Properties

Label 1728.4.c.j.1727.5
Level $1728$
Weight $4$
Character 1728.1727
Analytic conductor $101.955$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + 6854 x^{2} - 888 x + 9496\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.5
Root \(-0.453986 - 2.07664i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.4.c.j.1727.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.49508i q^{5} -26.1852i q^{7} +O(q^{10})\) \(q-1.49508i q^{5} -26.1852i q^{7} +56.3941 q^{11} +41.3170 q^{13} +51.0410i q^{17} +79.0640i q^{19} -27.3688 q^{23} +122.765 q^{25} +134.567i q^{29} -187.192i q^{31} -39.1491 q^{35} +196.585 q^{37} +298.015i q^{41} +465.576i q^{43} +373.845 q^{47} -342.667 q^{49} +620.093i q^{53} -84.3140i q^{55} -321.152 q^{59} -674.699 q^{61} -61.7724i q^{65} -576.075i q^{67} -223.813 q^{71} +70.1371 q^{73} -1476.69i q^{77} +1052.32i q^{79} +1219.05 q^{83} +76.3107 q^{85} -1340.64i q^{89} -1081.89i q^{91} +118.207 q^{95} -576.059 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q + 72q^{13} - 384q^{25} + 240q^{37} + 288q^{49} - 144q^{61} + 156q^{73} + 168q^{85} + 516q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.49508i − 0.133724i −0.997762 0.0668622i \(-0.978701\pi\)
0.997762 0.0668622i \(-0.0212988\pi\)
\(6\) 0 0
\(7\) − 26.1852i − 1.41387i −0.707279 0.706935i \(-0.750077\pi\)
0.707279 0.706935i \(-0.249923\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 56.3941 1.54577 0.772885 0.634546i \(-0.218813\pi\)
0.772885 + 0.634546i \(0.218813\pi\)
\(12\) 0 0
\(13\) 41.3170 0.881482 0.440741 0.897634i \(-0.354716\pi\)
0.440741 + 0.897634i \(0.354716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 51.0410i 0.728192i 0.931361 + 0.364096i \(0.118622\pi\)
−0.931361 + 0.364096i \(0.881378\pi\)
\(18\) 0 0
\(19\) 79.0640i 0.954659i 0.878724 + 0.477330i \(0.158395\pi\)
−0.878724 + 0.477330i \(0.841605\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −27.3688 −0.248121 −0.124061 0.992275i \(-0.539592\pi\)
−0.124061 + 0.992275i \(0.539592\pi\)
\(24\) 0 0
\(25\) 122.765 0.982118
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 134.567i 0.861674i 0.902430 + 0.430837i \(0.141782\pi\)
−0.902430 + 0.430837i \(0.858218\pi\)
\(30\) 0 0
\(31\) − 187.192i − 1.08454i −0.840204 0.542270i \(-0.817565\pi\)
0.840204 0.542270i \(-0.182435\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −39.1491 −0.189069
\(36\) 0 0
\(37\) 196.585 0.873469 0.436734 0.899590i \(-0.356135\pi\)
0.436734 + 0.899590i \(0.356135\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 298.015i 1.13517i 0.823313 + 0.567587i \(0.192123\pi\)
−0.823313 + 0.567587i \(0.807877\pi\)
\(42\) 0 0
\(43\) 465.576i 1.65115i 0.564289 + 0.825577i \(0.309150\pi\)
−0.564289 + 0.825577i \(0.690850\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 373.845 1.16023 0.580116 0.814534i \(-0.303007\pi\)
0.580116 + 0.814534i \(0.303007\pi\)
\(48\) 0 0
\(49\) −342.667 −0.999028
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 620.093i 1.60710i 0.595237 + 0.803550i \(0.297058\pi\)
−0.595237 + 0.803550i \(0.702942\pi\)
\(54\) 0 0
\(55\) − 84.3140i − 0.206707i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −321.152 −0.708652 −0.354326 0.935122i \(-0.615290\pi\)
−0.354326 + 0.935122i \(0.615290\pi\)
\(60\) 0 0
\(61\) −674.699 −1.41617 −0.708085 0.706127i \(-0.750441\pi\)
−0.708085 + 0.706127i \(0.750441\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 61.7724i − 0.117876i
\(66\) 0 0
\(67\) − 576.075i − 1.05043i −0.850970 0.525215i \(-0.823985\pi\)
0.850970 0.525215i \(-0.176015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −223.813 −0.374110 −0.187055 0.982349i \(-0.559894\pi\)
−0.187055 + 0.982349i \(0.559894\pi\)
\(72\) 0 0
\(73\) 70.1371 0.112451 0.0562255 0.998418i \(-0.482093\pi\)
0.0562255 + 0.998418i \(0.482093\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1476.69i − 2.18552i
\(78\) 0 0
\(79\) 1052.32i 1.49868i 0.662187 + 0.749338i \(0.269628\pi\)
−0.662187 + 0.749338i \(0.730372\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1219.05 1.61214 0.806070 0.591820i \(-0.201591\pi\)
