Properties

Label 8470.2.a.cz.1.5
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.10784448.1
Defining polynomial: \(x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.58124\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.81361 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.81361 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.91638 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.81361 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.81361 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.91638 q^{9} -1.00000 q^{10} +2.81361 q^{12} -0.500855 q^{13} +1.00000 q^{14} +2.81361 q^{15} +1.00000 q^{16} +2.39194 q^{17} -4.91638 q^{18} +3.43043 q^{19} +1.00000 q^{20} -2.81361 q^{21} +3.60363 q^{23} -2.81361 q^{24} +1.00000 q^{25} +0.500855 q^{26} +5.39194 q^{27} -1.00000 q^{28} -4.27139 q^{29} -2.81361 q^{30} +3.10278 q^{31} -1.00000 q^{32} -2.39194 q^{34} -1.00000 q^{35} +4.91638 q^{36} -2.27139 q^{37} -3.43043 q^{38} -1.40921 q^{39} -1.00000 q^{40} -0.0899376 q^{41} +2.81361 q^{42} +3.19923 q^{43} +4.91638 q^{45} -3.60363 q^{46} -1.29720 q^{47} +2.81361 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.72999 q^{51} -0.500855 q^{52} +8.60536 q^{53} -5.39194 q^{54} +1.00000 q^{56} +9.65187 q^{57} +4.27139 q^{58} +5.89109 q^{59} +2.81361 q^{60} -9.25687 q^{61} -3.10278 q^{62} -4.91638 q^{63} +1.00000 q^{64} -0.500855 q^{65} +3.90762 q^{67} +2.39194 q^{68} +10.1392 q^{69} +1.00000 q^{70} +15.5063 q^{71} -4.91638 q^{72} -7.66162 q^{73} +2.27139 q^{74} +2.81361 q^{75} +3.43043 q^{76} +1.40921 q^{78} +10.3606 q^{79} +1.00000 q^{80} +0.421663 q^{81} +0.0899376 q^{82} -4.19309 q^{83} -2.81361 q^{84} +2.39194 q^{85} -3.19923 q^{86} -12.0180 q^{87} -16.2856 q^{89} -4.91638 q^{90} +0.500855 q^{91} +3.60363 q^{92} +8.72999 q^{93} +1.29720 q^{94} +3.43043 q^{95} -2.81361 q^{96} +15.9907 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 4q^{3} + 6q^{4} + 6q^{5} - 4q^{6} - 6q^{7} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 4q^{3} + 6q^{4} + 6q^{5} - 4q^{6} - 6q^{7} - 6q^{8} + 2q^{9} - 6q^{10} + 4q^{12} + 6q^{14} + 4q^{15} + 6q^{16} - 2q^{17} - 2q^{18} + 6q^{20} - 4q^{21} + 4q^{23} - 4q^{24} + 6q^{25} + 16q^{27} - 6q^{28} - 8q^{29} - 4q^{30} + 4q^{31} - 6q^{32} + 2q^{34} - 6q^{35} + 2q^{36} + 4q^{37} - 6q^{40} - 12q^{41} + 4q^{42} + 6q^{43} + 2q^{45} - 4q^{46} + 16q^{47} + 4q^{48} + 6q^{49} - 6q^{50} + 12q^{53} - 16q^{54} + 6q^{56} - 8q^{57} + 8q^{58} + 22q^{59} + 4q^{60} + 4q^{61} - 4q^{62} - 2q^{63} + 6q^{64} + 20q^{67} - 2q^{68} + 12q^{69} + 6q^{70} + 14q^{71} - 2q^{72} - 18q^{73} - 4q^{74} + 4q^{75} - 32q^{79} + 6q^{80} + 6q^{81} + 12q^{82} + 16q^{83} - 4q^{84} - 2q^{85} - 6q^{86} + 4q^{87} + 4q^{89} - 2q^{90} + 4q^{92} + 12q^{93} - 16q^{94} - 4q^{96} + 4q^{97} - 6q^{98} + O(q^{100}) \)

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
\(3\) 2.81361 1.62444 0.812218 0.583354i \(-0.198260\pi\)
0.812218 + 0.583354i \(0.198260\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.81361 −1.14865
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 4.91638 1.63879
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.81361 0.812218
\(13\) −0.500855 −0.138912 −0.0694560 0.997585i \(-0.522126\pi\)
−0.0694560 + 0.997585i \(0.522126\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.81361 0.726470
\(16\) 1.00000 0.250000
\(17\) 2.39194 0.580132 0.290066 0.957007i \(-0.406323\pi\)
0.290066 + 0.957007i \(0.406323\pi\)
\(18\) −4.91638 −1.15880
\(19\) 3.43043 0.786994 0.393497 0.919326i \(-0.371265\pi\)
0.393497 + 0.919326i \(0.371265\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.81361 −0.613979
\(22\) 0 0
\(23\) 3.60363 0.751409 0.375704 0.926740i \(-0.377401\pi\)
0.375704 + 0.926740i \(0.377401\pi\)
\(24\) −2.81361 −0.574325
\(25\) 1.00000 0.200000
\(26\) 0.500855 0.0982257
\(27\) 5.39194 1.03768
\(28\) −1.00000 −0.188982
\(29\) −4.27139 −0.793177 −0.396589 0.917996i \(-0.629806\pi\)
−0.396589 + 0.917996i \(0.629806\pi\)
\(30\) −2.81361 −0.513692
\(31\) 3.10278 0.557275 0.278637 0.960396i \(-0.410117\pi\)
0.278637 + 0.960396i \(0.410117\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.39194 −0.410215
\(35\) −1.00000 −0.169031
\(36\) 4.91638 0.819397
\(37\) −2.27139 −0.373414 −0.186707 0.982416i \(-0.559782\pi\)
−0.186707 + 0.982416i \(0.559782\pi\)
\(38\) −3.43043 −0.556489
\(39\) −1.40921 −0.225654
\(40\) −1.00000 −0.158114
\(41\) −0.0899376 −0.0140459 −0.00702295 0.999975i \(-0.502235\pi\)
−0.00702295 + 0.999975i \(0.502235\pi\)
\(42\) 2.81361 0.434149
\(43\) 3.19923 0.487878 0.243939 0.969791i \(-0.421560\pi\)
0.243939 + 0.969791i \(0.421560\pi\)
\(44\) 0 0
\(45\) 4.91638 0.732891
\(46\) −3.60363 −0.531326
\(47\) −1.29720 −0.189216 −0.0946078 0.995515i \(-0.530160\pi\)
−0.0946078 + 0.995515i \(0.530160\pi\)
\(48\) 2.81361 0.406109
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 6.72999 0.942387
\(52\) −0.500855 −0.0694560
\(53\) 8.60536 1.18204 0.591019 0.806658i \(-0.298726\pi\)
0.591019 + 0.806658i \(0.298726\pi\)
\(54\) −5.39194 −0.733751
