LMFDB - Embedded newform 108.4.i.a.25.2

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			25.2
Character	$\chi$	$=$	108.25
Dual form			108.4.i.a.13.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.40571 + 2.75494i) q^{3} +(8.18465 - 6.86774i) q^{5} +(-24.4722 + 8.90714i) q^{7} +(11.8206 - 24.2749i) q^{9} +O(q^{10})$	\(q+(-4.40571 + 2.75494i) q^{3} +(8.18465 - 6.86774i) q^{5} +(-24.4722 + 8.90714i) q^{7} +(11.8206 - 24.2749i) q^{9} +(0.555929 + 0.466480i) q^{11} +(-12.2754 - 69.6174i) q^{13} +(-17.1390 + 52.8055i) q^{15} +(-42.6108 - 73.8041i) q^{17} +(75.7224 - 131.155i) q^{19} +(83.2788 - 106.662i) q^{21} +(-156.978 - 57.1353i) q^{23} +(-1.88334 + 10.6810i) q^{25} +(14.7977 + 139.514i) q^{27} +(-42.3122 + 239.964i) q^{29} +(57.5825 + 20.9583i) q^{31} +(-3.73439 - 0.523626i) q^{33} +(-139.124 + 240.970i) q^{35} +(7.30212 + 12.6477i) q^{37} +(245.874 + 272.896i) q^{39} +(-13.2323 - 75.0441i) q^{41} +(-152.644 - 128.083i) q^{43} +(-69.9662 - 279.863i) q^{45} +(252.257 - 91.8142i) q^{47} +(256.797 - 215.478i) q^{49} +(391.057 + 207.770i) q^{51} -52.6859 q^{53} +7.75374 q^{55} +(27.7129 + 786.442i) q^{57} +(-575.596 + 482.982i) q^{59} +(-34.9282 + 12.7128i) q^{61} +(-73.0561 + 699.349i) q^{63} +(-578.584 - 485.490i) q^{65} +(-88.9446 - 504.430i) q^{67} +(849.003 - 180.743i) q^{69} +(298.140 + 516.394i) q^{71} +(187.698 - 325.103i) q^{73} +(-21.1280 - 52.2458i) q^{75} +(-17.7598 - 6.46403i) q^{77} +(61.3493 - 347.929i) q^{79} +(-449.546 - 573.890i) q^{81} +(14.6810 - 83.2603i) q^{83} +(-855.622 - 311.421i) q^{85} +(-474.672 - 1173.78i) q^{87} +(-351.204 + 608.304i) q^{89} +(920.499 + 1594.35i) q^{91} +(-311.431 + 66.2999i) q^{93} +(-280.977 - 1593.50i) q^{95} +(10.3720 + 8.70312i) q^{97} +(17.8952 - 7.98106i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−4.40571	+	2.75494i	−0.847880	+	0.530188i
$3$	−4.40571	+	2.75494i	−0.847880	+	0.530188i
$4$		0			0
$5$	8.18465	−	6.86774i	0.732057	−	0.614269i	−0.198634	−	0.980074i	$-0.563651\pi$
$5$	8.18465	−	6.86774i	0.732057	−	0.614269i	0.930692	+	0.365805i	$0.119206\pi$
$6$		0			0
$7$	−24.4722	+	8.90714i	−1.32137	+	0.480941i	−0.903898	−	0.427748i	$-0.859307\pi$
$7$	−24.4722	+	8.90714i	−1.32137	+	0.480941i	−0.417475	+	0.908688i	$0.637085\pi$
$8$		0			0
$9$	11.8206	−	24.2749i	0.437801	−	0.899072i
$10$		0			0
$11$	0.555929	+	0.466480i	0.0152381	+	0.0127863i	0.650375	−	0.759613i	$-0.274612\pi$
$11$	0.555929	+	0.466480i	0.0152381	+	0.0127863i	−0.635137	+	0.772400i	$0.719056\pi$
$12$		0			0
$13$	−12.2754	−	69.6174i	−0.261892	−	1.48526i	−0.777742	−	0.628583i	$-0.783635\pi$
$13$	−12.2754	−	69.6174i	−0.261892	−	1.48526i	0.515851	−	0.856678i	$-0.327476\pi$
$14$		0			0
$15$	−17.1390	+	52.8055i	−0.295019	+	0.908954i
$16$		0			0
$17$	−42.6108	−	73.8041i	−0.607920	−	1.05295i	−0.991583	−	0.129475i	$-0.958671\pi$
$17$	−42.6108	−	73.8041i	−0.607920	−	1.05295i	0.383662	−	0.923473i	$-0.374663\pi$
$18$		0			0
$19$	75.7224	−	131.155i	0.914311	−	1.58363i	0.106404	−	0.994323i	$-0.466066\pi$
$19$	75.7224	−	131.155i	0.914311	−	1.58363i	0.807907	−	0.589310i	$-0.200600\pi$
$20$		0			0
$21$	83.2788	−	106.662i	0.865377	−	1.10836i
$22$		0			0
$23$	−156.978	−	57.1353i	−1.42314	−	0.517979i	−0.488180	−	0.872743i	$-0.662339\pi$
$23$	−156.978	−	57.1353i	−1.42314	−	0.517979i	−0.934956	+	0.354764i	$0.884561\pi$
$24$		0			0
$25$	−1.88334	+	10.6810i	−0.0150668	+	0.0854478i
$26$		0			0
$27$	14.7977	+	139.514i	0.105475	+	0.994422i
$28$		0			0
$29$	−42.3122	+	239.964i	−0.270937	+	1.53656i	0.480641	+	0.876918i	$0.340404\pi$
$29$	−42.3122	+	239.964i	−0.270937	+	1.53656i	−0.751578	+	0.659644i	$0.770707\pi$
$30$		0			0
$31$	57.5825	+	20.9583i	0.333617	+	0.121427i	0.503397	−	0.864055i	$-0.332083\pi$
$31$	57.5825	+	20.9583i	0.333617	+	0.121427i	−0.169780	+	0.985482i	$0.554306\pi$
$32$		0			0
$33$	−3.73439	−	0.523626i	−0.0196992	−	0.00276217i
$34$		0			0
$35$	−139.124	+	240.970i	−0.671894	+	1.16375i
$36$		0			0
$37$	7.30212	+	12.6477i	0.0324449	+	0.0561962i	0.881792	−	0.471639i	$-0.156337\pi$
$37$	7.30212	+	12.6477i	0.0324449	+	0.0561962i	−0.849347	+	0.527835i	$0.823004\pi$
$38$		0			0
$39$	245.874	+	272.896i	1.00952	+	1.12047i
$40$		0			0
$41$	−13.2323	−	75.0441i	−0.0504034	−	0.285852i	0.949179	−	0.314735i	$-0.101916\pi$
$41$	−13.2323	−	75.0441i	−0.0504034	−	0.285852i	−0.999583	+	0.0288839i	$0.990805\pi$
$42$		0			0
$43$	−152.644	−	128.083i	−0.541348	−	0.454245i	0.330650	−	0.943753i	$-0.392732\pi$
$43$	−152.644	−	128.083i	−0.541348	−	0.454245i	−0.871999	+	0.489508i	$0.837176\pi$
$44$		0			0
$45$	−69.9662	−	279.863i	−0.231777	−	0.927100i
$46$		0			0
$47$	252.257	−	91.8142i	0.782884	−	0.284946i	0.0805091	−	0.996754i	$-0.474345\pi$
$47$	252.257	−	91.8142i	0.782884	−	0.284946i	0.702375	+	0.711807i	$0.252123\pi$
$48$		0			0
$49$	256.797	−	215.478i	0.748679	−	0.628216i
$50$		0			0
$51$	391.057	+	207.770i	1.07370	+	0.570462i
$52$		0			0
$53$	−52.6859			−0.136546			−0.0682732	−	0.997667i	$-0.521749\pi$
$53$	−52.6859			−0.136546			−0.0682732	+	0.997667i	$0.521749\pi$
$54$		0			0
$55$	7.75374			0.0190093
$56$		0			0
$57$	27.7129	+	786.442i	0.0643975	+	1.82749i
$58$		0			0
$59$	−575.596	+	482.982i	−1.27010	+	1.06574i	−0.275576	+	0.961279i	$0.588869\pi$
$59$	−575.596	+	482.982i	−1.27010	+	1.06574i	−0.994529	+	0.104465i	$0.966687\pi$
$60$		0			0
$61$	−34.9282	+	12.7128i	−0.0733130	+	0.0266838i	−0.378416	−	0.925636i	$-0.623531\pi$
$61$	−34.9282	+	12.7128i	−0.0733130	+	0.0266838i	0.305103	+	0.952319i	$0.401309\pi$
$62$		0			0
$63$	−73.0561	+	699.349i	−0.146098	+	1.39857i
$64$		0			0
$65$	−578.584	−	485.490i	−1.10407	−	0.926424i
$66$		0			0
$67$	−88.9446	−	504.430i	−0.162184	−	0.919790i	−0.951921	−	0.306344i	$-0.900894\pi$
$67$	−88.9446	−	504.430i	−0.162184	−	0.919790i	0.789737	−	0.613445i	$-0.210217\pi$
$68$		0			0
$69$	849.003	−	180.743i	1.48128	−	0.315346i
$70$		0			0
$71$	298.140	+	516.394i	0.498349	+	0.863165i	0.999998	−	0.00190566i	$-0.000606591\pi$
$71$	298.140	+	516.394i	0.498349	+	0.863165i	−0.501649	+	0.865071i	$0.667273\pi$
$72$		0			0
$73$	187.698	−	325.103i	0.300937	−	0.521239i	−0.675411	−	0.737441i	$-0.736034\pi$
$73$	187.698	−	325.103i	0.300937	−	0.521239i	0.976349	+	0.216203i	$0.0693672\pi$
$74$		0			0
$75$	−21.1280	−	52.2458i	−0.0325286	−	0.0804377i
$76$		0			0
$77$	−17.7598	−	6.46403i	−0.0262846	−	0.00956682i
$78$		0			0
$79$	61.3493	−	347.929i	0.0873714	−	0.495508i	−0.909448	−	0.415817i	$-0.863496\pi$
$79$	61.3493	−	347.929i	0.0873714	−	0.495508i	0.996820	−	0.0796906i	$-0.0253932\pi$
$80$		0			0
$81$	−449.546	−	573.890i	−0.616661	−	0.787229i
$82$		0			0
$83$	14.6810	−	83.2603i	0.0194151	−	0.110109i	−0.973560	−	0.228431i	$-0.926641\pi$
