Properties

Label 4998.2.a.e.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.9092309302\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 714)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4998.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} +1.00000 q^{30} -3.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +1.00000 q^{41} -4.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} -9.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -1.00000 q^{51} +1.00000 q^{52} +1.00000 q^{54} -2.00000 q^{55} -4.00000 q^{57} -3.00000 q^{58} +11.0000 q^{59} -1.00000 q^{60} +3.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -2.00000 q^{66} -12.0000 q^{67} +1.00000 q^{68} -8.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -4.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} +9.00000 q^{83} +1.00000 q^{85} +4.00000 q^{86} -3.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} +8.00000 q^{92} +3.00000 q^{93} +9.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 1.00000 0.182574
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −4.00000 −0.648886
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) 1.00000 0.149071
\(46\) −8.00000 −1.17954
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) −1.00000 −0.140028
\(52\) 1.00000 0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −3.00000 −0.393919
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) −1.00000 −0.129099
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) −2.00000 −0.246183
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 1.00000 0.121268
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −4.00000 −0.464991
\(75\) 4.00000 0.461880
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −1.00000 −0.110432
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 4.00000 0.431331
\(87\) −3.00000 −0.321634
\(88\) 2.00000 0.213201
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 3.00000 0.311086
\(94\) 9.00000 0.928279
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −4.00000 −0.400000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 1.00000 0.0990148
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 2.00000 0.190693
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 4.00000 0.374634
\(115\) 8.00000 0.746004
\(116\) 3.00000 0.278543
\(117\) 1.00000 0.0924500
\(118\) −11.0000 −1.01263
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −1.00000 −0.0901670
\(124\) −3.00000 −0.269408
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) −1.00000 −0.0877058
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −1.00000 −0.0860663
\(136\) −1.00000 −0.0857493
\(137\) 20.0000 1.70872 0.854358 0.519685i \(-0.173951\pi\)
0.854358 + 0.519685i \(0.173951\pi\)
\(138\) 8.00000 0.681005
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −4.00000 −0.335673
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) 3.00000 0.249136
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) −4.00000 −0.326599
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −4.00000 −0.324443
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) −1.00000 −0.0800641
\(157\) 1.00000 0.0798087 0.0399043 0.999204i \(-0.487295\pi\)
0.0399043 + 0.999204i \(0.487295\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 1.00000 0.0780869
\(165\) 2.00000 0.155700
\(166\) −9.00000 −0.698535
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −1.00000 −0.0766965
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −11.0000 −0.826811
\(178\) 6.00000 0.449719
\(179\) −21.0000 −1.56961 −0.784807 0.619740i \(-0.787238\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(180\) 1.00000 0.0745356
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) 4.00000 0.294086
\(186\) −3.00000 −0.219971
\(187\) −2.00000 −0.146254
\(188\) −9.00000 −0.656392
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −8.00000 −0.574367
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 1.00000 0.0712470 0.0356235 0.999365i \(-0.488658\pi\)
0.0356235 + 0.999365i \(0.488658\pi\)
\(198\) 2.00000 0.142134
