Properties

Label 588.3.d.c.97.2
Level $588$
Weight $3$
Character 588.97
Analytic conductor $16.022$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.2
Root \(-1.60021 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 588.97
Dual form 588.3.d.c.97.7

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} -0.480662i q^{5} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} -0.480662i q^{5} -3.00000 q^{9} +5.76316 q^{11} +1.41991i q^{13} -0.832530 q^{15} -23.0671i q^{17} -10.6739i q^{19} -6.59980 q^{23} +24.7690 q^{25} +5.19615i q^{27} -6.20258 q^{29} -41.9320i q^{31} -9.98209i q^{33} -60.0929 q^{37} +2.45936 q^{39} -48.8250i q^{41} -51.5603 q^{43} +1.44198i q^{45} +19.2421i q^{47} -39.9534 q^{51} +82.1345 q^{53} -2.77013i q^{55} -18.4877 q^{57} -92.5800i q^{59} -4.99187i q^{61} +0.682497 q^{65} -2.20699 q^{67} +11.4312i q^{69} -80.5899 q^{71} +13.9088i q^{73} -42.9011i q^{75} -64.8885 q^{79} +9.00000 q^{81} -118.005i q^{83} -11.0875 q^{85} +10.7432i q^{87} +104.265i q^{89} -72.6284 q^{93} -5.13052 q^{95} +31.7875i q^{97} -17.2895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} - 16q^{23} + 72q^{25} + 80q^{29} + 128q^{37} - 112q^{43} - 96q^{51} - 144q^{53} - 192q^{57} + 240q^{65} - 64q^{67} + 224q^{71} - 432q^{79} + 72q^{81} - 96q^{85} - 96q^{93} - 272q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) − 0.480662i − 0.0961323i −0.998844 0.0480662i \(-0.984694\pi\)
0.998844 0.0480662i \(-0.0153058\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 5.76316 0.523924 0.261962 0.965078i \(-0.415631\pi\)
0.261962 + 0.965078i \(0.415631\pi\)
\(12\) 0 0
\(13\) 1.41991i 0.109224i 0.998508 + 0.0546120i \(0.0173922\pi\)
−0.998508 + 0.0546120i \(0.982608\pi\)
\(14\) 0 0
\(15\) −0.832530 −0.0555020
\(16\) 0 0
\(17\) − 23.0671i − 1.35689i −0.734651 0.678445i \(-0.762654\pi\)
0.734651 0.678445i \(-0.237346\pi\)
\(18\) 0 0
\(19\) − 10.6739i − 0.561783i −0.959740 0.280891i \(-0.909370\pi\)
0.959740 0.280891i \(-0.0906300\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.59980 −0.286948 −0.143474 0.989654i \(-0.545827\pi\)
−0.143474 + 0.989654i \(0.545827\pi\)
\(24\) 0 0
\(25\) 24.7690 0.990759
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) −6.20258 −0.213882 −0.106941 0.994265i \(-0.534106\pi\)
−0.106941 + 0.994265i \(0.534106\pi\)
\(30\) 0 0
\(31\) − 41.9320i − 1.35265i −0.736605 0.676323i \(-0.763572\pi\)
0.736605 0.676323i \(-0.236428\pi\)
\(32\) 0 0
\(33\) − 9.98209i − 0.302488i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −60.0929 −1.62413 −0.812066 0.583566i \(-0.801657\pi\)
−0.812066 + 0.583566i \(0.801657\pi\)
\(38\) 0 0
\(39\) 2.45936 0.0630605
\(40\) 0 0
\(41\) − 48.8250i − 1.19085i −0.803409 0.595427i \(-0.796983\pi\)
0.803409 0.595427i \(-0.203017\pi\)
\(42\) 0 0
\(43\) −51.5603 −1.19908 −0.599539 0.800346i \(-0.704649\pi\)
−0.599539 + 0.800346i \(0.704649\pi\)
\(44\) 0 0
\(45\) 1.44198i 0.0320441i
\(46\) 0 0
\(47\) 19.2421i 0.409406i 0.978824 + 0.204703i \(0.0656229\pi\)
−0.978824 + 0.204703i \(0.934377\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −39.9534 −0.783401
\(52\) 0 0
\(53\) 82.1345 1.54971 0.774854 0.632141i \(-0.217824\pi\)
0.774854 + 0.632141i \(0.217824\pi\)
\(54\) 0 0
\(55\) − 2.77013i − 0.0503660i
\(56\) 0 0
\(57\) −18.4877 −0.324345
\(58\) 0 0
\(59\) − 92.5800i − 1.56915i −0.620032 0.784576i \(-0.712881\pi\)
0.620032 0.784576i \(-0.287119\pi\)
\(60\) 0 0
\(61\) − 4.99187i − 0.0818340i −0.999163 0.0409170i \(-0.986972\pi\)
0.999163 0.0409170i \(-0.0130279\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.682497 0.0105000
\(66\) 0 0
\(67\) −2.20699 −0.0329402 −0.0164701 0.999864i \(-0.505243\pi\)
−0.0164701 + 0.999864i \(0.505243\pi\)
\(68\) 0 0
\(69\) 11.4312i 0.165669i
\(70\) 0 0
\(71\) −80.5899 −1.13507 −0.567535 0.823349i \(-0.692103\pi\)
−0.567535 + 0.823349i \(0.692103\pi\)
\(72\) 0 0
\(73\) 13.9088i 0.190531i 0.995452 + 0.0952657i \(0.0303701\pi\)
−0.995452 + 0.0952657i \(0.969630\pi\)
\(74\) 0 0
\(75\) − 42.9011i − 0.572015i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −64.8885 −0.821373 −0.410687 0.911777i \(-0.634711\pi\)
−0.410687 + 0.911777i \(0.634711\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) − 118.005i − 1.42174i −0.703322 0.710872i \(-0.748301\pi\)
