Properties

Label 8004.2.a.e.1.6
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.975163\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.900888 q^{5} +3.90501 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.900888 q^{5} +3.90501 q^{7} +1.00000 q^{9} -5.39807 q^{11} -2.18514 q^{13} -0.900888 q^{15} -2.64370 q^{17} +3.37158 q^{19} -3.90501 q^{21} -1.00000 q^{23} -4.18840 q^{25} -1.00000 q^{27} +1.00000 q^{29} +5.08447 q^{31} +5.39807 q^{33} +3.51798 q^{35} -3.79035 q^{37} +2.18514 q^{39} +8.82974 q^{41} +0.613535 q^{43} +0.900888 q^{45} +1.02676 q^{47} +8.24908 q^{49} +2.64370 q^{51} +5.94186 q^{53} -4.86306 q^{55} -3.37158 q^{57} -10.0058 q^{59} -3.09826 q^{61} +3.90501 q^{63} -1.96857 q^{65} -12.6836 q^{67} +1.00000 q^{69} -4.69605 q^{71} +9.03175 q^{73} +4.18840 q^{75} -21.0795 q^{77} -3.30647 q^{79} +1.00000 q^{81} -1.01449 q^{83} -2.38168 q^{85} -1.00000 q^{87} -17.4884 q^{89} -8.53300 q^{91} -5.08447 q^{93} +3.03742 q^{95} +4.91409 q^{97} -5.39807 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) 0.900888 0.402889 0.201445 0.979500i \(-0.435436\pi\)
0.201445 + 0.979500i \(0.435436\pi\)
\(6\) 0 0
\(7\) 3.90501 1.47595 0.737977 0.674826i \(-0.235781\pi\)
0.737977 + 0.674826i \(0.235781\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.39807 −1.62758 −0.813790 0.581159i \(-0.802600\pi\)
−0.813790 + 0.581159i \(0.802600\pi\)
\(12\) 0 0
\(13\) −2.18514 −0.606050 −0.303025 0.952983i \(-0.597997\pi\)
−0.303025 + 0.952983i \(0.597997\pi\)
\(14\) 0 0
\(15\) −0.900888 −0.232608
\(16\) 0 0
\(17\) −2.64370 −0.641191 −0.320595 0.947216i \(-0.603883\pi\)
−0.320595 + 0.947216i \(0.603883\pi\)
\(18\) 0 0
\(19\) 3.37158 0.773494 0.386747 0.922186i \(-0.373599\pi\)
0.386747 + 0.922186i \(0.373599\pi\)
\(20\) 0 0
\(21\) −3.90501 −0.852142
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.18840 −0.837680
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 5.08447 0.913197 0.456599 0.889673i \(-0.349068\pi\)
0.456599 + 0.889673i \(0.349068\pi\)
\(32\) 0 0
\(33\) 5.39807 0.939684
\(34\) 0 0
\(35\) 3.51798 0.594646
\(36\) 0 0
\(37\) −3.79035 −0.623130 −0.311565 0.950225i \(-0.600853\pi\)
−0.311565 + 0.950225i \(0.600853\pi\)
\(38\) 0 0
\(39\) 2.18514 0.349903
\(40\) 0 0
\(41\) 8.82974 1.37897 0.689487 0.724298i \(-0.257836\pi\)
0.689487 + 0.724298i \(0.257836\pi\)
\(42\) 0 0
\(43\) 0.613535 0.0935632 0.0467816 0.998905i \(-0.485104\pi\)
0.0467816 + 0.998905i \(0.485104\pi\)
\(44\) 0 0
\(45\) 0.900888 0.134296
\(46\) 0 0
\(47\) 1.02676 0.149768 0.0748839 0.997192i \(-0.476141\pi\)
0.0748839 + 0.997192i \(0.476141\pi\)
\(48\) 0 0
\(49\) 8.24908 1.17844
\(50\) 0 0
\(51\) 2.64370 0.370192
\(52\) 0 0
\(53\) 5.94186 0.816177 0.408088 0.912942i \(-0.366196\pi\)
0.408088 + 0.912942i \(0.366196\pi\)
\(54\) 0 0
\(55\) −4.86306 −0.655735
\(56\) 0 0
\(57\) −3.37158 −0.446577
\(58\) 0 0
\(59\) −10.0058 −1.30265 −0.651323 0.758800i \(-0.725786\pi\)
−0.651323 + 0.758800i \(0.725786\pi\)
\(60\) 0 0
\(61\) −3.09826 −0.396692 −0.198346 0.980132i \(-0.563557\pi\)
−0.198346 + 0.980132i \(0.563557\pi\)
\(62\) 0 0
\(63\) 3.90501 0.491985
\(64\) 0 0
\(65\) −1.96857 −0.244171
\(66\) 0 0
\(67\) −12.6836 −1.54954 −0.774771 0.632242i \(-0.782135\pi\)
−0.774771 + 0.632242i \(0.782135\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −4.69605 −0.557318 −0.278659 0.960390i \(-0.589890\pi\)
−0.278659 + 0.960390i \(0.589890\pi\)
\(72\) 0 0
\(73\) 9.03175 1.05709 0.528543 0.848906i \(-0.322738\pi\)
0.528543 + 0.848906i \(0.322738\pi\)
\(74\) 0 0
\(75\) 4.18840 0.483635
\(76\) 0 0
\(77\) −21.0795 −2.40223
\(78\) 0 0
\(79\) −3.30647 −0.372007 −0.186004 0.982549i \(-0.559554\pi\)
−0.186004 + 0.982549i \(0.559554\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.01449 −0.111354 −0.0556772 0.998449i \(-0.517732\pi\)
−0.0556772 + 0.998449i \(0.517732\pi\)
\(84\) 0 0
\(85\) −2.38168 −0.258329
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −17.4884 −1.85376 −0.926882 0.375352i \(-0.877522\pi\)
−0.926882 + 0.375352i \(0.877522\pi\)
\(90\) 0 0
