Properties

Label 147.4.a.f.1.1
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
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 \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} -3.00000 q^{3} +8.00000 q^{4} -18.0000 q^{5} -12.0000 q^{6} +9.00000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -3.00000 q^{3} +8.00000 q^{4} -18.0000 q^{5} -12.0000 q^{6} +9.00000 q^{9} -72.0000 q^{10} -50.0000 q^{11} -24.0000 q^{12} +36.0000 q^{13} +54.0000 q^{15} -64.0000 q^{16} -126.000 q^{17} +36.0000 q^{18} +72.0000 q^{19} -144.000 q^{20} -200.000 q^{22} +14.0000 q^{23} +199.000 q^{25} +144.000 q^{26} -27.0000 q^{27} +158.000 q^{29} +216.000 q^{30} +36.0000 q^{31} -256.000 q^{32} +150.000 q^{33} -504.000 q^{34} +72.0000 q^{36} -162.000 q^{37} +288.000 q^{38} -108.000 q^{39} +270.000 q^{41} -324.000 q^{43} -400.000 q^{44} -162.000 q^{45} +56.0000 q^{46} +72.0000 q^{47} +192.000 q^{48} +796.000 q^{50} +378.000 q^{51} +288.000 q^{52} -22.0000 q^{53} -108.000 q^{54} +900.000 q^{55} -216.000 q^{57} +632.000 q^{58} -468.000 q^{59} +432.000 q^{60} -792.000 q^{61} +144.000 q^{62} -512.000 q^{64} -648.000 q^{65} +600.000 q^{66} +232.000 q^{67} -1008.00 q^{68} -42.0000 q^{69} -734.000 q^{71} -180.000 q^{73} -648.000 q^{74} -597.000 q^{75} +576.000 q^{76} -432.000 q^{78} +236.000 q^{79} +1152.00 q^{80} +81.0000 q^{81} +1080.00 q^{82} -36.0000 q^{83} +2268.00 q^{85} -1296.00 q^{86} -474.000 q^{87} -234.000 q^{89} -648.000 q^{90} +112.000 q^{92} -108.000 q^{93} +288.000 q^{94} -1296.00 q^{95} +768.000 q^{96} -468.000 q^{97} -450.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −3.00000 −0.577350
\(4\) 8.00000 1.00000
\(5\) −18.0000 −1.60997 −0.804984 0.593296i \(-0.797826\pi\)
−0.804984 + 0.593296i \(0.797826\pi\)
\(6\) −12.0000 −0.816497
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) −72.0000 −2.27684
\(11\) −50.0000 −1.37051 −0.685253 0.728305i \(-0.740308\pi\)
−0.685253 + 0.728305i \(0.740308\pi\)
\(12\) −24.0000 −0.577350
\(13\) 36.0000 0.768046 0.384023 0.923323i \(-0.374538\pi\)
0.384023 + 0.923323i \(0.374538\pi\)
\(14\) 0 0
\(15\) 54.0000 0.929516
\(16\) −64.0000 −1.00000
\(17\) −126.000 −1.79762 −0.898808 0.438342i \(-0.855566\pi\)
−0.898808 + 0.438342i \(0.855566\pi\)
\(18\) 36.0000 0.471405
\(19\) 72.0000 0.869365 0.434682 0.900584i \(-0.356861\pi\)
0.434682 + 0.900584i \(0.356861\pi\)
\(20\) −144.000 −1.60997
\(21\) 0 0
\(22\) −200.000 −1.93819
\(23\) 14.0000 0.126922 0.0634609 0.997984i \(-0.479786\pi\)
0.0634609 + 0.997984i \(0.479786\pi\)
\(24\) 0 0
\(25\) 199.000 1.59200
\(26\) 144.000 1.08618
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 158.000 1.01172 0.505860 0.862616i \(-0.331175\pi\)
0.505860 + 0.862616i \(0.331175\pi\)
\(30\) 216.000 1.31453
\(31\) 36.0000 0.208574 0.104287 0.994547i \(-0.466744\pi\)
0.104287 + 0.994547i \(0.466744\pi\)
\(32\) −256.000 −1.41421
\(33\) 150.000 0.791262
\(34\) −504.000 −2.54221
\(35\) 0 0
\(36\) 72.0000 0.333333
\(37\) −162.000 −0.719801 −0.359900 0.932991i \(-0.617189\pi\)
−0.359900 + 0.932991i \(0.617189\pi\)
\(38\) 288.000 1.22947
\(39\) −108.000 −0.443432
\(40\) 0 0
\(41\) 270.000 1.02846 0.514231 0.857652i \(-0.328078\pi\)
0.514231 + 0.857652i \(0.328078\pi\)
\(42\) 0 0
\(43\) −324.000 −1.14906 −0.574529 0.818484i \(-0.694815\pi\)
−0.574529 + 0.818484i \(0.694815\pi\)
\(44\) −400.000 −1.37051
\(45\) −162.000 −0.536656
\(46\) 56.0000 0.179495
\(47\) 72.0000 0.223453 0.111726 0.993739i \(-0.464362\pi\)
0.111726 + 0.993739i \(0.464362\pi\)
\(48\) 192.000 0.577350
\(49\) 0 0
\(50\) 796.000 2.25143
\(51\) 378.000 1.03785
\(52\) 288.000 0.768046
\(53\) −22.0000 −0.0570176 −0.0285088 0.999594i \(-0.509076\pi\)
−0.0285088 + 0.999594i \(0.509076\pi\)
\(54\) −108.000 −0.272166
\(55\) 900.000 2.20647
\(56\) 0 0
\(57\) −216.000 −0.501928
\(58\) 632.000 1.43079
\(59\) −468.000 −1.03268 −0.516342 0.856382i \(-0.672707\pi\)
−0.516342 + 0.856382i \(0.672707\pi\)
\(60\) 432.000 0.929516
\(61\) −792.000 −1.66238 −0.831190 0.555988i \(-0.812340\pi\)
−0.831190 + 0.555988i \(0.812340\pi\)
\(62\) 144.000 0.294968
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) −648.000 −1.23653
\(66\) 600.000 1.11901
\(67\) 232.000 0.423034 0.211517 0.977374i \(-0.432160\pi\)
0.211517 + 0.977374i \(0.432160\pi\)
\(68\) −1008.00 −1.79762
\(69\) −42.0000 −0.0732783
\(70\) 0 0
\(71\) −734.000 −1.22690 −0.613449 0.789734i \(-0.710218\pi\)
−0.613449 + 0.789734i \(0.710218\pi\)
\(72\) 0 0
\(73\) −180.000 −0.288595 −0.144297 0.989534i \(-0.546092\pi\)
−0.144297 + 0.989534i \(0.546092\pi\)
\(74\) −648.000 −1.01795
\(75\) −597.000 −0.919142
\(76\) 576.000 0.869365
\(77\) 0 0
\(78\) −432.000 −0.627107
\(79\) 236.000 0.336102 0.168051 0.985778i \(-0.446253\pi\)
0.168051 + 0.985778i \(0.446253\pi\)
