Properties

Label 8470.2.a.r.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} -2.00000 q^{38} +4.00000 q^{39} -1.00000 q^{40} -12.0000 q^{41} +2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -12.0000 q^{51} -2.00000 q^{52} +4.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} +2.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +8.00000 q^{67} +6.00000 q^{68} +12.0000 q^{69} +1.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -4.00000 q^{74} -2.00000 q^{75} -2.00000 q^{76} +4.00000 q^{78} -14.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -12.0000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} -16.0000 q^{93} +12.0000 q^{94} +2.00000 q^{95} -2.00000 q^{96} +8.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 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
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −2.00000 −0.324443
\(39\) 4.00000 0.640513
\(40\) −1.00000 −0.158114
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 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
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −12.0000 −1.68034
\(52\) −2.00000 −0.277350
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 2.00000 0.258199
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 6.00000 0.727607
\(69\) 12.0000 1.44463
\(70\) 1.00000 0.119523
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −4.00000 −0.464991
\(75\) −2.00000 −0.230940
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) −12.0000 −1.32518
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) −16.0000 −1.65912
\(94\) 12.0000 1.23771
\(95\) 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
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −12.0000 −1.18818
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 4.00000 0.384900
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) −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
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 2.00000 0.182574
\(121\) 0 0
\(122\) −2.00000 −0.181071
\(123\) 24.0000 2.16401
\(124\) 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
\(131\) 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
\(134\) 8.00000 0.691095
\(135\) −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
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 1.00000 0.0845154
\(141\) −24.0000 −2.02116
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) −2.00000 −0.164957
\(148\) −4.00000 −0.328798
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −2.00000 −0.163299
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) −2.00000 −0.162221
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 4.00000 0.320256
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −14.0000 −1.11378
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 6.00000 0.472866
\(162\) −11.0000 −0.864242
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 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
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) 4.00000 0.295689
\(184\) −6.00000 −0.442326
\(185\) 4.00000 0.294086
\(186\) −16.0000 −1.17318
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) −4.00000 −0.290957
\(190\) 2.00000 0.145095
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −2.00000 −0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 8.00000 0.574367
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) −16.0000 −1.12855
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) 12.0000 0.838116
\(206\) −16.0000 −1.11477
\(207\) −6.00000 −0.417029
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) −24.0000 −1.64445
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 4.00000 0.272166
\(217\) −8.00000 −0.543075
\(218\) 16.0000 1.08366
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 8.00000 0.536925
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 28.0000 1.81880
\(238\) −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
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) 24.0000 1.53018
\(247\) 4.00000 0.254514
\(248\) 8.00000 0.508001
\(249\) 24.0000 1.52094
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) −8.00000 −0.498058
\(259\) 4.00000 0.248548
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 18.0000 1.11204
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) −12.0000 −0.734388
\(268\) 8.00000 0.488678
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −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
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −14.0000 −0.839664
\(279\) 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 \(0.557275\pi\)
\(282\) −24.0000 −1.42918
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 12.0000 0.712069
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) −2.00000 −0.115470
\(301\) −4.00000 −0.230556
\(302\) −14.0000 −0.805609
\(303\) −12.0000 −0.689382
\(304\) −2.00000 −0.114708
\(305\) 2.00000 0.114520
\(306\) 6.00000 0.342997
