Properties

Label 7728.2.a.ch.1.6
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 11 x^{4} + 23 x^{3} + 9 x^{2} - 23 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(4.26681\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.93128 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.93128 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.69846 q^{11} -1.59242 q^{13} +3.93128 q^{15} -1.68098 q^{17} -2.69846 q^{19} +1.00000 q^{21} +1.00000 q^{23} +10.4550 q^{25} +1.00000 q^{27} +4.39770 q^{29} +8.29927 q^{31} -2.69846 q^{33} +3.93128 q^{35} +6.24043 q^{37} -1.59242 q^{39} +2.60466 q^{41} -10.2306 q^{43} +3.93128 q^{45} +3.07632 q^{47} +1.00000 q^{49} -1.68098 q^{51} +2.38932 q^{53} -10.6084 q^{55} -2.69846 q^{57} -1.63053 q^{59} -1.29167 q^{61} +1.00000 q^{63} -6.26027 q^{65} -11.0771 q^{67} +1.00000 q^{69} +7.70607 q^{71} +9.36643 q^{73} +10.4550 q^{75} -2.69846 q^{77} +7.54355 q^{79} +1.00000 q^{81} -2.16174 q^{83} -6.60842 q^{85} +4.39770 q^{87} +5.84350 q^{89} -1.59242 q^{91} +8.29927 q^{93} -10.6084 q^{95} +7.64198 q^{97} -2.69846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} + 2q^{5} + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} + 2q^{5} + 6q^{7} + 6q^{9} + 3q^{11} + 2q^{15} + 6q^{17} + 3q^{19} + 6q^{21} + 6q^{23} + 10q^{25} + 6q^{27} + 5q^{29} + 4q^{31} + 3q^{33} + 2q^{35} + q^{37} + 12q^{41} + 6q^{43} + 2q^{45} + 6q^{47} + 6q^{49} + 6q^{51} + 10q^{53} - 3q^{55} + 3q^{57} + 14q^{59} + 4q^{61} + 6q^{63} + 27q^{65} - 7q^{67} + 6q^{69} - 7q^{71} + 10q^{73} + 10q^{75} + 3q^{77} - 14q^{79} + 6q^{81} - 14q^{83} + 21q^{85} + 5q^{87} + 25q^{89} + 4q^{93} - 3q^{95} + 11q^{97} + 3q^{99} + O(q^{100}) \)

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
\(5\) 3.93128 1.75812 0.879062 0.476708i \(-0.158170\pi\)
0.879062 + 0.476708i \(0.158170\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.69846 −0.813617 −0.406808 0.913513i \(-0.633358\pi\)
−0.406808 + 0.913513i \(0.633358\pi\)
\(12\) 0 0
\(13\) −1.59242 −0.441659 −0.220830 0.975312i \(-0.570876\pi\)
−0.220830 + 0.975312i \(0.570876\pi\)
\(14\) 0 0
\(15\) 3.93128 1.01505
\(16\) 0 0
\(17\) −1.68098 −0.407698 −0.203849 0.979002i \(-0.565345\pi\)
−0.203849 + 0.979002i \(0.565345\pi\)
\(18\) 0 0
\(19\) −2.69846 −0.619070 −0.309535 0.950888i \(-0.600173\pi\)
−0.309535 + 0.950888i \(0.600173\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 10.4550 2.09100
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.39770 0.816633 0.408317 0.912840i \(-0.366116\pi\)
0.408317 + 0.912840i \(0.366116\pi\)
\(30\) 0 0
\(31\) 8.29927 1.49059 0.745297 0.666733i \(-0.232308\pi\)
0.745297 + 0.666733i \(0.232308\pi\)
\(32\) 0 0
\(33\) −2.69846 −0.469742
\(34\) 0 0
\(35\) 3.93128 0.664508
\(36\) 0 0
\(37\) 6.24043 1.02592 0.512960 0.858412i \(-0.328549\pi\)
0.512960 + 0.858412i \(0.328549\pi\)
\(38\) 0 0
\(39\) −1.59242 −0.254992
\(40\) 0 0
\(41\) 2.60466 0.406779 0.203390 0.979098i \(-0.434804\pi\)
0.203390 + 0.979098i \(0.434804\pi\)
\(42\) 0 0
\(43\) −10.2306 −1.56015 −0.780073 0.625689i \(-0.784818\pi\)
−0.780073 + 0.625689i \(0.784818\pi\)
\(44\) 0 0
\(45\) 3.93128 0.586041
\(46\) 0 0
\(47\) 3.07632 0.448728 0.224364 0.974505i \(-0.427970\pi\)
0.224364 + 0.974505i \(0.427970\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.68098 −0.235385
\(52\) 0 0
\(53\) 2.38932 0.328198 0.164099 0.986444i \(-0.447528\pi\)
0.164099 + 0.986444i \(0.447528\pi\)
\(54\) 0 0
\(55\) −10.6084 −1.43044
\(56\) 0 0
\(57\) −2.69846 −0.357420
\(58\) 0 0
\(59\) −1.63053 −0.212276 −0.106138 0.994351i \(-0.533849\pi\)
−0.106138 + 0.994351i \(0.533849\pi\)
\(60\) 0 0
\(61\) −1.29167 −0.165381 −0.0826905 0.996575i \(-0.526351\pi\)
−0.0826905 + 0.996575i \(0.526351\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −6.26027 −0.776491
\(66\) 0 0
\(67\) −11.0771 −1.35329 −0.676643 0.736311i \(-0.736566\pi\)
−0.676643 + 0.736311i \(0.736566\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 7.70607 0.914542 0.457271 0.889327i \(-0.348827\pi\)
0.457271 + 0.889327i \(0.348827\pi\)
\(72\) 0 0
\(73\) 9.36643 1.09626 0.548129 0.836394i \(-0.315340\pi\)
