LMFDB - Embedded newform 138.8.a.d.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 684x - 5052 $ x^3 - x^2 - 684*x - 5052
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.33366$ 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)

$f(q)$	$=$	$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -149.775 q^{5} -216.000 q^{6} -286.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -149.775 q^{5} -216.000 q^{6} -286.000 q^{7} +512.000 q^{8} +729.000 q^{9} -1198.20 q^{10} +2112.48 q^{11} -1728.00 q^{12} +12277.9 q^{13} -2288.00 q^{14} +4043.91 q^{15} +4096.00 q^{16} -13108.8 q^{17} +5832.00 q^{18} -35142.7 q^{19} -9585.57 q^{20} +7722.00 q^{21} +16899.8 q^{22} -12167.0 q^{23} -13824.0 q^{24} -55692.6 q^{25} +98223.2 q^{26} -19683.0 q^{27} -18304.0 q^{28} -163671. q^{29} +32351.3 q^{30} -3512.82 q^{31} +32768.0 q^{32} -57036.9 q^{33} -104870. q^{34} +42835.5 q^{35} +46656.0 q^{36} -361526. q^{37} -281142. q^{38} -331503. q^{39} -76684.6 q^{40} +311126. q^{41} +61776.0 q^{42} +990934. q^{43} +135198. q^{44} -109186. q^{45} -97336.0 q^{46} -1.24008e6 q^{47} -110592. q^{48} -741747. q^{49} -445541. q^{50} +353937. q^{51} +785786. q^{52} -1.58660e6 q^{53} -157464. q^{54} -316395. q^{55} -146432. q^{56} +948854. q^{57} -1.30937e6 q^{58} -1.85291e6 q^{59} +258811. q^{60} -1.18405e6 q^{61} -28102.6 q^{62} -208494. q^{63} +262144. q^{64} -1.83892e6 q^{65} -456295. q^{66} -3.77863e6 q^{67} -838962. q^{68} +328509. q^{69} +342684. q^{70} +5.94088e6 q^{71} +373248. q^{72} +5.29393e6 q^{73} -2.89221e6 q^{74} +1.50370e6 q^{75} -2.24913e6 q^{76} -604168. q^{77} -2.65203e6 q^{78} -8.12232e6 q^{79} -613477. q^{80} +531441. q^{81} +2.48901e6 q^{82} +4.49158e6 q^{83} +494208. q^{84} +1.96336e6 q^{85} +7.92747e6 q^{86} +4.41912e6 q^{87} +1.08159e6 q^{88} -5.91682e6 q^{89} -873485. q^{90} -3.51148e6 q^{91} -778688. q^{92} +94846.3 q^{93} -9.92064e6 q^{94} +5.26349e6 q^{95} -884736. q^{96} -931842. q^{97} -5.93398e6 q^{98} +1.54000e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9}+O(q^{10}) $ 3 * q + 24 * q^2 - 81 * q^3 + 192 * q^4 + 92 * q^5 - 648 * q^6 - 858 * q^7 + 1536 * q^8 + 2187 * q^9	\( 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9} + 736 q^{10} - 1820 q^{11} - 5184 q^{12} - 3698 q^{13} - 6864 q^{14} - 2484 q^{15} + 12288 q^{16} + 3164 q^{17} + 17496 q^{18} - 14234 q^{19} + 5888 q^{20} + 23166 q^{21} - 14560 q^{22} - 36501 q^{23} - 41472 q^{24} - 170119 q^{25} - 29584 q^{26} - 59049 q^{27} - 54912 q^{28} - 171914 q^{29} - 19872 q^{30} - 124972 q^{31} + 98304 q^{32} + 49140 q^{33} + 25312 q^{34} - 26312 q^{35} + 139968 q^{36} - 679074 q^{37} - 113872 q^{38} + 99846 q^{39} + 47104 q^{40} - 331362 q^{41} + 185328 q^{42} - 145922 q^{43} - 116480 q^{44} + 67068 q^{45} - 292008 q^{46} - 1824192 q^{47} - 331776 q^{48} - 2225241 q^{49} - 1360952 q^{50} - 85428 q^{51} - 236672 q^{52} - 3442448 q^{53} - 472392 q^{54} - 1548176 q^{55} - 439296 q^{56} + 384318 q^{57} - 1375312 q^{58} - 3147660 q^{59} - 158976 q^{60} - 3444530 q^{61} - 999776 q^{62} - 625482 q^{63} + 786432 q^{64} - 2527960 q^{65} + 393120 q^{66} - 921870 q^{67} + 202496 q^{68} + 985527 q^{69} - 210496 q^{70} + 2594520 q^{71} + 1119744 q^{72} - 540826 q^{73} - 5432592 q^{74} + 4593213 q^{75} - 910976 q^{76} + 520520 q^{77} + 798768 q^{78} + 316994 q^{79} + 376832 q^{80} + 1594323 q^{81} - 2650896 q^{82} + 4119316 q^{83} + 1482624 q^{84} - 382312 q^{85} - 1167376 q^{86} + 4641678 q^{87} - 931840 q^{88} + 1417176 q^{89} + 536544 q^{90} + 1057628 q^{91} - 2336064 q^{92} + 3374244 q^{93} - 14593536 q^{94} + 12543632 q^{95} - 2654208 q^{96} - 15089326 q^{97} - 17801928 q^{98} - 1326780 q^{99}+O(q^{100}) \) 3 * q + 24 * q^2 - 81 * q^3 + 192 * q^4 + 92 * q^5 - 648 * q^6 - 858 * q^7 + 1536 * q^8 + 2187 * q^9 + 736 * q^10 - 1820 * q^11 - 5184 * q^12 - 3698 * q^13 - 6864 * q^14 - 2484 * q^15 + 12288 * q^16 + 3164 * q^17 + 17496 * q^18 - 14234 * q^19 + 5888 * q^20 + 23166 * q^21 - 14560 * q^22 - 36501 * q^23 - 41472 * q^24 - 170119 * q^25 - 29584 * q^26 - 59049 * q^27 - 54912 * q^28 - 171914 * q^29 - 19872 * q^30 - 124972 * q^31 + 98304 * q^32 + 49140 * q^33 + 25312 * q^34 - 26312 * q^35 + 139968 * q^36 - 679074 * q^37 - 113872 * q^38 + 99846 * q^39 + 47104 * q^40 - 331362 * q^41 + 185328 * q^42 - 145922 * q^43 - 116480 * q^44 + 67068 * q^45 - 292008 * q^46 - 1824192 * q^47 - 331776 * q^48 - 2225241 * q^49 - 1360952 * q^50 - 85428 * q^51 - 236672 * q^52 - 3442448 * q^53 - 472392 * q^54 - 1548176 * q^55 - 439296 * q^56 + 384318 * q^57 - 1375312 * q^58 - 3147660 * q^59 - 158976 * q^60 - 3444530 * q^61 - 999776 * q^62 - 625482 * q^63 + 786432 * q^64 - 2527960 * q^65 + 393120 * q^66 - 921870 * q^67 + 202496 * q^68 + 985527 * q^69 - 210496 * q^70 + 2594520 * q^71 + 1119744 * q^72 - 540826 * q^73 - 5432592 * q^74 + 4593213 * q^75 - 910976 * q^76 + 520520 * q^77 + 798768 * q^78 + 316994 * q^79 + 376832 * q^80 + 1594323 * q^81 - 2650896 * q^82 + 4119316 * q^83 + 1482624 * q^84 - 382312 * q^85 - 1167376 * q^86 + 4641678 * q^87 - 931840 * q^88 + 1417176 * q^89 + 536544 * q^90 + 1057628 * q^91 - 2336064 * q^92 + 3374244 * q^93 - 14593536 * q^94 + 12543632 * q^95 - 2654208 * q^96 - 15089326 * q^97 - 17801928 * q^98 - 1326780 * 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
$5$	−149.775	−0.535850	−0.267925	−	0.963440i	$-0.586338\pi$
$5$	−149.775	−0.535850	−0.267925	+	0.963440i	$0.586338\pi$
$6$	−216.000	−0.408248
$7$	−286.000	−0.315154	−0.157577	−	0.987507i	$-0.550368\pi$
$7$	−286.000	−0.315154	−0.157577	+	0.987507i	$0.550368\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−1198.20	−0.378903
$11$	2112.48	0.478539	0.239270	−	0.970953i	$-0.423092\pi$
$11$	2112.48	0.478539	0.239270	+	0.970953i	$0.423092\pi$
$12$	−1728.00	−0.288675
$13$	12277.9	1.54997	0.774983	−	0.631982i	$-0.217758\pi$
$13$	12277.9	1.54997	0.774983	+	0.631982i	$0.217758\pi$
$14$	−2288.00	−0.222848
$15$	4043.91	0.309373
$16$	4096.00	0.250000
$17$	−13108.8	−0.647129	−0.323565	−	0.946206i	$-0.604881\pi$
$17$	−13108.8	−0.647129	−0.323565	+	0.946206i	$0.604881\pi$
$18$	5832.00	0.235702
$19$	−35142.7	−1.17543	−0.587716	−	0.809067i	$-0.699973\pi$
$19$	−35142.7	−1.17543	−0.587716	+	0.809067i	$0.699973\pi$
$20$	−9585.57	−0.267925
$21$	7722.00	0.181954
$22$	16899.8	0.338378
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	−13824.0	−0.204124
$25$	−55692.6	−0.712865
$26$	98223.2	1.09599
$27$	−19683.0	−0.192450
$28$	−18304.0	−0.157577
$29$	−163671.	−1.24617	−0.623087	−	0.782152i	$-0.714122\pi$
$29$	−163671.	−1.24617	−0.623087	+	0.782152i	$0.714122\pi$
$30$	32351.3	0.218760
$31$	−3512.82	−0.0211783	−0.0105891	−	0.999944i	$-0.503371\pi$
