LMFDB - Embedded newform 138.5.c.a.47.20

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			47.20
Character	$\chi$	$=$	138.47
Dual form			138.5.c.a.47.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} +(-2.30705 + 8.69928i) q^{3} -8.00000 q^{4} +39.0683i q^{5} +(-24.6053 - 6.52531i) q^{6} -28.2665 q^{7} -22.6274i q^{8} +(-70.3551 - 40.1393i) q^{9} +O(q^{10})$	\(q+2.82843i q^{2} +(-2.30705 + 8.69928i) q^{3} -8.00000 q^{4} +39.0683i q^{5} +(-24.6053 - 6.52531i) q^{6} -28.2665 q^{7} -22.6274i q^{8} +(-70.3551 - 40.1393i) q^{9} -110.502 q^{10} +127.349i q^{11} +(18.4564 - 69.5943i) q^{12} +37.9299 q^{13} -79.9497i q^{14} +(-339.866 - 90.1323i) q^{15} +64.0000 q^{16} +101.902i q^{17} +(113.531 - 198.994i) q^{18} +475.144 q^{19} -312.546i q^{20} +(65.2121 - 245.898i) q^{21} -360.198 q^{22} -110.304i q^{23} +(196.842 + 52.2025i) q^{24} -901.330 q^{25} +107.282i q^{26} +(511.495 - 519.436i) q^{27} +226.132 q^{28} -606.567i q^{29} +(254.933 - 961.286i) q^{30} -957.969 q^{31} +181.019i q^{32} +(-1107.85 - 293.800i) q^{33} -288.222 q^{34} -1104.32i q^{35} +(562.841 + 321.114i) q^{36} +1514.57 q^{37} +1343.91i q^{38} +(-87.5061 + 329.963i) q^{39} +884.014 q^{40} +184.777i q^{41} +(695.506 + 184.448i) q^{42} +294.916 q^{43} -1018.79i q^{44} +(1568.17 - 2748.65i) q^{45} +311.987 q^{46} -2229.52i q^{47} +(-147.651 + 556.754i) q^{48} -1602.00 q^{49} -2549.35i q^{50} +(-886.473 - 235.092i) q^{51} -303.440 q^{52} +3846.40i q^{53} +(1469.19 + 1446.73i) q^{54} -4975.31 q^{55} +639.598i q^{56} +(-1096.18 + 4133.41i) q^{57} +1715.63 q^{58} +2944.84i q^{59} +(2718.93 + 721.058i) q^{60} -3909.45 q^{61} -2709.55i q^{62} +(1988.69 + 1134.60i) q^{63} -512.000 q^{64} +1481.86i q^{65} +(830.992 - 3133.46i) q^{66} -1206.44 q^{67} -815.215i q^{68} +(959.567 + 254.477i) q^{69} +3123.50 q^{70} +3510.34i q^{71} +(-908.248 + 1591.95i) q^{72} -9594.69 q^{73} +4283.86i q^{74} +(2079.41 - 7840.92i) q^{75} -3801.15 q^{76} -3599.71i q^{77} +(-933.277 - 247.505i) q^{78} +7997.14 q^{79} +2500.37i q^{80} +(3338.68 + 5648.01i) q^{81} -522.629 q^{82} +1980.40i q^{83} +(-521.697 + 1967.19i) q^{84} -3981.13 q^{85} +834.149i q^{86} +(5276.70 + 1399.38i) q^{87} +2881.58 q^{88} +14572.2i q^{89} +(7774.36 + 4435.46i) q^{90} -1072.15 q^{91} +882.433i q^{92} +(2210.08 - 8333.65i) q^{93} +6306.04 q^{94} +18563.0i q^{95} +(-1574.74 - 417.620i) q^{96} -6778.19 q^{97} -4531.15i q^{98} +(5111.70 - 8959.65i) 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
$3$	−2.30705	+	8.69928i	−0.256338	+	0.966587i
$3$	−2.30705	+	8.69928i	−0.256338	+	0.966587i
$4$	−8.00000			−0.500000
$5$			39.0683i			1.56273i	0.624074	+	0.781365i	$0.285476\pi$
$5$			39.0683i			1.56273i	−0.624074	+	0.781365i	$0.714524\pi$
$6$	−24.6053	−	6.52531i	−0.683480	−	0.181259i
$7$	−28.2665			−0.576867			−0.288434	−	0.957500i	$-0.593134\pi$
$7$	−28.2665			−0.576867			−0.288434	+	0.957500i	$0.593134\pi$
$8$		−	22.6274i		−	0.353553i
$9$	−70.3551	−	40.1393i	−0.868581	−	0.495547i
$10$	−110.502			−1.10502
$11$			127.349i			1.05247i	0.850339	+	0.526236i	$0.176397\pi$
$11$			127.349i			1.05247i	−0.850339	+	0.526236i	$0.823603\pi$
$12$	18.4564	−	69.5943i	0.128169	−	0.483294i
$13$	37.9299			0.224438			0.112219	−	0.993684i	$-0.464204\pi$
$13$	37.9299			0.224438			0.112219	+	0.993684i	$0.464204\pi$
$14$		−	79.9497i		−	0.407907i
$15$	−339.866	−	90.1323i	−1.51052	−	0.400588i
$16$	64.0000			0.250000
$17$			101.902i			0.352602i	0.984336	+	0.176301i	$0.0564131\pi$
$17$			101.902i			0.352602i	−0.984336	+	0.176301i	$0.943587\pi$
$18$	113.531	−	198.994i	0.350404	−	0.614180i
$19$	475.144			1.31619			0.658094	−	0.752936i	$-0.271363\pi$
$19$	475.144			1.31619			0.658094	+	0.752936i	$0.271363\pi$
$20$		−	312.546i		−	0.781365i
$21$	65.2121	−	245.898i	0.147873	−	0.557593i
$22$	−360.198			−0.744210
$23$		−	110.304i		−	0.208514i
$23$		−	110.304i		−	0.208514i
$24$	196.842	+	52.2025i	0.341740	+	0.0906293i
$25$	−901.330			−1.44213
$26$			107.282i			0.158701i
$27$	511.495	−	519.436i	0.701640	−	0.712532i
$28$	226.132			0.288434
$29$		−	606.567i		−	0.721245i	−0.932712	−	0.360623i	$-0.882564\pi$
$29$		−	606.567i		−	0.721245i	0.932712	−	0.360623i	$-0.117436\pi$
$30$	254.933	−	961.286i	0.283258	−	1.06810i
$31$	−957.969			−0.996846			−0.498423	−	0.866934i	$-0.666087\pi$
$31$	−957.969			−0.996846			−0.498423	+	0.866934i	$0.666087\pi$
$32$			181.019i			0.176777i
$33$	−1107.85	−	293.800i	−1.01731	−	0.269789i
$34$	−288.222			−0.249327
$35$		−	1104.32i		−	0.901488i
$36$	562.841	+	321.114i	0.434291	+	0.247773i
$37$	1514.57			1.10634			0.553168	−	0.833070i	$-0.313419\pi$
$37$	1514.57			1.10634			0.553168	+	0.833070i	$0.313419\pi$
$38$			1343.91i			0.930685i
$39$	−87.5061	+	329.963i	−0.0575320	+	0.216938i
$40$	884.014			0.552509
$41$			184.777i			0.109921i	0.998489	+	0.0549605i	$0.0175033\pi$
$41$			184.777i			0.109921i	−0.998489	+	0.0549605i	$0.982497\pi$
$42$	695.506	+	184.448i	0.394278	+	0.104562i
$43$	294.916			0.159500			0.0797502	−	0.996815i	$-0.474588\pi$
$43$	294.916			0.159500			0.0797502	+	0.996815i	$0.474588\pi$
$44$		−	1018.79i		−	0.526236i
$45$	1568.17	−	2748.65i	0.774406	−	1.35736i
$46$	311.987			0.147442
$47$		−	2229.52i		−	1.00929i	−0.863327	−	0.504645i	$-0.831623\pi$
$47$		−	2229.52i		−	1.00929i	0.863327	−	0.504645i	$-0.168377\pi$
$48$	−147.651	+	556.754i	−0.0640846	+	0.241647i
$49$	−1602.00			−0.667224
$50$		−	2549.35i		−	1.01974i
$51$	−886.473	−	235.092i	−0.340820	−	0.0903853i
$52$	−303.440			−0.112219
$53$			3846.40i			1.36931i	0.728866	+	0.684657i	$0.240048\pi$
$53$			3846.40i			1.36931i	−0.728866	+	0.684657i	$0.759952\pi$
$54$	1469.19	+	1446.73i	0.503836	+	0.496134i
$55$	−4975.31			−1.64473
$56$			639.598i			0.203953i
$57$	−1096.18	+	4133.41i	−0.337389	+	1.27221i
$58$	1715.63			0.509997
$59$			2944.84i			0.845975i	0.906136	+	0.422987i	$0.139019\pi$
$59$			2944.84i			0.845975i	−0.906136	+	0.422987i	$0.860981\pi$
$60$	2718.93	+	721.058i	0.755258	+	0.200294i
$61$	−3909.45			−1.05064			−0.525322	−	0.850903i	$-0.676055\pi$
$61$	−3909.45			−1.05064			−0.525322	+	0.850903i	$0.676055\pi$
$62$		−	2709.55i		−	0.704877i
$63$	1988.69	+	1134.60i	0.501056	+	0.285865i
$64$	−512.000			−0.125000
$65$			1481.86i			0.350735i
$66$	830.992	−	3133.46i	0.190770	−	0.719344i
$67$	−1206.44			−0.268756			−0.134378	−	0.990930i	$-0.542904\pi$
$67$	−1206.44			−0.268756			−0.134378	+	0.990930i	$0.542904\pi$
$68$		−	815.215i		−	0.176301i
$69$	959.567	+	254.477i	0.201547	+	0.0534502i
$70$	3123.50			0.637449
$71$			3510.34i			0.696357i	0.937428	+	0.348179i	$0.113200\pi$
$71$			3510.34i			0.696357i	−0.937428	+	0.348179i	$0.886800\pi$
$72$	−908.248	+	1591.95i	−0.175202	+	0.307090i
$73$	−9594.69			−1.80047			−0.900234	−	0.435407i	$-0.856604\pi$
$73$	−9594.69			−1.80047			−0.900234	+	0.435407i	$0.856604\pi$
$74$			4283.86i			0.782298i
$75$	2079.41	−	7840.92i	0.369673	−	1.39394i
$76$	−3801.15			−0.658094
$77$		−	3599.71i		−	0.607137i
$78$	−933.277	−	247.505i	−0.153399	−	0.0406812i
$79$	7997.14			1.28139			0.640694	−	0.767796i	$-0.278647\pi$
$79$	7997.14			1.28139			0.640694	+	0.767796i	$0.278647\pi$
$80$			2500.37i			0.390683i
