Properties

Label 91.2.a.b.1.1
Level $91$
Weight $2$
Character 91.1
Self dual yes
Analytic conductor $0.727$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.726638658394\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{12} +1.00000 q^{13} +6.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} -7.00000 q^{19} +6.00000 q^{20} -2.00000 q^{21} +3.00000 q^{23} +4.00000 q^{25} +4.00000 q^{27} -2.00000 q^{28} -9.00000 q^{29} +5.00000 q^{31} -3.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} -1.00000 q^{43} -3.00000 q^{45} +3.00000 q^{47} -8.00000 q^{48} +1.00000 q^{49} +12.0000 q^{51} -2.00000 q^{52} -9.00000 q^{53} +14.0000 q^{57} -12.0000 q^{60} -10.0000 q^{61} +1.00000 q^{63} -8.00000 q^{64} -3.00000 q^{65} +14.0000 q^{67} +12.0000 q^{68} -6.00000 q^{69} -6.00000 q^{71} +11.0000 q^{73} -8.00000 q^{75} +14.0000 q^{76} -1.00000 q^{79} -12.0000 q^{80} -11.0000 q^{81} +3.00000 q^{83} +4.00000 q^{84} +18.0000 q^{85} +18.0000 q^{87} +15.0000 q^{89} +1.00000 q^{91} -6.00000 q^{92} -10.0000 q^{93} +21.0000 q^{95} -1.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −2.00000 −1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 4.00000 1.15470
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 6.00000 1.54919
\(16\) 4.00000 1.00000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 6.00000 1.34164
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) −2.00000 −0.377964
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −8.00000 −1.15470
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) −2.00000 −0.277350
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.0000 1.85435
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −12.0000 −1.54919
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 12.0000 1.45521
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) 14.0000 1.60591
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −12.0000 −1.34164
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 4.00000 0.436436
\(85\) 18.0000 1.95237
\(86\) 0 0
\(87\) 18.0000 1.92980
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −6.00000 −0.625543
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 21.0000 2.15455
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −8.00000 −0.769800
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 4.00000 0.377964
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) 18.0000 1.67126
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 12.0000 1.08200
\(124\) −10.0000 −0.898027
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) −12.0000 −1.03280
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 6.00000 0.507093
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) 27.0000 2.24223
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) −4.00000 −0.328798
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −15.0000 −1.20483
\(156\) 4.00000 0.320256
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 0 0
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 2.00000 0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 6.00000 0.447214
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 20.0000 1.47844
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 16.0000 1.15470
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) −2.00000 −0.142857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) −28.0000 −1.97497
\(202\) 0 0
\(203\) −9.00000 −0.631676
\(204\) −24.0000 −1.68034
\(205\) 18.0000 1.25717
\(206\) 0 0
\(207\) 3.00000 0.208514
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 18.0000 1.23625
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) 0 0
\(219\) −22.0000 −1.48662
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −28.0000 −1.85435
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) 2.00000 0.129914
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 24.0000 1.54919
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 20.0000 1.28037
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −7.00000 −0.445399
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 0 0
\(255\) −36.0000 −2.25441
\(256\) 16.0000 1.00000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 6.00000 0.372104
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) 27.0000 1.65860
\(266\) 0 0
\(267\) −30.0000 −1.83597
\(268\) −28.0000 −1.71037
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −24.0000 −1.45521
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) −42.0000 −2.48787