\(91\) −8.53300 −0.894501
\(92\) 0 0
\(93\) −5.08447 −0.527235
\(94\) 0 0
\(95\) 3.03742 0.311632
\(96\) 0 0
\(97\) 4.91409 0.498950 0.249475 0.968381i \(-0.419742\pi\)
0.249475 + 0.968381i \(0.419742\pi\)
\(98\) 0 0
\(99\) −5.39807 −0.542527
\(100\) 0 0
\(101\) −12.7246 −1.26615 −0.633074 0.774091i \(-0.718207\pi\)
−0.633074 + 0.774091i \(0.718207\pi\)
\(102\) 0 0
\(103\) −8.55800 −0.843245 −0.421622 0.906772i \(-0.638539\pi\)
−0.421622 + 0.906772i \(0.638539\pi\)
\(104\) 0 0
\(105\) −3.51798 −0.343319
\(106\) 0 0
\(107\) −14.5746 −1.40898 −0.704489 0.709715i \(-0.748824\pi\)
−0.704489 + 0.709715i \(0.748824\pi\)
\(108\) 0 0
\(109\) 8.01651 0.767842 0.383921 0.923366i \(-0.374573\pi\)
0.383921 + 0.923366i \(0.374573\pi\)
\(110\) 0 0
\(111\) 3.79035 0.359764
\(112\) 0 0
\(113\) −14.3051 −1.34571 −0.672856 0.739774i \(-0.734933\pi\)
−0.672856 + 0.739774i \(0.734933\pi\)
\(114\) 0 0
\(115\) −0.900888 −0.0840083
\(116\) 0 0
\(117\) −2.18514 −0.202017
\(118\) 0 0
\(119\) −10.3237 −0.946368
\(120\) 0 0
\(121\) 18.1392 1.64902
\(122\) 0 0
\(123\) −8.82974 −0.796151
\(124\) 0 0
\(125\) −8.27772 −0.740382
\(126\) 0 0
\(127\) 14.7786 1.31139 0.655696 0.755025i \(-0.272375\pi\)
0.655696 + 0.755025i \(0.272375\pi\)
\(128\) 0 0
\(129\) −0.613535 −0.0540187
\(130\) 0 0
\(131\) 0.115094 0.0100558 0.00502789 0.999987i \(-0.498400\pi\)
0.00502789 + 0.999987i \(0.498400\pi\)
\(132\) 0 0
\(133\) 13.1660 1.14164
\(134\) 0 0
\(135\) −0.900888 −0.0775361
\(136\) 0 0
\(137\) 7.19790 0.614958 0.307479 0.951555i \(-0.400515\pi\)
0.307479 + 0.951555i \(0.400515\pi\)
\(138\) 0 0
\(139\) −4.33778 −0.367925 −0.183963 0.982933i \(-0.558893\pi\)
−0.183963 + 0.982933i \(0.558893\pi\)
\(140\) 0 0
\(141\) −1.02676 −0.0864684
\(142\) 0 0
\(143\) 11.7956 0.986395
\(144\) 0 0
\(145\) 0.900888 0.0748147
\(146\) 0 0
\(147\) −8.24908 −0.680373
\(148\) 0 0
\(149\) −11.5364 −0.945099 −0.472550 0.881304i \(-0.656666\pi\)
−0.472550 + 0.881304i \(0.656666\pi\)
\(150\) 0 0
\(151\) 13.6790 1.11318 0.556592 0.830786i \(-0.312109\pi\)
0.556592 + 0.830786i \(0.312109\pi\)
\(152\) 0 0
\(153\) −2.64370 −0.213730
\(154\) 0 0
\(155\) 4.58054 0.367918
\(156\) 0 0
\(157\) 0.0214029 0.00170814 0.000854068 1.00000i \(-0.499728\pi\)
0.000854068 1.00000i \(0.499728\pi\)
\(158\) 0 0
\(159\) −5.94186 −0.471220
\(160\) 0 0
\(161\) −3.90501 −0.307758
\(162\) 0 0
\(163\) −0.555776 −0.0435318 −0.0217659 0.999763i \(-0.506929\pi\)
−0.0217659 + 0.999763i \(0.506929\pi\)
\(164\) 0 0
\(165\) 4.86306 0.378589
\(166\) 0 0
\(167\) 6.90544 0.534359 0.267179 0.963647i \(-0.413908\pi\)
0.267179 + 0.963647i \(0.413908\pi\)
\(168\) 0 0
\(169\) −8.22515 −0.632704
\(170\) 0 0
\(171\) 3.37158 0.257831
\(172\) 0 0
\(173\) 5.27785 0.401267 0.200634 0.979666i \(-0.435700\pi\)
0.200634 + 0.979666i \(0.435700\pi\)
\(174\) 0 0
\(175\) −16.3557 −1.23638
\(176\) 0 0
\(177\) 10.0058 0.752083
\(178\) 0 0
\(179\) −6.77665 −0.506511 −0.253255 0.967399i \(-0.581501\pi\)
−0.253255 + 0.967399i \(0.581501\pi\)
\(180\) 0 0
\(181\) −22.0103 −1.63601 −0.818005 0.575211i \(-0.804920\pi\)
−0.818005 + 0.575211i \(0.804920\pi\)
\(182\) 0 0
\(183\) 3.09826 0.229030
\(184\) 0 0
\(185\) −3.41469 −0.251053
\(186\) 0 0
\(187\) 14.2709 1.04359
\(188\) 0 0
\(189\) −3.90501 −0.284047
\(190\) 0 0
\(191\) 11.7210 0.848103 0.424051 0.905638i \(-0.360608\pi\)
0.424051 + 0.905638i \(0.360608\pi\)
\(192\) 0 0
\(193\) 8.21648 0.591435 0.295717 0.955275i \(-0.404441\pi\)
0.295717 + 0.955275i \(0.404441\pi\)
\(194\) 0 0
\(195\) 1.96857 0.140972
\(196\) 0 0
\(197\) 0.510756 0.0363899 0.0181949 0.999834i \(-0.494208\pi\)
0.0181949 + 0.999834i \(0.494208\pi\)
\(198\) 0 0
\(199\) −9.23340 −0.654538 −0.327269 0.944931i \(-0.606128\pi\)
−0.327269 + 0.944931i \(0.606128\pi\)
\(200\) 0 0
\(201\) 12.6836 0.894629
\(202\) 0 0
\(203\) 3.90501 0.274078
\(204\) 0 0
\(205\) 7.95461 0.555574
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −18.2000 −1.25892
\(210\) 0 0
\(211\) 13.8507 0.953520 0.476760 0.879033i \(-0.341811\pi\)
0.476760 + 0.879033i \(0.341811\pi\)
