Properties

Label 7448.2.a.be.1.1
Level $7448$
Weight $2$
Character 7448.1
Self dual yes
Analytic conductor $59.473$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1064)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 7448.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} -5.27492 q^{11} -2.00000 q^{13} +2.00000 q^{15} +0.274917 q^{17} +1.00000 q^{19} +7.54983 q^{23} -4.00000 q^{25} -4.00000 q^{27} -4.00000 q^{29} +4.00000 q^{31} -10.5498 q^{33} +2.00000 q^{37} -4.00000 q^{39} -10.5498 q^{41} -1.00000 q^{43} +1.00000 q^{45} -5.27492 q^{47} +0.549834 q^{51} +8.54983 q^{53} -5.27492 q^{55} +2.00000 q^{57} +4.00000 q^{59} -4.72508 q^{61} -2.00000 q^{65} +15.0997 q^{69} -4.54983 q^{71} -11.2749 q^{73} -8.00000 q^{75} -14.5498 q^{79} -11.0000 q^{81} +3.54983 q^{83} +0.274917 q^{85} -8.00000 q^{87} -1.45017 q^{89} +8.00000 q^{93} +1.00000 q^{95} +6.54983 q^{97} -5.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 2 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{3} + 2 q^{5} + 2 q^{9} - 3 q^{11} - 4 q^{13} + 4 q^{15} - 7 q^{17} + 2 q^{19} - 8 q^{25} - 8 q^{27} - 8 q^{29} + 8 q^{31} - 6 q^{33} + 4 q^{37} - 8 q^{39} - 6 q^{41} - 2 q^{43} + 2 q^{45} - 3 q^{47} - 14 q^{51} + 2 q^{53} - 3 q^{55} + 4 q^{57} + 8 q^{59} - 17 q^{61} - 4 q^{65} + 6 q^{71} - 15 q^{73} - 16 q^{75} - 14 q^{79} - 22 q^{81} - 8 q^{83} - 7 q^{85} - 16 q^{87} - 18 q^{89} + 16 q^{93} + 2 q^{95} - 2 q^{97} - 3 q^{99} + O(q^{100}) \)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.27492 −1.59045 −0.795224 0.606316i \(-0.792647\pi\)
−0.795224 + 0.606316i \(0.792647\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) 0.274917 0.0666772 0.0333386 0.999444i \(-0.489386\pi\)
0.0333386 + 0.999444i \(0.489386\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.54983 1.57425 0.787125 0.616794i \(-0.211569\pi\)
0.787125 + 0.616794i \(0.211569\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −10.5498 −1.83649
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −10.5498 −1.64761 −0.823804 0.566875i \(-0.808152\pi\)
−0.823804 + 0.566875i \(0.808152\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −5.27492 −0.769426 −0.384713 0.923036i \(-0.625700\pi\)
−0.384713 + 0.923036i \(0.625700\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.549834 0.0769922
\(52\) 0 0
\(53\) 8.54983 1.17441 0.587205 0.809438i \(-0.300228\pi\)
0.587205 + 0.809438i \(0.300228\pi\)
\(54\) 0 0
\(55\) −5.27492 −0.711270
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −4.72508 −0.604985 −0.302492 0.953152i \(-0.597819\pi\)
−0.302492 + 0.953152i \(0.597819\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 15.0997 1.81779
\(70\) 0 0
\(71\) −4.54983 −0.539966 −0.269983 0.962865i \(-0.587018\pi\)
−0.269983 + 0.962865i \(0.587018\pi\)
\(72\) 0 0
\(73\) −11.2749 −1.31963 −0.659815 0.751428i \(-0.729365\pi\)
−0.659815 + 0.751428i \(0.729365\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.5498 −1.63698 −0.818492 0.574518i \(-0.805190\pi\)
−0.818492 + 0.574518i \(0.805190\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 3.54983 0.389645 0.194822 0.980839i \(-0.437587\pi\)
0.194822 + 0.980839i \(0.437587\pi\)
\(84\) 0 0
\(85\) 0.274917 0.0298190
\(86\) 0 0
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) −1.45017 −0.153717 −0.0768586 0.997042i \(-0.524489\pi\)
−0.0768586 + 0.997042i \(0.524489\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 6.54983 0.665035 0.332517 0.943097i \(-0.392102\pi\)
0.332517 + 0.943097i \(0.392102\pi\)
