Properties

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

$q$-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 \(-0.670628\pi\)
0.510738 0.859736i \(-0.329372\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 13.3544i 1.21404i 0.794687 + 0.607020i \(0.207635\pi\)
−0.794687 + 0.607020i \(0.792365\pi\)
\(12\) 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 \(0.0749944\pi\)
−0.972374 + 0.233428i \(0.925006\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 33.5947i 1.46064i 0.683107 + 0.730319i \(0.260628\pi\)
−0.683107 + 0.730319i \(0.739372\pi\)
\(24\) 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 \(-0.996971\pi\)
0.999955 0.00951650i \(-0.00302924\pi\)
\(32\) 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 \(0.0884267\pi\)
−0.961661 + 0.274241i \(0.911573\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 24.4804i − 0.520861i −0.965493 0.260430i \(-0.916136\pi\)
0.965493 0.260430i \(-0.0838644\pi\)
\(48\) 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 \(-0.845618\pi\)
0.884673 0.466212i \(-0.154382\pi\)
\(60\) 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 \(0.267084\pi\)
−0.668155 + 0.744022i \(0.732916\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 71.2023i − 1.00285i −0.865201 0.501425i \(-0.832809\pi\)
0.865201 0.501425i \(-0.167191\pi\)
\(72\) 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 \(-0.759654\pi\)
0.728223 0.685340i \(-0.240346\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 100.556i − 1.21152i −0.795648 0.605760i \(-0.792869\pi\)
0.795648 0.605760i \(-0.207131\pi\)
\(84\) 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 \(-0.771931\pi\)
0.754109 0.656750i \(-0.228069\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 29.5388i 0.276063i 0.990428 + 0.138032i \(0.0440776\pi\)
−0.990428 + 0.138032i \(0.955922\pi\)
\(108\) 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 \(0.0507786\pi\)
−0.987303 + 0.158850i \(0.949221\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 136.966i 1.04555i 0.852472 + 0.522773i \(0.175102\pi\)
−0.852472 + 0.522773i \(0.824898\pi\)
\(132\) 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 \(-0.808243\pi\)
0.823966 0.566639i \(-0.191757\pi\)
\(140\) 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 \(0.0101711\pi\)
−0.999490 + 0.0319481i \(0.989829\pi\)
\(152\) 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 \(0.126418\pi\)
−0.922166 + 0.386794i \(0.873582\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 302.797i 1.81316i 0.422039 + 0.906578i \(0.361315\pi\)
−0.422039 + 0.906578i \(0.638685\pi\)
\(168\) 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 \(0.353634\pi\)
−0.443789 + 0.896131i \(0.646366\pi\)
\(180\) 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 \(0.112870\pi\)
−0.937788 + 0.347208i \(0.887130\pi\)
\(192\) 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 \(0.181216\pi\)
−0.842274 + 0.539049i \(0.818784\pi\)
\(200\) 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 \(0.177736\pi\)
−0.848118 + 0.529808i \(0.822264\pi\)
\(212\) 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 \(-0.776957\pi\)
0.764384 0.644761i \(-0.223043\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 167.933i 0.739793i 0.929073 + 0.369896i \(0.120607\pi\)
−0.929073 + 0.369896i \(0.879393\pi\)
\(228\) 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 \(-0.980661\pi\)
0.998155 0.0607188i \(-0.0193393\pi\)
\(240\) 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 \(0.0714804\pi\)
−0.974892 + 0.222680i \(0.928520\pi\)
\(252\) 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 \(-0.864698\pi\)
0.911012 0.412380i \(-0.135302\pi\)
\(264\) 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 \(0.171547\pi\)
−0.858258 + 0.513218i \(0.828453\pi\)
\(272\) 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 \(-0.859331\pi\)
0.903930 0.427680i \(-0.140669\pi\)
\(284\) 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 \(-0.939549\pi\)
0.982021 0.188774i \(-0.0604515\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 394.948i 1.26993i 0.772541 + 0.634964i \(0.218985\pi\)
−0.772541 + 0.634964i \(0.781015\pi\)
\(312\) 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 \(-0.935686\pi\)
0.979657 0.200678i \(-0.0643144\pi\)
\(332\) 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 \(-0.914508\pi\)
0.964148 0.265364i \(-0.0854922\pi\)
\(348\) 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 \(0.125365\pi\)
−0.923440 + 0.383744i \(0.874635\pi\)
\(360\) 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 \(-0.699182\pi\)
0.585705 0.810525i \(-0.300818\pi\)
\(368\) 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 \(0.189615\pi\)
−0.827760 + 0.561083i \(0.810385\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 386.549i − 1.00927i −0.863334 0.504634i \(-0.831628\pi\)
0.863334 0.504634i \(-0.168372\pi\)
\(384\) 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 \(0.240094\pi\)
−0.728766 + 0.684763i \(0.759906\pi\)
\(420\) 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 \(0.124556\pi\)
−0.924412 + 0.381394i \(0.875444\pi\)
\(432\) 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 \(-0.824844\pi\)
0.852385 0.522915i \(-0.175156\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 463.025i − 1.04520i −0.852577 0.522602i \(-0.824961\pi\)
0.852577 0.522602i \(-0.175039\pi\)
\(444\) 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 \(-0.972897\pi\)
0.996377 0.0850452i \(-0.0271035\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 637.095i 1.36423i 0.731245 + 0.682114i \(0.238939\pi\)
−0.731245 + 0.682114i \(0.761061\pi\)
