Properties

Label 1728.3.b.i.1567.7
Level $1728$
Weight $3$
Character 1728.1567
Analytic conductor $47.085$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.116304318664704.2
Defining polynomial: \( x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.7
Root \(0.578188 - 1.29062i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1567
Dual form 1728.3.b.i.1567.6

$q$-expansion

\(f(q)\) \(=\) \(q+2.64040i q^{5} -8.30833i q^{7} +O(q^{10})\) \(q+2.64040i q^{5} -8.30833i q^{7} -18.7629 q^{11} +22.3117i q^{13} +23.4983 q^{17} +9.93333 q^{19} -32.3517i q^{23} +18.0283 q^{25} -8.45525i q^{29} +46.9532i q^{31} +21.9373 q^{35} -28.3219i q^{37} -77.7915 q^{41} -58.4797 q^{43} -54.2933i q^{47} -20.0283 q^{49} -5.81484i q^{53} -49.5416i q^{55} -47.5795 q^{59} -27.7128i q^{61} -58.9118 q^{65} -50.6336 q^{67} +7.34824i q^{71} -29.4383 q^{73} +155.888i q^{77} +43.2050i q^{79} -10.7858 q^{83} +62.0450i q^{85} -136.703 q^{89} +185.372 q^{91} +26.2280i q^{95} -54.6465 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{17} - 72 q^{25} - 336 q^{41} + 48 q^{49} - 912 q^{65} + 60 q^{73} - 1248 q^{89} - 204 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 2.64040i 0.528081i 0.964512 + 0.264040i \(0.0850552\pi\)
−0.964512 + 0.264040i \(0.914945\pi\)
\(6\) 0 0
\(7\) − 8.30833i − 1.18690i −0.804870 0.593452i \(-0.797765\pi\)
0.804870 0.593452i \(-0.202235\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −18.7629 −1.70572 −0.852859 0.522141i \(-0.825133\pi\)
−0.852859 + 0.522141i \(0.825133\pi\)
\(12\) 0 0
\(13\) 22.3117i 1.71628i 0.513415 + 0.858141i \(0.328380\pi\)
−0.513415 + 0.858141i \(0.671620\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 23.4983 1.38225 0.691126 0.722734i \(-0.257115\pi\)
0.691126 + 0.722734i \(0.257115\pi\)
\(18\) 0 0
\(19\) 9.93333 0.522807 0.261403 0.965230i \(-0.415815\pi\)
0.261403 + 0.965230i \(0.415815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 32.3517i − 1.40659i −0.710896 0.703297i \(-0.751710\pi\)
0.710896 0.703297i \(-0.248290\pi\)
\(24\) 0 0
\(25\) 18.0283 0.721131
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.45525i − 0.291560i −0.989317 0.145780i \(-0.953431\pi\)
0.989317 0.145780i \(-0.0465692\pi\)
\(30\) 0 0
\(31\) 46.9532i 1.51462i 0.653055 + 0.757310i \(0.273487\pi\)
−0.653055 + 0.757310i \(0.726513\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 21.9373 0.626781
\(36\) 0 0
\(37\) − 28.3219i − 0.765457i −0.923861 0.382728i \(-0.874984\pi\)
0.923861 0.382728i \(-0.125016\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −77.7915 −1.89735 −0.948677 0.316246i \(-0.897578\pi\)
−0.948677 + 0.316246i \(0.897578\pi\)
\(42\) 0 0
\(43\) −58.4797 −1.35999 −0.679997 0.733215i \(-0.738019\pi\)
−0.679997 + 0.733215i \(0.738019\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 54.2933i − 1.15518i −0.816329 0.577588i \(-0.803994\pi\)
0.816329 0.577588i \(-0.196006\pi\)
\(48\) 0 0
\(49\) −20.0283 −0.408740
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.81484i − 0.109714i −0.998494 0.0548570i \(-0.982530\pi\)
0.998494 0.0548570i \(-0.0174703\pi\)
\(54\) 0 0
\(55\) − 49.5416i − 0.900757i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −47.5795 −0.806432 −0.403216 0.915105i \(-0.632108\pi\)
−0.403216 + 0.915105i \(0.632108\pi\)
\(60\) 0 0
\(61\) − 27.7128i − 0.454308i −0.973859 0.227154i \(-0.927058\pi\)
0.973859 0.227154i \(-0.0729421\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −58.9118 −0.906335
\(66\) 0 0
\(67\) −50.6336 −0.755725 −0.377862 0.925862i \(-0.623341\pi\)
−0.377862 + 0.925862i \(0.623341\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.34824i 0.103496i 0.998660 + 0.0517482i \(0.0164793\pi\)
−0.998660 + 0.0517482i \(0.983521\pi\)
\(72\) 0 0
\(73\) −29.4383 −0.403265 −0.201632 0.979461i \(-0.564625\pi\)
−0.201632 + 0.979461i \(0.564625\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 155.888i 2.02452i
\(78\) 0 0
\(79\) 43.2050i 0.546899i 0.961886 + 0.273450i \(0.0881647\pi\)
−0.961886 + 0.273450i \(0.911835\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.7858 −0.129950 −0.0649748 0.997887i \(-0.520697\pi\)
