Properties

Label 8004.2.a.i.1.8
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.144883\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.144883 q^{5} -2.05535 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.144883 q^{5} -2.05535 q^{7} +1.00000 q^{9} -5.70628 q^{11} -4.46851 q^{13} -0.144883 q^{15} +4.26133 q^{17} +3.58278 q^{19} +2.05535 q^{21} +1.00000 q^{23} -4.97901 q^{25} -1.00000 q^{27} +1.00000 q^{29} -6.14680 q^{31} +5.70628 q^{33} -0.297786 q^{35} +9.53672 q^{37} +4.46851 q^{39} -7.46765 q^{41} -2.12219 q^{43} +0.144883 q^{45} -12.1076 q^{47} -2.77555 q^{49} -4.26133 q^{51} +6.19264 q^{53} -0.826745 q^{55} -3.58278 q^{57} -13.2421 q^{59} +5.14159 q^{61} -2.05535 q^{63} -0.647414 q^{65} -0.811030 q^{67} -1.00000 q^{69} +11.4670 q^{71} -13.7114 q^{73} +4.97901 q^{75} +11.7284 q^{77} +1.51002 q^{79} +1.00000 q^{81} -13.5147 q^{83} +0.617396 q^{85} -1.00000 q^{87} +4.10880 q^{89} +9.18435 q^{91} +6.14680 q^{93} +0.519085 q^{95} -4.29931 q^{97} -5.70628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) 0.144883 0.0647938 0.0323969 0.999475i \(-0.489686\pi\)
0.0323969 + 0.999475i \(0.489686\pi\)
\(6\) 0 0
\(7\) −2.05535 −0.776848 −0.388424 0.921481i \(-0.626980\pi\)
−0.388424 + 0.921481i \(0.626980\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.70628 −1.72051 −0.860254 0.509866i \(-0.829695\pi\)
−0.860254 + 0.509866i \(0.829695\pi\)
\(12\) 0 0
\(13\) −4.46851 −1.23934 −0.619671 0.784861i \(-0.712734\pi\)
−0.619671 + 0.784861i \(0.712734\pi\)
\(14\) 0 0
\(15\) −0.144883 −0.0374087
\(16\) 0 0
\(17\) 4.26133 1.03352 0.516762 0.856129i \(-0.327137\pi\)
0.516762 + 0.856129i \(0.327137\pi\)
\(18\) 0 0
\(19\) 3.58278 0.821946 0.410973 0.911648i \(-0.365189\pi\)
0.410973 + 0.911648i \(0.365189\pi\)
\(20\) 0 0
\(21\) 2.05535 0.448513
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.97901 −0.995802
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −6.14680 −1.10400 −0.551999 0.833845i \(-0.686135\pi\)
−0.551999 + 0.833845i \(0.686135\pi\)
\(32\) 0 0
\(33\) 5.70628 0.993336
\(34\) 0 0
\(35\) −0.297786 −0.0503350
\(36\) 0 0
\(37\) 9.53672 1.56783 0.783913 0.620870i \(-0.213221\pi\)
0.783913 + 0.620870i \(0.213221\pi\)
\(38\) 0 0
\(39\) 4.46851 0.715535
\(40\) 0 0
\(41\) −7.46765 −1.16625 −0.583125 0.812382i \(-0.698170\pi\)
−0.583125 + 0.812382i \(0.698170\pi\)
\(42\) 0 0
\(43\) −2.12219 −0.323630 −0.161815 0.986821i \(-0.551735\pi\)
−0.161815 + 0.986821i \(0.551735\pi\)
\(44\) 0 0
\(45\) 0.144883 0.0215979
\(46\) 0 0
\(47\) −12.1076 −1.76608 −0.883040 0.469298i \(-0.844507\pi\)
−0.883040 + 0.469298i \(0.844507\pi\)
\(48\) 0 0
\(49\) −2.77555 −0.396507
\(50\) 0 0
\(51\) −4.26133 −0.596705
\(52\) 0 0
\(53\) 6.19264 0.850625 0.425313 0.905047i \(-0.360164\pi\)
0.425313 + 0.905047i \(0.360164\pi\)
\(54\) 0 0
\(55\) −0.826745 −0.111478
\(56\) 0 0
\(57\) −3.58278 −0.474551
\(58\) 0 0
\(59\) −13.2421 −1.72397 −0.861987 0.506931i \(-0.830780\pi\)
−0.861987 + 0.506931i \(0.830780\pi\)
\(60\) 0 0
\(61\) 5.14159 0.658313 0.329157 0.944275i \(-0.393236\pi\)
0.329157 + 0.944275i \(0.393236\pi\)
\(62\) 0 0
\(63\) −2.05535 −0.258949
\(64\) 0 0
\(65\) −0.647414 −0.0803018
\(66\) 0 0
\(67\) −0.811030 −0.0990831 −0.0495415 0.998772i \(-0.515776\pi\)
−0.0495415 + 0.998772i \(0.515776\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 11.4670 1.36088 0.680442 0.732802i \(-0.261788\pi\)
0.680442 + 0.732802i \(0.261788\pi\)
\(72\) 0 0
\(73\) −13.7114 −1.60480 −0.802398 0.596790i \(-0.796443\pi\)
−0.802398 + 0.596790i \(0.796443\pi\)
\(74\) 0 0
\(75\) 4.97901 0.574926
\(76\) 0 0
\(77\) 11.7284 1.33657
\(78\) 0 0
\(79\) 1.51002 0.169891 0.0849455 0.996386i \(-0.472928\pi\)
0.0849455 + 0.996386i \(0.472928\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.5147 −1.48343 −0.741715 0.670715i \(-0.765988\pi\)
−0.741715 + 0.670715i \(0.765988\pi\)
\(84\) 0 0
\(85\) 0.617396 0.0669660
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 4.10880 0.435532 0.217766 0.976001i \(-0.430123\pi\)
0.217766 + 0.976001i \(0.430123\pi\)
\(90\) 0 0
