Properties

Label 786.2.a.a.1.1
Level 786
Weight 2
Character 786.1
Self dual yes
Analytic conductor 6.276
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 786.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.27624159887\)
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
Root \(0\)
Character \(\chi\) = 786.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} +3.00000 q^{13} -2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} -2.00000 q^{21} +3.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +9.00000 q^{29} -2.00000 q^{30} -5.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +5.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} -1.00000 q^{38} -3.00000 q^{39} +2.00000 q^{40} -12.0000 q^{41} +2.00000 q^{42} -6.00000 q^{43} -3.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +5.00000 q^{51} +3.00000 q^{52} +12.0000 q^{53} +1.00000 q^{54} +6.00000 q^{55} -2.00000 q^{56} -1.00000 q^{57} -9.00000 q^{58} -5.00000 q^{59} +2.00000 q^{60} -3.00000 q^{61} +5.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} -3.00000 q^{66} -5.00000 q^{68} -4.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +8.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} -6.00000 q^{77} +3.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -14.0000 q^{83} -2.00000 q^{84} +10.0000 q^{85} +6.00000 q^{86} -9.00000 q^{87} +3.00000 q^{88} -14.0000 q^{89} +2.00000 q^{90} +6.00000 q^{91} +4.00000 q^{92} +5.00000 q^{93} +8.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} +12.0000 q^{97} +3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) −2.00000 −0.534522
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −2.00000 −0.447214
\(21\) −2.00000 −0.436436
\(22\) 3.00000 0.639602
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −3.00000 −0.588348
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) −2.00000 −0.365148
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) 5.00000 0.857493
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −1.00000 −0.162221
\(39\) −3.00000 −0.480384
\(40\) 2.00000 0.316228
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 2.00000 0.308607
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −3.00000 −0.452267
\(45\) −2.00000 −0.298142
\(46\) −4.00000 −0.589768
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 5.00000 0.700140
\(52\) 3.00000 0.416025
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 1.00000 0.136083
\(55\) 6.00000 0.809040
\(56\) −2.00000 −0.267261
\(57\) −1.00000 −0.132453
\(58\) −9.00000 −1.18176
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 2.00000 0.258199
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 5.00000 0.635001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) −3.00000 −0.369274
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −5.00000 −0.606339
\(69\) −4.00000 −0.481543
\(70\) 4.00000 0.478091
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) 1.00000 0.114708
\(77\) −6.00000 −0.683763
\(78\) 3.00000 0.339683
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) −2.00000 −0.218218
\(85\) 10.0000 1.08465
\(86\) 6.00000 0.646997
\(87\) −9.00000 −0.964901
\(88\) 3.00000 0.319801
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 2.00000 0.210819
\(91\) 6.00000 0.628971
\(92\) 4.00000 0.417029
\(93\) 5.00000 0.518476
\(94\) 8.00000 0.825137
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 3.00000 0.303046
\(99\) −3.00000 −0.301511
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −5.00000 −0.495074
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −3.00000 −0.294174
\(105\) 4.00000 0.390360
\(106\) −12.0000 −1.16554
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −13.0000 −1.24517 −0.622587 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) −6.00000 −0.572078
\(111\) 8.00000 0.759326
\(112\) 2.00000 0.188982
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 1.00000 0.0936586
\(115\) −8.00000 −0.746004
