Properties

Label 1728.4.c.j.1727.4
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 1727.4
Root \(-1.29835 - 1.36719i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.4.c.j.1727.9

$q$-expansion

\(f(q)\) \(=\) \(q-5.83890i q^{5} +8.83113i q^{7} +O(q^{10})\) \(q-5.83890i q^{5} +8.83113i q^{7} +23.6146 q^{11} -54.6531 q^{13} +117.211i q^{17} +109.576i q^{19} +33.5763 q^{23} +90.9072 q^{25} +40.0490i q^{29} -292.510i q^{31} +51.5641 q^{35} -283.265 q^{37} -367.472i q^{41} -323.337i q^{43} -66.2249 q^{47} +265.011 q^{49} +158.506i q^{53} -137.883i q^{55} -848.630 q^{59} +348.716 q^{61} +319.114i q^{65} +194.285i q^{67} -939.761 q^{71} -473.826 q^{73} +208.544i q^{77} +273.221i q^{79} +338.366 q^{83} +684.386 q^{85} +739.884i q^{89} -482.648i q^{91} +639.805 q^{95} -448.629 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q + 72q^{13} - 384q^{25} + 240q^{37} + 288q^{49} - 144q^{61} + 156q^{73} + 168q^{85} + 516q^{97} + O(q^{100}) \)

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
\(5\) − 5.83890i − 0.522247i −0.965305 0.261124i \(-0.915907\pi\)
0.965305 0.261124i \(-0.0840930\pi\)
\(6\) 0 0
\(7\) 8.83113i 0.476836i 0.971163 + 0.238418i \(0.0766288\pi\)
−0.971163 + 0.238418i \(0.923371\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 23.6146 0.647279 0.323639 0.946180i \(-0.395094\pi\)
0.323639 + 0.946180i \(0.395094\pi\)
\(12\) 0 0
\(13\) −54.6531 −1.16600 −0.583001 0.812471i \(-0.698122\pi\)
−0.583001 + 0.812471i \(0.698122\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 117.211i 1.67223i 0.548553 + 0.836116i \(0.315179\pi\)
−0.548553 + 0.836116i \(0.684821\pi\)
\(18\) 0 0
\(19\) 109.576i 1.32308i 0.749910 + 0.661539i \(0.230097\pi\)
−0.749910 + 0.661539i \(0.769903\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 33.5763 0.304397 0.152199 0.988350i \(-0.451365\pi\)
0.152199 + 0.988350i \(0.451365\pi\)
\(24\) 0 0
\(25\) 90.9072 0.727258
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 40.0490i 0.256445i 0.991745 + 0.128223i \(0.0409272\pi\)
−0.991745 + 0.128223i \(0.959073\pi\)
\(30\) 0 0
\(31\) − 292.510i − 1.69472i −0.531020 0.847359i \(-0.678191\pi\)
0.531020 0.847359i \(-0.321809\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 51.5641 0.249026
\(36\) 0 0
\(37\) −283.265 −1.25861 −0.629304 0.777159i \(-0.716660\pi\)
−0.629304 + 0.777159i \(0.716660\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 367.472i − 1.39974i −0.714269 0.699871i \(-0.753241\pi\)
0.714269 0.699871i \(-0.246759\pi\)
\(42\) 0 0
\(43\) − 323.337i − 1.14671i −0.819308 0.573354i \(-0.805642\pi\)
0.819308 0.573354i \(-0.194358\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −66.2249 −0.205530 −0.102765 0.994706i \(-0.532769\pi\)
−0.102765 + 0.994706i \(0.532769\pi\)
\(48\) 0 0
\(49\) 265.011 0.772627
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 158.506i 0.410801i 0.978678 + 0.205401i \(0.0658498\pi\)
−0.978678 + 0.205401i \(0.934150\pi\)
\(54\) 0 0
\(55\) − 137.883i − 0.338040i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −848.630 −1.87258 −0.936290 0.351228i \(-0.885764\pi\)
−0.936290 + 0.351228i \(0.885764\pi\)
\(60\) 0 0
\(61\) 348.716 0.731943 0.365972 0.930626i \(-0.380737\pi\)
0.365972 + 0.930626i \(0.380737\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 319.114i 0.608942i
\(66\) 0 0
\(67\) 194.285i 0.354264i 0.984187 + 0.177132i \(0.0566819\pi\)
−0.984187 + 0.177132i \(0.943318\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −939.761 −1.57083 −0.785417 0.618968i \(-0.787551\pi\)
−0.785417 + 0.618968i \(0.787551\pi\)
\(72\) 0 0
\(73\) −473.826 −0.759686 −0.379843 0.925051i \(-0.624022\pi\)
−0.379843 + 0.925051i \(0.624022\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 208.544i 0.308646i
\(78\) 0 0
\(79\) 273.221i 0.389112i 0.980891 + 0.194556i \(0.0623265\pi\)
−0.980891 + 0.194556i \(0.937673\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 338.366 0.447476 0.223738 0.974649i \(-0.428174\pi\)
0.223738 + 0.974649i \(0.428174\pi\)
\(84\) 0 0
