Properties

Label 8470.2.a.bn.1.1
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: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.23607 q^{6} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.23607 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.23607 q^{6} +1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{9} +1.00000 q^{10} -2.23607 q^{12} +1.00000 q^{13} -1.00000 q^{14} +2.23607 q^{15} +1.00000 q^{16} -1.23607 q^{17} -2.00000 q^{18} -2.23607 q^{19} -1.00000 q^{20} -2.23607 q^{21} -3.76393 q^{23} +2.23607 q^{24} +1.00000 q^{25} -1.00000 q^{26} +2.23607 q^{27} +1.00000 q^{28} +4.47214 q^{29} -2.23607 q^{30} -5.23607 q^{31} -1.00000 q^{32} +1.23607 q^{34} -1.00000 q^{35} +2.00000 q^{36} +8.94427 q^{37} +2.23607 q^{38} -2.23607 q^{39} +1.00000 q^{40} -7.23607 q^{41} +2.23607 q^{42} +3.23607 q^{43} -2.00000 q^{45} +3.76393 q^{46} +7.70820 q^{47} -2.23607 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.76393 q^{51} +1.00000 q^{52} -5.70820 q^{53} -2.23607 q^{54} -1.00000 q^{56} +5.00000 q^{57} -4.47214 q^{58} +4.70820 q^{59} +2.23607 q^{60} -3.52786 q^{61} +5.23607 q^{62} +2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -7.23607 q^{67} -1.23607 q^{68} +8.41641 q^{69} +1.00000 q^{70} -1.52786 q^{71} -2.00000 q^{72} -7.23607 q^{73} -8.94427 q^{74} -2.23607 q^{75} -2.23607 q^{76} +2.23607 q^{78} +10.2361 q^{79} -1.00000 q^{80} -11.0000 q^{81} +7.23607 q^{82} -0.236068 q^{83} -2.23607 q^{84} +1.23607 q^{85} -3.23607 q^{86} -10.0000 q^{87} -17.2361 q^{89} +2.00000 q^{90} +1.00000 q^{91} -3.76393 q^{92} +11.7082 q^{93} -7.70820 q^{94} +2.23607 q^{95} +2.23607 q^{96} +3.52786 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + 4q^{9} + 2q^{10} + 2q^{13} - 2q^{14} + 2q^{16} + 2q^{17} - 4q^{18} - 2q^{20} - 12q^{23} + 2q^{25} - 2q^{26} + 2q^{28} - 6q^{31} - 2q^{32} - 2q^{34} - 2q^{35} + 4q^{36} + 2q^{40} - 10q^{41} + 2q^{43} - 4q^{45} + 12q^{46} + 2q^{47} + 2q^{49} - 2q^{50} + 10q^{51} + 2q^{52} + 2q^{53} - 2q^{56} + 10q^{57} - 4q^{59} - 16q^{61} + 6q^{62} + 4q^{63} + 2q^{64} - 2q^{65} - 10q^{67} + 2q^{68} - 10q^{69} + 2q^{70} - 12q^{71} - 4q^{72} - 10q^{73} + 16q^{79} - 2q^{80} - 22q^{81} + 10q^{82} + 4q^{83} - 2q^{85} - 2q^{86} - 20q^{87} - 30q^{89} + 4q^{90} + 2q^{91} - 12q^{92} + 10q^{93} - 2q^{94} + 16q^{97} - 2q^{98} + O(q^{100}) \)

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
\(3\) −2.23607 −1.29099 −0.645497 0.763763i \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.23607 0.912871
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 2.00000 0.666667
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −2.23607 −0.645497
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.23607 0.577350
\(16\) 1.00000 0.250000
\(17\) −1.23607 −0.299791 −0.149895 0.988702i \(-0.547894\pi\)
−0.149895 + 0.988702i \(0.547894\pi\)
\(18\) −2.00000 −0.471405
\(19\) −2.23607 −0.512989 −0.256495 0.966546i \(-0.582568\pi\)
−0.256495 + 0.966546i \(0.582568\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.23607 −0.487950
\(22\) 0 0
\(23\) −3.76393 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(24\) 2.23607 0.456435
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) 2.23607 0.430331
\(28\) 1.00000 0.188982
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) −2.23607 −0.408248
\(31\) −5.23607 −0.940426 −0.470213 0.882553i \(-0.655823\pi\)
−0.470213 + 0.882553i \(0.655823\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.23607 0.211984
\(35\) −1.00000 −0.169031
\(36\) 2.00000 0.333333
\(37\) 8.94427 1.47043 0.735215 0.677834i \(-0.237081\pi\)
0.735215 + 0.677834i \(0.237081\pi\)
\(38\) 2.23607 0.362738
\(39\) −2.23607 −0.358057
\(40\) 1.00000 0.158114
\(41\) −7.23607 −1.13008 −0.565042 0.825062i \(-0.691140\pi\)
−0.565042 + 0.825062i \(0.691140\pi\)
\(42\) 2.23607 0.345033
\(43\) 3.23607 0.493496 0.246748 0.969080i \(-0.420638\pi\)
0.246748 + 0.969080i \(0.420638\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 3.76393 0.554962
\(47\) 7.70820 1.12436 0.562179 0.827016i \(-0.309963\pi\)
0.562179 + 0.827016i \(0.309963\pi\)
\(48\) −2.23607 −0.322749
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 2.76393 0.387028
\(52\) 1.00000 0.138675
\(53\) −5.70820 −0.784082 −0.392041 0.919948i \(-0.628231\pi\)
−0.392041 + 0.919948i \(0.628231\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 5.00000 0.662266
\(58\) −4.47214 −0.587220
\(59\) 4.70820 0.612956 0.306478 0.951878i \(-0.400849\pi\)
0.306478 + 0.951878i \(0.400849\pi\)
\(60\) 2.23607 0.288675
\(61\) −3.52786 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(62\) 5.23607 0.664981
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −7.23607 −0.884026 −0.442013 0.897009i \(-0.645736\pi\)
−0.442013 + 0.897009i \(0.645736\pi\)
\(68\) −1.23607 −0.149895
\(69\) 8.41641 1.01322
\(70\) 1.00000 0.119523
\(71\) −1.52786 −0.181324 −0.0906621 0.995882i \(-0.528898\pi\)
