Properties

Label 3200.2.d.o.1601.1
Level $3200$
Weight $2$
Character 3200.1601
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1601.1
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 3200.1601
Dual form 3200.2.d.o.1601.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607i q^{3} -2.00000 q^{9} +O(q^{10})\) \(q-2.23607i q^{3} -2.00000 q^{9} +2.23607i q^{11} -4.00000i q^{13} -3.00000 q^{17} -2.23607i q^{19} -8.94427 q^{23} -2.23607i q^{27} +4.00000i q^{29} +8.94427 q^{31} +5.00000 q^{33} +8.00000i q^{37} -8.94427 q^{39} -5.00000 q^{41} -8.94427i q^{43} -8.94427 q^{47} -7.00000 q^{49} +6.70820i q^{51} +4.00000i q^{53} -5.00000 q^{57} -8.94427i q^{59} +8.00000i q^{61} -6.70820i q^{67} +20.0000i q^{69} -8.94427 q^{71} +9.00000 q^{73} -11.0000 q^{81} -6.70820i q^{83} +8.94427 q^{87} -15.0000 q^{89} -20.0000i q^{93} -2.00000 q^{97} -4.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9} + O(q^{10}) \) \( 4 q - 8 q^{9} - 12 q^{17} + 20 q^{33} - 20 q^{41} - 28 q^{49} - 20 q^{57} + 36 q^{73} - 44 q^{81} - 60 q^{89} - 8 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 2.23607i − 1.29099i −0.763763 0.645497i \(-0.776650\pi\)
0.763763 0.645497i \(-0.223350\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 2.23607i 0.674200i 0.941469 + 0.337100i \(0.109446\pi\)
−0.941469 + 0.337100i \(0.890554\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) − 2.23607i − 0.512989i −0.966546 0.256495i \(-0.917432\pi\)
0.966546 0.256495i \(-0.0825676\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.94427 −1.86501 −0.932505 0.361158i \(-0.882382\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 2.23607i − 0.430331i
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 8.94427 1.60644 0.803219 0.595683i \(-0.203119\pi\)
0.803219 + 0.595683i \(0.203119\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) −8.94427 −1.43223
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) − 8.94427i − 1.36399i −0.731357 0.681994i \(-0.761113\pi\)
0.731357 0.681994i \(-0.238887\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.94427 −1.30466 −0.652328 0.757937i \(-0.726208\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 6.70820i 0.939336i
\(52\) 0 0
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) − 8.94427i − 1.16445i −0.813029 0.582223i \(-0.802183\pi\)
0.813029 0.582223i \(-0.197817\pi\)
\(60\) 0 0
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.70820i − 0.819538i −0.912189 0.409769i \(-0.865609\pi\)
0.912189 0.409769i \(-0.134391\pi\)
\(68\) 0 0
\(69\) 20.0000i 2.40772i
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 6.70820i − 0.736321i −0.929762 0.368161i \(-0.879988\pi\)
0.929762 0.368161i \(-0.120012\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.94427 0.958927
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 20.0000i − 2.07390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) − 4.47214i − 0.449467i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −8.94427 −0.881305 −0.440653 0.897678i \(-0.645253\pi\)
−0.440653 + 0.897678i \(0.645253\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.23607i 0.216169i 0.994142 + 0.108084i \(0.0344717\pi\)
−0.994142 + 0.108084i \(0.965528\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 17.8885 1.69791
\(112\) 0 0
