Properties

Label 1620.2.a.g.1.1
Level $1620$
Weight $2$
Character 1620.1
Self dual yes
Analytic conductor $12.936$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9357651274\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1620.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -2.73205 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -2.73205 q^{7} +1.73205 q^{11} +5.46410 q^{13} -4.73205 q^{17} -4.46410 q^{19} +3.46410 q^{23} +1.00000 q^{25} -7.73205 q^{29} +5.92820 q^{31} +2.73205 q^{35} -6.19615 q^{37} -11.1962 q^{41} +3.26795 q^{43} -1.26795 q^{47} +0.464102 q^{49} -7.26795 q^{53} -1.73205 q^{55} -7.73205 q^{59} -4.00000 q^{61} -5.46410 q^{65} +6.39230 q^{67} -11.1962 q^{71} -0.196152 q^{73} -4.73205 q^{77} -14.3923 q^{79} -15.1244 q^{83} +4.73205 q^{85} +5.19615 q^{89} -14.9282 q^{91} +4.46410 q^{95} +0.732051 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 2 q^{25} - 12 q^{29} - 2 q^{31} + 2 q^{35} - 2 q^{37} - 12 q^{41} + 10 q^{43} - 6 q^{47} - 6 q^{49} - 18 q^{53} - 12 q^{59} - 8 q^{61} - 4 q^{65} - 8 q^{67} - 12 q^{71} + 10 q^{73} - 6 q^{77} - 8 q^{79} - 6 q^{83} + 6 q^{85} - 16 q^{91} + 2 q^{95} - 2 q^{97} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.73205 −1.03262 −0.516309 0.856402i \(-0.672694\pi\)
−0.516309 + 0.856402i \(0.672694\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.73205 0.522233 0.261116 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(12\) 0 0
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.73205 −1.14769 −0.573845 0.818964i \(-0.694549\pi\)
−0.573845 + 0.818964i \(0.694549\pi\)
\(18\) 0 0
\(19\) −4.46410 −1.02414 −0.512068 0.858945i \(-0.671120\pi\)
−0.512068 + 0.858945i \(0.671120\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.73205 −1.43581 −0.717903 0.696143i \(-0.754898\pi\)
−0.717903 + 0.696143i \(0.754898\pi\)
\(30\) 0 0
\(31\) 5.92820 1.06474 0.532368 0.846513i \(-0.321302\pi\)
0.532368 + 0.846513i \(0.321302\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.73205 0.461801
\(36\) 0 0
\(37\) −6.19615 −1.01864 −0.509321 0.860577i \(-0.670103\pi\)
−0.509321 + 0.860577i \(0.670103\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.1962 −1.74855 −0.874273 0.485435i \(-0.838661\pi\)
−0.874273 + 0.485435i \(0.838661\pi\)
\(42\) 0 0
\(43\) 3.26795 0.498358 0.249179 0.968458i \(-0.419839\pi\)
0.249179 + 0.968458i \(0.419839\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.26795 −0.184949 −0.0924747 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) 0 0
\(49\) 0.464102 0.0663002
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.26795 −0.998330 −0.499165 0.866507i \(-0.666360\pi\)
−0.499165 + 0.866507i \(0.666360\pi\)
\(54\) 0 0
\(55\) −1.73205 −0.233550
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.73205 −1.00663 −0.503314 0.864104i \(-0.667886\pi\)
−0.503314 + 0.864104i \(0.667886\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.46410 −0.677738
\(66\) 0 0
\(67\) 6.39230 0.780944 0.390472 0.920615i \(-0.372312\pi\)
0.390472 + 0.920615i \(0.372312\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.1962 −1.32874 −0.664369 0.747404i \(-0.731300\pi\)
−0.664369 + 0.747404i \(0.731300\pi\)
\(72\) 0 0
\(73\) −0.196152 −0.0229579 −0.0114790 0.999934i \(-0.503654\pi\)
−0.0114790 + 0.999934i \(0.503654\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.73205 −0.539267
\(78\) 0 0
\(79\) −14.3923 −1.61926 −0.809630 0.586940i \(-0.800332\pi\)
−0.809630 + 0.586940i \(0.800332\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.1244 −1.66011 −0.830057 0.557679i \(-0.811692\pi\)
−0.830057 + 0.557679i \(0.811692\pi\)
\(84\) 0 0
\(85\) 4.73205 0.513263
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −14.9282 −1.56490
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.46410 0.458007
