Properties

Label 546.8.a.r.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: \(x^{6} - x^{5} - 264981 x^{4} + 17519669 x^{3} + 15113237808 x^{2} - 1787613752904 x - 21984668630064\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-462.924\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +496.924 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +496.924 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +3975.39 q^{10} -3824.74 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -13417.0 q^{15} +4096.00 q^{16} -2147.73 q^{17} +5832.00 q^{18} -18544.8 q^{19} +31803.1 q^{20} +9261.00 q^{21} -30597.9 q^{22} -67563.5 q^{23} -13824.0 q^{24} +168809. q^{25} +17576.0 q^{26} -19683.0 q^{27} -21952.0 q^{28} +58711.4 q^{29} -107336. q^{30} +157539. q^{31} +32768.0 q^{32} +103268. q^{33} -17181.9 q^{34} -170445. q^{35} +46656.0 q^{36} +381564. q^{37} -148359. q^{38} -59319.0 q^{39} +254425. q^{40} +227136. q^{41} +74088.0 q^{42} +532865. q^{43} -244783. q^{44} +362258. q^{45} -540508. q^{46} +714364. q^{47} -110592. q^{48} +117649. q^{49} +1.35047e6 q^{50} +57988.8 q^{51} +140608. q^{52} -352284. q^{53} -157464. q^{54} -1.90061e6 q^{55} -175616. q^{56} +500710. q^{57} +469691. q^{58} -534961. q^{59} -858685. q^{60} +947554. q^{61} +1.26032e6 q^{62} -250047. q^{63} +262144. q^{64} +1.09174e6 q^{65} +826144. q^{66} +2.20515e6 q^{67} -137455. q^{68} +1.82421e6 q^{69} -1.36356e6 q^{70} -5.03775e6 q^{71} +373248. q^{72} +1.46454e6 q^{73} +3.05251e6 q^{74} -4.55783e6 q^{75} -1.18687e6 q^{76} +1.31189e6 q^{77} -474552. q^{78} +8.61725e6 q^{79} +2.03540e6 q^{80} +531441. q^{81} +1.81709e6 q^{82} -8.80720e6 q^{83} +592704. q^{84} -1.06726e6 q^{85} +4.26292e6 q^{86} -1.58521e6 q^{87} -1.95827e6 q^{88} +3.11870e6 q^{89} +2.89806e6 q^{90} -753571. q^{91} -4.32406e6 q^{92} -4.25357e6 q^{93} +5.71491e6 q^{94} -9.21537e6 q^{95} -884736. q^{96} +2.00663e6 q^{97} +941192. q^{98} -2.78824e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 1624 q^{10} - 2690 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} - 5481 q^{15} + 24576 q^{16} + 2910 q^{17} + 34992 q^{18} - 13055 q^{19} + 12992 q^{20} + 55566 q^{21} - 21520 q^{22} + 11581 q^{23} - 82944 q^{24} + 68081 q^{25} + 105456 q^{26} - 118098 q^{27} - 131712 q^{28} - 92335 q^{29} - 43848 q^{30} - 83081 q^{31} + 196608 q^{32} + 72630 q^{33} + 23280 q^{34} - 69629 q^{35} + 279936 q^{36} - 265114 q^{37} - 104440 q^{38} - 355914 q^{39} + 103936 q^{40} - 367468 q^{41} + 444528 q^{42} + 454955 q^{43} - 172160 q^{44} + 147987 q^{45} + 92648 q^{46} + 733973 q^{47} - 663552 q^{48} + 705894 q^{49} + 544648 q^{50} - 78570 q^{51} + 843648 q^{52} - 1577379 q^{53} - 944784 q^{54} + 2231118 q^{55} - 1053696 q^{56} + 352485 q^{57} - 738680 q^{58} + 2062708 q^{59} - 350784 q^{60} - 271270 q^{61} - 664648 q^{62} - 1500282 q^{63} + 1572864 q^{64} + 445991 q^{65} + 581040 q^{66} - 758674 q^{67} + 186240 q^{68} - 312687 q^{69} - 557032 q^{70} - 6138216 q^{71} + 2239488 q^{72} + 6361979 q^{73} - 2120912 q^{74} - 1838187 q^{75} - 835520 q^{76} + 922670 q^{77} - 2847312 q^{78} - 899781 q^{79} + 831488 q^{80} + 3188646 q^{81} - 2939744 q^{82} + 3313561 q^{83} + 3556224 q^{84} + 5307940 q^{85} + 3639640 q^{86} + 2493045 q^{87} - 1377280 q^{88} + 11210703 q^{89} + 1183896 q^{90} - 4521426 q^{91} + 741184 q^{92} + 2243187 q^{93} + 5871784 q^{94} + 12912395 q^{95} - 5308416 q^{96} + 28682643 q^{97} + 5647152 q^{98} - 1961010 q^{99} + O(q^{100}) \)

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
\(5\) 496.924 1.77785 0.888925 0.458053i \(-0.151453\pi\)
0.888925 + 0.458053i \(0.151453\pi\)
\(6\) −216.000 −0.408248
\(7\) −343.000 −0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 3975.39 1.25713
\(11\) −3824.74 −0.866418 −0.433209 0.901293i \(-0.642619\pi\)
−0.433209 + 0.901293i \(0.642619\pi\)
\(12\) −1728.00 −0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) −13417.0 −1.02644
\(16\) 4096.00 0.250000
\(17\) −2147.73 −0.106025 −0.0530126 0.998594i \(-0.516882\pi\)
−0.0530126 + 0.998594i \(0.516882\pi\)
\(18\) 5832.00 0.235702
\(19\) −18544.8 −0.620276 −0.310138 0.950691i \(-0.600375\pi\)
−0.310138 + 0.950691i \(0.600375\pi\)
\(20\) 31803.1 0.888925
\(21\) 9261.00 0.218218
\(22\) −30597.9 −0.612650
\(23\) −67563.5 −1.15788 −0.578941 0.815369i \(-0.696534\pi\)
−0.578941 + 0.815369i \(0.696534\pi\)
\(24\) −13824.0 −0.204124
\(25\) 168809. 2.16075
\(26\) 17576.0 0.196116
\(27\) −19683.0 −0.192450
\(28\) −21952.0 −0.188982
\(29\) 58711.4 0.447022 0.223511 0.974701i \(-0.428248\pi\)
0.223511 + 0.974701i \(0.428248\pi\)
\(30\) −107336. −0.725804
\(31\) 157539. 0.949781 0.474890 0.880045i \(-0.342488\pi\)
0.474890 + 0.880045i \(0.342488\pi\)
\(32\) 32768.0 0.176777
\(33\) 103268. 0.500227
\(34\) −17181.9 −0.0749712
\(35\) −170445. −0.671964
\(36\) 46656.0 0.166667
\(37\) 381564. 1.23840 0.619200 0.785233i \(-0.287457\pi\)
0.619200 + 0.785233i \(0.287457\pi\)
\(38\) −148359. −0.438602
\(39\) −59319.0 −0.160128
\(40\) 254425. 0.628565
\(41\) 227136. 0.514686 0.257343 0.966320i \(-0.417153\pi\)
0.257343 + 0.966320i \(0.417153\pi\)
\(42\) 74088.0 0.154303
\(43\) 532865. 1.02206 0.511031 0.859562i \(-0.329264\pi\)
0.511031 + 0.859562i \(0.329264\pi\)
\(44\) −244783. −0.433209
\(45\) 362258. 0.592617
\(46\) −540508. −0.818747
\(47\) 714364. 1.00364 0.501819 0.864973i \(-0.332664\pi\)
0.501819 + 0.864973i \(0.332664\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) 1.35047e6 1.52788
\(51\) 57988.8 0.0612137
\(52\) 140608. 0.138675
