Properties

Label 1728.2.c.f.1727.5
Level $1728$
Weight $2$
Character 1728.1727
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.5
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.f.1727.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.03528i q^{5} -0.267949i q^{7} +O(q^{10})\) \(q+1.03528i q^{5} -0.267949i q^{7} +3.86370 q^{11} +2.46410 q^{13} -6.69213i q^{17} +1.73205i q^{19} -5.93426 q^{23} +3.92820 q^{25} +2.07055i q^{29} +0.535898i q^{31} +0.277401 q^{35} +6.46410 q^{37} -2.07055i q^{41} -7.46410i q^{43} -9.52056 q^{47} +6.92820 q^{49} +13.3843i q^{53} +4.00000i q^{55} +7.45001 q^{59} +9.39230 q^{61} +2.55103i q^{65} +9.73205i q^{67} +11.3137 q^{71} +9.92820 q^{73} -1.03528i q^{77} -15.1962i q^{79} +7.72741 q^{83} +6.92820 q^{85} -6.69213i q^{89} -0.660254i q^{91} -1.79315 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{13} - 24q^{25} + 24q^{37} - 8q^{61} + 24q^{73} + 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.03528i 0.462990i 0.972836 + 0.231495i \(0.0743616\pi\)
−0.972836 + 0.231495i \(0.925638\pi\)
\(6\) 0 0
\(7\) − 0.267949i − 0.101275i −0.998717 0.0506376i \(-0.983875\pi\)
0.998717 0.0506376i \(-0.0161254\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.86370 1.16495 0.582475 0.812848i \(-0.302084\pi\)
0.582475 + 0.812848i \(0.302084\pi\)
\(12\) 0 0
\(13\) 2.46410 0.683419 0.341709 0.939806i \(-0.388994\pi\)
0.341709 + 0.939806i \(0.388994\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.69213i − 1.62308i −0.584297 0.811540i \(-0.698630\pi\)
0.584297 0.811540i \(-0.301370\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i 0.980064 + 0.198680i \(0.0636654\pi\)
−0.980064 + 0.198680i \(0.936335\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.93426 −1.23738 −0.618689 0.785636i \(-0.712336\pi\)
−0.618689 + 0.785636i \(0.712336\pi\)
\(24\) 0 0
\(25\) 3.92820 0.785641
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.07055i 0.384492i 0.981347 + 0.192246i \(0.0615771\pi\)
−0.981347 + 0.192246i \(0.938423\pi\)
\(30\) 0 0
\(31\) 0.535898i 0.0962502i 0.998841 + 0.0481251i \(0.0153246\pi\)
−0.998841 + 0.0481251i \(0.984675\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.277401 0.0468894
\(36\) 0 0
\(37\) 6.46410 1.06269 0.531346 0.847155i \(-0.321686\pi\)
0.531346 + 0.847155i \(0.321686\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 2.07055i − 0.323366i −0.986843 0.161683i \(-0.948308\pi\)
0.986843 0.161683i \(-0.0516922\pi\)
\(42\) 0 0
\(43\) − 7.46410i − 1.13826i −0.822246 0.569132i \(-0.807279\pi\)
0.822246 0.569132i \(-0.192721\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.52056 −1.38872 −0.694358 0.719630i \(-0.744312\pi\)
−0.694358 + 0.719630i \(0.744312\pi\)
\(48\) 0 0
\(49\) 6.92820 0.989743
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.3843i 1.83847i 0.393710 + 0.919235i \(0.371192\pi\)
−0.393710 + 0.919235i \(0.628808\pi\)
\(54\) 0 0
\(55\) 4.00000i 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.45001 0.969908 0.484954 0.874540i \(-0.338836\pi\)
0.484954 + 0.874540i \(0.338836\pi\)
\(60\) 0 0
\(61\) 9.39230 1.20256 0.601281 0.799038i \(-0.294657\pi\)
0.601281 + 0.799038i \(0.294657\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.55103i 0.316416i
\(66\) 0 0
\(67\) 9.73205i 1.18896i 0.804111 + 0.594480i \(0.202642\pi\)
−0.804111 + 0.594480i \(0.797358\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3137 1.34269 0.671345 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(72\) 0 0
\(73\) 9.92820 1.16201 0.581004 0.813901i \(-0.302660\pi\)
0.581004 + 0.813901i \(0.302660\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.03528i − 0.117981i
\(78\) 0 0
\(79\) − 15.1962i − 1.70970i −0.518875 0.854850i \(-0.673649\pi\)
0.518875 0.854850i \(-0.326351\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.72741 0.848193 0.424097 0.905617i \(-0.360592\pi\)
0.424097 + 0.905617i \(0.360592\pi\)
\(84\) 0 0
\(85\) 6.92820 0.751469
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 6.69213i − 0.709364i −0.934987 0.354682i \(-0.884589\pi\)
