Properties

Label 1568.2.a.q.1.1
Level $1568$
Weight $2$
Character 1568.1
Self dual yes
Analytic conductor $12.521$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5205430369\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1568.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{3} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{3} -1.00000 q^{9} -2.00000 q^{11} +2.82843 q^{13} +4.24264 q^{17} +4.24264 q^{19} -8.00000 q^{23} -5.00000 q^{25} +5.65685 q^{27} +6.00000 q^{29} -8.48528 q^{31} +2.82843 q^{33} -2.00000 q^{37} -4.00000 q^{39} -4.24264 q^{41} -6.00000 q^{43} +2.82843 q^{47} -6.00000 q^{51} +6.00000 q^{53} -6.00000 q^{57} -12.7279 q^{59} -5.65685 q^{61} -12.0000 q^{67} +11.3137 q^{69} +4.00000 q^{71} -1.41421 q^{73} +7.07107 q^{75} -12.0000 q^{79} -5.00000 q^{81} +9.89949 q^{83} -8.48528 q^{87} +4.24264 q^{89} +12.0000 q^{93} -18.3848 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 4q^{11} - 16q^{23} - 10q^{25} + 12q^{29} - 4q^{37} - 8q^{39} - 12q^{43} - 12q^{51} + 12q^{53} - 12q^{57} - 24q^{67} + 8q^{71} - 24q^{79} - 10q^{81} + 24q^{93} + 4q^{99} + O(q^{100}) \)

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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 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
\(17\) 4.24264 1.02899 0.514496 0.857493i \(-0.327979\pi\)
0.514496 + 0.857493i \(0.327979\pi\)
\(18\) 0 0
\(19\) 4.24264 0.973329 0.486664 0.873589i \(-0.338214\pi\)
0.486664 + 0.873589i \(0.338214\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.48528 −1.52400 −0.762001 0.647576i \(-0.775783\pi\)
−0.762001 + 0.647576i \(0.775783\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −4.24264 −0.662589 −0.331295 0.943527i \(-0.607485\pi\)
−0.331295 + 0.943527i \(0.607485\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −12.7279 −1.65703 −0.828517 0.559964i \(-0.810815\pi\)
−0.828517 + 0.559964i \(0.810815\pi\)
\(60\) 0 0
\(61\) −5.65685 −0.724286 −0.362143 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 11.3137 1.36201
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −1.41421 −0.165521 −0.0827606 0.996569i \(-0.526374\pi\)
−0.0827606 + 0.996569i \(0.526374\pi\)
\(74\) 0 0
\(75\) 7.07107 0.816497
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 9.89949 1.08661 0.543305 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.48528 −0.909718
\(88\) 0 0
\(89\) 4.24264 0.449719 0.224860 0.974391i \(-0.427808\pi\)
0.224860 + 0.974391i \(0.427808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.3848 −1.86669 −0.933346 0.358979i \(-0.883125\pi\)
−0.933346 + 0.358979i \(0.883125\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −16.9706 −1.68863 −0.844317 0.535844i \(-0.819994\pi\)
−0.844317 + 0.535844i \(0.819994\pi\)
\(102\) 0 0
\(103\) 8.48528 0.836080 0.418040 0.908429i \(-0.362717\pi\)
0.418040 + 0.908429i \(0.362717\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 2.82843 0.268462
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.82843 −0.261488
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 8.48528 0.747087
\(130\) 0 0
\(131\) 1.41421 0.123560 0.0617802 0.998090i \(-0.480322\pi\)
0.0617802 + 0.998090i \(0.480322\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −12.7279 −1.07957 −0.539784 0.841803i \(-0.681494\pi\)
−0.539784 + 0.841803i \(0.681494\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) −4.24264 −0.342997
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.82843 −0.225733 −0.112867 0.993610i \(-0.536003\pi\)
−0.112867 + 0.993610i \(0.536003\pi\)
\(158\) 0 0
\(159\) −8.48528 −0.672927
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.1421 1.09435 0.547176 0.837018i \(-0.315703\pi\)
0.547176 + 0.837018i \(0.315703\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) −4.24264 −0.324443
\(172\) 0 0
\(173\) −8.48528 −0.645124 −0.322562 0.946548i \(-0.604544\pi\)
−0.322562 + 0.946548i \(0.604544\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 18.0000 1.35296
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 14.1421 1.05118 0.525588 0.850739i \(-0.323845\pi\)
