Properties

Label 8470.2.a.bw.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} +4.46410 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -7.46410 q^{17} -2.00000 q^{18} +5.92820 q^{19} +1.00000 q^{20} -1.00000 q^{21} -4.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.46410 q^{26} +5.00000 q^{27} +1.00000 q^{28} -6.92820 q^{29} -1.00000 q^{30} -0.535898 q^{31} +1.00000 q^{32} -7.46410 q^{34} +1.00000 q^{35} -2.00000 q^{36} -8.92820 q^{37} +5.92820 q^{38} -4.46410 q^{39} +1.00000 q^{40} -8.39230 q^{41} -1.00000 q^{42} -12.3923 q^{43} -2.00000 q^{45} -4.46410 q^{46} -0.535898 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +7.46410 q^{51} +4.46410 q^{52} -7.46410 q^{53} +5.00000 q^{54} +1.00000 q^{56} -5.92820 q^{57} -6.92820 q^{58} +9.92820 q^{59} -1.00000 q^{60} +4.92820 q^{61} -0.535898 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.46410 q^{65} +6.39230 q^{67} -7.46410 q^{68} +4.46410 q^{69} +1.00000 q^{70} +8.00000 q^{71} -2.00000 q^{72} -0.535898 q^{73} -8.92820 q^{74} -1.00000 q^{75} +5.92820 q^{76} -4.46410 q^{78} -7.53590 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.39230 q^{82} -1.00000 q^{83} -1.00000 q^{84} -7.46410 q^{85} -12.3923 q^{86} +6.92820 q^{87} -17.4641 q^{89} -2.00000 q^{90} +4.46410 q^{91} -4.46410 q^{92} +0.535898 q^{93} -0.535898 q^{94} +5.92820 q^{95} -1.00000 q^{96} -1.07180 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + 2q^{8} - 4q^{9} + 2q^{10} - 2q^{12} + 2q^{13} + 2q^{14} - 2q^{15} + 2q^{16} - 8q^{17} - 4q^{18} - 2q^{19} + 2q^{20} - 2q^{21} - 2q^{23} - 2q^{24} + 2q^{25} + 2q^{26} + 10q^{27} + 2q^{28} - 2q^{30} - 8q^{31} + 2q^{32} - 8q^{34} + 2q^{35} - 4q^{36} - 4q^{37} - 2q^{38} - 2q^{39} + 2q^{40} + 4q^{41} - 2q^{42} - 4q^{43} - 4q^{45} - 2q^{46} - 8q^{47} - 2q^{48} + 2q^{49} + 2q^{50} + 8q^{51} + 2q^{52} - 8q^{53} + 10q^{54} + 2q^{56} + 2q^{57} + 6q^{59} - 2q^{60} - 4q^{61} - 8q^{62} - 4q^{63} + 2q^{64} + 2q^{65} - 8q^{67} - 8q^{68} + 2q^{69} + 2q^{70} + 16q^{71} - 4q^{72} - 8q^{73} - 4q^{74} - 2q^{75} - 2q^{76} - 2q^{78} - 22q^{79} + 2q^{80} + 2q^{81} + 4q^{82} - 2q^{83} - 2q^{84} - 8q^{85} - 4q^{86} - 28q^{89} - 4q^{90} + 2q^{91} - 2q^{92} + 8q^{93} - 8q^{94} - 2q^{95} - 2q^{96} - 16q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 4.46410 1.23812 0.619060 0.785344i \(-0.287514\pi\)
0.619060 + 0.785344i \(0.287514\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −7.46410 −1.81031 −0.905155 0.425081i \(-0.860246\pi\)
−0.905155 + 0.425081i \(0.860246\pi\)
\(18\) −2.00000 −0.471405
\(19\) 5.92820 1.36002 0.680012 0.733201i \(-0.261975\pi\)
0.680012 + 0.733201i \(0.261975\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.46410 −0.930830 −0.465415 0.885093i \(-0.654095\pi\)
−0.465415 + 0.885093i \(0.654095\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 4.46410 0.875482
\(27\) 5.00000 0.962250
\(28\) 1.00000 0.188982
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) −1.00000 −0.182574
\(31\) −0.535898 −0.0962502 −0.0481251 0.998841i \(-0.515325\pi\)
−0.0481251 + 0.998841i \(0.515325\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −7.46410 −1.28008
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) −8.92820 −1.46779 −0.733894 0.679264i \(-0.762299\pi\)
−0.733894 + 0.679264i \(0.762299\pi\)
\(38\) 5.92820 0.961682
\(39\) −4.46410 −0.714828
\(40\) 1.00000 0.158114
\(41\) −8.39230 −1.31066 −0.655329 0.755344i \(-0.727470\pi\)
−0.655329 + 0.755344i \(0.727470\pi\)
\(42\) −1.00000 −0.154303
\(43\) −12.3923 −1.88981 −0.944904 0.327346i \(-0.893846\pi\)
−0.944904 + 0.327346i \(0.893846\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −4.46410 −0.658196
\(47\) −0.535898 −0.0781688 −0.0390844 0.999236i \(-0.512444\pi\)
−0.0390844 + 0.999236i \(0.512444\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 7.46410 1.04518
\(52\) 4.46410 0.619060
\(53\) −7.46410 −1.02527 −0.512637 0.858606i \(-0.671331\pi\)
−0.512637 + 0.858606i \(0.671331\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −5.92820 −0.785210
\(58\) −6.92820 −0.909718
\(59\) 9.92820 1.29254 0.646271 0.763108i \(-0.276328\pi\)
0.646271 + 0.763108i \(0.276328\pi\)
\(60\) −1.00000 −0.129099
\(61\) 4.92820 0.630992 0.315496 0.948927i \(-0.397829\pi\)
0.315496 + 0.948927i \(0.397829\pi\)
\(62\) −0.535898 −0.0680592
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 4.46410 0.553704
\(66\) 0 0
\(67\) 6.39230 0.780944 0.390472 0.920615i \(-0.372312\pi\)
0.390472 + 0.920615i \(0.372312\pi\)
\(68\) −7.46410 −0.905155
\(69\) 4.46410 0.537415
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −2.00000 −0.235702
\(73\) −0.535898 −0.0627222 −0.0313611 0.999508i \(-0.509984\pi\)
−0.0313611 + 0.999508i \(0.509984\pi\)
\(74\) −8.92820 −1.03788
\(75\) −1.00000 −0.115470