$83$	14.6810	−	83.2603i	0.0194151	−	0.110109i	0.992975	+	0.118322i	$0.0377516\pi$
$84$		0			0
$85$	−855.622	−	311.421i	−1.09183	−	0.397392i
$86$		0			0
$87$	−474.672	−	1173.78i	−0.584945	−	1.44647i
$88$		0			0
$89$	−351.204	+	608.304i	−0.418287	+	0.724495i	−0.995767	−	0.0919098i	$-0.970703\pi$
$89$	−351.204	+	608.304i	−0.418287	+	0.724495i	0.577480	+	0.816405i	$0.304036\pi$
$90$		0			0
$91$	920.499	+	1594.35i	1.06038	+	1.83663i
$92$		0			0
$93$	−311.431	+	66.2999i	−0.347246	+	0.0739246i
$94$		0			0
$95$	−280.977	−	1593.50i	−0.303449	−	1.72094i
$96$		0			0
$97$	10.3720	+	8.70312i	0.0108569	+	0.00910998i	0.648200	−	0.761470i	$-0.275522\pi$
$97$	10.3720	+	8.70312i	0.0108569	+	0.00910998i	−0.637343	+	0.770580i	$0.719967\pi$
$98$		0			0
$99$	17.8952	−	7.98106i	0.0181670	−	0.00810229i
$100$		0			0
$101$	1713.61	−	623.703i	1.68822	−	0.614463i	0.693823	−	0.720146i	$-0.255925\pi$
$101$	1713.61	−	623.703i	1.68822	−	0.614463i	0.994400	+	0.105683i	$0.0337028\pi$
$102$		0			0
$103$	547.583	−	459.477i	0.523834	−	0.439549i	−0.342132	−	0.939652i	$-0.611149\pi$
$103$	547.583	−	459.477i	0.523834	−	0.439549i	0.865966	+	0.500103i	$0.166704\pi$
$104$		0			0
$105$	−50.9167	−	1444.92i	−0.0473234	−	1.34295i
$106$		0			0
$107$	−1919.61			−1.73435			−0.867175	−	0.498004i	$-0.834066\pi$
$107$	−1919.61			−1.73435			−0.867175	+	0.498004i	$0.834066\pi$
$108$		0			0
$109$	−49.6874			−0.0436623			−0.0218311	−	0.999762i	$-0.506950\pi$
$109$	−49.6874			−0.0436623			−0.0218311	+	0.999762i	$0.506950\pi$
$110$		0			0
$111$	−67.0146	−	35.6050i	−0.0573040	−	0.0304458i
$112$		0			0
$113$	1232.86	−	1034.49i	1.02635	−	0.861208i	0.0359359	−	0.999354i	$-0.488559\pi$
$113$	1232.86	−	1034.49i	1.02635	−	0.861208i	0.990412	+	0.138146i	$0.0441144\pi$
$114$		0			0
$115$	−1677.20	+	610.450i	−1.36000	+	0.494998i
$116$		0			0
$117$	−1835.06	−	524.936i	−1.45001	−	0.414790i
$118$		0			0
$119$	1700.16	+	1426.61i	1.30970	+	1.09896i
$120$		0			0
$121$	−231.034	−	1310.26i	−0.173579	−	0.984418i
$122$		0			0
$123$	265.040	+	294.169i	0.194291	+	0.215645i
$124$		0			0
$125$	725.708	+	1256.96i	0.519275	+	0.899410i
$126$		0			0
$127$	410.415	−	710.860i	0.286760	−	0.496682i	−0.686275	−	0.727342i	$-0.740755\pi$
$127$	410.415	−	710.860i	0.286760	−	0.496682i	0.973034	+	0.230660i	$0.0740886\pi$
$128$		0			0
$129$	1025.37	+	143.774i	0.699834	+	0.0981289i
$130$		0			0
$131$	−586.240	−	213.374i	−0.390993	−	0.142310i	0.139040	−	0.990287i	$-0.455598\pi$
$131$	−586.240	−	213.374i	−0.390993	−	0.142310i	−0.530033	+	0.847977i	$0.677820\pi$
$132$		0			0
$133$	−684.875	+	3884.12i	−0.446513	+	2.53230i
$134$		0			0
$135$	1079.26	+	1040.24i	0.688056	+	0.663184i
$136$		0			0
$137$	211.268	−	1198.16i	0.131751	−	0.747194i	−0.845317	−	0.534265i	$-0.820588\pi$
$137$	211.268	−	1198.16i	0.131751	−	0.747194i	0.977068	−	0.212929i	$-0.0683004\pi$
$138$		0			0
$139$	−26.9027	−	9.79178i	−0.0164162	−	0.00597502i	0.333799	−	0.942644i	$-0.391669\pi$
$139$	−26.9027	−	9.79178i	−0.0164162	−	0.00597502i	−0.350215	+	0.936669i	$0.613892\pi$
$140$		0			0
$141$	−858.432	+	1099.46i	−0.512716	+	0.656676i
$142$		0			0
$143$	25.6508	−	44.4286i	0.0150002	−	0.0259811i
$144$		0			0
$145$	1301.70	+	2254.61i	0.745520	+	1.29128i
$146$		0			0
$147$	−537.744	+	1656.79i	−0.301717	+	0.929593i
$148$		0			0
$149$	−154.781	−	877.807i	−0.0851017	−	0.482636i	−0.997335	−	0.0729639i	$-0.976754\pi$
$149$	−154.781	−	877.807i	−0.0851017	−	0.482636i	0.912233	−	0.409672i	$-0.134357\pi$
$150$		0			0
$151$	−222.292	−	186.525i	−0.119800	−	0.100525i	0.580919	−	0.813961i	$-0.302693\pi$
$151$	−222.292	−	186.525i	−0.119800	−	0.100525i	−0.700720	+	0.713437i	$0.747138\pi$
$152$		0			0
$153$	−2295.28	+	161.964i	−1.21282	+	0.0855820i
$154$		0			0
$155$	615.229	−	223.925i	0.318815	−	0.116039i
$156$		0			0
$157$	−1213.97	+	1018.64i	−0.617104	+	0.517812i	−0.896892	−	0.442250i	$-0.854180\pi$
$157$	−1213.97	+	1018.64i	−0.617104	+	0.517812i	0.279788	+	0.960062i	$0.409736\pi$
$158$		0			0
$159$	232.119	−	145.146i	0.115775	−	0.0723953i
$160$		0			0
$161$	4350.50			2.12961
$162$		0			0
$163$	3469.70			1.66729			0.833643	−	0.552304i	$-0.186251\pi$
$163$	3469.70			1.66729			0.833643	+	0.552304i	$0.186251\pi$
$164$		0			0
$165$	−34.1608	+	21.3611i	−0.0161176	+	0.0100785i
$166$		0			0
$167$	−1868.59	+	1567.93i	−0.865843	+	0.726529i	−0.963219	−	0.268719i	$-0.913400\pi$
$167$	−1868.59	+	1567.93i	−0.865843	+	0.726529i	0.0973754	+	0.995248i	$0.468955\pi$
$168$		0			0
$169$	−2631.39	+	957.749i	−1.19772	+	0.435935i
$170$		0			0
$171$	−2288.69	−	3388.49i	−1.02351	−	1.51535i
$172$		0			0
$173$	2917.03	+	2447.68i	1.28195	+	1.07569i	0.992972	+	0.118351i	$0.0377608\pi$
$173$	2917.03	+	2447.68i	1.28195	+	1.07569i	0.288981	+	0.957335i	$0.406684\pi$
$174$		0			0
$175$	−49.0475	−	278.162i	−0.0211865	−	0.120155i
$176$		0			0
$177$	1205.32	−	3713.61i	0.511851	−	1.57702i
$178$		0			0
$179$	−157.552	−	272.889i	−0.0657878	−	0.113948i	0.831255	−	0.555891i	$-0.187623\pi$
$179$	−157.552	−	272.889i	−0.0657878	−	0.113948i	−0.897043	+	0.441943i	$0.854289\pi$
$180$		0			0
$181$	−1920.99	+	3327.25i	−0.788873	+	1.36637i	0.137785	+	0.990462i	$0.456002\pi$
$181$	−1920.99	+	3327.25i	−0.788873	+	1.36637i	−0.926658	+	0.375906i	$0.877331\pi$
$182$		0			0
$183$	118.860	−	152.234i	0.0480132	−	0.0614943i
$184$		0			0
$185$	146.626	+	53.3675i	0.0582711	+	0.0212090i
$186$		0			0
$187$	10.7395	−	60.9069i	0.00419974	−	0.0238179i
$188$		0			0
$189$	−1604.80	−	3282.39i	−0.617629	−	1.26328i
$190$		0			0
$191$	436.206	−	2473.85i	0.165250	−	0.937180i	−0.783556	−	0.621321i	$-0.786596\pi$
$191$	436.206	−	2473.85i	0.165250	−	0.937180i	0.948806	−	0.315859i	$-0.102293\pi$
$192$		0			0
$193$	3588.62	+	1306.15i	1.33842	+	0.487144i	0.909314	−	0.416111i	$-0.136607\pi$
$193$	3588.62	+	1306.15i	1.33842	+	0.487144i	0.429104	+	0.903255i	$0.358829\pi$
$194$		0			0
$195$	3886.57	+	544.965i	1.42730	+	0.200132i
$196$		0			0
$197$	−1819.27	+	3151.06i	−0.657957	+	1.13961i	0.323187	+	0.946335i	$0.395246\pi$
$197$	−1819.27	+	3151.06i	−0.657957	+	1.13961i	−0.981144	+	0.193279i	$0.938088\pi$
$198$		0			0
$199$	−1745.94	−	3024.05i	−0.621941	−	1.07723i	−0.989124	−	0.147085i	$-0.953011\pi$
$199$	−1745.94	−	3024.05i	−0.621941	−	1.07723i	0.367183	−	0.930149i	$-0.380322\pi$
$200$		0			0
$201$	1781.54	+	1977.34i	0.625174	+	0.693883i
$202$		0			0
$203$	−1101.93	−	6249.33i	−0.380986	−	2.16068i
$204$		0			0
$205$	−623.685	−	523.334i	−0.212488	−	0.178299i
$206$		0			0
$207$	−3242.53	+	3135.25i	−1.08875	+	1.05273i
$208$		0			0
$209$	103.277	−	37.5899i	0.0341811	−	0.0124409i
$210$		0			0
$211$	3627.33	−	3043.69i	1.18349	−	0.993062i	0.183536	−	0.983013i	$-0.441246\pi$
