Properties

Label 3120.2.a.bj.1.3
Level $3120$
Weight $2$
Character 3120.1
Self dual yes
Analytic conductor $24.913$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 3120.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +2.71982 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +2.71982 q^{7} +1.00000 q^{9} +2.71982 q^{11} +1.00000 q^{13} -1.00000 q^{15} -2.83709 q^{17} +3.55691 q^{19} +2.71982 q^{21} +4.83709 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -7.55691 q^{31} +2.71982 q^{33} -2.71982 q^{35} -4.27674 q^{37} +1.00000 q^{39} +2.83709 q^{41} -11.1138 q^{43} -1.00000 q^{45} +11.5569 q^{47} +0.397442 q^{49} -2.83709 q^{51} +1.16291 q^{53} -2.71982 q^{55} +3.55691 q^{57} +2.11727 q^{59} +6.60256 q^{61} +2.71982 q^{63} -1.00000 q^{65} -1.88273 q^{67} +4.83709 q^{69} +6.71982 q^{71} +9.11383 q^{73} +1.00000 q^{75} +7.39744 q^{77} -10.2767 q^{79} +1.00000 q^{81} -2.11727 q^{83} +2.83709 q^{85} +6.00000 q^{87} +1.16291 q^{89} +2.71982 q^{91} -7.55691 q^{93} -3.55691 q^{95} -10.8371 q^{97} +2.71982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.71982 1.02800 0.513998 0.857791i \(-0.328164\pi\)
0.513998 + 0.857791i \(0.328164\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.71982 0.820058 0.410029 0.912073i \(-0.365519\pi\)
0.410029 + 0.912073i \(0.365519\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.83709 −0.688095 −0.344048 0.938952i \(-0.611798\pi\)
−0.344048 + 0.938952i \(0.611798\pi\)
\(18\) 0 0
\(19\) 3.55691 0.816012 0.408006 0.912979i \(-0.366224\pi\)
0.408006 + 0.912979i \(0.366224\pi\)
\(20\) 0 0
\(21\) 2.71982 0.593514
\(22\) 0 0
\(23\) 4.83709 1.00860 0.504302 0.863528i \(-0.331750\pi\)
0.504302 + 0.863528i \(0.331750\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −7.55691 −1.35726 −0.678631 0.734479i \(-0.737426\pi\)
−0.678631 + 0.734479i \(0.737426\pi\)
\(32\) 0 0
\(33\) 2.71982 0.473461
\(34\) 0 0
\(35\) −2.71982 −0.459734
\(36\) 0 0
\(37\) −4.27674 −0.703091 −0.351546 0.936171i \(-0.614344\pi\)
−0.351546 + 0.936171i \(0.614344\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 2.83709 0.443079 0.221540 0.975151i \(-0.428892\pi\)
0.221540 + 0.975151i \(0.428892\pi\)
\(42\) 0 0
\(43\) −11.1138 −1.69484 −0.847421 0.530921i \(-0.821846\pi\)
−0.847421 + 0.530921i \(0.821846\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 11.5569 1.68575 0.842875 0.538110i \(-0.180862\pi\)
0.842875 + 0.538110i \(0.180862\pi\)
\(48\) 0 0
\(49\) 0.397442 0.0567775
\(50\) 0 0
\(51\) −2.83709 −0.397272
\(52\) 0 0
\(53\) 1.16291 0.159738 0.0798690 0.996805i \(-0.474550\pi\)
0.0798690 + 0.996805i \(0.474550\pi\)
\(54\) 0 0
\(55\) −2.71982 −0.366741
\(56\) 0 0
\(57\) 3.55691 0.471125
\(58\) 0 0
\(59\) 2.11727 0.275645 0.137822 0.990457i \(-0.455990\pi\)
0.137822 + 0.990457i \(0.455990\pi\)
\(60\) 0 0
\(61\) 6.60256 0.845371 0.422685 0.906276i \(-0.361088\pi\)
0.422685 + 0.906276i \(0.361088\pi\)
\(62\) 0 0
\(63\) 2.71982 0.342666
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −1.88273 −0.230013 −0.115006 0.993365i \(-0.536689\pi\)
−0.115006 + 0.993365i \(0.536689\pi\)
\(68\) 0 0
\(69\) 4.83709 0.582317
\(70\) 0 0
\(71\) 6.71982 0.797496 0.398748 0.917060i \(-0.369445\pi\)
0.398748 + 0.917060i \(0.369445\pi\)
\(72\) 0 0
\(73\) 9.11383 1.06669 0.533346 0.845897i \(-0.320934\pi\)
0.533346 + 0.845897i \(0.320934\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 7.39744 0.843017
\(78\) 0 0
\(79\) −10.2767 −1.15622 −0.578112 0.815958i \(-0.696210\pi\)
−0.578112 + 0.815958i \(0.696210\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.11727 −0.232400 −0.116200 0.993226i \(-0.537071\pi\)
−0.116200 + 0.993226i \(0.537071\pi\)
\(84\) 0 0
\(85\) 2.83709 0.307726
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 1.16291 0.123268 0.0616341 0.998099i \(-0.480369\pi\)
0.0616341 + 0.998099i \(0.480369\pi\)
\(90\) 0 0
\(91\) 2.71982 0.285115
\(92\) 0 0
\(93\) −7.55691 −0.783616
\(94\) 0 0
\(95\) −3.55691 −0.364932
\(96\) 0 0
\(97\) −10.8371 −1.10034 −0.550170 0.835053i \(-0.685437\pi\)