$31$	−3512.82	−0.0211783	−0.0105891	+	0.999944i	$0.503371\pi$
$32$	32768.0	0.176777
$33$	−57036.9	−0.276285
$34$	−104870.	−0.457590
$35$	42835.5	0.168875
$36$	46656.0	0.166667
$37$	−361526.	−1.17337	−0.586683	−	0.809817i	$-0.699567\pi$
$37$	−361526.	−1.17337	−0.586683	+	0.809817i	$0.699567\pi$
$38$	−281142.	−0.831157
$39$	−331503.	−0.894874
$40$	−76684.6	−0.189452
$41$	311126.	0.705005	0.352503	−	0.935811i	$-0.385331\pi$
$41$	311126.	0.705005	0.352503	+	0.935811i	$0.385331\pi$
$42$	61776.0	0.128661
$43$	990934.	1.90066	0.950331	−	0.311241i	$-0.100744\pi$
$43$	990934.	1.90066	0.950331	+	0.311241i	$0.100744\pi$
$44$	135198.	0.239270
$45$	−109186.	−0.178617
$46$	−97336.0	−0.147442
$47$	−1.24008e6	−1.74224	−0.871118	−	0.491073i	$-0.836605\pi$
$47$	−1.24008e6	−1.74224	−0.871118	+	0.491073i	$0.836605\pi$
$48$	−110592.	−0.144338
$49$	−741747.	−0.900678
$50$	−445541.	−0.504072
$51$	353937.	0.373620
$52$	785786.	0.774983
$53$	−1.58660e6	−1.46387	−0.731933	−	0.681376i	$-0.761382\pi$
$53$	−1.58660e6	−1.46387	−0.731933	+	0.681376i	$0.761382\pi$
$54$	−157464.	−0.136083
$55$	−316395.	−0.256425
$56$	−146432.	−0.111424
$57$	948854.	0.678636
$58$	−1.30937e6	−0.881179
$59$	−1.85291e6	−1.17455	−0.587275	−	0.809388i	$-0.699799\pi$
$59$	−1.85291e6	−1.17455	−0.587275	+	0.809388i	$0.699799\pi$
$60$	258811.	0.154687
$61$	−1.18405e6	−0.667908	−0.333954	−	0.942589i	$-0.608383\pi$
$61$	−1.18405e6	−0.667908	−0.333954	+	0.942589i	$0.608383\pi$
$62$	−28102.6	−0.0149753
$63$	−208494.	−0.105051
$64$	262144.	0.125000
$65$	−1.83892e6	−0.830549
$66$	−456295.	−0.195363
$67$	−3.77863e6	−1.53487	−0.767436	−	0.641125i	$-0.778468\pi$
$67$	−3.77863e6	−1.53487	−0.767436	+	0.641125i	$0.778468\pi$
$68$	−838962.	−0.323565
$69$	328509.	0.120386
$70$	342684.	0.119413
$71$	5.94088e6	1.96991	0.984955	−	0.172810i	$-0.0552847\pi$
$71$	5.94088e6	1.96991	0.984955	+	0.172810i	$0.0552847\pi$
$72$	373248.	0.117851
$73$	5.29393e6	1.59275	0.796376	−	0.604802i	$-0.206748\pi$
$73$	5.29393e6	1.59275	0.796376	+	0.604802i	$0.206748\pi$
$74$	−2.89221e6	−0.829695
$75$	1.50370e6	0.411573
$76$	−2.24913e6	−0.587716
$77$	−604168.	−0.150814
$78$	−2.65203e6	−0.632771
$79$	−8.12232e6	−1.85347	−0.926734	−	0.375718i	$-0.877396\pi$
$79$	−8.12232e6	−1.85347	−0.926734	+	0.375718i	$0.877396\pi$
$80$	−613477.	−0.133962
$81$	531441.	0.111111
$82$	2.48901e6	0.498514
$83$	4.49158e6	0.862235	0.431118	−	0.902296i	$-0.358119\pi$
$83$	4.49158e6	0.862235	0.431118	+	0.902296i	$0.358119\pi$
$84$	494208.	0.0909771
$85$	1.96336e6	0.346764
$86$	7.92747e6	1.34397
$87$	4.41912e6	0.719479
$88$	1.08159e6	0.169189
$89$	−5.91682e6	−0.889658	−0.444829	−	0.895616i	$-0.646736\pi$
$89$	−5.91682e6	−0.889658	−0.444829	+	0.895616i	$0.646736\pi$
$90$	−873485.	−0.126301
$91$	−3.51148e6	−0.488478
$92$	−778688.	−0.104257
$93$	94846.3	0.0122273
$94$	−9.92064e6	−1.23195
$95$	5.26349e6	0.629856
$96$	−884736.	−0.102062
$97$	−931842.	−0.103667	−0.0518336	−	0.998656i	$-0.516507\pi$
$97$	−931842.	−0.103667	−0.0518336	+	0.998656i	$0.516507\pi$
$98$	−5.93398e6	−0.636875
$99$	1.54000e6	0.159513
$100$	−3.56432e6	−0.356432
$101$	1.19084e6	0.115008	0.0575042	−	0.998345i	$-0.481686\pi$
$101$	1.19084e6	0.115008	0.0575042	+	0.998345i	$0.481686\pi$
$102$	2.83150e6	0.264189
$103$	−135644.	−0.0122312	−0.00611562	−	0.999981i	$-0.501947\pi$
$103$	−135644.	−0.0122312	−0.00611562	+	0.999981i	$0.501947\pi$
$104$	6.28628e6	0.547996
$105$	−1.15656e6	−0.0975002
$106$	−1.26928e7	−1.03511
$107$	−1.05271e7	−0.830742	−0.415371	−	0.909652i	$-0.636348\pi$
$107$	−1.05271e7	−0.830742	−0.415371	+	0.909652i	$0.636348\pi$
$108$	−1.25971e6	−0.0962250
$109$	1.92882e7	1.42659	0.713296	−	0.700863i	$-0.247202\pi$
$109$	1.92882e7	1.42659	0.713296	+	0.700863i	$0.247202\pi$
$110$	−2.53116e6	−0.181320
$111$	9.76120e6	0.677443
$112$	−1.17146e6	−0.0787885
$113$	−1.64526e7	−1.07265	−0.536326	−	0.844011i	$-0.680188\pi$
$113$	−1.64526e7	−1.07265	−0.536326	+	0.844011i	$0.680188\pi$
$114$	7.59083e6	0.479868
$115$	1.82231e6	0.111732
$116$	−1.04749e7	−0.623087
$117$	8.95059e6	0.516656
$118$	−1.48233e7	−0.830532
$119$	3.74911e6	0.203945
$120$	2.07048e6	0.109380
$121$	−1.50246e7	−0.771000
$122$	−9.47242e6	−0.472282
$123$	−8.40039e6	−0.407035
$124$	−224821.	−0.0105891
$125$	2.00425e7	0.917838
$126$	−1.66795e6	−0.0742825
$127$	6.91607e6	0.299603	0.149802	−	0.988716i	$-0.452136\pi$
$127$	6.91607e6	0.299603	0.149802	+	0.988716i	$0.452136\pi$
$128$	2.09715e6	0.0883883
$129$	−2.67552e7	−1.09735
$130$	−1.47113e7	−0.587287
$131$	−1.51215e6	−0.0587686	−0.0293843	−	0.999568i	$-0.509355\pi$
$131$	−1.51215e6	−0.0587686	−0.0293843	+	0.999568i	$0.509355\pi$
$132$	−3.65036e6	−0.138142
$133$	1.00508e7	0.370442
$134$	−3.02290e7	−1.08532
$135$	2.94801e6	0.103124
$136$	−6.71170e6	−0.228795
$137$	3.75097e7	1.24630	0.623148	−	0.782104i	$-0.285853\pi$
$137$	3.75097e7	1.24630	0.623148	+	0.782104i	$0.285853\pi$
$138$	2.62807e6	0.0851257
$139$	−3.89553e7	−1.23031	−0.615155	−	0.788406i	$-0.710907\pi$
$139$	−3.89553e7	−1.23031	−0.615155	+	0.788406i	$0.710907\pi$
$140$	2.74147e6	0.0844376
$141$	3.34822e7	1.00588
$142$	4.75270e7	1.39294
$143$	2.59368e7	0.741720
$144$	2.98598e6	0.0833333
$145$	2.45138e7	0.667763
$146$	4.23514e7	1.12625
$147$	2.00272e7	0.520007
$148$	−2.31377e7	−0.586683
$149$	−4.08660e7	−1.01207	−0.506035	−	0.862513i	$-0.668889\pi$
$149$	−4.08660e7	−1.01207	−0.506035	+	0.862513i	$0.668889\pi$
$150$	1.20296e7	0.291026
$151$	5.64209e7	1.33359	0.666793	−	0.745243i	$-0.267667\pi$
$151$	5.64209e7	1.33359	0.666793	+	0.745243i	$0.267667\pi$
$152$	−1.79931e7	−0.415578
$153$	−9.55630e6	−0.215710
$154$	−4.83335e6	−0.106641
$155$	526132.	0.0113484
$156$	−2.12162e7	−0.447437
$157$	5.51988e7	1.13836	0.569181	−	0.822212i	$-0.307260\pi$
$157$	5.51988e7	1.13836	0.569181	+	0.822212i	$0.307260\pi$
$158$	−6.49785e7	−1.31060
$159$	4.28382e7	0.845164
$160$	−4.90781e6	−0.0947258
$161$	3.47976e6	0.0657142
$162$	4.25153e6	0.0785674
$163$	7.44029e7	1.34565	0.672827	−	0.739800i	$-0.265080\pi$
$163$	7.44029e7	1.34565	0.672827	+	0.739800i	$0.265080\pi$
$164$	1.99120e7	0.352503
$165$	8.54267e6	0.148047
$166$	3.59326e7	0.609692
$167$	7.02174e7	1.16664	0.583320	−	0.812242i	$-0.301753\pi$
$167$	7.02174e7	1.16664	0.583320	+	0.812242i	$0.301753\pi$
$168$	3.95366e6	0.0643306
$169$	8.79983e7	1.40240
$170$	1.57069e7	0.245199
$171$	−2.56190e7	−0.391811
$172$	6.34198e7	0.950331
$173$	−1.00685e8	−1.47844	−0.739218	−	0.673467i	$-0.764805\pi$
$173$	−1.00685e8	−1.47844	−0.739218	+	0.673467i	$0.764805\pi$
$174$	3.53530e7	0.508749
$175$	1.59281e7	0.224662
$176$	8.65270e6	0.119635
$177$	5.00285e7	0.678127
$178$	−4.73345e7	−0.629083
$179$	4.00269e7	0.521635	0.260817	−	0.965388i	$-0.416008\pi$