−0.0906621 + 0.995882i \(0.528898\pi\)
\(72\) −2.00000 −0.235702
\(73\) −7.23607 −0.846918 −0.423459 0.905915i \(-0.639184\pi\)
−0.423459 + 0.905915i \(0.639184\pi\)
\(74\) −8.94427 −1.03975
\(75\) −2.23607 −0.258199
\(76\) −2.23607 −0.256495
\(77\) 0 0
\(78\) 2.23607 0.253185
\(79\) 10.2361 1.15165 0.575824 0.817574i \(-0.304681\pi\)
0.575824 + 0.817574i \(0.304681\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) 7.23607 0.799090
\(83\) −0.236068 −0.0259118 −0.0129559 0.999916i \(-0.504124\pi\)
−0.0129559 + 0.999916i \(0.504124\pi\)
\(84\) −2.23607 −0.243975
\(85\) 1.23607 0.134070
\(86\) −3.23607 −0.348954
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) −17.2361 −1.82702 −0.913510 0.406817i \(-0.866639\pi\)
−0.913510 + 0.406817i \(0.866639\pi\)
\(90\) 2.00000 0.210819
\(91\) 1.00000 0.104828
\(92\) −3.76393 −0.392417
\(93\) 11.7082 1.21408
\(94\) −7.70820 −0.795041
\(95\) 2.23607 0.229416
\(96\) 2.23607 0.228218
\(97\) 3.52786 0.358200 0.179100 0.983831i \(-0.442681\pi\)
0.179100 + 0.983831i \(0.442681\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) −2.76393 −0.273670
\(103\) 14.1803 1.39723 0.698615 0.715498i \(-0.253800\pi\)
0.698615 + 0.715498i \(0.253800\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 2.23607 0.218218
\(106\) 5.70820 0.554430
\(107\) −2.47214 −0.238990 −0.119495 0.992835i \(-0.538128\pi\)
−0.119495 + 0.992835i \(0.538128\pi\)
\(108\) 2.23607 0.215166
\(109\) −3.70820 −0.355182 −0.177591 0.984104i \(-0.556830\pi\)
−0.177591 + 0.984104i \(0.556830\pi\)
\(110\) 0 0
\(111\) −20.0000 −1.89832
\(112\) 1.00000 0.0944911
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) −5.00000 −0.468293
\(115\) 3.76393 0.350988
\(116\) 4.47214 0.415227
\(117\) 2.00000 0.184900
\(118\) −4.70820 −0.433425
\(119\) −1.23607 −0.113310
\(120\) −2.23607 −0.204124
\(121\) 0 0
\(122\) 3.52786 0.319398
\(123\) 16.1803 1.45893
\(124\) −5.23607 −0.470213
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) 6.23607 0.553362 0.276681 0.960962i \(-0.410766\pi\)
0.276681 + 0.960962i \(0.410766\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −7.23607 −0.637100
\(130\) 1.00000 0.0877058
\(131\) −3.18034 −0.277868 −0.138934 0.990302i \(-0.544368\pi\)
−0.138934 + 0.990302i \(0.544368\pi\)
\(132\) 0 0
\(133\) −2.23607 −0.193892
\(134\) 7.23607 0.625101
\(135\) −2.23607 −0.192450
\(136\) 1.23607 0.105992
\(137\) 13.4721 1.15100 0.575501 0.817801i \(-0.304807\pi\)
0.575501 + 0.817801i \(0.304807\pi\)
\(138\) −8.41641 −0.716452
\(139\) 4.23607 0.359299 0.179649 0.983731i \(-0.442504\pi\)
0.179649 + 0.983731i \(0.442504\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −17.2361 −1.45154
\(142\) 1.52786 0.128216
\(143\) 0 0
\(144\) 2.00000 0.166667
\(145\) −4.47214 −0.371391
\(146\) 7.23607 0.598861
\(147\) −2.23607 −0.184428
\(148\) 8.94427 0.735215
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 2.23607 0.182574
\(151\) 12.2361 0.995757 0.497879 0.867247i \(-0.334113\pi\)
0.497879 + 0.867247i \(0.334113\pi\)
\(152\) 2.23607 0.181369
\(153\) −2.47214 −0.199860
\(154\) 0 0
\(155\) 5.23607 0.420571
\(156\) −2.23607 −0.179029
\(157\) 8.52786 0.680598 0.340299 0.940317i \(-0.389472\pi\)
0.340299 + 0.940317i \(0.389472\pi\)
\(158\) −10.2361 −0.814338
\(159\) 12.7639 1.01225
\(160\) 1.00000 0.0790569
\(161\) −3.76393 −0.296639
\(162\) 11.0000 0.864242
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −7.23607 −0.565042
\(165\) 0 0
\(166\) 0.236068 0.0183224
\(167\) 1.70820 0.132185 0.0660924 0.997814i \(-0.478947\pi\)
0.0660924 + 0.997814i \(0.478947\pi\)
\(168\) 2.23607 0.172516
\(169\) −12.0000 −0.923077
\(170\) −1.23607 −0.0948021
\(171\) −4.47214 −0.341993
\(172\) 3.23607 0.246748
\(173\) 19.8885 1.51210 0.756049 0.654515i \(-0.227127\pi\)
0.756049 + 0.654515i \(0.227127\pi\)
\(174\) 10.0000 0.758098
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −10.5279 −0.791323
\(178\) 17.2361 1.29190
\(179\) −11.2361 −0.839823 −0.419912 0.907565i \(-0.637939\pi\)
−0.419912 + 0.907565i \(0.637939\pi\)
\(180\) −2.00000 −0.149071
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 7.88854 0.583138
\(184\) 3.76393 0.277481
\(185\) −8.94427 −0.657596
\(186\) −11.7082 −0.858487
\(187\) 0 0
\(188\) 7.70820 0.562179
\(189\) 2.23607 0.162650
\(190\) −2.23607 −0.162221
\(191\) 25.1803 1.82199 0.910993 0.412422i \(-0.135317\pi\)
0.910993 + 0.412422i \(0.135317\pi\)
\(192\) −2.23607 −0.161374
\(193\) 7.94427 0.571841 0.285921 0.958253i \(-0.407701\pi\)
0.285921 + 0.958253i \(0.407701\pi\)
\(194\) −3.52786 −0.253286
\(195\) 2.23607 0.160128
\(196\) 1.00000 0.0714286
\(197\) −7.52786 −0.536338 −0.268169 0.963372i \(-0.586419\pi\)
−0.268169 + 0.963372i \(0.586419\pi\)
\(198\) 0 0
\(199\) −4.47214 −0.317021 −0.158511 0.987357i \(-0.550669\pi\)
−0.158511 + 0.987357i \(0.550669\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 16.1803 1.14127