−0.550170 + 0.835053i \(0.685437\pi\)
\(98\) 0 0
\(99\) 2.71982 0.273353
\(100\) 0 0
\(101\) 7.67418 0.763610 0.381805 0.924243i \(-0.375303\pi\)
0.381805 + 0.924243i \(0.375303\pi\)
\(102\) 0 0
\(103\) −3.76547 −0.371023 −0.185511 0.982642i \(-0.559394\pi\)
−0.185511 + 0.982642i \(0.559394\pi\)
\(104\) 0 0
\(105\) −2.71982 −0.265428
\(106\) 0 0
\(107\) 12.6026 1.21834 0.609168 0.793041i \(-0.291504\pi\)
0.609168 + 0.793041i \(0.291504\pi\)
\(108\) 0 0
\(109\) 11.4396 1.09572 0.547860 0.836570i \(-0.315443\pi\)
0.547860 + 0.836570i \(0.315443\pi\)
\(110\) 0 0
\(111\) −4.27674 −0.405930
\(112\) 0 0
\(113\) −13.1138 −1.23365 −0.616823 0.787102i \(-0.711580\pi\)
−0.616823 + 0.787102i \(0.711580\pi\)
\(114\) 0 0
\(115\) −4.83709 −0.451061
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −7.71639 −0.707360
\(120\) 0 0
\(121\) −3.60256 −0.327505
\(122\) 0 0
\(123\) 2.83709 0.255812
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 13.4396 1.19258 0.596288 0.802771i \(-0.296642\pi\)
0.596288 + 0.802771i \(0.296642\pi\)
\(128\) 0 0
\(129\) −11.1138 −0.978518
\(130\) 0 0
\(131\) 9.43965 0.824746 0.412373 0.911015i \(-0.364700\pi\)
0.412373 + 0.911015i \(0.364700\pi\)
\(132\) 0 0
\(133\) 9.67418 0.838858
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 1.76547 0.150834 0.0754170 0.997152i \(-0.475971\pi\)
0.0754170 + 0.997152i \(0.475971\pi\)
\(138\) 0 0
\(139\) 6.27674 0.532386 0.266193 0.963920i \(-0.414234\pi\)
0.266193 + 0.963920i \(0.414234\pi\)
\(140\) 0 0
\(141\) 11.5569 0.973268
\(142\) 0 0
\(143\) 2.71982 0.227443
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 0.397442 0.0327805
\(148\) 0 0
\(149\) −20.8302 −1.70648 −0.853239 0.521520i \(-0.825365\pi\)
−0.853239 + 0.521520i \(0.825365\pi\)
\(150\) 0 0
\(151\) −4.99656 −0.406614 −0.203307 0.979115i \(-0.565169\pi\)
−0.203307 + 0.979115i \(0.565169\pi\)
\(152\) 0 0
\(153\) −2.83709 −0.229365
\(154\) 0 0
\(155\) 7.55691 0.606986
\(156\) 0 0
\(157\) 8.87930 0.708645 0.354322 0.935123i \(-0.384712\pi\)
0.354322 + 0.935123i \(0.384712\pi\)
\(158\) 0 0
\(159\) 1.16291 0.0922247
\(160\) 0 0
\(161\) 13.1560 1.03684
\(162\) 0 0
\(163\) 13.8337 1.08354 0.541768 0.840528i \(-0.317755\pi\)
0.541768 + 0.840528i \(0.317755\pi\)
\(164\) 0 0
\(165\) −2.71982 −0.211738
\(166\) 0 0
\(167\) 9.88273 0.764749 0.382374 0.924007i \(-0.375106\pi\)
0.382374 + 0.924007i \(0.375106\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 3.55691 0.272004
\(172\) 0 0
\(173\) 13.1138 0.997026 0.498513 0.866882i \(-0.333880\pi\)
0.498513 + 0.866882i \(0.333880\pi\)
\(174\) 0 0
\(175\) 2.71982 0.205599
\(176\) 0 0
\(177\) 2.11727 0.159143
\(178\) 0 0
\(179\) −8.55348 −0.639317 −0.319658 0.947533i \(-0.603568\pi\)
−0.319658 + 0.947533i \(0.603568\pi\)
\(180\) 0 0
\(181\) 3.72326 0.276748 0.138374 0.990380i \(-0.455812\pi\)
0.138374 + 0.990380i \(0.455812\pi\)
\(182\) 0 0
\(183\) 6.60256 0.488075
\(184\) 0 0
\(185\) 4.27674 0.314432
\(186\) 0 0
\(187\) −7.71639 −0.564278
\(188\) 0 0
\(189\) 2.71982 0.197838
\(190\) 0 0
\(191\) 4.23453 0.306400 0.153200 0.988195i \(-0.451042\pi\)
0.153200 + 0.988195i \(0.451042\pi\)
\(192\) 0 0
\(193\) −23.3906 −1.68369 −0.841845 0.539719i \(-0.818530\pi\)
−0.841845 + 0.539719i \(0.818530\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 14.5535 1.03689 0.518446 0.855110i \(-0.326511\pi\)
0.518446 + 0.855110i \(0.326511\pi\)
\(198\) 0 0
\(199\) 15.1138 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(200\) 0 0
\(201\) −1.88273 −0.132798
\(202\) 0 0
\(203\) 16.3189 1.14537
\(204\) 0 0
\(205\) −2.83709 −0.198151
\(206\) 0 0
\(207\) 4.83709 0.336201
\(208\) 0 0
\(209\) 9.67418 0.669177
\(210\) 0 0
\(211\) 18.2277 1.25484 0.627422 0.778680i \(-0.284110\pi\)
0.627422 + 0.778680i \(0.284110\pi\)
\(212\) 0 0
\(213\) 6.71982 0.460435
\(214\) 0 0
\(215\) 11.1138 0.757957
\(216\) 0 0
\(217\) −20.5535 −1.39526
\(218\) 0 0
\(219\) 9.11383 0.615855
\(220\) 0 0
\(221\) −2.83709 −0.190843
\(222\) 0 0
\(223\) −10.1173 −0.677502 −0.338751 0.940876i \(-0.610004\pi\)
