Properties

Label 4864.2.a.bn.1.8
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 68 x^{3} - 62 x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.28103\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.13611 q^{3} -0.594041 q^{5} -3.48756 q^{7} +6.83520 q^{9} +O(q^{10})\) \(q+3.13611 q^{3} -0.594041 q^{5} -3.48756 q^{7} +6.83520 q^{9} -4.83520 q^{11} -0.215597 q^{13} -1.86298 q^{15} +1.29720 q^{17} +1.00000 q^{19} -10.9374 q^{21} +4.52815 q^{23} -4.64712 q^{25} +12.0276 q^{27} -9.41093 q^{29} +1.22031 q^{31} -15.1637 q^{33} +2.07176 q^{35} -5.62653 q^{37} -0.676135 q^{39} +0.450021 q^{41} +0.794359 q^{43} -4.06039 q^{45} -12.1986 q^{47} +5.16310 q^{49} +4.06817 q^{51} -2.56409 q^{53} +2.87231 q^{55} +3.13611 q^{57} +2.75191 q^{59} -7.76665 q^{61} -23.8382 q^{63} +0.128073 q^{65} -4.11631 q^{67} +14.2008 q^{69} -7.82788 q^{71} -3.08931 q^{73} -14.5739 q^{75} +16.8631 q^{77} +10.0731 q^{79} +17.2143 q^{81} +11.6296 q^{83} -0.770591 q^{85} -29.5137 q^{87} -13.7091 q^{89} +0.751907 q^{91} +3.82703 q^{93} -0.594041 q^{95} +2.08846 q^{97} -33.0495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} - 4q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} - 4q^{7} + 12q^{9} + 4q^{11} - 8q^{13} - 4q^{17} + 8q^{19} - 16q^{21} + 12q^{25} - 28q^{29} - 8q^{31} - 12q^{35} - 4q^{37} + 4q^{39} - 8q^{41} - 4q^{43} - 24q^{45} - 12q^{47} + 12q^{49} + 12q^{51} - 32q^{53} + 8q^{55} + 12q^{59} - 8q^{61} + 16q^{63} + 8q^{65} - 4q^{67} - 28q^{69} + 24q^{71} - 24q^{77} + 24q^{79} - 8q^{81} + 40q^{83} - 24q^{85} - 24q^{87} + 8q^{89} - 4q^{91} - 32q^{93} - 8q^{95} + 16q^{97} - 76q^{99} + O(q^{100}) \)

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
\(3\) 3.13611 1.81063 0.905317 0.424735i \(-0.139633\pi\)
0.905317 + 0.424735i \(0.139633\pi\)
\(4\) 0 0
\(5\) −0.594041 −0.265663 −0.132832 0.991139i \(-0.542407\pi\)
−0.132832 + 0.991139i \(0.542407\pi\)
\(6\) 0 0
\(7\) −3.48756 −1.31818 −0.659088 0.752066i \(-0.729057\pi\)
−0.659088 + 0.752066i \(0.729057\pi\)
\(8\) 0 0
\(9\) 6.83520 2.27840
\(10\) 0 0
\(11\) −4.83520 −1.45787 −0.728933 0.684585i \(-0.759984\pi\)
−0.728933 + 0.684585i \(0.759984\pi\)
\(12\) 0 0
\(13\) −0.215597 −0.0597957 −0.0298979 0.999553i \(-0.509518\pi\)
−0.0298979 + 0.999553i \(0.509518\pi\)
\(14\) 0 0
\(15\) −1.86298 −0.481019
\(16\) 0 0
\(17\) 1.29720 0.314618 0.157309 0.987549i \(-0.449718\pi\)
0.157309 + 0.987549i \(0.449718\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −10.9374 −2.38673
\(22\) 0 0
\(23\) 4.52815 0.944184 0.472092 0.881549i \(-0.343499\pi\)
0.472092 + 0.881549i \(0.343499\pi\)
\(24\) 0 0
\(25\) −4.64712 −0.929423
\(26\) 0 0
\(27\) 12.0276 2.31471
\(28\) 0 0
\(29\) −9.41093 −1.74757 −0.873783 0.486316i \(-0.838340\pi\)
−0.873783 + 0.486316i \(0.838340\pi\)
\(30\) 0 0
\(31\) 1.22031 0.219174 0.109587 0.993977i \(-0.465047\pi\)
0.109587 + 0.993977i \(0.465047\pi\)
\(32\) 0 0
\(33\) −15.1637 −2.63966
\(34\) 0 0
\(35\) 2.07176 0.350191
\(36\) 0 0
\(37\) −5.62653 −0.924995 −0.462498 0.886621i \(-0.653047\pi\)
−0.462498 + 0.886621i \(0.653047\pi\)
\(38\) 0 0
\(39\) −0.676135 −0.108268
\(40\) 0 0
\(41\) 0.450021 0.0702815 0.0351407 0.999382i \(-0.488812\pi\)
0.0351407 + 0.999382i \(0.488812\pi\)
\(42\) 0 0
\(43\) 0.794359 0.121139 0.0605693 0.998164i \(-0.480708\pi\)
0.0605693 + 0.998164i \(0.480708\pi\)
\(44\) 0 0
\(45\) −4.06039 −0.605287
\(46\) 0 0
\(47\) −12.1986 −1.77935 −0.889676 0.456593i \(-0.849070\pi\)
−0.889676 + 0.456593i \(0.849070\pi\)
\(48\) 0 0
\(49\) 5.16310 0.737586
\(50\) 0 0
\(51\) 4.06817 0.569658
\(52\) 0 0
\(53\) −2.56409 −0.352205 −0.176103 0.984372i \(-0.556349\pi\)
−0.176103 + 0.984372i \(0.556349\pi\)
\(54\) 0 0
\(55\) 2.87231 0.387302
\(56\) 0 0
\(57\) 3.13611 0.415388
\(58\) 0 0
\(59\) 2.75191 0.358268 0.179134 0.983825i \(-0.442670\pi\)
0.179134 + 0.983825i \(0.442670\pi\)
\(60\) 0 0
\(61\) −7.76665 −0.994417 −0.497209 0.867631i \(-0.665642\pi\)
−0.497209 + 0.867631i \(0.665642\pi\)
\(62\) 0 0
\(63\) −23.8382 −3.00333
\(64\) 0 0
\(65\) 0.128073 0.0158855
\(66\) 0 0
\(67\) −4.11631 −0.502887 −0.251443 0.967872i \(-0.580905\pi\)
−0.251443 + 0.967872i \(0.580905\pi\)
\(68\) 0 0
\(69\) 14.2008 1.70957
\(70\) 0 0
\(71\) −7.82788 −0.928999 −0.464499 0.885573i \(-0.653766\pi\)
