Properties

Label 4560.2.a.bt.1.1
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -4.38776 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -4.38776 q^{7} +1.00000 q^{9} -6.38776 q^{11} -1.13536 q^{13} -1.00000 q^{15} +1.72928 q^{17} -1.00000 q^{19} -4.38776 q^{21} -1.52311 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.65847 q^{29} +3.72928 q^{31} -6.38776 q^{33} +4.38776 q^{35} -4.59392 q^{37} -1.13536 q^{39} -9.64015 q^{41} -2.59392 q^{43} -1.00000 q^{45} +9.52311 q^{47} +12.2524 q^{49} +1.72928 q^{51} +11.5231 q^{53} +6.38776 q^{55} -1.00000 q^{57} -14.9817 q^{59} -1.79383 q^{61} -4.38776 q^{63} +1.13536 q^{65} +7.04623 q^{67} -1.52311 q^{69} +9.31695 q^{71} -8.50479 q^{73} +1.00000 q^{75} +28.0279 q^{77} +9.25240 q^{79} +1.00000 q^{81} -9.04623 q^{83} -1.72928 q^{85} +6.65847 q^{87} +10.5939 q^{89} +4.98168 q^{91} +3.72928 q^{93} +1.00000 q^{95} -4.59392 q^{97} -6.38776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} - 3 q^{15} - 3 q^{19} + 2 q^{21} + 8 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{29} + 6 q^{31} - 4 q^{33} - 2 q^{35} - 6 q^{37} - 6 q^{39} + 4 q^{41} - 3 q^{45} + 16 q^{47} + 19 q^{49} + 22 q^{53} + 4 q^{55} - 3 q^{57} - 22 q^{59} + 2 q^{61} + 2 q^{63} + 6 q^{65} - 4 q^{67} + 8 q^{69} + 8 q^{71} + 10 q^{73} + 3 q^{75} + 36 q^{77} + 10 q^{79} + 3 q^{81} - 2 q^{83} + 10 q^{87} + 24 q^{89} - 8 q^{91} + 6 q^{93} + 3 q^{95} - 6 q^{97} - 4 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.38776 −1.65842 −0.829208 0.558940i \(-0.811208\pi\)
−0.829208 + 0.558940i \(0.811208\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.38776 −1.92598 −0.962990 0.269536i \(-0.913130\pi\)
−0.962990 + 0.269536i \(0.913130\pi\)
\(12\) 0 0
\(13\) −1.13536 −0.314892 −0.157446 0.987528i \(-0.550326\pi\)
−0.157446 + 0.987528i \(0.550326\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.72928 0.419412 0.209706 0.977764i \(-0.432749\pi\)
0.209706 + 0.977764i \(0.432749\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.38776 −0.957487
\(22\) 0 0
\(23\) −1.52311 −0.317591 −0.158796 0.987311i \(-0.550761\pi\)
−0.158796 + 0.987311i \(0.550761\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.65847 1.23645 0.618224 0.786002i \(-0.287853\pi\)
0.618224 + 0.786002i \(0.287853\pi\)
\(30\) 0 0
\(31\) 3.72928 0.669799 0.334899 0.942254i \(-0.391298\pi\)
0.334899 + 0.942254i \(0.391298\pi\)
\(32\) 0 0
\(33\) −6.38776 −1.11197
\(34\) 0 0
\(35\) 4.38776 0.741666
\(36\) 0 0
\(37\) −4.59392 −0.755236 −0.377618 0.925961i \(-0.623257\pi\)
−0.377618 + 0.925961i \(0.623257\pi\)
\(38\) 0 0
\(39\) −1.13536 −0.181803
\(40\) 0 0
\(41\) −9.64015 −1.50554 −0.752769 0.658284i \(-0.771282\pi\)
−0.752769 + 0.658284i \(0.771282\pi\)
\(42\) 0 0
\(43\) −2.59392 −0.395569 −0.197785 0.980245i \(-0.563375\pi\)
−0.197785 + 0.980245i \(0.563375\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 9.52311 1.38909 0.694544 0.719450i \(-0.255606\pi\)
0.694544 + 0.719450i \(0.255606\pi\)
\(48\) 0 0
\(49\) 12.2524 1.75034
\(50\) 0 0
\(51\) 1.72928 0.242148
\(52\) 0 0
\(53\) 11.5231 1.58282 0.791411 0.611285i \(-0.209347\pi\)
0.791411 + 0.611285i \(0.209347\pi\)
\(54\) 0 0
\(55\) 6.38776 0.861325
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −14.9817 −1.95045 −0.975224 0.221219i \(-0.928996\pi\)
−0.975224 + 0.221219i \(0.928996\pi\)
\(60\) 0 0
\(61\) −1.79383 −0.229677 −0.114838 0.993384i \(-0.536635\pi\)
−0.114838 + 0.993384i \(0.536635\pi\)
\(62\) 0 0
\(63\) −4.38776 −0.552805
\(64\) 0 0
\(65\) 1.13536 0.140824
\(66\) 0 0
\(67\) 7.04623 0.860834 0.430417 0.902630i \(-0.358367\pi\)
0.430417 + 0.902630i \(0.358367\pi\)
\(68\) 0 0
\(69\) −1.52311 −0.183361
\(70\) 0 0
\(71\) 9.31695 1.10572 0.552859 0.833275i \(-0.313537\pi\)
0.552859 + 0.833275i \(0.313537\pi\)
\(72\) 0 0
\(73\) −8.50479 −0.995411 −0.497705 0.867346i \(-0.665824\pi\)
−0.497705 + 0.867346i \(0.665824\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 28.0279 3.19408
\(78\) 0 0
\(79\) 9.25240 1.04098 0.520488 0.853869i \(-0.325750\pi\)
