Properties

Label 1176.2.a.l.1.1
Level $1176$
Weight $2$
Character 1176.1
Self dual yes
Analytic conductor $9.390$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.39040727770\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1176.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} -4.82843 q^{11} +4.24264 q^{13} -0.585786 q^{15} +4.58579 q^{17} +1.17157 q^{19} +0.828427 q^{23} -4.65685 q^{25} -1.00000 q^{27} -2.82843 q^{29} +2.82843 q^{31} +4.82843 q^{33} +9.65685 q^{37} -4.24264 q^{39} +1.75736 q^{41} +11.3137 q^{43} +0.585786 q^{45} +12.4853 q^{47} -4.58579 q^{51} -2.00000 q^{53} -2.82843 q^{55} -1.17157 q^{57} -8.48528 q^{59} -3.07107 q^{61} +2.48528 q^{65} -11.3137 q^{67} -0.828427 q^{69} +6.48528 q^{71} +16.2426 q^{73} +4.65685 q^{75} +2.34315 q^{79} +1.00000 q^{81} -4.00000 q^{83} +2.68629 q^{85} +2.82843 q^{87} +14.2426 q^{89} -2.82843 q^{93} +0.686292 q^{95} +8.24264 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} + 2q^{9} - 4q^{11} - 4q^{15} + 12q^{17} + 8q^{19} - 4q^{23} + 2q^{25} - 2q^{27} + 4q^{33} + 8q^{37} + 12q^{41} + 4q^{45} + 8q^{47} - 12q^{51} - 4q^{53} - 8q^{57} + 8q^{61} - 12q^{65} + 4q^{69} - 4q^{71} + 24q^{73} - 2q^{75} + 16q^{79} + 2q^{81} - 8q^{83} + 28q^{85} + 20q^{89} + 24q^{95} + 8q^{97} - 4q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0.585786 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) 4.58579 1.11222 0.556108 0.831110i \(-0.312294\pi\)
0.556108 + 0.831110i \(0.312294\pi\)
\(18\) 0 0
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.828427 0.172739 0.0863695 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 0 0
\(33\) 4.82843 0.840521
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.65685 1.58758 0.793789 0.608194i \(-0.208106\pi\)
0.793789 + 0.608194i \(0.208106\pi\)
\(38\) 0 0
\(39\) −4.24264 −0.679366
\(40\) 0 0
\(41\) 1.75736 0.274453 0.137227 0.990540i \(-0.456181\pi\)
0.137227 + 0.990540i \(0.456181\pi\)
\(42\) 0 0
\(43\) 11.3137 1.72532 0.862662 0.505781i \(-0.168795\pi\)
0.862662 + 0.505781i \(0.168795\pi\)
\(44\) 0 0
\(45\) 0.585786 0.0873239
\(46\) 0 0
\(47\) 12.4853 1.82117 0.910583 0.413327i \(-0.135633\pi\)
0.910583 + 0.413327i \(0.135633\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.58579 −0.642139
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −2.82843 −0.381385
\(56\) 0 0
\(57\) −1.17157 −0.155179
\(58\) 0 0
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) −3.07107 −0.393210 −0.196605 0.980483i \(-0.562992\pi\)
−0.196605 + 0.980483i \(0.562992\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.48528 0.308261
\(66\) 0 0
\(67\) −11.3137 −1.38219 −0.691095 0.722764i \(-0.742871\pi\)
−0.691095 + 0.722764i \(0.742871\pi\)
\(68\) 0 0
\(69\) −0.828427 −0.0997309
\(70\) 0 0
\(71\) 6.48528 0.769661 0.384831 0.922987i \(-0.374260\pi\)
0.384831 + 0.922987i \(0.374260\pi\)
\(72\) 0 0
\(73\) 16.2426 1.90106 0.950529 0.310637i \(-0.100542\pi\)
0.950529 + 0.310637i \(0.100542\pi\)
\(74\) 0 0
\(75\) 4.65685 0.537727
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.68629 0.291369
\(86\) 0 0
\(87\) 2.82843 0.303239
\(88\) 0 0
\(89\) 14.2426 1.50972 0.754858 0.655888i \(-0.227706\pi\)
0.754858 + 0.655888i \(0.227706\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.82843 −0.293294
\(94\) 0 0
\(95\) 0.686292 0.0704120
\(96\) 0 0
\(97\) 8.24264 0.836913 0.418457 0.908237i \(-0.362571\pi\)
0.418457 + 0.908237i \(0.362571\pi\)
\(98\) 0 0
\(99\) −4.82843 −0.485275
\(100\) 0 0
\(101\) 14.7279 1.46548 0.732742 0.680507i \(-0.238240\pi\)
0.732742 + 0.680507i \(0.238240\pi\)
\(102\) 0 0
\(103\) −14.1421 −1.39347 −0.696733 0.717331i \(-0.745364\pi\)
−0.696733 + 0.717331i \(0.745364\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.17157 0.693302 0.346651 0.937994i \(-0.387319\pi\)
0.346651 + 0.937994i \(0.387319\pi\)
\(108\) 0 0
\(109\) 19.3137 1.84992 0.924959 0.380067i \(-0.124099\pi\)