$211$	3627.33	−	3043.69i	1.18349	−	0.993062i	0.999950	−	0.0100489i	$-0.00319870\pi$
$212$		0			0
$213$	−2736.16	−	1453.73i	−0.880180	−	0.467642i
$214$		0			0
$215$	−2128.98			−0.675327
$216$		0			0
$217$	−1595.85			−0.499232
$218$		0			0
$219$	68.6938	+	1949.41i	0.0211959	+	0.601501i
$220$		0			0
$221$	−4614.99	+	3872.43i	−1.40469	+	1.17868i
$222$		0			0
$223$	590.712	−	215.002i	0.177386	−	0.0645631i	−0.251800	−	0.967779i	$-0.581023\pi$
$223$	590.712	−	215.002i	0.177386	−	0.0645631i	0.429186	+	0.903216i	$0.358800\pi$
$224$		0			0
$225$	237.018	+	171.974i	0.0702275	+	0.0509552i
$226$		0			0
$227$	−994.390	−	834.392i	−0.290749	−	0.243967i	0.485732	−	0.874108i	$-0.338553\pi$
$227$	−994.390	−	834.392i	−0.290749	−	0.243967i	−0.776481	+	0.630140i	$0.782997\pi$
$228$		0			0
$229$	101.719	+	576.874i	0.0293526	+	0.166467i	0.995960	−	0.0897934i	$-0.0286207\pi$
$229$	101.719	+	576.874i	0.0293526	+	0.166467i	−0.966608	+	0.256260i	$0.917510\pi$
$230$		0			0
$231$	96.0525	−	20.4484i	0.0273584	−	0.00582428i
$232$		0			0
$233$	−423.007	−	732.669i	−0.118936	−	0.206003i	0.800410	−	0.599453i	$-0.204615\pi$
$233$	−423.007	−	732.669i	−0.118936	−	0.206003i	−0.919346	+	0.393449i	$0.871282\pi$
$234$		0			0
$235$	1434.08	−	2483.90i	0.398082	−	0.689498i
$236$		0			0
$237$	688.236	+	1701.89i	0.188632	+	0.466454i
$238$		0			0
$239$	3912.58	+	1424.06i	1.05893	+	0.385419i	0.812026	−	0.583621i	$-0.198365\pi$
$239$	3912.58	+	1424.06i	1.05893	+	0.385419i	0.246903	+	0.969040i	$0.420587\pi$
$240$		0			0
$241$	982.311	−	5570.96i	0.262557	−	1.48903i	−0.513346	−	0.858182i	$-0.671594\pi$
$241$	982.311	−	5570.96i	0.262557	−	1.48903i	0.775903	−	0.630852i	$-0.217295\pi$
$242$		0			0
$243$	3561.60	+	1289.92i	0.940234	+	0.340530i
$244$		0			0
$245$	621.945	−	3527.23i	0.162182	−	0.919780i
$246$		0			0
$247$	−10060.2	−	3661.61i	−2.59156	−	0.943251i
$248$		0			0
$249$	164.697	+	407.266i	0.0419166	+	0.103652i
$250$		0			0
$251$	−185.584	+	321.441i	−0.0466692	+	0.0808334i	−0.888416	−	0.459039i	$-0.848194\pi$
$251$	−185.584	+	321.441i	−0.0466692	+	0.0808334i	0.841747	+	0.539872i	$0.181527\pi$
$252$		0			0
$253$	−60.6160	−	104.990i	−0.0150628	−	0.0260896i
$254$		0			0
$255$	4627.57	−	985.154i	1.13643	−	0.241932i
$256$		0			0
$257$	−523.583	−	2969.39i	−0.127083	−	0.720721i	−0.980049	−	0.198755i	$-0.936310\pi$
$257$	−523.583	−	2969.39i	−0.127083	−	0.720721i	0.852967	−	0.521965i	$-0.174801\pi$
$258$		0			0
$259$	−291.353	−	244.474i	−0.0698989	−	0.0586521i
$260$		0			0
$261$	5324.97	+	3863.66i	1.26286	+	0.916300i
$262$		0			0
$263$	−3971.62	+	1445.55i	−0.931182	+	0.338922i	−0.762678	−	0.646778i	$-0.776116\pi$
$263$	−3971.62	+	1445.55i	−0.931182	+	0.338922i	−0.168504	+	0.985701i	$0.553894\pi$
$264$		0			0
$265$	−431.215	+	361.832i	−0.0999598	+	0.0838762i
$266$		0			0
$267$	−128.534	−	3647.56i	−0.0294612	−	0.836056i
$268$		0			0
$269$	−1948.75			−0.441700			−0.220850	−	0.975308i	$-0.570883\pi$
$269$	−1948.75			−0.441700			−0.220850	+	0.975308i	$0.570883\pi$
$270$		0			0
$271$	6235.43			1.39769			0.698847	−	0.715271i	$-0.253697\pi$
$271$	6235.43			1.39769			0.698847	+	0.715271i	$0.253697\pi$
$272$		0			0
$273$	−8447.79	−	4488.34i	−1.87283	−	0.995042i
$274$		0			0
$275$	−6.02946	+	5.05932i	−0.00132215	+	0.00110941i
$276$		0			0
$277$	2656.36	−	966.836i	0.576192	−	0.209717i	−0.0374534	−	0.999298i	$-0.511925\pi$
$277$	2656.36	−	966.836i	0.576192	−	0.209717i	0.613646	+	0.789582i	$0.289702\pi$
$278$		0			0
$279$	1189.42	−	1150.07i	0.255229	−	0.246785i
$280$		0			0
$281$	−1579.35	−	1325.24i	−0.335290	−	0.281341i	0.459561	−	0.888146i	$-0.348007\pi$
$281$	−1579.35	−	1325.24i	−0.335290	−	0.281341i	−0.794851	+	0.606805i	$0.792451\pi$
$282$		0			0
$283$	−1178.94	−	6686.12i	−0.247636	−	1.40441i	−0.814291	−	0.580457i	$-0.802874\pi$
$283$	−1178.94	−	6686.12i	−0.247636	−	1.40441i	0.566656	−	0.823955i	$-0.308237\pi$
$284$		0			0
$285$	5627.90	+	6246.43i	1.16971	+	1.29827i
$286$		0			0
$287$	992.252	+	1718.63i	0.204079	+	0.353476i
$288$		0			0
$289$	−1174.87	+	2034.93i	−0.239134	+	0.414192i
$290$		0			0
$291$	−69.6725	−	9.76931i	−0.0140353	−	0.00196800i
$292$		0			0
$293$	4133.18	+	1504.35i	0.824105	+	0.299950i	0.719437	−	0.694557i	$-0.244400\pi$
$293$	4133.18	+	1504.35i	0.824105	+	0.299950i	0.104668	+	0.994507i	$0.466622\pi$
$294$		0			0
$295$	−1394.05	+	7906.08i	−0.275136	+	1.56037i
$296$		0			0
$297$	−56.8538	+	84.4624i	−0.0111077	+	0.0165017i
$298$		0			0
$299$	−2050.64	+	11629.8i	−0.396627	+	2.24938i
$300$		0			0
$301$	4876.39	+	1774.86i	0.933788	+	0.339871i
$302$		0			0
$303$	−5831.41	+	7468.74i	−1.10563	+	1.41607i
$304$		0			0
$305$	−198.567	+	343.927i	−0.0372783	+	0.0645679i
$306$		0			0
$307$	489.447	+	847.746i	0.0909909	+	0.157601i	0.907928	−	0.419125i	$-0.137663\pi$
$307$	489.447	+	847.746i	0.0909909	+	0.157601i	−0.816937	+	0.576726i	$0.804330\pi$
$308$		0			0
$309$	−1146.66	+	3532.88i	−0.211105	+	0.650416i
$310$		0			0
$311$	571.383	+	3240.47i	0.104181	+	0.590837i	0.991544	+	0.129767i	$0.0414231\pi$
$311$	571.383	+	3240.47i	0.104181	+	0.590837i	−0.887364	+	0.461070i	$0.847466\pi$
$312$		0			0
$313$	−591.666	−	496.467i	−0.106847	−	0.0896549i	0.587799	−	0.809007i	$-0.299994\pi$
$313$	−591.666	−	496.467i	−0.106847	−	0.0896549i	−0.694646	+	0.719352i	$0.744439\pi$
$314$		0			0
$315$	4205.00	+	6225.65i	0.752143	+	1.11357i
$316$		0			0
$317$	657.562	−	239.333i	0.116506	−	0.0424046i	−0.283109	−	0.959088i	$-0.591366\pi$
$317$	657.562	−	239.333i	0.116506	−	0.0424046i	0.399615	+	0.916683i	$0.369144\pi$
$318$		0			0
$319$	−135.461	+	113.665i	−0.0237754	+	0.0199500i
$320$		0			0
$321$	8457.24	−	5288.40i	1.47052	−	0.919532i
$322$		0			0
$323$	−12906.4			−2.22331
$324$		0			0
$325$	766.701			0.130858
$326$		0			0
$327$	218.908	−	136.886i	0.0370204	−	0.0231492i
$328$		0			0
$329$	−5355.49	+	4493.79i	−0.897439	+	0.753041i
$330$		0			0
$331$	5019.41	−	1826.92i	0.833509	−	0.303373i	0.110211	−	0.993908i	$-0.464847\pi$
$331$	5019.41	−	1826.92i	0.833509	−	0.303373i	0.723298	+	0.690536i	$0.242625\pi$
$332$		0			0
$333$	393.337	−	27.7555i	0.0647289	−	0.00456754i
$334$		0			0
$335$	−4192.27	−	3517.73i	−0.683726	−	0.573714i
$336$		0			0
$337$	571.369	+	3240.40i	0.0923575	+	0.523785i	0.995525	+	0.0944973i	$0.0301244\pi$
$337$	571.369	+	3240.40i	0.0923575	+	0.523785i	−0.903168	+	0.429288i	$0.858765\pi$
$338$		0			0
$339$	−2581.66	+	7954.10i	−0.413617	+	1.27436i
$340$		0			0
$341$	22.2351	+	38.5124i	0.00353109	+	0.00611602i
$342$		0			0
$343$	101.247	−	175.365i	0.0159383	−	0.0276059i
$344$		0			0
$345$	5707.50	−	7310.05i	0.890671	−	1.14075i
$346$		0			0
$347$	4481.59	+	1631.16i	0.693326	+	0.252350i	0.664559	−	0.747236i	$-0.268620\pi$
$347$	4481.59	+	1631.16i	0.693326	+	0.252350i	0.0287674	+	0.999586i	$0.490842\pi$
