Properties

Label 210.2.a.e.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
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 210.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +14.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -1.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} -2.00000 q^{78} +16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -1.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -8.00000 q^{92} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) −1.00000 −0.377964
\(8\) 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
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 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
\(21\) −1.00000 −0.218218
\(22\) −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
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) −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 \(0.598656\pi\)
\(44\) −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
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) −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
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 1.00000 0.129099
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 4.00000 0.455842
\(78\) −2.00000 −0.226455
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −1.00000 −0.109109
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) −4.00000 −0.426401
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) −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
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) −1.00000 −0.0975900
\(106\) −10.0000 −0.971286
\(107\) 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 \(0.266089\pi\)
\(110\) −4.00000 −0.381385
\(111\) 6.00000 0.569495
\(112\) −1.00000 −0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 4.00000 0.374634
\(115\) −8.00000 −0.746004
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) 12.0000 1.10469
\(119\) −2.00000 −0.183340
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) 14.0000 1.26750
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) −4.00000 −0.348155
\(133\) −4.00000 −0.346844
\(134\) −12.0000 −1.03664
\(135\) 1.00000 0.0860663
\(136\) 2.00000 0.171499
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −8.00000 −0.681005
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 10.0000 0.827606
\(147\) 1.00000 0.0824786
\(148\) 6.00000 0.493197
\(149\) −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 \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 4.00000 0.324443
\(153\) 2.00000 0.161690
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 16.0000 1.27289
\(159\) −10.0000 −0.793052
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) 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
\(165\) −4.00000 −0.311400
\(166\) −12.0000 −0.931381
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −2.00000 −0.151620
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) 12.0000 0.901975
\(178\) 10.0000 0.749532
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 1.00000 0.0745356
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 2.00000 0.148250
\(183\) 14.0000 1.03491
\(184\) −8.00000 −0.589768
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 4.00000 0.290191
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −4.00000 −0.284268
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) 6.00000 0.422159
\(203\) 2.00000 0.140372
\(204\) 2.00000 0.140028
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) −8.00000 −0.556038
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) −1.00000 −0.0690066
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −10.0000 −0.686803
\(213\) −8.00000 −0.548151
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 10.0000 0.675737
\(220\) −4.00000 −0.269680
\(221\) −4.00000 −0.269069
\(222\) 6.00000 0.402694
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 18.0000 1.19734
\(227\) 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
\(231\) 4.00000 0.263181
\(232\) −2.00000 −0.131306
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 16.0000 1.03931
\(238\) −2.00000 −0.129641
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000 0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 1.00000 0.0638877
\(246\) −6.00000 −0.382546
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 32.0000 2.01182
\(254\) −16.0000 −1.00393
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −4.00000 −0.249029
\(259\) −6.00000 −0.372822
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) 20.0000 1.23560
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −4.00000 −0.246183
\(265\) −10.0000 −0.614295