−0.338751 + 0.940876i \(0.610004\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −11.3224 −0.751493 −0.375746 0.926723i \(-0.622614\pi\)
−0.375746 + 0.926723i \(0.622614\pi\)
\(228\) 0 0
\(229\) −6.23453 −0.411990 −0.205995 0.978553i \(-0.566043\pi\)
−0.205995 + 0.978553i \(0.566043\pi\)
\(230\) 0 0
\(231\) 7.39744 0.486716
\(232\) 0 0
\(233\) 6.83709 0.447913 0.223956 0.974599i \(-0.428103\pi\)
0.223956 + 0.974599i \(0.428103\pi\)
\(234\) 0 0
\(235\) −11.5569 −0.753890
\(236\) 0 0
\(237\) −10.2767 −0.667546
\(238\) 0 0
\(239\) −1.28018 −0.0828077 −0.0414039 0.999142i \(-0.513183\pi\)
−0.0414039 + 0.999142i \(0.513183\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.397442 −0.0253917
\(246\) 0 0
\(247\) 3.55691 0.226321
\(248\) 0 0
\(249\) −2.11727 −0.134176
\(250\) 0 0
\(251\) −18.2277 −1.15052 −0.575260 0.817971i \(-0.695099\pi\)
−0.575260 + 0.817971i \(0.695099\pi\)
\(252\) 0 0
\(253\) 13.1560 0.827113
\(254\) 0 0
\(255\) 2.83709 0.177665
\(256\) 0 0
\(257\) 1.11383 0.0694787 0.0347394 0.999396i \(-0.488940\pi\)
0.0347394 + 0.999396i \(0.488940\pi\)
\(258\) 0 0
\(259\) −11.6320 −0.722776
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −1.16291 −0.0714370
\(266\) 0 0
\(267\) 1.16291 0.0711689
\(268\) 0 0
\(269\) 15.6742 0.955672 0.477836 0.878449i \(-0.341421\pi\)
0.477836 + 0.878449i \(0.341421\pi\)
\(270\) 0 0
\(271\) −0.443086 −0.0269155 −0.0134578 0.999909i \(-0.504284\pi\)
−0.0134578 + 0.999909i \(0.504284\pi\)
\(272\) 0 0
\(273\) 2.71982 0.164611
\(274\) 0 0
\(275\) 2.71982 0.164012
\(276\) 0 0
\(277\) −4.87930 −0.293168 −0.146584 0.989198i \(-0.546828\pi\)
−0.146584 + 0.989198i \(0.546828\pi\)
\(278\) 0 0
\(279\) −7.55691 −0.452421
\(280\) 0 0
\(281\) −9.11383 −0.543685 −0.271843 0.962342i \(-0.587633\pi\)
−0.271843 + 0.962342i \(0.587633\pi\)
\(282\) 0 0
\(283\) −33.3415 −1.98195 −0.990973 0.134063i \(-0.957198\pi\)
−0.990973 + 0.134063i \(0.957198\pi\)
\(284\) 0 0
\(285\) −3.55691 −0.210693
\(286\) 0 0
\(287\) 7.71639 0.455484
\(288\) 0 0
\(289\) −8.95092 −0.526525
\(290\) 0 0
\(291\) −10.8371 −0.635281
\(292\) 0 0
\(293\) −29.4328 −1.71948 −0.859740 0.510731i \(-0.829375\pi\)
−0.859740 + 0.510731i \(0.829375\pi\)
\(294\) 0 0
\(295\) −2.11727 −0.123272
\(296\) 0 0
\(297\) 2.71982 0.157820
\(298\) 0 0
\(299\) 4.83709 0.279736
\(300\) 0 0
\(301\) −30.2277 −1.74229
\(302\) 0 0
\(303\) 7.67418 0.440870
\(304\) 0 0
\(305\) −6.60256 −0.378061
\(306\) 0 0
\(307\) 21.8337 1.24611 0.623056 0.782177i \(-0.285891\pi\)
0.623056 + 0.782177i \(0.285891\pi\)
\(308\) 0 0
\(309\) −3.76547 −0.214210
\(310\) 0 0
\(311\) −25.1070 −1.42368 −0.711842 0.702339i \(-0.752139\pi\)
−0.711842 + 0.702339i \(0.752139\pi\)
\(312\) 0 0
\(313\) 8.22766 0.465055 0.232527 0.972590i \(-0.425300\pi\)
0.232527 + 0.972590i \(0.425300\pi\)
\(314\) 0 0
\(315\) −2.71982 −0.153245
\(316\) 0 0
\(317\) 27.6742 1.55434 0.777168 0.629293i \(-0.216655\pi\)
0.777168 + 0.629293i \(0.216655\pi\)
\(318\) 0 0
\(319\) 16.3189 0.913685
\(320\) 0 0
\(321\) 12.6026 0.703406
\(322\) 0 0
\(323\) −10.0913 −0.561494
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 11.4396 0.632614
\(328\) 0 0
\(329\) 31.4328 1.73294
\(330\) 0 0
\(331\) 13.2311 0.727247 0.363623 0.931546i \(-0.381540\pi\)
0.363623 + 0.931546i \(0.381540\pi\)
\(332\) 0 0
\(333\) −4.27674 −0.234364
\(334\) 0 0
\(335\) 1.88273 0.102865
\(336\) 0 0
\(337\) −4.32582 −0.235642 −0.117821 0.993035i \(-0.537591\pi\)
−0.117821 + 0.993035i \(0.537591\pi\)
\(338\) 0 0
\(339\) −13.1138 −0.712245
\(340\) 0 0
\(341\) −20.5535 −1.11303
\(342\) 0 0
\(343\) −17.9578 −0.969630
\(344\) 0 0
\(345\) −4.83709 −0.260420
\(346\) 0 0
\(347\) −6.27674 −0.336953 −0.168476 0.985706i \(-0.553885\pi\)
−0.168476 + 0.985706i \(0.553885\pi\)
\(348\) 0 0
\(349\) 17.6673 0.945709 0.472855 0.881140i \(-0.343224\pi\)
0.472855 + 0.881140i \(0.343224\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −13.7655 −0.732662 −0.366331 0.930485i \(-0.619386\pi\)
−0.366331 + 0.930485i \(0.619386\pi\)
\(354\) 0 0
\(355\) −6.71982 −0.356651
\(356\) 0 0