\(91\) 9.18435 0.962781
\(92\) 0 0
\(93\) 6.14680 0.637393
\(94\) 0 0
\(95\) 0.519085 0.0532570
\(96\) 0 0
\(97\) −4.29931 −0.436529 −0.218264 0.975890i \(-0.570040\pi\)
−0.218264 + 0.975890i \(0.570040\pi\)
\(98\) 0 0
\(99\) −5.70628 −0.573503
\(100\) 0 0
\(101\) −8.47026 −0.842822 −0.421411 0.906870i \(-0.638465\pi\)
−0.421411 + 0.906870i \(0.638465\pi\)
\(102\) 0 0
\(103\) 6.06353 0.597458 0.298729 0.954338i \(-0.403437\pi\)
0.298729 + 0.954338i \(0.403437\pi\)
\(104\) 0 0
\(105\) 0.297786 0.0290609
\(106\) 0 0
\(107\) −2.86602 −0.277069 −0.138534 0.990358i \(-0.544239\pi\)
−0.138534 + 0.990358i \(0.544239\pi\)
\(108\) 0 0
\(109\) −9.68092 −0.927264 −0.463632 0.886028i \(-0.653454\pi\)
−0.463632 + 0.886028i \(0.653454\pi\)
\(110\) 0 0
\(111\) −9.53672 −0.905185
\(112\) 0 0
\(113\) 8.96674 0.843520 0.421760 0.906708i \(-0.361413\pi\)
0.421760 + 0.906708i \(0.361413\pi\)
\(114\) 0 0
\(115\) 0.144883 0.0135104
\(116\) 0 0
\(117\) −4.46851 −0.413114
\(118\) 0 0
\(119\) −8.75851 −0.802891
\(120\) 0 0
\(121\) 21.5616 1.96015
\(122\) 0 0
\(123\) 7.46765 0.673335
\(124\) 0 0
\(125\) −1.44579 −0.129316
\(126\) 0 0
\(127\) 18.8240 1.67036 0.835181 0.549976i \(-0.185363\pi\)
0.835181 + 0.549976i \(0.185363\pi\)
\(128\) 0 0
\(129\) 2.12219 0.186848
\(130\) 0 0
\(131\) 1.84666 0.161344 0.0806719 0.996741i \(-0.474293\pi\)
0.0806719 + 0.996741i \(0.474293\pi\)
\(132\) 0 0
\(133\) −7.36385 −0.638527
\(134\) 0 0
\(135\) −0.144883 −0.0124696
\(136\) 0 0
\(137\) 4.78041 0.408418 0.204209 0.978927i \(-0.434538\pi\)
0.204209 + 0.978927i \(0.434538\pi\)
\(138\) 0 0
\(139\) −11.1294 −0.943985 −0.471993 0.881602i \(-0.656465\pi\)
−0.471993 + 0.881602i \(0.656465\pi\)
\(140\) 0 0
\(141\) 12.1076 1.01965
\(142\) 0 0
\(143\) 25.4986 2.13230
\(144\) 0 0
\(145\) 0.144883 0.0120319
\(146\) 0 0
\(147\) 2.77555 0.228923
\(148\) 0 0
\(149\) 11.6153 0.951564 0.475782 0.879563i \(-0.342165\pi\)
0.475782 + 0.879563i \(0.342165\pi\)
\(150\) 0 0
\(151\) 13.1230 1.06793 0.533966 0.845506i \(-0.320701\pi\)
0.533966 + 0.845506i \(0.320701\pi\)
\(152\) 0 0
\(153\) 4.26133 0.344508
\(154\) 0 0
\(155\) −0.890569 −0.0715322
\(156\) 0 0
\(157\) 10.2659 0.819309 0.409655 0.912241i \(-0.365649\pi\)
0.409655 + 0.912241i \(0.365649\pi\)
\(158\) 0 0
\(159\) −6.19264 −0.491109
\(160\) 0 0
\(161\) −2.05535 −0.161984
\(162\) 0 0
\(163\) −2.89039 −0.226393 −0.113197 0.993573i \(-0.536109\pi\)
−0.113197 + 0.993573i \(0.536109\pi\)
\(164\) 0 0
\(165\) 0.826745 0.0643620
\(166\) 0 0
\(167\) −16.8680 −1.30528 −0.652641 0.757667i \(-0.726339\pi\)
−0.652641 + 0.757667i \(0.726339\pi\)
\(168\) 0 0
\(169\) 6.96762 0.535970
\(170\) 0 0
\(171\) 3.58278 0.273982
\(172\) 0 0
\(173\) 23.1674 1.76138 0.880692 0.473689i \(-0.157078\pi\)
0.880692 + 0.473689i \(0.157078\pi\)
\(174\) 0 0
\(175\) 10.2336 0.773587
\(176\) 0 0
\(177\) 13.2421 0.995337
\(178\) 0 0
\(179\) 20.8434 1.55791 0.778954 0.627081i \(-0.215750\pi\)
0.778954 + 0.627081i \(0.215750\pi\)
\(180\) 0 0
\(181\) −12.6412 −0.939612 −0.469806 0.882770i \(-0.655676\pi\)
−0.469806 + 0.882770i \(0.655676\pi\)
\(182\) 0 0
\(183\) −5.14159 −0.380077
\(184\) 0 0
\(185\) 1.38171 0.101585
\(186\) 0 0
\(187\) −24.3163 −1.77819
\(188\) 0 0
\(189\) 2.05535 0.149504
\(190\) 0 0
\(191\) 15.1359 1.09520 0.547599 0.836741i \(-0.315542\pi\)
0.547599 + 0.836741i \(0.315542\pi\)
\(192\) 0 0
\(193\) 0.714971 0.0514647 0.0257324 0.999669i \(-0.491808\pi\)
0.0257324 + 0.999669i \(0.491808\pi\)
\(194\) 0 0
\(195\) 0.647414 0.0463623
\(196\) 0 0
\(197\) −6.61412 −0.471237 −0.235618 0.971846i \(-0.575711\pi\)
−0.235618 + 0.971846i \(0.575711\pi\)
\(198\) 0 0
\(199\) 19.7823 1.40233 0.701164 0.713000i \(-0.252664\pi\)
0.701164 + 0.713000i \(0.252664\pi\)
\(200\) 0 0
\(201\) 0.811030 0.0572056
\(202\) 0 0
\(203\) −2.05535 −0.144257
\(204\) 0 0
\(205\) −1.08194 −0.0755659
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −20.4443 −1.41416
\(210\) 0 0
\(211\) 0.00743211 0.000511647 0 0.000255824 1.00000i \(-0.499919\pi\)
0.000255824 1.00000i \(0.499919\pi\)