$179$	4.00269e7	0.521635	0.260817	+	0.965388i	$0.416008\pi$
$180$	−6.98788e6	−0.0893083
$181$	1.41418e7	0.177268	0.0886340	−	0.996064i	$-0.471750\pi$
$181$	1.41418e7	0.177268	0.0886340	+	0.996064i	$0.471750\pi$
$182$	−2.80918e7	−0.345406
$183$	3.19694e7	0.385617
$184$	−6.22950e6	−0.0737210
$185$	5.41474e7	0.628748
$186$	758770.	0.00864599
$187$	−2.76920e7	−0.309677
$188$	−7.93651e7	−0.871118
$189$	5.62934e6	0.0606514
$190$	4.21079e7	0.445375
$191$	−7.93670e7	−0.824182	−0.412091	−	0.911143i	$-0.635201\pi$
$191$	−7.93670e7	−0.824182	−0.412091	+	0.911143i	$0.635201\pi$
$192$	−7.07789e6	−0.0721688
$193$	−1.60585e7	−0.160788	−0.0803940	−	0.996763i	$-0.525618\pi$
$193$	−1.60585e7	−0.160788	−0.0803940	+	0.996763i	$0.525618\pi$
$194$	−7.45474e6	−0.0733037
$195$	4.96508e7	0.479518
$196$	−4.74718e7	−0.450339
$197$	−1.79663e8	−1.67427	−0.837136	−	0.546995i	$-0.815772\pi$
$197$	−1.79663e8	−1.67427	−0.837136	+	0.546995i	$0.815772\pi$
$198$	1.23200e7	0.112793
$199$	6.19400e7	0.557167	0.278583	−	0.960412i	$-0.410135\pi$
$199$	6.19400e7	0.557167	0.278583	+	0.960412i	$0.410135\pi$
$200$	−2.85146e7	−0.252036
$201$	1.02023e8	0.886159
$202$	9.52674e6	0.0813233
$203$	4.68099e7	0.392737
$204$	2.26520e7	0.186810
$205$	−4.65987e7	−0.377777
$206$	−1.08515e6	−0.00864879
$207$	−8.86974e6	−0.0695048
$208$	5.02903e7	0.387492
$209$	−7.42382e7	−0.562491
$210$	−9.25248e6	−0.0689430
$211$	−1.45843e7	−0.106880	−0.0534400	−	0.998571i	$-0.517019\pi$
$211$	−1.45843e7	−0.106880	−0.0534400	+	0.998571i	$0.517019\pi$
$212$	−1.01542e8	−0.731933
$213$	−1.60404e8	−1.13733
$214$	−8.42170e7	−0.587423
$215$	−1.48417e8	−1.01847
$216$	−1.00777e7	−0.0680414
$217$	1.00467e6	0.00667442
$218$	1.54306e8	1.00875
$219$	−1.42936e8	−0.919575
$220$	−2.02493e7	−0.128213
$221$	−1.60948e8	−1.00303
$222$	7.80896e7	0.479025
$223$	2.08109e8	1.25668	0.628338	−	0.777940i	$-0.283735\pi$
$223$	2.08109e8	1.25668	0.628338	+	0.777940i	$0.283735\pi$
$224$	−9.37165e6	−0.0557119
$225$	−4.05999e7	−0.237622
$226$	−1.31621e8	−0.758480
$227$	−1.52634e8	−0.866086	−0.433043	−	0.901373i	$-0.642560\pi$
$227$	−1.52634e8	−0.866086	−0.433043	+	0.901373i	$0.642560\pi$
$228$	6.07266e7	0.339318
$229$	2.37130e8	1.30486	0.652429	−	0.757850i	$-0.273750\pi$
$229$	2.37130e8	1.30486	0.652429	+	0.757850i	$0.273750\pi$
$230$	1.45785e7	0.0790068
$231$	1.63125e7	0.0870722
$232$	−8.37996e7	−0.440589
$233$	−1.15431e8	−0.597830	−0.298915	−	0.954280i	$-0.596625\pi$
$233$	−1.15431e8	−0.597830	−0.298915	+	0.954280i	$0.596625\pi$
$234$	7.16047e7	0.365331
$235$	1.85732e8	0.933577
$236$	−1.18586e8	−0.587275
$237$	2.19303e8	1.07010
$238$	2.99929e7	0.144211
$239$	−2.14418e7	−0.101594	−0.0507972	−	0.998709i	$-0.516176\pi$
$239$	−2.14418e7	−0.101594	−0.0507972	+	0.998709i	$0.516176\pi$
$240$	1.65639e7	0.0773433
$241$	−3.47212e8	−1.59785	−0.798924	−	0.601432i	$-0.794597\pi$
$241$	−3.47212e8	−1.59785	−0.798924	+	0.601432i	$0.794597\pi$
$242$	−1.20197e8	−0.545180
$243$	−1.43489e7	−0.0641500
$244$	−7.57794e7	−0.333954
$245$	1.11095e8	0.482628
$246$	−6.72031e7	−0.287817
$247$	−4.31479e8	−1.82188
$248$	−1.79857e6	−0.00748765
$249$	−1.21273e8	−0.497812
$250$	1.60340e8	0.649010
$251$	1.65064e8	0.658861	0.329430	−	0.944180i	$-0.393143\pi$
$251$	1.65064e8	0.658861	0.329430	+	0.944180i	$0.393143\pi$
$252$	−1.33436e7	−0.0525257
$253$	−2.57025e7	−0.0997823
$254$	5.53286e7	0.211852
$255$	−5.30108e7	−0.200204
$256$	1.67772e7	0.0625000
$257$	−9.92319e7	−0.364658	−0.182329	−	0.983238i	$-0.558364\pi$
$257$	−9.92319e7	−0.364658	−0.182329	+	0.983238i	$0.558364\pi$
$258$	−2.14042e8	−0.775942
$259$	1.03396e8	0.369791
$260$	−1.17691e8	−0.415275
$261$	−1.19316e8	−0.415392
$262$	−1.20972e7	−0.0415557
$263$	−5.07473e8	−1.72016	−0.860078	−	0.510162i	$-0.829585\pi$
$263$	−5.07473e8	−1.72016	−0.860078	+	0.510162i	$0.829585\pi$
$264$	−2.92029e7	−0.0976814
$265$	2.37632e8	0.784413
$266$	8.04066e7	0.261942
$267$	1.59754e8	0.513644
$268$	−2.41832e8	−0.767436
$269$	−3.71687e8	−1.16424	−0.582122	−	0.813102i	$-0.697777\pi$
$269$	−3.71687e8	−1.16424	−0.582122	+	0.813102i	$0.697777\pi$
$270$	2.35841e7	0.0729199
$271$	5.79033e8	1.76730	0.883650	−	0.468147i	$-0.155078\pi$
$271$	5.79033e8	1.76730	0.883650	+	0.468147i	$0.155078\pi$
$272$	−5.36936e7	−0.161782
$273$	9.48099e7	0.282023
$274$	3.00077e8	0.881265
$275$	−1.17649e8	−0.341134
$276$	2.10246e7	0.0601929
$277$	−2.48531e8	−0.702588	−0.351294	−	0.936265i	$-0.614258\pi$
$277$	−2.48531e8	−0.702588	−0.351294	+	0.936265i	$0.614258\pi$
$278$	−3.11642e8	−0.869961
$279$	−2.56085e6	−0.00705942
$280$	2.19318e7	0.0597064
$281$	−5.33855e8	−1.43533	−0.717664	−	0.696390i	$-0.754788\pi$
$281$	−5.33855e8	−1.43533	−0.717664	+	0.696390i	$0.754788\pi$
$282$	2.67857e8	0.711265
$283$	2.53020e8	0.663593	0.331797	−	0.943351i	$-0.392345\pi$
$283$	2.53020e8	0.663593	0.331797	+	0.943351i	$0.392345\pi$
$284$	3.80216e8	0.984955
$285$	−1.42114e8	−0.363647
$286$	2.07494e8	0.524475
$287$	−8.89819e7	−0.222185
$288$	2.38879e7	0.0589256
$289$	−2.38499e8	−0.581224
$290$	1.96110e8	0.472180
$291$	2.51597e7	0.0598523
$292$	3.38811e8	0.796376
$293$	−8.20920e7	−0.190662	−0.0953310	−	0.995446i	$-0.530391\pi$
$293$	−8.20920e7	−0.190662	−0.0953310	+	0.995446i	$0.530391\pi$
$294$	1.60217e8	0.367700
$295$	2.77518e8	0.629382
$296$	−1.85101e8	−0.414848
$297$	−4.15799e7	−0.0920949
$298$	−3.26928e8	−0.715642
$299$	−1.49385e8	−0.323190
$300$	9.62368e7	0.205786
$301$	−2.83407e8	−0.599001
$302$	4.51368e8	0.942988
$303$	−3.21528e7	−0.0664002
$304$	−1.43945e8	−0.293858
$305$	1.77341e8	0.357899
$306$	−7.64504e7	−0.152530
$307$	7.59102e7	0.149732	0.0748662	−	0.997194i	$-0.476147\pi$
$307$	7.59102e7	0.149732	0.0748662	+	0.997194i	$0.476147\pi$
$308$	−3.86668e7	−0.0754068
$309$	3.66239e6	0.00706171
$310$	4.20906e6	0.00802451
$311$	−7.46880e7	−0.140796	−0.0703979	−	0.997519i	$-0.522427\pi$
$311$	−7.46880e7	−0.140796	−0.0703979	+	0.997519i	$0.522427\pi$
$312$	−1.69730e8	−0.316386
$313$	2.96236e7	0.0546051	0.0273026	−	0.999627i	$-0.491308\pi$
$313$	2.96236e7	0.0546051	0.0273026	+	0.999627i	$0.491308\pi$
$314$	4.41590e8	0.804944
$315$	3.12271e7	0.0562918
$316$	−5.19828e8	−0.926734
$317$	−6.12509e8	−1.07995	−0.539977	−	0.841680i	$-0.681567\pi$
$317$	−6.12509e8	−1.07995	−0.539977	+	0.841680i	$0.681567\pi$
$318$	3.42705e8	0.597621
$319$	−3.45751e8	−0.596343
$320$	−3.92625e7	−0.0669812
$321$	2.84232e8	0.479629
$322$	2.78381e7	0.0464669
$323$	4.60678e8	0.760657
$324$	3.40122e7	0.0555556
$325$	−6.83788e8	−1.10492
$326$	5.95223e8	0.951521
$327$	−5.20782e8	−0.823643
$328$	1.59296e8	0.249257
$329$	3.54663e8	0.549073
$330$	6.83414e7	0.104685
$331$	3.32926e8	0.504603	0.252302	−	0.967649i	$-0.418812\pi$
