Properties

Label 1600.2.f.c.1249.2
Level $1600$
Weight $2$
Character 1600.1249
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1249
Dual form 1600.2.f.c.1249.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.46410i q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.46410i q^{7} -2.00000 q^{9} +3.00000i q^{11} -3.46410 q^{13} +3.00000i q^{17} -1.00000i q^{19} +3.46410i q^{21} +5.00000 q^{27} +10.3923i q^{29} +6.92820 q^{31} -3.00000i q^{33} +10.3923 q^{37} +3.46410 q^{39} -9.00000 q^{41} +4.00000 q^{43} -10.3923i q^{47} -5.00000 q^{49} -3.00000i q^{51} +1.00000i q^{57} +12.0000i q^{59} +3.46410i q^{61} +6.92820i q^{63} +11.0000 q^{67} +10.3923 q^{71} -7.00000i q^{73} +10.3923 q^{77} +10.3923 q^{79} +1.00000 q^{81} -15.0000 q^{83} -10.3923i q^{87} +3.00000 q^{89} +12.0000i q^{91} -6.92820 q^{93} +14.0000i q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} - 8 q^{9} + 20 q^{27} - 36 q^{41} + 16 q^{43} - 20 q^{49} + 44 q^{67} + 4 q^{81} - 60 q^{83} + 12 q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.46410i − 1.30931i −0.755929 0.654654i \(-0.772814\pi\)
0.755929 0.654654i \(-0.227186\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i −0.993399 0.114708i \(-0.963407\pi\)
0.993399 0.114708i \(-0.0365932\pi\)
\(20\) 0 0
\(21\) 3.46410i 0.755929i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 10.3923i 1.92980i 0.262613 + 0.964901i \(0.415416\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) − 3.00000i − 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.3923 1.70848 0.854242 0.519875i \(-0.174022\pi\)
0.854242 + 0.519875i \(0.174022\pi\)
\(38\) 0 0
\(39\) 3.46410 0.554700
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 10.3923i − 1.51587i −0.652328 0.757937i \(-0.726208\pi\)
0.652328 0.757937i \(-0.273792\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) − 3.00000i − 0.420084i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i 0.975100 + 0.221766i \(0.0711822\pi\)
−0.975100 + 0.221766i \(0.928818\pi\)
\(62\) 0 0
\(63\) 6.92820i 0.872872i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923 1.23334 0.616670 0.787222i \(-0.288481\pi\)
0.616670 + 0.787222i \(0.288481\pi\)
\(72\) 0 0
\(73\) − 7.00000i − 0.819288i −0.912245 0.409644i \(-0.865653\pi\)
0.912245 0.409644i \(-0.134347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923 1.18431
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.0000 −1.64646 −0.823232 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 10.3923i − 1.11417i
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 12.0000i 1.25794i
\(92\) 0 0
\(93\) −6.92820 −0.718421
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 0 0
\(99\) − 6.00000i − 0.603023i
\(100\) 0 0
\(101\) 10.3923i 1.03407i 0.855963 + 0.517036i \(0.172965\pi\)
−0.855963 + 0.517036i \(0.827035\pi\)
\(102\) 0 0
\(103\) 10.3923i 1.02398i 0.858990 + 0.511992i \(0.171092\pi\)
−0.858990 + 0.511992i \(0.828908\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) − 3.46410i − 0.331801i −0.986143 0.165900i \(-0.946947\pi\)
0.986143 0.165900i \(-0.0530530\pi\)
\(110\) 0 0
\(111\) −10.3923 −0.986394
\(112\) 0 0
\(113\) 15.0000i 1.41108i 0.708669 + 0.705541i \(0.249296\pi\)
−0.708669 + 0.705541i \(0.750704\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.92820 0.640513
\(118\) 0 0
\(119\) 10.3923 0.952661
