Properties

Label 546.8.a.q.1.6
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 367021 x^{4} - 17702143 x^{3} + 34815194576 x^{2} + 1422988371620 x - 933871993059968\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(527.645\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +529.645 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +529.645 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +4237.16 q^{10} +3727.68 q^{11} -1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} -14300.4 q^{15} +4096.00 q^{16} +24121.3 q^{17} +5832.00 q^{18} +9818.46 q^{19} +33897.3 q^{20} -9261.00 q^{21} +29821.5 q^{22} -36040.5 q^{23} -13824.0 q^{24} +202398. q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} -159862. q^{29} -114403. q^{30} +222746. q^{31} +32768.0 q^{32} -100647. q^{33} +192971. q^{34} +181668. q^{35} +46656.0 q^{36} +219687. q^{37} +78547.7 q^{38} +59319.0 q^{39} +271178. q^{40} -29299.4 q^{41} -74088.0 q^{42} -79448.9 q^{43} +238572. q^{44} +386111. q^{45} -288324. q^{46} +652464. q^{47} -110592. q^{48} +117649. q^{49} +1.61919e6 q^{50} -651276. q^{51} -140608. q^{52} +586529. q^{53} -157464. q^{54} +1.97435e6 q^{55} +175616. q^{56} -265098. q^{57} -1.27890e6 q^{58} -1.11075e6 q^{59} -915226. q^{60} +370461. q^{61} +1.78197e6 q^{62} +250047. q^{63} +262144. q^{64} -1.16363e6 q^{65} -805180. q^{66} -3.54571e6 q^{67} +1.54377e6 q^{68} +973093. q^{69} +1.45334e6 q^{70} +3.51099e6 q^{71} +373248. q^{72} -6.04121e6 q^{73} +1.75749e6 q^{74} -5.46476e6 q^{75} +628381. q^{76} +1.27860e6 q^{77} +474552. q^{78} -4.72429e6 q^{79} +2.16942e6 q^{80} +531441. q^{81} -234395. q^{82} +2.27200e6 q^{83} -592704. q^{84} +1.27757e7 q^{85} -635591. q^{86} +4.31628e6 q^{87} +1.90857e6 q^{88} -2.77728e6 q^{89} +3.08889e6 q^{90} -753571. q^{91} -2.30659e6 q^{92} -6.01415e6 q^{93} +5.21971e6 q^{94} +5.20029e6 q^{95} -884736. q^{96} +3.86868e6 q^{97} +941192. q^{98} +2.71748e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 104 q^{10} + 10054 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} - 351 q^{15} + 24576 q^{16} + 21222 q^{17} + 34992 q^{18} + 9527 q^{19} + 832 q^{20} - 55566 q^{21} + 80432 q^{22} + 33229 q^{23} - 82944 q^{24} + 265321 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} + 174185 q^{29} - 2808 q^{30} + 119045 q^{31} + 196608 q^{32} - 271458 q^{33} + 169776 q^{34} + 4459 q^{35} + 279936 q^{36} + 56562 q^{37} + 76216 q^{38} + 355914 q^{39} + 6656 q^{40} + 101632 q^{41} - 444528 q^{42} + 441323 q^{43} + 643456 q^{44} + 9477 q^{45} + 265832 q^{46} - 892849 q^{47} - 663552 q^{48} + 705894 q^{49} + 2122568 q^{50} - 572994 q^{51} - 843648 q^{52} + 2093965 q^{53} - 944784 q^{54} - 331222 q^{55} + 1053696 q^{56} - 257229 q^{57} + 1393480 q^{58} - 136204 q^{59} - 22464 q^{60} - 3989946 q^{61} + 952360 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 28561 q^{65} - 2171664 q^{66} - 2218250 q^{67} + 1358208 q^{68} - 897183 q^{69} + 35672 q^{70} + 2045928 q^{71} + 2239488 q^{72} - 8557479 q^{73} + 452496 q^{74} - 7163667 q^{75} + 609728 q^{76} + 3448522 q^{77} + 2847312 q^{78} - 8559709 q^{79} + 53248 q^{80} + 3188646 q^{81} + 813056 q^{82} + 2496351 q^{83} - 3556224 q^{84} + 5335304 q^{85} + 3530584 q^{86} - 4702995 q^{87} + 5147648 q^{88} - 2446683 q^{89} + 75816 q^{90} - 4521426 q^{91} + 2126656 q^{92} - 3214215 q^{93} - 7142792 q^{94} + 16410211 q^{95} - 5308416 q^{96} + 5786889 q^{97} + 5647152 q^{98} + 7329366 q^{99} + O(q^{100}) \)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −27.0000 −0.577350
\(4\) 64.0000 0.500000
\(5\) 529.645 1.89491 0.947457 0.319883i \(-0.103644\pi\)
0.947457 + 0.319883i \(0.103644\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 4237.16 1.33991
\(11\) 3727.68 0.844432 0.422216 0.906495i \(-0.361252\pi\)
0.422216 + 0.906495i \(0.361252\pi\)
\(12\) −1728.00 −0.288675
\(13\) −2197.00 −0.277350
\(14\) 2744.00 0.267261
\(15\) −14300.4 −1.09403
\(16\) 4096.00 0.250000
\(17\) 24121.3 1.19078 0.595388 0.803438i \(-0.296998\pi\)
0.595388 + 0.803438i \(0.296998\pi\)
\(18\) 5832.00 0.235702
\(19\) 9818.46 0.328402 0.164201 0.986427i \(-0.447495\pi\)
0.164201 + 0.986427i \(0.447495\pi\)
\(20\) 33897.3 0.947457
\(21\) −9261.00 −0.218218
\(22\) 29821.5 0.597104
\(23\) −36040.5 −0.617651 −0.308825 0.951119i \(-0.599936\pi\)
−0.308825 + 0.951119i \(0.599936\pi\)
\(24\) −13824.0 −0.204124
\(25\) 202398. 2.59070
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) −159862. −1.21717 −0.608587 0.793487i \(-0.708263\pi\)
−0.608587 + 0.793487i \(0.708263\pi\)
\(30\) −114403. −0.773596
\(31\) 222746. 1.34290 0.671452 0.741048i \(-0.265671\pi\)
0.671452 + 0.741048i \(0.265671\pi\)
\(32\) 32768.0 0.176777
\(33\) −100647. −0.487533
\(34\) 192971. 0.842006
\(35\) 181668. 0.716210
\(36\) 46656.0 0.166667
\(37\) 219687. 0.713013 0.356507 0.934293i \(-0.383968\pi\)
0.356507 + 0.934293i \(0.383968\pi\)
\(38\) 78547.7 0.232215
\(39\) 59319.0 0.160128
\(40\) 271178. 0.669953
\(41\) −29299.4 −0.0663919 −0.0331959 0.999449i \(-0.510569\pi\)
−0.0331959 + 0.999449i \(0.510569\pi\)
\(42\) −74088.0 −0.154303
\(43\) −79448.9 −0.152387 −0.0761936 0.997093i \(-0.524277\pi\)
−0.0761936 + 0.997093i \(0.524277\pi\)
\(44\) 238572. 0.422216
\(45\) 386111. 0.631638
\(46\) −288324. −0.436745
\(47\) 652464. 0.916673 0.458336 0.888779i \(-0.348446\pi\)
0.458336 + 0.888779i \(0.348446\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) 1.61919e6 1.83190
\(51\) −651276. −0.687495
\(52\) −140608. −0.138675
