Properties

Label 6048.2.a.bg.1.2
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+2.41421 q^{5} -1.00000 q^{7} -2.41421 q^{11} +2.82843 q^{13} +2.82843 q^{17} -3.82843 q^{19} -5.24264 q^{23} +0.828427 q^{25} -2.82843 q^{29} -4.65685 q^{31} -2.41421 q^{35} +0.171573 q^{37} -2.07107 q^{41} -4.82843 q^{43} -0.343146 q^{47} +1.00000 q^{49} -4.00000 q^{53} -5.82843 q^{55} +3.65685 q^{59} -11.3137 q^{61} +6.82843 q^{65} +4.48528 q^{67} -5.58579 q^{71} -3.17157 q^{73} +2.41421 q^{77} +4.82843 q^{79} -10.8284 q^{83} +6.82843 q^{85} +1.24264 q^{89} -2.82843 q^{91} -9.24264 q^{95} -9.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + 2q^{5} - 2q^{7} - 2q^{11} - 2q^{19} - 2q^{23} - 4q^{25} + 2q^{31} - 2q^{35} + 6q^{37} + 10q^{41} - 4q^{43} - 12q^{47} + 2q^{49} - 8q^{53} - 6q^{55} - 4q^{59} + 8q^{65} - 8q^{67} - 14q^{71} - 12q^{73} + 2q^{77} + 4q^{79} - 16q^{83} + 8q^{85} - 6q^{89} - 10q^{95} - 8q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.41421 1.07967 0.539835 0.841771i \(-0.318487\pi\)
0.539835 + 0.841771i \(0.318487\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.41421 −0.727913 −0.363956 0.931416i \(-0.618574\pi\)
−0.363956 + 0.931416i \(0.618574\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) −3.82843 −0.878301 −0.439151 0.898413i \(-0.644721\pi\)
−0.439151 + 0.898413i \(0.644721\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.24264 −1.09317 −0.546583 0.837405i \(-0.684072\pi\)
−0.546583 + 0.837405i \(0.684072\pi\)
\(24\) 0 0
\(25\) 0.828427 0.165685
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) −4.65685 −0.836396 −0.418198 0.908356i \(-0.637338\pi\)
−0.418198 + 0.908356i \(0.637338\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.41421 −0.408077
\(36\) 0 0
\(37\) 0.171573 0.0282064 0.0141032 0.999901i \(-0.495511\pi\)
0.0141032 + 0.999901i \(0.495511\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.07107 −0.323446 −0.161723 0.986836i \(-0.551705\pi\)
−0.161723 + 0.986836i \(0.551705\pi\)
\(42\) 0 0
\(43\) −4.82843 −0.736328 −0.368164 0.929761i \(-0.620014\pi\)
−0.368164 + 0.929761i \(0.620014\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.343146 −0.0500530 −0.0250265 0.999687i \(-0.507967\pi\)
−0.0250265 + 0.999687i \(0.507967\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −5.82843 −0.785905
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.65685 0.476082 0.238041 0.971255i \(-0.423495\pi\)
0.238041 + 0.971255i \(0.423495\pi\)
\(60\) 0 0
\(61\) −11.3137 −1.44857 −0.724286 0.689500i \(-0.757830\pi\)
−0.724286 + 0.689500i \(0.757830\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.82843 0.846962
\(66\) 0 0
\(67\) 4.48528 0.547964 0.273982 0.961735i \(-0.411659\pi\)
0.273982 + 0.961735i \(0.411659\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.58579 −0.662911 −0.331455 0.943471i \(-0.607540\pi\)
−0.331455 + 0.943471i \(0.607540\pi\)
\(72\) 0 0
\(73\) −3.17157 −0.371205 −0.185602 0.982625i \(-0.559424\pi\)
−0.185602 + 0.982625i \(0.559424\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.41421 0.275125
\(78\) 0 0
\(79\) 4.82843 0.543240 0.271620 0.962405i \(-0.412441\pi\)
0.271620 + 0.962405i \(0.412441\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.8284 −1.18857 −0.594287 0.804253i \(-0.702566\pi\)
−0.594287 + 0.804253i \(0.702566\pi\)
\(84\) 0 0
\(85\) 6.82843 0.740647
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.24264 0.131720 0.0658598 0.997829i \(-0.479021\pi\)
0.0658598 + 0.997829i \(0.479021\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −9.24264 −0.948275
\(96\) 0 0
\(97\) −9.65685 −0.980505 −0.490252 0.871580i \(-0.663095\pi\)
−0.490252 + 0.871580i \(0.663095\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.4853 −1.24233 −0.621166 0.783679i \(-0.713341\pi\)
