Properties

Label 4704.2.a.bk.1.1
Level $4704$
Weight $2$
Character 4704.1
Self dual yes
Analytic conductor $37.562$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5616291108\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4704.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{9} -5.65685 q^{11} -5.65685 q^{13} -5.65685 q^{17} +4.00000 q^{19} -5.65685 q^{23} -5.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +8.00000 q^{31} +5.65685 q^{33} +2.00000 q^{37} +5.65685 q^{39} +5.65685 q^{41} +8.00000 q^{47} +5.65685 q^{51} -2.00000 q^{53} -4.00000 q^{57} +4.00000 q^{59} -5.65685 q^{61} +11.3137 q^{67} +5.65685 q^{69} +5.65685 q^{71} +11.3137 q^{73} +5.00000 q^{75} +11.3137 q^{79} +1.00000 q^{81} -12.0000 q^{83} +6.00000 q^{87} -5.65685 q^{89} -8.00000 q^{93} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 8q^{19} - 10q^{25} - 2q^{27} - 12q^{29} + 16q^{31} + 4q^{37} + 16q^{47} - 4q^{53} - 8q^{57} + 8q^{59} + 10q^{75} + 2q^{81} - 24q^{83} + 12q^{87} - 16q^{93} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.65685 −1.70561 −0.852803 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.65685 −1.37199 −0.685994 0.727607i \(-0.740633\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.65685 −1.17954 −0.589768 0.807573i \(-0.700781\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 5.65685 0.984732
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) 5.65685 0.883452 0.441726 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.65685 0.792118
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −5.65685 −0.724286 −0.362143 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.3137 1.38219 0.691095 0.722764i \(-0.257129\pi\)
0.691095 + 0.722764i \(0.257129\pi\)
\(68\) 0 0
\(69\) 5.65685 0.681005
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) 11.3137 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 0 0
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −5.65685 −0.599625 −0.299813 0.953998i \(-0.596924\pi\)
−0.299813 + 0.953998i \(0.596924\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −5.65685 −0.568535
\(100\) 0 0
\(101\) 11.3137 1.12576 0.562878 0.826540i \(-0.309694\pi\)
0.562878 + 0.826540i \(0.309694\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.65685 −0.546869 −0.273434 0.961891i \(-0.588160\pi\)
−0.273434 + 0.961891i \(0.588160\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.65685 −0.522976
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) −5.65685 −0.510061
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.3137 −1.00393 −0.501965 0.864888i \(-0.667389\pi\)
−0.501965 + 0.864888i \(0.667389\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 32.0000 2.67597
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −22.6274 −1.84139 −0.920697 0.390279i \(-0.872378\pi\)
−0.920697 + 0.390279i \(0.872378\pi\)
\(152\) 0 0
\(153\) −5.65685 −0.457330
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −16.9706 −1.35440 −0.677199 0.735800i \(-0.736806\pi\)
−0.677199 + 0.735800i \(0.736806\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −11.3137 −0.886158 −0.443079 0.896483i \(-0.646114\pi\)
−0.443079 + 0.896483i \(0.646114\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 22.6274 1.72033 0.860165 0.510015i \(-0.170360\pi\)
0.860165 + 0.510015i \(0.170360\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 5.65685 0.422813 0.211407 0.977398i \(-0.432196\pi\)
0.211407 + 0.977398i \(0.432196\pi\)
\(180\) 0 0
\(181\) 16.9706 1.26141 0.630706 0.776022i \(-0.282765\pi\)
0.630706 + 0.776022i \(0.282765\pi\)
\(182\) 0 0
\(183\) 5.65685 0.418167
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 32.0000 2.34007
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −11.3137 −0.798007
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.65685 −0.393179
\(208\) 0 0
\(209\) −22.6274 −1.56517
\(210\) 0 0
\(211\) 22.6274 1.55774 0.778868 0.627188i \(-0.215794\pi\)
0.778868 + 0.627188i \(0.215794\pi\)
\(212\) 0 0
\(213\) −5.65685 −0.387601
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −11.3137 −0.764510
\(220\) 0 0
\(221\) 32.0000 2.15255
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 16.9706 1.12145 0.560723 0.828003i \(-0.310523\pi\)
0.560723 + 0.828003i \(0.310523\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −11.3137 −0.734904
\(238\) 0 0
\(239\) −5.65685 −0.365911 −0.182956 0.983121i \(-0.558567\pi\)
−0.182956 + 0.983121i \(0.558567\pi\)
\(240\) 0 0
\(241\) −22.6274 −1.45756 −0.728780 0.684748i \(-0.759912\pi\)
−0.728780 + 0.684748i \(0.759912\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −22.6274 −1.43975
