LMFDB - Embedded newform 69.8.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, 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("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.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:	$21.5545667584$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 466x^{4} + 540x^{3} + 48973x^{2} - 77282x - 1061812 $ x^6 - 2x^5 - 466x^4 + 540x^3 + 48973x^2 - 77282*x - 1061812
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.35108$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.35108 q^{2} +27.0000 q^{3} -116.770 q^{4} +0.849066 q^{5} +90.4791 q^{6} +405.736 q^{7} -820.245 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+3.35108 q^{2} +27.0000 q^{3} -116.770 q^{4} +0.849066 q^{5} +90.4791 q^{6} +405.736 q^{7} -820.245 q^{8} +729.000 q^{9} +2.84529 q^{10} +2663.62 q^{11} -3152.80 q^{12} -9506.50 q^{13} +1359.65 q^{14} +22.9248 q^{15} +12197.9 q^{16} -36191.1 q^{17} +2442.94 q^{18} -35806.9 q^{19} -99.1457 q^{20} +10954.9 q^{21} +8925.99 q^{22} -12167.0 q^{23} -22146.6 q^{24} -78124.3 q^{25} -31857.0 q^{26} +19683.0 q^{27} -47377.9 q^{28} +128430. q^{29} +76.8228 q^{30} -32645.9 q^{31} +145867. q^{32} +71917.7 q^{33} -121279. q^{34} +344.497 q^{35} -85125.5 q^{36} -583405. q^{37} -119992. q^{38} -256676. q^{39} -696.442 q^{40} -756639. q^{41} +36710.6 q^{42} +769827. q^{43} -311031. q^{44} +618.969 q^{45} -40772.6 q^{46} -84751.0 q^{47} +329343. q^{48} -658922. q^{49} -261801. q^{50} -977159. q^{51} +1.11008e6 q^{52} -52581.8 q^{53} +65959.3 q^{54} +2261.59 q^{55} -332803. q^{56} -966787. q^{57} +430380. q^{58} +2.26778e6 q^{59} -2676.93 q^{60} +1.62312e6 q^{61} -109399. q^{62} +295781. q^{63} -1.07252e6 q^{64} -8071.65 q^{65} +241002. q^{66} +2.11923e6 q^{67} +4.22604e6 q^{68} -328509. q^{69} +1154.44 q^{70} -4.91652e6 q^{71} -597958. q^{72} +3.45746e6 q^{73} -1.95504e6 q^{74} -2.10936e6 q^{75} +4.18118e6 q^{76} +1.08072e6 q^{77} -860140. q^{78} +2.91026e6 q^{79} +10356.8 q^{80} +531441. q^{81} -2.53556e6 q^{82} +1.88250e6 q^{83} -1.27920e6 q^{84} -30728.6 q^{85} +2.57975e6 q^{86} +3.46762e6 q^{87} -2.18482e6 q^{88} -234356. q^{89} +2074.22 q^{90} -3.85713e6 q^{91} +1.42074e6 q^{92} -881439. q^{93} -284007. q^{94} -30402.4 q^{95} +3.93842e6 q^{96} +1.52882e7 q^{97} -2.20810e6 q^{98} +1.94178e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9}+O(q^{10}) $ 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9} - 13042 q^{10} - 14824 q^{11} + 4806 q^{12} - 756 q^{13} - 3926 q^{14} - 10044 q^{15} - 13022 q^{16} - 69484 q^{17} - 5832 q^{18} - 43864 q^{19} + 78886 q^{20} - 29808 q^{21} + 98204 q^{22} - 73002 q^{23} - 52812 q^{24} + 228018 q^{25} - 311956 q^{26} + 118098 q^{27} - 545442 q^{28} - 311100 q^{29} - 352134 q^{30} - 245248 q^{31} - 390156 q^{32} - 400248 q^{33} + 235834 q^{34} - 1331256 q^{35} + 129762 q^{36} - 630044 q^{37} + 80910 q^{38} - 20412 q^{39} - 2153982 q^{40} - 969204 q^{41} - 106002 q^{42} - 1770208 q^{43} - 1749140 q^{44} - 271188 q^{45} + 97336 q^{46} - 1400024 q^{47} - 351594 q^{48} + 1985598 q^{49} - 956660 q^{50} - 1876068 q^{51} + 3217272 q^{52} - 1573516 q^{53} - 157464 q^{54} - 431296 q^{55} + 7740702 q^{56} - 1184328 q^{57} + 5987188 q^{58} - 1410320 q^{59} + 2129922 q^{60} - 942172 q^{61} + 3334412 q^{62} - 804816 q^{63} + 1996866 q^{64} - 420944 q^{65} + 2651508 q^{66} - 452072 q^{67} - 9258254 q^{68} - 1971054 q^{69} + 21981136 q^{70} + 122928 q^{71} - 1425924 q^{72} + 16490716 q^{73} - 600104 q^{74} + 6156486 q^{75} + 7428658 q^{76} + 7239696 q^{77} - 8422812 q^{78} + 2458408 q^{79} + 19440230 q^{80} + 3188646 q^{81} + 20510784 q^{82} - 7566456 q^{83} - 14726934 q^{84} + 5817744 q^{85} - 669666 q^{86} - 8399700 q^{87} + 14775668 q^{88} - 20368036 q^{89} - 9507618 q^{90} + 8815576 q^{91} - 2165726 q^{92} - 6621696 q^{93} + 16952576 q^{94} + 5143832 q^{95} - 10534212 q^{96} + 12586972 q^{97} - 39164812 q^{98} - 10806696 q^{99}+O(q^{100}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9 - 13042 * q^10 - 14824 * q^11 + 4806 * q^12 - 756 * q^13 - 3926 * q^14 - 10044 * q^15 - 13022 * q^16 - 69484 * q^17 - 5832 * q^18 - 43864 * q^19 + 78886 * q^20 - 29808 * q^21 + 98204 * q^22 - 73002 * q^23 - 52812 * q^24 + 228018 * q^25 - 311956 * q^26 + 118098 * q^27 - 545442 * q^28 - 311100 * q^29 - 352134 * q^30 - 245248 * q^31 - 390156 * q^32 - 400248 * q^33 + 235834 * q^34 - 1331256 * q^35 + 129762 * q^36 - 630044 * q^37 + 80910 * q^38 - 20412 * q^39 - 2153982 * q^40 - 969204 * q^41 - 106002 * q^42 - 1770208 * q^43 - 1749140 * q^44 - 271188 * q^45 + 97336 * q^46 - 1400024 * q^47 - 351594 * q^48 + 1985598 * q^49 - 956660 * q^50 - 1876068 * q^51 + 3217272 * q^52 - 1573516 * q^53 - 157464 * q^54 - 431296 * q^55 + 7740702 * q^56 - 1184328 * q^57 + 5987188 * q^58 - 1410320 * q^59 + 2129922 * q^60 - 942172 * q^61 + 3334412 * q^62 - 804816 * q^63 + 1996866 * q^64 - 420944 * q^65 + 2651508 * q^66 - 452072 * q^67 - 9258254 * q^68 - 1971054 * q^69 + 21981136 * q^70 + 122928 * q^71 - 1425924 * q^72 + 16490716 * q^73 - 600104 * q^74 + 6156486 * q^75 + 7428658 * q^76 + 7239696 * q^77 - 8422812 * q^78 + 2458408 * q^79 + 19440230 * q^80 + 3188646 * q^81 + 20510784 * q^82 - 7566456 * q^83 - 14726934 * q^84 + 5817744 * q^85 - 669666 * q^86 - 8399700 * q^87 + 14775668 * q^88 - 20368036 * q^89 - 9507618 * q^90 + 8815576 * q^91 - 2165726 * q^92 - 6621696 * q^93 + 16952576 * q^94 + 5143832 * q^95 - 10534212 * q^96 + 12586972 * q^97 - 39164812 * q^98 - 10806696 * 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 $

$2$	3.35108	0.296196	0.148098	−	0.988973i	$-0.452685\pi$
$2$	3.35108	0.296196	0.148098	+	0.988973i	$0.452685\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−116.770	−0.912268
$5$	0.849066	0.00303771	0.00151886	−	0.999999i	$-0.499517\pi$
$5$	0.849066	0.00303771	0.00151886	+	0.999999i	$0.499517\pi$
$6$	90.4791	0.171009
$7$	405.736	0.447095	0.223548	−	0.974693i	$-0.428236\pi$
$7$	405.736	0.447095	0.223548	+	0.974693i	$0.428236\pi$
$8$	−820.245	−0.566407
$9$	729.000	0.333333
$10$	2.84529	0.000899759 0
$11$	2663.62	0.603389	0.301695	−	0.953405i	$-0.402448\pi$
$11$	2663.62	0.603389	0.301695	+	0.953405i	$0.402448\pi$
$12$	−3152.80	−0.526698
$13$	−9506.50	−1.20010	−0.600052	−	0.799961i	$-0.704854\pi$
$13$	−9506.50	−1.20010	−0.600052	+	0.799961i	$0.704854\pi$
$14$	1359.65	0.132428
$15$	22.9248	0.00175382
$16$	12197.9	0.744500
$17$	−36191.1	−1.78661	−0.893306	−	0.449448i	$-0.851621\pi$
$17$	−36191.1	−1.78661	−0.893306	+	0.449448i	$0.851621\pi$
$18$	2442.94	0.0987321
$19$	−35806.9	−1.19765	−0.598824	−	0.800881i	$-0.704365\pi$
$19$	−35806.9	−1.19765	−0.598824	+	0.800881i	$0.704365\pi$
$20$	−99.1457	−0.00277121
$21$	10954.9	0.258131
$22$	8925.99	0.178722
