LMFDB - Embedded newform 138.8.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,8,Mod(1,138)]

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([0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("138.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	138.a (trivial)

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:	yes
Analytic conductor:	$43.1091335168$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 $ x^4 - 2x^3 - 8367x^2 - 89140*x + 11077220
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$90.4389$ of defining polynomial
Character	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -10.0669 q^{5} +216.000 q^{6} +971.172 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -10.0669 q^{5} +216.000 q^{6} +971.172 q^{7} +512.000 q^{8} +729.000 q^{9} -80.5353 q^{10} -2066.60 q^{11} +1728.00 q^{12} +9535.57 q^{13} +7769.38 q^{14} -271.807 q^{15} +4096.00 q^{16} -20856.4 q^{17} +5832.00 q^{18} +41075.9 q^{19} -644.282 q^{20} +26221.6 q^{21} -16532.8 q^{22} -12167.0 q^{23} +13824.0 q^{24} -78023.7 q^{25} +76284.5 q^{26} +19683.0 q^{27} +62155.0 q^{28} +24659.9 q^{29} -2174.45 q^{30} +244895. q^{31} +32768.0 q^{32} -55798.1 q^{33} -166851. q^{34} -9776.71 q^{35} +46656.0 q^{36} +520379. q^{37} +328607. q^{38} +257460. q^{39} -5154.26 q^{40} +123395. q^{41} +209773. q^{42} +7429.38 q^{43} -132262. q^{44} -7338.78 q^{45} -97336.0 q^{46} -1.38129e6 q^{47} +110592. q^{48} +119632. q^{49} -624189. q^{50} -563122. q^{51} +610276. q^{52} +1.01252e6 q^{53} +157464. q^{54} +20804.2 q^{55} +497240. q^{56} +1.10905e6 q^{57} +197279. q^{58} +2.93571e6 q^{59} -17395.6 q^{60} -2.36149e6 q^{61} +1.95916e6 q^{62} +707984. q^{63} +262144. q^{64} -95993.7 q^{65} -446385. q^{66} +3.19614e6 q^{67} -1.33481e6 q^{68} -328509. q^{69} -78213.6 q^{70} -3.49894e6 q^{71} +373248. q^{72} +2.48917e6 q^{73} +4.16304e6 q^{74} -2.10664e6 q^{75} +2.62886e6 q^{76} -2.00702e6 q^{77} +2.05968e6 q^{78} -4.80101e6 q^{79} -41234.1 q^{80} +531441. q^{81} +987160. q^{82} +7.77498e6 q^{83} +1.67819e6 q^{84} +209959. q^{85} +59435.0 q^{86} +665817. q^{87} -1.05810e6 q^{88} +539607. q^{89} -58710.2 q^{90} +9.26068e6 q^{91} -778688. q^{92} +6.61217e6 q^{93} -1.10503e7 q^{94} -413508. q^{95} +884736. q^{96} -1.04621e7 q^{97} +957058. q^{98} -1.50655e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) $ 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9 + 2160 * q^10 + 4120 * q^11 + 6912 * q^12 + 8036 * q^13 + 16176 * q^14 + 7290 * q^15 + 16384 * q^16 + 37182 * q^17 + 23328 * q^18 + 5702 * q^19 + 17280 * q^20 + 54594 * q^21 + 32960 * q^22 - 48668 * q^23 + 55296 * q^24 + 121480 * q^25 + 64288 * q^26 + 78732 * q^27 + 129408 * q^28 + 217716 * q^29 + 58320 * q^30 + 222852 * q^31 + 131072 * q^32 + 111240 * q^33 + 297456 * q^34 + 68440 * q^35 + 186624 * q^36 + 486428 * q^37 + 45616 * q^38 + 216972 * q^39 + 138240 * q^40 + 338336 * q^41 + 436752 * q^42 + 730974 * q^43 + 263680 * q^44 + 196830 * q^45 - 389344 * q^46 + 338248 * q^47 + 442368 * q^48 - 310552 * q^49 + 971840 * q^50 + 1003914 * q^51 + 514304 * q^52 - 375502 * q^53 + 629856 * q^54 + 424840 * q^55 + 1035264 * q^56 + 153954 * q^57 + 1741728 * q^58 + 71392 * q^59 + 466560 * q^60 + 2101164 * q^61 + 1782816 * q^62 + 1474038 * q^63 + 1048576 * q^64 + 1578780 * q^65 + 889920 * q^66 + 4337162 * q^67 + 2379648 * q^68 - 1314036 * q^69 + 547520 * q^70 + 2288016 * q^71 + 1492992 * q^72 - 1107328 * q^73 + 3891424 * q^74 + 3279960 * q^75 + 364928 * q^76 + 5826200 * q^77 + 1735776 * q^78 + 60610 * q^79 + 1105920 * q^80 + 2125764 * q^81 + 2706688 * q^82 + 1485464 * q^83 + 3494016 * q^84 - 8843820 * q^85 + 5847792 * q^86 + 5878332 * q^87 + 2109440 * q^88 + 1485090 * q^89 + 1574640 * q^90 - 2898412 * q^91 - 3114752 * q^92 + 6017004 * q^93 + 2705984 * q^94 + 8545200 * q^95 + 3538944 * q^96 + 1935444 * q^97 - 2484416 * q^98 + 3003480 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	−10.0669	−0.0360165	−0.0180082	−	0.999838i	$-0.505733\pi$
$5$	−10.0669	−0.0360165	−0.0180082	+	0.999838i	$0.505733\pi$
$6$	216.000	0.408248
$7$	971.172	1.07017	0.535085	−	0.844798i	$-0.320279\pi$
$7$	971.172	1.07017	0.535085	+	0.844798i	$0.320279\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−80.5353	−0.0254675
$11$	−2066.60	−0.468146	−0.234073	−	0.972219i	$-0.575205\pi$
$11$	−2066.60	−0.468146	−0.234073	+	0.972219i	$0.575205\pi$
$12$	1728.00	0.288675
$13$	9535.57	1.20377	0.601887	−	0.798581i	$-0.294416\pi$
$13$	9535.57	1.20377	0.601887	+	0.798581i	$0.294416\pi$
$14$	7769.38	0.756725
$15$	−271.807	−0.0207941
$16$	4096.00	0.250000
$17$	−20856.4	−1.02960	−0.514799	−	0.857311i	$-0.672133\pi$
$17$	−20856.4	−1.02960	−0.514799	+	0.857311i	$0.672133\pi$
$18$	5832.00	0.235702
$19$	41075.9	1.37388	0.686942	−	0.726713i	$-0.258953\pi$
$19$	41075.9	1.37388	0.686942	+	0.726713i	$0.258953\pi$
$20$	−644.282	−0.0180082
$21$	26221.6	0.617863
$22$	−16532.8	−0.331029
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	13824.0	0.204124
$25$	−78023.7	−0.998703
$26$	76284.5	0.851196
$27$	19683.0	0.192450
$28$	62155.0	0.535085
$29$	24659.9	0.187758	0.0938789	−	0.995584i	$-0.470073\pi$
$29$	24659.9	0.187758	0.0938789	+	0.995584i	$0.470073\pi$
$30$	−2174.45	−0.0147037
$31$	244895.	1.47643	0.738217	−	0.674563i	$-0.235668\pi$
$31$	244895.	1.47643	0.738217	+	0.674563i	$0.235668\pi$
$32$	32768.0	0.176777
$33$	−55798.1	−0.270284
$34$	−166851.	−0.728036
$35$	−9776.71	−0.0385438
$36$	46656.0	0.166667
$37$	520379.	1.68894	0.844470	−	0.535603i	$-0.179916\pi$
$37$	520379.	1.68894	0.844470	+	0.535603i	$0.179916\pi$
$38$	328607.	0.971482
$39$	257460.	0.694999
$40$	−5154.26	−0.0127338
$41$	123395.	0.279611	0.139805	−	0.990179i	$-0.455352\pi$
$41$	123395.	0.279611	0.139805	+	0.990179i	$0.455352\pi$
$42$	209773.	0.436895
$43$	7429.38	0.0142499	0.00712497	−	0.999975i	$-0.497732\pi$
$43$	7429.38	0.0142499	0.00712497	+	0.999975i	$0.497732\pi$
$44$	−132262.	−0.234073
$45$	−7338.78	−0.0120055
$46$	−97336.0	−0.147442
$47$	−1.38129e6	−1.94063	−0.970316	−	0.241839i	$-0.922249\pi$
$47$	−1.38129e6	−1.94063	−0.970316	+	0.241839i	$0.922249\pi$
$48$	110592.	0.144338
$49$	119632.	0.145265
$50$	−624189.	−0.706190
$51$	−563122.	−0.594439
$52$	610276.	0.601887
$53$	1.01252e6	0.934192	0.467096	−	0.884206i	$-0.345300\pi$
