Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
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 \(0\)
Character \(\chi\) = 210.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +10.0000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} -10.0000 q^{58} +4.00000 q^{59} +1.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -6.00000 q^{68} +8.00000 q^{69} -1.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{77} -2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} +6.00000 q^{85} -8.00000 q^{86} -10.0000 q^{87} +4.00000 q^{88} +14.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -8.00000 q^{92} +8.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −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
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 6.00000 1.02899
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −1.00000 −0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −4.00000 −0.603023
\(45\) −1.00000 −0.149071
\(46\) 8.00000 1.17954
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 1.00000 0.129099
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −4.00000 −0.492366
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −6.00000 −0.727607
\(69\) 8.00000 0.963087
\(70\) −1.00000 −0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) −2.00000 −0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 1.00000 0.109109
\(85\) 6.00000 0.650791
\(86\) −8.00000 −0.862662
\(87\) −10.0000 −1.07211
\(88\) 4.00000 0.426401
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) −8.00000 −0.834058
\(93\) 8.00000 0.829561
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 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 \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −6.00000 −0.594089
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 2.00000 0.196116
\(105\) −1.00000 −0.0975900
\(106\) −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
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 10.0000 0.928477
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 6.00000 0.550019
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) 6.00000 0.543214
\(123\) 2.00000 0.180334
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) −2.00000 −0.175412
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 0 0
\(135\) 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 \(0.417494\pi\)
\(138\) −8.00000 −0.681005
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 1.00000 0.0845154
\(141\) −4.00000 −0.336861
\(142\) 12.0000 1.00702
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) −10.0000 −0.830455
\(146\) 6.00000 0.496564
\(147\) −1.00000 −0.0824786
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) −4.00000 −0.322329
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 8.00000 0.636446
\(159\) −10.0000 −0.793052
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −2.00000 −0.156174
\(165\) −4.00000 −0.311400
\(166\) 4.00000 0.310460
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 10.0000 0.758098
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) −4.00000 −0.300658
\(178\) −14.0000 −1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) −2.00000 −0.148250
\(183\) 6.00000 0.443533
\(184\) 8.00000 0.589768
\(185\) −2.00000 −0.147043
\(186\) −8.00000 −0.586588
\(187\) 24.0000 1.75505
\(188\) 4.00000 0.291730
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −2.00000 −0.143592
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 4.00000 0.284268
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −10.0000 −0.701862
\(204\) 6.00000 0.420084
\(205\) 2.00000 0.139686
\(206\) 16.0000 1.11477
\(207\) −8.00000 −0.556038
\(208\) −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 \(0.543966\pi\)
\(212\) 10.0000 0.686803
\(213\) 12.0000 0.822226
\(214\) 4.00000 0.273434
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 8.00000 0.543075
\(218\) 18.0000 1.21911
\(219\) 6.00000 0.405442
\(220\) 4.00000 0.269680
