LMFDB - Embedded newform 108.4.i.a.13.7

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			13.7
Character	$\chi$	$=$	108.13
Dual form			108.4.i.a.25.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.66172 + 4.46265i) q^{3} +(-12.8040 - 10.7439i) q^{5} +(-20.8033 - 7.57178i) q^{7} +(-12.8305 + 23.7566i) q^{9} +O(q^{10})$	\(q+(2.66172 + 4.46265i) q^{3} +(-12.8040 - 10.7439i) q^{5} +(-20.8033 - 7.57178i) q^{7} +(-12.8305 + 23.7566i) q^{9} +(-41.5462 + 34.8614i) q^{11} +(12.2534 - 69.4923i) q^{13} +(13.8654 - 85.7371i) q^{15} +(17.7250 - 30.7006i) q^{17} +(37.9595 + 65.7479i) q^{19} +(-21.5823 - 112.992i) q^{21} +(-161.185 + 58.6666i) q^{23} +(26.8068 + 152.029i) q^{25} +(-140.169 + 5.97534i) q^{27} +(-25.1440 - 142.599i) q^{29} +(160.113 - 58.2764i) q^{31} +(-266.158 - 92.6149i) q^{33} +(185.016 + 320.457i) q^{35} +(-180.731 + 313.036i) q^{37} +(342.735 - 130.286i) q^{39} +(-24.4277 + 138.536i) q^{41} +(269.808 - 226.396i) q^{43} +(419.521 - 166.332i) q^{45} +(-74.2539 - 27.0262i) q^{47} +(112.692 + 94.5596i) q^{49} +(184.185 - 2.61583i) q^{51} -195.786 q^{53} +906.505 q^{55} +(-192.372 + 344.402i) q^{57} +(-199.404 - 167.320i) q^{59} +(347.480 + 126.472i) q^{61} +(446.797 - 397.066i) q^{63} +(-903.509 + 758.134i) q^{65} +(-13.6574 + 77.4549i) q^{67} +(-690.838 - 563.159i) q^{69} +(-365.521 + 633.100i) q^{71} +(-361.799 - 626.655i) q^{73} +(-607.100 + 524.287i) q^{75} +(1128.26 - 410.653i) q^{77} +(52.6510 + 298.599i) q^{79} +(-399.756 - 609.620i) q^{81} +(8.20364 + 46.5252i) q^{83} +(-556.794 + 202.656i) q^{85} +(569.442 - 491.766i) q^{87} +(-340.199 - 589.241i) q^{89} +(-781.091 + 1352.89i) q^{91} +(686.243 + 559.413i) q^{93} +(220.351 - 1249.67i) q^{95} +(426.854 - 358.173i) q^{97} +(-295.130 - 1434.29i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{4}{9}\right)$	$1$

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
$3$	2.66172	+	4.46265i	0.512248	+	0.858838i
$3$	2.66172	+	4.46265i	0.512248	+	0.858838i
$4$		0			0
$5$	−12.8040	−	10.7439i	−1.14523	−	0.960961i	−0.145631	−	0.989339i	$-0.546521\pi$
$5$	−12.8040	−	10.7439i	−1.14523	−	0.960961i	−0.999597	+	0.0283783i	$0.990966\pi$
$6$		0			0
$7$	−20.8033	−	7.57178i	−1.12327	−	0.408838i	−0.287427	−	0.957803i	$-0.592800\pi$
$7$	−20.8033	−	7.57178i	−1.12327	−	0.408838i	−0.835845	+	0.548965i	$0.815022\pi$
$8$		0			0
$9$	−12.8305	+	23.7566i	−0.475204	+	0.879875i
$10$		0			0
$11$	−41.5462	+	34.8614i	−1.13879	+	0.955554i	−0.999398	−	0.0346836i	$-0.988958\pi$
$11$	−41.5462	+	34.8614i	−1.13879	+	0.955554i	−0.139387	+	0.990238i	$0.544513\pi$
$12$		0			0
$13$	12.2534	−	69.4923i	0.261421	−	1.48259i	−0.517615	−	0.855613i	$-0.673180\pi$
$13$	12.2534	−	69.4923i	0.261421	−	1.48259i	0.779036	−	0.626979i	$-0.215709\pi$
$14$		0			0
$15$	13.8654	−	85.7371i	0.238669	−	1.47582i
$16$		0			0
$17$	17.7250	−	30.7006i	0.252879	−	0.437999i	−0.711439	−	0.702748i	$-0.751956\pi$
$17$	17.7250	−	30.7006i	0.252879	−	0.437999i	0.964317	+	0.264750i	$0.0852893\pi$
$18$		0			0
$19$	37.9595	+	65.7479i	0.458343	+	0.793873i	0.998874	−	0.0474510i	$-0.0151098\pi$
$19$	37.9595	+	65.7479i	0.458343	+	0.793873i	−0.540531	+	0.841324i	$0.681776\pi$
$20$		0			0
$21$	−21.5823	−	112.992i	−0.224269	−	1.17413i
$22$		0			0
$23$	−161.185	+	58.6666i	−1.46128	+	0.531862i	−0.945717	−	0.324991i	$-0.894639\pi$
$23$	−161.185	+	58.6666i	−1.46128	+	0.531862i	−0.515562	+	0.856853i	$0.672417\pi$
$24$		0			0
$25$	26.8068	+	152.029i	0.214454	+	1.21623i
$26$		0			0
$27$	−140.169	+	5.97534i	−0.999093	+	0.0425909i
$28$		0			0
$29$	−25.1440	−	142.599i	−0.161004	−	0.913100i	−0.953089	−	0.302689i	$-0.902116\pi$
$29$	−25.1440	−	142.599i	−0.161004	−	0.913100i	0.792085	−	0.610411i	$-0.208996\pi$
$30$		0			0
$31$	160.113	−	58.2764i	0.927650	−	0.337637i	0.166372	−	0.986063i	$-0.446795\pi$
$31$	160.113	−	58.2764i	0.927650	−	0.337637i	0.761278	+	0.648426i	$0.224572\pi$
$32$		0			0
$33$	−266.158	−	92.6149i	−1.40401	−	0.488551i
$34$		0			0
$35$	185.016	+	320.457i	0.893526	+	1.54763i
$36$		0			0
$37$	−180.731	+	313.036i	−0.803029	+	1.39089i	0.114585	+	0.993413i	$0.463446\pi$
$37$	−180.731	+	313.036i	−0.803029	+	1.39089i	−0.917614	+	0.397473i	$0.869887\pi$
$38$		0			0
$39$	342.735	−	130.286i	1.40722	−	0.534936i
$40$		0			0
$41$	−24.4277	+	138.536i	−0.0930479	+	0.527701i	0.902280	+	0.431150i	$0.141892\pi$
$41$	−24.4277	+	138.536i	−0.0930479	+	0.527701i	−0.995328	+	0.0965508i	$0.969219\pi$
$42$		0			0
$43$	269.808	−	226.396i	0.956870	−	0.802909i	−0.0235712	−	0.999722i	$-0.507504\pi$
$43$	269.808	−	226.396i	0.956870	−	0.802909i	0.980441	+	0.196813i	$0.0630592\pi$
$44$		0			0
$45$	419.521	−	166.332i	1.38974	−	0.551006i
$46$		0			0
$47$	−74.2539	−	27.0262i	−0.230448	−	0.0838761i	0.224215	−	0.974540i	$-0.428018\pi$
$47$	−74.2539	−	27.0262i	−0.230448	−	0.0838761i	−0.454663	+	0.890663i	$0.650240\pi$
$48$		0			0
$49$	112.692	+	94.5596i	0.328547	+	0.275684i
$50$		0			0
$51$	184.185	−	2.61583i	0.505706	−	0.00718216i
$52$		0			0
$53$	−195.786			−0.507420			−0.253710	−	0.967280i	$-0.581651\pi$
$53$	−195.786			−0.507420			−0.253710	+	0.967280i	$0.581651\pi$
$54$		0			0
$55$	906.505			2.22242
$56$		0			0
$57$	−192.372	+	344.402i	−0.447023	+	0.800302i
$58$		0			0
$59$	−199.404	−	167.320i	−0.440003	−	0.369207i	0.395707	−	0.918377i	$-0.370499\pi$
$59$	−199.404	−	167.320i	−0.440003	−	0.369207i	−0.835711	+	0.549170i	$0.814944\pi$
$60$		0			0
$61$	347.480	+	126.472i	0.729348	+	0.265461i	0.679889	−	0.733315i	$-0.262028\pi$
$61$	347.480	+	126.472i	0.729348	+	0.265461i	0.0494591	+	0.998776i	$0.484250\pi$
$62$		0			0
$63$	446.797	−	397.066i	0.893510	−	0.794058i
$64$		0			0
$65$	−903.509	+	758.134i	−1.72410	+	1.44669i
$66$		0			0
$67$	−13.6574	+	77.4549i	−0.0249032	+	0.141233i	−0.994724	−	0.102585i	$-0.967289\pi$
$67$	−13.6574	+	77.4549i	−0.0249032	+	0.141233i	0.969821	+	0.243818i	$0.0783999\pi$
$68$		0			0
$69$	−690.838	−	563.159i	−1.20532	−	0.982556i
$70$		0			0
$71$	−365.521	+	633.100i	−0.610976	+	1.05824i	0.380100	+	0.924945i	$0.375890\pi$
$71$	−365.521	+	633.100i	−0.610976	+	1.05824i	−0.991076	+	0.133297i	$0.957444\pi$
$72$		0			0
$73$	−361.799	−	626.655i	−0.580074	−	1.00472i	−0.995470	−	0.0950773i	$-0.969690\pi$
$73$	−361.799	−	626.655i	−0.580074	−	1.00472i	0.415396	−	0.909641i	$-0.363643\pi$
$74$		0			0
$75$	−607.100	+	524.287i	−0.934691	+	0.807193i
$76$		0			0
$77$	1128.26	−	410.653i	1.66983	−	0.607769i
$78$		0			0
$79$	52.6510	+	298.599i	0.0749836	+	0.425253i	0.999072	+	0.0430739i	$0.0137151\pi$
$79$	52.6510	+	298.599i	0.0749836	+	0.425253i	−0.924088	+	0.382179i	$0.875174\pi$
$80$		0			0
$81$	−399.756	−	609.620i	−0.548362	−	0.836241i
$82$		0			0
