LMFDB - Embedded newform 471.8.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,8,Mod(1,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, 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("471.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	471.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:	$147.133347003$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.8012 q^{2} -27.0000 q^{3} +188.881 q^{4} -168.224 q^{5} +480.631 q^{6} -317.259 q^{7} -1083.76 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-17.8012 q^{2} -27.0000 q^{3} +188.881 q^{4} -168.224 q^{5} +480.631 q^{6} -317.259 q^{7} -1083.76 q^{8} +729.000 q^{9} +2994.59 q^{10} -3149.42 q^{11} -5099.80 q^{12} +12732.5 q^{13} +5647.58 q^{14} +4542.06 q^{15} -4884.62 q^{16} +18563.1 q^{17} -12977.0 q^{18} +19153.9 q^{19} -31774.5 q^{20} +8565.99 q^{21} +56063.3 q^{22} +55699.9 q^{23} +29261.5 q^{24} -49825.5 q^{25} -226654. q^{26} -19683.0 q^{27} -59924.3 q^{28} +248175. q^{29} -80853.9 q^{30} -48841.3 q^{31} +225673. q^{32} +85034.3 q^{33} -330444. q^{34} +53370.7 q^{35} +137695. q^{36} -47383.8 q^{37} -340961. q^{38} -343778. q^{39} +182315. q^{40} +582230. q^{41} -152485. q^{42} +785839. q^{43} -594867. q^{44} -122636. q^{45} -991522. q^{46} +145378. q^{47} +131885. q^{48} -722890. q^{49} +886953. q^{50} -501203. q^{51} +2.40494e6 q^{52} +2.01758e6 q^{53} +350380. q^{54} +529809. q^{55} +343833. q^{56} -517154. q^{57} -4.41780e6 q^{58} +1.30642e6 q^{59} +857911. q^{60} -876840. q^{61} +869433. q^{62} -231282. q^{63} -3.39202e6 q^{64} -2.14192e6 q^{65} -1.51371e6 q^{66} +3.03432e6 q^{67} +3.50622e6 q^{68} -1.50390e6 q^{69} -950061. q^{70} -2.08298e6 q^{71} -790062. q^{72} +1.73728e6 q^{73} +843486. q^{74} +1.34529e6 q^{75} +3.61781e6 q^{76} +999181. q^{77} +6.11965e6 q^{78} +1.60447e6 q^{79} +821712. q^{80} +531441. q^{81} -1.03644e7 q^{82} -338992. q^{83} +1.61796e6 q^{84} -3.12276e6 q^{85} -1.39888e7 q^{86} -6.70072e6 q^{87} +3.41322e6 q^{88} -1.07653e7 q^{89} +2.18306e6 q^{90} -4.03951e6 q^{91} +1.05207e7 q^{92} +1.31872e6 q^{93} -2.58790e6 q^{94} -3.22215e6 q^{95} -6.09318e6 q^{96} +1.00033e6 q^{97} +1.28683e7 q^{98} -2.29593e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9}+O(q^{10}) $ 48 * q + 2 * q^2 - 1296 * q^3 + 3214 * q^4 + 428 * q^5 - 54 * q^6 - 680 * q^7 + 2355 * q^8 + 34992 * q^9	\( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9} + 6185 q^{10} + 11989 q^{11} - 86778 q^{12} - 2393 q^{13} + 18201 q^{14} - 11556 q^{15} + 208150 q^{16} + 62538 q^{17} + 1458 q^{18} - 39882 q^{19} + 113423 q^{20} + 18360 q^{21} - 93716 q^{22} + 195618 q^{23} - 63585 q^{24} + 886490 q^{25} + 294399 q^{26} - 944784 q^{27} - 60819 q^{28} + 421501 q^{29} - 166995 q^{30} + 392689 q^{31} - 341578 q^{32} - 323703 q^{33} + 50837 q^{34} + 697874 q^{35} + 2343006 q^{36} - 410396 q^{37} + 677216 q^{38} + 64611 q^{39} + 3232376 q^{40} + 3832958 q^{41} - 491427 q^{42} - 1751932 q^{43} + 4888297 q^{44} + 312012 q^{45} + 1163150 q^{46} + 106461 q^{47} - 5620050 q^{48} + 8202048 q^{49} - 2159111 q^{50} - 1688526 q^{51} - 3605030 q^{52} + 1755534 q^{53} - 39366 q^{54} - 1220729 q^{55} - 4430622 q^{56} + 1076814 q^{57} - 10000202 q^{58} - 2037752 q^{59} - 3062421 q^{60} + 1274098 q^{61} + 97748 q^{62} - 495720 q^{63} + 15135201 q^{64} + 6139645 q^{65} + 2530332 q^{66} - 7751257 q^{67} + 1700631 q^{68} - 5281686 q^{69} - 20935703 q^{70} - 12592217 q^{71} + 1716795 q^{72} + 12508355 q^{73} - 14999956 q^{74} - 23935230 q^{75} - 23946874 q^{76} + 1874177 q^{77} - 7948773 q^{78} - 5103480 q^{79} + 3128449 q^{80} + 25509168 q^{81} + 11622426 q^{82} + 3040643 q^{83} + 1642113 q^{84} - 13756076 q^{85} + 964635 q^{86} - 11380527 q^{87} - 29653500 q^{88} + 28462995 q^{89} + 4508865 q^{90} + 3016621 q^{91} + 22938254 q^{92} - 10602603 q^{93} - 10070348 q^{94} - 2579984 q^{95} + 9222606 q^{96} + 16208760 q^{97} + 6323227 q^{98} + 8739981 q^{99}+O(q^{100}) \) 48 * q + 2 * q^2 - 1296 * q^3 + 3214 * q^4 + 428 * q^5 - 54 * q^6 - 680 * q^7 + 2355 * q^8 + 34992 * q^9 + 6185 * q^10 + 11989 * q^11 - 86778 * q^12 - 2393 * q^13 + 18201 * q^14 - 11556 * q^15 + 208150 * q^16 + 62538 * q^17 + 1458 * q^18 - 39882 * q^19 + 113423 * q^20 + 18360 * q^21 - 93716 * q^22 + 195618 * q^23 - 63585 * q^24 + 886490 * q^25 + 294399 * q^26 - 944784 * q^27 - 60819 * q^28 + 421501 * q^29 - 166995 * q^30 + 392689 * q^31 - 341578 * q^32 - 323703 * q^33 + 50837 * q^34 + 697874 * q^35 + 2343006 * q^36 - 410396 * q^37 + 677216 * q^38 + 64611 * q^39 + 3232376 * q^40 + 3832958 * q^41 - 491427 * q^42 - 1751932 * q^43 + 4888297 * q^44 + 312012 * q^45 + 1163150 * q^46 + 106461 * q^47 - 5620050 * q^48 + 8202048 * q^49 - 2159111 * q^50 - 1688526 * q^51 - 3605030 * q^52 + 1755534 * q^53 - 39366 * q^54 - 1220729 * q^55 - 4430622 * q^56 + 1076814 * q^57 - 10000202 * q^58 - 2037752 * q^59 - 3062421 * q^60 + 1274098 * q^61 + 97748 * q^62 - 495720 * q^63 + 15135201 * q^64 + 6139645 * q^65 + 2530332 * q^66 - 7751257 * q^67 + 1700631 * q^68 - 5281686 * q^69 - 20935703 * q^70 - 12592217 * q^71 + 1716795 * q^72 + 12508355 * q^73 - 14999956 * q^74 - 23935230 * q^75 - 23946874 * q^76 + 1874177 * q^77 - 7948773 * q^78 - 5103480 * q^79 + 3128449 * q^80 + 25509168 * q^81 + 11622426 * q^82 + 3040643 * q^83 + 1642113 * q^84 - 13756076 * q^85 + 964635 * q^86 - 11380527 * q^87 - 29653500 * q^88 + 28462995 * q^89 + 4508865 * q^90 + 3016621 * q^91 + 22938254 * q^92 - 10602603 * q^93 - 10070348 * q^94 - 2579984 * q^95 + 9222606 * q^96 + 16208760 * q^97 + 6323227 * q^98 + 8739981 * 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 $

$2$	−17.8012	−1.57342	−0.786708	−	0.617326i	$-0.788216\pi$
$2$	−17.8012	−1.57342	−0.786708	+	0.617326i	$0.788216\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	188.881	1.47564
$5$	−168.224	−0.601858	−0.300929	−	0.953647i	$-0.597297\pi$
$5$	−168.224	−0.601858	−0.300929	+	0.953647i	$0.597297\pi$
$6$	480.631	0.908412
$7$	−317.259	−0.349599	−0.174800	−	0.984604i	$-0.555928\pi$
$7$	−317.259	−0.349599	−0.174800	+	0.984604i	$0.555928\pi$
$8$	−1083.76	−0.748374
$9$	729.000	0.333333
$10$	2994.59	0.946973
$11$	−3149.42	−0.713438	−0.356719	−	0.934212i	$-0.616105\pi$
$11$	−3149.42	−0.713438	−0.356719	+	0.934212i	$0.616105\pi$
$12$	−5099.80	−0.851959
$13$	12732.5	1.60736	0.803680	−	0.595062i	$-0.202873\pi$
$13$	12732.5	1.60736	0.803680	+	0.595062i	$0.202873\pi$
$14$	5647.58	0.550065
$15$	4542.06	0.347483
$16$	−4884.62	−0.298134
$17$	18563.1	0.916386	0.458193	−	0.888853i	$-0.348497\pi$
$17$	18563.1	0.916386	0.458193	+	0.888853i	$0.348497\pi$
$18$	−12977.0	−0.524472
$19$	19153.9	0.640647	0.320323	−	0.947308i	$-0.396208\pi$
$19$	19153.9	0.640647	0.320323	+	0.947308i	$0.396208\pi$