0.703322 0.710872i \(-0.251699\pi\)
\(84\) 0 0
\(85\) −11.0875 −0.130441
\(86\) 0 0
\(87\) 10.7432i 0.123485i
\(88\) 0 0
\(89\) 104.265i 1.17151i 0.810487 + 0.585757i \(0.199203\pi\)
−0.810487 + 0.585757i \(0.800797\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −72.6284 −0.780951
\(94\) 0 0
\(95\) −5.13052 −0.0540055
\(96\) 0 0
\(97\) 31.7875i 0.327706i 0.986485 + 0.163853i \(0.0523923\pi\)
−0.986485 + 0.163853i \(0.947608\pi\)
\(98\) 0 0
\(99\) −17.2895 −0.174641
\(100\) 0 0
\(101\) − 72.5037i − 0.717859i −0.933365 0.358929i \(-0.883142\pi\)
0.933365 0.358929i \(-0.116858\pi\)
\(102\) 0 0
\(103\) − 106.963i − 1.03847i −0.854631 0.519236i \(-0.826217\pi\)
0.854631 0.519236i \(-0.173783\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 196.619 1.83756 0.918782 0.394765i \(-0.129174\pi\)
0.918782 + 0.394765i \(0.129174\pi\)
\(108\) 0 0
\(109\) 42.6922 0.391672 0.195836 0.980637i \(-0.437258\pi\)
0.195836 + 0.980637i \(0.437258\pi\)
\(110\) 0 0
\(111\) 104.084i 0.937693i
\(112\) 0 0
\(113\) 175.501 1.55310 0.776552 0.630053i \(-0.216967\pi\)
0.776552 + 0.630053i \(0.216967\pi\)
\(114\) 0 0
\(115\) 3.17227i 0.0275849i
\(116\) 0 0
\(117\) − 4.25973i − 0.0364080i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −87.7860 −0.725504
\(122\) 0 0
\(123\) −84.5674 −0.687540
\(124\) 0 0
\(125\) − 23.9220i − 0.191376i
\(126\) 0 0
\(127\) 31.0434 0.244436 0.122218 0.992503i \(-0.460999\pi\)
0.122218 + 0.992503i \(0.460999\pi\)
\(128\) 0 0
\(129\) 89.3051i 0.692288i
\(130\) 0 0
\(131\) 46.5095i 0.355034i 0.984118 + 0.177517i \(0.0568065\pi\)
−0.984118 + 0.177517i \(0.943194\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.49759 0.0185007
\(136\) 0 0
\(137\) 45.4654 0.331864 0.165932 0.986137i \(-0.446937\pi\)
0.165932 + 0.986137i \(0.446937\pi\)
\(138\) 0 0
\(139\) 138.075i 0.993343i 0.867939 + 0.496672i \(0.165445\pi\)
−0.867939 + 0.496672i \(0.834555\pi\)
\(140\) 0 0
\(141\) 33.3283 0.236371
\(142\) 0 0
\(143\) 8.18318i 0.0572250i
\(144\) 0 0
\(145\) 2.98134i 0.0205610i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 239.228 1.60556 0.802780 0.596276i \(-0.203354\pi\)
0.802780 + 0.596276i \(0.203354\pi\)
\(150\) 0 0
\(151\) −188.146 −1.24600 −0.623001 0.782221i \(-0.714087\pi\)
−0.623001 + 0.782221i \(0.714087\pi\)
\(152\) 0 0
\(153\) 69.2014i 0.452297i
\(154\) 0 0
\(155\) −20.1551 −0.130033
\(156\) 0 0
\(157\) 215.239i 1.37095i 0.728097 + 0.685474i \(0.240405\pi\)
−0.728097 + 0.685474i \(0.759595\pi\)
\(158\) 0 0
\(159\) − 142.261i − 0.894724i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 149.811 0.919083 0.459542 0.888156i \(-0.348014\pi\)
0.459542 + 0.888156i \(0.348014\pi\)
\(164\) 0 0
\(165\) −4.79801 −0.0290788
\(166\) 0 0
\(167\) − 137.195i − 0.821528i −0.911742 0.410764i \(-0.865262\pi\)
0.911742 0.410764i \(-0.134738\pi\)
\(168\) 0 0
\(169\) 166.984 0.988070
\(170\) 0 0
\(171\) 32.0216i 0.187261i
\(172\) 0 0
\(173\) 251.181i 1.45191i 0.687740 + 0.725957i \(0.258603\pi\)
−0.687740 + 0.725957i \(0.741397\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −160.353 −0.905951
\(178\) 0 0
\(179\) −116.793 −0.652477 −0.326239 0.945287i \(-0.605781\pi\)
−0.326239 + 0.945287i \(0.605781\pi\)
\(180\) 0 0
\(181\) 117.148i 0.647228i 0.946189 + 0.323614i \(0.104898\pi\)
−0.946189 + 0.323614i \(0.895102\pi\)
\(182\) 0 0
\(183\) −8.64618 −0.0472469
\(184\) 0 0
\(185\) 28.8843i 0.156131i
\(186\) 0 0
\(187\) − 132.940i − 0.710907i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 341.818 1.78962 0.894812 0.446443i \(-0.147309\pi\)
0.894812 + 0.446443i \(0.147309\pi\)
\(192\) 0 0
\(193\) −224.551 −1.16348 −0.581738 0.813376i \(-0.697627\pi\)
−0.581738 + 0.813376i \(0.697627\pi\)
\(194\) 0 0
\(195\) − 1.18212i − 0.00606215i
\(196\) 0 0
\(197\) 257.109 1.30512 0.652560 0.757737i \(-0.273695\pi\)
0.652560 + 0.757737i \(0.273695\pi\)
\(198\) 0 0
\(199\) 245.479i 1.23356i 0.787134 + 0.616782i \(0.211564\pi\)
−0.787134 + 0.616782i \(0.788436\pi\)
\(200\) 0 0
\(201\) 3.82262i 0.0190180i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −23.4683 −0.114480
\(206\) 0 0
\(207\) 19.7994 0.0956492
\(208\) 0 0
\(209\) − 61.5152i − 0.294331i
\(210\) 0 0