\(80\) 1152.00 1.60997
\(81\) 81.0000 0.111111
\(82\) 1080.00 1.45446
\(83\) −36.0000 −0.0476086 −0.0238043 0.999717i \(-0.507578\pi\)
−0.0238043 + 0.999717i \(0.507578\pi\)
\(84\) 0 0
\(85\) 2268.00 2.89411
\(86\) −1296.00 −1.62501
\(87\) −474.000 −0.584116
\(88\) 0 0
\(89\) −234.000 −0.278696 −0.139348 0.990243i \(-0.544501\pi\)
−0.139348 + 0.990243i \(0.544501\pi\)
\(90\) −648.000 −0.758947
\(91\) 0 0
\(92\) 112.000 0.126922
\(93\) −108.000 −0.120420
\(94\) 288.000 0.316010
\(95\) −1296.00 −1.39965
\(96\) 768.000 0.816497
\(97\) −468.000 −0.489878 −0.244939 0.969538i \(-0.578768\pi\)
−0.244939 + 0.969538i \(0.578768\pi\)
\(98\) 0 0
\(99\) −450.000 −0.456835
\(100\) 1592.00 1.59200
\(101\) 666.000 0.656133 0.328067 0.944655i \(-0.393603\pi\)
0.328067 + 0.944655i \(0.393603\pi\)
\(102\) 1512.00 1.46775
\(103\) 252.000 0.241071 0.120535 0.992709i \(-0.461539\pi\)
0.120535 + 0.992709i \(0.461539\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −88.0000 −0.0806351
\(107\) 670.000 0.605340 0.302670 0.953095i \(-0.402122\pi\)
0.302670 + 0.953095i \(0.402122\pi\)
\(108\) −216.000 −0.192450
\(109\) 162.000 0.142356 0.0711779 0.997464i \(-0.477324\pi\)
0.0711779 + 0.997464i \(0.477324\pi\)
\(110\) 3600.00 3.12042
\(111\) 486.000 0.415577
\(112\) 0 0
\(113\) −1390.00 −1.15717 −0.578585 0.815622i \(-0.696395\pi\)
−0.578585 + 0.815622i \(0.696395\pi\)
\(114\) −864.000 −0.709833
\(115\) −252.000 −0.204340
\(116\) 1264.00 1.01172
\(117\) 324.000 0.256015
\(118\) −1872.00 −1.46044
\(119\) 0 0
\(120\) 0 0
\(121\) 1169.00 0.878287
\(122\) −3168.00 −2.35096
\(123\) −810.000 −0.593782
\(124\) 288.000 0.208574
\(125\) −1332.00 −0.953102
\(126\) 0 0
\(127\) 916.000 0.640015 0.320007 0.947415i \(-0.396315\pi\)
0.320007 + 0.947415i \(0.396315\pi\)
\(128\) 0 0
\(129\) 972.000 0.663410
\(130\) −2592.00 −1.74872
\(131\) −2268.00 −1.51264 −0.756321 0.654201i \(-0.773005\pi\)
−0.756321 + 0.654201i \(0.773005\pi\)
\(132\) 1200.00 0.791262
\(133\) 0 0
\(134\) 928.000 0.598261
\(135\) 486.000 0.309839
\(136\) 0 0
\(137\) 806.000 0.502637 0.251318 0.967904i \(-0.419136\pi\)
0.251318 + 0.967904i \(0.419136\pi\)
\(138\) −168.000 −0.103631
\(139\) −2628.00 −1.60363 −0.801813 0.597575i \(-0.796131\pi\)
−0.801813 + 0.597575i \(0.796131\pi\)
\(140\) 0 0
\(141\) −216.000 −0.129011
\(142\) −2936.00 −1.73510
\(143\) −1800.00 −1.05261
\(144\) −576.000 −0.333333
\(145\) −2844.00 −1.62884
\(146\) −720.000 −0.408134
\(147\) 0 0
\(148\) −1296.00 −0.719801
\(149\) −2390.00 −1.31407 −0.657035 0.753860i \(-0.728190\pi\)
−0.657035 + 0.753860i \(0.728190\pi\)
\(150\) −2388.00 −1.29986
\(151\) 3240.00 1.74614 0.873071 0.487593i \(-0.162125\pi\)
0.873071 + 0.487593i \(0.162125\pi\)
\(152\) 0 0
\(153\) −1134.00 −0.599206
\(154\) 0 0
\(155\) −648.000 −0.335798
\(156\) −864.000 −0.443432
\(157\) −3024.00 −1.53721 −0.768603 0.639726i \(-0.779048\pi\)
−0.768603 + 0.639726i \(0.779048\pi\)
\(158\) 944.000 0.475320
\(159\) 66.0000 0.0329191
\(160\) 4608.00 2.27684
\(161\) 0 0
\(162\) 324.000 0.157135
\(163\) −1784.00 −0.857262 −0.428631 0.903480i \(-0.641004\pi\)
−0.428631 + 0.903480i \(0.641004\pi\)
\(164\) 2160.00 1.02846
\(165\) −2700.00 −1.27391
\(166\) −144.000 −0.0673287
\(167\) 3024.00 1.40122 0.700611 0.713543i \(-0.252911\pi\)
0.700611 + 0.713543i \(0.252911\pi\)
\(168\) 0 0
\(169\) −901.000 −0.410105
\(170\) 9072.00 4.09289
\(171\) 648.000 0.289788
\(172\) −2592.00 −1.14906
\(173\) −1566.00 −0.688213 −0.344106 0.938931i \(-0.611818\pi\)
−0.344106 + 0.938931i \(0.611818\pi\)
\(174\) −1896.00 −0.826065
\(175\) 0 0
\(176\) 3200.00 1.37051
\(177\) 1404.00 0.596221
\(178\) −936.000 −0.394136
\(179\) 3802.00 1.58757 0.793784 0.608199i \(-0.208108\pi\)
0.793784 + 0.608199i \(0.208108\pi\)
\(180\) −1296.00 −0.536656
\(181\) 468.000 0.192189 0.0960944 0.995372i \(-0.469365\pi\)
0.0960944 + 0.995372i \(0.469365\pi\)
\(182\) 0 0
\(183\) 2376.00 0.959776
\(184\) 0 0
\(185\) 2916.00 1.15886
\(186\) −432.000 −0.170300
\(187\) 6300.00 2.46365
\(188\) 576.000 0.223453
\(189\) 0 0
\(190\) −5184.00 −1.97940
\(191\) 482.000 0.182598 0.0912992 0.995824i \(-0.470898\pi\)
0.0912992 + 0.995824i \(0.470898\pi\)
\(192\) 1536.00 0.577350
\(193\) −810.000 −0.302099 −0.151049 0.988526i \(-0.548265\pi\)
−0.151049 + 0.988526i \(0.548265\pi\)
\(194\) −1872.00 −0.692793
\(195\) 1944.00 0.713911
\(196\) 0 0
\(197\) −2462.00 −0.890407 −0.445204 0.895429i \(-0.646869\pi\)
−0.445204 + 0.895429i \(0.646869\pi\)
\(198\) −1800.00 −0.646063
\(199\) 4536.00 1.61582 0.807911 0.589305i \(-0.200598\pi\)
0.807911 + 0.589305i \(0.200598\pi\)
\(200\) 0 0
\(201\) −696.000 −0.244239
\(202\) 2664.00 0.927913
\(203\) 0 0
\(204\) 3024.00 1.03785
\(205\) −4860.00 −1.65579
\(206\) 1008.00 0.340926
