Properties

Label 1728.3.g.k.703.4
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 703.4
Root \(-1.27597 - 0.609843i\) of defining polynomial
Character \(\chi\) \(=\) 1728.703
Dual form 1728.3.g.k.703.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.76102 q^{5} +12.0363i q^{7} +O(q^{10})\) \(q-1.76102 q^{5} +12.0363i q^{7} -13.3544i q^{11} -8.15828 q^{13} -4.15828 q^{17} -8.87027i q^{19} -33.5947i q^{23} -21.8988 q^{25} +36.4649 q^{29} +0.590023i q^{31} -21.1962i q^{35} +69.8156 q^{37} +57.5661 q^{41} -23.5847i q^{43} +24.4804i q^{47} -95.8727 q^{49} -29.9292 q^{53} +23.5174i q^{55} +55.0130i q^{59} +11.1645 q^{61} +14.3669 q^{65} -99.6989i q^{67} +71.2023i q^{71} +36.0508 q^{73} +160.738 q^{77} +108.284i q^{79} +100.556i q^{83} +7.32280 q^{85} +159.616 q^{89} -98.1956i q^{91} +15.6207i q^{95} -24.0242 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) 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
\(5\) −1.76102 −0.352204 −0.176102 0.984372i \(-0.556349\pi\)
−0.176102 + 0.984372i \(0.556349\pi\)
\(6\) 0 0
\(7\) 12.0363i 1.71947i 0.510738 + 0.859736i \(0.329372\pi\)
−0.510738 + 0.859736i \(0.670628\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 13.3544i − 1.21404i −0.794687 0.607020i \(-0.792365\pi\)
0.794687 0.607020i \(-0.207635\pi\)
\(12\) 0 0
\(13\) −8.15828 −0.627560 −0.313780 0.949496i \(-0.601595\pi\)
−0.313780 + 0.949496i \(0.601595\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.15828 −0.244605 −0.122302 0.992493i \(-0.539028\pi\)
−0.122302 + 0.992493i \(0.539028\pi\)
\(18\) 0 0
\(19\) − 8.87027i − 0.466856i −0.972374 0.233428i \(-0.925006\pi\)
0.972374 0.233428i \(-0.0749944\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 33.5947i − 1.46064i −0.683107 0.730319i \(-0.739372\pi\)
0.683107 0.730319i \(-0.260628\pi\)
\(24\) 0 0
\(25\) −21.8988 −0.875953
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 36.4649 1.25741 0.628706 0.777643i \(-0.283585\pi\)
0.628706 + 0.777643i \(0.283585\pi\)
\(30\) 0 0
\(31\) 0.590023i 0.0190330i 0.999955 + 0.00951650i \(0.00302924\pi\)
−0.999955 + 0.00951650i \(0.996971\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 21.1962i − 0.605604i
\(36\) 0 0
\(37\) 69.8156 1.88691 0.943455 0.331502i \(-0.107555\pi\)
0.943455 + 0.331502i \(0.107555\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 57.5661 1.40405 0.702025 0.712152i \(-0.252279\pi\)
0.702025 + 0.712152i \(0.252279\pi\)
\(42\) 0 0
\(43\) − 23.5847i − 0.548482i −0.961661 0.274241i \(-0.911573\pi\)
0.961661 0.274241i \(-0.0884267\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 24.4804i 0.520861i 0.965493 + 0.260430i \(0.0838644\pi\)
−0.965493 + 0.260430i \(0.916136\pi\)
\(48\) 0 0
\(49\) −95.8727 −1.95659
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −29.9292 −0.564702 −0.282351 0.959311i \(-0.591114\pi\)
−0.282351 + 0.959311i \(0.591114\pi\)
\(54\) 0 0
\(55\) 23.5174i 0.427589i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 55.0130i 0.932424i 0.884673 + 0.466212i \(0.154382\pi\)
−0.884673 + 0.466212i \(0.845618\pi\)
\(60\) 0 0
\(61\) 11.1645 0.183025 0.0915125 0.995804i \(-0.470830\pi\)
0.0915125 + 0.995804i \(0.470830\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.3669 0.221029
\(66\) 0 0
\(67\) − 99.6989i − 1.48804i −0.668155 0.744022i \(-0.732916\pi\)
0.668155 0.744022i \(-0.267084\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 71.2023i 1.00285i 0.865201 + 0.501425i \(0.167191\pi\)
−0.865201 + 0.501425i \(0.832809\pi\)
\(72\) 0 0
\(73\) 36.0508 0.493847 0.246924 0.969035i \(-0.420580\pi\)
0.246924 + 0.969035i \(0.420580\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 160.738 2.08751
\(78\) 0 0
\(79\) 108.284i 1.37068i 0.728223 + 0.685340i \(0.240346\pi\)
−0.728223 + 0.685340i \(0.759654\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 100.556i 1.21152i 0.795648 + 0.605760i \(0.207131\pi\)
−0.795648 + 0.605760i \(0.792869\pi\)
\(84\) 0 0
\(85\) 7.32280 0.0861506
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 159.616 1.79344 0.896721 0.442596i \(-0.145942\pi\)
