Properties

Label 1792.2.b.g.897.2
Level 1792
Weight 2
Character 1792.897
Analytic conductor 14.309
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1792.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 897.2
Root \(1.00000i\)
Character \(\chi\) = 1792.897
Dual form 1792.2.b.g.897.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{13} +6.00000 q^{17} -2.00000i q^{19} +2.00000i q^{21} +5.00000 q^{25} +4.00000i q^{27} -6.00000i q^{29} +4.00000 q^{31} -2.00000i q^{37} +8.00000 q^{39} -6.00000 q^{41} +8.00000i q^{43} +12.0000 q^{47} +1.00000 q^{49} +12.0000i q^{51} -6.00000i q^{53} +4.00000 q^{57} -6.00000i q^{59} +8.00000i q^{61} -1.00000 q^{63} +4.00000i q^{67} -2.00000 q^{73} +10.0000i q^{75} -8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} +12.0000 q^{87} +6.00000 q^{89} -4.00000i q^{91} +8.00000i q^{93} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{9} + 12q^{17} + 10q^{25} + 8q^{31} + 16q^{39} - 12q^{41} + 24q^{47} + 2q^{49} + 8q^{57} - 2q^{63} - 4q^{73} - 16q^{79} - 22q^{81} + 24q^{87} + 12q^{89} - 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 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
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 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 \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 12.0000i 1.68034i
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 10.0000i 1.15470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 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
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 4.00000i − 0.419314i
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 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
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 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
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) −16.0000 −1.40872
\(130\) 0 0
\(131\) − 18.0000i − 1.57267i −0.617802 0.786334i \(-0.711977\pi\)
0.617802 0.786334i \(-0.288023\pi\)
\(132\) 0 0
\(133\) − 2.00000i − 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 14.0000i 1.18746i 0.804663 + 0.593732i \(0.202346\pi\)
−0.804663 + 0.593732i \(0.797654\pi\)
\(140\) 0 0
\(141\) 24.0000i 2.02116i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.00000i 0.164957i
\(148\) 0 0
\(149\) 18.0000i 1.47462i 0.675556 + 0.737309i \(0.263904\pi\)
−0.675556 + 0.737309i \(0.736096\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 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
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 2.00000i 0.152944i
\(172\) 0 0
\(173\) − 12.0000i − 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) 0 0
\(175\) 5.00000 0.377964
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) − 20.0000i − 1.48659i −0.668965 0.743294i \(-0.733262\pi\)
0.668965 0.743294i \(-0.266738\pi\)
\(182\) 0 0
\(183\) −16.0000 −1.18275
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000i 0.290957i
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 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
\(211\) 4.00000i 0.275371i 0.990476 + 0.137686i \(0.0439664\pi\)
−0.990476 + 0.137686i \(0.956034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) − 4.00000i − 0.270295i
\(220\) 0 0
\(221\) − 24.0000i − 1.61441i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 0 0
\(229\) 4.00000i 0.264327i 0.991228 + 0.132164i \(0.0421925\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 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 \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 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
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) − 18.0000i − 1.13615i −0.822977 0.568075i \(-0.807688\pi\)
0.822977 0.568075i \(-0.192312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) − 2.00000i − 0.124274i
\(260\) 0 0
\(261\) 6.00000i 0.371391i
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 0 0
\(269\) − 12.0000i − 0.731653i −0.930683 0.365826i \(-0.880786\pi\)
0.930683 0.365826i \(-0.119214\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 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 \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 22.0000i − 1.30776i −0.756596 0.653882i \(-0.773139\pi\)
0.756596 0.653882i \(-0.226861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) − 20.0000i − 1.17242i
