Properties

Label 3800.2.d.d.3649.1
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.d.3649.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} -3.00000 q^{11} -4.00000i q^{13} -5.00000i q^{17} +1.00000 q^{19} +6.00000 q^{21} -4.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} +6.00000i q^{33} +10.0000i q^{37} -8.00000 q^{39} +6.00000 q^{41} -7.00000i q^{43} +9.00000i q^{47} -2.00000 q^{49} -10.0000 q^{51} -8.00000i q^{53} -2.00000i q^{57} -14.0000 q^{59} -5.00000 q^{61} -3.00000i q^{63} -6.00000 q^{71} -15.0000i q^{73} -9.00000i q^{77} +4.00000 q^{79} -11.0000 q^{81} +4.00000i q^{83} +4.00000i q^{87} +12.0000 q^{91} -16.0000i q^{93} -16.0000i q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{9} - 6 q^{11} + 2 q^{19} + 12 q^{21} - 4 q^{29} + 16 q^{31} - 16 q^{39} + 12 q^{41} - 4 q^{49} - 20 q^{51} - 28 q^{59} - 10 q^{61} - 12 q^{71} + 8 q^{79} - 22 q^{81} + 24 q^{91} + 6 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 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
\(17\) − 5.00000i − 1.21268i −0.795206 0.606339i \(-0.792637\pi\)
0.795206 0.606339i \(-0.207363\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 6.00000i 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 7.00000i − 1.06749i −0.845645 0.533745i \(-0.820784\pi\)
0.845645 0.533745i \(-0.179216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −10.0000 −1.40028
\(52\) 0 0
\(53\) − 8.00000i − 1.09888i −0.835532 0.549442i \(-0.814840\pi\)
0.835532 0.549442i \(-0.185160\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) − 3.00000i − 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) − 15.0000i − 1.75562i −0.479012 0.877809i \(-0.659005\pi\)
0.479012 0.877809i \(-0.340995\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 9.00000i − 1.02565i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.00000i 0.428845i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) − 16.0000i − 1.65912i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 16.0000i − 1.62455i −0.583272 0.812277i \(-0.698228\pi\)
0.583272 0.812277i \(-0.301772\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 10.0000i − 0.966736i −0.875417 0.483368i \(-0.839413\pi\)
0.875417 0.483368i \(-0.160587\pi\)
\(108\) 0 0
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 0 0
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000i 0.369800i
\(118\) 0 0
\(119\) 15.0000 1.37505
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) − 12.0000i − 1.08200i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.00000i 0.532414i 0.963916 + 0.266207i \(0.0857705\pi\)
−0.963916 + 0.266207i \(0.914230\pi\)
\(128\) 0 0
\(129\) −14.0000 −1.23263
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 21.0000i − 1.79415i −0.441877 0.897076i \(-0.645687\pi\)
0.441877 0.897076i \(-0.354313\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 18.0000 1.51587
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.00000i 0.329914i
\(148\) 0 0
\(149\) −17.0000 −1.39269 −0.696347 0.717705i \(-0.745193\pi\)
−0.696347 + 0.717705i \(0.745193\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 5.00000i 0.404226i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) −16.0000 −1.26888
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000i 0.154765i 0.997001 + 0.0773823i \(0.0246562\pi\)
−0.997001 + 0.0773823i \(0.975344\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 28.0000i 2.10461i
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 10.0000i 0.739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.0000i 1.09691i
\(188\) 0 0
\(189\) 12.0000 0.872872
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 0 0
\(193\) − 24.0000i − 1.72756i −0.503871 0.863779i \(-0.668091\pi\)
0.503871 0.863779i \(-0.331909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 15.0000 1.06332 0.531661 0.846957i \(-0.321568\pi\)
0.531661 + 0.846957i \(0.321568\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) 0 0
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.0000i 1.62923i
\(218\) 0 0
\(219\) −30.0000 −2.02721
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) 0 0
