Properties

Label 195.2.a.b.1.1
Level $195$
Weight $2$
Character 195.1
Self dual yes
Analytic conductor $1.557$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 195.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} +6.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} -1.00000 q^{17} +2.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} -2.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +6.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} -6.00000 q^{31} -8.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} +3.00000 q^{35} +2.00000 q^{36} +11.0000 q^{37} -4.00000 q^{38} +1.00000 q^{39} -5.00000 q^{41} -6.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} -10.0000 q^{47} +4.00000 q^{48} +2.00000 q^{49} +2.00000 q^{50} +1.00000 q^{51} -2.00000 q^{52} +11.0000 q^{53} -2.00000 q^{54} -1.00000 q^{55} +2.00000 q^{57} -4.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} +13.0000 q^{61} -12.0000 q^{62} +3.00000 q^{63} -8.00000 q^{64} -1.00000 q^{65} +2.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} +3.00000 q^{69} +6.00000 q^{70} -5.00000 q^{71} +10.0000 q^{73} +22.0000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -3.00000 q^{77} +2.00000 q^{78} -3.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -12.0000 q^{83} -6.00000 q^{84} -1.00000 q^{85} +8.00000 q^{86} +2.00000 q^{87} -15.0000 q^{89} +2.00000 q^{90} -3.00000 q^{91} -6.00000 q^{92} +6.00000 q^{93} -20.0000 q^{94} -2.00000 q^{95} +8.00000 q^{96} +17.0000 q^{97} +4.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214
\(6\) −2.00000 −0.816497
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −2.00000 −0.577350
\(13\) −1.00000 −0.277350
\(14\) 6.00000 1.60357
\(15\) −1.00000 −0.258199
\(16\) −4.00000 −1.00000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 2.00000 0.471405
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) −3.00000 −0.654654
\(22\) −2.00000 −0.426401
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 6.00000 1.13389
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.00000 −0.365148
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −8.00000 −1.41421
\(33\) 1.00000 0.174078
\(34\) −2.00000 −0.342997
\(35\) 3.00000 0.507093
\(36\) 2.00000 0.333333
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) −4.00000 −0.648886
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) −6.00000 −0.925820
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −2.00000 −0.301511
\(45\) 1.00000 0.149071
\(46\) −6.00000 −0.884652
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 4.00000 0.577350
\(49\) 2.00000 0.285714
\(50\) 2.00000 0.282843
\(51\) 1.00000 0.140028
\(52\) −2.00000 −0.277350
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) −2.00000 −0.272166
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −4.00000 −0.525226
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) −2.00000 −0.258199
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) −12.0000 −1.52400
\(63\) 3.00000 0.377964
\(64\) −8.00000 −1.00000
\(65\) −1.00000 −0.124035
\(66\) 2.00000 0.246183
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −2.00000 −0.242536
\(69\) 3.00000 0.361158
\(70\) 6.00000 0.717137
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 22.0000 2.55745
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) −3.00000 −0.341882
\(78\) 2.00000 0.226455
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −6.00000 −0.654654
\(85\) −1.00000 −0.108465
\(86\) 8.00000 0.862662
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 2.00000 0.210819
\(91\) −3.00000 −0.314485
\(92\) −6.00000 −0.625543
\(93\) 6.00000 0.622171
\(94\) −20.0000 −2.06284
\(95\) −2.00000 −0.205196
\(96\) 8.00000 0.816497
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 4.00000 0.404061
\(99\) −1.00000 −0.100504
\(100\) 2.00000 0.200000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 2.00000 0.198030