$20$	−31774.5	−0.888124
$21$	8565.99	0.201841
$22$	56063.3	1.12253
$23$	55699.9	0.954567	0.477284	−	0.878749i	$-0.341621\pi$
$23$	55699.9	0.954567	0.477284	+	0.878749i	$0.341621\pi$
$24$	29261.5	0.432074
$25$	−49825.5	−0.637767
$26$	−226654.	−2.52904
$27$	−19683.0	−0.192450
$28$	−59924.3	−0.515882
$29$	248175.	1.88958	0.944789	−	0.327679i	$-0.106267\pi$
$29$	248175.	1.88958	0.944789	+	0.327679i	$0.106267\pi$
$30$	−80853.9	−0.546735
$31$	−48841.3	−0.294457	−0.147228	−	0.989103i	$-0.547035\pi$
$31$	−48841.3	−0.294457	−0.147228	+	0.989103i	$0.547035\pi$
$32$	225673.	1.21746
$33$	85034.3	0.411903
$34$	−330444.	−1.44186
$35$	53370.7	0.210409
$36$	137695.	0.491879
$37$	−47383.8	−0.153788	−0.0768942	−	0.997039i	$-0.524500\pi$
$37$	−47383.8	−0.153788	−0.0768942	+	0.997039i	$0.524500\pi$
$38$	−340961.	−1.00800
$39$	−343778.	−0.928009
$40$	182315.	0.450415
$41$	582230.	1.31932	0.659662	−	0.751563i	$-0.270700\pi$
$41$	582230.	1.31932	0.659662	+	0.751563i	$0.270700\pi$
$42$	−152485.	−0.317580
$43$	785839.	1.50728	0.753640	−	0.657288i	$-0.228296\pi$
$43$	785839.	1.50728	0.753640	+	0.657288i	$0.228296\pi$
$44$	−594867.	−1.05277
$45$	−122636.	−0.200619
$46$	−991522.	−1.50193
$47$	145378.	0.204247	0.102124	−	0.994772i	$-0.467436\pi$
$47$	145378.	0.204247	0.102124	+	0.994772i	$0.467436\pi$
$48$	131885.	0.172127
$49$	−722890.	−0.877780
$50$	886953.	1.00347
$51$	−501203.	−0.529076
$52$	2.40494e6	2.37188
$53$	2.01758e6	1.86151	0.930755	−	0.365643i	$-0.119151\pi$
$53$	2.01758e6	1.86151	0.930755	+	0.365643i	$0.119151\pi$
$54$	350380.	0.302804
$55$	529809.	0.429388
$56$	343833.	0.261631
$57$	−517154.	−0.369878
$58$	−4.41780e6	−2.97309
$59$	1.30642e6	0.828133	0.414066	−	0.910247i	$-0.364108\pi$
$59$	1.30642e6	0.828133	0.414066	+	0.910247i	$0.364108\pi$
$60$	857911.	0.512758
$61$	−876840.	−0.494614	−0.247307	−	0.968937i	$-0.579546\pi$
$61$	−876840.	−0.494614	−0.247307	+	0.968937i	$0.579546\pi$
$62$	869433.	0.463303
$63$	−231282.	−0.116533
$64$	−3.39202e6	−1.61744
$65$	−2.14192e6	−0.967402
$66$	−1.51371e6	−0.648095
$67$	3.03432e6	1.23254	0.616268	−	0.787536i	$-0.288644\pi$
$67$	3.03432e6	1.23254	0.616268	+	0.787536i	$0.288644\pi$
$68$	3.50622e6	1.35225
$69$	−1.50390e6	−0.551120
$70$	−950061.	−0.331061
$71$	−2.08298e6	−0.690686	−0.345343	−	0.938477i	$-0.612237\pi$
$71$	−2.08298e6	−0.690686	−0.345343	+	0.938477i	$0.612237\pi$
$72$	−790062.	−0.249458
$73$	1.73728e6	0.522685	0.261342	−	0.965246i	$-0.415835\pi$
$73$	1.73728e6	0.522685	0.261342	+	0.965246i	$0.415835\pi$
$74$	843486.	0.241973
$75$	1.34529e6	0.368215
$76$	3.61781e6	0.945362
$77$	999181.	0.249417
$78$	6.11965e6	1.46014
$79$	1.60447e6	0.366132	0.183066	−	0.983101i	$-0.441398\pi$
$79$	1.60447e6	0.366132	0.183066	+	0.983101i	$0.441398\pi$
$80$	821712.	0.179434
$81$	531441.	0.111111
$82$	−1.03644e7	−2.07584
$83$	−338992.	−0.0650753	−0.0325377	−	0.999471i	$-0.510359\pi$
$83$	−338992.	−0.0650753	−0.0325377	+	0.999471i	$0.510359\pi$
$84$	1.61796e6	0.297844
$85$	−3.12276e6	−0.551534
$86$	−1.39888e7	−2.37158
$87$	−6.70072e6	−1.09095
$88$	3.41322e6	0.533918
$89$	−1.07653e7	−1.61869	−0.809343	−	0.587337i	$-0.800176\pi$
$89$	−1.07653e7	−1.61869	−0.809343	+	0.587337i	$0.800176\pi$
$90$	2.18306e6	0.315658
$91$	−4.03951e6	−0.561932
$92$	1.05207e7	1.40859
$93$	1.31872e6	0.170005
$94$	−2.58790e6	−0.321366
$95$	−3.22215e6	−0.385578
$96$	−6.09318e6	−0.702902
$97$	1.00033e6	0.111287	0.0556435	−	0.998451i	$-0.482279\pi$
$97$	1.00033e6	0.111287	0.0556435	+	0.998451i	$0.482279\pi$
$98$	1.28683e7	1.38111
$99$	−2.29593e6	−0.237813
$100$	−9.41112e6	−0.941112
$101$	1.48131e7	1.43061	0.715305	−	0.698812i	$-0.246288\pi$
$101$	1.48131e7	1.43061	0.715305	+	0.698812i	$0.246288\pi$
$102$	8.92200e6	0.832456
$103$	−8.31502e6	−0.749778	−0.374889	−	0.927070i	$-0.622319\pi$
$103$	−8.31502e6	−0.749778	−0.374889	+	0.927070i	$0.622319\pi$
$104$	−1.37990e7	−1.20291
$105$	−1.44101e6	−0.121480
$106$	−3.59153e7	−2.92893
$107$	−1.17986e7	−0.931078	−0.465539	−	0.885027i	$-0.654139\pi$
$107$	−1.17986e7	−0.931078	−0.465539	+	0.885027i	$0.654139\pi$
$108$	−3.71775e6	−0.283986
$109$	21348.1	0.00157895	0.000789473	−	1.00000i	$-0.499749\pi$
$109$	21348.1	0.00157895	0.000789473		1.00000i	$0.499749\pi$
$110$	−9.43122e6	−0.675606
$111$	1.27936e6	0.0887898
$112$	1.54969e6	0.104227
$113$	−1.92059e7	−1.25216	−0.626079	−	0.779760i	$-0.715341\pi$
$113$	−1.92059e7	−1.25216	−0.626079	+	0.779760i	$0.715341\pi$
$114$	9.20595e6	0.581971
$115$	−9.37008e6	−0.574514
$116$	4.68756e7	2.78833
$117$	9.28201e6	0.535786
$118$	−2.32558e7	−1.30300
$119$	−5.88930e6	−0.320368
$120$	−4.92251e6	−0.260047
$121$	−9.56833e6	−0.491007
$122$	1.56088e7	0.778233
$123$	−1.57202e7	−0.761712
$124$	−9.22522e6	−0.434511
$125$	2.15244e7	0.985703
$126$	4.11708e6	0.183355
$127$	−2.46911e7	−1.06961	−0.534807	−	0.844975i	$-0.679616\pi$
$127$	−2.46911e7	−1.06961	−0.534807	+	0.844975i	$0.679616\pi$
$128$	3.14957e7	1.32744
$129$	−2.12176e7	−0.870228
$130$	3.81287e7	1.52213
$131$	−4.15090e7	−1.61322	−0.806609	−	0.591085i	$-0.798700\pi$
$131$	−4.15090e7	−1.61322	−0.806609	+	0.591085i	$0.798700\pi$
$132$	1.60614e7	0.607820
$133$	−6.07673e6	−0.223970
$134$	−5.40145e7	−1.93929
$135$	3.31116e6	0.115828
$136$	−2.01179e7	−0.685799
$137$	1.93872e7	0.644160	0.322080	−	0.946712i	$-0.395618\pi$
$137$	1.93872e7	0.644160	0.322080	+	0.946712i	$0.395618\pi$
$138$	2.67711e7	0.867140
$139$	3.19008e7	1.00751	0.503755	−	0.863847i	$-0.331951\pi$
$139$	3.19008e7	1.00751	0.503755	+	0.863847i	$0.331951\pi$
$140$	1.00807e7	0.310487
$141$	−3.92520e6	−0.117922
$142$	3.70795e7	1.08674
$143$	−4.01001e7	−1.14675
$144$	−3.56089e6	−0.0993778
$145$	−4.17491e7	−1.13726
$146$	−3.09256e7	−0.822400
$147$	1.95180e7	0.506787
$148$	−8.94991e6	−0.226936
$149$	−2.68638e7	−0.665297	−0.332648	−	0.943051i	$-0.607942\pi$
$149$	−2.68638e7	−0.665297	−0.332648	+	0.943051i	$0.607942\pi$
$150$	−2.39477e7	−0.579355
$151$	−2.73254e7	−0.645874	−0.322937	−	0.946420i	$-0.604670\pi$
$151$	−2.73254e7	−0.645874	−0.322937	+	0.946420i	$0.604670\pi$
$152$	−2.07582e7	−0.479443
$153$	1.35325e7	0.305462
$154$	−1.77866e7	−0.392437
$155$	8.21630e6	0.177221
$156$	−6.49333e7	−1.36940
$157$	−3.86989e6	−0.0798087
$157$	−3.86989e6	−0.0798087
$158$	−2.85615e7	−0.576077
$159$	−5.44747e7	−1.07474
$160$	−3.79638e7	−0.732739
$161$	−1.76713e7	−0.333716
$162$	−9.46027e6	−0.174824
$163$	−3.18437e7	−0.575926	−0.287963	−	0.957641i	$-0.592978\pi$
$163$	−3.18437e7	−0.575926	−0.287963	+	0.957641i	$0.592978\pi$
$164$	1.09972e8	1.94684
$165$	−1.43048e7	−0.247907
$166$	6.03445e6	0.102391
$167$	−7.85827e7	−1.30563	−0.652814	−	0.757518i	$-0.726412\pi$
$167$	−7.85827e7	−1.30563	−0.652814	+	0.757518i	$0.726412\pi$
