Properties

Label 1470.2.a.i.1.1
Level $1470$
Weight $2$
Character 1470.1
Self dual yes
Analytic conductor $11.738$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -6.00000 q^{22} -1.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -8.00000 q^{29} -1.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} +6.00000 q^{39} -1.00000 q^{40} -10.0000 q^{41} -6.00000 q^{43} +6.00000 q^{44} +1.00000 q^{45} -2.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} +6.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} +6.00000 q^{55} -4.00000 q^{57} +8.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +14.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -6.00000 q^{66} +14.0000 q^{67} +8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -4.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -6.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -8.00000 q^{83} +6.00000 q^{86} -8.00000 q^{87} -6.00000 q^{88} -18.0000 q^{89} -1.00000 q^{90} +2.00000 q^{93} +2.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 4.00000 0.648886
\(39\) 6.00000 0.960769
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 6.00000 0.904534
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −1.00000 −0.136083
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 8.00000 1.05045
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) −6.00000 −0.738549
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −4.00000 −0.464991
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −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
\(82\) 10.0000 1.10432
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) −8.00000 −0.857690
\(88\) −6.00000 −0.639602
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 2.00000 0.206284
\(95\) −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 \(0.467625\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 1.00000 0.100000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −6.00000 −0.588348
\(105\) 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
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −6.00000 −0.572078
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 6.00000 0.554700
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 25.0000 2.27273
\(122\) −14.0000 −1.26750
\(123\) −10.0000 −0.901670
\(124\) 2.00000 0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.00000 −0.528271
\(130\) −6.00000 −0.526235
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) −14.0000 −1.20942
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) −8.00000 −0.671345
\(143\) 36.0000 3.01047
\(144\) 1.00000 0.0833333
\(145\) −8.00000 −0.664364
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −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
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 6.00000 0.480384
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 8.00000 0.636446
\(159\) 10.0000 0.793052
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −10.0000 −0.780869
\(165\) 6.00000 0.467099
\(166\) 8.00000 0.620920
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −6.00000 −0.457496
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) −4.00000 −0.300658
\(178\) 18.0000 1.34916
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 1.00000 0.0745356
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −2.00000 −0.143592
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −6.00000 −0.426401
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 14.0000 0.987484
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 10.0000 0.686803
\(213\) 8.00000 0.548151
\(214\) −4.00000 −0.273434
\(215\) −6.00000 −0.409197
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 6.00000 0.405442
\(220\) 6.00000 0.404520
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 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 \(0.268975\pi\)
\(228\) −4.00000 −0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −6.00000 −0.392232
\(235\) −2.00000 −0.130466
\(236\) −4.00000 −0.260378
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 1.00000 0.0645497
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) −25.0000 −1.60706
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) −24.0000 −1.52708
\(248\) −2.00000 −0.127000
\(249\) −8.00000 −0.506979
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 6.00000 0.373544
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) −8.00000 −0.495188
\(262\) −12.0000 −0.741362
\(263\) −32.0000 −1.97320 −0.986602 0.163144i \(-0.947836\pi\)
−0.986602 + 0.163144i \(0.947836\pi\)
\(264\) −6.00000 −0.369274
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 14.0000 0.855186
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −18.0000 −1.09342 −0.546711 0.837321i \(-0.684120\pi\)
−0.546711 + 0.837321i \(0.684120\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −8.00000 −0.479808
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 2.00000 0.119098
\(283\) 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
\(286\) −36.0000 −2.12872