\(76\) 5.92820 0.680012
\(77\) 0 0
\(78\) −4.46410 −0.505460
\(79\) −7.53590 −0.847855 −0.423927 0.905696i \(-0.639349\pi\)
−0.423927 + 0.905696i \(0.639349\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −8.39230 −0.926775
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) −1.00000 −0.109109
\(85\) −7.46410 −0.809595
\(86\) −12.3923 −1.33630
\(87\) 6.92820 0.742781
\(88\) 0 0
\(89\) −17.4641 −1.85119 −0.925596 0.378514i \(-0.876435\pi\)
−0.925596 + 0.378514i \(0.876435\pi\)
\(90\) −2.00000 −0.210819
\(91\) 4.46410 0.467965
\(92\) −4.46410 −0.465415
\(93\) 0.535898 0.0555701
\(94\) −0.535898 −0.0552737
\(95\) 5.92820 0.608221
\(96\) −1.00000 −0.102062
\(97\) −1.07180 −0.108824 −0.0544122 0.998519i \(-0.517329\pi\)
−0.0544122 + 0.998519i \(0.517329\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −11.5359 −1.14786 −0.573932 0.818903i \(-0.694583\pi\)
−0.573932 + 0.818903i \(0.694583\pi\)
\(102\) 7.46410 0.739056
\(103\) −1.46410 −0.144262 −0.0721311 0.997395i \(-0.522980\pi\)
−0.0721311 + 0.997395i \(0.522980\pi\)
\(104\) 4.46410 0.437741
\(105\) −1.00000 −0.0975900
\(106\) −7.46410 −0.724978
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 5.00000 0.481125
\(109\) 18.3923 1.76166 0.880832 0.473430i \(-0.156984\pi\)
0.880832 + 0.473430i \(0.156984\pi\)
\(110\) 0 0
\(111\) 8.92820 0.847428
\(112\) 1.00000 0.0944911
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) −5.92820 −0.555227
\(115\) −4.46410 −0.416280
\(116\) −6.92820 −0.643268
\(117\) −8.92820 −0.825413
\(118\) 9.92820 0.913965
\(119\) −7.46410 −0.684233
\(120\) −1.00000 −0.0912871
\(121\) 0 0
\(122\) 4.92820 0.446179
\(123\) 8.39230 0.756709
\(124\) −0.535898 −0.0481251
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) 13.3923 1.18837 0.594187 0.804327i \(-0.297474\pi\)
0.594187 + 0.804327i \(0.297474\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.3923 1.09108
\(130\) 4.46410 0.391528
\(131\) −10.8564 −0.948529 −0.474264 0.880383i \(-0.657286\pi\)
−0.474264 + 0.880383i \(0.657286\pi\)
\(132\) 0 0
\(133\) 5.92820 0.514040
\(134\) 6.39230 0.552211
\(135\) 5.00000 0.430331
\(136\) −7.46410 −0.640041
\(137\) 7.92820 0.677352 0.338676 0.940903i \(-0.390021\pi\)
0.338676 + 0.940903i \(0.390021\pi\)
\(138\) 4.46410 0.380010
\(139\) 16.8564 1.42974 0.714871 0.699256i \(-0.246485\pi\)
0.714871 + 0.699256i \(0.246485\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0.535898 0.0451308
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −6.92820 −0.575356
\(146\) −0.535898 −0.0443513
\(147\) −1.00000 −0.0824786
\(148\) −8.92820 −0.733894
\(149\) −3.07180 −0.251651 −0.125826 0.992052i \(-0.540158\pi\)
−0.125826 + 0.992052i \(0.540158\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −5.53590 −0.450505 −0.225253 0.974300i \(-0.572321\pi\)
−0.225253 + 0.974300i \(0.572321\pi\)
\(152\) 5.92820 0.480841
\(153\) 14.9282 1.20687
\(154\) 0 0
\(155\) −0.535898 −0.0430444
\(156\) −4.46410 −0.357414
\(157\) −20.3205 −1.62175 −0.810877 0.585217i \(-0.801009\pi\)
−0.810877 + 0.585217i \(0.801009\pi\)
\(158\) −7.53590 −0.599524
\(159\) 7.46410 0.591942
\(160\) 1.00000 0.0790569
\(161\) −4.46410 −0.351820
\(162\) 1.00000 0.0785674
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) −8.39230 −0.655329
\(165\) 0 0
\(166\) −1.00000 −0.0776151
\(167\) −4.39230 −0.339887 −0.169943 0.985454i \(-0.554358\pi\)
−0.169943 + 0.985454i \(0.554358\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 6.92820 0.532939
\(170\) −7.46410 −0.572470
\(171\) −11.8564 −0.906682
\(172\) −12.3923 −0.944904
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 6.92820 0.525226
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −9.92820 −0.746249
\(178\) −17.4641 −1.30899
\(179\) −8.39230 −0.627270 −0.313635 0.949544i \(-0.601547\pi\)
−0.313635 + 0.949544i \(0.601547\pi\)
\(180\) −2.00000 −0.149071
\(181\) 10.4641 0.777791 0.388895 0.921282i \(-0.372857\pi\)
0.388895 + 0.921282i \(0.372857\pi\)
\(182\) 4.46410 0.330901
\(183\) −4.92820 −0.364303
\(184\) −4.46410 −0.329098
\(185\) −8.92820 −0.656415
\(186\) 0.535898 0.0392940
\(187\) 0 0
\(188\) −0.535898 −0.0390844
\(189\) 5.00000 0.363696
\(190\) 5.92820 0.430077
\(191\) 3.39230 0.245459 0.122729 0.992440i \(-0.460835\pi\)
0.122729 + 0.992440i \(0.460835\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) −1.07180 −0.0769505
\(195\) −4.46410 −0.319681
\(196\) 1.00000 0.0714286
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 0 0
\(199\) −25.8564 −1.83291 −0.916456 0.400135i \(-0.868963\pi\)
−0.916456 + 0.400135i \(0.868963\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.39230 −0.450878
\(202\) −11.5359 −0.811663
\(203\) −6.92820 −0.486265
\(204\) 7.46410 0.522592
\(205\) −8.39230 −0.586144
\(206\) −1.46410 −0.102009
