LMFDB - Embedded newform 108.4.i.a.25.4

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			25.4
Character	$\chi$	$=$	108.25
Dual form			108.4.i.a.13.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.716497 + 5.14652i) q^{3} +(7.52940 - 6.31792i) q^{5} +(24.9076 - 9.06562i) q^{7} +(-25.9733 - 7.37493i) q^{9} +O(q^{10})$	\(q+(-0.716497 + 5.14652i) q^{3} +(7.52940 - 6.31792i) q^{5} +(24.9076 - 9.06562i) q^{7} +(-25.9733 - 7.37493i) q^{9} +(17.0456 + 14.3030i) q^{11} +(13.5975 + 77.1153i) q^{13} +(27.1205 + 43.2769i) q^{15} +(32.2518 + 55.8618i) q^{17} +(35.2151 - 60.9943i) q^{19} +(28.8101 + 134.683i) q^{21} +(-112.980 - 41.1215i) q^{23} +(-4.93023 + 27.9607i) q^{25} +(56.5650 - 128.388i) q^{27} +(30.3150 - 171.925i) q^{29} +(139.932 + 50.9312i) q^{31} +(-85.8237 + 77.4776i) q^{33} +(130.263 - 225.623i) q^{35} +(144.919 + 251.008i) q^{37} +(-406.617 + 14.7269i) q^{39} +(-76.1146 - 431.667i) q^{41} +(-76.4782 - 64.1728i) q^{43} +(-242.157 + 108.568i) q^{45} +(-290.224 + 105.633i) q^{47} +(275.449 - 231.129i) q^{49} +(-310.602 + 125.960i) q^{51} -533.670 q^{53} +218.708 q^{55} +(288.677 + 224.937i) q^{57} +(320.392 - 268.841i) q^{59} +(-87.3117 + 31.7789i) q^{61} +(-713.789 + 51.7720i) q^{63} +(589.589 + 494.724i) q^{65} +(-80.8767 - 458.675i) q^{67} +(292.583 - 551.992i) q^{69} +(-451.994 - 782.877i) q^{71} +(-86.0801 + 149.095i) q^{73} +(-140.368 - 45.4073i) q^{75} +(554.231 + 201.723i) q^{77} +(-27.8206 + 157.779i) q^{79} +(620.221 + 383.102i) q^{81} +(103.082 - 584.607i) q^{83} +(595.767 + 216.841i) q^{85} +(863.095 + 279.201i) q^{87} +(-818.257 + 1417.26i) q^{89} +(1037.78 + 1797.48i) q^{91} +(-362.379 + 683.672i) q^{93} +(-120.209 - 681.737i) q^{95} +(-323.876 - 271.765i) q^{97} +(-337.247 - 497.205i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{5}{9}\right)$	$1$

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
$3$	−0.716497	+	5.14652i	−0.137890	+	0.990448i
$3$	−0.716497	+	5.14652i	−0.137890	+	0.990448i
$4$		0			0
$5$	7.52940	−	6.31792i	0.673450	−	0.565092i	−0.240634	−	0.970616i	$-0.577355\pi$
$5$	7.52940	−	6.31792i	0.673450	−	0.565092i	0.914084	+	0.405524i	$0.132911\pi$
$6$		0			0
$7$	24.9076	−	9.06562i	1.34488	−	0.489497i	0.433536	−	0.901136i	$-0.357266\pi$
$7$	24.9076	−	9.06562i	1.34488	−	0.489497i	0.911347	+	0.411639i	$0.135044\pi$
$8$		0			0
$9$	−25.9733	−	7.37493i	−0.961973	−	0.273146i
$10$		0			0
$11$	17.0456	+	14.3030i	0.467223	+	0.392046i	0.845780	−	0.533531i	$-0.179135\pi$
$11$	17.0456	+	14.3030i	0.467223	+	0.392046i	−0.378558	+	0.925578i	$0.623580\pi$
$12$		0			0
$13$	13.5975	+	77.1153i	0.290098	+	1.64522i	0.686487	+	0.727142i	$0.259152\pi$
$13$	13.5975	+	77.1153i	0.290098	+	1.64522i	−0.396389	+	0.918083i	$0.629737\pi$
$14$		0			0
$15$	27.1205	+	43.2769i	0.466832	+	0.744937i
$16$		0			0
$17$	32.2518	+	55.8618i	0.460130	+	0.796969i	0.998967	−	0.0454411i	$-0.0144693\pi$
$17$	32.2518	+	55.8618i	0.460130	+	0.796969i	−0.538837	+	0.842410i	$0.681136\pi$
$18$		0			0
$19$	35.2151	−	60.9943i	0.425205	−	0.736477i	−0.571234	−	0.820787i	$-0.693535\pi$
$19$	35.2151	−	60.9943i	0.425205	−	0.736477i	0.996440	+	0.0843100i	$0.0268686\pi$
$20$		0			0
$21$	28.8101	+	134.683i	0.299375	+	1.39953i
$22$		0			0
$23$	−112.980	−	41.1215i	−1.02426	−	0.372801i	−0.225369	−	0.974274i	$-0.572359\pi$
$23$	−112.980	−	41.1215i	−1.02426	−	0.372801i	−0.798894	+	0.601472i	$0.794581\pi$
$24$		0			0
$25$	−4.93023	+	27.9607i	−0.0394419	+	0.223686i
$26$		0			0
$27$	56.5650	−	128.388i	0.403183	−	0.915119i
$28$		0			0
$29$	30.3150	−	171.925i	0.194116	−	1.10089i	−0.719556	−	0.694434i	$-0.755655\pi$
$29$	30.3150	−	171.925i	0.194116	−	1.10089i	0.913672	−	0.406452i	$-0.133234\pi$
$30$		0			0
$31$	139.932	+	50.9312i	0.810728	+	0.295081i	0.713925	−	0.700223i	$-0.246916\pi$
$31$	139.932	+	50.9312i	0.810728	+	0.295081i	0.0968036	+	0.995304i	$0.469138\pi$
$32$		0			0
$33$	−85.8237	+	77.4776i	−0.452727	+	0.408700i
$34$		0			0
$35$	130.263	−	225.623i	0.629100	−	1.08963i
$36$		0			0
$37$	144.919	+	251.008i	0.643908	+	1.11528i	0.984553	+	0.175090i	$0.0560215\pi$
$37$	144.919	+	251.008i	0.643908	+	1.11528i	−0.340644	+	0.940192i	$0.610645\pi$
$38$		0			0
$39$	−406.617	+	14.7269i	−1.66951	+	0.0604664i
$40$		0			0
$41$	−76.1146	−	431.667i	−0.289929	−	1.64427i	−0.687126	−	0.726538i	$-0.741128\pi$
$41$	−76.1146	−	431.667i	−0.289929	−	1.64427i	0.397197	−	0.917733i	$-0.369983\pi$
$42$		0			0
$43$	−76.4782	−	64.1728i	−0.271228	−	0.227588i	0.497021	−	0.867739i	$-0.334427\pi$
$43$	−76.4782	−	64.1728i	−0.271228	−	0.227588i	−0.768249	+	0.640151i	$0.778872\pi$
$44$		0			0
$45$	−242.157	+	108.568i	−0.802193	+	0.359653i
$46$		0			0
$47$	−290.224	+	105.633i	−0.900712	+	0.327832i	−0.750538	−	0.660827i	$-0.770206\pi$
$47$	−290.224	+	105.633i	−0.900712	+	0.327832i	−0.150174	+	0.988660i	$0.547983\pi$
$48$		0			0
$49$	275.449	−	231.129i	0.803057	−	0.673845i
$50$		0			0
$51$	−310.602	+	125.960i	−0.852804	+	0.345841i
$52$		0			0
$53$	−533.670			−1.38312			−0.691558	−	0.722321i	$-0.743075\pi$
$53$	−533.670			−1.38312			−0.691558	+	0.722321i	$0.743075\pi$
$54$		0			0
$55$	218.708			0.536193
$56$		0			0
$57$	288.677	+	224.937i	0.670810	+	0.522696i
$58$		0			0
$59$	320.392	−	268.841i	0.706975	−	0.593222i	−0.216774	−	0.976222i	$-0.569553\pi$
$59$	320.392	−	268.841i	0.706975	−	0.593222i	0.923748	+	0.383000i	$0.125109\pi$
$60$		0			0
$61$	−87.3117	+	31.7789i	−0.183264	+	0.0667027i	−0.432022	−	0.901863i	$-0.642200\pi$
$61$	−87.3117	+	31.7789i	−0.183264	+	0.0667027i	0.248758	+	0.968566i	$0.419978\pi$
$62$		0			0
$63$	−713.789	+	51.7720i	−1.42744	+	0.103534i
$64$		0			0
$65$	589.589	+	494.724i	1.12507	+	0.944045i
$66$		0			0
$67$	−80.8767	−	458.675i	−0.147473	−	0.836359i	−0.965349	−	0.260964i	$-0.915960\pi$
$67$	−80.8767	−	458.675i	−0.147473	−	0.836359i	0.817876	−	0.575395i	$-0.195152\pi$
$68$		0			0
$69$	292.583	−	551.992i	0.510475	−	0.963073i
$70$		0			0
$71$	−451.994	−	782.877i	−0.755519	−	1.30860i	−0.945116	−	0.326735i	$-0.894051\pi$
$71$	−451.994	−	782.877i	−0.755519	−	1.30860i	0.189597	−	0.981862i	$-0.439282\pi$
$72$		0			0
$73$	−86.0801	+	149.095i	−0.138012	+	0.239045i	−0.926744	−	0.375693i	$-0.877405\pi$
$73$	−86.0801	+	149.095i	−0.138012	+	0.239045i	0.788732	+	0.614738i	$0.210738\pi$
$74$		0			0
$75$	−140.368	−	45.4073i	−0.216111	−	0.0699091i
$76$		0			0
$77$	554.231	+	201.723i	0.820265	+	0.298552i
$78$		0			0
$79$	−27.8206	+	157.779i	−0.0396211	+	0.224702i	−0.998189	−	0.0601633i	$-0.980838\pi$
$79$	−27.8206	+	157.779i	−0.0396211	+	0.224702i	0.958567	+	0.284866i	$0.0919490\pi$
$80$		0			0
$81$	620.221	+	383.102i	0.850783	+	0.525517i
$82$		0			0
