Properties

Label 1764.3.d.h.685.4
Level $1764$
Weight $3$
Character 1764.685
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.4
Root \(-1.60021 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1764.685
Dual form 1764.3.d.h.685.5

$q$-expansion

\(f(q)\) \(=\) \(q-0.480662i q^{5} +O(q^{10})\) \(q-0.480662i q^{5} -5.76316 q^{11} -1.41991i q^{13} -23.0671i q^{17} +10.6739i q^{19} +6.59980 q^{23} +24.7690 q^{25} +6.20258 q^{29} +41.9320i q^{31} -60.0929 q^{37} -48.8250i q^{41} -51.5603 q^{43} +19.2421i q^{47} -82.1345 q^{53} +2.77013i q^{55} -92.5800i q^{59} +4.99187i q^{61} -0.682497 q^{65} -2.20699 q^{67} +80.5899 q^{71} -13.9088i q^{73} -64.8885 q^{79} -118.005i q^{83} -11.0875 q^{85} +104.265i q^{89} +5.13052 q^{95} -31.7875i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 16 q^{23} + 72 q^{25} - 80 q^{29} + 128 q^{37} - 112 q^{43} + 144 q^{53} - 240 q^{65} - 64 q^{67} - 224 q^{71} - 432 q^{79} - 96 q^{85} + 272 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 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
\(9\) 0 0
\(10\) 0 0
\(11\) −5.76316 −0.523924 −0.261962 0.965078i \(-0.584369\pi\)
−0.261962 + 0.965078i \(0.584369\pi\)
\(12\) 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 0
\(16\) 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 \(0.0906300\pi\)
−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 \(-0.454173\pi\)
0.143474 + 0.989654i \(0.454173\pi\)
\(24\) 0 0
\(25\) 24.7690 0.990759
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.20258 0.213882 0.106941 0.994265i \(-0.465894\pi\)
0.106941 + 0.994265i \(0.465894\pi\)
\(30\) 0 0
\(31\) 41.9320i 1.35265i 0.736605 + 0.676323i \(0.236428\pi\)
−0.736605 + 0.676323i \(0.763572\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(39\) 0 0
\(40\) 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
\(45\) 0 0
\(46\) 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
\(51\) 0 0
\(52\) 0 0
\(53\) −82.1345 −1.54971 −0.774854 0.632141i \(-0.782176\pi\)
−0.774854 + 0.632141i \(0.782176\pi\)
\(54\) 0 0
\(55\) 2.77013i 0.0503660i
\(56\) 0 0
\(57\) 0 0
\(58\) 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 \(0.0130279\pi\)
−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
\(69\) 0 0
\(70\) 0 0
\(71\) 80.5899 1.13507 0.567535 0.823349i \(-0.307897\pi\)
0.567535 + 0.823349i \(0.307897\pi\)
\(72\) 0 0
\(73\) − 13.9088i − 0.190531i −0.995452 0.0952657i \(-0.969630\pi\)
0.995452 0.0952657i \(-0.0303701\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 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
\(93\) 0 0
\(94\) 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
\(99\) 0 0
\(100\) 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 \(0.173783\pi\)
−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 \(-0.870826\pi\)
−0.918782 + 0.394765i \(0.870826\pi\)
\(108\) 0 0
\(109\) 42.6922 0.391672 0.195836 0.980637i \(-0.437258\pi\)
0.195836 + 0.980637i \(0.437258\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −175.501 −1.55310 −0.776552 0.630053i \(-0.783033\pi\)
−0.776552 + 0.630053i \(0.783033\pi\)
\(114\) 0 0
\(115\) − 3.17227i − 0.0275849i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −87.7860 −0.725504
\(122\) 0 0
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 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
\(135\) 0 0
\(136\) 0 0
\(137\) −45.4654 −0.331864 −0.165932 0.986137i \(-0.553063\pi\)
−0.165932 + 0.986137i \(0.553063\pi\)
\(138\) 0 0
\(139\) − 138.075i − 0.993343i −0.867939 0.496672i \(-0.834555\pi\)
0.867939 0.496672i \(-0.165445\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 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 \(-0.796646\pi\)