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 22.0000 2.13683
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) −2.00000 −0.192450
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −2.00000 −0.190693
\(111\) −11.0000 −1.04407
\(112\) −12.0000 −1.13389
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 4.00000 0.374634
\(115\) −3.00000 −0.279751
\(116\) −4.00000 −0.371391
\(117\) −1.00000 −0.0924500
\(118\) 16.0000 1.47292
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 26.0000 2.35393
\(123\) 5.00000 0.450835
\(124\) −12.0000 −1.07763
\(125\) 1.00000 0.0894427
\(126\) 6.00000 0.534522
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000 0.174078
\(133\) −6.00000 −0.520266
\(134\) 24.0000 2.07328
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 6.00000 0.510754
\(139\) −1.00000 −0.0848189 −0.0424094 0.999100i \(-0.513503\pi\)
−0.0424094 + 0.999100i \(0.513503\pi\)
\(140\) 6.00000 0.507093
\(141\) 10.0000 0.842152
\(142\) −10.0000 −0.839181
\(143\) 1.00000 0.0836242
\(144\) −4.00000 −0.333333
\(145\) −2.00000 −0.166091
\(146\) 20.0000 1.65521
\(147\) −2.00000 −0.164957
\(148\) 22.0000 1.80839
\(149\) 13.0000 1.06500 0.532501 0.846430i \(-0.321252\pi\)
0.532501 + 0.846430i \(0.321252\pi\)
\(150\) −2.00000 −0.163299
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) −6.00000 −0.483494
\(155\) −6.00000 −0.481932
\(156\) 2.00000 0.160128
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −6.00000 −0.477334
\(159\) −11.0000 −0.872357
\(160\) −8.00000 −0.632456
\(161\) −9.00000 −0.709299
\(162\) 2.00000 0.157135
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) −10.0000 −0.780869
\(165\) 1.00000 0.0778499
\(166\) −24.0000 −1.86276
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −2.00000 −0.153393
\(171\) −2.00000 −0.152944
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 4.00000 0.303239
\(175\) 3.00000 0.226779
\(176\) 4.00000 0.301511
\(177\) −8.00000 −0.601317
\(178\) −30.0000 −2.24860
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 2.00000 0.149071
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) −6.00000 −0.444750
\(183\) −13.0000 −0.960988
\(184\) 0 0
\(185\) 11.0000 0.808736
\(186\) 12.0000 0.879883
\(187\) 1.00000 0.0731272
\(188\) −20.0000 −1.45865
\(189\) −3.00000 −0.218218
\(190\) −4.00000 −0.290191
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 8.00000 0.577350
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 34.0000 2.44106
\(195\) 1.00000 0.0716115
\(196\) 4.00000 0.285714
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −2.00000 −0.142134
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 2.00000 0.140028
\(205\) −5.00000 −0.349215
\(206\) −32.0000 −2.22955
\(207\) −3.00000 −0.208514
\(208\) 4.00000 0.277350
\(209\) 2.00000 0.138343
\(210\) −6.00000 −0.414039
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 22.0000 1.51097
\(213\) 5.00000 0.342594
\(214\) 18.0000 1.23045
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −18.0000 −1.22192
\(218\) −32.0000 −2.16731
\(219\) −10.0000 −0.675737
\(220\) −2.00000 −0.134840
\(221\) 1.00000 0.0672673
\(222\) −22.0000 −1.47654
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −24.0000 −1.60357
\(225\) 1.00000 0.0666667
\(226\) 28.0000 1.86253
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 4.00000 0.264906
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) −6.00000 −0.395628
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −27.0000 −1.76883 −0.884414 0.466702i \(-0.845442\pi\)
−0.884414 + 0.466702i \(0.845442\pi\)
\(234\) −2.00000 −0.130744
\(235\) −10.0000 −0.652328
\(236\) 16.0000 1.04151
\(237\) 3.00000 0.194871
\(238\) −6.00000 −0.388922
\(239\) −13.0000 −0.840900 −0.420450 0.907316i \(-0.638128\pi\)
−0.420450 + 0.907316i \(0.638128\pi\)
\(240\) 4.00000 0.258199
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −20.0000 −1.28565
