Properties

Label 8470.2.a.cw.1.5
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.19898000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 7 x^{3} + 24 x^{2} - 15 x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.05906\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.05906 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.05906 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.23973 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.05906 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.05906 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.23973 q^{9} -1.00000 q^{10} +2.05906 q^{12} -2.27257 q^{13} -1.00000 q^{14} +2.05906 q^{15} +1.00000 q^{16} -7.44890 q^{17} -1.23973 q^{18} -0.851836 q^{19} +1.00000 q^{20} +2.05906 q^{21} +0.651398 q^{23} -2.05906 q^{24} +1.00000 q^{25} +2.27257 q^{26} -3.62450 q^{27} +1.00000 q^{28} +2.36641 q^{29} -2.05906 q^{30} -0.145026 q^{31} -1.00000 q^{32} +7.44890 q^{34} +1.00000 q^{35} +1.23973 q^{36} +2.32711 q^{37} +0.851836 q^{38} -4.67936 q^{39} -1.00000 q^{40} +4.64211 q^{41} -2.05906 q^{42} -11.7588 q^{43} +1.23973 q^{45} -0.651398 q^{46} -5.27536 q^{47} +2.05906 q^{48} +1.00000 q^{49} -1.00000 q^{50} -15.3377 q^{51} -2.27257 q^{52} -0.962643 q^{53} +3.62450 q^{54} -1.00000 q^{56} -1.75398 q^{57} -2.36641 q^{58} +1.04296 q^{59} +2.05906 q^{60} -3.58885 q^{61} +0.145026 q^{62} +1.23973 q^{63} +1.00000 q^{64} -2.27257 q^{65} -0.430333 q^{67} -7.44890 q^{68} +1.34127 q^{69} -1.00000 q^{70} +10.6384 q^{71} -1.23973 q^{72} -1.16253 q^{73} -2.32711 q^{74} +2.05906 q^{75} -0.851836 q^{76} +4.67936 q^{78} +10.7068 q^{79} +1.00000 q^{80} -11.1823 q^{81} -4.64211 q^{82} +1.08367 q^{83} +2.05906 q^{84} -7.44890 q^{85} +11.7588 q^{86} +4.87258 q^{87} +3.73563 q^{89} -1.23973 q^{90} -2.27257 q^{91} +0.651398 q^{92} -0.298618 q^{93} +5.27536 q^{94} -0.851836 q^{95} -2.05906 q^{96} -1.47983 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} - q^{12} - 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 9 q^{17} - 3 q^{18} - 12 q^{19} + 6 q^{20} - q^{21} + 4 q^{23} + q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} + 6 q^{28} - 15 q^{29} + q^{30} + 8 q^{31} - 6 q^{32} + 9 q^{34} + 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} + 19 q^{39} - 6 q^{40} - 4 q^{41} + q^{42} - 30 q^{43} + 3 q^{45} - 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 16 q^{51} - 9 q^{52} - 6 q^{53} + 4 q^{54} - 6 q^{56} + 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} + 14 q^{61} - 8 q^{62} + 3 q^{63} + 6 q^{64} - 9 q^{65} + 18 q^{67} - 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} - 3 q^{72} - 23 q^{73} + 4 q^{74} - q^{75} - 12 q^{76} - 19 q^{78} - 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} - 25 q^{83} - q^{84} - 9 q^{85} + 30 q^{86} - 14 q^{87} - 18 q^{89} - 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} + 7 q^{94} - 12 q^{95} + q^{96} + 7 q^{97} - 6 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.05906 1.18880 0.594400 0.804170i \(-0.297390\pi\)
0.594400 + 0.804170i \(0.297390\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.05906 −0.840608
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.23973 0.413245
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.05906 0.594400
\(13\) −2.27257 −0.630298 −0.315149 0.949042i \(-0.602054\pi\)
−0.315149 + 0.949042i \(0.602054\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.05906 0.531647
\(16\) 1.00000 0.250000
\(17\) −7.44890 −1.80662 −0.903312 0.428985i \(-0.858871\pi\)
−0.903312 + 0.428985i \(0.858871\pi\)
\(18\) −1.23973 −0.292208
\(19\) −0.851836 −0.195425 −0.0977123 0.995215i \(-0.531152\pi\)
−0.0977123 + 0.995215i \(0.531152\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.05906 0.449324
\(22\) 0 0
\(23\) 0.651398 0.135826 0.0679130 0.997691i \(-0.478366\pi\)
0.0679130 + 0.997691i \(0.478366\pi\)
\(24\) −2.05906 −0.420304
\(25\) 1.00000 0.200000
\(26\) 2.27257 0.445688
\(27\) −3.62450 −0.697534
\(28\) 1.00000 0.188982
\(29\) 2.36641 0.439431 0.219715 0.975564i \(-0.429487\pi\)
0.219715 + 0.975564i \(0.429487\pi\)
\(30\) −2.05906 −0.375931
\(31\) −0.145026 −0.0260475 −0.0130237 0.999915i \(-0.504146\pi\)
−0.0130237 + 0.999915i \(0.504146\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.44890 1.27748
\(35\) 1.00000 0.169031
\(36\) 1.23973 0.206622
\(37\) 2.32711 0.382574 0.191287 0.981534i \(-0.438734\pi\)
0.191287 + 0.981534i \(0.438734\pi\)
\(38\) 0.851836 0.138186
\(39\) −4.67936 −0.749298
\(40\) −1.00000 −0.158114
\(41\) 4.64211 0.724976 0.362488 0.931988i \(-0.381927\pi\)
0.362488 + 0.931988i \(0.381927\pi\)
\(42\) −2.05906 −0.317720
\(43\) −11.7588 −1.79320 −0.896602 0.442837i \(-0.853972\pi\)
−0.896602 + 0.442837i \(0.853972\pi\)
\(44\) 0 0
\(45\) 1.23973 0.184809
\(46\) −0.651398 −0.0960434
\(47\) −5.27536 −0.769491 −0.384746 0.923023i \(-0.625711\pi\)
−0.384746 + 0.923023i \(0.625711\pi\)
\(48\) 2.05906 0.297200
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −15.3377 −2.14771
\(52\) −2.27257 −0.315149
\(53\) −0.962643 −0.132229 −0.0661146 0.997812i \(-0.521060\pi\)