0.934987 0.354682i \(-0.115411\pi\)
\(90\) 0 0
\(91\) − 0.660254i − 0.0692134i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.79315 −0.183973
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 15.4548i − 1.53781i −0.639362 0.768906i \(-0.720802\pi\)
0.639362 0.768906i \(-0.279198\pi\)
\(102\) 0 0
\(103\) 17.5885i 1.73304i 0.499140 + 0.866521i \(0.333649\pi\)
−0.499140 + 0.866521i \(0.666351\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.1774 −1.46726 −0.733628 0.679551i \(-0.762174\pi\)
−0.733628 + 0.679551i \(0.762174\pi\)
\(108\) 0 0
\(109\) 0.928203 0.0889057 0.0444529 0.999011i \(-0.485846\pi\)
0.0444529 + 0.999011i \(0.485846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.76268i 0.824324i 0.911111 + 0.412162i \(0.135226\pi\)
−0.911111 + 0.412162i \(0.864774\pi\)
\(114\) 0 0
\(115\) − 6.14359i − 0.572893i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.79315 −0.164378
\(120\) 0 0
\(121\) 3.92820 0.357109
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.24316i 0.826733i
\(126\) 0 0
\(127\) − 3.46410i − 0.307389i −0.988118 0.153695i \(-0.950883\pi\)
0.988118 0.153695i \(-0.0491172\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.72741 −0.675147 −0.337573 0.941299i \(-0.609606\pi\)
−0.337573 + 0.941299i \(0.609606\pi\)
\(132\) 0 0
\(133\) 0.464102 0.0402427
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.8332i 0.925546i 0.886477 + 0.462773i \(0.153145\pi\)
−0.886477 + 0.462773i \(0.846855\pi\)
\(138\) 0 0
\(139\) − 1.19615i − 0.101456i −0.998712 0.0507282i \(-0.983846\pi\)
0.998712 0.0507282i \(-0.0161542\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.52056 0.796149
\(144\) 0 0
\(145\) −2.14359 −0.178016
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 11.3137i − 0.926855i −0.886135 0.463428i \(-0.846619\pi\)
0.886135 0.463428i \(-0.153381\pi\)
\(150\) 0 0
\(151\) − 8.26795i − 0.672836i −0.941713 0.336418i \(-0.890784\pi\)
0.941713 0.336418i \(-0.109216\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.554803 −0.0445628
\(156\) 0 0
\(157\) 3.07180 0.245156 0.122578 0.992459i \(-0.460884\pi\)
0.122578 + 0.992459i \(0.460884\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.59008i 0.125316i
\(162\) 0 0
\(163\) 15.5885i 1.22098i 0.792023 + 0.610491i \(0.209028\pi\)
−0.792023 + 0.610491i \(0.790972\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.34795 0.181690 0.0908451 0.995865i \(-0.471043\pi\)
0.0908451 + 0.995865i \(0.471043\pi\)
\(168\) 0 0
\(169\) −6.92820 −0.532939
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 9.24316i − 0.702744i −0.936236 0.351372i \(-0.885715\pi\)
0.936236 0.351372i \(-0.114285\pi\)
\(174\) 0 0
\(175\) − 1.05256i − 0.0795660i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −19.5959 −1.46467 −0.732334 0.680946i \(-0.761569\pi\)
−0.732334 + 0.680946i \(0.761569\pi\)
\(180\) 0 0
\(181\) 6.46410 0.480473 0.240236 0.970714i \(-0.422775\pi\)
0.240236 + 0.970714i \(0.422775\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.69213i 0.492015i
\(186\) 0 0
\(187\) − 25.8564i − 1.89081i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.6617 0.988523 0.494262 0.869313i \(-0.335438\pi\)
0.494262 + 0.869313i \(0.335438\pi\)
\(192\) 0 0
\(193\) 0.0717968 0.00516804 0.00258402 0.999997i \(-0.499177\pi\)
0.00258402 + 0.999997i \(0.499177\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 14.4195i − 1.02735i −0.857985 0.513675i \(-0.828284\pi\)
0.857985 0.513675i \(-0.171716\pi\)
\(198\) 0 0
\(199\) 25.5885i 1.81392i 0.421219 + 0.906959i \(0.361602\pi\)
−0.421219 + 0.906959i \(0.638398\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.554803 0.0389395
\(204\) 0 0
\(205\) 2.14359 0.149715
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.69213i 0.462904i
\(210\) 0 0
\(211\) 21.7321i 1.49610i 0.663645 + 0.748048i \(0.269009\pi\)
−0.663645 + 0.748048i \(0.730991\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.72741 0.527005
\(216\) 0 0
\(217\) 0.143594 0.00974776