$83$	8.20364	+	46.5252i	0.0108490	+	0.0615277i	0.989752	−	0.142800i	$-0.0456104\pi$
$83$	8.20364	+	46.5252i	0.0108490	+	0.0615277i	−0.978903	+	0.204327i	$0.934499\pi$
$84$		0			0
$85$	−556.794	+	202.656i	−0.710503	+	0.258602i
$86$		0			0
$87$	569.442	−	491.766i	0.701730	−	0.606010i
$88$		0			0
$89$	−340.199	−	589.241i	−0.405180	−	0.701792i	0.589163	−	0.808014i	$-0.299458\pi$
$89$	−340.199	−	589.241i	−0.405180	−	0.701792i	−0.994342	+	0.106223i	$0.966124\pi$
$90$		0			0
$91$	−781.091	+	1352.89i	−0.899786	+	1.55848i
$92$		0			0
$93$	686.243	+	559.413i	0.765162	+	0.623747i
$94$		0			0
$95$	220.351	−	1249.67i	0.237974	−	1.34962i
$96$		0			0
$97$	426.854	−	358.173i	0.446809	−	0.374917i	−0.391441	−	0.920203i	$-0.628023\pi$
$97$	426.854	−	358.173i	0.446809	−	0.374917i	0.838250	+	0.545286i	$0.183579\pi$
$98$		0			0
$99$	−295.130	−	1434.29i	−0.299613	−	1.45607i
$100$		0			0
$101$	42.5638	+	15.4920i	0.0419332	+	0.0152624i	0.362902	−	0.931827i	$-0.381786\pi$
$101$	42.5638	+	15.4920i	0.0419332	+	0.0152624i	−0.320968	+	0.947090i	$0.604008\pi$
$102$		0			0
$103$	−1512.45	−	1269.10i	−1.44686	−	1.21406i	−0.934833	−	0.355087i	$-0.884451\pi$
$103$	−1512.45	−	1269.10i	−1.44686	−	1.21406i	−0.512025	−	0.858971i	$-0.671104\pi$
$104$		0			0
$105$	−937.628	+	1678.63i	−0.871458	+	1.56017i
$106$		0			0
$107$	1552.11			1.40232			0.701160	−	0.713004i	$-0.252666\pi$
$107$	1552.11			1.40232			0.701160	+	0.713004i	$0.252666\pi$
$108$		0			0
$109$	−1881.98			−1.65377			−0.826887	−	0.562368i	$-0.809891\pi$
$109$	−1881.98			−1.65377			−0.826887	+	0.562368i	$0.809891\pi$
$110$		0			0
$111$	−1878.03	+	26.6722i	−1.60590	+	0.0228073i
$112$		0			0
$113$	−801.214	−	672.299i	−0.667008	−	0.559686i	0.245170	−	0.969480i	$-0.421156\pi$
$113$	−801.214	−	672.299i	−0.667008	−	0.559686i	−0.912178	+	0.409794i	$0.865601\pi$
$114$		0			0
$115$	2694.13	+	980.582i	2.18460	+	0.795128i
$116$		0			0
$117$	1493.69	+	1182.72i	1.18027	+	0.934552i
$118$		0			0
$119$	−601.195	+	504.463i	−0.463122	+	0.388605i
$120$		0			0
$121$	279.643	−	1585.93i	0.210100	−	1.19153i
$122$		0			0
$123$	−683.259	+	259.732i	−0.500873	+	0.190401i
$124$		0			0
$125$	245.487	−	425.196i	0.175656	−	0.304246i
$126$		0			0
$127$	−445.810	−	772.165i	−0.311490	−	0.539517i	0.667195	−	0.744883i	$-0.267495\pi$
$127$	−445.810	−	772.165i	−0.311490	−	0.539517i	−0.978685	+	0.205366i	$0.934161\pi$
$128$		0			0
$129$	1728.48	+	601.458i	1.17972	+	0.410507i
$130$		0			0
$131$	395.806	−	144.062i	0.263983	−	0.0960819i	−0.206638	−	0.978417i	$-0.566252\pi$
$131$	395.806	−	144.062i	0.263983	−	0.0960819i	0.470621	+	0.882336i	$0.344030\pi$
$132$		0			0
$133$	−291.855	−	1655.19i	−0.190279	−	1.07912i
$134$		0			0
$135$	1858.93	+	1429.45i	1.18512	+	0.911312i
$136$		0			0
$137$	29.8041	+	169.027i	0.0185864	+	0.105409i	0.992690	−	0.120695i	$-0.0385123\pi$
$137$	29.8041	+	169.027i	0.0185864	+	0.105409i	−0.974103	+	0.226104i	$0.927401\pi$
$138$		0			0
$139$	354.692	−	129.097i	0.216436	−	0.0787761i	−0.231527	−	0.972829i	$-0.574372\pi$
$139$	354.692	−	129.097i	0.216436	−	0.0787761i	0.447962	+	0.894052i	$0.352150\pi$
$140$		0			0
$141$	−77.0344	−	403.305i	−0.0460104	−	0.240883i
$142$		0			0
$143$	1913.52	+	3314.31i	1.11900	+	1.93816i
$144$		0			0
$145$	−1210.12	+	2095.98i	−0.693066	+	1.20043i
$146$		0			0
$147$	−122.033	+	754.595i	−0.0684701	+	0.423387i
$148$		0			0
$149$	177.014	−	1003.90i	0.0973257	−	0.551962i	−0.896684	−	0.442671i	$-0.854031\pi$
$149$	177.014	−	1003.90i	0.0973257	−	0.551962i	0.994010	−	0.109291i	$-0.0348579\pi$
$150$		0			0
$151$	−810.985	+	680.498i	−0.437067	+	0.366742i	−0.834610	−	0.550841i	$-0.814307\pi$
$151$	−810.985	+	680.498i	−0.437067	+	0.366742i	0.397544	+	0.917583i	$0.369863\pi$
$152$		0			0
$153$	501.921	+	814.990i	0.265215	+	0.430641i
$154$		0			0
$155$	−2676.21	−	974.060i	−1.38683	−	0.504764i
$156$		0			0
$157$	−682.838	−	572.969i	−0.347111	−	0.291261i	0.452518	−	0.891755i	$-0.350526\pi$
$157$	−682.838	−	572.969i	−0.347111	−	0.291261i	−0.799629	+	0.600495i	$0.794970\pi$
$158$		0			0
$159$	−521.127	−	873.725i	−0.259925	−	0.435792i
$160$		0			0
$161$	3797.39			1.85886
$162$		0			0
$163$	1261.47			0.606173			0.303086	−	0.952963i	$-0.401983\pi$
$163$	1261.47			0.606173			0.303086	+	0.952963i	$0.401983\pi$
$164$		0			0
$165$	2412.86	+	4045.41i	1.13843	+	1.90870i
$166$		0			0
$167$	−1362.74	−	1143.48i	−0.631450	−	0.529850i	0.269929	−	0.962880i	$-0.413000\pi$
$167$	−1362.74	−	1143.48i	−0.631450	−	0.529850i	−0.901379	+	0.433030i	$0.857444\pi$
$168$		0			0
$169$	−2614.53	−	951.612i	−1.19005	−	0.433141i
$170$		0			0
$171$	−2048.99	+	58.2122i	−0.916316	+	0.0260327i
$172$		0			0
$173$	2778.70	−	2331.60i	1.22116	−	1.02467i	0.222395	−	0.974957i	$-0.428613\pi$
$173$	2778.70	−	2331.60i	1.22116	−	1.02467i	0.998763	−	0.0497167i	$-0.0158319\pi$
$174$		0			0
$175$	593.459	−	3365.68i	0.256350	−	1.45384i
$176$		0			0
$177$	215.933	−	1335.23i	0.0916978	−	0.567017i
$178$		0			0
$179$	975.147	−	1689.00i	0.407184	−	0.705263i	−0.587389	−	0.809305i	$-0.699844\pi$
$179$	975.147	−	1689.00i	0.407184	−	0.705263i	0.994573	+	0.104042i	$0.0331775\pi$
$180$		0			0
$181$	1929.16	+	3341.40i	0.792226	+	1.37218i	0.924585	+	0.380975i	$0.124411\pi$
$181$	1929.16	+	3341.40i	0.792226	+	1.37218i	−0.132359	+	0.991202i	$0.542255\pi$
$182$		0			0
$183$	360.491	+	1887.31i	0.145619	+	0.762373i
$184$		0			0
$185$	5677.31	−	2066.37i	2.25624	−	0.821204i
$186$		0			0
$187$	333.859	+	1893.41i	0.130557	+	0.740426i
$188$		0			0
$189$	2961.22	+	937.020i	1.13967	+	0.360625i
$190$		0			0
$191$	236.098	+	1338.98i	0.0894421	+	0.507251i	0.996309	+	0.0858348i	$0.0273557\pi$
$191$	236.098	+	1338.98i	0.0894421	+	0.507251i	−0.906867	+	0.421416i	$0.861533\pi$
$192$		0			0
$193$	−1975.24	+	718.930i	−0.736690	+	0.268133i	−0.682994	−	0.730424i	$-0.739322\pi$
$193$	−1975.24	+	718.930i	−0.736690	+	0.268133i	−0.0536961	+	0.998557i	$0.517100\pi$
$194$		0			0
$195$	−5788.17	−	2014.11i	−2.12564	−	0.739657i
$196$		0			0
$197$	744.973	+	1290.33i	0.269427	+	0.466661i	0.968714	−	0.248180i	$-0.0798323\pi$
$197$	744.973	+	1290.33i	0.269427	+	0.466661i	−0.699287	+	0.714841i	$0.746499\pi$
$198$		0			0
$199$	−537.438	+	930.870i	−0.191447	+	0.331596i	−0.945730	−	0.324953i	$-0.894651\pi$
$199$	−537.438	+	930.870i	−0.191447	+	0.331596i	0.754283	+	0.656549i	$0.227985\pi$
$200$		0			0
$201$	−382.006	+	145.215i	−0.134053	+	0.0509586i
$202$		0			0
$203$	−556.647	+	3156.90i	−0.192458	+	1.09148i
$204$		0			0
$205$	1801.19	−	1511.38i	0.613661	−	0.514923i
$206$		0			0
$207$	674.367	−	4581.94i	0.226434	−	1.53849i
$208$		0			0
$209$	−3869.13	−	1408.25i	−1.28054	−	0.466080i
$210$		0			0
$211$	−2250.23	−	1888.17i	−0.734180	−	0.616051i	0.197087	−	0.980386i	$-0.436852\pi$