\(266\) −4.00000 −0.245256
\(267\) 10.0000 0.611990
\(268\) −12.0000 −0.733017
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 1.00000 0.0608581
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 2.00000 0.121268
\(273\) 2.00000 0.121046
\(274\) 10.0000 0.604122
\(275\) −4.00000 −0.241209
\(276\) −8.00000 −0.481543
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −8.00000 −0.474713
\(285\) 4.00000 0.236940
\(286\) 8.00000 0.473050
\(287\) 6.00000 0.354169
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 2.00000 0.117242
\(292\) 10.0000 0.585206
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 1.00000 0.0583212
\(295\) 12.0000 0.698667
\(296\) 6.00000 0.348743
\(297\) −4.00000 −0.232104
\(298\) −10.0000 −0.579284
\(299\) 16.0000 0.925304
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) 14.0000 0.801638
\(306\) 2.00000 0.114332
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 4.00000 0.227921
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −2.00000 −0.113228
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −18.0000 −1.01580
\(315\) −1.00000 −0.0563436
\(316\) 16.0000 0.900070
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −10.0000 −0.560772
\(319\) 8.00000 0.447914
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) 8.00000 0.445823
\(323\) 8.00000 0.445132
\(324\) 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
\(330\) −4.00000 −0.220193
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −12.0000 −0.658586
\(333\) 6.00000 0.328798
\(334\) −8.00000 −0.437741
\(335\) −12.0000 −0.655630
\(336\) −1.00000 −0.0545545
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −9.00000 −0.489535
\(339\) 18.0000 0.977626
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) −8.00000 −0.430706
\(346\) −18.0000 −0.967686
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −2.00000 −0.107211
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) −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
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) −2.00000 −0.105851
\(358\) 4.00000 0.211407
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 6.00000 0.315353
\(363\) 5.00000 0.262432
\(364\) 2.00000 0.104828
\(365\) 10.0000 0.523424
\(366\) 14.0000 0.731792
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −8.00000 −0.417029
\(369\) −6.00000 −0.312348
\(370\) 6.00000 0.311925
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −8.00000 −0.413670
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) −1.00000 −0.0514344
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) −16.0000 −0.818631
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 4.00000 0.203859
\(386\) 2.00000 0.101797
\(387\) −4.00000 −0.203331
\(388\) 2.00000 0.101535
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) −2.00000 −0.101274
\(391\) −16.0000 −0.809155
\(392\) 1.00000 0.0505076
\(393\) 20.0000 1.00887
\(394\) 6.00000 0.302276
\(395\) 16.0000 0.805047
\(396\) −4.00000 −0.201008
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −24.0000 −1.20301
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 2.00000 0.0992583
\(407\) −24.0000 −1.18964
\(408\) 2.00000 0.0990148
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) −6.00000 −0.296319
\(411\) 10.0000 0.493264
\(412\) −8.00000 −0.394132
\(413\) −12.0000 −0.590481
\(414\) −8.00000 −0.393179
\(415\) −12.0000 −0.589057
\(416\) −2.00000 −0.0980581
\(417\) −4.00000 −0.195881
\(418\) −16.0000 −0.782586
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 2.00000 0.0970143
\(426\) −8.00000 −0.387601
\(427\) −14.0000 −0.677507
\(428\) 12.0000 0.580042
\(429\) 8.00000 0.386244
\(430\) −4.00000 −0.192897
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 14.0000 0.670478
\(437\) −32.0000 −1.53077
\(438\) 10.0000 0.477818
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −4.00000 −0.190693
\(441\) 1.00000 0.0476190
\(442\) −4.00000 −0.190261
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 6.00000 0.284747
\(445\) 10.0000 0.474045
\(446\) −16.0000 −0.757622
\(447\) −10.0000 −0.472984
\(448\) −1.00000 −0.0472456
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 1.00000 0.0471405
\(451\) 24.0000 1.13012
\(452\) 18.0000 0.846649
\(453\) −8.00000 −0.375873
\(454\) 4.00000 0.187729
\(455\) 2.00000 0.0937614
\(456\) 4.00000 0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −10.0000 −0.467269
\(459\) 2.00000 0.0933520
\(460\) −8.00000 −0.373002
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 4.00000 0.186097
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 12.0000 0.552345
\(473\) 16.0000 0.735681
\(474\) 16.0000 0.734904