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 4.00000 0.282843
\(201\) 12.0000 0.846415
\(202\) −14.0000 −0.985037
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 1.00000 0.0698430
\(206\) −14.0000 −0.975426
\(207\) 8.00000 0.556038
\(208\) 1.00000 0.0693375
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 3.00000 0.206529 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 16.0000 1.08366
\(219\) −6.00000 −0.405442
\(220\) −2.00000 −0.134840
\(221\) 1.00000 0.0672673
\(222\) 4.00000 0.268462
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) −15.0000 −0.997785
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) −4.00000 −0.264906
\(229\) −9.00000 −0.594737 −0.297368 0.954763i \(-0.596109\pi\)
−0.297368 + 0.954763i \(0.596109\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) −11.0000 −0.720634 −0.360317 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −9.00000 −0.587095
\(236\) 11.0000 0.716039
\(237\) 0 0
\(238\) 0 0
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 1.00000 0.0637577
\(247\) 4.00000 0.254514
\(248\) 3.00000 0.190500
\(249\) −9.00000 −0.570352
\(250\) 9.00000 0.569210
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 8.00000 0.501965
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 1.00000 0.0620174
\(261\) 3.00000 0.185695
\(262\) 6.00000 0.370681
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) 1.00000 0.0608581
\(271\) 18.0000 1.09342 0.546711 0.837321i \(-0.315880\pi\)
0.546711 + 0.837321i \(0.315880\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −20.0000 −1.20824
\(275\) 8.00000 0.482418
\(276\) −8.00000 −0.481543
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) −9.00000 −0.535942
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 4.00000 0.237356
\(285\) −4.00000 −0.236940
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −3.00000 −0.176166
\(291\) −8.00000 −0.468968
\(292\) 6.00000 0.351123
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 11.0000 0.640445
\(296\) −4.00000 −0.232495
\(297\) 2.00000 0.116052
\(298\) 2.00000 0.115857
\(299\) 8.00000 0.462652
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 4.00000 0.230174
\(303\) −14.0000 −0.804279
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 3.00000 0.170389
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 1.00000 0.0566139
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −1.00000 −0.0564333
\(315\) 0 0
\(316\) 0 0
\(317\) −11.0000 −0.617822 −0.308911 0.951091i \(-0.599964\pi\)
−0.308911 + 0.951091i \(0.599964\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −16.0000 −0.886158
\(327\) 16.0000 0.884802
\(328\) −1.00000 −0.0552158
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) −22.0000 −1.20923 −0.604615 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 9.00000 0.493939
\(333\) 4.00000 0.219199
\(334\) −2.00000 −0.109435
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 12.0000 0.652714
\(339\) −15.0000 −0.814688
\(340\) 1.00000 0.0542326
\(341\) 6.00000 0.324918
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) −8.00000 −0.430706
\(346\) −21.0000 −1.12897
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −3.00000 −0.160817
\(349\) 27.0000 1.44528 0.722638 0.691226i \(-0.242929\pi\)
0.722638 + 0.691226i \(0.242929\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 2.00000 0.106600
\(353\) 28.0000 1.49029 0.745145 0.666903i \(-0.232380\pi\)
0.745145 + 0.666903i \(0.232380\pi\)
\(354\) 11.0000 0.584643
\(355\) 4.00000 0.212298
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 21.0000 1.10988
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −16.0000 −0.840941
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 8.00000 0.417029
\(369\) 1.00000 0.0520579
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 3.00000 0.155543
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 2.00000 0.103418
\(375\) 9.00000 0.464758
\(376\) 9.00000 0.464140
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 4.00000 0.205196
\(381\) 8.00000 0.409852
\(382\) −3.00000 −0.153493
