Properties

Label 7168.2.a.bb.1.3
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.2367681689\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.9433055232.1
Defining polynomial: \(x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 9 x^{4} - 76 x^{3} - 4 x^{2} + 48 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1792)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.21236\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.78579 q^{3} -4.18248 q^{5} +1.00000 q^{7} +0.189043 q^{9} +O(q^{10})\) \(q-1.78579 q^{3} -4.18248 q^{5} +1.00000 q^{7} +0.189043 q^{9} -4.50362 q^{11} -4.84427 q^{13} +7.46903 q^{15} +5.13834 q^{17} +2.12835 q^{19} -1.78579 q^{21} -7.11888 q^{23} +12.4932 q^{25} +5.01978 q^{27} -5.43932 q^{29} -0.831138 q^{31} +8.04251 q^{33} -4.18248 q^{35} +7.98492 q^{37} +8.65084 q^{39} -2.22639 q^{41} +2.28804 q^{43} -0.790669 q^{45} +7.83759 q^{47} +1.00000 q^{49} -9.17599 q^{51} +7.90235 q^{53} +18.8363 q^{55} -3.80079 q^{57} -2.62811 q^{59} -2.33370 q^{61} +0.189043 q^{63} +20.2611 q^{65} +8.16822 q^{67} +12.7128 q^{69} -6.04851 q^{71} -7.67177 q^{73} -22.3101 q^{75} -4.50362 q^{77} -1.90198 q^{79} -9.53139 q^{81} +11.2843 q^{83} -21.4910 q^{85} +9.71349 q^{87} -2.49938 q^{89} -4.84427 q^{91} +1.48424 q^{93} -8.90180 q^{95} -1.98784 q^{97} -0.851377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 8q^{5} + 8q^{7} + 12q^{9} - 12q^{11} - 20q^{13} + 4q^{17} - 4q^{19} + 8q^{23} + 12q^{25} + 12q^{27} - 8q^{29} - 4q^{31} + 8q^{33} - 8q^{35} - 8q^{37} - 16q^{39} - 12q^{41} + 4q^{43} - 52q^{45} + 20q^{47} + 8q^{49} - 32q^{51} - 40q^{53} + 24q^{55} - 4q^{57} - 4q^{59} + 8q^{61} + 12q^{63} + 36q^{65} - 28q^{67} - 4q^{69} - 16q^{71} + 16q^{73} + 28q^{75} - 12q^{77} + 20q^{81} + 8q^{83} - 16q^{85} + 20q^{87} + 16q^{89} - 20q^{91} - 16q^{93} - 40q^{95} - 36q^{97} + 4q^{99} + O(q^{100}) \)

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
\(3\) −1.78579 −1.03103 −0.515513 0.856882i \(-0.672399\pi\)
−0.515513 + 0.856882i \(0.672399\pi\)
\(4\) 0 0
\(5\) −4.18248 −1.87046 −0.935231 0.354037i \(-0.884809\pi\)
−0.935231 + 0.354037i \(0.884809\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0.189043 0.0630143
\(10\) 0 0
\(11\) −4.50362 −1.35789 −0.678946 0.734188i \(-0.737563\pi\)
−0.678946 + 0.734188i \(0.737563\pi\)
\(12\) 0 0
\(13\) −4.84427 −1.34356 −0.671779 0.740752i \(-0.734469\pi\)
−0.671779 + 0.740752i \(0.734469\pi\)
\(14\) 0 0
\(15\) 7.46903 1.92850
\(16\) 0 0
\(17\) 5.13834 1.24623 0.623115 0.782130i \(-0.285867\pi\)
0.623115 + 0.782130i \(0.285867\pi\)
\(18\) 0 0
\(19\) 2.12835 0.488278 0.244139 0.969740i \(-0.421495\pi\)
0.244139 + 0.969740i \(0.421495\pi\)
\(20\) 0 0
\(21\) −1.78579 −0.389691
\(22\) 0 0
\(23\) −7.11888 −1.48439 −0.742195 0.670184i \(-0.766215\pi\)
−0.742195 + 0.670184i \(0.766215\pi\)
\(24\) 0 0
\(25\) 12.4932 2.49863
\(26\) 0 0
\(27\) 5.01978 0.966056
\(28\) 0 0
\(29\) −5.43932 −1.01006 −0.505029 0.863103i \(-0.668518\pi\)
−0.505029 + 0.863103i \(0.668518\pi\)
\(30\) 0 0
\(31\) −0.831138 −0.149277 −0.0746384 0.997211i \(-0.523780\pi\)
−0.0746384 + 0.997211i \(0.523780\pi\)
\(32\) 0 0
\(33\) 8.04251 1.40002
\(34\) 0 0
\(35\) −4.18248 −0.706969
\(36\) 0 0
\(37\) 7.98492 1.31271 0.656357 0.754451i \(-0.272097\pi\)
0.656357 + 0.754451i \(0.272097\pi\)
\(38\) 0 0
\(39\) 8.65084 1.38524
\(40\) 0 0
\(41\) −2.22639 −0.347704 −0.173852 0.984772i \(-0.555621\pi\)
−0.173852 + 0.984772i \(0.555621\pi\)
\(42\) 0 0
\(43\) 2.28804 0.348923 0.174462 0.984664i \(-0.444182\pi\)
0.174462 + 0.984664i \(0.444182\pi\)
\(44\) 0 0
\(45\) −0.790669 −0.117866
\(46\) 0 0
\(47\) 7.83759 1.14323 0.571615 0.820522i \(-0.306317\pi\)
0.571615 + 0.820522i \(0.306317\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −9.17599 −1.28490
\(52\) 0 0
\(53\) 7.90235 1.08547 0.542736 0.839903i \(-0.317388\pi\)
0.542736 + 0.839903i \(0.317388\pi\)
\(54\) 0 0
\(55\) 18.8363 2.53989
\(56\) 0 0
\(57\) −3.80079 −0.503427
\(58\) 0 0
\(59\) −2.62811 −0.342150 −0.171075 0.985258i \(-0.554724\pi\)
−0.171075 + 0.985258i \(0.554724\pi\)
\(60\) 0 0
\(61\) −2.33370 −0.298799 −0.149400 0.988777i \(-0.547734\pi\)
−0.149400 + 0.988777i \(0.547734\pi\)
\(62\) 0 0
\(63\) 0.189043 0.0238172
\(64\) 0 0
\(65\) 20.2611 2.51307
\(66\) 0 0