$53$	1.01252e6	0.934192	0.467096	+	0.884206i	$0.345300\pi$
$54$	157464.	0.136083
$55$	20804.2	0.0168610
$56$	497240.	0.378362
$57$	1.10905e6	0.793212
$58$	197279.	0.132765
$59$	2.93571e6	1.86094	0.930468	−	0.366372i	$-0.119400\pi$
$59$	2.93571e6	1.86094	0.930468	+	0.366372i	$0.119400\pi$
$60$	−17395.6	−0.0103971
$61$	−2.36149e6	−1.33208	−0.666041	−	0.745915i	$-0.732012\pi$
$61$	−2.36149e6	−1.33208	−0.666041	+	0.745915i	$0.732012\pi$
$62$	1.95916e6	1.04400
$63$	707984.	0.356724
$64$	262144.	0.125000
$65$	−95993.7	−0.0433557
$66$	−446385.	−0.191120
$67$	3.19614e6	1.29827	0.649133	−	0.760675i	$-0.275132\pi$
$67$	3.19614e6	1.29827	0.649133	+	0.760675i	$0.275132\pi$
$68$	−1.33481e6	−0.514799
$69$	−328509.	−0.120386
$70$	−78213.6	−0.0272546
$71$	−3.49894e6	−1.16020	−0.580100	−	0.814545i	$-0.696987\pi$
$71$	−3.49894e6	−1.16020	−0.580100	+	0.814545i	$0.696987\pi$
$72$	373248.	0.117851
$73$	2.48917e6	0.748903	0.374451	−	0.927247i	$-0.377831\pi$
$73$	2.48917e6	0.748903	0.374451	+	0.927247i	$0.377831\pi$
$74$	4.16304e6	1.19426
$75$	−2.10664e6	−0.576601
$76$	2.62886e6	0.686942
$77$	−2.00702e6	−0.500996
$78$	2.05968e6	0.491438
$79$	−4.80101e6	−1.09556	−0.547782	−	0.836621i	$-0.684528\pi$
$79$	−4.80101e6	−1.09556	−0.547782	+	0.836621i	$0.684528\pi$
$80$	−41234.1	−0.00900412
$81$	531441.	0.111111
$82$	987160.	0.197715
$83$	7.77498e6	1.49254	0.746270	−	0.665644i	$-0.231843\pi$
$83$	7.77498e6	1.49254	0.746270	+	0.665644i	$0.231843\pi$
$84$	1.67819e6	0.308932
$85$	209959.	0.0370825
$86$	59435.0	0.0100762
$87$	665817.	0.108402
$88$	−1.05810e6	−0.165514
$89$	539607.	0.0811358	0.0405679	−	0.999177i	$-0.487083\pi$
$89$	539607.	0.0811358	0.0405679	+	0.999177i	$0.487083\pi$
$90$	−58710.2	−0.00848917
$91$	9.26068e6	1.28824
$92$	−778688.	−0.104257
$93$	6.61217e6	0.852420
$94$	−1.10503e7	−1.37223
$95$	−413508.	−0.0494824
$96$	884736.	0.102062
$97$	−1.04621e7	−1.16391	−0.581953	−	0.813222i	$-0.697711\pi$
$97$	−1.04621e7	−1.16391	−0.581953	+	0.813222i	$0.697711\pi$
$98$	957058.	0.102718
$99$	−1.50655e6	−0.156049
$100$	−4.99351e6	−0.499351
$101$	−4.89157e6	−0.472415	−0.236207	−	0.971703i	$-0.575904\pi$
$101$	−4.89157e6	−0.472415	−0.236207	+	0.971703i	$0.575904\pi$
$102$	−4.50498e6	−0.420332
$103$	6.99268e6	0.630541	0.315270	−	0.949002i	$-0.397905\pi$
$103$	6.99268e6	0.630541	0.315270	+	0.949002i	$0.397905\pi$
$104$	4.88221e6	0.425598
$105$	−263971.	−0.0222533
$106$	8.10013e6	0.660574
$107$	1.26394e7	0.997434	0.498717	−	0.866765i	$-0.333805\pi$
$107$	1.26394e7	0.997434	0.498717	+	0.866765i	$0.333805\pi$
$108$	1.25971e6	0.0962250
$109$	−1.88878e7	−1.39697	−0.698486	−	0.715623i	$-0.746143\pi$
$109$	−1.88878e7	−1.39697	−0.698486	+	0.715623i	$0.746143\pi$
$110$	166434.	0.0119225
$111$	1.40502e7	0.975110
$112$	3.97792e6	0.267543
$113$	−9.49674e6	−0.619156	−0.309578	−	0.950874i	$-0.600188\pi$
$113$	−9.49674e6	−0.619156	−0.309578	+	0.950874i	$0.600188\pi$
$114$	8.87240e6	0.560886
$115$	122484.	0.00750996
$116$	1.57823e6	0.0938789
$117$	6.95143e6	0.401258
$118$	2.34857e7	1.31588
$119$	−2.02551e7	−1.10185
$120$	−139165.	−0.00735183
$121$	−1.52164e7	−0.780840
$122$	−1.88919e7	−0.941924
$123$	3.33167e6	0.161433
$124$	1.56733e7	0.738217
$125$	1.57194e6	0.0719862
$126$	5.66388e6	0.252242
$127$	−1.65209e7	−0.715684	−0.357842	−	0.933782i	$-0.616487\pi$
$127$	−1.65209e7	−0.715684	−0.357842	+	0.933782i	$0.616487\pi$
$128$	2.09715e6	0.0883883
$129$	200593.	0.00822720
$130$	−767950.	−0.0306571
$131$	1.15691e7	0.449624	0.224812	−	0.974402i	$-0.427823\pi$
$131$	1.15691e7	0.449624	0.224812	+	0.974402i	$0.427823\pi$
$132$	−3.57108e6	−0.135142
$133$	3.98918e7	1.47029
$134$	2.55691e7	0.918013
$135$	−198147.	−0.00693138
$136$	−1.06785e7	−0.364018
$137$	−4.83328e7	−1.60590	−0.802952	−	0.596043i	$-0.796739\pi$
$137$	−4.83328e7	−1.60590	−0.802952	+	0.596043i	$0.796739\pi$
$138$	−2.62807e6	−0.0851257
$139$	−3.30182e7	−1.04280	−0.521401	−	0.853312i	$-0.674591\pi$
$139$	−3.30182e7	−1.04280	−0.521401	+	0.853312i	$0.674591\pi$
$140$	−625709.	−0.0192719
$141$	−3.72949e7	−1.12042
$142$	−2.79916e7	−0.820385
$143$	−1.97062e7	−0.563541
$144$	2.98598e6	0.0833333
$145$	−248249.	−0.00676237
$146$	1.99134e7	0.529554
$147$	3.23007e6	0.0838690
$148$	3.33043e7	0.844470
$149$	−6.04124e6	−0.149615	−0.0748074	−	0.997198i	$-0.523834\pi$
$149$	−6.04124e6	−0.149615	−0.0748074	+	0.997198i	$0.523834\pi$
$150$	−1.68531e7	−0.407719
$151$	−1.73272e7	−0.409551	−0.204775	−	0.978809i	$-0.565646\pi$
$151$	−1.73272e7	−0.409551	−0.204775	+	0.978809i	$0.565646\pi$
$152$	2.10309e7	0.485741
$153$	−1.52043e7	−0.343199
$154$	−1.60562e7	−0.354258
$155$	−2.46534e6	−0.0531760
$156$	1.64775e7	0.347499
$157$	−3.58063e7	−0.738433	−0.369216	−	0.929343i	$-0.620374\pi$
$157$	−3.58063e7	−0.738433	−0.369216	+	0.929343i	$0.620374\pi$
$158$	−3.84081e7	−0.774681
$159$	2.73379e7	0.539356
$160$	−329873.	−0.00636688
$161$	−1.18163e7	−0.223146
$162$	4.25153e6	0.0785674
$163$	−8.41945e7	−1.52275	−0.761373	−	0.648315i	$-0.775474\pi$
$163$	−8.41945e7	−1.52275	−0.761373	+	0.648315i	$0.775474\pi$
$164$	7.89728e6	0.139805
$165$	561714.	0.00973468
$166$	6.21998e7	1.05538
$167$	−1.15659e8	−1.92163	−0.960817	−	0.277182i	$-0.910599\pi$
$167$	−1.15659e8	−1.92163	−0.960817	+	0.277182i	$0.910599\pi$
$168$	1.34255e7	0.218448
$169$	2.81785e7	0.449071
$170$	1.67967e6	0.0262213
$171$	2.99444e7	0.457961
$172$	475480.	0.00712497
$173$	−2.95752e7	−0.434276	−0.217138	−	0.976141i	$-0.569672\pi$
$173$	−2.95752e7	−0.434276	−0.217138	+	0.976141i	$0.569672\pi$
$174$	5.32653e6	0.0766518
$175$	−7.57744e7	−1.06878
$176$	−8.46477e6	−0.117036
$177$	7.92643e7	1.07441
$178$	4.31686e6	0.0573717
$179$	−7.92741e7	−1.03311	−0.516554	−	0.856255i	$-0.672786\pi$
$179$	−7.92741e7	−1.03311	−0.516554	+	0.856255i	$0.672786\pi$
$180$	−469682.	−0.00600275
$181$	9.63504e7	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$181$	9.63504e7	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$182$	7.40854e7	0.910926
$183$	−6.37601e7	−0.769078
$184$	−6.22950e6	−0.0737210
$185$	−5.23862e6	−0.0608297
$186$	5.28973e7	0.602752
$187$	4.31017e7	0.482002
$188$	−8.84028e7	−0.970316
$189$	1.91156e7	0.205954
$190$	−3.30806e6	−0.0349894
$191$	−1.35606e8	−1.40819	−0.704094	−	0.710107i	$-0.748647\pi$
$191$	−1.35606e8	−1.40819	−0.704094	+	0.710107i	$0.748647\pi$