$348$		0			0
$349$	2217.09	−	12573.7i	0.340052	−	1.92853i	−0.0300695	−	0.999548i	$-0.509573\pi$
$349$	2217.09	−	12573.7i	0.340052	−	1.92853i	0.370122	−	0.928983i	$-0.379316\pi$
$350$		0			0
$351$	9530.93	−	2742.76i	1.44935	−	0.417088i
$352$		0			0
$353$	−337.891	+	1916.28i	−0.0509465	+	0.288932i	−0.999627	−	0.0273036i	$-0.991308\pi$
$353$	−337.891	+	1916.28i	−0.0509465	+	0.288932i	0.948681	+	0.316236i	$0.102419\pi$
$354$		0			0
$355$	5986.63	+	2178.96i	0.895035	+	0.325766i
$356$		0			0
$357$	−11420.6	−	1601.37i	−1.69312	−	0.237405i
$358$		0			0
$359$	3268.90	−	5661.90i	0.480573	−	0.832378i	−0.519178	−	0.854666i	$-0.673762\pi$
$359$	3268.90	−	5661.90i	0.480573	−	0.832378i	0.999752	+	0.0222884i	$0.00709521\pi$
$360$		0			0
$361$	−8038.26	−	13922.7i	−1.17193	−	2.02984i
$362$		0			0
$363$	4627.56	+	5136.15i	0.669101	+	0.742639i
$364$		0			0
$365$	−696.477	−	3949.91i	−0.0998774	−	0.566433i
$366$		0			0
$367$	4978.80	+	4177.71i	0.708150	+	0.594209i	0.924079	−	0.382200i	$-0.124834\pi$
$367$	4978.80	+	4177.71i	0.708150	+	0.594209i	−0.215929	+	0.976409i	$0.569278\pi$
$368$		0			0
$369$	−1978.10	−	565.855i	−0.279068	−	0.0798299i
$370$		0			0
$371$	1289.34	−	469.280i	0.180429	−	0.0656707i
$372$		0			0
$373$	−7933.62	+	6657.10i	−1.10131	+	0.924106i	−0.997512	−	0.0704931i	$-0.977543\pi$
$373$	−7933.62	+	6657.10i	−1.10131	+	0.924106i	−0.103794	+	0.994599i	$0.533098\pi$
$374$		0			0
$375$	−6660.12	−	3538.54i	−0.917139	−	0.487278i
$376$		0			0
$377$	17225.1			2.35315
$378$		0			0
$379$	−12857.0			−1.74254			−0.871268	−	0.490807i	$-0.836702\pi$
$379$	−12857.0			−1.74254			−0.871268	+	0.490807i	$0.836702\pi$
$380$		0			0
$381$	150.204	+	4262.51i	0.0201973	+	0.573163i
$382$		0			0
$383$	−1960.38	+	1644.96i	−0.261543	+	0.219461i	−0.764124	−	0.645070i	$-0.776828\pi$
$383$	−1960.38	+	1644.96i	−0.261543	+	0.219461i	0.502581	+	0.864530i	$0.332384\pi$
$384$		0			0
$385$	−189.751	+	69.0637i	−0.0251184	+	0.00914236i
$386$		0			0
$387$	−4913.56	+	2191.39i	−0.645402	+	0.287842i
$388$		0			0
$389$	−8568.05	−	7189.45i	−1.11675	−	0.937068i	−0.118319	−	0.992976i	$-0.537750\pi$
$389$	−8568.05	−	7189.45i	−1.11675	−	0.937068i	−0.998436	+	0.0559072i	$0.982195\pi$
$390$		0			0
$391$	2472.14	+	14020.2i	0.319748	+	1.81338i
$392$		0			0
$393$	3170.64	−	674.991i	0.406966	−	0.0866381i
$394$		0			0
$395$	−1887.36	−	3269.01i	−0.240414	−	0.416410i
$396$		0			0
$397$	−1143.34	+	1980.33i	−0.144541	+	0.250352i	−0.929202	−	0.369573i	$-0.879504\pi$
$397$	−1143.34	+	1980.33i	−0.144541	+	0.250352i	0.784661	+	0.619926i	$0.212837\pi$
$398$		0			0
$399$	−7683.15	−	18999.1i	−0.964006	−	2.38382i
$400$		0			0
$401$	12492.1	+	4546.76i	1.55568	+	0.566220i	0.969740	−	0.244138i	$-0.0785050\pi$
$401$	12492.1	+	4546.76i	1.55568	+	0.566220i	0.585935	+	0.810358i	$0.300727\pi$
$402$		0			0
$403$	752.214	−	4266.02i	0.0929788	−	0.527309i
$404$		0			0
$405$	−7620.70	−	1609.73i	−0.935001	−	0.197501i
$406$		0			0
$407$	−1.84041	+	10.4375i	−0.000224142	+	0.00127117i
$408$		0			0
$409$	4155.16	+	1512.36i	0.502346	+	0.182839i	0.580749	−	0.814083i	$-0.302760\pi$
$409$	4155.16	+	1512.36i	0.502346	+	0.182839i	−0.0784028	+	0.996922i	$0.524982\pi$
$410$		0			0
$411$	2370.07	+	5860.77i	0.284445	+	0.703384i
$412$		0			0
$413$	9784.09	−	16946.5i	1.16572	−	2.01909i
$414$		0			0
$415$	−451.651	−	782.282i	−0.0534233	−	0.0925318i
$416$		0			0
$417$	145.501	−	30.9755i	0.0170869	−	0.00363759i
$418$		0			0
$419$	−1861.24	−	10555.6i	−0.217011	−	1.23073i	−0.877383	−	0.479790i	$-0.840713\pi$
$419$	−1861.24	−	10555.6i	−0.217011	−	1.23073i	0.660373	−	0.750938i	$-0.270398\pi$
$420$		0			0
$421$	−3381.05	−	2837.04i	−0.391406	−	0.328429i	0.425754	−	0.904839i	$-0.360009\pi$
$421$	−3381.05	−	2837.04i	−0.391406	−	0.328429i	−0.817161	+	0.576410i	$0.804453\pi$
$422$		0			0
$423$	753.057	−	7208.84i	0.0865600	−	0.828619i
$424$		0			0
$425$	868.551	−	316.127i	0.0991316	−	0.0360809i
$426$		0			0
$427$	741.533	−	622.220i	0.0840406	−	0.0705184i
$428$		0			0
$429$	9.38769	+	266.406i	0.00105651	+	0.0299818i
$430$		0			0
$431$	12301.2			1.37477			0.687385	−	0.726293i	$-0.258758\pi$
$431$	12301.2			1.37477			0.687385	+	0.726293i	$0.258758\pi$
$432$		0			0
$433$	−8467.47			−0.939770			−0.469885	−	0.882728i	$-0.655705\pi$
$433$	−8467.47			−0.939770			−0.469885	+	0.882728i	$0.655705\pi$
$434$		0			0
$435$	−11946.2	−	6347.07i	−1.31673	−	0.699584i
$436$		0			0
$437$	−19380.3	+	16262.0i	−2.12148	+	1.78013i
$438$		0			0
$439$	−1628.97	+	592.898i	−0.177100	+	0.0644590i	−0.429048	−	0.903282i	$-0.641151\pi$
$439$	−1628.97	+	592.898i	−0.177100	+	0.0644590i	0.251948	+	0.967741i	$0.418929\pi$
$440$		0			0
$441$	−2195.22	−	8780.82i	−0.237039	−	0.948150i
$442$		0			0
$443$	6356.76	+	5333.96i	0.681758	+	0.572063i	0.916519	−	0.399990i	$-0.130986\pi$
$443$	6356.76	+	5333.96i	0.681758	+	0.572063i	−0.234761	+	0.972053i	$0.575431\pi$
$444$		0			0
$445$	1303.19	+	7390.73i	0.138825	+	0.787313i
$446$		0			0
$447$	3100.23	+	3440.95i	0.328044	+	0.364097i
$448$		0			0
$449$	−7397.73	−	12813.2i	−0.777551	−	1.34676i	−0.933350	−	0.358968i	$-0.883129\pi$
$449$	−7397.73	−	12813.2i	−0.777551	−	1.34676i	0.155799	−	0.987789i	$-0.450205\pi$
$450$		0			0
$451$	27.6503	−	47.8918i	0.00288692	−	0.00500030i
$452$		0			0
$453$	1493.22	+	209.376i	0.154873	+	0.0217159i
$454$		0			0
$455$	18483.5	+	6727.46i	1.90444	+	0.693161i
$456$		0			0
$457$	−1618.87	+	9181.09i	−0.165706	+	0.939767i	0.782627	+	0.622491i	$0.213879\pi$
$457$	−1618.87	+	9181.09i	−0.165706	+	0.939767i	−0.948333	+	0.317276i	$0.897232\pi$
$458$		0			0
$459$	9666.13	−	7036.92i	0.982955	−	0.715589i
$460$		0			0
$461$	−889.156	+	5042.65i	−0.0898311	+	0.509457i	0.906378	+	0.422467i	$0.138836\pi$
$461$	−889.156	+	5042.65i	−0.0898311	+	0.509457i	−0.996209	+	0.0869899i	$0.972275\pi$
$462$		0			0
$463$	−863.470	−	314.277i	−0.0866714	−	0.0315458i	0.298321	−	0.954466i	$-0.403574\pi$
$463$	−863.470	−	314.277i	−0.0866714	−	0.0315458i	−0.384992	+	0.922920i	$0.625796\pi$
$464$		0			0
$465$	−2093.62	+	2681.47i	−0.208794	+	0.267420i
$466$		0			0
$467$	−2802.50	+	4854.08i	−0.277697	+	0.480985i	−0.970812	−	0.239842i	$-0.922904\pi$
$467$	−2802.50	+	4854.08i	−0.277697	+	0.480985i	0.693115	+	0.720827i	$0.256238\pi$
$468$		0			0
$469$	6669.70	+	11552.3i	0.656669	+	1.13738i
$470$		0			0
$471$	2542.11	−	7832.26i	0.248693	−	0.766224i
$472$		0			0
$473$	−25.1108	−	142.411i	−0.00244101	−	0.0138436i
$474$		0			0
$475$	1258.25	+	1055.80i	0.121542	+	0.101986i
$476$		0			0
$477$	−622.780	+	1278.95i	−0.0597801	+	0.122765i
$478$		0			0
$479$	−8793.16	+	3200.45i	−0.838768	+	0.305286i	−0.725452	−	0.688273i	$-0.758369\pi$
