LMFDB - Embedded newform 546.8.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,8,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("546.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	546.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:	$170.562223914$
Analytic rank:	$0$
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} - x^{5} - 280157x^{4} + 23551285x^{3} + 13122885428x^{2} - 1144917710924x - 95027285980032 $ x^6 - x^5 - 280157x^4 + 23551285x^3 + 13122885428x^2 - 1144917710924x - 95027285980032
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-219.782$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -225.782 q^{5} +216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -225.782 q^{5} +216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +1806.25 q^{10} -1780.90 q^{11} -1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +6096.11 q^{15} +4096.00 q^{16} -32702.4 q^{17} -5832.00 q^{18} +865.768 q^{19} -14450.0 q^{20} +9261.00 q^{21} +14247.2 q^{22} -5666.24 q^{23} +13824.0 q^{24} -27147.6 q^{25} +17576.0 q^{26} -19683.0 q^{27} -21952.0 q^{28} -88969.2 q^{29} -48768.8 q^{30} +225935. q^{31} -32768.0 q^{32} +48084.4 q^{33} +261619. q^{34} +77443.1 q^{35} +46656.0 q^{36} -130826. q^{37} -6926.14 q^{38} +59319.0 q^{39} +115600. q^{40} -259349. q^{41} -74088.0 q^{42} -757448. q^{43} -113978. q^{44} -164595. q^{45} +45329.9 q^{46} -1.13346e6 q^{47} -110592. q^{48} +117649. q^{49} +217181. q^{50} +882965. q^{51} -140608. q^{52} +534117. q^{53} +157464. q^{54} +402095. q^{55} +175616. q^{56} -23375.7 q^{57} +711753. q^{58} -1.50458e6 q^{59} +390151. q^{60} +235291. q^{61} -1.80748e6 q^{62} -250047. q^{63} +262144. q^{64} +496042. q^{65} -384675. q^{66} -1.36233e6 q^{67} -2.09296e6 q^{68} +152988. q^{69} -619545. q^{70} +644414. q^{71} -373248. q^{72} -1.44736e6 q^{73} +1.04661e6 q^{74} +732986. q^{75} +55409.2 q^{76} +610850. q^{77} -474552. q^{78} -7.12802e6 q^{79} -924802. q^{80} +531441. q^{81} +2.07479e6 q^{82} -7.07931e6 q^{83} +592704. q^{84} +7.38361e6 q^{85} +6.05958e6 q^{86} +2.40217e6 q^{87} +911823. q^{88} +3.77715e6 q^{89} +1.31676e6 q^{90} +753571. q^{91} -362639. q^{92} -6.10024e6 q^{93} +9.06770e6 q^{94} -195475. q^{95} +884736. q^{96} +6.21867e6 q^{97} -941192. q^{98} -1.29828e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 35 q^{5} + 1296 q^{6} - 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) $ 6 * q - 48 * q^2 - 162 * q^3 + 384 * q^4 - 35 * q^5 + 1296 * q^6 - 2058 * q^7 - 3072 * q^8 + 4374 * q^9	\( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 35 q^{5} + 1296 q^{6} - 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 280 q^{10} + 5606 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} + 945 q^{15} + 24576 q^{16} + 22022 q^{17} - 34992 q^{18} - 5779 q^{19} - 2240 q^{20} + 55566 q^{21} - 44848 q^{22} - 110789 q^{23} + 82944 q^{24} + 91769 q^{25} + 105456 q^{26} - 118098 q^{27} - 131712 q^{28} + 30693 q^{29} - 7560 q^{30} + 180467 q^{31} - 196608 q^{32} - 151362 q^{33} - 176176 q^{34} + 12005 q^{35} + 279936 q^{36} - 322222 q^{37} + 46232 q^{38} + 355914 q^{39} + 17920 q^{40} - 212652 q^{41} - 444528 q^{42} - 329299 q^{43} + 358784 q^{44} - 25515 q^{45} + 886312 q^{46} + 1322861 q^{47} - 663552 q^{48} + 705894 q^{49} - 734152 q^{50} - 594594 q^{51} - 843648 q^{52} - 168719 q^{53} + 944784 q^{54} - 2252362 q^{55} + 1053696 q^{56} + 156033 q^{57} - 245544 q^{58} + 1943712 q^{59} + 60480 q^{60} - 1085922 q^{61} - 1443736 q^{62} - 1500282 q^{63} + 1572864 q^{64} + 76895 q^{65} + 1210896 q^{66} + 885066 q^{67} + 1409408 q^{68} + 2991303 q^{69} - 96040 q^{70} + 1626164 q^{71} - 2239488 q^{72} - 4750115 q^{73} + 2577776 q^{74} - 2477763 q^{75} - 369856 q^{76} - 1922858 q^{77} - 2847312 q^{78} + 1794289 q^{79} - 143360 q^{80} + 3188646 q^{81} + 1701216 q^{82} - 8454255 q^{83} + 3556224 q^{84} - 18529504 q^{85} + 2634392 q^{86} - 828711 q^{87} - 2870272 q^{88} - 6055411 q^{89} + 204120 q^{90} + 4521426 q^{91} - 7090496 q^{92} - 4872609 q^{93} - 10582888 q^{94} - 3766747 q^{95} + 5308416 q^{96} - 9823899 q^{97} - 5647152 q^{98} + 4086774 q^{99}+O(q^{100}) \) 6 * q - 48 * q^2 - 162 * q^3 + 384 * q^4 - 35 * q^5 + 1296 * q^6 - 2058 * q^7 - 3072 * q^8 + 4374 * q^9 + 280 * q^10 + 5606 * q^11 - 10368 * q^12 - 13182 * q^13 + 16464 * q^14 + 945 * q^15 + 24576 * q^16 + 22022 * q^17 - 34992 * q^18 - 5779 * q^19 - 2240 * q^20 + 55566 * q^21 - 44848 * q^22 - 110789 * q^23 + 82944 * q^24 + 91769 * q^25 + 105456 * q^26 - 118098 * q^27 - 131712 * q^28 + 30693 * q^29 - 7560 * q^30 + 180467 * q^31 - 196608 * q^32 - 151362 * q^33 - 176176 * q^34 + 12005 * q^35 + 279936 * q^36 - 322222 * q^37 + 46232 * q^38 + 355914 * q^39 + 17920 * q^40 - 212652 * q^41 - 444528 * q^42 - 329299 * q^43 + 358784 * q^44 - 25515 * q^45 + 886312 * q^46 + 1322861 * q^47 - 663552 * q^48 + 705894 * q^49 - 734152 * q^50 - 594594 * q^51 - 843648 * q^52 - 168719 * q^53 + 944784 * q^54 - 2252362 * q^55 + 1053696 * q^56 + 156033 * q^57 - 245544 * q^58 + 1943712 * q^59 + 60480 * q^60 - 1085922 * q^61 - 1443736 * q^62 - 1500282 * q^63 + 1572864 * q^64 + 76895 * q^65 + 1210896 * q^66 + 885066 * q^67 + 1409408 * q^68 + 2991303 * q^69 - 96040 * q^70 + 1626164 * q^71 - 2239488 * q^72 - 4750115 * q^73 + 2577776 * q^74 - 2477763 * q^75 - 369856 * q^76 - 1922858 * q^77 - 2847312 * q^78 + 1794289 * q^79 - 143360 * q^80 + 3188646 * q^81 + 1701216 * q^82 - 8454255 * q^83 + 3556224 * q^84 - 18529504 * q^85 + 2634392 * q^86 - 828711 * q^87 - 2870272 * q^88 - 6055411 * q^89 + 204120 * q^90 + 4521426 * q^91 - 7090496 * q^92 - 4872609 * q^93 - 10582888 * q^94 - 3766747 * q^95 + 5308416 * q^96 - 9823899 * q^97 - 5647152 * q^98 + 4086774 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	−225.782	−0.807781	−0.403891	−	0.914807i	$-0.632342\pi$
$5$	−225.782	−0.807781	−0.403891	+	0.914807i	$0.632342\pi$
$6$	216.000	0.408248
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	1806.25	0.571188
$11$	−1780.90	−0.403428	−0.201714	−	0.979444i	$-0.564651\pi$
$11$	−1780.90	−0.403428	−0.201714	+	0.979444i	$0.564651\pi$
$12$	−1728.00	−0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	2744.00	0.267261
$15$	6096.11	0.466373
$16$	4096.00	0.250000
$17$	−32702.4	−1.61439	−0.807196	−	0.590284i	$-0.799016\pi$
$17$	−32702.4	−1.61439	−0.807196	+	0.590284i	$0.799016\pi$
$18$	−5832.00	−0.235702
$19$	865.768	0.0289577	0.0144788	−	0.999895i	$-0.495391\pi$
$19$	865.768	0.0289577	0.0144788	+	0.999895i	$0.495391\pi$
$20$	−14450.0	−0.403891
$21$	9261.00	0.218218
$22$	14247.2	0.285267
$23$	−5666.24	−0.0971063	−0.0485531	−	0.998821i	$-0.515461\pi$
$23$	−5666.24	−0.0971063	−0.0485531	+	0.998821i	$0.515461\pi$
$24$	13824.0	0.204124
