Properties

Label 5070.2.a.bq.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +5.04892 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +5.04892 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.74094 q^{11} +1.00000 q^{12} -5.04892 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.08815 q^{17} -1.00000 q^{18} +4.44504 q^{19} +1.00000 q^{20} +5.04892 q^{21} +4.74094 q^{22} +2.69202 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +5.04892 q^{28} -7.93900 q^{29} -1.00000 q^{30} +7.85086 q^{31} -1.00000 q^{32} -4.74094 q^{33} +1.08815 q^{34} +5.04892 q^{35} +1.00000 q^{36} -7.14675 q^{37} -4.44504 q^{38} -1.00000 q^{40} -0.664874 q^{41} -5.04892 q^{42} +4.35690 q^{43} -4.74094 q^{44} +1.00000 q^{45} -2.69202 q^{46} +7.00000 q^{47} +1.00000 q^{48} +18.4916 q^{49} -1.00000 q^{50} -1.08815 q^{51} -10.8605 q^{53} -1.00000 q^{54} -4.74094 q^{55} -5.04892 q^{56} +4.44504 q^{57} +7.93900 q^{58} +12.2784 q^{59} +1.00000 q^{60} -11.0707 q^{61} -7.85086 q^{62} +5.04892 q^{63} +1.00000 q^{64} +4.74094 q^{66} +9.58211 q^{67} -1.08815 q^{68} +2.69202 q^{69} -5.04892 q^{70} +4.47219 q^{71} -1.00000 q^{72} -8.17390 q^{73} +7.14675 q^{74} +1.00000 q^{75} +4.44504 q^{76} -23.9366 q^{77} +14.1468 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.664874 q^{82} +4.86294 q^{83} +5.04892 q^{84} -1.08815 q^{85} -4.35690 q^{86} -7.93900 q^{87} +4.74094 q^{88} +14.8877 q^{89} -1.00000 q^{90} +2.69202 q^{92} +7.85086 q^{93} -7.00000 q^{94} +4.44504 q^{95} -1.00000 q^{96} +14.9269 q^{97} -18.4916 q^{98} -4.74094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 6q^{7} - 3q^{8} + 3q^{9} - 3q^{10} + 3q^{12} - 6q^{14} + 3q^{15} + 3q^{16} - 7q^{17} - 3q^{18} + 13q^{19} + 3q^{20} + 6q^{21} + 3q^{23} - 3q^{24} + 3q^{25} + 3q^{27} + 6q^{28} - 14q^{29} - 3q^{30} + 10q^{31} - 3q^{32} + 7q^{34} + 6q^{35} + 3q^{36} + 6q^{37} - 13q^{38} - 3q^{40} - 3q^{41} - 6q^{42} + 9q^{43} + 3q^{45} - 3q^{46} + 21q^{47} + 3q^{48} + 5q^{49} - 3q^{50} - 7q^{51} + 3q^{53} - 3q^{54} - 6q^{56} + 13q^{57} + 14q^{58} + 7q^{59} + 3q^{60} - 21q^{61} - 10q^{62} + 6q^{63} + 3q^{64} + 23q^{67} - 7q^{68} + 3q^{69} - 6q^{70} + 7q^{71} - 3q^{72} + 9q^{73} - 6q^{74} + 3q^{75} + 13q^{76} - 21q^{77} + 15q^{79} + 3q^{80} + 3q^{81} + 3q^{82} + 20q^{83} + 6q^{84} - 7q^{85} - 9q^{86} - 14q^{87} + 3q^{89} - 3q^{90} + 3q^{92} + 10q^{93} - 21q^{94} + 13q^{95} - 3q^{96} + 16q^{97} - 5q^{98} + O(q^{100}) \)

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
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 5.04892 1.90831 0.954156 0.299311i \(-0.0967567\pi\)
0.954156 + 0.299311i \(0.0967567\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.74094 −1.42945 −0.714723 0.699407i \(-0.753447\pi\)
−0.714723 + 0.699407i \(0.753447\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −5.04892 −1.34938
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −1.08815 −0.263914 −0.131957 0.991255i \(-0.542126\pi\)
−0.131957 + 0.991255i \(0.542126\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.44504 1.01976 0.509881 0.860245i \(-0.329689\pi\)
0.509881 + 0.860245i \(0.329689\pi\)
\(20\) 1.00000 0.223607
\(21\) 5.04892 1.10176
\(22\) 4.74094 1.01077
\(23\) 2.69202 0.561325 0.280663 0.959806i \(-0.409446\pi\)
0.280663 + 0.959806i \(0.409446\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 5.04892 0.954156
\(29\) −7.93900 −1.47424 −0.737118 0.675764i \(-0.763814\pi\)
−0.737118 + 0.675764i \(0.763814\pi\)
\(30\) −1.00000 −0.182574
\(31\) 7.85086 1.41006 0.705028 0.709180i \(-0.250935\pi\)
0.705028 + 0.709180i \(0.250935\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.74094 −0.825292
\(34\) 1.08815 0.186615
\(35\) 5.04892 0.853423
\(36\) 1.00000 0.166667
\(37\) −7.14675 −1.17492 −0.587459 0.809254i \(-0.699872\pi\)
−0.587459 + 0.809254i \(0.699872\pi\)
\(38\) −4.44504 −0.721081
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −0.664874 −0.103836 −0.0519180 0.998651i \(-0.516533\pi\)
−0.0519180 + 0.998651i \(0.516533\pi\)
\(42\) −5.04892 −0.779065
\(43\) 4.35690 0.664420 0.332210 0.943205i \(-0.392206\pi\)
0.332210 + 0.943205i \(0.392206\pi\)
\(44\) −4.74094 −0.714723
\(45\) 1.00000 0.149071
\(46\) −2.69202 −0.396917
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 1.00000 0.144338
\(49\) 18.4916 2.64165
\(50\) −1.00000 −0.141421
\(51\) −1.08815 −0.152371
\(52\) 0 0
\(53\) −10.8605 −1.49181 −0.745905 0.666052i \(-0.767983\pi\)
−0.745905 + 0.666052i \(0.767983\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.74094 −0.639268
\(56\) −5.04892 −0.674690
\(57\) 4.44504 0.588760
\(58\) 7.93900 1.04244
\(59\) 12.2784 1.59852 0.799258 0.600988i \(-0.205226\pi\)
0.799258 + 0.600988i \(0.205226\pi\)
\(60\) 1.00000 0.129099
\(61\) −11.0707 −1.41746 −0.708728 0.705482i \(-0.750731\pi\)
−0.708728 + 0.705482i \(0.750731\pi\)
\(62\) −7.85086 −0.997060
\(63\) 5.04892 0.636104
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.74094 0.583569
\(67\) 9.58211 1.17064 0.585320 0.810802i \(-0.300969\pi\)
0.585320 + 0.810802i \(0.300969\pi\)
