Properties

Label 1110.2.a.c.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -5.00000 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{20} -3.00000 q^{21} +3.00000 q^{22} -7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +5.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -5.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -1.00000 q^{38} +5.00000 q^{39} +1.00000 q^{40} +8.00000 q^{41} +3.00000 q^{42} +4.00000 q^{43} -3.00000 q^{44} -1.00000 q^{45} +7.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} -5.00000 q^{51} -5.00000 q^{52} -13.0000 q^{53} +1.00000 q^{54} +3.00000 q^{55} -3.00000 q^{56} -1.00000 q^{57} +6.00000 q^{58} +14.0000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -6.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +5.00000 q^{65} -3.00000 q^{66} -2.00000 q^{67} +5.00000 q^{68} +7.00000 q^{69} +3.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} -13.0000 q^{73} +1.00000 q^{74} -1.00000 q^{75} +1.00000 q^{76} -9.00000 q^{77} -5.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} -17.0000 q^{83} -3.00000 q^{84} -5.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} +3.00000 q^{88} +7.00000 q^{89} +1.00000 q^{90} -15.0000 q^{91} -7.00000 q^{92} -6.00000 q^{93} +12.0000 q^{94} -1.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} -2.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
\(6\) 1.00000 0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −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
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −3.00000 −0.801784
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) −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
\(21\) −3.00000 −0.654654
\(22\) 3.00000 0.639602
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 5.00000 0.980581
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) −5.00000 −0.857493
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −1.00000 −0.162221
\(39\) 5.00000 0.800641
\(40\) 1.00000 0.158114
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 3.00000 0.462910
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −3.00000 −0.452267
\(45\) −1.00000 −0.149071
\(46\) 7.00000 1.03209
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) −5.00000 −0.700140
\(52\) −5.00000 −0.693375
\(53\) −13.0000 −1.78569 −0.892844 0.450367i \(-0.851293\pi\)
−0.892844 + 0.450367i \(0.851293\pi\)
\(54\) 1.00000 0.136083
\(55\) 3.00000 0.404520
\(56\) −3.00000 −0.400892
\(57\) −1.00000 −0.132453
\(58\) 6.00000 0.787839
\(59\) 14.0000 1.82264 0.911322 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 1.00000 0.129099
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −6.00000 −0.762001
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 5.00000 0.620174
\(66\) −3.00000 −0.369274
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 5.00000 0.606339
\(69\) 7.00000 0.842701
\(70\) 3.00000 0.358569
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) −1.00000 −0.117851
\(73\) −13.0000 −1.52153 −0.760767 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 1.00000 0.116248
\(75\) −1.00000 −0.115470
\(76\) 1.00000 0.114708
\(77\) −9.00000 −1.02565
\(78\) −5.00000 −0.566139
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) −17.0000 −1.86599 −0.932996 0.359886i \(-0.882816\pi\)
−0.932996 + 0.359886i \(0.882816\pi\)
\(84\) −3.00000 −0.327327
\(85\) −5.00000 −0.542326
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) 3.00000 0.319801
\(89\) 7.00000 0.741999 0.370999 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) 1.00000 0.105409
\(91\) −15.0000 −1.57243
\(92\) −7.00000 −0.729800
\(93\) −6.00000 −0.622171
\(94\) 12.0000 1.23771
\(95\) −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 \(0.633125\pi\)
\(98\) −2.00000 −0.202031
\(99\) −3.00000 −0.301511
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 5.00000 0.495074
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 5.00000 0.490290
\(105\) 3.00000 0.292770
\(106\) 13.0000 1.26267
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) −3.00000 −0.286039