\(207\) 8.92820 0.620553
\(208\) 4.46410 0.309530
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) −24.0000 −1.65223 −0.826114 0.563503i \(-0.809453\pi\)
−0.826114 + 0.563503i \(0.809453\pi\)
\(212\) −7.46410 −0.512637
\(213\) −8.00000 −0.548151
\(214\) −6.00000 −0.410152
\(215\) −12.3923 −0.845148
\(216\) 5.00000 0.340207
\(217\) −0.535898 −0.0363792
\(218\) 18.3923 1.24568
\(219\) 0.535898 0.0362127
\(220\) 0 0
\(221\) −33.3205 −2.24138
\(222\) 8.92820 0.599222
\(223\) −3.46410 −0.231973 −0.115987 0.993251i \(-0.537003\pi\)
−0.115987 + 0.993251i \(0.537003\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.00000 −0.133333
\(226\) 1.00000 0.0665190
\(227\) −16.0000 −1.06196 −0.530979 0.847385i \(-0.678176\pi\)
−0.530979 + 0.847385i \(0.678176\pi\)
\(228\) −5.92820 −0.392605
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) −4.46410 −0.294354
\(231\) 0 0
\(232\) −6.92820 −0.454859
\(233\) −18.8564 −1.23532 −0.617662 0.786444i \(-0.711920\pi\)
−0.617662 + 0.786444i \(0.711920\pi\)
\(234\) −8.92820 −0.583655
\(235\) −0.535898 −0.0349582
\(236\) 9.92820 0.646271
\(237\) 7.53590 0.489509
\(238\) −7.46410 −0.483826
\(239\) −12.4641 −0.806236 −0.403118 0.915148i \(-0.632074\pi\)
−0.403118 + 0.915148i \(0.632074\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 4.92820 0.315496
\(245\) 1.00000 0.0638877
\(246\) 8.39230 0.535074
\(247\) 26.4641 1.68387
\(248\) −0.535898 −0.0340296
\(249\) 1.00000 0.0633724
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 13.3923 0.840308
\(255\) 7.46410 0.467420
\(256\) 1.00000 0.0625000
\(257\) 2.39230 0.149228 0.0746139 0.997212i \(-0.476228\pi\)
0.0746139 + 0.997212i \(0.476228\pi\)
\(258\) 12.3923 0.771511
\(259\) −8.92820 −0.554772
\(260\) 4.46410 0.276852
\(261\) 13.8564 0.857690
\(262\) −10.8564 −0.670711
\(263\) 16.4641 1.01522 0.507610 0.861587i \(-0.330529\pi\)
0.507610 + 0.861587i \(0.330529\pi\)
\(264\) 0 0
\(265\) −7.46410 −0.458516
\(266\) 5.92820 0.363481
\(267\) 17.4641 1.06879
\(268\) 6.39230 0.390472
\(269\) −10.3205 −0.629252 −0.314626 0.949216i \(-0.601879\pi\)
−0.314626 + 0.949216i \(0.601879\pi\)
\(270\) 5.00000 0.304290
\(271\) 29.8564 1.81365 0.906824 0.421510i \(-0.138500\pi\)
0.906824 + 0.421510i \(0.138500\pi\)
\(272\) −7.46410 −0.452578
\(273\) −4.46410 −0.270180
\(274\) 7.92820 0.478960
\(275\) 0 0
\(276\) 4.46410 0.268707
\(277\) 27.3205 1.64153 0.820765 0.571266i \(-0.193547\pi\)
0.820765 + 0.571266i \(0.193547\pi\)
\(278\) 16.8564 1.01098
\(279\) 1.07180 0.0641668
\(280\) 1.00000 0.0597614
\(281\) 17.0000 1.01413 0.507067 0.861906i \(-0.330729\pi\)
0.507067 + 0.861906i \(0.330729\pi\)
\(282\) 0.535898 0.0319123
\(283\) −13.7846 −0.819410 −0.409705 0.912218i \(-0.634368\pi\)
−0.409705 + 0.912218i \(0.634368\pi\)
\(284\) 8.00000 0.474713
\(285\) −5.92820 −0.351156
\(286\) 0 0
\(287\) −8.39230 −0.495382
\(288\) −2.00000 −0.117851
\(289\) 38.7128 2.27722
\(290\) −6.92820 −0.406838
\(291\) 1.07180 0.0628298
\(292\) −0.535898 −0.0313611
\(293\) −20.4641 −1.19553 −0.597763 0.801673i \(-0.703944\pi\)
−0.597763 + 0.801673i \(0.703944\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 9.92820 0.578042
\(296\) −8.92820 −0.518941
\(297\) 0 0
\(298\) −3.07180 −0.177944
\(299\) −19.9282 −1.15248
\(300\) −1.00000 −0.0577350
\(301\) −12.3923 −0.714281
\(302\) −5.53590 −0.318555
\(303\) 11.5359 0.662720
\(304\) 5.92820 0.340006
\(305\) 4.92820 0.282188
\(306\) 14.9282 0.853389
\(307\) −14.9282 −0.851998 −0.425999 0.904724i \(-0.640077\pi\)
−0.425999 + 0.904724i \(0.640077\pi\)
\(308\) 0 0
\(309\) 1.46410 0.0832898
\(310\) −0.535898 −0.0304370
\(311\) 23.3205 1.32238 0.661192 0.750216i \(-0.270051\pi\)
0.661192 + 0.750216i \(0.270051\pi\)
\(312\) −4.46410 −0.252730
\(313\) 14.7846 0.835676 0.417838 0.908522i \(-0.362788\pi\)
0.417838 + 0.908522i \(0.362788\pi\)
\(314\) −20.3205 −1.14675
\(315\) −2.00000 −0.112687
\(316\) −7.53590 −0.423927
\(317\) −22.3923 −1.25768 −0.628839 0.777536i \(-0.716469\pi\)
−0.628839 + 0.777536i \(0.716469\pi\)
\(318\) 7.46410 0.418566
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 6.00000 0.334887
\(322\) −4.46410 −0.248775
\(323\) −44.2487 −2.46206
\(324\) 1.00000 0.0555556
\(325\) 4.46410 0.247624
\(326\) −22.0000 −1.21847
\(327\) −18.3923 −1.01710
\(328\) −8.39230 −0.463388
\(329\) −0.535898 −0.0295450
\(330\) 0 0
\(331\) 1.46410 0.0804743 0.0402372 0.999190i \(-0.487189\pi\)
0.0402372 + 0.999190i \(0.487189\pi\)
\(332\) −1.00000 −0.0548821
\(333\) 17.8564 0.978525
\(334\) −4.39230 −0.240336
\(335\) 6.39230 0.349249
\(336\) −1.00000 −0.0545545
\(337\) 31.7846 1.73142 0.865709 0.500548i \(-0.166868\pi\)
0.865709 + 0.500548i \(0.166868\pi\)
\(338\) 6.92820 0.376845
\(339\) −1.00000 −0.0543125
\(340\) −7.46410 −0.404798
\(341\) 0 0