−0.464499 + 0.885573i \(0.653766\pi\)
\(72\) 0 0
\(73\) −3.08931 −0.361577 −0.180788 0.983522i \(-0.557865\pi\)
−0.180788 + 0.983522i \(0.557865\pi\)
\(74\) 0 0
\(75\) −14.5739 −1.68285
\(76\) 0 0
\(77\) 16.8631 1.92172
\(78\) 0 0
\(79\) 10.0731 1.13331 0.566654 0.823956i \(-0.308237\pi\)
0.566654 + 0.823956i \(0.308237\pi\)
\(80\) 0 0
\(81\) 17.2143 1.91270
\(82\) 0 0
\(83\) 11.6296 1.27651 0.638255 0.769825i \(-0.279657\pi\)
0.638255 + 0.769825i \(0.279657\pi\)
\(84\) 0 0
\(85\) −0.770591 −0.0835823
\(86\) 0 0
\(87\) −29.5137 −3.16420
\(88\) 0 0
\(89\) −13.7091 −1.45317 −0.726583 0.687079i \(-0.758893\pi\)
−0.726583 + 0.687079i \(0.758893\pi\)
\(90\) 0 0
\(91\) 0.751907 0.0788213
\(92\) 0 0
\(93\) 3.82703 0.396845
\(94\) 0 0
\(95\) −0.594041 −0.0609473
\(96\) 0 0
\(97\) 2.08846 0.212051 0.106026 0.994363i \(-0.466187\pi\)
0.106026 + 0.994363i \(0.466187\pi\)
\(98\) 0 0
\(99\) −33.0495 −3.32160
\(100\) 0 0
\(101\) −2.77074 −0.275699 −0.137849 0.990453i \(-0.544019\pi\)
−0.137849 + 0.990453i \(0.544019\pi\)
\(102\) 0 0
\(103\) −14.3363 −1.41260 −0.706301 0.707912i \(-0.749637\pi\)
−0.706301 + 0.707912i \(0.749637\pi\)
\(104\) 0 0
\(105\) 6.49726 0.634067
\(106\) 0 0
\(107\) 2.42388 0.234325 0.117163 0.993113i \(-0.462620\pi\)
0.117163 + 0.993113i \(0.462620\pi\)
\(108\) 0 0
\(109\) −0.00123810 −0.000118589 0 −5.92945e−5 1.00000i \(-0.500019\pi\)
−5.92945e−5 1.00000i \(0.500019\pi\)
\(110\) 0 0
\(111\) −17.6454 −1.67483
\(112\) 0 0
\(113\) 1.81614 0.170848 0.0854242 0.996345i \(-0.472775\pi\)
0.0854242 + 0.996345i \(0.472775\pi\)
\(114\) 0 0
\(115\) −2.68990 −0.250835
\(116\) 0 0
\(117\) −1.47365 −0.136239
\(118\) 0 0
\(119\) −4.52407 −0.414721
\(120\) 0 0
\(121\) 12.3791 1.12538
\(122\) 0 0
\(123\) 1.41132 0.127254
\(124\) 0 0
\(125\) 5.73078 0.512577
\(126\) 0 0
\(127\) 16.1269 1.43103 0.715517 0.698596i \(-0.246191\pi\)
0.715517 + 0.698596i \(0.246191\pi\)
\(128\) 0 0
\(129\) 2.49120 0.219338
\(130\) 0 0
\(131\) 11.1477 0.973983 0.486992 0.873407i \(-0.338094\pi\)
0.486992 + 0.873407i \(0.338094\pi\)
\(132\) 0 0
\(133\) −3.48756 −0.302410
\(134\) 0 0
\(135\) −7.14489 −0.614934
\(136\) 0 0
\(137\) −11.1666 −0.954025 −0.477013 0.878896i \(-0.658280\pi\)
−0.477013 + 0.878896i \(0.658280\pi\)
\(138\) 0 0
\(139\) −13.0534 −1.10718 −0.553589 0.832790i \(-0.686742\pi\)
−0.553589 + 0.832790i \(0.686742\pi\)
\(140\) 0 0
\(141\) −38.2562 −3.22176
\(142\) 0 0
\(143\) 1.04245 0.0871742
\(144\) 0 0
\(145\) 5.59048 0.464264
\(146\) 0 0
\(147\) 16.1921 1.33550
\(148\) 0 0
\(149\) −14.3811 −1.17814 −0.589072 0.808080i \(-0.700507\pi\)
−0.589072 + 0.808080i \(0.700507\pi\)
\(150\) 0 0
\(151\) 17.3489 1.41184 0.705918 0.708293i \(-0.250535\pi\)
0.705918 + 0.708293i \(0.250535\pi\)
\(152\) 0 0
\(153\) 8.86663 0.716825
\(154\) 0 0
\(155\) −0.724915 −0.0582266
\(156\) 0 0
\(157\) 9.63293 0.768791 0.384396 0.923168i \(-0.374410\pi\)
0.384396 + 0.923168i \(0.374410\pi\)
\(158\) 0 0
\(159\) −8.04128 −0.637715
\(160\) 0 0
\(161\) −15.7922 −1.24460
\(162\) 0 0
\(163\) −22.5365 −1.76520 −0.882598 0.470129i \(-0.844207\pi\)
−0.882598 + 0.470129i \(0.844207\pi\)
\(164\) 0 0
\(165\) 9.00787 0.701262
\(166\) 0 0
\(167\) 16.5108 1.27765 0.638823 0.769353i \(-0.279421\pi\)
0.638823 + 0.769353i \(0.279421\pi\)
\(168\) 0 0
\(169\) −12.9535 −0.996424
\(170\) 0 0
\(171\) 6.83520 0.522701
\(172\) 0 0
\(173\) −5.14911 −0.391479 −0.195740 0.980656i \(-0.562711\pi\)
−0.195740 + 0.980656i \(0.562711\pi\)
\(174\) 0 0
\(175\) 16.2071 1.22514
\(176\) 0 0
\(177\) 8.63029 0.648692
\(178\) 0 0
\(179\) −9.31580 −0.696295 −0.348148 0.937440i \(-0.613189\pi\)
−0.348148 + 0.937440i \(0.613189\pi\)
\(180\) 0 0
\(181\) 5.15517 0.383180 0.191590 0.981475i \(-0.438636\pi\)
0.191590 + 0.981475i \(0.438636\pi\)
\(182\) 0 0
\(183\) −24.3571 −1.80053
\(184\) 0 0
\(185\) 3.34239 0.245737
\(186\) 0 0
\(187\) −6.27223 −0.458671
\(188\) 0 0
\(189\) −41.9471 −3.05120
\(190\) 0 0
\(191\) 14.8805 1.07672 0.538359 0.842715i \(-0.319044\pi\)
0.538359 + 0.842715i \(0.319044\pi\)
\(192\) 0 0
\(193\) −21.2754 −1.53144 −0.765719 0.643175i \(-0.777617\pi\)
−0.765719 + 0.643175i \(0.777617\pi\)
\(194\) 0 0
\(195\) 0.401652 0.0287629
\(196\) 0 0
\(197\) 3.69169 0.263022 0.131511 0.991315i \(-0.458017\pi\)
