Properties

Label 2646.2.d.c.2645.2
Level $2646$
Weight $2$
Character 2646.2645
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2645.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2646.2645
Dual form 2646.2.d.c.2645.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000i q^{11} +3.46410i q^{13} +1.00000 q^{16} +3.00000 q^{22} -5.00000 q^{25} +3.46410 q^{26} -9.00000i q^{29} -1.73205i q^{31} -1.00000i q^{32} -8.00000 q^{37} -10.3923 q^{41} +4.00000 q^{43} -3.00000i q^{44} -10.3923 q^{47} +5.00000i q^{50} -3.46410i q^{52} -6.00000i q^{53} -9.00000 q^{58} -5.19615 q^{59} +13.8564i q^{61} -1.73205 q^{62} -1.00000 q^{64} +2.00000 q^{67} -12.0000i q^{71} +5.19615i q^{73} +8.00000i q^{74} -13.0000 q^{79} +10.3923i q^{82} +5.19615 q^{83} -4.00000i q^{86} -3.00000 q^{88} +10.3923 q^{89} +10.3923i q^{94} -8.66025i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + O(q^{10}) \) \( 4 q - 4 q^{4} + 4 q^{16} + 12 q^{22} - 20 q^{25} - 32 q^{37} + 16 q^{43} - 36 q^{58} - 4 q^{64} + 8 q^{67} - 52 q^{79} - 12 q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 3.46410 0.679366
\(27\) 0 0
\(28\) 0 0
\(29\) − 9.00000i − 1.67126i −0.549294 0.835629i \(-0.685103\pi\)
0.549294 0.835629i \(-0.314897\pi\)
\(30\) 0 0
\(31\) − 1.73205i − 0.311086i −0.987829 0.155543i \(-0.950287\pi\)
0.987829 0.155543i \(-0.0497126\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.3923 −1.62301 −0.811503 0.584349i \(-0.801350\pi\)
−0.811503 + 0.584349i \(0.801350\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) − 3.00000i − 0.452267i
\(45\) 0 0
\(46\) 0 0
\(47\) −10.3923 −1.51587 −0.757937 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 5.00000i 0.707107i
\(51\) 0 0
\(52\) − 3.46410i − 0.480384i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) −5.19615 −0.676481 −0.338241 0.941060i \(-0.609832\pi\)
−0.338241 + 0.941060i \(0.609832\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i 0.461644 + 0.887066i \(0.347260\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) −1.73205 −0.219971
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 12.0000i − 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 5.19615i 0.608164i 0.952646 + 0.304082i \(0.0983496\pi\)
−0.952646 + 0.304082i \(0.901650\pi\)
\(74\) 8.00000i 0.929981i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.3923i 1.14764i
\(83\) 5.19615 0.570352 0.285176 0.958475i \(-0.407948\pi\)
0.285176 + 0.958475i \(0.407948\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 4.00000i − 0.431331i
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 10.3923 1.10158 0.550791 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 10.3923i 1.07188i
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.66025i − 0.879316i −0.898165 0.439658i \(-0.855100\pi\)
0.898165 0.439658i \(-0.144900\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000 0.500000
\(101\) −5.19615 −0.517036 −0.258518 0.966006i \(-0.583234\pi\)
−0.258518 + 0.966006i \(0.583234\pi\)
\(102\) 0 0
\(103\) 17.3205i 1.70664i 0.521387 + 0.853320i \(0.325415\pi\)
−0.521387 + 0.853320i \(0.674585\pi\)
\(104\) −3.46410 −0.339683
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 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
\(116\) 9.00000i 0.835629i
\(117\) 0 0
\(118\) 5.19615i 0.478345i
\(119\) 0 0
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 13.8564 1.25450
\(123\) 0 0
\(124\) 1.73205i 0.155543i