0.548129 + 0.836394i \(0.315340\pi\)
\(74\) 0 0
\(75\) 10.4550 1.20724
\(76\) 0 0
\(77\) −2.69846 −0.307518
\(78\) 0 0
\(79\) 7.54355 0.848716 0.424358 0.905495i \(-0.360500\pi\)
0.424358 + 0.905495i \(0.360500\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.16174 −0.237282 −0.118641 0.992937i \(-0.537854\pi\)
−0.118641 + 0.992937i \(0.537854\pi\)
\(84\) 0 0
\(85\) −6.60842 −0.716784
\(86\) 0 0
\(87\) 4.39770 0.471483
\(88\) 0 0
\(89\) 5.84350 0.619410 0.309705 0.950833i \(-0.399770\pi\)
0.309705 + 0.950833i \(0.399770\pi\)
\(90\) 0 0
\(91\) −1.59242 −0.166931
\(92\) 0 0
\(93\) 8.29927 0.860595
\(94\) 0 0
\(95\) −10.6084 −1.08840
\(96\) 0 0
\(97\) 7.64198 0.775926 0.387963 0.921675i \(-0.373179\pi\)
0.387963 + 0.921675i \(0.373179\pi\)
\(98\) 0 0
\(99\) −2.69846 −0.271206
\(100\) 0 0
\(101\) −0.232042 −0.0230890 −0.0115445 0.999933i \(-0.503675\pi\)
−0.0115445 + 0.999933i \(0.503675\pi\)
\(102\) 0 0
\(103\) 10.2612 1.01106 0.505532 0.862808i \(-0.331296\pi\)
0.505532 + 0.862808i \(0.331296\pi\)
\(104\) 0 0
\(105\) 3.93128 0.383654
\(106\) 0 0
\(107\) 17.8420 1.72486 0.862428 0.506180i \(-0.168943\pi\)
0.862428 + 0.506180i \(0.168943\pi\)
\(108\) 0 0
\(109\) 19.7741 1.89402 0.947008 0.321210i \(-0.104090\pi\)
0.947008 + 0.321210i \(0.104090\pi\)
\(110\) 0 0
\(111\) 6.24043 0.592315
\(112\) 0 0
\(113\) 12.2412 1.15156 0.575778 0.817606i \(-0.304699\pi\)
0.575778 + 0.817606i \(0.304699\pi\)
\(114\) 0 0
\(115\) 3.93128 0.366594
\(116\) 0 0
\(117\) −1.59242 −0.147220
\(118\) 0 0
\(119\) −1.68098 −0.154095
\(120\) 0 0
\(121\) −3.71830 −0.338028
\(122\) 0 0
\(123\) 2.60466 0.234854
\(124\) 0 0
\(125\) 21.4451 1.91811
\(126\) 0 0
\(127\) 9.24804 0.820631 0.410315 0.911944i \(-0.365419\pi\)
0.410315 + 0.911944i \(0.365419\pi\)
\(128\) 0 0
\(129\) −10.2306 −0.900750
\(130\) 0 0
\(131\) 0.855736 0.0747660 0.0373830 0.999301i \(-0.488098\pi\)
0.0373830 + 0.999301i \(0.488098\pi\)
\(132\) 0 0
\(133\) −2.69846 −0.233986
\(134\) 0 0
\(135\) 3.93128 0.338351
\(136\) 0 0
\(137\) −2.46723 −0.210789 −0.105395 0.994430i \(-0.533611\pi\)
−0.105395 + 0.994430i \(0.533611\pi\)
\(138\) 0 0
\(139\) −1.50523 −0.127672 −0.0638358 0.997960i \(-0.520333\pi\)
−0.0638358 + 0.997960i \(0.520333\pi\)
\(140\) 0 0
\(141\) 3.07632 0.259073
\(142\) 0 0
\(143\) 4.29710 0.359341
\(144\) 0 0
\(145\) 17.2886 1.43574
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −1.63577 −0.134008 −0.0670038 0.997753i \(-0.521344\pi\)
−0.0670038 + 0.997753i \(0.521344\pi\)
\(150\) 0 0
\(151\) 0.726816 0.0591474 0.0295737 0.999563i \(-0.490585\pi\)
0.0295737 + 0.999563i \(0.490585\pi\)
\(152\) 0 0
\(153\) −1.68098 −0.135899
\(154\) 0 0
\(155\) 32.6268 2.62065
\(156\) 0 0
\(157\) 6.96278 0.555690 0.277845 0.960626i \(-0.410380\pi\)
0.277845 + 0.960626i \(0.410380\pi\)
\(158\) 0 0
\(159\) 2.38932 0.189485
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −4.19708 −0.328741 −0.164370 0.986399i \(-0.552559\pi\)
−0.164370 + 0.986399i \(0.552559\pi\)
\(164\) 0 0
\(165\) −10.6084 −0.825864
\(166\) 0 0
\(167\) 1.29691 0.100358 0.0501790 0.998740i \(-0.484021\pi\)
0.0501790 + 0.998740i \(0.484021\pi\)
\(168\) 0 0
\(169\) −10.4642 −0.804937
\(170\) 0 0
\(171\) −2.69846 −0.206357
\(172\) 0 0
\(173\) −14.9405 −1.13590 −0.567952 0.823062i \(-0.692264\pi\)
−0.567952 + 0.823062i \(0.692264\pi\)
\(174\) 0 0
\(175\) 10.4550 0.790323
\(176\) 0 0
\(177\) −1.63053 −0.122558
\(178\) 0 0
\(179\) −12.8474 −0.960263 −0.480132 0.877196i \(-0.659411\pi\)
−0.480132 + 0.877196i \(0.659411\pi\)
\(180\) 0 0
\(181\) 9.55867 0.710490 0.355245 0.934773i \(-0.384397\pi\)
0.355245 + 0.934773i \(0.384397\pi\)
\(182\) 0 0
\(183\) −1.29167 −0.0954827
\(184\) 0 0
\(185\) 24.5329 1.80369
\(186\) 0 0
\(187\) 4.53607 0.331710
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.2222 −0.884365 −0.442183 0.896925i \(-0.645796\pi\)
−0.442183 + 0.896925i \(0.645796\pi\)
\(192\) 0 0
\(193\) −1.67784 −0.120774 −0.0603868 0.998175i \(-0.519233\pi\)
−0.0603868 + 0.998175i \(0.519233\pi\)
\(194\) 0 0
\(195\) −6.26027 −0.448307
\(196\) 0 0
\(197\) −9.43525 −0.672234 −0.336117 0.941820i \(-0.609114\pi\)