\(98\) 0 0
\(99\) −5.27492 −0.530149
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.5498 −1.79328 −0.896640 0.442760i \(-0.853999\pi\)
−0.896640 + 0.442760i \(0.853999\pi\)
\(108\) 0 0
\(109\) 4.54983 0.435795 0.217898 0.975972i \(-0.430080\pi\)
0.217898 + 0.975972i \(0.430080\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 3.45017 0.324564 0.162282 0.986744i \(-0.448115\pi\)
0.162282 + 0.986744i \(0.448115\pi\)
\(114\) 0 0
\(115\) 7.54983 0.704026
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 16.8248 1.52952
\(122\) 0 0
\(123\) −21.0997 −1.90249
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 15.0997 1.33988 0.669939 0.742416i \(-0.266320\pi\)
0.669939 + 0.742416i \(0.266320\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −6.82475 −0.596281 −0.298141 0.954522i \(-0.596366\pi\)
−0.298141 + 0.954522i \(0.596366\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) 11.2749 0.963281 0.481641 0.876369i \(-0.340041\pi\)
0.481641 + 0.876369i \(0.340041\pi\)
\(138\) 0 0
\(139\) −11.8248 −1.00296 −0.501481 0.865169i \(-0.667211\pi\)
−0.501481 + 0.865169i \(0.667211\pi\)
\(140\) 0 0
\(141\) −10.5498 −0.888456
\(142\) 0 0
\(143\) 10.5498 0.882221
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.0997 −1.31894 −0.659468 0.751733i \(-0.729218\pi\)
−0.659468 + 0.751733i \(0.729218\pi\)
\(150\) 0 0
\(151\) 6.54983 0.533018 0.266509 0.963832i \(-0.414130\pi\)
0.266509 + 0.963832i \(0.414130\pi\)
\(152\) 0 0
\(153\) 0.274917 0.0222257
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −14.0997 −1.12528 −0.562638 0.826703i \(-0.690214\pi\)
−0.562638 + 0.826703i \(0.690214\pi\)
\(158\) 0 0
\(159\) 17.0997 1.35609
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −21.5498 −1.68791 −0.843957 0.536411i \(-0.819780\pi\)
−0.843957 + 0.536411i \(0.819780\pi\)
\(164\) 0 0
\(165\) −10.5498 −0.821303
\(166\) 0 0
\(167\) −13.0997 −1.01368 −0.506841 0.862039i \(-0.669187\pi\)
−0.506841 + 0.862039i \(0.669187\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −20.5498 −1.56237 −0.781187 0.624296i \(-0.785386\pi\)
−0.781187 + 0.624296i \(0.785386\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) −15.0997 −1.12860 −0.564301 0.825569i \(-0.690854\pi\)
−0.564301 + 0.825569i \(0.690854\pi\)
\(180\) 0 0
\(181\) −0.549834 −0.0408689 −0.0204344 0.999791i \(-0.506505\pi\)
−0.0204344 + 0.999791i \(0.506505\pi\)
\(182\) 0 0
\(183\) −9.45017 −0.698576
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −1.45017 −0.106047
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −21.0000 −1.51951 −0.759753 0.650211i \(-0.774680\pi\)
−0.759753 + 0.650211i \(0.774680\pi\)
\(192\) 0 0
\(193\) −3.09967 −0.223119 −0.111560 0.993758i \(-0.535585\pi\)
−0.111560 + 0.993758i \(0.535585\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 12.0997 0.862066 0.431033 0.902336i \(-0.358149\pi\)
0.431033 + 0.902336i \(0.358149\pi\)
\(198\) 0 0
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −10.5498 −0.736832
\(206\) 0 0
\(207\) 7.54983 0.524750
\(208\) 0 0
\(209\) −5.27492 −0.364874
\(210\) 0 0
\(211\) −25.6495 −1.76578 −0.882892 0.469576i \(-0.844407\pi\)
−0.882892 + 0.469576i \(0.844407\pi\)
\(212\) 0 0
\(213\) −9.09967 −0.623499
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −22.5498 −1.52378
\(220\) 0 0
\(221\) −0.549834 −0.0369859
\(222\) 0 0
\(223\) 17.0997 1.14508 0.572539 0.819877i \(-0.305958\pi\)
0.572539 + 0.819877i \(0.305958\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 9.45017 0.627230 0.313615 0.949550i \(-0.398460\pi\)