\(68\) −1.08815 −0.131957
\(69\) 2.69202 0.324081
\(70\) −5.04892 −0.603461
\(71\) 4.47219 0.530751 0.265376 0.964145i \(-0.414504\pi\)
0.265376 + 0.964145i \(0.414504\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.17390 −0.956683 −0.478341 0.878174i \(-0.658762\pi\)
−0.478341 + 0.878174i \(0.658762\pi\)
\(74\) 7.14675 0.830793
\(75\) 1.00000 0.115470
\(76\) 4.44504 0.509881
\(77\) −23.9366 −2.72783
\(78\) 0 0
\(79\) 14.1468 1.59163 0.795817 0.605537i \(-0.207042\pi\)
0.795817 + 0.605537i \(0.207042\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 0.664874 0.0734231
\(83\) 4.86294 0.533777 0.266888 0.963727i \(-0.414005\pi\)
0.266888 + 0.963727i \(0.414005\pi\)
\(84\) 5.04892 0.550882
\(85\) −1.08815 −0.118026
\(86\) −4.35690 −0.469816
\(87\) −7.93900 −0.851150
\(88\) 4.74094 0.505386
\(89\) 14.8877 1.57809 0.789046 0.614334i \(-0.210575\pi\)
0.789046 + 0.614334i \(0.210575\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 2.69202 0.280663
\(93\) 7.85086 0.814096
\(94\) −7.00000 −0.721995
\(95\) 4.44504 0.456052
\(96\) −1.00000 −0.102062
\(97\) 14.9269 1.51560 0.757800 0.652487i \(-0.226274\pi\)
0.757800 + 0.652487i \(0.226274\pi\)
\(98\) −18.4916 −1.86793
\(99\) −4.74094 −0.476482
\(100\) 1.00000 0.100000
\(101\) 11.4819 1.14249 0.571245 0.820780i \(-0.306461\pi\)
0.571245 + 0.820780i \(0.306461\pi\)
\(102\) 1.08815 0.107743
\(103\) 4.16421 0.410312 0.205156 0.978729i \(-0.434230\pi\)
0.205156 + 0.978729i \(0.434230\pi\)
\(104\) 0 0
\(105\) 5.04892 0.492724
\(106\) 10.8605 1.05487
\(107\) −0.796561 −0.0770065 −0.0385032 0.999258i \(-0.512259\pi\)
−0.0385032 + 0.999258i \(0.512259\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0.704103 0.0674408 0.0337204 0.999431i \(-0.489264\pi\)
0.0337204 + 0.999431i \(0.489264\pi\)
\(110\) 4.74094 0.452031
\(111\) −7.14675 −0.678340
\(112\) 5.04892 0.477078
\(113\) 14.4983 1.36388 0.681941 0.731407i \(-0.261136\pi\)
0.681941 + 0.731407i \(0.261136\pi\)
\(114\) −4.44504 −0.416316
\(115\) 2.69202 0.251032
\(116\) −7.93900 −0.737118
\(117\) 0 0
\(118\) −12.2784 −1.13032
\(119\) −5.49396 −0.503630
\(120\) −1.00000 −0.0912871
\(121\) 11.4765 1.04332
\(122\) 11.0707 1.00229
\(123\) −0.664874 −0.0599497
\(124\) 7.85086 0.705028
\(125\) 1.00000 0.0894427
\(126\) −5.04892 −0.449793
\(127\) −2.93900 −0.260794 −0.130397 0.991462i \(-0.541625\pi\)
−0.130397 + 0.991462i \(0.541625\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.35690 0.383603
\(130\) 0 0
\(131\) −3.56465 −0.311445 −0.155722 0.987801i \(-0.549771\pi\)
−0.155722 + 0.987801i \(0.549771\pi\)
\(132\) −4.74094 −0.412646
\(133\) 22.4426 1.94602
\(134\) −9.58211 −0.827768
\(135\) 1.00000 0.0860663
\(136\) 1.08815 0.0933077
\(137\) −9.47650 −0.809632 −0.404816 0.914398i \(-0.632664\pi\)
−0.404816 + 0.914398i \(0.632664\pi\)
\(138\) −2.69202 −0.229160
\(139\) −14.2078 −1.20509 −0.602543 0.798087i \(-0.705846\pi\)
−0.602543 + 0.798087i \(0.705846\pi\)
\(140\) 5.04892 0.426711
\(141\) 7.00000 0.589506
\(142\) −4.47219 −0.375298
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −7.93900 −0.659298
\(146\) 8.17390 0.676477
\(147\) 18.4916 1.52516
\(148\) −7.14675 −0.587459
\(149\) 10.2010 0.835702 0.417851 0.908516i \(-0.362783\pi\)
0.417851 + 0.908516i \(0.362783\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 3.86294 0.314361 0.157181 0.987570i \(-0.449760\pi\)
0.157181 + 0.987570i \(0.449760\pi\)
\(152\) −4.44504 −0.360541
\(153\) −1.08815 −0.0879714
\(154\) 23.9366 1.92887
\(155\) 7.85086 0.630596
\(156\) 0 0
\(157\) −11.2174 −0.895249 −0.447625 0.894222i \(-0.647730\pi\)
−0.447625 + 0.894222i \(0.647730\pi\)
\(158\) −14.1468 −1.12546
\(159\) −10.8605 −0.861297
\(160\) −1.00000 −0.0790569
\(161\) 13.5918 1.07118
\(162\) −1.00000 −0.0785674
\(163\) −6.50902 −0.509826 −0.254913 0.966964i \(-0.582047\pi\)
−0.254913 + 0.966964i \(0.582047\pi\)
\(164\) −0.664874 −0.0519180
\(165\) −4.74094 −0.369082
\(166\) −4.86294 −0.377437
\(167\) 9.67456 0.748640 0.374320 0.927300i \(-0.377876\pi\)
0.374320 + 0.927300i \(0.377876\pi\)
\(168\) −5.04892 −0.389532
\(169\) 0 0
\(170\) 1.08815 0.0834570
\(171\) 4.44504 0.339921
\(172\) 4.35690 0.332210
\(173\) −13.8388 −1.05214 −0.526071 0.850441i \(-0.676335\pi\)
−0.526071 + 0.850441i \(0.676335\pi\)
\(174\) 7.93900 0.601854
\(175\) 5.04892 0.381662
\(176\) −4.74094 −0.357362
\(177\) 12.2784 0.922904
\(178\) −14.8877 −1.11588
\(179\) −14.2078 −1.06194 −0.530969 0.847392i \(-0.678172\pi\)
−0.530969 + 0.847392i \(0.678172\pi\)
\(180\) 1.00000 0.0745356
\(181\) −24.9028 −1.85101 −0.925504 0.378739i \(-0.876358\pi\)
−0.925504 + 0.378739i \(0.876358\pi\)
\(182\) 0 0
\(183\) −11.0707 −0.818369
\(184\) −2.69202 −0.198458
\(185\) −7.14675 −0.525440
\(186\) −7.85086 −0.575653
\(187\) 5.15883 0.377251
\(188\) 7.00000 0.510527
\(189\) 5.04892 0.367255
\(190\) −4.44504 −0.322477
\(191\) 2.44504 0.176917 0.0884585 0.996080i \(-0.471806\pi\)
0.0884585 + 0.996080i \(0.471806\pi\)
\(192\) 1.00000 0.0721688
\(193\) −15.2959 −1.10102 −0.550511 0.834828i \(-0.685567\pi\)