\(342\) −11.8564 −0.641121
\(343\) 1.00000 0.0539949
\(344\) −12.3923 −0.668148
\(345\) 4.46410 0.240339
\(346\) 18.0000 0.967686
\(347\) 0.928203 0.0498286 0.0249143 0.999690i \(-0.492069\pi\)
0.0249143 + 0.999690i \(0.492069\pi\)
\(348\) 6.92820 0.371391
\(349\) −1.53590 −0.0822148 −0.0411074 0.999155i \(-0.513089\pi\)
−0.0411074 + 0.999155i \(0.513089\pi\)
\(350\) 1.00000 0.0534522
\(351\) 22.3205 1.19138
\(352\) 0 0
\(353\) −20.5359 −1.09302 −0.546508 0.837454i \(-0.684043\pi\)
−0.546508 + 0.837454i \(0.684043\pi\)
\(354\) −9.92820 −0.527678
\(355\) 8.00000 0.424596
\(356\) −17.4641 −0.925596
\(357\) 7.46410 0.395042
\(358\) −8.39230 −0.443547
\(359\) −19.7128 −1.04040 −0.520201 0.854044i \(-0.674143\pi\)
−0.520201 + 0.854044i \(0.674143\pi\)
\(360\) −2.00000 −0.105409
\(361\) 16.1436 0.849663
\(362\) 10.4641 0.549981
\(363\) 0 0
\(364\) 4.46410 0.233983
\(365\) −0.535898 −0.0280502
\(366\) −4.92820 −0.257601
\(367\) −21.4641 −1.12042 −0.560208 0.828352i \(-0.689279\pi\)
−0.560208 + 0.828352i \(0.689279\pi\)
\(368\) −4.46410 −0.232707
\(369\) 16.7846 0.873772
\(370\) −8.92820 −0.464155
\(371\) −7.46410 −0.387517
\(372\) 0.535898 0.0277850
\(373\) 24.9282 1.29073 0.645367 0.763873i \(-0.276705\pi\)
0.645367 + 0.763873i \(0.276705\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −0.535898 −0.0276368
\(377\) −30.9282 −1.59288
\(378\) 5.00000 0.257172
\(379\) 23.8564 1.22542 0.612711 0.790307i \(-0.290079\pi\)
0.612711 + 0.790307i \(0.290079\pi\)
\(380\) 5.92820 0.304110
\(381\) −13.3923 −0.686109
\(382\) 3.39230 0.173565
\(383\) −21.4641 −1.09676 −0.548382 0.836228i \(-0.684756\pi\)
−0.548382 + 0.836228i \(0.684756\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 19.0000 0.967075
\(387\) 24.7846 1.25987
\(388\) −1.07180 −0.0544122
\(389\) −13.3205 −0.675377 −0.337688 0.941258i \(-0.609645\pi\)
−0.337688 + 0.941258i \(0.609645\pi\)
\(390\) −4.46410 −0.226049
\(391\) 33.3205 1.68509
\(392\) 1.00000 0.0505076
\(393\) 10.8564 0.547633
\(394\) 4.00000 0.201517
\(395\) −7.53590 −0.379172
\(396\) 0 0
\(397\) −11.0718 −0.555678 −0.277839 0.960628i \(-0.589618\pi\)
−0.277839 + 0.960628i \(0.589618\pi\)
\(398\) −25.8564 −1.29606
\(399\) −5.92820 −0.296781
\(400\) 1.00000 0.0500000
\(401\) 4.14359 0.206921 0.103461 0.994634i \(-0.467008\pi\)
0.103461 + 0.994634i \(0.467008\pi\)
\(402\) −6.39230 −0.318819
\(403\) −2.39230 −0.119169
\(404\) −11.5359 −0.573932
\(405\) 1.00000 0.0496904
\(406\) −6.92820 −0.343841
\(407\) 0 0
\(408\) 7.46410 0.369528
\(409\) −31.7128 −1.56810 −0.784049 0.620699i \(-0.786849\pi\)
−0.784049 + 0.620699i \(0.786849\pi\)
\(410\) −8.39230 −0.414466
\(411\) −7.92820 −0.391069
\(412\) −1.46410 −0.0721311
\(413\) 9.92820 0.488535
\(414\) 8.92820 0.438797
\(415\) −1.00000 −0.0490881
\(416\) 4.46410 0.218871
\(417\) −16.8564 −0.825462
\(418\) 0 0
\(419\) 38.7128 1.89124 0.945622 0.325267i \(-0.105454\pi\)
0.945622 + 0.325267i \(0.105454\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 5.85641 0.285424 0.142712 0.989764i \(-0.454418\pi\)
0.142712 + 0.989764i \(0.454418\pi\)
\(422\) −24.0000 −1.16830
\(423\) 1.07180 0.0521125
\(424\) −7.46410 −0.362489
\(425\) −7.46410 −0.362062
\(426\) −8.00000 −0.387601
\(427\) 4.92820 0.238492
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −12.3923 −0.597610
\(431\) −10.3205 −0.497121 −0.248561 0.968616i \(-0.579958\pi\)
−0.248561 + 0.968616i \(0.579958\pi\)
\(432\) 5.00000 0.240563
\(433\) 22.2487 1.06920 0.534602 0.845104i \(-0.320461\pi\)
0.534602 + 0.845104i \(0.320461\pi\)
\(434\) −0.535898 −0.0257239
\(435\) 6.92820 0.332182
\(436\) 18.3923 0.880832
\(437\) −26.4641 −1.26595
\(438\) 0.535898 0.0256062
\(439\) −22.5359 −1.07558 −0.537790 0.843079i \(-0.680741\pi\)
−0.537790 + 0.843079i \(0.680741\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) −33.3205 −1.58489
\(443\) −15.7128 −0.746538 −0.373269 0.927723i \(-0.621763\pi\)
−0.373269 + 0.927723i \(0.621763\pi\)
\(444\) 8.92820 0.423714
\(445\) −17.4641 −0.827878
\(446\) −3.46410 −0.164030
\(447\) 3.07180 0.145291
\(448\) 1.00000 0.0472456
\(449\) −8.07180 −0.380932 −0.190466 0.981694i \(-0.561000\pi\)
−0.190466 + 0.981694i \(0.561000\pi\)
\(450\) −2.00000 −0.0942809
\(451\) 0 0
\(452\) 1.00000 0.0470360
\(453\) 5.53590 0.260099
\(454\) −16.0000 −0.750917
\(455\) 4.46410 0.209280
\(456\) −5.92820 −0.277614
\(457\) −39.6410 −1.85433 −0.927164 0.374655i \(-0.877761\pi\)
−0.927164 + 0.374655i \(0.877761\pi\)
\(458\) −18.0000 −0.841085
\(459\) −37.3205 −1.74197
\(460\) −4.46410 −0.208140
\(461\) 12.1436 0.565584 0.282792 0.959181i \(-0.408739\pi\)
0.282792 + 0.959181i \(0.408739\pi\)
\(462\) 0 0
\(463\) −8.46410 −0.393360 −0.196680 0.980468i \(-0.563016\pi\)
−0.196680 + 0.980468i \(0.563016\pi\)