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) 9.00000 0.811503
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −3.46410 −0.300376
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000i 0.768922i 0.923141 + 0.384461i \(0.125613\pi\)
−0.923141 + 0.384461i \(0.874387\pi\)
\(138\) 0 0
\(139\) 5.00000i 0.424094i 0.977259 + 0.212047i \(0.0680131\pi\)
−0.977259 + 0.212047i \(0.931987\pi\)
\(140\) 0 0
\(141\) 10.3923i 0.875190i
\(142\) 0 0
\(143\) − 10.3923i − 0.869048i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.00000 0.412393
\(148\) 0 0
\(149\) 10.3923i 0.851371i 0.904871 + 0.425685i \(0.139967\pi\)
−0.904871 + 0.425685i \(0.860033\pi\)
\(150\) 0 0
\(151\) −3.46410 −0.281905 −0.140952 0.990016i \(-0.545016\pi\)
−0.140952 + 0.990016i \(0.545016\pi\)
\(152\) 0 0
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.92820 0.552931 0.276465 0.961024i \(-0.410837\pi\)
0.276465 + 0.961024i \(0.410837\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −19.0000 −1.48819 −0.744097 0.668071i \(-0.767120\pi\)
−0.744097 + 0.668071i \(0.767120\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.3923i 0.804181i 0.915600 + 0.402090i \(0.131716\pi\)
−0.915600 + 0.402090i \(0.868284\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 2.00000i 0.152944i
\(172\) 0 0
\(173\) 20.7846 1.58022 0.790112 0.612962i \(-0.210022\pi\)
0.790112 + 0.612962i \(0.210022\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) 3.00000i 0.224231i 0.993695 + 0.112115i \(0.0357626\pi\)
−0.993695 + 0.112115i \(0.964237\pi\)
\(180\) 0 0
\(181\) − 13.8564i − 1.02994i −0.857209 0.514969i \(-0.827803\pi\)
0.857209 0.514969i \(-0.172197\pi\)
\(182\) 0 0
\(183\) − 3.46410i − 0.256074i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) − 17.3205i − 1.25988i
\(190\) 0 0
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 0 0
\(193\) − 7.00000i − 0.503871i −0.967744 0.251936i \(-0.918933\pi\)
0.967744 0.251936i \(-0.0810671\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.7846 −1.48084 −0.740421 0.672143i \(-0.765374\pi\)
−0.740421 + 0.672143i \(0.765374\pi\)
\(198\) 0 0
\(199\) −13.8564 −0.982255 −0.491127 0.871088i \(-0.663415\pi\)
−0.491127 + 0.871088i \(0.663415\pi\)
\(200\) 0 0
\(201\) −11.0000 −0.775880
\(202\) 0 0
\(203\) 36.0000 2.52670
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) − 25.0000i − 1.72107i −0.509390 0.860535i \(-0.670129\pi\)
0.509390 0.860535i \(-0.329871\pi\)
\(212\) 0 0
\(213\) −10.3923 −0.712069
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 24.0000i − 1.62923i
\(218\) 0 0
\(219\) 7.00000i 0.473016i
\(220\) 0 0
\(221\) − 10.3923i − 0.699062i
\(222\) 0 0
\(223\) 13.8564i 0.927894i 0.885863 + 0.463947i \(0.153567\pi\)
−0.885863 + 0.463947i \(0.846433\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i 0.973447 + 0.228914i \(0.0735176\pi\)
−0.973447 + 0.228914i \(0.926482\pi\)
\(230\) 0 0
\(231\) −10.3923 −0.683763
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10.3923 −0.675053
\(238\) 0 0
\(239\) −10.3923 −0.672222 −0.336111 0.941822i \(-0.609112\pi\)
−0.336111 + 0.941822i \(0.609112\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.46410i 0.220416i
\(248\) 0 0
\(249\) 15.0000 0.950586
\(250\) 0 0
\(251\) 9.00000i 0.568075i 0.958813 + 0.284037i \(0.0916740\pi\)