0.896721 + 0.442596i \(0.145942\pi\)
\(90\) 0 0
\(91\) − 98.1956i − 1.07907i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.6207i 0.164428i
\(96\) 0 0
\(97\) −24.0242 −0.247673 −0.123836 0.992303i \(-0.539520\pi\)
−0.123836 + 0.992303i \(0.539520\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 22.5078 0.222850 0.111425 0.993773i \(-0.464459\pi\)
0.111425 + 0.993773i \(0.464459\pi\)
\(102\) 0 0
\(103\) 135.290i 1.31350i 0.754109 + 0.656750i \(0.228069\pi\)
−0.754109 + 0.656750i \(0.771931\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 29.5388i − 0.276063i −0.990428 0.138032i \(-0.955922\pi\)
0.990428 0.138032i \(-0.0440776\pi\)
\(108\) 0 0
\(109\) 130.246 1.19492 0.597460 0.801898i \(-0.296176\pi\)
0.597460 + 0.801898i \(0.296176\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −134.421 −1.18957 −0.594785 0.803885i \(-0.702763\pi\)
−0.594785 + 0.803885i \(0.702763\pi\)
\(114\) 0 0
\(115\) 59.1608i 0.514442i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 50.0503i − 0.420591i
\(120\) 0 0
\(121\) −57.3408 −0.473891
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 82.5896 0.660717
\(126\) 0 0
\(127\) − 40.3479i − 0.317700i −0.987303 0.158850i \(-0.949221\pi\)
0.987303 0.158850i \(-0.0507786\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 136.966i − 1.04555i −0.852472 0.522773i \(-0.824898\pi\)
0.852472 0.522773i \(-0.175102\pi\)
\(132\) 0 0
\(133\) 106.765 0.802747
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 245.969 1.79539 0.897697 0.440614i \(-0.145239\pi\)
0.897697 + 0.440614i \(0.145239\pi\)
\(138\) 0 0
\(139\) 157.526i 1.13328i 0.823966 + 0.566639i \(0.191757\pi\)
−0.823966 + 0.566639i \(0.808243\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 108.949i 0.761882i
\(144\) 0 0
\(145\) −64.2154 −0.442865
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 184.731 1.23981 0.619904 0.784678i \(-0.287172\pi\)
0.619904 + 0.784678i \(0.287172\pi\)
\(150\) 0 0
\(151\) − 9.64832i − 0.0638961i −0.999490 0.0319481i \(-0.989829\pi\)
0.999490 0.0319481i \(-0.0101711\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 1.03904i − 0.00670349i
\(156\) 0 0
\(157\) 36.3166 0.231316 0.115658 0.993289i \(-0.463102\pi\)
0.115658 + 0.993289i \(0.463102\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 404.356 2.51153
\(162\) 0 0
\(163\) − 126.095i − 0.773588i −0.922166 0.386794i \(-0.873582\pi\)
0.922166 0.386794i \(-0.126418\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 302.797i − 1.81316i −0.422039 0.906578i \(-0.638685\pi\)
0.422039 0.906578i \(-0.361315\pi\)
\(168\) 0 0
\(169\) −102.443 −0.606169
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 204.895 1.18436 0.592181 0.805805i \(-0.298267\pi\)
0.592181 + 0.805805i \(0.298267\pi\)
\(174\) 0 0
\(175\) − 263.581i − 1.50618i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 320.815i − 1.79226i −0.443789 0.896131i \(-0.646366\pi\)
0.443789 0.896131i \(-0.353634\pi\)
\(180\) 0 0
\(181\) −285.364 −1.57660 −0.788299 0.615292i \(-0.789038\pi\)
−0.788299 + 0.615292i \(0.789038\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −122.947 −0.664576
\(186\) 0 0
\(187\) 55.5314i 0.296960i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 132.634i − 0.694417i −0.937788 0.347208i \(-0.887130\pi\)
0.937788 0.347208i \(-0.112870\pi\)
\(192\) 0 0
\(193\) 366.794 1.90049 0.950245 0.311504i \(-0.100833\pi\)
0.950245 + 0.311504i \(0.100833\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.3110 −0.0726445 −0.0363223 0.999340i \(-0.511564\pi\)
−0.0363223 + 0.999340i \(0.511564\pi\)
\(198\) 0 0
\(199\) − 214.542i − 1.07810i −0.842274 0.539049i \(-0.818784\pi\)
0.842274 0.539049i \(-0.181216\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 438.903i 2.16208i
\(204\) 0 0
\(205\) −101.375 −0.494512
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −118.457 −0.566782
\(210\) 0 0
\(211\) − 223.579i − 1.05962i −0.848118 0.529808i \(-0.822264\pi\)
