Properties

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

$q$-expansion

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

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 \(0.0153058\pi\)
−0.998844 + 0.0480662i \(0.984694\pi\)
\(6\) 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 \(-0.982608\pi\)
0.998508 0.0546120i \(-0.0173922\pi\)
\(14\) 0 0
\(15\) −0.832530 −0.0555020
\(16\) 0 0
\(17\) 23.0671i 1.35689i 0.734651 + 0.678445i \(0.237346\pi\)
−0.734651 + 0.678445i \(0.762654\pi\)
\(18\) 0 0
\(19\) 10.6739i 0.561783i 0.959740 + 0.280891i \(0.0906300\pi\)
−0.959740 + 0.280891i \(0.909370\pi\)
\(20\) 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 \(0.236428\pi\)
−0.736605 + 0.676323i \(0.763572\pi\)
\(32\) 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 \(0.203017\pi\)
−0.803409 + 0.595427i \(0.796983\pi\)
\(42\) 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 \(-0.934377\pi\)
0.978824 0.204703i \(-0.0656229\pi\)
\(48\) 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 \(0.287119\pi\)
−0.620032 + 0.784576i \(0.712881\pi\)
\(60\) 0 0
\(61\) 4.99187i 0.0818340i 0.999163 + 0.0409170i \(0.0130279\pi\)
−0.999163 + 0.0409170i \(0.986972\pi\)
\(62\) 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 \(-0.969630\pi\)
0.995452 0.0952657i \(-0.0303701\pi\)
\(74\) 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 \(0.251699\pi\)
−0.703322 + 0.710872i \(0.748301\pi\)
\(84\) 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 \(-0.800797\pi\)
0.810487 0.585757i \(-0.199203\pi\)
\(90\) 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 \(-0.947608\pi\)
0.986485 0.163853i \(-0.0523923\pi\)
\(98\) 0 0
\(99\) −17.2895 −0.174641
\(100\) 0 0
\(101\) 72.5037i 0.717859i 0.933365 + 0.358929i \(0.116858\pi\)
−0.933365 + 0.358929i \(0.883142\pi\)
\(102\) 0 0
\(103\) 106.963i 1.03847i 0.854631 + 0.519236i \(0.173783\pi\)
−0.854631 + 0.519236i \(0.826217\pi\)
\(104\) 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 \(-0.943194\pi\)
0.984118 0.177517i \(-0.0568065\pi\)
\(132\) 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 \(-0.834555\pi\)
0.867939 0.496672i \(-0.165445\pi\)
\(140\) 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 \(-0.759595\pi\)
0.728097 0.685474i \(-0.240405\pi\)
\(158\) 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 \(0.134738\pi\)
−0.911742 + 0.410764i \(0.865262\pi\)
\(168\) 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 \(-0.741397\pi\)
0.687740 0.725957i \(-0.258603\pi\)
\(174\) 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 \(-0.895102\pi\)
0.946189 0.323614i \(-0.104898\pi\)
\(182\) 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 \(-0.788436\pi\)
0.787134 0.616782i \(-0.211564\pi\)
\(200\) 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 \(-0.932253\pi\)
0.977436 0.211231i \(-0.0677473\pi\)
\(224\) 0 0
\(225\) −74.3069 −0.330253
\(226\) 0 0
\(227\) − 224.253i − 0.987899i −0.869491 0.493949i \(-0.835553\pi\)
0.869491 0.493949i \(-0.164447\pi\)
\(228\) 0 0
\(229\) − 366.724i − 1.60142i −0.599055 0.800708i \(-0.704457\pi\)
0.599055 0.800708i \(-0.295543\pi\)
\(230\) 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 \(-0.889913\pi\)
0.940788 0.338994i \(-0.110087\pi\)
\(242\) 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 \(0.228791\pi\)
−0.752617 + 0.658458i \(0.771209\pi\)
\(252\) 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 \(-0.898896\pi\)
0.949979 0.312313i \(-0.101104\pi\)
\(258\) 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 \(0.0933052\pi\)
−0.957345 + 0.288947i \(0.906695\pi\)
\(270\) 0 0
\(271\) 516.080i 1.90435i 0.305549 + 0.952176i \(0.401160\pi\)
−0.305549 + 0.952176i \(0.598840\pi\)
\(272\) 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 \(0.152372\pi\)