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −15.5885 −1.36197 −0.680985 0.732297i \(-0.738448\pi\)
−0.680985 + 0.732297i \(0.738448\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 2.00000i − 0.172774i
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) 6.92820i 0.587643i 0.955860 + 0.293821i \(0.0949270\pi\)
−0.955860 + 0.293821i \(0.905073\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −10.3923 −0.869048
\(144\) 0 0
\(145\) 0 0
\(146\) 5.19615 0.430037
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) 15.0000i 1.22885i 0.788976 + 0.614424i \(0.210612\pi\)
−0.788976 + 0.614424i \(0.789388\pi\)
\(150\) 0 0
\(151\) −23.0000 −1.87171 −0.935857 0.352381i \(-0.885372\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.3923i 0.829396i 0.909959 + 0.414698i \(0.136113\pi\)
−0.909959 + 0.414698i \(0.863887\pi\)
\(158\) 13.0000i 1.03422i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 10.3923 0.811503
\(165\) 0 0
\(166\) − 5.19615i − 0.403300i
\(167\) −10.3923 −0.804181 −0.402090 0.915600i \(-0.631716\pi\)
−0.402090 + 0.915600i \(0.631716\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 25.9808 1.97528 0.987640 0.156737i \(-0.0500975\pi\)
0.987640 + 0.156737i \(0.0500975\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000i 0.226134i
\(177\) 0 0
\(178\) − 10.3923i − 0.778936i
\(179\) − 9.00000i − 0.672692i −0.941739 0.336346i \(-0.890809\pi\)
0.941739 0.336346i \(-0.109191\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i 0.635071 + 0.772454i \(0.280971\pi\)
−0.635071 + 0.772454i \(0.719029\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 10.3923 0.757937
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000i 0.434145i 0.976156 + 0.217072i \(0.0696508\pi\)
−0.976156 + 0.217072i \(0.930349\pi\)
\(192\) 0 0
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) −8.66025 −0.621770
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.0000i − 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(198\) 0 0
\(199\) 1.73205i 0.122782i 0.998114 + 0.0613909i \(0.0195536\pi\)
−0.998114 + 0.0613909i \(0.980446\pi\)
\(200\) − 5.00000i − 0.353553i
\(201\) 0 0
\(202\) 5.19615i 0.365600i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 17.3205 1.20678
\(207\) 0 0
\(208\) 3.46410i 0.240192i
\(209\) 0 0
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 8.00000i 0.541828i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 5.19615i 0.347960i 0.984749 + 0.173980i \(0.0556628\pi\)
−0.984749 + 0.173980i \(0.944337\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) −25.9808 −1.72440 −0.862202 0.506565i \(-0.830915\pi\)
−0.862202 + 0.506565i \(0.830915\pi\)
\(228\) 0 0
\(229\) − 6.92820i − 0.457829i −0.973447 0.228914i \(-0.926482\pi\)
0.973447 0.228914i \(-0.0735176\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 5.19615 0.338241
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000i 0.776215i 0.921614 + 0.388108i \(0.126871\pi\)
−0.921614 + 0.388108i \(0.873129\pi\)
\(240\) 0 0
\(241\) − 12.1244i − 0.780998i −0.920603 0.390499i \(-0.872302\pi\)
0.920603 0.390499i \(-0.127698\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) 0 0
\(244\) − 13.8564i − 0.887066i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 1.73205 0.109985
\(249\) 0 0
\(250\) 0 0
\(251\) 25.9808 1.63989 0.819946 0.572441i \(-0.194004\pi\)