\(202\) 1.00000 0.0703598
\(203\) 4.47214 0.313882
\(204\) 2.76393 0.193514
\(205\) 7.23607 0.505389
\(206\) −14.1803 −0.987991
\(207\) −7.52786 −0.523223
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) −2.23607 −0.154303
\(211\) 3.52786 0.242868 0.121434 0.992599i \(-0.461251\pi\)
0.121434 + 0.992599i \(0.461251\pi\)
\(212\) −5.70820 −0.392041
\(213\) 3.41641 0.234088
\(214\) 2.47214 0.168992
\(215\) −3.23607 −0.220698
\(216\) −2.23607 −0.152145
\(217\) −5.23607 −0.355447
\(218\) 3.70820 0.251151
\(219\) 16.1803 1.09337
\(220\) 0 0
\(221\) −1.23607 −0.0831469
\(222\) 20.0000 1.34231
\(223\) −8.18034 −0.547796 −0.273898 0.961759i \(-0.588313\pi\)
−0.273898 + 0.961759i \(0.588313\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.00000 0.133333
\(226\) 1.00000 0.0665190
\(227\) −7.41641 −0.492244 −0.246122 0.969239i \(-0.579156\pi\)
−0.246122 + 0.969239i \(0.579156\pi\)
\(228\) 5.00000 0.331133
\(229\) 6.94427 0.458890 0.229445 0.973322i \(-0.426309\pi\)
0.229445 + 0.973322i \(0.426309\pi\)
\(230\) −3.76393 −0.248186
\(231\) 0 0
\(232\) −4.47214 −0.293610
\(233\) 10.4164 0.682402 0.341201 0.939990i \(-0.389166\pi\)
0.341201 + 0.939990i \(0.389166\pi\)
\(234\) −2.00000 −0.130744
\(235\) −7.70820 −0.502828
\(236\) 4.70820 0.306478
\(237\) −22.8885 −1.48677
\(238\) 1.23607 0.0801224
\(239\) 10.7082 0.692656 0.346328 0.938113i \(-0.387428\pi\)
0.346328 + 0.938113i \(0.387428\pi\)
\(240\) 2.23607 0.144338
\(241\) −22.3607 −1.44038 −0.720189 0.693778i \(-0.755945\pi\)
−0.720189 + 0.693778i \(0.755945\pi\)
\(242\) 0 0
\(243\) 17.8885 1.14755
\(244\) −3.52786 −0.225848
\(245\) −1.00000 −0.0638877
\(246\) −16.1803 −1.03162
\(247\) −2.23607 −0.142278
\(248\) 5.23607 0.332491
\(249\) 0.527864 0.0334520
\(250\) 1.00000 0.0632456
\(251\) −29.8885 −1.88655 −0.943274 0.332015i \(-0.892272\pi\)
−0.943274 + 0.332015i \(0.892272\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −6.23607 −0.391286
\(255\) −2.76393 −0.173084
\(256\) 1.00000 0.0625000
\(257\) −4.76393 −0.297166 −0.148583 0.988900i \(-0.547471\pi\)
−0.148583 + 0.988900i \(0.547471\pi\)
\(258\) 7.23607 0.450498
\(259\) 8.94427 0.555770
\(260\) −1.00000 −0.0620174
\(261\) 8.94427 0.553637
\(262\) 3.18034 0.196482
\(263\) 7.65248 0.471872 0.235936 0.971769i \(-0.424184\pi\)
0.235936 + 0.971769i \(0.424184\pi\)
\(264\) 0 0
\(265\) 5.70820 0.350652
\(266\) 2.23607 0.137102
\(267\) 38.5410 2.35867
\(268\) −7.23607 −0.442013
\(269\) −27.8328 −1.69700 −0.848498 0.529198i \(-0.822493\pi\)
−0.848498 + 0.529198i \(0.822493\pi\)
\(270\) 2.23607 0.136083
\(271\) 7.41641 0.450515 0.225257 0.974299i \(-0.427678\pi\)
0.225257 + 0.974299i \(0.427678\pi\)
\(272\) −1.23607 −0.0749476
\(273\) −2.23607 −0.135333
\(274\) −13.4721 −0.813881
\(275\) 0 0
\(276\) 8.41641 0.506608
\(277\) 23.5967 1.41779 0.708896 0.705313i \(-0.249194\pi\)
0.708896 + 0.705313i \(0.249194\pi\)
\(278\) −4.23607 −0.254062
\(279\) −10.4721 −0.626950
\(280\) 1.00000 0.0597614
\(281\) 25.8328 1.54106 0.770528 0.637406i \(-0.219992\pi\)
0.770528 + 0.637406i \(0.219992\pi\)
\(282\) 17.2361 1.02639
\(283\) 26.2361 1.55957 0.779786 0.626046i \(-0.215328\pi\)
0.779786 + 0.626046i \(0.215328\pi\)
\(284\) −1.52786 −0.0906621
\(285\) −5.00000 −0.296174
\(286\) 0 0
\(287\) −7.23607 −0.427132
\(288\) −2.00000 −0.117851
\(289\) −15.4721 −0.910126
\(290\) 4.47214 0.262613
\(291\) −7.88854 −0.462435
\(292\) −7.23607 −0.423459
\(293\) −5.00000 −0.292103 −0.146052 0.989277i \(-0.546657\pi\)
−0.146052 + 0.989277i \(0.546657\pi\)
\(294\) 2.23607 0.130410
\(295\) −4.70820 −0.274122
\(296\) −8.94427 −0.519875
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) −3.76393 −0.217674
\(300\) −2.23607 −0.129099
\(301\) 3.23607 0.186524
\(302\) −12.2361 −0.704107
\(303\) 2.23607 0.128459
\(304\) −2.23607 −0.128247
\(305\) 3.52786 0.202005
\(306\) 2.47214 0.141323
\(307\) 11.4164 0.651569 0.325784 0.945444i \(-0.394372\pi\)
0.325784 + 0.945444i \(0.394372\pi\)
\(308\) 0 0
\(309\) −31.7082 −1.80382
\(310\) −5.23607 −0.297389
\(311\) −27.1246 −1.53810 −0.769048 0.639191i \(-0.779269\pi\)
−0.769048 + 0.639191i \(0.779269\pi\)
\(312\) 2.23607 0.126592
\(313\) 19.5279 1.10378 0.551890 0.833917i \(-0.313907\pi\)
0.551890 + 0.833917i \(0.313907\pi\)
\(314\) −8.52786 −0.481255
\(315\) −2.00000 −0.112687
\(316\) 10.2361 0.575824
\(317\) 0.652476 0.0366467 0.0183233 0.999832i \(-0.494167\pi\)
0.0183233 + 0.999832i \(0.494167\pi\)
\(318\) −12.7639 −0.715766
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 5.52786 0.308535
\(322\) 3.76393 0.209756
\(323\) 2.76393 0.153789
\(324\) −11.0000 −0.611111
\(325\) 1.00000 0.0554700
\(326\) 6.00000 0.332309
\(327\) 8.29180 0.458537
\(328\) 7.23607 0.399545
\(329\) 7.70820 0.424967
\(330\) 0 0
\(331\) −21.7082 −1.19319 −0.596595 0.802542i \(-0.703480\pi\)
−0.596595 + 0.802542i \(0.703480\pi\)