$25$	−27147.6	−0.347490
$26$	17576.0	0.196116
$27$	−19683.0	−0.192450
$28$	−21952.0	−0.188982
$29$	−88969.2	−0.677402	−0.338701	−	0.940894i	$-0.609988\pi$
$29$	−88969.2	−0.677402	−0.338701	+	0.940894i	$0.609988\pi$
$30$	−48768.8	−0.329775
$31$	225935.	1.36213	0.681063	−	0.732225i	$-0.261518\pi$
$31$	225935.	1.36213	0.681063	+	0.732225i	$0.261518\pi$
$32$	−32768.0	−0.176777
$33$	48084.4	0.232919
$34$	261619.	1.14155
$35$	77443.1	0.305313
$36$	46656.0	0.166667
$37$	−130826.	−0.424609	−0.212305	−	0.977204i	$-0.568097\pi$
$37$	−130826.	−0.424609	−0.212305	+	0.977204i	$0.568097\pi$
$38$	−6926.14	−0.0204762
$39$	59319.0	0.160128
$40$	115600.	0.285594
$41$	−259349.	−0.587679	−0.293840	−	0.955855i	$-0.594933\pi$
$41$	−259349.	−0.587679	−0.293840	+	0.955855i	$0.594933\pi$
$42$	−74088.0	−0.154303
$43$	−757448.	−1.45282	−0.726412	−	0.687259i	$-0.758814\pi$
$43$	−757448.	−1.45282	−0.726412	+	0.687259i	$0.758814\pi$
$44$	−113978.	−0.201714
$45$	−164595.	−0.269260
$46$	45329.9	0.0686645
$47$	−1.13346e6	−1.59245	−0.796223	−	0.605004i	$-0.793172\pi$
$47$	−1.13346e6	−1.59245	−0.796223	+	0.605004i	$0.793172\pi$
$48$	−110592.	−0.144338
$49$	117649.	0.142857
$50$	217181.	0.245712
$51$	882965.	0.932069
$52$	−140608.	−0.138675
$53$	534117.	0.492800	0.246400	−	0.969168i	$-0.420752\pi$
$53$	534117.	0.492800	0.246400	+	0.969168i	$0.420752\pi$
$54$	157464.	0.136083
$55$	402095.	0.325882
$56$	175616.	0.133631
$57$	−23375.7	−0.0167187
$58$	711753.	0.478996
$59$	−1.50458e6	−0.953744	−0.476872	−	0.878973i	$-0.658230\pi$
$59$	−1.50458e6	−0.953744	−0.476872	+	0.878973i	$0.658230\pi$
$60$	390151.	0.233186
$61$	235291.	0.132724	0.0663621	−	0.997796i	$-0.478861\pi$
$61$	235291.	0.132724	0.0663621	+	0.997796i	$0.478861\pi$
$62$	−1.80748e6	−0.963169
$63$	−250047.	−0.125988
$64$	262144.	0.125000
$65$	496042.	0.224038
$66$	−384675.	−0.164699
$67$	−1.36233e6	−0.553377	−0.276689	−	0.960960i	$-0.589237\pi$
$67$	−1.36233e6	−0.553377	−0.276689	+	0.960960i	$0.589237\pi$
$68$	−2.09296e6	−0.807196
$69$	152988.	0.0560643
$70$	−619545.	−0.215889
$71$	644414.	0.213678	0.106839	−	0.994276i	$-0.465927\pi$
$71$	644414.	0.213678	0.106839	+	0.994276i	$0.465927\pi$
$72$	−373248.	−0.117851
$73$	−1.44736e6	−0.435458	−0.217729	−	0.976009i	$-0.569865\pi$
$73$	−1.44736e6	−0.435458	−0.217729	+	0.976009i	$0.569865\pi$
$74$	1.04661e6	0.300244
$75$	732986.	0.200623
$76$	55409.2	0.0144788
$77$	610850.	0.152481
$78$	−474552.	−0.113228
$79$	−7.12802e6	−1.62657	−0.813287	−	0.581863i	$-0.802324\pi$
$79$	−7.12802e6	−1.62657	−0.813287	+	0.581863i	$0.802324\pi$
$80$	−924802.	−0.201945
$81$	531441.	0.111111
$82$	2.07479e6	0.415552
$83$	−7.07931e6	−1.35899	−0.679497	−	0.733678i	$-0.737802\pi$
$83$	−7.07931e6	−1.35899	−0.679497	+	0.733678i	$0.737802\pi$
$84$	592704.	0.109109
$85$	7.38361e6	1.30407
$86$	6.05958e6	1.02730
$87$	2.40217e6	0.391098
$88$	911823.	0.142633
$89$	3.77715e6	0.567935	0.283968	−	0.958834i	$-0.408349\pi$
$89$	3.77715e6	0.567935	0.283968	+	0.958834i	$0.408349\pi$
$90$	1.31676e6	0.190396
$91$	753571.	0.104828
$92$	−362639.	−0.0485531
$93$	−6.10024e6	−0.786424
$94$	9.06770e6	1.12603
$95$	−195475.	−0.0233915
$96$	884736.	0.102062
$97$	6.21867e6	0.691826	0.345913	−	0.938267i	$-0.387569\pi$
$97$	6.21867e6	0.691826	0.345913	+	0.938267i	$0.387569\pi$
$98$	−941192.	−0.101015
$99$	−1.29828e6	−0.134476
$100$	−1.73745e6	−0.173745
$101$	−1.60189e7	−1.54707	−0.773533	−	0.633756i	$-0.781512\pi$
$101$	−1.60189e7	−1.54707	−0.773533	+	0.633756i	$0.781512\pi$
$102$	−7.06372e6	−0.659072
$103$	−7.28049e6	−0.656494	−0.328247	−	0.944592i	$-0.606458\pi$
$103$	−7.28049e6	−0.656494	−0.328247	+	0.944592i	$0.606458\pi$
$104$	1.12486e6	0.0980581
$105$	−2.09096e6	−0.176272
$106$	−4.27293e6	−0.348462
$107$	−1.45325e7	−1.14683	−0.573413	−	0.819267i	$-0.694381\pi$
$107$	−1.45325e7	−1.14683	−0.573413	+	0.819267i	$0.694381\pi$
$108$	−1.25971e6	−0.0962250
$109$	−7.78229e6	−0.575592	−0.287796	−	0.957692i	$-0.592922\pi$
$109$	−7.78229e6	−0.575592	−0.287796	+	0.957692i	$0.592922\pi$
$110$	−3.21676e6	−0.230433
$111$	3.53231e6	0.245148
$112$	−1.40493e6	−0.0944911
$113$	1.21707e7	0.793491	0.396746	−	0.917929i	$-0.370140\pi$
$113$	1.21707e7	0.793491	0.396746	+	0.917929i	$0.370140\pi$
$114$	187006.	0.0118219
$115$	1.27933e6	0.0784406
$116$	−5.69403e6	−0.338701
$117$	−1.60161e6	−0.0924500
$118$	1.20366e7	0.674399
$119$	1.12169e7	0.610182
$120$	−3.12121e6	−0.164888
$121$	−1.63156e7	−0.837246
$122$	−1.88232e6	−0.0938502
$123$	7.00241e6	0.339297
$124$	1.44598e7	0.681063
$125$	2.37686e7	1.08848
$126$	2.00038e6	0.0890871
$127$	−4.52100e7	−1.95849	−0.979246	−	0.202675i	$-0.935037\pi$
$127$	−4.52100e7	−1.95849	−0.979246	+	0.202675i	$0.935037\pi$
$128$	−2.09715e6	−0.0883883
$129$	2.04511e7	0.838788
$130$	−3.96834e6	−0.158419
$131$	−2.03895e7	−0.792421	−0.396211	−	0.918160i	$-0.629675\pi$
$131$	−2.03895e7	−0.792421	−0.396211	+	0.918160i	$0.629675\pi$
$132$	3.07740e6	0.116460
$133$	−296958.	−0.0109450
$134$	1.08987e7	0.391297
$135$	4.44406e6	0.155458
$136$	1.67436e7	0.570773
$137$	−5.35712e7	−1.77996	−0.889979	−	0.456002i	$-0.849281\pi$
$137$	−5.35712e7	−1.77996	−0.889979	+	0.456002i	$0.849281\pi$
$138$	−1.22391e6	−0.0396435
$139$	2.19598e7	0.693550	0.346775	−	0.937948i	$-0.387277\pi$
$139$	2.19598e7	0.693550	0.346775	+	0.937948i	$0.387277\pi$
$140$	4.95636e6	0.152656
$141$	3.06035e7	0.919399
$142$	−5.15531e6	−0.151094
$143$	3.91265e6	0.111891
$144$	2.98598e6	0.0833333
$145$	2.00876e7	0.547193
$146$	1.15789e7	0.307916
$147$	−3.17652e6	−0.0824786
$148$	−8.37289e6	−0.212305
$149$	−3.47109e7	−0.859634	−0.429817	−	0.902916i	$-0.641422\pi$
$149$	−3.47109e7	−0.859634	−0.429817	+	0.902916i	$0.641422\pi$
$150$	−5.86389e6	−0.141862
$151$	6.63339e6	0.156789	0.0783946	−	0.996922i	$-0.475021\pi$
$151$	6.63339e6	0.156789	0.0783946	+	0.996922i	$0.475021\pi$
$152$	−443273.	−0.0102381
$153$	−2.38401e7	−0.538130
$154$	−4.88680e6	−0.107821
$155$	−5.10119e7	−1.10030
$156$	3.79642e6	0.0800641
$157$	−2.07570e7	−0.428072	−0.214036	−	0.976826i	$-0.568661\pi$
$157$	−2.07570e7	−0.428072	−0.214036	+	0.976826i	$0.568661\pi$
$158$	5.70241e7	1.15016
$159$	−1.44212e7	−0.284518
$160$	7.39841e6	0.142797
$161$	1.94352e6	0.0367027
$162$	−4.25153e6	−0.0785674
$163$	3.34327e7	0.604665	0.302332	−	0.953203i	$-0.402235\pi$
$163$	3.34327e7	0.604665	0.302332	+	0.953203i	$0.402235\pi$
$164$	−1.65983e7	−0.293840
$165$	−1.08566e7	−0.188148
$166$	5.66345e7	0.960954
$167$	−3.16221e7	−0.525391	−0.262696	−	0.964879i	$-0.584612\pi$
$167$	−3.16221e7	−0.525391	−0.262696	+	0.964879i	$0.584612\pi$
$168$	−4.74163e6	−0.0771517
$169$	4.82681e6	0.0769231
$170$	−5.90689e7	−0.922120
$171$	631145.	0.00965256
$172$	−4.84767e7	−0.726412