0.924959 + 0.380067i \(0.124099\pi\)
\(110\) 0 0
\(111\) −9.65685 −0.916588
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0.485281 0.0452527
\(116\) 0 0
\(117\) 4.24264 0.392232
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) −1.75736 −0.158456
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.585786 −0.0504165
\(136\) 0 0
\(137\) −14.8284 −1.26688 −0.633439 0.773793i \(-0.718357\pi\)
−0.633439 + 0.773793i \(0.718357\pi\)
\(138\) 0 0
\(139\) 12.9706 1.10015 0.550074 0.835116i \(-0.314599\pi\)
0.550074 + 0.835116i \(0.314599\pi\)
\(140\) 0 0
\(141\) −12.4853 −1.05145
\(142\) 0 0
\(143\) −20.4853 −1.71307
\(144\) 0 0
\(145\) −1.65685 −0.137594
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −21.3137 −1.74609 −0.873044 0.487642i \(-0.837857\pi\)
−0.873044 + 0.487642i \(0.837857\pi\)
\(150\) 0 0
\(151\) 1.65685 0.134833 0.0674164 0.997725i \(-0.478524\pi\)
0.0674164 + 0.997725i \(0.478524\pi\)
\(152\) 0 0
\(153\) 4.58579 0.370739
\(154\) 0 0
\(155\) 1.65685 0.133082
\(156\) 0 0
\(157\) −8.24264 −0.657834 −0.328917 0.944359i \(-0.606684\pi\)
−0.328917 + 0.944359i \(0.606684\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −5.65685 −0.443079 −0.221540 0.975151i \(-0.571108\pi\)
−0.221540 + 0.975151i \(0.571108\pi\)
\(164\) 0 0
\(165\) 2.82843 0.220193
\(166\) 0 0
\(167\) −9.17157 −0.709718 −0.354859 0.934920i \(-0.615471\pi\)
−0.354859 + 0.934920i \(0.615471\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 1.17157 0.0895924
\(172\) 0 0
\(173\) −19.4142 −1.47604 −0.738018 0.674781i \(-0.764238\pi\)
−0.738018 + 0.674781i \(0.764238\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.48528 0.637793
\(178\) 0 0
\(179\) 10.4853 0.783707 0.391853 0.920028i \(-0.371834\pi\)
0.391853 + 0.920028i \(0.371834\pi\)
\(180\) 0 0
\(181\) 7.07107 0.525588 0.262794 0.964852i \(-0.415356\pi\)
0.262794 + 0.964852i \(0.415356\pi\)
\(182\) 0 0
\(183\) 3.07107 0.227020
\(184\) 0 0
\(185\) 5.65685 0.415900
\(186\) 0 0
\(187\) −22.1421 −1.61919
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.1421 −1.16800 −0.584002 0.811752i \(-0.698514\pi\)
−0.584002 + 0.811752i \(0.698514\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) −2.48528 −0.177975
\(196\) 0 0
\(197\) −25.3137 −1.80353 −0.901764 0.432230i \(-0.857727\pi\)
−0.901764 + 0.432230i \(0.857727\pi\)
\(198\) 0 0
\(199\) 5.65685 0.401004 0.200502 0.979693i \(-0.435743\pi\)
0.200502 + 0.979693i \(0.435743\pi\)
\(200\) 0 0
\(201\) 11.3137 0.798007
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.02944 0.0718990
\(206\) 0 0
\(207\) 0.828427 0.0575797
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) −9.65685 −0.664805 −0.332403 0.943138i \(-0.607859\pi\)
−0.332403 + 0.943138i \(0.607859\pi\)
\(212\) 0 0
\(213\) −6.48528 −0.444364
\(214\) 0 0
\(215\) 6.62742 0.451986
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −16.2426 −1.09758
\(220\) 0 0
\(221\) 19.4558 1.30874
\(222\) 0 0
\(223\) 2.34315 0.156909 0.0784543 0.996918i \(-0.475002\pi\)
0.0784543 + 0.996918i \(0.475002\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) 0 0
\(227\) −10.8284 −0.718708 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(228\) 0 0
\(229\) −22.5858 −1.49251 −0.746255 0.665660i \(-0.768150\pi\)
−0.746255 + 0.665660i \(0.768150\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.17157 −0.600850 −0.300425 0.953805i \(-0.597128\pi\)
−0.300425 + 0.953805i \(0.597128\pi\)
\(234\) 0 0
\(235\) 7.31371 0.477094
\(236\) 0 0
\(237\) −2.34315 −0.152204
\(238\) 0 0
\(239\) −7.17157 −0.463890 −0.231945 0.972729i \(-0.574509\pi\)
−0.231945 + 0.972729i \(0.574509\pi\)
\(240\) 0 0
\(241\) −5.89949 −0.380020 −0.190010 0.981782i \(-0.560852\pi\)
−0.190010 + 0.981782i \(0.560852\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.97056 0.316269
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 22.1421 1.39760 0.698800 0.715317i \(-0.253718\pi\)