\(223\) 22.0000i 1.47323i 0.676313 + 0.736614i \(0.263577\pi\)
−0.676313 + 0.736614i \(0.736423\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 0 0
\(231\) −18.0000 −1.18431
\(232\) 0 0
\(233\) − 13.0000i − 0.851658i −0.904804 0.425829i \(-0.859982\pi\)
0.904804 0.425829i \(-0.140018\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.00000i − 0.254514i
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −13.0000 −0.820553 −0.410276 0.911961i \(-0.634568\pi\)
−0.410276 + 0.911961i \(0.634568\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0000i 1.49708i 0.663090 + 0.748539i \(0.269245\pi\)
−0.663090 + 0.748539i \(0.730755\pi\)
\(258\) 0 0
\(259\) −30.0000 −1.86411
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) − 5.00000i − 0.308313i −0.988046 0.154157i \(-0.950734\pi\)
0.988046 0.154157i \(-0.0492660\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) − 24.0000i − 1.45255i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 9.00000i − 0.540758i −0.962754 0.270379i \(-0.912851\pi\)
0.962754 0.270379i \(-0.0871489\pi\)
\(278\) 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
\(283\) 13.0000i 0.772770i 0.922338 + 0.386385i \(0.126276\pi\)
−0.922338 + 0.386385i \(0.873724\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.0000i 1.06251i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −32.0000 −1.87587
\(292\) 0 0
\(293\) − 4.00000i − 0.233682i −0.993151 0.116841i \(-0.962723\pi\)
0.993151 0.116841i \(-0.0372769\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 12.0000i 0.696311i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 21.0000 1.21042
\(302\) 0 0
\(303\) 36.0000i 2.06815i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 0 0
\(309\) −28.0000 −1.59286
\(310\) 0 0
\(311\) 31.0000 1.75785 0.878924 0.476961i \(-0.158262\pi\)
0.878924 + 0.476961i \(0.158262\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) − 5.00000i − 0.278207i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 24.0000i 1.32720i
\(328\) 0 0
\(329\) −27.0000 −1.48856
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) − 10.0000i − 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 4.00000 0.217250
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 27.0000i − 1.44944i −0.689046 0.724718i \(-0.741970\pi\)
0.689046 0.724718i \(-0.258030\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) 2.00000i 0.106449i 0.998583 + 0.0532246i \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 30.0000i − 1.58777i
\(358\) 0 0
\(359\) 11.0000 0.580558 0.290279 0.956942i \(-0.406252\pi\)
0.290279 + 0.956942i \(0.406252\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 4.00000i 0.209946i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 16.0000i − 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000i 0.412021i
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.00000i 0.355830i
\(388\) 0 0
\(389\) 29.0000 1.47036 0.735179 0.677873i \(-0.237098\pi\)
0.735179 + 0.677873i \(0.237098\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 13.0000i − 0.652451i −0.945292 0.326226i \(-0.894223\pi\)
0.945292 0.326226i \(-0.105777\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) − 32.0000i − 1.59403i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 30.0000i − 1.48704i
\(408\) 0 0
\(409\) 8.00000 0.395575 0.197787 0.980245i \(-0.436624\pi\)
0.197787 + 0.980245i \(0.436624\pi\)
\(410\) 0 0
\(411\) −42.0000 −2.07171
\(412\) 0 0
\(413\) − 42.0000i − 2.06668i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 10.0000i 0.489702i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) − 9.00000i − 0.437595i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 15.0000i − 0.725901i
\(428\) 0 0
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 19.0000i 0.902717i 0.892343 + 0.451359i \(0.149060\pi\)
−0.892343 + 0.451359i \(0.850940\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 34.0000i 1.60814i
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) − 4.00000i − 0.187936i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.0000i 1.35656i 0.734802 + 0.678281i \(0.237275\pi\)
