Properties

Label 8470.2.a.t.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
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 \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -7.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} +6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -7.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{42} -1.00000 q^{43} -1.00000 q^{45} +6.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} +3.00000 q^{53} +4.00000 q^{54} +1.00000 q^{56} +14.0000 q^{57} -6.00000 q^{58} -3.00000 q^{59} +2.00000 q^{60} +11.0000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +5.00000 q^{67} +3.00000 q^{68} -12.0000 q^{69} -1.00000 q^{70} -15.0000 q^{71} +1.00000 q^{72} +11.0000 q^{73} -10.0000 q^{74} -2.00000 q^{75} -7.00000 q^{76} -4.00000 q^{78} -1.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -2.00000 q^{84} -3.00000 q^{85} -1.00000 q^{86} +12.0000 q^{87} +12.0000 q^{89} -1.00000 q^{90} +2.00000 q^{91} +6.00000 q^{92} -16.0000 q^{93} +7.00000 q^{95} -2.00000 q^{96} -1.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 4.00000 0.769800
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −7.00000 −1.13555
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.00000 −0.308607
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 14.0000 1.85435
\(58\) −6.00000 −0.787839
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 2.00000 0.258199
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 8.00000 1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 3.00000 0.363803
\(69\) −12.0000 −1.44463
\(70\) −1.00000 −0.119523
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −10.0000 −1.16248
\(75\) −2.00000 −0.230940
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.00000 −0.218218
\(85\) −3.00000 −0.325396
\(86\) −1.00000 −0.107833
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) 6.00000 0.625543
\(93\) −16.0000 −1.65912
\(94\) 0 0
\(95\) 7.00000 0.718185
\(96\) −2.00000 −0.204124
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 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
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 2.00000 0.196116
\(105\) 2.00000 0.195180
\(106\) 3.00000 0.291386
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 4.00000 0.384900
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 1.00000 0.0944911
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 14.0000 1.31122
\(115\) −6.00000 −0.559503
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) −3.00000 −0.276172
\(119\) 3.00000 0.275010
\(120\) 2.00000 0.182574
\(121\) 0 0
\(122\) 11.0000 0.995893
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) −22.0000 −1.95218 −0.976092 0.217357i \(-0.930256\pi\)
−0.976092 + 0.217357i \(0.930256\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 5.00000 0.431934
\(135\) −4.00000 −0.344265
\(136\) 3.00000 0.257248
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −12.0000 −1.02151
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −15.0000 −1.25877
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 11.0000 0.910366
\(147\) −2.00000 −0.164957
\(148\) −10.0000 −0.821995
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −2.00000 −0.163299
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −7.00000 −0.567775
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −4.00000 −0.320256
\(157\) −16.0000 −1.27694 −0.638470 0.769647i \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) −1.00000 −0.0795557
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 6.00000 0.472866
\(162\) −11.0000 −0.864242
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) −3.00000 −0.230089
\(171\) −7.00000 −0.535303
\(172\) −1.00000 −0.0762493
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 12.0000 0.909718
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 12.0000 0.899438
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 2.00000 0.148250
\(183\) −22.0000 −1.62629
\(184\) 6.00000 0.442326
\(185\) 10.0000 0.735215
\(186\) −16.0000 −1.17318
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 7.00000 0.507833