−0.958813 + 0.284037i \(0.908326\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) − 36.0000i − 2.23693i
\(260\) 0 0
\(261\) − 20.7846i − 1.28654i
\(262\) 0 0
\(263\) − 20.7846i − 1.28163i −0.767694 0.640817i \(-0.778596\pi\)
0.767694 0.640817i \(-0.221404\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −20.7846 −1.26258 −0.631288 0.775549i \(-0.717473\pi\)
−0.631288 + 0.775549i \(0.717473\pi\)
\(272\) 0 0
\(273\) − 12.0000i − 0.726273i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −17.3205 −1.04069 −0.520344 0.853957i \(-0.674196\pi\)
−0.520344 + 0.853957i \(0.674196\pi\)
\(278\) 0 0
\(279\) −13.8564 −0.829561
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 11.0000 0.653882 0.326941 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 31.1769i 1.84032i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) − 14.0000i − 0.820695i
\(292\) 0 0
\(293\) 10.3923 0.607125 0.303562 0.952812i \(-0.401824\pi\)
0.303562 + 0.952812i \(0.401824\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.0000i 0.870388i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 13.8564i − 0.798670i
\(302\) 0 0
\(303\) − 10.3923i − 0.597022i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −19.0000 −1.08439 −0.542194 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 0 0
\(309\) − 10.3923i − 0.591198i
\(310\) 0 0
\(311\) −10.3923 −0.589294 −0.294647 0.955606i \(-0.595202\pi\)
−0.294647 + 0.955606i \(0.595202\pi\)
\(312\) 0 0
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.7846 −1.16738 −0.583690 0.811977i \(-0.698392\pi\)
−0.583690 + 0.811977i \(0.698392\pi\)
\(318\) 0 0
\(319\) −31.1769 −1.74557
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.46410i 0.191565i
\(328\) 0 0
\(329\) −36.0000 −1.98474
\(330\) 0 0
\(331\) 19.0000i 1.04433i 0.852843 + 0.522167i \(0.174876\pi\)
−0.852843 + 0.522167i \(0.825124\pi\)
\(332\) 0 0
\(333\) −20.7846 −1.13899
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 7.00000i − 0.381314i −0.981657 0.190657i \(-0.938938\pi\)
0.981657 0.190657i \(-0.0610619\pi\)
\(338\) 0 0
\(339\) − 15.0000i − 0.814688i
\(340\) 0 0
\(341\) 20.7846i 1.12555i
\(342\) 0 0
\(343\) − 6.92820i − 0.374088i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −33.0000 −1.77153 −0.885766 0.464131i \(-0.846367\pi\)
−0.885766 + 0.464131i \(0.846367\pi\)
\(348\) 0 0
\(349\) − 10.3923i − 0.556287i −0.960539 0.278144i \(-0.910281\pi\)
0.960539 0.278144i \(-0.0897191\pi\)
\(350\) 0 0
\(351\) −17.3205 −0.924500
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.3923 −0.550019
\(358\) 0 0
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) 18.0000 0.947368
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6.92820i 0.361649i 0.983515 + 0.180825i \(0.0578766\pi\)
−0.983515 + 0.180825i \(0.942123\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 13.8564 0.717458 0.358729 0.933442i \(-0.383210\pi\)
0.358729 + 0.933442i \(0.383210\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 36.0000i − 1.85409i
\(378\) 0 0
\(379\) 19.0000i 0.975964i 0.872854 + 0.487982i \(0.162267\pi\)
−0.872854 + 0.487982i \(0.837733\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 31.1769i − 1.59307i −0.604595 0.796533i \(-0.706665\pi\)
0.604595 0.796533i \(-0.293335\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) − 20.7846i − 1.05382i −0.849921 0.526911i \(-0.823350\pi\)