\(207\) 126.000 0.0423073
\(208\) −2304.00 −0.768046
\(209\) −3600.00 −1.19147
\(210\) 0 0
\(211\) 2916.00 0.951402 0.475701 0.879607i \(-0.342195\pi\)
0.475701 + 0.879607i \(0.342195\pi\)
\(212\) −176.000 −0.0570176
\(213\) 2202.00 0.708350
\(214\) 2680.00 0.856080
\(215\) 5832.00 1.84995
\(216\) 0 0
\(217\) 0 0
\(218\) 648.000 0.201322
\(219\) 540.000 0.166620
\(220\) 7200.00 2.20647
\(221\) −4536.00 −1.38065
\(222\) 1944.00 0.587715
\(223\) 1080.00 0.324315 0.162157 0.986765i \(-0.448155\pi\)
0.162157 + 0.986765i \(0.448155\pi\)
\(224\) 0 0
\(225\) 1791.00 0.530667
\(226\) −5560.00 −1.63649
\(227\) 1332.00 0.389462 0.194731 0.980857i \(-0.437617\pi\)
0.194731 + 0.980857i \(0.437617\pi\)
\(228\) −1728.00 −0.501928
\(229\) −1620.00 −0.467479 −0.233739 0.972299i \(-0.575096\pi\)
−0.233739 + 0.972299i \(0.575096\pi\)
\(230\) −1008.00 −0.288981
\(231\) 0 0
\(232\) 0 0
\(233\) 6718.00 1.88889 0.944444 0.328673i \(-0.106601\pi\)
0.944444 + 0.328673i \(0.106601\pi\)
\(234\) 1296.00 0.362061
\(235\) −1296.00 −0.359752
\(236\) −3744.00 −1.03268
\(237\) −708.000 −0.194049
\(238\) 0 0
\(239\) −3578.00 −0.968375 −0.484187 0.874964i \(-0.660885\pi\)
−0.484187 + 0.874964i \(0.660885\pi\)
\(240\) −3456.00 −0.929516
\(241\) 756.000 0.202067 0.101034 0.994883i \(-0.467785\pi\)
0.101034 + 0.994883i \(0.467785\pi\)
\(242\) 4676.00 1.24209
\(243\) −243.000 −0.0641500
\(244\) −6336.00 −1.66238
\(245\) 0 0
\(246\) −3240.00 −0.839735
\(247\) 2592.00 0.667713
\(248\) 0 0
\(249\) 108.000 0.0274868
\(250\) −5328.00 −1.34789
\(251\) 6516.00 1.63859 0.819295 0.573372i \(-0.194365\pi\)
0.819295 + 0.573372i \(0.194365\pi\)
\(252\) 0 0
\(253\) −700.000 −0.173947
\(254\) 3664.00 0.905117
\(255\) −6804.00 −1.67091
\(256\) 4096.00 1.00000
\(257\) 6030.00 1.46358 0.731792 0.681528i \(-0.238684\pi\)
0.731792 + 0.681528i \(0.238684\pi\)
\(258\) 3888.00 0.938203
\(259\) 0 0
\(260\) −5184.00 −1.23653
\(261\) 1422.00 0.337240
\(262\) −9072.00 −2.13920
\(263\) 590.000 0.138331 0.0691653 0.997605i \(-0.477966\pi\)
0.0691653 + 0.997605i \(0.477966\pi\)
\(264\) 0 0
\(265\) 396.000 0.0917966
\(266\) 0 0
\(267\) 702.000 0.160905
\(268\) 1856.00 0.423034
\(269\) 990.000 0.224392 0.112196 0.993686i \(-0.464212\pi\)
0.112196 + 0.993686i \(0.464212\pi\)
\(270\) 1944.00 0.438178
\(271\) 3420.00 0.766606 0.383303 0.923623i \(-0.374787\pi\)
0.383303 + 0.923623i \(0.374787\pi\)
\(272\) 8064.00 1.79762
\(273\) 0 0
\(274\) 3224.00 0.710836
\(275\) −9950.00 −2.18185
\(276\) −336.000 −0.0732783
\(277\) −2734.00 −0.593033 −0.296516 0.955028i \(-0.595825\pi\)
−0.296516 + 0.955028i \(0.595825\pi\)
\(278\) −10512.0 −2.26787
\(279\) 324.000 0.0695246
\(280\) 0 0
\(281\) 598.000 0.126953 0.0634763 0.997983i \(-0.479781\pi\)
0.0634763 + 0.997983i \(0.479781\pi\)
\(282\) −864.000 −0.182448
\(283\) −3600.00 −0.756176 −0.378088 0.925770i \(-0.623418\pi\)
−0.378088 + 0.925770i \(0.623418\pi\)
\(284\) −5872.00 −1.22690
\(285\) 3888.00 0.808089
\(286\) −7200.00 −1.48862
\(287\) 0 0
\(288\) −2304.00 −0.471405
\(289\) 10963.0 2.23143
\(290\) −11376.0 −2.30352
\(291\) 1404.00 0.282831
\(292\) −1440.00 −0.288595
\(293\) −7902.00 −1.57556 −0.787781 0.615955i \(-0.788770\pi\)
−0.787781 + 0.615955i \(0.788770\pi\)
\(294\) 0 0
\(295\) 8424.00 1.66259
\(296\) 0 0
\(297\) 1350.00 0.263754
\(298\) −9560.00 −1.85838
\(299\) 504.000 0.0974818
\(300\) −4776.00 −0.919142
\(301\) 0 0
\(302\) 12960.0 2.46942
\(303\) −1998.00 −0.378819
\(304\) −4608.00 −0.869365
\(305\) 14256.0 2.67638
\(306\) −4536.00 −0.847405
\(307\) 10224.0 1.90070 0.950349 0.311185i \(-0.100726\pi\)
0.950349 + 0.311185i \(0.100726\pi\)
\(308\) 0 0
\(309\) −756.000 −0.139182
\(310\) −2592.00 −0.474889
\(311\) −3888.00 −0.708901 −0.354451 0.935075i \(-0.615332\pi\)
−0.354451 + 0.935075i \(0.615332\pi\)
\(312\) 0 0
\(313\) −5112.00 −0.923154 −0.461577 0.887100i \(-0.652716\pi\)
−0.461577 + 0.887100i \(0.652716\pi\)
\(314\) −12096.0 −2.17394
\(315\) 0 0
\(316\) 1888.00 0.336102
\(317\) −10102.0 −1.78986 −0.894929 0.446209i \(-0.852774\pi\)
−0.894929 + 0.446209i \(0.852774\pi\)
\(318\) 264.000 0.0465547
\(319\) −7900.00 −1.38657
\(320\) 9216.00 1.60997
\(321\) −2010.00 −0.349493
\(322\) 0 0
\(323\) −9072.00 −1.56279
\(324\) 648.000 0.111111
\(325\) 7164.00 1.22273
\(326\) −7136.00 −1.21235
\(327\) −486.000 −0.0821892
\(328\) 0 0
\(329\) 0 0
\(330\) −10800.0 −1.80158
\(331\) 5508.00 0.914644 0.457322 0.889301i \(-0.348809\pi\)
0.457322 + 0.889301i \(0.348809\pi\)
\(332\) −288.000 −0.0476086
\(333\) −1458.00 −0.239934
\(334\) 12096.0 1.98163
\(335\) −4176.00 −0.681072
\(336\) 0 0
\(337\) −9234.00 −1.49261 −0.746303 0.665607i \(-0.768173\pi\)
−0.746303 + 0.665607i \(0.768173\pi\)
\(338\) −3604.00 −0.579976
\(339\) 4170.00 0.668092