0.819946 + 0.572441i \(0.194004\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 8.00000i − 0.501965i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.7846 −1.29651 −0.648254 0.761424i \(-0.724501\pi\)
−0.648254 + 0.761424i \(0.724501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 15.5885i 0.963058i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) −15.5885 −0.950445 −0.475223 0.879866i \(-0.657632\pi\)
−0.475223 + 0.879866i \(0.657632\pi\)
\(270\) 0 0
\(271\) − 3.46410i − 0.210429i −0.994450 0.105215i \(-0.966447\pi\)
0.994450 0.105215i \(-0.0335529\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) − 15.0000i − 0.904534i
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 6.92820 0.415526
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 0 0
\(283\) 6.92820i 0.411839i 0.978569 + 0.205919i \(0.0660185\pi\)
−0.978569 + 0.205919i \(0.933982\pi\)
\(284\) 12.0000i 0.712069i
\(285\) 0 0
\(286\) 10.3923i 0.614510i
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) − 5.19615i − 0.304082i
\(293\) −5.19615 −0.303562 −0.151781 0.988414i \(-0.548501\pi\)
−0.151781 + 0.988414i \(0.548501\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 8.00000i − 0.464991i
\(297\) 0 0
\(298\) 15.0000 0.868927
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 23.0000i 1.32350i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 17.3205i − 0.988534i −0.869310 0.494267i \(-0.835437\pi\)
0.869310 0.494267i \(-0.164563\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 31.1769 1.76788 0.883940 0.467600i \(-0.154881\pi\)
0.883940 + 0.467600i \(0.154881\pi\)
\(312\) 0 0
\(313\) 20.7846i 1.17482i 0.809291 + 0.587408i \(0.199852\pi\)
−0.809291 + 0.587408i \(0.800148\pi\)
\(314\) 10.3923 0.586472
\(315\) 0 0
\(316\) 13.0000 0.731307
\(317\) 9.00000i 0.505490i 0.967533 + 0.252745i \(0.0813334\pi\)
−0.967533 + 0.252745i \(0.918667\pi\)
\(318\) 0 0
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) − 17.3205i − 0.960769i
\(326\) 10.0000i 0.553849i
\(327\) 0 0
\(328\) − 10.3923i − 0.573819i
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −5.19615 −0.285176
\(333\) 0 0
\(334\) 10.3923i 0.568642i
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) 5.19615 0.281387
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000i 0.215666i
\(345\) 0 0
\(346\) − 25.9808i − 1.39673i
\(347\) − 3.00000i − 0.161048i −0.996753 0.0805242i \(-0.974341\pi\)
0.996753 0.0805242i \(-0.0256594\pi\)
\(348\) 0 0
\(349\) − 27.7128i − 1.48343i −0.670714 0.741716i \(-0.734012\pi\)
0.670714 0.741716i \(-0.265988\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) 10.3923 0.553127 0.276563 0.960996i \(-0.410804\pi\)
0.276563 + 0.960996i \(0.410804\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.3923 −0.550791
\(357\) 0 0
\(358\) −9.00000 −0.475665
\(359\) 12.0000i 0.633336i 0.948536 + 0.316668i \(0.102564\pi\)
−0.948536 + 0.316668i \(0.897436\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 20.7846 1.09241
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.3923i 0.542474i 0.962513 + 0.271237i \(0.0874327\pi\)
−0.962513 + 0.271237i \(0.912567\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) − 10.3923i − 0.535942i
\(377\) 31.1769 1.60569
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6.00000 0.306987
\(383\) −20.7846 −1.06204 −0.531022 0.847358i \(-0.678192\pi\)
−0.531022 + 0.847358i \(0.678192\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 11.0000i 0.559885i
