Properties

Label 1440.2.h.b.1151.2
Level $1440$
Weight $2$
Character 1440.1151
Analytic conductor $11.498$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1440.1151
Dual form 1440.2.h.b.1151.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{5} -0.585786i q^{7} +O(q^{10})\) \(q-1.00000i q^{5} -0.585786i q^{7} -1.41421 q^{11} +2.24264 q^{13} -1.17157i q^{17} -5.65685i q^{19} -3.17157 q^{23} -1.00000 q^{25} -2.00000i q^{29} +3.17157i q^{31} -0.585786 q^{35} +4.58579 q^{37} -8.24264i q^{41} -6.82843i q^{43} -8.00000 q^{47} +6.65685 q^{49} -6.82843i q^{53} +1.41421i q^{55} -5.41421 q^{59} -3.17157 q^{61} -2.24264i q^{65} -11.3137i q^{67} -13.6569 q^{71} +0.828427 q^{73} +0.828427i q^{77} +6.48528i q^{79} +1.17157 q^{83} -1.17157 q^{85} -0.928932i q^{89} -1.31371i q^{91} -5.65685 q^{95} +10.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 8 q^{13} - 24 q^{23} - 4 q^{25} - 8 q^{35} + 24 q^{37} - 32 q^{47} + 4 q^{49} - 16 q^{59} - 24 q^{61} - 32 q^{71} - 8 q^{73} + 16 q^{83} - 16 q^{85} + 8 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) − 0.585786i − 0.221406i −0.993854 0.110703i \(-0.964690\pi\)
0.993854 0.110703i \(-0.0353103\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 2.24264 0.621997 0.310998 0.950410i \(-0.399337\pi\)
0.310998 + 0.950410i \(0.399337\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.17157i − 0.284148i −0.989856 0.142074i \(-0.954623\pi\)
0.989856 0.142074i \(-0.0453771\pi\)
\(18\) 0 0
\(19\) − 5.65685i − 1.29777i −0.760886 0.648886i \(-0.775235\pi\)
0.760886 0.648886i \(-0.224765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.17157 −0.661319 −0.330659 0.943750i \(-0.607271\pi\)
−0.330659 + 0.943750i \(0.607271\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.00000i − 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) 3.17157i 0.569631i 0.958582 + 0.284816i \(0.0919324\pi\)
−0.958582 + 0.284816i \(0.908068\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.585786 −0.0990160
\(36\) 0 0
\(37\) 4.58579 0.753899 0.376949 0.926234i \(-0.376973\pi\)
0.376949 + 0.926234i \(0.376973\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 8.24264i − 1.28728i −0.765327 0.643642i \(-0.777423\pi\)
0.765327 0.643642i \(-0.222577\pi\)
\(42\) 0 0
\(43\) − 6.82843i − 1.04133i −0.853762 0.520663i \(-0.825685\pi\)
0.853762 0.520663i \(-0.174315\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 6.65685 0.950979
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.82843i − 0.937957i −0.883210 0.468978i \(-0.844622\pi\)
0.883210 0.468978i \(-0.155378\pi\)
\(54\) 0 0
\(55\) 1.41421i 0.190693i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.41421 −0.704871 −0.352435 0.935836i \(-0.614646\pi\)
−0.352435 + 0.935836i \(0.614646\pi\)
\(60\) 0 0
\(61\) −3.17157 −0.406078 −0.203039 0.979171i \(-0.565082\pi\)
−0.203039 + 0.979171i \(0.565082\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 2.24264i − 0.278165i
\(66\) 0 0
\(67\) − 11.3137i − 1.38219i −0.722764 0.691095i \(-0.757129\pi\)
0.722764 0.691095i \(-0.242871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.6569 −1.62077 −0.810385 0.585897i \(-0.800742\pi\)
−0.810385 + 0.585897i \(0.800742\pi\)
\(72\) 0 0
\(73\) 0.828427 0.0969601 0.0484800 0.998824i \(-0.484562\pi\)
0.0484800 + 0.998824i \(0.484562\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.828427i 0.0944080i
\(78\) 0 0
\(79\) 6.48528i 0.729651i 0.931076 + 0.364826i \(0.118871\pi\)
−0.931076 + 0.364826i \(0.881129\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.17157 0.128597 0.0642984 0.997931i \(-0.479519\pi\)
0.0642984 + 0.997931i \(0.479519\pi\)
\(84\) 0 0
\(85\) −1.17157 −0.127075
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 0.928932i − 0.0984666i −0.998787 0.0492333i \(-0.984322\pi\)
0.998787 0.0492333i \(-0.0156778\pi\)
\(90\) 0 0
\(91\) − 1.31371i − 0.137714i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) 10.4853 1.06462 0.532310 0.846550i \(-0.321324\pi\)