\(340\) 18144.0 2.89411
\(341\) −1800.00 −0.285852
\(342\) 2592.00 0.409823
\(343\) 0 0
\(344\) 0 0
\(345\) 756.000 0.117976
\(346\) −6264.00 −0.973280
\(347\) −6494.00 −1.00466 −0.502329 0.864677i \(-0.667523\pi\)
−0.502329 + 0.864677i \(0.667523\pi\)
\(348\) −3792.00 −0.584116
\(349\) −10080.0 −1.54605 −0.773023 0.634378i \(-0.781256\pi\)
−0.773023 + 0.634378i \(0.781256\pi\)
\(350\) 0 0
\(351\) −972.000 −0.147811
\(352\) 12800.0 1.93819
\(353\) 738.000 0.111274 0.0556371 0.998451i \(-0.482281\pi\)
0.0556371 + 0.998451i \(0.482281\pi\)
\(354\) 5616.00 0.843184
\(355\) 13212.0 1.97527
\(356\) −1872.00 −0.278696
\(357\) 0 0
\(358\) 15208.0 2.24516
\(359\) 194.000 0.0285207 0.0142603 0.999898i \(-0.495461\pi\)
0.0142603 + 0.999898i \(0.495461\pi\)
\(360\) 0 0
\(361\) −1675.00 −0.244205
\(362\) 1872.00 0.271796
\(363\) −3507.00 −0.507079
\(364\) 0 0
\(365\) 3240.00 0.464628
\(366\) 9504.00 1.35733
\(367\) 4752.00 0.675892 0.337946 0.941165i \(-0.390268\pi\)
0.337946 + 0.941165i \(0.390268\pi\)
\(368\) −896.000 −0.126922
\(369\) 2430.00 0.342820
\(370\) 11664.0 1.63887
\(371\) 0 0
\(372\) −864.000 −0.120420
\(373\) −2306.00 −0.320108 −0.160054 0.987108i \(-0.551167\pi\)
−0.160054 + 0.987108i \(0.551167\pi\)
\(374\) 25200.0 3.48412
\(375\) 3996.00 0.550273
\(376\) 0 0
\(377\) 5688.00 0.777047
\(378\) 0 0
\(379\) −7452.00 −1.00998 −0.504991 0.863124i \(-0.668504\pi\)
−0.504991 + 0.863124i \(0.668504\pi\)
\(380\) −10368.0 −1.39965
\(381\) −2748.00 −0.369513
\(382\) 1928.00 0.258233
\(383\) 1152.00 0.153693 0.0768465 0.997043i \(-0.475515\pi\)
0.0768465 + 0.997043i \(0.475515\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3240.00 −0.427232
\(387\) −2916.00 −0.383020
\(388\) −3744.00 −0.489878
\(389\) 1894.00 0.246863 0.123431 0.992353i \(-0.460610\pi\)
0.123431 + 0.992353i \(0.460610\pi\)
\(390\) 7776.00 1.00962
\(391\) −1764.00 −0.228157
\(392\) 0 0
\(393\) 6804.00 0.873324
\(394\) −9848.00 −1.25923
\(395\) −4248.00 −0.541114
\(396\) −3600.00 −0.456835
\(397\) −9216.00 −1.16508 −0.582541 0.812801i \(-0.697942\pi\)
−0.582541 + 0.812801i \(0.697942\pi\)
\(398\) 18144.0 2.28512
\(399\) 0 0
\(400\) −12736.0 −1.59200
\(401\) −11650.0 −1.45081 −0.725403 0.688324i \(-0.758347\pi\)
−0.725403 + 0.688324i \(0.758347\pi\)
\(402\) −2784.00 −0.345406
\(403\) 1296.00 0.160194
\(404\) 5328.00 0.656133
\(405\) −1458.00 −0.178885
\(406\) 0 0
\(407\) 8100.00 0.986492
\(408\) 0 0
\(409\) 7524.00 0.909628 0.454814 0.890586i \(-0.349706\pi\)
0.454814 + 0.890586i \(0.349706\pi\)
\(410\) −19440.0 −2.34164
\(411\) −2418.00 −0.290197
\(412\) 2016.00 0.241071
\(413\) 0 0
\(414\) 504.000 0.0598315
\(415\) 648.000 0.0766484
\(416\) −9216.00 −1.08618
\(417\) 7884.00 0.925854
\(418\) −14400.0 −1.68499
\(419\) 3852.00 0.449123 0.224561 0.974460i \(-0.427905\pi\)
0.224561 + 0.974460i \(0.427905\pi\)
\(420\) 0 0
\(421\) 10402.0 1.20419 0.602093 0.798426i \(-0.294334\pi\)
0.602093 + 0.798426i \(0.294334\pi\)
\(422\) 11664.0 1.34548
\(423\) 648.000 0.0744843
\(424\) 0 0
\(425\) −25074.0 −2.86181
\(426\) 8808.00 1.00176
\(427\) 0 0
\(428\) 5360.00 0.605340
\(429\) 5400.00 0.607726
\(430\) 23328.0 2.61622
\(431\) −10390.0 −1.16118 −0.580590 0.814196i \(-0.697178\pi\)
−0.580590 + 0.814196i \(0.697178\pi\)
\(432\) 1728.00 0.192450
\(433\) −11232.0 −1.24659 −0.623297 0.781985i \(-0.714207\pi\)
−0.623297 + 0.781985i \(0.714207\pi\)
\(434\) 0 0
\(435\) 8532.00 0.940409
\(436\) 1296.00 0.142356
\(437\) 1008.00 0.110341
\(438\) 2160.00 0.235637
\(439\) −14616.0 −1.58903 −0.794514 0.607245i \(-0.792275\pi\)
−0.794514 + 0.607245i \(0.792275\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −18144.0 −1.95254
\(443\) 11938.0 1.28034 0.640171 0.768232i \(-0.278863\pi\)
0.640171 + 0.768232i \(0.278863\pi\)
\(444\) 3888.00 0.415577
\(445\) 4212.00 0.448692
\(446\) 4320.00 0.458650
\(447\) 7170.00 0.758679
\(448\) 0 0
\(449\) 8186.00 0.860404 0.430202 0.902733i \(-0.358442\pi\)
0.430202 + 0.902733i \(0.358442\pi\)
\(450\) 7164.00 0.750476
\(451\) −13500.0 −1.40951
\(452\) −11120.0 −1.15717
\(453\) −9720.00 −1.00814
\(454\) 5328.00 0.550783
\(455\) 0 0
\(456\) 0 0
\(457\) 2106.00 0.215568 0.107784 0.994174i \(-0.465625\pi\)
0.107784 + 0.994174i \(0.465625\pi\)
\(458\) −6480.00 −0.661115
\(459\) 3402.00 0.345952
\(460\) −2016.00 −0.204340
\(461\) −9486.00 −0.958367 −0.479183 0.877715i \(-0.659067\pi\)
−0.479183 + 0.877715i \(0.659067\pi\)
\(462\) 0 0
\(463\) −12652.0 −1.26995 −0.634977 0.772531i \(-0.718990\pi\)
−0.634977 + 0.772531i \(0.718990\pi\)
\(464\) −10112.0 −1.01172
\(465\) 1944.00 0.193873
\(466\) 26872.0 2.67129
\(467\) 3708.00 0.367421 0.183711 0.982980i \(-0.441189\pi\)
0.183711 + 0.982980i \(0.441189\pi\)
\(468\) 2592.00 0.256015
\(469\) 0 0
\(470\) −5184.00 −0.508766