$168$	−9.28349e6	−0.151053
$169$	9.93688e7	1.58360
$170$	5.55888e7	0.867793
$171$	1.39632e7	0.213549
$172$	1.48430e8	2.22420
$173$	1.17114e8	1.71968	0.859841	−	0.510561i	$-0.170562\pi$
$173$	1.17114e8	1.71968	0.859841	+	0.510561i	$0.170562\pi$
$174$	1.19281e8	1.71652
$175$	1.58076e7	0.222963
$176$	1.53837e7	0.212700
$177$	−3.52733e7	−0.478123
$178$	1.91635e8	2.54686
$179$	1.04133e8	1.35707	0.678535	−	0.734568i	$-0.262615\pi$
$179$	1.04133e8	1.35707	0.678535	+	0.734568i	$0.262615\pi$
$180$	−2.31636e7	−0.296041
$181$	−3.18761e7	−0.399567	−0.199784	−	0.979840i	$-0.564024\pi$
$181$	−3.18761e7	−0.399567	−0.199784	+	0.979840i	$0.564024\pi$
$182$	7.19080e7	0.884152
$183$	2.36747e7	0.285565
$184$	−6.03653e7	−0.714373
$185$	7.97111e6	0.0925588
$186$	−2.34747e7	−0.267488
$187$	−5.84629e7	−0.653784
$188$	2.74592e7	0.301394
$189$	6.24461e6	0.0672804
$190$	5.73580e7	0.606675
$191$	1.21929e8	1.26617	0.633084	−	0.774083i	$-0.281789\pi$
$191$	1.21929e8	1.26617	0.633084	+	0.774083i	$0.281789\pi$
$192$	9.15845e7	0.933829
$193$	−1.53928e8	−1.54123	−0.770615	−	0.637301i	$-0.780051\pi$
$193$	−1.53928e8	−1.54123	−0.770615	+	0.637301i	$0.780051\pi$
$194$	−1.78071e7	−0.175101
$195$	5.78319e7	0.558530
$196$	−1.36540e8	−1.29528
$197$	1.28857e8	1.20082	0.600409	−	0.799693i	$-0.295005\pi$
$197$	1.28857e8	1.20082	0.600409	+	0.799693i	$0.295005\pi$
$198$	4.08702e7	0.374178
$199$	8.30637e7	0.747180	0.373590	−	0.927594i	$-0.378127\pi$
$199$	8.30637e7	0.747180	0.373590	+	0.927594i	$0.378127\pi$
$200$	5.39990e7	0.477288
$201$	−8.19267e7	−0.711605
$202$	−2.63691e8	−2.25094
$203$	−7.87357e7	−0.660595
$204$	−9.46679e7	−0.780724
$205$	−9.79453e7	−0.794045
$206$	1.48017e8	1.17971
$207$	4.06052e7	0.318189
$208$	−6.21936e7	−0.479208
$209$	−6.03235e7	−0.457061
$210$	2.56516e7	0.191138
$211$	2.37442e7	0.174008	0.0870039	−	0.996208i	$-0.472271\pi$
$211$	2.37442e7	0.174008	0.0870039	+	0.996208i	$0.472271\pi$
$212$	3.81084e8	2.74691
$213$	5.62404e7	0.398768
$214$	2.10028e8	1.46497
$215$	−1.32197e8	−0.907168
$216$	2.13317e7	0.144025
$217$	1.54953e7	0.102942
$218$	−380022.	−0.00248434
$219$	−4.69066e7	−0.301772
$220$	1.00071e8	0.633621
$221$	2.36355e8	1.47296
$222$	−2.27741e7	−0.139703
$223$	−9.31460e7	−0.562467	−0.281234	−	0.959639i	$-0.590744\pi$
$223$	−9.31460e7	−0.562467	−0.281234	+	0.959639i	$0.590744\pi$
$224$	−7.15969e7	−0.425624
$225$	−3.63228e7	−0.212589
$226$	3.41887e8	1.97016
$227$	3.40075e8	1.92967	0.964837	−	0.262851i	$-0.0846626\pi$
$227$	3.40075e8	1.92967	0.964837	+	0.262851i	$0.0846626\pi$
$228$	−9.76808e7	−0.545805
$229$	−1.67948e8	−0.924169	−0.462084	−	0.886836i	$-0.652898\pi$
$229$	−1.67948e8	−0.924169	−0.462084	+	0.886836i	$0.652898\pi$
$230$	1.66798e8	0.903949
$231$	−2.69779e7	−0.144001
$232$	−2.68962e8	−1.41411
$233$	2.07253e8	1.07339	0.536693	−	0.843777i	$-0.319673\pi$
$233$	2.07253e8	1.07339	0.536693	+	0.843777i	$0.319673\pi$
$234$	−1.65231e8	−0.843015
$235$	−2.44561e7	−0.122928
$236$	2.46758e8	1.22202
$237$	−4.33207e7	−0.211386
$238$	1.04836e8	0.504072
$239$	−2.13265e7	−0.101048	−0.0505239	−	0.998723i	$-0.516089\pi$
$239$	−2.13265e7	−0.101048	−0.0505239	+	0.998723i	$0.516089\pi$
$240$	−2.21862e7	−0.103596
$241$	2.14983e8	0.989338	0.494669	−	0.869081i	$-0.335289\pi$
$241$	2.14983e8	0.989338	0.494669	+	0.869081i	$0.335289\pi$
$242$	1.70327e8	0.772558
$243$	−1.43489e7	−0.0641500
$244$	−1.65619e8	−0.729870
$245$	1.21608e8	0.528299
$246$	2.79838e8	1.19849
$247$	2.43877e8	1.02975
$248$	5.29323e7	0.220364
$249$	9.15279e6	0.0375713
$250$	−3.83159e8	−1.55092
$251$	−4.03236e7	−0.160954	−0.0804769	−	0.996756i	$-0.525644\pi$
$251$	−4.03236e7	−0.160954	−0.0804769	+	0.996756i	$0.525644\pi$
$252$	−4.36848e7	−0.171961
$253$	−1.75422e8	−0.681024
$254$	4.39530e8	1.68295
$255$	8.43146e7	0.318429
$256$	−1.26481e8	−0.471180
$257$	4.67831e8	1.71919	0.859593	−	0.510980i	$-0.170717\pi$
$257$	4.67831e8	1.71919	0.859593	+	0.510980i	$0.170717\pi$
$258$	3.77699e8	1.36923
$259$	1.50329e7	0.0537643
$260$	−4.04569e8	−1.42753
$261$	1.80919e8	0.629859
$262$	7.38909e8	2.53826
$263$	−2.23586e8	−0.757880	−0.378940	−	0.925421i	$-0.623711\pi$
$263$	−2.23586e8	−0.757880	−0.378940	+	0.925421i	$0.623711\pi$
$264$	−9.21569e7	−0.308258
$265$	−3.39406e8	−1.12036
$266$	1.08173e8	0.352397
$267$	2.90664e8	0.934549
$268$	5.73127e8	1.81878
$269$	3.83923e8	1.20257	0.601286	−	0.799034i	$-0.294655\pi$
$269$	3.83923e8	1.20257	0.601286	+	0.799034i	$0.294655\pi$
$270$	−5.89425e7	−0.182245
$271$	4.06868e8	1.24183	0.620913	−	0.783880i	$-0.286762\pi$
$271$	4.06868e8	1.24183	0.620913	+	0.783880i	$0.286762\pi$
$272$	−9.06735e7	−0.273205
$273$	1.09067e8	0.324432
$274$	−3.45115e8	−1.01353
$275$	1.56922e8	0.455007
$276$	−2.84058e8	−0.813252
$277$	2.56398e8	0.724829	0.362415	−	0.932017i	$-0.381952\pi$
$277$	2.56398e8	0.724829	0.362415	+	0.932017i	$0.381952\pi$
$278$	−5.67871e8	−1.58523
$279$	−3.56053e7	−0.0981523
$280$	−5.78411e7	−0.157465
$281$	2.24816e8	0.604443	0.302222	−	0.953238i	$-0.402272\pi$
$281$	2.24816e8	0.604443	0.302222	+	0.953238i	$0.402272\pi$
$282$	6.98732e7	0.185540
$283$	−6.52627e8	−1.71164	−0.855820	−	0.517273i	$-0.826947\pi$
$283$	−6.52627e8	−1.71164	−0.855820	+	0.517273i	$0.826947\pi$
$284$	−3.93436e8	−1.01920
$285$	8.69980e7	0.222614
$286$	7.13828e8	1.80432
$287$	−1.84718e8	−0.461235
$288$	1.64516e8	0.405821
$289$	−6.57511e7	−0.160236
$290$	7.43182e8	1.78938
$291$	−2.70090e7	−0.0642516
$292$	3.28140e8	0.771293
$293$	4.58871e7	0.106575	0.0532873	−	0.998579i	$-0.483030\pi$
$293$	4.58871e7	0.106575	0.0532873	+	0.998579i	$0.483030\pi$
$294$	−3.47444e8	−0.797386
$295$	−2.19771e8	−0.498418
$296$	5.13527e7	0.115091
$297$	6.19900e7	0.137301
$298$	4.78207e8	1.04679
$299$	7.09200e8	1.53433
$300$	2.54100e8	0.543351
$301$	−2.49314e8	−0.526944
$302$	4.86425e8	1.01623
$303$	−3.99954e8	−0.825963
$304$	−9.35593e7	−0.190998
$305$	1.47506e8	0.297687
$306$	−2.40894e8	−0.480619
$307$	−4.87671e8	−0.961929	−0.480964	−	0.876740i	$-0.659713\pi$
$307$	−4.87671e8	−0.961929	−0.480964	+	0.876740i	$0.659713\pi$
$308$	1.88727e8	0.368049
$309$	2.24505e8	0.432885
$310$	−1.46260e8	−0.278843
$311$	−6.69408e8	−1.26191	−0.630957	−	0.775818i	$-0.717337\pi$
$311$	−6.69408e8	−1.26191	−0.630957	+	0.775818i	$0.717337\pi$
$312$	3.72574e8	0.694498
$313$	−5.03772e8	−0.928600	−0.464300	−	0.885678i	$-0.653694\pi$
$313$	−5.03772e8	−0.928600	−0.464300	+	0.885678i	$0.653694\pi$
$314$	6.88886e7	0.125572
$315$	3.89072e7	0.0701364
$316$	3.03055e8	0.540277
$317$	1.93659e8	0.341452	0.170726	−	0.985319i	$-0.445389\pi$
$317$	1.93659e8	0.341452	0.170726	+	0.985319i	$0.445389\pi$
$318$	9.69713e8	1.69102
$319$	−7.81607e8	−1.34810
$320$	5.70620e8	0.973469
$321$	3.18561e8	0.537558
$322$	3.14569e8	0.525074