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −4.00000 −0.203331
\(388\) 8.00000 0.406138
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 1.00000 0.0506370
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) −1.00000 −0.0503793
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 5.00000 0.250627
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −12.0000 −0.598506
\(403\) −3.00000 −0.149441
\(404\) 14.0000 0.696526
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 1.00000 0.0495074
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −1.00000 −0.0493865
\(411\) −20.0000 −0.986527
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 9.00000 0.441793
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) −3.00000 −0.146038
\(423\) −9.00000 −0.437595
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 4.00000 0.193801
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 2.00000 0.0965609
\(430\) 4.00000 0.192897
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) −16.0000 −0.766261
\(437\) 32.0000 1.53077
\(438\) 6.00000 0.286691
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) −1.00000 −0.0475651
\(443\) 33.0000 1.56788 0.783939 0.620838i \(-0.213208\pi\)
0.783939 + 0.620838i \(0.213208\pi\)
\(444\) −4.00000 −0.189832
\(445\) −6.00000 −0.284427
\(446\) 16.0000 0.757622
\(447\) 2.00000 0.0945968
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 4.00000 0.188562
\(451\) −2.00000 −0.0941763
\(452\) 15.0000 0.705541
\(453\) 4.00000 0.187936
\(454\) −10.0000 −0.469323
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 9.00000 0.420542
\(459\) −1.00000 −0.0466760
\(460\) 8.00000 0.373002
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 0 0
\(463\) −18.0000 −0.836531 −0.418265 0.908325i \(-0.637362\pi\)
−0.418265 + 0.908325i \(0.637362\pi\)
\(464\) 3.00000 0.139272
\(465\) 3.00000 0.139122
\(466\) 11.0000 0.509565
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 9.00000 0.415139
\(471\) −1.00000 −0.0460776
\(472\) −11.0000 −0.506316
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 0 0
\(478\) −13.0000 −0.594606
\(479\) 26.0000 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 1.00000 0.0456435
\(481\) 4.00000 0.182384
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 8.00000 0.363261
\(486\) 1.00000 0.0453609
\(487\) −27.0000 −1.22349 −0.611743 0.791056i \(-0.709531\pi\)
−0.611743 + 0.791056i \(0.709531\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −1.00000 −0.0450835
\(493\) 3.00000 0.135113
\(494\) −4.00000 −0.179969
\(495\) −2.00000 −0.0898933
\(496\) −3.00000 −0.134704
\(497\) 0 0
\(498\) 9.00000 0.403300
\(499\) 13.0000 0.581960 0.290980 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) −9.00000 −0.402492
\(501\) −2.00000 −0.0893534
\(502\) −23.0000 −1.02654
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 16.0000 0.711287
\(507\) 12.0000 0.532939
\(508\) −8.00000 −0.354943
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 1.00000 0.0442807
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −14.0000 −0.617514
\(515\) 14.0000 0.616914
\(516\) 4.00000 0.176090
\(517\) 18.0000 0.791639
\(518\) 0 0
\(519\) −21.0000 −0.921798
\(520\) −1.00000 −0.0438529
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −3.00000 −0.131306
\(523\) −40.0000 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) −3.00000 −0.130682
\(528\) 2.00000 0.0870388
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 11.0000 0.477359
\(532\) 0 0
\(533\) 1.00000 0.0433148
\(534\) −6.00000 −0.259645
\(535\) −12.0000 −0.518805
\(536\) 12.0000 0.518321
\(537\) 21.0000 0.906217
\(538\) 5.00000 0.215565
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) −18.0000 −0.773166
\(543\) −16.0000 −0.686626
\(544\) −1.00000 −0.0428746
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) −41.0000 −1.75303 −0.876517 0.481371i \(-0.840139\pi\)
−0.876517 + 0.481371i \(0.840139\pi\)
\(548\) 20.0000 0.854358
\(549\) 0 0
\(550\) −8.00000 −0.341121
\(551\) 12.0000 0.511217
\(552\) 8.00000 0.340503
\(553\) 0 0
\(554\) −28.0000 −1.18961
