Properties

Label 4002.2.a.k.1.1
Level 4002
Weight 2
Character 4002.1
Self dual yes
Analytic conductor 31.956
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 4002.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +1.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -9.00000 q^{41} -1.00000 q^{42} -5.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +7.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} +3.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} -1.00000 q^{57} -1.00000 q^{58} -9.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -8.00000 q^{67} -3.00000 q^{68} -1.00000 q^{69} +1.00000 q^{70} -2.00000 q^{71} +1.00000 q^{72} +14.0000 q^{73} -1.00000 q^{74} +4.00000 q^{75} +1.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -8.00000 q^{83} -1.00000 q^{84} -3.00000 q^{85} -5.00000 q^{86} +1.00000 q^{87} -4.00000 q^{88} +18.0000 q^{89} +1.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} +2.00000 q^{93} +7.00000 q^{94} +1.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} -6.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −4.00000 −0.852803
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −1.00000 −0.185695
\(30\) −1.00000 −0.182574
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −3.00000 −0.514496
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 1.00000 0.162221
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) −1.00000 −0.154303
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) −4.00000 −0.603023
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) −4.00000 −0.565685
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) 1.00000 0.133631
\(57\) −1.00000 −0.132453
\(58\) −1.00000 −0.131306
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −2.00000 −0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −3.00000 −0.363803
\(69\) −1.00000 −0.120386
\(70\) 1.00000 0.119523
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −1.00000 −0.116248
\(75\) 4.00000 0.461880
\(76\) 1.00000 0.114708
\(77\) −4.00000 −0.455842
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) −1.00000 −0.109109
\(85\) −3.00000 −0.325396
\(86\) −5.00000 −0.539164
\(87\) 1.00000 0.107211
\(88\) −4.00000 −0.426401
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 1.00000 0.105409
\(91\) −2.00000 −0.209657
\(92\) 1.00000 0.104257
\(93\) 2.00000 0.207390
\(94\) 7.00000 0.721995
\(95\) 1.00000 0.102598
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −6.00000 −0.606092
\(99\) −4.00000 −0.402015
\(100\) −4.00000 −0.400000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 3.00000 0.297044
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) −2.00000 −0.196116
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) 13.0000 1.25676 0.628379 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) −4.00000 −0.381385
\(111\) 1.00000 0.0949158
\(112\) 1.00000 0.0944911
\(113\) 11.0000 1.03479 0.517396 0.855746i \(-0.326901\pi\)
0.517396 + 0.855746i \(0.326901\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 1.00000 0.0932505
\(116\) −1.00000 −0.0928477
\(117\) −2.00000 −0.184900
\(118\) −9.00000 −0.828517
\(119\) −3.00000 −0.275010
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 9.00000 0.811503
\(124\) −2.00000 −0.179605
\(125\) −9.00000 −0.804984
\(126\) 1.00000 0.0890871
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.00000 0.440225
\(130\) −2.00000 −0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 4.00000 0.348155
\(133\) 1.00000 0.0867110
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) −3.00000 −0.257248
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 1.00000 0.0845154
\(141\) −7.00000 −0.589506
\(142\) −2.00000 −0.167836
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) −1.00000 −0.0830455