0.806070 + 0.591820i \(0.201591\pi\)
\(84\) 0 0
\(85\) 76.3107 0.0973771
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1340.64i − 1.59672i −0.602183 0.798358i \(-0.705702\pi\)
0.602183 0.798358i \(-0.294298\pi\)
\(90\) 0 0
\(91\) − 1081.89i − 1.24630i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 118.207 0.127661
\(96\) 0 0
\(97\) −576.059 −0.602989 −0.301494 0.953468i \(-0.597485\pi\)
−0.301494 + 0.953468i \(0.597485\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 116.079i 0.114359i 0.998364 + 0.0571797i \(0.0182108\pi\)
−0.998364 + 0.0571797i \(0.981789\pi\)
\(102\) 0 0
\(103\) 165.074i 0.157915i 0.996878 + 0.0789573i \(0.0251591\pi\)
−0.996878 + 0.0789573i \(0.974841\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 936.718 0.846318 0.423159 0.906056i \(-0.360921\pi\)
0.423159 + 0.906056i \(0.360921\pi\)
\(108\) 0 0
\(109\) −346.957 −0.304885 −0.152443 0.988312i \(-0.548714\pi\)
−0.152443 + 0.988312i \(0.548714\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1462.22i 1.21729i 0.793443 + 0.608645i \(0.208287\pi\)
−0.793443 + 0.608645i \(0.791713\pi\)
\(114\) 0 0
\(115\) 40.9187i 0.0331799i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1336.52 1.02957
\(120\) 0 0
\(121\) 1849.30 1.38941
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 370.429i − 0.265058i
\(126\) 0 0
\(127\) − 265.004i − 0.185160i −0.995705 0.0925800i \(-0.970489\pi\)
0.995705 0.0925800i \(-0.0295114\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1151.86 0.768231 0.384115 0.923285i \(-0.374507\pi\)
0.384115 + 0.923285i \(0.374507\pi\)
\(132\) 0 0
\(133\) 2070.31 1.34976
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2348.67i − 1.46467i −0.680943 0.732337i \(-0.738430\pi\)
0.680943 0.732337i \(-0.261570\pi\)
\(138\) 0 0
\(139\) − 215.240i − 0.131341i −0.997841 0.0656706i \(-0.979081\pi\)
0.997841 0.0656706i \(-0.0209187\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2330.04 1.36257
\(144\) 0 0
\(145\) 201.190 0.115227
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 219.954i 0.120935i 0.998170 + 0.0604674i \(0.0192591\pi\)
−0.998170 + 0.0604674i \(0.980741\pi\)
\(150\) 0 0
\(151\) 1148.02i 0.618703i 0.950948 + 0.309351i \(0.100112\pi\)
−0.950948 + 0.309351i \(0.899888\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −279.869 −0.145030
\(156\) 0 0
\(157\) 276.320 0.140463 0.0702316 0.997531i \(-0.477626\pi\)
0.0702316 + 0.997531i \(0.477626\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 716.659i 0.350811i
\(162\) 0 0
\(163\) 1419.15i 0.681941i 0.940074 + 0.340971i \(0.110756\pi\)
−0.940074 + 0.340971i \(0.889244\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3508.69 1.62581 0.812906 0.582394i \(-0.197884\pi\)
0.812906 + 0.582394i \(0.197884\pi\)
\(168\) 0 0
\(169\) −489.908 −0.222990
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 330.965i − 0.145450i −0.997352 0.0727248i \(-0.976831\pi\)
0.997352 0.0727248i \(-0.0231695\pi\)
\(174\) 0 0
\(175\) − 3214.62i − 1.38859i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1136.75 0.474662 0.237331 0.971429i \(-0.423727\pi\)
0.237331 + 0.971429i \(0.423727\pi\)
\(180\) 0 0
\(181\) −2056.92 −0.844692 −0.422346 0.906435i \(-0.638793\pi\)
−0.422346 + 0.906435i \(0.638793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 293.911i − 0.116804i
\(186\) 0 0
\(187\) 2878.42i 1.12562i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1983.23 −0.751316 −0.375658 0.926758i \(-0.622583\pi\)
−0.375658 + 0.926758i \(0.622583\pi\)
\(192\) 0 0
\(193\) −189.908 −0.0708283 −0.0354141 0.999373i \(-0.511275\pi\)
−0.0354141 + 0.999373i \(0.511275\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2160.42i 0.781337i 0.920531 + 0.390669i \(0.127756\pi\)
−0.920531 + 0.390669i \(0.872244\pi\)
\(198\) 0 0
\(199\) − 2656.23i − 0.946205i −0.881007 0.473103i \(-0.843134\pi\)
0.881007 0.473103i \(-0.156866\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3523.68 1.21829
\(204\) 0 0
\(205\) 445.558 0.151801
