−1.59028
$162$		0			0
$163$	2562.11			1.23116			0.615582	−	0.788073i	$-0.288921\pi$
$163$	2562.11			1.23116			0.615582	+	0.788073i	$0.288921\pi$
$164$		0			0
$165$	−795.325	+	68.1943i	−0.375248	+	0.0321753i
$166$		0			0
$167$	−3356.67	+	1221.73i	−1.55537	+	0.566109i	−0.969670	−	0.244417i	$-0.921404\pi$
$167$	−3356.67	+	1221.73i	−1.55537	+	0.566109i	−0.585703	+	0.810526i	$0.699181\pi$
$168$		0			0
$169$	−364.420	−	2066.73i	−0.165872	−	0.940705i
$170$		0			0
$171$	−1024.94	−	1230.11i	−0.458359	−	0.550109i
$172$		0			0
$173$	−2140.09	−	778.930i	−0.940510	−	0.342318i	−0.174143	−	0.984720i	$-0.555715\pi$
$173$	−2140.09	−	778.930i	−0.940510	−	0.342318i	−0.766367	+	0.642403i	$0.777938\pi$
$174$		0			0
$175$	−2504.66	−	2101.66i	−1.08191	−	0.907830i
$176$		0			0
$177$	3846.73	−	1785.66i	1.63355	−	0.758294i
$178$		0			0
$179$	577.965	−	1001.07i	0.241336	−	0.418006i	−0.719759	−	0.694224i	$-0.755748\pi$
$179$	577.965	−	1001.07i	0.241336	−	0.418006i	0.961095	+	0.276218i	$0.0890811\pi$
$180$		0			0
$181$	1786.92	+	3095.03i	0.733815	+	1.27100i	0.955241	+	0.295827i	$0.0955952\pi$
$181$	1786.92	+	3095.03i	0.733815	+	1.27100i	−0.221427	+	0.975177i	$0.571071\pi$
$182$		0			0
$183$	−844.124	−	3172.27i	−0.340980	−	1.28143i
$184$		0			0
$185$	202.599	−	1149.00i	0.0805157	−	0.456627i
$186$		0			0
$187$	2058.98	−	1727.69i	0.805174	−	0.675621i
$188$		0			0
$189$	1939.20	+	4215.67i	0.746328	+	1.62246i
$190$		0			0
$191$	−615.799	+	516.717i	−0.233286	+	0.195750i	−0.751935	−	0.659237i	$-0.770880\pi$
$191$	−615.799	+	516.717i	−0.233286	+	0.195750i	0.518649	+	0.854987i	$0.326435\pi$
$192$		0			0
$193$	−155.121	+	879.737i	−0.0578543	+	0.328108i	−0.999974	−	0.00716788i	$-0.997718\pi$
$193$	−155.121	+	879.737i	−0.0578543	+	0.328108i	0.942120	+	0.335276i	$0.108829\pi$
$194$		0			0
$195$	−254.448	−	68.6517i	−0.0934432	−	0.0252115i
$196$		0			0
$197$	1162.98	+	2014.35i	0.420605	+	0.728509i	0.995999	−	0.0893676i	$-0.0284846\pi$
$197$	1162.98	+	2014.35i	0.420605	+	0.728509i	−0.575394	+	0.817876i	$0.695151\pi$
$198$		0			0
$199$	158.416	−	274.384i	0.0564311	−	0.0977415i	−0.836430	−	0.548074i	$-0.815361\pi$
$199$	158.416	−	274.384i	0.0564311	−	0.0977415i	0.892861	+	0.450333i	$0.148695\pi$
$200$		0			0
$201$	−2587.63	−	1818.56i	−0.908045	−	0.638166i
$202$		0			0
$203$	3574.90	+	2999.70i	1.23600	+	1.03713i
$204$		0			0
$205$	2244.35	+	816.878i	0.764646	+	0.278309i
$206$		0			0
$207$	−9.18465	+	2652.00i	−0.00308395	+	0.890466i
$208$		0			0
$209$	−309.377	−	1754.56i	−0.102393	−	0.580697i
$210$		0			0
$211$	−3958.48	+	1440.77i	−1.29153	+	0.470079i	−0.894229	−	0.447609i	$-0.852276\pi$
$211$	−3958.48	+	1440.77i	−1.29153	+	0.470079i	−0.397302	+	0.917688i	$0.630054\pi$
$212$		0			0
$213$	618.816	−	1321.08i	0.199064	−	0.424971i
$214$		0			0
$215$	55.0324			0.0174567
$216$		0			0
$217$	781.202			0.244384
$218$		0			0
$219$	3619.98	+	5188.97i	1.11697	+	1.60109i
$220$		0			0
$221$	833.886	−	303.510i	0.253816	−	0.0923813i
$222$		0			0
$223$	−92.7029	−	525.744i	−0.0278379	−	0.157876i	0.967720	−	0.252028i	$-0.0810974\pi$
$223$	−92.7029	−	525.744i	−0.0278379	−	0.157876i	−0.995558	+	0.0941513i	$0.969986\pi$
$224$		0			0
$225$	−1722.71	+	2038.66i	−0.510432	+	0.604048i
$226$		0			0
$227$	5420.84	+	1973.03i	1.58500	+	0.576891i	0.976283	−	0.216500i	$-0.0694640\pi$
$227$	5420.84	+	1973.03i	1.58500	+	0.576891i	0.608713	+	0.793391i	$0.291686\pi$
$228$		0			0
$229$	1142.36	+	958.554i	0.329647	+	0.276607i	0.792556	−	0.609799i	$-0.208750\pi$
$229$	1142.36	+	958.554i	0.329647	+	0.276607i	−0.462909	+	0.886406i	$0.653194\pi$
$230$		0			0
$231$	−458.923	+	5142.92i	−0.130714	+	1.46485i
$232$		0			0
$233$	225.811	−	391.116i	0.0634908	−	0.109969i	−0.832533	−	0.553976i	$-0.813110\pi$
$233$	225.811	−	391.116i	0.0634908	−	0.109969i	0.896024	+	0.444007i	$0.146443\pi$
$234$		0			0
$235$	972.558	+	1684.52i	0.269969	+	0.467600i
$236$		0			0
$237$	1338.97	−	1334.34i	0.366984	−	0.365716i
$238$		0			0
$239$	537.820	−	3050.13i	0.145559	−	0.825508i	−0.821357	−	0.570414i	$-0.806783\pi$
$239$	537.820	−	3050.13i	0.145559	−	0.825508i	0.966916	−	0.255093i	$-0.0821062\pi$
$240$		0			0
$241$	427.852	−	359.010i	0.114358	−	0.0959580i	−0.583815	−	0.811886i	$-0.698441\pi$
$241$	427.852	−	359.010i	0.114358	−	0.0959580i	0.698174	+	0.715928i	$0.253996\pi$
$242$		0			0
$243$	3446.82	−	1571.09i	0.909933	−	0.414756i
$244$		0			0
$245$	2941.51	−	2468.22i	0.767046	−	0.643628i
$246$		0			0
$247$	102.143	−	579.285i	0.0263127	−	0.149227i
$248$		0			0
$249$	2180.48	−	2172.94i	0.554949	−	0.553030i
$250$		0			0
$251$	−315.189	−	545.923i	−0.0792612	−	0.137284i	0.823670	−	0.567069i	$-0.191923\pi$
$251$	−315.189	−	545.923i	−0.0792612	−	0.137284i	−0.902931	+	0.429785i	$0.858589\pi$
$252$		0			0
$253$	−1475.48	+	2555.60i	−0.366650	+	0.635056i
$254$		0			0
$255$	−211.271	+	2367.61i	−0.0518835	+	0.581433i
$256$		0			0
$257$	−1711.18	−	1435.85i	−0.415333	−	0.348506i	0.411051	−	0.911612i	$-0.365162\pi$
$257$	−1711.18	−	1435.85i	−0.415333	−	0.348506i	−0.826384	+	0.563106i	$0.809606\pi$
$258$		0			0
$259$	−7091.71	−	2581.17i	−1.70138	−	0.619251i
$260$		0			0
$261$	2458.82	−	2909.79i	0.583132	−	0.690081i
$262$		0			0
$263$	115.370	+	654.298i	0.0270496	+	0.153406i	0.995341	−	0.0964178i	$-0.0307385\pi$
$263$	115.370	+	654.298i	0.0270496	+	0.153406i	−0.968291	+	0.249824i	$0.919627\pi$