\(67\) 8.16822 0.997907 0.498954 0.866629i \(-0.333718\pi\)
0.498954 + 0.866629i \(0.333718\pi\)
\(68\) 0 0
\(69\) 12.7128 1.53044
\(70\) 0 0
\(71\) −6.04851 −0.717826 −0.358913 0.933371i \(-0.616853\pi\)
−0.358913 + 0.933371i \(0.616853\pi\)
\(72\) 0 0
\(73\) −7.67177 −0.897913 −0.448957 0.893554i \(-0.648204\pi\)
−0.448957 + 0.893554i \(0.648204\pi\)
\(74\) 0 0
\(75\) −22.3101 −2.57615
\(76\) 0 0
\(77\) −4.50362 −0.513235
\(78\) 0 0
\(79\) −1.90198 −0.213989 −0.106995 0.994260i \(-0.534123\pi\)
−0.106995 + 0.994260i \(0.534123\pi\)
\(80\) 0 0
\(81\) −9.53139 −1.05904
\(82\) 0 0
\(83\) 11.2843 1.23861 0.619306 0.785149i \(-0.287414\pi\)
0.619306 + 0.785149i \(0.287414\pi\)
\(84\) 0 0
\(85\) −21.4910 −2.33103
\(86\) 0 0
\(87\) 9.71349 1.04139
\(88\) 0 0
\(89\) −2.49938 −0.264934 −0.132467 0.991187i \(-0.542290\pi\)
−0.132467 + 0.991187i \(0.542290\pi\)
\(90\) 0 0
\(91\) −4.84427 −0.507817
\(92\) 0 0
\(93\) 1.48424 0.153908
\(94\) 0 0
\(95\) −8.90180 −0.913306
\(96\) 0 0
\(97\) −1.98784 −0.201834 −0.100917 0.994895i \(-0.532178\pi\)
−0.100917 + 0.994895i \(0.532178\pi\)
\(98\) 0 0
\(99\) −0.851377 −0.0855666
\(100\) 0 0
\(101\) 15.7459 1.56678 0.783390 0.621531i \(-0.213489\pi\)
0.783390 + 0.621531i \(0.213489\pi\)
\(102\) 0 0
\(103\) 15.2106 1.49874 0.749371 0.662150i \(-0.230356\pi\)
0.749371 + 0.662150i \(0.230356\pi\)
\(104\) 0 0
\(105\) 7.46903 0.728903
\(106\) 0 0
\(107\) −1.26941 −0.122718 −0.0613592 0.998116i \(-0.519544\pi\)
−0.0613592 + 0.998116i \(0.519544\pi\)
\(108\) 0 0
\(109\) −4.02383 −0.385413 −0.192707 0.981256i \(-0.561727\pi\)
−0.192707 + 0.981256i \(0.561727\pi\)
\(110\) 0 0
\(111\) −14.2594 −1.35344
\(112\) 0 0
\(113\) 1.66487 0.156618 0.0783089 0.996929i \(-0.475048\pi\)
0.0783089 + 0.996929i \(0.475048\pi\)
\(114\) 0 0
\(115\) 29.7746 2.77650
\(116\) 0 0
\(117\) −0.915774 −0.0846634
\(118\) 0 0
\(119\) 5.13834 0.471031
\(120\) 0 0
\(121\) 9.28258 0.843871
\(122\) 0 0
\(123\) 3.97586 0.358492
\(124\) 0 0
\(125\) −31.3400 −2.80313
\(126\) 0 0
\(127\) 7.86069 0.697523 0.348762 0.937211i \(-0.386602\pi\)
0.348762 + 0.937211i \(0.386602\pi\)
\(128\) 0 0
\(129\) −4.08596 −0.359749
\(130\) 0 0
\(131\) 7.70009 0.672760 0.336380 0.941726i \(-0.390797\pi\)
0.336380 + 0.941726i \(0.390797\pi\)
\(132\) 0 0
\(133\) 2.12835 0.184552
\(134\) 0 0
\(135\) −20.9951 −1.80697
\(136\) 0 0
\(137\) −17.6977 −1.51201 −0.756007 0.654564i \(-0.772852\pi\)
−0.756007 + 0.654564i \(0.772852\pi\)
\(138\) 0 0
\(139\) 16.1930 1.37348 0.686738 0.726905i \(-0.259042\pi\)
0.686738 + 0.726905i \(0.259042\pi\)
\(140\) 0 0
\(141\) −13.9963 −1.17870
\(142\) 0 0
\(143\) 21.8167 1.82441
\(144\) 0 0
\(145\) 22.7499 1.88927
\(146\) 0 0
\(147\) −1.78579 −0.147289
\(148\) 0 0
\(149\) −12.1806 −0.997874 −0.498937 0.866638i \(-0.666276\pi\)
−0.498937 + 0.866638i \(0.666276\pi\)
\(150\) 0 0
\(151\) −17.7449 −1.44406 −0.722029 0.691863i \(-0.756790\pi\)
−0.722029 + 0.691863i \(0.756790\pi\)
\(152\) 0 0
\(153\) 0.971366 0.0785303
\(154\) 0 0
\(155\) 3.47622 0.279217
\(156\) 0 0
\(157\) 13.9789 1.11564 0.557818 0.829963i \(-0.311639\pi\)
0.557818 + 0.829963i \(0.311639\pi\)
\(158\) 0 0
\(159\) −14.1119 −1.11915
\(160\) 0 0
\(161\) −7.11888 −0.561047
\(162\) 0 0
\(163\) 13.5962 1.06494 0.532468 0.846450i \(-0.321265\pi\)
0.532468 + 0.846450i \(0.321265\pi\)
\(164\) 0 0
\(165\) −33.6377 −2.61869
\(166\) 0 0
\(167\) −6.70735 −0.519030 −0.259515 0.965739i \(-0.583563\pi\)
−0.259515 + 0.965739i \(0.583563\pi\)
\(168\) 0 0
\(169\) 10.4669 0.805147
\(170\) 0 0
\(171\) 0.402350 0.0307685
\(172\) 0 0
\(173\) −20.8358 −1.58411 −0.792057 0.610448i \(-0.790990\pi\)
−0.792057 + 0.610448i \(0.790990\pi\)
\(174\) 0 0
\(175\) 12.4932 0.944394
\(176\) 0 0
\(177\) 4.69325 0.352766
\(178\) 0 0
\(179\) 22.4086 1.67490 0.837448 0.546517i \(-0.184047\pi\)
0.837448 + 0.546517i \(0.184047\pi\)
\(180\) 0 0
\(181\) −4.42942 −0.329236 −0.164618 0.986357i \(-0.552639\pi\)
−0.164618 + 0.986357i \(0.552639\pi\)
\(182\) 0 0
\(183\) 4.16749 0.308070
\(184\) 0 0
\(185\) −33.3968 −2.45538
\(186\) 0 0
\(187\) −23.1411 −1.69225
\(188\) 0 0
\(189\) 5.01978 0.365135
\(190\) 0 0
\(191\) −7.38976 −0.534704 −0.267352 0.963599i \(-0.586149\pi\)
−0.267352 + 0.963599i \(0.586149\pi\)
\(192\) 0 0
\(193\) 0.139138 0.0100154 0.00500769 0.999987i \(-0.498406\pi\)
