Properties

Label 4598.2.a.k.1.1
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{12} -7.00000 q^{13} +1.00000 q^{16} +7.00000 q^{17} -6.00000 q^{18} -1.00000 q^{19} +8.00000 q^{23} -3.00000 q^{24} -5.00000 q^{25} +7.00000 q^{26} +9.00000 q^{27} +9.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -7.00000 q^{34} +6.00000 q^{36} +3.00000 q^{37} +1.00000 q^{38} -21.0000 q^{39} -10.0000 q^{41} +10.0000 q^{43} -8.00000 q^{46} +3.00000 q^{47} +3.00000 q^{48} -7.00000 q^{49} +5.00000 q^{50} +21.0000 q^{51} -7.00000 q^{52} +5.00000 q^{53} -9.00000 q^{54} -3.00000 q^{57} -9.00000 q^{58} +4.00000 q^{59} -6.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -3.00000 q^{67} +7.00000 q^{68} +24.0000 q^{69} +2.00000 q^{71} -6.00000 q^{72} -3.00000 q^{73} -3.00000 q^{74} -15.0000 q^{75} -1.00000 q^{76} +21.0000 q^{78} +14.0000 q^{79} +9.00000 q^{81} +10.0000 q^{82} -6.00000 q^{83} -10.0000 q^{86} +27.0000 q^{87} +6.00000 q^{89} +8.00000 q^{92} +6.00000 q^{93} -3.00000 q^{94} -3.00000 q^{96} +14.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −3.00000 −1.22474
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 0 0
\(12\) 3.00000 0.866025
\(13\) −7.00000 −1.94145 −0.970725 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) −6.00000 −1.41421
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −3.00000 −0.612372
\(25\) −5.00000 −1.00000
\(26\) 7.00000 1.37281
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −7.00000 −1.20049
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 1.00000 0.162221
\(39\) −21.0000 −3.36269
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 3.00000 0.433013
\(49\) −7.00000 −1.00000
\(50\) 5.00000 0.707107
\(51\) 21.0000 2.94059
\(52\) −7.00000 −0.970725
\(53\) 5.00000 0.686803 0.343401 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) −9.00000 −1.18176
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 7.00000 0.848875
\(69\) 24.0000 2.88926
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −6.00000 −0.707107
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) −3.00000 −0.348743
\(75\) −15.0000 −1.73205
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 21.0000 2.37778
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 10.0000 1.10432
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 27.0000 2.89470
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 6.00000 0.622171
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 7.00000 0.707107
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) −21.0000 −2.07931
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 7.00000 0.686406
\(105\) 0 0
\(106\) −5.00000 −0.485643
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 9.00000 0.866025
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) 0 0
\(113\) −20.0000 −1.88144 −0.940721 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) −42.0000 −3.88290
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 6.00000 0.543214
\(123\) −30.0000 −2.70501
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 30.0000 2.64135
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) −24.0000 −2.04302
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −2.00000 −0.167836
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) 3.00000 0.248282
\(147\) −21.0000 −1.73205
\(148\) 3.00000 0.246598
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) 15.0000 1.22474
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 1.00000 0.0811107
\(153\) 42.0000 3.39550
\(154\) 0 0
\(155\) 0 0
\(156\) −21.0000 −1.68135
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −14.0000 −1.11378
\(159\) 15.0000 1.18958
\(160\) 0 0
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) 36.0000 2.76923
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 10.0000 0.762493
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) −27.0000 −2.04686