\(212\) 0 0
\(213\) −11.4670 −0.785707
\(214\) 0 0
\(215\) −0.307470 −0.0209692
\(216\) 0 0
\(217\) 12.6338 0.857638
\(218\) 0 0
\(219\) 13.7114 0.926529
\(220\) 0 0
\(221\) −19.0418 −1.28089
\(222\) 0 0
\(223\) −15.8140 −1.05898 −0.529492 0.848315i \(-0.677618\pi\)
−0.529492 + 0.848315i \(0.677618\pi\)
\(224\) 0 0
\(225\) −4.97901 −0.331934
\(226\) 0 0
\(227\) 16.8842 1.12064 0.560321 0.828276i \(-0.310678\pi\)
0.560321 + 0.828276i \(0.310678\pi\)
\(228\) 0 0
\(229\) 29.8250 1.97089 0.985445 0.169996i \(-0.0543756\pi\)
0.985445 + 0.169996i \(0.0543756\pi\)
\(230\) 0 0
\(231\) −11.7284 −0.771671
\(232\) 0 0
\(233\) 29.2967 1.91929 0.959644 0.281216i \(-0.0907378\pi\)
0.959644 + 0.281216i \(0.0907378\pi\)
\(234\) 0 0
\(235\) −1.75420 −0.114431
\(236\) 0 0
\(237\) −1.51002 −0.0980867
\(238\) 0 0
\(239\) −17.7957 −1.15111 −0.575553 0.817764i \(-0.695213\pi\)
−0.575553 + 0.817764i \(0.695213\pi\)
\(240\) 0 0
\(241\) −3.01052 −0.193925 −0.0969623 0.995288i \(-0.530913\pi\)
−0.0969623 + 0.995288i \(0.530913\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.402131 −0.0256912
\(246\) 0 0
\(247\) −16.0097 −1.01867
\(248\) 0 0
\(249\) 13.5147 0.856459
\(250\) 0 0
\(251\) 1.62166 0.102358 0.0511791 0.998689i \(-0.483702\pi\)
0.0511791 + 0.998689i \(0.483702\pi\)
\(252\) 0 0
\(253\) −5.70628 −0.358751
\(254\) 0 0
\(255\) −0.617396 −0.0386628
\(256\) 0 0
\(257\) −5.82055 −0.363076 −0.181538 0.983384i \(-0.558108\pi\)
−0.181538 + 0.983384i \(0.558108\pi\)
\(258\) 0 0
\(259\) −19.6013 −1.21796
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 7.13534 0.439984 0.219992 0.975502i \(-0.429397\pi\)
0.219992 + 0.975502i \(0.429397\pi\)
\(264\) 0 0
\(265\) 0.897212 0.0551153
\(266\) 0 0
\(267\) −4.10880 −0.251454
\(268\) 0 0
\(269\) 24.1410 1.47191 0.735953 0.677033i \(-0.236735\pi\)
0.735953 + 0.677033i \(0.236735\pi\)
\(270\) 0 0
\(271\) 2.19676 0.133444 0.0667218 0.997772i \(-0.478746\pi\)
0.0667218 + 0.997772i \(0.478746\pi\)
\(272\) 0 0
\(273\) −9.18435 −0.555862
\(274\) 0 0
\(275\) 28.4116 1.71329
\(276\) 0 0
\(277\) 7.32361 0.440033 0.220017 0.975496i \(-0.429389\pi\)
0.220017 + 0.975496i \(0.429389\pi\)
\(278\) 0 0
\(279\) −6.14680 −0.367999
\(280\) 0 0
\(281\) 18.9450 1.13016 0.565082 0.825035i \(-0.308844\pi\)
0.565082 + 0.825035i \(0.308844\pi\)
\(282\) 0 0
\(283\) 8.58396 0.510263 0.255132 0.966906i \(-0.417881\pi\)
0.255132 + 0.966906i \(0.417881\pi\)
\(284\) 0 0
\(285\) −0.519085 −0.0307480
\(286\) 0 0
\(287\) 15.3486 0.906000
\(288\) 0 0
\(289\) 1.15893 0.0681723
\(290\) 0 0
\(291\) 4.29931 0.252030
\(292\) 0 0
\(293\) 20.1079 1.17471 0.587357 0.809328i \(-0.300169\pi\)
0.587357 + 0.809328i \(0.300169\pi\)
\(294\) 0 0
\(295\) −1.91856 −0.111703
\(296\) 0 0
\(297\) 5.70628 0.331112
\(298\) 0 0
\(299\) −4.46851 −0.258421
\(300\) 0 0
\(301\) 4.36183 0.251412
\(302\) 0 0
\(303\) 8.47026 0.486603
\(304\) 0 0
\(305\) 0.744931 0.0426546
\(306\) 0 0
\(307\) −12.3268 −0.703526 −0.351763 0.936089i \(-0.614418\pi\)
−0.351763 + 0.936089i \(0.614418\pi\)
\(308\) 0 0
\(309\) −6.06353 −0.344942
\(310\) 0 0
\(311\) −23.4348 −1.32887 −0.664433 0.747348i \(-0.731327\pi\)
−0.664433 + 0.747348i \(0.731327\pi\)
\(312\) 0 0
\(313\) −6.71056 −0.379303 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(314\) 0 0
\(315\) −0.297786 −0.0167783
\(316\) 0 0
\(317\) −16.8933 −0.948822 −0.474411 0.880303i \(-0.657339\pi\)
−0.474411 + 0.880303i \(0.657339\pi\)
\(318\) 0 0
\(319\) −5.70628 −0.319490
\(320\) 0 0
\(321\) 2.86602 0.159966
\(322\) 0 0
\(323\) 15.2674 0.849501
\(324\) 0 0
\(325\) 22.2488 1.23414
\(326\) 0 0
\(327\) 9.68092 0.535356
\(328\) 0 0
\(329\) 24.8854 1.37198
\(330\) 0 0
\(331\) −33.5980 −1.84671 −0.923356 0.383944i \(-0.874566\pi\)
−0.923356 + 0.383944i \(0.874566\pi\)
\(332\) 0 0
\(333\) 9.53672 0.522609
\(334\) 0 0
\(335\) −0.117505 −0.00641997
\(336\) 0 0
\(337\) −14.8324 −0.807971 −0.403986 0.914765i \(-0.632375\pi\)
−0.403986 + 0.914765i \(0.632375\pi\)
\(338\) 0 0
\(339\) −8.96674 −0.487006
\(340\) 0 0
\(341\) 35.0753 1.89944
\(342\) 0 0
