Properties

Label 210.2.a.b.1.1
Level $210$
Weight $2$
Character 210.1
Self dual yes
Analytic conductor $1.677$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.67685844245\)
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 \(\chi\) \(=\) 210.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} +8.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} -1.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +8.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} +1.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -4.00000 q^{93} +8.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.00000 0.258199
\(16\) 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
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −8.00000 −1.29777
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 8.00000 1.05963
\(58\) −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
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 10.0000 1.16248
\(75\) 1.00000 0.115470
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 1.00000 0.109109
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000 0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 1.00000 0.0975900
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) −6.00000 −0.550019
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 10.0000 0.827606
\(147\) 1.00000 0.0824786
\(148\) −10.0000 −0.821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −8.00000 −0.648886
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 2.00000 0.160128
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 8.00000 0.611775
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −6.00000 −0.454859
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 6.00000 0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −2.00000 −0.148250
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) −8.00000 −0.580381
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 10.0000 0.717958
\(195\) 2.00000 0.143223
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) −6.00000 −0.420084
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 12.0000 0.822226
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) −14.0000 −0.948200
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 10.0000 0.671156
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 8.00000 0.529813
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 30.0000 1.96537 0.982683 0.185296i \(-0.0593245\pi\)
0.982683 + 0.185296i \(0.0593245\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 8.00000 0.519656
\(238\) 6.00000 0.388922
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 1.00000 0.0645497
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 1.00000 0.0638877
\(246\) 6.00000 0.382546
\(247\) 16.0000 1.01806
\(248\) 4.00000 0.254000
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 4.00000 0.249029
\(259\) −10.0000 −0.621370
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −8.00000 −0.490511
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −6.00000 −0.363803
\(273\) 2.00000 0.121046
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 16.0000 0.959616
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) −10.0000 −0.585206
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −12.0000 −0.698667
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) −8.00000 −0.460348
\(303\) 6.00000 0.344691
\(304\) 8.00000 0.458831
\(305\) −10.0000 −0.572598
\(306\) 6.00000 0.342997
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 4.00000 0.227185
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −2.00000 −0.113228
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 22.0000 1.24153
\(315\) 1.00000 0.0563436
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −48.0000 −2.67079
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −20.0000 −1.10770
\(327\) 14.0000 0.774202
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 12.0000 0.658586
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 1.00000 0.0545545
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 12.0000 0.637793
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) −6.00000 −0.317554
\(358\) 24.0000 1.26844
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.0000 2.36842
\(362\) 10.0000 0.525588
\(363\) −11.0000 −0.577350
\(364\) 2.00000 0.104828
\(365\) −10.0000 −0.523424
\(366\) 10.0000 0.522708
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 10.0000 0.519875
\(371\) −6.00000 −0.311504
\(372\) −4.00000 −0.207390
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) −1.00000 −0.0514344
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 8.00000 0.410391
\(381\) 8.00000 0.409852
\(382\) −12.0000 −0.613973
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) −10.0000 −0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) −12.0000 −0.605320
\(394\) −18.0000 −0.906827
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −20.0000 −1.00251
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 4.00000 0.199502
\(403\) −8.00000 −0.398508
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 6.00000 0.296319
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) −2.00000 −0.0980581
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 1.00000 0.0487950
\(421\) 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
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) −12.0000 −0.581402
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 4.00000 0.192006
\(435\) 6.00000 0.287678
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) −8.00000 −0.378811
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 8.00000 0.375873
\(454\) −12.0000 −0.563188
\(455\) 2.00000 0.0937614
\(456\) −8.00000 −0.374634