−0.621166 + 0.783679i \(0.713341\pi\)
\(102\) 0 0
\(103\) 6.65685 0.655919 0.327960 0.944692i \(-0.393639\pi\)
0.327960 + 0.944692i \(0.393639\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.31371 0.513696 0.256848 0.966452i \(-0.417316\pi\)
0.256848 + 0.966452i \(0.417316\pi\)
\(108\) 0 0
\(109\) 18.3137 1.75414 0.877068 0.480367i \(-0.159497\pi\)
0.877068 + 0.480367i \(0.159497\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.1421 1.14224 0.571118 0.820868i \(-0.306510\pi\)
0.571118 + 0.820868i \(0.306510\pi\)
\(114\) 0 0
\(115\) −12.6569 −1.18026
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.82843 −0.259281
\(120\) 0 0
\(121\) −5.17157 −0.470143
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.0711 −0.900784
\(126\) 0 0
\(127\) 2.82843 0.250982 0.125491 0.992095i \(-0.459949\pi\)
0.125491 + 0.992095i \(0.459949\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 0 0
\(133\) 3.82843 0.331967
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.82843 −0.241649 −0.120824 0.992674i \(-0.538554\pi\)
−0.120824 + 0.992674i \(0.538554\pi\)
\(138\) 0 0
\(139\) −1.65685 −0.140533 −0.0702663 0.997528i \(-0.522385\pi\)
−0.0702663 + 0.997528i \(0.522385\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −6.82843 −0.571022
\(144\) 0 0
\(145\) −6.82843 −0.567070
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.17157 0.259825 0.129913 0.991525i \(-0.458530\pi\)
0.129913 + 0.991525i \(0.458530\pi\)
\(150\) 0 0
\(151\) −5.31371 −0.432423 −0.216212 0.976346i \(-0.569370\pi\)
−0.216212 + 0.976346i \(0.569370\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −11.2426 −0.903031
\(156\) 0 0
\(157\) −14.1421 −1.12867 −0.564333 0.825547i \(-0.690866\pi\)
−0.564333 + 0.825547i \(0.690866\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.24264 0.413178
\(162\) 0 0
\(163\) 16.4853 1.29123 0.645613 0.763664i \(-0.276602\pi\)
0.645613 + 0.763664i \(0.276602\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.51472 −0.117212 −0.0586062 0.998281i \(-0.518666\pi\)
−0.0586062 + 0.998281i \(0.518666\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.24264 0.702705 0.351352 0.936243i \(-0.385722\pi\)
0.351352 + 0.936243i \(0.385722\pi\)
\(174\) 0 0
\(175\) −0.828427 −0.0626232
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −19.6569 −1.46922 −0.734611 0.678488i \(-0.762635\pi\)
−0.734611 + 0.678488i \(0.762635\pi\)
\(180\) 0 0
\(181\) 19.6569 1.46108 0.730541 0.682869i \(-0.239268\pi\)
0.730541 + 0.682869i \(0.239268\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.414214 0.0304536
\(186\) 0 0
\(187\) −6.82843 −0.499344
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.4142 −0.898261 −0.449130 0.893466i \(-0.648266\pi\)
−0.449130 + 0.893466i \(0.648266\pi\)
\(192\) 0 0
\(193\) −3.65685 −0.263226 −0.131613 0.991301i \(-0.542016\pi\)
−0.131613 + 0.991301i \(0.542016\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.8284 −1.76895 −0.884476 0.466586i \(-0.845484\pi\)
−0.884476 + 0.466586i \(0.845484\pi\)
\(198\) 0 0
\(199\) 3.48528 0.247065 0.123533 0.992341i \(-0.460578\pi\)
0.123533 + 0.992341i \(0.460578\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.82843 0.198517
\(204\) 0 0
\(205\) −5.00000 −0.349215
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.24264 0.639327
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.6569 −0.794991
\(216\) 0 0
\(217\) 4.65685 0.316128
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 24.7990 1.66066 0.830332 0.557270i \(-0.188151\pi\)
0.830332 + 0.557270i \(0.188151\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.8284 −1.38243 −0.691216 0.722649i \(-0.742925\pi\)