\(221\) 12.0000 0.807207
\(222\) 2.00000 0.134231
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −8.00000 −0.527504
\(231\) −4.00000 −0.263181
\(232\) −10.0000 −0.656532
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 2.00000 0.130744
\(235\) −4.00000 −0.260931
\(236\) 4.00000 0.260378
\(237\) 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
\(241\) −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
\(244\) −6.00000 −0.384111
\(245\) −1.00000 −0.0638877
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 32.0000 2.01182
\(254\) −16.0000 −1.00393
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 8.00000 0.498058
\(259\) −2.00000 −0.124274
\(260\) 2.00000 0.124035
\(261\) 10.0000 0.618984
\(262\) −4.00000 −0.247121
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −4.00000 −0.246183
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 0 0
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −6.00000 −0.363803
\(273\) −2.00000 −0.121046
\(274\) −6.00000 −0.362473
\(275\) −4.00000 −0.241209
\(276\) 8.00000 0.481543
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 16.0000 0.959616
\(279\) −8.00000 −0.478947
\(280\) −1.00000 −0.0597614
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 4.00000 0.238197
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 2.00000 0.118056
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 10.0000 0.587220
\(291\) −2.00000 −0.117242
\(292\) −6.00000 −0.351123
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 1.00000 0.0583212
\(295\) −4.00000 −0.232889
\(296\) −2.00000 −0.116248
\(297\) 4.00000 0.232104
\(298\) 6.00000 0.347571
\(299\) 16.0000 0.925304
\(300\) −1.00000 −0.0577350
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 6.00000 0.342997
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 4.00000 0.227921
\(309\) 16.0000 0.910208
\(310\) −8.00000 −0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −2.00000 −0.113228
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 10.0000 0.564333
\(315\) 1.00000 0.0563436
\(316\) −8.00000 −0.450035
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 10.0000 0.560772
\(319\) −40.0000 −2.23957
\(320\) −1.00000 −0.0559017
\(321\) 4.00000 0.223258
\(322\) −8.00000 −0.445823
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −16.0000 −0.886158
\(327\) 18.0000 0.995402
\(328\) 2.00000 0.110432
\(329\) −4.00000 −0.220527
\(330\) 4.00000 0.220193
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −4.00000 −0.219529
\(333\) 2.00000 0.109599
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 6.00000 0.325396
\(341\) 32.0000 1.73290
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −8.00000 −0.431331
\(345\) −8.00000 −0.430706
\(346\) −18.0000 −0.967686
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −10.0000 −0.536056
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 4.00000 0.212598
\(355\) 12.0000 0.636894
\(356\) 14.0000 0.741999
\(357\) −6.00000 −0.317554
\(358\) 4.00000 0.211407
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 1.00000 0.0527046
\(361\) −19.0000 −1.00000
\(362\) −18.0000 −0.946059
\(363\) −5.00000 −0.262432
\(364\) 2.00000 0.104828
\(365\) 6.00000 0.314054
\(366\) −6.00000 −0.313625
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −8.00000 −0.417029
\(369\) −2.00000 −0.104116
\(370\) 2.00000 0.103975
\(371\) −10.0000 −0.519174
\(372\) 8.00000 0.414781
\(373\) −30.0000 −1.55334 −0.776671 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) −24.0000 −1.24101
\(375\) 1.00000 0.0516398
\(376\) −4.00000 −0.206284
\(377\) −20.0000 −1.03005
\(378\) −1.00000 −0.0514344
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −12.0000 −0.613973
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 1.00000 0.0510310
\(385\) −4.00000 −0.203859
\(386\) −10.0000 −0.508987
\(387\) 8.00000 0.406663
\(388\) 2.00000 0.101535
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 2.00000 0.101274
\(391\) 48.0000 2.42746
\(392\) −1.00000 −0.0505076
