Properties

Label 1568.2.a.n.1.2
Level $1568$
Weight $2$
Character 1568.1
Self dual yes
Analytic conductor $12.521$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1568.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{3} -1.00000 q^{5} +O(q^{10})\) \(q+1.73205 q^{3} -1.00000 q^{5} -5.19615 q^{11} -1.73205 q^{15} -5.00000 q^{17} +1.73205 q^{19} +1.73205 q^{23} -4.00000 q^{25} -5.19615 q^{27} +8.00000 q^{29} -8.66025 q^{31} -9.00000 q^{33} -5.00000 q^{37} -4.00000 q^{41} -6.92820 q^{43} +8.66025 q^{47} -8.66025 q^{51} -1.00000 q^{53} +5.19615 q^{55} +3.00000 q^{57} +1.73205 q^{59} -11.0000 q^{61} +12.1244 q^{67} +3.00000 q^{69} +13.8564 q^{71} -15.0000 q^{73} -6.92820 q^{75} +1.73205 q^{79} -9.00000 q^{81} +6.92820 q^{83} +5.00000 q^{85} +13.8564 q^{87} -7.00000 q^{89} -15.0000 q^{93} -1.73205 q^{95} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 10 q^{17} - 8 q^{25} + 16 q^{29} - 18 q^{33} - 10 q^{37} - 8 q^{41} - 2 q^{53} + 6 q^{57} - 22 q^{61} + 6 q^{69} - 30 q^{73} - 18 q^{81} + 10 q^{85} - 14 q^{89} - 30 q^{93} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.73205 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.19615 −1.56670 −0.783349 0.621582i \(-0.786490\pi\)
−0.783349 + 0.621582i \(0.786490\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.73205 −0.447214
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 1.73205 0.397360 0.198680 0.980064i \(-0.436335\pi\)
0.198680 + 0.980064i \(0.436335\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.73205 0.361158 0.180579 0.983561i \(-0.442203\pi\)
0.180579 + 0.983561i \(0.442203\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) −8.66025 −1.55543 −0.777714 0.628619i \(-0.783621\pi\)
−0.777714 + 0.628619i \(0.783621\pi\)
\(32\) 0 0
\(33\) −9.00000 −1.56670
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.66025 1.26323 0.631614 0.775283i \(-0.282393\pi\)
0.631614 + 0.775283i \(0.282393\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.66025 −1.21268
\(52\) 0 0
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) 5.19615 0.700649
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) 1.73205 0.225494 0.112747 0.993624i \(-0.464035\pi\)
0.112747 + 0.993624i \(0.464035\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.1244 1.48123 0.740613 0.671932i \(-0.234535\pi\)
0.740613 + 0.671932i \(0.234535\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) −15.0000 −1.75562 −0.877809 0.479012i \(-0.840995\pi\)
−0.877809 + 0.479012i \(0.840995\pi\)
\(74\) 0 0
\(75\) −6.92820 −0.800000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.73205 0.194871 0.0974355 0.995242i \(-0.468936\pi\)
0.0974355 + 0.995242i \(0.468936\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 6.92820 0.760469 0.380235 0.924890i \(-0.375843\pi\)
0.380235 + 0.924890i \(0.375843\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) 13.8564 1.48556
\(88\) 0 0
\(89\) −7.00000 −0.741999 −0.370999 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −15.0000 −1.55543
\(94\) 0 0
\(95\) −1.73205 −0.177705
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.0000 1.09454 0.547270 0.836956i \(-0.315667\pi\)
0.547270 + 0.836956i \(0.315667\pi\)
\(102\) 0 0
\(103\) −15.5885 −1.53598 −0.767988 0.640464i \(-0.778742\pi\)
−0.767988 + 0.640464i \(0.778742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.1244 1.17211 0.586053 0.810273i \(-0.300681\pi\)
0.586053 + 0.810273i \(0.300681\pi\)
\(108\) 0 0