−0.550511 + 0.834828i \(0.685567\pi\)
\(194\) −14.9269 −1.07169
\(195\) 0 0
\(196\) 18.4916 1.32083
\(197\) 3.44935 0.245756 0.122878 0.992422i \(-0.460788\pi\)
0.122878 + 0.992422i \(0.460788\pi\)
\(198\) 4.74094 0.336924
\(199\) −9.18598 −0.651177 −0.325588 0.945512i \(-0.605562\pi\)
−0.325588 + 0.945512i \(0.605562\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 9.58211 0.675870
\(202\) −11.4819 −0.807862
\(203\) −40.0834 −2.81330
\(204\) −1.08815 −0.0761855
\(205\) −0.664874 −0.0464368
\(206\) −4.16421 −0.290134
\(207\) 2.69202 0.187108
\(208\) 0 0
\(209\) −21.0737 −1.45770
\(210\) −5.04892 −0.348408
\(211\) −20.7875 −1.43107 −0.715534 0.698578i \(-0.753817\pi\)
−0.715534 + 0.698578i \(0.753817\pi\)
\(212\) −10.8605 −0.745905
\(213\) 4.47219 0.306429
\(214\) 0.796561 0.0544518
\(215\) 4.35690 0.297138
\(216\) −1.00000 −0.0680414
\(217\) 39.6383 2.69082
\(218\) −0.704103 −0.0476879
\(219\) −8.17390 −0.552341
\(220\) −4.74094 −0.319634
\(221\) 0 0
\(222\) 7.14675 0.479659
\(223\) 17.3502 1.16185 0.580927 0.813955i \(-0.302690\pi\)
0.580927 + 0.813955i \(0.302690\pi\)
\(224\) −5.04892 −0.337345
\(225\) 1.00000 0.0666667
\(226\) −14.4983 −0.964411
\(227\) −13.7573 −0.913106 −0.456553 0.889696i \(-0.650916\pi\)
−0.456553 + 0.889696i \(0.650916\pi\)
\(228\) 4.44504 0.294380
\(229\) 1.75541 0.116001 0.0580005 0.998317i \(-0.481528\pi\)
0.0580005 + 0.998317i \(0.481528\pi\)
\(230\) −2.69202 −0.177507
\(231\) −23.9366 −1.57491
\(232\) 7.93900 0.521221
\(233\) −5.49396 −0.359921 −0.179961 0.983674i \(-0.557597\pi\)
−0.179961 + 0.983674i \(0.557597\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) 12.2784 0.799258
\(237\) 14.1468 0.918930
\(238\) 5.49396 0.356120
\(239\) −17.4004 −1.12554 −0.562770 0.826613i \(-0.690264\pi\)
−0.562770 + 0.826613i \(0.690264\pi\)
\(240\) 1.00000 0.0645497
\(241\) −3.15452 −0.203201 −0.101600 0.994825i \(-0.532396\pi\)
−0.101600 + 0.994825i \(0.532396\pi\)
\(242\) −11.4765 −0.737737
\(243\) 1.00000 0.0641500
\(244\) −11.0707 −0.708728
\(245\) 18.4916 1.18138
\(246\) 0.664874 0.0423908
\(247\) 0 0
\(248\) −7.85086 −0.498530
\(249\) 4.86294 0.308176
\(250\) −1.00000 −0.0632456
\(251\) 23.9433 1.51129 0.755644 0.654982i \(-0.227324\pi\)
0.755644 + 0.654982i \(0.227324\pi\)
\(252\) 5.04892 0.318052
\(253\) −12.7627 −0.802385
\(254\) 2.93900 0.184409
\(255\) −1.08815 −0.0681423
\(256\) 1.00000 0.0625000
\(257\) 12.3381 0.769630 0.384815 0.922994i \(-0.374265\pi\)
0.384815 + 0.922994i \(0.374265\pi\)
\(258\) −4.35690 −0.271248
\(259\) −36.0834 −2.24211
\(260\) 0 0
\(261\) −7.93900 −0.491412
\(262\) 3.56465 0.220225
\(263\) 2.70709 0.166926 0.0834631 0.996511i \(-0.473402\pi\)
0.0834631 + 0.996511i \(0.473402\pi\)
\(264\) 4.74094 0.291785
\(265\) −10.8605 −0.667158
\(266\) −22.4426 −1.37605
\(267\) 14.8877 0.911112
\(268\) 9.58211 0.585320
\(269\) 20.9681 1.27845 0.639223 0.769022i \(-0.279256\pi\)
0.639223 + 0.769022i \(0.279256\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −7.99761 −0.485820 −0.242910 0.970049i \(-0.578102\pi\)
−0.242910 + 0.970049i \(0.578102\pi\)
\(272\) −1.08815 −0.0659785
\(273\) 0 0
\(274\) 9.47650 0.572496
\(275\) −4.74094 −0.285889
\(276\) 2.69202 0.162041
\(277\) −5.78554 −0.347620 −0.173810 0.984779i \(-0.555608\pi\)
−0.173810 + 0.984779i \(0.555608\pi\)
\(278\) 14.2078 0.852124
\(279\) 7.85086 0.470018
\(280\) −5.04892 −0.301731
\(281\) 19.5579 1.16673 0.583365 0.812210i \(-0.301736\pi\)
0.583365 + 0.812210i \(0.301736\pi\)
\(282\) −7.00000 −0.416844
\(283\) 31.0726 1.84707 0.923537 0.383508i \(-0.125284\pi\)
0.923537 + 0.383508i \(0.125284\pi\)
\(284\) 4.47219 0.265376
\(285\) 4.44504 0.263302
\(286\) 0 0
\(287\) −3.35690 −0.198151
\(288\) −1.00000 −0.0589256
\(289\) −15.8159 −0.930349
\(290\) 7.93900 0.466194
\(291\) 14.9269 0.875032
\(292\) −8.17390 −0.478341
\(293\) −10.0543 −0.587378 −0.293689 0.955901i \(-0.594883\pi\)
−0.293689 + 0.955901i \(0.594883\pi\)
\(294\) −18.4916 −1.07845
\(295\) 12.2784 0.714878
\(296\) 7.14675 0.415397
\(297\) −4.74094 −0.275097
\(298\) −10.2010 −0.590931
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 21.9976 1.26792
\(302\) −3.86294 −0.222287
\(303\) 11.4819 0.659617
\(304\) 4.44504 0.254941
\(305\) −11.0707 −0.633906
\(306\) 1.08815 0.0622052
\(307\) −23.6872 −1.35190 −0.675951 0.736947i \(-0.736267\pi\)
−0.675951 + 0.736947i \(0.736267\pi\)
\(308\) −23.9366 −1.36391
\(309\) 4.16421 0.236894
\(310\) −7.85086 −0.445899
\(311\) 6.77240 0.384027 0.192014 0.981392i \(-0.438498\pi\)
0.192014 + 0.981392i \(0.438498\pi\)
\(312\) 0 0
\(313\) 7.16123 0.404776 0.202388 0.979305i \(-0.435130\pi\)
0.202388 + 0.979305i \(0.435130\pi\)
\(314\) 11.2174 0.633037
\(315\) 5.04892 0.284474
\(316\) 14.1468 0.795817
\(317\) −1.20344 −0.0675919 −0.0337959 0.999429i \(-0.510760\pi\)
−0.0337959 + 0.999429i \(0.510760\pi\)
\(318\) 10.8605 0.609029
\(319\) 37.6383 2.10734
\(320\) 1.00000 0.0559017
\(321\) −0.796561 −0.0444597
\(322\) −13.5918 −0.757441
\(323\) −4.83685 −0.269130