0.131511 + 0.991315i \(0.458017\pi\)
\(198\) 0 0
\(199\) 11.6857 0.828377 0.414188 0.910191i \(-0.364065\pi\)
0.414188 + 0.910191i \(0.364065\pi\)
\(200\) 0 0
\(201\) −12.9092 −0.910545
\(202\) 0 0
\(203\) 32.8212 2.30360
\(204\) 0 0
\(205\) −0.267331 −0.0186712
\(206\) 0 0
\(207\) 30.9508 2.15123
\(208\) 0 0
\(209\) −4.83520 −0.334458
\(210\) 0 0
\(211\) −7.61611 −0.524315 −0.262157 0.965025i \(-0.584434\pi\)
−0.262157 + 0.965025i \(0.584434\pi\)
\(212\) 0 0
\(213\) −24.5491 −1.68208
\(214\) 0 0
\(215\) −0.471882 −0.0321821
\(216\) 0 0
\(217\) −4.25591 −0.288910
\(218\) 0 0
\(219\) −9.68844 −0.654684
\(220\) 0 0
\(221\) −0.279672 −0.0188128
\(222\) 0 0
\(223\) 1.95477 0.130901 0.0654505 0.997856i \(-0.479152\pi\)
0.0654505 + 0.997856i \(0.479152\pi\)
\(224\) 0 0
\(225\) −31.7640 −2.11760
\(226\) 0 0
\(227\) −13.3709 −0.887461 −0.443730 0.896160i \(-0.646345\pi\)
−0.443730 + 0.896160i \(0.646345\pi\)
\(228\) 0 0
\(229\) 11.5800 0.765225 0.382613 0.923909i \(-0.375024\pi\)
0.382613 + 0.923909i \(0.375024\pi\)
\(230\) 0 0
\(231\) 52.8844 3.47954
\(232\) 0 0
\(233\) −1.58872 −0.104080 −0.0520401 0.998645i \(-0.516572\pi\)
−0.0520401 + 0.998645i \(0.516572\pi\)
\(234\) 0 0
\(235\) 7.24648 0.472708
\(236\) 0 0
\(237\) 31.5903 2.05201
\(238\) 0 0
\(239\) −23.8219 −1.54091 −0.770455 0.637494i \(-0.779971\pi\)
−0.770455 + 0.637494i \(0.779971\pi\)
\(240\) 0 0
\(241\) 22.8554 1.47225 0.736123 0.676848i \(-0.236654\pi\)
0.736123 + 0.676848i \(0.236654\pi\)
\(242\) 0 0
\(243\) 17.9032 1.14849
\(244\) 0 0
\(245\) −3.06710 −0.195950
\(246\) 0 0
\(247\) −0.215597 −0.0137181
\(248\) 0 0
\(249\) 36.4716 2.31129
\(250\) 0 0
\(251\) 2.63524 0.166335 0.0831676 0.996536i \(-0.473496\pi\)
0.0831676 + 0.996536i \(0.473496\pi\)
\(252\) 0 0
\(253\) −21.8945 −1.37649
\(254\) 0 0
\(255\) −2.41666 −0.151337
\(256\) 0 0
\(257\) 20.0579 1.25118 0.625588 0.780153i \(-0.284859\pi\)
0.625588 + 0.780153i \(0.284859\pi\)
\(258\) 0 0
\(259\) 19.6229 1.21931
\(260\) 0 0
\(261\) −64.3256 −3.98165
\(262\) 0 0
\(263\) 4.03667 0.248912 0.124456 0.992225i \(-0.460282\pi\)
0.124456 + 0.992225i \(0.460282\pi\)
\(264\) 0 0
\(265\) 1.52318 0.0935679
\(266\) 0 0
\(267\) −42.9934 −2.63115
\(268\) 0 0
\(269\) −14.3095 −0.872467 −0.436233 0.899834i \(-0.643688\pi\)
−0.436233 + 0.899834i \(0.643688\pi\)
\(270\) 0 0
\(271\) −9.85034 −0.598366 −0.299183 0.954196i \(-0.596714\pi\)
−0.299183 + 0.954196i \(0.596714\pi\)
\(272\) 0 0
\(273\) 2.35806 0.142717
\(274\) 0 0
\(275\) 22.4697 1.35498
\(276\) 0 0
\(277\) 16.9641 1.01928 0.509638 0.860389i \(-0.329779\pi\)
0.509638 + 0.860389i \(0.329779\pi\)
\(278\) 0 0
\(279\) 8.34107 0.499367
\(280\) 0 0
\(281\) 10.2078 0.608944 0.304472 0.952521i \(-0.401520\pi\)
0.304472 + 0.952521i \(0.401520\pi\)
\(282\) 0 0
\(283\) −18.7709 −1.11581 −0.557907 0.829903i \(-0.688396\pi\)
−0.557907 + 0.829903i \(0.688396\pi\)
\(284\) 0 0
\(285\) −1.86298 −0.110353
\(286\) 0 0
\(287\) −1.56948 −0.0926433
\(288\) 0 0
\(289\) −15.3173 −0.901016
\(290\) 0 0
\(291\) 6.54966 0.383948
\(292\) 0 0
\(293\) −19.8271 −1.15831 −0.579155 0.815217i \(-0.696618\pi\)
−0.579155 + 0.815217i \(0.696618\pi\)
\(294\) 0 0
\(295\) −1.63475 −0.0951786
\(296\) 0 0
\(297\) −58.1559 −3.37454
\(298\) 0 0
\(299\) −0.976253 −0.0564582
\(300\) 0 0
\(301\) −2.77038 −0.159682
\(302\) 0 0
\(303\) −8.68935 −0.499190
\(304\) 0 0
\(305\) 4.61371 0.264180
\(306\) 0 0
\(307\) −18.3935 −1.04977 −0.524886 0.851173i \(-0.675892\pi\)
−0.524886 + 0.851173i \(0.675892\pi\)
\(308\) 0 0
\(309\) −44.9604 −2.55771
\(310\) 0 0
\(311\) 19.1747 1.08730 0.543648 0.839313i \(-0.317043\pi\)
0.543648 + 0.839313i \(0.317043\pi\)
\(312\) 0 0
\(313\) −14.2274 −0.804183 −0.402091 0.915600i \(-0.631717\pi\)
−0.402091 + 0.915600i \(0.631717\pi\)
\(314\) 0 0
\(315\) 14.1609 0.797874
\(316\) 0 0
\(317\) −0.777938 −0.0436934 −0.0218467 0.999761i \(-0.506955\pi\)
−0.0218467 + 0.999761i \(0.506955\pi\)
\(318\) 0 0
\(319\) 45.5037 2.54772
\(320\) 0 0
\(321\) 7.60156 0.424278
\(322\) 0 0
\(323\) 1.29720 0.0721782
\(324\) 0 0
\(325\) 1.00190 0.0555755
\(326\) 0 0
\(327\) −0.00388283 −0.000214721 0
\(328\) 0 0
\(329\) 42.5435 2.34550
\(330\) 0 0
\(331\) 16.3988 0.901359 0.450680 0.892686i \(-0.351182\pi\)