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −22.0000 −1.28745
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 16.0000 0.923760
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) −28.0000 −1.60591
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 2.00000 0.112509
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 24.0000 1.34164
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 42.0000 2.33694
\(324\) 22.0000 1.22222
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 32.0000 1.76960
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) −6.00000 −0.329293
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −42.0000 −2.29471
\(336\) −8.00000 −0.436436
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) −36.0000 −1.95237
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 18.0000 0.969087
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −36.0000 −1.92980
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 18.0000 0.955341
\(356\) −30.0000 −1.59000
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 22.0000 1.15470
\(364\) −2.00000 −0.104828
\(365\) −33.0000 −1.72730
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 12.0000 0.625543
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 20.0000 1.03695
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) −6.00000 −0.309839
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −42.0000 −2.15455
\(381\) 32.0000 1.63941
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 0 0
\(395\) 3.00000 0.150946
\(396\) 0 0
\(397\) 11.0000 0.552074 0.276037 0.961147i \(-0.410979\pi\)
0.276037 + 0.961147i \(0.410979\pi\)
\(398\) 0 0
\(399\) 14.0000 0.700877
\(400\) 16.0000 0.800000
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 5.00000 0.249068
\(404\) 0 0
\(405\) 33.0000 1.63978
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 23.0000 1.13728 0.568638 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) −12.0000 −0.585540
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 3.00000 0.145865
\(424\) 0 0
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 24.0000 1.16008
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 16.0000 0.769800
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) −54.0000 −2.58910
\(436\) 32.0000 1.53252
\(437\) −21.0000 −1.00457
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 8.00000 0.379663
\(445\) −45.0000 −2.13320
\(446\) 0 0
\(447\) 0 0
\(448\) −8.00000 −0.377964
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) −24.0000 −1.12022
\(460\) 18.0000 0.839254
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −36.0000 −1.67126
\(465\) 30.0000 1.39122
\(466\) 0 0
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −28.0000 −1.28473
\(476\) 12.0000 0.550019
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) 9.00000 0.411220 0.205610 0.978634i \(-0.434082\pi\)
0.205610 + 0.978634i \(0.434082\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 22.0000 1.00000
\(485\) 3.00000 0.136223
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) 32.0000 1.44709
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −24.0000 −1.08200
\(493\) 54.0000 2.43204
\(494\) 0 0
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) −6.00000 −0.268328
\(501\) 30.0000 1.34030
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.00000 −0.0888231
\(508\) 32.0000 1.41977
\(509\) 21.0000 0.930809 0.465404 0.885098i \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) 0 0
\(511\) 11.0000 0.486611
\(512\) 0 0
\(513\) −28.0000 −1.23623
\(514\) 0 0
\(515\) 12.0000 0.528783
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) 24.0000 1.04844
\(525\) −8.00000 −0.349149
\(526\) 0 0
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 14.0000 0.606977
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) 0 0
\(537\) −30.0000 −1.29460
\(538\) 0 0
\(539\) 0 0
\(540\) 24.0000 1.03280
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 32.0000 1.37325
\(544\) 0 0
\(545\) 48.0000 2.05609
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) −12.0000 −0.512615
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 63.0000 2.68389
\(552\) 0 0
\(553\) −1.00000 −0.0425243
\(554\) 0 0
\(555\) 12.0000 0.509372
\(556\) 8.00000 0.339276
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) −12.0000 −0.507093
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 12.0000 0.505291
\(565\) −27.0000 −1.13590
\(566\) 0 0
\(567\) −11.0000 −0.461957
\(568\) 0 0