\(343\) 20.0921 1.08487
\(344\) 0 0
\(345\) −0.144883 −0.00780026
\(346\) 0 0
\(347\) 9.55077 0.512712 0.256356 0.966582i \(-0.417478\pi\)
0.256356 + 0.966582i \(0.417478\pi\)
\(348\) 0 0
\(349\) −23.9806 −1.28365 −0.641827 0.766850i \(-0.721823\pi\)
−0.641827 + 0.766850i \(0.721823\pi\)
\(350\) 0 0
\(351\) 4.46851 0.238512
\(352\) 0 0
\(353\) 25.1127 1.33662 0.668308 0.743884i \(-0.267019\pi\)
0.668308 + 0.743884i \(0.267019\pi\)
\(354\) 0 0
\(355\) 1.66138 0.0881769
\(356\) 0 0
\(357\) 8.75851 0.463549
\(358\) 0 0
\(359\) −27.3475 −1.44335 −0.721674 0.692233i \(-0.756627\pi\)
−0.721674 + 0.692233i \(0.756627\pi\)
\(360\) 0 0
\(361\) −6.16369 −0.324405
\(362\) 0 0
\(363\) −21.5616 −1.13169
\(364\) 0 0
\(365\) −1.98655 −0.103981
\(366\) 0 0
\(367\) −18.3332 −0.956988 −0.478494 0.878091i \(-0.658817\pi\)
−0.478494 + 0.878091i \(0.658817\pi\)
\(368\) 0 0
\(369\) −7.46765 −0.388750
\(370\) 0 0
\(371\) −12.7280 −0.660806
\(372\) 0 0
\(373\) −1.00140 −0.0518504 −0.0259252 0.999664i \(-0.508253\pi\)
−0.0259252 + 0.999664i \(0.508253\pi\)
\(374\) 0 0
\(375\) 1.44579 0.0746604
\(376\) 0 0
\(377\) −4.46851 −0.230140
\(378\) 0 0
\(379\) 31.2117 1.60324 0.801618 0.597837i \(-0.203973\pi\)
0.801618 + 0.597837i \(0.203973\pi\)
\(380\) 0 0
\(381\) −18.8240 −0.964383
\(382\) 0 0
\(383\) −28.9277 −1.47813 −0.739067 0.673632i \(-0.764734\pi\)
−0.739067 + 0.673632i \(0.764734\pi\)
\(384\) 0 0
\(385\) 1.69925 0.0866017
\(386\) 0 0
\(387\) −2.12219 −0.107877
\(388\) 0 0
\(389\) 1.77323 0.0899065 0.0449532 0.998989i \(-0.485686\pi\)
0.0449532 + 0.998989i \(0.485686\pi\)
\(390\) 0 0
\(391\) 4.26133 0.215505
\(392\) 0 0
\(393\) −1.84666 −0.0931519
\(394\) 0 0
\(395\) 0.218778 0.0110079
\(396\) 0 0
\(397\) −24.7682 −1.24308 −0.621540 0.783382i \(-0.713493\pi\)
−0.621540 + 0.783382i \(0.713493\pi\)
\(398\) 0 0
\(399\) 7.36385 0.368654
\(400\) 0 0
\(401\) 16.6678 0.832351 0.416176 0.909284i \(-0.363370\pi\)
0.416176 + 0.909284i \(0.363370\pi\)
\(402\) 0 0
\(403\) 27.4670 1.36823
\(404\) 0 0
\(405\) 0.144883 0.00719932
\(406\) 0 0
\(407\) −54.4192 −2.69746
\(408\) 0 0
\(409\) 26.3335 1.30211 0.651054 0.759032i \(-0.274327\pi\)
0.651054 + 0.759032i \(0.274327\pi\)
\(410\) 0 0
\(411\) −4.78041 −0.235800
\(412\) 0 0
\(413\) 27.2171 1.33927
\(414\) 0 0
\(415\) −1.95806 −0.0961172
\(416\) 0 0
\(417\) 11.1294 0.545010
\(418\) 0 0
\(419\) 24.9933 1.22100 0.610501 0.792016i \(-0.290968\pi\)
0.610501 + 0.792016i \(0.290968\pi\)
\(420\) 0 0
\(421\) −4.22862 −0.206090 −0.103045 0.994677i \(-0.532859\pi\)
−0.103045 + 0.994677i \(0.532859\pi\)
\(422\) 0 0
\(423\) −12.1076 −0.588693
\(424\) 0 0
\(425\) −21.2172 −1.02919
\(426\) 0 0
\(427\) −10.5678 −0.511409
\(428\) 0 0
\(429\) −25.4986 −1.23108
\(430\) 0 0
\(431\) −23.7606 −1.14451 −0.572255 0.820076i \(-0.693931\pi\)
−0.572255 + 0.820076i \(0.693931\pi\)
\(432\) 0 0
\(433\) 35.3647 1.69952 0.849760 0.527171i \(-0.176747\pi\)
0.849760 + 0.527171i \(0.176747\pi\)
\(434\) 0 0
\(435\) −0.144883 −0.00694663
\(436\) 0 0
\(437\) 3.58278 0.171388
\(438\) 0 0
\(439\) −21.1935 −1.01151 −0.505756 0.862677i \(-0.668786\pi\)
−0.505756 + 0.862677i \(0.668786\pi\)
\(440\) 0 0
\(441\) −2.77555 −0.132169
\(442\) 0 0
\(443\) 16.0327 0.761738 0.380869 0.924629i \(-0.375625\pi\)
0.380869 + 0.924629i \(0.375625\pi\)
\(444\) 0 0
\(445\) 0.595297 0.0282198
\(446\) 0 0
\(447\) −11.6153 −0.549386
\(448\) 0 0
\(449\) −11.1698 −0.527136 −0.263568 0.964641i \(-0.584899\pi\)
−0.263568 + 0.964641i \(0.584899\pi\)
\(450\) 0 0
\(451\) 42.6125 2.00654
\(452\) 0 0
\(453\) −13.1230 −0.616571
\(454\) 0 0
\(455\) 1.33066 0.0623823
\(456\) 0 0
\(457\) −6.94861 −0.325042 −0.162521 0.986705i \(-0.551963\pi\)
−0.162521 + 0.986705i \(0.551963\pi\)
\(458\) 0 0
\(459\) −4.26133 −0.198902
\(460\) 0 0
\(461\) −8.55364 −0.398383 −0.199191 0.979961i \(-0.563832\pi\)
−0.199191 + 0.979961i \(0.563832\pi\)
\(462\) 0 0
\(463\) 13.2774 0.617052 0.308526 0.951216i \(-0.400164\pi\)
0.308526 + 0.951216i \(0.400164\pi\)
\(464\) 0 0
\(465\) 0.890569 0.0412992
\(466\) 0 0