$211$	−2250.23	−	1888.17i	−0.734180	−	0.616051i	−0.931268	+	0.364335i	$0.881296\pi$
$212$		0			0
$213$	−3798.22	+	53.9432i	−1.22183	+	0.0173527i
$214$		0			0
$215$	−5887.01			−1.86740
$216$		0			0
$217$	−3772.13			−1.18004
$218$		0			0
$219$	1833.54	−	3282.56i	0.565748	−	1.01285i
$220$		0			0
$221$	−1916.26	−	1607.93i	−0.583266	−	0.489418i
$222$		0			0
$223$	26.0168	+	9.46936i	0.00781263	+	0.00284357i	0.345924	−	0.938263i	$-0.387566\pi$
$223$	26.0168	+	9.46936i	0.00781263	+	0.00284357i	−0.338111	+	0.941106i	$0.609788\pi$
$224$		0			0
$225$	−3955.64	−	1313.77i	−1.17204	−	0.389265i
$226$		0			0
$227$	−2720.48	+	2282.76i	−0.795440	+	0.667453i	−0.947085	−	0.320982i	$-0.895987\pi$
$227$	−2720.48	+	2282.76i	−0.795440	+	0.667453i	0.151646	+	0.988435i	$0.451543\pi$
$228$		0			0
$229$	−449.344	+	2548.36i	−0.129666	+	0.735372i	0.848761	+	0.528777i	$0.177349\pi$
$229$	−449.344	+	2548.36i	−0.129666	+	0.735372i	−0.978427	+	0.206595i	$0.933762\pi$
$230$		0			0
$231$	4835.71	+	3941.99i	1.37734	+	1.12279i
$232$		0			0
$233$	−2151.96	+	3727.31i	−0.605064	+	1.04800i	0.386978	+	0.922089i	$0.373519\pi$
$233$	−2151.96	+	3727.31i	−0.605064	+	1.04800i	−0.992041	+	0.125912i	$0.959814\pi$
$234$		0			0
$235$	660.384	+	1143.82i	0.183314	+	0.317509i
$236$		0			0
$237$	−1192.40	+	1029.75i	−0.326813	+	0.282234i
$238$		0			0
$239$	−1776.21	+	646.488i	−0.480726	+	0.174970i	−0.571005	−	0.820947i	$-0.693446\pi$
$239$	−1776.21	+	646.488i	−0.480726	+	0.174970i	0.0902790	+	0.995917i	$0.471224\pi$
$240$		0			0
$241$	294.365	+	1669.43i	0.0786793	+	0.446212i	0.998542	+	0.0539728i	$0.0171884\pi$
$241$	294.365	+	1669.43i	0.0786793	+	0.446212i	−0.919863	+	0.392239i	$0.871700\pi$
$242$		0			0
$243$	1656.48	−	3406.61i	0.437298	−	0.899316i
$244$		0			0
$245$	−426.974	−	2421.49i	−0.111340	−	0.631442i
$246$		0			0
$247$	5034.10	−	1832.26i	1.29681	−	0.472001i
$248$		0			0
$249$	−185.790	+	160.447i	−0.0472850	+	0.0408350i
$250$		0			0
$251$	−81.6710	−	141.458i	−0.0205380	−	0.0355728i	0.855574	−	0.517681i	$-0.173205\pi$
$251$	−81.6710	−	141.458i	−0.0205380	−	0.0355728i	−0.876112	+	0.482108i	$0.839871\pi$
$252$		0			0
$253$	4651.42	−	8056.50i	1.15586	−	2.00201i
$254$		0			0
$255$	−2386.41	−	1945.36i	−0.586051	−	0.477739i
$256$		0			0
$257$	−312.000	+	1769.44i	−0.0757277	+	0.429473i	0.923247	+	0.384206i	$0.125525\pi$
$257$	−312.000	+	1769.44i	−0.0757277	+	0.429473i	−0.998975	+	0.0452667i	$0.985586\pi$
$258$		0			0
$259$	6130.05	−	5143.72i	1.47067	−	1.23404i
$260$		0			0
$261$	3710.27	+	1232.28i	0.879924	+	0.292245i
$262$		0			0
$263$	3362.81	+	1223.96i	0.788440	+	0.286969i	0.704687	−	0.709518i	$-0.251087\pi$
$263$	3362.81	+	1223.96i	0.788440	+	0.286969i	0.0837527	+	0.996487i	$0.473309\pi$
$264$		0			0
$265$	2506.85	+	2103.50i	0.581112	+	0.487611i
$266$		0			0
$267$	1724.07	−	3086.58i	0.395173	−	0.707475i
$268$		0			0
$269$	1080.51			0.244908			0.122454	−	0.992474i	$-0.460924\pi$
$269$	1080.51			0.244908			0.122454	+	0.992474i	$0.460924\pi$
$270$		0			0
$271$	−7858.41			−1.76149			−0.880746	−	0.473590i	$-0.842958\pi$
$271$	−7858.41			−1.76149			−0.880746	+	0.473590i	$0.842958\pi$
$272$		0			0
$273$	−8116.51	+	115.273i	−1.79939	+	0.0255554i
$274$		0			0
$275$	−6413.65	−	5381.70i	−1.40639	−	1.18010i
$276$		0			0
$277$	5715.93	+	2080.43i	1.23984	+	0.451267i	0.876958	−	0.480567i	$-0.159569\pi$
$277$	5715.93	+	2080.43i	1.23984	+	0.451267i	0.362887	+	0.931833i	$0.381791\pi$
$278$		0			0
$279$	−669.882	+	4551.46i	−0.143745	+	0.976663i
$280$		0			0
$281$	−4250.20	+	3566.34i	−0.902297	+	0.757117i	−0.970638	−	0.240545i	$-0.922674\pi$
$281$	−4250.20	+	3566.34i	−0.902297	+	0.757117i	0.0683411	+	0.997662i	$0.478229\pi$
$282$		0			0
$283$	−975.267	+	5531.01i	−0.204854	+	1.16178i	0.692816	+	0.721114i	$0.256370\pi$
$283$	−975.267	+	5531.01i	−0.204854	+	1.16178i	−0.897670	+	0.440669i	$0.854741\pi$
$284$		0			0
$285$	6163.36	−	2342.92i	1.28100	−	0.486957i
$286$		0			0
$287$	1557.14	−	2697.05i	0.320262	−	0.554710i
$288$		0			0
$289$	1828.15	+	3166.45i	0.372105	+	0.644504i
$290$		0			0
$291$	2734.57	+	951.546i	0.550870	+	0.191686i
$292$		0			0
$293$	5648.62	−	2055.93i	1.12626	−	0.409927i	0.289330	−	0.957229i	$-0.406567\pi$
$293$	5648.62	−	2055.93i	1.12626	−	0.409927i	0.836935	+	0.547302i	$0.184345\pi$
$294$		0			0
$295$	755.515	+	4284.74i	0.149111	+	0.845652i
$296$		0			0
$297$	5615.17	−	5134.73i	1.09705	−	1.00319i
$298$		0			0
$299$	2101.82	+	11920.0i	0.406525	+	2.30552i
$300$		0			0
$301$	−7327.12	+	2666.86i	−1.40308	+	0.510681i
$302$		0			0
$303$	44.1576	+	231.183i	0.00837224	+	0.0438320i
$304$		0			0
$305$	−3090.34	−	5352.63i	−0.580172	−	1.00489i
$306$		0			0
$307$	959.723	−	1662.29i	0.178418	−	0.309029i	−0.762921	−	0.646492i	$-0.776235\pi$
$307$	959.723	−	1662.29i	0.178418	−	0.309029i	0.941339	+	0.337463i	$0.109569\pi$
$308$		0			0
$309$	1637.82	−	10127.5i	0.301529	−	1.86451i
$310$		0			0
$311$	−1333.89	+	7564.89i	−0.243210	+	1.37931i	0.581404	+	0.813615i	$0.302504\pi$
$311$	−1333.89	+	7564.89i	−0.243210	+	1.37931i	−0.824614	+	0.565696i	$0.808608\pi$
$312$		0			0
$313$	−6883.88	+	5776.26i	−1.24313	+	1.04311i	−0.245858	+	0.969306i	$0.579070\pi$
$313$	−6883.88	+	5776.26i	−1.24313	+	1.04311i	−0.997273	+	0.0738052i	$0.976486\pi$
$314$		0			0
$315$	−9986.83	+	283.728i	−1.78633	+	0.0507500i
$316$		0			0
$317$	3964.88	+	1443.10i	0.702491	+	0.255686i	0.668474	−	0.743735i	$-0.266948\pi$
$317$	3964.88	+	1443.10i	0.702491	+	0.255686i	0.0340172	+	0.999421i	$0.489170\pi$
$318$		0			0
$319$	6015.82	+	5047.87i	1.05587	+	0.885976i
$320$		0			0
$321$	4131.28	+	6926.53i	0.718335	+	1.20436i
$322$		0			0
$323$	2691.33			0.463621
$324$		0			0
$325$	10893.3			1.85924
$326$		0			0
$327$	−5009.31	−	8398.64i	−0.847142	−	1.42032i
$328$		0			0
$329$	1340.09	+	1124.47i	0.224564	+	0.188431i
$330$		0			0
$331$	4463.91	+	1624.73i	0.741265	+	0.269798i	0.684925	−	0.728613i	$-0.259835\pi$
$331$	4463.91	+	1624.73i	0.741265	+	0.269798i	0.0563398	+	0.998412i	$0.482057\pi$
$332$		0			0
$333$	−5117.80	−	8309.98i	−0.842204	−	1.36752i
$334$		0			0
$335$	1007.04	−	845.003i	0.164239	−	0.137813i
$336$		0			0
$337$	1170.70	−	6639.39i	0.189235	−	1.07321i	−0.731156	−	0.682210i	$-0.761019\pi$
$337$	1170.70	−	6639.39i	0.189235	−	1.07321i	0.920392	−	0.390997i	$-0.127870\pi$
$338$		0			0
$339$	867.628	−	5365.01i	0.139006	−	0.859550i
$340$		0			0
$341$	−4620.49	+	8002.92i	−0.733764	+	1.27092i
$342$		0			0
$343$	2168.36	+	3755.71i	0.341343	+	0.591223i
$344$		0			0
$345$	2795.01	+	14633.0i	0.436169	+	2.28352i
$346$		0			0
$347$	5429.26	−	1976.09i	0.839936	−	0.305712i	0.114006	−	0.993480i	$-0.463632\pi$