\(212\) 0 0
\(213\) 4.69605 0.321768
\(214\) 0 0
\(215\) 0.552726 0.0376956
\(216\) 0 0
\(217\) 19.8549 1.34784
\(218\) 0 0
\(219\) −9.03175 −0.610309
\(220\) 0 0
\(221\) 5.77686 0.388593
\(222\) 0 0
\(223\) 0.400408 0.0268133 0.0134066 0.999910i \(-0.495732\pi\)
0.0134066 + 0.999910i \(0.495732\pi\)
\(224\) 0 0
\(225\) −4.18840 −0.279227
\(226\) 0 0
\(227\) 6.20230 0.411661 0.205830 0.978588i \(-0.434010\pi\)
0.205830 + 0.978588i \(0.434010\pi\)
\(228\) 0 0
\(229\) 7.12470 0.470813 0.235407 0.971897i \(-0.424358\pi\)
0.235407 + 0.971897i \(0.424358\pi\)
\(230\) 0 0
\(231\) 21.0795 1.38693
\(232\) 0 0
\(233\) −8.64191 −0.566150 −0.283075 0.959098i \(-0.591355\pi\)
−0.283075 + 0.959098i \(0.591355\pi\)
\(234\) 0 0
\(235\) 0.924992 0.0603398
\(236\) 0 0
\(237\) 3.30647 0.214778
\(238\) 0 0
\(239\) 15.2043 0.983486 0.491743 0.870741i \(-0.336360\pi\)
0.491743 + 0.870741i \(0.336360\pi\)
\(240\) 0 0
\(241\) −23.0248 −1.48316 −0.741580 0.670865i \(-0.765923\pi\)
−0.741580 + 0.670865i \(0.765923\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 7.43150 0.474781
\(246\) 0 0
\(247\) −7.36739 −0.468776
\(248\) 0 0
\(249\) 1.01449 0.0642905
\(250\) 0 0
\(251\) 18.5440 1.17049 0.585245 0.810857i \(-0.300998\pi\)
0.585245 + 0.810857i \(0.300998\pi\)
\(252\) 0 0
\(253\) 5.39807 0.339374
\(254\) 0 0
\(255\) 2.38168 0.149146
\(256\) 0 0
\(257\) −12.8335 −0.800531 −0.400265 0.916399i \(-0.631082\pi\)
−0.400265 + 0.916399i \(0.631082\pi\)
\(258\) 0 0
\(259\) −14.8014 −0.919712
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −14.8682 −0.916812 −0.458406 0.888743i \(-0.651579\pi\)
−0.458406 + 0.888743i \(0.651579\pi\)
\(264\) 0 0
\(265\) 5.35295 0.328829
\(266\) 0 0
\(267\) 17.4884 1.07027
\(268\) 0 0
\(269\) −15.3258 −0.934430 −0.467215 0.884144i \(-0.654743\pi\)
−0.467215 + 0.884144i \(0.654743\pi\)
\(270\) 0 0
\(271\) −25.3597 −1.54049 −0.770246 0.637747i \(-0.779867\pi\)
−0.770246 + 0.637747i \(0.779867\pi\)
\(272\) 0 0
\(273\) 8.53300 0.516441
\(274\) 0 0
\(275\) 22.6093 1.36339
\(276\) 0 0
\(277\) −14.5368 −0.873430 −0.436715 0.899600i \(-0.643858\pi\)
−0.436715 + 0.899600i \(0.643858\pi\)
\(278\) 0 0
\(279\) 5.08447 0.304399
\(280\) 0 0
\(281\) −27.7945 −1.65808 −0.829040 0.559189i \(-0.811113\pi\)
−0.829040 + 0.559189i \(0.811113\pi\)
\(282\) 0 0
\(283\) −4.01747 −0.238814 −0.119407 0.992845i \(-0.538099\pi\)
−0.119407 + 0.992845i \(0.538099\pi\)
\(284\) 0 0
\(285\) −3.03742 −0.179921
\(286\) 0 0
\(287\) 34.4802 2.03530
\(288\) 0 0
\(289\) −10.0109 −0.588875
\(290\) 0 0
\(291\) −4.91409 −0.288069
\(292\) 0 0
\(293\) −32.9948 −1.92757 −0.963787 0.266673i \(-0.914076\pi\)
−0.963787 + 0.266673i \(0.914076\pi\)
\(294\) 0 0
\(295\) −9.01412 −0.524823
\(296\) 0 0
\(297\) 5.39807 0.313228
\(298\) 0 0
\(299\) 2.18514 0.126370
\(300\) 0 0
\(301\) 2.39586 0.138095
\(302\) 0 0
\(303\) 12.7246 0.731011
\(304\) 0 0
\(305\) −2.79119 −0.159823
\(306\) 0 0
\(307\) −9.56599 −0.545960 −0.272980 0.962020i \(-0.588009\pi\)
−0.272980 + 0.962020i \(0.588009\pi\)
\(308\) 0 0
\(309\) 8.55800 0.486848
\(310\) 0 0
\(311\) 4.28315 0.242875 0.121438 0.992599i \(-0.461250\pi\)
0.121438 + 0.992599i \(0.461250\pi\)
\(312\) 0 0
\(313\) 0.673397 0.0380626 0.0190313 0.999819i \(-0.493942\pi\)
0.0190313 + 0.999819i \(0.493942\pi\)
\(314\) 0 0
\(315\) 3.51798 0.198215
\(316\) 0 0
\(317\) −17.0792 −0.959263 −0.479632 0.877470i \(-0.659230\pi\)
−0.479632 + 0.877470i \(0.659230\pi\)
\(318\) 0 0
\(319\) −5.39807 −0.302234
\(320\) 0 0
\(321\) 14.5746 0.813474
\(322\) 0 0
\(323\) −8.91344 −0.495957
\(324\) 0 0
\(325\) 9.15225 0.507676
\(326\) 0 0
\(327\) −8.01651 −0.443314
\(328\) 0 0
\(329\) 4.00949 0.221050
\(330\) 0 0
\(331\) 7.39283 0.406347 0.203173 0.979143i \(-0.434875\pi\)
0.203173 + 0.979143i \(0.434875\pi\)
\(332\) 0 0
\(333\) −3.79035 −0.207710
\(334\) 0 0
\(335\) −11.4265 −0.624294
\(336\) 0 0
\(337\) 14.4508 0.787186 0.393593 0.919285i \(-0.371232\pi\)
0.393593 + 0.919285i \(0.371232\pi\)
\(338\) 0 0
\(339\) 14.3051 0.776947
\(340\) 0 0
\(341\) −27.4463 −1.48630