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 16.4901i − 1.10924i
\(222\) 0 0
\(223\) − 23.4641i − 1.57127i −0.618689 0.785636i \(-0.712336\pi\)
0.618689 0.785636i \(-0.287664\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.14110 −0.274855 −0.137427 0.990512i \(-0.543883\pi\)
−0.137427 + 0.990512i \(0.543883\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 11.3137i − 0.741186i −0.928795 0.370593i \(-0.879155\pi\)
0.928795 0.370593i \(-0.120845\pi\)
\(234\) 0 0
\(235\) − 9.85641i − 0.642961i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.4548 0.999689 0.499844 0.866115i \(-0.333391\pi\)
0.499844 + 0.866115i \(0.333391\pi\)
\(240\) 0 0
\(241\) 5.92820 0.381869 0.190935 0.981603i \(-0.438848\pi\)
0.190935 + 0.981603i \(0.438848\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.17260i 0.458241i
\(246\) 0 0
\(247\) 4.26795i 0.271563i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.58630 0.226365 0.113183 0.993574i \(-0.463895\pi\)
0.113183 + 0.993574i \(0.463895\pi\)
\(252\) 0 0
\(253\) −22.9282 −1.44148
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.07055i 0.129158i 0.997913 + 0.0645788i \(0.0205704\pi\)
−0.997913 + 0.0645788i \(0.979430\pi\)
\(258\) 0 0
\(259\) − 1.73205i − 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.4548 −0.952985 −0.476492 0.879179i \(-0.658092\pi\)
−0.476492 + 0.879179i \(0.658092\pi\)
\(264\) 0 0
\(265\) −13.8564 −0.851192
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 3.10583i − 0.189366i −0.995507 0.0946829i \(-0.969816\pi\)
0.995507 0.0946829i \(-0.0301837\pi\)
\(270\) 0 0
\(271\) − 13.0526i − 0.792886i −0.918059 0.396443i \(-0.870244\pi\)
0.918059 0.396443i \(-0.129756\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 15.1774 0.915232
\(276\) 0 0
\(277\) −20.9282 −1.25745 −0.628727 0.777626i \(-0.716424\pi\)
−0.628727 + 0.777626i \(0.716424\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.5254i 1.04548i 0.852493 + 0.522738i \(0.175089\pi\)
−0.852493 + 0.522738i \(0.824911\pi\)
\(282\) 0 0
\(283\) 0.535898i 0.0318559i 0.999873 + 0.0159279i \(0.00507023\pi\)
−0.999873 + 0.0159279i \(0.994930\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.554803 −0.0327490
\(288\) 0 0
\(289\) −27.7846 −1.63439
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 27.8038i 1.62432i 0.583438 + 0.812158i \(0.301707\pi\)
−0.583438 + 0.812158i \(0.698293\pi\)
\(294\) 0 0
\(295\) 7.71281i 0.449057i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −14.6226 −0.845647
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.72363i 0.556773i
\(306\) 0 0
\(307\) 22.3923i 1.27800i 0.769208 + 0.638998i \(0.220651\pi\)
−0.769208 + 0.638998i \(0.779349\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.0754 0.571321 0.285661 0.958331i \(-0.407787\pi\)
0.285661 + 0.958331i \(0.407787\pi\)
\(312\) 0 0
\(313\) −19.9282 −1.12641 −0.563204 0.826318i \(-0.690432\pi\)
−0.563204 + 0.826318i \(0.690432\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.07055i 0.116294i 0.998308 + 0.0581469i \(0.0185192\pi\)
−0.998308 + 0.0581469i \(0.981481\pi\)
\(318\) 0 0
\(319\) 8.00000i 0.447914i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 11.5911 0.644947
\(324\) 0 0
\(325\) 9.67949 0.536922
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.55103i 0.140643i
\(330\) 0 0
\(331\) − 25.1962i − 1.38491i −0.721463 0.692453i \(-0.756530\pi\)
0.721463 0.692453i \(-0.243470\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −10.0754 −0.550476
\(336\) 0 0
\(337\) −14.8564 −0.809280 −0.404640 0.914476i \(-0.632603\pi\)
−0.404640 + 0.914476i \(0.632603\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.07055i 0.112127i
\(342\) 0 0
\(343\) − 3.73205i − 0.201512i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.4548 −0.829658 −0.414829 0.909899i \(-0.636159\pi\)
−0.414829 + 0.909899i \(0.636159\pi\)
\(348\) 0 0
\(349\) −10.3205 −0.552444 −0.276222 0.961094i \(-0.589083\pi\)
−0.276222 + 0.961094i \(0.589083\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 35.0507i − 1.86556i −0.360443 0.932781i \(-0.617375\pi\)