\(53\) 586529. 0.541158 0.270579 0.962698i \(-0.412785\pi\)
0.270579 + 0.962698i \(0.412785\pi\)
\(54\) −157464. −0.136083
\(55\) 1.97435e6 1.60013
\(56\) 175616. 0.133631
\(57\) −265098. −0.189603
\(58\) −1.27890e6 −0.860672
\(59\) −1.11075e6 −0.704100 −0.352050 0.935981i \(-0.614515\pi\)
−0.352050 + 0.935981i \(0.614515\pi\)
\(60\) −915226. −0.547015
\(61\) 370461. 0.208972 0.104486 0.994526i \(-0.466680\pi\)
0.104486 + 0.994526i \(0.466680\pi\)
\(62\) 1.78197e6 0.949576
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −1.16363e6 −0.525555
\(66\) −805180. −0.344738
\(67\) −3.54571e6 −1.44026 −0.720130 0.693839i \(-0.755918\pi\)
−0.720130 + 0.693839i \(0.755918\pi\)
\(68\) 1.54377e6 0.595388
\(69\) 973093. 0.356601
\(70\) 1.45334e6 0.506437
\(71\) 3.51099e6 1.16419 0.582097 0.813119i \(-0.302232\pi\)
0.582097 + 0.813119i \(0.302232\pi\)
\(72\) 373248. 0.117851
\(73\) −6.04121e6 −1.81758 −0.908791 0.417251i \(-0.862994\pi\)
−0.908791 + 0.417251i \(0.862994\pi\)
\(74\) 1.75749e6 0.504177
\(75\) −5.46476e6 −1.49574
\(76\) 628381. 0.164201
\(77\) 1.27860e6 0.319165
\(78\) 474552. 0.113228
\(79\) −4.72429e6 −1.07806 −0.539028 0.842288i \(-0.681208\pi\)
−0.539028 + 0.842288i \(0.681208\pi\)
\(80\) 2.16942e6 0.473729
\(81\) 531441. 0.111111
\(82\) −234395. −0.0469461
\(83\) 2.27200e6 0.436149 0.218075 0.975932i \(-0.430022\pi\)
0.218075 + 0.975932i \(0.430022\pi\)
\(84\) −592704. −0.109109
\(85\) 1.27757e7 2.25642
\(86\) −635591. −0.107754
\(87\) 4.31628e6 0.702735
\(88\) 1.90857e6 0.298552
\(89\) −2.77728e6 −0.417595 −0.208797 0.977959i \(-0.566955\pi\)
−0.208797 + 0.977959i \(0.566955\pi\)
\(90\) 3.08889e6 0.446636
\(91\) −753571. −0.104828
\(92\) −2.30659e6 −0.308825
\(93\) −6.01415e6 −0.775326
\(94\) 5.21971e6 0.648185
\(95\) 5.20029e6 0.622293
\(96\) −884736. −0.102062
\(97\) 3.86868e6 0.430390 0.215195 0.976571i \(-0.430961\pi\)
0.215195 + 0.976571i \(0.430961\pi\)
\(98\) 941192. 0.101015
\(99\) 2.71748e6 0.281477
\(100\) 1.29535e7 1.29535
\(101\) −4.17061e6 −0.402786 −0.201393 0.979511i \(-0.564547\pi\)
−0.201393 + 0.979511i \(0.564547\pi\)
\(102\) −5.21021e6 −0.486133
\(103\) 5.54746e6 0.500223 0.250111 0.968217i \(-0.419533\pi\)
0.250111 + 0.968217i \(0.419533\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) −4.90504e6 −0.413504
\(106\) 4.69223e6 0.382657
\(107\) −2.37587e7 −1.87490 −0.937452 0.348115i \(-0.886822\pi\)
−0.937452 + 0.348115i \(0.886822\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 1.37063e7 1.01374 0.506870 0.862022i \(-0.330802\pi\)
0.506870 + 0.862022i \(0.330802\pi\)
\(110\) 1.57948e7 1.13146
\(111\) −5.93154e6 −0.411658
\(112\) 1.40493e6 0.0944911
\(113\) −2.23664e6 −0.145821 −0.0729107 0.997338i \(-0.523229\pi\)
−0.0729107 + 0.997338i \(0.523229\pi\)
\(114\) −2.12079e6 −0.134070
\(115\) −1.90886e7 −1.17040
\(116\) −1.02312e7 −0.608587
\(117\) −1.60161e6 −0.0924500
\(118\) −8.88600e6 −0.497874
\(119\) 8.27362e6 0.450071
\(120\) −7.32181e6 −0.386798
\(121\) −5.59154e6 −0.286934
\(122\) 2.96369e6 0.147765
\(123\) 791083. 0.0383314
\(124\) 1.42558e7 0.671452
\(125\) 6.58208e7 3.01424
\(126\) 2.00038e6 0.0890871
\(127\) −9.41494e6 −0.407854 −0.203927 0.978986i \(-0.565370\pi\)
−0.203927 + 0.978986i \(0.565370\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 2.14512e6 0.0879808
\(130\) −9.30903e6 −0.371623
\(131\) −1.62654e7 −0.632145 −0.316072 0.948735i \(-0.602364\pi\)
−0.316072 + 0.948735i \(0.602364\pi\)
\(132\) −6.44144e6 −0.243767
\(133\) 3.36773e6 0.124124
\(134\) −2.83656e7 −1.01842
\(135\) −1.04250e7 −0.364676
\(136\) 1.23501e7 0.421003
\(137\) −2.09518e7 −0.696146 −0.348073 0.937467i \(-0.613164\pi\)
−0.348073 + 0.937467i \(0.613164\pi\)
\(138\) 7.78474e6 0.252155
\(139\) −2.42651e7 −0.766355 −0.383177 0.923675i \(-0.625170\pi\)
−0.383177 + 0.923675i \(0.625170\pi\)
\(140\) 1.16268e7 0.358105
\(141\) −1.76165e7 −0.529241
\(142\) 2.80879e7 0.823210
\(143\) −8.18972e6 −0.234203
\(144\) 2.98598e6 0.0833333
\(145\) −8.46701e7 −2.30644
\(146\) −4.83297e7 −1.28522
\(147\) −3.17652e6 −0.0824786
\(148\) 1.40599e7 0.356507
\(149\) 4.42973e7 1.09705 0.548523 0.836135i \(-0.315190\pi\)
0.548523 + 0.836135i \(0.315190\pi\)
\(150\) −4.37181e7 −1.05765
\(151\) −5.68350e7 −1.34337 −0.671687 0.740835i \(-0.734430\pi\)
−0.671687 + 0.740835i \(0.734430\pi\)
\(152\) 5.02705e6 0.116108
\(153\) 1.75845e7 0.396926
\(154\) 1.02288e7 0.225684
\(155\) 1.17976e8 2.54469
\(156\) 3.79642e6 0.0800641
\(157\) 7.78930e7 1.60639 0.803193 0.595719i \(-0.203133\pi\)
0.803193 + 0.595719i \(0.203133\pi\)
\(158\) −3.77943e7 −0.762301
\(159\) −1.58363e7 −0.312438
\(160\) 1.73554e7 0.334977
\(161\) −1.23619e7 −0.233450
\(162\) 4.25153e6 0.0785674
\(163\) −5.14450e7 −0.930436 −0.465218 0.885196i \(-0.654024\pi\)
−0.465218 + 0.885196i \(0.654024\pi\)
\(164\) −1.87516e6 −0.0331959
\(165\) −5.33074e7 −0.923834
\(166\) 1.81760e7 0.308404
\(167\) 2.40929e7 0.400296 0.200148 0.979766i \(-0.435858\pi\)
0.200148 + 0.979766i \(0.435858\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 1.02206e8 1.59553
\(171\) 7.15766e6 0.109467
\(172\) −5.08473e6 −0.0761936
\(173\) 8.78537e7 1.29003 0.645013 0.764171i \(-0.276852\pi\)
0.645013 + 0.764171i \(0.276852\pi\)
\(174\) 3.45302e7 0.496909
\(175\) 6.94227e7 0.979193
\(176\) 1.52686e7 0.211108
\(177\) 2.99903e7 0.406512
\(178\) −2.22183e7 −0.295284
\(179\) 7.47061e7 0.973578 0.486789 0.873520i \(-0.338168\pi\)
0.486789 + 0.873520i \(0.338168\pi\)
\(180\) 2.47111e7 0.315819
\(181\) 5.00109e7 0.626887 0.313444 0.949607i \(-0.398517\pi\)