0.525588 + 0.850739i \(0.323845\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.48528 −0.620505
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 25.4558 1.80452 0.902258 0.431196i \(-0.141908\pi\)
0.902258 + 0.431196i \(0.141908\pi\)
\(200\) 0 0
\(201\) 16.9706 1.19701
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) −8.48528 −0.586939
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −5.65685 −0.387601
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −16.9706 −1.13643 −0.568216 0.822879i \(-0.692366\pi\)
−0.568216 + 0.822879i \(0.692366\pi\)
\(224\) 0 0
\(225\) 5.00000 0.333333
\(226\) 0 0
\(227\) 12.7279 0.844782 0.422391 0.906414i \(-0.361191\pi\)
0.422391 + 0.906414i \(0.361191\pi\)
\(228\) 0 0
\(229\) 2.82843 0.186908 0.0934539 0.995624i \(-0.470209\pi\)
0.0934539 + 0.995624i \(0.470209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.9706 1.10236
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 18.3848 1.18427 0.592134 0.805840i \(-0.298286\pi\)
0.592134 + 0.805840i \(0.298286\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) −12.7279 −0.803379 −0.401690 0.915776i \(-0.631577\pi\)
−0.401690 + 0.915776i \(0.631577\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −21.2132 −1.32324 −0.661622 0.749838i \(-0.730131\pi\)
−0.661622 + 0.749838i \(0.730131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 20.0000 1.23325 0.616626 0.787256i \(-0.288499\pi\)
0.616626 + 0.787256i \(0.288499\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 25.4558 1.55207 0.776035 0.630690i \(-0.217228\pi\)
0.776035 + 0.630690i \(0.217228\pi\)
\(270\) 0 0
\(271\) 16.9706 1.03089 0.515444 0.856923i \(-0.327627\pi\)
0.515444 + 0.856923i \(0.327627\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.0000 0.603023
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) 8.48528 0.508001
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 21.2132 1.26099 0.630497 0.776192i \(-0.282851\pi\)
0.630497 + 0.776192i \(0.282851\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 26.0000 1.52415
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −11.3137 −0.656488
\(298\) 0 0
\(299\) −22.6274 −1.30858
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 24.0000 1.37876
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −21.2132 −1.21070 −0.605351 0.795959i \(-0.706967\pi\)
−0.605351 + 0.795959i \(0.706967\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 11.3137 0.641542 0.320771 0.947157i \(-0.396058\pi\)
0.320771 + 0.947157i \(0.396058\pi\)
\(312\) 0 0
\(313\) 18.3848 1.03917 0.519584 0.854419i \(-0.326087\pi\)
0.519584 + 0.854419i \(0.326087\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 5.65685 0.315735
\(322\) 0 0
\(323\) 18.0000 1.00155
\(324\) 0 0
\(325\) −14.1421 −0.784465
\(326\) 0 0
\(327\) −25.4558 −1.40771
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 18.0000 0.989369 0.494685 0.869072i \(-0.335284\pi\)
0.494685 + 0.869072i \(0.335284\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) 16.9706 0.921714
\(340\) 0 0
\(341\) 16.9706 0.919007
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.0000 0.536828 0.268414 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 0 0
\(349\) −19.7990 −1.05982 −0.529908 0.848055i \(-0.677773\pi\)
−0.529908 + 0.848055i \(0.677773\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) 21.2132 1.12906 0.564532 0.825411i \(-0.309057\pi\)
0.564532 + 0.825411i \(0.309057\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 9.89949 0.519589
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 33.9411 1.77171 0.885856 0.463960i \(-0.153572\pi\)
0.885856 + 0.463960i \(0.153572\pi\)
\(368\) 0 0
\(369\) 4.24264 0.220863
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.9706 0.874028
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) 0 0
\(381\) 16.9706 0.869428
\(382\) 0 0
\(383\) −25.4558 −1.30073 −0.650366 0.759621i \(-0.725385\pi\)
−0.650366 + 0.759621i \(0.725385\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −33.9411 −1.71648