0.848118 0.529808i \(-0.177736\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 41.5331i 0.193177i
\(216\) 0 0
\(217\) −7.10170 −0.0327267
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 33.9244 0.153504
\(222\) 0 0
\(223\) 287.564i 1.28952i 0.764384 + 0.644761i \(0.223043\pi\)
−0.764384 + 0.644761i \(0.776957\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 167.933i − 0.739793i −0.929073 0.369896i \(-0.879393\pi\)
0.929073 0.369896i \(-0.120607\pi\)
\(228\) 0 0
\(229\) −28.4693 −0.124320 −0.0621601 0.998066i \(-0.519799\pi\)
−0.0621601 + 0.998066i \(0.519799\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 158.334 0.679545 0.339773 0.940508i \(-0.389650\pi\)
0.339773 + 0.940508i \(0.389650\pi\)
\(234\) 0 0
\(235\) − 43.1105i − 0.183449i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 29.0236i 0.121438i 0.998155 + 0.0607188i \(0.0193393\pi\)
−0.998155 + 0.0607188i \(0.980661\pi\)
\(240\) 0 0
\(241\) 301.579 1.25137 0.625683 0.780078i \(-0.284820\pi\)
0.625683 + 0.780078i \(0.284820\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 168.834 0.689117
\(246\) 0 0
\(247\) 72.3661i 0.292980i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 111.785i − 0.445359i −0.974892 0.222680i \(-0.928520\pi\)
0.974892 0.222680i \(-0.0714804\pi\)
\(252\) 0 0
\(253\) −448.637 −1.77327
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 462.857 1.80100 0.900500 0.434856i \(-0.143201\pi\)
0.900500 + 0.434856i \(0.143201\pi\)
\(258\) 0 0
\(259\) 840.323i 3.24449i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 216.912i 0.824759i 0.911012 + 0.412380i \(0.135302\pi\)
−0.911012 + 0.412380i \(0.864698\pi\)
\(264\) 0 0
\(265\) 52.7059 0.198890
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −431.963 −1.60581 −0.802906 0.596106i \(-0.796714\pi\)
−0.802906 + 0.596106i \(0.796714\pi\)
\(270\) 0 0
\(271\) − 278.164i − 1.02644i −0.858258 0.513218i \(-0.828453\pi\)
0.858258 0.513218i \(-0.171547\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 292.446i 1.06344i
\(276\) 0 0
\(277\) −271.579 −0.980430 −0.490215 0.871602i \(-0.663082\pi\)
−0.490215 + 0.871602i \(0.663082\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −343.222 −1.22143 −0.610715 0.791850i \(-0.709118\pi\)
−0.610715 + 0.791850i \(0.709118\pi\)
\(282\) 0 0
\(283\) 242.067i 0.855361i 0.903930 + 0.427680i \(0.140669\pi\)
−0.903930 + 0.427680i \(0.859331\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 692.883i 2.41423i
\(288\) 0 0
\(289\) −271.709 −0.940169
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 471.585 1.60950 0.804752 0.593611i \(-0.202298\pi\)
0.804752 + 0.593611i \(0.202298\pi\)
\(294\) 0 0
\(295\) − 96.8789i − 0.328403i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 274.075i 0.916637i
\(300\) 0 0
\(301\) 283.873 0.943100
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −19.6609 −0.0644620
\(306\) 0 0
\(307\) 115.907i 0.377549i 0.982021 + 0.188774i \(0.0604515\pi\)
−0.982021 + 0.188774i \(0.939549\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 394.948i − 1.26993i −0.772541 0.634964i \(-0.781015\pi\)
0.772541 0.634964i \(-0.218985\pi\)
\(312\) 0 0
\(313\) 298.099 0.952394 0.476197 0.879339i \(-0.342015\pi\)
0.476197 + 0.879339i \(0.342015\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.5754 0.0491337 0.0245669 0.999698i \(-0.492179\pi\)
0.0245669 + 0.999698i \(0.492179\pi\)
\(318\) 0 0
\(319\) − 486.968i − 1.52655i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 36.8850i 0.114195i
\(324\) 0 0
\(325\) 178.657 0.549713
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −294.654 −0.895606
\(330\) 0 0
\(331\) 132.849i 0.401355i 0.979657 + 0.200678i \(0.0643144\pi\)
−0.979657 + 0.200678i \(0.935686\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 175.572i 0.524094i
\(336\) 0 0
\(337\) 64.2930 0.190781 0.0953903 0.995440i \(-0.469590\pi\)
0.0953903 + 0.995440i \(0.469590\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.87942 0.0231068
\(342\) 0 0