−0.336117 + 0.941820i \(0.609114\pi\)
\(198\) 0 0
\(199\) 9.55028 0.677001 0.338501 0.940966i \(-0.390080\pi\)
0.338501 + 0.940966i \(0.390080\pi\)
\(200\) 0 0
\(201\) −11.0771 −0.781320
\(202\) 0 0
\(203\) 4.39770 0.308658
\(204\) 0 0
\(205\) 10.2397 0.715168
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 7.28170 0.503685
\(210\) 0 0
\(211\) −23.7063 −1.63201 −0.816004 0.578046i \(-0.803815\pi\)
−0.816004 + 0.578046i \(0.803815\pi\)
\(212\) 0 0
\(213\) 7.70607 0.528011
\(214\) 0 0
\(215\) −40.2192 −2.74293
\(216\) 0 0
\(217\) 8.29927 0.563391
\(218\) 0 0
\(219\) 9.36643 0.632925
\(220\) 0 0
\(221\) 2.67684 0.180064
\(222\) 0 0
\(223\) 10.6313 0.711926 0.355963 0.934500i \(-0.384153\pi\)
0.355963 + 0.934500i \(0.384153\pi\)
\(224\) 0 0
\(225\) 10.4550 0.696999
\(226\) 0 0
\(227\) −12.2230 −0.811267 −0.405634 0.914036i \(-0.632949\pi\)
−0.405634 + 0.914036i \(0.632949\pi\)
\(228\) 0 0
\(229\) −15.6259 −1.03259 −0.516294 0.856411i \(-0.672689\pi\)
−0.516294 + 0.856411i \(0.672689\pi\)
\(230\) 0 0
\(231\) −2.69846 −0.177546
\(232\) 0 0
\(233\) −22.3685 −1.46541 −0.732705 0.680546i \(-0.761742\pi\)
−0.732705 + 0.680546i \(0.761742\pi\)
\(234\) 0 0
\(235\) 12.0939 0.788919
\(236\) 0 0
\(237\) 7.54355 0.490006
\(238\) 0 0
\(239\) −20.4193 −1.32081 −0.660406 0.750909i \(-0.729616\pi\)
−0.660406 + 0.750909i \(0.729616\pi\)
\(240\) 0 0
\(241\) 8.00158 0.515427 0.257714 0.966221i \(-0.417031\pi\)
0.257714 + 0.966221i \(0.417031\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.93128 0.251161
\(246\) 0 0
\(247\) 4.29710 0.273418
\(248\) 0 0
\(249\) −2.16174 −0.136995
\(250\) 0 0
\(251\) −31.4694 −1.98633 −0.993166 0.116707i \(-0.962766\pi\)
−0.993166 + 0.116707i \(0.962766\pi\)
\(252\) 0 0
\(253\) −2.69846 −0.169651
\(254\) 0 0
\(255\) −6.60842 −0.413835
\(256\) 0 0
\(257\) −6.89989 −0.430403 −0.215202 0.976570i \(-0.569041\pi\)
−0.215202 + 0.976570i \(0.569041\pi\)
\(258\) 0 0
\(259\) 6.24043 0.387761
\(260\) 0 0
\(261\) 4.39770 0.272211
\(262\) 0 0
\(263\) −3.80925 −0.234888 −0.117444 0.993079i \(-0.537470\pi\)
−0.117444 + 0.993079i \(0.537470\pi\)
\(264\) 0 0
\(265\) 9.39308 0.577012
\(266\) 0 0
\(267\) 5.84350 0.357616
\(268\) 0 0
\(269\) 17.8222 1.08664 0.543320 0.839526i \(-0.317167\pi\)
0.543320 + 0.839526i \(0.317167\pi\)
\(270\) 0 0
\(271\) −28.6062 −1.73770 −0.868851 0.495074i \(-0.835141\pi\)
−0.868851 + 0.495074i \(0.835141\pi\)
\(272\) 0 0
\(273\) −1.59242 −0.0963779
\(274\) 0 0
\(275\) −28.2124 −1.70127
\(276\) 0 0
\(277\) −0.806180 −0.0484387 −0.0242193 0.999707i \(-0.507710\pi\)
−0.0242193 + 0.999707i \(0.507710\pi\)
\(278\) 0 0
\(279\) 8.29927 0.496865
\(280\) 0 0
\(281\) 18.1136 1.08057 0.540285 0.841482i \(-0.318317\pi\)
0.540285 + 0.841482i \(0.318317\pi\)
\(282\) 0 0
\(283\) 3.23431 0.192260 0.0961298 0.995369i \(-0.469354\pi\)
0.0961298 + 0.995369i \(0.469354\pi\)
\(284\) 0 0
\(285\) −10.6084 −0.628389
\(286\) 0 0
\(287\) 2.60466 0.153748
\(288\) 0 0
\(289\) −14.1743 −0.833782
\(290\) 0 0
\(291\) 7.64198 0.447981
\(292\) 0 0
\(293\) 13.6703 0.798629 0.399315 0.916814i \(-0.369248\pi\)
0.399315 + 0.916814i \(0.369248\pi\)
\(294\) 0 0
\(295\) −6.41006 −0.373208
\(296\) 0 0
\(297\) −2.69846 −0.156581
\(298\) 0 0
\(299\) −1.59242 −0.0920923
\(300\) 0 0
\(301\) −10.2306 −0.589680
\(302\) 0 0
\(303\) −0.232042 −0.0133304
\(304\) 0 0
\(305\) −5.07791 −0.290760
\(306\) 0 0
\(307\) −15.8940 −0.907118 −0.453559 0.891226i \(-0.649846\pi\)
−0.453559 + 0.891226i \(0.649846\pi\)
\(308\) 0 0
\(309\) 10.2612 0.583738
\(310\) 0 0
\(311\) 12.9375 0.733620 0.366810 0.930296i \(-0.380450\pi\)
0.366810 + 0.930296i \(0.380450\pi\)
\(312\) 0 0
\(313\) −13.9670 −0.789464 −0.394732 0.918796i \(-0.629163\pi\)
−0.394732 + 0.918796i \(0.629163\pi\)
\(314\) 0 0
\(315\) 3.93128 0.221503
\(316\) 0 0
\(317\) 33.0258 1.85491 0.927457 0.373930i \(-0.121990\pi\)
0.927457 + 0.373930i \(0.121990\pi\)
\(318\) 0 0
\(319\) −11.8670 −0.664426
\(320\) 0 0
\(321\) 17.8420 0.995846
\(322\) 0 0
\(323\) 4.53607 0.252394
\(324\) 0 0
\(325\) −16.6488 −0.923508
\(326\) 0 0
\(327\) 19.7741 1.09351
\(328\) 0 0
\(329\) 3.07632 0.169603
\(330\) 0 0