\(116\) 9.00000 0.835629
\(117\) 3.00000 0.277350
\(118\) 5.00000 0.460287
\(119\) −10.0000 −0.916698
\(120\) −2.00000 −0.182574
\(121\) −2.00000 −0.181818
\(122\) 3.00000 0.271607
\(123\) 12.0000 1.08200
\(124\) −5.00000 −0.449013
\(125\) 12.0000 1.07331
\(126\) −2.00000 −0.178174
\(127\) 15.0000 1.33103 0.665517 0.746382i \(-0.268211\pi\)
0.665517 + 0.746382i \(0.268211\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.00000 0.528271
\(130\) 6.00000 0.526235
\(131\) −1.00000 −0.0873704
\(132\) 3.00000 0.261116
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) 2.00000 0.172133
\(136\) 5.00000 0.428746
\(137\) −7.00000 −0.598050 −0.299025 0.954245i \(-0.596661\pi\)
−0.299025 + 0.954245i \(0.596661\pi\)
\(138\) 4.00000 0.340503
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −4.00000 −0.338062
\(141\) 8.00000 0.673722
\(142\) 8.00000 0.671345
\(143\) −9.00000 −0.752618
\(144\) 1.00000 0.0833333
\(145\) −18.0000 −1.49482
\(146\) 2.00000 0.165521
\(147\) 3.00000 0.247436
\(148\) −8.00000 −0.657596
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −5.00000 −0.404226
\(154\) 6.00000 0.483494
\(155\) 10.0000 0.803219
\(156\) −3.00000 −0.240192
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 8.00000 0.636446
\(159\) −12.0000 −0.951662
\(160\) 2.00000 0.158114
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) −12.0000 −0.937043
\(165\) −6.00000 −0.467099
\(166\) 14.0000 1.08661
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 2.00000 0.154303
\(169\) −4.00000 −0.307692
\(170\) −10.0000 −0.766965
\(171\) 1.00000 0.0764719
\(172\) −6.00000 −0.457496
\(173\) −11.0000 −0.836315 −0.418157 0.908375i \(-0.637324\pi\)
−0.418157 + 0.908375i \(0.637324\pi\)
\(174\) 9.00000 0.682288
\(175\) −2.00000 −0.151186
\(176\) −3.00000 −0.226134
\(177\) 5.00000 0.375823
\(178\) 14.0000 1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −2.00000 −0.149071
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) −6.00000 −0.444750
\(183\) 3.00000 0.221766
\(184\) −4.00000 −0.294884
\(185\) 16.0000 1.17634
\(186\) −5.00000 −0.366618
\(187\) 15.0000 1.09691
\(188\) −8.00000 −0.583460
\(189\) −2.00000 −0.145479
\(190\) 2.00000 0.145095
\(191\) −5.00000 −0.361787 −0.180894 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) −12.0000 −0.861550
\(195\) 6.00000 0.429669
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 3.00000 0.213201
\(199\) −17.0000 −1.20510 −0.602549 0.798082i \(-0.705848\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) 18.0000 1.26335
\(204\) 5.00000 0.350070
\(205\) 24.0000 1.67623
\(206\) 8.00000 0.557386
\(207\) 4.00000 0.278019
\(208\) 3.00000 0.208013
\(209\) −3.00000 −0.207514
\(210\) −4.00000 −0.276026
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 12.0000 0.824163
\(213\) 8.00000 0.548151
\(214\) −3.00000 −0.205076
\(215\) 12.0000 0.818393
\(216\) 1.00000 0.0680414
\(217\) −10.0000 −0.678844
\(218\) 13.0000 0.880471
\(219\) 2.00000 0.135147
\(220\) 6.00000 0.404520
\(221\) −15.0000 −1.00901
\(222\) −8.00000 −0.536925
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) −2.00000 −0.133631
\(225\) −1.00000 −0.0666667
\(226\) −4.00000 −0.266076
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 8.00000 0.527504
\(231\) 6.00000 0.394771
\(232\) −9.00000 −0.590879
\(233\) 20.0000 1.31024 0.655122 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) −3.00000 −0.196116
\(235\) 16.0000 1.04372
\(236\) −5.00000 −0.325472
\(237\) 8.00000 0.519656
\(238\) 10.0000 0.648204
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 2.00000 0.129099
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) −3.00000 −0.192055
\(245\) 6.00000 0.383326
\(246\) −12.0000 −0.765092
\(247\) 3.00000 0.190885
\(248\) 5.00000 0.317500
\(249\) 14.0000 0.887214
\(250\) −12.0000 −0.758947
\(251\) −22.0000 −1.38863 −0.694314 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) 2.00000 0.125988
\(253\) −12.0000 −0.754434