\(332\) −0.236068 −0.0129559
\(333\) 17.8885 0.980286
\(334\) −1.70820 −0.0934688
\(335\) 7.23607 0.395349
\(336\) −2.23607 −0.121988
\(337\) −2.88854 −0.157349 −0.0786745 0.996900i \(-0.525069\pi\)
−0.0786745 + 0.996900i \(0.525069\pi\)
\(338\) 12.0000 0.652714
\(339\) 2.23607 0.121447
\(340\) 1.23607 0.0670352
\(341\) 0 0
\(342\) 4.47214 0.241825
\(343\) 1.00000 0.0539949
\(344\) −3.23607 −0.174477
\(345\) −8.41641 −0.453124
\(346\) −19.8885 −1.06921
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) −10.0000 −0.536056
\(349\) 0.416408 0.0222898 0.0111449 0.999938i \(-0.496452\pi\)
0.0111449 + 0.999938i \(0.496452\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.23607 0.119352
\(352\) 0 0
\(353\) −4.76393 −0.253559 −0.126779 0.991931i \(-0.540464\pi\)
−0.126779 + 0.991931i \(0.540464\pi\)
\(354\) 10.5279 0.559550
\(355\) 1.52786 0.0810906
\(356\) −17.2361 −0.913510
\(357\) 2.76393 0.146283
\(358\) 11.2361 0.593845
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 2.00000 0.105409
\(361\) −14.0000 −0.736842
\(362\) −7.00000 −0.367912
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 7.23607 0.378753
\(366\) −7.88854 −0.412341
\(367\) 8.18034 0.427010 0.213505 0.976942i \(-0.431512\pi\)
0.213505 + 0.976942i \(0.431512\pi\)
\(368\) −3.76393 −0.196209
\(369\) −14.4721 −0.753389
\(370\) 8.94427 0.464991
\(371\) −5.70820 −0.296355
\(372\) 11.7082 0.607042
\(373\) −5.52786 −0.286222 −0.143111 0.989707i \(-0.545711\pi\)
−0.143111 + 0.989707i \(0.545711\pi\)
\(374\) 0 0
\(375\) 2.23607 0.115470
\(376\) −7.70820 −0.397520
\(377\) 4.47214 0.230327
\(378\) −2.23607 −0.115011
\(379\) 11.4164 0.586421 0.293211 0.956048i \(-0.405276\pi\)
0.293211 + 0.956048i \(0.405276\pi\)
\(380\) 2.23607 0.114708
\(381\) −13.9443 −0.714387
\(382\) −25.1803 −1.28834
\(383\) −17.2361 −0.880722 −0.440361 0.897821i \(-0.645150\pi\)
−0.440361 + 0.897821i \(0.645150\pi\)
\(384\) 2.23607 0.114109
\(385\) 0 0
\(386\) −7.94427 −0.404353
\(387\) 6.47214 0.328997
\(388\) 3.52786 0.179100
\(389\) −18.2918 −0.927431 −0.463715 0.885984i \(-0.653484\pi\)
−0.463715 + 0.885984i \(0.653484\pi\)
\(390\) −2.23607 −0.113228
\(391\) 4.65248 0.235286
\(392\) −1.00000 −0.0505076
\(393\) 7.11146 0.358726
\(394\) 7.52786 0.379248
\(395\) −10.2361 −0.515032
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 4.47214 0.224168
\(399\) 5.00000 0.250313
\(400\) 1.00000 0.0500000
\(401\) −28.8328 −1.43984 −0.719921 0.694056i \(-0.755822\pi\)
−0.719921 + 0.694056i \(0.755822\pi\)
\(402\) −16.1803 −0.807002
\(403\) −5.23607 −0.260827
\(404\) −1.00000 −0.0497519
\(405\) 11.0000 0.546594
\(406\) −4.47214 −0.221948
\(407\) 0 0
\(408\) −2.76393 −0.136835
\(409\) 11.5279 0.570016 0.285008 0.958525i \(-0.408004\pi\)
0.285008 + 0.958525i \(0.408004\pi\)
\(410\) −7.23607 −0.357364
\(411\) −30.1246 −1.48594
\(412\) 14.1803 0.698615
\(413\) 4.70820 0.231676
\(414\) 7.52786 0.369974
\(415\) 0.236068 0.0115881
\(416\) −1.00000 −0.0490290
\(417\) −9.47214 −0.463852
\(418\) 0 0
\(419\) −28.2361 −1.37942 −0.689711 0.724085i \(-0.742262\pi\)
−0.689711 + 0.724085i \(0.742262\pi\)
\(420\) 2.23607 0.109109
\(421\) −3.05573 −0.148927 −0.0744635 0.997224i \(-0.523724\pi\)
−0.0744635 + 0.997224i \(0.523724\pi\)
\(422\) −3.52786 −0.171734
\(423\) 15.4164 0.749571
\(424\) 5.70820 0.277215
\(425\) −1.23607 −0.0599581
\(426\) −3.41641 −0.165526
\(427\) −3.52786 −0.170725
\(428\) −2.47214 −0.119495
\(429\) 0 0
\(430\) 3.23607 0.156057
\(431\) −33.6525 −1.62098 −0.810491 0.585751i \(-0.800800\pi\)
−0.810491 + 0.585751i \(0.800800\pi\)
\(432\) 2.23607 0.107583
\(433\) −12.6525 −0.608039 −0.304020 0.952666i \(-0.598329\pi\)
−0.304020 + 0.952666i \(0.598329\pi\)
\(434\) 5.23607 0.251339
\(435\) 10.0000 0.479463
\(436\) −3.70820 −0.177591
\(437\) 8.41641 0.402611
\(438\) −16.1803 −0.773127
\(439\) 15.1246 0.721858 0.360929 0.932593i \(-0.382460\pi\)
0.360929 + 0.932593i \(0.382460\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 1.23607 0.0587938
\(443\) −24.3607 −1.15741 −0.578705 0.815537i \(-0.696442\pi\)
−0.578705 + 0.815537i \(0.696442\pi\)
\(444\) −20.0000 −0.949158
\(445\) 17.2361 0.817068
\(446\) 8.18034 0.387350
\(447\) −8.94427 −0.423050
\(448\) 1.00000 0.0472456
\(449\) 23.8328 1.12474 0.562370 0.826886i \(-0.309890\pi\)
0.562370 + 0.826886i \(0.309890\pi\)
\(450\) −2.00000 −0.0942809
\(451\) 0 0
\(452\) −1.00000 −0.0470360
\(453\) −27.3607 −1.28552
\(454\) 7.41641 0.348069
\(455\) −1.00000 −0.0468807
\(456\) −5.00000 −0.234146
\(457\) −36.4164 −1.70349 −0.851744 0.523958i \(-0.824455\pi\)
−0.851744 + 0.523958i \(0.824455\pi\)
\(458\) −6.94427 −0.324485
\(459\) −2.76393 −0.129009
\(460\) 3.76393 0.175494
\(461\) −23.8885 −1.11260 −0.556300 0.830981i \(-0.687780\pi\)
−0.556300 + 0.830981i \(0.687780\pi\)
\(462\) 0 0
\(463\) 10.7082 0.497652 0.248826 0.968548i \(-0.419955\pi\)
0.248826 + 0.968548i \(0.419955\pi\)