\(569\) 27.0000 1.13190 0.565949 0.824440i \(-0.308510\pi\)
0.565949 + 0.824440i \(0.308510\pi\)
\(570\) 0 0
\(571\) 23.0000 0.962520 0.481260 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) 0 0
\(573\) −48.0000 −2.00523
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) −8.00000 −0.333333
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 0 0
\(579\) 44.0000 1.82858
\(580\) −54.0000 −2.24223
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.00000 −0.124035
\(586\) 0 0
\(587\) −9.00000 −0.371470 −0.185735 0.982600i \(-0.559467\pi\)
−0.185735 + 0.982600i \(0.559467\pi\)
\(588\) 4.00000 0.164957
\(589\) −35.0000 −1.44215
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 8.00000 0.328798
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 15.0000 0.612883 0.306442 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) 20.0000 0.813788
\(605\) 33.0000 1.34164
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) 18.0000 0.729397
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 12.0000 0.485071
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) −36.0000 −1.45166
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 30.0000 1.20483
\(621\) 12.0000 0.481543
\(622\) 0 0
\(623\) 15.0000 0.600962
\(624\) −8.00000 −0.320256
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) −28.0000 −1.11732
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 0 0
\(633\) −46.0000 −1.82834
\(634\) 0 0
\(635\) 48.0000 1.90482
\(636\) −36.0000 −1.42749
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −39.0000 −1.54041 −0.770204 0.637798i \(-0.779845\pi\)
−0.770204 + 0.637798i \(0.779845\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −6.00000 −0.236433
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 32.0000 1.25322
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 36.0000 1.40664
\(656\) −24.0000 −0.937043
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) −3.00000 −0.116863 −0.0584317 0.998291i \(-0.518610\pi\)
−0.0584317 + 0.998291i \(0.518610\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 21.0000 0.814345
\(666\) 0 0
\(667\) −27.0000 −1.04544
\(668\) 30.0000 1.16073
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) 16.0000 0.615840
\(676\) −2.00000 −0.0769231
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 14.0000 0.535303
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −28.0000 −1.06827
\(688\) −4.00000 −0.152499
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −37.0000 −1.40755 −0.703773 0.710425i \(-0.748503\pi\)
−0.703773 + 0.710425i \(0.748503\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) −8.00000 −0.302372
\(701\) 9.00000 0.339925 0.169963 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) 15.0000 0.561754
\(714\) 0 0
\(715\) 0 0
\(716\) −30.0000 −1.12115
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) −12.0000 −0.447214
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) 2.00000 0.0743808
\(724\) 32.0000 1.18927
\(725\) −36.0000 −1.33701
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) −40.0000 −1.47844
\(733\) −49.0000 −1.80986 −0.904928 0.425564i \(-0.860076\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 12.0000 0.441129
\(741\) 14.0000 0.514303
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.00000 0.109764
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) 12.0000 0.437595
\(753\) −24.0000 −0.874609
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) −8.00000 −0.290957
\(757\) −43.0000 −1.56286 −0.781431 0.623992i \(-0.785510\pi\)
−0.781431 + 0.623992i \(0.785510\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) −48.0000 −1.73658
\(765\) 18.0000 0.650791
\(766\) 0 0
\(767\) 0 0
\(768\) −32.0000 −1.15470
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 48.0000 1.72868
\(772\) 44.0000 1.58359
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) 42.0000 1.50481
\(780\) −12.0000 −0.429669
\(781\) 0 0
\(782\) 0 0
\(783\) −36.0000 −1.28654
\(784\) 4.00000 0.142857
\(785\) −42.0000 −1.49904
\(786\) 0 0
\(787\) 5.00000 0.178231 0.0891154 0.996021i \(-0.471596\pi\)
0.0891154 + 0.996021i \(0.471596\pi\)
\(788\) −12.0000 −0.427482
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) −54.0000 −1.91518
\(796\) −4.00000 −0.141776
\(797\) 54.0000 1.91278 0.956389 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 15.0000 0.529999
\(802\) 0 0
\(803\) 0 0
\(804\) 56.0000 1.97497
\(805\) −9.00000 −0.317208
\(806\) 0 0
\(807\) 48.0000 1.68968