\(393\) −4.00000 −0.201773
\(394\) 22.0000 1.10834
\(395\) 8.00000 0.402524
\(396\) −4.00000 −0.201008
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) 10.0000 0.496292
\(407\) −8.00000 −0.396545
\(408\) −6.00000 −0.297044
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −6.00000 −0.295958
\(412\) −16.0000 −0.788263
\(413\) −4.00000 −0.196827
\(414\) 8.00000 0.393179
\(415\) 4.00000 0.196352
\(416\) 2.00000 0.0980581
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 4.00000 0.194717
\(423\) 4.00000 0.194487
\(424\) −10.0000 −0.485643
\(425\) −6.00000 −0.291043
\(426\) −12.0000 −0.581402
\(427\) 6.00000 0.290360
\(428\) −4.00000 −0.193347
\(429\) −8.00000 −0.386244
\(430\) 8.00000 0.385794
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −8.00000 −0.384012
\(435\) 10.0000 0.479463
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −4.00000 −0.190693
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −14.0000 −0.663664
\(446\) 24.0000 1.13643
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 8.00000 0.376705
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 22.0000 1.02799
\(459\) 6.00000 0.280056
\(460\) 8.00000 0.373002
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 4.00000 0.186097
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 10.0000 0.464238
\(465\) −8.00000 −0.370991
\(466\) 18.0000 0.833834
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 4.00000 0.184506
\(471\) 10.0000 0.460776
\(472\) −4.00000 −0.184115
\(473\) −32.0000 −1.47136
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 10.0000 0.457869
\(478\) 12.0000 0.548867
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −4.00000 −0.182384
\(482\) 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 \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 6.00000 0.271607
\(489\) −16.0000 −0.723545
\(490\) 1.00000 0.0451754
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 2.00000 0.0901670
\(493\) −60.0000 −2.70226
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) −8.00000 −0.359211
\(497\) 12.0000 0.538274
\(498\) −4.00000 −0.179244
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 12.0000 0.536120
\(502\) 20.0000 0.892644
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 1.00000 0.0445435
\(505\) 6.00000 0.266996
\(506\) −32.0000 −1.42257
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 6.00000 0.265684
\(511\) 6.00000 0.265424
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 16.0000 0.705044
\(516\) −8.00000 −0.352180
\(517\) −16.0000 −0.703679
\(518\) 2.00000 0.0878750
\(519\) −18.0000 −0.790112
\(520\) −2.00000 −0.0877058
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −10.0000 −0.437688
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 4.00000 0.174741
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 48.0000 2.09091
\(528\) 4.00000 0.174078
\(529\) 41.0000 1.78261
\(530\) 10.0000 0.434372
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 14.0000 0.605839
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) −2.00000 −0.0862261
\(539\) −4.00000 −0.172292
\(540\) 1.00000 0.0430331
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −8.00000 −0.343629
\(543\) −18.0000 −0.772454
\(544\) 6.00000 0.257248
\(545\) 18.0000 0.771035
\(546\) 2.00000 0.0855921
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 6.00000 0.256307
\(549\) −6.00000 −0.256074
\(550\) 4.00000 0.170561
\(551\) 0 0
\(552\) −8.00000 −0.340503
\(553\) 8.00000 0.340195
\(554\) 22.0000 0.934690
\(555\) 2.00000 0.0848953
\(556\) −16.0000 −0.678551
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 8.00000 0.338667
\(559\) −16.0000 −0.676728
\(560\) 1.00000 0.0422577
\(561\) −24.0000 −1.01328
\(562\) 22.0000 0.928014
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −4.00000 −0.168430
\(565\) −6.00000 −0.252422
\(566\) −28.0000 −1.17693
\(567\) −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 \(0.540139\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000 0.334497