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −2.00000 −0.144338
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −1.00000 −0.0717958
\(195\) 4.00000 0.286446
\(196\) 1.00000 0.0714286
\(197\) −27.0000 −1.92367 −0.961835 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 1.00000 0.0707107
\(201\) −10.0000 −0.705346
\(202\) −6.00000 −0.422159
\(203\) −6.00000 −0.421117
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −13.0000 −0.905753
\(207\) 6.00000 0.417029
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 3.00000 0.206041
\(213\) 30.0000 2.05557
\(214\) 9.00000 0.615227
\(215\) 1.00000 0.0681994
\(216\) 4.00000 0.272166
\(217\) 8.00000 0.543075
\(218\) −4.00000 −0.270914
\(219\) −22.0000 −1.48662
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 20.0000 1.34231
\(223\) 23.0000 1.54019 0.770097 0.637927i \(-0.220208\pi\)
0.770097 + 0.637927i \(0.220208\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 14.0000 0.927173
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) 2.00000 0.129914
\(238\) 3.00000 0.194461
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 2.00000 0.129099
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 11.0000 0.704203
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −14.0000 −0.890799
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −22.0000 −1.38040
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) 2.00000 0.124515
\(259\) −10.0000 −0.621370
\(260\) −2.00000 −0.124035
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) −3.00000 −0.184289
\(266\) −7.00000 −0.429198
\(267\) −24.0000 −1.46878
\(268\) 5.00000 0.305424
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) −4.00000 −0.243432
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 3.00000 0.181902
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 11.0000 0.659736
\(279\) 8.00000 0.478947
\(280\) −1.00000 −0.0597614
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) −15.0000 −0.890086
\(285\) −14.0000 −0.829288
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 6.00000 0.352332
\(291\) 2.00000 0.117242
\(292\) 11.0000 0.643726
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) −2.00000 −0.116642
\(295\) 3.00000 0.174667
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) −2.00000 −0.115470
\(301\) −1.00000 −0.0576390
\(302\) 8.00000 0.460348
\(303\) 12.0000 0.689382
\(304\) −7.00000 −0.401478
\(305\) −11.0000 −0.629858
\(306\) 3.00000 0.171499
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 0 0
\(309\) 26.0000 1.47909
\(310\) −8.00000 −0.454369
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) −4.00000 −0.226455
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −16.0000 −0.902932
\(315\) −1.00000 −0.0563436
\(316\) −1.00000 −0.0562544
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −18.0000 −1.00466
\(322\) 6.00000 0.334367
\(323\) −21.0000 −1.16847
\(324\) −11.0000 −0.611111
\(325\) 2.00000 0.110940
\(326\) 11.0000 0.609234
\(327\) 8.00000 0.442401
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 21.0000 1.14907
\(335\) −5.00000 −0.273179
\(336\) −2.00000 −0.109109
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) −24.0000 −1.30350
\(340\) −3.00000 −0.162698
\(341\) 0 0
\(342\) −7.00000 −0.378517
\(343\) 1.00000 0.0539949
\(344\) −1.00000 −0.0539164
\(345\) 12.0000 0.646058
\(346\) 12.0000 0.645124
\(347\) −27.0000 −1.44944 −0.724718 0.689046i \(-0.758030\pi\)
−0.724718 + 0.689046i \(0.758030\pi\)
\(348\) 12.0000 0.643268
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 1.00000 0.0534522
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 6.00000 0.318896
\(355\) 15.0000 0.796117
\(356\) 12.0000 0.635999
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 30.0000 1.57895
\(362\) 11.0000 0.578147
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −11.0000 −0.575766
\(366\) −22.0000 −1.14996
\(367\) −7.00000 −0.365397 −0.182699 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) 3.00000 0.155752
\(372\) −16.0000 −0.829561
\(373\) 5.00000 0.258890 0.129445 0.991587i \(-0.458680\pi\)