0.698800 + 0.715317i \(0.253718\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −2.68629 −0.168222
\(256\) 0 0
\(257\) −0.585786 −0.0365404 −0.0182702 0.999833i \(-0.505816\pi\)
−0.0182702 + 0.999833i \(0.505816\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) 0 0
\(263\) 8.14214 0.502066 0.251033 0.967979i \(-0.419230\pi\)
0.251033 + 0.967979i \(0.419230\pi\)
\(264\) 0 0
\(265\) −1.17157 −0.0719691
\(266\) 0 0
\(267\) −14.2426 −0.871635
\(268\) 0 0
\(269\) −19.8995 −1.21329 −0.606647 0.794971i \(-0.707486\pi\)
−0.606647 + 0.794971i \(0.707486\pi\)
\(270\) 0 0
\(271\) −22.1421 −1.34504 −0.672519 0.740079i \(-0.734788\pi\)
−0.672519 + 0.740079i \(0.734788\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.4853 1.35591
\(276\) 0 0
\(277\) 16.6274 0.999045 0.499522 0.866301i \(-0.333509\pi\)
0.499522 + 0.866301i \(0.333509\pi\)
\(278\) 0 0
\(279\) 2.82843 0.169334
\(280\) 0 0
\(281\) 6.82843 0.407350 0.203675 0.979039i \(-0.434711\pi\)
0.203675 + 0.979039i \(0.434711\pi\)
\(282\) 0 0
\(283\) 2.14214 0.127337 0.0636684 0.997971i \(-0.479720\pi\)
0.0636684 + 0.997971i \(0.479720\pi\)
\(284\) 0 0
\(285\) −0.686292 −0.0406524
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.02944 0.237026
\(290\) 0 0
\(291\) −8.24264 −0.483192
\(292\) 0 0
\(293\) −10.2426 −0.598381 −0.299191 0.954193i \(-0.596717\pi\)
−0.299191 + 0.954193i \(0.596717\pi\)
\(294\) 0 0
\(295\) −4.97056 −0.289397
\(296\) 0 0
\(297\) 4.82843 0.280174
\(298\) 0 0
\(299\) 3.51472 0.203261
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −14.7279 −0.846097
\(304\) 0 0
\(305\) −1.79899 −0.103010
\(306\) 0 0
\(307\) −28.4853 −1.62574 −0.812870 0.582445i \(-0.802096\pi\)
−0.812870 + 0.582445i \(0.802096\pi\)
\(308\) 0 0
\(309\) 14.1421 0.804518
\(310\) 0 0
\(311\) −15.7990 −0.895879 −0.447939 0.894064i \(-0.647842\pi\)
−0.447939 + 0.894064i \(0.647842\pi\)
\(312\) 0 0
\(313\) 3.27208 0.184949 0.0924744 0.995715i \(-0.470522\pi\)
0.0924744 + 0.995715i \(0.470522\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 13.6569 0.764637
\(320\) 0 0
\(321\) −7.17157 −0.400278
\(322\) 0 0
\(323\) 5.37258 0.298939
\(324\) 0 0
\(325\) −19.7574 −1.09594
\(326\) 0 0
\(327\) −19.3137 −1.06805
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.3137 −0.841718 −0.420859 0.907126i \(-0.638271\pi\)
−0.420859 + 0.907126i \(0.638271\pi\)
\(332\) 0 0
\(333\) 9.65685 0.529192
\(334\) 0 0
\(335\) −6.62742 −0.362094
\(336\) 0 0
\(337\) 21.6569 1.17972 0.589862 0.807504i \(-0.299182\pi\)
0.589862 + 0.807504i \(0.299182\pi\)
\(338\) 0 0
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) −13.6569 −0.739560
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.485281 −0.0261267
\(346\) 0 0
\(347\) 14.4853 0.777611 0.388805 0.921320i \(-0.372888\pi\)
0.388805 + 0.921320i \(0.372888\pi\)
\(348\) 0 0
\(349\) 13.4142 0.718046 0.359023 0.933329i \(-0.383110\pi\)
0.359023 + 0.933329i \(0.383110\pi\)
\(350\) 0 0
\(351\) −4.24264 −0.226455
\(352\) 0 0
\(353\) 31.6985 1.68714 0.843570 0.537019i \(-0.180450\pi\)
0.843570 + 0.537019i \(0.180450\pi\)
\(354\) 0 0
\(355\) 3.79899 0.201629
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.1716 1.22295 0.611474 0.791264i \(-0.290577\pi\)
0.611474 + 0.791264i \(0.290577\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) 0 0
\(363\) −12.3137 −0.646302
\(364\) 0 0
\(365\) 9.51472 0.498023
\(366\) 0 0
\(367\) 27.3137 1.42576 0.712882 0.701284i \(-0.247390\pi\)
0.712882 + 0.701284i \(0.247390\pi\)
\(368\) 0 0
\(369\) 1.75736 0.0914845
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −17.3137 −0.896470 −0.448235 0.893916i \(-0.647947\pi\)
−0.448235 + 0.893916i \(0.647947\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) 0 0
\(381\) −20.0000 −1.02463
\(382\) 0 0
\(383\) 24.9706 1.27594 0.637968 0.770063i \(-0.279775\pi\)