\(111\) 1.00000 0.0949158
\(112\) 3.00000 0.283473
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 1.00000 0.0936586
\(115\) 7.00000 0.652753
\(116\) −6.00000 −0.557086
\(117\) −5.00000 −0.462250
\(118\) −14.0000 −1.28880
\(119\) 15.0000 1.37505
\(120\) −1.00000 −0.0912871
\(121\) −2.00000 −0.181818
\(122\) 6.00000 0.543214
\(123\) −8.00000 −0.721336
\(124\) 6.00000 0.538816
\(125\) −1.00000 −0.0894427
\(126\) −3.00000 −0.267261
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −5.00000 −0.438529
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 3.00000 0.261116
\(133\) 3.00000 0.260133
\(134\) 2.00000 0.172774
\(135\) 1.00000 0.0860663
\(136\) −5.00000 −0.428746
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −7.00000 −0.595880
\(139\) −18.0000 −1.52674 −0.763370 0.645961i \(-0.776457\pi\)
−0.763370 + 0.645961i \(0.776457\pi\)
\(140\) −3.00000 −0.253546
\(141\) 12.0000 1.01058
\(142\) 14.0000 1.17485
\(143\) 15.0000 1.25436
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 13.0000 1.07589
\(147\) −2.00000 −0.164957
\(148\) −1.00000 −0.0821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 5.00000 0.404226
\(154\) 9.00000 0.725241
\(155\) −6.00000 −0.481932
\(156\) 5.00000 0.400320
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 8.00000 0.636446
\(159\) 13.0000 1.03097
\(160\) 1.00000 0.0790569
\(161\) −21.0000 −1.65503
\(162\) −1.00000 −0.0785674
\(163\) 9.00000 0.704934 0.352467 0.935824i \(-0.385343\pi\)
0.352467 + 0.935824i \(0.385343\pi\)
\(164\) 8.00000 0.624695
\(165\) −3.00000 −0.233550
\(166\) 17.0000 1.31946
\(167\) −1.00000 −0.0773823 −0.0386912 0.999251i \(-0.512319\pi\)
−0.0386912 + 0.999251i \(0.512319\pi\)
\(168\) 3.00000 0.231455
\(169\) 12.0000 0.923077
\(170\) 5.00000 0.383482
\(171\) 1.00000 0.0764719
\(172\) 4.00000 0.304997
\(173\) −19.0000 −1.44454 −0.722272 0.691609i \(-0.756902\pi\)
−0.722272 + 0.691609i \(0.756902\pi\)
\(174\) −6.00000 −0.454859
\(175\) 3.00000 0.226779
\(176\) −3.00000 −0.226134
\(177\) −14.0000 −1.05230
\(178\) −7.00000 −0.524672
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 15.0000 1.11187
\(183\) 6.00000 0.443533
\(184\) 7.00000 0.516047
\(185\) 1.00000 0.0735215
\(186\) 6.00000 0.439941
\(187\) −15.0000 −1.09691
\(188\) −12.0000 −0.875190
\(189\) −3.00000 −0.218218
\(190\) 1.00000 0.0725476
\(191\) −25.0000 −1.80894 −0.904468 0.426541i \(-0.859732\pi\)
−0.904468 + 0.426541i \(0.859732\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 8.00000 0.574367
\(195\) −5.00000 −0.358057
\(196\) 2.00000 0.142857
\(197\) 9.00000 0.641223 0.320612 0.947211i \(-0.396112\pi\)
0.320612 + 0.947211i \(0.396112\pi\)
\(198\) 3.00000 0.213201
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.00000 0.141069
\(202\) 10.0000 0.703598
\(203\) −18.0000 −1.26335
\(204\) −5.00000 −0.350070
\(205\) −8.00000 −0.558744
\(206\) −8.00000 −0.557386
\(207\) −7.00000 −0.486534
\(208\) −5.00000 −0.346688
\(209\) −3.00000 −0.207514
\(210\) −3.00000 −0.207020
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) −13.0000 −0.892844
\(213\) 14.0000 0.959264
\(214\) 3.00000 0.205076
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) 18.0000 1.22192
\(218\) 5.00000 0.338643
\(219\) 13.0000 0.878459
\(220\) 3.00000 0.202260
\(221\) −25.0000 −1.68168
\(222\) −1.00000 −0.0671156
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −3.00000 −0.200446
\(225\) 1.00000 0.0666667
\(226\) −10.0000 −0.665190
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −7.00000 −0.461566
\(231\) 9.00000 0.592157
\(232\) 6.00000 0.393919
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 5.00000 0.326860
\(235\) 12.0000 0.782794
\(236\) 14.0000 0.911322
\(237\) 8.00000 0.519656
\(238\) −15.0000 −0.972306
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 1.00000 0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) −2.00000 −0.127775
\(246\) 8.00000 0.510061
\(247\) −5.00000 −0.318142
\(248\) −6.00000 −0.381000
\(249\) 17.0000 1.07733
\(250\) 1.00000 0.0632456