$81$	3338.68	+	5648.01i	0.508867	+	0.860845i
$82$	−522.629			−0.0777259
$83$			1980.40i			0.287472i	0.989616	+	0.143736i	$0.0459117\pi$
$83$			1980.40i			0.287472i	−0.989616	+	0.143736i	$0.954088\pi$
$84$	−521.697	+	1967.19i	−0.0739366	+	0.278796i
$85$	−3981.13			−0.551021
$86$			834.149i			0.112784i
$87$	5276.70	+	1399.38i	0.697146	+	0.184883i
$88$	2881.58			0.372105
$89$			14572.2i			1.83969i	0.392286	+	0.919843i	$0.371684\pi$
$89$			14572.2i			1.83969i	−0.392286	+	0.919843i	$0.628316\pi$
$90$	7774.36	+	4435.46i	0.959798	+	0.547588i
$91$	−1072.15			−0.129471
$92$			882.433i			0.104257i
$93$	2210.08	−	8333.65i	0.255530	−	0.963539i
$94$	6306.04			0.713676
$95$			18563.0i			2.05685i
$96$	−1574.74	−	417.620i	−0.170870	−	0.0453147i
$97$	−6778.19			−0.720394			−0.360197	−	0.932876i	$-0.617291\pi$
$97$	−6778.19			−0.720394			−0.360197	+	0.932876i	$0.617291\pi$
$98$		−	4531.15i		−	0.471799i
$99$	5111.70	−	8959.65i	0.521549	−	0.914157i
$100$	7210.64			0.721064
$101$		−	11019.2i		−	1.08021i	−0.841599	−	0.540103i	$-0.818385\pi$
$101$		−	11019.2i		−	1.08021i	0.841599	−	0.540103i	$-0.181615\pi$
$102$	664.941	−	2507.33i	0.0639121	−	0.240996i
$103$	536.877			0.0506058			0.0253029	−	0.999680i	$-0.491945\pi$
$103$	536.877			0.0506058			0.0253029	+	0.999680i	$0.491945\pi$
$104$		−	858.257i		−	0.0793507i
$105$	9606.82	+	2547.72i	0.871367	+	0.231086i
$106$	−10879.3			−0.968251
$107$			20217.2i			1.76585i	0.469513	+	0.882926i	$0.344429\pi$
$107$			20217.2i			1.76585i	−0.469513	+	0.882926i	$0.655571\pi$
$108$	−4091.96	+	4155.49i	−0.350820	+	0.356266i
$109$	−6395.83			−0.538324			−0.269162	−	0.963095i	$-0.586747\pi$
$109$	−6395.83			−0.538324			−0.269162	+	0.963095i	$0.586747\pi$
$110$		−	14072.3i		−	1.16300i
$111$	−3494.19	+	13175.7i	−0.283596	+	1.06937i
$112$	−1809.06			−0.144217
$113$		−	23229.6i		−	1.81922i	−0.415462	−	0.909611i	$-0.636380\pi$
$113$		−	23229.6i		−	1.81922i	0.415462	−	0.909611i	$-0.363620\pi$
$114$	−11691.1	−	3100.46i	−0.899588	−	0.238570i
$115$	4309.39			0.325852
$116$			4852.54i			0.360623i
$117$	−2668.56	−	1522.48i	−0.194942	−	0.111219i
$118$	−8329.26			−0.598195
$119$		−	2880.41i		−	0.203404i
$120$	−2039.46	+	7690.29i	−0.141629	+	0.534048i
$121$	−1576.78			−0.107696
$122$		−	11057.6i		−	0.742918i
$123$	−1607.43	−	426.290i	−0.106248	−	0.0281770i
$124$	7663.75			0.498423
$125$		−	10795.7i		−	0.690926i
$126$	−3209.13	+	5624.87i	−0.202137	+	0.354300i
$127$	−14695.9			−0.911150			−0.455575	−	0.890198i	$-0.650566\pi$
$127$	−14695.9			−0.911150			−0.455575	+	0.890198i	$0.650566\pi$
$128$		−	1448.15i		−	0.0883883i
$129$	−680.385	+	2565.56i	−0.0408861	+	0.154171i
$130$	−4191.33			−0.248007
$131$		−	16972.8i		−	0.989032i	−0.869169	−	0.494516i	$-0.835345\pi$
$131$		−	16972.8i		−	0.989032i	0.869169	−	0.494516i	$-0.164655\pi$
$132$	8862.76	+	2350.40i	0.508653	+	0.134894i
$133$	−13430.7			−0.759266
$134$		−	3412.34i		−	0.190039i
$135$	20293.5	+	19983.2i	1.11350	+	1.09647i
$136$	2305.78			0.124664
$137$		−	28996.8i		−	1.54493i	−0.635059	−	0.772464i	$-0.719024\pi$
$137$		−	28996.8i		−	1.54493i	0.635059	−	0.772464i	$-0.280976\pi$
$138$	−719.769	+	2714.07i	−0.0377950	+	0.142515i
$139$	14052.0			0.727291			0.363645	−	0.931537i	$-0.381532\pi$
$139$	14052.0			0.727291			0.363645	+	0.931537i	$0.381532\pi$
$140$			8834.59i			0.450744i
$141$	19395.3	+	5143.61i	0.975567	+	0.258720i
$142$	−9928.73			−0.492399
$143$			4830.34i			0.236214i
$144$	−4502.73	−	2568.91i	−0.217145	−	0.123887i
$145$	23697.5			1.12711
$146$		−	27137.9i		−	1.27312i
$147$	3695.90	−	13936.3i	0.171035	−	0.644930i
$148$	−12116.6			−0.553168
$149$			6226.09i			0.280442i	0.990120	+	0.140221i	$0.0447813\pi$
$149$			6226.09i			0.280442i	−0.990120	+	0.140221i	$0.955219\pi$
$150$	22177.5	+	5881.46i	0.985666	+	0.261398i
$151$	33388.9			1.46436			0.732182	−	0.681109i	$-0.238502\pi$
$151$	33388.9			1.46436			0.732182	+	0.681109i	$0.238502\pi$
$152$		−	10751.3i		−	0.465343i
$153$	4090.27	−	7169.31i	0.174731	−	0.306263i
$154$	10181.5			0.429310
$155$		−	37426.2i		−	1.55780i
$156$	700.049	−	2639.71i	0.0287660	−	0.108469i
$157$	−43245.0			−1.75443			−0.877216	−	0.480095i	$-0.840602\pi$
$157$	−43245.0			−1.75443			−0.877216	+	0.480095i	$0.840602\pi$
$158$			22619.3i			0.906078i
$159$	−33460.9	−	8873.82i	−1.32356	−	0.351008i
$160$	−7072.11			−0.276254
$161$			3117.91i			0.120285i
$162$	−15975.0	+	9443.20i	−0.608710	+	0.359823i
$163$	41977.2			1.57993			0.789966	−	0.613151i	$-0.210098\pi$
$163$	41977.2			1.57993			0.789966	+	0.613151i	$0.210098\pi$
$164$		−	1478.22i		−	0.0549605i
$165$	11478.3	−	43281.6i	0.421607	−	1.58977i
$166$	−5601.41			−0.203274
$167$			30551.7i			1.09547i	0.836651	+	0.547737i	$0.184511\pi$
$167$			30551.7i			1.09547i	−0.836651	+	0.547737i	$0.815489\pi$
$168$	−5564.04	−	1475.58i	−0.197139	−	0.0522811i
$169$	−27122.3			−0.949628
$170$		−	11260.3i		−	0.389631i
$171$	−33428.8	−	19071.9i	−1.14322	−	0.652233i
$172$	−2359.33			−0.0797502
$173$			20862.2i			0.697055i	0.937298	+	0.348528i	$0.113318\pi$
$173$			20862.2i			0.697055i	−0.937298	+	0.348528i	$0.886682\pi$
$174$	−3958.04	+	14924.8i	−0.130732	+	0.492957i
$175$	25477.4			0.831916
$176$			8150.34i			0.263118i
$177$	−25618.0	−	6793.88i	−0.817708	−	0.216856i
$178$	−41216.3			−1.30085
$179$		−	52789.5i		−	1.64756i	−0.566909	−	0.823781i	$-0.691861\pi$
$179$		−	52789.5i		−	1.64756i	0.566909	−	0.823781i	$-0.308139\pi$
$180$	−12545.4	+	21989.2i	−0.387203	+	0.678679i
$181$	−3617.49			−0.110420			−0.0552102	−	0.998475i	$-0.517583\pi$
$181$	−3617.49			−0.110420			−0.0552102	+	0.998475i	$0.517583\pi$
$182$		−	3032.49i		−	0.0915496i
$183$	9019.27	−	34009.4i	0.269320	−	1.01554i
$184$	−2495.90			−0.0737210
$185$			59171.8i			1.72891i
$186$	23571.1	+	6251.05i	0.681325	+	0.180687i
$187$	−12977.1			−0.371103
$188$			17836.2i			0.504645i
$189$	−14458.2	+	14682.6i	−0.404753	+	0.411036i
$190$	−52504.2			−1.45441
$191$		−	35812.0i		−	0.981661i	−0.871255	−	0.490831i	$-0.836693\pi$
$191$		−	35812.0i		−	0.981661i	0.871255	−	0.490831i	$-0.163307\pi$
$192$	1181.21	−	4454.03i	0.0320423	−	0.120823i
$193$	−32923.3			−0.883871			−0.441936	−	0.897047i	$-0.645708\pi$
$193$	−32923.3			−0.883871			−0.441936	+	0.897047i	$0.645708\pi$
$194$		−	19171.6i		−	0.509395i
$195$	−12891.1	−	3418.71i	−0.339016	−	0.0899070i
$196$	12816.0			0.333612
$197$			58745.0i			1.51370i	0.653591	+	0.756848i	$0.273261\pi$
$197$			58745.0i			1.51370i	−0.653591	+	0.756848i	$0.726739\pi$
$198$	25341.7	+	14458.1i	0.646407	+	0.368791i
$199$	62663.2			1.58236			0.791182	−	0.611581i	$-0.209466\pi$
$199$	62663.2			1.58236			0.791182	+	0.611581i	$0.209466\pi$
$200$			20394.8i			0.509869i
$201$	2783.32	−	10495.2i	0.0688924	−	0.259776i
$202$	31166.9			0.763820
$203$			17145.5i			0.416063i
$204$	7091.79	+	1880.74i	0.170410	+	0.0451927i
$205$	−7218.93			−0.171777
$206$			1518.52i			0.0357837i
$207$	−4427.53	+	7760.46i	−0.103329	+	0.181112i
$208$	2427.52			0.0561094
$209$			60509.1i			1.38525i
$210$	−7206.05	+	27172.2i	−0.163403	+	0.616150i
$211$	33032.4			0.741951			0.370975	−	0.928643i	$-0.379023\pi$
$211$	33032.4			0.741951			0.370975	+	0.928643i	$0.379023\pi$