\(357\) −7.71639 −0.408394
\(358\) 0 0
\(359\) −0.996562 −0.0525965 −0.0262983 0.999654i \(-0.508372\pi\)
−0.0262983 + 0.999654i \(0.508372\pi\)
\(360\) 0 0
\(361\) −6.34836 −0.334124
\(362\) 0 0
\(363\) −3.60256 −0.189085
\(364\) 0 0
\(365\) −9.11383 −0.477040
\(366\) 0 0
\(367\) −14.2277 −0.742678 −0.371339 0.928497i \(-0.621101\pi\)
−0.371339 + 0.928497i \(0.621101\pi\)
\(368\) 0 0
\(369\) 2.83709 0.147693
\(370\) 0 0
\(371\) 3.16291 0.164210
\(372\) 0 0
\(373\) 15.6742 0.811578 0.405789 0.913967i \(-0.366997\pi\)
0.405789 + 0.913967i \(0.366997\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) −26.2017 −1.34589 −0.672945 0.739693i \(-0.734971\pi\)
−0.672945 + 0.739693i \(0.734971\pi\)
\(380\) 0 0
\(381\) 13.4396 0.688534
\(382\) 0 0
\(383\) −22.4362 −1.14644 −0.573218 0.819403i \(-0.694305\pi\)
−0.573218 + 0.819403i \(0.694305\pi\)
\(384\) 0 0
\(385\) −7.39744 −0.377009
\(386\) 0 0
\(387\) −11.1138 −0.564948
\(388\) 0 0
\(389\) 31.6742 1.60594 0.802972 0.596016i \(-0.203251\pi\)
0.802972 + 0.596016i \(0.203251\pi\)
\(390\) 0 0
\(391\) −13.7233 −0.694015
\(392\) 0 0
\(393\) 9.43965 0.476167
\(394\) 0 0
\(395\) 10.2767 0.517079
\(396\) 0 0
\(397\) 17.9509 0.900931 0.450465 0.892794i \(-0.351258\pi\)
0.450465 + 0.892794i \(0.351258\pi\)
\(398\) 0 0
\(399\) 9.67418 0.484315
\(400\) 0 0
\(401\) 13.5829 0.678297 0.339149 0.940733i \(-0.389861\pi\)
0.339149 + 0.940733i \(0.389861\pi\)
\(402\) 0 0
\(403\) −7.55691 −0.376437
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −11.6320 −0.576576
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 1.76547 0.0870841
\(412\) 0 0
\(413\) 5.75859 0.283362
\(414\) 0 0
\(415\) 2.11727 0.103933
\(416\) 0 0
\(417\) 6.27674 0.307373
\(418\) 0 0
\(419\) −12.3189 −0.601820 −0.300910 0.953653i \(-0.597290\pi\)
−0.300910 + 0.953653i \(0.597290\pi\)
\(420\) 0 0
\(421\) 22.7880 1.11062 0.555310 0.831644i \(-0.312600\pi\)
0.555310 + 0.831644i \(0.312600\pi\)
\(422\) 0 0
\(423\) 11.5569 0.561916
\(424\) 0 0
\(425\) −2.83709 −0.137619
\(426\) 0 0
\(427\) 17.9578 0.869039
\(428\) 0 0
\(429\) 2.71982 0.131314
\(430\) 0 0
\(431\) 8.99656 0.433349 0.216675 0.976244i \(-0.430479\pi\)
0.216675 + 0.976244i \(0.430479\pi\)
\(432\) 0 0
\(433\) −20.3258 −0.976797 −0.488398 0.872621i \(-0.662419\pi\)
−0.488398 + 0.872621i \(0.662419\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) 0 0
\(437\) 17.2051 0.823032
\(438\) 0 0
\(439\) 25.3906 1.21183 0.605913 0.795531i \(-0.292808\pi\)
0.605913 + 0.795531i \(0.292808\pi\)
\(440\) 0 0
\(441\) 0.397442 0.0189258
\(442\) 0 0
\(443\) −10.9284 −0.519223 −0.259611 0.965713i \(-0.583594\pi\)
−0.259611 + 0.965713i \(0.583594\pi\)
\(444\) 0 0
\(445\) −1.16291 −0.0551272
\(446\) 0 0
\(447\) −20.8302 −0.985235
\(448\) 0 0
\(449\) 2.83709 0.133891 0.0669453 0.997757i \(-0.478675\pi\)
0.0669453 + 0.997757i \(0.478675\pi\)
\(450\) 0 0
\(451\) 7.71639 0.363350
\(452\) 0 0
\(453\) −4.99656 −0.234759
\(454\) 0 0
\(455\) −2.71982 −0.127507
\(456\) 0 0
\(457\) −13.7164 −0.641625 −0.320813 0.947143i \(-0.603956\pi\)
−0.320813 + 0.947143i \(0.603956\pi\)
\(458\) 0 0
\(459\) −2.83709 −0.132424
\(460\) 0 0
\(461\) −19.6251 −0.914032 −0.457016 0.889458i \(-0.651082\pi\)
−0.457016 + 0.889458i \(0.651082\pi\)
\(462\) 0 0
\(463\) −27.0388 −1.25660 −0.628299 0.777972i \(-0.716249\pi\)
−0.628299 + 0.777972i \(0.716249\pi\)
\(464\) 0 0
\(465\) 7.55691 0.350444
\(466\) 0 0
\(467\) −28.9215 −1.33833 −0.669164 0.743115i \(-0.733348\pi\)
−0.669164 + 0.743115i \(0.733348\pi\)
\(468\) 0 0
\(469\) −5.12070 −0.236452
\(470\) 0 0
\(471\) 8.87930 0.409136
\(472\) 0 0
\(473\) −30.2277 −1.38987
\(474\) 0 0
\(475\) 3.55691 0.163202
\(476\) 0 0
\(477\) 1.16291 0.0532460
\(478\) 0 0
\(479\) 12.1595 0.555580 0.277790 0.960642i \(-0.410398\pi\)
0.277790 + 0.960642i \(0.410398\pi\)
\(480\) 0 0
\(481\) −4.27674 −0.195002
\(482\) 0 0
\(483\) 13.1560 0.598620
\(484\) 0 0
\(485\) 10.8371 0.492087
\(486\) 0 0
\(487\) 0.159472 0.00722636 0.00361318 0.999993i \(-0.498850\pi\)
0.00361318 + 0.999993i \(0.498850\pi\)