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) 16.0000 0.677942 0.338971 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(558\) 3.00000 0.127000
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) −20.0000 −0.843649
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 9.00000 0.378968
\(565\) 15.0000 0.631055
\(566\) 13.0000 0.546431
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 4.00000 0.167542
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) −32.0000 −1.33449
\(576\) 1.00000 0.0416667
\(577\) −15.0000 −0.624458 −0.312229 0.950007i \(-0.601076\pi\)
−0.312229 + 0.950007i \(0.601076\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 14.0000 0.581820
\(580\) 3.00000 0.124568
\(581\) 0 0
\(582\) 8.00000 0.331611
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) −11.0000 −0.452863
\(591\) −1.00000 −0.0411345
\(592\) 4.00000 0.164399
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 5.00000 0.204636
\(598\) −8.00000 −0.327144
\(599\) 11.0000 0.449448 0.224724 0.974422i \(-0.427852\pi\)
0.224724 + 0.974422i \(0.427852\pi\)
\(600\) −4.00000 −0.163299
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) −4.00000 −0.162758
\(605\) −7.00000 −0.284590
\(606\) 14.0000 0.568711
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 1.00000 0.0404226
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −20.0000 −0.807134
\(615\) −1.00000 −0.0403239
\(616\) 0 0
\(617\) 17.0000 0.684394 0.342197 0.939628i \(-0.388829\pi\)
0.342197 + 0.939628i \(0.388829\pi\)
\(618\) 14.0000 0.563163
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) −3.00000 −0.120483
\(621\) −8.00000 −0.321029
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) −8.00000 −0.319744
\(627\) 8.00000 0.319489
\(628\) 1.00000 0.0399043
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 0 0
\(633\) −3.00000 −0.119239
\(634\) 11.0000 0.436866
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 0 0
\(638\) 6.00000 0.237542
\(639\) 4.00000 0.158238
\(640\) −1.00000 −0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −12.0000 −0.473602
\(643\) 45.0000 1.77463 0.887313 0.461167i \(-0.152569\pi\)
0.887313 + 0.461167i \(0.152569\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) −4.00000 −0.157378
\(647\) 33.0000 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −22.0000 −0.863576
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −16.0000 −0.625650
\(655\) −6.00000 −0.234439
\(656\) 1.00000 0.0390434
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 39.0000 1.51922 0.759612 0.650376i \(-0.225389\pi\)
0.759612 + 0.650376i \(0.225389\pi\)
\(660\) 2.00000 0.0778499
\(661\) 47.0000 1.82809 0.914044 0.405615i \(-0.132943\pi\)
0.914044 + 0.405615i \(0.132943\pi\)
\(662\) 22.0000 0.855054
\(663\) −1.00000 −0.0388368
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 24.0000 0.929284
\(668\) 2.00000 0.0773823
\(669\) 16.0000 0.618596
\(670\) 12.0000 0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −34.0000 −1.30963
\(675\) 4.00000 0.153960
\(676\) −12.0000 −0.461538
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 15.0000 0.576072
\(679\) 0 0
\(680\) −1.00000 −0.0383482
\(681\) −10.0000 −0.383201
\(682\) −6.00000 −0.229752
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 4.00000 0.152944
\(685\) 20.0000 0.764161
\(686\) 0 0
\(687\) 9.00000 0.343371
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 8.00000 0.304555
\(691\) −47.0000 −1.78796 −0.893982 0.448103i \(-0.852100\pi\)
−0.893982 + 0.448103i \(0.852100\pi\)
\(692\) 21.0000 0.798300
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 3.00000 0.113715
\(697\) 1.00000 0.0378777
\(698\) −27.0000 −1.02197
\(699\) 11.0000 0.416058
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 1.00000 0.0377426
\(703\) 16.0000 0.603451
\(704\) −2.00000 −0.0753778
\(705\) 9.00000 0.338960
\(706\) −28.0000 −1.05379
\(707\) 0 0
\(708\) −11.0000 −0.413405
\(709\) −42.0000 −1.57734 −0.788672 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) −4.00000 −0.150117
