Properties

Label 2960.2.a.u.1.1
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.31955\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.93923 q^{3} -1.00000 q^{5} +1.31955 q^{7} +5.63910 q^{9} +O(q^{10})\) \(q-2.93923 q^{3} -1.00000 q^{5} +1.31955 q^{7} +5.63910 q^{9} +0.258786 q^{11} -5.87847 q^{13} +2.93923 q^{15} +4.25879 q^{17} -2.93923 q^{19} -3.87847 q^{21} +8.51757 q^{23} +1.00000 q^{25} -7.75694 q^{27} -3.61968 q^{29} -3.95865 q^{31} -0.760632 q^{33} -1.31955 q^{35} -1.00000 q^{37} +17.2782 q^{39} -8.89789 q^{41} +5.61968 q^{43} -5.63910 q^{45} -5.57834 q^{47} -5.25879 q^{49} -12.5176 q^{51} +12.8979 q^{53} -0.258786 q^{55} +8.63910 q^{57} -9.57834 q^{59} +0.380316 q^{61} +7.44108 q^{63} +5.87847 q^{65} -5.57834 q^{67} -25.0351 q^{69} -11.2394 q^{71} +2.51757 q^{73} -2.93923 q^{75} +0.341481 q^{77} +7.69987 q^{79} +5.88216 q^{81} +6.17860 q^{83} -4.25879 q^{85} +10.6391 q^{87} +6.00000 q^{89} -7.75694 q^{91} +11.6354 q^{93} +2.93923 q^{95} -12.8979 q^{97} +1.45932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + q^{7} + 11 q^{9} - 11 q^{11} + q^{17} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 5 q^{29} - 3 q^{31} - 14 q^{33} - q^{35} - 3 q^{37} + 40 q^{39} - 9 q^{41} + 11 q^{43} - 11 q^{45} - 2 q^{47} - 4 q^{49} - 14 q^{51} + 21 q^{53} + 11 q^{55} + 20 q^{57} - 14 q^{59} + 7 q^{61} + 37 q^{63} - 2 q^{67} - 28 q^{69} - 22 q^{71} - 16 q^{73} + 7 q^{77} + 26 q^{79} + 47 q^{81} - 2 q^{83} - q^{85} + 26 q^{87} + 18 q^{89} + 12 q^{91} - 18 q^{93} - 21 q^{97} - 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.93923 −1.69697 −0.848484 0.529221i \(-0.822484\pi\)
−0.848484 + 0.529221i \(0.822484\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.31955 0.498743 0.249372 0.968408i \(-0.419776\pi\)
0.249372 + 0.968408i \(0.419776\pi\)
\(8\) 0 0
\(9\) 5.63910 1.87970
\(10\) 0 0
\(11\) 0.258786 0.0780268 0.0390134 0.999239i \(-0.487578\pi\)
0.0390134 + 0.999239i \(0.487578\pi\)
\(12\) 0 0
\(13\) −5.87847 −1.63039 −0.815197 0.579184i \(-0.803371\pi\)
−0.815197 + 0.579184i \(0.803371\pi\)
\(14\) 0 0
\(15\) 2.93923 0.758907
\(16\) 0 0
\(17\) 4.25879 1.03291 0.516454 0.856315i \(-0.327252\pi\)
0.516454 + 0.856315i \(0.327252\pi\)
\(18\) 0 0
\(19\) −2.93923 −0.674307 −0.337153 0.941450i \(-0.609464\pi\)
−0.337153 + 0.941450i \(0.609464\pi\)
\(20\) 0 0
\(21\) −3.87847 −0.846351
\(22\) 0 0
\(23\) 8.51757 1.77604 0.888018 0.459808i \(-0.152082\pi\)
0.888018 + 0.459808i \(0.152082\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −7.75694 −1.49282
\(28\) 0 0
\(29\) −3.61968 −0.672158 −0.336079 0.941834i \(-0.609101\pi\)
−0.336079 + 0.941834i \(0.609101\pi\)
\(30\) 0 0
\(31\) −3.95865 −0.710995 −0.355497 0.934677i \(-0.615689\pi\)
−0.355497 + 0.934677i \(0.615689\pi\)
\(32\) 0 0
\(33\) −0.760632 −0.132409
\(34\) 0 0
\(35\) −1.31955 −0.223045
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 17.2782 2.76673
\(40\) 0 0
\(41\) −8.89789 −1.38962 −0.694808 0.719195i \(-0.744511\pi\)
−0.694808 + 0.719195i \(0.744511\pi\)
\(42\) 0 0
\(43\) 5.61968 0.856994 0.428497 0.903543i \(-0.359043\pi\)
0.428497 + 0.903543i \(0.359043\pi\)
\(44\) 0 0
\(45\) −5.63910 −0.840628
\(46\) 0 0
\(47\) −5.57834 −0.813684 −0.406842 0.913499i \(-0.633370\pi\)
−0.406842 + 0.913499i \(0.633370\pi\)
\(48\) 0 0
\(49\) −5.25879 −0.751255
\(50\) 0 0
\(51\) −12.5176 −1.75281
\(52\) 0 0
\(53\) 12.8979 1.77166 0.885831 0.464009i \(-0.153589\pi\)
0.885831 + 0.464009i \(0.153589\pi\)
\(54\) 0 0
\(55\) −0.258786 −0.0348947
\(56\) 0 0
\(57\) 8.63910 1.14428
\(58\) 0 0
\(59\) −9.57834 −1.24699 −0.623497 0.781826i \(-0.714288\pi\)
−0.623497 + 0.781826i \(0.714288\pi\)
\(60\) 0 0
\(61\) 0.380316 0.0486945 0.0243472 0.999704i \(-0.492249\pi\)
0.0243472 + 0.999704i \(0.492249\pi\)
\(62\) 0 0
\(63\) 7.44108 0.937488
\(64\) 0 0
\(65\) 5.87847 0.729134
\(66\) 0 0
\(67\) −5.57834 −0.681502 −0.340751 0.940154i \(-0.610681\pi\)
−0.340751 + 0.940154i \(0.610681\pi\)
\(68\) 0 0
\(69\) −25.0351 −3.01388
\(70\) 0 0
\(71\) −11.2394 −1.33387 −0.666934 0.745117i \(-0.732394\pi\)
−0.666934 + 0.745117i \(0.732394\pi\)
\(72\) 0 0