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	−22146.6	−0.327015
$25$	−78124.3	−0.999991
$26$	−31857.0	−0.355467
$27$	19683.0	0.192450
$28$	−47377.9	−0.407871
$29$	128430.	0.977855	0.488927	−	0.872325i	$-0.337388\pi$
$29$	128430.	0.977855	0.488927	+	0.872325i	$0.337388\pi$
$30$	76.8228	0.000519476 0
$31$	−32645.9	−0.196817	−0.0984085	−	0.995146i	$-0.531375\pi$
$31$	−32645.9	−0.196817	−0.0984085	+	0.995146i	$0.531375\pi$
$32$	145867.	0.786925
$33$	71917.7	0.348367
$34$	−121279.	−0.529188
$35$	344.497	0.00135815
$36$	−85125.5	−0.304089
$37$	−583405.	−1.89349	−0.946747	−	0.321978i	$-0.895652\pi$
$37$	−583405.	−1.89349	−0.946747	+	0.321978i	$0.895652\pi$
$38$	−119992.	−0.354739
$39$	−256676.	−0.692881
$40$	−696.442	−0.00172058
$41$	−756639.	−1.71453	−0.857266	−	0.514874i	$-0.827839\pi$
$41$	−756639.	−1.71453	−0.857266	+	0.514874i	$0.827839\pi$
$42$	36710.6	0.0764573
$43$	769827.	1.47657	0.738284	−	0.674490i	$-0.235636\pi$
$43$	769827.	1.47657	0.738284	+	0.674490i	$0.235636\pi$
$44$	−311031.	−0.550452
$45$	618.969	0.00101257
$46$	−40772.6	−0.0617612
$47$	−84751.0	−0.119070	−0.0595350	−	0.998226i	$-0.518962\pi$
$47$	−84751.0	−0.119070	−0.0595350	+	0.998226i	$0.518962\pi$
$48$	329343.	0.429837
$49$	−658922.	−0.800106
$50$	−261801.	−0.296194
$51$	−977159.	−1.03150
$52$	1.11008e6	1.09482
$53$	−52581.8	−0.0485143	−0.0242572	−	0.999706i	$-0.507722\pi$
$53$	−52581.8	−0.0485143	−0.0242572	+	0.999706i	$0.507722\pi$
$54$	65959.3	0.0570030
$55$	2261.59	0.00183292
$56$	−332803.	−0.253238
$57$	−966787.	−0.691462
$58$	430380.	0.289637
$59$	2.26778e6	1.43753	0.718767	−	0.695251i	$-0.244707\pi$
$59$	2.26778e6	1.43753	0.718767	+	0.695251i	$0.244707\pi$
$60$	−2676.93	−0.00159996
$61$	1.62312e6	0.915578	0.457789	−	0.889061i	$-0.348642\pi$
$61$	1.62312e6	0.915578	0.457789	+	0.889061i	$0.348642\pi$
$62$	−109399.	−0.0582965
$63$	295781.	0.149032
$64$	−1.07252e6	−0.511416
$65$	−8071.65	−0.00364557
$66$	241002.	0.103185
$67$	2.11923e6	0.860827	0.430414	−	0.902632i	$-0.358368\pi$
$67$	2.11923e6	0.860827	0.430414	+	0.902632i	$0.358368\pi$
$68$	4.22604e6	1.62987
$69$	−328509.	−0.120386
$70$	1154.44	0.000402278 0
$71$	−4.91652e6	−1.63025	−0.815124	−	0.579286i	$-0.803331\pi$
$71$	−4.91652e6	−1.63025	−0.815124	+	0.579286i	$0.803331\pi$
$72$	−597958.	−0.188802
$73$	3.45746e6	1.04022	0.520112	−	0.854098i	$-0.325890\pi$
$73$	3.45746e6	1.04022	0.520112	+	0.854098i	$0.325890\pi$
$74$	−1.95504e6	−0.560846
$75$	−2.10936e6	−0.577345
$76$	4.18118e6	1.09258
$77$	1.08072e6	0.269772
$78$	−860140.	−0.205229
$79$	2.91026e6	0.664106	0.332053	−	0.943261i	$-0.392259\pi$
$79$	2.91026e6	0.664106	0.332053	+	0.943261i	$0.392259\pi$
$80$	10356.8	0.00226158
$81$	531441.	0.111111
$82$	−2.53556e6	−0.507838
$83$	1.88250e6	0.361379	0.180689	−	0.983540i	$-0.442167\pi$
$83$	1.88250e6	0.361379	0.180689	+	0.983540i	$0.442167\pi$
$84$	−1.27920e6	−0.235484
$85$	−30728.6	−0.00542721
$86$	2.57975e6	0.437354
$87$	3.46762e6	0.564565
$88$	−2.18482e6	−0.341764
$89$	−234356.	−0.0352380	−0.0176190	−	0.999845i	$-0.505609\pi$
$89$	−234356.	−0.0352380	−0.0176190	+	0.999845i	$0.505609\pi$
$90$	2074.22	0.000299920 0
$91$	−3.85713e6	−0.536561
$92$	1.42074e6	0.190221
$93$	−881439.	−0.113632
$94$	−284007.	−0.0352681
$95$	−30402.4	−0.00363811
$96$	3.93842e6	0.454331
$97$	1.52882e7	1.70081	0.850405	−	0.526128i	$-0.176357\pi$
$97$	1.52882e7	1.70081	0.850405	+	0.526128i	$0.176357\pi$
$98$	−2.20810e6	−0.236988
$99$	1.94178e6	0.201130
$100$	9.12259e6	0.912259
$101$	−1.36904e7	−1.32218	−0.661090	−	0.750307i	$-0.729906\pi$
$101$	−1.36904e7	−1.32218	−0.661090	+	0.750307i	$0.729906\pi$
$102$	−3.27454e6	−0.305527
$103$	5.79120e6	0.522202	0.261101	−	0.965311i	$-0.415914\pi$
$103$	5.79120e6	0.522202	0.261101	+	0.965311i	$0.415914\pi$
$104$	7.79766e6	0.679747
$105$	9301.41	0.000784126 0
$106$	−176206.	−0.0143698
$107$	7.59691e6	0.599506	0.299753	−	0.954017i	$-0.403096\pi$
$107$	7.59691e6	0.599506	0.299753	+	0.954017i	$0.403096\pi$
$108$	−2.29839e6	−0.175566
$109$	6.28572e6	0.464903	0.232452	−	0.972608i	$-0.425325\pi$
$109$	6.28572e6	0.464903	0.232452	+	0.972608i	$0.425325\pi$
$110$	7578.76	0.000542905 0
$111$	−1.57519e7	−1.09321
$112$	4.94912e6	0.332862
$113$	−2.80396e7	−1.82809	−0.914044	−	0.405615i	$-0.867057\pi$
$113$	−2.80396e7	−1.82809	−0.914044	+	0.405615i	$0.867057\pi$
$114$	−3.23978e6	−0.204809
$115$	−10330.6	−0.000633407 0
$116$	−1.49968e7	−0.892065
$117$	−6.93024e6	−0.400035
$118$	7.59950e6	0.425792
$119$	−1.46840e7	−0.798786
$120$	−18803.9	−0.000993378 0
$121$	−1.23923e7	−0.635922
$122$	5.43919e6	0.271191
$123$	−2.04293e7	−0.989885
$124$	3.81207e6	0.179550
$125$	−132666.	−0.00607540
$126$	991187.	0.0441427
$127$	−3.71325e7	−1.60858	−0.804288	−	0.594239i	$-0.797453\pi$
$127$	−3.71325e7	−1.60858	−0.804288	+	0.594239i	$0.797453\pi$
$128$	−2.22651e7	−0.938404
$129$	2.07853e7	0.852497
$130$	−27048.7	−0.00107981
$131$	4.87971e6	0.189646	0.0948232	−	0.995494i	$-0.469771\pi$
$131$	4.87971e6	0.189646	0.0948232	+	0.995494i	$0.469771\pi$
$132$	−8.39785e6	−0.317804
$133$	−1.45281e7	−0.535463
$134$	7.10170e6	0.254974
$135$	16712.2	0.000584608 0
$136$	2.96855e7	1.01195
$137$	−3.10114e7	−1.03038	−0.515192	−	0.857075i	$-0.672279\pi$
$137$	−3.10114e7	−1.03038	−0.515192	+	0.857075i	$0.672279\pi$
$138$	−1.10086e6	−0.0356579
$139$	4.11634e7	1.30005	0.650024	−	0.759914i	$-0.274759\pi$
$139$	4.11634e7	1.30005	0.650024	+	0.759914i	$0.274759\pi$
$140$	−40227.0	−0.00123899
$141$	−2.28828e6	−0.0687451
$142$	−1.64757e7	−0.482874
$143$	−2.53217e7	−0.724130
$144$	8.89226e6	0.248167
$145$	109046.	0.00297044
$146$	1.15862e7	0.308110
$147$	−1.77909e7	−0.461941
$148$	6.81243e7	1.72737
$149$	−5.31056e7	−1.31519	−0.657594	−	0.753372i	$-0.728426\pi$
$149$	−5.31056e7	−1.31519	−0.657594	+	0.753372i	$0.728426\pi$
$150$	−7.06862e6	−0.171007
$151$	4.43931e7	1.04929	0.524646	−	0.851320i	$-0.324198\pi$
$151$	4.43931e7	1.04929	0.524646	+	0.851320i	$0.324198\pi$
$152$	2.93704e7	0.678356
$153$	−2.63833e7	−0.595538
$154$	3.62159e6	0.0799056
$155$	−27718.5	−0.000597873 0
$156$	2.99721e7	0.632093
$157$	−8.18134e7	−1.68723	−0.843617	−	0.536945i	$-0.819578\pi$
$157$	−8.18134e7	−1.68723	−0.843617	+	0.536945i	$0.819578\pi$
$158$	9.75252e6	0.196706
$159$	−1.41971e6	−0.0280098
$160$	123851.	0.00239045
$161$	−4.93659e6	−0.0932258
$162$	1.78090e6	0.0329107
$163$	8.21430e7	1.48564	0.742820	−	0.669491i	$-0.233488\pi$
$163$	8.21430e7	1.48564	0.742820	+	0.669491i	$0.233488\pi$
$164$	8.83530e7	1.56411
$165$	61062.9	0.00105824
$166$	6.30842e6	0.107039
$167$	7.89460e7	1.31166	0.655831	−	0.754907i	$-0.272318\pi$
$167$	7.89460e7	1.31166	0.655831	+	0.754907i	$0.272318\pi$
$168$	−8.98567e6	−0.146207
$169$	2.76251e7	0.440251
$170$	−102974.	−0.00160752
$171$	−2.61032e7	−0.399216