$173$	−7.70230e7	−1.13099	−0.565495	−	0.824752i	$-0.691315\pi$
$173$	−7.70230e7	−1.13099	−0.565495	+	0.824752i	$0.691315\pi$
$174$	−1.92173e7	−0.276548
$175$	9.31164e6	0.131339
$176$	−7.29458e6	−0.100857
$177$	4.06235e7	0.550645
$178$	−3.02172e7	−0.401591
$179$	4.88537e6	0.0636666	0.0318333	−	0.999493i	$-0.489865\pi$
$179$	4.88537e6	0.0636666	0.0318333	+	0.999493i	$0.489865\pi$
$180$	−1.05341e7	−0.134630
$181$	6.45377e7	0.808982	0.404491	−	0.914542i	$-0.367449\pi$
$181$	6.45377e7	0.808982	0.404491	+	0.914542i	$0.367449\pi$
$182$	−6.02857e6	−0.0741249
$183$	−6.35285e6	−0.0766284
$184$	2.90111e6	0.0343323
$185$	2.95382e7	0.342991
$186$	4.88019e7	0.556086
$187$	5.82399e7	0.651291
$188$	−7.25416e7	−0.796223
$189$	6.75127e6	0.0727393
$190$	1.56380e6	0.0165403
$191$	1.71713e8	1.78315	0.891573	−	0.452877i	$-0.149602\pi$
$191$	1.71713e8	1.78315	0.891573	+	0.452877i	$0.149602\pi$
$192$	−7.07789e6	−0.0721688
$193$	4.16596e7	0.417123	0.208562	−	0.978009i	$-0.433122\pi$
$193$	4.16596e7	0.417123	0.208562	+	0.978009i	$0.433122\pi$
$194$	−4.97494e7	−0.489195
$195$	−1.33931e7	−0.129348
$196$	7.52954e6	0.0714286
$197$	−6.74946e7	−0.628980	−0.314490	−	0.949261i	$-0.601834\pi$
$197$	−6.74946e7	−0.628980	−0.314490	+	0.949261i	$0.601834\pi$
$198$	1.03862e7	0.0950889
$199$	−6.00800e7	−0.540436	−0.270218	−	0.962799i	$-0.587096\pi$
$199$	−6.00800e7	−0.540436	−0.270218	+	0.962799i	$0.587096\pi$
$200$	1.38996e7	0.122856
$201$	3.67830e7	0.319493
$202$	1.28152e8	1.09394
$203$	3.05164e7	0.256034
$204$	5.65098e7	0.466035
$205$	5.85562e7	0.474716
$206$	5.82439e7	0.464211
$207$	−4.13069e6	−0.0323688
$208$	−8.99891e6	−0.0693375
$209$	−1.54185e6	−0.0116823
$210$	1.67277e7	0.124643
$211$	1.55213e8	1.13747	0.568736	−	0.822520i	$-0.307433\pi$
$211$	1.55213e8	1.13747	0.568736	+	0.822520i	$0.307433\pi$
$212$	3.41835e7	0.246400
$213$	−1.73992e7	−0.123367
$214$	1.16260e8	0.810928
$215$	1.71018e8	1.17356
$216$	1.00777e7	0.0680414
$217$	−7.74957e7	−0.514835
$218$	6.22583e7	0.407005
$219$	3.90787e7	0.251412
$220$	2.57341e7	0.162941
$221$	7.18472e7	0.447752
$222$	−2.82585e7	−0.173346
$223$	−3.87501e7	−0.233995	−0.116997	−	0.993132i	$-0.537327\pi$
$223$	−3.87501e7	−0.233995	−0.116997	+	0.993132i	$0.537327\pi$
$224$	1.12394e7	0.0668153
$225$	−1.97906e7	−0.115830
$226$	−9.73658e7	−0.561083
$227$	−1.18488e8	−0.672333	−0.336167	−	0.941802i	$-0.609131\pi$
$227$	−1.18488e8	−0.672333	−0.336167	+	0.941802i	$0.609131\pi$
$228$	−1.49605e6	−0.00835937
$229$	8.34499e7	0.459200	0.229600	−	0.973285i	$-0.426258\pi$
$229$	8.34499e7	0.459200	0.229600	+	0.973285i	$0.426258\pi$
$230$	−1.02347e7	−0.0554659
$231$	−1.64929e7	−0.0880352
$232$	4.55522e7	0.239498
$233$	2.68526e8	1.39072	0.695360	−	0.718661i	$-0.255245\pi$
$233$	2.68526e8	1.39072	0.695360	+	0.718661i	$0.255245\pi$
$234$	1.28129e7	0.0653720
$235$	2.55915e8	1.28635
$236$	−9.62928e7	−0.476872
$237$	1.92456e8	0.939103
$238$	−8.97355e7	−0.431464
$239$	−3.69967e8	−1.75295	−0.876477	−	0.481443i	$-0.840113\pi$
$239$	−3.69967e8	−1.75295	−0.876477	+	0.481443i	$0.840113\pi$
$240$	2.49696e7	0.116593
$241$	3.88118e7	0.178609	0.0893046	−	0.996004i	$-0.471536\pi$
$241$	3.88118e7	0.178609	0.0893046	+	0.996004i	$0.471536\pi$
$242$	1.30524e8	0.592022
$243$	−1.43489e7	−0.0641500
$244$	1.50586e7	0.0663621
$245$	−2.65630e7	−0.115397
$246$	−5.60193e7	−0.239919
$247$	−1.90209e6	−0.00803142
$248$	−1.15679e8	−0.481584
$249$	1.91141e8	0.784615
$250$	−1.90149e8	−0.769669
$251$	1.41482e8	0.564733	0.282366	−	0.959307i	$-0.408881\pi$
$251$	1.41482e8	0.564733	0.282366	+	0.959307i	$0.408881\pi$
$252$	−1.60030e7	−0.0629941
$253$	1.00910e7	0.0391754
$254$	3.61680e8	1.38486
$255$	−1.99357e8	−0.752908
$256$	1.67772e7	0.0625000
$257$	2.70134e8	0.992689	0.496345	−	0.868126i	$-0.334675\pi$
$257$	2.70134e8	0.992689	0.496345	+	0.868126i	$0.334675\pi$
$258$	−1.63609e8	−0.593113
$259$	4.48735e7	0.160487
$260$	3.17467e7	0.112019
$261$	−6.48585e7	−0.225801
$262$	1.63116e8	0.560327
$263$	2.52684e8	0.856509	0.428255	−	0.903658i	$-0.359129\pi$
$263$	2.52684e8	0.856509	0.428255	+	0.903658i	$0.359129\pi$
$264$	−2.46192e7	−0.0823494
$265$	−1.20594e8	−0.398075
$266$	2.37567e6	0.00773927
$267$	−1.01983e8	−0.327898
$268$	−8.71893e7	−0.276689
$269$	3.08616e8	0.966687	0.483343	−	0.875431i	$-0.339422\pi$
$269$	3.08616e8	0.966687	0.483343	+	0.875431i	$0.339422\pi$
$270$	−3.55525e7	−0.109925
$271$	5.81853e8	1.77591	0.887955	−	0.459931i	$-0.152126\pi$
$271$	5.81853e8	1.77591	0.887955	+	0.459931i	$0.152126\pi$
$272$	−1.33949e8	−0.403598
$273$	−2.03464e7	−0.0605228
$274$	4.28570e8	1.25862
$275$	4.83473e7	0.140187
$276$	9.79126e6	0.0280322
$277$	−1.62144e8	−0.458375	−0.229188	−	0.973382i	$-0.573607\pi$
$277$	−1.62144e8	−0.458375	−0.229188	+	0.973382i	$0.573607\pi$
$278$	−1.75679e8	−0.490414
$279$	1.64706e8	0.454042
$280$	−3.96509e7	−0.107944
$281$	−6.88927e8	−1.85226	−0.926129	−	0.377208i	$-0.876884\pi$
$281$	−6.88927e8	−1.85226	−0.926129	+	0.377208i	$0.876884\pi$
$282$	−2.44828e8	−0.650113
$283$	2.09632e8	0.549801	0.274900	−	0.961473i	$-0.411355\pi$
$283$	2.09632e8	0.549801	0.274900	+	0.961473i	$0.411355\pi$
$284$	4.12425e7	0.106839
$285$	5.27781e6	0.0135051
$286$	−3.13012e7	−0.0791187
$287$	8.89566e7	0.222122
$288$	−2.38879e7	−0.0589256
$289$	6.59110e8	1.60626
$290$	−1.60701e8	−0.386924
$291$	−1.67904e8	−0.399426
$292$	−9.26311e7	−0.217729
$293$	−1.54418e8	−0.358642	−0.179321	−	0.983791i	$-0.557390\pi$
$293$	−1.54418e8	−0.358642	−0.179321	+	0.983791i	$0.557390\pi$
$294$	2.54122e7	0.0583212
$295$	3.39706e8	0.770417
$296$	6.69831e7	0.150122
$297$	3.50535e7	0.0776398
$298$	2.77687e8	0.607853
$299$	1.24487e7	0.0269324
$300$	4.69111e7	0.100312
$301$	2.59805e8	0.549116
$302$	−5.30671e7	−0.110867
$303$	4.32511e8	0.893199
$304$	3.54619e6	0.00723942
$305$	−5.31243e7	−0.107212
$306$	1.90721e8	0.380516
$307$	9.51145e8	1.87613	0.938064	−	0.346461i	$-0.112617\pi$
$307$	9.51145e8	1.87613	0.938064	+	0.346461i	$0.112617\pi$
$308$	3.90944e7	0.0762407
$309$	1.96573e8	0.379027
$310$	4.08096e8	0.778029
$311$	6.66311e8	1.25607	0.628037	−	0.778183i	$-0.283859\pi$
$311$	6.66311e8	1.25607	0.628037	+	0.778183i	$0.283859\pi$
$312$	−3.03713e7	−0.0566139
$313$	3.55765e8	0.655779	0.327890	−	0.944716i	$-0.393663\pi$
$313$	3.55765e8	0.655779	0.327890	+	0.944716i	$0.393663\pi$
$314$	1.66056e8	0.302692
$315$	5.64560e7	0.101771
$316$	−4.56193e8	−0.813287
$317$	−1.59496e8	−0.281218	−0.140609	−	0.990065i	$-0.544906\pi$
$317$	−1.59496e8	−0.281218	−0.140609	+	0.990065i	$0.544906\pi$
$318$	1.15369e8	0.201185
$319$	1.58446e8	0.273283
$320$	−5.91873e7	−0.100973
$321$	3.92378e8	0.662120
$322$	−1.55482e7	−0.0259527
$323$	−2.83127e7	−0.0467490
$324$	3.40122e7	0.0555556
$325$	5.96433e7	0.0963763
$326$	−2.67462e8	−0.427563