0.532310 + 0.846550i \(0.321324\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 11.1716i − 1.11161i −0.831312 0.555807i \(-0.812410\pi\)
0.831312 0.555807i \(-0.187590\pi\)
\(102\) 0 0
\(103\) 2.24264i 0.220974i 0.993878 + 0.110487i \(0.0352410\pi\)
−0.993878 + 0.110487i \(0.964759\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.1421 1.75387 0.876933 0.480612i \(-0.159586\pi\)
0.876933 + 0.480612i \(0.159586\pi\)
\(108\) 0 0
\(109\) 8.82843 0.845610 0.422805 0.906221i \(-0.361046\pi\)
0.422805 + 0.906221i \(0.361046\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 4.00000i − 0.376288i −0.982141 0.188144i \(-0.939753\pi\)
0.982141 0.188144i \(-0.0602472\pi\)
\(114\) 0 0
\(115\) 3.17157i 0.295751i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.686292 −0.0629122
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) − 12.5858i − 1.11681i −0.829569 0.558404i \(-0.811414\pi\)
0.829569 0.558404i \(-0.188586\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.89949 −0.515441 −0.257721 0.966219i \(-0.582971\pi\)
−0.257721 + 0.966219i \(0.582971\pi\)
\(132\) 0 0
\(133\) −3.31371 −0.287335
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.65685i 0.483298i 0.970364 + 0.241649i \(0.0776882\pi\)
−0.970364 + 0.241649i \(0.922312\pi\)
\(138\) 0 0
\(139\) − 19.6569i − 1.66727i −0.552314 0.833636i \(-0.686255\pi\)
0.552314 0.833636i \(-0.313745\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.17157 −0.265220
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 0.828427i − 0.0678674i −0.999424 0.0339337i \(-0.989196\pi\)
0.999424 0.0339337i \(-0.0108035\pi\)
\(150\) 0 0
\(151\) 22.9706i 1.86932i 0.355546 + 0.934659i \(0.384295\pi\)
−0.355546 + 0.934659i \(0.615705\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.17157 0.254747
\(156\) 0 0
\(157\) 19.8995 1.58815 0.794076 0.607818i \(-0.207955\pi\)
0.794076 + 0.607818i \(0.207955\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.85786i 0.146420i
\(162\) 0 0
\(163\) − 8.48528i − 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.48528 −0.192317 −0.0961584 0.995366i \(-0.530656\pi\)
−0.0961584 + 0.995366i \(0.530656\pi\)
\(168\) 0 0
\(169\) −7.97056 −0.613120
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 16.4853i 1.25335i 0.779280 + 0.626676i \(0.215585\pi\)
−0.779280 + 0.626676i \(0.784415\pi\)
\(174\) 0 0
\(175\) 0.585786i 0.0442813i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.5858 0.791219 0.395609 0.918419i \(-0.370533\pi\)
0.395609 + 0.918419i \(0.370533\pi\)
\(180\) 0 0
\(181\) −19.1716 −1.42501 −0.712506 0.701666i \(-0.752440\pi\)
−0.712506 + 0.701666i \(0.752440\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 4.58579i − 0.337154i
\(186\) 0 0
\(187\) 1.65685i 0.121161i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.82843 −0.494088 −0.247044 0.969004i \(-0.579459\pi\)
−0.247044 + 0.969004i \(0.579459\pi\)
\(192\) 0 0
\(193\) 15.6569 1.12701 0.563503 0.826114i \(-0.309454\pi\)
0.563503 + 0.826114i \(0.309454\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 4.34315i 0.307877i 0.988080 + 0.153939i \(0.0491958\pi\)
−0.988080 + 0.153939i \(0.950804\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.17157 −0.0822283
\(204\) 0 0
\(205\) −8.24264 −0.575691
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000i 0.553372i
\(210\) 0 0
\(211\) 6.00000i 0.413057i 0.978441 + 0.206529i \(0.0662166\pi\)
−0.978441 + 0.206529i \(0.933783\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.82843 −0.465695
\(216\) 0 0
\(217\) 1.85786 0.126120
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2.62742i − 0.176739i
\(222\) 0 0
\(223\) 23.8995i 1.60043i 0.599714 + 0.800214i \(0.295281\pi\)
−0.599714 + 0.800214i \(0.704719\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 28.2843 1.87729 0.938647 0.344881i \(-0.112081\pi\)
0.938647 + 0.344881i \(0.112081\pi\)
\(228\) 0 0
