LMFDB - Embedded newform 138.5.c.a.47.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,5,Mod(47,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("138.47");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	138.c (of order $2$, degree $1$, 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:	$14.2650549056$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.3
Character	$\chi$	$=$	138.47
Dual form			138.5.c.a.47.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} +(-7.96199 + 4.19604i) q^{3} -8.00000 q^{4} -18.2777i q^{5} +(11.8682 + 22.5199i) q^{6} -45.3951 q^{7} +22.6274i q^{8} +(45.7866 - 66.8176i) q^{9} +O(q^{10})$	\(q-2.82843i q^{2} +(-7.96199 + 4.19604i) q^{3} -8.00000 q^{4} -18.2777i q^{5} +(11.8682 + 22.5199i) q^{6} -45.3951 q^{7} +22.6274i q^{8} +(45.7866 - 66.8176i) q^{9} -51.6971 q^{10} +1.54496i q^{11} +(63.6959 - 33.5683i) q^{12} -189.031 q^{13} +128.397i q^{14} +(76.6938 + 145.527i) q^{15} +64.0000 q^{16} +356.899i q^{17} +(-188.989 - 129.504i) q^{18} +657.616 q^{19} +146.222i q^{20} +(361.435 - 190.479i) q^{21} +4.36981 q^{22} -110.304i q^{23} +(-94.9454 - 180.159i) q^{24} +290.926 q^{25} +534.661i q^{26} +(-84.1833 + 724.123i) q^{27} +363.160 q^{28} +1188.78i q^{29} +(411.612 - 216.923i) q^{30} +1288.49 q^{31} -181.019i q^{32} +(-6.48272 - 12.3010i) q^{33} +1009.46 q^{34} +829.717i q^{35} +(-366.293 + 534.541i) q^{36} -1780.37 q^{37} -1860.02i q^{38} +(1505.07 - 793.182i) q^{39} +413.577 q^{40} +1976.56i q^{41} +(-538.757 - 1022.29i) q^{42} +2184.87 q^{43} -12.3597i q^{44} +(-1221.27 - 836.873i) q^{45} -311.987 q^{46} -3223.28i q^{47} +(-509.567 + 268.546i) q^{48} -340.289 q^{49} -822.863i q^{50} +(-1497.56 - 2841.63i) q^{51} +1512.25 q^{52} +322.136i q^{53} +(2048.13 + 238.106i) q^{54} +28.2383 q^{55} -1027.17i q^{56} +(-5235.93 + 2759.38i) q^{57} +3362.37 q^{58} +2735.49i q^{59} +(-613.551 - 1164.21i) q^{60} +2379.36 q^{61} -3644.40i q^{62} +(-2078.48 + 3033.19i) q^{63} -512.000 q^{64} +3455.06i q^{65} +(-34.7924 + 18.3359i) q^{66} -1715.69 q^{67} -2855.20i q^{68} +(462.840 + 878.240i) q^{69} +2346.79 q^{70} +7193.14i q^{71} +(1511.91 + 1036.03i) q^{72} -8552.24 q^{73} +5035.65i q^{74} +(-2316.35 + 1220.74i) q^{75} -5260.93 q^{76} -70.1336i q^{77} +(-2243.46 - 4256.97i) q^{78} -5391.82 q^{79} -1169.77i q^{80} +(-2368.18 - 6118.70i) q^{81} +5590.56 q^{82} +2968.83i q^{83} +(-2891.48 + 1523.83i) q^{84} +6523.30 q^{85} -6179.76i q^{86} +(-4988.15 - 9465.02i) q^{87} -34.9585 q^{88} +3166.68i q^{89} +(-2367.03 + 3454.28i) q^{90} +8581.09 q^{91} +882.433i q^{92} +(-10258.9 + 5406.55i) q^{93} -9116.81 q^{94} -12019.7i q^{95} +(759.564 + 1441.27i) q^{96} +5212.12 q^{97} +962.483i q^{98} +(103.231 + 70.7385i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 8 q^{3} - 224 q^{4} - 32 q^{6} - 104 q^{7} + 80 q^{9}+O(q^{10}) $ 28 * q - 8 * q^3 - 224 * q^4 - 32 * q^6 - 104 * q^7 + 80 * q^9	$ 28 q - 8 q^{3} - 224 q^{4} - 32 q^{6} - 104 q^{7} + 80 q^{9} + 192 q^{10} + 64 q^{12} + 148 q^{15} + 1792 q^{16} + 448 q^{18} + 912 q^{19} - 1412 q^{21} - 1984 q^{22} + 256 q^{24} - 3028 q^{25} - 1700 q^{27} + 832 q^{28} + 768 q^{30} - 2400 q^{31} + 2772 q^{33} + 2944 q^{34} - 640 q^{36} - 2080 q^{37} + 468 q^{39} - 1536 q^{40} - 4480 q^{42} + 5536 q^{43} + 13852 q^{45} - 512 q^{48} + 13444 q^{49} - 16972 q^{51} + 6112 q^{54} - 624 q^{55} - 3304 q^{57} - 20160 q^{58} - 1184 q^{60} + 6376 q^{61} + 276 q^{63} - 14336 q^{64} + 960 q^{66} + 2168 q^{67} + 26624 q^{70} - 3584 q^{72} - 13568 q^{73} - 2464 q^{75} - 7296 q^{76} - 9920 q^{78} - 46064 q^{79} + 19184 q^{81} + 23168 q^{82} + 11296 q^{84} + 27584 q^{85} + 42140 q^{87} + 15872 q^{88} - 11008 q^{90} - 18040 q^{91} - 23892 q^{93} + 3840 q^{94} - 2048 q^{96} + 56848 q^{97} - 40092 q^{99}+O(q^{100}) $ 28 * q - 8 * q^3 - 224 * q^4 - 32 * q^6 - 104 * q^7 + 80 * q^9 + 192 * q^10 + 64 * q^12 + 148 * q^15 + 1792 * q^16 + 448 * q^18 + 912 * q^19 - 1412 * q^21 - 1984 * q^22 + 256 * q^24 - 3028 * q^25 - 1700 * q^27 + 832 * q^28 + 768 * q^30 - 2400 * q^31 + 2772 * q^33 + 2944 * q^34 - 640 * q^36 - 2080 * q^37 + 468 * q^39 - 1536 * q^40 - 4480 * q^42 + 5536 * q^43 + 13852 * q^45 - 512 * q^48 + 13444 * q^49 - 16972 * q^51 + 6112 * q^54 - 624 * q^55 - 3304 * q^57 - 20160 * q^58 - 1184 * q^60 + 6376 * q^61 + 276 * q^63 - 14336 * q^64 + 960 * q^66 + 2168 * q^67 + 26624 * q^70 - 3584 * q^72 - 13568 * q^73 - 2464 * q^75 - 7296 * q^76 - 9920 * q^78 - 46064 * q^79 + 19184 * q^81 + 23168 * q^82 + 11296 * q^84 + 27584 * q^85 + 42140 * q^87 + 15872 * q^88 - 11008 * q^90 - 18040 * q^91 - 23892 * q^93 + 3840 * q^94 - 2048 * q^96 + 56848 * q^97 - 40092 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	2.82843i		−	0.707107i
$2$		−	2.82843i		−	0.707107i
$3$	−7.96199	+	4.19604i	−0.884666	+	0.466226i
$3$	−7.96199	+	4.19604i	−0.884666	+	0.466226i
$4$	−8.00000			−0.500000
$5$		−	18.2777i		−	0.731108i	−0.930790	−	0.365554i	$-0.880880\pi$
$5$		−	18.2777i		−	0.731108i	0.930790	−	0.365554i	$-0.119120\pi$
$6$	11.8682	+	22.5199i	0.329672	+	0.625553i
$7$	−45.3951			−0.926430			−0.463215	−	0.886246i	$-0.653304\pi$
$7$	−45.3951			−0.926430			−0.463215	+	0.886246i	$0.653304\pi$
$8$			22.6274i			0.353553i
$9$	45.7866	−	66.8176i	0.565266	−	0.824908i
$10$	−51.6971			−0.516971
$11$			1.54496i			0.0127683i	0.999980	+	0.00638414i	$0.00203215\pi$
$11$			1.54496i			0.0127683i	−0.999980	+	0.00638414i	$0.997968\pi$
$12$	63.6959	−	33.5683i	0.442333	−	0.233113i
$13$	−189.031			−1.11853			−0.559264	−	0.828989i	$-0.688916\pi$
$13$	−189.031			−1.11853			−0.559264	+	0.828989i	$0.688916\pi$
$14$			128.397i			0.655085i
$15$	76.6938	+	145.527i	0.340862	+	0.646786i
$16$	64.0000			0.250000
$17$			356.899i			1.23495i	0.786592	+	0.617473i	$0.211844\pi$
$17$			356.899i			1.23495i	−0.786592	+	0.617473i	$0.788156\pi$
$18$	−188.989	−	129.504i	−0.583298	−	0.399704i
$19$	657.616			1.82165			0.910825	−	0.412792i	$-0.135446\pi$
$19$	657.616			1.82165			0.910825	+	0.412792i	$0.135446\pi$
$20$			146.222i			0.365554i
$21$	361.435	−	190.479i	0.819580	−	0.431926i
$22$	4.36981			0.00902854
$23$		−	110.304i		−	0.208514i
$23$		−	110.304i		−	0.208514i
$24$	−94.9454	−	180.159i	−0.164836	−	0.312777i
$25$	290.926			0.465482
$26$			534.661i			0.790919i
$27$	−84.1833	+	724.123i	−0.115478	+	0.993310i
$28$	363.160			0.463215
$29$			1188.78i			1.41353i	0.707450	+	0.706763i	$0.249845\pi$
$29$			1188.78i			1.41353i	−0.707450	+	0.706763i	$0.750155\pi$
$30$	411.612	−	216.923i	0.457347	−	0.241025i
$31$	1288.49			1.34078			0.670390	−	0.742009i	$-0.266127\pi$
$31$	1288.49			1.34078			0.670390	+	0.742009i	$0.266127\pi$
$32$		−	181.019i		−	0.176777i
$33$	−6.48272	−	12.3010i	−0.00595291	−	0.0112957i
$34$	1009.46			0.873239
$35$			829.717i			0.677320i
$36$	−366.293	+	534.541i	−0.282633	+	0.412454i
$37$	−1780.37			−1.30049			−0.650246	−	0.759724i	$-0.725334\pi$
$37$	−1780.37			−1.30049			−0.650246	+	0.759724i	$0.725334\pi$
$38$		−	1860.02i		−	1.28810i
$39$	1505.07	−	793.182i	0.989524	−	0.521487i
$40$	413.577			0.258486
$41$			1976.56i			1.17583i	0.808924	+	0.587913i	$0.200050\pi$
$41$			1976.56i			1.17583i	−0.808924	+	0.587913i	$0.799950\pi$
$42$	−538.757	−	1022.29i	−0.305418	−	0.579531i
$43$	2184.87			1.18165			0.590826	−	0.806799i	$-0.298802\pi$
$43$	2184.87			1.18165			0.590826	+	0.806799i	$0.298802\pi$
$44$		−	12.3597i		−	0.00638414i
$45$	−1221.27	−	836.873i	−0.603097	−	0.413271i
$46$	−311.987			−0.147442
$47$		−	3223.28i		−	1.45916i	−0.683897	−	0.729579i	$-0.739716\pi$
$47$		−	3223.28i		−	1.45916i	0.683897	−	0.729579i	$-0.260284\pi$
$48$	−509.567	+	268.546i	−0.221166	+	0.116557i
$49$	−340.289			−0.141728