\(468\) 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 \(-0.932497\pi\)
0.977598 0.210480i \(-0.0675028\pi\)
\(480\) 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 \(0.0538814\pi\)
−0.985707 + 0.168466i \(0.946119\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.3778i 0.0618692i 0.999521 + 0.0309346i \(0.00984836\pi\)
−0.999521 + 0.0309346i \(0.990152\pi\)
\(492\) 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 \(0.0120713\pi\)
−0.999281 + 0.0379140i \(0.987929\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 252.844i 0.502672i 0.967900 + 0.251336i \(0.0808699\pi\)
−0.967900 + 0.251336i \(0.919130\pi\)
\(504\) 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 \(0.0593209\pi\)
−0.982685 + 0.185285i \(0.940679\pi\)
\(524\) 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 \(-0.712255\pi\)
0.618490 0.785793i \(-0.287745\pi\)
\(548\) 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 \(0.0290084\pi\)
−0.995850 + 0.0910064i \(0.970992\pi\)
\(564\) 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 \(0.280526\pi\)
−0.636149 + 0.771566i \(0.719474\pi\)
\(572\) 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 \(-0.950589\pi\)
0.987976 0.154605i \(-0.0494106\pi\)
\(588\) 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 \(0.0258399\pi\)
−0.996707 + 0.0810893i \(0.974160\pi\)
\(600\) 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 \(-0.800721\pi\)
0.810345 0.585953i \(-0.199279\pi\)
\(608\) 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 \(-0.977414\pi\)
0.997484 0.0708962i \(-0.0225859\pi\)
\(620\) 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 \(-0.961907\pi\)
0.992848 0.119386i \(-0.0380927\pi\)
\(632\) 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 \(0.100791\pi\)
−0.950285 + 0.311381i \(0.899209\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 253.680i 0.392086i 0.980595 + 0.196043i \(0.0628092\pi\)
−0.980595 + 0.196043i \(0.937191\pi\)
\(648\) 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 \(0.117371\pi\)
−0.932785 + 0.360434i \(0.882629\pi\)
\(660\) 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 \(-0.721220\pi\)
0.640373 0.768064i \(-0.278780\pi\)
\(684\) 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 \(0.270884\pi\)
−0.659225 + 0.751946i \(0.729116\pi\)
\(692\) 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 \(0.181303\pi\)
−0.842128 + 0.539278i \(0.818697\pi\)
\(720\) 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 \(-0.797844\pi\)
0.805017 0.593252i \(-0.202156\pi\)
\(728\) 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 \(0.303539\pi\)
−0.578755 + 0.815502i \(0.696461\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 762.230i − 1.02588i −0.858424 0.512941i \(-0.828556\pi\)
0.858424 0.512941i \(-0.171444\pi\)
\(744\) 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 \(0.0259298\pi\)
−0.996684 + 0.0813709i \(0.974070\pi\)
\(752\) 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 \(-0.810722\pi\)
0.828354 0.560205i \(-0.189278\pi\)
\(788\) 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 \(-0.736806\pi\)
0.677198 0.735801i \(-0.263194\pi\)
\(812\) 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 \(0.248696\pi\)
−0.709999 + 0.704203i \(0.751304\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 160.002i 0.193473i 0.995310 + 0.0967364i \(0.0308404\pi\)
−0.995310 + 0.0967364i \(0.969160\pi\)
\(828\) 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 \(-0.983891\pi\)
0.998720 0.0505853i \(-0.0161087\pi\)
\(840\) 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 \(0.0320794\pi\)
−0.994926 + 0.100610i \(0.967921\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 66.7620i − 0.0773604i −0.999252 0.0386802i \(-0.987685\pi\)
0.999252 0.0386802i \(-0.0123154\pi\)
\(864\) 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 \(-0.606139\pi\)
0.327299 0.944921i \(-0.393861\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1155.43i 1.30262i 0.758810 + 0.651312i \(0.225781\pi\)
−0.758810 + 0.651312i \(0.774219\pi\)
\(888\) 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 \(0.0404618\pi\)
−0.991932 + 0.126772i \(0.959538\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 20.1180i − 0.0220835i −0.999939 0.0110417i \(-0.996485\pi\)
0.999939 0.0110417i \(-0.00351476\pi\)
\(912\) 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 \(-0.897672\pi\)
0.948771 0.315964i \(-0.102328\pi\)
\(920\) 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 \(0.0719294\pi\)
−0.974577 + 0.224055i \(0.928071\pi\)
\(948\) 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 \(0.115053\pi\)
−0.935386 + 0.353629i \(0.884947\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 983.197i − 1.01256i −0.862369 0.506281i \(-0.831020\pi\)
0.862369 0.506281i \(-0.168980\pi\)
\(972\) 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 \(0.326700\pi\)
−0.517936 + 0.855419i \(0.673300\pi\)
\(984\) 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 \(0.0130965\pi\)
−0.999154 + 0.0411323i \(0.986903\pi\)
\(992\) 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.703.3 8
3.2 odd 2 1728.3.g.n.703.5 8
4.3 odd 2 inner 1728.3.g.k.703.4 8
8.3 odd 2 864.3.g.c.703.6 yes 8
8.5 even 2 864.3.g.c.703.5 yes 8
12.11 even 2 1728.3.g.n.703.6 8
24.5 odd 2 864.3.g.a.703.3 8
24.11 even 2 864.3.g.a.703.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.a.703.3 8 24.5 odd 2
864.3.g.a.703.4 yes 8 24.11 even 2
864.3.g.c.703.5 yes 8 8.5 even 2
864.3.g.c.703.6 yes 8 8.3 odd 2
1728.3.g.k.703.3 8 1.1 even 1 trivial
1728.3.g.k.703.4 8 4.3 odd 2 inner
1728.3.g.n.703.5 8 3.2 odd 2
1728.3.g.n.703.6 8 12.11 even 2