\(96\) 0 0
\(97\) 0.732051 0.0743285 0.0371642 0.999309i \(-0.488168\pi\)
0.0371642 + 0.999309i \(0.488168\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.12436 0.609396 0.304698 0.952449i \(-0.401444\pi\)
0.304698 + 0.952449i \(0.401444\pi\)
\(102\) 0 0
\(103\) 18.3923 1.81225 0.906124 0.423013i \(-0.139027\pi\)
0.906124 + 0.423013i \(0.139027\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.46410 0.334887 0.167444 0.985882i \(-0.446449\pi\)
0.167444 + 0.985882i \(0.446449\pi\)
\(108\) 0 0
\(109\) −7.92820 −0.759384 −0.379692 0.925113i \(-0.623970\pi\)
−0.379692 + 0.925113i \(0.623970\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.339746 −0.0319606 −0.0159803 0.999872i \(-0.505087\pi\)
−0.0159803 + 0.999872i \(0.505087\pi\)
\(114\) 0 0
\(115\) −3.46410 −0.323029
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.9282 1.18513
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.19615 0.372348 0.186174 0.982517i \(-0.440391\pi\)
0.186174 + 0.982517i \(0.440391\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.2679 0.897115 0.448557 0.893754i \(-0.351938\pi\)
0.448557 + 0.893754i \(0.351938\pi\)
\(132\) 0 0
\(133\) 12.1962 1.05754
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.39230 0.375260 0.187630 0.982240i \(-0.439919\pi\)
0.187630 + 0.982240i \(0.439919\pi\)
\(138\) 0 0
\(139\) 15.3923 1.30556 0.652779 0.757548i \(-0.273603\pi\)
0.652779 + 0.757548i \(0.273603\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.46410 0.791428
\(144\) 0 0
\(145\) 7.73205 0.642112
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −17.3923 −1.41537 −0.707683 0.706530i \(-0.750259\pi\)
−0.707683 + 0.706530i \(0.750259\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.92820 −0.476165
\(156\) 0 0
\(157\) 3.26795 0.260811 0.130405 0.991461i \(-0.458372\pi\)
0.130405 + 0.991461i \(0.458372\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.46410 −0.745876
\(162\) 0 0
\(163\) 18.7321 1.46721 0.733604 0.679577i \(-0.237837\pi\)
0.733604 + 0.679577i \(0.237837\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −21.1244 −1.63465 −0.817326 0.576176i \(-0.804544\pi\)
−0.817326 + 0.576176i \(0.804544\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −24.2487 −1.84360 −0.921798 0.387671i \(-0.873280\pi\)
−0.921798 + 0.387671i \(0.873280\pi\)
\(174\) 0 0
\(175\) −2.73205 −0.206524
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.1244 −0.906217 −0.453108 0.891455i \(-0.649685\pi\)
−0.453108 + 0.891455i \(0.649685\pi\)
\(180\) 0 0
\(181\) −16.4641 −1.22377 −0.611884 0.790948i \(-0.709588\pi\)
−0.611884 + 0.790948i \(0.709588\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.19615 0.455550
\(186\) 0 0
\(187\) −8.19615 −0.599362
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0526 1.37859 0.689297 0.724479i \(-0.257919\pi\)
0.689297 + 0.724479i \(0.257919\pi\)
\(192\) 0 0
\(193\) −10.5885 −0.762174 −0.381087 0.924539i \(-0.624450\pi\)
−0.381087 + 0.924539i \(0.624450\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.8564 −0.987228 −0.493614 0.869681i \(-0.664324\pi\)
−0.493614 + 0.869681i \(0.664324\pi\)
\(198\) 0 0
\(199\) 15.8564 1.12403 0.562015 0.827127i \(-0.310026\pi\)
0.562015 + 0.827127i \(0.310026\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 21.1244 1.48264
\(204\) 0 0
\(205\) 11.1962 0.781973
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.73205 −0.534837
\(210\) 0 0
\(211\) −19.9282 −1.37191 −0.685957 0.727642i \(-0.740616\pi\)
−0.685957 + 0.727642i \(0.740616\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.26795 −0.222872
\(216\) 0 0
\(217\) −16.1962 −1.09947
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −25.8564 −1.73929
\(222\) 0 0
\(223\) −5.85641 −0.392174 −0.196087 0.980587i \(-0.562823\pi\)