−0.802780 + 0.596276i \(0.796646\pi\)
\(150\) 0 0
\(151\) −188.146 −1.24600 −0.623001 0.782221i \(-0.714087\pi\)
−0.623001 + 0.782221i \(0.714087\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 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
\(159\) 0 0
\(160\) 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
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 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
\(177\) 0 0
\(178\) 0 0
\(179\) 116.793 0.652477 0.326239 0.945287i \(-0.394219\pi\)
0.326239 + 0.945287i \(0.394219\pi\)
\(180\) 0 0
\(181\) − 117.148i − 0.647228i −0.946189 0.323614i \(-0.895102\pi\)
0.946189 0.323614i \(-0.104898\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 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 \(-0.852691\pi\)
−0.894812 + 0.446443i \(0.852691\pi\)
\(192\) 0 0
\(193\) −224.551 −1.16348 −0.581738 0.813376i \(-0.697627\pi\)
−0.581738 + 0.813376i \(0.697627\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −257.109 −1.30512 −0.652560 0.757737i \(-0.726305\pi\)
−0.652560 + 0.757737i \(0.726305\pi\)
\(198\) 0 0
\(199\) − 245.479i − 1.23356i −0.787134 0.616782i \(-0.788436\pi\)
0.787134 0.616782i \(-0.211564\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −23.4683 −0.114480
\(206\) 0 0
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) 24.7831i 0.115270i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 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
\(225\) 0 0
\(226\) 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 \(-0.704457\pi\)
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 \(-0.547771\pi\)
−0.149515 + 0.988759i \(0.547771\pi\)
\(234\) 0 0
\(235\) 9.24893 0.0393572
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 214.544 0.897674 0.448837 0.893614i \(-0.351838\pi\)
0.448837 + 0.893614i \(0.351838\pi\)
\(240\) 0 0
\(241\) − 163.395i − 0.677989i −0.940788 0.338994i \(-0.889913\pi\)
0.940788 0.338994i \(-0.110087\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.1560 0.0613601
\(248\) 0 0
\(249\) 0 0
\(250\) 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
\(255\) 0 0
\(256\) 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
\(261\) 0 0
\(262\) 0 0
\(263\) 106.515 0.405002 0.202501 0.979282i \(-0.435093\pi\)
0.202501 + 0.979282i \(0.435093\pi\)
\(264\) 0 0
\(265\) 39.4789i 0.148977i
\(266\) 0 0
\(267\) 0 0
\(268\) 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 \(0.401160\pi\)
−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
\(279\) 0 0
\(280\) 0 0
\(281\) −228.093 −0.811720 −0.405860 0.913935i \(-0.633028\pi\)
−0.405860 + 0.913935i \(0.633028\pi\)
\(282\) 0 0
\(283\) 260.710i 0.921236i 0.887598 + 0.460618i \(0.152372\pi\)
−0.887598 + 0.460618i \(0.847628\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −243.092 −0.841150
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 0 0
\(299\) − 9.37113i − 0.0313416i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 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
\(309\) 0 0
\(310\) 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 \(-0.920704\pi\)
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 \(-0.403884\pi\)
0.297390 + 0.954756i \(0.403884\pi\)
\(318\) 0 0
\(319\) −35.7465 −0.112058
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 246.216 0.762277
\(324\) 0 0
\(325\) − 35.1697i − 0.108215i
\(326\) 0 0
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 0 0
\(341\) − 241.661i − 0.708684i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −351.689 −1.01351 −0.506756 0.862089i \(-0.669156\pi\)
−0.506756 + 0.862089i \(0.669156\pi\)
\(348\) 0 0