\(109\) −3.00000 −0.287348 −0.143674 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 0 0
\(111\) −8.66025 −0.821995
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) −1.73205 −0.161515
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 16.0000 1.45455
\(122\) 0 0
\(123\) −6.92820 −0.624695
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −8.66025 −0.756650 −0.378325 0.925673i \(-0.623500\pi\)
−0.378325 + 0.925673i \(0.623500\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 5.19615 0.447214
\(136\) 0 0
\(137\) 5.00000 0.427179 0.213589 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) 15.0000 1.26323
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.00000 −0.573462 −0.286731 0.958011i \(-0.592569\pi\)
−0.286731 + 0.958011i \(0.592569\pi\)
\(150\) 0 0
\(151\) 8.66025 0.704761 0.352381 0.935857i \(-0.385372\pi\)
0.352381 + 0.935857i \(0.385372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.66025 0.695608
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 0 0
\(159\) −1.73205 −0.137361
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.66025 0.678323 0.339162 0.940728i \(-0.389857\pi\)
0.339162 + 0.940728i \(0.389857\pi\)
\(164\) 0 0
\(165\) 9.00000 0.700649
\(166\) 0 0
\(167\) −3.46410 −0.268060 −0.134030 0.990977i \(-0.542792\pi\)
−0.134030 + 0.990977i \(0.542792\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.00000 0.380143 0.190071 0.981770i \(-0.439128\pi\)
0.190071 + 0.981770i \(0.439128\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) 8.66025 0.647298 0.323649 0.946177i \(-0.395090\pi\)
0.323649 + 0.946177i \(0.395090\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −19.0526 −1.40841
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) 25.9808 1.89990
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.5167 −1.62925 −0.814624 0.579989i \(-0.803057\pi\)
−0.814624 + 0.579989i \(0.803057\pi\)
\(192\) 0 0
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) −8.66025 −0.613909 −0.306955 0.951724i \(-0.599310\pi\)
−0.306955 + 0.951724i \(0.599310\pi\)
\(200\) 0 0
\(201\) 21.0000 1.48123
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.00000 −0.622543
\(210\) 0 0
\(211\) 20.7846 1.43087 0.715436 0.698679i \(-0.246228\pi\)
0.715436 + 0.698679i \(0.246228\pi\)
\(212\) 0 0
\(213\) 24.0000 1.64445
\(214\) 0 0
\(215\) 6.92820 0.472500
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −25.9808 −1.75562
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 13.8564 0.927894 0.463947 0.885863i \(-0.346433\pi\)
0.463947 + 0.885863i \(0.346433\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.66025 −0.574801 −0.287401 0.957810i \(-0.592791\pi\)
−0.287401 + 0.957810i \(0.592791\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.00000 0.327561 0.163780 0.986497i \(-0.447631\pi\)
0.163780 + 0.986497i \(0.447631\pi\)
\(234\) 0 0
\(235\) −8.66025 −0.564933
\(236\) 0 0
\(237\) 3.00000 0.194871
\(238\) 0 0
\(239\) −17.3205 −1.12037 −0.560185 0.828367i \(-0.689270\pi\)
−0.560185 + 0.828367i \(0.689270\pi\)
\(240\) 0 0
\(241\) −15.0000 −0.966235 −0.483117 0.875556i \(-0.660496\pi\)
−0.483117 + 0.875556i \(0.660496\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −17.3205 −1.09326 −0.546630 0.837374i \(-0.684090\pi\)
−0.546630 + 0.837374i \(0.684090\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) 8.66025 0.542326
\(256\) 0 0
\(257\) −5.00000 −0.311891 −0.155946 0.987766i \(-0.549842\pi\)
−0.155946 + 0.987766i \(0.549842\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.73205 −0.106803 −0.0534014 0.998573i \(-0.517006\pi\)