$327$	2.10122e8	0.332318
$328$	1.32787e8	0.207776
$329$	3.88778e8	0.601888
$330$	8.68526e7	0.133041
$331$	3.48798e8	0.528659	0.264330	−	0.964432i	$-0.414849\pi$
$331$	3.48798e8	0.528659	0.264330	+	0.964432i	$0.414849\pi$
$332$	−4.53076e8	−0.679497
$333$	−9.53725e7	−0.141536
$334$	2.52977e8	0.371508
$335$	3.07590e8	0.447008
$336$	3.79331e7	0.0545545
$337$	4.05244e8	0.576782	0.288391	−	0.957513i	$-0.406880\pi$
$337$	4.05244e8	0.576782	0.288391	+	0.957513i	$0.406880\pi$
$338$	−3.86145e7	−0.0543928
$339$	−3.28610e8	−0.458122
$340$	4.72551e8	0.652037
$341$	−4.02368e8	−0.549520
$342$	−5.04916e6	−0.00682539
$343$	−4.03536e7	−0.0539949
$344$	3.87813e8	0.513651
$345$	−3.45420e7	−0.0452877
$346$	6.16184e8	0.799731
$347$	−3.13003e7	−0.0402157	−0.0201079	−	0.999798i	$-0.506401\pi$
$347$	−3.13003e7	−0.0402157	−0.0201079	+	0.999798i	$0.506401\pi$
$348$	1.53739e8	0.195549
$349$	−9.68097e8	−1.21907	−0.609537	−	0.792758i	$-0.708645\pi$
$349$	−9.68097e8	−1.21907	−0.609537	+	0.792758i	$0.708645\pi$
$350$	−7.44931e7	−0.0928705
$351$	4.32436e7	0.0533761
$352$	5.83567e7	0.0713167
$353$	6.87057e8	0.831345	0.415673	−	0.909514i	$-0.363546\pi$
$353$	6.87057e8	0.831345	0.415673	+	0.909514i	$0.363546\pi$
$354$	−3.24988e8	−0.389364
$355$	−1.45497e8	−0.172605
$356$	2.41737e8	0.283968
$357$	−3.02857e8	−0.352289
$358$	−3.90830e7	−0.0450191
$359$	6.46244e8	0.737167	0.368584	−	0.929595i	$-0.379843\pi$
$359$	6.46244e8	0.737167	0.368584	+	0.929595i	$0.379843\pi$
$360$	8.42726e7	0.0951979
$361$	−8.93122e8	−0.999161
$362$	−5.16302e8	−0.572036
$363$	4.40520e8	0.483384
$364$	4.82285e7	0.0524142
$365$	3.26787e8	0.351755
$366$	5.08228e7	0.0541844
$367$	−2.28391e8	−0.241184	−0.120592	−	0.992702i	$-0.538479\pi$
$367$	−2.28391e8	−0.241184	−0.120592	+	0.992702i	$0.538479\pi$
$368$	−2.32089e7	−0.0242766
$369$	−1.89065e8	−0.195893
$370$	−2.36306e8	−0.242532
$371$	−1.83202e8	−0.186261
$372$	−3.90415e8	−0.393212
$373$	−1.23068e9	−1.22790	−0.613951	−	0.789344i	$-0.710421\pi$
$373$	−1.23068e9	−1.22790	−0.613951	+	0.789344i	$0.710421\pi$
$374$	−4.65919e8	−0.460532
$375$	−6.41753e8	−0.628432
$376$	5.80333e8	0.563014
$377$	1.95465e8	0.187878
$378$	−5.40102e7	−0.0514344
$379$	−8.25512e8	−0.778908	−0.389454	−	0.921046i	$-0.627336\pi$
$379$	−8.25512e8	−0.778908	−0.389454	+	0.921046i	$0.627336\pi$
$380$	−1.25104e7	−0.0116957
$381$	1.22067e9	1.13074
$382$	−1.37371e9	−1.26087
$383$	−1.49154e9	−1.35656	−0.678280	−	0.734804i	$-0.737274\pi$
$383$	−1.49154e9	−1.35656	−0.678280	+	0.734804i	$0.737274\pi$
$384$	5.66231e7	0.0510310
$385$	−1.37919e8	−0.123172
$386$	−3.33277e8	−0.294951
$387$	−5.52179e8	−0.484275
$388$	3.97995e8	0.345913
$389$	−1.25333e9	−1.07954	−0.539772	−	0.841811i	$-0.681490\pi$
$389$	−1.25333e9	−1.07954	−0.539772	+	0.841811i	$0.681490\pi$
$390$	1.07145e8	0.0914632
$391$	1.85300e8	0.156768
$392$	−6.02363e7	−0.0505076
$393$	5.50515e8	0.457505
$394$	5.39957e8	0.444756
$395$	1.60938e9	1.31392
$396$	−8.30898e7	−0.0672380
$397$	1.00779e9	0.808360	0.404180	−	0.914679i	$-0.367557\pi$
$397$	1.00779e9	0.808360	0.404180	+	0.914679i	$0.367557\pi$
$398$	4.80640e8	0.382146
$399$	8.01788e6	0.00631909
$400$	−1.11197e8	−0.0868724
$401$	6.80591e8	0.527085	0.263542	−	0.964648i	$-0.415109\pi$
$401$	6.80591e8	0.527085	0.263542	+	0.964648i	$0.415109\pi$
$402$	−2.94264e8	−0.225915
$403$	−4.96379e8	−0.377786
$404$	−1.02521e9	−0.773533
$405$	−1.19990e8	−0.0897535
$406$	−2.44131e8	−0.181043
$407$	2.32989e8	0.171299
$408$	−4.52078e8	−0.329536
$409$	−1.36542e9	−0.986810	−0.493405	−	0.869800i	$-0.664248\pi$
$409$	−1.36542e9	−0.986810	−0.493405	+	0.869800i	$0.664248\pi$
$410$	−4.68449e8	−0.335675
$411$	1.44642e9	1.02766
$412$	−4.65952e8	−0.328247
$413$	5.16069e8	0.360481
$414$	3.30455e7	0.0228882
$415$	1.59838e9	1.09777
$416$	7.19913e7	0.0490290
$417$	−5.92916e8	−0.400421
$418$	1.23348e7	0.00826067
$419$	5.26150e8	0.349431	0.174715	−	0.984619i	$-0.444100\pi$
$419$	5.26150e8	0.349431	0.174715	+	0.984619i	$0.444100\pi$
$420$	−1.33822e8	−0.0881361
$421$	2.72206e9	1.77791	0.888957	−	0.457990i	$-0.151431\pi$
$421$	2.72206e9	1.77791	0.888957	+	0.457990i	$0.151431\pi$
$422$	−1.24171e9	−0.804314
$423$	−8.26294e8	−0.530815
$424$	−2.73468e8	−0.174231
$425$	8.87793e8	0.560984
$426$	1.39193e8	0.0872339
$427$	−8.07047e7	−0.0501650
$428$	−9.30081e8	−0.573413
$429$	−1.05641e8	−0.0646002
$430$	−1.36814e9	−0.829835
$431$	−2.34704e9	−1.41205	−0.706025	−	0.708187i	$-0.749513\pi$
$431$	−2.34704e9	−1.41205	−0.706025	+	0.708187i	$0.749513\pi$
$432$	−8.06216e7	−0.0481125
$433$	1.04705e9	0.619810	0.309905	−	0.950768i	$-0.399703\pi$
$433$	1.04705e9	0.619810	0.309905	+	0.950768i	$0.399703\pi$
$434$	6.19965e8	0.364043
$435$	−5.42366e8	−0.315922
$436$	−4.98066e8	−0.287796
$437$	−4.90565e6	−0.00281197
$438$	−3.12630e8	−0.177775
$439$	−3.24465e8	−0.183038	−0.0915190	−	0.995803i	$-0.529172\pi$
$439$	−3.24465e8	−0.183038	−0.0915190	+	0.995803i	$0.529172\pi$
$440$	−2.05873e8	−0.115217
$441$	8.57661e7	0.0476190
$442$	−5.74778e8	−0.316608
$443$	2.70182e9	1.47653	0.738266	−	0.674509i	$-0.235645\pi$
$443$	2.70182e9	1.47653	0.738266	+	0.674509i	$0.235645\pi$
$444$	2.26068e8	0.122574
$445$	−8.52811e8	−0.458767
$446$	3.10001e8	0.165459
$447$	9.37193e8	0.496310
$448$	−8.99154e7	−0.0472456
$449$	4.40718e8	0.229773	0.114886	−	0.993379i	$-0.463350\pi$
$449$	4.40718e8	0.229773	0.114886	+	0.993379i	$0.463350\pi$
$450$	1.58325e8	0.0819041
$451$	4.61875e8	0.237086
$452$	7.78927e8	0.396746
$453$	−1.79102e8	−0.0905223
$454$	9.47906e8	0.475412
$455$	−1.70143e8	−0.0846785
$456$	1.19684e7	0.00591096
$457$	1.74847e9	0.856941	0.428471	−	0.903556i	$-0.359053\pi$
$457$	1.74847e9	0.856941	0.428471	+	0.903556i	$0.359053\pi$
$458$	−6.67599e8	−0.324703
$459$	6.43682e8	0.310690
$460$	8.18773e7	0.0392203
$461$	1.16590e9	0.554251	0.277125	−	0.960834i	$-0.410618\pi$
$461$	1.16590e9	0.554251	0.277125	+	0.960834i	$0.410618\pi$
$462$	1.31944e8	0.0622503
$463$	−3.60900e9	−1.68987	−0.844934	−	0.534870i	$-0.820361\pi$
$463$	−3.60900e9	−1.68987	−0.844934	+	0.534870i	$0.820361\pi$
$464$	−3.64418e8	−0.169351
$465$	1.37732e9	0.635258
$466$	−2.14820e9	−0.983388
$467$	−1.23980e7	−0.00563306	−0.00281653	−	0.999996i	$-0.500897\pi$
$467$	−1.23980e7	−0.00563306	−0.00281653	+	0.999996i	$0.500897\pi$
$468$	−1.02503e8	−0.0462250
$469$	4.67280e8	0.209157
$470$	−2.04732e9	−0.909585
$471$	5.60440e8	0.247147
$472$	7.70343e8	0.337200
$473$	1.34894e9	0.586110
$474$	−1.53965e9	−0.664046
$475$	−2.35036e7	−0.0100625
$476$	7.17884e8	0.305091
$477$	3.89371e8	0.164267
$478$	2.95974e9	1.23953
$479$	−1.16840e9	−0.485755	−0.242877	−	0.970057i	$-0.578091\pi$
$479$	−1.16840e9	−0.485755	−0.242877	+	0.970057i	$0.578091\pi$
$480$	−1.99757e8	−0.0824438
$481$	2.87426e8	0.117765