\(229\) −3.65685 −0.241652 −0.120826 0.992674i \(-0.538554\pi\)
−0.120826 + 0.992674i \(0.538554\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.48528i 0.293841i 0.989148 + 0.146920i \(0.0469361\pi\)
−0.989148 + 0.146920i \(0.953064\pi\)
\(234\) 0 0
\(235\) 8.00000i 0.521862i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.1421 1.43226 0.716128 0.697969i \(-0.245913\pi\)
0.716128 + 0.697969i \(0.245913\pi\)
\(240\) 0 0
\(241\) −25.6569 −1.65270 −0.826352 0.563154i \(-0.809588\pi\)
−0.826352 + 0.563154i \(0.809588\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 6.65685i − 0.425291i
\(246\) 0 0
\(247\) − 12.6863i − 0.807209i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.72792 −0.298424 −0.149212 0.988805i \(-0.547674\pi\)
−0.149212 + 0.988805i \(0.547674\pi\)
\(252\) 0 0
\(253\) 4.48528 0.281987
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.9706i 1.55762i 0.627259 + 0.778810i \(0.284177\pi\)
−0.627259 + 0.778810i \(0.715823\pi\)
\(258\) 0 0
\(259\) − 2.68629i − 0.166918i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.1716 1.18217 0.591085 0.806609i \(-0.298700\pi\)
0.591085 + 0.806609i \(0.298700\pi\)
\(264\) 0 0
\(265\) −6.82843 −0.419467
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.9706i 1.40054i 0.713878 + 0.700270i \(0.246937\pi\)
−0.713878 + 0.700270i \(0.753063\pi\)
\(270\) 0 0
\(271\) 21.3137i 1.29472i 0.762186 + 0.647358i \(0.224126\pi\)
−0.762186 + 0.647358i \(0.775874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.41421 0.0852803
\(276\) 0 0
\(277\) −20.8701 −1.25396 −0.626980 0.779035i \(-0.715709\pi\)
−0.626980 + 0.779035i \(0.715709\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 17.8995i − 1.06779i −0.845549 0.533897i \(-0.820727\pi\)
0.845549 0.533897i \(-0.179273\pi\)
\(282\) 0 0
\(283\) − 7.31371i − 0.434755i −0.976088 0.217377i \(-0.930250\pi\)
0.976088 0.217377i \(-0.0697502\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.82843 −0.285013
\(288\) 0 0
\(289\) 15.6274 0.919260
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 2.00000i − 0.116841i −0.998292 0.0584206i \(-0.981394\pi\)
0.998292 0.0584206i \(-0.0186065\pi\)
\(294\) 0 0
\(295\) 5.41421i 0.315228i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.11270 −0.411338
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.17157i 0.181604i
\(306\) 0 0
\(307\) 3.31371i 0.189123i 0.995519 + 0.0945617i \(0.0301450\pi\)
−0.995519 + 0.0945617i \(0.969855\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.5147 0.652940 0.326470 0.945208i \(-0.394141\pi\)
0.326470 + 0.945208i \(0.394141\pi\)
\(312\) 0 0
\(313\) −34.2843 −1.93786 −0.968931 0.247332i \(-0.920446\pi\)
−0.968931 + 0.247332i \(0.920446\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.3137i 0.972435i 0.873838 + 0.486217i \(0.161624\pi\)
−0.873838 + 0.486217i \(0.838376\pi\)
\(318\) 0 0
\(319\) 2.82843i 0.158362i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.62742 −0.368759
\(324\) 0 0
\(325\) −2.24264 −0.124399
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.68629i 0.258364i
\(330\) 0 0
\(331\) 31.3137i 1.72116i 0.509318 + 0.860579i \(0.329898\pi\)
−0.509318 + 0.860579i \(0.670102\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) 16.8284 0.916703 0.458351 0.888771i \(-0.348440\pi\)
0.458351 + 0.888771i \(0.348440\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 4.48528i − 0.242892i
\(342\) 0 0
\(343\) − 8.00000i − 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.9706 1.12576 0.562879 0.826539i \(-0.309694\pi\)
0.562879 + 0.826539i \(0.309694\pi\)
\(348\) 0 0
\(349\) 8.82843 0.472575 0.236287 0.971683i \(-0.424069\pi\)
0.236287 + 0.971683i \(0.424069\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.5147i 0.825765i 0.910784 + 0.412883i \(0.135478\pi\)
−0.910784 + 0.412883i \(0.864522\pi\)
\(354\) 0 0
\(355\) 13.6569i 0.724831i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.1127 1.64207 0.821033 0.570881i \(-0.193398\pi\)
