Properties

Label 1728.3.g.k.703.1
Level $1728$
Weight $3$
Character 1728.703
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 703.1
Root \(1.20036 - 0.747754i\) of defining polynomial
Character \(\chi\) \(=\) 1728.703
Dual form 1728.3.g.k.703.2

$q$-expansion

\(f(q)\) \(=\) \(q-9.64469 q^{5} -1.12019i q^{7} +O(q^{10})\) \(q-9.64469 q^{5} -1.12019i q^{7} -13.8529i q^{11} +8.65671 q^{13} +12.6567 q^{17} -27.6817i q^{19} +27.5298i q^{23} +68.0201 q^{25} -6.15193 q^{29} +13.2481i q^{31} +10.8038i q^{35} +32.9316 q^{37} -60.2500 q^{41} +25.3479i q^{43} -66.7652i q^{47} +47.7452 q^{49} -13.1698 q^{53} +133.607i q^{55} +83.2556i q^{59} -101.413 q^{61} -83.4913 q^{65} -112.006i q^{67} -85.9221i q^{71} +25.0798 q^{73} -15.5178 q^{77} -63.4604i q^{79} -111.443i q^{83} -122.070 q^{85} -22.4278 q^{89} -9.69712i q^{91} +266.981i q^{95} -71.2151 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} + 16 q^{13} + 48 q^{17} + 48 q^{25} + 32 q^{29} + 96 q^{37} - 128 q^{41} - 168 q^{53} - 32 q^{61} - 112 q^{65} + 24 q^{73} + 440 q^{77} - 144 q^{85} + 624 q^{89} - 136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −9.64469 −1.92894 −0.964469 0.264195i \(-0.914894\pi\)
−0.964469 + 0.264195i \(0.914894\pi\)
\(6\) 0 0
\(7\) − 1.12019i − 0.160027i −0.996794 0.0800133i \(-0.974504\pi\)
0.996794 0.0800133i \(-0.0254963\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 13.8529i − 1.25935i −0.776858 0.629675i \(-0.783188\pi\)
0.776858 0.629675i \(-0.216812\pi\)
\(12\) 0 0
\(13\) 8.65671 0.665901 0.332950 0.942944i \(-0.391956\pi\)
0.332950 + 0.942944i \(0.391956\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 12.6567 0.744512 0.372256 0.928130i \(-0.378584\pi\)
0.372256 + 0.928130i \(0.378584\pi\)
\(18\) 0 0
\(19\) − 27.6817i − 1.45693i −0.685082 0.728466i \(-0.740234\pi\)
0.685082 0.728466i \(-0.259766\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 27.5298i 1.19695i 0.801143 + 0.598473i \(0.204226\pi\)
−0.801143 + 0.598473i \(0.795774\pi\)
\(24\) 0 0
\(25\) 68.0201 2.72080
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.15193 −0.212136 −0.106068 0.994359i \(-0.533826\pi\)
−0.106068 + 0.994359i \(0.533826\pi\)
\(30\) 0 0
\(31\) 13.2481i 0.427358i 0.976904 + 0.213679i \(0.0685446\pi\)
−0.976904 + 0.213679i \(0.931455\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.8038i 0.308681i
\(36\) 0 0
\(37\) 32.9316 0.890044 0.445022 0.895520i \(-0.353196\pi\)
0.445022 + 0.895520i \(0.353196\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −60.2500 −1.46951 −0.734756 0.678332i \(-0.762703\pi\)
−0.734756 + 0.678332i \(0.762703\pi\)
\(42\) 0 0
\(43\) 25.3479i 0.589487i 0.955576 + 0.294744i \(0.0952342\pi\)
−0.955576 + 0.294744i \(0.904766\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 66.7652i − 1.42054i −0.703931 0.710268i \(-0.748574\pi\)
0.703931 0.710268i \(-0.251426\pi\)
\(48\) 0 0
\(49\) 47.7452 0.974392
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.1698 −0.248487 −0.124243 0.992252i \(-0.539650\pi\)
−0.124243 + 0.992252i \(0.539650\pi\)
\(54\) 0 0
\(55\) 133.607i 2.42921i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 83.2556i 1.41111i 0.708655 + 0.705556i \(0.249302\pi\)
−0.708655 + 0.705556i \(0.750698\pi\)
\(60\) 0 0
\(61\) −101.413 −1.66251 −0.831257 0.555888i \(-0.812378\pi\)
−0.831257 + 0.555888i \(0.812378\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −83.4913 −1.28448
\(66\) 0 0
\(67\) − 112.006i − 1.67173i −0.548938 0.835863i \(-0.684968\pi\)
0.548938 0.835863i \(-0.315032\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 85.9221i − 1.21017i −0.796161 0.605085i \(-0.793139\pi\)
0.796161 0.605085i \(-0.206861\pi\)
\(72\) 0 0
\(73\) 25.0798 0.343559 0.171780 0.985135i \(-0.445048\pi\)
0.171780 + 0.985135i \(0.445048\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.5178 −0.201530
\(78\) 0 0
\(79\) − 63.4604i − 0.803296i −0.915794 0.401648i \(-0.868437\pi\)
0.915794 0.401648i \(-0.131563\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 111.443i − 1.34269i −0.741146 0.671344i \(-0.765717\pi\)