$482$	−3.10494e8	−0.126296
$483$	−5.24750e7	−0.0211903
$484$	−1.04420e9	−0.418623
$485$	−1.40406e9	−0.558844
$486$	1.14791e8	0.0453609
$487$	2.62481e8	0.102979	0.0514893	−	0.998674i	$-0.483603\pi$
$487$	2.62481e8	0.102979	0.0514893	+	0.998674i	$0.483603\pi$
$488$	−1.20469e8	−0.0469251
$489$	−9.02683e8	−0.349103
$490$	2.12504e8	0.0815982
$491$	−7.12500e8	−0.271644	−0.135822	−	0.990733i	$-0.543367\pi$
$491$	−7.12500e8	−0.271644	−0.135822	+	0.990733i	$0.543367\pi$
$492$	4.48154e8	0.169648
$493$	2.90951e9	1.09359
$494$	1.52167e7	0.00567907
$495$	2.93128e8	0.108627
$496$	9.25429e8	0.340531
$497$	−2.21034e8	−0.0807629
$498$	−1.52913e9	−0.554807
$499$	−1.08950e9	−0.392531	−0.196266	−	0.980551i	$-0.562881\pi$
$499$	−1.08950e9	−0.392531	−0.196266	+	0.980551i	$0.562881\pi$
$500$	1.52119e9	0.544238
$501$	8.53797e8	0.303335
$502$	−1.13186e9	−0.399326
$503$	−5.67265e9	−1.98746	−0.993729	−	0.111811i	$-0.964335\pi$
$503$	−5.67265e9	−1.98746	−0.993729	+	0.111811i	$0.964335\pi$
$504$	1.28024e8	0.0445435
$505$	3.61678e9	1.24969
$506$	−8.07282e7	−0.0277012
$507$	−1.30324e8	−0.0444116
$508$	−2.89344e9	−0.979246
$509$	2.40085e9	0.806961	0.403481	−	0.914988i	$-0.367800\pi$
$509$	2.40085e9	0.806961	0.403481	+	0.914988i	$0.367800\pi$
$510$	1.59486e9	0.532386
$511$	4.96445e8	0.164588
$512$	−1.34218e8	−0.0441942
$513$	−1.70409e7	−0.00557291
$514$	−2.16107e9	−0.701937
$515$	1.64380e9	0.530303
$516$	1.30887e9	0.419394
$517$	2.01859e9	0.642437
$518$	−3.58988e8	−0.113482
$519$	2.07962e9	0.652977
$520$	−2.53974e8	−0.0792095
$521$	4.65208e8	0.144117	0.0720585	−	0.997400i	$-0.477043\pi$
$521$	4.65208e8	0.144117	0.0720585	+	0.997400i	$0.477043\pi$
$522$	5.18868e8	0.159665
$523$	5.02595e9	1.53625	0.768126	−	0.640299i	$-0.221190\pi$
$523$	5.02595e9	1.53625	0.768126	+	0.640299i	$0.221190\pi$
$524$	−1.30493e9	−0.396211
$525$	−2.51414e8	−0.0758285
$526$	−2.02147e9	−0.605643
$527$	−7.38862e9	−2.19900
$528$	1.96954e8	0.0582298
$529$	−3.37272e9	−0.990570
$530$	9.64750e8	0.281481
$531$	−1.09684e9	−0.317915
$532$	−1.90053e7	−0.00547249
$533$	5.69789e8	0.162993
$534$	8.15864e8	0.231859
$535$	3.28117e9	0.926384
$536$	6.97514e8	0.195648
$537$	−1.31905e8	−0.0367580
$538$	−2.46893e9	−0.683551
$539$	−2.09522e8	−0.0576326
$540$	2.84420e8	0.0777288
$541$	−2.24754e9	−0.610264	−0.305132	−	0.952310i	$-0.598701\pi$
$541$	−2.24754e9	−0.610264	−0.305132	+	0.952310i	$0.598701\pi$
$542$	−4.65482e9	−1.25576
$543$	−1.74252e9	−0.467066
$544$	1.07159e9	0.285387
$545$	1.75710e9	0.464952
$546$	1.62771e8	0.0427960
$547$	8.33462e7	0.0217736	0.0108868	−	0.999941i	$-0.496535\pi$
$547$	8.33462e7	0.0217736	0.0108868	+	0.999941i	$0.496535\pi$
$548$	−3.42856e9	−0.889979
$549$	1.71527e8	0.0442414
$550$	−3.86779e8	−0.0991272
$551$	−7.70267e7	−0.0196160
$552$	−7.83301e7	−0.0198217
$553$	2.44491e9	0.614787
$554$	1.29715e9	0.324120
$555$	−7.97532e8	−0.198026
$556$	1.40543e9	0.346775
$557$	−5.53606e9	−1.35740	−0.678699	−	0.734417i	$-0.737456\pi$
$557$	−5.53606e9	−1.35740	−0.678699	+	0.734417i	$0.737456\pi$
$558$	−1.31765e9	−0.321056
$559$	1.66411e9	0.402941
$560$	3.17207e8	0.0763281
$561$	−1.57248e9	−0.376023
$562$	5.51142e9	1.30974
$563$	−3.84475e9	−0.908005	−0.454002	−	0.891000i	$-0.650004\pi$
$563$	−3.84475e9	−0.908005	−0.454002	+	0.891000i	$0.650004\pi$
$564$	1.95862e9	0.459699
$565$	−2.74793e9	−0.640967
$566$	−1.67706e9	−0.388768
$567$	−1.82284e8	−0.0419961
$568$	−3.29940e8	−0.0755468
$569$	1.88472e9	0.428897	0.214449	−	0.976735i	$-0.431205\pi$
$569$	1.88472e9	0.428897	0.214449	+	0.976735i	$0.431205\pi$
$570$	−4.22225e7	−0.00954953
$571$	1.83547e9	0.412592	0.206296	−	0.978490i	$-0.433859\pi$
$571$	1.83547e9	0.412592	0.206296	+	0.978490i	$0.433859\pi$
$572$	2.50409e8	0.0559454
$573$	−4.63626e9	−1.02950
$574$	−7.11653e8	−0.157064
$575$	1.53825e8	0.0337434
$576$	1.91103e8	0.0416667
$577$	2.56963e9	0.556871	0.278436	−	0.960455i	$-0.410184\pi$
$577$	2.56963e9	0.556871	0.278436	+	0.960455i	$0.410184\pi$
$578$	−5.27288e9	−1.13580
$579$	−1.12481e9	−0.240826
$580$	1.28561e9	0.273596
$581$	2.42820e9	0.513651
$582$	1.34323e9	0.282437
$583$	−9.51211e8	−0.198809
$584$	7.41048e8	0.153958
$585$	3.61615e8	0.0746794
$586$	1.23534e9	0.253598
$587$	−7.99244e9	−1.63097	−0.815485	−	0.578779i	$-0.803530\pi$
$587$	−7.99244e9	−1.63097	−0.815485	+	0.578779i	$0.803530\pi$
$588$	−2.03297e8	−0.0412393
$589$	1.95607e8	0.0394440
$590$	−2.71764e9	−0.544767
$591$	1.82235e9	0.363142
$592$	−5.35865e8	−0.106152
$593$	−7.81149e9	−1.53830	−0.769152	−	0.639066i	$-0.779321\pi$
$593$	−7.81149e9	−1.53830	−0.769152	+	0.639066i	$0.779321\pi$
$594$	−2.80428e8	−0.0548996
$595$	−2.53258e9	−0.492894
$596$	−2.22150e9	−0.429817
$597$	1.62216e9	0.312021
$598$	−9.95898e7	−0.0190441
$599$	3.11337e8	0.0591885	0.0295943	−	0.999562i	$-0.490578\pi$
$599$	3.11337e8	0.0591885	0.0295943	+	0.999562i	$0.490578\pi$
$600$	−3.75289e8	−0.0709310
$601$	−4.89586e8	−0.0919958	−0.0459979	−	0.998942i	$-0.514647\pi$
$601$	−4.89586e8	−0.0919958	−0.0459979	+	0.998942i	$0.514647\pi$
$602$	−2.07844e9	−0.388284
$603$	−9.93141e8	−0.184459
$604$	4.24537e8	0.0783946
$605$	3.68375e9	0.676311
$606$	−3.46009e9	−0.631587
$607$	8.48356e9	1.53964	0.769818	−	0.638264i	$-0.220347\pi$
$607$	8.48356e9	1.53964	0.769818	+	0.638264i	$0.220347\pi$
$608$	−2.83695e7	−0.00511905
$609$	−8.23944e8	−0.147821
$610$	4.24994e8	0.0758104
$611$	2.49022e9	0.441665
$612$	−1.52576e9	−0.269065
$613$	−3.29849e9	−0.578367	−0.289184	−	0.957274i	$-0.593384\pi$
$613$	−3.29849e9	−0.578367	−0.289184	+	0.957274i	$0.593384\pi$
$614$	−7.60916e9	−1.32662
$615$	−1.58102e9	−0.274078
$616$	−3.12755e8	−0.0539103
$617$	−7.63311e9	−1.30829	−0.654145	−	0.756370i	$-0.726971\pi$
$617$	−7.63311e9	−1.30829	−0.654145	+	0.756370i	$0.726971\pi$
$618$	−1.57259e9	−0.268012
$619$	−7.03379e9	−1.19199	−0.595995	−	0.802988i	$-0.703242\pi$
$619$	−7.03379e9	−1.19199	−0.595995	+	0.802988i	$0.703242\pi$
$620$	−3.26476e9	−0.550150
$621$	1.11529e8	0.0186881
$622$	−5.33049e9	−0.888179
$623$	−1.29556e9	−0.214659
$624$	2.42971e8	0.0400320
$625$	−3.24561e9	−0.531761
$626$	−2.84612e9	−0.463706
$627$	4.16299e7	0.00674481
$628$	−1.32845e9	−0.214036
$629$	4.27834e9	0.685486
$630$	−4.51648e8	−0.0719629
$631$	7.22947e9	1.14552	0.572761	−	0.819722i	$-0.305872\pi$
$631$	7.22947e9	1.14552	0.572761	+	0.819722i	$0.305872\pi$
$632$	3.64954e9	0.575081
$633$	−4.19076e9	−0.656720
$634$	1.27597e9	0.198851
$635$	1.02076e10	1.58203
$636$	−9.22954e8	−0.142259
$637$	−2.58475e8	−0.0396214
$638$	−1.26756e9	−0.193240
$639$	4.69778e8	0.0712262
$640$	4.73499e8	0.0713984
$641$	−1.19785e10	−1.79639	−0.898193	−	0.439600i	$-0.855120\pi$
$641$	−1.19785e10	−1.79639	−0.898193	+	0.439600i	$0.855120\pi$
$642$	−3.13902e9	−0.468190