\(292\) 0 0
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000i 0.461112i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.00000i − 0.114146i −0.998370 0.0570730i \(-0.981823\pi\)
0.998370 0.0570730i \(-0.0181768\pi\)
\(308\) 0 0
\(309\) − 8.00000i − 0.455104i
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) − 12.0000i − 0.667698i
\(324\) 0 0
\(325\) − 20.0000i − 1.10940i
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 8.00000i 0.439720i 0.975531 + 0.219860i \(0.0705600\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 12.0000i 0.651751i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 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
\(349\) − 28.0000i − 1.49881i −0.662114 0.749403i \(-0.730341\pi\)
0.662114 0.749403i \(-0.269659\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 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
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) − 6.00000i − 0.311504i
\(372\) 0 0
\(373\) − 14.0000i − 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) − 16.0000i − 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 0 0
\(381\) 32.0000i 1.63941i
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 8.00000i − 0.406663i
\(388\) 0 0
\(389\) − 18.0000i − 0.912636i −0.889817 0.456318i \(-0.849168\pi\)
0.889817 0.456318i \(-0.150832\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.0000i 1.00377i 0.864934 + 0.501886i \(0.167360\pi\)
−0.864934 + 0.501886i \(0.832640\pi\)
\(398\) 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
\(411\) − 36.0000i − 1.77575i
\(412\) 0 0
\(413\) − 6.00000i − 0.295241i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −28.0000 −1.37117
\(418\) 0 0
\(419\) − 6.00000i − 0.293119i −0.989202 0.146560i \(-0.953180\pi\)
0.989202 0.146560i \(-0.0468200\pi\)
\(420\) 0 0
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) 0 0
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) 30.0000 1.45521
\(426\) 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
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 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 \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −36.0000 −1.70274
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 24.0000i 1.12022i
\(460\) 0 0
\(461\) 12.0000i 0.558896i 0.960161 + 0.279448i \(0.0901514\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) 0 0
\(469\) 4.00000i 0.184703i
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 10.0000i − 0.458831i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) −32.0000 −1.44709
\(490\) 0 0
\(491\) − 12.0000i − 0.541552i −0.962642 0.270776i \(-0.912720\pi\)
0.962642 0.270776i \(-0.0872803\pi\)
\(492\) 0 0
\(493\) − 36.0000i − 1.62136i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) − 24.0000i − 1.07224i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6.00000i − 0.266469i
\(508\) 0 0
\(509\) 36.0000i 1.59567i 0.602875 + 0.797836i \(0.294022\pi\)
−0.602875 + 0.797836i \(0.705978\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 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 \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 2.00000i 0.0874539i 0.999044 + 0.0437269i \(0.0139232\pi\)
−0.999044 + 0.0437269i \(0.986077\pi\)
\(524\) 0 0
\(525\) 10.0000i 0.436436i
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) 0 0
\(533\) 24.0000i 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −24.0000 −1.03568
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000i 1.63375i 0.576816 + 0.816874i \(0.304295\pi\)
−0.576816 + 0.816874i \(0.695705\pi\)
\(542\) 0 0
\(543\) 40.0000 1.71656
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.00000i − 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 0 0
\(549\) − 8.00000i − 0.341432i
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.00000i 0.254228i 0.991888 + 0.127114i \(0.0405714\pi\)
−0.991888 + 0.127114i \(0.959429\pi\)
\(558\) 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
\(567\) −11.0000 −0.461957
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 32.0000i 1.33916i 0.742741 + 0.669579i \(0.233526\pi\)
−0.742741 + 0.669579i \(0.766474\pi\)
\(572\) 0 0
\(573\) − 48.0000i − 2.00523i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 28.0000i 1.16364i
\(580\) 0 0
\(581\) 6.00000i 0.248922i
\(582\) 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
