Properties

Label 7098.2.a.y.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} +1.00000 q^{18} +3.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -3.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -7.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +3.00000 q^{38} -1.00000 q^{40} +4.00000 q^{41} +1.00000 q^{42} +5.00000 q^{43} -3.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} -7.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} +3.00000 q^{55} +1.00000 q^{56} +3.00000 q^{57} -1.00000 q^{58} -10.0000 q^{59} -1.00000 q^{60} -13.0000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -8.00000 q^{67} -7.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} +13.0000 q^{73} +1.00000 q^{74} -4.00000 q^{75} +3.00000 q^{76} -3.00000 q^{77} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} +2.00000 q^{83} +1.00000 q^{84} +7.00000 q^{85} +5.00000 q^{86} -1.00000 q^{87} -3.00000 q^{88} -12.0000 q^{89} -1.00000 q^{90} -1.00000 q^{92} -8.00000 q^{93} -3.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} +1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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 \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −3.00000 −0.639602
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.00000 −0.522233
\(34\) −7.00000 −1.20049
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 3.00000 0.486664
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 1.00000 0.154303
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −3.00000 −0.452267
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 3.00000 0.404520
\(56\) 1.00000 0.133631
\(57\) 3.00000 0.397360
\(58\) −1.00000 −0.131306
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −1.00000 −0.129099
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) −8.00000 −1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −7.00000 −0.848875
\(69\) −1.00000 −0.120386
\(70\) −1.00000 −0.119523
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) 13.0000 1.52153 0.760767 0.649025i \(-0.224823\pi\)
0.760767 + 0.649025i \(0.224823\pi\)
\(74\) 1.00000 0.116248
\(75\) −4.00000 −0.461880
\(76\) 3.00000 0.344124
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 1.00000 0.109109
\(85\) 7.00000 0.759257
\(86\) 5.00000 0.539164
\(87\) −1.00000 −0.107211
\(88\) −3.00000 −0.319801
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −3.00000 −0.307794
\(96\) 1.00000 0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 1.00000 0.101015
\(99\) −3.00000 −0.301511
\(100\) −4.00000 −0.400000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −7.00000 −0.693103
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) −6.00000 −0.582772
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 3.00000 0.286039
\(111\) 1.00000 0.0949158
\(112\) 1.00000 0.0944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 3.00000 0.280976
\(115\) 1.00000 0.0932505
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) −7.00000 −0.641689
\(120\) −1.00000 −0.0912871
\(121\) −2.00000 −0.181818
\(122\) −13.0000 −1.17696
\(123\) 4.00000 0.360668
\(124\) −8.00000 −0.718421
\(125\) 9.00000 0.804984
\(126\) 1.00000 0.0890871
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) −17.0000 −1.48530 −0.742648 0.669681i \(-0.766431\pi\)
−0.742648 + 0.669681i \(0.766431\pi\)
\(132\) −3.00000 −0.261116
\(133\) 3.00000 0.260133
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) −7.00000 −0.600245
\(137\) 17.0000 1.45241 0.726204 0.687479i \(-0.241283\pi\)
0.726204 + 0.687479i \(0.241283\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 1.00000 0.0830455
