Properties

Label 2800.2.g.h.449.2
Level 2800
Weight 2
Character 2800.449
Analytic conductor 22.358
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2800.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\)
Character \(\chi\) = 2800.449
Dual form 2800.2.g.h.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} -4.00000i q^{13} -6.00000i q^{17} +2.00000 q^{19} -2.00000 q^{21} +4.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} -2.00000i q^{37} +8.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} -12.0000i q^{47} -1.00000 q^{49} +12.0000 q^{51} +6.00000i q^{53} +4.00000i q^{57} -6.00000 q^{59} +8.00000 q^{61} -1.00000i q^{63} -4.00000i q^{67} +2.00000i q^{73} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} +12.0000i q^{87} +6.00000 q^{89} +4.00000 q^{91} +8.00000i q^{93} +10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 4q^{19} - 4q^{21} + 12q^{29} + 8q^{31} + 16q^{39} + 12q^{41} - 2q^{49} + 24q^{51} - 12q^{59} + 16q^{61} + 16q^{79} - 22q^{81} + 12q^{89} + 8q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.0000i 1.28654i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000i 0.369800i
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 0 0
\(129\) 16.0000 1.40872
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.00000i − 0.164957i
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.00000i 0.319235i 0.987179 + 0.159617i \(0.0510260\pi\)
−0.987179 + 0.159617i \(0.948974\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) − 12.0000i − 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 16.0000i 1.18275i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000i 0.271538i
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) − 10.0000i − 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 8.00000i − 0.509028i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 0 0
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 8.00000i 0.484182i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 22.0000i 1.30776i 0.756596 + 0.653882i \(0.226861\pi\)
−0.756596 + 0.653882i \(0.773139\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −20.0000 −1.17242
\(292\) 0 0
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) − 12.0000i − 0.667698i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 4.00000i − 0.221201i
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 12.0000i 0.635107i
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) − 22.0000i − 1.15470i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 24.0000i − 1.23606i
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) 0 0
\(383\) − 36.0000i − 1.83951i −0.392488 0.919757i \(-0.628386\pi\)
0.392488 0.919757i \(-0.371614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) − 36.0000i − 1.81596i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 20.0000i − 1.00377i −0.864934 0.501886i \(-0.832640\pi\)
0.864934 0.501886i \(-0.167360\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) − 16.0000i − 0.797017i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) 0 0
\(413\) − 6.00000i − 0.295241i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 28.0000i 1.37117i
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 12.0000i 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 36.0000i 1.70274i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 16.0000i − 0.751746i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 0 0
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 6.00000i − 0.277647i −0.990317 0.138823i \(-0.955668\pi\)
0.990317 0.138823i \(-0.0443321\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −8.00000 −0.368621
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) 0 0
\(489\) −32.0000 −1.44709
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) − 36.0000i − 1.62136i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6.00000i − 0.266469i
\(508\) 0 0
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) 8.00000i 0.353209i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) − 2.00000i − 0.0874539i −0.999044 0.0437269i \(-0.986077\pi\)