−0.0661146 + 0.997812i \(0.521060\pi\)
\(54\) 3.62450 0.493231
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −1.75398 −0.232321
\(58\) −2.36641 −0.310724
\(59\) 1.04296 0.135782 0.0678911 0.997693i \(-0.478373\pi\)
0.0678911 + 0.997693i \(0.478373\pi\)
\(60\) 2.05906 0.265824
\(61\) −3.58885 −0.459505 −0.229752 0.973249i \(-0.573792\pi\)
−0.229752 + 0.973249i \(0.573792\pi\)
\(62\) 0.145026 0.0184183
\(63\) 1.23973 0.156192
\(64\) 1.00000 0.125000
\(65\) −2.27257 −0.281878
\(66\) 0 0
\(67\) −0.430333 −0.0525735 −0.0262868 0.999654i \(-0.508368\pi\)
−0.0262868 + 0.999654i \(0.508368\pi\)
\(68\) −7.44890 −0.903312
\(69\) 1.34127 0.161470
\(70\) −1.00000 −0.119523
\(71\) 10.6384 1.26255 0.631276 0.775558i \(-0.282532\pi\)
0.631276 + 0.775558i \(0.282532\pi\)
\(72\) −1.23973 −0.146104
\(73\) −1.16253 −0.136063 −0.0680316 0.997683i \(-0.521672\pi\)
−0.0680316 + 0.997683i \(0.521672\pi\)
\(74\) −2.32711 −0.270521
\(75\) 2.05906 0.237760
\(76\) −0.851836 −0.0977123
\(77\) 0 0
\(78\) 4.67936 0.529833
\(79\) 10.7068 1.20461 0.602303 0.798268i \(-0.294250\pi\)
0.602303 + 0.798268i \(0.294250\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.1823 −1.24247
\(82\) −4.64211 −0.512636
\(83\) 1.08367 0.118948 0.0594741 0.998230i \(-0.481058\pi\)
0.0594741 + 0.998230i \(0.481058\pi\)
\(84\) 2.05906 0.224662
\(85\) −7.44890 −0.807947
\(86\) 11.7588 1.26799
\(87\) 4.87258 0.522395
\(88\) 0 0
\(89\) 3.73563 0.395976 0.197988 0.980204i \(-0.436559\pi\)
0.197988 + 0.980204i \(0.436559\pi\)
\(90\) −1.23973 −0.130679
\(91\) −2.27257 −0.238230
\(92\) 0.651398 0.0679130
\(93\) −0.298618 −0.0309652
\(94\) 5.27536 0.544112
\(95\) −0.851836 −0.0873965
\(96\) −2.05906 −0.210152
\(97\) −1.47983 −0.150254 −0.0751270 0.997174i \(-0.523936\pi\)
−0.0751270 + 0.997174i \(0.523936\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −13.9523 −1.38830 −0.694151 0.719830i \(-0.744220\pi\)
−0.694151 + 0.719830i \(0.744220\pi\)
\(102\) 15.3377 1.51866
\(103\) −1.40558 −0.138496 −0.0692481 0.997599i \(-0.522060\pi\)
−0.0692481 + 0.997599i \(0.522060\pi\)
\(104\) 2.27257 0.222844
\(105\) 2.05906 0.200944
\(106\) 0.962643 0.0935001
\(107\) −18.9282 −1.82985 −0.914927 0.403618i \(-0.867752\pi\)
−0.914927 + 0.403618i \(0.867752\pi\)
\(108\) −3.62450 −0.348767
\(109\) −10.8423 −1.03850 −0.519252 0.854621i \(-0.673789\pi\)
−0.519252 + 0.854621i \(0.673789\pi\)
\(110\) 0 0
\(111\) 4.79166 0.454804
\(112\) 1.00000 0.0944911
\(113\) −17.1609 −1.61436 −0.807179 0.590307i \(-0.799007\pi\)
−0.807179 + 0.590307i \(0.799007\pi\)
\(114\) 1.75398 0.164276
\(115\) 0.651398 0.0607432
\(116\) 2.36641 0.219715
\(117\) −2.81738 −0.260467
\(118\) −1.04296 −0.0960125
\(119\) −7.44890 −0.682839
\(120\) −2.05906 −0.187966
\(121\) 0 0
\(122\) 3.58885 0.324919
\(123\) 9.55840 0.861852
\(124\) −0.145026 −0.0130237
\(125\) 1.00000 0.0894427
\(126\) −1.23973 −0.110444
\(127\) 11.1188 0.986631 0.493316 0.869850i \(-0.335785\pi\)
0.493316 + 0.869850i \(0.335785\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −24.2121 −2.13176
\(130\) 2.27257 0.199318
\(131\) −10.9162 −0.953749 −0.476874 0.878971i \(-0.658230\pi\)
−0.476874 + 0.878971i \(0.658230\pi\)
\(132\) 0 0
\(133\) −0.851836 −0.0738635
\(134\) 0.430333 0.0371751
\(135\) −3.62450 −0.311947
\(136\) 7.44890 0.638738
\(137\) −4.81357 −0.411251 −0.205626 0.978631i \(-0.565923\pi\)
−0.205626 + 0.978631i \(0.565923\pi\)
\(138\) −1.34127 −0.114176
\(139\) −10.1432 −0.860336 −0.430168 0.902749i \(-0.641546\pi\)
−0.430168 + 0.902749i \(0.641546\pi\)
\(140\) 1.00000 0.0845154
\(141\) −10.8623 −0.914771
\(142\) −10.6384 −0.892759
\(143\) 0 0
\(144\) 1.23973 0.103311
\(145\) 2.36641 0.196519
\(146\) 1.16253 0.0962113
\(147\) 2.05906 0.169829
\(148\) 2.32711 0.191287
\(149\) 1.88432 0.154370 0.0771848 0.997017i \(-0.475407\pi\)
0.0771848 + 0.997017i \(0.475407\pi\)
\(150\) −2.05906 −0.168122
\(151\) 14.3826 1.17044 0.585218 0.810876i \(-0.301009\pi\)
0.585218 + 0.810876i \(0.301009\pi\)
\(152\) 0.851836 0.0690930
\(153\) −9.23466 −0.746578
\(154\) 0 0
\(155\) −0.145026 −0.0116488
\(156\) −4.67936 −0.374649
\(157\) −12.6590 −1.01030 −0.505150 0.863031i \(-0.668563\pi\)
−0.505150 + 0.863031i \(0.668563\pi\)
\(158\) −10.7068 −0.851784
\(159\) −1.98214 −0.157194
\(160\) −1.00000 −0.0790569
\(161\) 0.651398 0.0513374
\(162\) 11.1823 0.878561
\(163\) −8.95282 −0.701239 −0.350620 0.936518i \(-0.614029\pi\)
−0.350620 + 0.936518i \(0.614029\pi\)
\(164\) 4.64211 0.362488
\(165\) 0 0
\(166\) −1.08367 −0.0841091
\(167\) 6.36465 0.492511 0.246256 0.969205i \(-0.420800\pi\)
0.246256 + 0.969205i \(0.420800\pi\)
\(168\) −2.05906 −0.158860
\(169\) −7.83543 −0.602725
\(170\) 7.44890 0.571305
\(171\) −1.05605 −0.0807582
\(172\) −11.7588 −0.896602
\(173\) −24.4230 −1.85684 −0.928421 0.371529i \(-0.878834\pi\)
−0.928421 + 0.371529i \(0.878834\pi\)
\(174\) −4.87258 −0.369389
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 2.14753 0.161418
\(178\) −3.73563 −0.279997
\(179\) 5.80767 0.434086 0.217043 0.976162i \(-0.430359\pi\)
0.217043 + 0.976162i \(0.430359\pi\)
\(180\) 1.23973 0.0924044