\(211\) −95.8210 −0.454128 −0.227064 0.973880i \(-0.572913\pi\)
−0.227064 + 0.973880i \(0.572913\pi\)
\(212\) 0 0
\(213\) 139.586i 0.655333i
\(214\) 0 0
\(215\) 24.7831i 0.115270i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 24.0907 0.110003
\(220\) 0 0
\(221\) 32.7533 0.148205
\(222\) 0 0
\(223\) 94.2091i 0.422462i 0.977436 + 0.211231i \(0.0677473\pi\)
−0.977436 + 0.211231i \(0.932253\pi\)
\(224\) 0 0
\(225\) −74.3069 −0.330253
\(226\) 0 0
\(227\) 224.253i 0.987899i 0.869491 + 0.493949i \(0.164447\pi\)
−0.869491 + 0.493949i \(0.835553\pi\)
\(228\) 0 0
\(229\) 366.724i 1.60142i 0.599055 + 0.800708i \(0.295543\pi\)
−0.599055 + 0.800708i \(0.704457\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 69.6741 0.299030 0.149515 0.988759i \(-0.452229\pi\)
0.149515 + 0.988759i \(0.452229\pi\)
\(234\) 0 0
\(235\) 9.24893 0.0393572
\(236\) 0 0
\(237\) 112.390i 0.474220i
\(238\) 0 0
\(239\) −214.544 −0.897674 −0.448837 0.893614i \(-0.648162\pi\)
−0.448837 + 0.893614i \(0.648162\pi\)
\(240\) 0 0
\(241\) 163.395i 0.677989i 0.940788 + 0.338994i \(0.110087\pi\)
−0.940788 + 0.338994i \(0.889913\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.1560 0.0613601
\(248\) 0 0
\(249\) −204.390 −0.820844
\(250\) 0 0
\(251\) − 330.546i − 1.31692i −0.752617 0.658458i \(-0.771209\pi\)
0.752617 0.658458i \(-0.228791\pi\)
\(252\) 0 0
\(253\) −38.0357 −0.150339
\(254\) 0 0
\(255\) 19.2041i 0.0753101i
\(256\) 0 0
\(257\) 160.529i 0.624626i 0.949979 + 0.312313i \(0.101104\pi\)
−0.949979 + 0.312313i \(0.898896\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 18.6077 0.0712940
\(262\) 0 0
\(263\) −106.515 −0.405002 −0.202501 0.979282i \(-0.564907\pi\)
−0.202501 + 0.979282i \(0.564907\pi\)
\(264\) 0 0
\(265\) − 39.4789i − 0.148977i
\(266\) 0 0
\(267\) 180.592 0.676374
\(268\) 0 0
\(269\) − 155.454i − 0.577895i −0.957345 0.288947i \(-0.906695\pi\)
0.957345 0.288947i \(-0.0933052\pi\)
\(270\) 0 0
\(271\) − 516.080i − 1.90435i −0.305549 0.952176i \(-0.598840\pi\)
0.305549 0.952176i \(-0.401160\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 142.748 0.519082
\(276\) 0 0
\(277\) 200.202 0.722750 0.361375 0.932421i \(-0.382308\pi\)
0.361375 + 0.932421i \(0.382308\pi\)
\(278\) 0 0
\(279\) 125.796i 0.450882i
\(280\) 0 0
\(281\) 228.093 0.811720 0.405860 0.913935i \(-0.366972\pi\)
0.405860 + 0.913935i \(0.366972\pi\)
\(282\) 0 0
\(283\) − 260.710i − 0.921236i −0.887598 0.460618i \(-0.847628\pi\)
0.887598 0.460618i \(-0.152372\pi\)
\(284\) 0 0
\(285\) 8.88632i 0.0311801i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −243.092 −0.841150
\(290\) 0 0
\(291\) 55.0576 0.189201
\(292\) 0 0
\(293\) 349.885i 1.19415i 0.802186 + 0.597074i \(0.203670\pi\)
−0.802186 + 0.597074i \(0.796330\pi\)
\(294\) 0 0
\(295\) −44.4996 −0.150846
\(296\) 0 0
\(297\) 29.9463i 0.100829i
\(298\) 0 0
\(299\) − 9.37113i − 0.0313416i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −125.580 −0.414456
\(304\) 0 0
\(305\) −2.39940 −0.00786689
\(306\) 0 0
\(307\) − 146.898i − 0.478495i −0.970959 0.239247i \(-0.923099\pi\)
0.970959 0.239247i \(-0.0769007\pi\)
\(308\) 0 0
\(309\) −185.265 −0.599562
\(310\) 0 0
\(311\) − 80.7340i − 0.259595i −0.991541 0.129797i \(-0.958567\pi\)
0.991541 0.129797i \(-0.0414327\pi\)
\(312\) 0 0
\(313\) 154.339i 0.493096i 0.969131 + 0.246548i \(0.0792963\pi\)
−0.969131 + 0.246548i \(0.920704\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −188.545 −0.594780 −0.297390 0.954756i \(-0.596116\pi\)
−0.297390 + 0.954756i \(0.596116\pi\)
\(318\) 0 0
\(319\) −35.7465 −0.112058
\(320\) 0 0
\(321\) − 340.555i − 1.06092i
\(322\) 0 0
\(323\) −246.216 −0.762277
\(324\) 0 0
\(325\) 35.1697i 0.108215i
\(326\) 0 0
\(327\) − 73.9451i − 0.226132i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 146.787 0.443466 0.221733 0.975107i \(-0.428829\pi\)
0.221733 + 0.975107i \(0.428829\pi\)
\(332\) 0 0
\(333\) 180.279 0.541377
\(334\) 0 0
\(335\) 1.06082i 0.00316662i
\(336\) 0 0
\(337\) 101.231 0.300388 0.150194 0.988657i \(-0.452010\pi\)
0.150194 + 0.988657i \(0.452010\pi\)
\(338\) 0 0
\(339\) − 303.976i − 0.896685i
\(340\) 0 0
\(341\) − 241.661i − 0.708684i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5.49453 0.0159262