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 9.65187 1.27842
\(58\) 4.27139 0.560861
\(59\) 5.89109 0.766954 0.383477 0.923550i \(-0.374726\pi\)
0.383477 + 0.923550i \(0.374726\pi\)
\(60\) 2.81361 0.363235
\(61\) −9.25687 −1.18522 −0.592610 0.805489i \(-0.701903\pi\)
−0.592610 + 0.805489i \(0.701903\pi\)
\(62\) −3.10278 −0.394053
\(63\) −4.91638 −0.619406
\(64\) 1.00000 0.125000
\(65\) −0.500855 −0.0621234
\(66\) 0 0
\(67\) 3.90762 0.477391 0.238696 0.971094i \(-0.423280\pi\)
0.238696 + 0.971094i \(0.423280\pi\)
\(68\) 2.39194 0.290066
\(69\) 10.1392 1.22062
\(70\) 1.00000 0.119523
\(71\) 15.5063 1.84026 0.920131 0.391610i \(-0.128082\pi\)
0.920131 + 0.391610i \(0.128082\pi\)
\(72\) −4.91638 −0.579401
\(73\) −7.66162 −0.896725 −0.448363 0.893852i \(-0.647993\pi\)
−0.448363 + 0.893852i \(0.647993\pi\)
\(74\) 2.27139 0.264044
\(75\) 2.81361 0.324887
\(76\) 3.43043 0.393497
\(77\) 0 0
\(78\) 1.40921 0.159561
\(79\) 10.3606 1.16566 0.582832 0.812593i \(-0.301945\pi\)
0.582832 + 0.812593i \(0.301945\pi\)
\(80\) 1.00000 0.111803
\(81\) 0.421663 0.0468514
\(82\) 0.0899376 0.00993194
\(83\) −4.19309 −0.460252 −0.230126 0.973161i \(-0.573914\pi\)
−0.230126 + 0.973161i \(0.573914\pi\)
\(84\) −2.81361 −0.306990
\(85\) 2.39194 0.259443
\(86\) −3.19923 −0.344982
\(87\) −12.0180 −1.28847
\(88\) 0 0
\(89\) −16.2856 −1.72627 −0.863135 0.504973i \(-0.831503\pi\)
−0.863135 + 0.504973i \(0.831503\pi\)
\(90\) −4.91638 −0.518232
\(91\) 0.500855 0.0525038
\(92\) 3.60363 0.375704
\(93\) 8.72999 0.905258
\(94\) 1.29720 0.133796
\(95\) 3.43043 0.351954
\(96\) −2.81361 −0.287163
\(97\) 15.9907 1.62360 0.811802 0.583932i \(-0.198487\pi\)
0.811802 + 0.583932i \(0.198487\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.2349 1.01841 0.509207 0.860644i \(-0.329939\pi\)
0.509207 + 0.860644i \(0.329939\pi\)
\(102\) −6.72999 −0.666368
\(103\) 2.15822 0.212656 0.106328 0.994331i \(-0.466091\pi\)
0.106328 + 0.994331i \(0.466091\pi\)
\(104\) 0.500855 0.0491128
\(105\) −2.81361 −0.274580
\(106\) −8.60536 −0.835826
\(107\) 4.14362 0.400579 0.200290 0.979737i \(-0.435812\pi\)
0.200290 + 0.979737i \(0.435812\pi\)
\(108\) 5.39194 0.518840
\(109\) 8.55011 0.818952 0.409476 0.912321i \(-0.365711\pi\)
0.409476 + 0.912321i \(0.365711\pi\)
\(110\) 0 0
\(111\) −6.39079 −0.606587
\(112\) −1.00000 −0.0944911
\(113\) −17.6102 −1.65663 −0.828313 0.560266i \(-0.810699\pi\)
−0.828313 + 0.560266i \(0.810699\pi\)
\(114\) −9.65187 −0.903981
\(115\) 3.60363 0.336040
\(116\) −4.27139 −0.396589
\(117\) −2.46239 −0.227648
\(118\) −5.89109 −0.542319
\(119\) −2.39194 −0.219269
\(120\) −2.81361 −0.256846
\(121\) 0 0
\(122\) 9.25687 0.838078
\(123\) −0.253049 −0.0228167
\(124\) 3.10278 0.278637
\(125\) 1.00000 0.0894427
\(126\) 4.91638 0.437986
\(127\) 12.6352 1.12120 0.560598 0.828088i \(-0.310571\pi\)
0.560598 + 0.828088i \(0.310571\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 9.00138 0.792527
\(130\) 0.500855 0.0439279
\(131\) 6.16025 0.538224 0.269112 0.963109i \(-0.413270\pi\)
0.269112 + 0.963109i \(0.413270\pi\)
\(132\) 0 0
\(133\) −3.43043 −0.297456
\(134\) −3.90762 −0.337567
\(135\) 5.39194 0.464065
\(136\) −2.39194 −0.205107
\(137\) 4.37378 0.373677 0.186839 0.982391i \(-0.440176\pi\)
0.186839 + 0.982391i \(0.440176\pi\)
\(138\) −10.1392 −0.863106
\(139\) −12.2300 −1.03734 −0.518670 0.854975i \(-0.673572\pi\)
−0.518670 + 0.854975i \(0.673572\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −3.64980 −0.307369
\(142\) −15.5063 −1.30126
\(143\) 0 0
\(144\) 4.91638 0.409698
\(145\) −4.27139 −0.354720
\(146\) 7.66162 0.634080
\(147\) 2.81361 0.232062
\(148\) −2.27139 −0.186707
\(149\) −7.45847 −0.611021 −0.305511 0.952189i \(-0.598827\pi\)
−0.305511 + 0.952189i \(0.598827\pi\)
\(150\) −2.81361 −0.229730
\(151\) −8.99272 −0.731817 −0.365908 0.930651i \(-0.619242\pi\)
−0.365908 + 0.930651i \(0.619242\pi\)
\(152\) −3.43043 −0.278244
\(153\) 11.7597 0.950716
\(154\) 0 0
\(155\) 3.10278 0.249221
\(156\) −1.40921 −0.112827
\(157\) −1.14513 −0.0913915 −0.0456958 0.998955i \(-0.514550\pi\)
−0.0456958 + 0.998955i \(0.514550\pi\)
\(158\) −10.3606 −0.824248
\(159\) 24.2121 1.92014
\(160\) −1.00000 −0.0790569
\(161\) −3.60363 −0.284006
\(162\) −0.421663 −0.0331290
\(163\) −5.36940 −0.420564 −0.210282 0.977641i \(-0.567438\pi\)
−0.210282 + 0.977641i \(0.567438\pi\)
\(164\) −0.0899376 −0.00702295
\(165\) 0 0
\(166\) 4.19309 0.325447
\(167\) 23.4652 1.81579 0.907897 0.419194i \(-0.137687\pi\)
0.907897 + 0.419194i \(0.137687\pi\)
\(168\) 2.81361 0.217074
\(169\) −12.7491 −0.980703
\(170\) −2.39194 −0.183454
\(171\) 16.8653 1.28972
\(172\) 3.19923 0.243939
\(173\) −14.0465 −1.06793 −0.533966 0.845506i \(-0.679299\pi\)
−0.533966 + 0.845506i \(0.679299\pi\)
\(174\) 12.0180 0.911083
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 16.5752 1.24587
\(178\) 16.2856 1.22066
\(179\) −6.46206 −0.482997 −0.241499 0.970401i \(-0.577639\pi\)
−0.241499 + 0.970401i \(0.577639\pi\)
\(180\) 4.91638 0.366445
\(181\) 13.1930 0.980629 0.490314 0.871546i \(-0.336882\pi\)