\(808\) 0 0
\(809\) −3.00000 −0.105474 −0.0527372 0.998608i \(-0.516795\pi\)
−0.0527372 + 0.998608i \(0.516795\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 18.0000 0.631676
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 48.0000 1.68137
\(816\) 48.0000 1.68034
\(817\) 7.00000 0.244899
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) −36.0000 −1.25717
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) −6.00000 −0.208514
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) 38.0000 1.31821
\(832\) −8.00000 −0.277350
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 45.0000 1.55729
\(836\) 0 0
\(837\) 20.0000 0.691301
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) −24.0000 −0.826604
\(844\) −46.0000 −1.58339
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) −36.0000 −1.23625
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) −24.0000 −0.822226
\(853\) 17.0000 0.582069 0.291034 0.956713i \(-0.406001\pi\)
0.291034 + 0.956713i \(0.406001\pi\)
\(854\) 0 0
\(855\) 21.0000 0.718185
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) −6.00000 −0.204598
\(861\) 12.0000 0.408959
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −38.0000 −1.29055
\(868\) −10.0000 −0.339422
\(869\) 0 0
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 44.0000 1.48662
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) −21.0000 −0.702738
\(894\) 0 0
\(895\) −45.0000 −1.50418
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) −45.0000 −1.50083
\(900\) −8.00000 −0.266667
\(901\) 54.0000 1.79900
\(902\) 0 0
\(903\) 2.00000 0.0665558
\(904\) 0 0
\(905\) 48.0000 1.59557
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) −48.0000 −1.59294
\(909\) 0 0
\(910\) 0 0
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) 56.0000 1.85435
\(913\) 0 0
\(914\) 0 0
\(915\) −60.0000 −1.98354
\(916\) −28.0000 −0.925146
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 0 0
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) −3.00000 −0.0984268 −0.0492134 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 18.0000 0.589610
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −16.0000 −0.522140
\(940\) 18.0000 0.587095
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) −12.0000 −0.390360
\(946\) 0 0
\(947\) 54.0000 1.75476 0.877382 0.479792i \(-0.159288\pi\)
0.877382 + 0.479792i \(0.159288\pi\)
\(948\) −4.00000 −0.129914
\(949\) 11.0000 0.357075
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 21.0000 0.680257 0.340128 0.940379i \(-0.389529\pi\)
0.340128 + 0.940379i \(0.389529\pi\)
\(954\) 0 0
\(955\) −72.0000 −2.32987
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) −48.0000 −1.54919
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) 66.0000 2.12462
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) −84.0000 −2.69847
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) −20.0000 −0.641500
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) −8.00000 −0.256205
\(976\) −40.0000 −1.28037
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.00000 0.191663
\(981\) −16.0000 −0.510841
\(982\) 0 0
\(983\) 57.0000 1.81802 0.909009 0.416777i \(-0.136840\pi\)
0.909009 + 0.416777i \(0.136840\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 14.0000 0.445399
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −52.0000 −1.65017
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) 12.0000 0.380235
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 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 91.2.a.b.1.1 1
3.2 odd 2 819.2.a.c.1.1 1
4.3 odd 2 1456.2.a.k.1.1 1
5.4 even 2 2275.2.a.d.1.1 1
7.2 even 3 637.2.e.c.508.1 2
7.3 odd 6 637.2.e.b.79.1 2
7.4 even 3 637.2.e.c.79.1 2
7.5 odd 6 637.2.e.b.508.1 2
7.6 odd 2 637.2.a.b.1.1 1
8.3 odd 2 5824.2.a.f.1.1 1
8.5 even 2 5824.2.a.bd.1.1 1
13.5 odd 4 1183.2.c.a.337.2 2
13.8 odd 4 1183.2.c.a.337.1 2
13.12 even 2 1183.2.a.a.1.1 1
21.20 even 2 5733.2.a.f.1.1 1
91.90 odd 2 8281.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.b.1.1 1 1.1 even 1 trivial
637.2.a.b.1.1 1 7.6 odd 2
637.2.e.b.79.1 2 7.3 odd 6
637.2.e.b.508.1 2 7.5 odd 6
637.2.e.c.79.1 2 7.4 even 3
637.2.e.c.508.1 2 7.2 even 3
819.2.a.c.1.1 1 3.2 odd 2
1183.2.a.a.1.1 1 13.12 even 2
1183.2.c.a.337.1 2 13.8 odd 4
1183.2.c.a.337.2 2 13.5 odd 4
1456.2.a.k.1.1 1 4.3 odd 2
2275.2.a.d.1.1 1 5.4 even 2
5733.2.a.f.1.1 1 21.20 even 2
5824.2.a.f.1.1 1 8.3 odd 2
5824.2.a.bd.1.1 1 8.5 even 2
8281.2.a.h.1.1 1 91.90 odd 2