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 32.0000 2.01182
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.65685 −0.352865 −0.176432 0.984313i \(-0.556456\pi\)
−0.176432 + 0.984313i \(0.556456\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 16.9706 1.04645 0.523225 0.852195i \(-0.324729\pi\)
0.523225 + 0.852195i \(0.324729\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5.65685 0.346194
\(268\) 0 0
\(269\) −11.3137 −0.689809 −0.344904 0.938638i \(-0.612089\pi\)
−0.344904 + 0.938638i \(0.612089\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 28.2843 1.70561
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −11.3137 −0.660954 −0.330477 0.943814i \(-0.607210\pi\)
−0.330477 + 0.943814i \(0.607210\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.65685 0.328244
\(298\) 0 0
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −11.3137 −0.649956
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 11.3137 0.639489 0.319744 0.947504i \(-0.396403\pi\)
0.319744 + 0.947504i \(0.396403\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 33.9411 1.90034
\(320\) 0 0
\(321\) 5.65685 0.315735
\(322\) 0 0
\(323\) −22.6274 −1.25902
\(324\) 0 0
\(325\) 28.2843 1.56893
\(326\) 0 0
\(327\) 6.00000 0.331801
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 22.6274 1.24372 0.621858 0.783130i \(-0.286378\pi\)
0.621858 + 0.783130i \(0.286378\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −45.2548 −2.45069
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.65685 0.303676 0.151838 0.988405i \(-0.451481\pi\)
0.151838 + 0.988405i \(0.451481\pi\)
\(348\) 0 0
\(349\) 5.65685 0.302804 0.151402 0.988472i \(-0.451621\pi\)
0.151402 + 0.988472i \(0.451621\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 0 0
\(353\) 28.2843 1.50542 0.752710 0.658352i \(-0.228746\pi\)
0.752710 + 0.658352i \(0.228746\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −21.0000 −1.10221
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) 5.65685 0.294484
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 33.9411 1.74806
\(378\) 0 0
\(379\) −22.6274 −1.16229 −0.581146 0.813799i \(-0.697396\pi\)
−0.581146 + 0.813799i \(0.697396\pi\)
\(380\) 0 0
\(381\) 11.3137 0.579619
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −16.9706 −0.851728 −0.425864 0.904787i \(-0.640030\pi\)
−0.425864 + 0.904787i \(0.640030\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −45.2548 −2.25430
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.3137 −0.560800
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 28.2843 1.37199
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −32.0000 −1.54497
\(430\) 0 0
\(431\) −16.9706 −0.817443 −0.408722 0.912659i \(-0.634025\pi\)
−0.408722 + 0.912659i \(0.634025\pi\)
\(432\) 0 0
\(433\) 11.3137 0.543702 0.271851 0.962339i \(-0.412364\pi\)
0.271851 + 0.962339i \(0.412364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −22.6274 −1.08242
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.9706 −0.806296 −0.403148 0.915135i \(-0.632084\pi\)
−0.403148 + 0.915135i \(0.632084\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.00000 0.0945968
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −32.0000 −1.50682
\(452\) 0 0
\(453\) 22.6274 1.06313
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 0 0
\(459\) 5.65685 0.264039
\(460\) 0 0
\(461\) 22.6274 1.05386 0.526932 0.849907i \(-0.323342\pi\)
0.526932 + 0.849907i \(0.323342\pi\)
\(462\) 0 0
\(463\) 33.9411 1.57738 0.788689 0.614792i \(-0.210760\pi\)
0.788689 + 0.614792i \(0.210760\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 16.9706 0.781962
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −20.0000 −0.917663
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −11.3137 −0.515861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 11.3137 0.511624
\(490\) 0 0
\(491\) 28.2843 1.27645 0.638226 0.769849i \(-0.279669\pi\)
0.638226 + 0.769849i \(0.279669\pi\)
\(492\) 0 0
\(493\) 33.9411 1.52863
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −22.6274 −1.01294 −0.506471 0.862257i \(-0.669050\pi\)
−0.506471 + 0.862257i \(0.669050\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −19.0000 −0.843820
\(508\) 0 0
\(509\) −11.3137 −0.501471 −0.250736 0.968056i \(-0.580672\pi\)
−0.250736 + 0.968056i \(0.580672\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −45.2548 −1.99031
\(518\) 0 0
\(519\) −22.6274 −0.993233
\(520\) 0 0
\(521\) −28.2843 −1.23916 −0.619578 0.784935i \(-0.712696\pi\)