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.00000i 0.739600i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.00000 0.545455
\(122\) 0 0
\(123\) 11.1803i 1.00810i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.94427 −0.793676 −0.396838 0.917889i \(-0.629892\pi\)
−0.396838 + 0.917889i \(0.629892\pi\)
\(128\) 0 0
\(129\) −20.0000 −1.76090
\(130\) 0 0
\(131\) − 8.94427i − 0.781465i −0.920504 0.390732i \(-0.872222\pi\)
0.920504 0.390732i \(-0.127778\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) 20.1246i 1.70695i 0.521136 + 0.853474i \(0.325508\pi\)
−0.521136 + 0.853474i \(0.674492\pi\)
\(140\) 0 0
\(141\) 20.0000i 1.68430i
\(142\) 0 0
\(143\) 8.94427 0.747958
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 15.6525i 1.29099i
\(148\) 0 0
\(149\) 20.0000i 1.63846i 0.573462 + 0.819232i \(0.305600\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −17.8885 −1.45575 −0.727875 0.685710i \(-0.759492\pi\)
−0.727875 + 0.685710i \(0.759492\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.0000i 0.957704i 0.877896 + 0.478852i \(0.158947\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 8.94427 0.709327
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.1803i 0.875712i 0.899045 + 0.437856i \(0.144262\pi\)
−0.899045 + 0.437856i \(0.855738\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.94427 0.692129 0.346064 0.938211i \(-0.387518\pi\)
0.346064 + 0.938211i \(0.387518\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 4.47214i 0.341993i
\(172\) 0 0
\(173\) − 24.0000i − 1.82469i −0.409426 0.912343i \(-0.634271\pi\)
0.409426 0.912343i \(-0.365729\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −20.0000 −1.50329
\(178\) 0 0
\(179\) 15.6525i 1.16992i 0.811062 + 0.584960i \(0.198890\pi\)
−0.811062 + 0.584960i \(0.801110\pi\)
\(180\) 0 0
\(181\) − 20.0000i − 1.48659i −0.668965 0.743294i \(-0.733262\pi\)
0.668965 0.743294i \(-0.266738\pi\)
\(182\) 0 0
\(183\) 17.8885 1.32236
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.70820i − 0.490552i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.94427 −0.647185 −0.323592 0.946197i \(-0.604891\pi\)
−0.323592 + 0.946197i \(0.604891\pi\)
\(192\) 0 0
\(193\) 21.0000 1.51161 0.755807 0.654795i \(-0.227245\pi\)
0.755807 + 0.654795i \(0.227245\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) 17.8885 1.26809 0.634043 0.773298i \(-0.281394\pi\)
0.634043 + 0.773298i \(0.281394\pi\)
\(200\) 0 0
\(201\) −15.0000 −1.05802
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 17.8885 1.24334
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) − 6.70820i − 0.461812i −0.972976 0.230906i \(-0.925831\pi\)
0.972976 0.230906i \(-0.0741690\pi\)
\(212\) 0 0
\(213\) 20.0000i 1.37038i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 20.1246i − 1.35990i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 17.8885 1.19791 0.598953 0.800784i \(-0.295584\pi\)
0.598953 + 0.800784i \(0.295584\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 8.94427i − 0.593652i −0.954932 0.296826i \(-0.904072\pi\)
0.954932 0.296826i \(-0.0959282\pi\)
\(228\) 0 0
\(229\) 16.0000i 1.05731i 0.848837 + 0.528655i \(0.177303\pi\)
−0.848837 + 0.528655i \(0.822697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.94427 0.578557 0.289278 0.957245i \(-0.406585\pi\)