0.00500769 + 0.999987i \(0.498406\pi\)
\(194\) 0 0
\(195\) −36.1820 −2.59104
\(196\) 0 0
\(197\) −10.4057 −0.741377 −0.370688 0.928757i \(-0.620878\pi\)
−0.370688 + 0.928757i \(0.620878\pi\)
\(198\) 0 0
\(199\) 5.09550 0.361211 0.180605 0.983556i \(-0.442194\pi\)
0.180605 + 0.983556i \(0.442194\pi\)
\(200\) 0 0
\(201\) −14.5867 −1.02887
\(202\) 0 0
\(203\) −5.43932 −0.381766
\(204\) 0 0
\(205\) 9.31184 0.650367
\(206\) 0 0
\(207\) −1.34577 −0.0935378
\(208\) 0 0
\(209\) −9.58529 −0.663029
\(210\) 0 0
\(211\) −12.5589 −0.864592 −0.432296 0.901732i \(-0.642296\pi\)
−0.432296 + 0.901732i \(0.642296\pi\)
\(212\) 0 0
\(213\) 10.8014 0.740097
\(214\) 0 0
\(215\) −9.56970 −0.652648
\(216\) 0 0
\(217\) −0.831138 −0.0564213
\(218\) 0 0
\(219\) 13.7002 0.925772
\(220\) 0 0
\(221\) −24.8915 −1.67438
\(222\) 0 0
\(223\) 9.66949 0.647517 0.323758 0.946140i \(-0.395054\pi\)
0.323758 + 0.946140i \(0.395054\pi\)
\(224\) 0 0
\(225\) 2.36174 0.157450
\(226\) 0 0
\(227\) 2.99594 0.198847 0.0994237 0.995045i \(-0.468300\pi\)
0.0994237 + 0.995045i \(0.468300\pi\)
\(228\) 0 0
\(229\) −8.99569 −0.594452 −0.297226 0.954807i \(-0.596061\pi\)
−0.297226 + 0.954807i \(0.596061\pi\)
\(230\) 0 0
\(231\) 8.04251 0.529158
\(232\) 0 0
\(233\) 7.53066 0.493350 0.246675 0.969098i \(-0.420662\pi\)
0.246675 + 0.969098i \(0.420662\pi\)
\(234\) 0 0
\(235\) −32.7806 −2.13837
\(236\) 0 0
\(237\) 3.39653 0.220629
\(238\) 0 0
\(239\) −1.87072 −0.121007 −0.0605034 0.998168i \(-0.519271\pi\)
−0.0605034 + 0.998168i \(0.519271\pi\)
\(240\) 0 0
\(241\) 14.4911 0.933454 0.466727 0.884401i \(-0.345433\pi\)
0.466727 + 0.884401i \(0.345433\pi\)
\(242\) 0 0
\(243\) 1.96173 0.125845
\(244\) 0 0
\(245\) −4.18248 −0.267209
\(246\) 0 0
\(247\) −10.3103 −0.656029
\(248\) 0 0
\(249\) −20.1514 −1.27704
\(250\) 0 0
\(251\) −6.33974 −0.400161 −0.200080 0.979779i \(-0.564120\pi\)
−0.200080 + 0.979779i \(0.564120\pi\)
\(252\) 0 0
\(253\) 32.0607 2.01564
\(254\) 0 0
\(255\) 38.3784 2.40335
\(256\) 0 0
\(257\) 12.1594 0.758483 0.379241 0.925298i \(-0.376185\pi\)
0.379241 + 0.925298i \(0.376185\pi\)
\(258\) 0 0
\(259\) 7.98492 0.496159
\(260\) 0 0
\(261\) −1.02827 −0.0636480
\(262\) 0 0
\(263\) 0.0299529 0.00184698 0.000923488 1.00000i \(-0.499706\pi\)
0.000923488 1.00000i \(0.499706\pi\)
\(264\) 0 0
\(265\) −33.0514 −2.03033
\(266\) 0 0
\(267\) 4.46336 0.273153
\(268\) 0 0
\(269\) 18.0622 1.10127 0.550637 0.834745i \(-0.314385\pi\)
0.550637 + 0.834745i \(0.314385\pi\)
\(270\) 0 0
\(271\) 10.0906 0.612958 0.306479 0.951877i \(-0.400849\pi\)
0.306479 + 0.951877i \(0.400849\pi\)
\(272\) 0 0
\(273\) 8.65084 0.523573
\(274\) 0 0
\(275\) −56.2644 −3.39287
\(276\) 0 0
\(277\) −8.86651 −0.532737 −0.266368 0.963871i \(-0.585824\pi\)
−0.266368 + 0.963871i \(0.585824\pi\)
\(278\) 0 0
\(279\) −0.157121 −0.00940657
\(280\) 0 0
\(281\) −11.6731 −0.696356 −0.348178 0.937428i \(-0.613200\pi\)
−0.348178 + 0.937428i \(0.613200\pi\)
\(282\) 0 0
\(283\) 12.5547 0.746297 0.373149 0.927772i \(-0.378278\pi\)
0.373149 + 0.927772i \(0.378278\pi\)
\(284\) 0 0
\(285\) 15.8967 0.941642
\(286\) 0 0
\(287\) −2.22639 −0.131420
\(288\) 0 0
\(289\) 9.40252 0.553090
\(290\) 0 0
\(291\) 3.54986 0.208096
\(292\) 0 0
\(293\) −24.3152 −1.42051 −0.710256 0.703944i \(-0.751421\pi\)
−0.710256 + 0.703944i \(0.751421\pi\)
\(294\) 0 0
\(295\) 10.9920 0.639980
\(296\) 0 0
\(297\) −22.6072 −1.31180
\(298\) 0 0
\(299\) 34.4858 1.99436
\(300\) 0 0
\(301\) 2.28804 0.131881
\(302\) 0 0
\(303\) −28.1189 −1.61539
\(304\) 0 0
\(305\) 9.76065 0.558893
\(306\) 0 0
\(307\) −27.2536 −1.55544 −0.777722 0.628608i \(-0.783625\pi\)
−0.777722 + 0.628608i \(0.783625\pi\)
\(308\) 0 0
\(309\) −27.1629 −1.54524
\(310\) 0 0
\(311\) −5.78650 −0.328122 −0.164061 0.986450i \(-0.552459\pi\)
−0.164061 + 0.986450i \(0.552459\pi\)
\(312\) 0 0
\(313\) 3.03673 0.171646 0.0858231 0.996310i \(-0.472648\pi\)
0.0858231 + 0.996310i \(0.472648\pi\)
\(314\) 0 0
\(315\) −0.790669 −0.0445491
\(316\) 0 0
\(317\) −4.69828 −0.263882 −0.131941 0.991258i \(-0.542121\pi\)
−0.131941 + 0.991258i \(0.542121\pi\)
\(318\) 0 0
\(319\) 24.4966 1.37155
\(320\) 0 0
\(321\) 2.26690 0.126526
\(322\) 0 0
\(323\) 10.9362 0.608507
\(324\) 0 0
\(325\) −60.5202 −3.35706
\(326\) 0 0
\(327\) 7.18572 0.397371
