Properties

Label 3450.2.d.r.2899.1
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.r.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -2.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -8.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} +4.00000i q^{38} -2.00000 q^{41} -2.00000i q^{43} -2.00000 q^{44} +1.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +2.00000 q^{66} +2.00000i q^{67} +6.00000i q^{68} -1.00000 q^{69} -10.0000 q^{71} -1.00000i q^{72} -10.0000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +1.00000 q^{81} +2.00000i q^{82} -4.00000i q^{83} -2.00000 q^{86} +2.00000i q^{88} -4.00000 q^{89} -1.00000i q^{92} -8.00000i q^{93} -4.00000 q^{94} +1.00000 q^{96} -16.0000i q^{97} -7.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} + 2q^{16} - 8q^{19} - 2q^{24} - 16q^{31} - 12q^{34} + 2q^{36} - 4q^{41} - 4q^{44} + 2q^{46} + 14q^{49} + 12q^{51} - 2q^{54} - 4q^{61} - 2q^{64} + 4q^{66} - 2q^{69} - 20q^{71} + 12q^{74} + 8q^{76} + 2q^{81} - 4q^{86} - 8q^{89} - 8q^{94} + 2q^{96} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.00000i − 0.426401i
\(23\) 1.00000i 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 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 \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.00000i 0.348155i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 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
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 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 \(0.540868\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 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
\(82\) 2.00000i 0.220863i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 2.00000i 0.213201i
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 1.00000i − 0.104257i
\(93\) − 8.00000i − 0.829561i
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 16.0000i − 1.62455i −0.583272 0.812277i \(-0.698228\pi\)
0.583272 0.812277i \(-0.301772\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) − 16.0000i − 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) −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
\(122\) 2.00000i 0.181071i
\(123\) − 2.00000i − 0.180334i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) − 6.00000i − 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) − 2.00000i − 0.174078i
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 2.00000i − 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 10.0000i 0.839181i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 7.00000i 0.577350i
\(148\) − 6.00000i − 0.493197i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 2.00000i 0.152499i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 4.00000i 0.299813i
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) − 2.00000i − 0.147844i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) − 12.0000i − 0.877527i
\(188\) 4.00000i 0.291730i
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 14.0000i − 0.997459i −0.866758 0.498729i \(-0.833800\pi\)
0.866758 0.498729i \(-0.166200\pi\)
\(198\) 2.00000i 0.142134i
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 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
\(207\) − 1.00000i − 0.0695048i
\(208\) 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 \(0.741699\pi\)
\(212\) 2.00000i 0.137361i
\(213\) − 10.0000i − 0.685189i
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000i 0.135457i
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 6.00000i 0.402694i
\(223\) − 14.0000i − 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) − 8.00000i − 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 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
\(233\) 22.0000i 1.44127i 0.693316 + 0.720634i \(0.256149\pi\)
−0.693316 + 0.720634i \(0.743851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 7.00000i 0.449977i
\(243\) 1.00000i 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) − 8.00000i − 0.508001i
\(249\) 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
\(253\) 2.00000i 0.125739i
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) − 2.00000i − 0.124515i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 4.00000i 0.247121i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) − 4.00000i − 0.244796i
\(268\) − 2.00000i − 0.122169i
\(269\) −28.0000 −1.70719 −0.853595 0.520937i \(-0.825583\pi\)
−0.853595 + 0.520937i \(0.825583\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) − 24.0000i − 1.44202i −0.692925 0.721010i \(-0.743678\pi\)
0.692925 0.721010i \(-0.256322\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) − 4.00000i − 0.238197i
\(283\) 10.0000i 0.594438i 0.954809 + 0.297219i \(0.0960592\pi\)
−0.954809 + 0.297219i \(0.903941\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 10.0000i 0.585206i
\(293\) 22.0000i 1.28525i 0.766179 + 0.642627i \(0.222155\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) − 2.00000i − 0.116052i
\(298\) 10.0000i 0.579284i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 20.0000i 1.15087i
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 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
\(313\) − 20.0000i − 1.13047i −0.824931 0.565233i \(-0.808786\pi\)
0.824931 0.565233i \(-0.191214\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) − 2.00000i − 0.112154i
\(319\) 0 0
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) − 2.00000i − 0.110600i
\(328\) − 2.00000i − 0.110432i
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000i 0.219529i
\(333\) − 6.00000i − 0.328798i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) − 28.0000i − 1.52526i −0.646837 0.762629i \(-0.723908\pi\)
0.646837 0.762629i \(-0.276092\pi\)
\(338\) − 13.0000i − 0.707107i
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) − 4.00000i − 0.216295i
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 2.00000i − 0.106600i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4.00000 0.212000