\(349\) − 88.3780i − 0.253232i −0.991952 0.126616i \(-0.959588\pi\)
0.991952 0.126616i \(-0.0404116\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 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 \(-0.245859\pi\)
0.716246 + 0.697848i \(0.245859\pi\)
\(360\) 0 0
\(361\) 247.069 0.684400
\(362\) 0 0
\(363\) 0 0
\(364\) 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
\(369\) 0 0
\(370\) 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
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) 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
\(387\) 0 0
\(388\) 0 0
\(389\) −401.271 −1.03155 −0.515773 0.856725i \(-0.672495\pi\)
−0.515773 + 0.856725i \(0.672495\pi\)
\(390\) 0 0
\(391\) − 152.238i − 0.389356i
\(392\) 0 0
\(393\) 0 0
\(394\) 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 \(-0.345436\pi\)
0.466718 + 0.884406i \(0.345436\pi\)
\(402\) 0 0
\(403\) 59.5398 0.147741
\(404\) 0 0
\(405\) 0 0
\(406\) 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
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −56.7203 −0.136675
\(416\) 0 0
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) − 571.349i − 1.34435i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 277.092 0.642906 0.321453 0.946926i \(-0.395829\pi\)
0.321453 + 0.946926i \(0.395829\pi\)
\(432\) 0 0
\(433\) 822.794i 1.90022i 0.311919 + 0.950109i \(0.399028\pi\)
−0.311919 + 0.950109i \(0.600972\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 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 \(-0.275953\pi\)
0.647169 + 0.762346i \(0.275953\pi\)
\(444\) 0 0
\(445\) 50.1161 0.112620
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −227.961 −0.507708 −0.253854 0.967243i \(-0.581698\pi\)
−0.253854 + 0.967243i \(0.581698\pi\)
\(450\) 0 0
\(451\) 281.386i 0.623917i
\(452\) 0 0
\(453\) 0 0
\(454\) 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
\(459\) 0 0
\(460\) 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
\(465\) 0 0
\(466\) 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
\(471\) 0 0
\(472\) 0 0
\(473\) 297.150 0.628225
\(474\) 0 0
\(475\) 264.381i 0.556591i
\(476\) 0 0
\(477\) 0 0
\(478\) 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
\(489\) 0 0
\(490\) 0 0
\(491\) 818.649 1.66731 0.833655 0.552286i \(-0.186244\pi\)
0.833655 + 0.552286i \(0.186244\pi\)
\(492\) 0 0
\(493\) − 143.076i − 0.290214i
\(494\) 0 0
\(495\) 0 0
\(496\) 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
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 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
\(513\) 0 0
\(514\) 0 0
\(515\) 51.4128 0.0998307
\(516\) 0 0
\(517\) − 110.895i − 0.214498i
\(518\) 0 0
\(519\) 0 0
\(520\) 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 \(-0.931384\pi\)
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
\(531\) 0 0
\(532\) 0 0
\(533\) −69.3272 −0.130070
\(534\) 0 0
\(535\) 94.5074i 0.176649i
\(536\) 0 0
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 0 0
\(551\) 66.2056i 0.120155i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −418.568 −0.751469 −0.375735 0.926727i \(-0.622610\pi\)
−0.375735 + 0.926727i \(0.622610\pi\)
\(558\) 0 0
\(559\) 73.2111i 0.130968i
\(560\) 0 0
\(561\) 0 0
\(562\) 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 \(-0.418353\pi\)
0.253699 + 0.967283i \(0.418353\pi\)
\(570\) 0 0
\(571\) −916.207 −1.60457 −0.802283 0.596944i \(-0.796382\pi\)
−0.802283 + 0.596944i \(0.796382\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 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
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 473.354 0.811929
\(584\) 0 0