−0.691216 + 0.722649i \(0.742925\pi\)
\(228\) 0 0
\(229\) 19.3137 1.27629 0.638143 0.769918i \(-0.279703\pi\)
0.638143 + 0.769918i \(0.279703\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.8284 −0.840418 −0.420209 0.907427i \(-0.638043\pi\)
−0.420209 + 0.907427i \(0.638043\pi\)
\(234\) 0 0
\(235\) −0.828427 −0.0540406
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.6569 −1.01276 −0.506379 0.862311i \(-0.669016\pi\)
−0.506379 + 0.862311i \(0.669016\pi\)
\(240\) 0 0
\(241\) −24.1421 −1.55513 −0.777566 0.628802i \(-0.783546\pi\)
−0.777566 + 0.628802i \(0.783546\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.41421 0.154238
\(246\) 0 0
\(247\) −10.8284 −0.688996
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.9706 −1.57613 −0.788064 0.615593i \(-0.788916\pi\)
−0.788064 + 0.615593i \(0.788916\pi\)
\(252\) 0 0
\(253\) 12.6569 0.795730
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.27208 0.391241 0.195621 0.980680i \(-0.437328\pi\)
0.195621 + 0.980680i \(0.437328\pi\)
\(258\) 0 0
\(259\) −0.171573 −0.0106610
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.3848 0.825341 0.412670 0.910880i \(-0.364596\pi\)
0.412670 + 0.910880i \(0.364596\pi\)
\(264\) 0 0
\(265\) −9.65685 −0.593216
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −15.5858 −0.950282 −0.475141 0.879910i \(-0.657603\pi\)
−0.475141 + 0.879910i \(0.657603\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 13.8284 0.830870 0.415435 0.909623i \(-0.363629\pi\)
0.415435 + 0.909623i \(0.363629\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 15.3137 0.910305 0.455153 0.890413i \(-0.349585\pi\)
0.455153 + 0.890413i \(0.349585\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.07107 0.122251
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.17157 0.535809 0.267905 0.963445i \(-0.413669\pi\)
0.267905 + 0.963445i \(0.413669\pi\)
\(294\) 0 0
\(295\) 8.82843 0.514011
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −14.8284 −0.857550
\(300\) 0 0
\(301\) 4.82843 0.278306
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −27.3137 −1.56398
\(306\) 0 0
\(307\) 13.8284 0.789230 0.394615 0.918847i \(-0.370878\pi\)
0.394615 + 0.918847i \(0.370878\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.485281 0.0275178 0.0137589 0.999905i \(-0.495620\pi\)
0.0137589 + 0.999905i \(0.495620\pi\)
\(312\) 0 0
\(313\) −18.8284 −1.06425 −0.532123 0.846667i \(-0.678606\pi\)
−0.532123 + 0.846667i \(0.678606\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.2843 1.92560 0.962798 0.270221i \(-0.0870968\pi\)
0.962798 + 0.270221i \(0.0870968\pi\)
\(318\) 0 0
\(319\) 6.82843 0.382319
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10.8284 −0.602510
\(324\) 0 0
\(325\) 2.34315 0.129974
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.343146 0.0189182
\(330\) 0 0
\(331\) 34.6274 1.90329 0.951647 0.307192i \(-0.0993895\pi\)
0.951647 + 0.307192i \(0.0993895\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.8284 0.591620
\(336\) 0 0
\(337\) 25.6274 1.39601 0.698007 0.716091i \(-0.254070\pi\)
0.698007 + 0.716091i \(0.254070\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 11.2426 0.608823
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.0711 −0.970106 −0.485053 0.874485i \(-0.661200\pi\)
−0.485053 + 0.874485i \(0.661200\pi\)
\(348\) 0 0
\(349\) 19.6569 1.05221 0.526104 0.850420i \(-0.323652\pi\)
0.526104 + 0.850420i \(0.323652\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.414214 0.0220464 0.0110232 0.999939i \(-0.496491\pi\)
0.0110232 + 0.999939i \(0.496491\pi\)
\(354\) 0 0
\(355\) −13.4853 −0.715724
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −21.3137 −1.12489 −0.562447 0.826833i \(-0.690140\pi\)