0.313615 + 0.949550i \(0.398460\pi\)
\(228\) 0 0
\(229\) 16.2749 1.07548 0.537738 0.843112i \(-0.319279\pi\)
0.537738 + 0.843112i \(0.319279\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.27492 0.280059 0.140030 0.990147i \(-0.455280\pi\)
0.140030 + 0.990147i \(0.455280\pi\)
\(234\) 0 0
\(235\) −5.27492 −0.344098
\(236\) 0 0
\(237\) −29.0997 −1.89023
\(238\) 0 0
\(239\) 23.3746 1.51198 0.755988 0.654585i \(-0.227157\pi\)
0.755988 + 0.654585i \(0.227157\pi\)
\(240\) 0 0
\(241\) −10.5498 −0.679575 −0.339787 0.940502i \(-0.610355\pi\)
−0.339787 + 0.940502i \(0.610355\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) 7.09967 0.449923
\(250\) 0 0
\(251\) 28.0997 1.77364 0.886818 0.462119i \(-0.152911\pi\)
0.886818 + 0.462119i \(0.152911\pi\)
\(252\) 0 0
\(253\) −39.8248 −2.50376
\(254\) 0 0
\(255\) 0.549834 0.0344320
\(256\) 0 0
\(257\) 22.5498 1.40662 0.703310 0.710883i \(-0.251705\pi\)
0.703310 + 0.710883i \(0.251705\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 0 0
\(263\) 22.8248 1.40743 0.703717 0.710480i \(-0.251522\pi\)
0.703717 + 0.710480i \(0.251522\pi\)
\(264\) 0 0
\(265\) 8.54983 0.525212
\(266\) 0 0
\(267\) −2.90033 −0.177497
\(268\) 0 0
\(269\) 24.5498 1.49683 0.748415 0.663231i \(-0.230815\pi\)
0.748415 + 0.663231i \(0.230815\pi\)
\(270\) 0 0
\(271\) −17.5498 −1.06608 −0.533038 0.846091i \(-0.678950\pi\)
−0.533038 + 0.846091i \(0.678950\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 21.0997 1.27236
\(276\) 0 0
\(277\) −0.725083 −0.0435660 −0.0217830 0.999763i \(-0.506934\pi\)
−0.0217830 + 0.999763i \(0.506934\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 0.549834 0.0328004 0.0164002 0.999866i \(-0.494779\pi\)
0.0164002 + 0.999866i \(0.494779\pi\)
\(282\) 0 0
\(283\) 17.0000 1.01055 0.505273 0.862960i \(-0.331392\pi\)
0.505273 + 0.862960i \(0.331392\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.9244 −0.995554
\(290\) 0 0
\(291\) 13.0997 0.767916
\(292\) 0 0
\(293\) −27.6495 −1.61530 −0.807651 0.589661i \(-0.799261\pi\)
−0.807651 + 0.589661i \(0.799261\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 21.0997 1.22433
\(298\) 0 0
\(299\) −15.0997 −0.873236
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −4.72508 −0.270557
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) −20.0000 −1.13776
\(310\) 0 0
\(311\) 10.2749 0.582637 0.291319 0.956626i \(-0.405906\pi\)
0.291319 + 0.956626i \(0.405906\pi\)
\(312\) 0 0
\(313\) 31.1993 1.76349 0.881745 0.471726i \(-0.156369\pi\)
0.881745 + 0.471726i \(0.156369\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.09967 0.511088 0.255544 0.966797i \(-0.417745\pi\)
0.255544 + 0.966797i \(0.417745\pi\)
\(318\) 0 0
\(319\) 21.0997 1.18135
\(320\) 0 0
\(321\) −37.0997 −2.07070
\(322\) 0 0
\(323\) 0.274917 0.0152968
\(324\) 0 0
\(325\) 8.00000 0.443760
\(326\) 0 0
\(327\) 9.09967 0.503213
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −14.0000 −0.769510 −0.384755 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.09967 0.495691 0.247845 0.968800i \(-0.420278\pi\)
0.247845 + 0.968800i \(0.420278\pi\)
\(338\) 0 0
\(339\) 6.90033 0.374775
\(340\) 0 0
\(341\) −21.0997 −1.14261
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 15.0997 0.812939
\(346\) 0 0
\(347\) 27.1993 1.46014 0.730068 0.683374i \(-0.239488\pi\)
0.730068 + 0.683374i \(0.239488\pi\)
\(348\) 0 0
\(349\) 9.37459 0.501810 0.250905 0.968012i \(-0.419272\pi\)