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4458.75i 1.47568i
\(210\) 0 0
\(211\) 1001.20i 0.326662i 0.986571 + 0.163331i \(0.0522239\pi\)
−0.986571 + 0.163331i \(0.947776\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 696.075 0.220800
\(216\) 0 0
\(217\) −4901.68 −1.53340
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2108.86i 0.641888i
\(222\) 0 0
\(223\) − 3193.62i − 0.959015i −0.877538 0.479507i \(-0.840815\pi\)
0.877538 0.479507i \(-0.159185\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1714.72 0.501366 0.250683 0.968069i \(-0.419345\pi\)
0.250683 + 0.968069i \(0.419345\pi\)
\(228\) 0 0
\(229\) 407.497 0.117590 0.0587951 0.998270i \(-0.481274\pi\)
0.0587951 + 0.998270i \(0.481274\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3210.54i 0.902702i 0.892346 + 0.451351i \(0.149058\pi\)
−0.892346 + 0.451351i \(0.850942\pi\)
\(234\) 0 0
\(235\) − 558.930i − 0.155151i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1561.19 0.422532 0.211266 0.977429i \(-0.432241\pi\)
0.211266 + 0.977429i \(0.432241\pi\)
\(240\) 0 0
\(241\) 1460.89 0.390475 0.195238 0.980756i \(-0.437452\pi\)
0.195238 + 0.980756i \(0.437452\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 512.316i 0.133594i
\(246\) 0 0
\(247\) 3266.69i 0.841515i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6868.44 1.72722 0.863609 0.504162i \(-0.168198\pi\)
0.863609 + 0.504162i \(0.168198\pi\)
\(252\) 0 0
\(253\) −1543.44 −0.383539
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 4450.84i − 1.08029i −0.841570 0.540147i \(-0.818368\pi\)
0.841570 0.540147i \(-0.181632\pi\)
\(258\) 0 0
\(259\) − 5147.62i − 1.23497i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6525.31 1.52991 0.764957 0.644081i \(-0.222760\pi\)
0.764957 + 0.644081i \(0.222760\pi\)
\(264\) 0 0
\(265\) 927.091 0.214909
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 3222.22i − 0.730343i −0.930940 0.365171i \(-0.881010\pi\)
0.930940 0.365171i \(-0.118990\pi\)
\(270\) 0 0
\(271\) 7368.93i 1.65177i 0.563837 + 0.825886i \(0.309325\pi\)
−0.563837 + 0.825886i \(0.690675\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6923.21 1.51813
\(276\) 0 0
\(277\) −9078.61 −1.96924 −0.984622 0.174699i \(-0.944105\pi\)
−0.984622 + 0.174699i \(0.944105\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2613.34i 0.554801i 0.960754 + 0.277400i \(0.0894729\pi\)
−0.960754 + 0.277400i \(0.910527\pi\)
\(282\) 0 0
\(283\) − 927.970i − 0.194919i −0.995239 0.0974596i \(-0.968928\pi\)
0.995239 0.0974596i \(-0.0310717\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7803.60 1.60499
\(288\) 0 0
\(289\) 2307.81 0.469736
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5351.73i 1.06707i 0.845778 + 0.533535i \(0.179137\pi\)
−0.845778 + 0.533535i \(0.820863\pi\)
\(294\) 0 0
\(295\) 480.150i 0.0947641i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1130.80 −0.218714
\(300\) 0 0
\(301\) 12191.2 2.33452
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1008.73i 0.189377i
\(306\) 0 0
\(307\) 1892.38i 0.351804i 0.984408 + 0.175902i \(0.0562842\pi\)
−0.984408 + 0.175902i \(0.943716\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5645.22 1.02930 0.514648 0.857402i \(-0.327923\pi\)
0.514648 + 0.857402i \(0.327923\pi\)
\(312\) 0 0
\(313\) −818.001 −0.147719 −0.0738597 0.997269i \(-0.523532\pi\)
−0.0738597 + 0.997269i \(0.523532\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1946.72i − 0.344917i −0.985017 0.172458i \(-0.944829\pi\)
0.985017 0.172458i \(-0.0551710\pi\)
\(318\) 0 0
\(319\) 7588.81i 1.33195i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4035.51 −0.695176
\(324\) 0 0
\(325\) 5072.27 0.865719
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 9789.22i − 1.64042i
\(330\) 0 0
\(331\) − 3404.83i − 0.565396i −0.959209 0.282698i \(-0.908771\pi\)
0.959209 0.282698i \(-0.0912295\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −861.281 −0.140468
\(336\) 0 0
\(337\) 7072.15 1.14316 0.571579 0.820547i \(-0.306331\pi\)
0.571579 + 0.820547i \(0.306331\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 10556.6i − 1.67645i
