Properties

Label 1089.2.a.h.1.1
Level $1089$
Weight $2$
Character 1089.1
Self dual yes
Analytic conductor $8.696$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1089.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} -4.00000 q^{10} +2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} +4.00000 q^{20} +4.00000 q^{23} +11.0000 q^{25} +2.00000 q^{26} -2.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -8.00000 q^{35} -6.00000 q^{37} +6.00000 q^{38} +12.0000 q^{40} +10.0000 q^{41} -6.00000 q^{43} +4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} +11.0000 q^{50} -2.00000 q^{52} -6.00000 q^{56} +6.00000 q^{58} +4.00000 q^{59} +6.00000 q^{61} +4.00000 q^{62} +7.00000 q^{64} -8.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} -8.00000 q^{70} +2.00000 q^{73} -6.00000 q^{74} -6.00000 q^{76} +10.0000 q^{79} +4.00000 q^{80} +10.0000 q^{82} -12.0000 q^{83} +8.00000 q^{85} -6.00000 q^{86} +4.00000 q^{91} -4.00000 q^{92} -8.00000 q^{94} -24.0000 q^{95} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) −4.00000 −1.26491
\(11\) 0 0
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 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 \(0.383046\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 12.0000 1.89737
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 11.0000 1.55563
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −11.0000 −1.10000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 0 0
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) −8.00000 −0.701646
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 8.00000 0.676123
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −24.0000 −1.99309
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) −18.0000 −1.45999
\(153\) 0 0
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) −20.0000 −1.58114
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 22.0000 1.66304
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 24.0000 1.76452
\(186\) 0 0
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −24.0000 −1.74114
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) −33.0000 −2.33345
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −40.0000 −2.79372
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −16.0000 −1.05501
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 32.0000 2.08745
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) 12.0000 0.766652
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) −12.0000 −0.762001
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 10.0000 0.627456
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 32.0000 1.97320 0.986602 0.163144i \(-0.0521635\pi\)
0.986602 + 0.163144i \(0.0521635\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −16.0000 −0.975537 −0.487769 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −10.0000 −0.599760
\(279\) 0 0
\(280\) 24.0000 1.43427
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 20.0000 1.18056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −24.0000 −1.40933
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.931556
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) −14.0000 −0.805609
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −16.0000 −0.908739
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −4.00000 −0.224662 −0.112331 0.993671i \(-0.535832\pi\)
−0.112331 + 0.993671i \(0.535832\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −28.0000 −1.56525
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 22.0000 1.22034
\(326\) −20.0000 −1.10770
\(327\) 0 0
\(328\) −30.0000 −1.65647
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) −32.0000 −1.74835
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 18.0000 0.970495
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 22.0000 1.17595
\(351\) 0 0
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 24.0000 1.24770
\(371\) 0 0
\(372\) 0 0
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 24.0000 1.23771
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 24.0000 1.23117
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) −40.0000 −2.01262
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) −11.0000 −0.550000
\(401\) 28.0000 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −40.0000 −1.97546
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 48.0000 2.35623
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) 0 0
\(419\) −32.0000 −1.56330 −0.781651 0.623716i \(-0.785622\pi\)
−0.781651 + 0.623716i \(0.785622\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 6.00000 0.292075
\(423\) 0 0
\(424\) 0 0
\(425\) −22.0000 −1.06716
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 14.0000 0.661438
\(449\) −32.0000 −1.51017 −0.755087 0.655625i \(-0.772405\pi\)
−0.755087 + 0.655625i \(0.772405\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −12.0000 −0.564433
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) −16.0000 −0.750092
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 16.0000 0.746004
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 32.0000 1.47605
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) 66.0000 3.02829
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 0 0
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −18.0000 −0.814822
\(489\) 0 0
\(490\) 12.0000 0.542105
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) 8.00000 0.357057
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) −56.0000 −2.49197
\(506\) 0 0
\(507\) 0 0
\(508\) −10.0000 −0.443678
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 8.00000 0.352865
\(515\) −32.0000 −1.41009
\(516\) 0 0
\(517\) 0 0
\(518\) −12.0000 −0.527250
\(519\) 0 0
\(520\) 24.0000 1.05247
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −38.0000 −1.66162 −0.830812 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) 48.0000 2.07522
\(536\) −24.0000 −1.03664
\(537\) 0 0
\(538\) −16.0000 −0.689809
\(539\) 0 0
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) −38.0000 −1.62476 −0.812381 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) −4.00000 −0.170872
\(549\) 0 0
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) −48.0000 −2.01938
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 34.0000 1.42286 0.711428 0.702759i \(-0.248049\pi\)
0.711428 + 0.702759i \(0.248049\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 20.0000 0.834784
\(575\) 44.0000 1.83493
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 24.0000 0.996546
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) −16.0000 −0.658710
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) −28.0000 −1.14405 −0.572024 0.820237i \(-0.693842\pi\)