\(589\) − 8.00000i − 0.329634i
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 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 \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) − 48.0000i − 1.94187i
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 26.0000i 1.04503i 0.852631 + 0.522514i \(0.175006\pi\)
−0.852631 + 0.522514i \(0.824994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 12.0000i − 0.478471i
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 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
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000i 0.313545i
\(652\) 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
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 24.0000i 0.934907i 0.884018 + 0.467454i \(0.154829\pi\)
−0.884018 + 0.467454i \(0.845171\pi\)
\(660\) 0 0
\(661\) 40.0000i 1.55582i 0.628376 + 0.777910i \(0.283720\pi\)
−0.628376 + 0.777910i \(0.716280\pi\)
\(662\) 0 0
\(663\) 48.0000 1.86417
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 16.0000i − 0.618596i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 0 0
\(675\) 20.0000i 0.769800i
\(676\) 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 \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 46.0000i 1.74992i 0.484193 + 0.874961i \(0.339113\pi\)
−0.484193 + 0.874961i \(0.660887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) 12.0000i 0.453882i
\(700\) 0 0
\(701\) 18.0000i 0.679851i 0.940452 + 0.339925i \(0.110402\pi\)
−0.940452 + 0.339925i \(0.889598\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 46.0000i 1.72757i 0.503864 + 0.863783i \(0.331911\pi\)
−0.503864 + 0.863783i \(0.668089\pi\)
\(710\) 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 \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) − 20.0000i − 0.743808i
\(724\) 0 0
\(725\) − 30.0000i − 1.11417i
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 48.0000i 1.77534i
\(732\) 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
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) 0 0
\(741\) − 16.0000i − 0.587775i
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 36.0000 1.31191
\(754\) 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 \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 2.00000i 0.0724049i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 36.0000i 1.29651i
\(772\) 0 0
\(773\) − 24.0000i − 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) 12.0000i 0.429945i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 32.0000 1.13635
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 12.0000i − 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 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
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 2.00000i 0.0702295i 0.999383 + 0.0351147i \(0.0111797\pi\)
−0.999383 + 0.0351147i \(0.988820\pi\)
\(812\) 0 0
\(813\) 32.0000i 1.12229i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 4.00000i 0.139771i
\(820\) 0 0
\(821\) − 6.00000i − 0.209401i −0.994504 0.104701i \(-0.966612\pi\)
0.994504 0.104701i \(-0.0333885\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 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
\(829\) 56.0000i 1.94496i 0.232986 + 0.972480i \(0.425151\pi\)
−0.232986 + 0.972480i \(0.574849\pi\)
\(830\) 0 0
\(831\) −20.0000 −0.693792
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 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
\(847\) 11.0000 0.377964
\(848\) 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 \(-0.728477\pi\)
0.657716 0.753266i \(-0.271523\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 14.0000i 0.477674i 0.971060 + 0.238837i \(0.0767661\pi\)
−0.971060 + 0.238837i \(0.923234\pi\)
\(860\) 0 0
\(861\) − 12.0000i − 0.408959i
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 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
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 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
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 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
\(899\) − 24.0000i − 0.800445i
\(900\) 0 0
\(901\) − 36.0000i − 1.19933i
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) 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
\(925\) − 10.0000i − 0.328798i