0.129445 + 0.991587i \(0.458680\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 4.00000 0.205738
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 7.00000 0.359092
\(381\) 44.0000 2.25419
\(382\) 24.0000 1.22795
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) −1.00000 −0.0508329
\(388\) −1.00000 −0.0507673
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 4.00000 0.202548
\(391\) 18.0000 0.910299
\(392\) 1.00000 0.0505076
\(393\) 24.0000 1.21064
\(394\) −27.0000 −1.36024
\(395\) 1.00000 0.0503155
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 14.0000 0.701757
\(399\) 14.0000 0.700877
\(400\) 1.00000 0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −10.0000 −0.498755
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) 11.0000 0.546594
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −13.0000 −0.640464
\(413\) −3.00000 −0.147620
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 2.00000 0.0975900
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 3.00000 0.145693
\(425\) 3.00000 0.145521
\(426\) 30.0000 1.45350
\(427\) 11.0000 0.532327
\(428\) 9.00000 0.435031
\(429\) 0 0
\(430\) 1.00000 0.0482243
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 4.00000 0.192450
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 8.00000 0.384012
\(435\) −12.0000 −0.575356
\(436\) −4.00000 −0.191565
\(437\) −42.0000 −2.00913
\(438\) −22.0000 −1.05120
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 6.00000 0.285391
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 20.0000 0.949158
\(445\) −12.0000 −0.568855
\(446\) 23.0000 1.08908
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −3.00000 −0.141579 −0.0707894 0.997491i \(-0.522552\pi\)
−0.0707894 + 0.997491i \(0.522552\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) −16.0000 −0.751746
\(454\) 12.0000 0.563188
\(455\) −2.00000 −0.0937614
\(456\) 14.0000 0.655610
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 5.00000 0.233635
\(459\) 12.0000 0.560112
\(460\) −6.00000 −0.279751
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) −6.00000 −0.278543
\(465\) 16.0000 0.741982
\(466\) 12.0000 0.555889
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 2.00000 0.0924500
\(469\) 5.00000 0.230879
\(470\) 0 0
\(471\) 32.0000 1.47448
\(472\) −3.00000 −0.138086
\(473\) 0 0
\(474\) 2.00000 0.0918630
\(475\) −7.00000 −0.321182
\(476\) 3.00000 0.137505
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 2.00000 0.0912871
\(481\) −20.0000 −0.911922
\(482\) 8.00000 0.364390
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) 1.00000 0.0454077
\(486\) 10.0000 0.453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 11.0000 0.497947
\(489\) −22.0000 −0.994874
\(490\) −1.00000 −0.0451754
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) −14.0000 −0.629890
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −15.0000 −0.672842
\(498\) 0 0
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −42.0000 −1.87642
\(502\) 15.0000 0.669483
\(503\) 33.0000 1.47140 0.735699 0.677309i \(-0.236854\pi\)
0.735699 + 0.677309i \(0.236854\pi\)
\(504\) 1.00000 0.0445435
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) −22.0000 −0.976092
\(509\) 33.0000 1.46270 0.731350 0.682003i \(-0.238891\pi\)
0.731350 + 0.682003i \(0.238891\pi\)
\(510\) 6.00000 0.265684
\(511\) 11.0000 0.486611
\(512\) 1.00000 0.0441942
\(513\) −28.0000 −1.23623
\(514\) 21.0000 0.926270
\(515\) 13.0000 0.572848
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) −10.0000 −0.439375
\(519\) −24.0000 −1.05348
\(520\) −2.00000 −0.0877058
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −6.00000 −0.262613
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) −12.0000 −0.524222
\(525\) −2.00000 −0.0872872
\(526\) 18.0000 0.784837
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −3.00000 −0.130312
\(531\) −3.00000 −0.130189
\(532\) −7.00000 −0.303488
\(533\) 0 0
\(534\) −24.0000 −1.03858
\(535\) −9.00000 −0.389104
\(536\) 5.00000 0.215967
\(537\) 0 0
\(538\) −21.0000 −0.905374
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 2.00000 0.0859074