−0.572024 + 0.820237i \(0.693842\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) −12.0000 −0.489083
\(603\) 0 0
\(604\) 14.0000 0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0000 0.405887 0.202944 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) 30.0000 1.21666
\(609\) 0 0
\(610\) −24.0000 −0.971732
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 46.0000 1.85792 0.928961 0.370177i \(-0.120703\pi\)
0.928961 + 0.370177i \(0.120703\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) −30.0000 −1.19334
\(633\) 0 0
\(634\) −4.00000 −0.158860
\(635\) −40.0000 −1.58735
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 0 0
\(640\) 12.0000 0.474342
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 22.0000 0.862911
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −16.0000 −0.626128 −0.313064 0.949732i \(-0.601356\pi\)
−0.313064 + 0.949732i \(0.601356\pi\)
\(654\) 0 0
\(655\) −48.0000 −1.87552
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) −16.0000 −0.623745
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 36.0000 1.39707
\(665\) −48.0000 −1.86136
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) −32.0000 −1.23627
\(671\) 0 0
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) −24.0000 −0.920358
\(681\) 0 0
\(682\) 0 0
\(683\) −32.0000 −1.22445 −0.612223 0.790685i \(-0.709725\pi\)
−0.612223 + 0.790685i \(0.709725\pi\)
\(684\) 0 0
\(685\) −16.0000 −0.611329
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 40.0000 1.51729
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) −18.0000 −0.681310
\(699\) 0 0
\(700\) −22.0000 −0.831522
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) −36.0000 −1.35777
\(704\) 0 0
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 28.0000 1.05305
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 66.0000 2.45118
\(726\) 0 0
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) −12.0000 −0.444750
\(729\) 0 0
\(730\) −8.00000 −0.296093
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) −24.0000 −0.882258
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) −34.0000 −1.24483
\(747\) 0 0
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 56.0000 2.03805
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 72.0000 2.61171
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 44.0000 1.58053
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) 36.0000 1.29066
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) 0 0
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 40.0000 1.42766
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) −40.0000 −1.42314
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) 28.0000 0.991811 0.495905 0.868377i \(-0.334836\pi\)
0.495905 + 0.868377i \(0.334836\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 55.0000 1.94454
\(801\) 0 0
\(802\) 28.0000 0.988714
\(803\) 0 0
\(804\) 0 0
\(805\) −32.0000 −1.12785
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) −42.0000 −1.47755
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) 0 0
\(815\) 80.0000 2.80228
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 40.0000 1.39686
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) −48.0000 −1.67317 −0.836587 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 48.0000 1.66610
\(831\) 0 0
\(832\) 14.0000 0.485363
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −32.0000 −1.10542
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) −6.00000 −0.206529
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −22.0000 −0.754594
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 12.0000 0.410632
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) −24.0000 −0.816024
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −10.0000 −0.337484
\(879\) 0 0
\(880\) 0 0
\(881\) −20.0000 −0.673817 −0.336909 0.941537i \(-0.609381\pi\)
−0.336909 + 0.941537i \(0.609381\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 96.0000 3.20893
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) −32.0000 −1.06785
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −36.0000 −1.19734
\(905\) −40.0000 −1.32964
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −16.0000 −0.530395
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 14.0000 0.461817 0.230909 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 48.0000 1.58251
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) 0 0
\(925\) −66.0000 −2.17007
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) 30.0000 0.984798
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) −32.0000 −1.04372
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 66.0000 2.14132
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) −64.0000 −2.07099
\(956\) 0 0
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −26.0000 −0.837404
\(965\) −56.0000 −1.80270
\(966\) 0 0
\(967\) 50.0000 1.60789 0.803946 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −8.00000 −0.256865
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −12.0000 −0.383326
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 8.00000 0.254901
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 20.0000 0.635001
\(993\) 0 0
\(994\) 0 0
\(995\) −48.0000 −1.52170
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) −16.0000 −0.506471
\(999\) 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 1089.2.a.h.1.1 1
3.2 odd 2 1089.2.a.d.1.1 1
11.10 odd 2 99.2.a.a.1.1 1
33.32 even 2 99.2.a.c.1.1 yes 1
44.43 even 2 1584.2.a.b.1.1 1
55.32 even 4 2475.2.c.b.199.1 2
55.43 even 4 2475.2.c.b.199.2 2
55.54 odd 2 2475.2.a.j.1.1 1
77.76 even 2 4851.2.a.g.1.1 1
88.21 odd 2 6336.2.a.cl.1.1 1
88.43 even 2 6336.2.a.cm.1.1 1
99.32 even 6 891.2.e.c.298.1 2
99.43 odd 6 891.2.e.j.595.1 2
99.65 even 6 891.2.e.c.595.1 2
99.76 odd 6 891.2.e.j.298.1 2
132.131 odd 2 1584.2.a.r.1.1 1
165.32 odd 4 2475.2.c.g.199.2 2
165.98 odd 4 2475.2.c.g.199.1 2
165.164 even 2 2475.2.a.c.1.1 1
231.230 odd 2 4851.2.a.o.1.1 1
264.131 odd 2 6336.2.a.f.1.1 1
264.197 even 2 6336.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.a.a.1.1 1 11.10 odd 2
99.2.a.c.1.1 yes 1 33.32 even 2
891.2.e.c.298.1 2 99.32 even 6
891.2.e.c.595.1 2 99.65 even 6
891.2.e.j.298.1 2 99.76 odd 6
891.2.e.j.595.1 2 99.43 odd 6
1089.2.a.d.1.1 1 3.2 odd 2
1089.2.a.h.1.1 1 1.1 even 1 trivial
1584.2.a.b.1.1 1 44.43 even 2
1584.2.a.r.1.1 1 132.131 odd 2
2475.2.a.c.1.1 1 165.164 even 2
2475.2.a.j.1.1 1 55.54 odd 2
2475.2.c.b.199.1 2 55.32 even 4
2475.2.c.b.199.2 2 55.43 even 4
2475.2.c.g.199.1 2 165.98 odd 4
2475.2.c.g.199.2 2 165.32 odd 4
4851.2.a.g.1.1 1 77.76 even 2
4851.2.a.o.1.1 1 231.230 odd 2
6336.2.a.b.1.1 1 264.197 even 2
6336.2.a.f.1.1 1 264.131 odd 2
6336.2.a.cl.1.1 1 88.21 odd 2
6336.2.a.cm.1.1 1 88.43 even 2