$643$	−4.92889e9	−0.731158	−0.365579	−	0.930780i	$-0.619129\pi$
$643$	−4.92889e9	−0.731158	−0.365579	+	0.930780i	$0.619129\pi$
$644$	1.24385e8	0.0183514
$645$	−4.61748e9	−0.677557
$646$	2.26502e8	0.0330566
$647$	−7.96919e9	−1.15678	−0.578388	−	0.815762i	$-0.696318\pi$
$647$	−7.96919e9	−1.15678	−0.578388	+	0.815762i	$0.696318\pi$
$648$	−2.72098e8	−0.0392837
$649$	2.67950e9	0.384767
$650$	−4.77147e8	−0.0681483
$651$	2.09238e9	0.297240
$652$	2.13969e9	0.302332
$653$	−4.61773e9	−0.648981	−0.324491	−	0.945889i	$-0.605193\pi$
$653$	−4.61773e9	−0.648981	−0.324491	+	0.945889i	$0.605193\pi$
$654$	−1.68097e9	−0.234984
$655$	4.60357e9	0.640103
$656$	−1.06229e9	−0.146920
$657$	−1.05513e9	−0.145153
$658$	−3.11022e9	−0.425599
$659$	−3.03840e9	−0.413567	−0.206783	−	0.978387i	$-0.566300\pi$
$659$	−3.03840e9	−0.413567	−0.206783	+	0.978387i	$0.566300\pi$
$660$	−6.94821e8	−0.0940739
$661$	−1.08742e10	−1.46450	−0.732252	−	0.681034i	$-0.761531\pi$
$661$	−1.08742e10	−1.46450	−0.732252	+	0.681034i	$0.761531\pi$
$662$	−2.79038e9	−0.373819
$663$	−1.93988e9	−0.258509
$664$	3.62461e9	0.480477
$665$	6.70478e7	0.00884115
$666$	7.62980e8	0.100081
$667$	5.04121e8	0.0657800
$668$	−2.02381e9	−0.262696
$669$	1.04625e9	0.135097
$670$	−2.46072e9	−0.316082
$671$	−4.19030e8	−0.0535447
$672$	−3.03464e8	−0.0385758
$673$	7.52134e9	0.951136	0.475568	−	0.879679i	$-0.342243\pi$
$673$	7.52134e9	0.951136	0.475568	+	0.879679i	$0.342243\pi$
$674$	−3.24195e9	−0.407847
$675$	5.34347e8	0.0668744
$676$	3.08916e8	0.0384615
$677$	6.41747e9	0.794884	0.397442	−	0.917627i	$-0.369898\pi$
$677$	6.41747e9	0.794884	0.397442	+	0.917627i	$0.369898\pi$
$678$	2.62888e9	0.323941
$679$	−2.13300e9	−0.261486
$680$	−3.78041e9	−0.461060
$681$	3.19918e9	0.388172
$682$	3.21895e9	0.388569
$683$	1.11914e10	1.34403	0.672017	−	0.740536i	$-0.265428\pi$
$683$	1.11914e10	1.34403	0.672017	+	0.740536i	$0.265428\pi$
$684$	4.03933e7	0.00482628
$685$	1.20954e10	1.43782
$686$	3.22829e8	0.0381802
$687$	−2.25315e9	−0.265119
$688$	−3.10251e9	−0.363206
$689$	−1.17345e9	−0.136678
$690$	2.76336e8	0.0320233
$691$	1.26387e10	1.45723	0.728614	−	0.684924i	$-0.240165\pi$
$691$	1.26387e10	1.45723	0.728614	+	0.684924i	$0.240165\pi$
$692$	−4.92947e9	−0.565495
$693$	4.45310e8	0.0508272
$694$	2.50403e8	0.0284368
$695$	−4.95813e9	−0.560236
$696$	−1.22991e9	−0.138274
$697$	8.48133e9	0.948744
$698$	7.74478e9	0.862015
$699$	−7.25019e9	−0.802933
$700$	5.95945e8	0.0656694
$701$	−8.69672e9	−0.953548	−0.476774	−	0.879026i	$-0.658194\pi$
$701$	−8.69672e9	−0.953548	−0.476774	+	0.879026i	$0.658194\pi$
$702$	−3.45948e8	−0.0377426
$703$	−1.13265e8	−0.0122957
$704$	−4.66853e8	−0.0504285
$705$	−6.90971e9	−0.742673
$706$	−5.49646e9	−0.587850
$707$	5.49450e9	0.584736
$708$	2.59991e9	0.275322
$709$	−6.57486e9	−0.692827	−0.346413	−	0.938082i	$-0.612601\pi$
$709$	−6.57486e9	−0.692827	−0.346413	+	0.938082i	$0.612601\pi$
$710$	1.16398e9	0.122050
$711$	−5.19632e9	−0.542191
$712$	−1.93390e9	−0.200795
$713$	−1.28020e9	−0.132271
$714$	2.42286e9	0.249106
$715$	−8.83404e8	−0.0903833
$716$	3.12664e8	0.0318333
$717$	9.98912e9	1.01207
$718$	−5.16995e9	−0.521256
$719$	−4.44764e9	−0.446251	−0.223125	−	0.974790i	$-0.571626\pi$
$719$	−4.44764e9	−0.446251	−0.223125	+	0.974790i	$0.571626\pi$
$720$	−6.74181e8	−0.0673151
$721$	2.49721e9	0.248131
$722$	7.14498e9	0.706514
$723$	−1.04792e9	−0.103120
$724$	4.13041e9	0.404491
$725$	2.41530e9	0.235390
$726$	−3.52416e9	−0.341804
$727$	3.47692e7	0.00335602	0.00167801	−	0.999999i	$-0.499466\pi$
$727$	3.47692e7	0.00335602	0.00167801	+	0.999999i	$0.499466\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	−2.61430e9	−0.248728
$731$	2.47704e10	2.34543
$732$	−4.06582e8	−0.0383142
$733$	−8.82255e9	−0.827428	−0.413714	−	0.910407i	$-0.635769\pi$
$733$	−8.82255e9	−0.827428	−0.413714	+	0.910407i	$0.635769\pi$
$734$	1.82713e9	0.170543
$735$	7.17201e8	0.0666247
$736$	1.85671e8	0.0171661
$737$	2.42618e9	0.223248
$738$	1.51252e9	0.138517
$739$	−1.20276e10	−1.09628	−0.548141	−	0.836386i	$-0.684664\pi$
$739$	−1.20276e10	−1.09628	−0.548141	+	0.836386i	$0.684664\pi$
$740$	1.89045e9	0.171496
$741$	5.13565e7	0.00463694
$742$	1.46562e9	0.131706
$743$	1.11854e9	0.100044	0.0500218	−	0.998748i	$-0.484071\pi$
$743$	1.11854e9	0.100044	0.0500218	+	0.998748i	$0.484071\pi$
$744$	3.12332e9	0.278043
$745$	7.83708e9	0.694396
$746$	9.84543e9	0.868258
$747$	−5.16082e9	−0.452998
$748$	3.72735e9	0.325645
$749$	4.98465e9	0.433459
$750$	5.13402e9	0.444369
$751$	1.83970e10	1.58492	0.792458	−	0.609926i	$-0.208801\pi$
$751$	1.83970e10	1.58492	0.792458	+	0.609926i	$0.208801\pi$
$752$	−4.64266e9	−0.398111
$753$	−3.82001e9	−0.326049
$754$	−1.56372e9	−0.132850
$755$	−1.49770e9	−0.126651
$756$	4.32081e8	0.0363696
$757$	4.08610e9	0.342353	0.171176	−	0.985240i	$-0.445243\pi$
$757$	4.08610e9	0.342353	0.171176	+	0.985240i	$0.445243\pi$
$758$	6.60410e9	0.550771
$759$	−2.72458e8	−0.0226179
$760$	1.00083e8	0.00827014
$761$	−1.52003e9	−0.125027	−0.0625136	−	0.998044i	$-0.519912\pi$
$761$	−1.52003e9	−0.125027	−0.0625136	+	0.998044i	$0.519912\pi$
$762$	−9.76537e9	−0.799551
$763$	2.66932e9	0.217553
$764$	1.09896e10	0.891573
$765$	5.38265e9	0.434692
$766$	1.19323e10	0.959232
$767$	3.30555e9	0.264521
$768$	−4.52985e8	−0.0360844
$769$	−7.76362e9	−0.615633	−0.307817	−	0.951446i	$-0.599598\pi$
$769$	−7.76362e9	−0.615633	−0.307817	+	0.951446i	$0.599598\pi$
$770$	1.10335e9	0.0870955
$771$	−7.29362e9	−0.573129
$772$	2.66621e9	0.208562
$773$	−5.08431e9	−0.395917	−0.197959	−	0.980210i	$-0.563431\pi$
$773$	−5.08431e9	−0.395917	−0.197959	+	0.980210i	$0.563431\pi$
$774$	4.41744e9	0.342434
$775$	−6.13360e9	−0.473325
$776$	−3.18396e9	−0.244597
$777$	−1.21158e9	−0.0926574
$778$	1.00266e10	0.763353
$779$	−2.24536e8	−0.0170178
$780$	−8.57161e8	−0.0646742
$781$	−1.14764e9	−0.0862039
$782$	−1.48240e9	−0.110851
$783$	1.75118e9	0.130366
$784$	4.81890e8	0.0357143
$785$	4.68656e9	0.345788
$786$	−4.40412e9	−0.323505
$787$	1.35026e10	0.987430	0.493715	−	0.869624i	$-0.335639\pi$
$787$	1.35026e10	0.987430	0.493715	+	0.869624i	$0.335639\pi$
$788$	−4.31965e9	−0.314490
$789$	−6.82246e9	−0.494506
$790$	−1.28750e10	−0.929079
$791$	−4.17456e9	−0.299911
$792$	6.64719e8	0.0475444
$793$	−5.16933e8	−0.0368111
$794$	−8.06235e9	−0.571597
$795$	3.25603e9	0.229828
$796$	−3.84512e9	−0.270218
$797$	−8.75923e8	−0.0612860	−0.0306430	−	0.999530i	$-0.509756\pi$
$797$	−8.75923e8	−0.0612860	−0.0306430	+	0.999530i	$0.509756\pi$
$798$	−6.41430e7	−0.00446827
$799$	3.70670e10	2.57083
$800$	8.89574e8	0.0614281
$801$	2.75354e9	0.189312
$802$	−5.44472e9	−0.372705
$803$	2.57761e9	0.175676
$804$	2.35411e9	0.159746
$805$	−4.38811e8	−0.0296478
$806$	3.97103e9	0.267135
$807$	−8.33264e9	−0.558117
$808$	8.20170e9	0.546971
