Properties

Label 8470.2.a.e.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} +2.00000 q^{38} +4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{42} +4.00000 q^{43} +1.00000 q^{45} +6.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +12.0000 q^{51} -2.00000 q^{52} +12.0000 q^{53} -4.00000 q^{54} +1.00000 q^{56} +4.00000 q^{57} +12.0000 q^{59} -2.00000 q^{60} +10.0000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +12.0000 q^{69} +1.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -8.00000 q^{74} -2.00000 q^{75} -2.00000 q^{76} -4.00000 q^{78} +10.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +2.00000 q^{84} -6.00000 q^{85} -4.00000 q^{86} -18.0000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} -16.0000 q^{93} -2.00000 q^{95} +2.00000 q^{96} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 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
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 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
\(34\) 6.00000 1.02899
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 2.00000 0.324443
\(39\) 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 \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 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
\(51\) 12.0000 1.68034
\(52\) −2.00000 −0.277350
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −2.00000 −0.258199
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −8.00000 −1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 12.0000 1.44463
\(70\) 1.00000 0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −8.00000 −0.929981
\(75\) −2.00000 −0.230940
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 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
\(85\) −6.00000 −0.650791
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) −16.0000 −1.65912
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 2.00000 0.204124
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) −12.0000 −1.18818
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) 2.00000 0.195180
\(106\) −12.0000 −1.16554
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 4.00000 0.384900
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) −16.0000 −1.51865
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 6.00000 0.550019
\(120\) 2.00000 0.182574
\(121\) 0 0
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) 2.00000 0.175412
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 4.00000 0.345547
\(135\) 4.00000 0.344265
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −12.0000 −1.02151
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) −2.00000 −0.164957
\(148\) 8.00000 0.657596
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 2.00000 0.163299
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 2.00000 0.162221
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 4.00000 0.320256
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −10.0000 −0.795557
\(159\) −24.0000 −1.90332
\(160\) −1.00000 −0.0790569
\(161\) 6.00000 0.472866
\(162\) 11.0000 0.864242
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) −2.00000 −0.152944
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −24.0000 −1.80395
\(178\) 18.0000 1.34916
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −2.00000 −0.148250
\(183\) −20.0000 −1.47844
\(184\) 6.00000 0.442326
\(185\) 8.00000 0.588172
\(186\) 16.0000 1.17318
\(187\) 0 0
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 2.00000 0.145095
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −2.00000 −0.144338
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −8.00000 −0.574367
\(195\) 4.00000 0.286446
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −6.00000 −0.417029
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 12.0000 0.824163
\(213\) 24.0000 1.64445
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) −4.00000 −0.272166
\(217\) −8.00000 −0.543075
\(218\) 8.00000 0.541828
\(219\) 28.0000 1.89206
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 16.0000 1.07385
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 4.00000 0.264906
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −20.0000 −1.29914
\(238\) −6.00000 −0.388922
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) −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
\(244\) 10.0000 0.640184
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 8.00000 0.498058
\(259\) −8.00000 −0.497096
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) −2.00000 −0.122628
\(267\) 36.0000 2.20316
\(268\) −4.00000 −0.244339
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −4.00000 −0.243432
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −6.00000 −0.363803
\(273\) −4.00000 −0.242091
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −10.0000 −0.599760
\(279\) 8.00000 0.478947
\(280\) 1.00000 0.0597614