\(243\) −1.00000 −0.0641500
\(244\) 26.0000 1.66448
\(245\) 2.00000 0.127775
\(246\) 10.0000 0.637577
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 2.00000 0.126491
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 6.00000 0.377964
\(253\) 3.00000 0.188608
\(254\) 20.0000 1.25491
\(255\) 1.00000 0.0626224
\(256\) 16.0000 1.00000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −8.00000 −0.498058
\(259\) 33.0000 2.05052
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) −12.0000 −0.741362
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 11.0000 0.675725
\(266\) −12.0000 −0.735767
\(267\) 15.0000 0.917985
\(268\) 24.0000 1.46603
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) −2.00000 −0.121716
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 4.00000 0.242536
\(273\) 3.00000 0.181568
\(274\) 36.0000 2.17484
\(275\) −1.00000 −0.0603023
\(276\) 6.00000 0.361158
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) −2.00000 −0.119952
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 20.0000 1.19098
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) −10.0000 −0.593391
\(285\) 2.00000 0.118470
\(286\) 2.00000 0.118262
\(287\) −15.0000 −0.885422
\(288\) −8.00000 −0.471405
\(289\) −16.0000 −0.941176
\(290\) −4.00000 −0.234888
\(291\) −17.0000 −0.996558
\(292\) 20.0000 1.17041
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) −4.00000 −0.233285
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 26.0000 1.50614
\(299\) 3.00000 0.173494
\(300\) −2.00000 −0.115470
\(301\) 12.0000 0.691669
\(302\) 32.0000 1.84139
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 13.0000 0.744378
\(306\) −2.00000 −0.114332
\(307\) −5.00000 −0.285365 −0.142683 0.989769i \(-0.545573\pi\)
−0.142683 + 0.989769i \(0.545573\pi\)
\(308\) −6.00000 −0.341882
\(309\) 16.0000 0.910208
\(310\) −12.0000 −0.681554
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −20.0000 −1.12867
\(315\) 3.00000 0.169031
\(316\) −6.00000 −0.337526
\(317\) 28.0000 1.57264 0.786318 0.617822i \(-0.211985\pi\)
0.786318 + 0.617822i \(0.211985\pi\)
\(318\) −22.0000 −1.23370
\(319\) 2.00000 0.111979
\(320\) −8.00000 −0.447214
\(321\) −9.00000 −0.502331
\(322\) −18.0000 −1.00310
\(323\) 2.00000 0.111283
\(324\) 2.00000 0.111111
\(325\) −1.00000 −0.0554700
\(326\) −26.0000 −1.44001
\(327\) 16.0000 0.884802
\(328\) 0 0
\(329\) −30.0000 −1.65395
\(330\) 2.00000 0.110096
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −24.0000 −1.31717
\(333\) 11.0000 0.602796
\(334\) −24.0000 −1.31322
\(335\) 12.0000 0.655630
\(336\) 12.0000 0.654654
\(337\) 4.00000 0.217894 0.108947 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) 2.00000 0.108786
\(339\) −14.0000 −0.760376
\(340\) −2.00000 −0.108465
\(341\) 6.00000 0.324918
\(342\) −4.00000 −0.216295
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) −12.0000 −0.645124
\(347\) −19.0000 −1.01997 −0.509987 0.860182i \(-0.670350\pi\)
−0.509987 + 0.860182i \(0.670350\pi\)
\(348\) 4.00000 0.214423
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 6.00000 0.320713
\(351\) 1.00000 0.0533761
\(352\) 8.00000 0.426401
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −16.0000 −0.850390
\(355\) −5.00000 −0.265372
\(356\) −30.0000 −1.59000
\(357\) 3.00000 0.158777
\(358\) 4.00000 0.211407
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −14.0000 −0.735824
\(363\) 10.0000 0.524864
\(364\) −6.00000 −0.314485
\(365\) 10.0000 0.523424
\(366\) −26.0000 −1.35904
\(367\) 36.0000 1.87918 0.939592 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) 12.0000 0.625543
\(369\) −5.00000 −0.260290
\(370\) 22.0000 1.14373
\(371\) 33.0000 1.71327
\(372\) 12.0000 0.622171
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 2.00000 0.103418
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) −6.00000 −0.308607
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) −4.00000 −0.205196