0.450680 + 0.892686i \(0.351182\pi\)
\(332\) 0 0
\(333\) −38.4584 −2.10751
\(334\) 0 0
\(335\) 2.44525 0.133599
\(336\) 0 0
\(337\) 24.6109 1.34064 0.670320 0.742072i \(-0.266157\pi\)
0.670320 + 0.742072i \(0.266157\pi\)
\(338\) 0 0
\(339\) 5.69563 0.309344
\(340\) 0 0
\(341\) −5.90045 −0.319527
\(342\) 0 0
\(343\) 6.40629 0.345907
\(344\) 0 0
\(345\) −8.43584 −0.454170
\(346\) 0 0
\(347\) −13.4635 −0.722760 −0.361380 0.932419i \(-0.617694\pi\)
−0.361380 + 0.932419i \(0.617694\pi\)
\(348\) 0 0
\(349\) 2.83422 0.151712 0.0758560 0.997119i \(-0.475831\pi\)
0.0758560 + 0.997119i \(0.475831\pi\)
\(350\) 0 0
\(351\) −2.59311 −0.138410
\(352\) 0 0
\(353\) −5.02227 −0.267308 −0.133654 0.991028i \(-0.542671\pi\)
−0.133654 + 0.991028i \(0.542671\pi\)
\(354\) 0 0
\(355\) 4.65008 0.246801
\(356\) 0 0
\(357\) −14.1880 −0.750909
\(358\) 0 0
\(359\) 23.6898 1.25030 0.625149 0.780505i \(-0.285038\pi\)
0.625149 + 0.780505i \(0.285038\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 38.8223 2.03764
\(364\) 0 0
\(365\) 1.83518 0.0960577
\(366\) 0 0
\(367\) 16.9338 0.883938 0.441969 0.897030i \(-0.354280\pi\)
0.441969 + 0.897030i \(0.354280\pi\)
\(368\) 0 0
\(369\) 3.07598 0.160129
\(370\) 0 0
\(371\) 8.94244 0.464268
\(372\) 0 0
\(373\) 12.8599 0.665860 0.332930 0.942952i \(-0.391963\pi\)
0.332930 + 0.942952i \(0.391963\pi\)
\(374\) 0 0
\(375\) 17.9724 0.928089
\(376\) 0 0
\(377\) 2.02896 0.104497
\(378\) 0 0
\(379\) 20.7810 1.06745 0.533723 0.845659i \(-0.320793\pi\)
0.533723 + 0.845659i \(0.320793\pi\)
\(380\) 0 0
\(381\) 50.5758 2.59108
\(382\) 0 0
\(383\) 15.6170 0.797993 0.398996 0.916953i \(-0.369359\pi\)
0.398996 + 0.916953i \(0.369359\pi\)
\(384\) 0 0
\(385\) −10.0173 −0.510531
\(386\) 0 0
\(387\) 5.42960 0.276002
\(388\) 0 0
\(389\) 10.1133 0.512764 0.256382 0.966576i \(-0.417470\pi\)
0.256382 + 0.966576i \(0.417470\pi\)
\(390\) 0 0
\(391\) 5.87392 0.297057
\(392\) 0 0
\(393\) 34.9606 1.76353
\(394\) 0 0
\(395\) −5.98381 −0.301078
\(396\) 0 0
\(397\) −4.60677 −0.231207 −0.115604 0.993295i \(-0.536880\pi\)
−0.115604 + 0.993295i \(0.536880\pi\)
\(398\) 0 0
\(399\) −10.9374 −0.547554
\(400\) 0 0
\(401\) −4.00112 −0.199806 −0.0999032 0.994997i \(-0.531853\pi\)
−0.0999032 + 0.994997i \(0.531853\pi\)
\(402\) 0 0
\(403\) −0.263095 −0.0131057
\(404\) 0 0
\(405\) −10.2260 −0.508135
\(406\) 0 0
\(407\) 27.2054 1.34852
\(408\) 0 0
\(409\) 14.5111 0.717529 0.358765 0.933428i \(-0.383198\pi\)
0.358765 + 0.933428i \(0.383198\pi\)
\(410\) 0 0
\(411\) −35.0196 −1.72739
\(412\) 0 0
\(413\) −9.59745 −0.472260
\(414\) 0 0
\(415\) −6.90843 −0.339122
\(416\) 0 0
\(417\) −40.9370 −2.00470
\(418\) 0 0
\(419\) −5.02078 −0.245281 −0.122641 0.992451i \(-0.539136\pi\)
−0.122641 + 0.992451i \(0.539136\pi\)
\(420\) 0 0
\(421\) 4.10152 0.199896 0.0999479 0.994993i \(-0.468132\pi\)
0.0999479 + 0.994993i \(0.468132\pi\)
\(422\) 0 0
\(423\) −83.3800 −4.05407
\(424\) 0 0
\(425\) −6.02825 −0.292413
\(426\) 0 0
\(427\) 27.0867 1.31082
\(428\) 0 0
\(429\) 3.26925 0.157841
\(430\) 0 0
\(431\) 16.2920 0.784760 0.392380 0.919803i \(-0.371652\pi\)
0.392380 + 0.919803i \(0.371652\pi\)
\(432\) 0 0
\(433\) 7.44858 0.357956 0.178978 0.983853i \(-0.442721\pi\)
0.178978 + 0.983853i \(0.442721\pi\)
\(434\) 0 0
\(435\) 17.5324 0.840612
\(436\) 0 0
\(437\) 4.52815 0.216611
\(438\) 0 0
\(439\) 15.8898 0.758379 0.379189 0.925319i \(-0.376203\pi\)
0.379189 + 0.925319i \(0.376203\pi\)
\(440\) 0 0
\(441\) 35.2908 1.68052
\(442\) 0 0
\(443\) −15.1742 −0.720949 −0.360474 0.932769i \(-0.617385\pi\)
−0.360474 + 0.932769i \(0.617385\pi\)
\(444\) 0 0
\(445\) 8.14379 0.386053
\(446\) 0 0
\(447\) −45.1007 −2.13319
\(448\) 0 0
\(449\) −24.3909 −1.15108 −0.575538 0.817775i \(-0.695207\pi\)
−0.575538 + 0.817775i \(0.695207\pi\)
\(450\) 0 0
\(451\) −2.17594 −0.102461
\(452\) 0 0
\(453\) 54.4082 2.55632
\(454\) 0 0
\(455\) −0.446664 −0.0209399
\(456\) 0 0
\(457\) 10.6860 0.499869 0.249934 0.968263i \(-0.419591\pi\)
0.249934 + 0.968263i \(0.419591\pi\)
\(458\) 0 0
\(459\) 15.6022 0.728250
\(460\) 0 0
\(461\) −21.8232 −1.01641 −0.508205 0.861236i \(-0.669691\pi\)
−0.508205 + 0.861236i \(0.669691\pi\)
\(462\) 0 0
\(463\) −0.0258442 −0.00120108 −0.000600541 1.00000i \(-0.500191\pi\)