\(464\) −6.92820 −0.321634
\(465\) 0.535898 0.0248517
\(466\) −18.8564 −0.873506
\(467\) −15.9282 −0.737069 −0.368535 0.929614i \(-0.620140\pi\)
−0.368535 + 0.929614i \(0.620140\pi\)
\(468\) −8.92820 −0.412706
\(469\) 6.39230 0.295169
\(470\) −0.535898 −0.0247191
\(471\) 20.3205 0.936320
\(472\) 9.92820 0.456983
\(473\) 0 0
\(474\) 7.53590 0.346135
\(475\) 5.92820 0.272005
\(476\) −7.46410 −0.342117
\(477\) 14.9282 0.683515
\(478\) −12.4641 −0.570095
\(479\) −2.53590 −0.115868 −0.0579341 0.998320i \(-0.518451\pi\)
−0.0579341 + 0.998320i \(0.518451\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −39.8564 −1.81730
\(482\) −4.00000 −0.182195
\(483\) 4.46410 0.203124
\(484\) 0 0
\(485\) −1.07180 −0.0486678
\(486\) −16.0000 −0.725775
\(487\) 2.32051 0.105152 0.0525761 0.998617i \(-0.483257\pi\)
0.0525761 + 0.998617i \(0.483257\pi\)
\(488\) 4.92820 0.223089
\(489\) 22.0000 0.994874
\(490\) 1.00000 0.0451754
\(491\) 23.4641 1.05892 0.529460 0.848335i \(-0.322395\pi\)
0.529460 + 0.848335i \(0.322395\pi\)
\(492\) 8.39230 0.378354
\(493\) 51.7128 2.32903
\(494\) 26.4641 1.19068
\(495\) 0 0
\(496\) −0.535898 −0.0240625
\(497\) 8.00000 0.358849
\(498\) 1.00000 0.0448111
\(499\) −8.92820 −0.399681 −0.199841 0.979828i \(-0.564042\pi\)
−0.199841 + 0.979828i \(0.564042\pi\)
\(500\) 1.00000 0.0447214
\(501\) 4.39230 0.196234
\(502\) 0 0
\(503\) 30.2487 1.34872 0.674362 0.738401i \(-0.264419\pi\)
0.674362 + 0.738401i \(0.264419\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −11.5359 −0.513341
\(506\) 0 0
\(507\) −6.92820 −0.307692
\(508\) 13.3923 0.594187
\(509\) −26.3205 −1.16664 −0.583318 0.812244i \(-0.698246\pi\)
−0.583318 + 0.812244i \(0.698246\pi\)
\(510\) 7.46410 0.330516
\(511\) −0.535898 −0.0237067
\(512\) 1.00000 0.0441942
\(513\) 29.6410 1.30868
\(514\) 2.39230 0.105520
\(515\) −1.46410 −0.0645160
\(516\) 12.3923 0.545541
\(517\) 0 0
\(518\) −8.92820 −0.392283
\(519\) −18.0000 −0.790112
\(520\) 4.46410 0.195764
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) 13.8564 0.606478
\(523\) −3.92820 −0.171768 −0.0858842 0.996305i \(-0.527371\pi\)
−0.0858842 + 0.996305i \(0.527371\pi\)
\(524\) −10.8564 −0.474264
\(525\) −1.00000 −0.0436436
\(526\) 16.4641 0.717869
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −3.07180 −0.133556
\(530\) −7.46410 −0.324220
\(531\) −19.8564 −0.861695
\(532\) 5.92820 0.257020
\(533\) −37.4641 −1.62275
\(534\) 17.4641 0.755746
\(535\) −6.00000 −0.259403
\(536\) 6.39230 0.276106
\(537\) 8.39230 0.362155
\(538\) −10.3205 −0.444949
\(539\) 0 0
\(540\) 5.00000 0.215166
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 29.8564 1.28244
\(543\) −10.4641 −0.449058
\(544\) −7.46410 −0.320021
\(545\) 18.3923 0.787840
\(546\) −4.46410 −0.191046
\(547\) −34.5359 −1.47665 −0.738324 0.674446i \(-0.764383\pi\)
−0.738324 + 0.674446i \(0.764383\pi\)
\(548\) 7.92820 0.338676
\(549\) −9.85641 −0.420661
\(550\) 0 0
\(551\) −41.0718 −1.74972
\(552\) 4.46410 0.190005
\(553\) −7.53590 −0.320459
\(554\) 27.3205 1.16074
\(555\) 8.92820 0.378981
\(556\) 16.8564 0.714871
\(557\) 21.3205 0.903379 0.451689 0.892175i \(-0.350822\pi\)
0.451689 + 0.892175i \(0.350822\pi\)
\(558\) 1.07180 0.0453728
\(559\) −55.3205 −2.33981
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 17.0000 0.717102
\(563\) 42.7128 1.80013 0.900065 0.435755i \(-0.143519\pi\)
0.900065 + 0.435755i \(0.143519\pi\)
\(564\) 0.535898 0.0225654
\(565\) 1.00000 0.0420703
\(566\) −13.7846 −0.579410
\(567\) 1.00000 0.0419961
\(568\) 8.00000 0.335673
\(569\) −32.7128 −1.37139 −0.685696 0.727888i \(-0.740502\pi\)
−0.685696 + 0.727888i \(0.740502\pi\)
\(570\) −5.92820 −0.248305
\(571\) 1.85641 0.0776882 0.0388441 0.999245i \(-0.487632\pi\)
0.0388441 + 0.999245i \(0.487632\pi\)
\(572\) 0 0
\(573\) −3.39230 −0.141716
\(574\) −8.39230 −0.350288
\(575\) −4.46410 −0.186166
\(576\) −2.00000 −0.0833333
\(577\) 35.3205 1.47041 0.735206 0.677844i \(-0.237085\pi\)
0.735206 + 0.677844i \(0.237085\pi\)
\(578\) 38.7128 1.61024
\(579\) −19.0000 −0.789613
\(580\) −6.92820 −0.287678
\(581\) −1.00000 −0.0414870
\(582\) 1.07180 0.0444274
\(583\) 0 0
\(584\) −0.535898 −0.0221756
\(585\) −8.92820 −0.369136
\(586\) −20.4641 −0.845364
\(587\) −28.8564 −1.19103 −0.595516 0.803344i \(-0.703052\pi\)
−0.595516 + 0.803344i \(0.703052\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −3.17691 −0.130902
\(590\) 9.92820 0.408738
\(591\) −4.00000 −0.164538
\(592\) −8.92820 −0.366947
\(593\) 28.0000 1.14982 0.574911 0.818216i \(-0.305037\pi\)
0.574911 + 0.818216i \(0.305037\pi\)
\(594\) 0 0
\(595\) −7.46410 −0.305998
\(596\) −3.07180 −0.125826
\(597\) 25.8564 1.05823
\(598\) −19.9282 −0.814925
\(599\) −25.3923 −1.03750 −0.518751 0.854926i \(-0.673603\pi\)