−0.0649748 + 0.997887i \(0.520697\pi\)
\(84\) 0 0
\(85\) 62.0450i 0.729941i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −136.703 −1.53599 −0.767996 0.640454i \(-0.778746\pi\)
−0.767996 + 0.640454i \(0.778746\pi\)
\(90\) 0 0
\(91\) 185.372 2.03706
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 26.2280i 0.276084i
\(96\) 0 0
\(97\) −54.6465 −0.563366 −0.281683 0.959508i \(-0.590893\pi\)
−0.281683 + 0.959508i \(0.590893\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 123.165i 1.21946i 0.792611 + 0.609728i \(0.208721\pi\)
−0.792611 + 0.609728i \(0.791279\pi\)
\(102\) 0 0
\(103\) − 82.0848i − 0.796940i −0.917181 0.398470i \(-0.869541\pi\)
0.917181 0.398470i \(-0.130459\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −180.466 −1.68660 −0.843299 0.537445i \(-0.819390\pi\)
−0.843299 + 0.537445i \(0.819390\pi\)
\(108\) 0 0
\(109\) 22.2137i 0.203796i 0.994795 + 0.101898i \(0.0324915\pi\)
−0.994795 + 0.101898i \(0.967509\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −139.381 −1.23346 −0.616732 0.787173i \(-0.711544\pi\)
−0.616732 + 0.787173i \(0.711544\pi\)
\(114\) 0 0
\(115\) 85.4215 0.742795
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 195.231i − 1.64060i
\(120\) 0 0
\(121\) 231.046 1.90947
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 113.612i 0.908896i
\(126\) 0 0
\(127\) − 80.3083i − 0.632349i −0.948701 0.316175i \(-0.897601\pi\)
0.948701 0.316175i \(-0.102399\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 55.7583 0.425636 0.212818 0.977092i \(-0.431736\pi\)
0.212818 + 0.977092i \(0.431736\pi\)
\(132\) 0 0
\(133\) − 82.5293i − 0.620521i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.28641 0.0312877 0.0156438 0.999878i \(-0.495020\pi\)
0.0156438 + 0.999878i \(0.495020\pi\)
\(138\) 0 0
\(139\) 7.84615 0.0564471 0.0282236 0.999602i \(-0.491015\pi\)
0.0282236 + 0.999602i \(0.491015\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 418.631i − 2.92749i
\(144\) 0 0
\(145\) 22.3253 0.153967
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 269.056i − 1.80574i −0.429912 0.902871i \(-0.641455\pi\)
0.429912 0.902871i \(-0.358545\pi\)
\(150\) 0 0
\(151\) 179.626i 1.18958i 0.803881 + 0.594789i \(0.202765\pi\)
−0.803881 + 0.594789i \(0.797235\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −123.976 −0.799842
\(156\) 0 0
\(157\) − 145.890i − 0.929238i −0.885511 0.464619i \(-0.846191\pi\)
0.885511 0.464619i \(-0.153809\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −268.788 −1.66949
\(162\) 0 0
\(163\) −100.496 −0.616540 −0.308270 0.951299i \(-0.599750\pi\)
−0.308270 + 0.951299i \(0.599750\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 154.410i − 0.924611i −0.886721 0.462306i \(-0.847022\pi\)
0.886721 0.462306i \(-0.152978\pi\)
\(168\) 0 0
\(169\) −328.810 −1.94562
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 57.6442i − 0.333203i −0.986024 0.166602i \(-0.946721\pi\)
0.986024 0.166602i \(-0.0532794\pi\)
\(174\) 0 0
\(175\) − 149.785i − 0.855913i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −83.2753 −0.465225 −0.232613 0.972569i \(-0.574727\pi\)
−0.232613 + 0.972569i \(0.574727\pi\)
\(180\) 0 0
\(181\) 227.811i 1.25862i 0.777153 + 0.629311i \(0.216663\pi\)
−0.777153 + 0.629311i \(0.783337\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 74.7813 0.404223
\(186\) 0 0
\(187\) −440.896 −2.35773
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 28.0652i − 0.146938i −0.997297 0.0734692i \(-0.976593\pi\)
0.997297 0.0734692i \(-0.0234071\pi\)
\(192\) 0 0
\(193\) −310.951 −1.61115 −0.805573 0.592496i \(-0.798142\pi\)
−0.805573 + 0.592496i \(0.798142\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 177.071i 0.898838i 0.893321 + 0.449419i \(0.148369\pi\)
−0.893321 + 0.449419i \(0.851631\pi\)
\(198\) 0 0
\(199\) 183.708i 0.923156i 0.887100 + 0.461578i \(0.152717\pi\)
−0.887100 + 0.461578i \(0.847283\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −70.2489 −0.346054
\(204\) 0 0