$323$	3.55554e8	0.587080
$324$	1.00379e8	0.163960
$325$	−6.34405e8	−1.02512
$326$	5.66855e8	0.906172
$327$	−576400.	−0.000911605 0
$328$	−6.30998e8	−0.987347
$329$	−4.61224e7	−0.0714047
$330$	2.54643e8	0.390061
$331$	9.32058e8	1.41268	0.706342	−	0.707871i	$-0.250344\pi$
$331$	9.32058e8	1.41268	0.706342	+	0.707871i	$0.250344\pi$
$332$	−6.40293e7	−0.0960275
$333$	−3.45428e7	−0.0512628
$334$	1.39886e9	2.05429
$335$	−5.10447e8	−0.741812
$336$	−4.18416e7	−0.0601757
$337$	−3.47540e8	−0.494652	−0.247326	−	0.968932i	$-0.579552\pi$
$337$	−3.47540e8	−0.494652	−0.247326	+	0.968932i	$0.579552\pi$
$338$	−1.76888e9	−2.49167
$339$	5.18558e8	0.722934
$340$	−5.89832e8	−0.813864
$341$	1.53822e8	0.210077
$342$	−2.48561e8	−0.336001
$343$	4.90620e8	0.656471
$344$	−8.51661e8	−1.12801
$345$	2.52992e8	0.331696
$346$	−2.08477e9	−2.70578
$347$	−1.30092e9	−1.67146	−0.835732	−	0.549137i	$-0.814956\pi$
$347$	−1.30092e9	−1.67146	−0.835732	+	0.549137i	$0.814956\pi$
$348$	−1.26564e9	−1.60984
$349$	3.48481e8	0.438824	0.219412	−	0.975632i	$-0.429586\pi$
$349$	3.48481e8	0.438824	0.219412	+	0.975632i	$0.429586\pi$
$350$	−2.81394e8	−0.350813
$351$	−2.50614e8	−0.309336
$352$	−7.10740e8	−0.868583
$353$	−7.93754e8	−0.960449	−0.480224	−	0.877146i	$-0.659445\pi$
$353$	−7.93754e8	−0.960449	−0.480224	+	0.877146i	$0.659445\pi$
$354$	6.27906e8	0.752286
$355$	3.50408e8	0.415695
$356$	−2.03337e9	−2.38859
$357$	1.59011e8	0.184965
$358$	−1.85369e9	−2.13524
$359$	−1.49660e9	−1.70717	−0.853584	−	0.520955i	$-0.825576\pi$
$359$	−1.49660e9	−1.70717	−0.853584	+	0.520955i	$0.825576\pi$
$360$	1.32908e8	0.150138
$361$	−5.27002e8	−0.589572
$362$	5.67431e8	0.628685
$363$	2.58345e8	0.283483
$364$	−7.62988e8	−0.829207
$365$	−2.92253e8	−0.314582
$366$	−4.21437e8	−0.449313
$367$	9.89904e8	1.04535	0.522675	−	0.852532i	$-0.324934\pi$
$367$	9.89904e8	1.04535	0.522675	+	0.852532i	$0.324934\pi$
$368$	−2.72073e8	−0.284589
$369$	4.24446e8	0.439774
$370$	−1.41895e8	−0.145633
$371$	−6.40096e8	−0.650783
$372$	2.49081e8	0.250865
$373$	1.75365e9	1.74969	0.874846	−	0.484402i	$-0.160963\pi$
$373$	1.75365e9	1.74969	0.874846	+	0.484402i	$0.160963\pi$
$374$	1.04071e9	1.02867
$375$	−5.81159e8	−0.569096
$376$	−1.57555e8	−0.152853
$377$	3.15989e9	3.03723
$378$	−1.11161e8	−0.105860
$379$	−1.86817e8	−0.176271	−0.0881354	−	0.996109i	$-0.528091\pi$
$379$	−1.86817e8	−0.176271	−0.0881354	+	0.996109i	$0.528091\pi$
$380$	−6.08604e8	−0.568973
$381$	6.66659e8	0.617541
$382$	−2.17048e9	−1.99221
$383$	−2.42488e8	−0.220543	−0.110272	−	0.993901i	$-0.535172\pi$
$383$	−2.42488e8	−0.220543	−0.110272	+	0.993901i	$0.535172\pi$
$384$	−8.50383e8	−0.766400
$385$	−1.68087e8	−0.150114
$386$	2.74010e9	2.42499
$387$	5.72876e8	0.502426
$388$	1.88945e8	0.164219
$389$	−2.95230e8	−0.254294	−0.127147	−	0.991884i	$-0.540582\pi$
$389$	−2.95230e8	−0.254294	−0.127147	+	0.991884i	$0.540582\pi$
$390$	−1.02948e9	−0.878799
$391$	1.03396e9	0.874752
$392$	7.83440e8	0.656908
$393$	1.12074e9	0.931392
$394$	−2.29381e9	−1.88938
$395$	−2.69911e8	−0.220359
$396$	−4.33658e8	−0.350925
$397$	1.85863e9	1.49082	0.745411	−	0.666605i	$-0.232253\pi$
$397$	1.85863e9	1.49082	0.745411	+	0.666605i	$0.232253\pi$
$398$	−1.47863e9	−1.17562
$399$	1.64072e8	0.129309
$400$	2.43379e8	0.190140
$401$	4.38529e8	0.339620	0.169810	−	0.985477i	$-0.445685\pi$
$401$	4.38529e8	0.339620	0.169810	+	0.985477i	$0.445685\pi$
$402$	1.45839e9	1.11965
$403$	−6.21874e8	−0.473298
$404$	2.79792e9	2.11106
$405$	−8.94014e7	−0.0668731
$406$	1.40159e9	1.03939
$407$	1.49231e8	0.109718
$408$	5.43184e8	0.395947
$409$	1.91083e9	1.38099	0.690496	−	0.723336i	$-0.257392\pi$
$409$	1.91083e9	1.38099	0.690496	+	0.723336i	$0.257392\pi$
$410$	1.74354e9	1.24936
$411$	−5.23455e8	−0.371906
$412$	−1.57055e9	−1.10640
$413$	−4.14473e8	−0.289515
$414$	−7.22820e8	−0.500644
$415$	5.70268e7	0.0391661
$416$	2.87339e9	1.95690
$417$	−8.61321e8	−0.581686
$418$	1.07383e9	0.719148
$419$	2.27930e9	1.51374	0.756872	−	0.653563i	$-0.226726\pi$
$419$	2.27930e9	1.51374	0.756872	+	0.653563i	$0.226726\pi$
$420$	−2.72180e8	−0.179260
$421$	−8.77176e8	−0.572927	−0.286464	−	0.958091i	$-0.592480\pi$
$421$	−8.77176e8	−0.572927	−0.286464	+	0.958091i	$0.592480\pi$
$422$	−4.22674e8	−0.273787
$423$	1.05980e8	0.0680824
$424$	−2.18658e9	−1.39311
$425$	−9.24915e8	−0.584441
$426$	−1.00115e9	−0.627427
$427$	2.78185e8	0.172917
$428$	−2.22853e9	−1.37393
$429$	1.08270e9	0.662077
$430$	2.35326e9	1.42735
$431$	1.66547e9	1.00200	0.500998	−	0.865449i	$-0.332967\pi$
$431$	1.66547e9	1.00200	0.500998	+	0.865449i	$0.332967\pi$
$432$	9.61440e7	0.0573758
$433$	1.37260e9	0.812526	0.406263	−	0.913756i	$-0.366832\pi$
$433$	1.37260e9	0.812526	0.406263	+	0.913756i	$0.366832\pi$
$434$	−2.75835e8	−0.161970
$435$	1.12723e9	0.656596
$436$	4.03227e6	0.00232995
$437$	1.06687e9	0.611541
$438$	8.34991e8	0.474813
$439$	−2.47419e9	−1.39575	−0.697874	−	0.716220i	$-0.745871\pi$
$439$	−2.47419e9	−1.39575	−0.697874	+	0.716220i	$0.745871\pi$
$440$	−5.74186e8	−0.321343
$441$	−5.26987e8	−0.292593
$442$	−4.20739e9	−2.31758
$443$	−1.67258e9	−0.914059	−0.457029	−	0.889452i	$-0.651087\pi$
$443$	−1.67258e9	−0.914059	−0.457029	+	0.889452i	$0.651087\pi$
$444$	2.41648e8	0.131021
$445$	1.81099e9	0.974219
$446$	1.65811e9	0.884995
$447$	7.25322e8	0.384109
$448$	1.07615e9	0.565456
$449$	−1.84092e8	−0.0959784	−0.0479892	−	0.998848i	$-0.515281\pi$
$449$	−1.84092e8	−0.0959784	−0.0479892	+	0.998848i	$0.515281\pi$
$450$	6.46589e8	0.334491
$451$	−1.83369e9	−0.941255
$452$	−3.62763e9	−1.84773
$453$	7.37787e8	0.372896
$454$	−6.05372e9	−3.03618
$455$	6.79544e8	0.338203
$456$	5.60471e8	0.276807
$457$	−5.82490e8	−0.285484	−0.142742	−	0.989760i	$-0.545592\pi$
$457$	−5.82490e8	−0.285484	−0.142742	+	0.989760i	$0.545592\pi$
$458$	2.98967e9	1.45410
$459$	−3.65377e8	−0.176359
$460$	−1.76983e9	−0.847774
$461$	−3.80880e9	−1.81065	−0.905327	−	0.424716i	$-0.860374\pi$
$461$	−3.80880e9	−1.81065	−0.905327	+	0.424716i	$0.860374\pi$
$462$	4.80238e8	0.226574
$463$	1.58897e7	0.00744014	0.00372007	−	0.999993i	$-0.498816\pi$
$463$	1.58897e7	0.00744014	0.00372007	+	0.999993i	$0.498816\pi$
$464$	−1.21224e9	−0.563347
$465$	−2.21840e8	−0.102319
$466$	−3.68935e9	−1.68888
$467$	5.79999e8	0.263523	0.131761	−	0.991281i	$-0.457937\pi$
$467$	5.79999e8	0.263523	0.131761	+	0.991281i	$0.457937\pi$
$468$	1.75320e9	0.790626
$469$	−9.62666e8	−0.430894
$470$	4.35347e8	0.193416
$471$	1.04487e8	0.0460776
$472$	−1.41584e9	−0.619753
$473$	−2.47493e9	−1.07535
$474$	7.71160e8	0.332598
$475$	−9.54351e8	−0.408583
$476$	−1.11238e9	−0.472747
$477$	1.47082e9	0.620503
$478$	3.79637e8	0.158990
$479$	2.43646e9	1.01294	0.506472	−	0.862256i	$-0.330949\pi$
$479$	2.43646e9	1.01294	0.506472	+	0.862256i	$0.330949\pi$