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) −6.00000 −0.449719
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −18.0000 −1.33060
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 3.00000 0.216506
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 5.00000 0.353553
\(201\) −9.00000 −0.634811
\(202\) −8.00000 −0.562878
\(203\) 0 0
\(204\) 21.0000 1.47029
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 48.0000 3.33623
\(208\) −7.00000 −0.485363
\(209\) 0 0
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 5.00000 0.343401
\(213\) 6.00000 0.411113
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) −49.0000 −3.29610
\(222\) −9.00000 −0.604040
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) −30.0000 −2.00000
\(226\) 20.0000 1.33038
\(227\) 17.0000 1.12833 0.564165 0.825662i \(-0.309198\pi\)
0.564165 + 0.825662i \(0.309198\pi\)
\(228\) −3.00000 −0.198680
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) −5.00000 −0.327561 −0.163780 0.986497i \(-0.552369\pi\)
−0.163780 + 0.986497i \(0.552369\pi\)
\(234\) 42.0000 2.74563
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 42.0000 2.72819
\(238\) 0 0
\(239\) 21.0000 1.35838 0.679189 0.733964i \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 30.0000 1.91273
\(247\) 7.00000 0.445399
\(248\) −2.00000 −0.127000
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −10.0000 −0.627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) −30.0000 −1.86772
\(259\) 0 0
\(260\) 0 0
\(261\) 54.0000 3.34252
\(262\) −12.0000 −0.741362
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) −3.00000 −0.183254
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) −29.0000 −1.76162 −0.880812 0.473466i \(-0.843003\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) 0 0
\(276\) 24.0000 1.44463
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 4.00000 0.239904
\(279\) 12.0000 0.718421
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −9.00000 −0.535942
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −6.00000 −0.353553
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 42.0000 2.46208
\(292\) −3.00000 −0.175562
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 21.0000 1.22474
\(295\) 0 0
\(296\) −3.00000 −0.174371
\(297\) 0 0
\(298\) −16.0000 −0.926855
\(299\) −56.0000 −3.23856
\(300\) −15.0000 −0.866025
\(301\) 0 0
\(302\) −6.00000 −0.345261
\(303\) 24.0000 1.37876
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −42.0000 −2.40098
\(307\) −19.0000 −1.08439 −0.542194 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −5.00000 −0.283524 −0.141762 0.989901i \(-0.545277\pi\)
−0.141762 + 0.989901i \(0.545277\pi\)
\(312\) 21.0000 1.18889
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −29.0000 −1.62880 −0.814401 0.580302i \(-0.802934\pi\)
−0.814401 + 0.580302i \(0.802934\pi\)
\(318\) −15.0000 −0.841158
\(319\) 0 0
\(320\) 0 0
\(321\) 36.0000 2.00932
\(322\) 0 0
\(323\) −7.00000 −0.389490
\(324\) 9.00000 0.500000
\(325\) 35.0000 1.94145
\(326\) 2.00000 0.110770
\(327\) 6.00000 0.331801
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −6.00000 −0.329293
\(333\) 18.0000 0.986394
\(334\) 14.0000 0.766046
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) −36.0000 −1.95814
\(339\) −60.0000 −3.25875
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 27.0000 1.44735
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −63.0000 −3.36269
\(352\) 0 0
\(353\) −15.0000 −0.798369 −0.399185 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 9.00000 0.475665
\(359\) −35.0000 −1.84723 −0.923615 0.383322i \(-0.874780\pi\)
−0.923615 + 0.383322i \(0.874780\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 18.0000 0.940875
\(367\) −23.0000 −1.20059 −0.600295 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(368\) 8.00000 0.417029