\(342\) 0 0
\(343\) − 8.73194i − 0.00137458i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5783.02 −0.894665 −0.447332 0.894368i \(-0.647626\pi\)
−0.447332 + 0.894368i \(0.647626\pi\)
\(348\) 0 0
\(349\) −1748.60 −0.268196 −0.134098 0.990968i \(-0.542814\pi\)
−0.134098 + 0.990968i \(0.542814\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 9552.31i − 1.44028i −0.693830 0.720139i \(-0.744078\pi\)
0.693830 0.720139i \(-0.255922\pi\)
\(354\) 0 0
\(355\) 334.620i 0.0500276i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 921.392 0.135457 0.0677287 0.997704i \(-0.478425\pi\)
0.0677287 + 0.997704i \(0.478425\pi\)
\(360\) 0 0
\(361\) 607.881 0.0886254
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 104.861i − 0.0150375i
\(366\) 0 0
\(367\) − 8490.66i − 1.20765i −0.797115 0.603827i \(-0.793642\pi\)
0.797115 0.603827i \(-0.206358\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 16237.3 2.27223
\(372\) 0 0
\(373\) −2824.92 −0.392142 −0.196071 0.980590i \(-0.562818\pi\)
−0.196071 + 0.980590i \(0.562818\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5559.92i 0.759550i
\(378\) 0 0
\(379\) − 5322.35i − 0.721348i −0.932692 0.360674i \(-0.882547\pi\)
0.932692 0.360674i \(-0.117453\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8320.49 −1.11007 −0.555035 0.831827i \(-0.687295\pi\)
−0.555035 + 0.831827i \(0.687295\pi\)
\(384\) 0 0
\(385\) −2207.78 −0.292257
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 8358.52i − 1.08944i −0.838617 0.544722i \(-0.816635\pi\)
0.838617 0.544722i \(-0.183365\pi\)
\(390\) 0 0
\(391\) − 1396.93i − 0.180680i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1573.31 0.200410
\(396\) 0 0
\(397\) −7283.17 −0.920735 −0.460367 0.887728i \(-0.652282\pi\)
−0.460367 + 0.887728i \(0.652282\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1549.41i 0.192952i 0.995335 + 0.0964759i \(0.0307571\pi\)
−0.995335 + 0.0964759i \(0.969243\pi\)
\(402\) 0 0
\(403\) − 7734.22i − 0.956003i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11086.2 1.35018
\(408\) 0 0
\(409\) 3291.67 0.397952 0.198976 0.980004i \(-0.436238\pi\)
0.198976 + 0.980004i \(0.436238\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8409.45i 1.00194i
\(414\) 0 0
\(415\) − 1822.58i − 0.215583i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5299.78 0.617927 0.308964 0.951074i \(-0.400018\pi\)
0.308964 + 0.951074i \(0.400018\pi\)
\(420\) 0 0
\(421\) 10681.4 1.23653 0.618265 0.785970i \(-0.287836\pi\)
0.618265 + 0.785970i \(0.287836\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6266.04i 0.715170i
\(426\) 0 0
\(427\) 17667.2i 2.00228i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9866.13 −1.10263 −0.551316 0.834296i \(-0.685874\pi\)
−0.551316 + 0.834296i \(0.685874\pi\)
\(432\) 0 0
\(433\) −12560.2 −1.39400 −0.697002 0.717069i \(-0.745483\pi\)
−0.697002 + 0.717069i \(0.745483\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2163.89i − 0.236871i
\(438\) 0 0
\(439\) − 929.228i − 0.101024i −0.998723 0.0505121i \(-0.983915\pi\)
0.998723 0.0505121i \(-0.0160854\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4373.85 −0.469092 −0.234546 0.972105i \(-0.575360\pi\)
−0.234546 + 0.972105i \(0.575360\pi\)
\(444\) 0 0
\(445\) −2004.37 −0.213520
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 14149.5i − 1.48721i −0.668618 0.743606i \(-0.733114\pi\)
0.668618 0.743606i \(-0.266886\pi\)
\(450\) 0 0
\(451\) 16806.3i 1.75472i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1617.52 −0.166661
\(456\) 0 0
\(457\) −12582.4 −1.28792 −0.643960 0.765059i \(-0.722710\pi\)
−0.643960 + 0.765059i \(0.722710\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10602.2i 1.07114i 0.844492 + 0.535568i \(0.179902\pi\)
−0.844492 + 0.535568i \(0.820098\pi\)
\(462\) 0 0
\(463\) − 5080.88i − 0.509997i −0.966941 0.254999i \(-0.917925\pi\)
0.966941 0.254999i \(-0.0820750\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6903.92 0.684102 0.342051 0.939681i \(-0.388879\pi\)