$480$	1.02502e9	0.423047
$481$	−6.03315e8	−0.247193
$482$	−3.82695e9	−1.55664
$483$	4.77124e8	0.192671
$484$	−1.80728e9	−0.724547
$485$	−1.68281e8	−0.0669789
$486$	2.55427e8	0.100935
$487$	3.24214e9	1.27198	0.635990	−	0.771697i	$-0.280592\pi$
$487$	3.24214e9	1.27198	0.635990	+	0.771697i	$0.280592\pi$
$488$	9.50285e8	0.370156
$489$	8.59780e8	0.332511
$490$	−2.16476e9	−0.831234
$491$	−2.76619e9	−1.05462	−0.527310	−	0.849673i	$-0.676800\pi$
$491$	−2.76619e9	−1.05462	−0.527310	+	0.849673i	$0.676800\pi$
$492$	−2.96926e9	−1.12401
$493$	4.60689e9	1.73158
$494$	−4.34130e9	−1.62022
$495$	3.86231e8	0.143129
$496$	2.38571e8	0.0877874
$497$	6.60844e8	0.241463
$498$	−1.62930e8	−0.0591152
$499$	8.06345e8	0.290515	0.145258	−	0.989394i	$-0.453599\pi$
$499$	8.06345e8	0.290515	0.145258	+	0.989394i	$0.453599\pi$
$500$	4.06556e9	1.45454
$501$	2.12173e9	0.753804
$502$	7.17807e8	0.253247
$503$	−5.04266e9	−1.76674	−0.883368	−	0.468680i	$-0.844730\pi$
$503$	−5.04266e9	−1.76674	−0.883368	+	0.468680i	$0.844730\pi$
$504$	2.50654e8	0.0872103
$505$	−2.49193e9	−0.861024
$506$	3.12272e9	1.07153
$507$	−2.68296e9	−0.914294
$508$	−4.66368e9	−1.57836
$509$	2.28362e9	0.767558	0.383779	−	0.923425i	$-0.374622\pi$
$509$	2.28362e9	0.767558	0.383779	+	0.923425i	$0.374622\pi$
$510$	−1.50090e9	−0.501020
$511$	−5.51168e8	−0.182730
$512$	−1.77993e9	−0.586081
$513$	−3.77005e8	−0.123293
$514$	−8.32793e9	−2.70499
$515$	1.39879e9	0.451260
$516$	−4.00762e9	−1.28414
$517$	−4.57856e8	−0.145718
$518$	−2.67603e8	−0.0845936
$519$	−3.16208e9	−0.992859
$520$	2.32133e9	0.723978
$521$	2.37077e9	0.734441	0.367220	−	0.930134i	$-0.380310\pi$
$521$	2.37077e9	0.734441	0.367220	+	0.930134i	$0.380310\pi$
$522$	−3.22058e9	−0.991031
$523$	−1.58006e9	−0.482966	−0.241483	−	0.970405i	$-0.577634\pi$
$523$	−1.58006e9	−0.482966	−0.241483	+	0.970405i	$0.577634\pi$
$524$	−7.84029e9	−2.38052
$525$	−4.26805e8	−0.128728
$526$	3.98010e9	1.19246
$527$	−9.06645e8	−0.269836
$528$	−4.15360e8	−0.122802
$529$	−3.02352e8	−0.0888010
$530$	6.04183e9	1.76280
$531$	9.52379e8	0.276044
$532$	−1.14778e9	−0.330498
$533$	7.41326e9	2.12063
$534$	−5.17416e9	−1.47043
$535$	1.98481e9	0.560377
$536$	−3.28848e9	−0.922398
$537$	−2.81159e9	−0.783505
$538$	−6.83428e9	−1.89215
$539$	2.27668e9	0.626241
$540$	6.25417e8	0.170919
$541$	−4.18017e9	−1.13502	−0.567510	−	0.823367i	$-0.692093\pi$
$541$	−4.18017e9	−1.13502	−0.567510	+	0.823367i	$0.692093\pi$
$542$	−7.24272e9	−1.95391
$543$	8.60654e8	0.230690
$544$	4.18919e9	1.11567
$545$	−3.59128e6	−0.000950302 0
$546$	−1.94151e9	−0.510466
$547$	−2.97423e9	−0.776996	−0.388498	−	0.921450i	$-0.627006\pi$
$547$	−2.97423e9	−0.776996	−0.388498	+	0.921450i	$0.627006\pi$
$548$	3.66189e9	0.950546
$549$	−6.39216e8	−0.164871
$550$	−2.79339e9	−0.715915
$551$	4.75351e9	1.21055
$552$	1.62986e9	0.412444
$553$	−5.09033e8	−0.127999
$554$	−4.56419e9	−1.14046
$555$	−2.15220e8	−0.0534388
$556$	6.02546e9	1.48672
$557$	−1.14215e9	−0.280045	−0.140023	−	0.990148i	$-0.544718\pi$
$557$	−1.14215e9	−0.280045	−0.140023	+	0.990148i	$0.544718\pi$
$558$	6.33816e8	0.154434
$559$	1.00057e10	2.42274
$560$	−2.60696e8	−0.0627300
$561$	1.57850e9	0.377463
$562$	−4.00199e9	−0.951041
$563$	6.84654e9	1.61693	0.808466	−	0.588543i	$-0.200298\pi$
$563$	6.84654e9	1.61693	0.808466	+	0.588543i	$0.200298\pi$
$564$	−7.41398e8	−0.174010
$565$	3.23089e9	0.753621
$566$	1.16175e10	2.69312
$567$	−1.68604e8	−0.0388444
$568$	2.25745e9	0.516891
$569$	−4.54096e8	−0.103337	−0.0516684	−	0.998664i	$-0.516454\pi$
$569$	−4.54096e8	−0.103337	−0.0516684	+	0.998664i	$0.516454\pi$
$570$	−1.54866e9	−0.350264
$571$	−3.97096e9	−0.892625	−0.446312	−	0.894877i	$-0.647263\pi$
$571$	−3.97096e9	−0.892625	−0.446312	+	0.894877i	$0.647263\pi$
$572$	−7.57416e9	−1.69219
$573$	−3.29209e9	−0.731023
$574$	3.28819e9	0.725714
$575$	−2.77528e9	−0.608792
$576$	−2.47278e9	−0.539147
$577$	−7.65858e9	−1.65971	−0.829857	−	0.557977i	$-0.811578\pi$
$577$	−7.65858e9	−1.65971	−0.829857	+	0.557977i	$0.811578\pi$
$578$	1.17045e9	0.252118
$579$	4.15606e9	0.889829
$580$	−7.88563e9	−1.67818
$581$	1.07548e8	0.0227503
$582$	4.80792e8	0.101094
$583$	−6.35421e9	−1.32807
$584$	−1.88280e9	−0.391164
$585$	−1.56146e9	−0.322467
$586$	−8.16844e8	−0.167686
$587$	−2.26433e9	−0.462068	−0.231034	−	0.972946i	$-0.574211\pi$
$587$	−2.26433e9	−0.462068	−0.231034	+	0.972946i	$0.574211\pi$
$588$	3.68659e9	0.747833
$589$	−9.35500e8	−0.188643
$590$	3.91219e9	0.784219
$591$	−3.47915e9	−0.693292
$592$	2.31452e8	0.0458495
$593$	−2.92886e9	−0.576776	−0.288388	−	0.957514i	$-0.593119\pi$
$593$	−2.92886e9	−0.576776	−0.288388	+	0.957514i	$0.593119\pi$
$594$	−1.10349e9	−0.216032
$595$	9.90724e8	0.192816
$596$	−5.07407e9	−0.981736
$597$	−2.24272e9	−0.431385
$598$	−1.26246e10	−2.41414
$599$	9.05370e9	1.72120	0.860602	−	0.509278i	$-0.170087\pi$
$599$	9.05370e9	1.72120	0.860602	+	0.509278i	$0.170087\pi$
$600$	−1.45797e9	−0.275562
$601$	7.20703e9	1.35424	0.677120	−	0.735872i	$-0.263228\pi$
$601$	7.20703e9	1.35424	0.677120	+	0.735872i	$0.263228\pi$
$602$	4.43809e9	0.829102
$603$	2.21202e9	0.410846
$604$	−5.16127e9	−0.953076
$605$	1.60963e9	0.295516
$606$	7.11965e9	1.29958
$607$	1.79417e9	0.325613	0.162807	−	0.986658i	$-0.447945\pi$
$607$	1.79417e9	0.325613	0.162807	+	0.986658i	$0.447945\pi$
$608$	4.32251e9	0.779963
$609$	2.12586e9	0.381395
$610$	−2.62578e9	−0.468386
$611$	1.85103e9	0.328298
$612$	2.55603e9	0.450751
$613$	6.04302e9	1.05960	0.529800	−	0.848122i	$-0.322267\pi$
$613$	6.04302e9	1.05960	0.529800	+	0.848122i	$0.322267\pi$
$614$	8.68112e9	1.51351
$615$	2.64452e9	0.458442
$616$	−1.08287e9	−0.186657
$617$	−7.25498e9	−1.24348	−0.621739	−	0.783225i	$-0.713573\pi$
$617$	−7.25498e9	−1.24348	−0.621739	+	0.783225i	$0.713573\pi$
$618$	−3.99646e9	−0.681108
$619$	5.22282e9	0.885090	0.442545	−	0.896746i	$-0.354076\pi$
$619$	5.22282e9	0.885090	0.442545	+	0.896746i	$0.354076\pi$
$620$	1.55191e9	0.261514
$621$	−1.09634e9	−0.183707
$622$	1.19162e10	1.98551
$623$	3.41540e9	0.565892
$624$	1.67923e9	0.276671
$625$	2.71690e8	0.0445137
$626$	8.96772e9	1.46107
$627$	1.62873e9	0.263885
$628$	−7.30951e8	−0.117769
$629$	−8.79588e8	−0.140930
$630$	−6.92594e8	−0.110354
$631$	8.34968e9	1.32302	0.661511	−	0.749936i	$-0.269916\pi$
$631$	8.34968e9	1.32302	0.661511	+	0.749936i	$0.269916\pi$
$632$	−1.73886e9	−0.274003
$633$	−6.41093e8	−0.100463
$634$	−3.44736e9	−0.537247
$635$	4.15364e9	0.643755
$636$	−1.02893e10	−1.58593
$637$	−9.20422e9	−1.41091
$638$	1.39135e10	2.12112
$639$	−1.51849e9	−0.230229
$640$	−5.29834e9	−0.798932
$641$	3.59096e9	0.538527	0.269263	−	0.963067i	$-0.413220\pi$
$641$	3.59096e9	0.538527	0.269263	+	0.963067i	$0.413220\pi$
$642$	−5.67076e9	−0.845802
$643$	−1.20633e10	−1.78948	−0.894741	−	0.446585i	$-0.852640\pi$