\(543\) −22.0000 −0.944110
\(544\) 3.00000 0.128624
\(545\) 4.00000 0.171341
\(546\) −4.00000 −0.171184
\(547\) −19.0000 −0.812381 −0.406191 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) 0 0
\(549\) 11.0000 0.469469
\(550\) 0 0
\(551\) 42.0000 1.78926
\(552\) −12.0000 −0.510754
\(553\) −1.00000 −0.0425243
\(554\) −10.0000 −0.424859
\(555\) −20.0000 −0.848953
\(556\) 11.0000 0.466504
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 8.00000 0.338667
\(559\) −2.00000 −0.0845910
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 3.00000 0.126547
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 32.0000 1.34506
\(567\) −11.0000 −0.461957
\(568\) −15.0000 −0.629386
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) −14.0000 −0.586395
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) −48.0000 −2.00523
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) −8.00000 −0.332756
\(579\) −40.0000 −1.66234
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) 0 0
\(584\) 11.0000 0.455183
\(585\) −2.00000 −0.0826898
\(586\) −24.0000 −0.991431
\(587\) −30.0000 −1.23823 −0.619116 0.785299i \(-0.712509\pi\)
−0.619116 + 0.785299i \(0.712509\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −56.0000 −2.30744
\(590\) 3.00000 0.123508
\(591\) 54.0000 2.22126
\(592\) −10.0000 −0.410997
\(593\) −33.0000 −1.35515 −0.677574 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) −28.0000 −1.14596
\(598\) 12.0000 0.490716
\(599\) 39.0000 1.59350 0.796748 0.604311i \(-0.206552\pi\)
0.796748 + 0.604311i \(0.206552\pi\)
\(600\) −2.00000 −0.0816497
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) −1.00000 −0.0407570
\(603\) 5.00000 0.203616
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) 17.0000 0.690009 0.345004 0.938601i \(-0.387877\pi\)
0.345004 + 0.938601i \(0.387877\pi\)
\(608\) −7.00000 −0.283887
\(609\) 12.0000 0.486265
\(610\) −11.0000 −0.445377
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 26.0000 1.04587
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −8.00000 −0.321288
\(621\) 24.0000 0.963087
\(622\) −12.0000 −0.481156
\(623\) 12.0000 0.480770
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) −30.0000 −1.19618
\(630\) −1.00000 −0.0398410
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −16.0000 −0.635943
\(634\) 3.00000 0.119145
\(635\) 22.0000 0.873043
\(636\) −6.00000 −0.237915
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) −15.0000 −0.593391
\(640\) −1.00000 −0.0395285
\(641\) −3.00000 −0.118493 −0.0592464 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) −18.0000 −0.710403
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 6.00000 0.236433
\(645\) −2.00000 −0.0787499
\(646\) −21.0000 −0.826234
\(647\) 9.00000 0.353827 0.176913 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) −16.0000 −0.627089
\(652\) 11.0000 0.430793
\(653\) 15.0000 0.586995 0.293498 0.955960i \(-0.405181\pi\)
0.293498 + 0.955960i \(0.405181\pi\)
\(654\) 8.00000 0.312825
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 50.0000 1.94477 0.972387 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) −10.0000 −0.388661
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) 7.00000 0.271448
\(666\) −10.0000 −0.387492
\(667\) −36.0000 −1.39393
\(668\) 21.0000 0.812514
\(669\) −46.0000 −1.77846
\(670\) −5.00000 −0.193167
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) −22.0000 −0.847408
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −24.0000 −0.921714
\(679\) −1.00000 −0.0383765
\(680\) −3.00000 −0.115045
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) −7.00000 −0.267652
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −10.0000 −0.381524
\(688\) −1.00000 −0.0381246
\(689\) 6.00000 0.228582
\(690\) 12.0000 0.456832
\(691\) 35.0000 1.33146 0.665731 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) −27.0000 −1.02491
\(695\) −11.0000 −0.417254
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) 2.00000 0.0757011
\(699\) −24.0000 −0.907763
\(700\) 1.00000 0.0377964