$83$	103.082	−	584.607i	0.136322	−	0.773120i	−0.837608	−	0.546272i	$-0.816047\pi$
$83$	103.082	−	584.607i	0.136322	−	0.773120i	0.973930	−	0.226849i	$-0.0728423\pi$
$84$		0			0
$85$	595.767	+	216.841i	0.760235	+	0.276703i
$86$		0			0
$87$	863.095	+	279.201i	1.06360	+	0.344063i
$88$		0			0
$89$	−818.257	+	1417.26i	−0.974551	+	1.68797i	−0.293143	+	0.956068i	$0.594701\pi$
$89$	−818.257	+	1417.26i	−0.974551	+	1.68797i	−0.681408	+	0.731904i	$0.738632\pi$
$90$		0			0
$91$	1037.78	+	1797.48i	1.19548	+	2.07063i
$92$		0			0
$93$	−362.379	+	683.672i	−0.404053	+	0.762295i
$94$		0			0
$95$	−120.209	−	681.737i	−0.129823	−	0.736260i
$96$		0			0
$97$	−323.876	−	271.765i	−0.339017	−	0.284469i	0.457345	−	0.889289i	$-0.348800\pi$
$97$	−323.876	−	271.765i	−0.339017	−	0.284469i	−0.796362	+	0.604820i	$0.793245\pi$
$98$		0			0
$99$	−337.247	−	497.205i	−0.342370	−	0.504758i
$100$		0			0
$101$	743.112	−	270.471i	0.732103	−	0.266464i	0.0510481	−	0.998696i	$-0.483744\pi$
$101$	743.112	−	270.471i	0.732103	−	0.266464i	0.681055	+	0.732233i	$0.261522\pi$
$102$		0			0
$103$	696.301	−	584.266i	0.666103	−	0.558927i	−0.245806	−	0.969319i	$-0.579053\pi$
$103$	696.301	−	584.266i	0.666103	−	0.558927i	0.911909	+	0.410392i	$0.134608\pi$
$104$		0			0
$105$	1067.84	+	832.060i	0.992478	+	0.773340i
$106$		0			0
$107$	225.538			0.203772			0.101886	−	0.994796i	$-0.467512\pi$
$107$	225.538			0.203772			0.101886	+	0.994796i	$0.467512\pi$
$108$		0			0
$109$	−1916.33			−1.68396			−0.841979	−	0.539510i	$-0.818610\pi$
$109$	−1916.33			−1.68396			−0.841979	+	0.539510i	$0.818610\pi$
$110$		0			0
$111$	−1395.65	+	565.984i	−1.19342	+	0.483971i
$112$		0			0
$113$	−496.602	+	416.698i	−0.413419	+	0.346900i	−0.825653	−	0.564178i	$-0.809193\pi$
$113$	−496.602	+	416.698i	−0.413419	+	0.346900i	0.412234	+	0.911078i	$0.364749\pi$
$114$		0			0
$115$	−1110.48	+	404.180i	−0.900456	+	0.327739i
$116$		0			0
$117$	215.548	−	2103.22i	0.170320	−	1.66190i
$118$		0			0
$119$	1309.74	+	1099.00i	1.00894	+	0.846597i
$120$		0			0
$121$	−145.148	−	823.172i	−0.109051	−	0.618462i
$122$		0			0
$123$	2276.12	−	82.4366i	1.66854	−	0.0604313i
$124$		0			0
$125$	753.840	+	1305.69i	0.539404	+	0.934275i
$126$		0			0
$127$	−180.673	+	312.934i	−0.126237	+	0.218649i	−0.922216	−	0.386676i	$-0.873623\pi$
$127$	−180.673	+	312.934i	−0.126237	+	0.218649i	0.795979	+	0.605324i	$0.206957\pi$
$128$		0			0
$129$	385.063	−	347.617i	0.262813	−	0.237255i
$130$		0			0
$131$	−659.456	−	240.022i	−0.439824	−	0.160083i	0.112611	−	0.993639i	$-0.464079\pi$
$131$	−659.456	−	240.022i	−0.439824	−	0.160083i	−0.552434	+	0.833556i	$0.686301\pi$
$132$		0			0
$133$	324.171	−	1838.47i	0.211348	−	1.19861i
$134$		0			0
$135$	−385.243	−	1324.06i	−0.245603	−	0.844122i
$136$		0			0
$137$	−490.451	+	2781.48i	−0.305854	+	1.73459i	0.313602	+	0.949555i	$0.398464\pi$
$137$	−490.451	+	2781.48i	−0.305854	+	1.73459i	−0.619456	+	0.785031i	$0.712647\pi$
$138$		0			0
$139$	−928.816	−	338.061i	−0.566771	−	0.206288i	0.0427113	−	0.999087i	$-0.486400\pi$
$139$	−928.816	−	338.061i	−0.566771	−	0.206288i	−0.609482	+	0.792800i	$0.708623\pi$
$140$		0			0
$141$	−335.696	−	1569.33i	−0.200502	−	0.937313i
$142$		0			0
$143$	−871.200	+	1508.96i	−0.509464	+	0.882418i
$144$		0			0
$145$	−857.954	−	1486.02i	−0.491374	−	0.851085i
$146$		0			0
$147$	992.150	+	1583.20i	0.556674	+	0.888302i
$148$		0			0
$149$	−339.478	−	1925.27i	−0.186652	−	1.05855i	−0.923815	−	0.382840i	$-0.874946\pi$
$149$	−339.478	−	1925.27i	−0.186652	−	1.05855i	0.737163	−	0.675715i	$-0.236165\pi$
$150$		0			0
$151$	−2176.29	−	1826.12i	−1.17287	−	0.984156i	−0.172872	−	0.984944i	$-0.555305\pi$
$151$	−2176.29	−	1826.12i	−1.17287	−	0.984156i	−1.00000		0.000788384i	$0.999749\pi$
$152$		0			0
$153$	−425.708	−	1688.77i	−0.224944	−	0.892345i
$154$		0			0
$155$	1375.38	−	500.599i	0.712733	−	0.259413i
$156$		0			0
$157$	544.235	−	456.668i	0.276654	−	0.232140i	−0.493894	−	0.869522i	$-0.664427\pi$
$157$	544.235	−	456.668i	0.276654	−	0.232140i	0.770548	+	0.637382i	$0.219983\pi$
$158$		0			0
$159$	382.373	−	2746.54i	0.190718	−	1.36990i
$160$		0			0
$161$	−3186.86			−1.56000
$162$		0			0
$163$	1340.84			0.644311			0.322156	−	0.946687i	$-0.395593\pi$
$163$	1340.84			0.644311			0.322156	+	0.946687i	$0.395593\pi$
$164$		0			0
$165$	−156.704	+	1125.59i	−0.0739357	+	0.531071i
$166$		0			0
$167$	1287.33	−	1080.20i	0.596508	−	0.500530i	−0.293813	−	0.955863i	$-0.594924\pi$
$167$	1287.33	−	1080.20i	0.596508	−	0.500530i	0.890321	+	0.455333i	$0.150480\pi$
$168$		0			0
$169$	−3697.37	+	1345.73i	−1.68292	+	0.612531i
$170$		0			0
$171$	−1364.48	+	1324.51i	−0.610201	+	0.592328i
$172$		0			0
$173$	−118.475	−	99.4124i	−0.0520664	−	0.0436889i	0.616383	−	0.787446i	$-0.288597\pi$
$173$	−118.475	−	99.4124i	−0.0520664	−	0.0436889i	−0.668450	+	0.743758i	$0.733042\pi$
$174$		0			0
$175$	130.681	+	741.130i	0.0564490	+	0.320138i
$176$		0			0
$177$	1154.03	+	1841.53i	0.490071	+	0.782021i
$178$		0			0
$179$	−426.813	−	739.262i	−0.178221	−	0.308687i	0.763050	−	0.646339i	$-0.223701\pi$
$179$	−426.813	−	739.262i	−0.178221	−	0.308687i	−0.941271	+	0.337652i	$0.890367\pi$
$180$		0			0
$181$	−985.862	+	1707.56i	−0.404854	+	0.701228i	−0.994304	−	0.106577i	$-0.966011\pi$
$181$	−985.862	+	1707.56i	−0.404854	+	0.701228i	0.589451	+	0.807804i	$0.299344\pi$
$182$		0			0
$183$	−100.992	−	472.121i	−0.0407953	−	0.190711i
$184$		0			0
$185$	2677.00	+	974.349i	1.06388	+	0.387219i
$186$		0			0
$187$	−249.238	+	1413.50i	−0.0974656	+	0.552755i
$188$		0			0
$189$	244.983	−	3710.62i	0.0942850	−	1.42809i
$190$		0			0
$191$	−887.225	+	5031.70i	−0.336112	+	1.90618i	0.0798581	+	0.996806i	$0.474553\pi$
$191$	−887.225	+	5031.70i	−0.336112	+	1.90618i	−0.415970	+	0.909378i	$0.636558\pi$
$192$		0			0
$193$	456.904	+	166.299i	0.170408	+	0.0620233i	0.425815	−	0.904810i	$-0.359987\pi$
$193$	456.904	+	166.299i	0.170408	+	0.0620233i	−0.255408	+	0.966833i	$0.582210\pi$
$194$		0			0
$195$	−2968.54	+	2679.86i	−1.09016	+	0.984147i
$196$		0			0
$197$	1019.37	−	1765.60i	0.368665	−	0.638546i	−0.620692	−	0.784054i	$-0.713148\pi$
$197$	1019.37	−	1765.60i	0.368665	−	0.638546i	0.989357	+	0.145508i	$0.0464817\pi$
$198$		0			0
$199$	2237.59	+	3875.62i	0.797079	+	1.38058i	0.921511	+	0.388353i	$0.126956\pi$
$199$	2237.59	+	3875.62i	0.797079	+	1.38058i	−0.124432	+	0.992228i	$0.539711\pi$
$200$		0			0
$201$	2418.53	−	87.5942i	0.848705	−	0.0307384i
$202$		0			0
$203$	−803.533	−	4557.06i	−0.277817	−	1.57558i
$204$		0			0
$205$	−3300.34	−	2769.31i	−1.12442	−	0.943498i
$206$		0			0
$207$	2631.20	+	1901.28i	0.883484	+	0.638397i
$208$		0			0
$209$	1472.66	−	536.006i	0.487399	−	0.177399i
$210$		0			0
$211$	3003.60	−	2520.32i	0.979983	−	0.822304i	−0.00410365	−	0.999992i	$-0.501306\pi$