\(343\) − 564.175i − 1.64482i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 184.163i 0.530728i 0.964148 + 0.265364i \(0.0854922\pi\)
−0.964148 + 0.265364i \(0.914508\pi\)
\(348\) 0 0
\(349\) 107.326 0.307525 0.153763 0.988108i \(-0.450861\pi\)
0.153763 + 0.988108i \(0.450861\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 444.637 1.25960 0.629798 0.776759i \(-0.283138\pi\)
0.629798 + 0.776759i \(0.283138\pi\)
\(354\) 0 0
\(355\) − 125.389i − 0.353207i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 275.528i − 0.767488i −0.923440 0.383744i \(-0.874635\pi\)
0.923440 0.383744i \(-0.125365\pi\)
\(360\) 0 0
\(361\) 282.318 0.782045
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −63.4862 −0.173935
\(366\) 0 0
\(367\) 594.925i 1.62105i 0.585705 + 0.810525i \(0.300818\pi\)
−0.585705 + 0.810525i \(0.699182\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 360.237i − 0.970990i
\(372\) 0 0
\(373\) −478.902 −1.28392 −0.641960 0.766738i \(-0.721878\pi\)
−0.641960 + 0.766738i \(0.721878\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −297.491 −0.789101
\(378\) 0 0
\(379\) − 425.301i − 1.12217i −0.827760 0.561083i \(-0.810385\pi\)
0.827760 0.561083i \(-0.189615\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 386.549i 1.00927i 0.863334 + 0.504634i \(0.168372\pi\)
−0.863334 + 0.504634i \(0.831628\pi\)
\(384\) 0 0
\(385\) −283.063 −0.735227
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −186.277 −0.478861 −0.239430 0.970914i \(-0.576961\pi\)
−0.239430 + 0.970914i \(0.576961\pi\)
\(390\) 0 0
\(391\) 139.696i 0.357279i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 190.689i − 0.482758i
\(396\) 0 0
\(397\) −221.030 −0.556750 −0.278375 0.960472i \(-0.589796\pi\)
−0.278375 + 0.960472i \(0.589796\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −712.734 −1.77739 −0.888695 0.458498i \(-0.848388\pi\)
−0.888695 + 0.458498i \(0.848388\pi\)
\(402\) 0 0
\(403\) − 4.81357i − 0.0119443i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 932.348i − 2.29078i
\(408\) 0 0
\(409\) 198.746 0.485933 0.242966 0.970035i \(-0.421880\pi\)
0.242966 + 0.970035i \(0.421880\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −662.154 −1.60328
\(414\) 0 0
\(415\) − 177.081i − 0.426701i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 573.831i − 1.36953i −0.728766 0.684763i \(-0.759906\pi\)
0.728766 0.684763i \(-0.240094\pi\)
\(420\) 0 0
\(421\) 83.4289 0.198168 0.0990842 0.995079i \(-0.468409\pi\)
0.0990842 + 0.995079i \(0.468409\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 91.0614 0.214262
\(426\) 0 0
\(427\) 134.380i 0.314706i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 328.762i − 0.762789i −0.924412 0.381394i \(-0.875444\pi\)
0.924412 0.381394i \(-0.124556\pi\)
\(432\) 0 0
\(433\) 445.063 1.02786 0.513930 0.857832i \(-0.328189\pi\)
0.513930 + 0.857832i \(0.328189\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −297.994 −0.681908
\(438\) 0 0
\(439\) 459.120i 1.04583i 0.852385 + 0.522915i \(0.175156\pi\)
−0.852385 + 0.522915i \(0.824844\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 463.025i 1.04520i 0.852577 + 0.522602i \(0.175039\pi\)
−0.852577 + 0.522602i \(0.824961\pi\)
\(444\) 0 0
\(445\) −281.087 −0.631657
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 271.385 0.604420 0.302210 0.953241i \(-0.402276\pi\)
0.302210 + 0.953241i \(0.402276\pi\)
\(450\) 0 0
\(451\) − 768.762i − 1.70457i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 172.924i 0.380053i
\(456\) 0 0
\(457\) 92.0749 0.201477 0.100738 0.994913i \(-0.467879\pi\)
0.100738 + 0.994913i \(0.467879\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −551.486 −1.19628 −0.598141 0.801391i \(-0.704094\pi\)
−0.598141 + 0.801391i \(0.704094\pi\)
\(462\) 0 0
\(463\) 78.7518i 0.170090i 0.996377 + 0.0850452i \(0.0271035\pi\)
−0.996377 + 0.0850452i \(0.972897\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 637.095i − 1.36423i −0.731245 0.682114i \(-0.761061\pi\)
0.731245 0.682114i \(-0.238939\pi\)
\(468\) 0 0
\(469\) 1200.01 2.55865