$192$	7.07789e6	0.0721688
$193$	−1.75736e8	−1.75959	−0.879795	−	0.475354i	$-0.842320\pi$
$193$	−1.75736e8	−1.75959	−0.879795	+	0.475354i	$0.842320\pi$
$194$	−8.36969e7	−0.823006
$195$	−2.59183e6	−0.0250314
$196$	7.65647e6	0.0726327
$197$	−1.22530e8	−1.14186	−0.570929	−	0.821000i	$-0.693417\pi$
$197$	−1.22530e8	−1.14186	−0.570929	+	0.821000i	$0.693417\pi$
$198$	−1.20524e7	−0.110343
$199$	2.14948e8	1.93352	0.966760	−	0.255686i	$-0.0823015\pi$
$199$	2.14948e8	1.93352	0.966760	+	0.255686i	$0.0823015\pi$
$200$	−3.99481e7	−0.353095
$201$	8.62958e7	0.749555
$202$	−3.91325e7	−0.334048
$203$	2.39490e7	0.200933
$204$	−3.60398e7	−0.297219
$205$	−1.24221e6	−0.0100706
$206$	5.59414e7	0.445860
$207$	−8.86974e6	−0.0695048
$208$	3.90577e7	0.300943
$209$	−8.48873e7	−0.643178
$210$	−2.11177e6	−0.0157354
$211$	2.30223e8	1.68717	0.843586	−	0.536994i	$-0.180440\pi$
$211$	2.30223e8	1.68717	0.843586	+	0.536994i	$0.180440\pi$
$212$	6.48010e7	0.467096
$213$	−9.44715e7	−0.669842
$214$	1.01115e8	0.705293
$215$	−74790.9	−0.000513233 0
$216$	1.00777e7	0.0680414
$217$	2.37835e8	1.58004
$218$	−1.51102e8	−0.987809
$219$	6.72077e7	0.432379
$220$	1.33147e6	0.00843048
$221$	−1.98877e8	−1.23940
$222$	1.12402e8	0.689507
$223$	2.16447e8	1.30703	0.653515	−	0.756914i	$-0.273294\pi$
$223$	2.16447e8	1.30703	0.653515	+	0.756914i	$0.273294\pi$
$224$	3.18234e7	0.189181
$225$	−5.68792e7	−0.332901
$226$	−7.59739e7	−0.437809
$227$	−2.11703e8	−1.20126	−0.600630	−	0.799527i	$-0.705083\pi$
$227$	−2.11703e8	−1.20126	−0.600630	+	0.799527i	$0.705083\pi$
$228$	7.09792e7	0.396606
$229$	−1.41898e7	−0.0780825	−0.0390412	−	0.999238i	$-0.512430\pi$
$229$	−1.41898e7	−0.0780825	−0.0390412	+	0.999238i	$0.512430\pi$
$230$	979873.	0.00531034
$231$	−5.41895e7	−0.289250
$232$	1.26259e7	0.0663824
$233$	2.97428e8	1.54041	0.770204	−	0.637798i	$-0.220154\pi$
$233$	2.97428e8	1.54041	0.770204	+	0.637798i	$0.220154\pi$
$234$	5.56114e7	0.283732
$235$	1.39054e7	0.0698948
$236$	1.87886e8	0.930468
$237$	−1.29627e8	−0.632525
$238$	−1.62041e8	−0.779122
$239$	2.69292e8	1.27594	0.637972	−	0.770060i	$-0.279774\pi$
$239$	2.69292e8	1.27594	0.637972	+	0.770060i	$0.279774\pi$
$240$	−1.11332e6	−0.00519853
$241$	3.06355e7	0.140982	0.0704912	−	0.997512i	$-0.477543\pi$
$241$	3.06355e7	0.140982	0.0704912	+	0.997512i	$0.477543\pi$
$242$	−1.21731e8	−0.552137
$243$	1.43489e7	0.0641500
$244$	−1.51135e8	−0.666041
$245$	−1.20433e6	−0.00523195
$246$	2.66533e7	0.114151
$247$	3.91682e8	1.65384
$248$	1.25386e8	0.521998
$249$	2.09924e8	0.861718
$250$	1.25755e7	0.0509020
$251$	−2.34129e8	−0.934539	−0.467270	−	0.884115i	$-0.654762\pi$
$251$	−2.34129e8	−0.934539	−0.467270	+	0.884115i	$0.654762\pi$
$252$	4.53110e7	0.178362
$253$	2.51443e7	0.0976151
$254$	−1.32167e8	−0.506065
$255$	5.66890e6	0.0214096
$256$	1.67772e7	0.0625000
$257$	3.87910e8	1.42549	0.712747	−	0.701421i	$-0.247451\pi$
$257$	3.87910e8	1.42549	0.712747	+	0.701421i	$0.247451\pi$
$258$	1.60475e6	0.00581751
$259$	5.05378e8	1.80745
$260$	−6.14360e6	−0.0216778
$261$	1.79770e7	0.0625859
$262$	9.25527e7	0.317932
$263$	−3.05352e8	−1.03504	−0.517518	−	0.855672i	$-0.673144\pi$
$263$	−3.05352e8	−1.03504	−0.517518	+	0.855672i	$0.673144\pi$
$264$	−2.85686e7	−0.0955598
$265$	−1.01929e7	−0.0336463
$266$	3.19134e8	1.03965
$267$	1.45694e7	0.0468438
$268$	2.04553e8	0.649133
$269$	−2.30284e8	−0.721324	−0.360662	−	0.932697i	$-0.617449\pi$
$269$	−2.30284e8	−0.721324	−0.360662	+	0.932697i	$0.617449\pi$
$270$	−1.58518e6	−0.00490122
$271$	−3.89654e7	−0.118929	−0.0594643	−	0.998230i	$-0.518939\pi$
$271$	−3.89654e7	−0.118929	−0.0594643	+	0.998230i	$0.518939\pi$
$272$	−8.54277e7	−0.257399
$273$	2.50038e8	0.743768
$274$	−3.86662e8	−1.13555
$275$	1.61243e8	0.467538
$276$	−2.10246e7	−0.0601929
$277$	3.37749e8	0.954804	0.477402	−	0.878685i	$-0.341579\pi$
$277$	3.37749e8	0.954804	0.477402	+	0.878685i	$0.341579\pi$
$278$	−2.64146e8	−0.737373
$279$	1.78529e8	0.492145
$280$	−5.00567e6	−0.0136273
$281$	4.32675e8	1.16329	0.581647	−	0.813441i	$-0.302408\pi$
$281$	4.32675e8	1.16329	0.581647	+	0.813441i	$0.302408\pi$
$282$	−2.98359e8	−0.792260
$283$	2.64590e8	0.693938	0.346969	−	0.937877i	$-0.387211\pi$
$283$	2.64590e8	0.693938	0.346969	+	0.937877i	$0.387211\pi$
$284$	−2.23932e8	−0.580100
$285$	−1.11647e7	−0.0285687
$286$	−1.57649e8	−0.398484
$287$	1.19838e8	0.299232
$288$	2.38879e7	0.0589256
$289$	2.46498e7	0.0600718
$290$	−1.98599e6	−0.00478172
$291$	−2.82477e8	−0.671982
$292$	1.59307e8	0.374451
$293$	−1.03680e8	−0.240802	−0.120401	−	0.992725i	$-0.538418\pi$
$293$	−1.03680e8	−0.240802	−0.120401	+	0.992725i	$0.538418\pi$
$294$	2.58406e7	0.0593044
$295$	−2.95536e7	−0.0670244
$296$	2.66434e8	0.597130
$297$	−4.06768e7	−0.0900947
$298$	−4.83299e7	−0.105794
$299$	−1.16019e8	−0.251004
$300$	−1.34825e8	−0.288301
$301$	7.21521e6	0.0152499
$302$	−1.38617e8	−0.289596
$303$	−1.32072e8	−0.272749
$304$	1.68247e8	0.343471
$305$	2.37729e7	0.0479769
$306$	−1.21634e8	−0.242679
$307$	−3.01316e8	−0.594344	−0.297172	−	0.954824i	$-0.596044\pi$
$307$	−3.01316e8	−0.594344	−0.297172	+	0.954824i	$0.596044\pi$
$308$	−1.28449e8	−0.250498
$309$	1.88802e8	0.364043
$310$	−1.97227e7	−0.0376011
$311$	6.35884e8	1.19872	0.599358	−	0.800481i	$-0.295422\pi$
$311$	6.35884e8	1.19872	0.599358	+	0.800481i	$0.295422\pi$
$312$	1.31820e8	0.245719
$313$	−2.65449e8	−0.489302	−0.244651	−	0.969611i	$-0.578673\pi$
$313$	−2.65449e8	−0.489302	−0.244651	+	0.969611i	$0.578673\pi$
$314$	−2.86451e8	−0.522151
$315$	−7.12722e6	−0.0128479
$316$	−3.07265e8	−0.547782
$317$	4.07834e8	0.719078	0.359539	−	0.933130i	$-0.382934\pi$
$317$	4.07834e8	0.719078	0.359539	+	0.933130i	$0.382934\pi$
$318$	2.18703e8	0.381382
$319$	−5.09620e7	−0.0878980
$320$	−2.63898e6	−0.00450206
$321$	3.41265e8	0.575869
$322$	−9.45300e7	−0.157788
$323$	−8.56695e8	−1.41455
$324$	3.40122e7	0.0555556
$325$	−7.44000e8	−1.20221
$326$	−6.73556e8	−1.07674
$327$	−5.09970e8	−0.806543
$328$	6.31783e7	0.0988574
$329$	−1.34147e9	−2.07681
$330$	4.49371e6	0.00688346
$331$	−2.72660e8	−0.413260	−0.206630	−	0.978419i	$-0.566250\pi$
$331$	−2.72660e8	−0.413260	−0.206630	+	0.978419i	$0.566250\pi$
$332$	4.97598e8	0.746270
$333$	3.79357e8	0.562980
$334$	−9.25270e8	−1.35880
$335$	−3.21753e7	−0.0467590
$336$	1.07404e8	0.154466
$337$	5.05642e8	0.719679	0.359839	−	0.933014i	$-0.382832\pi$