$211$	3003.60	−	2520.32i	0.979983	−	0.822304i	0.984087	+	0.177688i	$0.0568618\pi$
$212$		0			0
$213$	4352.94	−	1765.27i	1.40028	−	0.567860i
$214$		0			0
$215$	−981.273			−0.311266
$216$		0			0
$217$	3947.10			1.23478
$218$		0			0
$219$	−705.644	−	549.839i	−0.217731	−	0.169656i
$220$		0			0
$221$	−3869.25	+	3246.69i	−1.17771	+	0.988217i
$222$		0			0
$223$	1464.87	−	533.169i	0.439888	−	0.160106i	−0.112576	−	0.993643i	$-0.535910\pi$
$223$	1464.87	−	533.169i	0.439888	−	0.160106i	0.552464	+	0.833537i	$0.313688\pi$
$224$		0			0
$225$	334.263	−	689.871i	0.0990408	−	0.204406i
$226$		0			0
$227$	−195.605	−	164.132i	−0.0571927	−	0.0479904i	0.613743	−	0.789506i	$-0.289663\pi$
$227$	−195.605	−	164.132i	−0.0571927	−	0.0479904i	−0.670936	+	0.741515i	$0.734107\pi$
$228$		0			0
$229$	−897.549	−	5090.25i	−0.259003	−	1.46888i	−0.785584	−	0.618755i	$-0.787637\pi$
$229$	−897.549	−	5090.25i	−0.259003	−	1.46888i	0.526581	−	0.850125i	$-0.323474\pi$
$230$		0			0
$231$	−1435.28	+	2707.82i	−0.408807	+	0.771263i
$232$		0			0
$233$	2015.27	+	3490.54i	0.566629	+	0.981430i	0.996896	+	0.0787281i	$0.0250859\pi$
$233$	2015.27	+	3490.54i	0.566629	+	0.981430i	−0.430268	+	0.902701i	$0.641581\pi$
$234$		0			0
$235$	−1517.83	+	2628.96i	−0.421329	+	0.729764i
$236$		0			0
$237$	−792.077	−	256.227i	−0.217092	−	0.0702268i
$238$		0			0
$239$	5365.88	+	1953.02i	1.45226	+	0.528579i	0.943221	−	0.332166i	$-0.107779\pi$
$239$	5365.88	+	1953.02i	1.45226	+	0.528579i	0.509037	+	0.860745i	$0.330002\pi$
$240$		0			0
$241$	−438.520	+	2486.97i	−0.117210	+	0.664730i	0.868422	+	0.495825i	$0.165134\pi$
$241$	−438.520	+	2486.97i	−0.117210	+	0.664730i	−0.985632	+	0.168905i	$0.945977\pi$
$242$		0			0
$243$	−2416.03	+	2917.49i	−0.637812	+	0.770192i
$244$		0			0
$245$	613.710	−	3480.52i	0.160035	−	0.907602i
$246$		0			0
$247$	5182.43	+	1886.25i	1.33502	+	0.485908i
$248$		0			0
$249$	2934.83	+	949.383i	0.746938	+	0.241625i
$250$		0			0
$251$	−289.705	+	501.784i	−0.0728526	+	0.126184i	−0.900150	−	0.435579i	$-0.856544\pi$
$251$	−289.705	+	501.784i	−0.0728526	+	0.126184i	0.827298	+	0.561764i	$0.189877\pi$
$252$		0			0
$253$	−1337.66	−	2316.90i	−0.332403	−	0.575740i
$254$		0			0
$255$	−1542.84	+	2910.76i	−0.378889	+	0.714819i
$256$		0			0
$257$	−614.801	−	3486.71i	−0.149223	−	0.846284i	−0.963879	−	0.266340i	$-0.914186\pi$
$257$	−614.801	−	3486.71i	−0.149223	−	0.846284i	0.814657	−	0.579944i	$-0.196925\pi$
$258$		0			0
$259$	5885.13	+	4938.21i	1.41191	+	1.18473i
$260$		0			0
$261$	−2055.32	+	4241.88i	−0.487436	+	1.00600i
$262$		0			0
$263$	2173.81	−	791.204i	0.509670	−	0.185505i	−0.0743683	−	0.997231i	$-0.523694\pi$
$263$	2173.81	−	791.204i	0.509670	−	0.185505i	0.584038	+	0.811726i	$0.301472\pi$
$264$		0			0
$265$	−4018.21	+	3371.68i	−0.931460	+	0.781587i
$266$		0			0
$267$	−6707.69	−	5226.64i	−1.53747	−	1.19800i
$268$		0			0
$269$	−595.561			−0.134989			−0.0674944	−	0.997720i	$-0.521500\pi$
$269$	−595.561			−0.134989			−0.0674944	+	0.997720i	$0.521500\pi$
$270$		0			0
$271$	−498.234			−0.111681			−0.0558406	−	0.998440i	$-0.517784\pi$
$271$	−498.234			−0.111681			−0.0558406	+	0.998440i	$0.517784\pi$
$272$		0			0
$273$	−9994.35	+	4053.05i	−2.21570	+	0.898541i
$274$		0			0
$275$	−483.961	+	406.091i	−0.106123	+	0.0890481i
$276$		0			0
$277$	8126.82	−	2957.92i	1.76279	−	0.641604i	0.762804	−	0.646630i	$-0.223822\pi$
$277$	8126.82	−	2957.92i	1.76279	−	0.641604i	0.999987	+	0.00502614i	$0.00159988\pi$
$278$		0			0
$279$	−3258.88	−	2354.84i	−0.699298	−	0.505307i
$280$		0			0
$281$	−1183.18	−	992.805i	−0.251183	−	0.210768i	0.508498	−	0.861063i	$-0.330201\pi$
$281$	−1183.18	−	992.805i	−0.251183	−	0.210768i	−0.759682	+	0.650295i	$0.774645\pi$
$282$		0			0
$283$	1007.68	+	5714.81i	0.211661	+	1.20039i	0.886607	+	0.462523i	$0.153056\pi$
$283$	1007.68	+	5714.81i	0.211661	+	1.20039i	−0.674946	+	0.737867i	$0.735833\pi$
$284$		0			0
$285$	3594.70	−	130.193i	0.747128	−	0.0270595i
$286$		0			0
$287$	−5809.16	−	10061.8i	−1.19479	−	2.06943i
$288$		0			0
$289$	376.139	−	651.493i	0.0765600	−	0.132606i
$290$		0			0
$291$	1630.70	−	1472.12i	0.328499	−	0.296553i
$292$		0			0
$293$	−3902.90	−	1420.54i	−0.778191	−	0.283238i	−0.0777730	−	0.996971i	$-0.524781\pi$
$293$	−3902.90	−	1420.54i	−0.778191	−	0.283238i	−0.700418	+	0.713733i	$0.747003\pi$
$294$		0			0
$295$	713.846	−	4048.42i	0.140887	−	0.799011i
$296$		0			0
$297$	2800.51	−	1379.40i	0.547146	−	0.269498i
$298$		0			0
$299$	1634.84	−	9271.66i	0.316206	−	1.79329i
$300$		0			0
$301$	−2486.65	−	905.067i	−0.476174	−	0.173313i
$302$		0			0
$303$	859.544	+	4018.23i	0.162969	+	0.761852i
$304$		0			0
$305$	−456.629	+	790.904i	−0.0857261	+	0.148482i
$306$		0			0
$307$	−3125.08	−	5412.80i	−0.580970	−	1.00627i	−0.995365	−	0.0961718i	$-0.969340\pi$
$307$	−3125.08	−	5412.80i	−0.580970	−	1.00627i	0.414395	−	0.910097i	$-0.363993\pi$
$308$		0			0
$309$	2508.04	+	4002.15i	0.461739	+	0.736810i
$310$		0			0
$311$	−893.683	−	5068.33i	−0.162946	−	0.924111i	−0.951157	−	0.308707i	$-0.900104\pi$
$311$	−893.683	−	5068.33i	−0.162946	−	0.924111i	0.788212	−	0.615404i	$-0.211007\pi$
$312$		0			0
$313$	4309.20	+	3615.85i	0.778180	+	0.652971i	0.942790	−	0.333388i	$-0.108192\pi$
$313$	4309.20	+	3615.85i	0.778180	+	0.652971i	−0.164610	+	0.986359i	$0.552636\pi$
$314$		0			0
$315$	−5047.31	+	4899.47i	−0.902806	+	0.876362i
$316$		0			0
$317$	−2826.93	+	1028.92i	−0.500872	+	0.182302i	−0.580086	−	0.814555i	$-0.696981\pi$
$317$	−2826.93	+	1028.92i	−0.500872	+	0.182302i	0.0792143	+	0.996858i	$0.474759\pi$
$318$		0			0
$319$	2975.78	−	2496.98i	0.522294	−	0.438257i
$320$		0			0
$321$	−161.597	+	1160.73i	−0.0280981	+	0.201825i
$322$		0			0
$323$	4543.01			0.782599
$324$		0			0
$325$	−2223.24			−0.379456
$326$		0			0
$327$	1373.05	−	9862.44i	0.232201	−	1.66787i
$328$		0			0
$329$	−6271.14	+	5262.11i	−1.05088	+	0.881792i
$330$		0			0
$331$	−4837.44	+	1760.69i	−0.803293	+	0.292375i	−0.710850	−	0.703343i	$-0.751690\pi$
$331$	−4837.44	+	1760.69i	−0.803293	+	0.292375i	−0.0924426	+	0.995718i	$0.529467\pi$
$332$		0			0
$333$	−1912.87	−	7588.26i	−0.314788	−	1.24875i
$334$		0			0
$335$	−3506.82	−	2942.57i	−0.571935	−	0.479910i
$336$		0			0
$337$	935.450	+	5305.20i	0.151208	+	0.857545i	0.962171	+	0.272447i	$0.0878328\pi$
$337$	935.450	+	5305.20i	0.151208	+	0.857545i	−0.810963	+	0.585098i	$0.801056\pi$
$338$		0			0
$339$	−1788.73	−	2854.33i	−0.286580	−	0.457304i
$340$		0			0
$341$	1656.77	+	2869.60i	0.263105	+	0.455712i
$342$		0			0
$343$	219.637	−	380.422i	0.0345751	−	0.0598859i
$344$		0			0
$345$	−1284.47	−	6004.68i	−0.200445	−	0.937047i
$346$		0			0
$347$	9458.09	+	3442.46i	1.46322	+	0.532568i	0.946250	−	0.323436i	$-0.104838\pi$