0.741146 0.671344i \(-0.234283\pi\)
\(84\) 0 0
\(85\) −122.070 −1.43612
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −22.4278 −0.251998 −0.125999 0.992030i \(-0.540214\pi\)
−0.125999 + 0.992030i \(0.540214\pi\)
\(90\) 0 0
\(91\) − 9.69712i − 0.106562i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 266.981i 2.81033i
\(96\) 0 0
\(97\) −71.2151 −0.734177 −0.367088 0.930186i \(-0.619645\pi\)
−0.367088 + 0.930186i \(0.619645\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −138.475 −1.37104 −0.685521 0.728052i \(-0.740426\pi\)
−0.685521 + 0.728052i \(0.740426\pi\)
\(102\) 0 0
\(103\) 117.157i 1.13744i 0.822530 + 0.568721i \(0.192562\pi\)
−0.822530 + 0.568721i \(0.807438\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 39.2155i 0.366500i 0.983066 + 0.183250i \(0.0586618\pi\)
−0.983066 + 0.183250i \(0.941338\pi\)
\(108\) 0 0
\(109\) −153.735 −1.41042 −0.705208 0.709000i \(-0.749146\pi\)
−0.705208 + 0.709000i \(0.749146\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −113.527 −1.00466 −0.502332 0.864675i \(-0.667524\pi\)
−0.502332 + 0.864675i \(0.667524\pi\)
\(114\) 0 0
\(115\) − 265.516i − 2.30883i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 14.1779i − 0.119142i
\(120\) 0 0
\(121\) −70.9017 −0.585965
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −414.915 −3.31932
\(126\) 0 0
\(127\) − 120.302i − 0.947263i −0.880723 0.473632i \(-0.842943\pi\)
0.880723 0.473632i \(-0.157057\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 60.2137i − 0.459646i −0.973232 0.229823i \(-0.926185\pi\)
0.973232 0.229823i \(-0.0738148\pi\)
\(132\) 0 0
\(133\) −31.0086 −0.233148
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.0339 −0.102437 −0.0512186 0.998687i \(-0.516311\pi\)
−0.0512186 + 0.998687i \(0.516311\pi\)
\(138\) 0 0
\(139\) 234.127i 1.68436i 0.539194 + 0.842182i \(0.318729\pi\)
−0.539194 + 0.842182i \(0.681271\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 119.920i − 0.838603i
\(144\) 0 0
\(145\) 59.3335 0.409197
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 50.9812 0.342155 0.171078 0.985258i \(-0.445275\pi\)
0.171078 + 0.985258i \(0.445275\pi\)
\(150\) 0 0
\(151\) 158.996i 1.05295i 0.850190 + 0.526475i \(0.176487\pi\)
−0.850190 + 0.526475i \(0.823513\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 127.774i − 0.824347i
\(156\) 0 0
\(157\) 2.68658 0.0171120 0.00855600 0.999963i \(-0.497277\pi\)
0.00855600 + 0.999963i \(0.497277\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 30.8384 0.191543
\(162\) 0 0
\(163\) 146.659i 0.899746i 0.893093 + 0.449873i \(0.148531\pi\)
−0.893093 + 0.449873i \(0.851469\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 18.7844i − 0.112481i −0.998417 0.0562407i \(-0.982089\pi\)
0.998417 0.0562407i \(-0.0179114\pi\)
\(168\) 0 0
\(169\) −94.0614 −0.556576
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 170.434 0.985166 0.492583 0.870266i \(-0.336053\pi\)
0.492583 + 0.870266i \(0.336053\pi\)
\(174\) 0 0
\(175\) − 76.1951i − 0.435401i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 186.431i 1.04151i 0.853706 + 0.520756i \(0.174350\pi\)
−0.853706 + 0.520756i \(0.825650\pi\)
\(180\) 0 0
\(181\) −115.886 −0.640255 −0.320127 0.947375i \(-0.603726\pi\)
−0.320127 + 0.947375i \(0.603726\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −317.615 −1.71684
\(186\) 0 0
\(187\) − 175.332i − 0.937602i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 46.4872i − 0.243389i −0.992568 0.121694i \(-0.961167\pi\)
0.992568 0.121694i \(-0.0388328\pi\)
\(192\) 0 0
\(193\) 62.0795 0.321656 0.160828 0.986982i \(-0.448584\pi\)
0.160828 + 0.986982i \(0.448584\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −98.1691 −0.498320 −0.249160 0.968462i \(-0.580155\pi\)
−0.249160 + 0.968462i \(0.580155\pi\)
\(198\) 0 0
\(199\) 104.609i 0.525671i 0.964841 + 0.262836i \(0.0846577\pi\)
−0.964841 + 0.262836i \(0.915342\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.89131i 0.0339473i
\(204\) 0 0
\(205\) 581.092 2.83460