\(346\) 0 0
\(347\) 351.689 1.01351 0.506756 0.862089i \(-0.330844\pi\)
0.506756 + 0.862089i \(0.330844\pi\)
\(348\) 0 0
\(349\) 88.3780i 0.253232i 0.991952 + 0.126616i \(0.0404116\pi\)
−0.991952 + 0.126616i \(0.959588\pi\)
\(350\) 0 0
\(351\) −7.37808 −0.0210202
\(352\) 0 0
\(353\) − 520.542i − 1.47462i −0.675552 0.737312i \(-0.736095\pi\)
0.675552 0.737312i \(-0.263905\pi\)
\(354\) 0 0
\(355\) 38.7365i 0.109117i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −514.265 −1.43249 −0.716246 0.697848i \(-0.754141\pi\)
−0.716246 + 0.697848i \(0.754141\pi\)
\(360\) 0 0
\(361\) 247.069 0.684400
\(362\) 0 0
\(363\) 152.050i 0.418870i
\(364\) 0 0
\(365\) 6.68542 0.0183162
\(366\) 0 0
\(367\) − 417.686i − 1.13811i −0.822300 0.569054i \(-0.807310\pi\)
0.822300 0.569054i \(-0.192690\pi\)
\(368\) 0 0
\(369\) 146.475i 0.396951i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −140.314 −0.376176 −0.188088 0.982152i \(-0.560229\pi\)
−0.188088 + 0.982152i \(0.560229\pi\)
\(374\) 0 0
\(375\) −41.4342 −0.110491
\(376\) 0 0
\(377\) − 8.80712i − 0.0233611i
\(378\) 0 0
\(379\) 153.298 0.404480 0.202240 0.979336i \(-0.435178\pi\)
0.202240 + 0.979336i \(0.435178\pi\)
\(380\) 0 0
\(381\) − 53.7687i − 0.141125i
\(382\) 0 0
\(383\) − 121.717i − 0.317799i −0.987295 0.158900i \(-0.949205\pi\)
0.987295 0.158900i \(-0.0507946\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 154.681 0.399692
\(388\) 0 0
\(389\) 401.271 1.03155 0.515773 0.856725i \(-0.327505\pi\)
0.515773 + 0.856725i \(0.327505\pi\)
\(390\) 0 0
\(391\) 152.238i 0.389356i
\(392\) 0 0
\(393\) 80.5568 0.204979
\(394\) 0 0
\(395\) 31.1894i 0.0789605i
\(396\) 0 0
\(397\) − 60.2251i − 0.151700i −0.997119 0.0758502i \(-0.975833\pi\)
0.997119 0.0758502i \(-0.0241671\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −374.308 −0.933436 −0.466718 0.884406i \(-0.654564\pi\)
−0.466718 + 0.884406i \(0.654564\pi\)
\(402\) 0 0
\(403\) 59.5398 0.147741
\(404\) 0 0
\(405\) − 4.32595i − 0.0106814i
\(406\) 0 0
\(407\) −346.325 −0.850921
\(408\) 0 0
\(409\) − 614.609i − 1.50271i −0.659897 0.751356i \(-0.729400\pi\)
0.659897 0.751356i \(-0.270600\pi\)
\(410\) 0 0
\(411\) − 78.7484i − 0.191602i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −56.7203 −0.136675
\(416\) 0 0
\(417\) 239.152 0.573507
\(418\) 0 0
\(419\) − 129.067i − 0.308035i −0.988068 0.154017i \(-0.950779\pi\)
0.988068 0.154017i \(-0.0492212\pi\)
\(420\) 0 0
\(421\) −697.880 −1.65767 −0.828836 0.559492i \(-0.810996\pi\)
−0.828836 + 0.559492i \(0.810996\pi\)
\(422\) 0 0
\(423\) − 57.7263i − 0.136469i
\(424\) 0 0
\(425\) − 571.349i − 1.34435i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 14.1737 0.0330389
\(430\) 0 0
\(431\) −277.092 −0.642906 −0.321453 0.946926i \(-0.604171\pi\)
−0.321453 + 0.946926i \(0.604171\pi\)
\(432\) 0 0
\(433\) − 822.794i − 1.90022i −0.311919 0.950109i \(-0.600972\pi\)
0.311919 0.950109i \(-0.399028\pi\)
\(434\) 0 0
\(435\) 5.16384 0.0118709
\(436\) 0 0
\(437\) 70.4454i 0.161202i
\(438\) 0 0
\(439\) 365.827i 0.833320i 0.909062 + 0.416660i \(0.136799\pi\)
−0.909062 + 0.416660i \(0.863201\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −573.392 −1.29434 −0.647169 0.762346i \(-0.724047\pi\)
−0.647169 + 0.762346i \(0.724047\pi\)
\(444\) 0 0
\(445\) 50.1161 0.112620
\(446\) 0 0
\(447\) − 414.356i − 0.926970i
\(448\) 0 0
\(449\) 227.961 0.507708 0.253854 0.967243i \(-0.418302\pi\)
0.253854 + 0.967243i \(0.418302\pi\)
\(450\) 0 0
\(451\) − 281.386i − 0.623917i
\(452\) 0 0
\(453\) 325.879i 0.719380i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −343.124 −0.750818 −0.375409 0.926859i \(-0.622498\pi\)
−0.375409 + 0.926859i \(0.622498\pi\)
\(458\) 0 0
\(459\) 119.860 0.261134
\(460\) 0 0
\(461\) 611.314i 1.32606i 0.748593 + 0.663030i \(0.230730\pi\)
−0.748593 + 0.663030i \(0.769270\pi\)
\(462\) 0 0
\(463\) −67.2682 −0.145288 −0.0726439 0.997358i \(-0.523144\pi\)
−0.0726439 + 0.997358i \(0.523144\pi\)
\(464\) 0 0
\(465\) 34.9097i 0.0750746i
\(466\) 0 0
\(467\) 93.7498i 0.200749i 0.994950 + 0.100375i \(0.0320041\pi\)
−0.994950 + 0.100375i \(0.967996\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 372.804 0.791517
\(472\) 0 0
\(473\) −297.150 −0.628225
\(474\) 0 0