\(146\) 13.0000 1.07589
\(147\) 1.00000 0.0824786
\(148\) 1.00000 0.0821995
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) −4.00000 −0.326599
\(151\) 11.0000 0.895167 0.447584 0.894242i \(-0.352285\pi\)
0.447584 + 0.894242i \(0.352285\pi\)
\(152\) 3.00000 0.243332
\(153\) −7.00000 −0.565916
\(154\) −3.00000 −0.241747
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 19.0000 1.51637 0.758183 0.652042i \(-0.226088\pi\)
0.758183 + 0.652042i \(0.226088\pi\)
\(158\) −12.0000 −0.954669
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 4.00000 0.312348
\(165\) 3.00000 0.233550
\(166\) 2.00000 0.155230
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 7.00000 0.536875
\(171\) 3.00000 0.229416
\(172\) 5.00000 0.381246
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −1.00000 −0.0758098
\(175\) −4.00000 −0.302372
\(176\) −3.00000 −0.226134
\(177\) −10.0000 −0.751646
\(178\) −12.0000 −0.899438
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) −13.0000 −0.960988
\(184\) −1.00000 −0.0737210
\(185\) −1.00000 −0.0735215
\(186\) −8.00000 −0.586588
\(187\) 21.0000 1.53567
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) −3.00000 −0.217643
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 1.00000 0.0721688
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) −3.00000 −0.213201
\(199\) −9.00000 −0.637993 −0.318997 0.947756i \(-0.603346\pi\)
−0.318997 + 0.947756i \(0.603346\pi\)
\(200\) −4.00000 −0.282843
\(201\) −8.00000 −0.564276
\(202\) −14.0000 −0.985037
\(203\) −1.00000 −0.0701862
\(204\) −7.00000 −0.490098
\(205\) −4.00000 −0.279372
\(206\) 1.00000 0.0696733
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −9.00000 −0.622543
\(210\) −1.00000 −0.0690066
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) −6.00000 −0.412082
\(213\) 6.00000 0.411113
\(214\) 2.00000 0.136717
\(215\) −5.00000 −0.340997
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) 7.00000 0.474100
\(219\) 13.0000 0.878459
\(220\) 3.00000 0.202260
\(221\) 0 0
\(222\) 1.00000 0.0671156
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 1.00000 0.0668153
\(225\) −4.00000 −0.266667
\(226\) −10.0000 −0.665190
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 3.00000 0.198680
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 1.00000 0.0659380
\(231\) −3.00000 −0.197386
\(232\) −1.00000 −0.0656532
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) −12.0000 −0.779484
\(238\) −7.00000 −0.453743
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) −13.0000 −0.832240
\(245\) −1.00000 −0.0638877
\(246\) 4.00000 0.255031
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) 2.00000 0.126745
\(250\) 9.00000 0.569210
\(251\) −17.0000 −1.07303 −0.536515 0.843891i \(-0.680260\pi\)
−0.536515 + 0.843891i \(0.680260\pi\)
\(252\) 1.00000 0.0629941
\(253\) 3.00000 0.188608
\(254\) 4.00000 0.250982
\(255\) 7.00000 0.438357
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 5.00000 0.311286
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −17.0000 −1.05026
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −3.00000 −0.184637
\(265\) 6.00000 0.368577
\(266\) 3.00000 0.183942
\(267\) −12.0000 −0.734388
\(268\) −8.00000 −0.488678
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) 17.0000 1.02701
\(275\) 12.0000 0.723627
\(276\) −1.00000 −0.0601929
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) −8.00000 −0.479808
\(279\) −8.00000 −0.478947
\(280\) −1.00000 −0.0597614
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 6.00000 0.356034
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 1.00000 0.0589256