\(392\) 0 0
\(393\) −2.00000 −0.100887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −36.7696 −1.84541 −0.922705 0.385506i \(-0.874027\pi\)
−0.922705 + 0.385506i \(0.874027\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −35.3553 −1.74821 −0.874105 0.485738i \(-0.838551\pi\)
−0.874105 + 0.485738i \(0.838551\pi\)
\(410\) 0 0
\(411\) 16.9706 0.837096
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 18.0000 0.881464
\(418\) 0 0
\(419\) 12.7279 0.621800 0.310900 0.950443i \(-0.399370\pi\)
0.310900 + 0.950443i \(0.399370\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) −21.2132 −1.02899
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 0 0
\(433\) −1.41421 −0.0679628 −0.0339814 0.999422i \(-0.510819\pi\)
−0.0339814 + 0.999422i \(0.510819\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −33.9411 −1.62362
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8.48528 0.401340
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 8.48528 0.399556
\(452\) 0 0
\(453\) 16.9706 0.797347
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) −25.4558 −1.18560 −0.592798 0.805351i \(-0.701977\pi\)
−0.592798 + 0.805351i \(0.701977\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.7279 −0.588978 −0.294489 0.955655i \(-0.595149\pi\)
−0.294489 + 0.955655i \(0.595149\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) −21.2132 −0.973329
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −2.82843 −0.129234 −0.0646171 0.997910i \(-0.520583\pi\)
−0.0646171 + 0.997910i \(0.520583\pi\)
\(480\) 0 0
\(481\) −5.65685 −0.257930
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 8.48528 0.383718
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 25.4558 1.14647
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.07107 0.314037
\(508\) 0 0
\(509\) 42.4264 1.88052 0.940259 0.340461i \(-0.110583\pi\)
0.940259 + 0.340461i \(0.110583\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 24.0000 1.05963
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) −29.6985 −1.30111 −0.650557 0.759457i \(-0.725465\pi\)
−0.650557 + 0.759457i \(0.725465\pi\)
\(522\) 0 0
\(523\) 12.7279 0.556553 0.278277 0.960501i \(-0.410237\pi\)
0.278277 + 0.960501i \(0.410237\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −36.0000 −1.56818
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 12.7279 0.552345
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.65685 −0.244111
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) −20.0000 −0.858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −42.0000 −1.79579 −0.897895 0.440209i \(-0.854904\pi\)
−0.897895 + 0.440209i \(0.854904\pi\)
\(548\) 0 0
\(549\) 5.65685 0.241429
\(550\) 0 0
\(551\) 25.4558 1.08446
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) −16.9706 −0.717778
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) −12.7279 −0.536418 −0.268209 0.963361i \(-0.586432\pi\)
−0.268209 + 0.963361i \(0.586432\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −42.0000 −1.75765 −0.878823 0.477149i \(-0.841670\pi\)
−0.878823 + 0.477149i \(0.841670\pi\)
\(572\) 0 0
\(573\) −28.2843 −1.18159
\(574\) 0 0
\(575\) 40.0000 1.66812
\(576\) 0 0
\(577\) 1.41421 0.0588745 0.0294372 0.999567i \(-0.490628\pi\)
0.0294372 + 0.999567i \(0.490628\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 38.1838 1.57601 0.788006 0.615667i \(-0.211113\pi\)
0.788006 + 0.615667i \(0.211113\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 0 0
\(591\) −25.4558 −1.04711
\(592\) 0 0
\(593\) −29.6985 −1.21957 −0.609785 0.792567i \(-0.708744\pi\)
−0.609785 + 0.792567i \(0.708744\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −36.0000 −1.47338
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 18.3848 0.749931 0.374965 0.927039i \(-0.377655\pi\)
0.374965 + 0.927039i \(0.377655\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −4.24264 −0.170526 −0.0852631 0.996358i \(-0.527173\pi\)
−0.0852631 + 0.996358i \(0.527173\pi\)
\(620\) 0 0