\(475\) − 264.381i − 0.556591i
\(476\) 0 0
\(477\) −246.403 −0.516569
\(478\) 0 0
\(479\) 353.796i 0.738613i 0.929308 + 0.369306i \(0.120405\pi\)
−0.929308 + 0.369306i \(0.879595\pi\)
\(480\) 0 0
\(481\) − 85.3265i − 0.177394i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 15.2790 0.0315032
\(486\) 0 0
\(487\) 952.677 1.95621 0.978107 0.208101i \(-0.0667281\pi\)
0.978107 + 0.208101i \(0.0667281\pi\)
\(488\) 0 0
\(489\) − 259.479i − 0.530633i
\(490\) 0 0
\(491\) −818.649 −1.66731 −0.833655 0.552286i \(-0.813756\pi\)
−0.833655 + 0.552286i \(0.813756\pi\)
\(492\) 0 0
\(493\) 143.076i 0.290214i
\(494\) 0 0
\(495\) 8.31039i 0.0167887i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −24.0821 −0.0482607 −0.0241304 0.999709i \(-0.507682\pi\)
−0.0241304 + 0.999709i \(0.507682\pi\)
\(500\) 0 0
\(501\) −237.629 −0.474310
\(502\) 0 0
\(503\) 420.445i 0.835874i 0.908476 + 0.417937i \(0.137247\pi\)
−0.908476 + 0.417937i \(0.862753\pi\)
\(504\) 0 0
\(505\) −34.8498 −0.0690094
\(506\) 0 0
\(507\) − 289.225i − 0.570463i
\(508\) 0 0
\(509\) − 239.869i − 0.471255i −0.971843 0.235627i \(-0.924285\pi\)
0.971843 0.235627i \(-0.0757145\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 55.4631 0.108115
\(514\) 0 0
\(515\) −51.4128 −0.0998307
\(516\) 0 0
\(517\) 110.895i 0.214498i
\(518\) 0 0
\(519\) 435.059 0.838263
\(520\) 0 0
\(521\) 26.8565i 0.0515479i 0.999668 + 0.0257739i \(0.00820501\pi\)
−0.999668 + 0.0257739i \(0.991795\pi\)
\(522\) 0 0
\(523\) 223.737i 0.427796i 0.976856 + 0.213898i \(0.0686160\pi\)
−0.976856 + 0.213898i \(0.931384\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −967.252 −1.83539
\(528\) 0 0
\(529\) −485.443 −0.917661
\(530\) 0 0
\(531\) 277.740i 0.523051i
\(532\) 0 0
\(533\) 69.3272 0.130070
\(534\) 0 0
\(535\) − 94.5074i − 0.176649i
\(536\) 0 0
\(537\) 202.292i 0.376708i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 986.190 1.82290 0.911451 0.411410i \(-0.134963\pi\)
0.911451 + 0.411410i \(0.134963\pi\)
\(542\) 0 0
\(543\) 202.907 0.373678
\(544\) 0 0
\(545\) − 20.5205i − 0.0376523i
\(546\) 0 0
\(547\) 735.369 1.34437 0.672183 0.740385i \(-0.265357\pi\)
0.672183 + 0.740385i \(0.265357\pi\)
\(548\) 0 0
\(549\) 14.9756i 0.0272780i
\(550\) 0 0
\(551\) 66.2056i 0.120155i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 50.0291 0.0901426
\(556\) 0 0
\(557\) 418.568 0.751469 0.375735 0.926727i \(-0.377390\pi\)
0.375735 + 0.926727i \(0.377390\pi\)
\(558\) 0 0
\(559\) − 73.2111i − 0.130968i
\(560\) 0 0
\(561\) −230.258 −0.410442
\(562\) 0 0
\(563\) 746.738i 1.32635i 0.748462 + 0.663177i \(0.230792\pi\)
−0.748462 + 0.663177i \(0.769208\pi\)
\(564\) 0 0
\(565\) − 84.3565i − 0.149304i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −288.710 −0.507398 −0.253699 0.967283i \(-0.581647\pi\)
−0.253699 + 0.967283i \(0.581647\pi\)
\(570\) 0 0
\(571\) −916.207 −1.60457 −0.802283 0.596944i \(-0.796382\pi\)
−0.802283 + 0.596944i \(0.796382\pi\)
\(572\) 0 0
\(573\) − 592.047i − 1.03324i
\(574\) 0 0
\(575\) −163.470 −0.284296
\(576\) 0 0
\(577\) 992.068i 1.71936i 0.510836 + 0.859678i \(0.329336\pi\)
−0.510836 + 0.859678i \(0.670664\pi\)
\(578\) 0 0
\(579\) 388.934i 0.671733i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 473.354 0.811929
\(584\) 0 0
\(585\) −2.04749 −0.00349998
\(586\) 0 0
\(587\) 154.965i 0.263996i 0.991250 + 0.131998i \(0.0421392\pi\)
−0.991250 + 0.131998i \(0.957861\pi\)
\(588\) 0 0
\(589\) −447.577 −0.759893
\(590\) 0 0
\(591\) − 445.325i − 0.753512i
\(592\) 0 0
\(593\) − 1004.17i − 1.69337i −0.532094 0.846685i \(-0.678595\pi\)
0.532094 0.846685i \(-0.321405\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 425.182 0.712198
\(598\) 0 0
\(599\) −796.917 −1.33041 −0.665206 0.746660i \(-0.731656\pi\)
−0.665206 + 0.746660i \(0.731656\pi\)
\(600\) 0 0
\(601\) − 467.002i − 0.777041i −0.921440 0.388521i \(-0.872986\pi\)
0.921440 0.388521i \(-0.127014\pi\)
\(602\) 0 0
\(603\) 6.62098 0.0109801
\(604\) 0 0
\(605\) 42.1953i 0.0697444i
\(606\) 0 0
\(607\) 910.216i 1.49953i 0.661703 + 0.749766i \(0.269834\pi\)
−0.661703 + 0.749766i \(0.730166\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −27.3221 −0.0447170
\(612\) 0 0
\(613\) 775.800 1.26558 0.632790 0.774324i \(-0.281910\pi\)