\(328\) 0 0
\(329\) 7.83759 0.432101
\(330\) 0 0
\(331\) 36.0648 1.98230 0.991151 0.132738i \(-0.0423768\pi\)
0.991151 + 0.132738i \(0.0423768\pi\)
\(332\) 0 0
\(333\) 1.50949 0.0827197
\(334\) 0 0
\(335\) −34.1634 −1.86655
\(336\) 0 0
\(337\) −31.3282 −1.70656 −0.853279 0.521454i \(-0.825390\pi\)
−0.853279 + 0.521454i \(0.825390\pi\)
\(338\) 0 0
\(339\) −2.97311 −0.161477
\(340\) 0 0
\(341\) 3.74313 0.202702
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −53.1712 −2.86264
\(346\) 0 0
\(347\) −34.1006 −1.83062 −0.915308 0.402755i \(-0.868053\pi\)
−0.915308 + 0.402755i \(0.868053\pi\)
\(348\) 0 0
\(349\) −29.6604 −1.58769 −0.793843 0.608123i \(-0.791923\pi\)
−0.793843 + 0.608123i \(0.791923\pi\)
\(350\) 0 0
\(351\) −24.3171 −1.29795
\(352\) 0 0
\(353\) −0.605671 −0.0322366 −0.0161183 0.999870i \(-0.505131\pi\)
−0.0161183 + 0.999870i \(0.505131\pi\)
\(354\) 0 0
\(355\) 25.2978 1.34267
\(356\) 0 0
\(357\) −9.17599 −0.485645
\(358\) 0 0
\(359\) 11.6214 0.613354 0.306677 0.951814i \(-0.400783\pi\)
0.306677 + 0.951814i \(0.400783\pi\)
\(360\) 0 0
\(361\) −14.4701 −0.761585
\(362\) 0 0
\(363\) −16.5767 −0.870052
\(364\) 0 0
\(365\) 32.0871 1.67951
\(366\) 0 0
\(367\) 13.1299 0.685376 0.342688 0.939449i \(-0.388663\pi\)
0.342688 + 0.939449i \(0.388663\pi\)
\(368\) 0 0
\(369\) −0.420883 −0.0219103
\(370\) 0 0
\(371\) 7.90235 0.410270
\(372\) 0 0
\(373\) −18.4801 −0.956863 −0.478431 0.878125i \(-0.658794\pi\)
−0.478431 + 0.878125i \(0.658794\pi\)
\(374\) 0 0
\(375\) 55.9666 2.89010
\(376\) 0 0
\(377\) 26.3495 1.35707
\(378\) 0 0
\(379\) −4.13943 −0.212628 −0.106314 0.994333i \(-0.533905\pi\)
−0.106314 + 0.994333i \(0.533905\pi\)
\(380\) 0 0
\(381\) −14.0375 −0.719164
\(382\) 0 0
\(383\) 34.3667 1.75606 0.878029 0.478608i \(-0.158858\pi\)
0.878029 + 0.478608i \(0.158858\pi\)
\(384\) 0 0
\(385\) 18.8363 0.959987
\(386\) 0 0
\(387\) 0.432538 0.0219872
\(388\) 0 0
\(389\) 10.5339 0.534089 0.267044 0.963684i \(-0.413953\pi\)
0.267044 + 0.963684i \(0.413953\pi\)
\(390\) 0 0
\(391\) −36.5792 −1.84989
\(392\) 0 0
\(393\) −13.7507 −0.693633
\(394\) 0 0
\(395\) 7.95499 0.400259
\(396\) 0 0
\(397\) 11.2088 0.562554 0.281277 0.959627i \(-0.409242\pi\)
0.281277 + 0.959627i \(0.409242\pi\)
\(398\) 0 0
\(399\) −3.80079 −0.190278
\(400\) 0 0
\(401\) −15.4031 −0.769192 −0.384596 0.923085i \(-0.625659\pi\)
−0.384596 + 0.923085i \(0.625659\pi\)
\(402\) 0 0
\(403\) 4.02625 0.200562
\(404\) 0 0
\(405\) 39.8649 1.98090
\(406\) 0 0
\(407\) −35.9610 −1.78252
\(408\) 0 0
\(409\) 22.6029 1.11764 0.558820 0.829289i \(-0.311254\pi\)
0.558820 + 0.829289i \(0.311254\pi\)
\(410\) 0 0
\(411\) 31.6043 1.55892
\(412\) 0 0
\(413\) −2.62811 −0.129321
\(414\) 0 0
\(415\) −47.1964 −2.31678
\(416\) 0 0
\(417\) −28.9174 −1.41609
\(418\) 0 0
\(419\) −12.7468 −0.622720 −0.311360 0.950292i \(-0.600785\pi\)
−0.311360 + 0.950292i \(0.600785\pi\)
\(420\) 0 0
\(421\) −30.9413 −1.50799 −0.753994 0.656882i \(-0.771875\pi\)
−0.753994 + 0.656882i \(0.771875\pi\)
\(422\) 0 0
\(423\) 1.48164 0.0720399
\(424\) 0 0
\(425\) 64.1941 3.11387
\(426\) 0 0
\(427\) −2.33370 −0.112936
\(428\) 0 0
\(429\) −38.9601 −1.88101
\(430\) 0 0
\(431\) −13.1089 −0.631434 −0.315717 0.948853i \(-0.602245\pi\)
−0.315717 + 0.948853i \(0.602245\pi\)
\(432\) 0 0
\(433\) 35.1859 1.69093 0.845463 0.534033i \(-0.179324\pi\)
0.845463 + 0.534033i \(0.179324\pi\)
\(434\) 0 0
\(435\) −40.6265 −1.94789
\(436\) 0 0
\(437\) −15.1515 −0.724795
\(438\) 0 0
\(439\) 4.82251 0.230166 0.115083 0.993356i \(-0.463287\pi\)
0.115083 + 0.993356i \(0.463287\pi\)
\(440\) 0 0
\(441\) 0.189043 0.00900204
\(442\) 0 0
\(443\) 17.7618 0.843888 0.421944 0.906622i \(-0.361348\pi\)
0.421944 + 0.906622i \(0.361348\pi\)
\(444\) 0 0
\(445\) 10.4536 0.495548
\(446\) 0 0
\(447\) 21.7520 1.02883
\(448\) 0 0
\(449\) −29.9204 −1.41203 −0.706015 0.708197i \(-0.749509\pi\)
−0.706015 + 0.708197i \(0.749509\pi\)
\(450\) 0 0
\(451\) 10.0268 0.472144
\(452\) 0 0
\(453\) 31.6886 1.48886
\(454\) 0 0
\(455\) 20.2611 0.949853
\(456\) 0 0
\(457\) 29.1293 1.36261 0.681305 0.732000i \(-0.261413\pi\)
0.681305 + 0.732000i \(0.261413\pi\)
\(458\) 0 0
\(459\) 25.7933 1.20393
\(460\) 0 0
\(461\) −10.1181 −0.471247 −0.235624 0.971844i \(-0.575713\pi\)
−0.235624 + 0.971844i \(0.575713\pi\)