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −12.0000 −0.712069
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) −14.0000 −0.819288
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 2.00000 0.116642
\(295\) 12.0000 0.698667
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) −2.00000 −0.115470
\(301\) −4.00000 −0.230556
\(302\) −10.0000 −0.575435
\(303\) −36.0000 −2.06815
\(304\) −2.00000 −0.114708
\(305\) 10.0000 0.572598
\(306\) 6.00000 0.342997
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) −8.00000 −0.454369
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −4.00000 −0.226455
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) −2.00000 −0.112867
\(315\) −1.00000 −0.0563436
\(316\) 10.0000 0.562544
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 24.0000 1.34585
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 24.0000 1.33955
\(322\) −6.00000 −0.334367
\(323\) 12.0000 0.667698
\(324\) −11.0000 −0.611111
\(325\) −2.00000 −0.110940
\(326\) 4.00000 0.221540
\(327\) 16.0000 0.884802
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 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
\(339\) 12.0000 0.651751
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 12.0000 0.646058
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −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
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 24.0000 1.27559
\(355\) −12.0000 −0.636894
\(356\) −18.0000 −0.953998
\(357\) −12.0000 −0.635107
\(358\) 12.0000 0.634220
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −14.0000 −0.732793
\(366\) 20.0000 1.04542
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) −12.0000 −0.623009
\(372\) −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
\(376\) 0 0
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −2.00000 −0.102598
\(381\) −32.0000 −1.63941
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 4.00000 0.203331
\(388\) 8.00000 0.406138
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) −4.00000 −0.202548
\(391\) 36.0000 1.82060
\(392\) −1.00000 −0.0505076
\(393\) −36.0000 −1.81596
\(394\) 18.0000 0.906827
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −8.00000 −0.401004
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −8.00000 −0.399004
\(403\) −16.0000 −0.797017
\(404\) 18.0000 0.895533
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 0 0
\(408\) −12.0000 −0.594089
\(409\) −8.00000 −0.395575 −0.197787 0.980245i \(-0.563376\pi\)
−0.197787 + 0.980245i \(0.563376\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) −4.00000 −0.197066
\(413\) −12.0000 −0.590481
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 2.00000 0.0975900
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) −6.00000 −0.291043
\(426\) −24.0000 −1.16280
\(427\) −10.0000 −0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 4.00000 0.192450
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) 12.0000 0.574038
\(438\) −28.0000 −1.33789
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −16.0000 −0.759326
\(445\) −18.0000 −0.853282
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −20.0000 −0.939682
\(454\) −12.0000 −0.563188
\(455\) 2.00000 0.0937614
\(456\) −4.00000 −0.187317
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −26.0000 −1.21490
\(459\) −24.0000 −1.12022
\(460\) −6.00000 −0.279751
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) 0 0
\(465\) −16.0000 −0.741982
\(466\) −6.00000 −0.277945
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 20.0000 0.918630
\(475\) −2.00000 −0.0917663
\(476\) 6.00000 0.275010
\(477\) 12.0000 0.549442
\(478\) 18.0000 0.823301
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 2.00000 0.0912871
\(481\) −16.0000 −0.729537
\(482\) 8.00000 0.364390
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) −10.0000 −0.453609
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −10.0000 −0.452679
\(489\) 8.00000 0.361773
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 1.00000 0.0445435
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) 16.0000 0.709885
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) −12.0000 −0.531369
\(511\) 14.0000 0.619324
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) 24.0000 1.05859
\(515\) −4.00000 −0.176261
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 12.0000 0.526742
\(520\) 2.00000 0.0877058
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 18.0000 0.786334
\(525\) 2.00000 0.0872872
\(526\) −24.0000 −1.04645
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −12.0000 −0.521247
\(531\) 12.0000 0.520756
\(532\) 2.00000 0.0867110
\(533\) 0 0
\(534\) −36.0000 −1.55787
\(535\) −12.0000 −0.518805
\(536\) 4.00000 0.172774
\(537\) 24.0000 1.03568
\(538\) −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 \(0.641463\pi\)
\(542\) 20.0000 0.859074
\(543\) 20.0000 0.858282
\(544\) 6.00000 0.257248
\(545\) −8.00000 −0.342682