$479$	−8793.16	+	3200.45i	−0.838768	+	0.305286i	−0.113315	+	0.993559i	$0.536147\pi$
$480$		0			0
$481$	790.860	−	663.610i	0.0749691	−	0.0629065i
$482$		0			0
$483$	−19167.1	+	11985.4i	−1.80566	+	1.12909i
$484$		0			0
$485$	144.662			0.0135438
$486$		0			0
$487$	−5274.19			−0.490752			−0.245376	−	0.969428i	$-0.578911\pi$
$487$	−5274.19			−0.490752			−0.245376	+	0.969428i	$0.578911\pi$
$488$		0			0
$489$	−15286.5	+	9558.80i	−1.41366	+	0.883975i
$490$		0			0
$491$	10145.4	−	8513.01i	0.932497	−	0.782458i	−0.0437671	−	0.999042i	$-0.513936\pi$
$491$	10145.4	−	8513.01i	0.932497	−	0.782458i	0.976264	+	0.216584i	$0.0694915\pi$
$492$		0			0
$493$	19513.3	−	7102.27i	1.78263	−	0.648824i
$494$		0			0
$495$	91.6541	−	188.222i	0.00832231	−	0.0170908i
$496$		0			0
$497$	−11895.7	−	9981.71i	−1.07364	−	0.900887i
$498$		0			0
$499$	1209.51	+	6859.46i	0.108507	+	0.615374i	0.989761	+	0.142732i	$0.0455886\pi$
$499$	1209.51	+	6859.46i	0.108507	+	0.615374i	−0.881254	+	0.472642i	$0.843300\pi$
$500$		0			0
$501$	3912.91	−	12055.7i	0.348934	−	1.07507i
$502$		0			0
$503$	−297.634	−	515.517i	−0.0263834	−	0.0456974i	0.852532	−	0.522675i	$-0.175066\pi$
$503$	−297.634	−	515.517i	−0.0263834	−	0.0456974i	−0.878916	+	0.476977i	$0.841732\pi$
$504$		0			0
$505$	9741.86	−	16873.4i	0.858430	−	1.48684i
$506$		0			0
$507$	8954.63	−	11468.9i	0.784397	−	1.00464i
$508$		0			0
$509$	18230.3	+	6635.30i	1.58752	+	0.577809i	0.976821	−	0.214057i	$-0.0686678\pi$
$509$	18230.3	+	6635.30i	1.58752	+	0.577809i	0.610695	+	0.791866i	$0.290890\pi$
$510$		0			0
$511$	−1697.65	+	9627.83i	−0.146966	+	0.833484i
$512$		0			0
$513$	19418.4	+	8623.51i	1.67124	+	0.742178i
$514$		0			0
$515$	1326.21	−	7521.31i	0.113475	−	0.643550i
$516$		0			0
$517$	183.067	+	66.6308i	0.0155730	+	0.00566812i
$518$		0			0
$519$	−19594.8	−	2747.53i	−1.65726	−	0.232376i
$520$		0			0
$521$	−2437.03	+	4221.05i	−0.204929	+	0.354948i	−0.950110	−	0.311915i	$-0.899030\pi$
$521$	−2437.03	+	4221.05i	−0.204929	+	0.354948i	0.745181	+	0.666862i	$0.232363\pi$
$522$		0			0
$523$	4980.61	+	8626.67i	0.416419	+	0.721258i	0.995576	−	0.0939573i	$-0.0299517\pi$
$523$	4980.61	+	8626.67i	0.416419	+	0.721258i	−0.579158	+	0.815216i	$0.696618\pi$
$524$		0			0
$525$	982.408	+	1090.38i	0.0816682	+	0.0906439i
$526$		0			0
$527$	−906.828	−	5142.88i	−0.0749565	−	0.425099i
$528$		0			0
$529$	12057.1	+	10117.1i	0.990970	+	0.831523i
$530$		0			0
$531$	4920.46	+	19681.7i	0.402128	+	1.60850i
$532$		0			0
$533$	−5061.94	+	1842.40i	−0.411364	+	0.149724i
$534$		0			0
$535$	−15711.3	+	13183.4i	−1.26964	+	1.06536i
$536$		0			0
$537$	1445.92	+	768.223i	0.116194	+	0.0617342i
$538$		0			0
$539$	243.277			0.0194410
$540$		0			0
$541$	−20408.9			−1.62190			−0.810948	−	0.585118i	$-0.801048\pi$
$541$	−20408.9			−1.62190			−0.810948	+	0.585118i	$0.801048\pi$
$542$		0			0
$543$	−703.044	−	19951.1i	−0.0555626	−	1.57677i
$544$		0			0
$545$	−406.674	+	341.240i	−0.0319633	+	0.0268204i
$546$		0			0
$547$	−15260.6	+	5554.42i	−1.19287	+	0.434168i	−0.860729	−	0.509064i	$-0.829992\pi$
$547$	−15260.6	+	5554.42i	−1.19287	+	0.434168i	−0.332137	+	0.943231i	$0.607770\pi$
$548$		0			0
$549$	−104.270	+	998.153i	−0.00810590	+	0.0775958i
$550$		0			0
$551$	28268.6	+	23720.1i	2.18563	+	1.83396i
$552$		0			0
$553$	1597.71	+	9061.04i	0.122860	+	0.696771i
$554$		0			0
$555$	−793.017	+	168.824i	−0.0606517	+	0.0129120i
$556$		0			0
$557$	4689.68	+	8122.76i	0.356747	+	0.617904i	0.987415	−	0.158149i	$-0.0505525\pi$
$557$	4689.68	+	8122.76i	0.356747	+	0.617904i	−0.630668	+	0.776052i	$0.717219\pi$
$558$		0			0
$559$	−7043.07	+	12199.0i	−0.532898	+	0.923007i
$560$		0			0
$561$	120.479	+	297.925i	0.00906711	+	0.0224214i
$562$		0			0
$563$	10092.1	+	3673.21i	0.755471	+	0.274969i	0.690906	−	0.722945i	$-0.257212\pi$
$563$	10092.1	+	3673.21i	0.755471	+	0.274969i	0.0645648	+	0.997914i	$0.479434\pi$
$564$		0			0
$565$	2985.89	−	16933.9i	0.222332	−	1.26091i
$566$		0			0
$567$	16113.1	+	10040.2i	1.19345	+	0.743646i
$568$		0			0
$569$	13.7609	−	78.0417i	0.00101386	−	0.00574987i	−0.984297	−	0.176522i	$-0.943515\pi$
$569$	13.7609	−	78.0417i	0.00101386	−	0.00574987i	0.985310	+	0.170773i	$0.0546263\pi$
$570$		0			0
$571$	−17084.1	−	6218.11i	−1.25210	−	0.455726i	−0.370988	−	0.928638i	$-0.620981\pi$
$571$	−17084.1	−	6218.11i	−1.25210	−	0.455726i	−0.881110	+	0.472911i	$0.843203\pi$
$572$		0			0
$573$	4893.50	+	12100.8i	0.356769	+	0.882230i
$574$		0			0
$575$	905.904	−	1569.07i	0.0657023	−	0.113800i
$576$		0			0
$577$	−2781.75	−	4818.13i	−0.200703	−	0.347628i	0.748052	−	0.663640i	$-0.230989\pi$
$577$	−2781.75	−	4818.13i	−0.200703	−	0.347628i	−0.948755	+	0.316012i	$0.897656\pi$
$578$		0			0
$579$	−19408.8	+	4131.90i	−1.39310	+	0.296573i
$580$		0			0
$581$	382.335	+	2168.33i	0.0273011	+	0.154832i
$582$		0			0
$583$	−29.2896	−	24.5769i	−0.00208070	−	0.00174592i
$584$		0			0
$585$	−18624.5	+	8306.30i	−1.31628	+	0.587048i
$586$		0			0
$587$	−2307.80	+	839.972i	−0.162271	+	0.0590619i	−0.421879	−	0.906652i	$-0.638629\pi$
$587$	−2307.80	+	839.972i	−0.162271	+	0.0590619i	0.259607	+	0.965714i	$0.416407\pi$
$588$		0			0
$589$	7109.08	−	5965.22i	0.497325	−	0.417305i
$590$		0			0
$591$	−665.815	−	18894.7i	−0.0463418	−	1.31510i
$592$		0			0
$593$	−17323.9			−1.19967			−0.599837	−	0.800122i	$-0.704768\pi$
$593$	−17323.9			−1.19967			−0.599837	+	0.800122i	$0.704768\pi$
$594$		0			0
$595$	23712.8			1.63383
$596$		0			0
$597$	16023.2	+	8513.16i	1.09847	+	0.583619i
$598$		0			0
$599$	1054.57	−	884.892i	0.0719344	−	0.0603601i	−0.606112	−	0.795380i	$-0.707271\pi$
$599$	1054.57	−	884.892i	0.0719344	−	0.0603601i	0.678046	+	0.735020i	$0.262827\pi$
$600$		0			0
$601$	16841.2	−	6129.69i	1.14304	−	0.416032i	0.300029	−	0.953930i	$-0.403004\pi$
$601$	16841.2	−	6129.69i	1.14304	−	0.416032i	0.843010	+	0.537898i	$0.180781\pi$
$602$		0			0
$603$	−13296.4	−	3803.55i	−0.897961	−	0.256870i
$604$		0			0
$605$	−10889.5	−	9137.34i	−0.731768	−	0.614026i
$606$		0			0
$607$	−4434.77	−	25150.8i	−0.296543	−	1.68178i	−0.660862	−	0.750507i	$-0.729809\pi$
$607$	−4434.77	−	25150.8i	−0.296543	−	1.68178i	0.364319	−	0.931274i	$-0.381302\pi$
$608$		0			0
$609$	22071.3	+	24497.0i	1.46859	+	1.63000i
$610$		0			0
$611$	−9488.44	−	16434.5i	−0.628251	−	1.08816i
$612$		0			0
$613$	−6521.06	+	11294.8i	−0.429662	+	0.744197i	−0.996843	−	0.0793963i	$-0.974701\pi$
$613$	−6521.06	+	11294.8i	−0.429662	+	0.744197i	0.567181	+	0.823593i	$0.308034\pi$
$614$		0			0
$615$	4189.53	+	587.445i	0.274696	+	0.0385172i
$616$		0			0
$617$	−21565.7	−	7849.26i	−1.40713	−	0.512154i	−0.476845	−	0.878987i	$-0.658220\pi$
$617$	−21565.7	−	7849.26i	−1.40713	−	0.512154i	−0.930287	+	0.366833i	$0.880442\pi$
$618$		0			0
