LMFDB - Embedded newform 162.4.c.i.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,4,Mod(55,162)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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			55.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	162.55
Dual form			162.4.c.i.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$83$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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	−1.00000		0.000816748i	$-0.999740\pi$
$5$	−5.59808	−	9.69615i	−0.500707	−	0.867250i	0.499293	−	0.866433i	$-0.333593\pi$
$6$		0			0
$7$	−9.19615	+	15.9282i	−0.496546	+	0.860042i	−0.999992	−	0.00398426i	$-0.998732\pi$
$7$	−9.19615	+	15.9282i	−0.496546	+	0.860042i	0.503447	+	0.864026i	$0.332065\pi$
$8$	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	$-0.728636\pi$
$11$	11.7846	−	20.4115i	0.323018	−	0.559483i	0.981109	+	0.193455i	$0.0619694\pi$
$12$		0			0
$13$	33.8731	+	58.6699i	0.722669	+	1.25170i	0.959926	+	0.280253i	$0.0904183\pi$
$13$	33.8731	+	58.6699i	0.722669	+	1.25170i	−0.237257	+	0.971447i	$0.576248\pi$
$14$	−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	$-0.185043\pi$
$17$	117.158			1.67147			0.835733	+	0.549137i	$0.185043\pi$
$18$		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	$-0.268251\pi$
$23$	−34.6077	−	59.9423i	−0.313748	−	0.543427i	−0.979171	+	0.203039i	$0.934918\pi$
$24$		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	0.351370	+	0.936237i	$0.385716\pi$
$29$	−99.1865	+	171.796i	−0.635120	+	1.10006i	−0.986490	+	0.163823i	$0.947617\pi$
$30$		0			0
$31$	155.531	+	269.387i	0.901101	+	1.56075i	0.826066	+	0.563573i	$0.190574\pi$
$31$	155.531	+	269.387i	0.901101	+	1.56075i	0.0750350	+	0.997181i	$0.476093\pi$
$32$	−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	$-0.0853800\pi$
$41$	66.3154	+	114.862i	0.252603	+	0.437521i	−0.711639	+	0.702546i	$0.752047\pi$
$42$		0			0
$43$	167.588	−	290.272i	0.594349	−	1.02944i	−0.399290	−	0.916825i	$-0.630743\pi$
$43$	167.588	−	290.272i	0.594349	−	1.02944i	0.993638	−	0.112618i	$-0.0359235\pi$
$44$	−94.2769			−0.323018
$45$		0			0
$46$	138.431			0.443707
$47$	189.531	−	328.277i	0.588211	−	1.01881i	−0.406256	−	0.913759i	$-0.633166\pi$
$47$	189.531	−	328.277i	0.588211	−	1.01881i	0.994467	−	0.105051i	$-0.0335007\pi$
$48$		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	$-0.420428\pi$
$53$	190.908			0.494777			0.247388	+	0.968916i	$0.420428\pi$
$54$		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	$-0.0451301\pi$
$59$	168.862	+	292.477i	0.372609	+	0.645377i	−0.617357	+	0.786683i	$0.711797\pi$
$60$		0			0
$61$	−138.735	+	240.295i	−0.291199	+	0.504372i	−0.974094	−	0.226145i	$-0.927388\pi$
$61$	−138.735	+	240.295i	−0.291199	+	0.504372i	0.682894	+	0.730517i	$0.260721\pi$
$62$	−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	−0.991836	−	0.127518i	$-0.959299\pi$
$67$	−332.535	−	575.967i	−0.606352	−	1.05023i	0.385485	−	0.922714i	$-0.374034\pi$
$68$	−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	$-0.645604\pi$
$71$	−528.431			−0.883284			−0.441642	+	0.897191i	$0.645604\pi$
$72$		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	−0.643080	−	0.765799i	$-0.722344\pi$
$79$	239.904	−	415.526i	0.341662	−	0.591776i	0.984741	+	0.174024i	$0.0556769\pi$
$80$	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	$-0.795407\pi$
$83$	89.8846	−	155.685i	0.118869	−	0.205887i	0.919320	+	0.393512i	$0.128740\pi$
$84$		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	$-0.331836\pi$
$89$	846.458			1.00814			0.504069	+	0.863663i	$0.331836\pi$