\(254\) −15.0000 −0.941184
\(255\) −10.0000 −0.626224
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) −6.00000 −0.373544
\(259\) −16.0000 −0.994192
\(260\) −6.00000 −0.372104
\(261\) 9.00000 0.557086
\(262\) 1.00000 0.0617802
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) −3.00000 −0.184637
\(265\) −24.0000 −1.47431
\(266\) −2.00000 −0.122628
\(267\) 14.0000 0.856786
\(268\) 0 0
\(269\) −8.00000 −0.487769 −0.243884 0.969804i \(-0.578422\pi\)
−0.243884 + 0.969804i \(0.578422\pi\)
\(270\) −2.00000 −0.121716
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) −5.00000 −0.303170
\(273\) −6.00000 −0.363137
\(274\) 7.00000 0.422885
\(275\) 3.00000 0.180907
\(276\) −4.00000 −0.240772
\(277\) 3.00000 0.180253 0.0901263 0.995930i \(-0.471273\pi\)
0.0901263 + 0.995930i \(0.471273\pi\)
\(278\) −12.0000 −0.719712
\(279\) −5.00000 −0.299342
\(280\) 4.00000 0.239046
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) −8.00000 −0.476393
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −8.00000 −0.474713
\(285\) 2.00000 0.118470
\(286\) 9.00000 0.532181
\(287\) −24.0000 −1.41668
\(288\) −1.00000 −0.0589256
\(289\) 8.00000 0.470588
\(290\) 18.0000 1.05700
\(291\) −12.0000 −0.703452
\(292\) −2.00000 −0.117041
\(293\) 29.0000 1.69420 0.847099 0.531435i \(-0.178347\pi\)
0.847099 + 0.531435i \(0.178347\pi\)
\(294\) −3.00000 −0.174964
\(295\) 10.0000 0.582223
\(296\) 8.00000 0.464991
\(297\) 3.00000 0.174078
\(298\) −9.00000 −0.521356
\(299\) 12.0000 0.693978
\(300\) 1.00000 0.0577350
\(301\) −12.0000 −0.691669
\(302\) 12.0000 0.690522
\(303\) −14.0000 −0.804279
\(304\) 1.00000 0.0573539
\(305\) 6.00000 0.343559
\(306\) 5.00000 0.285831
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) −6.00000 −0.341882
\(309\) 8.00000 0.455104
\(310\) −10.0000 −0.567962
\(311\) −33.0000 −1.87126 −0.935629 0.352985i \(-0.885167\pi\)
−0.935629 + 0.352985i \(0.885167\pi\)
\(312\) 3.00000 0.169842
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −2.00000 −0.112867
\(315\) −4.00000 −0.225374
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 12.0000 0.672927
\(319\) −27.0000 −1.51171
\(320\) −2.00000 −0.111803
\(321\) −3.00000 −0.167444
\(322\) −8.00000 −0.445823
\(323\) −5.00000 −0.278207
\(324\) 1.00000 0.0555556
\(325\) −3.00000 −0.166410
\(326\) 1.00000 0.0553849
\(327\) 13.0000 0.718902
\(328\) 12.0000 0.662589
\(329\) −16.0000 −0.882109
\(330\) 6.00000 0.330289
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) −14.0000 −0.768350
\(333\) −8.00000 −0.438397
\(334\) −21.0000 −1.14907
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) 4.00000 0.217571
\(339\) −4.00000 −0.217250
\(340\) 10.0000 0.542326
\(341\) 15.0000 0.812296
\(342\) −1.00000 −0.0540738
\(343\) −20.0000 −1.07990
\(344\) 6.00000 0.323498
\(345\) 8.00000 0.430706
\(346\) 11.0000 0.591364
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) −9.00000 −0.482451
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 2.00000 0.106904
\(351\) −3.00000 −0.160128
\(352\) 3.00000 0.159901
\(353\) 28.0000 1.49029 0.745145 0.666903i \(-0.232380\pi\)
0.745145 + 0.666903i \(0.232380\pi\)
\(354\) −5.00000 −0.265747
\(355\) 16.0000 0.849192
\(356\) −14.0000 −0.741999
\(357\) 10.0000 0.529256
\(358\) 4.00000 0.211407
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 2.00000 0.105409
\(361\) −18.0000 −0.947368
\(362\) −20.0000 −1.05118
\(363\) 2.00000 0.104973
\(364\) 6.00000 0.314485
\(365\) 4.00000 0.209370
\(366\) −3.00000 −0.156813
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 4.00000 0.208514
\(369\) −12.0000 −0.624695
\(370\) −16.0000 −0.831800
\(371\) 24.0000 1.24602
\(372\) 5.00000 0.259238
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −15.0000 −0.775632
\(375\) −12.0000 −0.619677
\(376\) 8.00000 0.412568
\(377\) 27.0000 1.39057
\(378\) 2.00000 0.102869
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) −2.00000 −0.102598
\(381\) −15.0000 −0.768473
\(382\) 5.00000 0.255822