\(146\) 14.0000 1.15865
\(147\) 6.00000 0.494872
\(148\) −1.00000 −0.0821995
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 4.00000 0.326599
\(151\) −13.0000 −1.05792 −0.528962 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) 1.00000 0.0811107
\(153\) −3.00000 −0.242536
\(154\) −4.00000 −0.322329
\(155\) −2.00000 −0.160644
\(156\) 2.00000 0.160128
\(157\) 19.0000 1.51637 0.758183 0.652042i \(-0.226088\pi\)
0.758183 + 0.652042i \(0.226088\pi\)
\(158\) −4.00000 −0.318223
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) −9.00000 −0.702782
\(165\) 4.00000 0.311400
\(166\) −8.00000 −0.620920
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) −3.00000 −0.230089
\(171\) 1.00000 0.0764719
\(172\) −5.00000 −0.381246
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 1.00000 0.0758098
\(175\) −4.00000 −0.302372
\(176\) −4.00000 −0.301511
\(177\) 9.00000 0.676481
\(178\) 18.0000 1.34916
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 1.00000 0.0745356
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) −2.00000 −0.148250
\(183\) 10.0000 0.739221
\(184\) 1.00000 0.0737210
\(185\) −1.00000 −0.0735215
\(186\) 2.00000 0.146647
\(187\) 12.0000 0.877527
\(188\) 7.00000 0.510527
\(189\) −1.00000 −0.0727393
\(190\) 1.00000 0.0725476
\(191\) 19.0000 1.37479 0.687396 0.726283i \(-0.258754\pi\)
0.687396 + 0.726283i \(0.258754\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) −8.00000 −0.574367
\(195\) 2.00000 0.143223
\(196\) −6.00000 −0.428571
\(197\) −17.0000 −1.21120 −0.605600 0.795769i \(-0.707067\pi\)
−0.605600 + 0.795769i \(0.707067\pi\)
\(198\) −4.00000 −0.284268
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −4.00000 −0.282843
\(201\) 8.00000 0.564276
\(202\) 2.00000 0.140720
\(203\) −1.00000 −0.0701862
\(204\) 3.00000 0.210042
\(205\) −9.00000 −0.628587
\(206\) −1.00000 −0.0696733
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) −4.00000 −0.276686
\(210\) −1.00000 −0.0690066
\(211\) 3.00000 0.206529 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 6.00000 0.412082
\(213\) 2.00000 0.137038
\(214\) 13.0000 0.888662
\(215\) −5.00000 −0.340997
\(216\) −1.00000 −0.0680414
\(217\) −2.00000 −0.135769
\(218\) −8.00000 −0.541828
\(219\) −14.0000 −0.946032
\(220\) −4.00000 −0.269680
\(221\) 6.00000 0.403604
\(222\) 1.00000 0.0671156
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 1.00000 0.0668153
\(225\) −4.00000 −0.266667
\(226\) 11.0000 0.731709
\(227\) −11.0000 −0.730096 −0.365048 0.930989i \(-0.618947\pi\)
−0.365048 + 0.930989i \(0.618947\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −19.0000 −1.25556 −0.627778 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 1.00000 0.0659380
\(231\) 4.00000 0.263181
\(232\) −1.00000 −0.0656532
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) −2.00000 −0.130744
\(235\) 7.00000 0.456630
\(236\) −9.00000 −0.585850
\(237\) 4.00000 0.259828
\(238\) −3.00000 −0.194461
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 3.00000 0.193247 0.0966235 0.995321i \(-0.469196\pi\)
0.0966235 + 0.995321i \(0.469196\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) −6.00000 −0.383326
\(246\) 9.00000 0.573819
\(247\) −2.00000 −0.127257
\(248\) −2.00000 −0.127000
\(249\) 8.00000 0.506979
\(250\) −9.00000 −0.569210
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) 2.00000 0.125491
\(255\) 3.00000 0.187867
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 5.00000 0.311286
\(259\) −1.00000 −0.0621370
\(260\) −2.00000 −0.124035
\(261\) −1.00000 −0.0618984
\(262\) 6.00000 0.370681
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 4.00000 0.246183
\(265\) 6.00000 0.368577
\(266\) 1.00000 0.0613139
\(267\) −18.0000 −1.10158
\(268\) −8.00000 −0.488678