\(342\) 0 0
\(343\) 4.87767 0.263369
\(344\) 0 0
\(345\) 0.900888 0.0485022
\(346\) 0 0
\(347\) 16.0698 0.862669 0.431335 0.902192i \(-0.358043\pi\)
0.431335 + 0.902192i \(0.358043\pi\)
\(348\) 0 0
\(349\) 17.8796 0.957075 0.478537 0.878067i \(-0.341167\pi\)
0.478537 + 0.878067i \(0.341167\pi\)
\(350\) 0 0
\(351\) 2.18514 0.116634
\(352\) 0 0
\(353\) −31.2236 −1.66187 −0.830933 0.556373i \(-0.812193\pi\)
−0.830933 + 0.556373i \(0.812193\pi\)
\(354\) 0 0
\(355\) −4.23061 −0.224538
\(356\) 0 0
\(357\) 10.3237 0.546386
\(358\) 0 0
\(359\) −23.6558 −1.24851 −0.624253 0.781222i \(-0.714596\pi\)
−0.624253 + 0.781222i \(0.714596\pi\)
\(360\) 0 0
\(361\) −7.63244 −0.401708
\(362\) 0 0
\(363\) −18.1392 −0.952062
\(364\) 0 0
\(365\) 8.13660 0.425889
\(366\) 0 0
\(367\) 30.1044 1.57144 0.785719 0.618583i \(-0.212293\pi\)
0.785719 + 0.618583i \(0.212293\pi\)
\(368\) 0 0
\(369\) 8.82974 0.459658
\(370\) 0 0
\(371\) 23.2030 1.20464
\(372\) 0 0
\(373\) −25.6816 −1.32974 −0.664871 0.746958i \(-0.731513\pi\)
−0.664871 + 0.746958i \(0.731513\pi\)
\(374\) 0 0
\(375\) 8.27772 0.427460
\(376\) 0 0
\(377\) −2.18514 −0.112541
\(378\) 0 0
\(379\) 24.8901 1.27852 0.639260 0.768991i \(-0.279241\pi\)
0.639260 + 0.768991i \(0.279241\pi\)
\(380\) 0 0
\(381\) −14.7786 −0.757133
\(382\) 0 0
\(383\) 5.32682 0.272188 0.136094 0.990696i \(-0.456545\pi\)
0.136094 + 0.990696i \(0.456545\pi\)
\(384\) 0 0
\(385\) −18.9903 −0.967835
\(386\) 0 0
\(387\) 0.613535 0.0311877
\(388\) 0 0
\(389\) 31.6443 1.60443 0.802216 0.597034i \(-0.203654\pi\)
0.802216 + 0.597034i \(0.203654\pi\)
\(390\) 0 0
\(391\) 2.64370 0.133697
\(392\) 0 0
\(393\) −0.115094 −0.00580571
\(394\) 0 0
\(395\) −2.97876 −0.149878
\(396\) 0 0
\(397\) −1.70334 −0.0854881 −0.0427441 0.999086i \(-0.513610\pi\)
−0.0427441 + 0.999086i \(0.513610\pi\)
\(398\) 0 0
\(399\) −13.1660 −0.659127
\(400\) 0 0
\(401\) 24.1375 1.20537 0.602684 0.797980i \(-0.294098\pi\)
0.602684 + 0.797980i \(0.294098\pi\)
\(402\) 0 0
\(403\) −11.1103 −0.553443
\(404\) 0 0
\(405\) 0.900888 0.0447655
\(406\) 0 0
\(407\) 20.4606 1.01419
\(408\) 0 0
\(409\) −24.7975 −1.22616 −0.613078 0.790022i \(-0.710069\pi\)
−0.613078 + 0.790022i \(0.710069\pi\)
\(410\) 0 0
\(411\) −7.19790 −0.355046
\(412\) 0 0
\(413\) −39.0728 −1.92265
\(414\) 0 0
\(415\) −0.913939 −0.0448635
\(416\) 0 0
\(417\) 4.33778 0.212422
\(418\) 0 0
\(419\) 23.5959 1.15274 0.576368 0.817190i \(-0.304470\pi\)
0.576368 + 0.817190i \(0.304470\pi\)
\(420\) 0 0
\(421\) −34.6655 −1.68949 −0.844746 0.535167i \(-0.820249\pi\)
−0.844746 + 0.535167i \(0.820249\pi\)
\(422\) 0 0
\(423\) 1.02676 0.0499226
\(424\) 0 0
\(425\) 11.0729 0.537113
\(426\) 0 0
\(427\) −12.0987 −0.585499
\(428\) 0 0
\(429\) −11.7956 −0.569495
\(430\) 0 0
\(431\) −2.67396 −0.128800 −0.0644001 0.997924i \(-0.520513\pi\)
−0.0644001 + 0.997924i \(0.520513\pi\)
\(432\) 0 0
\(433\) 26.4396 1.27061 0.635304 0.772262i \(-0.280875\pi\)
0.635304 + 0.772262i \(0.280875\pi\)
\(434\) 0 0
\(435\) −0.900888 −0.0431943
\(436\) 0 0
\(437\) −3.37158 −0.161285
\(438\) 0 0
\(439\) −6.00149 −0.286436 −0.143218 0.989691i \(-0.545745\pi\)
−0.143218 + 0.989691i \(0.545745\pi\)
\(440\) 0 0
\(441\) 8.24908 0.392813
\(442\) 0 0
\(443\) 22.8584 1.08604 0.543019 0.839721i \(-0.317281\pi\)
0.543019 + 0.839721i \(0.317281\pi\)
\(444\) 0 0
\(445\) −15.7551 −0.746862
\(446\) 0 0
\(447\) 11.5364 0.545653
\(448\) 0 0
\(449\) 9.28471 0.438173 0.219086 0.975705i \(-0.429692\pi\)
0.219086 + 0.975705i \(0.429692\pi\)
\(450\) 0 0
\(451\) −47.6636 −2.24439
\(452\) 0 0
\(453\) −13.6790 −0.642697
\(454\) 0 0
\(455\) −7.68728 −0.360385
\(456\) 0 0
\(457\) 11.6566 0.545275 0.272637 0.962117i \(-0.412104\pi\)
0.272637 + 0.962117i \(0.412104\pi\)
\(458\) 0 0
\(459\) 2.64370 0.123397
\(460\) 0 0
\(461\) 14.0282 0.653359 0.326680 0.945135i \(-0.394070\pi\)
0.326680 + 0.945135i \(0.394070\pi\)
\(462\) 0 0
\(463\) 37.9064 1.76166 0.880829 0.473434i \(-0.156986\pi\)
0.880829 + 0.473434i \(0.156986\pi\)
\(464\) 0 0
\(465\) −4.58054 −0.212417
\(466\) 0 0