0.342051 + 0.939681i \(0.388879\pi\)
\(468\) 0 0
\(469\) −15084.7 −1.48517
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 26255.8i 2.55231i
\(474\) 0 0
\(475\) 9706.27i 0.937588i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5114.56 0.487871 0.243936 0.969791i \(-0.421561\pi\)
0.243936 + 0.969791i \(0.421561\pi\)
\(480\) 0 0
\(481\) 8122.29 0.769947
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 861.257i 0.0806344i
\(486\) 0 0
\(487\) − 15594.2i − 1.45101i −0.688218 0.725504i \(-0.741607\pi\)
0.688218 0.725504i \(-0.258393\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10521.0 −0.967023 −0.483511 0.875338i \(-0.660639\pi\)
−0.483511 + 0.875338i \(0.660639\pi\)
\(492\) 0 0
\(493\) −6868.46 −0.627464
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5860.61i 0.528942i
\(498\) 0 0
\(499\) − 8984.43i − 0.806009i −0.915198 0.403004i \(-0.867966\pi\)
0.915198 0.403004i \(-0.132034\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4299.12 0.381090 0.190545 0.981678i \(-0.438974\pi\)
0.190545 + 0.981678i \(0.438974\pi\)
\(504\) 0 0
\(505\) 173.548 0.0152926
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2984.83i 0.259922i 0.991519 + 0.129961i \(0.0414853\pi\)
−0.991519 + 0.129961i \(0.958515\pi\)
\(510\) 0 0
\(511\) − 1836.56i − 0.158991i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 246.799 0.0211170
\(516\) 0 0
\(517\) 21082.7 1.79345
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 12255.7i − 1.03058i −0.857016 0.515290i \(-0.827684\pi\)
0.857016 0.515290i \(-0.172316\pi\)
\(522\) 0 0
\(523\) − 15148.5i − 1.26654i −0.773932 0.633269i \(-0.781713\pi\)
0.773932 0.633269i \(-0.218287\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9554.49 0.789754
\(528\) 0 0
\(529\) −11417.9 −0.938436
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12313.1i 1.00064i
\(534\) 0 0
\(535\) − 1400.47i − 0.113173i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −19324.4 −1.54427
\(540\) 0 0
\(541\) −5723.84 −0.454875 −0.227437 0.973793i \(-0.573035\pi\)
−0.227437 + 0.973793i \(0.573035\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 518.730i 0.0407706i
\(546\) 0 0
\(547\) 8367.43i 0.654051i 0.945016 + 0.327025i \(0.106046\pi\)
−0.945016 + 0.327025i \(0.893954\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10639.4 −0.822605
\(552\) 0 0
\(553\) 27555.3 2.11893
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15824.2i 1.20375i 0.798589 + 0.601877i \(0.205580\pi\)
−0.798589 + 0.601877i \(0.794420\pi\)
\(558\) 0 0
\(559\) 19236.2i 1.45546i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2781.72 −0.208233 −0.104117 0.994565i \(-0.533202\pi\)
−0.104117 + 0.994565i \(0.533202\pi\)
\(564\) 0 0
\(565\) 2186.14 0.162781
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 4687.63i − 0.345370i −0.984977 0.172685i \(-0.944756\pi\)
0.984977 0.172685i \(-0.0552443\pi\)
\(570\) 0 0
\(571\) − 15169.4i − 1.11177i −0.831259 0.555885i \(-0.812379\pi\)
0.831259 0.555885i \(-0.187621\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3359.92 −0.243684
\(576\) 0 0
\(577\) −14452.3 −1.04273 −0.521367 0.853332i \(-0.674578\pi\)
−0.521367 + 0.853332i \(0.674578\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 31921.0i − 2.27936i
\(582\) 0 0
\(583\) 34969.6i 2.48421i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7405.55 −0.520715 −0.260358 0.965512i \(-0.583840\pi\)
−0.260358 + 0.965512i \(0.583840\pi\)
\(588\) 0 0
\(589\) 14800.2 1.03537
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19888.1i 1.37725i 0.725120 + 0.688623i \(0.241784\pi\)
−0.725120 + 0.688623i \(0.758216\pi\)
\(594\) 0 0
\(595\) − 1998.21i − 0.137679i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2335.49 0.159308 0.0796539 0.996823i \(-0.474618\pi\)
0.0796539 + 0.996823i \(0.474618\pi\)
\(600\) 0 0
\(601\) 24547.9 1.66611 0.833054 0.553192i \(-0.186590\pi\)
0.833054 + 0.553192i \(0.186590\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 2764.86i − 0.185798i
\(606\) 0 0
\(607\) − 18328.9i − 1.22561i −0.790233 0.612806i \(-0.790041\pi\)