$347$	5429.26	−	1976.09i	0.839936	−	0.305712i	0.725930	+	0.687768i	$0.241409\pi$
$348$		0			0
$349$	−980.754	−	5562.13i	−0.150426	−	0.853106i	−0.962849	−	0.270039i	$-0.912963\pi$
$349$	−980.754	−	5562.13i	−0.150426	−	0.853106i	0.812424	−	0.583067i	$-0.198148\pi$
$350$		0			0
$351$	−1302.30	+	9813.87i	−0.198039	+	1.49238i
$352$		0			0
$353$	−1941.90	−	11013.0i	−0.292795	−	1.66052i	−0.676030	−	0.736874i	$-0.736301\pi$
$353$	−1941.90	−	11013.0i	−0.292795	−	1.66052i	0.383235	−	0.923651i	$-0.374810\pi$
$354$		0			0
$355$	11482.1	−	4179.14i	1.71664	−	0.624805i
$356$		0			0
$357$	−3851.45	−	1340.19i	−0.570982	−	0.198684i
$358$		0			0
$359$	−4436.64	−	7684.48i	−0.652247	−	1.12973i	−0.982576	−	0.185860i	$-0.940493\pi$
$359$	−4436.64	−	7684.48i	−0.652247	−	1.12973i	0.330329	−	0.943866i	$-0.392840\pi$
$360$		0			0
$361$	547.646	−	948.551i	0.0798434	−	0.138293i
$362$		0			0
$363$	7821.80	−	2973.36i	1.13096	−	0.429920i
$364$		0			0
$365$	−2100.20	+	11910.8i	−0.301177	+	1.70806i
$366$		0			0
$367$	−4478.44	+	3757.85i	−0.636982	+	0.534491i	−0.903090	−	0.429452i	$-0.858707\pi$
$367$	−4478.44	+	3757.85i	−0.636982	+	0.534491i	0.266108	+	0.963943i	$0.414262\pi$
$368$		0			0
$369$	−2977.74	−	2357.81i	−0.420094	−	0.332636i
$370$		0			0
$371$	4072.99	+	1482.45i	0.569971	+	0.207452i
$372$		0			0
$373$	663.209	+	556.499i	0.0920635	+	0.0772504i	0.687658	−	0.726035i	$-0.258639\pi$
$373$	663.209	+	556.499i	0.0920635	+	0.0772504i	−0.595594	+	0.803285i	$0.703083\pi$
$374$		0			0
$375$	2550.92	−	36.2288i	0.351277	−	0.00498892i
$376$		0			0
$377$	−10217.6			−1.39584
$378$		0			0
$379$	−1536.47			−0.208240			−0.104120	−	0.994565i	$-0.533203\pi$
$379$	−1536.47			−0.208240			−0.104120	+	0.994565i	$0.533203\pi$
$380$		0			0
$381$	2259.28	−	4044.78i	0.303797	−	0.543886i
$382$		0			0
$383$	−999.899	−	839.015i	−0.133401	−	0.111936i	0.573646	−	0.819103i	$-0.305528\pi$
$383$	−999.899	−	839.015i	−0.133401	−	0.111936i	−0.707047	+	0.707167i	$0.749973\pi$
$384$		0			0
$385$	−18858.3	−	6863.85i	−2.49638	−	0.908609i
$386$		0			0
$387$	1916.63	+	9314.52i	0.251751	+	1.22347i
$388$		0			0
$389$	8814.64	−	7396.36i	1.14889	−	0.964037i	0.149201	−	0.988807i	$-0.452330\pi$
$389$	8814.64	−	7396.36i	1.14889	−	0.964037i	0.999693	+	0.0247702i	$0.00788542\pi$
$390$		0			0
$391$	−1055.90	+	5988.33i	−0.136571	+	0.774535i
$392$		0			0
$393$	1696.42	+	1382.89i	0.217744	+	0.177501i
$394$		0			0
$395$	2533.96	−	4388.95i	0.322778	−	0.559068i
$396$		0			0
$397$	−7192.58	−	12457.9i	−0.909283	−	1.57492i	−0.815062	−	0.579373i	$-0.803297\pi$
$397$	−7192.58	−	12457.9i	−0.909283	−	1.57492i	−0.0942206	−	0.995551i	$-0.530036\pi$
$398$		0			0
$399$	6609.71	−	5708.10i	0.829322	−	0.716197i
$400$		0			0
$401$	−11581.1	+	4215.16i	−1.44222	+	0.524926i	−0.940407	−	0.340050i	$-0.889556\pi$
$401$	−11581.1	+	4215.16i	−1.44222	+	0.524926i	−0.501814	+	0.864976i	$0.667334\pi$
$402$		0			0
$403$	−2087.84	−	11840.7i	−0.258071	−	1.46359i
$404$		0			0
$405$	−1431.19	+	12100.5i	−0.175596	+	1.48464i
$406$		0			0
$407$	−3404.17	−	19306.0i	−0.414590	−	2.35126i
$408$		0			0
$409$	−8895.19	+	3237.58i	−1.07540	+	0.391414i	−0.818194	−	0.574943i	$-0.805024\pi$
$409$	−8895.19	+	3237.58i	−1.07540	+	0.391414i	−0.257207	+	0.966356i	$0.582802\pi$
$410$		0			0
$411$	−674.980	+	582.908i	−0.0810080	+	0.0699580i
$412$		0			0
$413$	2881.35	+	4990.65i	0.343298	+	0.594609i
$414$		0			0
$415$	394.820	−	683.849i	0.0467011	−	0.0808888i
$416$		0			0
$417$	1520.20	+	1239.24i	0.178525	+	0.145530i
$418$		0			0
$419$	−2393.77	+	13575.7i	−0.279101	+	1.58286i	0.446526	+	0.894771i	$0.352661\pi$
$419$	−2393.77	+	13575.7i	−0.279101	+	1.58286i	−0.725627	+	0.688089i	$0.758450\pi$
$420$		0			0
$421$	−6924.37	+	5810.23i	−0.801598	+	0.672621i	−0.948587	−	0.316517i	$-0.897486\pi$
$421$	−6924.37	+	5810.23i	−0.801598	+	0.672621i	0.146988	+	0.989138i	$0.453042\pi$
$422$		0			0
$423$	1594.77	−	1417.26i	0.183310	−	0.162907i
$424$		0			0
$425$	5142.52	+	1871.72i	0.586939	+	0.213628i
$426$		0			0
$427$	−6271.10	−	5262.08i	−0.710726	−	0.596370i
$428$		0			0
$429$	−9697.36	+	17361.1i	−1.09136	+	1.95385i
$430$		0			0
$431$	7656.71			0.855709			0.427855	−	0.903848i	$-0.359269\pi$
$431$	7656.71			0.855709			0.427855	+	0.903848i	$0.359269\pi$
$432$		0			0
$433$	15823.7			1.75620			0.878102	−	0.478473i	$-0.158810\pi$
$433$	15823.7			1.75620			0.878102	+	0.478473i	$0.158810\pi$
$434$		0			0
$435$	−12574.6	+	178.588i	−1.38599	+	0.0196842i
$436$		0			0
$437$	−9975.71	−	8370.62i	−1.09200	−	0.916295i
$438$		0			0
$439$	−8402.95	−	3058.42i	−0.913556	−	0.332507i	−0.157884	−	0.987458i	$-0.550467\pi$
$439$	−8402.95	−	3058.42i	−0.913556	−	0.332507i	−0.755672	+	0.654951i	$0.772689\pi$
$440$		0			0
$441$	−3692.31	+	1463.93i	−0.398695	+	0.158074i
$442$		0			0
$443$	−25.8250	+	21.6698i	−0.00276972	+	0.00232407i	−0.644171	−	0.764881i	$-0.722798\pi$
$443$	−25.8250	+	21.6698i	−0.00276972	+	0.00232407i	0.641402	+	0.767205i	$0.278353\pi$
$444$		0			0
$445$	−1974.81	+	11199.7i	−0.210371	+	1.19307i
$446$		0			0
$447$	4951.20	−	1882.13i	0.523900	−	0.199154i
$448$		0			0
$449$	6809.18	−	11793.8i	0.715691	−	1.23961i	−0.247002	−	0.969015i	$-0.579445\pi$
$449$	6809.18	−	11793.8i	0.715691	−	1.23961i	0.962693	−	0.270597i	$-0.0872212\pi$
$450$		0			0
$451$	−3814.69	−	6607.24i	−0.398285	−	0.689851i
$452$		0			0
$453$	−5195.44	−	1807.85i	−0.538859	−	0.187506i
$454$		0			0
$455$	24536.4	−	8930.51i	2.52809	−	0.920151i
$456$		0			0
$457$	−1345.77	−	7632.27i	−0.137752	−	0.781230i	−0.972904	−	0.231211i	$-0.925731\pi$
$457$	−1345.77	−	7632.27i	−0.137752	−	0.781230i	0.835152	−	0.550020i	$-0.185380\pi$
$458$		0			0
$459$	−2301.04	+	4409.17i	−0.233994	+	0.448372i
$460$		0			0
$461$	−612.250	−	3472.24i	−0.0618553	−	0.350799i	−0.999990	−	0.00452774i	$-0.998559\pi$
$461$	−612.250	−	3472.24i	−0.0618553	−	0.350799i	0.938134	−	0.346271i	$-0.112552\pi$
$462$		0			0
$463$	430.400	−	156.653i	0.0432017	−	0.0157241i	−0.320329	−	0.947306i	$-0.603793\pi$
$463$	430.400	−	156.653i	0.0432017	−	0.0157241i	0.363531	+	0.931582i	$0.381571\pi$
$464$		0			0
$465$	−2776.42	−	14535.7i	−0.276889	−	1.44962i
$466$		0			0
$467$	2690.34	+	4659.80i	0.266582	+	0.461734i	0.967977	−	0.251039i	$-0.0807722\pi$
$467$	2690.34	+	4659.80i	0.266582	+	0.461734i	−0.701395	+	0.712773i	$0.747439\pi$
$468$		0			0
$469$	870.590	−	1507.91i	0.0857145	−	0.148462i
$470$		0			0
$471$	739.440	−	4572.35i	0.0723388	−	0.447310i
$472$		0			0
$473$	−3317.03	+	18811.8i	−0.322446	+	1.82868i
$474$		0			0
$475$	−8978.00	+	7533.44i	−0.867240	+	0.727701i
$476$		0			0
$477$	2512.04	−	4651.22i	0.241128	−	0.446467i
$478$		0			0
$479$	8832.30	+	3214.70i	0.842502	+	0.306646i	0.726979	−	0.686659i	$-0.240924\pi$