$172$	−8.98929e7	−1.34703
$173$	9.10823e7	1.33744	0.668718	−	0.743517i	$-0.266844\pi$
$173$	9.10823e7	1.33744	0.668718	+	0.743517i	$0.266844\pi$
$174$	1.16203e7	0.167222
$175$	−3.16978e7	−0.447091
$176$	3.24905e7	0.449223
$177$	6.12300e7	0.829961
$178$	−785346.	−0.0104374
$179$	−1.15252e8	−1.50197	−0.750985	−	0.660319i	$-0.770421\pi$
$179$	−1.15252e8	−1.50197	−0.750985	+	0.660319i	$0.770421\pi$
$180$	−72277.2	−0.000923736 0
$181$	4.18769e6	0.0524927	0.0262464	−	0.999656i	$-0.491645\pi$
$181$	4.18769e6	0.0524927	0.0262464	+	0.999656i	$0.491645\pi$
$182$	−1.29255e7	−0.158927
$183$	4.38241e7	0.528609
$184$	9.97992e6	0.118104
$185$	−495349.	−0.00575189
$186$	−2.95377e6	−0.0336575
$187$	−9.63992e7	−1.07802
$188$	9.89639e6	0.108624
$189$	7.98610e6	0.0860435
$190$	−101881.	−0.00107759
$191$	−1.27500e7	−0.132402	−0.0662010	−	0.997806i	$-0.521088\pi$
$191$	−1.27500e7	−0.132402	−0.0662010	+	0.997806i	$0.521088\pi$
$192$	−2.89579e7	−0.295266
$193$	−3.88747e7	−0.389240	−0.194620	−	0.980879i	$-0.562347\pi$
$193$	−3.88747e7	−0.389240	−0.194620	+	0.980879i	$0.562347\pi$
$194$	5.12321e7	0.503774
$195$	−217935.	−0.00210477
$196$	7.69424e7	0.729911
$197$	7.01957e7	0.654152	0.327076	−	0.944998i	$-0.393937\pi$
$197$	7.01957e7	0.654152	0.327076	+	0.944998i	$0.393937\pi$
$198$	6.50705e6	0.0595739
$199$	1.64934e7	0.148362	0.0741811	−	0.997245i	$-0.476366\pi$
$199$	1.64934e7	0.148362	0.0741811	+	0.997245i	$0.476366\pi$
$200$	6.40810e7	0.566402
$201$	5.72192e7	0.496999
$202$	−4.58775e7	−0.391625
$203$	5.21087e7	0.437194
$204$	1.14103e8	0.941005
$205$	−642437.	−0.00520825
$206$	1.94068e7	0.154674
$207$	−8.86974e6	−0.0695048
$208$	−1.15959e8	−0.893478
$209$	−9.53759e7	−0.722648
$210$	31169.8	0.000232255 0
$211$	−1.48167e8	−1.08583	−0.542917	−	0.839787i	$-0.682680\pi$
$211$	−1.48167e8	−1.08583	−0.542917	+	0.839787i	$0.682680\pi$
$212$	6.13999e6	0.0442581
$213$	−1.32746e8	−0.941224
$214$	2.54578e7	0.177571
$215$	653634.	0.00448539
$216$	−1.61449e7	−0.109005
$217$	−1.32456e7	−0.0879959
$218$	2.10640e7	0.137703
$219$	9.33513e7	0.600573
$220$	−264086.	−0.00167212
$221$	3.44051e8	2.14412
$222$	−5.27860e7	−0.323805
$223$	1.24032e8	0.748972	0.374486	−	0.927233i	$-0.377819\pi$
$223$	1.24032e8	0.748972	0.374486	+	0.927233i	$0.377819\pi$
$224$	5.91836e7	0.351830
$225$	−5.69526e7	−0.333330
$226$	−9.39629e7	−0.541473
$227$	−3.44311e8	−1.95371	−0.976855	−	0.213902i	$-0.931383\pi$
$227$	−3.44311e8	−1.95371	−0.976855	+	0.213902i	$0.931383\pi$
$228$	1.12892e8	0.630799
$229$	−1.13249e8	−0.623177	−0.311589	−	0.950217i	$-0.600861\pi$
$229$	−1.13249e8	−0.623177	−0.311589	+	0.950217i	$0.600861\pi$
$230$	−34618.6	−0.000187613 0
$231$	2.91796e7	0.155753
$232$	−1.05344e8	−0.553863
$233$	−1.91098e8	−0.989717	−0.494859	−	0.868973i	$-0.664780\pi$
$233$	−1.91098e8	−0.989717	−0.494859	+	0.868973i	$0.664780\pi$
$234$	−2.32238e7	−0.118489
$235$	−71959.2	−0.000361700 0
$236$	−2.64809e8	−1.31142
$237$	7.85771e7	0.383422
$238$	−4.92073e7	−0.236598
$239$	3.38240e7	0.160263	0.0801313	−	0.996784i	$-0.474466\pi$
$239$	3.38240e7	0.160263	0.0801313	+	0.996784i	$0.474466\pi$
$240$	279634.	0.00130572
$241$	−1.53606e8	−0.706883	−0.353441	−	0.935457i	$-0.614989\pi$
$241$	−1.53606e8	−0.706883	−0.353441	+	0.935457i	$0.614989\pi$
$242$	−4.15276e7	−0.188358
$243$	1.43489e7	0.0641500
$244$	−1.89532e8	−0.835252
$245$	−559468.	−0.00243049
$246$	−6.84601e7	−0.293200
$247$	3.40398e8	1.43730
$248$	2.67776e7	0.111478
$249$	5.08276e7	0.208642
$250$	−444574.	−0.00179951
$251$	1.31841e8	0.526250	0.263125	−	0.964762i	$-0.415247\pi$
$251$	1.31841e8	0.526250	0.263125	+	0.964762i	$0.415247\pi$
$252$	−3.45385e7	−0.135957
$253$	−3.24082e7	−0.125815
$254$	−1.24434e8	−0.476454
$255$	−829673.	−0.00313340
$256$	6.26700e7	0.233464
$257$	1.01425e8	0.372718	0.186359	−	0.982482i	$-0.440331\pi$
$257$	1.01425e8	0.372718	0.186359	+	0.982482i	$0.440331\pi$
$258$	6.96533e7	0.252507
$259$	−2.36708e8	−0.846572
$260$	942529.	0.00332574
$261$	9.36256e7	0.325952
$262$	1.63523e7	0.0561726
$263$	3.14970e8	1.06764	0.533819	−	0.845599i	$-0.320756\pi$
$263$	3.14970e8	1.06764	0.533819	+	0.845599i	$0.320756\pi$
$264$	−5.89901e7	−0.197317
$265$	−44645.5	−0.000147373 0
$266$	−4.86850e7	−0.158602
$267$	−6.32761e6	−0.0203447
$268$	−2.47463e8	−0.785305
$269$	2.25334e8	0.705819	0.352909	−	0.935657i	$-0.385192\pi$
$269$	2.25334e8	0.705819	0.352909	+	0.935657i	$0.385192\pi$
$270$	56003.8	0.000173159 0
$271$	3.42841e7	0.104641	0.0523204	−	0.998630i	$-0.483338\pi$
$271$	3.42841e7	0.104641	0.0523204	+	0.998630i	$0.483338\pi$
$272$	−4.41455e8	−1.33013
$273$	−1.04142e8	−0.309784
$274$	−1.03922e8	−0.305196
$275$	−2.08093e8	−0.603384
$276$	3.83601e7	0.109824
$277$	−3.88720e8	−1.09890	−0.549449	−	0.835527i	$-0.685162\pi$
$277$	−3.88720e8	−1.09890	−0.549449	+	0.835527i	$0.685162\pi$
$278$	1.37942e8	0.385070
$279$	−2.37988e7	−0.0656056
$280$	−282571.	−0.000769263 0
$281$	−2.65787e8	−0.714597	−0.357299	−	0.933990i	$-0.616302\pi$
$281$	−2.65787e8	−0.714597	−0.357299	+	0.933990i	$0.616302\pi$
$282$	−7.66820e6	−0.0203620
$283$	9.51931e7	0.249662	0.124831	−	0.992178i	$-0.460161\pi$
$283$	9.51931e7	0.249662	0.124831	+	0.992178i	$0.460161\pi$
$284$	5.74104e8	1.48722
$285$	−820866.	−0.00210046
$286$	−8.48550e7	−0.214485
$287$	−3.06996e8	−0.766559
$288$	1.06337e8	0.262308
$289$	8.99456e8	2.19198
$290$	365421.	0.000879834 0
$291$	4.12782e8	0.981964
$292$	−4.03728e8	−0.948962
$293$	−2.36355e8	−0.548944	−0.274472	−	0.961595i	$-0.588503\pi$
$293$	−2.36355e8	−0.548944	−0.274472	+	0.961595i	$0.588503\pi$
$294$	−5.96187e7	−0.136825
$295$	1.92549e6	0.00436682
$296$	4.78535e8	1.07249
$297$	5.24280e7	0.116122
$298$	−1.77961e8	−0.389554
$299$	1.15666e8	0.250239
$300$	2.46310e8	0.526693
$301$	3.12346e8	0.660167
$302$	1.48765e8	0.310797
$303$	−3.69640e8	−0.763360
$304$	−4.36769e8	−0.891649
$305$	1.37813e6	0.00278126
$306$	−8.84125e7	−0.176396
$307$	−4.36850e8	−0.861684	−0.430842	−	0.902427i	$-0.641783\pi$
$307$	−4.36850e8	−0.861684	−0.430842	+	0.902427i	$0.641783\pi$
$308$	−1.26197e8	−0.246105
$309$	1.56363e8	0.301493
$310$	−92886.9	−0.000177088 0
$311$	−1.43706e8	−0.270904	−0.135452	−	0.990784i	$-0.543249\pi$
$311$	−1.43706e8	−0.270904	−0.135452	+	0.990784i	$0.543249\pi$
$312$	2.10537e8	0.392452
$313$	7.98141e8	1.47121	0.735605	−	0.677411i	$-0.236898\pi$
$313$	7.98141e8	1.47121	0.735605	+	0.677411i	$0.236898\pi$
$314$	−2.74163e8	−0.499753
$315$	251138.	0.000452716 0
$316$	−3.39832e8	−0.605842
$317$	1.58206e8	0.278943	0.139472	−	0.990226i	$-0.455460\pi$
$317$	1.58206e8	0.278943	0.139472	+	0.990226i	$0.455460\pi$
$318$	−4.75756e6	−0.00829639
$319$	3.42089e8	0.590027
$320$	−910638.	−0.00155353
$321$	2.05116e8	0.346125
$322$	−1.65429e7	−0.0276131
$323$	1.29589e9	2.13973