\(181\) −12.8812 −0.957450 −0.478725 0.877965i \(-0.658901\pi\)
−0.478725 + 0.877965i \(0.658901\pi\)
\(182\) 2.27257 0.168454
\(183\) −7.38966 −0.546259
\(184\) −0.651398 −0.0480217
\(185\) 2.32711 0.171093
\(186\) 0.298618 0.0218957
\(187\) 0 0
\(188\) −5.27536 −0.384746
\(189\) −3.62450 −0.263643
\(190\) 0.851836 0.0617987
\(191\) −5.65871 −0.409450 −0.204725 0.978820i \(-0.565630\pi\)
−0.204725 + 0.978820i \(0.565630\pi\)
\(192\) 2.05906 0.148600
\(193\) 13.7878 0.992470 0.496235 0.868188i \(-0.334715\pi\)
0.496235 + 0.868188i \(0.334715\pi\)
\(194\) 1.47983 0.106246
\(195\) −4.67936 −0.335096
\(196\) 1.00000 0.0714286
\(197\) 2.82825 0.201504 0.100752 0.994912i \(-0.467875\pi\)
0.100752 + 0.994912i \(0.467875\pi\)
\(198\) 0 0
\(199\) −0.513194 −0.0363793 −0.0181897 0.999835i \(-0.505790\pi\)
−0.0181897 + 0.999835i \(0.505790\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −0.886082 −0.0624994
\(202\) 13.9523 0.981677
\(203\) 2.36641 0.166089
\(204\) −15.3377 −1.07386
\(205\) 4.64211 0.324219
\(206\) 1.40558 0.0979316
\(207\) 0.807561 0.0561294
\(208\) −2.27257 −0.157574
\(209\) 0 0
\(210\) −2.05906 −0.142089
\(211\) −20.4146 −1.40540 −0.702698 0.711488i \(-0.748021\pi\)
−0.702698 + 0.711488i \(0.748021\pi\)
\(212\) −0.962643 −0.0661146
\(213\) 21.9052 1.50092
\(214\) 18.9282 1.29390
\(215\) −11.7588 −0.801945
\(216\) 3.62450 0.246616
\(217\) −0.145026 −0.00984502
\(218\) 10.8423 0.734333
\(219\) −2.39371 −0.161752
\(220\) 0 0
\(221\) 16.9281 1.13871
\(222\) −4.79166 −0.321595
\(223\) 15.3296 1.02654 0.513272 0.858226i \(-0.328433\pi\)
0.513272 + 0.858226i \(0.328433\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.23973 0.0826490
\(226\) 17.1609 1.14152
\(227\) −28.8027 −1.91170 −0.955850 0.293856i \(-0.905061\pi\)
−0.955850 + 0.293856i \(0.905061\pi\)
\(228\) −1.75398 −0.116160
\(229\) 7.06918 0.467145 0.233572 0.972339i \(-0.424958\pi\)
0.233572 + 0.972339i \(0.424958\pi\)
\(230\) −0.651398 −0.0429519
\(231\) 0 0
\(232\) −2.36641 −0.155362
\(233\) 25.1088 1.64493 0.822466 0.568814i \(-0.192598\pi\)
0.822466 + 0.568814i \(0.192598\pi\)
\(234\) 2.81738 0.184178
\(235\) −5.27536 −0.344127
\(236\) 1.04296 0.0678911
\(237\) 22.0459 1.43203
\(238\) 7.44890 0.482840
\(239\) 9.09178 0.588098 0.294049 0.955790i \(-0.404997\pi\)
0.294049 + 0.955790i \(0.404997\pi\)
\(240\) 2.05906 0.132912
\(241\) 11.8409 0.762739 0.381370 0.924423i \(-0.375452\pi\)
0.381370 + 0.924423i \(0.375452\pi\)
\(242\) 0 0
\(243\) −12.1515 −0.779518
\(244\) −3.58885 −0.229752
\(245\) 1.00000 0.0638877
\(246\) −9.55840 −0.609421
\(247\) 1.93586 0.123176
\(248\) 0.145026 0.00920917
\(249\) 2.23134 0.141406
\(250\) −1.00000 −0.0632456
\(251\) −12.2824 −0.775260 −0.387630 0.921815i \(-0.626706\pi\)
−0.387630 + 0.921815i \(0.626706\pi\)
\(252\) 1.23973 0.0780959
\(253\) 0 0
\(254\) −11.1188 −0.697654
\(255\) −15.3377 −0.960487
\(256\) 1.00000 0.0625000
\(257\) −23.8811 −1.48966 −0.744830 0.667254i \(-0.767469\pi\)
−0.744830 + 0.667254i \(0.767469\pi\)
\(258\) 24.2121 1.50738
\(259\) 2.32711 0.144600
\(260\) −2.27257 −0.140939
\(261\) 2.93372 0.181592
\(262\) 10.9162 0.674402
\(263\) −10.5902 −0.653019 −0.326509 0.945194i \(-0.605872\pi\)
−0.326509 + 0.945194i \(0.605872\pi\)
\(264\) 0 0
\(265\) −0.962643 −0.0591347
\(266\) 0.851836 0.0522294
\(267\) 7.69189 0.470736
\(268\) −0.430333 −0.0262868
\(269\) 16.5719 1.01041 0.505203 0.863001i \(-0.331418\pi\)
0.505203 + 0.863001i \(0.331418\pi\)
\(270\) 3.62450 0.220580
\(271\) −29.4244 −1.78740 −0.893702 0.448661i \(-0.851901\pi\)
−0.893702 + 0.448661i \(0.851901\pi\)
\(272\) −7.44890 −0.451656
\(273\) −4.67936 −0.283208
\(274\) 4.81357 0.290799
\(275\) 0 0
\(276\) 1.34127 0.0807349
\(277\) 29.6933 1.78410 0.892049 0.451939i \(-0.149267\pi\)
0.892049 + 0.451939i \(0.149267\pi\)
\(278\) 10.1432 0.608349
\(279\) −0.179794 −0.0107640
\(280\) −1.00000 −0.0597614
\(281\) −23.8076 −1.42024 −0.710121 0.704080i \(-0.751360\pi\)
−0.710121 + 0.704080i \(0.751360\pi\)
\(282\) 10.8623 0.646841
\(283\) −17.5036 −1.04048 −0.520241 0.854019i \(-0.674158\pi\)
−0.520241 + 0.854019i \(0.674158\pi\)
\(284\) 10.6384 0.631276
\(285\) −1.75398 −0.103897
\(286\) 0 0
\(287\) 4.64211 0.274015
\(288\) −1.23973 −0.0730521
\(289\) 38.4861 2.26389
\(290\) −2.36641 −0.138960
\(291\) −3.04706 −0.178622
\(292\) −1.16253 −0.0680316
\(293\) 23.1174 1.35053 0.675267 0.737573i \(-0.264028\pi\)
0.675267 + 0.737573i \(0.264028\pi\)
\(294\) −2.05906 −0.120087
\(295\) 1.04296 0.0607237
\(296\) −2.32711 −0.135261
\(297\) 0 0
\(298\) −1.88432 −0.109156
\(299\) −1.48035 −0.0856107
\(300\) 2.05906 0.118880
\(301\) −11.7588 −0.677768
\(302\) −14.3826 −0.827623
\(303\) −28.7285 −1.65041
\(304\) −0.851836 −0.0488561
\(305\) −3.58885 −0.205497
\(306\) 9.23466 0.527910
\(307\) −17.1933 −0.981271 −0.490635 0.871365i \(-0.663235\pi\)
−0.490635 + 0.871365i \(0.663235\pi\)
\(308\) 0 0
\(309\) −2.89418 −0.164644
\(310\) 0.145026 0.00823693
\(311\) −13.9706 −0.792200 −0.396100 0.918207i \(-0.629637\pi\)
−0.396100 + 0.918207i \(0.629637\pi\)
\(312\) 4.67936 0.264917