\(488\) 0 0
\(489\) 13.8337 0.625579
\(490\) 0 0
\(491\) 42.2277 1.90571 0.952854 0.303430i \(-0.0981318\pi\)
0.952854 + 0.303430i \(0.0981318\pi\)
\(492\) 0 0
\(493\) −17.0225 −0.766657
\(494\) 0 0
\(495\) −2.71982 −0.122247
\(496\) 0 0
\(497\) 18.2767 0.819824
\(498\) 0 0
\(499\) −7.79145 −0.348793 −0.174397 0.984676i \(-0.555797\pi\)
−0.174397 + 0.984676i \(0.555797\pi\)
\(500\) 0 0
\(501\) 9.88273 0.441528
\(502\) 0 0
\(503\) −27.3484 −1.21940 −0.609702 0.792631i \(-0.708711\pi\)
−0.609702 + 0.792631i \(0.708711\pi\)
\(504\) 0 0
\(505\) −7.67418 −0.341497
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −33.4819 −1.48406 −0.742029 0.670368i \(-0.766136\pi\)
−0.742029 + 0.670368i \(0.766136\pi\)
\(510\) 0 0
\(511\) 24.7880 1.09656
\(512\) 0 0
\(513\) 3.55691 0.157042
\(514\) 0 0
\(515\) 3.76547 0.165926
\(516\) 0 0
\(517\) 31.4328 1.38241
\(518\) 0 0
\(519\) 13.1138 0.575633
\(520\) 0 0
\(521\) −17.3484 −0.760045 −0.380023 0.924977i \(-0.624084\pi\)
−0.380023 + 0.924977i \(0.624084\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 2.71982 0.118703
\(526\) 0 0
\(527\) 21.4396 0.933926
\(528\) 0 0
\(529\) 0.397442 0.0172801
\(530\) 0 0
\(531\) 2.11727 0.0918815
\(532\) 0 0
\(533\) 2.83709 0.122888
\(534\) 0 0
\(535\) −12.6026 −0.544856
\(536\) 0 0
\(537\) −8.55348 −0.369110
\(538\) 0 0
\(539\) 1.08097 0.0465608
\(540\) 0 0
\(541\) −32.6448 −1.40351 −0.701754 0.712419i \(-0.747599\pi\)
−0.701754 + 0.712419i \(0.747599\pi\)
\(542\) 0 0
\(543\) 3.72326 0.159780
\(544\) 0 0
\(545\) −11.4396 −0.490021
\(546\) 0 0
\(547\) −34.2277 −1.46347 −0.731734 0.681590i \(-0.761289\pi\)
−0.731734 + 0.681590i \(0.761289\pi\)
\(548\) 0 0
\(549\) 6.60256 0.281790
\(550\) 0 0
\(551\) 21.3415 0.909178
\(552\) 0 0
\(553\) −27.9509 −1.18859
\(554\) 0 0
\(555\) 4.27674 0.181537
\(556\) 0 0
\(557\) 6.65164 0.281839 0.140919 0.990021i \(-0.454994\pi\)
0.140919 + 0.990021i \(0.454994\pi\)
\(558\) 0 0
\(559\) −11.1138 −0.470065
\(560\) 0 0
\(561\) −7.71639 −0.325786
\(562\) 0 0
\(563\) −40.2699 −1.69717 −0.848586 0.529057i \(-0.822546\pi\)
−0.848586 + 0.529057i \(0.822546\pi\)
\(564\) 0 0
\(565\) 13.1138 0.551703
\(566\) 0 0
\(567\) 2.71982 0.114222
\(568\) 0 0
\(569\) −13.4328 −0.563131 −0.281566 0.959542i \(-0.590854\pi\)
−0.281566 + 0.959542i \(0.590854\pi\)
\(570\) 0 0
\(571\) 35.7164 1.49468 0.747342 0.664439i \(-0.231330\pi\)
0.747342 + 0.664439i \(0.231330\pi\)
\(572\) 0 0
\(573\) 4.23453 0.176900
\(574\) 0 0
\(575\) 4.83709 0.201721
\(576\) 0 0
\(577\) −13.7164 −0.571021 −0.285510 0.958376i \(-0.592163\pi\)
−0.285510 + 0.958376i \(0.592163\pi\)
\(578\) 0 0
\(579\) −23.3906 −0.972079
\(580\) 0 0
\(581\) −5.75859 −0.238907
\(582\) 0 0
\(583\) 3.16291 0.130994
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −30.6707 −1.26592 −0.632959 0.774186i \(-0.718160\pi\)
−0.632959 + 0.774186i \(0.718160\pi\)
\(588\) 0 0
\(589\) −26.8793 −1.10754
\(590\) 0 0
\(591\) 14.5535 0.598650
\(592\) 0 0
\(593\) −45.6673 −1.87533 −0.937666 0.347538i \(-0.887018\pi\)
−0.937666 + 0.347538i \(0.887018\pi\)
\(594\) 0 0
\(595\) 7.71639 0.316341
\(596\) 0 0
\(597\) 15.1138 0.618568
\(598\) 0 0
\(599\) 40.2208 1.64338 0.821688 0.569937i \(-0.193032\pi\)
0.821688 + 0.569937i \(0.193032\pi\)
\(600\) 0 0
\(601\) 17.3974 0.709656 0.354828 0.934932i \(-0.384539\pi\)
0.354828 + 0.934932i \(0.384539\pi\)
\(602\) 0 0
\(603\) −1.88273 −0.0766708
\(604\) 0 0
\(605\) 3.60256 0.146465
\(606\) 0 0
\(607\) 14.2277 0.577483 0.288741 0.957407i \(-0.406763\pi\)
0.288741 + 0.957407i \(0.406763\pi\)
\(608\) 0 0
\(609\) 16.3189 0.661277
\(610\) 0 0
\(611\) 11.5569 0.467543
\(612\) 0 0
\(613\) −40.8302 −1.64912 −0.824558 0.565777i \(-0.808576\pi\)
−0.824558 + 0.565777i \(0.808576\pi\)
\(614\) 0 0
\(615\) −2.83709 −0.114403
\(616\) 0 0
\(617\) 5.11383 0.205875 0.102937 0.994688i \(-0.467176\pi\)
0.102937 + 0.994688i \(0.467176\pi\)
\(618\) 0 0
\(619\) −11.5569 −0.464512 −0.232256 0.972655i \(-0.574611\pi\)
−0.232256 + 0.972655i \(0.574611\pi\)
\(620\) 0 0
\(621\) 4.83709 0.194106
\(622\) 0 0