−0.0534014 + 0.998573i \(0.517006\pi\)
\(264\) 0 0
\(265\) 1.00000 0.0614295
\(266\) 0 0
\(267\) −12.1244 −0.741999
\(268\) 0 0
\(269\) 17.0000 1.03651 0.518254 0.855227i \(-0.326582\pi\)
0.518254 + 0.855227i \(0.326582\pi\)
\(270\) 0 0
\(271\) 12.1244 0.736502 0.368251 0.929726i \(-0.379957\pi\)
0.368251 + 0.929726i \(0.379957\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 20.7846 1.25336
\(276\) 0 0
\(277\) 15.0000 0.901263 0.450631 0.892710i \(-0.351199\pi\)
0.450631 + 0.892710i \(0.351199\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) −8.66025 −0.514799 −0.257399 0.966305i \(-0.582866\pi\)
−0.257399 + 0.966305i \(0.582866\pi\)
\(284\) 0 0
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −20.7846 −1.21842
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) −1.73205 −0.100844
\(296\) 0 0
\(297\) 27.0000 1.56670
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 19.0526 1.09454
\(304\) 0 0
\(305\) 11.0000 0.629858
\(306\) 0 0
\(307\) −20.7846 −1.18624 −0.593120 0.805114i \(-0.702104\pi\)
−0.593120 + 0.805114i \(0.702104\pi\)
\(308\) 0 0
\(309\) −27.0000 −1.53598
\(310\) 0 0
\(311\) −12.1244 −0.687509 −0.343755 0.939060i \(-0.611699\pi\)
−0.343755 + 0.939060i \(0.611699\pi\)
\(312\) 0 0
\(313\) 25.0000 1.41308 0.706542 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.00000 −0.280828 −0.140414 0.990093i \(-0.544843\pi\)
−0.140414 + 0.990093i \(0.544843\pi\)
\(318\) 0 0
\(319\) −41.5692 −2.32743
\(320\) 0 0
\(321\) 21.0000 1.17211
\(322\) 0 0
\(323\) −8.66025 −0.481869
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −5.19615 −0.287348
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.66025 0.476011 0.238005 0.971264i \(-0.423506\pi\)
0.238005 + 0.971264i \(0.423506\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.1244 −0.662424
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 0 0
\(339\) 6.92820 0.376288
\(340\) 0 0
\(341\) 45.0000 2.43689
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3.00000 −0.161515
\(346\) 0 0
\(347\) 12.1244 0.650870 0.325435 0.945564i \(-0.394489\pi\)
0.325435 + 0.945564i \(0.394489\pi\)
\(348\) 0 0
\(349\) −32.0000 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.0000 0.585471 0.292735 0.956193i \(-0.405434\pi\)
0.292735 + 0.956193i \(0.405434\pi\)
\(354\) 0 0
\(355\) −13.8564 −0.735422
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.66025 −0.457071 −0.228535 0.973536i \(-0.573394\pi\)
−0.228535 + 0.973536i \(0.573394\pi\)
\(360\) 0 0
\(361\) −16.0000 −0.842105
\(362\) 0 0
\(363\) 27.7128 1.45455
\(364\) 0 0
\(365\) 15.0000 0.785136
\(366\) 0 0
\(367\) 29.4449 1.53701 0.768505 0.639844i \(-0.221001\pi\)
0.768505 + 0.639844i \(0.221001\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 35.0000 1.81223 0.906116 0.423030i \(-0.139034\pi\)
0.906116 + 0.423030i \(0.139034\pi\)
\(374\) 0 0
\(375\) 15.5885 0.804984
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 10.3923 0.533817 0.266908 0.963722i \(-0.413998\pi\)
0.266908 + 0.963722i \(0.413998\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −19.0526 −0.973540 −0.486770 0.873530i \(-0.661825\pi\)
−0.486770 + 0.873530i \(0.661825\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.00000 0.253510 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) −8.66025 −0.437968
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) −1.73205 −0.0871489
\(396\) 0 0