\(387\) 0 0
\(388\) 8.66025i 0.439658i
\(389\) 9.00000i 0.456318i 0.973624 + 0.228159i \(0.0732706\pi\)
−0.973624 + 0.228159i \(0.926729\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −15.0000 −0.755689
\(395\) 0 0
\(396\) 0 0
\(397\) − 24.2487i − 1.21701i −0.793551 0.608504i \(-0.791770\pi\)
0.793551 0.608504i \(-0.208230\pi\)
\(398\) 1.73205 0.0868199
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 30.0000i 1.49813i 0.662497 + 0.749064i \(0.269497\pi\)
−0.662497 + 0.749064i \(0.730503\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 5.19615 0.258518
\(405\) 0 0
\(406\) 0 0
\(407\) − 24.0000i − 1.18964i
\(408\) 0 0
\(409\) 6.92820i 0.342578i 0.985221 + 0.171289i \(0.0547931\pi\)
−0.985221 + 0.171289i \(0.945207\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 17.3205i − 0.853320i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 3.46410 0.169842
\(417\) 0 0
\(418\) 0 0
\(419\) −31.1769 −1.52309 −0.761546 0.648111i \(-0.775559\pi\)
−0.761546 + 0.648111i \(0.775559\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 2.00000i 0.0973585i
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) − 18.0000i − 0.867029i −0.901146 0.433515i \(-0.857273\pi\)
0.901146 0.433515i \(-0.142727\pi\)
\(432\) 0 0
\(433\) 12.1244i 0.582659i 0.956623 + 0.291330i \(0.0940977\pi\)
−0.956623 + 0.291330i \(0.905902\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 0 0
\(438\) 0 0
\(439\) − 15.5885i − 0.743996i −0.928233 0.371998i \(-0.878673\pi\)
0.928233 0.371998i \(-0.121327\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 27.0000i − 1.28281i −0.767203 0.641404i \(-0.778352\pi\)
0.767203 0.641404i \(-0.221648\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 5.19615 0.246045
\(447\) 0 0
\(448\) 0 0
\(449\) − 30.0000i − 1.41579i −0.706319 0.707894i \(-0.749646\pi\)
0.706319 0.707894i \(-0.250354\pi\)
\(450\) 0 0
\(451\) − 31.1769i − 1.46806i
\(452\) − 12.0000i − 0.564433i
\(453\) 0 0
\(454\) 25.9808i 1.21934i
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) −6.92820 −0.323734
\(459\) 0 0
\(460\) 0 0
\(461\) 15.5885 0.726027 0.363013 0.931784i \(-0.381748\pi\)
0.363013 + 0.931784i \(0.381748\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) − 9.00000i − 0.417815i
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −5.19615 −0.240449 −0.120225 0.992747i \(-0.538361\pi\)
−0.120225 + 0.992747i \(0.538361\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) − 5.19615i − 0.239172i
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) 10.3923 0.474837 0.237418 0.971408i \(-0.423699\pi\)
0.237418 + 0.971408i \(0.423699\pi\)
\(480\) 0 0
\(481\) − 27.7128i − 1.26360i
\(482\) −12.1244 −0.552249
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) −11.0000 −0.498458 −0.249229 0.968445i \(-0.580177\pi\)
−0.249229 + 0.968445i \(0.580177\pi\)
\(488\) −13.8564 −0.627250
\(489\) 0 0
\(490\) 0 0
\(491\) 36.0000i 1.62466i 0.583200 + 0.812329i \(0.301800\pi\)
−0.583200 + 0.812329i \(0.698200\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) − 1.73205i − 0.0777714i
\(497\) 0 0
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 25.9808i − 1.15958i
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 5.19615 0.230315 0.115158 0.993347i \(-0.463263\pi\)
0.115158 + 0.993347i \(0.463263\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 20.7846i 0.916770i
\(515\) 0 0
\(516\) 0 0
