LMFDB - Embedded newform 162.4.c.j.109.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,4,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.55"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,4,0,-8,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.55830942093$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	162.109
Dual form			162.4.c.j.55.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(5.59808 - 9.69615i) q^{5} +(-9.19615 - 15.9282i) q^{7} -8.00000 q^{8} +22.3923 q^{10} +(-11.7846 - 20.4115i) q^{11} +(33.8731 - 58.6699i) q^{13} +(18.3923 - 31.8564i) q^{14} +(-8.00000 - 13.8564i) q^{16} -117.158 q^{17} +110.315 q^{19} +(22.3923 + 38.7846i) q^{20} +(23.5692 - 40.8231i) q^{22} +(34.6077 - 59.9423i) q^{23} +(-0.176915 - 0.306425i) q^{25} +135.492 q^{26} +73.5692 q^{28} +(99.1865 + 171.796i) q^{29} +(155.531 - 269.387i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-117.158 - 202.923i) q^{34} -205.923 q^{35} -206.608 q^{37} +(110.315 + 191.072i) q^{38} +(-44.7846 + 77.5692i) q^{40} +(-66.3154 + 114.862i) q^{41} +(167.588 + 290.272i) q^{43} +94.2769 q^{44} +138.431 q^{46} +(-189.531 - 328.277i) q^{47} +(2.36156 - 4.09034i) q^{49} +(0.353829 - 0.612850i) q^{50} +(135.492 + 234.679i) q^{52} -190.908 q^{53} -263.885 q^{55} +(73.5692 + 127.426i) q^{56} +(-198.373 + 343.592i) q^{58} +(-168.862 + 292.477i) q^{59} +(-138.735 - 240.295i) q^{61} +622.123 q^{62} +64.0000 q^{64} +(-379.248 - 656.877i) q^{65} +(-332.535 + 575.967i) q^{67} +(234.315 - 405.846i) q^{68} +(-205.923 - 356.669i) q^{70} +528.431 q^{71} -73.8306 q^{73} +(-206.608 - 357.855i) q^{74} +(-220.631 + 382.144i) q^{76} +(-216.746 + 375.415i) q^{77} +(239.904 + 415.526i) q^{79} -179.138 q^{80} -265.261 q^{82} +(-89.8846 - 155.685i) q^{83} +(-655.858 + 1135.98i) q^{85} +(-335.177 + 580.543i) q^{86} +(94.2769 + 163.292i) q^{88} -846.458 q^{89} -1246.01 q^{91} +(138.431 + 239.769i) q^{92} +(379.061 - 656.554i) q^{94} +(617.554 - 1069.63i) q^{95} +(336.492 + 582.822i) q^{97} +9.44624 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 8 q^{4} + 12 q^{5} - 16 q^{7} - 32 q^{8} + 48 q^{10} + 36 q^{11} - 10 q^{13} + 32 q^{14} - 32 q^{16} - 240 q^{17} - 16 q^{19} + 48 q^{20} - 72 q^{22} + 180 q^{23} + 124 q^{25} - 40 q^{26}+ \cdots + 1368 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 8 * q^4 + 12 * q^5 - 16 * q^7 - 32 * q^8 + 48 * q^10 + 36 * q^11 - 10 * q^13 + 32 * q^14 - 32 * q^16 - 240 * q^17 - 16 * q^19 + 48 * q^20 - 72 * q^22 + 180 * q^23 + 124 * q^25 - 40 * q^26 + 128 * q^28 + 324 * q^29 + 248 * q^31 + 64 * q^32 - 240 * q^34 - 408 * q^35 - 868 * q^37 - 16 * q^38 - 96 * q^40 + 192 * q^41 + 608 * q^43 - 288 * q^44 + 720 * q^46 - 384 * q^47 + 342 * q^49 - 248 * q^50 - 40 * q^52 + 816 * q^53 - 432 * q^55 + 128 * q^56 - 648 * q^58 - 1008 * q^59 - 742 * q^61 + 992 * q^62 + 256 * q^64 - 696 * q^65 + 104 * q^67 + 480 * q^68 - 408 * q^70 + 2280 * q^71 + 1700 * q^73 - 868 * q^74 + 32 * q^76 - 576 * q^77 + 440 * q^79 - 384 * q^80 + 768 * q^82 + 264 * q^83 - 1314 * q^85 - 1216 * q^86 - 288 * q^88 - 1536 * q^89 - 2864 * q^91 + 720 * q^92 + 768 * q^94 + 1140 * q^95 + 764 * q^97 + 1368 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	5.59808	−	9.69615i	0.500707	−	0.867250i	−0.499293	−	0.866433i	$-0.666407\pi$
$5$	5.59808	−	9.69615i	0.500707	−	0.867250i	1.00000		0.000816748i	$-0.000259979\pi$
$6$		0			0
$7$	−9.19615	−	15.9282i	−0.496546	−	0.860042i	0.503447	−	0.864026i	$-0.332065\pi$
$7$	−9.19615	−	15.9282i	−0.496546	−	0.860042i	−0.999992	+	0.00398426i	$0.998732\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	22.3923			0.708107
$11$	−11.7846	−	20.4115i	−0.323018	−	0.559483i	0.658092	−	0.752938i	$-0.271364\pi$
$11$	−11.7846	−	20.4115i	−0.323018	−	0.559483i	−0.981109	+	0.193455i	$0.938031\pi$
$12$		0			0
$13$	33.8731	−	58.6699i	0.722669	−	1.25170i	−0.237257	−	0.971447i	$-0.576248\pi$
$13$	33.8731	−	58.6699i	0.722669	−	1.25170i	0.959926	−	0.280253i	$-0.0904183\pi$
$14$	18.3923	−	31.8564i	0.351111	−	0.608142i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−117.158			−1.67147			−0.835733	−	0.549137i	$-0.814957\pi$
$17$	−117.158			−1.67147			−0.835733	+	0.549137i	$0.814957\pi$
$18$		0			0
$19$	110.315			1.33200			0.666002	−	0.745950i	$-0.268004\pi$
$19$	110.315			1.33200			0.666002	+	0.745950i	$0.268004\pi$
$20$	22.3923	+	38.7846i	0.250354	+	0.433625i
$21$		0			0
$22$	23.5692	−	40.8231i	0.228408	−	0.395614i
$23$	34.6077	−	59.9423i	0.313748	−	0.543427i	−0.665423	−	0.746467i	$-0.731749\pi$
$23$	34.6077	−	59.9423i	0.313748	−	0.543427i	0.979171	+	0.203039i	$0.0650820\pi$
$24$		0			0
$25$	−0.176915	−	0.306425i	−0.00141532	−	0.00245140i
$26$	135.492			1.02201
$27$		0			0
$28$	73.5692			0.496546
$29$	99.1865	+	171.796i	0.635120	+	1.10006i	0.986490	+	0.163823i	$0.0523827\pi$
$29$	99.1865	+	171.796i	0.635120	+	1.10006i	−0.351370	+	0.936237i	$0.614284\pi$
$30$		0			0
$31$	155.531	−	269.387i	0.901101	−	1.56075i	0.0750350	−	0.997181i	$-0.476093\pi$
$31$	155.531	−	269.387i	0.901101	−	1.56075i	0.826066	−	0.563573i	$-0.190574\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	−117.158	−	202.923i	−0.590952	−	1.02356i
$35$	−205.923			−0.994496
$36$		0			0
$37$	−206.608			−0.918003			−0.459001	−	0.888436i	$-0.651793\pi$
$37$	−206.608			−0.918003			−0.459001	+	0.888436i	$0.651793\pi$
$38$	110.315	+	191.072i	0.470935	+	0.815683i
$39$		0			0
$40$	−44.7846	+	77.5692i	−0.177027	+	0.306619i
$41$	−66.3154	+	114.862i	−0.252603	+	0.437521i	−0.964242	−	0.265025i	$-0.914620\pi$
$41$	−66.3154	+	114.862i	−0.252603	+	0.437521i	0.711639	+	0.702546i	$0.247953\pi$
$42$		0			0
$43$	167.588	+	290.272i	0.594349	+	1.02944i	0.993638	+	0.112618i	$0.0359235\pi$
$43$	167.588	+	290.272i	0.594349	+	1.02944i	−0.399290	+	0.916825i	$0.630743\pi$
$44$	94.2769			0.323018
$45$		0			0
$46$	138.431			0.443707
$47$	−189.531	−	328.277i	−0.588211	−	1.01881i	−0.994467	−	0.105051i	$-0.966499\pi$
$47$	−189.531	−	328.277i	−0.588211	−	1.01881i	0.406256	−	0.913759i	$-0.366834\pi$
$48$		0			0
$49$	2.36156	−	4.09034i	0.00688502	−	0.0119252i
$50$	0.353829	−	0.612850i	0.00100078	−	0.00173340i
$51$		0			0
$52$	135.492	+	234.679i	0.361335	+	0.625850i
$53$	−190.908			−0.494777			−0.247388	−	0.968916i	$-0.579572\pi$
$53$	−190.908			−0.494777			−0.247388	+	0.968916i	$0.579572\pi$
$54$		0			0
$55$	−263.885			−0.646949
$56$	73.5692	+	127.426i	0.175555	+	0.304071i
$57$		0			0
$58$	−198.373	+	343.592i	−0.449098	+	0.777860i
$59$	−168.862	+	292.477i	−0.372609	+	0.645377i	−0.989966	−	0.141306i	$-0.954870\pi$
$59$	−168.862	+	292.477i	−0.372609	+	0.645377i	0.617357	+	0.786683i	$0.288203\pi$
$60$		0			0
$61$	−138.735	−	240.295i	−0.291199	−	0.504372i	0.682894	−	0.730517i	$-0.260721\pi$
$61$	−138.735	−	240.295i	−0.291199	−	0.504372i	−0.974094	+	0.226145i	$0.927388\pi$
$62$	622.123			1.27435
$63$		0			0
$64$	64.0000			0.125000
$65$	−379.248	−	656.877i	−0.723691	−	1.25347i
$66$		0			0
$67$	−332.535	+	575.967i	−0.606352	+	1.05023i	0.385485	+	0.922714i	$0.374034\pi$
$67$	−332.535	+	575.967i	−0.606352	+	1.05023i	−0.991836	+	0.127518i	$0.959299\pi$
$68$	234.315	−	405.846i	0.417866	−	0.723766i