$212$		−	30771.2i		−	0.684657i
$213$	−30537.4	−	8098.51i	−0.673090	−	0.178503i
$214$	−57183.0			−1.24865
$215$			11521.9i			0.249256i
$216$	−11753.5	−	11573.8i	−0.251918	−	0.248067i
$217$	27078.4			0.575048
$218$		−	18090.1i		−	0.380652i
$219$	22135.4	−	83466.9i	0.461529	−	1.74031i
$220$	39802.5			0.822365
$221$			3865.13i			0.0791370i
$222$	−37266.5	−	9883.06i	−0.756159	−	0.200533i
$223$	11753.1			0.236344			0.118172	−	0.992993i	$-0.462297\pi$
$223$	11753.1			0.236344			0.118172	+	0.992993i	$0.462297\pi$
$224$		−	5116.78i		−	0.101977i
$225$	63413.1	+	36178.7i	1.25260	+	0.714642i
$226$	65703.3			1.28638
$227$			75387.9i			1.46302i	0.681831	+	0.731510i	$0.261184\pi$
$227$			75387.9i			1.46302i	−0.681831	+	0.731510i	$0.738816\pi$
$228$	8769.43	−	33067.3i	0.168695	−	0.636105i
$229$	48478.3			0.924436			0.462218	−	0.886766i	$-0.347054\pi$
$229$	48478.3			0.924436			0.462218	+	0.886766i	$0.347054\pi$
$230$			12188.8i			0.230412i
$231$	31314.9	+	8304.70i	0.586850	+	0.155632i
$232$	−13725.0			−0.254999
$233$			59475.9i			1.09554i	0.836628	+	0.547771i	$0.184523\pi$
$233$			59475.9i			1.09554i	−0.836628	+	0.547771i	$0.815477\pi$
$234$	4306.23	−	7547.84i	0.0786439	−	0.137845i
$235$	87103.6			1.57725
$236$		−	23558.7i		−	0.422987i
$237$	−18449.8	+	69569.4i	−0.328469	+	1.23857i
$238$	8147.03			0.143829
$239$			84513.0i			1.47954i	0.672858	+	0.739772i	$0.265067\pi$
$239$			84513.0i			1.47954i	−0.672858	+	0.739772i	$0.734933\pi$
$240$	−21751.4	−	5768.47i	−0.377629	−	0.100147i
$241$	46786.0			0.805530			0.402765	−	0.915303i	$-0.368049\pi$
$241$	46786.0			0.805530			0.402765	+	0.915303i	$0.368049\pi$
$242$		−	4459.81i		−	0.0761528i
$243$	−56836.1	+	16013.9i	−0.962524	+	0.271196i
$244$	31275.6			0.525322
$245$		−	62587.6i		−	1.04269i
$246$	1205.73	−	4546.50i	0.0199241	−	0.0751289i
$247$	18022.2			0.295402
$248$			21676.4i			0.352438i
$249$	−17228.0	−	4568.87i	−0.277867	−	0.0736902i
$250$	30534.9			0.488559
$251$			54497.2i			0.865022i	0.901629	+	0.432511i	$0.142372\pi$
$251$			54497.2i			0.865022i	−0.901629	+	0.432511i	$0.857628\pi$
$252$	−15909.5	−	9076.78i	−0.250528	−	0.142932i
$253$	14047.1			0.219455
$254$		−	41566.4i		−	0.644280i
$255$	9184.65	−	34633.0i	0.141248	−	0.532610i
$256$	4096.00			0.0625000
$257$			6114.73i			0.0925787i	0.998928	+	0.0462894i	$0.0147396\pi$
$257$			6114.73i			0.0925787i	−0.998928	+	0.0462894i	$0.985260\pi$
$258$	−7256.50	−	1924.42i	−0.109015	−	0.0289108i
$259$	−42811.7			−0.638209
$260$		−	11854.9i		−	0.175368i
$261$	−24347.2	+	42675.1i	−0.357411	+	0.626460i
$262$	48006.3			0.699351
$263$		−	77736.5i		−	1.12386i	−0.827183	−	0.561932i	$-0.810058\pi$
$263$		−	77736.5i		−	1.12386i	0.827183	−	0.561932i	$-0.189942\pi$
$264$	−6647.94	+	25067.7i	−0.0953848	+	0.359672i
$265$	−150272.			−2.13987
$266$		−	37987.6i		−	0.536882i
$267$	−126767.	−	33618.6i	−1.77822	−	0.471582i
$268$	9651.56			0.134378
$269$			51171.1i			0.707164i	0.935404	+	0.353582i	$0.115036\pi$
$269$			51171.1i			0.707164i	−0.935404	+	0.353582i	$0.884964\pi$
$270$	−56521.1	+	57398.6i	−0.775324	+	0.787360i
$271$	−60019.7			−0.817250			−0.408625	−	0.912702i	$-0.633992\pi$
$271$	−60019.7			−0.817250			−0.408625	+	0.912702i	$0.633992\pi$
$272$			6521.72i			0.0881504i
$273$	2473.49	−	9326.91i	0.0331883	−	0.125145i
$274$	82015.2			1.09243
$275$		−	114783.i		−	1.51780i
$276$	−7676.54	−	2035.81i	−0.100774	−	0.0267251i
$277$	74216.3			0.967253			0.483626	−	0.875275i	$-0.339319\pi$
$277$	74216.3			0.967253			0.483626	+	0.875275i	$0.339319\pi$
$278$			39745.0i			0.514272i
$279$	67398.0	+	38452.2i	0.865842	+	0.493984i
$280$	−24988.0			−0.318724
$281$			73171.6i			0.926680i	0.886181	+	0.463340i	$0.153349\pi$
$281$			73171.6i			0.926680i	−0.886181	+	0.463340i	$0.846651\pi$
$282$	−14548.3	+	54858.1i	−0.182943	+	0.689830i
$283$	5495.42			0.0686164			0.0343082	−	0.999411i	$-0.489077\pi$
$283$	5495.42			0.0686164			0.0343082	+	0.999411i	$0.489077\pi$
$284$		−	28082.7i		−	0.348179i
$285$	−161485.	−	42825.8i	−1.98812	−	0.527249i
$286$	−13662.3			−0.167029
$287$		−	5223.01i		−	0.0634099i
$288$	7265.99	−	12735.6i	0.0876011	−	0.153545i
$289$	73137.0			0.875672
$290$			67026.7i			0.796989i
$291$	15637.6	−	58965.4i	0.184665	−	0.696324i
$292$	76757.5			0.900234
$293$		−	133178.i		−	1.55131i	−0.631159	−	0.775653i	$-0.717421\pi$
$293$		−	133178.i		−	1.55131i	0.631159	−	0.775653i	$-0.282579\pi$
$294$	39417.8	+	10453.6i	0.456034	+	0.120940i
$295$	−115050.			−1.32203
$296$		−	34270.9i		−	0.391149i
$297$	66149.6	+	65138.5i	0.749919	+	0.738456i
$298$	−17610.0			−0.198302
$299$		−	4183.83i		−	0.0467985i
$300$	−16635.3	+	62727.4i	−0.184836	+	0.696971i
$301$	−8336.25			−0.0920105
$302$			94438.2i			1.03546i
$303$	95858.9	+	25421.7i	1.04411	+	0.276898i
$304$	30409.2			0.329047
$305$		−	152735.i		−	1.64187i
$306$	20277.9	+	11569.0i	0.216561	+	0.123553i
$307$	−143773.			−1.52546			−0.762728	−	0.646719i	$-0.776141\pi$
$307$	−143773.			−1.52546			−0.762728	+	0.646719i	$0.776141\pi$
$308$			28797.7i			0.303568i
$309$	−1238.60	+	4670.45i	−0.0129722	+	0.0489149i
$310$	105857.			1.10153
$311$		−	115364.i		−	1.19275i	−0.802708	−	0.596373i	$-0.796608\pi$
$311$		−	115364.i		−	1.19275i	0.802708	−	0.596373i	$-0.203392\pi$
$312$	7466.22	+	1980.04i	0.0766993	+	0.0203406i
$313$	−144591.			−1.47589			−0.737945	−	0.674861i	$-0.764203\pi$
$313$	−144591.			−1.47589			−0.737945	+	0.674861i	$0.764203\pi$
$314$		−	122315.i		−	1.24057i
$315$	−44326.8	+	77694.8i	−0.446730	+	0.783016i
$316$	−63977.1			−0.640694
$317$			93443.7i			0.929890i	0.885340	+	0.464945i	$0.153926\pi$
$317$			93443.7i			0.929890i	−0.885340	+	0.464945i	$0.846074\pi$
$318$	25099.0	−	94641.8i	0.248200	−	0.935899i
$319$	77245.8			0.759090
$320$		−	20003.0i		−	0.195341i
$321$	−175875.	−	46642.1i	−1.70685	−	0.452655i
$322$	−8818.79			−0.0850545
$323$			48418.0i			0.464090i
$324$	−26709.4	−	45184.0i	−0.254433	−	0.430423i
$325$	−34187.4			−0.323668
$326$			118729.i			1.11718i
$327$	14755.5	−	55639.1i	0.137993	−	0.520337i
$328$	4181.03			0.0388630
$329$			63020.8i			0.582227i
$330$	122419.	+	32465.4i	1.12414	+	0.298121i
$331$	22676.9			0.206980			0.103490	−	0.994631i	$-0.466999\pi$
$331$	22676.9			0.206980			0.103490	+	0.994631i	$0.466999\pi$
$332$		−	15843.2i		−	0.143736i
$333$	−106558.	−	60793.9i	−0.960943	−	0.548241i
$334$	−86413.2			−0.774617
$335$		−	47133.7i		−	0.419993i
$336$	4173.57	−	15737.5i	0.0369683	−	0.139398i
$337$	−117566.			−1.03519			−0.517597	−	0.855624i	$-0.673173\pi$
$337$	−117566.			−1.03519			−0.517597	+	0.855624i	$0.673173\pi$
$338$		−	76713.5i		−	0.671488i
$339$	202081.	+	53591.8i	1.75844	+	0.466336i
$340$	31849.0			0.275511
$341$		−	121996.i		−	1.04915i
$342$	53943.6	−	94550.9i	0.461198	−	0.808376i
$343$	113151.			0.961767
$344$		−	6673.19i		−	0.0563919i
$345$	−9941.96	+	37488.6i	−0.0835283	+	0.314964i
$346$	−59007.1			−0.492893
$347$			13675.6i			0.113576i	0.998386	+	0.0567882i	$0.0180860\pi$
$347$			13675.6i			0.113576i	−0.998386	+	0.0567882i	$0.981914\pi$
$348$	−42213.6	−	11195.0i	−0.348573	−	0.0924414i
$349$	−83703.2			−0.687213			−0.343607	−	0.939114i	$-0.611649\pi$