0.289278 + 0.957245i \(0.406585\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 17.8885i 1.14755i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.94427 −0.569110
\(248\) 0 0
\(249\) −15.0000 −0.950586
\(250\) 0 0
\(251\) − 29.0689i − 1.83481i −0.397953 0.917406i \(-0.630279\pi\)
0.397953 0.917406i \(-0.369721\pi\)
\(252\) 0 0
\(253\) − 20.0000i − 1.25739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) − 8.00000i − 0.495188i
\(262\) 0 0
\(263\) 8.94427 0.551527 0.275764 0.961225i \(-0.411069\pi\)
0.275764 + 0.961225i \(0.411069\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 33.5410i 2.05268i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −26.8328 −1.62998 −0.814989 0.579477i \(-0.803257\pi\)
−0.814989 + 0.579477i \(0.803257\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 12.0000i 0.721010i 0.932757 + 0.360505i \(0.117396\pi\)
−0.932757 + 0.360505i \(0.882604\pi\)
\(278\) 0 0
\(279\) −17.8885 −1.07096
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 24.5967i 1.46212i 0.682311 + 0.731062i \(0.260975\pi\)
−0.682311 + 0.731062i \(0.739025\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 4.47214i 0.262161i
\(292\) 0 0
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 35.7771i 2.06904i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.23607i − 0.127619i −0.997962 0.0638096i \(-0.979675\pi\)
0.997962 0.0638096i \(-0.0203250\pi\)
\(308\) 0 0
\(309\) 20.0000i 1.13776i
\(310\) 0 0
\(311\) 17.8885 1.01437 0.507183 0.861838i \(-0.330687\pi\)
0.507183 + 0.861838i \(0.330687\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 32.0000i − 1.79730i −0.438667 0.898650i \(-0.644549\pi\)
0.438667 0.898650i \(-0.355451\pi\)
\(318\) 0 0
\(319\) −8.94427 −0.500783
\(320\) 0 0
\(321\) 5.00000 0.279073
\(322\) 0 0
\(323\) 6.70820i 0.373254i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.23607i 0.122905i 0.998110 + 0.0614527i \(0.0195733\pi\)
−0.998110 + 0.0614527i \(0.980427\pi\)
\(332\) 0 0
\(333\) − 16.0000i − 0.876795i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −33.0000 −1.79762 −0.898812 0.438334i \(-0.855569\pi\)
−0.898812 + 0.438334i \(0.855569\pi\)
\(338\) 0 0
\(339\) 2.23607i 0.121447i
\(340\) 0 0
\(341\) 20.0000i 1.08306i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 33.5410i − 1.80058i −0.435294 0.900288i \(-0.643356\pi\)
0.435294 0.900288i \(-0.356644\pi\)
\(348\) 0 0
\(349\) − 24.0000i − 1.28469i −0.766415 0.642345i \(-0.777962\pi\)
0.766415 0.642345i \(-0.222038\pi\)
\(350\) 0 0
\(351\) −8.94427 −0.477410
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.8885 0.944121 0.472061 0.881566i \(-0.343510\pi\)
0.472061 + 0.881566i \(0.343510\pi\)
\(360\) 0 0
\(361\) 14.0000 0.736842
\(362\) 0 0
\(363\) − 13.4164i − 0.704179i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) − 29.0689i − 1.49317i −0.665291 0.746584i \(-0.731693\pi\)
0.665291 0.746584i \(-0.268307\pi\)
\(380\) 0 0
\(381\) 20.0000i 1.02463i
\(382\) 0 0
\(383\) −17.8885 −0.914062 −0.457031 0.889451i \(-0.651087\pi\)
−0.457031 + 0.889451i \(0.651087\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 17.8885i 0.909326i
\(388\) 0 0
\(389\) − 20.0000i − 1.01404i −0.861934 0.507020i \(-0.830747\pi\)
0.861934 0.507020i \(-0.169253\pi\)
\(390\) 0 0
\(391\) 26.8328 1.35699
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 8.00000i − 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) − 35.7771i − 1.78218i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −17.8885 −0.886702