\(383\) −9.00000 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) 1.00000 0.0510310
\(385\) 12.0000 0.611577
\(386\) −19.0000 −0.967075
\(387\) −6.00000 −0.304997
\(388\) 12.0000 0.609208
\(389\) −21.0000 −1.06474 −0.532371 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(390\) −6.00000 −0.303822
\(391\) −20.0000 −1.01144
\(392\) 3.00000 0.151523
\(393\) 1.00000 0.0504433
\(394\) −6.00000 −0.302276
\(395\) 16.0000 0.805047
\(396\) −3.00000 −0.150756
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 17.0000 0.852133
\(399\) −2.00000 −0.100125
\(400\) −1.00000 −0.0500000
\(401\) −17.0000 −0.848939 −0.424470 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) 0 0
\(403\) −15.0000 −0.747203
\(404\) 14.0000 0.696526
\(405\) −2.00000 −0.0993808
\(406\) −18.0000 −0.893325
\(407\) 24.0000 1.18964
\(408\) −5.00000 −0.247537
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −24.0000 −1.18528
\(411\) 7.00000 0.345285
\(412\) −8.00000 −0.394132
\(413\) −10.0000 −0.492068
\(414\) −4.00000 −0.196589
\(415\) 28.0000 1.37447
\(416\) −3.00000 −0.147087
\(417\) −12.0000 −0.587643
\(418\) 3.00000 0.146735
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 4.00000 0.195180
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) −2.00000 −0.0973585
\(423\) −8.00000 −0.388973
\(424\) −12.0000 −0.582772
\(425\) 5.00000 0.242536
\(426\) −8.00000 −0.387601
\(427\) −6.00000 −0.290360
\(428\) 3.00000 0.145010
\(429\) 9.00000 0.434524
\(430\) −12.0000 −0.578691
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 10.0000 0.480015
\(435\) 18.0000 0.863034
\(436\) −13.0000 −0.622587
\(437\) 4.00000 0.191346
\(438\) −2.00000 −0.0955637
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) −6.00000 −0.286039
\(441\) −3.00000 −0.142857
\(442\) 15.0000 0.713477
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 8.00000 0.379663
\(445\) 28.0000 1.32733
\(446\) 19.0000 0.899676
\(447\) −9.00000 −0.425685
\(448\) 2.00000 0.0944911
\(449\) 23.0000 1.08544 0.542719 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 1.00000 0.0471405
\(451\) 36.0000 1.69517
\(452\) 4.00000 0.188144
\(453\) 12.0000 0.563809
\(454\) 6.00000 0.281594
\(455\) −12.0000 −0.562569
\(456\) 1.00000 0.0468293
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −20.0000 −0.934539
\(459\) 5.00000 0.233380
\(460\) −8.00000 −0.373002
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) −6.00000 −0.279145
\(463\) 13.0000 0.604161 0.302081 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) 9.00000 0.417815
\(465\) −10.0000 −0.463739
\(466\) −20.0000 −0.926482
\(467\) 25.0000 1.15686 0.578431 0.815731i \(-0.303665\pi\)
0.578431 + 0.815731i \(0.303665\pi\)
\(468\) 3.00000 0.138675
\(469\) 0 0
\(470\) −16.0000 −0.738025
\(471\) −2.00000 −0.0921551
\(472\) 5.00000 0.230144
\(473\) 18.0000 0.827641
\(474\) −8.00000 −0.367452
\(475\) −1.00000 −0.0458831
\(476\) −10.0000 −0.458349
\(477\) 12.0000 0.549442
\(478\) −8.00000 −0.365911
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −24.0000 −1.09431
\(482\) −8.00000 −0.364390
\(483\) −8.00000 −0.364013
\(484\) −2.00000 −0.0909091
\(485\) −24.0000 −1.08978
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 3.00000 0.135804
\(489\) 1.00000 0.0452216
\(490\) −6.00000 −0.271052
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 12.0000 0.541002
\(493\) −45.0000 −2.02670
\(494\) −3.00000 −0.134976
\(495\) 6.00000 0.269680
\(496\) −5.00000 −0.224507
\(497\) −16.0000 −0.717698
\(498\) −14.0000 −0.627355
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 12.0000 0.536656
\(501\) −21.0000 −0.938211
\(502\) 22.0000 0.981908
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −28.0000 −1.24598
\(506\) 12.0000 0.533465
\(507\) 4.00000 0.177646
\(508\) 15.0000 0.665517
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 10.0000 0.442807
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −26.0000 −1.14681
\(515\) 16.0000 0.705044