$331$	3.32926e8	0.504603	0.252302	+	0.967649i	$0.418812\pi$
$332$	2.87461e8	0.431118
$333$	−2.63553e8	−0.391122
$334$	5.61739e8	0.824939
$335$	5.65943e8	0.822461
$336$	3.16293e7	0.0454886
$337$	7.05571e8	1.00424	0.502118	−	0.864799i	$-0.332554\pi$
$337$	7.05571e8	1.00424	0.502118	+	0.864799i	$0.332554\pi$
$338$	7.03986e8	0.991644
$339$	4.44219e8	0.619296
$340$	1.25655e8	0.173382
$341$	−7.42076e6	−0.0101346
$342$	−2.04952e8	−0.277052
$343$	4.47673e8	0.599006
$344$	5.07358e8	0.671986
$345$	−4.92023e7	−0.0645087
$346$	−8.05478e8	−1.04541
$347$	3.08246e8	0.396045	0.198023	−	0.980197i	$-0.436548\pi$
$347$	3.08246e8	0.396045	0.198023	+	0.980197i	$0.436548\pi$
$348$	2.82824e8	0.359740
$349$	1.09169e9	1.37471	0.687353	−	0.726324i	$-0.258773\pi$
$349$	1.09169e9	1.37471	0.687353	+	0.726324i	$0.258773\pi$
$350$	1.27425e8	0.158860
$351$	−2.41666e8	−0.298291
$352$	6.92216e7	0.0845946
$353$	2.95230e8	0.357230	0.178615	−	0.983919i	$-0.442838\pi$
$353$	2.95230e8	0.357230	0.178615	+	0.983919i	$0.442838\pi$
$354$	4.00228e8	0.479508
$355$	−8.89793e8	−1.05558
$356$	−3.78676e8	−0.444829
$357$	−1.01226e8	−0.117748
$358$	3.20215e8	0.368851
$359$	−2.00913e7	−0.0229180	−0.0114590	−	0.999934i	$-0.503648\pi$
$359$	−2.00913e7	−0.0229180	−0.0114590	+	0.999934i	$0.503648\pi$
$360$	−5.59031e7	−0.0631505
$361$	3.41139e8	0.381642
$362$	1.13135e8	0.125347
$363$	4.05665e8	0.445137
$364$	−2.24735e8	−0.244239
$365$	−7.92896e8	−0.853476
$366$	2.55755e8	0.272672
$367$	−3.69496e8	−0.390192	−0.195096	−	0.980784i	$-0.562502\pi$
$367$	−3.69496e8	−0.390192	−0.195096	+	0.980784i	$0.562502\pi$
$368$	−4.98360e7	−0.0521286
$369$	2.26811e8	0.235002
$370$	4.33179e8	0.444592
$371$	4.53767e8	0.461344
$372$	6.07016e6	0.00611364
$373$	8.01257e8	0.799449	0.399725	−	0.916635i	$-0.369106\pi$
$373$	8.01257e8	0.799449	0.399725	+	0.916635i	$0.369106\pi$
$374$	−2.21536e8	−0.218975
$375$	−5.41147e8	−0.529914
$376$	−6.34921e8	−0.615974
$377$	−2.00954e9	−1.93153
$378$	4.50347e7	0.0428870
$379$	−1.11610e9	−1.05309	−0.526546	−	0.850147i	$-0.676513\pi$
$379$	−1.11610e9	−1.05309	−0.526546	+	0.850147i	$0.676513\pi$
$380$	3.36863e8	0.314928
$381$	−1.86734e8	−0.172976
$382$	−6.34936e8	−0.582785
$383$	−5.88465e8	−0.535211	−0.267605	−	0.963529i	$-0.586232\pi$
$383$	−5.88465e8	−0.535211	−0.267605	+	0.963529i	$0.586232\pi$
$384$	−5.66231e7	−0.0510310
$385$	9.04891e7	0.0808134
$386$	−1.28468e8	−0.113694
$387$	7.22391e8	0.633554
$388$	−5.96379e7	−0.0518336
$389$	2.20225e9	1.89689	0.948446	−	0.316940i	$-0.102655\pi$
$389$	2.20225e9	1.89689	0.948446	+	0.316940i	$0.102655\pi$
$390$	3.97206e8	0.339070
$391$	1.59495e8	0.134936
$392$	−3.79774e8	−0.318438
$393$	4.08281e7	0.0339301
$394$	−1.43730e9	−1.18389
$395$	1.21652e9	0.993181
$396$	9.85597e7	0.0797565
$397$	2.34167e9	1.87827	0.939137	−	0.343544i	$-0.111627\pi$
$397$	2.34167e9	1.87827	0.939137	+	0.343544i	$0.111627\pi$
$398$	4.95520e8	0.393976
$399$	−2.71372e8	−0.213875
$400$	−2.28117e8	−0.178216
$401$	9.94580e8	0.770255	0.385127	−	0.922863i	$-0.374158\pi$
$401$	9.94580e8	0.770255	0.385127	+	0.922863i	$0.374158\pi$
$402$	8.16184e8	0.626609
$403$	−4.31301e7	−0.0328256
$404$	7.62140e7	0.0575042
$405$	−7.95964e7	−0.0595389
$406$	3.74479e8	0.277707
$407$	−7.63715e8	−0.561502
$408$	1.81216e8	0.132095
$409$	−8.16035e8	−0.589763	−0.294881	−	0.955534i	$-0.595280\pi$
$409$	−8.16035e8	−0.589763	−0.294881	+	0.955534i	$0.595280\pi$
$410$	−3.72790e8	−0.267129
$411$	−1.01276e9	−0.719550
$412$	−8.68122e6	−0.00611562
$413$	5.29931e8	0.370164
$414$	−7.09579e7	−0.0491473
$415$	−6.72725e8	−0.462029
$416$	4.02322e8	0.273998
$417$	1.05179e9	0.710320
$418$	−5.93905e8	−0.397741
$419$	1.76484e9	1.17208	0.586040	−	0.810282i	$-0.300686\pi$
$419$	1.76484e9	1.17208	0.586040	+	0.810282i	$0.300686\pi$
$420$	−7.40198e7	−0.0487501
$421$	−1.50129e9	−0.980566	−0.490283	−	0.871563i	$-0.663107\pi$
$421$	−1.50129e9	−0.980566	−0.490283	+	0.871563i	$0.663107\pi$
$422$	−1.16674e8	−0.0755756
$423$	−9.04018e8	−0.580746
$424$	−8.12339e8	−0.517555
$425$	7.30062e8	0.461316
$426$	−1.28323e9	−0.804213
$427$	3.38639e8	0.210494
$428$	−6.73736e8	−0.415371
$429$	−7.00293e8	−0.428232
$430$	−1.18733e9	−0.720167
$431$	1.45381e9	0.874659	0.437329	−	0.899301i	$-0.355924\pi$
$431$	1.45381e9	0.874659	0.437329	+	0.899301i	$0.355924\pi$
$432$	−8.06216e7	−0.0481125
$433$	−2.63208e9	−1.55809	−0.779043	−	0.626970i	$-0.784295\pi$
$433$	−2.63208e9	−1.55809	−0.779043	+	0.626970i	$0.784295\pi$
$434$	8.03734e6	0.00471953
$435$	−6.61872e8	−0.385533
$436$	1.23445e9	0.713296
$437$	4.27582e8	0.245095
$438$	−1.14349e9	−0.650238
$439$	6.68744e8	0.377254	0.188627	−	0.982049i	$-0.439596\pi$
$439$	6.68744e8	0.377254	0.188627	+	0.982049i	$0.439596\pi$
$440$	−1.61994e8	−0.0906600
$441$	−5.40734e8	−0.300226
$442$	−1.28759e9	−0.709248
$443$	−1.29682e9	−0.708707	−0.354354	−	0.935112i	$-0.615299\pi$
$443$	−1.29682e9	−0.708707	−0.354354	+	0.935112i	$0.615299\pi$
$444$	6.24717e8	0.338722
$445$	8.86189e8	0.476723
$446$	1.66487e9	0.888605
$447$	1.10338e9	0.584319
$448$	−7.49732e7	−0.0393943
$449$	−1.52269e9	−0.793871	−0.396935	−	0.917847i	$-0.629926\pi$
$449$	−1.52269e9	−0.793871	−0.396935	+	0.917847i	$0.629926\pi$
$450$	−3.24799e8	−0.168024
$451$	6.57246e8	0.337373
$452$	−1.05296e9	−0.536326
$453$	−1.52337e9	−0.769946
$454$	−1.22107e9	−0.612415
$455$	5.25930e8	0.261751
$456$	4.85813e8	0.239934
$457$	−9.33160e8	−0.457351	−0.228675	−	0.973503i	$-0.573439\pi$
$457$	−9.33160e8	−0.457351	−0.228675	+	0.973503i	$0.573439\pi$
$458$	1.89704e9	0.922674
$459$	2.58020e8	0.124540
$460$	1.16628e8	0.0558662
$461$	−1.05955e9	−0.503696	−0.251848	−	0.967767i	$-0.581038\pi$
$461$	−1.05955e9	−0.503696	−0.251848	+	0.967767i	$0.581038\pi$
$462$	1.30500e8	0.0615694
$463$	−3.02601e7	−0.0141689	−0.00708446	−	0.999975i	$-0.502255\pi$
$463$	−3.02601e7	−0.0141689	−0.00708446	+	0.999975i	$0.502255\pi$
$464$	−6.70397e8	−0.311544
$465$	−1.42056e7	−0.00655199
$466$	−9.23449e8	−0.422729
$467$	1.60547e9	0.729448	0.364724	−	0.931116i	$-0.381163\pi$
$467$	1.60547e9	0.729448	0.364724	+	0.931116i	$0.381163\pi$
$468$	5.72838e8	0.258328
$469$	1.08069e9	0.483721
$470$	1.48586e9	0.660139
$471$	−1.49037e9	−0.657234
$472$	−9.48688e8	−0.415266
$473$	2.09332e9	0.909541
$474$	1.75442e9	0.756675
$475$	1.95719e9	0.837925
$476$	2.39943e8	0.101973
$477$	−1.15663e9	−0.487956
$478$	−1.71535e8	−0.0718380
$479$	8.49652e8	0.353237	0.176619	−	0.984279i	$-0.443484\pi$
$479$	8.49652e8	0.353237	0.176619	+	0.984279i	$0.443484\pi$
$480$	1.32511e8	0.0546900
$481$	−4.43878e9	−1.81868
$482$	−2.77770e9	−1.12985
$483$	−9.39536e7	−0.0379401
$484$	−9.61575e8	−0.385500
$485$	1.39566e8	0.0555500
$486$	−1.14791e8	−0.0453609
