Properties

Label 4800.2.f.bj.3649.1
Level $4800$
Weight $2$
Character 4800.3649
Analytic conductor $38.328$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.3281929702\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.3649
Dual form 4800.2.f.bj.3649.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +5.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +5.00000i q^{7} -1.00000 q^{9} +6.00000 q^{11} +3.00000i q^{13} +2.00000i q^{17} +1.00000 q^{19} +5.00000 q^{21} -2.00000i q^{23} +1.00000i q^{27} +6.00000 q^{29} +3.00000 q^{31} -6.00000i q^{33} -6.00000i q^{37} +3.00000 q^{39} +4.00000 q^{41} -11.0000i q^{43} +10.0000i q^{47} -18.0000 q^{49} +2.00000 q^{51} +8.00000i q^{53} -1.00000i q^{57} -6.00000 q^{59} -3.00000 q^{61} -5.00000i q^{63} -1.00000i q^{67} -2.00000 q^{69} -12.0000 q^{71} +10.0000i q^{73} +30.0000i q^{77} +8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -6.00000i q^{87} +16.0000 q^{89} -15.0000 q^{91} -3.00000i q^{93} +7.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 12q^{11} + 2q^{19} + 10q^{21} + 12q^{29} + 6q^{31} + 6q^{39} + 8q^{41} - 36q^{49} + 4q^{51} - 12q^{59} - 6q^{61} - 4q^{69} - 24q^{71} + 16q^{79} + 2q^{81} + 32q^{89} - 30q^{91} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 5.00000i 1.88982i 0.327327 + 0.944911i \(0.393852\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 3.00000i 0.832050i 0.909353 + 0.416025i \(0.136577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 0 0
\(23\) − 2.00000i − 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) − 6.00000i − 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) − 11.0000i − 1.67748i −0.544529 0.838742i \(-0.683292\pi\)
0.544529 0.838742i \(-0.316708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0000i 1.45865i 0.684167 + 0.729325i \(0.260166\pi\)
−0.684167 + 0.729325i \(0.739834\pi\)
\(48\) 0 0
\(49\) −18.0000 −2.57143
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 8.00000i 1.09888i 0.835532 + 0.549442i \(0.185160\pi\)
−0.835532 + 0.549442i \(0.814840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 0.132453i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 0 0
\(63\) − 5.00000i − 0.629941i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 1.00000i − 0.122169i −0.998133 0.0610847i \(-0.980544\pi\)
0.998133 0.0610847i \(-0.0194560\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 30.0000i 3.41882i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) −15.0000 −1.57243
\(92\) 0 0
\(93\) − 3.00000i − 0.311086i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000i 0.710742i 0.934725 + 0.355371i \(0.115646\pi\)
−0.934725 + 0.355371i \(0.884354\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.00000i − 0.277350i
\(118\) 0 0
\(119\) −10.0000 −0.916698
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) − 4.00000i − 0.360668i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 0 0
\(129\) −11.0000 −0.968496
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) 5.00000i 0.433555i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 0 0
\(143\) 18.0000i 1.50524i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 18.0000i 1.48461i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 0 0
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.00000i 0.558661i 0.960195 + 0.279330i \(0.0901125\pi\)
−0.960195 + 0.279330i \(0.909888\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) 10.0000 0.788110
\(162\) 0 0
\(163\) 7.00000i 0.548282i 0.961689 + 0.274141i \(0.0883936\pi\)
−0.961689 + 0.274141i \(0.911606\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) 0 0
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) 19.0000 1.41226 0.706129 0.708083i \(-0.250440\pi\)
0.706129 + 0.708083i \(0.250440\pi\)
\(182\) 0 0