\(573\) −12.0000 −0.501307
\(574\) −2.00000 −0.0834784
\(575\) −8.00000 −0.333623
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) −19.0000 −0.790296
\(579\) −10.0000 −0.415586
\(580\) −10.0000 −0.415227
\(581\) 4.00000 0.165948
\(582\) 2.00000 0.0829027
\(583\) −40.0000 −1.65663
\(584\) 6.00000 0.248282
\(585\) 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
\(589\) 0 0
\(590\) 4.00000 0.164677
\(591\) 22.0000 0.904959
\(592\) 2.00000 0.0821995
\(593\) −38.0000 −1.56047 −0.780236 0.625485i \(-0.784901\pi\)
−0.780236 + 0.625485i \(0.784901\pi\)
\(594\) −4.00000 −0.164122
\(595\) −6.00000 −0.245976
\(596\) −6.00000 −0.245770
\(597\) −24.0000 −0.982255
\(598\) −16.0000 −0.654289
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 8.00000 0.326056
\(603\) 0 0
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) −6.00000 −0.243733
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) −6.00000 −0.242933
\(611\) −8.00000 −0.323645
\(612\) −6.00000 −0.242536
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −20.0000 −0.807134
\(615\) −2.00000 −0.0806478
\(616\) −4.00000 −0.161165
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) −16.0000 −0.643614
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 8.00000 0.321288
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) −14.0000 −0.560898
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −12.0000 −0.478471
\(630\) −1.00000 −0.0398410
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 8.00000 0.318223
\(633\) 4.00000 0.158986
\(634\) 30.0000 1.19145
\(635\) −16.0000 −0.634941
\(636\) −10.0000 −0.396526
\(637\) −2.00000 −0.0792429
\(638\) 40.0000 1.58362
\(639\) −12.0000 −0.474713
\(640\) 1.00000 0.0395285
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −4.00000 −0.157867
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 8.00000 0.315244
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −16.0000 −0.628055
\(650\) 2.00000 0.0784465
\(651\) −8.00000 −0.313545
\(652\) 16.0000 0.626608
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −18.0000 −0.703856
\(655\) −4.00000 −0.156293
\(656\) −2.00000 −0.0780869
\(657\) −6.00000 −0.234082
\(658\) 4.00000 0.155936
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −4.00000 −0.155700
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) −20.0000 −0.777322
\(663\) −12.0000 −0.466041
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −80.0000 −3.09761
\(668\) −12.0000 −0.464294
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) −1.00000 −0.0385758
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 6.00000 0.231111
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 6.00000 0.230429
\(679\) −2.00000 −0.0767530
\(680\) −6.00000 −0.230089
\(681\) −20.0000 −0.766402
\(682\) −32.0000 −1.22534
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 1.00000 0.0381802
\(687\) 22.0000 0.839352
\(688\) 8.00000 0.304997
\(689\) −20.0000 −0.761939
\(690\) 8.00000 0.304555
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 18.0000 0.684257
\(693\) 4.00000 0.151947
\(694\) −20.0000 −0.759190
\(695\) 16.0000 0.606915
\(696\) 10.0000 0.379049
\(697\) 12.0000 0.454532
\(698\) −18.0000 −0.681310
\(699\) 18.0000 0.680823
\(700\) −1.00000 −0.0377964
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) 4.00000 0.150649
\(706\) −18.0000 −0.677439
\(707\) 6.00000 0.225653
\(708\) −4.00000 −0.150329
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) −12.0000 −0.450352
\(711\) −8.00000 −0.300023
\(712\) −14.0000 −0.524672
\(713\) 64.0000 2.39682
\(714\) 6.00000 0.224544
\(715\) −8.00000 −0.299183
\(716\) −4.00000 −0.149487
\(717\) 12.0000 0.448148
\(718\) 20.0000 0.746393
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.0000 0.595871
\(722\) 19.0000 0.707107
\(723\) 14.0000 0.520666
\(724\) 18.0000 0.668965
\(725\) 10.0000 0.371391
\(726\) 5.00000 0.185567
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) −48.0000 −1.77534
\(732\) 6.00000 0.221766