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 6.50902 0.360502
\(327\) 0.704103 0.0389370
\(328\) 0.664874 0.0367115
\(329\) 35.3424 1.94849
\(330\) 4.74094 0.260980
\(331\) 20.9933 1.15390 0.576948 0.816781i \(-0.304244\pi\)
0.576948 + 0.816781i \(0.304244\pi\)
\(332\) 4.86294 0.266888
\(333\) −7.14675 −0.391640
\(334\) −9.67456 −0.529369
\(335\) 9.58211 0.523526
\(336\) 5.04892 0.275441
\(337\) −3.01938 −0.164476 −0.0822380 0.996613i \(-0.526207\pi\)
−0.0822380 + 0.996613i \(0.526207\pi\)
\(338\) 0 0
\(339\) 14.4983 0.787438
\(340\) −1.08815 −0.0590130
\(341\) −37.2204 −2.01560
\(342\) −4.44504 −0.240360
\(343\) 58.0200 3.13278
\(344\) −4.35690 −0.234908
\(345\) 2.69202 0.144934
\(346\) 13.8388 0.743977
\(347\) 11.2610 0.604521 0.302261 0.953225i \(-0.402259\pi\)
0.302261 + 0.953225i \(0.402259\pi\)
\(348\) −7.93900 −0.425575
\(349\) −5.84010 −0.312613 −0.156307 0.987709i \(-0.549959\pi\)
−0.156307 + 0.987709i \(0.549959\pi\)
\(350\) −5.04892 −0.269876
\(351\) 0 0
\(352\) 4.74094 0.252693
\(353\) 0.675628 0.0359601 0.0179800 0.999838i \(-0.494276\pi\)
0.0179800 + 0.999838i \(0.494276\pi\)
\(354\) −12.2784 −0.652592
\(355\) 4.47219 0.237359
\(356\) 14.8877 0.789046
\(357\) −5.49396 −0.290771
\(358\) 14.2078 0.750903
\(359\) −8.52350 −0.449853 −0.224927 0.974376i \(-0.572214\pi\)
−0.224927 + 0.974376i \(0.572214\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 0.758397 0.0399156
\(362\) 24.9028 1.30886
\(363\) 11.4765 0.602360
\(364\) 0 0
\(365\) −8.17390 −0.427841
\(366\) 11.0707 0.578674
\(367\) −9.40044 −0.490699 −0.245349 0.969435i \(-0.578903\pi\)
−0.245349 + 0.969435i \(0.578903\pi\)
\(368\) 2.69202 0.140331
\(369\) −0.664874 −0.0346120
\(370\) 7.14675 0.371542
\(371\) −54.8340 −2.84684
\(372\) 7.85086 0.407048
\(373\) 23.6582 1.22497 0.612487 0.790481i \(-0.290169\pi\)
0.612487 + 0.790481i \(0.290169\pi\)
\(374\) −5.15883 −0.266757
\(375\) 1.00000 0.0516398
\(376\) −7.00000 −0.360997
\(377\) 0 0
\(378\) −5.04892 −0.259688
\(379\) 16.3437 0.839522 0.419761 0.907635i \(-0.362114\pi\)
0.419761 + 0.907635i \(0.362114\pi\)
\(380\) 4.44504 0.228026
\(381\) −2.93900 −0.150570
\(382\) −2.44504 −0.125099
\(383\) −4.79656 −0.245093 −0.122546 0.992463i \(-0.539106\pi\)
−0.122546 + 0.992463i \(0.539106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −23.9366 −1.21992
\(386\) 15.2959 0.778541
\(387\) 4.35690 0.221473
\(388\) 14.9269 0.757800
\(389\) 21.4534 1.08773 0.543865 0.839173i \(-0.316960\pi\)
0.543865 + 0.839173i \(0.316960\pi\)
\(390\) 0 0
\(391\) −2.92931 −0.148142
\(392\) −18.4916 −0.933965
\(393\) −3.56465 −0.179813
\(394\) −3.44935 −0.173776
\(395\) 14.1468 0.711800
\(396\) −4.74094 −0.238241
\(397\) 1.36765 0.0686404 0.0343202 0.999411i \(-0.489073\pi\)
0.0343202 + 0.999411i \(0.489073\pi\)
\(398\) 9.18598 0.460452
\(399\) 22.4426 1.12354
\(400\) 1.00000 0.0500000
\(401\) −18.8931 −0.943475 −0.471737 0.881739i \(-0.656373\pi\)
−0.471737 + 0.881739i \(0.656373\pi\)
\(402\) −9.58211 −0.477912
\(403\) 0 0
\(404\) 11.4819 0.571245
\(405\) 1.00000 0.0496904
\(406\) 40.0834 1.98930
\(407\) 33.8823 1.67948
\(408\) 1.08815 0.0538713
\(409\) −25.9638 −1.28383 −0.641913 0.766778i \(-0.721859\pi\)
−0.641913 + 0.766778i \(0.721859\pi\)
\(410\) 0.664874 0.0328358
\(411\) −9.47650 −0.467441
\(412\) 4.16421 0.205156
\(413\) 61.9928 3.05047
\(414\) −2.69202 −0.132306
\(415\) 4.86294 0.238712
\(416\) 0 0
\(417\) −14.2078 −0.695757
\(418\) 21.0737 1.03075
\(419\) −32.4282 −1.58422 −0.792110 0.610378i \(-0.791017\pi\)
−0.792110 + 0.610378i \(0.791017\pi\)
\(420\) 5.04892 0.246362
\(421\) −33.2868 −1.62230 −0.811150 0.584839i \(-0.801158\pi\)
−0.811150 + 0.584839i \(0.801158\pi\)
\(422\) 20.7875 1.01192
\(423\) 7.00000 0.340352
\(424\) 10.8605 0.527435
\(425\) −1.08815 −0.0527828
\(426\) −4.47219 −0.216678
\(427\) −55.8950 −2.70495
\(428\) −0.796561 −0.0385032
\(429\) 0 0
\(430\) −4.35690 −0.210108
\(431\) 11.1578 0.537451 0.268725 0.963217i \(-0.413398\pi\)
0.268725 + 0.963217i \(0.413398\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0828 0.676775 0.338387 0.941007i \(-0.390119\pi\)
0.338387 + 0.941007i \(0.390119\pi\)
\(434\) −39.6383 −1.90270
\(435\) −7.93900 −0.380646
\(436\) 0.704103 0.0337204
\(437\) 11.9661 0.572418
\(438\) 8.17390 0.390564
\(439\) −28.9933 −1.38377 −0.691887 0.722006i \(-0.743220\pi\)
−0.691887 + 0.722006i \(0.743220\pi\)
\(440\) 4.74094 0.226015
\(441\) 18.4916 0.880551
\(442\) 0 0
\(443\) 33.3080 1.58251 0.791255 0.611486i \(-0.209428\pi\)
0.791255 + 0.611486i \(0.209428\pi\)
\(444\) −7.14675 −0.339170
\(445\) 14.8877 0.705744
\(446\) −17.3502 −0.821555
\(447\) 10.2010 0.482493
\(448\) 5.04892 0.238539
\(449\) −37.7899 −1.78341 −0.891707 0.452614i \(-0.850492\pi\)
−0.891707 + 0.452614i \(0.850492\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 3.15213 0.148428
\(452\) 14.4983 0.681941
\(453\) 3.86294 0.181497
\(454\) 13.7573 0.645664
\(455\) 0 0
\(456\) −4.44504 −0.208158
\(457\) −10.5700 −0.494445 −0.247222 0.968959i \(-0.579518\pi\)