$349$	−83703.2			−0.687213			−0.343607	+	0.939114i	$0.611649\pi$
$350$			72061.1i			0.588254i
$351$	19401.0	−	19702.2i	0.157474	−	0.159919i
$352$	−23052.6			−0.186052
$353$			69071.4i			0.554305i	0.960826	+	0.277152i	$0.0893907\pi$
$353$			69071.4i			0.554305i	−0.960826	+	0.277152i	$0.910609\pi$
$354$	19216.0	−	72458.6i	0.153340	−	0.578207i
$355$	−137143.			−1.08822
$356$		−	116577.i		−	0.919843i
$357$	25057.5	+	6645.24i	0.196608	+	0.0521404i
$358$	149311.			1.16500
$359$			226822.i			1.75993i	0.475037	+	0.879966i	$0.342435\pi$
$359$			226822.i			1.75993i	−0.475037	+	0.879966i	$0.657565\pi$
$360$	−62194.9	−	35483.7i	−0.479899	−	0.273794i
$361$	95440.6			0.732350
$362$		−	10231.8i		−	0.0780791i
$363$	3637.71	−	13716.9i	0.0276067	−	0.104098i
$364$	8577.18			0.0647354
$365$		−	374848.i		−	2.81365i
$366$	96193.1	+	25510.4i	0.718095	+	0.190438i
$367$	117964.			0.875825			0.437912	−	0.899018i	$-0.355718\pi$
$367$	117964.			0.875825			0.437912	+	0.899018i	$0.355718\pi$
$368$		−	7059.46i		−	0.0521286i
$369$	7416.83	−	13000.0i	0.0544710	−	0.0954754i
$370$	−167363.			−1.22252
$371$		−	108724.i		−	0.789912i
$372$	−17680.6	+	66669.2i	−0.127765	+	0.481769i
$373$	−107943.			−0.775852			−0.387926	−	0.921691i	$-0.626808\pi$
$373$	−107943.			−0.775852			−0.387926	+	0.921691i	$0.626808\pi$
$374$		−	36704.8i		−	0.262410i
$375$	93915.0	+	24906.2i	0.667840	+	0.177111i
$376$	−50448.3			−0.356838
$377$		−	23007.1i		−	0.161874i
$378$	−41528.8	−	40893.9i	−0.290647	−	0.286204i
$379$	127154.			0.885219			0.442609	−	0.896714i	$-0.354053\pi$
$379$	127154.			0.885219			0.442609	+	0.896714i	$0.354053\pi$
$380$		−	148504.i		−	1.02842i
$381$	33904.2	−	127844.i	0.233563	−	0.880705i
$382$	101292.			0.694139
$383$		−	21301.0i		−	0.145212i	−0.997361	−	0.0726058i	$-0.976868\pi$
$383$		−	21301.0i		−	0.145212i	0.997361	−	0.0726058i	$-0.0231315\pi$
$384$	12597.9	+	3340.96i	0.0854350	+	0.0226573i
$385$	140635.			0.948791
$386$		−	93121.2i		−	0.624991i
$387$	−20748.8	−	11837.7i	−0.138539	−	0.0790399i
$388$	54225.5			0.360197
$389$			80896.8i			0.534604i	0.963613	+	0.267302i	$0.0861322\pi$
$389$			80896.8i			0.534604i	−0.963613	+	0.267302i	$0.913868\pi$
$390$	9669.58	−	36461.5i	0.0635738	−	0.239721i
$391$	11240.2			0.0735225
$392$			36249.2i			0.235899i
$393$	147651.	+	39157.0i	0.955986	+	0.253527i
$394$	−166156.			−1.07034
$395$			312435.i			2.00246i
$396$	−40893.6	+	71677.2i	−0.260774	+	0.457079i
$397$	242618.			1.53937			0.769684	−	0.638425i	$-0.220414\pi$
$397$	242618.			1.53937			0.769684	+	0.638425i	$0.220414\pi$
$398$			177238.i			1.11890i
$399$	30985.1	−	116837.i	0.194629	−	0.733897i
$400$	−57685.1			−0.360532
$401$		−	79078.6i		−	0.491779i	−0.969298	−	0.245890i	$-0.920920\pi$
$401$		−	79078.6i		−	0.491779i	0.969298	−	0.245890i	$-0.0790801\pi$
$402$	29684.9	+	7872.43i	0.183689	+	0.0487143i
$403$	−36335.7			−0.223730
$404$			88153.4i			0.540103i
$405$	−220658.	+	130436.i	−1.34527	+	0.795222i
$406$	−48494.9			−0.294201
$407$			192880.i			1.16439i
$408$	−5319.53	+	20058.6i	−0.0319560	+	0.120498i
$409$	220840.			1.32017			0.660086	−	0.751190i	$-0.270520\pi$
$409$	220840.			1.32017			0.660086	+	0.751190i	$0.270520\pi$
$410$		−	20418.2i		−	0.121465i
$411$	252251.	+	66896.8i	1.49331	+	0.396024i
$412$	−4295.02			−0.0253029
$413$		−	83240.3i		−	0.488015i
$414$	−21949.9	−	12522.9i	−0.128065	−	0.0730644i
$415$	−77370.7			−0.449242
$416$			6866.05i			0.0396753i
$417$	−32418.6	+	122242.i	−0.186433	+	0.702990i
$418$	−171146.			−0.979520
$419$		−	102964.i		−	0.586487i	−0.956038	−	0.293243i	$-0.905265\pi$
$419$		−	102964.i		−	0.586487i	0.956038	−	0.293243i	$-0.0947346\pi$
$420$	−76854.6	−	20381.8i	−0.435684	−	0.115543i
$421$	74121.9			0.418198			0.209099	−	0.977894i	$-0.432947\pi$
$421$	74121.9			0.418198			0.209099	+	0.977894i	$0.432947\pi$
$422$			93429.7i			0.524638i
$423$	−89491.4	+	156858.i	−0.500151	+	0.876651i
$424$	87034.1			0.484125
$425$		−	91847.2i		−	0.508496i
$426$	22906.0	−	86372.9i	0.126221	−	0.475947i
$427$	110506.			0.606082
$428$		−	161738.i		−	0.882926i
$429$	−42020.5	−	11143.8i	−0.228322	−	0.0605507i
$430$	−32588.7			−0.176251
$431$		−	156240.i		−	0.841078i	−0.907274	−	0.420539i	$-0.861841\pi$
$431$		−	156240.i		−	0.841078i	0.907274	−	0.420539i	$-0.138159\pi$
$432$	32735.7	−	33243.9i	0.175410	−	0.178133i
$433$	−217223.			−1.15859			−0.579295	−	0.815118i	$-0.696672\pi$
$433$	−217223.			−1.15859			−0.579295	+	0.815118i	$0.696672\pi$
$434$			76589.4i			0.406620i
$435$	−54671.3	+	206152.i	−0.288922	+	1.08945i
$436$	51166.6			0.269162
$437$		−	52410.3i		−	0.274444i
$438$	236080.	+	62608.3i	1.23058	+	0.326350i
$439$	−85838.8			−0.445405			−0.222702	−	0.974886i	$-0.571488\pi$
$439$	−85838.8			−0.445405			−0.222702	+	0.974886i	$0.571488\pi$
$440$			112578.i			0.581500i
$441$	112709.	+	64303.3i	0.579538	+	0.330641i
$442$	−10932.2			−0.0559583
$443$			215458.i			1.09788i	0.835861	+	0.548941i	$0.184969\pi$
$443$			215458.i			1.09788i	−0.835861	+	0.548941i	$0.815031\pi$
$444$	27953.5	−	105406.i	0.141798	−	0.534685i
$445$	−569309.			−2.87493
$446$			33242.9i			0.167120i
$447$	−54162.5	−	14363.9i	−0.271072	−	0.0718880i
$448$	14472.4			0.0721084
$449$		−	44955.9i		−	0.222995i	−0.993765	−	0.111497i	$-0.964435\pi$
$449$		−	44955.9i		−	0.222995i	0.993765	−	0.111497i	$-0.0355646\pi$
$450$	−102329.	+	179359.i	−0.505328	+	0.885725i
$451$	−23531.2			−0.115689
$452$			185837.i			0.909611i
$453$	−77029.8	+	290460.i	−0.375373	+	1.41543i
$454$	−213229.			−1.03451
$455$		−	41886.9i		−	0.202328i
$456$	93528.4	+	24803.7i	0.449794	+	0.119285i
$457$	−118411.			−0.566970			−0.283485	−	0.958977i	$-0.591491\pi$
$457$	−118411.			−0.566970			−0.283485	+	0.958977i	$0.591491\pi$
$458$			137117.i			0.653675i
$459$	52931.5	+	52122.3i	0.251240	+	0.247399i
$460$	−34475.1			−0.162926
$461$			20708.9i			0.0974441i	0.998812	+	0.0487221i	$0.0155149\pi$
$461$			20708.9i			0.0974441i	−0.998812	+	0.0487221i	$0.984485\pi$
$462$	−23489.2	+	88572.0i	−0.110049	+	0.414966i
$463$	−169822.			−0.792192			−0.396096	−	0.918209i	$-0.629635\pi$
$463$	−169822.			−0.792192			−0.396096	+	0.918209i	$0.629635\pi$
$464$		−	38820.3i		−	0.180311i
$465$	325581.	+	86343.9i	1.50575	+	0.399324i
$466$	−168223.			−0.774665
$467$			259984.i			1.19210i	0.802947	+	0.596051i	$0.203264\pi$
$467$			259984.i			1.19210i	−0.802947	+	0.596051i	$0.796736\pi$
$468$	21348.5	+	12179.8i	0.0974711	+	0.0556096i
$469$	34102.0			0.155036
$470$			246366.i			1.11528i
$471$	99768.2	−	376201.i	0.449728	−	1.69581i
$472$	66634.1			0.299097
$473$			37557.3i			0.167870i
$474$	−196772.	−	52183.8i	−0.875803	−	0.232263i
$475$	−428261.			−1.89811
$476$			23043.3i			0.101702i
$477$	154392.	−	270614.i	0.678559	−	1.18936i
$478$	−239039.			−1.04620
$479$		−	34067.5i		−	0.148480i	−0.997240	−	0.0742402i	$-0.976347\pi$
$479$		−	34067.5i		−	0.148480i	0.997240	−	0.0742402i	$-0.0236532\pi$
$480$	16315.7	−	61522.3i	0.0708146	−	0.267024i
$481$	57447.7			0.248303
$482$			132331.i			0.569596i
$483$	−27123.6	−	7193.16i	−0.116266	−	0.0308337i
$484$	12614.2			0.0538481
$485$		−	264812.i		−	1.12578i
$486$	−45294.1	−	160757.i	−0.191765	−	0.680607i