$90$		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	−0.634416	−	0.772992i	$-0.718759\pi$
$97$	336.492	−	582.822i	0.352223	−	0.610068i	0.986639	+	0.162924i	$0.0520926\pi$
$98$	−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	$-0.865772\pi$
$101$	−103.261	+	178.854i	−0.101732	+	0.176204i	0.810667	+	0.585508i	$0.199105\pi$
$102$		0			0
$103$	685.885	+	1187.99i	0.656138	+	1.13646i	0.981607	+	0.190912i	$0.0611444\pi$
$103$	685.885	+	1187.99i	0.656138	+	1.13646i	−0.325469	+	0.945553i	$0.605522\pi$
$104$	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	$-0.693973\pi$
$107$	−1267.00			−1.14472			−0.572362	+	0.820001i	$0.693973\pi$
$108$		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	0.959240	+	0.282594i	$0.0911949\pi$
$113$	870.098	+	1507.05i	0.724353	+	1.25462i	−0.234886	+	0.972023i	$0.575472\pi$
$114$		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	−0.985425	−	0.170108i	$-0.945588\pi$
$131$	−959.638	−	1662.14i	−0.640030	−	1.10857i	0.345395	−	0.938457i	$-0.387745\pi$
$132$		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	0.0894453	+	0.995992i	$0.471491\pi$
$137$	−1311.43	+	2271.46i	−0.817832	+	1.41653i	−0.907277	+	0.420534i	$0.861843\pi$
$138$		0			0
$139$	314.284	+	544.357i	0.191779	+	0.332171i	0.945840	−	0.324634i	$-0.105241\pi$
$139$	314.284	+	544.357i	0.191779	+	0.332171i	−0.754061	+	0.656804i	$0.771908\pi$
$140$	−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	$-0.216602\pi$
$149$	−284.156	−	492.173i	−0.156235	−	0.270606i	−0.933508	+	0.358557i	$0.883269\pi$
$150$		0			0
$151$	−178.715	+	309.544i	−0.0963155	+	0.166823i	−0.910157	−	0.414264i	$-0.864039\pi$
$151$	−178.715	+	309.544i	−0.0963155	+	0.166823i	0.813841	+	0.581087i	$0.197372\pi$
$152$	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	0.943524	−	0.331304i	$-0.107489\pi$
$157$	363.627	+	629.820i	0.184844	+	0.320160i	−0.758680	+	0.651464i	$0.774155\pi$
$158$	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	−0.954101	−	0.299484i	$-0.903185\pi$
$167$	−1589.26	−	2752.68i	−0.736411	−	1.27550i	0.217690	−	0.976018i	$-0.430148\pi$
$168$		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	$-0.990167\pi$
$173$	−1076.32	+	1864.25i	−0.473014	+	0.819285i	0.526509	+	0.850170i	$0.323501\pi$
$174$		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	$-0.886854\pi$
$179$	−4490.29			−1.87497			−0.937487	+	0.348022i	$0.886854\pi$
$180$		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	$-0.880064\pi$
$191$	−386.138	+	668.811i	−0.146283	+	0.253369i	0.783568	+	0.621306i	$0.213398\pi$
$192$		0			0
$193$	−1826.34	−	3163.31i	−0.681154	−	1.17979i	−0.974629	−	0.223826i	$-0.928145\pi$
$193$	−1826.34	−	3163.31i	−0.681154	−	1.17979i	0.293475	−	0.955967i	$-0.405188\pi$
$194$	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	$-0.341097\pi$
$197$	2647.40			0.957460			0.478730	+	0.877962i	$0.341097\pi$
$198$		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	0.713109	−	0.701054i	$-0.247287\pi$
$211$	−768.003	−	1330.22i	−0.250576	−	0.434010i	−0.963685	+	0.267043i	$0.913953\pi$
$212$	−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	−0.714342	−	0.699797i	$-0.753274\pi$
$223$	828.765	−	1435.46i	0.248871	−	0.431057i	0.963213	+	0.268740i	$0.0866071\pi$
$224$	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	$-0.762278\pi$
$227$	757.131	−	1311.39i	0.221377	−	0.383436i	0.955226	+	0.295876i	$0.0956116\pi$
$228$		0			0
$229$	−2149.52	−	3723.08i	−0.620280	−	1.07436i	−0.989433	−	0.144988i	$-0.953685\pi$