0.821033 + 0.570881i \(0.193398\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 0.828427i − 0.0433619i
\(366\) 0 0
\(367\) − 18.9289i − 0.988082i −0.869439 0.494041i \(-0.835519\pi\)
0.869439 0.494041i \(-0.164481\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 2.92893 0.151654 0.0758272 0.997121i \(-0.475840\pi\)
0.0758272 + 0.997121i \(0.475840\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.48528i − 0.231004i
\(378\) 0 0
\(379\) − 6.68629i − 0.343452i −0.985145 0.171726i \(-0.945066\pi\)
0.985145 0.171726i \(-0.0549343\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 30.6274 1.56499 0.782494 0.622658i \(-0.213947\pi\)
0.782494 + 0.622658i \(0.213947\pi\)
\(384\) 0 0
\(385\) 0.828427 0.0422206
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.65685i 0.185410i 0.995694 + 0.0927049i \(0.0295513\pi\)
−0.995694 + 0.0927049i \(0.970449\pi\)
\(390\) 0 0
\(391\) 3.71573i 0.187912i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.48528 0.326310
\(396\) 0 0
\(397\) 20.3848 1.02308 0.511541 0.859259i \(-0.329075\pi\)
0.511541 + 0.859259i \(0.329075\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.72792i 0.435852i 0.975965 + 0.217926i \(0.0699291\pi\)
−0.975965 + 0.217926i \(0.930071\pi\)
\(402\) 0 0
\(403\) 7.11270i 0.354309i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.48528 −0.321463
\(408\) 0 0
\(409\) 0.343146 0.0169675 0.00848373 0.999964i \(-0.497300\pi\)
0.00848373 + 0.999964i \(0.497300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.17157i 0.156063i
\(414\) 0 0
\(415\) − 1.17157i − 0.0575103i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −27.7574 −1.35604 −0.678018 0.735045i \(-0.737161\pi\)
−0.678018 + 0.735045i \(0.737161\pi\)
\(420\) 0 0
\(421\) −39.9411 −1.94661 −0.973306 0.229513i \(-0.926287\pi\)
−0.973306 + 0.229513i \(0.926287\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.17157i 0.0568296i
\(426\) 0 0
\(427\) 1.85786i 0.0899084i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.48528 −0.216048 −0.108024 0.994148i \(-0.534452\pi\)
−0.108024 + 0.994148i \(0.534452\pi\)
\(432\) 0 0
\(433\) 12.8284 0.616495 0.308247 0.951306i \(-0.400258\pi\)
0.308247 + 0.951306i \(0.400258\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17.9411i 0.858240i
\(438\) 0 0
\(439\) 15.6569i 0.747261i 0.927578 + 0.373630i \(0.121887\pi\)
−0.927578 + 0.373630i \(0.878113\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.31371 0.347485 0.173742 0.984791i \(-0.444414\pi\)
0.173742 + 0.984791i \(0.444414\pi\)
\(444\) 0 0
\(445\) −0.928932 −0.0440356
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 25.2132i 1.18988i 0.803768 + 0.594942i \(0.202825\pi\)
−0.803768 + 0.594942i \(0.797175\pi\)
\(450\) 0 0
\(451\) 11.6569i 0.548900i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.31371 −0.0615876
\(456\) 0 0
\(457\) 3.85786 0.180463 0.0902316 0.995921i \(-0.471239\pi\)
0.0902316 + 0.995921i \(0.471239\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 5.79899i − 0.270086i −0.990840 0.135043i \(-0.956883\pi\)
0.990840 0.135043i \(-0.0431172\pi\)
\(462\) 0 0
\(463\) − 16.3848i − 0.761465i −0.924685 0.380733i \(-0.875672\pi\)
0.924685 0.380733i \(-0.124328\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.7990 1.28638 0.643192 0.765705i \(-0.277610\pi\)
0.643192 + 0.765705i \(0.277610\pi\)
\(468\) 0 0
\(469\) −6.62742 −0.306026
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.65685i 0.444023i
\(474\) 0 0
\(475\) 5.65685i 0.259554i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 10.2843 0.468922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 10.4853i − 0.476112i
\(486\) 0 0
\(487\) − 42.0416i − 1.90509i −0.304401 0.952544i \(-0.598456\pi\)
0.304401 0.952544i \(-0.401544\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −27.0711 −1.22170 −0.610850 0.791746i \(-0.709172\pi\)
−0.610850 + 0.791746i \(0.709172\pi\)
\(492\) 0 0
\(493\) −2.34315 −0.105530