\(289\) 32.0000 1.88235
\(290\) 1.00000 0.0587220
\(291\) 6.00000 0.351726
\(292\) 13.0000 0.760767
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 1.00000 0.0583212
\(295\) 10.0000 0.582223
\(296\) 1.00000 0.0581238
\(297\) −3.00000 −0.174078
\(298\) −22.0000 −1.27443
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) 5.00000 0.288195
\(302\) 11.0000 0.632979
\(303\) −14.0000 −0.804279
\(304\) 3.00000 0.172062
\(305\) 13.0000 0.744378
\(306\) −7.00000 −0.400163
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) −3.00000 −0.170941
\(309\) 1.00000 0.0568880
\(310\) 8.00000 0.454369
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 19.0000 1.07223
\(315\) −1.00000 −0.0563436
\(316\) −12.0000 −0.675053
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) −6.00000 −0.336463
\(319\) 3.00000 0.167968
\(320\) −1.00000 −0.0559017
\(321\) 2.00000 0.111629
\(322\) −1.00000 −0.0557278
\(323\) −21.0000 −1.16847
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 7.00000 0.387101
\(328\) 4.00000 0.220863
\(329\) 0 0
\(330\) 3.00000 0.165145
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 2.00000 0.109764
\(333\) 1.00000 0.0547997
\(334\) −15.0000 −0.820763
\(335\) 8.00000 0.437087
\(336\) 1.00000 0.0545545
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 7.00000 0.379628
\(341\) 24.0000 1.29967
\(342\) 3.00000 0.162221
\(343\) 1.00000 0.0539949
\(344\) 5.00000 0.269582
\(345\) 1.00000 0.0538382
\(346\) 6.00000 0.322562
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −16.0000 −0.851594 −0.425797 0.904819i \(-0.640006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) −10.0000 −0.531494
\(355\) −6.00000 −0.318447
\(356\) −12.0000 −0.635999
\(357\) −7.00000 −0.370479
\(358\) 6.00000 0.317110
\(359\) 26.0000 1.37223 0.686114 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −10.0000 −0.526316
\(362\) 26.0000 1.36653
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −13.0000 −0.680451
\(366\) −13.0000 −0.679521
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 4.00000 0.208232
\(370\) −1.00000 −0.0519875
\(371\) −6.00000 −0.311504
\(372\) −8.00000 −0.414781
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 21.0000 1.08588
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −3.00000 −0.153897
\(381\) 4.00000 0.204926
\(382\) −9.00000 −0.460480
\(383\) 17.0000 0.868659 0.434330 0.900754i \(-0.356985\pi\)
0.434330 + 0.900754i \(0.356985\pi\)
\(384\) 1.00000 0.0510310
\(385\) 3.00000 0.152894
\(386\) 8.00000 0.407189
\(387\) 5.00000 0.254164
\(388\) 6.00000 0.304604
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 7.00000 0.354005
\(392\) 1.00000 0.0505076
\(393\) −17.0000 −0.857537
\(394\) 14.0000 0.705310
\(395\) 12.0000 0.603786
\(396\) −3.00000 −0.150756
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) −9.00000 −0.451129
\(399\) 3.00000 0.150188
\(400\) −4.00000 −0.200000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) −1.00000 −0.0496904
\(406\) −1.00000 −0.0496292
\(407\) −3.00000 −0.148704
\(408\) −7.00000 −0.346552
\(409\) −29.0000 −1.43396 −0.716979 0.697095i \(-0.754476\pi\)
−0.716979 + 0.697095i \(0.754476\pi\)
\(410\) −4.00000 −0.197546
\(411\) 17.0000 0.838548
\(412\) 1.00000 0.0492665
\(413\) −10.0000 −0.492068
\(414\) −1.00000 −0.0491473
\(415\) −2.00000 −0.0981761
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) −9.00000 −0.440204
\(419\) −11.0000 −0.537385 −0.268693 0.963226i \(-0.586592\pi\)
−0.268693 + 0.963226i \(0.586592\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 15.0000 0.730189
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 28.0000 1.35820