\(408\) 0 0
\(409\) 19.0000 0.939490 0.469745 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 0 0
\(411\) 29.0689i 1.43386i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 45.0000 2.20366
\(418\) 0 0
\(419\) − 20.1246i − 0.983152i −0.870835 0.491576i \(-0.836421\pi\)
0.870835 0.491576i \(-0.163579\pi\)
\(420\) 0 0
\(421\) − 32.0000i − 1.55958i −0.626038 0.779792i \(-0.715325\pi\)
0.626038 0.779792i \(-0.284675\pi\)
\(422\) 0 0
\(423\) 17.8885 0.869771
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) − 20.0000i − 0.965609i
\(430\) 0 0
\(431\) −17.8885 −0.861661 −0.430830 0.902433i \(-0.641779\pi\)
−0.430830 + 0.902433i \(0.641779\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.0000i 0.956730i
\(438\) 0 0
\(439\) 26.8328 1.28066 0.640330 0.768100i \(-0.278798\pi\)
0.640330 + 0.768100i \(0.278798\pi\)
\(440\) 0 0
\(441\) 14.0000 0.666667
\(442\) 0 0
\(443\) − 11.1803i − 0.531194i −0.964084 0.265597i \(-0.914431\pi\)
0.964084 0.265597i \(-0.0855691\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 44.7214 2.11525
\(448\) 0 0
\(449\) −11.0000 −0.519122 −0.259561 0.965727i \(-0.583578\pi\)
−0.259561 + 0.965727i \(0.583578\pi\)
\(450\) 0 0
\(451\) − 11.1803i − 0.526462i
\(452\) 0 0
\(453\) 40.0000i 1.87936i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 0 0
\(459\) 6.70820i 0.313112i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −35.7771 −1.66270 −0.831351 0.555748i \(-0.812432\pi\)
−0.831351 + 0.555748i \(0.812432\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 8.94427i − 0.413892i −0.978352 0.206946i \(-0.933648\pi\)
0.978352 0.206946i \(-0.0663524\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 26.8328 1.23639
\(472\) 0 0
\(473\) 20.0000 0.919601
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 8.00000i − 0.366295i
\(478\) 0 0
\(479\) −26.8328 −1.22602 −0.613011 0.790074i \(-0.710042\pi\)
−0.613011 + 0.790074i \(0.710042\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −26.8328 −1.21591 −0.607955 0.793971i \(-0.708010\pi\)
−0.607955 + 0.793971i \(0.708010\pi\)
\(488\) 0 0
\(489\) 25.0000 1.13054
\(490\) 0 0
\(491\) 26.8328i 1.21095i 0.795865 + 0.605474i \(0.207016\pi\)
−0.795865 + 0.605474i \(0.792984\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 26.8328i 1.20120i 0.799549 + 0.600601i \(0.205072\pi\)
−0.799549 + 0.600601i \(0.794928\pi\)
\(500\) 0 0
\(501\) − 20.0000i − 0.893534i
\(502\) 0 0
\(503\) −17.8885 −0.797611 −0.398805 0.917036i \(-0.630575\pi\)
−0.398805 + 0.917036i \(0.630575\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.70820i 0.297922i
\(508\) 0 0
\(509\) − 4.00000i − 0.177297i −0.996063 0.0886484i \(-0.971745\pi\)
0.996063 0.0886484i \(-0.0282548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 20.0000i − 0.879599i
\(518\) 0 0
\(519\) −53.6656 −2.35566
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) − 33.5410i − 1.46665i −0.679880 0.733323i \(-0.737968\pi\)
0.679880 0.733323i \(-0.262032\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26.8328 −1.16886
\(528\) 0 0
\(529\) 57.0000 2.47826
\(530\) 0 0
\(531\) 17.8885i 0.776297i
\(532\) 0 0
\(533\) 20.0000i 0.866296i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 35.0000 1.51036
\(538\) 0 0
\(539\) − 15.6525i − 0.674200i