\(251\) 26.0000 1.64111 0.820553 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) 3.00000 0.188982
\(253\) 21.0000 1.32026
\(254\) 3.00000 0.188237
\(255\) 5.00000 0.313112
\(256\) 1.00000 0.0625000
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 4.00000 0.249029
\(259\) −3.00000 −0.186411
\(260\) 5.00000 0.310087
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −3.00000 −0.184637
\(265\) 13.0000 0.798584
\(266\) −3.00000 −0.183942
\(267\) −7.00000 −0.428393
\(268\) −2.00000 −0.122169
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 5.00000 0.303170
\(273\) 15.0000 0.907841
\(274\) 6.00000 0.362473
\(275\) −3.00000 −0.180907
\(276\) 7.00000 0.421350
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) 18.0000 1.07957
\(279\) 6.00000 0.359211
\(280\) 3.00000 0.179284
\(281\) −21.0000 −1.25275 −0.626377 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(282\) −12.0000 −0.714590
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) −14.0000 −0.830747
\(285\) 1.00000 0.0592349
\(286\) −15.0000 −0.886969
\(287\) 24.0000 1.41668
\(288\) −1.00000 −0.0589256
\(289\) 8.00000 0.470588
\(290\) −6.00000 −0.352332
\(291\) 8.00000 0.468968
\(292\) −13.0000 −0.760767
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 2.00000 0.116642
\(295\) −14.0000 −0.815112
\(296\) 1.00000 0.0581238
\(297\) 3.00000 0.174078
\(298\) −6.00000 −0.347571
\(299\) 35.0000 2.02410
\(300\) −1.00000 −0.0577350
\(301\) 12.0000 0.691669
\(302\) −19.0000 −1.09333
\(303\) 10.0000 0.574485
\(304\) 1.00000 0.0573539
\(305\) 6.00000 0.343559
\(306\) −5.00000 −0.285831
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) −9.00000 −0.512823
\(309\) −8.00000 −0.455104
\(310\) 6.00000 0.340777
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) −5.00000 −0.283069
\(313\) −20.0000 −1.13047 −0.565233 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) 8.00000 0.451466
\(315\) −3.00000 −0.169031
\(316\) −8.00000 −0.450035
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) −13.0000 −0.729004
\(319\) 18.0000 1.00781
\(320\) −1.00000 −0.0559017
\(321\) 3.00000 0.167444
\(322\) 21.0000 1.17028
\(323\) 5.00000 0.278207
\(324\) 1.00000 0.0555556
\(325\) −5.00000 −0.277350
\(326\) −9.00000 −0.498464
\(327\) 5.00000 0.276501
\(328\) −8.00000 −0.441726
\(329\) −36.0000 −1.98474
\(330\) 3.00000 0.165145
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −17.0000 −0.932996
\(333\) −1.00000 −0.0547997
\(334\) 1.00000 0.0547176
\(335\) 2.00000 0.109272
\(336\) −3.00000 −0.163663
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) −12.0000 −0.652714
\(339\) −10.0000 −0.543125
\(340\) −5.00000 −0.271163
\(341\) −18.0000 −0.974755
\(342\) −1.00000 −0.0540738
\(343\) −15.0000 −0.809924
\(344\) −4.00000 −0.215666
\(345\) −7.00000 −0.376867
\(346\) 19.0000 1.02145
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) −3.00000 −0.160357
\(351\) 5.00000 0.266880
\(352\) 3.00000 0.159901
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 14.0000 0.744092
\(355\) 14.0000 0.743043
\(356\) 7.00000 0.370999
\(357\) −15.0000 −0.793884
\(358\) 4.00000 0.211407
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −18.0000 −0.947368
\(362\) 14.0000 0.735824
\(363\) 2.00000 0.104973
\(364\) −15.0000 −0.786214
\(365\) 13.0000 0.680451
\(366\) −6.00000 −0.313625
\(367\) −21.0000 −1.09619 −0.548096 0.836416i \(-0.684647\pi\)
−0.548096 + 0.836416i \(0.684647\pi\)
\(368\) −7.00000 −0.364900
\(369\) 8.00000 0.416463
\(370\) −1.00000 −0.0519875
\(371\) −39.0000 −2.02478
\(372\) −6.00000 −0.311086
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 15.0000 0.775632
\(375\) 1.00000 0.0516398
\(376\) 12.0000 0.618853
\(377\) 30.0000 1.54508
\(378\) 3.00000 0.154303
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 3.00000 0.153695
\(382\) 25.0000 1.27911
\(383\) 3.00000 0.153293 0.0766464 0.997058i \(-0.475579\pi\)
0.0766464 + 0.997058i \(0.475579\pi\)
\(384\) 1.00000 0.0510310
\(385\) 9.00000 0.458682
\(386\) 14.0000 0.712581
\(387\) 4.00000 0.203331
\(388\) −8.00000 −0.406138
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 5.00000 0.253185