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) −21.0000 −0.784807
\(717\) −13.0000 −0.485494
\(718\) −15.0000 −0.559795
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −18.0000 −0.669427
\(724\) 16.0000 0.594635
\(725\) −12.0000 −0.445669
\(726\) −7.00000 −0.259794
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 24.0000 0.884051
\(738\) −1.00000 −0.0368105
\(739\) 42.0000 1.54499 0.772497 0.635018i \(-0.219007\pi\)
0.772497 + 0.635018i \(0.219007\pi\)
\(740\) 4.00000 0.147043
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) −3.00000 −0.109985
\(745\) −2.00000 −0.0732743
\(746\) −19.0000 −0.695639
\(747\) 9.00000 0.329293
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) −27.0000 −0.985244 −0.492622 0.870243i \(-0.663961\pi\)
−0.492622 + 0.870243i \(0.663961\pi\)
\(752\) −9.00000 −0.328196
\(753\) −23.0000 −0.838167
\(754\) −3.00000 −0.109254
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) 27.0000 0.981332 0.490666 0.871348i \(-0.336754\pi\)
0.490666 + 0.871348i \(0.336754\pi\)
\(758\) −8.00000 −0.290573
\(759\) 16.0000 0.580763
\(760\) −4.00000 −0.145095
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 3.00000 0.108536
\(765\) 1.00000 0.0361551
\(766\) −24.0000 −0.867155
\(767\) 11.0000 0.397187
\(768\) −1.00000 −0.0360844
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −14.0000 −0.503871
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) 4.00000 0.143777
\(775\) 12.0000 0.431053
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) 2.00000 0.0717035
\(779\) 4.00000 0.143315
\(780\) −1.00000 −0.0358057
\(781\) −8.00000 −0.286263
\(782\) −8.00000 −0.286079
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) 1.00000 0.0356915
\(786\) −6.00000 −0.214013
\(787\) 19.0000 0.677277 0.338638 0.940917i \(-0.390034\pi\)
0.338638 + 0.940917i \(0.390034\pi\)
\(788\) 1.00000 0.0356235
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) 0 0
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) −5.00000 −0.177220
\(797\) 40.0000 1.41687 0.708436 0.705775i \(-0.249401\pi\)
0.708436 + 0.705775i \(0.249401\pi\)
\(798\) 0 0
\(799\) −9.00000 −0.318397
\(800\) 4.00000 0.141421
\(801\) −6.00000 −0.212000
\(802\) −30.0000 −1.05934
\(803\) −12.0000 −0.423471
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 3.00000 0.105670
\(807\) 5.00000 0.176008
\(808\) −14.0000 −0.492518
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 11.0000 0.386262 0.193131 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(812\) 0 0
\(813\) −18.0000 −0.631288
\(814\) 8.00000 0.280400
\(815\) 16.0000 0.560456
\(816\) −1.00000 −0.0350070
\(817\) −16.0000 −0.559769
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 1.00000 0.0349215
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 20.0000 0.697580
\(823\) 35.0000 1.22002 0.610012 0.792392i \(-0.291165\pi\)
0.610012 + 0.792392i \(0.291165\pi\)
\(824\) −14.0000 −0.487713
\(825\) −8.00000 −0.278524
\(826\) 0 0
\(827\) −46.0000 −1.59958 −0.799788 0.600282i \(-0.795055\pi\)
−0.799788 + 0.600282i \(0.795055\pi\)
\(828\) 8.00000 0.278019
\(829\) −54.0000 −1.87550 −0.937749 0.347314i \(-0.887094\pi\)
−0.937749 + 0.347314i \(0.887094\pi\)
\(830\) −9.00000 −0.312395
\(831\) −28.0000 −0.971309
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 0 0
\(835\) 2.00000 0.0692129
\(836\) −8.00000 −0.276686
\(837\) 3.00000 0.103695
\(838\) −4.00000 −0.138178
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −17.0000 −0.585859
\(843\) −20.0000 −0.688837
\(844\) 3.00000 0.103264
\(845\) −12.0000 −0.412813
\(846\) 9.00000 0.309426
\(847\) 0 0
\(848\) 0 0
\(849\) 13.0000 0.446159
\(850\) 4.00000 0.137199
\(851\) 32.0000 1.09695
\(852\) −4.00000 −0.137038
\(853\) 16.0000 0.547830 0.273915 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 12.0000 0.410152
\(857\) −21.0000 −0.717346 −0.358673 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 21.0000 0.714021
\(866\) −19.0000 −0.645646
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 3.00000 0.101710
\(871\) −12.0000 −0.406604
\(872\) 16.0000 0.541828
\(873\) 8.00000 0.270759