0.849921 0.526911i \(-0.176650\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −27.7128 −1.39087 −0.695433 0.718591i \(-0.744787\pi\)
−0.695433 + 0.718591i \(0.744787\pi\)
\(398\) 0 0
\(399\) 3.46410 0.173422
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 31.1769i 1.54538i
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) − 9.00000i − 0.443937i
\(412\) 0 0
\(413\) 41.5692 2.04549
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 5.00000i − 0.244851i
\(418\) 0 0
\(419\) − 21.0000i − 1.02592i −0.858413 0.512959i \(-0.828549\pi\)
0.858413 0.512959i \(-0.171451\pi\)
\(420\) 0 0
\(421\) − 20.7846i − 1.01298i −0.862246 0.506490i \(-0.830943\pi\)
0.862246 0.506490i \(-0.169057\pi\)
\(422\) 0 0
\(423\) 20.7846i 1.01058i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 0 0
\(429\) 10.3923i 0.501745i
\(430\) 0 0
\(431\) 10.3923 0.500580 0.250290 0.968171i \(-0.419474\pi\)
0.250290 + 0.968171i \(0.419474\pi\)
\(432\) 0 0
\(433\) 1.00000i 0.0480569i 0.999711 + 0.0240285i \(0.00764923\pi\)
−0.999711 + 0.0240285i \(0.992351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −17.3205 −0.826663 −0.413331 0.910581i \(-0.635635\pi\)
−0.413331 + 0.910581i \(0.635635\pi\)
\(440\) 0 0
\(441\) 10.0000 0.476190
\(442\) 0 0
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 10.3923i − 0.491539i
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) − 27.0000i − 1.27138i
\(452\) 0 0
\(453\) 3.46410 0.162758
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.0000i 1.73079i 0.501093 + 0.865393i \(0.332931\pi\)
−0.501093 + 0.865393i \(0.667069\pi\)
\(458\) 0 0
\(459\) 15.0000i 0.700140i
\(460\) 0 0
\(461\) − 20.7846i − 0.968036i −0.875058 0.484018i \(-0.839177\pi\)
0.875058 0.484018i \(-0.160823\pi\)
\(462\) 0 0
\(463\) 20.7846i 0.965943i 0.875636 + 0.482971i \(0.160442\pi\)
−0.875636 + 0.482971i \(0.839558\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) − 38.1051i − 1.75953i
\(470\) 0 0
\(471\) −6.92820 −0.319235
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 41.5692 1.89935 0.949673 0.313243i \(-0.101415\pi\)
0.949673 + 0.313243i \(0.101415\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.8564i 0.627894i 0.949441 + 0.313947i \(0.101651\pi\)
−0.949441 + 0.313947i \(0.898349\pi\)
\(488\) 0 0
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) 12.0000i 0.541552i 0.962642 + 0.270776i \(0.0872803\pi\)
−0.962642 + 0.270776i \(0.912720\pi\)
\(492\) 0 0
\(493\) −31.1769 −1.40414
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 36.0000i − 1.61482i
\(498\) 0 0
\(499\) 16.0000i 0.716258i 0.933672 + 0.358129i \(0.116585\pi\)
−0.933672 + 0.358129i \(0.883415\pi\)
\(500\) 0 0
\(501\) − 10.3923i − 0.464294i
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 20.7846i 0.921262i 0.887592 + 0.460631i \(0.152377\pi\)
−0.887592 + 0.460631i \(0.847623\pi\)
\(510\) 0 0
\(511\) −24.2487 −1.07270
\(512\) 0 0
\(513\) − 5.00000i − 0.220755i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 31.1769 1.37116
\(518\) 0 0
\(519\) −20.7846 −0.912343
\(520\) 0 0
\(521\) −21.0000 −0.920027 −0.460013 0.887912i \(-0.652155\pi\)
−0.460013 + 0.887912i \(0.652155\pi\)
\(522\) 0 0
\(523\) 1.00000 0.0437269 0.0218635 0.999761i \(-0.493040\pi\)
0.0218635 + 0.999761i \(0.493040\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.7846i 0.905392i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) − 24.0000i − 1.04151i
