Properties

Label 930.2.a.o.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 930.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +6.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +4.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +1.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +6.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000 q^{83} +2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} -8.00000 q^{92} -1.00000 q^{93} +4.00000 q^{95} +1.00000 q^{96} +18.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) 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
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 6.00000 0.960769
\(40\) 1.00000 0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) 1.00000 0.149071
\(46\) −8.00000 −1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) 6.00000 0.832050
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) −4.00000 −0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) −4.00000 −0.426401
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) −7.00000 −0.707107
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 2.00000 0.198030
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) −4.00000 −0.381385
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 4.00000 0.374634
\(115\) −8.00000 −0.746004
\(116\) 6.00000 0.557086
\(117\) 6.00000 0.554700
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 10.0000 0.901670
\(124\) −1.00000 −0.0898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 6.00000 0.526235
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 1.00000 0.0860663
\(136\) 2.00000 0.171499
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) −8.00000 −0.681005
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −24.0000 −2.00698
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 2.00000 0.165521
\(147\) −7.00000 −0.577350
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000 0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) 6.00000 0.480384
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 10.0000 0.780869
\(165\) −4.00000 −0.311400
\(166\) 4.00000 0.310460
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −12.0000 −0.901975
\(178\) −14.0000 −1.04934
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −8.00000 −0.589768
\(185\) −2.00000 −0.147043
\(186\) −1.00000 −0.0733236
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 18.0000 1.29232
\(195\) 6.00000 0.429669
\(196\) −7.00000 −0.500000
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −4.00000 −0.284268
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 10.0000 0.698430
\(206\) 16.0000 1.11477
\(207\) −8.00000 −0.556038
\(208\) 6.00000 0.416025
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) 2.00000 0.135147
\(220\) −4.00000 −0.269680
\(221\) 12.0000 0.807207
\(222\) −2.00000 −0.134231
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000 0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) −7.00000 −0.447214
\(246\) 10.0000 0.637577
\(247\) 24.0000 1.52708
\(248\) −1.00000 −0.0635001
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 32.0000 2.01182
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) 6.00000 0.371391
\(262\) −4.00000 −0.247121
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) −4.00000 −0.246183
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) −4.00000 −0.244339
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −22.0000 −1.32907
\(275\) −4.00000 −0.241209
\(276\) −8.00000 −0.481543
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −4.00000 −0.239904
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) −24.0000 −1.41915
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 6.00000 0.352332
\(291\) 18.0000 1.05518
\(292\) 2.00000 0.117041
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) −7.00000 −0.408248
\(295\) −12.0000 −0.698667
\(296\) −2.00000 −0.116248
\(297\) −4.00000 −0.232104
\(298\) −10.0000 −0.579284
\(299\) −48.0000 −2.77591
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) −2.00000 −0.114520
\(306\) 2.00000 0.114332
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) −1.00000 −0.0567962
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 6.00000 0.339683
\(313\) 18.0000 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −10.0000 −0.560772
\(319\) −24.0000 −1.34374
\(320\) 1.00000 0.0559017
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) −4.00000 −0.221540
\(327\) −18.0000 −0.995402
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000 0.219529
\(333\) −2.00000 −0.109599
\(334\) 24.0000 1.31322
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 23.0000 1.25104
\(339\) −14.0000 −0.760376
\(340\) 2.00000 0.108465
\(341\) 4.00000 0.216612
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) −8.00000 −0.430706
\(346\) 14.0000 0.752645
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 6.00000 0.321634
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) −4.00000 −0.213201
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) −2.00000 −0.104542
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −8.00000 −0.417029
\(369\) 10.0000 0.520579
\(370\) −2.00000 −0.103975
\(371\) 0 0
\(372\) −1.00000 −0.0518476
\(373\) 38.0000 1.96757 0.983783 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) −8.00000 −0.413670
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −4.00000 −0.203331
\(388\) 18.0000 0.913812
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 6.00000 0.303822
\(391\) −16.0000 −0.809155
\(392\) −7.00000 −0.353553
\(393\) −4.00000 −0.201773
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 26.0000 1.29838 0.649189 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) −4.00000 −0.199502
\(403\) −6.00000 −0.298881