\(397\) −17.0000 −0.853206 −0.426603 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −19.0000 −0.948815 −0.474407 0.880305i \(-0.657338\pi\)
−0.474407 + 0.880305i \(0.657338\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) 25.9808 1.28782
\(408\) 0 0
\(409\) 3.00000 0.148340 0.0741702 0.997246i \(-0.476369\pi\)
0.0741702 + 0.997246i \(0.476369\pi\)
\(410\) 0 0
\(411\) 8.66025 0.427179
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.92820 −0.340092
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 34.6410 1.69232 0.846162 0.532925i \(-0.178907\pi\)
0.846162 + 0.532925i \(0.178907\pi\)
\(420\) 0 0
\(421\) −24.0000 −1.16969 −0.584844 0.811146i \(-0.698844\pi\)
−0.584844 + 0.811146i \(0.698844\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.5167 1.08459 0.542295 0.840188i \(-0.317556\pi\)
0.542295 + 0.840188i \(0.317556\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) −13.8564 −0.664364
\(436\) 0 0
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) 8.66025 0.413331 0.206666 0.978412i \(-0.433739\pi\)
0.206666 + 0.978412i \(0.433739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.5885 −0.740630 −0.370315 0.928906i \(-0.620750\pi\)
−0.370315 + 0.928906i \(0.620750\pi\)
\(444\) 0 0
\(445\) 7.00000 0.331832
\(446\) 0 0
\(447\) −12.1244 −0.573462
\(448\) 0 0
\(449\) 28.0000 1.32140 0.660701 0.750649i \(-0.270259\pi\)
0.660701 + 0.750649i \(0.270259\pi\)
\(450\) 0 0
\(451\) 20.7846 0.978709
\(452\) 0 0
\(453\) 15.0000 0.704761
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.0000 0.701670 0.350835 0.936437i \(-0.385898\pi\)
0.350835 + 0.936437i \(0.385898\pi\)
\(458\) 0 0
\(459\) 25.9808 1.21268
\(460\) 0 0
\(461\) 40.0000 1.86299 0.931493 0.363760i \(-0.118507\pi\)
0.931493 + 0.363760i \(0.118507\pi\)
\(462\) 0 0
\(463\) 27.7128 1.28792 0.643962 0.765058i \(-0.277290\pi\)
0.643962 + 0.765058i \(0.277290\pi\)
\(464\) 0 0
\(465\) 15.0000 0.695608
\(466\) 0 0
\(467\) 5.19615 0.240449 0.120225 0.992747i \(-0.461639\pi\)
0.120225 + 0.992747i \(0.461639\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −12.1244 −0.558661
\(472\) 0 0
\(473\) 36.0000 1.65528
\(474\) 0 0
\(475\) −6.92820 −0.317888
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −25.9808 −1.18709 −0.593546 0.804800i \(-0.702272\pi\)
−0.593546 + 0.804800i \(0.702272\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) 25.9808 1.17730 0.588650 0.808388i \(-0.299659\pi\)
0.588650 + 0.808388i \(0.299659\pi\)
\(488\) 0 0
\(489\) 15.0000 0.678323
\(490\) 0 0
\(491\) −17.3205 −0.781664 −0.390832 0.920462i \(-0.627813\pi\)
−0.390832 + 0.920462i \(0.627813\pi\)
\(492\) 0 0
\(493\) −40.0000 −1.80151
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −19.0526 −0.852910 −0.426455 0.904509i \(-0.640238\pi\)
−0.426455 + 0.904509i \(0.640238\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 27.7128 1.23565 0.617827 0.786314i \(-0.288013\pi\)
0.617827 + 0.786314i \(0.288013\pi\)
\(504\) 0 0
\(505\) −11.0000 −0.489494
\(506\) 0 0
\(507\) −22.5167 −1.00000
\(508\) 0 0
\(509\) −17.0000 −0.753512 −0.376756 0.926313i \(-0.622960\pi\)
−0.376756 + 0.926313i \(0.622960\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −9.00000 −0.397360
\(514\) 0 0
\(515\) 15.5885 0.686909
\(516\) 0 0
\(517\) −45.0000 −1.97910
\(518\) 0 0
\(519\) 8.66025 0.380143
\(520\) 0 0
\(521\) 25.0000 1.09527 0.547635 0.836717i \(-0.315528\pi\)
0.547635 + 0.836717i \(0.315528\pi\)
\(522\) 0 0
\(523\) −25.9808 −1.13606 −0.568030 0.823008i \(-0.692294\pi\)