\(73\) 2.51757 0.294659 0.147330 0.989087i \(-0.452932\pi\)
0.147330 + 0.989087i \(0.452932\pi\)
\(74\) 0 0
\(75\) −2.93923 −0.339394
\(76\) 0 0
\(77\) 0.341481 0.0389154
\(78\) 0 0
\(79\) 7.69987 0.866303 0.433151 0.901321i \(-0.357402\pi\)
0.433151 + 0.901321i \(0.357402\pi\)
\(80\) 0 0
\(81\) 5.88216 0.653574
\(82\) 0 0
\(83\) 6.17860 0.678190 0.339095 0.940752i \(-0.389879\pi\)
0.339095 + 0.940752i \(0.389879\pi\)
\(84\) 0 0
\(85\) −4.25879 −0.461930
\(86\) 0 0
\(87\) 10.6391 1.14063
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −7.75694 −0.813148
\(92\) 0 0
\(93\) 11.6354 1.20654
\(94\) 0 0
\(95\) 2.93923 0.301559
\(96\) 0 0
\(97\) −12.8979 −1.30958 −0.654791 0.755810i \(-0.727243\pi\)
−0.654791 + 0.755810i \(0.727243\pi\)
\(98\) 0 0
\(99\) 1.45932 0.146667
\(100\) 0 0
\(101\) 10.3960 1.03444 0.517222 0.855851i \(-0.326966\pi\)
0.517222 + 0.855851i \(0.326966\pi\)
\(102\) 0 0
\(103\) 13.8785 1.36749 0.683743 0.729723i \(-0.260351\pi\)
0.683743 + 0.729723i \(0.260351\pi\)
\(104\) 0 0
\(105\) 3.87847 0.378500
\(106\) 0 0
\(107\) 4.21744 0.407715 0.203858 0.979001i \(-0.434652\pi\)
0.203858 + 0.979001i \(0.434652\pi\)
\(108\) 0 0
\(109\) 16.1373 1.54567 0.772834 0.634608i \(-0.218838\pi\)
0.772834 + 0.634608i \(0.218838\pi\)
\(110\) 0 0
\(111\) 2.93923 0.278980
\(112\) 0 0
\(113\) 1.01942 0.0958987 0.0479494 0.998850i \(-0.484731\pi\)
0.0479494 + 0.998850i \(0.484731\pi\)
\(114\) 0 0
\(115\) −8.51757 −0.794268
\(116\) 0 0
\(117\) −33.1493 −3.06465
\(118\) 0 0
\(119\) 5.61968 0.515156
\(120\) 0 0
\(121\) −10.9330 −0.993912
\(122\) 0 0
\(123\) 26.1530 2.35813
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.45681 −0.661685 −0.330842 0.943686i \(-0.607333\pi\)
−0.330842 + 0.943686i \(0.607333\pi\)
\(128\) 0 0
\(129\) −16.5176 −1.45429
\(130\) 0 0
\(131\) −2.42166 −0.211582 −0.105791 0.994388i \(-0.533737\pi\)
−0.105791 + 0.994388i \(0.533737\pi\)
\(132\) 0 0
\(133\) −3.87847 −0.336306
\(134\) 0 0
\(135\) 7.75694 0.667611
\(136\) 0 0
\(137\) 15.7958 1.34952 0.674762 0.738035i \(-0.264246\pi\)
0.674762 + 0.738035i \(0.264246\pi\)
\(138\) 0 0
\(139\) 11.4155 0.968247 0.484123 0.875000i \(-0.339139\pi\)
0.484123 + 0.875000i \(0.339139\pi\)
\(140\) 0 0
\(141\) 16.3960 1.38080
\(142\) 0 0
\(143\) −1.52126 −0.127214
\(144\) 0 0
\(145\) 3.61968 0.300598
\(146\) 0 0
\(147\) 15.4568 1.27486
\(148\) 0 0
\(149\) 18.3960 1.50706 0.753531 0.657412i \(-0.228349\pi\)
0.753531 + 0.657412i \(0.228349\pi\)
\(150\) 0 0
\(151\) −5.87847 −0.478383 −0.239192 0.970972i \(-0.576882\pi\)
−0.239192 + 0.970972i \(0.576882\pi\)
\(152\) 0 0
\(153\) 24.0157 1.94156
\(154\) 0 0
\(155\) 3.95865 0.317967
\(156\) 0 0
\(157\) 3.61968 0.288882 0.144441 0.989513i \(-0.453862\pi\)
0.144441 + 0.989513i \(0.453862\pi\)
\(158\) 0 0
\(159\) −37.9099 −3.00645
\(160\) 0 0
\(161\) 11.2394 0.885786
\(162\) 0 0
\(163\) 22.0546 1.72745 0.863723 0.503966i \(-0.168126\pi\)
0.863723 + 0.503966i \(0.168126\pi\)
\(164\) 0 0
\(165\) 0.760632 0.0592151
\(166\) 0 0
\(167\) 0.600267 0.0464500 0.0232250 0.999730i \(-0.492607\pi\)
0.0232250 + 0.999730i \(0.492607\pi\)
\(168\) 0 0
\(169\) 21.5564 1.65819
\(170\) 0 0
\(171\) −16.5746 −1.26749
\(172\) 0 0
\(173\) −22.6937 −1.72537 −0.862684 0.505744i \(-0.831218\pi\)
−0.862684 + 0.505744i \(0.831218\pi\)
\(174\) 0 0
\(175\) 1.31955 0.0997487
\(176\) 0 0
\(177\) 28.1530 2.11611
\(178\) 0 0
\(179\) 8.73501 0.652885 0.326443 0.945217i \(-0.394150\pi\)
0.326443 + 0.945217i \(0.394150\pi\)
\(180\) 0 0
\(181\) 10.6391 0.790798 0.395399 0.918509i \(-0.370606\pi\)
0.395399 + 0.918509i \(0.370606\pi\)
\(182\) 0 0
\(183\) −1.11784 −0.0826330
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 1.10211 0.0805945
\(188\) 0 0
\(189\) −10.2357 −0.744536
\(190\) 0 0
\(191\) −22.1116 −1.59994 −0.799971 0.600039i \(-0.795152\pi\)
−0.799971 + 0.600039i \(0.795152\pi\)
\(192\) 0 0
\(193\) 7.03514 0.506401 0.253200 0.967414i \(-0.418517\pi\)
0.253200 + 0.967414i \(0.418517\pi\)
\(194\) 0 0
\(195\) −17.2782 −1.23732
\(196\) 0 0
\(197\) 25.7569 1.83511 0.917553 0.397614i \(-0.130162\pi\)
0.917553 + 0.397614i \(0.130162\pi\)
\(198\) 0 0