0.490314 + 0.871546i \(0.336882\pi\)
\(182\) −0.500855 −0.0371258
\(183\) −26.0452 −1.92532
\(184\) −3.60363 −0.265663
\(185\) −2.27139 −0.166996
\(186\) −8.72999 −0.640114
\(187\) 0 0
\(188\) −1.29720 −0.0946078
\(189\) −5.39194 −0.392206
\(190\) −3.43043 −0.248869
\(191\) −12.3673 −0.894867 −0.447434 0.894317i \(-0.647662\pi\)
−0.447434 + 0.894317i \(0.647662\pi\)
\(192\) 2.81361 0.203055
\(193\) 12.2658 0.882909 0.441455 0.897284i \(-0.354463\pi\)
0.441455 + 0.897284i \(0.354463\pi\)
\(194\) −15.9907 −1.14806
\(195\) −1.40921 −0.100915
\(196\) 1.00000 0.0714286
\(197\) 8.65314 0.616511 0.308255 0.951304i \(-0.400255\pi\)
0.308255 + 0.951304i \(0.400255\pi\)
\(198\) 0 0
\(199\) 12.4838 0.884954 0.442477 0.896780i \(-0.354100\pi\)
0.442477 + 0.896780i \(0.354100\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 10.9945 0.775492
\(202\) −10.2349 −0.720128
\(203\) 4.27139 0.299793
\(204\) 6.72999 0.471193
\(205\) −0.0899376 −0.00628151
\(206\) −2.15822 −0.150370
\(207\) 17.7168 1.23140
\(208\) −0.500855 −0.0347280
\(209\) 0 0
\(210\) 2.81361 0.194157
\(211\) −6.76072 −0.465427 −0.232714 0.972545i \(-0.574761\pi\)
−0.232714 + 0.972545i \(0.574761\pi\)
\(212\) 8.60536 0.591019
\(213\) 43.6287 2.98939
\(214\) −4.14362 −0.283252
\(215\) 3.19923 0.218186
\(216\) −5.39194 −0.366875
\(217\) −3.10278 −0.210630
\(218\) −8.55011 −0.579087
\(219\) −21.5568 −1.45667
\(220\) 0 0
\(221\) −1.19802 −0.0805873
\(222\) 6.39079 0.428922
\(223\) −6.14794 −0.411696 −0.205848 0.978584i \(-0.565995\pi\)
−0.205848 + 0.978584i \(0.565995\pi\)
\(224\) 1.00000 0.0668153
\(225\) 4.91638 0.327759
\(226\) 17.6102 1.17141
\(227\) 11.2377 0.745870 0.372935 0.927858i \(-0.378351\pi\)
0.372935 + 0.927858i \(0.378351\pi\)
\(228\) 9.65187 0.639211
\(229\) −17.2115 −1.13737 −0.568685 0.822555i \(-0.692548\pi\)
−0.568685 + 0.822555i \(0.692548\pi\)
\(230\) −3.60363 −0.237616
\(231\) 0 0
\(232\) 4.27139 0.280430
\(233\) 7.59338 0.497459 0.248729 0.968573i \(-0.419987\pi\)
0.248729 + 0.968573i \(0.419987\pi\)
\(234\) 2.46239 0.160972
\(235\) −1.29720 −0.0846198
\(236\) 5.89109 0.383477
\(237\) 29.1508 1.89355
\(238\) 2.39194 0.155047
\(239\) 28.8381 1.86538 0.932691 0.360676i \(-0.117454\pi\)
0.932691 + 0.360676i \(0.117454\pi\)
\(240\) 2.81361 0.181618
\(241\) 15.3781 0.990589 0.495295 0.868725i \(-0.335060\pi\)
0.495295 + 0.868725i \(0.335060\pi\)
\(242\) 0 0
\(243\) −14.9894 −0.961573
\(244\) −9.25687 −0.592610
\(245\) 1.00000 0.0638877
\(246\) 0.253049 0.0161338
\(247\) −1.71815 −0.109323
\(248\) −3.10278 −0.197026
\(249\) −11.7977 −0.747650
\(250\) −1.00000 −0.0632456
\(251\) 9.22254 0.582122 0.291061 0.956705i \(-0.405992\pi\)
0.291061 + 0.956705i \(0.405992\pi\)
\(252\) −4.91638 −0.309703
\(253\) 0 0
\(254\) −12.6352 −0.792806
\(255\) 6.72999 0.421448
\(256\) 1.00000 0.0625000
\(257\) 0.000868886 0 5.41996e−5 0 2.70998e−5 1.00000i \(-0.499991\pi\)
2.70998e−5 1.00000i \(0.499991\pi\)
\(258\) −9.00138 −0.560401
\(259\) 2.27139 0.141137
\(260\) −0.500855 −0.0310617
\(261\) −20.9998 −1.29985
\(262\) −6.16025 −0.380582
\(263\) 5.23650 0.322896 0.161448 0.986881i \(-0.448384\pi\)
0.161448 + 0.986881i \(0.448384\pi\)
\(264\) 0 0
\(265\) 8.60536 0.528623
\(266\) 3.43043 0.210333
\(267\) −45.8213 −2.80422
\(268\) 3.90762 0.238696
\(269\) −27.8809 −1.69993 −0.849963 0.526842i \(-0.823376\pi\)
−0.849963 + 0.526842i \(0.823376\pi\)
\(270\) −5.39194 −0.328143
\(271\) −7.08659 −0.430479 −0.215240 0.976561i \(-0.569053\pi\)
−0.215240 + 0.976561i \(0.569053\pi\)
\(272\) 2.39194 0.145033
\(273\) 1.40921 0.0852891
\(274\) −4.37378 −0.264230
\(275\) 0 0
\(276\) 10.1392 0.610308
\(277\) 21.2777 1.27846 0.639228 0.769018i \(-0.279254\pi\)
0.639228 + 0.769018i \(0.279254\pi\)
\(278\) 12.2300 0.733509
\(279\) 15.2544 0.913259
\(280\) 1.00000 0.0597614
\(281\) 30.4417 1.81600 0.908001 0.418969i \(-0.137608\pi\)
0.908001 + 0.418969i \(0.137608\pi\)
\(282\) 3.64980 0.217343
\(283\) 14.8159 0.880711 0.440356 0.897823i \(-0.354852\pi\)
0.440356 + 0.897823i \(0.354852\pi\)
\(284\) 15.5063 0.920131
\(285\) 9.65187 0.571728
\(286\) 0 0
\(287\) 0.0899376 0.00530885
\(288\) −4.91638 −0.289701
\(289\) −11.2786 −0.663447
\(290\) 4.27139 0.250825
\(291\) 44.9914 2.63744
\(292\) −7.66162 −0.448363
\(293\) −31.4790 −1.83902 −0.919511 0.393065i \(-0.871415\pi\)
−0.919511 + 0.393065i \(0.871415\pi\)
\(294\) −2.81361 −0.164093
\(295\) 5.89109 0.342992
\(296\) 2.27139 0.132022
\(297\) 0 0
\(298\) 7.45847 0.432057
\(299\) −1.80489 −0.104380
\(300\) 2.81361 0.162444
\(301\) −3.19923 −0.184401
\(302\) 8.99272 0.517473
\(303\) 28.7971 1.65435
\(304\) 3.43043 0.196748
\(305\) −9.25687 −0.530047
\(306\) −11.7597 −0.672258
\(307\) −10.6499 −0.607823 −0.303911 0.952700i \(-0.598293\pi\)
−0.303911 + 0.952700i \(0.598293\pi\)
\(308\) 0 0
\(309\) 6.07239 0.345446
\(310\) −3.10278 −0.176226
\(311\) 0.450689 0.0255562 0.0127781 0.999918i \(-0.495932\pi\)
0.0127781 + 0.999918i \(0.495932\pi\)
\(312\) 1.40921 0.0797807
\(313\) −26.5206 −1.49903 −0.749517 0.661985i \(-0.769714\pi\)