$619$	1776.31	−	10073.9i	0.115341	−	0.654129i	−0.871240	−	0.490857i	$-0.836684\pi$
$619$	1776.31	−	10073.9i	0.115341	−	0.654129i	0.986581	−	0.163273i	$-0.0522050\pi$
$620$		0			0
$621$	5648.23	−	22746.0i	0.364985	−	1.46983i
$622$		0			0
$623$	3176.49	−	18014.7i	0.204275	−	1.15850i
$624$		0			0
$625$	13298.2	+	4840.15i	0.851085	+	0.309770i
$626$		0			0
$627$	−351.453	+	450.133i	−0.0223854	+	0.0286708i
$628$		0			0
$629$	622.299	−	1077.85i	0.0394478	−	0.0683257i
$630$		0			0
$631$	−3681.07	−	6375.79i	−0.232236	−	0.402245i	0.726230	−	0.687452i	$-0.241271\pi$
$631$	−3681.07	−	6375.79i	−0.232236	−	0.402245i	−0.958466	+	0.285207i	$0.907937\pi$
$632$		0			0
$633$	−7595.78	+	23402.7i	−0.476944	+	1.46947i
$634$		0			0
$635$	−1522.89	−	8636.76i	−0.0951720	−	0.539747i
$636$		0			0
$637$	−18153.3	−	15232.5i	−1.12914	−	0.947460i
$638$		0			0
$639$	16059.7	−	1133.24i	0.994225	−	0.0701567i
$640$		0			0
$641$	−27630.6	+	10056.7i	−1.70256	+	0.619683i	−0.996114	−	0.0880788i	$-0.971927\pi$
$641$	−27630.6	+	10056.7i	−1.70256	+	0.619683i	−0.706451	+	0.707762i	$0.749705\pi$
$642$		0			0
$643$	15587.5	−	13079.5i	0.956004	−	0.802182i	−0.0242947	−	0.999705i	$-0.507734\pi$
$643$	15587.5	−	13079.5i	0.956004	−	0.802182i	0.980298	+	0.197523i	$0.0632896\pi$
$644$		0			0
$645$	9379.67	−	5865.21i	0.572596	−	0.358050i
$646$		0			0
$647$	−6586.91			−0.400244			−0.200122	−	0.979771i	$-0.564134\pi$
$647$	−6586.91			−0.400244			−0.200122	+	0.979771i	$0.564134\pi$
$648$		0			0
$649$	−545.291			−0.0329808
$650$		0			0
$651$	7030.85	−	4396.46i	0.423288	−	0.264687i
$652$		0			0
$653$	−15535.4	+	13035.7i	−0.931004	+	0.781205i	−0.975997	−	0.217782i	$-0.930118\pi$
$653$	−15535.4	+	13035.7i	−0.931004	+	0.781205i	0.0449934	+	0.998987i	$0.485673\pi$
$654$		0			0
$655$	−6263.56	+	2279.75i	−0.373645	+	0.135996i
$656$		0			0
$657$	−5673.14	−	8399.28i	−0.336880	−	0.498763i
$658$		0			0
$659$	−8105.40	−	6801.23i	−0.479122	−	0.402031i	0.370987	−	0.928638i	$-0.379019\pi$
$659$	−8105.40	−	6801.23i	−0.479122	−	0.402031i	−0.850109	+	0.526607i	$0.823464\pi$
$660$		0			0
$661$	−2105.13	−	11938.8i	−0.123873	−	0.702518i	−0.981971	−	0.189032i	$-0.939465\pi$
$661$	−2105.13	−	11938.8i	−0.123873	−	0.702518i	0.858098	−	0.513486i	$-0.171646\pi$
$662$		0			0
$663$	9663.99	−	29774.8i	0.566091	−	1.74413i
$664$		0			0
$665$	21069.6	+	36493.7i	1.22864	+	2.12807i
$666$		0			0
$667$	20352.5	−	35251.6i	1.18149	−	2.04640i
$668$		0			0
$669$	−2010.19	+	2574.61i	−0.116171	+	0.148790i
$670$		0			0
$671$	−25.3478	−	9.22586i	−0.00145833	−	0.000530790i
$672$		0			0
$673$	1997.23	−	11326.8i	0.114395	−	0.648764i	−0.872654	−	0.488340i	$-0.837603\pi$
$673$	1997.23	−	11326.8i	0.114395	−	0.648764i	0.987048	−	0.160424i	$-0.0512862\pi$
$674$		0			0
$675$	−1518.01	−	104.698i	−0.0865604	−	0.00597014i
$676$		0			0
$677$	1826.50	−	10358.6i	0.103690	−	0.588055i	−0.888045	−	0.459756i	$-0.847937\pi$
$677$	1826.50	−	10358.6i	0.103690	−	0.588055i	0.991735	−	0.128300i	$-0.0409519\pi$
$678$		0			0
$679$	−331.345	−	120.600i	−0.0187273	−	0.00681619i
$680$		0			0
$681$	6679.70	+	936.610i	0.375869	+	0.0527033i
$682$		0			0
$683$	2489.30	−	4311.59i	0.139459	−	0.241550i	−0.787833	−	0.615889i	$-0.788797\pi$
$683$	2489.30	−	4311.59i	0.139459	−	0.241550i	0.927292	+	0.374339i	$0.122130\pi$
$684$		0			0
$685$	−6499.49	−	11257.4i	−0.362529	−	0.627919i
$686$		0			0
$687$	−2037.40	−	2261.31i	−0.113146	−	0.125582i
$688$		0			0
$689$	646.741	+	3667.85i	0.0357604	+	0.202807i
$690$		0			0
$691$	5474.46	+	4593.62i	0.301387	+	0.252893i	0.780921	−	0.624630i	$-0.214750\pi$
$691$	5474.46	+	4593.62i	0.301387	+	0.252893i	−0.479534	+	0.877523i	$0.659194\pi$
$692$		0			0
$693$	−366.846	+	354.709i	−0.0201087	+	0.0194434i
$694$		0			0
$695$	−287.436	+	104.618i	−0.0156879	+	0.00570993i
$696$		0			0
$697$	−4974.72	+	4174.29i	−0.270346	+	0.226847i
$698$		0			0
$699$	3882.11	+	2062.57i	0.210064	+	0.111608i
$700$		0			0
$701$	−2134.56			−0.115009			−0.0575044	−	0.998345i	$-0.518314\pi$
$701$	−2134.56			−0.115009			−0.0575044	+	0.998345i	$0.518314\pi$
$702$		0			0
$703$	2211.74			0.118659
$704$		0			0
$705$	524.845	+	14894.2i	0.0280380	+	0.795670i
$706$		0			0
$707$	−36380.3	+	30526.7i	−1.93525	+	1.62387i
$708$		0			0
$709$	−10970.4	+	3992.90i	−0.581103	+	0.211504i	−0.615812	−	0.787893i	$-0.711172\pi$
$709$	−10970.4	+	3992.90i	−0.581103	+	0.211504i	0.0347089	+	0.999397i	$0.488950\pi$
$710$		0			0
$711$	−7720.78	−	5601.99i	−0.407246	−	0.295487i
$712$		0			0
$713$	−7841.72	−	6579.98i	−0.411886	−	0.345613i
$714$		0			0
$715$	−95.1805	−	539.795i	−0.00497839	−	0.0282338i
$716$		0			0
$717$	−21160.9	+	4504.91i	−1.10219	+	0.234643i
$718$		0			0
$719$	−3341.98	−	5788.48i	−0.173345	−	0.300242i	0.766242	−	0.642552i	$-0.222124\pi$
$719$	−3341.98	−	5788.48i	−0.173345	−	0.300242i	−0.939587	+	0.342310i	$0.888791\pi$
$720$		0			0
$721$	−9307.92	+	16121.8i	−0.480784	+	0.832742i
$722$		0			0
$723$	11019.9	+	27250.3i	0.566851	+	1.40173i
$724$		0			0
$725$	−2483.37	−	903.872i	−0.127214	−	0.0463020i
$726$		0			0
$727$	−4870.25	+	27620.6i	−0.248456	+	1.40907i	0.563871	+	0.825863i	$0.309312\pi$
$727$	−4870.25	+	27620.6i	−0.248456	+	1.40907i	−0.812327	+	0.583202i	$0.801799\pi$
$728$		0			0
$729$	−19245.1	+	4128.95i	−0.977750	+	0.209772i
$730$		0			0
$731$	−2948.80	+	16723.5i	−0.149200	+	0.846157i
$732$		0			0
$733$	27055.3	+	9847.32i	1.36331	+	0.496206i	0.917078	−	0.398709i	$-0.130542\pi$
$733$	27055.3	+	9847.32i	1.36331	+	0.496206i	0.446237	+	0.894915i	$0.352764\pi$
$734$		0			0
$735$	6977.18	+	17253.4i	0.350146	+	0.865851i
$736$		0			0
$737$	185.859	−	321.918i	0.00928930	−	0.0160895i
$738$		0			0
$739$	2190.75	+	3794.49i	0.109050	+	0.188880i	0.915386	−	0.402578i	$-0.131886\pi$
$739$	2190.75	+	3794.49i	0.109050	+	0.188880i	−0.806336	+	0.591458i	$0.798552\pi$
$740$		0			0
$741$	54409.9	−	11583.2i	2.69743	−	0.574251i
$742$		0			0
$743$	−5769.87	−	32722.6i	−0.284894	−	1.61571i	−0.705664	−	0.708546i	$-0.749351\pi$
$743$	−5769.87	−	32722.6i	−0.284894	−	1.61571i	0.420770	−	0.907167i	$-0.361760\pi$
$744$		0			0
$745$	−7295.38	−	6121.55i	−0.358768	−	0.301042i
$746$		0			0
$747$	−1847.60	−	1340.57i	−0.0904955	−	0.0656612i
$748$		0			0
$749$	46977.0	−	17098.2i	2.29172	−	0.834119i
$750$		0			0
$751$	−4666.53	+	3915.68i	−0.226743	+	0.190260i	−0.749081	−	0.662479i	$-0.769504\pi$
$751$	−4666.53	+	3915.68i	−0.226743	+	0.190260i	0.522338	+	0.852739i	$0.325060\pi$
$752$		0			0
$753$	−67.9200	−	1927.45i	−0.00328704	−	0.0932804i
$754$		0			0
$755$	−3100.39			−0.149450
$756$		0			0
$757$	−234.359			−0.0112522			−0.00562610	−	0.999984i	$-0.501791\pi$
$757$	−234.359			−0.0112522			−0.00562610	+	0.999984i	$0.501791\pi$
$758$		0			0
$759$	556.298	+	295.563i	0.0266039	+	0.0141347i