−0.562447 + 0.826833i \(0.690140\pi\)
\(360\) 0 0
\(361\) −4.34315 −0.228587
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −7.65685 −0.400778
\(366\) 0 0
\(367\) 2.51472 0.131267 0.0656336 0.997844i \(-0.479093\pi\)
0.0656336 + 0.997844i \(0.479093\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) 3.68629 0.190869 0.0954345 0.995436i \(-0.469576\pi\)
0.0954345 + 0.995436i \(0.469576\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) 14.4853 0.744059 0.372029 0.928221i \(-0.378662\pi\)
0.372029 + 0.928221i \(0.378662\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −35.4558 −1.81171 −0.905855 0.423589i \(-0.860770\pi\)
−0.905855 + 0.423589i \(0.860770\pi\)
\(384\) 0 0
\(385\) 5.82843 0.297044
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −11.3137 −0.573628 −0.286814 0.957986i \(-0.592596\pi\)
−0.286814 + 0.957986i \(0.592596\pi\)
\(390\) 0 0
\(391\) −14.8284 −0.749906
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.6569 0.586520
\(396\) 0 0
\(397\) −3.65685 −0.183532 −0.0917661 0.995781i \(-0.529251\pi\)
−0.0917661 + 0.995781i \(0.529251\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.8284 1.14000 0.569999 0.821646i \(-0.306944\pi\)
0.569999 + 0.821646i \(0.306944\pi\)
\(402\) 0 0
\(403\) −13.1716 −0.656123
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.414214 −0.0205318
\(408\) 0 0
\(409\) −34.4853 −1.70519 −0.852594 0.522574i \(-0.824972\pi\)
−0.852594 + 0.522574i \(0.824972\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.65685 −0.179942
\(414\) 0 0
\(415\) −26.1421 −1.28327
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.142136 0.00694378 0.00347189 0.999994i \(-0.498895\pi\)
0.00347189 + 0.999994i \(0.498895\pi\)
\(420\) 0 0
\(421\) 10.5147 0.512456 0.256228 0.966616i \(-0.417520\pi\)
0.256228 + 0.966616i \(0.417520\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.34315 0.113659
\(426\) 0 0
\(427\) 11.3137 0.547509
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.2132 1.06997 0.534986 0.844861i \(-0.320317\pi\)
0.534986 + 0.844861i \(0.320317\pi\)
\(432\) 0 0
\(433\) 21.4558 1.03110 0.515551 0.856859i \(-0.327587\pi\)
0.515551 + 0.856859i \(0.327587\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.0711 0.960129
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.0710678 0.00337653 0.00168827 0.999999i \(-0.499463\pi\)
0.00168827 + 0.999999i \(0.499463\pi\)
\(444\) 0 0
\(445\) 3.00000 0.142214
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −28.8284 −1.36050 −0.680249 0.732981i \(-0.738128\pi\)
−0.680249 + 0.732981i \(0.738128\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.82843 −0.320122
\(456\) 0 0
\(457\) −8.65685 −0.404951 −0.202475 0.979287i \(-0.564899\pi\)
−0.202475 + 0.979287i \(0.564899\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.2426 1.17567 0.587833 0.808982i \(-0.299981\pi\)
0.587833 + 0.808982i \(0.299981\pi\)
\(462\) 0 0
\(463\) −12.3431 −0.573635 −0.286817 0.957985i \(-0.592597\pi\)
−0.286817 + 0.957985i \(0.592597\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.6274 0.769425 0.384713 0.923036i \(-0.374301\pi\)
0.384713 + 0.923036i \(0.374301\pi\)
\(468\) 0 0
\(469\) −4.48528 −0.207111
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.6569 0.535983
\(474\) 0 0
\(475\) −3.17157 −0.145522
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −36.9706 −1.68923 −0.844614 0.535376i \(-0.820170\pi\)
−0.844614 + 0.535376i \(0.820170\pi\)
\(480\) 0 0
\(481\) 0.485281 0.0221269
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −23.3137 −1.05862
\(486\) 0 0
\(487\) −22.4853 −1.01891 −0.509453 0.860499i \(-0.670152\pi\)