−0.000600541 1.00000i \(0.500191\pi\)
\(464\) 0 0
\(465\) −2.27341 −0.105427
\(466\) 0 0
\(467\) 4.66985 0.216095 0.108047 0.994146i \(-0.465540\pi\)
0.108047 + 0.994146i \(0.465540\pi\)
\(468\) 0 0
\(469\) 14.3559 0.662893
\(470\) 0 0
\(471\) 30.2099 1.39200
\(472\) 0 0
\(473\) −3.84088 −0.176604
\(474\) 0 0
\(475\) −4.64712 −0.213224
\(476\) 0 0
\(477\) −17.5261 −0.802464
\(478\) 0 0
\(479\) 30.5428 1.39554 0.697768 0.716324i \(-0.254177\pi\)
0.697768 + 0.716324i \(0.254177\pi\)
\(480\) 0 0
\(481\) 1.21306 0.0553108
\(482\) 0 0
\(483\) −49.5261 −2.25352
\(484\) 0 0
\(485\) −1.24063 −0.0563342
\(486\) 0 0
\(487\) −2.37133 −0.107455 −0.0537277 0.998556i \(-0.517110\pi\)
−0.0537277 + 0.998556i \(0.517110\pi\)
\(488\) 0 0
\(489\) −70.6770 −3.19613
\(490\) 0 0
\(491\) 41.3206 1.86477 0.932385 0.361466i \(-0.117724\pi\)
0.932385 + 0.361466i \(0.117724\pi\)
\(492\) 0 0
\(493\) −12.2079 −0.549815
\(494\) 0 0
\(495\) 19.6328 0.882427
\(496\) 0 0
\(497\) 27.3002 1.22458
\(498\) 0 0
\(499\) 15.8610 0.710035 0.355018 0.934860i \(-0.384475\pi\)
0.355018 + 0.934860i \(0.384475\pi\)
\(500\) 0 0
\(501\) 51.7798 2.31335
\(502\) 0 0
\(503\) −18.6879 −0.833254 −0.416627 0.909078i \(-0.636788\pi\)
−0.416627 + 0.909078i \(0.636788\pi\)
\(504\) 0 0
\(505\) 1.64593 0.0732431
\(506\) 0 0
\(507\) −40.6237 −1.80416
\(508\) 0 0
\(509\) −33.2172 −1.47233 −0.736163 0.676804i \(-0.763364\pi\)
−0.736163 + 0.676804i \(0.763364\pi\)
\(510\) 0 0
\(511\) 10.7742 0.476622
\(512\) 0 0
\(513\) 12.0276 0.531032
\(514\) 0 0
\(515\) 8.51637 0.375276
\(516\) 0 0
\(517\) 58.9827 2.59406
\(518\) 0 0
\(519\) −16.1482 −0.708826
\(520\) 0 0
\(521\) 10.9831 0.481178 0.240589 0.970627i \(-0.422659\pi\)
0.240589 + 0.970627i \(0.422659\pi\)
\(522\) 0 0
\(523\) 8.00433 0.350005 0.175002 0.984568i \(-0.444007\pi\)
0.175002 + 0.984568i \(0.444007\pi\)
\(524\) 0 0
\(525\) 50.8273 2.21829
\(526\) 0 0
\(527\) 1.58299 0.0689561
\(528\) 0 0
\(529\) −2.49589 −0.108517
\(530\) 0 0
\(531\) 18.8098 0.816277
\(532\) 0 0
\(533\) −0.0970230 −0.00420253
\(534\) 0 0
\(535\) −1.43988 −0.0622516
\(536\) 0 0
\(537\) −29.2154 −1.26074
\(538\) 0 0
\(539\) −24.9646 −1.07530
\(540\) 0 0
\(541\) 24.4359 1.05058 0.525291 0.850923i \(-0.323956\pi\)
0.525291 + 0.850923i \(0.323956\pi\)
\(542\) 0 0
\(543\) 16.1672 0.693800
\(544\) 0 0
\(545\) 0.000735485 0 3.15047e−5 0
\(546\) 0 0
\(547\) 5.25284 0.224595 0.112298 0.993675i \(-0.464179\pi\)
0.112298 + 0.993675i \(0.464179\pi\)
\(548\) 0 0
\(549\) −53.0866 −2.26568
\(550\) 0 0
\(551\) −9.41093 −0.400919
\(552\) 0 0
\(553\) −35.1305 −1.49390
\(554\) 0 0
\(555\) 10.4821 0.444940
\(556\) 0 0
\(557\) 3.56791 0.151177 0.0755886 0.997139i \(-0.475916\pi\)
0.0755886 + 0.997139i \(0.475916\pi\)
\(558\) 0 0
\(559\) −0.171261 −0.00724357
\(560\) 0 0
\(561\) −19.6704 −0.830485
\(562\) 0 0
\(563\) 17.0555 0.718805 0.359403 0.933183i \(-0.382980\pi\)
0.359403 + 0.933183i \(0.382980\pi\)
\(564\) 0 0
\(565\) −1.07886 −0.0453881
\(566\) 0 0
\(567\) −60.0361 −2.52128
\(568\) 0 0
\(569\) 0.858816 0.0360034 0.0180017 0.999838i \(-0.494270\pi\)
0.0180017 + 0.999838i \(0.494270\pi\)
\(570\) 0 0
\(571\) 40.5440 1.69671 0.848356 0.529426i \(-0.177593\pi\)
0.848356 + 0.529426i \(0.177593\pi\)
\(572\) 0 0
\(573\) 46.6671 1.94954
\(574\) 0 0
\(575\) −21.0428 −0.877546
\(576\) 0 0
\(577\) 36.5405 1.52120 0.760600 0.649220i \(-0.224905\pi\)
0.760600 + 0.649220i \(0.224905\pi\)
\(578\) 0 0
\(579\) −66.7221 −2.77288
\(580\) 0 0
\(581\) −40.5588 −1.68266
\(582\) 0 0
\(583\) 12.3979 0.513468
\(584\) 0 0
\(585\) 0.875406 0.0361936
\(586\) 0 0
\(587\) 30.4022 1.25483 0.627416 0.778684i \(-0.284113\pi\)
0.627416 + 0.778684i \(0.284113\pi\)
\(588\) 0 0
\(589\) 1.22031 0.0502821
\(590\) 0 0
\(591\) 11.5776 0.476237
\(592\) 0 0
\(593\) −16.3814 −0.672703 −0.336351 0.941737i \(-0.609193\pi\)
−0.336351 + 0.941737i \(0.609193\pi\)
\(594\) 0 0
\(595\) 2.68749 0.110176
\(596\) 0 0
\(597\) 36.6476 1.49989
\(598\) 0 0
\(599\) −2.24749 −0.0918298 −0.0459149 0.998945i \(-0.514620\pi\)
−0.0459149 + 0.998945i \(0.514620\pi\)
\(600\) 0 0
\(601\) 33.1805 1.35346 0.676730 0.736231i \(-0.263396\pi\)
0.676730 + 0.736231i \(0.263396\pi\)
\(602\) 0 0
\(603\) −28.1358 −1.14578
\(604\) 0 0
\(605\) −7.35371 −0.298971
\(606\) 0 0