\(391\) −35.0000 −1.77003
\(392\) −2.00000 −0.101015
\(393\) −12.0000 −0.605320
\(394\) −9.00000 −0.453413
\(395\) 8.00000 0.402524
\(396\) −3.00000 −0.150756
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) −16.0000 −0.802008
\(399\) −3.00000 −0.150188
\(400\) 1.00000 0.0500000
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −30.0000 −1.49441
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) 18.0000 0.893325
\(407\) 3.00000 0.148704
\(408\) 5.00000 0.247537
\(409\) −40.0000 −1.97787 −0.988936 0.148340i \(-0.952607\pi\)
−0.988936 + 0.148340i \(0.952607\pi\)
\(410\) 8.00000 0.395092
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) 42.0000 2.06668
\(414\) 7.00000 0.344031
\(415\) 17.0000 0.834497
\(416\) 5.00000 0.245145
\(417\) 18.0000 0.881464
\(418\) 3.00000 0.146735
\(419\) −5.00000 −0.244266 −0.122133 0.992514i \(-0.538973\pi\)
−0.122133 + 0.992514i \(0.538973\pi\)
\(420\) 3.00000 0.146385
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −22.0000 −1.07094
\(423\) −12.0000 −0.583460
\(424\) 13.0000 0.631336
\(425\) 5.00000 0.242536
\(426\) −14.0000 −0.678302
\(427\) −18.0000 −0.871081
\(428\) −3.00000 −0.145010
\(429\) −15.0000 −0.724207
\(430\) 4.00000 0.192897
\(431\) 17.0000 0.818861 0.409431 0.912341i \(-0.365727\pi\)
0.409431 + 0.912341i \(0.365727\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 33.0000 1.58588 0.792939 0.609301i \(-0.208550\pi\)
0.792939 + 0.609301i \(0.208550\pi\)
\(434\) −18.0000 −0.864028
\(435\) −6.00000 −0.287678
\(436\) −5.00000 −0.239457
\(437\) −7.00000 −0.334855
\(438\) −13.0000 −0.621164
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) −3.00000 −0.143019
\(441\) 2.00000 0.0952381
\(442\) 25.0000 1.18913
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 1.00000 0.0474579
\(445\) −7.00000 −0.331832
\(446\) −16.0000 −0.757622
\(447\) −6.00000 −0.283790
\(448\) 3.00000 0.141737
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −24.0000 −1.13012
\(452\) 10.0000 0.470360
\(453\) −19.0000 −0.892698
\(454\) −12.0000 −0.563188
\(455\) 15.0000 0.703211
\(456\) 1.00000 0.0468293
\(457\) 28.0000 1.30978 0.654892 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) −14.0000 −0.654177
\(459\) −5.00000 −0.233380
\(460\) 7.00000 0.326377
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) −9.00000 −0.418718
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −6.00000 −0.278543
\(465\) 6.00000 0.278243
\(466\) −22.0000 −1.01913
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −5.00000 −0.231125
\(469\) −6.00000 −0.277054
\(470\) −12.0000 −0.553519
\(471\) 8.00000 0.368621
\(472\) −14.0000 −0.644402
\(473\) −12.0000 −0.551761
\(474\) −8.00000 −0.367452
\(475\) 1.00000 0.0458831
\(476\) 15.0000 0.687524
\(477\) −13.0000 −0.595229
\(478\) −20.0000 −0.914779
\(479\) −11.0000 −0.502603 −0.251301 0.967909i \(-0.580859\pi\)
−0.251301 + 0.967909i \(0.580859\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 5.00000 0.227980
\(482\) 14.0000 0.637683
\(483\) 21.0000 0.955533
\(484\) −2.00000 −0.0909091
\(485\) 8.00000 0.363261
\(486\) 1.00000 0.0453609
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 6.00000 0.271607
\(489\) −9.00000 −0.406994
\(490\) 2.00000 0.0903508
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) −8.00000 −0.360668
\(493\) −30.0000 −1.35113
\(494\) 5.00000 0.224961
\(495\) 3.00000 0.134840
\(496\) 6.00000 0.269408
\(497\) −42.0000 −1.88396
\(498\) −17.0000 −0.761788
\(499\) −17.0000 −0.761025 −0.380512 0.924776i \(-0.624252\pi\)
−0.380512 + 0.924776i \(0.624252\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 1.00000 0.0446767
\(502\) −26.0000 −1.16044
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) −3.00000 −0.133631
\(505\) 10.0000 0.444994
\(506\) −21.0000 −0.933564
\(507\) −12.0000 −0.532939
\(508\) −3.00000 −0.133103
\(509\) 21.0000 0.930809 0.465404 0.885098i \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) −5.00000 −0.221404
\(511\) −39.0000 −1.72526
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 7.00000 0.308757
\(515\) −8.00000 −0.352522