$643$	−1.20633e10	−1.78948	−0.894741	+	0.446585i	$0.852640\pi$
$644$	−3.33778e9	−0.492444
$645$	3.56933e9	0.523754
$646$	−6.32928e9	−0.923721
$647$	8.62316e9	1.25170	0.625851	−	0.779942i	$-0.284752\pi$
$647$	8.62316e9	1.25170	0.625851	+	0.779942i	$0.284752\pi$
$648$	−5.75955e8	−0.0831526
$649$	−4.11446e9	−0.590821
$650$	1.12932e10	1.61294
$651$	−4.18374e8	−0.0594335
$652$	−6.01469e9	−0.849858
$653$	2.70263e9	0.379831	0.189916	−	0.981800i	$-0.439179\pi$
$653$	2.70263e9	0.379831	0.189916	+	0.981800i	$0.439179\pi$
$654$	1.02606e7	0.00143433
$655$	6.98283e9	0.970928
$656$	−2.84397e9	−0.393334
$657$	1.26648e9	0.174228
$658$	8.21033e8	0.112349
$659$	1.28231e10	1.74539	0.872697	−	0.488262i	$-0.162369\pi$
$659$	1.28231e10	1.74539	0.872697	+	0.488262i	$0.162369\pi$
$660$	−2.70192e9	−0.365821
$661$	−9.80158e9	−1.32005	−0.660026	−	0.751243i	$-0.729455\pi$
$661$	−9.80158e9	−1.32005	−0.660026	+	0.751243i	$0.729455\pi$
$662$	−1.65917e10	−2.22274
$663$	−6.38158e9	−0.850415
$664$	3.67386e8	0.0487007
$665$	1.02225e9	0.134798
$666$	6.14901e8	0.0806577
$667$	1.38233e10	1.80373
$668$	−1.48428e10	−1.92663
$669$	2.51494e9	0.324741
$670$	9.08655e9	1.16718
$671$	2.76154e9	0.352876
$672$	1.93312e9	0.245734
$673$	6.64004e9	0.839688	0.419844	−	0.907596i	$-0.362085\pi$
$673$	6.64004e9	0.839688	0.419844	+	0.907596i	$0.362085\pi$
$674$	6.18661e9	0.778294
$675$	9.80716e8	0.122738
$676$	1.87689e10	2.33682
$677$	9.66007e9	1.19652	0.598260	−	0.801302i	$-0.295859\pi$
$677$	9.66007e9	1.19652	0.598260	+	0.801302i	$0.295859\pi$
$678$	−9.23094e9	−1.13748
$679$	−3.17365e8	−0.0389059
$680$	3.38433e9	0.412754
$681$	−9.18202e9	−1.11410
$682$	−2.73821e9	−0.330538
$683$	6.00040e9	0.720623	0.360311	−	0.932832i	$-0.382670\pi$
$683$	6.00040e9	0.720623	0.360311	+	0.932832i	$0.382670\pi$
$684$	2.63738e9	0.315121
$685$	−3.26140e9	−0.387693
$686$	−8.73360e9	−1.03290
$687$	4.53460e9	0.533569
$688$	−3.83852e9	−0.449371
$689$	2.56889e10	2.99212
$690$	−4.50355e9	−0.521895
$691$	−2.66239e9	−0.306972	−0.153486	−	0.988151i	$-0.549050\pi$
$691$	−2.66239e9	−0.306972	−0.153486	+	0.988151i	$0.549050\pi$
$692$	2.21207e10	2.53763
$693$	7.28403e8	0.0831391
$694$	2.31579e10	2.62991
$695$	−5.36649e9	−0.606378
$696$	7.26198e9	0.816437
$697$	1.08080e10	1.20901
$698$	−6.20338e9	−0.690453
$699$	−5.59584e9	−0.619720
$700$	2.98576e9	0.329012
$701$	−1.25846e10	−1.37984	−0.689919	−	0.723887i	$-0.742354\pi$
$701$	−1.25846e10	−1.37984	−0.689919	+	0.723887i	$0.742354\pi$
$702$	4.46123e9	0.486715
$703$	−9.07582e8	−0.0985240
$704$	1.06829e10	1.15394
$705$	6.60315e8	0.0709724
$706$	1.41297e10	1.51119
$707$	−4.69959e9	−0.500141
$708$	−6.66247e9	−0.705535
$709$	9.49660e7	0.0100071	0.00500353	−	0.999987i	$-0.498407\pi$
$709$	9.49660e7	0.0100071	0.00500353	+	0.999987i	$0.498407\pi$
$710$	−6.23767e9	−0.654061
$711$	1.16966e9	0.122044
$712$	1.16670e10	1.21138
$713$	−2.72045e9	−0.281079
$714$	−2.83058e9	−0.291026
$715$	6.74581e9	0.690181
$716$	1.96688e10	2.00254
$717$	5.75815e8	0.0583400
$718$	2.66413e10	2.68608
$719$	−1.53194e10	−1.53706	−0.768529	−	0.639815i	$-0.779011\pi$
$719$	−1.53194e10	−1.53706	−0.768529	+	0.639815i	$0.779011\pi$
$720$	5.99028e8	0.0598113
$721$	2.63801e9	0.262122
$722$	9.38124e9	0.927641
$723$	−5.80454e9	−0.571195
$724$	−6.02080e9	−0.589616
$725$	−1.23654e10	−1.20511
$726$	−4.59884e9	−0.446036
$727$	8.16977e8	0.0788568	0.0394284	−	0.999222i	$-0.487446\pi$
$727$	8.16977e8	0.0788568	0.0394284	+	0.999222i	$0.487446\pi$
$728$	4.37786e9	0.420535
$729$	3.87420e8	0.0370370
$730$	5.20244e9	0.494968
$731$	1.45876e10	1.38125
$732$	4.47171e9	0.421391
$733$	9.45795e8	0.0887019	0.0443509	−	0.999016i	$-0.485878\pi$
$733$	9.45795e8	0.0887019	0.0443509	+	0.999016i	$0.485878\pi$
$734$	−1.76214e10	−1.64477
$735$	−3.28341e9	−0.305014
$736$	1.25700e10	1.16215
$737$	−9.55635e9	−0.879338
$738$	−7.55563e9	−0.691948
$739$	1.28908e9	0.117496	0.0587481	−	0.998273i	$-0.481289\pi$
$739$	1.28908e9	0.117496	0.0587481	+	0.998273i	$0.481289\pi$
$740$	1.50559e9	0.136583
$741$	−6.58468e9	−0.594526
$742$	1.13944e10	1.02395
$743$	−9.99618e9	−0.894073	−0.447037	−	0.894516i	$-0.647521\pi$
$743$	−9.99618e9	−0.894073	−0.447037	+	0.894516i	$0.647521\pi$
$744$	−1.42917e9	−0.127227
$745$	4.51915e9	0.400414
$746$	−3.12170e10	−2.75299
$747$	−2.47125e8	−0.0216918
$748$	−1.10426e10	−0.964748
$749$	3.74320e9	0.325504
$750$	1.03453e10	0.895424
$751$	1.75051e10	1.50808	0.754040	−	0.656828i	$-0.228102\pi$
$751$	1.75051e10	1.50808	0.754040	+	0.656828i	$0.228102\pi$
$752$	−7.10116e8	−0.0608929
$753$	1.08874e9	0.0929267
$754$	−5.62498e10	−4.77883
$755$	4.59681e9	0.388725
$756$	1.17949e9	0.0992815
$757$	1.22670e10	1.02779	0.513894	−	0.857854i	$-0.328203\pi$
$757$	1.22670e10	1.02779	0.513894	+	0.857854i	$0.328203\pi$
$758$	3.32557e9	0.277347
$759$	4.73640e9	0.393190
$760$	3.49204e9	0.288557
$761$	−1.65232e10	−1.35909	−0.679546	−	0.733633i	$-0.737823\pi$
$761$	−1.65232e10	−1.35909	−0.679546	+	0.733633i	$0.737823\pi$
$762$	−1.18673e10	−0.971649
$763$	−6.77289e6	−0.000551999 0
$764$	2.30302e10	1.86840
$765$	−2.27649e9	−0.183845
$766$	4.31656e9	0.347006
$767$	1.66340e10	1.33111
$768$	3.41500e9	0.272036
$769$	−2.15030e10	−1.70513	−0.852565	−	0.522621i	$-0.824954\pi$
$769$	−2.15030e10	−1.70513	−0.852565	+	0.522621i	$0.824954\pi$
$770$	2.99214e9	0.236191
$771$	−1.26314e10	−0.992572
$772$	−2.90742e10	−2.27429
$773$	−1.63814e10	−1.27562	−0.637811	−	0.770193i	$-0.720160\pi$
$773$	−1.63814e10	−1.27562	−0.637811	+	0.770193i	$0.720160\pi$
$774$	−1.01979e10	−0.790526
$775$	2.43355e9	0.187795
$776$	−1.08412e9	−0.0832842
$777$	−4.05889e8	−0.0310408
$778$	5.25543e9	0.400110
$779$	1.11520e10	0.845220
$780$	1.09234e10	0.824187
$781$	6.56017e9	0.492761
$782$	−1.84057e10	−1.37635
$783$	−4.88483e9	−0.363649
$784$	3.53104e9	0.261696
$785$	6.51011e8	0.0480335
$786$	−1.99505e10	−1.46547
$787$	−1.28196e10	−0.937481	−0.468741	−	0.883336i	$-0.655292\pi$
$787$	−1.28196e10	−0.937481	−0.468741	+	0.883336i	$0.655292\pi$
$788$	2.43387e10	1.77197
$789$	6.03683e9	0.437562
$790$	4.80474e9	0.346717
$791$	6.09323e9	0.437754
$792$	2.48824e9	0.177973
$793$	−1.11644e10	−0.795022
$794$	−3.30858e10	−2.34568
$795$	9.16397e9	0.646843
$796$	1.56892e10	1.10257
$797$	2.61499e10	1.82964	0.914819	−	0.403863i	$-0.132333\pi$
$797$	2.61499e10	1.82964	0.914819	+	0.403863i	$0.132333\pi$
$798$	−2.92067e9	−0.203457
$799$	2.69866e9	0.187169
$800$	−1.12443e10	−0.776457
$801$	−7.84793e9	−0.539562
$802$	−7.80632e9	−0.534363
$803$	−5.47142e9	−0.372903
$804$	−1.54744e10	−1.05007
$805$	2.97274e9	0.200850
$806$	1.10701e10	0.744694
$807$	−1.03659e10	−0.694305
$808$	−1.60539e10	−1.07063
$809$	1.46683e10	0.974003	0.487002	−	0.873401i	$-0.338091\pi$