\(426\) 6.00000 0.290701
\(427\) −13.0000 −0.629114
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) −5.00000 −0.241121
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 1.00000 0.0481125
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) −8.00000 −0.384012
\(435\) 1.00000 0.0479463
\(436\) 7.00000 0.335239
\(437\) −3.00000 −0.143509
\(438\) 13.0000 0.621164
\(439\) −27.0000 −1.28864 −0.644320 0.764756i \(-0.722859\pi\)
−0.644320 + 0.764756i \(0.722859\pi\)
\(440\) 3.00000 0.143019
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 1.00000 0.0474579
\(445\) 12.0000 0.568855
\(446\) 4.00000 0.189405
\(447\) −22.0000 −1.04056
\(448\) 1.00000 0.0472456
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) −4.00000 −0.188562
\(451\) −12.0000 −0.565058
\(452\) −10.0000 −0.470360
\(453\) 11.0000 0.516825
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) 28.0000 1.30978 0.654892 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) −10.0000 −0.467269
\(459\) −7.00000 −0.326732
\(460\) 1.00000 0.0466252
\(461\) −27.0000 −1.25752 −0.628758 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) −3.00000 −0.139573
\(463\) 15.0000 0.697109 0.348555 0.937288i \(-0.386673\pi\)
0.348555 + 0.937288i \(0.386673\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 8.00000 0.370991
\(466\) −18.0000 −0.833834
\(467\) −9.00000 −0.416470 −0.208235 0.978079i \(-0.566772\pi\)
−0.208235 + 0.978079i \(0.566772\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 19.0000 0.875474
\(472\) −10.0000 −0.460287
\(473\) −15.0000 −0.689701
\(474\) −12.0000 −0.551178
\(475\) −12.0000 −0.550598
\(476\) −7.00000 −0.320844
\(477\) −6.00000 −0.274721
\(478\) 8.00000 0.365911
\(479\) 3.00000 0.137073 0.0685367 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −26.0000 −1.18427
\(483\) −1.00000 −0.0455016
\(484\) −2.00000 −0.0909091
\(485\) −6.00000 −0.272446
\(486\) 1.00000 0.0453609
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) −13.0000 −0.588482
\(489\) −4.00000 −0.180886
\(490\) −1.00000 −0.0451754
\(491\) 40.0000 1.80517 0.902587 0.430507i \(-0.141665\pi\)
0.902587 + 0.430507i \(0.141665\pi\)
\(492\) 4.00000 0.180334
\(493\) 7.00000 0.315264
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) −8.00000 −0.359211
\(497\) 6.00000 0.269137
\(498\) 2.00000 0.0896221
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 9.00000 0.402492
\(501\) −15.0000 −0.670151
\(502\) −17.0000 −0.758747
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 1.00000 0.0445435
\(505\) 14.0000 0.622992
\(506\) 3.00000 0.133366
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −27.0000 −1.19675 −0.598377 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 7.00000 0.309965
\(511\) 13.0000 0.575086
\(512\) 1.00000 0.0441942
\(513\) 3.00000 0.132453
\(514\) 6.00000 0.264649
\(515\) −1.00000 −0.0440653
\(516\) 5.00000 0.220113
\(517\) 0 0
\(518\) 1.00000 0.0439375
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −17.0000 −0.742648
\(525\) −4.00000 −0.174574
\(526\) −24.0000 −1.04645
\(527\) 56.0000 2.43940
\(528\) −3.00000 −0.130558
\(529\) −22.0000 −0.956522
\(530\) 6.00000 0.260623
\(531\) −10.0000 −0.433963
\(532\) 3.00000 0.130066
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) −2.00000 −0.0864675
\(536\) −8.00000 −0.345547
\(537\) 6.00000 0.258919
\(538\) −4.00000 −0.172452
\(539\) −3.00000 −0.129219
\(540\) −1.00000 −0.0430331
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) −20.0000 −0.859074
\(543\) 26.0000 1.11577
\(544\) −7.00000 −0.300123
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 17.0000 0.726204