\(467\) −12.6994 −0.587658 −0.293829 0.955858i \(-0.594930\pi\)
−0.293829 + 0.955858i \(0.594930\pi\)
\(468\) 0 0
\(469\) 1.66695 0.0769725
\(470\) 0 0
\(471\) −10.2659 −0.473028
\(472\) 0 0
\(473\) 12.1098 0.556809
\(474\) 0 0
\(475\) −17.8387 −0.818495
\(476\) 0 0
\(477\) 6.19264 0.283542
\(478\) 0 0
\(479\) −6.81769 −0.311508 −0.155754 0.987796i \(-0.549781\pi\)
−0.155754 + 0.987796i \(0.549781\pi\)
\(480\) 0 0
\(481\) −42.6149 −1.94307
\(482\) 0 0
\(483\) 2.05535 0.0935215
\(484\) 0 0
\(485\) −0.622899 −0.0282844
\(486\) 0 0
\(487\) −4.77935 −0.216573 −0.108287 0.994120i \(-0.534536\pi\)
−0.108287 + 0.994120i \(0.534536\pi\)
\(488\) 0 0
\(489\) 2.89039 0.130708
\(490\) 0 0
\(491\) 29.1826 1.31699 0.658496 0.752585i \(-0.271193\pi\)
0.658496 + 0.752585i \(0.271193\pi\)
\(492\) 0 0
\(493\) 4.26133 0.191921
\(494\) 0 0
\(495\) −0.826745 −0.0371594
\(496\) 0 0
\(497\) −23.5687 −1.05720
\(498\) 0 0
\(499\) 33.7998 1.51309 0.756544 0.653943i \(-0.226886\pi\)
0.756544 + 0.653943i \(0.226886\pi\)
\(500\) 0 0
\(501\) 16.8680 0.753605
\(502\) 0 0
\(503\) 11.4418 0.510164 0.255082 0.966919i \(-0.417898\pi\)
0.255082 + 0.966919i \(0.417898\pi\)
\(504\) 0 0
\(505\) −1.22720 −0.0546097
\(506\) 0 0
\(507\) −6.96762 −0.309443
\(508\) 0 0
\(509\) 33.3677 1.47900 0.739498 0.673158i \(-0.235063\pi\)
0.739498 + 0.673158i \(0.235063\pi\)
\(510\) 0 0
\(511\) 28.1816 1.24668
\(512\) 0 0
\(513\) −3.58278 −0.158184
\(514\) 0 0
\(515\) 0.878505 0.0387116
\(516\) 0 0
\(517\) 69.0896 3.03856
\(518\) 0 0
\(519\) −23.1674 −1.01694
\(520\) 0 0
\(521\) −35.0119 −1.53390 −0.766949 0.641708i \(-0.778226\pi\)
−0.766949 + 0.641708i \(0.778226\pi\)
\(522\) 0 0
\(523\) 1.52408 0.0666436 0.0333218 0.999445i \(-0.489391\pi\)
0.0333218 + 0.999445i \(0.489391\pi\)
\(524\) 0 0
\(525\) −10.2336 −0.446630
\(526\) 0 0
\(527\) −26.1935 −1.14101
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −13.2421 −0.574658
\(532\) 0 0
\(533\) 33.3693 1.44538
\(534\) 0 0
\(535\) −0.415239 −0.0179523
\(536\) 0 0
\(537\) −20.8434 −0.899459
\(538\) 0 0
\(539\) 15.8381 0.682194
\(540\) 0 0
\(541\) 0.523796 0.0225197 0.0112599 0.999937i \(-0.496416\pi\)
0.0112599 + 0.999937i \(0.496416\pi\)
\(542\) 0 0
\(543\) 12.6412 0.542485
\(544\) 0 0
\(545\) −1.40261 −0.0600810
\(546\) 0 0
\(547\) −28.2220 −1.20668 −0.603342 0.797483i \(-0.706165\pi\)
−0.603342 + 0.797483i \(0.706165\pi\)
\(548\) 0 0
\(549\) 5.14159 0.219438
\(550\) 0 0
\(551\) 3.58278 0.152632
\(552\) 0 0
\(553\) −3.10362 −0.131980
\(554\) 0 0
\(555\) −1.38171 −0.0586504
\(556\) 0 0
\(557\) −34.1829 −1.44838 −0.724188 0.689602i \(-0.757785\pi\)
−0.724188 + 0.689602i \(0.757785\pi\)
\(558\) 0 0
\(559\) 9.48302 0.401089
\(560\) 0 0
\(561\) 24.3163 1.02664
\(562\) 0 0
\(563\) 23.3468 0.983951 0.491976 0.870609i \(-0.336275\pi\)
0.491976 + 0.870609i \(0.336275\pi\)
\(564\) 0 0
\(565\) 1.29913 0.0546549
\(566\) 0 0
\(567\) −2.05535 −0.0863165
\(568\) 0 0
\(569\) 25.9873 1.08945 0.544723 0.838616i \(-0.316635\pi\)
0.544723 + 0.838616i \(0.316635\pi\)
\(570\) 0 0
\(571\) 39.4492 1.65090 0.825448 0.564478i \(-0.190922\pi\)
0.825448 + 0.564478i \(0.190922\pi\)
\(572\) 0 0
\(573\) −15.1359 −0.632313
\(574\) 0 0
\(575\) −4.97901 −0.207639
\(576\) 0 0
\(577\) −4.44605 −0.185091 −0.0925457 0.995708i \(-0.529500\pi\)
−0.0925457 + 0.995708i \(0.529500\pi\)
\(578\) 0 0
\(579\) −0.714971 −0.0297132
\(580\) 0 0
\(581\) 27.7774 1.15240
\(582\) 0 0
\(583\) −35.3370 −1.46351
\(584\) 0 0
\(585\) −0.647414 −0.0267673
\(586\) 0 0
\(587\) 28.7877 1.18819 0.594097 0.804393i \(-0.297509\pi\)
0.594097 + 0.804393i \(0.297509\pi\)
\(588\) 0 0
\(589\) −22.0226 −0.907426
\(590\) 0 0
\(591\) 6.61412 0.272069
\(592\) 0 0
\(593\) 15.2194 0.624985 0.312493 0.949920i \(-0.398836\pi\)
0.312493 + 0.949920i \(0.398836\pi\)
\(594\) 0 0
\(595\) −1.26896 −0.0520224
\(596\) 0 0
\(597\) −19.7823 −0.809635
\(598\) 0 0
\(599\) −25.0977 −1.02546 −0.512732 0.858549i \(-0.671367\pi\)
−0.512732 + 0.858549i \(0.671367\pi\)
\(600\) 0 0
\(601\) 14.8712 0.606607 0.303304 0.952894i \(-0.401910\pi\)