−0.196087 + 0.980587i \(0.562823\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 26.1962 1.73870 0.869350 0.494197i \(-0.164538\pi\)
0.869350 + 0.494197i \(0.164538\pi\)
\(228\) 0 0
\(229\) −17.8564 −1.17998 −0.589992 0.807409i \(-0.700869\pi\)
−0.589992 + 0.807409i \(0.700869\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.5885 1.61084 0.805422 0.592702i \(-0.201939\pi\)
0.805422 + 0.592702i \(0.201939\pi\)
\(234\) 0 0
\(235\) 1.26795 0.0827119
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.53590 0.552141 0.276071 0.961137i \(-0.410968\pi\)
0.276071 + 0.961137i \(0.410968\pi\)
\(240\) 0 0
\(241\) 10.3205 0.664802 0.332401 0.943138i \(-0.392141\pi\)
0.332401 + 0.943138i \(0.392141\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.464102 −0.0296504
\(246\) 0 0
\(247\) −24.3923 −1.55205
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.5359 1.29621 0.648107 0.761549i \(-0.275561\pi\)
0.648107 + 0.761549i \(0.275561\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.4641 0.964624 0.482312 0.875999i \(-0.339797\pi\)
0.482312 + 0.875999i \(0.339797\pi\)
\(258\) 0 0
\(259\) 16.9282 1.05187
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.2487 −1.49524 −0.747620 0.664127i \(-0.768803\pi\)
−0.747620 + 0.664127i \(0.768803\pi\)
\(264\) 0 0
\(265\) 7.26795 0.446467
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.2679 −0.991874 −0.495937 0.868358i \(-0.665175\pi\)
−0.495937 + 0.868358i \(0.665175\pi\)
\(270\) 0 0
\(271\) 16.7846 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.73205 0.104447
\(276\) 0 0
\(277\) 5.12436 0.307893 0.153946 0.988079i \(-0.450802\pi\)
0.153946 + 0.988079i \(0.450802\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 29.3205 1.74911 0.874557 0.484922i \(-0.161152\pi\)
0.874557 + 0.484922i \(0.161152\pi\)
\(282\) 0 0
\(283\) 16.5359 0.982957 0.491479 0.870890i \(-0.336457\pi\)
0.491479 + 0.870890i \(0.336457\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 30.5885 1.80558
\(288\) 0 0
\(289\) 5.39230 0.317194
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.5885 1.43647 0.718237 0.695799i \(-0.244950\pi\)
0.718237 + 0.695799i \(0.244950\pi\)
\(294\) 0 0
\(295\) 7.73205 0.450177
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 18.9282 1.09465
\(300\) 0 0
\(301\) −8.92820 −0.514613
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −1.80385 −0.102951 −0.0514755 0.998674i \(-0.516392\pi\)
−0.0514755 + 0.998674i \(0.516392\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.5167 −0.936574 −0.468287 0.883576i \(-0.655129\pi\)
−0.468287 + 0.883576i \(0.655129\pi\)
\(312\) 0 0
\(313\) 14.9282 0.843792 0.421896 0.906644i \(-0.361365\pi\)
0.421896 + 0.906644i \(0.361365\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.12436 0.175481 0.0877406 0.996143i \(-0.472035\pi\)
0.0877406 + 0.996143i \(0.472035\pi\)
\(318\) 0 0
\(319\) −13.3923 −0.749825
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 21.1244 1.17539
\(324\) 0 0
\(325\) 5.46410 0.303094
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.46410 0.190982
\(330\) 0 0
\(331\) −29.3923 −1.61555 −0.807774 0.589493i \(-0.799328\pi\)
−0.807774 + 0.589493i \(0.799328\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.39230 −0.349249
\(336\) 0 0
\(337\) −34.2487 −1.86565 −0.932823 0.360335i \(-0.882662\pi\)
−0.932823 + 0.360335i \(0.882662\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.2679 0.556041
\(342\) 0 0
\(343\) 17.8564 0.964155
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.8038 1.17049 0.585246 0.810856i \(-0.300998\pi\)
0.585246 + 0.810856i \(0.300998\pi\)
\(348\) 0 0
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −28.9808 −1.54249 −0.771245 0.636538i \(-0.780366\pi\)
−0.771245 + 0.636538i \(0.780366\pi\)