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 32.0000 1.82042
\(310\) −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
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 14.0000 0.790066
\(315\) 1.00000 0.0563436
\(316\) −14.0000 −0.787562
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 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
\(326\) 8.00000 0.443079
\(327\) −32.0000 −1.76960
\(328\) −12.0000 −0.662589
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −12.0000 −0.658586
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 2.00000 0.109109
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −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
\(346\) 18.0000 0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) −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 \(0.720526\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 6.00000 0.317999
\(357\) 12.0000 0.635107
\(358\) −12.0000 −0.634220
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 2.00000 0.104685
\(366\) 4.00000 0.209083
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −6.00000 −0.312772
\(369\) −12.0000 −0.624695
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) −16.0000 −0.829561
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 12.0000 0.618853
\(377\) 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
\(381\) 16.0000 0.819705
\(382\) −24.0000 −1.22795
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 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
\(394\) −6.00000 −0.302276
\(395\) 14.0000 0.704416
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) −16.0000 −0.802008
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −16.0000 −0.798007
\(403\) −16.0000 −0.797017
\(404\) 6.00000 0.298511
\(405\) 11.0000 0.546594
\(406\) 0 0
\(407\) 0 0
\(408\) −12.0000 −0.594089
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 12.0000 0.592638
\(411\) 12.0000 0.591916
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 12.0000 0.589057
\(416\) −2.00000 −0.0980581
\(417\) 28.0000 1.37117
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) −24.0000 −1.16280
\(427\) 2.00000 0.0967868
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 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
\(436\) 16.0000 0.766261
\(437\) 12.0000 0.574038
\(438\) 4.00000 0.191127
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 8.00000 0.379663
\(445\) −6.00000 −0.284427
\(446\) 8.00000 0.378811
\(447\) 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
\(453\) 28.0000 1.31555
\(454\) −24.0000 −1.12638
\(455\) −2.00000 −0.0937614
\(456\) 4.00000 0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −10.0000 −0.467269
\(459\) 24.0000 1.12022
\(460\) 6.00000 0.279751
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) −6.00000 −0.277945
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) −12.0000 −0.553519
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) 0 0
\(474\) 28.0000 1.28608
\(475\) −2.00000 −0.0917663
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) −18.0000 −0.823301
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 2.00000 0.0912871
\(481\) 8.00000 0.364769
\(482\) −20.0000 −0.910975
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 10.0000 0.453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −16.0000 −0.723545
\(490\) −1.00000 −0.0451754
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 24.0000 1.08200
\(493\) 0 0
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) 24.0000 1.07547
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 18.0000 0.799408
\(508\) −8.00000 −0.354943
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 12.0000 0.531369
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) 24.0000 1.05859
\(515\) 16.0000 0.705044
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) 4.00000 0.175750
\(519\) −36.0000 −1.58022
\(520\) 2.00000 0.0877058
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 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
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) 24.0000 1.03956
\(534\) −12.0000 −0.519291
\(535\) 12.0000 0.518805
\(536\) 8.00000 0.345547
\(537\) 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
\(543\) −4.00000 −0.171656
\(544\) 6.00000 0.257248
\(545\) −16.0000 −0.685365
\(546\) −4.00000 −0.171184
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 12.0000 0.510754
\(553\) 14.0000 0.595341
\(554\) 22.0000 0.934690
\(555\) −8.00000 −0.339581
\(556\) −14.0000 −0.593732
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 8.00000 0.338667
\(559\) −8.00000 −0.338364
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −24.0000 −1.01058
\(565\) 6.00000 0.252422
\(566\) 16.0000 0.672530
\(567\) 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
\(573\) 48.0000 2.00523
\(574\) 12.0000 0.500870
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) 8.00000 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 19.0000 0.790296