$229$	−2149.52	−	3723.08i	−0.620280	−	1.07436i	0.369153	−	0.929369i	$-0.379648\pi$
$230$	−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	$-0.439822\pi$
$233$	1336.78			0.375860			0.187930	+	0.982182i	$0.439822\pi$
$234$		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	0.781889	+	0.623418i	$0.214256\pi$
$239$	3439.31	+	5957.07i	0.930840	+	1.61226i	0.148951	+	0.988845i	$0.452410\pi$
$240$		0			0
$241$	−765.646	+	1326.14i	−0.204646	+	0.354457i	−0.950020	−	0.312190i	$-0.898938\pi$
$241$	−765.646	+	1326.14i	−0.204646	+	0.354457i	0.745374	+	0.666646i	$0.232271\pi$
$242$	−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	$-0.452533\pi$
$251$	1181.60			0.297139			0.148570	+	0.988902i	$0.452533\pi$
$252$		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	−0.999315	−	0.0370161i	$-0.988215\pi$
$257$	−2190.68	−	3794.36i	−0.531714	−	0.920956i	0.467600	−	0.883940i	$-0.345119\pi$
$258$		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	$-0.726874\pi$
$263$	1400.04	−	2424.94i	0.328251	−	0.568548i	0.982165	+	0.188021i	$0.0602074\pi$
$264$		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	$-0.397075\pi$
$269$	2803.73			0.635489			0.317745	+	0.948176i	$0.397075\pi$
$270$		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	−0.911936	−	0.410332i	$-0.865413\pi$
$277$	−463.831	+	803.378i	−0.100610	+	0.174261i	0.811326	+	0.584594i	$0.198746\pi$
$278$	−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	$-0.871998\pi$
$281$	−570.764	+	988.592i	−0.121170	+	0.209873i	0.799059	+	0.601253i	$0.205331\pi$
$282$		0			0
$283$	−1805.52	−	3127.26i	−0.379248	−	0.656877i	0.611705	−	0.791086i	$-0.290484\pi$
$283$	−1805.52	−	3127.26i	−0.379248	−	0.656877i	−0.990953	+	0.134209i	$0.957151\pi$
$284$	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	$-0.241099\pi$
$293$	−1162.11	−	2012.83i	−0.231710	−	0.401333i	−0.958311	+	0.285726i	$0.907765\pi$
$294$		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	−0.995707	−	0.0925637i	$-0.970494\pi$
$311$	−3170.15	−	5490.87i	−0.578016	−	1.00115i	0.417691	−	0.908589i	$-0.362840\pi$
$312$		0			0
$313$	−1194.42	+	2068.79i	−0.215694	+	0.373594i	−0.953487	−	0.301434i	$-0.902535\pi$
$313$	−1194.42	+	2068.79i	−0.215694	+	0.373594i	0.737793	+	0.675027i	$0.235868\pi$
$314$	−1454.51			−0.261409
$315$		0			0
$316$	−1919.23			−0.341662
$317$	2930.63	−	5076.00i	0.519245	−	0.899359i	−0.480505	−	0.876992i	$-0.659547\pi$
$317$	2930.63	−	5076.00i	0.519245	−	0.899359i	0.999750	−	0.0223668i	$-0.00712015\pi$
$318$		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	−0.995134	−	0.0985339i	$-0.968585\pi$
$331$	−2482.48	+	4299.78i	−0.412234	+	0.714010i	0.582900	+	0.812544i	$0.301918\pi$
$332$	−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	0.963624	−	0.267262i	$-0.0861188\pi$
$337$	1548.83	+	2682.65i	0.250357	+	0.433630i	−0.713267	+	0.700892i	$0.752785\pi$
$338$	−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	−0.989067	−	0.147464i	$-0.952889\pi$
$347$	−4022.10	−	6966.48i	−0.622241	−	1.07775i	0.366827	−	0.930289i	$-0.380444\pi$
$348$		0			0
$349$	572.353	−	991.345i	0.0877862	−	0.152050i	−0.818789	−	0.574095i	$-0.805354\pi$
$349$	572.353	−	991.345i	0.0877862	−	0.152050i	0.906575	+	0.422045i	$0.138687\pi$
$350$	13.0155			0.00198773
$351$		0			0
$352$	−754.215			−0.114204
$353$	−3900.63	+	6756.09i	−0.588129	+	1.01867i	0.406348	+	0.913718i	$0.366802\pi$
$353$	−3900.63	+	6756.09i	−0.588129	+	1.01867i	−0.994477	+	0.104951i	$0.966531\pi$
$354$		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	$-0.492013\pi$
$359$	341.307			0.0501769			0.0250885	+	0.999685i	$0.492013\pi$