\(621\) −45.2548 −1.81601
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 12.0000 0.479234
\(628\) 0 0
\(629\) −8.48528 −0.338330
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 0 0
\(633\) −16.9706 −0.674519
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −12.7279 −0.501940 −0.250970 0.967995i \(-0.580750\pi\)
−0.250970 + 0.967995i \(0.580750\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.7990 −0.778379 −0.389189 0.921158i \(-0.627245\pi\)
−0.389189 + 0.921158i \(0.627245\pi\)
\(648\) 0 0
\(649\) 25.4558 0.999229
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.41421 0.0551737
\(658\) 0 0
\(659\) −34.0000 −1.32445 −0.662226 0.749304i \(-0.730388\pi\)
−0.662226 + 0.749304i \(0.730388\pi\)
\(660\) 0 0
\(661\) 22.6274 0.880105 0.440052 0.897972i \(-0.354960\pi\)
0.440052 + 0.897972i \(0.354960\pi\)
\(662\) 0 0
\(663\) −16.9706 −0.659082
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) 0 0
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 11.3137 0.436761
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) −28.2843 −1.08866
\(676\) 0 0
\(677\) 42.4264 1.63058 0.815290 0.579053i \(-0.196578\pi\)
0.815290 + 0.579053i \(0.196578\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) 16.9706 0.646527
\(690\) 0 0
\(691\) 46.6690 1.77537 0.887687 0.460447i \(-0.152311\pi\)
0.887687 + 0.460447i \(0.152311\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) 33.9411 1.28377
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −8.48528 −0.320028
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 67.8823 2.54221
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.3137 0.422518
\(718\) 0 0
\(719\) 25.4558 0.949343 0.474671 0.880163i \(-0.342567\pi\)
0.474671 + 0.880163i \(0.342567\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −26.0000 −0.966950
\(724\) 0 0
\(725\) −30.0000 −1.11417
\(726\) 0 0
\(727\) −8.48528 −0.314702 −0.157351 0.987543i \(-0.550295\pi\)
−0.157351 + 0.987543i \(0.550295\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) −25.4558 −0.941518
\(732\) 0 0
\(733\) 28.2843 1.04470 0.522352 0.852730i \(-0.325055\pi\)
0.522352 + 0.852730i \(0.325055\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) −16.9706 −0.623429
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −9.89949 −0.362204
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 0 0
\(759\) −22.6274 −0.821323
\(760\) 0 0
\(761\) −12.7279 −0.461387 −0.230693 0.973026i \(-0.574099\pi\)
−0.230693 + 0.973026i \(0.574099\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −36.0000 −1.29988
\(768\) 0 0
\(769\) 35.3553 1.27495 0.637473 0.770473i \(-0.279980\pi\)
0.637473 + 0.770473i \(0.279980\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 42.4264 1.52400
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 33.9411 1.21296
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −12.7279 −0.453701 −0.226851 0.973930i \(-0.572843\pi\)
−0.226851 + 0.973930i \(0.572843\pi\)
\(788\) 0 0
\(789\) −28.2843 −1.00695
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.4264 −1.50282 −0.751410 0.659835i \(-0.770626\pi\)
−0.751410 + 0.659835i \(0.770626\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) −4.24264 −0.149906
\(802\) 0 0
\(803\) 2.82843 0.0998130
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −36.0000 −1.26726
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −29.6985 −1.04285 −0.521427 0.853296i \(-0.674600\pi\)
−0.521427 + 0.853296i \(0.674600\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −25.4558 −0.890587
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) −14.1421 −0.492366
\(826\) 0 0
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) 39.5980 1.37529 0.687647 0.726045i \(-0.258644\pi\)
0.687647 + 0.726045i \(0.258644\pi\)
\(830\) 0 0
\(831\) 25.4558 0.883053
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −48.0000 −1.65912
\(838\) 0 0
\(839\) 48.0833 1.66002 0.830009 0.557750i \(-0.188335\pi\)
0.830009 + 0.557750i \(0.188335\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −30.0000 −1.02960