$337$	5.05642e8	0.719679	0.359839	+	0.933014i	$0.382832\pi$
$338$	2.25428e8	0.317541
$339$	−2.56412e8	−0.357470
$340$	1.34374e7	0.0185412
$341$	−5.06099e8	−0.691186
$342$	2.39555e8	0.323827
$343$	−6.83618e8	−0.914712
$344$	3.80384e6	0.00503811
$345$	3.30707e6	0.00433588
$346$	−2.36601e8	−0.307079
$347$	−1.05304e9	−1.35297	−0.676487	−	0.736454i	$-0.736499\pi$
$347$	−1.05304e9	−1.35297	−0.676487	+	0.736454i	$0.736499\pi$
$348$	4.26123e7	0.0542010
$349$	−1.26100e8	−0.158792	−0.0793958	−	0.996843i	$-0.525299\pi$
$349$	−1.26100e8	−0.158792	−0.0793958	+	0.996843i	$0.525299\pi$
$350$	−6.06195e8	−0.755743
$351$	1.87689e8	0.231666
$352$	−6.77182e7	−0.0827572
$353$	−3.34219e8	−0.404407	−0.202204	−	0.979343i	$-0.564810\pi$
$353$	−3.34219e8	−0.404407	−0.202204	+	0.979343i	$0.564810\pi$
$354$	6.34114e8	0.759724
$355$	3.52236e7	0.0417863
$356$	3.45349e7	0.0405679
$357$	−5.46889e8	−0.636151
$358$	−6.34193e8	−0.730518
$359$	5.37104e8	0.612671	0.306336	−	0.951924i	$-0.400897\pi$
$359$	5.37104e8	0.612671	0.306336	+	0.951924i	$0.400897\pi$
$360$	−3.75746e6	−0.00424458
$361$	7.93361e8	0.887555
$362$	7.70804e8	0.854011
$363$	−4.10842e8	−0.450818
$364$	5.92683e8	0.644122
$365$	−2.50583e7	−0.0269728
$366$	−5.10081e8	−0.543820
$367$	−2.86339e8	−0.302378	−0.151189	−	0.988505i	$-0.548310\pi$
$367$	−2.86339e8	−0.302378	−0.151189	+	0.988505i	$0.548310\pi$
$368$	−4.98360e7	−0.0521286
$369$	8.99550e7	0.0932037
$370$	−4.19089e7	−0.0430131
$371$	9.83327e8	0.999745
$372$	4.23179e8	0.426210
$373$	1.16695e9	1.16432	0.582159	−	0.813075i	$-0.302208\pi$
$373$	1.16695e9	1.16432	0.582159	+	0.813075i	$0.302208\pi$
$374$	3.44813e8	0.340827
$375$	4.24422e7	0.0415613
$376$	−7.07222e8	−0.686117
$377$	2.35146e8	0.226018
$378$	1.52925e8	0.145632
$379$	−7.02743e8	−0.663070	−0.331535	−	0.943443i	$-0.607566\pi$
$379$	−7.02743e8	−0.663070	−0.331535	+	0.943443i	$0.607566\pi$
$380$	−2.64645e7	−0.0247412
$381$	−4.46065e8	−0.413200
$382$	−1.08484e9	−0.995739
$383$	−4.77966e8	−0.434712	−0.217356	−	0.976092i	$-0.569743\pi$
$383$	−4.77966e8	−0.434712	−0.217356	+	0.976092i	$0.569743\pi$
$384$	5.66231e7	0.0510310
$385$	2.02045e7	0.0180441
$386$	−1.40589e9	−1.24422
$387$	5.41602e6	0.00474998
$388$	−6.69575e8	−0.581953
$389$	1.64085e9	1.41334	0.706668	−	0.707545i	$-0.250197\pi$
$389$	1.64085e9	1.41334	0.706668	+	0.707545i	$0.250197\pi$
$390$	−2.07346e7	−0.0176999
$391$	2.53760e8	0.214686
$392$	6.12517e7	0.0513591
$393$	3.12365e8	0.259590
$394$	−9.80243e8	−0.807415
$395$	4.83314e7	0.0394584
$396$	−9.64191e7	−0.0780243
$397$	9.42161e8	0.755716	0.377858	−	0.925864i	$-0.376661\pi$
$397$	9.42161e8	0.755716	0.377858	+	0.925864i	$0.376661\pi$
$398$	1.71959e9	1.36720
$399$	1.07708e9	0.848872
$400$	−3.19585e8	−0.249676
$401$	1.49153e8	0.115512	0.0577561	−	0.998331i	$-0.481605\pi$
$401$	1.49153e8	0.115512	0.0577561	+	0.998331i	$0.481605\pi$
$402$	6.90366e8	0.530015
$403$	2.33521e9	1.77729
$404$	−3.13060e8	−0.236207
$405$	−5.34997e6	−0.00400183
$406$	1.91592e8	0.142081
$407$	−1.07541e9	−0.790670
$408$	−2.88319e8	−0.210166
$409$	−1.15135e9	−0.832101	−0.416051	−	0.909341i	$-0.636586\pi$
$409$	−1.15135e9	−0.832101	−0.416051	+	0.909341i	$0.636586\pi$
$410$	−9.93766e6	−0.00712099
$411$	−1.30499e9	−0.927170
$412$	4.47531e8	0.315270
$413$	2.85108e9	1.99152
$414$	−7.09579e7	−0.0491473
$415$	−7.82700e7	−0.0537560
$416$	3.12461e8	0.212799
$417$	−8.91492e8	−0.602062
$418$	−6.79099e8	−0.454795
$419$	3.94962e8	0.262305	0.131152	−	0.991362i	$-0.458132\pi$
$419$	3.94962e8	0.262305	0.131152	+	0.991362i	$0.458132\pi$
$420$	−1.68941e7	−0.0111266
$421$	−3.31295e8	−0.216385	−0.108193	−	0.994130i	$-0.534506\pi$
$421$	−3.31295e8	−0.216385	−0.108193	+	0.994130i	$0.534506\pi$
$422$	1.84178e9	1.19301
$423$	−1.00696e9	−0.646878
$424$	5.18408e8	0.330287
$425$	1.62729e9	1.02826
$426$	−7.55772e8	−0.473650
$427$	−2.29341e9	−1.42556
$428$	8.08924e8	0.498717
$429$	−5.32066e8	−0.325361
$430$	−598327.	−0.000362910 0
$431$	−6.07719e8	−0.365622	−0.182811	−	0.983148i	$-0.558520\pi$
$431$	−6.07719e8	−0.365622	−0.182811	+	0.983148i	$0.558520\pi$
$432$	8.06216e7	0.0481125
$433$	1.70910e9	1.01172	0.505860	−	0.862615i	$-0.331175\pi$
$433$	1.70910e9	1.01172	0.505860	+	0.862615i	$0.331175\pi$
$434$	1.90268e9	1.11725
$435$	−6.70272e6	−0.00390426
$436$	−1.20882e9	−0.698486
$437$	−4.99771e8	−0.286474
$438$	5.37662e8	0.305738
$439$	−1.60133e9	−0.903349	−0.451674	−	0.892183i	$-0.649173\pi$
$439$	−1.60133e9	−0.903349	−0.451674	+	0.892183i	$0.649173\pi$
$440$	1.06518e7	0.00596125
$441$	8.72120e7	0.0484218
$442$	−1.59102e9	−0.876390
$443$	−2.60952e9	−1.42609	−0.713045	−	0.701118i	$-0.752685\pi$
$443$	−2.60952e9	−1.42609	−0.713045	+	0.701118i	$0.752685\pi$
$444$	8.99216e8	0.487555
$445$	−5.43218e6	−0.00292223
$446$	1.73158e9	0.924210
$447$	−1.63114e8	−0.0863801
$448$	2.54587e8	0.133771
$449$	3.28207e8	0.171114	0.0855571	−	0.996333i	$-0.472733\pi$
$449$	3.28207e8	0.171114	0.0855571	+	0.996333i	$0.472733\pi$
$450$	−4.55034e8	−0.235397
$451$	−2.55008e8	−0.130899
$452$	−6.07791e8	−0.309578
$453$	−4.67833e8	−0.236454
$454$	−1.69363e9	−0.849419
$455$	−9.32264e7	−0.0463980
$456$	5.67834e8	0.280443
$457$	1.35864e9	0.665884	0.332942	−	0.942947i	$-0.391959\pi$
$457$	1.35864e9	0.665884	0.332942	+	0.942947i	$0.391959\pi$
$458$	−1.13519e8	−0.0552126
$459$	−4.10516e8	−0.198146
$460$	7.83898e6	0.00375498
$461$	1.49808e9	0.712168	0.356084	−	0.934454i	$-0.384112\pi$
$461$	1.49808e9	0.712168	0.356084	+	0.934454i	$0.384112\pi$
$462$	−4.33516e8	−0.204531
$463$	−3.81745e9	−1.78748	−0.893738	−	0.448590i	$-0.851927\pi$
$463$	−3.81745e9	−1.78748	−0.893738	+	0.448590i	$0.851927\pi$
$464$	1.01007e8	0.0469394
$465$	−6.65641e7	−0.0307012
$466$	2.37942e9	1.08923
$467$	2.11226e9	0.959706	0.479853	−	0.877349i	$-0.340690\pi$
$467$	2.11226e9	0.959706	0.479853	+	0.877349i	$0.340690\pi$
$468$	4.44891e8	0.200629
$469$	3.10400e9	1.38937
$470$	1.11243e8	0.0494231
$471$	−9.66771e8	−0.426334
$472$	1.50309e9	0.657940
$473$	−1.53535e7	−0.00667105
$474$	−1.03702e9	−0.447262
$475$	−3.20489e9	−1.37210
$476$	−1.29633e9	−0.550923
$477$	7.38124e8	0.311397
$478$	2.15434e9	0.902228
$479$	3.66057e9	1.52186	0.760929	−	0.648836i	$-0.224744\pi$
$479$	3.66057e9	1.52186	0.760929	+	0.648836i	$0.224744\pi$
$480$	−8.90656e6	−0.00367592