0.303304 + 0.952894i \(0.401910\pi\)
\(602\) 0 0
\(603\) −0.811030 −0.0330277
\(604\) 0 0
\(605\) 3.12392 0.127006
\(606\) 0 0
\(607\) −2.01004 −0.0815852 −0.0407926 0.999168i \(-0.512988\pi\)
−0.0407926 + 0.999168i \(0.512988\pi\)
\(608\) 0 0
\(609\) 2.05535 0.0832869
\(610\) 0 0
\(611\) 54.1031 2.18878
\(612\) 0 0
\(613\) 27.1099 1.09496 0.547479 0.836819i \(-0.315588\pi\)
0.547479 + 0.836819i \(0.315588\pi\)
\(614\) 0 0
\(615\) 1.08194 0.0436280
\(616\) 0 0
\(617\) 38.4508 1.54797 0.773986 0.633202i \(-0.218260\pi\)
0.773986 + 0.633202i \(0.218260\pi\)
\(618\) 0 0
\(619\) 40.1717 1.61464 0.807319 0.590115i \(-0.200918\pi\)
0.807319 + 0.590115i \(0.200918\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −8.44500 −0.338342
\(624\) 0 0
\(625\) 24.6856 0.987423
\(626\) 0 0
\(627\) 20.4443 0.816468
\(628\) 0 0
\(629\) 40.6391 1.62039
\(630\) 0 0
\(631\) −1.82352 −0.0725932 −0.0362966 0.999341i \(-0.511556\pi\)
−0.0362966 + 0.999341i \(0.511556\pi\)
\(632\) 0 0
\(633\) −0.00743211 −0.000295400 0
\(634\) 0 0
\(635\) 2.72729 0.108229
\(636\) 0 0
\(637\) 12.4026 0.491408
\(638\) 0 0
\(639\) 11.4670 0.453628
\(640\) 0 0
\(641\) 1.12386 0.0443900 0.0221950 0.999754i \(-0.492935\pi\)
0.0221950 + 0.999754i \(0.492935\pi\)
\(642\) 0 0
\(643\) −19.8557 −0.783032 −0.391516 0.920171i \(-0.628049\pi\)
−0.391516 + 0.920171i \(0.628049\pi\)
\(644\) 0 0
\(645\) 0.307470 0.0121066
\(646\) 0 0
\(647\) 47.5287 1.86855 0.934274 0.356557i \(-0.116049\pi\)
0.934274 + 0.356557i \(0.116049\pi\)
\(648\) 0 0
\(649\) 75.5631 2.96611
\(650\) 0 0
\(651\) −12.6338 −0.495158
\(652\) 0 0
\(653\) 19.1825 0.750668 0.375334 0.926890i \(-0.377528\pi\)
0.375334 + 0.926890i \(0.377528\pi\)
\(654\) 0 0
\(655\) 0.267551 0.0104541
\(656\) 0 0
\(657\) −13.7114 −0.534932
\(658\) 0 0
\(659\) 29.3549 1.14350 0.571752 0.820427i \(-0.306264\pi\)
0.571752 + 0.820427i \(0.306264\pi\)
\(660\) 0 0
\(661\) −37.8387 −1.47175 −0.735877 0.677115i \(-0.763230\pi\)
−0.735877 + 0.677115i \(0.763230\pi\)
\(662\) 0 0
\(663\) 19.0418 0.739523
\(664\) 0 0
\(665\) −1.06690 −0.0413726
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 15.8140 0.611405
\(670\) 0 0
\(671\) −29.3394 −1.13263
\(672\) 0 0
\(673\) −22.4296 −0.864597 −0.432298 0.901731i \(-0.642297\pi\)
−0.432298 + 0.901731i \(0.642297\pi\)
\(674\) 0 0
\(675\) 4.97901 0.191642
\(676\) 0 0
\(677\) 40.1866 1.54450 0.772249 0.635320i \(-0.219132\pi\)
0.772249 + 0.635320i \(0.219132\pi\)
\(678\) 0 0
\(679\) 8.83657 0.339117
\(680\) 0 0
\(681\) −16.8842 −0.647003
\(682\) 0 0
\(683\) 0.397782 0.0152207 0.00761035 0.999971i \(-0.497578\pi\)
0.00761035 + 0.999971i \(0.497578\pi\)
\(684\) 0 0
\(685\) 0.692602 0.0264630
\(686\) 0 0
\(687\) −29.8250 −1.13789
\(688\) 0 0
\(689\) −27.6719 −1.05422
\(690\) 0 0
\(691\) 35.7272 1.35913 0.679564 0.733617i \(-0.262169\pi\)
0.679564 + 0.733617i \(0.262169\pi\)
\(692\) 0 0
\(693\) 11.7284 0.445524
\(694\) 0 0
\(695\) −1.61247 −0.0611644
\(696\) 0 0
\(697\) −31.8221 −1.20535
\(698\) 0 0
\(699\) −29.2967 −1.10810
\(700\) 0 0
\(701\) −24.4728 −0.924325 −0.462163 0.886795i \(-0.652926\pi\)
−0.462163 + 0.886795i \(0.652926\pi\)
\(702\) 0 0
\(703\) 34.1679 1.28867
\(704\) 0 0
\(705\) 1.75420 0.0660668
\(706\) 0 0
\(707\) 17.4093 0.654745
\(708\) 0 0
\(709\) 12.6182 0.473888 0.236944 0.971523i \(-0.423854\pi\)
0.236944 + 0.971523i \(0.423854\pi\)
\(710\) 0 0
\(711\) 1.51002 0.0566304
\(712\) 0 0
\(713\) −6.14680 −0.230199
\(714\) 0 0
\(715\) 3.69432 0.138160
\(716\) 0 0
\(717\) 17.7957 0.664592
\(718\) 0 0
\(719\) 37.7271 1.40698 0.703492 0.710704i \(-0.251623\pi\)
0.703492 + 0.710704i \(0.251623\pi\)
\(720\) 0 0
\(721\) −12.4627 −0.464134
\(722\) 0 0
\(723\) 3.01052 0.111962
\(724\) 0 0
\(725\) −4.97901 −0.184916
\(726\) 0 0
\(727\) −9.91752 −0.367820 −0.183910 0.982943i \(-0.558876\pi\)
−0.183910 + 0.982943i \(0.558876\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −9.04333 −0.334480
\(732\) 0 0
\(733\) −32.9733 −1.21790 −0.608949 0.793209i \(-0.708409\pi\)
−0.608949 + 0.793209i \(0.708409\pi\)
\(734\) 0 0