0.790233 0.612806i \(-0.209959\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15446.1 1.02272
\(612\) 0 0
\(613\) −6450.01 −0.424981 −0.212491 0.977163i \(-0.568157\pi\)
−0.212491 + 0.977163i \(0.568157\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6724.96i − 0.438795i −0.975636 0.219398i \(-0.929591\pi\)
0.975636 0.219398i \(-0.0704092\pi\)
\(618\) 0 0
\(619\) − 3136.55i − 0.203665i −0.994802 0.101832i \(-0.967529\pi\)
0.994802 0.101832i \(-0.0324705\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −35105.0 −2.25755
\(624\) 0 0
\(625\) 14791.8 0.946673
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10033.9i 0.636053i
\(630\) 0 0
\(631\) − 6061.57i − 0.382420i −0.981549 0.191210i \(-0.938759\pi\)
0.981549 0.191210i \(-0.0612412\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −396.204 −0.0247604
\(636\) 0 0
\(637\) −14157.9 −0.880625
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 18603.3i − 1.14631i −0.819446 0.573157i \(-0.805719\pi\)
0.819446 0.573157i \(-0.194281\pi\)
\(642\) 0 0
\(643\) − 21602.4i − 1.32491i −0.749103 0.662453i \(-0.769515\pi\)
0.749103 0.662453i \(-0.230485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −672.754 −0.0408790 −0.0204395 0.999791i \(-0.506507\pi\)
−0.0204395 + 0.999791i \(0.506507\pi\)
\(648\) 0 0
\(649\) −18111.1 −1.09541
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 3322.88i − 0.199134i −0.995031 0.0995668i \(-0.968254\pi\)
0.995031 0.0995668i \(-0.0317457\pi\)
\(654\) 0 0
\(655\) − 1722.12i − 0.102731i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −25093.7 −1.48332 −0.741662 0.670773i \(-0.765962\pi\)
−0.741662 + 0.670773i \(0.765962\pi\)
\(660\) 0 0
\(661\) 28966.3 1.70447 0.852237 0.523157i \(-0.175246\pi\)
0.852237 + 0.523157i \(0.175246\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 3095.29i − 0.180496i
\(666\) 0 0
\(667\) − 3682.95i − 0.213800i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −38049.1 −2.18907
\(672\) 0 0
\(673\) −4710.36 −0.269793 −0.134897 0.990860i \(-0.543070\pi\)
−0.134897 + 0.990860i \(0.543070\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4784.53i 0.271617i 0.990735 + 0.135808i \(0.0433631\pi\)
−0.990735 + 0.135808i \(0.956637\pi\)
\(678\) 0 0
\(679\) 15084.2i 0.852548i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18019.8 −1.00953 −0.504764 0.863258i \(-0.668420\pi\)
−0.504764 + 0.863258i \(0.668420\pi\)
\(684\) 0 0
\(685\) −3511.46 −0.195863
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 25620.4i 1.41663i
\(690\) 0 0
\(691\) 16956.5i 0.933511i 0.884386 + 0.466756i \(0.154577\pi\)
−0.884386 + 0.466756i \(0.845423\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −321.802 −0.0175635
\(696\) 0 0
\(697\) −15211.0 −0.826625
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 30620.5i − 1.64982i −0.565266 0.824909i \(-0.691226\pi\)
0.565266 0.824909i \(-0.308774\pi\)
\(702\) 0 0
\(703\) 15542.8i 0.833865i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3039.56 0.161689
\(708\) 0 0
\(709\) −24192.8 −1.28149 −0.640747 0.767752i \(-0.721375\pi\)
−0.640747 + 0.767752i \(0.721375\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5123.23i 0.269098i
\(714\) 0 0
\(715\) − 3483.60i − 0.182209i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −37146.6 −1.92675 −0.963377 0.268151i \(-0.913588\pi\)
−0.963377 + 0.268151i \(0.913588\pi\)
\(720\) 0 0
\(721\) 4322.49 0.223271
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 16520.1i 0.846265i
\(726\) 0 0
\(727\) − 14614.3i − 0.745551i −0.927922 0.372775i \(-0.878406\pi\)
0.927922 0.372775i \(-0.121594\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −23763.5 −1.20236
\(732\) 0 0
\(733\) 8044.73 0.405374 0.202687 0.979244i \(-0.435033\pi\)
0.202687 + 0.979244i \(0.435033\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 32487.3i − 1.62372i
\(738\) 0 0
\(739\) − 7025.01i − 0.349688i −0.984596 0.174844i \(-0.944058\pi\)
0.984596 0.174844i \(-0.0559420\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27063.4 1.33629 0.668143 0.744033i \(-0.267089\pi\)