−0.509453 + 0.860499i \(0.670152\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 27.8701 1.25776 0.628879 0.777503i \(-0.283514\pi\)
0.628879 + 0.777503i \(0.283514\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.58579 0.250557
\(498\) 0 0
\(499\) −34.7696 −1.55650 −0.778249 0.627955i \(-0.783892\pi\)
−0.778249 + 0.627955i \(0.783892\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 10.1421 0.452215 0.226108 0.974102i \(-0.427400\pi\)
0.226108 + 0.974102i \(0.427400\pi\)
\(504\) 0 0
\(505\) −30.1421 −1.34131
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 32.4853 1.43988 0.719942 0.694034i \(-0.244168\pi\)
0.719942 + 0.694034i \(0.244168\pi\)
\(510\) 0 0
\(511\) 3.17157 0.140302
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.0711 0.708176
\(516\) 0 0
\(517\) 0.828427 0.0364342
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −27.2426 −1.19352 −0.596761 0.802419i \(-0.703546\pi\)
−0.596761 + 0.802419i \(0.703546\pi\)
\(522\) 0 0
\(523\) −11.9706 −0.523436 −0.261718 0.965144i \(-0.584289\pi\)
−0.261718 + 0.965144i \(0.584289\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.1716 −0.573763
\(528\) 0 0
\(529\) 4.48528 0.195012
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.85786 −0.253732
\(534\) 0 0
\(535\) 12.8284 0.554621
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.41421 −0.103988
\(540\) 0 0
\(541\) −24.7990 −1.06619 −0.533096 0.846055i \(-0.678972\pi\)
−0.533096 + 0.846055i \(0.678972\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 44.2132 1.89389
\(546\) 0 0
\(547\) −16.9706 −0.725609 −0.362804 0.931865i \(-0.618181\pi\)
−0.362804 + 0.931865i \(0.618181\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.8284 0.461307
\(552\) 0 0
\(553\) −4.82843 −0.205326
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.7696 1.30375 0.651874 0.758327i \(-0.273983\pi\)
0.651874 + 0.758327i \(0.273983\pi\)
\(558\) 0 0
\(559\) −13.6569 −0.577623
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −23.7990 −1.00301 −0.501504 0.865155i \(-0.667220\pi\)
−0.501504 + 0.865155i \(0.667220\pi\)
\(564\) 0 0
\(565\) 29.3137 1.23324
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.79899 −0.410795 −0.205398 0.978679i \(-0.565849\pi\)
−0.205398 + 0.978679i \(0.565849\pi\)
\(570\) 0 0
\(571\) −12.6274 −0.528441 −0.264220 0.964462i \(-0.585115\pi\)
−0.264220 + 0.964462i \(0.585115\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.34315 −0.181122
\(576\) 0 0
\(577\) −39.9411 −1.66277 −0.831385 0.555696i \(-0.812452\pi\)
−0.831385 + 0.555696i \(0.812452\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.8284 0.449239
\(582\) 0 0
\(583\) 9.65685 0.399946
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −31.4558 −1.29832 −0.649161 0.760651i \(-0.724880\pi\)
−0.649161 + 0.760651i \(0.724880\pi\)
\(588\) 0 0
\(589\) 17.8284 0.734608
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −19.2426 −0.790201 −0.395100 0.918638i \(-0.629290\pi\)
−0.395100 + 0.918638i \(0.629290\pi\)
\(594\) 0 0
\(595\) −6.82843 −0.279938
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −27.0416 −1.10489 −0.552446 0.833549i \(-0.686305\pi\)
−0.552446 + 0.833549i \(0.686305\pi\)
\(600\) 0 0
\(601\) −9.85786 −0.402111 −0.201055 0.979580i \(-0.564437\pi\)
−0.201055 + 0.979580i \(0.564437\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.4853 −0.507599
\(606\) 0 0
\(607\) 25.6569 1.04138 0.520690 0.853746i \(-0.325675\pi\)
0.520690 + 0.853746i \(0.325675\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.970563 −0.0392648
\(612\) 0 0
\(613\) 8.02944 0.324306 0.162153 0.986766i \(-0.448156\pi\)
0.162153 + 0.986766i \(0.448156\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −46.2843 −1.86333 −0.931667 0.363313i \(-0.881646\pi\)