\(464\) 4.47214 0.207614
\(465\) −11.7082 −0.542955
\(466\) −10.4164 −0.482531
\(467\) −2.81966 −0.130478 −0.0652392 0.997870i \(-0.520781\pi\)
−0.0652392 + 0.997870i \(0.520781\pi\)
\(468\) 2.00000 0.0924500
\(469\) −7.23607 −0.334131
\(470\) 7.70820 0.355553
\(471\) −19.0689 −0.878648
\(472\) −4.70820 −0.216713
\(473\) 0 0
\(474\) 22.8885 1.05131
\(475\) −2.23607 −0.102598
\(476\) −1.23607 −0.0566551
\(477\) −11.4164 −0.522721
\(478\) −10.7082 −0.489782
\(479\) −27.5967 −1.26093 −0.630464 0.776219i \(-0.717135\pi\)
−0.630464 + 0.776219i \(0.717135\pi\)
\(480\) −2.23607 −0.102062
\(481\) 8.94427 0.407824
\(482\) 22.3607 1.01850
\(483\) 8.41641 0.382960
\(484\) 0 0
\(485\) −3.52786 −0.160192
\(486\) −17.8885 −0.811441
\(487\) 2.23607 0.101326 0.0506630 0.998716i \(-0.483867\pi\)
0.0506630 + 0.998716i \(0.483867\pi\)
\(488\) 3.52786 0.159699
\(489\) 13.4164 0.606711
\(490\) 1.00000 0.0451754
\(491\) 31.2361 1.40966 0.704832 0.709374i \(-0.251022\pi\)
0.704832 + 0.709374i \(0.251022\pi\)
\(492\) 16.1803 0.729466
\(493\) −5.52786 −0.248962
\(494\) 2.23607 0.100605
\(495\) 0 0
\(496\) −5.23607 −0.235106
\(497\) −1.52786 −0.0685341
\(498\) −0.527864 −0.0236542
\(499\) 26.8328 1.20120 0.600601 0.799549i \(-0.294928\pi\)
0.600601 + 0.799549i \(0.294928\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −3.81966 −0.170650
\(502\) 29.8885 1.33399
\(503\) 25.5967 1.14130 0.570651 0.821192i \(-0.306691\pi\)
0.570651 + 0.821192i \(0.306691\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 1.00000 0.0444994
\(506\) 0 0
\(507\) 26.8328 1.19169
\(508\) 6.23607 0.276681
\(509\) 28.8885 1.28046 0.640231 0.768182i \(-0.278839\pi\)
0.640231 + 0.768182i \(0.278839\pi\)
\(510\) 2.76393 0.122389
\(511\) −7.23607 −0.320105
\(512\) −1.00000 −0.0441942
\(513\) −5.00000 −0.220755
\(514\) 4.76393 0.210128
\(515\) −14.1803 −0.624860
\(516\) −7.23607 −0.318550
\(517\) 0 0
\(518\) −8.94427 −0.392989
\(519\) −44.4721 −1.95211
\(520\) 1.00000 0.0438529
\(521\) −1.05573 −0.0462523 −0.0231261 0.999733i \(-0.507362\pi\)
−0.0231261 + 0.999733i \(0.507362\pi\)
\(522\) −8.94427 −0.391480
\(523\) −35.6525 −1.55897 −0.779487 0.626418i \(-0.784520\pi\)
−0.779487 + 0.626418i \(0.784520\pi\)
\(524\) −3.18034 −0.138934
\(525\) −2.23607 −0.0975900
\(526\) −7.65248 −0.333664
\(527\) 6.47214 0.281931
\(528\) 0 0
\(529\) −8.83282 −0.384035
\(530\) −5.70820 −0.247949
\(531\) 9.41641 0.408637
\(532\) −2.23607 −0.0969458
\(533\) −7.23607 −0.313429
\(534\) −38.5410 −1.66783
\(535\) 2.47214 0.106880
\(536\) 7.23607 0.312551
\(537\) 25.1246 1.08421
\(538\) 27.8328 1.19996
\(539\) 0 0
\(540\) −2.23607 −0.0962250
\(541\) −35.3050 −1.51788 −0.758939 0.651161i \(-0.774282\pi\)
−0.758939 + 0.651161i \(0.774282\pi\)
\(542\) −7.41641 −0.318562
\(543\) −15.6525 −0.671712
\(544\) 1.23607 0.0529960
\(545\) 3.70820 0.158842
\(546\) 2.23607 0.0956949
\(547\) 6.65248 0.284439 0.142220 0.989835i \(-0.454576\pi\)
0.142220 + 0.989835i \(0.454576\pi\)
\(548\) 13.4721 0.575501
\(549\) −7.05573 −0.301131
\(550\) 0 0
\(551\) −10.0000 −0.426014
\(552\) −8.41641 −0.358226
\(553\) 10.2361 0.435282
\(554\) −23.5967 −1.00253
\(555\) 20.0000 0.848953
\(556\) 4.23607 0.179649
\(557\) −31.1246 −1.31879 −0.659396 0.751796i \(-0.729188\pi\)
−0.659396 + 0.751796i \(0.729188\pi\)
\(558\) 10.4721 0.443321
\(559\) 3.23607 0.136871
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −25.8328 −1.08969
\(563\) −36.5967 −1.54237 −0.771185 0.636612i \(-0.780335\pi\)
−0.771185 + 0.636612i \(0.780335\pi\)
\(564\) −17.2361 −0.725769
\(565\) 1.00000 0.0420703
\(566\) −26.2361 −1.10278
\(567\) −11.0000 −0.461957
\(568\) 1.52786 0.0641078
\(569\) 7.47214 0.313248 0.156624 0.987658i \(-0.449939\pi\)
0.156624 + 0.987658i \(0.449939\pi\)
\(570\) 5.00000 0.209427
\(571\) 36.3607 1.52165 0.760824 0.648959i \(-0.224795\pi\)
0.760824 + 0.648959i \(0.224795\pi\)
\(572\) 0 0
\(573\) −56.3050 −2.35217
\(574\) 7.23607 0.302028
\(575\) −3.76393 −0.156967
\(576\) 2.00000 0.0833333
\(577\) −35.0132 −1.45762 −0.728808 0.684718i \(-0.759925\pi\)
−0.728808 + 0.684718i \(0.759925\pi\)
\(578\) 15.4721 0.643556
\(579\) −17.7639 −0.738244
\(580\) −4.47214 −0.185695
\(581\) −0.236068 −0.00979375
\(582\) 7.88854 0.326991
\(583\) 0 0
\(584\) 7.23607 0.299431
\(585\) −2.00000 −0.0826898
\(586\) 5.00000 0.206548
\(587\) −23.0689 −0.952155 −0.476077 0.879403i \(-0.657942\pi\)
−0.476077 + 0.879403i \(0.657942\pi\)
\(588\) −2.23607 −0.0922139
\(589\) 11.7082 0.482428
\(590\) 4.70820 0.193834
\(591\) 16.8328 0.692410
\(592\) 8.94427 0.367607
\(593\) 36.9443 1.51712 0.758560 0.651604i \(-0.225903\pi\)
0.758560 + 0.651604i \(0.225903\pi\)
\(594\) 0 0
\(595\) 1.23607 0.0506738
\(596\) 4.00000 0.163846
\(597\) 10.0000 0.409273
\(598\) 3.76393 0.153919
\(599\) 27.0689 1.10600 0.553002 0.833180i \(-0.313482\pi\)
0.553002 + 0.833180i \(0.313482\pi\)
\(600\) 2.23607 0.0912871