0.360443 0.932781i \(-0.382625\pi\)
\(354\) 0 0
\(355\) 11.7128i 0.621652i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −32.1480 −1.69671 −0.848353 0.529432i \(-0.822405\pi\)
−0.848353 + 0.529432i \(0.822405\pi\)
\(360\) 0 0
\(361\) 16.0000 0.842105
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.2784i 0.537998i
\(366\) 0 0
\(367\) − 17.0526i − 0.890136i −0.895497 0.445068i \(-0.853179\pi\)
0.895497 0.445068i \(-0.146821\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.58630 0.186192
\(372\) 0 0
\(373\) −13.2487 −0.685992 −0.342996 0.939337i \(-0.611442\pi\)
−0.342996 + 0.939337i \(0.611442\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.10205i 0.262769i
\(378\) 0 0
\(379\) − 9.19615i − 0.472375i −0.971708 0.236187i \(-0.924102\pi\)
0.971708 0.236187i \(-0.0758979\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −35.0507 −1.79101 −0.895504 0.445053i \(-0.853185\pi\)
−0.895504 + 0.445053i \(0.853185\pi\)
\(384\) 0 0
\(385\) 1.07180 0.0546238
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14.4195i 0.731100i 0.930792 + 0.365550i \(0.119119\pi\)
−0.930792 + 0.365550i \(0.880881\pi\)
\(390\) 0 0
\(391\) 39.7128i 2.00836i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.7322 0.791574
\(396\) 0 0
\(397\) −26.7846 −1.34428 −0.672141 0.740424i \(-0.734625\pi\)
−0.672141 + 0.740424i \(0.734625\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 22.6274i − 1.12996i −0.825105 0.564980i \(-0.808884\pi\)
0.825105 0.564980i \(-0.191116\pi\)
\(402\) 0 0
\(403\) 1.32051i 0.0657792i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.9754 1.23798
\(408\) 0 0
\(409\) −13.0000 −0.642809 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 1.99622i − 0.0982277i
\(414\) 0 0
\(415\) 8.00000i 0.392705i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.7322 0.768569 0.384284 0.923215i \(-0.374448\pi\)
0.384284 + 0.923215i \(0.374448\pi\)
\(420\) 0 0
\(421\) −9.53590 −0.464751 −0.232376 0.972626i \(-0.574650\pi\)
−0.232376 + 0.972626i \(0.574650\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 26.2880i − 1.27516i
\(426\) 0 0
\(427\) − 2.51666i − 0.121790i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.37945 −0.259119 −0.129560 0.991572i \(-0.541356\pi\)
−0.129560 + 0.991572i \(0.541356\pi\)
\(432\) 0 0
\(433\) −26.7846 −1.28719 −0.643593 0.765368i \(-0.722557\pi\)
−0.643593 + 0.765368i \(0.722557\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 10.2784i − 0.491684i
\(438\) 0 0
\(439\) − 21.3205i − 1.01757i −0.860893 0.508786i \(-0.830094\pi\)
0.860893 0.508786i \(-0.169906\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 27.3233 1.29817 0.649085 0.760716i \(-0.275152\pi\)
0.649085 + 0.760716i \(0.275152\pi\)
\(444\) 0 0
\(445\) 6.92820 0.328428
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.9038i 0.608967i 0.952518 + 0.304484i \(0.0984839\pi\)
−0.952518 + 0.304484i \(0.901516\pi\)
\(450\) 0 0
\(451\) − 8.00000i − 0.376705i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.683545 0.0320451
\(456\) 0 0
\(457\) −34.7846 −1.62716 −0.813578 0.581456i \(-0.802483\pi\)
−0.813578 + 0.581456i \(0.802483\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.7332i 1.19852i 0.800556 + 0.599258i \(0.204538\pi\)
−0.800556 + 0.599258i \(0.795462\pi\)
\(462\) 0 0
\(463\) − 31.1962i − 1.44981i −0.688850 0.724904i \(-0.741884\pi\)
0.688850 0.724904i \(-0.258116\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 38.3596 1.77507 0.887536 0.460738i \(-0.152415\pi\)
0.887536 + 0.460738i \(0.152415\pi\)
\(468\) 0 0
\(469\) 2.60770 0.120412
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 28.8391i − 1.32602i
\(474\) 0 0
\(475\) 6.80385i 0.312182i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.9000 0.680799 0.340399 0.940281i \(-0.389438\pi\)
0.340399 + 0.940281i \(0.389438\pi\)
\(480\) 0 0
\(481\) 15.9282 0.726264
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.24693i 0.329066i
\(486\) 0 0
\(487\) 23.4449i 1.06239i 0.847250 + 0.531194i \(0.178257\pi\)