−0.247222 + 0.968959i \(0.579518\pi\)
\(458\) −1.75541 −0.0820251
\(459\) −1.08815 −0.0507903
\(460\) 2.69202 0.125516
\(461\) −20.1420 −0.938105 −0.469052 0.883170i \(-0.655404\pi\)
−0.469052 + 0.883170i \(0.655404\pi\)
\(462\) 23.9366 1.11363
\(463\) 15.9065 0.739237 0.369618 0.929184i \(-0.379488\pi\)
0.369618 + 0.929184i \(0.379488\pi\)
\(464\) −7.93900 −0.368559
\(465\) 7.85086 0.364075
\(466\) 5.49396 0.254503
\(467\) −25.6136 −1.18525 −0.592627 0.805477i \(-0.701909\pi\)
−0.592627 + 0.805477i \(0.701909\pi\)
\(468\) 0 0
\(469\) 48.3793 2.23395
\(470\) −7.00000 −0.322886
\(471\) −11.2174 −0.516872
\(472\) −12.2784 −0.565161
\(473\) −20.6558 −0.949754
\(474\) −14.1468 −0.649782
\(475\) 4.44504 0.203953
\(476\) −5.49396 −0.251815
\(477\) −10.8605 −0.497270
\(478\) 17.4004 0.795877
\(479\) −7.59658 −0.347097 −0.173548 0.984825i \(-0.555523\pi\)
−0.173548 + 0.984825i \(0.555523\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 3.15452 0.143685
\(483\) 13.5918 0.618448
\(484\) 11.4765 0.521659
\(485\) 14.9269 0.677796
\(486\) −1.00000 −0.0453609
\(487\) 0.779103 0.0353045 0.0176523 0.999844i \(-0.494381\pi\)
0.0176523 + 0.999844i \(0.494381\pi\)
\(488\) 11.0707 0.501146
\(489\) −6.50902 −0.294348
\(490\) −18.4916 −0.835364
\(491\) 11.3948 0.514240 0.257120 0.966379i \(-0.417226\pi\)
0.257120 + 0.966379i \(0.417226\pi\)
\(492\) −0.664874 −0.0299749
\(493\) 8.63879 0.389072
\(494\) 0 0
\(495\) −4.74094 −0.213089
\(496\) 7.85086 0.352514
\(497\) 22.5797 1.01284
\(498\) −4.86294 −0.217913
\(499\) 23.5555 1.05449 0.527246 0.849713i \(-0.323225\pi\)
0.527246 + 0.849713i \(0.323225\pi\)
\(500\) 1.00000 0.0447214
\(501\) 9.67456 0.432228
\(502\) −23.9433 −1.06864
\(503\) −12.6165 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(504\) −5.04892 −0.224897
\(505\) 11.4819 0.510937
\(506\) 12.7627 0.567372
\(507\) 0 0
\(508\) −2.93900 −0.130397
\(509\) −8.45473 −0.374749 −0.187375 0.982289i \(-0.559998\pi\)
−0.187375 + 0.982289i \(0.559998\pi\)
\(510\) 1.08815 0.0481839
\(511\) −41.2693 −1.82565
\(512\) −1.00000 −0.0441942
\(513\) 4.44504 0.196253
\(514\) −12.3381 −0.544211
\(515\) 4.16421 0.183497
\(516\) 4.35690 0.191802
\(517\) −33.1866 −1.45954
\(518\) 36.0834 1.58541
\(519\) −13.8388 −0.607455
\(520\) 0 0
\(521\) 34.7254 1.52135 0.760674 0.649134i \(-0.224869\pi\)
0.760674 + 0.649134i \(0.224869\pi\)
\(522\) 7.93900 0.347481
\(523\) 29.3163 1.28191 0.640957 0.767577i \(-0.278538\pi\)
0.640957 + 0.767577i \(0.278538\pi\)
\(524\) −3.56465 −0.155722
\(525\) 5.04892 0.220353
\(526\) −2.70709 −0.118035
\(527\) −8.54288 −0.372134
\(528\) −4.74094 −0.206323
\(529\) −15.7530 −0.684914
\(530\) 10.8605 0.471752
\(531\) 12.2784 0.532839
\(532\) 22.4426 0.973012
\(533\) 0 0
\(534\) −14.8877 −0.644253
\(535\) −0.796561 −0.0344383
\(536\) −9.58211 −0.413884
\(537\) −14.2078 −0.613110
\(538\) −20.9681 −0.903998
\(539\) −87.6674 −3.77610
\(540\) 1.00000 0.0430331
\(541\) 29.1032 1.25124 0.625622 0.780126i \(-0.284845\pi\)
0.625622 + 0.780126i \(0.284845\pi\)
\(542\) 7.99761 0.343527
\(543\) −24.9028 −1.06868
\(544\) 1.08815 0.0466539
\(545\) 0.704103 0.0301605
\(546\) 0 0
\(547\) −2.37004 −0.101336 −0.0506678 0.998716i \(-0.516135\pi\)
−0.0506678 + 0.998716i \(0.516135\pi\)
\(548\) −9.47650 −0.404816
\(549\) −11.0707 −0.472485
\(550\) 4.74094 0.202154
\(551\) −35.2892 −1.50337
\(552\) −2.69202 −0.114580
\(553\) 71.4258 3.03733
\(554\) 5.78554 0.245804
\(555\) −7.14675 −0.303363
\(556\) −14.2078 −0.602543
\(557\) 28.8364 1.22184 0.610918 0.791694i \(-0.290800\pi\)
0.610918 + 0.791694i \(0.290800\pi\)
\(558\) −7.85086 −0.332353
\(559\) 0 0
\(560\) 5.04892 0.213356
\(561\) 5.15883 0.217806
\(562\) −19.5579 −0.825002
\(563\) −35.7133 −1.50514 −0.752568 0.658514i \(-0.771185\pi\)
−0.752568 + 0.658514i \(0.771185\pi\)
\(564\) 7.00000 0.294753
\(565\) 14.4983 0.609947
\(566\) −31.0726 −1.30608
\(567\) 5.04892 0.212035
\(568\) −4.47219 −0.187649
\(569\) 42.1618 1.76752 0.883758 0.467945i \(-0.155005\pi\)
0.883758 + 0.467945i \(0.155005\pi\)
\(570\) −4.44504 −0.186182
\(571\) 23.1497 0.968786 0.484393 0.874850i \(-0.339040\pi\)
0.484393 + 0.874850i \(0.339040\pi\)
\(572\) 0 0
\(573\) 2.44504 0.102143
\(574\) 3.35690 0.140114
\(575\) 2.69202 0.112265
\(576\) 1.00000 0.0416667
\(577\) −41.7318 −1.73732 −0.868660 0.495409i \(-0.835018\pi\)
−0.868660 + 0.495409i \(0.835018\pi\)
\(578\) 15.8159 0.657856
\(579\) −15.2959 −0.635676
\(580\) −7.93900 −0.329649
\(581\) 24.5526 1.01861
\(582\) −14.9269 −0.618741
\(583\) 51.4892 2.13246
\(584\) 8.17390 0.338238
\(585\) 0 0
\(586\) 10.0543 0.415339
\(587\) −3.52648 −0.145554 −0.0727768 0.997348i \(-0.523186\pi\)
−0.0727768 + 0.997348i \(0.523186\pi\)
\(588\) 18.4916 0.762579
\(589\) 34.8974 1.43792
\(590\) −12.2784 −0.505495
\(591\) 3.44935 0.141887
\(592\) −7.14675 −0.293730
\(593\) 7.53617 0.309473 0.154737 0.987956i \(-0.450547\pi\)
0.154737 + 0.987956i \(0.450547\pi\)
\(594\) 4.74094 0.194523
\(595\) −5.49396 −0.225230
\(596\) 10.2010 0.417851
\(597\) −9.18598 −0.375957