$487$	415744.			1.75294			0.876472	−	0.481454i	$-0.159891\pi$
$487$	415744.			1.75294			0.876472	+	0.481454i	$0.159891\pi$
$488$			88460.7i			0.371459i
$489$	−96843.3	+	365172.i	−0.404997	+	1.52714i
$490$	177024.			0.737294
$491$		−	44906.8i		−	0.186273i	−0.995653	−	0.0931364i	$-0.970311\pi$
$491$		−	44906.8i		−	0.186273i	0.995653	−	0.0931364i	$-0.0296893\pi$
$492$	12859.4	+	3410.32i	0.0531241	+	0.0140885i
$493$	61810.3			0.254312
$494$			50974.4i			0.208881i
$495$	350038.	+	199705.i	1.42858	+	0.815040i
$496$	−61310.0			−0.249212
$497$		−	99225.0i		−	0.401706i
$498$	12922.7	−	48728.3i	0.0521069	−	0.196482i
$499$	365759.			1.46890			0.734452	−	0.678661i	$-0.237439\pi$
$499$	365759.			1.46890			0.734452	+	0.678661i	$0.237439\pi$
$500$			86365.8i			0.345463i
$501$	−265778.	−	70484.1i	−1.05887	−	0.280812i
$502$	−154141.			−0.611663
$503$		−	113371.i		−	0.448091i	−0.974579	−	0.224046i	$-0.928073\pi$
$503$		−	113371.i		−	0.448091i	0.974579	−	0.224046i	$-0.0719265\pi$
$504$	25673.0	−	44999.0i	0.101068	−	0.177150i
$505$	430500.			1.68807
$506$			39731.3i			0.155178i
$507$	62572.4	−	235945.i	0.243426	−	0.917898i
$508$	117567.			0.455575
$509$			209002.i			0.806706i	0.915044	+	0.403353i	$0.132155\pi$
$509$			209002.i			0.806706i	−0.915044	+	0.403353i	$0.867845\pi$
$510$	97956.9	+	25978.1i	0.376612	+	0.0998774i
$511$	271208.			1.03863
$512$			11585.2i			0.0441942i
$513$	243034.	−	246807.i	0.923490	−	0.937826i
$514$	−17295.1			−0.0654630
$515$			20974.9i			0.0790833i
$516$	5443.08	−	20524.5i	0.0204430	−	0.0770855i
$517$	283928.			1.06225
$518$		−	121090.i		−	0.451282i
$519$	−181486.	−	48130.0i	−0.673765	−	0.178682i
$520$	33530.6			0.124004
$521$			285798.i			1.05289i	0.850209	+	0.526445i	$0.176476\pi$
$521$			285798.i			1.05289i	−0.850209	+	0.526445i	$0.823524\pi$
$522$	−120703.	−	68864.2i	−0.442974	−	0.252728i
$523$	−157539.			−0.575948			−0.287974	−	0.957638i	$-0.592982\pi$
$523$	−157539.			−0.575948			−0.287974	+	0.957638i	$0.592982\pi$
$524$			135782.i			0.494516i
$525$	−58777.6	+	221635.i	−0.213252	+	0.804120i
$526$	219872.			0.794692
$527$		−	97618.8i		−	0.351490i
$528$	−70902.1	−	18803.2i	−0.254326	−	0.0674472i
$529$	−12167.0			−0.0434783
$530$		−	425034.i		−	1.51312i
$531$	118204.	−	207184.i	0.419220	−	0.734798i
$532$	107445.			0.379633
$533$			7008.59i			0.0246704i
$534$	95087.8	−	358552.i	0.333459	−	1.25739i
$535$	−789852.			−2.75955
$536$			27298.7i			0.0950195i
$537$	459231.	+	121788.i	1.59251	+	0.422333i
$538$	−144734.			−0.500040
$539$		−	204014.i		−	0.702234i
$540$	−162348.	−	159866.i	−0.556748	−	0.548237i
$541$	327517.			1.11902			0.559512	−	0.828822i	$-0.310989\pi$
$541$	327517.			1.11902			0.559512	+	0.828822i	$0.310989\pi$
$542$		−	169761.i		−	0.577883i
$543$	8345.70	−	31469.5i	0.0283050	−	0.106731i
$544$	−18446.2			−0.0623318
$545$		−	249874.i		−	0.841255i
$546$	26380.5	+	6996.09i	0.0884907	+	0.0234677i
$547$	−546792.			−1.82746			−0.913729	−	0.406324i	$-0.866810\pi$
$547$	−546792.			−1.82746			−0.913729	+	0.406324i	$0.866810\pi$
$548$			231974.i			0.772464i
$549$	275050.	+	156922.i	0.912570	+	0.520643i
$550$	324657.			1.07325
$551$		−	288207.i		−	0.949294i
$552$	5758.15	−	21712.5i	0.0188975	−	0.0712577i
$553$	−226051.			−0.739191
$554$			209915.i			0.683951i
$555$	−514752.	−	136512.i	−1.67114	−	0.443185i
$556$	−112416.			−0.363645
$557$		−	192387.i		−	0.620104i	−0.950720	−	0.310052i	$-0.899654\pi$
$557$		−	192387.i		−	0.620104i	0.950720	−	0.310052i	$-0.100346\pi$
$558$	−108759.	+	190630.i	−0.349299	+	0.612243i
$559$	11186.2			0.0357979
$560$		−	70676.7i		−	0.225372i
$561$	29938.8	−	112892.i	0.0951280	−	0.358704i
$562$	−206960.			−0.655262
$563$			321358.i			1.01385i	0.861991	+	0.506924i	$0.169217\pi$
$563$			321358.i			1.01385i	−0.861991	+	0.506924i	$0.830783\pi$
$564$	−155162.	−	41148.9i	−0.487784	−	0.129360i
$565$	907542.			2.84295
$566$			15543.4i			0.0485191i
$567$	−94372.7	−	159649.i	−0.293549	−	0.496594i
$568$	79429.9			0.246200
$569$			73720.9i			0.227702i	0.993498	+	0.113851i	$0.0363186\pi$
$569$			73720.9i			0.227702i	−0.993498	+	0.113851i	$0.963681\pi$
$570$	121130.	−	456749.i	0.372821	−	1.40581i
$571$	−248373.			−0.761786			−0.380893	−	0.924619i	$-0.624383\pi$
$571$	−248373.			−0.761786			−0.380893	+	0.924619i	$0.624383\pi$
$572$		−	38642.7i		−	0.118107i
$573$	311539.	+	82619.9i	0.948861	+	0.251637i
$574$	14772.9			0.0448376
$575$			99420.4i			0.300704i
$576$	36021.8	+	20551.3i	0.108573	+	0.0619433i
$577$	−120518.			−0.361994			−0.180997	−	0.983484i	$-0.557932\pi$
$577$	−120518.			−0.361994			−0.180997	+	0.983484i	$0.557932\pi$
$578$			206863.i			0.619194i
$579$	75955.6	−	286409.i	0.226570	−	0.854339i
$580$	−189580.			−0.563556
$581$		−	55978.9i		−	0.165834i
$582$	166779.	+	44229.8i	0.492375	+	0.130578i
$583$	−489836.			−1.44116
$584$			217103.i			0.636561i
$585$	59480.7	−	104256.i	0.173806	−	0.304642i
$586$	376685.			1.09694
$587$			571617.i			1.65893i	0.558555	+	0.829467i	$0.311356\pi$
$587$			571617.i			1.65893i	−0.558555	+	0.829467i	$0.688644\pi$
$588$	−29567.2	+	111490.i	−0.0855176	+	0.322465i
$589$	−455173.			−1.31204
$590$		−	325410.i		−	0.934817i
$591$	−511039.	−	135527.i	−1.46312	−	0.388018i
$592$	96932.7			0.276584
$593$		−	341236.i		−	0.970387i	−0.874407	−	0.485194i	$-0.838749\pi$
$593$		−	341236.i		−	0.970387i	0.874407	−	0.485194i	$-0.161251\pi$
$594$	−184239.	+	187099.i	−0.522167	+	0.530273i
$595$	112533.			0.317866
$596$		−	49808.7i		−	0.140221i
$597$	−144567.	+	545125.i	−0.405620	+	1.52949i
$598$	11833.7			0.0330915
$599$		−	2524.45i		−	0.00703581i	−0.999994	−	0.00351790i	$-0.998880\pi$
$599$		−	2524.45i		−	0.00703581i	0.999994	−	0.00351790i	$-0.00111979\pi$
$600$	−177420.	−	47051.6i	−0.492833	−	0.130699i
$601$	−522859.			−1.44756			−0.723778	−	0.690033i	$-0.757596\pi$
$601$	−522859.			−1.44756			−0.723778	+	0.690033i	$0.757596\pi$
$602$		−	23578.5i		−	0.0650613i
$603$	84879.5	+	48425.8i	0.233436	+	0.133181i
$604$	−267112.			−0.732182
$605$		−	61602.1i		−	0.168300i
$606$	−71903.5	+	271130.i	−0.195797	+	0.738299i
$607$	258807.			0.702422			0.351211	−	0.936296i	$-0.385770\pi$
$607$	258807.			0.702422			0.351211	+	0.936296i	$0.385770\pi$
$608$			86010.2i			0.232671i
$609$	−149154.	−	39555.5i	−0.402161	−	0.106653i
$610$	432001.			1.16098
$611$		−	84565.7i		−	0.226523i
$612$	−32722.1	+	57354.5i	−0.0873653	+	0.153132i
$613$	43337.7			0.115331			0.0576653	−	0.998336i	$-0.481634\pi$
$613$	43337.7			0.115331			0.0576653	+	0.998336i	$0.481634\pi$
$614$		−	406651.i		−	1.07866i
$615$	16654.4	−	62799.5i	0.0440330	−	0.166037i
$616$	−81452.2			−0.214655
$617$			492569.i			1.29389i	0.762537	+	0.646944i	$0.223953\pi$
$617$			492569.i			1.29389i	−0.762537	+	0.646944i	$0.776047\pi$
$618$	−13210.0	−	3503.29i	−0.0345881	−	0.00917274i
$619$	−62545.1			−0.163234			−0.0816172	−	0.996664i	$-0.526008\pi$
$619$	−62545.1			−0.163234			−0.0816172	+	0.996664i	$0.526008\pi$
$620$			299410.i			0.778901i
$621$	−57295.9	−	56420.1i	−0.148573	−	0.146302i
$622$	326297.			0.843398
$623$		−	411904.i		−	1.06126i
$624$	−5600.39	+	21117.7i	−0.0143830	+	0.0542346i