$264$		0			0
$265$	975.328	−	354.990i	0.226090	−	0.0822901i
$266$		0			0
$267$	−2135.45	−	3061.00i	−0.489466	−	0.701612i
$268$		0			0
$269$	−6952.64			−1.57587			−0.787936	−	0.615757i	$-0.788850\pi$
$269$	−6952.64			−1.57587			−0.787936	+	0.615757i	$0.788850\pi$
$270$		0			0
$271$	−8783.48			−1.96885			−0.984426	−	0.175802i	$-0.943748\pi$
$271$	−8783.48			−1.96885			−0.984426	+	0.175802i	$0.943748\pi$
$272$		0			0
$273$	−723.123	+	1543.76i	−0.160313	+	0.342244i
$274$		0			0
$275$	−2790.81	+	1015.77i	−0.611972	+	0.222740i
$276$		0			0
$277$	1226.41	+	6955.33i	0.266021	+	1.50868i	0.766111	+	0.642709i	$0.222189\pi$
$277$	1226.41	+	6955.33i	0.266021	+	1.50868i	−0.500089	+	0.865974i	$0.666699\pi$
$278$		0			0
$279$	2.20859	−	637.712i	0.000473924	−	0.136842i
$280$		0			0
$281$	−6216.63	−	2262.67i	−1.31976	−	0.480354i	−0.416379	−	0.909191i	$-0.636701\pi$
$281$	−6216.63	−	2262.67i	−1.31976	−	0.480354i	−0.903382	+	0.428837i	$0.858923\pi$
$282$		0			0
$283$	4086.69	+	3429.14i	0.858405	+	0.720288i	0.961624	−	0.274371i	$-0.0884696\pi$
$283$	4086.69	+	3429.14i	0.858405	+	0.720288i	−0.103218	+	0.994659i	$0.532914\pi$
$284$		0			0
$285$	1289.11	+	905.973i	0.267930	+	0.188299i
$286$		0			0
$287$	7724.54	−	13379.3i	1.58873	−	2.75176i
$288$		0			0
$289$	−1545.39	−	2676.69i	−0.314551	−	0.544818i
$290$		0			0
$291$	4144.91	+	1118.32i	0.834980	+	0.225283i
$292$		0			0
$293$	−50.5304	+	286.572i	−0.0100751	+	0.0571390i	−0.989431	−	0.145006i	$-0.953680\pi$
$293$	−50.5304	+	286.572i	−0.0100751	+	0.0571390i	0.979356	+	0.202145i	$0.0647911\pi$
$294$		0			0
$295$	−3196.99	+	2682.59i	−0.630968	+	0.529445i
$296$		0			0
$297$	4196.98	+	389.168i	0.819979	+	0.0760332i
$298$		0			0
$299$	−746.344	+	626.257i	−0.144355	+	0.121128i
$300$		0			0
$301$	61.8139	−	350.564i	0.0118369	−	0.0671302i
$302$		0			0
$303$	1322.59	+	4970.37i	0.250761	+	0.942377i
$304$		0			0
$305$	1615.17	+	2797.55i	0.303227	+	0.525204i
$306$		0			0
$307$	572.764	−	992.056i	0.106480	−	0.184429i	−0.807862	−	0.589372i	$-0.799375\pi$
$307$	572.764	−	992.056i	0.106480	−	0.184429i	0.914342	+	0.404943i	$0.132709\pi$
$308$		0			0
$309$	3239.94	−	1503.98i	0.596485	−	0.276889i
$310$		0			0
$311$	−7195.48	−	6037.72i	−1.31196	−	1.10086i	−0.987945	−	0.154808i	$-0.950524\pi$
$311$	−7195.48	−	6037.72i	−1.31196	−	1.10086i	−0.324011	−	0.946053i	$-0.605031\pi$
$312$		0			0
$313$	−2912.59	−	1060.10i	−0.525972	−	0.191438i	0.0653667	−	0.997861i	$-0.479178\pi$
$313$	−2912.59	−	1060.10i	−0.525972	−	0.191438i	−0.591339	+	0.806423i	$0.701401\pi$
$314$		0			0
$315$	−2923.04	−	3508.15i	−0.522840	−	0.627498i
$316$		0			0
$317$	116.213	+	659.077i	0.0205905	+	0.116774i	0.993371	−	0.114956i	$-0.0366726\pi$
$317$	116.213	+	659.077i	0.0205905	+	0.116774i	−0.972780	+	0.231730i	$0.925561\pi$
$318$		0			0
$319$	3983.33	−	1449.81i	0.699134	−	0.254464i
$320$		0			0
$321$	−6723.84	+	576.529i	−1.16912	+	0.100245i
$322$		0			0
$323$	−5305.36			−0.913926
$324$		0			0
$325$	−980.545			−0.167356
$326$		0			0
$327$	2983.97	−	255.858i	0.504630	−	0.0432690i
$328$		0			0
$329$	11823.0	−	4303.23i	1.98123	−	0.721109i
$330$		0			0
$331$	50.4286	+	285.995i	0.00837404	+	0.0474915i	0.988708	−	0.149852i	$-0.0478797\pi$
$331$	50.4286	+	285.995i	0.00837404	+	0.0474915i	−0.980334	+	0.197343i	$0.936769\pi$
$332$		0			0
$333$	−2127.11	+	5781.82i	−0.350046	+	0.951476i
$334$		0			0
$335$	2924.63	+	1064.48i	0.476984	+	0.173608i
$336$		0			0
$337$	−6338.56	−	5318.68i	−1.02458	−	0.859724i	−0.0343831	−	0.999409i	$-0.510947\pi$
$337$	−6338.56	−	5318.68i	−1.02458	−	0.859724i	−0.990196	+	0.139685i	$0.955391\pi$
$338$		0			0
$339$	−9252.63	+	4295.08i	−1.48240	+	0.688132i
$340$		0			0
$341$	354.800	−	614.532i	0.0563446	−	0.0975917i
$342$		0			0
$343$	−6746.56	−	11685.4i	−1.06204	−	1.83951i
$344$		0			0
$345$	−671.083	−	2521.97i	−0.104724	−	0.393561i
$346$		0			0
$347$	−698.081	+	3959.02i	−0.107997	+	0.612482i	0.881984	+	0.471280i	$0.156208\pi$
$347$	−698.081	+	3959.02i	−0.107997	+	0.612482i	−0.989981	+	0.141202i	$0.954903\pi$
$348$		0			0
$349$	−6291.10	+	5278.86i	−0.964914	+	0.809659i	−0.981745	−	0.190200i	$-0.939086\pi$
$349$	−6291.10	+	5278.86i	−0.964914	+	0.809659i	0.0168317	+	0.999858i	$0.494642\pi$
$350$		0			0
$351$	1258.16	+	594.666i	0.191326	+	0.0904299i
$352$		0			0
$353$	−3542.55	+	2972.56i	−0.534139	+	0.448196i	−0.869528	−	0.493884i	$-0.835577\pi$
$353$	−3542.55	+	2972.56i	−0.534139	+	0.448196i	0.335389	+	0.942080i	$0.391132\pi$
$354$		0			0
$355$	−249.285	+	1413.76i	−0.0372694	+	0.211366i
$356$		0			0
$357$	14844.7	+	4005.18i	2.20074	+	0.593772i
$358$		0			0
$359$	6159.42	+	10668.4i	0.905521	+	1.56841i	0.820217	+	0.572053i	$0.193853\pi$
$359$	6159.42	+	10668.4i	0.905521	+	1.56841i	0.0853041	+	0.996355i	$0.472814\pi$
$360$		0			0
$361$	1671.15	−	2894.52i	0.243644	−	0.422004i
$362$		0			0
$363$	−1821.19	−	1279.92i	−0.263327	−	0.185064i
$364$		0			0
$365$	−4769.42	−	4002.02i	−0.683953	−	0.573905i
$366$		0			0
$367$	8080.39	+	2941.02i	1.14930	+	0.418311i	0.845263	−	0.534350i	$-0.179444\pi$
$367$	8080.39	+	2941.02i	1.14930	+	0.418311i	0.304036	+	0.952661i	$0.401666\pi$
$368$		0			0
$369$	−10900.0	−	6343.54i	−1.53775	−	0.894936i
$370$		0			0
$371$	−1165.82	−	6611.70i	−0.163144	−	0.925236i
$372$		0			0
$373$	−2385.97	+	868.421i	−0.331208	+	0.120550i	−0.502271	−	0.864710i	$-0.667502\pi$