\(540\) 0 0
\(541\) − 40.0000i − 1.71973i −0.510518 0.859867i \(-0.670546\pi\)
0.510518 0.859867i \(-0.329454\pi\)
\(542\) 0 0
\(543\) −44.7214 −1.91918
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.1246i − 0.860466i −0.902718 0.430233i \(-0.858431\pi\)
0.902718 0.430233i \(-0.141569\pi\)
\(548\) 0 0
\(549\) − 16.0000i − 0.682863i
\(550\) 0 0
\(551\) 8.94427 0.381039
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 12.0000i − 0.508456i −0.967144 0.254228i \(-0.918179\pi\)
0.967144 0.254228i \(-0.0818214\pi\)
\(558\) 0 0
\(559\) −35.7771 −1.51321
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) − 8.94427i − 0.376956i −0.982077 0.188478i \(-0.939645\pi\)
0.982077 0.188478i \(-0.0603554\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.0000 0.461144 0.230572 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(570\) 0 0
\(571\) − 44.7214i − 1.87153i −0.352623 0.935765i \(-0.614710\pi\)
0.352623 0.935765i \(-0.385290\pi\)
\(572\) 0 0
\(573\) 20.0000i 0.835512i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 23.0000 0.957503 0.478751 0.877951i \(-0.341090\pi\)
0.478751 + 0.877951i \(0.341090\pi\)
\(578\) 0 0
\(579\) − 46.9574i − 1.95148i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −8.94427 −0.370434
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.70820i 0.276877i 0.990371 + 0.138439i \(0.0442084\pi\)
−0.990371 + 0.138439i \(0.955792\pi\)
\(588\) 0 0
\(589\) − 20.0000i − 0.824086i
\(590\) 0 0
\(591\) 26.8328 1.10375
\(592\) 0 0
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 40.0000i − 1.63709i
\(598\) 0 0
\(599\) 35.7771 1.46181 0.730906 0.682478i \(-0.239098\pi\)
0.730906 + 0.682478i \(0.239098\pi\)
\(600\) 0 0
\(601\) 25.0000 1.01977 0.509886 0.860242i \(-0.329688\pi\)
0.509886 + 0.860242i \(0.329688\pi\)
\(602\) 0 0
\(603\) 13.4164i 0.546358i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −17.8885 −0.726074 −0.363037 0.931775i \(-0.618260\pi\)
−0.363037 + 0.931775i \(0.618260\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35.7771i 1.44739i
\(612\) 0 0
\(613\) − 4.00000i − 0.161558i −0.996732 0.0807792i \(-0.974259\pi\)
0.996732 0.0807792i \(-0.0257409\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) − 8.94427i − 0.359501i −0.983712 0.179750i \(-0.942471\pi\)
0.983712 0.179750i \(-0.0575290\pi\)
\(620\) 0 0
\(621\) 20.0000i 0.802572i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 11.1803i − 0.446500i
\(628\) 0 0
\(629\) − 24.0000i − 0.956943i
\(630\) 0 0
\(631\) 8.94427 0.356066 0.178033 0.984025i \(-0.443027\pi\)
0.178033 + 0.984025i \(0.443027\pi\)
\(632\) 0 0
\(633\) −15.0000 −0.596196
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 28.0000i 1.10940i
\(638\) 0 0
\(639\) 17.8885 0.707660
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) − 8.94427i − 0.352728i −0.984325 0.176364i \(-0.943566\pi\)
0.984325 0.176364i \(-0.0564335\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17.8885 −0.703271 −0.351636 0.936137i \(-0.614374\pi\)
−0.351636 + 0.936137i \(0.614374\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −18.0000 −0.702247
\(658\) 0 0
\(659\) − 6.70820i − 0.261315i −0.991428 0.130657i \(-0.958291\pi\)
0.991428 0.130657i \(-0.0417087\pi\)
\(660\) 0 0
\(661\) − 28.0000i − 1.08907i −0.838737 0.544537i \(-0.816705\pi\)
0.838737 0.544537i \(-0.183295\pi\)
\(662\) 0 0