$487$	1.48989e9	0.584525	0.292262	−	0.956338i	$-0.405592\pi$
$487$	1.48989e9	0.584525	0.292262	+	0.956338i	$0.405592\pi$
$488$	−6.06235e8	−0.236141
$489$	−2.00888e9	−0.776914
$490$	8.88759e8	0.341270
$491$	4.17131e9	1.59033	0.795165	−	0.606394i	$-0.207384\pi$
$491$	4.17131e9	1.59033	0.795165	+	0.606394i	$0.207384\pi$
$492$	−5.37625e8	−0.203517
$493$	2.14553e9	0.806436
$494$	−3.45183e9	−1.28826
$495$	−2.30652e8	−0.0854751
$496$	−1.43885e7	−0.00529457
$497$	−1.69909e9	−0.620825
$498$	−9.70181e8	−0.352006
$499$	4.59203e9	1.65445	0.827224	−	0.561873i	$-0.189919\pi$
$499$	4.59203e9	1.65445	0.827224	+	0.561873i	$0.189919\pi$
$500$	1.28272e9	0.458919
$501$	−1.89587e9	−0.673560
$502$	1.32051e9	0.465885
$503$	−3.28143e9	−1.14967	−0.574837	−	0.818268i	$-0.694935\pi$
$503$	−3.28143e9	−1.14967	−0.574837	+	0.818268i	$0.694935\pi$
$504$	−1.06749e8	−0.0371413
$505$	−1.78358e8	−0.0616273
$506$	−2.05620e8	−0.0705567
$507$	−2.37595e9	−0.809674
$508$	4.42629e8	0.149802
$509$	−1.95964e9	−0.658665	−0.329332	−	0.944214i	$-0.606824\pi$
$509$	−1.95964e9	−0.658665	−0.329332	+	0.944214i	$0.606824\pi$
$510$	−4.24086e8	−0.141566
$511$	−1.51406e9	−0.501962
$512$	1.34218e8	0.0441942
$513$	6.91714e8	0.226212
$514$	−7.93855e8	−0.257852
$515$	2.03160e7	0.00655411
$516$	−1.71233e9	−0.548674
$517$	−2.61964e9	−0.833728
$518$	8.27172e8	0.261482
$519$	2.71849e9	0.853575
$520$	−9.41526e8	−0.293644
$521$	−7.29024e6	−0.00225845	−0.00112922	−	0.999999i	$-0.500359\pi$
$521$	−7.29024e6	−0.00225845	−0.00112922	+	0.999999i	$0.500359\pi$
$522$	−9.54530e8	−0.293726
$523$	2.62342e8	0.0801884	0.0400942	−	0.999196i	$-0.487234\pi$
$523$	2.62342e8	0.0801884	0.0400942	+	0.999196i	$0.487234\pi$
$524$	−9.67777e7	−0.0293843
$525$	−4.30058e8	−0.129709
$526$	−4.05979e9	−1.21633
$527$	4.60489e7	0.0137051
$528$	−2.33623e8	−0.0690712
$529$	1.48036e8	0.0434783
$530$	1.90106e9	0.554664
$531$	−1.35077e9	−0.391517
$532$	6.43252e8	0.185221
$533$	3.81997e9	1.09273
$534$	1.27803e9	0.363201
$535$	1.57670e9	0.445153
$536$	−1.93466e9	−0.542659
$537$	−1.08073e9	−0.301166
$538$	−2.97349e9	−0.823245
$539$	−1.56692e9	−0.431010
$540$	1.88673e8	0.0515622
$541$	4.96003e9	1.34677	0.673386	−	0.739291i	$-0.264839\pi$
$541$	4.96003e9	1.34677	0.673386	+	0.739291i	$0.264839\pi$
$542$	4.63226e9	1.24967
$543$	−3.81829e8	−0.102346
$544$	−4.29549e8	−0.114397
$545$	−2.88889e9	−0.764439
$546$	7.58479e8	0.199420
$547$	−4.88993e9	−1.27746	−0.638729	−	0.769431i	$-0.720540\pi$
$547$	−4.88993e9	−1.27746	−0.638729	+	0.769431i	$0.720540\pi$
$548$	2.40062e9	0.623148
$549$	−8.63175e8	−0.222636
$550$	−9.41194e8	−0.241218
$551$	5.75185e9	1.46479
$552$	1.68197e8	0.0425628
$553$	2.32298e9	0.584128
$554$	−1.98825e9	−0.496805
$555$	−1.46198e9	−0.363008
$556$	−2.49314e9	−0.615155
$557$	−7.49454e9	−1.83760	−0.918802	−	0.394719i	$-0.870842\pi$
$557$	−7.49454e9	−1.83760	−0.918802	+	0.394719i	$0.870842\pi$
$558$	−2.04868e7	−0.00499177
$559$	1.21666e10	2.94596
$560$	1.75454e8	0.0422188
$561$	7.47684e8	0.178792
$562$	−4.27084e9	−1.01493
$563$	6.63382e9	1.56669	0.783347	−	0.621584i	$-0.213511\pi$
$563$	6.63382e9	1.56669	0.783347	+	0.621584i	$0.213511\pi$
$564$	2.14286e9	0.502940
$565$	2.46418e9	0.574781
$566$	2.02416e9	0.469231
$567$	−1.51992e8	−0.0350171
$568$	3.04173e9	0.696468
$569$	−2.76499e9	−0.629218	−0.314609	−	0.949221i	$-0.601873\pi$
$569$	−2.76499e9	−0.629218	−0.314609	+	0.949221i	$0.601873\pi$
$570$	−1.13691e9	−0.257137
$571$	−1.14949e9	−0.258393	−0.129196	−	0.991619i	$-0.541240\pi$
$571$	−1.14949e9	−0.258393	−0.129196	+	0.991619i	$0.541240\pi$
$572$	1.65995e9	0.370860
$573$	2.14291e9	0.475842
$574$	−7.11856e8	−0.157109
$575$	6.77611e8	0.148643
$576$	1.91103e8	0.0416667
$577$	−5.62286e9	−1.21855	−0.609273	−	0.792961i	$-0.708539\pi$
$577$	−5.62286e9	−1.21855	−0.609273	+	0.792961i	$0.708539\pi$
$578$	−1.90799e9	−0.410987
$579$	4.33579e8	0.0928310
$580$	1.56888e9	0.333881
$581$	−1.28459e9	−0.271737
$582$	2.01278e8	0.0423219
$583$	−3.35165e9	−0.700518
$584$	2.71049e9	0.563123
$585$	−1.34057e9	−0.276850
$586$	−6.56736e8	−0.134818
$587$	3.44922e9	0.703862	0.351931	−	0.936026i	$-0.385525\pi$
$587$	3.44922e9	0.703862	0.351931	+	0.936026i	$0.385525\pi$
$588$	1.28174e9	0.260003
$589$	1.23450e8	0.0248936
$590$	2.22015e9	0.445041
$591$	4.85089e9	0.966641
$592$	−1.48081e9	−0.293342
$593$	−4.35770e9	−0.858156	−0.429078	−	0.903267i	$-0.641162\pi$
$593$	−4.35770e9	−0.858156	−0.429078	+	0.903267i	$0.641162\pi$
$594$	−3.32639e8	−0.0651209
$595$	−5.61522e8	−0.109284
$596$	−2.61543e9	−0.506035
$597$	−1.67238e9	−0.321680
$598$	−1.19508e9	−0.228530
$599$	−8.25103e9	−1.56861	−0.784304	−	0.620377i	$-0.786980\pi$
$599$	−8.25103e9	−1.56861	−0.784304	+	0.620377i	$0.786980\pi$
$600$	7.69894e8	0.145513
$601$	9.29399e9	1.74639	0.873196	−	0.487370i	$-0.162044\pi$
$601$	9.29399e9	1.74639	0.873196	+	0.487370i	$0.162044\pi$
$602$	−2.26726e9	−0.423558
$603$	−2.75462e9	−0.511624
$604$	3.61094e9	0.666793
$605$	2.25031e9	0.413140
$606$	−2.57222e8	−0.0469520
$607$	−2.65995e9	−0.482740	−0.241370	−	0.970433i	$-0.577597\pi$
$607$	−2.65995e9	−0.482740	−0.241370	+	0.970433i	$0.577597\pi$
$608$	−1.15156e9	−0.207789
$609$	−1.26387e9	−0.226747
$610$	1.41873e9	0.253072
$611$	−1.52256e10	−2.70041
$612$	−6.11603e8	−0.107855
$613$	−1.36100e9	−0.238641	−0.119321	−	0.992856i	$-0.538072\pi$
$613$	−1.36100e9	−0.238641	−0.119321	+	0.992856i	$0.538072\pi$
$614$	6.07282e8	0.105877
$615$	1.25817e9	0.218110
$616$	−3.09334e8	−0.0533206
$617$	5.15057e9	0.882790	0.441395	−	0.897313i	$-0.354484\pi$
$617$	5.15057e9	0.882790	0.441395	+	0.897313i	$0.354484\pi$
$618$	2.92991e7	0.00499338
$619$	1.02764e10	1.74150	0.870750	−	0.491727i	$-0.163634\pi$
$619$	1.02764e10	1.74150	0.870750	+	0.491727i	$0.163634\pi$
$620$	3.36724e7	0.00567419
$621$	2.39483e8	0.0401286
$622$	−5.97504e8	−0.0995577
$623$	1.69221e9	0.280379
$624$	−1.35784e9	−0.223718
$625$	1.34913e9	0.221041
$626$	2.36989e8	0.0386116
$627$	2.00443e9	0.324754
$628$	3.53272e9	0.569181
$629$	4.73917e9	0.759320
$630$	2.49817e8	0.0398043
$631$	−4.01683e9	−0.636474	−0.318237	−	0.948011i	$-0.603091\pi$
$631$	−4.01683e9	−0.636474	−0.318237	+	0.948011i	$0.603091\pi$
$632$	−4.15863e9	−0.655300
$633$	3.93776e8	0.0617072
$634$	−4.90007e9	−0.763643
$635$	−1.03585e9	−0.160542
$636$	2.74164e9	0.422582
$637$	−9.10709e9	−1.39602
$638$	−2.76601e9	−0.421678
$639$	4.33090e9	0.656637
$640$	−3.14100e8	−0.0473629
$641$	3.68266e9	0.552279	0.276140	−	0.961118i	$-0.410945\pi$
$641$	3.68266e9	0.552279	0.276140	+	0.961118i	$0.410945\pi$
$642$	2.27386e9	0.339149
$643$	8.97517e9	1.33139	0.665694	−	0.746225i	$-0.268136\pi$
$643$	8.97517e9	1.33139	0.665694	+	0.746225i	$0.268136\pi$
