Properties

Label 1932.2.a.g.1.1
Level $1932$
Weight $2$
Character 1932.1
Self dual yes
Analytic conductor $15.427$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1932.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.4270976705\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1932.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +0.618034 q^{13} -1.61803 q^{15} +6.70820 q^{17} -4.23607 q^{19} -1.00000 q^{21} -1.00000 q^{23} -2.38197 q^{25} +1.00000 q^{27} -1.76393 q^{29} -3.00000 q^{31} -3.00000 q^{33} +1.61803 q^{35} +9.47214 q^{37} +0.618034 q^{39} -11.9443 q^{41} -9.09017 q^{43} -1.61803 q^{45} -9.23607 q^{47} +1.00000 q^{49} +6.70820 q^{51} -9.32624 q^{53} +4.85410 q^{55} -4.23607 q^{57} -9.61803 q^{59} +5.85410 q^{61} -1.00000 q^{63} -1.00000 q^{65} +6.56231 q^{67} -1.00000 q^{69} -4.61803 q^{71} +1.76393 q^{73} -2.38197 q^{75} +3.00000 q^{77} -6.70820 q^{79} +1.00000 q^{81} -7.47214 q^{83} -10.8541 q^{85} -1.76393 q^{87} -16.7984 q^{89} -0.618034 q^{91} -3.00000 q^{93} +6.85410 q^{95} -12.4164 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} - q^{13} - q^{15} - 4 q^{19} - 2 q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} - 8 q^{29} - 6 q^{31} - 6 q^{33} + q^{35} + 10 q^{37} - q^{39} - 6 q^{41} - 7 q^{43} - q^{45} - 14 q^{47} + 2 q^{49} - 3 q^{53} + 3 q^{55} - 4 q^{57} - 17 q^{59} + 5 q^{61} - 2 q^{63} - 2 q^{65} - 7 q^{67} - 2 q^{69} - 7 q^{71} + 8 q^{73} - 7 q^{75} + 6 q^{77} + 2 q^{81} - 6 q^{83} - 15 q^{85} - 8 q^{87} - 9 q^{89} + q^{91} - 6 q^{93} + 7 q^{95} + 2 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.61803 −0.723607 −0.361803 0.932254i \(-0.617839\pi\)
−0.361803 + 0.932254i \(0.617839\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 0.618034 0.171412 0.0857059 0.996320i \(-0.472685\pi\)
0.0857059 + 0.996320i \(0.472685\pi\)
\(14\) 0 0
\(15\) −1.61803 −0.417775
\(16\) 0 0
\(17\) 6.70820 1.62698 0.813489 0.581580i \(-0.197565\pi\)
0.813489 + 0.581580i \(0.197565\pi\)
\(18\) 0 0
\(19\) −4.23607 −0.971821 −0.485910 0.874009i \(-0.661512\pi\)
−0.485910 + 0.874009i \(0.661512\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −2.38197 −0.476393
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.76393 −0.327554 −0.163777 0.986497i \(-0.552368\pi\)
−0.163777 + 0.986497i \(0.552368\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 1.61803 0.273498
\(36\) 0 0
\(37\) 9.47214 1.55721 0.778605 0.627515i \(-0.215928\pi\)
0.778605 + 0.627515i \(0.215928\pi\)
\(38\) 0 0
\(39\) 0.618034 0.0989646
\(40\) 0 0
\(41\) −11.9443 −1.86538 −0.932691 0.360677i \(-0.882546\pi\)
−0.932691 + 0.360677i \(0.882546\pi\)
\(42\) 0 0
\(43\) −9.09017 −1.38624 −0.693119 0.720823i \(-0.743764\pi\)
−0.693119 + 0.720823i \(0.743764\pi\)
\(44\) 0 0
\(45\) −1.61803 −0.241202
\(46\) 0 0
\(47\) −9.23607 −1.34722 −0.673609 0.739087i \(-0.735257\pi\)
−0.673609 + 0.739087i \(0.735257\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.70820 0.939336
\(52\) 0 0
\(53\) −9.32624 −1.28106 −0.640529 0.767934i \(-0.721285\pi\)
−0.640529 + 0.767934i \(0.721285\pi\)
\(54\) 0 0
\(55\) 4.85410 0.654527
\(56\) 0 0
\(57\) −4.23607 −0.561081
\(58\) 0 0
\(59\) −9.61803 −1.25216 −0.626081 0.779758i \(-0.715342\pi\)
−0.626081 + 0.779758i \(0.715342\pi\)
\(60\) 0 0
\(61\) 5.85410 0.749541 0.374770 0.927118i \(-0.377722\pi\)
0.374770 + 0.927118i \(0.377722\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 6.56231 0.801713 0.400857 0.916141i \(-0.368713\pi\)
0.400857 + 0.916141i \(0.368713\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −4.61803 −0.548060 −0.274030 0.961721i \(-0.588357\pi\)
−0.274030 + 0.961721i \(0.588357\pi\)
\(72\) 0 0
\(73\) 1.76393 0.206453 0.103226 0.994658i \(-0.467083\pi\)
0.103226 + 0.994658i \(0.467083\pi\)
\(74\) 0 0
\(75\) −2.38197 −0.275046
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −6.70820 −0.754732 −0.377366 0.926064i \(-0.623170\pi\)
−0.377366 + 0.926064i \(0.623170\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.47214 −0.820173 −0.410087 0.912047i \(-0.634502\pi\)