−0.847250 + 0.531194i \(0.821743\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.7322 −0.709985 −0.354992 0.934869i \(-0.615517\pi\)
−0.354992 + 0.934869i \(0.615517\pi\)
\(492\) 0 0
\(493\) 13.8564 0.624061
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 3.03150i − 0.135981i
\(498\) 0 0
\(499\) 12.5359i 0.561184i 0.959827 + 0.280592i \(0.0905308\pi\)
−0.959827 + 0.280592i \(0.909469\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −29.1165 −1.29824 −0.649120 0.760686i \(-0.724863\pi\)
−0.649120 + 0.760686i \(0.724863\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.5606i 0.822686i 0.911481 + 0.411343i \(0.134940\pi\)
−0.911481 + 0.411343i \(0.865060\pi\)
\(510\) 0 0
\(511\) − 2.66025i − 0.117683i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −18.2089 −0.802380
\(516\) 0 0
\(517\) −36.7846 −1.61779
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 37.6018i 1.64736i 0.567053 + 0.823681i \(0.308084\pi\)
−0.567053 + 0.823681i \(0.691916\pi\)
\(522\) 0 0
\(523\) 10.8038i 0.472419i 0.971702 + 0.236210i \(0.0759052\pi\)
−0.971702 + 0.236210i \(0.924095\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.58630 0.156222
\(528\) 0 0
\(529\) 12.2154 0.531104
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 5.10205i − 0.220994i
\(534\) 0 0
\(535\) − 15.7128i − 0.679324i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 26.7685 1.15300
\(540\) 0 0
\(541\) −14.6077 −0.628034 −0.314017 0.949417i \(-0.601675\pi\)
−0.314017 + 0.949417i \(0.601675\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.960947i 0.0411624i
\(546\) 0 0
\(547\) 5.73205i 0.245085i 0.992463 + 0.122542i \(0.0391047\pi\)
−0.992463 + 0.122542i \(0.960895\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.58630 −0.152782
\(552\) 0 0
\(553\) −4.07180 −0.173150
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 10.2784i − 0.435511i −0.976003 0.217756i \(-0.930126\pi\)
0.976003 0.217756i \(-0.0698736\pi\)
\(558\) 0 0
\(559\) − 18.3923i − 0.777912i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.0411 −0.802487 −0.401244 0.915971i \(-0.631422\pi\)
−0.401244 + 0.915971i \(0.631422\pi\)
\(564\) 0 0
\(565\) −9.07180 −0.381653
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 38.5627i − 1.61663i −0.588748 0.808316i \(-0.700379\pi\)
0.588748 0.808316i \(-0.299621\pi\)
\(570\) 0 0
\(571\) − 23.0526i − 0.964720i −0.875973 0.482360i \(-0.839780\pi\)
0.875973 0.482360i \(-0.160220\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −23.3110 −0.972134
\(576\) 0 0
\(577\) −26.0718 −1.08538 −0.542692 0.839932i \(-0.682595\pi\)
−0.542692 + 0.839932i \(0.682595\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 2.07055i − 0.0859010i
\(582\) 0 0
\(583\) 51.7128i 2.14173i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.277401 −0.0114496 −0.00572479 0.999984i \(-0.501822\pi\)
−0.00572479 + 0.999984i \(0.501822\pi\)
\(588\) 0 0
\(589\) −0.928203 −0.0382459
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 25.8076i − 1.05979i −0.848063 0.529895i \(-0.822231\pi\)
0.848063 0.529895i \(-0.177769\pi\)
\(594\) 0 0
\(595\) − 1.85641i − 0.0761052i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.8685 −0.484934 −0.242467 0.970160i \(-0.577957\pi\)
−0.242467 + 0.970160i \(0.577957\pi\)
\(600\) 0 0
\(601\) 6.78461 0.276750 0.138375 0.990380i \(-0.455812\pi\)
0.138375 + 0.990380i \(0.455812\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.06678i 0.165338i
\(606\) 0 0
\(607\) 10.6603i 0.432686i 0.976317 + 0.216343i \(0.0694130\pi\)
−0.976317 + 0.216343i \(0.930587\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.4596 −0.949075
\(612\) 0 0
\(613\) 37.3923 1.51026 0.755130 0.655575i \(-0.227573\pi\)
0.755130 + 0.655575i \(0.227573\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 9.72363i − 0.391459i −0.980658 0.195729i \(-0.937293\pi\)
0.980658 0.195729i \(-0.0627074\pi\)
\(618\) 0 0
\(619\) 0.660254i 0.0265379i 0.999912 + 0.0132689i \(0.00422375\pi\)
−0.999912 + 0.0132689i \(0.995776\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.79315 −0.0718411