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −3.00000 −0.181902
\(273\) 2.00000 0.121046
\(274\) −18.0000 −1.08742
\(275\) 16.0000 0.964836
\(276\) −1.00000 −0.0601929
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 6.00000 0.359856
\(279\) −2.00000 −0.119737
\(280\) 1.00000 0.0597614
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −7.00000 −0.416844
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −2.00000 −0.118678
\(285\) −1.00000 −0.0592349
\(286\) 8.00000 0.473050
\(287\) −9.00000 −0.531253
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) −1.00000 −0.0587220
\(291\) 8.00000 0.468968
\(292\) 14.0000 0.819288
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 6.00000 0.349927
\(295\) −9.00000 −0.524000
\(296\) −1.00000 −0.0581238
\(297\) 4.00000 0.232104
\(298\) −9.00000 −0.521356
\(299\) −2.00000 −0.115663
\(300\) 4.00000 0.230940
\(301\) −5.00000 −0.288195
\(302\) −13.0000 −0.748066
\(303\) −2.00000 −0.114897
\(304\) 1.00000 0.0573539
\(305\) −10.0000 −0.572598
\(306\) −3.00000 −0.171499
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −4.00000 −0.227921
\(309\) 1.00000 0.0568880
\(310\) −2.00000 −0.113592
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) 2.00000 0.113228
\(313\) 17.0000 0.960897 0.480448 0.877023i \(-0.340474\pi\)
0.480448 + 0.877023i \(0.340474\pi\)
\(314\) 19.0000 1.07223
\(315\) 1.00000 0.0563436
\(316\) −4.00000 −0.225018
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −6.00000 −0.336463
\(319\) 4.00000 0.223957
\(320\) 1.00000 0.0559017
\(321\) −13.0000 −0.725589
\(322\) 1.00000 0.0557278
\(323\) −3.00000 −0.166924
\(324\) 1.00000 0.0555556
\(325\) 8.00000 0.443760
\(326\) 1.00000 0.0553849
\(327\) 8.00000 0.442401
\(328\) −9.00000 −0.496942
\(329\) 7.00000 0.385922
\(330\) 4.00000 0.220193
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) −8.00000 −0.439057
\(333\) −1.00000 −0.0547997
\(334\) −14.0000 −0.766046
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) −9.00000 −0.489535
\(339\) −11.0000 −0.597438
\(340\) −3.00000 −0.162698
\(341\) 8.00000 0.433224
\(342\) 1.00000 0.0540738
\(343\) −13.0000 −0.701934
\(344\) −5.00000 −0.269582
\(345\) −1.00000 −0.0538382
\(346\) −15.0000 −0.806405
\(347\) −11.0000 −0.590511 −0.295255 0.955418i \(-0.595405\pi\)
−0.295255 + 0.955418i \(0.595405\pi\)
\(348\) 1.00000 0.0536056
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) −4.00000 −0.213809
\(351\) 2.00000 0.106752
\(352\) −4.00000 −0.213201
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 9.00000 0.478345
\(355\) −2.00000 −0.106149
\(356\) 18.0000 0.953998
\(357\) 3.00000 0.158777
\(358\) 20.0000 1.05703
\(359\) −31.0000 −1.63612 −0.818059 0.575135i \(-0.804950\pi\)
−0.818059 + 0.575135i \(0.804950\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.0000 −0.947368
\(362\) −18.0000 −0.946059
\(363\) −5.00000 −0.262432
\(364\) −2.00000 −0.104828
\(365\) 14.0000 0.732793
\(366\) 10.0000 0.522708
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 1.00000 0.0521286
\(369\) −9.00000 −0.468521
\(370\) −1.00000 −0.0519875
\(371\) 6.00000 0.311504
\(372\) 2.00000 0.103695
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 12.0000 0.620505
\(375\) 9.00000 0.464758
\(376\) 7.00000 0.360997
\(377\) 2.00000 0.103005
\(378\) −1.00000 −0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 1.00000 0.0512989
\(381\) −2.00000 −0.102463
\(382\) 19.0000 0.972125
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −4.00000 −0.203859
\(386\) −12.0000 −0.610784
\(387\) −5.00000 −0.254164
\(388\) −8.00000 −0.406138
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 2.00000 0.101274
\(391\) −3.00000 −0.151717
\(392\) −6.00000 −0.303046
\(393\) −6.00000 −0.302660
\(394\) −17.0000 −0.856448
\(395\) −4.00000 −0.201262
\(396\) −4.00000 −0.201008