\(517\) − 31.1769i − 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −20.7846 −0.910590 −0.455295 0.890341i \(-0.650466\pi\)
−0.455295 + 0.890341i \(0.650466\pi\)
\(522\) 0 0
\(523\) − 38.1051i − 1.66622i −0.553107 0.833110i \(-0.686558\pi\)
0.553107 0.833110i \(-0.313442\pi\)
\(524\) 15.5885 0.680985
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 36.0000i − 1.55933i
\(534\) 0 0
\(535\) 0 0
\(536\) 2.00000i 0.0863868i
\(537\) 0 0
\(538\) 15.5885i 0.672066i
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −3.46410 −0.148796
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.0000 0.598597 0.299298 0.954160i \(-0.403247\pi\)
0.299298 + 0.954160i \(0.403247\pi\)
\(548\) 12.0000i 0.512615i
\(549\) 0 0
\(550\) −15.0000 −0.639602
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) − 14.0000i − 0.594803i
\(555\) 0 0
\(556\) − 6.92820i − 0.293821i
\(557\) 21.0000i 0.889799i 0.895581 + 0.444899i \(0.146761\pi\)
−0.895581 + 0.444899i \(0.853239\pi\)
\(558\) 0 0
\(559\) 13.8564i 0.586064i
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −31.1769 −1.31395 −0.656975 0.753912i \(-0.728164\pi\)
−0.656975 + 0.753912i \(0.728164\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.92820 0.291214
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 12.0000i 0.503066i 0.967849 + 0.251533i \(0.0809347\pi\)
−0.967849 + 0.251533i \(0.919065\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 10.3923 0.434524
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 8.66025i − 0.360531i −0.983618 0.180266i \(-0.942304\pi\)
0.983618 0.180266i \(-0.0576957\pi\)
\(578\) 17.0000i 0.707107i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) −5.19615 −0.215018
\(585\) 0 0
\(586\) 5.19615i 0.214651i
\(587\) 10.3923 0.428936 0.214468 0.976731i \(-0.431198\pi\)
0.214468 + 0.976731i \(0.431198\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) 20.7846 0.853522 0.426761 0.904365i \(-0.359655\pi\)
0.426761 + 0.904365i \(0.359655\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 15.0000i − 0.614424i
\(597\) 0 0
\(598\) 0 0
\(599\) − 30.0000i − 1.22577i −0.790173 0.612883i \(-0.790010\pi\)
0.790173 0.612883i \(-0.209990\pi\)
\(600\) 0 0
\(601\) 34.6410i 1.41304i 0.707695 + 0.706518i \(0.249735\pi\)
−0.707695 + 0.706518i \(0.750265\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 23.0000 0.935857
\(605\) 0 0
\(606\) 0 0
\(607\) 5.19615i 0.210905i 0.994424 + 0.105453i \(0.0336291\pi\)
−0.994424 + 0.105453i \(0.966371\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 36.0000i − 1.45640i
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) −17.3205 −0.698999
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 34.6410i 1.39234i 0.717877 + 0.696170i \(0.245114\pi\)
−0.717877 + 0.696170i \(0.754886\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 31.1769i − 1.25008i
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 20.7846 0.830720
\(627\) 0 0
\(628\) − 10.3923i − 0.414698i
\(629\) 0 0
\(630\) 0 0
\(631\) −37.0000 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(632\) − 13.0000i − 0.517112i
\(633\) 0 0
\(634\) 9.00000 0.357436
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) − 27.0000i − 1.06894i
\(639\) 0 0
\(640\) 0 0
\(641\) 42.0000i 1.65890i 0.558581 + 0.829450i \(0.311346\pi\)
−0.558581 + 0.829450i \(0.688654\pi\)
\(642\) 0 0
\(643\) − 3.46410i − 0.136611i −0.997664 0.0683054i \(-0.978241\pi\)