$50$		−	822.863i		−	0.329145i
$51$	−1497.56	−	2841.63i	−0.575764	−	1.09251i
$52$	1512.25			0.559264
$53$			322.136i			0.114680i	0.998355	+	0.0573400i	$0.0182619\pi$
$53$			322.136i			0.114680i	−0.998355	+	0.0573400i	$0.981738\pi$
$54$	2048.13	+	238.106i	0.702376	+	0.0816551i
$55$	28.2383			0.00933499
$56$		−	1027.17i		−	0.327542i
$57$	−5235.93	+	2759.38i	−1.61155	+	0.849301i
$58$	3362.37			0.999514
$59$			2735.49i			0.785835i	0.919574	+	0.392918i	$0.128534\pi$
$59$			2735.49i			0.785835i	−0.919574	+	0.392918i	$0.871466\pi$
$60$	−613.551	−	1164.21i	−0.170431	−	0.323393i
$61$	2379.36			0.639441			0.319720	−	0.947512i	$-0.396411\pi$
$61$	2379.36			0.639441			0.319720	+	0.947512i	$0.396411\pi$
$62$		−	3644.40i		−	0.948075i
$63$	−2078.48	+	3033.19i	−0.523679	+	0.764220i
$64$	−512.000			−0.125000
$65$			3455.06i			0.817765i
$66$	−34.7924	+	18.3359i	−0.00798724	+	0.00420934i
$67$	−1715.69			−0.382199			−0.191100	−	0.981571i	$-0.561205\pi$
$67$	−1715.69			−0.382199			−0.191100	+	0.981571i	$0.561205\pi$
$68$		−	2855.20i		−	0.617473i
$69$	462.840	+	878.240i	0.0972149	+	0.184466i
$70$	2346.79			0.478937
$71$			7193.14i			1.42693i	0.700692	+	0.713464i	$0.252875\pi$
$71$			7193.14i			1.42693i	−0.700692	+	0.713464i	$0.747125\pi$
$72$	1511.91	+	1036.03i	0.291649	+	0.199852i
$73$	−8552.24			−1.60485			−0.802424	−	0.596754i	$-0.796457\pi$
$73$	−8552.24			−1.60485			−0.802424	+	0.596754i	$0.796457\pi$
$74$			5035.65i			0.919586i
$75$	−2316.35	+	1220.74i	−0.411796	+	0.217020i
$76$	−5260.93			−0.910825
$77$		−	70.1336i		−	0.0118289i
$78$	−2243.46	−	4256.97i	−0.368747	−	0.699699i
$79$	−5391.82			−0.863935			−0.431968	−	0.901889i	$-0.642181\pi$
$79$	−5391.82			−0.863935			−0.431968	+	0.901889i	$0.642181\pi$
$80$		−	1169.77i		−	0.182777i
$81$	−2368.18	−	6118.70i	−0.360948	−	0.932586i
$82$	5590.56			0.831434
$83$			2968.83i			0.430952i	0.976509	+	0.215476i	$0.0691302\pi$
$83$			2968.83i			0.430952i	−0.976509	+	0.215476i	$0.930870\pi$
$84$	−2891.48	+	1523.83i	−0.409790	+	0.215963i
$85$	6523.30			0.902879
$86$		−	6179.76i		−	0.835554i
$87$	−4988.15	−	9465.02i	−0.659023	−	1.25050i
$88$	−34.9585			−0.00451427
$89$			3166.68i			0.399783i	0.979818	+	0.199891i	$0.0640589\pi$
$89$			3166.68i			0.399783i	−0.979818	+	0.199891i	$0.935941\pi$
$90$	−2367.03	+	3454.28i	−0.292226	+	0.426454i
$91$	8581.09			1.03624
$92$			882.433i			0.104257i
$93$	−10258.9	+	5406.55i	−1.18614	+	0.625107i
$94$	−9116.81			−1.03178
$95$		−	12019.7i		−	1.33182i
$96$	759.564	+	1441.27i	0.0824179	+	0.156388i
$97$	5212.12			0.553950			0.276975	−	0.960877i	$-0.410668\pi$
$97$	5212.12			0.553950			0.276975	+	0.960877i	$0.410668\pi$
$98$			962.483i			0.100217i
$99$	103.231	+	70.7385i	0.0105327	+	0.00721748i
$100$	−2327.41			−0.232741
$101$		−	11535.1i		−	1.13078i	−0.824823	−	0.565391i	$-0.808725\pi$
$101$		−	11535.1i		−	1.13078i	0.824823	−	0.565391i	$-0.191275\pi$
$102$	−8037.34	+	4235.75i	−0.772524	+	0.407127i
$103$	9200.40			0.867226			0.433613	−	0.901099i	$-0.357239\pi$
$103$	9200.40			0.867226			0.433613	+	0.901099i	$0.357239\pi$
$104$		−	4277.29i		−	0.395460i
$105$	−3481.52	−	6606.20i	−0.315784	−	0.599201i
$106$	911.138			0.0810909
$107$			5552.54i			0.484981i	0.970154	+	0.242490i	$0.0779643\pi$
$107$			5552.54i			0.484981i	−0.970154	+	0.242490i	$0.922036\pi$
$108$	673.466	−	5792.98i	0.0577389	−	0.496655i
$109$	5972.05			0.502655			0.251328	−	0.967902i	$-0.419133\pi$
$109$	5972.05			0.502655			0.251328	+	0.967902i	$0.419133\pi$
$110$		−	79.8701i		−	0.00660083i
$111$	14175.3	−	7470.51i	1.15050	−	0.606323i
$112$	−2905.28			−0.231607
$113$			17662.1i			1.38320i	0.722279	+	0.691601i	$0.243095\pi$
$113$			17662.1i			1.38320i	−0.722279	+	0.691601i	$0.756905\pi$
$114$	7804.70	+	14809.4i	0.600547	+	1.13954i
$115$	−2016.10			−0.152446
$116$		−	9510.21i		−	0.706763i
$117$	−8655.10	+	12630.6i	−0.632266	+	0.922684i
$118$	7737.14			0.555670
$119$		−	16201.5i		−	1.14409i
$120$	−3292.90	+	1735.38i	−0.228673	+	0.120513i
$121$	14638.6			0.999837
$122$		−	6729.84i		−	0.452153i
$123$	−8293.73	−	15737.4i	−0.548201	−	1.04021i
$124$	−10307.9			−0.670390
$125$		−	16741.0i		−	1.07142i
$126$	8579.15	+	5878.84i	0.540385	+	0.370297i
$127$	6008.85			0.372549			0.186275	−	0.982498i	$-0.440359\pi$
$127$	6008.85			0.372549			0.186275	+	0.982498i	$0.440359\pi$
$128$			1448.15i			0.0883883i
$129$	−17395.9	+	9167.81i	−1.04537	+	0.550917i
$130$	9772.37			0.578247
$131$			11993.9i			0.698906i	0.936954	+	0.349453i	$0.113633\pi$
$131$			11993.9i			0.698906i	−0.936954	+	0.349453i	$0.886367\pi$
$132$	51.8617	+	98.4078i	0.00297645	+	0.00564783i
$133$	−29852.5			−1.68763
$134$			4852.71i			0.270256i
$135$	13235.3	+	1538.68i	0.726217	+	0.0844267i
$136$	−8075.71			−0.436619
$137$		−	19062.0i		−	1.01561i	−0.861472	−	0.507805i	$-0.830457\pi$
$137$		−	19062.0i		−	1.01561i	0.861472	−	0.507805i	$-0.169543\pi$
$138$	2484.04	−	1309.11i	0.130437	−	0.0687413i
$139$	−6644.57			−0.343904			−0.171952	−	0.985105i	$-0.555007\pi$
$139$	−6644.57			−0.343904			−0.171952	+	0.985105i	$0.555007\pi$
$140$		−	6637.73i		−	0.338660i
$141$	13525.0	+	25663.7i	0.680298	+	1.29087i
$142$	20345.3			1.00899
$143$		−	292.046i		−	0.0142817i
$144$	2930.34	−	4276.33i	0.141317	−	0.206227i
$145$	21728.1			1.03344
$146$			24189.4i			1.13480i
$147$	2709.38	−	1427.87i	0.125382	−	0.0660773i
$148$	14243.0			0.650246
$149$			27818.6i			1.25303i	0.779409	+	0.626516i	$0.215520\pi$
$149$			27818.6i			1.25303i	−0.779409	+	0.626516i	$0.784480\pi$
$150$	3452.76	+	6551.63i	0.153456	+	0.291183i
$151$	27526.9			1.20727			0.603633	−	0.797262i	$-0.293719\pi$
$151$	27526.9			1.20727			0.603633	+	0.797262i	$0.293719\pi$
$152$			14880.1i			0.644051i
$153$	23847.2	+	16341.2i	1.01872	+	0.698074i
$154$	−198.368			−0.00836431
$155$		−	23550.6i		−	0.980255i
$156$	−12040.5	+	6345.46i	−0.494762	+	0.260744i
$157$	30473.0			1.23628			0.618139	−	0.786069i	$-0.287887\pi$
$157$	30473.0			1.23628			0.618139	+	0.786069i	$0.287887\pi$
$158$			15250.4i			0.610895i
$159$	−1351.69	−	2564.84i	−0.0534668	−	0.101453i
$160$	−3308.62			−0.129243
$161$			5007.26i			0.193174i
$162$	−17306.3	+	6698.22i	−0.659438	+	0.255229i
$163$	−5089.19			−0.191546			−0.0957731	−	0.995403i	$-0.530532\pi$
$163$	−5089.19			−0.191546			−0.0957731	+	0.995403i	$0.530532\pi$
$164$		−	15812.5i		−	0.587913i
$165$	−224.833	+	118.489i	−0.00825834	+	0.00435222i
$166$	8397.11			0.304729
$167$			4792.30i			0.171835i	0.996302	+	0.0859174i	$0.0273821\pi$
$167$			4792.30i			0.171835i	−0.996302	+	0.0859174i	$0.972618\pi$
$168$	4310.05	+	8178.34i	0.152709	+	0.289765i
$169$	7171.84			0.251106
$170$		−	18450.7i		−	0.638432i
$171$	30110.0	−	43940.3i	1.02972	−	1.50269i
$172$	−17479.0			−0.590826
$173$			24355.8i			0.813785i	0.913476	+	0.406892i	$0.133388\pi$
$173$			24355.8i			0.813785i	−0.913476	+	0.406892i	$0.866612\pi$
$174$	−26771.1	+	14108.6i	−0.884236	+	0.466000i
$175$	−13206.6			−0.431236
$176$			98.8776i			0.00319207i
$177$	−11478.2	−	21780.0i	−0.366377	−	0.695202i
$178$	8956.72			0.282689
$179$			44832.8i			1.39923i	0.714518	+	0.699617i	$0.246646\pi$
$179$			44832.8i			1.39923i	−0.714518	+	0.699617i	$0.753354\pi$
$180$	9770.17	+	6694.98i	0.301548	+	0.206635i
$181$	−64934.7			−1.98207			−0.991037	−	0.133588i	$-0.957350\pi$
$181$	−64934.7			−1.98207			−0.991037	+	0.133588i	$0.957350\pi$
$182$		−	24271.0i		−	0.732731i
$183$	−18944.4	+	9983.88i	−0.565691	+	0.298124i
$184$	2495.90			0.0737210
$185$			32541.1i			0.950799i
$186$	15292.0	+	29016.7i	0.442017	+	0.838729i
$187$	−551.396			−0.0157681
$188$			25786.2i			0.729579i
$189$	3821.50	−	32871.6i	0.106982	−	0.920232i
$190$	−33996.8			−0.941741
