Properties

Label 8470.2.a.d.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $2$
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: \(2\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -8.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} +8.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} +2.00000 q^{42} -4.00000 q^{43} -1.00000 q^{45} +6.00000 q^{46} -6.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} -4.00000 q^{54} -1.00000 q^{56} +16.0000 q^{57} +8.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} +4.00000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{67} -2.00000 q^{68} +12.0000 q^{69} +1.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +10.0000 q^{74} -2.00000 q^{75} -8.00000 q^{76} -4.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -4.00000 q^{83} -2.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} +16.0000 q^{87} -2.00000 q^{89} +1.00000 q^{90} -2.00000 q^{91} -6.00000 q^{92} +16.0000 q^{93} +6.00000 q^{94} +8.00000 q^{95} +2.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 4.00000 0.769800
\(28\) 1.00000 0.188982
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −2.00000 −0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −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 \(0.807139\pi\)
\(38\) 8.00000 1.29777
\(39\) 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
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 16.0000 2.11925
\(58\) 8.00000 1.05045
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 2.00000 0.258199
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 8.00000 1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −2.00000 −0.242536
\(69\) 12.0000 1.44463
\(70\) 1.00000 0.119523
\(71\) 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 \(0.385800\pi\)
\(74\) 10.0000 1.16248
\(75\) −2.00000 −0.230940
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.00000 −0.218218
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 16.0000 1.71538
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 1.00000 0.105409
\(91\) −2.00000 −0.209657
\(92\) −6.00000 −0.625543
\(93\) 16.0000 1.65912
\(94\) 6.00000 0.618853
\(95\) 8.00000 0.820783
\(96\) 2.00000 0.204124
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) −4.00000 −0.396059
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 2.00000 0.196116
\(105\) 2.00000 0.195180
\(106\) 2.00000 0.194257
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 4.00000 0.384900
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 1.00000 0.0944911
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) −16.0000 −1.49854
\(115\) 6.00000 0.559503
\(116\) −8.00000 −0.742781
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) −2.00000 −0.183340
\(120\) −2.00000 −0.182574
\(121\) 0 0
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) −2.00000 −0.175412
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −2.00000 −0.172774
\(135\) −4.00000 −0.344265
\(136\) 2.00000 0.171499
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −12.0000 −1.02151
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 12.0000 1.01058
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) −6.00000 −0.496564
\(147\) −2.00000 −0.164957
\(148\) −10.0000 −0.821995
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 2.00000 0.163299
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 8.00000 0.648886
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 4.00000 0.320256
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 4.00000 0.318223
\(159\) 4.00000 0.317221
\(160\) 1.00000 0.0790569
\(161\) −6.00000 −0.472866
\(162\) 11.0000 0.864242
\(163\) 18.0000 1.40987 0.704934 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) −8.00000 −0.611775
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −16.0000 −1.21296
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 2.00000 0.149906
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 2.00000 0.148250
\(183\) −8.00000 −0.591377
\(184\) 6.00000 0.442326
\(185\) 10.0000 0.735215
\(186\) −16.0000 −1.17318
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 4.00000 0.290957
\(190\) −8.00000 −0.580381
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) −2.00000 −0.144338
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) −8.00000 −0.562878
\(203\) −8.00000 −0.561490
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) −6.00000 −0.417029
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −2.00000 −0.137361
\(213\) −16.0000 −1.09630
\(214\) 20.0000 1.36717
\(215\) 4.00000 0.272798
\(216\) −4.00000 −0.272166
\(217\) −8.00000 −0.543075
\(218\) 12.0000 0.812743