\(624\) 0 0
\(625\) 10.0718 0.402872
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 43.2586i − 1.72483i
\(630\) 0 0
\(631\) − 19.9808i − 0.795422i −0.917511 0.397711i \(-0.869805\pi\)
0.917511 0.397711i \(-0.130195\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.58630 0.142318
\(636\) 0 0
\(637\) 17.0718 0.676409
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.2233i 1.66772i 0.551975 + 0.833861i \(0.313874\pi\)
−0.551975 + 0.833861i \(0.686126\pi\)
\(642\) 0 0
\(643\) − 17.6077i − 0.694380i −0.937795 0.347190i \(-0.887136\pi\)
0.937795 0.347190i \(-0.112864\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 23.1822 0.911387 0.455694 0.890137i \(-0.349391\pi\)
0.455694 + 0.890137i \(0.349391\pi\)
\(648\) 0 0
\(649\) 28.7846 1.12989
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.3528i 0.405135i 0.979268 + 0.202567i \(0.0649285\pi\)
−0.979268 + 0.202567i \(0.935071\pi\)
\(654\) 0 0
\(655\) − 8.00000i − 0.312586i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −34.4959 −1.34377 −0.671885 0.740655i \(-0.734515\pi\)
−0.671885 + 0.740655i \(0.734515\pi\)
\(660\) 0 0
\(661\) 43.2487 1.68218 0.841090 0.540895i \(-0.181914\pi\)
0.841090 + 0.540895i \(0.181914\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.480473i 0.0186320i
\(666\) 0 0
\(667\) − 12.2872i − 0.475762i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 36.2891 1.40092
\(672\) 0 0
\(673\) 20.8564 0.803955 0.401978 0.915649i \(-0.368323\pi\)
0.401978 + 0.915649i \(0.368323\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.5921i 0.829853i 0.909855 + 0.414927i \(0.136193\pi\)
−0.909855 + 0.414927i \(0.863807\pi\)
\(678\) 0 0
\(679\) − 1.87564i − 0.0719806i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33.9411 1.29872 0.649361 0.760481i \(-0.275037\pi\)
0.649361 + 0.760481i \(0.275037\pi\)
\(684\) 0 0
\(685\) −11.2154 −0.428518
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 32.9802i 1.25644i
\(690\) 0 0
\(691\) 30.3923i 1.15618i 0.815974 + 0.578089i \(0.196201\pi\)
−0.815974 + 0.578089i \(0.803799\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.23835 0.0469732
\(696\) 0 0
\(697\) −13.8564 −0.524849
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 49.4703i − 1.86847i −0.356663 0.934233i \(-0.616086\pi\)
0.356663 0.934233i \(-0.383914\pi\)
\(702\) 0 0
\(703\) 11.1962i 0.422271i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.14110 −0.155742
\(708\) 0 0
\(709\) −12.1769 −0.457314 −0.228657 0.973507i \(-0.573433\pi\)
−0.228657 + 0.973507i \(0.573433\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 3.18016i − 0.119098i
\(714\) 0 0
\(715\) 9.85641i 0.368609i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.58630 0.133746 0.0668732 0.997761i \(-0.478698\pi\)
0.0668732 + 0.997761i \(0.478698\pi\)
\(720\) 0 0
\(721\) 4.71281 0.175514
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.13355i 0.302072i
\(726\) 0 0
\(727\) − 21.3205i − 0.790734i −0.918523 0.395367i \(-0.870617\pi\)
0.918523 0.395367i \(-0.129383\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −49.9507 −1.84749
\(732\) 0 0
\(733\) 32.9282 1.21623 0.608115 0.793849i \(-0.291926\pi\)
0.608115 + 0.793849i \(0.291926\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 37.6018i 1.38508i
\(738\) 0 0
\(739\) 14.3923i 0.529429i 0.964327 + 0.264715i \(0.0852778\pi\)
−0.964327 + 0.264715i \(0.914722\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.5302 0.936611 0.468306 0.883567i \(-0.344865\pi\)
0.468306 + 0.883567i \(0.344865\pi\)
\(744\) 0 0
\(745\) 11.7128 0.429124
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.06678i 0.148597i
\(750\) 0 0
\(751\) − 3.19615i − 0.116629i −0.998298 0.0583146i \(-0.981427\pi\)
0.998298 0.0583146i \(-0.0185727\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.55961 0.311516
\(756\) 0 0
\(757\) 5.39230 0.195987 0.0979933 0.995187i \(-0.468758\pi\)
0.0979933 + 0.995187i \(0.468758\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 43.8134i 1.58824i 0.607764 + 0.794118i \(0.292067\pi\)