\(623\) 3.16291 0.126719
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 9.67418 0.386350
\(628\) 0 0
\(629\) 12.1335 0.483794
\(630\) 0 0
\(631\) −35.2242 −1.40225 −0.701127 0.713036i \(-0.747319\pi\)
−0.701127 + 0.713036i \(0.747319\pi\)
\(632\) 0 0
\(633\) 18.2277 0.724484
\(634\) 0 0
\(635\) −13.4396 −0.533336
\(636\) 0 0
\(637\) 0.397442 0.0157472
\(638\) 0 0
\(639\) 6.71982 0.265832
\(640\) 0 0
\(641\) −21.9018 −0.865071 −0.432535 0.901617i \(-0.642381\pi\)
−0.432535 + 0.901617i \(0.642381\pi\)
\(642\) 0 0
\(643\) −7.50783 −0.296080 −0.148040 0.988981i \(-0.547296\pi\)
−0.148040 + 0.988981i \(0.547296\pi\)
\(644\) 0 0
\(645\) 11.1138 0.437607
\(646\) 0 0
\(647\) 14.0422 0.552056 0.276028 0.961150i \(-0.410982\pi\)
0.276028 + 0.961150i \(0.410982\pi\)
\(648\) 0 0
\(649\) 5.75859 0.226044
\(650\) 0 0
\(651\) −20.5535 −0.805554
\(652\) 0 0
\(653\) 7.99312 0.312795 0.156398 0.987694i \(-0.450012\pi\)
0.156398 + 0.987694i \(0.450012\pi\)
\(654\) 0 0
\(655\) −9.43965 −0.368838
\(656\) 0 0
\(657\) 9.11383 0.355564
\(658\) 0 0
\(659\) 25.3415 0.987164 0.493582 0.869699i \(-0.335687\pi\)
0.493582 + 0.869699i \(0.335687\pi\)
\(660\) 0 0
\(661\) 27.4396 1.06728 0.533639 0.845712i \(-0.320824\pi\)
0.533639 + 0.845712i \(0.320824\pi\)
\(662\) 0 0
\(663\) −2.83709 −0.110183
\(664\) 0 0
\(665\) −9.67418 −0.375149
\(666\) 0 0
\(667\) 29.0225 1.12376
\(668\) 0 0
\(669\) −10.1173 −0.391156
\(670\) 0 0
\(671\) 17.9578 0.693253
\(672\) 0 0
\(673\) 27.1070 1.04490 0.522448 0.852671i \(-0.325019\pi\)
0.522448 + 0.852671i \(0.325019\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −36.5957 −1.40649 −0.703243 0.710949i \(-0.748265\pi\)
−0.703243 + 0.710949i \(0.748265\pi\)
\(678\) 0 0
\(679\) −29.4750 −1.13115
\(680\) 0 0
\(681\) −11.3224 −0.433875
\(682\) 0 0
\(683\) 13.4656 0.515248 0.257624 0.966245i \(-0.417060\pi\)
0.257624 + 0.966245i \(0.417060\pi\)
\(684\) 0 0
\(685\) −1.76547 −0.0674550
\(686\) 0 0
\(687\) −6.23453 −0.237862
\(688\) 0 0
\(689\) 1.16291 0.0443033
\(690\) 0 0
\(691\) −29.5500 −1.12414 −0.562068 0.827091i \(-0.689994\pi\)
−0.562068 + 0.827091i \(0.689994\pi\)
\(692\) 0 0
\(693\) 7.39744 0.281006
\(694\) 0 0
\(695\) −6.27674 −0.238090
\(696\) 0 0
\(697\) −8.04908 −0.304881
\(698\) 0 0
\(699\) 6.83709 0.258603
\(700\) 0 0
\(701\) 43.6604 1.64903 0.824516 0.565839i \(-0.191448\pi\)
0.824516 + 0.565839i \(0.191448\pi\)
\(702\) 0 0
\(703\) −15.2120 −0.573731
\(704\) 0 0
\(705\) −11.5569 −0.435259
\(706\) 0 0
\(707\) 20.8724 0.784988
\(708\) 0 0
\(709\) −26.7880 −1.00604 −0.503022 0.864273i \(-0.667779\pi\)
−0.503022 + 0.864273i \(0.667779\pi\)
\(710\) 0 0
\(711\) −10.2767 −0.385408
\(712\) 0 0
\(713\) −36.5535 −1.36894
\(714\) 0 0
\(715\) −2.71982 −0.101716
\(716\) 0 0
\(717\) −1.28018 −0.0478091
\(718\) 0 0
\(719\) 34.8793 1.30078 0.650389 0.759601i \(-0.274606\pi\)
0.650389 + 0.759601i \(0.274606\pi\)
\(720\) 0 0
\(721\) −10.2414 −0.381410
\(722\) 0 0
\(723\) −6.00000 −0.223142
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −37.4396 −1.38856 −0.694280 0.719705i \(-0.744277\pi\)
−0.694280 + 0.719705i \(0.744277\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 31.5309 1.16621
\(732\) 0 0
\(733\) −47.1560 −1.74175 −0.870874 0.491506i \(-0.836446\pi\)
−0.870874 + 0.491506i \(0.836446\pi\)
\(734\) 0 0
\(735\) −0.397442 −0.0146599
\(736\) 0 0
\(737\) −5.12070 −0.188624
\(738\) 0 0
\(739\) 31.7914 1.16947 0.584734 0.811225i \(-0.301199\pi\)
0.584734 + 0.811225i \(0.301199\pi\)
\(740\) 0 0
\(741\) 3.55691 0.130667
\(742\) 0 0
\(743\) −16.6776 −0.611842 −0.305921 0.952057i \(-0.598964\pi\)
−0.305921 + 0.952057i \(0.598964\pi\)
\(744\) 0 0
\(745\) 20.8302 0.763160
\(746\) 0 0
\(747\) −2.11727 −0.0774667
\(748\) 0 0
\(749\) 34.2767 1.25244
\(750\) 0 0
\(751\) −16.1855 −0.590616 −0.295308 0.955402i \(-0.595422\pi\)
−0.295308 + 0.955402i \(0.595422\pi\)
\(752\) 0 0
\(753\) −18.2277 −0.664253
\(754\) 0 0
\(755\) 4.99656 0.181844
\(756\) 0 0
\(757\) 12.3258 0.447990 0.223995 0.974590i \(-0.428090\pi\)