\(549\) −13.0000 −0.554826
\(550\) 12.0000 0.511682
\(551\) −3.00000 −0.127804
\(552\) −1.00000 −0.0425628
\(553\) −12.0000 −0.510292
\(554\) −28.0000 −1.18961
\(555\) −1.00000 −0.0424476
\(556\) −8.00000 −0.339276
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) −1.00000 −0.0422577
\(561\) 21.0000 0.886621
\(562\) 10.0000 0.421825
\(563\) −41.0000 −1.72794 −0.863972 0.503540i \(-0.832031\pi\)
−0.863972 + 0.503540i \(0.832031\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 14.0000 0.588464
\(567\) 1.00000 0.0419961
\(568\) 6.00000 0.251754
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) −3.00000 −0.125656
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) −9.00000 −0.375980
\(574\) 4.00000 0.166957
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 32.0000 1.33102
\(579\) 8.00000 0.332469
\(580\) 1.00000 0.0415227
\(581\) 2.00000 0.0829740
\(582\) 6.00000 0.248708
\(583\) 18.0000 0.745484
\(584\) 13.0000 0.537944
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 1.00000 0.0412393
\(589\) −24.0000 −0.988903
\(590\) 10.0000 0.411693
\(591\) 14.0000 0.575883
\(592\) 1.00000 0.0410997
\(593\) 16.0000 0.657041 0.328521 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) −3.00000 −0.123091
\(595\) 7.00000 0.286972
\(596\) −22.0000 −0.901155
\(597\) −9.00000 −0.368345
\(598\) 0 0
\(599\) −3.00000 −0.122577 −0.0612883 0.998120i \(-0.519521\pi\)
−0.0612883 + 0.998120i \(0.519521\pi\)
\(600\) −4.00000 −0.163299
\(601\) 4.00000 0.163163 0.0815817 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) 5.00000 0.203785
\(603\) −8.00000 −0.325785
\(604\) 11.0000 0.447584
\(605\) 2.00000 0.0813116
\(606\) −14.0000 −0.568711
\(607\) −11.0000 −0.446476 −0.223238 0.974764i \(-0.571663\pi\)
−0.223238 + 0.974764i \(0.571663\pi\)
\(608\) 3.00000 0.121666
\(609\) −1.00000 −0.0405220
\(610\) 13.0000 0.526355
\(611\) 0 0
\(612\) −7.00000 −0.282958
\(613\) 31.0000 1.25208 0.626039 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) −16.0000 −0.645707
\(615\) −4.00000 −0.161296
\(616\) −3.00000 −0.120873
\(617\) −37.0000 −1.48956 −0.744782 0.667308i \(-0.767447\pi\)
−0.744782 + 0.667308i \(0.767447\pi\)
\(618\) 1.00000 0.0402259
\(619\) 23.0000 0.924448 0.462224 0.886763i \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(620\) 8.00000 0.321288
\(621\) −1.00000 −0.0401286
\(622\) 4.00000 0.160385
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −26.0000 −1.03917
\(627\) −9.00000 −0.359425
\(628\) 19.0000 0.758183
\(629\) −7.00000 −0.279108
\(630\) −1.00000 −0.0398410
\(631\) 47.0000 1.87104 0.935520 0.353273i \(-0.114931\pi\)
0.935520 + 0.353273i \(0.114931\pi\)
\(632\) −12.0000 −0.477334
\(633\) 15.0000 0.596196
\(634\) 24.0000 0.953162
\(635\) −4.00000 −0.158735
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 3.00000 0.118771
\(639\) 6.00000 0.237356
\(640\) −1.00000 −0.0395285
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 2.00000 0.0789337
\(643\) −39.0000 −1.53801 −0.769005 0.639243i \(-0.779248\pi\)
−0.769005 + 0.639243i \(0.779248\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −5.00000 −0.196875
\(646\) −21.0000 −0.826234
\(647\) −22.0000 −0.864909 −0.432455 0.901656i \(-0.642352\pi\)
−0.432455 + 0.901656i \(0.642352\pi\)
\(648\) 1.00000 0.0392837
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) −4.00000 −0.156652
\(653\) −35.0000 −1.36966 −0.684828 0.728705i \(-0.740123\pi\)
−0.684828 + 0.728705i \(0.740123\pi\)
\(654\) 7.00000 0.273722
\(655\) 17.0000 0.664245
\(656\) 4.00000 0.156174
\(657\) 13.0000 0.507178
\(658\) 0 0
\(659\) 26.0000 1.01282 0.506408 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) 3.00000 0.116775
