Properties

Label 1728.4.c.j.1727.7
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 1727.7
Root \(-2.18604 + 2.07664i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.4.c.j.1727.6

$q$-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 \(0.0212988\pi\)
−0.997762 + 0.0668622i \(0.978701\pi\)
\(6\) 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 \(-0.781187\pi\)
−0.772885 + 0.634546i \(0.781187\pi\)
\(12\) 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 \(-0.881378\pi\)
0.931361 0.364096i \(-0.118622\pi\)
\(18\) 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 \(-0.460408\pi\)
0.124061 + 0.992275i \(0.460408\pi\)
\(24\) 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 \(-0.858218\pi\)
0.902430 0.430837i \(-0.141782\pi\)
\(30\) 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 \(-0.807877\pi\)
0.823313 0.567587i \(-0.192123\pi\)
\(42\) 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 \(-0.696993\pi\)
−0.580116 + 0.814534i \(0.696993\pi\)
\(48\) 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 \(-0.702942\pi\)
0.595237 0.803550i \(-0.297058\pi\)
\(54\) 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 \(-0.384710\pi\)
0.354326 + 0.935122i \(0.384710\pi\)
\(60\) 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 \(-0.440106\pi\)
0.187055 + 0.982349i \(0.440106\pi\)
\(72\) 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 \(-0.798409\pi\)
−0.806070 + 0.591820i \(0.798409\pi\)
\(84\) 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 \(0.294298\pi\)
−0.602183 + 0.798358i \(0.705702\pi\)
\(90\) 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 \(-0.981789\pi\)
0.998364 0.0571797i \(-0.0182108\pi\)
\(102\) 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 \(-0.639079\pi\)
−0.423159 + 0.906056i \(0.639079\pi\)
\(108\) 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 \(-0.791713\pi\)
0.793443 0.608645i \(-0.208287\pi\)
\(114\) 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 \(-0.625493\pi\)
−0.384115 + 0.923285i \(0.625493\pi\)
\(132\) 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 \(0.261570\pi\)
−0.680943 + 0.732337i \(0.738430\pi\)
\(138\) 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 \(-0.980741\pi\)
0.998170 0.0604674i \(-0.0192591\pi\)
\(150\) 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 \(-0.802116\pi\)
−0.812906 + 0.582394i \(0.802116\pi\)
\(168\) 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 \(0.0231695\pi\)
−0.997352 + 0.0727248i \(0.976831\pi\)
\(174\) 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 \(-0.576273\pi\)
−0.237331 + 0.971429i \(0.576273\pi\)
\(180\) 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 \(-0.377417\pi\)
0.375658 + 0.926758i \(0.377417\pi\)
\(192\) 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 \(-0.872244\pi\)
0.920531 0.390669i \(-0.127756\pi\)
\(198\) 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 \(-0.580655\pi\)
−0.250683 + 0.968069i \(0.580655\pi\)
\(228\) 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 \(-0.850942\pi\)
0.892346 0.451351i \(-0.149058\pi\)
\(234\) 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 \(-0.567759\pi\)
−0.211266 + 0.977429i \(0.567759\pi\)
\(240\) 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 \(-0.831802\pi\)
−0.863609 + 0.504162i \(0.831802\pi\)
\(252\) 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 \(0.181632\pi\)
−0.841570 + 0.540147i \(0.818368\pi\)
\(258\) 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 \(-0.777240\pi\)
−0.764957 + 0.644081i \(0.777240\pi\)
\(264\) 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 \(0.118990\pi\)
−0.930940 + 0.365171i \(0.881010\pi\)
\(270\) 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 \(-0.910527\pi\)
0.960754 0.277400i \(-0.0894729\pi\)
\(282\) 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 \(-0.820863\pi\)
0.845778 0.533535i \(-0.179137\pi\)
\(294\) 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 \(-0.672077\pi\)
−0.514648 + 0.857402i \(0.672077\pi\)
\(312\) 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 \(0.0551710\pi\)
−0.985017 + 0.172458i \(0.944829\pi\)
\(318\) 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 \(-0.352374\pi\)
0.447332 + 0.894368i \(0.352374\pi\)
\(348\) 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 \(0.255922\pi\)
−0.693830 + 0.720139i \(0.744078\pi\)
\(354\) 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 \(-0.521575\pi\)
−0.0677287 + 0.997704i \(0.521575\pi\)
\(360\) 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 \(-0.312705\pi\)
0.555035 + 0.831827i \(0.312705\pi\)
\(384\) 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 \(0.183365\pi\)
−0.838617 + 0.544722i \(0.816635\pi\)
\(390\) 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 \(-0.969243\pi\)
0.995335 0.0964759i \(-0.0307571\pi\)
\(402\) 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 \(-0.599982\pi\)
−0.308964 + 0.951074i \(0.599982\pi\)
\(420\) 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 \(-0.314126\pi\)
0.551316 + 0.834296i \(0.314126\pi\)
\(432\) 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 \(-0.424640\pi\)
0.234546 + 0.972105i \(0.424640\pi\)
\(444\) 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 \(0.266886\pi\)
−0.668618 + 0.743606i \(0.733114\pi\)
\(450\) 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 \(-0.820098\pi\)
0.844492 0.535568i \(-0.179902\pi\)