$625$	−141561.			−0.362396
$626$		−	408966.i		−	1.04361i
$627$	−526386.	−	139597.i	−1.33896	−	0.355093i
$628$	345960.			0.877216
$629$			154338.i			0.390096i
$630$	−219754.	−	125375.i	−0.553676	−	0.315886i
$631$	384384.			0.965399			0.482700	−	0.875786i	$-0.339656\pi$
$631$	384384.			0.965399			0.482700	+	0.875786i	$0.339656\pi$
$632$		−	180955.i		−	0.453039i
$633$	−76207.2	+	287358.i	−0.190190	+	0.717160i
$634$	−264299.			−0.657532
$635$		−	574145.i		−	1.42388i
$636$	267688.	+	70990.6i	0.661780	+	0.175504i
$637$	−60764.0			−0.149750
$638$			218484.i			0.536758i
$639$	140902.	−	246970.i	0.345078	−	0.604843i
$640$	56576.9			0.138127
$641$		−	663358.i		−	1.61448i	−0.590226	−	0.807238i	$-0.700961\pi$
$641$		−	663358.i		−	1.61448i	0.590226	−	0.807238i	$-0.299039\pi$
$642$	131924.	−	497451.i	0.320076	−	1.20692i
$643$	9675.87			0.0234028			0.0117014	−	0.999932i	$-0.496275\pi$
$643$	9675.87			0.0234028			0.0117014	+	0.999932i	$0.496275\pi$
$644$		−	24943.3i		−	0.0601426i
$645$	−100232.	−	26581.5i	−0.240928	−	0.0638939i
$646$	−136947.			−0.328161
$647$		−	194291.i		−	0.464135i	−0.972700	−	0.232067i	$-0.925451\pi$
$647$		−	194291.i		−	0.464135i	0.972700	−	0.232067i	$-0.0745490\pi$
$648$	127800.	−	75545.6i	0.304355	−	0.179912i
$649$	−375022.			−0.890364
$650$		−	96696.5i		−	0.228868i
$651$	−62471.2	+	235563.i	−0.147407	+	0.555834i
$652$	−335818.			−0.789966
$653$		−	625764.i		−	1.46752i	−0.679408	−	0.733761i	$-0.737763\pi$
$653$		−	625764.i		−	1.46752i	0.679408	−	0.733761i	$-0.262237\pi$
$654$	157371.	+	41734.7i	0.367934	+	0.0975758i
$655$	663097.			1.54559
$656$			11825.7i			0.0274803i
$657$	675035.	+	385124.i	1.56385	+	0.892216i
$658$	−178250.			−0.411696
$659$			815370.i			1.87752i	0.344574	+	0.938759i	$0.388023\pi$
$659$			815370.i			1.87752i	−0.344574	+	0.938759i	$0.611977\pi$
$660$	−91826.1	+	346253.i	−0.210804	+	0.794887i
$661$	94326.3			0.215889			0.107944	−	0.994157i	$-0.465573\pi$
$661$	94326.3			0.215889			0.107944	+	0.994157i	$0.465573\pi$
$662$			64139.9i			0.146357i
$663$	−33623.9	−	8917.04i	−0.0764928	−	0.0202859i
$664$	44811.3			0.101637
$665$		−	524712.i		−	1.18653i
$666$	171951.	−	301391.i	0.387665	−	0.679489i
$667$	−66906.9			−0.150390
$668$		−	244413.i		−	0.547737i
$669$	−27115.0	+	102244.i	−0.0605839	+	0.228447i
$670$	133314.			0.296980
$671$		−	497864.i		−	1.10577i
$672$	44512.4	+	11804.7i	0.0985694	+	0.0261405i
$673$	144103.			0.318159			0.159079	−	0.987266i	$-0.449147\pi$
$673$	144103.			0.318159			0.159079	+	0.987266i	$0.449147\pi$
$674$		−	332527.i		−	0.731993i
$675$	−461026.	+	468183.i	−1.01185	+	1.02756i
$676$	216979.			0.474814
$677$		−	256803.i		−	0.560302i	−0.959956	−	0.280151i	$-0.909615\pi$
$677$		−	256803.i		−	0.560302i	0.959956	−	0.280151i	$-0.0903846\pi$
$678$	−151581.	+	571572.i	−0.329750	+	1.24340i
$679$	191596.			0.415572
$680$			90082.7i			0.194815i
$681$	−655821.	−	173923.i	−1.41414	−	0.375028i
$682$	345058.			0.741863
$683$			180538.i			0.387015i	0.981099	+	0.193508i	$0.0619864\pi$
$683$			180538.i			0.387015i	−0.981099	+	0.193508i	$0.938014\pi$
$684$	267430.	+	152575.i	0.571608	+	0.326116i
$685$	1.13285e6			2.41431
$686$			320039.i			0.680072i
$687$	−111842.	+	421727.i	−0.236968	+	0.893548i
$688$	18874.6			0.0398751
$689$			145894.i			0.307325i
$690$	−106034.	−	28120.1i	−0.222713	−	0.0590635i
$691$	−284045.			−0.594881			−0.297441	−	0.954740i	$-0.596133\pi$
$691$	−284045.			−0.594881			−0.297441	+	0.954740i	$0.596133\pi$
$692$		−	166897.i		−	0.348528i
$693$	−144490.	+	253258.i	−0.300865	+	0.527347i
$694$	−38680.5			−0.0803106
$695$			548987.i			1.13656i
$696$	31664.3	−	119398.i	0.0653660	−	0.246478i
$697$	−18829.2			−0.0387583
$698$		−	236748.i		−	0.485933i
$699$	−517397.	−	137214.i	−1.05894	−	0.280829i
$700$	−203820.			−0.415958
$701$			481182.i			0.979204i	0.871946	+	0.489602i	$0.162858\pi$
$701$			481182.i			0.979204i	−0.871946	+	0.489602i	$0.837142\pi$
$702$	55726.1	+	54874.3i	0.113080	+	0.111351i
$703$	719640.			1.45615
$704$		−	65202.7i		−	0.131559i
$705$	−200952.	+	757739.i	−0.404310	+	1.52455i
$706$	−195363.			−0.391953
$707$			311474.i			0.623135i
$708$	204944.	+	54351.0i	0.408854	+	0.108428i
$709$	−27679.9			−0.0550646			−0.0275323	−	0.999621i	$-0.508765\pi$
$709$	−27679.9			−0.0550646			−0.0275323	+	0.999621i	$0.508765\pi$
$710$		−	387898.i		−	0.769487i
$711$	−562640.	−	321000.i	−1.11299	−	0.634988i
$712$	329730.			0.650427
$713$			105668.i			0.207857i
$714$	−18795.6	+	70873.3i	−0.0368688	+	0.139023i
$715$	−188713.			−0.369139
$716$			422316.i			0.823781i
$717$	−735203.	−	194975.i	−1.43011	−	0.379264i
$718$	−641549.			−1.24446
$719$			113948.i			0.220420i	0.993908	+	0.110210i	$0.0351523\pi$
$719$			113948.i			0.220420i	−0.993908	+	0.110210i	$0.964848\pi$
$720$	100363.	−	175914.i	0.193602	−	0.339340i
$721$	−15175.6			−0.0291928
$722$			269947.i			0.517850i
$723$	−107937.	+	407005.i	−0.206488	+	0.778615i
$724$	28939.9			0.0552102
$725$			546717.i			1.04013i
$726$	38797.2	+	10289.0i	0.0736083	+	0.0195209i
$727$	−649074.			−1.22808			−0.614038	−	0.789276i	$-0.710456\pi$
$727$	−649074.			−1.22808			−0.614038	+	0.789276i	$0.710456\pi$
$728$			24259.9i			0.0457748i
$729$	−8185.86	−	531378.i	−0.0154031	−	0.999881i
$730$	1.06023e6			1.98955
$731$			30052.5i			0.0562401i
$732$	−72154.2	+	272075.i	−0.134660	+	0.507770i
$733$	629058.			1.17080			0.585400	−	0.810745i	$-0.300937\pi$
$733$	629058.			1.17080			0.585400	+	0.810745i	$0.300937\pi$
$734$			333652.i			0.619302i
$735$	544467.	+	144392.i	1.00785	+	0.267282i
$736$	19967.2			0.0368605
$737$		−	153640.i		−	0.282858i
$738$	36769.6	+	20978.0i	0.0675113	+	0.0385168i
$739$	−782594.			−1.43301			−0.716503	−	0.697585i	$-0.754258\pi$
$739$	−782594.			−1.43301			−0.716503	+	0.697585i	$0.754258\pi$
$740$		−	473374.i		−	0.864453i
$741$	−41578.0	+	156780.i	−0.0757229	+	0.285532i
$742$	307519.			0.558552
$743$			428785.i			0.776716i	0.921509	+	0.388358i	$0.126958\pi$
$743$			428785.i			0.776716i	−0.921509	+	0.388358i	$0.873042\pi$
$744$	−188569.	−	50008.4i	−0.340662	−	0.0903435i
$745$	−243243.			−0.438255
$746$		−	305310.i		−	0.548610i
$747$	79491.8	−	139331.i	0.142456	−	0.249693i
$748$	103817.			0.185552
$749$		−	571470.i		−	1.01866i
$750$	−70445.4	+	265632.i	−0.125236	+	0.472234i
$751$	−78870.0			−0.139840			−0.0699201	−	0.997553i	$-0.522274\pi$
$751$	−78870.0			−0.139840			−0.0699201	+	0.997553i	$0.522274\pi$
$752$		−	142689.i		−	0.252323i
$753$	−474087.	−	125728.i	−0.836119	−	0.221738i
$754$	65073.8			0.114463
$755$			1.30445e6i			2.28841i
$756$	115665.	−	117461.i	0.202377	−	0.205518i
$757$	1.06312e6			1.85520			0.927602	−	0.373569i	$-0.121866\pi$
$757$	1.06312e6			1.85520			0.927602	+	0.373569i	$0.121866\pi$
$758$			359645.i			0.625944i
$759$	−32407.4	+	122200.i	−0.0562549	+	0.212123i
$760$	420034.			0.727205
$761$			121344.i			0.209531i	0.994497	+	0.104766i	$0.0334093\pi$
$761$			121344.i			0.209531i	−0.994497	+	0.104766i	$0.966591\pi$
$762$	361598.	+	95895.5i	0.622753	+	0.165154i
$763$	180788.			0.310541
$764$			286496.i			0.490831i
$765$	280093.	+	159800.i	0.478607	+	0.273057i
$766$	60248.2			0.102680