−0.568030 + 0.823008i \(0.692294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 43.3013 1.88623
\(528\) 0 0
\(529\) −20.0000 −0.869565
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −12.1244 −0.524182
\(536\) 0 0
\(537\) 15.0000 0.647298
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9.00000 0.386940 0.193470 0.981106i \(-0.438026\pi\)
0.193470 + 0.981106i \(0.438026\pi\)
\(542\) 0 0
\(543\) 10.3923 0.445976
\(544\) 0 0
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) 3.46410 0.148114 0.0740571 0.997254i \(-0.476405\pi\)
0.0740571 + 0.997254i \(0.476405\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 13.8564 0.590303
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.66025 0.367607
\(556\) 0 0
\(557\) −43.0000 −1.82197 −0.910984 0.412441i \(-0.864676\pi\)
−0.910984 + 0.412441i \(0.864676\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 45.0000 1.89990
\(562\) 0 0
\(563\) 15.5885 0.656975 0.328488 0.944508i \(-0.393461\pi\)
0.328488 + 0.944508i \(0.393461\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −25.0000 −1.04805 −0.524027 0.851701i \(-0.675571\pi\)
−0.524027 + 0.851701i \(0.675571\pi\)
\(570\) 0 0
\(571\) −5.19615 −0.217452 −0.108726 0.994072i \(-0.534677\pi\)
−0.108726 + 0.994072i \(0.534677\pi\)
\(572\) 0 0
\(573\) −39.0000 −1.62925
\(574\) 0 0
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) −23.0000 −0.957503 −0.478751 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) 0 0
\(579\) −19.0526 −0.791797
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 5.19615 0.215203
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −34.6410 −1.42979 −0.714894 0.699233i \(-0.753525\pi\)
−0.714894 + 0.699233i \(0.753525\pi\)
\(588\) 0 0
\(589\) −15.0000 −0.618064
\(590\) 0 0
\(591\) −13.8564 −0.569976
\(592\) 0 0
\(593\) −29.0000 −1.19089 −0.595444 0.803397i \(-0.703024\pi\)
−0.595444 + 0.803397i \(0.703024\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −15.0000 −0.613909
\(598\) 0 0
\(599\) −43.3013 −1.76924 −0.884621 0.466311i \(-0.845583\pi\)
−0.884621 + 0.466311i \(0.845583\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16.0000 −0.650493
\(606\) 0 0
\(607\) −5.19615 −0.210905 −0.105453 0.994424i \(-0.533629\pi\)
−0.105453 + 0.994424i \(0.533629\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 5.00000 0.201948 0.100974 0.994889i \(-0.467804\pi\)
0.100974 + 0.994889i \(0.467804\pi\)
\(614\) 0 0
\(615\) 6.92820 0.279372
\(616\) 0 0
\(617\) −20.0000 −0.805170 −0.402585 0.915383i \(-0.631888\pi\)
−0.402585 + 0.915383i \(0.631888\pi\)
\(618\) 0 0
\(619\) 15.5885 0.626553 0.313276 0.949662i \(-0.398573\pi\)
0.313276 + 0.949662i \(0.398573\pi\)
\(620\) 0 0
\(621\) −9.00000 −0.361158
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −15.5885 −0.622543
\(628\) 0 0
\(629\) 25.0000 0.996815
\(630\) 0 0
\(631\) −13.8564 −0.551615 −0.275807 0.961213i \(-0.588945\pi\)
−0.275807 + 0.961213i \(0.588945\pi\)
\(632\) 0 0
\(633\) 36.0000 1.43087
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.00000 −0.0394976 −0.0197488 0.999805i \(-0.506287\pi\)
−0.0197488 + 0.999805i \(0.506287\pi\)
\(642\) 0 0
\(643\) −34.6410 −1.36611 −0.683054 0.730368i \(-0.739349\pi\)
−0.683054 + 0.730368i \(0.739349\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) 29.4449 1.15760 0.578799 0.815471i \(-0.303522\pi\)
0.578799 + 0.815471i \(0.303522\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5.00000 −0.195665 −0.0978326 0.995203i \(-0.531191\pi\)