0.637968 + 0.770063i \(0.279775\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.3137 0.575108
\(388\) 0 0
\(389\) 3.79899 0.192616 0.0963082 0.995352i \(-0.469297\pi\)
0.0963082 + 0.995352i \(0.469297\pi\)
\(390\) 0 0
\(391\) 3.79899 0.192123
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) 1.37258 0.0690621
\(396\) 0 0
\(397\) −7.75736 −0.389331 −0.194665 0.980870i \(-0.562362\pi\)
−0.194665 + 0.980870i \(0.562362\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.82843 0.340995 0.170498 0.985358i \(-0.445462\pi\)
0.170498 + 0.985358i \(0.445462\pi\)
\(402\) 0 0
\(403\) 12.0000 0.597763
\(404\) 0 0
\(405\) 0.585786 0.0291080
\(406\) 0 0
\(407\) −46.6274 −2.31124
\(408\) 0 0
\(409\) 0.242641 0.0119978 0.00599890 0.999982i \(-0.498090\pi\)
0.00599890 + 0.999982i \(0.498090\pi\)
\(410\) 0 0
\(411\) 14.8284 0.731432
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −2.34315 −0.115021
\(416\) 0 0
\(417\) −12.9706 −0.635171
\(418\) 0 0
\(419\) 24.4853 1.19618 0.598092 0.801427i \(-0.295926\pi\)
0.598092 + 0.801427i \(0.295926\pi\)
\(420\) 0 0
\(421\) −2.68629 −0.130922 −0.0654609 0.997855i \(-0.520852\pi\)
−0.0654609 + 0.997855i \(0.520852\pi\)
\(422\) 0 0
\(423\) 12.4853 0.607055
\(424\) 0 0
\(425\) −21.3553 −1.03589
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 20.4853 0.989039
\(430\) 0 0
\(431\) 20.1421 0.970213 0.485106 0.874455i \(-0.338781\pi\)
0.485106 + 0.874455i \(0.338781\pi\)
\(432\) 0 0
\(433\) 15.0711 0.724269 0.362135 0.932126i \(-0.382048\pi\)
0.362135 + 0.932126i \(0.382048\pi\)
\(434\) 0 0
\(435\) 1.65685 0.0794401
\(436\) 0 0
\(437\) 0.970563 0.0464283
\(438\) 0 0
\(439\) 35.3137 1.68543 0.842716 0.538359i \(-0.180956\pi\)
0.842716 + 0.538359i \(0.180956\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.8284 −0.799543 −0.399771 0.916615i \(-0.630910\pi\)
−0.399771 + 0.916615i \(0.630910\pi\)
\(444\) 0 0
\(445\) 8.34315 0.395503
\(446\) 0 0
\(447\) 21.3137 1.00810
\(448\) 0 0
\(449\) 28.6274 1.35101 0.675506 0.737355i \(-0.263925\pi\)
0.675506 + 0.737355i \(0.263925\pi\)
\(450\) 0 0
\(451\) −8.48528 −0.399556
\(452\) 0 0
\(453\) −1.65685 −0.0778458
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.6274 0.964910 0.482455 0.875921i \(-0.339745\pi\)
0.482455 + 0.875921i \(0.339745\pi\)
\(458\) 0 0
\(459\) −4.58579 −0.214046
\(460\) 0 0
\(461\) −20.3848 −0.949414 −0.474707 0.880144i \(-0.657446\pi\)
−0.474707 + 0.880144i \(0.657446\pi\)
\(462\) 0 0
\(463\) −9.65685 −0.448792 −0.224396 0.974498i \(-0.572041\pi\)
−0.224396 + 0.974498i \(0.572041\pi\)
\(464\) 0 0
\(465\) −1.65685 −0.0768348
\(466\) 0 0
\(467\) −13.1716 −0.609508 −0.304754 0.952431i \(-0.598574\pi\)
−0.304754 + 0.952431i \(0.598574\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 8.24264 0.379801
\(472\) 0 0
\(473\) −54.6274 −2.51177
\(474\) 0 0
\(475\) −5.45584 −0.250331
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 17.1716 0.784589 0.392295 0.919840i \(-0.371681\pi\)
0.392295 + 0.919840i \(0.371681\pi\)
\(480\) 0 0
\(481\) 40.9706 1.86810
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.82843 0.219248
\(486\) 0 0
\(487\) −28.9706 −1.31278 −0.656391 0.754421i \(-0.727918\pi\)
−0.656391 + 0.754421i \(0.727918\pi\)
\(488\) 0 0
\(489\) 5.65685 0.255812
\(490\) 0 0
\(491\) 24.8284 1.12049 0.560246 0.828327i \(-0.310707\pi\)
0.560246 + 0.828327i \(0.310707\pi\)
\(492\) 0 0
\(493\) −12.9706 −0.584165
\(494\) 0 0
\(495\) −2.82843 −0.127128
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 8.97056 0.401578 0.200789 0.979635i \(-0.435650\pi\)
0.200789 + 0.979635i \(0.435650\pi\)
\(500\) 0 0
\(501\) 9.17157 0.409756
\(502\) 0 0
\(503\) 3.31371 0.147751 0.0738755 0.997267i \(-0.476463\pi\)
0.0738755 + 0.997267i \(0.476463\pi\)
\(504\) 0 0
\(505\) 8.62742 0.383915
\(506\) 0 0
\(507\) −5.00000 −0.222058
\(508\) 0 0
\(509\) 0.384776 0.0170549 0.00852746 0.999964i \(-0.497286\pi\)