−0.410087 + 0.912047i \(0.634502\pi\)
\(84\) 0 0
\(85\) −10.8541 −1.17729
\(86\) 0 0
\(87\) −1.76393 −0.189113
\(88\) 0 0
\(89\) −16.7984 −1.78062 −0.890312 0.455351i \(-0.849514\pi\)
−0.890312 + 0.455351i \(0.849514\pi\)
\(90\) 0 0
\(91\) −0.618034 −0.0647876
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) 6.85410 0.703216
\(96\) 0 0
\(97\) −12.4164 −1.26070 −0.630348 0.776313i \(-0.717088\pi\)
−0.630348 + 0.776313i \(0.717088\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 5.32624 0.529980 0.264990 0.964251i \(-0.414631\pi\)
0.264990 + 0.964251i \(0.414631\pi\)
\(102\) 0 0
\(103\) 19.4164 1.91316 0.956578 0.291477i \(-0.0941468\pi\)
0.956578 + 0.291477i \(0.0941468\pi\)
\(104\) 0 0
\(105\) 1.61803 0.157904
\(106\) 0 0
\(107\) 10.6180 1.02648 0.513242 0.858244i \(-0.328444\pi\)
0.513242 + 0.858244i \(0.328444\pi\)
\(108\) 0 0
\(109\) −2.09017 −0.200202 −0.100101 0.994977i \(-0.531917\pi\)
−0.100101 + 0.994977i \(0.531917\pi\)
\(110\) 0 0
\(111\) 9.47214 0.899055
\(112\) 0 0
\(113\) 3.32624 0.312906 0.156453 0.987685i \(-0.449994\pi\)
0.156453 + 0.987685i \(0.449994\pi\)
\(114\) 0 0
\(115\) 1.61803 0.150882
\(116\) 0 0
\(117\) 0.618034 0.0571373
\(118\) 0 0
\(119\) −6.70820 −0.614940
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −11.9443 −1.07698
\(124\) 0 0
\(125\) 11.9443 1.06833
\(126\) 0 0
\(127\) 5.56231 0.493575 0.246787 0.969070i \(-0.420625\pi\)
0.246787 + 0.969070i \(0.420625\pi\)
\(128\) 0 0
\(129\) −9.09017 −0.800345
\(130\) 0 0
\(131\) 5.94427 0.519353 0.259677 0.965696i \(-0.416384\pi\)
0.259677 + 0.965696i \(0.416384\pi\)
\(132\) 0 0
\(133\) 4.23607 0.367314
\(134\) 0 0
\(135\) −1.61803 −0.139258
\(136\) 0 0
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) 6.85410 0.581357 0.290679 0.956821i \(-0.406119\pi\)
0.290679 + 0.956821i \(0.406119\pi\)
\(140\) 0 0
\(141\) −9.23607 −0.777817
\(142\) 0 0
\(143\) −1.85410 −0.155048
\(144\) 0 0
\(145\) 2.85410 0.237020
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −1.23607 −0.101263 −0.0506313 0.998717i \(-0.516123\pi\)
−0.0506313 + 0.998717i \(0.516123\pi\)
\(150\) 0 0
\(151\) −19.1246 −1.55634 −0.778169 0.628054i \(-0.783852\pi\)
−0.778169 + 0.628054i \(0.783852\pi\)
\(152\) 0 0
\(153\) 6.70820 0.542326
\(154\) 0 0
\(155\) 4.85410 0.389891
\(156\) 0 0
\(157\) 8.76393 0.699438 0.349719 0.936855i \(-0.386277\pi\)
0.349719 + 0.936855i \(0.386277\pi\)
\(158\) 0 0
\(159\) −9.32624 −0.739619
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 10.8541 0.850159 0.425079 0.905156i \(-0.360246\pi\)
0.425079 + 0.905156i \(0.360246\pi\)
\(164\) 0 0
\(165\) 4.85410 0.377891
\(166\) 0 0
\(167\) −10.5279 −0.814671 −0.407335 0.913279i \(-0.633542\pi\)
−0.407335 + 0.913279i \(0.633542\pi\)
\(168\) 0 0
\(169\) −12.6180 −0.970618
\(170\) 0 0
\(171\) −4.23607 −0.323940
\(172\) 0 0
\(173\) 13.4721 1.02427 0.512134 0.858906i \(-0.328855\pi\)
0.512134 + 0.858906i \(0.328855\pi\)
\(174\) 0 0
\(175\) 2.38197 0.180060
\(176\) 0 0
\(177\) −9.61803 −0.722936
\(178\) 0 0
\(179\) 11.0344 0.824753 0.412376 0.911014i \(-0.364699\pi\)
0.412376 + 0.911014i \(0.364699\pi\)
\(180\) 0 0
\(181\) 17.4721 1.29869 0.649347 0.760492i \(-0.275042\pi\)
0.649347 + 0.760492i \(0.275042\pi\)
\(182\) 0 0
\(183\) 5.85410 0.432748
\(184\) 0 0
\(185\) −15.3262 −1.12681
\(186\) 0 0
\(187\) −20.1246 −1.47166
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −20.6525 −1.49436 −0.747180 0.664621i \(-0.768593\pi\)
−0.747180 + 0.664621i \(0.768593\pi\)
\(192\) 0 0
\(193\) −5.70820 −0.410886 −0.205443 0.978669i \(-0.565863\pi\)
−0.205443 + 0.978669i \(0.565863\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 1.38197 0.0984610 0.0492305 0.998787i \(-0.484323\pi\)
0.0492305 + 0.998787i \(0.484323\pi\)
\(198\) 0 0
\(199\) 22.5623 1.59940 0.799700 0.600400i \(-0.204992\pi\)
0.799700 + 0.600400i \(0.204992\pi\)
\(200\) 0 0
\(201\) 6.56231 0.462869
\(202\) 0 0
\(203\) 1.76393 0.123804
\(204\) 0 0
\(205\) 19.3262 1.34980
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 12.7082 0.879045
\(210\) 0 0