−0.0978326 + 0.995203i \(0.531191\pi\)
\(654\) 0 0
\(655\) 8.66025 0.338384
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.92820 0.269884 0.134942 0.990853i \(-0.456915\pi\)
0.134942 + 0.990853i \(0.456915\pi\)
\(660\) 0 0
\(661\) −31.0000 −1.20576 −0.602880 0.797832i \(-0.705980\pi\)
−0.602880 + 0.797832i \(0.705980\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.8564 0.536522
\(668\) 0 0
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 57.1577 2.20655
\(672\) 0 0
\(673\) 20.0000 0.770943 0.385472 0.922720i \(-0.374039\pi\)
0.385472 + 0.922720i \(0.374039\pi\)
\(674\) 0 0
\(675\) 20.7846 0.800000
\(676\) 0 0
\(677\) −25.0000 −0.960828 −0.480414 0.877042i \(-0.659514\pi\)
−0.480414 + 0.877042i \(0.659514\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −15.0000 −0.574801
\(682\) 0 0
\(683\) −32.9090 −1.25923 −0.629613 0.776909i \(-0.716787\pi\)
−0.629613 + 0.776909i \(0.716787\pi\)
\(684\) 0 0
\(685\) −5.00000 −0.191040
\(686\) 0 0
\(687\) 25.9808 0.991228
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 5.19615 0.197671 0.0988355 0.995104i \(-0.468488\pi\)
0.0988355 + 0.995104i \(0.468488\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.92820 −0.262802
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) 8.66025 0.327561
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 0 0
\(703\) −8.66025 −0.326628
\(704\) 0 0
\(705\) −15.0000 −0.564933
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 27.0000 1.01401 0.507003 0.861944i \(-0.330753\pi\)
0.507003 + 0.861944i \(0.330753\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.0000 −0.561754
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −30.0000 −1.12037
\(718\) 0 0
\(719\) −15.5885 −0.581351 −0.290676 0.956822i \(-0.593880\pi\)
−0.290676 + 0.956822i \(0.593880\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −25.9808 −0.966235
\(724\) 0 0
\(725\) −32.0000 −1.18845
\(726\) 0 0
\(727\) −13.8564 −0.513906 −0.256953 0.966424i \(-0.582719\pi\)
−0.256953 + 0.966424i \(0.582719\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 34.6410 1.28124
\(732\) 0 0
\(733\) −51.0000 −1.88373 −0.941864 0.335994i \(-0.890928\pi\)
−0.941864 + 0.335994i \(0.890928\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −63.0000 −2.32063
\(738\) 0 0
\(739\) −19.0526 −0.700860 −0.350430 0.936589i \(-0.613964\pi\)
−0.350430 + 0.936589i \(0.613964\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −27.7128 −1.01668 −0.508342 0.861155i \(-0.669742\pi\)
−0.508342 + 0.861155i \(0.669742\pi\)
\(744\) 0 0
\(745\) 7.00000 0.256460
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −8.66025 −0.316017 −0.158009 0.987438i \(-0.550507\pi\)
−0.158009 + 0.987438i \(0.550507\pi\)
\(752\) 0 0
\(753\) −30.0000 −1.09326
\(754\) 0 0
\(755\) −8.66025 −0.315179
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) −15.5885 −0.565825
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) −8.66025 −0.311891
\(772\) 0 0
\(773\) −5.00000 −0.179838 −0.0899188 0.995949i \(-0.528661\pi\)
−0.0899188 + 0.995949i \(0.528661\pi\)
\(774\) 0 0
\(775\) 34.6410 1.24434
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.92820 −0.248229
\(780\) 0 0
\(781\) −72.0000 −2.57636
\(782\) 0 0
\(783\) −41.5692 −1.48556
\(784\) 0 0
\(785\) 7.00000 0.249841
\(786\) 0 0
\(787\) −39.8372 −1.42004 −0.710021 0.704181i \(-0.751315\pi\)
−0.710021 + 0.704181i \(0.751315\pi\)
\(788\) 0 0
\(789\) −3.00000 −0.106803
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 1.73205 0.0614295