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −314.961 −0.665879
\(474\) 0 0
\(475\) 194.248i 0.408944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 201.640i 0.420961i 0.977598 + 0.210480i \(0.0675028\pi\)
−0.977598 + 0.210480i \(0.932497\pi\)
\(480\) 0 0
\(481\) −569.575 −1.18415
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 42.3071 0.0872312
\(486\) 0 0
\(487\) − 164.086i − 0.336932i −0.985707 0.168466i \(-0.946119\pi\)
0.985707 0.168466i \(-0.0538814\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 30.3778i − 0.0618692i −0.999521 0.0309346i \(-0.990152\pi\)
0.999521 0.0309346i \(-0.00984836\pi\)
\(492\) 0 0
\(493\) −151.631 −0.307569
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −857.013 −1.72437
\(498\) 0 0
\(499\) − 37.8381i − 0.0758279i −0.999281 0.0379140i \(-0.987929\pi\)
0.999281 0.0379140i \(-0.0120713\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 252.844i − 0.502672i −0.967900 0.251336i \(-0.919130\pi\)
0.967900 0.251336i \(-0.0808699\pi\)
\(504\) 0 0
\(505\) −39.6366 −0.0784884
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 38.9887 0.0765986 0.0382993 0.999266i \(-0.487806\pi\)
0.0382993 + 0.999266i \(0.487806\pi\)
\(510\) 0 0
\(511\) 433.919i 0.849157i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 238.249i − 0.462619i
\(516\) 0 0
\(517\) 326.922 0.632345
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −254.943 −0.489334 −0.244667 0.969607i \(-0.578679\pi\)
−0.244667 + 0.969607i \(0.578679\pi\)
\(522\) 0 0
\(523\) − 193.808i − 0.370571i −0.982685 0.185285i \(-0.940679\pi\)
0.982685 0.185285i \(-0.0593209\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 2.45348i − 0.00465556i
\(528\) 0 0
\(529\) −599.601 −1.13346
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −469.640 −0.881126
\(534\) 0 0
\(535\) 52.0183i 0.0972305i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1280.33i 2.37537i
\(540\) 0 0
\(541\) 519.448 0.960162 0.480081 0.877224i \(-0.340607\pi\)
0.480081 + 0.877224i \(0.340607\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −229.366 −0.420855
\(546\) 0 0
\(547\) 859.658i 1.57159i 0.618490 + 0.785793i \(0.287745\pi\)
−0.618490 + 0.785793i \(0.712255\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 323.454i − 0.587030i
\(552\) 0 0
\(553\) −1303.34 −2.35685
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 603.829 1.08407 0.542037 0.840354i \(-0.317653\pi\)
0.542037 + 0.840354i \(0.317653\pi\)
\(558\) 0 0
\(559\) 192.411i 0.344206i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 102.473i − 0.182013i −0.995850 0.0910064i \(-0.970992\pi\)
0.995850 0.0910064i \(-0.0290084\pi\)
\(564\) 0 0
\(565\) 236.718 0.418971
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −237.219 −0.416905 −0.208453 0.978032i \(-0.566843\pi\)
−0.208453 + 0.978032i \(0.566843\pi\)
\(570\) 0 0
\(571\) − 881.129i − 1.54313i −0.636149 0.771566i \(-0.719474\pi\)
0.636149 0.771566i \(-0.280526\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 735.683i 1.27945i
\(576\) 0 0
\(577\) −197.356 −0.342038 −0.171019 0.985268i \(-0.554706\pi\)
−0.171019 + 0.985268i \(0.554706\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1210.32 −2.08317
\(582\) 0 0
\(583\) 399.687i 0.685570i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 181.507i 0.309210i 0.987976 + 0.154605i \(0.0494106\pi\)
−0.987976 + 0.154605i \(0.950589\pi\)
\(588\) 0 0
\(589\) 5.23366 0.00888568
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −828.164 −1.39657 −0.698283 0.715822i \(-0.746052\pi\)
−0.698283 + 0.715822i \(0.746052\pi\)
\(594\) 0 0
\(595\) 88.1395i 0.148134i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 97.1450i − 0.162179i −0.996707 0.0810893i \(-0.974160\pi\)
0.996707 0.0810893i \(-0.0258399\pi\)
\(600\) 0 0
\(601\) −729.916 −1.21450 −0.607251 0.794510i \(-0.707728\pi\)
−0.607251 + 0.794510i \(0.707728\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 100.978 0.166906
\(606\) 0 0
\(607\) 711.346i 1.17191i 0.810345 + 0.585953i \(0.199279\pi\)