0.997664 0.0683054i \(-0.0217592\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.3923 −0.408564 −0.204282 0.978912i \(-0.565486\pi\)
−0.204282 + 0.978912i \(0.565486\pi\)
\(648\) 0 0
\(649\) − 15.5885i − 0.611900i
\(650\) −17.3205 −0.679366
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −10.3923 −0.405751
\(657\) 0 0
\(658\) 0 0
\(659\) 15.0000i 0.584317i 0.956370 + 0.292159i \(0.0943735\pi\)
−0.956370 + 0.292159i \(0.905627\pi\)
\(660\) 0 0
\(661\) 10.3923i 0.404214i 0.979363 + 0.202107i \(0.0647788\pi\)
−0.979363 + 0.202107i \(0.935221\pi\)
\(662\) − 28.0000i − 1.08825i
\(663\) 0 0
\(664\) 5.19615i 0.201650i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 10.3923 0.402090
\(669\) 0 0
\(670\) 0 0
\(671\) −41.5692 −1.60476
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) − 13.0000i − 0.500741i
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) −15.5885 −0.599113 −0.299557 0.954079i \(-0.596839\pi\)
−0.299557 + 0.954079i \(0.596839\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) − 5.19615i − 0.198971i
\(683\) − 21.0000i − 0.803543i −0.915740 0.401771i \(-0.868395\pi\)
0.915740 0.401771i \(-0.131605\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 20.7846 0.791831
\(690\) 0 0
\(691\) − 13.8564i − 0.527123i −0.964643 0.263561i \(-0.915103\pi\)
0.964643 0.263561i \(-0.0848971\pi\)
\(692\) −25.9808 −0.987640
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −27.7128 −1.04895
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000i 0.226617i 0.993560 + 0.113308i \(0.0361448\pi\)
−0.993560 + 0.113308i \(0.963855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) − 3.00000i − 0.113067i
\(705\) 0 0
\(706\) − 10.3923i − 0.391120i
\(707\) 0 0
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.3923i 0.389468i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 9.00000i 0.336346i
\(717\) 0 0
\(718\) 12.0000 0.447836
\(719\) 10.3923 0.387568 0.193784 0.981044i \(-0.437924\pi\)
0.193784 + 0.981044i \(0.437924\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 19.0000i − 0.707107i
\(723\) 0 0
\(724\) − 20.7846i − 0.772454i
\(725\) 45.0000i 1.67126i
\(726\) 0 0
\(727\) − 24.2487i − 0.899335i −0.893196 0.449667i \(-0.851542\pi\)
0.893196 0.449667i \(-0.148458\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 13.8564i − 0.511798i −0.966704 0.255899i \(-0.917629\pi\)
0.966704 0.255899i \(-0.0823715\pi\)
\(734\) 10.3923 0.383587
\(735\) 0 0
\(736\) 0 0
\(737\) 6.00000i 0.221013i
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.00000i 0.220119i 0.993925 + 0.110059i \(0.0351041\pi\)
−0.993925 + 0.110059i \(0.964896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000i 0.146450i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) −10.3923 −0.378968
\(753\) 0 0
\(754\) − 31.1769i − 1.13540i
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 26.0000i 0.944363i
\(759\) 0 0
\(760\) 0 0
\(761\) 41.5692 1.50688 0.753442 0.657515i \(-0.228392\pi\)
0.753442 + 0.657515i \(0.228392\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 6.00000i − 0.217072i
\(765\) 0 0
\(766\) 20.7846i 0.750978i
\(767\) − 18.0000i − 0.649942i
\(768\) 0 0
\(769\) − 15.5885i − 0.562134i −0.959688 0.281067i \(-0.909312\pi\)
0.959688 0.281067i \(-0.0906883\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 11.0000 0.395899
\(773\) 41.5692 1.49514 0.747570 0.664183i \(-0.231220\pi\)
0.747570 + 0.664183i \(0.231220\pi\)