$481$	4.96211e9	2.03310
$482$	2.45084e8	0.0996896
$483$	−3.19039e8	−0.128833
$484$	−9.73847e8	−0.390420
$485$	1.05321e8	0.0419198
$486$	1.14791e8	0.0453609
$487$	−1.88367e9	−0.739014	−0.369507	−	0.929228i	$-0.620473\pi$
$487$	−1.88367e9	−0.739014	−0.369507	+	0.929228i	$0.620473\pi$
$488$	−1.20908e9	−0.470962
$489$	−2.27325e9	−0.879157
$490$	−9.63463e6	−0.00369955
$491$	−1.56036e8	−0.0594892	−0.0297446	−	0.999558i	$-0.509469\pi$
$491$	−1.56036e8	−0.0594892	−0.0297446	+	0.999558i	$0.509469\pi$
$492$	2.13227e8	0.0807167
$493$	−5.14316e8	−0.193315
$494$	3.13346e9	1.16944
$495$	1.51663e7	0.00562032
$496$	1.00309e9	0.369109
$497$	−3.39808e9	−1.24161
$498$	1.67939e9	0.609327
$499$	−2.59287e9	−0.934174	−0.467087	−	0.884211i	$-0.654697\pi$
$499$	−2.59287e9	−0.934174	−0.467087	+	0.884211i	$0.654697\pi$
$500$	1.00604e8	0.0359931
$501$	−3.12279e9	−1.10946
$502$	−1.87303e9	−0.660819
$503$	1.41030e8	0.0494109	0.0247055	−	0.999695i	$-0.492135\pi$
$503$	1.41030e8	0.0494109	0.0247055	+	0.999695i	$0.492135\pi$
$504$	3.62488e8	0.126121
$505$	4.92430e7	0.0170147
$506$	2.01154e8	0.0690243
$507$	7.60820e8	0.259271
$508$	−1.05734e9	−0.357842
$509$	−1.57066e9	−0.527924	−0.263962	−	0.964533i	$-0.585029\pi$
$509$	−1.57066e9	−0.527924	−0.263962	+	0.964533i	$0.585029\pi$
$510$	4.53512e7	0.0151389
$511$	2.41742e9	0.801454
$512$	1.34218e8	0.0441942
$513$	8.08498e8	0.264404
$514$	3.10328e9	1.00798
$515$	−7.03947e7	−0.0227099
$516$	1.28380e7	0.00411360
$517$	2.85457e9	0.908499
$518$	4.04302e9	1.27806
$519$	−7.98529e8	−0.250729
$520$	−4.91488e7	−0.0153286
$521$	1.08322e9	0.335572	0.167786	−	0.985823i	$-0.446338\pi$
$521$	1.08322e9	0.335572	0.167786	+	0.985823i	$0.446338\pi$
$522$	1.43816e8	0.0442549
$523$	4.46212e9	1.36391	0.681954	−	0.731395i	$-0.261130\pi$
$523$	4.46212e9	1.36391	0.681954	+	0.731395i	$0.261130\pi$
$524$	7.40421e8	0.224812
$525$	−2.04591e9	−0.617062
$526$	−2.44282e9	−0.731881
$527$	−5.10762e9	−1.52013
$528$	−2.28549e8	−0.0675710
$529$	1.48036e8	0.0434783
$530$	−8.15433e7	−0.0237915
$531$	2.14013e9	0.620312
$532$	2.55308e9	0.735145
$533$	1.17664e9	0.336588
$534$	1.16555e8	0.0331236
$535$	−1.27240e8	−0.0359241
$536$	1.63642e9	0.459007
$537$	−2.14040e9	−0.596465
$538$	−1.84227e9	−0.510053
$539$	−2.47232e8	−0.0680054
$540$	−1.26814e7	−0.00346569
$541$	3.63866e9	0.987985	0.493993	−	0.869466i	$-0.335537\pi$
$541$	3.63866e9	0.987985	0.493993	+	0.869466i	$0.335537\pi$
$542$	−3.11723e8	−0.0840952
$543$	2.60146e9	0.697297
$544$	−6.83422e8	−0.182009
$545$	1.90142e8	0.0503141
$546$	2.00031e9	0.525923
$547$	2.28626e9	0.597269	0.298635	−	0.954368i	$-0.403469\pi$
$547$	2.28626e9	0.597269	0.298635	+	0.954368i	$0.403469\pi$
$548$	−3.09330e9	−0.802952
$549$	−1.72152e9	−0.444027
$550$	1.28995e9	0.330600
$551$	1.01293e9	0.257957
$552$	−1.68197e8	−0.0425628
$553$	−4.66261e9	−1.17244
$554$	2.70199e9	0.675149
$555$	−1.41443e8	−0.0351200
$556$	−2.11317e9	−0.521401
$557$	−2.29871e9	−0.563626	−0.281813	−	0.959469i	$-0.590936\pi$
$557$	−2.29871e9	−0.563626	−0.281813	+	0.959469i	$0.590936\pi$
$558$	1.42823e9	0.347999
$559$	7.08433e7	0.0171537
$560$	−4.00454e7	−0.00963595
$561$	1.16375e9	0.278284
$562$	3.46140e9	0.822573
$563$	3.55963e8	0.0840669	0.0420334	−	0.999116i	$-0.486616\pi$
$563$	3.55963e8	0.0840669	0.0420334	+	0.999116i	$0.486616\pi$
$564$	−2.38687e9	−0.560212
$565$	9.56029e7	0.0222998
$566$	2.11672e9	0.490688
$567$	5.16121e8	0.118908
$568$	−1.79146e9	−0.410193
$569$	6.89772e9	1.56968	0.784842	−	0.619696i	$-0.212744\pi$
$569$	6.89772e9	1.56968	0.784842	+	0.619696i	$0.212744\pi$
$570$	−8.93177e7	−0.0202011
$571$	9.04590e8	0.203341	0.101671	−	0.994818i	$-0.467581\pi$
$571$	9.04590e8	0.203341	0.101671	+	0.994818i	$0.467581\pi$
$572$	−1.26119e9	−0.281771
$573$	−3.66135e9	−0.813018
$574$	9.58703e8	0.211589
$575$	9.49314e8	0.208244
$576$	1.91103e8	0.0416667
$577$	−3.27571e9	−0.709889	−0.354945	−	0.934887i	$-0.615500\pi$
$577$	−3.27571e9	−0.709889	−0.354945	+	0.934887i	$0.615500\pi$
$578$	1.97198e8	0.0424772
$579$	−4.74488e9	−1.01590
$580$	−1.58879e7	−0.00338119
$581$	7.55084e9	1.59727
$582$	−2.25982e9	−0.475163
$583$	−2.09246e9	−0.437338
$584$	1.27446e9	0.264777
$585$	−6.99794e7	−0.0144519
$586$	−8.29442e8	−0.170272
$587$	7.50283e8	0.153106	0.0765529	−	0.997066i	$-0.475609\pi$
$587$	7.50283e8	0.153106	0.0765529	+	0.997066i	$0.475609\pi$
$588$	2.06725e8	0.0419345
$589$	1.00593e10	2.02845
$590$	−2.36429e8	−0.0473934
$591$	−3.30832e9	−0.659252
$592$	2.13147e9	0.422235
$593$	1.36947e9	0.269688	0.134844	−	0.990867i	$-0.456947\pi$
$593$	1.36947e9	0.269688	0.134844	+	0.990867i	$0.456947\pi$
$594$	−3.25414e8	−0.0637066
$595$	2.03907e8	0.0396846
$596$	−3.86640e8	−0.0748074
$597$	5.80361e9	1.11632
$598$	−9.28154e8	−0.177487
$599$	−5.06032e9	−0.962020	−0.481010	−	0.876715i	$-0.659730\pi$
$599$	−5.06032e9	−0.962020	−0.481010	+	0.876715i	$0.659730\pi$
$600$	−1.07860e9	−0.203859
$601$	−9.68350e9	−1.81958	−0.909791	−	0.415067i	$-0.863758\pi$
$601$	−9.68350e9	−1.81958	−0.909791	+	0.415067i	$0.863758\pi$
$602$	5.77216e7	0.0107833
$603$	2.32999e9	0.432756
$604$	−1.10894e9	−0.204775
$605$	1.53182e8	0.0281231
$606$	−1.05658e9	−0.192862
$607$	−5.82449e9	−1.05705	−0.528527	−	0.848916i	$-0.677256\pi$
$607$	−5.82449e9	−1.05705	−0.528527	+	0.848916i	$0.677256\pi$
$608$	1.34598e9	0.242871
$609$	6.46623e8	0.116009
$610$	1.90183e8	0.0339248
$611$	−1.31714e10	−2.33608
$612$	−9.73075e8	−0.171600
$613$	−2.39156e9	−0.419343	−0.209671	−	0.977772i	$-0.567239\pi$
$613$	−2.39156e9	−0.419343	−0.209671	+	0.977772i	$0.567239\pi$
$614$	−2.41053e9	−0.420265
$615$	−3.35396e7	−0.00581427
$616$	−1.02759e9	−0.177129
$617$	−6.83517e9	−1.17152	−0.585762	−	0.810483i	$-0.699205\pi$
$617$	−6.83517e9	−1.17152	−0.585762	+	0.810483i	$0.699205\pi$
$618$	1.51042e9	0.257417
$619$	−8.54071e9	−1.44736	−0.723680	−	0.690135i	$-0.757551\pi$
$619$	−8.54071e9	−1.44736	−0.723680	+	0.690135i	$0.757551\pi$
$620$	−1.57782e8	−0.0265880
$621$	−2.39483e8	−0.0401286
$622$	5.08707e9	0.847621
$623$	5.24051e8	0.0868292
$624$	1.05456e9	0.173750
$625$	6.07977e9	0.996110
$626$	−2.12360e9	−0.345988
$627$	−2.29196e9	−0.371339
$628$	−2.29161e9	−0.369216
$629$	−1.08532e10	−1.73893
$630$	−5.70177e7	−0.00908486
$631$	1.16903e10	1.85236	0.926178	−	0.377086i	$-0.123074\pi$