\(516\) 6.00000 0.264135
\(517\) 24.0000 1.05552
\(518\) 16.0000 0.703000
\(519\) 11.0000 0.482846
\(520\) 6.00000 0.263117
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) −9.00000 −0.393919
\(523\) −23.0000 −1.00572 −0.502860 0.864368i \(-0.667719\pi\)
−0.502860 + 0.864368i \(0.667719\pi\)
\(524\) −1.00000 −0.0436852
\(525\) 2.00000 0.0872872
\(526\) 21.0000 0.915644
\(527\) 25.0000 1.08902
\(528\) 3.00000 0.130558
\(529\) −7.00000 −0.304348
\(530\) 24.0000 1.04249
\(531\) −5.00000 −0.216982
\(532\) 2.00000 0.0867110
\(533\) −36.0000 −1.55933
\(534\) −14.0000 −0.605839
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 8.00000 0.344904
\(539\) 9.00000 0.387657
\(540\) 2.00000 0.0860663
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −4.00000 −0.171815
\(543\) −20.0000 −0.858282
\(544\) 5.00000 0.214373
\(545\) 26.0000 1.11372
\(546\) 6.00000 0.256776
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) −7.00000 −0.299025
\(549\) −3.00000 −0.128037
\(550\) −3.00000 −0.127920
\(551\) 9.00000 0.383413
\(552\) 4.00000 0.170251
\(553\) −16.0000 −0.680389
\(554\) −3.00000 −0.127458
\(555\) −16.0000 −0.679162
\(556\) 12.0000 0.508913
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 5.00000 0.211667
\(559\) −18.0000 −0.761319
\(560\) −4.00000 −0.169031
\(561\) −15.0000 −0.633300
\(562\) 10.0000 0.421825
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 8.00000 0.336861
\(565\) −8.00000 −0.336563
\(566\) 14.0000 0.588464
\(567\) 2.00000 0.0839921
\(568\) 8.00000 0.335673
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 23.0000 0.962520 0.481260 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) −9.00000 −0.376309
\(573\) 5.00000 0.208878
\(574\) 24.0000 1.00174
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 5.00000 0.208153 0.104076 0.994569i \(-0.466811\pi\)
0.104076 + 0.994569i \(0.466811\pi\)
\(578\) −8.00000 −0.332756
\(579\) −19.0000 −0.789613
\(580\) −18.0000 −0.747409
\(581\) −28.0000 −1.16164
\(582\) 12.0000 0.497416
\(583\) −36.0000 −1.49097
\(584\) 2.00000 0.0827606
\(585\) −6.00000 −0.248069
\(586\) −29.0000 −1.19798
\(587\) −33.0000 −1.36206 −0.681028 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 3.00000 0.123718
\(589\) −5.00000 −0.206021
\(590\) −10.0000 −0.411693
\(591\) −6.00000 −0.246807
\(592\) −8.00000 −0.328798
\(593\) 19.0000 0.780236 0.390118 0.920765i \(-0.372434\pi\)
0.390118 + 0.920765i \(0.372434\pi\)
\(594\) −3.00000 −0.123091
\(595\) 20.0000 0.819920
\(596\) 9.00000 0.368654
\(597\) 17.0000 0.695764
\(598\) −12.0000 −0.490716
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) 4.00000 0.162623
\(606\) 14.0000 0.568711
\(607\) 44.0000 1.78590 0.892952 0.450151i \(-0.148630\pi\)
0.892952 + 0.450151i \(0.148630\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −18.0000 −0.729397
\(610\) −6.00000 −0.242933
\(611\) −24.0000 −0.970936
\(612\) −5.00000 −0.202113
\(613\) −3.00000 −0.121169 −0.0605844 0.998163i \(-0.519296\pi\)
−0.0605844 + 0.998163i \(0.519296\pi\)
\(614\) −10.0000 −0.403567
\(615\) −24.0000 −0.967773
\(616\) 6.00000 0.241747
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) −8.00000 −0.321807
\(619\) −31.0000 −1.24600 −0.622998 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) 10.0000 0.401610
\(621\) −4.00000 −0.160514
\(622\) 33.0000 1.32318
\(623\) −28.0000 −1.12180
\(624\) −3.00000 −0.120096
\(625\) −19.0000 −0.760000
\(626\) −8.00000 −0.319744
\(627\) 3.00000 0.119808
\(628\) 2.00000 0.0798087
\(629\) 40.0000 1.59490
\(630\) 4.00000 0.159364
\(631\) 26.0000 1.03504 0.517522 0.855670i \(-0.326855\pi\)
0.517522 + 0.855670i \(0.326855\pi\)
\(632\) 8.00000 0.318223
\(633\) −2.00000 −0.0794929
\(634\) −18.0000 −0.714871
\(635\) −30.0000 −1.19051
\(636\) −12.0000 −0.475831
\(637\) −9.00000 −0.356593
\(638\) 27.0000 1.06894
\(639\) −8.00000 −0.316475
\(640\) 2.00000 0.0790569
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 3.00000 0.118401