\(601\) 26.6525 1.08718 0.543589 0.839352i \(-0.317065\pi\)
0.543589 + 0.839352i \(0.317065\pi\)
\(602\) −3.23607 −0.131892
\(603\) −14.4721 −0.589351
\(604\) 12.2361 0.497879
\(605\) 0 0
\(606\) −2.23607 −0.0908341
\(607\) 21.0557 0.854626 0.427313 0.904104i \(-0.359460\pi\)
0.427313 + 0.904104i \(0.359460\pi\)
\(608\) 2.23607 0.0906845
\(609\) −10.0000 −0.405220
\(610\) −3.52786 −0.142839
\(611\) 7.70820 0.311841
\(612\) −2.47214 −0.0999302
\(613\) 0.875388 0.0353566 0.0176783 0.999844i \(-0.494373\pi\)
0.0176783 + 0.999844i \(0.494373\pi\)
\(614\) −11.4164 −0.460729
\(615\) −16.1803 −0.652454
\(616\) 0 0
\(617\) −43.8885 −1.76689 −0.883443 0.468538i \(-0.844781\pi\)
−0.883443 + 0.468538i \(0.844781\pi\)
\(618\) 31.7082 1.27549
\(619\) −47.5410 −1.91083 −0.955417 0.295258i \(-0.904594\pi\)
−0.955417 + 0.295258i \(0.904594\pi\)
\(620\) 5.23607 0.210286
\(621\) −8.41641 −0.337739
\(622\) 27.1246 1.08760
\(623\) −17.2361 −0.690548
\(624\) −2.23607 −0.0895144
\(625\) 1.00000 0.0400000
\(626\) −19.5279 −0.780490
\(627\) 0 0
\(628\) 8.52786 0.340299
\(629\) −11.0557 −0.440821
\(630\) 2.00000 0.0796819
\(631\) −16.3607 −0.651308 −0.325654 0.945489i \(-0.605585\pi\)
−0.325654 + 0.945489i \(0.605585\pi\)
\(632\) −10.2361 −0.407169
\(633\) −7.88854 −0.313541
\(634\) −0.652476 −0.0259131
\(635\) −6.23607 −0.247471
\(636\) 12.7639 0.506123
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −3.05573 −0.120883
\(640\) 1.00000 0.0395285
\(641\) −23.8328 −0.941340 −0.470670 0.882309i \(-0.655988\pi\)
−0.470670 + 0.882309i \(0.655988\pi\)
\(642\) −5.52786 −0.218167
\(643\) 6.11146 0.241012 0.120506 0.992713i \(-0.461548\pi\)
0.120506 + 0.992713i \(0.461548\pi\)
\(644\) −3.76393 −0.148320
\(645\) 7.23607 0.284920
\(646\) −2.76393 −0.108745
\(647\) −9.23607 −0.363107 −0.181554 0.983381i \(-0.558113\pi\)
−0.181554 + 0.983381i \(0.558113\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) −1.00000 −0.0392232
\(651\) 11.7082 0.458881
\(652\) −6.00000 −0.234978
\(653\) 26.2918 1.02888 0.514439 0.857527i \(-0.328000\pi\)
0.514439 + 0.857527i \(0.328000\pi\)
\(654\) −8.29180 −0.324235
\(655\) 3.18034 0.124266
\(656\) −7.23607 −0.282521
\(657\) −14.4721 −0.564612
\(658\) −7.70820 −0.300497
\(659\) −20.7639 −0.808848 −0.404424 0.914572i \(-0.632528\pi\)
−0.404424 + 0.914572i \(0.632528\pi\)
\(660\) 0 0
\(661\) 4.41641 0.171778 0.0858892 0.996305i \(-0.472627\pi\)
0.0858892 + 0.996305i \(0.472627\pi\)
\(662\) 21.7082 0.843713
\(663\) 2.76393 0.107342
\(664\) 0.236068 0.00916121
\(665\) 2.23607 0.0867110
\(666\) −17.8885 −0.693167
\(667\) −16.8328 −0.651769
\(668\) 1.70820 0.0660924
\(669\) 18.2918 0.707202
\(670\) −7.23607 −0.279554
\(671\) 0 0
\(672\) 2.23607 0.0862582
\(673\) −10.0557 −0.387620 −0.193810 0.981039i \(-0.562085\pi\)
−0.193810 + 0.981039i \(0.562085\pi\)
\(674\) 2.88854 0.111263
\(675\) 2.23607 0.0860663
\(676\) −12.0000 −0.461538
\(677\) −45.8328 −1.76150 −0.880749 0.473583i \(-0.842960\pi\)
−0.880749 + 0.473583i \(0.842960\pi\)
\(678\) −2.23607 −0.0858757
\(679\) 3.52786 0.135387
\(680\) −1.23607 −0.0474010
\(681\) 16.5836 0.635485
\(682\) 0 0
\(683\) −6.36068 −0.243385 −0.121692 0.992568i \(-0.538832\pi\)
−0.121692 + 0.992568i \(0.538832\pi\)
\(684\) −4.47214 −0.170996
\(685\) −13.4721 −0.514744
\(686\) −1.00000 −0.0381802
\(687\) −15.5279 −0.592425
\(688\) 3.23607 0.123374
\(689\) −5.70820 −0.217465
\(690\) 8.41641 0.320407
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 19.8885 0.756049
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) −4.23607 −0.160683
\(696\) 10.0000 0.379049
\(697\) 8.94427 0.338788
\(698\) −0.416408 −0.0157613
\(699\) −23.2918 −0.880977
\(700\) 1.00000 0.0377964
\(701\) −17.5967 −0.664620 −0.332310 0.943170i \(-0.607828\pi\)
−0.332310 + 0.943170i \(0.607828\pi\)
\(702\) −2.23607 −0.0843949
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 17.2361 0.649148
\(706\) 4.76393 0.179293
\(707\) −1.00000 −0.0376089
\(708\) −10.5279 −0.395661
\(709\) 19.0557 0.715653 0.357826 0.933788i \(-0.383518\pi\)
0.357826 + 0.933788i \(0.383518\pi\)
\(710\) −1.52786 −0.0573397
\(711\) 20.4721 0.767765
\(712\) 17.2361 0.645949
\(713\) 19.7082 0.738078
\(714\) −2.76393 −0.103438
\(715\) 0 0
\(716\) −11.2361 −0.419912
\(717\) −23.9443 −0.894215
\(718\) −16.0000 −0.597115
\(719\) 6.83282 0.254821 0.127411 0.991850i \(-0.459333\pi\)
0.127411 + 0.991850i \(0.459333\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 14.1803 0.528103
\(722\) 14.0000 0.521026
\(723\) 50.0000 1.85952
\(724\) 7.00000 0.260153
\(725\) 4.47214 0.166091
\(726\) 0 0
\(727\) −39.3050 −1.45774 −0.728870 0.684652i \(-0.759954\pi\)
−0.728870 + 0.684652i \(0.759954\pi\)
\(728\) −1.00000 −0.0370625
\(729\) −7.00000 −0.259259
\(730\) −7.23607 −0.267819
\(731\) −4.00000 −0.147945
\(732\) 7.88854 0.291569
\(733\) −48.4164 −1.78830 −0.894150 0.447767i \(-0.852220\pi\)