\(183\) 3.00000i 0.221766i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 3.00000i 0.215945i 0.994154 + 0.107972i \(0.0344358\pi\)
−0.994154 + 0.107972i \(0.965564\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) −21.0000 −1.48865 −0.744325 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(200\) 0 0
\(201\) −1.00000 −0.0705346
\(202\) 0 0
\(203\) 30.0000i 2.10559i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.00000i 0.139010i
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 0 0
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 15.0000i 1.01827i
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 3.00000i 0.200895i 0.994942 + 0.100447i \(0.0320274\pi\)
−0.994942 + 0.100447i \(0.967973\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) −3.00000 −0.198246 −0.0991228 0.995075i \(-0.531604\pi\)
−0.0991228 + 0.995075i \(0.531604\pi\)
\(230\) 0 0
\(231\) 30.0000 1.97386
\(232\) 0 0
\(233\) 20.0000i 1.31024i 0.755523 + 0.655122i \(0.227383\pi\)
−0.755523 + 0.655122i \(0.772617\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.00000i 0.190885i
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) − 12.0000i − 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 0 0
\(259\) 30.0000 1.86411
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 16.0000i − 0.979184i
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 15.0000i 0.907841i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.00000i 0.0600842i 0.999549 + 0.0300421i \(0.00956413\pi\)
−0.999549 + 0.0300421i \(0.990436\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) − 13.0000i − 0.772770i −0.922338 0.386385i \(-0.873724\pi\)
0.922338 0.386385i \(-0.126276\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 20.0000i 1.18056i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 7.00000 0.410347
\(292\) 0 0
\(293\) 2.00000i 0.116841i 0.998292 + 0.0584206i \(0.0186065\pi\)
−0.998292 + 0.0584206i \(0.981394\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 6.00000i 0.348155i
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 55.0000 3.17015
\(302\) 0 0
\(303\) − 8.00000i − 0.459588i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 13.0000i 0.741949i 0.928643 + 0.370975i \(0.120976\pi\)
−0.928643 + 0.370975i \(0.879024\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) 0 0
\(313\) − 29.0000i − 1.63918i −0.572953 0.819588i \(-0.694202\pi\)
0.572953 0.819588i \(-0.305798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 16.0000i − 0.898650i −0.893368 0.449325i \(-0.851665\pi\)
0.893368 0.449325i \(-0.148335\pi\)
\(318\) 0 0
\(319\) 36.0000 2.01561
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 2.00000i 0.111283i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 7.00000i 0.387101i
\(328\) 0 0
\(329\) −50.0000 −2.75659
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000i 1.25289i 0.779466 + 0.626445i \(0.215491\pi\)
−0.779466 + 0.626445i \(0.784509\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 18.0000 0.974755
\(342\) 0 0
\(343\) − 55.0000i − 2.96972i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 18.0000i − 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 10.0000i 0.529256i
\(358\) 0 0
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) − 25.0000i − 1.31216i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 13.0000i − 0.678594i −0.940679 0.339297i \(-0.889811\pi\)
0.940679 0.339297i \(-0.110189\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) −40.0000 −2.07670
\(372\) 0 0
\(373\) 25.0000i 1.29445i 0.762299 + 0.647225i \(0.224071\pi\)
−0.762299 + 0.647225i \(0.775929\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0000i 0.927047i
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 20.0000i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.0000i 0.559161i