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 10.0000 0.493865
\(411\) −22.0000 −1.08518
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 4.00000 0.196352
\(416\) 6.00000 0.294174
\(417\) −4.00000 −0.195881
\(418\) −16.0000 −0.782586
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −24.0000 −1.15873
\(430\) −4.00000 −0.192897
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) −18.0000 −0.862044
\(437\) −32.0000 −1.53077
\(438\) 2.00000 0.0955637
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) 12.0000 0.570782
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −14.0000 −0.663664
\(446\) −16.0000 −0.757622
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) 42.0000 1.98210 0.991051 0.133482i \(-0.0426157\pi\)
0.991051 + 0.133482i \(0.0426157\pi\)
\(450\) 1.00000 0.0471405
\(451\) −40.0000 −1.88353
\(452\) −14.0000 −0.658505
\(453\) 8.00000 0.375873
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −10.0000 −0.467269
\(459\) 2.00000 0.0933520
\(460\) −8.00000 −0.373002
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 6.00000 0.278543
\(465\) −1.00000 −0.0463739
\(466\) 10.0000 0.463241
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −12.0000 −0.552345
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) −12.0000 −0.547153
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 18.0000 0.817338
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −4.00000 −0.180886
\(490\) −7.00000 −0.316228
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 10.0000 0.450835
\(493\) 12.0000 0.540453
\(494\) 24.0000 1.07981
\(495\) −4.00000 −0.179787
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.0000 1.07224
\(502\) 12.0000 0.535586
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 32.0000 1.42257
\(507\) 23.0000 1.02147
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 2.00000 0.0882162
\(515\) 16.0000 0.705044
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 6.00000 0.263117
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 6.00000 0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) −2.00000 −0.0871214
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 60.0000 2.59889
\(534\) −14.0000 −0.605839
\(535\) −4.00000 −0.172935
\(536\) −4.00000 −0.172774
\(537\) 4.00000 0.172613
\(538\) −10.0000 −0.431131
\(539\) 28.0000 1.20605
\(540\) 1.00000 0.0430331
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −16.0000 −0.687259
\(543\) −10.0000 −0.429141
\(544\) 2.00000 0.0857493
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −22.0000 −0.939793
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) 24.0000 1.02243
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) −2.00000 −0.0848953
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 10.0000 0.421825
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 4.00000 0.167542
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −24.0000 −1.00349
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −13.0000 −0.540729
\(579\) −14.0000 −0.581820
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 18.0000 0.746124
\(583\) 40.0000 1.65663
\(584\) 2.00000 0.0827606
\(585\) 6.00000 0.248069
\(586\) −26.0000 −1.07405
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −7.00000 −0.288675
\(589\) −4.00000 −0.164817
\(590\) −12.0000 −0.494032
\(591\) −10.0000 −0.411345
\(592\) −2.00000 −0.0821995
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −8.00000 −0.327418
\(598\) −48.0000 −1.96287
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 1.00000 0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 8.00000 0.325515
\(605\) 5.00000 0.203279
\(606\) 6.00000 0.243733
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 28.0000 1.12999
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 16.0000 0.643614
\(619\) 12.0000 0.482321 0.241160 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) −1.00000 −0.0401610
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) −16.0000 −0.638978
\(628\) 14.0000 0.558661
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) −42.0000 −1.66410
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −4.00000 −0.157867
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 8.00000 0.314756
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 48.0000 1.88416
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −18.0000 −0.703856
\(655\) −4.00000 −0.156293
\(656\) 10.0000 0.390434
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) −4.00000 −0.155700
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −20.0000 −0.777322
\(663\) 12.0000 0.466041
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −48.0000 −1.85857
\(668\) 24.0000 0.928588
\(669\) −16.0000 −0.618596
\(670\) −4.00000 −0.154533
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 26.0000 1.00148
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) −14.0000 −0.537667
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 4.00000 0.153280
\(682\) 4.00000 0.153168
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 4.00000 0.152944
\(685\) −22.0000 −0.840577
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) −60.0000 −2.28582
\(690\) −8.00000 −0.304555
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) −4.00000 −0.151729
\(696\) 6.00000 0.227429
\(697\) 20.0000 0.757554
\(698\) 30.0000 1.13552
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 6.00000 0.226455
\(703\) −8.00000 −0.301726
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 18.0000 0.669427
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) 5.00000 0.185567
\(727\) −48.0000 −1.78022 −0.890111 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) −8.00000 −0.295891
\(732\) −2.00000 −0.0739221
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 32.0000 1.18114
\(735\) −7.00000 −0.258199