$479$	8832.30	+	3214.70i	0.842502	+	0.306646i	0.115522	+	0.993305i	$0.463146\pi$
$480$		0			0
$481$	19539.0	+	16395.2i	1.85219	+	1.55417i
$482$		0			0
$483$	10107.6	+	16946.4i	0.952196	+	1.59646i
$484$		0			0
$485$	−9313.63			−0.871979
$486$		0			0
$487$	−8531.94			−0.793879			−0.396939	−	0.917845i	$-0.629928\pi$
$487$	−8531.94			−0.793879			−0.396939	+	0.917845i	$0.629928\pi$
$488$		0			0
$489$	3357.68	+	5629.51i	0.310511	+	0.520604i
$490$		0			0
$491$	−13067.3	−	10964.8i	−1.20106	−	1.00781i	−0.999599	−	0.0283132i	$-0.990986\pi$
$491$	−13067.3	−	10964.8i	−1.20106	−	1.00781i	−0.201462	−	0.979496i	$-0.564569\pi$
$492$		0			0
$493$	−4823.53	−	1755.62i	−0.440651	−	0.160384i
$494$		0			0
$495$	−11630.9	+	21535.5i	−1.05610	+	1.95545i
$496$		0			0
$497$	12397.7	−	10402.9i	1.11894	−	0.938904i
$498$		0			0
$499$	678.059	−	3845.46i	0.0608299	−	0.344983i	−0.939169	−	0.343455i	$-0.888403\pi$
$499$	678.059	−	3845.46i	0.0608299	−	0.344983i	0.999999	−	0.00152782i	$-0.000486322\pi$
$500$		0			0
$501$	1475.70	−	9125.06i	0.131596	−	0.813728i
$502$		0			0
$503$	8186.51	−	14179.5i	0.725683	−	1.25692i	−0.233010	−	0.972474i	$-0.574857\pi$
$503$	8186.51	−	14179.5i	0.725683	−	1.25692i	0.958692	−	0.284445i	$-0.0918093\pi$
$504$		0			0
$505$	−378.545	−	655.660i	−0.0333565	−	0.0577752i
$506$		0			0
$507$	−2712.43	−	14200.7i	−0.237600	−	1.24393i
$508$		0			0
$509$	13457.6	−	4898.15i	1.17190	−	0.426536i	0.318564	−	0.947901i	$-0.396799\pi$
$509$	13457.6	−	4898.15i	1.17190	−	0.426536i	0.853333	+	0.521366i	$0.174577\pi$
$510$		0			0
$511$	2781.73	+	15776.0i	0.240815	+	1.36573i
$512$		0			0
$513$	−5713.61	−	8988.98i	−0.491739	−	0.773632i
$514$		0			0
$515$	5730.48	+	32499.2i	0.490321	+	2.78075i
$516$		0			0
$517$	4027.14	−	1465.76i	0.342579	−	0.124688i
$518$		0			0
$519$	17801.2	+	6194.28i	1.50556	+	0.523890i
$520$		0			0
$521$	−6002.38	−	10396.4i	−0.504739	−	0.874234i	−0.999985	−	0.00548128i	$-0.998255\pi$
$521$	−6002.38	−	10396.4i	−0.504739	−	0.874234i	0.495246	−	0.868753i	$-0.335078\pi$
$522$		0			0
$523$	−1823.72	+	3158.77i	−0.152477	+	0.264099i	−0.932138	−	0.362104i	$-0.882058\pi$
$523$	−1823.72	+	3158.77i	−0.152477	+	0.264099i	0.779660	+	0.626203i	$0.215392\pi$
$524$		0			0
$525$	16599.5	−	6310.08i	1.37992	−	0.524561i
$526$		0			0
$527$	1048.88	−	5948.51i	0.0866983	−	0.491691i
$528$		0			0
$529$	13218.4	−	11091.6i	1.08641	−	0.911610i
$530$		0			0
$531$	6533.41	−	2590.37i	0.533947	−	0.211700i
$532$		0			0
$533$	9327.89	+	3395.07i	0.758041	+	0.275904i
$534$		0			0
$535$	−19873.3	−	16675.7i	−1.60598	−	1.34757i
$536$		0			0
$537$	10133.0	−	143.911i	0.814286	−	0.0115647i
$538$		0			0
$539$	−7978.38			−0.637576
$540$		0			0
$541$	7624.55			0.605924			0.302962	−	0.953003i	$-0.402024\pi$
$541$	7624.55			0.605924			0.302962	+	0.953003i	$0.402024\pi$
$542$		0			0
$543$	−9776.61	+	17503.0i	−0.772661	+	1.38329i
$544$		0			0
$545$	24097.0	+	20219.8i	1.89395	+	1.58921i
$546$		0			0
$547$	3960.55	+	1441.52i	0.309582	+	0.112678i	0.492139	−	0.870517i	$-0.336215\pi$
$547$	3960.55	+	1441.52i	0.309582	+	0.112678i	−0.182557	+	0.983195i	$0.558437\pi$
$548$		0			0
$549$	−7462.90	+	6632.25i	−0.580162	+	0.515587i
$550$		0			0
$551$	8421.10	−	7066.14i	0.651090	−	0.546330i
$552$		0			0
$553$	1165.61	−	6610.50i	0.0896324	−	0.508331i
$554$		0			0
$555$	24332.9	+	19835.8i	1.86103	+	1.51708i
$556$		0			0
$557$	−8126.05	+	14074.7i	−0.618154	+	1.07067i	0.371669	+	0.928365i	$0.378786\pi$
$557$	−8126.05	+	14074.7i	−0.618154	+	1.07067i	−0.989822	+	0.142308i	$0.954548\pi$
$558$		0			0
$559$	−12426.7	−	21523.7i	−0.940241	−	1.62855i
$560$		0			0
$561$	−7560.98	+	6529.61i	−0.569028	+	0.491409i
$562$		0			0
$563$	−13643.5	+	4965.82i	−1.02132	+	0.371731i	−0.797770	−	0.602962i	$-0.793987\pi$
$563$	−13643.5	+	4965.82i	−1.02132	+	0.371731i	−0.223551	+	0.974692i	$0.571765\pi$
$564$		0			0
$565$	3035.70	+	17216.3i	0.226040	+	1.28194i
$566$		0			0
$567$	3700.33	+	15709.0i	0.274073	+	1.16352i
$568$		0			0
$569$	669.121	+	3794.77i	0.0492988	+	0.279587i	0.999485	−	0.0320961i	$-0.0102183\pi$
$569$	669.121	+	3794.77i	0.0492988	+	0.279587i	−0.950186	+	0.311683i	$0.899107\pi$
$570$		0			0
$571$	−9021.71	+	3283.63i	−0.661203	+	0.240658i	−0.650756	−	0.759287i	$-0.725548\pi$
$571$	−9021.71	+	3283.63i	−0.661203	+	0.240658i	−0.0104473	+	0.999945i	$0.503326\pi$
$572$		0			0
$573$	−5346.96	+	4617.60i	−0.389830	+	0.336654i
$574$		0			0
$575$	−13239.9	−	22932.1i	−0.960245	−	1.66319i
$576$		0			0
$577$	−2519.05	+	4363.13i	−0.181750	+	0.314800i	−0.942476	−	0.334273i	$-0.891509\pi$
$577$	−2519.05	+	4363.13i	−0.181750	+	0.314800i	0.760727	+	0.649072i	$0.224843\pi$
$578$		0			0
$579$	−8465.88	−	6901.24i	−0.607651	−	0.495346i
$580$		0			0
$581$	181.615	−	1029.99i	0.0129685	−	0.0735478i
$582$		0			0
$583$	8134.16	−	6825.37i	0.577843	−	0.484868i
$584$		0			0
$585$	−6418.23	−	31191.6i	−0.453609	−	2.20447i
$586$		0			0
$587$	−26587.6	−	9677.08i	−1.86948	−	0.680436i	−0.969806	−	0.243878i	$-0.921580\pi$
$587$	−26587.6	−	9677.08i	−1.86948	−	0.680436i	−0.899676	−	0.436558i	$-0.856197\pi$
$588$		0			0
$589$	9909.36	+	8314.94i	0.693223	+	0.581683i
$590$		0			0
$591$	−3775.39	+	6759.05i	−0.262773	+	0.470440i
$592$		0			0
$593$	−6418.01			−0.444446			−0.222223	−	0.974996i	$-0.571331\pi$
$593$	−6418.01			−0.444446			−0.222223	+	0.974996i	$0.571331\pi$
$594$		0			0
$595$	13117.6			0.903815
$596$		0			0
$597$	−5584.65	+	79.3146i	−0.382855	+	0.00543740i
$598$		0			0
$599$	−19148.7	−	16067.6i	−1.30616	−	1.09600i	−0.989045	−	0.147617i	$-0.952840\pi$
$599$	−19148.7	−	16067.6i	−1.30616	−	1.09600i	−0.317120	−	0.948385i	$-0.602716\pi$
$600$		0			0
$601$	4731.73	+	1722.21i	0.321150	+	0.116889i	0.497565	−	0.867427i	$-0.334228\pi$
$601$	4731.73	+	1722.21i	0.321150	+	0.116889i	−0.176415	+	0.984316i	$0.556450\pi$
$602$		0			0
$603$	−1664.84	−	1318.24i	−0.112434	−	0.0890264i
$604$		0			0
$605$	−20619.6	+	17301.9i	−1.38563	+	1.16268i
$606$		0			0
$607$	−2916.95	+	16542.9i	−0.195050	+	1.10618i	0.717297	+	0.696768i	$0.245379\pi$
$607$	−2916.95	+	16542.9i	−0.195050	+	1.10618i	−0.912347	+	0.409417i	$0.865732\pi$
$608$		0			0
$609$	−15569.8	+	5918.66i	−1.03599	+	0.393820i
$610$		0			0
$611$	−2787.97	+	4828.91i	−0.184598	+	0.319733i
$612$		0			0
$613$	−6512.11	−	11279.3i	−0.429073	−	0.743176i	0.567718	−	0.823223i	$-0.307826\pi$
$613$	−6512.11	−	11279.3i	−0.429073	−	0.743176i	−0.996791	+	0.0800468i	$0.974493\pi$
$614$		0			0
$615$	11539.0	+	4015.22i	0.756582	+	0.263267i
$616$		0			0
$617$	19391.5	−	7057.92i	1.26527	−	0.460520i	0.379735	−	0.925095i	$-0.376015\pi$
$617$	19391.5	−	7057.92i	1.26527	−	0.460520i	0.885534	+	0.464575i	$0.153793\pi$