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 4.00000 0.200502
\(399\) −1.00000 −0.0500626
\(400\) −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
\(403\) 4.00000 0.199254
\(404\) 2.00000 0.0995037
\(405\) 1.00000 0.0496904
\(406\) −1.00000 −0.0496292
\(407\) 4.00000 0.198273
\(408\) 3.00000 0.148522
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −9.00000 −0.444478
\(411\) 18.0000 0.887875
\(412\) −1.00000 −0.0492665
\(413\) −9.00000 −0.442861
\(414\) 1.00000 0.0491473
\(415\) −8.00000 −0.392705
\(416\) −2.00000 −0.0980581
\(417\) −6.00000 −0.293821
\(418\) −4.00000 −0.195646
\(419\) −5.00000 −0.244266 −0.122133 0.992514i \(-0.538973\pi\)
−0.122133 + 0.992514i \(0.538973\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 3.00000 0.146038
\(423\) 7.00000 0.340352
\(424\) 6.00000 0.291386
\(425\) 12.0000 0.582086
\(426\) 2.00000 0.0969003
\(427\) −10.0000 −0.483934
\(428\) 13.0000 0.628379
\(429\) −8.00000 −0.386244
\(430\) −5.00000 −0.241121
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 1.00000 0.0479463
\(436\) −8.00000 −0.383131
\(437\) 1.00000 0.0478365
\(438\) −14.0000 −0.668946
\(439\) 33.0000 1.57500 0.787502 0.616312i \(-0.211374\pi\)
0.787502 + 0.616312i \(0.211374\pi\)
\(440\) −4.00000 −0.190693
\(441\) −6.00000 −0.285714
\(442\) 6.00000 0.285391
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 1.00000 0.0474579
\(445\) 18.0000 0.853282
\(446\) 16.0000 0.757622
\(447\) 9.00000 0.425685
\(448\) 1.00000 0.0472456
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) −4.00000 −0.188562
\(451\) 36.0000 1.69517
\(452\) 11.0000 0.517396
\(453\) 13.0000 0.610793
\(454\) −11.0000 −0.516256
\(455\) −2.00000 −0.0937614
\(456\) −1.00000 −0.0468293
\(457\) 37.0000 1.73079 0.865393 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) −19.0000 −0.887812
\(459\) 3.00000 0.140028
\(460\) 1.00000 0.0466252
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 4.00000 0.186097
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 2.00000 0.0927478
\(466\) 4.00000 0.185296
\(467\) 10.0000 0.462745 0.231372 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) 7.00000 0.322886
\(471\) −19.0000 −0.875474
\(472\) −9.00000 −0.414259
\(473\) 20.0000 0.919601
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) −3.00000 −0.137505
\(477\) 6.00000 0.274721
\(478\) −6.00000 −0.274434
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 2.00000 0.0911922
\(482\) 3.00000 0.136646
\(483\) −1.00000 −0.0455016
\(484\) 5.00000 0.227273
\(485\) −8.00000 −0.363261
\(486\) −1.00000 −0.0453609
\(487\) −3.00000 −0.135943 −0.0679715 0.997687i \(-0.521653\pi\)
−0.0679715 + 0.997687i \(0.521653\pi\)
\(488\) −10.0000 −0.452679
\(489\) −1.00000 −0.0452216
\(490\) −6.00000 −0.271052
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 9.00000 0.405751
\(493\) 3.00000 0.135113
\(494\) −2.00000 −0.0899843
\(495\) −4.00000 −0.179787
\(496\) −2.00000 −0.0898027
\(497\) −2.00000 −0.0897123
\(498\) 8.00000 0.358489
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) −9.00000 −0.402492
\(501\) 14.0000 0.625474
\(502\) 8.00000 0.357057
\(503\) 27.0000 1.20387 0.601935 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 1.00000 0.0445435
\(505\) 2.00000 0.0889988
\(506\) −4.00000 −0.177822
\(507\) 9.00000 0.399704
\(508\) 2.00000 0.0887357
\(509\) 33.0000 1.46270 0.731350 0.682003i \(-0.238891\pi\)
0.731350 + 0.682003i \(0.238891\pi\)
\(510\) 3.00000 0.132842
\(511\) 14.0000 0.619324
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −2.00000 −0.0882162
\(515\) −1.00000 −0.0440653
\(516\) 5.00000 0.220113
\(517\) −28.0000 −1.23144
\(518\) −1.00000 −0.0439375
\(519\) 15.0000 0.658427
\(520\) −2.00000 −0.0877058