$644$	2.22705e8	0.0328571
$645$	4.00725e9	0.588014
$646$	3.68543e9	0.537866
$647$	1.33756e10	1.94155	0.970773	−	0.240000i	$-0.0771475\pi$
$647$	1.33756e10	1.94155	0.970773	+	0.240000i	$0.0771475\pi$
$648$	2.72098e8	0.0392837
$649$	−3.91422e9	−0.562068
$650$	−5.47030e9	−0.781294
$651$	−2.71260e7	−0.00385348
$652$	4.76179e9	0.672827
$653$	−5.70945e9	−0.802413	−0.401207	−	0.915988i	$-0.631409\pi$
$653$	−5.70945e9	−0.802413	−0.401207	+	0.915988i	$0.631409\pi$
$654$	−4.16626e9	−0.582404
$655$	2.26482e8	0.0314912
$656$	1.27437e9	0.176251
$657$	3.85927e9	0.530917
$658$	2.83730e9	0.388253
$659$	8.18369e9	1.11391	0.556955	−	0.830543i	$-0.311969\pi$
$659$	8.18369e9	1.11391	0.556955	+	0.830543i	$0.311969\pi$
$660$	5.46731e8	0.0740236
$661$	−5.77617e9	−0.777920	−0.388960	−	0.921255i	$-0.627166\pi$
$661$	−5.77617e9	−0.777920	−0.388960	+	0.921255i	$0.627166\pi$
$662$	2.66341e9	0.356808
$663$	4.34560e9	0.579099
$664$	2.29969e9	0.304846
$665$	−1.50536e9	−0.198502
$666$	−2.10842e9	−0.276565
$667$	1.99139e9	0.259845
$668$	4.49391e9	0.583320
$669$	−5.61894e9	−0.725543
$670$	4.52754e9	0.581568
$671$	−2.50128e9	−0.319620
$672$	2.53034e8	0.0321653
$673$	−8.18822e9	−1.03547	−0.517734	−	0.855542i	$-0.673224\pi$
$673$	−8.18822e9	−1.03547	−0.517734	+	0.855542i	$0.673224\pi$
$674$	5.64457e9	0.710103
$675$	1.09620e9	0.137191
$676$	5.63189e9	0.701198
$677$	−9.71590e8	−0.120343	−0.0601717	−	0.998188i	$-0.519165\pi$
$677$	−9.71590e8	−0.120343	−0.0601717	+	0.998188i	$0.519165\pi$
$678$	3.55375e9	0.437909
$679$	2.66507e8	0.0326711
$680$	1.00524e9	0.122600
$681$	4.12112e9	0.500035
$682$	−5.93661e7	−0.00716627
$683$	8.74030e9	1.04967	0.524836	−	0.851203i	$-0.324127\pi$
$683$	8.74030e9	1.04967	0.524836	+	0.851203i	$0.324127\pi$
$684$	−1.63962e9	−0.195905
$685$	−5.61800e9	−0.667828
$686$	3.58138e9	0.423561
$687$	−6.40252e9	−0.753360
$688$	4.05886e9	0.475166
$689$	−1.94801e10	−2.26894
$690$	−3.93618e8	−0.0456146
$691$	1.15731e10	1.33437	0.667184	−	0.744893i	$-0.267500\pi$
$691$	1.15731e10	1.33437	0.667184	+	0.744893i	$0.267500\pi$
$692$	−6.44382e9	−0.739218
$693$	−4.40439e8	−0.0502712
$694$	2.46597e9	0.280046
$695$	5.83451e9	0.659262
$696$	2.26259e9	0.254374
$697$	−4.07848e9	−0.456230
$698$	8.73351e9	0.972064
$699$	3.11664e9	0.345157
$700$	1.01940e9	0.112331
$701$	−1.05225e10	−1.15374	−0.576869	−	0.816836i	$-0.695726\pi$
$701$	−1.05225e10	−1.15374	−0.576869	+	0.816836i	$0.695726\pi$
$702$	−1.93333e9	−0.210924
$703$	1.27050e10	1.37921
$704$	5.53773e8	0.0598174
$705$	−5.01478e9	−0.539001
$706$	2.36184e9	0.252600
$707$	−3.40581e8	−0.0362454
$708$	3.20182e9	0.339063
$709$	1.22260e10	1.28832	0.644159	−	0.764892i	$-0.277208\pi$
$709$	1.22260e10	1.28832	0.644159	+	0.764892i	$0.277208\pi$
$710$	−7.11834e9	−0.746405
$711$	−5.92117e9	−0.617823
$712$	−3.02941e9	−0.314542
$713$	4.27405e7	0.00441598
$714$	−8.09808e8	−0.0832604
$715$	−3.88467e9	−0.397450
$716$	2.56172e9	0.260817
$717$	5.78930e8	0.0586555
$718$	−1.60730e8	−0.0162055
$719$	1.59878e10	1.60412	0.802059	−	0.597244i	$-0.203738\pi$
$719$	1.59878e10	1.60412	0.802059	+	0.597244i	$0.203738\pi$
$720$	−4.47225e8	−0.0446542
$721$	3.87942e7	0.00385472
$722$	2.72911e9	0.269862
$723$	9.37473e9	0.922518
$724$	9.05076e8	0.0886340
$725$	9.11526e9	0.888354
$726$	3.24532e9	0.314760
$727$	−9.16952e9	−0.885067	−0.442534	−	0.896752i	$-0.645920\pi$
$727$	−9.16952e9	−0.885067	−0.442534	+	0.896752i	$0.645920\pi$
$728$	−1.79788e9	−0.172703
$729$	3.87420e8	0.0370370
$730$	−6.34317e9	−0.603498
$731$	−1.29899e10	−1.22997
$732$	2.04604e9	0.192808
$733$	7.27345e9	0.682145	0.341072	−	0.940037i	$-0.389210\pi$
$733$	7.27345e9	0.682145	0.341072	+	0.940037i	$0.389210\pi$
$734$	−2.95597e9	−0.275907
$735$	−2.99956e9	−0.278646
$736$	−3.98688e8	−0.0368605
$737$	−7.98226e9	−0.734497
$738$	1.81449e9	0.166171
$739$	−9.36951e9	−0.854007	−0.427004	−	0.904250i	$-0.640431\pi$
$739$	−9.36951e9	−0.854007	−0.427004	+	0.904250i	$0.640431\pi$
$740$	3.46544e9	0.314374
$741$	1.16499e10	1.05186
$742$	3.63014e9	0.326219
$743$	−9.03180e9	−0.807818	−0.403909	−	0.914799i	$-0.632349\pi$
$743$	−9.03180e9	−0.807818	−0.403909	+	0.914799i	$0.632349\pi$
$744$	4.85613e7	0.00432300
$745$	6.12070e9	0.542318
$746$	6.41006e9	0.565296
$747$	3.27436e9	0.287412
$748$	−1.77229e9	−0.154838
$749$	3.01076e9	0.261812
$750$	−4.32917e9	−0.374706
$751$	−4.48125e9	−0.386064	−0.193032	−	0.981192i	$-0.561832\pi$
$751$	−4.48125e9	−0.386064	−0.193032	+	0.981192i	$0.561832\pi$
$752$	−5.07937e9	−0.435559
$753$	−4.45672e9	−0.380394
$754$	−1.60763e10	−1.36580
$755$	−8.45042e9	−0.714602
$756$	3.60278e8	0.0303257
$757$	−1.95365e10	−1.63686	−0.818431	−	0.574604i	$-0.805156\pi$
$757$	−1.95365e10	−1.63686	−0.818431	+	0.574604i	$0.805156\pi$
$758$	−8.92881e9	−0.744649
$759$	6.93968e8	0.0576093
$760$	2.69491e9	0.222688
$761$	8.66885e8	0.0713042	0.0356521	−	0.999364i	$-0.488649\pi$
$761$	8.66885e8	0.0713042	0.0356521	+	0.999364i	$0.488649\pi$
$762$	−1.49387e9	−0.122313
$763$	−5.51644e9	−0.449596
$764$	−5.07949e9	−0.412091
$765$	1.43129e9	0.115588
$766$	−4.70772e9	−0.378451
$767$	−2.27498e10	−1.82051
$768$	−4.52985e8	−0.0360844
$769$	−1.53184e10	−1.21471	−0.607353	−	0.794432i	$-0.707769\pi$
$769$	−1.53184e10	−1.21471	−0.607353	+	0.794432i	$0.707769\pi$
$770$	7.23913e8	0.0571437
$771$	2.67926e9	0.210535
$772$	−1.02774e9	−0.0803940
$773$	−1.11265e10	−0.866423	−0.433212	−	0.901292i	$-0.642620\pi$
$773$	−1.11265e10	−0.866423	−0.433212	+	0.901292i	$0.642620\pi$
$774$	5.77912e9	0.447990
$775$	1.95638e8	0.0150972
$776$	−4.77103e8	−0.0366519
$777$	−2.79170e9	−0.213499
$778$	1.76180e10	1.34130
$779$	−1.09338e10	−0.828686
$780$	3.17765e9	0.239759
$781$	1.25500e10	0.942679
$782$	1.27596e9	0.0954140
$783$	3.22154e9	0.239826
$784$	−3.03820e9	−0.225169
$785$	−8.26737e9	−0.609991
$786$	3.26625e8	0.0239922
$787$	7.04550e7	0.00515229	0.00257615	−	0.999997i	$-0.499180\pi$
$787$	7.04550e7	0.00515229	0.00257615	+	0.999997i	$0.499180\pi$
$788$	−1.14984e10	−0.837136
$789$	1.37018e10	0.993133
$790$	9.73213e9	0.702285
$791$	4.70543e9	0.338051
$792$	7.88478e8	0.0563964
$793$	−1.45377e10	−1.03524
$794$	1.87334e10	1.32814
$795$	−6.41607e9	−0.452881
$796$	3.96416e9	0.278583
$797$	−4.98535e9	−0.348812	−0.174406	−	0.984674i	$-0.555801\pi$
$797$	−4.98535e9	−0.348812	−0.174406	+	0.984674i	$0.555801\pi$
$798$	−2.17098e9	−0.151232
$799$	1.62559e10	1.12745
$800$	−1.82493e9	−0.126018
$801$	−4.31336e9	−0.296553
$802$	7.95664e9	0.544652
$803$	1.11833e10	0.762194
$804$	6.52947e9	0.443080
$805$	−5.21180e8	−0.0352129
$806$	−3.45041e8	−0.0232112
$807$	1.00355e10	0.672176
$808$	6.09712e8	0.0406616
$809$	1.02894e10	0.683233	0.341617	−	0.939839i	$-0.389026\pi$