$809$	−2.07237e10	−1.37609	−0.688047	−	0.725666i	$-0.741532\pi$
$809$	−2.07237e10	−1.37609	−0.688047	+	0.725666i	$0.741532\pi$
$810$	9.59917e8	0.0634653
$811$	6.65486e9	0.438092	0.219046	−	0.975714i	$-0.429705\pi$
$811$	6.65486e9	0.438092	0.219046	+	0.975714i	$0.429705\pi$
$812$	1.95305e9	0.128017
$813$	−1.57100e10	−1.02532
$814$	−1.86391e9	−0.121127
$815$	−7.54849e9	−0.488437
$816$	3.61663e9	0.233017
$817$	−6.55774e8	−0.0420704
$818$	1.09233e10	0.697780
$819$	5.49353e8	0.0349428
$820$	3.74760e9	0.237358
$821$	−1.54891e10	−0.976845	−0.488422	−	0.872607i	$-0.662427\pi$
$821$	−1.54891e10	−0.976845	−0.488422	+	0.872607i	$0.662427\pi$
$822$	−1.15714e10	−0.726664
$823$	6.28904e9	0.393265	0.196632	−	0.980477i	$-0.436999\pi$
$823$	6.28904e9	0.393265	0.196632	+	0.980477i	$0.436999\pi$
$824$	3.72761e9	0.232106
$825$	−1.30538e9	−0.0809370
$826$	−4.12856e9	−0.254899
$827$	1.98057e10	1.21765	0.608824	−	0.793305i	$-0.291642\pi$
$827$	1.98057e10	1.21765	0.608824	+	0.793305i	$0.291642\pi$
$828$	−2.64364e8	−0.0161844
$829$	2.55286e10	1.55628	0.778138	−	0.628093i	$-0.216164\pi$
$829$	2.55286e10	1.55628	0.778138	+	0.628093i	$0.216164\pi$
$830$	−1.27870e10	−0.776240
$831$	4.37788e9	0.264643
$832$	−5.75930e8	−0.0346688
$833$	−3.84741e9	−0.230627
$834$	4.74333e9	0.283140
$835$	7.13969e9	0.424401
$836$	−9.86784e7	−0.00584117
$837$	−4.44708e9	−0.262141
$838$	−4.20920e9	−0.247085
$839$	1.55885e10	0.911247	0.455624	−	0.890173i	$-0.349416\pi$
$839$	1.55885e10	0.911247	0.455624	+	0.890173i	$0.349416\pi$
$840$	1.07057e9	0.0623217
$841$	−9.33436e9	−0.541126
$842$	−2.17765e10	−1.25718
$843$	1.86010e10	1.06940
$844$	9.93366e9	0.568736
$845$	−1.08981e9	−0.0621370
$846$	6.61035e9	0.375343
$847$	5.59623e9	0.316449
$848$	2.18774e9	0.123200
$849$	−5.66007e9	−0.317428
$850$	−7.10235e9	−0.396676
$851$	7.41294e8	0.0412322
$852$	−1.11355e9	−0.0616837
$853$	8.59640e9	0.474236	0.237118	−	0.971481i	$-0.423797\pi$
$853$	8.59640e9	0.474236	0.237118	+	0.971481i	$0.423797\pi$
$854$	6.45637e8	0.0354720
$855$	−1.42501e8	−0.00779716
$856$	7.44065e9	0.405464
$857$	8.70079e9	0.472200	0.236100	−	0.971729i	$-0.424131\pi$
$857$	8.70079e9	0.472200	0.236100	+	0.971729i	$0.424131\pi$
$858$	8.45131e8	0.0456792
$859$	2.35439e10	1.26737	0.633684	−	0.773592i	$-0.281542\pi$
$859$	2.35439e10	1.26737	0.633684	+	0.773592i	$0.281542\pi$
$860$	1.09451e10	0.586782
$861$	−2.40183e9	−0.128242
$862$	1.87763e10	0.998470
$863$	3.45528e9	0.182998	0.0914988	−	0.995805i	$-0.470834\pi$
$863$	3.45528e9	0.182998	0.0914988	+	0.995805i	$0.470834\pi$
$864$	6.44973e8	0.0340207
$865$	1.73904e10	0.913593
$866$	−8.37637e9	−0.438272
$867$	−1.77960e10	−0.927374
$868$	−4.95972e9	−0.257418
$869$	1.26943e10	0.656206
$870$	4.33892e9	0.223390
$871$	2.99305e9	0.153479
$872$	3.98453e9	0.203502
$873$	4.53341e9	0.230609
$874$	3.92452e7	0.00198837
$875$	−8.15264e9	−0.411406
$876$	2.50104e9	0.125706
$877$	2.13614e10	1.06938	0.534688	−	0.845049i	$-0.320429\pi$
$877$	2.13614e10	1.06938	0.534688	+	0.845049i	$0.320429\pi$
$878$	2.59572e9	0.129427
$879$	4.16928e9	0.207062
$880$	1.64698e9	0.0814704
$881$	1.61338e10	0.794916	0.397458	−	0.917620i	$-0.369892\pi$
$881$	1.61338e10	0.794916	0.397458	+	0.917620i	$0.369892\pi$
$882$	−6.86129e8	−0.0336718
$883$	−1.39709e10	−0.682909	−0.341455	−	0.939898i	$-0.610920\pi$
$883$	−1.39709e10	−0.682909	−0.341455	+	0.939898i	$0.610920\pi$
$884$	4.59822e9	0.223876
$885$	−9.17205e9	−0.444800
$886$	−2.16145e10	−1.04407
$887$	2.88913e10	1.39006	0.695032	−	0.718979i	$-0.255390\pi$
$887$	2.88913e10	1.39006	0.695032	+	0.718979i	$0.255390\pi$
$888$	−1.80854e9	−0.0866730
$889$	1.55070e10	0.740240
$890$	6.82249e9	0.324398
$891$	−9.46445e8	−0.0448253
$892$	−2.48001e9	−0.116997
$893$	−9.81315e8	−0.0461135
$894$	−7.49755e9	−0.350944
$895$	−1.10303e9	−0.0514287
$896$	7.19323e8	0.0334077
$897$	−3.36116e8	−0.0155495
$898$	−3.52574e9	−0.162474
$899$	−2.01012e10	−0.922707
$900$	−1.26660e9	−0.0579149
$901$	−1.74669e10	−0.795572
$902$	−3.69500e9	−0.167645
$903$	−7.01472e9	−0.317032
$904$	−6.23141e9	−0.280541
$905$	−1.45714e10	−0.653480
$906$	1.43281e9	0.0640089
$907$	−3.63947e10	−1.61962	−0.809809	−	0.586694i	$-0.800429\pi$
$907$	−3.63947e10	−1.61962	−0.809809	+	0.586694i	$0.800429\pi$
$908$	−7.58325e9	−0.336167
$909$	−1.16778e10	−0.515689
$910$	1.36114e9	0.0598767
$911$	3.52825e10	1.54613	0.773063	−	0.634329i	$-0.218723\pi$
$911$	3.52825e10	1.54613	0.773063	+	0.634329i	$0.218723\pi$
$912$	−9.57470e7	−0.00417968
$913$	1.26076e10	0.548256
$914$	−1.39877e10	−0.605949
$915$	1.43436e9	0.0618989
$916$	5.34079e9	0.229600
$917$	6.99358e9	0.299507
$918$	−5.14945e9	−0.219691
$919$	−3.49903e10	−1.48711	−0.743556	−	0.668674i	$-0.766862\pi$
$919$	−3.49903e10	−1.48711	−0.743556	+	0.668674i	$0.766862\pi$
$920$	−6.55018e8	−0.0277329
$921$	−2.56809e10	−1.08318
$922$	−9.32716e9	−0.391914
$923$	−1.41578e9	−0.0592637
$924$	−1.05555e9	−0.0440176
$925$	3.55163e9	0.147547
$926$	2.88720e10	1.19492
$927$	−5.30748e9	−0.218831
$928$	2.91534e9	0.119749
$929$	2.56138e10	1.04814	0.524070	−	0.851675i	$-0.324413\pi$
$929$	2.56138e10	1.04814	0.524070	+	0.851675i	$0.324413\pi$
$930$	−1.10186e10	−0.449195
$931$	1.01857e8	0.00413681
$932$	1.71856e10	0.695360
$933$	−1.79904e10	−0.725195
$934$	9.91843e7	0.00398317
$935$	−1.31495e10	−0.526100
$936$	8.20026e8	0.0326860
$937$	−1.30392e10	−0.517802	−0.258901	−	0.965904i	$-0.583360\pi$
$937$	−1.30392e10	−0.517802	−0.258901	+	0.965904i	$0.583360\pi$
$938$	−3.73824e9	−0.147896
$939$	−9.60564e9	−0.378614
$940$	1.63786e10	0.643174
$941$	−2.66227e10	−1.04157	−0.520786	−	0.853688i	$-0.674361\pi$
$941$	−2.66227e10	−1.04157	−0.520786	+	0.853688i	$0.674361\pi$
$942$	−4.48352e9	−0.174760
$943$	1.46953e9	0.0570674
$944$	−6.16274e9	−0.238436
$945$	−1.52431e9	−0.0587574
$946$	−1.07915e10	−0.414442
$947$	−3.77003e10	−1.44251	−0.721256	−	0.692668i	$-0.756435\pi$
$947$	−3.77003e10	−1.44251	−0.721256	+	0.692668i	$0.756435\pi$
$948$	1.23172e10	0.469551
$949$	3.17985e9	0.120774
$950$	1.88028e8	0.00711526
$951$	4.30639e9	0.162361
$952$	−5.74307e9	−0.215732
$953$	6.19400e9	0.231818	0.115909	−	0.993260i	$-0.463022\pi$
$953$	6.19400e9	0.231818	0.115909	+	0.993260i	$0.463022\pi$
$954$	−3.11497e9	−0.116154
$955$	−3.87697e10	−1.44039
$956$	−2.36779e10	−0.876477
$957$	−4.27803e9	−0.157780
$958$	9.34719e9	0.343480
$959$	1.83749e10	0.672761
$960$	1.59806e9	0.0582966
$961$	2.35339e10	0.855387
$962$	−2.29941e9	−0.0832727
$963$	−1.05942e10	−0.382275
$964$	2.48395e9	0.0893046
$965$	−9.40597e9	−0.336944
$966$	4.19800e8	0.0149838
$967$	3.96074e10	1.40859	0.704293	−	0.709909i	$-0.251264\pi$
$967$	3.96074e10	1.40859	0.704293	+	0.709909i	$0.251264\pi$
$968$	8.35356e9	0.296011
$969$	7.64443e8	0.0269906
$970$	1.12325e10	0.395162