\(85\) 684.386 0.873318
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 739.884i 0.881208i 0.897702 + 0.440604i \(0.145236\pi\)
−0.897702 + 0.440604i \(0.854764\pi\)
\(90\) 0 0
\(91\) − 482.648i − 0.555992i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 639.805 0.690974
\(96\) 0 0
\(97\) −448.629 −0.469602 −0.234801 0.972044i \(-0.575444\pi\)
−0.234801 + 0.972044i \(0.575444\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1152.87i − 1.13580i −0.823099 0.567898i \(-0.807757\pi\)
0.823099 0.567898i \(-0.192243\pi\)
\(102\) 0 0
\(103\) 1389.90i 1.32962i 0.747013 + 0.664809i \(0.231487\pi\)
−0.747013 + 0.664809i \(0.768513\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 245.479 0.221788 0.110894 0.993832i \(-0.464629\pi\)
0.110894 + 0.993832i \(0.464629\pi\)
\(108\) 0 0
\(109\) 644.998 0.566785 0.283393 0.959004i \(-0.408540\pi\)
0.283393 + 0.959004i \(0.408540\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 825.442i − 0.687177i −0.939120 0.343589i \(-0.888357\pi\)
0.939120 0.343589i \(-0.111643\pi\)
\(114\) 0 0
\(115\) − 196.049i − 0.158971i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1035.11 −0.797380
\(120\) 0 0
\(121\) −773.351 −0.581030
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1260.66i − 0.902056i
\(126\) 0 0
\(127\) 754.649i 0.527278i 0.964621 + 0.263639i \(0.0849228\pi\)
−0.964621 + 0.263639i \(0.915077\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2026.43 −1.35153 −0.675765 0.737117i \(-0.736186\pi\)
−0.675765 + 0.737117i \(0.736186\pi\)
\(132\) 0 0
\(133\) −967.681 −0.630892
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1882.88i − 1.17420i −0.809515 0.587099i \(-0.800270\pi\)
0.809515 0.587099i \(-0.199730\pi\)
\(138\) 0 0
\(139\) − 93.7277i − 0.0571934i −0.999591 0.0285967i \(-0.990896\pi\)
0.999591 0.0285967i \(-0.00910385\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1290.61 −0.754729
\(144\) 0 0
\(145\) 233.842 0.133928
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2764.69i 1.52008i 0.649874 + 0.760042i \(0.274822\pi\)
−0.649874 + 0.760042i \(0.725178\pi\)
\(150\) 0 0
\(151\) − 3694.11i − 1.99088i −0.0954051 0.995439i \(-0.530415\pi\)
0.0954051 0.995439i \(-0.469585\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1707.94 −0.885062
\(156\) 0 0
\(157\) −812.401 −0.412972 −0.206486 0.978450i \(-0.566203\pi\)
−0.206486 + 0.978450i \(0.566203\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 296.516i 0.145148i
\(162\) 0 0
\(163\) 1034.18i 0.496953i 0.968638 + 0.248477i \(0.0799299\pi\)
−0.968638 + 0.248477i \(0.920070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1453.29 −0.673409 −0.336704 0.941610i \(-0.609312\pi\)
−0.336704 + 0.941610i \(0.609312\pi\)
\(168\) 0 0
\(169\) 789.957 0.359562
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 634.902i 0.279022i 0.990221 + 0.139511i \(0.0445530\pi\)
−0.990221 + 0.139511i \(0.955447\pi\)
\(174\) 0 0
\(175\) 802.813i 0.346783i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2909.35 −1.21483 −0.607417 0.794383i \(-0.707794\pi\)
−0.607417 + 0.794383i \(0.707794\pi\)
\(180\) 0 0
\(181\) −1354.46 −0.556222 −0.278111 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1653.96i 0.657305i
\(186\) 0 0
\(187\) 2767.90i 1.08240i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3488.40 −1.32153 −0.660765 0.750593i \(-0.729768\pi\)
−0.660765 + 0.750593i \(0.729768\pi\)
\(192\) 0 0
\(193\) 2912.99 1.08643 0.543217 0.839592i \(-0.317206\pi\)
0.543217 + 0.839592i \(0.317206\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2432.66i 0.879795i 0.898048 + 0.439897i \(0.144985\pi\)
−0.898048 + 0.439897i \(0.855015\pi\)
\(198\) 0 0
\(199\) 563.190i 0.200621i 0.994956 + 0.100310i \(0.0319836\pi\)
−0.994956 + 0.100310i \(0.968016\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −353.678 −0.122282
\(204\) 0 0
\(205\) −2145.63 −0.731012
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2587.60i 0.856401i
\(210\) 0 0
\(211\) 1447.62i 0.472314i 0.971715 + 0.236157i \(0.0758879\pi\)