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) 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
\(597\) 0 0
\(598\) 0 0
\(599\) 796.917 1.33041 0.665206 0.746660i \(-0.268344\pi\)
0.665206 + 0.746660i \(0.268344\pi\)
\(600\) 0 0
\(601\) 467.002i 0.777041i 0.921440 + 0.388521i \(0.127014\pi\)
−0.921440 + 0.388521i \(0.872986\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 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
\(615\) 0 0
\(616\) 0 0
\(617\) −974.803 −1.57991 −0.789954 0.613166i \(-0.789896\pi\)
−0.789954 + 0.613166i \(0.789896\pi\)
\(618\) 0 0
\(619\) 199.260i 0.321906i 0.986962 + 0.160953i \(0.0514568\pi\)
−0.986962 + 0.160953i \(0.948543\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 607.726 0.972361
\(626\) 0 0
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 0 0
\(635\) − 14.9214i − 0.0234982i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1138.28 1.77578 0.887891 0.460053i \(-0.152170\pi\)
0.887891 + 0.460053i \(0.152170\pi\)
\(642\) 0 0
\(643\) − 647.823i − 1.00750i −0.863849 0.503751i \(-0.831953\pi\)
0.863849 0.503751i \(-0.168047\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 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 \(-0.637056\pi\)
−0.417394 + 0.908726i \(0.637056\pi\)
\(654\) 0 0
\(655\) 22.3553 0.0341302
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −698.290 −1.05962 −0.529810 0.848116i \(-0.677737\pi\)
−0.529810 + 0.848116i \(0.677737\pi\)
\(660\) 0 0
\(661\) 571.725i 0.864940i 0.901648 + 0.432470i \(0.142358\pi\)
−0.901648 + 0.432470i \(0.857642\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 40.9358 0.0613730
\(668\) 0 0
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) 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
\(681\) 0 0
\(682\) 0 0
\(683\) −215.960 −0.316193 −0.158097 0.987424i \(-0.550536\pi\)
−0.158097 + 0.987424i \(0.550536\pi\)
\(684\) 0 0
\(685\) 21.8535i 0.0319029i
\(686\) 0 0
\(687\) 0 0
\(688\) 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
\(699\) 0 0
\(700\) 0 0
\(701\) 821.973 1.17257 0.586286 0.810104i \(-0.300589\pi\)
0.586286 + 0.810104i \(0.300589\pi\)
\(702\) 0 0
\(703\) − 641.423i − 0.912409i
\(704\) 0 0
\(705\) 0 0
\(706\) 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
\(711\) 0 0
\(712\) 0 0
\(713\) 276.743i 0.388139i
\(714\) 0 0
\(715\) 3.93334 0.00550117
\(716\) 0 0
\(717\) 0 0
\(718\) 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
\(723\) 0 0
\(724\) 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
\(729\) 0 0
\(730\) 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
\(741\) 0 0
\(742\) 0 0
\(743\) −444.584 −0.598363 −0.299181 0.954196i \(-0.596714\pi\)
−0.299181 + 0.954196i \(0.596714\pi\)
\(744\) 0 0
\(745\) 114.988i 0.154346i
\(746\) 0 0
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 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
\(759\) 0 0
\(760\) 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
\(765\) 0 0
\(766\) 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
\(771\) 0 0
\(772\) 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
\(783\) 0 0
\(784\) 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
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7.08802 0.00893823
\(794\) 0 0
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) 0 0
\(803\) 80.1586i 0.0998239i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 550.274 0.680191 0.340095 0.940391i \(-0.389541\pi\)
0.340095 + 0.940391i \(0.389541\pi\)
\(810\) 0 0
\(811\) 415.532i 0.512370i 0.966628 + 0.256185i \(0.0824656\pi\)
−0.966628 + 0.256185i \(0.917534\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 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 \(-0.677564\pi\)