$760$		0			0
$761$	5554.52	−	4660.79i	0.264587	−	0.222015i	−0.500836	−	0.865542i	$-0.666974\pi$
$761$	5554.52	−	4660.79i	0.264587	−	0.222015i	0.765423	+	0.643527i	$0.222530\pi$
$762$		0			0
$763$	1215.96	−	442.573i	0.0576942	−	0.0209990i
$764$		0			0
$765$	−17673.7	+	17089.0i	−0.835287	+	0.807651i
$766$		0			0
$767$	40689.7	+	34142.7i	1.91554	+	1.60733i
$768$		0			0
$769$	−2924.93	−	16588.1i	−0.137160	−	0.777871i	−0.973331	−	0.229403i	$-0.926323\pi$
$769$	−2924.93	−	16588.1i	−0.137160	−	0.777871i	0.836172	−	0.548467i	$-0.184789\pi$
$770$		0			0
$771$	10487.2	+	11639.8i	0.489868	+	0.543707i
$772$		0			0
$773$	−95.1463	−	164.798i	−0.00442713	−	0.00766802i	0.863803	−	0.503829i	$-0.168076\pi$
$773$	−95.1463	−	164.798i	−0.00442713	−	0.00766802i	−0.868230	+	0.496161i	$0.834743\pi$
$774$		0			0
$775$	−332.303	+	575.566i	−0.0154022	+	0.0266773i
$776$		0			0
$777$	1957.13	+	274.424i	0.0903625	+	0.0126704i
$778$		0			0
$779$	−10844.4	−	3947.04i	−0.498769	−	0.181537i
$780$		0			0
$781$	−75.1426	+	426.155i	−0.00344278	+	0.0195250i
$782$		0			0
$783$	−34104.4	−	2352.21i	−1.55657	−	0.107358i
$784$		0			0
$785$	−2940.16	+	16674.5i	−0.133680	+	0.758136i
$786$		0			0
$787$	−25173.0	−	9162.21i	−1.14018	−	0.414990i	−0.298200	−	0.954503i	$-0.596386\pi$
$787$	−25173.0	−	9162.21i	−1.14018	−	0.414990i	−0.841977	+	0.539513i	$0.818608\pi$
$788$		0			0
$789$	13515.4	−	17310.3i	0.609838	−	0.781067i
$790$		0			0
$791$	−20956.3	+	36297.4i	−0.941999	+	1.63159i
$792$		0			0
$793$	1313.79	+	2275.55i	0.0588324	+	0.101901i
$794$		0			0
$795$	902.984	−	2782.10i	0.0402837	−	0.124114i
$796$		0			0
$797$	−4724.22	−	26792.4i	−0.209963	−	1.19076i	−0.889437	−	0.457058i	$-0.848903\pi$
$797$	−4724.22	−	26792.4i	−0.209963	−	1.19076i	0.679474	−	0.733700i	$-0.262208\pi$
$798$		0			0
$799$	−17525.2	−	14705.4i	−0.775965	−	0.651112i
$800$		0			0
$801$	10615.1	+	15716.0i	0.468247	+	0.693255i
$802$		0			0
$803$	256.001	−	93.1766i	0.0112504	−	0.00409481i
$804$		0			0
$805$	35607.3	−	29878.1i	1.55900	−	1.30815i
$806$		0			0
$807$	8585.63	−	5368.68i	0.374509	−	0.234184i
$808$		0			0
$809$	623.745			0.0271072			0.0135536	−	0.999908i	$-0.495686\pi$
$809$	623.745			0.0271072			0.0135536	+	0.999908i	$0.495686\pi$
$810$		0			0
$811$	43605.5			1.88803			0.944017	−	0.329896i	$-0.107014\pi$
$811$	43605.5			1.88803			0.944017	+	0.329896i	$0.107014\pi$
$812$		0			0
$813$	−27471.5	+	17178.2i	−1.18508	+	0.741041i
$814$		0			0
$815$	28398.2	−	23828.9i	1.22055	−	1.02416i
$816$		0			0
$817$	−28357.4	+	10321.2i	−1.21432	+	0.441976i
$818$		0			0
$819$	49583.6	−	3498.83i	2.11550	−	0.149278i
$820$		0			0
$821$	3888.07	+	3262.48i	0.165279	+	0.138686i	0.721676	−	0.692231i	$-0.243372\pi$
$821$	3888.07	+	3262.48i	0.165279	+	0.138686i	−0.556397	+	0.830917i	$0.687816\pi$
$822$		0			0
$823$	2079.11	+	11791.2i	0.0880598	+	0.499412i	0.996654	+	0.0817325i	$0.0260453\pi$
$823$	2079.11	+	11791.2i	0.0880598	+	0.499412i	−0.908595	+	0.417679i	$0.862844\pi$
$824$		0			0
$825$	12.6260	−	38.9007i	0.000532824	−	0.00164164i
$826$		0			0
$827$	−5343.27	−	9254.81i	−0.224672	−	0.389143i	0.731549	−	0.681789i	$-0.238798\pi$
$827$	−5343.27	−	9254.81i	−0.224672	−	0.389143i	−0.956221	+	0.292646i	$0.905464\pi$
$828$		0			0
$829$	−162.955	+	282.246i	−0.00682710	+	0.0118249i	−0.869419	−	0.494076i	$-0.835506\pi$
$829$	−162.955	+	282.246i	−0.00682710	+	0.0118249i	0.862592	+	0.505901i	$0.168840\pi$
$830$		0			0
$831$	−9039.59	+	11577.7i	−0.377352	+	0.483305i
$832$		0			0
$833$	−26845.5	−	9770.97i	−1.11662	−	0.406415i
$834$		0			0
$835$	−4525.60	+	25666.0i	−0.187563	+	1.06372i
$836$		0			0
$837$	−2071.88	+	8343.68i	−0.0855612	+	0.344564i
$838$		0			0
$839$	−5536.19	+	31397.3i	−0.227808	+	1.29196i	0.629437	+	0.777051i	$0.283286\pi$
$839$	−5536.19	+	31397.3i	−0.227808	+	1.29196i	−0.857245	+	0.514909i	$0.827826\pi$
$840$		0			0
$841$	−32874.5	−	11965.3i	−1.34792	−	0.490604i
$842$		0			0
$843$	10609.1	+	1487.59i	0.433449	+	0.0607772i
$844$		0			0
$845$	−14959.5	+	25910.6i	−0.609019	+	1.05485i
$846$		0			0
$847$	17324.6	+	30007.1i	0.702810	+	1.21730i
$848$		0			0
$849$	23613.9	+	26209.2i	0.954568	+	1.05948i
$850$		0			0
$851$	−423.645	−	2402.61i	−0.0170651	−	0.0967807i
$852$		0			0
$853$	15404.7	+	12926.1i	0.618344	+	0.518852i	0.897282	−	0.441457i	$-0.145538\pi$
$853$	15404.7	+	12926.1i	0.618344	+	0.518852i	−0.278939	+	0.960309i	$0.589983\pi$
$854$		0			0
$855$	−42003.4	−	12015.5i	−1.68010	−	0.480608i
$856$		0			0
$857$	29990.8	−	10915.8i	1.19541	−	0.435094i	0.333790	−	0.942647i	$-0.391672\pi$
$857$	29990.8	−	10915.8i	1.19541	−	0.435094i	0.861620	+	0.507554i	$0.169450\pi$
$858$		0			0
$859$	−3000.06	+	2517.35i	−0.119163	+	0.0999893i	−0.700422	−	0.713729i	$-0.747005\pi$
$859$	−3000.06	+	2517.35i	−0.119163	+	0.0999893i	0.581259	+	0.813719i	$0.302560\pi$
$860$		0			0
$861$	−9106.30	−	4838.20i	−0.360443	−	0.191505i
$862$		0			0
$863$	10871.7			0.428826			0.214413	−	0.976743i	$-0.431216\pi$
$863$	10871.7			0.428826			0.214413	+	0.976743i	$0.431216\pi$
$864$		0			0
$865$	40684.9			1.59922
$866$		0			0
$867$	−429.977	−	12202.0i	−0.0168429	−	0.477971i
$868$		0			0
$869$	196.408	−	164.806i	0.00766706	−	0.00643343i
$870$		0			0
$871$	−34025.3	+	12384.2i	−1.32365	+	0.481770i
$872$		0			0
$873$	333.871	−	148.903i	0.0129437	−	0.00577273i
$874$		0			0
$875$	−28955.6	−	24296.6i	−1.11872	−	0.938716i
$876$		0			0
$877$	−583.987	−	3311.96i	−0.0224856	−	0.127522i	0.971499	−	0.237045i	$-0.0761788\pi$
$877$	−583.987	−	3311.96i	−0.0224856	−	0.127522i	−0.993984	+	0.109523i	$0.965068\pi$
$878$		0			0
$879$	−22354.0	+	4758.90i	−0.857772	+	0.182609i
$880$		0			0
$881$	3132.17	+	5425.07i	0.119779	+	0.207464i	0.919680	−	0.392669i	$-0.128448\pi$
$881$	3132.17	+	5425.07i	0.119779	+	0.207464i	−0.799901	+	0.600132i	$0.795115\pi$
$882$		0			0
$883$	−11471.9	+	19869.9i	−0.437214	+	0.757277i	−0.997473	−	0.0710403i	$-0.977368\pi$
$883$	−11471.9	+	19869.9i	−0.437214	+	0.757277i	0.560259	+	0.828317i	$0.310701\pi$
$884$		0			0
$885$	−15639.0	−	38672.4i	−0.594009	−	1.46888i
$886$		0			0
$887$	29550.4	+	10755.5i	1.11861	+	0.407140i	0.834144	−	0.551547i	$-0.185962\pi$
$887$	29550.4	+	10755.5i	1.11861	+	0.407140i	0.284464	+	0.958687i	$0.408185\pi$
$888$		0			0
$889$	−3712.02	+	21051.9i	−0.140042	+	0.794217i
$890$		0			0
$891$	17.7927	−	528.746i	0.000668999	−	0.0198806i
$892$		0			0
$893$	7059.65	−	40037.2i	0.264549	−	1.50033i
$894$		0			0
$895$	−3163.64	−	1151.47i	−0.118155	−	0.0430049i
$896$		0			0
$897$	−23004.7	−	56886.7i	−0.856305	−	2.11749i
$898$		0			0
$899$	−7465.70	+	12931.0i	−0.276969	+	0.479724i
$900$		0			0
$901$	2244.99	+	3888.43i	0.0830093	+	0.143776i
$902$		0			0