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −14.0000 −0.654177
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 6.00000 0.278543
\(465\) −4.00000 −0.185496
\(466\) −30.0000 −1.38972
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 2.00000 0.0924500
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 8.00000 0.367065
\(476\) −6.00000 −0.275010
\(477\) −6.00000 −0.274721
\(478\) 12.0000 0.548867
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −20.0000 −0.911922
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −10.0000 −0.454077
\(486\) −1.00000 −0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 10.0000 0.452679
\(489\) 20.0000 0.904431
\(490\) −1.00000 −0.0451754
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −6.00000 −0.270501
\(493\) −36.0000 −1.62136
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 12.0000 0.538274
\(498\) −12.0000 −0.537733
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 6.00000 0.265684
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 8.00000 0.353209
\(514\) 30.0000 1.32324
\(515\) 8.00000 0.352522
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) 6.00000 0.263371
\(520\) −2.00000 −0.0877058
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −6.00000 −0.262613
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −12.0000 −0.524222
\(525\) 1.00000 0.0436436
\(526\) 24.0000 1.04645
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) −12.0000 −0.520756
\(532\) 8.00000 0.346844
\(533\) −12.0000 −0.519778
\(534\) 6.00000 0.259645
\(535\) 12.0000 0.518805
\(536\) 4.00000 0.172774
\(537\) −24.0000 −1.03568
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −20.0000 −0.859074
\(543\) −10.0000 −0.429141
\(544\) 6.00000 0.257248
\(545\) 14.0000 0.599694
\(546\) −2.00000 −0.0855921
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 48.0000 2.04487
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 10.0000 0.424859
\(555\) −10.0000 −0.424476
\(556\) −16.0000 −0.678551
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 4.00000 0.169334
\(559\) −8.00000 −0.338364
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −12.0000 −0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −8.00000 −0.335083
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −19.0000 −0.790296
\(579\) 2.00000 0.0831172
\(580\) 6.00000 0.249136
\(581\) 12.0000 0.497844
\(582\) 10.0000 0.414513
\(583\) 0 0
\(584\) 10.0000 0.413803
\(585\) 2.00000 0.0826898
\(586\) −30.0000 −1.23929
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 1.00000 0.0412393
\(589\) −32.0000 −1.31854
\(590\) 12.0000 0.494032
\(591\) 18.0000 0.740421
\(592\) −10.0000 −0.410997
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 6.00000 0.245770
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) 8.00000 0.325515
\(605\) −11.0000 −0.447214
\(606\) −6.00000 −0.243733
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −8.00000 −0.324443
\(609\) 6.00000 0.243132
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 4.00000 0.161427
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −8.00000 −0.321807
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −6.00000 −0.240385
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 60.0000 2.39236
\(630\) −1.00000 −0.0398410
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −8.00000 −0.318223
\(633\) −4.00000 −0.158986
\(634\) −18.0000 −0.714871
\(635\) 8.00000 0.317470
\(636\) −6.00000 −0.237915
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) −1.00000 −0.0395285
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −12.0000 −0.473602
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 48.0000 1.88853
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) −4.00000 −0.156772
\(652\) 20.0000 0.783260
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −14.0000 −0.547443
\(655\) −12.0000 −0.468879
\(656\) −6.00000 −0.234261
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) 28.0000 1.08825
\(663\) −12.0000 −0.466041
\(664\) −12.0000 −0.465690
\(665\) 8.00000 0.310227
\(666\) 10.0000 0.387492
\(667\) 0 0
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −6.00000 −0.230429
\(679\) −10.0000 −0.383765
\(680\) 6.00000 0.230089
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 8.00000 0.305888
\(685\) 6.00000 0.229248
\(686\) −1.00000 −0.0381802
\(687\) 14.0000 0.534133
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −16.0000 −0.606915
\(696\) −6.00000 −0.227429
\(697\) 36.0000 1.36360
\(698\) 10.0000 0.378506
\(699\) 30.0000 1.13470
\(700\) 1.00000 0.0377964
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −80.0000 −3.01726
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 6.00000 0.225653
\(708\) −12.0000 −0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −12.0000 −0.450352
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) −12.0000 −0.448148
\(718\) 12.0000 0.447836
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) −45.0000 −1.67473
\(723\) 26.0000 0.966950
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) 11.0000 0.408248
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 24.0000 0.887672
\(732\) −10.0000 −0.369611
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −8.00000 −0.295285
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −10.0000 −0.367607
\(741\) 16.0000 0.587775
\(742\) 6.00000 0.220267
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 4.00000 0.146647
\(745\) 6.00000 0.219823
\(746\) 34.0000 1.24483
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) −1.00000 −0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) −12.0000 −0.437014
\(755\) 8.00000 0.291150
\(756\) 1.00000 0.0363696
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −8.00000 −0.289809
\(763\) 14.0000 0.506834
\(764\) 12.0000 0.434145