$191$			2588.15i			0.0709451i	0.999371	+	0.0354725i	$0.0112936\pi$
$191$			2588.15i			0.0709451i	−0.999371	+	0.0354725i	$0.988706\pi$
$192$	4076.54	−	2148.37i	0.110583	−	0.0582783i
$193$	18676.8			0.501403			0.250702	−	0.968064i	$-0.419339\pi$
$193$	18676.8			0.501403			0.250702	+	0.968064i	$0.419339\pi$
$194$		−	14742.1i		−	0.391702i
$195$	−14497.5	−	27509.1i	−0.381263	−	0.723448i
$196$	2722.31			0.0708640
$197$			26864.9i			0.692234i	0.938191	+	0.346117i	$0.112500\pi$
$197$			26864.9i			0.692234i	−0.938191	+	0.346117i	$0.887500\pi$
$198$	200.079	−	291.980i	0.00510353	−	0.00744772i
$199$	10764.3			0.271819			0.135909	−	0.990721i	$-0.456604\pi$
$199$	10764.3			0.271819			0.135909	+	0.990721i	$0.456604\pi$
$200$			6582.90i			0.164573i
$201$	13660.3	−	7199.11i	0.338118	−	0.178191i
$202$	−32626.2			−0.799584
$203$		−	53964.5i		−	1.30953i
$204$	11980.5	+	22733.0i	0.287882	+	0.546257i
$205$	36127.0			0.859655
$206$		−	26022.7i		−	0.613221i
$207$	−7370.26	−	5050.45i	−0.172005	−	0.117866i
$208$	−12098.0			−0.279632
$209$			1015.99i			0.0232593i
$210$	−18685.1	+	9847.23i	−0.423699	+	0.223293i
$211$	10284.6			0.231005			0.115502	−	0.993307i	$-0.463152\pi$
$211$	10284.6			0.231005			0.115502	+	0.993307i	$0.463152\pi$
$212$		−	2577.09i		−	0.0573400i
$213$	−30182.7	−	57271.7i	−0.665271	−	1.26235i
$214$	15705.0			0.342933
$215$		−	39934.4i		−	0.863914i
$216$	−16385.0	−	1904.85i	−0.351188	−	0.0408275i
$217$	−58491.1			−1.24214
$218$		−	16891.5i		−	0.355431i
$219$	68092.8	−	35885.5i	1.41975	−	0.748222i
$220$	−225.907			−0.00466749
$221$		−	67465.2i		−	1.38132i
$222$	−21129.8	−	40093.8i	−0.428735	−	0.813526i
$223$	−31633.9			−0.636127			−0.318063	−	0.948069i	$-0.603032\pi$
$223$	−31633.9			−0.636127			−0.318063	+	0.948069i	$0.603032\pi$
$224$			8217.38i			0.163771i
$225$	13320.5	−	19439.0i	0.263121	−	0.383980i
$226$	49956.0			0.978072
$227$			30830.8i			0.598319i	0.954203	+	0.299159i	$0.0967062\pi$
$227$			30830.8i			0.598319i	−0.954203	+	0.299159i	$0.903294\pi$
$228$	41887.4	−	22075.0i	0.805776	−	0.424651i
$229$	−66473.9			−1.26759			−0.633797	−	0.773500i	$-0.718504\pi$
$229$	−66473.9			−1.26759			−0.633797	+	0.773500i	$0.718504\pi$
$230$			5702.41i			0.107796i
$231$	294.283	+	558.403i	0.00551495	+	0.0104646i
$232$	−26898.9			−0.499757
$233$			86271.7i			1.58912i	0.607186	+	0.794559i	$0.292298\pi$
$233$			86271.7i			1.58912i	−0.607186	+	0.794559i	$0.707702\pi$
$234$	35724.8	+	24480.3i	0.652436	+	0.447080i
$235$	−58914.1			−1.06680
$236$		−	21883.9i		−	0.392918i
$237$	42929.6	−	22624.3i	0.764294	−	0.402789i
$238$	−45824.7			−0.808994
$239$		−	39974.8i		−	0.699826i	−0.936782	−	0.349913i	$-0.886211\pi$
$239$		−	39974.8i		−	0.699826i	0.936782	−	0.349913i	$-0.113789\pi$
$240$	4908.41	+	9313.71i	0.0852154	+	0.161696i
$241$	38531.6			0.663411			0.331706	−	0.943383i	$-0.392376\pi$
$241$	38531.6			0.663411			0.331706	+	0.943383i	$0.392376\pi$
$242$		−	41404.3i		−	0.706992i
$243$	44529.7	+	38780.0i	0.754114	+	0.656743i
$244$	−19034.9			−0.319720
$245$			6219.70i			0.103618i
$246$	−44512.0	+	23458.2i	−0.735541	+	0.387636i
$247$	−124310.			−2.03757
$248$			29155.2i			0.474037i
$249$	−12457.3	−	23637.8i	−0.200921	−	0.381248i
$250$	−47350.7			−0.757612
$251$			62073.9i			0.985285i	0.870232	+	0.492642i	$0.163969\pi$
$251$			62073.9i			0.985285i	−0.870232	+	0.492642i	$0.836031\pi$
$252$	16627.9	−	24265.5i	0.261840	−	0.382110i
$253$	170.416			0.00266237
$254$		−	16995.6i		−	0.263432i
$255$	−51938.4	+	27372.0i	−0.798746	+	0.420946i
$256$	4096.00			0.0625000
$257$		−	63107.2i		−	0.955461i	−0.878507	−	0.477730i	$-0.841460\pi$
$257$		−	63107.2i		−	0.955461i	0.878507	−	0.477730i	$-0.158540\pi$
$258$	25930.5	+	49203.2i	0.389557	+	0.739186i
$259$	80820.1			1.20481
$260$		−	27640.4i		−	0.408882i
$261$	79431.1	+	54430.0i	1.16603	+	0.799019i
$262$	33923.9			0.494201
$263$			37758.1i			0.545882i	0.962031	+	0.272941i	$0.0879964\pi$
$263$			37758.1i			0.545882i	−0.962031	+	0.272941i	$0.912004\pi$
$264$	278.339	−	146.687i	0.00399362	−	0.00210467i
$265$	5887.90			0.0838434
$266$			84435.6i			1.19334i
$267$	−13287.5	−	25213.1i	−0.186389	−	0.353674i
$268$	13725.5			0.191100
$269$			46003.8i			0.635754i	0.948132	+	0.317877i	$0.102970\pi$
$269$			46003.8i			0.635754i	−0.948132	+	0.317877i	$0.897030\pi$
$270$	4352.03	−	37435.1i	0.0596987	−	0.513513i
$271$	−99526.0			−1.35518			−0.677591	−	0.735439i	$-0.736976\pi$
$271$	−99526.0			−1.35518			−0.677591	+	0.735439i	$0.736976\pi$
$272$			22841.6i			0.308737i
$273$	−68322.5	+	36006.5i	−0.916724	+	0.483121i
$274$	−53915.4			−0.718145
$275$			449.470i			0.00594340i
$276$	−3702.72	−	7025.92i	−0.0486074	−	0.0922328i
$277$	−137776.			−1.79562			−0.897811	−	0.440382i	$-0.854843\pi$
$277$	−137776.			−1.79562			−0.897811	+	0.440382i	$0.854843\pi$
$278$			18793.7i			0.243177i
$279$	58995.5	−	86093.8i	0.757898	−	1.10602i
$280$	−18774.3			−0.239469
$281$			53635.7i			0.679268i	0.940558	+	0.339634i	$0.110303\pi$
$281$			53635.7i			0.679268i	−0.940558	+	0.339634i	$0.889697\pi$
$282$	72588.0	−	38254.5i	0.912781	−	0.481043i
$283$	138263.			1.72636			0.863181	−	0.504895i	$-0.168469\pi$
$283$	138263.			1.72636			0.863181	+	0.504895i	$0.168469\pi$
$284$		−	57545.1i		−	0.713464i
$285$	50435.1	+	95700.7i	0.620930	+	1.17822i
$286$	−826.031			−0.0100987
$287$		−	89726.2i		−	1.08932i
$288$	−12095.3	−	8288.25i	−0.145825	−	0.0999259i
$289$	−43856.2			−0.525092
$290$		−	61456.3i		−	0.730752i
$291$	−41498.8	+	21870.2i	−0.490061	+	0.258266i
$292$	68417.9			0.802424
$293$			94395.3i			1.09955i	0.835313	+	0.549775i	$0.185287\pi$
$293$			94395.3i			1.09955i	−0.835313	+	0.549775i	$0.814713\pi$
$294$	−4038.61	−	7663.28i	−0.0467237	−	0.0886584i
$295$	49998.5			0.574530
$296$		−	40285.2i		−	0.459793i
$297$	−1118.74	−	130.060i	−0.0126829	−	0.00147445i
$298$	78682.8			0.886028
$299$			20850.9i			0.233229i
$300$	18530.8	−	9765.89i	0.205898	−	0.108510i
$301$	−99182.4			−1.09472
$302$		−	77857.7i		−	0.853666i
$303$	48401.7	+	91842.4i	0.527200	+	1.00036i
$304$	42087.4			0.455413
$305$		−	43489.2i		−	0.467500i
$306$	46219.9	−	67450.0i	0.493613	−	0.720342i
$307$	−24755.3			−0.262659			−0.131329	−	0.991339i	$-0.541925\pi$
$307$	−24755.3			−0.262659			−0.131329	+	0.991339i	$0.541925\pi$
$308$			561.069i			0.00591446i
$309$	−73253.5	+	38605.2i	−0.767205	+	0.404323i
$310$	−66611.2			−0.693145
$311$		−	127346.i		−	1.31663i	−0.752740	−	0.658317i	$-0.771268\pi$
$311$		−	127346.i		−	1.31663i	0.752740	−	0.658317i	$-0.228732\pi$
$312$	17947.7	+	34055.7i	0.184374	+	0.349849i
$313$	109290.			1.11556			0.557778	−	0.829990i	$-0.311654\pi$
$313$	109290.			1.11556			0.557778	+	0.829990i	$0.311654\pi$
$314$		−	86190.8i		−	0.874181i
$315$	55439.7	+	37989.9i	0.558727	+	0.382866i
$316$	43134.6			0.431968
$317$			36468.1i			0.362906i	0.983400	+	0.181453i	$0.0580800\pi$
$317$			36468.1i			0.362906i	−0.983400	+	0.181453i	$0.941920\pi$
$318$	−7254.47	+	3823.17i	−0.0717384	+	0.0378067i
$319$	−1836.61			−0.0180483
$320$			9358.18i			0.0913885i
$321$	−23298.7	−	44209.3i	−0.226111	−	0.429046i
$322$	14162.7			0.136595
$323$			234703.i			2.24964i
$324$	18945.4	+	48949.6i	0.180474	+	0.466293i
$325$	−54994.1			−0.520654
$326$			14394.4i			0.135444i
$327$	−47549.4	+	25058.9i	−0.444682	+	0.234351i
$328$	−44724.5			−0.415717
$329$			146321.i			1.35181i
$330$	335.138	+	635.925i	0.00307748	+	0.00583953i
$331$	45783.7			0.417883			0.208941	−	0.977928i	$-0.432998\pi$
$331$	45783.7			0.417883			0.208941	+	0.977928i	$0.432998\pi$
$332$		−	23750.6i		−	0.215476i
$333$	−81517.2	+	118960.i	−0.735124	+	1.07279i
$334$	13554.7			0.121506
$335$			31358.9i			0.279429i
$336$	23131.8	−	12190.7i	0.204895	−	0.107981i