0.223995 + 0.974590i \(0.428090\pi\)
\(758\) 0 0
\(759\) 13.1560 0.477534
\(760\) 0 0
\(761\) −0.00687569 −0.000249244 0 −0.000124622 1.00000i \(-0.500040\pi\)
−0.000124622 1.00000i \(0.500040\pi\)
\(762\) 0 0
\(763\) 31.1138 1.12640
\(764\) 0 0
\(765\) 2.83709 0.102575
\(766\) 0 0
\(767\) 2.11727 0.0764501
\(768\) 0 0
\(769\) −20.3258 −0.732968 −0.366484 0.930424i \(-0.619439\pi\)
−0.366484 + 0.930424i \(0.619439\pi\)
\(770\) 0 0
\(771\) 1.11383 0.0401136
\(772\) 0 0
\(773\) 9.90184 0.356144 0.178072 0.984017i \(-0.443014\pi\)
0.178072 + 0.984017i \(0.443014\pi\)
\(774\) 0 0
\(775\) −7.55691 −0.271452
\(776\) 0 0
\(777\) −11.6320 −0.417295
\(778\) 0 0
\(779\) 10.0913 0.361558
\(780\) 0 0
\(781\) 18.2767 0.653993
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −8.87930 −0.316916
\(786\) 0 0
\(787\) −36.3449 −1.29556 −0.647778 0.761829i \(-0.724302\pi\)
−0.647778 + 0.761829i \(0.724302\pi\)
\(788\) 0 0
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) −35.6673 −1.26818
\(792\) 0 0
\(793\) 6.60256 0.234464
\(794\) 0 0
\(795\) −1.16291 −0.0412442
\(796\) 0 0
\(797\) 18.8371 0.667244 0.333622 0.942707i \(-0.391729\pi\)
0.333622 + 0.942707i \(0.391729\pi\)
\(798\) 0 0
\(799\) −32.7880 −1.15996
\(800\) 0 0
\(801\) 1.16291 0.0410894
\(802\) 0 0
\(803\) 24.7880 0.874750
\(804\) 0 0
\(805\) −13.1560 −0.463689
\(806\) 0 0
\(807\) 15.6742 0.551757
\(808\) 0 0
\(809\) 32.2277 1.13306 0.566532 0.824040i \(-0.308285\pi\)
0.566532 + 0.824040i \(0.308285\pi\)
\(810\) 0 0
\(811\) 23.0034 0.807760 0.403880 0.914812i \(-0.367661\pi\)
0.403880 + 0.914812i \(0.367661\pi\)
\(812\) 0 0
\(813\) −0.443086 −0.0155397
\(814\) 0 0
\(815\) −13.8337 −0.484572
\(816\) 0 0
\(817\) −39.5309 −1.38301
\(818\) 0 0
\(819\) 2.71982 0.0950383
\(820\) 0 0
\(821\) 49.9372 1.74282 0.871410 0.490556i \(-0.163206\pi\)
0.871410 + 0.490556i \(0.163206\pi\)
\(822\) 0 0
\(823\) −28.2345 −0.984194 −0.492097 0.870540i \(-0.663769\pi\)
−0.492097 + 0.870540i \(0.663769\pi\)
\(824\) 0 0
\(825\) 2.71982 0.0946921
\(826\) 0 0
\(827\) 9.55004 0.332087 0.166044 0.986118i \(-0.446901\pi\)
0.166044 + 0.986118i \(0.446901\pi\)
\(828\) 0 0
\(829\) −37.9862 −1.31932 −0.659658 0.751565i \(-0.729299\pi\)
−0.659658 + 0.751565i \(0.729299\pi\)
\(830\) 0 0
\(831\) −4.87930 −0.169261
\(832\) 0 0
\(833\) −1.12758 −0.0390683
\(834\) 0 0
\(835\) −9.88273 −0.342006
\(836\) 0 0
\(837\) −7.55691 −0.261205
\(838\) 0 0
\(839\) 4.72670 0.163184 0.0815919 0.996666i \(-0.474000\pi\)
0.0815919 + 0.996666i \(0.474000\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −9.11383 −0.313897
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −9.79832 −0.336674
\(848\) 0 0
\(849\) −33.3415 −1.14428
\(850\) 0 0
\(851\) −20.6870 −0.709140
\(852\) 0 0
\(853\) −48.3611 −1.65585 −0.827927 0.560836i \(-0.810480\pi\)
−0.827927 + 0.560836i \(0.810480\pi\)
\(854\) 0 0
\(855\) −3.55691 −0.121644
\(856\) 0 0
\(857\) 6.83709 0.233551 0.116775 0.993158i \(-0.462744\pi\)
0.116775 + 0.993158i \(0.462744\pi\)
\(858\) 0 0
\(859\) −12.6026 −0.429994 −0.214997 0.976615i \(-0.568974\pi\)
−0.214997 + 0.976615i \(0.568974\pi\)
\(860\) 0 0
\(861\) 7.71639 0.262974
\(862\) 0 0
\(863\) −8.20855 −0.279422 −0.139711 0.990192i \(-0.544617\pi\)
−0.139711 + 0.990192i \(0.544617\pi\)
\(864\) 0 0
\(865\) −13.1138 −0.445884
\(866\) 0 0
\(867\) −8.95092 −0.303989
\(868\) 0 0
\(869\) −27.9509 −0.948170
\(870\) 0 0
\(871\) −1.88273 −0.0637940
\(872\) 0 0
\(873\) −10.8371 −0.366780
\(874\) 0 0
\(875\) −2.71982 −0.0919468
\(876\) 0 0
\(877\) 13.5309 0.456907 0.228454 0.973555i \(-0.426633\pi\)
0.228454 + 0.973555i \(0.426633\pi\)
\(878\) 0 0
\(879\) −29.4328 −0.992743
\(880\) 0 0
\(881\) −9.34836 −0.314954 −0.157477 0.987523i \(-0.550336\pi\)
−0.157477 + 0.987523i \(0.550336\pi\)
\(882\) 0 0
\(883\) 55.1001 1.85427 0.927133 0.374733i \(-0.122266\pi\)
0.927133 + 0.374733i \(0.122266\pi\)
\(884\) 0 0
\(885\) −2.11727 −0.0711711
\(886\) 0 0
\(887\) −0.133492 −0.00448223 −0.00224112 0.999997i \(-0.500713\pi\)
−0.00224112 + 0.999997i \(0.500713\pi\)
\(888\) 0 0