\(211\) −26.1246 −1.79849 −0.899246 0.437443i \(-0.855884\pi\)
−0.899246 + 0.437443i \(0.855884\pi\)
\(212\) 0 0
\(213\) −4.61803 −0.316422
\(214\) 0 0
\(215\) 14.7082 1.00309
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) 0 0
\(219\) 1.76393 0.119195
\(220\) 0 0
\(221\) 4.14590 0.278883
\(222\) 0 0
\(223\) 11.7984 0.790078 0.395039 0.918664i \(-0.370731\pi\)
0.395039 + 0.918664i \(0.370731\pi\)
\(224\) 0 0
\(225\) −2.38197 −0.158798
\(226\) 0 0
\(227\) 26.4508 1.75560 0.877802 0.479023i \(-0.159009\pi\)
0.877802 + 0.479023i \(0.159009\pi\)
\(228\) 0 0
\(229\) 24.2705 1.60384 0.801920 0.597431i \(-0.203812\pi\)
0.801920 + 0.597431i \(0.203812\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −12.0902 −0.792053 −0.396027 0.918239i \(-0.629611\pi\)
−0.396027 + 0.918239i \(0.629611\pi\)
\(234\) 0 0
\(235\) 14.9443 0.974857
\(236\) 0 0
\(237\) −6.70820 −0.435745
\(238\) 0 0
\(239\) −17.6180 −1.13962 −0.569808 0.821778i \(-0.692982\pi\)
−0.569808 + 0.821778i \(0.692982\pi\)
\(240\) 0 0
\(241\) 21.0000 1.35273 0.676364 0.736567i \(-0.263554\pi\)
0.676364 + 0.736567i \(0.263554\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.61803 −0.103372
\(246\) 0 0
\(247\) −2.61803 −0.166582
\(248\) 0 0
\(249\) −7.47214 −0.473527
\(250\) 0 0
\(251\) −22.6525 −1.42981 −0.714906 0.699221i \(-0.753530\pi\)
−0.714906 + 0.699221i \(0.753530\pi\)
\(252\) 0 0
\(253\) 3.00000 0.188608
\(254\) 0 0
\(255\) −10.8541 −0.679710
\(256\) 0 0
\(257\) 16.6525 1.03875 0.519376 0.854546i \(-0.326164\pi\)
0.519376 + 0.854546i \(0.326164\pi\)
\(258\) 0 0
\(259\) −9.47214 −0.588570
\(260\) 0 0
\(261\) −1.76393 −0.109185
\(262\) 0 0
\(263\) 15.1803 0.936060 0.468030 0.883713i \(-0.344964\pi\)
0.468030 + 0.883713i \(0.344964\pi\)
\(264\) 0 0
\(265\) 15.0902 0.926982
\(266\) 0 0
\(267\) −16.7984 −1.02804
\(268\) 0 0
\(269\) −22.7984 −1.39004 −0.695021 0.718990i \(-0.744605\pi\)
−0.695021 + 0.718990i \(0.744605\pi\)
\(270\) 0 0
\(271\) −21.1803 −1.28661 −0.643307 0.765608i \(-0.722438\pi\)
−0.643307 + 0.765608i \(0.722438\pi\)
\(272\) 0 0
\(273\) −0.618034 −0.0374051
\(274\) 0 0
\(275\) 7.14590 0.430914
\(276\) 0 0
\(277\) 24.0344 1.44409 0.722045 0.691846i \(-0.243202\pi\)
0.722045 + 0.691846i \(0.243202\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) −12.7639 −0.761432 −0.380716 0.924692i \(-0.624323\pi\)
−0.380716 + 0.924692i \(0.624323\pi\)
\(282\) 0 0
\(283\) −14.8541 −0.882985 −0.441492 0.897265i \(-0.645551\pi\)
−0.441492 + 0.897265i \(0.645551\pi\)
\(284\) 0 0
\(285\) 6.85410 0.406002
\(286\) 0 0
\(287\) 11.9443 0.705048
\(288\) 0 0
\(289\) 28.0000 1.64706
\(290\) 0 0
\(291\) −12.4164 −0.727863
\(292\) 0 0
\(293\) −0.944272 −0.0551650 −0.0275825 0.999620i \(-0.508781\pi\)
−0.0275825 + 0.999620i \(0.508781\pi\)
\(294\) 0 0
\(295\) 15.5623 0.906072
\(296\) 0 0
\(297\) −3.00000 −0.174078
\(298\) 0 0
\(299\) −0.618034 −0.0357418
\(300\) 0 0
\(301\) 9.09017 0.523949
\(302\) 0 0
\(303\) 5.32624 0.305984
\(304\) 0 0
\(305\) −9.47214 −0.542373
\(306\) 0 0
\(307\) −28.4164 −1.62181 −0.810905 0.585178i \(-0.801025\pi\)
−0.810905 + 0.585178i \(0.801025\pi\)
\(308\) 0 0
\(309\) 19.4164 1.10456
\(310\) 0 0
\(311\) −15.2705 −0.865911 −0.432956 0.901415i \(-0.642529\pi\)
−0.432956 + 0.901415i \(0.642529\pi\)
\(312\) 0 0
\(313\) 17.5279 0.990733 0.495367 0.868684i \(-0.335034\pi\)
0.495367 + 0.868684i \(0.335034\pi\)
\(314\) 0 0
\(315\) 1.61803 0.0911659
\(316\) 0 0
\(317\) 5.61803 0.315540 0.157770 0.987476i \(-0.449570\pi\)
0.157770 + 0.987476i \(0.449570\pi\)
\(318\) 0 0
\(319\) 5.29180 0.296284
\(320\) 0 0
\(321\) 10.6180 0.592641
\(322\) 0 0
\(323\) −28.4164 −1.58113
\(324\) 0 0
\(325\) −1.47214 −0.0816594
\(326\) 0 0
\(327\) −2.09017 −0.115587
\(328\) 0 0
\(329\) 9.23607 0.509201
\(330\) 0 0
\(331\) 7.88854 0.433594 0.216797 0.976217i \(-0.430439\pi\)
0.216797 + 0.976217i \(0.430439\pi\)
\(332\) 0 0
\(333\) 9.47214 0.519070
\(334\) 0 0
\(335\) −10.6180 −0.580125
\(336\) 0 0
\(337\) −12.6180 −0.687348 −0.343674 0.939089i \(-0.611672\pi\)