\(354\) 0 0
\(355\) 11.1962 0.594230
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.33975 0.492933 0.246466 0.969151i \(-0.420730\pi\)
0.246466 + 0.969151i \(0.420730\pi\)
\(360\) 0 0
\(361\) 0.928203 0.0488528
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.196152 0.0102671
\(366\) 0 0
\(367\) −26.3923 −1.37767 −0.688834 0.724920i \(-0.741877\pi\)
−0.688834 + 0.724920i \(0.741877\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 19.8564 1.03089
\(372\) 0 0
\(373\) −14.0526 −0.727614 −0.363807 0.931474i \(-0.618523\pi\)
−0.363807 + 0.931474i \(0.618523\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −42.2487 −2.17592
\(378\) 0 0
\(379\) 4.53590 0.232993 0.116497 0.993191i \(-0.462834\pi\)
0.116497 + 0.993191i \(0.462834\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 18.2487 0.932466 0.466233 0.884662i \(-0.345611\pi\)
0.466233 + 0.884662i \(0.345611\pi\)
\(384\) 0 0
\(385\) 4.73205 0.241168
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 27.4641 1.39249 0.696243 0.717807i \(-0.254854\pi\)
0.696243 + 0.717807i \(0.254854\pi\)
\(390\) 0 0
\(391\) −16.3923 −0.828994
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.3923 0.724155
\(396\) 0 0
\(397\) 37.3205 1.87306 0.936531 0.350584i \(-0.114017\pi\)
0.936531 + 0.350584i \(0.114017\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.7846 −0.738308 −0.369154 0.929368i \(-0.620353\pi\)
−0.369154 + 0.929368i \(0.620353\pi\)
\(402\) 0 0
\(403\) 32.3923 1.61358
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.7321 −0.531968
\(408\) 0 0
\(409\) −17.8564 −0.882942 −0.441471 0.897275i \(-0.645543\pi\)
−0.441471 + 0.897275i \(0.645543\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 21.1244 1.03946
\(414\) 0 0
\(415\) 15.1244 0.742425
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.4641 1.04859 0.524295 0.851537i \(-0.324329\pi\)
0.524295 + 0.851537i \(0.324329\pi\)
\(420\) 0 0
\(421\) 13.7846 0.671821 0.335910 0.941894i \(-0.390956\pi\)
0.335910 + 0.941894i \(0.390956\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.73205 −0.229538
\(426\) 0 0
\(427\) 10.9282 0.528853
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −33.5885 −1.61790 −0.808950 0.587878i \(-0.799963\pi\)
−0.808950 + 0.587878i \(0.799963\pi\)
\(432\) 0 0
\(433\) 11.4641 0.550930 0.275465 0.961311i \(-0.411168\pi\)
0.275465 + 0.961311i \(0.411168\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −15.4641 −0.739748
\(438\) 0 0
\(439\) 6.60770 0.315368 0.157684 0.987490i \(-0.449597\pi\)
0.157684 + 0.987490i \(0.449597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 37.2679 1.77065 0.885327 0.464969i \(-0.153935\pi\)
0.885327 + 0.464969i \(0.153935\pi\)
\(444\) 0 0
\(445\) −5.19615 −0.246321
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.1244 1.13850 0.569249 0.822165i \(-0.307234\pi\)
0.569249 + 0.822165i \(0.307234\pi\)
\(450\) 0 0
\(451\) −19.3923 −0.913148
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 14.9282 0.699845
\(456\) 0 0
\(457\) 22.1962 1.03829 0.519146 0.854686i \(-0.326250\pi\)
0.519146 + 0.854686i \(0.326250\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.58846 0.446579 0.223289 0.974752i \(-0.428320\pi\)
0.223289 + 0.974752i \(0.428320\pi\)
\(462\) 0 0
\(463\) −2.39230 −0.111180 −0.0555899 0.998454i \(-0.517704\pi\)
−0.0555899 + 0.998454i \(0.517704\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.1962 1.21221 0.606107 0.795383i \(-0.292730\pi\)
0.606107 + 0.795383i \(0.292730\pi\)
\(468\) 0 0
\(469\) −17.4641 −0.806417
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.66025 0.260259
\(474\) 0 0
\(475\) −4.46410 −0.204827
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.33975 −0.152597 −0.0762984 0.997085i \(-0.524310\pi\)
−0.0762984 + 0.997085i \(0.524310\pi\)