−0.887598 + 0.460618i \(0.847628\pi\)
\(284\) 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 \(-0.796330\pi\)
0.802186 0.597074i \(-0.203670\pi\)
\(294\) 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 \(0.0769007\pi\)
−0.970959 + 0.239247i \(0.923099\pi\)
\(308\) 0 0
\(309\) −185.265 −0.599562
\(310\) 0 0
\(311\) 80.7340i 0.259595i 0.991541 + 0.129797i \(0.0414327\pi\)
−0.991541 + 0.129797i \(0.958567\pi\)
\(312\) 0 0
\(313\) − 154.339i − 0.493096i −0.969131 0.246548i \(-0.920704\pi\)
0.969131 0.246548i \(-0.0792963\pi\)
\(314\) 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 \(-0.959588\pi\)
0.991952 0.126616i \(-0.0404116\pi\)
\(350\) 0 0
\(351\) −7.37808 −0.0210202
\(352\) 0 0
\(353\) 520.542i 1.47462i 0.675552 + 0.737312i \(0.263905\pi\)
−0.675552 + 0.737312i \(0.736095\pi\)
\(354\) 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 \(0.192690\pi\)
−0.822300 + 0.569054i \(0.807310\pi\)
\(368\) 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 \(0.0507946\pi\)
−0.987295 + 0.158900i \(0.949205\pi\)
\(384\) 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 \(0.0241671\pi\)
−0.997119 + 0.0758502i \(0.975833\pi\)
\(398\) 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 \(0.270600\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(410\) 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 \(0.0492212\pi\)
−0.988068 + 0.154017i \(0.950779\pi\)
\(420\) 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 \(0.399028\pi\)
−0.311919 + 0.950109i \(0.600972\pi\)
\(434\) 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 \(-0.863201\pi\)
0.909062 0.416660i \(-0.136799\pi\)
\(440\) 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 \(-0.769270\pi\)
0.748593 0.663030i \(-0.230730\pi\)
\(462\) 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 \(-0.967996\pi\)
0.994950 0.100375i \(-0.0320041\pi\)
\(468\) 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 \(-0.879595\pi\)
0.929308 0.369306i \(-0.120405\pi\)
\(480\) 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 \(-0.862753\pi\)
0.908476 0.417937i \(-0.137247\pi\)
\(504\) 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 \(0.0757145\pi\)
−0.971843 + 0.235627i \(0.924285\pi\)
\(510\) 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 \(-0.991795\pi\)
0.999668 0.0257739i \(-0.00820501\pi\)
\(522\) 0 0
\(523\) − 223.737i − 0.427796i −0.976856 0.213898i \(-0.931384\pi\)
0.976856 0.213898i \(-0.0686160\pi\)
\(524\) 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 \(-0.769208\pi\)
0.748462 0.663177i \(-0.230792\pi\)
\(564\) 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 \(-0.670664\pi\)
0.510836 0.859678i \(-0.329336\pi\)
\(578\) 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 \(-0.957861\pi\)
0.991250 0.131998i \(-0.0421392\pi\)
\(588\) 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 \(0.321405\pi\)
−0.532094 + 0.846685i \(0.678595\pi\)
\(594\) 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 \(0.127014\pi\)
−0.921440 + 0.388521i \(0.872986\pi\)
\(602\) 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 \(-0.730166\pi\)
0.661703 0.749766i \(-0.269834\pi\)
\(608\) 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 \(0.0514568\pi\)
−0.986962 + 0.160953i \(0.948543\pi\)
\(620\) 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 \(-0.831953\pi\)
0.863849 0.503751i \(-0.168047\pi\)
\(644\) 0 0
\(645\) 42.9255 0.0665512
\(646\) 0 0
\(647\) − 882.943i − 1.36467i −0.731039 0.682336i \(-0.760964\pi\)
0.731039 0.682336i \(-0.239036\pi\)
\(648\) 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 \(0.142358\pi\)
−0.901648 + 0.432470i \(0.857642\pi\)
\(662\) 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 \(0.113007\pi\)