$324$	−6.20565e7	−0.101363
$325$	7.42689e8	1.20009
$326$	2.75268e8	0.440041
$327$	1.69714e8	0.268412
$328$	6.20629e8	0.971122
$329$	−3.43865e7	−0.0532356
$330$	204627.	0.000313446 0
$331$	5.55118e8	0.841370	0.420685	−	0.907207i	$-0.361790\pi$
$331$	5.55118e8	0.841370	0.420685	+	0.907207i	$0.361790\pi$
$332$	−2.19821e8	−0.329674
$333$	−4.25302e8	−0.631165
$334$	2.64554e8	0.388510
$335$	1.79937e6	0.00261495
$336$	1.33626e8	0.192178
$337$	−2.24145e8	−0.319025	−0.159512	−	0.987196i	$-0.550992\pi$
$337$	−2.24145e8	−0.319025	−0.159512	+	0.987196i	$0.550992\pi$
$338$	9.25738e7	0.130401
$339$	−7.57069e8	−1.05545
$340$	3.58819e6	0.00495107
$341$	−8.69561e7	−0.118757
$342$	−8.74740e7	−0.118246
$343$	−6.01489e8	−0.804819
$344$	−6.31447e8	−0.836339
$345$	−278926.	−0.000365698 0
$346$	3.05224e8	0.396143
$347$	3.33457e7	0.0428437	0.0214219	−	0.999771i	$-0.493181\pi$
$347$	3.33457e7	0.0428437	0.0214219	+	0.999771i	$0.493181\pi$
$348$	−4.04915e8	−0.515034
$349$	9.08225e8	1.14368	0.571840	−	0.820365i	$-0.306230\pi$
$349$	9.08225e8	1.14368	0.571840	+	0.820365i	$0.306230\pi$
$350$	−1.06222e8	−0.132427
$351$	−1.87116e8	−0.230960
$352$	3.88535e8	0.474822
$353$	3.91774e8	0.474051	0.237025	−	0.971503i	$-0.423828\pi$
$353$	3.91774e8	0.474051	0.237025	+	0.971503i	$0.423828\pi$
$354$	2.05186e8	0.245831
$355$	−4.17445e6	−0.00495222
$356$	2.73658e7	0.0321465
$357$	−3.96468e8	−0.461179
$358$	−3.86217e8	−0.444878
$359$	−1.07880e9	−1.23058	−0.615288	−	0.788302i	$-0.710960\pi$
$359$	−1.07880e9	−1.23058	−0.615288	+	0.788302i	$0.710960\pi$
$360$	−507706.	−0.000573527 0
$361$	3.88263e8	0.434361
$362$	1.40333e7	0.0155482
$363$	−3.34592e8	−0.367149
$364$	4.50398e8	0.489487
$365$	2.93561e6	0.00315990
$366$	1.46858e8	0.156572
$367$	1.24645e9	1.31626	0.658131	−	0.752903i	$-0.271347\pi$
$367$	1.24645e9	1.31626	0.658131	+	0.752903i	$0.271347\pi$
$368$	−1.48412e8	−0.155239
$369$	−5.51590e8	−0.571510
$370$	−1.65996e6	−0.00170369
$371$	−2.13343e7	−0.0216905
$372$	1.02926e8	0.103663
$373$	4.12114e8	0.411184	0.205592	−	0.978638i	$-0.434088\pi$
$373$	4.12114e8	0.411184	0.205592	+	0.978638i	$0.434088\pi$
$374$	−3.23041e8	−0.319306
$375$	−3.58198e6	−0.00350763
$376$	6.95165e7	0.0674420
$377$	−1.22092e9	−1.17353
$378$	2.67620e7	0.0254858
$379$	−1.27834e9	−1.20617	−0.603085	−	0.797677i	$-0.706062\pi$
$379$	−1.27834e9	−1.20617	−0.603085	+	0.797677i	$0.706062\pi$
$380$	3.55010e6	0.00331893
$381$	−1.00258e9	−0.928712
$382$	−4.27264e7	−0.0392170
$383$	−9.34297e8	−0.849746	−0.424873	−	0.905253i	$-0.639681\pi$
$383$	−9.34297e8	−0.849746	−0.424873	+	0.905253i	$0.639681\pi$
$384$	−6.01158e8	−0.541788
$385$	917607.	0.000819491 0
$386$	−1.30272e8	−0.115291
$387$	5.61204e8	0.492190
$388$	−1.78521e9	−1.55159
$389$	3.03608e8	0.261511	0.130755	−	0.991415i	$-0.458260\pi$
$389$	3.03608e8	0.261511	0.130755	+	0.991415i	$0.458260\pi$
$390$	−730316.	−0.000623426 0
$391$	4.40337e8	0.372534
$392$	5.40477e8	0.453185
$393$	1.31752e8	0.109492
$394$	2.35231e8	0.193757
$395$	2.47100e6	0.00201736
$396$	−2.26742e8	−0.183484
$397$	−4.05735e8	−0.325443	−0.162722	−	0.986672i	$-0.552027\pi$
$397$	−4.05735e8	−0.325443	−0.162722	+	0.986672i	$0.552027\pi$
$398$	5.52706e7	0.0439444
$399$	−3.92260e8	−0.309150
$400$	−9.52951e8	−0.744493
$401$	−8.42816e8	−0.652721	−0.326360	−	0.945245i	$-0.605822\pi$
$401$	−8.42816e8	−0.652721	−0.326360	+	0.945245i	$0.605822\pi$
$402$	1.91746e8	0.147209
$403$	3.10348e8	0.236201
$404$	1.59863e9	1.20618
$405$	451229.	0.000337524 0
$406$	1.74621e8	0.129495
$407$	−1.55397e9	−1.14251
$408$	8.01510e8	0.584249
$409$	9.52820e8	0.688619	0.344310	−	0.938856i	$-0.388113\pi$
$409$	9.52820e8	0.688619	0.344310	+	0.938856i	$0.388113\pi$
$410$	−2.15286e6	−0.00154267
$411$	−8.37308e8	−0.594893
$412$	−6.76240e8	−0.476388
$413$	9.20118e8	0.642715
$414$	−2.97232e7	−0.0205871
$415$	1.59837e6	0.00109776
$416$	−1.38669e9	−0.944392
$417$	1.11141e9	0.750583
$418$	−3.19612e8	−0.214046
$419$	−1.41758e9	−0.941454	−0.470727	−	0.882279i	$-0.656008\pi$
$419$	−1.41758e9	−0.941454	−0.470727	+	0.882279i	$0.656008\pi$
$420$	−1.08613e6	−0.000715333 0
$421$	1.59148e9	1.03948	0.519738	−	0.854326i	$-0.326030\pi$
$421$	1.59148e9	1.03948	0.519738	+	0.854326i	$0.326030\pi$
$422$	−4.96520e8	−0.321620
$423$	−6.17835e7	−0.0396900
$424$	4.31300e7	0.0274788
$425$	2.82740e9	1.78660
$426$	−4.44843e8	−0.278787
$427$	6.58556e8	0.409351
$428$	−8.87093e8	−0.546910
$429$	−6.83686e8	−0.418077
$430$	2.19038e6	0.00132856
$431$	−2.04772e9	−1.23197	−0.615984	−	0.787759i	$-0.711241\pi$
$431$	−2.04772e9	−1.23197	−0.615984	+	0.787759i	$0.711241\pi$
$432$	2.40091e8	0.143279
$433$	1.15795e9	0.685458	0.342729	−	0.939434i	$-0.388649\pi$
$433$	1.15795e9	0.685458	0.342729	+	0.939434i	$0.388649\pi$
$434$	−4.43870e7	−0.0260641
$435$	2.94424e6	0.00171498
$436$	−7.33985e8	−0.424116
$437$	4.35663e8	0.249727
$438$	3.12828e8	0.177888
$439$	1.71302e8	0.0966356	0.0483178	−	0.998832i	$-0.484614\pi$
$439$	1.71302e8	0.0966356	0.0483178	+	0.998832i	$0.484614\pi$
$440$	−1.85506e6	−0.00103818
$441$	−4.80354e8	−0.266702
$442$	1.15294e9	0.635081
$443$	−2.01064e9	−1.09881	−0.549404	−	0.835557i	$-0.685145\pi$
$443$	−2.01064e9	−1.09881	−0.549404	+	0.835557i	$0.685145\pi$
$444$	1.83936e9	0.997300
$445$	−198984.	−0.000107043 0
$446$	4.15640e8	0.221843
$447$	−1.43385e9	−0.759324
$448$	−4.35158e8	−0.228652
$449$	1.66574e9	0.868451	0.434225	−	0.900804i	$-0.357022\pi$
$449$	1.66574e9	0.868451	0.434225	+	0.900804i	$0.357022\pi$
$450$	−1.90853e8	−0.0987312
$451$	−2.01540e9	−1.03453
$452$	3.27419e9	1.66771
$453$	1.19861e9	0.605809
$454$	−1.15381e9	−0.578682
$455$	−3.27496e6	−0.00162992
$456$	7.93001e8	0.391649
$457$	−1.76044e9	−0.862811	−0.431405	−	0.902158i	$-0.641982\pi$
$457$	−1.76044e9	−0.862811	−0.431405	+	0.902158i	$0.641982\pi$
$458$	−3.79508e8	−0.184583
$459$	−7.12349e8	−0.343834
$460$	1.20631e6	0.000577837 0
$461$	−2.02454e9	−0.962440	−0.481220	−	0.876600i	$-0.659806\pi$
$461$	−2.02454e9	−0.962440	−0.481220	+	0.876600i	$0.659806\pi$
$462$	9.77831e7	0.0461335
$463$	−2.65240e9	−1.24196	−0.620978	−	0.783828i	$-0.713264\pi$
$463$	−2.65240e9	−1.24196	−0.620978	+	0.783828i	$0.713264\pi$
$464$	1.56658e9	0.728013
$465$	−748400.	−0.000345182 0
$466$	−6.40386e8	−0.293151
$467$	2.57500e9	1.16995	0.584977	−	0.811050i	$-0.301103\pi$
$467$	2.57500e9	1.16995	0.584977	+	0.811050i	$0.301103\pi$
$468$	8.09246e8	0.364939
$469$	8.59847e8	0.384872
$470$	−241141.	−0.000107134 0
$471$	−2.20896e9	−0.974125
$472$	−1.86013e9	−0.814229
$473$	2.05053e9	0.890946
$474$	2.63318e8	0.113568
$475$	2.79739e9	1.19764
$476$	1.71466e9	0.728707
$477$	−3.83321e7	−0.0161714
$478$	1.13347e8	0.0474692
$479$	−8.23862e8	−0.342515	−0.171258	−	0.985226i	$-0.554783\pi$
$479$	−8.23862e8	−0.342515	−0.171258	+	0.985226i	$0.554783\pi$