\(53\) −352284. −0.325033 −0.162516 0.986706i \(-0.551961\pi\)
−0.162516 + 0.986706i \(0.551961\pi\)
\(54\) −157464. −0.136083
\(55\) −1.90061e6 −1.54036
\(56\) −175616. −0.133631
\(57\) 500710. 0.358117
\(58\) 469691. 0.316093
\(59\) −534961. −0.339109 −0.169555 0.985521i \(-0.554233\pi\)
−0.169555 + 0.985521i \(0.554233\pi\)
\(60\) −858685. −0.513221
\(61\) 947554. 0.534502 0.267251 0.963627i \(-0.413885\pi\)
0.267251 + 0.963627i \(0.413885\pi\)
\(62\) 1.26032e6 0.671596
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) 1.09174e6 0.493087
\(66\) 826144. 0.353714
\(67\) 2.20515e6 0.895728 0.447864 0.894102i \(-0.352185\pi\)
0.447864 + 0.894102i \(0.352185\pi\)
\(68\) −137455. −0.0530126
\(69\) 1.82421e6 0.668504
\(70\) −1.36356e6 −0.475150
\(71\) −5.03775e6 −1.67044 −0.835222 0.549912i \(-0.814661\pi\)
−0.835222 + 0.549912i \(0.814661\pi\)
\(72\) 373248. 0.117851
\(73\) 1.46454e6 0.440628 0.220314 0.975429i \(-0.429292\pi\)
0.220314 + 0.975429i \(0.429292\pi\)
\(74\) 3.05251e6 0.875681
\(75\) −4.55783e6 −1.24751
\(76\) −1.18687e6 −0.310138
\(77\) 1.31189e6 0.327475
\(78\) −474552. −0.113228
\(79\) 8.61725e6 1.96641 0.983204 0.182509i \(-0.0584217\pi\)
0.983204 + 0.182509i \(0.0584217\pi\)
\(80\) 2.03540e6 0.444462
\(81\) 531441. 0.111111
\(82\) 1.81709e6 0.363938
\(83\) −8.80720e6 −1.69069 −0.845346 0.534219i \(-0.820606\pi\)
−0.845346 + 0.534219i \(0.820606\pi\)
\(84\) 592704. 0.109109
\(85\) −1.06726e6 −0.188497
\(86\) 4.26292e6 0.722707
\(87\) −1.58521e6 −0.258089
\(88\) −1.95827e6 −0.306325
\(89\) 3.11870e6 0.468930 0.234465 0.972125i \(-0.424666\pi\)
0.234465 + 0.972125i \(0.424666\pi\)
\(90\) 2.89806e6 0.419043
\(91\) −753571. −0.104828
\(92\) −4.32406e6 −0.578941
\(93\) −4.25357e6 −0.548356
\(94\) 5.71491e6 0.709679
\(95\) −9.21537e6 −1.10276
\(96\) −884736. −0.102062
\(97\) 2.00663e6 0.223238 0.111619 0.993751i \(-0.464396\pi\)
0.111619 + 0.993751i \(0.464396\pi\)
\(98\) 941192. 0.101015
\(99\) −2.78824e6 −0.288806
\(100\) 1.08037e7 1.08037
\(101\) 1.91316e7 1.84767 0.923837 0.382785i \(-0.125035\pi\)
0.923837 + 0.382785i \(0.125035\pi\)
\(102\) 463910. 0.0432846
\(103\) 1.37161e6 0.123680 0.0618400 0.998086i \(-0.480303\pi\)
0.0618400 + 0.998086i \(0.480303\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) 4.60201e6 0.387959
\(106\) −2.81827e6 −0.229833
\(107\) 2.35974e7 1.86218 0.931088 0.364795i \(-0.118861\pi\)
0.931088 + 0.364795i \(0.118861\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 8.39526e6 0.620928 0.310464 0.950585i \(-0.399516\pi\)
0.310464 + 0.950585i \(0.399516\pi\)
\(110\) −1.52048e7 −1.08920
\(111\) −1.03022e7 −0.714991
\(112\) −1.40493e6 −0.0944911
\(113\) −7.06983e6 −0.460930 −0.230465 0.973081i \(-0.574025\pi\)
−0.230465 + 0.973081i \(0.574025\pi\)
\(114\) 4.00568e6 0.253227
\(115\) −3.35739e7 −2.05854
\(116\) 3.75753e6 0.223511
\(117\) 1.60161e6 0.0924500
\(118\) −4.27969e6 −0.239787
\(119\) 736673. 0.0400738
\(120\) −6.86948e6 −0.362902
\(121\) −4.85853e6 −0.249319
\(122\) 7.58043e6 0.377950
\(123\) −6.13267e6 −0.297154
\(124\) 1.00825e7 0.474890
\(125\) 4.50629e7 2.06364
\(126\) −2.00038e6 −0.0890871
\(127\) −985521. −0.0426926 −0.0213463 0.999772i \(-0.506795\pi\)
−0.0213463 + 0.999772i \(0.506795\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −1.43873e7 −0.590088
\(130\) 8.73394e6 0.348665
\(131\) −3.54561e7 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(132\) 6.60915e6 0.250113
\(133\) 6.36088e6 0.234442
\(134\) 1.76412e7 0.633375
\(135\) −9.78096e6 −0.342147
\(136\) −1.09964e6 −0.0374856
\(137\) 2.66368e7 0.885034 0.442517 0.896760i \(-0.354086\pi\)
0.442517 + 0.896760i \(0.354086\pi\)
\(138\) 1.45937e7 0.472704
\(139\) −5.67858e7 −1.79345 −0.896723 0.442592i \(-0.854059\pi\)
−0.896723 + 0.442592i \(0.854059\pi\)
\(140\) −1.09085e7 −0.335982
\(141\) −1.92878e7 −0.579451
\(142\) −4.03020e7 −1.18118
\(143\) −8.40296e6 −0.240301
\(144\) 2.98598e6 0.0833333
\(145\) 2.91751e7 0.794739
\(146\) 1.17163e7 0.311571
\(147\) −3.17652e6 −0.0824786
\(148\) 2.44201e7 0.619200
\(149\) −7.19892e7 −1.78285 −0.891427 0.453165i \(-0.850295\pi\)
−0.891427 + 0.453165i \(0.850295\pi\)
\(150\) −3.64627e7 −0.882122
\(151\) −3.22126e7 −0.761388 −0.380694 0.924701i \(-0.624315\pi\)
−0.380694 + 0.924701i \(0.624315\pi\)
\(152\) −9.49495e6 −0.219301
\(153\) −1.56570e6 −0.0353417
\(154\) 1.04951e7 0.231560
\(155\) 7.82852e7 1.68857
\(156\) −3.79642e6 −0.0800641
\(157\) −8.76407e7 −1.80741 −0.903706 0.428153i \(-0.859164\pi\)
−0.903706 + 0.428153i \(0.859164\pi\)
\(158\) 6.89380e7 1.39046
\(159\) 9.51167e6 0.187658
\(160\) 1.62832e7 0.314282
\(161\) 2.31743e7 0.437639
\(162\) 4.25153e6 0.0785674
\(163\) 7.24820e7 1.31091 0.655456 0.755233i \(-0.272476\pi\)
0.655456 + 0.755233i \(0.272476\pi\)
\(164\) 1.45367e7 0.257343
\(165\) 5.13164e7 0.889328
\(166\) −7.04576e7 −1.19550
\(167\) 7.83523e7 1.30180 0.650900 0.759163i \(-0.274392\pi\)
0.650900 + 0.759163i \(0.274392\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −8.53808e6 −0.133287
\(171\) −1.35192e7 −0.206759
\(172\) 3.41033e7 0.511031
\(173\) 2.63684e7 0.387189 0.193595 0.981082i \(-0.437985\pi\)
0.193595 + 0.981082i \(0.437985\pi\)
\(174\) −1.26817e7 −0.182496
\(175\) −5.79013e7 −0.816687
\(176\) −1.56661e7 −0.216605
\(177\) 1.44439e7 0.195785
\(178\) 2.49496e7 0.331584
\(179\) 1.43940e8 1.87584 0.937920 0.346852i \(-0.112749\pi\)
0.937920 + 0.346852i \(0.112749\pi\)
\(180\) 2.31845e7 0.296308
\(181\) 7.83164e7 0.981698 0.490849 0.871245i \(-0.336687\pi\)