−0.894150 + 0.447767i \(0.852220\pi\)
\(734\) −8.18034 −0.301942
\(735\) 2.23607 0.0824786
\(736\) 3.76393 0.138740
\(737\) 0 0
\(738\) 14.4721 0.532727
\(739\) 18.0689 0.664675 0.332337 0.943161i \(-0.392163\pi\)
0.332337 + 0.943161i \(0.392163\pi\)
\(740\) −8.94427 −0.328798
\(741\) 5.00000 0.183680
\(742\) 5.70820 0.209555
\(743\) 28.0000 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) −11.7082 −0.429244
\(745\) −4.00000 −0.146549
\(746\) 5.52786 0.202389
\(747\) −0.472136 −0.0172746
\(748\) 0 0
\(749\) −2.47214 −0.0903299
\(750\) −2.23607 −0.0816497
\(751\) −11.1803 −0.407976 −0.203988 0.978973i \(-0.565390\pi\)
−0.203988 + 0.978973i \(0.565390\pi\)
\(752\) 7.70820 0.281089
\(753\) 66.8328 2.43552
\(754\) −4.47214 −0.162866
\(755\) −12.2361 −0.445316
\(756\) 2.23607 0.0813250
\(757\) −42.6525 −1.55023 −0.775115 0.631820i \(-0.782308\pi\)
−0.775115 + 0.631820i \(0.782308\pi\)
\(758\) −11.4164 −0.414663
\(759\) 0 0
\(760\) −2.23607 −0.0811107
\(761\) −30.5410 −1.10711 −0.553556 0.832812i \(-0.686729\pi\)
−0.553556 + 0.832812i \(0.686729\pi\)
\(762\) 13.9443 0.505148
\(763\) −3.70820 −0.134246
\(764\) 25.1803 0.910993
\(765\) 2.47214 0.0893803
\(766\) 17.2361 0.622764
\(767\) 4.70820 0.170003
\(768\) −2.23607 −0.0806872
\(769\) −12.0000 −0.432731 −0.216366 0.976312i \(-0.569420\pi\)
−0.216366 + 0.976312i \(0.569420\pi\)
\(770\) 0 0
\(771\) 10.6525 0.383640
\(772\) 7.94427 0.285921
\(773\) −27.9443 −1.00509 −0.502543 0.864552i \(-0.667602\pi\)
−0.502543 + 0.864552i \(0.667602\pi\)
\(774\) −6.47214 −0.232636
\(775\) −5.23607 −0.188085
\(776\) −3.52786 −0.126643
\(777\) −20.0000 −0.717496
\(778\) 18.2918 0.655793
\(779\) 16.1803 0.579721
\(780\) 2.23607 0.0800641
\(781\) 0 0
\(782\) −4.65248 −0.166372
\(783\) 10.0000 0.357371
\(784\) 1.00000 0.0357143
\(785\) −8.52786 −0.304373
\(786\) −7.11146 −0.253657
\(787\) −31.7771 −1.13273 −0.566365 0.824154i \(-0.691651\pi\)
−0.566365 + 0.824154i \(0.691651\pi\)
\(788\) −7.52786 −0.268169
\(789\) −17.1115 −0.609184
\(790\) 10.2361 0.364183
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) −3.52786 −0.125278
\(794\) −6.00000 −0.212932
\(795\) −12.7639 −0.452690
\(796\) −4.47214 −0.158511
\(797\) 28.8885 1.02328 0.511642 0.859199i \(-0.329037\pi\)
0.511642 + 0.859199i \(0.329037\pi\)
\(798\) −5.00000 −0.176998
\(799\) −9.52786 −0.337072
\(800\) −1.00000 −0.0353553
\(801\) −34.4721 −1.21801
\(802\) 28.8328 1.01812
\(803\) 0 0
\(804\) 16.1803 0.570637
\(805\) 3.76393 0.132661
\(806\) 5.23607 0.184433
\(807\) 62.2361 2.19081
\(808\) 1.00000 0.0351799
\(809\) −1.63932 −0.0576354 −0.0288177 0.999585i \(-0.509174\pi\)
−0.0288177 + 0.999585i \(0.509174\pi\)
\(810\) −11.0000 −0.386501
\(811\) 2.47214 0.0868084 0.0434042 0.999058i \(-0.486180\pi\)
0.0434042 + 0.999058i \(0.486180\pi\)
\(812\) 4.47214 0.156941
\(813\) −16.5836 −0.581612
\(814\) 0 0
\(815\) 6.00000 0.210171
\(816\) 2.76393 0.0967570
\(817\) −7.23607 −0.253158
\(818\) −11.5279 −0.403062
\(819\) 2.00000 0.0698857
\(820\) 7.23607 0.252694
\(821\) 13.0557 0.455648 0.227824 0.973702i \(-0.426839\pi\)
0.227824 + 0.973702i \(0.426839\pi\)
\(822\) 30.1246 1.05072
\(823\) 16.3607 0.570297 0.285149 0.958483i \(-0.407957\pi\)
0.285149 + 0.958483i \(0.407957\pi\)
\(824\) −14.1803 −0.493996
\(825\) 0 0
\(826\) −4.70820 −0.163819
\(827\) 12.1803 0.423552 0.211776 0.977318i \(-0.432075\pi\)
0.211776 + 0.977318i \(0.432075\pi\)
\(828\) −7.52786 −0.261611
\(829\) −11.9443 −0.414842 −0.207421 0.978252i \(-0.566507\pi\)
−0.207421 + 0.978252i \(0.566507\pi\)
\(830\) −0.236068 −0.00819404
\(831\) −52.7639 −1.83036
\(832\) 1.00000 0.0346688
\(833\) −1.23607 −0.0428272
\(834\) 9.47214 0.327993
\(835\) −1.70820 −0.0591148
\(836\) 0 0
\(837\) −11.7082 −0.404695
\(838\) 28.2361 0.975399
\(839\) −13.5967 −0.469412 −0.234706 0.972066i \(-0.575413\pi\)
−0.234706 + 0.972066i \(0.575413\pi\)
\(840\) −2.23607 −0.0771517
\(841\) −9.00000 −0.310345
\(842\) 3.05573 0.105307
\(843\) −57.7639 −1.98950
\(844\) 3.52786 0.121434
\(845\) 12.0000 0.412813
\(846\) −15.4164 −0.530027
\(847\) 0 0
\(848\) −5.70820 −0.196021
\(849\) −58.6656 −2.01340
\(850\) 1.23607 0.0423968
\(851\) −33.6656 −1.15404
\(852\) 3.41641 0.117044
\(853\) 3.47214 0.118884 0.0594418 0.998232i \(-0.481068\pi\)
0.0594418 + 0.998232i \(0.481068\pi\)
\(854\) 3.52786 0.120721
\(855\) 4.47214 0.152944
\(856\) 2.47214 0.0844959
\(857\) 10.8328 0.370042 0.185021 0.982735i \(-0.440765\pi\)
0.185021 + 0.982735i \(0.440765\pi\)
\(858\) 0 0
\(859\) 0.944272 0.0322181 0.0161091 0.999870i \(-0.494872\pi\)
0.0161091 + 0.999870i \(0.494872\pi\)
\(860\) −3.23607 −0.110349
\(861\) 16.1803 0.551425
\(862\) 33.6525 1.14621
\(863\) −56.7214 −1.93082 −0.965409 0.260741i \(-0.916033\pi\)
−0.965409 + 0.260741i \(0.916033\pi\)
\(864\) −2.23607 −0.0760726
\(865\) −19.8885 −0.676231
\(866\) 12.6525 0.429949
\(867\) 34.5967 1.17497