0.313444 + 0.949607i \(0.398517\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) −1.00024e7 −0.120650
\(184\) −1.84527e7 −0.218373
\(185\) 1.16356e8 1.35110
\(186\) −4.81132e7 −0.548238
\(187\) 8.99168e7 1.00553
\(188\) 4.17577e7 0.458336
\(189\) −6.75127e6 −0.0727393
\(190\) 4.16023e7 0.440028
\(191\) 1.78752e8 1.85624 0.928119 0.372284i \(-0.121425\pi\)
0.928119 + 0.372284i \(0.121425\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 2.51808e7 0.252127 0.126063 0.992022i \(-0.459766\pi\)
0.126063 + 0.992022i \(0.459766\pi\)
\(194\) 3.09495e7 0.304332
\(195\) 3.14180e7 0.303429
\(196\) 7.52954e6 0.0714286
\(197\) 1.23584e8 1.15168 0.575839 0.817563i \(-0.304675\pi\)
0.575839 + 0.817563i \(0.304675\pi\)
\(198\) 2.17399e7 0.199035
\(199\) 2.90019e7 0.260880 0.130440 0.991456i \(-0.458361\pi\)
0.130440 + 0.991456i \(0.458361\pi\)
\(200\) 1.03628e8 0.915951
\(201\) 9.57341e7 0.831535
\(202\) −3.33649e7 −0.284813
\(203\) −5.48327e7 −0.460048
\(204\) −4.16817e7 −0.343748
\(205\) −1.55183e7 −0.125807
\(206\) 4.43797e7 0.353711
\(207\) −2.62735e7 −0.205884
\(208\) −8.99891e6 −0.0693375
\(209\) 3.66001e7 0.277313
\(210\) −3.92403e7 −0.292392
\(211\) −6.62659e7 −0.485625 −0.242813 0.970073i \(-0.578070\pi\)
−0.242813 + 0.970073i \(0.578070\pi\)
\(212\) 3.75379e7 0.270579
\(213\) −9.47968e7 −0.672148
\(214\) −1.90069e8 −1.32576
\(215\) −4.20797e7 −0.288761
\(216\) −1.00777e7 −0.0680414
\(217\) 7.64020e7 0.507570
\(218\) 1.09650e8 0.716823
\(219\) 1.63113e8 1.04938
\(220\) 1.26358e8 0.800063
\(221\) −5.29946e7 −0.330262
\(222\) −4.74523e7 −0.291086
\(223\) 2.82014e8 1.70296 0.851479 0.524389i \(-0.175706\pi\)
0.851479 + 0.524389i \(0.175706\pi\)
\(224\) 1.12394e7 0.0668153
\(225\) 1.47548e8 0.863567
\(226\) −1.78931e7 −0.103111
\(227\) −1.19805e8 −0.679805 −0.339903 0.940461i \(-0.610394\pi\)
−0.339903 + 0.940461i \(0.610394\pi\)
\(228\) −1.69663e7 −0.0948015
\(229\) 1.03961e7 0.0572064 0.0286032 0.999591i \(-0.490894\pi\)
0.0286032 + 0.999591i \(0.490894\pi\)
\(230\) −1.52709e8 −0.827595
\(231\) −3.45221e7 −0.184270
\(232\) −8.18494e7 −0.430336
\(233\) −1.76469e8 −0.913952 −0.456976 0.889479i \(-0.651067\pi\)
−0.456976 + 0.889479i \(0.651067\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 3.45574e8 1.73702
\(236\) −7.10880e7 −0.352050
\(237\) 1.27556e8 0.622416
\(238\) 6.61890e7 0.318248
\(239\) 2.48570e8 1.17776 0.588878 0.808222i \(-0.299570\pi\)
0.588878 + 0.808222i \(0.299570\pi\)
\(240\) −5.85745e7 −0.273507
\(241\) −3.65205e7 −0.168065 −0.0840326 0.996463i \(-0.526780\pi\)
−0.0840326 + 0.996463i \(0.526780\pi\)
\(242\) −4.47323e7 −0.202893
\(243\) −1.43489e7 −0.0641500
\(244\) 2.37095e7 0.104486
\(245\) 6.23122e7 0.270702
\(246\) 6.32867e6 0.0271044
\(247\) −2.15712e7 −0.0910823
\(248\) 1.14046e8 0.474788
\(249\) −6.13440e7 −0.251811
\(250\) 5.26566e8 2.13139
\(251\) 3.43704e8 1.37191 0.685957 0.727643i \(-0.259384\pi\)
0.685957 + 0.727643i \(0.259384\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) −1.34348e8 −0.521564
\(254\) −7.53195e7 −0.288396
\(255\) −3.44945e8 −1.30274
\(256\) 1.67772e7 0.0625000
\(257\) −2.72290e8 −1.00061 −0.500306 0.865849i \(-0.666779\pi\)
−0.500306 + 0.865849i \(0.666779\pi\)
\(258\) 1.71610e7 0.0622118
\(259\) 7.53525e7 0.269494
\(260\) −7.44723e7 −0.262777
\(261\) −1.16539e8 −0.405724
\(262\) −1.30124e8 −0.446994
\(263\) 3.55067e8 1.20355 0.601777 0.798664i \(-0.294460\pi\)
0.601777 + 0.798664i \(0.294460\pi\)
\(264\) −5.15315e7 −0.172369
\(265\) 3.10652e8 1.02545
\(266\) 2.69418e7 0.0877691
\(267\) 7.49867e7 0.241099
\(268\) −2.26925e8 −0.720130
\(269\) 2.87140e8 0.899417 0.449709 0.893175i \(-0.351528\pi\)
0.449709 + 0.893175i \(0.351528\pi\)
\(270\) −8.34000e7 −0.257865
\(271\) −5.05877e8 −1.54402 −0.772009 0.635611i \(-0.780748\pi\)
−0.772009 + 0.635611i \(0.780748\pi\)
\(272\) 9.88010e7 0.297694
\(273\) 2.03464e7 0.0605228
\(274\) −1.67615e8 −0.492250
\(275\) 7.54478e8 2.18767
\(276\) 6.22779e7 0.178300
\(277\) 3.57253e8 1.00994 0.504971 0.863136i \(-0.331503\pi\)
0.504971 + 0.863136i \(0.331503\pi\)
\(278\) −1.94121e8 −0.541895
\(279\) 1.62382e8 0.447635
\(280\) 9.30141e7 0.253219
\(281\) −3.11274e8 −0.836893 −0.418447 0.908241i \(-0.637425\pi\)
−0.418447 + 0.908241i \(0.637425\pi\)
\(282\) −1.40932e8 −0.374230
\(283\) −3.45216e8 −0.905396 −0.452698 0.891664i \(-0.649538\pi\)
−0.452698 + 0.891664i \(0.649538\pi\)
\(284\) 2.24703e8 0.582097
\(285\) −1.40408e8 −0.359281
\(286\) −6.55178e7 −0.165607
\(287\) −1.00497e7 −0.0250938
\(288\) 2.38879e7 0.0589256
\(289\) 1.71501e8 0.417949
\(290\) −6.77361e8 −1.63090
\(291\) −1.04454e8 −0.248486
\(292\) −3.86638e8 −0.908791
\(293\) −7.03438e8 −1.63376 −0.816882 0.576805i \(-0.804299\pi\)
−0.816882 + 0.576805i \(0.804299\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −5.88303e8 −1.33421
\(296\) 1.12480e8 0.252088
\(297\) −7.33720e7 −0.162511
\(298\) 3.54378e8 0.775729
\(299\) 7.91809e7 0.171306
\(300\) −3.49745e8 −0.747871
\(301\) −2.72510e7 −0.0575969
\(302\) −4.54680e8 −0.949909
\(303\) 1.12606e8 0.232549
\(304\) 4.02164e7 0.0821005
\(305\) 1.96213e8 0.395984
\(306\) 1.40676e8 0.280669
\(307\) 3.64374e8 0.718726 0.359363 0.933198i \(-0.382994\pi\)
0.359363 + 0.933198i \(0.382994\pi\)
\(308\) 8.18301e7 0.159583
\(309\) −1.49781e8 −0.288804
\(310\) 9.43812e8 1.79937
\(311\) −1.04616e9 −1.97214 −0.986071 0.166325i \(-0.946810\pi\)
−0.986071 + 0.166325i \(0.946810\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 5.12995e8 0.945601 0.472800 0.881170i \(-0.343243\pi\)