\(598\) 0 0
\(599\) −44.2161 −1.80662 −0.903311 0.428987i \(-0.858871\pi\)
−0.903311 + 0.428987i \(0.858871\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −40.7251 −1.66121 −0.830606 0.556860i \(-0.812006\pi\)
−0.830606 + 0.556860i \(0.812006\pi\)
\(602\) −21.9976 −0.896556
\(603\) 9.58211 0.390213
\(604\) 3.86294 0.157181
\(605\) 11.4765 0.466586
\(606\) −11.4819 −0.466419
\(607\) 40.9788 1.66328 0.831640 0.555316i \(-0.187403\pi\)
0.831640 + 0.555316i \(0.187403\pi\)
\(608\) −4.44504 −0.180270
\(609\) −40.0834 −1.62426
\(610\) 11.0707 0.448239
\(611\) 0 0
\(612\) −1.08815 −0.0439857
\(613\) −40.5491 −1.63776 −0.818882 0.573963i \(-0.805405\pi\)
−0.818882 + 0.573963i \(0.805405\pi\)
\(614\) 23.6872 0.955939
\(615\) −0.664874 −0.0268103
\(616\) 23.9366 0.964433
\(617\) −3.06829 −0.123525 −0.0617624 0.998091i \(-0.519672\pi\)
−0.0617624 + 0.998091i \(0.519672\pi\)
\(618\) −4.16421 −0.167509
\(619\) −31.0847 −1.24940 −0.624700 0.780865i \(-0.714779\pi\)
−0.624700 + 0.780865i \(0.714779\pi\)
\(620\) 7.85086 0.315298
\(621\) 2.69202 0.108027
\(622\) −6.77240 −0.271548
\(623\) 75.1667 3.01149
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −7.16123 −0.286220
\(627\) −21.0737 −0.841601
\(628\) −11.2174 −0.447625
\(629\) 7.77671 0.310078
\(630\) −5.04892 −0.201154
\(631\) 23.7101 0.943883 0.471942 0.881630i \(-0.343553\pi\)
0.471942 + 0.881630i \(0.343553\pi\)
\(632\) −14.1468 −0.562728
\(633\) −20.7875 −0.826227
\(634\) 1.20344 0.0477947
\(635\) −2.93900 −0.116631
\(636\) −10.8605 −0.430649
\(637\) 0 0
\(638\) −37.6383 −1.49012
\(639\) 4.47219 0.176917
\(640\) −1.00000 −0.0395285
\(641\) −48.3866 −1.91115 −0.955577 0.294742i \(-0.904766\pi\)
−0.955577 + 0.294742i \(0.904766\pi\)
\(642\) 0.796561 0.0314378
\(643\) 20.4209 0.805321 0.402660 0.915349i \(-0.368086\pi\)
0.402660 + 0.915349i \(0.368086\pi\)
\(644\) 13.5918 0.535592
\(645\) 4.35690 0.171553
\(646\) 4.83685 0.190303
\(647\) −0.292913 −0.0115156 −0.00575780 0.999983i \(-0.501833\pi\)
−0.00575780 + 0.999983i \(0.501833\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −58.2113 −2.28499
\(650\) 0 0
\(651\) 39.6383 1.55355
\(652\) −6.50902 −0.254913
\(653\) 21.8323 0.854365 0.427183 0.904165i \(-0.359506\pi\)
0.427183 + 0.904165i \(0.359506\pi\)
\(654\) −0.704103 −0.0275326
\(655\) −3.56465 −0.139282
\(656\) −0.664874 −0.0259590
\(657\) −8.17390 −0.318894
\(658\) −35.3424 −1.37779
\(659\) −17.0519 −0.664248 −0.332124 0.943236i \(-0.607765\pi\)
−0.332124 + 0.943236i \(0.607765\pi\)
\(660\) −4.74094 −0.184541
\(661\) −31.8098 −1.23726 −0.618629 0.785683i \(-0.712312\pi\)
−0.618629 + 0.785683i \(0.712312\pi\)
\(662\) −20.9933 −0.815928
\(663\) 0 0
\(664\) −4.86294 −0.188719
\(665\) 22.4426 0.870289
\(666\) 7.14675 0.276931
\(667\) −21.3720 −0.827526
\(668\) 9.67456 0.374320
\(669\) 17.3502 0.670797
\(670\) −9.58211 −0.370189
\(671\) 52.4855 2.02618
\(672\) −5.04892 −0.194766
\(673\) −4.10513 −0.158241 −0.0791206 0.996865i \(-0.525211\pi\)
−0.0791206 + 0.996865i \(0.525211\pi\)
\(674\) 3.01938 0.116302
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 11.0543 0.424851 0.212426 0.977177i \(-0.431864\pi\)
0.212426 + 0.977177i \(0.431864\pi\)
\(678\) −14.4983 −0.556803
\(679\) 75.3648 2.89223
\(680\) 1.08815 0.0417285
\(681\) −13.7573 −0.527182
\(682\) 37.2204 1.42524
\(683\) −9.06590 −0.346897 −0.173449 0.984843i \(-0.555491\pi\)
−0.173449 + 0.984843i \(0.555491\pi\)
\(684\) 4.44504 0.169960
\(685\) −9.47650 −0.362078
\(686\) −58.0200 −2.21521
\(687\) 1.75541 0.0669732
\(688\) 4.35690 0.166105
\(689\) 0 0
\(690\) −2.69202 −0.102484
\(691\) −12.4282 −0.472790 −0.236395 0.971657i \(-0.575966\pi\)
−0.236395 + 0.971657i \(0.575966\pi\)
\(692\) −13.8388 −0.526071
\(693\) −23.9366 −0.909277
\(694\) −11.2610 −0.427461
\(695\) −14.2078 −0.538931
\(696\) 7.93900 0.300927
\(697\) 0.723480 0.0274038
\(698\) 5.84010 0.221051
\(699\) −5.49396 −0.207801
\(700\) 5.04892 0.190831
\(701\) −38.5418 −1.45570 −0.727852 0.685734i \(-0.759481\pi\)
−0.727852 + 0.685734i \(0.759481\pi\)
\(702\) 0 0
\(703\) −31.7676 −1.19814
\(704\) −4.74094 −0.178681
\(705\) 7.00000 0.263635
\(706\) −0.675628 −0.0254276
\(707\) 57.9711 2.18023
\(708\) 12.2784 0.461452
\(709\) 38.7851 1.45660 0.728302 0.685256i \(-0.240310\pi\)
0.728302 + 0.685256i \(0.240310\pi\)
\(710\) −4.47219 −0.167838
\(711\) 14.1468 0.530545
\(712\) −14.8877 −0.557940
\(713\) 21.1347 0.791500
\(714\) 5.49396 0.205606
\(715\) 0 0
\(716\) −14.2078 −0.530969
\(717\) −17.4004 −0.649831
\(718\) 8.52350 0.318094
\(719\) −12.1021 −0.451334 −0.225667 0.974205i \(-0.572456\pi\)
−0.225667 + 0.974205i \(0.572456\pi\)
\(720\) 1.00000 0.0372678
\(721\) 21.0248 0.783003
\(722\) −0.758397 −0.0282246
\(723\) −3.15452 −0.117318
\(724\) −24.9028 −0.925504
\(725\) −7.93900 −0.294847
\(726\) −11.4765 −0.425933
\(727\) 40.6088 1.50610 0.753048 0.657965i \(-0.228583\pi\)
0.753048 + 0.657965i \(0.228583\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.17390 0.302530
\(731\) −4.74094 −0.175350