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) − 20.9706i − 0.938771i −0.882993 0.469386i \(-0.844475\pi\)
0.882993 0.469386i \(-0.155525\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −11.1716 −0.497128
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 8.14214i − 0.360894i −0.983585 0.180447i \(-0.942246\pi\)
0.983585 0.180447i \(-0.0577544\pi\)
\(510\) 0 0
\(511\) − 0.485281i − 0.0214676i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.24264 0.0988226
\(516\) 0 0
\(517\) 11.3137 0.497576
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 32.7279i − 1.43384i −0.697157 0.716918i \(-0.745552\pi\)
0.697157 0.716918i \(-0.254448\pi\)
\(522\) 0 0
\(523\) 11.7990i 0.515934i 0.966154 + 0.257967i \(0.0830526\pi\)
−0.966154 + 0.257967i \(0.916947\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.71573 0.161860
\(528\) 0 0
\(529\) −12.9411 −0.562658
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 18.4853i − 0.800686i
\(534\) 0 0
\(535\) − 18.1421i − 0.784353i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9.41421 −0.405499
\(540\) 0 0
\(541\) 24.1421 1.03795 0.518976 0.854789i \(-0.326313\pi\)
0.518976 + 0.854789i \(0.326313\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 8.82843i − 0.378168i
\(546\) 0 0
\(547\) 43.7990i 1.87271i 0.351055 + 0.936355i \(0.385823\pi\)
−0.351055 + 0.936355i \(0.614177\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) 3.79899 0.161549
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.51472i 0.148923i 0.997224 + 0.0744617i \(0.0237239\pi\)
−0.997224 + 0.0744617i \(0.976276\pi\)
\(558\) 0 0
\(559\) − 15.3137i − 0.647701i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −33.6569 −1.41847 −0.709234 0.704974i \(-0.750959\pi\)
−0.709234 + 0.704974i \(0.750959\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 37.4142i − 1.56849i −0.620454 0.784243i \(-0.713052\pi\)
0.620454 0.784243i \(-0.286948\pi\)
\(570\) 0 0
\(571\) 10.6274i 0.444744i 0.974962 + 0.222372i \(0.0713799\pi\)
−0.974962 + 0.222372i \(0.928620\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.17157 0.132264
\(576\) 0 0
\(577\) 23.4558 0.976480 0.488240 0.872710i \(-0.337639\pi\)
0.488240 + 0.872710i \(0.337639\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 0.686292i − 0.0284722i
\(582\) 0 0
\(583\) 9.65685i 0.399946i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.4558 0.555382 0.277691 0.960670i \(-0.410431\pi\)
0.277691 + 0.960670i \(0.410431\pi\)
\(588\) 0 0
\(589\) 17.9411 0.739251
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 30.6274i − 1.25772i −0.777520 0.628859i \(-0.783522\pi\)
0.777520 0.628859i \(-0.216478\pi\)
\(594\) 0 0
\(595\) 0.686292i 0.0281352i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 42.1421 1.72188 0.860940 0.508706i \(-0.169876\pi\)
0.860940 + 0.508706i \(0.169876\pi\)
\(600\) 0 0
\(601\) −21.9411 −0.894997 −0.447499 0.894285i \(-0.647685\pi\)
−0.447499 + 0.894285i \(0.647685\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.00000i 0.365902i
\(606\) 0 0
\(607\) − 38.0416i − 1.54406i −0.635585 0.772031i \(-0.719241\pi\)
0.635585 0.772031i \(-0.280759\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −17.9411 −0.725820
\(612\) 0 0
\(613\) −6.92893 −0.279857 −0.139928 0.990162i \(-0.544687\pi\)
−0.139928 + 0.990162i \(0.544687\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.8284i − 0.758004i −0.925396 0.379002i \(-0.876267\pi\)
0.925396 0.379002i \(-0.123733\pi\)
\(618\) 0 0
\(619\) 6.97056i 0.280171i 0.990139 + 0.140085i \(0.0447377\pi\)
−0.990139 + 0.140085i \(0.955262\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.544156 −0.0218011
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 5.37258i − 0.214219i
\(630\) 0 0
\(631\) − 12.1421i − 0.483371i −0.970355 0.241685i \(-0.922300\pi\)
0.970355 0.241685i \(-0.0777002\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12.5858 −0.499452
\(636\) 0 0
\(637\) 14.9289 0.591506