$347$	9458.09	+	3442.46i	1.46322	+	0.532568i	0.516969	+	0.856004i	$0.327060\pi$
$348$		0			0
$349$	197.712	−	1121.28i	0.0303246	−	0.171979i	−0.965884	−	0.258975i	$-0.916615\pi$
$349$	197.712	−	1121.28i	0.0303246	−	0.171979i	0.996209	+	0.0869952i	$0.0277265\pi$
$350$		0			0
$351$	10669.8	+	2616.27i	1.62254	+	0.397852i
$352$		0			0
$353$	910.038	−	5161.08i	0.137214	−	0.778177i	−0.836079	−	0.548609i	$-0.815158\pi$
$353$	910.038	−	5161.08i	0.137214	−	0.778177i	0.973293	−	0.229568i	$-0.0737314\pi$
$354$		0			0
$355$	−8349.40	−	3038.93i	−1.24828	−	0.454337i
$356$		0			0
$357$	−6594.44	+	5953.15i	−0.977632	+	0.882560i
$358$		0			0
$359$	324.335	−	561.764i	0.0476817	−	0.0825871i	−0.841200	−	0.540725i	$-0.818150\pi$
$359$	324.335	−	561.764i	0.0476817	−	0.0825871i	0.888881	+	0.458138i	$0.151483\pi$
$360$		0			0
$361$	949.293	+	1644.22i	0.138401	+	0.239718i
$362$		0			0
$363$	4340.47	−	157.203i	0.627591	−	0.0227301i
$364$		0			0
$365$	293.839	+	1666.44i	0.0421376	+	0.238974i
$366$		0			0
$367$	−4084.64	−	3427.42i	−0.580971	−	0.487493i	0.304295	−	0.952578i	$-0.401579\pi$
$367$	−4084.64	−	3427.42i	−0.580971	−	0.487493i	−0.885266	+	0.465085i	$0.846024\pi$
$368$		0			0
$369$	−1206.57	+	11773.2i	−0.170221	+	1.66094i
$370$		0			0
$371$	−13292.4	+	4838.04i	−1.86013	+	0.677032i
$372$		0			0
$373$	−5635.34	+	4728.61i	−0.782270	+	0.656403i	−0.943819	−	0.330462i	$-0.892795\pi$
$373$	−5635.34	+	4728.61i	−0.782270	+	0.656403i	0.161549	+	0.986865i	$0.448351\pi$
$374$		0			0
$375$	−7259.88	+	2944.13i	−0.999729	+	0.405424i
$376$		0			0
$377$	13670.3			1.86752
$378$		0			0
$379$	12285.0			1.66501			0.832507	−	0.554015i	$-0.186905\pi$
$379$	12285.0			1.66501			0.832507	+	0.554015i	$0.186905\pi$
$380$		0			0
$381$	−1481.07	−	1154.05i	−0.199153	−	0.155181i
$382$		0			0
$383$	−9861.87	+	8275.10i	−1.31571	+	1.10401i	−0.328518	+	0.944498i	$0.606549\pi$
$383$	−9861.87	+	8275.10i	−1.31571	+	1.10401i	−0.987195	+	0.159517i	$0.949006\pi$
$384$		0			0
$385$	5447.50	−	1982.73i	0.721117	−	0.262465i
$386$		0			0
$387$	1513.12	+	2230.80i	0.198750	+	0.293018i
$388$		0			0
$389$	3905.60	+	3277.18i	0.509053	+	0.427146i	0.860796	−	0.508951i	$-0.169966\pi$
$389$	3905.60	+	3277.18i	0.509053	+	0.427146i	−0.351743	+	0.936097i	$0.614411\pi$
$390$		0			0
$391$	−1346.70	−	7637.53i	−0.174183	−	0.987843i
$392$		0			0
$393$	1707.78	−	3221.92i	0.219201	−	0.413549i
$394$		0			0
$395$	787.359	+	1363.75i	0.100295	+	0.173715i
$396$		0			0
$397$	3842.87	−	6656.05i	0.485814	−	0.841455i	−0.514053	−	0.857759i	$-0.671856\pi$
$397$	3842.87	−	6656.05i	0.485814	−	0.841455i	0.999867	+	0.0163035i	$0.00518981\pi$
$398$		0			0
$399$	9229.44	+	2985.61i	1.15802	+	0.374605i
$400$		0			0
$401$	10580.7	+	3851.08i	1.31765	+	0.479585i	0.902704	−	0.430262i	$-0.141579\pi$
$401$	10580.7	+	3851.08i	1.31765	+	0.479585i	0.414944	+	0.909847i	$0.363801\pi$
$402$		0			0
$403$	−2024.84	+	11483.4i	−0.250284	+	1.41943i
$404$		0			0
$405$	7090.30	−	1033.98i	0.869925	−	0.126861i
$406$		0			0
$407$	−1119.92	+	6351.37i	−0.136394	+	0.773527i
$408$		0			0
$409$	9890.09	+	3599.70i	1.19568	+	0.435192i	0.861715	−	0.507393i	$-0.169391\pi$
$409$	9890.09	+	3599.70i	1.19568	+	0.435192i	0.333966	+	0.942585i	$0.391613\pi$
$410$		0			0
$411$	−13963.6	−	4517.04i	−1.67584	−	0.542115i
$412$		0			0
$413$	5542.98	−	9600.73i	0.660417	−	1.14388i
$414$		0			0
$415$	−2917.35	−	5053.01i	−0.345078	−	0.597692i
$416$		0			0
$417$	2405.33	−	4537.95i	0.282469	−	0.532912i
$418$		0			0
$419$	−605.417	−	3433.49i	−0.0705884	−	0.400327i	−0.999546	−	0.0301399i	$-0.990405\pi$
$419$	−605.417	−	3433.49i	−0.0705884	−	0.400327i	0.928957	−	0.370187i	$-0.120706\pi$
$420$		0			0
$421$	6215.42	+	5215.36i	0.719527	+	0.603755i	0.927255	−	0.374432i	$-0.122162\pi$
$421$	6215.42	+	5215.36i	0.719527	+	0.603755i	−0.207727	+	0.978187i	$0.566607\pi$
$422$		0			0
$423$	8317.09	−	603.249i	0.956006	−	0.0693403i
$424$		0			0
$425$	−1720.95	+	626.373i	−0.196419	+	0.0714907i
$426$		0			0
$427$	−1886.63	+	1583.07i	−0.213818	+	0.179415i
$428$		0			0
$429$	−7141.69	−	5564.81i	−0.803739	−	0.626274i
$430$		0			0
$431$	−3793.19			−0.423925			−0.211962	−	0.977278i	$-0.567985\pi$
$431$	−3793.19			−0.423925			−0.211962	+	0.977278i	$0.567985\pi$
$432$		0			0
$433$	−13262.0			−1.47190			−0.735948	−	0.677038i	$-0.763263\pi$
$433$	−13262.0			−1.47190			−0.735948	+	0.677038i	$0.763263\pi$
$434$		0			0
$435$	8262.55	−	3350.75i	0.910710	−	0.369324i
$436$		0			0
$437$	−6486.79	+	5443.07i	−0.710081	+	0.595829i
$438$		0			0
$439$	−9674.73	+	3521.31i	−1.05182	+	0.382832i	−0.809350	−	0.587327i	$-0.800180\pi$
$439$	−9674.73	+	3521.31i	−1.05182	+	0.382832i	−0.242471	+	0.970159i	$0.577958\pi$
$440$		0			0
$441$	−8858.86	+	3971.75i	−0.956577	+	0.428869i
$442$		0			0
$443$	6433.59	+	5398.43i	0.689998	+	0.578977i	0.918909	−	0.394470i	$-0.129072\pi$
$443$	6433.59	+	5398.43i	0.689998	+	0.578977i	−0.228910	+	0.973447i	$0.573516\pi$
$444$		0			0
$445$	2793.16	+	15840.8i	0.297547	+	1.68748i
$446$		0			0
$447$	10151.7	−	367.674i	1.07418	−	0.0389047i
$448$		0			0
$449$	5426.60	+	9399.15i	0.570373	+	0.987914i	0.996528	+	0.0832641i	$0.0265345\pi$
$449$	5426.60	+	9399.15i	0.570373	+	0.987914i	−0.426155	+	0.904650i	$0.640132\pi$
$450$		0			0
$451$	4876.71	−	8446.71i	0.509169	−	0.881907i
$452$		0			0
$453$	10957.5	−	9891.88i	1.13648	−	1.02596i
$454$		0			0
$455$	19170.2	+	6977.38i	1.97519	+	0.718911i
$456$		0			0
$457$	2521.85	−	14302.1i	0.258134	−	1.46395i	−0.529765	−	0.848144i	$-0.677720\pi$
$457$	2521.85	−	14302.1i	0.258134	−	1.46395i	0.787899	−	0.615805i	$-0.211169\pi$
$458$		0			0
$459$	8996.29	−	980.917i	0.914839	−	0.0997501i
$460$		0			0
$461$	−1197.28	+	6790.14i	−0.120961	+	0.686005i	0.862664	+	0.505778i	$0.168794\pi$
$461$	−1197.28	+	6790.14i	−0.120961	+	0.686005i	−0.983625	+	0.180227i	$0.942317\pi$
$462$		0			0
$463$	−16976.9	−	6179.10i	−1.70407	−	0.620231i	−0.707792	−	0.706421i	$-0.750309\pi$
$463$	−16976.9	−	6179.10i	−1.70407	−	0.620231i	−0.996279	+	0.0861899i	$0.972531\pi$
$464$		0			0
$465$	1590.88	+	7437.12i	0.158657	+	0.741695i
$466$		0			0
$467$	4432.93	−	7678.07i	0.439254	−	0.760811i	−0.558378	−	0.829587i	$-0.688576\pi$
$467$	4432.93	−	7678.07i	0.439254	−	0.760811i	0.997632	+	0.0687761i	$0.0219094\pi$
$468$		0			0
$469$	−6172.61	−	10691.3i	−0.607729	−	1.05262i
$470$		0			0
$471$	1960.30	+	3128.12i	0.191775	+	0.306021i
$472$		0			0
$473$	−385.756	−	2187.73i	−0.0374991	−	0.212668i
$474$		0			0
$475$	1531.83	+	1285.36i	0.147969	+	0.124160i
$476$		0			0
$477$	13861.1	+	3935.78i	1.33052	+	0.377792i
$478$		0			0
$479$	−13232.7	+	4816.31i	−1.26225	+	0.459422i	−0.884524	−	0.466495i	$-0.845516\pi$