$360$		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	−0.870239	−	0.492630i	$-0.836036\pi$
$367$	−59.6839	+	103.376i	−0.00848903	+	0.0147034i	0.861750	+	0.507334i	$0.169369\pi$
$368$	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	0.976953	−	0.213456i	$-0.0684718\pi$
$373$	2187.22	+	3788.37i	0.303619	+	0.525883i	−0.673334	+	0.739338i	$0.735139\pi$
$374$	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	$-0.171051\pi$
$383$	−103.232	−	178.802i	−0.0137726	−	0.0238548i	−0.872830	+	0.488025i	$0.837717\pi$
$384$		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	$-0.791121\pi$
$389$	1014.47	−	1757.11i	0.132225	−	0.229021i	0.924534	+	0.381099i	$0.124454\pi$
$390$		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	$-0.166634\pi$
$401$	0.817240	+	1.41550i	0.000101773	+	0.000176276i	−0.865975	+	0.500088i	$0.833301\pi$
$402$		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	0.990324	−	0.138771i	$-0.0443152\pi$
$409$	3101.68	+	5372.26i	0.374983	+	0.649490i	−0.615342	+	0.788261i	$0.710982\pi$
$410$	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	0.996722	+	0.0808979i	$0.0257788\pi$
$419$	4875.18	+	8444.07i	0.568421	+	0.984534i	−0.428302	+	0.903636i	$0.640888\pi$
$420$		0			0
$421$	5030.69	−	8713.41i	0.582377	−	1.00871i	−0.412820	−	0.910813i	$-0.635456\pi$
$421$	5030.69	−	8713.41i	0.582377	−	1.00871i	0.995197	−	0.0978939i	$-0.0312106\pi$
$422$	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	$-0.623363\pi$
$431$	−6763.21			−0.755853			−0.377926	+	0.925836i	$0.623363\pi$
$432$		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	−0.290751	−	0.956799i	$-0.593905\pi$
$439$	6284.46	−	10885.0i	0.683237	−	1.18340i	0.973987	−	0.226602i	$-0.0727616\pi$
$440$	−2111.08			−0.228731
$441$		0			0
$442$	−15874.0			−1.70825
$443$	5127.71	−	8881.46i	0.549944	−	0.952530i	−0.448334	−	0.893866i	$-0.647983\pi$
$443$	5127.71	−	8881.46i	0.549944	−	0.952530i	0.998278	−	0.0586643i	$-0.0186842\pi$
$444$		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	$-0.568789\pi$
$449$	−4080.23			−0.428860			−0.214430	+	0.976739i	$0.568789\pi$
$450$		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	−0.804735	−	0.593634i	$-0.797693\pi$
$457$	1091.60	−	1890.71i	0.111735	−	0.193531i	0.916470	+	0.400104i	$0.131026\pi$
$458$	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	$-0.885833\pi$
$461$	−1625.13	+	2814.81i	−0.164186	+	0.284379i	0.772180	+	0.635404i	$0.219166\pi$
$462$		0			0
$463$	−9495.57	−	16446.8i	−0.953124	−	1.65086i	−0.738604	−	0.674140i	$-0.764515\pi$
$463$	−9495.57	−	16446.8i	−0.953124	−	1.65086i	−0.214520	−	0.976720i	$-0.568819\pi$
$464$	−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	$-0.388835\pi$
$467$	6906.52			0.684359			0.342180	+	0.939635i	$0.388835\pi$
$468$		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	$-0.947831\pi$
$479$	−3690.14	+	6391.50i	−0.351997	+	0.609677i	0.634602	+	0.772839i	$0.281164\pi$
$480$		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	−0.998667	−	0.0516124i	$-0.983564\pi$
$491$	−5918.97	−	10252.0i	−0.544031	−	0.942290i	0.454636	−	0.890677i	$-0.349769\pi$
$492$		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	0.606950	−	0.794740i	$-0.292393\pi$
$499$	−4289.19	−	7429.09i	−0.384791	−	0.666477i	−0.991740	+	0.128264i	$0.959060\pi$
$500$	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	$-0.551167\pi$
$503$	−3611.17			−0.320107			−0.160054	+	0.987108i	$0.551167\pi$
$504$		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	$-0.295737\pi$