$480$	3.34398e6	0.00138013
$481$	5.54614e9	2.27239
$482$	−5.14745e8	−0.209376
$483$	−1.33288e8	−0.0538239
$484$	1.44705e9	0.580131
$485$	1.29807e7	0.00516657
$486$	4.80843e7	0.0190010
$487$	2.67396e9	1.04907	0.524533	−	0.851390i	$-0.324240\pi$
$487$	2.67396e9	1.04907	0.524533	+	0.851390i	$0.324240\pi$
$488$	−1.33135e9	−0.518589
$489$	2.21786e9	0.857735
$490$	−1.87482e6	−0.000719903 0
$491$	−1.23865e9	−0.472239	−0.236120	−	0.971724i	$-0.575876\pi$
$491$	−1.23865e9	−0.472239	−0.236120	+	0.971724i	$0.575876\pi$
$492$	2.38553e9	0.903040
$493$	−4.64803e9	−1.74705
$494$	1.14070e9	0.425724
$495$	1.64870e6	0.000610974 0
$496$	−3.98211e8	−0.146530
$497$	−1.99481e9	−0.728876
$498$	1.70327e8	0.0617990
$499$	−2.12672e9	−0.766229	−0.383115	−	0.923701i	$-0.625149\pi$
$499$	−2.12672e9	−0.766229	−0.383115	+	0.923701i	$0.625149\pi$
$500$	1.54914e7	0.00554239
$501$	2.13154e9	0.757289
$502$	4.41809e8	0.155873
$503$	2.18140e8	0.0764271	0.0382135	−	0.999270i	$-0.487833\pi$
$503$	2.18140e8	0.0764271	0.0382135	+	0.999270i	$0.487833\pi$
$504$	−2.42613e8	−0.0844126
$505$	−1.16240e7	−0.00401640
$506$	−1.08603e8	−0.0372660
$507$	7.45877e8	0.254179
$508$	4.33598e9	1.46745
$509$	2.84720e9	0.956985	0.478492	−	0.878092i	$-0.341183\pi$
$509$	2.84720e9	0.956985	0.478492	+	0.878092i	$0.341183\pi$
$510$	−2.78030e6	−0.000928103 0
$511$	1.40281e9	0.465079
$512$	3.05995e9	1.00756
$513$	−7.04787e8	−0.230487
$514$	3.39884e8	0.110398
$515$	4.91712e6	0.00158630
$516$	−2.42711e9	−0.777706
$517$	−2.25744e8	−0.0718455
$518$	−7.93228e8	−0.250752
$519$	2.45922e9	0.772169
$520$	6.62073e6	0.00206488
$521$	−3.03891e9	−0.941426	−0.470713	−	0.882286i	$-0.656003\pi$
$521$	−3.03891e9	−0.941426	−0.470713	+	0.882286i	$0.656003\pi$
$522$	3.13747e8	0.0965457
$523$	−3.95904e8	−0.121013	−0.0605067	−	0.998168i	$-0.519272\pi$
$523$	−3.95904e8	−0.121013	−0.0605067	+	0.998168i	$0.519272\pi$
$524$	−5.69805e8	−0.173008
$525$	−8.55841e8	−0.258128
$526$	1.05549e9	0.316230
$527$	1.18149e9	0.351636
$528$	8.77244e8	0.259359
$529$	1.48036e8	0.0434783
$530$	−149610.	−4.36512e−5 0
$531$	1.65321e9	0.479178
$532$	1.69645e9	0.488485
$533$	7.19299e9	2.05762
$534$	−2.12043e7	−0.00602601
$535$	6.45028e6	0.00182113
$536$	−1.73829e9	−0.487578
$537$	−3.11179e9	−0.867163
$538$	7.55111e8	0.209061
$539$	−1.75512e9	−0.482775
$540$	−1.95148e6	−0.000533319 0
$541$	−5.18940e9	−1.40905	−0.704525	−	0.709679i	$-0.748840\pi$
$541$	−5.18940e9	−1.40905	−0.704525	+	0.709679i	$0.748840\pi$
$542$	1.14889e8	0.0309942
$543$	1.13068e8	0.0303067
$544$	−5.27910e9	−1.40593
$545$	5.33700e6	0.00141224
$546$	−3.48990e8	−0.0917568
$547$	−7.01245e9	−1.83195	−0.915976	−	0.401232i	$-0.868582\pi$
$547$	−7.01245e9	−1.83195	−0.915976	+	0.401232i	$0.868582\pi$
$548$	3.62121e9	0.939987
$549$	1.18325e9	0.305193
$550$	−6.97337e8	−0.178720
$551$	−4.59869e9	−1.17113
$552$	2.69458e8	0.0681874
$553$	1.18080e9	0.296919
$554$	−1.30263e9	−0.325490
$555$	−1.33744e7	−0.00332086
$556$	−4.80666e9	−1.18599
$557$	−2.56612e9	−0.629193	−0.314597	−	0.949225i	$-0.601869\pi$
$557$	−2.56612e9	−0.629193	−0.314597	+	0.949225i	$0.601869\pi$
$558$	−7.97518e7	−0.0194322
$559$	−7.31836e9	−1.77204
$560$	4.20213e6	0.00101114
$561$	−2.60278e9	−0.622397
$562$	−8.90673e8	−0.211661
$563$	−8.98840e8	−0.212277	−0.106139	−	0.994351i	$-0.533849\pi$
$563$	−8.98840e8	−0.212277	−0.106139	+	0.994351i	$0.533849\pi$
$564$	2.67203e8	0.0627139
$565$	−2.38075e7	−0.00555320
$566$	3.18999e8	0.0739490
$567$	2.15625e8	0.0496773
$568$	4.03275e9	0.923384
$569$	−3.86463e9	−0.879458	−0.439729	−	0.898130i	$-0.644926\pi$
$569$	−3.86463e9	−0.879458	−0.439729	+	0.898130i	$0.644926\pi$
$570$	−2.75079e6	−0.000622150 0
$571$	4.00586e9	0.900471	0.450235	−	0.892910i	$-0.351340\pi$
$571$	4.00586e9	0.900471	0.450235	+	0.892910i	$0.351340\pi$
$572$	2.95682e9	0.660600
$573$	−3.44251e8	−0.0764424
$574$	−1.02877e9	−0.227052
$575$	9.50538e8	0.208512
$576$	−7.81865e8	−0.170472
$577$	−5.38283e9	−1.16653	−0.583264	−	0.812282i	$-0.698225\pi$
$577$	−5.38283e9	−1.16653	−0.583264	+	0.812282i	$0.698225\pi$
$578$	3.01415e9	0.649258
$579$	−1.04962e9	−0.224728
$580$	−1.27333e7	−0.00270984
$581$	7.63799e8	0.161571
$582$	1.38327e9	0.290854
$583$	−1.40058e8	−0.0292730
$584$	−2.83596e9	−0.589190
$585$	−5.88423e6	−0.00121519
$586$	−7.92044e8	−0.162595
$587$	4.39683e9	0.897234	0.448617	−	0.893724i	$-0.351917\pi$
$587$	4.39683e9	0.897234	0.448617	+	0.893724i	$0.351917\pi$
$588$	2.07745e9	0.421414
$589$	1.16895e9	0.235717
$590$	6.45248e6	0.00129343
$591$	1.89528e9	0.377675
$592$	−7.11631e9	−1.40971
$593$	−1.78104e9	−0.350737	−0.175368	−	0.984503i	$-0.556112\pi$
$593$	−1.78104e9	−0.350737	−0.175368	+	0.984503i	$0.556112\pi$
$594$	1.75690e8	0.0343950
$595$	−1.24677e7	−0.00242648
$596$	6.20115e9	1.19980
$597$	4.45321e8	0.0856570
$598$	3.87605e8	0.0741199
$599$	−3.87905e9	−0.737449	−0.368724	−	0.929539i	$-0.620205\pi$
$599$	−3.87905e9	−0.737449	−0.368724	+	0.929539i	$0.620205\pi$
$600$	1.73019e9	0.327012
$601$	3.87683e9	0.728478	0.364239	−	0.931305i	$-0.381329\pi$
$601$	3.87683e9	0.728478	0.364239	+	0.931305i	$0.381329\pi$
$602$	1.04670e9	0.195539
$603$	1.54492e9	0.286942
$604$	−5.18380e9	−0.957236
$605$	−1.05219e7	−0.00193175
$606$	−1.23869e9	−0.226105
$607$	−6.73366e9	−1.22205	−0.611027	−	0.791610i	$-0.709243\pi$
$607$	−6.73366e9	−1.22205	−0.611027	+	0.791610i	$0.709243\pi$
$608$	−5.22306e9	−0.942459
$609$	1.40694e9	0.252414
$610$	4.61824e6	0.000823800 0
$611$	8.05685e8	0.142896
$612$	3.08079e9	0.543290
$613$	6.28531e9	1.10208	0.551042	−	0.834477i	$-0.314230\pi$
$613$	6.28531e9	1.10208	0.551042	+	0.834477i	$0.314230\pi$
$614$	−1.46392e9	−0.255228
$615$	−1.73458e7	−0.00300699
$616$	−8.86459e8	−0.152801
$617$	−6.71956e9	−1.15171	−0.575854	−	0.817552i	$-0.695330\pi$
$617$	−6.71956e9	−1.15171	−0.575854	+	0.817552i	$0.695330\pi$
$618$	5.23983e8	0.0893013
$619$	1.62081e9	0.274672	0.137336	−	0.990525i	$-0.456146\pi$
$619$	1.62081e9	0.274672	0.137336	+	0.990525i	$0.456146\pi$
$620$	3.23670e6	0.000545420 0
$621$	−2.39483e8	−0.0401286
$622$	−4.81572e8	−0.0802407
$623$	−9.50866e7	−0.0157547
$624$	−3.13090e9	−0.515850
$625$	6.10335e9	0.999972
$626$	2.67463e9	0.435767
$627$	−2.57515e9	−0.417221
$628$	9.55337e9	1.53921
$629$	2.11141e10	3.38294
$630$	841583.	0.000134093 0
$631$	−8.43752e9	−1.33694	−0.668470	−	0.743739i	$-0.733051\pi$
$631$	−8.43752e9	−1.33694	−0.668470	+	0.743739i	$0.733051\pi$
$632$	−2.38713e9	−0.376154
$633$	−4.00051e9	−0.626906
$634$	5.30162e8	0.0826220
$635$	−3.15280e7	−0.00488639
$636$	1.65780e8	0.0255524
$637$	6.26404e9	0.960211
$638$	1.14637e9	0.174764
$639$	−3.58414e9	−0.543416
$640$	−1.89046e7	−0.00285060
$641$	−1.15180e10	−1.72733	−0.863663	−	0.504069i	$-0.831836\pi$
$641$	−1.15180e10	−1.72733	−0.863663	+	0.504069i	$0.831836\pi$