\(607\) −35.7847 −1.45246 −0.726228 0.687454i \(-0.758728\pi\)
−0.726228 + 0.687454i \(0.758728\pi\)
\(608\) 0 0
\(609\) 102.931 4.17098
\(610\) 0 0
\(611\) 2.62998 0.106398
\(612\) 0 0
\(613\) −42.2856 −1.70790 −0.853950 0.520355i \(-0.825800\pi\)
−0.853950 + 0.520355i \(0.825800\pi\)
\(614\) 0 0
\(615\) −0.838380 −0.0338067
\(616\) 0 0
\(617\) 26.2079 1.05509 0.527545 0.849527i \(-0.323113\pi\)
0.527545 + 0.849527i \(0.323113\pi\)
\(618\) 0 0
\(619\) −40.7494 −1.63786 −0.818929 0.573896i \(-0.805432\pi\)
−0.818929 + 0.573896i \(0.805432\pi\)
\(620\) 0 0
\(621\) 54.4628 2.18552
\(622\) 0 0
\(623\) 47.8115 1.91553
\(624\) 0 0
\(625\) 19.8313 0.793250
\(626\) 0 0
\(627\) −15.1637 −0.605581
\(628\) 0 0
\(629\) −7.29874 −0.291020
\(630\) 0 0
\(631\) 42.3804 1.68714 0.843569 0.537021i \(-0.180450\pi\)
0.843569 + 0.537021i \(0.180450\pi\)
\(632\) 0 0
\(633\) −23.8850 −0.949343
\(634\) 0 0
\(635\) −9.58005 −0.380173
\(636\) 0 0
\(637\) −1.11315 −0.0441045
\(638\) 0 0
\(639\) −53.5051 −2.11663
\(640\) 0 0
\(641\) −40.0356 −1.58131 −0.790656 0.612261i \(-0.790260\pi\)
−0.790656 + 0.612261i \(0.790260\pi\)
\(642\) 0 0
\(643\) 2.25369 0.0888767 0.0444384 0.999012i \(-0.485850\pi\)
0.0444384 + 0.999012i \(0.485850\pi\)
\(644\) 0 0
\(645\) −1.47987 −0.0582700
\(646\) 0 0
\(647\) −6.89272 −0.270981 −0.135490 0.990779i \(-0.543261\pi\)
−0.135490 + 0.990779i \(0.543261\pi\)
\(648\) 0 0
\(649\) −13.3060 −0.522307
\(650\) 0 0
\(651\) −13.3470 −0.523111
\(652\) 0 0
\(653\) 5.75398 0.225170 0.112585 0.993642i \(-0.464087\pi\)
0.112585 + 0.993642i \(0.464087\pi\)
\(654\) 0 0
\(655\) −6.62222 −0.258751
\(656\) 0 0
\(657\) −21.1161 −0.823817
\(658\) 0 0
\(659\) −34.1024 −1.32844 −0.664220 0.747537i \(-0.731236\pi\)
−0.664220 + 0.747537i \(0.731236\pi\)
\(660\) 0 0
\(661\) −46.3767 −1.80384 −0.901922 0.431900i \(-0.857843\pi\)
−0.901922 + 0.431900i \(0.857843\pi\)
\(662\) 0 0
\(663\) −0.877084 −0.0340631
\(664\) 0 0
\(665\) 2.07176 0.0803392
\(666\) 0 0
\(667\) −42.6141 −1.65002
\(668\) 0 0
\(669\) 6.13038 0.237014
\(670\) 0 0
\(671\) 37.5533 1.44973
\(672\) 0 0
\(673\) −29.1579 −1.12395 −0.561977 0.827153i \(-0.689959\pi\)
−0.561977 + 0.827153i \(0.689959\pi\)
\(674\) 0 0
\(675\) −55.8937 −2.15135
\(676\) 0 0
\(677\) 10.0436 0.386006 0.193003 0.981198i \(-0.438177\pi\)
0.193003 + 0.981198i \(0.438177\pi\)
\(678\) 0 0
\(679\) −7.28365 −0.279521
\(680\) 0 0
\(681\) −41.9328 −1.60687
\(682\) 0 0
\(683\) −25.5866 −0.979042 −0.489521 0.871991i \(-0.662828\pi\)
−0.489521 + 0.871991i \(0.662828\pi\)
\(684\) 0 0
\(685\) 6.63341 0.253449
\(686\) 0 0
\(687\) 36.3161 1.38554
\(688\) 0 0
\(689\) 0.552809 0.0210604
\(690\) 0 0
\(691\) −42.2386 −1.60683 −0.803417 0.595417i \(-0.796987\pi\)
−0.803417 + 0.595417i \(0.796987\pi\)
\(692\) 0 0
\(693\) 115.262 4.37845
\(694\) 0 0
\(695\) 7.75428 0.294136
\(696\) 0 0
\(697\) 0.583768 0.0221118
\(698\) 0 0
\(699\) −4.98239 −0.188451
\(700\) 0 0
\(701\) −43.9811 −1.66114 −0.830572 0.556911i \(-0.811986\pi\)
−0.830572 + 0.556911i \(0.811986\pi\)
\(702\) 0 0
\(703\) −5.62653 −0.212208
\(704\) 0 0
\(705\) 22.7258 0.855902
\(706\) 0 0
\(707\) 9.66314 0.363420
\(708\) 0 0
\(709\) 11.1307 0.418023 0.209012 0.977913i \(-0.432975\pi\)
0.209012 + 0.977913i \(0.432975\pi\)
\(710\) 0 0
\(711\) 68.8514 2.58213
\(712\) 0 0
\(713\) 5.52575 0.206941
\(714\) 0 0
\(715\) −0.619259 −0.0231590
\(716\) 0 0
\(717\) −74.7081 −2.79003
\(718\) 0 0
\(719\) −17.8961 −0.667413 −0.333706 0.942677i \(-0.608299\pi\)
−0.333706 + 0.942677i \(0.608299\pi\)
\(720\) 0 0
\(721\) 49.9989 1.86206
\(722\) 0 0
\(723\) 71.6771 2.66570
\(724\) 0 0
\(725\) 43.7337 1.62423
\(726\) 0 0
\(727\) −35.8772 −1.33061 −0.665306 0.746571i \(-0.731699\pi\)
−0.665306 + 0.746571i \(0.731699\pi\)
\(728\) 0 0
\(729\) 4.50357 0.166799
\(730\) 0 0
\(731\) 1.03044 0.0381123
\(732\) 0 0
\(733\) 24.4618 0.903516 0.451758 0.892141i \(-0.350797\pi\)
0.451758 + 0.892141i \(0.350797\pi\)
\(734\) 0 0
\(735\) −9.61875 −0.354793
\(736\) 0 0
\(737\) 19.9032 0.733142
\(738\) 0 0
\(739\) −26.7820 −0.985193 −0.492596 0.870258i \(-0.663952\pi\)
−0.492596 + 0.870258i \(0.663952\pi\)
\(740\) 0 0
\(741\) −0.676135 −0.0248384
\(742\) 0 0
\(743\) 23.7835 0.872534 0.436267 0.899817i \(-0.356300\pi\)