\(471\) 9072.00 0.887507
\(472\) 0 0
\(473\) 16200.0 1.57479
\(474\) −2832.00 −0.274426
\(475\) 14328.0 1.38403
\(476\) 0 0
\(477\) −198.000 −0.0190059
\(478\) −14312.0 −1.36949
\(479\) 8064.00 0.769214 0.384607 0.923080i \(-0.374337\pi\)
0.384607 + 0.923080i \(0.374337\pi\)
\(480\) −13824.0 −1.31453
\(481\) −5832.00 −0.552841
\(482\) 3024.00 0.285766
\(483\) 0 0
\(484\) 9352.00 0.878287
\(485\) 8424.00 0.788689
\(486\) −972.000 −0.0907218
\(487\) −11664.0 −1.08531 −0.542655 0.839955i \(-0.682581\pi\)
−0.542655 + 0.839955i \(0.682581\pi\)
\(488\) 0 0
\(489\) 5352.00 0.494940
\(490\) 0 0
\(491\) −9814.00 −0.902036 −0.451018 0.892515i \(-0.648939\pi\)
−0.451018 + 0.892515i \(0.648939\pi\)
\(492\) −6480.00 −0.593782
\(493\) −19908.0 −1.81868
\(494\) 10368.0 0.944288
\(495\) 8100.00 0.735491
\(496\) −2304.00 −0.208574
\(497\) 0 0
\(498\) 432.000 0.0388723
\(499\) −15228.0 −1.36613 −0.683065 0.730358i \(-0.739353\pi\)
−0.683065 + 0.730358i \(0.739353\pi\)
\(500\) −10656.0 −0.953102
\(501\) −9072.00 −0.808996
\(502\) 26064.0 2.31732
\(503\) 11088.0 0.982882 0.491441 0.870911i \(-0.336470\pi\)
0.491441 + 0.870911i \(0.336470\pi\)
\(504\) 0 0
\(505\) −11988.0 −1.05635
\(506\) −2800.00 −0.245998
\(507\) 2703.00 0.236774
\(508\) 7328.00 0.640015
\(509\) 5814.00 0.506289 0.253144 0.967429i \(-0.418535\pi\)
0.253144 + 0.967429i \(0.418535\pi\)
\(510\) −27216.0 −2.36303
\(511\) 0 0
\(512\) 16384.0 1.41421
\(513\) −1944.00 −0.167309
\(514\) 24120.0 2.06982
\(515\) −4536.00 −0.388117
\(516\) 7776.00 0.663410
\(517\) −3600.00 −0.306243
\(518\) 0 0
\(519\) 4698.00 0.397340
\(520\) 0 0
\(521\) 11682.0 0.982337 0.491169 0.871065i \(-0.336570\pi\)
0.491169 + 0.871065i \(0.336570\pi\)
\(522\) 5688.00 0.476929
\(523\) 2988.00 0.249820 0.124910 0.992168i \(-0.460136\pi\)
0.124910 + 0.992168i \(0.460136\pi\)
\(524\) −18144.0 −1.51264
\(525\) 0 0
\(526\) 2360.00 0.195629
\(527\) −4536.00 −0.374936
\(528\) −9600.00 −0.791262
\(529\) −11971.0 −0.983891
\(530\) 1584.00 0.129820
\(531\) −4212.00 −0.344228
\(532\) 0 0
\(533\) 9720.00 0.789906
\(534\) 2808.00 0.227554
\(535\) −12060.0 −0.974578
\(536\) 0 0
\(537\) −11406.0 −0.916583
\(538\) 3960.00 0.317338
\(539\) 0 0
\(540\) 3888.00 0.309839
\(541\) 7130.00 0.566622 0.283311 0.959028i \(-0.408567\pi\)
0.283311 + 0.959028i \(0.408567\pi\)
\(542\) 13680.0 1.08414
\(543\) −1404.00 −0.110960
\(544\) 32256.0 2.54221
\(545\) −2916.00 −0.229188
\(546\) 0 0
\(547\) −5488.00 −0.428976 −0.214488 0.976727i \(-0.568808\pi\)
−0.214488 + 0.976727i \(0.568808\pi\)
\(548\) 6448.00 0.502637
\(549\) −7128.00 −0.554127
\(550\) −39800.0 −3.08560
\(551\) 11376.0 0.879553
\(552\) 0 0
\(553\) 0 0
\(554\) −10936.0 −0.838675
\(555\) −8748.00 −0.669067
\(556\) −21024.0 −1.60363
\(557\) 5746.00 0.437102 0.218551 0.975826i \(-0.429867\pi\)
0.218551 + 0.975826i \(0.429867\pi\)
\(558\) 1296.00 0.0983227
\(559\) −11664.0 −0.882531
\(560\) 0 0
\(561\) −18900.0 −1.42239
\(562\) 2392.00 0.179538
\(563\) −13068.0 −0.978243 −0.489121 0.872216i \(-0.662682\pi\)
−0.489121 + 0.872216i \(0.662682\pi\)
\(564\) −1728.00 −0.129011
\(565\) 25020.0 1.86301
\(566\) −14400.0 −1.06939
\(567\) 0 0
\(568\) 0 0
\(569\) −1130.00 −0.0832549 −0.0416275 0.999133i \(-0.513254\pi\)
−0.0416275 + 0.999133i \(0.513254\pi\)
\(570\) 15552.0 1.14281
\(571\) 16864.0 1.23597 0.617983 0.786192i \(-0.287950\pi\)
0.617983 + 0.786192i \(0.287950\pi\)
\(572\) −14400.0 −1.05261
\(573\) −1446.00 −0.105423
\(574\) 0 0
\(575\) 2786.00 0.202060
\(576\) −4608.00 −0.333333
\(577\) 2088.00 0.150649 0.0753246 0.997159i \(-0.476001\pi\)
0.0753246 + 0.997159i \(0.476001\pi\)
\(578\) 43852.0 3.15571
\(579\) 2430.00 0.174417
\(580\) −22752.0 −1.62884
\(581\) 0 0
\(582\) 5616.00 0.399984
\(583\) 1100.00 0.0781430
\(584\) 0 0
\(585\) −5832.00 −0.412177
\(586\) −31608.0 −2.22818
\(587\) −10260.0 −0.721423 −0.360712 0.932677i \(-0.617466\pi\)
−0.360712 + 0.932677i \(0.617466\pi\)
\(588\) 0 0
\(589\) 2592.00 0.181327
\(590\) 33696.0 2.35126
\(591\) 7386.00 0.514077
\(592\) 10368.0 0.719801
\(593\) −3582.00 −0.248052 −0.124026 0.992279i \(-0.539581\pi\)
−0.124026 + 0.992279i \(0.539581\pi\)
\(594\) 5400.00 0.373005
\(595\) 0 0
\(596\) −19120.0 −1.31407
\(597\) −13608.0 −0.932895
\(598\) 2016.00 0.137860
\(599\) 7034.00 0.479802 0.239901 0.970797i \(-0.422885\pi\)
0.239901 + 0.970797i \(0.422885\pi\)
\(600\) 0 0
\(601\) 18072.0 1.22658 0.613288 0.789859i \(-0.289846\pi\)
0.613288 + 0.789859i \(0.289846\pi\)
\(602\) 0 0
\(603\) 2088.00 0.141011
\(604\) 25920.0 1.74614
\(605\) −21042.0 −1.41401
\(606\) −7992.00 −0.535731
\(607\) −28584.0 −1.91135 −0.955674 0.294425i \(-0.904872\pi\)
−0.955674 + 0.294425i \(0.904872\pi\)
\(608\) −18432.0 −1.22947
\(609\) 0 0
\(610\) 57024.0 3.78497