−0.810345 + 0.585953i \(0.800721\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 199.718i − 0.326871i
\(612\) 0 0
\(613\) −317.107 −0.517304 −0.258652 0.965971i \(-0.583278\pi\)
−0.258652 + 0.965971i \(0.583278\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 682.859 1.10674 0.553371 0.832935i \(-0.313341\pi\)
0.553371 + 0.832935i \(0.313341\pi\)
\(618\) 0 0
\(619\) 87.7695i 0.141792i 0.997484 + 0.0708962i \(0.0225859\pi\)
−0.997484 + 0.0708962i \(0.977414\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1921.19i 3.08378i
\(624\) 0 0
\(625\) 402.029 0.643246
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −290.313 −0.461547
\(630\) 0 0
\(631\) 150.666i 0.238773i 0.992848 + 0.119386i \(0.0380927\pi\)
−0.992848 + 0.119386i \(0.961907\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 71.0534i 0.111895i
\(636\) 0 0
\(637\) 782.156 1.22788
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −447.618 −0.698312 −0.349156 0.937065i \(-0.613532\pi\)
−0.349156 + 0.937065i \(0.613532\pi\)
\(642\) 0 0
\(643\) − 400.436i − 0.622762i −0.950285 0.311381i \(-0.899209\pi\)
0.950285 0.311381i \(-0.100791\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 253.680i − 0.392086i −0.980595 0.196043i \(-0.937191\pi\)
0.980595 0.196043i \(-0.0628092\pi\)
\(648\) 0 0
\(649\) 734.668 1.13200
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −719.542 −1.10190 −0.550951 0.834537i \(-0.685735\pi\)
−0.550951 + 0.834537i \(0.685735\pi\)
\(654\) 0 0
\(655\) 241.200i 0.368245i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 475.052i − 0.720868i −0.932785 0.360434i \(-0.882629\pi\)
0.932785 0.360434i \(-0.117371\pi\)
\(660\) 0 0
\(661\) −132.074 −0.199809 −0.0999046 0.994997i \(-0.531854\pi\)
−0.0999046 + 0.994997i \(0.531854\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −188.016 −0.282730
\(666\) 0 0
\(667\) − 1225.03i − 1.83662i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 149.096i − 0.222199i
\(672\) 0 0
\(673\) 919.525 1.36631 0.683154 0.730275i \(-0.260608\pi\)
0.683154 + 0.730275i \(0.260608\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −200.276 −0.295829 −0.147914 0.989000i \(-0.547256\pi\)
−0.147914 + 0.989000i \(0.547256\pi\)
\(678\) 0 0
\(679\) − 289.163i − 0.425866i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1049.18i 1.53613i 0.640373 + 0.768064i \(0.278780\pi\)
−0.640373 + 0.768064i \(0.721220\pi\)
\(684\) 0 0
\(685\) −433.156 −0.632344
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 244.171 0.354384
\(690\) 0 0
\(691\) − 1039.19i − 1.50389i −0.659225 0.751946i \(-0.729116\pi\)
0.659225 0.751946i \(-0.270884\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 277.405i − 0.399145i
\(696\) 0 0
\(697\) −239.376 −0.343437
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 903.635 1.28907 0.644533 0.764576i \(-0.277052\pi\)
0.644533 + 0.764576i \(0.277052\pi\)
\(702\) 0 0
\(703\) − 619.284i − 0.880915i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 270.911i 0.383184i
\(708\) 0 0
\(709\) −990.096 −1.39647 −0.698234 0.715869i \(-0.746031\pi\)
−0.698234 + 0.715869i \(0.746031\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 19.8216 0.0278003
\(714\) 0 0
\(715\) − 191.861i − 0.268338i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 775.482i − 1.07856i −0.842128 0.539278i \(-0.818697\pi\)
0.842128 0.539278i \(-0.181303\pi\)
\(720\) 0 0
\(721\) −1628.40 −2.25853
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −798.539 −1.10143
\(726\) 0 0
\(727\) 862.588i 1.18650i 0.805017 + 0.593252i \(0.202156\pi\)
−0.805017 + 0.593252i \(0.797844\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 98.0719i 0.134161i
\(732\) 0 0
\(733\) 1013.12 1.38215 0.691075 0.722783i \(-0.257137\pi\)
0.691075 + 0.722783i \(0.257137\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1331.42 −1.80654
\(738\) 0 0
\(739\) − 1205.31i − 1.63100i −0.578755 0.815502i \(-0.696461\pi\)
0.578755 0.815502i \(-0.303539\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 762.230i 1.02588i 0.858424 + 0.512941i \(0.171444\pi\)