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −30.0000 −1.31181 −0.655904 0.754844i \(-0.727712\pi\)
−0.655904 + 0.754844i \(0.727712\pi\)
\(524\) 6.00000 0.262111
\(525\) 4.00000 0.174574
\(526\) 21.0000 0.915644
\(527\) 6.00000 0.261364
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) −9.00000 −0.390567
\(532\) 1.00000 0.0433555
\(533\) 18.0000 0.779667
\(534\) −18.0000 −0.778936
\(535\) 13.0000 0.562039
\(536\) −8.00000 −0.345547
\(537\) −20.0000 −0.863064
\(538\) 12.0000 0.517357
\(539\) 24.0000 1.03375
\(540\) −1.00000 −0.0430331
\(541\) 45.0000 1.93470 0.967351 0.253442i \(-0.0815627\pi\)
0.967351 + 0.253442i \(0.0815627\pi\)
\(542\) −20.0000 −0.859074
\(543\) 18.0000 0.772454
\(544\) −3.00000 −0.128624
\(545\) −8.00000 −0.342682
\(546\) 2.00000 0.0855921
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −18.0000 −0.768922
\(549\) −10.0000 −0.426790
\(550\) 16.0000 0.682242
\(551\) −1.00000 −0.0426014
\(552\) −1.00000 −0.0425628
\(553\) −4.00000 −0.170097
\(554\) −24.0000 −1.01966
\(555\) 1.00000 0.0424476
\(556\) 6.00000 0.254457
\(557\) 1.00000 0.0423714 0.0211857 0.999776i \(-0.493256\pi\)
0.0211857 + 0.999776i \(0.493256\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 10.0000 0.422955
\(560\) 1.00000 0.0422577
\(561\) −12.0000 −0.506640
\(562\) −22.0000 −0.928014
\(563\) 14.0000 0.590030 0.295015 0.955493i \(-0.404675\pi\)
0.295015 + 0.955493i \(0.404675\pi\)
\(564\) −7.00000 −0.294753
\(565\) 11.0000 0.462773
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) −2.00000 −0.0839181
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) −1.00000 −0.0418854
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 8.00000 0.334497
\(573\) −19.0000 −0.793736
\(574\) −9.00000 −0.375653
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −8.00000 −0.332756
\(579\) 12.0000 0.498703
\(580\) −1.00000 −0.0415227
\(581\) −8.00000 −0.331896
\(582\) 8.00000 0.331611
\(583\) −24.0000 −0.993978
\(584\) 14.0000 0.579324
\(585\) −2.00000 −0.0826898
\(586\) −26.0000 −1.07405
\(587\) −7.00000 −0.288921 −0.144460 0.989511i \(-0.546145\pi\)
−0.144460 + 0.989511i \(0.546145\pi\)
\(588\) 6.00000 0.247436
\(589\) −2.00000 −0.0824086
\(590\) −9.00000 −0.370524
\(591\) 17.0000 0.699287
\(592\) −1.00000 −0.0410997
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 4.00000 0.164122
\(595\) −3.00000 −0.122988
\(596\) −9.00000 −0.368654
\(597\) −4.00000 −0.163709
\(598\) −2.00000 −0.0817861
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 4.00000 0.163299
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) −5.00000 −0.203785
\(603\) −8.00000 −0.325785
\(604\) −13.0000 −0.528962
\(605\) 5.00000 0.203279
\(606\) −2.00000 −0.0812444
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 1.00000 0.0405554
\(609\) 1.00000 0.0405220
\(610\) −10.0000 −0.404888
\(611\) −14.0000 −0.566379
\(612\) −3.00000 −0.121268
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 20.0000 0.807134
\(615\) 9.00000 0.362915
\(616\) −4.00000 −0.161165
\(617\) −45.0000 −1.81163 −0.905816 0.423672i \(-0.860741\pi\)
−0.905816 + 0.423672i \(0.860741\pi\)
\(618\) 1.00000 0.0402259
\(619\) 19.0000 0.763674 0.381837 0.924230i \(-0.375291\pi\)
0.381837 + 0.924230i \(0.375291\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −1.00000 −0.0401286
\(622\) 15.0000 0.601445
\(623\) 18.0000 0.721155
\(624\) 2.00000 0.0800641
\(625\) 11.0000 0.440000
\(626\) 17.0000 0.679457
\(627\) 4.00000 0.159745
\(628\) 19.0000 0.758183
\(629\) 3.00000 0.119618
\(630\) 1.00000 0.0398410
\(631\) 39.0000 1.55257 0.776283 0.630385i \(-0.217103\pi\)
0.776283 + 0.630385i \(0.217103\pi\)
\(632\) −4.00000 −0.159111
\(633\) −3.00000 −0.119239
\(634\) −6.00000 −0.238290
\(635\) 2.00000 0.0793676
\(636\) −6.00000 −0.237915
\(637\) 12.0000 0.475457
\(638\) 4.00000 0.158362