\(796\) 0 0
\(797\) −40.0000 −1.41687 −0.708436 0.705775i \(-0.750599\pi\)
−0.708436 + 0.705775i \(0.750599\pi\)
\(798\) 0 0
\(799\) −43.3013 −1.53189
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 77.9423 2.75052
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 29.4449 1.03651
\(808\) 0 0
\(809\) 7.00000 0.246107 0.123053 0.992400i \(-0.460731\pi\)
0.123053 + 0.992400i \(0.460731\pi\)
\(810\) 0 0
\(811\) 34.6410 1.21641 0.608205 0.793780i \(-0.291890\pi\)
0.608205 + 0.793780i \(0.291890\pi\)
\(812\) 0 0
\(813\) 21.0000 0.736502
\(814\) 0 0
\(815\) −8.66025 −0.303355
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.0000 0.872506 0.436253 0.899824i \(-0.356305\pi\)
0.436253 + 0.899824i \(0.356305\pi\)
\(822\) 0 0
\(823\) −15.5885 −0.543379 −0.271690 0.962385i \(-0.587582\pi\)
−0.271690 + 0.962385i \(0.587582\pi\)
\(824\) 0 0
\(825\) 36.0000 1.25336
\(826\) 0 0
\(827\) 20.7846 0.722752 0.361376 0.932420i \(-0.382307\pi\)
0.361376 + 0.932420i \(0.382307\pi\)
\(828\) 0 0
\(829\) −45.0000 −1.56291 −0.781457 0.623959i \(-0.785523\pi\)
−0.781457 + 0.623959i \(0.785523\pi\)
\(830\) 0 0
\(831\) 25.9808 0.901263
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3.46410 0.119880
\(836\) 0 0
\(837\) 45.0000 1.55543
\(838\) 0 0
\(839\) −41.5692 −1.43513 −0.717564 0.696492i \(-0.754743\pi\)
−0.717564 + 0.696492i \(0.754743\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −34.6410 −1.19310
\(844\) 0 0
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −15.0000 −0.514799
\(850\) 0 0
\(851\) −8.66025 −0.296870
\(852\) 0 0
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.00000 −0.239115 −0.119558 0.992827i \(-0.538148\pi\)
−0.119558 + 0.992827i \(0.538148\pi\)
\(858\) 0 0
\(859\) −25.9808 −0.886452 −0.443226 0.896410i \(-0.646166\pi\)
−0.443226 + 0.896410i \(0.646166\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −53.6936 −1.82775 −0.913875 0.405995i \(-0.866925\pi\)
−0.913875 + 0.405995i \(0.866925\pi\)
\(864\) 0 0
\(865\) −5.00000 −0.170005
\(866\) 0 0
\(867\) 13.8564 0.470588
\(868\) 0 0
\(869\) −9.00000 −0.305304
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.00000 −0.236373 −0.118187 0.992991i \(-0.537708\pi\)
−0.118187 + 0.992991i \(0.537708\pi\)
\(878\) 0 0
\(879\) 17.3205 0.584206
\(880\) 0 0
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 0 0
\(883\) −34.6410 −1.16576 −0.582882 0.812557i \(-0.698075\pi\)
−0.582882 + 0.812557i \(0.698075\pi\)
\(884\) 0 0
\(885\) −3.00000 −0.100844
\(886\) 0 0
\(887\) −46.7654 −1.57023 −0.785114 0.619352i \(-0.787396\pi\)
−0.785114 + 0.619352i \(0.787396\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 46.7654 1.56670
\(892\) 0 0
\(893\) 15.0000 0.501956
\(894\) 0 0
\(895\) −8.66025 −0.289480
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −69.2820 −2.31069
\(900\) 0 0
\(901\) 5.00000 0.166574
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 39.8372 1.32277 0.661386 0.750046i \(-0.269969\pi\)
0.661386 + 0.750046i \(0.269969\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −13.8564 −0.459083 −0.229542 0.973299i \(-0.573723\pi\)
−0.229542 + 0.973299i \(0.573723\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) 19.0526 0.629858
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −36.3731 −1.19984 −0.599918 0.800061i \(-0.704800\pi\)
−0.599918 + 0.800061i \(0.704800\pi\)
\(920\) 0 0
\(921\) −36.0000 −1.18624
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −47.0000 −1.54202 −0.771010 0.636823i \(-0.780248\pi\)