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) −16.0000 −0.663221
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 2.00000 0.0826898
\(586\) 6.00000 0.247858
\(587\) −30.0000 −1.23823 −0.619116 0.785299i \(-0.712509\pi\)
−0.619116 + 0.785299i \(0.712509\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −4.00000 −0.164399
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 0 0
\(597\) 32.0000 1.30967
\(598\) 12.0000 0.490716
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −2.00000 −0.0816497
\(601\) −32.0000 −1.30531 −0.652654 0.757656i \(-0.726344\pi\)
−0.652654 + 0.757656i \(0.726344\pi\)
\(602\) −4.00000 −0.163028
\(603\) 8.00000 0.325785
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −24.0000 −0.970936
\(612\) 6.00000 0.242536
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 4.00000 0.161427
\(615\) −24.0000 −0.967773
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 32.0000 1.28723
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) −8.00000 −0.321288
\(621\) −24.0000 −0.963087
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) −24.0000 −0.956943
\(630\) 1.00000 0.0398410
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −14.0000 −0.556890
\(633\) −8.00000 −0.317971
\(634\) −12.0000 −0.476581
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −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
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 6.00000 0.236433
\(645\) 8.00000 0.315000
\(646\) −12.0000 −0.472134
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 16.0000 0.627089
\(652\) 8.00000 0.313304
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −32.0000 −1.25130
\(655\) −18.0000 −0.703318
\(656\) −12.0000 −0.468521
\(657\) −2.00000 −0.0780274
\(658\) −12.0000 −0.467809
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −28.0000 −1.08825
\(663\) 24.0000 0.932083
\(664\) −12.0000 −0.465690
\(665\) −2.00000 −0.0775567
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −14.0000 −0.539260
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 12.0000 0.460857
\(679\) −8.00000 −0.307012
\(680\) −6.00000 −0.230089
\(681\) 48.0000 1.83936
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 6.00000 0.229248
\(686\) −1.00000 −0.0381802
\(687\) 20.0000 0.763048
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) −12.0000 −0.456832
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) −72.0000 −2.72719
\(698\) −14.0000 −0.529908
\(699\) 12.0000 0.453882
\(700\) −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
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 24.0000 0.903892
\(706\) −24.0000 −0.903252
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) −12.0000 −0.450352
\(711\) −14.0000 −0.525041
\(712\) 6.00000 0.224860
\(713\) −48.0000 −1.79761
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 36.0000 1.34444
\(718\) −30.0000 −1.11959
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.0000 0.595871
\(722\) −15.0000 −0.558242
\(723\) 40.0000 1.48762
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) 20.0000 0.741759 0.370879 0.928681i \(-0.379056\pi\)
0.370879 + 0.928681i \(0.379056\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) 2.00000 0.0740233
\(731\) 24.0000 0.887672
\(732\) 4.00000 0.147844
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 8.00000 0.295285
\(735\) 2.00000 0.0737711
\(736\) −6.00000 −0.221163
\(737\) 0 0
\(738\) −12.0000 −0.441726
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 4.00000 0.147043
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −16.0000 −0.586588
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 2.00000 0.0730297
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 14.0000 0.509512
\(756\) −4.00000 −0.145479
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 16.0000 0.579619
\(763\) −16.0000 −0.579239
\(764\) −24.0000 −0.868290
\(765\) −6.00000 −0.216930
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −48.0000 −1.72868
\(772\) −2.00000 −0.0719816
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) 8.00000 0.287183
\(777\) −8.00000 −0.286998
\(778\) −6.00000 −0.215110
\(779\) 24.0000 0.859889
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) −36.0000 −1.28736
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −14.0000 −0.499681
\(786\) −36.0000 −1.28408
\(787\) 16.0000 0.570338 0.285169 0.958477i \(-0.407950\pi\)
0.285169 + 0.958477i \(0.407950\pi\)
\(788\) −6.00000 −0.213741
\(789\) 48.0000 1.70885
\(790\) 14.0000 0.498098
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 38.0000 1.34857
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) −4.00000 −0.141598
\(799\) 72.0000 2.54718
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) −16.0000 −0.564276
\(805\) −6.00000 −0.211472
\(806\) −16.0000 −0.563576
\(807\) 36.0000 1.26726
\(808\) 6.00000 0.211079
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 11.0000 0.386501