\(205\) − 205.401i − 1.00196i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −186.378 −0.891761
\(210\) 0 0
\(211\) 198.513 0.940821 0.470411 0.882448i \(-0.344106\pi\)
0.470411 + 0.882448i \(0.344106\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 154.410i − 0.718186i
\(216\) 0 0
\(217\) 390.103 1.79771
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 524.286i 2.37233i
\(222\) 0 0
\(223\) 2.68500i 0.0120403i 0.999982 + 0.00602017i \(0.00191629\pi\)
−0.999982 + 0.00602017i \(0.998084\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −337.120 −1.48511 −0.742555 0.669786i \(-0.766386\pi\)
−0.742555 + 0.669786i \(0.766386\pi\)
\(228\) 0 0
\(229\) − 46.9618i − 0.205073i −0.994729 0.102537i \(-0.967304\pi\)
0.994729 0.102537i \(-0.0326959\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 86.6965 0.372088 0.186044 0.982541i \(-0.440433\pi\)
0.186044 + 0.982541i \(0.440433\pi\)
\(234\) 0 0
\(235\) 143.356 0.610026
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 149.408i − 0.625138i −0.949895 0.312569i \(-0.898811\pi\)
0.949895 0.312569i \(-0.101189\pi\)
\(240\) 0 0
\(241\) 338.559 1.40481 0.702405 0.711778i \(-0.252110\pi\)
0.702405 + 0.711778i \(0.252110\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 52.8827i − 0.215848i
\(246\) 0 0
\(247\) 221.629i 0.897284i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 117.462 0.467976 0.233988 0.972239i \(-0.424822\pi\)
0.233988 + 0.972239i \(0.424822\pi\)
\(252\) 0 0
\(253\) 607.011i 2.39925i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −79.7319 −0.310241 −0.155120 0.987896i \(-0.549577\pi\)
−0.155120 + 0.987896i \(0.549577\pi\)
\(258\) 0 0
\(259\) −235.308 −0.908524
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 70.1112i 0.266582i 0.991077 + 0.133291i \(0.0425546\pi\)
−0.991077 + 0.133291i \(0.957445\pi\)
\(264\) 0 0
\(265\) 15.3535 0.0579379
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 128.891i 0.479147i 0.970878 + 0.239574i \(0.0770077\pi\)
−0.970878 + 0.239574i \(0.922992\pi\)
\(270\) 0 0
\(271\) − 334.953i − 1.23599i −0.786182 0.617995i \(-0.787945\pi\)
0.786182 0.617995i \(-0.212055\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −338.262 −1.23005
\(276\) 0 0
\(277\) − 490.465i − 1.77063i −0.464991 0.885315i \(-0.653942\pi\)
0.464991 0.885315i \(-0.346058\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −496.474 −1.76681 −0.883406 0.468608i \(-0.844756\pi\)
−0.883406 + 0.468608i \(0.844756\pi\)
\(282\) 0 0
\(283\) 323.319 1.14247 0.571235 0.820787i \(-0.306465\pi\)
0.571235 + 0.820787i \(0.306465\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 646.317i 2.25198i
\(288\) 0 0
\(289\) 263.170 0.910621
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 238.720i − 0.814742i −0.913263 0.407371i \(-0.866446\pi\)
0.913263 0.407371i \(-0.133554\pi\)
\(294\) 0 0
\(295\) − 125.629i − 0.425861i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 721.819 2.41411
\(300\) 0 0
\(301\) 485.868i 1.61418i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 73.1730 0.239912
\(306\) 0 0
\(307\) 196.695 0.640699 0.320349 0.947299i \(-0.396200\pi\)
0.320349 + 0.947299i \(0.396200\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 20.0012i − 0.0643126i −0.999483 0.0321563i \(-0.989763\pi\)
0.999483 0.0321563i \(-0.0102374\pi\)
\(312\) 0 0
\(313\) −341.608 −1.09140 −0.545700 0.837981i \(-0.683736\pi\)
−0.545700 + 0.837981i \(0.683736\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 481.401i 1.51861i 0.650732 + 0.759307i \(0.274462\pi\)
−0.650732 + 0.759307i \(0.725538\pi\)
\(318\) 0 0
\(319\) 158.645i 0.497319i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 233.416 0.722651
\(324\) 0 0
\(325\) 402.240i 1.23766i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −451.086 −1.37108
\(330\) 0 0
\(331\) −91.4065 −0.276152 −0.138076 0.990422i \(-0.544092\pi\)
−0.138076 + 0.990422i \(0.544092\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 133.693i − 0.399084i
\(336\) 0 0
\(337\) −194.220 −0.576320 −0.288160 0.957582i \(-0.593044\pi\)