0.520488 + 0.853869i \(0.325750\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.04623 −0.992953 −0.496476 0.868050i \(-0.665373\pi\)
−0.496476 + 0.868050i \(0.665373\pi\)
\(84\) 0 0
\(85\) −1.72928 −0.187567
\(86\) 0 0
\(87\) 6.65847 0.713863
\(88\) 0 0
\(89\) 10.5939 1.12295 0.561477 0.827492i \(-0.310233\pi\)
0.561477 + 0.827492i \(0.310233\pi\)
\(90\) 0 0
\(91\) 4.98168 0.522222
\(92\) 0 0
\(93\) 3.72928 0.386709
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −4.59392 −0.466442 −0.233221 0.972424i \(-0.574927\pi\)
−0.233221 + 0.972424i \(0.574927\pi\)
\(98\) 0 0
\(99\) −6.38776 −0.641994
\(100\) 0 0
\(101\) 1.22449 0.121841 0.0609206 0.998143i \(-0.480596\pi\)
0.0609206 + 0.998143i \(0.480596\pi\)
\(102\) 0 0
\(103\) 7.04623 0.694286 0.347143 0.937812i \(-0.387152\pi\)
0.347143 + 0.937812i \(0.387152\pi\)
\(104\) 0 0
\(105\) 4.38776 0.428201
\(106\) 0 0
\(107\) 3.45856 0.334352 0.167176 0.985927i \(-0.446535\pi\)
0.167176 + 0.985927i \(0.446535\pi\)
\(108\) 0 0
\(109\) 12.9817 1.24342 0.621710 0.783248i \(-0.286438\pi\)
0.621710 + 0.783248i \(0.286438\pi\)
\(110\) 0 0
\(111\) −4.59392 −0.436036
\(112\) 0 0
\(113\) 13.7938 1.29761 0.648807 0.760953i \(-0.275268\pi\)
0.648807 + 0.760953i \(0.275268\pi\)
\(114\) 0 0
\(115\) 1.52311 0.142031
\(116\) 0 0
\(117\) −1.13536 −0.104964
\(118\) 0 0
\(119\) −7.58767 −0.695560
\(120\) 0 0
\(121\) 29.8034 2.70940
\(122\) 0 0
\(123\) −9.64015 −0.869223
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.04623 −0.625252 −0.312626 0.949876i \(-0.601209\pi\)
−0.312626 + 0.949876i \(0.601209\pi\)
\(128\) 0 0
\(129\) −2.59392 −0.228382
\(130\) 0 0
\(131\) 0.117037 0.0102256 0.00511279 0.999987i \(-0.498373\pi\)
0.00511279 + 0.999987i \(0.498373\pi\)
\(132\) 0 0
\(133\) 4.38776 0.380467
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 11.2524 0.961357 0.480679 0.876897i \(-0.340390\pi\)
0.480679 + 0.876897i \(0.340390\pi\)
\(138\) 0 0
\(139\) −13.3169 −1.12953 −0.564764 0.825252i \(-0.691033\pi\)
−0.564764 + 0.825252i \(0.691033\pi\)
\(140\) 0 0
\(141\) 9.52311 0.801991
\(142\) 0 0
\(143\) 7.25240 0.606476
\(144\) 0 0
\(145\) −6.65847 −0.552956
\(146\) 0 0
\(147\) 12.2524 1.01056
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −10.2341 −0.832837 −0.416419 0.909173i \(-0.636715\pi\)
−0.416419 + 0.909173i \(0.636715\pi\)
\(152\) 0 0
\(153\) 1.72928 0.139804
\(154\) 0 0
\(155\) −3.72928 −0.299543
\(156\) 0 0
\(157\) 12.0279 0.959931 0.479966 0.877287i \(-0.340649\pi\)
0.479966 + 0.877287i \(0.340649\pi\)
\(158\) 0 0
\(159\) 11.5231 0.913842
\(160\) 0 0
\(161\) 6.68305 0.526698
\(162\) 0 0
\(163\) 14.4157 1.12912 0.564561 0.825391i \(-0.309046\pi\)
0.564561 + 0.825391i \(0.309046\pi\)
\(164\) 0 0
\(165\) 6.38776 0.497286
\(166\) 0 0
\(167\) 3.79383 0.293576 0.146788 0.989168i \(-0.453107\pi\)
0.146788 + 0.989168i \(0.453107\pi\)
\(168\) 0 0
\(169\) −11.7110 −0.900843
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −12.2986 −0.935047 −0.467524 0.883981i \(-0.654854\pi\)
−0.467524 + 0.883981i \(0.654854\pi\)
\(174\) 0 0
\(175\) −4.38776 −0.331683
\(176\) 0 0
\(177\) −14.9817 −1.12609
\(178\) 0 0
\(179\) 2.02791 0.151573 0.0757864 0.997124i \(-0.475853\pi\)
0.0757864 + 0.997124i \(0.475853\pi\)
\(180\) 0 0
\(181\) −17.5231 −1.30248 −0.651241 0.758871i \(-0.725751\pi\)
−0.651241 + 0.758871i \(0.725751\pi\)
\(182\) 0 0
\(183\) −1.79383 −0.132604
\(184\) 0 0
\(185\) 4.59392 0.337752
\(186\) 0 0
\(187\) −11.0462 −0.807780
\(188\) 0 0
\(189\) −4.38776 −0.319162
\(190\) 0 0
\(191\) 10.3878 0.751632 0.375816 0.926694i \(-0.377363\pi\)
0.375816 + 0.926694i \(0.377363\pi\)
\(192\) 0 0
\(193\) 1.13536 0.0817249 0.0408625 0.999165i \(-0.486989\pi\)
0.0408625 + 0.999165i \(0.486989\pi\)
\(194\) 0 0
\(195\) 1.13536 0.0813048
\(196\) 0 0
\(197\) 14.5693 1.03802 0.519011 0.854767i \(-0.326300\pi\)
0.519011 + 0.854767i \(0.326300\pi\)
\(198\) 0 0
\(199\) −9.18785 −0.651309 −0.325655 0.945489i \(-0.605585\pi\)
−0.325655 + 0.945489i \(0.605585\pi\)
\(200\) 0 0
\(201\) 7.04623 0.497003
\(202\) 0 0
\(203\) −29.2158 −2.05054
\(204\) 0 0
\(205\) 9.64015 0.673297