−0.858424 + 0.512941i \(0.828556\pi\)
\(744\) 0 0
\(745\) −325.315 −0.436664
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 355.538 0.474684
\(750\) 0 0
\(751\) − 122.219i − 0.162742i −0.996684 0.0813709i \(-0.974070\pi\)
0.996684 0.0813709i \(-0.0259298\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.9909i 0.0225044i
\(756\) 0 0
\(757\) 279.583 0.369330 0.184665 0.982802i \(-0.440880\pi\)
0.184665 + 0.982802i \(0.440880\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 250.171 0.328740 0.164370 0.986399i \(-0.447441\pi\)
0.164370 + 0.986399i \(0.447441\pi\)
\(762\) 0 0
\(763\) 1567.69i 2.05463i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 448.812i − 0.585152i
\(768\) 0 0
\(769\) 993.027 1.29132 0.645662 0.763624i \(-0.276582\pi\)
0.645662 + 0.763624i \(0.276582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −631.978 −0.817565 −0.408782 0.912632i \(-0.634047\pi\)
−0.408782 + 0.912632i \(0.634047\pi\)
\(774\) 0 0
\(775\) − 12.9208i − 0.0166720i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 510.627i − 0.655490i
\(780\) 0 0
\(781\) 950.867 1.21750
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −63.9541 −0.0814702
\(786\) 0 0
\(787\) 881.763i 1.12041i 0.828354 + 0.560205i \(0.189278\pi\)
−0.828354 + 0.560205i \(0.810722\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1617.94i − 2.04543i
\(792\) 0 0
\(793\) −91.0833 −0.114859
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −442.306 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(798\) 0 0
\(799\) − 101.796i − 0.127405i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 481.439i − 0.599550i
\(804\) 0 0
\(805\) −712.077 −0.884568
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −665.367 −0.822456 −0.411228 0.911533i \(-0.634900\pi\)
−0.411228 + 0.911533i \(0.634900\pi\)
\(810\) 0 0
\(811\) 1193.47i 1.47160i 0.677198 + 0.735801i \(0.263194\pi\)
−0.677198 + 0.735801i \(0.736806\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 222.055i 0.272461i
\(816\) 0 0
\(817\) −209.203 −0.256062
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −359.355 −0.437704 −0.218852 0.975758i \(-0.570231\pi\)
−0.218852 + 0.975758i \(0.570231\pi\)
\(822\) 0 0
\(823\) − 1159.12i − 1.40841i −0.709999 0.704203i \(-0.751304\pi\)
0.709999 0.704203i \(-0.248696\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 160.002i − 0.193473i −0.995310 0.0967364i \(-0.969160\pi\)
0.995310 0.0967364i \(-0.0308404\pi\)
\(828\) 0 0
\(829\) −992.384 −1.19709 −0.598543 0.801091i \(-0.704253\pi\)
−0.598543 + 0.801091i \(0.704253\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 398.665 0.478590
\(834\) 0 0
\(835\) 533.231i 0.638600i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 84.8821i 0.101171i 0.998720 + 0.0505853i \(0.0161087\pi\)
−0.998720 + 0.0505853i \(0.983891\pi\)
\(840\) 0 0
\(841\) 488.691 0.581083
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 180.403 0.213495
\(846\) 0 0
\(847\) − 690.172i − 0.814843i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2345.43i − 2.75609i
\(852\) 0 0
\(853\) 626.933 0.734974 0.367487 0.930029i \(-0.380218\pi\)
0.367487 + 0.930029i \(0.380218\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1560.76 1.82119 0.910594 0.413302i \(-0.135625\pi\)
0.910594 + 0.413302i \(0.135625\pi\)
\(858\) 0 0
\(859\) − 172.848i − 0.201220i −0.994926 0.100610i \(-0.967921\pi\)
0.994926 0.100610i \(-0.0320794\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 66.7620i 0.0773604i 0.999252 + 0.0386802i \(0.0123154\pi\)
−0.999252 + 0.0386802i \(0.987685\pi\)
\(864\) 0 0
\(865\) −360.823 −0.417136
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1446.07 1.66406
\(870\) 0 0
\(871\) 813.372i 0.933836i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 994.074i 1.13609i
\(876\) 0 0
\(877\) 903.899 1.03067 0.515336 0.856988i \(-0.327667\pi\)
0.515336 + 0.856988i \(0.327667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 514.281 0.583747 0.291873 0.956457i \(-0.405721\pi\)