0.668143 + 0.744033i \(0.267089\pi\)
\(744\) 0 0
\(745\) 328.849 0.0161719
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 24528.2i − 1.19658i
\(750\) 0 0
\(751\) 11434.2i 0.555579i 0.960642 + 0.277789i \(0.0896017\pi\)
−0.960642 + 0.277789i \(0.910398\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1716.38 0.0827357
\(756\) 0 0
\(757\) −22087.2 −1.06046 −0.530232 0.847853i \(-0.677895\pi\)
−0.530232 + 0.847853i \(0.677895\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 35524.0i − 1.69218i −0.533043 0.846088i \(-0.678952\pi\)
0.533043 0.846088i \(-0.321048\pi\)
\(762\) 0 0
\(763\) 9085.16i 0.431068i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13269.0 −0.624664
\(768\) 0 0
\(769\) 21213.0 0.994745 0.497372 0.867537i \(-0.334298\pi\)
0.497372 + 0.867537i \(0.334298\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14370.7i 0.668663i 0.942456 + 0.334332i \(0.108511\pi\)
−0.942456 + 0.334332i \(0.891489\pi\)
\(774\) 0 0
\(775\) − 22980.6i − 1.06515i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −23562.3 −1.08371
\(780\) 0 0
\(781\) −12621.8 −0.578287
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 413.122i − 0.0187834i
\(786\) 0 0
\(787\) 41642.9i 1.88616i 0.332564 + 0.943081i \(0.392086\pi\)
−0.332564 + 0.943081i \(0.607914\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 38288.5 1.72109
\(792\) 0 0
\(793\) −27876.5 −1.24833
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 23527.1i − 1.04564i −0.852445 0.522818i \(-0.824881\pi\)
0.852445 0.522818i \(-0.175119\pi\)
\(798\) 0 0
\(799\) 19081.4i 0.844872i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3955.32 0.173823
\(804\) 0 0
\(805\) 1071.47 0.0469120
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7508.22i 0.326298i 0.986601 + 0.163149i \(0.0521651\pi\)
−0.986601 + 0.163149i \(0.947835\pi\)
\(810\) 0 0
\(811\) 15834.0i 0.685581i 0.939412 + 0.342791i \(0.111372\pi\)
−0.939412 + 0.342791i \(0.888628\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2121.75 0.0911923
\(816\) 0 0
\(817\) −36810.3 −1.57629
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 13279.1i − 0.564486i −0.959343 0.282243i \(-0.908922\pi\)
0.959343 0.282243i \(-0.0910784\pi\)
\(822\) 0 0
\(823\) − 27997.4i − 1.18582i −0.805270 0.592909i \(-0.797979\pi\)
0.805270 0.592909i \(-0.202021\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35177.8 −1.47914 −0.739571 0.673078i \(-0.764972\pi\)
−0.739571 + 0.673078i \(0.764972\pi\)
\(828\) 0 0
\(829\) 13390.3 0.560996 0.280498 0.959855i \(-0.409500\pi\)
0.280498 + 0.959855i \(0.409500\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 17490.1i − 0.727484i
\(834\) 0 0
\(835\) − 5245.79i − 0.217411i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8466.38 −0.348381 −0.174191 0.984712i \(-0.555731\pi\)
−0.174191 + 0.984712i \(0.555731\pi\)
\(840\) 0 0
\(841\) 6280.62 0.257518
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 732.454i 0.0298192i
\(846\) 0 0
\(847\) − 48424.4i − 1.96444i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5380.29 −0.216726
\(852\) 0 0
\(853\) 16448.2 0.660229 0.330114 0.943941i \(-0.392913\pi\)
0.330114 + 0.943941i \(0.392913\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13593.7i 0.541832i 0.962603 + 0.270916i \(0.0873266\pi\)
−0.962603 + 0.270916i \(0.912673\pi\)
\(858\) 0 0
\(859\) 15107.9i 0.600086i 0.953926 + 0.300043i \(0.0970011\pi\)
−0.953926 + 0.300043i \(0.902999\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −43717.5 −1.72440 −0.862202 0.506565i \(-0.830915\pi\)
−0.862202 + 0.506565i \(0.830915\pi\)
\(864\) 0 0
\(865\) −494.820 −0.0194502
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 59344.8i 2.31661i
\(870\) 0 0
\(871\) − 23801.7i − 0.925935i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9699.78 −0.374757
\(876\) 0 0
\(877\) 23868.3 0.919012 0.459506 0.888175i \(-0.348026\pi\)
0.459506 + 0.888175i \(0.348026\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48039.2i 1.83710i 0.395310 + 0.918548i \(0.370637\pi\)