−0.734802 + 0.678281i \(0.762725\pi\)
\(458\) 0 0
\(459\) −20.0000 −0.933520
\(460\) 0 0
\(461\) 21.0000 0.978068 0.489034 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) 0 0
\(463\) − 37.0000i − 1.71954i −0.510685 0.859768i \(-0.670608\pi\)
0.510685 0.859768i \(-0.329392\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.0000i 0.601568i 0.953692 + 0.300784i \(0.0972484\pi\)
−0.953692 + 0.300784i \(0.902752\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) 21.0000i 0.965581i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.00000i 0.366295i
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 40.0000 1.82384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 10.0000i 0.450377i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 18.0000i − 0.807410i
\(498\) 0 0
\(499\) 29.0000 1.29822 0.649109 0.760695i \(-0.275142\pi\)
0.649109 + 0.760695i \(0.275142\pi\)
\(500\) 0 0
\(501\) 4.00000 0.178707
\(502\) 0 0
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.00000i 0.266469i
\(508\) 0 0
\(509\) 8.00000 0.354594 0.177297 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(510\) 0 0
\(511\) 45.0000 1.99068
\(512\) 0 0
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 27.0000i − 1.18746i
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) 0 0
\(523\) 26.0000i 1.13690i 0.822718 + 0.568450i \(0.192457\pi\)
−0.822718 + 0.568450i \(0.807543\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 40.0000i − 1.74243i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 14.0000 0.607548
\(532\) 0 0
\(533\) − 24.0000i − 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 36.0000i − 1.55351i
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 19.0000 0.816874 0.408437 0.912787i \(-0.366074\pi\)
0.408437 + 0.912787i \(0.366074\pi\)
\(542\) 0 0
\(543\) − 4.00000i − 0.171656i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) −2.00000 −0.0852029
\(552\) 0 0
\(553\) 12.0000i 0.510292i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.00000i 0.296600i 0.988942 + 0.148300i \(0.0473800\pi\)
−0.988942 + 0.148300i \(0.952620\pi\)
\(558\) 0 0
\(559\) −28.0000 −1.18427
\(560\) 0 0
\(561\) 30.0000 1.26660
\(562\) 0 0
\(563\) 14.0000i 0.590030i 0.955493 + 0.295015i \(0.0953246\pi\)
−0.955493 + 0.295015i \(0.904675\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 33.0000i − 1.38587i
\(568\) 0 0
\(569\) −8.00000 −0.335377 −0.167689 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 30.0000i 1.25327i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 37.0000i 1.54033i 0.637845 + 0.770165i \(0.279826\pi\)
−0.637845 + 0.770165i \(0.720174\pi\)
\(578\) 0 0
\(579\) −48.0000 −1.99481
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 24.0000i 0.993978i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.0000i 1.19696i 0.801138 + 0.598479i \(0.204228\pi\)
−0.801138 + 0.598479i \(0.795772\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 30.0000i − 1.22782i
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) 0 0
\(613\) − 31.0000i − 1.25208i −0.779792 0.626039i \(-0.784675\pi\)
0.779792 0.626039i \(-0.215325\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 33.0000i − 1.32853i −0.747497 0.664265i \(-0.768745\pi\)
0.747497 0.664265i \(-0.231255\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.00000i 0.239617i
\(628\) 0 0
\(629\) 50.0000 1.99363
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.00000i 0.316972i
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) − 43.0000i − 1.69575i −0.530193 0.847877i \(-0.677880\pi\)
0.530193 0.847877i \(-0.322120\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 49.0000i − 1.92639i −0.268806 0.963194i \(-0.586629\pi\)
0.268806 0.963194i \(-0.413371\pi\)
\(648\) 0 0
\(649\) 42.0000 1.64864
\(650\) 0 0
\(651\) 48.0000 1.88127
\(652\) 0 0
\(653\) 21.0000i 0.821794i 0.911682 + 0.410897i \(0.134784\pi\)
−0.911682 + 0.410897i \(0.865216\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 15.0000i 0.585206i
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 0 0
\(663\) 40.0000i 1.55347i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 44.0000 1.70114