$479$	−13232.7	+	4816.31i	−1.26225	+	0.459422i	−0.377727	+	0.925917i	$0.623294\pi$
$480$		0			0
$481$	−17386.0	+	14588.6i	−1.64809	+	1.38291i
$482$		0			0
$483$	2283.38	−	16401.2i	0.215108	−	1.54510i
$484$		0			0
$485$	−4155.58			−0.389062
$486$		0			0
$487$	−19389.4			−1.80414			−0.902072	−	0.431586i	$-0.857954\pi$
$487$	−19389.4			−1.80414			−0.902072	+	0.431586i	$0.857954\pi$
$488$		0			0
$489$	−960.709	+	6900.66i	−0.0888441	+	0.638157i
$490$		0			0
$491$	−13194.3	+	11071.3i	−1.21273	+	1.01760i	−0.213556	+	0.976931i	$0.568505\pi$
$491$	−13194.3	+	11071.3i	−1.21273	+	1.01760i	−0.999173	+	0.0406697i	$0.987051\pi$
$492$		0			0
$493$	10581.8	−	3851.45i	0.966691	−	0.351847i
$494$		0			0
$495$	−5680.57	−	1612.96i	−0.515803	−	0.146459i
$496$		0			0
$497$	−18355.3	−	15402.0i	−1.65664	−	1.39009i
$498$		0			0
$499$	−2323.05	−	13174.7i	−0.208405	−	1.18192i	−0.891991	−	0.452054i	$-0.850691\pi$
$499$	−2323.05	−	13174.7i	−0.208405	−	1.18192i	0.683585	−	0.729871i	$-0.260420\pi$
$500$		0			0
$501$	4636.90	+	7399.24i	0.413496	+	0.659828i
$502$		0			0
$503$	6922.31	+	11989.8i	0.613620	+	1.06282i	0.990625	+	0.136609i	$0.0436205\pi$
$503$	6922.31	+	11989.8i	0.613620	+	1.06282i	−0.377005	+	0.926211i	$0.623046\pi$
$504$		0			0
$505$	3886.38	−	6731.40i	0.342458	−	0.593155i
$506$		0			0
$507$	−4276.67	−	19992.8i	−0.374623	−	1.75130i
$508$		0			0
$509$	−14055.9	−	5115.94i	−1.22400	−	0.445501i	−0.352464	−	0.935825i	$-0.614656\pi$
$509$	−14055.9	−	5115.94i	−1.22400	−	0.445501i	−0.871540	+	0.490324i	$0.836878\pi$
$510$		0			0
$511$	−792.407	+	4493.96i	−0.0685989	+	0.389044i
$512$		0			0
$513$	−5838.98	−	7971.33i	−0.502529	−	0.686048i
$514$		0			0
$515$	1551.39	−	8798.34i	0.132742	−	0.752818i
$516$		0			0
$517$	−6457.91	−	2350.49i	−0.549359	−	0.199950i
$518$		0			0
$519$	596.514	−	538.505i	0.0504510	−	0.0455448i
$520$		0			0
$521$	−3924.43	+	6797.31i	−0.330004	+	0.571584i	−0.982512	−	0.186198i	$-0.940384\pi$
$521$	−3924.43	+	6797.31i	−0.330004	+	0.571584i	0.652508	+	0.757782i	$0.273717\pi$
$522$		0			0
$523$	7321.97	+	12682.0i	0.612175	+	1.06032i	0.990873	+	0.134798i	$0.0430385\pi$
$523$	7321.97	+	12682.0i	0.612175	+	1.06032i	−0.378698	+	0.925520i	$0.623628\pi$
$524$		0			0
$525$	−3907.87	+	141.535i	−0.324864	+	0.0117659i
$526$		0			0
$527$	1667.96	+	9459.49i	0.137870	+	0.781901i
$528$		0			0
$529$	1753.12	+	1471.05i	0.144088	+	0.120905i
$530$		0			0
$531$	−10304.3	+	4619.81i	−0.842126	+	0.377556i
$532$		0			0
$533$	32253.2	−	11739.2i	2.62109	−	0.953998i
$534$		0			0
$535$	1698.17	−	1424.93i	0.137230	−	0.115150i
$536$		0			0
$537$	4110.43	−	1666.92i	0.330313	−	0.133953i
$538$		0			0
$539$	8001.03			0.639385
$540$		0			0
$541$	−12240.4			−0.972745			−0.486372	−	0.873752i	$-0.661680\pi$
$541$	−12240.4			−0.972745			−0.486372	+	0.873752i	$0.661680\pi$
$542$		0			0
$543$	−8081.64	−	6297.22i	−0.638704	−	0.497679i
$544$		0			0
$545$	−14428.8	+	12107.2i	−1.13406	+	0.951591i
$546$		0			0
$547$	13010.6	−	4735.49i	1.01699	−	0.370155i	0.220879	−	0.975301i	$-0.429108\pi$
$547$	13010.6	−	4735.49i	1.01699	−	0.370155i	0.796114	+	0.605146i	$0.206885\pi$
$548$		0			0
$549$	2502.14	−	181.483i	0.194515	−	0.0141084i
$550$		0			0
$551$	−9418.91	−	7903.41i	−0.728238	−	0.611064i
$552$		0			0
$553$	737.416	+	4182.09i	0.0567054	+	0.321593i
$554$		0			0
$555$	−6932.57	+	13079.1i	−0.530218	+	1.00032i
$556$		0			0
$557$	−2049.36	−	3549.59i	−0.155896	−	0.270020i	0.777489	−	0.628897i	$-0.216493\pi$
$557$	−2049.36	−	3549.59i	−0.155896	−	0.270020i	−0.933385	+	0.358877i	$0.883160\pi$
$558$		0			0
$559$	3908.79	−	6770.22i	0.295750	−	0.512254i
$560$		0			0
$561$	−7096.01	−	2295.47i	−0.534035	−	0.172754i
$562$		0			0
$563$	19726.1	+	7179.71i	1.47665	+	0.537458i	0.949898	−	0.312559i	$-0.101186\pi$
$563$	19726.1	+	7179.71i	1.47665	+	0.537458i	0.526755	+	0.850017i	$0.323408\pi$
$564$		0			0
$565$	−1106.45	+	6274.97i	−0.0823869	+	0.467239i
$566$		0			0
$567$	18921.3	+	3919.46i	1.40144	+	0.290303i
$568$		0			0
$569$	3207.06	−	18188.1i	0.236286	−	1.34005i	−0.603602	−	0.797286i	$-0.706268\pi$
$569$	3207.06	−	18188.1i	0.236286	−	1.34005i	0.839888	−	0.542760i	$-0.182621\pi$
$570$		0			0
$571$	8850.02	+	3221.15i	0.648620	+	0.236078i	0.645315	−	0.763916i	$-0.276726\pi$
$571$	8850.02	+	3221.15i	0.648620	+	0.236078i	0.00330448	+	0.999995i	$0.498948\pi$
$572$		0			0
$573$	−25260.0	−	8171.32i	−1.84163	−	0.595745i
$574$		0			0
$575$	1706.81	−	2956.28i	0.123789	−	0.214409i
$576$		0			0
$577$	−6534.09	−	11317.4i	−0.471435	−	0.816549i	0.528031	−	0.849225i	$-0.322930\pi$
$577$	−6534.09	−	11317.4i	−0.471435	−	0.816549i	−0.999466	+	0.0326762i	$0.989597\pi$
$578$		0			0
$579$	−1183.23	+	2232.31i	−0.0849283	+	0.160227i
$580$		0			0
$581$	−2732.30	−	15495.7i	−0.195103	−	1.10649i
$582$		0			0
$583$	−9096.74	−	7633.07i	−0.646223	−	0.542246i
$584$		0			0
$585$	−11665.0	−	17197.8i	−0.824424	−	1.21545i
$586$		0			0
$587$	−2383.29	+	867.448i	−0.167579	+	0.0609939i	−0.424447	−	0.905453i	$-0.639532\pi$
$587$	−2383.29	+	867.448i	−0.167579	+	0.0609939i	0.256868	+	0.966446i	$0.417309\pi$
$588$		0			0
$589$	8034.24	−	6741.53i	0.562046	−	0.471613i
$590$		0			0
$591$	8356.30	+	6511.24i	0.581611	+	0.453192i
$592$		0			0
$593$	8431.98			0.583912			0.291956	−	0.956432i	$-0.405694\pi$
$593$	8431.98			0.583912			0.291956	+	0.956432i	$0.405694\pi$
$594$		0			0
$595$	16804.9			1.15787
$596$		0			0
$597$	−21549.2	+	8738.93i	−1.47730	+	0.599097i
$598$		0			0
$599$	−285.369	+	239.453i	−0.0194655	+	0.0163335i	−0.652468	−	0.757816i	$-0.726267\pi$
$599$	−285.369	+	239.453i	−0.0194655	+	0.0163335i	0.633003	+	0.774149i	$0.281822\pi$
$600$		0			0
$601$	3783.87	−	1377.22i	0.256818	−	0.0934740i	−0.210403	−	0.977615i	$-0.567478\pi$
$601$	3783.87	−	1377.22i	0.256818	−	0.0934740i	0.467221	+	0.884141i	$0.345255\pi$
$602$		0			0
$603$	−1282.06	+	12509.7i	−0.0865831	+	0.844836i
$604$		0			0
$605$	−6293.61	−	5280.96i	−0.422928	−	0.354879i
$606$		0			0
$607$	−240.592	−	1364.47i	−0.0160879	−	0.0912388i	0.975707	−	0.219081i	$-0.0703060\pi$
$607$	−240.592	−	1364.47i	−0.0160879	−	0.0912388i	−0.991795	+	0.127842i	$0.959195\pi$
$608$		0			0
$609$	24028.7	−	870.273i	1.59884	−	0.0579068i
$610$		0			0
$611$	−12092.2	−	20944.3i	−0.800652	−	1.38677i
$612$		0			0
$613$	−6625.82	+	11476.3i	−0.436565	+	0.756153i	−0.997422	−	0.0717601i	$-0.977138\pi$
$613$	−6625.82	+	11476.3i	−0.436565	+	0.756153i	0.560857	+	0.827913i	$0.310472\pi$
$614$		0			0
$615$	16617.0	−	15001.0i	1.08953	−	0.983577i
$616$		0			0
$617$	5742.14	+	2089.97i	0.374667	+	0.136368i	0.522489	−	0.852646i	$-0.325004\pi$
$617$	5742.14	+	2089.97i	0.374667	+	0.136368i	−0.147821	+	0.989014i	$0.547226\pi$
$618$		0			0