−0.607764 + 0.794118i \(0.707933\pi\)
\(762\) 0 0
\(763\) − 0.248711i − 0.00900395i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.3576 0.662853
\(768\) 0 0
\(769\) 12.8564 0.463614 0.231807 0.972762i \(-0.425536\pi\)
0.231807 + 0.972762i \(0.425536\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20.5569i 0.739379i 0.929155 + 0.369690i \(0.120536\pi\)
−0.929155 + 0.369690i \(0.879464\pi\)
\(774\) 0 0
\(775\) 2.10512i 0.0756181i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.58630 0.128493
\(780\) 0 0
\(781\) 43.7128 1.56417
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.18016i 0.113505i
\(786\) 0 0
\(787\) − 30.2679i − 1.07894i −0.842006 0.539468i \(-0.818625\pi\)
0.842006 0.539468i \(-0.181375\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.34795 0.0834836
\(792\) 0 0
\(793\) 23.1436 0.821853
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.03150i 0.107381i 0.998558 + 0.0536906i \(0.0170985\pi\)
−0.998558 + 0.0536906i \(0.982902\pi\)
\(798\) 0 0
\(799\) 63.7128i 2.25400i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 38.3596 1.35368
\(804\) 0 0
\(805\) −1.64617 −0.0580199
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.5959i 0.688956i 0.938794 + 0.344478i \(0.111944\pi\)
−0.938794 + 0.344478i \(0.888056\pi\)
\(810\) 0 0
\(811\) − 15.4641i − 0.543018i −0.962436 0.271509i \(-0.912477\pi\)
0.962436 0.271509i \(-0.0875227\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16.1384 −0.565302
\(816\) 0 0
\(817\) 12.9282 0.452301
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 21.5921i − 0.753571i −0.926301 0.376785i \(-0.877029\pi\)
0.926301 0.376785i \(-0.122971\pi\)
\(822\) 0 0
\(823\) − 39.9808i − 1.39364i −0.717245 0.696821i \(-0.754597\pi\)
0.717245 0.696821i \(-0.245403\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −54.3692 −1.89060 −0.945302 0.326196i \(-0.894233\pi\)
−0.945302 + 0.326196i \(0.894233\pi\)
\(828\) 0 0
\(829\) −15.6795 −0.544571 −0.272286 0.962216i \(-0.587780\pi\)
−0.272286 + 0.962216i \(0.587780\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 46.3644i − 1.60643i
\(834\) 0 0
\(835\) 2.43078i 0.0841206i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 52.9822 1.82915 0.914575 0.404416i \(-0.132525\pi\)
0.914575 + 0.404416i \(0.132525\pi\)
\(840\) 0 0
\(841\) 24.7128 0.852166
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 7.17260i − 0.246745i
\(846\) 0 0
\(847\) − 1.05256i − 0.0361664i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −38.3596 −1.31495
\(852\) 0 0
\(853\) −26.6077 −0.911030 −0.455515 0.890228i \(-0.650545\pi\)
−0.455515 + 0.890228i \(0.650545\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.0822i 1.30086i 0.759564 + 0.650432i \(0.225412\pi\)
−0.759564 + 0.650432i \(0.774588\pi\)
\(858\) 0 0
\(859\) 18.5167i 0.631780i 0.948796 + 0.315890i \(0.102303\pi\)
−0.948796 + 0.315890i \(0.897697\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.6302 −0.361855 −0.180927 0.983496i \(-0.557910\pi\)
−0.180927 + 0.983496i \(0.557910\pi\)
\(864\) 0 0
\(865\) 9.56922 0.325363
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 58.7134i − 1.99172i
\(870\) 0 0
\(871\) 23.9808i 0.812557i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.47670 0.0837276
\(876\) 0 0
\(877\) −25.2487 −0.852588 −0.426294 0.904585i \(-0.640181\pi\)
−0.426294 + 0.904585i \(0.640181\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 12.9038i − 0.434740i −0.976089 0.217370i \(-0.930252\pi\)
0.976089 0.217370i \(-0.0697478\pi\)
\(882\) 0 0
\(883\) − 24.1244i − 0.811849i −0.913907 0.405925i \(-0.866950\pi\)
0.913907 0.405925i \(-0.133050\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.8685 0.398506 0.199253 0.979948i \(-0.436149\pi\)
0.199253 + 0.979948i \(0.436149\pi\)
\(888\) 0 0
\(889\) −0.928203 −0.0311309
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 16.4901i − 0.551820i
\(894\) 0 0
\(895\) − 20.2872i − 0.678126i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.10961 −0.0370074