0.490849 + 0.871245i \(0.336687\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) −2.55840e7 −0.308595
\(184\) −3.45925e7 −0.409373
\(185\) 1.89608e8 2.20169
\(186\) −3.40285e7 −0.387746
\(187\) 8.21452e6 0.0918622
\(188\) 4.57193e7 0.501819
\(189\) 6.75127e6 0.0727393
\(190\) −7.37230e7 −0.779768
\(191\) 6.34840e7 0.659246 0.329623 0.944113i \(-0.393078\pi\)
0.329623 + 0.944113i \(0.393078\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 9.55888e7 0.957098 0.478549 0.878061i \(-0.341163\pi\)
0.478549 + 0.878061i \(0.341163\pi\)
\(194\) 1.60531e7 0.157853
\(195\) −2.94770e7 −0.284684
\(196\) 7.52954e6 0.0714286
\(197\) 8.28275e7 0.771867 0.385934 0.922527i \(-0.373879\pi\)
0.385934 + 0.922527i \(0.373879\pi\)
\(198\) −2.23059e7 −0.204217
\(199\) −8.13880e7 −0.732107 −0.366053 0.930594i \(-0.619291\pi\)
−0.366053 + 0.930594i \(0.619291\pi\)
\(200\) 8.64300e7 0.763940
\(201\) −5.95390e7 −0.517149
\(202\) 1.53052e8 1.30650
\(203\) −2.01380e7 −0.168959
\(204\) 3.71128e6 0.0306068
\(205\) 1.12869e8 0.915035
\(206\) 1.09729e7 0.0874550
\(207\) −4.92538e7 −0.385961
\(208\) 8.99891e6 0.0693375
\(209\) 7.09292e7 0.537419
\(210\) 3.68161e7 0.274328
\(211\) 1.89709e8 1.39027 0.695134 0.718880i \(-0.255345\pi\)
0.695134 + 0.718880i \(0.255345\pi\)
\(212\) −2.25462e7 −0.162516
\(213\) 1.36019e8 0.964432
\(214\) 1.88779e8 1.31676
\(215\) 2.64793e8 1.81707
\(216\) −1.00777e7 −0.0680414
\(217\) −5.40360e7 −0.358983
\(218\) 6.71621e7 0.439063
\(219\) −3.95426e7 −0.254396
\(220\) −1.21639e8 −0.770181
\(221\) −4.71857e6 −0.0294061
\(222\) −8.24178e7 −0.505575
\(223\) 4.32655e7 0.261261 0.130630 0.991431i \(-0.458300\pi\)
0.130630 + 0.991431i \(0.458300\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) 1.23061e8 0.720250
\(226\) −5.65587e7 −0.325926
\(227\) 1.76524e7 0.100164 0.0500821 0.998745i \(-0.484052\pi\)
0.0500821 + 0.998745i \(0.484052\pi\)
\(228\) 3.20455e7 0.179058
\(229\) 2.28130e8 1.25533 0.627664 0.778484i \(-0.284011\pi\)
0.627664 + 0.778484i \(0.284011\pi\)
\(230\) −2.68591e8 −1.45561
\(231\) −3.54209e7 −0.189068
\(232\) 3.00602e7 0.158046
\(233\) −9.21660e7 −0.477337 −0.238668 0.971101i \(-0.576711\pi\)
−0.238668 + 0.971101i \(0.576711\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) 3.54985e8 1.78432
\(236\) −3.42375e7 −0.169555
\(237\) −2.32666e8 −1.13531
\(238\) 5.89338e6 0.0283364
\(239\) 1.66857e8 0.790591 0.395296 0.918554i \(-0.370642\pi\)
0.395296 + 0.918554i \(0.370642\pi\)
\(240\) −5.49558e7 −0.256611
\(241\) 2.35779e8 1.08504 0.542519 0.840043i \(-0.317471\pi\)
0.542519 + 0.840043i \(0.317471\pi\)
\(242\) −3.88682e7 −0.176295
\(243\) −1.43489e7 −0.0641500
\(244\) 6.06435e7 0.267251
\(245\) 5.84626e7 0.253979
\(246\) −4.90614e7 −0.210120
\(247\) −4.07430e7 −0.172034
\(248\) 8.06602e7 0.335798
\(249\) 2.37794e8 0.976121
\(250\) 3.60503e8 1.45921
\(251\) −1.87788e8 −0.749564 −0.374782 0.927113i \(-0.622282\pi\)
−0.374782 + 0.927113i \(0.622282\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) 2.58413e8 1.00321
\(254\) −7.88417e6 −0.0301883
\(255\) 2.88160e7 0.108829
\(256\) 1.67772e7 0.0625000
\(257\) 2.09400e8 0.769502 0.384751 0.923020i \(-0.374287\pi\)
0.384751 + 0.923020i \(0.374287\pi\)
\(258\) −1.15099e8 −0.417255
\(259\) −1.30876e8 −0.468071
\(260\) 6.98715e7 0.246543
\(261\) 4.28006e7 0.149007
\(262\) −2.83649e8 −0.974376
\(263\) −3.30860e8 −1.12150 −0.560749 0.827986i \(-0.689487\pi\)
−0.560749 + 0.827986i \(0.689487\pi\)
\(264\) 5.28732e7 0.176857
\(265\) −1.75058e8 −0.577860
\(266\) 5.08870e7 0.165776
\(267\) −8.42048e7 −0.270737
\(268\) 1.41130e8 0.447864
\(269\) 3.78010e8 1.18405 0.592025 0.805920i \(-0.298329\pi\)
0.592025 + 0.805920i \(0.298329\pi\)
\(270\) −7.82477e7 −0.241935
\(271\) 6.09647e6 0.0186074 0.00930371 0.999957i \(-0.497038\pi\)
0.00930371 + 0.999957i \(0.497038\pi\)
\(272\) −8.79712e6 −0.0265063
\(273\) 2.03464e7 0.0605228
\(274\) 2.13094e8 0.625813
\(275\) −6.45649e8 −1.87211
\(276\) 1.16750e8 0.334252
\(277\) −3.85685e8 −1.09032 −0.545159 0.838333i \(-0.683531\pi\)
−0.545159 + 0.838333i \(0.683531\pi\)
\(278\) −4.54287e8 −1.26816
\(279\) 1.14846e8 0.316594
\(280\) −8.72678e7 −0.237575
\(281\) −6.82120e7 −0.183396 −0.0916978 0.995787i \(-0.529229\pi\)
−0.0916978 + 0.995787i \(0.529229\pi\)
\(282\) −1.54303e8 −0.409733
\(283\) −6.19700e7 −0.162528 −0.0812642 0.996693i \(-0.525896\pi\)
−0.0812642 + 0.996693i \(0.525896\pi\)
\(284\) −3.22416e8 −0.835222
\(285\) 2.48815e8 0.636678
\(286\) −6.72236e7 −0.169919
\(287\) −7.79076e7 −0.194533
\(288\) 2.38879e7 0.0589256
\(289\) −4.05726e8 −0.988759
\(290\) 2.33401e8 0.561965
\(291\) −5.41791e7 −0.128886
\(292\) 9.37307e7 0.220314
\(293\) −4.88598e6 −0.0113479 −0.00567395 0.999984i \(-0.501806\pi\)
−0.00567395 + 0.999984i \(0.501806\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −2.65835e8 −0.602886
\(296\) 1.95361e8 0.437841
\(297\) 7.52824e7 0.166742
\(298\) −5.75914e8 −1.26067
\(299\) −1.48437e8 −0.321139
\(300\) −2.91701e8 −0.623755
\(301\) −1.82773e8 −0.386303
\(302\) −2.57701e8 −0.538383
\(303\) −5.16552e8 −1.06676
\(304\) −7.59596e7 −0.155069
\(305\) 4.70862e8 0.950265
\(306\) −1.25256e7 −0.0249904
\(307\) −2.64128e8 −0.520992 −0.260496 0.965475i \(-0.583886\pi\)
−0.260496 + 0.965475i \(0.583886\pi\)
\(308\) 8.39607e7 0.163738
\(309\) −3.70334e7 −0.0714067
\(310\) 6.26281e8 1.19400
\(311\) 5.56532e8 1.04913 0.524564 0.851371i \(-0.324228\pi\)
0.524564 + 0.851371i \(0.324228\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) −4.42785e8 −0.816183 −0.408091 0.912941i \(-0.633806\pi\)