\(462\) 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 \(-0.611121\pi\)
−0.342051 + 0.939681i \(0.611121\pi\)
\(468\) 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 \(-0.578439\pi\)
−0.243936 + 0.969791i \(0.578439\pi\)
\(480\) 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 \(-0.339361\pi\)
0.483511 + 0.875338i \(0.339361\pi\)
\(492\) 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 \(-0.561026\pi\)
−0.190545 + 0.981678i \(0.561026\pi\)
\(504\) 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 \(-0.958515\pi\)
0.991519 0.129961i \(-0.0414853\pi\)
\(510\) 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 \(0.172316\pi\)
−0.857016 + 0.515290i \(0.827684\pi\)
\(522\) 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 \(-0.794420\pi\)
0.798589 0.601877i \(-0.205580\pi\)
\(558\) 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 \(-0.466798\pi\)
0.104117 + 0.994565i \(0.466798\pi\)
\(564\) 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 \(0.0552443\pi\)
−0.984977 + 0.172685i \(0.944756\pi\)
\(570\) 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 \(-0.416160\pi\)
0.260358 + 0.965512i \(0.416160\pi\)
\(588\) 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 \(-0.758216\pi\)
0.725120 0.688623i \(-0.241784\pi\)
\(594\) 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 \(-0.525382\pi\)
−0.0796539 + 0.996823i \(0.525382\pi\)
\(600\) 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 \(0.0704092\pi\)
−0.975636 + 0.219398i \(0.929591\pi\)
\(618\) 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 \(0.194281\pi\)
−0.819446 + 0.573157i \(0.805719\pi\)
\(642\) 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 \(-0.493493\pi\)
0.0204395 + 0.999791i \(0.493493\pi\)
\(648\) 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 \(0.0317457\pi\)
−0.995031 + 0.0995668i \(0.968254\pi\)
\(654\) 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 \(-0.234038\pi\)
0.741662 + 0.670773i \(0.234038\pi\)
\(660\) 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 \(-0.956637\pi\)
0.990735 0.135808i \(-0.0433631\pi\)
\(678\) 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 \(-0.331580\pi\)
0.504764 + 0.863258i \(0.331580\pi\)
\(684\) 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 \(0.308774\pi\)
−0.565266 + 0.824909i \(0.691226\pi\)
\(702\) 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 \(-0.0864124\pi\)
0.963377 + 0.268151i \(0.0864124\pi\)
\(720\) 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 \(-0.732911\pi\)
−0.668143 + 0.744033i \(0.732911\pi\)
\(744\) 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 \(0.321048\pi\)
−0.533043 + 0.846088i \(0.678952\pi\)
\(762\) 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 \(-0.891489\pi\)
0.942456 0.334332i \(-0.108511\pi\)
\(774\) 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 \(0.175119\pi\)
−0.852445 + 0.522818i \(0.824881\pi\)
\(798\) 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 \(-0.947835\pi\)
0.986601 0.163149i \(-0.0521651\pi\)
\(810\) 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 \(0.0910784\pi\)
−0.959343 + 0.282243i \(0.908922\pi\)
\(822\) 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 \(-0.235028\pi\)
0.739571 + 0.673078i \(0.235028\pi\)
\(828\) 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 \(-0.444269\pi\)
0.174191 + 0.984712i \(0.444269\pi\)
\(840\) 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 \(-0.912673\pi\)
0.962603 0.270916i \(-0.0873266\pi\)
\(858\) 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 \(-0.169085\pi\)
0.862202 + 0.506565i \(0.169085\pi\)
\(864\) 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 \(-0.629363\pi\)
0.395310 0.918548i \(-0.370637\pi\)
\(882\) 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 \(-0.244917\pi\)
0.718307 + 0.695726i \(0.244917\pi\)
\(888\) 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 \(-0.569032\pi\)
−0.215174 + 0.976576i \(0.569032\pi\)
\(912\) 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 \(-0.692595\pi\)
0.568807 0.822471i \(-0.307405\pi\)
\(930\) 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 \(-0.810034\pi\)
0.827141 0.561994i \(-0.189966\pi\)
\(942\) 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 \(-0.625136\pi\)
−0.383077 + 0.923716i \(0.625136\pi\)
\(948\) 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 \(-0.916187\pi\)
0.965535 0.260273i \(-0.0838126\pi\)
\(954\) 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 \(-0.117304\pi\)
0.932861 + 0.360237i \(0.117304\pi\)
\(972\) 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 \(0.0386483\pi\)
−0.992638 + 0.121119i \(0.961352\pi\)
\(978\) 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 \(-0.595742\pi\)
−0.296266 + 0.955105i \(0.595742\pi\)
\(984\) 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.1727.7 12
3.2 odd 2 inner 1728.4.c.j.1727.5 12
4.3 odd 2 inner 1728.4.c.j.1727.8 12
8.3 odd 2 108.4.b.b.107.8 yes 12
8.5 even 2 108.4.b.b.107.6 yes 12
12.11 even 2 inner 1728.4.c.j.1727.6 12
24.5 odd 2 108.4.b.b.107.7 yes 12
24.11 even 2 108.4.b.b.107.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.b.b.107.5 12 24.11 even 2
108.4.b.b.107.6 yes 12 8.5 even 2
108.4.b.b.107.7 yes 12 24.5 odd 2
108.4.b.b.107.8 yes 12 8.3 odd 2
1728.4.c.j.1727.5 12 3.2 odd 2 inner
1728.4.c.j.1727.6 12 12.11 even 2 inner
1728.4.c.j.1727.7 12 1.1 even 1 trivial
1728.4.c.j.1727.8 12 4.3 odd 2 inner