−0.343674 + 0.939089i \(0.611672\pi\)
\(338\) 0 0
\(339\) 3.32624 0.180656
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 1.61803 0.0871120
\(346\) 0 0
\(347\) −13.4721 −0.723222 −0.361611 0.932329i \(-0.617773\pi\)
−0.361611 + 0.932329i \(0.617773\pi\)
\(348\) 0 0
\(349\) 13.1459 0.703684 0.351842 0.936059i \(-0.385555\pi\)
0.351842 + 0.936059i \(0.385555\pi\)
\(350\) 0 0
\(351\) 0.618034 0.0329882
\(352\) 0 0
\(353\) −15.0000 −0.798369 −0.399185 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) 0 0
\(355\) 7.47214 0.396580
\(356\) 0 0
\(357\) −6.70820 −0.355036
\(358\) 0 0
\(359\) 27.0344 1.42682 0.713412 0.700745i \(-0.247149\pi\)
0.713412 + 0.700745i \(0.247149\pi\)
\(360\) 0 0
\(361\) −1.05573 −0.0555646
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −2.85410 −0.149391
\(366\) 0 0
\(367\) −30.7984 −1.60766 −0.803831 0.594858i \(-0.797208\pi\)
−0.803831 + 0.594858i \(0.797208\pi\)
\(368\) 0 0
\(369\) −11.9443 −0.621794
\(370\) 0 0
\(371\) 9.32624 0.484194
\(372\) 0 0
\(373\) −6.88854 −0.356675 −0.178338 0.983969i \(-0.557072\pi\)
−0.178338 + 0.983969i \(0.557072\pi\)
\(374\) 0 0
\(375\) 11.9443 0.616800
\(376\) 0 0
\(377\) −1.09017 −0.0561466
\(378\) 0 0
\(379\) −14.4721 −0.743384 −0.371692 0.928356i \(-0.621222\pi\)
−0.371692 + 0.928356i \(0.621222\pi\)
\(380\) 0 0
\(381\) 5.56231 0.284966
\(382\) 0 0
\(383\) 12.0557 0.616019 0.308009 0.951383i \(-0.400337\pi\)
0.308009 + 0.951383i \(0.400337\pi\)
\(384\) 0 0
\(385\) −4.85410 −0.247388
\(386\) 0 0
\(387\) −9.09017 −0.462079
\(388\) 0 0
\(389\) −21.6525 −1.09782 −0.548912 0.835880i \(-0.684958\pi\)
−0.548912 + 0.835880i \(0.684958\pi\)
\(390\) 0 0
\(391\) −6.70820 −0.339248
\(392\) 0 0
\(393\) 5.94427 0.299849
\(394\) 0 0
\(395\) 10.8541 0.546129
\(396\) 0 0
\(397\) −15.1246 −0.759083 −0.379541 0.925175i \(-0.623918\pi\)
−0.379541 + 0.925175i \(0.623918\pi\)
\(398\) 0 0
\(399\) 4.23607 0.212069
\(400\) 0 0
\(401\) −16.2361 −0.810791 −0.405395 0.914141i \(-0.632866\pi\)
−0.405395 + 0.914141i \(0.632866\pi\)
\(402\) 0 0
\(403\) −1.85410 −0.0923594
\(404\) 0 0
\(405\) −1.61803 −0.0804008
\(406\) 0 0
\(407\) −28.4164 −1.40855
\(408\) 0 0
\(409\) 36.4164 1.80068 0.900338 0.435192i \(-0.143319\pi\)
0.900338 + 0.435192i \(0.143319\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) 9.61803 0.473273
\(414\) 0 0
\(415\) 12.0902 0.593483
\(416\) 0 0
\(417\) 6.85410 0.335647
\(418\) 0 0
\(419\) 27.2148 1.32953 0.664765 0.747053i \(-0.268532\pi\)
0.664765 + 0.747053i \(0.268532\pi\)
\(420\) 0 0
\(421\) −13.3820 −0.652197 −0.326099 0.945336i \(-0.605734\pi\)
−0.326099 + 0.945336i \(0.605734\pi\)
\(422\) 0 0
\(423\) −9.23607 −0.449073
\(424\) 0 0
\(425\) −15.9787 −0.775081
\(426\) 0 0
\(427\) −5.85410 −0.283300
\(428\) 0 0
\(429\) −1.85410 −0.0895169
\(430\) 0 0
\(431\) 31.8541 1.53436 0.767179 0.641433i \(-0.221660\pi\)
0.767179 + 0.641433i \(0.221660\pi\)
\(432\) 0 0
\(433\) 14.2918 0.686820 0.343410 0.939186i \(-0.388418\pi\)
0.343410 + 0.939186i \(0.388418\pi\)
\(434\) 0 0
\(435\) 2.85410 0.136844
\(436\) 0 0
\(437\) 4.23607 0.202639
\(438\) 0 0
\(439\) −4.88854 −0.233317 −0.116659 0.993172i \(-0.537218\pi\)
−0.116659 + 0.993172i \(0.537218\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 7.41641 0.352364 0.176182 0.984358i \(-0.443625\pi\)
0.176182 + 0.984358i \(0.443625\pi\)
\(444\) 0 0
\(445\) 27.1803 1.28847
\(446\) 0 0
\(447\) −1.23607 −0.0584640
\(448\) 0 0
\(449\) 12.0902 0.570570 0.285285 0.958443i \(-0.407912\pi\)
0.285285 + 0.958443i \(0.407912\pi\)
\(450\) 0 0
\(451\) 35.8328 1.68730
\(452\) 0 0
\(453\) −19.1246 −0.898553
\(454\) 0 0
\(455\) 1.00000 0.0468807
\(456\) 0 0
\(457\) 39.9787 1.87013 0.935063 0.354482i \(-0.115343\pi\)
0.935063 + 0.354482i \(0.115343\pi\)
\(458\) 0 0
\(459\) 6.70820 0.313112
\(460\) 0 0
\(461\) 31.7984 1.48100 0.740499 0.672058i \(-0.234589\pi\)
0.740499 + 0.672058i \(0.234589\pi\)
\(462\) 0 0
\(463\) 15.8328 0.735813 0.367907 0.929863i \(-0.380075\pi\)
0.367907 + 0.929863i \(0.380075\pi\)
\(464\) 0 0
\(465\) 4.85410 0.225104
\(466\) 0 0