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −20.0000 −1.34231
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 18.0000 1.19734
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 16.0000 1.05963
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 2.00000 0.130744
\(235\) 6.00000 0.391397
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) 2.00000 0.129641
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 2.00000 0.129099
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 4.00000 0.256074
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 8.00000 0.508001
\(249\) 8.00000 0.506979
\(250\) 1.00000 0.0632456
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) −8.00000 −0.498058
\(259\) −10.0000 −0.621370
\(260\) 2.00000 0.124035
\(261\) −8.00000 −0.495188
\(262\) −20.0000 −1.23560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 8.00000 0.490511
\(267\) 4.00000 0.244796
\(268\) 2.00000 0.122169
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 4.00000 0.243432
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −2.00000 −0.121268
\(273\) 4.00000 0.242091
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) 1.00000 0.0597614
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) −12.0000 −0.714590
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 8.00000 0.474713
\(285\) −16.0000 −0.947758
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −8.00000 −0.469776
\(291\) 4.00000 0.234484
\(292\) 6.00000 0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 2.00000 0.116642
\(295\) −4.00000 −0.232889
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) 12.0000 0.693978
\(300\) −2.00000 −0.115470
\(301\) −4.00000 −0.230556
\(302\) 16.0000 0.920697
\(303\) −16.0000 −0.919176
\(304\) −8.00000 −0.458831
\(305\) −4.00000 −0.229039
\(306\) 2.00000 0.114332
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 28.0000 1.59286
\(310\) −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
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −10.0000 −0.564333
\(315\) −1.00000 −0.0563436
\(316\) −4.00000 −0.225018
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −4.00000 −0.224309
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 40.0000 2.23258
\(322\) 6.00000 0.334367
\(323\) 16.0000 0.890264
\(324\) −11.0000 −0.611111
\(325\) −2.00000 −0.110940
\(326\) −18.0000 −0.996928
\(327\) 24.0000 1.32720
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −4.00000 −0.219529
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) −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
\(339\) 36.0000 1.95525
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) −12.0000 −0.646058
\(346\) 14.0000 0.752645
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 16.0000 0.857690
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −8.00000 −0.427008
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 8.00000 0.425195
\(355\) −8.00000 −0.424596
\(356\) −2.00000 −0.106000
\(357\) 4.00000 0.211702
\(358\) −20.0000 −1.05703
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 1.00000 0.0527046
\(361\) 45.0000 2.36842
\(362\) −26.0000 −1.36653
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −6.00000 −0.314054
\(366\) 8.00000 0.418167
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) −10.0000 −0.519875
\(371\) −2.00000 −0.103835
\(372\) 16.0000 0.829561
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 6.00000 0.309426
\(377\) 16.0000 0.824042
\(378\) −4.00000 −0.205738
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 8.00000 0.410391
\(381\) 16.0000 0.819705
\(382\) −20.0000 −1.02329
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 4.00000 0.202548
\(391\) 12.0000 0.606866
\(392\) −1.00000 −0.0505076
\(393\) −40.0000 −2.01773
\(394\) −6.00000 −0.302276
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 24.0000 1.20301
\(399\) 16.0000 0.801002
\(400\) 1.00000 0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 4.00000 0.199502
\(403\) 16.0000 0.797017
\(404\) 8.00000 0.398015
\(405\) 11.0000 0.546594
\(406\) 8.00000 0.397033
\(407\) 0 0
\(408\) −4.00000 −0.198030
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −14.0000 −0.689730
\(413\) 4.00000 0.196827
\(414\) 6.00000 0.294884
\(415\) 4.00000 0.196352
\(416\) 2.00000 0.0980581
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 2.00000 0.0975900
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 12.0000 0.584151
\(423\) −6.00000 −0.291730
\(424\) 2.00000 0.0971286
\(425\) −2.00000 −0.0970143
\(426\) 16.0000 0.775203
\(427\) 4.00000 0.193574
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 4.00000 0.192450
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 8.00000 0.384012