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 8.00000 0.301941
\(703\) 70.0000 2.64010
\(704\) 0 0
\(705\) 0 0
\(706\) 21.0000 0.790345
\(707\) −6.00000 −0.225653
\(708\) 6.00000 0.225494
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 15.0000 0.562940
\(711\) −1.00000 −0.0375029
\(712\) 12.0000 0.449719
\(713\) 48.0000 1.79761
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 9.00000 0.335877
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −13.0000 −0.484145
\(722\) 30.0000 1.11648
\(723\) −16.0000 −0.595046
\(724\) 11.0000 0.408812
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) −11.0000 −0.407128
\(731\) −3.00000 −0.110959
\(732\) −22.0000 −0.813143
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −7.00000 −0.258375
\(735\) 2.00000 0.0737711
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 10.0000 0.367607
\(741\) 28.0000 1.02861
\(742\) 3.00000 0.110133
\(743\) −42.0000 −1.54083 −0.770415 0.637542i \(-0.779951\pi\)
−0.770415 + 0.637542i \(0.779951\pi\)
\(744\) −16.0000 −0.586588
\(745\) 0 0
\(746\) 5.00000 0.183063
\(747\) 0 0
\(748\) 0 0
\(749\) 9.00000 0.328853
\(750\) 2.00000 0.0730297
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) −30.0000 −1.09326
\(754\) −12.0000 −0.437014
\(755\) −8.00000 −0.291150
\(756\) 4.00000 0.145479
\(757\) −43.0000 −1.56286 −0.781431 0.623992i \(-0.785510\pi\)
−0.781431 + 0.623992i \(0.785510\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 7.00000 0.253917
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 44.0000 1.59395
\(763\) −4.00000 −0.144810
\(764\) 24.0000 0.868290
\(765\) −3.00000 −0.108465
\(766\) 9.00000 0.325183
\(767\) −6.00000 −0.216647
\(768\) −2.00000 −0.0721688
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) −42.0000 −1.51259
\(772\) 20.0000 0.719816
\(773\) −48.0000 −1.72644 −0.863220 0.504828i \(-0.831556\pi\)
−0.863220 + 0.504828i \(0.831556\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 8.00000 0.287368
\(776\) −1.00000 −0.0358979
\(777\) 20.0000 0.717496
\(778\) 12.0000 0.430221
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 18.0000 0.643679
\(783\) −24.0000 −0.857690
\(784\) 1.00000 0.0357143
\(785\) 16.0000 0.571064
\(786\) 24.0000 0.856052
\(787\) 14.0000 0.499046 0.249523 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) −27.0000 −0.961835
\(789\) −36.0000 −1.28163
\(790\) 1.00000 0.0355784
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) 20.0000 0.709773
\(795\) 6.00000 0.212798
\(796\) 14.0000 0.496217
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 14.0000 0.495595
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 12.0000 0.423999
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) −10.0000 −0.352673
\(805\) −6.00000 −0.211472
\(806\) 16.0000 0.563576
\(807\) 42.0000 1.47847
\(808\) −6.00000 −0.211079
\(809\) 27.0000 0.949269 0.474635 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(810\) 11.0000 0.386501
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −6.00000 −0.210559
\(813\) −4.00000 −0.140286
\(814\) 0 0
\(815\) −11.0000 −0.385313
\(816\) −6.00000 −0.210042
\(817\) 7.00000 0.244899
\(818\) 20.0000 0.699284
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) −46.0000 −1.60346 −0.801730 0.597687i \(-0.796087\pi\)
−0.801730 + 0.597687i \(0.796087\pi\)
\(824\) −13.0000 −0.452876
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 6.00000 0.208514
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) 2.00000 0.0693375
\(833\) 3.00000 0.103944
\(834\) −22.0000 −0.761798
\(835\) −21.0000 −0.726735
\(836\) 0 0
\(837\) 32.0000 1.10608
\(838\) −9.00000 −0.310900
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) 20.0000 0.689246
\(843\) −6.00000 −0.206651
\(844\) 8.00000 0.275371
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 3.00000 0.103020
\(849\) −64.0000 −2.19647
\(850\) 3.00000 0.102899
\(851\) −60.0000 −2.05677
\(852\) 30.0000 1.02778
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 11.0000 0.376412
\(855\) 7.00000 0.239395
\(856\) 9.00000 0.307614
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 17.0000 0.580033 0.290016 0.957022i \(-0.406339\pi\)