\(467\) 10.8885 0.503862 0.251931 0.967745i \(-0.418934\pi\)
0.251931 + 0.967745i \(0.418934\pi\)
\(468\) 0 0
\(469\) −6.56231 −0.303019
\(470\) 0 0
\(471\) 8.76393 0.403821
\(472\) 0 0
\(473\) 27.2705 1.25390
\(474\) 0 0
\(475\) 10.0902 0.462969
\(476\) 0 0
\(477\) −9.32624 −0.427019
\(478\) 0 0
\(479\) 18.2361 0.833227 0.416614 0.909084i \(-0.363217\pi\)
0.416614 + 0.909084i \(0.363217\pi\)
\(480\) 0 0
\(481\) 5.85410 0.266924
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 20.0902 0.912248
\(486\) 0 0
\(487\) −29.0000 −1.31412 −0.657058 0.753840i \(-0.728199\pi\)
−0.657058 + 0.753840i \(0.728199\pi\)
\(488\) 0 0
\(489\) 10.8541 0.490839
\(490\) 0 0
\(491\) −5.09017 −0.229716 −0.114858 0.993382i \(-0.536641\pi\)
−0.114858 + 0.993382i \(0.536641\pi\)
\(492\) 0 0
\(493\) −11.8328 −0.532923
\(494\) 0 0
\(495\) 4.85410 0.218176
\(496\) 0 0
\(497\) 4.61803 0.207147
\(498\) 0 0
\(499\) −3.50658 −0.156976 −0.0784880 0.996915i \(-0.525009\pi\)
−0.0784880 + 0.996915i \(0.525009\pi\)
\(500\) 0 0
\(501\) −10.5279 −0.470350
\(502\) 0 0
\(503\) −11.2148 −0.500042 −0.250021 0.968240i \(-0.580438\pi\)
−0.250021 + 0.968240i \(0.580438\pi\)
\(504\) 0 0
\(505\) −8.61803 −0.383497
\(506\) 0 0
\(507\) −12.6180 −0.560387
\(508\) 0 0
\(509\) 21.2361 0.941272 0.470636 0.882327i \(-0.344024\pi\)
0.470636 + 0.882327i \(0.344024\pi\)
\(510\) 0 0
\(511\) −1.76393 −0.0780318
\(512\) 0 0
\(513\) −4.23607 −0.187027
\(514\) 0 0
\(515\) −31.4164 −1.38437
\(516\) 0 0
\(517\) 27.7082 1.21861
\(518\) 0 0
\(519\) 13.4721 0.591361
\(520\) 0 0
\(521\) 22.3607 0.979639 0.489820 0.871824i \(-0.337063\pi\)
0.489820 + 0.871824i \(0.337063\pi\)
\(522\) 0 0
\(523\) 18.0000 0.787085 0.393543 0.919306i \(-0.371249\pi\)
0.393543 + 0.919306i \(0.371249\pi\)
\(524\) 0 0
\(525\) 2.38197 0.103958
\(526\) 0 0
\(527\) −20.1246 −0.876642
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.61803 −0.417387
\(532\) 0 0
\(533\) −7.38197 −0.319748
\(534\) 0 0
\(535\) −17.1803 −0.742771
\(536\) 0 0
\(537\) 11.0344 0.476171
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −3.12461 −0.134338 −0.0671688 0.997742i \(-0.521397\pi\)
−0.0671688 + 0.997742i \(0.521397\pi\)
\(542\) 0 0
\(543\) 17.4721 0.749801
\(544\) 0 0
\(545\) 3.38197 0.144868
\(546\) 0 0
\(547\) −11.1459 −0.476564 −0.238282 0.971196i \(-0.576584\pi\)
−0.238282 + 0.971196i \(0.576584\pi\)
\(548\) 0 0
\(549\) 5.85410 0.249847
\(550\) 0 0
\(551\) 7.47214 0.318324
\(552\) 0 0
\(553\) 6.70820 0.285262
\(554\) 0 0
\(555\) −15.3262 −0.650563
\(556\) 0 0
\(557\) 23.2361 0.984544 0.492272 0.870441i \(-0.336167\pi\)
0.492272 + 0.870441i \(0.336167\pi\)
\(558\) 0 0
\(559\) −5.61803 −0.237618
\(560\) 0 0
\(561\) −20.1246 −0.849662
\(562\) 0 0
\(563\) −30.6180 −1.29040 −0.645198 0.764015i \(-0.723225\pi\)
−0.645198 + 0.764015i \(0.723225\pi\)
\(564\) 0 0
\(565\) −5.38197 −0.226421
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −31.4164 −1.31704 −0.658522 0.752561i \(-0.728818\pi\)
−0.658522 + 0.752561i \(0.728818\pi\)
\(570\) 0 0
\(571\) −12.7082 −0.531822 −0.265911 0.963998i \(-0.585673\pi\)
−0.265911 + 0.963998i \(0.585673\pi\)
\(572\) 0 0
\(573\) −20.6525 −0.862770
\(574\) 0 0
\(575\) 2.38197 0.0993348
\(576\) 0 0
\(577\) −33.7771 −1.40616 −0.703079 0.711111i \(-0.748192\pi\)
−0.703079 + 0.711111i \(0.748192\pi\)
\(578\) 0 0
\(579\) −5.70820 −0.237225
\(580\) 0 0
\(581\) 7.47214 0.309996
\(582\) 0 0
\(583\) 27.9787 1.15876
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 24.6180 1.01610 0.508048 0.861329i \(-0.330367\pi\)
0.508048 + 0.861329i \(0.330367\pi\)
\(588\) 0 0
\(589\) 12.7082 0.523632
\(590\) 0 0
\(591\) 1.38197 0.0568465
\(592\) 0 0
\(593\) 6.65248 0.273184 0.136592 0.990627i \(-0.456385\pi\)
0.136592 + 0.990627i \(0.456385\pi\)
\(594\) 0 0
\(595\) 10.8541 0.444975
\(596\) 0 0
\(597\) 22.5623 0.923414
\(598\) 0 0
\(599\) 33.7426 1.37869 0.689344 0.724435i \(-0.257899\pi\)
0.689344 + 0.724435i \(0.257899\pi\)
\(600\) 0 0
\(601\) 29.7984 1.21550 0.607751 0.794128i \(-0.292072\pi\)
0.607751 + 0.794128i \(0.292072\pi\)