\(736\) −8.00000 −0.294884
\(737\) 16.0000 0.589368
\(738\) 10.0000 0.368105
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −1.00000 −0.0366618
\(745\) −10.0000 −0.366372
\(746\) 38.0000 1.39128
\(747\) 4.00000 0.146352
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 36.0000 1.31104
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 12.0000 0.435860
\(759\) 32.0000 1.16153
\(760\) 4.00000 0.145095
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) 2.00000 0.0723102
\(766\) −32.0000 −1.15621
\(767\) −72.0000 −2.59977
\(768\) 1.00000 0.0360844
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) −14.0000 −0.503871
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −4.00000 −0.143777
\(775\) −1.00000 −0.0359211
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 40.0000 1.43315
\(780\) 6.00000 0.214834
\(781\) 0 0
\(782\) −16.0000 −0.572159
\(783\) 6.00000 0.214423
\(784\) −7.00000 −0.250000
\(785\) 14.0000 0.499681
\(786\) −4.00000 −0.142675
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −10.0000 −0.356235
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) −12.0000 −0.426132
\(794\) −18.0000 −0.638796
\(795\) −10.0000 −0.354663
\(796\) −8.00000 −0.283552
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) 26.0000 0.918092
\(803\) −8.00000 −0.282314
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) −10.0000 −0.352017
\(808\) 6.00000 0.211079
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 1.00000 0.0351364
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 8.00000 0.280400
\(815\) −4.00000 −0.140114
\(816\) 2.00000 0.0700140
\(817\) −16.0000 −0.559769
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −22.0000 −0.767338
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 16.0000 0.557386
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −8.00000 −0.278019
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 4.00000 0.138842
\(831\) −2.00000 −0.0693792
\(832\) 6.00000 0.208013
\(833\) −14.0000 −0.485071
\(834\) −4.00000 −0.138509
\(835\) 24.0000 0.830554
\(836\) −16.0000 −0.553372
\(837\) −1.00000 −0.0345651
\(838\) −20.0000 −0.690889
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 38.0000 1.30957
\(843\) 10.0000 0.344418
\(844\) 4.00000 0.137686
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 20.0000 0.686398
\(850\) 2.00000 0.0685994
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) −4.00000 −0.136717
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) −24.0000 −0.819346
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) −22.0000 −0.747590
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 6.00000 0.203419
\(871\) −24.0000 −0.813209
\(872\) −18.0000 −0.609557
\(873\) 18.0000 0.609208
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) −8.00000 −0.269987
\(879\) −26.0000 −0.876958
\(880\) −4.00000 −0.134840
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) −7.00000 −0.235702
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 12.0000 0.403604
\(885\) −12.0000 −0.403376
\(886\) 28.0000 0.940678
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) −14.0000 −0.469281
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) −48.0000 −1.60267
\(898\) 42.0000 1.40156
\(899\) −6.00000 −0.200111
\(900\) 1.00000 0.0333333
\(901\) −20.0000 −0.666297
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −10.0000 −0.332411
\(906\) 8.00000 0.265782
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 4.00000 0.132745
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 4.00000 0.132453
\(913\) −16.0000 −0.529523
\(914\) 2.00000 0.0661541
\(915\) −2.00000 −0.0661180
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) −8.00000 −0.263752
\(921\) 28.0000 0.922631
\(922\) 6.00000 0.197599
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −16.0000 −0.525793
\(927\) 16.0000 0.525509
\(928\) 6.00000 0.196960
\(929\) −38.0000 −1.24674 −0.623370 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) −1.00000 −0.0327913
\(931\) −28.0000 −0.917663
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) −8.00000 −0.261628
\(936\) 6.00000 0.196116
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 14.0000 0.456145
\(943\) −80.0000 −2.60516
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 4.00000 0.129777
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −10.0000 −0.323762
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) −24.0000 −0.775810
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) 1.00000 0.0322581
\(962\) −12.0000 −0.386896
\(963\) −4.00000 −0.128898
\(964\) 18.0000 0.579741
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 5.00000 0.160706
\(969\) 8.00000 0.256997
\(970\) 18.0000 0.577945
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 6.00000 0.192154
\(976\) −2.00000 −0.0640184
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −4.00000 −0.127906
\(979\) 56.0000 1.78977
\(980\) −7.00000 −0.223607
\(981\) −18.0000 −0.574696
\(982\) −20.0000 −0.638226
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 10.0000 0.318788
\(985\) −10.0000 −0.318626
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) 32.0000 1.01754
\(990\) −4.00000 −0.127128
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 4.00000 0.126745
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 4.00000 0.126618
\(999\) −2.00000 −0.0632772
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 930.2.a.o.1.1 1
3.2 odd 2 2790.2.a.c.1.1 1
4.3 odd 2 7440.2.a.j.1.1 1
5.2 odd 4 4650.2.d.n.3349.2 2
5.3 odd 4 4650.2.d.n.3349.1 2
5.4 even 2 4650.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.o.1.1 1 1.1 even 1 trivial
2790.2.a.c.1.1 1 3.2 odd 2
4650.2.a.h.1.1 1 5.4 even 2
4650.2.d.n.3349.1 2 5.3 odd 4
4650.2.d.n.3349.2 2 5.2 odd 4
7440.2.a.j.1.1 1 4.3 odd 2