\(868\) −5.23607 −0.177724
\(869\) 0 0
\(870\) −10.0000 −0.339032
\(871\) −7.23607 −0.245185
\(872\) 3.70820 0.125576
\(873\) 7.05573 0.238800
\(874\) −8.41641 −0.284689
\(875\) −1.00000 −0.0338062
\(876\) 16.1803 0.546683
\(877\) −51.9574 −1.75448 −0.877239 0.480054i \(-0.840617\pi\)
−0.877239 + 0.480054i \(0.840617\pi\)
\(878\) −15.1246 −0.510431
\(879\) 11.1803 0.377104
\(880\) 0 0
\(881\) −4.94427 −0.166577 −0.0832884 0.996525i \(-0.526542\pi\)
−0.0832884 + 0.996525i \(0.526542\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −22.4721 −0.756248 −0.378124 0.925755i \(-0.623431\pi\)
−0.378124 + 0.925755i \(0.623431\pi\)
\(884\) −1.23607 −0.0415735
\(885\) 10.5279 0.353890
\(886\) 24.3607 0.818413
\(887\) 5.23607 0.175810 0.0879050 0.996129i \(-0.471983\pi\)
0.0879050 + 0.996129i \(0.471983\pi\)
\(888\) 20.0000 0.671156
\(889\) 6.23607 0.209151
\(890\) −17.2361 −0.577754
\(891\) 0 0
\(892\) −8.18034 −0.273898
\(893\) −17.2361 −0.576783
\(894\) 8.94427 0.299141
\(895\) 11.2361 0.375580
\(896\) −1.00000 −0.0334077
\(897\) 8.41641 0.281016
\(898\) −23.8328 −0.795311
\(899\) −23.4164 −0.780981
\(900\) 2.00000 0.0666667
\(901\) 7.05573 0.235060
\(902\) 0 0
\(903\) −7.23607 −0.240801
\(904\) 1.00000 0.0332595
\(905\) −7.00000 −0.232688
\(906\) 27.3607 0.908998
\(907\) 40.3607 1.34015 0.670077 0.742291i \(-0.266261\pi\)
0.670077 + 0.742291i \(0.266261\pi\)
\(908\) −7.41641 −0.246122
\(909\) −2.00000 −0.0663358
\(910\) 1.00000 0.0331497
\(911\) −43.6525 −1.44627 −0.723136 0.690706i \(-0.757300\pi\)
−0.723136 + 0.690706i \(0.757300\pi\)
\(912\) 5.00000 0.165567
\(913\) 0 0
\(914\) 36.4164 1.20455
\(915\) −7.88854 −0.260787
\(916\) 6.94427 0.229445
\(917\) −3.18034 −0.105024
\(918\) 2.76393 0.0912234
\(919\) 12.9443 0.426992 0.213496 0.976944i \(-0.431515\pi\)
0.213496 + 0.976944i \(0.431515\pi\)
\(920\) −3.76393 −0.124093
\(921\) −25.5279 −0.841172
\(922\) 23.8885 0.786727
\(923\) −1.52786 −0.0502903
\(924\) 0 0
\(925\) 8.94427 0.294086
\(926\) −10.7082 −0.351893
\(927\) 28.3607 0.931487
\(928\) −4.47214 −0.146805
\(929\) −42.8328 −1.40530 −0.702650 0.711536i \(-0.748000\pi\)
−0.702650 + 0.711536i \(0.748000\pi\)
\(930\) 11.7082 0.383927
\(931\) −2.23607 −0.0732842
\(932\) 10.4164 0.341201
\(933\) 60.6525 1.98567
\(934\) 2.81966 0.0922621
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 20.4721 0.668796 0.334398 0.942432i \(-0.391467\pi\)
0.334398 + 0.942432i \(0.391467\pi\)
\(938\) 7.23607 0.236266
\(939\) −43.6656 −1.42497
\(940\) −7.70820 −0.251414
\(941\) −55.8885 −1.82191 −0.910957 0.412501i \(-0.864655\pi\)
−0.910957 + 0.412501i \(0.864655\pi\)
\(942\) 19.0689 0.621298
\(943\) 27.2361 0.886928
\(944\) 4.70820 0.153239
\(945\) −2.23607 −0.0727393
\(946\) 0 0
\(947\) 10.0689 0.327195 0.163597 0.986527i \(-0.447690\pi\)
0.163597 + 0.986527i \(0.447690\pi\)
\(948\) −22.8885 −0.743385
\(949\) −7.23607 −0.234893
\(950\) 2.23607 0.0725476
\(951\) −1.45898 −0.0473107
\(952\) 1.23607 0.0400612
\(953\) 24.8885 0.806219 0.403110 0.915152i \(-0.367929\pi\)
0.403110 + 0.915152i \(0.367929\pi\)
\(954\) 11.4164 0.369620
\(955\) −25.1803 −0.814817
\(956\) 10.7082 0.346328
\(957\) 0 0
\(958\) 27.5967 0.891610
\(959\) 13.4721 0.435038
\(960\) 2.23607 0.0721688
\(961\) −3.58359 −0.115600
\(962\) −8.94427 −0.288375
\(963\) −4.94427 −0.159327
\(964\) −22.3607 −0.720189
\(965\) −7.94427 −0.255735
\(966\) −8.41641 −0.270793
\(967\) 55.1935 1.77490 0.887452 0.460901i \(-0.152474\pi\)
0.887452 + 0.460901i \(0.152474\pi\)
\(968\) 0 0
\(969\) −6.18034 −0.198541
\(970\) 3.52786 0.113273
\(971\) −16.1246 −0.517463 −0.258732 0.965949i \(-0.583305\pi\)
−0.258732 + 0.965949i \(0.583305\pi\)
\(972\) 17.8885 0.573775
\(973\) 4.23607 0.135802
\(974\) −2.23607 −0.0716482
\(975\) −2.23607 −0.0716115
\(976\) −3.52786 −0.112924
\(977\) −45.9443 −1.46989 −0.734944 0.678128i \(-0.762791\pi\)
−0.734944 + 0.678128i \(0.762791\pi\)
\(978\) −13.4164 −0.429009
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −7.41641 −0.236788
\(982\) −31.2361 −0.996783
\(983\) 43.1935 1.37766 0.688829 0.724924i \(-0.258125\pi\)
0.688829 + 0.724924i \(0.258125\pi\)
\(984\) −16.1803 −0.515810
\(985\) 7.52786 0.239858
\(986\) 5.52786 0.176043
\(987\) −17.2361 −0.548630
\(988\) −2.23607 −0.0711388
\(989\) −12.1803 −0.387312
\(990\) 0 0
\(991\) 19.2918 0.612824 0.306412 0.951899i \(-0.400871\pi\)
0.306412 + 0.951899i \(0.400871\pi\)
\(992\) 5.23607 0.166245
\(993\) 48.5410 1.54040
\(994\) 1.52786 0.0484609
\(995\) 4.47214 0.141776
\(996\) 0.527864 0.0167260
\(997\) −59.8328 −1.89492 −0.947462 0.319868i \(-0.896361\pi\)
−0.947462 + 0.319868i \(0.896361\pi\)
\(998\) −26.8328 −0.849378
\(999\) 20.0000 0.632772
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.bn.1.1 2
11.10 odd 2 8470.2.a.by.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.bn.1.1 2 1.1 even 1 trivial
8470.2.a.by.1.1 yes 2 11.10 odd 2