$373$	−2385.97	+	868.421i	−0.331208	+	0.120550i	0.171063	+	0.985260i	$0.445280\pi$
$374$		0			0
$375$	2522.94	−	5386.09i	0.347424	−	0.741697i
$376$		0			0
$377$	1399.53			0.191193
$378$		0			0
$379$	11211.7			1.51955			0.759774	−	0.650187i	$-0.225309\pi$
$379$	11211.7			1.51955			0.759774	+	0.650187i	$0.225309\pi$
$380$		0			0
$381$	−1462.62	−	2096.56i	−0.196673	−	0.281916i
$382$		0			0
$383$	7480.47	−	2722.67i	0.998001	−	0.363243i	0.209188	−	0.977876i	$-0.432918\pi$
$383$	7480.47	−	2722.67i	0.998001	−	0.363243i	0.788813	+	0.614633i	$0.210696\pi$
$384$		0			0
$385$	−882.314	−	5003.85i	−0.116797	−	0.662389i
$386$		0			0
$387$	−285.998	−	51.4511i	−0.0375662	−	0.00675816i
$388$		0			0
$389$	−12432.6	−	4525.09i	−1.62045	−	0.589797i	−0.636986	−	0.770875i	$-0.719819\pi$
$389$	−12432.6	−	4525.09i	−1.62045	−	0.589797i	−0.983469	+	0.181078i	$0.942041\pi$
$390$		0			0
$391$	6731.52	+	5648.41i	0.870659	+	0.730569i
$392$		0			0
$393$	−1005.24	+	11265.3i	−0.129027	+	1.44595i
$394$		0			0
$395$	−930.091	+	1610.96i	−0.118476	+	0.205206i
$396$		0			0
$397$	6065.99	+	10506.6i	0.766859	+	1.32824i	0.939258	+	0.343212i	$0.111515\pi$
$397$	6065.99	+	10506.6i	0.766859	+	1.32824i	−0.172399	+	0.985027i	$0.555152\pi$
$398$		0			0
$399$	7219.13	−	7194.17i	0.905785	−	0.902654i
$400$		0			0
$401$	533.798	−	3027.32i	0.0664753	−	0.377000i	−0.933362	−	0.358938i	$-0.883139\pi$
$401$	533.798	−	3027.32i	0.0664753	−	0.377000i	0.999837	−	0.0180626i	$-0.00574981\pi$
$402$		0			0
$403$	179.469	−	150.593i	0.0221837	−	0.0186143i
$404$		0			0
$405$	−2872.04	+	2376.22i	−0.352377	+	0.291544i
$406$		0			0
$407$	−5251.33	+	4406.39i	−0.639555	+	0.536651i
$408$		0			0
$409$	−1787.94	+	10139.9i	−0.216156	+	1.22588i	0.662733	+	0.748856i	$0.269396\pi$
$409$	−1787.94	+	10139.9i	−0.216156	+	1.22588i	−0.878889	+	0.477026i	$0.841715\pi$
$410$		0			0
$411$	5117.54	−	5099.85i	0.614184	−	0.612061i
$412$		0			0
$413$	13497.5	+	23378.4i	1.60816	+	2.78541i
$414$		0			0
$415$	−1514.63	+	2623.42i	−0.179157	+	0.310310i
$416$		0			0
$417$	508.738	−	5701.18i	0.0597434	−	0.669515i
$418$		0			0
$419$	3312.78	+	2779.75i	0.386252	+	0.324104i	0.815151	−	0.579248i	$-0.196654\pi$
$419$	3312.78	+	2779.75i	0.386252	+	0.324104i	−0.428899	+	0.903353i	$0.641098\pi$
$420$		0			0
$421$	−5518.54	−	2008.58i	−0.638853	−	0.232523i	0.00222710	−	0.999998i	$-0.499291\pi$
$421$	−5518.54	−	2008.58i	−0.638853	−	0.232523i	−0.641080	+	0.767474i	$0.721513\pi$
$422$		0			0
$423$	−3479.39	−	9663.56i	−0.399938	−	1.11078i
$424$		0			0
$425$	1535.72	+	8709.50i	0.175278	+	0.994054i
$426$		0			0
$427$	19635.0	−	7146.54i	2.22530	−	0.809942i
$428$		0			0
$429$	885.974	+	1269.98i	0.0997092	+	0.142926i
$430$		0			0
$431$	−187.346			−0.0209377			−0.0104688	−	0.999945i	$-0.503332\pi$
$431$	−187.346			−0.0209377			−0.0104688	+	0.999945i	$0.503332\pi$
$432$		0			0
$433$	−313.829			−0.0348307			−0.0174153	−	0.999848i	$-0.505544\pi$
$433$	−313.829			−0.0348307			−0.0174153	+	0.999848i	$0.505544\pi$
$434$		0			0
$435$	−1590.20	+	3394.84i	−0.175274	+	0.374184i
$436$		0			0
$437$	5473.49	−	1992.19i	0.599159	−	0.218076i
$438$		0			0
$439$	2543.97	+	14427.5i	0.276576	+	1.56854i	0.733911	+	0.679246i	$0.237693\pi$
$439$	2543.97	+	14427.5i	0.276576	+	1.56854i	−0.457335	+	0.889295i	$0.651196\pi$
$440$		0			0
$441$	−17594.4	+	10077.0i	−1.89983	+	1.08811i
$442$		0			0
$443$	−3295.46	−	1199.45i	−0.353436	−	0.128640i	0.159200	−	0.987246i	$-0.449109\pi$
$443$	−3295.46	−	1199.45i	−0.353436	−	0.128640i	−0.512636	+	0.858606i	$0.671331\pi$
$444$		0			0
$445$	2813.51	+	2360.82i	0.299715	+	0.251491i
$446$		0			0
$447$	8321.01	+	5847.94i	0.880471	+	0.618787i
$448$		0			0
$449$	−905.931	+	1569.12i	−0.0952194	+	0.164925i	−0.909700	−	0.415266i	$-0.863689\pi$
$449$	−905.931	+	1569.12i	−0.0952194	+	0.164925i	0.814481	+	0.580191i	$0.197022\pi$
$450$		0			0
$451$	−7016.55	−	12153.0i	−0.732586	−	1.26888i
$452$		0			0
$453$	3296.48	+	889.411i	0.341904	+	0.0922477i
$454$		0			0
$455$	291.304	−	1652.07i	0.0300143	−	0.170220i
$456$		0			0
$457$	11433.8	−	9594.07i	1.17035	−	0.982039i	0.170354	−	0.985383i	$-0.445509\pi$
$457$	11433.8	−	9594.07i	1.17035	−	0.982039i	0.999994	+	0.00334372i	$0.00106434\pi$
$458$		0			0
$459$	3311.49	−	12106.7i	0.336747	−	1.23114i
$460$		0			0
$461$	−6096.59	+	5115.65i	−0.615936	+	0.516832i	−0.896523	−	0.442997i	$-0.853915\pi$
$461$	−6096.59	+	5115.65i	−0.615936	+	0.516832i	0.280587	+	0.959829i	$0.409471\pi$
$462$		0			0
$463$	2533.18	−	14366.4i	0.254270	−	1.44204i	−0.543669	−	0.839300i	$-0.682965\pi$
$463$	2533.18	−	14366.4i	0.254270	−	1.44204i	0.797939	−	0.602738i	$-0.205924\pi$
$464$		0			0
$465$	161.372	+	606.447i	0.0160934	+	0.0604802i
$466$		0			0
$467$	4109.70	+	7118.22i	0.407226	+	0.705336i	0.994578	−	0.103996i	$-0.0331629\pi$
$467$	4109.70	+	7118.22i	0.407226	+	0.705336i	−0.587352	+	0.809332i	$0.699830\pi$
$468$		0			0
$469$	10065.9	−	17434.6i	0.991044	−	1.71654i
$470$		0			0
$471$	−6255.70	+	2903.90i	−0.611990	+	0.284086i
$472$		0			0
$473$	−247.697	−	207.842i	−0.0240785	−	0.0202042i
$474$		0			0
$475$	5508.68	+	2004.99i	0.532117	+	0.193675i
$476$		0			0
$477$	−5400.57	+	932.993i	−0.518397	+	0.0895573i
$478$		0			0
$479$	729.906	+	4139.50i	0.0696247	+	0.394861i	0.999627	+	0.0273078i	$0.00869343\pi$
$479$	729.906	+	4139.50i	0.0696247	+	0.394861i	−0.930002	+	0.367554i	$0.880195\pi$