−0.408091 + 0.912941i \(0.633806\pi\)
\(314\) −7.01126e8 −1.27803
\(315\) −1.24254e8 −0.223988
\(316\) 5.51504e8 0.983204
\(317\) 2.33443e8 0.411599 0.205799 0.978594i \(-0.434021\pi\)
0.205799 + 0.978594i \(0.434021\pi\)
\(318\) 7.60933e7 0.132694
\(319\) −2.24556e8 −0.387308
\(320\) 1.30266e8 0.222231
\(321\) −6.37130e8 −1.07513
\(322\) 1.85394e8 0.309457
\(323\) 3.98294e7 0.0657649
\(324\) 3.40122e7 0.0555556
\(325\) 3.70872e8 0.599284
\(326\) 5.79856e8 0.926955
\(327\) −2.26672e8 −0.358493
\(328\) 1.16294e8 0.181969
\(329\) −2.45027e8 −0.379339
\(330\) 4.10531e8 0.628850
\(331\) 1.18852e9 1.80139 0.900693 0.434455i \(-0.143059\pi\)
0.900693 + 0.434455i \(0.143059\pi\)
\(332\) −5.63661e8 −0.845346
\(333\) 2.78160e8 0.412800
\(334\) 6.26819e8 0.920512
\(335\) 1.09579e9 1.59247
\(336\) 3.79331e7 0.0545545
\(337\) 3.54795e8 0.504978 0.252489 0.967600i \(-0.418751\pi\)
0.252489 + 0.967600i \(0.418751\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) 1.90885e8 0.266118
\(340\) −6.83047e7 −0.0942485
\(341\) −6.02548e8 −0.822908
\(342\) −1.08153e8 −0.146201
\(343\) −4.03536e7 −0.0539949
\(344\) 2.72827e8 0.361354
\(345\) 9.06496e8 1.18850
\(346\) 2.10948e8 0.273784
\(347\) −4.52712e8 −0.581659 −0.290830 0.956775i \(-0.593931\pi\)
−0.290830 + 0.956775i \(0.593931\pi\)
\(348\) −1.01453e8 −0.129044
\(349\) 3.15340e8 0.397091 0.198545 0.980092i \(-0.436378\pi\)
0.198545 + 0.980092i \(0.436378\pi\)
\(350\) −4.63211e8 −0.577485
\(351\) −4.32436e7 −0.0533761
\(352\) −1.25329e8 −0.153163
\(353\) 7.96217e8 0.963429 0.481715 0.876328i \(-0.340014\pi\)
0.481715 + 0.876328i \(0.340014\pi\)
\(354\) 1.15552e8 0.138441
\(355\) −2.50338e9 −2.96980
\(356\) 1.99597e8 0.234465
\(357\) −1.98902e7 −0.0231366
\(358\) 1.15152e9 1.32642
\(359\) −1.65187e9 −1.88428 −0.942142 0.335214i \(-0.891191\pi\)
−0.942142 + 0.335214i \(0.891191\pi\)
\(360\) 1.85476e8 0.209522
\(361\) −5.49961e8 −0.615257
\(362\) 6.26531e8 0.694165
\(363\) 1.31180e8 0.143945
\(364\) −4.82285e7 −0.0524142
\(365\) 7.27766e8 0.783370
\(366\) −2.04672e8 −0.218210
\(367\) 3.40603e8 0.359681 0.179840 0.983696i \(-0.442442\pi\)
0.179840 + 0.983696i \(0.442442\pi\)
\(368\) −2.76740e8 −0.289471
\(369\) 1.65582e8 0.171562
\(370\) 1.51687e9 1.55683
\(371\) 1.20833e8 0.122851
\(372\) −2.72228e8 −0.274178
\(373\) −5.88708e8 −0.587380 −0.293690 0.955901i \(-0.594883\pi\)
−0.293690 + 0.955901i \(0.594883\pi\)
\(374\) 6.57162e7 0.0649564
\(375\) −1.21670e9 −1.19144
\(376\) 3.65754e8 0.354840
\(377\) 1.28989e8 0.123982
\(378\) 5.40102e7 0.0514344
\(379\) 8.60219e6 0.00811655 0.00405828 0.999992i \(-0.498708\pi\)
0.00405828 + 0.999992i \(0.498708\pi\)
\(380\) −5.89784e8 −0.551379
\(381\) 2.66091e7 0.0246486
\(382\) 5.07872e8 0.466157
\(383\) −1.84752e9 −1.68032 −0.840161 0.542338i \(-0.817539\pi\)
−0.840161 + 0.542338i \(0.817539\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 6.51908e8 0.582202
\(386\) 7.64710e8 0.676770
\(387\) 3.88458e8 0.340687
\(388\) 1.28425e8 0.111619
\(389\) 1.35859e8 0.117021 0.0585105 0.998287i \(-0.481365\pi\)
0.0585105 + 0.998287i \(0.481365\pi\)
\(390\) −2.35816e8 −0.201302
\(391\) 1.45108e8 0.122765
\(392\) 6.02363e7 0.0505076
\(393\) 9.57315e8 0.795574
\(394\) 6.62620e8 0.545793
\(395\) 4.28212e9 3.49598
\(396\) −1.78447e8 −0.144403
\(397\) 6.42588e8 0.515425 0.257713 0.966222i \(-0.417031\pi\)
0.257713 + 0.966222i \(0.417031\pi\)
\(398\) −6.51104e8 −0.517678
\(399\) −1.71744e8 −0.135355
\(400\) 6.91440e8 0.540187
\(401\) −2.24032e9 −1.73502 −0.867511 0.497418i \(-0.834281\pi\)
−0.867511 + 0.497418i \(0.834281\pi\)
\(402\) −4.76312e8 −0.365679
\(403\) 3.46114e8 0.263422
\(404\) 1.22442e9 0.923837
\(405\) 2.64086e8 0.197539
\(406\) −1.61104e8 −0.119472
\(407\) −1.45938e9 −1.07297
\(408\) 2.96903e7 0.0216423
\(409\) 7.64379e8 0.552430 0.276215 0.961096i \(-0.410920\pi\)
0.276215 + 0.961096i \(0.410920\pi\)
\(410\) 9.02955e8 0.647027
\(411\) −7.19193e8 −0.510975
\(412\) 8.77829e7 0.0618400
\(413\) 1.83492e8 0.128171
\(414\) −3.94030e8 −0.272916
\(415\) −4.37651e9 −3.00580
\(416\) 7.19913e7 0.0490290
\(417\) 1.53322e9 1.03545
\(418\) 5.67433e8 0.380012
\(419\) −5.63647e8 −0.374333 −0.187166 0.982328i \(-0.559930\pi\)
−0.187166 + 0.982328i \(0.559930\pi\)
\(420\) 2.94529e8 0.193979
\(421\) −1.67916e8 −0.109674 −0.0548371 0.998495i \(-0.517464\pi\)
−0.0548371 + 0.998495i \(0.517464\pi\)
\(422\) 1.51767e9 0.983068
\(423\) 5.20771e8 0.334546
\(424\) −1.80369e8 −0.114917
\(425\) −3.62556e8 −0.229094
\(426\) 1.08815e9 0.681956
\(427\) −3.25011e8 −0.202023
\(428\) 1.51023e9 0.931088
\(429\) 2.26880e8 0.138738
\(430\) 2.11835e9 1.28486
\(431\) 4.07362e8 0.245081 0.122541 0.992463i \(-0.460896\pi\)
0.122541 + 0.992463i \(0.460896\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) −1.43524e9 −0.849604 −0.424802 0.905286i \(-0.639656\pi\)
−0.424802 + 0.905286i \(0.639656\pi\)
\(434\) −4.32288e8 −0.253840
\(435\) −7.87728e8 −0.458843
\(436\) 5.37297e8 0.310464
\(437\) 1.25295e9 0.718207
\(438\) −3.16341e8 −0.179885
\(439\) 8.80200e8 0.496541 0.248271 0.968691i \(-0.420138\pi\)
0.248271 + 0.968691i \(0.420138\pi\)
\(440\) −9.73110e8 −0.544600
\(441\) 8.57661e7 0.0476190
\(442\) −3.77486e7 −0.0207933
\(443\) 1.78113e9 0.973382 0.486691 0.873574i \(-0.338204\pi\)
0.486691 + 0.873574i \(0.338204\pi\)
\(444\) −6.59342e8 −0.357495
\(445\) 1.54976e9 0.833687
\(446\) 3.46124e8 0.184739
\(447\) 1.94371e9 1.02933
\(448\) −8.99154e7 −0.0472456
\(449\) −1.92118e7 −0.0100162 −0.00500812 0.999987i \(-0.501594\pi\)