$337$	−76913.7			−0.677242			−0.338621	−	0.940923i	$-0.609960\pi$
$337$	−76913.7			−0.677242			−0.338621	+	0.940923i	$0.609960\pi$
$338$		−	20285.0i		−	0.177559i
$339$	−74110.9	−	140626.i	−0.644885	−	1.22367i
$340$	−52186.4			−0.451439
$341$			1990.67i			0.0171195i
$342$	−124282.	−	85163.9i	−1.06257	−	0.728120i
$343$	124441.			1.05773
$344$			49438.0i			0.417777i
$345$	16052.2	−	8459.65i	0.134864	−	0.0710745i
$346$	68888.5			0.575433
$347$		−	180596.i		−	1.49985i	−0.661521	−	0.749927i	$-0.730089\pi$
$347$		−	180596.i		−	1.49985i	0.661521	−	0.749927i	$-0.269911\pi$
$348$	39905.2	+	75720.2i	0.329512	+	0.625249i
$349$	73687.6			0.604984			0.302492	−	0.953152i	$-0.402181\pi$
$349$	73687.6			0.604984			0.302492	+	0.953152i	$0.402181\pi$
$350$			37353.9i			0.304930i
$351$	15913.3	−	136882.i	0.129165	−	1.11105i
$352$	279.668			0.00225713
$353$		−	183142.i		−	1.46974i	−0.678210	−	0.734868i	$-0.737244\pi$
$353$		−	183142.i		−	1.46974i	0.678210	−	0.734868i	$-0.262756\pi$
$354$	−61603.1	+	32465.3i	−0.491582	+	0.259068i
$355$	131474.			1.04324
$356$		−	25333.4i		−	0.199891i
$357$	67981.9	+	128996.i	0.533405	+	1.01214i
$358$	126806.			0.989407
$359$			82203.7i			0.637826i	0.947784	+	0.318913i	$0.103318\pi$
$359$			82203.7i			0.637826i	−0.947784	+	0.318913i	$0.896682\pi$
$360$	18936.3	−	27634.2i	0.146113	−	0.213227i
$361$	302137.			2.31841
$362$			183663.i			1.40154i
$363$	−116552.	+	61424.1i	−0.884521	+	0.466150i
$364$	−68648.7			−0.518119
$365$			156315.i			1.17332i
$366$	28238.7	+	53583.0i	0.210806	+	0.400004i
$367$	172823.			1.28313			0.641564	−	0.767070i	$-0.278286\pi$
$367$	172823.			1.28313			0.641564	+	0.767070i	$0.278286\pi$
$368$		−	7059.46i		−	0.0521286i
$369$	132069.	+	90500.0i	0.969949	+	0.664655i
$370$	92040.1			0.672317
$371$		−	14623.4i		−	0.106243i
$372$	82071.6	−	43252.4i	0.593071	−	0.312553i
$373$	−56731.1			−0.407759			−0.203879	−	0.978996i	$-0.565355\pi$
$373$	−56731.1			−0.407759			−0.203879	+	0.978996i	$0.565355\pi$
$374$			1559.58i			0.0111498i
$375$	70245.9	+	133292.i	0.499526	+	0.947853i
$376$	72934.5			0.515890
$377$		−	224716.i		−	1.58107i
$378$	−92974.9	−	10808.8i	−0.650702	−	0.0756477i
$379$	−191202.			−1.33111			−0.665555	−	0.746349i	$-0.731805\pi$
$379$	−191202.			−1.33111			−0.665555	+	0.746349i	$0.731805\pi$
$380$			96157.6i			0.665911i
$381$	−47842.4	+	25213.3i	−0.329582	+	0.173692i
$382$	7320.39			0.0501657
$383$		−	156216.i		−	1.06494i	−0.846447	−	0.532472i	$-0.821263\pi$
$383$		−	156216.i		−	1.06494i	0.846447	−	0.532472i	$-0.178737\pi$
$384$	−6076.51	−	11530.2i	−0.0412090	−	0.0781941i
$385$	−1281.88			−0.00864821
$386$		−	52825.9i		−	0.354546i
$387$	100038.	−	145988.i	0.667948	−	0.974754i
$388$	−41696.9			−0.276975
$389$			145789.i			0.963439i	0.876325	+	0.481720i	$0.159988\pi$
$389$			145789.i			0.963439i	−0.876325	+	0.481720i	$0.840012\pi$
$390$	−77807.5	+	41005.2i	−0.511555	+	0.269594i
$391$	39367.5			0.257504
$392$		−	7699.86i		−	0.0501084i
$393$	−50326.9	−	95495.5i	−0.325848	−	0.618298i
$394$	75985.4			0.489483
$395$			98550.0i			0.631630i
$396$	−825.845	−	565.908i	−0.00526633	−	0.00360874i
$397$	−57457.2			−0.364555			−0.182278	−	0.983247i	$-0.558347\pi$
$397$	−57457.2			−0.364555			−0.182278	+	0.983247i	$0.558347\pi$
$398$		−	30446.0i		−	0.192205i
$399$	237685.	−	125262.i	1.49299	−	0.786818i
$400$	18619.3			0.116370
$401$		−	142941.i		−	0.888929i	−0.895797	−	0.444464i	$-0.853394\pi$
$401$		−	142941.i		−	0.888929i	0.895797	−	0.444464i	$-0.146606\pi$
$402$	−20362.1	−	38637.2i	−0.126000	−	0.239086i
$403$	−243565.			−1.49970
$404$			92280.8i			0.565391i
$405$	−111836.	+	43284.9i	−0.681821	+	0.263892i
$406$	−152635.			−0.925980
$407$		−	2750.61i		−	0.0166050i
$408$	64298.7	−	33886.0i	0.386262	−	0.203563i
$409$	102598.			0.613328			0.306664	−	0.951818i	$-0.400787\pi$
$409$	102598.			0.613328			0.306664	+	0.951818i	$0.400787\pi$
$410$		−	102183.i		−	0.607868i
$411$	79984.7	+	151771.i	0.473504	+	0.898475i
$412$	−73603.2			−0.433613
$413$		−	124178.i		−	0.728021i
$414$	−14284.8	+	20846.2i	−0.0833440	+	0.121626i
$415$	54263.3			0.315072
$416$			34218.3i			0.197730i
$417$	52904.0	−	27880.8i	0.304240	−	0.160337i
$418$	2873.66			0.0164468
$419$		−	222876.i		−	1.26951i	−0.772715	−	0.634753i	$-0.781102\pi$
$419$		−	222876.i		−	1.26951i	0.772715	−	0.634753i	$-0.218898\pi$
$420$	27852.2	+	52849.6i	0.157892	+	0.299601i
$421$	−60883.7			−0.343508			−0.171754	−	0.985140i	$-0.554943\pi$
$421$	−60883.7			−0.343508			−0.171754	+	0.985140i	$0.554943\pi$
$422$		−	29089.1i		−	0.163345i
$423$	−215372.	−	147583.i	−1.20367	−	0.824813i
$424$	−7289.10			−0.0405455
$425$			103831.i			0.574845i
$426$	−161989.	+	85369.5i	−0.892619	+	0.470418i
$427$	−108011.			−0.592397
$428$		−	44420.3i		−	0.242490i
$429$	1225.44	+	2325.27i	0.00665850	+	0.0126345i
$430$	−112952.			−0.610880
$431$		−	134375.i		−	0.723374i	−0.932300	−	0.361687i	$-0.882201\pi$
$431$		−	134375.i		−	0.723374i	0.932300	−	0.361687i	$-0.117799\pi$
$432$	−5387.73	+	46343.9i	−0.0288694	+	0.248328i
$433$	−99398.7			−0.530157			−0.265079	−	0.964227i	$-0.585398\pi$
$433$	−99398.7			−0.530157			−0.265079	+	0.964227i	$0.585398\pi$
$434$			165438.i			0.878325i
$435$	−172999.	+	91171.8i	−0.914249	+	0.481817i
$436$	−47776.4			−0.251328
$437$		−	72537.7i		−	0.379840i
$438$	−101499.	−	192596.i	−0.529073	−	1.00392i
$439$	89608.5			0.464965			0.232482	−	0.972601i	$-0.425315\pi$
$439$	89608.5			0.464965			0.232482	+	0.972601i	$0.425315\pi$
$440$			638.961i			0.00330042i
$441$	−15580.7	+	22737.3i	−0.0801141	+	0.116913i
$442$	−190820.			−0.976743
$443$		−	110636.i		−	0.563751i	−0.959451	−	0.281876i	$-0.909043\pi$
$443$		−	110636.i		−	0.563751i	0.959451	−	0.281876i	$-0.0909566\pi$
$444$	−113402.	+	59764.1i	−0.575250	+	0.303162i
$445$	57879.6			0.292284
$446$			89474.3i			0.449809i
$447$	−116728.	−	221491.i	−0.584196	−	1.10851i
$448$	23242.3			0.115804
$449$			70980.1i			0.352082i	0.984383	+	0.176041i	$0.0563291\pi$
$449$			70980.1i			0.352082i	−0.984383	+	0.176041i	$0.943671\pi$
$450$	−54981.7	−	37676.1i	−0.271515	−	0.186055i
$451$	−3053.71			−0.0150133
$452$		−	141297.i		−	0.691601i
$453$	−219169.	+	115504.i	−1.06803	+	0.562859i
$454$	87202.6			0.423075
$455$		−	156842.i		−	0.757601i
$456$	−62437.6	−	118476.i	−0.300273	−	0.569769i
$457$	−290199.			−1.38951			−0.694757	−	0.719244i	$-0.744488\pi$
$457$	−290199.			−1.38951			−0.694757	+	0.719244i	$0.744488\pi$
$458$			188017.i			0.896324i
$459$	−258439.	−	30045.0i	−1.22668	−	0.142609i
$460$	16128.8			0.0762232
$461$		−	99167.6i		−	0.466625i	−0.972402	−	0.233313i	$-0.925043\pi$
$461$		−	99167.6i		−	0.466625i	0.972402	−	0.233313i	$-0.0749565\pi$
$462$	1579.40	−	832.359i	0.00739961	−	0.00389966i
$463$	289464.			1.35031			0.675154	−	0.737676i	$-0.264077\pi$
$463$	289464.			1.35031			0.675154	+	0.737676i	$0.264077\pi$
$464$			76081.6i			0.353382i
$465$	98819.2	+	187510.i	0.457020	+	0.867198i
$466$	244013.			1.12368
$467$			176411.i			0.808892i	0.914562	+	0.404446i	$0.132536\pi$
$467$			176411.i			0.808892i	−0.914562	+	0.404446i	$0.867464\pi$
$468$	69240.8	−	101045.i	0.316133	−	0.461342i
$469$	77883.9			0.354081
$470$			166634.i			0.754343i
$471$	−242626.	+	127866.i	−1.09369	+	0.576385i
$472$	−61897.1			−0.277835
$473$			3375.55i			0.0150877i
$474$	−63991.1	−	121423.i	−0.284815	−	0.540437i
$475$	191318.			0.847945
$476$			129612.i			0.572045i
$477$	21524.3	+	14749.5i	0.0946004	+	0.0648247i
$478$	−113066.			−0.494852
$479$			19728.7i			0.0859860i	0.999075	+	0.0429930i	$0.0136893\pi$
$479$			19728.7i			0.0859860i	−0.999075	+	0.0429930i	$0.986311\pi$
$480$	26343.2	−	13883.1i	0.114337	−	0.0602564i