0.250905 + 0.968012i \(0.419272\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 7.09967 0.377877 0.188939 0.981989i \(-0.439495\pi\)
0.188939 + 0.981989i \(0.439495\pi\)
\(354\) 0 0
\(355\) −4.54983 −0.241480
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −25.2749 −1.33396 −0.666980 0.745076i \(-0.732413\pi\)
−0.666980 + 0.745076i \(0.732413\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 33.6495 1.76614
\(364\) 0 0
\(365\) −11.2749 −0.590156
\(366\) 0 0
\(367\) −34.1993 −1.78519 −0.892595 0.450858i \(-0.851118\pi\)
−0.892595 + 0.450858i \(0.851118\pi\)
\(368\) 0 0
\(369\) −10.5498 −0.549202
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 8.54983 0.442694 0.221347 0.975195i \(-0.428955\pi\)
0.221347 + 0.975195i \(0.428955\pi\)
\(374\) 0 0
\(375\) −18.0000 −0.929516
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −27.0997 −1.39202 −0.696008 0.718034i \(-0.745042\pi\)
−0.696008 + 0.718034i \(0.745042\pi\)
\(380\) 0 0
\(381\) 30.1993 1.54716
\(382\) 0 0
\(383\) −22.0000 −1.12415 −0.562074 0.827087i \(-0.689996\pi\)
−0.562074 + 0.827087i \(0.689996\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) 15.1752 0.769416 0.384708 0.923038i \(-0.374302\pi\)
0.384708 + 0.923038i \(0.374302\pi\)
\(390\) 0 0
\(391\) 2.07558 0.104967
\(392\) 0 0
\(393\) −13.6495 −0.688526
\(394\) 0 0
\(395\) −14.5498 −0.732082
\(396\) 0 0
\(397\) 28.2749 1.41908 0.709539 0.704666i \(-0.248903\pi\)
0.709539 + 0.704666i \(0.248903\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.09967 −0.454416 −0.227208 0.973846i \(-0.572960\pi\)
−0.227208 + 0.973846i \(0.572960\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −10.5498 −0.522936
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 22.5498 1.11230
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3.54983 0.174255
\(416\) 0 0
\(417\) −23.6495 −1.15812
\(418\) 0 0
\(419\) −4.45017 −0.217405 −0.108702 0.994074i \(-0.534670\pi\)
−0.108702 + 0.994074i \(0.534670\pi\)
\(420\) 0 0
\(421\) 3.45017 0.168151 0.0840754 0.996459i \(-0.473206\pi\)
0.0840754 + 0.996459i \(0.473206\pi\)
\(422\) 0 0
\(423\) −5.27492 −0.256475
\(424\) 0 0
\(425\) −1.09967 −0.0533418
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 21.0997 1.01870
\(430\) 0 0
\(431\) −21.0997 −1.01634 −0.508168 0.861258i \(-0.669677\pi\)
−0.508168 + 0.861258i \(0.669677\pi\)
\(432\) 0 0
\(433\) 5.09967 0.245074 0.122537 0.992464i \(-0.460897\pi\)
0.122537 + 0.992464i \(0.460897\pi\)
\(434\) 0 0
\(435\) −8.00000 −0.383571
\(436\) 0 0
\(437\) 7.54983 0.361158
\(438\) 0 0
\(439\) 26.1993 1.25043 0.625213 0.780454i \(-0.285012\pi\)
0.625213 + 0.780454i \(0.285012\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 22.8248 1.08444 0.542218 0.840238i \(-0.317585\pi\)
0.542218 + 0.840238i \(0.317585\pi\)
\(444\) 0 0
\(445\) −1.45017 −0.0687444
\(446\) 0 0
\(447\) −32.1993 −1.52298
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 55.6495 2.62043
\(452\) 0 0
\(453\) 13.0997 0.615476
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.72508 0.221030 0.110515 0.993874i \(-0.464750\pi\)
0.110515 + 0.993874i \(0.464750\pi\)
\(458\) 0 0
\(459\) −1.09967 −0.0513281
\(460\) 0 0
\(461\) −23.2749 −1.08402 −0.542010 0.840372i \(-0.682337\pi\)
−0.542010 + 0.840372i \(0.682337\pi\)
\(462\) 0 0
\(463\) −19.8248 −0.921334 −0.460667 0.887573i \(-0.652390\pi\)
−0.460667 + 0.887573i \(0.652390\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 10.7251 0.496298 0.248149 0.968722i \(-0.420178\pi\)
0.248149 + 0.968722i \(0.420178\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −28.1993 −1.29936