−0.00500812 + 0.999987i \(0.501594\pi\)
\(450\) 9.84492e8 0.509294
\(451\) −8.68736e8 −0.445934
\(452\) −4.52469e8 −0.230465
\(453\) 8.69740e8 0.439588
\(454\) 1.41219e8 0.0708268
\(455\) −3.74468e8 −0.186369
\(456\) 2.56364e8 0.126613
\(457\) −1.53351e9 −0.751591 −0.375795 0.926703i \(-0.622630\pi\)
−0.375795 + 0.926703i \(0.622630\pi\)
\(458\) 1.82504e9 0.887651
\(459\) 4.22738e7 0.0204046
\(460\) −2.14873e9 −1.02927
\(461\) −1.15224e9 −0.547757 −0.273879 0.961764i \(-0.588307\pi\)
−0.273879 + 0.961764i \(0.588307\pi\)
\(462\) −2.83367e8 −0.133691
\(463\) −9.11439e8 −0.426770 −0.213385 0.976968i \(-0.568449\pi\)
−0.213385 + 0.976968i \(0.568449\pi\)
\(464\) 2.40482e8 0.111756
\(465\) −2.11370e9 −0.974895
\(466\) −7.37328e8 −0.337528
\(467\) −2.27214e9 −1.03235 −0.516174 0.856484i \(-0.672644\pi\)
−0.516174 + 0.856484i \(0.672644\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) −7.56366e8 −0.338553
\(470\) 2.83988e9 1.26170
\(471\) 2.36630e9 1.04351
\(472\) −2.73900e8 −0.119893
\(473\) −2.03807e9 −0.885533
\(474\) −1.86133e9 −0.802783
\(475\) −3.13053e9 −1.34026
\(476\) 4.71470e7 0.0200369
\(477\) −2.56815e8 −0.108344
\(478\) 1.33486e9 0.559032
\(479\) 6.50250e8 0.270337 0.135169 0.990823i \(-0.456842\pi\)
0.135169 + 0.990823i \(0.456842\pi\)
\(480\) −4.39647e8 −0.181451
\(481\) 8.38296e8 0.343471
\(482\) 1.88623e9 0.767238
\(483\) −6.25705e8 −0.252671
\(484\) −3.10946e8 −0.124660
\(485\) 9.97145e8 0.396883
\(486\) −1.14791e8 −0.0453609
\(487\) −1.88694e9 −0.740299 −0.370150 0.928972i \(-0.620694\pi\)
−0.370150 + 0.928972i \(0.620694\pi\)
\(488\) 4.85148e8 0.188975
\(489\) −1.95701e9 −0.756856
\(490\) 4.67701e8 0.179590
\(491\) 1.64654e9 0.627750 0.313875 0.949464i \(-0.398373\pi\)
0.313875 + 0.949464i \(0.398373\pi\)
\(492\) −3.92491e8 −0.148577
\(493\) −1.26096e8 −0.0473957
\(494\) −3.25944e8 −0.121646
\(495\) −1.38554e9 −0.513454
\(496\) 6.45282e8 0.237445
\(497\) 1.72795e9 0.631369
\(498\) 1.90235e9 0.690222
\(499\) −4.13719e9 −1.49057 −0.745287 0.666744i \(-0.767687\pi\)
−0.745287 + 0.666744i \(0.767687\pi\)
\(500\) 2.88402e9 1.03182
\(501\) −2.11551e9 −0.751595
\(502\) −1.50230e9 −0.530022
\(503\) −2.42528e9 −0.849717 −0.424858 0.905260i \(-0.639676\pi\)
−0.424858 + 0.905260i \(0.639676\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 9.50693e9 3.28489
\(506\) 2.06730e9 0.709377
\(507\) −1.30324e8 −0.0444116
\(508\) −6.30734e7 −0.0213463
\(509\) −1.44039e9 −0.484135 −0.242068 0.970259i \(-0.577826\pi\)
−0.242068 + 0.970259i \(0.577826\pi\)
\(510\) 2.30528e8 0.0769535
\(511\) −5.02338e8 −0.166542
\(512\) 1.34218e8 0.0441942
\(513\) 3.65018e8 0.119372
\(514\) 1.67520e9 0.544120
\(515\) 6.81585e8 0.219884
\(516\) −9.20790e8 −0.295044
\(517\) −2.73226e9 −0.869570
\(518\) −1.04701e9 −0.330976
\(519\) −7.11948e8 −0.223544
\(520\) 5.58972e8 0.174333
\(521\) 1.57419e9 0.487668 0.243834 0.969817i \(-0.421595\pi\)
0.243834 + 0.969817i \(0.421595\pi\)
\(522\) 3.42405e8 0.105364
\(523\) 3.48172e9 1.06424 0.532118 0.846670i \(-0.321396\pi\)
0.532118 + 0.846670i \(0.321396\pi\)
\(524\) −2.26919e9 −0.688988
\(525\) 1.56334e9 0.471514
\(526\) −2.64688e9 −0.793020
\(527\) −3.38353e8 −0.100701
\(528\) 4.22986e8 0.125057
\(529\) 1.16000e9 0.340693
\(530\) −1.40047e9 −0.408609
\(531\) −3.89986e8 −0.113036
\(532\) 4.07096e8 0.117221
\(533\) 4.99018e8 0.142748
\(534\) −6.73638e8 −0.191440
\(535\) 1.17261e10 3.31067
\(536\) 1.12904e9 0.316688
\(537\) −3.88638e9 −1.08302
\(538\) 3.02408e9 0.837250
\(539\) −4.49977e8 −0.123774
\(540\) −6.25981e8 −0.171074
\(541\) −5.58813e9 −1.51731 −0.758657 0.651490i \(-0.774144\pi\)
−0.758657 + 0.651490i \(0.774144\pi\)
\(542\) 4.87718e7 0.0131574
\(543\) −2.11454e9 −0.566783
\(544\) −7.03769e7 −0.0187428
\(545\) 4.17181e9 1.10392
\(546\) 1.62771e8 0.0427960
\(547\) −8.36867e8 −0.218625 −0.109313 0.994007i \(-0.534865\pi\)
−0.109313 + 0.994007i \(0.534865\pi\)
\(548\) 1.70475e9 0.442517
\(549\) 6.90767e8 0.178167
\(550\) −5.16519e9 −1.32378
\(551\) −1.08879e9 −0.277277
\(552\) 9.33998e8 0.236352
\(553\) −2.95572e9 −0.743233
\(554\) −3.08548e9 −0.770971
\(555\) −5.11942e9 −1.27115
\(556\) −3.63429e9 −0.896723
\(557\) −2.13091e9 −0.522483 −0.261242 0.965273i \(-0.584132\pi\)
−0.261242 + 0.965273i \(0.584132\pi\)
\(558\) 9.18770e8 0.223865
\(559\) 1.17070e9 0.283469
\(560\) −6.98143e8 −0.167991
\(561\) −2.21792e8 −0.0530367
\(562\) −5.45696e8 −0.129680
\(563\) −4.67407e9 −1.10386 −0.551932 0.833889i \(-0.686109\pi\)
−0.551932 + 0.833889i \(0.686109\pi\)
\(564\) −1.23442e9 −0.289725
\(565\) −3.51317e9 −0.819464
\(566\) −4.95760e8 −0.114925
\(567\) −1.82284e8 −0.0419961
\(568\) −2.57933e9 −0.590591
\(569\) −3.10951e9 −0.707618 −0.353809 0.935318i \(-0.615114\pi\)
−0.353809 + 0.935318i \(0.615114\pi\)
\(570\) 1.99052e9 0.450199
\(571\) 1.90010e9 0.427121 0.213560 0.976930i \(-0.431494\pi\)
0.213560 + 0.976930i \(0.431494\pi\)
\(572\) −5.37789e8 −0.120151
\(573\) −1.71407e9 −0.380616
\(574\) −6.23261e8 −0.137556
\(575\) −1.14053e10 −2.50190
\(576\) 1.91103e8 0.0416667
\(577\) 2.04455e8 0.0443081 0.0221541 0.999755i \(-0.492948\pi\)
0.0221541 + 0.999755i \(0.492948\pi\)
\(578\) −3.24581e9 −0.699158
\(579\) −2.58090e9 −0.552581
\(580\) 1.86721e9 0.397369
\(581\) 3.02087e9 0.639021
\(582\) −4.33433e8 −0.0911363
\(583\) 1.34740e9 0.281615
\(584\) 7.49845e8 0.155785
\(585\) 7.95880e8 0.164362
\(586\) −3.90879e7 −0.00802417
\(587\) −2.35979e9 −0.481548 −0.240774 0.970581i \(-0.577401\pi\)