\(643\) 21.0000 0.828159 0.414080 0.910241i \(-0.364104\pi\)
0.414080 + 0.910241i \(0.364104\pi\)
\(644\) 8.00000 0.315244
\(645\) −12.0000 −0.472500
\(646\) 5.00000 0.196722
\(647\) −31.0000 −1.21874 −0.609368 0.792888i \(-0.708577\pi\)
−0.609368 + 0.792888i \(0.708577\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 15.0000 0.588802
\(650\) 3.00000 0.117670
\(651\) 10.0000 0.391931
\(652\) −1.00000 −0.0391630
\(653\) −38.0000 −1.48705 −0.743527 0.668705i \(-0.766849\pi\)
−0.743527 + 0.668705i \(0.766849\pi\)
\(654\) −13.0000 −0.508340
\(655\) 2.00000 0.0781465
\(656\) −12.0000 −0.468521
\(657\) −2.00000 −0.0780274
\(658\) 16.0000 0.623745
\(659\) 13.0000 0.506408 0.253204 0.967413i \(-0.418516\pi\)
0.253204 + 0.967413i \(0.418516\pi\)
\(660\) −6.00000 −0.233550
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 17.0000 0.660724
\(663\) 15.0000 0.582552
\(664\) 14.0000 0.543305
\(665\) −4.00000 −0.155113
\(666\) 8.00000 0.309994
\(667\) 36.0000 1.39393
\(668\) 21.0000 0.812514
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) 9.00000 0.347441
\(672\) 2.00000 0.0771517
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) −1.00000 −0.0385186
\(675\) 1.00000 0.0384900
\(676\) −4.00000 −0.153846
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 4.00000 0.153619
\(679\) 24.0000 0.921035
\(680\) −10.0000 −0.383482
\(681\) 6.00000 0.229920
\(682\) −15.0000 −0.574380
\(683\) 41.0000 1.56882 0.784411 0.620242i \(-0.212966\pi\)
0.784411 + 0.620242i \(0.212966\pi\)
\(684\) 1.00000 0.0382360
\(685\) 14.0000 0.534913
\(686\) 20.0000 0.763604
\(687\) −20.0000 −0.763048
\(688\) −6.00000 −0.228748
\(689\) 36.0000 1.37149
\(690\) −8.00000 −0.304555
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −11.0000 −0.418157
\(693\) −6.00000 −0.227921
\(694\) 30.0000 1.13878
\(695\) −24.0000 −0.910372
\(696\) 9.00000 0.341144
\(697\) 60.0000 2.27266
\(698\) 30.0000 1.13552
\(699\) −20.0000 −0.756469
\(700\) −2.00000 −0.0755929
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 3.00000 0.113228
\(703\) −8.00000 −0.301726
\(704\) −3.00000 −0.113067
\(705\) −16.0000 −0.602595
\(706\) −28.0000 −1.05379
\(707\) 28.0000 1.05305
\(708\) 5.00000 0.187912
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) −16.0000 −0.600469
\(711\) −8.00000 −0.300023
\(712\) 14.0000 0.524672
\(713\) −20.0000 −0.749006
\(714\) −10.0000 −0.374241
\(715\) 18.0000 0.673162
\(716\) −4.00000 −0.149487
\(717\) −8.00000 −0.298765
\(718\) 14.0000 0.522475
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −16.0000 −0.595871
\(722\) 18.0000 0.669891
\(723\) −8.00000 −0.297523
\(724\) 20.0000 0.743294
\(725\) −9.00000 −0.334252
\(726\) −2.00000 −0.0742270
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 30.0000 1.10959
\(732\) 3.00000 0.110883
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 20.0000 0.738213
\(735\) −6.00000 −0.221313
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) 12.0000 0.441726
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 16.0000 0.588172
\(741\) −3.00000 −0.110208
\(742\) −24.0000 −0.881068
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) −5.00000 −0.183309
\(745\) −18.0000 −0.659469
\(746\) −10.0000 −0.366126
\(747\) −14.0000 −0.512233
\(748\) 15.0000 0.548454
\(749\) 6.00000 0.219235
\(750\) 12.0000 0.438178
\(751\) 37.0000 1.35015 0.675075 0.737749i \(-0.264111\pi\)
0.675075 + 0.737749i \(0.264111\pi\)
\(752\) −8.00000 −0.291730
\(753\) 22.0000 0.801725
\(754\) −27.0000 −0.983282
\(755\) 24.0000 0.873449
\(756\) −2.00000 −0.0727393
\(757\) 19.0000 0.690567 0.345283 0.938498i \(-0.387783\pi\)
0.345283 + 0.938498i \(0.387783\pi\)
\(758\) −10.0000 −0.363216
\(759\) 12.0000 0.435572
\(760\) 2.00000 0.0725476
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) 15.0000 0.543393
\(763\) −26.0000 −0.941263
\(764\) −5.00000 −0.180894
\(765\) 10.0000 0.361551
\(766\) 9.00000 0.325183
\(767\) −15.0000 −0.541619
\(768\) −1.00000 −0.0360844