\(199\) 12.7350 0.902761 0.451380 0.892332i \(-0.350932\pi\)
0.451380 + 0.892332i \(0.350932\pi\)
\(200\) 0 0
\(201\) 16.3960 1.15649
\(202\) 0 0
\(203\) −4.77636 −0.335235
\(204\) 0 0
\(205\) 8.89789 0.621455
\(206\) 0 0
\(207\) 48.0315 3.33842
\(208\) 0 0
\(209\) −0.760632 −0.0526140
\(210\) 0 0
\(211\) 6.65483 0.458137 0.229069 0.973410i \(-0.426432\pi\)
0.229069 + 0.973410i \(0.426432\pi\)
\(212\) 0 0
\(213\) 33.0351 2.26353
\(214\) 0 0
\(215\) −5.61968 −0.383259
\(216\) 0 0
\(217\) −5.22364 −0.354604
\(218\) 0 0
\(219\) −7.39973 −0.500028
\(220\) 0 0
\(221\) −25.0351 −1.68405
\(222\) 0 0
\(223\) 16.5589 1.10887 0.554434 0.832228i \(-0.312935\pi\)
0.554434 + 0.832228i \(0.312935\pi\)
\(224\) 0 0
\(225\) 5.63910 0.375940
\(226\) 0 0
\(227\) 4.01573 0.266533 0.133267 0.991080i \(-0.457453\pi\)
0.133267 + 0.991080i \(0.457453\pi\)
\(228\) 0 0
\(229\) 25.5139 1.68600 0.843002 0.537910i \(-0.180786\pi\)
0.843002 + 0.537910i \(0.180786\pi\)
\(230\) 0 0
\(231\) −1.00369 −0.0660381
\(232\) 0 0
\(233\) 7.96116 0.521553 0.260777 0.965399i \(-0.416021\pi\)
0.260777 + 0.965399i \(0.416021\pi\)
\(234\) 0 0
\(235\) 5.57834 0.363891
\(236\) 0 0
\(237\) −22.6317 −1.47009
\(238\) 0 0
\(239\) −11.7983 −0.763168 −0.381584 0.924334i \(-0.624621\pi\)
−0.381584 + 0.924334i \(0.624621\pi\)
\(240\) 0 0
\(241\) 24.9963 1.61015 0.805077 0.593171i \(-0.202124\pi\)
0.805077 + 0.593171i \(0.202124\pi\)
\(242\) 0 0
\(243\) 5.98176 0.383730
\(244\) 0 0
\(245\) 5.25879 0.335971
\(246\) 0 0
\(247\) 17.2782 1.09939
\(248\) 0 0
\(249\) −18.1604 −1.15087
\(250\) 0 0
\(251\) 8.81770 0.556569 0.278284 0.960499i \(-0.410234\pi\)
0.278284 + 0.960499i \(0.410234\pi\)
\(252\) 0 0
\(253\) 2.20423 0.138578
\(254\) 0 0
\(255\) 12.5176 0.783881
\(256\) 0 0
\(257\) 11.8396 0.738536 0.369268 0.929323i \(-0.379608\pi\)
0.369268 + 0.929323i \(0.379608\pi\)
\(258\) 0 0
\(259\) −1.31955 −0.0819929
\(260\) 0 0
\(261\) −20.4118 −1.26346
\(262\) 0 0
\(263\) 9.15919 0.564780 0.282390 0.959300i \(-0.408873\pi\)
0.282390 + 0.959300i \(0.408873\pi\)
\(264\) 0 0
\(265\) −12.8979 −0.792311
\(266\) 0 0
\(267\) −17.6354 −1.07927
\(268\) 0 0
\(269\) 18.3960 1.12163 0.560813 0.827942i \(-0.310489\pi\)
0.560813 + 0.827942i \(0.310489\pi\)
\(270\) 0 0
\(271\) 27.2394 1.65467 0.827337 0.561706i \(-0.189855\pi\)
0.827337 + 0.561706i \(0.189855\pi\)
\(272\) 0 0
\(273\) 22.7995 1.37989
\(274\) 0 0
\(275\) 0.258786 0.0156054
\(276\) 0 0
\(277\) −21.8785 −1.31455 −0.657275 0.753651i \(-0.728291\pi\)
−0.657275 + 0.753651i \(0.728291\pi\)
\(278\) 0 0
\(279\) −22.3232 −1.33646
\(280\) 0 0
\(281\) −20.9963 −1.25253 −0.626267 0.779608i \(-0.715418\pi\)
−0.626267 + 0.779608i \(0.715418\pi\)
\(282\) 0 0
\(283\) −18.9136 −1.12430 −0.562149 0.827036i \(-0.690025\pi\)
−0.562149 + 0.827036i \(0.690025\pi\)
\(284\) 0 0
\(285\) −8.63910 −0.511736
\(286\) 0 0
\(287\) −11.7412 −0.693062
\(288\) 0 0
\(289\) 1.13726 0.0668974
\(290\) 0 0
\(291\) 37.9099 2.22232
\(292\) 0 0
\(293\) −12.8979 −0.753503 −0.376751 0.926314i \(-0.622959\pi\)
−0.376751 + 0.926314i \(0.622959\pi\)
\(294\) 0 0
\(295\) 9.57834 0.557672
\(296\) 0 0
\(297\) −2.00738 −0.116480
\(298\) 0 0
\(299\) −50.0703 −2.89564
\(300\) 0 0
\(301\) 7.41546 0.427420
\(302\) 0 0
\(303\) −30.5564 −1.75542
\(304\) 0 0
\(305\) −0.380316 −0.0217768
\(306\) 0 0
\(307\) 3.78256 0.215882 0.107941 0.994157i \(-0.465574\pi\)
0.107941 + 0.994157i \(0.465574\pi\)
\(308\) 0 0
\(309\) −40.7921 −2.32058
\(310\) 0 0
\(311\) 15.2807 0.866490 0.433245 0.901276i \(-0.357369\pi\)
0.433245 + 0.901276i \(0.357369\pi\)
\(312\) 0 0
\(313\) 19.0351 1.07593 0.537965 0.842967i \(-0.319193\pi\)
0.537965 + 0.842967i \(0.319193\pi\)
\(314\) 0 0
\(315\) −7.44108 −0.419257
\(316\) 0 0
\(317\) −24.1373 −1.35568 −0.677842 0.735208i \(-0.737085\pi\)
−0.677842 + 0.735208i \(0.737085\pi\)
\(318\) 0 0
\(319\) −0.936722 −0.0524464
\(320\) 0 0
\(321\) −12.3960 −0.691880
\(322\) 0 0
\(323\) −12.5176 −0.696496
\(324\) 0 0
\(325\) −5.87847 −0.326079
\(326\) 0 0
\(327\) −47.4312 −2.62295
\(328\) 0 0
\(329\) −7.36090 −0.405819
\(330\) 0 0
\(331\) −0.817705 −0.0449451 −0.0224726 0.999747i \(-0.507154\pi\)