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −383.471 −1.83479
\(210\) 0 0
\(211\) 44.1261i 0.209128i 0.994518 + 0.104564i \(0.0333447\pi\)
−0.994518 + 0.104564i \(0.966655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 244.473i − 1.13708i
\(216\) 0 0
\(217\) 14.8403 0.0683886
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 109.565 0.495771
\(222\) 0 0
\(223\) − 353.445i − 1.58496i −0.609901 0.792478i \(-0.708791\pi\)
0.609901 0.792478i \(-0.291209\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 192.295i 0.847116i 0.905869 + 0.423558i \(0.139219\pi\)
−0.905869 + 0.423558i \(0.860781\pi\)
\(228\) 0 0
\(229\) 356.455 1.55657 0.778286 0.627909i \(-0.216089\pi\)
0.778286 + 0.627909i \(0.216089\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −190.754 −0.818687 −0.409344 0.912380i \(-0.634242\pi\)
−0.409344 + 0.912380i \(0.634242\pi\)
\(234\) 0 0
\(235\) 643.930i 2.74013i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 230.647i 0.965051i 0.875882 + 0.482526i \(0.160280\pi\)
−0.875882 + 0.482526i \(0.839720\pi\)
\(240\) 0 0
\(241\) 120.413 0.499639 0.249820 0.968292i \(-0.419629\pi\)
0.249820 + 0.968292i \(0.419629\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −460.488 −1.87954
\(246\) 0 0
\(247\) − 239.632i − 0.970171i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 453.709i 1.80761i 0.427948 + 0.903803i \(0.359237\pi\)
−0.427948 + 0.903803i \(0.640763\pi\)
\(252\) 0 0
\(253\) 381.366 1.50737
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −109.467 −0.425940 −0.212970 0.977059i \(-0.568314\pi\)
−0.212970 + 0.977059i \(0.568314\pi\)
\(258\) 0 0
\(259\) − 36.8895i − 0.142431i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 99.7622i 0.379324i 0.981849 + 0.189662i \(0.0607392\pi\)
−0.981849 + 0.189662i \(0.939261\pi\)
\(264\) 0 0
\(265\) 127.019 0.479316
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −320.980 −1.19324 −0.596618 0.802525i \(-0.703489\pi\)
−0.596618 + 0.802525i \(0.703489\pi\)
\(270\) 0 0
\(271\) 297.292i 1.09702i 0.836145 + 0.548509i \(0.184804\pi\)
−0.836145 + 0.548509i \(0.815196\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 942.273i − 3.42645i
\(276\) 0 0
\(277\) −90.4130 −0.326401 −0.163200 0.986593i \(-0.552182\pi\)
−0.163200 + 0.986593i \(0.552182\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 318.187 1.13234 0.566169 0.824289i \(-0.308425\pi\)
0.566169 + 0.824289i \(0.308425\pi\)
\(282\) 0 0
\(283\) − 365.829i − 1.29268i −0.763049 0.646340i \(-0.776299\pi\)
0.763049 0.646340i \(-0.223701\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 67.4912i 0.235161i
\(288\) 0 0
\(289\) −128.808 −0.445702
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 54.6252 0.186434 0.0932171 0.995646i \(-0.470285\pi\)
0.0932171 + 0.995646i \(0.470285\pi\)
\(294\) 0 0
\(295\) − 802.974i − 2.72195i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 238.317i 0.797047i
\(300\) 0 0
\(301\) 28.3944 0.0943336
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 978.100 3.20689
\(306\) 0 0
\(307\) 25.2901i 0.0823781i 0.999151 + 0.0411891i \(0.0131146\pi\)
−0.999151 + 0.0411891i \(0.986885\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 449.554i − 1.44551i −0.691103 0.722756i \(-0.742875\pi\)
0.691103 0.722756i \(-0.257125\pi\)
\(312\) 0 0
\(313\) −278.962 −0.891253 −0.445627 0.895219i \(-0.647019\pi\)
−0.445627 + 0.895219i \(0.647019\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 363.560 1.14688 0.573439 0.819248i \(-0.305609\pi\)
0.573439 + 0.819248i \(0.305609\pi\)
\(318\) 0 0
\(319\) 85.2219i 0.267153i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 350.359i − 1.08470i
\(324\) 0 0
\(325\) 588.830 1.81178
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −74.7895 −0.227324
\(330\) 0 0
\(331\) 209.510i 0.632962i 0.948599 + 0.316481i \(0.102501\pi\)
−0.948599 + 0.316481i \(0.897499\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1080.26i 3.22466i
\(336\) 0 0
\(337\) −446.770 −1.32573 −0.662864 0.748740i \(-0.730659\pi\)
