Properties

Label 3450.2.a.c.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{27} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{41} +2.00000 q^{43} +2.00000 q^{44} +1.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +6.00000 q^{51} +2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{66} +2.00000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -10.0000 q^{71} -1.00000 q^{72} +10.0000 q^{73} -6.00000 q^{74} +4.00000 q^{76} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{86} -2.00000 q^{88} +4.00000 q^{89} -1.00000 q^{92} +8.00000 q^{93} +4.00000 q^{94} +1.00000 q^{96} -16.0000 q^{97} +7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −6.00000 −0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 7.00000 0.707107
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −6.00000 −0.594089
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) 2.00000 0.180334
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 10.0000 0.839181
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 7.00000 0.577350
\(148\) 6.00000 0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) −4.00000 −0.324443
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 2.00000 0.152499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −4.00000 −0.299813
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) −12.0000 −0.877527
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) −2.00000 −0.142134
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 2.00000 0.137361
\(213\) 10.0000 0.685189
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 6.00000 0.402694
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) −4.00000 −0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 6.00000 0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −4.00000 −0.244796
\(268\) 2.00000 0.122169
\(269\) 28.0000 1.70719 0.853595 0.520937i \(-0.174417\pi\)
0.853595 + 0.520937i \(0.174417\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) −12.0000 −0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) −4.00000 −0.238197
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 10.0000 0.585206
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) −2.00000 −0.116052
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 2.00000 0.112154
\(319\) 0 0
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) −2.00000 −0.110600
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) 13.0000 0.707107
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4.00000 0.212000
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 0 0
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 8.00000 0.409316
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 2.00000 0.101666
\(388\) −16.0000 −0.812277
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 7.00000 0.353553
\(393\) 4.00000 0.201773
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 2.00000 0.0997509
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) −6.00000 −0.297044
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) −8.00000 −0.391293
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 20.0000 0.973585
\(423\) −4.00000 −0.194487
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) −10.0000 −0.484502
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −32.0000 −1.53782 −0.768911 0.639356i \(-0.779201\pi\)
−0.768911 + 0.639356i \(0.779201\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −4.00000 −0.191346
\(438\) 10.0000 0.477818
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) −10.0000 −0.470360
\(453\) 20.0000 0.939682
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 14.0000 0.654177
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 0 0
\(463\) −34.0000 −1.58011 −0.790057 0.613033i \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 26.0000 1.18921
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 2.00000 0.0905357
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000 0.0901670
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 18.0000 0.803379
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 13.0000 0.577350
\(508\) −6.00000 −0.266207
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 48.0000 2.09091
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 4.00000 0.173097
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 24.0000 1.03568
\(538\) −28.0000 −1.20717
\(539\) −14.0000 −0.603023
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 8.00000 0.343629
\(543\) −6.00000 −0.257485
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 24.0000 1.01966
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 8.00000 0.338667
\(559\) 0 0
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) −12.0000 −0.506189
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −19.0000 −0.790296
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) −16.0000 −0.663221
\(583\) 4.00000 0.165663
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 7.00000 0.288675
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 6.00000 0.246598
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 26.0000 1.06233 0.531166 0.847268i \(-0.321754\pi\)
0.531166 + 0.847268i \(0.321754\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 16.0000 0.643614
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 2.00000 0.0801927
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) −8.00000 −0.319489
\(628\) 6.00000 0.239426
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) 0 0
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) −8.00000 −0.315735
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 18.0000 0.701180 0.350590 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) −8.00000 −0.309529
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 28.0000 1.07852
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −10.0000 −0.384048
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 16.0000 0.612672
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 2.00000 0.0762493
\(689\) 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
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) −26.0000 −0.984115
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4.00000 −0.149906
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 26.0000 0.970988
\(718\) 32.0000 1.19423
\(719\) −22.0000 −0.820462 −0.410231 0.911982i \(-0.634552\pi\)
−0.410231 + 0.911982i \(0.634552\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −2.00000 −0.0743808
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 2.00000 0.0739221
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 4.00000 0.147342
\(738\) 2.00000 0.0736210
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 4.00000 0.146352
\(748\) −12.0000 −0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −4.00000 −0.145865
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 8.00000 0.290573
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) −6.00000 −0.217357
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 14.0000 0.503871
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 16.0000 0.574367
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) −6.00000 −0.214560
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) −14.0000 −0.498729
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 0 0
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) 20.0000 0.705785
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) −28.0000 −0.985647
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 8.00000 0.279885
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) 44.0000 1.53561 0.767805 0.640683i \(-0.221349\pi\)
0.767805 + 0.640683i \(0.221349\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 10.0000 0.348578 0.174289 0.984695i \(-0.444237\pi\)
0.174289 + 0.984695i \(0.444237\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) 0 0
\(833\) 42.0000 1.45521
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) 8.00000 0.276520
\(838\) 6.00000 0.207267
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −22.0000 −0.758170
\(843\) −12.0000 −0.413302
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 10.0000 0.343199
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 10.0000 0.342594
\(853\) 16.0000 0.547830 0.273915 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 32.0000 1.08740
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −2.00000 −0.0677285
\(873\) −16.0000 −0.541518
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 12.0000 0.404980
\(879\) 22.0000 0.742042
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 7.00000 0.235702
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 14.0000 0.468755
\(893\) −16.0000 −0.535420
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) −8.00000 −0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) −4.00000 −0.132453
\(913\) 8.00000 0.264761
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −32.0000 −1.05386
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 34.0000 1.11731
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) −22.0000 −0.720634
\(933\) 2.00000 0.0654771
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) 0 0
\(937\) −40.0000 −1.30674 −0.653372 0.757037i \(-0.726646\pi\)
−0.653372 + 0.757037i \(0.726646\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 6.00000 0.195491
\(943\) 2.00000 0.0651290
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −26.0000 −0.840900
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 38.0000 1.22200 0.610999 0.791632i \(-0.290768\pi\)
0.610999 + 0.791632i \(0.290768\pi\)
\(968\) 7.00000 0.224989
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 8.00000 0.255812
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 8.00000 0.254000
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) 16.0000 0.506725 0.253363 0.967371i \(-0.418463\pi\)
0.253363 + 0.967371i \(0.418463\pi\)
\(998\) 36.0000 1.13956
\(999\) −6.00000 −0.189832
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 3450.2.a.c.1.1 1
5.2 odd 4 3450.2.d.r.2899.1 2
5.3 odd 4 3450.2.d.r.2899.2 2
5.4 even 2 690.2.a.i.1.1 1
15.14 odd 2 2070.2.a.g.1.1 1
20.19 odd 2 5520.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.i.1.1 1 5.4 even 2
2070.2.a.g.1.1 1 15.14 odd 2
3450.2.a.c.1.1 1 1.1 even 1 trivial
3450.2.d.r.2899.1 2 5.2 odd 4
3450.2.d.r.2899.2 2 5.3 odd 4
5520.2.a.d.1.1 1 20.19 odd 2