\(331\) 9.51818 0.523166 0.261583 0.965181i \(-0.415755\pi\)
0.261583 + 0.965181i \(0.415755\pi\)
\(332\) 0 0
\(333\) 6.24043 0.341973
\(334\) 0 0
\(335\) −43.5473 −2.37925
\(336\) 0 0
\(337\) −23.2420 −1.26607 −0.633037 0.774121i \(-0.718192\pi\)
−0.633037 + 0.774121i \(0.718192\pi\)
\(338\) 0 0
\(339\) 12.2412 0.664851
\(340\) 0 0
\(341\) −22.3953 −1.21277
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 3.93128 0.211653
\(346\) 0 0
\(347\) 25.2712 1.35663 0.678314 0.734772i \(-0.262711\pi\)
0.678314 + 0.734772i \(0.262711\pi\)
\(348\) 0 0
\(349\) −3.31241 −0.177309 −0.0886546 0.996062i \(-0.528257\pi\)
−0.0886546 + 0.996062i \(0.528257\pi\)
\(350\) 0 0
\(351\) −1.59242 −0.0849973
\(352\) 0 0
\(353\) −27.7291 −1.47587 −0.737936 0.674871i \(-0.764199\pi\)
−0.737936 + 0.674871i \(0.764199\pi\)
\(354\) 0 0
\(355\) 30.2947 1.60788
\(356\) 0 0
\(357\) −1.68098 −0.0889670
\(358\) 0 0
\(359\) −14.2809 −0.753717 −0.376858 0.926271i \(-0.622996\pi\)
−0.376858 + 0.926271i \(0.622996\pi\)
\(360\) 0 0
\(361\) −11.7183 −0.616753
\(362\) 0 0
\(363\) −3.71830 −0.195160
\(364\) 0 0
\(365\) 36.8221 1.92736
\(366\) 0 0
\(367\) 35.9110 1.87454 0.937269 0.348607i \(-0.113345\pi\)
0.937269 + 0.348607i \(0.113345\pi\)
\(368\) 0 0
\(369\) 2.60466 0.135593
\(370\) 0 0
\(371\) 2.38932 0.124047
\(372\) 0 0
\(373\) 7.16923 0.371208 0.185604 0.982625i \(-0.440576\pi\)
0.185604 + 0.982625i \(0.440576\pi\)
\(374\) 0 0
\(375\) 21.4451 1.10742
\(376\) 0 0
\(377\) −7.00301 −0.360673
\(378\) 0 0
\(379\) −20.8434 −1.07066 −0.535328 0.844644i \(-0.679812\pi\)
−0.535328 + 0.844644i \(0.679812\pi\)
\(380\) 0 0
\(381\) 9.24804 0.473791
\(382\) 0 0
\(383\) −13.0398 −0.666304 −0.333152 0.942873i \(-0.608112\pi\)
−0.333152 + 0.942873i \(0.608112\pi\)
\(384\) 0 0
\(385\) −10.6084 −0.540655
\(386\) 0 0
\(387\) −10.2306 −0.520048
\(388\) 0 0
\(389\) 8.35563 0.423647 0.211824 0.977308i \(-0.432060\pi\)
0.211824 + 0.977308i \(0.432060\pi\)
\(390\) 0 0
\(391\) −1.68098 −0.0850109
\(392\) 0 0
\(393\) 0.855736 0.0431662
\(394\) 0 0
\(395\) 29.6558 1.49215
\(396\) 0 0
\(397\) −19.2960 −0.968439 −0.484219 0.874947i \(-0.660896\pi\)
−0.484219 + 0.874947i \(0.660896\pi\)
\(398\) 0 0
\(399\) −2.69846 −0.135092
\(400\) 0 0
\(401\) 26.6589 1.33128 0.665640 0.746273i \(-0.268158\pi\)
0.665640 + 0.746273i \(0.268158\pi\)
\(402\) 0 0
\(403\) −13.2160 −0.658334
\(404\) 0 0
\(405\) 3.93128 0.195347
\(406\) 0 0
\(407\) −16.8396 −0.834706
\(408\) 0 0
\(409\) −4.60910 −0.227905 −0.113953 0.993486i \(-0.536351\pi\)
−0.113953 + 0.993486i \(0.536351\pi\)
\(410\) 0 0
\(411\) −2.46723 −0.121699
\(412\) 0 0
\(413\) −1.63053 −0.0802329
\(414\) 0 0
\(415\) −8.49842 −0.417171
\(416\) 0 0
\(417\) −1.50523 −0.0737112
\(418\) 0 0
\(419\) −22.7226 −1.11007 −0.555037 0.831826i \(-0.687296\pi\)
−0.555037 + 0.831826i \(0.687296\pi\)
\(420\) 0 0
\(421\) 8.03667 0.391683 0.195842 0.980636i \(-0.437256\pi\)
0.195842 + 0.980636i \(0.437256\pi\)
\(422\) 0 0
\(423\) 3.07632 0.149576
\(424\) 0 0
\(425\) −17.5747 −0.852496
\(426\) 0 0
\(427\) −1.29167 −0.0625081
\(428\) 0 0
\(429\) 4.29710 0.207466
\(430\) 0 0
\(431\) −16.3215 −0.786178 −0.393089 0.919500i \(-0.628594\pi\)
−0.393089 + 0.919500i \(0.628594\pi\)
\(432\) 0 0
\(433\) −36.3297 −1.74590 −0.872948 0.487814i \(-0.837795\pi\)
−0.872948 + 0.487814i \(0.837795\pi\)
\(434\) 0 0
\(435\) 17.2886 0.828926
\(436\) 0 0
\(437\) −2.69846 −0.129085
\(438\) 0 0
\(439\) −32.7223 −1.56175 −0.780875 0.624687i \(-0.785227\pi\)
−0.780875 + 0.624687i \(0.785227\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 22.8360 1.08497 0.542486 0.840065i \(-0.317483\pi\)
0.542486 + 0.840065i \(0.317483\pi\)
\(444\) 0 0
\(445\) 22.9725 1.08900
\(446\) 0 0
\(447\) −1.63577 −0.0773693
\(448\) 0 0
\(449\) 25.8511 1.21999 0.609995 0.792405i \(-0.291171\pi\)
0.609995 + 0.792405i \(0.291171\pi\)
\(450\) 0 0
\(451\) −7.02857 −0.330963
\(452\) 0 0
\(453\) 0.726816 0.0341488
\(454\) 0 0
\(455\) −6.26027 −0.293486
\(456\) 0 0
\(457\) 26.6361 1.24598 0.622991 0.782229i \(-0.285917\pi\)
0.622991 + 0.782229i \(0.285917\pi\)
\(458\) 0 0
\(459\) −1.68098 −0.0784615