−0.662864 + 0.748740i \(0.730659\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 183.524 0.538193
\(342\) 0 0
\(343\) − 108.373i − 0.315955i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 109.320i − 0.315043i −0.987516 0.157522i \(-0.949650\pi\)
0.987516 0.157522i \(-0.0503504\pi\)
\(348\) 0 0
\(349\) −669.523 −1.91840 −0.959201 0.282724i \(-0.908762\pi\)
−0.959201 + 0.282724i \(0.908762\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −472.139 −1.33751 −0.668753 0.743485i \(-0.733171\pi\)
−0.668753 + 0.743485i \(0.733171\pi\)
\(354\) 0 0
\(355\) 828.692i 2.33434i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 148.271i − 0.413012i −0.978445 0.206506i \(-0.933791\pi\)
0.978445 0.206506i \(-0.0662094\pi\)
\(360\) 0 0
\(361\) −405.276 −1.12265
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −241.887 −0.662705
\(366\) 0 0
\(367\) − 248.314i − 0.676604i −0.941038 0.338302i \(-0.890147\pi\)
0.941038 0.338302i \(-0.109853\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.7526i 0.0397645i
\(372\) 0 0
\(373\) −168.343 −0.451322 −0.225661 0.974206i \(-0.572454\pi\)
−0.225661 + 0.974206i \(0.572454\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −53.2555 −0.141261
\(378\) 0 0
\(379\) − 517.108i − 1.36440i −0.731165 0.682201i \(-0.761023\pi\)
0.731165 0.682201i \(-0.238977\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 303.786i − 0.793175i −0.917997 0.396588i \(-0.870194\pi\)
0.917997 0.396588i \(-0.129806\pi\)
\(384\) 0 0
\(385\) 149.664 0.388738
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 114.745 0.294974 0.147487 0.989064i \(-0.452881\pi\)
0.147487 + 0.989064i \(0.452881\pi\)
\(390\) 0 0
\(391\) 348.436i 0.891141i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 612.056i 1.54951i
\(396\) 0 0
\(397\) −602.700 −1.51814 −0.759068 0.651011i \(-0.774345\pi\)
−0.759068 + 0.651011i \(0.774345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −336.618 −0.839447 −0.419724 0.907652i \(-0.637873\pi\)
−0.419724 + 0.907652i \(0.637873\pi\)
\(402\) 0 0
\(403\) 114.685i 0.284578i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 456.197i − 1.12088i
\(408\) 0 0
\(409\) −312.211 −0.763353 −0.381676 0.924296i \(-0.624653\pi\)
−0.381676 + 0.924296i \(0.624653\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 93.2617 0.225815
\(414\) 0 0
\(415\) 1074.83i 2.58996i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 40.3367i − 0.0962690i −0.998841 0.0481345i \(-0.984672\pi\)
0.998841 0.0481345i \(-0.0153276\pi\)
\(420\) 0 0
\(421\) −170.177 −0.404221 −0.202110 0.979363i \(-0.564780\pi\)
−0.202110 + 0.979363i \(0.564780\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 860.910 2.02567
\(426\) 0 0
\(427\) 113.602i 0.266046i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 172.213i − 0.399565i −0.979840 0.199783i \(-0.935976\pi\)
0.979840 0.199783i \(-0.0640236\pi\)
\(432\) 0 0
\(433\) −759.997 −1.75519 −0.877595 0.479403i \(-0.840853\pi\)
−0.877595 + 0.479403i \(0.840853\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 762.070 1.74387
\(438\) 0 0
\(439\) − 558.574i − 1.27238i −0.771534 0.636189i \(-0.780510\pi\)
0.771534 0.636189i \(-0.219490\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 857.477i 1.93562i 0.251691 + 0.967808i \(0.419013\pi\)
−0.251691 + 0.967808i \(0.580987\pi\)
\(444\) 0 0
\(445\) 216.309 0.486089
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 64.5889 0.143851 0.0719253 0.997410i \(-0.477086\pi\)
0.0719253 + 0.997410i \(0.477086\pi\)
\(450\) 0 0
\(451\) 834.634i 1.85063i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 93.5258i 0.205551i
\(456\) 0 0
\(457\) 446.676 0.977410 0.488705 0.872449i \(-0.337469\pi\)
0.488705 + 0.872449i \(0.337469\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 591.058 1.28212 0.641060 0.767491i \(-0.278495\pi\)
0.641060 + 0.767491i \(0.278495\pi\)
\(462\) 0 0
\(463\) 117.571i 0.253933i 0.991907 + 0.126967i \(0.0405241\pi\)
−0.991907 + 0.126967i \(0.959476\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.1484i 0.0559923i 0.999608 + 0.0279962i \(0.00891262\pi\)