−0.971715 + 0.236157i \(0.924112\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1887.93 −0.598865
\(216\) 0 0
\(217\) 2583.19 0.808103
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 6405.96i − 1.94983i
\(222\) 0 0
\(223\) 5049.28i 1.51625i 0.652107 + 0.758127i \(0.273885\pi\)
−0.652107 + 0.758127i \(0.726115\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5716.54 −1.67145 −0.835727 0.549146i \(-0.814953\pi\)
−0.835727 + 0.549146i \(0.814953\pi\)
\(228\) 0 0
\(229\) 2582.55 0.745240 0.372620 0.927984i \(-0.378460\pi\)
0.372620 + 0.927984i \(0.378460\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 587.204i 0.165103i 0.996587 + 0.0825516i \(0.0263069\pi\)
−0.996587 + 0.0825516i \(0.973693\pi\)
\(234\) 0 0
\(235\) 386.681i 0.107337i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5583.00 −1.51102 −0.755512 0.655135i \(-0.772612\pi\)
−0.755512 + 0.655135i \(0.772612\pi\)
\(240\) 0 0
\(241\) −2056.95 −0.549791 −0.274895 0.961474i \(-0.588643\pi\)
−0.274895 + 0.961474i \(0.588643\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 1547.37i − 0.403503i
\(246\) 0 0
\(247\) − 5988.67i − 1.54271i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1256.79 −0.316047 −0.158023 0.987435i \(-0.550512\pi\)
−0.158023 + 0.987435i \(0.550512\pi\)
\(252\) 0 0
\(253\) 792.890 0.197030
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 643.773i − 0.156255i −0.996943 0.0781273i \(-0.975106\pi\)
0.996943 0.0781273i \(-0.0248941\pi\)
\(258\) 0 0
\(259\) − 2501.55i − 0.600150i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −917.108 −0.215024 −0.107512 0.994204i \(-0.534288\pi\)
−0.107512 + 0.994204i \(0.534288\pi\)
\(264\) 0 0
\(265\) 925.501 0.214540
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6577.00i 1.49073i 0.666656 + 0.745366i \(0.267725\pi\)
−0.666656 + 0.745366i \(0.732275\pi\)
\(270\) 0 0
\(271\) − 4656.21i − 1.04371i −0.853035 0.521854i \(-0.825241\pi\)
0.853035 0.521854i \(-0.174759\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2146.74 0.470739
\(276\) 0 0
\(277\) 2881.42 0.625010 0.312505 0.949916i \(-0.398832\pi\)
0.312505 + 0.949916i \(0.398832\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 2604.93i − 0.553015i −0.961012 0.276507i \(-0.910823\pi\)
0.961012 0.276507i \(-0.0891771\pi\)
\(282\) 0 0
\(283\) − 5786.02i − 1.21535i −0.794187 0.607674i \(-0.792103\pi\)
0.794187 0.607674i \(-0.207897\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3245.19 0.667448
\(288\) 0 0
\(289\) −8825.50 −1.79636
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 3279.55i − 0.653902i −0.945041 0.326951i \(-0.893979\pi\)
0.945041 0.326951i \(-0.106021\pi\)
\(294\) 0 0
\(295\) 4955.07i 0.977950i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1835.05 −0.354928
\(300\) 0 0
\(301\) 2855.43 0.546792
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 2036.12i − 0.382255i
\(306\) 0 0
\(307\) − 3014.47i − 0.560407i −0.959941 0.280203i \(-0.909598\pi\)
0.959941 0.280203i \(-0.0904019\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6230.80 −1.13606 −0.568032 0.823006i \(-0.692295\pi\)
−0.568032 + 0.823006i \(0.692295\pi\)
\(312\) 0 0
\(313\) −3922.49 −0.708346 −0.354173 0.935180i \(-0.615238\pi\)
−0.354173 + 0.935180i \(0.615238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6882.19i 1.21938i 0.792641 + 0.609689i \(0.208705\pi\)
−0.792641 + 0.609689i \(0.791295\pi\)
\(318\) 0 0
\(319\) 945.741i 0.165992i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12843.6 −2.21249
\(324\) 0 0
\(325\) −4968.36 −0.847984
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 584.841i − 0.0980040i
\(330\) 0 0
\(331\) − 9489.08i − 1.57573i −0.615847 0.787866i \(-0.711186\pi\)
0.615847 0.787866i \(-0.288814\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1134.41 0.185013
\(336\) 0 0
\(337\) 4578.52 0.740083 0.370041 0.929015i \(-0.379344\pi\)
0.370041 + 0.929015i \(0.379344\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 6907.50i − 1.09696i
\(342\) 0 0