\(532\) 0 0
\(533\) 31.1769 1.35042
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 3.00000i − 0.129460i
\(538\) 0 0
\(539\) − 15.0000i − 0.646096i
\(540\) 0 0
\(541\) 6.92820i 0.297867i 0.988847 + 0.148933i \(0.0475840\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 0 0
\(543\) 13.8564i 0.594635i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.00000 0.0427569 0.0213785 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 0 0
\(549\) − 6.92820i − 0.295689i
\(550\) 0 0
\(551\) 10.3923 0.442727
\(552\) 0 0
\(553\) − 36.0000i − 1.53088i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.3923 0.440336 0.220168 0.975462i \(-0.429339\pi\)
0.220168 + 0.975462i \(0.429339\pi\)
\(558\) 0 0
\(559\) −13.8564 −0.586064
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 3.46410i − 0.145479i
\(568\) 0 0
\(569\) −3.00000 −0.125767 −0.0628833 0.998021i \(-0.520030\pi\)
−0.0628833 + 0.998021i \(0.520030\pi\)
\(570\) 0 0
\(571\) − 20.0000i − 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) −20.7846 −0.868290
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 17.0000i 0.707719i 0.935299 + 0.353860i \(0.115131\pi\)
−0.935299 + 0.353860i \(0.884869\pi\)
\(578\) 0 0
\(579\) 7.00000i 0.290910i
\(580\) 0 0
\(581\) 51.9615i 2.15573i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.0000 −0.619116 −0.309558 0.950881i \(-0.600181\pi\)
−0.309558 + 0.950881i \(0.600181\pi\)
\(588\) 0 0
\(589\) − 6.92820i − 0.285472i
\(590\) 0 0
\(591\) 20.7846 0.854965
\(592\) 0 0
\(593\) 3.00000i 0.123195i 0.998101 + 0.0615976i \(0.0196196\pi\)
−0.998101 + 0.0615976i \(0.980380\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13.8564 0.567105
\(598\) 0 0
\(599\) 31.1769 1.27385 0.636927 0.770924i \(-0.280205\pi\)
0.636927 + 0.770924i \(0.280205\pi\)
\(600\) 0 0
\(601\) 41.0000 1.67242 0.836212 0.548406i \(-0.184765\pi\)
0.836212 + 0.548406i \(0.184765\pi\)
\(602\) 0 0
\(603\) −22.0000 −0.895909
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 20.7846i − 0.843621i −0.906684 0.421811i \(-0.861395\pi\)
0.906684 0.421811i \(-0.138605\pi\)
\(608\) 0 0
\(609\) −36.0000 −1.45879
\(610\) 0 0
\(611\) 36.0000i 1.45640i
\(612\) 0 0
\(613\) −20.7846 −0.839482 −0.419741 0.907644i \(-0.637879\pi\)
−0.419741 + 0.907644i \(0.637879\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 10.3923i − 0.416359i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) 0 0
\(629\) 31.1769i 1.24310i
\(630\) 0 0
\(631\) 13.8564 0.551615 0.275807 0.961213i \(-0.411055\pi\)
0.275807 + 0.961213i \(0.411055\pi\)
\(632\) 0 0
\(633\) 25.0000i 0.993661i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 17.3205 0.686264
\(638\) 0 0
\(639\) −20.7846 −0.822226
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 41.5692i − 1.63425i −0.576457 0.817127i \(-0.695565\pi\)
0.576457 0.817127i \(-0.304435\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 24.0000i 0.940634i
\(652\) 0 0
\(653\) 31.1769 1.22005 0.610023 0.792383i \(-0.291160\pi\)
0.610023 + 0.792383i \(0.291160\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) 33.0000i 1.28550i 0.766077 + 0.642749i \(0.222206\pi\)
−0.766077 + 0.642749i \(0.777794\pi\)
\(660\) 0 0
\(661\) − 10.3923i − 0.404214i −0.979363 0.202107i \(-0.935221\pi\)