−0.749517 + 0.661985i \(0.769714\pi\)
\(314\) 1.14513 0.0646236
\(315\) −4.91638 −0.277007
\(316\) 10.3606 0.582832
\(317\) 29.2492 1.64280 0.821399 0.570354i \(-0.193194\pi\)
0.821399 + 0.570354i \(0.193194\pi\)
\(318\) −24.2121 −1.35775
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 11.6585 0.650715
\(322\) 3.60363 0.200822
\(323\) 8.20539 0.456560
\(324\) 0.421663 0.0234257
\(325\) −0.500855 −0.0277824
\(326\) 5.36940 0.297383
\(327\) 24.0567 1.33034
\(328\) 0.0899376 0.00496597
\(329\) 1.29720 0.0715168
\(330\) 0 0
\(331\) −16.5049 −0.907191 −0.453596 0.891208i \(-0.649859\pi\)
−0.453596 + 0.891208i \(0.649859\pi\)
\(332\) −4.19309 −0.230126
\(333\) −11.1670 −0.611949
\(334\) −23.4652 −1.28396
\(335\) 3.90762 0.213496
\(336\) −2.81361 −0.153495
\(337\) −26.9655 −1.46890 −0.734452 0.678660i \(-0.762561\pi\)
−0.734452 + 0.678660i \(0.762561\pi\)
\(338\) 12.7491 0.693462
\(339\) −49.5481 −2.69108
\(340\) 2.39194 0.129721
\(341\) 0 0
\(342\) −16.8653 −0.911970
\(343\) −1.00000 −0.0539949
\(344\) −3.19923 −0.172491
\(345\) 10.1392 0.545876
\(346\) 14.0465 0.755142
\(347\) 11.4888 0.616751 0.308375 0.951265i \(-0.400215\pi\)
0.308375 + 0.951265i \(0.400215\pi\)
\(348\) −12.0180 −0.644233
\(349\) 12.0767 0.646452 0.323226 0.946322i \(-0.395233\pi\)
0.323226 + 0.946322i \(0.395233\pi\)
\(350\) 1.00000 0.0534522
\(351\) −2.70058 −0.144146
\(352\) 0 0
\(353\) −1.27099 −0.0676477 −0.0338239 0.999428i \(-0.510769\pi\)
−0.0338239 + 0.999428i \(0.510769\pi\)
\(354\) −16.5752 −0.880962
\(355\) 15.5063 0.822990
\(356\) −16.2856 −0.863135
\(357\) −6.72999 −0.356189
\(358\) 6.46206 0.341530
\(359\) 0.866556 0.0457351 0.0228675 0.999739i \(-0.492720\pi\)
0.0228675 + 0.999739i \(0.492720\pi\)
\(360\) −4.91638 −0.259116
\(361\) −7.23217 −0.380641
\(362\) −13.1930 −0.693409
\(363\) 0 0
\(364\) 0.500855 0.0262519
\(365\) −7.66162 −0.401028
\(366\) 26.0452 1.36140
\(367\) 18.3256 0.956588 0.478294 0.878200i \(-0.341255\pi\)
0.478294 + 0.878200i \(0.341255\pi\)
\(368\) 3.60363 0.187852
\(369\) −0.442167 −0.0230183
\(370\) 2.27139 0.118084
\(371\) −8.60536 −0.446768
\(372\) 8.72999 0.452629
\(373\) 19.1453 0.991308 0.495654 0.868520i \(-0.334928\pi\)
0.495654 + 0.868520i \(0.334928\pi\)
\(374\) 0 0
\(375\) 2.81361 0.145294
\(376\) 1.29720 0.0668978
\(377\) 2.13934 0.110182
\(378\) 5.39194 0.277332
\(379\) −6.30548 −0.323891 −0.161945 0.986800i \(-0.551777\pi\)
−0.161945 + 0.986800i \(0.551777\pi\)
\(380\) 3.43043 0.175977
\(381\) 35.5506 1.82131
\(382\) 12.3673 0.632767
\(383\) −8.05894 −0.411792 −0.205896 0.978574i \(-0.566011\pi\)
−0.205896 + 0.978574i \(0.566011\pi\)
\(384\) −2.81361 −0.143581
\(385\) 0 0
\(386\) −12.2658 −0.624311
\(387\) 15.7286 0.799532
\(388\) 15.9907 0.811802
\(389\) 33.7643 1.71192 0.855958 0.517046i \(-0.172968\pi\)
0.855958 + 0.517046i \(0.172968\pi\)
\(390\) 1.40921 0.0713580
\(391\) 8.61968 0.435916
\(392\) −1.00000 −0.0505076
\(393\) 17.3325 0.874310
\(394\) −8.65314 −0.435939
\(395\) 10.3606 0.521300
\(396\) 0 0
\(397\) 4.64243 0.232997 0.116498 0.993191i \(-0.462833\pi\)
0.116498 + 0.993191i \(0.462833\pi\)
\(398\) −12.4838 −0.625757
\(399\) −9.65187 −0.483198
\(400\) 1.00000 0.0500000
\(401\) 18.2561 0.911668 0.455834 0.890065i \(-0.349341\pi\)
0.455834 + 0.890065i \(0.349341\pi\)
\(402\) −10.9945 −0.548356
\(403\) −1.55404 −0.0774122
\(404\) 10.2349 0.509207
\(405\) 0.421663 0.0209526
\(406\) −4.27139 −0.211985
\(407\) 0 0
\(408\) −6.72999 −0.333184
\(409\) 7.94680 0.392944 0.196472 0.980509i \(-0.437052\pi\)
0.196472 + 0.980509i \(0.437052\pi\)
\(410\) 0.0899376 0.00444170
\(411\) 12.3061 0.607015
\(412\) 2.15822 0.106328
\(413\) −5.89109 −0.289882
\(414\) −17.7168 −0.870734
\(415\) −4.19309 −0.205831
\(416\) 0.500855 0.0245564
\(417\) −34.4105 −1.68509
\(418\) 0 0
\(419\) −21.4201 −1.04644 −0.523221 0.852197i \(-0.675270\pi\)
−0.523221 + 0.852197i \(0.675270\pi\)
\(420\) −2.81361 −0.137290
\(421\) 0.894942 0.0436168 0.0218084 0.999762i \(-0.493058\pi\)
0.0218084 + 0.999762i \(0.493058\pi\)
\(422\) 6.76072 0.329107
\(423\) −6.37751 −0.310085
\(424\) −8.60536 −0.417913
\(425\) 2.39194 0.116026
\(426\) −43.6287 −2.11382
\(427\) 9.25687 0.447971
\(428\) 4.14362 0.200290
\(429\) 0 0
\(430\) −3.19923 −0.154281
\(431\) −37.2082 −1.79226 −0.896128 0.443796i \(-0.853631\pi\)
−0.896128 + 0.443796i \(0.853631\pi\)
\(432\) 5.39194 0.259420
\(433\) 37.0912 1.78249 0.891245 0.453522i \(-0.149833\pi\)
0.891245 + 0.453522i \(0.149833\pi\)
\(434\) 3.10278 0.148938
\(435\) −12.0180 −0.576219
\(436\) 8.55011 0.409476
\(437\) 12.3620 0.591354
\(438\) 21.5568 1.03002
\(439\) −16.6322 −0.793813 −0.396907 0.917859i \(-0.629916\pi\)
−0.396907 + 0.917859i \(0.629916\pi\)
\(440\) 0 0
\(441\) 4.91638 0.234113
\(442\) 1.19802 0.0569838
\(443\) 18.9936 0.902411 0.451206 0.892420i \(-0.350994\pi\)
0.451206 + 0.892420i \(0.350994\pi\)
\(444\) −6.39079 −0.303294
\(445\) −16.2856 −0.772012
\(446\) 6.14794 0.291113
\(447\) −20.9852 −0.992565
\(448\) −1.00000 −0.0472456
\(449\) −15.0632 −0.710877 −0.355439 0.934700i \(-0.615669\pi\)