\(460\) 0 0
\(461\) 0.647907 0.0301760 0.0150880 0.999886i \(-0.495197\pi\)
0.0150880 + 0.999886i \(0.495197\pi\)
\(462\) 0 0
\(463\) 7.94371 0.369176 0.184588 0.982816i \(-0.440905\pi\)
0.184588 + 0.982816i \(0.440905\pi\)
\(464\) 0 0
\(465\) 32.6268 1.51303
\(466\) 0 0
\(467\) 5.54750 0.256707 0.128354 0.991728i \(-0.459031\pi\)
0.128354 + 0.991728i \(0.459031\pi\)
\(468\) 0 0
\(469\) −11.0771 −0.511494
\(470\) 0 0
\(471\) 6.96278 0.320828
\(472\) 0 0
\(473\) 27.6068 1.26936
\(474\) 0 0
\(475\) −28.2124 −1.29447
\(476\) 0 0
\(477\) 2.38932 0.109399
\(478\) 0 0
\(479\) −1.71594 −0.0784034 −0.0392017 0.999231i \(-0.512481\pi\)
−0.0392017 + 0.999231i \(0.512481\pi\)
\(480\) 0 0
\(481\) −9.93741 −0.453107
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 30.0428 1.36417
\(486\) 0 0
\(487\) −17.7399 −0.803873 −0.401936 0.915668i \(-0.631663\pi\)
−0.401936 + 0.915668i \(0.631663\pi\)
\(488\) 0 0
\(489\) −4.19708 −0.189799
\(490\) 0 0
\(491\) 11.0497 0.498666 0.249333 0.968418i \(-0.419789\pi\)
0.249333 + 0.968418i \(0.419789\pi\)
\(492\) 0 0
\(493\) −7.39246 −0.332940
\(494\) 0 0
\(495\) −10.6084 −0.476813
\(496\) 0 0
\(497\) 7.70607 0.345664
\(498\) 0 0
\(499\) 19.0954 0.854828 0.427414 0.904056i \(-0.359425\pi\)
0.427414 + 0.904056i \(0.359425\pi\)
\(500\) 0 0
\(501\) 1.29691 0.0579417
\(502\) 0 0
\(503\) 8.59024 0.383020 0.191510 0.981491i \(-0.438662\pi\)
0.191510 + 0.981491i \(0.438662\pi\)
\(504\) 0 0
\(505\) −0.912222 −0.0405933
\(506\) 0 0
\(507\) −10.4642 −0.464731
\(508\) 0 0
\(509\) −42.7427 −1.89454 −0.947269 0.320440i \(-0.896169\pi\)
−0.947269 + 0.320440i \(0.896169\pi\)
\(510\) 0 0
\(511\) 9.36643 0.414347
\(512\) 0 0
\(513\) −2.69846 −0.119140
\(514\) 0 0
\(515\) 40.3396 1.77757
\(516\) 0 0
\(517\) −8.30134 −0.365093
\(518\) 0 0
\(519\) −14.9405 −0.655814
\(520\) 0 0
\(521\) 15.2763 0.669265 0.334633 0.942349i \(-0.391388\pi\)
0.334633 + 0.942349i \(0.391388\pi\)
\(522\) 0 0
\(523\) −3.77637 −0.165129 −0.0825645 0.996586i \(-0.526311\pi\)
−0.0825645 + 0.996586i \(0.526311\pi\)
\(524\) 0 0
\(525\) 10.4550 0.456293
\(526\) 0 0
\(527\) −13.9509 −0.607712
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −1.63053 −0.0707588
\(532\) 0 0
\(533\) −4.14772 −0.179658
\(534\) 0 0
\(535\) 70.1421 3.03251
\(536\) 0 0
\(537\) −12.8474 −0.554408
\(538\) 0 0
\(539\) −2.69846 −0.116231
\(540\) 0 0
\(541\) −22.2494 −0.956578 −0.478289 0.878203i \(-0.658743\pi\)
−0.478289 + 0.878203i \(0.658743\pi\)
\(542\) 0 0
\(543\) 9.55867 0.410202
\(544\) 0 0
\(545\) 77.7376 3.32991
\(546\) 0 0
\(547\) 28.2224 1.20670 0.603351 0.797476i \(-0.293832\pi\)
0.603351 + 0.797476i \(0.293832\pi\)
\(548\) 0 0
\(549\) −1.29167 −0.0551270
\(550\) 0 0
\(551\) −11.8670 −0.505553
\(552\) 0 0
\(553\) 7.54355 0.320784
\(554\) 0 0
\(555\) 24.5329 1.04136
\(556\) 0 0
\(557\) −18.6304 −0.789397 −0.394699 0.918811i \(-0.629151\pi\)
−0.394699 + 0.918811i \(0.629151\pi\)
\(558\) 0 0
\(559\) 16.2914 0.689052
\(560\) 0 0
\(561\) 4.53607 0.191513
\(562\) 0 0
\(563\) −11.4535 −0.482709 −0.241354 0.970437i \(-0.577592\pi\)
−0.241354 + 0.970437i \(0.577592\pi\)
\(564\) 0 0
\(565\) 48.1237 2.02458
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −2.98194 −0.125009 −0.0625046 0.998045i \(-0.519909\pi\)
−0.0625046 + 0.998045i \(0.519909\pi\)
\(570\) 0 0
\(571\) −30.6458 −1.28249 −0.641243 0.767338i \(-0.721581\pi\)
−0.641243 + 0.767338i \(0.721581\pi\)
\(572\) 0 0
\(573\) −12.2222 −0.510588
\(574\) 0 0
\(575\) 10.4550 0.436003
\(576\) 0 0
\(577\) −19.0107 −0.791424 −0.395712 0.918375i \(-0.629502\pi\)
−0.395712 + 0.918375i \(0.629502\pi\)
\(578\) 0 0
\(579\) −1.67784 −0.0697286
\(580\) 0 0
\(581\) −2.16174 −0.0896842
\(582\) 0 0
\(583\) −6.44748 −0.267027
\(584\) 0 0
\(585\) −6.26027 −0.258830
\(586\) 0 0
\(587\) −18.8626 −0.778544 −0.389272 0.921123i \(-0.627273\pi\)
−0.389272 + 0.921123i \(0.627273\pi\)
\(588\) 0 0
\(589\) −22.3953 −0.922781
\(590\) 0 0
\(591\) −9.43525 −0.388114
\(592\) 0 0
\(593\) −45.7993 −1.88075 −0.940376 0.340137i \(-0.889526\pi\)
−0.940376 + 0.340137i \(0.889526\pi\)
\(594\) 0 0
\(595\) −6.60842 −0.270919
\(596\) 0 0