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) −3.00000 −0.116335
\(666\) 1.00000 0.0387492
\(667\) 1.00000 0.0387202
\(668\) −15.0000 −0.580367
\(669\) 4.00000 0.154649
\(670\) 8.00000 0.309067
\(671\) 39.0000 1.50558
\(672\) 1.00000 0.0385758
\(673\) 3.00000 0.115642 0.0578208 0.998327i \(-0.481585\pi\)
0.0578208 + 0.998327i \(0.481585\pi\)
\(674\) 23.0000 0.885927
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −10.0000 −0.384048
\(679\) 6.00000 0.230259
\(680\) 7.00000 0.268438
\(681\) 20.0000 0.766402
\(682\) 24.0000 0.919007
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 3.00000 0.114708
\(685\) −17.0000 −0.649537
\(686\) 1.00000 0.0381802
\(687\) −10.0000 −0.381524
\(688\) 5.00000 0.190623
\(689\) 0 0
\(690\) 1.00000 0.0380693
\(691\) −16.0000 −0.608669 −0.304334 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 6.00000 0.228086
\(693\) −3.00000 −0.113961
\(694\) −24.0000 −0.911028
\(695\) 8.00000 0.303457
\(696\) −1.00000 −0.0379049
\(697\) −28.0000 −1.06058
\(698\) 16.0000 0.605609
\(699\) −18.0000 −0.680823
\(700\) −4.00000 −0.151186
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 3.00000 0.113147
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −16.0000 −0.602168
\(707\) −14.0000 −0.526524
\(708\) −10.0000 −0.375823
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) −6.00000 −0.225176
\(711\) −12.0000 −0.450035
\(712\) −12.0000 −0.449719
\(713\) 8.00000 0.299602
\(714\) −7.00000 −0.261968
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 8.00000 0.298765
\(718\) 26.0000 0.970311
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 1.00000 0.0372419
\(722\) −10.0000 −0.372161
\(723\) −26.0000 −0.966950
\(724\) 26.0000 0.966282
\(725\) 4.00000 0.148556
\(726\) −2.00000 −0.0742270
\(727\) −35.0000 −1.29808 −0.649039 0.760755i \(-0.724829\pi\)
−0.649039 + 0.760755i \(0.724829\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −13.0000 −0.481152
\(731\) −35.0000 −1.29452
\(732\) −13.0000 −0.480494
\(733\) −8.00000 −0.295487 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) 32.0000 1.18114
\(735\) −1.00000 −0.0368856
\(736\) −1.00000 −0.0368605
\(737\) 24.0000 0.884051
\(738\) 4.00000 0.147242
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) −8.00000 −0.293294
\(745\) 22.0000 0.806018
\(746\) 6.00000 0.219676
\(747\) 2.00000 0.0731762
\(748\) 21.0000 0.767836
\(749\) 2.00000 0.0730784
\(750\) 9.00000 0.328634
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) −17.0000 −0.619514
\(754\) 0 0
\(755\) −11.0000 −0.400331
\(756\) 1.00000 0.0363696
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 4.00000 0.145287
\(759\) 3.00000 0.108893
\(760\) −3.00000 −0.108821
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 4.00000 0.144905
\(763\) 7.00000 0.253417
\(764\) −9.00000 −0.325609
\(765\) 7.00000 0.253086
\(766\) 17.0000 0.614235
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 39.0000 1.40638 0.703188 0.711004i \(-0.251759\pi\)
0.703188 + 0.711004i \(0.251759\pi\)
\(770\) 3.00000 0.108112
\(771\) 6.00000 0.216085
\(772\) 8.00000 0.287926
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 5.00000 0.179721
\(775\) 32.0000 1.14947
\(776\) 6.00000 0.215387
\(777\) 1.00000 0.0358748
\(778\) 18.0000 0.645331
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 7.00000 0.250319
\(783\) −1.00000 −0.0357371
\(784\) 1.00000 0.0357143
\(785\) −19.0000 −0.678139
\(786\) −17.0000 −0.606370
\(787\) −17.0000 −0.605985 −0.302992 0.952993i \(-0.597986\pi\)
−0.302992 + 0.952993i \(0.597986\pi\)
\(788\) 14.0000 0.498729