−0.518751 + 0.854926i \(0.673603\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 2.67949 0.109299 0.0546494 0.998506i \(-0.482596\pi\)
0.0546494 + 0.998506i \(0.482596\pi\)
\(602\) −12.3923 −0.505073
\(603\) −12.7846 −0.520630
\(604\) −5.53590 −0.225253
\(605\) 0 0
\(606\) 11.5359 0.468614
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 5.92820 0.240420
\(609\) 6.92820 0.280745
\(610\) 4.92820 0.199537
\(611\) −2.39230 −0.0967823
\(612\) 14.9282 0.603437
\(613\) 26.5359 1.07177 0.535887 0.844289i \(-0.319977\pi\)
0.535887 + 0.844289i \(0.319977\pi\)
\(614\) −14.9282 −0.602453
\(615\) 8.39230 0.338410
\(616\) 0 0
\(617\) −1.21539 −0.0489298 −0.0244649 0.999701i \(-0.507788\pi\)
−0.0244649 + 0.999701i \(0.507788\pi\)
\(618\) 1.46410 0.0588948
\(619\) 6.07180 0.244046 0.122023 0.992527i \(-0.461062\pi\)
0.122023 + 0.992527i \(0.461062\pi\)
\(620\) −0.535898 −0.0215222
\(621\) −22.3205 −0.895691
\(622\) 23.3205 0.935067
\(623\) −17.4641 −0.699684
\(624\) −4.46410 −0.178707
\(625\) 1.00000 0.0400000
\(626\) 14.7846 0.590912
\(627\) 0 0
\(628\) −20.3205 −0.810877
\(629\) 66.6410 2.65715
\(630\) −2.00000 −0.0796819
\(631\) 38.9282 1.54971 0.774854 0.632141i \(-0.217824\pi\)
0.774854 + 0.632141i \(0.217824\pi\)
\(632\) −7.53590 −0.299762
\(633\) 24.0000 0.953914
\(634\) −22.3923 −0.889312
\(635\) 13.3923 0.531457
\(636\) 7.46410 0.295971
\(637\) 4.46410 0.176874
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 1.00000 0.0395285
\(641\) −34.8564 −1.37675 −0.688373 0.725357i \(-0.741675\pi\)
−0.688373 + 0.725357i \(0.741675\pi\)
\(642\) 6.00000 0.236801
\(643\) −41.5692 −1.63933 −0.819665 0.572843i \(-0.805840\pi\)
−0.819665 + 0.572843i \(0.805840\pi\)
\(644\) −4.46410 −0.175910
\(645\) 12.3923 0.487947
\(646\) −44.2487 −1.74094
\(647\) −11.3205 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 4.46410 0.175096
\(651\) 0.535898 0.0210035
\(652\) −22.0000 −0.861586
\(653\) −25.4641 −0.996487 −0.498244 0.867037i \(-0.666021\pi\)
−0.498244 + 0.867037i \(0.666021\pi\)
\(654\) −18.3923 −0.719196
\(655\) −10.8564 −0.424195
\(656\) −8.39230 −0.327664
\(657\) 1.07180 0.0418148
\(658\) −0.535898 −0.0208915
\(659\) −29.3205 −1.14216 −0.571082 0.820893i \(-0.693476\pi\)
−0.571082 + 0.820893i \(0.693476\pi\)
\(660\) 0 0
\(661\) −0.607695 −0.0236366 −0.0118183 0.999930i \(-0.503762\pi\)
−0.0118183 + 0.999930i \(0.503762\pi\)
\(662\) 1.46410 0.0569039
\(663\) 33.3205 1.29406
\(664\) −1.00000 −0.0388075
\(665\) 5.92820 0.229886
\(666\) 17.8564 0.691922
\(667\) 30.9282 1.19754
\(668\) −4.39230 −0.169943
\(669\) 3.46410 0.133930
\(670\) 6.39230 0.246956
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) 38.8564 1.49780 0.748902 0.662681i \(-0.230581\pi\)
0.748902 + 0.662681i \(0.230581\pi\)
\(674\) 31.7846 1.22430
\(675\) 5.00000 0.192450
\(676\) 6.92820 0.266469
\(677\) −33.3923 −1.28337 −0.641685 0.766968i \(-0.721764\pi\)
−0.641685 + 0.766968i \(0.721764\pi\)
\(678\) −1.00000 −0.0384048
\(679\) −1.07180 −0.0411318
\(680\) −7.46410 −0.286235
\(681\) 16.0000 0.613121
\(682\) 0 0
\(683\) −32.7846 −1.25447 −0.627234 0.778831i \(-0.715813\pi\)
−0.627234 + 0.778831i \(0.715813\pi\)
\(684\) −11.8564 −0.453341
\(685\) 7.92820 0.302921
\(686\) 1.00000 0.0381802
\(687\) 18.0000 0.686743
\(688\) −12.3923 −0.472452
\(689\) −33.3205 −1.26941
\(690\) 4.46410 0.169945
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 0.928203 0.0352341
\(695\) 16.8564 0.639400
\(696\) 6.92820 0.262613
\(697\) 62.6410 2.37270
\(698\) −1.53590 −0.0581346
\(699\) 18.8564 0.713215
\(700\) 1.00000 0.0377964
\(701\) 30.3923 1.14790 0.573951 0.818890i \(-0.305410\pi\)
0.573951 + 0.818890i \(0.305410\pi\)
\(702\) 22.3205 0.842433
\(703\) −52.9282 −1.99622
\(704\) 0 0
\(705\) 0.535898 0.0201831
\(706\) −20.5359 −0.772879
\(707\) −11.5359 −0.433852
\(708\) −9.92820 −0.373125
\(709\) 19.8564 0.745723 0.372861 0.927887i \(-0.378377\pi\)
0.372861 + 0.927887i \(0.378377\pi\)
\(710\) 8.00000 0.300235
\(711\) 15.0718 0.565237
\(712\) −17.4641 −0.654495
\(713\) 2.39230 0.0895925
\(714\) 7.46410 0.279337
\(715\) 0 0
\(716\) −8.39230 −0.313635
\(717\) 12.4641 0.465480
\(718\) −19.7128 −0.735676
\(719\) 39.7128 1.48104 0.740519 0.672035i \(-0.234580\pi\)
0.740519 + 0.672035i \(0.234580\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −1.46410 −0.0545260
\(722\) 16.1436 0.600802
\(723\) 4.00000 0.148762
\(724\) 10.4641 0.388895
\(725\) −6.92820 −0.257307
\(726\) 0 0
\(727\) −20.7846 −0.770859 −0.385429 0.922737i \(-0.625947\pi\)
−0.385429 + 0.922737i \(0.625947\pi\)
\(728\) 4.46410 0.165451
\(729\) 13.0000 0.481481
\(730\) −0.535898 −0.0198345
\(731\) 92.4974 3.42114
\(732\) −4.92820 −0.182152
\(733\) −48.1769 −1.77945 −0.889727 0.456492i \(-0.849106\pi\)
−0.889727 + 0.456492i \(0.849106\pi\)