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) 40.0000 1.40286
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) −12.0000 −0.420084
\(817\) −8.00000 −0.279885
\(818\) 4.00000 0.139857
\(819\) 2.00000 0.0698857
\(820\) 12.0000 0.419058
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 12.0000 0.418548
\(823\) 50.0000 1.74289 0.871445 0.490493i \(-0.163183\pi\)
0.871445 + 0.490493i \(0.163183\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 12.0000 0.416526
\(831\) −44.0000 −1.52634
\(832\) −2.00000 −0.0693375
\(833\) 6.00000 0.207888
\(834\) 28.0000 0.969561
\(835\) 0 0
\(836\) 0 0
\(837\) 32.0000 1.10608
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −29.0000 −1.00000
\(842\) −10.0000 −0.344623
\(843\) 12.0000 0.413302
\(844\) 4.00000 0.137686
\(845\) 9.00000 0.309609
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) 0 0
\(849\) −32.0000 −1.09824
\(850\) 6.00000 0.205798
\(851\) 24.0000 0.822709
\(852\) −24.0000 −0.822226
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 2.00000 0.0684386
\(855\) 2.00000 0.0683986
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −16.0000 −0.545913 −0.272956 0.962026i \(-0.588002\pi\)
−0.272956 + 0.962026i \(0.588002\pi\)
\(860\) −4.00000 −0.136399
\(861\) −24.0000 −0.817918
\(862\) −30.0000 −1.02180
\(863\) 18.0000 0.612727 0.306364 0.951915i \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) 4.00000 0.136083
\(865\) −18.0000 −0.612018
\(866\) −4.00000 −0.135926
\(867\) −38.0000 −1.29055
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 16.0000 0.541828
\(873\) 8.00000 0.270759
\(874\) 12.0000 0.405906
\(875\) 1.00000 0.0338062
\(876\) 4.00000 0.135147
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 28.0000 0.944954
\(879\) −12.0000 −0.404750
\(880\) 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
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 8.00000 0.268462
\(889\) 8.00000 0.268311
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −24.0000 −0.803129
\(894\) 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
\(901\) 0 0
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) −6.00000 −0.199557
\(905\) −2.00000 −0.0664822
\(906\) 28.0000 0.930238
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) −24.0000 −0.796468
\(909\) 6.00000 0.199007
\(910\) −2.00000 −0.0662994
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) −4.00000 −0.132236
\(916\) −10.0000 −0.330409
\(917\) −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
\(921\) −8.00000 −0.263609
\(922\) 18.0000 0.592798
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −22.0000 −0.722965
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 16.0000 0.524661
\(931\) −2.00000 −0.0655474
\(932\) −6.00000 −0.196537
\(933\) −48.0000 −1.57145
\(934\) −6.00000 −0.196326
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −8.00000 −0.261209
\(939\) −16.0000 −0.522140
\(940\) −12.0000 −0.391397
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −28.0000 −0.912289
\(943\) 72.0000 2.34464
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 28.0000 0.909398
\(949\) 4.00000 0.129845
\(950\) −2.00000 −0.0648886
\(951\) 24.0000 0.778253
\(952\) −6.00000 −0.194461
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) −18.0000 −0.582162
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 6.00000 0.193750
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) 8.00000 0.257930
\(963\) −12.0000 −0.386695
\(964\) −20.0000 −0.644157
\(965\) 2.00000 0.0643823
\(966\) −12.0000 −0.386094
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) −8.00000 −0.256865
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 10.0000 0.320750
\(973\) 14.0000 0.448819
\(974\) 2.00000 0.0640841
\(975\) 4.00000 0.128103
\(976\) −2.00000 −0.0640184
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −16.0000 −0.511624
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 16.0000 0.510841
\(982\) 36.0000 1.14881
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 24.0000 0.765092
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 4.00000 0.127257
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 8.00000 0.254000
\(993\) 56.0000 1.77711
\(994\) −12.0000 −0.380617
\(995\) 16.0000 0.507234
\(996\) 24.0000 0.760469
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −4.00000 −0.126618
\(999\) −16.0000 −0.506218
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.r.1.1 1
11.10 odd 2 770.2.a.a.1.1 1
33.32 even 2 6930.2.a.bm.1.1 1
44.43 even 2 6160.2.a.k.1.1 1
55.32 even 4 3850.2.c.o.1849.1 2
55.43 even 4 3850.2.c.o.1849.2 2
55.54 odd 2 3850.2.a.ba.1.1 1
77.76 even 2 5390.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.a.1.1 1 11.10 odd 2
3850.2.a.ba.1.1 1 55.54 odd 2
3850.2.c.o.1849.1 2 55.32 even 4
3850.2.c.o.1849.2 2 55.43 even 4
5390.2.a.r.1.1 1 77.76 even 2
6160.2.a.k.1.1 1 44.43 even 2
6930.2.a.bm.1.1 1 33.32 even 2
8470.2.a.r.1.1 1 1.1 even 1 trivial