−0.240774 + 0.970581i \(0.577401\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) −2.92154e9 −0.589127
\(590\) −2.12668e9 −0.426304
\(591\) −2.23634e9 −0.445638
\(592\) 1.56289e9 0.309600
\(593\) 1.48166e9 0.291780 0.145890 0.989301i \(-0.453395\pi\)
0.145890 + 0.989301i \(0.453395\pi\)
\(594\) 6.02259e8 0.117905
\(595\) 3.66070e8 0.0712451
\(596\) −4.60731e9 −0.891427
\(597\) 2.19747e9 0.422682
\(598\) −1.18750e9 −0.227080
\(599\) −3.55355e9 −0.675568 −0.337784 0.941224i \(-0.609677\pi\)
−0.337784 + 0.941224i \(0.609677\pi\)
\(600\) −2.33361e9 −0.441061
\(601\) 5.40134e9 1.01494 0.507470 0.861669i \(-0.330581\pi\)
0.507470 + 0.861669i \(0.330581\pi\)
\(602\) −1.46218e9 −0.273158
\(603\) 1.60755e9 0.298576
\(604\) −2.06160e9 −0.380694
\(605\) −2.41432e9 −0.443252
\(606\) −4.13242e9 −0.754310
\(607\) 4.42281e9 0.802672 0.401336 0.915931i \(-0.368546\pi\)
0.401336 + 0.915931i \(0.368546\pi\)
\(608\) −6.07677e8 −0.109650
\(609\) 5.43726e8 0.0975483
\(610\) 3.76690e9 0.671939
\(611\) 1.56946e9 0.278359
\(612\) −1.00205e8 −0.0176709
\(613\) 4.76865e9 0.836149 0.418075 0.908413i \(-0.362705\pi\)
0.418075 + 0.908413i \(0.362705\pi\)
\(614\) −2.11303e9 −0.368397
\(615\) −3.04747e9 −0.528295
\(616\) 6.71686e8 0.115780
\(617\) 1.10275e10 1.89008 0.945040 0.326955i \(-0.106023\pi\)
0.945040 + 0.326955i \(0.106023\pi\)
\(618\) −2.96267e8 −0.0504922
\(619\) −1.49822e9 −0.253897 −0.126949 0.991909i \(-0.540518\pi\)
−0.126949 + 0.991909i \(0.540518\pi\)
\(620\) 5.01025e9 0.844284
\(621\) 1.32985e9 0.222835
\(622\) 4.45225e9 0.741846
\(623\) −1.06971e9 −0.177239
\(624\) −2.42971e8 −0.0400320
\(625\) 9.20465e9 1.50809
\(626\) −3.54228e9 −0.577129
\(627\) −1.91509e9 −0.310279
\(628\) −5.60901e9 −0.903706
\(629\) −8.19497e8 −0.131302
\(630\) −9.94035e8 −0.158383
\(631\) 6.65589e9 1.05464 0.527319 0.849668i \(-0.323197\pi\)
0.527319 + 0.849668i \(0.323197\pi\)
\(632\) 4.41203e9 0.695230
\(633\) −5.12213e9 −0.802672
\(634\) 1.86755e9 0.291044
\(635\) −4.89729e8 −0.0759011
\(636\) 6.08747e8 0.0938289
\(637\) 2.58475e8 0.0396214
\(638\) −1.79645e9 −0.273868
\(639\) −3.67252e9 −0.556815
\(640\) 1.04213e9 0.157141
\(641\) 6.55404e9 0.982892 0.491446 0.870908i \(-0.336469\pi\)
0.491446 + 0.870908i \(0.336469\pi\)
\(642\) −5.09704e9 −0.760230
\(643\) −3.63452e9 −0.539149 −0.269574 0.962980i \(-0.586883\pi\)
−0.269574 + 0.962980i \(0.586883\pi\)
\(644\) 1.48315e9 0.218819
\(645\) −7.14942e9 −1.04909
\(646\) 3.18635e8 0.0465028
\(647\) 8.91495e9 1.29406 0.647029 0.762465i \(-0.276011\pi\)
0.647029 + 0.762465i \(0.276011\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 2.04609e9 0.293811
\(650\) 2.96698e9 0.423758
\(651\) 1.45897e9 0.207259
\(652\) 4.63885e9 0.655456
\(653\) 3.92610e9 0.551779 0.275889 0.961189i \(-0.411028\pi\)
0.275889 + 0.961189i \(0.411028\pi\)
\(654\) −1.81338e9 −0.253493
\(655\) −1.76190e10 −2.44983
\(656\) 9.30349e8 0.128672
\(657\) 1.06765e9 0.146876
\(658\) −1.96021e9 −0.268233
\(659\) 3.58393e8 0.0487822 0.0243911 0.999702i \(-0.492235\pi\)
0.0243911 + 0.999702i \(0.492235\pi\)
\(660\) 3.28425e9 0.444664
\(661\) −1.07820e10 −1.45210 −0.726049 0.687643i \(-0.758646\pi\)
−0.726049 + 0.687643i \(0.758646\pi\)
\(662\) 9.50813e9 1.27377
\(663\) 1.27401e8 0.0169776
\(664\) −4.50929e9 −0.597750
\(665\) 3.16087e9 0.416803
\(666\) 2.22528e9 0.291894
\(667\) −3.96675e9 −0.517600
\(668\) 5.01455e9 0.650900
\(669\) −1.16817e9 −0.150839
\(670\) 8.76633e9 1.12605
\(671\) −3.62415e9 −0.463103
\(672\) 3.03464e8 0.0385758
\(673\) −4.39366e9 −0.555615 −0.277808 0.960637i \(-0.589608\pi\)
−0.277808 + 0.960637i \(0.589608\pi\)
\(674\) 2.83836e9 0.357073
\(675\) −3.32266e9 −0.415836
\(676\) 3.08916e8 0.0384615
\(677\) 8.11370e9 1.00498 0.502491 0.864582i \(-0.332417\pi\)
0.502491 + 0.864582i \(0.332417\pi\)
\(678\) 1.52708e9 0.188174
\(679\) −6.88276e8 −0.0843758
\(680\) −5.46437e8 −0.0666437
\(681\) −4.76614e8 −0.0578298
\(682\) −4.82038e9 −0.581884
\(683\) −1.09490e10 −1.31493 −0.657466 0.753484i \(-0.728372\pi\)
−0.657466 + 0.753484i \(0.728372\pi\)
\(684\) −8.65228e8 −0.103379
\(685\) 1.32365e10 1.57346
\(686\) −3.22829e8 −0.0381802
\(687\) −6.15950e9 −0.724764
\(688\) 2.18261e9 0.255516
\(689\) −7.73968e8 −0.0901479
\(690\) 7.25197e9 0.840396
\(691\) −1.34406e9 −0.154969 −0.0774844 0.996994i \(-0.524689\pi\)
−0.0774844 + 0.996994i \(0.524689\pi\)
\(692\) 1.68758e9 0.193595
\(693\) 9.56365e8 0.109158
\(694\) −3.62169e9 −0.411295
\(695\) −2.82183e10 −3.18848
\(696\) −8.11626e8 −0.0912481
\(697\) −4.87828e8 −0.0545697
\(698\) 2.52272e9 0.280786
\(699\) 2.48848e9 0.275591
\(700\) −3.70569e9 −0.408343
\(701\) −1.54593e10 −1.69503 −0.847516 0.530769i \(-0.821903\pi\)
−0.847516 + 0.530769i \(0.821903\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) −7.07604e9 −0.768151
\(704\) −1.00263e9 −0.108302
\(705\) −9.58459e9 −1.03018
\(706\) 6.36973e9 0.681247
\(707\) −6.56212e9 −0.698355
\(708\) 9.24412e8 0.0978924
\(709\) −1.07079e10 −1.12835 −0.564176 0.825655i \(-0.690806\pi\)
−0.564176 + 0.825655i \(0.690806\pi\)
\(710\) −2.00270e10 −2.09997
\(711\) 6.28197e9 0.655470
\(712\) 1.59677e9 0.165792
\(713\) −1.06439e10 −1.09973
\(714\) −1.59121e8 −0.0163600
\(715\) −4.17563e9 −0.427219
\(716\) 9.21216e9 0.937920
\(717\) −4.50514e9 −0.456448
\(718\) −1.32150e10 −1.33239
\(719\) −1.74028e10 −1.74610 −0.873050 0.487631i \(-0.837861\pi\)
−0.873050 + 0.487631i \(0.837861\pi\)
\(720\) 1.48381e9 0.148154
\(721\) −4.70461e8 −0.0467466
\(722\) −4.39969e9 −0.435053
\(723\) −6.36603e9 −0.626447
\(724\) 5.01225e9 0.490849