\(388\) 0 0
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 16.0000i 0.807093i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.0000i 0.552074i 0.961147 + 0.276037i \(0.0890213\pi\)
−0.961147 + 0.276037i \(0.910979\pi\)
\(398\) 0 0
\(399\) 5.00000 0.250313
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 9.00000i 0.448322i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 36.0000i − 1.78445i
\(408\) 0 0
\(409\) 11.0000 0.543915 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) − 30.0000i − 1.47620i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) 32.0000 1.56330 0.781651 0.623716i \(-0.214378\pi\)
0.781651 + 0.623716i \(0.214378\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 0 0
\(423\) − 10.0000i − 0.486217i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 15.0000i − 0.725901i
\(428\) 0 0
\(429\) 18.0000 0.869048
\(430\) 0 0
\(431\) 22.0000 1.05970 0.529851 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(432\) 0 0
\(433\) 19.0000i 0.913082i 0.889702 + 0.456541i \(0.150912\pi\)
−0.889702 + 0.456541i \(0.849088\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2.00000i − 0.0956730i
\(438\) 0 0
\(439\) 5.00000 0.238637 0.119318 0.992856i \(-0.461929\pi\)
0.119318 + 0.992856i \(0.461929\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) − 9.00000i − 0.422857i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 16.0000 0.745194 0.372597 0.927993i \(-0.378467\pi\)
0.372597 + 0.927993i \(0.378467\pi\)
\(462\) 0 0
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 14.0000i − 0.647843i −0.946084 0.323921i \(-0.894999\pi\)
0.946084 0.323921i \(-0.105001\pi\)
\(468\) 0 0
\(469\) 5.00000 0.230879
\(470\) 0 0
\(471\) 7.00000 0.322543
\(472\) 0 0
\(473\) − 66.0000i − 3.03468i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 8.00000i − 0.366295i
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) − 10.0000i − 0.455016i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.00000i 0.135943i 0.997687 + 0.0679715i \(0.0216527\pi\)
−0.997687 + 0.0679715i \(0.978347\pi\)
\(488\) 0 0
\(489\) 7.00000 0.316551
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 60.0000i − 2.69137i
\(498\) 0 0
\(499\) 17.0000 0.761025 0.380512 0.924776i \(-0.375748\pi\)
0.380512 + 0.924776i \(0.375748\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 0 0
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 4.00000i − 0.177646i
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −50.0000 −2.21187
\(512\) 0 0
\(513\) 1.00000i 0.0441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 60.0000i 2.63880i
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) − 21.0000i − 0.918266i −0.888368 0.459133i \(-0.848160\pi\)
0.888368 0.459133i \(-0.151840\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000i 0.261364i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 2.00000i − 0.0863064i
\(538\) 0 0
\(539\) −108.000 −4.65189
\(540\) 0 0
\(541\) −17.0000 −0.730887 −0.365444 0.930834i \(-0.619083\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 0 0
\(543\) − 19.0000i − 0.815368i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 16.0000i 0.684111i 0.939680 + 0.342055i \(0.111123\pi\)
−0.939680 + 0.342055i \(0.888877\pi\)
\(548\) 0 0
\(549\) 3.00000 0.128037
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 40.0000i 1.70097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 14.0000i − 0.593199i −0.955002 0.296600i \(-0.904147\pi\)
0.955002 0.296600i \(-0.0958526\pi\)
\(558\) 0 0
\(559\) 33.0000 1.39575
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 18.0000i 0.758610i 0.925272 + 0.379305i \(0.123837\pi\)
−0.925272 + 0.379305i \(0.876163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 5.00000i 0.209980i