0.632790 + 0.774324i \(0.281910\pi\)
\(614\) 0 0
\(615\) 40.6483i 0.0660948i
\(616\) 0 0
\(617\) 974.803 1.57991 0.789954 0.613166i \(-0.210104\pi\)
0.789954 + 0.613166i \(0.210104\pi\)
\(618\) 0 0
\(619\) − 199.260i − 0.321906i −0.986962 0.160953i \(-0.948543\pi\)
0.986962 0.160953i \(-0.0514568\pi\)
\(620\) 0 0
\(621\) − 34.2935i − 0.0552231i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 607.726 0.972361
\(626\) 0 0
\(627\) −106.548 −0.169932
\(628\) 0 0
\(629\) 1386.17i 2.20377i
\(630\) 0 0
\(631\) 751.062 1.19027 0.595136 0.803625i \(-0.297098\pi\)
0.595136 + 0.803625i \(0.297098\pi\)
\(632\) 0 0
\(633\) 165.967i 0.262191i
\(634\) 0 0
\(635\) − 14.9214i − 0.0234982i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 241.770 0.378356
\(640\) 0 0
\(641\) −1138.28 −1.77578 −0.887891 0.460053i \(-0.847830\pi\)
−0.887891 + 0.460053i \(0.847830\pi\)
\(642\) 0 0
\(643\) 647.823i 1.00750i 0.863849 + 0.503751i \(0.168047\pi\)
−0.863849 + 0.503751i \(0.831953\pi\)
\(644\) 0 0
\(645\) 42.9255 0.0665512
\(646\) 0 0
\(647\) 882.943i 1.36467i 0.731039 + 0.682336i \(0.239036\pi\)
−0.731039 + 0.682336i \(0.760964\pi\)
\(648\) 0 0
\(649\) − 533.554i − 0.822116i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 545.117 0.834788 0.417394 0.908726i \(-0.362944\pi\)
0.417394 + 0.908726i \(0.362944\pi\)
\(654\) 0 0
\(655\) 22.3553 0.0341302
\(656\) 0 0
\(657\) − 41.7264i − 0.0635104i
\(658\) 0 0
\(659\) 698.290 1.05962 0.529810 0.848116i \(-0.322263\pi\)
0.529810 + 0.848116i \(0.322263\pi\)
\(660\) 0 0
\(661\) − 571.725i − 0.864940i −0.901648 0.432470i \(-0.857642\pi\)
0.901648 0.432470i \(-0.142358\pi\)
\(662\) 0 0
\(663\) − 56.7303i − 0.0855661i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 40.9358 0.0613730
\(668\) 0 0
\(669\) 163.175 0.243909
\(670\) 0 0
\(671\) − 28.7690i − 0.0428748i
\(672\) 0 0
\(673\) 221.015 0.328403 0.164202 0.986427i \(-0.447495\pi\)
0.164202 + 0.986427i \(0.447495\pi\)
\(674\) 0 0
\(675\) 128.703i 0.190672i
\(676\) 0 0
\(677\) − 470.666i − 0.695222i −0.937639 0.347611i \(-0.886993\pi\)
0.937639 0.347611i \(-0.113007\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 388.418 0.570364
\(682\) 0 0
\(683\) 215.960 0.316193 0.158097 0.987424i \(-0.449464\pi\)
0.158097 + 0.987424i \(0.449464\pi\)
\(684\) 0 0
\(685\) − 21.8535i − 0.0319029i
\(686\) 0 0
\(687\) 635.185 0.924578
\(688\) 0 0
\(689\) 116.624i 0.169265i
\(690\) 0 0
\(691\) − 80.1141i − 0.115939i −0.998318 0.0579697i \(-0.981537\pi\)
0.998318 0.0579697i \(-0.0184627\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 66.3672 0.0954924
\(696\) 0 0
\(697\) −1126.25 −1.61586
\(698\) 0 0
\(699\) − 120.679i − 0.172645i
\(700\) 0 0
\(701\) −821.973 −1.17257 −0.586286 0.810104i \(-0.699411\pi\)
−0.586286 + 0.810104i \(0.699411\pi\)
\(702\) 0 0
\(703\) 641.423i 0.912409i
\(704\) 0 0
\(705\) − 16.0196i − 0.0227229i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 575.456 0.811644 0.405822 0.913952i \(-0.366985\pi\)
0.405822 + 0.913952i \(0.366985\pi\)
\(710\) 0 0
\(711\) 194.665 0.273791
\(712\) 0 0
\(713\) 276.743i 0.388139i
\(714\) 0 0
\(715\) 3.93334 0.00550117
\(716\) 0 0
\(717\) 371.601i 0.518272i
\(718\) 0 0
\(719\) 1213.93i 1.68836i 0.536063 + 0.844178i \(0.319911\pi\)
−0.536063 + 0.844178i \(0.680089\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 283.009 0.391437
\(724\) 0 0
\(725\) −153.632 −0.211906
\(726\) 0 0
\(727\) 379.498i 0.522005i 0.965338 + 0.261003i \(0.0840532\pi\)
−0.965338 + 0.261003i \(0.915947\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 1189.35i 1.62702i
\(732\) 0 0
\(733\) 1250.46i 1.70595i 0.521954 + 0.852973i \(0.325203\pi\)
−0.521954 + 0.852973i \(0.674797\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.7193 −0.0172582
\(738\) 0 0
\(739\) 501.234 0.678260 0.339130 0.940740i \(-0.389867\pi\)
0.339130 + 0.940740i \(0.389867\pi\)
\(740\) 0 0
\(741\) − 26.2509i − 0.0354263i
\(742\) 0 0
\(743\) 444.584 0.598363 0.299181 0.954196i \(-0.403286\pi\)
0.299181 + 0.954196i \(0.403286\pi\)
\(744\) 0 0
\(745\) − 114.988i − 0.154346i
\(746\) 0 0
\(747\) 354.014i 0.473914i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 239.335 0.318688 0.159344 0.987223i \(-0.449062\pi\)