−0.288160 + 0.957582i \(0.593044\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 880.979i − 2.58352i
\(342\) 0 0
\(343\) − 240.707i − 0.701768i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 68.5829 0.197645 0.0988226 0.995105i \(-0.468492\pi\)
0.0988226 + 0.995105i \(0.468492\pi\)
\(348\) 0 0
\(349\) − 536.933i − 1.53849i −0.638955 0.769244i \(-0.720633\pi\)
0.638955 0.769244i \(-0.279367\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 486.832 1.37913 0.689563 0.724226i \(-0.257803\pi\)
0.689563 + 0.724226i \(0.257803\pi\)
\(354\) 0 0
\(355\) −19.4023 −0.0546544
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.1271i 0.0867049i 0.999060 + 0.0433525i \(0.0138038\pi\)
−0.999060 + 0.0433525i \(0.986196\pi\)
\(360\) 0 0
\(361\) −262.329 −0.726673
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 77.7291i − 0.212956i
\(366\) 0 0
\(367\) 656.463i 1.78873i 0.447340 + 0.894364i \(0.352371\pi\)
−0.447340 + 0.894364i \(0.647629\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −48.3116 −0.130220
\(372\) 0 0
\(373\) − 285.162i − 0.764508i −0.924057 0.382254i \(-0.875148\pi\)
0.924057 0.382254i \(-0.124852\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 188.651 0.500399
\(378\) 0 0
\(379\) 96.1985 0.253822 0.126911 0.991914i \(-0.459494\pi\)
0.126911 + 0.991914i \(0.459494\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 542.017i − 1.41519i −0.706619 0.707594i \(-0.749780\pi\)
0.706619 0.707594i \(-0.250220\pi\)
\(384\) 0 0
\(385\) −411.608 −1.06911
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 251.966i − 0.647729i −0.946104 0.323864i \(-0.895018\pi\)
0.946104 0.323864i \(-0.104982\pi\)
\(390\) 0 0
\(391\) − 760.209i − 1.94427i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −114.079 −0.288807
\(396\) 0 0
\(397\) − 534.951i − 1.34748i −0.738966 0.673742i \(-0.764686\pi\)
0.738966 0.673742i \(-0.235314\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 586.919 1.46364 0.731819 0.681499i \(-0.238672\pi\)
0.731819 + 0.681499i \(0.238672\pi\)
\(402\) 0 0
\(403\) −1047.60 −2.59951
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 531.401i 1.30565i
\(408\) 0 0
\(409\) 364.507 0.891214 0.445607 0.895229i \(-0.352988\pi\)
0.445607 + 0.895229i \(0.352988\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 395.306i 0.957157i
\(414\) 0 0
\(415\) − 28.4789i − 0.0686239i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 181.773 0.433827 0.216913 0.976191i \(-0.430401\pi\)
0.216913 + 0.976191i \(0.430401\pi\)
\(420\) 0 0
\(421\) − 168.535i − 0.400320i −0.979763 0.200160i \(-0.935854\pi\)
0.979763 0.200160i \(-0.0641462\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 423.633 0.996785
\(426\) 0 0
\(427\) −230.247 −0.539220
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 373.929i 0.867585i 0.901013 + 0.433792i \(0.142825\pi\)
−0.901013 + 0.433792i \(0.857175\pi\)
\(432\) 0 0
\(433\) 483.636 1.11694 0.558471 0.829524i \(-0.311388\pi\)
0.558471 + 0.829524i \(0.311388\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 321.360i − 0.735377i
\(438\) 0 0
\(439\) 449.117i 1.02305i 0.859270 + 0.511523i \(0.170919\pi\)
−0.859270 + 0.511523i \(0.829081\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −361.663 −0.816396 −0.408198 0.912893i \(-0.633843\pi\)
−0.408198 + 0.912893i \(0.633843\pi\)
\(444\) 0 0
\(445\) − 360.952i − 0.811128i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 589.191 1.31223 0.656115 0.754661i \(-0.272199\pi\)
0.656115 + 0.754661i \(0.272199\pi\)
\(450\) 0 0
\(451\) 1459.59 3.23635
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 489.458i 1.07573i
\(456\) 0 0
\(457\) −452.703 −0.990597 −0.495299 0.868723i \(-0.664941\pi\)
−0.495299 + 0.868723i \(0.664941\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 290.060i 0.629197i 0.949225 + 0.314598i \(0.101870\pi\)
−0.949225 + 0.314598i \(0.898130\pi\)
\(462\) 0 0
\(463\) 389.267i 0.840749i 0.907351 + 0.420374i \(0.138101\pi\)
−0.907351 + 0.420374i \(0.861899\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −175.827 −0.376504 −0.188252 0.982121i \(-0.560282\pi\)