\(313\) 10.4793 0.592327 0.296163 0.955137i \(-0.404293\pi\)
0.296163 + 0.955137i \(0.404293\pi\)
\(314\) 12.6590 0.714390
\(315\) 1.23973 0.0698511
\(316\) 10.7068 0.602303
\(317\) −32.2756 −1.81278 −0.906390 0.422442i \(-0.861173\pi\)
−0.906390 + 0.422442i \(0.861173\pi\)
\(318\) 1.98214 0.111153
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −38.9743 −2.17533
\(322\) −0.651398 −0.0363010
\(323\) 6.34524 0.353059
\(324\) −11.1823 −0.621237
\(325\) −2.27257 −0.126060
\(326\) 8.95282 0.495851
\(327\) −22.3250 −1.23457
\(328\) −4.64211 −0.256318
\(329\) −5.27536 −0.290840
\(330\) 0 0
\(331\) 27.1275 1.49106 0.745532 0.666470i \(-0.232196\pi\)
0.745532 + 0.666470i \(0.232196\pi\)
\(332\) 1.08367 0.0594741
\(333\) 2.88500 0.158097
\(334\) −6.36465 −0.348258
\(335\) −0.430333 −0.0235116
\(336\) 2.05906 0.112331
\(337\) 25.8297 1.40703 0.703517 0.710678i \(-0.251612\pi\)
0.703517 + 0.710678i \(0.251612\pi\)
\(338\) 7.83543 0.426191
\(339\) −35.3353 −1.91915
\(340\) −7.44890 −0.403973
\(341\) 0 0
\(342\) 1.05605 0.0571047
\(343\) 1.00000 0.0539949
\(344\) 11.7588 0.633993
\(345\) 1.34127 0.0722115
\(346\) 24.4230 1.31299
\(347\) 6.20850 0.333290 0.166645 0.986017i \(-0.446707\pi\)
0.166645 + 0.986017i \(0.446707\pi\)
\(348\) 4.87258 0.261198
\(349\) −10.3551 −0.554295 −0.277148 0.960827i \(-0.589389\pi\)
−0.277148 + 0.960827i \(0.589389\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 8.23692 0.439654
\(352\) 0 0
\(353\) 27.4798 1.46260 0.731300 0.682055i \(-0.238914\pi\)
0.731300 + 0.682055i \(0.238914\pi\)
\(354\) −2.14753 −0.114140
\(355\) 10.6384 0.564630
\(356\) 3.73563 0.197988
\(357\) −15.3377 −0.811759
\(358\) −5.80767 −0.306945
\(359\) 0.0307419 0.00162250 0.000811248 1.00000i \(-0.499742\pi\)
0.000811248 1.00000i \(0.499742\pi\)
\(360\) −1.23973 −0.0653397
\(361\) −18.2744 −0.961809
\(362\) 12.8812 0.677019
\(363\) 0 0
\(364\) −2.27257 −0.119115
\(365\) −1.16253 −0.0608493
\(366\) 7.38966 0.386264
\(367\) −10.2929 −0.537285 −0.268643 0.963240i \(-0.586575\pi\)
−0.268643 + 0.963240i \(0.586575\pi\)
\(368\) 0.651398 0.0339565
\(369\) 5.75499 0.299593
\(370\) −2.32711 −0.120981
\(371\) −0.962643 −0.0499779
\(372\) −0.298618 −0.0154826
\(373\) −12.2144 −0.632438 −0.316219 0.948686i \(-0.602413\pi\)
−0.316219 + 0.948686i \(0.602413\pi\)
\(374\) 0 0
\(375\) 2.05906 0.106329
\(376\) 5.27536 0.272056
\(377\) −5.37782 −0.276972
\(378\) 3.62450 0.186424
\(379\) 0.699608 0.0359364 0.0179682 0.999839i \(-0.494280\pi\)
0.0179682 + 0.999839i \(0.494280\pi\)
\(380\) −0.851836 −0.0436983
\(381\) 22.8942 1.17291
\(382\) 5.65871 0.289525
\(383\) −37.4527 −1.91374 −0.956871 0.290513i \(-0.906174\pi\)
−0.956871 + 0.290513i \(0.906174\pi\)
\(384\) −2.05906 −0.105076
\(385\) 0 0
\(386\) −13.7878 −0.701783
\(387\) −14.5778 −0.741032
\(388\) −1.47983 −0.0751270
\(389\) −8.64880 −0.438512 −0.219256 0.975667i \(-0.570363\pi\)
−0.219256 + 0.975667i \(0.570363\pi\)
\(390\) 4.67936 0.236949
\(391\) −4.85220 −0.245386
\(392\) −1.00000 −0.0505076
\(393\) −22.4770 −1.13382
\(394\) −2.82825 −0.142485
\(395\) 10.7068 0.538716
\(396\) 0 0
\(397\) 9.10554 0.456994 0.228497 0.973545i \(-0.426619\pi\)
0.228497 + 0.973545i \(0.426619\pi\)
\(398\) 0.513194 0.0257241
\(399\) −1.75398 −0.0878090
\(400\) 1.00000 0.0500000
\(401\) 19.2528 0.961437 0.480718 0.876875i \(-0.340376\pi\)
0.480718 + 0.876875i \(0.340376\pi\)
\(402\) 0.886082 0.0441938
\(403\) 0.329582 0.0164177
\(404\) −13.9523 −0.694151
\(405\) −11.1823 −0.555651
\(406\) −2.36641 −0.117443
\(407\) 0 0
\(408\) 15.3377 0.759331
\(409\) 26.7614 1.32327 0.661634 0.749827i \(-0.269863\pi\)
0.661634 + 0.749827i \(0.269863\pi\)
\(410\) −4.64211 −0.229258
\(411\) −9.91144 −0.488895
\(412\) −1.40558 −0.0692481
\(413\) 1.04296 0.0513209
\(414\) −0.807561 −0.0396894
\(415\) 1.08367 0.0531953
\(416\) 2.27257 0.111422
\(417\) −20.8855 −1.02277
\(418\) 0 0
\(419\) −24.4437 −1.19415 −0.597075 0.802185i \(-0.703671\pi\)
−0.597075 + 0.802185i \(0.703671\pi\)
\(420\) 2.05906 0.100472
\(421\) −7.85402 −0.382781 −0.191391 0.981514i \(-0.561300\pi\)
−0.191391 + 0.981514i \(0.561300\pi\)
\(422\) 20.4146 0.993765
\(423\) −6.54005 −0.317988
\(424\) 0.962643 0.0467501
\(425\) −7.44890 −0.361325
\(426\) −21.9052 −1.06131
\(427\) −3.58885 −0.173677
\(428\) −18.9282 −0.914927
\(429\) 0 0
\(430\) 11.7588 0.567061
\(431\) −23.8170 −1.14722 −0.573612 0.819127i \(-0.694458\pi\)
−0.573612 + 0.819127i \(0.694458\pi\)
\(432\) −3.62450 −0.174384
\(433\) 15.4390 0.741952 0.370976 0.928642i \(-0.379023\pi\)
0.370976 + 0.928642i \(0.379023\pi\)
\(434\) 0.145026 0.00696148
\(435\) 4.87258 0.233622
\(436\) −10.8423 −0.519252
\(437\) −0.554884 −0.0265437
\(438\) 2.39371 0.114376
\(439\) 29.0621 1.38706 0.693530 0.720428i \(-0.256054\pi\)
0.693530 + 0.720428i \(0.256054\pi\)
\(440\) 0 0
\(441\) 1.23973 0.0590350
\(442\) −16.9281 −0.805190
\(443\) 34.7126 1.64924 0.824622 0.565685i \(-0.191388\pi\)
0.824622 + 0.565685i \(0.191388\pi\)
\(444\) 4.79166 0.227402
\(445\) 3.73563 0.177086
\(446\) −15.3296 −0.725877
\(447\) 3.87993 0.183515
\(448\) 1.00000 0.0472456