$69$		0			0
$70$	−205.923	−	356.669i	−0.351607	−	0.609002i
$71$	528.431			0.883284			0.441642	−	0.897191i	$-0.354396\pi$
$71$	528.431			0.883284			0.441642	+	0.897191i	$0.354396\pi$
$72$		0			0
$73$	−73.8306			−0.118373			−0.0591865	−	0.998247i	$-0.518851\pi$
$73$	−73.8306			−0.118373			−0.0591865	+	0.998247i	$0.518851\pi$
$74$	−206.608	−	357.855i	−0.324563	−	0.562159i
$75$		0			0
$76$	−220.631	+	382.144i	−0.333001	+	0.576775i
$77$	−216.746	+	375.415i	−0.320786	+	0.555617i
$78$		0			0
$79$	239.904	+	415.526i	0.341662	+	0.591776i	0.984741	−	0.174024i	$-0.0556769\pi$
$79$	239.904	+	415.526i	0.341662	+	0.591776i	−0.643080	+	0.765799i	$0.722344\pi$
$80$	−179.138			−0.250354
$81$		0			0
$82$	−265.261			−0.357234
$83$	−89.8846	−	155.685i	−0.118869	−	0.205887i	0.800451	−	0.599398i	$-0.204593\pi$
$83$	−89.8846	−	155.685i	−0.118869	−	0.205887i	−0.919320	+	0.393512i	$0.871260\pi$
$84$		0			0
$85$	−655.858	+	1135.98i	−0.836915	+	1.44958i
$86$	−335.177	+	580.543i	−0.420268	+	0.727926i
$87$		0			0
$88$	94.2769	+	163.292i	0.114204	+	0.197807i
$89$	−846.458			−1.00814			−0.504069	−	0.863663i	$-0.668164\pi$
$89$	−846.458			−1.00814			−0.504069	+	0.863663i	$0.668164\pi$
$90$		0			0
$91$	−1246.01			−1.43535
$92$	138.431	+	239.769i	0.156874	+	0.271714i
$93$		0			0
$94$	379.061	−	656.554i	0.415928	−	0.720408i
$95$	617.554	−	1069.63i	0.666944	−	1.15518i
$96$		0			0
$97$	336.492	+	582.822i	0.352223	+	0.610068i	0.986639	−	0.162924i	$-0.0520926\pi$
$97$	336.492	+	582.822i	0.352223	+	0.610068i	−0.634416	+	0.772992i	$0.718759\pi$
$98$	9.44624			0.00973689
$99$		0			0
$100$	1.41532			0.00141532
$101$	103.261	+	178.854i	0.101732	+	0.176204i	0.912398	−	0.409304i	$-0.134228\pi$
$101$	103.261	+	178.854i	0.101732	+	0.176204i	−0.810667	+	0.585508i	$0.800895\pi$
$102$		0			0
$103$	685.885	−	1187.99i	0.656138	−	1.13646i	−0.325469	−	0.945553i	$-0.605522\pi$
$103$	685.885	−	1187.99i	0.656138	−	1.13646i	0.981607	−	0.190912i	$-0.0611444\pi$
$104$	−270.985	+	469.359i	−0.255502	+	0.442543i
$105$		0			0
$106$	−190.908	−	330.662i	−0.174930	−	0.302988i
$107$	1267.00			1.14472			0.572362	−	0.820001i	$-0.306027\pi$
$107$	1267.00			1.14472			0.572362	+	0.820001i	$0.306027\pi$
$108$		0			0
$109$	1725.13			1.51594			0.757970	−	0.652289i	$-0.226191\pi$
$109$	1725.13			1.51594			0.757970	+	0.652289i	$0.226191\pi$
$110$	−263.885	−	457.061i	−0.228731	−	0.396174i
$111$		0			0
$112$	−147.138	+	254.851i	−0.124136	+	0.215011i
$113$	−870.098	+	1507.05i	−0.724353	+	1.25462i	0.234886	+	0.972023i	$0.424528\pi$
$113$	−870.098	+	1507.05i	−0.724353	+	1.25462i	−0.959240	+	0.282594i	$0.908805\pi$
$114$		0			0
$115$	−387.473	−	671.123i	−0.314192	−	0.544196i
$116$	−793.492			−0.635120
$117$		0			0
$118$	−675.446			−0.526948
$119$	1077.40	+	1866.11i	0.829959	+	1.43753i
$120$		0			0
$121$	387.746	−	671.596i	0.291319	−	0.504580i
$122$	277.469	−	480.591i	0.205909	−	0.356645i
$123$		0			0
$124$	622.123	+	1077.55i	0.450551	+	0.780377i
$125$	1395.56			0.998580
$126$		0			0
$127$	492.131			0.343855			0.171927	−	0.985110i	$-0.445001\pi$
$127$	492.131			0.343855			0.171927	+	0.985110i	$0.445001\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	758.496	−	1313.75i	0.511727	−	0.886337i
$131$	959.638	−	1662.14i	0.640030	−	1.10857i	−0.345395	−	0.938457i	$-0.612255\pi$
$131$	959.638	−	1662.14i	0.640030	−	1.10857i	0.985425	−	0.170108i	$-0.0544116\pi$
$132$		0			0
$133$	−1014.48	−	1757.13i	−0.661401	−	1.14558i
$134$	−1330.14			−0.857511
$135$		0			0
$136$	937.261			0.590952
$137$	1311.43	+	2271.46i	0.817832	+	1.41653i	0.907277	+	0.420534i	$0.138157\pi$
$137$	1311.43	+	2271.46i	0.817832	+	1.41653i	−0.0894453	+	0.995992i	$0.528509\pi$
$138$		0			0
$139$	314.284	−	544.357i	0.191779	−	0.332171i	−0.754061	−	0.656804i	$-0.771908\pi$
$139$	314.284	−	544.357i	0.191779	−	0.332171i	0.945840	+	0.324634i	$0.105241\pi$
$140$	411.846	−	713.338i	0.248624	−	0.430629i
$141$		0			0
$142$	528.431	+	915.269i	0.312288	+	0.540899i
$143$	−1596.72			−0.933739
$144$		0			0
$145$	2221.02			1.27204
$146$	−73.8306	−	127.878i	−0.0418511	−	0.0724883i
$147$		0			0
$148$	413.215	−	715.710i	0.229501	−	0.397507i
$149$	284.156	−	492.173i	0.156235	−	0.270606i	−0.777273	−	0.629163i	$-0.783398\pi$
$149$	284.156	−	492.173i	0.156235	−	0.270606i	0.933508	+	0.358557i	$0.116731\pi$
$150$		0			0
$151$	−178.715	−	309.544i	−0.0963155	−	0.166823i	0.813841	−	0.581087i	$-0.197372\pi$
$151$	−178.715	−	309.544i	−0.0963155	−	0.166823i	−0.910157	+	0.414264i	$0.864039\pi$
$152$	−882.523			−0.470935
$153$		0			0
$154$	−866.985			−0.453660
$155$	−1741.35	−	3016.10i	−0.902376	−	1.56296i
$156$		0			0
$157$	363.627	−	629.820i	0.184844	−	0.320160i	−0.758680	−	0.651464i	$-0.774155\pi$
$157$	363.627	−	629.820i	0.184844	−	0.320160i	0.943524	+	0.331304i	$0.107489\pi$
$158$	−479.808	+	831.051i	−0.241591	+	0.418449i
$159$		0			0
$160$	−179.138	−	310.277i	−0.0885134	−	0.153310i
$161$	−1273.03			−0.623161
$162$		0			0
$163$	−396.554			−0.190555			−0.0952776	−	0.995451i	$-0.530374\pi$
$163$	−396.554			−0.190555			−0.0952776	+	0.995451i	$0.530374\pi$
$164$	−265.261	−	459.446i	−0.126301	−	0.218761i
$165$		0			0
$166$	179.769	−	311.369i	0.0840530	−	0.145584i
$167$	1589.26	−	2752.68i	0.736411	−	1.27550i	−0.217690	−	0.976018i	$-0.569852\pi$
$167$	1589.26	−	2752.68i	0.736411	−	1.27550i	0.954101	−	0.299484i	$-0.0968145\pi$
$168$		0			0
$169$	−1196.27	−	2072.00i	−0.544501	−	0.943104i
$170$	−2623.43			−1.18358
$171$		0			0
$172$	−1340.71			−0.594349
$173$	1076.32	+	1864.25i	0.473014	+	0.819285i	0.999523	−	0.0308850i	$-0.00983256\pi$
$173$	1076.32	+	1864.25i	0.473014	+	0.819285i	−0.526509	+	0.850170i	$0.676499\pi$
$174$		0			0
$175$	−3.25387	+	5.63586i	−0.00140554	+	0.00243446i
$176$	−188.554	+	326.585i	−0.0807544	+	0.139871i
$177$		0			0
$178$	−846.458	−	1466.11i	−0.356431	−	0.617356i
$179$	4490.29			1.87497			0.937487	−	0.348022i	$-0.113146\pi$
$179$	4490.29			1.87497			0.937487	+	0.348022i	$0.113146\pi$
$180$		0			0
$181$	1407.32			0.577931			0.288966	−	0.957340i	$-0.406689\pi$
$181$	1407.32			0.577931			0.288966	+	0.957340i	$0.406689\pi$
$182$	−1246.01	−	2158.15i	−0.507474	−	0.878970i
$183$		0			0
$184$	−276.862	+	479.538i	−0.110927	+	0.192131i
$185$	−1156.61	+	2003.30i	−0.459650	+	0.796138i
$186$		0			0
$187$	1380.66	+	2391.37i	0.539913	+	0.935156i
$188$	1516.25			0.588211
$189$		0			0
$190$	2470.22			0.943201
$191$	386.138	+	668.811i	0.146283	+	0.253369i	0.929851	−	0.367937i	$-0.119936\pi$
$191$	386.138	+	668.811i	0.146283	+	0.253369i	−0.783568	+	0.621306i	$0.786602\pi$
$192$		0			0
$193$	−1826.34	+	3163.31i	−0.681154	+	1.17979i	0.293475	+	0.955967i	$0.405188\pi$
$193$	−1826.34	+	3163.31i	−0.681154	+	1.17979i	−0.974629	+	0.223826i	$0.928145\pi$
$194$	−672.985	+	1165.64i	−0.249059	+	0.431383i
$195$		0			0
$196$	9.44624	+	16.3614i	0.00344251	+	0.00596260i
$197$	−2647.40			−0.957460			−0.478730	−	0.877962i	$-0.658903\pi$
$197$	−2647.40			−0.957460			−0.478730	+	0.877962i	$0.658903\pi$
$198$		0			0
$199$	1470.22			0.523723			0.261861	−	0.965106i	$-0.415664\pi$
$199$	1470.22			0.523723			0.261861	+	0.965106i	$0.415664\pi$