\(597\) 9.55028 0.390867
\(598\) 0 0
\(599\) −7.42959 −0.303565 −0.151782 0.988414i \(-0.548501\pi\)
−0.151782 + 0.988414i \(0.548501\pi\)
\(600\) 0 0
\(601\) −26.7887 −1.09273 −0.546367 0.837546i \(-0.683990\pi\)
−0.546367 + 0.837546i \(0.683990\pi\)
\(602\) 0 0
\(603\) −11.0771 −0.451096
\(604\) 0 0
\(605\) −14.6177 −0.594294
\(606\) 0 0
\(607\) −23.1182 −0.938339 −0.469170 0.883108i \(-0.655447\pi\)
−0.469170 + 0.883108i \(0.655447\pi\)
\(608\) 0 0
\(609\) 4.39770 0.178204
\(610\) 0 0
\(611\) −4.89881 −0.198185
\(612\) 0 0
\(613\) −45.5827 −1.84107 −0.920533 0.390665i \(-0.872245\pi\)
−0.920533 + 0.390665i \(0.872245\pi\)
\(614\) 0 0
\(615\) 10.2397 0.412903
\(616\) 0 0
\(617\) −20.3459 −0.819093 −0.409547 0.912289i \(-0.634313\pi\)
−0.409547 + 0.912289i \(0.634313\pi\)
\(618\) 0 0
\(619\) 33.7563 1.35678 0.678391 0.734701i \(-0.262678\pi\)
0.678391 + 0.734701i \(0.262678\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 5.84350 0.234115
\(624\) 0 0
\(625\) 32.0319 1.28128
\(626\) 0 0
\(627\) 7.28170 0.290803
\(628\) 0 0
\(629\) −10.4901 −0.418266
\(630\) 0 0
\(631\) −3.54510 −0.141128 −0.0705642 0.997507i \(-0.522480\pi\)
−0.0705642 + 0.997507i \(0.522480\pi\)
\(632\) 0 0
\(633\) −23.7063 −0.942240
\(634\) 0 0
\(635\) 36.3567 1.44277
\(636\) 0 0
\(637\) −1.59242 −0.0630941
\(638\) 0 0
\(639\) 7.70607 0.304847
\(640\) 0 0
\(641\) 22.3914 0.884407 0.442204 0.896915i \(-0.354197\pi\)
0.442204 + 0.896915i \(0.354197\pi\)
\(642\) 0 0
\(643\) 2.35721 0.0929594 0.0464797 0.998919i \(-0.485200\pi\)
0.0464797 + 0.998919i \(0.485200\pi\)
\(644\) 0 0
\(645\) −40.2192 −1.58363
\(646\) 0 0
\(647\) 12.4923 0.491123 0.245561 0.969381i \(-0.421028\pi\)
0.245561 + 0.969381i \(0.421028\pi\)
\(648\) 0 0
\(649\) 4.39991 0.172712
\(650\) 0 0
\(651\) 8.29927 0.325274
\(652\) 0 0
\(653\) −40.2732 −1.57601 −0.788007 0.615666i \(-0.788887\pi\)
−0.788007 + 0.615666i \(0.788887\pi\)
\(654\) 0 0
\(655\) 3.36414 0.131448
\(656\) 0 0
\(657\) 9.36643 0.365419
\(658\) 0 0
\(659\) 32.5709 1.26878 0.634391 0.773013i \(-0.281251\pi\)
0.634391 + 0.773013i \(0.281251\pi\)
\(660\) 0 0
\(661\) −33.7307 −1.31197 −0.655985 0.754774i \(-0.727747\pi\)
−0.655985 + 0.754774i \(0.727747\pi\)
\(662\) 0 0
\(663\) 2.67684 0.103960
\(664\) 0 0
\(665\) −10.6084 −0.411377
\(666\) 0 0
\(667\) 4.39770 0.170280
\(668\) 0 0
\(669\) 10.6313 0.411031
\(670\) 0 0
\(671\) 3.48551 0.134557
\(672\) 0 0
\(673\) 13.9469 0.537613 0.268806 0.963194i \(-0.413371\pi\)
0.268806 + 0.963194i \(0.413371\pi\)
\(674\) 0 0
\(675\) 10.4550 0.402413
\(676\) 0 0
\(677\) −1.23877 −0.0476100 −0.0238050 0.999717i \(-0.507578\pi\)
−0.0238050 + 0.999717i \(0.507578\pi\)
\(678\) 0 0
\(679\) 7.64198 0.293272
\(680\) 0 0
\(681\) −12.2230 −0.468385
\(682\) 0 0
\(683\) 41.9616 1.60562 0.802808 0.596237i \(-0.203338\pi\)
0.802808 + 0.596237i \(0.203338\pi\)
\(684\) 0 0
\(685\) −9.69937 −0.370594
\(686\) 0 0
\(687\) −15.6259 −0.596165
\(688\) 0 0
\(689\) −3.80481 −0.144952
\(690\) 0 0
\(691\) −18.8108 −0.715597 −0.357799 0.933799i \(-0.616473\pi\)
−0.357799 + 0.933799i \(0.616473\pi\)
\(692\) 0 0
\(693\) −2.69846 −0.102506
\(694\) 0 0
\(695\) −5.91747 −0.224462
\(696\) 0 0
\(697\) −4.37839 −0.165843
\(698\) 0 0
\(699\) −22.3685 −0.846055
\(700\) 0 0
\(701\) 26.0550 0.984085 0.492042 0.870571i \(-0.336251\pi\)
0.492042 + 0.870571i \(0.336251\pi\)
\(702\) 0 0
\(703\) −16.8396 −0.635116
\(704\) 0 0
\(705\) 12.0939 0.455483
\(706\) 0 0
\(707\) −0.232042 −0.00872683
\(708\) 0 0
\(709\) 44.1870 1.65948 0.829739 0.558151i \(-0.188489\pi\)
0.829739 + 0.558151i \(0.188489\pi\)
\(710\) 0 0
\(711\) 7.54355 0.282905
\(712\) 0 0
\(713\) 8.29927 0.310810
\(714\) 0 0
\(715\) 16.8931 0.631766
\(716\) 0 0
\(717\) −20.4193 −0.762571
\(718\) 0 0
\(719\) 33.8677 1.26305 0.631526 0.775354i \(-0.282429\pi\)
0.631526 + 0.775354i \(0.282429\pi\)
\(720\) 0 0
\(721\) 10.2612 0.382146
\(722\) 0 0
\(723\) 8.00158 0.297582
\(724\) 0 0
\(725\) 45.9780 1.70758
\(726\) 0 0
\(727\) 40.8066 1.51343 0.756717 0.653743i \(-0.226802\pi\)
0.756717 + 0.653743i \(0.226802\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 17.1974 0.636068