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 38.8701i 1.53527i 0.640884 + 0.767637i \(0.278568\pi\)
−0.640884 + 0.767637i \(0.721432\pi\)
\(642\) 0 0
\(643\) − 6.82843i − 0.269287i −0.990894 0.134643i \(-0.957011\pi\)
0.990894 0.134643i \(-0.0429889\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −48.9706 −1.92523 −0.962616 0.270871i \(-0.912688\pi\)
−0.962616 + 0.270871i \(0.912688\pi\)
\(648\) 0 0
\(649\) 7.65685 0.300558
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.2843i 1.02858i 0.857615 + 0.514292i \(0.171945\pi\)
−0.857615 + 0.514292i \(0.828055\pi\)
\(654\) 0 0
\(655\) 5.89949i 0.230512i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −44.0416 −1.71562 −0.857809 0.513968i \(-0.828175\pi\)
−0.857809 + 0.513968i \(0.828175\pi\)
\(660\) 0 0
\(661\) 0.142136 0.00552844 0.00276422 0.999996i \(-0.499120\pi\)
0.00276422 + 0.999996i \(0.499120\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.31371i 0.128500i
\(666\) 0 0
\(667\) 6.34315i 0.245608i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.48528 0.173152
\(672\) 0 0
\(673\) 30.7696 1.18608 0.593040 0.805173i \(-0.297928\pi\)
0.593040 + 0.805173i \(0.297928\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 29.3137i − 1.12662i −0.826247 0.563309i \(-0.809528\pi\)
0.826247 0.563309i \(-0.190472\pi\)
\(678\) 0 0
\(679\) − 6.14214i − 0.235714i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −19.7990 −0.757587 −0.378794 0.925481i \(-0.623661\pi\)
−0.378794 + 0.925481i \(0.623661\pi\)
\(684\) 0 0
\(685\) 5.65685 0.216137
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 15.3137i − 0.583406i
\(690\) 0 0
\(691\) 18.3431i 0.697806i 0.937159 + 0.348903i \(0.113446\pi\)
−0.937159 + 0.348903i \(0.886554\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −19.6569 −0.745627
\(696\) 0 0
\(697\) −9.65685 −0.365779
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 38.4853i − 1.45357i −0.686866 0.726785i \(-0.741014\pi\)
0.686866 0.726785i \(-0.258986\pi\)
\(702\) 0 0
\(703\) − 25.9411i − 0.978388i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.54416 −0.246118
\(708\) 0 0
\(709\) −20.3431 −0.764003 −0.382001 0.924162i \(-0.624765\pi\)
−0.382001 + 0.924162i \(0.624765\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 10.0589i − 0.376708i
\(714\) 0 0
\(715\) 3.17157i 0.118610i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 11.7990 0.440028 0.220014 0.975497i \(-0.429390\pi\)
0.220014 + 0.975497i \(0.429390\pi\)
\(720\) 0 0
\(721\) 1.31371 0.0489251
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000i 0.0742781i
\(726\) 0 0
\(727\) − 18.2426i − 0.676582i −0.941042 0.338291i \(-0.890151\pi\)
0.941042 0.338291i \(-0.109849\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 6.24264 0.230577 0.115289 0.993332i \(-0.463221\pi\)
0.115289 + 0.993332i \(0.463221\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) − 12.0000i − 0.441427i −0.975339 0.220714i \(-0.929161\pi\)
0.975339 0.220714i \(-0.0708386\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 40.9706 1.50306 0.751532 0.659697i \(-0.229315\pi\)
0.751532 + 0.659697i \(0.229315\pi\)
\(744\) 0 0
\(745\) −0.828427 −0.0303512
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 10.6274i − 0.388317i
\(750\) 0 0
\(751\) − 17.7990i − 0.649494i −0.945801 0.324747i \(-0.894721\pi\)
0.945801 0.324747i \(-0.105279\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22.9706 0.835984
\(756\) 0 0
\(757\) −0.585786 −0.0212908 −0.0106454 0.999943i \(-0.503389\pi\)
−0.0106454 + 0.999943i \(0.503389\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 14.8701i − 0.539039i −0.962995 0.269520i \(-0.913135\pi\)
0.962995 0.269520i \(-0.0868649\pi\)
\(762\) 0 0
\(763\) − 5.17157i − 0.187224i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.1421 −0.438427
\(768\) 0 0
\(769\) 47.5980 1.71643 0.858214 0.513293i \(-0.171575\pi\)