\(381\) −10.0000 −0.512316
\(382\) −16.0000 −0.818631
\(383\) −30.0000 −1.53293 −0.766464 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) −26.0000 −1.32337
\(387\) 4.00000 0.203331
\(388\) 34.0000 1.72609
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 2.00000 0.101274
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) −3.00000 −0.150946
\(396\) −2.00000 −0.100504
\(397\) −29.0000 −1.45547 −0.727734 0.685859i \(-0.759427\pi\)
−0.727734 + 0.685859i \(0.759427\pi\)
\(398\) 8.00000 0.401004
\(399\) 6.00000 0.300376
\(400\) −4.00000 −0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −24.0000 −1.19701
\(403\) 6.00000 0.298881
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) −12.0000 −0.595550
\(407\) −11.0000 −0.545250
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −10.0000 −0.493865
\(411\) −18.0000 −0.887875
\(412\) −32.0000 −1.57653
\(413\) 24.0000 1.18096
\(414\) −6.00000 −0.294884
\(415\) −12.0000 −0.589057
\(416\) 8.00000 0.392232
\(417\) 1.00000 0.0489702
\(418\) 4.00000 0.195646
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) −6.00000 −0.292770
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) −8.00000 −0.389434
\(423\) −10.0000 −0.486217
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) 10.0000 0.484502
\(427\) 39.0000 1.88734
\(428\) 18.0000 0.870063
\(429\) −1.00000 −0.0482805
\(430\) 8.00000 0.385794
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 4.00000 0.192450
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) −36.0000 −1.72806
\(435\) 2.00000 0.0958927
\(436\) −32.0000 −1.53252
\(437\) 6.00000 0.287019
\(438\) −20.0000 −0.955637
\(439\) −17.0000 −0.811366 −0.405683 0.914014i \(-0.632966\pi\)
−0.405683 + 0.914014i \(0.632966\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 2.00000 0.0951303
\(443\) 9.00000 0.427603 0.213801 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) −22.0000 −1.04407
\(445\) −15.0000 −0.711068
\(446\) 16.0000 0.757622
\(447\) −13.0000 −0.614879
\(448\) −24.0000 −1.13389
\(449\) −13.0000 −0.613508 −0.306754 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) 2.00000 0.0942809
\(451\) 5.00000 0.235441
\(452\) 28.0000 1.31701
\(453\) −16.0000 −0.751746
\(454\) 44.0000 2.06502
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 36.0000 1.68217
\(459\) 1.00000 0.0466760
\(460\) −6.00000 −0.279751
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 6.00000 0.279145
\(463\) 27.0000 1.25480 0.627398 0.778699i \(-0.284120\pi\)
0.627398 + 0.778699i \(0.284120\pi\)
\(464\) 8.00000 0.371391
\(465\) 6.00000 0.278243
\(466\) −54.0000 −2.50150
\(467\) −23.0000 −1.06431 −0.532157 0.846646i \(-0.678618\pi\)
−0.532157 + 0.846646i \(0.678618\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 36.0000 1.66233
\(470\) −20.0000 −0.922531
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 6.00000 0.275589
\(475\) −2.00000 −0.0917663
\(476\) −6.00000 −0.275010
\(477\) 11.0000 0.503655
\(478\) −26.0000 −1.18921
\(479\) 9.00000 0.411220 0.205610 0.978634i \(-0.434082\pi\)
0.205610 + 0.978634i \(0.434082\pi\)
\(480\) 8.00000 0.365148
\(481\) −11.0000 −0.501557
\(482\) −4.00000 −0.182195
\(483\) 9.00000 0.409514
\(484\) −20.0000 −0.909091
\(485\) 17.0000 0.771930
\(486\) −2.00000 −0.0907218
\(487\) −7.00000 −0.317200 −0.158600 0.987343i \(-0.550698\pi\)
−0.158600 + 0.987343i \(0.550698\pi\)
\(488\) 0 0
\(489\) 13.0000 0.587880
\(490\) 4.00000 0.180702
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 10.0000 0.450835
\(493\) 2.00000 0.0900755
\(494\) 4.00000 0.179969
\(495\) −1.00000 −0.0449467
\(496\) 24.0000 1.07763
\(497\) −15.0000 −0.672842
\(498\) 24.0000 1.07547
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 2.00000 0.0894427
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 28.0000 1.24846 0.624229 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) −1.00000 −0.0444116
\(508\) 20.0000 0.887357