\(475\) 4.00000 0.183533
\(476\) −2.00000 −0.0916698
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 1.00000 0.0456435
\(481\) −12.0000 −0.547153
\(482\) −14.0000 −0.637683
\(483\) 8.00000 0.364013
\(484\) 5.00000 0.227273
\(485\) 2.00000 0.0908153
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 14.0000 0.633750
\(489\) 20.0000 0.904431
\(490\) 1.00000 0.0451754
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) −6.00000 −0.270501
\(493\) −4.00000 −0.180151
\(494\) −8.00000 −0.359937
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) −12.0000 −0.537733
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 1.00000 0.0447214
\(501\) −8.00000 −0.357414
\(502\) 12.0000 0.535586
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 6.00000 0.266996
\(506\) 32.0000 1.42257
\(507\) −9.00000 −0.399704
\(508\) −16.0000 −0.709885
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 2.00000 0.0885615
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −14.0000 −0.617514
\(515\) −8.00000 −0.352522
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) −18.0000 −0.790112
\(520\) −2.00000 −0.0877058
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 20.0000 0.873704
\(525\) −1.00000 −0.0436436
\(526\) 8.00000 0.348817
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) 12.0000 0.520756
\(532\) −4.00000 −0.173422
\(533\) 12.0000 0.519778
\(534\) 10.0000 0.432742
\(535\) 12.0000 0.518805
\(536\) −12.0000 −0.518321
\(537\) 4.00000 0.172613
\(538\) −18.0000 −0.776035
\(539\) −4.00000 −0.172292
\(540\) 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
\(543\) 6.00000 0.257485
\(544\) 2.00000 0.0857493
\(545\) 14.0000 0.599694
\(546\) 2.00000 0.0855921
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 10.0000 0.427179
\(549\) 14.0000 0.597505
\(550\) −4.00000 −0.170561
\(551\) −8.00000 −0.340811
\(552\) −8.00000 −0.340503
\(553\) −16.0000 −0.680389
\(554\) 22.0000 0.934690
\(555\) 6.00000 0.254686
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −1.00000 −0.0422577
\(561\) −8.00000 −0.337760
\(562\) −6.00000 −0.253095
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 28.0000 1.17693
\(567\) −1.00000 −0.0419961
\(568\) −8.00000 −0.335673
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 4.00000 0.167542
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 8.00000 0.334497
\(573\) −16.0000 −0.668410
\(574\) 6.00000 0.250435
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) −13.0000 −0.540729
\(579\) 2.00000 0.0831172
\(580\) −2.00000 −0.0830455
\(581\) 12.0000 0.497844
\(582\) 2.00000 0.0829027
\(583\) 40.0000 1.65663
\(584\) 10.0000 0.413803
\(585\) −2.00000 −0.0826898
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) 6.00000 0.246807
\(592\) 6.00000 0.246598
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −4.00000 −0.164122
\(595\) −2.00000 −0.0819920
\(596\) −10.0000 −0.409616
\(597\) −24.0000 −0.982255
\(598\) 16.0000 0.654289
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000 0.0408248
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 4.00000 0.163028
\(603\) −12.0000 −0.488678
\(604\) −8.00000 −0.325515
\(605\) 5.00000 0.203279
\(606\) 6.00000 0.243733
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 4.00000 0.162221
\(609\) 2.00000 0.0810441
\(610\) 14.0000 0.566843
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 20.0000 0.807134
\(615\) −6.00000 −0.241943
\(616\) 4.00000 0.161165
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) −8.00000 −0.321807
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) −8.00000 −0.320771
\(623\) −10.0000 −0.400642
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) −16.0000 −0.638978
\(628\) −18.0000 −0.718278
\(629\) 12.0000 0.478471
\(630\) −1.00000 −0.0398410
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 16.0000 0.636446
\(633\) −12.0000 −0.476957
\(634\) −2.00000 −0.0794301
\(635\) −16.0000 −0.634941
\(636\) −10.0000 −0.396526
\(637\) −2.00000 −0.0792429
\(638\) 8.00000 0.316723
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 12.0000 0.473602
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 8.00000 0.315244
\(645\) −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
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 14.0000 0.547443
\(655\) 20.0000 0.781465
\(656\) −6.00000 −0.234261
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) −4.00000 −0.155700
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 12.0000 0.466393
\(663\) −4.00000 −0.155347
\(664\) −12.0000 −0.465690
\(665\) −4.00000 −0.155113
\(666\) 6.00000 0.232495
\(667\) 16.0000 0.619522
\(668\) −8.00000 −0.309529
\(669\) −16.0000 −0.618596