\(435\) −16.0000 −0.767141
\(436\) −12.0000 −0.574696
\(437\) 48.0000 2.29615
\(438\) 12.0000 0.573382
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −4.00000 −0.190261
\(443\) 26.0000 1.23530 0.617649 0.786454i \(-0.288085\pi\)
0.617649 + 0.786454i \(0.288085\pi\)
\(444\) 20.0000 0.949158
\(445\) 2.00000 0.0948091
\(446\) 2.00000 0.0947027
\(447\) −8.00000 −0.378387
\(448\) 1.00000 0.0472456
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) 32.0000 1.50349
\(454\) 12.0000 0.563188
\(455\) 2.00000 0.0937614
\(456\) −16.0000 −0.749269
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) −6.00000 −0.280362
\(459\) −8.00000 −0.373408
\(460\) 6.00000 0.279751
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) −8.00000 −0.371391
\(465\) −16.0000 −0.741982
\(466\) 26.0000 1.20443
\(467\) −2.00000 −0.0925490 −0.0462745 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 2.00000 0.0923514
\(470\) −6.00000 −0.276759
\(471\) −20.0000 −0.921551
\(472\) −4.00000 −0.184115
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) −8.00000 −0.367065
\(476\) −2.00000 −0.0916698
\(477\) −2.00000 −0.0915737
\(478\) 20.0000 0.914779
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) −10.0000 −0.453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −4.00000 −0.181071
\(489\) −36.0000 −1.62798
\(490\) 1.00000 0.0451754
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 8.00000 0.358849
\(498\) −8.00000 −0.358489
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −8.00000 −0.357057
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) −8.00000 −0.354943
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 4.00000 0.177123
\(511\) 6.00000 0.265424
\(512\) −1.00000 −0.0441942
\(513\) −32.0000 −1.41283
\(514\) 22.0000 0.970378
\(515\) 14.0000 0.616914
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) 28.0000 1.22906
\(520\) −2.00000 −0.0877058
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 8.00000 0.350150
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 20.0000 0.873704
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −2.00000 −0.0868744
\(531\) 4.00000 0.173585
\(532\) −8.00000 −0.346844
\(533\) 0 0
\(534\) −4.00000 −0.173097
\(535\) 20.0000 0.864675
\(536\) −2.00000 −0.0863868
\(537\) −40.0000 −1.72613
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) −20.0000 −0.859867 −0.429934 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 8.00000 0.343629
\(543\) −52.0000 −2.23153
\(544\) 2.00000 0.0857493
\(545\) 12.0000 0.514024
\(546\) −4.00000 −0.171184
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 6.00000 0.256307
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 64.0000 2.72649
\(552\) −12.0000 −0.510754
\(553\) −4.00000 −0.170097
\(554\) −18.0000 −0.764747
\(555\) −20.0000 −0.848953
\(556\) −4.00000 −0.169638
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 8.00000 0.338667
\(559\) 8.00000 0.338364
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 12.0000 0.505291
\(565\) 18.0000 0.757266
\(566\) −12.0000 −0.504398
\(567\) −11.0000 −0.461957
\(568\) −8.00000 −0.335673
\(569\) −32.0000 −1.34151 −0.670755 0.741679i \(-0.734030\pi\)
−0.670755 + 0.741679i \(0.734030\pi\)
\(570\) 16.0000 0.670166
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) −40.0000 −1.67102
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 13.0000 0.540729
\(579\) 44.0000 1.82858
\(580\) 8.00000 0.332182
\(581\) −4.00000 −0.165948
\(582\) −4.00000 −0.165805
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 2.00000 0.0826898
\(586\) −6.00000 −0.247858
\(587\) −26.0000 −1.07313 −0.536567 0.843857i \(-0.680279\pi\)
−0.536567 + 0.843857i \(0.680279\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 64.0000 2.63707
\(590\) 4.00000 0.164677
\(591\) −12.0000 −0.493614
\(592\) −10.0000 −0.410997
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 4.00000 0.163846
\(597\) 48.0000 1.96451
\(598\) −12.0000 −0.490716
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 2.00000 0.0816497
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 4.00000 0.163028
\(603\) 2.00000 0.0814463
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 16.0000 0.649956
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 8.00000 0.324443
\(609\) 16.0000 0.648353
\(610\) 4.00000 0.161955
\(611\) 12.0000 0.485468
\(612\) −2.00000 −0.0808452
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −28.0000 −1.12633
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 8.00000 0.321288
\(621\) −24.0000 −0.963087
\(622\) 12.0000 0.481156
\(623\) −2.00000 −0.0801283
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 20.0000 0.797452