$480$		0			0
$481$	−2126.79	+	774.088i	−0.201607	+	0.0733791i
$482$		0			0
$483$	−16819.1	+	1442.14i	−1.58446	+	0.135858i
$484$		0			0
$485$	−4224.69			−0.395532
$486$		0			0
$487$	−13008.8			−1.21044			−0.605219	−	0.796059i	$-0.706914\pi$
$487$	−13008.8			−1.21044			−0.605219	+	0.796059i	$0.706914\pi$
$488$		0			0
$489$	13264.4	−	1137.34i	1.22666	−	0.105179i
$490$		0			0
$491$	−13256.1	+	4824.83i	−1.21841	+	0.443465i	−0.869614	−	0.493733i	$-0.835632\pi$
$491$	−13256.1	+	4824.83i	−1.21841	+	0.443465i	−0.348797	+	0.937198i	$0.613410\pi$
$492$		0			0
$493$	−2191.93	−	12431.1i	−0.200243	−	1.13563i
$494$		0			0
$495$	−4087.25	+	706.106i	−0.371128	+	0.0641153i
$496$		0			0
$497$	8725.86	+	3175.95i	0.787542	+	0.286642i
$498$		0			0
$499$	7229.72	+	6066.46i	0.648591	+	0.544232i	0.906643	−	0.421899i	$-0.138636\pi$
$499$	7229.72	+	6066.46i	0.648591	+	0.544232i	−0.258052	+	0.966131i	$0.583081\pi$
$500$		0			0
$501$	−16835.7	+	7815.15i	−1.50132	+	0.696916i
$502$		0			0
$503$	4547.85	−	7877.10i	0.403138	−	0.698256i	−0.590965	−	0.806697i	$-0.701253\pi$
$503$	4547.85	−	7877.10i	0.403138	−	0.698256i	0.994103	+	0.108442i	$0.0345861\pi$
$504$		0			0
$505$	−2530.67	−	4383.24i	−0.222996	−	0.386241i
$506$		0			0
$507$	−2804.10	−	10538.0i	−0.245630	−	0.923095i
$508$		0			0
$509$	1245.45	−	7063.30i	0.108455	−	0.615079i	−0.881329	−	0.472503i	$-0.843350\pi$
$509$	1245.45	−	7063.30i	0.108455	−	0.615079i	0.989784	−	0.142576i	$-0.0455384\pi$
$510$		0			0
$511$	−30850.6	+	25886.7i	−2.67074	+	2.24102i
$512$		0			0
$513$	−5852.35	−	5913.47i	−0.503679	−	0.508940i
$514$		0			0
$515$	−2692.69	+	2259.43i	−0.230396	+	0.193325i
$516$		0			0
$517$	1984.56	−	11255.0i	0.168822	−	0.957435i
$518$		0			0
$519$	−11425.4	−	3082.63i	−0.966316	−	0.260718i
$520$		0			0
$521$	4043.78	+	7004.04i	0.340041	+	0.588968i	0.984440	−	0.175721i	$-0.0562257\pi$
$521$	4043.78	+	7004.04i	0.340041	+	0.588968i	−0.644399	+	0.764689i	$0.722892\pi$
$522$		0			0
$523$	7678.53	−	13299.6i	0.641986	−	1.11195i	−0.343003	−	0.939334i	$-0.611444\pi$
$523$	7678.53	−	13299.6i	0.641986	−	1.11195i	0.984989	−	0.172618i	$-0.0552226\pi$
$524$		0			0
$525$	−13899.9	−	9768.76i	−1.15551	−	0.812083i
$526$		0			0
$527$	−1618.69	−	1358.25i	−0.133798	−	0.112270i
$528$		0			0
$529$	2367.38	+	861.657i	0.194574	+	0.0708192i
$530$		0			0
$531$	19122.4	−	10952.2i	1.56279	−	0.895077i
$532$		0			0
$533$	−804.538	−	4562.76i	−0.0653816	−	0.370798i
$534$		0			0
$535$	6240.43	−	2271.33i	0.504294	−	0.183548i
$536$		0			0
$537$	2547.83	−	5439.24i	0.204743	−	0.437096i
$538$		0			0
$539$	−22561.3			−1.80294
$540$		0			0
$541$	2309.43			0.183531			0.0917654	−	0.995781i	$-0.470749\pi$
$541$	2309.43			0.183531			0.0917654	+	0.995781i	$0.470749\pi$
$542$		0			0
$543$	10625.1	+	15230.2i	0.839714	+	1.20367i
$544$		0			0
$545$	−2769.44	+	1007.99i	−0.217669	+	0.0792252i
$546$		0			0
$547$	2461.77	+	13961.4i	0.192427	+	1.09131i	0.916035	+	0.401098i	$0.131371\pi$
$547$	2461.77	+	13961.4i	0.192427	+	1.09131i	−0.723608	+	0.690212i	$0.757517\pi$
$548$		0			0
$549$	−5778.36	−	16048.6i	−0.449207	−	1.24761i
$550$		0			0
$551$	−7862.54	−	2861.73i	−0.607905	−	0.221259i
$552$		0			0
$553$	9217.36	+	7734.28i	0.708792	+	0.594747i
$554$		0			0
$555$	538.837	−	6038.48i	0.0412114	−	0.461836i
$556$		0			0
$557$	11430.0	−	19797.4i	0.869489	−	1.50600i	0.00697005	−	0.999976i	$-0.497781\pi$
$557$	11430.0	−	19797.4i	0.869489	−	1.50600i	0.862519	−	0.506024i	$-0.168885\pi$
$558$		0			0
$559$	−53.3776	−	92.4528i	−0.00403870	−	0.00699524i
$560$		0			0
$561$	9892.72	−	9858.52i	0.744511	−	0.741937i
$562$		0			0
$563$	1033.46	−	5861.04i	0.0773626	−	0.438745i	−0.921382	−	0.388658i	$-0.872939\pi$
$563$	1033.46	−	5861.04i	0.0773626	−	0.438745i	0.998745	−	0.0500875i	$-0.0159500\pi$
$564$		0			0
$565$	7689.79	−	6452.50i	0.572587	−	0.480458i
$566$		0			0
$567$	11910.9	+	20964.3i	0.882207	+	1.55277i
$568$		0			0
$569$	−8500.63	+	7132.88i	−0.626301	+	0.525529i	−0.899777	−	0.436350i	$-0.856271\pi$
$569$	−8500.63	+	7132.88i	−0.626301	+	0.525529i	0.273476	+	0.961879i	$0.411826\pi$
$570$		0			0
$571$	−1804.91	+	10236.2i	−0.132282	+	0.750211i	0.844431	+	0.535664i	$0.179939\pi$
$571$	−1804.91	+	10236.2i	−0.132282	+	0.750211i	−0.976714	+	0.214547i	$0.931173\pi$
$572$		0			0
$573$	−2958.71	+	2948.48i	−0.215710	+	0.214964i
$574$		0			0
$575$	−4854.85	−	8408.85i	−0.352106	−	0.609866i
$576$		0			0
$577$	−10062.0	+	17427.8i	−0.725971	+	1.25742i	0.232602	+	0.972572i	$0.425276\pi$
$577$	−10062.0	+	17427.8i	−0.725971	+	1.25742i	−0.958573	+	0.284846i	$0.908057\pi$
$578$		0			0
$579$	−412.563	+	4623.39i	−0.0296123	+	0.331851i
$580$		0			0
$581$	15010.3	+	12595.1i	1.07183	+	0.899368i
$582$		0			0
$583$	−5730.58	−	2085.76i	−0.407095	−	0.148170i
$584$		0			0
$585$	−1347.79	−	242.468i	−0.0952554	−	0.0171365i
$586$		0			0
$587$	2679.51	+	15196.3i	0.188408	+	1.06851i	0.921498	+	0.388382i	$0.126966\pi$
$587$	2679.51	+	15196.3i	0.188408	+	1.06851i	−0.733091	+	0.680131i	$0.761923\pi$
$588$		0			0
$589$	−1316.18	+	479.052i	−0.0920754	+	0.0335127i
$590$		0			0
$591$	6915.14	+	9912.32i	0.481304	+	0.689913i
$592$		0			0
$593$	8549.53			0.592052			0.296026	−	0.955180i	$-0.404338\pi$
$593$	8549.53			0.592052			0.296026	+	0.955180i	$0.404338\pi$
$594$		0			0
$595$	−15130.4			−1.04250
$596$		0			0