\(467\) −25.5002 −1.18001 −0.590004 0.807400i \(-0.700874\pi\)
−0.590004 + 0.807400i \(0.700874\pi\)
\(468\) 0 0
\(469\) −49.5294 −2.28705
\(470\) 0 0
\(471\) −0.0214029 −0.000986192 0
\(472\) 0 0
\(473\) −3.31191 −0.152282
\(474\) 0 0
\(475\) −14.1215 −0.647940
\(476\) 0 0
\(477\) 5.94186 0.272059
\(478\) 0 0
\(479\) −36.0071 −1.64521 −0.822603 0.568616i \(-0.807479\pi\)
−0.822603 + 0.568616i \(0.807479\pi\)
\(480\) 0 0
\(481\) 8.28247 0.377648
\(482\) 0 0
\(483\) 3.90501 0.177684
\(484\) 0 0
\(485\) 4.42704 0.201022
\(486\) 0 0
\(487\) −20.6281 −0.934747 −0.467373 0.884060i \(-0.654800\pi\)
−0.467373 + 0.884060i \(0.654800\pi\)
\(488\) 0 0
\(489\) 0.555776 0.0251331
\(490\) 0 0
\(491\) −16.8265 −0.759370 −0.379685 0.925116i \(-0.623968\pi\)
−0.379685 + 0.925116i \(0.623968\pi\)
\(492\) 0 0
\(493\) −2.64370 −0.119066
\(494\) 0 0
\(495\) −4.86306 −0.218578
\(496\) 0 0
\(497\) −18.3381 −0.822576
\(498\) 0 0
\(499\) 39.1885 1.75432 0.877160 0.480198i \(-0.159435\pi\)
0.877160 + 0.480198i \(0.159435\pi\)
\(500\) 0 0
\(501\) −6.90544 −0.308512
\(502\) 0 0
\(503\) −33.7518 −1.50492 −0.752458 0.658640i \(-0.771132\pi\)
−0.752458 + 0.658640i \(0.771132\pi\)
\(504\) 0 0
\(505\) −11.4635 −0.510118
\(506\) 0 0
\(507\) 8.22515 0.365292
\(508\) 0 0
\(509\) 17.6294 0.781407 0.390704 0.920517i \(-0.372232\pi\)
0.390704 + 0.920517i \(0.372232\pi\)
\(510\) 0 0
\(511\) 35.2690 1.56021
\(512\) 0 0
\(513\) −3.37158 −0.148859
\(514\) 0 0
\(515\) −7.70980 −0.339734
\(516\) 0 0
\(517\) −5.54250 −0.243759
\(518\) 0 0
\(519\) −5.27785 −0.231672
\(520\) 0 0
\(521\) −17.1971 −0.753418 −0.376709 0.926332i \(-0.622944\pi\)
−0.376709 + 0.926332i \(0.622944\pi\)
\(522\) 0 0
\(523\) 18.5539 0.811304 0.405652 0.914028i \(-0.367044\pi\)
0.405652 + 0.914028i \(0.367044\pi\)
\(524\) 0 0
\(525\) 16.3557 0.713823
\(526\) 0 0
\(527\) −13.4418 −0.585534
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −10.0058 −0.434215
\(532\) 0 0
\(533\) −19.2942 −0.835726
\(534\) 0 0
\(535\) −13.1301 −0.567662
\(536\) 0 0
\(537\) 6.77665 0.292434
\(538\) 0 0
\(539\) −44.5292 −1.91801
\(540\) 0 0
\(541\) 3.64277 0.156615 0.0783074 0.996929i \(-0.475048\pi\)
0.0783074 + 0.996929i \(0.475048\pi\)
\(542\) 0 0
\(543\) 22.0103 0.944551
\(544\) 0 0
\(545\) 7.22198 0.309355
\(546\) 0 0
\(547\) −0.827186 −0.0353679 −0.0176840 0.999844i \(-0.505629\pi\)
−0.0176840 + 0.999844i \(0.505629\pi\)
\(548\) 0 0
\(549\) −3.09826 −0.132231
\(550\) 0 0
\(551\) 3.37158 0.143634
\(552\) 0 0
\(553\) −12.9118 −0.549065
\(554\) 0 0
\(555\) 3.41469 0.144945
\(556\) 0 0
\(557\) −31.7654 −1.34594 −0.672972 0.739668i \(-0.734983\pi\)
−0.672972 + 0.739668i \(0.734983\pi\)
\(558\) 0 0
\(559\) −1.34066 −0.0567039
\(560\) 0 0
\(561\) −14.2709 −0.602517
\(562\) 0 0
\(563\) −21.1838 −0.892789 −0.446395 0.894836i \(-0.647292\pi\)
−0.446395 + 0.894836i \(0.647292\pi\)
\(564\) 0 0
\(565\) −12.8873 −0.542173
\(566\) 0 0
\(567\) 3.90501 0.163995
\(568\) 0 0
\(569\) 11.4066 0.478188 0.239094 0.970996i \(-0.423150\pi\)
0.239094 + 0.970996i \(0.423150\pi\)
\(570\) 0 0
\(571\) −20.0422 −0.838740 −0.419370 0.907815i \(-0.637749\pi\)
−0.419370 + 0.907815i \(0.637749\pi\)
\(572\) 0 0
\(573\) −11.7210 −0.489652
\(574\) 0 0
\(575\) 4.18840 0.174668
\(576\) 0 0
\(577\) 35.4444 1.47557 0.737785 0.675036i \(-0.235872\pi\)
0.737785 + 0.675036i \(0.235872\pi\)
\(578\) 0 0
\(579\) −8.21648 −0.341465
\(580\) 0 0
\(581\) −3.96158 −0.164354
\(582\) 0 0
\(583\) −32.0746 −1.32839
\(584\) 0 0
\(585\) −1.96857 −0.0813904
\(586\) 0 0
\(587\) −18.2013 −0.751249 −0.375625 0.926772i \(-0.622572\pi\)
−0.375625 + 0.926772i \(0.622572\pi\)
\(588\) 0 0
\(589\) 17.1427 0.706352
\(590\) 0 0
\(591\) −0.510756 −0.0210097
\(592\) 0 0
\(593\) 32.2026 1.32240 0.661200 0.750209i \(-0.270047\pi\)
0.661200 + 0.750209i \(0.270047\pi\)
\(594\) 0 0
\(595\) −9.30046 −0.381282
\(596\) 0 0
\(597\) 9.23340 0.377898
\(598\) 0 0
\(599\) 9.58646 0.391692 0.195846 0.980635i \(-0.437255\pi\)
0.195846 + 0.980635i \(0.437255\pi\)
\(600\) 0 0
\(601\) −11.3781 −0.464122 −0.232061 0.972701i \(-0.574547\pi\)