\(611\) 2592.00 0.171622
\(612\) −9072.00 −0.599206
\(613\) −10910.0 −0.718843 −0.359421 0.933175i \(-0.617026\pi\)
−0.359421 + 0.933175i \(0.617026\pi\)
\(614\) 40896.0 2.68799
\(615\) 14580.0 0.955971
\(616\) 0 0
\(617\) −5522.00 −0.360304 −0.180152 0.983639i \(-0.557659\pi\)
−0.180152 + 0.983639i \(0.557659\pi\)
\(618\) −3024.00 −0.196834
\(619\) −2412.00 −0.156618 −0.0783089 0.996929i \(-0.524952\pi\)
−0.0783089 + 0.996929i \(0.524952\pi\)
\(620\) −5184.00 −0.335798
\(621\) −378.000 −0.0244261
\(622\) −15552.0 −1.00254
\(623\) 0 0
\(624\) 6912.00 0.443432
\(625\) −899.000 −0.0575360
\(626\) −20448.0 −1.30554
\(627\) 10800.0 0.687895
\(628\) −24192.0 −1.53721
\(629\) 20412.0 1.29393
\(630\) 0 0
\(631\) 24676.0 1.55679 0.778396 0.627773i \(-0.216034\pi\)
0.778396 + 0.627773i \(0.216034\pi\)
\(632\) 0 0
\(633\) −8748.00 −0.549292
\(634\) −40408.0 −2.53124
\(635\) −16488.0 −1.03040
\(636\) 528.000 0.0329191
\(637\) 0 0
\(638\) −31600.0 −1.96090
\(639\) −6606.00 −0.408966
\(640\) 0 0
\(641\) −27482.0 −1.69341 −0.846703 0.532065i \(-0.821416\pi\)
−0.846703 + 0.532065i \(0.821416\pi\)
\(642\) −8040.00 −0.494258
\(643\) 22752.0 1.39541 0.697707 0.716383i \(-0.254204\pi\)
0.697707 + 0.716383i \(0.254204\pi\)
\(644\) 0 0
\(645\) −17496.0 −1.06807
\(646\) −36288.0 −2.21011
\(647\) −14832.0 −0.901246 −0.450623 0.892714i \(-0.648798\pi\)
−0.450623 + 0.892714i \(0.648798\pi\)
\(648\) 0 0
\(649\) 23400.0 1.41530
\(650\) 28656.0 1.72920
\(651\) 0 0
\(652\) −14272.0 −0.857262
\(653\) 2822.00 0.169117 0.0845585 0.996419i \(-0.473052\pi\)
0.0845585 + 0.996419i \(0.473052\pi\)
\(654\) −1944.00 −0.116233
\(655\) 40824.0 2.43531
\(656\) −17280.0 −1.02846
\(657\) −1620.00 −0.0961982
\(658\) 0 0
\(659\) −15826.0 −0.935498 −0.467749 0.883861i \(-0.654935\pi\)
−0.467749 + 0.883861i \(0.654935\pi\)
\(660\) −21600.0 −1.27391
\(661\) 23832.0 1.40236 0.701178 0.712986i \(-0.252658\pi\)
0.701178 + 0.712986i \(0.252658\pi\)
\(662\) 22032.0 1.29350
\(663\) 13608.0 0.797121
\(664\) 0 0
\(665\) 0 0
\(666\) −5832.00 −0.339317
\(667\) 2212.00 0.128409
\(668\) 24192.0 1.40122
\(669\) −3240.00 −0.187243
\(670\) −16704.0 −0.963182
\(671\) 39600.0 2.27830
\(672\) 0 0
\(673\) 13770.0 0.788699 0.394350 0.918961i \(-0.370970\pi\)
0.394350 + 0.918961i \(0.370970\pi\)
\(674\) −36936.0 −2.11086
\(675\) −5373.00 −0.306381
\(676\) −7208.00 −0.410105
\(677\) −8334.00 −0.473119 −0.236560 0.971617i \(-0.576020\pi\)
−0.236560 + 0.971617i \(0.576020\pi\)
\(678\) 16680.0 0.944825
\(679\) 0 0
\(680\) 0 0
\(681\) −3996.00 −0.224856
\(682\) −7200.00 −0.404255
\(683\) −18598.0 −1.04192 −0.520961 0.853580i \(-0.674426\pi\)
−0.520961 + 0.853580i \(0.674426\pi\)
\(684\) 5184.00 0.289788
\(685\) −14508.0 −0.809229
\(686\) 0 0
\(687\) 4860.00 0.269899
\(688\) 20736.0 1.14906
\(689\) −792.000 −0.0437922
\(690\) 3024.00 0.166843
\(691\) −8964.00 −0.493497 −0.246749 0.969080i \(-0.579362\pi\)
−0.246749 + 0.969080i \(0.579362\pi\)
\(692\) −12528.0 −0.688213
\(693\) 0 0
\(694\) −25976.0 −1.42080
\(695\) 47304.0 2.58179
\(696\) 0 0
\(697\) −34020.0 −1.84878
\(698\) −40320.0 −2.18644
\(699\) −20154.0 −1.09055
\(700\) 0 0
\(701\) 3542.00 0.190841 0.0954205 0.995437i \(-0.469580\pi\)
0.0954205 + 0.995437i \(0.469580\pi\)
\(702\) −3888.00 −0.209036
\(703\) −11664.0 −0.625770
\(704\) 25600.0 1.37051
\(705\) 3888.00 0.207703
\(706\) 2952.00 0.157365
\(707\) 0 0
\(708\) 11232.0 0.596221
\(709\) −486.000 −0.0257435 −0.0128717 0.999917i \(-0.504097\pi\)
−0.0128717 + 0.999917i \(0.504097\pi\)
\(710\) 52848.0 2.79345
\(711\) 2124.00 0.112034
\(712\) 0 0
\(713\) 504.000 0.0264726
\(714\) 0 0
\(715\) 32400.0 1.69467
\(716\) 30416.0 1.58757
\(717\) 10734.0 0.559091
\(718\) 776.000 0.0403343
\(719\) 26928.0 1.39672 0.698362 0.715744i \(-0.253912\pi\)
0.698362 + 0.715744i \(0.253912\pi\)
\(720\) 10368.0 0.536656
\(721\) 0 0
\(722\) −6700.00 −0.345358
\(723\) −2268.00 −0.116664
\(724\) 3744.00 0.192189
\(725\) 31442.0 1.61066
\(726\) −14028.0 −0.717118
\(727\) −20628.0 −1.05234 −0.526169 0.850380i \(-0.676372\pi\)
−0.526169 + 0.850380i \(0.676372\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 12960.0 0.657084
\(731\) 40824.0 2.06557
\(732\) 19008.0 0.959776
\(733\) 9756.00 0.491604 0.245802 0.969320i \(-0.420949\pi\)
0.245802 + 0.969320i \(0.420949\pi\)
\(734\) 19008.0 0.955856
\(735\) 0 0
\(736\) −3584.00 −0.179495
\(737\) −11600.0 −0.579771
\(738\) 9720.00 0.484821
\(739\) 19064.0 0.948959 0.474479 0.880267i \(-0.342636\pi\)
0.474479 + 0.880267i \(0.342636\pi\)
\(740\) 23328.0 1.15886
\(741\) −7776.00 −0.385504
\(742\) 0 0
\(743\) −3766.00 −0.185950 −0.0929752 0.995668i \(-0.529638\pi\)
−0.0929752 + 0.995668i \(0.529638\pi\)
\(744\) 0 0
\(745\) 43020.0 2.11561
\(746\) −9224.00 −0.452701