\(732\) −11.0707 −0.409184
\(733\) 32.6601 1.20633 0.603164 0.797617i \(-0.293906\pi\)
0.603164 + 0.797617i \(0.293906\pi\)
\(734\) 9.40044 0.346976
\(735\) 18.4916 0.682072
\(736\) −2.69202 −0.0992292
\(737\) −45.4282 −1.67337
\(738\) 0.664874 0.0244744
\(739\) 3.72348 0.136970 0.0684852 0.997652i \(-0.478183\pi\)
0.0684852 + 0.997652i \(0.478183\pi\)
\(740\) −7.14675 −0.262720
\(741\) 0 0
\(742\) 54.8340 2.01302
\(743\) −31.3099 −1.14865 −0.574324 0.818628i \(-0.694735\pi\)
−0.574324 + 0.818628i \(0.694735\pi\)
\(744\) −7.85086 −0.287826
\(745\) 10.2010 0.373737
\(746\) −23.6582 −0.866187
\(747\) 4.86294 0.177926
\(748\) 5.15883 0.188626
\(749\) −4.02177 −0.146952
\(750\) −1.00000 −0.0365148
\(751\) 2.66296 0.0971726 0.0485863 0.998819i \(-0.484528\pi\)
0.0485863 + 0.998819i \(0.484528\pi\)
\(752\) 7.00000 0.255264
\(753\) 23.9433 0.872543
\(754\) 0 0
\(755\) 3.86294 0.140587
\(756\) 5.04892 0.183627
\(757\) 32.2553 1.17234 0.586170 0.810188i \(-0.300635\pi\)
0.586170 + 0.810188i \(0.300635\pi\)
\(758\) −16.3437 −0.593632
\(759\) −12.7627 −0.463257
\(760\) −4.44504 −0.161239
\(761\) 27.2228 0.986826 0.493413 0.869795i \(-0.335749\pi\)
0.493413 + 0.869795i \(0.335749\pi\)
\(762\) 2.93900 0.106469
\(763\) 3.55496 0.128698
\(764\) 2.44504 0.0884585
\(765\) −1.08815 −0.0393420
\(766\) 4.79656 0.173307
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −5.62133 −0.202710 −0.101355 0.994850i \(-0.532318\pi\)
−0.101355 + 0.994850i \(0.532318\pi\)
\(770\) 23.9366 0.862615
\(771\) 12.3381 0.444346
\(772\) −15.2959 −0.550511
\(773\) 35.7918 1.28734 0.643670 0.765303i \(-0.277411\pi\)
0.643670 + 0.765303i \(0.277411\pi\)
\(774\) −4.35690 −0.156605
\(775\) 7.85086 0.282011
\(776\) −14.9269 −0.535845
\(777\) −36.0834 −1.29448
\(778\) −21.4534 −0.769142
\(779\) −2.95539 −0.105888
\(780\) 0 0
\(781\) −21.2024 −0.758681
\(782\) 2.92931 0.104752
\(783\) −7.93900 −0.283717
\(784\) 18.4916 0.660413
\(785\) −11.2174 −0.400368
\(786\) 3.56465 0.127147
\(787\) −18.5442 −0.661030 −0.330515 0.943801i \(-0.607222\pi\)
−0.330515 + 0.943801i \(0.607222\pi\)
\(788\) 3.44935 0.122878
\(789\) 2.70709 0.0963748
\(790\) −14.1468 −0.503319
\(791\) 73.2006 2.60271
\(792\) 4.74094 0.168462
\(793\) 0 0
\(794\) −1.36765 −0.0485361
\(795\) −10.8605 −0.385184
\(796\) −9.18598 −0.325588
\(797\) −33.0898 −1.17210 −0.586050 0.810275i \(-0.699318\pi\)
−0.586050 + 0.810275i \(0.699318\pi\)
\(798\) −22.4426 −0.794461
\(799\) −7.61702 −0.269471
\(800\) −1.00000 −0.0353553
\(801\) 14.8877 0.526031
\(802\) 18.8931 0.667137
\(803\) 38.7520 1.36753
\(804\) 9.58211 0.337935
\(805\) 13.5918 0.479048
\(806\) 0 0
\(807\) 20.9681 0.738111
\(808\) −11.4819 −0.403931
\(809\) 11.4004 0.400818 0.200409 0.979712i \(-0.435773\pi\)
0.200409 + 0.979712i \(0.435773\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 38.9724 1.36851 0.684253 0.729245i \(-0.260129\pi\)
0.684253 + 0.729245i \(0.260129\pi\)
\(812\) −40.0834 −1.40665
\(813\) −7.99761 −0.280488
\(814\) −33.8823 −1.18757
\(815\) −6.50902 −0.228001
\(816\) −1.08815 −0.0380927
\(817\) 19.3666 0.677551
\(818\) 25.9638 0.907801
\(819\) 0 0
\(820\) −0.664874 −0.0232184
\(821\) −7.12498 −0.248664 −0.124332 0.992241i \(-0.539679\pi\)
−0.124332 + 0.992241i \(0.539679\pi\)
\(822\) 9.47650 0.330531
\(823\) 19.1323 0.666909 0.333455 0.942766i \(-0.391786\pi\)
0.333455 + 0.942766i \(0.391786\pi\)
\(824\) −4.16421 −0.145067
\(825\) −4.74094 −0.165058
\(826\) −61.9928 −2.15701
\(827\) 42.0538 1.46235 0.731177 0.682187i \(-0.238971\pi\)
0.731177 + 0.682187i \(0.238971\pi\)
\(828\) 2.69202 0.0935542
\(829\) −17.6364 −0.612537 −0.306269 0.951945i \(-0.599081\pi\)
−0.306269 + 0.951945i \(0.599081\pi\)
\(830\) −4.86294 −0.168795
\(831\) −5.78554 −0.200698
\(832\) 0 0
\(833\) −20.1215 −0.697169
\(834\) 14.2078 0.491974
\(835\) 9.67456 0.334802
\(836\) −21.0737 −0.728848
\(837\) 7.85086 0.271365
\(838\) 32.4282 1.12021
\(839\) −44.4905 −1.53598 −0.767991 0.640460i \(-0.778744\pi\)
−0.767991 + 0.640460i \(0.778744\pi\)
\(840\) −5.04892 −0.174204
\(841\) 34.0277 1.17337
\(842\) 33.2868 1.14714
\(843\) 19.5579 0.673611
\(844\) −20.7875 −0.715534
\(845\) 0 0
\(846\) −7.00000 −0.240665
\(847\) 57.9439 1.99098
\(848\) −10.8605 −0.372953
\(849\) 31.0726 1.06641
\(850\) 1.08815 0.0373231
\(851\) −19.2392 −0.659512
\(852\) 4.47219 0.153215
\(853\) 7.59743 0.260131 0.130066 0.991505i \(-0.458481\pi\)
0.130066 + 0.991505i \(0.458481\pi\)
\(854\) 55.8950 1.91269
\(855\) 4.44504 0.152017
\(856\) 0.796561 0.0272259
\(857\) −34.2978 −1.17159 −0.585796 0.810459i \(-0.699218\pi\)
−0.585796 + 0.810459i \(0.699218\pi\)
\(858\) 0 0
\(859\) −30.8412 −1.05229 −0.526144 0.850396i \(-0.676363\pi\)
−0.526144 + 0.850396i \(0.676363\pi\)
\(860\) 4.35690 0.148569
\(861\) −3.35690 −0.114403
\(862\) −11.1578 −0.380035
\(863\) −16.1075 −0.548306 −0.274153 0.961686i \(-0.588398\pi\)
−0.274153 + 0.961686i \(0.588398\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −13.8388 −0.470532
\(866\) −14.0828 −0.478552
\(867\) −15.8159 −0.537137