$642$	6.87362e8	0.102521
$643$	−1.06978e10	−1.58692	−0.793461	−	0.608621i	$-0.791723\pi$
$643$	−1.06978e10	−1.58692	−0.793461	+	0.608621i	$0.791723\pi$
$644$	5.76447e8	0.0850469
$645$	1.76481e7	0.00258964
$646$	4.34263e9	0.633781
$647$	1.28732e10	1.86862	0.934311	−	0.356459i	$-0.116016\pi$
$647$	1.28732e10	1.86862	0.934311	+	0.356459i	$0.116016\pi$
$648$	−4.35912e8	−0.0629341
$649$	6.04049e9	0.867393
$650$	2.48881e9	0.355463
$651$	−3.57631e8	−0.0508045
$652$	−9.59186e9	−1.35530
$653$	−4.92927e9	−0.692765	−0.346383	−	0.938093i	$-0.612590\pi$
$653$	−4.92927e9	−0.692765	−0.346383	+	0.938093i	$0.612590\pi$
$654$	5.68727e8	0.0795026
$655$	4.14320e6	0.000576091 0
$656$	−9.22940e9	−1.27647
$657$	2.52049e9	0.346741
$658$	−1.15232e8	−0.0157682
$659$	−7.40103e9	−1.00738	−0.503690	−	0.863885i	$-0.668024\pi$
$659$	−7.40103e9	−1.00738	−0.503690	+	0.863885i	$0.668024\pi$
$660$	−7.13033e6	−0.000965397 0
$661$	2.57426e9	0.346695	0.173348	−	0.984861i	$-0.444542\pi$
$661$	2.57426e9	0.346695	0.173348	+	0.984861i	$0.444542\pi$
$662$	1.86024e9	0.249211
$663$	9.28937e9	1.23791
$664$	−1.54411e9	−0.204687
$665$	−1.23354e7	−0.00162658
$666$	−1.42522e9	−0.186949
$667$	−1.56261e9	−0.203897
$668$	−9.21854e9	−1.19659
$669$	3.34886e9	0.432419
$670$	6.02982e6	0.000774537 0
$671$	4.32336e9	0.552450
$672$	1.59796e9	0.203129
$673$	1.33767e9	0.169159	0.0845795	−	0.996417i	$-0.473045\pi$
$673$	1.33767e9	0.169159	0.0845795	+	0.996417i	$0.473045\pi$
$674$	−7.51127e8	−0.0944939
$675$	−1.53772e9	−0.192448
$676$	−3.22579e9	−0.401626
$677$	5.47938e9	0.678689	0.339345	−	0.940662i	$-0.389795\pi$
$677$	5.47938e9	0.678689	0.339345	+	0.940662i	$0.389795\pi$
$678$	−2.53700e9	−0.312620
$679$	6.20298e9	0.760425
$680$	2.52050e7	0.00307401
$681$	−9.29639e9	−1.12798
$682$	−2.91397e8	−0.0351754
$683$	9.90832e9	1.18995	0.594973	−	0.803745i	$-0.297163\pi$
$683$	9.90832e9	1.18995	0.594973	+	0.803745i	$0.297163\pi$
$684$	3.04808e9	0.364192
$685$	−2.63307e7	−0.00313001
$686$	−2.01564e9	−0.238384
$687$	−3.05773e9	−0.359792
$688$	9.39027e9	1.09931
$689$	4.99869e8	0.0582223
$690$	−934703.	−0.000108318 0
$691$	−9.22230e9	−1.06332	−0.531662	−	0.846956i	$-0.678432\pi$
$691$	−9.22230e9	−1.06332	−0.531662	+	0.846956i	$0.678432\pi$
$692$	−1.06357e10	−1.22010
$693$	7.87848e8	0.0899241
$694$	1.11744e8	0.0126902
$695$	3.49505e7	0.00394917
$696$	−2.84429e9	−0.319773
$697$	2.73836e10	3.06320
$698$	3.04353e9	0.338754
$699$	−5.15966e9	−0.571414
$700$	3.70136e9	0.407867
$701$	1.26162e9	0.138329	0.0691646	−	0.997605i	$-0.477967\pi$
$701$	1.26162e9	0.138329	0.0691646	+	0.997605i	$0.477967\pi$
$702$	−6.27042e8	−0.0684096
$703$	2.08899e10	2.26774
$704$	−2.85677e9	−0.308583
$705$	−1.94290e6	−0.000208828 0
$706$	1.31287e9	0.140412
$707$	−5.55467e9	−0.591140
$708$	−7.14984e9	−0.757146
$709$	5.11962e9	0.539481	0.269740	−	0.962933i	$-0.413062\pi$
$709$	5.11962e9	0.539481	0.269740	+	0.962933i	$0.413062\pi$
$710$	−1.39889e7	−0.00146683
$711$	2.12158e9	0.221369
$712$	1.92229e8	0.0199590
$713$	3.97202e8	0.0410392
$714$	−1.32860e9	−0.136600
$715$	−2.14998e7	−0.00219970
$716$	1.34580e10	1.37020
$717$	9.13248e8	0.0925277
$718$	−3.61513e9	−0.364492
$719$	3.58731e9	0.359930	0.179965	−	0.983673i	$-0.442402\pi$
$719$	3.58731e9	0.359930	0.179965	+	0.983673i	$0.442402\pi$
$720$	7.55012e6	0.000753859 0
$721$	2.34970e9	0.233474
$722$	1.30110e9	0.128656
$723$	−4.14735e9	−0.408119
$724$	−4.88997e8	−0.0478874
$725$	−1.00335e10	−0.977846
$726$	−1.12125e9	−0.108748
$727$	1.00951e10	0.974406	0.487203	−	0.873289i	$-0.338017\pi$
$727$	1.00951e10	0.974406	0.487203	+	0.873289i	$0.338017\pi$
$728$	3.16379e9	0.303912
$729$	3.87420e8	0.0370370
$730$	9.83746e6	0.000935951 0
$731$	−2.78609e10	−2.63806
$732$	−5.11736e9	−0.482233
$733$	5.17030e9	0.484900	0.242450	−	0.970164i	$-0.422049\pi$
$733$	5.17030e9	0.484900	0.242450	+	0.970164i	$0.422049\pi$
$734$	4.17694e9	0.389872
$735$	−1.51056e7	−0.00140324
$736$	−1.77477e9	−0.164085
$737$	5.64482e9	0.519414
$738$	−1.84842e9	−0.169279
$739$	1.13371e10	1.03335	0.516675	−	0.856182i	$-0.327170\pi$
$739$	1.13371e10	1.03335	0.516675	+	0.856182i	$0.327170\pi$
$740$	5.78421e7	0.00524726
$741$	9.19076e9	0.829827
$742$	−7.14930e7	−0.00642466
$743$	7.35179e9	0.657555	0.328778	−	0.944407i	$-0.393363\pi$
$743$	7.35179e9	0.657555	0.328778	+	0.944407i	$0.393363\pi$
$744$	7.22995e8	0.0643621
$745$	−4.50901e7	−0.00399516
$746$	1.38103e9	0.121791
$747$	1.37235e9	0.120460
$748$	1.12566e10	0.983445
$749$	3.08234e9	0.268036
$750$	−1.20035e7	−0.00103895
$751$	2.57204e8	0.0221584	0.0110792	−	0.999939i	$-0.496473\pi$
$751$	2.57204e8	0.0221584	0.0110792	+	0.999939i	$0.496473\pi$
$752$	−1.03378e9	−0.0886476
$753$	3.55970e9	0.303830
$754$	−4.09141e9	−0.347595
$755$	3.76927e7	0.00318745
$756$	−9.32539e8	−0.0784947
$757$	2.62859e9	0.220236	0.110118	−	0.993919i	$-0.464877\pi$
$757$	2.62859e9	0.220236	0.110118	+	0.993919i	$0.464877\pi$
$758$	−4.28381e9	−0.357263
$759$	−8.75022e8	−0.0726395
$760$	2.49374e7	0.00206065
$761$	−3.73461e9	−0.307184	−0.153592	−	0.988134i	$-0.549084\pi$
$761$	−3.73461e9	−0.307184	−0.153592	+	0.988134i	$0.549084\pi$
$762$	−3.35972e9	−0.275081
$763$	2.55034e9	0.207856
$764$	1.48883e9	0.120786
$765$	−2.24012e7	−0.00180907
$766$	−3.13090e9	−0.251692
$767$	−2.15586e10	−1.72519
$768$	1.69209e9	0.134790
$769$	−1.96580e10	−1.55882	−0.779412	−	0.626512i	$-0.784482\pi$
$769$	−1.96580e10	−1.55882	−0.779412	+	0.626512i	$0.784482\pi$
$770$	3.07497e6	0.000242730 0
$771$	2.73848e9	0.215189
$772$	4.53941e9	0.355091
$773$	1.22575e9	0.0954497	0.0477248	−	0.998861i	$-0.484803\pi$
$773$	1.22575e9	0.0954497	0.0477248	+	0.998861i	$0.484803\pi$
$774$	1.88064e9	0.145785
$775$	2.55044e9	0.196815
$776$	−1.25401e10	−0.963351
$777$	−6.39112e9	−0.488769
$778$	1.01741e9	0.0774585
$779$	2.70929e10	2.05341
$780$	2.54483e7	0.00192012
$781$	−1.30957e10	−0.983674
$782$	1.47560e9	0.110343
$783$	2.52789e9	0.188188
$784$	−8.03745e9	−0.595679
$785$	−6.94650e7	−0.00512533
$786$	4.41512e8	0.0324312
$787$	−1.08127e10	−0.790721	−0.395361	−	0.918526i	$-0.629380\pi$
$787$	−1.08127e10	−0.790721	−0.395361	+	0.918526i	$0.629380\pi$
$788$	−8.19677e9	−0.596762
$789$	8.50419e9	0.616401
$790$	8.28053e6	0.000597535 0
$791$	−1.13767e10	−0.817329
$792$	−1.59273e9	−0.113921
$793$	−1.54302e10	−1.09879
$794$	−1.35965e9	−0.0963951
$795$	−1.20543e6	−8.50856e−5 0
$796$	−1.92593e9	−0.135346
$797$	−1.95009e10	−1.36443	−0.682215	−	0.731151i	$-0.738983\pi$
$797$	−1.95009e10	−1.36443	−0.682215	+	0.731151i	$0.738983\pi$
$798$	−1.31449e9	−0.0915690
$799$	3.06723e9	0.212732
$800$	−1.13958e10	−0.786918
$801$	−1.70846e8	−0.0117460
$802$	−2.82434e9	−0.193333
$803$	9.20934e9	0.627660
$804$	−6.68150e9	−0.453396
$805$	−4.19149e6	−0.000283193 0