\(639\) −2.00000 −0.0791188
\(640\) 1.00000 0.0395285
\(641\) −29.0000 −1.14543 −0.572716 0.819754i \(-0.694110\pi\)
−0.572716 + 0.819754i \(0.694110\pi\)
\(642\) −13.0000 −0.513069
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 1.00000 0.0394055
\(645\) 5.00000 0.196875
\(646\) −3.00000 −0.118033
\(647\) 44.0000 1.72982 0.864909 0.501928i \(-0.167376\pi\)
0.864909 + 0.501928i \(0.167376\pi\)
\(648\) 1.00000 0.0392837
\(649\) 36.0000 1.41312
\(650\) 8.00000 0.313786
\(651\) 2.00000 0.0783862
\(652\) 1.00000 0.0391630
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 8.00000 0.312825
\(655\) 6.00000 0.234439
\(656\) −9.00000 −0.351391
\(657\) 14.0000 0.546192
\(658\) 7.00000 0.272888
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 4.00000 0.155700
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 17.0000 0.660724
\(663\) −6.00000 −0.233021
\(664\) −8.00000 −0.310460
\(665\) 1.00000 0.0387783
\(666\) −1.00000 −0.0387492
\(667\) −1.00000 −0.0387202
\(668\) −14.0000 −0.541676
\(669\) −16.0000 −0.618596
\(670\) −8.00000 −0.309067
\(671\) 40.0000 1.54418
\(672\) −1.00000 −0.0385758
\(673\) 51.0000 1.96591 0.982953 0.183858i \(-0.0588587\pi\)
0.982953 + 0.183858i \(0.0588587\pi\)
\(674\) 30.0000 1.15556
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −11.0000 −0.422452
\(679\) −8.00000 −0.307012
\(680\) −3.00000 −0.115045
\(681\) 11.0000 0.421521
\(682\) 8.00000 0.306336
\(683\) 35.0000 1.33924 0.669619 0.742705i \(-0.266457\pi\)
0.669619 + 0.742705i \(0.266457\pi\)
\(684\) 1.00000 0.0382360
\(685\) −18.0000 −0.687745
\(686\) −13.0000 −0.496342
\(687\) 19.0000 0.724895
\(688\) −5.00000 −0.190623
\(689\) −12.0000 −0.457164
\(690\) −1.00000 −0.0380693
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) −15.0000 −0.570214
\(693\) −4.00000 −0.151947
\(694\) −11.0000 −0.417554
\(695\) 6.00000 0.227593
\(696\) 1.00000 0.0379049
\(697\) 27.0000 1.02270
\(698\) 6.00000 0.227103
\(699\) −4.00000 −0.151294
\(700\) −4.00000 −0.151186
\(701\) 9.00000 0.339925 0.169963 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) 2.00000 0.0754851
\(703\) −1.00000 −0.0377157
\(704\) −4.00000 −0.150756
\(705\) −7.00000 −0.263635
\(706\) −18.0000 −0.677439
\(707\) 2.00000 0.0752177
\(708\) 9.00000 0.338241
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) −2.00000 −0.0750587
\(711\) −4.00000 −0.150012
\(712\) 18.0000 0.674579
\(713\) −2.00000 −0.0749006
\(714\) 3.00000 0.112272
\(715\) 8.00000 0.299183
\(716\) 20.0000 0.747435
\(717\) 6.00000 0.224074
\(718\) −31.0000 −1.15691
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 1.00000 0.0372678
\(721\) −1.00000 −0.0372419
\(722\) −18.0000 −0.669891
\(723\) −3.00000 −0.111571
\(724\) −18.0000 −0.668965
\(725\) 4.00000 0.148556
\(726\) −5.00000 −0.185567
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 14.0000 0.518163
\(731\) 15.0000 0.554795
\(732\) 10.0000 0.369611
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) −4.00000 −0.147643
\(735\) 6.00000 0.221313
\(736\) 1.00000 0.0368605
\(737\) 32.0000 1.17874
\(738\) −9.00000 −0.331295
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 2.00000 0.0734718
\(742\) 6.00000 0.220267
\(743\) 27.0000 0.990534 0.495267 0.868741i \(-0.335070\pi\)
0.495267 + 0.868741i \(0.335070\pi\)
\(744\) 2.00000 0.0733236
\(745\) −9.00000 −0.329734
\(746\) −4.00000 −0.146450
\(747\) −8.00000 −0.292705
\(748\) 12.0000 0.438763
\(749\) 13.0000 0.475010
\(750\) 9.00000 0.328634
\(751\) −46.0000 −1.67856 −0.839282 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) 7.00000 0.255264
\(753\) −8.00000 −0.291536
\(754\) 2.00000 0.0728357
\(755\) −13.0000 −0.473118
\(756\) −1.00000 −0.0363696
\(757\) 13.0000 0.472493 0.236247 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) −20.0000 −0.726433