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 8.00000 0.469776
\(291\) 2.00000 0.117242
\(292\) 6.00000 0.351123
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) −4.00000 −0.232495
\(297\) 6.00000 0.348155
\(298\) 0 0
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −14.0000 −0.804279
\(304\) −4.00000 −0.229416
\(305\) 14.0000 0.801638
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) −2.00000 −0.113592
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) −6.00000 −0.339683
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) −10.0000 −0.560772
\(319\) −48.0000 −2.68748
\(320\) 1.00000 0.0559017
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) −2.00000 −0.110770
\(327\) 6.00000 0.331801
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) −6.00000 −0.330289
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −8.00000 −0.439057
\(333\) 4.00000 0.219199
\(334\) 18.0000 0.984916
\(335\) 14.0000 0.764902
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −23.0000 −1.25104
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) −8.00000 −0.428845
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) −6.00000 −0.319801
\(353\) 4.00000 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 4.00000 0.212598
\(355\) 8.00000 0.424596
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) −14.0000 −0.731792
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 2.00000 0.103142
\(377\) −48.0000 −2.47213
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −4.00000 −0.205196
\(381\) 4.00000 0.204926
\(382\) 20.0000 1.02329
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) −6.00000 −0.304997
\(388\) 2.00000 0.101535
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) −6.00000 −0.303822
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) −6.00000 −0.302276
\(395\) −8.00000 −0.402524
\(396\) 6.00000 0.301511
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 18.0000 0.902258
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −14.0000 −0.698257
\(403\) 12.0000 0.597763
\(404\) −14.0000 −0.696526
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 10.0000 0.493865
\(411\) 10.0000 0.493264
\(412\) −12.0000 −0.591198
\(413\) 0 0
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) −6.00000 −0.294174
\(417\) 8.00000 0.391762
\(418\) 24.0000 1.17388
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) −12.0000 −0.584151
\(423\) −2.00000 −0.0972433
\(424\) −10.0000 −0.485643
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 36.0000 1.73810
\(430\) 6.00000 0.289346
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) −8.00000 −0.383571
\(436\) 6.00000 0.287348
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) −6.00000 −0.286039
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 4.00000 0.189832
\(445\) −18.0000 −0.853282
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −60.0000 −2.82529
\(452\) 6.00000 0.282216
\(453\) −8.00000 −0.375873
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −8.00000 −0.371391
\(465\) 2.00000 0.0927478
\(466\) 10.0000 0.463241
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 2.00000 0.0922531
\(471\) −14.0000 −0.645086
\(472\) 4.00000 0.184115
\(473\) −36.0000 −1.65528
\(474\) 8.00000 0.367452
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 4.00000 0.182956
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 24.0000 1.09431
\(482\) 28.0000 1.27537
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 2.00000 0.0908153
\(486\) −1.00000 −0.0453609
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) −14.0000 −0.633750
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −22.0000 −0.992846 −0.496423 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) 24.0000 1.07981
\(495\) 6.00000 0.269680
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 8.00000 0.358489
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 1.00000 0.0447214
\(501\) −18.0000 −0.804181
\(502\) 20.0000 0.892644
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) 4.00000 0.177471
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 8.00000 0.352865
\(515\) −12.0000 −0.528783
\(516\) −6.00000 −0.264135
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) −6.00000 −0.263117
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 8.00000 0.350150
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) 0 0
\(528\) 6.00000 0.261116
\(529\) −23.0000 −1.00000
\(530\) −10.0000 −0.434372
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −60.0000 −2.59889
\(534\) 18.0000 0.778936
\(535\) 4.00000 0.172935
\(536\) −14.0000 −0.604708
\(537\) 2.00000 0.0863064
\(538\) −22.0000 −0.948487
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 18.0000 0.773166
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 30.0000 1.28271 0.641354 0.767245i \(-0.278373\pi\)
0.641354 + 0.767245i \(0.278373\pi\)
\(548\) 10.0000 0.427179
\(549\) 14.0000 0.597505
\(550\) −6.00000 −0.255841
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) 0 0