\(663\) 26.8328 1.04210
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 35.7771i − 1.38529i
\(668\) 0 0
\(669\) − 40.0000i − 1.54649i
\(670\) 0 0
\(671\) −17.8885 −0.690580
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 8.00000i − 0.307465i −0.988113 0.153732i \(-0.950871\pi\)
0.988113 0.153732i \(-0.0491294\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) − 15.6525i − 0.598925i −0.954108 0.299463i \(-0.903193\pi\)
0.954108 0.299463i \(-0.0968074\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 35.7771 1.36498
\(688\) 0 0
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) 15.6525i 0.595448i 0.954652 + 0.297724i \(0.0962276\pi\)
−0.954652 + 0.297724i \(0.903772\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15.0000 0.568166
\(698\) 0 0
\(699\) − 13.4164i − 0.507455i
\(700\) 0 0
\(701\) 12.0000i 0.453234i 0.973984 + 0.226617i \(0.0727665\pi\)
−0.973984 + 0.226617i \(0.927233\pi\)
\(702\) 0 0
\(703\) 17.8885 0.674679
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 16.0000i − 0.600893i −0.953799 0.300446i \(-0.902864\pi\)
0.953799 0.300446i \(-0.0971356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −80.0000 −2.99602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 20.0000i − 0.746914i
\(718\) 0 0
\(719\) −17.8885 −0.667130 −0.333565 0.942727i \(-0.608252\pi\)
−0.333565 + 0.942727i \(0.608252\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 11.1803i − 0.415801i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 26.8328 0.995174 0.497587 0.867414i \(-0.334220\pi\)
0.497587 + 0.867414i \(0.334220\pi\)
\(728\) 0 0
\(729\) 7.00000 0.259259
\(730\) 0 0
\(731\) 26.8328i 0.992448i
\(732\) 0 0
\(733\) − 4.00000i − 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 15.0000 0.552532
\(738\) 0 0
\(739\) 26.8328i 0.987061i 0.869728 + 0.493531i \(0.164294\pi\)
−0.869728 + 0.493531i \(0.835706\pi\)
\(740\) 0 0
\(741\) 20.0000i 0.734718i
\(742\) 0 0
\(743\) −8.94427 −0.328134 −0.164067 0.986449i \(-0.552461\pi\)
−0.164067 + 0.986449i \(0.552461\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 13.4164i 0.490881i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −26.8328 −0.979143 −0.489572 0.871963i \(-0.662847\pi\)
−0.489572 + 0.871963i \(0.662847\pi\)
\(752\) 0 0
\(753\) −65.0000 −2.36873
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 8.00000i − 0.290765i −0.989376 0.145382i \(-0.953559\pi\)
0.989376 0.145382i \(-0.0464413\pi\)
\(758\) 0 0
\(759\) −44.7214 −1.62328
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −35.7771 −1.29184
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) − 40.2492i − 1.44954i
\(772\) 0 0
\(773\) − 4.00000i − 0.143870i −0.997409 0.0719350i \(-0.977083\pi\)
0.997409 0.0719350i \(-0.0229174\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 11.1803i 0.400577i
\(780\) 0 0
\(781\) − 20.0000i − 0.715656i
\(782\) 0 0
\(783\) 8.94427 0.319642
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 26.8328i 0.956487i 0.878227 + 0.478243i \(0.158726\pi\)
−0.878227 + 0.478243i \(0.841274\pi\)
\(788\) 0 0
\(789\) − 20.0000i − 0.712019i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 32.0000 1.13635
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 52.0000i − 1.84193i −0.389640 0.920967i \(-0.627401\pi\)
0.389640 0.920967i \(-0.372599\pi\)
\(798\) 0 0