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) − 2.00000i − 0.0655474i
\(932\) 0 0
\(933\) − 48.0000i − 1.57145i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 20.0000i 0.652675i
\(940\) 0 0
\(941\) − 24.0000i − 0.782378i −0.920310 0.391189i \(-0.872064\pi\)
0.920310 0.391189i \(-0.127936\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 0 0
\(949\) 8.00000i 0.259691i
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 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
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) − 6.00000i − 0.192549i −0.995355 0.0962746i \(-0.969307\pi\)
0.995355 0.0962746i \(-0.0306927\pi\)
\(972\) 0 0
\(973\) 14.0000i 0.448819i
\(974\) 0 0
\(975\) 40.0000 1.28103
\(976\) 0 0
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 2.00000i − 0.0638551i
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 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
\(993\) −16.0000 −0.507745
\(994\) 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 1792.2.b.g.897.2 2
4.3 odd 2 1792.2.b.c.897.1 2
8.3 odd 2 1792.2.b.c.897.2 2
8.5 even 2 inner 1792.2.b.g.897.1 2
16.3 odd 4 448.2.a.g.1.1 1
16.5 even 4 112.2.a.c.1.1 1
16.11 odd 4 14.2.a.a.1.1 1
16.13 even 4 448.2.a.a.1.1 1
48.5 odd 4 1008.2.a.h.1.1 1
48.11 even 4 126.2.a.b.1.1 1
48.29 odd 4 4032.2.a.r.1.1 1
48.35 even 4 4032.2.a.w.1.1 1
80.27 even 4 350.2.c.d.99.1 2
80.37 odd 4 2800.2.g.h.449.1 2
80.43 even 4 350.2.c.d.99.2 2
80.53 odd 4 2800.2.g.h.449.2 2
80.59 odd 4 350.2.a.f.1.1 1
80.69 even 4 2800.2.a.g.1.1 1
112.5 odd 12 784.2.i.i.753.1 2
112.11 odd 12 98.2.c.b.79.1 2
112.13 odd 4 3136.2.a.z.1.1 1
112.27 even 4 98.2.a.a.1.1 1
112.37 even 12 784.2.i.c.753.1 2
112.53 even 12 784.2.i.c.177.1 2
112.59 even 12 98.2.c.a.79.1 2
112.69 odd 4 784.2.a.b.1.1 1
112.75 even 12 98.2.c.a.67.1 2
112.83 even 4 3136.2.a.e.1.1 1
112.101 odd 12 784.2.i.i.177.1 2
112.107 odd 12 98.2.c.b.67.1 2
144.11 even 12 1134.2.f.f.757.1 2
144.43 odd 12 1134.2.f.l.757.1 2
144.59 even 12 1134.2.f.f.379.1 2
144.139 odd 12 1134.2.f.l.379.1 2
176.43 even 4 1694.2.a.e.1.1 1
208.155 odd 4 2366.2.a.j.1.1 1
208.187 even 4 2366.2.d.b.337.2 2
208.203 even 4 2366.2.d.b.337.1 2
240.59 even 4 3150.2.a.i.1.1 1
240.107 odd 4 3150.2.g.j.2899.2 2
240.203 odd 4 3150.2.g.j.2899.1 2
272.203 odd 4 4046.2.a.f.1.1 1
304.75 even 4 5054.2.a.c.1.1 1
336.11 even 12 882.2.g.c.667.1 2
336.59 odd 12 882.2.g.d.667.1 2
336.107 even 12 882.2.g.c.361.1 2
336.251 odd 4 882.2.a.i.1.1 1
336.293 even 4 7056.2.a.bd.1.1 1
336.299 odd 12 882.2.g.d.361.1 2
368.91 even 4 7406.2.a.a.1.1 1
560.27 odd 4 2450.2.c.c.99.1 2
560.139 even 4 2450.2.a.t.1.1 1
560.363 odd 4 2450.2.c.c.99.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 16.11 odd 4
98.2.a.a.1.1 1 112.27 even 4
98.2.c.a.67.1 2 112.75 even 12
98.2.c.a.79.1 2 112.59 even 12
98.2.c.b.67.1 2 112.107 odd 12
98.2.c.b.79.1 2 112.11 odd 12
112.2.a.c.1.1 1 16.5 even 4
126.2.a.b.1.1 1 48.11 even 4
350.2.a.f.1.1 1 80.59 odd 4
350.2.c.d.99.1 2 80.27 even 4
350.2.c.d.99.2 2 80.43 even 4
448.2.a.a.1.1 1 16.13 even 4
448.2.a.g.1.1 1 16.3 odd 4
784.2.a.b.1.1 1 112.69 odd 4
784.2.i.c.177.1 2 112.53 even 12
784.2.i.c.753.1 2 112.37 even 12
784.2.i.i.177.1 2 112.101 odd 12
784.2.i.i.753.1 2 112.5 odd 12
882.2.a.i.1.1 1 336.251 odd 4
882.2.g.c.361.1 2 336.107 even 12
882.2.g.c.667.1 2 336.11 even 12
882.2.g.d.361.1 2 336.299 odd 12
882.2.g.d.667.1 2 336.59 odd 12
1008.2.a.h.1.1 1 48.5 odd 4
1134.2.f.f.379.1 2 144.59 even 12
1134.2.f.f.757.1 2 144.11 even 12
1134.2.f.l.379.1 2 144.139 odd 12
1134.2.f.l.757.1 2 144.43 odd 12
1694.2.a.e.1.1 1 176.43 even 4
1792.2.b.c.897.1 2 4.3 odd 2
1792.2.b.c.897.2 2 8.3 odd 2
1792.2.b.g.897.1 2 8.5 even 2 inner
1792.2.b.g.897.2 2 1.1 even 1 trivial
2366.2.a.j.1.1 1 208.155 odd 4
2366.2.d.b.337.1 2 208.203 even 4
2366.2.d.b.337.2 2 208.187 even 4
2450.2.a.t.1.1 1 560.139 even 4
2450.2.c.c.99.1 2 560.27 odd 4
2450.2.c.c.99.2 2 560.363 odd 4
2800.2.a.g.1.1 1 80.69 even 4
2800.2.g.h.449.1 2 80.37 odd 4
2800.2.g.h.449.2 2 80.53 odd 4
3136.2.a.e.1.1 1 112.83 even 4
3136.2.a.z.1.1 1 112.13 odd 4
3150.2.a.i.1.1 1 240.59 even 4
3150.2.g.j.2899.1 2 240.203 odd 4
3150.2.g.j.2899.2 2 240.107 odd 4
4032.2.a.r.1.1 1 48.29 odd 4
4032.2.a.w.1.1 1 48.35 even 4
4046.2.a.f.1.1 1 272.203 odd 4
5054.2.a.c.1.1 1 304.75 even 4
7056.2.a.bd.1.1 1 336.293 even 4
7406.2.a.a.1.1 1 368.91 even 4