−0.619578 + 0.784935i \(0.712696\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −45.2548 −1.97133
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −32.0000 −1.38607
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.65685 −0.244111
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) −16.9706 −0.728277
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −33.9411 −1.45122 −0.725609 0.688107i \(-0.758442\pi\)
−0.725609 + 0.688107i \(0.758442\pi\)
\(548\) 0 0
\(549\) −5.65685 −0.241429
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −32.0000 −1.35104
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 33.9411 1.42039 0.710196 0.704004i \(-0.248606\pi\)
0.710196 + 0.704004i \(0.248606\pi\)
\(572\) 0 0
\(573\) −5.65685 −0.236318
\(574\) 0 0
\(575\) 28.2843 1.17954
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.3137 0.468566
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) 5.65685 0.232299 0.116150 0.993232i \(-0.462945\pi\)
0.116150 + 0.993232i \(0.462945\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) −16.9706 −0.693398 −0.346699 0.937976i \(-0.612698\pi\)
−0.346699 + 0.937976i \(0.612698\pi\)
\(600\) 0 0
\(601\) −45.2548 −1.84598 −0.922992 0.384820i \(-0.874263\pi\)
−0.922992 + 0.384820i \(0.874263\pi\)
\(602\) 0 0
\(603\) 11.3137 0.460730
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −45.2548 −1.83081
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) 5.65685 0.227002
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 22.6274 0.903652
\(628\) 0 0
\(629\) −11.3137 −0.451107
\(630\) 0 0
\(631\) 11.3137 0.450392 0.225196 0.974314i \(-0.427698\pi\)
0.225196 + 0.974314i \(0.427698\pi\)
\(632\) 0 0
\(633\) −22.6274 −0.899359
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 5.65685 0.223782
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) −22.6274 −0.888204
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 11.3137 0.441390
\(658\) 0 0
\(659\) 39.5980 1.54252 0.771259 0.636521i \(-0.219627\pi\)
0.771259 + 0.636521i \(0.219627\pi\)
\(660\) 0 0
\(661\) −50.9117 −1.98024 −0.990118 0.140240i \(-0.955213\pi\)
−0.990118 + 0.140240i \(0.955213\pi\)
\(662\) 0 0
\(663\) −32.0000 −1.24278
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 33.9411 1.31421
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 32.0000 1.23535
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) 45.2548 1.73928 0.869642 0.493682i \(-0.164349\pi\)
0.869642 + 0.493682i \(0.164349\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 5.65685 0.216454 0.108227 0.994126i \(-0.465483\pi\)
0.108227 + 0.994126i \(0.465483\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −16.9706 −0.647467
\(688\) 0 0
\(689\) 11.3137 0.431018
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −32.0000 −1.21209
\(698\) 0 0
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) 11.3137 0.424297
\(712\) 0 0
\(713\) −45.2548 −1.69481
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.65685 0.211259
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 22.6274 0.841523
\(724\) 0 0
\(725\) 30.0000 1.11417
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 5.65685 0.208941 0.104470 0.994528i \(-0.466685\pi\)
0.104470 + 0.994528i \(0.466685\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −64.0000 −2.35747
\(738\) 0 0
\(739\) −11.3137 −0.416181 −0.208091 0.978110i \(-0.566725\pi\)
−0.208091 + 0.978110i \(0.566725\pi\)
\(740\) 0 0
\(741\) 22.6274 0.831239
\(742\) 0 0
\(743\) −39.5980 −1.45271 −0.726354 0.687320i \(-0.758787\pi\)
−0.726354 + 0.687320i \(0.758787\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 22.6274 0.825686 0.412843 0.910802i \(-0.364536\pi\)
0.412843 + 0.910802i \(0.364536\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) 0 0
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 5.65685 0.205061 0.102530 0.994730i \(-0.467306\pi\)
0.102530 + 0.994730i \(0.467306\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) 33.9411 1.22395 0.611974 0.790878i \(-0.290376\pi\)
0.611974 + 0.790878i \(0.290376\pi\)
\(770\) 0 0
\(771\) 5.65685 0.203727
\(772\) 0 0
\(773\) 11.3137 0.406926 0.203463 0.979083i \(-0.434780\pi\)
0.203463 + 0.979083i \(0.434780\pi\)
\(774\) 0 0
\(775\) −40.0000 −1.43684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 22.6274 0.810711
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) 0 0
\(789\) −16.9706 −0.604168
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 32.0000 1.13635
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.6274 0.801504 0.400752 0.916187i \(-0.368749\pi\)