0.00852746 + 0.999964i \(0.497286\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.17157 −0.0517262
\(514\) 0 0
\(515\) −8.28427 −0.365049
\(516\) 0 0
\(517\) −60.2843 −2.65130
\(518\) 0 0
\(519\) 19.4142 0.852189
\(520\) 0 0
\(521\) 5.75736 0.252234 0.126117 0.992015i \(-0.459748\pi\)
0.126117 + 0.992015i \(0.459748\pi\)
\(522\) 0 0
\(523\) 28.9706 1.26679 0.633397 0.773827i \(-0.281660\pi\)
0.633397 + 0.773827i \(0.281660\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.9706 0.565007
\(528\) 0 0
\(529\) −22.3137 −0.970161
\(530\) 0 0
\(531\) −8.48528 −0.368230
\(532\) 0 0
\(533\) 7.45584 0.322948
\(534\) 0 0
\(535\) 4.20101 0.181626
\(536\) 0 0
\(537\) −10.4853 −0.452473
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) −7.07107 −0.303449
\(544\) 0 0
\(545\) 11.3137 0.484626
\(546\) 0 0
\(547\) −36.9706 −1.58075 −0.790374 0.612625i \(-0.790114\pi\)
−0.790374 + 0.612625i \(0.790114\pi\)
\(548\) 0 0
\(549\) −3.07107 −0.131070
\(550\) 0 0
\(551\) −3.31371 −0.141169
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −5.65685 −0.240120
\(556\) 0 0
\(557\) −12.6274 −0.535041 −0.267520 0.963552i \(-0.586204\pi\)
−0.267520 + 0.963552i \(0.586204\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) 22.1421 0.934842
\(562\) 0 0
\(563\) 30.1421 1.27034 0.635170 0.772373i \(-0.280930\pi\)
0.635170 + 0.772373i \(0.280930\pi\)
\(564\) 0 0
\(565\) −5.85786 −0.246442
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −34.1421 −1.43131 −0.715656 0.698453i \(-0.753872\pi\)
−0.715656 + 0.698453i \(0.753872\pi\)
\(570\) 0 0
\(571\) −30.3431 −1.26982 −0.634911 0.772586i \(-0.718963\pi\)
−0.634911 + 0.772586i \(0.718963\pi\)
\(572\) 0 0
\(573\) 16.1421 0.674347
\(574\) 0 0
\(575\) −3.85786 −0.160884
\(576\) 0 0
\(577\) −29.2132 −1.21616 −0.608081 0.793875i \(-0.708060\pi\)
−0.608081 + 0.793875i \(0.708060\pi\)
\(578\) 0 0
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 9.65685 0.399946
\(584\) 0 0
\(585\) 2.48528 0.102754
\(586\) 0 0
\(587\) 3.79899 0.156801 0.0784005 0.996922i \(-0.475019\pi\)
0.0784005 + 0.996922i \(0.475019\pi\)
\(588\) 0 0
\(589\) 3.31371 0.136539
\(590\) 0 0
\(591\) 25.3137 1.04127
\(592\) 0 0
\(593\) 44.5858 1.83092 0.915459 0.402410i \(-0.131827\pi\)
0.915459 + 0.402410i \(0.131827\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.65685 −0.231520
\(598\) 0 0
\(599\) −27.4558 −1.12182 −0.560908 0.827878i \(-0.689548\pi\)
−0.560908 + 0.827878i \(0.689548\pi\)
\(600\) 0 0
\(601\) −3.75736 −0.153266 −0.0766329 0.997059i \(-0.524417\pi\)
−0.0766329 + 0.997059i \(0.524417\pi\)
\(602\) 0 0
\(603\) −11.3137 −0.460730
\(604\) 0 0
\(605\) 7.21320 0.293258
\(606\) 0 0
\(607\) −8.97056 −0.364104 −0.182052 0.983289i \(-0.558274\pi\)
−0.182052 + 0.983289i \(0.558274\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 52.9706 2.14296
\(612\) 0 0
\(613\) −21.6569 −0.874712 −0.437356 0.899288i \(-0.644085\pi\)
−0.437356 + 0.899288i \(0.644085\pi\)
\(614\) 0 0
\(615\) −1.02944 −0.0415109
\(616\) 0 0
\(617\) 12.4853 0.502639 0.251319 0.967904i \(-0.419136\pi\)
0.251319 + 0.967904i \(0.419136\pi\)
\(618\) 0 0
\(619\) 22.3431 0.898047 0.449023 0.893520i \(-0.351772\pi\)
0.449023 + 0.893520i \(0.351772\pi\)
\(620\) 0 0
\(621\) −0.828427 −0.0332436
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 5.65685 0.225913
\(628\) 0 0
\(629\) 44.2843 1.76573
\(630\) 0 0
\(631\) −36.9706 −1.47177 −0.735887 0.677104i \(-0.763235\pi\)
−0.735887 + 0.677104i \(0.763235\pi\)
\(632\) 0 0
\(633\) 9.65685 0.383825
\(634\) 0 0
\(635\) 11.7157 0.464925
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.48528 0.256554
\(640\) 0 0
\(641\) −27.1127 −1.07089 −0.535444 0.844571i \(-0.679856\pi\)
−0.535444 + 0.844571i \(0.679856\pi\)
\(642\) 0 0
\(643\) −7.79899 −0.307562 −0.153781 0.988105i \(-0.549145\pi\)
−0.153781 + 0.988105i \(0.549145\pi\)
\(644\) 0 0
\(645\) −6.62742 −0.260954
