Properties

Label 8470.2.a.b.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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) −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 \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −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 \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 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 \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) −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 \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 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 \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) −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 \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −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 \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 4.00000 0.384900
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 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 \(-0.0697436\pi\)
0.976092 + 0.217357i \(0.0697436\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) −2.00000 −0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 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 \(-0.654487\pi\)
−0.466504 + 0.884519i \(0.654487\pi\)
\(140\) 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 \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −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 \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) −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 \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 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 \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 1.00000 0.0717958
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) 27.0000 1.92367 0.961835 0.273629i \(-0.0882242\pi\)
0.961835 + 0.273629i \(0.0882242\pi\)
\(198\) 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 \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 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 \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −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 \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 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 \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 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 \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 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 \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) −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 \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 11.0000 0.659736
\(279\) 8.00000 0.478947
\(280\) −1.00000 −0.0597614
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) −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 \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 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 \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 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 \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 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 \(-0.241970\pi\)
0.724718 + 0.689046i \(0.241970\pi\)
\(348\) −12.0000 −0.643268
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 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 \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 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 \(-0.541320\pi\)
−0.129445 + 0.991587i \(0.541320\pi\)
\(374\) 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 \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 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 \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 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 \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 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 \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) −5.00000 −0.233635
\(459\) −12.0000 −0.560112
\(460\) −6.00000 −0.279751
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 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 \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) −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 \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 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 \(-0.763146\pi\)
−0.735699 + 0.677309i \(0.763146\pi\)
\(504\) 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 \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 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 \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 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 \(-0.366857\pi\)
0.406191 + 0.913788i \(0.366857\pi\)
\(548\) 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 \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −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 \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 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 \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) −14.0000 −0.586395
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 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 \(-0.263031\pi\)
0.677574 + 0.735455i \(0.263031\pi\)
\(594\) 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 \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 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 \(-0.612123\pi\)
−0.345004 + 0.938601i \(0.612123\pi\)
\(608\) −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 \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) 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 \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 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 \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) −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 \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 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 \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 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 \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 10.0000 0.367607
\(741\) 28.0000 1.02861
\(742\) 3.00000 0.110133
\(743\) 42.0000 1.54083 0.770415 0.637542i \(-0.220049\pi\)
0.770415 + 0.637542i \(0.220049\pi\)
\(744\) 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 \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 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 \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 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 \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 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 \(-0.657420\pi\)
−0.474635 + 0.880183i \(0.657420\pi\)
\(810\) −11.0000 −0.386501
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −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 \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 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 \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 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 \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −11.0000 −0.376412
\(855\) −7.00000 −0.239395
\(856\) 9.00000 0.307614
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 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 \(-0.285221\pi\)
0.624701 + 0.780864i \(0.285221\pi\)
\(878\) 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 \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 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 \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 5.00000 0.163256
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) −21.0000 −0.684580 −0.342290 0.939594i \(-0.611203\pi\)
−0.342290 + 0.939594i \(0.611203\pi\)
\(942\) −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 \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) −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 \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 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 \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −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.b.1.1 1
11.10 odd 2 8470.2.a.t.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.b.1.1 1 1.1 even 1 trivial
8470.2.a.t.1.1 yes 1 11.10 odd 2