\(789\) −24.0000 −0.854423
\(790\) 12.0000 0.426941
\(791\) −10.0000 −0.355559
\(792\) −3.00000 −0.106600
\(793\) 0 0
\(794\) −8.00000 −0.283909
\(795\) 6.00000 0.212798
\(796\) −9.00000 −0.318997
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 3.00000 0.106199
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) −12.0000 −0.423999
\(802\) 6.00000 0.211867
\(803\) −39.0000 −1.37628
\(804\) −8.00000 −0.282138
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) −4.00000 −0.140807
\(808\) −14.0000 −0.492518
\(809\) 52.0000 1.82822 0.914111 0.405463i \(-0.132890\pi\)
0.914111 + 0.405463i \(0.132890\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 7.00000 0.245803 0.122902 0.992419i \(-0.460780\pi\)
0.122902 + 0.992419i \(0.460780\pi\)
\(812\) −1.00000 −0.0350931
\(813\) −20.0000 −0.701431
\(814\) −3.00000 −0.105150
\(815\) 4.00000 0.140114
\(816\) −7.00000 −0.245049
\(817\) 15.0000 0.524784
\(818\) −29.0000 −1.01396
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 17.0000 0.592943
\(823\) −12.0000 −0.418294 −0.209147 0.977884i \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) 1.00000 0.0348367
\(825\) 12.0000 0.417786
\(826\) −10.0000 −0.347945
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 15.0000 0.520972 0.260486 0.965478i \(-0.416117\pi\)
0.260486 + 0.965478i \(0.416117\pi\)
\(830\) −2.00000 −0.0694210
\(831\) −28.0000 −0.971309
\(832\) 0 0
\(833\) −7.00000 −0.242536
\(834\) −8.00000 −0.277017
\(835\) 15.0000 0.519096
\(836\) −9.00000 −0.311272
\(837\) −8.00000 −0.276520
\(838\) −11.0000 −0.379989
\(839\) 28.0000 0.966667 0.483334 0.875436i \(-0.339426\pi\)
0.483334 + 0.875436i \(0.339426\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −28.0000 −0.965517
\(842\) −34.0000 −1.17172
\(843\) 10.0000 0.344418
\(844\) 15.0000 0.516321
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) −6.00000 −0.206041
\(849\) 14.0000 0.480479
\(850\) 28.0000 0.960392
\(851\) −1.00000 −0.0342796
\(852\) 6.00000 0.205557
\(853\) −32.0000 −1.09566 −0.547830 0.836590i \(-0.684546\pi\)
−0.547830 + 0.836590i \(0.684546\pi\)
\(854\) −13.0000 −0.444851
\(855\) −3.00000 −0.102598
\(856\) 2.00000 0.0683586
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 0 0
\(859\) 22.0000 0.750630 0.375315 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) −5.00000 −0.170499
\(861\) 4.00000 0.136320
\(862\) −14.0000 −0.476842
\(863\) 28.0000 0.953131 0.476566 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) 8.00000 0.271851
\(867\) 32.0000 1.08678
\(868\) −8.00000 −0.271538
\(869\) 36.0000 1.22122
\(870\) 1.00000 0.0339032
\(871\) 0 0
\(872\) 7.00000 0.237050
\(873\) 6.00000 0.203069
\(874\) −3.00000 −0.101477
\(875\) 9.00000 0.304256
\(876\) 13.0000 0.439229
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −27.0000 −0.911206
\(879\) 14.0000 0.472208
\(880\) 3.00000 0.101130
\(881\) −51.0000 −1.71823 −0.859117 0.511780i \(-0.828986\pi\)
−0.859117 + 0.511780i \(0.828986\pi\)
\(882\) 1.00000 0.0336718
\(883\) −3.00000 −0.100958 −0.0504790 0.998725i \(-0.516075\pi\)
−0.0504790 + 0.998725i \(0.516075\pi\)
\(884\) 0 0
\(885\) 10.0000 0.336146
\(886\) −34.0000 −1.14225
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 1.00000 0.0335578
\(889\) 4.00000 0.134156
\(890\) 12.0000 0.402241
\(891\) −3.00000 −0.100504
\(892\) 4.00000 0.133930
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) −6.00000 −0.200558
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 9.00000 0.300334
\(899\) 8.00000 0.266815
\(900\) −4.00000 −0.133333
\(901\) 42.0000 1.39922
\(902\) −12.0000 −0.399556
\(903\) 5.00000 0.166390
\(904\) −10.0000 −0.332595