−0.999608 + 0.0279962i \(0.991087\pi\)
\(468\) 0 0
\(469\) −125.467 −0.267521
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 351.142 0.742371
\(474\) 0 0
\(475\) − 1882.91i − 3.96402i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 342.053i − 0.714098i −0.934086 0.357049i \(-0.883783\pi\)
0.934086 0.357049i \(-0.116217\pi\)
\(480\) 0 0
\(481\) 285.079 0.592680
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 686.848 1.41618
\(486\) 0 0
\(487\) 58.9556i 0.121059i 0.998166 + 0.0605293i \(0.0192789\pi\)
−0.998166 + 0.0605293i \(0.980721\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 253.074i − 0.515426i −0.966222 0.257713i \(-0.917031\pi\)
0.966222 0.257713i \(-0.0829688\pi\)
\(492\) 0 0
\(493\) −77.8632 −0.157938
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −96.2487 −0.193659
\(498\) 0 0
\(499\) − 440.150i − 0.882063i −0.897492 0.441032i \(-0.854613\pi\)
0.897492 0.441032i \(-0.145387\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 299.245i − 0.594921i −0.954734 0.297461i \(-0.903860\pi\)
0.954734 0.297461i \(-0.0961397\pi\)
\(504\) 0 0
\(505\) 1335.55 2.64466
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 380.415 0.747378 0.373689 0.927554i \(-0.378093\pi\)
0.373689 + 0.927554i \(0.378093\pi\)
\(510\) 0 0
\(511\) − 28.0941i − 0.0549786i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1129.94i − 2.19406i
\(516\) 0 0
\(517\) −924.889 −1.78895
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −523.369 −1.00455 −0.502273 0.864709i \(-0.667503\pi\)
−0.502273 + 0.864709i \(0.667503\pi\)
\(522\) 0 0
\(523\) 641.527i 1.22663i 0.789839 + 0.613315i \(0.210164\pi\)
−0.789839 + 0.613315i \(0.789836\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 167.677i 0.318173i
\(528\) 0 0
\(529\) −228.887 −0.432680
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −521.566 −0.978548
\(534\) 0 0
\(535\) − 378.222i − 0.706956i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 661.407i − 1.22710i
\(540\) 0 0
\(541\) −366.964 −0.678307 −0.339154 0.940731i \(-0.610141\pi\)
−0.339154 + 0.940731i \(0.610141\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1482.73 2.72061
\(546\) 0 0
\(547\) − 550.755i − 1.00686i −0.864035 0.503432i \(-0.832070\pi\)
0.864035 0.503432i \(-0.167930\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 170.296i 0.309067i
\(552\) 0 0
\(553\) −71.0875 −0.128549
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −498.904 −0.895698 −0.447849 0.894109i \(-0.647810\pi\)
−0.447849 + 0.894109i \(0.647810\pi\)
\(558\) 0 0
\(559\) 219.430i 0.392540i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 823.441i 1.46260i 0.682058 + 0.731298i \(0.261085\pi\)
−0.682058 + 0.731298i \(0.738915\pi\)
\(564\) 0 0
\(565\) 1094.93 1.93793
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 372.094 0.653944 0.326972 0.945034i \(-0.393972\pi\)
0.326972 + 0.945034i \(0.393972\pi\)
\(570\) 0 0
\(571\) − 800.325i − 1.40162i −0.713348 0.700810i \(-0.752822\pi\)
0.713348 0.700810i \(-0.247178\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1872.58i 3.25665i
\(576\) 0 0
\(577\) 404.343 0.700768 0.350384 0.936606i \(-0.386051\pi\)
0.350384 + 0.936606i \(0.386051\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −124.837 −0.214866
\(582\) 0 0
\(583\) 182.439i 0.312932i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.7837i 0.0405173i 0.999795 + 0.0202586i \(0.00644897\pi\)
−0.999795 + 0.0202586i \(0.993551\pi\)
\(588\) 0 0
\(589\) 366.729 0.622631
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 568.416 0.958544 0.479272 0.877667i \(-0.340901\pi\)
0.479272 + 0.877667i \(0.340901\pi\)
\(594\) 0 0
\(595\) 136.741i 0.229817i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 244.953i − 0.408936i −0.978873 0.204468i \(-0.934454\pi\)
0.978873 0.204468i \(-0.0655465\pi\)
\(600\) 0 0
\(601\) −0.682820 −0.00113614 −0.000568070 1.00000i \(-0.500181\pi\)
−0.000568070 1.00000i \(0.500181\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 683.825 1.13029
\(606\) 0 0