\(630\) 1.00000 0.0398410
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 4.00000 0.159111
\(633\) 24.0000 0.953914
\(634\) 22.0000 0.873732
\(635\) 8.00000 0.317470
\(636\) 4.00000 0.158610
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 1.00000 0.0395285
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −40.0000 −1.57867
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) −6.00000 −0.236433
\(645\) −8.00000 −0.315000
\(646\) −16.0000 −0.629512
\(647\) 38.0000 1.49393 0.746967 0.664861i \(-0.231509\pi\)
0.746967 + 0.664861i \(0.231509\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 16.0000 0.627089
\(652\) 18.0000 0.704934
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −24.0000 −0.938474
\(655\) −20.0000 −0.781465
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 6.00000 0.233904
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 8.00000 0.310929
\(663\) −8.00000 −0.310694
\(664\) 4.00000 0.155230
\(665\) 8.00000 0.310227
\(666\) 10.0000 0.387492
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 2.00000 0.0772667
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 22.0000 0.847408
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −36.0000 −1.38257
\(679\) −2.00000 −0.0767530
\(680\) −2.00000 −0.0766965
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) −18.0000 −0.688751 −0.344375 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) −8.00000 −0.305888
\(685\) −6.00000 −0.229248
\(686\) −1.00000 −0.0381802
\(687\) −12.0000 −0.457829
\(688\) −4.00000 −0.152499
\(689\) 4.00000 0.152388
\(690\) 12.0000 0.456832
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 4.00000 0.151729
\(696\) −16.0000 −0.606478
\(697\) 0 0
\(698\) 4.00000 0.151402
\(699\) 52.0000 1.96682
\(700\) 1.00000 0.0377964
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 8.00000 0.301941
\(703\) 80.0000 3.01726
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) −26.0000 −0.978523
\(707\) 8.00000 0.300871
\(708\) −8.00000 −0.300658
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 8.00000 0.300235
\(711\) −4.00000 −0.150012
\(712\) 2.00000 0.0749532
\(713\) 48.0000 1.79761
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 40.0000 1.49383
\(718\) −4.00000 −0.149279
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −14.0000 −0.521387
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) 6.00000 0.222070
\(731\) 8.00000 0.295891
\(732\) −8.00000 −0.295689
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 22.0000 0.812035
\(735\) 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
\(741\) −32.0000 −1.17555
\(742\) 2.00000 0.0734223
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) −16.0000 −0.586588
\(745\) −4.00000 −0.146549
\(746\) −22.0000 −0.805477
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −20.0000 −0.730784
\(750\) −2.00000 −0.0730297
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −6.00000 −0.218797
\(753\) −16.0000 −0.583072
\(754\) −16.0000 −0.582686
\(755\) 16.0000 0.582300
\(756\) 4.00000 0.145479
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −28.0000 −1.01500 −0.507500 0.861652i \(-0.669430\pi\)
−0.507500 + 0.861652i \(0.669430\pi\)
\(762\) −16.0000 −0.579619
\(763\) −12.0000 −0.434429
\(764\) 20.0000 0.723575
\(765\) 2.00000 0.0723102
\(766\) −18.0000 −0.650366
\(767\) −8.00000 −0.288863
\(768\) −2.00000 −0.0721688
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 44.0000 1.58462
\(772\) −22.0000 −0.791797
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 4.00000 0.143777
\(775\) −8.00000 −0.287368
\(776\) 2.00000 0.0717958
\(777\) 20.0000 0.717496
\(778\) 2.00000 0.0717035
\(779\) 0 0
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) −12.0000 −0.429119
\(783\) −32.0000 −1.14359
\(784\) 1.00000 0.0357143
\(785\) −10.0000 −0.356915
\(786\) 40.0000 1.42675
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) −10.0000 −0.354887
\(795\) −4.00000 −0.141865
\(796\) −24.0000 −0.850657
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) −16.0000 −0.566394
\(799\) 12.0000 0.424529
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 6.00000 0.211472
\(806\) −16.0000 −0.563576
\(807\) −36.0000 −1.26726
\(808\) −8.00000 −0.281439
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) −11.0000 −0.386501
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) −8.00000 −0.280745
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) −18.0000 −0.630512
\(816\) 4.00000 0.140028
\(817\) 32.0000 1.11954
\(818\) 20.0000 0.699284
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 12.0000 0.418548
\(823\) 54.0000 1.88232 0.941161 0.337959i \(-0.109737\pi\)