\(905\) −26.0000 −0.864269
\(906\) 11.0000 0.365451
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 20.0000 0.663723
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 57.0000 1.88849 0.944247 0.329238i \(-0.106792\pi\)
0.944247 + 0.329238i \(0.106792\pi\)
\(912\) 3.00000 0.0993399
\(913\) −6.00000 −0.198571
\(914\) 28.0000 0.926158
\(915\) 13.0000 0.429767
\(916\) −10.0000 −0.330409
\(917\) −17.0000 −0.561389
\(918\) −7.00000 −0.231034
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 1.00000 0.0329690
\(921\) −16.0000 −0.527218
\(922\) −27.0000 −0.889198
\(923\) 0 0
\(924\) −3.00000 −0.0986928
\(925\) −4.00000 −0.131519
\(926\) 15.0000 0.492931
\(927\) 1.00000 0.0328443
\(928\) −1.00000 −0.0328266
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 8.00000 0.262330
\(931\) 3.00000 0.0983210
\(932\) −18.0000 −0.589610
\(933\) 4.00000 0.130954
\(934\) −9.00000 −0.294489
\(935\) −21.0000 −0.686773
\(936\) 0 0
\(937\) 4.00000 0.130674 0.0653372 0.997863i \(-0.479188\pi\)
0.0653372 + 0.997863i \(0.479188\pi\)
\(938\) −8.00000 −0.261209
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 19.0000 0.619053
\(943\) −4.00000 −0.130258
\(944\) −10.0000 −0.325472
\(945\) −1.00000 −0.0325300
\(946\) −15.0000 −0.487692
\(947\) 25.0000 0.812391 0.406195 0.913786i \(-0.366855\pi\)
0.406195 + 0.913786i \(0.366855\pi\)
\(948\) −12.0000 −0.389742
\(949\) 0 0
\(950\) −12.0000 −0.389331
\(951\) 24.0000 0.778253
\(952\) −7.00000 −0.226871
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) −6.00000 −0.194257
\(955\) 9.00000 0.291233
\(956\) 8.00000 0.258738
\(957\) 3.00000 0.0969762
\(958\) 3.00000 0.0969256
\(959\) 17.0000 0.548959
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) −26.0000 −0.837404
\(965\) −8.00000 −0.257529
\(966\) −1.00000 −0.0321745
\(967\) −43.0000 −1.38279 −0.691393 0.722478i \(-0.743003\pi\)
−0.691393 + 0.722478i \(0.743003\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −21.0000 −0.674617
\(970\) −6.00000 −0.192648
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) −8.00000 −0.256468
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) −13.0000 −0.416120
\(977\) 15.0000 0.479893 0.239946 0.970786i \(-0.422870\pi\)
0.239946 + 0.970786i \(0.422870\pi\)
\(978\) −4.00000 −0.127906
\(979\) 36.0000 1.15056
\(980\) −1.00000 −0.0319438
\(981\) 7.00000 0.223493
\(982\) 40.0000 1.27645
\(983\) 51.0000 1.62665 0.813324 0.581811i \(-0.197656\pi\)
0.813324 + 0.581811i \(0.197656\pi\)
\(984\) 4.00000 0.127515
\(985\) −14.0000 −0.446077
\(986\) 7.00000 0.222925
\(987\) 0 0
\(988\) 0 0
\(989\) −5.00000 −0.158991
\(990\) 3.00000 0.0953463
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −8.00000 −0.254000
\(993\) 8.00000 0.253872
\(994\) 6.00000 0.190308
\(995\) 9.00000 0.285319
\(996\) 2.00000 0.0633724
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) 14.0000 0.443162
\(999\) 1.00000 0.0316386
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 7098.2.a.y.1.1 1
13.5 odd 4 546.2.c.c.337.1 2
13.8 odd 4 546.2.c.c.337.2 yes 2
13.12 even 2 7098.2.a.o.1.1 1
39.5 even 4 1638.2.c.e.883.2 2
39.8 even 4 1638.2.c.e.883.1 2
52.31 even 4 4368.2.h.h.337.2 2
52.47 even 4 4368.2.h.h.337.1 2
91.34 even 4 3822.2.c.b.883.2 2
91.83 even 4 3822.2.c.b.883.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.c.337.1 2 13.5 odd 4
546.2.c.c.337.2 yes 2 13.8 odd 4
1638.2.c.e.883.1 2 39.8 even 4
1638.2.c.e.883.2 2 39.5 even 4
3822.2.c.b.883.1 2 91.83 even 4
3822.2.c.b.883.2 2 91.34 even 4
4368.2.h.h.337.1 2 52.47 even 4
4368.2.h.h.337.2 2 52.31 even 4
7098.2.a.o.1.1 1 13.12 even 2
7098.2.a.y.1.1 1 1.1 even 1 trivial