$509$	−4529.87	−	7845.96i	−0.394465	−	0.683234i	−0.993033	+	0.117838i	$0.962404\pi$
$510$		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	$-0.681428\pi$
$521$	−12834.1			−1.07922			−0.539609	+	0.841916i	$0.681428\pi$
$522$		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	−0.982998	−	0.183617i	$-0.941219\pi$
$547$	−4253.52	+	7367.32i	−0.332482	+	0.575875i	0.650516	+	0.759492i	$0.274553\pi$
$548$	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	$-0.507707\pi$
$557$	−636.486			−0.0484179			−0.0242090	+	0.999707i	$0.507707\pi$
$558$		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	−0.999961	−	0.00883838i	$-0.997187\pi$
$563$	−6781.31	−	11745.6i	−0.507635	−	0.879249i	0.492326	−	0.870411i	$-0.336147\pi$
$564$		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	−0.155236	−	0.987877i	$-0.549614\pi$
$569$	10558.4	−	18287.6i	0.777909	−	1.34738i	0.933145	−	0.359500i	$-0.117053\pi$
$570$		0			0
$571$	7263.55	+	12580.8i	0.532347	+	0.922052i	0.999287	+	0.0377630i	$0.0120232\pi$
$571$	7263.55	+	12580.8i	0.532347	+	0.922052i	−0.466940	+	0.884289i	$0.654643\pi$
$572$	−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	0.339507	+	0.940603i	$0.389740\pi$
$587$	−9170.74	+	15884.2i	−0.644833	+	1.11688i	−0.984340	+	0.176280i	$0.943594\pi$
$588$		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	$-0.0593504\pi$
$593$	28380.4			1.96534			0.982668	+	0.185376i	$0.0593504\pi$
$594$		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	0.909695	+	0.415276i	$0.136315\pi$
$599$	11940.6	+	20681.7i	0.814487	+	1.41073i	−0.0952079	+	0.995457i	$0.530352\pi$
$600$		0			0
$601$	−8310.38	+	14394.0i	−0.564039	+	0.976944i	0.433100	+	0.901346i	$0.357420\pi$
$601$	−8310.38	+	14394.0i	−0.564039	+	0.976944i	−0.997138	+	0.0755978i	$0.975914\pi$
$602$	−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	0.891089	−	0.453829i	$-0.149942\pi$
$607$	785.389	+	1360.33i	0.0525172	+	0.0909624i	−0.838572	+	0.544791i	$0.816609\pi$
$608$	−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	$-0.239607\pi$
$617$	−3481.28	−	6029.76i	−0.227149	−	0.393434i	−0.956962	+	0.290213i	$0.906274\pi$
$618$		0			0
$619$	−1330.21	+	2303.99i	−0.0863741	+	0.149604i	−0.905976	−	0.423329i	$-0.860861\pi$
$619$	−1330.21	+	2303.99i	−0.0863741	+	0.149604i	0.819602	+	0.572934i	$0.194195\pi$
$620$	−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	0.114385	+	0.993436i	$0.463510\pi$
$641$	−13034.2	+	22575.8i	−0.803149	+	1.39109i	−0.917534	+	0.397658i	$0.869823\pi$
$642$		0			0
$643$	−10585.2	−	18334.1i	−0.649207	−	1.12446i	−0.983313	−	0.181923i	$-0.941768\pi$
$643$	−10585.2	−	18334.1i	−0.649207	−	1.12446i	0.334106	−	0.942535i	$-0.391566\pi$
$644$	−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	$-0.418942\pi$
$647$	8291.45			0.503818			0.251909	+	0.967751i	$0.418942\pi$
$648$		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	−0.952648	−	0.304077i	$-0.901652\pi$
$653$	−12342.5	−	21377.8i	−0.739662	−	1.28113i	0.212985	−	0.977055i	$-0.431681\pi$
$654$		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	0.0399839	−	0.999200i	$-0.487269\pi$
$659$	14977.2	−	25941.3i	0.885325	−	1.53343i	0.845341	−	0.534227i	$-0.179397\pi$
$660$		0			0
$661$	2063.38	+	3573.88i	0.121416	+	0.210299i	0.920326	−	0.391151i	$-0.127923\pi$
$661$	2063.38	+	3573.88i	0.121416	+	0.210299i	−0.798910	+	0.601450i	$0.794590\pi$
$662$	−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	−0.191812	−	0.981432i	$-0.561436\pi$
$673$	13164.9	−	22802.2i	0.754039	−	1.30603i	0.945851	−	0.324602i	$-0.105230\pi$