\(509\) −7.00000 −0.310270 −0.155135 0.987893i \(-0.549581\pi\)
−0.155135 + 0.987893i \(0.549581\pi\)
\(510\) 2.00000 0.0885615
\(511\) 30.0000 1.32712
\(512\) 32.0000 1.41421
\(513\) 2.00000 0.0883022
\(514\) −36.0000 −1.58789
\(515\) −16.0000 −0.705044
\(516\) −8.00000 −0.352180
\(517\) 10.0000 0.439799
\(518\) 66.0000 2.89987
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) −4.00000 −0.175075
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −12.0000 −0.524222
\(525\) −3.00000 −0.130931
\(526\) −16.0000 −0.697633
\(527\) 6.00000 0.261364
\(528\) −4.00000 −0.174078
\(529\) −14.0000 −0.608696
\(530\) 22.0000 0.955619
\(531\) 8.00000 0.347170
\(532\) −12.0000 −0.520266
\(533\) 5.00000 0.216574
\(534\) 30.0000 1.29823
\(535\) 9.00000 0.389104
\(536\) 0 0
\(537\) −2.00000 −0.0863064
\(538\) −8.00000 −0.344904
\(539\) −2.00000 −0.0861461
\(540\) −2.00000 −0.0860663
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 44.0000 1.88996
\(543\) 7.00000 0.300399
\(544\) 8.00000 0.342997
\(545\) −16.0000 −0.685365
\(546\) 6.00000 0.256776
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 36.0000 1.53784
\(549\) 13.0000 0.554826
\(550\) −2.00000 −0.0852803
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) −9.00000 −0.382719
\(554\) −36.0000 −1.52949
\(555\) −11.0000 −0.466924
\(556\) −2.00000 −0.0848189
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −12.0000 −0.508001
\(559\) −4.00000 −0.169182
\(560\) −12.0000 −0.507093
\(561\) −1.00000 −0.0422200
\(562\) 60.0000 2.53095
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 20.0000 0.842152
\(565\) 14.0000 0.588984
\(566\) −24.0000 −1.00880
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 4.00000 0.167542
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 2.00000 0.0836242
\(573\) 8.00000 0.334205
\(574\) −30.0000 −1.25218
\(575\) −3.00000 −0.125109
\(576\) −8.00000 −0.333333
\(577\) −19.0000 −0.790980 −0.395490 0.918470i \(-0.629425\pi\)
−0.395490 + 0.918470i \(0.629425\pi\)
\(578\) −32.0000 −1.33102
\(579\) 13.0000 0.540262
\(580\) −4.00000 −0.166091
\(581\) −36.0000 −1.49353
\(582\) −34.0000 −1.40935
\(583\) −11.0000 −0.455573
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 48.0000 1.98286
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) −4.00000 −0.164957
\(589\) 12.0000 0.494451
\(590\) 16.0000 0.658710
\(591\) 0 0
\(592\) −44.0000 −1.80839
\(593\) 4.00000 0.164260 0.0821302 0.996622i \(-0.473828\pi\)
0.0821302 + 0.996622i \(0.473828\pi\)
\(594\) 2.00000 0.0820610
\(595\) −3.00000 −0.122988
\(596\) 26.0000 1.06500
\(597\) −4.00000 −0.163709
\(598\) 6.00000 0.245358
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 24.0000 0.978167
\(603\) 12.0000 0.488678
\(604\) 32.0000 1.30206
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 16.0000 0.648886
\(609\) 6.00000 0.243132
\(610\) 26.0000 1.05271
\(611\) 10.0000 0.404557
\(612\) −2.00000 −0.0808452
\(613\) 13.0000 0.525065 0.262533 0.964923i \(-0.415442\pi\)
0.262533 + 0.964923i \(0.415442\pi\)
\(614\) −10.0000 −0.403567
\(615\) 5.00000 0.201619
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 32.0000 1.28723
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) −12.0000 −0.481932
\(621\) 3.00000 0.120386
\(622\) 48.0000 1.92462
\(623\) −45.0000 −1.80289
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −20.0000 −0.799361
\(627\) −2.00000 −0.0798723
\(628\) −20.0000 −0.798087
\(629\) −11.0000 −0.438599
\(630\) 6.00000 0.239046
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 56.0000 2.22404
\(635\) 10.0000 0.396838
\(636\) −22.0000 −0.872357
\(637\) −2.00000 −0.0792429
\(638\) 4.00000 0.158362
\(639\) −5.00000 −0.197797
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) −18.0000 −0.710403
\(643\) 15.0000 0.591542 0.295771 0.955259i \(-0.404423\pi\)
0.295771 + 0.955259i \(0.404423\pi\)
\(644\) −18.0000 −0.709299
\(645\) −4.00000 −0.157500