$597$	698.340	−	1490.85i	0.0478746	−	0.102205i
$598$		0			0
$599$	−15101.5	+	5496.51i	−1.03010	+	0.374927i	−0.801120	−	0.598504i	$-0.795762\pi$
$599$	−15101.5	+	5496.51i	−1.03010	+	0.374927i	−0.228983	+	0.973430i	$0.573540\pi$
$600$		0			0
$601$	−1363.83	−	7734.68i	−0.0925656	−	0.524966i	−0.995466	−	0.0951161i	$-0.969678\pi$
$601$	−1363.83	−	7734.68i	−0.0925656	−	0.524966i	0.902901	−	0.429849i	$-0.141433\pi$
$602$		0			0
$603$	−14203.8	−	8266.30i	−0.959244	−	0.558258i
$604$		0			0
$605$	2058.38	+	749.189i	0.138322	+	0.0503453i
$606$		0			0
$607$	1421.95	+	1193.16i	0.0950829	+	0.0797840i	0.689090	−	0.724676i	$-0.258010\pi$
$607$	1421.95	+	1193.16i	0.0950829	+	0.0797840i	−0.594007	+	0.804460i	$0.702455\pi$
$608$		0			0
$609$	19839.4	+	13943.0i	1.32009	+	0.927746i
$610$		0			0
$611$	1886.63	−	3267.74i	0.124918	−	0.216364i
$612$		0			0
$613$	−3861.72	−	6688.70i	−0.254443	−	0.440708i	0.710301	−	0.703898i	$-0.248559\pi$
$613$	−3861.72	−	6688.70i	−0.254443	−	0.440708i	−0.964744	+	0.263190i	$0.915225\pi$
$614$		0			0
$615$	11982.0	+	3232.81i	0.785627	+	0.211967i
$616$		0			0
$617$	3164.80	−	17948.4i	0.206499	−	1.17111i	−0.688564	−	0.725175i	$-0.741759\pi$
$617$	3164.80	−	17948.4i	0.206499	−	1.17111i	0.895063	−	0.445939i	$-0.147130\pi$
$618$		0			0
$619$	1605.52	−	1347.19i	0.104251	−	0.0874770i	−0.589172	−	0.808007i	$-0.700546\pi$
$619$	1605.52	−	1347.19i	0.104251	−	0.0874770i	0.693423	+	0.720530i	$0.256102\pi$
$620$		0			0
$621$	1129.70	+	13733.9i	0.0730003	+	0.887474i
$622$		0			0
$623$	18198.9	−	15270.7i	1.17034	−	0.982035i
$624$		0			0
$625$	1129.38	−	6405.05i	0.0722806	−	0.409923i
$626$		0			0
$627$	−2380.56	−	8946.32i	−0.151628	−	0.569827i
$628$		0			0
$629$	10206.6	+	17678.4i	0.647004	+	1.12064i
$630$		0			0
$631$	−9208.35	+	15949.3i	−0.580948	+	1.00623i	0.414419	+	0.910086i	$0.363985\pi$
$631$	−9208.35	+	15949.3i	−0.580948	+	1.00623i	−0.995367	+	0.0961459i	$0.969348\pi$
$632$		0			0
$633$	−19854.1	+	9216.30i	−1.24665	+	0.578697i
$634$		0			0
$635$	1927.04	+	1616.98i	0.120429	+	0.101052i
$636$		0			0
$637$	−6999.60	−	2547.65i	−0.435376	−	0.158464i
$638$		0			0
$639$	2617.27	−	7114.13i	0.162031	−	0.440423i
$640$		0			0
$641$	−4147.90	−	23523.9i	−0.255589	−	1.44951i	−0.794557	−	0.607189i	$-0.792297\pi$
$641$	−4147.90	−	23523.9i	−0.255589	−	1.44951i	0.538969	−	0.842326i	$-0.318814\pi$
$642$		0			0
$643$	−3014.06	+	1097.03i	−0.184857	+	0.0672824i	−0.432790	−	0.901495i	$-0.642471\pi$
$643$	−3014.06	+	1097.03i	−0.184857	+	0.0672824i	0.247933	+	0.968777i	$0.420249\pi$
$644$		0			0
$645$	284.912	−	24.4295i	0.0173928	−	0.00149133i
$646$		0			0
$647$	−8439.89			−0.512838			−0.256419	−	0.966566i	$-0.582543\pi$
$647$	−8439.89			−0.512838			−0.256419	+	0.966566i	$0.582543\pi$
$648$		0			0
$649$	24520.8			1.48309
$650$		0			0
$651$	4044.40	−	346.783i	0.243491	−	0.0208779i
$652$		0			0
$653$	26279.5	−	9564.97i	1.57488	−	0.573210i	0.600798	−	0.799401i	$-0.294849\pi$
$653$	26279.5	−	9564.97i	1.57488	−	0.573210i	0.974083	+	0.226191i	$0.0726272\pi$
$654$		0			0
$655$	−1932.66	−	10960.6i	−0.115290	−	0.653844i
$656$		0			0
$657$	21044.6	+	25257.2i	1.24966	+	1.49981i
$658$		0			0
$659$	−4419.19	−	1608.45i	−0.261225	−	0.0950781i	0.208088	−	0.978110i	$-0.433276\pi$
$659$	−4419.19	−	1608.45i	−0.261225	−	0.0950781i	−0.469313	+	0.883032i	$0.655498\pi$
$660$		0			0
$661$	−24298.4	−	20388.8i	−1.42980	−	1.19975i	−0.945829	−	0.324664i	$-0.894749\pi$
$661$	−24298.4	−	20388.8i	−1.42980	−	1.19975i	−0.483973	−	0.875083i	$-0.660807\pi$
$662$		0			0
$663$	4182.43	−	1941.49i	0.244995	−	0.113727i
$664$		0			0
$665$	−5014.64	+	8685.61i	−0.292420	+	0.506486i
$666$		0			0
$667$	6929.33	+	12002.0i	0.402256	+	0.696728i
$668$		0			0
$669$	−713.320	−	2680.71i	−0.0412236	−	0.154921i
$670$		0			0
$671$	3295.83	−	18691.6i	0.189619	−	1.07538i
$672$		0			0
$673$	1397.36	−	1172.52i	0.0800358	−	0.0671580i	−0.601892	−	0.798577i	$-0.705586\pi$
$673$	1397.36	−	1172.52i	0.0800358	−	0.0671580i	0.681928	+	0.731419i	$0.261142\pi$
$674$		0			0
$675$	−8013.75	+	11319.2i	−0.456962	+	0.645446i
$676$		0			0
$677$	7383.95	−	6195.87i	0.419185	−	0.351738i	−0.408668	−	0.912683i	$-0.634007\pi$
$677$	7383.95	−	6195.87i	0.419185	−	0.351738i	0.827853	+	0.560945i	$0.189562\pi$
$678$		0			0
$679$	−4745.28	+	26911.8i	−0.268199	+	1.52103i
$680$		0			0
$681$	28940.4	+	7808.29i	1.62849	+	0.439375i
$682$		0			0
$683$	3985.79	+	6903.58i	0.223297	+	0.386762i	0.955807	−	0.293994i	$-0.0949847\pi$
$683$	3985.79	+	6903.58i	0.223297	+	0.386762i	−0.732510	+	0.680756i	$0.761651\pi$
$684$		0			0
$685$	−3554.81	+	6157.11i	−0.198281	+	0.343432i
$686$		0			0
$687$	6339.69	+	4455.48i	0.352073	+	0.247434i
$688$		0			0
$689$	−1542.37	−	1294.20i	−0.0852826	−	0.0715606i
$690$		0			0
$691$	18824.4	+	6851.54i	1.03635	+	0.377199i	0.803494	−	0.595313i	$-0.202972\pi$
$691$	18824.4	+	6851.54i	1.03635	+	0.377199i	0.232853	+	0.972512i	$0.425194\pi$
$692$		0			0
$693$	−92.9184	+	26829.4i	−0.00509333	+	1.47066i
$694$		0			0
$695$	978.087	+	5547.01i	0.0533827	+	0.302748i
$696$		0			0
$697$	−39267.8	+	14292.3i	−2.13396	+	0.776699i
$698$		0			0
$699$	995.437	−	2125.11i	0.0538639	−	0.114991i
$700$		0			0
$701$	−14959.1			−0.805987			−0.402993	−	0.915203i	$-0.632030\pi$
$701$	−14959.1			−0.805987			−0.402993	+	0.915203i	$0.632030\pi$
$702$		0			0
$703$	13531.1			0.725938
$704$		0			0