\(516\) −4.00000 −0.176090
\(517\) 36.0000 1.58328
\(518\) 3.00000 0.131812
\(519\) 19.0000 0.834007
\(520\) −5.00000 −0.219265
\(521\) −32.0000 −1.40195 −0.700973 0.713188i \(-0.747251\pi\)
−0.700973 + 0.713188i \(0.747251\pi\)
\(522\) 6.00000 0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 12.0000 0.524222
\(525\) −3.00000 −0.130931
\(526\) −24.0000 −1.04645
\(527\) 30.0000 1.30682
\(528\) 3.00000 0.130558
\(529\) 26.0000 1.13043
\(530\) −13.0000 −0.564684
\(531\) 14.0000 0.607548
\(532\) 3.00000 0.130066
\(533\) −40.0000 −1.73259
\(534\) 7.00000 0.302920
\(535\) 3.00000 0.129701
\(536\) 2.00000 0.0863868
\(537\) 4.00000 0.172613
\(538\) 5.00000 0.215565
\(539\) −6.00000 −0.258438
\(540\) 1.00000 0.0430331
\(541\) −11.0000 −0.472927 −0.236463 0.971640i \(-0.575988\pi\)
−0.236463 + 0.971640i \(0.575988\pi\)
\(542\) 20.0000 0.859074
\(543\) 14.0000 0.600798
\(544\) −5.00000 −0.214373
\(545\) 5.00000 0.214176
\(546\) −15.0000 −0.641941
\(547\) −3.00000 −0.128271 −0.0641354 0.997941i \(-0.520429\pi\)
−0.0641354 + 0.997941i \(0.520429\pi\)
\(548\) −6.00000 −0.256307
\(549\) −6.00000 −0.256074
\(550\) 3.00000 0.127920
\(551\) −6.00000 −0.255609
\(552\) −7.00000 −0.297940
\(553\) −24.0000 −1.02058
\(554\) −13.0000 −0.552317
\(555\) −1.00000 −0.0424476
\(556\) −18.0000 −0.763370
\(557\) 16.0000 0.677942 0.338971 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(558\) −6.00000 −0.254000
\(559\) −20.0000 −0.845910
\(560\) −3.00000 −0.126773
\(561\) 15.0000 0.633300
\(562\) 21.0000 0.885832
\(563\) 30.0000 1.26435 0.632175 0.774826i \(-0.282163\pi\)
0.632175 + 0.774826i \(0.282163\pi\)
\(564\) 12.0000 0.505291
\(565\) −10.0000 −0.420703
\(566\) −5.00000 −0.210166
\(567\) 3.00000 0.125988
\(568\) 14.0000 0.587427
\(569\) 33.0000 1.38343 0.691716 0.722170i \(-0.256855\pi\)
0.691716 + 0.722170i \(0.256855\pi\)
\(570\) −1.00000 −0.0418854
\(571\) −18.0000 −0.753277 −0.376638 0.926360i \(-0.622920\pi\)
−0.376638 + 0.926360i \(0.622920\pi\)
\(572\) 15.0000 0.627182
\(573\) 25.0000 1.04439
\(574\) −24.0000 −1.00174
\(575\) −7.00000 −0.291920
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −8.00000 −0.332756
\(579\) 14.0000 0.581820
\(580\) 6.00000 0.249136
\(581\) −51.0000 −2.11584
\(582\) −8.00000 −0.331611
\(583\) 39.0000 1.61521
\(584\) 13.0000 0.537944
\(585\) 5.00000 0.206725
\(586\) 21.0000 0.867502
\(587\) −40.0000 −1.65098 −0.825488 0.564419i \(-0.809100\pi\)
−0.825488 + 0.564419i \(0.809100\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 6.00000 0.247226
\(590\) 14.0000 0.576371
\(591\) −9.00000 −0.370211
\(592\) −1.00000 −0.0410997
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) −3.00000 −0.123091
\(595\) −15.0000 −0.614940
\(596\) 6.00000 0.245770
\(597\) −16.0000 −0.654836
\(598\) −35.0000 −1.43126
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 1.00000 0.0408248
\(601\) 11.0000 0.448699 0.224350 0.974509i \(-0.427974\pi\)
0.224350 + 0.974509i \(0.427974\pi\)
\(602\) −12.0000 −0.489083
\(603\) −2.00000 −0.0814463
\(604\) 19.0000 0.773099
\(605\) 2.00000 0.0813116
\(606\) −10.0000 −0.406222
\(607\) 26.0000 1.05531 0.527654 0.849460i \(-0.323072\pi\)
0.527654 + 0.849460i \(0.323072\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 18.0000 0.729397
\(610\) −6.00000 −0.242933
\(611\) 60.0000 2.42734
\(612\) 5.00000 0.202113
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −6.00000 −0.242140
\(615\) 8.00000 0.322591
\(616\) 9.00000 0.362620
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 8.00000 0.321807
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) −6.00000 −0.240966
\(621\) 7.00000 0.280900
\(622\) −32.0000 −1.28308
\(623\) 21.0000 0.841347
\(624\) 5.00000 0.200160
\(625\) 1.00000 0.0400000
\(626\) 20.0000 0.799361
\(627\) 3.00000 0.119808
\(628\) −8.00000 −0.319235
\(629\) −5.00000 −0.199363
\(630\) 3.00000 0.119523
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 8.00000 0.318223
\(633\) −22.0000 −0.874421
\(634\) −26.0000 −1.03259
\(635\) 3.00000 0.119051
\(636\) 13.0000 0.515484
\(637\) −10.0000 −0.396214