\(765\) −6.00000 −0.216930
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 2.00000 0.0719816
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) 10.0000 0.358979
\(777\) −10.0000 −0.358748
\(778\) −6.00000 −0.215110
\(779\) −48.0000 −1.71978
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) −22.0000 −0.785214
\(786\) 12.0000 0.428026
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 18.0000 0.641223
\(789\) −24.0000 −0.854423
\(790\) −8.00000 −0.284627
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) −2.00000 −0.0709773
\(795\) −6.00000 −0.212798
\(796\) 20.0000 0.708881
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) −8.00000 −0.283197
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −18.0000 −0.633630
\(808\) −6.00000 −0.211079
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 6.00000 0.210559
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) 20.0000 0.700569
\(816\) −6.00000 −0.210042
\(817\) −32.0000 −1.11954
\(818\) −26.0000 −0.909069
\(819\) 2.00000 0.0698857
\(820\) −6.00000 −0.209529
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −6.00000 −0.209274
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −12.0000 −0.416526
\(831\) −10.0000 −0.346896
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 12.0000 0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) −38.0000 −1.30957
\(843\) 18.0000 0.619953
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) −6.00000 −0.206041
\(849\) −4.00000 −0.137280
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 12.0000 0.411113
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 10.0000 0.342193
\(855\) 8.00000 0.273594
\(856\) −12.0000 −0.410152
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −4.00000 −0.136399
\(861\) −6.00000 −0.204479
\(862\) 12.0000 0.408722
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) 10.0000 0.339814
\(867\) 19.0000 0.645274
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) −8.00000 −0.271070
\(872\) −14.0000 −0.474100
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −10.0000 −0.337869
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 28.0000 0.944954
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −12.0000 −0.403604
\(885\) −12.0000 −0.403376
\(886\) 12.0000 0.403148
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 10.0000 0.335578
\(889\) 8.00000 0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) −24.0000 −0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −24.0000 −0.800445
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) −6.00000 −0.199557
\(905\) −10.0000 −0.332411
\(906\) −8.00000 −0.265782
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 12.0000 0.398234
\(909\) 6.00000 0.199007
\(910\) −2.00000 −0.0662994
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) −10.0000 −0.330590
\(916\) 14.0000 0.462573
\(917\) −12.0000 −0.396275
\(918\) 6.00000 0.198030
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 18.0000 0.592798
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 40.0000 1.31448
\(927\) 8.00000 0.262754
\(928\) −6.00000 −0.196960
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 4.00000 0.131165
\(931\) 8.00000 0.262189
\(932\) 30.0000 0.982683
\(933\) 24.0000 0.785725
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 4.00000 0.130605
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 22.0000 0.716799
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 8.00000 0.259828
\(949\) −20.0000 −0.649227
\(950\) −8.00000 −0.259554
\(951\) 18.0000 0.583690
\(952\) 6.00000 0.194461
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 6.00000 0.194257
\(955\) 12.0000 0.388311
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 20.0000 0.644826
\(963\) 12.0000 0.386695
\(964\) 26.0000 0.837404
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 11.0000 0.353553
\(969\) −48.0000 −1.54198
\(970\) 10.0000 0.321081
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) −32.0000 −1.02535
\(975\) 2.00000 0.0640513
\(976\) −10.0000 −0.320092
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) −20.0000 −0.639529
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 6.00000 0.191273
\(985\) 18.0000 0.573528
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 4.00000 0.127000
\(993\) −28.0000 −0.888553
\(994\) −12.0000 −0.380617
\(995\) 20.0000 0.634043
\(996\) 12.0000 0.380235
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −20.0000 −0.633089
\(999\) −10.0000 −0.316386
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 210.2.a.b.1.1 1
3.2 odd 2 630.2.a.h.1.1 1
4.3 odd 2 1680.2.a.g.1.1 1
5.2 odd 4 1050.2.g.c.799.1 2
5.3 odd 4 1050.2.g.c.799.2 2
5.4 even 2 1050.2.a.k.1.1 1
7.2 even 3 1470.2.i.l.361.1 2
7.3 odd 6 1470.2.i.s.961.1 2
7.4 even 3 1470.2.i.l.961.1 2
7.5 odd 6 1470.2.i.s.361.1 2
7.6 odd 2 1470.2.a.b.1.1 1
8.3 odd 2 6720.2.a.bi.1.1 1
8.5 even 2 6720.2.a.n.1.1 1
12.11 even 2 5040.2.a.g.1.1 1
15.2 even 4 3150.2.g.i.2899.2 2
15.8 even 4 3150.2.g.i.2899.1 2
15.14 odd 2 3150.2.a.f.1.1 1
20.19 odd 2 8400.2.a.cm.1.1 1
21.20 even 2 4410.2.a.bi.1.1 1
35.34 odd 2 7350.2.a.cs.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.b.1.1 1 1.1 even 1 trivial
630.2.a.h.1.1 1 3.2 odd 2
1050.2.a.k.1.1 1 5.4 even 2
1050.2.g.c.799.1 2 5.2 odd 4
1050.2.g.c.799.2 2 5.3 odd 4
1470.2.a.b.1.1 1 7.6 odd 2
1470.2.i.l.361.1 2 7.2 even 3
1470.2.i.l.961.1 2 7.4 even 3
1470.2.i.s.361.1 2 7.5 odd 6
1470.2.i.s.961.1 2 7.3 odd 6
1680.2.a.g.1.1 1 4.3 odd 2
3150.2.a.f.1.1 1 15.14 odd 2
3150.2.g.i.2899.1 2 15.8 even 4
3150.2.g.i.2899.2 2 15.2 even 4
4410.2.a.bi.1.1 1 21.20 even 2
5040.2.a.g.1.1 1 12.11 even 2
6720.2.a.n.1.1 1 8.5 even 2
6720.2.a.bi.1.1 1 8.3 odd 2
7350.2.a.cs.1.1 1 35.34 odd 2
8400.2.a.cm.1.1 1 20.19 odd 2