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 30.0000 1.01303 0.506514 0.862232i \(-0.330934\pi\)
0.506514 + 0.862232i \(0.330934\pi\)
\(878\) 1.00000 0.0337484
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) −35.0000 −1.17918 −0.589590 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 1.00000 0.0336336
\(885\) −11.0000 −0.369761
\(886\) −33.0000 −1.10866
\(887\) −26.0000 −0.872995 −0.436497 0.899706i \(-0.643781\pi\)
−0.436497 + 0.899706i \(0.643781\pi\)
\(888\) 4.00000 0.134231
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) −2.00000 −0.0670025
\(892\) −16.0000 −0.535720
\(893\) −36.0000 −1.20469
\(894\) −2.00000 −0.0668900
\(895\) −21.0000 −0.701953
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) 6.00000 0.200223
\(899\) −9.00000 −0.300167
\(900\) −4.00000 −0.133333
\(901\) 0 0
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −15.0000 −0.498893
\(905\) 16.0000 0.531858
\(906\) −4.00000 −0.132891
\(907\) 47.0000 1.56061 0.780305 0.625400i \(-0.215064\pi\)
0.780305 + 0.625400i \(0.215064\pi\)
\(908\) 10.0000 0.331862
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) −4.00000 −0.132453
\(913\) −18.0000 −0.595713
\(914\) −1.00000 −0.0330771
\(915\) 0 0
\(916\) −9.00000 −0.297368
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) 58.0000 1.91324 0.956622 0.291333i \(-0.0940987\pi\)
0.956622 + 0.291333i \(0.0940987\pi\)
\(920\) −8.00000 −0.263752
\(921\) −20.0000 −0.659022
\(922\) 20.0000 0.658665
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) 18.0000 0.591517
\(927\) 14.0000 0.459820
\(928\) −3.00000 −0.0984798
\(929\) −5.00000 −0.164045 −0.0820223 0.996630i \(-0.526138\pi\)
−0.0820223 + 0.996630i \(0.526138\pi\)
\(930\) −3.00000 −0.0983739
\(931\) 0 0
\(932\) −11.0000 −0.360317
\(933\) 18.0000 0.589294
\(934\) 3.00000 0.0981630
\(935\) −2.00000 −0.0654070
\(936\) −1.00000 −0.0326860
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) −8.00000 −0.261070
\(940\) −9.00000 −0.293548
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 1.00000 0.0325818
\(943\) 8.00000 0.260516
\(944\) 11.0000 0.358020
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 14.0000 0.454939 0.227469 0.973785i \(-0.426955\pi\)
0.227469 + 0.973785i \(0.426955\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) 16.0000 0.519109
\(951\) 11.0000 0.356699
\(952\) 0 0
\(953\) 20.0000 0.647864 0.323932 0.946080i \(-0.394995\pi\)
0.323932 + 0.946080i \(0.394995\pi\)
\(954\) 0 0
\(955\) 3.00000 0.0970777
\(956\) 13.0000 0.420450
\(957\) 6.00000 0.193952
\(958\) −26.0000 −0.840022
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −22.0000 −0.709677
\(962\) −4.00000 −0.128965
\(963\) −12.0000 −0.386695
\(964\) 18.0000 0.579741
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) 7.00000 0.224989
\(969\) −4.00000 −0.128499
\(970\) −8.00000 −0.256865
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 27.0000 0.865136
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) −52.0000 −1.66363 −0.831814 0.555055i \(-0.812697\pi\)
−0.831814 + 0.555055i \(0.812697\pi\)
\(978\) 16.0000 0.511624
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −28.0000 −0.893516
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 1.00000 0.0318788
\(985\) 1.00000 0.0318626
\(986\) −3.00000 −0.0955395
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −32.0000 −1.01754
\(990\) 2.00000 0.0635642
\(991\) 25.0000 0.794151 0.397076 0.917786i \(-0.370025\pi\)
0.397076 + 0.917786i \(0.370025\pi\)
\(992\) 3.00000 0.0952501
\(993\) 22.0000 0.698149
\(994\) 0 0
\(995\) −5.00000 −0.158511
\(996\) −9.00000 −0.285176
\(997\) −12.0000 −0.380044 −0.190022 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) −13.0000 −0.411508
\(999\) −4.00000 −0.126554
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 4998.2.a.e.1.1 1
7.2 even 3 714.2.i.i.613.1 yes 2
7.4 even 3 714.2.i.i.205.1 2
7.6 odd 2 4998.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.i.i.205.1 2 7.4 even 3
714.2.i.i.613.1 yes 2 7.2 even 3
4998.2.a.e.1.1 1 1.1 even 1 trivial
4998.2.a.m.1.1 1 7.6 odd 2