\(638\) −18.0000 −0.712627
\(639\) −14.0000 −0.553831
\(640\) 1.00000 0.0395285
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) −3.00000 −0.118401
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) −21.0000 −0.827516
\(645\) 4.00000 0.157500
\(646\) −5.00000 −0.196722
\(647\) −15.0000 −0.589711 −0.294855 0.955542i \(-0.595271\pi\)
−0.294855 + 0.955542i \(0.595271\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −42.0000 −1.64864
\(650\) 5.00000 0.196116
\(651\) −18.0000 −0.705476
\(652\) 9.00000 0.352467
\(653\) 20.0000 0.782660 0.391330 0.920250i \(-0.372015\pi\)
0.391330 + 0.920250i \(0.372015\pi\)
\(654\) −5.00000 −0.195515
\(655\) −12.0000 −0.468879
\(656\) 8.00000 0.312348
\(657\) −13.0000 −0.507178
\(658\) 36.0000 1.40343
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) −3.00000 −0.116775
\(661\) 33.0000 1.28355 0.641776 0.766892i \(-0.278198\pi\)
0.641776 + 0.766892i \(0.278198\pi\)
\(662\) −20.0000 −0.777322
\(663\) 25.0000 0.970920
\(664\) 17.0000 0.659728
\(665\) −3.00000 −0.116335
\(666\) 1.00000 0.0387492
\(667\) 42.0000 1.62625
\(668\) −1.00000 −0.0386912
\(669\) −16.0000 −0.618596
\(670\) −2.00000 −0.0772667
\(671\) 18.0000 0.694882
\(672\) 3.00000 0.115728
\(673\) 15.0000 0.578208 0.289104 0.957298i \(-0.406643\pi\)
0.289104 + 0.957298i \(0.406643\pi\)
\(674\) 5.00000 0.192593
\(675\) −1.00000 −0.0384900
\(676\) 12.0000 0.461538
\(677\) 41.0000 1.57576 0.787879 0.615830i \(-0.211179\pi\)
0.787879 + 0.615830i \(0.211179\pi\)
\(678\) 10.0000 0.384048
\(679\) −24.0000 −0.921035
\(680\) 5.00000 0.191741
\(681\) −12.0000 −0.459841
\(682\) 18.0000 0.689256
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 1.00000 0.0382360
\(685\) 6.00000 0.229248
\(686\) 15.0000 0.572703
\(687\) −14.0000 −0.534133
\(688\) 4.00000 0.152499
\(689\) 65.0000 2.47630
\(690\) 7.00000 0.266485
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) −19.0000 −0.722272
\(693\) −9.00000 −0.341882
\(694\) 12.0000 0.455514
\(695\) 18.0000 0.682779
\(696\) −6.00000 −0.227429
\(697\) 40.0000 1.51511
\(698\) 16.0000 0.605609
\(699\) −22.0000 −0.832116
\(700\) 3.00000 0.113389
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) −5.00000 −0.188713
\(703\) −1.00000 −0.0377157
\(704\) −3.00000 −0.113067
\(705\) −12.0000 −0.451946
\(706\) −2.00000 −0.0752710
\(707\) −30.0000 −1.12827
\(708\) −14.0000 −0.526152
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) −14.0000 −0.525411
\(711\) −8.00000 −0.300023
\(712\) −7.00000 −0.262336
\(713\) −42.0000 −1.57291
\(714\) 15.0000 0.561361
\(715\) −15.0000 −0.560968
\(716\) −4.00000 −0.149487
\(717\) −20.0000 −0.746914
\(718\) −24.0000 −0.895672
\(719\) −10.0000 −0.372937 −0.186469 0.982461i \(-0.559704\pi\)
−0.186469 + 0.982461i \(0.559704\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 24.0000 0.893807
\(722\) 18.0000 0.669891
\(723\) 14.0000 0.520666
\(724\) −14.0000 −0.520306
\(725\) −6.00000 −0.222834
\(726\) −2.00000 −0.0742270
\(727\) −30.0000 −1.11264 −0.556319 0.830969i \(-0.687787\pi\)
−0.556319 + 0.830969i \(0.687787\pi\)
\(728\) 15.0000 0.555937
\(729\) 1.00000 0.0370370
\(730\) −13.0000 −0.481152
\(731\) 20.0000 0.739727
\(732\) 6.00000 0.221766
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 21.0000 0.775124
\(735\) 2.00000 0.0737711
\(736\) 7.00000 0.258023
\(737\) 6.00000 0.221013
\(738\) −8.00000 −0.294484
\(739\) −38.0000 −1.39785 −0.698926 0.715194i \(-0.746338\pi\)
−0.698926 + 0.715194i \(0.746338\pi\)
\(740\) 1.00000 0.0367607
\(741\) 5.00000 0.183680
\(742\) 39.0000 1.43174
\(743\) −34.0000 −1.24734 −0.623670 0.781688i \(-0.714359\pi\)
−0.623670 + 0.781688i \(0.714359\pi\)
\(744\) 6.00000 0.219971
\(745\) −6.00000 −0.219823
\(746\) −6.00000 −0.219676
\(747\) −17.0000 −0.621997
\(748\) −15.0000 −0.548454
\(749\) −9.00000 −0.328853
\(750\) −1.00000 −0.0365148
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −12.0000 −0.437595
\(753\) −26.0000 −0.947493
\(754\) −30.0000 −1.09254
\(755\) −19.0000 −0.691481
\(756\) −3.00000 −0.109109
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) −6.00000 −0.217930