\(449\) −30.7259 −1.45005 −0.725024 0.688724i \(-0.758171\pi\)
−0.725024 + 0.688724i \(0.758171\pi\)
\(450\) −1.23973 −0.0584416
\(451\) 0 0
\(452\) −17.1609 −0.807179
\(453\) 29.6146 1.39141
\(454\) 28.8027 1.35178
\(455\) −2.27257 −0.106540
\(456\) 1.75398 0.0821378
\(457\) 20.4932 0.958633 0.479317 0.877642i \(-0.340885\pi\)
0.479317 + 0.877642i \(0.340885\pi\)
\(458\) −7.06918 −0.330321
\(459\) 26.9985 1.26018
\(460\) 0.651398 0.0303716
\(461\) 17.5629 0.817988 0.408994 0.912537i \(-0.365880\pi\)
0.408994 + 0.912537i \(0.365880\pi\)
\(462\) 0 0
\(463\) 13.6361 0.633721 0.316861 0.948472i \(-0.397371\pi\)
0.316861 + 0.948472i \(0.397371\pi\)
\(464\) 2.36641 0.109858
\(465\) −0.298618 −0.0138481
\(466\) −25.1088 −1.16314
\(467\) −30.7210 −1.42160 −0.710799 0.703395i \(-0.751666\pi\)
−0.710799 + 0.703395i \(0.751666\pi\)
\(468\) −2.81738 −0.130234
\(469\) −0.430333 −0.0198709
\(470\) 5.27536 0.243334
\(471\) −26.0657 −1.20105
\(472\) −1.04296 −0.0480063
\(473\) 0 0
\(474\) −22.0459 −1.01260
\(475\) −0.851836 −0.0390849
\(476\) −7.44890 −0.341420
\(477\) −1.19342 −0.0546430
\(478\) −9.09178 −0.415848
\(479\) 34.6829 1.58470 0.792352 0.610064i \(-0.208856\pi\)
0.792352 + 0.610064i \(0.208856\pi\)
\(480\) −2.05906 −0.0939829
\(481\) −5.28852 −0.241136
\(482\) −11.8409 −0.539338
\(483\) 1.34127 0.0610298
\(484\) 0 0
\(485\) −1.47983 −0.0671956
\(486\) 12.1515 0.551202
\(487\) 35.5251 1.60980 0.804899 0.593412i \(-0.202220\pi\)
0.804899 + 0.593412i \(0.202220\pi\)
\(488\) 3.58885 0.162460
\(489\) −18.4344 −0.833633
\(490\) −1.00000 −0.0451754
\(491\) 26.0609 1.17611 0.588055 0.808821i \(-0.299894\pi\)
0.588055 + 0.808821i \(0.299894\pi\)
\(492\) 9.55840 0.430926
\(493\) −17.6271 −0.793886
\(494\) −1.93586 −0.0870983
\(495\) 0 0
\(496\) −0.145026 −0.00651187
\(497\) 10.6384 0.477200
\(498\) −2.23134 −0.0999889
\(499\) 13.6585 0.611439 0.305719 0.952122i \(-0.401103\pi\)
0.305719 + 0.952122i \(0.401103\pi\)
\(500\) 1.00000 0.0447214
\(501\) 13.1052 0.585497
\(502\) 12.2824 0.548191
\(503\) 33.6925 1.50227 0.751136 0.660147i \(-0.229506\pi\)
0.751136 + 0.660147i \(0.229506\pi\)
\(504\) −1.23973 −0.0552222
\(505\) −13.9523 −0.620867
\(506\) 0 0
\(507\) −16.1336 −0.716519
\(508\) 11.1188 0.493316
\(509\) 25.0801 1.11166 0.555829 0.831297i \(-0.312401\pi\)
0.555829 + 0.831297i \(0.312401\pi\)
\(510\) 15.3377 0.679167
\(511\) −1.16253 −0.0514271
\(512\) −1.00000 −0.0441942
\(513\) 3.08748 0.136315
\(514\) 23.8811 1.05335
\(515\) −1.40558 −0.0619374
\(516\) −24.2121 −1.06588
\(517\) 0 0
\(518\) −2.32711 −0.102247
\(519\) −50.2884 −2.20741
\(520\) 2.27257 0.0996588
\(521\) −23.2513 −1.01866 −0.509330 0.860572i \(-0.670107\pi\)
−0.509330 + 0.860572i \(0.670107\pi\)
\(522\) −2.93372 −0.128405
\(523\) 0.198677 0.00868753 0.00434376 0.999991i \(-0.498617\pi\)
0.00434376 + 0.999991i \(0.498617\pi\)
\(524\) −10.9162 −0.476874
\(525\) 2.05906 0.0898648
\(526\) 10.5902 0.461754
\(527\) 1.08029 0.0470580
\(528\) 0 0
\(529\) −22.5757 −0.981551
\(530\) 0.962643 0.0418145
\(531\) 1.29300 0.0561113
\(532\) −0.851836 −0.0369318
\(533\) −10.5495 −0.456951
\(534\) −7.69189 −0.332861
\(535\) −18.9282 −0.818336
\(536\) 0.430333 0.0185876
\(537\) 11.9584 0.516041
\(538\) −16.5719 −0.714465
\(539\) 0 0
\(540\) −3.62450 −0.155973
\(541\) 32.5444 1.39919 0.699597 0.714538i \(-0.253363\pi\)
0.699597 + 0.714538i \(0.253363\pi\)
\(542\) 29.4244 1.26389
\(543\) −26.5231 −1.13822
\(544\) 7.44890 0.319369
\(545\) −10.8423 −0.464433
\(546\) 4.67936 0.200258
\(547\) −24.7703 −1.05910 −0.529550 0.848279i \(-0.677639\pi\)
−0.529550 + 0.848279i \(0.677639\pi\)
\(548\) −4.81357 −0.205626
\(549\) −4.44922 −0.189888
\(550\) 0 0
\(551\) −2.01579 −0.0858755
\(552\) −1.34127 −0.0570882
\(553\) 10.7068 0.455298
\(554\) −29.6933 −1.26155
\(555\) 4.79166 0.203395
\(556\) −10.1432 −0.430168
\(557\) 37.3936 1.58442 0.792208 0.610251i \(-0.208931\pi\)
0.792208 + 0.610251i \(0.208931\pi\)
\(558\) 0.179794 0.00761128
\(559\) 26.7228 1.13025
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 23.8076 1.00426
\(563\) 17.2949 0.728894 0.364447 0.931224i \(-0.381258\pi\)
0.364447 + 0.931224i \(0.381258\pi\)
\(564\) −10.8623 −0.457385
\(565\) −17.1609 −0.721963
\(566\) 17.5036 0.735732
\(567\) −11.1823 −0.469611
\(568\) −10.6384 −0.446379
\(569\) 38.4117 1.61030 0.805152 0.593068i \(-0.202084\pi\)
0.805152 + 0.593068i \(0.202084\pi\)
\(570\) 1.75398 0.0734662
\(571\) −5.08249 −0.212696 −0.106348 0.994329i \(-0.533916\pi\)
−0.106348 + 0.994329i \(0.533916\pi\)
\(572\) 0 0
\(573\) −11.6516 −0.486754
\(574\) −4.64211 −0.193758
\(575\) 0.651398 0.0271652
\(576\) 1.23973 0.0516556
\(577\) 41.3365 1.72086 0.860430 0.509568i \(-0.170195\pi\)
0.860430 + 0.509568i \(0.170195\pi\)
\(578\) −38.4861 −1.60081
\(579\) 28.3900 1.17985
\(580\) 2.36641 0.0982597
\(581\) 1.08367 0.0449582
\(582\) 3.04706 0.126305
\(583\) 0 0
\(584\) 1.16253 0.0481056
\(585\) −2.81738 −0.116484
\(586\) −23.1174 −0.954972
\(587\) −6.92122 −0.285669 −0.142835 0.989747i \(-0.545622\pi\)
−0.142835 + 0.989747i \(0.545622\pi\)
\(588\) 2.05906 0.0849143
\(589\) 0.123538 0.00509032