\(472\) 0 0
\(473\) 5.27492 0.242541
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 8.54983 0.391470
\(478\) 0 0
\(479\) 14.4502 0.660245 0.330122 0.943938i \(-0.392910\pi\)
0.330122 + 0.943938i \(0.392910\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.54983 0.297413
\(486\) 0 0
\(487\) 7.09967 0.321717 0.160858 0.986978i \(-0.448574\pi\)
0.160858 + 0.986978i \(0.448574\pi\)
\(488\) 0 0
\(489\) −43.0997 −1.94903
\(490\) 0 0
\(491\) 5.54983 0.250461 0.125230 0.992128i \(-0.460033\pi\)
0.125230 + 0.992128i \(0.460033\pi\)
\(492\) 0 0
\(493\) −1.09967 −0.0495266
\(494\) 0 0
\(495\) −5.27492 −0.237090
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 31.1993 1.39667 0.698337 0.715769i \(-0.253924\pi\)
0.698337 + 0.715769i \(0.253924\pi\)
\(500\) 0 0
\(501\) −26.1993 −1.17050
\(502\) 0 0
\(503\) 10.6495 0.474838 0.237419 0.971407i \(-0.423699\pi\)
0.237419 + 0.971407i \(0.423699\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) 0 0
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) 27.8248 1.22373
\(518\) 0 0
\(519\) −41.0997 −1.80408
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) −3.64950 −0.159582 −0.0797908 0.996812i \(-0.525425\pi\)
−0.0797908 + 0.996812i \(0.525425\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.09967 0.0479023
\(528\) 0 0
\(529\) 34.0000 1.47826
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 21.0997 0.913928
\(534\) 0 0
\(535\) −18.5498 −0.801979
\(536\) 0 0
\(537\) −30.1993 −1.30320
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −17.0000 −0.730887 −0.365444 0.930834i \(-0.619083\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 0 0
\(543\) −1.09967 −0.0471913
\(544\) 0 0
\(545\) 4.54983 0.194893
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 0 0
\(549\) −4.72508 −0.201662
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) −28.0997 −1.19062 −0.595311 0.803496i \(-0.702971\pi\)
−0.595311 + 0.803496i \(0.702971\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) −2.90033 −0.122452
\(562\) 0 0
\(563\) 26.5498 1.11894 0.559471 0.828850i \(-0.311004\pi\)
0.559471 + 0.828850i \(0.311004\pi\)
\(564\) 0 0
\(565\) 3.45017 0.145150
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 44.1993 1.85293 0.926466 0.376378i \(-0.122830\pi\)
0.926466 + 0.376378i \(0.122830\pi\)
\(570\) 0 0
\(571\) −35.5498 −1.48771 −0.743857 0.668339i \(-0.767006\pi\)
−0.743857 + 0.668339i \(0.767006\pi\)
\(572\) 0 0
\(573\) −42.0000 −1.75458
\(574\) 0 0
\(575\) −30.1993 −1.25940
\(576\) 0 0
\(577\) 24.3746 1.01473 0.507364 0.861732i \(-0.330620\pi\)
0.507364 + 0.861732i \(0.330620\pi\)
\(578\) 0 0
\(579\) −6.19934 −0.257636
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −45.0997 −1.86784
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) 10.2749 0.424091 0.212046 0.977260i \(-0.431987\pi\)
0.212046 + 0.977260i \(0.431987\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 24.1993 0.995428
\(592\) 0 0
\(593\) −3.00000 −0.123195 −0.0615976 0.998101i \(-0.519620\pi\)
−0.0615976 + 0.998101i \(0.519620\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.00000 −0.245564
\(598\) 0 0
\(599\) −29.6495 −1.21145 −0.605723 0.795676i \(-0.707116\pi\)
−0.605723 + 0.795676i \(0.707116\pi\)
\(600\) 0 0
\(601\) 26.5498 1.08299 0.541495 0.840704i \(-0.317858\pi\)
0.541495 + 0.840704i \(0.317858\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16.8248 0.684023
\(606\) 0 0
\(607\) 19.6495 0.797549 0.398774 0.917049i \(-0.369436\pi\)