\(799\) 26.8328 0.949277
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) 0 0
\(803\) 20.1246i 0.710182i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 26.8328i 0.942228i 0.882072 + 0.471114i \(0.156148\pi\)
−0.882072 + 0.471114i \(0.843852\pi\)
\(812\) 0 0
\(813\) 60.0000i 2.10429i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28.0000i 0.977207i 0.872506 + 0.488603i \(0.162493\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) −35.7771 −1.24711 −0.623555 0.781779i \(-0.714312\pi\)
−0.623555 + 0.781779i \(0.714312\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 24.5967i 0.855313i 0.903941 + 0.427656i \(0.140661\pi\)
−0.903941 + 0.427656i \(0.859339\pi\)
\(828\) 0 0
\(829\) − 20.0000i − 0.694629i −0.937749 0.347314i \(-0.887094\pi\)
0.937749 0.347314i \(-0.112906\pi\)
\(830\) 0 0
\(831\) 26.8328 0.930820
\(832\) 0 0
\(833\) 21.0000 0.727607
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 20.0000i − 0.691301i
\(838\) 0 0
\(839\) 35.7771 1.23516 0.617581 0.786507i \(-0.288113\pi\)
0.617581 + 0.786507i \(0.288113\pi\)
\(840\) 0 0
\(841\) 13.0000 0.448276
\(842\) 0 0
\(843\) 22.3607i 0.770143i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 55.0000 1.88760
\(850\) 0 0
\(851\) − 71.5542i − 2.45285i
\(852\) 0 0
\(853\) − 4.00000i − 0.136957i −0.997653 0.0684787i \(-0.978185\pi\)
0.997653 0.0684787i \(-0.0218145\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.0000 0.922302 0.461151 0.887322i \(-0.347437\pi\)
0.461151 + 0.887322i \(0.347437\pi\)
\(858\) 0 0
\(859\) 38.0132i 1.29699i 0.761218 + 0.648496i \(0.224602\pi\)
−0.761218 + 0.648496i \(0.775398\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 26.8328 0.913400 0.456700 0.889621i \(-0.349031\pi\)
0.456700 + 0.889621i \(0.349031\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 17.8885i 0.607527i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −26.8328 −0.909195
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) 0 0
\(879\) 53.6656 1.81010
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 33.5410i 1.12875i 0.825520 + 0.564373i \(0.190882\pi\)
−0.825520 + 0.564373i \(0.809118\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 17.8885 0.600639 0.300319 0.953839i \(-0.402907\pi\)
0.300319 + 0.953839i \(0.402907\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 24.5967i − 0.824022i
\(892\) 0 0
\(893\) 20.0000i 0.669274i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 80.0000 2.67112
\(898\) 0 0
\(899\) 35.7771i 1.19323i
\(900\) 0 0
\(901\) − 12.0000i − 0.399778i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 8.94427i − 0.296990i −0.988913 0.148495i \(-0.952557\pi\)
0.988913 0.148495i \(-0.0474428\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 35.7771 1.18535 0.592674 0.805443i \(-0.298072\pi\)
0.592674 + 0.805443i \(0.298072\pi\)
\(912\) 0 0
\(913\) 15.0000 0.496428
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.94427 −0.295044 −0.147522 0.989059i \(-0.547130\pi\)
−0.147522 + 0.989059i \(0.547130\pi\)
\(920\) 0 0
\(921\) −5.00000 −0.164756
\(922\) 0 0
\(923\) 35.7771i 1.17762i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 17.8885 0.587537
\(928\) 0 0
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 15.6525i 0.512989i
\(932\) 0 0
\(933\) − 40.0000i − 1.30954i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −23.0000 −0.751377 −0.375689 0.926746i \(-0.622594\pi\)