\(357\) 0 0
\(358\) − 24.0000i − 1.26844i
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 6.00000i − 0.315353i
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) − 16.0000i − 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000i 0.414781i
\(373\) − 6.00000i − 0.310668i −0.987862 0.155334i \(-0.950355\pi\)
0.987862 0.155334i \(-0.0496454\pi\)
\(374\) −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 \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 8.00000i 0.409316i
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 2.00000i 0.101666i
\(388\) 16.0000i 0.812277i
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 7.00000i 0.353553i
\(393\) − 4.00000i − 0.201773i
\(394\) −14.0000 −0.705310
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) − 8.00000i − 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 2.00000i 0.0997509i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 6.00000i 0.297044i
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 16.0000i 0.788263i
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 0 0
\(417\) − 12.0000i − 0.587643i
\(418\) 8.00000i 0.391293i
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 4.00000i 0.194487i
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −10.0000 −0.484502
\(427\) 0 0
\(428\) 8.00000i 0.386695i
\(429\) 0 0
\(430\) 0 0
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 32.0000i 1.53782i 0.639356 + 0.768911i \(0.279201\pi\)
−0.639356 + 0.768911i \(0.720799\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 4.00000i − 0.191346i
\(438\) − 10.0000i − 0.477818i
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) − 10.0000i − 0.472984i
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) − 10.0000i − 0.470360i
\(453\) − 20.0000i − 0.939682i
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) −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
\(463\) 34.0000i 1.58011i 0.613033 + 0.790057i \(0.289949\pi\)
−0.613033 + 0.790057i \(0.710051\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 4.00000i 0.185098i 0.995708 + 0.0925490i \(0.0295015\pi\)
−0.995708 + 0.0925490i \(0.970499\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) − 4.00000i − 0.183920i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) − 26.0000i − 1.18921i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 2.00000i − 0.0910975i
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) − 4.00000i − 0.179244i
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 18.0000i 0.803379i
\(503\) − 24.0000i − 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 13.0000i 0.577350i
\(508\) 6.00000i 0.266207i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) − 8.00000i − 0.351840i
\(518\) 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
\(523\) 34.0000i 1.48672i 0.668894 + 0.743358i \(0.266768\pi\)
−0.668894 + 0.743358i \(0.733232\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 48.0000i 2.09091i
\(528\) 2.00000i 0.0870388i
\(529\) −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
\(537\) 24.0000i 1.03568i
\(538\) 28.0000i 1.20717i
\(539\) 14.0000 0.603023
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 8.00000i 0.343629i
\(543\) 6.00000i 0.257485i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) − 1.00000i − 0.0425628i
\(553\) 0 0
\(554\) −24.0000 −1.01966
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 0 0
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) − 12.0000i − 0.506189i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) 0 0
\(568\) − 10.0000i − 0.419591i
\(569\) 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
\(573\) − 8.00000i − 0.334205i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) − 16.0000i − 0.663221i
\(583\) − 4.00000i − 0.165663i
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 6.00000i 0.246598i
\(593\) − 42.0000i − 1.72473i −0.506284 0.862367i \(-0.668981\pi\)
0.506284 0.862367i \(-0.331019\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) − 24.0000i − 0.982255i
\(598\) 0 0
\(599\) −26.0000 −1.06233 −0.531166 0.847268i \(-0.678246\pi\)
−0.531166 + 0.847268i \(0.678246\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) − 2.00000i − 0.0814463i
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) − 6.00000i − 0.242536i
\(613\) − 10.0000i − 0.403896i −0.979396 0.201948i \(-0.935273\pi\)
0.979396 0.201948i \(-0.0647272\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 2.00000i 0.0801927i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) − 8.00000i − 0.319489i
\(628\) − 6.00000i − 0.239426i
\(629\) 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
\(633\) − 20.0000i − 0.794929i
\(634\) 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 \(0.474829\pi\)
\(642\) − 8.00000i − 0.315735i
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 36.0000i 1.41531i 0.706560 + 0.707653i \(0.250246\pi\)
−0.706560 + 0.707653i \(0.749754\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 8.00000i 0.313304i
\(653\) − 14.0000i − 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) 8.00000i 0.309529i
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) − 10.0000i − 0.385472i −0.981251 0.192736i \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 10.0000i 0.384048i
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 16.0000i 0.612672i
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 14.0000i 0.534133i
\(688\) − 2.00000i − 0.0762493i
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 26.0000i 0.984115i
\(699\) −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
\(703\) − 24.0000i − 0.905177i
\(704\) −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 \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 4.00000i − 0.149906i
\(713\) − 8.00000i − 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 26.0000i 0.970988i
\(718\) − 32.0000i − 1.19423i
\(719\) 22.0000 0.820462 0.410231 0.911982i \(-0.365448\pi\)