−0.0224726 + 0.999747i \(0.507154\pi\)
\(332\) 0 0
\(333\) −5.63910 −0.309021
\(334\) 0 0
\(335\) 5.57834 0.304777
\(336\) 0 0
\(337\) −19.7958 −1.07834 −0.539172 0.842195i \(-0.681263\pi\)
−0.539172 + 0.842195i \(0.681263\pi\)
\(338\) 0 0
\(339\) −2.99631 −0.162737
\(340\) 0 0
\(341\) −1.02444 −0.0554767
\(342\) 0 0
\(343\) −16.1761 −0.873427
\(344\) 0 0
\(345\) 25.0351 1.34785
\(346\) 0 0
\(347\) −15.7569 −0.845877 −0.422938 0.906158i \(-0.639001\pi\)
−0.422938 + 0.906158i \(0.639001\pi\)
\(348\) 0 0
\(349\) −0.160365 −0.00858416 −0.00429208 0.999991i \(-0.501366\pi\)
−0.00429208 + 0.999991i \(0.501366\pi\)
\(350\) 0 0
\(351\) 45.5989 2.43389
\(352\) 0 0
\(353\) −30.2075 −1.60779 −0.803893 0.594775i \(-0.797241\pi\)
−0.803893 + 0.594775i \(0.797241\pi\)
\(354\) 0 0
\(355\) 11.2394 0.596524
\(356\) 0 0
\(357\) −16.5176 −0.874203
\(358\) 0 0
\(359\) 7.39973 0.390543 0.195271 0.980749i \(-0.437441\pi\)
0.195271 + 0.980749i \(0.437441\pi\)
\(360\) 0 0
\(361\) −10.3609 −0.545310
\(362\) 0 0
\(363\) 32.1347 1.68664
\(364\) 0 0
\(365\) −2.51757 −0.131776
\(366\) 0 0
\(367\) 13.9198 0.726609 0.363304 0.931671i \(-0.381649\pi\)
0.363304 + 0.931671i \(0.381649\pi\)
\(368\) 0 0
\(369\) −50.1761 −2.61206
\(370\) 0 0
\(371\) 17.0194 0.883604
\(372\) 0 0
\(373\) 20.2357 1.04776 0.523882 0.851791i \(-0.324483\pi\)
0.523882 + 0.851791i \(0.324483\pi\)
\(374\) 0 0
\(375\) 2.93923 0.151781
\(376\) 0 0
\(377\) 21.2782 1.09588
\(378\) 0 0
\(379\) 27.9488 1.43563 0.717816 0.696233i \(-0.245142\pi\)
0.717816 + 0.696233i \(0.245142\pi\)
\(380\) 0 0
\(381\) 21.9173 1.12286
\(382\) 0 0
\(383\) −11.7569 −0.600752 −0.300376 0.953821i \(-0.597112\pi\)
−0.300376 + 0.953821i \(0.597112\pi\)
\(384\) 0 0
\(385\) −0.341481 −0.0174035
\(386\) 0 0
\(387\) 31.6900 1.61089
\(388\) 0 0
\(389\) −9.58085 −0.485768 −0.242884 0.970055i \(-0.578093\pi\)
−0.242884 + 0.970055i \(0.578093\pi\)
\(390\) 0 0
\(391\) 36.2745 1.83448
\(392\) 0 0
\(393\) 7.11784 0.359047
\(394\) 0 0
\(395\) −7.69987 −0.387422
\(396\) 0 0
\(397\) −32.4787 −1.63006 −0.815031 0.579418i \(-0.803280\pi\)
−0.815031 + 0.579418i \(0.803280\pi\)
\(398\) 0 0
\(399\) 11.3997 0.570700
\(400\) 0 0
\(401\) 7.27820 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(402\) 0 0
\(403\) 23.2708 1.15920
\(404\) 0 0
\(405\) −5.88216 −0.292287
\(406\) 0 0
\(407\) −0.258786 −0.0128275
\(408\) 0 0
\(409\) −32.9963 −1.63156 −0.815781 0.578361i \(-0.803693\pi\)
−0.815781 + 0.578361i \(0.803693\pi\)
\(410\) 0 0
\(411\) −46.4275 −2.29010
\(412\) 0 0
\(413\) −12.6391 −0.621930
\(414\) 0 0
\(415\) −6.17860 −0.303296
\(416\) 0 0
\(417\) −33.5527 −1.64308
\(418\) 0 0
\(419\) −25.7131 −1.25617 −0.628083 0.778146i \(-0.716160\pi\)
−0.628083 + 0.778146i \(0.716160\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) −31.4568 −1.52948
\(424\) 0 0
\(425\) 4.25879 0.206581
\(426\) 0 0
\(427\) 0.501846 0.0242860
\(428\) 0 0
\(429\) 4.47135 0.215879
\(430\) 0 0
\(431\) −17.9198 −0.863167 −0.431584 0.902073i \(-0.642045\pi\)
−0.431584 + 0.902073i \(0.642045\pi\)
\(432\) 0 0
\(433\) 4.96486 0.238596 0.119298 0.992859i \(-0.461936\pi\)
0.119298 + 0.992859i \(0.461936\pi\)
\(434\) 0 0
\(435\) −10.6391 −0.510106
\(436\) 0 0
\(437\) −25.0351 −1.19759
\(438\) 0 0
\(439\) 13.8371 0.660410 0.330205 0.943909i \(-0.392882\pi\)
0.330205 + 0.943909i \(0.392882\pi\)
\(440\) 0 0
\(441\) −29.6548 −1.41213
\(442\) 0 0
\(443\) −14.3390 −0.681265 −0.340632 0.940197i \(-0.610641\pi\)
−0.340632 + 0.940197i \(0.610641\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) −54.0703 −2.55744
\(448\) 0 0
\(449\) 12.4787 0.588908 0.294454 0.955666i \(-0.404862\pi\)
0.294454 + 0.955666i \(0.404862\pi\)
\(450\) 0 0
\(451\) −2.30265 −0.108427
\(452\) 0 0
\(453\) 17.2782 0.811801
\(454\) 0 0
\(455\) 7.75694 0.363651
\(456\) 0 0
\(457\) 21.6585 1.01314 0.506571 0.862198i \(-0.330913\pi\)
0.506571 + 0.862198i \(0.330913\pi\)
\(458\) 0 0
\(459\) −33.0351 −1.54195
\(460\) 0 0
\(461\) −29.4155 −1.37001 −0.685007 0.728536i \(-0.740201\pi\)
−0.685007 + 0.728536i \(0.740201\pi\)
\(462\) 0 0
\(463\) 9.03514 0.419899 0.209949 0.977712i \(-0.432670\pi\)