\(206\) 0 0
\(207\) −1.52311 −0.105864
\(208\) 0 0
\(209\) 6.38776 0.441850
\(210\) 0 0
\(211\) 14.5048 0.998551 0.499276 0.866443i \(-0.333600\pi\)
0.499276 + 0.866443i \(0.333600\pi\)
\(212\) 0 0
\(213\) 9.31695 0.638387
\(214\) 0 0
\(215\) 2.59392 0.176904
\(216\) 0 0
\(217\) −16.3632 −1.11080
\(218\) 0 0
\(219\) −8.50479 −0.574701
\(220\) 0 0
\(221\) −1.96336 −0.132070
\(222\) 0 0
\(223\) 19.4586 1.30304 0.651521 0.758631i \(-0.274131\pi\)
0.651521 + 0.758631i \(0.274131\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 11.2524 0.746848 0.373424 0.927661i \(-0.378184\pi\)
0.373424 + 0.927661i \(0.378184\pi\)
\(228\) 0 0
\(229\) 10.2062 0.674443 0.337221 0.941425i \(-0.390513\pi\)
0.337221 + 0.941425i \(0.390513\pi\)
\(230\) 0 0
\(231\) 28.0279 1.84410
\(232\) 0 0
\(233\) −29.3449 −1.92245 −0.961223 0.275774i \(-0.911066\pi\)
−0.961223 + 0.275774i \(0.911066\pi\)
\(234\) 0 0
\(235\) −9.52311 −0.621219
\(236\) 0 0
\(237\) 9.25240 0.601008
\(238\) 0 0
\(239\) −20.1170 −1.30126 −0.650631 0.759394i \(-0.725496\pi\)
−0.650631 + 0.759394i \(0.725496\pi\)
\(240\) 0 0
\(241\) 3.55102 0.228741 0.114371 0.993438i \(-0.463515\pi\)
0.114371 + 0.993438i \(0.463515\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −12.2524 −0.782777
\(246\) 0 0
\(247\) 1.13536 0.0722412
\(248\) 0 0
\(249\) −9.04623 −0.573281
\(250\) 0 0
\(251\) 0.836734 0.0528142 0.0264071 0.999651i \(-0.491593\pi\)
0.0264071 + 0.999651i \(0.491593\pi\)
\(252\) 0 0
\(253\) 9.72928 0.611675
\(254\) 0 0
\(255\) −1.72928 −0.108292
\(256\) 0 0
\(257\) 16.2986 1.01668 0.508340 0.861156i \(-0.330259\pi\)
0.508340 + 0.861156i \(0.330259\pi\)
\(258\) 0 0
\(259\) 20.1570 1.25250
\(260\) 0 0
\(261\) 6.65847 0.412149
\(262\) 0 0
\(263\) 0.206167 0.0127128 0.00635641 0.999980i \(-0.497977\pi\)
0.00635641 + 0.999980i \(0.497977\pi\)
\(264\) 0 0
\(265\) −11.5231 −0.707859
\(266\) 0 0
\(267\) 10.5939 0.648338
\(268\) 0 0
\(269\) −24.7509 −1.50909 −0.754545 0.656248i \(-0.772143\pi\)
−0.754545 + 0.656248i \(0.772143\pi\)
\(270\) 0 0
\(271\) −18.3265 −1.11326 −0.556629 0.830761i \(-0.687905\pi\)
−0.556629 + 0.830761i \(0.687905\pi\)
\(272\) 0 0
\(273\) 4.98168 0.301505
\(274\) 0 0
\(275\) −6.38776 −0.385196
\(276\) 0 0
\(277\) 7.25240 0.435754 0.217877 0.975976i \(-0.430087\pi\)
0.217877 + 0.975976i \(0.430087\pi\)
\(278\) 0 0
\(279\) 3.72928 0.223266
\(280\) 0 0
\(281\) 17.4061 1.03836 0.519180 0.854665i \(-0.326238\pi\)
0.519180 + 0.854665i \(0.326238\pi\)
\(282\) 0 0
\(283\) 4.32320 0.256988 0.128494 0.991710i \(-0.458986\pi\)
0.128494 + 0.991710i \(0.458986\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 42.2986 2.49681
\(288\) 0 0
\(289\) −14.0096 −0.824093
\(290\) 0 0
\(291\) −4.59392 −0.269301
\(292\) 0 0
\(293\) 17.2524 1.00790 0.503948 0.863734i \(-0.331880\pi\)
0.503948 + 0.863734i \(0.331880\pi\)
\(294\) 0 0
\(295\) 14.9817 0.872267
\(296\) 0 0
\(297\) −6.38776 −0.370655
\(298\) 0 0
\(299\) 1.72928 0.100007
\(300\) 0 0
\(301\) 11.3815 0.656019
\(302\) 0 0
\(303\) 1.22449 0.0703451
\(304\) 0 0
\(305\) 1.79383 0.102715
\(306\) 0 0
\(307\) 5.31695 0.303454 0.151727 0.988422i \(-0.451517\pi\)
0.151727 + 0.988422i \(0.451517\pi\)
\(308\) 0 0
\(309\) 7.04623 0.400846
\(310\) 0 0
\(311\) 8.65847 0.490977 0.245488 0.969400i \(-0.421052\pi\)
0.245488 + 0.969400i \(0.421052\pi\)
\(312\) 0 0
\(313\) −24.7389 −1.39832 −0.699162 0.714964i \(-0.746443\pi\)
−0.699162 + 0.714964i \(0.746443\pi\)
\(314\) 0 0
\(315\) 4.38776 0.247222
\(316\) 0 0
\(317\) −18.0279 −1.01255 −0.506274 0.862373i \(-0.668978\pi\)
−0.506274 + 0.862373i \(0.668978\pi\)
\(318\) 0 0
\(319\) −42.5327 −2.38137
\(320\) 0 0
\(321\) 3.45856 0.193038
\(322\) 0 0
\(323\) −1.72928 −0.0962198
\(324\) 0 0
\(325\) −1.13536 −0.0629784
\(326\) 0 0
\(327\) 12.9817 0.717888
\(328\) 0 0
\(329\) −41.7851 −2.30369
\(330\) 0 0
\(331\) −12.9817 −0.713538 −0.356769 0.934193i \(-0.616122\pi\)
−0.356769 + 0.934193i \(0.616122\pi\)
\(332\) 0 0
\(333\) −4.59392 −0.251745
\(334\) 0 0
\(335\) −7.04623 −0.384977
\(336\) 0 0
\(337\) −14.8646 −0.809729 −0.404864 0.914377i \(-0.632681\pi\)