0.410231 + 0.911982i \(0.365448\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000i 0.111648i
\(723\) 2.00000i 0.0743808i
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 2.00000i 0.0739221i
\(733\) − 30.0000i − 1.10808i −0.832492 0.554038i \(-0.813086\pi\)
0.832492 0.554038i \(-0.186914\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 4.00000i 0.147342i
\(738\) − 2.00000i − 0.0736210i
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 4.00000i 0.146352i
\(748\) 12.0000i 0.438763i
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) − 4.00000i − 0.145865i
\(753\) − 18.0000i − 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000i 1.52652i 0.646094 + 0.763258i \(0.276401\pi\)
−0.646094 + 0.763258i \(0.723599\pi\)
\(758\) − 8.00000i − 0.290573i
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) − 6.00000i − 0.217357i
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 14.0000i 0.503871i
\(773\) 38.0000i 1.36677i 0.730061 + 0.683383i \(0.239492\pi\)
−0.730061 + 0.683383i \(0.760508\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) 16.0000 0.574367
\(777\) 0 0
\(778\) − 18.0000i − 0.645331i
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) − 6.00000i − 0.214560i
\(783\) 0 0
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) − 18.0000i − 0.641631i −0.947142 0.320815i \(-0.896043\pi\)
0.947142 0.320815i \(-0.103957\pi\)
\(788\) 14.0000i 0.498729i
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) − 2.00000i − 0.0710669i
\(793\) 0 0
\(794\) −8.00000 −0.283909
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) − 20.0000i − 0.705785i
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) − 28.0000i − 0.985647i
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 8.00000i 0.279885i
\(818\) 22.0000i 0.769212i
\(819\) 0 0
\(820\) 0 0
\(821\) 44.0000 1.53561 0.767805 0.640683i \(-0.221349\pi\)
0.767805 + 0.640683i \(0.221349\pi\)
\(822\) − 2.00000i − 0.0697580i
\(823\) − 10.0000i − 0.348578i −0.984695 0.174289i \(-0.944237\pi\)
0.984695 0.174289i \(-0.0557627\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) 32.0000i 1.11275i 0.830932 + 0.556375i \(0.187808\pi\)
−0.830932 + 0.556375i \(0.812192\pi\)
\(828\) 1.00000i 0.0347524i
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) 0 0
\(833\) − 42.0000i − 1.45521i
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) 8.00000i 0.276520i
\(838\) − 6.00000i − 0.207267i
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) − 22.0000i − 0.758170i
\(843\) 12.0000i 0.413302i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) − 2.00000i − 0.0686803i
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 10.0000i 0.342594i
\(853\) − 16.0000i − 0.547830i −0.961754 0.273915i \(-0.911681\pi\)
0.961754 0.273915i \(-0.0883186\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 40.0000i 1.36241i
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 32.0000 1.08740
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) − 2.00000i − 0.0677285i
\(873\) 16.0000i 0.541518i
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) − 12.0000i − 0.404980i
\(879\) −22.0000 −0.742042
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 28.0000i 0.940148i 0.882627 + 0.470074i \(0.155773\pi\)
−0.882627 + 0.470074i \(0.844227\pi\)
\(888\) − 6.00000i − 0.201347i
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 14.0000i 0.468755i
\(893\) 16.0000i 0.535420i
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) − 2.00000i − 0.0667409i
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 4.00000i 0.133185i
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) − 38.0000i − 1.26177i −0.775877 0.630885i \(-0.782692\pi\)
0.775877 0.630885i \(-0.217308\pi\)
\(908\) 8.00000i 0.265489i
\(909\) 0 0
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 8.00000i − 0.264761i
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 6.00000i 0.198030i
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) − 32.0000i − 1.05386i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 34.0000 1.11731
\(927\) 16.0000i 0.525509i
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) − 22.0000i − 0.720634i
\(933\) − 2.00000i − 0.0654771i
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) 0 0
\(937\) − 40.0000i − 1.30674i −0.757037 0.653372i \(-0.773354\pi\)
0.757037 0.653372i \(-0.226646\pi\)
\(938\) 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 \(0.712617\pi\)
\(942\) 6.00000i 0.195491i
\(943\) − 2.00000i − 0.0651290i
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) − 22.0000i − 0.712650i −0.934362 0.356325i \(-0.884030\pi\)
0.934362 0.356325i \(-0.115970\pi\)
\(954\) 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
\(963\) 8.00000i 0.257796i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 38.0000i 1.22200i 0.791632 + 0.610999i \(0.209232\pi\)
−0.791632 + 0.610999i \(0.790768\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 38.0000i − 1.21573i −0.794041 0.607864i \(-0.792027\pi\)
0.794041 0.607864i \(-0.207973\pi\)
\(978\) − 8.00000i − 0.255812i
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 12.0000i 0.382935i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 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 \(0.693750\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 4.00000i 0.126936i
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) 16.0000i 0.506725i 0.967371 + 0.253363i \(0.0815366\pi\)
−0.967371 + 0.253363i \(0.918463\pi\)
\(998\) − 36.0000i − 1.13956i
\(999\) 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.d.r.2899.1 2
5.2 odd 4 690.2.a.i.1.1 1
5.3 odd 4 3450.2.a.c.1.1 1
5.4 even 2 inner 3450.2.d.r.2899.2 2
15.2 even 4 2070.2.a.g.1.1 1
20.7 even 4 5520.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.i.1.1 1 5.2 odd 4
2070.2.a.g.1.1 1 15.2 even 4
3450.2.a.c.1.1 1 5.3 odd 4
3450.2.d.r.2899.1 2 1.1 even 1 trivial
3450.2.d.r.2899.2 2 5.4 even 2 inner
5520.2.a.d.1.1 1 20.7 even 4