$618$		0			0
$619$	−905.701	−	5136.48i	−0.0588097	−	0.333526i	0.941181	−	0.337904i	$-0.109718\pi$
$619$	−905.701	−	5136.48i	−0.0588097	−	0.333526i	−0.999990	+	0.00437732i	$0.998607\pi$
$620$		0			0
$621$	22242.6	−	9186.36i	1.43730	−	0.593617i
$622$		0			0
$623$	2615.65	+	14834.1i	0.168208	+	0.953956i
$624$		0			0
$625$	10421.6	−	3793.13i	0.666979	−	0.242761i
$626$		0			0
$627$	−4014.02	−	21015.0i	−0.255669	−	1.33853i
$628$		0			0
$629$	6406.92	+	11097.1i	0.406138	+	0.703451i
$630$		0			0
$631$	−2190.76	+	3794.50i	−0.138213	+	0.239393i	−0.926820	−	0.375505i	$-0.877469\pi$
$631$	−2190.76	+	3794.50i	−0.138213	+	0.239393i	0.788607	+	0.614898i	$0.210803\pi$
$632$		0			0
$633$	2436.75	−	15067.7i	0.153005	−	0.946112i
$634$		0			0
$635$	−2587.87	+	14676.6i	−0.161727	+	0.917199i
$636$		0			0
$637$	7952.02	−	6672.53i	0.494616	−	0.415032i
$638$		0			0
$639$	−10350.5	−	16806.5i	−0.640783	−	1.04046i
$640$		0			0
$641$	−4363.85	−	1588.31i	−0.268895	−	0.0978697i	0.204054	−	0.978960i	$-0.434588\pi$
$641$	−4363.85	−	1588.31i	−0.268895	−	0.0978697i	−0.472949	+	0.881090i	$0.656810\pi$
$642$		0			0
$643$	7650.57	+	6419.59i	0.469221	+	0.393723i	0.846510	−	0.532372i	$-0.178699\pi$
$643$	7650.57	+	6419.59i	0.469221	+	0.393723i	−0.377289	+	0.926095i	$0.623144\pi$
$644$		0			0
$645$	−15669.6	−	26271.7i	−0.956571	−	1.60379i
$646$		0			0
$647$	27979.8			1.70016			0.850078	−	0.526658i	$-0.176555\pi$
$647$	27979.8			1.70016			0.850078	+	0.526658i	$0.176555\pi$
$648$		0			0
$649$	14117.5			0.853866
$650$		0			0
$651$	−10040.4	−	16833.7i	−0.604474	−	1.01346i
$652$		0			0
$653$	−17499.4	−	14683.7i	−1.04871	−	0.879968i	−0.0557484	−	0.998445i	$-0.517754\pi$
$653$	−17499.4	−	14683.7i	−1.04871	−	0.879968i	−0.992957	+	0.118477i	$0.962199\pi$
$654$		0			0
$655$	−6615.70	−	2407.92i	−0.394652	−	0.143641i
$656$		0			0
$657$	19529.3	−	554.831i	1.15968	−	0.0329468i
$658$		0			0
$659$	21442.2	−	17992.2i	1.26748	−	1.06354i	0.272638	−	0.962117i	$-0.412104\pi$
$659$	21442.2	−	17992.2i	1.26748	−	1.06354i	0.994843	−	0.101427i	$-0.0323407\pi$
$660$		0			0
$661$	−5005.68	+	28388.6i	−0.294551	+	1.67048i	0.374469	+	0.927239i	$0.377825\pi$
$661$	−5005.68	+	28388.6i	−0.294551	+	1.67048i	−0.669021	+	0.743244i	$0.733286\pi$
$662$		0			0
$663$	2075.10	−	12831.5i	0.121554	−	0.751634i
$664$		0			0
$665$	−14046.2	+	24328.8i	−0.819083	+	1.41869i
$666$		0			0
$667$	12418.6	+	21509.7i	0.720915	+	1.24866i
$668$		0			0
$669$	26.9911	+	141.309i	0.00155984	+	0.00816639i
$670$		0			0
$671$	−18845.4	+	6859.18i	−1.08423	+	0.394629i
$672$		0			0
$673$	−3476.92	−	19718.6i	−0.199146	−	1.12941i	−0.906390	−	0.422443i	$-0.861173\pi$
$673$	−3476.92	−	19718.6i	−0.199146	−	1.12941i	0.707244	−	0.706970i	$-0.249938\pi$
$674$		0			0
$675$	−4665.90	−	21149.5i	−0.266060	−	1.20599i
$676$		0			0
$677$	−4675.19	−	26514.3i	−0.265409	−	1.50521i	−0.767867	−	0.640609i	$-0.778682\pi$
$677$	−4675.19	−	26514.3i	−0.265409	−	1.50521i	0.502458	−	0.864602i	$-0.332429\pi$
$678$		0			0
$679$	−11592.0	+	4219.14i	−0.655169	+	0.238462i
$680$		0			0
$681$	−17428.3	−	6064.51i	−0.980696	−	0.341252i
$682$		0			0
$683$	10823.6	+	18747.1i	0.606376	+	1.05027i	0.991832	+	0.127548i	$0.0407108\pi$
$683$	10823.6	+	18747.1i	0.606376	+	1.05027i	−0.385456	+	0.922726i	$0.625956\pi$
$684$		0			0
$685$	1434.39	−	2484.44i	0.0800079	−	0.138578i
$686$		0			0
$687$	−12568.5	+	4777.74i	−0.697987	+	0.265331i
$688$		0			0
$689$	−2399.04	+	13605.6i	−0.132650	+	0.752297i
$690$		0			0
$691$	−10158.7	+	8524.19i	−0.559271	+	0.469284i	−0.878066	−	0.478539i	$-0.841166\pi$
$691$	−10158.7	+	8524.19i	−0.559271	+	0.469284i	0.318795	+	0.947824i	$0.396722\pi$
$692$		0			0
$693$	−4720.42	+	32072.5i	−0.258750	+	1.75806i
$694$		0			0
$695$	−5928.49	−	2157.79i	−0.323569	−	0.117769i
$696$		0			0
$697$	3820.16	+	3205.50i	0.207603	+	0.174199i
$698$		0			0
$699$	−22361.6	+	317.585i	−1.21001	+	0.0171848i
$700$		0			0
$701$	−20645.6			−1.11237			−0.556187	−	0.831057i	$-0.687736\pi$
$701$	−20645.6			−1.11237			−0.556187	+	0.831057i	$0.687736\pi$
$702$		0			0
$703$	−27441.9			−1.47225
$704$		0			0
$705$	−3346.71	+	5991.59i	−0.178786	+	0.320080i
$706$		0			0
$707$	−768.165	−	644.567i	−0.0408626	−	0.0342878i
$708$		0			0
$709$	7122.25	+	2592.29i	0.377266	+	0.137314i	0.523691	−	0.851908i	$-0.324555\pi$
$709$	7122.25	+	2592.29i	0.377266	+	0.137314i	−0.146425	+	0.989222i	$0.546777\pi$
$710$		0			0
$711$	−7769.24	−	2580.36i	−0.409802	−	0.136106i
$712$		0			0
$713$	−22389.0	+	18786.6i	−1.17598	+	0.986763i
$714$		0			0
$715$	11107.7	−	62995.1i	0.580987	−	3.29494i
$716$		0			0
$717$	−7612.82	−	6205.84i	−0.396522	−	0.323238i
$718$		0			0
$719$	−947.326	+	1640.82i	−0.0491367	+	0.0851073i	−0.889548	−	0.456842i	$-0.848980\pi$
$719$	−947.326	+	1640.82i	−0.0491367	+	0.0851073i	0.840411	+	0.541950i	$0.182314\pi$
$720$		0			0
$721$	21854.7	+	37853.4i	1.12886	+	1.95525i
$722$		0			0
$723$	−6666.55	+	5757.19i	−0.342921	+	0.296144i
$724$		0			0
$725$	21005.1	−	7645.22i	1.07601	−	0.391636i
$726$		0			0
$727$	3900.92	+	22123.2i	0.199006	+	1.12862i	0.906598	+	0.421996i	$0.138670\pi$
$727$	3900.92	+	22123.2i	0.199006	+	1.12862i	−0.707592	+	0.706621i	$0.750219\pi$
$728$		0			0
$729$	19611.6	−	1675.11i	0.996372	−	0.0851046i
$730$		0			0
$731$	−2168.14	−	12296.1i	−0.109701	−	0.622146i
$732$		0			0
$733$	4541.24	−	1652.88i	0.228833	−	0.0832883i	−0.225059	−	0.974345i	$-0.572257\pi$
$733$	4541.24	−	1652.88i	0.228833	−	0.0832883i	0.453892	+	0.891057i	$0.350035\pi$
$734$		0			0
$735$	9669.78	−	8350.76i	0.485272	−	0.419078i
$736$		0			0
$737$	−2132.77	−	3694.07i	−0.106597	−	0.184631i
$738$		0			0
$739$	−3014.98	+	5222.09i	−0.150078	+	0.259943i	−0.931256	−	0.364366i	$-0.881286\pi$
$739$	−3014.98	+	5222.09i	−0.150078	+	0.259943i	0.781178	+	0.624309i	$0.214619\pi$
$740$		0			0
$741$	21576.1	+	17588.5i	1.06966	+	0.871969i
$742$		0			0
$743$	2481.91	−	14075.6i	0.122547	−	0.694999i	−0.860187	−	0.509978i	$-0.829654\pi$
$743$	2481.91	−	14075.6i	0.122547	−	0.694999i	0.982735	−	0.185021i	$-0.0592354\pi$
$744$		0			0
$745$	−13052.2	+	10952.1i	−0.641874	+	0.538596i
$746$		0			0
$747$	−1210.54	−	402.051i	−0.0592922	−	0.0196925i
$748$		0			0
$749$	−32289.0	−	11752.2i	−1.57519	−	0.573321i
$750$		0			0
$751$	125.354	+	105.185i	0.00609088	+	0.00511085i	0.645828	−	0.763483i	$-0.276512\pi$
$751$	125.354	+	105.185i	0.00609088	+	0.00511085i	−0.639737	+	0.768594i	$0.720957\pi$
$752$		0			0
$753$	413.894	−	740.991i	0.0200307	−	0.0358608i
$754$		0			0
$755$	17695.1			0.852966
$756$		0			0
$757$	1044.73			0.0501604			0.0250802	−	0.999685i	$-0.492016\pi$
$757$	1044.73			0.0501604			0.0250802	+	0.999685i	$0.492016\pi$
$758$		0			0