\(546\) 4.00000 0.171184
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) −6.00000 −0.256307
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) −12.0000 −0.510754
\(553\) −10.0000 −0.425243
\(554\) −10.0000 −0.424859
\(555\) −16.0000 −0.679162
\(556\) 10.0000 0.424094
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) −8.00000 −0.338667
\(559\) −8.00000 −0.338364
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −4.00000 −0.168133
\(567\) 11.0000 0.461957
\(568\) 12.0000 0.503509
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −4.00000 −0.167542
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −16.0000 −0.666089 −0.333044 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) −19.0000 −0.790296
\(579\) 28.0000 1.16364
\(580\) 0 0
\(581\) 0 0
\(582\) 16.0000 0.663221
\(583\) 0 0
\(584\) 14.0000 0.579324
\(585\) −2.00000 −0.0826898
\(586\) 18.0000 0.743573
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −16.0000 −0.659269
\(590\) −12.0000 −0.494032
\(591\) 36.0000 1.48084
\(592\) 8.00000 0.328798
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 0 0
\(597\) −16.0000 −0.654836
\(598\) −12.0000 −0.490716
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 2.00000 0.0816497
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 36.0000 1.46240
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) −8.00000 −0.321807
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 8.00000 0.321288
\(621\) −24.0000 −0.963087
\(622\) −24.0000 −0.962312
\(623\) 18.0000 0.721155
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 16.0000 0.639489
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −48.0000 −1.91389
\(630\) 1.00000 0.0398410
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −10.0000 −0.397779
\(633\) 40.0000 1.58986
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) −24.0000 −0.951662
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −24.0000 −0.947204
\(643\) 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
\(646\) −12.0000 −0.472134
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 16.0000 0.627089
\(652\) −4.00000 −0.156652
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −16.0000 −0.625650
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −20.0000 −0.777322
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) 2.00000 0.0775567
\(666\) −8.00000 −0.309994
\(667\) 0 0
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −22.0000 −0.847408
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −12.0000 −0.460857
\(679\) −8.00000 −0.307012
\(680\) 6.00000 0.230089
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −6.00000 −0.229248
\(686\) 1.00000 0.0381802
\(687\) −52.0000 −1.98392
\(688\) 4.00000 0.152499
\(689\) −24.0000 −0.914327
\(690\) −12.0000 −0.456832
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) 0 0
\(698\) 2.00000 0.0757011
\(699\) −12.0000 −0.453882
\(700\) −1.00000 −0.0377964
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 8.00000 0.301941
\(703\) −16.0000 −0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) −18.0000 −0.676960
\(708\) −24.0000 −0.901975
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 12.0000 0.450352
\(711\) 10.0000 0.375029
\(712\) 18.0000 0.674579
\(713\) −48.0000 −1.79761
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 36.0000 1.34444
\(718\) −18.0000 −0.671754
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) 4.00000 0.148968
\(722\) 15.0000 0.558242
\(723\) 16.0000 0.595046
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 13.0000 0.481481
\(730\) 14.0000 0.518163
\(731\) −24.0000 −0.887672
\(732\) −20.0000 −0.739221
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −20.0000 −0.738213
\(735\) −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 \(0.619908\pi\)
\(740\) 8.00000 0.294086
\(741\) −8.00000 −0.293887
\(742\) 12.0000 0.440534
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 16.0000 0.586588
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 2.00000 0.0730297
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 10.0000 0.363937
\(756\) −4.00000 −0.145479
\(757\) −52.0000 −1.88997 −0.944986 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 32.0000 1.15924
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 12.0000 0.433578
\(767\) −24.0000 −0.866590
\(768\) −2.00000 −0.0721688
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) 48.0000 1.72868
\(772\) −14.0000 −0.503871
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −4.00000 −0.143777
\(775\) 8.00000 0.287368
\(776\) −8.00000 −0.287183
\(777\) 16.0000 0.573997
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) −36.0000 −1.28736
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 2.00000 0.0713831
\(786\) 36.0000 1.28408
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −18.0000 −0.641223
\(789\) −48.0000 −1.70885
\(790\) −10.0000 −0.355784
