Properties

Label 1050.2.a.q.1.1
Level $1050$
Weight $2$
Character 1050.1
Self dual yes
Analytic conductor $8.384$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1050.1

$q$-expansion

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

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −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 \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 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 \(0.121119\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −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 \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 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
\(55\) 0 0
\(56\) 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 \(0.416153\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) −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
\(70\) 0 0
\(71\) −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 \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 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 \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −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 \(0.233878\pi\)
\(90\) 0 0
\(91\) 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 \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −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 \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) −2.00000 −0.180334
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 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
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 8.00000 0.681005
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) −12.0000 −1.00702
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 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 \(0.579041\pi\)
\(150\) 0 0
\(151\) 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
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −8.00000 −0.636446
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 1.00000 0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 10.0000 0.758098
\(175\) 0 0
\(176\) −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 \(0.547762\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −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 \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) −4.00000 −0.284268
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 10.0000 0.701862
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 8.00000 0.556038
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −18.0000 −1.21911
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −2.00000 −0.134231
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 10.0000 0.656532
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 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 \(0.626871\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −8.00000 −0.498058
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 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
\(265\) 0 0
\(266\) 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 \(0.480580\pi\)
\(270\) 0 0
\(271\) 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
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −16.0000 −0.959616
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −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 \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −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
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 6.00000 0.351123
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −4.00000 −0.227921
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 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 \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −10.0000 −0.560772
\(319\) −40.0000 −2.23957
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 8.00000 0.445823
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) −18.0000 −0.995402
\(328\) −2.00000 −0.110432
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 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 \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 10.0000 0.536056
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −4.00000 −0.213201
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 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 \(0.676975\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 18.0000 0.946059
\(363\) 5.00000 0.262432
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 8.00000 0.417029
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) −8.00000 −0.414781
\(373\) 30.0000 1.55334 0.776671 0.629907i \(-0.216907\pi\)
0.776671 + 0.629907i \(0.216907\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) −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 \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −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 \(0.418415\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 1.00000 0.0505076
\(393\) 4.00000 0.201773
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) 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 \(0.277769\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 16.0000 0.788263
\(413\) 4.00000 0.196827
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 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 \(0.260263\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) −6.00000 −0.290360
\(428\) 4.00000 0.193347
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −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 \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −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 \(0.223411\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 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 \(0.289758\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −22.0000 −1.02799
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 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 \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 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 \(0.558505\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −14.0000 −0.637683
\(483\) 8.00000 0.364013
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −6.00000 −0.271607
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) −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 \(0.284395\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) −20.0000 −0.892644
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −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 \(0.485887\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 16.0000 0.703679
\(518\) −2.00000 −0.0878750
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −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 \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 2.00000 0.0862261
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −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
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) −6.00000 −0.256307
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) 8.00000 0.340503
\(553\) −8.00000 −0.340195
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −8.00000 −0.338667
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) −22.0000 −0.928014
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −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
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 19.0000 0.790296
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) −2.00000 −0.0829027
\(583\) 40.0000 1.65663