\(369\) −60.0000 −3.12348
\(370\) 0 0
\(371\) 0 0
\(372\) 6.00000 0.311086
\(373\) −5.00000 −0.258890 −0.129445 0.991587i \(-0.541320\pi\)
−0.129445 + 0.991587i \(0.541320\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) −63.0000 −3.24467
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 30.0000 1.53695
\(382\) 0 0
\(383\) −30.0000 −1.53293 −0.766464 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) 60.0000 3.04997
\(388\) 14.0000 0.710742
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 56.0000 2.83204
\(392\) 7.00000 0.353553
\(393\) 36.0000 1.81596
\(394\) −8.00000 −0.403034
\(395\) 0 0
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −7.00000 −0.350878
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 9.00000 0.448879
\(403\) −14.0000 −0.697390
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −21.0000 −1.03965
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −27.0000 −1.33181
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) −48.0000 −2.35907
\(415\) 0 0
\(416\) 7.00000 0.343203
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 1.00000 0.0486792
\(423\) 18.0000 0.875190
\(424\) −5.00000 −0.242821
\(425\) −35.0000 −1.69775
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 9.00000 0.433013
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −8.00000 −0.382692
\(438\) 9.00000 0.430037
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) −42.0000 −2.00000
\(442\) 49.0000 2.33069
\(443\) −2.00000 −0.0950229 −0.0475114 0.998871i \(-0.515129\pi\)
−0.0475114 + 0.998871i \(0.515129\pi\)
\(444\) 9.00000 0.427121
\(445\) 0 0
\(446\) −22.0000 −1.04173
\(447\) 48.0000 2.27032
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 30.0000 1.41421
\(451\) 0 0
\(452\) −20.0000 −0.940721
\(453\) 18.0000 0.845714
\(454\) −17.0000 −0.797850
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −4.00000 −0.186908
\(459\) 63.0000 2.94059
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 5.00000 0.231621
\(467\) −32.0000 −1.48078 −0.740392 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) −42.0000 −1.94145
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) −4.00000 −0.184115
\(473\) 0 0
\(474\) −42.0000 −1.92912
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) 30.0000 1.37361
\(478\) −21.0000 −0.960518
\(479\) 25.0000 1.14228 0.571140 0.820853i \(-0.306501\pi\)
0.571140 + 0.820853i \(0.306501\pi\)
\(480\) 0 0
\(481\) −21.0000 −0.957518
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.00000 0.271886 0.135943 0.990717i \(-0.456594\pi\)
0.135943 + 0.990717i \(0.456594\pi\)
\(488\) 6.00000 0.271607
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 14.0000 0.631811 0.315906 0.948791i \(-0.397692\pi\)
0.315906 + 0.948791i \(0.397692\pi\)
\(492\) −30.0000 −1.35250
\(493\) 63.0000 2.83738
\(494\) −7.00000 −0.314945
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 18.0000 0.806599
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) −42.0000 −1.87642
\(502\) −12.0000 −0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 108.000 4.79645
\(508\) 10.0000 0.443678
\(509\) −5.00000 −0.221621 −0.110811 0.993842i \(-0.535345\pi\)
−0.110811 + 0.993842i \(0.535345\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −9.00000 −0.397360
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 30.0000 1.32068
\(517\) 0 0
\(518\) 0 0
\(519\) −45.0000 −1.97528
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −54.0000 −2.36352
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 14.0000 0.609850
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 70.0000 3.03204
\(534\) −18.0000 −0.778936
\(535\) 0 0
\(536\) 3.00000 0.129580
\(537\) −27.0000 −1.16514
\(538\) 21.0000 0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 29.0000 1.24566
\(543\) 30.0000 1.28742
\(544\) −7.00000 −0.300123
\(545\) 0 0
\(546\) 0 0