0.472800 + 0.881170i \(0.343243\pi\)
\(314\) 6.23144e8 1.13589
\(315\) 1.32436e8 0.238737
\(316\) −3.02354e8 −0.539028
\(317\) 9.43433e8 1.66343 0.831714 0.555205i \(-0.187360\pi\)
0.831714 + 0.555205i \(0.187360\pi\)
\(318\) −1.26690e8 −0.220927
\(319\) −5.95915e8 −1.02782
\(320\) 1.38843e8 0.236864
\(321\) 6.41484e8 1.08248
\(322\) −9.88951e7 −0.165074
\(323\) 2.36834e8 0.391053
\(324\) 3.40122e7 0.0555556
\(325\) −4.44669e8 −0.718531
\(326\) −4.11560e8 −0.657917
\(327\) −3.70069e8 −0.585283
\(328\) −1.50013e7 −0.0234731
\(329\) 2.23795e8 0.346470
\(330\) −4.26459e8 −0.653249
\(331\) −6.52039e8 −0.988270 −0.494135 0.869385i \(-0.664515\pi\)
−0.494135 + 0.869385i \(0.664515\pi\)
\(332\) 1.45408e8 0.218075
\(333\) 1.60152e8 0.237671
\(334\) 1.92743e8 0.283052
\(335\) −1.87796e9 −2.72917
\(336\) −3.79331e7 −0.0545545
\(337\) −5.31387e8 −0.756322 −0.378161 0.925740i \(-0.623443\pi\)
−0.378161 + 0.925740i \(0.623443\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) 6.03893e7 0.0841901
\(340\) 8.17648e8 1.12821
\(341\) 8.30329e8 1.13399
\(342\) 5.72612e7 0.0774051
\(343\) 4.03536e7 0.0539949
\(344\) −4.06779e7 −0.0538770
\(345\) 5.15393e8 0.675728
\(346\) 7.02830e8 0.912187
\(347\) −4.68034e8 −0.601345 −0.300673 0.953727i \(-0.597211\pi\)
−0.300673 + 0.953727i \(0.597211\pi\)
\(348\) 2.76242e8 0.351368
\(349\) 5.87418e7 0.0739704 0.0369852 0.999316i \(-0.488225\pi\)
0.0369852 + 0.999316i \(0.488225\pi\)
\(350\) 5.55381e8 0.692394
\(351\) 4.32436e7 0.0533761
\(352\) 1.22149e8 0.149276
\(353\) 6.26384e8 0.757930 0.378965 0.925411i \(-0.376280\pi\)
0.378965 + 0.925411i \(0.376280\pi\)
\(354\) 2.39922e8 0.287448
\(355\) 1.85958e9 2.20605
\(356\) −1.77746e8 −0.208797
\(357\) −2.23388e8 −0.259849
\(358\) 5.97649e8 0.688423
\(359\) −6.26958e8 −0.715167 −0.357584 0.933881i \(-0.616399\pi\)
−0.357584 + 0.933881i \(0.616399\pi\)
\(360\) 1.97689e8 0.223318
\(361\) −7.97470e8 −0.892152
\(362\) 4.00087e8 0.443276
\(363\) 1.50971e8 0.165662
\(364\) −4.82285e7 −0.0524142
\(365\) −3.19970e9 −3.44416
\(366\) −8.00195e7 −0.0853124
\(367\) −1.88762e8 −0.199335 −0.0996676 0.995021i \(-0.531778\pi\)
−0.0996676 + 0.995021i \(0.531778\pi\)
\(368\) −1.47622e8 −0.154413
\(369\) −2.13593e7 −0.0221306
\(370\) 9.30847e8 0.955371
\(371\) 2.01180e8 0.204539
\(372\) −3.84906e8 −0.387663
\(373\) −5.38574e8 −0.537358 −0.268679 0.963230i \(-0.586587\pi\)
−0.268679 + 0.963230i \(0.586587\pi\)
\(374\) 7.19334e8 0.711017
\(375\) −1.77716e9 −1.74027
\(376\) 3.34062e8 0.324093
\(377\) 3.51217e8 0.337583
\(378\) −5.40102e7 −0.0514344
\(379\) −1.41681e9 −1.33682 −0.668411 0.743792i \(-0.733025\pi\)
−0.668411 + 0.743792i \(0.733025\pi\)
\(380\) 3.32819e8 0.311147
\(381\) 2.54203e8 0.235474
\(382\) 1.43001e9 1.31256
\(383\) −2.11756e8 −0.192593 −0.0962965 0.995353i \(-0.530700\pi\)
−0.0962965 + 0.995353i \(0.530700\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 6.77201e8 0.604791
\(386\) 2.01447e8 0.178281
\(387\) −5.79183e7 −0.0507957
\(388\) 2.47596e8 0.215195
\(389\) 1.97233e9 1.69885 0.849427 0.527706i \(-0.176948\pi\)
0.849427 + 0.527706i \(0.176948\pi\)
\(390\) 2.51344e8 0.214557
\(391\) −8.69345e8 −0.735484
\(392\) 6.02363e7 0.0505076
\(393\) 4.39167e8 0.364969
\(394\) 9.88674e8 0.814360
\(395\) −2.50219e9 −2.04282
\(396\) 1.73919e8 0.140739
\(397\) −1.50693e9 −1.20872 −0.604359 0.796712i \(-0.706571\pi\)
−0.604359 + 0.796712i \(0.706571\pi\)
\(398\) 2.32015e8 0.184470
\(399\) −9.09287e7 −0.0716632
\(400\) 8.29024e8 0.647675
\(401\) 1.47465e9 1.14205 0.571023 0.820934i \(-0.306547\pi\)
0.571023 + 0.820934i \(0.306547\pi\)
\(402\) 7.65873e8 0.587984
\(403\) −4.89374e8 −0.372455
\(404\) −2.66919e8 −0.201393
\(405\) 2.81475e8 0.210546
\(406\) −4.38662e8 −0.325303
\(407\) 8.18923e8 0.602091
\(408\) −3.33454e8 −0.243066
\(409\) 6.90014e8 0.498685 0.249342 0.968415i \(-0.419786\pi\)
0.249342 + 0.968415i \(0.419786\pi\)
\(410\) −1.24146e8 −0.0889589
\(411\) 5.65700e8 0.401920
\(412\) 3.55037e8 0.250111
\(413\) −3.80987e8 −0.266125
\(414\) −2.10188e8 −0.145582
\(415\) 1.20335e9 0.826465
\(416\) −7.19913e7 −0.0490290
\(417\) 6.55157e8 0.442455
\(418\) 2.92801e8 0.196090
\(419\) 2.48002e9 1.64704 0.823522 0.567284i \(-0.192006\pi\)
0.823522 + 0.567284i \(0.192006\pi\)
\(420\) −3.13922e8 −0.206752
\(421\) 6.39837e8 0.417909 0.208955 0.977925i \(-0.432994\pi\)
0.208955 + 0.977925i \(0.432994\pi\)
\(422\) −5.30127e8 −0.343389
\(423\) 4.75646e8 0.305558
\(424\) 3.00303e8 0.191328
\(425\) 4.88212e9 3.08495
\(426\) −7.58374e8 −0.475281
\(427\) 1.27068e8 0.0789840
\(428\) −1.52056e9 −0.937452
\(429\) 2.21123e8 0.135217
\(430\) −3.36638e8 −0.204185
\(431\) −9.03517e8 −0.543583 −0.271792 0.962356i \(-0.587616\pi\)
−0.271792 + 0.962356i \(0.587616\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 6.85293e8 0.405666 0.202833 0.979213i \(-0.434985\pi\)
0.202833 + 0.979213i \(0.434985\pi\)
\(434\) 6.11216e8 0.358906
\(435\) 2.28609e9 1.33162
\(436\) 8.77202e8 0.506870
\(437\) −3.53862e8 −0.202838
\(438\) 1.30490e9 0.742025
\(439\) 6.71521e8 0.378821 0.189410 0.981898i \(-0.439342\pi\)
0.189410 + 0.981898i \(0.439342\pi\)
\(440\) 1.01087e9 0.565730
\(441\) 8.57661e7 0.0476190
\(442\) −4.23957e8 −0.233531
\(443\) −2.23297e9 −1.22031 −0.610155 0.792282i \(-0.708893\pi\)
−0.610155 + 0.792282i \(0.708893\pi\)
\(444\) −3.79619e8 −0.205829
\(445\) −1.47097e9 −0.791307
\(446\) 2.25611e9 1.20417
\(447\) −1.19603e9 −0.633380
\(448\) 8.99154e7 0.0472456