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −1.00000 −0.0418487 −0.0209243 0.999781i \(-0.506661\pi\)
−0.0209243 + 0.999781i \(0.506661\pi\)
\(572\) 0 0
\(573\) 10.0000i 0.417756i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 43.0000i − 1.79011i −0.445952 0.895057i \(-0.647135\pi\)
0.445952 0.895057i \(-0.352865\pi\)
\(578\) 0 0
\(579\) 3.00000 0.124676
\(580\) 0 0
\(581\) −30.0000 −1.24461
\(582\) 0 0
\(583\) 48.0000i 1.98796i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.0000i 0.990586i 0.868726 + 0.495293i \(0.164939\pi\)
−0.868726 + 0.495293i \(0.835061\pi\)
\(588\) 0 0
\(589\) 3.00000 0.123613
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) 4.00000i 0.164260i 0.996622 + 0.0821302i \(0.0261723\pi\)
−0.996622 + 0.0821302i \(0.973828\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 21.0000i 0.859473i
\(598\) 0 0
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) 1.00000i 0.0407231i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 30.0000 1.21566
\(610\) 0 0
\(611\) −30.0000 −1.21367
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.0000i 1.28827i 0.764911 + 0.644136i \(0.222783\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) −19.0000 −0.763674 −0.381837 0.924230i \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) 0 0
\(623\) 80.0000i 3.20513i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 6.00000i − 0.239617i
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 31.0000 1.23409 0.617045 0.786928i \(-0.288330\pi\)
0.617045 + 0.786928i \(0.288330\pi\)
\(632\) 0 0
\(633\) − 15.0000i − 0.596196i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 54.0000i − 2.13956i
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 12.0000i − 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 15.0000 0.587896
\(652\) 0 0
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 0 0
\(663\) 6.00000i 0.233021i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 12.0000i − 0.464642i
\(668\) 0 0
\(669\) 3.00000 0.115987
\(670\) 0 0
\(671\) −18.0000 −0.694882
\(672\) 0 0
\(673\) − 34.0000i − 1.31060i −0.755367 0.655302i \(-0.772541\pi\)
0.755367 0.655302i \(-0.227459\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 0 0
\(679\) −35.0000 −1.34318
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 3.00000i 0.114457i
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 0 0
\(693\) − 30.0000i − 1.13961i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.00000i 0.303022i
\(698\) 0 0
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 0 0
\(703\) − 6.00000i − 0.226294i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 40.0000i 1.50435i
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) − 6.00000i − 0.224702i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.0000i 0.896296i
\(718\) 0 0
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) −20.0000 −0.744839
\(722\) 0 0
\(723\) 7.00000i 0.260333i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 13.0000i 0.482143i 0.970507 + 0.241072i \(0.0774989\pi\)
−0.970507 + 0.241072i \(0.922501\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 22.0000 0.813699
\(732\) 0 0
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 6.00000i − 0.221013i
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 3.00000 0.110208
\(742\) 0 0
\(743\) − 12.0000i − 0.440237i −0.975473 0.220119i \(-0.929356\pi\)
0.975473 0.220119i \(-0.0706445\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) − 20.0000i − 0.728841i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 3.00000i − 0.109037i −0.998513 0.0545184i \(-0.982638\pi\)
0.998513 0.0545184i \(-0.0173624\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −52.0000 −1.88500 −0.942499 0.334208i \(-0.891531\pi\)
−0.942499 + 0.334208i \(0.891531\pi\)
\(762\) 0 0