\(735\) 0.402131 0.0148328
\(736\) 0 0
\(737\) 4.62796 0.170473
\(738\) 0 0
\(739\) −42.1806 −1.55164 −0.775819 0.630956i \(-0.782663\pi\)
−0.775819 + 0.630956i \(0.782663\pi\)
\(740\) 0 0
\(741\) 16.0097 0.588131
\(742\) 0 0
\(743\) −13.7038 −0.502745 −0.251373 0.967890i \(-0.580882\pi\)
−0.251373 + 0.967890i \(0.580882\pi\)
\(744\) 0 0
\(745\) 1.68287 0.0616555
\(746\) 0 0
\(747\) −13.5147 −0.494477
\(748\) 0 0
\(749\) 5.89066 0.215240
\(750\) 0 0
\(751\) 7.36385 0.268711 0.134355 0.990933i \(-0.457104\pi\)
0.134355 + 0.990933i \(0.457104\pi\)
\(752\) 0 0
\(753\) −1.62166 −0.0590965
\(754\) 0 0
\(755\) 1.90130 0.0691954
\(756\) 0 0
\(757\) 38.9064 1.41408 0.707039 0.707174i \(-0.250030\pi\)
0.707039 + 0.707174i \(0.250030\pi\)
\(758\) 0 0
\(759\) 5.70628 0.207125
\(760\) 0 0
\(761\) 43.0038 1.55889 0.779444 0.626472i \(-0.215502\pi\)
0.779444 + 0.626472i \(0.215502\pi\)
\(762\) 0 0
\(763\) 19.8976 0.720343
\(764\) 0 0
\(765\) 0.617396 0.0223220
\(766\) 0 0
\(767\) 59.1725 2.13659
\(768\) 0 0
\(769\) −26.9722 −0.972641 −0.486320 0.873781i \(-0.661661\pi\)
−0.486320 + 0.873781i \(0.661661\pi\)
\(770\) 0 0
\(771\) 5.82055 0.209622
\(772\) 0 0
\(773\) 9.21412 0.331409 0.165704 0.986175i \(-0.447010\pi\)
0.165704 + 0.986175i \(0.447010\pi\)
\(774\) 0 0
\(775\) 30.6050 1.09936
\(776\) 0 0
\(777\) 19.6013 0.703191
\(778\) 0 0
\(779\) −26.7549 −0.958595
\(780\) 0 0
\(781\) −65.4340 −2.34141
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 1.48736 0.0530862
\(786\) 0 0
\(787\) 29.9823 1.06875 0.534377 0.845246i \(-0.320546\pi\)
0.534377 + 0.845246i \(0.320546\pi\)
\(788\) 0 0
\(789\) −7.13534 −0.254025
\(790\) 0 0
\(791\) −18.4298 −0.655287
\(792\) 0 0
\(793\) −22.9753 −0.815876
\(794\) 0 0
\(795\) −0.897212 −0.0318208
\(796\) 0 0
\(797\) −27.1582 −0.961992 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(798\) 0 0
\(799\) −51.5946 −1.82529
\(800\) 0 0
\(801\) 4.10880 0.145177
\(802\) 0 0
\(803\) 78.2410 2.76106
\(804\) 0 0
\(805\) −0.297786 −0.0104956
\(806\) 0 0
\(807\) −24.1410 −0.849805
\(808\) 0 0
\(809\) −9.80455 −0.344710 −0.172355 0.985035i \(-0.555138\pi\)
−0.172355 + 0.985035i \(0.555138\pi\)
\(810\) 0 0
\(811\) −42.3639 −1.48760 −0.743799 0.668404i \(-0.766978\pi\)
−0.743799 + 0.668404i \(0.766978\pi\)
\(812\) 0 0
\(813\) −2.19676 −0.0770437
\(814\) 0 0
\(815\) −0.418770 −0.0146689
\(816\) 0 0
\(817\) −7.60332 −0.266007
\(818\) 0 0
\(819\) 9.18435 0.320927
\(820\) 0 0
\(821\) 24.6899 0.861685 0.430842 0.902427i \(-0.358216\pi\)
0.430842 + 0.902427i \(0.358216\pi\)
\(822\) 0 0
\(823\) −14.7338 −0.513586 −0.256793 0.966466i \(-0.582666\pi\)
−0.256793 + 0.966466i \(0.582666\pi\)
\(824\) 0 0
\(825\) −28.4116 −0.989166
\(826\) 0 0
\(827\) −10.7964 −0.375429 −0.187714 0.982224i \(-0.560108\pi\)
−0.187714 + 0.982224i \(0.560108\pi\)
\(828\) 0 0
\(829\) −12.5085 −0.434439 −0.217220 0.976123i \(-0.569699\pi\)
−0.217220 + 0.976123i \(0.569699\pi\)
\(830\) 0 0
\(831\) −7.32361 −0.254053
\(832\) 0 0
\(833\) −11.8275 −0.409800
\(834\) 0 0
\(835\) −2.44389 −0.0845743
\(836\) 0 0
\(837\) 6.14680 0.212464
\(838\) 0 0
\(839\) 12.8777 0.444588 0.222294 0.974980i \(-0.428646\pi\)
0.222294 + 0.974980i \(0.428646\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −18.9450 −0.652501
\(844\) 0 0
\(845\) 1.00949 0.0347276
\(846\) 0 0
\(847\) −44.3166 −1.52274
\(848\) 0 0
\(849\) −8.58396 −0.294601
\(850\) 0 0
\(851\) 9.53672 0.326914
\(852\) 0 0
\(853\) −16.3317 −0.559186 −0.279593 0.960119i \(-0.590200\pi\)
−0.279593 + 0.960119i \(0.590200\pi\)
\(854\) 0 0
\(855\) 0.519085 0.0177523
\(856\) 0 0
\(857\) −39.4820 −1.34868 −0.674340 0.738421i \(-0.735572\pi\)
−0.674340 + 0.738421i \(0.735572\pi\)
\(858\) 0 0
\(859\) −39.0864 −1.33361 −0.666806 0.745231i \(-0.732339\pi\)
−0.666806 + 0.745231i \(0.732339\pi\)
\(860\) 0 0
\(861\) −15.3486 −0.523079
\(862\) 0 0
\(863\) −21.0938 −0.718041 −0.359021 0.933330i \(-0.616889\pi\)
−0.359021 + 0.933330i \(0.616889\pi\)
\(864\) 0 0
\(865\) 3.35657 0.114127
\(866\) 0 0
\(867\) −1.15893 −0.0393593
\(868\) 0 0