\(670\) −12.0000 −0.463600
\(671\) −56.0000 −2.16186
\(672\) −1.00000 −0.0385758
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 18.0000 0.691286
\(679\) −2.00000 −0.0767530
\(680\) 2.00000 0.0766965
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 4.00000 0.152944
\(685\) 10.0000 0.382080
\(686\) −1.00000 −0.0381802
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) 20.0000 0.761939
\(690\) −8.00000 −0.304555
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) −18.0000 −0.684257
\(693\) 4.00000 0.151947
\(694\) −4.00000 −0.151838
\(695\) −4.00000 −0.151729
\(696\) −2.00000 −0.0758098
\(697\) −12.0000 −0.454532
\(698\) 14.0000 0.529908
\(699\) −22.0000 −0.832116
\(700\) −1.00000 −0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 24.0000 0.905177
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) −6.00000 −0.225653
\(708\) 12.0000 0.450988
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −8.00000 −0.300235
\(711\) 16.0000 0.600047
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 8.00000 0.299183
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) −14.0000 −0.520666
\(724\) 6.00000 0.222988
\(725\) −2.00000 −0.0742781
\(726\) 5.00000 0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) −8.00000 −0.295891
\(732\) 14.0000 0.517455
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) −32.0000 −1.18114
\(735\) 1.00000 0.0368856
\(736\) −8.00000 −0.294884
\(737\) 48.0000 1.76810
\(738\) −6.00000 −0.220863
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 6.00000 0.220564
\(741\) −8.00000 −0.293887
\(742\) 10.0000 0.367112
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −10.0000 −0.366126
\(747\) −12.0000 −0.439057
\(748\) −8.00000 −0.292509
\(749\) −12.0000 −0.438470
\(750\) 1.00000 0.0365148
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) −1.00000 −0.0363696
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 28.0000 1.01701
\(759\) 32.0000 1.16153
\(760\) 4.00000 0.145095
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −16.0000 −0.579619
\(763\) −14.0000 −0.506834
\(764\) −16.0000 −0.578860
\(765\) 2.00000 0.0723102
\(766\) 16.0000 0.578103
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 4.00000 0.144150
\(771\) −14.0000 −0.504198
\(772\) 2.00000 0.0719816
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) −6.00000 −0.215249
\(778\) −26.0000 −0.932145
\(779\) −24.0000 −0.859889
\(780\) −2.00000 −0.0716115
\(781\) 32.0000 1.14505
\(782\) −16.0000 −0.572159
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) −18.0000 −0.642448
\(786\) 20.0000 0.713376
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 6.00000 0.213741
\(789\) 8.00000 0.284808
\(790\) 16.0000 0.569254
\(791\) −18.0000 −0.640006
\(792\) −4.00000 −0.142134
\(793\) −28.0000 −0.994309
\(794\) 30.0000 1.06466
\(795\) −10.0000 −0.354663
\(796\) −24.0000 −0.850657
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) −14.0000 −0.494357
\(803\) −40.0000 −1.41157
\(804\) −12.0000 −0.423207
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 6.00000 0.211079
\(809\) −22.0000 −0.773479 −0.386739 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) 1.00000 0.0351364
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 2.00000 0.0701862
\(813\) 16.0000 0.561144
\(814\) −24.0000 −0.841200
\(815\) 20.0000 0.700569
\(816\) 2.00000 0.0700140
\(817\) −16.0000 −0.559769
\(818\) −38.0000 −1.32864
\(819\) 2.00000 0.0698857
\(820\) −6.00000 −0.209529
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 10.0000 0.348790
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) −8.00000 −0.278693
\(825\) −4.00000 −0.139262
\(826\) −12.0000 −0.417533
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) −8.00000 −0.278019
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) −12.0000 −0.416526
\(831\) 22.0000 0.763172
\(832\) −2.00000 −0.0693375
\(833\) 2.00000 0.0692959
\(834\) −4.00000 −0.138509
\(835\) −8.00000 −0.276851
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −25.0000 −0.862069
\(842\) −26.0000 −0.896019
\(843\) −6.00000 −0.206651
\(844\) −12.0000 −0.413057
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) −10.0000 −0.343401
\(849\) 28.0000 0.960958
\(850\) 2.00000 0.0685994
\(851\) −48.0000 −1.64542
\(852\) −8.00000 −0.274075
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −14.0000 −0.479070
\(855\) 4.00000 0.136797
\(856\) 12.0000 0.410152
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 8.00000 0.273115
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −4.00000 −0.136399
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.0000 −0.612018
\(866\) −14.0000 −0.475739