$481$	336546.			1.45464
$482$		−	108984.i		−	0.469103i
$483$	−21010.6	−	39867.8i	−0.0900627	−	0.170894i
$484$	−117109.			−0.499918
$485$		−	95265.5i		−	0.404997i
$486$	109687.	−	125949.i	0.464388	−	0.533239i
$487$	−12724.9			−0.0536535			−0.0268267	−	0.999640i	$-0.508540\pi$
$487$	−12724.9			−0.0536535			−0.0268267	+	0.999640i	$0.508540\pi$
$488$			53838.8i			0.226076i
$489$	40520.1	−	21354.4i	0.169454	−	0.0893039i
$490$	17592.0			0.0732693
$491$			345235.i			1.43203i	0.698084	+	0.716015i	$0.254036\pi$
$491$			345235.i			1.43203i	−0.698084	+	0.716015i	$0.745964\pi$
$492$	66349.8	+	125899.i	0.274100	+	0.520106i
$493$	−424273.			−1.74563
$494$			351602.i			1.44078i
$495$	1292.94	−	1886.82i	0.00527675	−	0.00770051i
$496$	82463.4			0.335195
$497$		−	326533.i		−	1.32195i
$498$	−66857.7	+	35234.6i	−0.269583	+	0.142073i
$499$	251942.			1.01181			0.505905	−	0.862589i	$-0.331159\pi$
$499$	251942.			1.01181			0.505905	+	0.862589i	$0.331159\pi$
$500$			133928.i			0.535712i
$501$	−20108.7	−	38156.2i	−0.0801139	−	0.152016i
$502$	175572.			0.696701
$503$			235840.i			0.932140i	0.884748	+	0.466070i	$0.154330\pi$
$503$			235840.i			0.932140i	−0.884748	+	0.466070i	$0.845670\pi$
$504$	−68633.2	−	47030.7i	−0.270192	−	0.185149i
$505$	−210835.			−0.826723
$506$		−	482.008i		−	0.00188258i
$507$	−57102.1	+	30093.3i	−0.222145	+	0.117072i
$508$	−48070.8			−0.186275
$509$		−	509273.i		−	1.96569i	−0.184428	−	0.982846i	$-0.559043\pi$
$509$		−	509273.i		−	1.96569i	0.184428	−	0.982846i	$-0.440957\pi$
$510$	77419.7	+	146904.i	0.297654	+	0.564798i
$511$	388229.			1.48678
$512$		−	11585.2i		−	0.0441942i
$513$	−55360.2	+	476195.i	−0.210360	+	1.80946i
$514$	−178494.			−0.675613
$515$		−	168162.i		−	0.634035i
$516$	139168.	−	73342.5i	0.522683	−	0.275458i
$517$	4979.84			0.0186309
$518$		−	228594.i		−	0.851932i
$519$	−102198.	−	193920.i	−0.379408	−	0.719927i
$520$	−78179.0			−0.289123
$521$		−	301357.i		−	1.11021i	−0.831780	−	0.555106i	$-0.812678\pi$
$521$		−	301357.i		−	1.11021i	0.831780	−	0.555106i	$-0.187322\pi$
$522$	153951.	−	224665.i	0.564992	−	0.824508i
$523$	−442177.			−1.61656			−0.808282	−	0.588795i	$-0.799603\pi$
$523$	−442177.			−1.61656			−0.808282	+	0.588795i	$0.799603\pi$
$524$		−	95951.4i		−	0.349453i
$525$	105151.	−	55415.4i	0.381500	−	0.201054i
$526$	106796.			0.385997
$527$			459861.i			1.65579i
$528$	−414.894	−	787.262i	−0.00148823	−	0.00282391i
$529$	−12167.0			−0.0434783
$530$		−	16653.5i		−	0.0592862i
$531$	182779.	+	125249.i	0.648242	+	0.444206i
$532$	238820.			0.843815
$533$		−	373632.i		−	1.31519i
$534$	−71313.3	+	37582.7i	−0.250085	+	0.131797i
$535$	101488.			0.354573
$536$		−	38821.7i		−	0.135128i
$537$	−188120.	−	356959.i	−0.652359	−	1.23785i
$538$	130118.			0.449546
$539$		−	525.734i		−	0.00180962i
$540$	−105882.	−	12309.4i	−0.363108	−	0.0422133i
$541$	81041.9			0.276895			0.138447	−	0.990370i	$-0.455789\pi$
$541$	81041.9			0.276895			0.138447	+	0.990370i	$0.455789\pi$
$542$			281502.i			0.958259i
$543$	517010.	−	272468.i	1.75347	−	0.924095i
$544$	64605.7			0.218310
$545$		−	109155.i		−	0.367495i
$546$	101842.	+	193245.i	0.341618	+	0.648222i
$547$	148791.			0.497282			0.248641	−	0.968596i	$-0.420016\pi$
$547$	148791.			0.497282			0.248641	+	0.968596i	$0.420016\pi$
$548$			152496.i			0.507805i
$549$	108943.	−	158983.i	0.361454	−	0.527480i
$550$	1271.29			0.00420262
$551$			781758.i			2.57495i
$552$	−19872.3	+	10472.9i	−0.0652184	+	0.0343706i
$553$	244762.			0.800375
$554$			389690.i			1.26970i
$555$	−136544.	−	259092.i	−0.443287	−	0.841139i
$556$	53156.5			0.171952
$557$			451850.i			1.45641i	0.685359	+	0.728206i	$0.259645\pi$
$557$			451850.i			1.45641i	−0.685359	+	0.728206i	$0.740355\pi$
$558$	−243510.	−	166865.i	−0.782075	−	0.535915i
$559$	−413009.			−1.32171
$560$			53101.9i			0.169330i
$561$	4390.21	−	2313.68i	0.0139495	−	0.00735152i
$562$	151705.			0.480315
$563$			146607.i			0.462529i	0.972891	+	0.231264i	$0.0742863\pi$
$563$			146607.i			0.462529i	−0.972891	+	0.231264i	$0.925714\pi$
$564$	−108200.	−	205310.i	−0.340149	−	0.645433i
$565$	322823.			1.01127
$566$		−	391066.i		−	1.22072i
$567$	107504.	+	277759.i	0.334393	+	0.863975i
$568$	−162762.			−0.504495
$569$		−	173628.i		−	0.536286i	−0.963379	−	0.268143i	$-0.913590\pi$
$569$		−	173628.i		−	0.536286i	0.963379	−	0.268143i	$-0.0864099\pi$
$570$	270682.	−	142652.i	0.833126	−	0.439064i
$571$	133864.			0.410573			0.205287	−	0.978702i	$-0.434187\pi$
$571$	133864.			0.410573			0.205287	+	0.978702i	$0.434187\pi$
$572$			2336.37i			0.00714084i
$573$	−10860.0	−	20606.8i	−0.0330765	−	0.0627627i
$574$	−253784.			−0.770265
$575$		−	32090.3i		−	0.0970596i
$576$	−23442.7	+	34210.6i	−0.0706583	+	0.103114i
$577$	−207665.			−0.623752			−0.311876	−	0.950123i	$-0.600957\pi$
$577$	−207665.			−0.623752			−0.311876	+	0.950123i	$0.600957\pi$
$578$			124044.i			0.371296i
$579$	−148704.	+	78368.4i	−0.443574	+	0.233767i
$580$	−173825.			−0.516720
$581$		−	134770.i		−	0.399246i
$582$	61858.4	+	117376.i	0.182622	+	0.346525i
$583$	−497.688			−0.00146427
$584$		−	193515.i		−	0.567400i
$585$	230858.	+	158195.i	0.674581	+	0.462255i
$586$	266990.			0.777499
$587$			547417.i			1.58870i	0.607461	+	0.794350i	$0.292188\pi$
$587$			547417.i			1.58870i	−0.607461	+	0.794350i	$0.707812\pi$
$588$	−21675.0	+	11422.9i	−0.0626910	+	0.0330387i
$589$	847331.			2.44243
$590$		−	141417.i		−	0.406254i
$591$	−112726.	−	213898.i	−0.322738	−	0.612395i
$592$	−113944.			−0.325123
$593$		−	374187.i		−	1.06409i	−0.846715	−	0.532046i	$-0.821423\pi$
$593$		−	374187.i		−	1.06409i	0.846715	−	0.532046i	$-0.178577\pi$
$594$	−367.865	+	3164.28i	−0.00104260	+	0.00896814i
$595$	−296125.			−0.836454
$596$		−	222549.i		−	0.626516i
$597$	−85705.2	+	45167.3i	−0.240469	+	0.126729i
$598$	58975.3			0.164918
$599$		−	453167.i		−	1.26300i	−0.775374	−	0.631502i	$-0.782439\pi$
$599$		−	453167.i		−	1.26300i	0.775374	−	0.631502i	$-0.217561\pi$
$600$	−27622.1	−	52413.0i	−0.0767281	−	0.145592i
$601$	461976.			1.27900			0.639501	−	0.768791i	$-0.279141\pi$
$601$	461976.			1.27900			0.639501	+	0.768791i	$0.279141\pi$
$602$			280530.i			0.774082i
$603$	−78555.7	+	114638.i	−0.216044	+	0.315279i
$604$	−220215.			−0.603633
$605$		−	267560.i		−	0.730988i
$606$	259770.	−	136901.i	0.707364	−	0.372787i
$607$	14225.3			0.0386086			0.0193043	−	0.999814i	$-0.493855\pi$
$607$	14225.3			0.0386086			0.0193043	+	0.999814i	$0.493855\pi$
$608$		−	119041.i		−	0.322025i
$609$	226437.	+	429665.i	0.610538	+	1.15850i
$610$	−123006.			−0.330572
$611$			609301.i			1.63211i
$612$	−190777.	−	130730.i	−0.509359	−	0.349037i
$613$	454261.			1.20888			0.604441	−	0.796650i	$-0.293396\pi$
$613$	454261.			1.20888			0.604441	+	0.796650i	$0.293396\pi$
$614$			70018.6i			0.185728i
$615$	−287643.	+	151590.i	−0.760507	+	0.400794i
$616$	1586.94			0.00418215
$617$		−	319738.i		−	0.839894i	−0.907549	−	0.419947i	$-0.862049\pi$
$617$		−	319738.i		−	0.839894i	0.907549	−	0.419947i	$-0.137951\pi$
$618$	109192.	+	207192.i	0.285900	+	0.542496i
$619$	−664043.			−1.73306			−0.866532	−	0.499121i	$-0.833656\pi$
$619$	−664043.			−1.73306			−0.866532	+	0.499121i	$0.833656\pi$
$620$			188405.i			0.490127i
$621$	79873.8	+	9285.76i	0.207119	+	0.0240788i
$622$	−360190.			−0.931001
$623$		−	143752.i		−	0.370371i
$624$	96324.2	−	50763.7i	0.247381	−	0.130372i
$625$	−124158.			−0.317845
$626$		−	309119.i		−	0.788817i
$627$	−4263.14	−	8089.31i	−0.0108441	−	0.0205767i
$628$	−243784.			−0.618139
$629$		−	635414.i		−	1.60604i
$630$	107452.	−	156807.i	0.270727	−	0.395080i
$631$	−151974.			−0.381689			−0.190844	−	0.981620i	$-0.561123\pi$
$631$	−151974.			−0.381689			−0.190844	+	0.981620i	$0.561123\pi$