−0.931667 + 0.363313i \(0.881646\pi\)
\(618\) 0 0
\(619\) −40.3137 −1.62034 −0.810172 0.586192i \(-0.800627\pi\)
−0.810172 + 0.586192i \(0.800627\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.24264 −0.0497853
\(624\) 0 0
\(625\) −28.4558 −1.13823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.485281 0.0193494
\(630\) 0 0
\(631\) −3.85786 −0.153579 −0.0767896 0.997047i \(-0.524467\pi\)
−0.0767896 + 0.997047i \(0.524467\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.82843 0.270978
\(636\) 0 0
\(637\) 2.82843 0.112066
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5.51472 −0.217818 −0.108909 0.994052i \(-0.534736\pi\)
−0.108909 + 0.994052i \(0.534736\pi\)
\(642\) 0 0
\(643\) 3.48528 0.137446 0.0687230 0.997636i \(-0.478108\pi\)
0.0687230 + 0.997636i \(0.478108\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.6863 0.420121 0.210061 0.977688i \(-0.432634\pi\)
0.210061 + 0.977688i \(0.432634\pi\)
\(648\) 0 0
\(649\) −8.82843 −0.346546
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.02944 0.196817 0.0984085 0.995146i \(-0.468625\pi\)
0.0984085 + 0.995146i \(0.468625\pi\)
\(654\) 0 0
\(655\) −4.82843 −0.188662
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.8995 1.43740 0.718700 0.695320i \(-0.244737\pi\)
0.718700 + 0.695320i \(0.244737\pi\)
\(660\) 0 0
\(661\) 10.8284 0.421177 0.210589 0.977575i \(-0.432462\pi\)
0.210589 + 0.977575i \(0.432462\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9.24264 0.358414
\(666\) 0 0
\(667\) 14.8284 0.574159
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 27.3137 1.05443
\(672\) 0 0
\(673\) 34.2843 1.32156 0.660781 0.750579i \(-0.270225\pi\)
0.660781 + 0.750579i \(0.270225\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.04163 −0.193766 −0.0968828 0.995296i \(-0.530887\pi\)
−0.0968828 + 0.995296i \(0.530887\pi\)
\(678\) 0 0
\(679\) 9.65685 0.370596
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.8995 −0.493585 −0.246793 0.969068i \(-0.579377\pi\)
−0.246793 + 0.969068i \(0.579377\pi\)
\(684\) 0 0
\(685\) −6.82843 −0.260901
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −11.3137 −0.431018
\(690\) 0 0
\(691\) −24.6863 −0.939111 −0.469555 0.882903i \(-0.655586\pi\)
−0.469555 + 0.882903i \(0.655586\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −5.85786 −0.221882
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31.5147 −1.19029 −0.595147 0.803617i \(-0.702906\pi\)
−0.595147 + 0.803617i \(0.702906\pi\)
\(702\) 0 0
\(703\) −0.656854 −0.0247737
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.4853 0.469557
\(708\) 0 0
\(709\) 26.3137 0.988232 0.494116 0.869396i \(-0.335492\pi\)
0.494116 + 0.869396i \(0.335492\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.4142 0.914319
\(714\) 0 0
\(715\) −16.4853 −0.616515
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.65685 −0.285552 −0.142776 0.989755i \(-0.545603\pi\)
−0.142776 + 0.989755i \(0.545603\pi\)
\(720\) 0 0
\(721\) −6.65685 −0.247914
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.34315 −0.0870222
\(726\) 0 0
\(727\) −6.62742 −0.245797 −0.122899 0.992419i \(-0.539219\pi\)
−0.122899 + 0.992419i \(0.539219\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.6569 −0.505117
\(732\) 0 0
\(733\) −0.828427 −0.0305987 −0.0152993 0.999883i \(-0.504870\pi\)
−0.0152993 + 0.999883i \(0.504870\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.8284 −0.398870
\(738\) 0 0
\(739\) 30.9706 1.13927 0.569635 0.821898i \(-0.307084\pi\)
0.569635 + 0.821898i \(0.307084\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.8995 −0.619982 −0.309991 0.950739i \(-0.600326\pi\)
−0.309991 + 0.950739i \(0.600326\pi\)
\(744\) 0 0
\(745\) 7.65685 0.280525