\(734\) −21.4641 −0.792254
\(735\) −1.00000 −0.0368856
\(736\) −4.46410 −0.164549
\(737\) 0 0
\(738\) 16.7846 0.617850
\(739\) −20.2487 −0.744861 −0.372430 0.928060i \(-0.621475\pi\)
−0.372430 + 0.928060i \(0.621475\pi\)
\(740\) −8.92820 −0.328207
\(741\) −26.4641 −0.972183
\(742\) −7.46410 −0.274016
\(743\) 33.0718 1.21329 0.606643 0.794974i \(-0.292516\pi\)
0.606643 + 0.794974i \(0.292516\pi\)
\(744\) 0.535898 0.0196470
\(745\) −3.07180 −0.112542
\(746\) 24.9282 0.912686
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) −1.00000 −0.0365148
\(751\) −26.1769 −0.955209 −0.477605 0.878575i \(-0.658495\pi\)
−0.477605 + 0.878575i \(0.658495\pi\)
\(752\) −0.535898 −0.0195422
\(753\) 0 0
\(754\) −30.9282 −1.12634
\(755\) −5.53590 −0.201472
\(756\) 5.00000 0.181848
\(757\) −27.3205 −0.992981 −0.496490 0.868042i \(-0.665378\pi\)
−0.496490 + 0.868042i \(0.665378\pi\)
\(758\) 23.8564 0.866504
\(759\) 0 0
\(760\) 5.92820 0.215039
\(761\) 48.2487 1.74901 0.874507 0.485013i \(-0.161185\pi\)
0.874507 + 0.485013i \(0.161185\pi\)
\(762\) −13.3923 −0.485152
\(763\) 18.3923 0.665846
\(764\) 3.39230 0.122729
\(765\) 14.9282 0.539730
\(766\) −21.4641 −0.775530
\(767\) 44.3205 1.60032
\(768\) −1.00000 −0.0360844
\(769\) −22.7846 −0.821634 −0.410817 0.911718i \(-0.634756\pi\)
−0.410817 + 0.911718i \(0.634756\pi\)
\(770\) 0 0
\(771\) −2.39230 −0.0861568
\(772\) 19.0000 0.683825
\(773\) 2.46410 0.0886276 0.0443138 0.999018i \(-0.485890\pi\)
0.0443138 + 0.999018i \(0.485890\pi\)
\(774\) 24.7846 0.890864
\(775\) −0.535898 −0.0192500
\(776\) −1.07180 −0.0384753
\(777\) 8.92820 0.320298
\(778\) −13.3205 −0.477563
\(779\) −49.7513 −1.78252
\(780\) −4.46410 −0.159840
\(781\) 0 0
\(782\) 33.3205 1.19154
\(783\) −34.6410 −1.23797
\(784\) 1.00000 0.0357143
\(785\) −20.3205 −0.725270
\(786\) 10.8564 0.387235
\(787\) 48.4974 1.72875 0.864373 0.502851i \(-0.167715\pi\)
0.864373 + 0.502851i \(0.167715\pi\)
\(788\) 4.00000 0.142494
\(789\) −16.4641 −0.586138
\(790\) −7.53590 −0.268115
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) −11.0718 −0.392923
\(795\) 7.46410 0.264724
\(796\) −25.8564 −0.916456
\(797\) −21.3923 −0.757754 −0.378877 0.925447i \(-0.623690\pi\)
−0.378877 + 0.925447i \(0.623690\pi\)
\(798\) −5.92820 −0.209856
\(799\) 4.00000 0.141510
\(800\) 1.00000 0.0353553
\(801\) 34.9282 1.23413
\(802\) 4.14359 0.146315
\(803\) 0 0
\(804\) −6.39230 −0.225439
\(805\) −4.46410 −0.157339
\(806\) −2.39230 −0.0842653
\(807\) 10.3205 0.363299
\(808\) −11.5359 −0.405831
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 1.00000 0.0351364
\(811\) 35.7128 1.25405 0.627023 0.779001i \(-0.284273\pi\)
0.627023 + 0.779001i \(0.284273\pi\)
\(812\) −6.92820 −0.243132
\(813\) −29.8564 −1.04711
\(814\) 0 0
\(815\) −22.0000 −0.770626
\(816\) 7.46410 0.261296
\(817\) −73.4641 −2.57018
\(818\) −31.7128 −1.10881
\(819\) −8.92820 −0.311977
\(820\) −8.39230 −0.293072
\(821\) 36.9282 1.28880 0.644402 0.764687i \(-0.277107\pi\)
0.644402 + 0.764687i \(0.277107\pi\)
\(822\) −7.92820 −0.276528
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −1.46410 −0.0510044
\(825\) 0 0
\(826\) 9.92820 0.345446
\(827\) −18.3923 −0.639563 −0.319782 0.947491i \(-0.603610\pi\)
−0.319782 + 0.947491i \(0.603610\pi\)
\(828\) 8.92820 0.310277
\(829\) −21.3923 −0.742985 −0.371493 0.928436i \(-0.621154\pi\)
−0.371493 + 0.928436i \(0.621154\pi\)
\(830\) −1.00000 −0.0347105
\(831\) −27.3205 −0.947738
\(832\) 4.46410 0.154765
\(833\) −7.46410 −0.258616
\(834\) −16.8564 −0.583690
\(835\) −4.39230 −0.152002
\(836\) 0 0
\(837\) −2.67949 −0.0926168
\(838\) 38.7128 1.33731
\(839\) 6.24871 0.215729 0.107865 0.994166i \(-0.465599\pi\)
0.107865 + 0.994166i \(0.465599\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 19.0000 0.655172
\(842\) 5.85641 0.201825
\(843\) −17.0000 −0.585511
\(844\) −24.0000 −0.826114
\(845\) 6.92820 0.238337
\(846\) 1.07180 0.0368491
\(847\) 0 0
\(848\) −7.46410 −0.256318
\(849\) 13.7846 0.473087
\(850\) −7.46410 −0.256017
\(851\) 39.8564 1.36626
\(852\) −8.00000 −0.274075
\(853\) 9.53590 0.326503 0.163251 0.986584i \(-0.447802\pi\)
0.163251 + 0.986584i \(0.447802\pi\)
\(854\) 4.92820 0.168640
\(855\) −11.8564 −0.405481
\(856\) −6.00000 −0.205076
\(857\) 6.92820 0.236663 0.118331 0.992974i \(-0.462245\pi\)
0.118331 + 0.992974i \(0.462245\pi\)
\(858\) 0 0
\(859\) −26.6410 −0.908980 −0.454490 0.890752i \(-0.650178\pi\)
−0.454490 + 0.890752i \(0.650178\pi\)
\(860\) −12.3923 −0.422574
\(861\) 8.39230 0.286009
\(862\) −10.3205 −0.351518
\(863\) 0.784610 0.0267084 0.0133542 0.999911i \(-0.495749\pi\)
0.0133542 + 0.999911i \(0.495749\pi\)
\(864\) 5.00000 0.170103
\(865\) 18.0000 0.612018
\(866\) 22.2487 0.756042
\(867\) −38.7128 −1.31476
\(868\) −0.535898 −0.0181896
\(869\) 0 0
\(870\) 6.92820 0.234888