−0.188252 + 0.982121i \(0.560282\pi\)
\(468\) 0 0
\(469\) 420.680i 0.896972i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1097.25 2.31976
\(474\) 0 0
\(475\) 179.081 0.377012
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 321.473i 0.671134i 0.942016 + 0.335567i \(0.108928\pi\)
−0.942016 + 0.335567i \(0.891072\pi\)
\(480\) 0 0
\(481\) 631.909 1.31374
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 144.289i − 0.297503i
\(486\) 0 0
\(487\) − 244.014i − 0.501055i −0.968109 0.250528i \(-0.919396\pi\)
0.968109 0.250528i \(-0.0806041\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 239.563 0.487909 0.243955 0.969787i \(-0.421555\pi\)
0.243955 + 0.969787i \(0.421555\pi\)
\(492\) 0 0
\(493\) − 198.684i − 0.403010i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 61.0516 0.122840
\(498\) 0 0
\(499\) −134.164 −0.268865 −0.134433 0.990923i \(-0.542921\pi\)
−0.134433 + 0.990923i \(0.542921\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 214.317i − 0.426078i −0.977044 0.213039i \(-0.931664\pi\)
0.977044 0.213039i \(-0.0683362\pi\)
\(504\) 0 0
\(505\) −325.206 −0.643971
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 453.973i − 0.891892i −0.895060 0.445946i \(-0.852867\pi\)
0.895060 0.445946i \(-0.147133\pi\)
\(510\) 0 0
\(511\) 244.583i 0.478637i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 216.737 0.420849
\(516\) 0 0
\(517\) 1018.70i 1.97040i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −97.5418 −0.187220 −0.0936102 0.995609i \(-0.529841\pi\)
−0.0936102 + 0.995609i \(0.529841\pi\)
\(522\) 0 0
\(523\) 476.762 0.911590 0.455795 0.890085i \(-0.349355\pi\)
0.455795 + 0.890085i \(0.349355\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1103.32i 2.09359i
\(528\) 0 0
\(529\) −517.630 −0.978507
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 1735.66i − 3.25639i
\(534\) 0 0
\(535\) − 476.503i − 0.890660i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 375.788 0.697195
\(540\) 0 0
\(541\) 987.446i 1.82522i 0.408826 + 0.912612i \(0.365938\pi\)
−0.408826 + 0.912612i \(0.634062\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −58.6532 −0.107621
\(546\) 0 0
\(547\) −396.486 −0.724837 −0.362418 0.932016i \(-0.618049\pi\)
−0.362418 + 0.932016i \(0.618049\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 83.9888i − 0.152430i
\(552\) 0 0
\(553\) 358.961 0.649117
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 488.922i 0.877778i 0.898541 + 0.438889i \(0.144628\pi\)
−0.898541 + 0.438889i \(0.855372\pi\)
\(558\) 0 0
\(559\) − 1304.78i − 2.33413i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 417.788 0.742075 0.371037 0.928618i \(-0.379002\pi\)
0.371037 + 0.928618i \(0.379002\pi\)
\(564\) 0 0
\(565\) − 368.023i − 0.651369i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 493.590 0.867469 0.433735 0.901041i \(-0.357196\pi\)
0.433735 + 0.901041i \(0.357196\pi\)
\(570\) 0 0
\(571\) −687.730 −1.20443 −0.602215 0.798334i \(-0.705715\pi\)
−0.602215 + 0.798334i \(0.705715\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 583.244i − 1.01434i
\(576\) 0 0
\(577\) 206.939 0.358647 0.179324 0.983790i \(-0.442609\pi\)
0.179324 + 0.983790i \(0.442609\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 89.6120i 0.154238i
\(582\) 0 0
\(583\) 109.103i 0.187141i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 879.623 1.49851 0.749253 0.662284i \(-0.230413\pi\)
0.749253 + 0.662284i \(0.230413\pi\)
\(588\) 0 0
\(589\) 466.402i 0.791854i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −444.195 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(594\) 0 0
\(595\) 515.490 0.866369
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1025.55i 1.71211i 0.516888 + 0.856053i \(0.327090\pi\)
−0.516888 + 0.856053i \(0.672910\pi\)
\(600\) 0 0
\(601\) −18.5617 −0.0308846 −0.0154423 0.999881i \(-0.504916\pi\)
−0.0154423 + 0.999881i \(0.504916\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 610.055i 1.00836i
\(606\) 0 0
\(607\) 648.184i 1.06785i 0.845533 + 0.533924i \(0.179283\pi\)