\(747\) −324.000 −0.0158695
\(748\) 50400.0 2.46365
\(749\) 0 0
\(750\) 15984.0 0.778204
\(751\) −11664.0 −0.566745 −0.283372 0.959010i \(-0.591453\pi\)
−0.283372 + 0.959010i \(0.591453\pi\)
\(752\) −4608.00 −0.223453
\(753\) −19548.0 −0.946041
\(754\) 22752.0 1.09891
\(755\) −58320.0 −2.81123
\(756\) 0 0
\(757\) −34182.0 −1.64117 −0.820585 0.571524i \(-0.806352\pi\)
−0.820585 + 0.571524i \(0.806352\pi\)
\(758\) −29808.0 −1.42833
\(759\) 2100.00 0.100428
\(760\) 0 0
\(761\) 4734.00 0.225502 0.112751 0.993623i \(-0.464034\pi\)
0.112751 + 0.993623i \(0.464034\pi\)
\(762\) −10992.0 −0.522570
\(763\) 0 0
\(764\) 3856.00 0.182598
\(765\) 20412.0 0.964703
\(766\) 4608.00 0.217355
\(767\) −16848.0 −0.793150
\(768\) −12288.0 −0.577350
\(769\) 30240.0 1.41805 0.709026 0.705182i \(-0.249135\pi\)
0.709026 + 0.705182i \(0.249135\pi\)
\(770\) 0 0
\(771\) −18090.0 −0.845001
\(772\) −6480.00 −0.302099
\(773\) −27702.0 −1.28897 −0.644484 0.764618i \(-0.722928\pi\)
−0.644484 + 0.764618i \(0.722928\pi\)
\(774\) −11664.0 −0.541672
\(775\) 7164.00 0.332050
\(776\) 0 0
\(777\) 0 0
\(778\) 7576.00 0.349117
\(779\) 19440.0 0.894108
\(780\) 15552.0 0.713911
\(781\) 36700.0 1.68147
\(782\) −7056.00 −0.322662
\(783\) −4266.00 −0.194705
\(784\) 0 0
\(785\) 54432.0 2.47486
\(786\) 27216.0 1.23507
\(787\) 22644.0 1.02563 0.512815 0.858499i \(-0.328603\pi\)
0.512815 + 0.858499i \(0.328603\pi\)
\(788\) −19696.0 −0.890407
\(789\) −1770.00 −0.0798652
\(790\) −16992.0 −0.765251
\(791\) 0 0
\(792\) 0 0
\(793\) −28512.0 −1.27679
\(794\) −36864.0 −1.64768
\(795\) −1188.00 −0.0529988
\(796\) 36288.0 1.61582
\(797\) −30150.0 −1.33998 −0.669992 0.742368i \(-0.733703\pi\)
−0.669992 + 0.742368i \(0.733703\pi\)
\(798\) 0 0
\(799\) −9072.00 −0.401682
\(800\) −50944.0 −2.25143
\(801\) −2106.00 −0.0928987
\(802\) −46600.0 −2.05175
\(803\) 9000.00 0.395521
\(804\) −5568.00 −0.244239
\(805\) 0 0
\(806\) 5184.00 0.226549
\(807\) −2970.00 −0.129553
\(808\) 0 0
\(809\) −11318.0 −0.491866 −0.245933 0.969287i \(-0.579094\pi\)
−0.245933 + 0.969287i \(0.579094\pi\)
\(810\) −5832.00 −0.252982
\(811\) 29628.0 1.28284 0.641418 0.767192i \(-0.278347\pi\)
0.641418 + 0.767192i \(0.278347\pi\)
\(812\) 0 0
\(813\) −10260.0 −0.442600
\(814\) 32400.0 1.39511
\(815\) 32112.0 1.38016
\(816\) −24192.0 −1.03785
\(817\) −23328.0 −0.998952
\(818\) 30096.0 1.28641
\(819\) 0 0
\(820\) −38880.0 −1.65579
\(821\) 17770.0 0.755393 0.377696 0.925930i \(-0.376716\pi\)
0.377696 + 0.925930i \(0.376716\pi\)
\(822\) −9672.00 −0.410401
\(823\) 7868.00 0.333246 0.166623 0.986021i \(-0.446714\pi\)
0.166623 + 0.986021i \(0.446714\pi\)
\(824\) 0 0
\(825\) 29850.0 1.25969
\(826\) 0 0
\(827\) 35726.0 1.50219 0.751097 0.660192i \(-0.229525\pi\)
0.751097 + 0.660192i \(0.229525\pi\)
\(828\) 1008.00 0.0423073
\(829\) −27108.0 −1.13571 −0.567853 0.823130i \(-0.692226\pi\)
−0.567853 + 0.823130i \(0.692226\pi\)
\(830\) 2592.00 0.108397
\(831\) 8202.00 0.342388
\(832\) −18432.0 −0.768046
\(833\) 0 0
\(834\) 31536.0 1.30936
\(835\) −54432.0 −2.25592
\(836\) −28800.0 −1.19147
\(837\) −972.000 −0.0401401
\(838\) 15408.0 0.635156
\(839\) −23256.0 −0.956956 −0.478478 0.878099i \(-0.658811\pi\)
−0.478478 + 0.878099i \(0.658811\pi\)
\(840\) 0 0
\(841\) 575.000 0.0235762
\(842\) 41608.0 1.70298
\(843\) −1794.00 −0.0732961
\(844\) 23328.0 0.951402
\(845\) 16218.0 0.660256
\(846\) 2592.00 0.105337
\(847\) 0 0
\(848\) 1408.00 0.0570176
\(849\) 10800.0 0.436578
\(850\) −100296. −4.04721
\(851\) −2268.00 −0.0913584
\(852\) 17616.0 0.708350
\(853\) −35280.0 −1.41614 −0.708068 0.706144i \(-0.750433\pi\)
−0.708068 + 0.706144i \(0.750433\pi\)
\(854\) 0 0
\(855\) −11664.0 −0.466550
\(856\) 0 0
\(857\) 19710.0 0.785625 0.392813 0.919619i \(-0.371502\pi\)
0.392813 + 0.919619i \(0.371502\pi\)
\(858\) 21600.0 0.859454
\(859\) −3888.00 −0.154432 −0.0772159 0.997014i \(-0.524603\pi\)
−0.0772159 + 0.997014i \(0.524603\pi\)
\(860\) 46656.0 1.84995
\(861\) 0 0
\(862\) −41560.0 −1.64216
\(863\) −36634.0 −1.44500 −0.722500 0.691370i \(-0.757007\pi\)
−0.722500 + 0.691370i \(0.757007\pi\)
\(864\) 6912.00 0.272166
\(865\) 28188.0 1.10800
\(866\) −44928.0 −1.76295
\(867\) −32889.0 −1.28831
\(868\) 0 0
\(869\) −11800.0 −0.460630
\(870\) 34128.0 1.32994
\(871\) 8352.00 0.324910
\(872\) 0 0
\(873\) −4212.00 −0.163293
\(874\) 4032.00 0.156046
\(875\) 0 0
\(876\) 4320.00 0.166620
\(877\) 1226.00 0.0472053 0.0236027 0.999721i \(-0.492486\pi\)
0.0236027 + 0.999721i \(0.492486\pi\)
\(878\) −58464.0 −2.24723
\(879\) 23706.0 0.909651
\(880\) −57600.0 −2.20647
\(881\) 38538.0 1.47376 0.736878 0.676026i \(-0.236299\pi\)
0.736878 + 0.676026i \(0.236299\pi\)
\(882\) 0 0
\(883\) −37260.0 −1.42004 −0.710022 0.704180i \(-0.751315\pi\)