\(774\) 0 0
\(775\) 8.66025i 0.311086i
\(776\) 8.66025 0.310885
\(777\) 0 0
\(778\) 9.00000 0.322666
\(779\) 0 0
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 24.2487i − 0.864373i −0.901784 0.432187i \(-0.857742\pi\)
0.901784 0.432187i \(-0.142258\pi\)
\(788\) 15.0000i 0.534353i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) −24.2487 −0.860555
\(795\) 0 0
\(796\) − 1.73205i − 0.0613909i
\(797\) −15.5885 −0.552171 −0.276086 0.961133i \(-0.589037\pi\)
−0.276086 + 0.961133i \(0.589037\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000i 0.176777i
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) −15.5885 −0.550105
\(804\) 0 0
\(805\) 0 0
\(806\) − 6.00000i − 0.211341i
\(807\) 0 0
\(808\) − 5.19615i − 0.182800i
\(809\) 6.00000i 0.210949i 0.994422 + 0.105474i \(0.0336361\pi\)
−0.994422 + 0.105474i \(0.966364\pi\)
\(810\) 0 0
\(811\) − 10.3923i − 0.364923i −0.983213 0.182462i \(-0.941593\pi\)
0.983213 0.182462i \(-0.0584065\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 6.92820 0.242239
\(819\) 0 0
\(820\) 0 0
\(821\) − 33.0000i − 1.15171i −0.817553 0.575854i \(-0.804670\pi\)
0.817553 0.575854i \(-0.195330\pi\)
\(822\) 0 0
\(823\) 11.0000 0.383436 0.191718 0.981450i \(-0.438594\pi\)
0.191718 + 0.981450i \(0.438594\pi\)
\(824\) −17.3205 −0.603388
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00000i 0.104320i 0.998639 + 0.0521601i \(0.0166106\pi\)
−0.998639 + 0.0521601i \(0.983389\pi\)
\(828\) 0 0
\(829\) − 31.1769i − 1.08282i −0.840759 0.541409i \(-0.817891\pi\)
0.840759 0.541409i \(-0.182109\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 3.46410i − 0.120096i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 31.1769i 1.07699i
\(839\) −20.7846 −0.717564 −0.358782 0.933421i \(-0.616808\pi\)
−0.358782 + 0.933421i \(0.616808\pi\)
\(840\) 0 0
\(841\) −52.0000 −1.79310
\(842\) 10.0000i 0.344623i
\(843\) 0 0
\(844\) 2.00000 0.0688428
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) − 6.00000i − 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.3923i 0.355826i 0.984046 + 0.177913i \(0.0569345\pi\)
−0.984046 + 0.177913i \(0.943065\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −10.3923 −0.354994 −0.177497 0.984121i \(-0.556800\pi\)
−0.177497 + 0.984121i \(0.556800\pi\)
\(858\) 0 0
\(859\) 48.4974i 1.65471i 0.561679 + 0.827355i \(0.310156\pi\)
−0.561679 + 0.827355i \(0.689844\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 12.1244 0.412002
\(867\) 0 0
\(868\) 0 0
\(869\) − 39.0000i − 1.32298i
\(870\) 0 0
\(871\) 6.92820i 0.234753i
\(872\) − 8.00000i − 0.270914i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −15.5885 −0.526085
\(879\) 0 0
\(880\) 0 0
\(881\) 31.1769 1.05038 0.525188 0.850986i \(-0.323995\pi\)
0.525188 + 0.850986i \(0.323995\pi\)
\(882\) 0 0
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −27.0000 −0.907083
\(887\) −31.1769 −1.04682 −0.523409 0.852081i \(-0.675340\pi\)
−0.523409 + 0.852081i \(0.675340\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) − 5.19615i − 0.173980i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −15.5885 −0.519904
\(900\) 0 0
\(901\) 0 0
\(902\) −31.1769 −1.03808
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) 22.0000 0.730498 0.365249 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) 25.9808 0.862202
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000i 1.19273i 0.802712 + 0.596367i \(0.203390\pi\)