$200$	1.41532	+	2.45140i	0.000500390	+	0.000866701i
$201$		0			0
$202$	−206.523	+	357.708i	−0.0719351	+	0.124595i
$203$	1824.27	−	3159.73i	0.630732	−	1.09246i
$204$		0			0
$205$	742.477	+	1286.01i	0.252960	+	0.438140i
$206$	2743.54			0.927919
$207$		0			0
$208$	−1083.94			−0.361335
$209$	−1300.02	−	2251.71i	−0.430261	−	0.745233i
$210$		0			0
$211$	−768.003	+	1330.22i	−0.250576	+	0.434010i	−0.963685	−	0.267043i	$-0.913953\pi$
$211$	−768.003	+	1330.22i	−0.250576	+	0.434010i	0.713109	+	0.701054i	$0.247287\pi$
$212$	381.815	−	661.323i	0.123694	−	0.214245i
$213$		0			0
$214$	1267.00	+	2194.51i	0.404721	+	0.700997i
$215$	3752.69			1.19038
$216$		0			0
$217$	−5721.14			−1.78975
$218$	1725.13	+	2988.01i	0.535966	+	0.928320i
$219$		0			0
$220$	527.769	−	914.123i	0.161737	−	0.280137i
$221$	−3968.49	+	6873.63i	−1.20792	+	2.09217i
$222$		0			0
$223$	828.765	+	1435.46i	0.248871	+	0.431057i	0.963213	−	0.268740i	$-0.0866071\pi$
$223$	828.765	+	1435.46i	0.248871	+	0.431057i	−0.714342	+	0.699797i	$0.753274\pi$
$224$	−588.554			−0.175555
$225$		0			0
$226$	−3480.39			−1.02439
$227$	−757.131	−	1311.39i	−0.221377	−	0.383436i	0.733850	−	0.679312i	$-0.237722\pi$
$227$	−757.131	−	1311.39i	−0.221377	−	0.383436i	−0.955226	+	0.295876i	$0.904388\pi$
$228$		0			0
$229$	−2149.52	+	3723.08i	−0.620280	+	1.07436i	0.369153	+	0.929369i	$0.379648\pi$
$229$	−2149.52	+	3723.08i	−0.620280	+	1.07436i	−0.989433	+	0.144988i	$0.953685\pi$
$230$	774.946	−	1342.25i	0.222167	−	0.384805i
$231$		0			0
$232$	−793.492	−	1374.37i	−0.224549	−	0.388930i
$233$	−1336.78			−0.375860			−0.187930	−	0.982182i	$-0.560178\pi$
$233$	−1336.78			−0.375860			−0.187930	+	0.982182i	$0.560178\pi$
$234$		0			0
$235$	−4244.03			−1.17809
$236$	−675.446	−	1169.91i	−0.186304	−	0.322688i
$237$		0			0
$238$	−2154.80	+	3732.22i	−0.586869	+	1.01649i
$239$	−3439.31	+	5957.07i	−0.930840	+	1.61226i	−0.148951	+	0.988845i	$0.547590\pi$
$239$	−3439.31	+	5957.07i	−0.930840	+	1.61226i	−0.781889	+	0.623418i	$0.785744\pi$
$240$		0			0
$241$	−765.646	−	1326.14i	−0.204646	−	0.354457i	0.745374	−	0.666646i	$-0.232271\pi$
$241$	−765.646	−	1326.14i	−0.204646	−	0.354457i	−0.950020	+	0.312190i	$0.898938\pi$
$242$	1550.98			0.411988
$243$		0			0
$244$	1109.88			0.291199
$245$	−26.4404	−	45.7961i	−0.00689476	−	0.0119421i
$246$		0			0
$247$	3736.72	−	6472.19i	0.962598	−	1.66727i
$248$	−1244.25	+	2155.10i	−0.318587	+	0.551810i
$249$		0			0
$250$	1395.56	+	2417.18i	0.353051	+	0.611503i
$251$	−1181.60			−0.297139			−0.148570	−	0.988902i	$-0.547467\pi$
$251$	−1181.60			−0.297139			−0.148570	+	0.988902i	$0.547467\pi$
$252$		0			0
$253$	−1631.35			−0.405384
$254$	492.131	+	852.395i	0.121571	+	0.210567i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	2190.68	−	3794.36i	0.531714	−	0.920956i	−0.467600	−	0.883940i	$-0.654881\pi$
$257$	2190.68	−	3794.36i	0.531714	−	0.920956i	0.999315	−	0.0370161i	$-0.0117853\pi$
$258$		0			0
$259$	1900.00	+	3290.89i	0.455830	+	0.789521i
$260$	3033.98			0.723691
$261$		0			0
$262$	3838.55			0.905140
$263$	−1400.04	−	2424.94i	−0.328251	−	0.568548i	0.653914	−	0.756569i	$-0.273126\pi$
$263$	−1400.04	−	2424.94i	−0.328251	−	0.568548i	−0.982165	+	0.188021i	$0.939793\pi$
$264$		0			0
$265$	−1068.72	+	1851.07i	−0.247738	+	0.429095i
$266$	2028.95	−	3514.25i	0.467681	−	0.810047i
$267$		0			0
$268$	−1330.14	−	2303.87i	−0.303176	−	0.525116i
$269$	−2803.73			−0.635489			−0.317745	−	0.948176i	$-0.602925\pi$
$269$	−2803.73			−0.635489			−0.317745	+	0.948176i	$0.602925\pi$
$270$		0			0
$271$	−6332.36			−1.41942			−0.709711	−	0.704493i	$-0.751175\pi$
$271$	−6332.36			−1.41942			−0.709711	+	0.704493i	$0.751175\pi$
$272$	937.261	+	1623.38i	0.208933	+	0.361883i
$273$		0			0
$274$	−2622.86	+	4542.92i	−0.578294	+	1.00163i
$275$	−4.16974	+	7.22220i	−0.000914344	+	0.00158369i
$276$		0			0
$277$	−463.831	−	803.378i	−0.100610	−	0.174261i	0.811326	−	0.584594i	$-0.198746\pi$
$277$	−463.831	−	803.378i	−0.100610	−	0.174261i	−0.911936	+	0.410332i	$0.865413\pi$
$278$	1257.14			0.271216
$279$		0			0
$280$	1647.38			0.351607
$281$	570.764	+	988.592i	0.121170	+	0.209873i	0.920230	−	0.391379i	$-0.128002\pi$
$281$	570.764	+	988.592i	0.121170	+	0.209873i	−0.799059	+	0.601253i	$0.794669\pi$
$282$		0			0
$283$	−1805.52	+	3127.26i	−0.379248	+	0.656877i	−0.990953	−	0.134209i	$-0.957151\pi$
$283$	−1805.52	+	3127.26i	−0.379248	+	0.656877i	0.611705	+	0.791086i	$0.290484\pi$
$284$	−1056.86	+	1830.54i	−0.220821	+	0.382473i
$285$		0			0
$286$	−1596.72	−	2765.61i	−0.330127	−	0.571796i
$287$	2439.38			0.501715
$288$		0			0
$289$	8812.92			1.79380
$290$	2221.02	+	3846.91i	0.449733	+	0.778960i
$291$		0			0
$292$	147.661	−	255.757i	0.0295932	−	0.0512570i
$293$	1162.11	−	2012.83i	0.231710	−	0.401333i	−0.726602	−	0.687059i	$-0.758901\pi$
$293$	1162.11	−	2012.83i	0.231710	−	0.401333i	0.958311	+	0.285726i	$0.0922347\pi$
$294$		0			0
$295$	1890.60	+	3274.61i	0.373136	+	0.646290i
$296$	1652.86			0.324563
$297$		0			0
$298$	1136.62			0.220949
$299$	−2344.54	−	4060.86i	−0.453472	−	0.785436i
$300$		0			0
$301$	3082.34	−	5338.77i	0.590243	−	1.02233i
$302$	357.430	−	619.088i	0.0681053	−	0.117962i
$303$		0			0
$304$	−882.523	−	1528.57i	−0.166501	−	0.288387i
$305$	−3106.59			−0.583222
$306$		0			0
$307$	6968.51			1.29548			0.647742	−	0.761860i	$-0.275713\pi$
$307$	6968.51			1.29548			0.647742	+	0.761860i	$0.275713\pi$
$308$	−866.985	−	1501.66i	−0.160393	−	0.277809i
$309$		0			0
$310$	3482.69	−	6032.20i	0.638076	−	1.10518i
$311$	3170.15	−	5490.87i	0.578016	−	1.00115i	−0.417691	−	0.908589i	$-0.637160\pi$
$311$	3170.15	−	5490.87i	0.578016	−	1.00115i	0.995707	−	0.0925637i	$-0.0295062\pi$
$312$		0			0
$313$	−1194.42	−	2068.79i	−0.215694	−	0.373594i	0.737793	−	0.675027i	$-0.235868\pi$
$313$	−1194.42	−	2068.79i	−0.215694	−	0.373594i	−0.953487	+	0.301434i	$0.902535\pi$
$314$	1454.51			0.261409
$315$		0			0
$316$	−1919.23			−0.341662
$317$	−2930.63	−	5076.00i	−0.519245	−	0.899359i	−0.999750	−	0.0223668i	$-0.992880\pi$
$317$	−2930.63	−	5076.00i	−0.519245	−	0.899359i	0.480505	−	0.876992i	$-0.340453\pi$
$318$		0			0
$319$	2337.75	−	4049.10i	0.410310	−	0.710677i
$320$	358.277	−	620.554i	0.0625884	−	0.108406i
$321$		0			0
$322$	−1273.03	−	2204.95i	−0.220321	−	0.381606i
$323$	−12924.3			−2.22640
$324$		0			0
$325$	−23.9706			−0.00409122
$326$	−396.554	−	686.851i	−0.0673714	−	0.116691i
$327$		0			0
$328$	530.523	−	918.892i	0.0893086	−	0.154687i
$329$	−3485.91	+	6037.77i	−0.584147	+	1.01177i
$330$		0			0
$331$	−2482.48	−	4299.78i	−0.412234	−	0.714010i	0.582900	−	0.812544i	$-0.301918\pi$
$331$	−2482.48	−	4299.78i	−0.412234	−	0.714010i	−0.995134	+	0.0985339i	$0.968585\pi$
$332$	719.077			0.118869
$333$		0			0
$334$	6357.04			1.04144
$335$	3723.11	+	6448.61i	0.607209	+	1.05172i
$336$		0			0
$337$	1548.83	−	2682.65i	0.250357	−	0.433630i	−0.713267	−	0.700892i	$-0.752785\pi$
$337$	1548.83	−	2682.65i	0.250357	−	0.433630i	0.963624	+	0.267262i	$0.0861188\pi$
$338$	2392.54	−	4144.00i	0.385021	−	0.666875i
$339$		0			0
$340$	−2623.43	−	4543.91i	−0.418457	−	0.724789i
$341$	−7331.48			−1.16429
$342$		0			0
$343$	−6395.43			−1.00677
$344$	−1340.71	−	2322.17i	−0.210134	−	0.363963i