\(602\) 0 0
\(603\) 6.56231 0.267238
\(604\) 0 0
\(605\) 3.23607 0.131565
\(606\) 0 0
\(607\) 15.5623 0.631655 0.315827 0.948817i \(-0.397718\pi\)
0.315827 + 0.948817i \(0.397718\pi\)
\(608\) 0 0
\(609\) 1.76393 0.0714781
\(610\) 0 0
\(611\) −5.70820 −0.230929
\(612\) 0 0
\(613\) −22.7082 −0.917176 −0.458588 0.888649i \(-0.651645\pi\)
−0.458588 + 0.888649i \(0.651645\pi\)
\(614\) 0 0
\(615\) 19.3262 0.779309
\(616\) 0 0
\(617\) 16.1459 0.650009 0.325005 0.945712i \(-0.394634\pi\)
0.325005 + 0.945712i \(0.394634\pi\)
\(618\) 0 0
\(619\) −37.8541 −1.52148 −0.760742 0.649054i \(-0.775165\pi\)
−0.760742 + 0.649054i \(0.775165\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 16.7984 0.673013
\(624\) 0 0
\(625\) −7.41641 −0.296656
\(626\) 0 0
\(627\) 12.7082 0.507517
\(628\) 0 0
\(629\) 63.5410 2.53355
\(630\) 0 0
\(631\) 30.7771 1.22522 0.612608 0.790387i \(-0.290120\pi\)
0.612608 + 0.790387i \(0.290120\pi\)
\(632\) 0 0
\(633\) −26.1246 −1.03836
\(634\) 0 0
\(635\) −9.00000 −0.357154
\(636\) 0 0
\(637\) 0.618034 0.0244874
\(638\) 0 0
\(639\) −4.61803 −0.182687
\(640\) 0 0
\(641\) −40.3262 −1.59279 −0.796395 0.604776i \(-0.793262\pi\)
−0.796395 + 0.604776i \(0.793262\pi\)
\(642\) 0 0
\(643\) −8.32624 −0.328355 −0.164177 0.986431i \(-0.552497\pi\)
−0.164177 + 0.986431i \(0.552497\pi\)
\(644\) 0 0
\(645\) 14.7082 0.579135
\(646\) 0 0
\(647\) −47.3951 −1.86329 −0.931647 0.363364i \(-0.881628\pi\)
−0.931647 + 0.363364i \(0.881628\pi\)
\(648\) 0 0
\(649\) 28.8541 1.13262
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) 0 0
\(653\) −19.1459 −0.749237 −0.374618 0.927179i \(-0.622226\pi\)
−0.374618 + 0.927179i \(0.622226\pi\)
\(654\) 0 0
\(655\) −9.61803 −0.375808
\(656\) 0 0
\(657\) 1.76393 0.0688175
\(658\) 0 0
\(659\) −40.3050 −1.57006 −0.785029 0.619459i \(-0.787352\pi\)
−0.785029 + 0.619459i \(0.787352\pi\)
\(660\) 0 0
\(661\) 42.7771 1.66384 0.831918 0.554899i \(-0.187243\pi\)
0.831918 + 0.554899i \(0.187243\pi\)
\(662\) 0 0
\(663\) 4.14590 0.161013
\(664\) 0 0
\(665\) −6.85410 −0.265791
\(666\) 0 0
\(667\) 1.76393 0.0682997
\(668\) 0 0
\(669\) 11.7984 0.456151
\(670\) 0 0
\(671\) −17.5623 −0.677985
\(672\) 0 0
\(673\) 18.3050 0.705604 0.352802 0.935698i \(-0.385229\pi\)
0.352802 + 0.935698i \(0.385229\pi\)
\(674\) 0 0
\(675\) −2.38197 −0.0916819
\(676\) 0 0
\(677\) −11.0902 −0.426230 −0.213115 0.977027i \(-0.568361\pi\)
−0.213115 + 0.977027i \(0.568361\pi\)
\(678\) 0 0
\(679\) 12.4164 0.476498
\(680\) 0 0
\(681\) 26.4508 1.01360
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) 14.5623 0.556397
\(686\) 0 0
\(687\) 24.2705 0.925978
\(688\) 0 0
\(689\) −5.76393 −0.219588
\(690\) 0 0
\(691\) 17.3820 0.661241 0.330621 0.943764i \(-0.392742\pi\)
0.330621 + 0.943764i \(0.392742\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) −11.0902 −0.420674
\(696\) 0 0
\(697\) −80.1246 −3.03494
\(698\) 0 0
\(699\) −12.0902 −0.457292
\(700\) 0 0
\(701\) 1.90983 0.0721333 0.0360666 0.999349i \(-0.488517\pi\)
0.0360666 + 0.999349i \(0.488517\pi\)
\(702\) 0 0
\(703\) −40.1246 −1.51333
\(704\) 0 0
\(705\) 14.9443 0.562834
\(706\) 0 0
\(707\) −5.32624 −0.200314
\(708\) 0 0
\(709\) −11.0902 −0.416500 −0.208250 0.978076i \(-0.566777\pi\)
−0.208250 + 0.978076i \(0.566777\pi\)
\(710\) 0 0
\(711\) −6.70820 −0.251577
\(712\) 0 0
\(713\) 3.00000 0.112351
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 0 0
\(717\) −17.6180 −0.657958
\(718\) 0 0
\(719\) 27.6525 1.03126 0.515632 0.856810i \(-0.327557\pi\)
0.515632 + 0.856810i \(0.327557\pi\)
\(720\) 0 0
\(721\) −19.4164 −0.723105
\(722\) 0 0
\(723\) 21.0000 0.780998
\(724\) 0 0
\(725\) 4.20163 0.156044
\(726\) 0 0
\(727\) −32.7082 −1.21308 −0.606540 0.795053i \(-0.707443\pi\)
−0.606540 + 0.795053i \(0.707443\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −60.9787 −2.25538
\(732\) 0 0
\(733\) 47.5967 1.75803 0.879013 0.476798i \(-0.158203\pi\)
0.879013 + 0.476798i \(0.158203\pi\)
\(734\) 0 0
\(735\) −1.61803 −0.0596821
\(736\) 0 0
\(737\) −19.6869 −0.725177
\(738\) 0 0