\(343\) 5369.42i 0.845253i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 942.483 0.145807 0.0729037 0.997339i \(-0.476773\pi\)
0.0729037 + 0.997339i \(0.476773\pi\)
\(348\) 0 0
\(349\) 2124.84 0.325903 0.162951 0.986634i \(-0.447899\pi\)
0.162951 + 0.986634i \(0.447899\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 3881.60i − 0.585260i −0.956226 0.292630i \(-0.905470\pi\)
0.956226 0.292630i \(-0.0945304\pi\)
\(354\) 0 0
\(355\) 5487.18i 0.820363i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8050.44 −1.18353 −0.591763 0.806112i \(-0.701568\pi\)
−0.591763 + 0.806112i \(0.701568\pi\)
\(360\) 0 0
\(361\) −5147.94 −0.750537
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2766.62i 0.396744i
\(366\) 0 0
\(367\) − 6890.96i − 0.980123i −0.871688 0.490062i \(-0.836974\pi\)
0.871688 0.490062i \(-0.163026\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1399.79 −0.195885
\(372\) 0 0
\(373\) 10512.5 1.45930 0.729649 0.683822i \(-0.239684\pi\)
0.729649 + 0.683822i \(0.239684\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 2188.80i − 0.299016i
\(378\) 0 0
\(379\) 3139.69i 0.425528i 0.977104 + 0.212764i \(0.0682466\pi\)
−0.977104 + 0.212764i \(0.931753\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5117.53 −0.682751 −0.341375 0.939927i \(-0.610893\pi\)
−0.341375 + 0.939927i \(0.610893\pi\)
\(384\) 0 0
\(385\) 1217.67 0.161190
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 787.004i − 0.102578i −0.998684 0.0512888i \(-0.983667\pi\)
0.998684 0.0512888i \(-0.0163329\pi\)
\(390\) 0 0
\(391\) 3935.52i 0.509023i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1595.31 0.203212
\(396\) 0 0
\(397\) −11160.6 −1.41092 −0.705458 0.708752i \(-0.749259\pi\)
−0.705458 + 0.708752i \(0.749259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13224.6i 1.64689i 0.567394 + 0.823447i \(0.307952\pi\)
−0.567394 + 0.823447i \(0.692048\pi\)
\(402\) 0 0
\(403\) 15986.5i 1.97605i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6689.20 −0.814671
\(408\) 0 0
\(409\) 8153.09 0.985683 0.492841 0.870119i \(-0.335958\pi\)
0.492841 + 0.870119i \(0.335958\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 7494.36i − 0.892914i
\(414\) 0 0
\(415\) − 1975.69i − 0.233693i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11682.5 −1.36212 −0.681061 0.732226i \(-0.738481\pi\)
−0.681061 + 0.732226i \(0.738481\pi\)
\(420\) 0 0
\(421\) −9595.77 −1.11085 −0.555426 0.831566i \(-0.687445\pi\)
−0.555426 + 0.831566i \(0.687445\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10655.4i 1.21614i
\(426\) 0 0
\(427\) 3079.56i 0.349017i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4294.48 0.479948 0.239974 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(432\) 0 0
\(433\) 168.392 0.0186892 0.00934460 0.999956i \(-0.497025\pi\)
0.00934460 + 0.999956i \(0.497025\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3679.16i 0.402742i
\(438\) 0 0
\(439\) − 1459.53i − 0.158677i −0.996848 0.0793387i \(-0.974719\pi\)
0.996848 0.0793387i \(-0.0252809\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 333.411 0.0357581 0.0178790 0.999840i \(-0.494309\pi\)
0.0178790 + 0.999840i \(0.494309\pi\)
\(444\) 0 0
\(445\) 4320.11 0.460209
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 11454.3i − 1.20393i −0.798523 0.601964i \(-0.794385\pi\)
0.798523 0.601964i \(-0.205615\pi\)
\(450\) 0 0
\(451\) − 8677.70i − 0.906024i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2818.14 −0.290365
\(456\) 0 0
\(457\) 9680.26 0.990861 0.495431 0.868648i \(-0.335010\pi\)
0.495431 + 0.868648i \(0.335010\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 14806.9i − 1.49593i −0.663737 0.747966i \(-0.731030\pi\)
0.663737 0.747966i \(-0.268970\pi\)
\(462\) 0 0
\(463\) 3658.08i 0.367183i 0.983003 + 0.183591i \(0.0587723\pi\)
−0.983003 + 0.183591i \(0.941228\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9852.37 0.976260 0.488130 0.872771i \(-0.337679\pi\)
0.488130 + 0.872771i \(0.337679\pi\)
\(468\) 0 0
\(469\) −1715.75 −0.168926