\(763\) − 35.0000i − 1.26709i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 18.0000i − 0.649942i
\(768\) 0 0
\(769\) −27.0000 −0.973645 −0.486822 0.873501i \(-0.661844\pi\)
−0.486822 + 0.873501i \(0.661844\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 0 0
\(773\) − 28.0000i − 1.00709i −0.863969 0.503545i \(-0.832029\pi\)
0.863969 0.503545i \(-0.167971\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 30.0000i − 1.07624i
\(778\) 0 0
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) −72.0000 −2.57636
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 3.00000i − 0.106938i −0.998569 0.0534692i \(-0.982972\pi\)
0.998569 0.0534692i \(-0.0170279\pi\)
\(788\) 0 0
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) −60.0000 −2.13335
\(792\) 0 0
\(793\) − 9.00000i − 0.319599i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.0000i 0.991811i 0.868377 + 0.495905i \(0.165164\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(798\) 0 0
\(799\) −20.0000 −0.707549
\(800\) 0 0
\(801\) −16.0000 −0.565332
\(802\) 0 0
\(803\) 60.0000i 2.11735i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 30.0000i 1.05605i
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) −9.00000 −0.316033 −0.158016 0.987436i \(-0.550510\pi\)
−0.158016 + 0.987436i \(0.550510\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 11.0000i − 0.384841i
\(818\) 0 0
\(819\) 15.0000 0.524142
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) − 49.0000i − 1.70803i −0.520246 0.854016i \(-0.674160\pi\)
0.520246 0.854016i \(-0.325840\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 32.0000i − 1.11275i −0.830932 0.556375i \(-0.812192\pi\)
0.830932 0.556375i \(-0.187808\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) 1.00000 0.0346896
\(832\) 0 0
\(833\) − 36.0000i − 1.24733i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.00000i 0.103695i
\(838\) 0 0
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) − 26.0000i − 0.895488i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 125.000i 4.29505i
\(848\) 0 0
\(849\) −13.0000 −0.446159
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) − 55.0000i − 1.88316i −0.336784 0.941582i \(-0.609339\pi\)
0.336784 0.941582i \(-0.390661\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 52.0000i 1.77629i 0.459567 + 0.888143i \(0.348005\pi\)
−0.459567 + 0.888143i \(0.651995\pi\)
\(858\) 0 0
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) 0 0
\(863\) − 8.00000i − 0.272323i −0.990687 0.136162i \(-0.956523\pi\)
0.990687 0.136162i \(-0.0434766\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 0 0
\(873\) − 7.00000i − 0.236914i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 17.0000i − 0.574049i −0.957923 0.287025i \(-0.907334\pi\)
0.957923 0.287025i \(-0.0926662\pi\)
\(878\) 0 0
\(879\) 2.00000 0.0674583
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) − 19.0000i − 0.639401i −0.947519 0.319700i \(-0.896418\pi\)
0.947519 0.319700i \(-0.103582\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 42.0000i − 1.41022i −0.709097 0.705111i \(-0.750897\pi\)
0.709097 0.705111i \(-0.249103\pi\)
\(888\) 0 0
\(889\) 40.0000 1.34156
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) 0 0
\(893\) 10.0000i 0.334637i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 6.00000i − 0.200334i
\(898\) 0 0
\(899\) 18.0000 0.600334
\(900\) 0 0
\(901\) −16.0000 −0.533037
\(902\) 0 0
\(903\) − 55.0000i − 1.83029i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 44.0000i − 1.46100i −0.682915 0.730498i \(-0.739288\pi\)
0.682915 0.730498i \(-0.260712\pi\)
\(908\) 0 0
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) −14.0000 −0.463841 −0.231920 0.972735i \(-0.574501\pi\)
−0.231920 + 0.972735i \(0.574501\pi\)
\(912\) 0 0