0.398774 + 0.917049i \(0.369436\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.5498 0.426801
\(612\) 0 0
\(613\) −15.1752 −0.612923 −0.306461 0.951883i \(-0.599145\pi\)
−0.306461 + 0.951883i \(0.599145\pi\)
\(614\) 0 0
\(615\) −21.0997 −0.850821
\(616\) 0 0
\(617\) 6.09967 0.245563 0.122782 0.992434i \(-0.460818\pi\)
0.122782 + 0.992434i \(0.460818\pi\)
\(618\) 0 0
\(619\) 3.54983 0.142680 0.0713399 0.997452i \(-0.477272\pi\)
0.0713399 + 0.997452i \(0.477272\pi\)
\(620\) 0 0
\(621\) −30.1993 −1.21186
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −10.5498 −0.421320
\(628\) 0 0
\(629\) 0.549834 0.0219233
\(630\) 0 0
\(631\) −17.0000 −0.676759 −0.338380 0.941010i \(-0.609879\pi\)
−0.338380 + 0.941010i \(0.609879\pi\)
\(632\) 0 0
\(633\) −51.2990 −2.03895
\(634\) 0 0
\(635\) 15.0997 0.599212
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.54983 −0.179989
\(640\) 0 0
\(641\) 31.4502 1.24221 0.621103 0.783729i \(-0.286685\pi\)
0.621103 + 0.783729i \(0.286685\pi\)
\(642\) 0 0
\(643\) 22.8248 0.900120 0.450060 0.892998i \(-0.351403\pi\)
0.450060 + 0.892998i \(0.351403\pi\)
\(644\) 0 0
\(645\) −2.00000 −0.0787499
\(646\) 0 0
\(647\) −16.0997 −0.632943 −0.316472 0.948602i \(-0.602498\pi\)
−0.316472 + 0.948602i \(0.602498\pi\)
\(648\) 0 0
\(649\) −21.0997 −0.828234
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8.27492 −0.323823 −0.161911 0.986805i \(-0.551766\pi\)
−0.161911 + 0.986805i \(0.551766\pi\)
\(654\) 0 0
\(655\) −6.82475 −0.266665
\(656\) 0 0
\(657\) −11.2749 −0.439876
\(658\) 0 0
\(659\) −40.5498 −1.57960 −0.789799 0.613366i \(-0.789815\pi\)
−0.789799 + 0.613366i \(0.789815\pi\)
\(660\) 0 0
\(661\) 29.4502 1.14548 0.572739 0.819738i \(-0.305881\pi\)
0.572739 + 0.819738i \(0.305881\pi\)
\(662\) 0 0
\(663\) −1.09967 −0.0427076
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −30.1993 −1.16932
\(668\) 0 0
\(669\) 34.1993 1.32222
\(670\) 0 0
\(671\) 24.9244 0.962197
\(672\) 0 0
\(673\) −4.54983 −0.175383 −0.0876916 0.996148i \(-0.527949\pi\)
−0.0876916 + 0.996148i \(0.527949\pi\)
\(674\) 0 0
\(675\) 16.0000 0.615840
\(676\) 0 0
\(677\) −6.90033 −0.265201 −0.132601 0.991170i \(-0.542333\pi\)
−0.132601 + 0.991170i \(0.542333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 18.9003 0.724262
\(682\) 0 0
\(683\) 27.6495 1.05798 0.528989 0.848628i \(-0.322571\pi\)
0.528989 + 0.848628i \(0.322571\pi\)
\(684\) 0 0
\(685\) 11.2749 0.430792
\(686\) 0 0
\(687\) 32.5498 1.24185
\(688\) 0 0
\(689\) −17.0997 −0.651446
\(690\) 0 0
\(691\) −15.9244 −0.605794 −0.302897 0.953023i \(-0.597954\pi\)
−0.302897 + 0.953023i \(0.597954\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.8248 −0.448538
\(696\) 0 0
\(697\) −2.90033 −0.109858
\(698\) 0 0
\(699\) 8.54983 0.323384
\(700\) 0 0
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) −10.5498 −0.397330
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 0 0
\(711\) −14.5498 −0.545661
\(712\) 0 0
\(713\) 30.1993 1.13097
\(714\) 0 0
\(715\) 10.5498 0.394541
\(716\) 0 0
\(717\) 46.7492 1.74588
\(718\) 0 0
\(719\) −34.2749 −1.27824 −0.639119 0.769108i \(-0.720701\pi\)
−0.639119 + 0.769108i \(0.720701\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −21.0997 −0.784705
\(724\) 0 0
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) 44.0997 1.63557 0.817783 0.575527i \(-0.195203\pi\)
0.817783 + 0.575527i \(0.195203\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −0.274917 −0.0101682