\(607\) 1075.29i 1.77149i 0.464173 + 0.885744i \(0.346351\pi\)
−0.464173 + 0.885744i \(0.653649\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 577.967i − 0.945936i
\(612\) 0 0
\(613\) −284.127 −0.463503 −0.231752 0.972775i \(-0.574446\pi\)
−0.231752 + 0.972775i \(0.574446\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 740.574 1.20028 0.600141 0.799894i \(-0.295111\pi\)
0.600141 + 0.799894i \(0.295111\pi\)
\(618\) 0 0
\(619\) − 16.2690i − 0.0262827i −0.999914 0.0131413i \(-0.995817\pi\)
0.999914 0.0131413i \(-0.00418314\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 25.1233i 0.0403264i
\(624\) 0 0
\(625\) 2301.23 3.68197
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 416.806 0.662648
\(630\) 0 0
\(631\) 600.999i 0.952455i 0.879322 + 0.476227i \(0.157996\pi\)
−0.879322 + 0.476227i \(0.842004\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1160.28i 1.82721i
\(636\) 0 0
\(637\) 413.316 0.648848
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 733.908 1.14494 0.572471 0.819925i \(-0.305985\pi\)
0.572471 + 0.819925i \(0.305985\pi\)
\(642\) 0 0
\(643\) 269.442i 0.419039i 0.977804 + 0.209520i \(0.0671900\pi\)
−0.977804 + 0.209520i \(0.932810\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 685.853i − 1.06005i −0.847982 0.530026i \(-0.822182\pi\)
0.847982 0.530026i \(-0.177818\pi\)
\(648\) 0 0
\(649\) 1153.33 1.77708
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 264.611 0.405223 0.202612 0.979259i \(-0.435057\pi\)
0.202612 + 0.979259i \(0.435057\pi\)
\(654\) 0 0
\(655\) 580.742i 0.886629i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 798.019i − 1.21095i −0.795863 0.605477i \(-0.792982\pi\)
0.795863 0.605477i \(-0.207018\pi\)
\(660\) 0 0
\(661\) −395.442 −0.598248 −0.299124 0.954214i \(-0.596694\pi\)
−0.299124 + 0.954214i \(0.596694\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 299.069 0.449727
\(666\) 0 0
\(667\) − 169.361i − 0.253915i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1404.86i 2.09369i
\(672\) 0 0
\(673\) −36.0196 −0.0535209 −0.0267604 0.999642i \(-0.508519\pi\)
−0.0267604 + 0.999642i \(0.508519\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 107.190 0.158331 0.0791654 0.996861i \(-0.474774\pi\)
0.0791654 + 0.996861i \(0.474774\pi\)
\(678\) 0 0
\(679\) 79.7742i 0.117488i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 529.124i 0.774705i 0.921932 + 0.387353i \(0.126610\pi\)
−0.921932 + 0.387353i \(0.873390\pi\)
\(684\) 0 0
\(685\) 135.353 0.197595
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −114.007 −0.165467
\(690\) 0 0
\(691\) − 1134.76i − 1.64220i −0.570788 0.821098i \(-0.693362\pi\)
0.570788 0.821098i \(-0.306638\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 2258.08i − 3.24903i
\(696\) 0 0
\(697\) −762.566 −1.09407
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1092.79 1.55890 0.779451 0.626463i \(-0.215498\pi\)
0.779451 + 0.626463i \(0.215498\pi\)
\(702\) 0 0
\(703\) − 911.603i − 1.29673i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 155.118i 0.219403i
\(708\) 0 0
\(709\) −494.678 −0.697713 −0.348856 0.937176i \(-0.613430\pi\)
−0.348856 + 0.937176i \(0.613430\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −364.717 −0.511524
\(714\) 0 0
\(715\) 1156.59i 1.61761i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 431.893i − 0.600686i −0.953831 0.300343i \(-0.902899\pi\)
0.953831 0.300343i \(-0.0971011\pi\)
\(720\) 0 0
\(721\) 131.237 0.182021
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −418.455 −0.577179
\(726\) 0 0
\(727\) 273.895i 0.376747i 0.982097 + 0.188374i \(0.0603216\pi\)
−0.982097 + 0.188374i \(0.939678\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 320.822i 0.438880i
\(732\) 0 0
\(733\) 667.160 0.910177 0.455089 0.890446i \(-0.349608\pi\)
0.455089 + 0.890446i \(0.349608\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1551.60 −2.10529
\(738\) 0 0
\(739\) 202.851i 0.274494i 0.990537 + 0.137247i \(0.0438254\pi\)
−0.990537 + 0.137247i \(0.956175\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 23.4215i − 0.0315229i −0.999876 0.0157614i \(-0.994983\pi\)