0.400752 + 0.916187i \(0.368749\pi\)
\(798\) 0 0
\(799\) −45.2548 −1.60100
\(800\) 0 0
\(801\) −5.65685 −0.199875
\(802\) 0 0
\(803\) −64.0000 −2.25851
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 11.3137 0.398261
\(808\) 0 0
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 33.9411 1.18311 0.591557 0.806263i \(-0.298514\pi\)
0.591557 + 0.806263i \(0.298514\pi\)
\(824\) 0 0
\(825\) −28.2843 −0.984732
\(826\) 0 0
\(827\) −28.2843 −0.983540 −0.491770 0.870725i \(-0.663650\pi\)
−0.491770 + 0.870725i \(0.663650\pi\)
\(828\) 0 0
\(829\) −16.9706 −0.589412 −0.294706 0.955588i \(-0.595222\pi\)
−0.294706 + 0.955588i \(0.595222\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) 56.0000 1.93333 0.966667 0.256036i \(-0.0824164\pi\)
0.966667 + 0.256036i \(0.0824164\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 10.0000 0.344418
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −11.3137 −0.387829
\(852\) 0 0
\(853\) 50.9117 1.74318 0.871592 0.490233i \(-0.163088\pi\)
0.871592 + 0.490233i \(0.163088\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.2843 0.966172 0.483086 0.875573i \(-0.339516\pi\)
0.483086 + 0.875573i \(0.339516\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.9706 0.577685 0.288842 0.957377i \(-0.406730\pi\)
0.288842 + 0.957377i \(0.406730\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −15.0000 −0.509427
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) −64.0000 −2.16856
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) 11.3137 0.381602
\(880\) 0 0
\(881\) 5.65685 0.190584 0.0952921 0.995449i \(-0.469621\pi\)
0.0952921 + 0.995449i \(0.469621\pi\)
\(882\) 0 0
\(883\) 22.6274 0.761473 0.380737 0.924684i \(-0.375670\pi\)
0.380737 + 0.924684i \(0.375670\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40.0000 −1.34307 −0.671534 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5.65685 −0.189512
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −32.0000 −1.06845
\(898\) 0 0
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 11.3137 0.376914
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 11.3137 0.375252
\(910\) 0 0
\(911\) −16.9706 −0.562260 −0.281130 0.959670i \(-0.590709\pi\)
−0.281130 + 0.959670i \(0.590709\pi\)
\(912\) 0 0
\(913\) 67.8823 2.24657
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −22.6274 −0.746410 −0.373205 0.927749i \(-0.621741\pi\)
−0.373205 + 0.927749i \(0.621741\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −39.5980 −1.29917 −0.649584 0.760290i \(-0.725057\pi\)
−0.649584 + 0.760290i \(0.725057\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.6274 0.739205 0.369603 0.929190i \(-0.379494\pi\)
0.369603 + 0.929190i \(0.379494\pi\)
\(938\) 0 0
\(939\) −11.3137 −0.369209
\(940\) 0 0
\(941\) 45.2548 1.47527 0.737633 0.675202i \(-0.235944\pi\)
0.737633 + 0.675202i \(0.235944\pi\)
\(942\) 0 0
\(943\) −32.0000 −1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.9706 0.551469 0.275735 0.961234i \(-0.411079\pi\)
0.275735 + 0.961234i \(0.411079\pi\)
\(948\) 0 0
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −33.9411 −1.09716
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −5.65685 −0.182290
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 33.9411 1.09147 0.545737 0.837957i \(-0.316250\pi\)
0.545737 + 0.837957i \(0.316250\pi\)
\(968\) 0 0
\(969\) 22.6274 0.726897
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −28.2843 −0.905822
\(976\) 0 0
\(977\) −34.0000 −1.08776 −0.543878 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) 0 0
\(979\) 32.0000 1.02272
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) −8.00000 −0.255160 −0.127580 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −22.6274 −0.718784 −0.359392 0.933187i \(-0.617016\pi\)
−0.359392 + 0.933187i \(0.617016\pi\)
\(992\) 0 0
\(993\) −22.6274 −0.718059
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −5.65685 −0.179154 −0.0895772 0.995980i \(-0.528552\pi\)
−0.0895772 + 0.995980i \(0.528552\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 4704.2.a.bk.1.1 2
4.3 odd 2 4704.2.a.bp.1.2 yes 2
7.6 odd 2 4704.2.a.bp.1.1 yes 2
8.3 odd 2 9408.2.a.dm.1.1 2
8.5 even 2 9408.2.a.dz.1.2 2
28.27 even 2 inner 4704.2.a.bk.1.2 yes 2
56.13 odd 2 9408.2.a.dm.1.2 2
56.27 even 2 9408.2.a.dz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bk.1.1 2 1.1 even 1 trivial
4704.2.a.bk.1.2 yes 2 28.27 even 2 inner
4704.2.a.bp.1.1 yes 2 7.6 odd 2
4704.2.a.bp.1.2 yes 2 4.3 odd 2
9408.2.a.dm.1.1 2 8.3 odd 2
9408.2.a.dm.1.2 2 56.13 odd 2
9408.2.a.dz.1.1 2 56.27 even 2
9408.2.a.dz.1.2 2 8.5 even 2