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) −9.00000 −0.384461
\(549\) −36.0000 −1.53644
\(550\) 0 0
\(551\) −9.00000 −0.383413
\(552\) −24.0000 −1.02151
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) −12.0000 −0.508001
\(559\) −70.0000 −2.96068
\(560\) 0 0
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −19.0000 −0.800755 −0.400377 0.916350i \(-0.631121\pi\)
−0.400377 + 0.916350i \(0.631121\pi\)
\(564\) 9.00000 0.378968
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) −2.00000 −0.0839181
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) −14.0000 −0.585882 −0.292941 0.956131i \(-0.594634\pi\)
−0.292941 + 0.956131i \(0.594634\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −40.0000 −1.66812
\(576\) 6.00000 0.250000
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) −32.0000 −1.33102
\(579\) −48.0000 −1.99481
\(580\) 0 0
\(581\) 0 0
\(582\) −42.0000 −1.74096
\(583\) 0 0
\(584\) 3.00000 0.124141
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 46.0000 1.89862 0.949312 0.314337i \(-0.101782\pi\)
0.949312 + 0.314337i \(0.101782\pi\)
\(588\) −21.0000 −0.866025
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 3.00000 0.123299
\(593\) −1.00000 −0.0410651 −0.0205325 0.999789i \(-0.506536\pi\)
−0.0205325 + 0.999789i \(0.506536\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.0000 0.655386
\(597\) 21.0000 0.859473
\(598\) 56.0000 2.29001
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 15.0000 0.612372
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 0 0
\(603\) −18.0000 −0.733017
\(604\) 6.00000 0.244137
\(605\) 0 0
\(606\) −24.0000 −0.974933
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −21.0000 −0.849569
\(612\) 42.0000 1.69775
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 19.0000 0.766778
\(615\) 0 0
\(616\) 0 0
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 12.0000 0.482711
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 72.0000 2.88926
\(622\) 5.00000 0.200482
\(623\) 0 0
\(624\) −21.0000 −0.840673
\(625\) 25.0000 1.00000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) −14.0000 −0.556890
\(633\) −3.00000 −0.119239
\(634\) 29.0000 1.15174
\(635\) 0 0
\(636\) 15.0000 0.594789
\(637\) 49.0000 1.94145
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) −36.0000 −1.42081
\(643\) 46.0000 1.81406 0.907031 0.421063i \(-0.138343\pi\)
0.907031 + 0.421063i \(0.138343\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 7.00000 0.275411
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) −9.00000 −0.353553
\(649\) 0 0
\(650\) −35.0000 −1.37281
\(651\) 0 0
\(652\) −2.00000 −0.0783260
\(653\) 8.00000 0.313064 0.156532 0.987673i \(-0.449969\pi\)
0.156532 + 0.987673i \(0.449969\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) −18.0000 −0.702247
\(658\) 0 0
\(659\) −25.0000 −0.973862 −0.486931 0.873441i \(-0.661884\pi\)
−0.486931 + 0.873441i \(0.661884\pi\)
\(660\) 0 0
\(661\) 7.00000 0.272268 0.136134 0.990690i \(-0.456532\pi\)
0.136134 + 0.990690i \(0.456532\pi\)
\(662\) 20.0000 0.777322
\(663\) −147.000 −5.70901
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −18.0000 −0.697486
\(667\) 72.0000 2.78785
\(668\) −14.0000 −0.541676
\(669\) 66.0000 2.55171
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −16.0000 −0.616297
\(675\) −45.0000 −1.73205
\(676\) 36.0000 1.38462
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 60.0000 2.30429
\(679\) 0 0
\(680\) 0 0
\(681\) 51.0000 1.95432
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 10.0000 0.381246
\(689\) −35.0000 −1.33339
\(690\) 0 0
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) −15.0000 −0.570214
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) −27.0000 −1.02343
\(697\) −70.0000 −2.65144
\(698\) 26.0000 0.984115
\(699\) −15.0000 −0.567352
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 63.0000 2.37778