0.209949 + 0.977712i \(0.432670\pi\)
\(464\) 0 0
\(465\) −11.6354 −0.539579
\(466\) 0 0
\(467\) −7.05825 −0.326617 −0.163308 0.986575i \(-0.552217\pi\)
−0.163308 + 0.986575i \(0.552217\pi\)
\(468\) 0 0
\(469\) −7.36090 −0.339895
\(470\) 0 0
\(471\) −10.6391 −0.490224
\(472\) 0 0
\(473\) 1.45429 0.0668685
\(474\) 0 0
\(475\) −2.93923 −0.134861
\(476\) 0 0
\(477\) 72.7325 3.33019
\(478\) 0 0
\(479\) 29.3353 1.34036 0.670181 0.742197i \(-0.266216\pi\)
0.670181 + 0.742197i \(0.266216\pi\)
\(480\) 0 0
\(481\) 5.87847 0.268035
\(482\) 0 0
\(483\) −33.0351 −1.50315
\(484\) 0 0
\(485\) 12.8979 0.585663
\(486\) 0 0
\(487\) 37.9099 1.71786 0.858931 0.512091i \(-0.171129\pi\)
0.858931 + 0.512091i \(0.171129\pi\)
\(488\) 0 0
\(489\) −64.8235 −2.93142
\(490\) 0 0
\(491\) 14.5564 0.656921 0.328461 0.944518i \(-0.393470\pi\)
0.328461 + 0.944518i \(0.393470\pi\)
\(492\) 0 0
\(493\) −15.4155 −0.694277
\(494\) 0 0
\(495\) −1.45932 −0.0655915
\(496\) 0 0
\(497\) −14.8309 −0.665258
\(498\) 0 0
\(499\) −23.9744 −1.07324 −0.536620 0.843824i \(-0.680299\pi\)
−0.536620 + 0.843824i \(0.680299\pi\)
\(500\) 0 0
\(501\) −1.76432 −0.0788242
\(502\) 0 0
\(503\) 43.7569 1.95103 0.975513 0.219943i \(-0.0705871\pi\)
0.975513 + 0.219943i \(0.0705871\pi\)
\(504\) 0 0
\(505\) −10.3960 −0.462618
\(506\) 0 0
\(507\) −63.3593 −2.81389
\(508\) 0 0
\(509\) 11.7958 0.522839 0.261419 0.965225i \(-0.415809\pi\)
0.261419 + 0.965225i \(0.415809\pi\)
\(510\) 0 0
\(511\) 3.32206 0.146959
\(512\) 0 0
\(513\) 22.7995 1.00662
\(514\) 0 0
\(515\) −13.8785 −0.611558
\(516\) 0 0
\(517\) −1.44359 −0.0634892
\(518\) 0 0
\(519\) 66.7020 2.92789
\(520\) 0 0
\(521\) 14.9806 0.656311 0.328156 0.944624i \(-0.393573\pi\)
0.328156 + 0.944624i \(0.393573\pi\)
\(522\) 0 0
\(523\) 27.5139 1.20310 0.601549 0.798836i \(-0.294550\pi\)
0.601549 + 0.798836i \(0.294550\pi\)
\(524\) 0 0
\(525\) −3.87847 −0.169270
\(526\) 0 0
\(527\) −16.8591 −0.734392
\(528\) 0 0
\(529\) 49.5490 2.15431
\(530\) 0 0
\(531\) −54.0132 −2.34397
\(532\) 0 0
\(533\) 52.3060 2.26562
\(534\) 0 0
\(535\) −4.21744 −0.182336
\(536\) 0 0
\(537\) −25.6742 −1.10793
\(538\) 0 0
\(539\) −1.36090 −0.0586180
\(540\) 0 0
\(541\) −11.0351 −0.474438 −0.237219 0.971456i \(-0.576236\pi\)
−0.237219 + 0.971456i \(0.576236\pi\)
\(542\) 0 0
\(543\) −31.2708 −1.34196
\(544\) 0 0
\(545\) −16.1373 −0.691244
\(546\) 0 0
\(547\) 16.1761 0.691640 0.345820 0.938301i \(-0.387601\pi\)
0.345820 + 0.938301i \(0.387601\pi\)
\(548\) 0 0
\(549\) 2.14464 0.0915310
\(550\) 0 0
\(551\) 10.6391 0.453241
\(552\) 0 0
\(553\) 10.1604 0.432063
\(554\) 0 0
\(555\) −2.93923 −0.124764
\(556\) 0 0
\(557\) 11.1567 0.472723 0.236362 0.971665i \(-0.424045\pi\)
0.236362 + 0.971665i \(0.424045\pi\)
\(558\) 0 0
\(559\) −33.0351 −1.39724
\(560\) 0 0
\(561\) −3.23937 −0.136766
\(562\) 0 0
\(563\) 41.7288 1.75866 0.879330 0.476213i \(-0.157991\pi\)
0.879330 + 0.476213i \(0.157991\pi\)
\(564\) 0 0
\(565\) −1.01942 −0.0428872
\(566\) 0 0
\(567\) 7.76181 0.325965
\(568\) 0 0
\(569\) −24.0703 −1.00908 −0.504539 0.863389i \(-0.668338\pi\)
−0.504539 + 0.863389i \(0.668338\pi\)
\(570\) 0 0
\(571\) −37.6511 −1.57565 −0.787825 0.615899i \(-0.788793\pi\)
−0.787825 + 0.615899i \(0.788793\pi\)
\(572\) 0 0
\(573\) 64.9913 2.71505
\(574\) 0 0
\(575\) 8.51757 0.355207
\(576\) 0 0
\(577\) 23.4312 0.975453 0.487726 0.872996i \(-0.337826\pi\)
0.487726 + 0.872996i \(0.337826\pi\)
\(578\) 0 0
\(579\) −20.6779 −0.859346
\(580\) 0 0
\(581\) 8.15298 0.338243
\(582\) 0 0
\(583\) 3.33779 0.138237
\(584\) 0 0
\(585\) 33.1493 1.37055
\(586\) 0 0
\(587\) −6.30265 −0.260138 −0.130069 0.991505i \(-0.541520\pi\)
−0.130069 + 0.991505i \(0.541520\pi\)
\(588\) 0 0
\(589\) 11.6354 0.479429
\(590\) 0 0
\(591\) −75.7057 −3.11412
\(592\) 0 0
\(593\) −34.0315 −1.39750 −0.698752 0.715364i \(-0.746261\pi\)
−0.698752 + 0.715364i \(0.746261\pi\)
\(594\) 0 0
\(595\) −5.61968 −0.230385
\(596\) 0 0
\(597\) −37.4312 −1.53196
\(598\) 0 0
\(599\) −34.8359 −1.42336 −0.711679 0.702505i \(-0.752065\pi\)
−0.711679 + 0.702505i \(0.752065\pi\)
\(600\) 0 0
\(601\) 15.2236 0.620985 0.310493 0.950576i \(-0.399506\pi\)