\(449\) −3.27633e9 −1.70815 −0.854075 0.520150i \(-0.825876\pi\)
−0.854075 + 0.520150i \(0.825876\pi\)
\(450\) 1.18039e9 0.610634
\(451\) −1.09219e8 −0.0560634
\(452\) −1.43145e8 −0.0729107
\(453\) 1.53455e9 0.775597
\(454\) −9.58440e8 −0.480695
\(455\) −3.99125e8 −0.198641
\(456\) −1.35730e8 −0.0670348
\(457\) 3.24483e9 1.59032 0.795161 0.606398i \(-0.207386\pi\)
0.795161 + 0.606398i \(0.207386\pi\)
\(458\) 8.31685e7 0.0404510
\(459\) −4.74780e8 −0.229165
\(460\) −1.22167e9 −0.585198
\(461\) 2.24985e9 1.06955 0.534775 0.844994i \(-0.320396\pi\)
0.534775 + 0.844994i \(0.320396\pi\)
\(462\) −2.76177e8 −0.130299
\(463\) −1.87496e9 −0.877929 −0.438964 0.898504i \(-0.644655\pi\)
−0.438964 + 0.898504i \(0.644655\pi\)
\(464\) −6.54795e8 −0.304293
\(465\) −3.18536e9 −1.46918
\(466\) −1.41175e9 −0.646262
\(467\) −1.48428e9 −0.674385 −0.337192 0.941436i \(-0.609477\pi\)
−0.337192 + 0.941436i \(0.609477\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) −1.21618e9 −0.544367
\(470\) 2.76459e9 1.22826
\(471\) −2.10311e9 −0.927447
\(472\) −5.68704e8 −0.248937
\(473\) −2.96161e8 −0.128681
\(474\) 1.02045e9 0.440115
\(475\) 1.98724e9 0.850791
\(476\) 5.29512e8 0.225036
\(477\) 4.27580e8 0.180386
\(478\) 1.98856e9 0.832799
\(479\) −1.16314e9 −0.483568 −0.241784 0.970330i \(-0.577733\pi\)
−0.241784 + 0.970330i \(0.577733\pi\)
\(480\) −4.68596e8 −0.193399
\(481\) −4.82652e8 −0.197754
\(482\) −2.92164e8 −0.118840
\(483\) 3.33771e8 0.134782
\(484\) −3.57858e8 −0.143467
\(485\) 2.04903e9 0.815552
\(486\) −1.14791e8 −0.0453609
\(487\) 1.04706e9 0.410790 0.205395 0.978679i \(-0.434152\pi\)
0.205395 + 0.978679i \(0.434152\pi\)
\(488\) 1.89676e8 0.0738827
\(489\) 1.38901e9 0.537187
\(490\) 4.98497e8 0.191415
\(491\) 2.44817e8 0.0933374 0.0466687 0.998910i \(-0.485139\pi\)
0.0466687 + 0.998910i \(0.485139\pi\)
\(492\) 5.06293e7 0.0191657
\(493\) −3.85609e9 −1.44938
\(494\) −1.72569e8 −0.0644049
\(495\) 1.43930e9 0.533376
\(496\) 9.12370e8 0.335726
\(497\) 1.20427e9 0.440024
\(498\) −4.90752e8 −0.178057
\(499\) −1.10896e9 −0.399543 −0.199771 0.979843i \(-0.564020\pi\)
−0.199771 + 0.979843i \(0.564020\pi\)
\(500\) 4.21253e9 1.50712
\(501\) −6.50509e8 −0.231111
\(502\) 2.74963e9 0.970089
\(503\) −3.69487e9 −1.29453 −0.647264 0.762266i \(-0.724087\pi\)
−0.647264 + 0.762266i \(0.724087\pi\)
\(504\) 1.28024e8 0.0445435
\(505\) −2.20894e9 −0.763245
\(506\) −1.07478e9 −0.368802
\(507\) −1.30324e8 −0.0444116
\(508\) −6.02556e8 −0.203927
\(509\) 2.94308e8 0.0989213 0.0494606 0.998776i \(-0.484250\pi\)
0.0494606 + 0.998776i \(0.484250\pi\)
\(510\) −2.75956e9 −0.921180
\(511\) −2.07214e9 −0.686982
\(512\) 1.34218e8 0.0441942
\(513\) −1.93257e8 −0.0632010
\(514\) −2.17832e9 −0.707539
\(515\) 2.93818e9 0.947880
\(516\) 1.37288e8 0.0439904
\(517\) 2.43218e9 0.774068
\(518\) 6.02820e8 0.190561
\(519\) −2.37205e9 −0.744797
\(520\) −5.95778e8 −0.185812
\(521\) −4.14682e9 −1.28465 −0.642323 0.766434i \(-0.722029\pi\)
−0.642323 + 0.766434i \(0.722029\pi\)
\(522\) −9.32316e8 −0.286891
\(523\) −1.27950e9 −0.391096 −0.195548 0.980694i \(-0.562649\pi\)
−0.195548 + 0.980694i \(0.562649\pi\)
\(524\) −1.04099e9 −0.316072
\(525\) −1.87441e9 −0.565337
\(526\) 2.84054e9 0.851041
\(527\) 5.37295e9 1.59910
\(528\) −4.12252e8 −0.121883
\(529\) −2.10591e9 −0.618507
\(530\) 2.48522e9 0.725101
\(531\) −8.09737e8 −0.234700
\(532\) 2.15535e8 0.0620621
\(533\) 6.43707e7 0.0184138
\(534\) 5.99893e8 0.170482
\(535\) −1.25837e10 −3.55278
\(536\) −1.81540e9 −0.509209
\(537\) −2.01707e9 −0.562095
\(538\) 2.29712e9 0.635984
\(539\) 4.38558e8 0.120633
\(540\) −6.67200e8 −0.182338
\(541\) 1.56702e9 0.425486 0.212743 0.977108i \(-0.431760\pi\)
0.212743 + 0.977108i \(0.431760\pi\)
\(542\) −4.04702e9 −1.09179
\(543\) −1.35029e9 −0.361934
\(544\) 7.90408e8 0.210502
\(545\) 7.25946e9 1.92095
\(546\) 1.62771e8 0.0427960
\(547\) −1.56165e8 −0.0407971 −0.0203986 0.999792i \(-0.506494\pi\)
−0.0203986 + 0.999792i \(0.506494\pi\)
\(548\) −1.34092e9 −0.348073
\(549\) 2.70066e8 0.0696573
\(550\) 6.03582e9 1.54692
\(551\) −1.56960e9 −0.399722
\(552\) 4.98224e8 0.126077
\(553\) −1.62043e9 −0.407467
\(554\) 2.85802e9 0.714137
\(555\) −3.14161e9 −0.780057
\(556\) −1.55296e9 −0.383177
\(557\) 3.69047e9 0.904875 0.452438 0.891796i \(-0.350554\pi\)
0.452438 + 0.891796i \(0.350554\pi\)
\(558\) 1.29906e9 0.316525
\(559\) 1.74549e8 0.0422646
\(560\) 7.44113e8 0.179053
\(561\) −2.42775e9 −0.580543
\(562\) −2.49019e9 −0.591773
\(563\) −4.93440e9 −1.16535 −0.582673 0.812706i \(-0.697993\pi\)
−0.582673 + 0.812706i \(0.697993\pi\)
\(564\) −1.12746e9 −0.264621
\(565\) −1.18462e9 −0.276319
\(566\) −2.76173e9 −0.640212
\(567\) 1.82284e8 0.0419961
\(568\) 1.79763e9 0.411605
\(569\) −7.09618e9 −1.61485 −0.807424 0.589972i \(-0.799139\pi\)
−0.807424 + 0.589972i \(0.799139\pi\)
\(570\) −1.12326e9 −0.254050
\(571\) 4.29669e9 0.965845 0.482922 0.875663i \(-0.339575\pi\)
0.482922 + 0.875663i \(0.339575\pi\)
\(572\) −5.24142e8 −0.117102
\(573\) −4.82630e9 −1.07170
\(574\) −8.03975e7 −0.0177440
\(575\) −7.29454e9 −1.60015
\(576\) 1.91103e8 0.0416667
\(577\) 7.52264e9 1.63025 0.815126 0.579284i \(-0.196668\pi\)
0.815126 + 0.579284i \(0.196668\pi\)
\(578\) 1.37201e9 0.295535
\(579\) −6.79882e8 −0.145566
\(580\) −5.41889e9 −1.15322
\(581\) 7.79296e8 0.164849
\(582\) −8.35635e8 −0.175706
\(583\) 2.18640e9 0.456971
\(584\) −3.09310e9 −0.642612
\(585\) −8.48286e8 −0.175185
\(586\) −5.62750e9 −1.15525
\(587\) −6.87873e9 −1.40370 −0.701851 0.712324i \(-0.747643\pi\)