−0.404864 + 0.914377i \(0.632681\pi\)
\(338\) 0 0
\(339\) 13.7938 0.749178
\(340\) 0 0
\(341\) −23.8217 −1.29002
\(342\) 0 0
\(343\) −23.0462 −1.24438
\(344\) 0 0
\(345\) 1.52311 0.0820017
\(346\) 0 0
\(347\) 28.5048 1.53022 0.765109 0.643901i \(-0.222685\pi\)
0.765109 + 0.643901i \(0.222685\pi\)
\(348\) 0 0
\(349\) −24.5048 −1.31171 −0.655856 0.754886i \(-0.727692\pi\)
−0.655856 + 0.754886i \(0.727692\pi\)
\(350\) 0 0
\(351\) −1.13536 −0.0606010
\(352\) 0 0
\(353\) 0.206167 0.0109732 0.00548659 0.999985i \(-0.498254\pi\)
0.00548659 + 0.999985i \(0.498254\pi\)
\(354\) 0 0
\(355\) −9.31695 −0.494492
\(356\) 0 0
\(357\) −7.58767 −0.401582
\(358\) 0 0
\(359\) −21.6122 −1.14065 −0.570325 0.821419i \(-0.693183\pi\)
−0.570325 + 0.821419i \(0.693183\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 29.8034 1.56427
\(364\) 0 0
\(365\) 8.50479 0.445161
\(366\) 0 0
\(367\) −21.3973 −1.11693 −0.558466 0.829527i \(-0.688610\pi\)
−0.558466 + 0.829527i \(0.688610\pi\)
\(368\) 0 0
\(369\) −9.64015 −0.501846
\(370\) 0 0
\(371\) −50.5606 −2.62498
\(372\) 0 0
\(373\) −1.67680 −0.0868212 −0.0434106 0.999057i \(-0.513822\pi\)
−0.0434106 + 0.999057i \(0.513822\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −7.55976 −0.389347
\(378\) 0 0
\(379\) −7.01832 −0.360507 −0.180253 0.983620i \(-0.557692\pi\)
−0.180253 + 0.983620i \(0.557692\pi\)
\(380\) 0 0
\(381\) −7.04623 −0.360989
\(382\) 0 0
\(383\) 28.5972 1.46125 0.730626 0.682778i \(-0.239229\pi\)
0.730626 + 0.682778i \(0.239229\pi\)
\(384\) 0 0
\(385\) −28.0279 −1.42843
\(386\) 0 0
\(387\) −2.59392 −0.131856
\(388\) 0 0
\(389\) 24.6339 1.24899 0.624494 0.781030i \(-0.285305\pi\)
0.624494 + 0.781030i \(0.285305\pi\)
\(390\) 0 0
\(391\) −2.63389 −0.133202
\(392\) 0 0
\(393\) 0.117037 0.00590374
\(394\) 0 0
\(395\) −9.25240 −0.465539
\(396\) 0 0
\(397\) −10.8401 −0.544047 −0.272024 0.962291i \(-0.587693\pi\)
−0.272024 + 0.962291i \(0.587693\pi\)
\(398\) 0 0
\(399\) 4.38776 0.219663
\(400\) 0 0
\(401\) 16.6864 0.833278 0.416639 0.909072i \(-0.363208\pi\)
0.416639 + 0.909072i \(0.363208\pi\)
\(402\) 0 0
\(403\) −4.23407 −0.210914
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 29.3449 1.45457
\(408\) 0 0
\(409\) 24.3265 1.20287 0.601435 0.798922i \(-0.294596\pi\)
0.601435 + 0.798922i \(0.294596\pi\)
\(410\) 0 0
\(411\) 11.2524 0.555040
\(412\) 0 0
\(413\) 65.7359 3.23465
\(414\) 0 0
\(415\) 9.04623 0.444062
\(416\) 0 0
\(417\) −13.3169 −0.652134
\(418\) 0 0
\(419\) 13.6681 0.667728 0.333864 0.942621i \(-0.391647\pi\)
0.333864 + 0.942621i \(0.391647\pi\)
\(420\) 0 0
\(421\) −15.7938 −0.769744 −0.384872 0.922970i \(-0.625754\pi\)
−0.384872 + 0.922970i \(0.625754\pi\)
\(422\) 0 0
\(423\) 9.52311 0.463030
\(424\) 0 0
\(425\) 1.72928 0.0838825
\(426\) 0 0
\(427\) 7.87090 0.380899
\(428\) 0 0
\(429\) 7.25240 0.350149
\(430\) 0 0
\(431\) 30.7389 1.48064 0.740320 0.672255i \(-0.234674\pi\)
0.740320 + 0.672255i \(0.234674\pi\)
\(432\) 0 0
\(433\) 24.0525 1.15589 0.577944 0.816076i \(-0.303855\pi\)
0.577944 + 0.816076i \(0.303855\pi\)
\(434\) 0 0
\(435\) −6.65847 −0.319249
\(436\) 0 0
\(437\) 1.52311 0.0728604
\(438\) 0 0
\(439\) −9.38150 −0.447754 −0.223877 0.974617i \(-0.571871\pi\)
−0.223877 + 0.974617i \(0.571871\pi\)
\(440\) 0 0
\(441\) 12.2524 0.583447
\(442\) 0 0
\(443\) −36.1974 −1.71979 −0.859896 0.510469i \(-0.829472\pi\)
−0.859896 + 0.510469i \(0.829472\pi\)
\(444\) 0 0
\(445\) −10.5939 −0.502200
\(446\) 0 0
\(447\) 14.0000 0.662177
\(448\) 0 0
\(449\) 29.3328 1.38430 0.692150 0.721754i \(-0.256664\pi\)
0.692150 + 0.721754i \(0.256664\pi\)
\(450\) 0 0
\(451\) 61.5789 2.89964
\(452\) 0 0
\(453\) −10.2341 −0.480839
\(454\) 0 0
\(455\) −4.98168 −0.233545
\(456\) 0 0
\(457\) −32.1974 −1.50613 −0.753066 0.657945i \(-0.771426\pi\)
−0.753066 + 0.657945i \(0.771426\pi\)
\(458\) 0 0
\(459\) 1.72928 0.0807160
\(460\) 0 0
\(461\) −16.6339 −0.774718 −0.387359 0.921929i \(-0.626613\pi\)
−0.387359 + 0.921929i \(0.626613\pi\)
\(462\) 0 0
\(463\) 32.2095 1.49690 0.748451 0.663190i \(-0.230798\pi\)