0.858214 + 0.513293i \(0.171575\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 47.2548i − 1.69964i −0.527074 0.849819i \(-0.676711\pi\)
0.527074 0.849819i \(-0.323289\pi\)
\(774\) 0 0
\(775\) − 3.17157i − 0.113926i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −46.6274 −1.67060
\(780\) 0 0
\(781\) 19.3137 0.691099
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 19.8995i − 0.710243i
\(786\) 0 0
\(787\) 2.14214i 0.0763589i 0.999271 + 0.0381794i \(0.0121558\pi\)
−0.999271 + 0.0381794i \(0.987844\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.34315 −0.0833127
\(792\) 0 0
\(793\) −7.11270 −0.252579
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 16.4853i − 0.583939i −0.956428 0.291969i \(-0.905689\pi\)
0.956428 0.291969i \(-0.0943105\pi\)
\(798\) 0 0
\(799\) 9.37258i 0.331578i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.17157 −0.0413439
\(804\) 0 0
\(805\) 1.85786 0.0654811
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 29.8995i 1.05121i 0.850729 + 0.525605i \(0.176161\pi\)
−0.850729 + 0.525605i \(0.823839\pi\)
\(810\) 0 0
\(811\) − 32.6274i − 1.14570i −0.819659 0.572852i \(-0.805837\pi\)
0.819659 0.572852i \(-0.194163\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.48528 −0.297226
\(816\) 0 0
\(817\) −38.6274 −1.35140
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17.3137i 0.604253i 0.953268 + 0.302126i \(0.0976964\pi\)
−0.953268 + 0.302126i \(0.902304\pi\)
\(822\) 0 0
\(823\) − 16.8701i − 0.588053i −0.955797 0.294027i \(-0.905005\pi\)
0.955797 0.294027i \(-0.0949954\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.37258 −0.186823 −0.0934115 0.995628i \(-0.529777\pi\)
−0.0934115 + 0.995628i \(0.529777\pi\)
\(828\) 0 0
\(829\) 30.4853 1.05880 0.529399 0.848373i \(-0.322418\pi\)
0.529399 + 0.848373i \(0.322418\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 7.79899i − 0.270219i
\(834\) 0 0
\(835\) 2.48528i 0.0860067i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7.51472 −0.259437 −0.129718 0.991551i \(-0.541407\pi\)
−0.129718 + 0.991551i \(0.541407\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.97056i 0.274196i
\(846\) 0 0
\(847\) 5.27208i 0.181151i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −14.5442 −0.498567
\(852\) 0 0
\(853\) 4.78680 0.163897 0.0819484 0.996637i \(-0.473886\pi\)
0.0819484 + 0.996637i \(0.473886\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 31.5147i − 1.07652i −0.842778 0.538261i \(-0.819081\pi\)
0.842778 0.538261i \(-0.180919\pi\)
\(858\) 0 0
\(859\) − 10.0000i − 0.341196i −0.985341 0.170598i \(-0.945430\pi\)
0.985341 0.170598i \(-0.0545699\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −22.7696 −0.775085 −0.387542 0.921852i \(-0.626676\pi\)
−0.387542 + 0.921852i \(0.626676\pi\)
\(864\) 0 0
\(865\) 16.4853 0.560516
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 9.17157i − 0.311124i
\(870\) 0 0
\(871\) − 25.3726i − 0.859717i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.585786 0.0198032
\(876\) 0 0
\(877\) 7.41421 0.250360 0.125180 0.992134i \(-0.460049\pi\)
0.125180 + 0.992134i \(0.460049\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 22.5858i − 0.760934i −0.924794 0.380467i \(-0.875763\pi\)
0.924794 0.380467i \(-0.124237\pi\)
\(882\) 0 0
\(883\) − 17.4558i − 0.587436i −0.955892 0.293718i \(-0.905107\pi\)
0.955892 0.293718i \(-0.0948926\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.142136 −0.00477245 −0.00238622 0.999997i \(-0.500760\pi\)
−0.00238622 + 0.999997i \(0.500760\pi\)
\(888\) 0 0
\(889\) −7.37258 −0.247268
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 45.2548i 1.51440i
\(894\) 0 0
\(895\) − 10.5858i − 0.353844i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.34315 0.211556
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.1716i 0.637285i
\(906\) 0 0
\(907\) 31.1127i 1.03308i 0.856263 + 0.516540i \(0.172780\pi\)
−0.856263 + 0.516540i \(0.827220\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.2843 0.672048 0.336024 0.941853i \(-0.390918\pi\)