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 7635.47i − 0.742240i
\(474\) 0 0
\(475\) 9961.26i 0.962219i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11107.9 1.05957 0.529784 0.848133i \(-0.322273\pi\)
0.529784 + 0.848133i \(0.322273\pi\)
\(480\) 0 0
\(481\) 15481.3 1.46754
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2619.50i 0.245248i
\(486\) 0 0
\(487\) − 11703.7i − 1.08900i −0.838761 0.544500i \(-0.816719\pi\)
0.838761 0.544500i \(-0.183281\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2851.25 0.262067 0.131034 0.991378i \(-0.458170\pi\)
0.131034 + 0.991378i \(0.458170\pi\)
\(492\) 0 0
\(493\) −4694.20 −0.428836
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8299.15i − 0.749030i
\(498\) 0 0
\(499\) − 14936.0i − 1.33994i −0.742389 0.669969i \(-0.766307\pi\)
0.742389 0.669969i \(-0.233693\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9047.91 −0.802041 −0.401020 0.916069i \(-0.631344\pi\)
−0.401020 + 0.916069i \(0.631344\pi\)
\(504\) 0 0
\(505\) −6731.52 −0.593166
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15173.5i 1.32132i 0.750683 + 0.660662i \(0.229724\pi\)
−0.750683 + 0.660662i \(0.770276\pi\)
\(510\) 0 0
\(511\) − 4184.41i − 0.362246i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8115.47 0.694389
\(516\) 0 0
\(517\) −1563.87 −0.133035
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7375.96i 0.620243i 0.950697 + 0.310121i \(0.100370\pi\)
−0.950697 + 0.310121i \(0.899630\pi\)
\(522\) 0 0
\(523\) 8514.96i 0.711918i 0.934502 + 0.355959i \(0.115846\pi\)
−0.934502 + 0.355959i \(0.884154\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 34285.4 2.83396
\(528\) 0 0
\(529\) −11039.6 −0.907342
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20083.5i 1.63210i
\(534\) 0 0
\(535\) − 1433.33i − 0.115828i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6258.13 0.500105
\(540\) 0 0
\(541\) 2214.98 0.176025 0.0880125 0.996119i \(-0.471948\pi\)
0.0880125 + 0.996119i \(0.471948\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 3766.08i − 0.296002i
\(546\) 0 0
\(547\) − 3906.55i − 0.305360i −0.988276 0.152680i \(-0.951210\pi\)
0.988276 0.152680i \(-0.0487904\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4388.41 −0.339297
\(552\) 0 0
\(553\) −2412.85 −0.185542
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2978.14i 0.226549i 0.993564 + 0.113275i \(0.0361340\pi\)
−0.993564 + 0.113275i \(0.963866\pi\)
\(558\) 0 0
\(559\) 17671.4i 1.33706i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8786.44 −0.657734 −0.328867 0.944376i \(-0.606667\pi\)
−0.328867 + 0.944376i \(0.606667\pi\)
\(564\) 0 0
\(565\) −4819.67 −0.358876
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19271.0i 1.41983i 0.704288 + 0.709914i \(0.251266\pi\)
−0.704288 + 0.709914i \(0.748734\pi\)
\(570\) 0 0
\(571\) 15535.7i 1.13861i 0.822125 + 0.569307i \(0.192788\pi\)
−0.822125 + 0.569307i \(0.807212\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3052.33 0.221375
\(576\) 0 0
\(577\) 3783.58 0.272985 0.136493 0.990641i \(-0.456417\pi\)
0.136493 + 0.990641i \(0.456417\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2988.15i 0.213373i
\(582\) 0 0
\(583\) 3743.06i 0.265903i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15515.4 1.09095 0.545477 0.838126i \(-0.316349\pi\)
0.545477 + 0.838126i \(0.316349\pi\)
\(588\) 0 0
\(589\) 32052.1 2.24225
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 14098.1i 0.976292i 0.872762 + 0.488146i \(0.162327\pi\)
−0.872762 + 0.488146i \(0.837673\pi\)
\(594\) 0 0
\(595\) 6043.90i 0.416430i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −572.539 −0.0390539 −0.0195270 0.999809i \(-0.506216\pi\)
−0.0195270 + 0.999809i \(0.506216\pi\)
\(600\) 0 0
\(601\) −23463.4 −1.59250 −0.796249 0.604969i \(-0.793185\pi\)
−0.796249 + 0.604969i \(0.793185\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4515.52i 0.303441i
\(606\) 0 0
\(607\) − 1980.51i − 0.132433i −0.997805 0.0662163i \(-0.978907\pi\)