0.941161 + 0.337959i \(0.109737\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −6.00000 −0.208514
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) −4.00000 −0.138842
\(831\) −36.0000 −1.24883
\(832\) −2.00000 −0.0693375
\(833\) −2.00000 −0.0692959
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) −32.0000 −1.10608
\(838\) −24.0000 −0.829066
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 35.0000 1.20690
\(842\) 6.00000 0.206774
\(843\) 24.0000 0.826604
\(844\) −12.0000 −0.413057
\(845\) 9.00000 0.309609
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) −24.0000 −0.823678
\(850\) 2.00000 0.0685994
\(851\) 60.0000 2.05677
\(852\) −16.0000 −0.548151
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) −4.00000 −0.136877
\(855\) 8.00000 0.273594
\(856\) 20.0000 0.683586
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −4.00000 −0.136083
\(865\) 14.0000 0.476014
\(866\) 10.0000 0.339814
\(867\) 26.0000 0.883006
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 16.0000 0.542451
\(871\) −4.00000 −0.135535
\(872\) 12.0000 0.406371
\(873\) −2.00000 −0.0676897
\(874\) −48.0000 −1.62362
\(875\) −1.00000 −0.0338062
\(876\) −12.0000 −0.405442
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −40.0000 −1.34993
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 4.00000 0.134535
\(885\) 8.00000 0.268917
\(886\) −26.0000 −0.873487
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −20.0000 −0.671156
\(889\) −8.00000 −0.268311
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 48.0000 1.60626
\(894\) 8.00000 0.267560
\(895\) −20.0000 −0.668526
\(896\) −1.00000 −0.0334077
\(897\) −24.0000 −0.801337
\(898\) −2.00000 −0.0667409
\(899\) 64.0000 2.13452
\(900\) 1.00000 0.0333333
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 18.0000 0.598671
\(905\) −26.0000 −0.864269
\(906\) −32.0000 −1.06313
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) −12.0000 −0.398234
\(909\) 8.00000 0.265343
\(910\) −2.00000 −0.0662994
\(911\) 4.00000 0.132526 0.0662630 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 16.0000 0.529813
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 8.00000 0.264472
\(916\) 6.00000 0.198246
\(917\) 20.0000 0.660458
\(918\) 8.00000 0.264039
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −6.00000 −0.197814
\(921\) 40.0000 1.31804
\(922\) −32.0000 −1.05386
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 22.0000 0.722965
\(927\) −14.0000 −0.459820
\(928\) 8.00000 0.262613
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 16.0000 0.524661
\(931\) −8.00000 −0.262189
\(932\) −26.0000 −0.851658
\(933\) 24.0000 0.785725
\(934\) 2.00000 0.0654420
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) −2.00000 −0.0653023
\(939\) 44.0000 1.43589
\(940\) 6.00000 0.195698
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 20.0000 0.651635
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −14.0000 −0.454939 −0.227469 0.973785i \(-0.573045\pi\)
−0.227469 + 0.973785i \(0.573045\pi\)
\(948\) 8.00000 0.259828
\(949\) −12.0000 −0.389536
\(950\) 8.00000 0.259554
\(951\) 44.0000 1.42680
\(952\) 2.00000 0.0648204
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 2.00000 0.0647524
\(955\) −20.0000 −0.647185
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) −20.0000 −0.646171
\(959\) 6.00000 0.193750
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) −20.0000 −0.644826
\(963\) −20.0000 −0.644491
\(964\) 0 0
\(965\) 22.0000 0.708205
\(966\) −12.0000 −0.386094
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 0 0
\(969\) −32.0000 −1.02799
\(970\) −2.00000 −0.0642161
\(971\) 44.0000 1.41203 0.706014 0.708198i \(-0.250492\pi\)
0.706014 + 0.708198i \(0.250492\pi\)
\(972\) 10.0000 0.320750
\(973\) −4.00000 −0.128234
\(974\) 2.00000 0.0640841
\(975\) 4.00000 0.128103
\(976\) 4.00000 0.128037
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 36.0000 1.15115
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −12.0000 −0.383131
\(982\) −8.00000 −0.255290
\(983\) 42.0000 1.33959 0.669796 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) −16.0000 −0.509544
\(987\) 12.0000 0.381964
\(988\) 16.0000 0.509028
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 8.00000 0.254000
\(993\) 16.0000 0.507745
\(994\) −8.00000 −0.253745
\(995\) 24.0000 0.760851
\(996\) 8.00000 0.253490
\(997\) 34.0000 1.07679 0.538395 0.842692i \(-0.319031\pi\)
0.538395 + 0.842692i \(0.319031\pi\)
\(998\) 24.0000 0.759707
\(999\) −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.d.1.1 1
11.10 odd 2 8470.2.a.s.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.d.1.1 1 1.1 even 1 trivial
8470.2.a.s.1.1 yes 1 11.10 odd 2