\(646\) 0 0
\(647\) −39.7990 −1.56466 −0.782330 0.622864i \(-0.785969\pi\)
−0.782330 + 0.622864i \(0.785969\pi\)
\(648\) 0 0
\(649\) 40.9706 1.60824
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.8579 0.698832 0.349416 0.936968i \(-0.386380\pi\)
0.349416 + 0.936968i \(0.386380\pi\)
\(654\) 0 0
\(655\) −2.34315 −0.0915543
\(656\) 0 0
\(657\) 16.2426 0.633686
\(658\) 0 0
\(659\) 38.4853 1.49917 0.749587 0.661906i \(-0.230252\pi\)
0.749587 + 0.661906i \(0.230252\pi\)
\(660\) 0 0
\(661\) 19.0711 0.741779 0.370889 0.928677i \(-0.379053\pi\)
0.370889 + 0.928677i \(0.379053\pi\)
\(662\) 0 0
\(663\) −19.4558 −0.755602
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.34315 −0.0907270
\(668\) 0 0
\(669\) −2.34315 −0.0905912
\(670\) 0 0
\(671\) 14.8284 0.572445
\(672\) 0 0
\(673\) −0.686292 −0.0264546 −0.0132273 0.999913i \(-0.504211\pi\)
−0.0132273 + 0.999913i \(0.504211\pi\)
\(674\) 0 0
\(675\) 4.65685 0.179242
\(676\) 0 0
\(677\) −31.6985 −1.21827 −0.609136 0.793066i \(-0.708484\pi\)
−0.609136 + 0.793066i \(0.708484\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.8284 0.414946
\(682\) 0 0
\(683\) 37.7990 1.44634 0.723169 0.690671i \(-0.242685\pi\)
0.723169 + 0.690671i \(0.242685\pi\)
\(684\) 0 0
\(685\) −8.68629 −0.331886
\(686\) 0 0
\(687\) 22.5858 0.861701
\(688\) 0 0
\(689\) −8.48528 −0.323263
\(690\) 0 0
\(691\) 31.3137 1.19123 0.595615 0.803270i \(-0.296908\pi\)
0.595615 + 0.803270i \(0.296908\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.59798 0.288208
\(696\) 0 0
\(697\) 8.05887 0.305252
\(698\) 0 0
\(699\) 9.17157 0.346901
\(700\) 0 0
\(701\) 22.1421 0.836297 0.418148 0.908379i \(-0.362679\pi\)
0.418148 + 0.908379i \(0.362679\pi\)
\(702\) 0 0
\(703\) 11.3137 0.426705
\(704\) 0 0
\(705\) −7.31371 −0.275450
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −38.6274 −1.45068 −0.725342 0.688389i \(-0.758318\pi\)
−0.725342 + 0.688389i \(0.758318\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 0 0
\(713\) 2.34315 0.0877515
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 7.17157 0.267827
\(718\) 0 0
\(719\) 38.6274 1.44056 0.720280 0.693684i \(-0.244013\pi\)
0.720280 + 0.693684i \(0.244013\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5.89949 0.219405
\(724\) 0 0
\(725\) 13.1716 0.489180
\(726\) 0 0
\(727\) −38.1421 −1.41461 −0.707307 0.706907i \(-0.750090\pi\)
−0.707307 + 0.706907i \(0.750090\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 51.8823 1.91893
\(732\) 0 0
\(733\) 40.0416 1.47897 0.739486 0.673172i \(-0.235069\pi\)
0.739486 + 0.673172i \(0.235069\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 54.6274 2.01223
\(738\) 0 0
\(739\) −0.970563 −0.0357027 −0.0178514 0.999841i \(-0.505683\pi\)
−0.0178514 + 0.999841i \(0.505683\pi\)
\(740\) 0 0
\(741\) −4.97056 −0.182598
\(742\) 0 0
\(743\) 0.828427 0.0303920 0.0151960 0.999885i \(-0.495163\pi\)
0.0151960 + 0.999885i \(0.495163\pi\)
\(744\) 0 0
\(745\) −12.4853 −0.457425
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −36.2843 −1.32403 −0.662016 0.749490i \(-0.730299\pi\)
−0.662016 + 0.749490i \(0.730299\pi\)
\(752\) 0 0
\(753\) −22.1421 −0.806904
\(754\) 0 0
\(755\) 0.970563 0.0353224
\(756\) 0 0
\(757\) −25.9411 −0.942846 −0.471423 0.881907i \(-0.656260\pi\)
−0.471423 + 0.881907i \(0.656260\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −12.1005 −0.438643 −0.219321 0.975653i \(-0.570384\pi\)
−0.219321 + 0.975653i \(0.570384\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.68629 0.0971231
\(766\) 0 0
\(767\) −36.0000 −1.29988
\(768\) 0 0
\(769\) −18.8701 −0.680472 −0.340236 0.940340i \(-0.610507\pi\)
−0.340236 + 0.940340i \(0.610507\pi\)
\(770\) 0 0
\(771\) 0.585786 0.0210966
\(772\) 0 0
\(773\) −37.3553 −1.34358 −0.671789 0.740742i \(-0.734474\pi\)
−0.671789 + 0.740742i \(0.734474\pi\)
\(774\) 0 0
\(775\) −13.1716 −0.473137