\(869\) −8.61662 −0.292299
\(870\) 0 0
\(871\) 3.62410 0.122798
\(872\) 0 0
\(873\) −4.29931 −0.145510
\(874\) 0 0
\(875\) 2.97161 0.100459
\(876\) 0 0
\(877\) −14.5956 −0.492858 −0.246429 0.969161i \(-0.579257\pi\)
−0.246429 + 0.969161i \(0.579257\pi\)
\(878\) 0 0
\(879\) −20.1079 −0.678221
\(880\) 0 0
\(881\) 44.4253 1.49673 0.748363 0.663289i \(-0.230840\pi\)
0.748363 + 0.663289i \(0.230840\pi\)
\(882\) 0 0
\(883\) −14.5763 −0.490532 −0.245266 0.969456i \(-0.578875\pi\)
−0.245266 + 0.969456i \(0.578875\pi\)
\(884\) 0 0
\(885\) 1.91856 0.0644917
\(886\) 0 0
\(887\) −3.36549 −0.113002 −0.0565010 0.998403i \(-0.517994\pi\)
−0.0565010 + 0.998403i \(0.517994\pi\)
\(888\) 0 0
\(889\) −38.6899 −1.29762
\(890\) 0 0
\(891\) −5.70628 −0.191168
\(892\) 0 0
\(893\) −43.3790 −1.45162
\(894\) 0 0
\(895\) 3.01986 0.100943
\(896\) 0 0
\(897\) 4.46851 0.149199
\(898\) 0 0
\(899\) −6.14680 −0.205007
\(900\) 0 0
\(901\) 26.3889 0.879142
\(902\) 0 0
\(903\) −4.36183 −0.145153
\(904\) 0 0
\(905\) −1.83150 −0.0608811
\(906\) 0 0
\(907\) −5.65151 −0.187655 −0.0938276 0.995588i \(-0.529910\pi\)
−0.0938276 + 0.995588i \(0.529910\pi\)
\(908\) 0 0
\(909\) −8.47026 −0.280941
\(910\) 0 0
\(911\) 5.39906 0.178879 0.0894394 0.995992i \(-0.471492\pi\)
0.0894394 + 0.995992i \(0.471492\pi\)
\(912\) 0 0
\(913\) 77.1187 2.55225
\(914\) 0 0
\(915\) −0.744931 −0.0246267
\(916\) 0 0
\(917\) −3.79554 −0.125340
\(918\) 0 0
\(919\) 14.2548 0.470224 0.235112 0.971968i \(-0.424454\pi\)
0.235112 + 0.971968i \(0.424454\pi\)
\(920\) 0 0
\(921\) 12.3268 0.406181
\(922\) 0 0
\(923\) −51.2405 −1.68660
\(924\) 0 0
\(925\) −47.4834 −1.56124
\(926\) 0 0
\(927\) 6.06353 0.199153
\(928\) 0 0
\(929\) 10.8621 0.356374 0.178187 0.983997i \(-0.442977\pi\)
0.178187 + 0.983997i \(0.442977\pi\)
\(930\) 0 0
\(931\) −9.94418 −0.325907
\(932\) 0 0
\(933\) 23.4348 0.767221
\(934\) 0 0
\(935\) −3.52303 −0.115216
\(936\) 0 0
\(937\) −23.1322 −0.755697 −0.377848 0.925868i \(-0.623336\pi\)
−0.377848 + 0.925868i \(0.623336\pi\)
\(938\) 0 0
\(939\) 6.71056 0.218991
\(940\) 0 0
\(941\) −8.86711 −0.289060 −0.144530 0.989500i \(-0.546167\pi\)
−0.144530 + 0.989500i \(0.546167\pi\)
\(942\) 0 0
\(943\) −7.46765 −0.243180
\(944\) 0 0
\(945\) 0.297786 0.00968697
\(946\) 0 0
\(947\) 0.341452 0.0110957 0.00554784 0.999985i \(-0.498234\pi\)
0.00554784 + 0.999985i \(0.498234\pi\)
\(948\) 0 0
\(949\) 61.2695 1.98889
\(950\) 0 0
\(951\) 16.8933 0.547803
\(952\) 0 0
\(953\) −2.70632 −0.0876662 −0.0438331 0.999039i \(-0.513957\pi\)
−0.0438331 + 0.999039i \(0.513957\pi\)
\(954\) 0 0
\(955\) 2.19295 0.0709621
\(956\) 0 0
\(957\) 5.70628 0.184458
\(958\) 0 0
\(959\) −9.82539 −0.317279
\(960\) 0 0
\(961\) 6.78312 0.218810
\(962\) 0 0
\(963\) −2.86602 −0.0923562
\(964\) 0 0
\(965\) 0.103587 0.00333460
\(966\) 0 0
\(967\) −39.6891 −1.27631 −0.638157 0.769906i \(-0.720303\pi\)
−0.638157 + 0.769906i \(0.720303\pi\)
\(968\) 0 0
\(969\) −15.2674 −0.490460
\(970\) 0 0
\(971\) 34.1509 1.09596 0.547978 0.836493i \(-0.315398\pi\)
0.547978 + 0.836493i \(0.315398\pi\)
\(972\) 0 0
\(973\) 22.8748 0.733333
\(974\) 0 0
\(975\) −22.2488 −0.712531
\(976\) 0 0
\(977\) −33.9568 −1.08637 −0.543187 0.839612i \(-0.682782\pi\)
−0.543187 + 0.839612i \(0.682782\pi\)
\(978\) 0 0
\(979\) −23.4460 −0.749336
\(980\) 0 0
\(981\) −9.68092 −0.309088
\(982\) 0 0
\(983\) −41.7522 −1.33169 −0.665844 0.746091i \(-0.731928\pi\)
−0.665844 + 0.746091i \(0.731928\pi\)
\(984\) 0 0
\(985\) −0.958276 −0.0305332
\(986\) 0 0
\(987\) −24.8854 −0.792111
\(988\) 0 0
\(989\) −2.12219 −0.0674816
\(990\) 0 0
\(991\) −3.93753 −0.125080 −0.0625398 0.998042i \(-0.519920\pi\)
−0.0625398 + 0.998042i \(0.519920\pi\)
\(992\) 0 0
\(993\) 33.5980 1.06620
\(994\) 0 0
\(995\) 2.86612 0.0908622
\(996\) 0 0
\(997\) −4.45768 −0.141176 −0.0705880 0.997506i \(-0.522488\pi\)
−0.0705880 + 0.997506i \(0.522488\pi\)
\(998\) 0 0
\(999\) −9.53672 −0.301728
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 8004.2.a.i.1.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.8 16 1.1 even 1 trivial