0.310493 + 0.950576i \(0.399506\pi\)
\(602\) 0 0
\(603\) −31.4568 −1.28102
\(604\) 0 0
\(605\) 10.9330 0.444491
\(606\) 0 0
\(607\) 30.8309 1.25139 0.625694 0.780068i \(-0.284816\pi\)
0.625694 + 0.780068i \(0.284816\pi\)
\(608\) 0 0
\(609\) 14.0388 0.568882
\(610\) 0 0
\(611\) 32.7921 1.32663
\(612\) 0 0
\(613\) −11.2112 −0.452817 −0.226409 0.974032i \(-0.572698\pi\)
−0.226409 + 0.974032i \(0.572698\pi\)
\(614\) 0 0
\(615\) −26.1530 −1.05459
\(616\) 0 0
\(617\) 26.5176 1.06756 0.533779 0.845624i \(-0.320772\pi\)
0.533779 + 0.845624i \(0.320772\pi\)
\(618\) 0 0
\(619\) 3.41546 0.137279 0.0686394 0.997642i \(-0.478134\pi\)
0.0686394 + 0.997642i \(0.478134\pi\)
\(620\) 0 0
\(621\) −66.0703 −2.65131
\(622\) 0 0
\(623\) 7.91730 0.317200
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.23568 0.0892843
\(628\) 0 0
\(629\) −4.25879 −0.169809
\(630\) 0 0
\(631\) −3.11533 −0.124019 −0.0620096 0.998076i \(-0.519751\pi\)
−0.0620096 + 0.998076i \(0.519751\pi\)
\(632\) 0 0
\(633\) −19.5601 −0.777444
\(634\) 0 0
\(635\) 7.45681 0.295914
\(636\) 0 0
\(637\) 30.9136 1.22484
\(638\) 0 0
\(639\) −63.3799 −2.50727
\(640\) 0 0
\(641\) −20.5018 −0.809774 −0.404887 0.914367i \(-0.632689\pi\)
−0.404887 + 0.914367i \(0.632689\pi\)
\(642\) 0 0
\(643\) −5.70238 −0.224880 −0.112440 0.993659i \(-0.535867\pi\)
−0.112440 + 0.993659i \(0.535867\pi\)
\(644\) 0 0
\(645\) 16.5176 0.650379
\(646\) 0 0
\(647\) 41.0351 1.61326 0.806629 0.591058i \(-0.201290\pi\)
0.806629 + 0.591058i \(0.201290\pi\)
\(648\) 0 0
\(649\) −2.47874 −0.0972989
\(650\) 0 0
\(651\) 15.3535 0.601752
\(652\) 0 0
\(653\) 14.9575 0.585331 0.292666 0.956215i \(-0.405458\pi\)
0.292666 + 0.956215i \(0.405458\pi\)
\(654\) 0 0
\(655\) 2.42166 0.0946222
\(656\) 0 0
\(657\) 14.1968 0.553872
\(658\) 0 0
\(659\) 15.7569 0.613803 0.306902 0.951741i \(-0.400708\pi\)
0.306902 + 0.951741i \(0.400708\pi\)
\(660\) 0 0
\(661\) 11.6197 0.451953 0.225977 0.974133i \(-0.427443\pi\)
0.225977 + 0.974133i \(0.427443\pi\)
\(662\) 0 0
\(663\) 73.5842 2.85777
\(664\) 0 0
\(665\) 3.87847 0.150401
\(666\) 0 0
\(667\) −30.8309 −1.19378
\(668\) 0 0
\(669\) −48.6706 −1.88171
\(670\) 0 0
\(671\) 0.0984203 0.00379947
\(672\) 0 0
\(673\) 12.7218 0.490389 0.245195 0.969474i \(-0.421148\pi\)
0.245195 + 0.969474i \(0.421148\pi\)
\(674\) 0 0
\(675\) −7.75694 −0.298565
\(676\) 0 0
\(677\) −21.5139 −0.826846 −0.413423 0.910539i \(-0.635667\pi\)
−0.413423 + 0.910539i \(0.635667\pi\)
\(678\) 0 0
\(679\) −17.0194 −0.653145
\(680\) 0 0
\(681\) −11.8032 −0.452298
\(682\) 0 0
\(683\) −47.6900 −1.82481 −0.912403 0.409293i \(-0.865775\pi\)
−0.912403 + 0.409293i \(0.865775\pi\)
\(684\) 0 0
\(685\) −15.7958 −0.603526
\(686\) 0 0
\(687\) −74.9913 −2.86110
\(688\) 0 0
\(689\) −75.8198 −2.88851
\(690\) 0 0
\(691\) −30.5721 −1.16302 −0.581509 0.813540i \(-0.697538\pi\)
−0.581509 + 0.813540i \(0.697538\pi\)
\(692\) 0 0
\(693\) 1.92565 0.0731492
\(694\) 0 0
\(695\) −11.4155 −0.433013
\(696\) 0 0
\(697\) −37.8942 −1.43534
\(698\) 0 0
\(699\) −23.3997 −0.885059
\(700\) 0 0
\(701\) 32.0703 1.21128 0.605639 0.795740i \(-0.292918\pi\)
0.605639 + 0.795740i \(0.292918\pi\)
\(702\) 0 0
\(703\) 2.93923 0.110855
\(704\) 0 0
\(705\) −16.3960 −0.617511
\(706\) 0 0
\(707\) 13.7181 0.515922
\(708\) 0 0
\(709\) 10.6160 0.398692 0.199346 0.979929i \(-0.436118\pi\)
0.199346 + 0.979929i \(0.436118\pi\)
\(710\) 0 0
\(711\) 43.4203 1.62839
\(712\) 0 0
\(713\) −33.7181 −1.26275
\(714\) 0 0
\(715\) 1.52126 0.0568920
\(716\) 0 0
\(717\) 34.6779 1.29507
\(718\) 0 0
\(719\) 32.4663 1.21079 0.605395 0.795925i \(-0.293015\pi\)
0.605395 + 0.795925i \(0.293015\pi\)
\(720\) 0 0
\(721\) 18.3133 0.682025
\(722\) 0 0
\(723\) −73.4700 −2.73238
\(724\) 0 0
\(725\) −3.61968 −0.134432
\(726\) 0 0
\(727\) −31.5139 −1.16879 −0.584393 0.811471i \(-0.698667\pi\)
−0.584393 + 0.811471i \(0.698667\pi\)
\(728\) 0 0
\(729\) −35.2283 −1.30475
\(730\) 0 0
\(731\) 23.9330 0.885195
\(732\) 0 0
\(733\) −16.8202 −0.621269 −0.310634 0.950529i \(-0.600541\pi\)
−0.310634 + 0.950529i \(0.600541\pi\)
\(734\) 0 0
\(735\) −15.4568 −0.570133
\(736\) 0 0
\(737\) −1.44359 −0.0531755
\(738\) 0 0