\(670\) 0 0
\(671\) 15.0000 0.579069
\(672\) 0 0
\(673\) 6.00000i 0.231283i 0.993291 + 0.115642i \(0.0368924\pi\)
−0.993291 + 0.115642i \(0.963108\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 38.0000i 1.46046i 0.683202 + 0.730229i \(0.260587\pi\)
−0.683202 + 0.730229i \(0.739413\pi\)
\(678\) 0 0
\(679\) 48.0000 1.84207
\(680\) 0 0
\(681\) 40.0000 1.53280
\(682\) 0 0
\(683\) − 28.0000i − 1.07139i −0.844411 0.535695i \(-0.820050\pi\)
0.844411 0.535695i \(-0.179950\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 0 0
\(689\) −32.0000 −1.21910
\(690\) 0 0
\(691\) 15.0000 0.570627 0.285313 0.958434i \(-0.407902\pi\)
0.285313 + 0.958434i \(0.407902\pi\)
\(692\) 0 0
\(693\) 9.00000i 0.341882i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 30.0000i − 1.13633i
\(698\) 0 0
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 10.0000i 0.377157i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 54.0000i − 2.03088i
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 6.00000i − 0.224074i
\(718\) 0 0
\(719\) −29.0000 −1.08152 −0.540759 0.841178i \(-0.681863\pi\)
−0.540759 + 0.841178i \(0.681863\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 0 0
\(723\) − 36.0000i − 1.33885i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 7.00000i − 0.259616i −0.991539 0.129808i \(-0.958564\pi\)
0.991539 0.129808i \(-0.0414360\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −35.0000 −1.29452
\(732\) 0 0
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) 30.0000 1.09618
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 26.0000i 0.947493i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 37.0000i 1.34479i 0.740193 + 0.672394i \(0.234734\pi\)
−0.740193 + 0.672394i \(0.765266\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.00000 −0.253750 −0.126875 0.991919i \(-0.540495\pi\)
−0.126875 + 0.991919i \(0.540495\pi\)
\(762\) 0 0
\(763\) − 36.0000i − 1.30329i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 56.0000i 2.02204i
\(768\) 0 0
\(769\) 1.00000 0.0360609 0.0180305 0.999837i \(-0.494260\pi\)
0.0180305 + 0.999837i \(0.494260\pi\)
\(770\) 0 0
\(771\) 48.0000 1.72868
\(772\) 0 0
\(773\) 34.0000i 1.22290i 0.791285 + 0.611448i \(0.209412\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 60.0000i 2.15249i
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 8.00000i 0.285897i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) 0 0
\(789\) −10.0000 −0.356009
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 20.0000i 0.710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 45.0000 1.59199
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 45.0000i 1.58802i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.00000i 0.281613i
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 7.00000i − 0.244899i
\(818\) 0 0
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) −11.0000 −0.383903 −0.191951 0.981404i \(-0.561482\pi\)
−0.191951 + 0.981404i \(0.561482\pi\)
\(822\) 0 0
\(823\) 37.0000i 1.28974i 0.764293 + 0.644869i \(0.223088\pi\)
−0.764293 + 0.644869i \(0.776912\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 8.00000i − 0.278187i −0.990279 0.139094i \(-0.955581\pi\)
0.990279 0.139094i \(-0.0444189\pi\)
\(828\) 0 0
\(829\) −24.0000 −0.833554 −0.416777 0.909009i \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) 10.0000i 0.346479i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 32.0000i − 1.10608i
\(838\) 0 0
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 20.0000i 0.688837i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 6.00000i − 0.206162i
\(848\) 0 0
\(849\) 26.0000 0.892318
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 30.0000i − 1.02718i −0.858036 0.513590i \(-0.828315\pi\)
0.858036 0.513590i \(-0.171685\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.0000i 1.29806i 0.760765 + 0.649028i \(0.224824\pi\)
−0.760765 + 0.649028i \(0.775176\pi\)
\(858\) 0 0
\(859\) −17.0000 −0.580033 −0.290016 0.957022i \(-0.593661\pi\)
−0.290016 + 0.957022i \(0.593661\pi\)
\(860\) 0 0
\(861\) 36.0000 1.22688
\(862\) 0 0