0.436267 + 0.899817i \(0.356300\pi\)
\(744\) 0 0
\(745\) 8.54295 0.312990
\(746\) 0 0
\(747\) 79.4903 2.90840
\(748\) 0 0
\(749\) −8.45344 −0.308882
\(750\) 0 0
\(751\) −22.4303 −0.818494 −0.409247 0.912424i \(-0.634209\pi\)
−0.409247 + 0.912424i \(0.634209\pi\)
\(752\) 0 0
\(753\) 8.26442 0.301172
\(754\) 0 0
\(755\) −10.3060 −0.375073
\(756\) 0 0
\(757\) −32.2421 −1.17186 −0.585930 0.810362i \(-0.699271\pi\)
−0.585930 + 0.810362i \(0.699271\pi\)
\(758\) 0 0
\(759\) −68.6635 −2.49233
\(760\) 0 0
\(761\) −16.3559 −0.592902 −0.296451 0.955048i \(-0.595803\pi\)
−0.296451 + 0.955048i \(0.595803\pi\)
\(762\) 0 0
\(763\) 0.00431797 0.000156321 0
\(764\) 0 0
\(765\) −5.26714 −0.190434
\(766\) 0 0
\(767\) −0.593302 −0.0214229
\(768\) 0 0
\(769\) −19.6832 −0.709794 −0.354897 0.934905i \(-0.615484\pi\)
−0.354897 + 0.934905i \(0.615484\pi\)
\(770\) 0 0
\(771\) 62.9038 2.26542
\(772\) 0 0
\(773\) −33.0819 −1.18987 −0.594936 0.803773i \(-0.702823\pi\)
−0.594936 + 0.803773i \(0.702823\pi\)
\(774\) 0 0
\(775\) −5.67093 −0.203706
\(776\) 0 0
\(777\) 61.5395 2.20772
\(778\) 0 0
\(779\) 0.450021 0.0161237
\(780\) 0 0
\(781\) 37.8494 1.35436
\(782\) 0 0
\(783\) −113.191 −4.04512
\(784\) 0 0
\(785\) −5.72235 −0.204240
\(786\) 0 0
\(787\) 35.1617 1.25338 0.626690 0.779269i \(-0.284409\pi\)
0.626690 + 0.779269i \(0.284409\pi\)
\(788\) 0 0
\(789\) 12.6594 0.450688
\(790\) 0 0
\(791\) −6.33392 −0.225208
\(792\) 0 0
\(793\) 1.67446 0.0594619
\(794\) 0 0
\(795\) 4.77685 0.169417
\(796\) 0 0
\(797\) −3.75101 −0.132868 −0.0664338 0.997791i \(-0.521162\pi\)
−0.0664338 + 0.997791i \(0.521162\pi\)
\(798\) 0 0
\(799\) −15.8241 −0.559815
\(800\) 0 0
\(801\) −93.7046 −3.31089
\(802\) 0 0
\(803\) 14.9374 0.527131
\(804\) 0 0
\(805\) 9.38121 0.330644
\(806\) 0 0
\(807\) −44.8763 −1.57972
\(808\) 0 0
\(809\) −23.9158 −0.840836 −0.420418 0.907331i \(-0.638117\pi\)
−0.420418 + 0.907331i \(0.638117\pi\)
\(810\) 0 0
\(811\) −52.0538 −1.82785 −0.913927 0.405878i \(-0.866966\pi\)
−0.913927 + 0.405878i \(0.866966\pi\)
\(812\) 0 0
\(813\) −30.8918 −1.08342
\(814\) 0 0
\(815\) 13.3876 0.468947
\(816\) 0 0
\(817\) 0.794359 0.0277911
\(818\) 0 0
\(819\) 5.13943 0.179586
\(820\) 0 0
\(821\) 0.703590 0.0245555 0.0122777 0.999925i \(-0.496092\pi\)
0.0122777 + 0.999925i \(0.496092\pi\)
\(822\) 0 0
\(823\) 23.4559 0.817622 0.408811 0.912619i \(-0.365943\pi\)
0.408811 + 0.912619i \(0.365943\pi\)
\(824\) 0 0
\(825\) 70.4676 2.45337
\(826\) 0 0
\(827\) 29.5282 1.02680 0.513399 0.858150i \(-0.328386\pi\)
0.513399 + 0.858150i \(0.328386\pi\)
\(828\) 0 0
\(829\) 4.42946 0.153841 0.0769207 0.997037i \(-0.475491\pi\)
0.0769207 + 0.997037i \(0.475491\pi\)
\(830\) 0 0
\(831\) 53.2014 1.84554
\(832\) 0 0
\(833\) 6.69759 0.232058
\(834\) 0 0
\(835\) −9.80811 −0.339424
\(836\) 0 0
\(837\) 14.6774 0.507326
\(838\) 0 0
\(839\) −36.4506 −1.25841 −0.629207 0.777237i \(-0.716620\pi\)
−0.629207 + 0.777237i \(0.716620\pi\)
\(840\) 0 0
\(841\) 59.5656 2.05399
\(842\) 0 0
\(843\) 32.0127 1.10257
\(844\) 0 0
\(845\) 7.69492 0.264713
\(846\) 0 0
\(847\) −43.1730 −1.48344
\(848\) 0 0
\(849\) −58.8677 −2.02033
\(850\) 0 0
\(851\) −25.4777 −0.873366
\(852\) 0 0
\(853\) −52.7043 −1.80456 −0.902281 0.431149i \(-0.858108\pi\)
−0.902281 + 0.431149i \(0.858108\pi\)
\(854\) 0 0
\(855\) −4.06039 −0.138862
\(856\) 0 0
\(857\) −13.7199 −0.468662 −0.234331 0.972157i \(-0.575290\pi\)
−0.234331 + 0.972157i \(0.575290\pi\)
\(858\) 0 0
\(859\) 55.0757 1.87916 0.939580 0.342330i \(-0.111216\pi\)
0.939580 + 0.342330i \(0.111216\pi\)
\(860\) 0 0
\(861\) −4.92206 −0.167743
\(862\) 0 0
\(863\) −38.3384 −1.30505 −0.652527 0.757766i \(-0.726291\pi\)
−0.652527 + 0.757766i \(0.726291\pi\)
\(864\) 0 0
\(865\) 3.05878 0.104002
\(866\) 0 0
\(867\) −48.0367 −1.63141
\(868\) 0 0
\(869\) −48.7053 −1.65221
\(870\) 0 0
\(871\) 0.887462 0.0300705
\(872\) 0 0
\(873\) 14.2751 0.483138
\(874\) 0 0
\(875\) −19.9865 −0.675666
\(876\) 0 0
\(877\) −21.4788 −0.725288 −0.362644 0.931928i \(-0.618126\pi\)
−0.362644 + 0.931928i \(0.618126\pi\)
\(878\) 0 0
\(879\) −62.1800 −2.09728
\(880\) 0 0
\(881\) −11.2383 −0.378629 −0.189314 0.981917i \(-0.560627\pi\)
−0.189314 + 0.981917i \(0.560627\pi\)
\(882\) 0 0
\(883\) 15.7137 0.528809 0.264405 0.964412i \(-0.414825\pi\)