0.999044 0.0437269i \(-0.0139232\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 24.0000i − 1.04546i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) − 24.0000i − 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 24.0000i − 1.03568i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 40.0000i 1.71656i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 6.00000i − 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 30.0000i − 1.26435i −0.774826 0.632175i \(-0.782163\pi\)
0.774826 0.632175i \(-0.217837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 11.0000i − 0.461957i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) − 48.0000i − 2.00523i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) −28.0000 −1.16364
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 42.0000i − 1.73353i −0.498721 0.866763i \(-0.666197\pi\)
0.498721 0.866763i \(-0.333803\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 40.0000i 1.63709i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000i 0.240385i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) 8.00000i 0.317971i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.00000i 0.158486i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 12.0000i − 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 0 0
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 2.00000i − 0.0780274i
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 0 0
\(663\) − 48.0000i − 1.86417i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 26.0000i 1.00223i 0.865382 + 0.501113i \(0.167076\pi\)
−0.865382 + 0.501113i \(0.832924\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −36.0000 −1.37952
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.00000i 0.305219i
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 46.0000 1.74992 0.874961 0.484193i \(-0.160887\pi\)
0.874961 + 0.484193i \(0.160887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 36.0000i − 1.36360i
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 48.0000i 1.79259i
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) − 20.0000i − 0.743808i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 44.0000i 1.63187i 0.578144 + 0.815935i \(0.303777\pi\)
−0.578144 + 0.815935i \(0.696223\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) − 40.0000i − 1.47743i −0.674016 0.738717i \(-0.735432\pi\)
0.674016 0.738717i \(-0.264568\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 36.0000i 1.31191i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) − 2.00000i − 0.0724049i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 36.0000 1.29651
\(772\) 0 0
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 4.00000i 0.143499i
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000i 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) − 32.0000i − 1.13635i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0000i 0.844840i
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 32.0000i 1.12229i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 40.0000i 1.39431i 0.716919 + 0.697156i \(0.245552\pi\)
−0.716919 + 0.697156i \(0.754448\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 0 0
\(829\) −56.0000 −1.94496 −0.972480 0.232986i \(-0.925151\pi\)
−0.972480 + 0.232986i \(0.925151\pi\)
\(830\) 0 0
\(831\) −20.0000 −0.693792
\(832\) 0 0
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 16.0000i 0.553041i
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) − 12.0000i − 0.413302i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 11.0000i − 0.377964i
\(848\) 0 0
\(849\) −44.0000 −1.51008
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 44.0000i 1.50653i 0.657716 + 0.753266i \(0.271523\pi\)
−0.657716 + 0.753266i \(0.728477\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 38.0000i − 1.29055i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) − 10.0000i − 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 0 0
\(879\) −48.0000 −1.61900
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 36.0000i − 1.20876i −0.796696 0.604381i \(-0.793421\pi\)
0.796696 0.604381i \(-0.206579\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 24.0000i − 0.803129i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 16.0000i 0.532447i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 18.0000i − 0.594412i
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.00000i − 0.131377i