$619$	158.970	−	901.563i	0.0103224	−	0.0585410i	−0.979211	−	0.202842i	$-0.934982\pi$
$619$	158.970	−	901.563i	0.0103224	−	0.0585410i	0.989534	+	0.144301i	$0.0460933\pi$
$620$		0			0
$621$	−11670.2	+	12179.3i	−0.754122	+	0.787016i
$622$		0			0
$623$	−7532.44	+	42718.6i	−0.484399	+	2.74716i
$624$		0			0
$625$	10590.2	+	3854.53i	0.677774	+	0.246690i
$626$		0			0
$627$	1703.40	+	7963.14i	0.108497	+	0.507204i
$628$		0			0
$629$	−9347.83	+	16190.9i	−0.592563	+	1.02635i
$630$		0			0
$631$	3125.11	+	5412.85i	0.197161	+	0.341493i	0.947607	−	0.319439i	$-0.103495\pi$
$631$	3125.11	+	5412.85i	0.197161	+	0.341493i	−0.750446	+	0.660932i	$0.770161\pi$
$632$		0			0
$633$	10818.8	+	17263.9i	0.679319	+	1.08401i
$634$		0			0
$635$	616.735	+	3497.68i	0.0385423	+	0.218585i
$636$		0			0
$637$	21569.0	+	18098.5i	1.34159	+	1.12573i
$638$		0			0
$639$	5966.10	+	23667.3i	0.369351	+	1.46520i
$640$		0			0
$641$	7732.55	−	2814.42i	0.476470	−	0.173421i	−0.0926108	−	0.995702i	$-0.529521\pi$
$641$	7732.55	−	2814.42i	0.476470	−	0.173421i	0.569081	+	0.822281i	$0.307299\pi$
$642$		0			0
$643$	−1394.39	+	1170.03i	−0.0855201	+	0.0717599i	−0.684545	−	0.728970i	$-0.739999\pi$
$643$	−1394.39	+	1170.03i	−0.0855201	+	0.0717599i	0.599025	+	0.800730i	$0.295555\pi$
$644$		0			0
$645$	703.080	−	5050.14i	0.0429205	−	0.308293i
$646$		0			0
$647$	−14438.2			−0.877318			−0.438659	−	0.898654i	$-0.644546\pi$
$647$	−14438.2			−0.877318			−0.438659	+	0.898654i	$0.644546\pi$
$648$		0			0
$649$	9306.51			0.562885
$650$		0			0
$651$	−2828.08	+	20313.8i	−0.170263	+	1.22298i
$652$		0			0
$653$	15367.0	−	12894.4i	0.920914	−	0.772739i	−0.0532495	−	0.998581i	$-0.516958\pi$
$653$	15367.0	−	12894.4i	0.920914	−	0.772739i	0.974164	+	0.225842i	$0.0725134\pi$
$654$		0			0
$655$	−6481.75	+	2359.16i	−0.386661	+	0.140733i
$656$		0			0
$657$	3335.35	−	3237.65i	0.198058	−	0.192257i
$658$		0			0
$659$	14421.1	+	12100.7i	0.852451	+	0.715291i	0.960328	−	0.278873i	$-0.0899608\pi$
$659$	14421.1	+	12100.7i	0.852451	+	0.715291i	−0.107877	+	0.994164i	$0.534405\pi$
$660$		0			0
$661$	387.406	+	2197.09i	0.0227963	+	0.129284i	0.994082	−	0.108632i	$-0.0346469\pi$
$661$	387.406	+	2197.09i	0.0227963	+	0.129284i	−0.971286	+	0.237916i	$0.923536\pi$
$662$		0			0
$663$	−13936.8	−	22239.4i	−0.816382	−	1.30273i
$664$		0			0
$665$	−9174.47	−	15890.6i	−0.534993	−	0.926636i
$666$		0			0
$667$	−10494.8	+	18177.6i	−0.609237	+	1.05523i
$668$		0			0
$669$	1694.39	+	7920.99i	0.0979206	+	0.457763i
$670$		0			0
$671$	−1942.82	−	707.127i	−0.111776	−	0.0406831i
$672$		0			0
$673$	−2561.47	+	14526.8i	−0.146712	+	0.832048i	0.819264	+	0.573417i	$0.194382\pi$
$673$	−2561.47	+	14526.8i	−0.146712	+	0.832048i	−0.965976	+	0.258631i	$0.916729\pi$
$674$		0			0
$675$	3310.94	+	2214.58i	0.188797	+	0.126280i
$676$		0			0
$677$	666.714	−	3781.13i	0.0378492	−	0.214654i	−0.960017	−	0.279941i	$-0.909685\pi$
$677$	666.714	−	3781.13i	0.0378492	−	0.214654i	0.997867	+	0.0652871i	$0.0207963\pi$
$678$		0			0
$679$	−10530.7	−	3832.86i	−0.595185	−	0.216630i
$680$		0			0
$681$	984.857	−	889.083i	0.0554182	−	0.0500290i
$682$		0			0
$683$	6452.62	−	11176.3i	0.361497	−	0.626132i	−0.626710	−	0.779252i	$-0.715599\pi$
$683$	6452.62	−	11176.3i	0.361497	−	0.626132i	0.988207	+	0.153121i	$0.0489323\pi$
$684$		0			0
$685$	13880.4	+	24041.5i	0.774222	+	1.34099i
$686$		0			0
$687$	26840.2	−	972.098i	1.49056	−	0.0539852i
$688$		0			0
$689$	−7256.57	−	41154.1i	−0.401239	−	2.27554i
$690$		0			0
$691$	−5853.56	−	4911.72i	−0.322257	−	0.270406i	0.467279	−	0.884110i	$-0.345234\pi$
$691$	−5853.56	−	4911.72i	−0.322257	−	0.270406i	−0.789536	+	0.613704i	$0.789679\pi$
$692$		0			0
$693$	−12907.5	−	9326.83i	−0.707525	−	0.511251i
$694$		0			0
$695$	−9129.27	+	3322.78i	−0.498263	+	0.181353i
$696$		0			0
$697$	21658.9	−	18174.0i	1.17703	−	0.987644i
$698$		0			0
$699$	−19408.1	+	7870.63i	−1.05019	+	0.425887i
$700$		0			0
$701$	16562.4			0.892373			0.446187	−	0.894940i	$-0.352782\pi$
$701$	16562.4			0.892373			0.446187	+	0.894940i	$0.352782\pi$
$702$		0			0
$703$	20413.4			1.09517
$704$		0			0
$705$	−12442.5	−	9695.18i	−0.664695	−	0.517932i
$706$		0			0
$707$	16057.1	−	13473.5i	0.854159	−	0.716725i
$708$		0			0
$709$	866.193	−	315.268i	0.0458823	−	0.0166998i	−0.318977	−	0.947762i	$-0.603339\pi$
$709$	866.193	−	315.268i	0.0458823	−	0.0166998i	0.364859	+	0.931063i	$0.381117\pi$
$710$		0			0
$711$	1886.20	−	3892.85i	0.0994908	−	0.205335i
$712$		0			0
$713$	−13715.2	−	11508.4i	−0.720392	−	0.604481i
$714$		0			0
$715$	2973.89	+	16865.8i	0.155548	+	0.882159i
$716$		0			0
$717$	−13895.9	+	26216.2i	−0.723781	+	1.36550i
$718$		0			0
$719$	1108.55	+	1920.06i	0.0574992	+	0.0995915i	0.893342	−	0.449377i	$-0.148354\pi$
$719$	1108.55	+	1920.06i	0.0574992	+	0.0995915i	−0.835843	+	0.548969i	$0.815021\pi$
$720$		0			0
$721$	12046.4	−	20865.0i	0.622237	−	1.07775i
$722$		0			0
$723$	−12485.0	−	4038.76i	−0.642218	−	0.207750i
$724$		0			0
$725$	4657.69	+	1695.26i	0.238596	+	0.0868420i
$726$		0			0
$727$	2305.90	−	13077.4i	0.117636	−	0.667145i	−0.867776	−	0.496956i	$-0.834451\pi$
$727$	2305.90	−	13077.4i	0.117636	−	0.667145i	0.985411	−	0.170189i	$-0.0544378\pi$
$728$		0			0
$729$	−13283.8	−	14524.5i	−0.674887	−	0.737921i
$730$		0			0
$731$	1118.25	−	6341.90i	0.0565799	−	0.320880i
$732$		0			0
$733$	18468.5	+	6721.99i	0.930628	+	0.338721i	0.762458	−	0.647037i	$-0.223992\pi$
$733$	18468.5	+	6721.99i	0.930628	+	0.338721i	0.168169	+	0.985758i	$0.446214\pi$
$734$		0			0
$735$	17472.8	+	5652.25i	0.876865	+	0.283655i
$736$		0			0
$737$	5181.82	−	8975.18i	0.258989	−	0.448582i
$738$		0			0
$739$	6917.27	+	11981.1i	0.344325	+	0.596388i	0.985231	−	0.171231i	$-0.0547746\pi$
$739$	6917.27	+	11981.1i	0.344325	+	0.596388i	−0.640906	+	0.767619i	$0.721441\pi$
$740$		0			0
$741$	−13420.8	+	25320.0i	−0.665352	+	1.25527i
$742$		0			0
$743$	−3872.43	−	21961.6i	−0.191206	−	1.08438i	−0.917720	−	0.397229i	$-0.869972\pi$
$743$	−3872.43	−	21961.6i	−0.191206	−	1.08438i	0.726514	−	0.687152i	$-0.241139\pi$
$744$		0			0
$745$	−14719.8	−	12351.4i	−0.723881	−	0.607408i
$746$		0			0
$747$	−6988.82	+	14423.9i	−0.342312	+	0.706485i
$748$		0			0
$749$	5617.60	−	2044.64i	0.274049	−	0.0997457i
$750$		0			0
$751$	18445.2	−	15477.4i	0.896238	−	0.752033i	−0.0732136	−	0.997316i	$-0.523325\pi$
$751$	18445.2	−	15477.4i	0.896238	−	0.752033i	0.969451	+	0.245283i	$0.0788810\pi$
$752$		0			0
$753$	−2374.87	−	1850.50i	−0.114933	−	0.0895563i
$754$		0			0
$755$	−27923.4			−1.34601
$756$		0			0
$757$	15873.5			0.762132			0.381066	−	0.924548i	$-0.375557\pi$
$757$	15873.5			0.762132			0.381066	+	0.924548i	$0.375557\pi$
$758$		0			0
$759$	12882.4	−	5224.25i	0.616075	−	0.249839i