$767$			111698.i			0.189869i
$768$	−9449.66	+	35632.3i	−0.0160211	+	0.0604117i
$769$	45888.9			0.0775987			0.0387994	−	0.999247i	$-0.487647\pi$
$769$	45888.9			0.0775987			0.0387994	+	0.999247i	$0.487647\pi$
$770$			397775.i			0.670896i
$771$	−53193.8	−	14107.0i	−0.0894854	−	0.0237315i
$772$	263387.			0.441936
$773$			764605.i			1.27961i	0.768537	+	0.639806i	$0.220985\pi$
$773$			764605.i			1.27961i	−0.768537	+	0.639806i	$0.779015\pi$
$774$	33482.1	−	58686.6i	0.0558896	−	0.0979619i
$775$	863446.			1.43758
$776$			153373.i			0.254698i
$777$	98768.6	−	372431.i	0.163598	−	0.616885i
$778$	−228811.			−0.378022
$779$			87795.8i			0.144677i
$780$	103129.	+	27349.7i	0.169508	+	0.0449535i
$781$	−447038.			−0.732896
$782$			31792.1i			0.0519883i
$783$	−315073.	−	310256.i	−0.513910	−	0.506054i
$784$	−102528.			−0.166806
$785$		−	1.68951e6i		−	2.74171i
$786$	−110753.	+	417620.i	−0.179271	+	0.675984i
$787$	972804.			1.57064			0.785319	−	0.619091i	$-0.212499\pi$
$787$	972804.			1.57064			0.785319	+	0.619091i	$0.212499\pi$
$788$		−	469960.i		−	0.756848i
$789$	676252.	+	179342.i	1.08631	+	0.288089i
$790$	−883698.			−1.41596
$791$			656621.i			1.04945i
$792$	−202734.	−	115665.i	−0.323203	−	0.184395i
$793$	−148285.			−0.235804
$794$			686228.i			1.08850i
$795$	346685.	−	1.30726e6i	0.548530	−	2.06837i
$796$	−501305.			−0.791182
$797$			701236.i			1.10395i	0.833862	+	0.551973i	$0.186125\pi$
$797$			701236.i			1.10395i	−0.833862	+	0.551973i	$0.813875\pi$
$798$	330465.	+	87639.2i	0.518943	+	0.137623i
$799$	227193.			0.355877
$800$		−	163158.i		−	0.254935i
$801$	584916.	−	1.02523e6i	0.911651	−	1.59792i
$802$	223668.			0.347740
$803$		−	1.22187e6i		−	1.89494i
$804$	−22266.6	+	83961.7i	−0.0344462	+	0.129888i
$805$	−121811.			−0.187973
$806$		−	102773.i		−	0.158201i
$807$	−445152.	−	118054.i	−0.683535	−	0.181273i
$808$	−249335.			−0.381910
$809$			459770.i			0.702495i	0.936283	+	0.351248i	$0.114242\pi$
$809$			459770.i			0.702495i	−0.936283	+	0.351248i	$0.885758\pi$
$810$	−368929.	−	624115.i	−0.562307	−	0.951249i
$811$	−890608.			−1.35408			−0.677041	−	0.735945i	$-0.736738\pi$
$811$	−890608.			−1.35408			−0.677041	+	0.735945i	$0.736738\pi$
$812$		−	137164.i		−	0.208031i
$813$	138468.	−	522128.i	0.209493	−	0.789944i
$814$	−545546.			−0.823346
$815$			1.63998e6i			2.46901i
$816$	−56734.3	−	15045.9i	−0.0852050	−	0.0225963i
$817$	140128.			0.209932
$818$			624629.i			0.933502i
$819$	75431.0	+	43035.2i	0.112456	+	0.0641588i
$820$	57751.4			0.0858885
$821$		−	546956.i		−	0.811458i	−0.913993	−	0.405729i	$-0.867018\pi$
$821$		−	546956.i		−	0.811458i	0.913993	−	0.405729i	$-0.132982\pi$
$822$	−189213.	+	713473.i	−0.280031	+	1.05593i
$823$	−143279.			−0.211535			−0.105767	−	0.994391i	$-0.533730\pi$
$823$	−143279.			−0.211535			−0.105767	+	0.994391i	$0.533730\pi$
$824$		−	12148.1i		−	0.0178919i
$825$	998534.	+	264811.i	1.46708	+	0.389070i
$826$	235439.			0.345079
$827$		−	540215.i		−	0.789870i	−0.918709	−	0.394935i	$-0.870767\pi$
$827$		−	540215.i		−	0.789870i	0.918709	−	0.394935i	$-0.129233\pi$
$828$	35420.2	−	62083.6i	0.0516643	−	0.0905559i
$829$	−170745.			−0.248449			−0.124225	−	0.992254i	$-0.539644\pi$
$829$	−170745.			−0.248449			−0.124225	+	0.992254i	$0.539644\pi$
$830$		−	218837.i		−	0.317662i
$831$	−171220.	+	645629.i	−0.247944	+	0.934934i
$832$	−19420.1			−0.0280547
$833$		−	163247.i		−	0.235264i
$834$	−345753.	−	91693.5i	−0.497089	−	0.131828i
$835$	−1.19360e6			−1.71193
$836$		−	484073.i		−	0.692625i
$837$	−489997.	+	497603.i	−0.699427	+	0.710285i
$838$	291227.			0.414709
$839$			229314.i			0.325767i	0.986645	+	0.162884i	$0.0520795\pi$
$839$			229314.i			0.325767i	−0.986645	+	0.162884i	$0.947921\pi$
$840$	57648.4	−	217378.i	0.0817013	−	0.308075i
$841$	339357.			0.479805
$842$			209648.i			0.295711i
$843$	−636540.	−	168810.i	−0.895717	−	0.237544i
$844$	−264259.			−0.370975
$845$		−	1.05962e6i		−	1.48401i
$846$	−443662.	−	253120.i	−0.619886	−	0.353660i
$847$	44570.1			0.0621265
$848$			246170.i			0.342328i
$849$	−12678.2	+	47806.2i	−0.0175890	+	0.0663237i
$850$	259783.			0.359561
$851$		−	167064.i		−	0.230687i
$852$	244299.	+	64788.1i	0.336545	+	0.0892516i
$853$	795040.			1.09268			0.546338	−	0.837565i	$-0.316022\pi$
$853$	795040.			1.09268			0.546338	+	0.837565i	$0.316022\pi$
$854$			312559.i			0.428565i
$855$	745107.	−	1.30600e6i	1.01926	−	1.78654i
$856$	457464.			0.624323
$857$			955473.i			1.30094i	0.759533	+	0.650469i	$0.225428\pi$
$857$			955473.i			1.30094i	−0.759533	+	0.650469i	$0.774572\pi$
$858$	31519.5	−	118852.i	0.0428158	−	0.161448i
$859$	673351.			0.912547			0.456273	−	0.889840i	$-0.349184\pi$
$859$	673351.			0.912547			0.456273	+	0.889840i	$0.349184\pi$
$860$		−	92174.9i		−	0.124628i
$861$	45436.4	+	12049.7i	0.0612912	+	0.0162544i
$862$	441912.			0.594732
$863$		−	529092.i		−	0.710410i	−0.934788	−	0.355205i	$-0.884411\pi$
$863$		−	529092.i		−	0.710410i	0.934788	−	0.355205i	$-0.115589\pi$
$864$	94027.9	+	92590.6i	0.125959	+	0.124034i
$865$	−815049.			−1.08931
$866$		−	614399.i		−	0.819247i
$867$	−168730.	+	636240.i	−0.224468	+	0.846413i
$868$	−216627.			−0.287524
$869$			1.01843e6i			1.34862i
$870$	−583085.	−	154634.i	−0.770359	−	0.204299i
$871$	−45760.4			−0.0603189
$872$			144721.i			0.190326i
$873$	476880.	+	272072.i	0.625721	+	0.356989i
$874$	148239.			0.194061
$875$			305157.i			0.398573i
$876$	−177083.	+	667735.i	−0.230764	+	0.870154i
$877$	−936003.			−1.21696			−0.608482	−	0.793567i	$-0.708221\pi$
$877$	−936003.			−1.21696			−0.608482	+	0.793567i	$0.708221\pi$
$878$		−	242789.i		−	0.314949i
$879$	1.15855e6	+	307248.i	1.49947	+	0.397659i
$880$	−318420.			−0.411182
$881$		−	302438.i		−	0.389658i	−0.980837	−	0.194829i	$-0.937585\pi$
$881$		−	302438.i		−	0.389658i	0.980837	−	0.194829i	$-0.0624153\pi$
$882$	−181877.	+	318790.i	−0.233798	+	0.409795i
$883$	−1.25078e6			−1.60420			−0.802099	−	0.597191i	$-0.796283\pi$
$883$	−1.25078e6			−1.60420			−0.802099	+	0.597191i	$0.796283\pi$
$884$		−	30921.1i		−	0.0395685i
$885$	265425.	−	1.00085e6i	0.338887	−	1.27786i
$886$	−609408.			−0.776320
$887$		−	326597.i		−	0.415112i	−0.978223	−	0.207556i	$-0.933449\pi$
$887$		−	326597.i		−	0.415112i	0.978223	−	0.207556i	$-0.0665509\pi$
$888$	298132.	+	79064.5i	0.378079	+	0.100266i
$889$	415403.			0.525612
$890$		−	1.61025e6i		−	2.03289i
$891$	−719268.	+	425177.i	−0.906015	+	0.535568i
$892$	−94025.0			−0.118172
$893$		−	1.05934e6i		−	1.32842i
$894$	40627.2	−	153195.i	0.0508325	−	0.191677i
$895$	2.06239e6			2.57469
$896$			40934.3i			0.0509884i
$897$	36396.3	+	9652.28i	0.0452348	+	0.0119962i
$898$	127155.			0.157681
$899$			581073.i			0.718970i
$900$	−507305.	−	289430.i	−0.626302	−	0.357321i
$901$	−391956.			−0.482822
$902$		−	66556.3i		−	0.0818043i
$903$	19232.1	−	72519.4i	0.0235858	−	0.0889362i
$904$	−525627.			−0.643192
$905$		−	141329.i		−	0.172557i
$906$	−821545.	−	217873.i	−1.00086	−	0.265428i
$907$	−923345.			−1.12240			−0.561202	−	0.827679i	$-0.689661\pi$
$907$	−923345.			−1.12240			−0.561202	+	0.827679i	$0.689661\pi$
$908$		−	603103.i		−	0.731510i
$909$	−442302.	+	775255.i	−0.535292	+	0.938246i