\(480\) 0 0
\(481\) −33.8564 −1.54372
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.732051 −0.0332407
\(486\) 0 0
\(487\) −30.5359 −1.38371 −0.691857 0.722035i \(-0.743207\pi\)
−0.691857 + 0.722035i \(0.743207\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.26795 −0.192610 −0.0963049 0.995352i \(-0.530702\pi\)
−0.0963049 + 0.995352i \(0.530702\pi\)
\(492\) 0 0
\(493\) 36.5885 1.64786
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 30.5885 1.37208
\(498\) 0 0
\(499\) 27.3923 1.22625 0.613124 0.789987i \(-0.289913\pi\)
0.613124 + 0.789987i \(0.289913\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −35.3205 −1.57486 −0.787432 0.616402i \(-0.788590\pi\)
−0.787432 + 0.616402i \(0.788590\pi\)
\(504\) 0 0
\(505\) −6.12436 −0.272530
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −26.7846 −1.18721 −0.593603 0.804758i \(-0.702295\pi\)
−0.593603 + 0.804758i \(0.702295\pi\)
\(510\) 0 0
\(511\) 0.535898 0.0237067
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −18.3923 −0.810462
\(516\) 0 0
\(517\) −2.19615 −0.0965867
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) 0 0
\(523\) 3.60770 0.157753 0.0788767 0.996884i \(-0.474867\pi\)
0.0788767 + 0.996884i \(0.474867\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −28.0526 −1.22199
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −61.1769 −2.64987
\(534\) 0 0
\(535\) −3.46410 −0.149766
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.803848 0.0346242
\(540\) 0 0
\(541\) 2.46410 0.105940 0.0529700 0.998596i \(-0.483131\pi\)
0.0529700 + 0.998596i \(0.483131\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.92820 0.339607
\(546\) 0 0
\(547\) 4.78461 0.204575 0.102288 0.994755i \(-0.467384\pi\)
0.102288 + 0.994755i \(0.467384\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 34.5167 1.47046
\(552\) 0 0
\(553\) 39.3205 1.67208
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.39230 −0.186108 −0.0930540 0.995661i \(-0.529663\pi\)
−0.0930540 + 0.995661i \(0.529663\pi\)
\(558\) 0 0
\(559\) 17.8564 0.755246
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.7321 0.958042 0.479021 0.877804i \(-0.340992\pi\)
0.479021 + 0.877804i \(0.340992\pi\)
\(564\) 0 0
\(565\) 0.339746 0.0142932
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −43.0526 −1.80486 −0.902429 0.430839i \(-0.858218\pi\)
−0.902429 + 0.430839i \(0.858218\pi\)
\(570\) 0 0
\(571\) 0.856406 0.0358395 0.0179197 0.999839i \(-0.494296\pi\)
0.0179197 + 0.999839i \(0.494296\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.46410 0.144463
\(576\) 0 0
\(577\) 11.8038 0.491401 0.245700 0.969346i \(-0.420982\pi\)
0.245700 + 0.969346i \(0.420982\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 41.3205 1.71426
\(582\) 0 0
\(583\) −12.5885 −0.521361
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.80385 0.157002 0.0785008 0.996914i \(-0.474987\pi\)
0.0785008 + 0.996914i \(0.474987\pi\)
\(588\) 0 0
\(589\) −26.4641 −1.09043
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −23.0718 −0.947445 −0.473723 0.880674i \(-0.657090\pi\)
−0.473723 + 0.880674i \(0.657090\pi\)
\(594\) 0 0
\(595\) −12.9282 −0.530005
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.4115 −0.588840 −0.294420 0.955676i \(-0.595126\pi\)
−0.294420 + 0.955676i \(0.595126\pi\)
\(600\) 0 0
\(601\) −24.3205 −0.992054 −0.496027 0.868307i \(-0.665208\pi\)
−0.496027 + 0.868307i \(0.665208\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.00000 0.325246
\(606\) 0 0
\(607\) 2.58846 0.105062 0.0525311 0.998619i \(-0.483271\pi\)
0.0525311 + 0.998619i \(0.483271\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6.92820 −0.280285
\(612\) 0 0
\(613\) 24.3923 0.985196 0.492598 0.870257i \(-0.336047\pi\)