\(646\) 4.00000 0.157378
\(647\) 47.0000 1.84776 0.923880 0.382682i \(-0.124999\pi\)
0.923880 + 0.382682i \(0.124999\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) −2.00000 −0.0784465
\(651\) 18.0000 0.705476
\(652\) −26.0000 −1.01824
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 32.0000 1.25130
\(655\) −6.00000 −0.234439
\(656\) 20.0000 0.780869
\(657\) 10.0000 0.390137
\(658\) −60.0000 −2.33904
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 2.00000 0.0778499
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 0 0
\(663\) −1.00000 −0.0388368
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 22.0000 0.852483
\(667\) 6.00000 0.232321
\(668\) −24.0000 −0.928588
\(669\) −8.00000 −0.309298
\(670\) 24.0000 0.927201
\(671\) −13.0000 −0.501859
\(672\) 24.0000 0.925820
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 8.00000 0.308148
\(675\) −1.00000 −0.0384900
\(676\) 2.00000 0.0769231
\(677\) 3.00000 0.115299 0.0576497 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) −28.0000 −1.07533
\(679\) 51.0000 1.95720
\(680\) 0 0
\(681\) −22.0000 −0.843042
\(682\) 12.0000 0.459504
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −4.00000 −0.152944
\(685\) 18.0000 0.687745
\(686\) −30.0000 −1.14541
\(687\) −18.0000 −0.686743
\(688\) −16.0000 −0.609994
\(689\) −11.0000 −0.419067
\(690\) 6.00000 0.228416
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) −12.0000 −0.456172
\(693\) −3.00000 −0.113961
\(694\) −38.0000 −1.44246
\(695\) −1.00000 −0.0379322
\(696\) 0 0
\(697\) 5.00000 0.189389
\(698\) −16.0000 −0.605609
\(699\) 27.0000 1.02123
\(700\) 6.00000 0.226779
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 2.00000 0.0754851
\(703\) −22.0000 −0.829746
\(704\) 8.00000 0.301511
\(705\) 10.0000 0.376622
\(706\) 12.0000 0.451626
\(707\) 0 0
\(708\) −16.0000 −0.601317
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) −10.0000 −0.375293
\(711\) −3.00000 −0.112509
\(712\) 0 0
\(713\) 18.0000 0.674105
\(714\) 6.00000 0.224544
\(715\) 1.00000 0.0373979
\(716\) 4.00000 0.149487
\(717\) 13.0000 0.485494
\(718\) 48.0000 1.79134
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −4.00000 −0.149071
\(721\) −48.0000 −1.78761
\(722\) −30.0000 −1.11648
\(723\) 2.00000 0.0743808
\(724\) −14.0000 −0.520306
\(725\) −2.00000 −0.0742781
\(726\) 20.0000 0.742270
\(727\) 38.0000 1.40934 0.704671 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 20.0000 0.740233
\(731\) −4.00000 −0.147945
\(732\) −26.0000 −0.960988
\(733\) −49.0000 −1.80986 −0.904928 0.425564i \(-0.860076\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(734\) 72.0000 2.65757
\(735\) −2.00000 −0.0737711
\(736\) 24.0000 0.884652
\(737\) −12.0000 −0.442026
\(738\) −10.0000 −0.368105
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 22.0000 0.808736
\(741\) −2.00000 −0.0734718
\(742\) 66.0000 2.42294
\(743\) −34.0000 −1.24734 −0.623670 0.781688i \(-0.714359\pi\)
−0.623670 + 0.781688i \(0.714359\pi\)
\(744\) 0 0
\(745\) 13.0000 0.476283
\(746\) 8.00000 0.292901
\(747\) −12.0000 −0.439057
\(748\) 2.00000 0.0731272
\(749\) 27.0000 0.986559
\(750\) −2.00000 −0.0730297
\(751\) −5.00000 −0.182453 −0.0912263 0.995830i \(-0.529079\pi\)
−0.0912263 + 0.995830i \(0.529079\pi\)
\(752\) 40.0000 1.45865
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 16.0000 0.582300
\(756\) −6.00000 −0.218218
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) −28.0000 −1.01701
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −20.0000 −0.724524
\(763\) −48.0000 −1.73772
\(764\) −16.0000 −0.578860
\(765\) −1.00000 −0.0361551
\(766\) −60.0000 −2.16789
\(767\) −8.00000 −0.288863
\(768\) −16.0000 −0.577350
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) −6.00000 −0.216225
\(771\) 18.0000 0.648254
\(772\) −26.0000 −0.935760
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 8.00000 0.287554