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) −2.00000 −0.0678064
\(871\) 24.0000 0.813209
\(872\) 14.0000 0.474100
\(873\) 2.00000 0.0676897
\(874\) −32.0000 −1.08242
\(875\) −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 \(0.424041\pi\)
\(878\) −8.00000 −0.269987
\(879\) 6.00000 0.202375
\(880\) −4.00000 −0.134840
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 1.00000 0.0336718
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) −4.00000 −0.134535
\(885\) 12.0000 0.403376
\(886\) −36.0000 −1.20944
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 6.00000 0.201347
\(889\) 16.0000 0.536623
\(890\) 10.0000 0.335201
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 4.00000 0.133705
\(896\) −1.00000 −0.0334077
\(897\) 16.0000 0.534224
\(898\) 2.00000 0.0667409
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −20.0000 −0.666297
\(902\) 24.0000 0.799113
\(903\) 4.00000 0.133112
\(904\) 18.0000 0.598671
\(905\) 6.00000 0.199447
\(906\) −8.00000 −0.265782
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 4.00000 0.132745
\(909\) 6.00000 0.199007
\(910\) 2.00000 0.0662994
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 4.00000 0.132453
\(913\) 48.0000 1.58857
\(914\) 10.0000 0.330771
\(915\) 14.0000 0.462826
\(916\) −10.0000 −0.330409
\(917\) −20.0000 −0.660458
\(918\) 2.00000 0.0660098
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) −8.00000 −0.263752
\(921\) 20.0000 0.659022
\(922\) 14.0000 0.461065
\(923\) 16.0000 0.526646
\(924\) 4.00000 0.131590
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) −2.00000 −0.0656532
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) −22.0000 −0.720634
\(933\) −8.00000 −0.261908
\(934\) −12.0000 −0.392652
\(935\) −8.00000 −0.261628
\(936\) −2.00000 −0.0653720
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 12.0000 0.391814
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −18.0000 −0.586472
\(943\) 48.0000 1.56310
\(944\) 12.0000 0.390567
\(945\) −1.00000 −0.0325300
\(946\) 16.0000 0.520205
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 16.0000 0.519656
\(949\) −20.0000 −0.649227
\(950\) 4.00000 0.129777
\(951\) −2.00000 −0.0648544
\(952\) −2.00000 −0.0648204
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) −10.0000 −0.323762
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) 8.00000 0.258603
\(958\) 32.0000 1.03387
\(959\) −10.0000 −0.322917
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) −12.0000 −0.386896
\(963\) 12.0000 0.386695
\(964\) −14.0000 −0.450910
\(965\) 2.00000 0.0643823
\(966\) 8.00000 0.257396
\(967\) −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
\(970\) 2.00000 0.0642161
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 1.00000 0.0320750
\(973\) 4.00000 0.128234
\(974\) −8.00000 −0.256337
\(975\) −2.00000 −0.0640513
\(976\) 14.0000 0.448129
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 20.0000 0.639529
\(979\) −40.0000 −1.27841
\(980\) 1.00000 0.0319438
\(981\) 14.0000 0.446986
\(982\) −36.0000 −1.14881
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −6.00000 −0.191273
\(985\) 6.00000 0.191176
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 32.0000 1.01754
\(990\) −4.00000 −0.127128
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 8.00000 0.253745
\(995\) −24.0000 −0.760851
\(996\) −12.0000 −0.380235
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) −12.0000 −0.379853
\(999\) 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 210.2.a.e.1.1 1
3.2 odd 2 630.2.a.a.1.1 1
4.3 odd 2 1680.2.a.j.1.1 1
5.2 odd 4 1050.2.g.g.799.2 2
5.3 odd 4 1050.2.g.g.799.1 2
5.4 even 2 1050.2.a.c.1.1 1
7.2 even 3 1470.2.i.a.361.1 2
7.3 odd 6 1470.2.i.j.961.1 2
7.4 even 3 1470.2.i.a.961.1 2
7.5 odd 6 1470.2.i.j.361.1 2
7.6 odd 2 1470.2.a.j.1.1 1
8.3 odd 2 6720.2.a.bq.1.1 1
8.5 even 2 6720.2.a.j.1.1 1
12.11 even 2 5040.2.a.k.1.1 1
15.2 even 4 3150.2.g.q.2899.1 2
15.8 even 4 3150.2.g.q.2899.2 2
15.14 odd 2 3150.2.a.bp.1.1 1
20.19 odd 2 8400.2.a.ce.1.1 1
21.20 even 2 4410.2.a.t.1.1 1
35.34 odd 2 7350.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.e.1.1 1 1.1 even 1 trivial
630.2.a.a.1.1 1 3.2 odd 2
1050.2.a.c.1.1 1 5.4 even 2
1050.2.g.g.799.1 2 5.3 odd 4
1050.2.g.g.799.2 2 5.2 odd 4
1470.2.a.j.1.1 1 7.6 odd 2
1470.2.i.a.361.1 2 7.2 even 3
1470.2.i.a.961.1 2 7.4 even 3
1470.2.i.j.361.1 2 7.5 odd 6
1470.2.i.j.961.1 2 7.3 odd 6
1680.2.a.j.1.1 1 4.3 odd 2
3150.2.a.bp.1.1 1 15.14 odd 2
3150.2.g.q.2899.1 2 15.2 even 4
3150.2.g.q.2899.2 2 15.8 even 4
4410.2.a.t.1.1 1 21.20 even 2
5040.2.a.k.1.1 1 12.11 even 2
6720.2.a.j.1.1 1 8.5 even 2
6720.2.a.bq.1.1 1 8.3 odd 2
7350.2.a.w.1.1 1 35.34 odd 2
8400.2.a.ce.1.1 1 20.19 odd 2