0.336024 + 0.941853i \(0.390918\pi\)
\(912\) 0 0
\(913\) −1.65685 −0.0548339
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.45584i 0.114122i
\(918\) 0 0
\(919\) 9.51472i 0.313862i 0.987610 + 0.156931i \(0.0501600\pi\)
−0.987610 + 0.156931i \(0.949840\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −30.6274 −1.00811
\(924\) 0 0
\(925\) −4.58579 −0.150780
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.27208i 0.238589i 0.992859 + 0.119295i \(0.0380633\pi\)
−0.992859 + 0.119295i \(0.961937\pi\)
\(930\) 0 0
\(931\) − 37.6569i − 1.23415i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.65685 0.0541849
\(936\) 0 0
\(937\) 14.2843 0.466647 0.233323 0.972399i \(-0.425040\pi\)
0.233323 + 0.972399i \(0.425040\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 13.0294i − 0.424748i −0.977189 0.212374i \(-0.931881\pi\)
0.977189 0.212374i \(-0.0681194\pi\)
\(942\) 0 0
\(943\) 26.1421i 0.851305i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.0294 0.358409 0.179204 0.983812i \(-0.442648\pi\)
0.179204 + 0.983812i \(0.442648\pi\)
\(948\) 0 0
\(949\) 1.85786 0.0603088
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 13.6569i 0.442389i 0.975230 + 0.221194i \(0.0709955\pi\)
−0.975230 + 0.221194i \(0.929004\pi\)
\(954\) 0 0
\(955\) 6.82843i 0.220963i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3.31371 0.107005
\(960\) 0 0
\(961\) 20.9411 0.675520
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 15.6569i − 0.504012i
\(966\) 0 0
\(967\) − 0.100505i − 0.00323202i −0.999999 0.00161601i \(-0.999486\pi\)
0.999999 0.00161601i \(-0.000514393\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 29.6985 0.953070 0.476535 0.879156i \(-0.341893\pi\)
0.476535 + 0.879156i \(0.341893\pi\)
\(972\) 0 0
\(973\) −11.5147 −0.369145
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 7.51472i − 0.240417i −0.992749 0.120209i \(-0.961644\pi\)
0.992749 0.120209i \(-0.0383563\pi\)
\(978\) 0 0
\(979\) 1.31371i 0.0419863i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.31371 0.105691 0.0528454 0.998603i \(-0.483171\pi\)
0.0528454 + 0.998603i \(0.483171\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.6569i 0.688648i
\(990\) 0 0
\(991\) 11.6569i 0.370292i 0.982711 + 0.185146i \(0.0592758\pi\)
−0.982711 + 0.185146i \(0.940724\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.34315 0.137687
\(996\) 0 0
\(997\) −25.7574 −0.815744 −0.407872 0.913039i \(-0.633729\pi\)
−0.407872 + 0.913039i \(0.633729\pi\)
\(998\) 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 1440.2.h.b.1151.2 4
3.2 odd 2 1440.2.h.c.1151.4 yes 4
4.3 odd 2 1440.2.h.c.1151.1 yes 4
5.2 odd 4 7200.2.o.l.7199.1 4
5.3 odd 4 7200.2.o.c.7199.4 4
5.4 even 2 7200.2.h.f.1151.3 4
8.3 odd 2 2880.2.h.c.1151.3 4
8.5 even 2 2880.2.h.b.1151.4 4
12.11 even 2 inner 1440.2.h.b.1151.3 yes 4
15.2 even 4 7200.2.o.k.7199.1 4
15.8 even 4 7200.2.o.d.7199.4 4
15.14 odd 2 7200.2.h.e.1151.3 4
20.3 even 4 7200.2.o.k.7199.2 4
20.7 even 4 7200.2.o.d.7199.3 4
20.19 odd 2 7200.2.h.e.1151.2 4
24.5 odd 2 2880.2.h.c.1151.2 4
24.11 even 2 2880.2.h.b.1151.1 4
60.23 odd 4 7200.2.o.l.7199.2 4
60.47 odd 4 7200.2.o.c.7199.3 4
60.59 even 2 7200.2.h.f.1151.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.h.b.1151.2 4 1.1 even 1 trivial
1440.2.h.b.1151.3 yes 4 12.11 even 2 inner
1440.2.h.c.1151.1 yes 4 4.3 odd 2
1440.2.h.c.1151.4 yes 4 3.2 odd 2
2880.2.h.b.1151.1 4 24.11 even 2
2880.2.h.b.1151.4 4 8.5 even 2
2880.2.h.c.1151.2 4 24.5 odd 2
2880.2.h.c.1151.3 4 8.3 odd 2
7200.2.h.e.1151.2 4 20.19 odd 2
7200.2.h.e.1151.3 4 15.14 odd 2
7200.2.h.f.1151.2 4 60.59 even 2
7200.2.h.f.1151.3 4 5.4 even 2
7200.2.o.c.7199.3 4 60.47 odd 4
7200.2.o.c.7199.4 4 5.3 odd 4
7200.2.o.d.7199.3 4 20.7 even 4
7200.2.o.d.7199.4 4 15.8 even 4
7200.2.o.k.7199.1 4 15.2 even 4
7200.2.o.k.7199.2 4 20.3 even 4
7200.2.o.l.7199.1 4 5.2 odd 4
7200.2.o.l.7199.2 4 60.23 odd 4