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) −2.00000 −0.0821995
\(593\) 38.0000 1.56047 0.780236 0.625485i \(-0.215099\pi\)
0.780236 + 0.625485i \(0.215099\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −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 \(0.366021\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 6.00000 0.242536
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 16.0000 0.643614
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 14.0000 0.560898
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) 2.00000 0.0792429
\(638\) −40.0000 −1.58362
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −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 \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) −16.0000 −0.626608
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −18.0000 −0.703856
\(655\) 0 0
\(656\) −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 \(0.575092\pi\)
\(660\) 0 0
\(661\) −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 \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −6.00000 −0.230429
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 32.0000 1.22534
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −22.0000 −0.839352
\(688\) −8.00000 −0.304997
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) 10.0000 0.379049
\(697\) −12.0000 −0.454532
\(698\) 18.0000 0.681310
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 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
\(705\) 0 0
\(706\) −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 \(0.415324\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 14.0000 0.524672
\(713\) −64.0000 −2.39682
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) −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 \(0.452338\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −19.0000 −0.707107
\(723\) −14.0000 −0.520666
\(724\) 18.0000 0.668965
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) −6.00000 −0.221766
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 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 \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 30.0000 1.09838
\(747\) 4.00000 0.146352
\(748\) −24.0000 −0.877527
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −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 \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −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
\(765\) 0 0
\(766\) 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 \(0.442292\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −10.0000 −0.359908
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) −2.00000 −0.0717496
\(778\) 10.0000 0.358517
\(779\) 0 0
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 48.0000 1.71648
\(783\) 10.0000 0.357371
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 4.00000 0.142675
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 22.0000 0.783718
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −4.00000 −0.142134
\(793\) −12.0000 −0.426132
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 34.0000 1.20058
\(803\) −24.0000 −0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) −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 \(0.235619\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) 26.0000 0.909069
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 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
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 8.00000 0.278019
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 2.00000 0.0693375
\(833\) 6.00000 0.207888
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 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 \(0.455902\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −26.0000 −0.896019
\(843\) −22.0000 −0.757720
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) 5.00000 0.171802
\(848\) −10.0000 −0.343401
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) −12.0000 −0.411113
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) −8.00000 −0.273115
\(859\) 24.0000 0.818869 0.409435 0.912339i \(-0.365726\pi\)
0.409435 + 0.912339i \(0.365726\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) −36.0000 −1.22616
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 19.0000 0.645274
\(868\) −8.00000 −0.271538
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 0 0
\(872\) −18.0000 −0.609557
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 32.0000 1.07995
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) −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 \(-0.799269\pi\)
−0.807664 + 0.589643i \(0.799269\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) −20.0000 −0.671534 −0.335767 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 24.0000 0.803579
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 16.0000 0.534224
\(898\) 26.0000 0.867631
\(899\) −80.0000 −2.66815
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) 8.00000 0.266371
\(903\) −8.00000 −0.266223
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −20.0000 −0.663723
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −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
\(915\) 0 0
\(916\) −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 \(0.629548\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 18.0000 0.592798
\(923\) −24.0000 −0.789970
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −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 \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 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
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) −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 \(-0.199560\pi\)
0.809829 + 0.586665i \(0.199560\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) −40.0000 −1.29302
\(958\) −8.00000 −0.258468
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −4.00000 −0.128965
\(963\) 4.00000 0.128898
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) −16.0000 −0.511624
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 20.0000 0.638226
\(983\) −28.0000 −0.893061 −0.446531 0.894768i \(-0.647341\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 60.0000 1.91079
\(987\) −4.00000 −0.127321
\(988\) 0 0
\(989\) −64.0000 −2.03508
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 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 1050.2.a.q.1.1 1
3.2 odd 2 3150.2.a.t.1.1 1
4.3 odd 2 8400.2.a.m.1.1 1
5.2 odd 4 1050.2.g.f.799.2 2
5.3 odd 4 1050.2.g.f.799.1 2
5.4 even 2 210.2.a.a.1.1 1
7.6 odd 2 7350.2.a.bo.1.1 1
15.2 even 4 3150.2.g.t.2899.1 2
15.8 even 4 3150.2.g.t.2899.2 2
15.14 odd 2 630.2.a.i.1.1 1
20.19 odd 2 1680.2.a.o.1.1 1
35.4 even 6 1470.2.i.t.961.1 2
35.9 even 6 1470.2.i.t.361.1 2
35.19 odd 6 1470.2.i.n.361.1 2
35.24 odd 6 1470.2.i.n.961.1 2
35.34 odd 2 1470.2.a.g.1.1 1
40.19 odd 2 6720.2.a.z.1.1 1
40.29 even 2 6720.2.a.cg.1.1 1
60.59 even 2 5040.2.a.bg.1.1 1
105.104 even 2 4410.2.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.a.1.1 1 5.4 even 2
630.2.a.i.1.1 1 15.14 odd 2
1050.2.a.q.1.1 1 1.1 even 1 trivial
1050.2.g.f.799.1 2 5.3 odd 4
1050.2.g.f.799.2 2 5.2 odd 4
1470.2.a.g.1.1 1 35.34 odd 2
1470.2.i.n.361.1 2 35.19 odd 6
1470.2.i.n.961.1 2 35.24 odd 6
1470.2.i.t.361.1 2 35.9 even 6
1470.2.i.t.961.1 2 35.4 even 6
1680.2.a.o.1.1 1 20.19 odd 2
3150.2.a.t.1.1 1 3.2 odd 2
3150.2.g.t.2899.1 2 15.2 even 4
3150.2.g.t.2899.2 2 15.8 even 4
4410.2.a.bc.1.1 1 105.104 even 2
5040.2.a.bg.1.1 1 60.59 even 2
6720.2.a.z.1.1 1 40.19 odd 2
6720.2.a.cg.1.1 1 40.29 even 2
7350.2.a.bo.1.1 1 7.6 odd 2
8400.2.a.m.1.1 1 4.3 odd 2