\(554\) 8.00000 0.339887
\(555\) 4.00000 0.169791
\(556\) 8.00000 0.339276
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) −2.00000 −0.0846668
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −2.00000 −0.0842152
\(565\) 6.00000 0.252422
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) −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
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 36.0000 1.50524
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 17.0000 0.707107
\(579\) 22.0000 0.914289
\(580\) −8.00000 −0.332182
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 60.0000 2.48495
\(584\) −6.00000 −0.248282
\(585\) 6.00000 0.248069
\(586\) −22.0000 −0.908812
\(587\) 32.0000 1.32078 0.660391 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 4.00000 0.164677
\(591\) 6.00000 0.246807
\(592\) 4.00000 0.164399
\(593\) −16.0000 −0.657041 −0.328521 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) 0 0
\(597\) −18.0000 −0.736691
\(598\) 0 0
\(599\) 44.0000 1.79779 0.898896 0.438163i \(-0.144371\pi\)
0.898896 + 0.438163i \(0.144371\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 12.0000 0.489490 0.244745 0.969587i \(-0.421296\pi\)
0.244745 + 0.969587i \(0.421296\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) −8.00000 −0.325515
\(605\) 25.0000 1.01639
\(606\) 14.0000 0.568711
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 36.0000 1.45403 0.727013 0.686624i \(-0.240908\pi\)
0.727013 + 0.686624i \(0.240908\pi\)
\(614\) 20.0000 0.807134
\(615\) −10.0000 −0.403239
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 12.0000 0.482711
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 1.00000 0.0400000
\(626\) −26.0000 −1.03917
\(627\) −24.0000 −0.958468
\(628\) −14.0000 −0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 8.00000 0.318223
\(633\) 12.0000 0.476957
\(634\) −10.0000 −0.397151
\(635\) 4.00000 0.158735
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 48.0000 1.90034
\(639\) 8.00000 0.316475
\(640\) −1.00000 −0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −4.00000 −0.157867
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) −46.0000 −1.80845 −0.904223 0.427060i \(-0.859549\pi\)
−0.904223 + 0.427060i \(0.859549\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.0000 −0.942082
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −6.00000 −0.234619
\(655\) 12.0000 0.468879
\(656\) −10.0000 −0.390434
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 6.00000 0.233550
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) −18.0000 −0.696441
\(669\) 16.0000 0.618596
\(670\) −14.0000 −0.540867
\(671\) 84.0000 3.24278
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 26.0000 1.00148
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) −12.0000 −0.459504
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −4.00000 −0.152944
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) −6.00000 −0.228748
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 8.00000 0.303457
\(696\) 8.00000 0.303239
\(697\) 0 0
\(698\) −2.00000 −0.0757011
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) −6.00000 −0.226455
\(703\) −16.0000 −0.603451
\(704\) 6.00000 0.226134
\(705\) −2.00000 −0.0753244
\(706\) −4.00000 −0.150542
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) −8.00000 −0.300235
\(711\) −8.00000 −0.300023
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) 2.00000 0.0747435
\(717\) −4.00000 −0.149383
\(718\) 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
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −28.0000 −1.04133
\(724\) 10.0000 0.371647
\(725\) −8.00000 −0.297113
\(726\) −25.0000 −0.927837
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) 0 0
\(732\) 14.0000 0.517455
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 0 0
\(737\) 84.0000 3.09418
\(738\) 10.0000 0.368105
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 4.00000 0.147043
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) 8.00000 0.292901
\(747\) −8.00000 −0.292705
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −2.00000 −0.0729325
\(753\) −20.0000 −0.728841
\(754\) 48.0000 1.74806
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −40.0000 −1.45382 −0.726912 0.686730i \(-0.759045\pi\)
−0.726912 + 0.686730i \(0.759045\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −4.00000 −0.144905
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) 14.0000 0.505841
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) 22.0000 0.791797
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 6.00000 0.215666
\(775\) 2.00000 0.0718421
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 40.0000 1.43315
\(780\) 6.00000 0.214834
\(781\) 48.0000 1.71758
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) −12.0000 −0.428026
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 6.00000 0.213741
\(789\) −32.0000 −1.13923
\(790\) 8.00000 0.284627
\(791\) 0 0
\(792\) −6.00000 −0.213201
\(793\) 84.0000 2.98293
\(794\) 14.0000 0.496841