\(590\) −1.04296 −0.0429381
\(591\) 5.82354 0.239548
\(592\) 2.32711 0.0956436
\(593\) 22.7439 0.933979 0.466989 0.884263i \(-0.345339\pi\)
0.466989 + 0.884263i \(0.345339\pi\)
\(594\) 0 0
\(595\) −7.44890 −0.305375
\(596\) 1.88432 0.0771848
\(597\) −1.05670 −0.0432477
\(598\) 1.48035 0.0605359
\(599\) 4.13028 0.168759 0.0843794 0.996434i \(-0.473109\pi\)
0.0843794 + 0.996434i \(0.473109\pi\)
\(600\) −2.05906 −0.0840608
\(601\) 3.87084 0.157895 0.0789474 0.996879i \(-0.474844\pi\)
0.0789474 + 0.996879i \(0.474844\pi\)
\(602\) 11.7588 0.479254
\(603\) −0.533499 −0.0217257
\(604\) 14.3826 0.585218
\(605\) 0 0
\(606\) 28.7285 1.16702
\(607\) 29.7726 1.20843 0.604216 0.796820i \(-0.293486\pi\)
0.604216 + 0.796820i \(0.293486\pi\)
\(608\) 0.851836 0.0345465
\(609\) 4.87258 0.197447
\(610\) 3.58885 0.145308
\(611\) 11.9886 0.485008
\(612\) −9.23466 −0.373289
\(613\) −26.1253 −1.05519 −0.527595 0.849496i \(-0.676906\pi\)
−0.527595 + 0.849496i \(0.676906\pi\)
\(614\) 17.1933 0.693863
\(615\) 9.55840 0.385432
\(616\) 0 0
\(617\) 13.3021 0.535524 0.267762 0.963485i \(-0.413716\pi\)
0.267762 + 0.963485i \(0.413716\pi\)
\(618\) 2.89418 0.116421
\(619\) −9.29198 −0.373476 −0.186738 0.982410i \(-0.559792\pi\)
−0.186738 + 0.982410i \(0.559792\pi\)
\(620\) −0.145026 −0.00582439
\(621\) −2.36099 −0.0947432
\(622\) 13.9706 0.560170
\(623\) 3.73563 0.149665
\(624\) −4.67936 −0.187324
\(625\) 1.00000 0.0400000
\(626\) −10.4793 −0.418838
\(627\) 0 0
\(628\) −12.6590 −0.505150
\(629\) −17.3344 −0.691168
\(630\) −1.23973 −0.0493922
\(631\) −29.7104 −1.18275 −0.591376 0.806396i \(-0.701415\pi\)
−0.591376 + 0.806396i \(0.701415\pi\)
\(632\) −10.7068 −0.425892
\(633\) −42.0348 −1.67073
\(634\) 32.2756 1.28183
\(635\) 11.1188 0.441235
\(636\) −1.98214 −0.0785970
\(637\) −2.27257 −0.0900425
\(638\) 0 0
\(639\) 13.1889 0.521743
\(640\) −1.00000 −0.0395285
\(641\) −17.1408 −0.677020 −0.338510 0.940963i \(-0.609923\pi\)
−0.338510 + 0.940963i \(0.609923\pi\)
\(642\) 38.9743 1.53819
\(643\) −25.7316 −1.01476 −0.507378 0.861724i \(-0.669385\pi\)
−0.507378 + 0.861724i \(0.669385\pi\)
\(644\) 0.651398 0.0256687
\(645\) −24.2121 −0.953352
\(646\) −6.34524 −0.249650
\(647\) −21.6382 −0.850686 −0.425343 0.905032i \(-0.639847\pi\)
−0.425343 + 0.905032i \(0.639847\pi\)
\(648\) 11.1823 0.439281
\(649\) 0 0
\(650\) 2.27257 0.0891375
\(651\) −0.298618 −0.0117038
\(652\) −8.95282 −0.350620
\(653\) 38.5661 1.50921 0.754604 0.656181i \(-0.227829\pi\)
0.754604 + 0.656181i \(0.227829\pi\)
\(654\) 22.3250 0.872975
\(655\) −10.9162 −0.426529
\(656\) 4.64211 0.181244
\(657\) −1.44122 −0.0562274
\(658\) 5.27536 0.205655
\(659\) −41.3104 −1.60923 −0.804613 0.593799i \(-0.797627\pi\)
−0.804613 + 0.593799i \(0.797627\pi\)
\(660\) 0 0
\(661\) −17.6230 −0.685456 −0.342728 0.939435i \(-0.611351\pi\)
−0.342728 + 0.939435i \(0.611351\pi\)
\(662\) −27.1275 −1.05434
\(663\) 34.8561 1.35370
\(664\) −1.08367 −0.0420546
\(665\) −0.851836 −0.0330328
\(666\) −2.88500 −0.111791
\(667\) 1.54147 0.0596861
\(668\) 6.36465 0.246256
\(669\) 31.5645 1.22036
\(670\) 0.430333 0.0166252
\(671\) 0 0
\(672\) −2.05906 −0.0794300
\(673\) −23.9060 −0.921508 −0.460754 0.887528i \(-0.652421\pi\)
−0.460754 + 0.887528i \(0.652421\pi\)
\(674\) −25.8297 −0.994923
\(675\) −3.62450 −0.139507
\(676\) −7.83543 −0.301363
\(677\) −4.64307 −0.178448 −0.0892238 0.996012i \(-0.528439\pi\)
−0.0892238 + 0.996012i \(0.528439\pi\)
\(678\) 35.3353 1.35704
\(679\) −1.47983 −0.0567906
\(680\) 7.44890 0.285652
\(681\) −59.3064 −2.27263
\(682\) 0 0
\(683\) −25.5818 −0.978859 −0.489430 0.872043i \(-0.662795\pi\)
−0.489430 + 0.872043i \(0.662795\pi\)
\(684\) −1.05605 −0.0403791
\(685\) −4.81357 −0.183917
\(686\) −1.00000 −0.0381802
\(687\) 14.5559 0.555341
\(688\) −11.7588 −0.448301
\(689\) 2.18767 0.0833437
\(690\) −1.34127 −0.0510612
\(691\) 22.5390 0.857424 0.428712 0.903441i \(-0.358967\pi\)
0.428712 + 0.903441i \(0.358967\pi\)
\(692\) −24.4230 −0.928421
\(693\) 0 0
\(694\) −6.20850 −0.235671
\(695\) −10.1432 −0.384754
\(696\) −4.87258 −0.184695
\(697\) −34.5786 −1.30976
\(698\) 10.3551 0.391946
\(699\) 51.7006 1.95550
\(700\) 1.00000 0.0377964
\(701\) 34.9336 1.31942 0.659711 0.751519i \(-0.270679\pi\)
0.659711 + 0.751519i \(0.270679\pi\)
\(702\) −8.23692 −0.310882
\(703\) −1.98232 −0.0747645
\(704\) 0 0
\(705\) −10.8623 −0.409098
\(706\) −27.4798 −1.03422
\(707\) −13.9523 −0.524729
\(708\) 2.14753 0.0807089
\(709\) −2.18225 −0.0819563 −0.0409781 0.999160i \(-0.513047\pi\)
−0.0409781 + 0.999160i \(0.513047\pi\)
\(710\) −10.6384 −0.399254
\(711\) 13.2735 0.497797
\(712\) −3.73563 −0.139999
\(713\) −0.0944698 −0.00353792
\(714\) 15.3377 0.574001
\(715\) 0 0
\(716\) 5.80767 0.217043
\(717\) 18.7205 0.699131
\(718\) −0.0307419 −0.00114728
\(719\) 23.1198 0.862224 0.431112 0.902299i \(-0.358121\pi\)
0.431112 + 0.902299i \(0.358121\pi\)
\(720\) 1.23973 0.0462022
\(721\) −1.40558 −0.0523466
\(722\) 18.2744 0.680102
\(723\) 24.3811 0.906745
\(724\) −12.8812 −0.478725
\(725\) 2.36641 0.0878861
\(726\) 0 0
\(727\) −48.6753 −1.80527 −0.902633 0.430410i \(-0.858369\pi\)