−0.845533 + 0.533924i \(0.820717\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1211.37 1.98261
\(612\) 0 0
\(613\) − 616.049i − 1.00497i −0.864585 0.502487i \(-0.832418\pi\)
0.864585 0.502487i \(-0.167582\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 321.805 0.521564 0.260782 0.965398i \(-0.416020\pi\)
0.260782 + 0.965398i \(0.416020\pi\)
\(618\) 0 0
\(619\) −769.275 −1.24277 −0.621386 0.783505i \(-0.713430\pi\)
−0.621386 + 0.783505i \(0.713430\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1135.78i 1.82307i
\(624\) 0 0
\(625\) 150.725 0.241160
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 665.516i − 1.05805i
\(630\) 0 0
\(631\) 766.730i 1.21510i 0.794280 + 0.607551i \(0.207848\pi\)
−0.794280 + 0.607551i \(0.792152\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 212.046 0.333931
\(636\) 0 0
\(637\) − 446.864i − 0.701513i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −436.825 −0.681474 −0.340737 0.940159i \(-0.610677\pi\)
−0.340737 + 0.940159i \(0.610677\pi\)
\(642\) 0 0
\(643\) 276.284 0.429680 0.214840 0.976649i \(-0.431077\pi\)
0.214840 + 0.976649i \(0.431077\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 795.622i 1.22971i 0.788640 + 0.614855i \(0.210785\pi\)
−0.788640 + 0.614855i \(0.789215\pi\)
\(648\) 0 0
\(649\) 892.729 1.37555
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 305.664i 0.468091i 0.972226 + 0.234046i \(0.0751965\pi\)
−0.972226 + 0.234046i \(0.924804\pi\)
\(654\) 0 0
\(655\) 147.224i 0.224770i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 153.889 0.233519 0.116760 0.993160i \(-0.462749\pi\)
0.116760 + 0.993160i \(0.462749\pi\)
\(660\) 0 0
\(661\) − 38.0847i − 0.0576167i −0.999585 0.0288084i \(-0.990829\pi\)
0.999585 0.0288084i \(-0.00917126\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 217.911 0.327685
\(666\) 0 0
\(667\) −273.541 −0.410107
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 519.973i 0.774922i
\(672\) 0 0
\(673\) 392.195 0.582757 0.291378 0.956608i \(-0.405886\pi\)
0.291378 + 0.956608i \(0.405886\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 917.262i − 1.35489i −0.735573 0.677446i \(-0.763087\pi\)
0.735573 0.677446i \(-0.236913\pi\)
\(678\) 0 0
\(679\) 454.021i 0.668661i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1280.10 −1.87424 −0.937119 0.349010i \(-0.886518\pi\)
−0.937119 + 0.349010i \(0.886518\pi\)
\(684\) 0 0
\(685\) 11.3179i 0.0165224i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 129.739 0.188300
\(690\) 0 0
\(691\) −1147.79 −1.66106 −0.830531 0.556972i \(-0.811963\pi\)
−0.830531 + 0.556972i \(0.811963\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20.7170i 0.0298086i
\(696\) 0 0
\(697\) −1827.97 −2.62262
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 450.531i 0.642697i 0.946961 + 0.321349i \(0.104136\pi\)
−0.946961 + 0.321349i \(0.895864\pi\)
\(702\) 0 0
\(703\) − 281.331i − 0.400186i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1023.30 1.44738
\(708\) 0 0
\(709\) − 1071.57i − 1.51139i −0.654926 0.755693i \(-0.727300\pi\)
0.654926 0.755693i \(-0.272700\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1519.02 2.13046
\(714\) 0 0
\(715\) 1105.36 1.54595
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 569.979i 0.792739i 0.918091 + 0.396370i \(0.129730\pi\)
−0.918091 + 0.396370i \(0.870270\pi\)
\(720\) 0 0
\(721\) −681.987 −0.945891
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 152.433i − 0.210253i
\(726\) 0 0
\(727\) 699.311i 0.961914i 0.876744 + 0.480957i \(0.159711\pi\)
−0.876744 + 0.480957i \(0.840289\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1374.17 −1.87985
\(732\) 0 0
\(733\) − 175.235i − 0.239065i −0.992830 0.119533i \(-0.961860\pi\)
0.992830 0.119533i \(-0.0381396\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 950.032 1.28905
\(738\) 0 0
\(739\) −975.643 −1.32022 −0.660110 0.751169i \(-0.729490\pi\)
−0.660110 + 0.751169i \(0.729490\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 880.230i − 1.18470i −0.805682 0.592349i \(-0.798201\pi\)