$674$	−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	$-0.893356\pi$
$677$	−3301.93	+	5719.11i	−0.187450	+	0.324673i	0.756949	+	0.653473i	$0.226689\pi$
$678$		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	$-0.384165\pi$
$683$	12706.4			0.711854			0.355927	+	0.934514i	$0.384165\pi$
$684$		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	−0.564131	−	0.825686i	$-0.690789\pi$
$691$	7865.10	−	13622.8i	0.432999	−	0.749977i	0.997130	+	0.0757086i	$0.0241219\pi$
$692$	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	$-0.505600\pi$
$701$	−652.959			−0.0351811			−0.0175905	+	0.999845i	$0.505600\pi$
$702$		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	−0.321793	−	0.946810i	$-0.604285\pi$
$709$	12442.2	−	21550.6i	0.659065	−	1.14153i	0.980858	−	0.194724i	$-0.0623812\pi$
$710$	−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	$-0.313475\pi$
$719$	21323.8			1.10604			0.553020	+	0.833168i	$0.313475\pi$
$720$		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	−0.950093	−	0.311965i	$-0.899013\pi$
$727$	−4016.00	+	6955.92i	−0.204877	+	0.354857i	0.745217	+	0.666822i	$0.232346\pi$
$728$	−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	−0.962189	−	0.272381i	$-0.912189\pi$
$733$	−14228.7	−	24644.8i	−0.716984	−	1.24185i	0.245205	−	0.969471i	$-0.421145\pi$
$734$	−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	$-0.203312\pi$
$743$	−2326.45	−	4029.54i	−0.114871	−	0.198963i	−0.917728	+	0.397209i	$0.869979\pi$
$744$		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	0.996106	−	0.0881649i	$-0.0281003\pi$
$751$	8678.87	+	15032.3i	0.421700	+	0.730406i	−0.574406	+	0.818571i	$0.694767\pi$
$752$	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	$-0.0991194\pi$
$761$	4421.51	+	7658.28i	0.210617	+	0.364799i	−0.741291	+	0.671184i	$0.765786\pi$
$762$		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	0.832725	−	0.553687i	$-0.186779\pi$
$769$	−1346.55	−	2332.30i	−0.0631442	−	0.109369i	−0.895869	+	0.444318i	$0.853446\pi$
$770$	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	$-0.638489\pi$
$773$	−18116.5			−0.842958			−0.421479	+	0.906838i	$0.638489\pi$
$774$		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	−0.961404	−	0.275140i	$-0.911276\pi$
$787$	−15873.7	−	27494.1i	−0.718980	−	1.24531i	0.242424	−	0.970170i	$-0.422057\pi$
$788$	−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	0.987928	+	0.154912i	$0.0495095\pi$
$797$	14132.9	+	24478.9i	0.628122	+	1.08794i	−0.359806	+	0.933027i	$0.617157\pi$
$798$		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	$-0.875650\pi$
$809$	−42553.4			−1.84932			−0.924659	+	0.380795i	$0.875650\pi$
$810$		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	$-0.990971\pi$
$821$	−11179.6	+	19363.6i	−0.475238	+	0.823136i	0.524360	+	0.851497i	$0.324305\pi$
$822$		0			0
$823$	−395.785	−	685.520i	−0.0167633	−	0.0290349i	0.857522	−	0.514447i	$-0.172003\pi$
$823$	−395.785	−	685.520i	−0.0167633	−	0.0290349i	−0.874285	+	0.485412i	$0.838670\pi$
$824$	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	$-0.339303\pi$
$827$	23005.9			0.967343			0.483671	+	0.875250i	$0.339303\pi$
$828$		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	−0.212163	+	0.977234i	$0.568051\pi$
$839$	−23145.1	+	40088.4i	−0.952391	+	1.64959i	−0.740228	+	0.672356i	$0.765282\pi$
$840$		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	0.437496	+	0.899220i	$0.355865\pi$
$853$	−13951.2	+	24164.2i	−0.559999	+	0.969947i	−0.997496	+	0.0707272i	$0.977468\pi$
$854$	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	$-0.875023\pi$