\(739\) −9.69735 −0.356723 −0.178361 0.983965i \(-0.557080\pi\)
−0.178361 + 0.983965i \(0.557080\pi\)
\(740\) 0 0
\(741\) −50.7847 −1.86562
\(742\) 0 0
\(743\) 39.1153 1.43500 0.717501 0.696557i \(-0.245286\pi\)
0.717501 + 0.696557i \(0.245286\pi\)
\(744\) 0 0
\(745\) −18.3960 −0.673979
\(746\) 0 0
\(747\) 34.8418 1.27479
\(748\) 0 0
\(749\) 5.56512 0.203345
\(750\) 0 0
\(751\) 26.1530 0.954336 0.477168 0.878812i \(-0.341663\pi\)
0.477168 + 0.878812i \(0.341663\pi\)
\(752\) 0 0
\(753\) −25.9173 −0.944479
\(754\) 0 0
\(755\) 5.87847 0.213939
\(756\) 0 0
\(757\) −33.5915 −1.22091 −0.610453 0.792053i \(-0.709013\pi\)
−0.610453 + 0.792053i \(0.709013\pi\)
\(758\) 0 0
\(759\) −6.47874 −0.235163
\(760\) 0 0
\(761\) −51.3766 −1.86240 −0.931201 0.364507i \(-0.881237\pi\)
−0.931201 + 0.364507i \(0.881237\pi\)
\(762\) 0 0
\(763\) 21.2939 0.770892
\(764\) 0 0
\(765\) −24.0157 −0.868290
\(766\) 0 0
\(767\) 56.3060 2.03309
\(768\) 0 0
\(769\) 27.5527 0.993576 0.496788 0.867872i \(-0.334513\pi\)
0.496788 + 0.867872i \(0.334513\pi\)
\(770\) 0 0
\(771\) −34.7995 −1.25327
\(772\) 0 0
\(773\) 13.4155 0.482521 0.241260 0.970460i \(-0.422439\pi\)
0.241260 + 0.970460i \(0.422439\pi\)
\(774\) 0 0
\(775\) −3.95865 −0.142199
\(776\) 0 0
\(777\) 3.87847 0.139139
\(778\) 0 0
\(779\) 26.1530 0.937028
\(780\) 0 0
\(781\) −2.90859 −0.104077
\(782\) 0 0
\(783\) 28.0777 1.00341
\(784\) 0 0
\(785\) −3.61968 −0.129192
\(786\) 0 0
\(787\) 25.4956 0.908821 0.454411 0.890792i \(-0.349850\pi\)
0.454411 + 0.890792i \(0.349850\pi\)
\(788\) 0 0
\(789\) −26.9210 −0.958413
\(790\) 0 0
\(791\) 1.34517 0.0478289
\(792\) 0 0
\(793\) −2.23568 −0.0793912
\(794\) 0 0
\(795\) 37.9099 1.34453
\(796\) 0 0
\(797\) 48.4663 1.71677 0.858383 0.513010i \(-0.171470\pi\)
0.858383 + 0.513010i \(0.171470\pi\)
\(798\) 0 0
\(799\) −23.7569 −0.840460
\(800\) 0 0
\(801\) 33.8346 1.19549
\(802\) 0 0
\(803\) 0.651511 0.0229913
\(804\) 0 0
\(805\) −11.2394 −0.396136
\(806\) 0 0
\(807\) −54.0703 −1.90336
\(808\) 0 0
\(809\) −12.9649 −0.455820 −0.227910 0.973682i \(-0.573189\pi\)
−0.227910 + 0.973682i \(0.573189\pi\)
\(810\) 0 0
\(811\) −0.0776702 −0.00272737 −0.00136368 0.999999i \(-0.500434\pi\)
−0.00136368 + 0.999999i \(0.500434\pi\)
\(812\) 0 0
\(813\) −80.0629 −2.80793
\(814\) 0 0
\(815\) −22.0546 −0.772538
\(816\) 0 0
\(817\) −16.5176 −0.577877
\(818\) 0 0
\(819\) −43.7422 −1.52848
\(820\) 0 0
\(821\) 36.3448 1.26844 0.634221 0.773152i \(-0.281321\pi\)
0.634221 + 0.773152i \(0.281321\pi\)
\(822\) 0 0
\(823\) 23.0996 0.805201 0.402601 0.915376i \(-0.368106\pi\)
0.402601 + 0.915376i \(0.368106\pi\)
\(824\) 0 0
\(825\) −0.760632 −0.0264818
\(826\) 0 0
\(827\) −15.2551 −0.530472 −0.265236 0.964184i \(-0.585450\pi\)
−0.265236 + 0.964184i \(0.585450\pi\)
\(828\) 0 0
\(829\) 28.1373 0.977247 0.488624 0.872495i \(-0.337499\pi\)
0.488624 + 0.872495i \(0.337499\pi\)
\(830\) 0 0
\(831\) 64.3060 2.23075
\(832\) 0 0
\(833\) −22.3960 −0.775977
\(834\) 0 0
\(835\) −0.600267 −0.0207731
\(836\) 0 0
\(837\) 30.7070 1.06139
\(838\) 0 0
\(839\) −4.24306 −0.146487 −0.0732434 0.997314i \(-0.523335\pi\)
−0.0732434 + 0.997314i \(0.523335\pi\)
\(840\) 0 0
\(841\) −15.8979 −0.548203
\(842\) 0 0
\(843\) 61.7131 2.12551
\(844\) 0 0
\(845\) −21.5564 −0.741563
\(846\) 0 0
\(847\) −14.4267 −0.495707
\(848\) 0 0
\(849\) 55.5915 1.90790
\(850\) 0 0
\(851\) −8.51757 −0.291979
\(852\) 0 0
\(853\) 36.4787 1.24901 0.624504 0.781022i \(-0.285301\pi\)
0.624504 + 0.781022i \(0.285301\pi\)
\(854\) 0 0
\(855\) 16.5746 0.566841
\(856\) 0 0
\(857\) 23.7288 0.810561 0.405280 0.914192i \(-0.367174\pi\)
0.405280 + 0.914192i \(0.367174\pi\)
\(858\) 0 0
\(859\) −2.77887 −0.0948138 −0.0474069 0.998876i \(-0.515096\pi\)
−0.0474069 + 0.998876i \(0.515096\pi\)
\(860\) 0 0
\(861\) 34.5102 1.17610
\(862\) 0 0
\(863\) −10.5151 −0.357937 −0.178968 0.983855i \(-0.557276\pi\)
−0.178968 + 0.983855i \(0.557276\pi\)
\(864\) 0 0
\(865\) 22.6937 0.771608
\(866\) 0 0
\(867\) −3.34266 −0.113523
\(868\) 0 0
\(869\) 1.99262 0.0675948
\(870\) 0 0
\(871\) 32.7921 1.11112
\(872\) 0 0
\(873\) −72.7325 −2.46162
\(874\) 0 0
\(875\) −1.31955 −0.0446090
\(876\) 0 0