−0.529349 + 0.848404i \(0.677564\pi\)
\(822\) 0 0
\(823\) −421.783 −0.512494 −0.256247 0.966611i \(-0.582486\pi\)
−0.256247 + 0.966611i \(0.582486\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −70.7290 −0.0855248 −0.0427624 0.999085i \(-0.513616\pi\)
−0.0427624 + 0.999085i \(0.513616\pi\)
\(828\) 0 0
\(829\) − 1268.85i − 1.53058i −0.643685 0.765290i \(-0.722595\pi\)
0.643685 0.765290i \(-0.277405\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −65.9445 −0.0789754
\(836\) 0 0
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 0 0
\(845\) − 80.2627i − 0.0949855i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 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
\(855\) 0 0
\(856\) 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 \(-0.863255\pi\)
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 \(-0.436373\pi\)
0.198561 + 0.980089i \(0.436373\pi\)
\(864\) 0 0
\(865\) 120.733 0.139576
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 373.963 0.430337
\(870\) 0 0
\(871\) 3.13374i 0.00359786i
\(872\) 0 0
\(873\) 0 0
\(874\) 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
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) 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
\(891\) 0 0
\(892\) 0 0
\(893\) −205.388 −0.229997
\(894\) 0 0
\(895\) − 56.1381i − 0.0627241i
\(896\) 0 0
\(897\) 0 0
\(898\) 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
\(909\) 0 0
\(910\) 0 0
\(911\) 1778.49 1.95224 0.976118 0.217241i \(-0.0697057\pi\)
0.976118 + 0.217241i \(0.0697057\pi\)
\(912\) 0 0
\(913\) 680.080i 0.744885i
\(914\) 0 0
\(915\) 0 0
\(916\) 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
\(921\) 0 0
\(922\) 0 0
\(923\) − 114.431i − 0.123977i
\(924\) 0 0
\(925\) −1488.44 −1.60912
\(926\) 0 0
\(927\) 0 0
\(928\) 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
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 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 \(-0.353974\pi\)
0.442830 + 0.896606i \(0.353974\pi\)
\(948\) 0 0
\(949\) −19.7492 −0.0208106
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1377.68 −1.44562 −0.722812 0.691045i \(-0.757151\pi\)
−0.722812 + 0.691045i \(0.757151\pi\)
\(954\) 0 0
\(955\) 164.299i 0.172041i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −797.295 −0.829652
\(962\) 0 0
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 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
\(975\) 0 0
\(976\) 0 0
\(977\) −644.168 −0.659332 −0.329666 0.944098i \(-0.606936\pi\)
−0.329666 + 0.944098i \(0.606936\pi\)
\(978\) 0 0
\(979\) − 600.895i − 0.613784i
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(993\) 0 0
\(994\) 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
\(999\) 0 0
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 1764.3.d.h.685.4 8
3.2 odd 2 588.3.d.c.97.7 yes 8
7.2 even 3 1764.3.z.m.325.2 8
7.3 odd 6 1764.3.z.m.901.2 8
7.4 even 3 1764.3.z.l.901.3 8
7.5 odd 6 1764.3.z.l.325.3 8
7.6 odd 2 inner 1764.3.d.h.685.5 8
12.11 even 2 2352.3.f.j.97.3 8
21.2 odd 6 588.3.m.e.325.3 8
21.5 even 6 588.3.m.f.325.2 8
21.11 odd 6 588.3.m.f.313.2 8
21.17 even 6 588.3.m.e.313.3 8
21.20 even 2 588.3.d.c.97.2 8
84.83 odd 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 21.20 even 2
588.3.d.c.97.7 yes 8 3.2 odd 2
588.3.m.e.313.3 8 21.17 even 6
588.3.m.e.325.3 8 21.2 odd 6
588.3.m.f.313.2 8 21.11 odd 6
588.3.m.f.325.2 8 21.5 even 6
1764.3.d.h.685.4 8 1.1 even 1 trivial
1764.3.d.h.685.5 8 7.6 odd 2 inner
1764.3.z.l.325.3 8 7.5 odd 6
1764.3.z.l.901.3 8 7.4 even 3
1764.3.z.m.325.2 8 7.2 even 3
1764.3.z.m.901.2 8 7.3 odd 6
2352.3.f.j.97.3 8 12.11 even 2
2352.3.f.j.97.6 8 84.83 odd 2