$345$		0			0
$346$	−2152.65	+	3728.50i	−0.334472	+	0.579322i
$347$	4022.10	−	6966.48i	0.622241	−	1.07775i	−0.366827	−	0.930289i	$-0.619556\pi$
$347$	4022.10	−	6966.48i	0.622241	−	1.07775i	0.989067	−	0.147464i	$-0.0471109\pi$
$348$		0			0
$349$	572.353	+	991.345i	0.0877862	+	0.152050i	0.906575	−	0.422045i	$-0.138687\pi$
$349$	572.353	+	991.345i	0.0877862	+	0.152050i	−0.818789	+	0.574095i	$0.805354\pi$
$350$	−13.0155			−0.00198773
$351$		0			0
$352$	−754.215			−0.114204
$353$	3900.63	+	6756.09i	0.588129	+	1.01867i	0.994477	+	0.104951i	$0.0334687\pi$
$353$	3900.63	+	6756.09i	0.588129	+	1.01867i	−0.406348	+	0.913718i	$0.633198\pi$
$354$		0			0
$355$	2958.20	−	5123.75i	0.442267	−	0.766029i
$356$	1692.92	−	2932.22i	0.252035	−	0.436537i
$357$		0			0
$358$	4490.29	+	7777.41i	0.662903	+	1.14818i
$359$	−341.307			−0.0501769			−0.0250885	−	0.999685i	$-0.507987\pi$
$359$	−341.307			−0.0501769			−0.0250885	+	0.999685i	$0.507987\pi$
$360$		0			0
$361$	5310.48			0.774235
$362$	1407.32	+	2437.56i	0.204329	+	0.353909i
$363$		0			0
$364$	2492.02	−	4316.30i	0.358838	−	0.621526i
$365$	−413.310	+	715.873i	−0.0592702	+	0.102659i
$366$		0			0
$367$	−59.6839	−	103.376i	−0.00848903	−	0.0147034i	0.861750	−	0.507334i	$-0.169369\pi$
$367$	−59.6839	−	103.376i	−0.00848903	−	0.0147034i	−0.870239	+	0.492630i	$0.836036\pi$
$368$	−1107.45			−0.156874
$369$		0			0
$370$	−4626.42			−0.650044
$371$	1755.62	+	3040.81i	0.245679	+	0.425529i
$372$		0			0
$373$	2187.22	−	3788.37i	0.303619	−	0.525883i	−0.673334	−	0.739338i	$-0.735139\pi$
$373$	2187.22	−	3788.37i	0.303619	−	0.525883i	0.976953	+	0.213456i	$0.0684718\pi$
$374$	−2761.31	+	4782.74i	−0.381776	+	0.661255i
$375$		0			0
$376$	1516.25	+	2626.22i	0.207964	+	0.360204i
$377$	13439.0			1.83593
$378$		0			0
$379$	8949.46			1.21294			0.606468	−	0.795108i	$-0.292586\pi$
$379$	8949.46			1.21294			0.606468	+	0.795108i	$0.292586\pi$
$380$	2470.22	+	4278.54i	0.333472	+	0.577590i
$381$		0			0
$382$	−772.277	+	1337.62i	−0.103437	+	0.179159i
$383$	103.232	−	178.802i	0.0137726	−	0.0238548i	−0.859057	−	0.511880i	$-0.828949\pi$
$383$	103.232	−	178.802i	0.0137726	−	0.0238548i	0.872830	+	0.488025i	$0.162283\pi$
$384$		0			0
$385$	2426.72	+	4203.21i	0.321240	+	0.556403i
$386$	−7305.35			−0.963297
$387$		0			0
$388$	−2691.94			−0.352223
$389$	−1014.47	−	1757.11i	−0.132225	−	0.229021i	0.792309	−	0.610120i	$-0.208879\pi$
$389$	−1014.47	−	1757.11i	−0.132225	−	0.229021i	−0.924534	+	0.381099i	$0.875546\pi$
$390$		0			0
$391$	−4054.56	+	7022.70i	−0.524419	+	0.908320i
$392$	−18.8925	+	32.7228i	−0.00243422	+	0.00421620i
$393$		0			0
$394$	−2647.40	−	4585.44i	−0.338513	−	0.586322i
$395$	5372.00			0.684290
$396$		0			0
$397$	−6646.07			−0.840193			−0.420097	−	0.907479i	$-0.638004\pi$
$397$	−6646.07			−0.840193			−0.420097	+	0.907479i	$0.638004\pi$
$398$	1470.22	+	2546.49i	0.185164	+	0.320713i
$399$		0			0
$400$	−2.83063	+	4.90280i	−0.000353829	+	0.000612850i
$401$	−0.817240	+	1.41550i	−0.000101773	+	0.000176276i	−0.866076	−	0.499912i	$-0.833366\pi$
$401$	−0.817240	+	1.41550i	−0.000101773	+	0.000176276i	0.865975	+	0.500088i	$0.166699\pi$
$402$		0			0
$403$	−10536.6	−	18249.9i	−1.30240	−	2.25582i
$404$	−826.091			−0.101732
$405$		0			0
$406$	7297.08			0.891990
$407$	2434.79	+	4217.18i	0.296531	+	0.513607i
$408$		0			0
$409$	3101.68	−	5372.26i	0.374983	−	0.649490i	−0.615342	−	0.788261i	$-0.710982\pi$
$409$	3101.68	−	5372.26i	0.374983	−	0.649490i	0.990324	+	0.138771i	$0.0443152\pi$
$410$	−1484.95	+	2572.02i	−0.178870	+	0.309812i
$411$		0			0
$412$	2743.54	+	4751.95i	0.328069	+	0.568232i
$413$	6211.51			0.740068
$414$		0			0
$415$	−2012.72			−0.238074
$416$	−1083.94	−	1877.44i	−0.127751	−	0.221271i
$417$		0			0
$418$	2600.05	−	4503.41i	0.304240	−	0.526960i
$419$	−4875.18	+	8444.07i	−0.568421	+	0.984534i	0.428302	+	0.903636i	$0.359112\pi$
$419$	−4875.18	+	8444.07i	−0.568421	+	0.984534i	−0.996722	+	0.0808979i	$0.974221\pi$
$420$		0			0
$421$	5030.69	+	8713.41i	0.582377	+	1.00871i	0.995197	+	0.0978939i	$0.0312106\pi$
$421$	5030.69	+	8713.41i	0.582377	+	1.00871i	−0.412820	+	0.910813i	$0.635456\pi$
$422$	−3072.01			−0.354368
$423$		0			0
$424$	1527.26			0.174930
$425$	20.7269	+	35.9000i	0.00236565	+	0.00409743i
$426$		0			0
$427$	−2551.65	+	4419.59i	−0.289187	+	0.500887i
$428$	−2534.00	+	4389.02i	−0.286181	+	0.495680i
$429$		0			0
$430$	3752.69	+	6499.85i	0.420862	+	0.728955i
$431$	6763.21			0.755853			0.377926	−	0.925836i	$-0.376637\pi$
$431$	6763.21			0.755853			0.377926	+	0.925836i	$0.376637\pi$
$432$		0			0
$433$	−10601.4			−1.17660			−0.588302	−	0.808641i	$-0.700204\pi$
$433$	−10601.4			−1.17660			−0.588302	+	0.808641i	$0.700204\pi$
$434$	−5721.14	−	9909.30i	−0.632773	−	1.09599i
$435$		0			0
$436$	−3450.26	+	5976.03i	−0.378985	+	0.656422i
$437$	3817.76	−	6612.55i	0.417914	−	0.723848i
$438$		0			0
$439$	6284.46	+	10885.0i	0.683237	+	1.18340i	0.973987	+	0.226602i	$0.0727616\pi$
$439$	6284.46	+	10885.0i	0.683237	+	1.18340i	−0.290751	+	0.956799i	$0.593905\pi$
$440$	2111.08			0.228731
$441$		0			0
$442$	−15874.0			−1.70825
$443$	−5127.71	−	8881.46i	−0.549944	−	0.952530i	−0.998278	−	0.0586643i	$-0.981316\pi$
$443$	−5127.71	−	8881.46i	−0.549944	−	0.952530i	0.448334	−	0.893866i	$-0.352017\pi$
$444$		0			0
$445$	−4738.53	+	8207.38i	−0.504782	+	0.874308i
$446$	−1657.53	+	2870.93i	−0.175978	+	0.304803i
$447$		0			0
$448$	−588.554	−	1019.41i	−0.0620682	−	0.107505i
$449$	4080.23			0.428860			0.214430	−	0.976739i	$-0.431211\pi$
$449$	4080.23			0.428860			0.214430	+	0.976739i	$0.431211\pi$
$450$		0			0
$451$	3126.00			0.326381
$452$	−3480.39	−	6028.22i	−0.362177	−	0.627308i
$453$		0			0
$454$	1514.26	−	2622.78i	0.156537	−	0.271130i
$455$	−6975.25	+	12081.5i	−0.718691	+	1.24481i
$456$		0			0
$457$	1091.60	+	1890.71i	0.111735	+	0.193531i	0.916470	−	0.400104i	$-0.131026\pi$
$457$	1091.60	+	1890.71i	0.111735	+	0.193531i	−0.804735	+	0.593634i	$0.797693\pi$
$458$	−8598.08			−0.877209
$459$		0			0
$460$	3099.78			0.314192
$461$	1625.13	+	2814.81i	0.164186	+	0.284379i	0.936366	−	0.351025i	$-0.114167\pi$
$461$	1625.13	+	2814.81i	0.164186	+	0.284379i	−0.772180	+	0.635404i	$0.780834\pi$
$462$		0			0
$463$	−9495.57	+	16446.8i	−0.953124	+	1.65086i	−0.214520	+	0.976720i	$0.568819\pi$
$463$	−9495.57	+	16446.8i	−0.953124	+	1.65086i	−0.738604	+	0.674140i	$0.764515\pi$
$464$	1586.98	−	2748.74i	0.158780	−	0.275015i
$465$		0			0
$466$	−1336.78	−	2315.37i	−0.132887	−	0.230166i
$467$	−6906.52			−0.684359			−0.342180	−	0.939635i	$-0.611165\pi$
$467$	−6906.52			−0.684359			−0.342180	+	0.939635i	$0.611165\pi$
$468$		0			0
$469$	12232.2			1.20432
$470$	−4244.03	−	7350.88i	−0.416516	−	0.721427i
$471$		0			0
$472$	1350.89	−	2339.81i	0.131737	−	0.228175i
$473$	3949.93	−	6841.48i	0.383970	−	0.665056i
$474$		0			0
$475$	−19.5164	−	33.8034i	−0.00188521	−	0.00326527i
$476$	−8619.20			−0.829959
$477$		0			0
$478$	−13757.3			−1.31641
$479$	3690.14	+	6391.50i	0.351997	+	0.609677i	0.986599	−	0.163162i	$-0.0521693\pi$
$479$	3690.14	+	6391.50i	0.351997	+	0.609677i	−0.634602	+	0.772839i	$0.718836\pi$
$480$		0			0
$481$	−6998.44	+	12121.6i	−0.663412	+	1.14906i