−0.355439 + 0.934700i \(0.615669\pi\)
\(450\) −4.91638 −0.231760
\(451\) 0 0
\(452\) −17.6102 −0.828313
\(453\) −25.3020 −1.18879
\(454\) −11.2377 −0.527410
\(455\) 0.500855 0.0234804
\(456\) −9.65187 −0.451990
\(457\) −13.0743 −0.611590 −0.305795 0.952097i \(-0.598922\pi\)
−0.305795 + 0.952097i \(0.598922\pi\)
\(458\) 17.2115 0.804242
\(459\) 12.8972 0.601991
\(460\) 3.60363 0.168020
\(461\) 40.0901 1.86718 0.933591 0.358342i \(-0.116658\pi\)
0.933591 + 0.358342i \(0.116658\pi\)
\(462\) 0 0
\(463\) 4.38625 0.203846 0.101923 0.994792i \(-0.467500\pi\)
0.101923 + 0.994792i \(0.467500\pi\)
\(464\) −4.27139 −0.198294
\(465\) 8.72999 0.404844
\(466\) −7.59338 −0.351756
\(467\) −2.76704 −0.128043 −0.0640217 0.997949i \(-0.520393\pi\)
−0.0640217 + 0.997949i \(0.520393\pi\)
\(468\) −2.46239 −0.113824
\(469\) −3.90762 −0.180437
\(470\) 1.29720 0.0598352
\(471\) −3.22195 −0.148460
\(472\) −5.89109 −0.271159
\(473\) 0 0
\(474\) −29.1508 −1.33894
\(475\) 3.43043 0.157399
\(476\) −2.39194 −0.109635
\(477\) 42.3072 1.93712
\(478\) −28.8381 −1.31902
\(479\) 2.13584 0.0975889 0.0487945 0.998809i \(-0.484462\pi\)
0.0487945 + 0.998809i \(0.484462\pi\)
\(480\) −2.81361 −0.128423
\(481\) 1.13764 0.0518717
\(482\) −15.3781 −0.700452
\(483\) −10.1392 −0.461349
\(484\) 0 0
\(485\) 15.9907 0.726098
\(486\) 14.9894 0.679935
\(487\) −30.0879 −1.36341 −0.681707 0.731625i \(-0.738762\pi\)
−0.681707 + 0.731625i \(0.738762\pi\)
\(488\) 9.25687 0.419039
\(489\) −15.1074 −0.683179
\(490\) −1.00000 −0.0451754
\(491\) −14.1922 −0.640484 −0.320242 0.947336i \(-0.603764\pi\)
−0.320242 + 0.947336i \(0.603764\pi\)
\(492\) −0.253049 −0.0114083
\(493\) −10.2169 −0.460147
\(494\) 1.71815 0.0773030
\(495\) 0 0
\(496\) 3.10278 0.139319
\(497\) −15.5063 −0.695554
\(498\) 11.7977 0.528668
\(499\) 17.9339 0.802831 0.401416 0.915896i \(-0.368518\pi\)
0.401416 + 0.915896i \(0.368518\pi\)
\(500\) 1.00000 0.0447214
\(501\) 66.0219 2.94964
\(502\) −9.22254 −0.411622
\(503\) −4.82298 −0.215046 −0.107523 0.994203i \(-0.534292\pi\)
−0.107523 + 0.994203i \(0.534292\pi\)
\(504\) 4.91638 0.218993
\(505\) 10.2349 0.455449
\(506\) 0 0
\(507\) −35.8711 −1.59309
\(508\) 12.6352 0.560598
\(509\) 35.9415 1.59308 0.796540 0.604586i \(-0.206662\pi\)
0.796540 + 0.604586i \(0.206662\pi\)
\(510\) −6.72999 −0.298009
\(511\) 7.66162 0.338930
\(512\) −1.00000 −0.0441942
\(513\) 18.4967 0.816648
\(514\) −0.000868886 0 −3.83249e−5 0
\(515\) 2.15822 0.0951026
\(516\) 9.00138 0.396264
\(517\) 0 0
\(518\) −2.27139 −0.0997991
\(519\) −39.5212 −1.73479
\(520\) 0.500855 0.0219639
\(521\) 14.6257 0.640766 0.320383 0.947288i \(-0.396188\pi\)
0.320383 + 0.947288i \(0.396188\pi\)
\(522\) 20.9998 0.919135
\(523\) −22.6277 −0.989439 −0.494720 0.869053i \(-0.664729\pi\)
−0.494720 + 0.869053i \(0.664729\pi\)
\(524\) 6.16025 0.269112
\(525\) −2.81361 −0.122796
\(526\) −5.23650 −0.228322
\(527\) 7.42166 0.323293
\(528\) 0 0
\(529\) −10.0139 −0.435385
\(530\) −8.60536 −0.373793
\(531\) 28.9628 1.25688
\(532\) −3.43043 −0.148728
\(533\) 0.0450457 0.00195114
\(534\) 45.8213 1.98288
\(535\) 4.14362 0.179144
\(536\) −3.90762 −0.168783
\(537\) −18.1817 −0.784598
\(538\) 27.8809 1.20203
\(539\) 0 0
\(540\) 5.39194 0.232032
\(541\) 25.8522 1.11147 0.555737 0.831358i \(-0.312436\pi\)
0.555737 + 0.831358i \(0.312436\pi\)
\(542\) 7.08659 0.304395
\(543\) 37.1199 1.59297
\(544\) −2.39194 −0.102554
\(545\) 8.55011 0.366247
\(546\) −1.40921 −0.0603085
\(547\) 32.9722 1.40979 0.704896 0.709311i \(-0.250994\pi\)
0.704896 + 0.709311i \(0.250994\pi\)
\(548\) 4.37378 0.186839
\(549\) −45.5103 −1.94233
\(550\) 0 0
\(551\) −14.6527 −0.624226
\(552\) −10.1392 −0.431553
\(553\) −10.3606 −0.440579
\(554\) −21.2777 −0.904004
\(555\) −6.39079 −0.271274
\(556\) −12.2300 −0.518670
\(557\) −10.6402 −0.450840 −0.225420 0.974262i \(-0.572375\pi\)
−0.225420 + 0.974262i \(0.572375\pi\)
\(558\) −15.2544 −0.645771
\(559\) −1.60235 −0.0677722
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −30.4417 −1.28411
\(563\) −27.8178 −1.17238 −0.586190 0.810174i \(-0.699373\pi\)
−0.586190 + 0.810174i \(0.699373\pi\)
\(564\) −3.64980 −0.153684
\(565\) −17.6102 −0.740866
\(566\) −14.8159 −0.622757
\(567\) −0.421663 −0.0177082
\(568\) −15.5063 −0.650631
\(569\) −23.2943 −0.976547 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(570\) −9.65187 −0.404272
\(571\) −22.9549 −0.960633 −0.480317 0.877095i \(-0.659478\pi\)
−0.480317 + 0.877095i \(0.659478\pi\)
\(572\) 0 0
\(573\) −34.7968 −1.45366
\(574\) −0.0899376 −0.00375392
\(575\) 3.60363 0.150282
\(576\) 4.91638 0.204849
\(577\) −45.7343 −1.90394 −0.951972 0.306184i \(-0.900948\pi\)
−0.951972 + 0.306184i \(0.900948\pi\)
\(578\) 11.2786 0.469128
\(579\) 34.5110 1.43423
\(580\) −4.27139 −0.177360
\(581\) 4.19309 0.173959
\(582\) −44.9914 −1.86495
\(583\) 0 0
\(584\) 7.66162 0.317040
\(585\) −2.46239 −0.101807
\(586\) 31.4790 1.30038
\(587\) −29.5719 −1.22056 −0.610282 0.792184i \(-0.708944\pi\)
−0.610282 + 0.792184i \(0.708944\pi\)
\(588\) 2.81361 0.116031