0.748451 + 0.663190i \(0.230798\pi\)
\(464\) 0 0
\(465\) −3.72928 −0.172941
\(466\) 0 0
\(467\) −8.81215 −0.407778 −0.203889 0.978994i \(-0.565358\pi\)
−0.203889 + 0.978994i \(0.565358\pi\)
\(468\) 0 0
\(469\) −30.9171 −1.42762
\(470\) 0 0
\(471\) 12.0279 0.554217
\(472\) 0 0
\(473\) 16.5693 0.761859
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 11.5231 0.527607
\(478\) 0 0
\(479\) −1.30488 −0.0596216 −0.0298108 0.999556i \(-0.509490\pi\)
−0.0298108 + 0.999556i \(0.509490\pi\)
\(480\) 0 0
\(481\) 5.21575 0.237818
\(482\) 0 0
\(483\) 6.68305 0.304089
\(484\) 0 0
\(485\) 4.59392 0.208599
\(486\) 0 0
\(487\) 24.3632 1.10400 0.552000 0.833844i \(-0.313865\pi\)
0.552000 + 0.833844i \(0.313865\pi\)
\(488\) 0 0
\(489\) 14.4157 0.651899
\(490\) 0 0
\(491\) −39.7605 −1.79437 −0.897183 0.441658i \(-0.854390\pi\)
−0.897183 + 0.441658i \(0.854390\pi\)
\(492\) 0 0
\(493\) 11.5144 0.518581
\(494\) 0 0
\(495\) 6.38776 0.287108
\(496\) 0 0
\(497\) −40.8805 −1.83374
\(498\) 0 0
\(499\) −42.3265 −1.89480 −0.947398 0.320058i \(-0.896298\pi\)
−0.947398 + 0.320058i \(0.896298\pi\)
\(500\) 0 0
\(501\) 3.79383 0.169496
\(502\) 0 0
\(503\) 38.8959 1.73428 0.867141 0.498063i \(-0.165955\pi\)
0.867141 + 0.498063i \(0.165955\pi\)
\(504\) 0 0
\(505\) −1.22449 −0.0544891
\(506\) 0 0
\(507\) −11.7110 −0.520102
\(508\) 0 0
\(509\) 11.4340 0.506802 0.253401 0.967361i \(-0.418451\pi\)
0.253401 + 0.967361i \(0.418451\pi\)
\(510\) 0 0
\(511\) 37.3169 1.65080
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −7.04623 −0.310494
\(516\) 0 0
\(517\) −60.8313 −2.67536
\(518\) 0 0
\(519\) −12.2986 −0.539850
\(520\) 0 0
\(521\) 41.5910 1.82213 0.911067 0.412258i \(-0.135260\pi\)
0.911067 + 0.412258i \(0.135260\pi\)
\(522\) 0 0
\(523\) 32.8313 1.43561 0.717807 0.696242i \(-0.245146\pi\)
0.717807 + 0.696242i \(0.245146\pi\)
\(524\) 0 0
\(525\) −4.38776 −0.191497
\(526\) 0 0
\(527\) 6.44898 0.280922
\(528\) 0 0
\(529\) −20.6801 −0.899136
\(530\) 0 0
\(531\) −14.9817 −0.650149
\(532\) 0 0
\(533\) 10.9450 0.474082
\(534\) 0 0
\(535\) −3.45856 −0.149527
\(536\) 0 0
\(537\) 2.02791 0.0875106
\(538\) 0 0
\(539\) −78.2653 −3.37113
\(540\) 0 0
\(541\) −0.0924575 −0.00397506 −0.00198753 0.999998i \(-0.500633\pi\)
−0.00198753 + 0.999998i \(0.500633\pi\)
\(542\) 0 0
\(543\) −17.5231 −0.751989
\(544\) 0 0
\(545\) −12.9817 −0.556074
\(546\) 0 0
\(547\) −8.77551 −0.375214 −0.187607 0.982244i \(-0.560073\pi\)
−0.187607 + 0.982244i \(0.560073\pi\)
\(548\) 0 0
\(549\) −1.79383 −0.0765589
\(550\) 0 0
\(551\) −6.65847 −0.283661
\(552\) 0 0
\(553\) −40.5972 −1.72637
\(554\) 0 0
\(555\) 4.59392 0.195001
\(556\) 0 0
\(557\) −44.5606 −1.88809 −0.944047 0.329812i \(-0.893015\pi\)
−0.944047 + 0.329812i \(0.893015\pi\)
\(558\) 0 0
\(559\) 2.94503 0.124562
\(560\) 0 0
\(561\) −11.0462 −0.466372
\(562\) 0 0
\(563\) 36.6743 1.54564 0.772819 0.634626i \(-0.218846\pi\)
0.772819 + 0.634626i \(0.218846\pi\)
\(564\) 0 0
\(565\) −13.7938 −0.580311
\(566\) 0 0
\(567\) −4.38776 −0.184268
\(568\) 0 0
\(569\) −22.5939 −0.947187 −0.473593 0.880744i \(-0.657043\pi\)
−0.473593 + 0.880744i \(0.657043\pi\)
\(570\) 0 0
\(571\) −16.1050 −0.673972 −0.336986 0.941510i \(-0.609408\pi\)
−0.336986 + 0.941510i \(0.609408\pi\)
\(572\) 0 0
\(573\) 10.3878 0.433955
\(574\) 0 0
\(575\) −1.52311 −0.0635183
\(576\) 0 0
\(577\) 8.32653 0.346638 0.173319 0.984866i \(-0.444551\pi\)
0.173319 + 0.984866i \(0.444551\pi\)
\(578\) 0 0
\(579\) 1.13536 0.0471839
\(580\) 0 0
\(581\) 39.6926 1.64673
\(582\) 0 0
\(583\) −73.6068 −3.04848
\(584\) 0 0
\(585\) 1.13536 0.0469413
\(586\) 0 0
\(587\) 25.9508 1.07111 0.535553 0.844502i \(-0.320103\pi\)
0.535553 + 0.844502i \(0.320103\pi\)
\(588\) 0 0
\(589\) −3.72928 −0.153662
\(590\) 0 0
\(591\) 14.5693 0.599303
\(592\) 0 0
\(593\) 7.38150 0.303122 0.151561 0.988448i \(-0.451570\pi\)
0.151561 + 0.988448i \(0.451570\pi\)
\(594\) 0 0
\(595\) 7.58767 0.311064
\(596\) 0 0
\(597\) −9.18785 −0.376033
\(598\) 0 0
\(599\) 25.4219 1.03871 0.519356 0.854558i \(-0.326172\pi\)
0.519356 + 0.854558i \(0.326172\pi\)
\(600\) 0 0