$809$	1.02894e10	0.683233	0.341617	+	0.939839i	$0.389026\pi$
$810$	−6.36771e8	−0.0421003
$811$	−2.35153e10	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$811$	−2.35153e10	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$812$	2.99584e9	0.196369
$813$	−1.56339e10	−1.02035
$814$	−6.10972e9	−0.397042
$815$	−1.11437e10	−0.721069
$816$	1.44973e9	0.0934051
$817$	−3.48241e10	−2.23410
$818$	−6.52828e9	−0.417025
$819$	−2.55987e9	−0.162826
$820$	−2.98232e9	−0.188889
$821$	−2.30200e9	−0.145179	−0.0725897	−	0.997362i	$-0.523126\pi$
$821$	−2.30200e9	−0.145179	−0.0725897	+	0.997362i	$0.523126\pi$
$822$	−8.10209e9	−0.508798
$823$	−1.58831e10	−0.993199	−0.496600	−	0.867980i	$-0.665418\pi$
$823$	−1.58831e10	−0.993199	−0.496600	+	0.867980i	$0.665418\pi$
$824$	−6.94497e7	−0.00432439
$825$	3.17653e9	0.196954
$826$	4.23945e9	0.261746
$827$	−1.95575e10	−1.20239	−0.601194	−	0.799103i	$-0.705308\pi$
$827$	−1.95575e10	−1.20239	−0.601194	+	0.799103i	$0.705308\pi$
$828$	−5.67664e8	−0.0347524
$829$	−1.64026e10	−0.999933	−0.499967	−	0.866045i	$-0.666654\pi$
$829$	−1.64026e10	−0.999933	−0.499967	+	0.866045i	$0.666654\pi$
$830$	−5.38180e9	−0.326704
$831$	6.71033e9	0.405639
$832$	3.21858e9	0.193746
$833$	9.72340e9	0.582855
$834$	8.41434e9	0.502272
$835$	−1.05168e10	−0.625144
$836$	−4.75124e9	−0.281245
$837$	6.91429e7	0.00407576
$838$	1.41187e10	0.828785
$839$	−1.59773e10	−0.933980	−0.466990	−	0.884263i	$-0.654662\pi$
$839$	−1.59773e10	−0.933980	−0.466990	+	0.884263i	$0.654662\pi$
$840$	−5.92158e8	−0.0344715
$841$	9.53835e9	0.552952
$842$	−1.20103e10	−0.693365
$843$	1.44141e10	0.828687
$844$	−9.33394e8	−0.0534400
$845$	−1.31799e10	−0.751474
$846$	−7.23215e9	−0.410649
$847$	4.29704e9	0.242984
$848$	−6.49871e9	−0.365967
$849$	−6.83154e9	−0.383126
$850$	5.84049e9	0.326200
$851$	4.39869e9	0.244664
$852$	−1.02658e10	−0.568664
$853$	−1.09738e9	−0.0605393	−0.0302696	−	0.999542i	$-0.509637\pi$
$853$	−1.09738e9	−0.0605393	−0.0302696	+	0.999542i	$0.509637\pi$
$854$	2.70911e9	0.148842
$855$	3.83708e9	0.209952
$856$	−5.38988e9	−0.293712
$857$	2.70258e10	1.46672	0.733359	−	0.679842i	$-0.237952\pi$
$857$	2.70258e10	1.46672	0.733359	+	0.679842i	$0.237952\pi$
$858$	−5.60234e9	−0.302806
$859$	1.76483e10	0.950008	0.475004	−	0.879984i	$-0.342447\pi$
$859$	1.76483e10	0.950008	0.475004	+	0.879984i	$0.342447\pi$
$860$	−9.49867e9	−0.509235
$861$	2.40251e9	0.128279
$862$	1.16305e10	0.618477
$863$	−2.01882e10	−1.06920	−0.534601	−	0.845105i	$-0.679538\pi$
$863$	−2.01882e10	−1.06920	−0.534601	+	0.845105i	$0.679538\pi$
$864$	−6.44973e8	−0.0340207
$865$	1.50800e10	0.792219
$866$	−2.10566e10	−1.10173
$867$	6.43946e9	0.335570
$868$	6.42987e7	0.00333721
$869$	−1.71582e10	−0.886957
$870$	−5.29497e9	−0.272613
$871$	−4.63936e10	−2.37900
$872$	9.87558e9	0.504376
$873$	−6.79313e8	−0.0345557
$874$	3.42065e9	0.173308
$875$	−5.73215e9	−0.289261
$876$	−9.14791e9	−0.459788
$877$	3.86814e10	1.93644	0.968219	−	0.250103i	$-0.0804645\pi$
$877$	3.86814e10	1.93644	0.968219	+	0.250103i	$0.0804645\pi$
$878$	5.34995e9	0.266759
$879$	2.21648e9	0.110079
$880$	−1.29596e9	−0.0641063
$881$	2.19786e10	1.08289	0.541444	−	0.840737i	$-0.317878\pi$
$881$	2.19786e10	1.08289	0.541444	+	0.840737i	$0.317878\pi$
$882$	−4.32587e9	−0.212292
$883$	2.41724e10	1.18156	0.590782	−	0.806831i	$-0.298819\pi$
$883$	2.41724e10	1.18156	0.590782	+	0.806831i	$0.298819\pi$
$884$	−1.03007e10	−0.501514
$885$	−7.49299e9	−0.363374
$886$	−1.03746e10	−0.501132
$887$	1.75161e10	0.842759	0.421380	−	0.906884i	$-0.361546\pi$
$887$	1.75161e10	0.842759	0.421380	+	0.906884i	$0.361546\pi$
$888$	4.99774e9	0.239512
$889$	−1.97800e9	−0.0944212
$890$	7.08951e9	0.337094
$891$	1.12266e9	0.0531710
$892$	1.33190e10	0.628338
$893$	4.35798e10	2.04788
$894$	8.82706e9	0.413176
$895$	−5.99502e9	−0.279518
$896$	−5.99785e8	−0.0278559
$897$	4.03340e9	0.186594
$898$	−1.21815e10	−0.561351
$899$	5.74948e8	0.0263918
$900$	−2.59839e9	−0.118811
$901$	2.07984e10	0.947311
$902$	5.25797e9	0.238558
$903$	7.65199e9	0.345834
$904$	−8.42371e9	−0.379240
$905$	−2.11809e9	−0.0949890
$906$	−1.21869e10	−0.544434
$907$	−2.79488e9	−0.124376	−0.0621882	−	0.998064i	$-0.519808\pi$
$907$	−2.79488e9	−0.124376	−0.0621882	+	0.998064i	$0.519808\pi$
$908$	−9.76858e9	−0.433043
$909$	8.68125e8	0.0383361
$910$	4.20744e9	0.185086
$911$	−2.94448e9	−0.129031	−0.0645156	−	0.997917i	$-0.520550\pi$
$911$	−2.94448e9	−0.129031	−0.0645156	+	0.997917i	$0.520550\pi$
$912$	3.88650e9	0.169659
$913$	9.48836e9	0.412613
$914$	−7.46528e9	−0.323396
$915$	−4.78821e9	−0.206633
$916$	1.51763e10	0.652429
$917$	4.32475e8	0.0185212
$918$	2.06416e9	0.0880632
$919$	−2.83038e10	−1.20293	−0.601465	−	0.798899i	$-0.705416\pi$
$919$	−2.83038e10	−1.20293	−0.601465	+	0.798899i	$0.705416\pi$
$920$	9.33021e8	0.0395034
$921$	−2.04958e9	−0.0864480
$922$	−8.47640e9	−0.356167
$923$	7.29415e10	3.05329
$924$	1.04400e9	0.0435361
$925$	2.01343e10	0.836452
$926$	−2.42081e8	−0.0100189
$927$	−9.88845e7	−0.00407708
$928$	−5.36317e9	−0.220295
$929$	−2.06249e10	−0.843989	−0.421994	−	0.906598i	$-0.638670\pi$
$929$	−2.06249e10	−0.843989	−0.421994	+	0.906598i	$0.638670\pi$
$930$	−1.13644e8	−0.00463295
$931$	2.60670e10	1.05869
$932$	−7.38760e9	−0.298915
$933$	2.01658e9	0.0812885
$934$	1.28438e10	0.515798
$935$	4.14756e9	0.165940
$936$	4.58270e9	0.182665
$937$	2.19379e10	0.871175	0.435588	−	0.900146i	$-0.356541\pi$
$937$	2.19379e10	0.871175	0.435588	+	0.900146i	$0.356541\pi$
$938$	8.64550e9	0.342043
$939$	−7.99838e8	−0.0315263
$940$	1.18869e10	0.466789
$941$	−2.50179e10	−0.978785	−0.489393	−	0.872064i	$-0.662781\pi$
$941$	−2.50179e10	−0.978785	−0.489393	+	0.872064i	$0.662781\pi$
$942$	−1.19229e10	−0.464735
$943$	−3.78547e9	−0.147004
$944$	−7.58950e9	−0.293637
$945$	−8.43132e8	−0.0325001
$946$	1.67466e10	0.643143
$947$	4.08898e10	1.56455	0.782276	−	0.622932i	$-0.214059\pi$
$947$	4.08898e10	1.56455	0.782276	+	0.622932i	$0.214059\pi$
$948$	1.40354e10	0.535050
$949$	6.49983e10	2.46871
$950$	1.56575e10	0.592502
$951$	1.65377e10	0.623512
$952$	1.91955e9	0.0721056
$953$	−1.90087e10	−0.711423	−0.355711	−	0.934596i	$-0.615761\pi$
$953$	−1.90087e10	−0.711423	−0.355711	+	0.934596i	$0.615761\pi$
$954$	−9.25304e9	−0.345037
$955$	1.18872e10	0.441638
$956$	−1.37228e9	−0.0507972
$957$	9.33529e9	0.344299
$958$	6.79722e9	0.249777
$959$	−1.07278e10	−0.392775
$960$	1.06009e9	0.0386716
$961$	−2.75003e10	−0.999551
$962$	−3.55102e10	−1.28600
$963$	−7.67427e9	−0.276914
$964$	−2.22216e10	−0.798924
$965$	2.40515e9	0.0861582
$966$	−7.51629e8	−0.0268277
$967$	−1.11251e10	−0.395651	−0.197825	−	0.980237i	$-0.563388\pi$
$967$	−1.11251e10	−0.395651	−0.197825	+	0.980237i	$0.563388\pi$
$968$	−7.69260e9	−0.272590