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.05887 0.0737668
\(780\) 0 0
\(781\) −31.3137 −1.12049
\(782\) 0 0
\(783\) 2.82843 0.101080
\(784\) 0 0
\(785\) −4.82843 −0.172334
\(786\) 0 0
\(787\) −7.31371 −0.260706 −0.130353 0.991468i \(-0.541611\pi\)
−0.130353 + 0.991468i \(0.541611\pi\)
\(788\) 0 0
\(789\) −8.14214 −0.289868
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −13.0294 −0.462689
\(794\) 0 0
\(795\) 1.17157 0.0415514
\(796\) 0 0
\(797\) −36.5858 −1.29594 −0.647968 0.761668i \(-0.724381\pi\)
−0.647968 + 0.761668i \(0.724381\pi\)
\(798\) 0 0
\(799\) 57.2548 2.02553
\(800\) 0 0
\(801\) 14.2426 0.503239
\(802\) 0 0
\(803\) −78.4264 −2.76761
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.8995 0.700495
\(808\) 0 0
\(809\) 40.6274 1.42838 0.714192 0.699950i \(-0.246794\pi\)
0.714192 + 0.699950i \(0.246794\pi\)
\(810\) 0 0
\(811\) 14.3431 0.503656 0.251828 0.967772i \(-0.418968\pi\)
0.251828 + 0.967772i \(0.418968\pi\)
\(812\) 0 0
\(813\) 22.1421 0.776559
\(814\) 0 0
\(815\) −3.31371 −0.116074
\(816\) 0 0
\(817\) 13.2548 0.463728
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −33.3137 −1.16266 −0.581328 0.813669i \(-0.697467\pi\)
−0.581328 + 0.813669i \(0.697467\pi\)
\(822\) 0 0
\(823\) 8.97056 0.312694 0.156347 0.987702i \(-0.450028\pi\)
0.156347 + 0.987702i \(0.450028\pi\)
\(824\) 0 0
\(825\) −22.4853 −0.782837
\(826\) 0 0
\(827\) 11.8579 0.412338 0.206169 0.978516i \(-0.433900\pi\)
0.206169 + 0.978516i \(0.433900\pi\)
\(828\) 0 0
\(829\) −2.38478 −0.0828267 −0.0414134 0.999142i \(-0.513186\pi\)
−0.0414134 + 0.999142i \(0.513186\pi\)
\(830\) 0 0
\(831\) −16.6274 −0.576799
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −5.37258 −0.185926
\(836\) 0 0
\(837\) −2.82843 −0.0977647
\(838\) 0 0
\(839\) −15.7990 −0.545442 −0.272721 0.962093i \(-0.587924\pi\)
−0.272721 + 0.962093i \(0.587924\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) −6.82843 −0.235184
\(844\) 0 0
\(845\) 2.92893 0.100758
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −2.14214 −0.0735179
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 19.0711 0.652981 0.326490 0.945200i \(-0.394134\pi\)
0.326490 + 0.945200i \(0.394134\pi\)
\(854\) 0 0
\(855\) 0.686292 0.0234707
\(856\) 0 0
\(857\) 31.2132 1.06622 0.533111 0.846045i \(-0.321023\pi\)
0.533111 + 0.846045i \(0.321023\pi\)
\(858\) 0 0
\(859\) 53.4558 1.82389 0.911945 0.410313i \(-0.134580\pi\)
0.911945 + 0.410313i \(0.134580\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −42.0833 −1.43253 −0.716265 0.697828i \(-0.754150\pi\)
−0.716265 + 0.697828i \(0.754150\pi\)
\(864\) 0 0
\(865\) −11.3726 −0.386679
\(866\) 0 0
\(867\) −4.02944 −0.136847
\(868\) 0 0
\(869\) −11.3137 −0.383791
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 8.24264 0.278971
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28.2843 0.955092 0.477546 0.878607i \(-0.341526\pi\)
0.477546 + 0.878607i \(0.341526\pi\)
\(878\) 0 0
\(879\) 10.2426 0.345476
\(880\) 0 0
\(881\) −18.0416 −0.607838 −0.303919 0.952698i \(-0.598295\pi\)
−0.303919 + 0.952698i \(0.598295\pi\)
\(882\) 0 0
\(883\) −21.6569 −0.728811 −0.364406 0.931240i \(-0.618728\pi\)
−0.364406 + 0.931240i \(0.618728\pi\)
\(884\) 0 0
\(885\) 4.97056 0.167084
\(886\) 0 0
\(887\) −0.201010 −0.00674926 −0.00337463 0.999994i \(-0.501074\pi\)
−0.00337463 + 0.999994i \(0.501074\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.82843 −0.161758
\(892\) 0 0
\(893\) 14.6274 0.489488
\(894\) 0 0
\(895\) 6.14214 0.205309
\(896\) 0 0
\(897\) −3.51472 −0.117353
\(898\) 0 0
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −9.17157 −0.305549
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.14214 0.137689
\(906\) 0 0
\(907\) 42.3431 1.40598 0.702991 0.711199i \(-0.251848\pi\)
0.702991 + 0.711199i \(0.251848\pi\)
\(908\) 0 0
\(909\) 14.7279 0.488494
\(910\) 0 0
\(911\) −10.4853 −0.347393 −0.173696 0.984799i \(-0.555571\pi\)