$760$		0			0
$761$	−21981.7	+	18444.8i	−1.04709	+	0.878613i	−0.992785	−	0.119912i	$-0.961739\pi$
$761$	−21981.7	+	18444.8i	−1.04709	+	0.878613i	−0.0543053	+	0.998524i	$0.517294\pi$
$762$		0			0
$763$	−47731.2	+	17372.7i	−2.26473	+	0.824293i
$764$		0			0
$765$	−13874.8	−	10025.8i	−0.655746	−	0.473836i
$766$		0			0
$767$	25088.3	+	21051.6i	1.18108	+	0.991040i
$768$		0			0
$769$	−2217.08	−	12573.7i	−0.103966	−	0.589620i	−0.991628	−	0.129125i	$-0.958783\pi$
$769$	−2217.08	−	12573.7i	−0.103966	−	0.589620i	0.887662	−	0.460495i	$-0.152328\pi$
$770$		0			0
$771$	18384.9	−	665.865i	0.858776	−	0.0311032i
$772$		0			0
$773$	3017.93	+	5227.21i	0.140424	+	0.243221i	0.927656	−	0.373435i	$-0.121820\pi$
$773$	3017.93	+	5227.21i	0.140424	+	0.243221i	−0.787233	+	0.616656i	$0.788487\pi$
$774$		0			0
$775$	−2113.97	+	3661.51i	−0.0979821	+	0.169710i
$776$		0			0
$777$	−29631.3	+	26749.7i	−1.36810	+	1.23506i
$778$		0			0
$779$	−29009.7	−	10558.7i	−1.33425	−	0.485626i
$780$		0			0
$781$	3492.95	−	19809.5i	0.160035	−	0.907605i
$782$		0			0
$783$	−20358.3	−	13617.0i	−0.929178	−	0.621497i
$784$		0			0
$785$	1212.58	−	6876.87i	0.0551321	−	0.312670i
$786$		0			0
$787$	20849.4	+	7588.56i	0.944346	+	0.343714i	0.767881	−	0.640593i	$-0.221311\pi$
$787$	20849.4	+	7588.56i	0.944346	+	0.343714i	0.176465	+	0.984307i	$0.443534\pi$
$788$		0			0
$789$	2514.41	+	11754.5i	0.113454	+	0.530380i
$790$		0			0
$791$	−8591.51	+	14880.9i	−0.386194	+	0.668907i
$792$		0			0
$793$	−3637.86	−	6300.95i	−0.162905	−	0.282161i
$794$		0			0
$795$	−14473.4	−	23095.6i	−0.645682	−	1.03033i
$796$		0			0
$797$	2619.80	+	14857.6i	0.116434	+	0.660332i	0.986030	+	0.166567i	$0.0532684\pi$
$797$	2619.80	+	14857.6i	0.116434	+	0.660332i	−0.869596	+	0.493764i	$0.835621\pi$
$798$		0			0
$799$	−15261.1	−	12805.6i	−0.675717	−	0.566994i
$800$		0			0
$801$	31705.0	−	30776.3i	1.39855	−	1.35759i
$802$		0			0
$803$	−3599.79	+	1310.22i	−0.158199	+	0.0575798i
$804$		0			0
$805$	−23995.1	+	20134.3i	−1.05058	+	0.881542i
$806$		0			0
$807$	426.718	−	3065.06i	0.0186136	−	0.133699i
$808$		0			0
$809$	−4962.96			−0.215684			−0.107842	−	0.994168i	$-0.534394\pi$
$809$	−4962.96			−0.215684			−0.107842	+	0.994168i	$0.534394\pi$
$810$		0			0
$811$	−18808.2			−0.814358			−0.407179	−	0.913348i	$-0.633488\pi$
$811$	−18808.2			−0.814358			−0.407179	+	0.913348i	$0.633488\pi$
$812$		0			0
$813$	356.984	−	2564.17i	0.0153997	−	0.110614i
$814$		0			0
$815$	10095.7	−	8471.32i	0.433912	−	0.364095i
$816$		0			0
$817$	−6607.37	+	2404.88i	−0.282941	+	0.102982i
$818$		0			0
$819$	−13698.2	−	54340.1i	−0.584435	−	2.31843i
$820$		0			0
$821$	10726.0	+	9000.16i	0.455955	+	0.382592i	0.841640	−	0.540039i	$-0.181590\pi$
$821$	10726.0	+	9000.16i	0.455955	+	0.382592i	−0.385685	+	0.922630i	$0.626035\pi$
$822$		0			0
$823$	−6387.96	−	36227.9i	−0.270559	−	1.53442i	−0.752724	−	0.658336i	$-0.771260\pi$
$823$	−6387.96	−	36227.9i	−0.270559	−	1.53442i	0.482165	−	0.876081i	$-0.339851\pi$
$824$		0			0
$825$	−1743.20	−	2781.68i	−0.0735641	−	0.117389i
$826$		0			0
$827$	10316.7	+	17869.0i	0.433791	+	0.751349i	0.997196	−	0.0748324i	$-0.0238422\pi$
$827$	10316.7	+	17869.0i	0.433791	+	0.751349i	−0.563405	+	0.826181i	$0.690509\pi$
$828$		0			0
$829$	−19800.1	+	34294.7i	−0.829535	+	1.43680i	0.0688680	+	0.997626i	$0.478061\pi$
$829$	−19800.1	+	34294.7i	−0.829535	+	1.43680i	−0.898403	+	0.439171i	$0.855272\pi$
$830$		0			0
$831$	9400.14	+	43944.1i	0.392403	+	1.83442i
$832$		0			0
$833$	21795.0	+	7932.73i	0.906545	+	0.329955i
$834$		0			0
$835$	2868.23	−	16266.5i	0.118873	−	0.674163i
$836$		0			0
$837$	14454.2	−	15084.7i	0.596906	−	0.622942i
$838$		0			0
$839$	3742.81	−	21226.5i	0.154012	−	0.873446i	−0.805671	−	0.592363i	$-0.798195\pi$
$839$	3742.81	−	21226.5i	0.154012	−	0.873446i	0.959683	−	0.281083i	$-0.0906937\pi$
$840$		0			0
$841$	−5721.07	−	2082.30i	−0.234576	−	0.0853786i
$842$		0			0
$843$	5957.23	−	5377.91i	0.243390	−	0.219721i
$844$		0			0
$845$	−19336.7	+	33492.2i	−0.787223	+	1.36351i
$846$		0			0
$847$	−11077.8	−	19187.4i	−0.449397	−	0.778378i
$848$		0			0
$849$	−30133.4	+	1091.37i	−1.21811	+	0.0441175i
$850$		0			0
$851$	−6051.23	−	34318.3i	−0.243753	−	1.38239i
$852$		0			0
$853$	−10060.5	−	8441.78i	−0.403828	−	0.338852i	0.418143	−	0.908381i	$-0.362681\pi$
$853$	−10060.5	−	8441.78i	−0.403828	−	0.338852i	−0.821971	+	0.569529i	$0.807126\pi$
$854$		0			0
$855$	−1905.55	+	18593.5i	−0.0762205	+	0.743723i
$856$		0			0
$857$	−6420.15	+	2336.74i	−0.255902	+	0.0931408i	−0.466786	−	0.884370i	$-0.654588\pi$
$857$	−6420.15	+	2336.74i	−0.255902	+	0.0931408i	0.210884	+	0.977511i	$0.432366\pi$
$858$		0			0
$859$	2844.30	−	2386.65i	0.112976	−	0.0947979i	−0.584550	−	0.811358i	$-0.698729\pi$
$859$	2844.30	−	2386.65i	0.112976	−	0.0947979i	0.697525	+	0.716560i	$0.254284\pi$
$860$		0			0
$861$	55945.3	−	22687.7i	2.21441	−	0.898020i
$862$		0			0
$863$	−28016.1			−1.10507			−0.552537	−	0.833488i	$-0.686340\pi$
$863$	−28016.1			−1.10507			−0.552537	+	0.833488i	$0.686340\pi$
$864$		0			0
$865$	−1520.13			−0.0597524
$866$		0			0
$867$	3083.41	+	2402.60i	0.120782	+	0.0941137i
$868$		0			0
$869$	−2730.92	+	2291.52i	−0.106606	+	0.0894527i
$870$		0			0
$871$	34271.1	−	12473.7i	1.33322	−	0.485251i
$872$		0			0
$873$	6407.88	+	9447.18i	0.248424	+	0.366253i
$874$		0			0
$875$	30613.2	+	25687.5i	1.18276	+	0.992454i
$876$		0			0
$877$	4156.37	+	23571.9i	0.160035	+	0.907602i	0.954037	+	0.299688i	$0.0968825\pi$
$877$	4156.37	+	23571.9i	0.160035	+	0.907602i	−0.794003	+	0.607914i	$0.792006\pi$
$878$		0			0
$879$	10107.3	−	19068.5i	0.387838	−	0.731702i
$880$		0			0
$881$	6076.95	+	10525.6i	0.232392	+	0.402515i	0.958512	−	0.285054i	$-0.0920114\pi$
$881$	6076.95	+	10525.6i	0.232392	+	0.402515i	−0.726119	+	0.687569i	$0.758678\pi$
$882$		0			0
$883$	−14612.5	+	25309.6i	−0.556909	+	0.964595i	0.440843	+	0.897584i	$0.354679\pi$
$883$	−14612.5	+	25309.6i	−0.556909	+	0.964595i	−0.997752	+	0.0670105i	$0.978654\pi$
$884$		0			0
$885$	20323.8	+	6574.50i	0.771951	+	0.249717i
$886$		0			0
$887$	−31069.7	−	11308.4i	−1.17612	−	0.428072i	−0.321290	−	0.946981i	$-0.604116\pi$
$887$	−31069.7	−	11308.4i	−1.17612	−	0.428072i	−0.854830	+	0.518909i	$0.826338\pi$
$888$		0			0
$889$	−1663.17	+	9432.33i	−0.0627459	+	0.355850i
$890$		0			0
$891$	5092.55	+	15401.2i	0.191478	+	0.579080i
$892$		0			0
$893$	−3777.25	+	21421.9i	−0.141546	+	0.802750i
$894$		0			0
$895$	−7884.24	−	2869.63i	−0.294459	−	0.107174i
$896$		0			0
$897$	46545.4	+	15056.9i	1.73256	+	0.560462i
$898$		0			0
$899$	12998.4	−	22513.9i	0.482226	−	0.835239i
$900$		0			0
$901$	−17211.8	−	29811.7i	−0.636414	−	1.10230i
$902$		0			0