$969$	−1.24383e10	−0.439166
$970$	1.11653e9	0.0392798
$971$	8.16030e9	0.286048	0.143024	−	0.989719i	$-0.454317\pi$
$971$	8.16030e9	0.286048	0.143024	+	0.989719i	$0.454317\pi$
$972$	−9.18330e8	−0.0320750
$973$	1.11412e10	0.387737
$974$	1.19191e10	0.413321
$975$	1.84623e10	0.637924
$976$	−4.84988e9	−0.166977
$977$	1.81735e10	0.623458	0.311729	−	0.950171i	$-0.399092\pi$
$977$	1.81735e10	0.623458	0.311729	+	0.950171i	$0.399092\pi$
$978$	−1.60710e10	−0.549361
$979$	−1.24991e10	−0.425736
$980$	7.11007e9	0.241314
$981$	1.40611e10	0.475531
$982$	3.33705e10	1.12453
$983$	3.84865e10	1.29232	0.646162	−	0.763201i	$-0.276373\pi$
$983$	3.84865e10	1.29232	0.646162	+	0.763201i	$0.276373\pi$
$984$	−4.30100e9	−0.143909
$985$	2.69089e10	0.897158
$986$	1.71642e10	0.570237
$987$	−9.57590e9	−0.317007
$988$	−2.76146e10	−0.910941
$989$	−1.20567e10	−0.396315
$990$	−1.84522e9	−0.0604400
$991$	4.51099e10	1.47236	0.736180	−	0.676786i	$-0.236628\pi$
$991$	4.51099e10	1.47236	0.736180	+	0.676786i	$0.236628\pi$
$992$	−1.15108e8	−0.00374383
$993$	−8.98901e9	−0.291333
$994$	−1.35927e10	−0.438990
$995$	−9.27703e9	−0.298558
$996$	−7.76145e9	−0.248906
$997$	6.99070e9	0.223402	0.111701	−	0.993742i	$-0.464370\pi$
$997$	6.99070e9	0.223402	0.111701	+	0.993742i	$0.464370\pi$
$998$	3.67363e10	1.16987
$999$	7.11592e9	0.225814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.8.a.d.1.1	✓	3
3.2	odd	2		414.8.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.d.1.1	✓	3	1.1	even	1	trivial
414.8.a.c.1.3		3	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)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -149.775 q^{5} -216.000 q^{6} -286.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -149.775 q^{5} -216.000 q^{6} -286.000 q^{7} +512.000 q^{8} +729.000 q^{9} -1198.20 q^{10} +2112.48 q^{11} -1728.00 q^{12} +12277.9 q^{13} -2288.00 q^{14} +4043.91 q^{15} +4096.00 q^{16} -13108.8 q^{17} +5832.00 q^{18} -35142.7 q^{19} -9585.57 q^{20} +7722.00 q^{21} +16899.8 q^{22} -12167.0 q^{23} -13824.0 q^{24} -55692.6 q^{25} +98223.2 q^{26} -19683.0 q^{27} -18304.0 q^{28} -163671. q^{29} +32351.3 q^{30} -3512.82 q^{31} +32768.0 q^{32} -57036.9 q^{33} -104870. q^{34} +42835.5 q^{35} +46656.0 q^{36} -361526. q^{37} -281142. q^{38} -331503. q^{39} -76684.6 q^{40} +311126. q^{41} +61776.0 q^{42} +990934. q^{43} +135198. q^{44} -109186. q^{45} -97336.0 q^{46} -1.24008e6 q^{47} -110592. q^{48} -741747. q^{49} -445541. q^{50} +353937. q^{51} +785786. q^{52} -1.58660e6 q^{53} -157464. q^{54} -316395. q^{55} -146432. q^{56} +948854. q^{57} -1.30937e6 q^{58} -1.85291e6 q^{59} +258811. q^{60} -1.18405e6 q^{61} -28102.6 q^{62} -208494. q^{63} +262144. q^{64} -1.83892e6 q^{65} -456295. q^{66} -3.77863e6 q^{67} -838962. q^{68} +328509. q^{69} +342684. q^{70} +5.94088e6 q^{71} +373248. q^{72} +5.29393e6 q^{73} -2.89221e6 q^{74} +1.50370e6 q^{75} -2.24913e6 q^{76} -604168. q^{77} -2.65203e6 q^{78} -8.12232e6 q^{79} -613477. q^{80} +531441. q^{81} +2.48901e6 q^{82} +4.49158e6 q^{83} +494208. q^{84} +1.96336e6 q^{85} +7.92747e6 q^{86} +4.41912e6 q^{87} +1.08159e6 q^{88} -5.91682e6 q^{89} -873485. q^{90} -3.51148e6 q^{91} -778688. q^{92} +94846.3 q^{93} -9.92064e6 q^{94} +5.26349e6 q^{95} -884736. q^{96} -931842. q^{97} -5.93398e6 q^{98} +1.54000e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9}+O(q^{10}) \) 3 * q + 24 * q^2 - 81 * q^3 + 192 * q^4 + 92 * q^5 - 648 * q^6 - 858 * q^7 + 1536 * q^8 + 2187 * q^9	\( 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9} + 736 q^{10} - 1820 q^{11} - 5184 q^{12} - 3698 q^{13} - 6864 q^{14} - 2484 q^{15} + 12288 q^{16} + 3164 q^{17} + 17496 q^{18} - 14234 q^{19} + 5888 q^{20} + 23166 q^{21} - 14560 q^{22} - 36501 q^{23} - 41472 q^{24} - 170119 q^{25} - 29584 q^{26} - 59049 q^{27} - 54912 q^{28} - 171914 q^{29} - 19872 q^{30} - 124972 q^{31} + 98304 q^{32} + 49140 q^{33} + 25312 q^{34} - 26312 q^{35} + 139968 q^{36} - 679074 q^{37} - 113872 q^{38} + 99846 q^{39} + 47104 q^{40} - 331362 q^{41} + 185328 q^{42} - 145922 q^{43} - 116480 q^{44} + 67068 q^{45} - 292008 q^{46} - 1824192 q^{47} - 331776 q^{48} - 2225241 q^{49} - 1360952 q^{50} - 85428 q^{51} - 236672 q^{52} - 3442448 q^{53} - 472392 q^{54} - 1548176 q^{55} - 439296 q^{56} + 384318 q^{57} - 1375312 q^{58} - 3147660 q^{59} - 158976 q^{60} - 3444530 q^{61} - 999776 q^{62} - 625482 q^{63} + 786432 q^{64} - 2527960 q^{65} + 393120 q^{66} - 921870 q^{67} + 202496 q^{68} + 985527 q^{69} - 210496 q^{70} + 2594520 q^{71} + 1119744 q^{72} - 540826 q^{73} - 5432592 q^{74} + 4593213 q^{75} - 910976 q^{76} + 520520 q^{77} + 798768 q^{78} + 316994 q^{79} + 376832 q^{80} + 1594323 q^{81} - 2650896 q^{82} + 4119316 q^{83} + 1482624 q^{84} - 382312 q^{85} - 1167376 q^{86} + 4641678 q^{87} - 931840 q^{88} + 1417176 q^{89} + 536544 q^{90} + 1057628 q^{91} - 2336064 q^{92} + 3374244 q^{93} - 14593536 q^{94} + 12543632 q^{95} - 2654208 q^{96} - 15089326 q^{97} - 17801928 q^{98} - 1326780 q^{99}+O(q^{100}) \) 3 * q + 24 * q^2 - 81 * q^3 + 192 * q^4 + 92 * q^5 - 648 * q^6 - 858 * q^7 + 1536 * q^8 + 2187 * q^9 + 736 * q^10 - 1820 * q^11 - 5184 * q^12 - 3698 * q^13 - 6864 * q^14 - 2484 * q^15 + 12288 * q^16 + 3164 * q^17 + 17496 * q^18 - 14234 * q^19 + 5888 * q^20 + 23166 * q^21 - 14560 * q^22 - 36501 * q^23 - 41472 * q^24 - 170119 * q^25 - 29584 * q^26 - 59049 * q^27 - 54912 * q^28 - 171914 * q^29 - 19872 * q^30 - 124972 * q^31 + 98304 * q^32 + 49140 * q^33 + 25312 * q^34 - 26312 * q^35 + 139968 * q^36 - 679074 * q^37 - 113872 * q^38 + 99846 * q^39 + 47104 * q^40 - 331362 * q^41 + 185328 * q^42 - 145922 * q^43 - 116480 * q^44 + 67068 * q^45 - 292008 * q^46 - 1824192 * q^47 - 331776 * q^48 - 2225241 * q^49 - 1360952 * q^50 - 85428 * q^51 - 236672 * q^52 - 3442448 * q^53 - 472392 * q^54 - 1548176 * q^55 - 439296 * q^56 + 384318 * q^57 - 1375312 * q^58 - 3147660 * q^59 - 158976 * q^60 - 3444530 * q^61 - 999776 * q^62 - 625482 * q^63 + 786432 * q^64 - 2527960 * q^65 + 393120 * q^66 - 921870 * q^67 + 202496 * q^68 + 985527 * q^69 - 210496 * q^70 + 2594520 * q^71 + 1119744 * q^72 - 540826 * q^73 - 5432592 * q^74 + 4593213 * q^75 - 910976 * q^76 + 520520 * q^77 + 798768 * q^78 + 316994 * q^79 + 376832 * q^80 + 1594323 * q^81 - 2650896 * q^82 + 4119316 * q^83 + 1482624 * q^84 - 382312 * q^85 - 1167376 * q^86 + 4641678 * q^87 - 931840 * q^88 + 1417176 * q^89 + 536544 * q^90 + 1057628 * q^91 - 2336064 * q^92 + 3374244 * q^93 - 14593536 * q^94 + 12543632 * q^95 - 2654208 * q^96 - 15089326 * q^97 - 17801928 * q^98 - 1326780 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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