$705$	5782.86	+	8289.30i	0.308929	+	0.442827i
$706$		0			0
$707$	−30764.4	+	11197.3i	−1.63651	+	0.595641i
$708$		0			0
$709$	−1068.39	−	6059.12i	−0.0565925	−	0.320952i	0.943349	−	0.331803i	$-0.107657\pi$
$709$	−1068.39	−	6059.12i	−0.0565925	−	0.320952i	−0.999941	+	0.0108511i	$0.996546\pi$
$710$		0			0
$711$	6339.72	−	7502.46i	0.334400	−	0.395730i
$712$		0			0
$713$	2180.02	+	793.463i	0.114506	+	0.0416766i
$714$		0			0
$715$	−1167.29	−	979.477i	−0.0610550	−	0.0512313i
$716$		0			0
$717$	1430.39	−	16029.7i	0.0745036	−	0.834925i
$718$		0			0
$719$	10762.6	−	18641.4i	0.558244	−	0.966908i	−0.439399	−	0.898292i	$-0.644809\pi$
$719$	10762.6	−	18641.4i	0.558244	−	0.966908i	0.997643	−	0.0686155i	$-0.0218582\pi$
$720$		0			0
$721$	11368.4	+	19690.7i	0.587214	+	1.01708i
$722$		0			0
$723$	2055.69	−	2048.58i	0.105743	−	0.105377i
$724$		0			0
$725$	−2422.00	+	13735.9i	−0.124070	+	0.703637i
$726$		0			0
$727$	9630.04	−	8080.56i	0.491277	−	0.412230i	−0.363207	−	0.931709i	$-0.618318\pi$
$727$	9630.04	−	8080.56i	0.491277	−	0.412230i	0.854484	+	0.519478i	$0.173874\pi$
$728$		0			0
$729$	17147.3	−	9663.87i	0.871173	−	0.490975i
$730$		0			0
$731$	−737.594	+	618.915i	−0.0373200	+	0.0313152i
$732$		0			0
$733$	−644.806	+	3656.88i	−0.0324918	+	0.184270i	−0.996734	−	0.0807515i	$-0.974268\pi$
$733$	−644.806	+	3656.88i	−0.0324918	+	0.184270i	0.964242	+	0.265022i	$0.0853791\pi$
$734$		0			0
$735$	14133.0	−	14084.1i	0.709257	−	0.706805i
$736$		0			0
$737$	−9143.30	−	15836.7i	−0.456985	−	0.791521i
$738$		0			0
$739$	−2061.21	+	3570.13i	−0.102602	+	0.177712i	−0.912756	−	0.408505i	$-0.866050\pi$
$739$	−2061.21	+	3570.13i	−0.102602	+	0.177712i	0.810154	+	0.586217i	$0.199384\pi$
$740$		0			0
$741$	271.662	−	3044.39i	0.0134680	−	0.150929i
$742$		0			0
$743$	7603.08	+	6379.74i	0.375411	+	0.315007i	0.810898	−	0.585188i	$-0.198979\pi$
$743$	7603.08	+	6379.74i	0.375411	+	0.315007i	−0.435487	+	0.900195i	$0.643424\pi$
$744$		0			0
$745$	−9404.72	−	3423.04i	−0.462500	−	0.168336i
$746$		0			0
$747$	10324.1	−	12217.6i	0.505674	−	0.598418i
$748$		0			0
$749$	−7459.27	−	42303.6i	−0.363893	−	2.06374i
$750$		0			0
$751$	−10461.9	+	3807.83i	−0.508337	+	0.185020i	−0.583440	−	0.812156i	$-0.698294\pi$
$751$	−10461.9	+	3807.83i	−0.508337	+	0.185020i	0.0751030	+	0.997176i	$0.476071\pi$
$752$		0			0
$753$	−1874.12	−	2686.41i	−0.0906997	−	0.130011i
$754$		0			0
$755$	−3359.93			−0.161961
$756$		0			0
$757$	−32325.4			−1.55203			−0.776015	−	0.630715i	$-0.782762\pi$
$757$	−32325.4			−1.55203			−0.776015	+	0.630715i	$0.782762\pi$
$758$		0			0
$759$	−6504.31	+	13885.7i	−0.311056	+	0.664057i
$760$		0			0
$761$	13060.1	−	4753.49i	0.622114	−	0.226431i	−0.0116814	−	0.999932i	$-0.503718\pi$
$761$	13060.1	−	4753.49i	0.622114	−	0.226431i	0.633795	+	0.773501i	$0.281496\pi$
$762$		0			0
$763$	3310.35	+	18773.9i	0.157068	+	0.890775i
$764$		0			0
$765$	−42.7761	+	12351.3i	−0.00202167	+	0.583740i
$766$		0			0
$767$	7607.52	+	2768.91i	0.358138	+	0.130351i
$768$		0			0
$769$	−7019.28	−	5889.88i	−0.329157	−	0.276196i	0.463199	−	0.886254i	$-0.346701\pi$
$769$	−7019.28	−	5889.88i	−0.329157	−	0.276196i	−0.792356	+	0.610059i	$0.791146\pi$
$770$		0			0
$771$	−9496.45	−	6674.02i	−0.443588	−	0.311750i
$772$		0			0
$773$	−1545.03	+	2676.06i	−0.0718897	+	0.124517i	−0.899729	−	0.436448i	$-0.856236\pi$
$773$	−1545.03	+	2676.06i	−0.0718897	+	0.124517i	0.827840	+	0.560965i	$0.189570\pi$
$774$		0			0
$775$	1167.42	+	2022.03i	0.0541097	+	0.0937207i
$776$		0			0
$777$	−37860.7	−	10215.0i	−1.74806	−	0.471638i
$778$		0			0
$779$	−4809.95	+	27278.6i	−0.221225	+	1.25463i
$780$		0			0
$781$	6461.41	−	5421.76i	0.296040	−	0.248407i
$782$		0			0
$783$	11438.0	−	16155.9i	0.522046	−	0.737376i
$784$		0			0
$785$	5199.06	−	4362.53i	0.236385	−	0.198351i
$786$		0			0
$787$	3688.64	−	20919.3i	0.167072	−	0.947512i	−0.779830	−	0.625991i	$-0.784695\pi$
$787$	3688.64	−	20919.3i	0.167072	−	0.947512i	0.946902	−	0.321521i	$-0.104194\pi$
$788$		0			0
$789$	887.739	+	3336.19i	0.0400562	+	0.150534i
$790$		0			0
$791$	−32465.9	−	56232.6i	−1.45936	−	2.52769i
$792$		0			0
$793$	3133.20	−	5426.86i	0.140307	−	0.243018i
$794$		0			0
$795$	4891.84	−	2270.80i	0.218234	−	0.101304i
$796$		0			0
$797$	−16702.1	−	14014.7i	−0.742307	−	0.622870i	0.191149	−	0.981561i	$-0.438779\pi$
$797$	−16702.1	−	14014.7i	−0.742307	−	0.622870i	−0.933456	+	0.358691i	$0.883223\pi$
$798$		0			0
$799$	−31979.8	−	11639.7i	−1.41598	−	0.515373i
$800$		0			0
$801$	−12414.4	−	14899.3i	−0.547615	−	0.657232i
$802$		0			0
$803$	6352.29	+	36025.6i	0.279162	+	1.58321i
$804$		0			0
$805$	15609.9	−	5681.54i	0.683449	−	0.248755i
$806$		0			0
$807$	−35994.9	+	3086.35i	−1.57011	+	0.134628i
$808$		0			0
$809$	17728.5			0.770460			0.385230	−	0.922821i	$-0.374122\pi$
$809$	17728.5			0.770460			0.385230	+	0.922821i	$0.374122\pi$
$810$		0			0
$811$	3211.20			0.139039			0.0695194	−	0.997581i	$-0.477853\pi$
$811$	3211.20			0.139039			0.0695194	+	0.997581i	$0.477853\pi$
$812$		0			0
$813$	−45473.5	+	3899.08i	−1.96165	+	0.168200i
$814$		0			0
$815$	−12310.8	+	4480.76i	−0.529114	+	0.192582i
$816$		0			0
$817$	110.829	+	628.543i	0.00474592	+	0.0269155i
$818$		0			0
$819$	−3058.43	+	8313.28i	−0.130489	+	0.354688i
$820$		0			0
$821$	11961.2	+	4353.53i	0.508465	+	0.185066i	0.583497	−	0.812115i	$-0.301684\pi$