$857$	−3276.46	+	5675.00i	−0.130597	+	0.226201i	0.793310	+	0.608818i	$0.208356\pi$
$858$		0			0
$859$	−17428.8	−	30187.6i	−0.692275	−	1.19906i	−0.971091	−	0.238711i	$-0.923275\pi$
$859$	−17428.8	−	30187.6i	−0.692275	−	1.19906i	0.278815	−	0.960345i	$-0.410058\pi$
$860$	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	$-0.568871\pi$
$863$	−10885.2			−0.429359			−0.214680	+	0.976685i	$0.568871\pi$
$864$		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	0.999044	+	0.0437269i	$0.0139231\pi$
$877$	13956.9	+	24174.1i	0.537390	+	0.930787i	−0.461653	+	0.887061i	$0.652744\pi$
$878$	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	$-0.434447\pi$
$881$	10694.5			0.408975			0.204488	+	0.978869i	$0.434447\pi$
$882$		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	$-0.223558\pi$
$887$	−4696.44	−	8134.47i	−0.177780	−	0.307924i	−0.941120	+	0.338073i	$0.890225\pi$
$888$		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	−0.962919	−	0.269789i	$-0.913046\pi$
$907$	−6769.24	+	11724.7i	−0.247816	+	0.429229i	0.715104	+	0.699018i	$0.246379\pi$
$908$	−6057.04			−0.221377
$909$		0			0
$910$	−27901.0			−1.01638
$911$	−18903.2	+	32741.3i	−0.687476	+	1.19074i	0.285175	+	0.958475i	$0.407948\pi$
$911$	−18903.2	+	32741.3i	−0.687476	+	1.19074i	−0.972652	+	0.232269i	$0.925385\pi$
$912$		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	$-0.998716\pi$
$929$	−14058.7	+	24350.4i	−0.496502	+	0.859967i	0.503490	+	0.864001i	$0.332049\pi$
$930$		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	$-0.154185\pi$
$941$	1131.57	+	1959.93i	0.0392009	+	0.0678980i	−0.845759	+	0.533565i	$0.820852\pi$
$942$		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	$-0.795092\pi$
$947$	3492.67	−	6049.48i	0.119848	−	0.207583i	0.919707	+	0.392604i	$0.128426\pi$
$948$		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	$-0.351672\pi$
$953$	26436.8			0.898608			0.449304	+	0.893379i	$0.351672\pi$
$954$		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	0.866857	−	0.498557i	$-0.166137\pi$
$967$	50.0781	+	86.7379i	0.00166536	+	0.00288449i	−0.865192	+	0.501442i	$0.832803\pi$
$968$	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	$-0.496433\pi$
$971$	678.145			0.0224127			0.0112063	+	0.999937i	$0.496433\pi$
$972$		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	$-0.115644\pi$
$977$	4874.13	+	8442.24i	0.159608	+	0.276449i	−0.775119	+	0.631815i	$0.782310\pi$
$978$		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	$-0.815168\pi$
$983$	1757.84	−	3044.67i	0.0570361	−	0.0987894i	0.893134	+	0.449791i	$0.148502\pi$
$984$		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	−0.988714	−	0.149817i	$-0.952131\pi$
$997$	−11478.2	+	19880.8i	−0.364611	+	0.631525i	0.624102	+	0.781343i	$0.285465\pi$
$998$	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.i.55.1		4
3.2	odd	2		162.4.c.j.55.2		4
9.2	odd	6		162.4.a.e.1.1	✓	2
9.4	even	3	inner	162.4.c.i.109.1		4
9.5	odd	6		162.4.c.j.109.2		4
9.7	even	3		162.4.a.h.1.2	yes	2
36.7	odd	6		1296.4.a.s.1.2		2
36.11	even	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	9.2	odd	6
162.4.a.h.1.2	yes	2	9.7	even	3
162.4.c.i.55.1		4	1.1	even	1	trivial
162.4.c.i.109.1		4	9.4	even	3	inner
162.4.c.j.55.2		4	3.2	odd	2
162.4.c.j.109.2		4	9.5	odd	6
1296.4.a.j.1.1		2	36.11	even	6
1296.4.a.s.1.2		2	36.7	odd	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 $\F_{q}$

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

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

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