0.979363 0.202107i \(-0.0647788\pi\)
\(662\) 0 0
\(663\) 10.3923i 0.403604i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 13.8564i − 0.535720i
\(670\) 0 0
\(671\) −10.3923 −0.401190
\(672\) 0 0
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20.7846 −0.798817 −0.399409 0.916773i \(-0.630785\pi\)
−0.399409 + 0.916773i \(0.630785\pi\)
\(678\) 0 0
\(679\) 48.4974 1.86116
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 6.92820i − 0.264327i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 37.0000i 1.40755i 0.710425 + 0.703773i \(0.248503\pi\)
−0.710425 + 0.703773i \(0.751497\pi\)
\(692\) 0 0
\(693\) −20.7846 −0.789542
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 27.0000i − 1.02270i
\(698\) 0 0
\(699\) − 6.00000i − 0.226941i
\(700\) 0 0
\(701\) 31.1769i 1.17754i 0.808302 + 0.588768i \(0.200387\pi\)
−0.808302 + 0.588768i \(0.799613\pi\)
\(702\) 0 0
\(703\) − 10.3923i − 0.391953i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.0000 1.35392
\(708\) 0 0
\(709\) 34.6410i 1.30097i 0.759519 + 0.650485i \(0.225434\pi\)
−0.759519 + 0.650485i \(0.774566\pi\)
\(710\) 0 0
\(711\) −20.7846 −0.779484
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 10.3923 0.388108
\(718\) 0 0
\(719\) 10.3923 0.387568 0.193784 0.981044i \(-0.437924\pi\)
0.193784 + 0.981044i \(0.437924\pi\)
\(720\) 0 0
\(721\) 36.0000 1.34071
\(722\) 0 0
\(723\) −5.00000 −0.185952
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 45.0333i − 1.67019i −0.550103 0.835097i \(-0.685412\pi\)
0.550103 0.835097i \(-0.314588\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 12.0000i 0.443836i
\(732\) 0 0
\(733\) −31.1769 −1.15155 −0.575773 0.817610i \(-0.695299\pi\)
−0.575773 + 0.817610i \(0.695299\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.0000i 1.21557i
\(738\) 0 0
\(739\) − 20.0000i − 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) − 3.46410i − 0.127257i
\(742\) 0 0
\(743\) − 41.5692i − 1.52503i −0.646972 0.762513i \(-0.723965\pi\)
0.646972 0.762513i \(-0.276035\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 30.0000 1.09764
\(748\) 0 0
\(749\) 31.1769i 1.13918i
\(750\) 0 0
\(751\) 51.9615 1.89610 0.948051 0.318117i \(-0.103050\pi\)
0.948051 + 0.318117i \(0.103050\pi\)
\(752\) 0 0
\(753\) − 9.00000i − 0.327978i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.92820 0.251810 0.125905 0.992042i \(-0.459817\pi\)
0.125905 + 0.992042i \(0.459817\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) −12.0000 −0.434429
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 41.5692i − 1.50098i
\(768\) 0 0
\(769\) 7.00000 0.252426 0.126213 0.992003i \(-0.459718\pi\)
0.126213 + 0.992003i \(0.459718\pi\)
\(770\) 0 0
\(771\) 6.00000i 0.216085i
\(772\) 0 0
\(773\) −10.3923 −0.373785 −0.186893 0.982380i \(-0.559842\pi\)
−0.186893 + 0.982380i \(0.559842\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 36.0000i 1.29149i
\(778\) 0 0
\(779\) 9.00000i 0.322458i
\(780\) 0 0
\(781\) 31.1769i 1.11560i
\(782\) 0 0
\(783\) 51.9615i 1.85695i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.0000 0.570338 0.285169 0.958477i \(-0.407950\pi\)
0.285169 + 0.958477i \(0.407950\pi\)
\(788\) 0 0
\(789\) 20.7846i 0.739952i
\(790\) 0 0
\(791\) 51.9615 1.84754
\(792\) 0 0
\(793\) − 12.0000i − 0.426132i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.7846 0.736229 0.368114 0.929781i \(-0.380004\pi\)