$821$	11961.2	+	4353.53i	0.508465	+	0.185066i	−0.0750324	+	0.997181i	$0.523906\pi$
$822$		0			0
$823$	−31351.4	−	26306.9i	−1.32787	−	1.11422i	−0.984570	−	0.174994i	$-0.944009\pi$
$823$	−31351.4	−	26306.9i	−1.32787	−	1.11422i	−0.343304	−	0.939224i	$-0.611546\pi$
$824$		0			0
$825$	−13997.6	+	6497.68i	−0.590706	+	0.274206i
$826$		0			0
$827$	14546.9	−	25195.9i	0.611662	−	1.05943i	−0.379298	−	0.925274i	$-0.623835\pi$
$827$	14546.9	−	25195.9i	0.611662	−	1.05943i	0.990960	−	0.134155i	$-0.0428321\pi$
$828$		0			0
$829$	−4034.95	−	6988.75i	−0.169047	−	0.292798i	0.769038	−	0.639203i	$-0.220736\pi$
$829$	−4034.95	−	6988.75i	−0.169047	−	0.292798i	−0.938085	+	0.346405i	$0.887402\pi$
$830$		0			0
$831$	9436.87	+	35464.4i	0.393937	+	1.48044i
$832$		0			0
$833$	−11666.3	+	66162.7i	−0.485249	+	2.75198i
$834$		0			0
$835$	13992.0	−	11740.7i	0.579896	−	0.486591i
$836$		0			0
$837$	−271.653	−	3302.51i	−0.0112183	−	0.136382i
$838$		0			0
$839$	32834.5	−	27551.4i	1.35110	−	1.13371i	0.372475	−	0.928042i	$-0.378509\pi$
$839$	32834.5	−	27551.4i	1.35110	−	1.13371i	0.978623	−	0.205664i	$-0.0659353\pi$
$840$		0			0
$841$	−778.180	+	4413.28i	−0.0319070	+	0.180954i
$842$		0			0
$843$	−33188.9	−	8954.55i	−1.35597	−	0.365850i
$844$		0			0
$845$	5365.43	+	9293.20i	0.218434	+	0.378338i
$846$		0			0
$847$	7084.46	−	12270.7i	0.287397	−	0.497786i
$848$		0			0
$849$	22679.7	+	15939.1i	0.916802	+	0.644320i
$850$		0			0
$851$	−17168.4	−	14406.0i	−0.691570	−	0.580296i
$852$		0			0
$853$	25788.2	+	9386.15i	1.03514	+	0.376759i	0.803035	−	0.595932i	$-0.203217\pi$
$853$	25788.2	+	9386.15i	1.03514	+	0.376759i	0.232103	+	0.972691i	$0.425439\pi$
$854$		0			0
$855$	7076.07	+	4118.11i	0.283037	+	0.164721i
$856$		0			0
$857$	3298.00	+	18703.9i	0.131456	+	0.745522i	0.977263	+	0.212033i	$0.0680083\pi$
$857$	3298.00	+	18703.9i	0.131456	+	0.745522i	−0.845807	+	0.533489i	$0.820881\pi$
$858$		0			0
$859$	12590.2	−	4582.47i	0.500085	−	0.182016i	−0.0796470	−	0.996823i	$-0.525379\pi$
$859$	12590.2	−	4582.47i	0.500085	−	0.182016i	0.579732	+	0.814807i	$0.303157\pi$
$860$		0			0
$861$	34051.9	−	72695.7i	1.34784	−	2.87743i
$862$		0			0
$863$	−3158.44			−0.124582			−0.0622912	−	0.998058i	$-0.519841\pi$
$863$	−3158.44			−0.124582			−0.0622912	+	0.998058i	$0.519841\pi$
$864$		0			0
$865$	11645.3			0.457747
$866$		0			0
$867$	−9188.93	−	13171.6i	−0.359945	−	0.515954i
$868$		0			0
$869$	10270.4	−	3738.13i	0.400921	−	0.145923i
$870$		0			0
$871$	−1048.40	−	5945.76i	−0.0407849	−	0.231302i
$872$		0			0
$873$	21955.3	+	3949.76i	0.851173	+	0.153126i
$874$		0			0
$875$	35575.7	+	12948.5i	1.37449	+	0.500273i
$876$		0			0
$877$	5090.44	+	4271.38i	0.196000	+	0.164463i	0.735506	−	0.677518i	$-0.236945\pi$
$877$	5090.44	+	4271.38i	0.196000	+	0.164463i	−0.539506	+	0.841982i	$0.681389\pi$
$878$		0			0
$879$	−134.391	+	1506.06i	−0.00515690	+	0.0577908i
$880$		0			0
$881$	−20849.2	+	36111.8i	−0.797305	+	1.38097i	0.124060	+	0.992275i	$0.460408\pi$
$881$	−20849.2	+	36111.8i	−0.797305	+	1.38097i	−0.921365	+	0.388698i	$0.872925\pi$
$882$		0			0
$883$	8379.42	+	14513.6i	0.319354	+	0.553138i	0.980353	−	0.197249i	$-0.0632006\pi$
$883$	8379.42	+	14513.6i	0.319354	+	0.553138i	−0.660999	+	0.750387i	$0.729867\pi$
$884$		0			0
$885$	−15360.5	+	15307.4i	−0.583431	+	0.581414i
$886$		0			0
$887$	−4201.60	+	23828.5i	−0.159049	+	0.902009i	0.795942	+	0.605373i	$0.206976\pi$
$887$	−4201.60	+	23828.5i	−0.159049	+	0.902009i	−0.954991	+	0.296636i	$0.904135\pi$
$888$		0			0
$889$	12464.9	−	10459.3i	0.470257	−	0.394593i
$890$		0			0
$891$	21901.2	+	151.703i	0.823476	+	0.00570396i
$892$		0			0
$893$	−17280.8	+	14500.3i	−0.647570	+	0.543376i
$894$		0			0
$895$	−1026.37	+	5820.84i	−0.0383328	+	0.217396i
$896$		0			0
$897$	−3585.94	+	3573.54i	−0.133479	+	0.133018i
$898$		0			0
$899$	−1666.26	−	2886.05i	−0.0618164	−	0.107069i
$900$		0			0
$901$	−9079.87	+	15726.8i	−0.335732	+	0.581504i
$902$		0			0
$903$	164.401	−	1842.36i	0.00605862	−	0.0678960i
$904$		0			0
$905$	−13998.8	−	11746.4i	−0.514183	−	0.431451i
$906$		0			0
$907$	17284.4	+	6291.01i	0.632766	+	0.230308i	0.638435	−	0.769676i	$-0.279582\pi$
$907$	17284.4	+	6291.01i	0.632766	+	0.230308i	−0.00566855	+	0.999984i	$0.501804\pi$
$908$		0			0
$909$	9053.63	+	25145.3i	0.330352	+	0.917509i
$910$		0			0
$911$	397.004	+	2251.52i	0.0144383	+	0.0818839i	0.991176	−	0.132556i	$-0.0423185\pi$
$911$	397.004	+	2251.52i	0.0144383	+	0.0818839i	−0.976737	+	0.214440i	$0.931207\pi$
$912$		0			0
$913$	16725.2	−	6087.47i	0.606268	−	0.220664i
$914$		0			0
$915$	9603.82	+	13766.4i	0.346986	+	0.497379i
$916$		0			0
$917$	−71991.5			−2.59255
$918$		0			0
$919$	4568.58			0.163986			0.0819932	−	0.996633i	$-0.473871\pi$
$919$	4568.58			0.163986			0.0819932	+	0.996633i	$0.473871\pi$
$920$		0			0
$921$	2524.90	−	5390.29i	0.0903349	−	0.192851i
$922$		0			0
$923$	2616.87	−	952.462i	0.0933209	−	0.0339660i
$924$		0			0
$925$	−3916.78	−	22213.2i	−0.139225	−	0.789584i
$926$		0			0
$927$	16106.0	−	9224.60i	0.570649	−	0.326835i
$928$		0			0
$929$	27954.9	+	10174.8i	0.987268	+	0.359336i	0.784662	−	0.619924i	$-0.212837\pi$
$929$	27954.9	+	10174.8i	0.987268	+	0.359336i	0.202606	+	0.979260i	$0.435059\pi$
$930$		0			0
$931$	34114.2	+	28625.2i	1.20091	+	1.00768i
$932$		0			0
$933$	−39932.3	−	28064.1i	−1.40121	−	0.984756i
$934$		0			0
$935$	−6871.80	+	11902.3i	−0.240355	+	0.416307i