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 19.7990 0.677905 0.338952 0.940804i \(-0.389927\pi\)
0.338952 + 0.940804i \(0.389927\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.7279 0.434778 0.217389 0.976085i \(-0.430246\pi\)
0.217389 + 0.976085i \(0.430246\pi\)
\(858\) 0 0
\(859\) −4.24264 −0.144757 −0.0723785 0.997377i \(-0.523059\pi\)
−0.0723785 + 0.997377i \(0.523059\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.41421 −0.0480292
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) −33.9411 −1.15005
\(872\) 0 0
\(873\) 18.3848 0.622230
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.24264 −0.142938 −0.0714691 0.997443i \(-0.522769\pi\)
−0.0714691 + 0.997443i \(0.522769\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.7696 −1.23460 −0.617300 0.786728i \(-0.711774\pi\)
−0.617300 + 0.786728i \(0.711774\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 10.0000 0.335013
\(892\) 0 0
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 32.0000 1.06845
\(898\) 0 0
\(899\) −50.9117 −1.69800
\(900\) 0 0
\(901\) 25.4558 0.848057
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 0 0
\(909\) 16.9706 0.562878
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) −19.7990 −0.655251
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 30.0000 0.988534
\(922\) 0 0
\(923\) 11.3137 0.372395
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 0 0
\(927\) −8.48528 −0.278693
\(928\) 0 0
\(929\) 4.24264 0.139197 0.0695983 0.997575i \(-0.477828\pi\)
0.0695983 + 0.997575i \(0.477828\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24.0416 −0.785406 −0.392703 0.919665i \(-0.628460\pi\)
−0.392703 + 0.919665i \(0.628460\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 33.9411 1.10645 0.553225 0.833032i \(-0.313397\pi\)
0.553225 + 0.833032i \(0.313397\pi\)
\(942\) 0 0
\(943\) 33.9411 1.10528
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.0000 −1.10485 −0.552426 0.833562i \(-0.686298\pi\)
−0.552426 + 0.833562i \(0.686298\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) −25.4558 −0.825462
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 16.9706 0.548580
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) −25.4558 −0.817760
\(970\) 0 0
\(971\) 43.8406 1.40691 0.703456 0.710739i \(-0.251639\pi\)
0.703456 + 0.710739i \(0.251639\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 20.0000 0.640513
\(976\) 0 0
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) −8.48528 −0.271191
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 0 0
\(983\) −25.4558 −0.811915 −0.405958 0.913892i \(-0.633062\pi\)
−0.405958 + 0.913892i \(0.633062\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 0 0
\(993\) −25.4558 −0.807817
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −56.5685 −1.79154 −0.895772 0.444514i \(-0.853376\pi\)
−0.895772 + 0.444514i \(0.853376\pi\)
\(998\) 0 0
\(999\) −11.3137 −0.357950
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 1568.2.a.q.1.1 2
4.3 odd 2 1568.2.a.r.1.2 yes 2
7.2 even 3 1568.2.i.r.1537.2 4
7.3 odd 6 1568.2.i.r.961.1 4
7.4 even 3 1568.2.i.r.961.2 4
7.5 odd 6 1568.2.i.r.1537.1 4
7.6 odd 2 inner 1568.2.a.q.1.2 yes 2
8.3 odd 2 3136.2.a.bl.1.1 2
8.5 even 2 3136.2.a.bo.1.2 2
28.3 even 6 1568.2.i.q.961.2 4
28.11 odd 6 1568.2.i.q.961.1 4
28.19 even 6 1568.2.i.q.1537.2 4
28.23 odd 6 1568.2.i.q.1537.1 4
28.27 even 2 1568.2.a.r.1.1 yes 2
56.13 odd 2 3136.2.a.bo.1.1 2
56.27 even 2 3136.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1568.2.a.q.1.1 2 1.1 even 1 trivial
1568.2.a.q.1.2 yes 2 7.6 odd 2 inner
1568.2.a.r.1.1 yes 2 28.27 even 2
1568.2.a.r.1.2 yes 2 4.3 odd 2
1568.2.i.q.961.1 4 28.11 odd 6
1568.2.i.q.961.2 4 28.3 even 6
1568.2.i.q.1537.1 4 28.23 odd 6
1568.2.i.q.1537.2 4 28.19 even 6
1568.2.i.r.961.1 4 7.3 odd 6
1568.2.i.r.961.2 4 7.4 even 3
1568.2.i.r.1537.1 4 7.5 odd 6
1568.2.i.r.1537.2 4 7.2 even 3
3136.2.a.bl.1.1 2 8.3 odd 2
3136.2.a.bl.1.2 2 56.27 even 2
3136.2.a.bo.1.1 2 56.13 odd 2
3136.2.a.bo.1.2 2 8.5 even 2