0.368114 + 0.929781i \(0.380004\pi\)
\(798\) 0 0
\(799\) 31.1769 1.10296
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 21.0000 0.741074
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) − 28.0000i − 0.983213i −0.870817 0.491606i \(-0.836410\pi\)
0.870817 0.491606i \(-0.163590\pi\)
\(812\) 0 0
\(813\) 20.7846 0.728948
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 4.00000i − 0.139942i
\(818\) 0 0
\(819\) − 24.0000i − 0.838628i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 34.6410i 1.20751i 0.797170 + 0.603755i \(0.206329\pi\)
−0.797170 + 0.603755i \(0.793671\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.00000 −0.104320 −0.0521601 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 0 0
\(829\) 10.3923i 0.360940i 0.983581 + 0.180470i \(0.0577618\pi\)
−0.983581 + 0.180470i \(0.942238\pi\)
\(830\) 0 0
\(831\) 17.3205 0.600842
\(832\) 0 0
\(833\) − 15.0000i − 0.519719i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 34.6410 1.19737
\(838\) 0 0
\(839\) −41.5692 −1.43513 −0.717564 0.696492i \(-0.754743\pi\)
−0.717564 + 0.696492i \(0.754743\pi\)
\(840\) 0 0
\(841\) −79.0000 −2.72414
\(842\) 0 0
\(843\) −18.0000 −0.619953
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 6.92820i − 0.238056i
\(848\) 0 0
\(849\) −11.0000 −0.377519
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −55.4256 −1.89774 −0.948869 0.315671i \(-0.897770\pi\)
−0.948869 + 0.315671i \(0.897770\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 39.0000i − 1.33221i −0.745856 0.666107i \(-0.767959\pi\)
0.745856 0.666107i \(-0.232041\pi\)
\(858\) 0 0
\(859\) 13.0000i 0.443554i 0.975097 + 0.221777i \(0.0711857\pi\)
−0.975097 + 0.221777i \(0.928814\pi\)
\(860\) 0 0
\(861\) − 31.1769i − 1.06251i
\(862\) 0 0
\(863\) − 10.3923i − 0.353758i −0.984233 0.176879i \(-0.943400\pi\)
0.984233 0.176879i \(-0.0566002\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 31.1769i 1.05760i
\(870\) 0 0
\(871\) −38.1051 −1.29114
\(872\) 0 0
\(873\) − 28.0000i − 0.947656i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 48.4974 1.63764 0.818821 0.574049i \(-0.194628\pi\)
0.818821 + 0.574049i \(0.194628\pi\)
\(878\) 0 0
\(879\) −10.3923 −0.350524
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 35.0000 1.17784 0.588922 0.808190i \(-0.299553\pi\)
0.588922 + 0.808190i \(0.299553\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3.00000i 0.100504i
\(892\) 0 0
\(893\) −10.3923 −0.347765
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 72.0000i 2.40133i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 13.8564i 0.461112i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 0 0
\(909\) − 20.7846i − 0.689382i
\(910\) 0 0
\(911\) 20.7846 0.688625 0.344312 0.938855i \(-0.388112\pi\)
0.344312 + 0.938855i \(0.388112\pi\)
\(912\) 0 0
\(913\) − 45.0000i − 1.48928i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −27.7128 −0.914161 −0.457081 0.889425i \(-0.651105\pi\)
−0.457081 + 0.889425i \(0.651105\pi\)
\(920\) 0 0
\(921\) 19.0000 0.626071
\(922\) 0 0
\(923\) −36.0000 −1.18495
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 20.7846i − 0.682656i
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 5.00000i 0.163868i
\(932\) 0 0
\(933\) 10.3923 0.340229
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.0000i 1.14340i 0.820463 + 0.571700i \(0.193716\pi\)