0.264405 + 0.964412i \(0.414825\pi\)
\(884\) 0 0
\(885\) −5.12674 −0.172334
\(886\) 0 0
\(887\) 4.48008 0.150426 0.0752131 0.997167i \(-0.476036\pi\)
0.0752131 + 0.997167i \(0.476036\pi\)
\(888\) 0 0
\(889\) −56.2437 −1.88635
\(890\) 0 0
\(891\) −83.2347 −2.78847
\(892\) 0 0
\(893\) −12.1986 −0.408211
\(894\) 0 0
\(895\) 5.53396 0.184980
\(896\) 0 0
\(897\) −3.06164 −0.102225
\(898\) 0 0
\(899\) −11.4843 −0.383022
\(900\) 0 0
\(901\) −3.32614 −0.110810
\(902\) 0 0
\(903\) −8.68822 −0.289126
\(904\) 0 0
\(905\) −3.06238 −0.101797
\(906\) 0 0
\(907\) 6.37021 0.211519 0.105760 0.994392i \(-0.466273\pi\)
0.105760 + 0.994392i \(0.466273\pi\)
\(908\) 0 0
\(909\) −18.9386 −0.628152
\(910\) 0 0
\(911\) 30.5275 1.01142 0.505711 0.862703i \(-0.331230\pi\)
0.505711 + 0.862703i \(0.331230\pi\)
\(912\) 0 0
\(913\) −56.2312 −1.86098
\(914\) 0 0
\(915\) 14.4691 0.478334
\(916\) 0 0
\(917\) −38.8785 −1.28388
\(918\) 0 0
\(919\) −48.0650 −1.58552 −0.792758 0.609536i \(-0.791356\pi\)
−0.792758 + 0.609536i \(0.791356\pi\)
\(920\) 0 0
\(921\) −57.6840 −1.90075
\(922\) 0 0
\(923\) 1.68767 0.0555502
\(924\) 0 0
\(925\) 26.1471 0.859712
\(926\) 0 0
\(927\) −97.9917 −3.21847
\(928\) 0 0
\(929\) 47.2953 1.55171 0.775854 0.630912i \(-0.217319\pi\)
0.775854 + 0.630912i \(0.217319\pi\)
\(930\) 0 0
\(931\) 5.16310 0.169214
\(932\) 0 0
\(933\) 60.1339 1.96870
\(934\) 0 0
\(935\) 3.72596 0.121852
\(936\) 0 0
\(937\) 8.30237 0.271227 0.135613 0.990762i \(-0.456699\pi\)
0.135613 + 0.990762i \(0.456699\pi\)
\(938\) 0 0
\(939\) −44.6189 −1.45608
\(940\) 0 0
\(941\) 31.3955 1.02347 0.511733 0.859145i \(-0.329004\pi\)
0.511733 + 0.859145i \(0.329004\pi\)
\(942\) 0 0
\(943\) 2.03776 0.0663586
\(944\) 0 0
\(945\) 24.9183 0.810591
\(946\) 0 0
\(947\) 2.14805 0.0698024 0.0349012 0.999391i \(-0.488888\pi\)
0.0349012 + 0.999391i \(0.488888\pi\)
\(948\) 0 0
\(949\) 0.666046 0.0216208
\(950\) 0 0
\(951\) −2.43970 −0.0791127
\(952\) 0 0
\(953\) −25.3661 −0.821689 −0.410844 0.911705i \(-0.634766\pi\)
−0.410844 + 0.911705i \(0.634766\pi\)
\(954\) 0 0
\(955\) −8.83965 −0.286044
\(956\) 0 0
\(957\) 142.705 4.61299
\(958\) 0 0
\(959\) 38.9442 1.25757
\(960\) 0 0
\(961\) −29.5108 −0.951963
\(962\) 0 0
\(963\) 16.5677 0.533887
\(964\) 0 0
\(965\) 12.6385 0.406847
\(966\) 0 0
\(967\) −12.8395 −0.412892 −0.206446 0.978458i \(-0.566190\pi\)
−0.206446 + 0.978458i \(0.566190\pi\)
\(968\) 0 0
\(969\) 4.06817 0.130688
\(970\) 0 0
\(971\) −8.77909 −0.281734 −0.140867 0.990028i \(-0.544989\pi\)
−0.140867 + 0.990028i \(0.544989\pi\)
\(972\) 0 0
\(973\) 45.5247 1.45945
\(974\) 0 0
\(975\) 3.14208 0.100627
\(976\) 0 0
\(977\) −45.9701 −1.47071 −0.735357 0.677680i \(-0.762986\pi\)
−0.735357 + 0.677680i \(0.762986\pi\)
\(978\) 0 0
\(979\) 66.2864 2.11852
\(980\) 0 0
\(981\) −0.00846269 −0.000270193 0
\(982\) 0 0
\(983\) −51.4992 −1.64257 −0.821284 0.570519i \(-0.806742\pi\)
−0.821284 + 0.570519i \(0.806742\pi\)
\(984\) 0 0
\(985\) −2.19302 −0.0698753
\(986\) 0 0
\(987\) 133.421 4.24684
\(988\) 0 0
\(989\) 3.59697 0.114377
\(990\) 0 0
\(991\) 29.4762 0.936343 0.468171 0.883638i \(-0.344913\pi\)
0.468171 + 0.883638i \(0.344913\pi\)
\(992\) 0 0
\(993\) 51.4284 1.63203
\(994\) 0 0
\(995\) −6.94178 −0.220069
\(996\) 0 0
\(997\) 54.3591 1.72157 0.860786 0.508968i \(-0.169973\pi\)
0.860786 + 0.508968i \(0.169973\pi\)
\(998\) 0 0
\(999\) −67.6737 −2.14110
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 4864.2.a.bn.1.8 8
4.3 odd 2 4864.2.a.bo.1.1 8
8.3 odd 2 4864.2.a.bq.1.8 8
8.5 even 2 4864.2.a.bp.1.1 8
16.3 odd 4 152.2.c.b.77.6 yes 16
16.5 even 4 608.2.c.b.305.1 16
16.11 odd 4 152.2.c.b.77.5 16
16.13 even 4 608.2.c.b.305.16 16
48.5 odd 4 5472.2.g.b.2737.9 16
48.11 even 4 1368.2.g.b.685.12 16
48.29 odd 4 5472.2.g.b.2737.8 16
48.35 even 4 1368.2.g.b.685.11 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.c.b.77.5 16 16.11 odd 4
152.2.c.b.77.6 yes 16 16.3 odd 4
608.2.c.b.305.1 16 16.5 even 4
608.2.c.b.305.16 16 16.13 even 4
1368.2.g.b.685.11 16 48.35 even 4
1368.2.g.b.685.12 16 48.11 even 4
4864.2.a.bn.1.8 8 1.1 even 1 trivial
4864.2.a.bo.1.1 8 4.3 odd 2
4864.2.a.bp.1.1 8 8.5 even 2
4864.2.a.bq.1.8 8 8.3 odd 2
5472.2.g.b.2737.8 16 48.29 odd 4
5472.2.g.b.2737.9 16 48.5 odd 4