\(889\) 36.5535 1.22596
\(890\) 0 0
\(891\) 2.71982 0.0911175
\(892\) 0 0
\(893\) 41.1070 1.37559
\(894\) 0 0
\(895\) 8.55348 0.285911
\(896\) 0 0
\(897\) 4.83709 0.161506
\(898\) 0 0
\(899\) −45.3415 −1.51222
\(900\) 0 0
\(901\) −3.29928 −0.109915
\(902\) 0 0
\(903\) −30.2277 −1.00591
\(904\) 0 0
\(905\) −3.72326 −0.123765
\(906\) 0 0
\(907\) 58.5466 1.94401 0.972004 0.234964i \(-0.0754973\pi\)
0.972004 + 0.234964i \(0.0754973\pi\)
\(908\) 0 0
\(909\) 7.67418 0.254537
\(910\) 0 0
\(911\) −50.4622 −1.67189 −0.835943 0.548816i \(-0.815079\pi\)
−0.835943 + 0.548816i \(0.815079\pi\)
\(912\) 0 0
\(913\) −5.75859 −0.190582
\(914\) 0 0
\(915\) −6.60256 −0.218274
\(916\) 0 0
\(917\) 25.6742 0.847836
\(918\) 0 0
\(919\) −56.9735 −1.87938 −0.939691 0.342026i \(-0.888887\pi\)
−0.939691 + 0.342026i \(0.888887\pi\)
\(920\) 0 0
\(921\) 21.8337 0.719443
\(922\) 0 0
\(923\) 6.71982 0.221186
\(924\) 0 0
\(925\) −4.27674 −0.140618
\(926\) 0 0
\(927\) −3.76547 −0.123674
\(928\) 0 0
\(929\) −36.5957 −1.20067 −0.600333 0.799750i \(-0.704965\pi\)
−0.600333 + 0.799750i \(0.704965\pi\)
\(930\) 0 0
\(931\) 1.41367 0.0463311
\(932\) 0 0
\(933\) −25.1070 −0.821965
\(934\) 0 0
\(935\) 7.71639 0.252353
\(936\) 0 0
\(937\) −47.1070 −1.53892 −0.769459 0.638697i \(-0.779474\pi\)
−0.769459 + 0.638697i \(0.779474\pi\)
\(938\) 0 0
\(939\) 8.22766 0.268499
\(940\) 0 0
\(941\) −36.3611 −1.18534 −0.592670 0.805446i \(-0.701926\pi\)
−0.592670 + 0.805446i \(0.701926\pi\)
\(942\) 0 0
\(943\) 13.7233 0.446891
\(944\) 0 0
\(945\) −2.71982 −0.0884759
\(946\) 0 0
\(947\) −44.5795 −1.44864 −0.724319 0.689465i \(-0.757846\pi\)
−0.724319 + 0.689465i \(0.757846\pi\)
\(948\) 0 0
\(949\) 9.11383 0.295847
\(950\) 0 0
\(951\) 27.6742 0.897397
\(952\) 0 0
\(953\) 19.8596 0.643317 0.321658 0.946856i \(-0.395760\pi\)
0.321658 + 0.946856i \(0.395760\pi\)
\(954\) 0 0
\(955\) −4.23453 −0.137026
\(956\) 0 0
\(957\) 16.3189 0.527517
\(958\) 0 0
\(959\) 4.80176 0.155057
\(960\) 0 0
\(961\) 26.1070 0.842160
\(962\) 0 0
\(963\) 12.6026 0.406112
\(964\) 0 0
\(965\) 23.3906 0.752969
\(966\) 0 0
\(967\) −47.4068 −1.52450 −0.762250 0.647283i \(-0.775905\pi\)
−0.762250 + 0.647283i \(0.775905\pi\)
\(968\) 0 0
\(969\) −10.0913 −0.324179
\(970\) 0 0
\(971\) 10.6448 0.341607 0.170803 0.985305i \(-0.445364\pi\)
0.170803 + 0.985305i \(0.445364\pi\)
\(972\) 0 0
\(973\) 17.0716 0.547291
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) −1.21199 −0.0387750 −0.0193875 0.999812i \(-0.506172\pi\)
−0.0193875 + 0.999812i \(0.506172\pi\)
\(978\) 0 0
\(979\) 3.16291 0.101087
\(980\) 0 0
\(981\) 11.4396 0.365240
\(982\) 0 0
\(983\) −51.8759 −1.65458 −0.827291 0.561773i \(-0.810119\pi\)
−0.827291 + 0.561773i \(0.810119\pi\)
\(984\) 0 0
\(985\) −14.5535 −0.463712
\(986\) 0 0
\(987\) 31.4328 1.00052
\(988\) 0 0
\(989\) −53.7586 −1.70942
\(990\) 0 0
\(991\) 21.6251 0.686944 0.343472 0.939163i \(-0.388397\pi\)
0.343472 + 0.939163i \(0.388397\pi\)
\(992\) 0 0
\(993\) 13.2311 0.419876
\(994\) 0 0
\(995\) −15.1138 −0.479141
\(996\) 0 0
\(997\) 23.2051 0.734913 0.367457 0.930041i \(-0.380229\pi\)
0.367457 + 0.930041i \(0.380229\pi\)
\(998\) 0 0
\(999\) −4.27674 −0.135310
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 3120.2.a.bj.1.3 3
3.2 odd 2 9360.2.a.dd.1.3 3
4.3 odd 2 195.2.a.e.1.3 3
12.11 even 2 585.2.a.n.1.1 3
20.3 even 4 975.2.c.i.274.1 6
20.7 even 4 975.2.c.i.274.6 6
20.19 odd 2 975.2.a.o.1.1 3
28.27 even 2 9555.2.a.bq.1.3 3
52.51 odd 2 2535.2.a.bc.1.1 3
60.23 odd 4 2925.2.c.w.2224.6 6
60.47 odd 4 2925.2.c.w.2224.1 6
60.59 even 2 2925.2.a.bh.1.3 3
156.155 even 2 7605.2.a.bx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.3 3 4.3 odd 2
585.2.a.n.1.1 3 12.11 even 2
975.2.a.o.1.1 3 20.19 odd 2
975.2.c.i.274.1 6 20.3 even 4
975.2.c.i.274.6 6 20.7 even 4
2535.2.a.bc.1.1 3 52.51 odd 2
2925.2.a.bh.1.3 3 60.59 even 2
2925.2.c.w.2224.1 6 60.47 odd 4
2925.2.c.w.2224.6 6 60.23 odd 4
3120.2.a.bj.1.3 3 1.1 even 1 trivial
7605.2.a.bx.1.3 3 156.155 even 2
9360.2.a.dd.1.3 3 3.2 odd 2
9555.2.a.bq.1.3 3 28.27 even 2