−0.701851 + 0.712324i \(0.747643\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) 2.18703e9 0.441012
\(590\) −4.70642e9 −0.943429
\(591\) −3.33677e9 −0.664922
\(592\) 8.99837e8 0.178253
\(593\) 7.09887e9 1.39797 0.698985 0.715137i \(-0.253636\pi\)
0.698985 + 0.715137i \(0.253636\pi\)
\(594\) −5.86976e8 −0.114913
\(595\) 4.38208e9 0.852847
\(596\) 2.83503e9 0.548523
\(597\) −7.83051e8 −0.150619
\(598\) 6.33447e8 0.121131
\(599\) 6.05066e9 1.15029 0.575147 0.818050i \(-0.304945\pi\)
0.575147 + 0.818050i \(0.304945\pi\)
\(600\) −2.79796e9 −0.528824
\(601\) −2.45316e9 −0.460962 −0.230481 0.973077i \(-0.574030\pi\)
−0.230481 + 0.973077i \(0.574030\pi\)
\(602\) −2.18008e8 −0.0407272
\(603\) −2.58482e9 −0.480087
\(604\) −3.63744e9 −0.671687
\(605\) −2.96153e9 −0.543716
\(606\) 9.00851e8 0.164437
\(607\) −9.44990e9 −1.71501 −0.857506 0.514475i \(-0.827987\pi\)
−0.857506 + 0.514475i \(0.827987\pi\)
\(608\) 3.21731e8 0.0580538
\(609\) 1.48048e9 0.265609
\(610\) 1.56970e9 0.280003
\(611\) −1.43346e9 −0.254239
\(612\) 1.12541e9 0.198463
\(613\) 3.84994e9 0.675060 0.337530 0.941315i \(-0.390409\pi\)
0.337530 + 0.941315i \(0.390409\pi\)
\(614\) 2.91500e9 0.508216
\(615\) 4.18993e8 0.0726347
\(616\) 6.54641e8 0.112842
\(617\) 7.81668e9 1.33975 0.669876 0.742473i \(-0.266347\pi\)
0.669876 + 0.742473i \(0.266347\pi\)
\(618\) −1.19825e9 −0.204215
\(619\) 4.99558e9 0.846582 0.423291 0.905994i \(-0.360875\pi\)
0.423291 + 0.905994i \(0.360875\pi\)
\(620\) 7.55049e9 1.27234
\(621\) 7.09385e8 0.118867
\(622\) −8.36931e9 −1.39452
\(623\) −9.52608e8 −0.157836
\(624\) 2.42971e8 0.0400320
\(625\) 1.90492e10 3.12103
\(626\) 4.10396e9 0.668641
\(627\) −9.88203e8 −0.160107
\(628\) 4.98515e9 0.803193
\(629\) 5.29914e9 0.849040
\(630\) 1.05949e9 0.168812
\(631\) 3.39241e9 0.537533 0.268767 0.963205i \(-0.413384\pi\)
0.268767 + 0.963205i \(0.413384\pi\)
\(632\) −2.41883e9 −0.381150
\(633\) 1.78918e9 0.280376
\(634\) 7.54747e9 1.17622
\(635\) −4.98657e9 −0.772848
\(636\) −1.01352e9 −0.156219
\(637\) −2.58475e8 −0.0396214
\(638\) −4.76732e9 −0.726779
\(639\) 2.55951e9 0.388065
\(640\) 1.11075e9 0.167488
\(641\) 8.25496e9 1.23798 0.618988 0.785401i \(-0.287543\pi\)
0.618988 + 0.785401i \(0.287543\pi\)
\(642\) 5.13187e9 0.765426
\(643\) −3.80038e9 −0.563753 −0.281876 0.959451i \(-0.590957\pi\)
−0.281876 + 0.959451i \(0.590957\pi\)
\(644\) −7.91161e8 −0.116725
\(645\) 1.13615e9 0.166716
\(646\) 1.89468e9 0.276516
\(647\) −6.58053e9 −0.955204 −0.477602 0.878576i \(-0.658494\pi\)
−0.477602 + 0.878576i \(0.658494\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −4.14053e9 −0.594565
\(650\) −3.55736e9 −0.508078
\(651\) −2.06286e9 −0.293046
\(652\) −3.29248e9 −0.465218
\(653\) −6.68693e9 −0.939790 −0.469895 0.882722i \(-0.655708\pi\)
−0.469895 + 0.882722i \(0.655708\pi\)
\(654\) −2.96056e9 −0.413858
\(655\) −8.61491e9 −1.19786
\(656\) −1.20010e8 −0.0165980
\(657\) −4.40404e9 −0.605861
\(658\) 1.79036e9 0.244991
\(659\) 8.02087e8 0.109175 0.0545874 0.998509i \(-0.482616\pi\)
0.0545874 + 0.998509i \(0.482616\pi\)
\(660\) −3.41167e9 −0.461917
\(661\) 7.97403e9 1.07392 0.536961 0.843607i \(-0.319572\pi\)
0.536961 + 0.843607i \(0.319572\pi\)
\(662\) −5.21631e9 −0.698812
\(663\) 1.43085e9 0.190677
\(664\) 1.16326e9 0.154202
\(665\) 1.78370e9 0.235205
\(666\) 1.28121e9 0.168059
\(667\) 5.76151e9 0.751788
\(668\) 1.54195e9 0.200148
\(669\) −7.61438e9 −0.983203
\(670\) −1.50237e10 −1.92981
\(671\) 1.38096e9 0.176463
\(672\) −3.03464e8 −0.0385758
\(673\) −1.27130e10 −1.60766 −0.803831 0.594857i \(-0.797209\pi\)
−0.803831 + 0.594857i \(0.797209\pi\)
\(674\) −4.25110e9 −0.534800
\(675\) −3.98381e9 −0.498580
\(676\) 3.08916e8 0.0384615
\(677\) −5.15085e9 −0.637997 −0.318999 0.947755i \(-0.603347\pi\)
−0.318999 + 0.947755i \(0.603347\pi\)
\(678\) 4.83114e8 0.0595314
\(679\) 1.32696e9 0.162672
\(680\) 6.54118e9 0.797765
\(681\) 3.23474e9 0.392486
\(682\) 6.64263e9 0.801853
\(683\) −1.06047e10 −1.27358 −0.636790 0.771037i \(-0.719738\pi\)
−0.636790 + 0.771037i \(0.719738\pi\)
\(684\) 4.58090e8 0.0547337
\(685\) −1.10970e10 −1.31914
\(686\) 3.22829e8 0.0381802
\(687\) −2.80694e8 −0.0330281
\(688\) −3.25423e8 −0.0380968
\(689\) −1.28860e9 −0.150090
\(690\) 4.12315e9 0.477812
\(691\) −8.18703e9 −0.943959 −0.471979 0.881610i \(-0.656460\pi\)
−0.471979 + 0.881610i \(0.656460\pi\)
\(692\) 5.62264e9 0.645013
\(693\) 9.32096e8 0.106388
\(694\) −3.74427e9 −0.425215
\(695\) −1.28519e10 −1.45218
\(696\) 2.20993e9 0.248454
\(697\) −7.06741e8 −0.0790579
\(698\) 4.69934e8 0.0523050
\(699\) 4.76467e9 0.527670
\(700\) 4.44305e9 0.489596
\(701\) −1.01784e10 −1.11601 −0.558005 0.829838i \(-0.688433\pi\)
−0.558005 + 0.829838i \(0.688433\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 2.15698e9 0.234155
\(704\) 9.77190e8 0.105554
\(705\) −9.33050e9 −1.00287
\(706\) 5.01107e9 0.535937
\(707\) −1.43052e9 −0.152239
\(708\) 1.91938e9 0.203256
\(709\) 6.47872e9 0.682696 0.341348 0.939937i \(-0.389117\pi\)
0.341348 + 0.939937i \(0.389117\pi\)
\(710\) 1.48766e10 1.55991
\(711\) −3.44400e9 −0.359352
\(712\) −1.42197e9 −0.147642
\(713\) −8.02789e9 −0.829446
\(714\) −1.78710e9 −0.183741
\(715\) −4.33764e9 −0.443795
\(716\) 4.78119e9 0.486789
\(717\) −6.71138e9 −0.679978
\(718\) −5.01566e9 −0.505700
\(719\) 2.52939e9 0.253784 0.126892 0.991917i \(-0.459500\pi\)
0.126892 + 0.991917i \(0.459500\pi\)
\(720\) 1.58151e9 0.157910
\(721\) 1.90278e9 0.189067
\(722\) −6.37976e9 −0.630847
\(723\) 9.86055e8 0.0970325
\(724\) 3.20070e9 0.313444