−0.395310 + 0.918548i \(0.629363\pi\)
\(882\) 0 0
\(883\) − 10091.3i − 0.384595i −0.981337 0.192298i \(-0.938406\pi\)
0.981337 0.192298i \(-0.0615939\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −37951.2 −1.43661 −0.718307 0.695726i \(-0.755083\pi\)
−0.718307 + 0.695726i \(0.755083\pi\)
\(888\) 0 0
\(889\) −6939.20 −0.261792
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 29557.7i 1.10763i
\(894\) 0 0
\(895\) − 1699.53i − 0.0634739i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 25190.0 0.934520
\(900\) 0 0
\(901\) −31650.2 −1.17028
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3075.26i 0.112956i
\(906\) 0 0
\(907\) 23697.4i 0.867540i 0.901024 + 0.433770i \(0.142817\pi\)
−0.901024 + 0.433770i \(0.857183\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 11833.0 0.430347 0.215174 0.976576i \(-0.430968\pi\)
0.215174 + 0.976576i \(0.430968\pi\)
\(912\) 0 0
\(913\) 68747.0 2.49200
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 30161.6i − 1.08618i
\(918\) 0 0
\(919\) 51576.2i 1.85130i 0.378386 + 0.925648i \(0.376479\pi\)
−0.378386 + 0.925648i \(0.623521\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −9247.29 −0.329771
\(924\) 0 0
\(925\) 24133.7 0.857849
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46577.3i 1.64494i 0.568807 + 0.822471i \(0.307405\pi\)
−0.568807 + 0.822471i \(0.692595\pi\)
\(930\) 0 0
\(931\) − 27092.6i − 0.953731i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4303.47 0.150523
\(936\) 0 0
\(937\) 11409.2 0.397783 0.198891 0.980022i \(-0.436266\pi\)
0.198891 + 0.980022i \(0.436266\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 32444.9i 1.12399i 0.827141 + 0.561994i \(0.189966\pi\)
−0.827141 + 0.561994i \(0.810034\pi\)
\(942\) 0 0
\(943\) − 8156.32i − 0.281661i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22327.6 0.766154 0.383077 0.923716i \(-0.374864\pi\)
0.383077 + 0.923716i \(0.374864\pi\)
\(948\) 0 0
\(949\) 2897.85 0.0991236
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 15314.3i 0.520546i 0.965535 + 0.260273i \(0.0838126\pi\)
−0.965535 + 0.260273i \(0.916187\pi\)
\(954\) 0 0
\(955\) 2965.09i 0.100469i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −61500.4 −2.07086
\(960\) 0 0
\(961\) −5250.01 −0.176228
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 283.928i 0.00947147i
\(966\) 0 0
\(967\) 16913.6i 0.562465i 0.959640 + 0.281232i \(0.0907431\pi\)
−0.959640 + 0.281232i \(0.909257\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −56451.5 −1.86572 −0.932861 0.360237i \(-0.882696\pi\)
−0.932861 + 0.360237i \(0.882696\pi\)
\(972\) 0 0
\(973\) −5636.12 −0.185699
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 7397.50i − 0.242238i −0.992638 0.121119i \(-0.961352\pi\)
0.992638 0.121119i \(-0.0386483\pi\)
\(978\) 0 0
\(979\) − 75604.3i − 2.46816i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18261.7 0.592532 0.296266 0.955105i \(-0.404258\pi\)
0.296266 + 0.955105i \(0.404258\pi\)
\(984\) 0 0
\(985\) 3230.01 0.104484
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 12742.3i − 0.409687i
\(990\) 0 0
\(991\) 7020.98i 0.225054i 0.993649 + 0.112527i \(0.0358945\pi\)
−0.993649 + 0.112527i \(0.964105\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3971.28 −0.126531
\(996\) 0 0
\(997\) −6983.34 −0.221830 −0.110915 0.993830i \(-0.535378\pi\)
−0.110915 + 0.993830i \(0.535378\pi\)
\(998\) 0 0
\(999\) 0 0
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 1728.4.c.j.1727.5 12
3.2 odd 2 inner 1728.4.c.j.1727.7 12
4.3 odd 2 inner 1728.4.c.j.1727.6 12
8.3 odd 2 108.4.b.b.107.5 12
8.5 even 2 108.4.b.b.107.7 yes 12
12.11 even 2 inner 1728.4.c.j.1727.8 12
24.5 odd 2 108.4.b.b.107.6 yes 12
24.11 even 2 108.4.b.b.107.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.b.b.107.5 12 8.3 odd 2
108.4.b.b.107.6 yes 12 24.5 odd 2
108.4.b.b.107.7 yes 12 8.5 even 2
108.4.b.b.107.8 yes 12 24.11 even 2
1728.4.c.j.1727.5 12 1.1 even 1 trivial
1728.4.c.j.1727.6 12 4.3 odd 2 inner
1728.4.c.j.1727.7 12 3.2 odd 2 inner
1728.4.c.j.1727.8 12 12.11 even 2 inner