$971$	−4.45444e9	−0.156144	−0.0780721	−	0.996948i	$-0.524876\pi$
$971$	−4.45444e9	−0.156144	−0.0780721	+	0.996948i	$0.524876\pi$
$972$	−9.18330e8	−0.0320750
$973$	−7.53223e9	−0.262137
$974$	−2.09985e9	−0.0728169
$975$	−1.61037e9	−0.0556429
$976$	9.63750e8	0.0331811
$977$	3.32279e10	1.13991	0.569956	−	0.821675i	$-0.306960\pi$
$977$	3.32279e10	1.13991	0.569956	+	0.821675i	$0.306960\pi$
$978$	7.22146e9	0.246853
$979$	−6.72674e9	−0.229121
$980$	−1.70003e9	−0.0576987
$981$	−5.67329e9	−0.191864
$982$	5.70000e9	0.192081
$983$	−5.00142e9	−0.167941	−0.0839704	−	0.996468i	$-0.526760\pi$
$983$	−5.00142e9	−0.167941	−0.0839704	+	0.996468i	$0.526760\pi$
$984$	−3.58524e9	−0.119960
$985$	1.52390e10	0.508078
$986$	−2.32761e10	−0.773286
$987$	−1.04970e10	−0.347500
$988$	−1.21734e8	−0.00401571
$989$	4.29188e9	0.141078
$990$	−2.34502e9	−0.0768110
$991$	−5.43776e9	−0.177485	−0.0887427	−	0.996055i	$-0.528285\pi$
$991$	−5.43776e9	−0.177485	−0.0887427	+	0.996055i	$0.528285\pi$
$992$	−7.40343e9	−0.240792
$993$	−9.41754e9	−0.305222
$994$	1.76827e9	0.0571080
$995$	1.35650e10	0.436554
$996$	1.22330e10	0.392308
$997$	−2.32030e10	−0.741501	−0.370750	−	0.928733i	$-0.620899\pi$
$997$	−2.32030e10	−0.741501	−0.370750	+	0.928733i	$0.620899\pi$
$998$	8.71598e9	0.277562
$999$	2.57506e9	0.0817161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.o.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.o.1.2	✓	6	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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -225.782 q^{5} +216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -225.782 q^{5} +216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +1806.25 q^{10} -1780.90 q^{11} -1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +6096.11 q^{15} +4096.00 q^{16} -32702.4 q^{17} -5832.00 q^{18} +865.768 q^{19} -14450.0 q^{20} +9261.00 q^{21} +14247.2 q^{22} -5666.24 q^{23} +13824.0 q^{24} -27147.6 q^{25} +17576.0 q^{26} -19683.0 q^{27} -21952.0 q^{28} -88969.2 q^{29} -48768.8 q^{30} +225935. q^{31} -32768.0 q^{32} +48084.4 q^{33} +261619. q^{34} +77443.1 q^{35} +46656.0 q^{36} -130826. q^{37} -6926.14 q^{38} +59319.0 q^{39} +115600. q^{40} -259349. q^{41} -74088.0 q^{42} -757448. q^{43} -113978. q^{44} -164595. q^{45} +45329.9 q^{46} -1.13346e6 q^{47} -110592. q^{48} +117649. q^{49} +217181. q^{50} +882965. q^{51} -140608. q^{52} +534117. q^{53} +157464. q^{54} +402095. q^{55} +175616. q^{56} -23375.7 q^{57} +711753. q^{58} -1.50458e6 q^{59} +390151. q^{60} +235291. q^{61} -1.80748e6 q^{62} -250047. q^{63} +262144. q^{64} +496042. q^{65} -384675. q^{66} -1.36233e6 q^{67} -2.09296e6 q^{68} +152988. q^{69} -619545. q^{70} +644414. q^{71} -373248. q^{72} -1.44736e6 q^{73} +1.04661e6 q^{74} +732986. q^{75} +55409.2 q^{76} +610850. q^{77} -474552. q^{78} -7.12802e6 q^{79} -924802. q^{80} +531441. q^{81} +2.07479e6 q^{82} -7.07931e6 q^{83} +592704. q^{84} +7.38361e6 q^{85} +6.05958e6 q^{86} +2.40217e6 q^{87} +911823. q^{88} +3.77715e6 q^{89} +1.31676e6 q^{90} +753571. q^{91} -362639. q^{92} -6.10024e6 q^{93} +9.06770e6 q^{94} -195475. q^{95} +884736. q^{96} +6.21867e6 q^{97} -941192. q^{98} -1.29828e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 35 q^{5} + 1296 q^{6} - 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) 6 * q - 48 * q^2 - 162 * q^3 + 384 * q^4 - 35 * q^5 + 1296 * q^6 - 2058 * q^7 - 3072 * q^8 + 4374 * q^9	\( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 35 q^{5} + 1296 q^{6} - 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 280 q^{10} + 5606 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} + 945 q^{15} + 24576 q^{16} + 22022 q^{17} - 34992 q^{18} - 5779 q^{19} - 2240 q^{20} + 55566 q^{21} - 44848 q^{22} - 110789 q^{23} + 82944 q^{24} + 91769 q^{25} + 105456 q^{26} - 118098 q^{27} - 131712 q^{28} + 30693 q^{29} - 7560 q^{30} + 180467 q^{31} - 196608 q^{32} - 151362 q^{33} - 176176 q^{34} + 12005 q^{35} + 279936 q^{36} - 322222 q^{37} + 46232 q^{38} + 355914 q^{39} + 17920 q^{40} - 212652 q^{41} - 444528 q^{42} - 329299 q^{43} + 358784 q^{44} - 25515 q^{45} + 886312 q^{46} + 1322861 q^{47} - 663552 q^{48} + 705894 q^{49} - 734152 q^{50} - 594594 q^{51} - 843648 q^{52} - 168719 q^{53} + 944784 q^{54} - 2252362 q^{55} + 1053696 q^{56} + 156033 q^{57} - 245544 q^{58} + 1943712 q^{59} + 60480 q^{60} - 1085922 q^{61} - 1443736 q^{62} - 1500282 q^{63} + 1572864 q^{64} + 76895 q^{65} + 1210896 q^{66} + 885066 q^{67} + 1409408 q^{68} + 2991303 q^{69} - 96040 q^{70} + 1626164 q^{71} - 2239488 q^{72} - 4750115 q^{73} + 2577776 q^{74} - 2477763 q^{75} - 369856 q^{76} - 1922858 q^{77} - 2847312 q^{78} + 1794289 q^{79} - 143360 q^{80} + 3188646 q^{81} + 1701216 q^{82} - 8454255 q^{83} + 3556224 q^{84} - 18529504 q^{85} + 2634392 q^{86} - 828711 q^{87} - 2870272 q^{88} - 6055411 q^{89} + 204120 q^{90} + 4521426 q^{91} - 7090496 q^{92} - 4872609 q^{93} - 10582888 q^{94} - 3766747 q^{95} + 5308416 q^{96} - 9823899 q^{97} - 5647152 q^{98} + 4086774 q^{99}+O(q^{100}) \) 6 * q - 48 * q^2 - 162 * q^3 + 384 * q^4 - 35 * q^5 + 1296 * q^6 - 2058 * q^7 - 3072 * q^8 + 4374 * q^9 + 280 * q^10 + 5606 * q^11 - 10368 * q^12 - 13182 * q^13 + 16464 * q^14 + 945 * q^15 + 24576 * q^16 + 22022 * q^17 - 34992 * q^18 - 5779 * q^19 - 2240 * q^20 + 55566 * q^21 - 44848 * q^22 - 110789 * q^23 + 82944 * q^24 + 91769 * q^25 + 105456 * q^26 - 118098 * q^27 - 131712 * q^28 + 30693 * q^29 - 7560 * q^30 + 180467 * q^31 - 196608 * q^32 - 151362 * q^33 - 176176 * q^34 + 12005 * q^35 + 279936 * q^36 - 322222 * q^37 + 46232 * q^38 + 355914 * q^39 + 17920 * q^40 - 212652 * q^41 - 444528 * q^42 - 329299 * q^43 + 358784 * q^44 - 25515 * q^45 + 886312 * q^46 + 1322861 * q^47 - 663552 * q^48 + 705894 * q^49 - 734152 * q^50 - 594594 * q^51 - 843648 * q^52 - 168719 * q^53 + 944784 * q^54 - 2252362 * q^55 + 1053696 * q^56 + 156033 * q^57 - 245544 * q^58 + 1943712 * q^59 + 60480 * q^60 - 1085922 * q^61 - 1443736 * q^62 - 1500282 * q^63 + 1572864 * q^64 + 76895 * q^65 + 1210896 * q^66 + 885066 * q^67 + 1409408 * q^68 + 2991303 * q^69 - 96040 * q^70 + 1626164 * q^71 - 2239488 * q^72 - 4750115 * q^73 + 2577776 * q^74 - 2477763 * q^75 - 369856 * q^76 - 1922858 * q^77 - 2847312 * q^78 + 1794289 * q^79 - 143360 * q^80 + 3188646 * q^81 + 1701216 * q^82 - 8454255 * q^83 + 3556224 * q^84 - 18529504 * q^85 + 2634392 * q^86 - 828711 * q^87 - 2870272 * q^88 - 6055411 * q^89 + 204120 * q^90 + 4521426 * q^91 - 7090496 * q^92 - 4872609 * q^93 - 10582888 * q^94 - 3766747 * q^95 + 5308416 * q^96 - 9823899 * q^97 - 5647152 * q^98 + 4086774 * q^99

Introduction

L-functions

Modular forms

Varieties

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