\(759\) −21.0000 −0.762252
\(760\) 1.00000 0.0362738
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) −3.00000 −0.108679
\(763\) −15.0000 −0.543036
\(764\) −25.0000 −0.904468
\(765\) −5.00000 −0.180775
\(766\) −3.00000 −0.108394
\(767\) −70.0000 −2.52755
\(768\) −1.00000 −0.0360844
\(769\) −38.0000 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) −9.00000 −0.324337
\(771\) 7.00000 0.252099
\(772\) −14.0000 −0.503871
\(773\) 37.0000 1.33080 0.665399 0.746488i \(-0.268262\pi\)
0.665399 + 0.746488i \(0.268262\pi\)
\(774\) −4.00000 −0.143777
\(775\) 6.00000 0.215526
\(776\) 8.00000 0.287183
\(777\) 3.00000 0.107624
\(778\) −6.00000 −0.215110
\(779\) 8.00000 0.286630
\(780\) −5.00000 −0.179029
\(781\) 42.0000 1.50288
\(782\) 35.0000 1.25160
\(783\) 6.00000 0.214423
\(784\) 2.00000 0.0714286
\(785\) 8.00000 0.285532
\(786\) 12.0000 0.428026
\(787\) 46.0000 1.63972 0.819861 0.572562i \(-0.194050\pi\)
0.819861 + 0.572562i \(0.194050\pi\)
\(788\) 9.00000 0.320612
\(789\) −24.0000 −0.854423
\(790\) −8.00000 −0.284627
\(791\) 30.0000 1.06668
\(792\) 3.00000 0.106600
\(793\) 30.0000 1.06533
\(794\) 0 0
\(795\) −13.0000 −0.461062
\(796\) 16.0000 0.567105
\(797\) −54.0000 −1.91278 −0.956389 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) 3.00000 0.106199
\(799\) −60.0000 −2.12265
\(800\) −1.00000 −0.0353553
\(801\) 7.00000 0.247333
\(802\) −15.0000 −0.529668
\(803\) 39.0000 1.37628
\(804\) 2.00000 0.0705346
\(805\) 21.0000 0.740153
\(806\) 30.0000 1.05670
\(807\) 5.00000 0.176008
\(808\) 10.0000 0.351799
\(809\) 19.0000 0.668004 0.334002 0.942572i \(-0.391601\pi\)
0.334002 + 0.942572i \(0.391601\pi\)
\(810\) 1.00000 0.0351364
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) −18.0000 −0.631676
\(813\) 20.0000 0.701431
\(814\) −3.00000 −0.105150
\(815\) −9.00000 −0.315256
\(816\) −5.00000 −0.175035
\(817\) 4.00000 0.139942
\(818\) 40.0000 1.39857
\(819\) −15.0000 −0.524142
\(820\) −8.00000 −0.279372
\(821\) 13.0000 0.453703 0.226852 0.973929i \(-0.427157\pi\)
0.226852 + 0.973929i \(0.427157\pi\)
\(822\) −6.00000 −0.209274
\(823\) −27.0000 −0.941161 −0.470580 0.882357i \(-0.655955\pi\)
−0.470580 + 0.882357i \(0.655955\pi\)
\(824\) −8.00000 −0.278693
\(825\) 3.00000 0.104447
\(826\) −42.0000 −1.46137
\(827\) −16.0000 −0.556375 −0.278187 0.960527i \(-0.589734\pi\)
−0.278187 + 0.960527i \(0.589734\pi\)
\(828\) −7.00000 −0.243267
\(829\) 31.0000 1.07667 0.538337 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) −17.0000 −0.590079
\(831\) −13.0000 −0.450965
\(832\) −5.00000 −0.173344
\(833\) 10.0000 0.346479
\(834\) −18.0000 −0.623289
\(835\) 1.00000 0.0346064
\(836\) −3.00000 −0.103757
\(837\) −6.00000 −0.207390
\(838\) 5.00000 0.172722
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) −3.00000 −0.103510
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) 21.0000 0.723278
\(844\) 22.0000 0.757271
\(845\) −12.0000 −0.412813
\(846\) 12.0000 0.412568
\(847\) −6.00000 −0.206162
\(848\) −13.0000 −0.446422
\(849\) −5.00000 −0.171600
\(850\) −5.00000 −0.171499
\(851\) 7.00000 0.239957
\(852\) 14.0000 0.479632
\(853\) −25.0000 −0.855984 −0.427992 0.903783i \(-0.640779\pi\)
−0.427992 + 0.903783i \(0.640779\pi\)
\(854\) 18.0000 0.615947
\(855\) −1.00000 −0.0341993
\(856\) 3.00000 0.102538
\(857\) 39.0000 1.33221 0.666107 0.745856i \(-0.267959\pi\)
0.666107 + 0.745856i \(0.267959\pi\)
\(858\) 15.0000 0.512092
\(859\) 3.00000 0.102359 0.0511793 0.998689i \(-0.483702\pi\)
0.0511793 + 0.998689i \(0.483702\pi\)
\(860\) −4.00000 −0.136399
\(861\) −24.0000 −0.817918
\(862\) −17.0000 −0.579022
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 19.0000 0.646019
\(866\) −33.0000 −1.12139
\(867\) −8.00000 −0.271694
\(868\) 18.0000 0.610960
\(869\) 24.0000 0.814144
\(870\) 6.00000 0.203419
\(871\) 10.0000 0.338837
\(872\) 5.00000 0.169321
\(873\) −8.00000 −0.270759
\(874\) 7.00000 0.236779
\(875\) −3.00000 −0.101419
\(876\) 13.0000 0.439229
\(877\) −12.0000 −0.405211 −0.202606 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) 12.0000 0.404980