0.291873 + 0.956457i \(0.405721\pi\)
\(882\) 0 0
\(883\) 1668.73i 1.88984i 0.327299 + 0.944921i \(0.393861\pi\)
−0.327299 + 0.944921i \(0.606139\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1155.43i − 1.30262i −0.758810 0.651312i \(-0.774219\pi\)
0.758810 0.651312i \(-0.225781\pi\)
\(888\) 0 0
\(889\) 485.640 0.546277
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 217.148 0.243167
\(894\) 0 0
\(895\) 564.961i 0.631241i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 21.5151i 0.0239323i
\(900\) 0 0
\(901\) 124.454 0.138129
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 502.531 0.555283
\(906\) 0 0
\(907\) − 229.965i − 0.253545i −0.991932 0.126772i \(-0.959538\pi\)
0.991932 0.126772i \(-0.0404618\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.1180i 0.0220835i 0.999939 + 0.0110417i \(0.00351476\pi\)
−0.999939 + 0.0110417i \(0.996485\pi\)
\(912\) 0 0
\(913\) 1342.87 1.47083
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1648.57 1.79779
\(918\) 0 0
\(919\) 580.742i 0.631929i 0.948771 + 0.315964i \(0.102328\pi\)
−0.948771 + 0.315964i \(0.897672\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 580.889i − 0.629348i
\(924\) 0 0
\(925\) −1528.88 −1.65284
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 375.347 0.404033 0.202017 0.979382i \(-0.435250\pi\)
0.202017 + 0.979382i \(0.435250\pi\)
\(930\) 0 0
\(931\) 850.417i 0.913445i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 97.7918i − 0.104590i
\(936\) 0 0
\(937\) 339.330 0.362145 0.181073 0.983470i \(-0.442043\pi\)
0.181073 + 0.983470i \(0.442043\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 459.867 0.488700 0.244350 0.969687i \(-0.421425\pi\)
0.244350 + 0.969687i \(0.421425\pi\)
\(942\) 0 0
\(943\) − 1933.91i − 2.05081i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 424.359i − 0.448109i −0.974577 0.224055i \(-0.928071\pi\)
0.974577 0.224055i \(-0.0719294\pi\)
\(948\) 0 0
\(949\) −294.113 −0.309919
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −140.791 −0.147734 −0.0738670 0.997268i \(-0.523534\pi\)
−0.0738670 + 0.997268i \(0.523534\pi\)
\(954\) 0 0
\(955\) 233.570i 0.244576i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2960.56i 3.08713i
\(960\) 0 0
\(961\) 960.652 0.999638
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −645.932 −0.669359
\(966\) 0 0
\(967\) − 683.919i − 0.707258i −0.935386 0.353629i \(-0.884947\pi\)
0.935386 0.353629i \(-0.115053\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 983.197i 1.01256i 0.862369 + 0.506281i \(0.168980\pi\)
−0.862369 + 0.506281i \(0.831020\pi\)
\(972\) 0 0
\(973\) −1896.03 −1.94864
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1008.88 1.03263 0.516315 0.856399i \(-0.327303\pi\)
0.516315 + 0.856399i \(0.327303\pi\)
\(978\) 0 0
\(979\) − 2131.59i − 2.17731i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1681.75i − 1.71084i −0.517936 0.855419i \(-0.673300\pi\)
0.517936 0.855419i \(-0.326700\pi\)
\(984\) 0 0
\(985\) 25.2019 0.0255857
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −792.321 −0.801134
\(990\) 0 0
\(991\) − 81.5243i − 0.0822647i −0.999154 0.0411323i \(-0.986903\pi\)
0.999154 0.0411323i \(-0.0130965\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 377.812i 0.379710i
\(996\) 0 0
\(997\) −549.116 −0.550768 −0.275384 0.961334i \(-0.588805\pi\)
−0.275384 + 0.961334i \(0.588805\pi\)
\(998\) 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.703.4 8
3.2 odd 2 1728.3.g.n.703.6 8
4.3 odd 2 inner 1728.3.g.k.703.3 8
8.3 odd 2 864.3.g.c.703.5 yes 8
8.5 even 2 864.3.g.c.703.6 yes 8
12.11 even 2 1728.3.g.n.703.5 8
24.5 odd 2 864.3.g.a.703.4 yes 8
24.11 even 2 864.3.g.a.703.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.a.703.3 8 24.11 even 2
864.3.g.a.703.4 yes 8 24.5 odd 2
864.3.g.c.703.5 yes 8 8.3 odd 2
864.3.g.c.703.6 yes 8 8.5 even 2
1728.3.g.k.703.3 8 4.3 odd 2 inner
1728.3.g.k.703.4 8 1.1 even 1 trivial
1728.3.g.n.703.5 8 12.11 even 2
1728.3.g.n.703.6 8 3.2 odd 2