−0.375689 + 0.926746i \(0.622594\pi\)
\(938\) 0 0
\(939\) 13.4164i 0.437828i
\(940\) 0 0
\(941\) − 60.0000i − 1.95594i −0.208736 0.977972i \(-0.566935\pi\)
0.208736 0.977972i \(-0.433065\pi\)
\(942\) 0 0
\(943\) 44.7214 1.45633
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 8.94427i − 0.290650i −0.989384 0.145325i \(-0.953577\pi\)
0.989384 0.145325i \(-0.0464227\pi\)
\(948\) 0 0
\(949\) − 36.0000i − 1.16861i
\(950\) 0 0
\(951\) −71.5542 −2.32030
\(952\) 0 0
\(953\) −21.0000 −0.680257 −0.340128 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.0000i 0.646508i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 49.0000 1.58065
\(962\) 0 0
\(963\) − 4.47214i − 0.144113i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 53.6656 1.72577 0.862885 0.505400i \(-0.168655\pi\)
0.862885 + 0.505400i \(0.168655\pi\)
\(968\) 0 0
\(969\) 15.0000 0.481869
\(970\) 0 0
\(971\) − 51.4296i − 1.65045i −0.564802 0.825227i \(-0.691047\pi\)
0.564802 0.825227i \(-0.308953\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.0000 −0.863807 −0.431903 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) 0 0
\(979\) − 33.5410i − 1.07198i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 17.8885 0.570556 0.285278 0.958445i \(-0.407914\pi\)
0.285278 + 0.958445i \(0.407914\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 80.0000i 2.54385i
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 5.00000 0.158670
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.00000i 0.253363i 0.991943 + 0.126681i \(0.0404325\pi\)
−0.991943 + 0.126681i \(0.959567\pi\)
\(998\) 0 0
\(999\) 17.8885 0.565968
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 3200.2.d.o.1601.1 4
4.3 odd 2 inner 3200.2.d.o.1601.3 yes 4
5.2 odd 4 3200.2.f.m.449.2 4
5.3 odd 4 3200.2.f.n.449.4 4
5.4 even 2 3200.2.d.p.1601.4 yes 4
8.3 odd 2 inner 3200.2.d.o.1601.2 yes 4
8.5 even 2 inner 3200.2.d.o.1601.4 yes 4
16.3 odd 4 6400.2.a.bs.1.1 2
16.5 even 4 6400.2.a.bq.1.1 2
16.11 odd 4 6400.2.a.bq.1.2 2
16.13 even 4 6400.2.a.bs.1.2 2
20.3 even 4 3200.2.f.n.449.1 4
20.7 even 4 3200.2.f.m.449.3 4
20.19 odd 2 3200.2.d.p.1601.2 yes 4
40.3 even 4 3200.2.f.m.449.4 4
40.13 odd 4 3200.2.f.m.449.1 4
40.19 odd 2 3200.2.d.p.1601.3 yes 4
40.27 even 4 3200.2.f.n.449.2 4
40.29 even 2 3200.2.d.p.1601.1 yes 4
40.37 odd 4 3200.2.f.n.449.3 4
80.19 odd 4 6400.2.a.br.1.2 2
80.29 even 4 6400.2.a.br.1.1 2
80.59 odd 4 6400.2.a.bt.1.1 2
80.69 even 4 6400.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.o.1601.1 4 1.1 even 1 trivial
3200.2.d.o.1601.2 yes 4 8.3 odd 2 inner
3200.2.d.o.1601.3 yes 4 4.3 odd 2 inner
3200.2.d.o.1601.4 yes 4 8.5 even 2 inner
3200.2.d.p.1601.1 yes 4 40.29 even 2
3200.2.d.p.1601.2 yes 4 20.19 odd 2
3200.2.d.p.1601.3 yes 4 40.19 odd 2
3200.2.d.p.1601.4 yes 4 5.4 even 2
3200.2.f.m.449.1 4 40.13 odd 4
3200.2.f.m.449.2 4 5.2 odd 4
3200.2.f.m.449.3 4 20.7 even 4
3200.2.f.m.449.4 4 40.3 even 4
3200.2.f.n.449.1 4 20.3 even 4
3200.2.f.n.449.2 4 40.27 even 4
3200.2.f.n.449.3 4 40.37 odd 4
3200.2.f.n.449.4 4 5.3 odd 4
6400.2.a.bq.1.1 2 16.5 even 4
6400.2.a.bq.1.2 2 16.11 odd 4
6400.2.a.br.1.1 2 80.29 even 4
6400.2.a.br.1.2 2 80.19 odd 4
6400.2.a.bs.1.1 2 16.3 odd 4
6400.2.a.bs.1.2 2 16.13 even 4
6400.2.a.bt.1.1 2 80.59 odd 4
6400.2.a.bt.1.2 2 80.69 even 4