\(725\) −3.23558e10 −3.15333
\(726\) 1.20777e9 0.117140
\(727\) −1.55617e10 −1.50205 −0.751027 0.660272i \(-0.770441\pi\)
−0.751027 + 0.660272i \(0.770441\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −2.55976e10 −2.43539
\(731\) −1.91642e9 −0.181459
\(732\) −6.40156e8 −0.0603250
\(733\) −9.41674e9 −0.883155 −0.441577 0.897223i \(-0.645581\pi\)
−0.441577 + 0.897223i \(0.645581\pi\)
\(734\) −1.51010e9 −0.140951
\(735\) −1.68243e9 −0.156290
\(736\) −1.18097e9 −0.109186
\(737\) −1.32173e10 −1.21620
\(738\) −1.70874e8 −0.0156487
\(739\) 2.34008e8 0.0213292 0.0106646 0.999943i \(-0.496605\pi\)
0.0106646 + 0.999943i \(0.496605\pi\)
\(740\) 7.44677e9 0.675550
\(741\) 5.82421e8 0.0525864
\(742\) 1.60944e9 0.144631
\(743\) 1.99362e10 1.78312 0.891561 0.452901i \(-0.149611\pi\)
0.891561 + 0.452901i \(0.149611\pi\)
\(744\) −3.07925e9 −0.274119
\(745\) 2.34618e10 2.07881
\(746\) −4.30859e9 −0.379970
\(747\) 1.65629e9 0.145383
\(748\) 5.75467e9 0.502765
\(749\) −8.14923e9 −0.708647
\(750\) −1.42173e10 −1.23056
\(751\) 1.01576e10 0.875091 0.437545 0.899196i \(-0.355848\pi\)
0.437545 + 0.899196i \(0.355848\pi\)
\(752\) 2.67249e9 0.229168
\(753\) −9.28001e9 −0.792074
\(754\) 2.80974e9 0.238707
\(755\) −3.01024e10 −2.54558
\(756\) −4.32081e8 −0.0363696
\(757\) −4.40537e9 −0.369102 −0.184551 0.982823i \(-0.559083\pi\)
−0.184551 + 0.982823i \(0.559083\pi\)
\(758\) −1.13345e10 −0.945276
\(759\) 3.62738e9 0.301125
\(760\) 2.66255e9 0.220014
\(761\) −2.28127e9 −0.187642 −0.0938210 0.995589i \(-0.529908\pi\)
−0.0938210 + 0.995589i \(0.529908\pi\)
\(762\) 2.03363e9 0.166506
\(763\) 4.70125e9 0.383158
\(764\) 1.14401e10 0.928119
\(765\) 9.31352e9 0.752140
\(766\) −1.69405e9 −0.136184
\(767\) 2.44032e9 0.195282
\(768\) −4.52985e8 −0.0360844
\(769\) −1.11493e10 −0.884112 −0.442056 0.896987i \(-0.645751\pi\)
−0.442056 + 0.896987i \(0.645751\pi\)
\(770\) 5.41761e9 0.427652
\(771\) 7.35182e9 0.577703
\(772\) 1.61157e9 0.126063
\(773\) −2.11163e10 −1.64433 −0.822167 0.569247i \(-0.807235\pi\)
−0.822167 + 0.569247i \(0.807235\pi\)
\(774\) −4.63346e8 −0.0359180
\(775\) 4.50835e10 3.47906
\(776\) 1.98076e9 0.152166
\(777\) −2.03452e9 −0.155592
\(778\) 1.57786e10 1.20127
\(779\) −2.87675e8 −0.0218032
\(780\) 2.01075e9 0.151715
\(781\) 1.30879e10 0.983084
\(782\) −6.95476e9 −0.520066
\(783\) 3.14657e9 0.234245
\(784\) 4.81890e8 0.0357143
\(785\) 4.12556e10 3.04396
\(786\) 3.51334e9 0.258072
\(787\) −6.54202e9 −0.478410 −0.239205 0.970969i \(-0.576887\pi\)
−0.239205 + 0.970969i \(0.576887\pi\)
\(788\) 7.90939e9 0.575839
\(789\) −9.58681e9 −0.694872
\(790\) −2.00175e10 −1.44449
\(791\) −7.67167e8 −0.0551153
\(792\) 1.39135e9 0.0995173
\(793\) −8.13903e8 −0.0579584
\(794\) −1.20554e10 −0.854693
\(795\) −8.38761e9 −0.592043
\(796\) 1.85612e9 0.130440
\(797\) 9.86813e9 0.690447 0.345224 0.938520i \(-0.387803\pi\)
0.345224 + 0.938520i \(0.387803\pi\)
\(798\) −7.27430e8 −0.0506735
\(799\) 1.57383e10 1.09155
\(800\) 6.63219e9 0.457975
\(801\) −2.02464e9 −0.139198
\(802\) 1.17972e10 0.807549
\(803\) −2.25197e10 −1.53483
\(804\) 6.12698e9 0.415767
\(805\) −6.54741e9 −0.442368
\(806\) −3.91499e9 −0.263365
\(807\) −7.75279e9 −0.519279
\(808\) −2.13535e9 −0.142406
\(809\) −2.46408e10 −1.63619 −0.818097 0.575081i \(-0.804971\pi\)
−0.818097 + 0.575081i \(0.804971\pi\)
\(810\) 2.25180e9 0.148879
\(811\) 2.37028e9 0.156037 0.0780184 0.996952i \(-0.475141\pi\)
0.0780184 + 0.996952i \(0.475141\pi\)
\(812\) −3.50929e9 −0.230024
\(813\) 1.36587e10 0.891439
\(814\) 6.55138e9 0.425743
\(815\) −2.72476e10 −1.76310
\(816\) −2.66763e9 −0.171874
\(817\) −7.80066e8 −0.0500442
\(818\) 5.52011e9 0.352623
\(819\) −5.49353e8 −0.0349428
\(820\) −9.93169e8 −0.0629035
\(821\) −1.17171e10 −0.738956 −0.369478 0.929239i \(-0.620464\pi\)
−0.369478 + 0.929239i \(0.620464\pi\)
\(822\) 4.52560e9 0.284200
\(823\) 1.94300e10 1.21499 0.607496 0.794323i \(-0.292174\pi\)
0.607496 + 0.794323i \(0.292174\pi\)
\(824\) 2.84030e9 0.176856
\(825\) −2.03709e10 −1.26305
\(826\) −3.04790e9 −0.188179
\(827\) −1.04477e10 −0.642319 −0.321159 0.947025i \(-0.604073\pi\)
−0.321159 + 0.947025i \(0.604073\pi\)
\(828\) −1.68150e9 −0.102942
\(829\) 8.73946e9 0.532775 0.266387 0.963866i \(-0.414170\pi\)
0.266387 + 0.963866i \(0.414170\pi\)
\(830\) 9.62682e9 0.584399
\(831\) −9.64582e9 −0.583090
\(832\) −5.75930e8 −0.0346688
\(833\) 2.83785e9 0.170111
\(834\) 5.24126e9 0.312863
\(835\) 1.27607e10 0.758527
\(836\) 2.34241e9 0.138657
\(837\) −4.38432e9 −0.258442
\(838\) 1.98401e10 1.16464
\(839\) 9.85221e9 0.575926 0.287963 0.957642i \(-0.407022\pi\)
0.287963 + 0.957642i \(0.407022\pi\)
\(840\) −2.51138e9 −0.146196
\(841\) 8.30601e9 0.481511
\(842\) 5.11870e9 0.295506
\(843\) 8.40439e9 0.483181
\(844\) −4.24102e9 −0.242813
\(845\) 2.55649e9 0.145763
\(846\) 3.80517e9 0.216062
\(847\) −1.91790e9 −0.108451
\(848\) 2.40242e9 0.135290
\(849\) 9.32084e9 0.522731
\(850\) 3.90570e10 2.18139
\(851\) −7.91761e9 −0.440393
\(852\) −6.06699e9 −0.336074
\(853\) −2.69371e10 −1.48604 −0.743019 0.669270i \(-0.766607\pi\)
−0.743019 + 0.669270i \(0.766607\pi\)
\(854\) 1.01654e9 0.0558501
\(855\) 3.79101e9 0.207431
\(856\) −1.21644e10 −0.662879
\(857\) −2.42559e10 −1.31639 −0.658194 0.752848i \(-0.728680\pi\)
−0.658194 + 0.752848i \(0.728680\pi\)
\(858\) 1.76898e9 0.0956131
\(859\) −1.59346e10 −0.857761 −0.428880 0.903361i \(-0.641092\pi\)
−0.428880 + 0.903361i \(0.641092\pi\)
\(860\) −2.69310e9 −0.144380
\(861\) 2.71342e8 0.0144879
\(862\) −7.22814e9 −0.384371