\(703\) −3.00000 −0.113147
\(704\) 0 0
\(705\) 0 0
\(706\) 15.0000 0.564532
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 84.0000 3.15025
\(712\) −6.00000 −0.224860
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) −9.00000 −0.336346
\(717\) 63.0000 2.35278
\(718\) 35.0000 1.30619
\(719\) −17.0000 −0.633993 −0.316997 0.948427i \(-0.602674\pi\)
−0.316997 + 0.948427i \(0.602674\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 30.0000 1.11571
\(724\) 10.0000 0.371647
\(725\) −45.0000 −1.67126
\(726\) 0 0
\(727\) −35.0000 −1.29808 −0.649039 0.760755i \(-0.724829\pi\)
−0.649039 + 0.760755i \(0.724829\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 70.0000 2.58904
\(732\) −18.0000 −0.665299
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 23.0000 0.848945
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 0 0
\(738\) 60.0000 2.20863
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 21.0000 0.771454
\(742\) 0 0
\(743\) 34.0000 1.24734 0.623670 0.781688i \(-0.285641\pi\)
0.623670 + 0.781688i \(0.285641\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) 5.00000 0.183063
\(747\) −36.0000 −1.31717
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 3.00000 0.109399
\(753\) 36.0000 1.31191
\(754\) 63.0000 2.29432
\(755\) 0 0
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −30.0000 −1.08679
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 30.0000 1.08394
\(767\) −28.0000 −1.01102
\(768\) 3.00000 0.108253
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −16.0000 −0.575853
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −60.0000 −2.15666
\(775\) −10.0000 −0.359211
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) −16.0000 −0.573628
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 0 0
\(782\) −56.0000 −2.00256
\(783\) 81.0000 2.89470
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −36.0000 −1.28408
\(787\) 31.0000 1.10503 0.552515 0.833503i \(-0.313668\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) 8.00000 0.284988
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 42.0000 1.49146
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 7.00000 0.248108
\(797\) 1.00000 0.0354218 0.0177109 0.999843i \(-0.494362\pi\)
0.0177109 + 0.999843i \(0.494362\pi\)
\(798\) 0 0
\(799\) 21.0000 0.742927
\(800\) 5.00000 0.176777
\(801\) 36.0000 1.27200
\(802\) 24.0000 0.847469
\(803\) 0 0
\(804\) −9.00000 −0.317406
\(805\) 0 0
\(806\) 14.0000 0.493129
\(807\) −63.0000 −2.21771
\(808\) −8.00000 −0.281439
\(809\) −3.00000 −0.105474 −0.0527372 0.998608i \(-0.516795\pi\)
−0.0527372 + 0.998608i \(0.516795\pi\)
\(810\) 0 0
\(811\) −47.0000 −1.65039 −0.825197 0.564846i \(-0.808936\pi\)
−0.825197 + 0.564846i \(0.808936\pi\)
\(812\) 0 0
\(813\) −87.0000 −3.05122
\(814\) 0 0
\(815\) 0 0
\(816\) 21.0000 0.735147
\(817\) −10.0000 −0.349856
\(818\) 20.0000 0.699284
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 27.0000 0.941733
\(823\) −25.0000 −0.871445 −0.435723 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −3.00000 −0.104320 −0.0521601 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 48.0000 1.66812
\(829\) −53.0000 −1.84077 −0.920383 0.391018i \(-0.872123\pi\)
−0.920383 + 0.391018i \(0.872123\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) −7.00000 −0.242681
\(833\) −49.0000 −1.69775
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 18.0000 0.622171
\(838\) −14.0000 −0.483622
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 10.0000 0.344623
\(843\) −54.0000 −1.85986
\(844\) −1.00000 −0.0344214
\(845\) 0 0
\(846\) −18.0000 −0.618853
\(847\) 0 0
\(848\) 5.00000 0.171701
\(849\) −48.0000 −1.64736
\(850\) 35.0000 1.20049
\(851\) 24.0000 0.822709
\(852\) 6.00000 0.205557
\(853\) −36.0000 −1.23262 −0.616308 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) −9.00000 −0.306186