\(769\) −17.0000 −0.613036 −0.306518 0.951865i \(-0.599164\pi\)
−0.306518 + 0.951865i \(0.599164\pi\)
\(770\) −12.0000 −0.432450
\(771\) −26.0000 −0.936367
\(772\) 19.0000 0.683825
\(773\) −53.0000 −1.90628 −0.953139 0.302534i \(-0.902168\pi\)
−0.953139 + 0.302534i \(0.902168\pi\)
\(774\) 6.00000 0.215666
\(775\) 5.00000 0.179605
\(776\) −12.0000 −0.430775
\(777\) 16.0000 0.573997
\(778\) 21.0000 0.752886
\(779\) −12.0000 −0.429945
\(780\) 6.00000 0.214834
\(781\) 24.0000 0.858788
\(782\) 20.0000 0.715199
\(783\) −9.00000 −0.321634
\(784\) −3.00000 −0.107143
\(785\) −4.00000 −0.142766
\(786\) −1.00000 −0.0356688
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 6.00000 0.213741
\(789\) 21.0000 0.747620
\(790\) −16.0000 −0.569254
\(791\) 8.00000 0.284447
\(792\) 3.00000 0.106600
\(793\) −9.00000 −0.319599
\(794\) 14.0000 0.496841
\(795\) 24.0000 0.851192
\(796\) −17.0000 −0.602549
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 2.00000 0.0707992
\(799\) 40.0000 1.41510
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) 17.0000 0.600291
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 15.0000 0.528352
\(807\) 8.00000 0.281613
\(808\) −14.0000 −0.492518
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 2.00000 0.0702728
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) 18.0000 0.631676
\(813\) −4.00000 −0.140286
\(814\) −24.0000 −0.841200
\(815\) 2.00000 0.0700569
\(816\) 5.00000 0.175035
\(817\) −6.00000 −0.209913
\(818\) 22.0000 0.769212
\(819\) 6.00000 0.209657
\(820\) 24.0000 0.838116
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) −7.00000 −0.244153
\(823\) 47.0000 1.63832 0.819159 0.573567i \(-0.194441\pi\)
0.819159 + 0.573567i \(0.194441\pi\)
\(824\) 8.00000 0.278693
\(825\) −3.00000 −0.104447
\(826\) 10.0000 0.347945
\(827\) 33.0000 1.14752 0.573761 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(828\) 4.00000 0.139010
\(829\) −15.0000 −0.520972 −0.260486 0.965478i \(-0.583883\pi\)
−0.260486 + 0.965478i \(0.583883\pi\)
\(830\) −28.0000 −0.971894
\(831\) −3.00000 −0.104069
\(832\) 3.00000 0.104006
\(833\) 15.0000 0.519719
\(834\) 12.0000 0.415526
\(835\) −42.0000 −1.45347
\(836\) −3.00000 −0.103757
\(837\) 5.00000 0.172825
\(838\) −12.0000 −0.414533
\(839\) −29.0000 −1.00119 −0.500596 0.865681i \(-0.666886\pi\)
−0.500596 + 0.865681i \(0.666886\pi\)
\(840\) −4.00000 −0.138013
\(841\) 52.0000 1.79310
\(842\) 27.0000 0.930481
\(843\) 10.0000 0.344418
\(844\) 2.00000 0.0688428
\(845\) 8.00000 0.275208
\(846\) 8.00000 0.275046
\(847\) −4.00000 −0.137442
\(848\) 12.0000 0.412082
\(849\) 14.0000 0.480479
\(850\) −5.00000 −0.171499
\(851\) −32.0000 −1.09695
\(852\) 8.00000 0.274075
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 6.00000 0.205316
\(855\) −2.00000 −0.0683986
\(856\) −3.00000 −0.102538
\(857\) −45.0000 −1.53717 −0.768585 0.639747i \(-0.779039\pi\)
−0.768585 + 0.639747i \(0.779039\pi\)
\(858\) −9.00000 −0.307255
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 12.0000 0.409197
\(861\) 24.0000 0.817918
\(862\) −28.0000 −0.953684
\(863\) 5.00000 0.170202 0.0851010 0.996372i \(-0.472879\pi\)
0.0851010 + 0.996372i \(0.472879\pi\)
\(864\) 1.00000 0.0340207
\(865\) 22.0000 0.748022
\(866\) −6.00000 −0.203888
\(867\) −8.00000 −0.271694
\(868\) −10.0000 −0.339422
\(869\) 24.0000 0.814144
\(870\) −18.0000 −0.610257
\(871\) 0 0
\(872\) 13.0000 0.440236
\(873\) 12.0000 0.406138
\(874\) −4.00000 −0.135302
\(875\) 24.0000 0.811348
\(876\) 2.00000 0.0675737
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −10.0000 −0.337484
\(879\) −29.0000 −0.978146
\(880\) 6.00000 0.202260
\(881\) −50.0000 −1.68454 −0.842271 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 3.00000 0.101015
\(883\) −17.0000 −0.572096 −0.286048 0.958215i \(-0.592342\pi\)
−0.286048 + 0.958215i \(0.592342\pi\)
\(884\) −15.0000 −0.504505
\(885\) −10.0000 −0.336146
\(886\) −6.00000 −0.201574