\(863\) 1.82541e10 0.966767 0.483384 0.875409i \(-0.339408\pi\)
0.483384 + 0.875409i \(0.339408\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 4.65312e10 2.44449
\(866\) 5.48234e9 0.286849
\(867\) −4.63052e9 −0.241303
\(868\) 4.88973e9 0.253785
\(869\) −1.76107e10 −0.910345
\(870\) 1.82887e10 0.941600
\(871\) 7.78992e9 0.399456
\(872\) 7.01761e9 0.358411
\(873\) 2.82027e9 0.143463
\(874\) −2.83090e9 −0.143428
\(875\) 2.25765e10 1.13928
\(876\) 1.04392e10 0.524691
\(877\) 2.98698e10 1.49532 0.747660 0.664082i \(-0.231177\pi\)
0.747660 + 0.664082i \(0.231177\pi\)
\(878\) 5.37217e9 0.267867
\(879\) 1.89928e10 0.943254
\(880\) 8.08693e9 0.400032
\(881\) −3.44603e10 −1.69787 −0.848933 0.528501i \(-0.822754\pi\)
−0.848933 + 0.528501i \(0.822754\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 2.69143e10 1.31559 0.657794 0.753197i \(-0.271490\pi\)
0.657794 + 0.753197i \(0.271490\pi\)
\(884\) −3.39165e9 −0.165131
\(885\) 1.58842e10 0.770306
\(886\) −1.78638e10 −0.862889
\(887\) −3.92756e10 −1.88969 −0.944844 0.327522i \(-0.893787\pi\)
−0.944844 + 0.327522i \(0.893787\pi\)
\(888\) −3.03695e9 −0.145543
\(889\) −3.22932e9 −0.154154
\(890\) −1.17678e10 −0.559538
\(891\) 1.98104e9 0.0938258
\(892\) 1.80489e10 0.851479
\(893\) 6.40619e9 0.301037
\(894\) −9.56821e9 −0.447867
\(895\) 3.95677e10 1.84485
\(896\) 7.19323e8 0.0334077
\(897\) −2.13789e9 −0.0989033
\(898\) −2.62107e10 −1.20784
\(899\) −3.56087e10 −1.63455
\(900\) 9.44310e9 0.431783
\(901\) 1.41479e10 0.644398
\(902\) −8.73751e8 −0.0396428
\(903\) 7.35777e8 0.0332536
\(904\) −1.14516e9 −0.0515557
\(905\) 2.64880e10 1.18790
\(906\) 1.22764e10 0.548430
\(907\) 7.28341e9 0.324123 0.162061 0.986781i \(-0.448186\pi\)
0.162061 + 0.986781i \(0.448186\pi\)
\(908\) −7.66752e9 −0.339903
\(909\) −3.04037e9 −0.134262
\(910\) −3.19300e9 −0.140460
\(911\) 3.07735e10 1.34854 0.674269 0.738486i \(-0.264459\pi\)
0.674269 + 0.738486i \(0.264459\pi\)
\(912\) −1.08584e9 −0.0474007
\(913\) 8.46930e9 0.368298
\(914\) 2.59586e10 1.12453
\(915\) −5.29774e9 −0.228621
\(916\) 6.65348e8 0.0286032
\(917\) −5.57905e9 −0.238928
\(918\) −3.79824e9 −0.162044
\(919\) 1.31495e10 0.558863 0.279431 0.960166i \(-0.409854\pi\)
0.279431 + 0.960166i \(0.409854\pi\)
\(920\) −9.77339e9 −0.413797
\(921\) −9.83811e9 −0.414957
\(922\) 1.79988e10 0.756286
\(923\) −7.71365e9 −0.322890
\(924\) −2.20941e9 −0.0921351
\(925\) 4.44642e10 1.84720
\(926\) −1.49997e10 −0.620790
\(927\) 4.04410e9 0.166741
\(928\) −5.23836e9 −0.215168
\(929\) 5.08773e9 0.208195 0.104097 0.994567i \(-0.466805\pi\)
0.104097 + 0.994567i \(0.466805\pi\)
\(930\) −2.54829e10 −1.03886
\(931\) 1.15513e9 0.0469146
\(932\) −1.12940e10 −0.456976
\(933\) 2.82464e10 1.13862
\(934\) −1.18743e10 −0.476862
\(935\) 4.76239e10 1.90539
\(936\) −8.20026e8 −0.0326860
\(937\) −2.49041e10 −0.988968 −0.494484 0.869187i \(-0.664643\pi\)
−0.494484 + 0.869187i \(0.664643\pi\)
\(938\) −9.72942e9 −0.384926
\(939\) −1.38509e10 −0.545943
\(940\) 2.21168e10 0.868508
\(941\) −4.73818e9 −0.185374 −0.0926868 0.995695i \(-0.529546\pi\)
−0.0926868 + 0.995695i \(0.529546\pi\)
\(942\) −1.68249e10 −0.655804
\(943\) 1.05596e9 0.0410070
\(944\) −4.54963e9 −0.176025
\(945\) −3.57577e9 −0.137835
\(946\) −2.36928e9 −0.0909910
\(947\) 3.83034e10 1.46559 0.732794 0.680450i \(-0.238216\pi\)
0.732794 + 0.680450i \(0.238216\pi\)
\(948\) 8.16357e9 0.311208
\(949\) 1.32725e10 0.504107
\(950\) 1.58979e10 0.601600
\(951\) −2.54727e10 −0.960380
\(952\) 4.23609e9 0.159124
\(953\) 2.03440e10 0.761397 0.380698 0.924699i \(-0.375684\pi\)
0.380698 + 0.924699i \(0.375684\pi\)
\(954\) 3.42064e9 0.127552
\(955\) 9.46749e10 3.51741
\(956\) 1.59085e10 0.588878
\(957\) 1.60897e10 0.593412
\(958\) −9.30512e9 −0.341934
\(959\) −7.18648e9 −0.263118
\(960\) −3.74877e9 −0.136754
\(961\) 2.21034e10 0.803391
\(962\) −3.86121e9 −0.139833
\(963\) −1.73201e10 −0.624968
\(964\) −2.33732e9 −0.0840326
\(965\) 1.33369e10 0.477759
\(966\) 2.67017e9 0.0953056
\(967\) 1.63326e10 0.580848 0.290424 0.956898i \(-0.406204\pi\)
0.290424 + 0.956898i \(0.406204\pi\)
\(968\) −2.86287e9 −0.101447
\(969\) −6.39453e9 −0.225775
\(970\) 1.63922e10 0.576682
\(971\) −2.46533e10 −0.864188 −0.432094 0.901829i \(-0.642225\pi\)
−0.432094 + 0.901829i \(0.642225\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) −8.32292e9 −0.289655
\(974\) 8.37648e9 0.290473
\(975\) 1.20061e10 0.414844
\(976\) 1.51741e9 0.0522430
\(977\) 2.98530e10 1.02413 0.512067 0.858946i \(-0.328880\pi\)
0.512067 + 0.858946i \(0.328880\pi\)
\(978\) 1.11121e10 0.379849
\(979\) −1.03528e10 −0.352631
\(980\) 3.98798e9 0.135351
\(981\) 9.99187e9 0.337913
\(982\) 1.95853e9 0.0659995
\(983\) −2.65165e10 −0.890389 −0.445194 0.895434i \(-0.646865\pi\)
−0.445194 + 0.895434i \(0.646865\pi\)
\(984\) 4.05035e8 0.0135522
\(985\) 6.54557e10 2.18233
\(986\) −3.08487e10 −1.02487
\(987\) −6.04247e9 −0.200034
\(988\) −1.38055e9 −0.0455411
\(989\) 2.86338e9 0.0941221
\(990\) 1.15144e10 0.377153
\(991\) −3.95353e10 −1.29041 −0.645205 0.764010i \(-0.723228\pi\)
−0.645205 + 0.764010i \(0.723228\pi\)
\(992\) 7.29896e9 0.237394
\(993\) 1.76051e10 0.570578
\(994\) 9.63416e9 0.311144
\(995\) 1.53607e10 0.494345
\(996\) −3.92602e9 −0.125905
\(997\) 2.19110e10 0.700211 0.350105 0.936710i \(-0.386146\pi\)
0.350105 + 0.936710i \(0.386146\pi\)
\(998\) −8.87167e9 −0.282520
\(999\) −4.32409e9 −0.137219
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 546.8.a.q.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.q.1.6 6 1.1 even 1 trivial