$632$		−	122003.i		−	0.305447i
$633$	−81885.6	+	43154.4i	−0.204362	+	0.107701i
$634$	103147.			0.256613
$635$		−	109828.i		−	0.272374i
$636$	10813.5	+	20518.7i	0.0267334	+	0.0507267i
$637$	64325.3			0.158527
$638$			5194.73i			0.0127621i
$639$	480628.	+	329349.i	1.17708	+	0.806594i
$640$	26468.9			0.0646214
$641$			684347.i			1.66556i	0.553605	+	0.832780i	$0.313252\pi$
$641$			684347.i			1.66556i	−0.553605	+	0.832780i	$0.686748\pi$
$642$	−125043.	+	65898.6i	−0.303381	+	0.159884i
$643$	−654785.			−1.58371			−0.791857	−	0.610706i	$-0.790886\pi$
$643$	−654785.			−1.58371			−0.791857	+	0.610706i	$0.790886\pi$
$644$		−	40058.1i		−	0.0965870i
$645$	167566.	+	317958.i	0.402779	+	0.764275i
$646$	663840.			1.59074
$647$		−	325775.i		−	0.778232i	−0.921189	−	0.389116i	$-0.872781\pi$
$647$		−	325775.i		−	0.778232i	0.921189	−	0.389116i	$-0.127219\pi$
$648$	138450.	−	53585.8i	0.329719	−	0.127614i
$649$	−4226.23			−0.0100338
$650$			155547.i			0.368158i
$651$	465705.	−	245431.i	1.09888	−	0.579118i
$652$	40713.5			0.0957731
$653$			548909.i			1.28728i	0.765327	+	0.643641i	$0.222577\pi$
$653$			548909.i			1.28728i	−0.765327	+	0.643641i	$0.777423\pi$
$654$	70877.4	+	134490.i	0.165711	+	0.314438i
$655$	219221.			0.510976
$656$			126500.i			0.293956i
$657$	−391578.	+	571440.i	−0.907167	+	1.32385i
$658$	413858.			0.955872
$659$			109541.i			0.252236i	0.992015	+	0.126118i	$0.0402518\pi$
$659$			109541.i			0.252236i	−0.992015	+	0.126118i	$0.959748\pi$
$660$	1798.67	−	947.912i	0.00412917	−	0.00217611i
$661$	−327299.			−0.749104			−0.374552	−	0.927206i	$-0.622203\pi$
$661$	−327299.			−0.749104			−0.374552	+	0.927206i	$0.622203\pi$
$662$		−	129496.i		−	0.295488i
$663$	283086.	+	537157.i	0.644009	+	1.22201i
$664$	−67176.9			−0.152364
$665$			545635.i			1.23384i
$666$	336470.	+	230565.i	0.758575	+	0.519811i
$667$	131127.			0.294741
$668$		−	38338.4i		−	0.0859174i
$669$	251869.	−	132737.i	0.562759	−	0.296579i
$670$	88696.3			0.197586
$671$			3676.02i			0.00816456i
$672$	−34480.4	−	65426.7i	−0.0763544	−	0.144883i
$673$	439385.			0.970097			0.485048	−	0.874487i	$-0.338802\pi$
$673$	439385.			0.970097			0.485048	+	0.874487i	$0.338802\pi$
$674$			217545.i			0.478882i
$675$	−24491.1	+	210666.i	−0.0537528	+	0.462368i
$676$	−57374.7			−0.125553
$677$		−	701438.i		−	1.53043i	−0.643777	−	0.765213i	$-0.722634\pi$
$677$		−	701438.i		−	1.53043i	0.643777	−	0.765213i	$-0.277366\pi$
$678$	−397749.	+	209617.i	−0.865267	+	0.456003i
$679$	−236604.			−0.513196
$680$			147605.i			0.319216i
$681$	−129367.	−	245474.i	−0.278952	−	0.529312i
$682$	5630.46			0.0121053
$683$			739653.i			1.58557i	0.609499	+	0.792787i	$0.291371\pi$
$683$			739653.i			1.58557i	−0.609499	+	0.792787i	$0.708629\pi$
$684$	−240880.	+	351522.i	−0.514859	+	0.751347i
$685$	−348409.			−0.742520
$686$		−	351972.i		−	0.747929i
$687$	529264.	−	278927.i	1.12140	−	0.590985i
$688$	139832.			0.295413
$689$		−	60893.8i		−	0.128273i
$690$	−23927.5	−	45402.5i	−0.0502573	−	0.0953634i
$691$	307540.			0.644089			0.322044	−	0.946725i	$-0.395630\pi$
$691$	307540.			0.644089			0.322044	+	0.946725i	$0.395630\pi$
$692$		−	194846.i		−	0.406892i
$693$	−4686.16	−	3211.18i	−0.00975777	−	0.00668649i
$694$	−510802.			−1.06056
$695$			121447.i			0.251431i
$696$	214169.	−	112869.i	0.442118	−	0.233000i
$697$	−705434.			−1.45208
$698$		−	208420.i		−	0.427788i
$699$	−361999.	−	686894.i	−0.740889	−	1.40584i
$700$	105653.			0.215618
$701$			68869.9i			0.140150i	0.997542	+	0.0700751i	$0.0223239\pi$
$701$			68869.9i			0.140150i	−0.997542	+	0.0700751i	$0.977676\pi$
$702$	−387161.	−	45009.5i	−0.785628	−	0.0913335i
$703$	−1.17080e6			−2.36904
$704$		−	791.021i		−	0.00159604i
$705$	469074.	−	247206.i	0.943762	−	0.497371i
$706$	−518005.			−1.03926
$707$			523637.i			1.04759i
$708$	91825.8	+	174240.i	0.183189	+	0.347601i
$709$	−611666.			−1.21681			−0.608404	−	0.793627i	$-0.708190\pi$
$709$	−611666.			−1.21681			−0.608404	+	0.793627i	$0.708190\pi$
$710$		−	371865.i		−	0.737680i
$711$	−246873.	+	360268.i	−0.488354	+	0.712668i
$712$	−71653.8			−0.141345
$713$		−	142126.i		−	0.279572i
$714$	364856.	−	192282.i	0.715689	−	0.377174i
$715$	−5337.93			−0.0104414
$716$		−	358663.i		−	0.699617i
$717$	167735.	+	318279.i	0.326277	+	0.619112i
$718$	232507.			0.451011
$719$		−	1.01275e6i		−	1.95904i	−0.201355	−	0.979518i	$-0.564535\pi$
$719$		−	1.01275e6i		−	1.95904i	0.201355	−	0.979518i	$-0.435465\pi$
$720$	−78161.4	−	53559.9i	−0.150774	−	0.103318i
$721$	−417653.			−0.803424
$722$		−	854574.i		−	1.63936i
$723$	−306788.	+	161680.i	−0.586897	+	0.309300i
$724$	519478.			0.991037
$725$			345846.i			0.657971i
$726$	173734.	+	329660.i	0.329618	+	0.625451i
$727$	−627113.			−1.18653			−0.593263	−	0.805009i	$-0.702160\pi$
$727$	−627113.			−1.18653			−0.593263	+	0.805009i	$0.702160\pi$
$728$			194168.i			0.366365i
$729$	−517267.	−	121918.i	−0.973330	−	0.229410i
$730$	442126.			0.829660
$731$			779780.i			1.45928i
$732$	151555.	−	79871.0i	0.282846	−	0.149062i
$733$	86692.4			0.161352			0.0806758	−	0.996740i	$-0.474292\pi$
$733$	86692.4			0.161352			0.0806758	+	0.996740i	$0.474292\pi$
$734$		−	488818.i		−	0.907308i
$735$	−26098.1	−	49521.2i	−0.0483096	−	0.0916677i
$736$	−19967.2			−0.0368605
$737$		−	2650.68i		−	0.00488003i
$738$	255973.	−	373548.i	0.469982	−	0.685857i
$739$	−719297.			−1.31710			−0.658551	−	0.752537i	$-0.728830\pi$
$739$	−719297.			−1.31710			−0.658551	+	0.752537i	$0.728830\pi$
$740$		−	260329.i		−	0.475400i
$741$	989755.	−	521609.i	1.80257	−	0.949967i
$742$	−41361.1			−0.0751251
$743$			118644.i			0.214915i	0.994210	+	0.107457i	$0.0342710\pi$
$743$			118644.i			0.214915i	−0.994210	+	0.107457i	$0.965729\pi$
$744$	−122336.	−	232133.i	−0.221009	−	0.419365i
$745$	508459.			0.916101
$746$			160460.i			0.288329i
$747$	198370.	+	135932.i	0.355496	+	0.243602i
$748$	4411.17			0.00788407
$749$		−	252058.i		−	0.449300i
$750$	377006.	−	198685.i	0.670233	−	0.353218i
$751$	−167909.			−0.297711			−0.148856	−	0.988859i	$-0.547559\pi$
$751$	−167909.			−0.297711			−0.148856	+	0.988859i	$0.547559\pi$
$752$		−	206290.i		−	0.364789i
$753$	−260464.	−	494232.i	−0.459365	−	0.871647i
$754$	−635592.			−1.11799
$755$		−	503128.i		−	0.882641i
$756$	−30572.0	+	262973.i	−0.0534910	+	0.460116i
$757$	−745852.			−1.30155			−0.650775	−	0.759271i	$-0.725556\pi$
$757$	−745852.			−1.30155			−0.650775	+	0.759271i	$0.725556\pi$
$758$			540801.i			0.941237i
$759$	−1356.85	+	715.070i	−0.00235531	+	0.00124127i
$760$	271975.			0.470870
$761$		−	639284.i		−	1.10389i	−0.833882	−	0.551943i	$-0.813887\pi$
$761$		−	639284.i		−	1.10389i	0.833882	−	0.551943i	$-0.186113\pi$
$762$	71314.1	+	135319.i	0.122819	+	0.233049i
$763$	−271102.			−0.465675
$764$		−	20705.2i		−	0.0354725i
$765$	298679.	−	435871.i	0.510367	−	0.744792i
$766$	−441845.			−0.753030
$767$		−	517094.i		−	0.878979i
$768$	−32612.3	+	17187.0i	−0.0552916	+	0.0291391i
$769$	220967.			0.373659			0.186829	−	0.982392i	$-0.440179\pi$
$769$	220967.			0.373659			0.186829	+	0.982392i	$0.440179\pi$
$770$			3625.71i			0.00611521i
$771$	264800.	+	502459.i	0.445461	+	0.845263i
$772$	−149414.			−0.250702
$773$		−	981825.i		−	1.64314i	−0.570106	−	0.821571i	$-0.693098\pi$
$773$		−	981825.i		−	1.64314i	0.570106	−	0.821571i	$-0.306902\pi$
$774$	−412916.	−	282950.i	−0.689255	−	0.472310i
$775$	374855.			0.624109
$776$			117937.i			0.195851i
$777$	−643489.	+	339124.i	−1.06586	+	0.561716i
$778$	412352.			0.681255
$779$			1.29982e6i			2.14194i
$780$	115980.	+	220073.i	0.190632	+	0.361724i
$781$	−11113.1			−0.0182194
$782$		−	111348.i		−	0.182083i
$783$	−860820.	−	100075.i	−1.40407	−	0.163231i
$784$	−21778.5			−0.0354320