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) −33.0000 −1.18387
\(778\) 0 0
\(779\) 10.0000 0.358287
\(780\) 2.00000 0.0716115
\(781\) 5.00000 0.178914
\(782\) 6.00000 0.214560
\(783\) 2.00000 0.0714742
\(784\) −8.00000 −0.285714
\(785\) −10.0000 −0.356915
\(786\) 12.0000 0.428026
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 0 0
\(789\) 8.00000 0.284808
\(790\) −6.00000 −0.213470
\(791\) 42.0000 1.49335
\(792\) 0 0
\(793\) −13.0000 −0.461644
\(794\) −58.0000 −2.05834
\(795\) −11.0000 −0.390130
\(796\) 8.00000 0.283552
\(797\) −47.0000 −1.66483 −0.832413 0.554156i \(-0.813041\pi\)
−0.832413 + 0.554156i \(0.813041\pi\)
\(798\) 12.0000 0.424795
\(799\) 10.0000 0.353775
\(800\) −8.00000 −0.282843
\(801\) −15.0000 −0.529999
\(802\) 60.0000 2.11867
\(803\) −10.0000 −0.352892
\(804\) −24.0000 −0.846415
\(805\) −9.00000 −0.317208
\(806\) 12.0000 0.422682
\(807\) 4.00000 0.140807
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 2.00000 0.0702728
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) −12.0000 −0.421117
\(813\) −22.0000 −0.771574
\(814\) −22.0000 −0.771100
\(815\) −13.0000 −0.455370
\(816\) −4.00000 −0.140028
\(817\) −8.00000 −0.279885
\(818\) 4.00000 0.139857
\(819\) −3.00000 −0.104828
\(820\) −10.0000 −0.349215
\(821\) 27.0000 0.942306 0.471153 0.882051i \(-0.343838\pi\)
0.471153 + 0.882051i \(0.343838\pi\)
\(822\) −36.0000 −1.25564
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 48.0000 1.67013
\(827\) 26.0000 0.904109 0.452054 0.891990i \(-0.350691\pi\)
0.452054 + 0.891990i \(0.350691\pi\)
\(828\) −6.00000 −0.208514
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) −24.0000 −0.833052
\(831\) 18.0000 0.624413
\(832\) 8.00000 0.277350
\(833\) −2.00000 −0.0692959
\(834\) 2.00000 0.0692543
\(835\) −12.0000 −0.415277
\(836\) 4.00000 0.138343
\(837\) 6.00000 0.207390
\(838\) −52.0000 −1.79631
\(839\) −5.00000 −0.172619 −0.0863096 0.996268i \(-0.527507\pi\)
−0.0863096 + 0.996268i \(0.527507\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 8.00000 0.275698
\(843\) −30.0000 −1.03325
\(844\) −8.00000 −0.275371
\(845\) 1.00000 0.0344010
\(846\) −20.0000 −0.687614
\(847\) −30.0000 −1.03081
\(848\) −44.0000 −1.51097
\(849\) 12.0000 0.411839
\(850\) −2.00000 −0.0685994
\(851\) −33.0000 −1.13123
\(852\) 10.0000 0.342594
\(853\) 45.0000 1.54077 0.770385 0.637579i \(-0.220064\pi\)
0.770385 + 0.637579i \(0.220064\pi\)
\(854\) 78.0000 2.66911
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) 29.0000 0.990621 0.495311 0.868716i \(-0.335054\pi\)
0.495311 + 0.868716i \(0.335054\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −29.0000 −0.989467 −0.494734 0.869045i \(-0.664734\pi\)
−0.494734 + 0.869045i \(0.664734\pi\)
\(860\) 8.00000 0.272798
\(861\) 15.0000 0.511199
\(862\) −48.0000 −1.63489
\(863\) 34.0000 1.15737 0.578687 0.815550i \(-0.303565\pi\)
0.578687 + 0.815550i \(0.303565\pi\)
\(864\) 8.00000 0.272166
\(865\) −6.00000 −0.204006
\(866\) 8.00000 0.271851
\(867\) 16.0000 0.543388
\(868\) −36.0000 −1.22192
\(869\) 3.00000 0.101768
\(870\) 4.00000 0.135613
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 17.0000 0.575363
\(874\) 12.0000 0.405906
\(875\) 3.00000 0.101419
\(876\) −20.0000 −0.675737
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −34.0000 −1.14744
\(879\) −24.0000 −0.809500
\(880\) 4.00000 0.134840
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 4.00000 0.134687
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 2.00000 0.0672673
\(885\) −8.00000 −0.268917
\(886\) 18.0000 0.604722
\(887\) −21.0000 −0.705111 −0.352555 0.935791i \(-0.614687\pi\)
−0.352555 + 0.935791i \(0.614687\pi\)
\(888\) 0 0
\(889\) 30.0000 1.00617
\(890\) −30.0000 −1.00560
\(891\) −1.00000 −0.0335013
\(892\) 16.0000 0.535720
\(893\) 20.0000 0.669274
\(894\) −26.0000 −0.869570
\(895\) 2.00000 0.0668526
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) −26.0000 −0.867631
\(899\) 12.0000 0.400222