0.805682 0.592349i \(-0.201799\pi\)
\(744\) 0 0
\(745\) 710.415 0.953578
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1499.37i 2.00183i
\(750\) 0 0
\(751\) 48.7538i 0.0649186i 0.999473 + 0.0324593i \(0.0103339\pi\)
−0.999473 + 0.0324593i \(0.989666\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −474.286 −0.628194
\(756\) 0 0
\(757\) 1194.56i 1.57802i 0.614382 + 0.789009i \(0.289406\pi\)
−0.614382 + 0.789009i \(0.710594\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1032.85 −1.35722 −0.678612 0.734497i \(-0.737418\pi\)
−0.678612 + 0.734497i \(0.737418\pi\)
\(762\) 0 0
\(763\) 184.559 0.241886
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1061.58i − 1.38406i
\(768\) 0 0
\(769\) −200.830 −0.261157 −0.130579 0.991438i \(-0.541684\pi\)
−0.130579 + 0.991438i \(0.541684\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 949.438i − 1.22825i −0.789208 0.614126i \(-0.789509\pi\)
0.789208 0.614126i \(-0.210491\pi\)
\(774\) 0 0
\(775\) 846.486i 1.09224i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −772.729 −0.991950
\(780\) 0 0
\(781\) − 137.874i − 0.176536i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 385.210 0.490713
\(786\) 0 0
\(787\) 646.331 0.821259 0.410630 0.911802i \(-0.365309\pi\)
0.410630 + 0.911802i \(0.365309\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1158.03i 1.46400i
\(792\) 0 0
\(793\) 618.319 0.779721
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.5148i 0.0470700i 0.999723 + 0.0235350i \(0.00749211\pi\)
−0.999723 + 0.0235350i \(0.992508\pi\)
\(798\) 0 0
\(799\) − 1275.80i − 1.59674i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 552.348 0.687856
\(804\) 0 0
\(805\) − 709.709i − 0.881626i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 572.790 0.708023 0.354011 0.935241i \(-0.384817\pi\)
0.354011 + 0.935241i \(0.384817\pi\)
\(810\) 0 0
\(811\) 486.350 0.599692 0.299846 0.953988i \(-0.403065\pi\)
0.299846 + 0.953988i \(0.403065\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 265.350i − 0.325583i
\(816\) 0 0
\(817\) −580.898 −0.711014
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 311.021i 0.378832i 0.981897 + 0.189416i \(0.0606595\pi\)
−0.981897 + 0.189416i \(0.939341\pi\)
\(822\) 0 0
\(823\) − 824.124i − 1.00137i −0.865631 0.500683i \(-0.833082\pi\)
0.865631 0.500683i \(-0.166918\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 918.441 1.11057 0.555285 0.831660i \(-0.312609\pi\)
0.555285 + 0.831660i \(0.312609\pi\)
\(828\) 0 0
\(829\) 937.997i 1.13148i 0.824584 + 0.565740i \(0.191409\pi\)
−0.824584 + 0.565740i \(0.808591\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −470.630 −0.564982
\(834\) 0 0
\(835\) 407.705 0.488269
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 612.844i − 0.730446i −0.930920 0.365223i \(-0.880993\pi\)
0.930920 0.365223i \(-0.119007\pi\)
\(840\) 0 0
\(841\) 769.509 0.914993
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 868.191i − 1.02744i
\(846\) 0 0
\(847\) − 1919.61i − 2.26636i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −916.261 −1.07669
\(852\) 0 0
\(853\) 491.372i 0.576051i 0.957623 + 0.288026i \(0.0929989\pi\)
−0.957623 + 0.288026i \(0.907001\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −642.820 −0.750082 −0.375041 0.927008i \(-0.622371\pi\)
−0.375041 + 0.927008i \(0.622371\pi\)
\(858\) 0 0
\(859\) 443.260 0.516018 0.258009 0.966142i \(-0.416934\pi\)
0.258009 + 0.966142i \(0.416934\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 565.494i − 0.655265i −0.944805 0.327632i \(-0.893749\pi\)
0.944805 0.327632i \(-0.106251\pi\)
\(864\) 0 0
\(865\) 152.204 0.175958
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 810.652i − 0.932856i
\(870\) 0 0
\(871\) − 1129.72i − 1.29704i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 943.925 1.07877
\(876\) 0 0
\(877\) 330.215i 0.376528i 0.982118 + 0.188264i \(0.0602861\pi\)
−0.982118 + 0.188264i \(0.939714\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 358.037 0.406398 0.203199 0.979137i \(-0.434866\pi\)
0.203199 + 0.979137i \(0.434866\pi\)