0.159344 + 0.987223i \(0.449062\pi\)
\(752\) 0 0
\(753\) −572.522 −0.760322
\(754\) 0 0
\(755\) 90.4347i 0.119781i
\(756\) 0 0
\(757\) 249.486 0.329572 0.164786 0.986329i \(-0.447307\pi\)
0.164786 + 0.986329i \(0.447307\pi\)
\(758\) 0 0
\(759\) 65.8797i 0.0867981i
\(760\) 0 0
\(761\) − 1375.68i − 1.80773i −0.427821 0.903864i \(-0.640719\pi\)
0.427821 0.903864i \(-0.359281\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 33.2624 0.0434803
\(766\) 0 0
\(767\) 131.455 0.171389
\(768\) 0 0
\(769\) − 528.594i − 0.687379i −0.939083 0.343689i \(-0.888323\pi\)
0.939083 0.343689i \(-0.111677\pi\)
\(770\) 0 0
\(771\) 278.044 0.360628
\(772\) 0 0
\(773\) − 61.9934i − 0.0801984i −0.999196 0.0400992i \(-0.987233\pi\)
0.999196 0.0400992i \(-0.0127674\pi\)
\(774\) 0 0
\(775\) − 1038.61i − 1.34015i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −521.152 −0.669001
\(780\) 0 0
\(781\) −464.453 −0.594690
\(782\) 0 0
\(783\) − 32.2296i − 0.0411616i
\(784\) 0 0
\(785\) 103.457 0.131792
\(786\) 0 0
\(787\) 163.568i 0.207837i 0.994586 + 0.103919i \(0.0331381\pi\)
−0.994586 + 0.103919i \(0.966862\pi\)
\(788\) 0 0
\(789\) 184.490i 0.233828i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7.08802 0.00893823
\(794\) 0 0
\(795\) −68.3794 −0.0860119
\(796\) 0 0
\(797\) 533.394i 0.669253i 0.942351 + 0.334626i \(0.108610\pi\)
−0.942351 + 0.334626i \(0.891390\pi\)
\(798\) 0 0
\(799\) 443.860 0.555519
\(800\) 0 0
\(801\) − 312.794i − 0.390505i
\(802\) 0 0
\(803\) 80.1586i 0.0998239i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −269.254 −0.333648
\(808\) 0 0
\(809\) −550.274 −0.680191 −0.340095 0.940391i \(-0.610459\pi\)
−0.340095 + 0.940391i \(0.610459\pi\)
\(810\) 0 0
\(811\) − 415.532i − 0.512370i −0.966628 0.256185i \(-0.917534\pi\)
0.966628 0.256185i \(-0.0824656\pi\)
\(812\) 0 0
\(813\) −893.876 −1.09948
\(814\) 0 0
\(815\) − 72.0082i − 0.0883536i
\(816\) 0 0
\(817\) 550.348i 0.673621i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 869.191 1.05870 0.529349 0.848404i \(-0.322436\pi\)
0.529349 + 0.848404i \(0.322436\pi\)
\(822\) 0 0
\(823\) −421.783 −0.512494 −0.256247 0.966611i \(-0.582486\pi\)
−0.256247 + 0.966611i \(0.582486\pi\)
\(824\) 0 0
\(825\) − 247.246i − 0.299692i
\(826\) 0 0
\(827\) 70.7290 0.0855248 0.0427624 0.999085i \(-0.486384\pi\)
0.0427624 + 0.999085i \(0.486384\pi\)
\(828\) 0 0
\(829\) 1268.85i 1.53058i 0.643685 + 0.765290i \(0.277405\pi\)
−0.643685 + 0.765290i \(0.722595\pi\)
\(830\) 0 0
\(831\) − 346.759i − 0.417280i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −65.9445 −0.0789754
\(836\) 0 0
\(837\) 217.885 0.260317
\(838\) 0 0
\(839\) − 613.254i − 0.730935i −0.930824 0.365467i \(-0.880909\pi\)
0.930824 0.365467i \(-0.119091\pi\)
\(840\) 0 0
\(841\) −802.528 −0.954254
\(842\) 0 0
\(843\) − 395.069i − 0.468647i
\(844\) 0 0
\(845\) − 80.2627i − 0.0949855i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −451.563 −0.531876
\(850\) 0 0
\(851\) 396.601 0.466041
\(852\) 0 0
\(853\) 668.244i 0.783404i 0.920092 + 0.391702i \(0.128114\pi\)
−0.920092 + 0.391702i \(0.871886\pi\)
\(854\) 0 0
\(855\) 15.3916 0.0180018
\(856\) 0 0
\(857\) − 519.712i − 0.606431i −0.952922 0.303216i \(-0.901940\pi\)
0.952922 0.303216i \(-0.0980602\pi\)
\(858\) 0 0
\(859\) 715.553i 0.833007i 0.909134 + 0.416504i \(0.136745\pi\)
−0.909134 + 0.416504i \(0.863255\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −342.716 −0.397122 −0.198561 0.980089i \(-0.563627\pi\)
−0.198561 + 0.980089i \(0.563627\pi\)
\(864\) 0 0
\(865\) 120.733 0.139576
\(866\) 0 0
\(867\) 421.048i 0.485638i
\(868\) 0 0
\(869\) −373.963 −0.430337
\(870\) 0 0
\(871\) − 3.13374i − 0.00359786i
\(872\) 0 0
\(873\) − 95.3625i − 0.109235i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −158.550 −0.180787 −0.0903934 0.995906i \(-0.528812\pi\)
−0.0903934 + 0.995906i \(0.528812\pi\)
\(878\) 0 0
\(879\) 606.019 0.689442
\(880\) 0 0
\(881\) − 1100.63i − 1.24930i −0.780906 0.624648i \(-0.785242\pi\)
0.780906 0.624648i \(-0.214758\pi\)
\(882\) 0 0
\(883\) 255.888 0.289794 0.144897 0.989447i \(-0.453715\pi\)
0.144897 + 0.989447i \(0.453715\pi\)
\(884\) 0 0
\(885\) 77.0756i 0.0870911i
\(886\) 0 0
\(887\) 1569.23i 1.76915i 0.466400 + 0.884574i \(0.345551\pi\)