−0.902633 + 0.430410i \(0.858369\pi\)
\(728\) 2.27257 0.0842270
\(729\) 8.52614 0.315783
\(730\) 1.16253 0.0430270
\(731\) 87.5903 3.23964
\(732\) −7.38966 −0.273130
\(733\) 21.8686 0.807735 0.403867 0.914818i \(-0.367666\pi\)
0.403867 + 0.914818i \(0.367666\pi\)
\(734\) 10.2929 0.379918
\(735\) 2.05906 0.0759496
\(736\) −0.651398 −0.0240109
\(737\) 0 0
\(738\) −5.75499 −0.211844
\(739\) −0.178113 −0.00655200 −0.00327600 0.999995i \(-0.501043\pi\)
−0.00327600 + 0.999995i \(0.501043\pi\)
\(740\) 2.32711 0.0855463
\(741\) 3.98605 0.146431
\(742\) 0.962643 0.0353397
\(743\) −37.0632 −1.35972 −0.679859 0.733343i \(-0.737959\pi\)
−0.679859 + 0.733343i \(0.737959\pi\)
\(744\) 0.298618 0.0109479
\(745\) 1.88432 0.0690362
\(746\) 12.2144 0.447201
\(747\) 1.34346 0.0491548
\(748\) 0 0
\(749\) −18.9282 −0.691620
\(750\) −2.05906 −0.0751863
\(751\) −26.4258 −0.964291 −0.482146 0.876091i \(-0.660142\pi\)
−0.482146 + 0.876091i \(0.660142\pi\)
\(752\) −5.27536 −0.192373
\(753\) −25.2903 −0.921628
\(754\) 5.37782 0.195849
\(755\) 14.3826 0.523435
\(756\) −3.62450 −0.131822
\(757\) 9.40549 0.341848 0.170924 0.985284i \(-0.445325\pi\)
0.170924 + 0.985284i \(0.445325\pi\)
\(758\) −0.699608 −0.0254109
\(759\) 0 0
\(760\) 0.851836 0.0308993
\(761\) −22.7380 −0.824253 −0.412127 0.911127i \(-0.635214\pi\)
−0.412127 + 0.911127i \(0.635214\pi\)
\(762\) −22.8942 −0.829371
\(763\) −10.8423 −0.392517
\(764\) −5.65871 −0.204725
\(765\) −9.23466 −0.333880
\(766\) 37.4527 1.35322
\(767\) −2.37021 −0.0855832
\(768\) 2.05906 0.0743000
\(769\) 13.3469 0.481300 0.240650 0.970612i \(-0.422639\pi\)
0.240650 + 0.970612i \(0.422639\pi\)
\(770\) 0 0
\(771\) −49.1726 −1.77091
\(772\) 13.7878 0.496235
\(773\) −2.14317 −0.0770845 −0.0385422 0.999257i \(-0.512271\pi\)
−0.0385422 + 0.999257i \(0.512271\pi\)
\(774\) 14.5778 0.523989
\(775\) −0.145026 −0.00520949
\(776\) 1.47983 0.0531228
\(777\) 4.79166 0.171900
\(778\) 8.64880 0.310074
\(779\) −3.95432 −0.141678
\(780\) −4.67936 −0.167548
\(781\) 0 0
\(782\) 4.85220 0.173514
\(783\) −8.57703 −0.306518
\(784\) 1.00000 0.0357143
\(785\) −12.6590 −0.451820
\(786\) 22.4770 0.801729
\(787\) 39.7496 1.41692 0.708461 0.705750i \(-0.249390\pi\)
0.708461 + 0.705750i \(0.249390\pi\)
\(788\) 2.82825 0.100752
\(789\) −21.8058 −0.776308
\(790\) −10.7068 −0.380930
\(791\) −17.1609 −0.610170
\(792\) 0 0
\(793\) 8.15591 0.289625
\(794\) −9.10554 −0.323144
\(795\) −1.98214 −0.0702993
\(796\) −0.513194 −0.0181897
\(797\) −45.8902 −1.62551 −0.812757 0.582603i \(-0.802034\pi\)
−0.812757 + 0.582603i \(0.802034\pi\)
\(798\) 1.75398 0.0620903
\(799\) 39.2957 1.39018
\(800\) −1.00000 −0.0353553
\(801\) 4.63119 0.163635
\(802\) −19.2528 −0.679838
\(803\) 0 0
\(804\) −0.886082 −0.0312497
\(805\) 0.651398 0.0229588
\(806\) −0.329582 −0.0116090
\(807\) 34.1225 1.20117
\(808\) 13.9523 0.490839
\(809\) 3.54263 0.124552 0.0622761 0.998059i \(-0.480164\pi\)
0.0622761 + 0.998059i \(0.480164\pi\)
\(810\) 11.1823 0.392905
\(811\) 31.0933 1.09183 0.545916 0.837840i \(-0.316182\pi\)
0.545916 + 0.837840i \(0.316182\pi\)
\(812\) 2.36641 0.0830446
\(813\) −60.5866 −2.12487
\(814\) 0 0
\(815\) −8.95282 −0.313604
\(816\) −15.3377 −0.536928
\(817\) 10.0166 0.350436
\(818\) −26.7614 −0.935692
\(819\) −2.81738 −0.0984473
\(820\) 4.64211 0.162110
\(821\) −12.0289 −0.419812 −0.209906 0.977722i \(-0.567316\pi\)
−0.209906 + 0.977722i \(0.567316\pi\)
\(822\) 9.91144 0.345701
\(823\) 8.79883 0.306708 0.153354 0.988171i \(-0.450993\pi\)
0.153354 + 0.988171i \(0.450993\pi\)
\(824\) 1.40558 0.0489658
\(825\) 0 0
\(826\) −1.04296 −0.0362893
\(827\) −27.7899 −0.966349 −0.483175 0.875524i \(-0.660516\pi\)
−0.483175 + 0.875524i \(0.660516\pi\)
\(828\) 0.807561 0.0280647
\(829\) −13.8407 −0.480708 −0.240354 0.970685i \(-0.577263\pi\)
−0.240354 + 0.970685i \(0.577263\pi\)
\(830\) −1.08367 −0.0376148
\(831\) 61.1403 2.12094
\(832\) −2.27257 −0.0787872
\(833\) −7.44890 −0.258089
\(834\) 20.8855 0.723205
\(835\) 6.36465 0.220258
\(836\) 0 0
\(837\) 0.525647 0.0181690
\(838\) 24.4437 0.844392
\(839\) 34.3543 1.18604 0.593022 0.805187i \(-0.297935\pi\)
0.593022 + 0.805187i \(0.297935\pi\)
\(840\) −2.05906 −0.0710444
\(841\) −23.4001 −0.806901
\(842\) 7.85402 0.270667
\(843\) −49.0213 −1.68838
\(844\) −20.4146 −0.702698
\(845\) −7.83543 −0.269547
\(846\) 6.54005 0.224852
\(847\) 0 0
\(848\) −0.962643 −0.0330573
\(849\) −36.0410 −1.23693
\(850\) 7.44890 0.255495
\(851\) 1.51588 0.0519635
\(852\) 21.9052 0.750460
\(853\) −40.0952 −1.37283 −0.686416 0.727209i \(-0.740817\pi\)
−0.686416 + 0.727209i \(0.740817\pi\)
\(854\) 3.58885 0.122808
\(855\) −1.05605 −0.0361162
\(856\) 18.9282 0.646951
\(857\) 26.0287 0.889123 0.444562 0.895748i \(-0.353359\pi\)
0.444562 + 0.895748i \(0.353359\pi\)
\(858\) 0 0
\(859\) 29.9861 1.02311 0.511556 0.859250i \(-0.329069\pi\)
0.511556 + 0.859250i \(0.329069\pi\)
\(860\) −11.7588 −0.400973
\(861\) 9.55840 0.325749
\(862\) 23.8170 0.811210
\(863\) 28.1310 0.957590 0.478795 0.877927i \(-0.341074\pi\)
0.478795 + 0.877927i \(0.341074\pi\)
\(864\) 3.62450 0.123308
\(865\) −24.4230 −0.830405