\(462\) 0 0
\(463\) −40.1547 −1.86615 −0.933074 0.359686i \(-0.882884\pi\)
−0.933074 + 0.359686i \(0.882884\pi\)
\(464\) 0 0
\(465\) −6.20779 −0.287880
\(466\) 0 0
\(467\) 40.3127 1.86545 0.932726 0.360587i \(-0.117423\pi\)
0.932726 + 0.360587i \(0.117423\pi\)
\(468\) 0 0
\(469\) 8.16822 0.377174
\(470\) 0 0
\(471\) −24.9633 −1.15025
\(472\) 0 0
\(473\) −10.3045 −0.473800
\(474\) 0 0
\(475\) 26.5899 1.22003
\(476\) 0 0
\(477\) 1.49388 0.0684002
\(478\) 0 0
\(479\) 12.6994 0.580249 0.290124 0.956989i \(-0.406303\pi\)
0.290124 + 0.956989i \(0.406303\pi\)
\(480\) 0 0
\(481\) −38.6811 −1.76371
\(482\) 0 0
\(483\) 12.7128 0.578454
\(484\) 0 0
\(485\) 8.31409 0.377523
\(486\) 0 0
\(487\) −34.2373 −1.55144 −0.775721 0.631076i \(-0.782614\pi\)
−0.775721 + 0.631076i \(0.782614\pi\)
\(488\) 0 0
\(489\) −24.2799 −1.09798
\(490\) 0 0
\(491\) 30.2843 1.36671 0.683356 0.730085i \(-0.260520\pi\)
0.683356 + 0.730085i \(0.260520\pi\)
\(492\) 0 0
\(493\) −27.9491 −1.25876
\(494\) 0 0
\(495\) 3.56087 0.160049
\(496\) 0 0
\(497\) −6.04851 −0.271313
\(498\) 0 0
\(499\) 22.2171 0.994574 0.497287 0.867586i \(-0.334330\pi\)
0.497287 + 0.867586i \(0.334330\pi\)
\(500\) 0 0
\(501\) 11.9779 0.535134
\(502\) 0 0
\(503\) −28.6238 −1.27627 −0.638137 0.769923i \(-0.720294\pi\)
−0.638137 + 0.769923i \(0.720294\pi\)
\(504\) 0 0
\(505\) −65.8571 −2.93060
\(506\) 0 0
\(507\) −18.6917 −0.830128
\(508\) 0 0
\(509\) −21.2622 −0.942431 −0.471215 0.882018i \(-0.656185\pi\)
−0.471215 + 0.882018i \(0.656185\pi\)
\(510\) 0 0
\(511\) −7.67177 −0.339379
\(512\) 0 0
\(513\) 10.6839 0.471704
\(514\) 0 0
\(515\) −63.6180 −2.80334
\(516\) 0 0
\(517\) −35.2975 −1.55238
\(518\) 0 0
\(519\) 37.2083 1.63326
\(520\) 0 0
\(521\) −23.7033 −1.03846 −0.519230 0.854635i \(-0.673781\pi\)
−0.519230 + 0.854635i \(0.673781\pi\)
\(522\) 0 0
\(523\) −19.6640 −0.859847 −0.429923 0.902865i \(-0.641459\pi\)
−0.429923 + 0.902865i \(0.641459\pi\)
\(524\) 0 0
\(525\) −22.3101 −0.973694
\(526\) 0 0
\(527\) −4.27067 −0.186033
\(528\) 0 0
\(529\) 27.6785 1.20341
\(530\) 0 0
\(531\) −0.496825 −0.0215604
\(532\) 0 0
\(533\) 10.7852 0.467160
\(534\) 0 0
\(535\) 5.30928 0.229540
\(536\) 0 0
\(537\) −40.0170 −1.72686
\(538\) 0 0
\(539\) −4.50362 −0.193985
\(540\) 0 0
\(541\) 12.2602 0.527106 0.263553 0.964645i \(-0.415106\pi\)
0.263553 + 0.964645i \(0.415106\pi\)
\(542\) 0 0
\(543\) 7.91002 0.339451
\(544\) 0 0
\(545\) 16.8296 0.720901
\(546\) 0 0
\(547\) 5.75986 0.246274 0.123137 0.992390i \(-0.460705\pi\)
0.123137 + 0.992390i \(0.460705\pi\)
\(548\) 0 0
\(549\) −0.441169 −0.0188286
\(550\) 0 0
\(551\) −11.5768 −0.493188
\(552\) 0 0
\(553\) −1.90198 −0.0808804
\(554\) 0 0
\(555\) 59.6396 2.53156
\(556\) 0 0
\(557\) −2.57936 −0.109291 −0.0546455 0.998506i \(-0.517403\pi\)
−0.0546455 + 0.998506i \(0.517403\pi\)
\(558\) 0 0
\(559\) −11.0839 −0.468798
\(560\) 0 0
\(561\) 41.3252 1.74475
\(562\) 0 0
\(563\) 15.1195 0.637212 0.318606 0.947887i \(-0.396785\pi\)
0.318606 + 0.947887i \(0.396785\pi\)
\(564\) 0 0
\(565\) −6.96329 −0.292948
\(566\) 0 0
\(567\) −9.53139 −0.400281
\(568\) 0 0
\(569\) 11.0034 0.461288 0.230644 0.973038i \(-0.425917\pi\)
0.230644 + 0.973038i \(0.425917\pi\)
\(570\) 0 0
\(571\) −40.9982 −1.71572 −0.857861 0.513881i \(-0.828207\pi\)
−0.857861 + 0.513881i \(0.828207\pi\)
\(572\) 0 0
\(573\) 13.1966 0.551294
\(574\) 0 0
\(575\) −88.9373 −3.70894
\(576\) 0 0
\(577\) 2.85596 0.118895 0.0594476 0.998231i \(-0.481066\pi\)
0.0594476 + 0.998231i \(0.481066\pi\)
\(578\) 0 0
\(579\) −0.248472 −0.0103261
\(580\) 0 0
\(581\) 11.2843 0.468152
\(582\) 0 0
\(583\) −35.5892 −1.47395
\(584\) 0 0
\(585\) 3.83021 0.158360
\(586\) 0 0
\(587\) −24.8280 −1.02476 −0.512381 0.858758i \(-0.671237\pi\)
−0.512381 + 0.858758i \(0.671237\pi\)
\(588\) 0 0
\(589\) −1.76895 −0.0728885
\(590\) 0 0
\(591\) 18.5824 0.764379
\(592\) 0 0
\(593\) −7.72713 −0.317315 −0.158658 0.987334i \(-0.550717\pi\)
−0.158658 + 0.987334i \(0.550717\pi\)
\(594\) 0 0
\(595\) −21.4910 −0.881045
\(596\) 0 0
\(597\) −9.09950 −0.372418
\(598\) 0 0
\(599\) 7.28771 0.297768 0.148884 0.988855i \(-0.452432\pi\)
0.148884 + 0.988855i \(0.452432\pi\)
\(600\) 0 0
\(601\) 20.0313 0.817095 0.408547 0.912737i \(-0.366035\pi\)
0.408547 + 0.912737i \(0.366035\pi\)
\(602\) 0 0