$809$	1.46683e10	0.974003	0.487002	+	0.873401i	$0.338091\pi$
$810$	1.59145e9	0.105219
$811$	8.98477e9	0.591472	0.295736	−	0.955270i	$-0.404435\pi$
$811$	8.98477e9	0.591472	0.295736	+	0.955270i	$0.404435\pi$
$812$	−1.48717e10	−0.974799
$813$	−1.09854e10	−0.716969
$814$	−2.65649e9	−0.172633
$815$	5.35689e9	0.346626
$816$	2.44819e9	0.157735
$817$	1.50518e10	0.965634
$818$	−3.40151e10	−2.17287
$819$	−2.94480e9	−0.187311
$820$	−1.85001e10	−1.17172
$821$	1.91635e10	1.20857	0.604287	−	0.796766i	$-0.293458\pi$
$821$	1.91635e10	1.20857	0.604287	+	0.796766i	$0.293458\pi$
$822$	9.31811e9	0.585162
$823$	−1.09346e10	−0.683762	−0.341881	−	0.939743i	$-0.611064\pi$
$823$	−1.09346e10	−0.683762	−0.341881	+	0.939743i	$0.611064\pi$
$824$	9.01149e9	0.561115
$825$	−4.23688e9	−0.262698
$826$	7.37810e9	0.455527
$827$	−6.53139e9	−0.401547	−0.200773	−	0.979638i	$-0.564345\pi$
$827$	−6.53139e9	−0.401547	−0.200773	+	0.979638i	$0.564345\pi$
$828$	7.66957e9	0.469532
$829$	8.76423e9	0.534285	0.267143	−	0.963657i	$-0.413921\pi$
$829$	8.76423e9	0.534285	0.267143	+	0.963657i	$0.413921\pi$
$830$	−1.01514e9	−0.0616245
$831$	−6.92275e9	−0.418480
$832$	−4.31890e10	−2.59981
$833$	−1.34191e10	−0.804386
$834$	1.53325e10	0.915234
$835$	1.32195e10	0.785802
$836$	−1.13940e10	−0.674457
$837$	9.61344e8	0.0566682
$838$	−4.05742e10	−2.38175
$839$	2.43012e9	0.142056	0.0710282	−	0.997474i	$-0.477372\pi$
$839$	2.43012e9	0.142056	0.0710282	+	0.997474i	$0.477372\pi$
$840$	1.56171e9	0.0909123
$841$	4.43409e10	2.57051
$842$	1.56148e10	0.901453
$843$	−6.07004e9	−0.348976
$844$	4.48484e9	0.256772
$845$	−1.67163e10	−0.953105
$846$	−1.88658e9	−0.107122
$847$	3.03564e9	0.171656
$848$	−9.85511e9	−0.554979
$849$	1.76209e10	0.988216
$850$	1.64646e10	0.919568
$851$	−2.63927e9	−0.146801
$852$	1.06228e10	0.588436
$853$	1.58234e10	0.872930	0.436465	−	0.899721i	$-0.356230\pi$
$853$	1.58234e10	0.872930	0.436465	+	0.899721i	$0.356230\pi$
$854$	−4.95202e9	−0.272070
$855$	−2.34894e9	−0.128526
$856$	1.27868e10	0.696794
$857$	1.22300e10	0.663734	0.331867	−	0.943326i	$-0.392321\pi$
$857$	1.22300e10	0.663734	0.331867	+	0.943326i	$0.392321\pi$
$858$	−1.92734e10	−1.04172
$859$	−3.47554e9	−0.187088	−0.0935442	−	0.995615i	$-0.529820\pi$
$859$	−3.47554e9	−0.187088	−0.0935442	+	0.995615i	$0.529820\pi$
$860$	−2.49696e10	−1.33865
$861$	4.98738e9	0.266294
$862$	−2.96473e10	−1.57655
$863$	−3.47810e10	−1.84206	−0.921030	−	0.389491i	$-0.872651\pi$
$863$	−3.47810e10	−1.84206	−0.921030	+	0.389491i	$0.872651\pi$
$864$	−4.44193e9	−0.234301
$865$	−1.97015e10	−1.03500
$866$	−2.44339e10	−1.27844
$867$	1.77528e9	0.0925124
$868$	2.92678e9	0.151905
$869$	−5.05315e9	−0.261212
$870$	−2.00659e10	−1.03310
$871$	3.86346e10	1.98113
$872$	−2.31363e7	−0.00118164
$873$	7.29244e8	0.0370957
$874$	−1.89915e10	−0.962207
$875$	−6.82881e9	−0.344601
$876$	−8.85978e9	−0.445306
$877$	4.29703e9	0.215114	0.107557	−	0.994199i	$-0.465697\pi$
$877$	4.29703e9	0.215114	0.107557	+	0.994199i	$0.465697\pi$
$878$	4.40435e10	2.19609
$879$	−1.23895e9	−0.0615309
$880$	−2.58792e9	−0.128015
$881$	−1.70280e10	−0.838971	−0.419485	−	0.907762i	$-0.637789\pi$
$881$	−1.70280e10	−0.838971	−0.419485	+	0.907762i	$0.637789\pi$
$882$	9.38098e9	0.460371
$883$	−3.38818e10	−1.65617	−0.828083	−	0.560606i	$-0.810568\pi$
$883$	−3.38818e10	−1.65617	−0.828083	+	0.560606i	$0.810568\pi$
$884$	4.46430e10	2.17356
$885$	5.93383e9	0.287762
$886$	2.97739e10	1.43819
$887$	−6.81742e8	−0.0328010	−0.0164005	−	0.999866i	$-0.505221\pi$
$887$	−6.81742e8	−0.0328010	−0.0164005	+	0.999866i	$0.505221\pi$
$888$	−1.38652e9	−0.0664479
$889$	7.83346e9	0.373936
$890$	−3.22378e10	−1.53285
$891$	−1.67373e9	−0.0792708
$892$	−1.75936e10	−0.829997
$893$	2.78455e9	0.130850
$894$	−1.29116e10	−0.604363
$895$	−1.75177e10	−0.816763
$896$	−9.99228e9	−0.464073
$897$	−1.91484e10	−0.885847
$898$	3.27706e9	0.151014
$899$	−1.21212e10	−0.556399
$900$	−6.86071e9	−0.313704
$901$	3.74525e10	1.70586
$902$	3.26418e10	1.48098
$903$	6.73149e9	0.304231
$904$	2.08146e10	0.937082
$905$	5.36234e9	0.240483
$906$	−1.31335e10	−0.586720
$907$	−1.35686e10	−0.603822	−0.301911	−	0.953336i	$-0.597625\pi$
$907$	−1.35686e10	−0.603822	−0.301911	+	0.953336i	$0.597625\pi$
$908$	6.42338e10	2.84750
$909$	1.07988e10	0.476870
$910$	−1.20967e10	−0.532134
$911$	2.68538e10	1.17677	0.588385	−	0.808581i	$-0.299764\pi$
$911$	2.68538e10	1.17677	0.588385	+	0.808581i	$0.299764\pi$
$912$	2.52610e9	0.110273
$913$	1.06763e9	0.0464272
$914$	1.03690e10	0.449185
$915$	−3.98266e9	−0.171870
$916$	−3.17223e10	−1.36374
$917$	1.31691e10	0.563980
$918$	6.50413e9	0.277485
$919$	2.03037e10	0.862921	0.431461	−	0.902132i	$-0.357998\pi$
$919$	2.03037e10	0.862921	0.431461	+	0.902132i	$0.357998\pi$
$920$	1.01549e10	0.429951
$921$	1.31671e10	0.555370
$922$	6.78011e10	2.84891
$923$	−2.65216e10	−1.11018
$924$	−5.09562e9	−0.212493
$925$	2.36092e9	0.0980811
$926$	−2.82854e8	−0.0117064
$927$	−6.06165e9	−0.249926
$928$	5.60065e10	2.30049
$929$	−1.31416e10	−0.537765	−0.268883	−	0.963173i	$-0.586654\pi$
$929$	−1.31416e10	−0.537765	−0.268883	+	0.963173i	$0.586654\pi$
$930$	3.94901e9	0.160990
$931$	−1.38461e10	−0.562347
$932$	3.91463e10	1.58393
$933$	1.80740e10	0.728566
$934$	−1.03247e10	−0.414631
$935$	9.83488e9	0.393485
$936$	−1.00595e10	−0.400969
$937$	−3.76543e10	−1.49529	−0.747646	−	0.664098i	$-0.768816\pi$
$937$	−3.76543e10	−1.49529	−0.747646	+	0.664098i	$0.768816\pi$
$938$	1.71366e10	0.677975
$939$	1.36018e10	0.536127
$940$	−4.61931e9	−0.181397
$941$	−7.32880e9	−0.286727	−0.143364	−	0.989670i	$-0.545792\pi$
$941$	−7.32880e9	−0.286727	−0.143364	+	0.989670i	$0.545792\pi$
$942$	−1.85999e9	−0.0724992
$943$	3.24301e10	1.25938
$944$	−6.38135e9	−0.246894
$945$	−1.05050e9	−0.0404933
$946$	4.40567e10	1.69197
$947$	−4.94313e10	−1.89137	−0.945686	−	0.325081i	$-0.894609\pi$
$947$	−4.94313e10	−1.89137	−0.945686	+	0.325081i	$0.894609\pi$
$948$	−8.18248e9	−0.311929
$949$	2.21200e10	0.840142
$950$	1.69886e10	0.642871
$951$	−5.22879e9	−0.197138
$952$	6.38259e9	0.239755
$953$	−3.01679e10	−1.12907	−0.564533	−	0.825411i	$-0.690944\pi$
$953$	−3.01679e10	−1.12907	−0.564533	+	0.825411i	$0.690944\pi$
$954$	−2.61822e10	−0.976310
$955$	−2.05115e10	−0.762054
$956$	−4.02818e9	−0.149110
$957$	2.11034e10	0.778324
$958$	−4.33719e10	−1.59378
$959$	−6.15077e9	−0.225198
$960$	−1.54067e10	−0.562033
$961$	−2.51271e10	−0.913295
$962$	1.07397e10	0.388938
$963$	−8.60115e9	−0.310359
$964$	4.06063e10	1.45990
$965$	2.58945e10	0.927601
$966$	−8.49337e9	−0.303152
$967$	2.77653e7	0.000987436 0	0.000493718	−	1.00000i	$-0.499843\pi$
$967$	2.77653e7	0.000987436 0	0.000493718		1.00000i	$0.499843\pi$
$968$	1.03698e10	0.367457
$969$	−9.59997e9	−0.338951
$970$	2.99559e9	0.105386