$631$	1.16903e10	1.85236	0.926178	+	0.377086i	$0.123074\pi$
$632$	−2.45812e9	−0.387341
$633$	6.21601e9	0.974089
$634$	3.26267e9	0.508465
$635$	1.66315e8	0.0257764
$636$	1.74963e9	0.269678
$637$	1.14076e9	0.174867
$638$	−4.07696e8	−0.0621533
$639$	−2.55073e9	−0.386733
$640$	−2.11118e7	−0.00318344
$641$	−4.35893e9	−0.653697	−0.326849	−	0.945077i	$-0.605987\pi$
$641$	−4.35893e9	−0.653697	−0.326849	+	0.945077i	$0.605987\pi$
$642$	2.73012e9	0.407201
$643$	3.50189e9	0.519474	0.259737	−	0.965679i	$-0.416364\pi$
$643$	3.50189e9	0.519474	0.259737	+	0.965679i	$0.416364\pi$
$644$	−7.56240e8	−0.111573
$645$	−2.01935e6	−0.000296315 0
$646$	−6.85356e9	−1.00024
$647$	−4.71652e9	−0.684632	−0.342316	−	0.939585i	$-0.611211\pi$
$647$	−4.71652e9	−0.684632	−0.342316	+	0.939585i	$0.611211\pi$
$648$	2.72098e8	0.0392837
$649$	−6.06693e9	−0.871189
$650$	−5.95200e9	−0.850092
$651$	6.42155e9	0.912235
$652$	−5.38845e9	−0.761373
$653$	−2.98375e9	−0.419340	−0.209670	−	0.977772i	$-0.567239\pi$
$653$	−2.98375e9	−0.419340	−0.209670	+	0.977772i	$0.567239\pi$
$654$	−4.07976e9	−0.570312
$655$	−1.16465e8	−0.0161939
$656$	5.05426e8	0.0699027
$657$	1.81461e9	0.249634
$658$	−1.07318e10	−1.46853
$659$	−5.13716e8	−0.0699237	−0.0349619	−	0.999389i	$-0.511131\pi$
$659$	−5.13716e8	−0.0699237	−0.0349619	+	0.999389i	$0.511131\pi$
$660$	3.59497e7	0.00486734
$661$	−2.64445e9	−0.356148	−0.178074	−	0.984017i	$-0.556987\pi$
$661$	−2.64445e9	−0.356148	−0.178074	+	0.984017i	$0.556987\pi$
$662$	−2.18128e9	−0.292219
$663$	−5.36969e9	−0.715569
$664$	3.98079e9	0.527692
$665$	−4.01587e8	−0.0529547
$666$	3.03485e9	0.398087
$667$	−3.00037e8	−0.0391502
$668$	−7.40216e9	−0.960817
$669$	5.84408e9	0.754614
$670$	−2.57402e8	−0.0330636
$671$	4.88024e9	0.623608
$672$	8.59231e8	0.109224
$673$	8.08792e9	1.02278	0.511392	−	0.859348i	$-0.329130\pi$
$673$	8.08792e9	1.02278	0.511392	+	0.859348i	$0.329130\pi$
$674$	4.04514e9	0.508890
$675$	−1.53574e9	−0.192200
$676$	1.80343e9	0.224535
$677$	9.80601e9	1.21460	0.607298	−	0.794474i	$-0.292253\pi$
$677$	9.80601e9	1.21460	0.607298	+	0.794474i	$0.292253\pi$
$678$	−2.05130e9	−0.252769
$679$	−1.01605e10	−1.24558
$680$	1.07499e8	0.0131106
$681$	−5.71599e9	−0.693548
$682$	−4.04879e9	−0.488743
$683$	1.16162e10	1.39505	0.697526	−	0.716559i	$-0.254284\pi$
$683$	1.16162e10	1.39505	0.697526	+	0.716559i	$0.254284\pi$
$684$	1.91644e9	0.228981
$685$	4.86562e8	0.0578391
$686$	−5.46895e9	−0.646799
$687$	−3.83126e8	−0.0450809
$688$	3.04307e7	0.00356248
$689$	9.65491e9	1.12456
$690$	2.64566e7	0.00306593
$691$	1.04349e9	0.120313	0.0601566	−	0.998189i	$-0.480840\pi$
$691$	1.04349e9	0.120313	0.0601566	+	0.998189i	$0.480840\pi$
$692$	−1.89281e9	−0.217138
$693$	−1.46312e9	−0.166999
$694$	−8.42428e9	−0.956698
$695$	3.32392e8	0.0375581
$696$	3.40898e8	0.0383259
$697$	−2.57357e9	−0.287887
$698$	−1.00880e9	−0.112283
$699$	8.03055e9	0.889355
$700$	−4.84956e9	−0.534391
$701$	−3.23469e9	−0.354666	−0.177333	−	0.984151i	$-0.556747\pi$
$701$	−3.23469e9	−0.354666	−0.177333	+	0.984151i	$0.556747\pi$
$702$	1.50151e9	0.163813
$703$	2.13751e10	2.32041
$704$	−5.41746e8	−0.0585182
$705$	3.75445e8	0.0403538
$706$	−2.67375e9	−0.285959
$707$	−4.75056e9	−0.505564
$708$	5.07291e9	0.537206
$709$	1.27054e10	1.33883	0.669416	−	0.742888i	$-0.266544\pi$
$709$	1.27054e10	1.33883	0.669416	+	0.742888i	$0.266544\pi$
$710$	2.81789e8	0.0295474
$711$	−3.49994e9	−0.365188
$712$	2.76279e8	0.0286858
$713$	−2.97964e9	−0.307858
$714$	−4.37511e9	−0.449827
$715$	1.98380e8	0.0202968
$716$	−5.07354e9	−0.516554
$717$	7.27089e9	0.736666
$718$	4.29683e9	0.433224
$719$	−1.09711e10	−1.10077	−0.550387	−	0.834910i	$-0.685520\pi$
$719$	−1.09711e10	−1.10077	−0.550387	+	0.834910i	$0.685520\pi$
$720$	−3.00596e7	−0.00300137
$721$	6.79110e9	0.674787
$722$	6.34689e9	0.627596
$723$	8.27158e8	0.0813962
$724$	6.16643e9	0.603877
$725$	−1.92405e9	−0.187514
$726$	−3.28673e9	−0.318776
$727$	−1.65278e10	−1.59531	−0.797654	−	0.603116i	$-0.793926\pi$
$727$	−1.65278e10	−1.59531	−0.797654	+	0.603116i	$0.793926\pi$
$728$	4.74147e9	0.455463
$729$	3.87420e8	0.0370370
$730$	−2.00466e8	−0.0190727
$731$	−1.54950e8	−0.0146717
$732$	−4.08065e9	−0.384539
$733$	−1.79161e9	−0.168028	−0.0840138	−	0.996465i	$-0.526774\pi$
$733$	−1.79161e9	−0.168028	−0.0840138	+	0.996465i	$0.526774\pi$
$734$	−2.29071e9	−0.213813
$735$	−3.25169e7	−0.00302067
$736$	−3.98688e8	−0.0368605
$737$	−6.60513e9	−0.607778
$738$	7.19640e8	0.0659049
$739$	−1.29335e10	−1.17886	−0.589429	−	0.807820i	$-0.700647\pi$
$739$	−1.29335e10	−1.17886	−0.589429	+	0.807820i	$0.700647\pi$
$740$	−3.35271e8	−0.0304148
$741$	1.05754e10	0.954848
$742$	7.86662e9	0.706927
$743$	−1.98134e10	−1.77214	−0.886069	−	0.463554i	$-0.846574\pi$
$743$	−1.98134e10	−1.77214	−0.886069	+	0.463554i	$0.846574\pi$
$744$	3.38543e9	0.301376
$745$	6.08167e7	0.00538860
$746$	9.33560e9	0.823297
$747$	5.66796e9	0.497513
$748$	2.75851e9	0.241001
$749$	1.22751e10	1.06743
$750$	3.39538e8	0.0293883
$751$	1.43776e10	1.23864	0.619322	−	0.785137i	$-0.287408\pi$
$751$	1.43776e10	1.23864	0.619322	+	0.785137i	$0.287408\pi$
$752$	−5.65778e9	−0.485158
$753$	−6.32149e9	−0.539556
$754$	1.88117e9	0.159819
$755$	1.74431e8	0.0147506
$756$	1.22340e9	0.102977
$757$	1.22101e10	1.02302	0.511508	−	0.859278i	$-0.329087\pi$
$757$	1.22101e10	1.02302	0.511508	+	0.859278i	$0.329087\pi$
$758$	−5.62194e9	−0.468861
$759$	6.78895e8	0.0563581
$760$	−2.11716e8	−0.0174947
$761$	−6.00939e8	−0.0494293	−0.0247146	−	0.999695i	$-0.507868\pi$
$761$	−6.00939e8	−0.0494293	−0.0247146	+	0.999695i	$0.507868\pi$
$762$	−3.56852e9	−0.292177
$763$	−1.83433e10	−1.49500
$764$	−8.67875e9	−0.704094
$765$	1.53060e8	0.0123608
$766$	−3.82373e9	−0.307388
$767$	2.79937e10	2.24015
$768$	4.52985e8	0.0360844
$769$	1.27908e10	1.01427	0.507137	−	0.861866i	$-0.330704\pi$
$769$	1.27908e10	1.01427	0.507137	+	0.861866i	$0.330704\pi$
$770$	1.61636e8	0.0127591
$771$	1.04736e10	0.823009
$772$	−1.12471e10	−0.879795
$773$	−5.32783e9	−0.414880	−0.207440	−	0.978248i	$-0.566513\pi$
$773$	−5.32783e9	−0.414880	−0.207440	+	0.978248i	$0.566513\pi$
$774$	4.33281e7	0.00335874
$775$	−1.91076e10	−1.47452
$776$	−5.35660e9	−0.411503
$777$	1.36452e10	1.04353
$778$	1.31268e10	0.999380
$779$	5.06857e9	0.384153
$780$	−1.65877e8	−0.0125157
$781$	7.23090e9	0.543143
$782$	2.03008e9	0.151806