\(603\) 1.54414 0.0628824
\(604\) 0 0
\(605\) −38.8242 −1.57843
\(606\) 0 0
\(607\) −17.2219 −0.699014 −0.349507 0.936934i \(-0.613651\pi\)
−0.349507 + 0.936934i \(0.613651\pi\)
\(608\) 0 0
\(609\) 9.71349 0.393610
\(610\) 0 0
\(611\) −37.9674 −1.53600
\(612\) 0 0
\(613\) 37.3993 1.51054 0.755271 0.655412i \(-0.227505\pi\)
0.755271 + 0.655412i \(0.227505\pi\)
\(614\) 0 0
\(615\) −16.6290 −0.670545
\(616\) 0 0
\(617\) −22.1036 −0.889856 −0.444928 0.895566i \(-0.646771\pi\)
−0.444928 + 0.895566i \(0.646771\pi\)
\(618\) 0 0
\(619\) −30.4908 −1.22553 −0.612764 0.790266i \(-0.709942\pi\)
−0.612764 + 0.790266i \(0.709942\pi\)
\(620\) 0 0
\(621\) −35.7352 −1.43400
\(622\) 0 0
\(623\) −2.49938 −0.100135
\(624\) 0 0
\(625\) 68.6132 2.74453
\(626\) 0 0
\(627\) 17.1173 0.683600
\(628\) 0 0
\(629\) 41.0292 1.63594
\(630\) 0 0
\(631\) 40.3151 1.60492 0.802458 0.596708i \(-0.203525\pi\)
0.802458 + 0.596708i \(0.203525\pi\)
\(632\) 0 0
\(633\) 22.4276 0.891417
\(634\) 0 0
\(635\) −32.8772 −1.30469
\(636\) 0 0
\(637\) −4.84427 −0.191937
\(638\) 0 0
\(639\) −1.14343 −0.0452333
\(640\) 0 0
\(641\) −0.875535 −0.0345815 −0.0172908 0.999851i \(-0.505504\pi\)
−0.0172908 + 0.999851i \(0.505504\pi\)
\(642\) 0 0
\(643\) −40.4431 −1.59492 −0.797459 0.603373i \(-0.793823\pi\)
−0.797459 + 0.603373i \(0.793823\pi\)
\(644\) 0 0
\(645\) 17.0895 0.672897
\(646\) 0 0
\(647\) 3.32973 0.130905 0.0654526 0.997856i \(-0.479151\pi\)
0.0654526 + 0.997856i \(0.479151\pi\)
\(648\) 0 0
\(649\) 11.8360 0.464603
\(650\) 0 0
\(651\) 1.48424 0.0581718
\(652\) 0 0
\(653\) −29.3589 −1.14890 −0.574452 0.818539i \(-0.694785\pi\)
−0.574452 + 0.818539i \(0.694785\pi\)
\(654\) 0 0
\(655\) −32.2055 −1.25837
\(656\) 0 0
\(657\) −1.45029 −0.0565814
\(658\) 0 0
\(659\) −29.2836 −1.14073 −0.570364 0.821392i \(-0.693198\pi\)
−0.570364 + 0.821392i \(0.693198\pi\)
\(660\) 0 0
\(661\) −13.7308 −0.534066 −0.267033 0.963687i \(-0.586043\pi\)
−0.267033 + 0.963687i \(0.586043\pi\)
\(662\) 0 0
\(663\) 44.4509 1.72633
\(664\) 0 0
\(665\) −8.90180 −0.345197
\(666\) 0 0
\(667\) 38.7219 1.49932
\(668\) 0 0
\(669\) −17.2677 −0.667606
\(670\) 0 0
\(671\) 10.5101 0.405737
\(672\) 0 0
\(673\) 49.3202 1.90115 0.950577 0.310490i \(-0.100493\pi\)
0.950577 + 0.310490i \(0.100493\pi\)
\(674\) 0 0
\(675\) 62.7129 2.41382
\(676\) 0 0
\(677\) 17.3000 0.664892 0.332446 0.943122i \(-0.392126\pi\)
0.332446 + 0.943122i \(0.392126\pi\)
\(678\) 0 0
\(679\) −1.98784 −0.0762861
\(680\) 0 0
\(681\) −5.35011 −0.205017
\(682\) 0 0
\(683\) 5.37135 0.205529 0.102764 0.994706i \(-0.467231\pi\)
0.102764 + 0.994706i \(0.467231\pi\)
\(684\) 0 0
\(685\) 74.0202 2.82816
\(686\) 0 0
\(687\) 16.0644 0.612895
\(688\) 0 0
\(689\) −38.2811 −1.45839
\(690\) 0 0
\(691\) −44.8853 −1.70752 −0.853759 0.520668i \(-0.825683\pi\)
−0.853759 + 0.520668i \(0.825683\pi\)
\(692\) 0 0
\(693\) −0.851377 −0.0323411
\(694\) 0 0
\(695\) −67.7271 −2.56904
\(696\) 0 0
\(697\) −11.4399 −0.433319
\(698\) 0 0
\(699\) −13.4482 −0.508657
\(700\) 0 0
\(701\) 8.61013 0.325200 0.162600 0.986692i \(-0.448012\pi\)
0.162600 + 0.986692i \(0.448012\pi\)
\(702\) 0 0
\(703\) 16.9947 0.640969
\(704\) 0 0
\(705\) 58.5392 2.20472
\(706\) 0 0
\(707\) 15.7459 0.592187
\(708\) 0 0
\(709\) −3.07942 −0.115650 −0.0578250 0.998327i \(-0.518417\pi\)
−0.0578250 + 0.998327i \(0.518417\pi\)
\(710\) 0 0
\(711\) −0.359556 −0.0134844
\(712\) 0 0
\(713\) 5.91677 0.221585
\(714\) 0 0
\(715\) −91.2481 −3.41248
\(716\) 0 0
\(717\) 3.34071 0.124761
\(718\) 0 0
\(719\) 42.5677 1.58751 0.793754 0.608239i \(-0.208124\pi\)
0.793754 + 0.608239i \(0.208124\pi\)
\(720\) 0 0
\(721\) 15.2106 0.566471
\(722\) 0 0
\(723\) −25.8781 −0.962416
\(724\) 0 0
\(725\) −67.9543 −2.52376
\(726\) 0 0
\(727\) −3.44163 −0.127643 −0.0638214 0.997961i \(-0.520329\pi\)
−0.0638214 + 0.997961i \(0.520329\pi\)
\(728\) 0 0
\(729\) 25.0909 0.929294
\(730\) 0 0
\(731\) 11.7567 0.434839
\(732\) 0 0
\(733\) −12.1501 −0.448775 −0.224388 0.974500i \(-0.572038\pi\)
−0.224388 + 0.974500i \(0.572038\pi\)
\(734\) 0 0
\(735\) 7.46903 0.275499
\(736\) 0 0
\(737\) −36.7866 −1.35505
\(738\) 0 0
\(739\) −27.9556 −1.02836 −0.514181 0.857682i \(-0.671904\pi\)
−0.514181 + 0.857682i \(0.671904\pi\)
\(740\) 0 0
\(741\) 18.4120 0.676383
\(742\) 0 0