$903$	6439.62	−	12149.1i	0.237317	−	0.447727i
$904$		0			0
$905$	3365.29	+	19085.5i	0.123609	+	0.701021i
$906$		0			0
$907$	5150.43	+	4321.72i	0.188553	+	0.158214i	0.732177	−	0.681114i	$-0.238504\pi$
$907$	5150.43	+	4321.72i	0.188553	+	0.158214i	−0.543625	+	0.839328i	$0.682949\pi$
$908$		0			0
$909$	−21295.7	+	1544.61i	−0.777046	+	0.0563601i
$910$		0			0
$911$	7244.57	−	2636.81i	0.263472	−	0.0958961i	−0.206906	−	0.978361i	$-0.566340\pi$
$911$	7244.57	−	2636.81i	0.263472	−	0.0958961i	0.470379	+	0.882465i	$0.344117\pi$
$912$		0			0
$913$	10118.7	−	8490.62i	0.366792	−	0.307775i
$914$		0			0
$915$	−3743.23	−	2916.73i	−0.135243	−	0.105381i
$916$		0			0
$917$	−18601.4			−0.669871
$918$		0			0
$919$	22717.1			0.815418			0.407709	−	0.913112i	$-0.366328\pi$
$919$	22717.1			0.815418			0.407709	+	0.913112i	$0.366328\pi$
$920$		0			0
$921$	30096.2	−	12205.0i	1.07677	−	0.436666i
$922$		0			0
$923$	54225.8	−	45500.8i	1.93376	−	1.62262i
$924$		0			0
$925$	−7732.85	+	2814.53i	−0.274870	+	0.100044i
$926$		0			0
$927$	−22394.1	+	10040.1i	−0.793441	+	0.355729i
$928$		0			0
$929$	29601.6	+	24838.7i	1.04542	+	0.877213i	0.992605	−	0.121392i	$-0.0387357\pi$
$929$	29601.6	+	24838.7i	1.04542	+	0.877213i	0.0528166	+	0.998604i	$0.483180\pi$
$930$		0			0
$931$	−4397.60	−	24940.0i	−0.154807	−	0.877955i
$932$		0			0
$933$	26724.6	−	967.911i	0.937752	−	0.0339635i
$934$		0			0
$935$	7053.75	+	12217.4i	0.246719	+	0.427330i
$936$		0			0
$937$	−15175.5	+	26284.7i	−0.529095	+	0.916419i	0.470330	+	0.882491i	$0.344135\pi$
$937$	−15175.5	+	26284.7i	−0.529095	+	0.916419i	−0.999424	+	0.0339279i	$0.989198\pi$
$938$		0			0
$939$	−21696.5	+	19586.6i	−0.754036	+	0.680708i
$940$		0			0
$941$	−22898.9	−	8334.50i	−0.793285	−	0.288732i	−0.0865842	−	0.996245i	$-0.527595\pi$
$941$	−22898.9	−	8334.50i	−0.793285	−	0.288732i	−0.706701	+	0.707512i	$0.749817\pi$
$942$		0			0
$943$	−9151.35	+	51899.9i	−0.316022	+	1.79225i
$944$		0			0
$945$	−21598.8	−	29486.5i	−0.743503	−	1.01502i
$946$		0			0
$947$	1088.11	−	6170.97i	0.0373377	−	0.211752i	−0.960431	−	0.278518i	$-0.910157\pi$
$947$	1088.11	−	6170.97i	0.0373377	−	0.211752i	0.997769	+	0.0667655i	$0.0212679\pi$
$948$		0			0
$949$	−12668.0	−	4610.77i	−0.433319	−	0.157715i
$950$		0			0
$951$	−3269.86	−	15286.1i	−0.111496	−	0.521225i
$952$		0			0
$953$	−3531.00	+	6115.87i	−0.120021	+	0.207883i	−0.919776	−	0.392444i	$-0.871630\pi$
$953$	−3531.00	+	6115.87i	−0.120021	+	0.207883i	0.799755	+	0.600327i	$0.204963\pi$
$954$		0			0
$955$	25109.6	+	43491.1i	0.850814	+	1.47365i
$956$		0			0
$957$	10718.6	+	17104.0i	0.362051	+	0.577736i
$958$		0			0
$959$	12999.9	+	73726.3i	0.437737	+	2.48253i
$960$		0			0
$961$	−5834.18	−	4895.46i	−0.195837	−	0.164327i
$962$		0			0
$963$	−5857.96	−	1663.33i	−0.196023	−	0.0556593i
$964$		0			0
$965$	4490.88	−	1634.55i	0.149810	−	0.0545263i
$966$		0			0
$967$	8819.45	−	7400.40i	0.293293	−	0.246102i	−0.484253	−	0.874928i	$-0.660909\pi$
$967$	8819.45	−	7400.40i	0.293293	−	0.246102i	0.777546	+	0.628826i	$0.216464\pi$
$968$		0			0
$969$	−3255.05	+	23380.7i	−0.107913	+	0.775124i
$970$		0			0
$971$	−25765.5			−0.851549			−0.425774	−	0.904829i	$-0.639998\pi$
$971$	−25765.5			−0.851549			−0.425774	+	0.904829i	$0.639998\pi$
$972$		0			0
$973$	−26199.3			−0.863218
$974$		0			0
$975$	1592.94	−	11441.9i	0.0523231	−	0.375831i
$976$		0			0
$977$	12494.6	−	10484.2i	0.409149	−	0.343317i	−0.414868	−	0.909882i	$-0.636172\pi$
$977$	12494.6	−	10484.2i	0.409149	−	0.343317i	0.824017	+	0.566565i	$0.191728\pi$
$978$		0			0
$979$	−34218.8	+	12454.6i	−1.11710	+	0.406590i
$980$		0			0
$981$	49773.4	+	14132.8i	1.61992	+	0.459966i
$982$		0			0
$983$	14280.0	+	11982.4i	0.463338	+	0.388787i	0.844358	−	0.535780i	$-0.179982\pi$
$983$	14280.0	+	11982.4i	0.463338	+	0.388787i	−0.381019	+	0.924567i	$0.624427\pi$
$984$		0			0
$985$	−3479.67	−	19734.2i	−0.112560	−	0.638358i
$986$		0			0
$987$	−22588.3	−	36044.8i	−0.728463	−	1.16243i
$988$		0			0
$989$	6001.65	+	10395.2i	0.192964	+	0.334223i
$990$		0			0
$991$	3160.07	−	5473.39i	0.101294	−	0.175447i	−0.810924	−	0.585152i	$-0.801035\pi$
$991$	3160.07	−	5473.39i	0.101294	−	0.175447i	0.912218	+	0.409705i	$0.134368\pi$
$992$		0			0
$993$	−5595.38	−	26157.5i	−0.178816	−	0.835935i
$994$		0			0
$995$	41333.6	+	15044.2i	1.31695	+	0.479330i
$996$		0			0
$997$	5035.77	−	28559.3i	0.159964	−	0.907203i	−0.794141	−	0.607733i	$-0.792079\pi$
$997$	5035.77	−	28559.3i	0.159964	−	0.907203i	0.954106	−	0.299470i	$-0.0968099\pi$
$998$		0			0
$999$	40423.7	−	4407.62i	1.28023	−	0.139591i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.25.4	yes	54
3.2	odd	2		324.4.i.a.73.3		54
27.13	even	9	inner	108.4.i.a.13.4	✓	54
27.14	odd	18		324.4.i.a.253.3		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.13.4	✓	54	27.13	even	9	inner
108.4.i.a.25.4	yes	54	1.1	even	1	trivial
324.4.i.a.73.3		54	3.2	odd	2
324.4.i.a.253.3		54	27.14	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)

\(f(q)\)	\(=\)	\(q+(-0.716497 + 5.14652i) q^{3} +(7.52940 - 6.31792i) q^{5} +(24.9076 - 9.06562i) q^{7} +(-25.9733 - 7.37493i) q^{9} +O(q^{10})\)	\(q+(-0.716497 + 5.14652i) q^{3} +(7.52940 - 6.31792i) q^{5} +(24.9076 - 9.06562i) q^{7} +(-25.9733 - 7.37493i) q^{9} +(17.0456 + 14.3030i) q^{11} +(13.5975 + 77.1153i) q^{13} +(27.1205 + 43.2769i) q^{15} +(32.2518 + 55.8618i) q^{17} +(35.2151 - 60.9943i) q^{19} +(28.8101 + 134.683i) q^{21} +(-112.980 - 41.1215i) q^{23} +(-4.93023 + 27.9607i) q^{25} +(56.5650 - 128.388i) q^{27} +(30.3150 - 171.925i) q^{29} +(139.932 + 50.9312i) q^{31} +(-85.8237 + 77.4776i) q^{33} +(130.263 - 225.623i) q^{35} +(144.919 + 251.008i) q^{37} +(-406.617 + 14.7269i) q^{39} +(-76.1146 - 431.667i) q^{41} +(-76.4782 - 64.1728i) q^{43} +(-242.157 + 108.568i) q^{45} +(-290.224 + 105.633i) q^{47} +(275.449 - 231.129i) q^{49} +(-310.602 + 125.960i) q^{51} -533.670 q^{53} +218.708 q^{55} +(288.677 + 224.937i) q^{57} +(320.392 - 268.841i) q^{59} +(-87.3117 + 31.7789i) q^{61} +(-713.789 + 51.7720i) q^{63} +(589.589 + 494.724i) q^{65} +(-80.8767 - 458.675i) q^{67} +(292.583 - 551.992i) q^{69} +(-451.994 - 782.877i) q^{71} +(-86.0801 + 149.095i) q^{73} +(-140.368 - 45.4073i) q^{75} +(554.231 + 201.723i) q^{77} +(-27.8206 + 157.779i) q^{79} +(620.221 + 383.102i) q^{81} +(103.082 - 584.607i) q^{83} +(595.767 + 216.841i) q^{85} +(863.095 + 279.201i) q^{87} +(-818.257 + 1417.26i) q^{89} +(1037.78 + 1797.48i) q^{91} +(-362.379 + 683.672i) q^{93} +(-120.209 - 681.737i) q^{95} +(-323.876 - 271.765i) q^{97} +(-337.247 - 497.205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 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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