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) 48.0000i 1.57145i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000i 0.779895i 0.920837 + 0.389948i \(0.127507\pi\)
−0.920837 + 0.389948i \(0.872493\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) − 54.0000i − 1.74923i −0.484817 0.874616i \(-0.661114\pi\)
0.484817 0.874616i \(-0.338886\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 0 0
\(973\) 14.0000i 0.448819i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.00000i 0.191957i 0.995383 + 0.0959785i \(0.0305980\pi\)
−0.995383 + 0.0959785i \(0.969402\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 24.0000i 0.763928i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) − 16.0000i − 0.507745i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 8.00000i − 0.253363i −0.991943 0.126681i \(-0.959567\pi\)
0.991943 0.126681i \(-0.0404325\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
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 2800.2.g.h.449.2 2
4.3 odd 2 350.2.c.d.99.2 2
5.2 odd 4 112.2.a.c.1.1 1
5.3 odd 4 2800.2.a.g.1.1 1
5.4 even 2 inner 2800.2.g.h.449.1 2
12.11 even 2 3150.2.g.j.2899.1 2
15.2 even 4 1008.2.a.h.1.1 1
20.3 even 4 350.2.a.f.1.1 1
20.7 even 4 14.2.a.a.1.1 1
20.19 odd 2 350.2.c.d.99.1 2
28.27 even 2 2450.2.c.c.99.2 2
35.2 odd 12 784.2.i.c.753.1 2
35.12 even 12 784.2.i.i.753.1 2
35.17 even 12 784.2.i.i.177.1 2
35.27 even 4 784.2.a.b.1.1 1
35.32 odd 12 784.2.i.c.177.1 2
40.27 even 4 448.2.a.g.1.1 1
40.37 odd 4 448.2.a.a.1.1 1
60.23 odd 4 3150.2.a.i.1.1 1
60.47 odd 4 126.2.a.b.1.1 1
60.59 even 2 3150.2.g.j.2899.2 2
80.27 even 4 1792.2.b.c.897.2 2
80.37 odd 4 1792.2.b.g.897.1 2
80.67 even 4 1792.2.b.c.897.1 2
80.77 odd 4 1792.2.b.g.897.2 2
105.62 odd 4 7056.2.a.bd.1.1 1
120.77 even 4 4032.2.a.r.1.1 1
120.107 odd 4 4032.2.a.w.1.1 1
140.27 odd 4 98.2.a.a.1.1 1
140.47 odd 12 98.2.c.a.67.1 2
140.67 even 12 98.2.c.b.79.1 2
140.83 odd 4 2450.2.a.t.1.1 1
140.87 odd 12 98.2.c.a.79.1 2
140.107 even 12 98.2.c.b.67.1 2
140.139 even 2 2450.2.c.c.99.1 2
180.7 even 12 1134.2.f.l.757.1 2
180.47 odd 12 1134.2.f.f.757.1 2
180.67 even 12 1134.2.f.l.379.1 2
180.167 odd 12 1134.2.f.f.379.1 2
220.87 odd 4 1694.2.a.e.1.1 1
260.47 odd 4 2366.2.d.b.337.1 2
260.187 odd 4 2366.2.d.b.337.2 2
260.207 even 4 2366.2.a.j.1.1 1
280.27 odd 4 3136.2.a.e.1.1 1
280.237 even 4 3136.2.a.z.1.1 1
340.67 even 4 4046.2.a.f.1.1 1
380.227 odd 4 5054.2.a.c.1.1 1
420.47 even 12 882.2.g.d.361.1 2
420.107 odd 12 882.2.g.c.361.1 2
420.167 even 4 882.2.a.i.1.1 1
420.227 even 12 882.2.g.d.667.1 2
420.347 odd 12 882.2.g.c.667.1 2
460.367 odd 4 7406.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 20.7 even 4
98.2.a.a.1.1 1 140.27 odd 4
98.2.c.a.67.1 2 140.47 odd 12
98.2.c.a.79.1 2 140.87 odd 12
98.2.c.b.67.1 2 140.107 even 12
98.2.c.b.79.1 2 140.67 even 12
112.2.a.c.1.1 1 5.2 odd 4
126.2.a.b.1.1 1 60.47 odd 4
350.2.a.f.1.1 1 20.3 even 4
350.2.c.d.99.1 2 20.19 odd 2
350.2.c.d.99.2 2 4.3 odd 2
448.2.a.a.1.1 1 40.37 odd 4
448.2.a.g.1.1 1 40.27 even 4
784.2.a.b.1.1 1 35.27 even 4
784.2.i.c.177.1 2 35.32 odd 12
784.2.i.c.753.1 2 35.2 odd 12
784.2.i.i.177.1 2 35.17 even 12
784.2.i.i.753.1 2 35.12 even 12
882.2.a.i.1.1 1 420.167 even 4
882.2.g.c.361.1 2 420.107 odd 12
882.2.g.c.667.1 2 420.347 odd 12
882.2.g.d.361.1 2 420.47 even 12
882.2.g.d.667.1 2 420.227 even 12
1008.2.a.h.1.1 1 15.2 even 4
1134.2.f.f.379.1 2 180.167 odd 12
1134.2.f.f.757.1 2 180.47 odd 12
1134.2.f.l.379.1 2 180.67 even 12
1134.2.f.l.757.1 2 180.7 even 12
1694.2.a.e.1.1 1 220.87 odd 4
1792.2.b.c.897.1 2 80.67 even 4
1792.2.b.c.897.2 2 80.27 even 4
1792.2.b.g.897.1 2 80.37 odd 4
1792.2.b.g.897.2 2 80.77 odd 4
2366.2.a.j.1.1 1 260.207 even 4
2366.2.d.b.337.1 2 260.47 odd 4
2366.2.d.b.337.2 2 260.187 odd 4
2450.2.a.t.1.1 1 140.83 odd 4
2450.2.c.c.99.1 2 140.139 even 2
2450.2.c.c.99.2 2 28.27 even 2
2800.2.a.g.1.1 1 5.3 odd 4
2800.2.g.h.449.1 2 5.4 even 2 inner
2800.2.g.h.449.2 2 1.1 even 1 trivial
3136.2.a.e.1.1 1 280.27 odd 4
3136.2.a.z.1.1 1 280.237 even 4
3150.2.a.i.1.1 1 60.23 odd 4
3150.2.g.j.2899.1 2 12.11 even 2
3150.2.g.j.2899.2 2 60.59 even 2
4032.2.a.r.1.1 1 120.77 even 4
4032.2.a.w.1.1 1 120.107 odd 4
4046.2.a.f.1.1 1 340.67 even 4
5054.2.a.c.1.1 1 380.227 odd 4
7056.2.a.bd.1.1 1 105.62 odd 4
7406.2.a.a.1.1 1 460.367 odd 4