\(743\) −14.0786 −0.516495 −0.258248 0.966079i \(-0.583145\pi\)
−0.258248 + 0.966079i \(0.583145\pi\)
\(744\) 0 0
\(745\) 50.9452 1.86649
\(746\) 0 0
\(747\) 2.13322 0.0780503
\(748\) 0 0
\(749\) −1.26941 −0.0463832
\(750\) 0 0
\(751\) 27.6318 1.00830 0.504148 0.863617i \(-0.331806\pi\)
0.504148 + 0.863617i \(0.331806\pi\)
\(752\) 0 0
\(753\) 11.3214 0.412576
\(754\) 0 0
\(755\) 74.2176 2.70106
\(756\) 0 0
\(757\) 2.76084 0.100344 0.0501722 0.998741i \(-0.484023\pi\)
0.0501722 + 0.998741i \(0.484023\pi\)
\(758\) 0 0
\(759\) −57.2537 −2.07818
\(760\) 0 0
\(761\) 34.5598 1.25279 0.626396 0.779505i \(-0.284529\pi\)
0.626396 + 0.779505i \(0.284529\pi\)
\(762\) 0 0
\(763\) −4.02383 −0.145673
\(764\) 0 0
\(765\) −4.06272 −0.146888
\(766\) 0 0
\(767\) 12.7313 0.459699
\(768\) 0 0
\(769\) 49.7370 1.79356 0.896781 0.442474i \(-0.145899\pi\)
0.896781 + 0.442474i \(0.145899\pi\)
\(770\) 0 0
\(771\) −21.7141 −0.782015
\(772\) 0 0
\(773\) −14.1791 −0.509986 −0.254993 0.966943i \(-0.582073\pi\)
−0.254993 + 0.966943i \(0.582073\pi\)
\(774\) 0 0
\(775\) −10.3835 −0.372988
\(776\) 0 0
\(777\) −14.2594 −0.511553
\(778\) 0 0
\(779\) −4.73855 −0.169776
\(780\) 0 0
\(781\) 27.2402 0.974730
\(782\) 0 0
\(783\) −27.3042 −0.975772
\(784\) 0 0
\(785\) −58.4664 −2.08675
\(786\) 0 0
\(787\) 13.1039 0.467104 0.233552 0.972344i \(-0.424965\pi\)
0.233552 + 0.972344i \(0.424965\pi\)
\(788\) 0 0
\(789\) −0.0534896 −0.00190428
\(790\) 0 0
\(791\) 1.66487 0.0591960
\(792\) 0 0
\(793\) 11.3050 0.401454
\(794\) 0 0
\(795\) 59.0229 2.09333
\(796\) 0 0
\(797\) −30.2441 −1.07130 −0.535651 0.844440i \(-0.679934\pi\)
−0.535651 + 0.844440i \(0.679934\pi\)
\(798\) 0 0
\(799\) 40.2722 1.42473
\(800\) 0 0
\(801\) −0.472490 −0.0166946
\(802\) 0 0
\(803\) 34.5507 1.21927
\(804\) 0 0
\(805\) 29.7746 1.04942
\(806\) 0 0
\(807\) −32.2553 −1.13544
\(808\) 0 0
\(809\) −1.36939 −0.0481450 −0.0240725 0.999710i \(-0.507663\pi\)
−0.0240725 + 0.999710i \(0.507663\pi\)
\(810\) 0 0
\(811\) 29.7374 1.04422 0.522110 0.852878i \(-0.325145\pi\)
0.522110 + 0.852878i \(0.325145\pi\)
\(812\) 0 0
\(813\) −18.0196 −0.631976
\(814\) 0 0
\(815\) −56.8659 −1.99192
\(816\) 0 0
\(817\) 4.86976 0.170371
\(818\) 0 0
\(819\) −0.915774 −0.0319997
\(820\) 0 0
\(821\) 18.1869 0.634727 0.317363 0.948304i \(-0.397203\pi\)
0.317363 + 0.948304i \(0.397203\pi\)
\(822\) 0 0
\(823\) 41.5715 1.44909 0.724546 0.689226i \(-0.242049\pi\)
0.724546 + 0.689226i \(0.242049\pi\)
\(824\) 0 0
\(825\) 100.476 3.49814
\(826\) 0 0
\(827\) −46.0708 −1.60204 −0.801019 0.598639i \(-0.795708\pi\)
−0.801019 + 0.598639i \(0.795708\pi\)
\(828\) 0 0
\(829\) 0.574936 0.0199683 0.00998417 0.999950i \(-0.496822\pi\)
0.00998417 + 0.999950i \(0.496822\pi\)
\(830\) 0 0
\(831\) 15.8337 0.549265
\(832\) 0 0
\(833\) 5.13834 0.178033
\(834\) 0 0
\(835\) 28.0534 0.970827
\(836\) 0 0
\(837\) −4.17213 −0.144210
\(838\) 0 0
\(839\) 28.6700 0.989799 0.494899 0.868950i \(-0.335205\pi\)
0.494899 + 0.868950i \(0.335205\pi\)
\(840\) 0 0
\(841\) 0.586242 0.0202152
\(842\) 0 0
\(843\) 20.8456 0.717961
\(844\) 0 0
\(845\) −43.7777 −1.50600
\(846\) 0 0
\(847\) 9.28258 0.318953
\(848\) 0 0
\(849\) −22.4200 −0.769452
\(850\) 0 0
\(851\) −56.8437 −1.94858
\(852\) 0 0
\(853\) 11.8044 0.404175 0.202087 0.979367i \(-0.435227\pi\)
0.202087 + 0.979367i \(0.435227\pi\)
\(854\) 0 0
\(855\) −1.68282 −0.0575513
\(856\) 0 0
\(857\) 48.6998 1.66355 0.831777 0.555110i \(-0.187324\pi\)
0.831777 + 0.555110i \(0.187324\pi\)
\(858\) 0 0
\(859\) −0.0381681 −0.00130228 −0.000651140 1.00000i \(-0.500207\pi\)
−0.000651140 1.00000i \(0.500207\pi\)
\(860\) 0 0
\(861\) 3.97586 0.135497
\(862\) 0 0
\(863\) −1.64855 −0.0561173 −0.0280587 0.999606i \(-0.508933\pi\)
−0.0280587 + 0.999606i \(0.508933\pi\)
\(864\) 0 0
\(865\) 87.1452 2.96303
\(866\) 0 0
\(867\) −16.7909 −0.570250
\(868\) 0 0
\(869\) 8.56579 0.290574
\(870\) 0 0
\(871\) −39.5690 −1.34075
\(872\) 0 0
\(873\) −0.375786 −0.0127184
\(874\) 0 0
\(875\) −31.3400 −1.05949
\(876\) 0 0
\(877\) −44.2259 −1.49340 −0.746701 0.665160i \(-0.768363\pi\)
−0.746701 + 0.665160i \(0.768363\pi\)
\(878\) 0 0
\(879\) 43.4219 1.46458
\(880\) 0 0
\(881\) 1.02249 0.0344486 0.0172243 0.999852i \(-0.494517\pi\)
0.0172243 + 0.999852i \(0.494517\pi\)
\(882\) 0 0