$759$	48334.1	−	686.453i	2.31149	−	0.0328283i
$760$		0			0
$761$	10258.8	+	8608.15i	0.488674	+	0.410046i	0.853551	−	0.521009i	$-0.174444\pi$
$761$	10258.8	+	8608.15i	0.488674	+	0.410046i	−0.364877	+	0.931056i	$0.618889\pi$
$762$		0			0
$763$	39151.5	+	14250.0i	1.85764	+	0.676125i
$764$		0			0
$765$	2329.52	−	15827.7i	0.110097	−	0.748043i
$766$		0			0
$767$	−14070.8	+	11806.8i	−0.662409	+	0.555827i
$768$		0			0
$769$	1313.03	−	7446.58i	0.0615724	−	0.349194i	−0.938421	−	0.345495i	$-0.887711\pi$
$769$	1313.03	−	7446.58i	0.0615724	−	0.349194i	0.999993	−	0.00369990i	$-0.00117772\pi$
$770$		0			0
$771$	−8726.84	+	3317.40i	−0.407639	+	0.154959i
$772$		0			0
$773$	9922.23	−	17185.8i	0.461679	−	0.799652i	−0.537366	−	0.843349i	$-0.680581\pi$
$773$	9922.23	−	17185.8i	0.461679	−	0.799652i	0.999045	+	0.0436977i	$0.0139138\pi$
$774$		0			0
$775$	13151.8	+	22779.6i	0.609583	+	1.05583i
$776$		0			0
$777$	39271.1	+	13665.1i	1.81318	+	0.630931i
$778$		0			0
$779$	−10035.7	+	3652.71i	−0.461576	+	0.168000i
$780$		0			0
$781$	−6884.76	−	39045.4i	−0.315437	−	1.78893i
$782$		0			0
$783$	4376.48	+	19837.6i	0.199748	+	0.905414i
$784$		0			0
$785$	2587.18	+	14672.6i	0.117631	+	0.667120i
$786$		0			0
$787$	32270.1	−	11745.3i	1.46163	−	0.531990i	0.515817	−	0.856699i	$-0.327489\pi$
$787$	32270.1	−	11745.3i	1.46163	−	0.531990i	0.945814	+	0.324709i	$0.105266\pi$
$788$		0			0
$789$	3488.73	+	18264.9i	0.157417	+	0.824141i
$790$		0			0
$791$	11577.4	+	20052.6i	0.520411	+	0.901378i
$792$		0			0
$793$	13046.6	−	22597.5i	0.584237	−	1.01193i
$794$		0			0
$795$	−2714.65	+	16786.1i	−0.121105	+	0.748859i
$796$		0			0
$797$	−3574.73	+	20273.3i	−0.158875	+	0.901024i	0.796282	+	0.604925i	$0.206797\pi$
$797$	−3574.73	+	20273.3i	−0.158875	+	0.901024i	−0.955157	+	0.296099i	$0.904314\pi$
$798$		0			0
$799$	−2145.87	+	1800.60i	−0.0950130	+	0.0797253i
$800$		0			0
$801$	18363.3	−	521.705i	0.810032	−	0.0230132i
$802$		0			0
$803$	36877.4	+	13422.3i	1.62064	+	0.589866i
$804$		0			0
$805$	−48621.9	−	40798.7i	−2.12882	−	1.78629i
$806$		0			0
$807$	2876.02	+	4821.96i	0.125453	+	0.210336i
$808$		0			0
$809$	13618.5			0.591845			0.295922	−	0.955212i	$-0.404373\pi$
$809$	13618.5			0.591845			0.295922	+	0.955212i	$0.404373\pi$
$810$		0			0
$811$	−23405.4			−1.01341			−0.506704	−	0.862120i	$-0.669136\pi$
$811$	−23405.4			−1.01341			−0.506704	+	0.862120i	$0.669136\pi$
$812$		0			0
$813$	−20916.9	−	35069.3i	−0.902320	−	1.51284i
$814$		0			0
$815$	−16151.9	−	13553.1i	−0.694206	−	0.582508i
$816$		0			0
$817$	25126.9	+	9145.43i	1.07598	+	0.391626i
$818$		0			0
$819$	−22118.3	−	35914.3i	−0.943682	−	1.53229i
$820$		0			0
$821$	−16317.5	+	13692.0i	−0.693647	+	0.582039i	−0.919958	−	0.392016i	$-0.871778\pi$
$821$	−16317.5	+	13692.0i	−0.693647	+	0.582039i	0.226311	+	0.974055i	$0.427333\pi$
$822$		0			0
$823$	5338.93	−	30278.6i	0.226128	−	1.28244i	−0.634389	−	0.773014i	$-0.718748\pi$
$823$	5338.93	−	30278.6i	0.226128	−	1.28244i	0.860517	−	0.509422i	$-0.170141\pi$
$824$		0			0
$825$	6945.29	−	42946.5i	0.293096	−	1.81237i
$826$		0			0
$827$	−9905.43	+	17156.7i	−0.416500	+	0.721399i	−0.995585	−	0.0938682i	$-0.970077\pi$
$827$	−9905.43	+	17156.7i	−0.416500	+	0.721399i	0.579085	+	0.815267i	$0.303410\pi$
$828$		0			0
$829$	17037.8	+	29510.4i	0.713811	+	1.23636i	0.963416	+	0.268009i	$0.0863656\pi$
$829$	17037.8	+	29510.4i	0.713811	+	1.23636i	−0.249606	+	0.968348i	$0.580301\pi$
$830$		0			0
$831$	5929.97	+	31045.7i	0.247543	+	1.29599i
$832$		0			0
$833$	4900.49	−	1783.63i	0.203832	−	0.0741887i
$834$		0			0
$835$	5163.25	+	29282.3i	0.213990	+	1.21360i
$836$		0			0
$837$	−22094.6	+	9125.26i	−0.912428	+	0.376840i
$838$		0			0
$839$	−1352.26	−	7669.05i	−0.0556439	−	0.315572i	0.944263	−	0.329191i	$-0.106776\pi$
$839$	−1352.26	−	7669.05i	−0.0556439	−	0.315572i	−0.999907	+	0.0136189i	$0.995665\pi$
$840$		0			0
$841$	3216.03	−	1170.54i	0.131864	−	0.0479945i
$842$		0			0
$843$	−27228.1	−	9474.56i	−1.11244	−	0.387095i
$844$		0			0
$845$	23252.6	+	40274.7i	0.946643	+	1.63963i
$846$		0			0
$847$	−17825.8	+	30875.2i	−0.723143	+	1.25252i
$848$		0			0
$849$	−27278.9	+	10369.7i	−1.10272	+	0.419185i
$850$		0			0
$851$	10766.5	−	61059.6i	0.433689	−	2.45957i
$852$		0			0
$853$	28259.9	−	23712.9i	1.13435	−	0.951834i	0.135112	−	0.990830i	$-0.456861\pi$
$853$	28259.9	−	23712.9i	1.13435	−	0.951834i	0.999239	+	0.0389967i	$0.0124162\pi$
$854$		0			0
$855$	26860.8	+	21268.7i	1.07441	+	0.850730i
$856$		0			0
$857$	3050.77	+	1110.39i	0.121601	+	0.0442592i	0.402104	−	0.915594i	$-0.368279\pi$
$857$	3050.77	+	1110.39i	0.121601	+	0.0442592i	−0.280503	+	0.959853i	$0.590501\pi$
$858$		0			0
$859$	3091.57	+	2594.14i	0.122798	+	0.103039i	0.702118	−	0.712060i	$-0.252238\pi$
$859$	3091.57	+	2594.14i	0.122798	+	0.103039i	−0.579321	+	0.815100i	$0.696682\pi$
$860$		0			0
$861$	16180.7	−	229.802i	0.640460	−	0.00909596i
$862$		0			0
$863$	24524.1			0.967334			0.483667	−	0.875252i	$-0.339305\pi$
$863$	24524.1			0.967334			0.483667	+	0.875252i	$0.339305\pi$
$864$		0			0
$865$	−60629.0			−2.38318
$866$		0			0
$867$	−9264.74	+	16586.6i	−0.362915	+	0.649724i
$868$		0			0
$869$	−12597.0	−	10570.1i	−0.491742	−	0.412621i
$870$		0			0
$871$	5215.17	+	1898.17i	0.202881	+	0.0738426i
$872$		0			0
$873$	3032.23	+	14736.2i	0.117555	+	0.571299i
$874$		0			0
$875$	−8326.43	+	6986.71i	−0.321697	+	0.269936i
$876$		0			0
$877$	−6912.31	+	39201.6i	−0.266148	+	1.50940i	0.499597	+	0.866258i	$0.333481\pi$
$877$	−6912.31	+	39201.6i	−0.266148	+	1.50940i	−0.765746	+	0.643144i	$0.777630\pi$
$878$		0			0
$879$	24209.9	+	19735.5i	0.928987	+	0.757295i
$880$		0			0
$881$	24739.8	−	42850.6i	0.946090	−	1.63868i	0.192536	−	0.981290i	$-0.438329\pi$
$881$	24739.8	−	42850.6i	0.946090	−	1.63868i	0.753554	−	0.657386i	$-0.228338\pi$
$882$		0			0
$883$	−13884.6	−	24048.9i	−0.529167	−	0.916544i	−0.999421	−	0.0340133i	$-0.989171\pi$
$883$	−13884.6	−	24048.9i	−0.529167	−	0.916544i	0.470254	−	0.882531i	$-0.344162\pi$
$884$		0			0
$885$	−17110.3	+	14776.4i	−0.649896	+	0.561246i
$886$		0			0
$887$	4131.95	−	1503.91i	0.156412	−	0.0569292i	−0.262628	−	0.964897i	$-0.584589\pi$
$887$	4131.95	−	1503.91i	0.156412	−	0.0569292i	0.419039	+	0.907968i	$0.362367\pi$
$888$		0			0
$889$	3427.65	+	19439.2i	0.129313	+	0.733373i
$890$		0			0
$891$	37860.5	+	11391.3i	1.42354	+	0.428310i
$892$		0			0
$893$	−1041.73	−	5907.94i	−0.0390371	−	0.221390i
$894$		0			0
$895$	−30632.3	+	11149.2i	−1.14405	+	0.416400i
$896$		0			0
$897$	−47600.3	+	41107.3i	−1.77183	+	1.53014i
$898$		0			0
$899$	−12336.0	−	21366.6i	−0.457652	−	0.792676i
$900$		0			0
$901$	−3470.30	+	6010.74i	−0.128316	+	0.222249i
$902$		0			0