$482$	1531.29	−	2652.28i	0.144706	−	0.250639i
$483$		0			0
$484$	1550.98	+	2686.38i	0.145660	+	0.252290i
$485$	7534.84			0.705442
$486$		0			0
$487$	−8756.51			−0.814774			−0.407387	−	0.913256i	$-0.633560\pi$
$487$	−8756.51			−0.814774			−0.407387	+	0.913256i	$0.633560\pi$
$488$	1109.88	+	1922.36i	0.102954	+	0.178322i
$489$		0			0
$490$	52.8808	−	91.5922i	0.00487533	−	0.00844432i
$491$	5918.97	−	10252.0i	0.544031	−	0.942290i	−0.454636	−	0.890677i	$-0.650231\pi$
$491$	5918.97	−	10252.0i	0.544031	−	0.942290i	0.998667	−	0.0516124i	$-0.0164360\pi$
$492$		0			0
$493$	−11620.5	−	20127.2i	−1.06158	−	1.83871i
$494$	14946.9			1.36132
$495$		0			0
$496$	−4976.98			−0.450551
$497$	−4859.53	−	8416.95i	−0.438591	−	0.759662i
$498$		0			0
$499$	−4289.19	+	7429.09i	−0.384791	+	0.666477i	−0.991740	−	0.128264i	$-0.959060\pi$
$499$	−4289.19	+	7429.09i	−0.384791	+	0.666477i	0.606950	+	0.794740i	$0.292393\pi$
$500$	−2791.12	+	4834.35i	−0.249645	+	0.432398i
$501$		0			0
$502$	−1181.60	−	2046.59i	−0.105055	−	0.181960i
$503$	3611.17			0.320107			0.160054	−	0.987108i	$-0.448833\pi$
$503$	3611.17			0.320107			0.160054	+	0.987108i	$0.448833\pi$
$504$		0			0
$505$	2312.26			0.203751
$506$	−1631.35	−	2825.59i	−0.143325	−	0.248246i
$507$		0			0
$508$	−984.261	+	1704.79i	−0.0859636	+	0.148893i
$509$	4529.87	−	7845.96i	0.394465	−	0.683234i	−0.598567	−	0.801072i	$-0.704263\pi$
$509$	4529.87	−	7845.96i	0.394465	−	0.683234i	0.993033	+	0.117838i	$0.0375965\pi$
$510$		0			0
$511$	678.958	+	1175.99i	0.0587775	+	0.101806i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	8762.70			0.751957
$515$	−7679.27	−	13300.9i	−0.657066	−	1.13807i
$516$		0			0
$517$	−4467.09	+	7737.23i	−0.380005	+	0.658188i
$518$	−3799.99	+	6581.78i	−0.322321	+	0.558276i
$519$		0			0
$520$	3033.98	+	5255.01i	0.255863	+	0.443169i
$521$	12834.1			1.07922			0.539609	−	0.841916i	$-0.318572\pi$
$521$	12834.1			1.07922			0.539609	+	0.841916i	$0.318572\pi$
$522$		0			0
$523$	7061.21			0.590373			0.295187	−	0.955440i	$-0.404618\pi$
$523$	7061.21			0.590373			0.295187	+	0.955440i	$0.404618\pi$
$524$	3838.55	+	6648.57i	0.320015	+	0.554283i
$525$		0			0
$526$	2800.08	−	4849.88i	0.232109	−	0.402024i
$527$	−18221.6	+	31560.8i	−1.50616	+	2.60875i
$528$		0			0
$529$	3688.11	+	6388.00i	0.303124	+	0.525027i
$530$	−4274.86			−0.350355
$531$		0			0
$532$	8115.81			0.661401
$533$	4492.61	+	7781.43i	0.365097	+	0.632366i
$534$		0			0
$535$	7092.76	−	12285.0i	0.573172	−	0.992762i
$536$	2660.28	−	4607.73i	0.214378	−	0.371313i
$537$		0			0
$538$	−2803.73	−	4856.21i	−0.224679	−	0.389156i
$539$	−111.320			−0.00889593
$540$		0			0
$541$	−21645.6			−1.72018			−0.860091	−	0.510141i	$-0.829593\pi$
$541$	−21645.6			−1.72018			−0.860091	+	0.510141i	$0.829593\pi$
$542$	−6332.36	−	10968.0i	−0.501842	−	0.869215i
$543$		0			0
$544$	−1874.52	+	3246.77i	−0.147738	+	0.255890i
$545$	9657.41	−	16727.1i	0.759042	−	1.31470i
$546$		0			0
$547$	−4253.52	−	7367.32i	−0.332482	−	0.575875i	0.650516	−	0.759492i	$-0.274553\pi$
$547$	−4253.52	−	7367.32i	−0.332482	−	0.575875i	−0.982998	+	0.183617i	$0.941219\pi$
$548$	−10491.4			−0.817832
$549$		0			0
$550$	−16.6790			−0.00129308
$551$	10941.8	+	18951.7i	0.845982	+	1.46528i
$552$		0			0
$553$	4412.38	−	7642.47i	0.339301	−	0.587687i
$554$	927.661	−	1606.76i	0.0711418	−	0.123221i
$555$		0			0
$556$	1257.14	+	2177.43i	0.0958894	+	0.166085i
$557$	636.486			0.0484179			0.0242090	−	0.999707i	$-0.492293\pi$
$557$	636.486			0.0484179			0.0242090	+	0.999707i	$0.492293\pi$
$558$		0			0
$559$	22706.9			1.71807
$560$	1647.38	+	2853.35i	0.124312	+	0.215315i
$561$		0			0
$562$	−1141.53	+	1977.18i	−0.0856805	+	0.148403i
$563$	6781.31	−	11745.6i	0.507635	−	0.879249i	−0.492326	−	0.870411i	$-0.663853\pi$
$563$	6781.31	−	11745.6i	0.507635	−	0.879249i	0.999961	−	0.00883838i	$-0.00281338\pi$
$564$		0			0
$565$	9741.75	+	16873.2i	0.725378	+	1.25639i
$566$	−7222.09			−0.536338
$567$		0			0
$568$	−4227.45			−0.312288
$569$	−10558.4	−	18287.6i	−0.777909	−	1.34738i	−0.933145	−	0.359500i	$-0.882947\pi$
$569$	−10558.4	−	18287.6i	−0.777909	−	1.34738i	0.155236	−	0.987877i	$-0.450386\pi$
$570$		0			0
$571$	7263.55	−	12580.8i	0.532347	−	0.922052i	−0.466940	−	0.884289i	$-0.654643\pi$
$571$	7263.55	−	12580.8i	0.532347	−	0.922052i	0.999287	−	0.0377630i	$-0.0120232\pi$
$572$	3193.45	−	5531.21i	0.233435	−	0.404321i
$573$		0			0
$574$	2439.38	+	4225.14i	0.177383	+	0.307237i
$575$	−24.4904			−0.00177621
$576$		0			0
$577$	−14590.9			−1.05273			−0.526366	−	0.850258i	$-0.676446\pi$
$577$	−14590.9			−1.05273			−0.526366	+	0.850258i	$0.676446\pi$
$578$	8812.92	+	15264.4i	0.634203	+	1.09847i
$579$		0			0
$580$	−4442.03	+	7693.82i	−0.318009	+	0.550808i
$581$	−1653.18	+	2863.40i	−0.118048	+	0.204464i
$582$		0			0
$583$	2249.77	+	3896.72i	0.159822	+	0.276819i
$584$	590.645			0.0418511
$585$		0			0
$586$	4648.42			0.327687
$587$	9170.74	+	15884.2i	0.644833	+	1.11688i	0.984340	+	0.176280i	$0.0564064\pi$
$587$	9170.74	+	15884.2i	0.644833	+	1.11688i	−0.339507	+	0.940603i	$0.610260\pi$
$588$		0			0
$589$	17157.4	−	29717.5i	1.20027	−	2.07893i
$590$	−3781.20	+	6549.23i	−0.263847	+	0.456996i
$591$		0			0
$592$	1652.86	+	2862.84i	0.114750	+	0.198753i
$593$	−28380.4			−1.96534			−0.982668	−	0.185376i	$-0.940650\pi$
$593$	−28380.4			−1.96534			−0.982668	+	0.185376i	$0.940650\pi$
$594$		0			0
$595$	24125.5			1.66227
$596$	1136.62	+	1968.69i	0.0781173	+	0.135303i
$597$		0			0
$598$	4689.08	−	8121.72i	0.320653	−	0.555387i
$599$	−11940.6	+	20681.7i	−0.814487	+	1.41073i	0.0952079	+	0.995457i	$0.469648\pi$
$599$	−11940.6	+	20681.7i	−0.814487	+	1.41073i	−0.909695	+	0.415276i	$0.863685\pi$
$600$		0			0
$601$	−8310.38	−	14394.0i	−0.564039	−	0.976944i	−0.997138	−	0.0755978i	$-0.975914\pi$
$601$	−8310.38	−	14394.0i	−0.564039	−	0.976944i	0.433100	−	0.901346i	$-0.357420\pi$
$602$	12329.4			0.834729
$603$		0			0
$604$	1429.72			0.0963155
$605$	−4341.26	−	7519.29i	−0.291731	−	0.505293i
$606$		0			0
$607$	785.389	−	1360.33i	0.0525172	−	0.0909624i	−0.838572	−	0.544791i	$-0.816609\pi$
$607$	785.389	−	1360.33i	0.0525172	−	0.0909624i	0.891089	+	0.453829i	$0.149942\pi$
$608$	1765.05	−	3057.15i	0.117734	−	0.203921i
$609$		0			0
$610$	−3106.59	−	5380.77i	−0.206200	−	0.357149i
$611$	−25680.0			−1.70033
$612$		0			0
$613$	−5766.40			−0.379939			−0.189969	−	0.981790i	$-0.560839\pi$
$613$	−5766.40			−0.379939			−0.189969	+	0.981790i	$0.560839\pi$
$614$	6968.51	+	12069.8i	0.458023	+	0.793319i
$615$		0			0
$616$	1733.97	−	3003.32i	0.113415	−	0.196440i
$617$	3481.28	−	6029.76i	0.227149	−	0.393434i	−0.729813	−	0.683647i	$-0.760393\pi$
$617$	3481.28	−	6029.76i	0.227149	−	0.393434i	0.956962	+	0.290213i	$0.0937261\pi$
$618$		0			0
$619$	−1330.21	−	2303.99i	−0.0863741	−	0.149604i	0.819602	−	0.572934i	$-0.194195\pi$
$619$	−1330.21	−	2303.99i	−0.0863741	−	0.149604i	−0.905976	+	0.423329i	$0.860861\pi$
$620$	13930.8			0.902376
$621$		0			0
$622$	12680.6			0.817438
$623$	7784.15	+	13482.5i	0.500587	+	0.867042i
$624$		0			0
$625$	7834.55	−	13569.8i	0.501411	−	0.868470i
$626$	2388.83	−	4137.58i	0.152519	−	0.264171i
$627$		0			0
$628$	1454.51	+	2519.28i	0.0924222	+	0.160080i