\(739\) 28.7082 1.05605 0.528024 0.849229i \(-0.322933\pi\)
0.528024 + 0.849229i \(0.322933\pi\)
\(740\) 0 0
\(741\) −2.61803 −0.0961759
\(742\) 0 0
\(743\) −20.3820 −0.747742 −0.373871 0.927481i \(-0.621970\pi\)
−0.373871 + 0.927481i \(0.621970\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) 0 0
\(747\) −7.47214 −0.273391
\(748\) 0 0
\(749\) −10.6180 −0.387975
\(750\) 0 0
\(751\) 38.2705 1.39651 0.698255 0.715849i \(-0.253960\pi\)
0.698255 + 0.715849i \(0.253960\pi\)
\(752\) 0 0
\(753\) −22.6525 −0.825502
\(754\) 0 0
\(755\) 30.9443 1.12618
\(756\) 0 0
\(757\) −17.4721 −0.635036 −0.317518 0.948252i \(-0.602849\pi\)
−0.317518 + 0.948252i \(0.602849\pi\)
\(758\) 0 0
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) −26.4721 −0.959614 −0.479807 0.877374i \(-0.659293\pi\)
−0.479807 + 0.877374i \(0.659293\pi\)
\(762\) 0 0
\(763\) 2.09017 0.0756692
\(764\) 0 0
\(765\) −10.8541 −0.392431
\(766\) 0 0
\(767\) −5.94427 −0.214635
\(768\) 0 0
\(769\) 5.12461 0.184798 0.0923991 0.995722i \(-0.470546\pi\)
0.0923991 + 0.995722i \(0.470546\pi\)
\(770\) 0 0
\(771\) 16.6525 0.599724
\(772\) 0 0
\(773\) −16.0557 −0.577484 −0.288742 0.957407i \(-0.593237\pi\)
−0.288742 + 0.957407i \(0.593237\pi\)
\(774\) 0 0
\(775\) 7.14590 0.256688
\(776\) 0 0
\(777\) −9.47214 −0.339811
\(778\) 0 0
\(779\) 50.5967 1.81282
\(780\) 0 0
\(781\) 13.8541 0.495739
\(782\) 0 0
\(783\) −1.76393 −0.0630378
\(784\) 0 0
\(785\) −14.1803 −0.506118
\(786\) 0 0
\(787\) 24.0902 0.858722 0.429361 0.903133i \(-0.358739\pi\)
0.429361 + 0.903133i \(0.358739\pi\)
\(788\) 0 0
\(789\) 15.1803 0.540434
\(790\) 0 0
\(791\) −3.32624 −0.118267
\(792\) 0 0
\(793\) 3.61803 0.128480
\(794\) 0 0
\(795\) 15.0902 0.535193
\(796\) 0 0
\(797\) 21.8197 0.772892 0.386446 0.922312i \(-0.373703\pi\)
0.386446 + 0.922312i \(0.373703\pi\)
\(798\) 0 0
\(799\) −61.9574 −2.19190
\(800\) 0 0
\(801\) −16.7984 −0.593541
\(802\) 0 0
\(803\) −5.29180 −0.186743
\(804\) 0 0
\(805\) −1.61803 −0.0570282
\(806\) 0 0
\(807\) −22.7984 −0.802541
\(808\) 0 0
\(809\) −42.2148 −1.48419 −0.742096 0.670293i \(-0.766168\pi\)
−0.742096 + 0.670293i \(0.766168\pi\)
\(810\) 0 0
\(811\) −40.7082 −1.42946 −0.714729 0.699401i \(-0.753450\pi\)
−0.714729 + 0.699401i \(0.753450\pi\)
\(812\) 0 0
\(813\) −21.1803 −0.742827
\(814\) 0 0
\(815\) −17.5623 −0.615181
\(816\) 0 0
\(817\) 38.5066 1.34717
\(818\) 0 0
\(819\) −0.618034 −0.0215959
\(820\) 0 0
\(821\) −51.7082 −1.80463 −0.902314 0.431079i \(-0.858133\pi\)
−0.902314 + 0.431079i \(0.858133\pi\)
\(822\) 0 0
\(823\) −27.9098 −0.972876 −0.486438 0.873715i \(-0.661704\pi\)
−0.486438 + 0.873715i \(0.661704\pi\)
\(824\) 0 0
\(825\) 7.14590 0.248788
\(826\) 0 0
\(827\) 50.9230 1.77077 0.885383 0.464863i \(-0.153896\pi\)
0.885383 + 0.464863i \(0.153896\pi\)
\(828\) 0 0
\(829\) 7.00000 0.243120 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 0 0
\(831\) 24.0344 0.833746
\(832\) 0 0
\(833\) 6.70820 0.232425
\(834\) 0 0
\(835\) 17.0344 0.589501
\(836\) 0 0
\(837\) −3.00000 −0.103695
\(838\) 0 0
\(839\) −11.0902 −0.382875 −0.191438 0.981505i \(-0.561315\pi\)
−0.191438 + 0.981505i \(0.561315\pi\)
\(840\) 0 0
\(841\) −25.8885 −0.892708
\(842\) 0 0
\(843\) −12.7639 −0.439613
\(844\) 0 0
\(845\) 20.4164 0.702346
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −14.8541 −0.509791
\(850\) 0 0
\(851\) −9.47214 −0.324701
\(852\) 0 0
\(853\) 2.29180 0.0784696 0.0392348 0.999230i \(-0.487508\pi\)
0.0392348 + 0.999230i \(0.487508\pi\)
\(854\) 0 0
\(855\) 6.85410 0.234405
\(856\) 0 0
\(857\) −12.1803 −0.416072 −0.208036 0.978121i \(-0.566707\pi\)
−0.208036 + 0.978121i \(0.566707\pi\)
\(858\) 0 0
\(859\) −35.3050 −1.20459 −0.602295 0.798274i \(-0.705747\pi\)
−0.602295 + 0.798274i \(0.705747\pi\)
\(860\) 0 0
\(861\) 11.9443 0.407060
\(862\) 0 0
\(863\) 50.6525 1.72423 0.862115 0.506712i \(-0.169139\pi\)
0.862115 + 0.506712i \(0.169139\pi\)
\(864\) 0 0
\(865\) −21.7984 −0.741167
\(866\) 0 0
\(867\) 28.0000 0.950930
\(868\) 0 0
\(869\) 20.1246 0.682681
\(870\) 0 0
\(871\) 4.05573 0.137423
\(872\) 0 0