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.31371 −0.194159
\(750\) 0 0
\(751\) −50.4853 −1.84223 −0.921117 0.389286i \(-0.872722\pi\)
−0.921117 + 0.389286i \(0.872722\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.8284 −0.466874
\(756\) 0 0
\(757\) 30.2843 1.10070 0.550350 0.834934i \(-0.314494\pi\)
0.550350 + 0.834934i \(0.314494\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 47.7990 1.73271 0.866356 0.499427i \(-0.166456\pi\)
0.866356 + 0.499427i \(0.166456\pi\)
\(762\) 0 0
\(763\) −18.3137 −0.663001
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.3431 0.373469
\(768\) 0 0
\(769\) 1.85786 0.0669963 0.0334982 0.999439i \(-0.489335\pi\)
0.0334982 + 0.999439i \(0.489335\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −38.4142 −1.38166 −0.690832 0.723016i \(-0.742755\pi\)
−0.690832 + 0.723016i \(0.742755\pi\)
\(774\) 0 0
\(775\) −3.85786 −0.138579
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.92893 0.284083
\(780\) 0 0
\(781\) 13.4853 0.482541
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −34.1421 −1.21859
\(786\) 0 0
\(787\) 25.9411 0.924701 0.462351 0.886697i \(-0.347006\pi\)
0.462351 + 0.886697i \(0.347006\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.1421 −0.431725
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 40.8995 1.44873 0.724367 0.689414i \(-0.242132\pi\)
0.724367 + 0.689414i \(0.242132\pi\)
\(798\) 0 0
\(799\) −0.970563 −0.0343360
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.65685 0.270205
\(804\) 0 0
\(805\) 12.6569 0.446095
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −48.4853 −1.70465 −0.852326 0.523011i \(-0.824809\pi\)
−0.852326 + 0.523011i \(0.824809\pi\)
\(810\) 0 0
\(811\) 9.34315 0.328082 0.164041 0.986454i \(-0.447547\pi\)
0.164041 + 0.986454i \(0.447547\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 39.7990 1.39410
\(816\) 0 0
\(817\) 18.4853 0.646718
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −6.28427 −0.219323 −0.109661 0.993969i \(-0.534977\pi\)
−0.109661 + 0.993969i \(0.534977\pi\)
\(822\) 0 0
\(823\) 48.6274 1.69505 0.847523 0.530759i \(-0.178093\pi\)
0.847523 + 0.530759i \(0.178093\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −7.58579 −0.263784 −0.131892 0.991264i \(-0.542105\pi\)
−0.131892 + 0.991264i \(0.542105\pi\)
\(828\) 0 0
\(829\) 8.14214 0.282788 0.141394 0.989953i \(-0.454842\pi\)
0.141394 + 0.989953i \(0.454842\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.82843 0.0979992
\(834\) 0 0
\(835\) −3.65685 −0.126551
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 34.1421 1.17872 0.589359 0.807871i \(-0.299380\pi\)
0.589359 + 0.807871i \(0.299380\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.0711 −0.415257
\(846\) 0 0
\(847\) 5.17157 0.177697
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −0.899495 −0.0308343
\(852\) 0 0
\(853\) 11.0294 0.377641 0.188820 0.982012i \(-0.439534\pi\)
0.188820 + 0.982012i \(0.439534\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.44365 −0.254270 −0.127135 0.991885i \(-0.540578\pi\)
−0.127135 + 0.991885i \(0.540578\pi\)
\(858\) 0 0
\(859\) 46.1127 1.57334 0.786672 0.617371i \(-0.211802\pi\)
0.786672 + 0.617371i \(0.211802\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 35.6569 1.21377 0.606887 0.794788i \(-0.292418\pi\)
0.606887 + 0.794788i \(0.292418\pi\)
\(864\) 0 0
\(865\) 22.3137 0.758689
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −11.6569 −0.395432
\(870\) 0 0
\(871\) 12.6863 0.429859
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.0711 0.340464
\(876\) 0 0
\(877\) 39.9411 1.34872 0.674358 0.738405i \(-0.264420\pi\)
0.674358 + 0.738405i \(0.264420\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 31.7279 1.06894 0.534470 0.845187i \(-0.320511\pi\)