$629$	24205.7			1.53441
$630$		0			0
$631$	−20432.5			−1.28908			−0.644538	−	0.764573i	$-0.722950\pi$
$631$	−20432.5			−1.28908			−0.644538	+	0.764573i	$0.722950\pi$
$632$	−1919.23	−	3324.20i	−0.120796	−	0.209224i
$633$		0			0
$634$	5861.26	−	10152.0i	0.367162	−	0.635943i
$635$	2754.98	−	4771.77i	0.172170	−	0.298208i
$636$		0			0
$637$	−159.987	−	277.105i	−0.00995118	−	0.0172359i
$638$	9351.00			0.580266
$639$		0			0
$640$	1433.11			0.0885134
$641$	13034.2	+	22575.8i	0.803149	+	1.39109i	0.917534	+	0.397658i	$0.130177\pi$
$641$	13034.2	+	22575.8i	0.803149	+	1.39109i	−0.114385	+	0.993436i	$0.536490\pi$
$642$		0			0
$643$	−10585.2	+	18334.1i	−0.649207	+	1.12446i	0.334106	+	0.942535i	$0.391566\pi$
$643$	−10585.2	+	18334.1i	−0.649207	+	1.12446i	−0.983313	+	0.181923i	$0.941768\pi$
$644$	2546.06	−	4409.91i	0.155790	−	0.269836i
$645$		0			0
$646$	−12924.3	−	22385.5i	−0.787151	−	1.36339i
$647$	−8291.45			−0.503818			−0.251909	−	0.967751i	$-0.581058\pi$
$647$	−8291.45			−0.503818			−0.251909	+	0.967751i	$0.581058\pi$
$648$		0			0
$649$	7959.87			0.481436
$650$	−23.9706	−	41.5182i	−0.00144647	−	0.00250535i
$651$		0			0
$652$	793.108	−	1373.70i	0.0476388	−	0.0825128i
$653$	12342.5	−	21377.8i	0.739662	−	1.28113i	−0.212985	−	0.977055i	$-0.568319\pi$
$653$	12342.5	−	21377.8i	0.739662	−	1.28113i	0.952648	−	0.304077i	$-0.0983480\pi$
$654$		0			0
$655$	−10744.3	−	18609.6i	−0.640936	−	1.11013i
$656$	2122.09			0.126301
$657$		0			0
$658$	−13943.6			−0.826108
$659$	−14977.2	−	25941.3i	−0.885325	−	1.53343i	−0.845341	−	0.534227i	$-0.820603\pi$
$659$	−14977.2	−	25941.3i	−0.885325	−	1.53343i	−0.0399839	−	0.999200i	$-0.512731\pi$
$660$		0			0
$661$	2063.38	−	3573.88i	0.121416	−	0.210299i	−0.798910	−	0.601450i	$-0.794590\pi$
$661$	2063.38	−	3573.88i	0.121416	−	0.210299i	0.920326	+	0.391151i	$0.127923\pi$
$662$	4964.96	−	8599.56i	0.291493	−	0.504881i
$663$		0			0
$664$	719.077	+	1245.48i	0.0420265	+	0.0727920i
$665$	−22716.5			−1.32467
$666$		0			0
$667$	13730.5			0.797070
$668$	6357.04	+	11010.7i	0.368206	+	0.637751i
$669$		0			0
$670$	−7446.21	+	12897.2i	−0.429362	+	0.743676i
$671$	−3269.87	+	5663.58i	−0.188125	+	0.325842i
$672$		0			0
$673$	13164.9	+	22802.2i	0.754039	+	1.30603i	0.945851	+	0.324602i	$0.105230\pi$
$673$	13164.9	+	22802.2i	0.754039	+	1.30603i	−0.191812	+	0.981432i	$0.561436\pi$
$674$	6195.32			0.354058
$675$		0			0
$676$	9570.15			0.544501
$677$	3301.93	+	5719.11i	0.187450	+	0.324673i	0.944399	−	0.328801i	$-0.106644\pi$
$677$	3301.93	+	5719.11i	0.187450	+	0.324673i	−0.756949	+	0.653473i	$0.773311\pi$
$678$		0			0
$679$	6188.87	−	10719.4i	0.349789	−	0.605853i
$680$	5246.86	−	9087.83i	0.295894	−	0.512503i
$681$		0			0
$682$	−7331.48	−	12698.5i	−0.411637	−	0.712977i
$683$	−12706.4			−0.711854			−0.355927	−	0.934514i	$-0.615835\pi$
$683$	−12706.4			−0.711854			−0.355927	+	0.934514i	$0.615835\pi$
$684$		0			0
$685$	29365.9			1.63798
$686$	−6395.43	−	11077.2i	−0.355946	−	0.616516i
$687$		0			0
$688$	2681.42	−	4644.35i	0.148587	−	0.257361i
$689$	−6466.63	+	11200.5i	−0.357560	+	0.619312i
$690$		0			0
$691$	7865.10	+	13622.8i	0.432999	+	0.749977i	0.997130	−	0.0757086i	$-0.0241219\pi$
$691$	7865.10	+	13622.8i	0.432999	+	0.749977i	−0.564131	+	0.825686i	$0.690789\pi$
$692$	−8610.60			−0.473014
$693$		0			0
$694$	16088.4			0.879982
$695$	−3518.78	−	6094.70i	−0.192050	−	0.332640i
$696$		0			0
$697$	7769.35	−	13456.9i	0.422217	−	0.731301i
$698$	−1144.71	+	1982.69i	−0.0620742	+	0.107516i
$699$		0			0
$700$	−13.0155	−	22.5434i	−0.000702769	−	0.00121723i
$701$	652.959			0.0351811			0.0175905	−	0.999845i	$-0.494400\pi$
$701$	652.959			0.0351811			0.0175905	+	0.999845i	$0.494400\pi$
$702$		0			0
$703$	−22792.0			−1.22278
$704$	−754.215	−	1306.34i	−0.0403772	−	0.0699354i
$705$		0			0
$706$	−7801.26	+	13512.2i	−0.415870	+	0.720308i
$707$	1899.22	−	3289.54i	0.101029	−	0.174987i
$708$		0			0
$709$	12442.2	+	21550.6i	0.659065	+	1.14153i	0.980858	+	0.194724i	$0.0623812\pi$
$709$	12442.2	+	21550.6i	0.659065	+	1.14153i	−0.321793	+	0.946810i	$0.604285\pi$
$710$	11832.8			0.625460
$711$		0			0
$712$	6771.66			0.356431
$713$	−10765.1	−	18645.7i	−0.565438	−	0.979367i
$714$		0			0
$715$	−8938.58	+	15482.1i	−0.467530	+	0.809786i
$716$	−8980.58	+	15554.8i	−0.468743	+	0.811887i
$717$		0			0
$718$	−341.307	−	591.162i	−0.0177402	−	0.0307270i
$719$	−21323.8			−1.10604			−0.553020	−	0.833168i	$-0.686525\pi$
$719$	−21323.8			−1.10604			−0.553020	+	0.833168i	$0.686525\pi$
$720$		0			0
$721$	−25230.0			−1.30321
$722$	5310.48	+	9198.02i	0.273733	+	0.474120i
$723$		0			0
$724$	−2814.65	+	4875.11i	−0.144483	+	0.250251i
$725$	35.0951	−	60.7865i	0.00179779	−	0.00311387i
$726$		0			0
$727$	−4016.00	−	6955.92i	−0.204877	−	0.354857i	0.745217	−	0.666822i	$-0.232346\pi$
$727$	−4016.00	−	6955.92i	−0.204877	−	0.354857i	−0.950093	+	0.311965i	$0.899013\pi$
$728$	9968.06			0.507474
$729$		0			0
$730$	−1653.24			−0.0838207
$731$	−19634.3	−	34007.6i	−0.993433	−	1.72068i
$732$		0			0
$733$	−14228.7	+	24644.8i	−0.716984	+	1.24185i	0.245205	+	0.969471i	$0.421145\pi$
$733$	−14228.7	+	24644.8i	−0.716984	+	1.24185i	−0.962189	+	0.272381i	$0.912189\pi$
$734$	119.368	−	206.751i	0.00600265	−	0.0103969i
$735$		0			0
$736$	−1107.45	−	1918.15i	−0.0554633	−	0.0960653i
$737$	15675.2			0.783449
$738$		0			0
$739$	−11006.3			−0.547868			−0.273934	−	0.961748i	$-0.588325\pi$
$739$	−11006.3			−0.547868			−0.273934	+	0.961748i	$0.588325\pi$
$740$	−4626.42	−	8013.20i	−0.229825	−	0.398069i
$741$		0			0
$742$	−3511.23	+	6081.63i	−0.173721	+	0.300894i
$743$	2326.45	−	4029.54i	0.114871	−	0.198963i	−0.802857	−	0.596172i	$-0.796688\pi$
$743$	2326.45	−	4029.54i	0.114871	−	0.198963i	0.917728	+	0.397209i	$0.130021\pi$
$744$		0			0
$745$	−3181.45	−	5510.44i	−0.156456	−	0.270989i
$746$	8748.86			0.429381
$747$		0			0
$748$	−11045.3			−0.539913
$749$	−11651.5	−	20181.0i	−0.568408	−	0.984511i
$750$		0			0
$751$	8678.87	−	15032.3i	0.421700	−	0.730406i	−0.574406	−	0.818571i	$-0.694767\pi$
$751$	8678.87	−	15032.3i	0.421700	−	0.730406i	0.996106	+	0.0881649i	$0.0281003\pi$
$752$	−3032.49	+	5252.43i	−0.147053	+	0.254703i
$753$		0			0
$754$	13439.0	+	23277.0i	0.649098	+	1.12427i
$755$	−4001.85			−0.192903
$756$		0			0
$757$	119.139			0.00572019			0.00286010	−	0.999996i	$-0.499090\pi$
$757$	119.139			0.00572019			0.00286010	+	0.999996i	$0.499090\pi$
$758$	8949.46	+	15500.9i	0.428838	+	0.742769i
$759$		0			0
$760$	−4940.43	+	8557.08i	−0.235800	+	0.408418i
$761$	−4421.51	+	7658.28i	−0.210617	+	0.364799i	−0.951908	−	0.306385i	$-0.900881\pi$
$761$	−4421.51	+	7658.28i	−0.210617	+	0.364799i	0.741291	+	0.671184i	$0.234214\pi$
$762$		0			0
$763$	−15864.6	−	27478.2i	−0.752734	−	1.30377i
$764$	−3089.11			−0.146283
$765$		0			0
$766$	412.926			0.0194773
$767$	11439.7	+	19814.2i	0.538545	+	0.932788i
$768$		0			0
$769$	−1346.55	+	2332.30i	−0.0631442	+	0.109369i	−0.895869	−	0.444318i	$-0.853446\pi$
$769$	−1346.55	+	2332.30i	−0.0631442	+	0.109369i	0.832725	+	0.553687i	$0.186779\pi$
$770$	−4853.45	+	8406.41i	−0.227151	+	0.393437i
$771$		0			0
$772$	−7305.35	−	12653.2i	−0.340577	−	0.589897i