\(795\) 10.0000 0.354663
\(796\) −18.0000 −0.637993
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −18.0000 −0.635999
\(802\) −2.00000 −0.0706225
\(803\) 36.0000 1.27041
\(804\) 14.0000 0.493742
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) 22.0000 0.774437
\(808\) 14.0000 0.492518
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) −18.0000 −0.631288
\(814\) −24.0000 −0.841200
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 16.0000 0.559427
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −10.0000 −0.348790
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 12.0000 0.418040
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) 0 0
\(829\) 22.0000 0.764092 0.382046 0.924143i \(-0.375220\pi\)
0.382046 + 0.924143i \(0.375220\pi\)
\(830\) 8.00000 0.277684
\(831\) −8.00000 −0.277517
\(832\) 6.00000 0.208013
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) −18.0000 −0.622916
\(836\) −24.0000 −0.830057
\(837\) 2.00000 0.0691301
\(838\) −12.0000 −0.414533
\(839\) −44.0000 −1.51905 −0.759524 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 34.0000 1.17172
\(843\) 6.00000 0.206651
\(844\) 12.0000 0.413057
\(845\) 23.0000 0.791224
\(846\) 2.00000 0.0687614
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) −4.00000 −0.136717
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) −36.0000 −1.22902
\(859\) 16.0000 0.545913 0.272956 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −6.00000 −0.204006
\(866\) −18.0000 −0.611665
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) −48.0000 −1.62829
\(870\) 8.00000 0.271225
\(871\) 84.0000 2.84623
\(872\) −6.00000 −0.203186
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) −10.0000 −0.337484
\(879\) 22.0000 0.742042
\(880\) 6.00000 0.202260
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 14.0000 0.471138 0.235569 0.971858i \(-0.424305\pi\)
0.235569 + 0.971858i \(0.424305\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 12.0000 0.403148
\(887\) 26.0000 0.872995 0.436497 0.899706i \(-0.356219\pi\)
0.436497 + 0.899706i \(0.356219\pi\)
\(888\) −4.00000 −0.134231
\(889\) 0 0
\(890\) 18.0000 0.603361
\(891\) 6.00000 0.201008
\(892\) 16.0000 0.535720
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 2.00000 0.0668526
\(896\) 0 0
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) −16.0000 −0.533630
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 60.0000 1.99778
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 10.0000 0.332411
\(906\) 8.00000 0.265782
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 20.0000 0.663723
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −4.00000 −0.132453
\(913\) −48.0000 −1.58857
\(914\) −38.0000 −1.25693
\(915\) 14.0000 0.462826
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 2.00000 0.0658665
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) −8.00000 −0.262896
\(927\) −12.0000 −0.394132
\(928\) 8.00000 0.262613
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) −2.00000 −0.0655826
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) −2.00000 −0.0652328
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 14.0000 0.456145
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 36.0000 1.17046
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −8.00000 −0.259828
\(949\) 36.0000 1.16861
\(950\) 4.00000 0.129777
\(951\) 10.0000 0.324272
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −10.0000 −0.323762
\(955\) −20.0000 −0.647185
\(956\) −4.00000 −0.129369
\(957\) −48.0000 −1.55162
\(958\) 36.0000 1.16311
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) −27.0000 −0.870968
\(962\) −24.0000 −0.773791
\(963\) 4.00000 0.128898
\(964\) −28.0000 −0.901819
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 6.00000 0.192154
\(976\) 14.0000 0.448129
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) −2.00000 −0.0639529
\(979\) −108.000 −3.45169
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 22.0000 0.702048
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 10.0000 0.318788
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 0 0
\(990\) −6.00000 −0.190693
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −18.0000 −0.570638
\(996\) −8.00000 −0.253490
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 32.0000 1.01294
\(999\) 4.00000 0.126554
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 1470.2.a.i.1.1 yes 1
3.2 odd 2 4410.2.a.v.1.1 1
5.4 even 2 7350.2.a.cf.1.1 1
7.2 even 3 1470.2.i.k.361.1 2
7.3 odd 6 1470.2.i.r.961.1 2
7.4 even 3 1470.2.i.k.961.1 2
7.5 odd 6 1470.2.i.r.361.1 2
7.6 odd 2 1470.2.a.c.1.1 1
21.20 even 2 4410.2.a.be.1.1 1
35.34 odd 2 7350.2.a.da.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1470.2.a.c.1.1 1 7.6 odd 2
1470.2.a.i.1.1 yes 1 1.1 even 1 trivial
1470.2.i.k.361.1 2 7.2 even 3
1470.2.i.k.961.1 2 7.4 even 3
1470.2.i.r.361.1 2 7.5 odd 6
1470.2.i.r.961.1 2 7.3 odd 6
4410.2.a.v.1.1 1 3.2 odd 2
4410.2.a.be.1.1 1 21.20 even 2
7350.2.a.cf.1.1 1 5.4 even 2
7350.2.a.da.1.1 1 35.34 odd 2