−0.771010 + 0.636823i \(0.780248\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −21.0000 −0.687509
\(934\) 0 0
\(935\) −25.9808 −0.849662
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 43.3013 1.41308
\(940\) 0 0
\(941\) 25.0000 0.814977 0.407488 0.913210i \(-0.366405\pi\)
0.407488 + 0.913210i \(0.366405\pi\)
\(942\) 0 0
\(943\) −6.92820 −0.225613
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.19615 −0.168852 −0.0844261 0.996430i \(-0.526906\pi\)
−0.0844261 + 0.996430i \(0.526906\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −8.66025 −0.280828
\(952\) 0 0
\(953\) 20.0000 0.647864 0.323932 0.946080i \(-0.394995\pi\)
0.323932 + 0.946080i \(0.394995\pi\)
\(954\) 0 0
\(955\) 22.5167 0.728622
\(956\) 0 0
\(957\) −72.0000 −2.32743
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 44.0000 1.41935
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.0000 0.354103
\(966\) 0 0
\(967\) −13.8564 −0.445592 −0.222796 0.974865i \(-0.571518\pi\)
−0.222796 + 0.974865i \(0.571518\pi\)
\(968\) 0 0
\(969\) −15.0000 −0.481869
\(970\) 0 0
\(971\) −8.66025 −0.277921 −0.138960 0.990298i \(-0.544376\pi\)
−0.138960 + 0.990298i \(0.544376\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37.0000 1.18373 0.591867 0.806035i \(-0.298391\pi\)
0.591867 + 0.806035i \(0.298391\pi\)
\(978\) 0 0
\(979\) 36.3731 1.16249
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 32.9090 1.04963 0.524816 0.851215i \(-0.324134\pi\)
0.524816 + 0.851215i \(0.324134\pi\)
\(984\) 0 0
\(985\) 8.00000 0.254901
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 25.9808 0.825306 0.412653 0.910888i \(-0.364602\pi\)
0.412653 + 0.910888i \(0.364602\pi\)
\(992\) 0 0
\(993\) 15.0000 0.476011
\(994\) 0 0
\(995\) 8.66025 0.274549
\(996\) 0 0
\(997\) −23.0000 −0.728417 −0.364209 0.931317i \(-0.618661\pi\)
−0.364209 + 0.931317i \(0.618661\pi\)
\(998\) 0 0
\(999\) 25.9808 0.821995
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.n.1.2 2
4.3 odd 2 inner 1568.2.a.n.1.1 2
7.2 even 3 1568.2.i.u.1537.1 4
7.3 odd 6 224.2.i.b.65.2 yes 4
7.4 even 3 1568.2.i.u.961.1 4
7.5 odd 6 224.2.i.b.193.2 yes 4
7.6 odd 2 1568.2.a.s.1.1 2
8.3 odd 2 3136.2.a.bu.1.2 2
8.5 even 2 3136.2.a.bu.1.1 2
21.5 even 6 2016.2.s.r.865.1 4
21.17 even 6 2016.2.s.r.289.1 4
28.3 even 6 224.2.i.b.65.1 4
28.11 odd 6 1568.2.i.u.961.2 4
28.19 even 6 224.2.i.b.193.1 yes 4
28.23 odd 6 1568.2.i.u.1537.2 4
28.27 even 2 1568.2.a.s.1.2 2
56.3 even 6 448.2.i.i.65.2 4
56.5 odd 6 448.2.i.i.193.1 4
56.13 odd 2 3136.2.a.bh.1.2 2
56.19 even 6 448.2.i.i.193.2 4
56.27 even 2 3136.2.a.bh.1.1 2
56.45 odd 6 448.2.i.i.65.1 4
84.47 odd 6 2016.2.s.r.865.2 4
84.59 odd 6 2016.2.s.r.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.i.b.65.1 4 28.3 even 6
224.2.i.b.65.2 yes 4 7.3 odd 6
224.2.i.b.193.1 yes 4 28.19 even 6
224.2.i.b.193.2 yes 4 7.5 odd 6
448.2.i.i.65.1 4 56.45 odd 6
448.2.i.i.65.2 4 56.3 even 6
448.2.i.i.193.1 4 56.5 odd 6
448.2.i.i.193.2 4 56.19 even 6
1568.2.a.n.1.1 2 4.3 odd 2 inner
1568.2.a.n.1.2 2 1.1 even 1 trivial
1568.2.a.s.1.1 2 7.6 odd 2
1568.2.a.s.1.2 2 28.27 even 2
1568.2.i.u.961.1 4 7.4 even 3
1568.2.i.u.961.2 4 28.11 odd 6
1568.2.i.u.1537.1 4 7.2 even 3
1568.2.i.u.1537.2 4 28.23 odd 6
2016.2.s.r.289.1 4 21.17 even 6
2016.2.s.r.289.2 4 84.59 odd 6
2016.2.s.r.865.1 4 21.5 even 6
2016.2.s.r.865.2 4 84.47 odd 6
3136.2.a.bh.1.1 2 56.27 even 2
3136.2.a.bh.1.2 2 56.13 odd 2
3136.2.a.bu.1.1 2 8.5 even 2
3136.2.a.bu.1.2 2 8.3 odd 2