$910$	118474.			0.143067
$911$		−	456460.i		−	0.550004i	−0.961444	−	0.275002i	$-0.911321\pi$
$911$		−	456460.i		−	0.550004i	0.961444	−	0.275002i	$-0.0886786\pi$
$912$	−70155.4	+	264538.i	−0.0843474	+	0.318053i
$913$	−252202.			−0.302557
$914$		−	334917.i		−	0.400908i
$915$	1.32869e6	+	352367.i	1.58701	+	0.420875i
$916$	−387827.			−0.462218
$917$			479761.i			0.570540i
$918$	−147424.	+	149713.i	−0.174938	+	0.177653i
$919$	846467.			1.00226			0.501128	−	0.865373i	$-0.332918\pi$
$919$	846467.			1.00226			0.501128	+	0.865373i	$0.332918\pi$
$920$		−	97510.4i		−	0.115206i
$921$	331690.	−	1.25072e6i	0.391033	−	1.47449i
$922$	−58573.7			−0.0689034
$923$			133147.i			0.156289i
$924$	−250519.	−	66437.6i	−0.293425	−	0.0778162i
$925$	−1.36513e6			−1.59548
$926$		−	480328.i		−	0.560165i
$927$	−37772.0	−	21549.9i	−0.0439553	−	0.0250775i
$928$	109800.			0.127499
$929$			1.04998e6i			1.21660i	0.793707	+	0.608300i	$0.208148\pi$
$929$			1.04998e6i			1.21660i	−0.793707	+	0.608300i	$0.791852\pi$
$930$	−244218.	+	920882.i	−0.282365	+	1.06473i
$931$	−761183.			−0.878192
$932$		−	475807.i		−	0.547771i
$933$	1.00358e6	+	266149.i	1.15289	+	0.305746i
$934$	−735346.			−0.842943
$935$		−	506993.i		−	0.579934i
$936$	−34449.8	+	60382.7i	−0.0393220	+	0.0689225i
$937$	727096.			0.828157			0.414078	−	0.910241i	$-0.364104\pi$
$937$	727096.			0.828157			0.414078	+	0.910241i	$0.364104\pi$
$938$			96455.0i			0.109627i
$939$	333579.	−	1.25784e6i	0.378327	−	1.42658i
$940$	−696829.			−0.788625
$941$		−	452998.i		−	0.511584i	−0.966732	−	0.255792i	$-0.917664\pi$
$941$		−	452998.i		−	0.511584i	0.966732	−	0.255792i	$-0.0823362\pi$
$942$	1.06406e6	+	282187.i	1.19912	+	0.318006i
$943$	20381.7			0.0229201
$944$			188470.i			0.211494i
$945$	−573625.	−	564856.i	−0.642339	−	0.632520i
$946$	−106228.			−0.118702
$947$			721302.i			0.804298i	0.915574	+	0.402149i	$0.131737\pi$
$947$			721302.i			0.804298i	−0.915574	+	0.402149i	$0.868263\pi$
$948$	147598.	−	556555.i	0.164234	−	0.619287i
$949$	−363926.			−0.404092
$950$		−	1.21131e6i		−	1.34217i
$951$	−812894.	−	215579.i	−0.898820	−	0.238367i
$952$	−65176.2			−0.0719143
$953$			383887.i			0.422685i	0.977412	+	0.211343i	$0.0677836\pi$
$953$			383887.i			0.422685i	−0.977412	+	0.211343i	$0.932216\pi$
$954$	765412.	+	436686.i	0.841005	+	0.479814i
$955$	1.39911e6			1.53407
$956$		−	676104.i		−	0.739772i
$957$	−178209.	+	671983.i	−0.194584	+	0.733727i
$958$	96357.4			0.104991
$959$			819637.i			0.891219i
$960$	174011.	+	46147.7i	0.188814	+	0.0500735i
$961$	−5816.22			−0.00629788
$962$			162487.i			0.175577i
$963$	811505.	−	1.42238e6i	0.875062	−	1.53379i
$964$	−374288.			−0.402765
$965$		−	1.28626e6i		−	1.38125i
$966$	20345.3	−	76717.1i	0.0218027	−	0.0822125i
$967$	961589.			1.02834			0.514170	−	0.857688i	$-0.328100\pi$
$967$	961589.			1.02834			0.514170	+	0.857688i	$0.328100\pi$
$968$			35678.5i			0.0380764i
$969$	−421202.	−	111703.i	−0.448583	−	0.118964i
$970$	749002.			0.796048
$971$		−	1.31187e6i		−	1.39140i	−0.718334	−	0.695698i	$-0.755095\pi$
$971$		−	1.31187e6i		−	1.39140i	0.718334	−	0.695698i	$-0.244905\pi$
$972$	454689.	−	128111.i	0.481262	−	0.135598i
$973$	−397200.			−0.419550
$974$			1.17590e6i			1.23952i
$975$	78871.8	−	297406.i	0.0829684	−	0.312853i
$976$	−250205.			−0.262661
$977$		−	933151.i		−	0.977604i	−0.872395	−	0.488802i	$-0.837434\pi$
$977$		−	933151.i		−	0.977604i	0.872395	−	0.488802i	$-0.162566\pi$
$978$	−1.03286e6	−	273914.i	−1.07985	−	0.286376i
$979$	−1.85575e6			−1.93622
$980$			500700.i			0.521346i
$981$	449979.	+	256724.i	0.467578	+	0.266765i
$982$	127016.			0.131715
$983$		−	1.44240e6i		−	1.49272i	−0.665540	−	0.746362i	$-0.731799\pi$
$983$		−	1.44240e6i		−	1.49272i	0.665540	−	0.746362i	$-0.268201\pi$
$984$	−9645.83	+	36372.0i	−0.00996207	+	0.0375644i
$985$	−2.29507e6			−2.36550
$986$			174826.i			0.179826i
$987$	−548236.	−	145392.i	−0.562773	−	0.149247i
$988$	−144177.			−0.147701
$989$		−	32530.5i		−	0.0332581i
$990$	−564852.	+	990057.i	−0.576321	+	1.01016i
$991$	915217.			0.931916			0.465958	−	0.884807i	$-0.345710\pi$
$991$	915217.			0.931916			0.465958	+	0.884807i	$0.345710\pi$
$992$		−	173411.i		−	0.176219i
$993$	−52316.6	+	197273.i	−0.0530568	+	0.200064i
$994$	280651.			0.284049
$995$			2.44814e6i			2.47281i
$996$	137824.	+	36550.9i	0.138934	+	0.0368451i
$997$	1.06708e6			1.07352			0.536758	−	0.843736i	$-0.319649\pi$
$997$	1.06708e6			1.07352			0.536758	+	0.843736i	$0.319649\pi$
$998$			1.03452e6i			1.03867i
$999$	774698.	−	786724.i	0.776249	−	0.788300i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.c.a.47.20	yes	28
3.2	odd	2	inner	138.5.c.a.47.6	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.c.a.47.6	✓	28	3.2	odd	2	inner
138.5.c.a.47.20	yes	28	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} +(-2.30705 + 8.69928i) q^{3} -8.00000 q^{4} +39.0683i q^{5} +(-24.6053 - 6.52531i) q^{6} -28.2665 q^{7} -22.6274i q^{8} +(-70.3551 - 40.1393i) q^{9} +O(q^{10})\)	\(q+2.82843i q^{2} +(-2.30705 + 8.69928i) q^{3} -8.00000 q^{4} +39.0683i q^{5} +(-24.6053 - 6.52531i) q^{6} -28.2665 q^{7} -22.6274i q^{8} +(-70.3551 - 40.1393i) q^{9} -110.502 q^{10} +127.349i q^{11} +(18.4564 - 69.5943i) q^{12} +37.9299 q^{13} -79.9497i q^{14} +(-339.866 - 90.1323i) q^{15} +64.0000 q^{16} +101.902i q^{17} +(113.531 - 198.994i) q^{18} +475.144 q^{19} -312.546i q^{20} +(65.2121 - 245.898i) q^{21} -360.198 q^{22} -110.304i q^{23} +(196.842 + 52.2025i) q^{24} -901.330 q^{25} +107.282i q^{26} +(511.495 - 519.436i) q^{27} +226.132 q^{28} -606.567i q^{29} +(254.933 - 961.286i) q^{30} -957.969 q^{31} +181.019i q^{32} +(-1107.85 - 293.800i) q^{33} -288.222 q^{34} -1104.32i q^{35} +(562.841 + 321.114i) q^{36} +1514.57 q^{37} +1343.91i q^{38} +(-87.5061 + 329.963i) q^{39} +884.014 q^{40} +184.777i q^{41} +(695.506 + 184.448i) q^{42} +294.916 q^{43} -1018.79i q^{44} +(1568.17 - 2748.65i) q^{45} +311.987 q^{46} -2229.52i q^{47} +(-147.651 + 556.754i) q^{48} -1602.00 q^{49} -2549.35i q^{50} +(-886.473 - 235.092i) q^{51} -303.440 q^{52} +3846.40i q^{53} +(1469.19 + 1446.73i) q^{54} -4975.31 q^{55} +639.598i q^{56} +(-1096.18 + 4133.41i) q^{57} +1715.63 q^{58} +2944.84i q^{59} +(2718.93 + 721.058i) q^{60} -3909.45 q^{61} -2709.55i q^{62} +(1988.69 + 1134.60i) q^{63} -512.000 q^{64} +1481.86i q^{65} +(830.992 - 3133.46i) q^{66} -1206.44 q^{67} -815.215i q^{68} +(959.567 + 254.477i) q^{69} +3123.50 q^{70} +3510.34i q^{71} +(-908.248 + 1591.95i) q^{72} -9594.69 q^{73} +4283.86i q^{74} +(2079.41 - 7840.92i) q^{75} -3801.15 q^{76} -3599.71i q^{77} +(-933.277 - 247.505i) q^{78} +7997.14 q^{79} +2500.37i q^{80} +(3338.68 + 5648.01i) q^{81} -522.629 q^{82} +1980.40i q^{83} +(-521.697 + 1967.19i) q^{84} -3981.13 q^{85} +834.149i q^{86} +(5276.70 + 1399.38i) q^{87} +2881.58 q^{88} +14572.2i q^{89} +(7774.36 + 4435.46i) q^{90} -1072.15 q^{91} +882.433i q^{92} +(2210.08 - 8333.65i) q^{93} +6306.04 q^{94} +18563.0i q^{95} +(-1574.74 - 417.620i) q^{96} -6778.19 q^{97} -4531.15i q^{98} +(5111.70 - 8959.65i) 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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