0.492598 + 0.870257i \(0.336047\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −14.1962 −0.568757
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 29.3205 1.16909
\(630\) 0 0
\(631\) −0.0717968 −0.00285818 −0.00142909 0.999999i \(-0.500455\pi\)
−0.00142909 + 0.999999i \(0.500455\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.19615 −0.166519
\(636\) 0 0
\(637\) 2.53590 0.100476
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12.8038 −0.505722 −0.252861 0.967503i \(-0.581371\pi\)
−0.252861 + 0.967503i \(0.581371\pi\)
\(642\) 0 0
\(643\) 14.5885 0.575313 0.287656 0.957734i \(-0.407124\pi\)
0.287656 + 0.957734i \(0.407124\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.248711 0.00977785 0.00488893 0.999988i \(-0.498444\pi\)
0.00488893 + 0.999988i \(0.498444\pi\)
\(648\) 0 0
\(649\) −13.3923 −0.525694
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.53590 −0.0992374 −0.0496187 0.998768i \(-0.515801\pi\)
−0.0496187 + 0.998768i \(0.515801\pi\)
\(654\) 0 0
\(655\) −10.2679 −0.401202
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.53590 −0.0987846 −0.0493923 0.998779i \(-0.515728\pi\)
−0.0493923 + 0.998779i \(0.515728\pi\)
\(660\) 0 0
\(661\) 15.3923 0.598691 0.299346 0.954145i \(-0.403232\pi\)
0.299346 + 0.954145i \(0.403232\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.1962 −0.472947
\(666\) 0 0
\(667\) −26.7846 −1.03710
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) −38.3923 −1.47991 −0.739957 0.672654i \(-0.765154\pi\)
−0.739957 + 0.672654i \(0.765154\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 40.6410 1.56196 0.780981 0.624555i \(-0.214720\pi\)
0.780981 + 0.624555i \(0.214720\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 21.4641 0.821301 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(684\) 0 0
\(685\) −4.39230 −0.167821
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −39.7128 −1.51294
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.3923 −0.583863
\(696\) 0 0
\(697\) 52.9808 2.00679
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17.8756 −0.675154 −0.337577 0.941298i \(-0.609607\pi\)
−0.337577 + 0.941298i \(0.609607\pi\)
\(702\) 0 0
\(703\) 27.6603 1.04323
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.7321 −0.629274
\(708\) 0 0
\(709\) 10.5359 0.395684 0.197842 0.980234i \(-0.436607\pi\)
0.197842 + 0.980234i \(0.436607\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.5359 0.769075
\(714\) 0 0
\(715\) −9.46410 −0.353937
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17.1962 −0.641308 −0.320654 0.947196i \(-0.603903\pi\)
−0.320654 + 0.947196i \(0.603903\pi\)
\(720\) 0 0
\(721\) −50.2487 −1.87136
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.73205 −0.287161
\(726\) 0 0
\(727\) −29.1769 −1.08211 −0.541056 0.840987i \(-0.681975\pi\)
−0.541056 + 0.840987i \(0.681975\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −15.4641 −0.571960
\(732\) 0 0
\(733\) 43.5692 1.60927 0.804633 0.593773i \(-0.202362\pi\)
0.804633 + 0.593773i \(0.202362\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.0718 0.407835
\(738\) 0 0
\(739\) −26.1769 −0.962933 −0.481467 0.876464i \(-0.659896\pi\)
−0.481467 + 0.876464i \(0.659896\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.41154 −0.198530 −0.0992651 0.995061i \(-0.531649\pi\)
−0.0992651 + 0.995061i \(0.531649\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.46410 −0.345811
\(750\) 0 0
\(751\) 40.7846 1.48825 0.744126 0.668040i \(-0.232866\pi\)
0.744126 + 0.668040i \(0.232866\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 17.3923 0.632971
\(756\) 0 0
\(757\) −20.3923 −0.741171 −0.370585 0.928798i \(-0.620843\pi\)
−0.370585 + 0.928798i \(0.620843\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.1244 −1.09201 −0.546004 0.837783i \(-0.683851\pi\)