\(589\) 10.6438 0.438572
\(590\) −5.89109 −0.242532
\(591\) 24.3465 1.00148
\(592\) −2.27139 −0.0933535
\(593\) 30.8187 1.26557 0.632786 0.774327i \(-0.281911\pi\)
0.632786 + 0.774327i \(0.281911\pi\)
\(594\) 0 0
\(595\) −2.39194 −0.0980601
\(596\) −7.45847 −0.305511
\(597\) 35.1246 1.43755
\(598\) 1.80489 0.0738076
\(599\) 0.426966 0.0174454 0.00872268 0.999962i \(-0.497223\pi\)
0.00872268 + 0.999962i \(0.497223\pi\)
\(600\) −2.81361 −0.114865
\(601\) −34.6294 −1.41256 −0.706282 0.707931i \(-0.749629\pi\)
−0.706282 + 0.707931i \(0.749629\pi\)
\(602\) 3.19923 0.130391
\(603\) 19.2113 0.782346
\(604\) −8.99272 −0.365908
\(605\) 0 0
\(606\) −28.7971 −1.16980
\(607\) −8.77207 −0.356047 −0.178024 0.984026i \(-0.556970\pi\)
−0.178024 + 0.984026i \(0.556970\pi\)
\(608\) −3.43043 −0.139122
\(609\) 12.0180 0.486994
\(610\) 9.25687 0.374800
\(611\) 0.649707 0.0262843
\(612\) 11.7597 0.475358
\(613\) 14.3859 0.581043 0.290521 0.956868i \(-0.406171\pi\)
0.290521 + 0.956868i \(0.406171\pi\)
\(614\) 10.6499 0.429795
\(615\) −0.253049 −0.0102039
\(616\) 0 0
\(617\) −28.5656 −1.15001 −0.575005 0.818150i \(-0.695000\pi\)
−0.575005 + 0.818150i \(0.695000\pi\)
\(618\) −6.07239 −0.244267
\(619\) 14.5070 0.583084 0.291542 0.956558i \(-0.405832\pi\)
0.291542 + 0.956558i \(0.405832\pi\)
\(620\) 3.10278 0.124610
\(621\) 19.4306 0.779722
\(622\) −0.450689 −0.0180710
\(623\) 16.2856 0.652469
\(624\) −1.40921 −0.0564135
\(625\) 1.00000 0.0400000
\(626\) 26.5206 1.05998
\(627\) 0 0
\(628\) −1.14513 −0.0456958
\(629\) −5.43303 −0.216629
\(630\) 4.91638 0.195873
\(631\) −29.3973 −1.17029 −0.585143 0.810930i \(-0.698962\pi\)
−0.585143 + 0.810930i \(0.698962\pi\)
\(632\) −10.3606 −0.412124
\(633\) −19.0220 −0.756057
\(634\) −29.2492 −1.16163
\(635\) 12.6352 0.501414
\(636\) 24.2121 0.960072
\(637\) −0.500855 −0.0198446
\(638\) 0 0
\(639\) 76.2350 3.01581
\(640\) −1.00000 −0.0395285
\(641\) −21.4821 −0.848491 −0.424245 0.905547i \(-0.639461\pi\)
−0.424245 + 0.905547i \(0.639461\pi\)
\(642\) −11.6585 −0.460125
\(643\) 6.62789 0.261379 0.130689 0.991423i \(-0.458281\pi\)
0.130689 + 0.991423i \(0.458281\pi\)
\(644\) −3.60363 −0.142003
\(645\) 9.00138 0.354429
\(646\) −8.20539 −0.322837
\(647\) −40.0796 −1.57569 −0.787847 0.615871i \(-0.788804\pi\)
−0.787847 + 0.615871i \(0.788804\pi\)
\(648\) −0.421663 −0.0165645
\(649\) 0 0
\(650\) 0.500855 0.0196451
\(651\) −8.72999 −0.342155
\(652\) −5.36940 −0.210282
\(653\) 1.08840 0.0425923 0.0212962 0.999773i \(-0.493221\pi\)
0.0212962 + 0.999773i \(0.493221\pi\)
\(654\) −24.0567 −0.940690
\(655\) 6.16025 0.240701
\(656\) −0.0899376 −0.00351147
\(657\) −37.6675 −1.46955
\(658\) −1.29720 −0.0505700
\(659\) −38.5368 −1.50118 −0.750591 0.660767i \(-0.770231\pi\)
−0.750591 + 0.660767i \(0.770231\pi\)
\(660\) 0 0
\(661\) 23.8017 0.925777 0.462889 0.886416i \(-0.346813\pi\)
0.462889 + 0.886416i \(0.346813\pi\)
\(662\) 16.5049 0.641481
\(663\) −3.37075 −0.130909
\(664\) 4.19309 0.162724
\(665\) −3.43043 −0.133026
\(666\) 11.1670 0.432713
\(667\) −15.3925 −0.596000
\(668\) 23.4652 0.907897
\(669\) −17.2979 −0.668775
\(670\) −3.90762 −0.150964
\(671\) 0 0
\(672\) 2.81361 0.108537
\(673\) −9.60964 −0.370424 −0.185212 0.982699i \(-0.559297\pi\)
−0.185212 + 0.982699i \(0.559297\pi\)
\(674\) 26.9655 1.03867
\(675\) 5.39194 0.207536
\(676\) −12.7491 −0.490352
\(677\) −4.45989 −0.171407 −0.0857037 0.996321i \(-0.527314\pi\)
−0.0857037 + 0.996321i \(0.527314\pi\)
\(678\) 49.5481 1.90288
\(679\) −15.9907 −0.613665
\(680\) −2.39194 −0.0917268
\(681\) 31.6184 1.21162
\(682\) 0 0
\(683\) −21.0977 −0.807281 −0.403641 0.914918i \(-0.632255\pi\)
−0.403641 + 0.914918i \(0.632255\pi\)
\(684\) 16.8653 0.644860
\(685\) 4.37378 0.167114
\(686\) 1.00000 0.0381802
\(687\) −48.4265 −1.84759
\(688\) 3.19923 0.121970
\(689\) −4.31003 −0.164199
\(690\) −10.1392 −0.385993
\(691\) −36.7936 −1.39969 −0.699847 0.714293i \(-0.746748\pi\)
−0.699847 + 0.714293i \(0.746748\pi\)
\(692\) −14.0465 −0.533966
\(693\) 0 0
\(694\) −11.4888 −0.436109
\(695\) −12.2300 −0.463912
\(696\) 12.0180 0.455541
\(697\) −0.215126 −0.00814846
\(698\) −12.0767 −0.457110
\(699\) 21.3648 0.808090
\(700\) −1.00000 −0.0377964
\(701\) −38.3577 −1.44875 −0.724374 0.689407i \(-0.757871\pi\)
−0.724374 + 0.689407i \(0.757871\pi\)
\(702\) 2.70058 0.101927
\(703\) −7.79183 −0.293875
\(704\) 0 0
\(705\) −3.64980 −0.137459
\(706\) 1.27099 0.0478342
\(707\) −10.2349 −0.384924
\(708\) 16.5752 0.622934
\(709\) −37.3501 −1.40271 −0.701357 0.712811i \(-0.747422\pi\)
−0.701357 + 0.712811i \(0.747422\pi\)
\(710\) −15.5063 −0.581942
\(711\) 50.9369 1.91028
\(712\) 16.2856 0.610329
\(713\) 11.1813 0.418741
\(714\) 6.72999 0.251863
\(715\) 0 0
\(716\) −6.46206 −0.241499
\(717\) 81.1391 3.03020
\(718\) −0.866556 −0.0323396
\(719\) −33.1232 −1.23529 −0.617644 0.786458i \(-0.711913\pi\)
−0.617644 + 0.786458i \(0.711913\pi\)
\(720\) 4.91638 0.183223
\(721\) −2.15822 −0.0803764
\(722\) 7.23217 0.269153
\(723\) 43.2679 1.60915
\(724\) 13.1930 0.490314
\(725\) −4.27139 −0.158635