0.290016 + 0.957022i \(0.406339\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) −33.0000 −1.12398
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 4.00000 0.136083
\(865\) −12.0000 −0.408012
\(866\) −22.0000 −0.747590
\(867\) 16.0000 0.543388
\(868\) 8.00000 0.271538
\(869\) 0 0
\(870\) −12.0000 −0.406838
\(871\) 10.0000 0.338837
\(872\) −4.00000 −0.135457
\(873\) −1.00000 −0.0338449
\(874\) −42.0000 −1.42067
\(875\) −1.00000 −0.0338062
\(876\) −22.0000 −0.743311
\(877\) −37.0000 −1.24940 −0.624701 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(878\) 14.0000 0.472477
\(879\) 48.0000 1.61900
\(880\) 0 0
\(881\) −48.0000 −1.61716 −0.808581 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 1.00000 0.0336718
\(883\) 11.0000 0.370179 0.185090 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) 6.00000 0.201802
\(885\) −6.00000 −0.201688
\(886\) 12.0000 0.403148
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 20.0000 0.671156
\(889\) −22.0000 −0.737856
\(890\) −12.0000 −0.402241
\(891\) 0 0
\(892\) 23.0000 0.770097
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −24.0000 −0.801337
\(898\) −3.00000 −0.100111
\(899\) −48.0000 −1.60089
\(900\) 1.00000 0.0333333
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 2.00000 0.0665558
\(904\) 12.0000 0.399114
\(905\) −11.0000 −0.365652
\(906\) −16.0000 −0.531564
\(907\) 41.0000 1.36138 0.680691 0.732570i \(-0.261680\pi\)
0.680691 + 0.732570i \(0.261680\pi\)
\(908\) 12.0000 0.398234
\(909\) −6.00000 −0.199007
\(910\) −2.00000 −0.0662994
\(911\) 39.0000 1.29213 0.646064 0.763283i \(-0.276414\pi\)
0.646064 + 0.763283i \(0.276414\pi\)
\(912\) 14.0000 0.463586
\(913\) 0 0
\(914\) −28.0000 −0.926158
\(915\) 22.0000 0.727298
\(916\) 5.00000 0.165205
\(917\) −12.0000 −0.396275
\(918\) 12.0000 0.396059
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) −6.00000 −0.197814
\(921\) −28.0000 −0.922631
\(922\) 3.00000 0.0987997
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 26.0000 0.854413
\(927\) −13.0000 −0.426976
\(928\) −6.00000 −0.196960
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 16.0000 0.524661
\(931\) −7.00000 −0.229416
\(932\) 12.0000 0.393073
\(933\) 24.0000 0.785725
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 5.00000 0.163256
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) 21.0000 0.684580 0.342290 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) 32.0000 1.04262
\(943\) 0 0
\(944\) −3.00000 −0.0976417
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 2.00000 0.0649570
\(949\) 22.0000 0.714150
\(950\) −7.00000 −0.227110
\(951\) −6.00000 −0.194563
\(952\) 3.00000 0.0972306
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 3.00000 0.0971286
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) −20.0000 −0.644826
\(963\) 9.00000 0.290021
\(964\) 8.00000 0.257663
\(965\) −20.0000 −0.643823
\(966\) −12.0000 −0.386094
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) 0 0
\(969\) 42.0000 1.34923
\(970\) 1.00000 0.0321081
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 10.0000 0.320750
\(973\) 11.0000 0.352644
\(974\) −4.00000 −0.128168
\(975\) −4.00000 −0.128103
\(976\) 11.0000 0.352101
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −22.0000 −0.703482
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −4.00000 −0.127710
\(982\) 24.0000 0.765871
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 0 0
\(985\) 27.0000 0.860292
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) −14.0000 −0.445399
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −55.0000 −1.74713 −0.873566 0.486705i \(-0.838199\pi\)
−0.873566 + 0.486705i \(0.838199\pi\)
\(992\) 8.00000 0.254000
\(993\) 20.0000 0.634681
\(994\) −15.0000 −0.475771
\(995\) −14.0000 −0.443830
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 44.0000 1.39280
\(999\) −40.0000 −1.26554
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 8470.2.a.t.1.1 yes 1
11.10 odd 2 8470.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.b.1.1 1 11.10 odd 2
8470.2.a.t.1.1 yes 1 1.1 even 1 trivial