−0.173696 + 0.984799i \(0.555571\pi\)
\(912\) 0 0
\(913\) 19.3137 0.639190
\(914\) 0 0
\(915\) 1.79899 0.0594728
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.68629 0.286534 0.143267 0.989684i \(-0.454239\pi\)
0.143267 + 0.989684i \(0.454239\pi\)
\(920\) 0 0
\(921\) 28.4853 0.938622
\(922\) 0 0
\(923\) 27.5147 0.905658
\(924\) 0 0
\(925\) −44.9706 −1.47862
\(926\) 0 0
\(927\) −14.1421 −0.464489
\(928\) 0 0
\(929\) 3.89949 0.127938 0.0639691 0.997952i \(-0.479624\pi\)
0.0639691 + 0.997952i \(0.479624\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 15.7990 0.517236
\(934\) 0 0
\(935\) −12.9706 −0.424183
\(936\) 0 0
\(937\) −19.3553 −0.632311 −0.316156 0.948707i \(-0.602392\pi\)
−0.316156 + 0.948707i \(0.602392\pi\)
\(938\) 0 0
\(939\) −3.27208 −0.106780
\(940\) 0 0
\(941\) 10.7279 0.349720 0.174860 0.984593i \(-0.444053\pi\)
0.174860 + 0.984593i \(0.444053\pi\)
\(942\) 0 0
\(943\) 1.45584 0.0474088
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.7696 0.479946 0.239973 0.970780i \(-0.422861\pi\)
0.239973 + 0.970780i \(0.422861\pi\)
\(948\) 0 0
\(949\) 68.9117 2.23697
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) −9.45584 −0.305984
\(956\) 0 0
\(957\) −13.6569 −0.441463
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 7.17157 0.231101
\(964\) 0 0
\(965\) 1.17157 0.0377143
\(966\) 0 0
\(967\) −8.68629 −0.279332 −0.139666 0.990199i \(-0.544603\pi\)
−0.139666 + 0.990199i \(0.544603\pi\)
\(968\) 0 0
\(969\) −5.37258 −0.172592
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 19.7574 0.632742
\(976\) 0 0
\(977\) 31.7990 1.01734 0.508670 0.860962i \(-0.330137\pi\)
0.508670 + 0.860962i \(0.330137\pi\)
\(978\) 0 0
\(979\) −68.7696 −2.19788
\(980\) 0 0
\(981\) 19.3137 0.616639
\(982\) 0 0
\(983\) 46.6274 1.48718 0.743592 0.668634i \(-0.233121\pi\)
0.743592 + 0.668634i \(0.233121\pi\)
\(984\) 0 0
\(985\) −14.8284 −0.472473
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.37258 0.298031
\(990\) 0 0
\(991\) 26.6274 0.845848 0.422924 0.906165i \(-0.361004\pi\)
0.422924 + 0.906165i \(0.361004\pi\)
\(992\) 0 0
\(993\) 15.3137 0.485966
\(994\) 0 0
\(995\) 3.31371 0.105052
\(996\) 0 0
\(997\) −17.6152 −0.557880 −0.278940 0.960309i \(-0.589983\pi\)
−0.278940 + 0.960309i \(0.589983\pi\)
\(998\) 0 0
\(999\) −9.65685 −0.305529
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 1176.2.a.l.1.1 2
3.2 odd 2 3528.2.a.bc.1.2 2
4.3 odd 2 2352.2.a.bg.1.1 2
7.2 even 3 1176.2.q.n.361.2 4
7.3 odd 6 1176.2.q.m.961.1 4
7.4 even 3 1176.2.q.n.961.2 4
7.5 odd 6 1176.2.q.m.361.1 4
7.6 odd 2 1176.2.a.m.1.2 yes 2
8.3 odd 2 9408.2.a.dh.1.2 2
8.5 even 2 9408.2.a.dv.1.2 2
12.11 even 2 7056.2.a.ce.1.2 2
21.2 odd 6 3528.2.s.bl.361.1 4
21.5 even 6 3528.2.s.bc.361.2 4
21.11 odd 6 3528.2.s.bl.3313.1 4
21.17 even 6 3528.2.s.bc.3313.2 4
21.20 even 2 3528.2.a.bm.1.1 2
28.3 even 6 2352.2.q.bg.961.1 4
28.11 odd 6 2352.2.q.ba.961.2 4
28.19 even 6 2352.2.q.bg.1537.1 4
28.23 odd 6 2352.2.q.ba.1537.2 4
28.27 even 2 2352.2.a.z.1.2 2
56.13 odd 2 9408.2.a.dr.1.1 2
56.27 even 2 9408.2.a.ed.1.1 2
84.83 odd 2 7056.2.a.cw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.l.1.1 2 1.1 even 1 trivial
1176.2.a.m.1.2 yes 2 7.6 odd 2
1176.2.q.m.361.1 4 7.5 odd 6
1176.2.q.m.961.1 4 7.3 odd 6
1176.2.q.n.361.2 4 7.2 even 3
1176.2.q.n.961.2 4 7.4 even 3
2352.2.a.z.1.2 2 28.27 even 2
2352.2.a.bg.1.1 2 4.3 odd 2
2352.2.q.ba.961.2 4 28.11 odd 6
2352.2.q.ba.1537.2 4 28.23 odd 6
2352.2.q.bg.961.1 4 28.3 even 6
2352.2.q.bg.1537.1 4 28.19 even 6
3528.2.a.bc.1.2 2 3.2 odd 2
3528.2.a.bm.1.1 2 21.20 even 2
3528.2.s.bc.361.2 4 21.5 even 6
3528.2.s.bc.3313.2 4 21.17 even 6
3528.2.s.bl.361.1 4 21.2 odd 6
3528.2.s.bl.3313.1 4 21.11 odd 6
7056.2.a.ce.1.2 2 12.11 even 2
7056.2.a.cw.1.1 2 84.83 odd 2
9408.2.a.dh.1.2 2 8.3 odd 2
9408.2.a.dr.1.1 2 56.13 odd 2
9408.2.a.dv.1.2 2 8.5 even 2
9408.2.a.ed.1.1 2 56.27 even 2