$903$	−31404.0	−	25600.0i	−1.15732	−	0.943427i
$904$		0			0
$905$	11198.5	−	63510.0i	0.411327	−	2.33275i
$906$		0			0
$907$	9794.91	−	8218.90i	0.358583	−	0.300887i	−0.445643	−	0.895211i	$-0.647025\pi$
$907$	9794.91	−	8218.90i	0.358583	−	0.300887i	0.804226	+	0.594324i	$0.202580\pi$
$908$		0			0
$909$	−914.152	+	812.403i	−0.0333559	+	0.0296432i
$910$		0			0
$911$	−22355.4	−	8136.70i	−0.813027	−	0.295918i	−0.0981533	−	0.995171i	$-0.531294\pi$
$911$	−22355.4	−	8136.70i	−0.813027	−	0.295918i	−0.714874	+	0.699254i	$0.753516\pi$
$912$		0			0
$913$	−1962.76	−	1646.95i	−0.0711478	−	0.0597001i
$914$		0			0
$915$	15661.3	−	28038.3i	0.565844	−	1.01303i
$916$		0			0
$917$	−9324.88			−0.335807
$918$		0			0
$919$	−15288.8			−0.548782			−0.274391	−	0.961618i	$-0.588476\pi$
$919$	−15288.8			−0.548782			−0.274391	+	0.961618i	$0.588476\pi$
$920$		0			0
$921$	9972.72	−	141.635i	0.356800	−	0.00506735i
$922$		0			0
$923$	39516.7	+	33158.5i	1.40922	+	1.18248i
$924$		0			0
$925$	−52435.3	−	19084.9i	−1.86385	−	0.678387i
$926$		0			0
$927$	49555.1	−	19647.6i	1.75577	−	0.696129i
$928$		0			0
$929$	5701.58	−	4784.19i	0.201359	−	0.168961i	−0.536532	−	0.843880i	$-0.680266\pi$
$929$	5701.58	−	4784.19i	0.201359	−	0.168961i	0.737892	+	0.674919i	$0.235822\pi$
$930$		0			0
$931$	−1939.36	+	10998.7i	−0.0682707	+	0.387183i
$932$		0			0
$933$	−37309.9	+	14182.9i	−1.30919	+	0.497671i
$934$		0			0
$935$	16067.8	−	27830.2i	0.562002	−	0.973417i
$936$		0			0
$937$	−25673.1	−	44467.1i	−0.895095	−	1.55035i	−0.833687	−	0.552237i	$-0.813774\pi$
$937$	−25673.1	−	44467.1i	−0.895095	−	1.55035i	−0.0614078	−	0.998113i	$-0.519559\pi$
$938$		0			0
$939$	−44100.4	−	15345.6i	−1.53265	−	0.533317i
$940$		0			0
$941$	18838.6	−	6856.68i	0.652625	−	0.237536i	0.00557575	−	0.999984i	$-0.498225\pi$
$941$	18838.6	−	6856.68i	0.652625	−	0.237536i	0.647049	+	0.762449i	$0.276003\pi$
$942$		0			0
$943$	−4190.07	−	23763.1i	−0.144695	−	0.820607i
$944$		0			0
$945$	−27848.3	−	43812.6i	−0.958630	−	1.50817i
$946$		0			0
$947$	5554.84	+	31503.0i	0.190610	+	1.08100i	0.918533	+	0.395345i	$0.129375\pi$
$947$	5554.84	+	31503.0i	0.190610	+	1.08100i	−0.727922	+	0.685660i	$0.759514\pi$
$948$		0			0
$949$	−47981.0	+	17463.6i	−1.64123	+	0.597359i
$950$		0			0
$951$	4113.35	+	21535.0i	0.140257	+	0.734301i
$952$		0			0
$953$	13941.4	+	24147.2i	0.473878	+	0.820780i	0.999553	−	0.0299051i	$-0.00952051\pi$
$953$	13941.4	+	24147.2i	0.473878	+	0.820780i	−0.525675	+	0.850686i	$0.676187\pi$
$954$		0			0
$955$	11362.8	−	19680.9i	0.385017	−	0.666869i
$956$		0			0
$957$	−6514.48	+	40282.5i	−0.220045	+	1.36066i
$958$		0			0
$959$	659.814	−	3741.99i	0.0222174	−	0.126001i
$960$		0			0
$961$	−581.184	+	487.672i	−0.0195087	+	0.0163698i
$962$		0			0
$963$	−19914.4	+	36872.9i	−0.666388	+	1.23387i
$964$		0			0
$965$	33015.2	+	12016.5i	1.10134	+	0.400856i
$966$		0			0
$967$	−25894.2	−	21727.8i	−0.861119	−	0.722565i	0.101090	−	0.994877i	$-0.467767\pi$
$967$	−25894.2	−	21727.8i	−0.861119	−	0.722565i	−0.962209	+	0.272312i	$0.912212\pi$
$968$		0			0
$969$	7163.55	+	12010.5i	0.237489	+	0.398175i
$970$		0			0
$971$	−10379.3			−0.343036			−0.171518	−	0.985181i	$-0.554867\pi$
$971$	−10379.3			−0.343036			−0.171518	+	0.985181i	$0.554867\pi$
$972$		0			0
$973$	−8356.25			−0.275323
$974$		0			0
$975$	28994.9	+	48613.1i	0.952390	+	1.59678i
$976$		0			0
$977$	−10729.8	−	9003.36i	−0.351358	−	0.294824i	0.449977	−	0.893040i	$-0.351432\pi$
$977$	−10729.8	−	9003.36i	−0.351358	−	0.294824i	−0.801335	+	0.598216i	$0.795876\pi$
$978$		0			0
$979$	34675.7	+	12620.9i	1.13201	+	0.412019i
$980$		0			0
$981$	24146.8	−	44709.6i	0.785881	−	1.45512i
$982$		0			0
$983$	−24025.0	+	20159.4i	−0.779532	+	0.654105i	−0.943131	−	0.332422i	$-0.892134\pi$
$983$	−24025.0	+	20159.4i	−0.779532	+	0.654105i	0.163599	+	0.986527i	$0.447690\pi$
$984$		0			0
$985$	4324.48	−	24525.3i	0.139888	−	0.793343i
$986$		0			0
$987$	−1451.17	+	8973.37i	−0.0467997	+	0.289387i
$988$		0			0
$989$	−30207.2	+	52320.4i	−0.971217	+	1.68220i
$990$		0			0
$991$	−13380.0	−	23174.8i	−0.428889	−	0.742858i	0.567885	−	0.823108i	$-0.307762\pi$
$991$	−13380.0	−	23174.8i	−0.428889	−	0.742858i	−0.996775	+	0.0802493i	$0.974428\pi$
$992$		0			0
$993$	4631.06	+	24245.5i	0.147998	+	0.774830i
$994$		0			0
$995$	16882.5	−	6144.73i	0.537901	−	0.195780i
$996$		0			0
$997$	2457.75	+	13938.6i	0.0780718	+	0.442767i	0.998638	+	0.0521799i	$0.0166169\pi$
$997$	2457.75	+	13938.6i	0.0780718	+	0.442767i	−0.920566	+	0.390587i	$0.872272\pi$
$998$		0			0
$999$	23462.4	−	44957.8i	0.743061	−	1.42383i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.13.7	✓	54
3.2	odd	2		324.4.i.a.253.8		54
27.2	odd	18		324.4.i.a.73.8		54
27.25	even	9	inner	108.4.i.a.25.7	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.13.7	✓	54	1.1	even	1	trivial
108.4.i.a.25.7	yes	54	27.25	even	9	inner
324.4.i.a.73.8		54	27.2	odd	18
324.4.i.a.253.8		54	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q+(2.66172 + 4.46265i) q^{3} +(-12.8040 - 10.7439i) q^{5} +(-20.8033 - 7.57178i) q^{7} +(-12.8305 + 23.7566i) q^{9} +O(q^{10})\)	\(q+(2.66172 + 4.46265i) q^{3} +(-12.8040 - 10.7439i) q^{5} +(-20.8033 - 7.57178i) q^{7} +(-12.8305 + 23.7566i) q^{9} +(-41.5462 + 34.8614i) q^{11} +(12.2534 - 69.4923i) q^{13} +(13.8654 - 85.7371i) q^{15} +(17.7250 - 30.7006i) q^{17} +(37.9595 + 65.7479i) q^{19} +(-21.5823 - 112.992i) q^{21} +(-161.185 + 58.6666i) q^{23} +(26.8068 + 152.029i) q^{25} +(-140.169 + 5.97534i) q^{27} +(-25.1440 - 142.599i) q^{29} +(160.113 - 58.2764i) q^{31} +(-266.158 - 92.6149i) q^{33} +(185.016 + 320.457i) q^{35} +(-180.731 + 313.036i) q^{37} +(342.735 - 130.286i) q^{39} +(-24.4277 + 138.536i) q^{41} +(269.808 - 226.396i) q^{43} +(419.521 - 166.332i) q^{45} +(-74.2539 - 27.0262i) q^{47} +(112.692 + 94.5596i) q^{49} +(184.185 - 2.61583i) q^{51} -195.786 q^{53} +906.505 q^{55} +(-192.372 + 344.402i) q^{57} +(-199.404 - 167.320i) q^{59} +(347.480 + 126.472i) q^{61} +(446.797 - 397.066i) q^{63} +(-903.509 + 758.134i) q^{65} +(-13.6574 + 77.4549i) q^{67} +(-690.838 - 563.159i) q^{69} +(-365.521 + 633.100i) q^{71} +(-361.799 - 626.655i) q^{73} +(-607.100 + 524.287i) q^{75} +(1128.26 - 410.653i) q^{77} +(52.6510 + 298.599i) q^{79} +(-399.756 - 609.620i) q^{81} +(8.20364 + 46.5252i) q^{83} +(-556.794 + 202.656i) q^{85} +(569.442 - 491.766i) q^{87} +(-340.199 - 589.241i) q^{89} +(-781.091 + 1352.89i) q^{91} +(686.243 + 559.413i) q^{93} +(220.351 - 1249.67i) q^{95} +(426.854 - 358.173i) q^{97} +(-295.130 - 1434.29i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 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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