$785$		−	556977.i		−	0.903853i
$786$	−270102.	+	142346.i	−0.437203	+	0.230410i
$787$	−1.06943e6			−1.72664			−0.863321	−	0.504655i	$-0.831620\pi$
$787$	−1.06943e6			−1.72664			−0.863321	+	0.504655i	$0.831620\pi$
$788$		−	214919.i		−	0.346117i
$789$	−158434.	−	300630.i	−0.254504	−	0.482923i
$790$	278742.			0.446630
$791$		−	801773.i		−	1.28144i
$792$	−1600.63	+	2335.84i	−0.00255176	+	0.00372386i
$793$	−449773.			−0.715233
$794$			162514.i			0.257780i
$795$	−46879.4	+	24705.8i	−0.0741733	+	0.0390900i
$796$	−86114.3			−0.135909
$797$			1.08625e6i			1.71007i	0.518571	+	0.855035i	$0.326464\pi$
$797$			1.08625e6i			1.71007i	−0.518571	+	0.855035i	$0.673536\pi$
$798$	−354295.	−	672276.i	−0.556364	−	1.05570i
$799$	1.15039e6			1.80198
$800$		−	52663.2i		−	0.0822863i
$801$	211590.	+	144991.i	0.329784	+	0.225984i
$802$	−404297.			−0.628567
$803$		−	13212.9i		−	0.0204912i
$804$	−109283.	+	57592.8i	−0.169059	+	0.0890956i
$805$	91521.2			0.141231
$806$			688906.i			1.06045i
$807$	−193034.	−	366282.i	−0.296405	−	0.562430i
$808$	261010.			0.399792
$809$			19313.3i			0.0295093i	0.999891	+	0.0147547i	$0.00469672\pi$
$809$			19313.3i			0.0295093i	−0.999891	+	0.0147547i	$0.995303\pi$
$810$	122428.	+	316319.i	0.186600	+	0.482120i
$811$	40707.8			0.0618922			0.0309461	−	0.999521i	$-0.490148\pi$
$811$	40707.8			0.0618922			0.0309461	+	0.999521i	$0.490148\pi$
$812$			431716.i			0.654766i
$813$	792425.	−	417615.i	1.19888	−	0.631822i
$814$	−7779.90			−0.0117415
$815$			93018.7i			0.140041i
$816$	−95844.0	−	181864.i	−0.143941	−	0.273129i
$817$	1.43681e6			2.15256
$818$		−	290191.i		−	0.433688i
$819$	392899.	−	573367.i	0.585750	−	0.854801i
$820$	−289016.			−0.429828
$821$			394072.i			0.584641i	0.956320	+	0.292321i	$0.0944274\pi$
$821$			394072.i			0.584641i	−0.956320	+	0.292321i	$0.905573\pi$
$822$	429274.	−	226231.i	0.635318	−	0.334818i
$823$	1.01839e6			1.50354			0.751768	−	0.659428i	$-0.229201\pi$
$823$	1.01839e6			1.50354			0.751768	+	0.659428i	$0.229201\pi$
$824$			208181.i			0.306611i
$825$	−1885.99	−	3578.67i	−0.00277097	−	0.00525792i
$826$	−351228.			−0.514789
$827$			423630.i			0.619406i	0.950833	+	0.309703i	$0.100230\pi$
$827$			423630.i			0.619406i	−0.950833	+	0.309703i	$0.899770\pi$
$828$	58962.0	+	40403.6i	0.0860027	+	0.0589331i
$829$	273122.			0.397418			0.198709	−	0.980059i	$-0.436325\pi$
$829$	273122.			0.397418			0.198709	+	0.980059i	$0.436325\pi$
$830$		−	153480.i		−	0.222790i
$831$	1.09697e6	−	578114.i	1.58852	−	0.837166i
$832$	96784.0			0.139816
$833$		−	121449.i		−	0.175027i
$834$	−78858.9	−	149635.i	−0.113375	−	0.215130i
$835$	87592.2			0.125630
$836$		−	8127.93i		−	0.0116297i
$837$	−108469.	+	933025.i	−0.154830	+	1.33181i
$838$	−630388.			−0.897676
$839$		−	323399.i		−	0.459425i	−0.973259	−	0.229712i	$-0.926221\pi$
$839$		−	323399.i		−	0.459425i	0.973259	−	0.229712i	$-0.0737785\pi$
$840$	149481.	−	78777.8i	0.211850	−	0.111647i
$841$	−705907.			−0.998057
$842$			172205.i			0.242897i
$843$	−225057.	−	427047.i	−0.316692	−	0.600925i
$844$	−82276.5			−0.115502
$845$		−	131085.i		−	0.183585i
$846$	−417428.	+	609163.i	−0.583231	+	0.851124i
$847$	−664521.			−0.926279
$848$			20616.7i			0.0286700i
$849$	−1.10085e6	+	580155.i	−1.52725	+	0.804875i
$850$	293679.			0.406477
$851$			196382.i			0.271171i
$852$	241461.	+	458174.i	0.332635	+	0.631177i
$853$	399583.			0.549173			0.274586	−	0.961562i	$-0.411459\pi$
$853$	399583.			0.549173			0.274586	+	0.961562i	$0.411459\pi$
$854$			305502.i			0.418888i
$855$	−803127.	−	550341.i	−1.09863	−	0.752834i
$856$	−125640.			−0.171467
$857$			63655.3i			0.0866708i	0.999061	+	0.0433354i	$0.0137984\pi$
$857$			63655.3i			0.0866708i	−0.999061	+	0.0433354i	$0.986202\pi$
$858$	6576.85	−	3466.06i	0.00893395	−	0.00470827i
$859$	876302.			1.18759			0.593796	−	0.804615i	$-0.297628\pi$
$859$	876302.			1.18759			0.593796	+	0.804615i	$0.297628\pi$
$860$			319476.i			0.431957i
$861$	376494.	+	714399.i	0.507869	+	0.963684i
$862$	−380069.			−0.511503
$863$			756177.i			1.01532i	0.861558	+	0.507659i	$0.169489\pi$
$863$			756177.i			1.01532i	−0.861558	+	0.507659i	$0.830511\pi$
$864$	131080.	+	15238.8i	0.175594	+	0.0204138i
$865$	445167.			0.594964
$866$			281142.i			0.374878i
$867$	349183.	−	184022.i	0.464531	−	0.244812i
$868$	467929.			0.621069
$869$		−	8330.16i		−	0.0110310i
$870$	257873.	+	489314.i	0.340696	+	0.646472i
$871$	324320.			0.427501
$872$			135132.i			0.177716i
$873$	238645.	−	348261.i	0.313129	−	0.456958i
$874$	−205168.			−0.268588
$875$			759959.i			0.992600i
$876$	−544743.	+	287084.i	−0.709877	+	0.374111i
$877$	−164075.			−0.213326			−0.106663	−	0.994295i	$-0.534017\pi$
$877$	−164075.			−0.213326			−0.106663	+	0.994295i	$0.534017\pi$
$878$		−	253451.i		−	0.328780i
$879$	−396086.	−	751574.i	−0.512639	−	0.972734i
$880$	1807.25			0.00233375
$881$			808718.i			1.04195i	0.853573	+	0.520973i	$0.174431\pi$
$881$			808718.i			1.04195i	−0.853573	+	0.520973i	$0.825569\pi$
$882$	64310.8	+	44068.8i	0.0826698	+	0.0566492i
$883$	1.02898e6			1.31973			0.659866	−	0.751383i	$-0.270613\pi$
$883$	1.02898e6			1.31973			0.659866	+	0.751383i	$0.270613\pi$
$884$			539721.i			0.690661i
$885$	−398088.	+	209795.i	−0.508267	+	0.267861i
$886$	−312925.			−0.398632
$887$			1.00246e6i			1.27415i	0.770801	+	0.637076i	$0.219856\pi$
$887$			1.00246e6i			1.27415i	−0.770801	+	0.637076i	$0.780144\pi$
$888$	169038.	+	320751.i	0.214368	+	0.406763i
$889$	−272772.			−0.345141
$890$		−	163708.i		−	0.206676i
$891$	9453.15	−	3658.75i	0.0119075	−	0.00460868i
$892$	253072.			0.318063
$893$		−	2.11968e6i		−	2.65808i
$894$	−626472.	+	330156.i	−0.783838	+	0.413089i
$895$	819441.			1.02299
$896$		−	65739.1i		−	0.0818856i
$897$	−87491.3	−	166015.i	−0.108738	−	0.206330i
$898$	200762.			0.248960
$899$			1.53173e6i			1.89523i
$900$	−106564.	+	155512.i	−0.131561	+	0.191990i
$901$	−114970.			−0.141624
$902$			8637.21i			0.0106160i
$903$	789690.	−	416173.i	0.968458	−	0.510386i
$904$	−399648.			−0.489036
$905$			1.18686e6i			1.44911i
$906$	326694.	+	619902.i	0.398001	+	0.755209i
$907$	1.39431e6			1.69491			0.847454	−	0.530869i	$-0.178134\pi$
$907$	1.39431e6			1.69491			0.847454	+	0.530869i	$0.178134\pi$
$908$		−	246646.i		−	0.299159i
$909$	−770748.	−	528153.i	−0.932792	−	0.639193i
$910$	−443617.			−0.535705
$911$		−	403508.i		−	0.486201i	−0.970001	−	0.243101i	$-0.921836\pi$
$911$		−	403508.i		−	0.486201i	0.970001	−	0.243101i	$-0.0781645\pi$
$912$	−335100.	+	176600.i	−0.402888	+	0.212325i
$913$	−4586.72			−0.00550251
$914$			820806.i			0.982535i
$915$	182482.	+	346261.i	0.217961	+	0.413581i
$916$	531791.			0.633797
$917$		−	544465.i		−	0.647487i
$918$	−84980.0	+	730976.i	−0.100840	+	0.867397i
$919$	303091.			0.358874			0.179437	−	0.983769i	$-0.442572\pi$
$919$	303091.			0.358874			0.179437	+	0.983769i	$0.442572\pi$
$920$		−	45619.2i		−	0.0538980i
$921$	197102.	−	103874.i	0.232365	−	0.122458i
$922$	−280488.			−0.329954
$923$		−	1.35973e6i		−	1.59606i
$924$	−2354.27	−	4467.23i	−0.00275747	−	0.00523232i
$925$	−517957.			−0.605355
$926$		−	818729.i		−	0.954812i
$927$	421255.	−	614748.i	0.490214	−	0.715382i
$928$	215191.			0.249879
$929$		−	378781.i		−	0.438891i	−0.975625	−	0.219446i	$-0.929575\pi$
$929$		−	378781.i		−	0.438891i	0.975625	−	0.219446i	$-0.0704249\pi$
$930$	530358.	−	279503.i	0.613201	−	0.323162i
$931$	−223779.			−0.258179
$932$		−	690173.i		−	0.794559i
$933$	534349.	+	1.01393e6i	0.613850	+	1.16478i
$934$	498964.			0.571973
$935$			10078.2i			0.0115282i
$936$	−285798.	−	195842.i	−0.326218	−	0.223540i
$937$	−548454.			−0.624685			−0.312343	−	0.949970i	$-0.601114\pi$
$937$	−548454.			−0.624685			−0.312343	+	0.949970i	$0.601114\pi$