\(759\) 4.00000 0.145191
\(760\) 1.00000 0.0362738
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) −2.00000 −0.0724524
\(763\) −8.00000 −0.289619
\(764\) 19.0000 0.687396
\(765\) −3.00000 −0.108465
\(766\) 8.00000 0.289052
\(767\) 18.0000 0.649942
\(768\) −1.00000 −0.0360844
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) −4.00000 −0.144150
\(771\) 2.00000 0.0720282
\(772\) −12.0000 −0.431889
\(773\) 32.0000 1.15096 0.575480 0.817816i \(-0.304815\pi\)
0.575480 + 0.817816i \(0.304815\pi\)
\(774\) −5.00000 −0.179721
\(775\) 8.00000 0.287368
\(776\) −8.00000 −0.287183
\(777\) 1.00000 0.0358748
\(778\) 24.0000 0.860442
\(779\) −9.00000 −0.322458
\(780\) 2.00000 0.0716115
\(781\) 8.00000 0.286263
\(782\) −3.00000 −0.107280
\(783\) 1.00000 0.0357371
\(784\) −6.00000 −0.214286
\(785\) 19.0000 0.678139
\(786\) −6.00000 −0.214013
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) −17.0000 −0.605600
\(789\) −21.0000 −0.747620
\(790\) −4.00000 −0.142314
\(791\) 11.0000 0.391115
\(792\) −4.00000 −0.142134
\(793\) 20.0000 0.710221
\(794\) −20.0000 −0.709773
\(795\) −6.00000 −0.212798
\(796\) 4.00000 0.141776
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) −1.00000 −0.0353996
\(799\) −21.0000 −0.742927
\(800\) −4.00000 −0.141421
\(801\) 18.0000 0.635999
\(802\) 6.00000 0.211867
\(803\) −56.0000 −1.97620
\(804\) 8.00000 0.282138
\(805\) 1.00000 0.0352454
\(806\) 4.00000 0.140894
\(807\) −12.0000 −0.422420
\(808\) 2.00000 0.0703598
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 1.00000 0.0351364
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) −1.00000 −0.0350931
\(813\) 20.0000 0.701431
\(814\) 4.00000 0.140200
\(815\) 1.00000 0.0350285
\(816\) 3.00000 0.105021
\(817\) −5.00000 −0.174928
\(818\) −6.00000 −0.209785
\(819\) −2.00000 −0.0698857
\(820\) −9.00000 −0.314294
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 18.0000 0.627822
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −16.0000 −0.557048
\(826\) −9.00000 −0.313150
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 1.00000 0.0347524
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) −8.00000 −0.277684
\(831\) 24.0000 0.832551
\(832\) −2.00000 −0.0693375
\(833\) 18.0000 0.623663
\(834\) −6.00000 −0.207763
\(835\) −14.0000 −0.484490
\(836\) −4.00000 −0.138343
\(837\) 2.00000 0.0691301
\(838\) −5.00000 −0.172722
\(839\) 21.0000 0.725001 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 1.00000 0.0344828
\(842\) −2.00000 −0.0689246
\(843\) 22.0000 0.757720
\(844\) 3.00000 0.103264
\(845\) −9.00000 −0.309609
\(846\) 7.00000 0.240665
\(847\) 5.00000 0.171802
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 12.0000 0.411597
\(851\) −1.00000 −0.0342796
\(852\) 2.00000 0.0685189
\(853\) −37.0000 −1.26686 −0.633428 0.773802i \(-0.718353\pi\)
−0.633428 + 0.773802i \(0.718353\pi\)
\(854\) −10.0000 −0.342193
\(855\) 1.00000 0.0341993
\(856\) 13.0000 0.444331
\(857\) 32.0000 1.09310 0.546550 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) −8.00000 −0.273115
\(859\) −7.00000 −0.238837 −0.119418 0.992844i \(-0.538103\pi\)
−0.119418 + 0.992844i \(0.538103\pi\)
\(860\) −5.00000 −0.170499
\(861\) 9.00000 0.306719
\(862\) 30.0000 1.02180
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −15.0000 −0.510015
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) −2.00000 −0.0678844
\(869\) 16.0000 0.542763
\(870\) 1.00000 0.0339032
\(871\) 16.0000 0.542139
\(872\) −8.00000 −0.270914
\(873\) −8.00000 −0.270759
\(874\) 1.00000 0.0338255
\(875\) −9.00000 −0.304256
\(876\) −14.0000 −0.473016
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 33.0000 1.11370
\(879\) 26.0000 0.876958
\(880\) −4.00000 −0.134840
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −6.00000 −0.202031