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) −2.00000 −0.0709773
\(795\) −24.0000 −0.851192
\(796\) 8.00000 0.283552
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −18.0000 −0.635999
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 6.00000 0.211472
\(806\) 16.0000 0.563576
\(807\) −36.0000 −1.26726
\(808\) −18.0000 −0.633238
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 11.0000 0.386501
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 0 0
\(813\) 40.0000 1.40286
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 12.0000 0.420084
\(817\) −8.00000 −0.279885
\(818\) 8.00000 0.279713
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −36.0000 −1.25641 −0.628204 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) −12.0000 −0.418548
\(823\) 2.00000 0.0697156 0.0348578 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −6.00000 −0.208514
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −20.0000 −0.693792
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) 32.0000 1.10608
\(838\) 12.0000 0.414533
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −29.0000 −1.00000
\(842\) −14.0000 −0.482472
\(843\) 12.0000 0.413302
\(844\) −20.0000 −0.688428
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 12.0000 0.412082
\(849\) −8.00000 −0.274559
\(850\) 6.00000 0.205798
\(851\) −48.0000 −1.64542
\(852\) 24.0000 0.822226
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 10.0000 0.342193
\(855\) −2.00000 −0.0683986
\(856\) 12.0000 0.410152
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) −4.00000 −0.136083
\(865\) −6.00000 −0.204006
\(866\) 4.00000 0.135926
\(867\) −38.0000 −1.29055
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 8.00000 0.270914
\(873\) 8.00000 0.270759
\(874\) −12.0000 −0.405906
\(875\) −1.00000 −0.0338062
\(876\) 28.0000 0.946032
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 20.0000 0.674967
\(879\) 36.0000 1.21425
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 12.0000 0.403604
\(885\) −24.0000 −0.806751
\(886\) −12.0000 −0.403148
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 16.0000 0.536925
\(889\) −16.0000 −0.536623
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) −24.0000 −0.801337
\(898\) 18.0000 0.600668
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −72.0000 −2.39867
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 6.00000 0.199557
\(905\) −10.0000 −0.332411
\(906\) 20.0000 0.664455
\(907\) 32.0000 1.06254 0.531271 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 12.0000 0.398234
\(909\) 18.0000 0.597022
\(910\) −2.00000 −0.0662994
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) −20.0000 −0.661180
\(916\) 26.0000 0.859064
\(917\) −18.0000 −0.594412
\(918\) 24.0000 0.792118
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 6.00000 0.197814
\(921\) 64.0000 2.10887
\(922\) −30.0000 −0.987997
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −2.00000 −0.0657241
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 16.0000 0.524661
\(931\) −2.00000 −0.0655474
\(932\) 6.00000 0.196537
\(933\) −48.0000 −1.57145
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −4.00000 −0.130605
\(939\) 32.0000 1.04428
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 4.00000 0.130327
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −20.0000 −0.649570
\(949\) 28.0000 0.908918
\(950\) 2.00000 0.0648886
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) −18.0000 −0.582162
\(957\) 0 0
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) −2.00000 −0.0645497
\(961\) 33.0000 1.06452
\(962\) 16.0000 0.515861
\(963\) −12.0000 −0.386695
\(964\) −8.00000 −0.257663
\(965\) −14.0000 −0.450676
\(966\) 12.0000 0.386094
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) −8.00000 −0.256865
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 10.0000 0.320750
\(973\) −10.0000 −0.320585
\(974\) 22.0000 0.704925
\(975\) 4.00000 0.128103
\(976\) 10.0000 0.320092
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −8.00000 −0.255812
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −8.00000 −0.255420
\(982\) 12.0000 0.382935
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) −8.00000 −0.254000
\(993\) −40.0000 −1.26936
\(994\) −12.0000 −0.380617
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −20.0000 −0.633089
\(999\) 32.0000 1.01244
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.e.1.1 1
11.10 odd 2 770.2.a.g.1.1 1
33.32 even 2 6930.2.a.f.1.1 1
44.43 even 2 6160.2.a.n.1.1 1
55.32 even 4 3850.2.c.a.1849.2 2
55.43 even 4 3850.2.c.a.1849.1 2
55.54 odd 2 3850.2.a.j.1.1 1
77.76 even 2 5390.2.a.bh.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.g.1.1 1 11.10 odd 2
3850.2.a.j.1.1 1 55.54 odd 2
3850.2.c.a.1849.1 2 55.43 even 4
3850.2.c.a.1849.2 2 55.32 even 4
5390.2.a.bh.1.1 1 77.76 even 2
6160.2.a.n.1.1 1 44.43 even 2
6930.2.a.f.1.1 1 33.32 even 2
8470.2.a.e.1.1 1 1.1 even 1 trivial