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −16.0000 −0.590571
\(735\) 1.00000 0.0368856
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) 10.0000 0.367112
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −8.00000 −0.293294
\(745\) 6.00000 0.219823
\(746\) 30.0000 1.09838
\(747\) −4.00000 −0.146352
\(748\) 24.0000 0.877527
\(749\) 4.00000 0.146157
\(750\) −1.00000 −0.0365148
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 4.00000 0.145865
\(753\) 20.0000 0.728841
\(754\) 20.0000 0.728357
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 28.0000 1.01701
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 16.0000 0.579619
\(763\) 18.0000 0.651644
\(764\) 12.0000 0.434145
\(765\) 6.00000 0.216930
\(766\) 28.0000 1.01168
\(767\) −8.00000 −0.288863
\(768\) −1.00000 −0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 4.00000 0.144150
\(771\) 14.0000 0.504198
\(772\) 10.0000 0.359908
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −8.00000 −0.287554
\(775\) −8.00000 −0.287368
\(776\) −2.00000 −0.0717958
\(777\) 2.00000 0.0717496
\(778\) −10.0000 −0.358517
\(779\) 0 0
\(780\) −2.00000 −0.0716115
\(781\) 48.0000 1.71758
\(782\) −48.0000 −1.71648
\(783\) −10.0000 −0.357371
\(784\) 1.00000 0.0357143
\(785\) 10.0000 0.356915
\(786\) 4.00000 0.142675
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −22.0000 −0.783718
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −6.00000 −0.213335
\(792\) 4.00000 0.142134
\(793\) 12.0000 0.426132
\(794\) −14.0000 −0.496841
\(795\) 10.0000 0.354663
\(796\) 24.0000 0.850657
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 14.0000 0.494666
\(802\) −34.0000 −1.20058
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) −16.0000 −0.563576
\(807\) −2.00000 −0.0704033
\(808\) 6.00000 0.211079
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 1.00000 0.0351364
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) −10.0000 −0.350931
\(813\) −8.00000 −0.280572
\(814\) 8.00000 0.280400
\(815\) −16.0000 −0.560456
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) −26.0000 −0.909069
\(819\) 2.00000 0.0698857
\(820\) 2.00000 0.0698430
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 6.00000 0.209274
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 16.0000 0.557386
\(825\) 4.00000 0.139262
\(826\) 4.00000 0.139178
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) −8.00000 −0.278019
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) −4.00000 −0.138842
\(831\) 22.0000 0.763172
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) −16.0000 −0.554035
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −28.0000 −0.967244
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 1.00000 0.0345033
\(841\) 71.0000 2.44828
\(842\) 26.0000 0.896019
\(843\) 22.0000 0.757720
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) −4.00000 −0.137523
\(847\) −5.00000 −0.171802
\(848\) 10.0000 0.343401
\(849\) −28.0000 −0.960958
\(850\) 6.00000 0.205798
\(851\) −16.0000 −0.548473
\(852\) 12.0000 0.411113
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 8.00000 0.273115
\(859\) 24.0000 0.818869 0.409435 0.912339i \(-0.365726\pi\)
0.409435 + 0.912339i \(0.365726\pi\)
\(860\) −8.00000 −0.272798
\(861\) −2.00000 −0.0681598
\(862\) 36.0000 1.22616
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.0000 −0.612018
\(866\) −34.0000 −1.15537
\(867\) −19.0000 −0.645274
\(868\) 8.00000 0.271538
\(869\) 32.0000 1.08553
\(870\) −10.0000 −0.339032
\(871\) 0 0
\(872\) 18.0000 0.609557
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 6.00000 0.202721
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −32.0000 −1.07995
\(879\) 30.0000 1.01187
\(880\) 4.00000 0.134840
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 12.0000 0.403604
\(885\) 4.00000 0.134459
\(886\) −4.00000 −0.134383
\(887\) 20.0000 0.671534 0.335767 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 2.00000 0.0671156
\(889\) −16.0000 −0.536623
\(890\) 14.0000 0.469281