$773$	18116.5			0.842958			0.421479	−	0.906838i	$-0.361511\pi$
$773$	18116.5			0.842958			0.421479	+	0.906838i	$0.361511\pi$
$774$		0			0
$775$	−110.063			−0.00510137
$776$	−2691.94	−	4662.57i	−0.124530	−	0.215692i
$777$		0			0
$778$	2028.94	−	3514.23i	0.0934975	−	0.161942i
$779$	−7315.60	+	12671.0i	−0.336468	+	0.582780i
$780$		0			0
$781$	−6227.35	−	10786.1i	−0.285316	−	0.494183i
$782$	−16218.2			−0.741640
$783$		0			0
$784$	−75.5700			−0.00344251
$785$	−4071.22	−	7051.56i	−0.185106	−	0.320613i
$786$		0			0
$787$	−15873.7	+	27494.1i	−0.718980	+	1.24531i	0.242424	+	0.970170i	$0.422057\pi$
$787$	−15873.7	+	27494.1i	−0.718980	+	1.24531i	−0.961404	+	0.275140i	$0.911276\pi$
$788$	5294.81	−	9170.88i	0.239365	−	0.414593i
$789$		0			0
$790$	5372.00	+	9304.58i	0.241933	+	0.419040i
$791$	32006.2			1.43870
$792$		0			0
$793$	−18797.5			−0.841763
$794$	−6646.07	−	11511.3i	−0.297053	−	0.514511i
$795$		0			0
$796$	−2940.43	+	5092.97i	−0.130931	+	0.226778i
$797$	−14132.9	+	24478.9i	−0.628122	+	1.08794i	0.359806	+	0.933027i	$0.382843\pi$
$797$	−14132.9	+	24478.9i	−0.628122	+	1.08794i	−0.987928	+	0.154912i	$0.950490\pi$
$798$		0			0
$799$	22205.0	+	38460.2i	0.983174	+	1.70291i
$800$	−11.3225			−0.000500390
$801$		0			0
$802$	−3.26896			−0.000143929
$803$	870.065	+	1507.00i	0.0382365	+	0.0662276i
$804$		0			0
$805$	−7126.52	+	12343.5i	−0.312021	+	0.540436i
$806$	21073.2	−	36499.9i	0.920933	−	1.59510i
$807$		0			0
$808$	−826.091	−	1430.83i	−0.0359676	−	0.0622976i
$809$	42553.4			1.84932			0.924659	−	0.380795i	$-0.124350\pi$
$809$	42553.4			1.84932			0.924659	+	0.380795i	$0.124350\pi$
$810$		0			0
$811$	−6900.03			−0.298758			−0.149379	−	0.988780i	$-0.547727\pi$
$811$	−6900.03			−0.298758			−0.149379	+	0.988780i	$0.547727\pi$
$812$	7297.08	+	12638.9i	0.315366	+	0.546230i
$813$		0			0
$814$	−4869.58	+	8434.36i	−0.209679	+	0.363175i
$815$	−2219.94	+	3845.05i	−0.0954123	+	0.165259i
$816$		0			0
$817$	18487.6	+	32021.4i	0.791675	+	1.37122i
$818$	12406.7			0.530306
$819$		0			0
$820$	−5939.81			−0.252960
$821$	11179.6	+	19363.6i	0.475238	+	0.823136i	0.999598	−	0.0283606i	$-0.00902868\pi$
$821$	11179.6	+	19363.6i	0.475238	+	0.823136i	−0.524360	+	0.851497i	$0.675695\pi$
$822$		0			0
$823$	−395.785	+	685.520i	−0.0167633	+	0.0290349i	−0.874285	−	0.485412i	$-0.838670\pi$
$823$	−395.785	+	685.520i	−0.0167633	+	0.0290349i	0.857522	+	0.514447i	$0.172003\pi$
$824$	−5487.08	+	9503.90i	−0.231980	+	0.401801i
$825$		0			0
$826$	6211.51	+	10758.6i	0.261654	+	0.453198i
$827$	−23005.9			−0.967343			−0.483671	−	0.875250i	$-0.660697\pi$
$827$	−23005.9			−0.967343			−0.483671	+	0.875250i	$0.660697\pi$
$828$		0			0
$829$	−15420.2			−0.646040			−0.323020	−	0.946392i	$-0.604698\pi$
$829$	−15420.2			−0.646040			−0.323020	+	0.946392i	$0.604698\pi$
$830$	−2012.72	−	3486.14i	−0.0841718	−	0.145790i
$831$		0			0
$832$	2167.88	−	3754.87i	0.0903336	−	0.156462i
$833$	−276.675	+	479.215i	−0.0115081	+	0.0199326i
$834$		0			0
$835$	−17793.6	−	30819.4i	−0.737453	−	1.27731i
$836$	10400.2			0.430261
$837$		0			0
$838$	−19500.7			−0.803869
$839$	23145.1	+	40088.4i	0.952391	+	1.64959i	0.740228	+	0.672356i	$0.234718\pi$
$839$	23145.1	+	40088.4i	0.952391	+	1.64959i	0.212163	+	0.977234i	$0.431949\pi$
$840$		0			0
$841$	−7481.44	+	12958.2i	−0.306755	+	0.531314i
$842$	−10061.4	+	17426.8i	−0.411803	+	0.713263i
$843$		0			0
$844$	−3072.01	−	5320.88i	−0.125288	−	0.217005i
$845$	−26787.2			−1.09054
$846$		0			0
$847$	−14263.1			−0.578613
$848$	1527.26	+	2645.29i	0.0618471	+	0.107122i
$849$		0			0
$850$	−41.4538	+	71.8001i	−0.00167277	+	0.00289732i
$851$	−7150.22	+	12384.5i	−0.288021	+	0.498868i
$852$		0			0
$853$	−13951.2	−	24164.2i	−0.559999	−	0.969947i	−0.997496	−	0.0707272i	$-0.977468\pi$
$853$	−13951.2	−	24164.2i	−0.559999	−	0.969947i	0.437496	−	0.899220i	$-0.355865\pi$
$854$	−10206.6			−0.408973
$855$		0			0
$856$	−10136.0			−0.404721
$857$	3276.46	+	5675.00i	0.130597	+	0.226201i	0.923907	−	0.382617i	$-0.124977\pi$
$857$	3276.46	+	5675.00i	0.130597	+	0.226201i	−0.793310	+	0.608818i	$0.791644\pi$
$858$		0			0
$859$	−17428.8	+	30187.6i	−0.692275	+	1.19906i	0.278815	+	0.960345i	$0.410058\pi$
$859$	−17428.8	+	30187.6i	−0.692275	+	1.19906i	−0.971091	+	0.238711i	$0.923275\pi$
$860$	−7505.38	+	12999.7i	−0.297595	+	0.515449i
$861$		0			0
$862$	6763.21	+	11714.2i	0.267234	+	0.462863i
$863$	10885.2			0.429359			0.214680	−	0.976685i	$-0.431129\pi$
$863$	10885.2			0.429359			0.214680	+	0.976685i	$0.431129\pi$
$864$		0			0
$865$	24101.4			0.947367
$866$	−10601.4	−	18362.1i	−0.415993	−	0.720520i
$867$		0			0
$868$	11442.3	−	19818.6i	0.447438	−	0.774985i
$869$	5654.35	−	9793.61i	0.220726	−	0.382308i
$870$		0			0
$871$	22527.9	+	39019.5i	0.876383	+	1.51794i
$872$	−13801.0			−0.535966
$873$		0			0
$874$	15271.0			0.591019
$875$	−12833.8	−	22228.7i	−0.495840	−	0.858821i
$876$		0			0
$877$	13956.9	−	24174.1i	0.537390	−	0.930787i	−0.461653	−	0.887061i	$-0.652744\pi$
$877$	13956.9	−	24174.1i	0.537390	−	0.930787i	0.999044	−	0.0437269i	$-0.0139231\pi$
$878$	−12568.9	+	21770.0i	−0.483121	+	0.836791i
$879$		0			0
$880$	2111.08	+	3656.49i	0.0808686	+	0.140069i
$881$	−10694.5			−0.408975			−0.204488	−	0.978869i	$-0.565553\pi$
$881$	−10694.5			−0.408975			−0.204488	+	0.978869i	$0.565553\pi$
$882$		0			0
$883$	3265.74			0.124463			0.0622315	−	0.998062i	$-0.480178\pi$
$883$	3265.74			0.124463			0.0622315	+	0.998062i	$0.480178\pi$
$884$	−15874.0	−	27494.5i	−0.603958	−	1.04609i
$885$		0			0
$886$	10255.4	−	17762.9i	0.388869	−	0.673541i
$887$	4696.44	−	8134.47i	0.177780	−	0.307924i	−0.763340	−	0.645997i	$-0.776442\pi$
$887$	4696.44	−	8134.47i	0.177780	−	0.307924i	0.941120	+	0.338073i	$0.109775\pi$
$888$		0			0
$889$	−4525.71	−	7838.75i	−0.170739	−	0.295729i
$890$	−18954.1			−0.713870
$891$		0			0
$892$	−6630.12			−0.248871
$893$	−20908.2	−	36214.0i	−0.783499	−	1.35706i
$894$		0			0
$895$	25137.0	−	43538.6i	0.938812	−	1.62607i
$896$	1177.11	−	2038.81i	0.0438888	−	0.0760177i
$897$		0			0
$898$	4080.23	+	7067.17i	0.151625	+	0.262622i
$899$	61706.2			2.28923
$900$		0			0
$901$	22366.3			0.827002
$902$	3126.00	+	5414.39i	0.115393	+	0.199867i
$903$		0			0
$904$	6960.78	−	12056.4i	0.256098	−	0.443574i
$905$	7878.30	−	13645.6i	0.289374	−	0.501211i
$906$		0			0
$907$	−6769.24	−	11724.7i	−0.247816	−	0.429229i	0.715104	−	0.699018i	$-0.246379\pi$
$907$	−6769.24	−	11724.7i	−0.247816	−	0.429229i	−0.962919	+	0.269789i	$0.913046\pi$
$908$	6057.04			0.221377
$909$		0			0
$910$	−27901.0			−1.01638
$911$	18903.2	+	32741.3i	0.687476	+	1.19074i	0.972652	+	0.232269i	$0.0746149\pi$
$911$	18903.2	+	32741.3i	0.687476	+	1.19074i	−0.285175	+	0.958475i	$0.592052\pi$
$912$		0			0
$913$	−2118.51	+	3669.37i	−0.0767935	+	0.133010i
$914$	−2183.20	+	3781.41i	−0.0790086	+	0.136847i
$915$		0			0
$916$	−8598.08	−	14892.3i	−0.310140	−	0.537179i
$917$	−35299.9			−1.27122
$918$		0			0
$919$	30674.2			1.10103			0.550515	−	0.834825i	$-0.314431\pi$
$919$	30674.2			1.10103			0.550515	+	0.834825i	$0.314431\pi$
$920$	3099.78	+	5368.98i	0.111084	+	0.192402i
$921$		0			0
$922$	−3250.26	+	5629.62i	−0.116097	+	0.201086i
$923$	17899.6	−	31003.0i	0.638322	−	1.10561i