0.997805 0.0662163i \(-0.0210927\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3619.39 0.239648
\(612\) 0 0
\(613\) −2479.01 −0.163338 −0.0816691 0.996660i \(-0.526025\pi\)
−0.0816691 + 0.996660i \(0.526025\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 20002.8i − 1.30516i −0.757722 0.652578i \(-0.773688\pi\)
0.757722 0.652578i \(-0.226312\pi\)
\(618\) 0 0
\(619\) − 22292.4i − 1.44751i −0.690058 0.723754i \(-0.742415\pi\)
0.690058 0.723754i \(-0.257585\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6534.01 −0.420192
\(624\) 0 0
\(625\) 4002.52 0.256161
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 33201.9i − 2.10469i
\(630\) 0 0
\(631\) 24203.8i 1.52700i 0.645807 + 0.763501i \(0.276521\pi\)
−0.645807 + 0.763501i \(0.723479\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4406.32 0.275370
\(636\) 0 0
\(637\) −14483.7 −0.900885
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 4423.87i − 0.272593i −0.990668 0.136297i \(-0.956480\pi\)
0.990668 0.136297i \(-0.0435200\pi\)
\(642\) 0 0
\(643\) − 11961.5i − 0.733619i −0.930296 0.366810i \(-0.880450\pi\)
0.930296 0.366810i \(-0.119550\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6287.16 0.382030 0.191015 0.981587i \(-0.438822\pi\)
0.191015 + 0.981587i \(0.438822\pi\)
\(648\) 0 0
\(649\) −20040.1 −1.21208
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 18793.2i − 1.12624i −0.826376 0.563119i \(-0.809601\pi\)
0.826376 0.563119i \(-0.190399\pi\)
\(654\) 0 0
\(655\) 11832.2i 0.705833i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −20827.2 −1.23113 −0.615563 0.788088i \(-0.711071\pi\)
−0.615563 + 0.788088i \(0.711071\pi\)
\(660\) 0 0
\(661\) 6038.16 0.355306 0.177653 0.984093i \(-0.443150\pi\)
0.177653 + 0.984093i \(0.443150\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5650.20i 0.329482i
\(666\) 0 0
\(667\) 1344.70i 0.0780612i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8234.79 0.473771
\(672\) 0 0
\(673\) −19811.5 −1.13474 −0.567368 0.823464i \(-0.692038\pi\)
−0.567368 + 0.823464i \(0.692038\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 22962.5i − 1.30357i −0.758402 0.651787i \(-0.774020\pi\)
0.758402 0.651787i \(-0.225980\pi\)
\(678\) 0 0
\(679\) − 3961.90i − 0.223923i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33097.3 1.85422 0.927109 0.374791i \(-0.122286\pi\)
0.927109 + 0.374791i \(0.122286\pi\)
\(684\) 0 0
\(685\) −10993.9 −0.613222
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 8662.84i − 0.478995i
\(690\) 0 0
\(691\) − 21151.0i − 1.16443i −0.813034 0.582216i \(-0.802186\pi\)
0.813034 0.582216i \(-0.197814\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −547.267 −0.0298691
\(696\) 0 0
\(697\) 43071.9 2.34069
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18752.3i 1.01036i 0.863013 + 0.505182i \(0.168575\pi\)
−0.863013 + 0.505182i \(0.831425\pi\)
\(702\) 0 0
\(703\) − 31039.1i − 1.66524i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10181.2 0.541588
\(708\) 0 0
\(709\) 25964.1 1.37532 0.687660 0.726033i \(-0.258638\pi\)
0.687660 + 0.726033i \(0.258638\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 9821.38i − 0.515868i
\(714\) 0 0
\(715\) 7535.75i 0.394155i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 11741.1 0.608998 0.304499 0.952513i \(-0.401511\pi\)
0.304499 + 0.952513i \(0.401511\pi\)
\(720\) 0 0
\(721\) −12274.4 −0.634010
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3640.74i 0.186502i
\(726\) 0 0
\(727\) − 5637.87i − 0.287616i −0.989606 0.143808i \(-0.954065\pi\)
0.989606 0.143808i \(-0.0459348\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 37898.8 1.91756
\(732\) 0 0
\(733\) 23807.3 1.19965 0.599825 0.800131i \(-0.295237\pi\)
0.599825 + 0.800131i \(0.295237\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4587.96i 0.229307i
\(738\) 0 0
\(739\) − 3556.31i − 0.177024i −0.996075 0.0885122i \(-0.971789\pi\)
0.996075 0.0885122i \(-0.0282112\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24054.6 1.18772 0.593860 0.804568i \(-0.297603\pi\)