\(879\) 21.0000 0.708312
\(880\) 3.00000 0.101130
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −19.0000 −0.639401 −0.319700 0.947519i \(-0.603582\pi\)
−0.319700 + 0.947519i \(0.603582\pi\)
\(884\) −25.0000 −0.840841
\(885\) 14.0000 0.470605
\(886\) −20.0000 −0.671913
\(887\) −10.0000 −0.335767 −0.167884 0.985807i \(-0.553693\pi\)
−0.167884 + 0.985807i \(0.553693\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −9.00000 −0.301850
\(890\) 7.00000 0.234641
\(891\) −3.00000 −0.100504
\(892\) 16.0000 0.535720
\(893\) −12.0000 −0.401565
\(894\) 6.00000 0.200670
\(895\) 4.00000 0.133705
\(896\) −3.00000 −0.100223
\(897\) −35.0000 −1.16862
\(898\) 22.0000 0.734150
\(899\) −36.0000 −1.20067
\(900\) 1.00000 0.0333333
\(901\) −65.0000 −2.16546
\(902\) 24.0000 0.799113
\(903\) −12.0000 −0.399335
\(904\) −10.0000 −0.332595
\(905\) 14.0000 0.465376
\(906\) 19.0000 0.631233
\(907\) 13.0000 0.431658 0.215829 0.976431i \(-0.430755\pi\)
0.215829 + 0.976431i \(0.430755\pi\)
\(908\) 12.0000 0.398234
\(909\) −10.0000 −0.331679
\(910\) −15.0000 −0.497245
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 51.0000 1.68785
\(914\) −28.0000 −0.926158
\(915\) −6.00000 −0.198354
\(916\) 14.0000 0.462573
\(917\) 36.0000 1.18882
\(918\) 5.00000 0.165025
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −7.00000 −0.230783
\(921\) −6.00000 −0.197707
\(922\) 10.0000 0.329332
\(923\) 70.0000 2.30408
\(924\) 9.00000 0.296078
\(925\) −1.00000 −0.0328798
\(926\) 8.00000 0.262896
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) 10.0000 0.328089 0.164045 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) −6.00000 −0.196748
\(931\) 2.00000 0.0655474
\(932\) 22.0000 0.720634
\(933\) −32.0000 −1.04763
\(934\) 12.0000 0.392652
\(935\) 15.0000 0.490552
\(936\) 5.00000 0.163430
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 6.00000 0.195907
\(939\) 20.0000 0.652675
\(940\) 12.0000 0.391397
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −8.00000 −0.260654
\(943\) −56.0000 −1.82361
\(944\) 14.0000 0.455661
\(945\) 3.00000 0.0975900
\(946\) 12.0000 0.390154
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) 8.00000 0.259828
\(949\) 65.0000 2.10999
\(950\) −1.00000 −0.0324443
\(951\) −26.0000 −0.843108
\(952\) −15.0000 −0.486153
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 13.0000 0.420891
\(955\) 25.0000 0.808981
\(956\) 20.0000 0.646846
\(957\) −18.0000 −0.581857
\(958\) 11.0000 0.355394
\(959\) −18.0000 −0.581250
\(960\) 1.00000 0.0322749
\(961\) 5.00000 0.161290
\(962\) −5.00000 −0.161206
\(963\) −3.00000 −0.0966736
\(964\) −14.0000 −0.450910
\(965\) 14.0000 0.450676
\(966\) −21.0000 −0.675664
\(967\) −50.0000 −1.60789 −0.803946 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) 2.00000 0.0642824
\(969\) −5.00000 −0.160623
\(970\) −8.00000 −0.256865
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −54.0000 −1.73116
\(974\) −22.0000 −0.704925
\(975\) 5.00000 0.160128
\(976\) −6.00000 −0.192055
\(977\) −15.0000 −0.479893 −0.239946 0.970786i \(-0.577130\pi\)
−0.239946 + 0.970786i \(0.577130\pi\)
\(978\) 9.00000 0.287788
\(979\) −21.0000 −0.671163
\(980\) −2.00000 −0.0638877
\(981\) −5.00000 −0.159638
\(982\) −9.00000 −0.287202
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 8.00000 0.255031
\(985\) −9.00000 −0.286764
\(986\) 30.0000 0.955395
\(987\) 36.0000 1.14589
\(988\) −5.00000 −0.159071
\(989\) −28.0000 −0.890348
\(990\) −3.00000 −0.0953463
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −6.00000 −0.190500
\(993\) −20.0000 −0.634681
\(994\) 42.0000 1.33216
\(995\) −16.0000 −0.507234
\(996\) 17.0000 0.538666
\(997\) 21.0000 0.665077 0.332538 0.943090i \(-0.392095\pi\)
0.332538 + 0.943090i \(0.392095\pi\)
\(998\) 17.0000 0.538126
\(999\) 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 1110.2.a.c.1.1 1
3.2 odd 2 3330.2.a.z.1.1 1
4.3 odd 2 8880.2.a.p.1.1 1
5.4 even 2 5550.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.c.1.1 1 1.1 even 1 trivial
3330.2.a.z.1.1 1 3.2 odd 2
5550.2.a.bg.1.1 1 5.4 even 2
8880.2.a.p.1.1 1 4.3 odd 2