−0.802712 + 0.596367i \(0.796610\pi\)
\(912\) 0 0
\(913\) 15.5885i 0.515903i
\(914\) − 22.0000i − 0.727695i
\(915\) 0 0
\(916\) 6.92820i 0.228914i
\(917\) 0 0
\(918\) 0 0
\(919\) −35.0000 −1.15454 −0.577272 0.816552i \(-0.695883\pi\)
−0.577272 + 0.816552i \(0.695883\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 15.5885i − 0.513378i
\(923\) 41.5692 1.36827
\(924\) 0 0
\(925\) 40.0000 1.31519
\(926\) − 5.00000i − 0.164310i
\(927\) 0 0
\(928\) −9.00000 −0.295439
\(929\) 41.5692 1.36384 0.681921 0.731426i \(-0.261145\pi\)
0.681921 + 0.731426i \(0.261145\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 18.0000i − 0.589610i
\(933\) 0 0
\(934\) 5.19615i 0.170023i
\(935\) 0 0
\(936\) 0 0
\(937\) − 34.6410i − 1.13167i −0.824518 0.565836i \(-0.808553\pi\)
0.824518 0.565836i \(-0.191447\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −5.19615 −0.169390 −0.0846949 0.996407i \(-0.526992\pi\)
−0.0846949 + 0.996407i \(0.526992\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −5.19615 −0.169120
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) − 9.00000i − 0.292461i −0.989251 0.146230i \(-0.953286\pi\)
0.989251 0.146230i \(-0.0467141\pi\)
\(948\) 0 0
\(949\) −18.0000 −0.584305
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) − 12.0000i − 0.388108i
\(957\) 0 0
\(958\) − 10.3923i − 0.335760i
\(959\) 0 0
\(960\) 0 0
\(961\) 28.0000 0.903226
\(962\) −27.7128 −0.893497
\(963\) 0 0
\(964\) 12.1244i 0.390499i
\(965\) 0 0
\(966\) 0 0
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 0 0
\(970\) 0 0
\(971\) −31.1769 −1.00051 −0.500257 0.865877i \(-0.666761\pi\)
−0.500257 + 0.865877i \(0.666761\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 11.0000i 0.352463i
\(975\) 0 0
\(976\) 13.8564i 0.443533i
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 0 0
\(979\) 31.1769i 0.996419i
\(980\) 0 0
\(981\) 0 0
\(982\) 36.0000 1.14881
\(983\) 20.7846 0.662926 0.331463 0.943468i \(-0.392458\pi\)
0.331463 + 0.943468i \(0.392458\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −1.73205 −0.0549927
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.1051i 1.20680i 0.797438 + 0.603401i \(0.206188\pi\)
−0.797438 + 0.603401i \(0.793812\pi\)
\(998\) − 16.0000i − 0.506471i
\(999\) 0 0
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 2646.2.d.c.2645.2 4
3.2 odd 2 inner 2646.2.d.c.2645.4 4
7.4 even 3 378.2.k.a.215.2 yes 4
7.5 odd 6 378.2.k.a.269.1 yes 4
7.6 odd 2 inner 2646.2.d.c.2645.1 4
21.5 even 6 378.2.k.a.269.2 yes 4
21.11 odd 6 378.2.k.a.215.1 4
21.20 even 2 inner 2646.2.d.c.2645.3 4
63.4 even 3 1134.2.t.a.593.1 4
63.5 even 6 1134.2.l.d.269.2 4
63.11 odd 6 1134.2.l.d.215.2 4
63.25 even 3 1134.2.l.d.215.1 4
63.32 odd 6 1134.2.t.a.593.2 4
63.40 odd 6 1134.2.l.d.269.1 4
63.47 even 6 1134.2.t.a.1025.1 4
63.61 odd 6 1134.2.t.a.1025.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.a.215.1 4 21.11 odd 6
378.2.k.a.215.2 yes 4 7.4 even 3
378.2.k.a.269.1 yes 4 7.5 odd 6
378.2.k.a.269.2 yes 4 21.5 even 6
1134.2.l.d.215.1 4 63.25 even 3
1134.2.l.d.215.2 4 63.11 odd 6
1134.2.l.d.269.1 4 63.40 odd 6
1134.2.l.d.269.2 4 63.5 even 6
1134.2.t.a.593.1 4 63.4 even 3
1134.2.t.a.593.2 4 63.32 odd 6
1134.2.t.a.1025.1 4 63.47 even 6
1134.2.t.a.1025.2 4 63.61 odd 6
2646.2.d.c.2645.1 4 7.6 odd 2 inner
2646.2.d.c.2645.2 4 1.1 even 1 trivial
2646.2.d.c.2645.3 4 21.20 even 2 inner
2646.2.d.c.2645.4 4 3.2 odd 2 inner