\(868\) 39.6383 1.34541
\(869\) −67.0689 −2.27516
\(870\) 7.93900 0.269157
\(871\) 0 0
\(872\) −0.704103 −0.0238439
\(873\) 14.9269 0.505200
\(874\) −11.9661 −0.404761
\(875\) 5.04892 0.170685
\(876\) −8.17390 −0.276170
\(877\) −32.7342 −1.10536 −0.552678 0.833395i \(-0.686394\pi\)
−0.552678 + 0.833395i \(0.686394\pi\)
\(878\) 28.9933 0.978476
\(879\) −10.0543 −0.339123
\(880\) −4.74094 −0.159817
\(881\) −9.32736 −0.314247 −0.157123 0.987579i \(-0.550222\pi\)
−0.157123 + 0.987579i \(0.550222\pi\)
\(882\) −18.4916 −0.622643
\(883\) 45.1135 1.51819 0.759095 0.650980i \(-0.225642\pi\)
0.759095 + 0.650980i \(0.225642\pi\)
\(884\) 0 0
\(885\) 12.2784 0.412735
\(886\) −33.3080 −1.11900
\(887\) −37.4383 −1.25706 −0.628528 0.777787i \(-0.716342\pi\)
−0.628528 + 0.777787i \(0.716342\pi\)
\(888\) 7.14675 0.239829
\(889\) −14.8388 −0.497676
\(890\) −14.8877 −0.499037
\(891\) −4.74094 −0.158827
\(892\) 17.3502 0.580927
\(893\) 31.1153 1.04123
\(894\) −10.2010 −0.341174
\(895\) −14.2078 −0.474913
\(896\) −5.04892 −0.168672
\(897\) 0 0
\(898\) 37.7899 1.26106
\(899\) −62.3279 −2.07875
\(900\) 1.00000 0.0333333
\(901\) 11.8179 0.393710
\(902\) −3.15213 −0.104954
\(903\) 21.9976 0.732035
\(904\) −14.4983 −0.482205
\(905\) −24.9028 −0.827796
\(906\) −3.86294 −0.128337
\(907\) −35.9004 −1.19205 −0.596026 0.802965i \(-0.703254\pi\)
−0.596026 + 0.802965i \(0.703254\pi\)
\(908\) −13.7573 −0.456553
\(909\) 11.4819 0.380830
\(910\) 0 0
\(911\) 34.0025 1.12655 0.563277 0.826268i \(-0.309541\pi\)
0.563277 + 0.826268i \(0.309541\pi\)
\(912\) 4.44504 0.147190
\(913\) −23.0549 −0.763005
\(914\) 10.5700 0.349625
\(915\) −11.0707 −0.365986
\(916\) 1.75541 0.0580005
\(917\) −17.9976 −0.594333
\(918\) 1.08815 0.0359142
\(919\) −27.7778 −0.916304 −0.458152 0.888874i \(-0.651489\pi\)
−0.458152 + 0.888874i \(0.651489\pi\)
\(920\) −2.69202 −0.0887533
\(921\) −23.6872 −0.780521
\(922\) 20.1420 0.663340
\(923\) 0 0
\(924\) −23.9366 −0.787457
\(925\) −7.14675 −0.234984
\(926\) −15.9065 −0.522719
\(927\) 4.16421 0.136771
\(928\) 7.93900 0.260610
\(929\) −44.9342 −1.47424 −0.737122 0.675760i \(-0.763816\pi\)
−0.737122 + 0.675760i \(0.763816\pi\)
\(930\) −7.85086 −0.257440
\(931\) 82.1958 2.69386
\(932\) −5.49396 −0.179961
\(933\) 6.77240 0.221718
\(934\) 25.6136 0.838101
\(935\) 5.15883 0.168712
\(936\) 0 0
\(937\) −4.50843 −0.147284 −0.0736421 0.997285i \(-0.523462\pi\)
−0.0736421 + 0.997285i \(0.523462\pi\)
\(938\) −48.3793 −1.57964
\(939\) 7.16123 0.233698
\(940\) 7.00000 0.228315
\(941\) −18.4926 −0.602843 −0.301421 0.953491i \(-0.597461\pi\)
−0.301421 + 0.953491i \(0.597461\pi\)
\(942\) 11.2174 0.365484
\(943\) −1.78986 −0.0582857
\(944\) 12.2784 0.399629
\(945\) 5.04892 0.164241
\(946\) 20.6558 0.671577
\(947\) −60.3062 −1.95969 −0.979844 0.199766i \(-0.935982\pi\)
−0.979844 + 0.199766i \(0.935982\pi\)
\(948\) 14.1468 0.459465
\(949\) 0 0
\(950\) −4.44504 −0.144216
\(951\) −1.20344 −0.0390242
\(952\) 5.49396 0.178060
\(953\) 23.6394 0.765755 0.382877 0.923799i \(-0.374933\pi\)
0.382877 + 0.923799i \(0.374933\pi\)
\(954\) 10.8605 0.351623
\(955\) 2.44504 0.0791197
\(956\) −17.4004 −0.562770
\(957\) 37.6383 1.21667
\(958\) 7.59658 0.245434
\(959\) −47.8461 −1.54503
\(960\) 1.00000 0.0322749
\(961\) 30.6359 0.988256
\(962\) 0 0
\(963\) −0.796561 −0.0256688
\(964\) −3.15452 −0.101600
\(965\) −15.2959 −0.492392
\(966\) −13.5918 −0.437309
\(967\) −22.4910 −0.723261 −0.361631 0.932321i \(-0.617780\pi\)
−0.361631 + 0.932321i \(0.617780\pi\)
\(968\) −11.4765 −0.368869
\(969\) −4.83685 −0.155382
\(970\) −14.9269 −0.479275
\(971\) −15.4257 −0.495033 −0.247517 0.968884i \(-0.579614\pi\)
−0.247517 + 0.968884i \(0.579614\pi\)
\(972\) 1.00000 0.0320750
\(973\) −71.7338 −2.29968
\(974\) −0.779103 −0.0249641
\(975\) 0 0
\(976\) −11.0707 −0.354364
\(977\) 38.5018 1.23178 0.615892 0.787831i \(-0.288796\pi\)
0.615892 + 0.787831i \(0.288796\pi\)
\(978\) 6.50902 0.208136
\(979\) −70.5816 −2.25580
\(980\) 18.4916 0.590691
\(981\) 0.704103 0.0224803
\(982\) −11.3948 −0.363623
\(983\) 3.83340 0.122266 0.0611332 0.998130i \(-0.480529\pi\)
0.0611332 + 0.998130i \(0.480529\pi\)
\(984\) 0.664874 0.0211954
\(985\) 3.44935 0.109906
\(986\) −8.63879 −0.275115
\(987\) 35.3424 1.12496
\(988\) 0 0
\(989\) 11.7289 0.372956
\(990\) 4.74094 0.150677
\(991\) −46.7670 −1.48560 −0.742802 0.669512i \(-0.766503\pi\)
−0.742802 + 0.669512i \(0.766503\pi\)
\(992\) −7.85086 −0.249265
\(993\) 20.9933 0.666202
\(994\) −22.5797 −0.716185
\(995\) −9.18598 −0.291215
\(996\) 4.86294 0.154088
\(997\) −42.8920 −1.35840 −0.679202 0.733952i \(-0.737674\pi\)
−0.679202 + 0.733952i \(0.737674\pi\)
\(998\) −23.5555 −0.745638
\(999\) −7.14675 −0.226113
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 5070.2.a.bq.1.3 3
13.5 odd 4 5070.2.b.y.1351.6 6
13.8 odd 4 5070.2.b.y.1351.1 6
13.12 even 2 5070.2.a.bv.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bq.1.3 3 1.1 even 1 trivial
5070.2.a.bv.1.1 yes 3 13.12 even 2
5070.2.b.y.1351.1 6 13.8 odd 4
5070.2.b.y.1351.6 6 13.5 odd 4