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −26.8248 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 28.7492 1.05470 0.527352 0.849647i \(-0.323185\pi\)
0.527352 + 0.849647i \(0.323185\pi\)
\(744\) 0 0
\(745\) −16.0997 −0.589846
\(746\) 0 0
\(747\) 3.54983 0.129882
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 56.1993 2.04802
\(754\) 0 0
\(755\) 6.54983 0.238373
\(756\) 0 0
\(757\) −12.3746 −0.449762 −0.224881 0.974386i \(-0.572199\pi\)
−0.224881 + 0.974386i \(0.572199\pi\)
\(758\) 0 0
\(759\) −79.6495 −2.89109
\(760\) 0 0
\(761\) 43.0000 1.55875 0.779374 0.626559i \(-0.215537\pi\)
0.779374 + 0.626559i \(0.215537\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.274917 0.00993965
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) −17.1993 −0.620224 −0.310112 0.950700i \(-0.600367\pi\)
−0.310112 + 0.950700i \(0.600367\pi\)
\(770\) 0 0
\(771\) 45.0997 1.62422
\(772\) 0 0
\(773\) 14.0000 0.503545 0.251773 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.5498 −0.377987
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) 16.0000 0.571793
\(784\) 0 0
\(785\) −14.0997 −0.503239
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 0 0
\(789\) 45.6495 1.62517
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 9.45017 0.335585
\(794\) 0 0
\(795\) 17.0997 0.606463
\(796\) 0 0
\(797\) 1.45017 0.0513675 0.0256837 0.999670i \(-0.491824\pi\)
0.0256837 + 0.999670i \(0.491824\pi\)
\(798\) 0 0
\(799\) −1.45017 −0.0513032
\(800\) 0 0
\(801\) −1.45017 −0.0512391
\(802\) 0 0
\(803\) 59.4743 2.09880
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 49.0997 1.72839
\(808\) 0 0
\(809\) 12.0997 0.425402 0.212701 0.977117i \(-0.431774\pi\)
0.212701 + 0.977117i \(0.431774\pi\)
\(810\) 0 0
\(811\) −52.7492 −1.85227 −0.926137 0.377187i \(-0.876891\pi\)
−0.926137 + 0.377187i \(0.876891\pi\)
\(812\) 0 0
\(813\) −35.0997 −1.23100
\(814\) 0 0
\(815\) −21.5498 −0.754858
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −26.9244 −0.939669 −0.469834 0.882755i \(-0.655686\pi\)
−0.469834 + 0.882755i \(0.655686\pi\)
\(822\) 0 0
\(823\) 4.92442 0.171655 0.0858273 0.996310i \(-0.472647\pi\)
0.0858273 + 0.996310i \(0.472647\pi\)
\(824\) 0 0
\(825\) 42.1993 1.46919
\(826\) 0 0
\(827\) −41.6495 −1.44830 −0.724148 0.689645i \(-0.757767\pi\)
−0.724148 + 0.689645i \(0.757767\pi\)
\(828\) 0 0
\(829\) 14.9003 0.517510 0.258755 0.965943i \(-0.416688\pi\)
0.258755 + 0.965943i \(0.416688\pi\)
\(830\) 0 0
\(831\) −1.45017 −0.0503057
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −13.0997 −0.453333
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) −37.6495 −1.29981 −0.649903 0.760018i \(-0.725190\pi\)
−0.649903 + 0.760018i \(0.725190\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 1.09967 0.0378746
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 34.0000 1.16688
\(850\) 0 0
\(851\) 15.0997 0.517610
\(852\) 0 0
\(853\) 42.0997 1.44147 0.720733 0.693213i \(-0.243806\pi\)
0.720733 + 0.693213i \(0.243806\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −41.6495 −1.42272 −0.711360 0.702828i \(-0.751920\pi\)
−0.711360 + 0.702828i \(0.751920\pi\)
\(858\) 0 0
\(859\) 10.0997 0.344596 0.172298 0.985045i \(-0.444881\pi\)
0.172298 + 0.985045i \(0.444881\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.90033 −0.0987284 −0.0493642 0.998781i \(-0.515720\pi\)
−0.0493642 + 0.998781i \(0.515720\pi\)
\(864\) 0 0
\(865\) −20.5498 −0.698715
\(866\) 0 0
\(867\) −33.8488 −1.14957
\(868\) 0 0
\(869\) 76.7492 2.60354
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 6.54983 0.221678