$783$	4.85380e8	0.0361340
$784$	4.90014e8	0.0363164
$785$	3.60459e8	0.0265958
$786$	2.49892e9	0.183558
$787$	−1.10076e9	−0.0804971	−0.0402485	−	0.999190i	$-0.512815\pi$
$787$	−1.10076e9	−0.0804971	−0.0402485	+	0.999190i	$0.512815\pi$
$788$	−7.84194e9	−0.570929
$789$	−8.24451e9	−0.597579
$790$	3.86651e8	0.0279013
$791$	−9.22297e9	−0.662603
$792$	−7.71352e8	−0.0551715
$793$	−2.25181e10	−1.60353
$794$	7.53729e9	0.534372
$795$	−2.75209e8	−0.0194257
$796$	1.37567e10	0.966760
$797$	1.41736e10	0.991694	0.495847	−	0.868410i	$-0.334858\pi$
$797$	1.41736e10	0.991694	0.495847	+	0.868410i	$0.334858\pi$
$798$	8.61663e9	0.600243
$799$	2.88088e10	1.99807
$800$	−2.55668e9	−0.176547
$801$	3.93374e8	0.0270453
$802$	1.19323e9	0.0816795
$803$	−5.14412e9	−0.350595
$804$	5.52293e9	0.374777
$805$	1.18953e8	0.00803694
$806$	1.86817e10	1.25674
$807$	−6.21766e9	−0.416457
$808$	−2.50448e9	−0.167024
$809$	4.00127e9	0.265692	0.132846	−	0.991137i	$-0.457588\pi$
$809$	4.00127e9	0.265692	0.132846	+	0.991137i	$0.457588\pi$
$810$	−4.27998e7	−0.00282972
$811$	−2.05460e10	−1.35255	−0.676275	−	0.736649i	$-0.736407\pi$
$811$	−2.05460e10	−1.35255	−0.676275	+	0.736649i	$0.736407\pi$
$812$	1.53273e9	0.100466
$813$	−1.05207e9	−0.0686635
$814$	−8.60331e9	−0.559088
$815$	8.47579e8	0.0548439
$816$	−2.30655e9	−0.148610
$817$	3.05169e8	0.0195778
$818$	−9.21081e9	−0.588385
$819$	6.75103e9	0.429414
$820$	−7.95013e7	−0.00503530
$821$	1.60430e10	1.01177	0.505887	−	0.862600i	$-0.331165\pi$
$821$	1.60430e10	1.01177	0.505887	+	0.862600i	$0.331165\pi$
$822$	−1.04399e10	−0.655608
$823$	2.08789e10	1.30560	0.652798	−	0.757532i	$-0.273595\pi$
$823$	2.08789e10	1.30560	0.652798	+	0.757532i	$0.273595\pi$
$824$	3.58025e9	0.222930
$825$	4.35357e9	0.269933
$826$	2.28087e10	1.40822
$827$	1.50170e9	0.0923241	0.0461621	−	0.998934i	$-0.485301\pi$
$827$	1.50170e9	0.0923241	0.0461621	+	0.998934i	$0.485301\pi$
$828$	−5.67664e8	−0.0347524
$829$	2.73315e10	1.66618	0.833090	−	0.553137i	$-0.186570\pi$
$829$	2.73315e10	1.66618	0.833090	+	0.553137i	$0.186570\pi$
$830$	−6.26160e8	−0.0380112
$831$	9.11921e9	0.551256
$832$	2.49969e9	0.150472
$833$	−2.49510e9	−0.149565
$834$	−7.13194e9	−0.425722
$835$	1.16433e9	0.0692105
$836$	−5.43279e9	−0.321589
$837$	4.82027e9	0.284140
$838$	3.15969e9	0.185477
$839$	2.62962e10	1.53718	0.768591	−	0.639740i	$-0.220958\pi$
$839$	2.62962e10	1.53718	0.768591	+	0.639740i	$0.220958\pi$
$840$	−1.35153e8	−0.00786772
$841$	−1.66418e10	−0.964747
$842$	−2.65036e9	−0.153008
$843$	1.16822e10	0.671628
$844$	1.47342e10	0.843586
$845$	−2.83671e8	−0.0161740
$846$	−8.05570e9	−0.457411
$847$	−1.47777e10	−0.835632
$848$	4.14727e9	0.233548
$849$	7.14393e9	0.400645
$850$	1.30183e10	0.727091
$851$	−6.33146e9	−0.352168
$852$	−6.04618e9	−0.334921
$853$	−1.09100e10	−0.601871	−0.300936	−	0.953644i	$-0.597299\pi$
$853$	−1.09100e10	−0.601871	−0.300936	+	0.953644i	$0.597299\pi$
$854$	−1.83473e10	−1.00802
$855$	−3.01447e8	−0.0164941
$856$	6.47139e9	0.352646
$857$	1.11091e10	0.602900	0.301450	−	0.953482i	$-0.402529\pi$
$857$	1.11091e10	0.602900	0.301450	+	0.953482i	$0.402529\pi$
$858$	−4.25653e9	−0.230065
$859$	−1.92427e10	−1.03584	−0.517918	−	0.855430i	$-0.673293\pi$
$859$	−1.92427e10	−1.03584	−0.517918	+	0.855430i	$0.673293\pi$
$860$	−4.78662e6	−0.000256616 0
$861$	3.23562e9	0.172761
$862$	−4.86175e9	−0.258534
$863$	1.41798e10	0.750985	0.375493	−	0.926825i	$-0.377474\pi$
$863$	1.41798e10	0.750985	0.375493	+	0.926825i	$0.377474\pi$
$864$	6.44973e8	0.0340207
$865$	2.97730e8	0.0156411
$866$	1.36728e10	0.715395
$867$	6.65544e8	0.0346825
$868$	1.52215e10	0.790018
$869$	9.92175e9	0.512884
$870$	−5.36217e7	−0.00276073
$871$	3.04770e10	1.56282
$872$	−9.67054e9	−0.493905
$873$	−7.62688e9	−0.387969
$874$	−3.99817e9	−0.202568
$875$	1.52662e9	0.0770376
$876$	4.30129e9	0.216190
$877$	2.27875e10	1.14077	0.570385	−	0.821377i	$-0.306794\pi$
$877$	2.27875e10	1.14077	0.570385	+	0.821377i	$0.306794\pi$
$878$	−1.28107e10	−0.638764
$879$	−2.79937e9	−0.139027
$880$	8.52141e7	0.00421524
$881$	−3.16523e10	−1.55952	−0.779758	−	0.626081i	$-0.784658\pi$
$881$	−3.16523e10	−1.55952	−0.779758	+	0.626081i	$0.784658\pi$
$882$	6.97696e8	0.0342394
$883$	−1.28325e10	−0.627263	−0.313631	−	0.949545i	$-0.601546\pi$
$883$	−1.28325e10	−0.627263	−0.313631	+	0.949545i	$0.601546\pi$
$884$	−1.27282e10	−0.619701
$885$	−7.97946e8	−0.0386966
$886$	−2.08761e10	−1.00840
$887$	6.07730e9	0.292400	0.146200	−	0.989255i	$-0.453296\pi$
$887$	6.07730e9	0.292400	0.146200	+	0.989255i	$0.453296\pi$
$888$	7.19373e9	0.344753
$889$	−1.60447e10	−0.765904
$890$	−4.34574e7	−0.00206633
$891$	−1.09827e9	−0.0520162
$892$	1.38526e10	0.653515
$893$	−5.67379e10	−2.66620
$894$	−1.30491e9	−0.0610800
$895$	7.98046e8	0.0372089
$896$	2.03670e9	0.0945906
$897$	−3.13252e9	−0.144917
$898$	2.62566e9	0.120996
$899$	6.03908e9	0.277212
$900$	−3.64027e9	−0.166450
$901$	−2.11174e10	−0.961843
$902$	−2.04006e9	−0.0925593
$903$	1.94811e8	0.00880451
$904$	−4.86233e9	−0.218905
$905$	−9.69952e8	−0.0434991
$906$	−3.74267e9	−0.167198
$907$	−6.33378e9	−0.281863	−0.140931	−	0.990019i	$-0.545010\pi$
$907$	−6.33378e9	−0.281863	−0.140931	+	0.990019i	$0.545010\pi$
$908$	−1.35490e10	−0.600630
$909$	−3.56595e9	−0.157472
$910$	−7.45811e8	−0.0328083
$911$	−4.31202e10	−1.88958	−0.944792	−	0.327670i	$-0.893736\pi$
$911$	−4.31202e10	−1.88958	−0.944792	+	0.327670i	$0.893736\pi$
$912$	4.54267e9	0.198303
$913$	−1.60677e10	−0.698726
$914$	1.08691e10	0.470851
$915$	6.41868e8	0.0276995
$916$	−9.08150e8	−0.0390412
$917$	1.12356e10	0.481174
$918$	−3.28413e9	−0.140111
$919$	2.66851e10	1.13414	0.567068	−	0.823671i	$-0.308078\pi$
$919$	2.66851e10	1.13414	0.567068	+	0.823671i	$0.308078\pi$
$920$	6.27119e7	0.00265517
$921$	−8.13554e9	−0.343145
$922$	1.19847e10	0.503579
$923$	−3.33644e10	−1.39662
$924$	−3.46813e9	−0.144625
$925$	−4.06019e10	−1.68675
$926$	−3.05396e10	−1.26394
$927$	5.09766e9	0.210180
$928$	8.08055e8	0.0331912
$929$	1.03414e10	0.423178	0.211589	−	0.977359i	$-0.432136\pi$
$929$	1.03414e10	0.423178	0.211589	+	0.977359i	$0.432136\pi$
$930$	−5.32513e8	−0.0217090
$931$	4.91401e9	0.199578
$932$	1.90354e10	0.770204
$933$	1.71689e10	0.692079
$934$	1.68981e10	0.678615
$935$	−4.33901e8	−0.0173600
$936$	3.55913e9	0.141866
$937$	2.29344e10	0.910748	0.455374	−	0.890300i	$-0.349506\pi$