\(726\) 0 0
\(727\) 5.85088 0.216997 0.108498 0.994097i \(-0.465396\pi\)
0.108498 + 0.994097i \(0.465396\pi\)
\(728\) −0.500855 −0.0185629
\(729\) −43.4394 −1.60887
\(730\) 7.66162 0.283569
\(731\) 7.65238 0.283033
\(732\) −26.0452 −0.962658
\(733\) 5.37537 0.198544 0.0992719 0.995060i \(-0.468349\pi\)
0.0992719 + 0.995060i \(0.468349\pi\)
\(734\) −18.3256 −0.676410
\(735\) 2.81361 0.103781
\(736\) −3.60363 −0.132832
\(737\) 0 0
\(738\) 0.442167 0.0162764
\(739\) −25.0957 −0.923159 −0.461580 0.887099i \(-0.652717\pi\)
−0.461580 + 0.887099i \(0.652717\pi\)
\(740\) −2.27139 −0.0834979
\(741\) −4.83418 −0.177588
\(742\) 8.60536 0.315913
\(743\) −28.8312 −1.05772 −0.528858 0.848711i \(-0.677379\pi\)
−0.528858 + 0.848711i \(0.677379\pi\)
\(744\) −8.72999 −0.320057
\(745\) −7.45847 −0.273257
\(746\) −19.1453 −0.700960
\(747\) −20.6149 −0.754258
\(748\) 0 0
\(749\) −4.14362 −0.151405
\(750\) −2.81361 −0.102738
\(751\) 29.2067 1.06577 0.532884 0.846188i \(-0.321108\pi\)
0.532884 + 0.846188i \(0.321108\pi\)
\(752\) −1.29720 −0.0473039
\(753\) 25.9486 0.945620
\(754\) −2.13934 −0.0779103
\(755\) −8.99272 −0.327278
\(756\) −5.39194 −0.196103
\(757\) −9.52403 −0.346157 −0.173078 0.984908i \(-0.555371\pi\)
−0.173078 + 0.984908i \(0.555371\pi\)
\(758\) 6.30548 0.229025
\(759\) 0 0
\(760\) −3.43043 −0.124435
\(761\) −29.0101 −1.05162 −0.525808 0.850604i \(-0.676237\pi\)
−0.525808 + 0.850604i \(0.676237\pi\)
\(762\) −35.5506 −1.28786
\(763\) −8.55011 −0.309535
\(764\) −12.3673 −0.447434
\(765\) 11.7597 0.425173
\(766\) 8.05894 0.291181
\(767\) −2.95058 −0.106539
\(768\) 2.81361 0.101527
\(769\) 6.81244 0.245663 0.122831 0.992428i \(-0.460803\pi\)
0.122831 + 0.992428i \(0.460803\pi\)
\(770\) 0 0
\(771\) 0.00244470 8.80438e−5 0
\(772\) 12.2658 0.441455
\(773\) 40.1042 1.44245 0.721224 0.692702i \(-0.243580\pi\)
0.721224 + 0.692702i \(0.243580\pi\)
\(774\) −15.7286 −0.565354
\(775\) 3.10278 0.111455
\(776\) −15.9907 −0.574031
\(777\) 6.39079 0.229268
\(778\) −33.7643 −1.21051
\(779\) −0.308524 −0.0110540
\(780\) −1.40921 −0.0504577
\(781\) 0 0
\(782\) −8.61968 −0.308239
\(783\) −23.0311 −0.823064
\(784\) 1.00000 0.0357143
\(785\) −1.14513 −0.0408715
\(786\) −17.3325 −0.618231
\(787\) 24.0127 0.855962 0.427981 0.903788i \(-0.359225\pi\)
0.427981 + 0.903788i \(0.359225\pi\)
\(788\) 8.65314 0.308255
\(789\) 14.7334 0.524525
\(790\) −10.3606 −0.368615
\(791\) 17.6102 0.626146
\(792\) 0 0
\(793\) 4.63635 0.164641
\(794\) −4.64243 −0.164754
\(795\) 24.2121 0.858715
\(796\) 12.4838 0.442477
\(797\) −25.6941 −0.910133 −0.455066 0.890458i \(-0.650384\pi\)
−0.455066 + 0.890458i \(0.650384\pi\)
\(798\) 9.65187 0.341673
\(799\) −3.10282 −0.109770
\(800\) −1.00000 −0.0353553
\(801\) −80.0662 −2.82900
\(802\) −18.2561 −0.644647
\(803\) 0 0
\(804\) 10.9945 0.387746
\(805\) −3.60363 −0.127011
\(806\) 1.55404 0.0547387
\(807\) −78.4458 −2.76142
\(808\) −10.2349 −0.360064
\(809\) 15.0751 0.530013 0.265007 0.964247i \(-0.414626\pi\)
0.265007 + 0.964247i \(0.414626\pi\)
\(810\) −0.421663 −0.0148157
\(811\) −32.4904 −1.14089 −0.570447 0.821335i \(-0.693230\pi\)
−0.570447 + 0.821335i \(0.693230\pi\)
\(812\) 4.27139 0.149896
\(813\) −19.9389 −0.699287
\(814\) 0 0
\(815\) −5.36940 −0.188082
\(816\) 6.72999 0.235597
\(817\) 10.9747 0.383957
\(818\) −7.94680 −0.277853
\(819\) 2.46239 0.0860430
\(820\) −0.0899376 −0.00314076
\(821\) 39.6858 1.38504 0.692521 0.721398i \(-0.256500\pi\)
0.692521 + 0.721398i \(0.256500\pi\)
\(822\) −12.3061 −0.429225
\(823\) 47.5189 1.65641 0.828203 0.560429i \(-0.189364\pi\)
0.828203 + 0.560429i \(0.189364\pi\)
\(824\) −2.15822 −0.0751852
\(825\) 0 0
\(826\) 5.89109 0.204977
\(827\) −10.6297 −0.369630 −0.184815 0.982773i \(-0.559169\pi\)
−0.184815 + 0.982773i \(0.559169\pi\)
\(828\) 17.7168 0.615702
\(829\) −4.00818 −0.139210 −0.0696050 0.997575i \(-0.522174\pi\)
−0.0696050 + 0.997575i \(0.522174\pi\)
\(830\) 4.19309 0.145544
\(831\) 59.8672 2.07677
\(832\) −0.500855 −0.0173640
\(833\) 2.39194 0.0828759
\(834\) 34.4105 1.19154
\(835\) 23.4652 0.812048
\(836\) 0 0
\(837\) 16.7300 0.578273
\(838\) 21.4201 0.739946
\(839\) 24.5933 0.849055 0.424527 0.905415i \(-0.360440\pi\)
0.424527 + 0.905415i \(0.360440\pi\)
\(840\) 2.81361 0.0970786
\(841\) −10.7552 −0.370870
\(842\) −0.894942 −0.0308417
\(843\) 85.6510 2.94998
\(844\) −6.76072 −0.232714
\(845\) −12.7491 −0.438584
\(846\) 6.37751 0.219263
\(847\) 0 0
\(848\) 8.60536 0.295509
\(849\) 41.6860 1.43066
\(850\) −2.39194 −0.0820430
\(851\) −8.18524 −0.280587
\(852\) 43.6287 1.49469
\(853\) −9.44511 −0.323395 −0.161697 0.986840i \(-0.551697\pi\)
−0.161697 + 0.986840i \(0.551697\pi\)
\(854\) −9.25687 −0.316764
\(855\) 16.8653 0.576781
\(856\) −4.14362 −0.141626
\(857\) 5.17154 0.176656 0.0883282 0.996091i \(-0.471848\pi\)
0.0883282 + 0.996091i \(0.471848\pi\)
\(858\) 0 0
\(859\) −34.8136 −1.18782 −0.593912 0.804530i \(-0.702417\pi\)
−0.593912 + 0.804530i \(0.702417\pi\)
\(860\) 3.19923 0.109093
\(861\) 0.253049 0.00862389
\(862\) 37.2082 1.26732
\(863\) −12.0780 −0.411140 −0.205570 0.978642i \(-0.565905\pi\)