0.534470 + 0.845187i \(0.320511\pi\)
\(882\) 0 0
\(883\) 16.6863 0.561538 0.280769 0.959775i \(-0.409410\pi\)
0.280769 + 0.959775i \(0.409410\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.7696 1.10029 0.550147 0.835068i \(-0.314572\pi\)
0.550147 + 0.835068i \(0.314572\pi\)
\(888\) 0 0
\(889\) −2.82843 −0.0948624
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.31371 0.0439616
\(894\) 0 0
\(895\) −47.4558 −1.58627
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.1716 0.439297
\(900\) 0 0
\(901\) −11.3137 −0.376914
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 47.4558 1.57749
\(906\) 0 0
\(907\) −10.6863 −0.354832 −0.177416 0.984136i \(-0.556774\pi\)
−0.177416 + 0.984136i \(0.556774\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 9.02944 0.299159 0.149579 0.988750i \(-0.452208\pi\)
0.149579 + 0.988750i \(0.452208\pi\)
\(912\) 0 0
\(913\) 26.1421 0.865178
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.00000 0.0660458
\(918\) 0 0
\(919\) −30.3431 −1.00093 −0.500464 0.865757i \(-0.666837\pi\)
−0.500464 + 0.865757i \(0.666837\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −15.7990 −0.520030
\(924\) 0 0
\(925\) 0.142136 0.00467339
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46.1421 1.51387 0.756937 0.653488i \(-0.226695\pi\)
0.756937 + 0.653488i \(0.226695\pi\)
\(930\) 0 0
\(931\) −3.82843 −0.125472
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −16.4853 −0.539126
\(936\) 0 0
\(937\) −11.3137 −0.369603 −0.184801 0.982776i \(-0.559164\pi\)
−0.184801 + 0.982776i \(0.559164\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7.38478 0.240737 0.120368 0.992729i \(-0.461592\pi\)
0.120368 + 0.992729i \(0.461592\pi\)
\(942\) 0 0
\(943\) 10.8579 0.353581
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.72792 0.121141 0.0605706 0.998164i \(-0.480708\pi\)
0.0605706 + 0.998164i \(0.480708\pi\)
\(948\) 0 0
\(949\) −8.97056 −0.291197
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −53.1716 −1.72240 −0.861198 0.508269i \(-0.830285\pi\)
−0.861198 + 0.508269i \(0.830285\pi\)
\(954\) 0 0
\(955\) −29.9706 −0.969825
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.82843 0.0913347
\(960\) 0 0
\(961\) −9.31371 −0.300442
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.82843 −0.284197
\(966\) 0 0
\(967\) −10.2843 −0.330720 −0.165360 0.986233i \(-0.552879\pi\)
−0.165360 + 0.986233i \(0.552879\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 41.1716 1.32126 0.660629 0.750712i \(-0.270290\pi\)
0.660629 + 0.750712i \(0.270290\pi\)
\(972\) 0 0
\(973\) 1.65685 0.0531163
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.0294 0.480834 0.240417 0.970670i \(-0.422716\pi\)
0.240417 + 0.970670i \(0.422716\pi\)
\(978\) 0 0
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 25.5980 0.816449 0.408224 0.912882i \(-0.366148\pi\)
0.408224 + 0.912882i \(0.366148\pi\)
\(984\) 0 0
\(985\) −59.9411 −1.90988
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25.3137 0.804929
\(990\) 0 0
\(991\) 37.4558 1.18982 0.594912 0.803791i \(-0.297187\pi\)
0.594912 + 0.803791i \(0.297187\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.41421 0.266749
\(996\) 0 0
\(997\) 22.6863 0.718482 0.359241 0.933245i \(-0.383036\pi\)
0.359241 + 0.933245i \(0.383036\pi\)
\(998\) 0 0
\(999\) 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 6048.2.a.bg.1.2 yes 2
3.2 odd 2 6048.2.a.z.1.1 2
4.3 odd 2 6048.2.a.bj.1.2 yes 2
12.11 even 2 6048.2.a.ba.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.z.1.1 2 3.2 odd 2
6048.2.a.ba.1.1 yes 2 12.11 even 2
6048.2.a.bg.1.2 yes 2 1.1 even 1 trivial
6048.2.a.bj.1.2 yes 2 4.3 odd 2