−0.820463 + 0.571700i \(0.806284\pi\)
\(938\) 0 0
\(939\) − 22.0000i − 0.717943i
\(940\) 0 0
\(941\) 51.9615i 1.69390i 0.531675 + 0.846949i \(0.321563\pi\)
−0.531675 + 0.846949i \(0.678437\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 24.2487i 0.787146i
\(950\) 0 0
\(951\) 20.7846 0.673987
\(952\) 0 0
\(953\) 39.0000i 1.26333i 0.775240 + 0.631667i \(0.217629\pi\)
−0.775240 + 0.631667i \(0.782371\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 31.1769 1.00781
\(958\) 0 0
\(959\) 31.1769 1.00676
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) 18.0000 0.580042
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 20.7846i − 0.668388i −0.942504 0.334194i \(-0.891536\pi\)
0.942504 0.334194i \(-0.108464\pi\)
\(968\) 0 0
\(969\) −3.00000 −0.0963739
\(970\) 0 0
\(971\) − 57.0000i − 1.82922i −0.404341 0.914609i \(-0.632499\pi\)
0.404341 0.914609i \(-0.367501\pi\)
\(972\) 0 0
\(973\) 17.3205 0.555270
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27.0000i 0.863807i 0.901920 + 0.431903i \(0.142158\pi\)
−0.901920 + 0.431903i \(0.857842\pi\)
\(978\) 0 0
\(979\) 9.00000i 0.287641i
\(980\) 0 0
\(981\) 6.92820i 0.221201i
\(982\) 0 0
\(983\) − 10.3923i − 0.331463i −0.986171 0.165732i \(-0.947001\pi\)
0.986171 0.165732i \(-0.0529985\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 36.0000 1.14589
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −24.2487 −0.770286 −0.385143 0.922857i \(-0.625848\pi\)
−0.385143 + 0.922857i \(0.625848\pi\)
\(992\) 0 0
\(993\) − 19.0000i − 0.602947i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 0 0
\(999\) 51.9615 1.64399
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 1600.2.f.c.1249.2 4
4.3 odd 2 1600.2.f.g.1249.3 4
5.2 odd 4 1600.2.d.e.801.4 yes 4
5.3 odd 4 1600.2.d.f.801.1 yes 4
5.4 even 2 1600.2.f.g.1249.4 4
8.3 odd 2 inner 1600.2.f.c.1249.4 4
8.5 even 2 1600.2.f.g.1249.1 4
20.3 even 4 1600.2.d.f.801.4 yes 4
20.7 even 4 1600.2.d.e.801.1 4
20.19 odd 2 inner 1600.2.f.c.1249.1 4
40.3 even 4 1600.2.d.f.801.2 yes 4
40.13 odd 4 1600.2.d.f.801.3 yes 4
40.19 odd 2 1600.2.f.g.1249.2 4
40.27 even 4 1600.2.d.e.801.3 yes 4
40.29 even 2 inner 1600.2.f.c.1249.3 4
40.37 odd 4 1600.2.d.e.801.2 yes 4
80.3 even 4 6400.2.a.ba.1.1 2
80.13 odd 4 6400.2.a.cg.1.2 2
80.27 even 4 6400.2.a.bb.1.2 2
80.37 odd 4 6400.2.a.cf.1.1 2
80.43 even 4 6400.2.a.cg.1.1 2
80.53 odd 4 6400.2.a.ba.1.2 2
80.67 even 4 6400.2.a.cf.1.2 2
80.77 odd 4 6400.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.d.e.801.1 4 20.7 even 4
1600.2.d.e.801.2 yes 4 40.37 odd 4
1600.2.d.e.801.3 yes 4 40.27 even 4
1600.2.d.e.801.4 yes 4 5.2 odd 4
1600.2.d.f.801.1 yes 4 5.3 odd 4
1600.2.d.f.801.2 yes 4 40.3 even 4
1600.2.d.f.801.3 yes 4 40.13 odd 4
1600.2.d.f.801.4 yes 4 20.3 even 4
1600.2.f.c.1249.1 4 20.19 odd 2 inner
1600.2.f.c.1249.2 4 1.1 even 1 trivial
1600.2.f.c.1249.3 4 40.29 even 2 inner
1600.2.f.c.1249.4 4 8.3 odd 2 inner
1600.2.f.g.1249.1 4 8.5 even 2
1600.2.f.g.1249.2 4 40.19 odd 2
1600.2.f.g.1249.3 4 4.3 odd 2
1600.2.f.g.1249.4 4 5.4 even 2
6400.2.a.ba.1.1 2 80.3 even 4
6400.2.a.ba.1.2 2 80.53 odd 4
6400.2.a.bb.1.1 2 80.77 odd 4
6400.2.a.bb.1.2 2 80.27 even 4
6400.2.a.cf.1.1 2 80.37 odd 4
6400.2.a.cf.1.2 2 80.67 even 4
6400.2.a.cg.1.1 2 80.43 even 4
6400.2.a.cg.1.2 2 80.13 odd 4