0.593860 + 0.804568i \(0.297603\pi\)
\(744\) 0 0
\(745\) 16142.8 0.793860
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2167.86i 0.105757i
\(750\) 0 0
\(751\) − 15373.0i − 0.746960i −0.927638 0.373480i \(-0.878164\pi\)
0.927638 0.373480i \(-0.121836\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −21569.6 −1.03973
\(756\) 0 0
\(757\) −26000.8 −1.24837 −0.624184 0.781277i \(-0.714569\pi\)
−0.624184 + 0.781277i \(0.714569\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9461.72i 0.450706i 0.974277 + 0.225353i \(0.0723535\pi\)
−0.974277 + 0.225353i \(0.927646\pi\)
\(762\) 0 0
\(763\) 5696.06i 0.270264i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 46380.2 2.18343
\(768\) 0 0
\(769\) 11196.2 0.525028 0.262514 0.964928i \(-0.415448\pi\)
0.262514 + 0.964928i \(0.415448\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39575.0i 1.84141i 0.390255 + 0.920707i \(0.372387\pi\)
−0.390255 + 0.920707i \(0.627613\pi\)
\(774\) 0 0
\(775\) − 26591.2i − 1.23250i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 40266.2 1.85197
\(780\) 0 0
\(781\) −22192.1 −1.01677
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4743.53i 0.215674i
\(786\) 0 0
\(787\) 25069.2i 1.13548i 0.823209 + 0.567739i \(0.192182\pi\)
−0.823209 + 0.567739i \(0.807818\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7289.58 0.327671
\(792\) 0 0
\(793\) −19058.4 −0.853448
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 5639.39i − 0.250637i −0.992117 0.125318i \(-0.960005\pi\)
0.992117 0.125318i \(-0.0399952\pi\)
\(798\) 0 0
\(799\) − 7762.31i − 0.343693i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11189.2 −0.491729
\(804\) 0 0
\(805\) 1731.33 0.0758030
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 34032.0i − 1.47899i −0.673163 0.739494i \(-0.735065\pi\)
0.673163 0.739494i \(-0.264935\pi\)
\(810\) 0 0
\(811\) − 20119.5i − 0.871137i −0.900156 0.435569i \(-0.856547\pi\)
0.900156 0.435569i \(-0.143453\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6038.49 0.259532
\(816\) 0 0
\(817\) 35430.0 1.51718
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9057.81i 0.385042i 0.981293 + 0.192521i \(0.0616664\pi\)
−0.981293 + 0.192521i \(0.938334\pi\)
\(822\) 0 0
\(823\) 26277.7i 1.11298i 0.830853 + 0.556491i \(0.187853\pi\)
−0.830853 + 0.556491i \(0.812147\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21737.7 0.914018 0.457009 0.889462i \(-0.348921\pi\)
0.457009 + 0.889462i \(0.348921\pi\)
\(828\) 0 0
\(829\) −27552.6 −1.15433 −0.577167 0.816626i \(-0.695842\pi\)
−0.577167 + 0.816626i \(0.695842\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 31062.3i 1.29201i
\(834\) 0 0
\(835\) 8485.65i 0.351686i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12231.7 −0.503320 −0.251660 0.967816i \(-0.580977\pi\)
−0.251660 + 0.967816i \(0.580977\pi\)
\(840\) 0 0
\(841\) 22785.1 0.934236
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 4612.48i − 0.187780i
\(846\) 0 0
\(847\) − 6829.56i − 0.277056i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −9511.00 −0.383117
\(852\) 0 0
\(853\) 2257.34 0.0906093 0.0453046 0.998973i \(-0.485574\pi\)
0.0453046 + 0.998973i \(0.485574\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 24908.2i − 0.992820i −0.868088 0.496410i \(-0.834651\pi\)
0.868088 0.496410i \(-0.165349\pi\)
\(858\) 0 0
\(859\) − 11850.7i − 0.470710i −0.971909 0.235355i \(-0.924375\pi\)
0.971909 0.235355i \(-0.0756253\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 43145.0 1.70182 0.850910 0.525311i \(-0.176051\pi\)
0.850910 + 0.525311i \(0.176051\pi\)
\(864\) 0 0
\(865\) 3707.13 0.145718
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6452.01i 0.251864i
\(870\) 0 0
\(871\) − 10618.3i − 0.413072i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11133.1 0.430133
\(876\) 0 0
\(877\) 14333.4 0.551886 0.275943 0.961174i \(-0.411010\pi\)
0.275943 + 0.961174i \(0.411010\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 5066.51i − 0.193752i −0.995296 0.0968758i \(-0.969115\pi\)