$938$		−	220289.i		−	0.250373i
$939$	−870165.	+	458584.i	−0.986894	+	0.520101i
$940$	471313.			0.533401
$941$			41864.8i			0.0472791i	0.999721	+	0.0236396i	$0.00752541\pi$
$941$			41864.8i			0.0472791i	−0.999721	+	0.0236396i	$0.992475\pi$
$942$	361659.	+	686250.i	0.407566	+	0.773358i
$943$	218023.			0.245177
$944$			175072.i			0.196459i
$945$	−600817.	−	69848.3i	−0.672789	−	0.0782154i
$946$	9547.49			0.0106686
$947$		−	894372.i		−	0.997283i	−0.866808	−	0.498641i	$-0.833833\pi$
$947$		−	894372.i		−	0.997283i	0.866808	−	0.498641i	$-0.166167\pi$
$948$	−343437.	+	180994.i	−0.382147	+	0.201395i
$949$	1.61664e6			1.79507
$950$		−	541128.i		−	0.599587i
$951$	−153021.	−	290358.i	−0.169196	−	0.321050i
$952$	366597.			0.404497
$953$			1.54402e6i			1.70007i	0.526729	+	0.850033i	$0.323418\pi$
$953$			1.54402e6i			1.70007i	−0.526729	+	0.850033i	$0.676582\pi$
$954$	41717.9	−	60880.0i	0.0458380	−	0.0668926i
$955$	47305.4			0.0518685
$956$			319798.i			0.349913i
$957$	14623.1	−	7706.49i	0.0159667	−	0.00841459i
$958$	55801.2			0.0608013
$959$			865320.i			0.940891i
$960$	−39267.2	−	74509.7i	−0.0426077	−	0.0808482i
$961$	736686.			0.797692
$962$		−	951896.i		−	1.02858i
$963$	371008.	+	254232.i	0.400065	+	0.274143i
$964$	−308253.			−0.331706
$965$		−	341368.i		−	0.366580i
$966$	−112763.	+	59427.1i	−0.120841	+	0.0636840i
$967$	−1.22682e6			−1.31198			−0.655991	−	0.754768i	$-0.727749\pi$
$967$	−1.22682e6			−1.31198			−0.655991	+	0.754768i	$0.727749\pi$
$968$			331234.i			0.353496i
$969$	−984821.	−	1.86870e6i	−1.04884	−	1.99018i
$970$	−269451.			−0.286376
$971$			1.26734e6i			1.34417i	0.740475	+	0.672084i	$0.234601\pi$
$971$			1.26734e6i			1.34417i	−0.740475	+	0.672084i	$0.765399\pi$
$972$	−356238.	−	310240.i	−0.377057	−	0.328372i
$973$	301630.			0.318603
$974$			35991.6i			0.0379387i
$975$	437863.	−	230757.i	0.460605	−	0.242743i
$976$	152279.			0.159860
$977$			140943.i			0.147657i	0.997271	+	0.0738287i	$0.0235218\pi$
$977$			140943.i			0.147657i	−0.997271	+	0.0738287i	$0.976478\pi$
$978$	−60399.5	−	114608.i	−0.0631474	−	0.119822i
$979$	−4892.40			−0.00510454
$980$		−	49757.6i		−	0.0518092i
$981$	273440.	−	399038.i	0.284134	−	0.414645i
$982$	976473.			1.01260
$983$		−	239228.i		−	0.247573i	−0.992309	−	0.123787i	$-0.960496\pi$
$983$		−	239228.i		−	0.247573i	0.992309	−	0.123787i	$-0.0395039\pi$
$984$	356096.	−	187666.i	0.367771	−	0.193818i
$985$	491028.			0.506097
$986$			1.20003e6i			1.23435i
$987$	−613968.	−	1.16501e6i	−0.630248	−	1.19590i
$988$	994480.			1.01878
$989$		−	241001.i		−	0.246391i
$990$	−5336.73	−	3656.98i	−0.00544508	−	0.00373123i
$991$	924819.			0.941693			0.470847	−	0.882215i	$-0.343949\pi$
$991$	924819.			0.941693			0.470847	+	0.882215i	$0.343949\pi$
$992$		−	233242.i		−	0.237019i
$993$	−364529.	+	192110.i	−0.369687	+	0.194828i
$994$	−923575.			−0.934758
$995$		−	196746.i		−	0.198729i
$996$	99658.4	+	189102.i	0.100460	+	0.190624i
$997$	−636920.			−0.640759			−0.320379	−	0.947289i	$-0.603810\pi$
$997$	−636920.			−0.640759			−0.320379	+	0.947289i	$0.603810\pi$
$998$		−	712599.i		−	0.715458i
$999$	149878.	−	1.28921e6i	0.150178	−	1.29179i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.c.a.47.3	✓	28
3.2	odd	2	inner	138.5.c.a.47.17	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.c.a.47.3	✓	28	1.1	even	1	trivial
138.5.c.a.47.17	yes	28	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} +(-7.96199 + 4.19604i) q^{3} -8.00000 q^{4} -18.2777i q^{5} +(11.8682 + 22.5199i) q^{6} -45.3951 q^{7} +22.6274i q^{8} +(45.7866 - 66.8176i) q^{9} +O(q^{10})\)	\(q-2.82843i q^{2} +(-7.96199 + 4.19604i) q^{3} -8.00000 q^{4} -18.2777i q^{5} +(11.8682 + 22.5199i) q^{6} -45.3951 q^{7} +22.6274i q^{8} +(45.7866 - 66.8176i) q^{9} -51.6971 q^{10} +1.54496i q^{11} +(63.6959 - 33.5683i) q^{12} -189.031 q^{13} +128.397i q^{14} +(76.6938 + 145.527i) q^{15} +64.0000 q^{16} +356.899i q^{17} +(-188.989 - 129.504i) q^{18} +657.616 q^{19} +146.222i q^{20} +(361.435 - 190.479i) q^{21} +4.36981 q^{22} -110.304i q^{23} +(-94.9454 - 180.159i) q^{24} +290.926 q^{25} +534.661i q^{26} +(-84.1833 + 724.123i) q^{27} +363.160 q^{28} +1188.78i q^{29} +(411.612 - 216.923i) q^{30} +1288.49 q^{31} -181.019i q^{32} +(-6.48272 - 12.3010i) q^{33} +1009.46 q^{34} +829.717i q^{35} +(-366.293 + 534.541i) q^{36} -1780.37 q^{37} -1860.02i q^{38} +(1505.07 - 793.182i) q^{39} +413.577 q^{40} +1976.56i q^{41} +(-538.757 - 1022.29i) q^{42} +2184.87 q^{43} -12.3597i q^{44} +(-1221.27 - 836.873i) q^{45} -311.987 q^{46} -3223.28i q^{47} +(-509.567 + 268.546i) q^{48} -340.289 q^{49} -822.863i q^{50} +(-1497.56 - 2841.63i) q^{51} +1512.25 q^{52} +322.136i q^{53} +(2048.13 + 238.106i) q^{54} +28.2383 q^{55} -1027.17i q^{56} +(-5235.93 + 2759.38i) q^{57} +3362.37 q^{58} +2735.49i q^{59} +(-613.551 - 1164.21i) q^{60} +2379.36 q^{61} -3644.40i q^{62} +(-2078.48 + 3033.19i) q^{63} -512.000 q^{64} +3455.06i q^{65} +(-34.7924 + 18.3359i) q^{66} -1715.69 q^{67} -2855.20i q^{68} +(462.840 + 878.240i) q^{69} +2346.79 q^{70} +7193.14i q^{71} +(1511.91 + 1036.03i) q^{72} -8552.24 q^{73} +5035.65i q^{74} +(-2316.35 + 1220.74i) q^{75} -5260.93 q^{76} -70.1336i q^{77} +(-2243.46 - 4256.97i) q^{78} -5391.82 q^{79} -1169.77i q^{80} +(-2368.18 - 6118.70i) q^{81} +5590.56 q^{82} +2968.83i q^{83} +(-2891.48 + 1523.83i) q^{84} +6523.30 q^{85} -6179.76i q^{86} +(-4988.15 - 9465.02i) q^{87} -34.9585 q^{88} +3166.68i q^{89} +(-2367.03 + 3454.28i) q^{90} +8581.09 q^{91} +882.433i q^{92} +(-10258.9 + 5406.55i) q^{93} -9116.81 q^{94} -12019.7i q^{95} +(759.564 + 1441.27i) q^{96} +5212.12 q^{97} +962.483i q^{98} +(103.231 + 70.7385i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 8 q^{3} - 224 q^{4} - 32 q^{6} - 104 q^{7} + 80 q^{9}+O(q^{10}) \) 28 * q - 8 * q^3 - 224 * q^4 - 32 * q^6 - 104 * q^7 + 80 * q^9	\( 28 q - 8 q^{3} - 224 q^{4} - 32 q^{6} - 104 q^{7} + 80 q^{9} + 192 q^{10} + 64 q^{12} + 148 q^{15} + 1792 q^{16} + 448 q^{18} + 912 q^{19} - 1412 q^{21} - 1984 q^{22} + 256 q^{24} - 3028 q^{25} - 1700 q^{27} + 832 q^{28} + 768 q^{30} - 2400 q^{31} + 2772 q^{33} + 2944 q^{34} - 640 q^{36} - 2080 q^{37} + 468 q^{39} - 1536 q^{40} - 4480 q^{42} + 5536 q^{43} + 13852 q^{45} - 512 q^{48} + 13444 q^{49} - 16972 q^{51} + 6112 q^{54} - 624 q^{55} - 3304 q^{57} - 20160 q^{58} - 1184 q^{60} + 6376 q^{61} + 276 q^{63} - 14336 q^{64} + 960 q^{66} + 2168 q^{67} + 26624 q^{70} - 3584 q^{72} - 13568 q^{73} - 2464 q^{75} - 7296 q^{76} - 9920 q^{78} - 46064 q^{79} + 19184 q^{81} + 23168 q^{82} + 11296 q^{84} + 27584 q^{85} + 42140 q^{87} + 15872 q^{88} - 11008 q^{90} - 18040 q^{91} - 23892 q^{93} + 3840 q^{94} - 2048 q^{96} + 56848 q^{97} - 40092 q^{99}+O(q^{100}) \) 28 * q - 8 * q^3 - 224 * q^4 - 32 * q^6 - 104 * q^7 + 80 * q^9 + 192 * q^10 + 64 * q^12 + 148 * q^15 + 1792 * q^16 + 448 * q^18 + 912 * q^19 - 1412 * q^21 - 1984 * q^22 + 256 * q^24 - 3028 * q^25 - 1700 * q^27 + 832 * q^28 + 768 * q^30 - 2400 * q^31 + 2772 * q^33 + 2944 * q^34 - 640 * q^36 - 2080 * q^37 + 468 * q^39 - 1536 * q^40 - 4480 * q^42 + 5536 * q^43 + 13852 * q^45 - 512 * q^48 + 13444 * q^49 - 16972 * q^51 + 6112 * q^54 - 624 * q^55 - 3304 * q^57 - 20160 * q^58 - 1184 * q^60 + 6376 * q^61 + 276 * q^63 - 14336 * q^64 + 960 * q^66 + 2168 * q^67 + 26624 * q^70 - 3584 * q^72 - 13568 * q^73 - 2464 * q^75 - 7296 * q^76 - 9920 * q^78 - 46064 * q^79 + 19184 * q^81 + 23168 * q^82 + 11296 * q^84 + 27584 * q^85 + 42140 * q^87 + 15872 * q^88 - 11008 * q^90 - 18040 * q^91 - 23892 * q^93 + 3840 * q^94 - 2048 * q^96 + 56848 * q^97 - 40092 * q^99

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