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) 96.0000 3.26033
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 21.0000 0.711558
\(872\) −2.00000 −0.0677285
\(873\) 84.0000 2.84297
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) −9.00000 −0.304082
\(877\) −33.0000 −1.11433 −0.557165 0.830402i \(-0.688111\pi\)
−0.557165 + 0.830402i \(0.688111\pi\)
\(878\) −22.0000 −0.742464
\(879\) −27.0000 −0.910687
\(880\) 0 0
\(881\) −43.0000 −1.44871 −0.724353 0.689429i \(-0.757862\pi\)
−0.724353 + 0.689429i \(0.757862\pi\)
\(882\) 42.0000 1.41421
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) −49.0000 −1.64805
\(885\) 0 0
\(886\) 2.00000 0.0671913
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −9.00000 −0.302020
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 22.0000 0.736614
\(893\) −3.00000 −0.100391
\(894\) −48.0000 −1.60536
\(895\) 0 0
\(896\) 0 0
\(897\) −168.000 −5.60936
\(898\) −8.00000 −0.266963
\(899\) 18.0000 0.600334
\(900\) −30.0000 −1.00000
\(901\) 35.0000 1.16602
\(902\) 0 0
\(903\) 0 0
\(904\) 20.0000 0.665190
\(905\) 0 0
\(906\) −18.0000 −0.598010
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 17.0000 0.564165
\(909\) 48.0000 1.59206
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) −3.00000 −0.0993399
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) 0 0
\(918\) −63.0000 −2.07931
\(919\) 9.00000 0.296883 0.148441 0.988921i \(-0.452574\pi\)
0.148441 + 0.988921i \(0.452574\pi\)
\(920\) 0 0
\(921\) −57.0000 −1.87821
\(922\) −18.0000 −0.592798
\(923\) −14.0000 −0.460816
\(924\) 0 0
\(925\) −15.0000 −0.493197
\(926\) −32.0000 −1.05159
\(927\) −24.0000 −0.788263
\(928\) −9.00000 −0.295439
\(929\) −19.0000 −0.623370 −0.311685 0.950186i \(-0.600893\pi\)
−0.311685 + 0.950186i \(0.600893\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) −5.00000 −0.163780
\(933\) −15.0000 −0.491078
\(934\) 32.0000 1.04707
\(935\) 0 0
\(936\) 42.0000 1.37281
\(937\) −43.0000 −1.40475 −0.702374 0.711808i \(-0.747877\pi\)
−0.702374 + 0.711808i \(0.747877\pi\)
\(938\) 0 0
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) −27.0000 −0.880175 −0.440087 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 18.0000 0.586472
\(943\) −80.0000 −2.60516
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 42.0000 1.36410
\(949\) 21.0000 0.681689
\(950\) −5.00000 −0.162221
\(951\) −87.0000 −2.82117
\(952\) 0 0
\(953\) 28.0000 0.907009 0.453504 0.891254i \(-0.350174\pi\)
0.453504 + 0.891254i \(0.350174\pi\)
\(954\) −30.0000 −0.971286
\(955\) 0 0
\(956\) 21.0000 0.679189
\(957\) 0 0
\(958\) −25.0000 −0.807713
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 21.0000 0.677067
\(963\) 72.0000 2.32017
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −43.0000 −1.38279 −0.691393 0.722478i \(-0.743003\pi\)
−0.691393 + 0.722478i \(0.743003\pi\)
\(968\) 0 0
\(969\) −21.0000 −0.674617
\(970\) 0 0
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −6.00000 −0.192252
\(975\) 105.000 3.36269
\(976\) −6.00000 −0.192055
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 6.00000 0.191859
\(979\) 0 0
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) −14.0000 −0.446758
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 30.0000 0.956365
\(985\) 0 0
\(986\) −63.0000 −2.00633
\(987\) 0 0
\(988\) 7.00000 0.222700
\(989\) 80.0000 2.54385
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −60.0000 −1.90404
\(994\) 0 0
\(995\) 0 0
\(996\) −18.0000 −0.570352
\(997\) −32.0000 −1.01345 −0.506725 0.862108i \(-0.669144\pi\)
−0.506725 + 0.862108i \(0.669144\pi\)
\(998\) 40.0000 1.26618
\(999\) 27.0000 0.854242
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 4598.2.a.k.1.1 1
11.10 odd 2 4598.2.a.s.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.k.1.1 1 1.1 even 1 trivial
4598.2.a.s.1.1 yes 1 11.10 odd 2