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −8.00000 −0.268462
\(889\) 30.0000 1.00617
\(890\) −28.0000 −0.938562
\(891\) −3.00000 −0.100504
\(892\) −19.0000 −0.636167
\(893\) −8.00000 −0.267710
\(894\) 9.00000 0.301005
\(895\) 8.00000 0.267411
\(896\) −2.00000 −0.0668153
\(897\) −12.0000 −0.400668
\(898\) −23.0000 −0.767520
\(899\) −45.0000 −1.50083
\(900\) −1.00000 −0.0333333
\(901\) −60.0000 −1.99889
\(902\) −36.0000 −1.19867
\(903\) 12.0000 0.399335
\(904\) −4.00000 −0.133038
\(905\) −40.0000 −1.32964
\(906\) −12.0000 −0.398673
\(907\) −34.0000 −1.12895 −0.564476 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) −6.00000 −0.199117
\(909\) 14.0000 0.464351
\(910\) 12.0000 0.397796
\(911\) 31.0000 1.02708 0.513538 0.858067i \(-0.328335\pi\)
0.513538 + 0.858067i \(0.328335\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 42.0000 1.39000
\(914\) 10.0000 0.330771
\(915\) −6.00000 −0.198354
\(916\) 20.0000 0.660819
\(917\) −2.00000 −0.0660458
\(918\) −5.00000 −0.165025
\(919\) 25.0000 0.824674 0.412337 0.911031i \(-0.364713\pi\)
0.412337 + 0.911031i \(0.364713\pi\)
\(920\) 8.00000 0.263752
\(921\) −10.0000 −0.329511
\(922\) 6.00000 0.197599
\(923\) −24.0000 −0.789970
\(924\) 6.00000 0.197386
\(925\) 8.00000 0.263038
\(926\) −13.0000 −0.427207
\(927\) −8.00000 −0.262754
\(928\) −9.00000 −0.295439
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 10.0000 0.327913
\(931\) −3.00000 −0.0983210
\(932\) 20.0000 0.655122
\(933\) 33.0000 1.08037
\(934\) −25.0000 −0.818025
\(935\) −30.0000 −0.981105
\(936\) −3.00000 −0.0980581
\(937\) 17.0000 0.555366 0.277683 0.960673i \(-0.410434\pi\)
0.277683 + 0.960673i \(0.410434\pi\)
\(938\) 0 0
\(939\) −8.00000 −0.261070
\(940\) 16.0000 0.521862
\(941\) −49.0000 −1.59735 −0.798677 0.601760i \(-0.794466\pi\)
−0.798677 + 0.601760i \(0.794466\pi\)
\(942\) 2.00000 0.0651635
\(943\) −48.0000 −1.56310
\(944\) −5.00000 −0.162736
\(945\) 4.00000 0.130120
\(946\) −18.0000 −0.585230
\(947\) −40.0000 −1.29983 −0.649913 0.760009i \(-0.725195\pi\)
−0.649913 + 0.760009i \(0.725195\pi\)
\(948\) 8.00000 0.259828
\(949\) −6.00000 −0.194768
\(950\) 1.00000 0.0324443
\(951\) −18.0000 −0.583690
\(952\) 10.0000 0.324102
\(953\) 8.00000 0.259145 0.129573 0.991570i \(-0.458639\pi\)
0.129573 + 0.991570i \(0.458639\pi\)
\(954\) −12.0000 −0.388514
\(955\) 10.0000 0.323592
\(956\) 8.00000 0.258738
\(957\) 27.0000 0.872786
\(958\) 12.0000 0.387702
\(959\) −14.0000 −0.452084
\(960\) 2.00000 0.0645497
\(961\) −6.00000 −0.193548
\(962\) 24.0000 0.773791
\(963\) 3.00000 0.0966736
\(964\) 8.00000 0.257663
\(965\) −38.0000 −1.22326
\(966\) 8.00000 0.257396
\(967\) 33.0000 1.06121 0.530604 0.847620i \(-0.321965\pi\)
0.530604 + 0.847620i \(0.321965\pi\)
\(968\) 2.00000 0.0642824
\(969\) 5.00000 0.160623
\(970\) 24.0000 0.770594
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 24.0000 0.769405
\(974\) −28.0000 −0.897178
\(975\) 3.00000 0.0960769
\(976\) −3.00000 −0.0960277
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) −1.00000 −0.0319765
\(979\) 42.0000 1.34233
\(980\) 6.00000 0.191663
\(981\) −13.0000 −0.415058
\(982\) −18.0000 −0.574403
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −12.0000 −0.382546
\(985\) −12.0000 −0.382352
\(986\) 45.0000 1.43309
\(987\) 16.0000 0.509286
\(988\) 3.00000 0.0954427
\(989\) −24.0000 −0.763156
\(990\) −6.00000 −0.190693
\(991\) −46.0000 −1.46124 −0.730619 0.682785i \(-0.760768\pi\)
−0.730619 + 0.682785i \(0.760768\pi\)
\(992\) 5.00000 0.158750
\(993\) 17.0000 0.539479
\(994\) 16.0000 0.507489
\(995\) 34.0000 1.07787
\(996\) 14.0000 0.443607
\(997\) 35.0000 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) −40.0000 −1.26618
\(999\) 8.00000 0.253109
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 786.2.a.a.1.1 1
3.2 odd 2 2358.2.a.w.1.1 1
4.3 odd 2 6288.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
786.2.a.a.1.1 1 1.1 even 1 trivial
2358.2.a.w.1.1 1 3.2 odd 2
6288.2.a.i.1.1 1 4.3 odd 2