$971$	1.17886e10	0.413233	0.206617	−	0.978422i	$-0.433755\pi$
$971$	1.17886e10	0.413233	0.206617	+	0.978422i	$0.433755\pi$
$972$	−2.71024e9	−0.0946621
$973$	−1.01208e10	−0.352225
$974$	−5.77139e10	−2.00135
$975$	1.71289e10	0.591854
$976$	4.28303e9	0.147461
$977$	1.71267e9	0.0587547	0.0293773	−	0.999568i	$-0.490648\pi$
$977$	1.71267e9	0.0587547	0.0293773	+	0.999568i	$0.490648\pi$
$978$	−1.53051e10	−0.523178
$979$	3.39045e10	1.15483
$980$	2.29694e10	0.779577
$981$	1.55628e7	0.000526315 0
$982$	4.92414e10	1.65936
$983$	4.97438e10	1.67033	0.835164	−	0.550001i	$-0.185373\pi$
$983$	4.97438e10	1.67033	0.835164	+	0.550001i	$0.185373\pi$
$984$	1.70370e10	0.570045
$985$	−2.16769e10	−0.722722
$986$	−8.20080e10	−2.72450
$987$	1.24531e9	0.0412255
$988$	4.60639e10	1.51954
$989$	4.37711e10	1.43880
$990$	−6.87536e9	−0.225202
$991$	1.60822e10	0.524912	0.262456	−	0.964944i	$-0.415468\pi$
$991$	1.60822e10	0.524912	0.262456	+	0.964944i	$0.415468\pi$
$992$	−1.10222e10	−0.358490
$993$	−2.51656e10	−0.815613
$994$	−1.17638e10	−0.379922
$995$	−1.39733e10	−0.449696
$996$	1.72879e9	0.0554415
$997$	−5.25619e10	−1.67972	−0.839862	−	0.542800i	$-0.817364\pi$
$997$	−5.25619e10	−1.67972	−0.839862	+	0.542800i	$0.817364\pi$
$998$	−1.43539e10	−0.457101
$999$	9.32655e8	0.0295966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.8.a.c.1.6	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.8.a.c.1.6	✓	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q-17.8012 q^{2} -27.0000 q^{3} +188.881 q^{4} -168.224 q^{5} +480.631 q^{6} -317.259 q^{7} -1083.76 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-17.8012 q^{2} -27.0000 q^{3} +188.881 q^{4} -168.224 q^{5} +480.631 q^{6} -317.259 q^{7} -1083.76 q^{8} +729.000 q^{9} +2994.59 q^{10} -3149.42 q^{11} -5099.80 q^{12} +12732.5 q^{13} +5647.58 q^{14} +4542.06 q^{15} -4884.62 q^{16} +18563.1 q^{17} -12977.0 q^{18} +19153.9 q^{19} -31774.5 q^{20} +8565.99 q^{21} +56063.3 q^{22} +55699.9 q^{23} +29261.5 q^{24} -49825.5 q^{25} -226654. q^{26} -19683.0 q^{27} -59924.3 q^{28} +248175. q^{29} -80853.9 q^{30} -48841.3 q^{31} +225673. q^{32} +85034.3 q^{33} -330444. q^{34} +53370.7 q^{35} +137695. q^{36} -47383.8 q^{37} -340961. q^{38} -343778. q^{39} +182315. q^{40} +582230. q^{41} -152485. q^{42} +785839. q^{43} -594867. q^{44} -122636. q^{45} -991522. q^{46} +145378. q^{47} +131885. q^{48} -722890. q^{49} +886953. q^{50} -501203. q^{51} +2.40494e6 q^{52} +2.01758e6 q^{53} +350380. q^{54} +529809. q^{55} +343833. q^{56} -517154. q^{57} -4.41780e6 q^{58} +1.30642e6 q^{59} +857911. q^{60} -876840. q^{61} +869433. q^{62} -231282. q^{63} -3.39202e6 q^{64} -2.14192e6 q^{65} -1.51371e6 q^{66} +3.03432e6 q^{67} +3.50622e6 q^{68} -1.50390e6 q^{69} -950061. q^{70} -2.08298e6 q^{71} -790062. q^{72} +1.73728e6 q^{73} +843486. q^{74} +1.34529e6 q^{75} +3.61781e6 q^{76} +999181. q^{77} +6.11965e6 q^{78} +1.60447e6 q^{79} +821712. q^{80} +531441. q^{81} -1.03644e7 q^{82} -338992. q^{83} +1.61796e6 q^{84} -3.12276e6 q^{85} -1.39888e7 q^{86} -6.70072e6 q^{87} +3.41322e6 q^{88} -1.07653e7 q^{89} +2.18306e6 q^{90} -4.03951e6 q^{91} +1.05207e7 q^{92} +1.31872e6 q^{93} -2.58790e6 q^{94} -3.22215e6 q^{95} -6.09318e6 q^{96} +1.00033e6 q^{97} +1.28683e7 q^{98} -2.29593e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9}+O(q^{10}) \) 48 * q + 2 * q^2 - 1296 * q^3 + 3214 * q^4 + 428 * q^5 - 54 * q^6 - 680 * q^7 + 2355 * q^8 + 34992 * q^9	\( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9} + 6185 q^{10} + 11989 q^{11} - 86778 q^{12} - 2393 q^{13} + 18201 q^{14} - 11556 q^{15} + 208150 q^{16} + 62538 q^{17} + 1458 q^{18} - 39882 q^{19} + 113423 q^{20} + 18360 q^{21} - 93716 q^{22} + 195618 q^{23} - 63585 q^{24} + 886490 q^{25} + 294399 q^{26} - 944784 q^{27} - 60819 q^{28} + 421501 q^{29} - 166995 q^{30} + 392689 q^{31} - 341578 q^{32} - 323703 q^{33} + 50837 q^{34} + 697874 q^{35} + 2343006 q^{36} - 410396 q^{37} + 677216 q^{38} + 64611 q^{39} + 3232376 q^{40} + 3832958 q^{41} - 491427 q^{42} - 1751932 q^{43} + 4888297 q^{44} + 312012 q^{45} + 1163150 q^{46} + 106461 q^{47} - 5620050 q^{48} + 8202048 q^{49} - 2159111 q^{50} - 1688526 q^{51} - 3605030 q^{52} + 1755534 q^{53} - 39366 q^{54} - 1220729 q^{55} - 4430622 q^{56} + 1076814 q^{57} - 10000202 q^{58} - 2037752 q^{59} - 3062421 q^{60} + 1274098 q^{61} + 97748 q^{62} - 495720 q^{63} + 15135201 q^{64} + 6139645 q^{65} + 2530332 q^{66} - 7751257 q^{67} + 1700631 q^{68} - 5281686 q^{69} - 20935703 q^{70} - 12592217 q^{71} + 1716795 q^{72} + 12508355 q^{73} - 14999956 q^{74} - 23935230 q^{75} - 23946874 q^{76} + 1874177 q^{77} - 7948773 q^{78} - 5103480 q^{79} + 3128449 q^{80} + 25509168 q^{81} + 11622426 q^{82} + 3040643 q^{83} + 1642113 q^{84} - 13756076 q^{85} + 964635 q^{86} - 11380527 q^{87} - 29653500 q^{88} + 28462995 q^{89} + 4508865 q^{90} + 3016621 q^{91} + 22938254 q^{92} - 10602603 q^{93} - 10070348 q^{94} - 2579984 q^{95} + 9222606 q^{96} + 16208760 q^{97} + 6323227 q^{98} + 8739981 q^{99}+O(q^{100}) \) 48 * q + 2 * q^2 - 1296 * q^3 + 3214 * q^4 + 428 * q^5 - 54 * q^6 - 680 * q^7 + 2355 * q^8 + 34992 * q^9 + 6185 * q^10 + 11989 * q^11 - 86778 * q^12 - 2393 * q^13 + 18201 * q^14 - 11556 * q^15 + 208150 * q^16 + 62538 * q^17 + 1458 * q^18 - 39882 * q^19 + 113423 * q^20 + 18360 * q^21 - 93716 * q^22 + 195618 * q^23 - 63585 * q^24 + 886490 * q^25 + 294399 * q^26 - 944784 * q^27 - 60819 * q^28 + 421501 * q^29 - 166995 * q^30 + 392689 * q^31 - 341578 * q^32 - 323703 * q^33 + 50837 * q^34 + 697874 * q^35 + 2343006 * q^36 - 410396 * q^37 + 677216 * q^38 + 64611 * q^39 + 3232376 * q^40 + 3832958 * q^41 - 491427 * q^42 - 1751932 * q^43 + 4888297 * q^44 + 312012 * q^45 + 1163150 * q^46 + 106461 * q^47 - 5620050 * q^48 + 8202048 * q^49 - 2159111 * q^50 - 1688526 * q^51 - 3605030 * q^52 + 1755534 * q^53 - 39366 * q^54 - 1220729 * q^55 - 4430622 * q^56 + 1076814 * q^57 - 10000202 * q^58 - 2037752 * q^59 - 3062421 * q^60 + 1274098 * q^61 + 97748 * q^62 - 495720 * q^63 + 15135201 * q^64 + 6139645 * q^65 + 2530332 * q^66 - 7751257 * q^67 + 1700631 * q^68 - 5281686 * q^69 - 20935703 * q^70 - 12592217 * q^71 + 1716795 * q^72 + 12508355 * q^73 - 14999956 * q^74 - 23935230 * q^75 - 23946874 * q^76 + 1874177 * q^77 - 7948773 * q^78 - 5103480 * q^79 + 3128449 * q^80 + 25509168 * q^81 + 11622426 * q^82 + 3040643 * q^83 + 1642113 * q^84 - 13756076 * q^85 + 964635 * q^86 - 11380527 * q^87 - 29653500 * q^88 + 28462995 * q^89 + 4508865 * q^90 + 3016621 * q^91 + 22938254 * q^92 - 10602603 * q^93 - 10070348 * q^94 - 2579984 * q^95 + 9222606 * q^96 + 16208760 * q^97 + 6323227 * q^98 + 8739981 * q^99

Introduction

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