\(883\) −18.0000 −0.605748 −0.302874 0.953031i \(-0.597946\pi\)
−0.302874 + 0.953031i \(0.597946\pi\)
\(884\) 6.00000 0.201802
\(885\) 9.00000 0.302532
\(886\) 36.0000 1.20944
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 1.00000 0.0335578
\(889\) 2.00000 0.0670778
\(890\) 18.0000 0.603361
\(891\) −4.00000 −0.134005
\(892\) 16.0000 0.535720
\(893\) 7.00000 0.234246
\(894\) 9.00000 0.301005
\(895\) 20.0000 0.668526
\(896\) 1.00000 0.0334077
\(897\) 2.00000 0.0667781
\(898\) −9.00000 −0.300334
\(899\) 2.00000 0.0667037
\(900\) −4.00000 −0.133333
\(901\) −18.0000 −0.599667
\(902\) 36.0000 1.19867
\(903\) 5.00000 0.166390
\(904\) 11.0000 0.365855
\(905\) −18.0000 −0.598340
\(906\) 13.0000 0.431896
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −11.0000 −0.365048
\(909\) 2.00000 0.0663358
\(910\) −2.00000 −0.0662994
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 32.0000 1.05905
\(914\) 37.0000 1.22385
\(915\) 10.0000 0.330590
\(916\) −19.0000 −0.627778
\(917\) 6.00000 0.198137
\(918\) 3.00000 0.0990148
\(919\) −19.0000 −0.626752 −0.313376 0.949629i \(-0.601460\pi\)
−0.313376 + 0.949629i \(0.601460\pi\)
\(920\) 1.00000 0.0329690
\(921\) −20.0000 −0.659022
\(922\) 14.0000 0.461065
\(923\) 4.00000 0.131662
\(924\) 4.00000 0.131590
\(925\) 4.00000 0.131519
\(926\) −24.0000 −0.788689
\(927\) −1.00000 −0.0328443
\(928\) −1.00000 −0.0328266
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 2.00000 0.0655826
\(931\) −6.00000 −0.196642
\(932\) 4.00000 0.131024
\(933\) −15.0000 −0.491078
\(934\) 10.0000 0.327210
\(935\) 12.0000 0.392442
\(936\) −2.00000 −0.0653720
\(937\) 5.00000 0.163343 0.0816714 0.996659i \(-0.473974\pi\)
0.0816714 + 0.996659i \(0.473974\pi\)
\(938\) −8.00000 −0.261209
\(939\) −17.0000 −0.554774
\(940\) 7.00000 0.228315
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) −19.0000 −0.619053
\(943\) −9.00000 −0.293080
\(944\) −9.00000 −0.292925
\(945\) −1.00000 −0.0325300
\(946\) 20.0000 0.650256
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 4.00000 0.129914
\(949\) −28.0000 −0.908918
\(950\) −4.00000 −0.129777
\(951\) 6.00000 0.194563
\(952\) −3.00000 −0.0972306
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 6.00000 0.194257
\(955\) 19.0000 0.614826
\(956\) −6.00000 −0.194054
\(957\) −4.00000 −0.129302
\(958\) 24.0000 0.775405
\(959\) −18.0000 −0.581250
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) 2.00000 0.0644826
\(963\) 13.0000 0.418919
\(964\) 3.00000 0.0966235
\(965\) −12.0000 −0.386294
\(966\) −1.00000 −0.0321745
\(967\) 18.0000 0.578841 0.289420 0.957202i \(-0.406537\pi\)
0.289420 + 0.957202i \(0.406537\pi\)
\(968\) 5.00000 0.160706
\(969\) 3.00000 0.0963739
\(970\) −8.00000 −0.256865
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 6.00000 0.192351
\(974\) −3.00000 −0.0961262
\(975\) −8.00000 −0.256205
\(976\) −10.0000 −0.320092
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) −1.00000 −0.0319765
\(979\) −72.0000 −2.30113
\(980\) −6.00000 −0.191663
\(981\) −8.00000 −0.255420
\(982\) 36.0000 1.14881
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 9.00000 0.286910
\(985\) −17.0000 −0.541665
\(986\) 3.00000 0.0955395
\(987\) −7.00000 −0.222812
\(988\) −2.00000 −0.0636285
\(989\) −5.00000 −0.158991
\(990\) −4.00000 −0.127128
\(991\) 49.0000 1.55654 0.778268 0.627932i \(-0.216098\pi\)
0.778268 + 0.627932i \(0.216098\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −17.0000 −0.539479
\(994\) −2.00000 −0.0634361
\(995\) 4.00000 0.126809
\(996\) 8.00000 0.253490
\(997\) −53.0000 −1.67853 −0.839263 0.543725i \(-0.817013\pi\)
−0.839263 + 0.543725i \(0.817013\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 4002.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.k.1.1 1 1.1 even 1 trivial