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.7492 1.10586 0.552930 0.833227i \(-0.313510\pi\)
0.552930 + 0.833227i \(0.313510\pi\)
\(878\) 0 0
\(879\) −55.2990 −1.86519
\(880\) 0 0
\(881\) −49.9244 −1.68200 −0.840998 0.541038i \(-0.818032\pi\)
−0.840998 + 0.541038i \(0.818032\pi\)
\(882\) 0 0
\(883\) −41.0241 −1.38057 −0.690285 0.723537i \(-0.742515\pi\)
−0.690285 + 0.723537i \(0.742515\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) 52.1993 1.75268 0.876341 0.481691i \(-0.159977\pi\)
0.876341 + 0.481691i \(0.159977\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 58.0241 1.94388
\(892\) 0 0
\(893\) −5.27492 −0.176518
\(894\) 0 0
\(895\) −15.0997 −0.504726
\(896\) 0 0
\(897\) −30.1993 −1.00833
\(898\) 0 0
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 2.35050 0.0783064
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.549834 −0.0182771
\(906\) 0 0
\(907\) 6.90033 0.229122 0.114561 0.993416i \(-0.463454\pi\)
0.114561 + 0.993416i \(0.463454\pi\)
\(908\) 0 0
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) 11.4502 0.379361 0.189680 0.981846i \(-0.439255\pi\)
0.189680 + 0.981846i \(0.439255\pi\)
\(912\) 0 0
\(913\) −18.7251 −0.619710
\(914\) 0 0
\(915\) −9.45017 −0.312413
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −39.5498 −1.30463 −0.652314 0.757949i \(-0.726202\pi\)
−0.652314 + 0.757949i \(0.726202\pi\)
\(920\) 0 0
\(921\) 52.0000 1.71346
\(922\) 0 0
\(923\) 9.09967 0.299519
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 0 0
\(927\) −10.0000 −0.328443
\(928\) 0 0
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 20.5498 0.672771
\(934\) 0 0
\(935\) −1.45017 −0.0474255
\(936\) 0 0
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) 62.3987 2.03630
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) −79.6495 −2.59374
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) 0 0
\(949\) 22.5498 0.731999
\(950\) 0 0
\(951\) 18.1993 0.590154
\(952\) 0 0
\(953\) 43.0997 1.39614 0.698068 0.716032i \(-0.254043\pi\)
0.698068 + 0.716032i \(0.254043\pi\)
\(954\) 0 0
\(955\) −21.0000 −0.679544
\(956\) 0 0
\(957\) 42.1993 1.36411
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −18.5498 −0.597760
\(964\) 0 0
\(965\) −3.09967 −0.0997819
\(966\) 0 0
\(967\) 42.1993 1.35704 0.678520 0.734582i \(-0.262622\pi\)
0.678520 + 0.734582i \(0.262622\pi\)
\(968\) 0 0
\(969\) 0.549834 0.0176632
\(970\) 0 0
\(971\) −21.4502 −0.688369 −0.344184 0.938902i \(-0.611844\pi\)
−0.344184 + 0.938902i \(0.611844\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 16.0000 0.512410
\(976\) 0 0
\(977\) −35.2990 −1.12932 −0.564658 0.825325i \(-0.690992\pi\)
−0.564658 + 0.825325i \(0.690992\pi\)
\(978\) 0 0
\(979\) 7.64950 0.244479
\(980\) 0 0
\(981\) 4.54983 0.145265
\(982\) 0 0
\(983\) 52.1993 1.66490 0.832450 0.554100i \(-0.186937\pi\)
0.832450 + 0.554100i \(0.186937\pi\)
\(984\) 0 0
\(985\) 12.0997 0.385528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7.54983 −0.240071
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) −3.00000 −0.0951064
\(996\) 0 0
\(997\) 7.17525 0.227242 0.113621 0.993524i \(-0.463755\pi\)
0.113621 + 0.993524i \(0.463755\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 7448.2.a.be.1.1 2
7.3 odd 6 1064.2.q.l.457.1 yes 4
7.5 odd 6 1064.2.q.l.305.1 4
7.6 odd 2 7448.2.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.l.305.1 4 7.5 odd 6
1064.2.q.l.457.1 yes 4 7.3 odd 6
7448.2.a.w.1.1 2 7.6 odd 2
7448.2.a.be.1.1 2 1.1 even 1 trivial