−0.546004 + 0.837783i \(0.683851\pi\)
\(762\) 0 0
\(763\) 21.6603 0.784154
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −42.2487 −1.52551
\(768\) 0 0
\(769\) 35.2487 1.27110 0.635551 0.772059i \(-0.280773\pi\)
0.635551 + 0.772059i \(0.280773\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.6603 0.635195 0.317598 0.948226i \(-0.397124\pi\)
0.317598 + 0.948226i \(0.397124\pi\)
\(774\) 0 0
\(775\) 5.92820 0.212947
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 49.9808 1.79075
\(780\) 0 0
\(781\) −19.3923 −0.693911
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.26795 −0.116638
\(786\) 0 0
\(787\) −36.1962 −1.29025 −0.645127 0.764076i \(-0.723195\pi\)
−0.645127 + 0.764076i \(0.723195\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.928203 0.0330031
\(792\) 0 0
\(793\) −21.8564 −0.776144
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23.3205 0.826055 0.413027 0.910719i \(-0.364471\pi\)
0.413027 + 0.910719i \(0.364471\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.339746 −0.0119894
\(804\) 0 0
\(805\) 9.46410 0.333566
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8.41154 −0.295734 −0.147867 0.989007i \(-0.547241\pi\)
−0.147867 + 0.989007i \(0.547241\pi\)
\(810\) 0 0
\(811\) −25.2487 −0.886602 −0.443301 0.896373i \(-0.646193\pi\)
−0.443301 + 0.896373i \(0.646193\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −18.7321 −0.656155
\(816\) 0 0
\(817\) −14.5885 −0.510386
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.33975 0.116558 0.0582790 0.998300i \(-0.481439\pi\)
0.0582790 + 0.998300i \(0.481439\pi\)
\(822\) 0 0
\(823\) −16.9282 −0.590080 −0.295040 0.955485i \(-0.595333\pi\)
−0.295040 + 0.955485i \(0.595333\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −45.4641 −1.58094 −0.790471 0.612500i \(-0.790164\pi\)
−0.790471 + 0.612500i \(0.790164\pi\)
\(828\) 0 0
\(829\) −51.7846 −1.79855 −0.899277 0.437380i \(-0.855907\pi\)
−0.899277 + 0.437380i \(0.855907\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.19615 −0.0760922
\(834\) 0 0
\(835\) 21.1244 0.731038
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8.41154 −0.290399 −0.145199 0.989402i \(-0.546382\pi\)
−0.145199 + 0.989402i \(0.546382\pi\)
\(840\) 0 0
\(841\) 30.7846 1.06154
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −16.8564 −0.579878
\(846\) 0 0
\(847\) 21.8564 0.750995
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −21.4641 −0.735780
\(852\) 0 0
\(853\) −0.196152 −0.00671613 −0.00335807 0.999994i \(-0.501069\pi\)
−0.00335807 + 0.999994i \(0.501069\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29.0718 −0.993074 −0.496537 0.868016i \(-0.665395\pi\)
−0.496537 + 0.868016i \(0.665395\pi\)
\(858\) 0 0
\(859\) 7.78461 0.265607 0.132804 0.991142i \(-0.457602\pi\)
0.132804 + 0.991142i \(0.457602\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −49.5167 −1.68557 −0.842783 0.538253i \(-0.819085\pi\)
−0.842783 + 0.538253i \(0.819085\pi\)
\(864\) 0 0
\(865\) 24.2487 0.824481
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −24.9282 −0.845631
\(870\) 0 0
\(871\) 34.9282 1.18350
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.73205 0.0923602
\(876\) 0 0
\(877\) 14.2487 0.481145 0.240572 0.970631i \(-0.422665\pi\)
0.240572 + 0.970631i \(0.422665\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.80385 0.229227 0.114614 0.993410i \(-0.463437\pi\)
0.114614 + 0.993410i \(0.463437\pi\)
\(882\) 0 0
\(883\) −46.8372 −1.57620 −0.788098 0.615550i \(-0.788934\pi\)
−0.788098 + 0.615550i \(0.788934\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.2679 0.445494 0.222747 0.974876i \(-0.428498\pi\)
0.222747 + 0.974876i \(0.428498\pi\)
\(888\) 0 0
\(889\) −11.4641 −0.384494
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.66025 0.189413