$936$		0			0
$937$	19606.1	+	33958.8i	0.683568	+	1.18397i	0.973885	+	0.227044i	$0.0729060\pi$
$937$	19606.1	+	33958.8i	0.683568	+	1.18397i	−0.290317	+	0.956931i	$0.593761\pi$
$938$		0			0
$939$	−15549.5	−	4195.36i	−0.540404	−	0.145804i
$940$		0			0
$941$	−3310.43	+	18774.4i	−0.114683	+	0.650400i	0.872223	+	0.489108i	$0.162677\pi$
$941$	−3310.43	+	18774.4i	−0.114683	+	0.650400i	−0.986907	+	0.161293i	$0.948434\pi$
$942$		0			0
$943$	35145.4	−	29490.5i	1.21367	−	1.01839i
$944$		0			0
$945$	−16690.3	−	16864.7i	−0.574536	−	0.580537i
$946$		0			0
$947$	−4780.70	+	4011.49i	−0.164046	+	0.137651i	−0.721115	−	0.692815i	$-0.756370\pi$
$947$	−4780.70	+	4011.49i	−0.164046	+	0.137651i	0.557069	+	0.830466i	$0.311926\pi$
$948$		0			0
$949$	−2097.26	+	11894.2i	−0.0717387	+	0.406850i
$950$		0			0
$951$	894.224	+	3360.56i	0.0304913	+	0.114588i
$952$		0			0
$953$	−2401.20	−	4159.01i	−0.0816187	−	0.141368i	0.822327	−	0.569016i	$-0.192676\pi$
$953$	−2401.20	−	4159.01i	−0.0816187	−	0.141368i	−0.903945	+	0.427648i	$0.859342\pi$
$954$		0			0
$955$	2055.22	−	3559.74i	0.0696390	−	0.120618i
$956$		0			0
$957$	19978.7	−	9274.15i	0.674839	−	0.313261i
$958$		0			0
$959$	35228.8	+	29560.5i	1.18623	+	0.995367i
$960$		0			0
$961$	27470.2	+	9998.32i	0.922096	+	0.335616i
$962$		0			0
$963$	−34554.4	+	5969.56i	−1.15628	+	0.199758i
$964$		0			0
$965$	−793.184	−	4498.37i	−0.0264596	−	0.150060i
$966$		0			0
$967$	−41151.1	+	14977.8i	−1.36849	+	0.498089i	−0.918668	−	0.395030i	$-0.870734\pi$
$967$	−41151.1	+	14977.8i	−1.36849	+	0.498089i	−0.449820	+	0.893119i	$0.648512\pi$
$968$		0			0
$969$	−27466.7	+	2355.10i	−0.910584	+	0.0780772i
$970$		0			0
$971$	54379.6			1.79725			0.898623	−	0.438723i	$-0.144569\pi$
$971$	54379.6			1.79725			0.898623	+	0.438723i	$0.144569\pi$
$972$		0			0
$973$	36433.8			1.20043
$974$		0			0
$975$	−5076.43	+	435.274i	−0.166745	+	0.0142974i
$976$		0			0
$977$	11761.8	−	4280.96i	0.385153	−	0.140184i	−0.142184	−	0.989840i	$-0.545413\pi$
$977$	11761.8	−	4280.96i	0.385153	−	0.140184i	0.527337	+	0.849656i	$0.323190\pi$
$978$		0			0
$979$	−3747.25	−	21251.7i	−0.122332	−	0.693777i
$980$		0			0
$981$	15334.9	−	2649.23i	0.499089	−	0.0862217i
$982$		0			0
$983$	−4043.60	−	1471.75i	−0.131201	−	0.0477534i	0.275585	−	0.961277i	$-0.411128\pi$
$983$	−4043.60	−	1471.75i	−0.131201	−	0.0477534i	−0.406786	+	0.913523i	$0.633351\pi$
$984$		0			0
$985$	−9110.88	−	7644.93i	−0.294717	−	0.247297i
$986$		0			0
$987$	59299.4	−	27526.8i	1.91238	−	0.887730i
$988$		0			0
$989$	528.564	−	915.500i	0.0169943	−	0.0294350i
$990$		0			0
$991$	4071.74	+	7052.46i	0.130518	+	0.226063i	0.923876	−	0.382691i	$-0.125003\pi$
$991$	4071.74	+	7052.46i	0.130518	+	0.226063i	−0.793359	+	0.608755i	$0.791669\pi$
$992$		0			0
$993$	388.033	+	1458.25i	0.0124006	+	0.0466025i
$994$		0			0
$995$	−281.320	+	1595.45i	−0.00896326	+	0.0508332i
$996$		0			0
$997$	−1755.63	+	1473.15i	−0.0557687	+	0.0467955i	−0.670246	−	0.742139i	$-0.733811\pi$
$997$	−1755.63	+	1473.15i	−0.0557687	+	0.0467955i	0.614477	+	0.788935i	$0.289367\pi$
$998$		0			0
$999$	−8445.80	+	30877.6i	−0.267481	+	0.977902i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.49.9	✓	54
3.2	odd	2		324.4.i.a.37.6		54
27.11	odd	18		324.4.i.a.289.6		54
27.16	even	9	inner	108.4.i.a.97.9	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.9	✓	54	1.1	even	1	trivial
108.4.i.a.97.9	yes	54	27.16	even	9	inner
324.4.i.a.37.6		54	3.2	odd	2
324.4.i.a.289.6		54	27.11	odd	18

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(5.17716 - 0.443910i) q^{3} +(-4.80495 + 1.74886i) q^{5} +(5.74341 + 32.5725i) q^{7} +(26.6059 - 4.59638i) q^{9} +O(q^{10})\)	\(q+(5.17716 - 0.443910i) q^{3} +(-4.80495 + 1.74886i) q^{5} +(5.74341 + 32.5725i) q^{7} +(26.6059 - 4.59638i) q^{9} +(28.2317 + 10.2755i) q^{11} +(7.59849 + 6.37589i) q^{13} +(-24.0996 + 11.1871i) q^{15} +(44.7319 - 77.4779i) q^{17} +(-29.6509 - 51.3568i) q^{19} +(44.1938 + 166.083i) q^{21} +(-17.0562 + 96.7304i) q^{23} +(-75.7266 + 63.5421i) q^{25} +(135.702 - 35.6068i) q^{27} +(108.085 - 90.6938i) q^{29} +(4.10141 - 23.2603i) q^{31} +(150.721 + 40.6655i) q^{33} +(-84.5615 - 146.465i) q^{35} +(-114.087 + 197.604i) q^{37} +(42.1689 + 29.6359i) q^{39} +(-357.814 - 300.241i) q^{41} +(-10.1135 - 3.68102i) q^{43} +(-119.801 + 68.6153i) q^{45} +(-66.0561 - 374.623i) q^{47} +(-705.667 + 256.842i) q^{49} +(197.191 - 420.972i) q^{51} -202.984 q^{53} -153.622 q^{55} +(-176.305 - 252.720i) q^{57} +(766.955 - 279.149i) q^{59} +(-109.702 - 622.151i) q^{61} +(302.524 + 840.222i) q^{63} +(-47.6609 - 17.3471i) q^{65} +(-466.269 - 391.246i) q^{67} +(-45.3629 + 508.360i) q^{69} +(140.376 - 243.138i) q^{71} +(608.806 + 1054.48i) q^{73} +(-363.841 + 362.583i) q^{75} +(-172.552 + 978.592i) q^{77} +(278.680 - 233.841i) q^{79} +(686.746 - 244.582i) q^{81} +(453.825 - 380.804i) q^{83} +(-79.4366 + 450.507i) q^{85} +(519.311 - 517.516i) q^{87} +(-359.138 - 622.046i) q^{89} +(-164.037 + 284.121i) q^{91} +(10.9082 - 122.243i) q^{93} +(232.286 + 194.911i) q^{95} +(776.386 + 282.581i) q^{97} +(798.358 + 143.625i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 54 * q + 12 * q^5 - 48 * q^9	\( 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) \) 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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