\(891\) −4.00000 −0.134005
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 4.00000 0.133705
\(896\) 1.00000 0.0334077
\(897\) −16.0000 −0.534224
\(898\) −26.0000 −0.867631
\(899\) −80.0000 −2.66815
\(900\) 1.00000 0.0333333
\(901\) −60.0000 −1.99889
\(902\) −8.00000 −0.266371
\(903\) 8.00000 0.266223
\(904\) −6.00000 −0.199557
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 20.0000 0.663723
\(909\) −6.00000 −0.199007
\(910\) 2.00000 0.0662994
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 6.00000 0.198462
\(915\) −6.00000 −0.198354
\(916\) −22.0000 −0.726900
\(917\) −4.00000 −0.132092
\(918\) −6.00000 −0.198030
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −8.00000 −0.263752
\(921\) −20.0000 −0.659022
\(922\) −18.0000 −0.592798
\(923\) 24.0000 0.789970
\(924\) −4.00000 −0.131590
\(925\) 2.00000 0.0657596
\(926\) −8.00000 −0.262896
\(927\) −16.0000 −0.525509
\(928\) −10.0000 −0.328266
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) −24.0000 −0.784884
\(936\) 2.00000 0.0653720
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) −4.00000 −0.130466
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −10.0000 −0.325818
\(943\) 16.0000 0.521032
\(944\) 4.00000 0.130189
\(945\) −1.00000 −0.0325300
\(946\) 32.0000 1.04041
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 8.00000 0.259828
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) −6.00000 −0.194461
\(953\) −50.0000 −1.61966 −0.809829 0.586665i \(-0.800440\pi\)
−0.809829 + 0.586665i \(0.800440\pi\)
\(954\) −10.0000 −0.323762
\(955\) −12.0000 −0.388311
\(956\) −12.0000 −0.388108
\(957\) 40.0000 1.29302
\(958\) 8.00000 0.258468
\(959\) −6.00000 −0.193750
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) −4.00000 −0.128898
\(964\) −14.0000 −0.450910
\(965\) −10.0000 −0.321911
\(966\) 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
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.0000 0.512936
\(974\) −8.00000 −0.256337
\(975\) 2.00000 0.0640513
\(976\) −6.00000 −0.192055
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 16.0000 0.511624
\(979\) −56.0000 −1.78977
\(980\) −1.00000 −0.0319438
\(981\) −18.0000 −0.574696
\(982\) −20.0000 −0.638226
\(983\) 28.0000 0.893061 0.446531 0.894768i \(-0.352659\pi\)
0.446531 + 0.894768i \(0.352659\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 22.0000 0.700978
\(986\) 60.0000 1.91079
\(987\) 4.00000 0.127321
\(988\) 0 0
\(989\) −64.0000 −2.03508
\(990\) −4.00000 −0.127128
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000 0.254000
\(993\) −20.0000 −0.634681
\(994\) −12.0000 −0.380617
\(995\) −24.0000 −0.760851
\(996\) 4.00000 0.126745
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −28.0000 −0.886325
\(999\) −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 210.2.a.a.1.1 1
3.2 odd 2 630.2.a.i.1.1 1
4.3 odd 2 1680.2.a.o.1.1 1
5.2 odd 4 1050.2.g.f.799.1 2
5.3 odd 4 1050.2.g.f.799.2 2
5.4 even 2 1050.2.a.q.1.1 1
7.2 even 3 1470.2.i.t.361.1 2
7.3 odd 6 1470.2.i.n.961.1 2
7.4 even 3 1470.2.i.t.961.1 2
7.5 odd 6 1470.2.i.n.361.1 2
7.6 odd 2 1470.2.a.g.1.1 1
8.3 odd 2 6720.2.a.z.1.1 1
8.5 even 2 6720.2.a.cg.1.1 1
12.11 even 2 5040.2.a.bg.1.1 1
15.2 even 4 3150.2.g.t.2899.2 2
15.8 even 4 3150.2.g.t.2899.1 2
15.14 odd 2 3150.2.a.t.1.1 1
20.19 odd 2 8400.2.a.m.1.1 1
21.20 even 2 4410.2.a.bc.1.1 1
35.34 odd 2 7350.2.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.a.1.1 1 1.1 even 1 trivial
630.2.a.i.1.1 1 3.2 odd 2
1050.2.a.q.1.1 1 5.4 even 2
1050.2.g.f.799.1 2 5.2 odd 4
1050.2.g.f.799.2 2 5.3 odd 4
1470.2.a.g.1.1 1 7.6 odd 2
1470.2.i.n.361.1 2 7.5 odd 6
1470.2.i.n.961.1 2 7.3 odd 6
1470.2.i.t.361.1 2 7.2 even 3
1470.2.i.t.961.1 2 7.4 even 3
1680.2.a.o.1.1 1 4.3 odd 2
3150.2.a.t.1.1 1 15.14 odd 2
3150.2.g.t.2899.1 2 15.8 even 4
3150.2.g.t.2899.2 2 15.2 even 4
4410.2.a.bc.1.1 1 21.20 even 2
5040.2.a.bg.1.1 1 12.11 even 2
6720.2.a.z.1.1 1 8.3 odd 2
6720.2.a.cg.1.1 1 8.5 even 2
7350.2.a.bo.1.1 1 35.34 odd 2
8400.2.a.m.1.1 1 20.19 odd 2