\(866\) −15.4390 −0.524639
\(867\) 79.2453 2.69131
\(868\) −0.145026 −0.00492251
\(869\) 0 0
\(870\) −4.87258 −0.165196
\(871\) 0.977962 0.0331370
\(872\) 10.8423 0.367166
\(873\) −1.83460 −0.0620917
\(874\) 0.554884 0.0187692
\(875\) 1.00000 0.0338062
\(876\) −2.39371 −0.0808760
\(877\) −15.9142 −0.537383 −0.268691 0.963226i \(-0.586591\pi\)
−0.268691 + 0.963226i \(0.586591\pi\)
\(878\) −29.0621 −0.980800
\(879\) 47.6002 1.60551
\(880\) 0 0
\(881\) 42.3031 1.42523 0.712614 0.701556i \(-0.247511\pi\)
0.712614 + 0.701556i \(0.247511\pi\)
\(882\) −1.23973 −0.0417440
\(883\) −38.5685 −1.29793 −0.648967 0.760817i \(-0.724799\pi\)
−0.648967 + 0.760817i \(0.724799\pi\)
\(884\) 16.9281 0.569355
\(885\) 2.14753 0.0721883
\(886\) −34.7126 −1.16619
\(887\) 11.4973 0.386041 0.193021 0.981195i \(-0.438172\pi\)
0.193021 + 0.981195i \(0.438172\pi\)
\(888\) −4.79166 −0.160798
\(889\) 11.1188 0.372912
\(890\) −3.73563 −0.125219
\(891\) 0 0
\(892\) 15.3296 0.513272
\(893\) 4.49375 0.150377
\(894\) −3.87993 −0.129764
\(895\) 5.80767 0.194129
\(896\) −1.00000 −0.0334077
\(897\) −3.04813 −0.101774
\(898\) 30.7259 1.02534
\(899\) −0.343191 −0.0114461
\(900\) 1.23973 0.0413245
\(901\) 7.17063 0.238888
\(902\) 0 0
\(903\) −24.2121 −0.805730
\(904\) 17.1609 0.570762
\(905\) −12.8812 −0.428185
\(906\) −29.6146 −0.983878
\(907\) 35.9537 1.19382 0.596911 0.802308i \(-0.296395\pi\)
0.596911 + 0.802308i \(0.296395\pi\)
\(908\) −28.8027 −0.955850
\(909\) −17.2971 −0.573708
\(910\) 2.27257 0.0753350
\(911\) 13.3132 0.441086 0.220543 0.975377i \(-0.429217\pi\)
0.220543 + 0.975377i \(0.429217\pi\)
\(912\) −1.75398 −0.0580802
\(913\) 0 0
\(914\) −20.4932 −0.677856
\(915\) −7.38966 −0.244295
\(916\) 7.06918 0.233572
\(917\) −10.9162 −0.360483
\(918\) −26.9985 −0.891083
\(919\) 57.6449 1.90153 0.950764 0.309915i \(-0.100301\pi\)
0.950764 + 0.309915i \(0.100301\pi\)
\(920\) −0.651398 −0.0214760
\(921\) −35.4020 −1.16653
\(922\) −17.5629 −0.578405
\(923\) −24.1766 −0.795783
\(924\) 0 0
\(925\) 2.32711 0.0765149
\(926\) −13.6361 −0.448109
\(927\) −1.74255 −0.0572328
\(928\) −2.36641 −0.0776811
\(929\) 0.549043 0.0180135 0.00900675 0.999959i \(-0.497133\pi\)
0.00900675 + 0.999959i \(0.497133\pi\)
\(930\) 0.298618 0.00979206
\(931\) −0.851836 −0.0279178
\(932\) 25.1088 0.822466
\(933\) −28.7663 −0.941767
\(934\) 30.7210 1.00522
\(935\) 0 0
\(936\) 2.81738 0.0920891
\(937\) 56.7821 1.85499 0.927495 0.373835i \(-0.121957\pi\)
0.927495 + 0.373835i \(0.121957\pi\)
\(938\) 0.430333 0.0140509
\(939\) 21.5776 0.704158
\(940\) −5.27536 −0.172063
\(941\) −29.2904 −0.954840 −0.477420 0.878675i \(-0.658428\pi\)
−0.477420 + 0.878675i \(0.658428\pi\)
\(942\) 26.0657 0.849267
\(943\) 3.02386 0.0984706
\(944\) 1.04296 0.0339456
\(945\) −3.62450 −0.117905
\(946\) 0 0
\(947\) 24.8536 0.807632 0.403816 0.914840i \(-0.367684\pi\)
0.403816 + 0.914840i \(0.367684\pi\)
\(948\) 22.0459 0.716017
\(949\) 2.64192 0.0857603
\(950\) 0.851836 0.0276372
\(951\) −66.4575 −2.15503
\(952\) 7.44890 0.241420
\(953\) −43.8496 −1.42043 −0.710214 0.703986i \(-0.751402\pi\)
−0.710214 + 0.703986i \(0.751402\pi\)
\(954\) 1.19342 0.0386384
\(955\) −5.65871 −0.183111
\(956\) 9.09178 0.294049
\(957\) 0 0
\(958\) −34.6829 −1.12056
\(959\) −4.81357 −0.155438
\(960\) 2.05906 0.0664559
\(961\) −30.9790 −0.999322
\(962\) 5.28852 0.170509
\(963\) −23.4659 −0.756178
\(964\) 11.8409 0.381370
\(965\) 13.7878 0.443846
\(966\) −1.34127 −0.0431546
\(967\) 40.0914 1.28925 0.644627 0.764497i \(-0.277013\pi\)
0.644627 + 0.764497i \(0.277013\pi\)
\(968\) 0 0
\(969\) 13.0652 0.419716
\(970\) 1.47983 0.0475145
\(971\) −51.9047 −1.66570 −0.832850 0.553498i \(-0.813292\pi\)
−0.832850 + 0.553498i \(0.813292\pi\)
\(972\) −12.1515 −0.389759
\(973\) −10.1432 −0.325176
\(974\) −35.5251 −1.13830
\(975\) −4.67936 −0.149860
\(976\) −3.58885 −0.114876
\(977\) −16.3448 −0.522918 −0.261459 0.965215i \(-0.584204\pi\)
−0.261459 + 0.965215i \(0.584204\pi\)
\(978\) 18.4344 0.589467
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −13.4416 −0.429156
\(982\) −26.0609 −0.831636
\(983\) −39.4682 −1.25884 −0.629420 0.777065i \(-0.716708\pi\)
−0.629420 + 0.777065i \(0.716708\pi\)
\(984\) −9.55840 −0.304711
\(985\) 2.82825 0.0901155
\(986\) 17.6271 0.561362
\(987\) −10.8623 −0.345751
\(988\) 1.93586 0.0615878
\(989\) −7.65968 −0.243564
\(990\) 0 0
\(991\) 20.8327 0.661771 0.330886 0.943671i \(-0.392653\pi\)
0.330886 + 0.943671i \(0.392653\pi\)
\(992\) 0.145026 0.00460459
\(993\) 55.8572 1.77258
\(994\) −10.6384 −0.337431
\(995\) −0.513194 −0.0162693
\(996\) 2.23134 0.0707028
\(997\) 1.23042 0.0389678 0.0194839 0.999810i \(-0.493798\pi\)
0.0194839 + 0.999810i \(0.493798\pi\)
\(998\) −13.6585 −0.432353
\(999\) −8.43460 −0.266859
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 8470.2.a.cw.1.5 6
11.5 even 5 770.2.n.j.421.3 12
11.9 even 5 770.2.n.j.631.3 yes 12
11.10 odd 2 8470.2.a.dc.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.j.421.3 12 11.5 even 5
770.2.n.j.631.3 yes 12 11.9 even 5
8470.2.a.cw.1.5 6 1.1 even 1 trivial
8470.2.a.dc.1.5 6 11.10 odd 2