−0.232061 + 0.972701i \(0.574547\pi\)
\(602\) 0 0
\(603\) −12.6836 −0.516514
\(604\) 0 0
\(605\) 16.3414 0.664372
\(606\) 0 0
\(607\) −42.0701 −1.70757 −0.853787 0.520623i \(-0.825700\pi\)
−0.853787 + 0.520623i \(0.825700\pi\)
\(608\) 0 0
\(609\) −3.90501 −0.158239
\(610\) 0 0
\(611\) −2.24361 −0.0907667
\(612\) 0 0
\(613\) −25.5846 −1.03335 −0.516676 0.856181i \(-0.672831\pi\)
−0.516676 + 0.856181i \(0.672831\pi\)
\(614\) 0 0
\(615\) −7.95461 −0.320761
\(616\) 0 0
\(617\) −21.8215 −0.878502 −0.439251 0.898364i \(-0.644756\pi\)
−0.439251 + 0.898364i \(0.644756\pi\)
\(618\) 0 0
\(619\) 23.7623 0.955087 0.477544 0.878608i \(-0.341527\pi\)
0.477544 + 0.878608i \(0.341527\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −68.2922 −2.73607
\(624\) 0 0
\(625\) 13.4847 0.539388
\(626\) 0 0
\(627\) 18.2000 0.726840
\(628\) 0 0
\(629\) 10.0205 0.399545
\(630\) 0 0
\(631\) 19.0191 0.757138 0.378569 0.925573i \(-0.376416\pi\)
0.378569 + 0.925573i \(0.376416\pi\)
\(632\) 0 0
\(633\) −13.8507 −0.550515
\(634\) 0 0
\(635\) 13.3139 0.528346
\(636\) 0 0
\(637\) −18.0254 −0.714193
\(638\) 0 0
\(639\) −4.69605 −0.185773
\(640\) 0 0
\(641\) −11.2100 −0.442767 −0.221383 0.975187i \(-0.571057\pi\)
−0.221383 + 0.975187i \(0.571057\pi\)
\(642\) 0 0
\(643\) −22.7058 −0.895430 −0.447715 0.894176i \(-0.647762\pi\)
−0.447715 + 0.894176i \(0.647762\pi\)
\(644\) 0 0
\(645\) −0.552726 −0.0217636
\(646\) 0 0
\(647\) −13.7388 −0.540127 −0.270064 0.962842i \(-0.587045\pi\)
−0.270064 + 0.962842i \(0.587045\pi\)
\(648\) 0 0
\(649\) 54.0121 2.12016
\(650\) 0 0
\(651\) −19.8549 −0.778174
\(652\) 0 0
\(653\) −3.31245 −0.129626 −0.0648130 0.997897i \(-0.520645\pi\)
−0.0648130 + 0.997897i \(0.520645\pi\)
\(654\) 0 0
\(655\) 0.103687 0.00405137
\(656\) 0 0
\(657\) 9.03175 0.352362
\(658\) 0 0
\(659\) 4.02358 0.156736 0.0783681 0.996924i \(-0.475029\pi\)
0.0783681 + 0.996924i \(0.475029\pi\)
\(660\) 0 0
\(661\) −15.0724 −0.586249 −0.293125 0.956074i \(-0.594695\pi\)
−0.293125 + 0.956074i \(0.594695\pi\)
\(662\) 0 0
\(663\) −5.77686 −0.224355
\(664\) 0 0
\(665\) 11.8611 0.459955
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −0.400408 −0.0154807
\(670\) 0 0
\(671\) 16.7246 0.645648
\(672\) 0 0
\(673\) −15.0200 −0.578978 −0.289489 0.957181i \(-0.593485\pi\)
−0.289489 + 0.957181i \(0.593485\pi\)
\(674\) 0 0
\(675\) 4.18840 0.161212
\(676\) 0 0
\(677\) −20.0315 −0.769872 −0.384936 0.922943i \(-0.625776\pi\)
−0.384936 + 0.922943i \(0.625776\pi\)
\(678\) 0 0
\(679\) 19.1895 0.736427
\(680\) 0 0
\(681\) −6.20230 −0.237673
\(682\) 0 0
\(683\) 32.3988 1.23971 0.619853 0.784718i \(-0.287192\pi\)
0.619853 + 0.784718i \(0.287192\pi\)
\(684\) 0 0
\(685\) 6.48450 0.247760
\(686\) 0 0
\(687\) −7.12470 −0.271824
\(688\) 0 0
\(689\) −12.9838 −0.494644
\(690\) 0 0
\(691\) −31.9042 −1.21369 −0.606847 0.794818i \(-0.707566\pi\)
−0.606847 + 0.794818i \(0.707566\pi\)
\(692\) 0 0
\(693\) −21.0795 −0.800745
\(694\) 0 0
\(695\) −3.90785 −0.148233
\(696\) 0 0
\(697\) −23.3432 −0.884185
\(698\) 0 0
\(699\) 8.64191 0.326867
\(700\) 0 0
\(701\) −41.2753 −1.55894 −0.779472 0.626437i \(-0.784513\pi\)
−0.779472 + 0.626437i \(0.784513\pi\)
\(702\) 0 0
\(703\) −12.7795 −0.481987
\(704\) 0 0
\(705\) −0.924992 −0.0348372
\(706\) 0 0
\(707\) −49.6898 −1.86878
\(708\) 0 0
\(709\) −11.4863 −0.431379 −0.215689 0.976462i \(-0.569200\pi\)
−0.215689 + 0.976462i \(0.569200\pi\)
\(710\) 0 0
\(711\) −3.30647 −0.124002
\(712\) 0 0
\(713\) −5.08447 −0.190415
\(714\) 0 0
\(715\) 10.6265 0.397408
\(716\) 0 0
\(717\) −15.2043 −0.567816
\(718\) 0 0
\(719\) 45.5003 1.69687 0.848437 0.529297i \(-0.177544\pi\)
0.848437 + 0.529297i \(0.177544\pi\)
\(720\) 0 0
\(721\) −33.4190 −1.24459
\(722\) 0 0
\(723\) 23.0248 0.856302
\(724\) 0 0
\(725\) −4.18840 −0.155553
\(726\) 0 0
\(727\) 2.52414 0.0936152 0.0468076 0.998904i \(-0.485095\pi\)
0.0468076 + 0.998904i \(0.485095\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.62200 −0.0599918
\(732\) 0 0
\(733\) −22.9871 −0.849049 −0.424524 0.905417i \(-0.639559\pi\)