\(900\) 2.00000 0.0666667
\(901\) −11.0000 −0.366463
\(902\) 10.0000 0.332964
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) −7.00000 −0.232688
\(906\) −32.0000 −1.06313
\(907\) 6.00000 0.199227 0.0996134 0.995026i \(-0.468239\pi\)
0.0996134 + 0.995026i \(0.468239\pi\)
\(908\) 44.0000 1.46019
\(909\) 0 0
\(910\) −6.00000 −0.198898
\(911\) 44.0000 1.45779 0.728893 0.684628i \(-0.240035\pi\)
0.728893 + 0.684628i \(0.240035\pi\)
\(912\) −8.00000 −0.264906
\(913\) 12.0000 0.397142
\(914\) −22.0000 −0.727695
\(915\) −13.0000 −0.429767
\(916\) 36.0000 1.18947
\(917\) −18.0000 −0.594412
\(918\) 2.00000 0.0660098
\(919\) 37.0000 1.22052 0.610259 0.792202i \(-0.291065\pi\)
0.610259 + 0.792202i \(0.291065\pi\)
\(920\) 0 0
\(921\) 5.00000 0.164756
\(922\) 30.0000 0.987997
\(923\) 5.00000 0.164577
\(924\) 6.00000 0.197386
\(925\) 11.0000 0.361678
\(926\) 54.0000 1.77455
\(927\) −16.0000 −0.525509
\(928\) 16.0000 0.525226
\(929\) 1.00000 0.0328089 0.0164045 0.999865i \(-0.494778\pi\)
0.0164045 + 0.999865i \(0.494778\pi\)
\(930\) 12.0000 0.393496
\(931\) −4.00000 −0.131095
\(932\) −54.0000 −1.76883
\(933\) −24.0000 −0.785725
\(934\) −46.0000 −1.50517
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 72.0000 2.35088
\(939\) 10.0000 0.326338
\(940\) −20.0000 −0.652328
\(941\) 37.0000 1.20617 0.603083 0.797679i \(-0.293939\pi\)
0.603083 + 0.797679i \(0.293939\pi\)
\(942\) 20.0000 0.651635
\(943\) 15.0000 0.488467
\(944\) −32.0000 −1.04151
\(945\) −3.00000 −0.0975900
\(946\) −8.00000 −0.260102
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 6.00000 0.194871
\(949\) −10.0000 −0.324614
\(950\) −4.00000 −0.129777
\(951\) −28.0000 −0.907962
\(952\) 0 0
\(953\) −1.00000 −0.0323932 −0.0161966 0.999869i \(-0.505156\pi\)
−0.0161966 + 0.999869i \(0.505156\pi\)
\(954\) 22.0000 0.712276
\(955\) −8.00000 −0.258874
\(956\) −26.0000 −0.840900
\(957\) −2.00000 −0.0646508
\(958\) 18.0000 0.581554
\(959\) 54.0000 1.74375
\(960\) 8.00000 0.258199
\(961\) 5.00000 0.161290
\(962\) −22.0000 −0.709308
\(963\) 9.00000 0.290021
\(964\) −4.00000 −0.128831
\(965\) −13.0000 −0.418485
\(966\) 18.0000 0.579141
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) −2.00000 −0.0642493
\(970\) 34.0000 1.09167
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −3.00000 −0.0961756
\(974\) −14.0000 −0.448589
\(975\) 1.00000 0.0320256
\(976\) −52.0000 −1.66448
\(977\) 32.0000 1.02377 0.511885 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) 26.0000 0.831388
\(979\) 15.0000 0.479402
\(980\) 4.00000 0.127775
\(981\) −16.0000 −0.510841
\(982\) 56.0000 1.78703
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 30.0000 0.954911
\(988\) 4.00000 0.127257
\(989\) −12.0000 −0.381578
\(990\) −2.00000 −0.0635642
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 48.0000 1.52400
\(993\) 0 0
\(994\) −30.0000 −0.951542
\(995\) 4.00000 0.126809
\(996\) 24.0000 0.760469
\(997\) −36.0000 −1.14013 −0.570066 0.821599i \(-0.693082\pi\)
−0.570066 + 0.821599i \(0.693082\pi\)
\(998\) −28.0000 −0.886325
\(999\) −11.0000 −0.348025
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 195.2.a.b.1.1 1
3.2 odd 2 585.2.a.b.1.1 1
4.3 odd 2 3120.2.a.u.1.1 1
5.2 odd 4 975.2.c.a.274.2 2
5.3 odd 4 975.2.c.a.274.1 2
5.4 even 2 975.2.a.c.1.1 1
7.6 odd 2 9555.2.a.v.1.1 1
12.11 even 2 9360.2.a.d.1.1 1
13.12 even 2 2535.2.a.a.1.1 1
15.2 even 4 2925.2.c.c.2224.1 2
15.8 even 4 2925.2.c.c.2224.2 2
15.14 odd 2 2925.2.a.q.1.1 1
39.38 odd 2 7605.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.b.1.1 1 1.1 even 1 trivial
585.2.a.b.1.1 1 3.2 odd 2
975.2.a.c.1.1 1 5.4 even 2
975.2.c.a.274.1 2 5.3 odd 4
975.2.c.a.274.2 2 5.2 odd 4
2535.2.a.a.1.1 1 13.12 even 2
2925.2.a.q.1.1 1 15.14 odd 2
2925.2.c.c.2224.1 2 15.2 even 4
2925.2.c.c.2224.2 2 15.8 even 4
3120.2.a.u.1.1 1 4.3 odd 2
7605.2.a.u.1.1 1 39.38 odd 2
9360.2.a.d.1.1 1 12.11 even 2
9555.2.a.v.1.1 1 7.6 odd 2