$903$	−26373.6	+	5614.62i	−0.971936	+	0.206913i
$904$		0			0
$905$	7128.06	+	40425.2i	0.261817	+	1.48484i
$906$		0			0
$907$	36556.9	+	30674.9i	1.33832	+	1.12298i	0.982055	+	0.188594i	$0.0603932\pi$
$907$	36556.9	+	30674.9i	1.33832	+	1.12298i	0.356261	+	0.934386i	$0.384051\pi$
$908$		0			0
$909$	5115.59	−	48970.3i	0.186659	−	1.78685i
$910$		0			0
$911$	36942.9	−	13446.1i	1.34355	−	0.489012i	0.432620	−	0.901576i	$-0.357589\pi$
$911$	36942.9	−	13446.1i	1.34355	−	0.489012i	0.910928	+	0.412564i	$0.135367\pi$
$912$		0			0
$913$	47.0008	−	39.4384i	0.00170373	−	0.00142960i
$914$		0			0
$915$	−72.6713	−	2062.28i	−0.00262562	−	0.0745104i
$916$		0			0
$917$	16247.1			0.585090
$918$		0			0
$919$	7346.12			0.263684			0.131842	−	0.991271i	$-0.457911\pi$
$919$	7346.12			0.263684			0.131842	+	0.991271i	$0.457911\pi$
$920$		0			0
$921$	−4491.85	−	2386.53i	−0.160707	−	0.0853843i
$922$		0			0
$923$	32290.2	−	27094.7i	1.15151	−	0.966234i
$924$		0			0
$925$	−148.842	+	54.1740i	−0.00529069	+	0.00192565i
$926$		0			0
$927$	−4680.99	−	18723.8i	−0.165851	−	0.663400i
$928$		0			0
$929$	−11537.2	−	9680.83i	−0.407451	−	0.341892i	0.415914	−	0.909404i	$-0.363462\pi$
$929$	−11537.2	−	9680.83i	−0.407451	−	0.341892i	−0.823365	+	0.567512i	$0.807906\pi$
$930$		0			0
$931$	−8815.77	−	49996.7i	−0.310339	−	1.76002i
$932$		0			0
$933$	−11444.7	−	12702.5i	−0.401587	−	0.445724i
$934$		0			0
$935$	−330.393	−	572.258i	−0.0115562	−	0.0200159i
$936$		0			0
$937$	−16943.9	+	29347.8i	−0.590752	+	1.02321i	0.403379	+	0.915033i	$0.367836\pi$
$937$	−16943.9	+	29347.8i	−0.590752	+	1.02321i	−0.994131	+	0.108180i	$0.965498\pi$
$938$		0			0
$939$	3974.45	+	557.287i	0.138127	+	0.0193678i
$940$		0			0
$941$	−48190.4	−	17539.9i	−1.66946	−	0.607634i	−0.677654	−	0.735381i	$-0.737003\pi$
$941$	−48190.4	−	17539.9i	−1.66946	−	0.607634i	−0.991807	+	0.127746i	$0.959226\pi$
$942$		0			0
$943$	−2210.49	+	12536.3i	−0.0763344	+	0.432914i
$944$		0			0
$945$	−35677.3	−	15843.9i	−1.22813	−	0.545400i
$946$		0			0
$947$	1862.24	−	10561.3i	0.0639015	−	0.362404i	−0.936043	−	0.351885i	$-0.885541\pi$
$947$	1862.24	−	10561.3i	0.0639015	−	0.362404i	0.999945	−	0.0105183i	$-0.00334815\pi$
$948$		0			0
$949$	−24936.9	−	9076.29i	−0.852989	−	0.310462i
$950$		0			0
$951$	−2237.68	+	2865.97i	−0.0763005	+	0.0977240i
$952$		0			0
$953$	17911.1	−	31022.9i	0.608810	−	1.05449i	−0.382626	−	0.923903i	$-0.624980\pi$
$953$	17911.1	−	31022.9i	0.608810	−	1.05449i	0.991437	−	0.130587i	$-0.0416863\pi$
$954$		0			0
$955$	−13419.5	−	23243.3i	−0.454708	−	0.787577i
$956$		0			0
$957$	283.662	−	873.964i	0.00958149	−	0.0295206i
$958$		0			0
$959$	5501.99	+	31203.3i	0.185264	+	1.05069i
$960$		0			0
$961$	−19944.7	−	16735.6i	−0.669489	−	0.561768i
$962$		0			0
$963$	−22691.0	+	46598.3i	−0.759300	+	1.55930i
$964$		0			0
$965$	38341.9	−	13955.3i	1.27904	−	0.465531i
$966$		0			0
$967$	−14813.8	+	12430.3i	−0.492637	+	0.413372i	−0.854970	−	0.518677i	$-0.826425\pi$
$967$	−14813.8	+	12430.3i	−0.492637	+	0.413372i	0.362333	+	0.932049i	$0.381980\pi$
$968$		0			0
$969$	56861.8	−	35556.3i	1.88510	−	1.17877i
$970$		0			0
$971$	−13651.7			−0.451190			−0.225595	−	0.974221i	$-0.572433\pi$
$971$	−13651.7			−0.451190			−0.225595	+	0.974221i	$0.572433\pi$
$972$		0			0
$973$	745.584			0.0245656
$974$		0			0
$975$	−3377.87	+	2112.21i	−0.110952	+	0.0693795i
$976$		0			0
$977$	35656.0	−	29918.9i	1.16759	−	0.979724i	0.167609	−	0.985854i	$-0.446395\pi$
$977$	35656.0	−	29918.9i	1.16759	−	0.979724i	0.999981	+	0.00612912i	$0.00195097\pi$
$978$		0			0
$979$	−479.006	+	174.344i	−0.0156375	+	0.00569158i
$980$		0			0
$981$	−587.336	+	1206.16i	−0.0191154	+	0.0392555i
$982$		0			0
$983$	7929.62	+	6653.74i	0.257290	+	0.215892i	0.762304	−	0.647219i	$-0.224068\pi$
$983$	7929.62	+	6653.74i	0.257290	+	0.215892i	−0.505014	+	0.863111i	$0.668513\pi$
$984$		0			0
$985$	6750.61	+	38284.6i	0.218368	+	1.23842i
$986$		0			0
$987$	11214.6	−	34552.4i	0.361667	−	1.11430i
$988$		0			0
$989$	16643.6	+	28827.6i	0.535123	+	0.926860i
$990$		0			0
$991$	−12434.4	+	21537.0i	−0.398579	+	0.690359i	−0.993551	−	0.113387i	$-0.963830\pi$
$991$	−12434.4	+	21537.0i	−0.398579	+	0.690359i	0.594972	+	0.803747i	$0.297163\pi$
$992$		0			0
$993$	−17081.0	+	21877.0i	−0.545871	+	0.699140i
$994$		0			0
$995$	−35058.3	−	12760.2i	−1.11701	−	0.406557i
$996$		0			0
$997$	6107.02	−	34634.6i	0.193993	−	1.10019i	−0.719852	−	0.694128i	$-0.755790\pi$
$997$	6107.02	−	34634.6i	0.193993	−	1.10019i	0.913845	−	0.406063i	$-0.133099\pi$
$998$		0			0
$999$	−1656.46	+	1205.90i	−0.0524607	+	0.0381912i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.25.2	yes	54
3.2	odd	2		324.4.i.a.73.2		54
27.13	even	9	inner	108.4.i.a.13.2	✓	54
27.14	odd	18		324.4.i.a.253.2		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.13.2	✓	54	27.13	even	9	inner
108.4.i.a.25.2	yes	54	1.1	even	1	trivial
324.4.i.a.73.2		54	3.2	odd	2
324.4.i.a.253.2		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)

\(f(q)\)	\(=\)	\(q+(-4.40571 + 2.75494i) q^{3} +(8.18465 - 6.86774i) q^{5} +(-24.4722 + 8.90714i) q^{7} +(11.8206 - 24.2749i) q^{9} +O(q^{10})\)	\(q+(-4.40571 + 2.75494i) q^{3} +(8.18465 - 6.86774i) q^{5} +(-24.4722 + 8.90714i) q^{7} +(11.8206 - 24.2749i) q^{9} +(0.555929 + 0.466480i) q^{11} +(-12.2754 - 69.6174i) q^{13} +(-17.1390 + 52.8055i) q^{15} +(-42.6108 - 73.8041i) q^{17} +(75.7224 - 131.155i) q^{19} +(83.2788 - 106.662i) q^{21} +(-156.978 - 57.1353i) q^{23} +(-1.88334 + 10.6810i) q^{25} +(14.7977 + 139.514i) q^{27} +(-42.3122 + 239.964i) q^{29} +(57.5825 + 20.9583i) q^{31} +(-3.73439 - 0.523626i) q^{33} +(-139.124 + 240.970i) q^{35} +(7.30212 + 12.6477i) q^{37} +(245.874 + 272.896i) q^{39} +(-13.2323 - 75.0441i) q^{41} +(-152.644 - 128.083i) q^{43} +(-69.9662 - 279.863i) q^{45} +(252.257 - 91.8142i) q^{47} +(256.797 - 215.478i) q^{49} +(391.057 + 207.770i) q^{51} -52.6859 q^{53} +7.75374 q^{55} +(27.7129 + 786.442i) q^{57} +(-575.596 + 482.982i) q^{59} +(-34.9282 + 12.7128i) q^{61} +(-73.0561 + 699.349i) q^{63} +(-578.584 - 485.490i) q^{65} +(-88.9446 - 504.430i) q^{67} +(849.003 - 180.743i) q^{69} +(298.140 + 516.394i) q^{71} +(187.698 - 325.103i) q^{73} +(-21.1280 - 52.2458i) q^{75} +(-17.7598 - 6.46403i) q^{77} +(61.3493 - 347.929i) q^{79} +(-449.546 - 573.890i) q^{81} +(14.6810 - 83.2603i) q^{83} +(-855.622 - 311.421i) q^{85} +(-474.672 - 1173.78i) q^{87} +(-351.204 + 608.304i) q^{89} +(920.499 + 1594.35i) q^{91} +(-311.431 + 66.2999i) q^{93} +(-280.977 - 1593.50i) q^{95} +(10.3720 + 8.70312i) q^{97} +(17.8952 - 7.98106i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 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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