−0.424524 + 0.905417i \(0.639559\pi\)
\(734\) 0 0
\(735\) −7.43150 −0.274115
\(736\) 0 0
\(737\) 68.4668 2.52201
\(738\) 0 0
\(739\) 28.4478 1.04647 0.523234 0.852189i \(-0.324725\pi\)
0.523234 + 0.852189i \(0.324725\pi\)
\(740\) 0 0
\(741\) 7.36739 0.270648
\(742\) 0 0
\(743\) 9.46727 0.347321 0.173660 0.984806i \(-0.444441\pi\)
0.173660 + 0.984806i \(0.444441\pi\)
\(744\) 0 0
\(745\) −10.3930 −0.380771
\(746\) 0 0
\(747\) −1.01449 −0.0371181
\(748\) 0 0
\(749\) −56.9138 −2.07959
\(750\) 0 0
\(751\) −11.2455 −0.410355 −0.205177 0.978725i \(-0.565777\pi\)
−0.205177 + 0.978725i \(0.565777\pi\)
\(752\) 0 0
\(753\) −18.5440 −0.675782
\(754\) 0 0
\(755\) 12.3233 0.448490
\(756\) 0 0
\(757\) 5.68039 0.206457 0.103229 0.994658i \(-0.467083\pi\)
0.103229 + 0.994658i \(0.467083\pi\)
\(758\) 0 0
\(759\) −5.39807 −0.195938
\(760\) 0 0
\(761\) −14.1168 −0.511734 −0.255867 0.966712i \(-0.582361\pi\)
−0.255867 + 0.966712i \(0.582361\pi\)
\(762\) 0 0
\(763\) 31.3045 1.13330
\(764\) 0 0
\(765\) −2.38168 −0.0861097
\(766\) 0 0
\(767\) 21.8641 0.789469
\(768\) 0 0
\(769\) −1.36141 −0.0490938 −0.0245469 0.999699i \(-0.507814\pi\)
−0.0245469 + 0.999699i \(0.507814\pi\)
\(770\) 0 0
\(771\) 12.8335 0.462187
\(772\) 0 0
\(773\) −36.4036 −1.30935 −0.654673 0.755912i \(-0.727194\pi\)
−0.654673 + 0.755912i \(0.727194\pi\)
\(774\) 0 0
\(775\) −21.2958 −0.764967
\(776\) 0 0
\(777\) 14.8014 0.530996
\(778\) 0 0
\(779\) 29.7702 1.06663
\(780\) 0 0
\(781\) 25.3496 0.907080
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 0.0192816 0.000688190 0
\(786\) 0 0
\(787\) −53.5148 −1.90760 −0.953799 0.300447i \(-0.902864\pi\)
−0.953799 + 0.300447i \(0.902864\pi\)
\(788\) 0 0
\(789\) 14.8682 0.529322
\(790\) 0 0
\(791\) −55.8616 −1.98621
\(792\) 0 0
\(793\) 6.77015 0.240415
\(794\) 0 0
\(795\) −5.35295 −0.189850
\(796\) 0 0
\(797\) 43.8576 1.55351 0.776757 0.629800i \(-0.216863\pi\)
0.776757 + 0.629800i \(0.216863\pi\)
\(798\) 0 0
\(799\) −2.71443 −0.0960297
\(800\) 0 0
\(801\) −17.4884 −0.617921
\(802\) 0 0
\(803\) −48.7541 −1.72049
\(804\) 0 0
\(805\) −3.51798 −0.123992
\(806\) 0 0
\(807\) 15.3258 0.539493
\(808\) 0 0
\(809\) −20.4766 −0.719920 −0.359960 0.932968i \(-0.617210\pi\)
−0.359960 + 0.932968i \(0.617210\pi\)
\(810\) 0 0
\(811\) 41.0750 1.44234 0.721170 0.692758i \(-0.243605\pi\)
0.721170 + 0.692758i \(0.243605\pi\)
\(812\) 0 0
\(813\) 25.3597 0.889403
\(814\) 0 0
\(815\) −0.500692 −0.0175385
\(816\) 0 0
\(817\) 2.06858 0.0723705
\(818\) 0 0
\(819\) −8.53300 −0.298167
\(820\) 0 0
\(821\) −34.0314 −1.18770 −0.593852 0.804574i \(-0.702393\pi\)
−0.593852 + 0.804574i \(0.702393\pi\)
\(822\) 0 0
\(823\) −54.7566 −1.90870 −0.954348 0.298698i \(-0.903448\pi\)
−0.954348 + 0.298698i \(0.903448\pi\)
\(824\) 0 0
\(825\) −22.6093 −0.787155
\(826\) 0 0
\(827\) 4.40663 0.153233 0.0766167 0.997061i \(-0.475588\pi\)
0.0766167 + 0.997061i \(0.475588\pi\)
\(828\) 0 0
\(829\) −27.7917 −0.965244 −0.482622 0.875829i \(-0.660315\pi\)
−0.482622 + 0.875829i \(0.660315\pi\)
\(830\) 0 0
\(831\) 14.5368 0.504275
\(832\) 0 0
\(833\) −21.8081 −0.755605
\(834\) 0 0
\(835\) 6.22103 0.215287
\(836\) 0 0
\(837\) −5.08447 −0.175745
\(838\) 0 0
\(839\) 35.7491 1.23420 0.617098 0.786886i \(-0.288308\pi\)
0.617098 + 0.786886i \(0.288308\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 27.7945 0.957293
\(844\) 0 0
\(845\) −7.40994 −0.254910
\(846\) 0 0
\(847\) 70.8337 2.43388
\(848\) 0 0
\(849\) 4.01747 0.137879
\(850\) 0 0
\(851\) 3.79035 0.129932
\(852\) 0 0
\(853\) 20.2840 0.694509 0.347255 0.937771i \(-0.387114\pi\)
0.347255 + 0.937771i \(0.387114\pi\)
\(854\) 0 0
\(855\) 3.03742 0.103877
\(856\) 0 0
\(857\) 11.5100 0.393175 0.196588 0.980486i \(-0.437014\pi\)
0.196588 + 0.980486i \(0.437014\pi\)
\(858\) 0 0
\(859\) −44.2221 −1.50884 −0.754419 0.656393i \(-0.772081\pi\)
−0.754419 + 0.656393i \(0.772081\pi\)
\(860\) 0 0
\(861\) −34.4802 −1.17508
\(862\) 0 0
\(863\) −26.5666 −0.904337 −0.452169 0.891932i \(-0.649349\pi\)
−0.452169 + 0.891932i \(0.649349\pi\)
\(864\) 0 0
\(865\) 4.75475 0.161666