\(732\) 0 0
\(733\) 13.0460 0.481863 0.240932 0.970542i \(-0.422547\pi\)
0.240932 + 0.970542i \(0.422547\pi\)
\(734\) 0 0
\(735\) 3.93128 0.145008
\(736\) 0 0
\(737\) 29.8912 1.10106
\(738\) 0 0
\(739\) 46.5452 1.71219 0.856096 0.516816i \(-0.172883\pi\)
0.856096 + 0.516816i \(0.172883\pi\)
\(740\) 0 0
\(741\) 4.29710 0.157858
\(742\) 0 0
\(743\) 1.36944 0.0502398 0.0251199 0.999684i \(-0.492003\pi\)
0.0251199 + 0.999684i \(0.492003\pi\)
\(744\) 0 0
\(745\) −6.43068 −0.235602
\(746\) 0 0
\(747\) −2.16174 −0.0790940
\(748\) 0 0
\(749\) 17.8420 0.651934
\(750\) 0 0
\(751\) −10.5582 −0.385276 −0.192638 0.981270i \(-0.561704\pi\)
−0.192638 + 0.981270i \(0.561704\pi\)
\(752\) 0 0
\(753\) −31.4694 −1.14681
\(754\) 0 0
\(755\) 2.85732 0.103988
\(756\) 0 0
\(757\) −34.1021 −1.23946 −0.619731 0.784814i \(-0.712758\pi\)
−0.619731 + 0.784814i \(0.712758\pi\)
\(758\) 0 0
\(759\) −2.69846 −0.0979480
\(760\) 0 0
\(761\) 20.4495 0.741293 0.370646 0.928774i \(-0.379136\pi\)
0.370646 + 0.928774i \(0.379136\pi\)
\(762\) 0 0
\(763\) 19.7741 0.715871
\(764\) 0 0
\(765\) −6.60842 −0.238928
\(766\) 0 0
\(767\) 2.59649 0.0937538
\(768\) 0 0
\(769\) −53.6421 −1.93438 −0.967192 0.254046i \(-0.918238\pi\)
−0.967192 + 0.254046i \(0.918238\pi\)
\(770\) 0 0
\(771\) −6.89989 −0.248493
\(772\) 0 0
\(773\) −0.609681 −0.0219287 −0.0109644 0.999940i \(-0.503490\pi\)
−0.0109644 + 0.999940i \(0.503490\pi\)
\(774\) 0 0
\(775\) 86.7688 3.11683
\(776\) 0 0
\(777\) 6.24043 0.223874
\(778\) 0 0
\(779\) −7.02857 −0.251825
\(780\) 0 0
\(781\) −20.7945 −0.744087
\(782\) 0 0
\(783\) 4.39770 0.157161
\(784\) 0 0
\(785\) 27.3727 0.976972
\(786\) 0 0
\(787\) −6.68256 −0.238208 −0.119104 0.992882i \(-0.538002\pi\)
−0.119104 + 0.992882i \(0.538002\pi\)
\(788\) 0 0
\(789\) −3.80925 −0.135613
\(790\) 0 0
\(791\) 12.2412 0.435247
\(792\) 0 0
\(793\) 2.05688 0.0730420
\(794\) 0 0
\(795\) 9.39308 0.333138
\(796\) 0 0
\(797\) 6.15012 0.217848 0.108924 0.994050i \(-0.465259\pi\)
0.108924 + 0.994050i \(0.465259\pi\)
\(798\) 0 0
\(799\) −5.17125 −0.182946
\(800\) 0 0
\(801\) 5.84350 0.206470
\(802\) 0 0
\(803\) −25.2750 −0.891934
\(804\) 0 0
\(805\) 3.93128 0.138560
\(806\) 0 0
\(807\) 17.8222 0.627371
\(808\) 0 0
\(809\) 1.64664 0.0578928 0.0289464 0.999581i \(-0.490785\pi\)
0.0289464 + 0.999581i \(0.490785\pi\)
\(810\) 0 0
\(811\) 53.6438 1.88369 0.941844 0.336050i \(-0.109091\pi\)
0.941844 + 0.336050i \(0.109091\pi\)
\(812\) 0 0
\(813\) −28.6062 −1.00326
\(814\) 0 0
\(815\) −16.4999 −0.577967
\(816\) 0 0
\(817\) 27.6068 0.965839
\(818\) 0 0
\(819\) −1.59242 −0.0556438
\(820\) 0 0
\(821\) −17.1819 −0.599652 −0.299826 0.953994i \(-0.596929\pi\)
−0.299826 + 0.953994i \(0.596929\pi\)
\(822\) 0 0
\(823\) 40.0941 1.39759 0.698797 0.715320i \(-0.253719\pi\)
0.698797 + 0.715320i \(0.253719\pi\)
\(824\) 0 0
\(825\) −28.2124 −0.982230
\(826\) 0 0
\(827\) 9.90076 0.344283 0.172142 0.985072i \(-0.444931\pi\)
0.172142 + 0.985072i \(0.444931\pi\)
\(828\) 0 0
\(829\) −43.3433 −1.50538 −0.752688 0.658377i \(-0.771243\pi\)
−0.752688 + 0.658377i \(0.771243\pi\)
\(830\) 0 0
\(831\) −0.806180 −0.0279661
\(832\) 0 0
\(833\) −1.68098 −0.0582426
\(834\) 0 0
\(835\) 5.09852 0.176442
\(836\) 0 0
\(837\) 8.29927 0.286865
\(838\) 0 0
\(839\) −14.0056 −0.483527 −0.241763 0.970335i \(-0.577726\pi\)
−0.241763 + 0.970335i \(0.577726\pi\)
\(840\) 0 0
\(841\) −9.66020 −0.333110
\(842\) 0 0
\(843\) 18.1136 0.623867
\(844\) 0 0
\(845\) −41.1377 −1.41518
\(846\) 0 0
\(847\) −3.71830 −0.127762
\(848\) 0 0
\(849\) 3.23431 0.111001
\(850\) 0 0
\(851\) 6.24043 0.213919
\(852\) 0 0
\(853\) −5.58205 −0.191126 −0.0955628 0.995423i \(-0.530465\pi\)
−0.0955628 + 0.995423i \(0.530465\pi\)
\(854\) 0 0
\(855\) −10.6084 −0.362800
\(856\) 0 0
\(857\) 32.5663 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(858\) 0 0
\(859\) 13.4543 0.459055 0.229527 0.973302i \(-0.426282\pi\)
0.229527 + 0.973302i \(0.426282\pi\)
\(860\) 0 0
\(861\) 2.60466 0.0887665
\(862\) 0 0
\(863\) 9.98498 0.339893 0.169946 0.985453i \(-0.445641\pi\)
0.169946 + 0.985453i \(0.445641\pi\)
\(864\) 0 0
\(865\) −58.7352 −1.99706