\(894\) 0 0
\(895\) 12.1244 0.405273
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −45.8372 −1.52876
\(900\) 0 0
\(901\) 34.3923 1.14577
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.4641 0.547285
\(906\) 0 0
\(907\) −1.21539 −0.0403564 −0.0201782 0.999796i \(-0.506423\pi\)
−0.0201782 + 0.999796i \(0.506423\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.80385 −0.225422 −0.112711 0.993628i \(-0.535953\pi\)
−0.112711 + 0.993628i \(0.535953\pi\)
\(912\) 0 0
\(913\) −26.1962 −0.866966
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −28.0526 −0.926377
\(918\) 0 0
\(919\) 51.3923 1.69528 0.847638 0.530575i \(-0.178024\pi\)
0.847638 + 0.530575i \(0.178024\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −61.1769 −2.01366
\(924\) 0 0
\(925\) −6.19615 −0.203728
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.48334 0.0486668 0.0243334 0.999704i \(-0.492254\pi\)
0.0243334 + 0.999704i \(0.492254\pi\)
\(930\) 0 0
\(931\) −2.07180 −0.0679004
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.19615 0.268043
\(936\) 0 0
\(937\) 19.0718 0.623048 0.311524 0.950238i \(-0.399160\pi\)
0.311524 + 0.950238i \(0.399160\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −8.53590 −0.278262 −0.139131 0.990274i \(-0.544431\pi\)
−0.139131 + 0.990274i \(0.544431\pi\)
\(942\) 0 0
\(943\) −38.7846 −1.26300
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52.6410 −1.71060 −0.855302 0.518130i \(-0.826628\pi\)
−0.855302 + 0.518130i \(0.826628\pi\)
\(948\) 0 0
\(949\) −1.07180 −0.0347920
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.53590 −0.0821458 −0.0410729 0.999156i \(-0.513078\pi\)
−0.0410729 + 0.999156i \(0.513078\pi\)
\(954\) 0 0
\(955\) −19.0526 −0.616526
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 4.14359 0.133664
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 10.5885 0.340854
\(966\) 0 0
\(967\) 7.41154 0.238339 0.119170 0.992874i \(-0.461977\pi\)
0.119170 + 0.992874i \(0.461977\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.2679 0.714612 0.357306 0.933987i \(-0.383695\pi\)
0.357306 + 0.933987i \(0.383695\pi\)
\(972\) 0 0
\(973\) −42.0526 −1.34814
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.39230 0.140522 0.0702611 0.997529i \(-0.477617\pi\)
0.0702611 + 0.997529i \(0.477617\pi\)
\(978\) 0 0
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −52.7321 −1.68189 −0.840946 0.541120i \(-0.818001\pi\)
−0.840946 + 0.541120i \(0.818001\pi\)
\(984\) 0 0
\(985\) 13.8564 0.441502
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.3205 0.359971
\(990\) 0 0
\(991\) 55.7846 1.77206 0.886028 0.463631i \(-0.153454\pi\)
0.886028 + 0.463631i \(0.153454\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −15.8564 −0.502682
\(996\) 0 0
\(997\) 52.1962 1.65307 0.826534 0.562886i \(-0.190309\pi\)
0.826534 + 0.562886i \(0.190309\pi\)
\(998\) 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 1620.2.a.g.1.1 2
3.2 odd 2 1620.2.a.h.1.1 yes 2
4.3 odd 2 6480.2.a.bh.1.2 2
5.2 odd 4 8100.2.d.m.649.1 4
5.3 odd 4 8100.2.d.m.649.4 4
5.4 even 2 8100.2.a.t.1.2 2
9.2 odd 6 1620.2.i.m.1081.2 4
9.4 even 3 1620.2.i.n.541.2 4
9.5 odd 6 1620.2.i.m.541.2 4
9.7 even 3 1620.2.i.n.1081.2 4
12.11 even 2 6480.2.a.bp.1.2 2
15.2 even 4 8100.2.d.l.649.1 4
15.8 even 4 8100.2.d.l.649.4 4
15.14 odd 2 8100.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.a.g.1.1 2 1.1 even 1 trivial
1620.2.a.h.1.1 yes 2 3.2 odd 2
1620.2.i.m.541.2 4 9.5 odd 6
1620.2.i.m.1081.2 4 9.2 odd 6
1620.2.i.n.541.2 4 9.4 even 3
1620.2.i.n.1081.2 4 9.7 even 3
6480.2.a.bh.1.2 2 4.3 odd 2
6480.2.a.bp.1.2 2 12.11 even 2
8100.2.a.s.1.2 2 15.14 odd 2
8100.2.a.t.1.2 2 5.4 even 2
8100.2.d.l.649.1 4 15.2 even 4
8100.2.d.l.649.4 4 15.8 even 4
8100.2.d.m.649.1 4 5.2 odd 4
8100.2.d.m.649.4 4 5.3 odd 4