−0.710022 + 0.704180i \(0.751315\pi\)
\(884\) −36288.0 −1.38065
\(885\) −25272.0 −0.959897
\(886\) 47752.0 1.81068
\(887\) −26640.0 −1.00844 −0.504219 0.863576i \(-0.668219\pi\)
−0.504219 + 0.863576i \(0.668219\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 16848.0 0.634546
\(891\) −4050.00 −0.152278
\(892\) 8640.00 0.324315
\(893\) 5184.00 0.194262
\(894\) 28680.0 1.07293
\(895\) −68436.0 −2.55594
\(896\) 0 0
\(897\) −1512.00 −0.0562812
\(898\) 32744.0 1.21679
\(899\) 5688.00 0.211018
\(900\) 14328.0 0.530667
\(901\) 2772.00 0.102496
\(902\) −54000.0 −1.99335
\(903\) 0 0
\(904\) 0 0
\(905\) −8424.00 −0.309418
\(906\) −38880.0 −1.42572
\(907\) −12636.0 −0.462593 −0.231296 0.972883i \(-0.574297\pi\)
−0.231296 + 0.972883i \(0.574297\pi\)
\(908\) 10656.0 0.389462
\(909\) 5994.00 0.218711
\(910\) 0 0
\(911\) −33638.0 −1.22336 −0.611678 0.791107i \(-0.709505\pi\)
−0.611678 + 0.791107i \(0.709505\pi\)
\(912\) 13824.0 0.501928
\(913\) 1800.00 0.0652479
\(914\) 8424.00 0.304859
\(915\) −42768.0 −1.54521
\(916\) −12960.0 −0.467479
\(917\) 0 0
\(918\) 13608.0 0.489249
\(919\) 36936.0 1.32580 0.662898 0.748710i \(-0.269326\pi\)
0.662898 + 0.748710i \(0.269326\pi\)
\(920\) 0 0
\(921\) −30672.0 −1.09737
\(922\) −37944.0 −1.35534
\(923\) −26424.0 −0.942315
\(924\) 0 0
\(925\) −32238.0 −1.14592
\(926\) −50608.0 −1.79598
\(927\) 2268.00 0.0803570
\(928\) −40448.0 −1.43079
\(929\) 22302.0 0.787626 0.393813 0.919191i \(-0.371156\pi\)
0.393813 + 0.919191i \(0.371156\pi\)
\(930\) 7776.00 0.274178
\(931\) 0 0
\(932\) 53744.0 1.88889
\(933\) 11664.0 0.409284
\(934\) 14832.0 0.519612
\(935\) −113400. −3.96639
\(936\) 0 0
\(937\) −13824.0 −0.481975 −0.240987 0.970528i \(-0.577471\pi\)
−0.240987 + 0.970528i \(0.577471\pi\)
\(938\) 0 0
\(939\) 15336.0 0.532983
\(940\) −10368.0 −0.359752
\(941\) −13554.0 −0.469551 −0.234776 0.972050i \(-0.575436\pi\)
−0.234776 + 0.972050i \(0.575436\pi\)
\(942\) 36288.0 1.25512
\(943\) 3780.00 0.130534
\(944\) 29952.0 1.03268
\(945\) 0 0
\(946\) 64800.0 2.22709
\(947\) 44878.0 1.53996 0.769978 0.638070i \(-0.220267\pi\)
0.769978 + 0.638070i \(0.220267\pi\)
\(948\) −5664.00 −0.194049
\(949\) −6480.00 −0.221654
\(950\) 57312.0 1.95731
\(951\) 30306.0 1.03337
\(952\) 0 0
\(953\) 38362.0 1.30395 0.651976 0.758239i \(-0.273940\pi\)
0.651976 + 0.758239i \(0.273940\pi\)
\(954\) −792.000 −0.0268784
\(955\) −8676.00 −0.293978
\(956\) −28624.0 −0.968375
\(957\) 23700.0 0.800535
\(958\) 32256.0 1.08783
\(959\) 0 0
\(960\) −27648.0 −0.929516
\(961\) −28495.0 −0.956497
\(962\) −23328.0 −0.781835
\(963\) 6030.00 0.201780
\(964\) 6048.00 0.202067
\(965\) 14580.0 0.486370
\(966\) 0 0
\(967\) 26444.0 0.879402 0.439701 0.898144i \(-0.355084\pi\)
0.439701 + 0.898144i \(0.355084\pi\)
\(968\) 0 0
\(969\) 27216.0 0.902274
\(970\) 33696.0 1.11537
\(971\) −17820.0 −0.588951 −0.294475 0.955659i \(-0.595145\pi\)
−0.294475 + 0.955659i \(0.595145\pi\)
\(972\) −1944.00 −0.0641500
\(973\) 0 0
\(974\) −46656.0 −1.53486
\(975\) −21492.0 −0.705943
\(976\) 50688.0 1.66238
\(977\) 34438.0 1.12771 0.563853 0.825875i \(-0.309318\pi\)
0.563853 + 0.825875i \(0.309318\pi\)
\(978\) 21408.0 0.699951
\(979\) 11700.0 0.381955
\(980\) 0 0
\(981\) 1458.00 0.0474519
\(982\) −39256.0 −1.27567
\(983\) −26064.0 −0.845689 −0.422845 0.906202i \(-0.638968\pi\)
−0.422845 + 0.906202i \(0.638968\pi\)
\(984\) 0 0
\(985\) 44316.0 1.43353
\(986\) −79632.0 −2.57201
\(987\) 0 0
\(988\) 20736.0 0.667713
\(989\) −4536.00 −0.145841
\(990\) 32400.0 1.04014
\(991\) 33696.0 1.08011 0.540055 0.841630i \(-0.318403\pi\)
0.540055 + 0.841630i \(0.318403\pi\)
\(992\) −9216.00 −0.294968
\(993\) −16524.0 −0.528070
\(994\) 0 0
\(995\) −81648.0 −2.60142
\(996\) 864.000 0.0274868
\(997\) −36072.0 −1.14585 −0.572925 0.819608i \(-0.694191\pi\)
−0.572925 + 0.819608i \(0.694191\pi\)
\(998\) −60912.0 −1.93200
\(999\) 4374.00 0.138526
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 147.4.a.f.1.1 1
3.2 odd 2 441.4.a.c.1.1 1
4.3 odd 2 2352.4.a.t.1.1 1
7.2 even 3 147.4.e.d.67.1 2
7.3 odd 6 147.4.e.a.79.1 2
7.4 even 3 147.4.e.d.79.1 2
7.5 odd 6 147.4.e.a.67.1 2
7.6 odd 2 147.4.a.h.1.1 yes 1
21.2 odd 6 441.4.e.l.361.1 2
21.5 even 6 441.4.e.o.361.1 2
21.11 odd 6 441.4.e.l.226.1 2
21.17 even 6 441.4.e.o.226.1 2
21.20 even 2 441.4.a.a.1.1 1
28.27 even 2 2352.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.f.1.1 1 1.1 even 1 trivial
147.4.a.h.1.1 yes 1 7.6 odd 2
147.4.e.a.67.1 2 7.5 odd 6
147.4.e.a.79.1 2 7.3 odd 6
147.4.e.d.67.1 2 7.2 even 3
147.4.e.d.79.1 2 7.4 even 3
441.4.a.a.1.1 1 21.20 even 2
441.4.a.c.1.1 1 3.2 odd 2
441.4.e.l.226.1 2 21.11 odd 6
441.4.e.l.361.1 2 21.2 odd 6
441.4.e.o.226.1 2 21.17 even 6
441.4.e.o.361.1 2 21.5 even 6
2352.4.a.s.1.1 1 28.27 even 2
2352.4.a.t.1.1 1 4.3 odd 2