\(873\) −12.4164 −0.420232
\(874\) 0 0
\(875\) −11.9443 −0.403790
\(876\) 0 0
\(877\) −3.41641 −0.115364 −0.0576819 0.998335i \(-0.518371\pi\)
−0.0576819 + 0.998335i \(0.518371\pi\)
\(878\) 0 0
\(879\) −0.944272 −0.0318495
\(880\) 0 0
\(881\) −9.70820 −0.327078 −0.163539 0.986537i \(-0.552291\pi\)
−0.163539 + 0.986537i \(0.552291\pi\)
\(882\) 0 0
\(883\) −21.6738 −0.729380 −0.364690 0.931129i \(-0.618825\pi\)
−0.364690 + 0.931129i \(0.618825\pi\)
\(884\) 0 0
\(885\) 15.5623 0.523121
\(886\) 0 0
\(887\) −3.27051 −0.109813 −0.0549065 0.998492i \(-0.517486\pi\)
−0.0549065 + 0.998492i \(0.517486\pi\)
\(888\) 0 0
\(889\) −5.56231 −0.186554
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 39.1246 1.30926
\(894\) 0 0
\(895\) −17.8541 −0.596797
\(896\) 0 0
\(897\) −0.618034 −0.0206356
\(898\) 0 0
\(899\) 5.29180 0.176491
\(900\) 0 0
\(901\) −62.5623 −2.08425
\(902\) 0 0
\(903\) 9.09017 0.302502
\(904\) 0 0
\(905\) −28.2705 −0.939744
\(906\) 0 0
\(907\) −52.5623 −1.74530 −0.872651 0.488344i \(-0.837601\pi\)
−0.872651 + 0.488344i \(0.837601\pi\)
\(908\) 0 0
\(909\) 5.32624 0.176660
\(910\) 0 0
\(911\) −33.8197 −1.12050 −0.560248 0.828325i \(-0.689294\pi\)
−0.560248 + 0.828325i \(0.689294\pi\)
\(912\) 0 0
\(913\) 22.4164 0.741875
\(914\) 0 0
\(915\) −9.47214 −0.313139
\(916\) 0 0
\(917\) −5.94427 −0.196297
\(918\) 0 0
\(919\) −26.3050 −0.867720 −0.433860 0.900980i \(-0.642849\pi\)
−0.433860 + 0.900980i \(0.642849\pi\)
\(920\) 0 0
\(921\) −28.4164 −0.936352
\(922\) 0 0
\(923\) −2.85410 −0.0939439
\(924\) 0 0
\(925\) −22.5623 −0.741844
\(926\) 0 0
\(927\) 19.4164 0.637719
\(928\) 0 0
\(929\) −6.96556 −0.228533 −0.114266 0.993450i \(-0.536452\pi\)
−0.114266 + 0.993450i \(0.536452\pi\)
\(930\) 0 0
\(931\) −4.23607 −0.138832
\(932\) 0 0
\(933\) −15.2705 −0.499934
\(934\) 0 0
\(935\) 32.5623 1.06490
\(936\) 0 0
\(937\) −27.2918 −0.891584 −0.445792 0.895137i \(-0.647078\pi\)
−0.445792 + 0.895137i \(0.647078\pi\)
\(938\) 0 0
\(939\) 17.5279 0.572000
\(940\) 0 0
\(941\) −36.0000 −1.17357 −0.586783 0.809744i \(-0.699606\pi\)
−0.586783 + 0.809744i \(0.699606\pi\)
\(942\) 0 0
\(943\) 11.9443 0.388959
\(944\) 0 0
\(945\) 1.61803 0.0526346
\(946\) 0 0
\(947\) 7.30495 0.237379 0.118690 0.992931i \(-0.462131\pi\)
0.118690 + 0.992931i \(0.462131\pi\)
\(948\) 0 0
\(949\) 1.09017 0.0353884
\(950\) 0 0
\(951\) 5.61803 0.182177
\(952\) 0 0
\(953\) −18.9098 −0.612549 −0.306275 0.951943i \(-0.599083\pi\)
−0.306275 + 0.951943i \(0.599083\pi\)
\(954\) 0 0
\(955\) 33.4164 1.08133
\(956\) 0 0
\(957\) 5.29180 0.171059
\(958\) 0 0
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 10.6180 0.342161
\(964\) 0 0
\(965\) 9.23607 0.297320
\(966\) 0 0
\(967\) −54.0000 −1.73652 −0.868261 0.496107i \(-0.834762\pi\)
−0.868261 + 0.496107i \(0.834762\pi\)
\(968\) 0 0
\(969\) −28.4164 −0.912867
\(970\) 0 0
\(971\) −4.79837 −0.153987 −0.0769936 0.997032i \(-0.524532\pi\)
−0.0769936 + 0.997032i \(0.524532\pi\)
\(972\) 0 0
\(973\) −6.85410 −0.219732
\(974\) 0 0
\(975\) −1.47214 −0.0471461
\(976\) 0 0
\(977\) 50.2837 1.60872 0.804358 0.594144i \(-0.202509\pi\)
0.804358 + 0.594144i \(0.202509\pi\)
\(978\) 0 0
\(979\) 50.3951 1.61064
\(980\) 0 0
\(981\) −2.09017 −0.0667340
\(982\) 0 0
\(983\) −36.5279 −1.16506 −0.582529 0.812810i \(-0.697937\pi\)
−0.582529 + 0.812810i \(0.697937\pi\)
\(984\) 0 0
\(985\) −2.23607 −0.0712470
\(986\) 0 0
\(987\) 9.23607 0.293987
\(988\) 0 0
\(989\) 9.09017 0.289051
\(990\) 0 0
\(991\) 0.686918 0.0218207 0.0109103 0.999940i \(-0.496527\pi\)
0.0109103 + 0.999940i \(0.496527\pi\)
\(992\) 0 0
\(993\) 7.88854 0.250335
\(994\) 0 0
\(995\) −36.5066 −1.15734
\(996\) 0 0
\(997\) −6.52786 −0.206740 −0.103370 0.994643i \(-0.532962\pi\)
−0.103370 + 0.994643i \(0.532962\pi\)
\(998\) 0 0
\(999\) 9.47214 0.299685
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 1932.2.a.g.1.1 2
3.2 odd 2 5796.2.a.l.1.2 2
4.3 odd 2 7728.2.a.ba.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.g.1.1 2 1.1 even 1 trivial
5796.2.a.l.1.2 2 3.2 odd 2
7728.2.a.ba.1.1 2 4.3 odd 2