$806$	1.04000e9	0.0699618
$807$	6.08401e9	0.407505
$808$	1.12294e10	0.748891
$809$	2.59821e10	1.72526	0.862631	−	0.505833i	$-0.168815\pi$
$809$	2.59821e10	1.72526	0.862631	+	0.505833i	$0.168815\pi$
$810$	1.51210e6	9.99733e−5 0
$811$	2.97565e9	0.195888	0.0979441	−	0.995192i	$-0.468773\pi$
$811$	2.97565e9	0.195888	0.0979441	+	0.995192i	$0.468773\pi$
$812$	−6.08475e9	−0.398838
$813$	9.25672e8	0.0604144
$814$	−5.20747e9	−0.338408
$815$	6.97448e7	0.00451295
$816$	−1.19193e10	−0.767953
$817$	−2.75651e10	−1.76841
$818$	3.19297e9	0.203966
$819$	−2.81185e9	−0.178854
$820$	7.50175e7	0.00475132
$821$	3.94381e9	0.248722	0.124361	−	0.992237i	$-0.460312\pi$
$821$	3.94381e9	0.248722	0.124361	+	0.992237i	$0.460312\pi$
$822$	−2.80588e9	−0.176205
$823$	−1.36446e10	−0.853218	−0.426609	−	0.904436i	$-0.640292\pi$
$823$	−1.36446e10	−0.853218	−0.426609	+	0.904436i	$0.640292\pi$
$824$	−4.75020e9	−0.295779
$825$	−5.61852e9	−0.348364
$826$	3.08339e9	0.190370
$827$	−2.13778e10	−1.31430	−0.657149	−	0.753760i	$-0.728238\pi$
$827$	−2.13778e10	−1.31430	−0.657149	+	0.753760i	$0.728238\pi$
$828$	1.03572e9	0.0634070
$829$	1.14474e10	0.697856	0.348928	−	0.937150i	$-0.386546\pi$
$829$	1.14474e10	0.697856	0.348928	+	0.937150i	$0.386546\pi$
$830$	5.35627e6	0.000325154 0
$831$	−1.04954e10	−0.634449
$832$	1.01959e10	0.613752
$833$	2.38471e10	1.42948
$834$	3.72443e9	0.222320
$835$	6.70304e7	0.00398445
$836$	1.11371e10	0.659248
$837$	−6.42569e8	−0.0378774
$838$	−4.75043e9	−0.278855
$839$	−3.27466e10	−1.91426	−0.957128	−	0.289665i	$-0.906456\pi$
$839$	−3.27466e10	−1.91426	−0.957128	+	0.289665i	$0.906456\pi$
$840$	−7.62943e6	−0.000444134 0
$841$	−7.55549e8	−0.0438003
$842$	5.33318e9	0.307889
$843$	−7.17625e9	−0.412573
$844$	1.73015e10	0.990571
$845$	2.34555e7	0.00133735
$846$	−2.07041e8	−0.0117560
$847$	−5.02800e9	−0.284318
$848$	−6.41387e8	−0.0361189
$849$	2.57021e9	0.144143
$850$	9.47485e9	0.529183
$851$	7.09829e9	0.394821
$852$	1.55008e10	0.858649
$853$	1.38359e10	0.763281	0.381641	−	0.924311i	$-0.375359\pi$
$853$	1.38359e10	0.763281	0.381641	+	0.924311i	$0.375359\pi$
$854$	2.20687e9	0.121248
$855$	−2.21634e7	−0.00121270
$856$	−6.23132e9	−0.339564
$857$	−1.99079e10	−1.08042	−0.540209	−	0.841531i	$-0.681655\pi$
$857$	−1.99079e10	−1.08042	−0.540209	+	0.841531i	$0.681655\pi$
$858$	−2.29108e9	−0.123833
$859$	−2.25943e10	−1.21625	−0.608124	−	0.793842i	$-0.708078\pi$
$859$	−2.25943e10	−1.21625	−0.608124	+	0.793842i	$0.708078\pi$
$860$	−7.63251e7	−0.00409188
$861$	−8.28888e9	−0.442573
$862$	−6.86206e9	−0.364904
$863$	2.03862e10	1.07969	0.539844	−	0.841765i	$-0.318483\pi$
$863$	2.03862e10	1.07969	0.539844	+	0.841765i	$0.318483\pi$
$864$	2.87111e9	0.151444
$865$	7.73349e7	0.00406274
$866$	3.88037e9	0.203030
$867$	2.42853e10	1.26554
$868$	1.54669e9	0.0802758
$869$	7.75182e9	0.400714
$870$	9.86637e6	0.000507972 0
$871$	−2.01465e10	−1.03308
$872$	−5.15583e9	−0.263324
$873$	1.11451e10	0.566937
$874$	1.45994e9	0.0739682
$875$	−5.38273e7	−0.00271628
$876$	−1.09007e10	−0.547884
$877$	−5.24578e8	−0.0262610	−0.0131305	−	0.999914i	$-0.504180\pi$
$877$	−5.24578e8	−0.0262610	−0.0131305	+	0.999914i	$0.504180\pi$
$878$	5.74047e8	0.0286231
$879$	−6.38158e9	−0.316933
$880$	2.75866e7	0.00136461
$881$	3.10939e10	1.53200	0.766000	−	0.642840i	$-0.222244\pi$
$881$	3.10939e10	1.53200	0.766000	+	0.642840i	$0.222244\pi$
$882$	−1.60970e9	−0.0789961
$883$	2.38047e10	1.16359	0.581794	−	0.813336i	$-0.302351\pi$
$883$	2.38047e10	1.16359	0.581794	+	0.813336i	$0.302351\pi$
$884$	−4.01749e10	−1.95601
$885$	5.19883e7	0.00252118
$886$	−6.73782e9	−0.325463
$887$	3.21183e10	1.54533	0.772663	−	0.634817i	$-0.218925\pi$
$887$	3.21183e10	1.54533	0.772663	+	0.634817i	$0.218925\pi$
$888$	1.29204e10	0.619201
$889$	−1.50660e10	−0.719187
$890$	−666810.	−3.17057e−5 0
$891$	1.41556e9	0.0670432
$892$	−1.44832e10	−0.683263
$893$	3.03467e9	0.142604
$894$	−4.80494e9	−0.224909
$895$	−9.78562e7	−0.00456255
$896$	−9.03375e9	−0.419556
$897$	3.12297e9	0.144476
$898$	5.58203e9	0.257232
$899$	−4.19272e9	−0.192458
$900$	6.65037e9	0.304086
$901$	1.90299e9	0.0866763
$902$	−6.75376e9	−0.306424
$903$	8.43335e9	0.381148
$904$	2.29993e10	1.03544
$905$	3.55562e6	0.000159458 0
$906$	4.01665e9	0.179439
$907$	−4.96459e9	−0.220931	−0.110466	−	0.993880i	$-0.535234\pi$
$907$	−4.96459e9	−0.220931	−0.110466	+	0.993880i	$0.535234\pi$
$908$	4.02053e10	1.78231
$909$	−9.98028e9	−0.440726
$910$	−1.09746e7	−0.000482776 0
$911$	−2.58578e10	−1.13312	−0.566562	−	0.824019i	$-0.691727\pi$
$911$	−2.58578e10	−1.13312	−0.566562	+	0.824019i	$0.691727\pi$
$912$	−1.17928e10	−0.514794
$913$	5.01427e9	0.218052
$914$	−5.89939e9	−0.255561
$915$	3.72096e7	0.00160576
$916$	1.32242e10	0.568505
$917$	1.97987e9	0.0847900
$918$	−2.38714e9	−0.101842
$919$	−4.44902e10	−1.89087	−0.945433	−	0.325818i	$-0.894360\pi$
$919$	−4.44902e10	−1.89087	−0.945433	+	0.325818i	$0.894360\pi$
$920$	8.47361e6	0.000358766 0
$921$	−1.17949e10	−0.497493
$922$	−6.78440e9	−0.285071
$923$	4.67389e10	1.95647
$924$	−3.40731e9	−0.142089
$925$	4.55781e10	1.89348
$926$	−8.88841e9	−0.367863
$927$	4.22179e9	0.174067
$928$	1.87338e10	0.769498
$929$	1.94659e9	0.0796562	0.0398281	−	0.999207i	$-0.487319\pi$
$929$	1.94659e9	0.0796562	0.0398281	+	0.999207i	$0.487319\pi$
$930$	−2.50795e6	−0.000102242 0
$931$	2.35939e10	0.958245
$932$	2.23146e10	0.902887
$933$	−3.88007e9	−0.156406
$934$	8.62905e9	0.346536
$935$	−8.18493e7	−0.00327472
$936$	5.68449e9	0.226582
$937$	−3.37108e10	−1.33869	−0.669346	−	0.742951i	$-0.733426\pi$
$937$	−3.37108e10	−1.33869	−0.669346	+	0.742951i	$0.733426\pi$
$938$	2.88141e9	0.113998
$939$	2.15498e10	0.849403
$940$	8.40269e6	0.000329967 0
$941$	1.04003e10	0.406895	0.203448	−	0.979086i	$-0.434785\pi$
$941$	1.04003e10	0.406895	0.203448	+	0.979086i	$0.434785\pi$
$942$	−7.40240e9	−0.288532
$943$	9.20603e9	0.357504
$944$	2.76621e10	1.07024
$945$	6.78073e6	0.000261375 0
$946$	6.87147e9	0.263895
$947$	−1.40028e10	−0.535783	−0.267892	−	0.963449i	$-0.586327\pi$
$947$	−1.40028e10	−0.535783	−0.267892	+	0.963449i	$0.586327\pi$
$948$	−9.17546e9	−0.349783
$949$	−3.28683e10	−1.24838
$950$	9.37427e9	0.354736
$951$	4.27157e9	0.161048
$952$	1.20445e10	0.452438
$953$	4.94805e9	0.185186	0.0925931	−	0.995704i	$-0.470484\pi$
$953$	4.94805e9	0.185186	0.0925931	+	0.995704i	$0.470484\pi$
$954$	−1.28454e8	−0.00478992
$955$	−1.08256e7	−0.000402199 0
$956$	−3.94964e9	−0.146202
$957$	9.23640e9	0.340652
$958$	−2.76083e9	−0.101452
$959$	−1.25824e10	−0.460680
$960$	−2.45872e7	−0.000896933 0
$961$	−2.64469e10	−0.961263
$962$	1.85856e10	0.673074
$963$	5.53814e9	0.199835
$964$	1.79366e10	0.644866
$965$	−3.30072e7	−0.00118240
$966$	−4.46658e8	−0.0159425
$967$	−1.79902e10	−0.639798	−0.319899	−	0.947452i	$-0.603649\pi$