\(601\) 25.8217 1.05329 0.526645 0.850085i \(-0.323450\pi\)
0.526645 + 0.850085i \(0.323450\pi\)
\(602\) 0 0
\(603\) 7.04623 0.286945
\(604\) 0 0
\(605\) −29.8034 −1.21168
\(606\) 0 0
\(607\) −5.31695 −0.215808 −0.107904 0.994161i \(-0.534414\pi\)
−0.107904 + 0.994161i \(0.534414\pi\)
\(608\) 0 0
\(609\) −29.2158 −1.18388
\(610\) 0 0
\(611\) −10.8122 −0.437413
\(612\) 0 0
\(613\) −37.4498 −1.51258 −0.756292 0.654234i \(-0.772991\pi\)
−0.756292 + 0.654234i \(0.772991\pi\)
\(614\) 0 0
\(615\) 9.64015 0.388728
\(616\) 0 0
\(617\) 36.2341 1.45873 0.729364 0.684125i \(-0.239816\pi\)
0.729364 + 0.684125i \(0.239816\pi\)
\(618\) 0 0
\(619\) −37.7293 −1.51647 −0.758234 0.651983i \(-0.773938\pi\)
−0.758234 + 0.651983i \(0.773938\pi\)
\(620\) 0 0
\(621\) −1.52311 −0.0611205
\(622\) 0 0
\(623\) −46.4835 −1.86232
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.38776 0.255102
\(628\) 0 0
\(629\) −7.94419 −0.316755
\(630\) 0 0
\(631\) 19.4586 0.774633 0.387317 0.921947i \(-0.373402\pi\)
0.387317 + 0.921947i \(0.373402\pi\)
\(632\) 0 0
\(633\) 14.5048 0.576514
\(634\) 0 0
\(635\) 7.04623 0.279621
\(636\) 0 0
\(637\) −13.9109 −0.551169
\(638\) 0 0
\(639\) 9.31695 0.368573
\(640\) 0 0
\(641\) 43.1912 1.70595 0.852974 0.521953i \(-0.174796\pi\)
0.852974 + 0.521953i \(0.174796\pi\)
\(642\) 0 0
\(643\) −18.4157 −0.726243 −0.363121 0.931742i \(-0.618289\pi\)
−0.363121 + 0.931742i \(0.618289\pi\)
\(644\) 0 0
\(645\) 2.59392 0.102136
\(646\) 0 0
\(647\) −18.2986 −0.719393 −0.359697 0.933069i \(-0.617120\pi\)
−0.359697 + 0.933069i \(0.617120\pi\)
\(648\) 0 0
\(649\) 95.6993 3.75653
\(650\) 0 0
\(651\) −16.3632 −0.641323
\(652\) 0 0
\(653\) −14.1570 −0.554007 −0.277003 0.960869i \(-0.589341\pi\)
−0.277003 + 0.960869i \(0.589341\pi\)
\(654\) 0 0
\(655\) −0.117037 −0.00457301
\(656\) 0 0
\(657\) −8.50479 −0.331804
\(658\) 0 0
\(659\) 40.3544 1.57199 0.785993 0.618236i \(-0.212152\pi\)
0.785993 + 0.618236i \(0.212152\pi\)
\(660\) 0 0
\(661\) −48.0837 −1.87024 −0.935120 0.354331i \(-0.884709\pi\)
−0.935120 + 0.354331i \(0.884709\pi\)
\(662\) 0 0
\(663\) −1.96336 −0.0762504
\(664\) 0 0
\(665\) −4.38776 −0.170150
\(666\) 0 0
\(667\) −10.1416 −0.392685
\(668\) 0 0
\(669\) 19.4586 0.752312
\(670\) 0 0
\(671\) 11.4586 0.442353
\(672\) 0 0
\(673\) −2.86464 −0.110424 −0.0552119 0.998475i \(-0.517583\pi\)
−0.0552119 + 0.998475i \(0.517583\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 33.4865 1.28699 0.643495 0.765450i \(-0.277484\pi\)
0.643495 + 0.765450i \(0.277484\pi\)
\(678\) 0 0
\(679\) 20.1570 0.773555
\(680\) 0 0
\(681\) 11.2524 0.431193
\(682\) 0 0
\(683\) 32.5972 1.24730 0.623650 0.781704i \(-0.285649\pi\)
0.623650 + 0.781704i \(0.285649\pi\)
\(684\) 0 0
\(685\) −11.2524 −0.429932
\(686\) 0 0
\(687\) 10.2062 0.389390
\(688\) 0 0
\(689\) −13.0829 −0.498418
\(690\) 0 0
\(691\) 29.3728 1.11739 0.558696 0.829372i \(-0.311302\pi\)
0.558696 + 0.829372i \(0.311302\pi\)
\(692\) 0 0
\(693\) 28.0279 1.06469
\(694\) 0 0
\(695\) 13.3169 0.505141
\(696\) 0 0
\(697\) −16.6705 −0.631442
\(698\) 0 0
\(699\) −29.3449 −1.10992
\(700\) 0 0
\(701\) 11.4952 0.434168 0.217084 0.976153i \(-0.430345\pi\)
0.217084 + 0.976153i \(0.430345\pi\)
\(702\) 0 0
\(703\) 4.59392 0.173263
\(704\) 0 0
\(705\) −9.52311 −0.358661
\(706\) 0 0
\(707\) −5.37276 −0.202063
\(708\) 0 0
\(709\) −31.7572 −1.19267 −0.596333 0.802737i \(-0.703376\pi\)
−0.596333 + 0.802737i \(0.703376\pi\)
\(710\) 0 0
\(711\) 9.25240 0.346992
\(712\) 0 0
\(713\) −5.68012 −0.212722
\(714\) 0 0
\(715\) −7.25240 −0.271224
\(716\) 0 0
\(717\) −20.1170 −0.751285
\(718\) 0 0
\(719\) 24.4802 0.912958 0.456479 0.889734i \(-0.349110\pi\)
0.456479 + 0.889734i \(0.349110\pi\)
\(720\) 0 0
\(721\) −30.9171 −1.15141
\(722\) 0 0
\(723\) 3.55102 0.132064
\(724\) 0 0
\(725\) 6.65847 0.247289
\(726\) 0 0
\(727\) 23.4340 0.869118 0.434559 0.900643i \(-0.356904\pi\)
0.434559 + 0.900643i \(0.356904\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.48562 −0.165907
\(732\) 0 0
\(733\) 45.8863 1.69485 0.847424 0.530916i \(-0.178152\pi\)