−0.937639 + 0.347611i \(0.886993\pi\)
\(678\) 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 \(0.0184627\pi\)
−0.998318 + 0.0579697i \(0.981537\pi\)
\(692\) 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 \(-0.680089\pi\)
0.536063 0.844178i \(-0.319911\pi\)
\(720\) 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 \(-0.915947\pi\)
0.965338 0.261003i \(-0.0840532\pi\)
\(728\) 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 \(-0.674797\pi\)
0.521954 0.852973i \(-0.325203\pi\)
\(734\) 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 \(0.359281\pi\)
−0.427821 + 0.903864i \(0.640719\pi\)
\(762\) 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 \(0.111677\pi\)
−0.939083 + 0.343689i \(0.888323\pi\)
\(770\) 0 0
\(771\) 278.044 0.360628
\(772\) 0 0
\(773\) 61.9934i 0.0801984i 0.999196 + 0.0400992i \(0.0127674\pi\)
−0.999196 + 0.0400992i \(0.987233\pi\)
\(774\) 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 \(-0.966862\pi\)
0.994586 0.103919i \(-0.0331381\pi\)
\(788\) 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 \(-0.891390\pi\)
0.942351 0.334626i \(-0.108610\pi\)
\(798\) 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 \(0.0824656\pi\)
−0.966628 + 0.256185i \(0.917534\pi\)
\(812\) 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 \(-0.722595\pi\)
0.643685 0.765290i \(-0.277405\pi\)
\(830\) 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 \(0.119091\pi\)
−0.930824 + 0.365467i \(0.880909\pi\)
\(840\) 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 \(-0.871886\pi\)
0.920092 0.391702i \(-0.128114\pi\)
\(854\) 0 0
\(855\) 15.3916 0.0180018
\(856\) 0 0
\(857\) 519.712i 0.606431i 0.952922 + 0.303216i \(0.0980602\pi\)
−0.952922 + 0.303216i \(0.901940\pi\)
\(858\) 0 0
\(859\) − 715.553i − 0.833007i −0.909134 0.416504i \(-0.863255\pi\)
0.909134 0.416504i \(-0.136745\pi\)
\(860\) 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 \(0.214758\pi\)
−0.780906 + 0.624648i \(0.785242\pi\)
\(882\) 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 \(-0.654449\pi\)
0.466400 0.884574i \(-0.345551\pi\)
\(888\) 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 \(-0.860323\pi\)
0.905259 0.424860i \(-0.139677\pi\)
\(930\) 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 \(0.280566\pi\)
−0.636053 + 0.771645i \(0.719434\pi\)
\(938\) 0 0
\(939\) 267.323 0.284689
\(940\) 0 0
\(941\) 1201.10i 1.27640i 0.769869 + 0.638202i \(0.220322\pi\)
−0.769869 + 0.638202i \(0.779678\pi\)
\(942\) 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 \(0.225184\pi\)
−0.760031 + 0.649887i \(0.774816\pi\)
\(972\) 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 \(0.211805\pi\)
−0.786667 + 0.617378i \(0.788195\pi\)
\(984\) 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 \(0.191698\pi\)
−0.824071 + 0.566487i \(0.808302\pi\)
\(998\) 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.97.7 yes 8
3.2 odd 2 1764.3.d.h.685.4 8
4.3 odd 2 2352.3.f.j.97.3 8
7.2 even 3 588.3.m.e.325.3 8
7.3 odd 6 588.3.m.e.313.3 8
7.4 even 3 588.3.m.f.313.2 8
7.5 odd 6 588.3.m.f.325.2 8
7.6 odd 2 inner 588.3.d.c.97.2 8
21.2 odd 6 1764.3.z.m.325.2 8
21.5 even 6 1764.3.z.l.325.3 8
21.11 odd 6 1764.3.z.l.901.3 8
21.17 even 6 1764.3.z.m.901.2 8
21.20 even 2 1764.3.d.h.685.5 8
28.27 even 2 2352.3.f.j.97.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.3.d.c.97.2 8 7.6 odd 2 inner
588.3.d.c.97.7 yes 8 1.1 even 1 trivial
588.3.m.e.313.3 8 7.3 odd 6
588.3.m.e.325.3 8 7.2 even 3
588.3.m.f.313.2 8 7.4 even 3
588.3.m.f.325.2 8 7.5 odd 6
1764.3.d.h.685.4 8 3.2 odd 2
1764.3.d.h.685.5 8 21.20 even 2
1764.3.z.l.325.3 8 21.5 even 6
1764.3.z.l.901.3 8 21.11 odd 6
1764.3.z.m.325.2 8 21.2 odd 6
1764.3.z.m.901.2 8 21.17 even 6
2352.3.f.j.97.3 8 4.3 odd 2
2352.3.f.j.97.6 8 28.27 even 2