$967$	−1.79902e10	−0.639798	−0.319899	+	0.947452i	$0.603649\pi$
$968$	1.01647e10	0.360190
$969$	3.49891e10	1.23538
$970$	4.34994e7	0.00153032
$971$	−1.54029e10	−0.539929	−0.269964	−	0.962870i	$-0.587012\pi$
$971$	−1.54029e10	−0.539929	−0.269964	+	0.962870i	$0.587012\pi$
$972$	−1.67553e9	−0.0585220
$973$	1.67015e10	0.581245
$974$	8.96064e9	0.310730
$975$	2.00526e10	0.692874
$976$	1.97986e10	0.681648
$977$	−1.43057e10	−0.490769	−0.245384	−	0.969426i	$-0.578914\pi$
$977$	−1.43057e10	−0.490769	−0.245384	+	0.969426i	$0.578914\pi$
$978$	7.43223e9	0.254058
$979$	−6.24235e8	−0.0212622
$980$	6.53292e7	0.00221726
$981$	4.58229e9	0.154968
$982$	−4.15080e9	−0.139876
$983$	1.72656e10	0.579756	0.289878	−	0.957064i	$-0.406385\pi$
$983$	1.72656e10	0.579756	0.289878	+	0.957064i	$0.406385\pi$
$984$	1.67570e10	0.560678
$985$	5.96008e7	0.00198712
$986$	−1.55759e10	−0.517469
$987$	−9.28435e8	−0.0307356
$988$	−3.97484e10	−1.31120
$989$	−9.36649e9	−0.307886
$990$	5.52492e6	0.000180968 0
$991$	−1.25900e10	−0.410929	−0.205464	−	0.978665i	$-0.565870\pi$
$991$	−1.25900e10	−0.410929	−0.205464	+	0.978665i	$0.565870\pi$
$992$	−4.76197e9	−0.154880
$993$	1.49882e10	0.485765
$994$	−6.68476e9	−0.215891
$995$	1.40040e7	0.000450682 0
$996$	−5.93515e9	−0.190337
$997$	−2.34721e10	−0.750100	−0.375050	−	0.927005i	$-0.622374\pi$
$997$	−2.34721e10	−0.750100	−0.375050	+	0.927005i	$0.622374\pi$
$998$	−7.12681e9	−0.226954
$999$	−1.14832e10	−0.364403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.b.1.4	✓	6
3.2	odd	2		207.8.a.c.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.b.1.4	✓	6	1.1	even	1	trivial
207.8.a.c.1.3		6	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+3.35108 q^{2} +27.0000 q^{3} -116.770 q^{4} +0.849066 q^{5} +90.4791 q^{6} +405.736 q^{7} -820.245 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+3.35108 q^{2} +27.0000 q^{3} -116.770 q^{4} +0.849066 q^{5} +90.4791 q^{6} +405.736 q^{7} -820.245 q^{8} +729.000 q^{9} +2.84529 q^{10} +2663.62 q^{11} -3152.80 q^{12} -9506.50 q^{13} +1359.65 q^{14} +22.9248 q^{15} +12197.9 q^{16} -36191.1 q^{17} +2442.94 q^{18} -35806.9 q^{19} -99.1457 q^{20} +10954.9 q^{21} +8925.99 q^{22} -12167.0 q^{23} -22146.6 q^{24} -78124.3 q^{25} -31857.0 q^{26} +19683.0 q^{27} -47377.9 q^{28} +128430. q^{29} +76.8228 q^{30} -32645.9 q^{31} +145867. q^{32} +71917.7 q^{33} -121279. q^{34} +344.497 q^{35} -85125.5 q^{36} -583405. q^{37} -119992. q^{38} -256676. q^{39} -696.442 q^{40} -756639. q^{41} +36710.6 q^{42} +769827. q^{43} -311031. q^{44} +618.969 q^{45} -40772.6 q^{46} -84751.0 q^{47} +329343. q^{48} -658922. q^{49} -261801. q^{50} -977159. q^{51} +1.11008e6 q^{52} -52581.8 q^{53} +65959.3 q^{54} +2261.59 q^{55} -332803. q^{56} -966787. q^{57} +430380. q^{58} +2.26778e6 q^{59} -2676.93 q^{60} +1.62312e6 q^{61} -109399. q^{62} +295781. q^{63} -1.07252e6 q^{64} -8071.65 q^{65} +241002. q^{66} +2.11923e6 q^{67} +4.22604e6 q^{68} -328509. q^{69} +1154.44 q^{70} -4.91652e6 q^{71} -597958. q^{72} +3.45746e6 q^{73} -1.95504e6 q^{74} -2.10936e6 q^{75} +4.18118e6 q^{76} +1.08072e6 q^{77} -860140. q^{78} +2.91026e6 q^{79} +10356.8 q^{80} +531441. q^{81} -2.53556e6 q^{82} +1.88250e6 q^{83} -1.27920e6 q^{84} -30728.6 q^{85} +2.57975e6 q^{86} +3.46762e6 q^{87} -2.18482e6 q^{88} -234356. q^{89} +2074.22 q^{90} -3.85713e6 q^{91} +1.42074e6 q^{92} -881439. q^{93} -284007. q^{94} -30402.4 q^{95} +3.93842e6 q^{96} +1.52882e7 q^{97} -2.20810e6 q^{98} +1.94178e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9}+O(q^{10}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9} - 13042 q^{10} - 14824 q^{11} + 4806 q^{12} - 756 q^{13} - 3926 q^{14} - 10044 q^{15} - 13022 q^{16} - 69484 q^{17} - 5832 q^{18} - 43864 q^{19} + 78886 q^{20} - 29808 q^{21} + 98204 q^{22} - 73002 q^{23} - 52812 q^{24} + 228018 q^{25} - 311956 q^{26} + 118098 q^{27} - 545442 q^{28} - 311100 q^{29} - 352134 q^{30} - 245248 q^{31} - 390156 q^{32} - 400248 q^{33} + 235834 q^{34} - 1331256 q^{35} + 129762 q^{36} - 630044 q^{37} + 80910 q^{38} - 20412 q^{39} - 2153982 q^{40} - 969204 q^{41} - 106002 q^{42} - 1770208 q^{43} - 1749140 q^{44} - 271188 q^{45} + 97336 q^{46} - 1400024 q^{47} - 351594 q^{48} + 1985598 q^{49} - 956660 q^{50} - 1876068 q^{51} + 3217272 q^{52} - 1573516 q^{53} - 157464 q^{54} - 431296 q^{55} + 7740702 q^{56} - 1184328 q^{57} + 5987188 q^{58} - 1410320 q^{59} + 2129922 q^{60} - 942172 q^{61} + 3334412 q^{62} - 804816 q^{63} + 1996866 q^{64} - 420944 q^{65} + 2651508 q^{66} - 452072 q^{67} - 9258254 q^{68} - 1971054 q^{69} + 21981136 q^{70} + 122928 q^{71} - 1425924 q^{72} + 16490716 q^{73} - 600104 q^{74} + 6156486 q^{75} + 7428658 q^{76} + 7239696 q^{77} - 8422812 q^{78} + 2458408 q^{79} + 19440230 q^{80} + 3188646 q^{81} + 20510784 q^{82} - 7566456 q^{83} - 14726934 q^{84} + 5817744 q^{85} - 669666 q^{86} - 8399700 q^{87} + 14775668 q^{88} - 20368036 q^{89} - 9507618 q^{90} + 8815576 q^{91} - 2165726 q^{92} - 6621696 q^{93} + 16952576 q^{94} + 5143832 q^{95} - 10534212 q^{96} + 12586972 q^{97} - 39164812 q^{98} - 10806696 q^{99}+O(q^{100}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9 - 13042 * q^10 - 14824 * q^11 + 4806 * q^12 - 756 * q^13 - 3926 * q^14 - 10044 * q^15 - 13022 * q^16 - 69484 * q^17 - 5832 * q^18 - 43864 * q^19 + 78886 * q^20 - 29808 * q^21 + 98204 * q^22 - 73002 * q^23 - 52812 * q^24 + 228018 * q^25 - 311956 * q^26 + 118098 * q^27 - 545442 * q^28 - 311100 * q^29 - 352134 * q^30 - 245248 * q^31 - 390156 * q^32 - 400248 * q^33 + 235834 * q^34 - 1331256 * q^35 + 129762 * q^36 - 630044 * q^37 + 80910 * q^38 - 20412 * q^39 - 2153982 * q^40 - 969204 * q^41 - 106002 * q^42 - 1770208 * q^43 - 1749140 * q^44 - 271188 * q^45 + 97336 * q^46 - 1400024 * q^47 - 351594 * q^48 + 1985598 * q^49 - 956660 * q^50 - 1876068 * q^51 + 3217272 * q^52 - 1573516 * q^53 - 157464 * q^54 - 431296 * q^55 + 7740702 * q^56 - 1184328 * q^57 + 5987188 * q^58 - 1410320 * q^59 + 2129922 * q^60 - 942172 * q^61 + 3334412 * q^62 - 804816 * q^63 + 1996866 * q^64 - 420944 * q^65 + 2651508 * q^66 - 452072 * q^67 - 9258254 * q^68 - 1971054 * q^69 + 21981136 * q^70 + 122928 * q^71 - 1425924 * q^72 + 16490716 * q^73 - 600104 * q^74 + 6156486 * q^75 + 7428658 * q^76 + 7239696 * q^77 - 8422812 * q^78 + 2458408 * q^79 + 19440230 * q^80 + 3188646 * q^81 + 20510784 * q^82 - 7566456 * q^83 - 14726934 * q^84 + 5817744 * q^85 - 669666 * q^86 - 8399700 * q^87 + 14775668 * q^88 - 20368036 * q^89 - 9507618 * q^90 + 8815576 * q^91 - 2165726 * q^92 - 6621696 * q^93 + 16952576 * q^94 + 5143832 * q^95 - 10534212 * q^96 + 12586972 * q^97 - 39164812 * q^98 - 10806696 * q^99

Introduction

L-functions

Modular forms

Varieties

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