0.847424 + 0.530916i \(0.178152\pi\)
\(734\) 0 0
\(735\) −12.2524 −0.451936
\(736\) 0 0
\(737\) −45.0096 −1.65795
\(738\) 0 0
\(739\) 31.1020 1.14411 0.572054 0.820216i \(-0.306147\pi\)
0.572054 + 0.820216i \(0.306147\pi\)
\(740\) 0 0
\(741\) 1.13536 0.0417085
\(742\) 0 0
\(743\) −14.5048 −0.532129 −0.266065 0.963955i \(-0.585723\pi\)
−0.266065 + 0.963955i \(0.585723\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 0 0
\(747\) −9.04623 −0.330984
\(748\) 0 0
\(749\) −15.1753 −0.554495
\(750\) 0 0
\(751\) 49.4094 1.80297 0.901487 0.432805i \(-0.142476\pi\)
0.901487 + 0.432805i \(0.142476\pi\)
\(752\) 0 0
\(753\) 0.836734 0.0304923
\(754\) 0 0
\(755\) 10.2341 0.372456
\(756\) 0 0
\(757\) 25.1108 0.912667 0.456333 0.889809i \(-0.349162\pi\)
0.456333 + 0.889809i \(0.349162\pi\)
\(758\) 0 0
\(759\) 9.72928 0.353151
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −56.9604 −2.06211
\(764\) 0 0
\(765\) −1.72928 −0.0625223
\(766\) 0 0
\(767\) 17.0096 0.614181
\(768\) 0 0
\(769\) 15.9634 0.575653 0.287827 0.957683i \(-0.407067\pi\)
0.287827 + 0.957683i \(0.407067\pi\)
\(770\) 0 0
\(771\) 16.2986 0.586981
\(772\) 0 0
\(773\) 15.6522 0.562971 0.281486 0.959565i \(-0.409173\pi\)
0.281486 + 0.959565i \(0.409173\pi\)
\(774\) 0 0
\(775\) 3.72928 0.133960
\(776\) 0 0
\(777\) 20.1570 0.723129
\(778\) 0 0
\(779\) 9.64015 0.345394
\(780\) 0 0
\(781\) −59.5144 −2.12959
\(782\) 0 0
\(783\) 6.65847 0.237954
\(784\) 0 0
\(785\) −12.0279 −0.429294
\(786\) 0 0
\(787\) −38.4556 −1.37080 −0.685398 0.728169i \(-0.740372\pi\)
−0.685398 + 0.728169i \(0.740372\pi\)
\(788\) 0 0
\(789\) 0.206167 0.00733975
\(790\) 0 0
\(791\) −60.5240 −2.15198
\(792\) 0 0
\(793\) 2.03664 0.0723234
\(794\) 0 0
\(795\) −11.5231 −0.408683
\(796\) 0 0
\(797\) 28.2986 1.00239 0.501194 0.865335i \(-0.332894\pi\)
0.501194 + 0.865335i \(0.332894\pi\)
\(798\) 0 0
\(799\) 16.4681 0.582601
\(800\) 0 0
\(801\) 10.5939 0.374318
\(802\) 0 0
\(803\) 54.3265 1.91714
\(804\) 0 0
\(805\) −6.68305 −0.235547
\(806\) 0 0
\(807\) −24.7509 −0.871274
\(808\) 0 0
\(809\) 36.0925 1.26894 0.634472 0.772946i \(-0.281218\pi\)
0.634472 + 0.772946i \(0.281218\pi\)
\(810\) 0 0
\(811\) 8.05581 0.282878 0.141439 0.989947i \(-0.454827\pi\)
0.141439 + 0.989947i \(0.454827\pi\)
\(812\) 0 0
\(813\) −18.3265 −0.642740
\(814\) 0 0
\(815\) −14.4157 −0.504959
\(816\) 0 0
\(817\) 2.59392 0.0907499
\(818\) 0 0
\(819\) 4.98168 0.174074
\(820\) 0 0
\(821\) −47.9142 −1.67222 −0.836108 0.548564i \(-0.815175\pi\)
−0.836108 + 0.548564i \(0.815175\pi\)
\(822\) 0 0
\(823\) 13.1633 0.458843 0.229421 0.973327i \(-0.426317\pi\)
0.229421 + 0.973327i \(0.426317\pi\)
\(824\) 0 0
\(825\) −6.38776 −0.222393
\(826\) 0 0
\(827\) 23.2524 0.808565 0.404283 0.914634i \(-0.367521\pi\)
0.404283 + 0.914634i \(0.367521\pi\)
\(828\) 0 0
\(829\) −10.3478 −0.359393 −0.179697 0.983722i \(-0.557512\pi\)
−0.179697 + 0.983722i \(0.557512\pi\)
\(830\) 0 0
\(831\) 7.25240 0.251583
\(832\) 0 0
\(833\) 21.1878 0.734115
\(834\) 0 0
\(835\) −3.79383 −0.131291
\(836\) 0 0
\(837\) 3.72928 0.128903
\(838\) 0 0
\(839\) −31.1512 −1.07546 −0.537729 0.843117i \(-0.680718\pi\)
−0.537729 + 0.843117i \(0.680718\pi\)
\(840\) 0 0
\(841\) 15.3353 0.528802
\(842\) 0 0
\(843\) 17.4061 0.599497
\(844\) 0 0
\(845\) 11.7110 0.402869
\(846\) 0 0
\(847\) −130.770 −4.49331
\(848\) 0 0
\(849\) 4.32320 0.148372
\(850\) 0 0
\(851\) 6.99707 0.239856
\(852\) 0 0
\(853\) 7.07414 0.242214 0.121107 0.992639i \(-0.461356\pi\)
0.121107 + 0.992639i \(0.461356\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −36.4769 −1.24603 −0.623013 0.782211i \(-0.714092\pi\)
−0.623013 + 0.782211i \(0.714092\pi\)
\(858\) 0 0
\(859\) −3.58767 −0.122410 −0.0612048 0.998125i \(-0.519494\pi\)
−0.0612048 + 0.998125i \(0.519494\pi\)
\(860\) 0 0
\(861\) 42.2986 1.44153
\(862\) 0 0
\(863\) 28.5972 0.973462 0.486731 0.873552i \(-0.338189\pi\)
0.486731 + 0.873552i \(0.338189\pi\)
\(864\) 0 0
\(865\) 12.2986 0.418166
\(866\) 0 0
\(867\) −14.0096 −0.475790
\(868\) 0 0
\(869\) −59.1020 −2.00490
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −4.59392 −0.155481