\(866\) 0 0
\(867\) −14.1743 −0.481384
\(868\) 0 0
\(869\) −20.3560 −0.690529
\(870\) 0 0
\(871\) 17.6395 0.597691
\(872\) 0 0
\(873\) 7.64198 0.258642
\(874\) 0 0
\(875\) 21.4451 0.724977
\(876\) 0 0
\(877\) −55.8448 −1.88575 −0.942873 0.333152i \(-0.891888\pi\)
−0.942873 + 0.333152i \(0.891888\pi\)
\(878\) 0 0
\(879\) 13.6703 0.461089
\(880\) 0 0
\(881\) −13.6444 −0.459692 −0.229846 0.973227i \(-0.573822\pi\)
−0.229846 + 0.973227i \(0.573822\pi\)
\(882\) 0 0
\(883\) 35.6072 1.19828 0.599138 0.800646i \(-0.295510\pi\)
0.599138 + 0.800646i \(0.295510\pi\)
\(884\) 0 0
\(885\) −6.41006 −0.215472
\(886\) 0 0
\(887\) 20.4118 0.685362 0.342681 0.939452i \(-0.388665\pi\)
0.342681 + 0.939452i \(0.388665\pi\)
\(888\) 0 0
\(889\) 9.24804 0.310169
\(890\) 0 0
\(891\) −2.69846 −0.0904019
\(892\) 0 0
\(893\) −8.30134 −0.277794
\(894\) 0 0
\(895\) −50.5070 −1.68826
\(896\) 0 0
\(897\) −1.59242 −0.0531695
\(898\) 0 0
\(899\) 36.4977 1.21727
\(900\) 0 0
\(901\) −4.01640 −0.133806
\(902\) 0 0
\(903\) −10.2306 −0.340452
\(904\) 0 0
\(905\) 37.5778 1.24913
\(906\) 0 0
\(907\) −46.4107 −1.54104 −0.770521 0.637414i \(-0.780004\pi\)
−0.770521 + 0.637414i \(0.780004\pi\)
\(908\) 0 0
\(909\) −0.232042 −0.00769634
\(910\) 0 0
\(911\) 21.2375 0.703630 0.351815 0.936070i \(-0.385565\pi\)
0.351815 + 0.936070i \(0.385565\pi\)
\(912\) 0 0
\(913\) 5.83338 0.193057
\(914\) 0 0
\(915\) −5.07791 −0.167870
\(916\) 0 0
\(917\) 0.855736 0.0282589
\(918\) 0 0
\(919\) −18.0974 −0.596979 −0.298490 0.954413i \(-0.596483\pi\)
−0.298490 + 0.954413i \(0.596483\pi\)
\(920\) 0 0
\(921\) −15.8940 −0.523725
\(922\) 0 0
\(923\) −12.2713 −0.403916
\(924\) 0 0
\(925\) 65.2436 2.14520
\(926\) 0 0
\(927\) 10.2612 0.337021
\(928\) 0 0
\(929\) −23.6583 −0.776204 −0.388102 0.921616i \(-0.626869\pi\)
−0.388102 + 0.921616i \(0.626869\pi\)
\(930\) 0 0
\(931\) −2.69846 −0.0884385
\(932\) 0 0
\(933\) 12.9375 0.423556
\(934\) 0 0
\(935\) 17.8326 0.583187
\(936\) 0 0
\(937\) 4.66452 0.152383 0.0761917 0.997093i \(-0.475724\pi\)
0.0761917 + 0.997093i \(0.475724\pi\)
\(938\) 0 0
\(939\) −13.9670 −0.455797
\(940\) 0 0
\(941\) 21.0877 0.687439 0.343719 0.939072i \(-0.388313\pi\)
0.343719 + 0.939072i \(0.388313\pi\)
\(942\) 0 0
\(943\) 2.60466 0.0848194
\(944\) 0 0
\(945\) 3.93128 0.127885
\(946\) 0 0
\(947\) 4.95508 0.161018 0.0805092 0.996754i \(-0.474345\pi\)
0.0805092 + 0.996754i \(0.474345\pi\)
\(948\) 0 0
\(949\) −14.9153 −0.484172
\(950\) 0 0
\(951\) 33.0258 1.07094
\(952\) 0 0
\(953\) 39.8355 1.29040 0.645199 0.764015i \(-0.276775\pi\)
0.645199 + 0.764015i \(0.276775\pi\)
\(954\) 0 0
\(955\) −48.0488 −1.55482
\(956\) 0 0
\(957\) −11.8670 −0.383607
\(958\) 0 0
\(959\) −2.46723 −0.0796709
\(960\) 0 0
\(961\) 37.8779 1.22187
\(962\) 0 0
\(963\) 17.8420 0.574952
\(964\) 0 0
\(965\) −6.59606 −0.212335
\(966\) 0 0
\(967\) −61.3251 −1.97208 −0.986041 0.166504i \(-0.946752\pi\)
−0.986041 + 0.166504i \(0.946752\pi\)
\(968\) 0 0
\(969\) 4.53607 0.145719
\(970\) 0 0
\(971\) 29.7183 0.953705 0.476852 0.878983i \(-0.341778\pi\)
0.476852 + 0.878983i \(0.341778\pi\)
\(972\) 0 0
\(973\) −1.50523 −0.0482553
\(974\) 0 0
\(975\) −16.6488 −0.533188
\(976\) 0 0
\(977\) 16.4741 0.527054 0.263527 0.964652i \(-0.415114\pi\)
0.263527 + 0.964652i \(0.415114\pi\)
\(978\) 0 0
\(979\) −15.7685 −0.503962
\(980\) 0 0
\(981\) 19.7741 0.631339
\(982\) 0 0
\(983\) −32.8673 −1.04830 −0.524152 0.851625i \(-0.675617\pi\)
−0.524152 + 0.851625i \(0.675617\pi\)
\(984\) 0 0
\(985\) −37.0926 −1.18187
\(986\) 0 0
\(987\) 3.07632 0.0979205
\(988\) 0 0
\(989\) −10.2306 −0.325313
\(990\) 0 0
\(991\) −58.4640 −1.85717 −0.928585 0.371120i \(-0.878974\pi\)
−0.928585 + 0.371120i \(0.878974\pi\)
\(992\) 0 0
\(993\) 9.51818 0.302050
\(994\) 0 0
\(995\) 37.5449 1.19025
\(996\) 0 0
\(997\) −34.5053 −1.09279 −0.546396 0.837527i \(-0.684001\pi\)
−0.546396 + 0.837527i \(0.684001\pi\)
\(998\) 0 0
\(999\) 6.24043 0.197438
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 7728.2.a.ch.1.6 6
4.3 odd 2 3864.2.a.w.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.w.1.6 6 4.3 odd 2
7728.2.a.ch.1.6 6 1.1 even 1 trivial