\(883\) 10.6948 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(884\) 0 0
\(885\) −19.6294 −0.659836
\(886\) 0 0
\(887\) −4.15374 −0.139469 −0.0697344 0.997566i \(-0.522215\pi\)
−0.0697344 + 0.997566i \(0.522215\pi\)
\(888\) 0 0
\(889\) 7.86069 0.263639
\(890\) 0 0
\(891\) 42.9257 1.43807
\(892\) 0 0
\(893\) 16.6812 0.558214
\(894\) 0 0
\(895\) −93.7235 −3.13283
\(896\) 0 0
\(897\) −61.5843 −2.05624
\(898\) 0 0
\(899\) 4.52083 0.150778
\(900\) 0 0
\(901\) 40.6050 1.35275
\(902\) 0 0
\(903\) −4.08596 −0.135972
\(904\) 0 0
\(905\) 18.5260 0.615825
\(906\) 0 0
\(907\) −4.85902 −0.161341 −0.0806706 0.996741i \(-0.525706\pi\)
−0.0806706 + 0.996741i \(0.525706\pi\)
\(908\) 0 0
\(909\) 2.97666 0.0987295
\(910\) 0 0
\(911\) 21.2511 0.704082 0.352041 0.935985i \(-0.385488\pi\)
0.352041 + 0.935985i \(0.385488\pi\)
\(912\) 0 0
\(913\) −50.8202 −1.68190
\(914\) 0 0
\(915\) −17.4305 −0.576233
\(916\) 0 0
\(917\) 7.70009 0.254279
\(918\) 0 0
\(919\) −14.2571 −0.470297 −0.235149 0.971959i \(-0.575558\pi\)
−0.235149 + 0.971959i \(0.575558\pi\)
\(920\) 0 0
\(921\) 48.6692 1.60370
\(922\) 0 0
\(923\) 29.3006 0.964441
\(924\) 0 0
\(925\) 99.7569 3.27999
\(926\) 0 0
\(927\) 2.87545 0.0944422
\(928\) 0 0
\(929\) 31.9394 1.04790 0.523948 0.851750i \(-0.324458\pi\)
0.523948 + 0.851750i \(0.324458\pi\)
\(930\) 0 0
\(931\) 2.12835 0.0697540
\(932\) 0 0
\(933\) 10.3335 0.338303
\(934\) 0 0
\(935\) 96.7873 3.16528
\(936\) 0 0
\(937\) 4.91109 0.160438 0.0802191 0.996777i \(-0.474438\pi\)
0.0802191 + 0.996777i \(0.474438\pi\)
\(938\) 0 0
\(939\) −5.42296 −0.176972
\(940\) 0 0
\(941\) 55.0908 1.79591 0.897954 0.440090i \(-0.145053\pi\)
0.897954 + 0.440090i \(0.145053\pi\)
\(942\) 0 0
\(943\) 15.8494 0.516128
\(944\) 0 0
\(945\) −20.9951 −0.682972
\(946\) 0 0
\(947\) 49.0597 1.59423 0.797113 0.603830i \(-0.206359\pi\)
0.797113 + 0.603830i \(0.206359\pi\)
\(948\) 0 0
\(949\) 37.1641 1.20640
\(950\) 0 0
\(951\) 8.39014 0.272069
\(952\) 0 0
\(953\) −56.6312 −1.83446 −0.917232 0.398354i \(-0.869582\pi\)
−0.917232 + 0.398354i \(0.869582\pi\)
\(954\) 0 0
\(955\) 30.9075 1.00014
\(956\) 0 0
\(957\) −43.7458 −1.41410
\(958\) 0 0
\(959\) −17.6977 −0.571487
\(960\) 0 0
\(961\) −30.3092 −0.977716
\(962\) 0 0
\(963\) −0.239973 −0.00773302
\(964\) 0 0
\(965\) −0.581943 −0.0187334
\(966\) 0 0
\(967\) 47.2953 1.52092 0.760458 0.649388i \(-0.224975\pi\)
0.760458 + 0.649388i \(0.224975\pi\)
\(968\) 0 0
\(969\) −19.5298 −0.627386
\(970\) 0 0
\(971\) 2.80126 0.0898966 0.0449483 0.998989i \(-0.485688\pi\)
0.0449483 + 0.998989i \(0.485688\pi\)
\(972\) 0 0
\(973\) 16.1930 0.519125
\(974\) 0 0
\(975\) 108.076 3.46121
\(976\) 0 0
\(977\) −22.0570 −0.705665 −0.352832 0.935687i \(-0.614781\pi\)
−0.352832 + 0.935687i \(0.614781\pi\)
\(978\) 0 0
\(979\) 11.2562 0.359751
\(980\) 0 0
\(981\) −0.760677 −0.0242865
\(982\) 0 0
\(983\) 55.5608 1.77212 0.886058 0.463575i \(-0.153434\pi\)
0.886058 + 0.463575i \(0.153434\pi\)
\(984\) 0 0
\(985\) 43.5217 1.38672
\(986\) 0 0
\(987\) −13.9963 −0.445507
\(988\) 0 0
\(989\) −16.2883 −0.517938
\(990\) 0 0
\(991\) −13.9880 −0.444345 −0.222173 0.975007i \(-0.571315\pi\)
−0.222173 + 0.975007i \(0.571315\pi\)
\(992\) 0 0
\(993\) −64.4042 −2.04380
\(994\) 0 0
\(995\) −21.3119 −0.675631
\(996\) 0 0
\(997\) 13.9439 0.441608 0.220804 0.975318i \(-0.429132\pi\)
0.220804 + 0.975318i \(0.429132\pi\)
\(998\) 0 0
\(999\) 40.0825 1.26816
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 7168.2.a.bb.1.3 8
4.3 odd 2 7168.2.a.ba.1.6 8
8.3 odd 2 7168.2.a.be.1.3 8
8.5 even 2 7168.2.a.bf.1.6 8
32.3 odd 8 1792.2.m.f.1345.7 yes 16
32.5 even 8 1792.2.m.e.449.7 16
32.11 odd 8 1792.2.m.f.449.7 yes 16
32.13 even 8 1792.2.m.e.1345.7 yes 16
32.19 odd 8 1792.2.m.g.1345.2 yes 16
32.21 even 8 1792.2.m.h.449.2 yes 16
32.27 odd 8 1792.2.m.g.449.2 yes 16
32.29 even 8 1792.2.m.h.1345.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.7 16 32.5 even 8
1792.2.m.e.1345.7 yes 16 32.13 even 8
1792.2.m.f.449.7 yes 16 32.11 odd 8
1792.2.m.f.1345.7 yes 16 32.3 odd 8
1792.2.m.g.449.2 yes 16 32.27 odd 8
1792.2.m.g.1345.2 yes 16 32.19 odd 8
1792.2.m.h.449.2 yes 16 32.21 even 8
1792.2.m.h.1345.2 yes 16 32.29 even 8
7168.2.a.ba.1.6 8 4.3 odd 2
7168.2.a.bb.1.3 8 1.1 even 1 trivial
7168.2.a.be.1.3 8 8.3 odd 2
7168.2.a.bf.1.6 8 8.5 even 2