\(725\) 9.91098e9 0.965904
\(726\) 1.04944e9 0.101784
\(727\) 1.21160e10 1.16947 0.584733 0.811226i \(-0.301199\pi\)
0.584733 + 0.811226i \(0.301199\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 5.82213e9 0.553926
\(731\) −1.14445e9 −0.108364
\(732\) −1.63737e9 −0.154298
\(733\) 1.12175e10 1.05204 0.526021 0.850471i \(-0.323683\pi\)
0.526021 + 0.850471i \(0.323683\pi\)
\(734\) 2.72483e9 0.254333
\(735\) −1.57849e9 −0.146635
\(736\) −2.21392e9 −0.204687
\(737\) −8.43412e9 −0.776075
\(738\) 1.32466e9 0.121313
\(739\) −9.44143e9 −0.860562 −0.430281 0.902695i \(-0.641586\pi\)
−0.430281 + 0.902695i \(0.641586\pi\)
\(740\) 1.21349e10 1.10085
\(741\) 1.10006e9 0.0993237
\(742\) 9.66667e8 0.0868687
\(743\) 1.86509e10 1.66817 0.834083 0.551639i \(-0.185997\pi\)
0.834083 + 0.551639i \(0.185997\pi\)
\(744\) −2.17783e9 −0.193873
\(745\) −3.57732e10 −3.16965
\(746\) −4.70967e9 −0.415340
\(747\) −6.42045e9 −0.563564
\(748\) 5.25730e8 0.0459311
\(749\) −8.09390e9 −0.703836
\(750\) −9.73358e9 −0.842477
\(751\) 6.87622e8 0.0592394 0.0296197 0.999561i \(-0.490570\pi\)
0.0296197 + 0.999561i \(0.490570\pi\)
\(752\) 2.92603e9 0.250909
\(753\) 5.07026e9 0.432761
\(754\) 1.03191e9 0.0876683
\(755\) −1.60072e10 −1.35363
\(756\) 4.32081e8 0.0363696
\(757\) 6.41867e9 0.537786 0.268893 0.963170i \(-0.413342\pi\)
0.268893 + 0.963170i \(0.413342\pi\)
\(758\) 6.88175e7 0.00573927
\(759\) −6.97715e9 −0.579204
\(760\) −4.71827e9 −0.389884
\(761\) −5.67316e9 −0.466636 −0.233318 0.972400i \(-0.574958\pi\)
−0.233318 + 0.972400i \(0.574958\pi\)
\(762\) 2.12873e8 0.0174292
\(763\) −2.87957e9 −0.234689
\(764\) 4.06298e9 0.329623
\(765\) −7.78033e8 −0.0628323
\(766\) −1.47801e10 −1.18817
\(767\) −1.17531e9 −0.0940520
\(768\) −4.52985e8 −0.0360844
\(769\) 2.70852e9 0.214778 0.107389 0.994217i \(-0.465751\pi\)
0.107389 + 0.994217i \(0.465751\pi\)
\(770\) 5.21526e9 0.411679
\(771\) −5.65379e9 −0.444272
\(772\) 6.11768e9 0.478549
\(773\) 9.88293e8 0.0769587 0.0384793 0.999259i \(-0.487749\pi\)
0.0384793 + 0.999259i \(0.487749\pi\)
\(774\) 3.10767e9 0.240902
\(775\) 2.65940e10 2.05224
\(776\) 1.02740e9 0.0789264
\(777\) 3.53366e9 0.270241
\(778\) 1.08687e9 0.0827464
\(779\) −4.21220e9 −0.319248
\(780\) −1.88653e9 −0.142342
\(781\) 1.92681e10 1.44730
\(782\) 1.16087e9 0.0868078
\(783\) −1.15562e9 −0.0860295
\(784\) 4.81890e8 0.0357143
\(785\) −4.35508e10 −3.21331
\(786\) 7.65852e9 0.562556
\(787\) 1.24048e10 0.907146 0.453573 0.891219i \(-0.350149\pi\)
0.453573 + 0.891219i \(0.350149\pi\)
\(788\) 5.30096e9 0.385934
\(789\) 8.93321e9 0.647498
\(790\) 3.42569e10 2.47203
\(791\) 2.42495e9 0.174215
\(792\) −1.42758e9 −0.102108
\(793\) 2.08178e9 0.148244
\(794\) 5.14070e9 0.364461
\(795\) 4.72658e9 0.333627
\(796\) −5.20883e9 −0.366053
\(797\) −9.25871e9 −0.647808 −0.323904 0.946090i \(-0.604995\pi\)
−0.323904 + 0.946090i \(0.604995\pi\)
\(798\) −1.37395e9 −0.0957107
\(799\) −1.53426e9 −0.106411
\(800\) 5.53152e9 0.381970
\(801\) 2.27353e9 0.156310
\(802\) −1.79226e10 −1.22685
\(803\) −5.60149e9 −0.381768
\(804\) −3.81050e9 −0.258574
\(805\) 1.15159e10 0.778056
\(806\) 2.76891e9 0.186267
\(807\) −1.02063e10 −0.683611
\(808\) 9.79536e9 0.653252
\(809\) −2.49825e10 −1.65888 −0.829441 0.558595i \(-0.811341\pi\)
−0.829441 + 0.558595i \(0.811341\pi\)
\(810\) 2.11269e9 0.139681
\(811\) 2.50005e10 1.64579 0.822896 0.568192i \(-0.192357\pi\)
0.822896 + 0.568192i \(0.192357\pi\)
\(812\) −1.28883e9 −0.0844793
\(813\) −1.64605e8 −0.0107430
\(814\) −1.16751e10 −0.758706
\(815\) 3.60181e10 2.33061
\(816\) 2.37522e8 0.0153034
\(817\) −9.88188e9 −0.633961
\(818\) 6.11503e9 0.390627
\(819\) −5.49353e8 −0.0349428
\(820\) 7.22364e9 0.457517
\(821\) −1.73071e10 −1.09150 −0.545749 0.837949i \(-0.683755\pi\)
−0.545749 + 0.837949i \(0.683755\pi\)
\(822\) −5.75355e9 −0.361314
\(823\) 2.31932e10 1.45031 0.725155 0.688586i \(-0.241768\pi\)
0.725155 + 0.688586i \(0.241768\pi\)
\(824\) 7.02263e8 0.0437275
\(825\) 1.74325e10 1.08087
\(826\) 1.46793e9 0.0906308
\(827\) −2.05547e10 −1.26369 −0.631846 0.775094i \(-0.717702\pi\)
−0.631846 + 0.775094i \(0.717702\pi\)
\(828\) −3.15224e9 −0.192980
\(829\) 2.49310e10 1.51984 0.759921 0.650015i \(-0.225237\pi\)
0.759921 + 0.650015i \(0.225237\pi\)
\(830\) −3.50121e10 −2.12542
\(831\) 1.04135e10 0.629495
\(832\) 5.75930e8 0.0346688
\(833\) −2.52679e8 −0.0151465
\(834\) 1.22657e10 0.732171
\(835\) 3.89352e10 2.31441
\(836\) 4.53947e9 0.268709
\(837\) −3.10085e9 −0.182785
\(838\) −4.50918e9 −0.264693
\(839\) 1.12662e10 0.658582 0.329291 0.944228i \(-0.393190\pi\)
0.329291 + 0.944228i \(0.393190\pi\)
\(840\) 2.35623e9 0.137164
\(841\) −1.38028e10 −0.800171
\(842\) −1.34333e9 −0.0775514
\(843\) 1.84173e9 0.105884
\(844\) 1.21414e10 0.695134
\(845\) 2.39856e9 0.136758
\(846\) 4.16617e9 0.236560
\(847\) 1.66647e9 0.0942338
\(848\) −1.44296e9 −0.0812582
\(849\) 1.67319e9 0.0938358
\(850\) −2.90045e9 −0.161994
\(851\) −2.57798e10 −1.43392
\(852\) 8.70523e9 0.482216
\(853\) −1.90222e10 −1.04940 −0.524699 0.851288i \(-0.675822\pi\)
−0.524699 + 0.851288i \(0.675822\pi\)
\(854\) −2.60009e9 −0.142852
\(855\) −6.71801e9 −0.367586
\(856\) 1.20819e10 0.658379
\(857\) 2.03465e10 1.10422 0.552110 0.833771i \(-0.313823\pi\)
0.552110 + 0.833771i \(0.313823\pi\)
\(858\) 1.81504e9 0.0981026
\(859\) 2.83480e10 1.52597 0.762987 0.646414i \(-0.223732\pi\)
0.762987 + 0.646414i \(0.223732\pi\)
\(860\) 1.69468e10 0.908537
\(861\) 2.10351e9 0.112314
\(862\) 3.25890e9 0.173299