\(900\) 0 0
\(901\) 89.5692 2.98398
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.69213i 0.222454i
\(906\) 0 0
\(907\) 54.5167i 1.81020i 0.425203 + 0.905098i \(0.360203\pi\)
−0.425203 + 0.905098i \(0.639797\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.554803 0.0183814 0.00919072 0.999958i \(-0.497074\pi\)
0.00919072 + 0.999958i \(0.497074\pi\)
\(912\) 0 0
\(913\) 29.8564 0.988103
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.07055i 0.0683757i
\(918\) 0 0
\(919\) − 57.0333i − 1.88136i −0.339301 0.940678i \(-0.610191\pi\)
0.339301 0.940678i \(-0.389809\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 27.8781 0.917620
\(924\) 0 0
\(925\) 25.3923 0.834894
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 22.6274i − 0.742381i −0.928557 0.371191i \(-0.878950\pi\)
0.928557 0.371191i \(-0.121050\pi\)
\(930\) 0 0
\(931\) 12.0000i 0.393284i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 26.7685 0.875424
\(936\) 0 0
\(937\) 6.21539 0.203048 0.101524 0.994833i \(-0.467628\pi\)
0.101524 + 0.994833i \(0.467628\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 31.9449i − 1.04137i −0.853748 0.520687i \(-0.825676\pi\)
0.853748 0.520687i \(-0.174324\pi\)
\(942\) 0 0
\(943\) 12.2872i 0.400126i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.277401 0.00901433 0.00450717 0.999990i \(-0.498565\pi\)
0.00450717 + 0.999990i \(0.498565\pi\)
\(948\) 0 0
\(949\) 24.4641 0.794138
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 12.9038i 0.417995i 0.977916 + 0.208997i \(0.0670200\pi\)
−0.977916 + 0.208997i \(0.932980\pi\)
\(954\) 0 0
\(955\) 14.1436i 0.457676i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.90276 0.0937349
\(960\) 0 0
\(961\) 30.7128 0.990736
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.0743295i 0.00239275i
\(966\) 0 0
\(967\) 11.7321i 0.377277i 0.982047 + 0.188639i \(0.0604075\pi\)
−0.982047 + 0.188639i \(0.939593\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −46.6418 −1.49681 −0.748404 0.663243i \(-0.769180\pi\)
−0.748404 + 0.663243i \(0.769180\pi\)
\(972\) 0 0
\(973\) −0.320508 −0.0102750
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.24316i 0.295715i 0.989009 + 0.147857i \(0.0472377\pi\)
−0.989009 + 0.147857i \(0.952762\pi\)
\(978\) 0 0
\(979\) − 25.8564i − 0.826374i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −44.0165 −1.40391 −0.701954 0.712222i \(-0.747689\pi\)
−0.701954 + 0.712222i \(0.747689\pi\)
\(984\) 0 0
\(985\) 14.9282 0.475652
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 44.2939i 1.40846i
\(990\) 0 0
\(991\) − 7.48334i − 0.237716i −0.992911 0.118858i \(-0.962077\pi\)
0.992911 0.118858i \(-0.0379233\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −26.4911 −0.839825
\(996\) 0 0
\(997\) 6.78461 0.214871 0.107435 0.994212i \(-0.465736\pi\)
0.107435 + 0.994212i \(0.465736\pi\)
\(998\) 0 0
\(999\) 0 0
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 1728.2.c.f.1727.5 8
3.2 odd 2 inner 1728.2.c.f.1727.3 8
4.3 odd 2 inner 1728.2.c.f.1727.6 8
8.3 odd 2 864.2.c.b.863.4 yes 8
8.5 even 2 864.2.c.b.863.3 8
12.11 even 2 inner 1728.2.c.f.1727.4 8
24.5 odd 2 864.2.c.b.863.5 yes 8
24.11 even 2 864.2.c.b.863.6 yes 8
72.5 odd 6 2592.2.s.c.1727.3 8
72.11 even 6 2592.2.s.c.863.2 8
72.13 even 6 2592.2.s.c.1727.2 8
72.29 odd 6 2592.2.s.g.863.2 8
72.43 odd 6 2592.2.s.c.863.3 8
72.59 even 6 2592.2.s.g.1727.3 8
72.61 even 6 2592.2.s.g.863.3 8
72.67 odd 6 2592.2.s.g.1727.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.c.b.863.3 8 8.5 even 2
864.2.c.b.863.4 yes 8 8.3 odd 2
864.2.c.b.863.5 yes 8 24.5 odd 2
864.2.c.b.863.6 yes 8 24.11 even 2
1728.2.c.f.1727.3 8 3.2 odd 2 inner
1728.2.c.f.1727.4 8 12.11 even 2 inner
1728.2.c.f.1727.5 8 1.1 even 1 trivial
1728.2.c.f.1727.6 8 4.3 odd 2 inner
2592.2.s.c.863.2 8 72.11 even 6
2592.2.s.c.863.3 8 72.43 odd 6
2592.2.s.c.1727.2 8 72.13 even 6
2592.2.s.c.1727.3 8 72.5 odd 6
2592.2.s.g.863.2 8 72.29 odd 6
2592.2.s.g.863.3 8 72.61 even 6
2592.2.s.g.1727.2 8 72.67 odd 6
2592.2.s.g.1727.3 8 72.59 even 6