\(871\) 28.5359 0.966902
\(872\) 18.3923 0.622842
\(873\) 2.14359 0.0725496
\(874\) −26.4641 −0.895162
\(875\) 1.00000 0.0338062
\(876\) 0.535898 0.0181063
\(877\) −8.39230 −0.283388 −0.141694 0.989911i \(-0.545255\pi\)
−0.141694 + 0.989911i \(0.545255\pi\)
\(878\) −22.5359 −0.760550
\(879\) 20.4641 0.690237
\(880\) 0 0
\(881\) −5.71281 −0.192470 −0.0962348 0.995359i \(-0.530680\pi\)
−0.0962348 + 0.995359i \(0.530680\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −34.7846 −1.17060 −0.585298 0.810819i \(-0.699022\pi\)
−0.585298 + 0.810819i \(0.699022\pi\)
\(884\) −33.3205 −1.12069
\(885\) −9.92820 −0.333733
\(886\) −15.7128 −0.527882
\(887\) −26.5359 −0.890988 −0.445494 0.895285i \(-0.646972\pi\)
−0.445494 + 0.895285i \(0.646972\pi\)
\(888\) 8.92820 0.299611
\(889\) 13.3923 0.449163
\(890\) −17.4641 −0.585398
\(891\) 0 0
\(892\) −3.46410 −0.115987
\(893\) −3.17691 −0.106311
\(894\) 3.07180 0.102736
\(895\) −8.39230 −0.280524
\(896\) 1.00000 0.0334077
\(897\) 19.9282 0.665383
\(898\) −8.07180 −0.269359
\(899\) 3.71281 0.123829
\(900\) −2.00000 −0.0666667
\(901\) 55.7128 1.85606
\(902\) 0 0
\(903\) 12.3923 0.412390
\(904\) 1.00000 0.0332595
\(905\) 10.4641 0.347839
\(906\) 5.53590 0.183918
\(907\) −42.6410 −1.41587 −0.707936 0.706277i \(-0.750373\pi\)
−0.707936 + 0.706277i \(0.750373\pi\)
\(908\) −16.0000 −0.530979
\(909\) 23.0718 0.765243
\(910\) 4.46410 0.147984
\(911\) 9.39230 0.311181 0.155590 0.987822i \(-0.450272\pi\)
0.155590 + 0.987822i \(0.450272\pi\)
\(912\) −5.92820 −0.196302
\(913\) 0 0
\(914\) −39.6410 −1.31121
\(915\) −4.92820 −0.162921
\(916\) −18.0000 −0.594737
\(917\) −10.8564 −0.358510
\(918\) −37.3205 −1.23176
\(919\) 3.71281 0.122474 0.0612372 0.998123i \(-0.480495\pi\)
0.0612372 + 0.998123i \(0.480495\pi\)
\(920\) −4.46410 −0.147177
\(921\) 14.9282 0.491901
\(922\) 12.1436 0.399928
\(923\) 35.7128 1.17550
\(924\) 0 0
\(925\) −8.92820 −0.293558
\(926\) −8.46410 −0.278148
\(927\) 2.92820 0.0961748
\(928\) −6.92820 −0.227429
\(929\) −25.8564 −0.848321 −0.424161 0.905587i \(-0.639431\pi\)
−0.424161 + 0.905587i \(0.639431\pi\)
\(930\) 0.535898 0.0175728
\(931\) 5.92820 0.194289
\(932\) −18.8564 −0.617662
\(933\) −23.3205 −0.763479
\(934\) −15.9282 −0.521187
\(935\) 0 0
\(936\) −8.92820 −0.291827
\(937\) 6.14359 0.200702 0.100351 0.994952i \(-0.468003\pi\)
0.100351 + 0.994952i \(0.468003\pi\)
\(938\) 6.39230 0.208716
\(939\) −14.7846 −0.482478
\(940\) −0.535898 −0.0174791
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 20.3205 0.662078
\(943\) 37.4641 1.22000
\(944\) 9.92820 0.323135
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) 29.6077 0.962121 0.481060 0.876687i \(-0.340252\pi\)
0.481060 + 0.876687i \(0.340252\pi\)
\(948\) 7.53590 0.244755
\(949\) −2.39230 −0.0776575
\(950\) 5.92820 0.192336
\(951\) 22.3923 0.726120
\(952\) −7.46410 −0.241913
\(953\) 32.5692 1.05502 0.527510 0.849549i \(-0.323126\pi\)
0.527510 + 0.849549i \(0.323126\pi\)
\(954\) 14.9282 0.483318
\(955\) 3.39230 0.109772
\(956\) −12.4641 −0.403118
\(957\) 0 0
\(958\) −2.53590 −0.0819312
\(959\) 7.92820 0.256015
\(960\) −1.00000 −0.0322749
\(961\) −30.7128 −0.990736
\(962\) −39.8564 −1.28502
\(963\) 12.0000 0.386695
\(964\) −4.00000 −0.128831
\(965\) 19.0000 0.611632
\(966\) 4.46410 0.143630
\(967\) 11.7128 0.376659 0.188329 0.982106i \(-0.439693\pi\)
0.188329 + 0.982106i \(0.439693\pi\)
\(968\) 0 0
\(969\) 44.2487 1.42147
\(970\) −1.07180 −0.0344133
\(971\) 12.7128 0.407974 0.203987 0.978974i \(-0.434610\pi\)
0.203987 + 0.978974i \(0.434610\pi\)
\(972\) −16.0000 −0.513200
\(973\) 16.8564 0.540392
\(974\) 2.32051 0.0743539
\(975\) −4.46410 −0.142966
\(976\) 4.92820 0.157748
\(977\) 18.8564 0.603270 0.301635 0.953423i \(-0.402468\pi\)
0.301635 + 0.953423i \(0.402468\pi\)
\(978\) 22.0000 0.703482
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −36.7846 −1.17444
\(982\) 23.4641 0.748770
\(983\) 14.7846 0.471556 0.235778 0.971807i \(-0.424236\pi\)
0.235778 + 0.971807i \(0.424236\pi\)
\(984\) 8.39230 0.267537
\(985\) 4.00000 0.127451
\(986\) 51.7128 1.64687
\(987\) 0.535898 0.0170578
\(988\) 26.4641 0.841935
\(989\) 55.3205 1.75909
\(990\) 0 0
\(991\) 43.1051 1.36928 0.684640 0.728882i \(-0.259960\pi\)
0.684640 + 0.728882i \(0.259960\pi\)
\(992\) −0.535898 −0.0170148
\(993\) −1.46410 −0.0464619
\(994\) 8.00000 0.253745
\(995\) −25.8564 −0.819703
\(996\) 1.00000 0.0316862
\(997\) −30.3205 −0.960260 −0.480130 0.877197i \(-0.659411\pi\)
−0.480130 + 0.877197i \(0.659411\pi\)
\(998\) −8.92820 −0.282617
\(999\) −44.6410 −1.41238
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.bw.1.2 yes 2
11.10 odd 2 8470.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bi.1.1 2 11.10 odd 2
8470.2.a.bw.1.2 yes 2 1.1 even 1 trivial