$937$	2.29344e10	0.910748	0.455374	+	0.890300i	$0.349506\pi$
$938$	2.48320e10	0.982431
$939$	−7.16713e9	−0.282498
$940$	8.89943e8	0.0349474
$941$	5.44364e9	0.212973	0.106487	−	0.994314i	$-0.466040\pi$
$941$	5.44364e9	0.212973	0.106487	+	0.994314i	$0.466040\pi$
$942$	−7.73417e9	−0.301464
$943$	−1.50135e9	−0.0583029
$944$	1.20247e10	0.465234
$945$	−1.92435e8	−0.00741776
$946$	−1.22828e8	−0.00471714
$947$	−8.75674e9	−0.335056	−0.167528	−	0.985867i	$-0.553578\pi$
$947$	−8.75674e9	−0.335056	−0.167528	+	0.985867i	$0.553578\pi$
$948$	−8.29615e9	−0.316262
$949$	2.37357e10	0.901509
$950$	−2.56392e10	−0.970222
$951$	1.10115e10	0.415160
$952$	−1.03706e10	−0.389561
$953$	−3.41231e10	−1.27710	−0.638549	−	0.769582i	$-0.720465\pi$
$953$	−3.41231e10	−1.27710	−0.638549	+	0.769582i	$0.720465\pi$
$954$	5.90499e9	0.220191
$955$	1.36513e9	0.0507180
$956$	1.72347e10	0.637972
$957$	−1.37597e9	−0.0507479
$958$	2.92845e10	1.07612
$959$	−4.69395e10	−1.71859
$960$	−7.12525e7	−0.00259927
$961$	3.24610e10	1.17986
$962$	3.96969e10	1.43762
$963$	9.21415e9	0.332478
$964$	1.96067e9	0.0704912
$965$	1.76912e9	0.0633742
$966$	−2.55231e9	−0.0910990
$967$	−3.56029e10	−1.26617	−0.633087	−	0.774081i	$-0.718212\pi$
$967$	−3.56029e10	−1.26617	−0.633087	+	0.774081i	$0.718212\pi$
$968$	−7.79077e9	−0.276069
$969$	−2.31308e10	−0.816689
$970$	8.42569e8	0.0296418
$971$	4.01429e10	1.40715	0.703577	−	0.710619i	$-0.251585\pi$
$971$	4.01429e10	1.40715	0.703577	+	0.710619i	$0.251585\pi$
$972$	9.18330e8	0.0320750
$973$	−3.20664e10	−1.11598
$974$	−1.50693e10	−0.522561
$975$	−2.00880e10	−0.694097
$976$	−9.67265e9	−0.333021
$977$	3.55282e10	1.21883	0.609413	−	0.792853i	$-0.291405\pi$
$977$	3.55282e10	1.21883	0.609413	+	0.792853i	$0.291405\pi$
$978$	−1.81860e10	−0.621658
$979$	−1.11515e9	−0.0379834
$980$	−7.70770e7	−0.00261597
$981$	−1.37692e10	−0.465658
$982$	−1.24828e9	−0.0420652
$983$	−2.12262e10	−0.712746	−0.356373	−	0.934344i	$-0.615987\pi$
$983$	−2.12262e10	−0.712746	−0.356373	+	0.934344i	$0.615987\pi$
$984$	1.70581e9	0.0570754
$985$	1.23350e9	0.0411257
$986$	−4.11453e9	−0.136694
$987$	−3.62198e10	−1.19905
$988$	2.50677e10	0.826922
$989$	−9.03933e7	−0.00297132
$990$	1.21330e8	0.00397417
$991$	−2.77326e10	−0.905175	−0.452588	−	0.891720i	$-0.649499\pi$
$991$	−2.77326e10	−0.905175	−0.452588	+	0.891720i	$0.649499\pi$
$992$	8.02472e9	0.260999
$993$	−7.36183e9	−0.238596
$994$	−2.71846e10	−0.877952
$995$	−2.16387e9	−0.0696386
$996$	1.34352e10	0.430859
$997$	−4.52372e10	−1.44565	−0.722823	−	0.691033i	$-0.757156\pi$
$997$	−4.52372e10	−1.44565	−0.722823	+	0.691033i	$0.757156\pi$
$998$	−2.07429e10	−0.660561
$999$	1.02426e10	0.325037

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.8.a.h.1.2	✓	4
3.2	odd	2		414.8.a.i.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.h.1.2	✓	4	1.1	even	1	trivial
414.8.a.i.1.3		4	3.2	odd	2

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -10.0669 q^{5} +216.000 q^{6} +971.172 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -10.0669 q^{5} +216.000 q^{6} +971.172 q^{7} +512.000 q^{8} +729.000 q^{9} -80.5353 q^{10} -2066.60 q^{11} +1728.00 q^{12} +9535.57 q^{13} +7769.38 q^{14} -271.807 q^{15} +4096.00 q^{16} -20856.4 q^{17} +5832.00 q^{18} +41075.9 q^{19} -644.282 q^{20} +26221.6 q^{21} -16532.8 q^{22} -12167.0 q^{23} +13824.0 q^{24} -78023.7 q^{25} +76284.5 q^{26} +19683.0 q^{27} +62155.0 q^{28} +24659.9 q^{29} -2174.45 q^{30} +244895. q^{31} +32768.0 q^{32} -55798.1 q^{33} -166851. q^{34} -9776.71 q^{35} +46656.0 q^{36} +520379. q^{37} +328607. q^{38} +257460. q^{39} -5154.26 q^{40} +123395. q^{41} +209773. q^{42} +7429.38 q^{43} -132262. q^{44} -7338.78 q^{45} -97336.0 q^{46} -1.38129e6 q^{47} +110592. q^{48} +119632. q^{49} -624189. q^{50} -563122. q^{51} +610276. q^{52} +1.01252e6 q^{53} +157464. q^{54} +20804.2 q^{55} +497240. q^{56} +1.10905e6 q^{57} +197279. q^{58} +2.93571e6 q^{59} -17395.6 q^{60} -2.36149e6 q^{61} +1.95916e6 q^{62} +707984. q^{63} +262144. q^{64} -95993.7 q^{65} -446385. q^{66} +3.19614e6 q^{67} -1.33481e6 q^{68} -328509. q^{69} -78213.6 q^{70} -3.49894e6 q^{71} +373248. q^{72} +2.48917e6 q^{73} +4.16304e6 q^{74} -2.10664e6 q^{75} +2.62886e6 q^{76} -2.00702e6 q^{77} +2.05968e6 q^{78} -4.80101e6 q^{79} -41234.1 q^{80} +531441. q^{81} +987160. q^{82} +7.77498e6 q^{83} +1.67819e6 q^{84} +209959. q^{85} +59435.0 q^{86} +665817. q^{87} -1.05810e6 q^{88} +539607. q^{89} -58710.2 q^{90} +9.26068e6 q^{91} -778688. q^{92} +6.61217e6 q^{93} -1.10503e7 q^{94} -413508. q^{95} +884736. q^{96} -1.04621e7 q^{97} +957058. q^{98} -1.50655e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 + 270 * q^5 + 864 * q^6 + 2022 * q^7 + 2048 * q^8 + 2916 * q^9 + 2160 * q^10 + 4120 * q^11 + 6912 * q^12 + 8036 * q^13 + 16176 * q^14 + 7290 * q^15 + 16384 * q^16 + 37182 * q^17 + 23328 * q^18 + 5702 * q^19 + 17280 * q^20 + 54594 * q^21 + 32960 * q^22 - 48668 * q^23 + 55296 * q^24 + 121480 * q^25 + 64288 * q^26 + 78732 * q^27 + 129408 * q^28 + 217716 * q^29 + 58320 * q^30 + 222852 * q^31 + 131072 * q^32 + 111240 * q^33 + 297456 * q^34 + 68440 * q^35 + 186624 * q^36 + 486428 * q^37 + 45616 * q^38 + 216972 * q^39 + 138240 * q^40 + 338336 * q^41 + 436752 * q^42 + 730974 * q^43 + 263680 * q^44 + 196830 * q^45 - 389344 * q^46 + 338248 * q^47 + 442368 * q^48 - 310552 * q^49 + 971840 * q^50 + 1003914 * q^51 + 514304 * q^52 - 375502 * q^53 + 629856 * q^54 + 424840 * q^55 + 1035264 * q^56 + 153954 * q^57 + 1741728 * q^58 + 71392 * q^59 + 466560 * q^60 + 2101164 * q^61 + 1782816 * q^62 + 1474038 * q^63 + 1048576 * q^64 + 1578780 * q^65 + 889920 * q^66 + 4337162 * q^67 + 2379648 * q^68 - 1314036 * q^69 + 547520 * q^70 + 2288016 * q^71 + 1492992 * q^72 - 1107328 * q^73 + 3891424 * q^74 + 3279960 * q^75 + 364928 * q^76 + 5826200 * q^77 + 1735776 * q^78 + 60610 * q^79 + 1105920 * q^80 + 2125764 * q^81 + 2706688 * q^82 + 1485464 * q^83 + 3494016 * q^84 - 8843820 * q^85 + 5847792 * q^86 + 5878332 * q^87 + 2109440 * q^88 + 1485090 * q^89 + 1574640 * q^90 - 2898412 * q^91 - 3114752 * q^92 + 6017004 * q^93 + 2705984 * q^94 + 8545200 * q^95 + 3538944 * q^96 + 1935444 * q^97 - 2484416 * q^98 + 3003480 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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