\(877\) 27.6197 0.932650 0.466325 0.884613i \(-0.345578\pi\)
0.466325 + 0.884613i \(0.345578\pi\)
\(878\) 0 0
\(879\) 37.9099 1.27867
\(880\) 0 0
\(881\) 10.2200 0.344319 0.172159 0.985069i \(-0.444926\pi\)
0.172159 + 0.985069i \(0.444926\pi\)
\(882\) 0 0
\(883\) −53.5684 −1.80272 −0.901361 0.433069i \(-0.857431\pi\)
−0.901361 + 0.433069i \(0.857431\pi\)
\(884\) 0 0
\(885\) −28.1530 −0.946352
\(886\) 0 0
\(887\) −36.5904 −1.22858 −0.614292 0.789079i \(-0.710558\pi\)
−0.614292 + 0.789079i \(0.710558\pi\)
\(888\) 0 0
\(889\) −9.83963 −0.330011
\(890\) 0 0
\(891\) 1.52222 0.0509963
\(892\) 0 0
\(893\) 16.3960 0.548673
\(894\) 0 0
\(895\) −8.73501 −0.291979
\(896\) 0 0
\(897\) 147.168 4.91381
\(898\) 0 0
\(899\) 14.3291 0.477901
\(900\) 0 0
\(901\) 54.9293 1.82996
\(902\) 0 0
\(903\) −21.7958 −0.725318
\(904\) 0 0
\(905\) −10.6391 −0.353656
\(906\) 0 0
\(907\) −7.64279 −0.253775 −0.126887 0.991917i \(-0.540499\pi\)
−0.126887 + 0.991917i \(0.540499\pi\)
\(908\) 0 0
\(909\) 58.6243 1.94445
\(910\) 0 0
\(911\) −22.0132 −0.729330 −0.364665 0.931139i \(-0.618817\pi\)
−0.364665 + 0.931139i \(0.618817\pi\)
\(912\) 0 0
\(913\) 1.59893 0.0529170
\(914\) 0 0
\(915\) 1.11784 0.0369546
\(916\) 0 0
\(917\) −3.19551 −0.105525
\(918\) 0 0
\(919\) −23.6999 −0.781786 −0.390893 0.920436i \(-0.627834\pi\)
−0.390893 + 0.920436i \(0.627834\pi\)
\(920\) 0 0
\(921\) −11.1178 −0.366345
\(922\) 0 0
\(923\) 66.0703 2.17473
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 78.2621 2.57046
\(928\) 0 0
\(929\) 8.82022 0.289382 0.144691 0.989477i \(-0.453781\pi\)
0.144691 + 0.989477i \(0.453781\pi\)
\(930\) 0 0
\(931\) 15.4568 0.506576
\(932\) 0 0
\(933\) −44.9136 −1.47041
\(934\) 0 0
\(935\) −1.10211 −0.0360429
\(936\) 0 0
\(937\) 40.0703 1.30904 0.654520 0.756045i \(-0.272871\pi\)
0.654520 + 0.756045i \(0.272871\pi\)
\(938\) 0 0
\(939\) −55.9488 −1.82582
\(940\) 0 0
\(941\) −31.4312 −1.02463 −0.512314 0.858798i \(-0.671211\pi\)
−0.512314 + 0.858798i \(0.671211\pi\)
\(942\) 0 0
\(943\) −75.7884 −2.46801
\(944\) 0 0
\(945\) 10.2357 0.332967
\(946\) 0 0
\(947\) 53.5370 1.73972 0.869859 0.493300i \(-0.164210\pi\)
0.869859 + 0.493300i \(0.164210\pi\)
\(948\) 0 0
\(949\) −14.7995 −0.480411
\(950\) 0 0
\(951\) 70.9451 2.30055
\(952\) 0 0
\(953\) 5.23937 0.169720 0.0848599 0.996393i \(-0.472956\pi\)
0.0848599 + 0.996393i \(0.472956\pi\)
\(954\) 0 0
\(955\) 22.1116 0.715516
\(956\) 0 0
\(957\) 2.75325 0.0889998
\(958\) 0 0
\(959\) 20.8433 0.673066
\(960\) 0 0
\(961\) −15.3291 −0.494486
\(962\) 0 0
\(963\) 23.7826 0.766382
\(964\) 0 0
\(965\) −7.03514 −0.226469
\(966\) 0 0
\(967\) 46.5929 1.49833 0.749163 0.662386i \(-0.230456\pi\)
0.749163 + 0.662386i \(0.230456\pi\)
\(968\) 0 0
\(969\) 36.7921 1.18193
\(970\) 0 0
\(971\) −38.1373 −1.22388 −0.611941 0.790903i \(-0.709611\pi\)
−0.611941 + 0.790903i \(0.709611\pi\)
\(972\) 0 0
\(973\) 15.0633 0.482907
\(974\) 0 0
\(975\) 17.2782 0.553345
\(976\) 0 0
\(977\) 40.8979 1.30844 0.654220 0.756305i \(-0.272997\pi\)
0.654220 + 0.756305i \(0.272997\pi\)
\(978\) 0 0
\(979\) 1.55271 0.0496250
\(980\) 0 0
\(981\) 90.9996 2.90539
\(982\) 0 0
\(983\) −50.4638 −1.60955 −0.804773 0.593583i \(-0.797713\pi\)
−0.804773 + 0.593583i \(0.797713\pi\)
\(984\) 0 0
\(985\) −25.7569 −0.820684
\(986\) 0 0
\(987\) 21.6354 0.688663
\(988\) 0 0
\(989\) 47.8661 1.52205
\(990\) 0 0
\(991\) 15.9587 0.506943 0.253472 0.967343i \(-0.418428\pi\)
0.253472 + 0.967343i \(0.418428\pi\)
\(992\) 0 0
\(993\) 2.40343 0.0762704
\(994\) 0 0
\(995\) −12.7350 −0.403727
\(996\) 0 0
\(997\) −9.23937 −0.292614 −0.146307 0.989239i \(-0.546739\pi\)
−0.146307 + 0.989239i \(0.546739\pi\)
\(998\) 0 0
\(999\) 7.75694 0.245419
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 2960.2.a.u.1.1 3
4.3 odd 2 370.2.a.g.1.3 3
12.11 even 2 3330.2.a.bg.1.2 3
20.3 even 4 1850.2.b.o.149.3 6
20.7 even 4 1850.2.b.o.149.4 6
20.19 odd 2 1850.2.a.z.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.3 3 4.3 odd 2
1850.2.a.z.1.1 3 20.19 odd 2
1850.2.b.o.149.3 6 20.3 even 4
1850.2.b.o.149.4 6 20.7 even 4
2960.2.a.u.1.1 3 1.1 even 1 trivial
3330.2.a.bg.1.2 3 12.11 even 2