0.999876 0.0157614i \(-0.00501723\pi\)
\(744\) 0 0
\(745\) −491.698 −0.659997
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 43.9287 0.0586498
\(750\) 0 0
\(751\) − 131.874i − 0.175598i −0.996138 0.0877988i \(-0.972017\pi\)
0.996138 0.0877988i \(-0.0279833\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 1533.46i − 2.03108i
\(756\) 0 0
\(757\) 771.905 1.01969 0.509845 0.860266i \(-0.329703\pi\)
0.509845 + 0.860266i \(0.329703\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −727.077 −0.955423 −0.477711 0.878517i \(-0.658534\pi\)
−0.477711 + 0.878517i \(0.658534\pi\)
\(762\) 0 0
\(763\) 172.212i 0.225704i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 720.719i 0.939660i
\(768\) 0 0
\(769\) −609.801 −0.792979 −0.396490 0.918039i \(-0.629772\pi\)
−0.396490 + 0.918039i \(0.629772\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1108.68 1.43426 0.717130 0.696939i \(-0.245455\pi\)
0.717130 + 0.696939i \(0.245455\pi\)
\(774\) 0 0
\(775\) 901.136i 1.16276i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1667.82i 2.14098i
\(780\) 0 0
\(781\) −1190.27 −1.52403
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −25.9113 −0.0330080
\(786\) 0 0
\(787\) − 822.206i − 1.04473i −0.852721 0.522367i \(-0.825049\pi\)
0.852721 0.522367i \(-0.174951\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 127.171i 0.160773i
\(792\) 0 0
\(793\) −877.906 −1.10707
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1310.57 −1.64438 −0.822189 0.569215i \(-0.807247\pi\)
−0.822189 + 0.569215i \(0.807247\pi\)
\(798\) 0 0
\(799\) − 845.028i − 1.05761i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 347.428i − 0.432662i
\(804\) 0 0
\(805\) −297.427 −0.369475
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −645.853 −0.798335 −0.399168 0.916878i \(-0.630701\pi\)
−0.399168 + 0.916878i \(0.630701\pi\)
\(810\) 0 0
\(811\) 1.06590i 0.00131430i 1.00000 0.000657151i \(0.000209178\pi\)
−1.00000 0.000657151i \(0.999791\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1414.48i − 1.73555i
\(816\) 0 0
\(817\) 701.674 0.858842
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 727.892 0.886592 0.443296 0.896375i \(-0.353809\pi\)
0.443296 + 0.896375i \(0.353809\pi\)
\(822\) 0 0
\(823\) − 197.736i − 0.240262i −0.992758 0.120131i \(-0.961669\pi\)
0.992758 0.120131i \(-0.0383315\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 287.442i − 0.347571i −0.984783 0.173786i \(-0.944400\pi\)
0.984783 0.173786i \(-0.0556000\pi\)
\(828\) 0 0
\(829\) −1080.80 −1.30374 −0.651871 0.758330i \(-0.726016\pi\)
−0.651871 + 0.758330i \(0.726016\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 604.297 0.725446
\(834\) 0 0
\(835\) 181.170i 0.216970i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 120.216i − 0.143284i −0.997430 0.0716422i \(-0.977176\pi\)
0.997430 0.0716422i \(-0.0228240\pi\)
\(840\) 0 0
\(841\) −803.154 −0.954998
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 907.193 1.07360
\(846\) 0 0
\(847\) 79.4231i 0.0937699i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 906.599i 1.06533i
\(852\) 0 0
\(853\) −1270.01 −1.48888 −0.744438 0.667691i \(-0.767283\pi\)
−0.744438 + 0.667691i \(0.767283\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 76.0067 0.0886892 0.0443446 0.999016i \(-0.485880\pi\)
0.0443446 + 0.999016i \(0.485880\pi\)
\(858\) 0 0
\(859\) 424.136i 0.493756i 0.969047 + 0.246878i \(0.0794047\pi\)
−0.969047 + 0.246878i \(0.920595\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1036.76i 1.20134i 0.799495 + 0.600672i \(0.205100\pi\)
−0.799495 + 0.600672i \(0.794900\pi\)
\(864\) 0 0
\(865\) −1643.78 −1.90032
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −879.108 −1.01163
\(870\) 0 0
\(871\) − 969.600i − 1.11320i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 464.782i 0.531180i
\(876\) 0 0
\(877\) 41.6471 0.0474881 0.0237441 0.999718i \(-0.492441\pi\)
0.0237441 + 0.999718i \(0.492441\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1196.29 1.35788 0.678940 0.734194i \(-0.262440\pi\)