\(913\) 36.0000i 1.19143i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 80.0000i − 2.64183i
\(918\) 0 0
\(919\) −1.00000 −0.0329870 −0.0164935 0.999864i \(-0.505250\pi\)
−0.0164935 + 0.999864i \(0.505250\pi\)
\(920\) 0 0
\(921\) 13.0000 0.428365
\(922\) 0 0
\(923\) − 36.0000i − 1.18495i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.00000i − 0.131377i
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 0 0
\(933\) 14.0000i 0.458339i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 23.0000i − 0.751377i −0.926746 0.375689i \(-0.877406\pi\)
0.926746 0.375689i \(-0.122594\pi\)
\(938\) 0 0
\(939\) −29.0000 −0.946379
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) − 8.00000i − 0.260516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 2.00000i − 0.0649913i −0.999472 0.0324956i \(-0.989654\pi\)
0.999472 0.0324956i \(-0.0103455\pi\)
\(948\) 0 0
\(949\) −30.0000 −0.973841
\(950\) 0 0
\(951\) −16.0000 −0.518836
\(952\) 0 0
\(953\) 24.0000i 0.777436i 0.921357 + 0.388718i \(0.127082\pi\)
−0.921357 + 0.388718i \(0.872918\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 36.0000i − 1.16371i
\(958\) 0 0
\(959\) 30.0000 0.968751
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 8.00000i 0.257796i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 56.0000i − 1.80084i −0.435023 0.900419i \(-0.643260\pi\)
0.435023 0.900419i \(-0.356740\pi\)
\(968\) 0 0
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) −46.0000 −1.47621 −0.738105 0.674686i \(-0.764279\pi\)
−0.738105 + 0.674686i \(0.764279\pi\)
\(972\) 0 0
\(973\) − 60.0000i − 1.92351i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 10.0000i − 0.319928i −0.987123 0.159964i \(-0.948862\pi\)
0.987123 0.159964i \(-0.0511379\pi\)
\(978\) 0 0
\(979\) 96.0000 3.06817
\(980\) 0 0
\(981\) 7.00000 0.223493
\(982\) 0 0
\(983\) − 36.0000i − 1.14822i −0.818778 0.574111i \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 50.0000i 1.59152i
\(988\) 0 0
\(989\) −22.0000 −0.699559
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 0 0
\(993\) − 20.0000i − 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 54.0000i − 1.71020i −0.518465 0.855099i \(-0.673497\pi\)
0.518465 0.855099i \(-0.326503\pi\)
\(998\) 0 0
\(999\) 6.00000 0.189832
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 4800.2.f.bj.3649.1 2
4.3 odd 2 4800.2.f.a.3649.2 2
5.2 odd 4 4800.2.a.b.1.1 1
5.3 odd 4 4800.2.a.ct.1.1 1
5.4 even 2 inner 4800.2.f.bj.3649.2 2
8.3 odd 2 1200.2.f.i.49.1 2
8.5 even 2 600.2.f.a.49.2 2
20.3 even 4 4800.2.a.a.1.1 1
20.7 even 4 4800.2.a.cs.1.1 1
20.19 odd 2 4800.2.f.a.3649.1 2
24.5 odd 2 1800.2.f.k.649.2 2
24.11 even 2 3600.2.f.b.2449.1 2
40.3 even 4 1200.2.a.j.1.1 1
40.13 odd 4 600.2.a.e.1.1 1
40.19 odd 2 1200.2.f.i.49.2 2
40.27 even 4 1200.2.a.i.1.1 1
40.29 even 2 600.2.f.a.49.1 2
40.37 odd 4 600.2.a.f.1.1 yes 1
120.29 odd 2 1800.2.f.k.649.1 2
120.53 even 4 1800.2.a.x.1.1 1
120.59 even 2 3600.2.f.b.2449.2 2
120.77 even 4 1800.2.a.a.1.1 1
120.83 odd 4 3600.2.a.a.1.1 1
120.107 odd 4 3600.2.a.bq.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.a.e.1.1 1 40.13 odd 4
600.2.a.f.1.1 yes 1 40.37 odd 4
600.2.f.a.49.1 2 40.29 even 2
600.2.f.a.49.2 2 8.5 even 2
1200.2.a.i.1.1 1 40.27 even 4
1200.2.a.j.1.1 1 40.3 even 4
1200.2.f.i.49.1 2 8.3 odd 2
1200.2.f.i.49.2 2 40.19 odd 2
1800.2.a.a.1.1 1 120.77 even 4
1800.2.a.x.1.1 1 120.53 even 4
1800.2.f.k.649.1 2 120.29 odd 2
1800.2.f.k.649.2 2 24.5 odd 2
3600.2.a.a.1.1 1 120.83 odd 4
3600.2.a.bq.1.1 1 120.107 odd 4
3600.2.f.b.2449.1 2 24.11 even 2
3600.2.f.b.2449.2 2 120.59 even 2
4800.2.a.a.1.1 1 20.3 even 4
4800.2.a.b.1.1 1 5.2 odd 4
4800.2.a.cs.1.1 1 20.7 even 4
4800.2.a.ct.1.1 1 5.3 odd 4
4800.2.f.a.3649.1 2 20.19 odd 2
4800.2.f.a.3649.2 2 4.3 odd 2
4800.2.f.bj.3649.1 2 1.1 even 1 trivial
4800.2.f.bj.3649.2 2 5.4 even 2 inner