0.995296 0.0968758i \(-0.0308850\pi\)
\(882\) 0 0
\(883\) 7046.42i 0.268551i 0.990944 + 0.134276i \(0.0428708\pi\)
−0.990944 + 0.134276i \(0.957129\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −52122.7 −1.97307 −0.986534 0.163559i \(-0.947703\pi\)
−0.986534 + 0.163559i \(0.947703\pi\)
\(888\) 0 0
\(889\) −6664.41 −0.251425
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 7256.67i − 0.271932i
\(894\) 0 0
\(895\) 16987.4i 0.634444i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 11714.7 0.434602
\(900\) 0 0
\(901\) −18578.7 −0.686955
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7908.56i 0.290486i
\(906\) 0 0
\(907\) 41205.1i 1.50848i 0.656599 + 0.754240i \(0.271994\pi\)
−0.656599 + 0.754240i \(0.728006\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 34305.8 1.24764 0.623821 0.781567i \(-0.285580\pi\)
0.623821 + 0.781567i \(0.285580\pi\)
\(912\) 0 0
\(913\) 7990.37 0.289642
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 17895.7i − 0.644458i
\(918\) 0 0
\(919\) 43537.1i 1.56274i 0.624068 + 0.781370i \(0.285479\pi\)
−0.624068 + 0.781370i \(0.714521\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 51360.8 1.83160
\(924\) 0 0
\(925\) −25750.9 −0.915333
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8764.21i 0.309520i 0.987952 + 0.154760i \(0.0494605\pi\)
−0.987952 + 0.154760i \(0.950539\pi\)
\(930\) 0 0
\(931\) 29038.9i 1.02225i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 16161.5 0.565281
\(936\) 0 0
\(937\) −22745.0 −0.793006 −0.396503 0.918033i \(-0.629776\pi\)
−0.396503 + 0.918033i \(0.629776\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 24740.5i − 0.857084i −0.903522 0.428542i \(-0.859027\pi\)
0.903522 0.428542i \(-0.140973\pi\)
\(942\) 0 0
\(943\) − 12338.3i − 0.426078i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1886.69 −0.0647405 −0.0323703 0.999476i \(-0.510306\pi\)
−0.0323703 + 0.999476i \(0.510306\pi\)
\(948\) 0 0
\(949\) 25896.0 0.885796
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36292.1i 1.23360i 0.787122 + 0.616798i \(0.211570\pi\)
−0.787122 + 0.616798i \(0.788430\pi\)
\(954\) 0 0
\(955\) 20368.5i 0.690165i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16627.9 0.559900
\(960\) 0 0
\(961\) −55770.8 −1.87207
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 17008.7i − 0.567387i
\(966\) 0 0
\(967\) − 12599.4i − 0.418996i −0.977809 0.209498i \(-0.932817\pi\)
0.977809 0.209498i \(-0.0671830\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −41974.3 −1.38725 −0.693625 0.720336i \(-0.743988\pi\)
−0.693625 + 0.720336i \(0.743988\pi\)
\(972\) 0 0
\(973\) 827.721 0.0272719
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33488.8i 1.09662i 0.836274 + 0.548312i \(0.184729\pi\)
−0.836274 + 0.548312i \(0.815271\pi\)
\(978\) 0 0
\(979\) 17472.1i 0.570387i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3461.54 −0.112315 −0.0561576 0.998422i \(-0.517885\pi\)
−0.0561576 + 0.998422i \(0.517885\pi\)
\(984\) 0 0
\(985\) 14204.0 0.459470
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 10856.5i − 0.349055i
\(990\) 0 0
\(991\) 20067.6i 0.643256i 0.946866 + 0.321628i \(0.104230\pi\)
−0.946866 + 0.321628i \(0.895770\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3288.41 0.104774
\(996\) 0 0
\(997\) −821.726 −0.0261026 −0.0130513 0.999915i \(-0.504154\pi\)
−0.0130513 + 0.999915i \(0.504154\pi\)
\(998\) 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.1727.4 12
3.2 odd 2 inner 1728.4.c.j.1727.10 12
4.3 odd 2 inner 1728.4.c.j.1727.3 12
8.3 odd 2 108.4.b.b.107.4 yes 12
8.5 even 2 108.4.b.b.107.10 yes 12
12.11 even 2 inner 1728.4.c.j.1727.9 12
24.5 odd 2 108.4.b.b.107.3 12
24.11 even 2 108.4.b.b.107.9 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.b.b.107.3 12 24.5 odd 2
108.4.b.b.107.4 yes 12 8.3 odd 2
108.4.b.b.107.9 yes 12 24.11 even 2
108.4.b.b.107.10 yes 12 8.5 even 2
1728.4.c.j.1727.3 12 4.3 odd 2 inner
1728.4.c.j.1727.4 12 1.1 even 1 trivial
1728.4.c.j.1727.9 12 12.11 even 2 inner
1728.4.c.j.1727.10 12 3.2 odd 2 inner