−0.466400 + 0.884574i \(0.654449\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 51.8685 0.0582138
\(892\) 0 0
\(893\) 205.388 0.229997
\(894\) 0 0
\(895\) 56.1381i 0.0627241i
\(896\) 0 0
\(897\) −16.2313 −0.0180951
\(898\) 0 0
\(899\) 260.087i 0.289307i
\(900\) 0 0
\(901\) − 1894.61i − 2.10278i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 56.3087 0.0622196
\(906\) 0 0
\(907\) 876.460 0.966328 0.483164 0.875530i \(-0.339487\pi\)
0.483164 + 0.875530i \(0.339487\pi\)
\(908\) 0 0
\(909\) 217.511i 0.239286i
\(910\) 0 0
\(911\) −1778.49 −1.95224 −0.976118 0.217241i \(-0.930294\pi\)
−0.976118 + 0.217241i \(0.930294\pi\)
\(912\) 0 0
\(913\) − 680.080i − 0.744885i
\(914\) 0 0
\(915\) 4.15589i 0.00454195i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −264.863 −0.288208 −0.144104 0.989563i \(-0.546030\pi\)
−0.144104 + 0.989563i \(0.546030\pi\)
\(920\) 0 0
\(921\) −254.435 −0.276259
\(922\) 0 0
\(923\) − 114.431i − 0.123977i
\(924\) 0 0
\(925\) −1488.44 −1.60912
\(926\) 0 0
\(927\) 320.888i 0.346157i
\(928\) 0 0
\(929\) 789.390i 0.849720i 0.905259 + 0.424860i \(0.139677\pi\)
−0.905259 + 0.424860i \(0.860323\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −139.835 −0.149877
\(934\) 0 0
\(935\) −63.8989 −0.0683411
\(936\) 0 0
\(937\) − 1446.06i − 1.54329i −0.636053 0.771645i \(-0.719434\pi\)
0.636053 0.771645i \(-0.280566\pi\)
\(938\) 0 0
\(939\) 267.323 0.284689
\(940\) 0 0
\(941\) − 1201.10i − 1.27640i −0.769869 0.638202i \(-0.779678\pi\)
0.769869 0.638202i \(-0.220322\pi\)
\(942\) 0 0
\(943\) 322.235i 0.341713i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −838.720 −0.885660 −0.442830 0.896606i \(-0.646026\pi\)
−0.442830 + 0.896606i \(0.646026\pi\)
\(948\) 0 0
\(949\) −19.7492 −0.0208106
\(950\) 0 0
\(951\) 326.570i 0.343396i
\(952\) 0 0
\(953\) 1377.68 1.44562 0.722812 0.691045i \(-0.242849\pi\)
0.722812 + 0.691045i \(0.242849\pi\)
\(954\) 0 0
\(955\) − 164.299i − 0.172041i
\(956\) 0 0
\(957\) 61.9147i 0.0646967i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −797.295 −0.829652
\(962\) 0 0
\(963\) −589.858 −0.612521
\(964\) 0 0
\(965\) 107.933i 0.111848i
\(966\) 0 0
\(967\) 848.834 0.877802 0.438901 0.898536i \(-0.355368\pi\)
0.438901 + 0.898536i \(0.355368\pi\)
\(968\) 0 0
\(969\) 426.458i 0.440101i
\(970\) 0 0
\(971\) − 1262.08i − 1.29977i −0.760031 0.649887i \(-0.774816\pi\)
0.760031 0.649887i \(-0.225184\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 60.9158 0.0624777
\(976\) 0 0
\(977\) 644.168 0.659332 0.329666 0.944098i \(-0.393064\pi\)
0.329666 + 0.944098i \(0.393064\pi\)
\(978\) 0 0
\(979\) 600.895i 0.613784i
\(980\) 0 0
\(981\) −128.077 −0.130557
\(982\) 0 0
\(983\) − 1213.77i − 1.23476i −0.786667 0.617378i \(-0.788195\pi\)
0.786667 0.617378i \(-0.211805\pi\)
\(984\) 0 0
\(985\) − 123.582i − 0.125464i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 340.288 0.344072
\(990\) 0 0
\(991\) 1691.12 1.70648 0.853241 0.521517i \(-0.174634\pi\)
0.853241 + 0.521517i \(0.174634\pi\)
\(992\) 0 0
\(993\) − 254.243i − 0.256035i
\(994\) 0 0
\(995\) 117.992 0.118585
\(996\) 0 0
\(997\) − 1129.58i − 1.13297i −0.824071 0.566487i \(-0.808302\pi\)
0.824071 0.566487i \(-0.191698\pi\)
\(998\) 0 0
\(999\) − 312.252i − 0.312564i
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 588.3.d.c.97.2 8
3.2 odd 2 1764.3.d.h.685.5 8
4.3 odd 2 2352.3.f.j.97.6 8
7.2 even 3 588.3.m.f.325.2 8
7.3 odd 6 588.3.m.f.313.2 8
7.4 even 3 588.3.m.e.313.3 8
7.5 odd 6 588.3.m.e.325.3 8
7.6 odd 2 inner 588.3.d.c.97.7 yes 8
21.2 odd 6 1764.3.z.l.325.3 8
21.5 even 6 1764.3.z.m.325.2 8
21.11 odd 6 1764.3.z.m.901.2 8
21.17 even 6 1764.3.z.l.901.3 8
21.20 even 2 1764.3.d.h.685.4 8
28.27 even 2 2352.3.f.j.97.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.3.d.c.97.2 8 1.1 even 1 trivial
588.3.d.c.97.7 yes 8 7.6 odd 2 inner
588.3.m.e.313.3 8 7.4 even 3
588.3.m.e.325.3 8 7.5 odd 6
588.3.m.f.313.2 8 7.3 odd 6
588.3.m.f.325.2 8 7.2 even 3
1764.3.d.h.685.4 8 21.20 even 2
1764.3.d.h.685.5 8 3.2 odd 2
1764.3.z.l.325.3 8 21.2 odd 6
1764.3.z.l.901.3 8 21.17 even 6
1764.3.z.m.325.2 8 21.5 even 6
1764.3.z.m.901.2 8 21.11 odd 6
2352.3.f.j.97.3 8 28.27 even 2
2352.3.f.j.97.6 8 4.3 odd 2