0.678940 + 0.734194i \(0.262440\pi\)
\(882\) 0 0
\(883\) − 106.778i − 0.120926i −0.998170 0.0604629i \(-0.980742\pi\)
0.998170 0.0604629i \(-0.0192577\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 216.134i 0.243668i 0.992550 + 0.121834i \(0.0388776\pi\)
−0.992550 + 0.121834i \(0.961122\pi\)
\(888\) 0 0
\(889\) −134.761 −0.151587
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1848.17 −2.06962
\(894\) 0 0
\(895\) − 1798.07i − 2.00901i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 81.5014i − 0.0906578i
\(900\) 0 0
\(901\) −166.686 −0.185001
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1117.69 1.23501
\(906\) 0 0
\(907\) 1672.90i 1.84444i 0.386670 + 0.922218i \(0.373625\pi\)
−0.386670 + 0.922218i \(0.626375\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 238.389i − 0.261678i −0.991404 0.130839i \(-0.958233\pi\)
0.991404 0.130839i \(-0.0417672\pi\)
\(912\) 0 0
\(913\) −1543.81 −1.69091
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −67.4505 −0.0735556
\(918\) 0 0
\(919\) 771.672i 0.839687i 0.907596 + 0.419844i \(0.137915\pi\)
−0.907596 + 0.419844i \(0.862085\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 743.802i − 0.805853i
\(924\) 0 0
\(925\) 2240.01 2.42163
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 97.4649 0.104914 0.0524569 0.998623i \(-0.483295\pi\)
0.0524569 + 0.998623i \(0.483295\pi\)
\(930\) 0 0
\(931\) − 1321.67i − 1.41962i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1691.02i 1.80858i
\(936\) 0 0
\(937\) −1399.93 −1.49406 −0.747028 0.664793i \(-0.768520\pi\)
−0.747028 + 0.664793i \(0.768520\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 253.338 0.269222 0.134611 0.990898i \(-0.457021\pi\)
0.134611 + 0.990898i \(0.457021\pi\)
\(942\) 0 0
\(943\) − 1658.67i − 1.75893i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 223.848i − 0.236376i −0.992991 0.118188i \(-0.962292\pi\)
0.992991 0.118188i \(-0.0377085\pi\)
\(948\) 0 0
\(949\) 217.109 0.228776
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 968.543 1.01631 0.508155 0.861266i \(-0.330328\pi\)
0.508155 + 0.861266i \(0.330328\pi\)
\(954\) 0 0
\(955\) 448.355i 0.469482i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 15.7206i 0.0163927i
\(960\) 0 0
\(961\) 785.488 0.817365
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −598.738 −0.620454
\(966\) 0 0
\(967\) 1198.55i 1.23945i 0.784818 + 0.619726i \(0.212756\pi\)
−0.784818 + 0.619726i \(0.787244\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1759.74i − 1.81230i −0.422961 0.906148i \(-0.639009\pi\)
0.422961 0.906148i \(-0.360991\pi\)
\(972\) 0 0
\(973\) 262.265 0.269543
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1502.13 −1.53749 −0.768746 0.639554i \(-0.779119\pi\)
−0.768746 + 0.639554i \(0.779119\pi\)
\(978\) 0 0
\(979\) 310.689i 0.317354i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 410.455i 0.417554i 0.977963 + 0.208777i \(0.0669483\pi\)
−0.977963 + 0.208777i \(0.933052\pi\)
\(984\) 0 0
\(985\) 946.811 0.961229
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −697.823 −0.705584
\(990\) 0 0
\(991\) 210.418i 0.212329i 0.994349 + 0.106164i \(0.0338570\pi\)
−0.994349 + 0.106164i \(0.966143\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1008.92i − 1.01399i
\(996\) 0 0
\(997\) −450.591 −0.451947 −0.225973 0.974134i \(-0.572556\pi\)
−0.225973 + 0.974134i \(0.572556\pi\)
\(998\) 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 1728.3.g.k.703.1 8
3.2 odd 2 1728.3.g.n.703.7 8
4.3 odd 2 inner 1728.3.g.k.703.2 8
8.3 odd 2 864.3.g.c.703.8 yes 8
8.5 even 2 864.3.g.c.703.7 yes 8
12.11 even 2 1728.3.g.n.703.8 8
24.5 odd 2 864.3.g.a.703.1 8
24.11 even 2 864.3.g.a.703.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.a.703.1 8 24.5 odd 2
864.3.g.a.703.2 yes 8 24.11 even 2
864.3.g.c.703.7 yes 8 8.5 even 2
864.3.g.c.703.8 yes 8 8.3 odd 2
1728.3.g.k.703.1 8 1.1 even 1 trivial
1728.3.g.k.703.2 8 4.3 odd 2 inner
1728.3.g.n.703.7 8 3.2 odd 2
1728.3.g.n.703.8 8 12.11 even 2