\(882\) 0 0
\(883\) 266.351 0.301643 0.150822 0.988561i \(-0.451808\pi\)
0.150822 + 0.988561i \(0.451808\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1375.40i − 1.55062i −0.631581 0.775310i \(-0.717594\pi\)
0.631581 0.775310i \(-0.282406\pi\)
\(888\) 0 0
\(889\) −667.228 −0.750537
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 539.313i − 0.603934i
\(894\) 0 0
\(895\) − 219.880i − 0.245676i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 397.001 0.441603
\(900\) 0 0
\(901\) − 136.639i − 0.151652i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −601.512 −0.664654
\(906\) 0 0
\(907\) 106.524 0.117446 0.0587230 0.998274i \(-0.481297\pi\)
0.0587230 + 0.998274i \(0.481297\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 842.877i − 0.925222i −0.886561 0.462611i \(-0.846913\pi\)
0.886561 0.462611i \(-0.153087\pi\)
\(912\) 0 0
\(913\) 202.373 0.221657
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 463.258i − 0.505189i
\(918\) 0 0
\(919\) 1388.07i 1.51041i 0.655489 + 0.755205i \(0.272463\pi\)
−0.655489 + 0.755205i \(0.727537\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −163.951 −0.177629
\(924\) 0 0
\(925\) − 510.595i − 0.551995i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 557.068 0.599643 0.299821 0.953995i \(-0.403073\pi\)
0.299821 + 0.953995i \(0.403073\pi\)
\(930\) 0 0
\(931\) −198.947 −0.213692
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 1164.14i − 1.24507i
\(936\) 0 0
\(937\) 876.585 0.935523 0.467761 0.883855i \(-0.345061\pi\)
0.467761 + 0.883855i \(0.345061\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1290.14i 1.37103i 0.728058 + 0.685515i \(0.240423\pi\)
−0.728058 + 0.685515i \(0.759577\pi\)
\(942\) 0 0
\(943\) 2516.69i 2.66881i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −850.851 −0.898470 −0.449235 0.893414i \(-0.648303\pi\)
−0.449235 + 0.893414i \(0.648303\pi\)
\(948\) 0 0
\(949\) − 656.818i − 0.692116i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −297.237 −0.311896 −0.155948 0.987765i \(-0.549843\pi\)
−0.155948 + 0.987765i \(0.549843\pi\)
\(954\) 0 0
\(955\) 74.1036 0.0775954
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 35.6129i − 0.0371355i
\(960\) 0 0
\(961\) −1243.61 −1.29408
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 821.037i − 0.850815i
\(966\) 0 0
\(967\) − 943.377i − 0.975570i −0.872964 0.487785i \(-0.837805\pi\)
0.872964 0.487785i \(-0.162195\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 709.360 0.730546 0.365273 0.930900i \(-0.380976\pi\)
0.365273 + 0.930900i \(0.380976\pi\)
\(972\) 0 0
\(973\) − 65.1884i − 0.0669973i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 295.503 0.302459 0.151230 0.988499i \(-0.451677\pi\)
0.151230 + 0.988499i \(0.451677\pi\)
\(978\) 0 0
\(979\) 2564.95 2.61997
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 559.879i 0.569561i 0.958593 + 0.284781i \(0.0919208\pi\)
−0.958593 + 0.284781i \(0.908079\pi\)
\(984\) 0 0
\(985\) −467.539 −0.474659
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1891.92i 1.91296i
\(990\) 0 0
\(991\) 1327.50i 1.33955i 0.742563 + 0.669776i \(0.233610\pi\)
−0.742563 + 0.669776i \(0.766390\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −485.064 −0.487501
\(996\) 0 0
\(997\) 878.699i 0.881343i 0.897668 + 0.440672i \(0.145260\pi\)
−0.897668 + 0.440672i \(0.854740\pi\)
\(998\) 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 1728.3.b.i.1567.7 yes 12
3.2 odd 2 1728.3.b.j.1567.5 yes 12
4.3 odd 2 inner 1728.3.b.i.1567.8 yes 12
8.3 odd 2 inner 1728.3.b.i.1567.6 yes 12
8.5 even 2 inner 1728.3.b.i.1567.5 12
12.11 even 2 1728.3.b.j.1567.6 yes 12
24.5 odd 2 1728.3.b.j.1567.7 yes 12
24.11 even 2 1728.3.b.j.1567.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.i.1567.5 12 8.5 even 2 inner
1728.3.b.i.1567.6 yes 12 8.3 odd 2 inner
1728.3.b.i.1567.7 yes 12 1.1 even 1 trivial
1728.3.b.i.1567.8 yes 12 4.3 odd 2 inner
1728.3.b.j.1567.5 yes 12 3.2 odd 2
1728.3.b.j.1567.6 yes 12 12.11 even 2
1728.3.b.j.1567.7 yes 12 24.5 odd 2
1728.3.b.j.1567.8 yes 12 24.11 even 2