\(863\) − 30.0000i − 1.02121i −0.859815 0.510606i \(-0.829421\pi\)
0.859815 0.510606i \(-0.170579\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 16.0000i 0.543388i
\(868\) 0 0
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 16.0000i 0.541518i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 34.0000i − 1.14810i −0.818821 0.574049i \(-0.805372\pi\)
0.818821 0.574049i \(-0.194628\pi\)
\(878\) 0 0
\(879\) −8.00000 −0.269833
\(880\) 0 0
\(881\) 37.0000 1.24656 0.623281 0.781998i \(-0.285799\pi\)
0.623281 + 0.781998i \(0.285799\pi\)
\(882\) 0 0
\(883\) 41.0000i 1.37976i 0.723924 + 0.689880i \(0.242337\pi\)
−0.723924 + 0.689880i \(0.757663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 38.0000i − 1.27592i −0.770072 0.637958i \(-0.779780\pi\)
0.770072 0.637958i \(-0.220220\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) 33.0000 1.10554
\(892\) 0 0
\(893\) 9.00000i 0.301174i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) − 42.0000i − 1.39767i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 4.00000i − 0.132818i −0.997792 0.0664089i \(-0.978846\pi\)
0.997792 0.0664089i \(-0.0211542\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −2.00000 −0.0662630 −0.0331315 0.999451i \(-0.510548\pi\)
−0.0331315 + 0.999451i \(0.510548\pi\)
\(912\) 0 0
\(913\) − 12.0000i − 0.397142i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 27.0000i − 0.891619i
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) −32.0000 −1.05444
\(922\) 0 0
\(923\) 24.0000i 0.789970i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 14.0000i 0.459820i
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) − 62.0000i − 2.02979i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.00000i 0.228680i 0.993442 + 0.114340i \(0.0364753\pi\)
−0.993442 + 0.114340i \(0.963525\pi\)
\(938\) 0 0
\(939\) 28.0000 0.913745
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 20.0000i − 0.649913i −0.945729 0.324956i \(-0.894650\pi\)
0.945729 0.324956i \(-0.105350\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) 0 0
\(953\) 4.00000i 0.129573i 0.997899 + 0.0647864i \(0.0206366\pi\)
−0.997899 + 0.0647864i \(0.979363\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 12.0000i − 0.387905i
\(958\) 0 0
\(959\) 63.0000 2.03438
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 10.0000i 0.322245i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 48.0000i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) 0 0
\(969\) −10.0000 −0.321246
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) − 15.0000i − 0.480878i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 52.0000i − 1.66363i −0.555055 0.831814i \(-0.687303\pi\)
0.555055 0.831814i \(-0.312697\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) 0 0
\(983\) 20.0000i 0.637901i 0.947771 + 0.318950i \(0.103330\pi\)
−0.947771 + 0.318950i \(0.896670\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 54.0000i 1.71884i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −10.0000 −0.317660 −0.158830 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) 0 0
\(993\) 16.0000i 0.507745i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 13.0000i − 0.411714i −0.978582 0.205857i \(-0.934002\pi\)
0.978582 0.205857i \(-0.0659982\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 3800.2.d.d.3649.1 2
5.2 odd 4 152.2.a.a.1.1 1
5.3 odd 4 3800.2.a.i.1.1 1
5.4 even 2 inner 3800.2.d.d.3649.2 2
15.2 even 4 1368.2.a.h.1.1 1
20.3 even 4 7600.2.a.b.1.1 1
20.7 even 4 304.2.a.e.1.1 1
35.27 even 4 7448.2.a.s.1.1 1
40.27 even 4 1216.2.a.d.1.1 1
40.37 odd 4 1216.2.a.p.1.1 1
60.47 odd 4 2736.2.a.p.1.1 1
95.37 even 4 2888.2.a.f.1.1 1
380.227 odd 4 5776.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.a.1.1 1 5.2 odd 4
304.2.a.e.1.1 1 20.7 even 4
1216.2.a.d.1.1 1 40.27 even 4
1216.2.a.p.1.1 1 40.37 odd 4
1368.2.a.h.1.1 1 15.2 even 4
2736.2.a.p.1.1 1 60.47 odd 4
2888.2.a.f.1.1 1 95.37 even 4
3800.2.a.i.1.1 1 5.3 odd 4
3800.2.d.d.3649.1 2 1.1 even 1 trivial
3800.2.d.d.3649.2 2 5.4 even 2 inner
5776.2.a.b.1.1 1 380.227 odd 4
7448.2.a.s.1.1 1 35.27 even 4
7600.2.a.b.1.1 1 20.3 even 4