\(866\) 0 0
\(867\) 10.0109 0.339987
\(868\) 0 0
\(869\) 17.8486 0.605471
\(870\) 0 0
\(871\) 27.7154 0.939100
\(872\) 0 0
\(873\) 4.91409 0.166317
\(874\) 0 0
\(875\) −32.3246 −1.09277
\(876\) 0 0
\(877\) 23.1214 0.780754 0.390377 0.920655i \(-0.372345\pi\)
0.390377 + 0.920655i \(0.372345\pi\)
\(878\) 0 0
\(879\) 32.9948 1.11289
\(880\) 0 0
\(881\) 21.4796 0.723666 0.361833 0.932243i \(-0.382151\pi\)
0.361833 + 0.932243i \(0.382151\pi\)
\(882\) 0 0
\(883\) −11.9649 −0.402650 −0.201325 0.979525i \(-0.564525\pi\)
−0.201325 + 0.979525i \(0.564525\pi\)
\(884\) 0 0
\(885\) 9.01412 0.303006
\(886\) 0 0
\(887\) 43.8213 1.47138 0.735688 0.677320i \(-0.236859\pi\)
0.735688 + 0.677320i \(0.236859\pi\)
\(888\) 0 0
\(889\) 57.7107 1.93556
\(890\) 0 0
\(891\) −5.39807 −0.180842
\(892\) 0 0
\(893\) 3.46179 0.115844
\(894\) 0 0
\(895\) −6.10501 −0.204068
\(896\) 0 0
\(897\) −2.18514 −0.0729598
\(898\) 0 0
\(899\) 5.08447 0.169576
\(900\) 0 0
\(901\) −15.7085 −0.523325
\(902\) 0 0
\(903\) −2.39586 −0.0797291
\(904\) 0 0
\(905\) −19.8288 −0.659131
\(906\) 0 0
\(907\) 37.5669 1.24739 0.623694 0.781668i \(-0.285631\pi\)
0.623694 + 0.781668i \(0.285631\pi\)
\(908\) 0 0
\(909\) −12.7246 −0.422050
\(910\) 0 0
\(911\) −6.78171 −0.224688 −0.112344 0.993669i \(-0.535836\pi\)
−0.112344 + 0.993669i \(0.535836\pi\)
\(912\) 0 0
\(913\) 5.47627 0.181238
\(914\) 0 0
\(915\) 2.79119 0.0922738
\(916\) 0 0
\(917\) 0.449442 0.0148419
\(918\) 0 0
\(919\) −35.3208 −1.16513 −0.582563 0.812786i \(-0.697950\pi\)
−0.582563 + 0.812786i \(0.697950\pi\)
\(920\) 0 0
\(921\) 9.56599 0.315210
\(922\) 0 0
\(923\) 10.2615 0.337762
\(924\) 0 0
\(925\) 15.8755 0.521984
\(926\) 0 0
\(927\) −8.55800 −0.281082
\(928\) 0 0
\(929\) −25.8804 −0.849108 −0.424554 0.905403i \(-0.639569\pi\)
−0.424554 + 0.905403i \(0.639569\pi\)
\(930\) 0 0
\(931\) 27.8124 0.911516
\(932\) 0 0
\(933\) −4.28315 −0.140224
\(934\) 0 0
\(935\) 12.8565 0.420451
\(936\) 0 0
\(937\) 29.1283 0.951582 0.475791 0.879558i \(-0.342162\pi\)
0.475791 + 0.879558i \(0.342162\pi\)
\(938\) 0 0
\(939\) −0.673397 −0.0219755
\(940\) 0 0
\(941\) −2.68696 −0.0875925 −0.0437963 0.999040i \(-0.513945\pi\)
−0.0437963 + 0.999040i \(0.513945\pi\)
\(942\) 0 0
\(943\) −8.82974 −0.287536
\(944\) 0 0
\(945\) −3.51798 −0.114440
\(946\) 0 0
\(947\) 39.4768 1.28282 0.641412 0.767197i \(-0.278349\pi\)
0.641412 + 0.767197i \(0.278349\pi\)
\(948\) 0 0
\(949\) −19.7357 −0.640647
\(950\) 0 0
\(951\) 17.0792 0.553831
\(952\) 0 0
\(953\) 19.7605 0.640106 0.320053 0.947400i \(-0.396299\pi\)
0.320053 + 0.947400i \(0.396299\pi\)
\(954\) 0 0
\(955\) 10.5593 0.341692
\(956\) 0 0
\(957\) 5.39807 0.174495
\(958\) 0 0
\(959\) 28.1078 0.907649
\(960\) 0 0
\(961\) −5.14819 −0.166071
\(962\) 0 0
\(963\) −14.5746 −0.469659
\(964\) 0 0
\(965\) 7.40213 0.238283
\(966\) 0 0
\(967\) 55.6598 1.78990 0.894949 0.446169i \(-0.147212\pi\)
0.894949 + 0.446169i \(0.147212\pi\)
\(968\) 0 0
\(969\) 8.91344 0.286341
\(970\) 0 0
\(971\) −21.7277 −0.697276 −0.348638 0.937257i \(-0.613356\pi\)
−0.348638 + 0.937257i \(0.613356\pi\)
\(972\) 0 0
\(973\) −16.9390 −0.543041
\(974\) 0 0
\(975\) −9.15225 −0.293107
\(976\) 0 0
\(977\) 8.02371 0.256701 0.128351 0.991729i \(-0.459032\pi\)
0.128351 + 0.991729i \(0.459032\pi\)
\(978\) 0 0
\(979\) 94.4036 3.01715
\(980\) 0 0
\(981\) 8.01651 0.255947
\(982\) 0 0
\(983\) 35.6083 1.13573 0.567865 0.823122i \(-0.307770\pi\)
0.567865 + 0.823122i \(0.307770\pi\)
\(984\) 0 0
\(985\) 0.460134 0.0146611
\(986\) 0 0
\(987\) −4.00949 −0.127623
\(988\) 0 0
\(989\) −0.613535 −0.0195093
\(990\) 0 0
\(991\) 27.9781 0.888755 0.444377 0.895840i \(-0.353425\pi\)
0.444377 + 0.895840i \(0.353425\pi\)
\(992\) 0 0
\(993\) −7.39283 −0.234604
\(994\) 0 0
\(995\) −8.31826 −0.263707
\(996\) 0 0
\(997\) 57.5748 1.82341 0.911707 0.410842i \(-0.134765\pi\)
0.911707 + 0.410842i \(0.134765\pi\)
\(998\) 0 0
\(999\) 3.79035 0.119921
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.e.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.6 9 1.1 even 1 trivial