$924$		0			0
$925$	36.5519	+	63.3098i	0.00129926	+	0.00225039i
$926$	−37982.3			−1.34792
$927$		0			0
$928$	6347.94			0.224549
$929$	14058.7	+	24350.4i	0.496502	+	0.859967i	0.999992	−	0.00403418i	$-0.00128412\pi$
$929$	14058.7	+	24350.4i	0.496502	+	0.859967i	−0.503490	+	0.864001i	$0.667951\pi$
$930$		0			0
$931$	260.516	−	451.228i	0.00917087	−	0.0158844i
$932$	2673.56	−	4630.74i	0.0939650	−	0.162752i
$933$		0			0
$934$	−6906.52	−	11962.4i	−0.241957	−	0.419083i
$935$	30916.1			1.08135
$936$		0			0
$937$	31859.0			1.11077			0.555384	−	0.831594i	$-0.312571\pi$
$937$	31859.0			1.11077			0.555384	+	0.831594i	$0.312571\pi$
$938$	12232.2	+	21186.7i	0.425793	+	0.737495i
$939$		0			0
$940$	8488.06	−	14701.8i	0.294521	−	0.510126i
$941$	−1131.57	+	1959.93i	−0.0392009	+	0.0678980i	−0.884960	−	0.465667i	$-0.845815\pi$
$941$	−1131.57	+	1959.93i	−0.0392009	+	0.0678980i	0.845759	+	0.533565i	$0.179148\pi$
$942$		0			0
$943$	4590.04	+	7950.19i	0.158507	+	0.274543i
$944$	5403.57			0.186304
$945$		0			0
$946$	15799.7			0.543016
$947$	−3492.67	−	6049.48i	−0.119848	−	0.207583i	0.799859	−	0.600188i	$-0.204908\pi$
$947$	−3492.67	−	6049.48i	−0.119848	−	0.207583i	−0.919707	+	0.392604i	$0.871574\pi$
$948$		0			0
$949$	−2500.87	+	4331.63i	−0.0855444	+	0.148167i
$950$	39.0328	−	67.6068i	0.00133304	−	0.00230890i
$951$		0			0
$952$	−8619.20	−	14928.9i	−0.293435	−	0.508244i
$953$	−26436.8			−0.898608			−0.449304	−	0.893379i	$-0.648328\pi$
$953$	−26436.8			−0.898608			−0.449304	+	0.893379i	$0.648328\pi$
$954$		0			0
$955$	8646.53			0.292979
$956$	−13757.3	−	23828.3i	−0.465420	−	0.806131i
$957$		0			0
$958$	−7380.27	+	12783.0i	−0.248900	+	0.431107i
$959$	24120.2	−	41777.4i	0.812181	−	1.40674i
$960$		0			0
$961$	−33484.1	−	57996.2i	−1.12397	−	1.94677i
$962$	−27993.7			−0.938206
$963$		0			0
$964$	6125.17			0.204646
$965$	20448.0	+	35416.9i	0.682117	+	1.18146i
$966$		0			0
$967$	50.0781	−	86.7379i	0.00166536	−	0.00288449i	−0.865192	−	0.501442i	$-0.832803\pi$
$967$	50.0781	−	86.7379i	0.00166536	−	0.00288449i	0.866857	+	0.498557i	$0.166137\pi$
$968$	−3101.97	+	5372.77i	−0.102997	+	0.178396i
$969$		0			0
$970$	7534.84	+	13050.7i	0.249411	+	0.431993i
$971$	−678.145			−0.0224127			−0.0112063	−	0.999937i	$-0.503567\pi$
$971$	−678.145			−0.0224127			−0.0112063	+	0.999937i	$0.503567\pi$
$972$		0			0
$973$	−11560.8			−0.380908
$974$	−8756.51	−	15166.7i	−0.288066	−	0.498945i
$975$		0			0
$976$	−2219.75	+	3844.73i	−0.0727998	+	0.126093i
$977$	−4874.13	+	8442.24i	−0.159608	+	0.276449i	−0.934727	−	0.355366i	$-0.884356\pi$
$977$	−4874.13	+	8442.24i	−0.159608	+	0.276449i	0.775119	+	0.631815i	$0.217690\pi$
$978$		0			0
$979$	9975.17	+	17277.5i	0.325646	+	0.564036i
$980$	211.523			0.00689476
$981$		0			0
$982$	23675.9			0.769376
$983$	−1757.84	−	3044.67i	−0.0570361	−	0.0987894i	0.836098	−	0.548581i	$-0.184832\pi$
$983$	−1757.84	−	3044.67i	−0.0570361	−	0.0987894i	−0.893134	+	0.449791i	$0.851498\pi$
$984$		0			0
$985$	−14820.4	+	25669.6i	−0.479407	+	0.830358i
$986$	23240.9	−	40254.5i	0.750651	−	1.30017i
$987$		0			0
$988$	14946.9	+	25888.8i	0.481299	+	0.833635i
$989$	23199.4			0.745903
$990$		0			0
$991$	−612.517			−0.0196339			−0.00981697	−	0.999952i	$-0.503125\pi$
$991$	−612.517			−0.0196339			−0.00981697	+	0.999952i	$0.503125\pi$
$992$	−4976.98	−	8620.39i	−0.159294	−	0.275905i
$993$		0			0
$994$	9719.06	−	16833.9i	0.310131	−	0.537162i
$995$	8230.38	−	14255.4i	0.262232	−	0.454198i
$996$		0			0
$997$	−11478.2	−	19880.8i	−0.364611	−	0.631525i	0.624102	−	0.781343i	$-0.285465\pi$
$997$	−11478.2	−	19880.8i	−0.364611	−	0.631525i	−0.988714	+	0.149817i	$0.952131\pi$
$998$	−17156.8			−0.544176
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.c.j.109.2		4
3.2	odd	2		162.4.c.i.109.1		4
9.2	odd	6		162.4.c.i.55.1		4
9.4	even	3		162.4.a.e.1.1	✓	2
9.5	odd	6		162.4.a.h.1.2	yes	2
9.7	even	3	inner	162.4.c.j.55.2		4
36.23	even	6		1296.4.a.s.1.2		2
36.31	odd	6		1296.4.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.a.e.1.1	✓	2	9.4	even	3
162.4.a.h.1.2	yes	2	9.5	odd	6
162.4.c.i.55.1		4	9.2	odd	6
162.4.c.i.109.1		4	3.2	odd	2
162.4.c.j.55.2		4	9.7	even	3	inner
162.4.c.j.109.2		4	1.1	even	1	trivial
1296.4.a.j.1.1		2	36.31	odd	6
1296.4.a.s.1.2		2	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(5.59808 - 9.69615i) q^{5} +(-9.19615 - 15.9282i) q^{7} -8.00000 q^{8} +22.3923 q^{10} +(-11.7846 - 20.4115i) q^{11} +(33.8731 - 58.6699i) q^{13} +(18.3923 - 31.8564i) q^{14} +(-8.00000 - 13.8564i) q^{16} -117.158 q^{17} +110.315 q^{19} +(22.3923 + 38.7846i) q^{20} +(23.5692 - 40.8231i) q^{22} +(34.6077 - 59.9423i) q^{23} +(-0.176915 - 0.306425i) q^{25} +135.492 q^{26} +73.5692 q^{28} +(99.1865 + 171.796i) q^{29} +(155.531 - 269.387i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-117.158 - 202.923i) q^{34} -205.923 q^{35} -206.608 q^{37} +(110.315 + 191.072i) q^{38} +(-44.7846 + 77.5692i) q^{40} +(-66.3154 + 114.862i) q^{41} +(167.588 + 290.272i) q^{43} +94.2769 q^{44} +138.431 q^{46} +(-189.531 - 328.277i) q^{47} +(2.36156 - 4.09034i) q^{49} +(0.353829 - 0.612850i) q^{50} +(135.492 + 234.679i) q^{52} -190.908 q^{53} -263.885 q^{55} +(73.5692 + 127.426i) q^{56} +(-198.373 + 343.592i) q^{58} +(-168.862 + 292.477i) q^{59} +(-138.735 - 240.295i) q^{61} +622.123 q^{62} +64.0000 q^{64} +(-379.248 - 656.877i) q^{65} +(-332.535 + 575.967i) q^{67} +(234.315 - 405.846i) q^{68} +(-205.923 - 356.669i) q^{70} +528.431 q^{71} -73.8306 q^{73} +(-206.608 - 357.855i) q^{74} +(-220.631 + 382.144i) q^{76} +(-216.746 + 375.415i) q^{77} +(239.904 + 415.526i) q^{79} -179.138 q^{80} -265.261 q^{82} +(-89.8846 - 155.685i) q^{83} +(-655.858 + 1135.98i) q^{85} +(-335.177 + 580.543i) q^{86} +(94.2769 + 163.292i) q^{88} -846.458 q^{89} -1246.01 q^{91} +(138.431 + 239.769i) q^{92} +(379.061 - 656.554i) q^{94} +(617.554 - 1069.63i) q^{95} +(336.492 + 582.822i) q^{97} +9.44624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{4} + 12 q^{5} - 16 q^{7} - 32 q^{8} + 48 q^{10} + 36 q^{11} - 10 q^{13} + 32 q^{14} - 32 q^{16} - 240 q^{17} - 16 q^{19} + 48 q^{20} - 72 q^{22} + 180 q^{23} + 124 q^{25} - 40 q^{26}+ \cdots + 1368 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 8 * q^4 + 12 * q^5 - 16 * q^7 - 32 * q^8 + 48 * q^10 + 36 * q^11 - 10 * q^13 + 32 * q^14 - 32 * q^16 - 240 * q^17 - 16 * q^19 + 48 * q^20 - 72 * q^22 + 180 * q^23 + 124 * q^25 - 40 * q^26 + 128 * q^28 + 324 * q^29 + 248 * q^31 + 64 * q^32 - 240 * q^34 - 408 * q^35 - 868 * q^37 - 16 * q^38 - 96 * q^40 + 192 * q^41 + 608 * q^43 - 288 * q^44 + 720 * q^46 - 384 * q^47 + 342 * q^49 - 248 * q^50 - 40 * q^52 + 816 * q^53 - 432 * q^55 + 128 * q^56 - 648 * q^58 - 1008 * q^59 - 742 * q^61 + 992 * q^62 + 256 * q^64 - 696 * q^65 + 104 * q^67 + 480 * q^68 - 408 * q^70 + 2280 * q^71 + 1700 * q^73 - 868 * q^74 + 32 * q^76 - 576 * q^77 + 440 * q^79 - 384 * q^80 + 768 * q^82 + 264 * q^83 - 1314 * q^85 - 1216 * q^86 - 288 * q^88 - 1536 * q^89 - 2864 * q^91 + 720 * q^92 + 768 * q^94 + 1140 * q^95 + 764 * q^97 + 1368 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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