−0.205570 + 0.978642i \(0.565905\pi\)
\(864\) −5.39194 −0.183438
\(865\) −14.0465 −0.477594
\(866\) −37.0912 −1.26041
\(867\) −31.7336 −1.07773
\(868\) −3.10278 −0.105315
\(869\) 0 0
\(870\) 12.0180 0.407449
\(871\) −1.95715 −0.0663154
\(872\) −8.55011 −0.289543
\(873\) 78.6161 2.66075
\(874\) −12.3620 −0.418151
\(875\) −1.00000 −0.0338062
\(876\) −21.5568 −0.728337
\(877\) 22.5704 0.762149 0.381074 0.924544i \(-0.375554\pi\)
0.381074 + 0.924544i \(0.375554\pi\)
\(878\) 16.6322 0.561311
\(879\) −88.5695 −2.98737
\(880\) 0 0
\(881\) 26.5330 0.893920 0.446960 0.894554i \(-0.352507\pi\)
0.446960 + 0.894554i \(0.352507\pi\)
\(882\) −4.91638 −0.165543
\(883\) −19.5994 −0.659572 −0.329786 0.944056i \(-0.606977\pi\)
−0.329786 + 0.944056i \(0.606977\pi\)
\(884\) −1.19802 −0.0402936
\(885\) 16.5752 0.557169
\(886\) −18.9936 −0.638101
\(887\) −16.1610 −0.542635 −0.271317 0.962490i \(-0.587459\pi\)
−0.271317 + 0.962490i \(0.587459\pi\)
\(888\) 6.39079 0.214461
\(889\) −12.6352 −0.423772
\(890\) 16.2856 0.545895
\(891\) 0 0
\(892\) −6.14794 −0.205848
\(893\) −4.44994 −0.148912
\(894\) 20.9852 0.701850
\(895\) −6.46206 −0.216003
\(896\) 1.00000 0.0334077
\(897\) −5.07826 −0.169558
\(898\) 15.0632 0.502666
\(899\) −13.2532 −0.442018
\(900\) 4.91638 0.163879
\(901\) 20.5835 0.685737
\(902\) 0 0
\(903\) −9.00138 −0.299547
\(904\) 17.6102 0.585706
\(905\) 13.1930 0.438551
\(906\) 25.3020 0.840601
\(907\) 49.8041 1.65372 0.826859 0.562410i \(-0.190126\pi\)
0.826859 + 0.562410i \(0.190126\pi\)
\(908\) 11.2377 0.372935
\(909\) 50.3189 1.66897
\(910\) −0.500855 −0.0166032
\(911\) 44.4515 1.47274 0.736372 0.676577i \(-0.236538\pi\)
0.736372 + 0.676577i \(0.236538\pi\)
\(912\) 9.65187 0.319605
\(913\) 0 0
\(914\) 13.0743 0.432459
\(915\) −26.0452 −0.861028
\(916\) −17.2115 −0.568685
\(917\) −6.16025 −0.203429
\(918\) −12.8972 −0.425672
\(919\) 25.2876 0.834162 0.417081 0.908869i \(-0.363053\pi\)
0.417081 + 0.908869i \(0.363053\pi\)
\(920\) −3.60363 −0.118808
\(921\) −29.9647 −0.987369
\(922\) −40.0901 −1.32030
\(923\) −7.76642 −0.255635
\(924\) 0 0
\(925\) −2.27139 −0.0746828
\(926\) −4.38625 −0.144141
\(927\) 10.6106 0.348499
\(928\) 4.27139 0.140215
\(929\) 58.4872 1.91890 0.959452 0.281873i \(-0.0909558\pi\)
0.959452 + 0.281873i \(0.0909558\pi\)
\(930\) −8.72999 −0.286268
\(931\) 3.43043 0.112428
\(932\) 7.59338 0.248729
\(933\) 1.26806 0.0415144
\(934\) 2.76704 0.0905404
\(935\) 0 0
\(936\) 2.46239 0.0804858
\(937\) −4.14373 −0.135370 −0.0676849 0.997707i \(-0.521561\pi\)
−0.0676849 + 0.997707i \(0.521561\pi\)
\(938\) 3.90762 0.127588
\(939\) −74.6186 −2.43509
\(940\) −1.29720 −0.0423099
\(941\) 5.11048 0.166597 0.0832985 0.996525i \(-0.473455\pi\)
0.0832985 + 0.996525i \(0.473455\pi\)
\(942\) 3.22195 0.104977
\(943\) −0.324102 −0.0105542
\(944\) 5.89109 0.191739
\(945\) −5.39194 −0.175400
\(946\) 0 0
\(947\) 14.4259 0.468779 0.234389 0.972143i \(-0.424691\pi\)
0.234389 + 0.972143i \(0.424691\pi\)
\(948\) 29.1508 0.946773
\(949\) 3.83736 0.124566
\(950\) −3.43043 −0.111298
\(951\) 82.2957 2.66862
\(952\) 2.39194 0.0775233
\(953\) −37.3047 −1.20842 −0.604208 0.796827i \(-0.706510\pi\)
−0.604208 + 0.796827i \(0.706510\pi\)
\(954\) −42.3072 −1.36975
\(955\) −12.3673 −0.400197
\(956\) 28.8381 0.932691
\(957\) 0 0
\(958\) −2.13584 −0.0690058
\(959\) −4.37378 −0.141237
\(960\) 2.81361 0.0908088
\(961\) −21.3728 −0.689445
\(962\) −1.13764 −0.0366788
\(963\) 20.3716 0.656467
\(964\) 15.3781 0.495295
\(965\) 12.2658 0.394849
\(966\) 10.1392 0.326223
\(967\) 55.7048 1.79135 0.895673 0.444712i \(-0.146694\pi\)
0.895673 + 0.444712i \(0.146694\pi\)
\(968\) 0 0
\(969\) 23.0867 0.741653
\(970\) −15.9907 −0.513429
\(971\) 0.975158 0.0312943 0.0156471 0.999878i \(-0.495019\pi\)
0.0156471 + 0.999878i \(0.495019\pi\)
\(972\) −14.9894 −0.480786
\(973\) 12.2300 0.392077
\(974\) 30.0879 0.964079
\(975\) −1.40921 −0.0451308
\(976\) −9.25687 −0.296305
\(977\) 14.8373 0.474688 0.237344 0.971426i \(-0.423723\pi\)
0.237344 + 0.971426i \(0.423723\pi\)
\(978\) 15.1074 0.483080
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 42.0356 1.34209
\(982\) 14.1922 0.452891
\(983\) −9.04686 −0.288550 −0.144275 0.989538i \(-0.546085\pi\)
−0.144275 + 0.989538i \(0.546085\pi\)
\(984\) 0.253049 0.00806691
\(985\) 8.65314 0.275712
\(986\) 10.2169 0.325373
\(987\) 3.64980 0.116174
\(988\) −1.71815 −0.0546615
\(989\) 11.5288 0.366596
\(990\) 0 0
\(991\) −34.7717 −1.10456 −0.552280 0.833659i \(-0.686242\pi\)
−0.552280 + 0.833659i \(0.686242\pi\)
\(992\) −3.10278 −0.0985132
\(993\) −46.4383 −1.47367
\(994\) 15.5063 0.491831
\(995\) 12.4838 0.395764
\(996\) −11.7977 −0.373825
\(997\) −23.5849 −0.746942 −0.373471 0.927642i \(-0.621832\pi\)
−0.373471 + 0.927642i \(0.621832\pi\)
\(998\) −17.9339 −0.567688
\(999\) −12.2472 −0.387484
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.cz.1.5 6
11.10 odd 2 8470.2.a.df.1.6 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cz.1.5 6 1.1 even 1 trivial
8470.2.a.df.1.6 yes 6 11.10 odd 2