\(874\) 0 0
\(875\) 4.38776 0.148333
\(876\) 0 0
\(877\) 15.6402 0.528130 0.264065 0.964505i \(-0.414937\pi\)
0.264065 + 0.964505i \(0.414937\pi\)
\(878\) 0 0
\(879\) 17.2524 0.581909
\(880\) 0 0
\(881\) −47.7851 −1.60992 −0.804960 0.593329i \(-0.797813\pi\)
−0.804960 + 0.593329i \(0.797813\pi\)
\(882\) 0 0
\(883\) −9.09871 −0.306196 −0.153098 0.988211i \(-0.548925\pi\)
−0.153098 + 0.988211i \(0.548925\pi\)
\(884\) 0 0
\(885\) 14.9817 0.503604
\(886\) 0 0
\(887\) 41.5510 1.39515 0.697573 0.716513i \(-0.254263\pi\)
0.697573 + 0.716513i \(0.254263\pi\)
\(888\) 0 0
\(889\) 30.9171 1.03693
\(890\) 0 0
\(891\) −6.38776 −0.213998
\(892\) 0 0
\(893\) −9.52311 −0.318679
\(894\) 0 0
\(895\) −2.02791 −0.0677854
\(896\) 0 0
\(897\) 1.72928 0.0577390
\(898\) 0 0
\(899\) 24.8313 0.828171
\(900\) 0 0
\(901\) 19.9267 0.663855
\(902\) 0 0
\(903\) 11.3815 0.378753
\(904\) 0 0
\(905\) 17.5231 0.582488
\(906\) 0 0
\(907\) −0.775511 −0.0257504 −0.0128752 0.999917i \(-0.504098\pi\)
−0.0128752 + 0.999917i \(0.504098\pi\)
\(908\) 0 0
\(909\) 1.22449 0.0406138
\(910\) 0 0
\(911\) 5.90754 0.195726 0.0978628 0.995200i \(-0.468799\pi\)
0.0978628 + 0.995200i \(0.468799\pi\)
\(912\) 0 0
\(913\) 57.7851 1.91241
\(914\) 0 0
\(915\) 1.79383 0.0593023
\(916\) 0 0
\(917\) −0.513530 −0.0169582
\(918\) 0 0
\(919\) 38.1483 1.25840 0.629198 0.777245i \(-0.283384\pi\)
0.629198 + 0.777245i \(0.283384\pi\)
\(920\) 0 0
\(921\) 5.31695 0.175199
\(922\) 0 0
\(923\) −10.5781 −0.348182
\(924\) 0 0
\(925\) −4.59392 −0.151047
\(926\) 0 0
\(927\) 7.04623 0.231429
\(928\) 0 0
\(929\) −30.2341 −0.991948 −0.495974 0.868337i \(-0.665189\pi\)
−0.495974 + 0.868337i \(0.665189\pi\)
\(930\) 0 0
\(931\) −12.2524 −0.401556
\(932\) 0 0
\(933\) 8.65847 0.283466
\(934\) 0 0
\(935\) 11.0462 0.361250
\(936\) 0 0
\(937\) 21.6926 0.708668 0.354334 0.935119i \(-0.384708\pi\)
0.354334 + 0.935119i \(0.384708\pi\)
\(938\) 0 0
\(939\) −24.7389 −0.807322
\(940\) 0 0
\(941\) −4.08039 −0.133017 −0.0665085 0.997786i \(-0.521186\pi\)
−0.0665085 + 0.997786i \(0.521186\pi\)
\(942\) 0 0
\(943\) 14.6831 0.478146
\(944\) 0 0
\(945\) 4.38776 0.142734
\(946\) 0 0
\(947\) −28.5606 −0.928095 −0.464047 0.885810i \(-0.653603\pi\)
−0.464047 + 0.885810i \(0.653603\pi\)
\(948\) 0 0
\(949\) 9.65599 0.313447
\(950\) 0 0
\(951\) −18.0279 −0.584595
\(952\) 0 0
\(953\) 36.8401 1.19337 0.596683 0.802477i \(-0.296485\pi\)
0.596683 + 0.802477i \(0.296485\pi\)
\(954\) 0 0
\(955\) −10.3878 −0.336140
\(956\) 0 0
\(957\) −42.5327 −1.37489
\(958\) 0 0
\(959\) −49.3728 −1.59433
\(960\) 0 0
\(961\) −17.0925 −0.551370
\(962\) 0 0
\(963\) 3.45856 0.111451
\(964\) 0 0
\(965\) −1.13536 −0.0365485
\(966\) 0 0
\(967\) −23.1999 −0.746059 −0.373029 0.927820i \(-0.621681\pi\)
−0.373029 + 0.927820i \(0.621681\pi\)
\(968\) 0 0
\(969\) −1.72928 −0.0555525
\(970\) 0 0
\(971\) 1.79383 0.0575668 0.0287834 0.999586i \(-0.490837\pi\)
0.0287834 + 0.999586i \(0.490837\pi\)
\(972\) 0 0
\(973\) 58.4315 1.87323
\(974\) 0 0
\(975\) −1.13536 −0.0363606
\(976\) 0 0
\(977\) 50.4961 1.61551 0.807756 0.589517i \(-0.200682\pi\)
0.807756 + 0.589517i \(0.200682\pi\)
\(978\) 0 0
\(979\) −67.6714 −2.16279
\(980\) 0 0
\(981\) 12.9817 0.414473
\(982\) 0 0
\(983\) −32.8034 −1.04627 −0.523133 0.852251i \(-0.675237\pi\)
−0.523133 + 0.852251i \(0.675237\pi\)
\(984\) 0 0
\(985\) −14.5693 −0.464218
\(986\) 0 0
\(987\) −41.7851 −1.33003
\(988\) 0 0
\(989\) 3.95084 0.125629
\(990\) 0 0
\(991\) −32.4277 −1.03010 −0.515050 0.857160i \(-0.672227\pi\)
−0.515050 + 0.857160i \(0.672227\pi\)
\(992\) 0 0
\(993\) −12.9817 −0.411961
\(994\) 0 0
\(995\) 9.18785 0.291274
\(996\) 0 0
\(997\) 5.93545 0.187978 0.0939888 0.995573i \(-0.470038\pi\)
0.0939888 + 0.995573i \(0.470038\pi\)
\(998\) 0 0
\(999\) −4.59392 −0.145345
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 4560.2.a.bt.1.1 3
4.3 odd 2 2280.2.a.r.1.3 3
12.11 even 2 6840.2.a.bk.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.r.1.3 3 4.3 odd 2
4560.2.a.bt.1.1 3 1.1 even 1 trivial
6840.2.a.bk.1.3 3 12.11 even 2