\(863\) −2.07145e10 −1.09708 −0.548539 0.836125i \(-0.684816\pi\)
−0.548539 + 0.836125i \(0.684816\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 1.31031e10 0.688364
\(866\) −1.14819e10 −0.600761
\(867\) 1.09546e10 0.570860
\(868\) −3.45831e9 −0.179492
\(869\) −3.29587e10 −1.70373
\(870\) −6.30182e9 −0.324451
\(871\) 4.84471e9 0.248430
\(872\) 4.29837e9 0.219531
\(873\) 1.46284e9 0.0744125
\(874\) 1.00236e10 0.507849
\(875\) −1.54566e10 −0.779982
\(876\) −2.53073e9 −0.127198
\(877\) −1.44434e10 −0.723052 −0.361526 0.932362i \(-0.617744\pi\)
−0.361526 + 0.932362i \(0.617744\pi\)
\(878\) 7.04160e9 0.351108
\(879\) 1.31922e8 0.00655171
\(880\) −7.78488e9 −0.385090
\(881\) 6.80586e9 0.335326 0.167663 0.985844i \(-0.446378\pi\)
0.167663 + 0.985844i \(0.446378\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) −2.73622e10 −1.33748 −0.668741 0.743495i \(-0.733167\pi\)
−0.668741 + 0.743495i \(0.733167\pi\)
\(884\) −3.01989e8 −0.0147031
\(885\) 7.17754e9 0.348076
\(886\) 1.42491e10 0.688285
\(887\) 1.70834e10 0.821941 0.410970 0.911649i \(-0.365190\pi\)
0.410970 + 0.911649i \(0.365190\pi\)
\(888\) −5.27474e9 −0.252787
\(889\) 3.38034e8 0.0161363
\(890\) 1.23980e10 0.589506
\(891\) −2.03262e9 −0.0962687
\(892\) 2.76899e9 0.130630
\(893\) −1.32478e10 −0.622533
\(894\) 1.55497e10 0.727847
\(895\) 7.15272e10 3.33496
\(896\) −7.19323e8 −0.0334077
\(897\) 4.00780e9 0.185410
\(898\) −1.53694e8 −0.00708255
\(899\) 9.24936e9 0.424573
\(900\) 7.87593e9 0.360125
\(901\) 7.56612e8 0.0344617
\(902\) −6.94989e9 −0.315323
\(903\) 4.93486e9 0.223032
\(904\) −3.61975e9 −0.162963
\(905\) 3.89173e10 1.74531
\(906\) 6.95792e9 0.310835
\(907\) −1.38221e10 −0.615103 −0.307551 0.951531i \(-0.599510\pi\)
−0.307551 + 0.951531i \(0.599510\pi\)
\(908\) 1.12975e9 0.0500821
\(909\) 1.39469e10 0.615892
\(910\) −2.99574e9 −0.131783
\(911\) 2.21664e10 0.971360 0.485680 0.874137i \(-0.338572\pi\)
0.485680 + 0.874137i \(0.338572\pi\)
\(912\) 2.05091e9 0.0895292
\(913\) 3.36853e10 1.46485
\(914\) −1.22681e10 −0.531455
\(915\) −1.27133e10 −0.548636
\(916\) 1.46003e10 0.627664
\(917\) 1.21614e10 0.520826
\(918\) 3.38191e8 0.0144282
\(919\) 1.73285e9 0.0736474 0.0368237 0.999322i \(-0.488276\pi\)
0.0368237 + 0.999322i \(0.488276\pi\)
\(920\) −1.71898e10 −0.727804
\(921\) 7.13147e9 0.300795
\(922\) −9.21789e9 −0.387323
\(923\) −1.10679e10 −0.463298
\(924\) −2.26694e9 −0.0945340
\(925\) 6.44112e10 2.67587
\(926\) −7.29151e9 −0.301772
\(927\) 9.99902e8 0.0412267
\(928\) 1.92385e9 0.0790231
\(929\) 4.89668e9 0.200377 0.100188 0.994968i \(-0.468055\pi\)
0.100188 + 0.994968i \(0.468055\pi\)
\(930\) −1.69096e10 −0.689355
\(931\) −2.18178e9 −0.0886109
\(932\) −5.89862e9 −0.238668
\(933\) −1.50264e10 −0.605714
\(934\) −1.81771e10 −0.729981
\(935\) 4.08200e9 0.163317
\(936\) 8.20026e8 0.0326860
\(937\) 8.45195e9 0.335636 0.167818 0.985818i \(-0.446328\pi\)
0.167818 + 0.985818i \(0.446328\pi\)
\(938\) −6.05093e9 −0.239393
\(939\) 1.19552e10 0.471223
\(940\) 2.27190e10 0.892159
\(941\) 3.53018e10 1.38113 0.690563 0.723273i \(-0.257363\pi\)
0.690563 + 0.723273i \(0.257363\pi\)
\(942\) 1.89304e10 0.737873
\(943\) −1.53461e10 −0.595946
\(944\) −2.19120e9 −0.0847773
\(945\) 3.35487e9 0.129320
\(946\) −1.63046e10 −0.626167
\(947\) −1.87980e10 −0.719263 −0.359631 0.933094i \(-0.617098\pi\)
−0.359631 + 0.933094i \(0.617098\pi\)
\(948\) −1.48906e10 −0.567653
\(949\) 3.21760e9 0.122208
\(950\) −2.50442e10 −0.947708
\(951\) −6.30297e9 −0.237637
\(952\) 3.77176e8 0.0141682
\(953\) −3.46583e10 −1.29712 −0.648562 0.761162i \(-0.724629\pi\)
−0.648562 + 0.761162i \(0.724629\pi\)
\(954\) −2.05452e9 −0.0766110
\(955\) 3.15467e10 1.17204
\(956\) 1.06789e10 0.395296
\(957\) 6.06301e9 0.223613
\(958\) 5.20200e9 0.191157
\(959\) −9.13642e9 −0.334511
\(960\) −3.51717e9 −0.128305
\(961\) −2.69393e9 −0.0979163
\(962\) 6.70637e9 0.242870
\(963\) 1.72025e10 0.620725
\(964\) 1.50898e10 0.542519
\(965\) 4.75004e10 1.70158
\(966\) −5.00564e9 −0.178665
\(967\) 4.51149e10 1.60446 0.802228 0.597018i \(-0.203648\pi\)
0.802228 + 0.597018i \(0.203648\pi\)
\(968\) −2.48757e9 −0.0881477
\(969\) −1.07539e9 −0.0379694
\(970\) 7.97716e9 0.280638
\(971\) 5.39377e10 1.89071 0.945355 0.326044i \(-0.105716\pi\)
0.945355 + 0.326044i \(0.105716\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 1.94775e10 0.677859
\(974\) −1.50955e10 −0.523471
\(975\) −1.00136e10 −0.345997
\(976\) 3.88118e9 0.133626
\(977\) −5.29565e9 −0.181672 −0.0908360 0.995866i \(-0.528954\pi\)
−0.0908360 + 0.995866i \(0.528954\pi\)
\(978\) −1.56561e10 −0.535178
\(979\) −1.19282e10 −0.406290
\(980\) 3.74161e9 0.126989
\(981\) 6.12015e9 0.206976
\(982\) 1.31723e10 0.443886
\(983\) −3.48690e10 −1.17085 −0.585426 0.810726i \(-0.699073\pi\)
−0.585426 + 0.810726i \(0.699073\pi\)
\(984\) −3.13993e9 −0.105060
\(985\) 4.11590e10 1.37226
\(986\) −1.00877e9 −0.0335138
\(987\) 6.61572e9 0.219012
\(988\) −2.60755e9 −0.0860169
\(989\) −3.60022e10 −1.18343
\(990\) −1.10843e10 −0.363067
\(991\) −1.20597e10 −0.393622 −0.196811 0.980441i \(-0.563058\pi\)
−0.196811 + 0.980441i \(0.563058\pi\)
\(992\) 5.16225e9 0.167899
\(993\) −3.20899e10 −1.04003
\(994\) 1.38236e10 0.446445
\(995\) −4.04436e10 −1.30158
\(996\) 1.52188e10 0.488061
\(997\) −3.70942e10 −1.18542 −0.592710 0.805416i \(-0.701942\pi\)
−0.592710 + 0.805416i \(0.701942\pi\)
\(998\) −3.30975e10 −1.05399
\(999\) −7.51032e9 −0.238330
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.r.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.r.1.6 6 1.1 even 1 trivial