Properties

Label 1728.3.g.k.703.5
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 703.5
Root \(1.40994 + 0.109843i\) of defining polynomial
Character \(\chi\) \(=\) 1728.703
Dual form 1728.3.g.k.703.6

$q$-expansion

\(f(q)\) \(=\) \(q+3.22512 q^{5} -6.57221i q^{7} +O(q^{10})\) \(q+3.22512 q^{5} -6.57221i q^{7} +13.8903i q^{11} +19.0865 q^{13} +23.0865 q^{17} -18.8425i q^{19} +13.5947i q^{23} -14.5986 q^{25} -0.752110 q^{29} +46.4433i q^{31} -21.1962i q^{35} +2.68178 q^{37} -34.1405 q^{41} -20.9127i q^{43} -15.4086i q^{47} +5.80609 q^{49} +46.8189 q^{53} +44.7980i q^{55} +40.4126i q^{59} +105.543 q^{61} +61.5562 q^{65} -27.9369i q^{67} +24.0130i q^{71} -120.117 q^{73} +91.2901 q^{77} +95.6394i q^{79} -115.446i q^{83} +74.4567 q^{85} +169.589 q^{89} -125.440i q^{91} -60.7694i q^{95} -93.1142 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) 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
\(5\) 3.22512 0.645024 0.322512 0.946565i \(-0.395473\pi\)
0.322512 + 0.946565i \(0.395473\pi\)
\(6\) 0 0
\(7\) − 6.57221i − 0.938887i −0.882963 0.469443i \(-0.844455\pi\)
0.882963 0.469443i \(-0.155545\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 13.8903i 1.26276i 0.775475 + 0.631379i \(0.217511\pi\)
−0.775475 + 0.631379i \(0.782489\pi\)
\(12\) 0 0
\(13\) 19.0865 1.46819 0.734095 0.679046i \(-0.237606\pi\)
0.734095 + 0.679046i \(0.237606\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 23.0865 1.35803 0.679014 0.734125i \(-0.262407\pi\)
0.679014 + 0.734125i \(0.262407\pi\)
\(18\) 0 0
\(19\) − 18.8425i − 0.991713i −0.868405 0.495856i \(-0.834854\pi\)
0.868405 0.495856i \(-0.165146\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 13.5947i 0.591072i 0.955332 + 0.295536i \(0.0954982\pi\)
−0.955332 + 0.295536i \(0.904502\pi\)
\(24\) 0 0
\(25\) −14.5986 −0.583944
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.752110 −0.0259348 −0.0129674 0.999916i \(-0.504128\pi\)
−0.0129674 + 0.999916i \(0.504128\pi\)
\(30\) 0 0
\(31\) 46.4433i 1.49817i 0.662473 + 0.749085i \(0.269507\pi\)
−0.662473 + 0.749085i \(0.730493\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 21.1962i − 0.605604i
\(36\) 0 0
\(37\) 2.68178 0.0724807 0.0362403 0.999343i \(-0.488462\pi\)
0.0362403 + 0.999343i \(0.488462\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −34.1405 −0.832694 −0.416347 0.909206i \(-0.636690\pi\)
−0.416347 + 0.909206i \(0.636690\pi\)
\(42\) 0 0
\(43\) − 20.9127i − 0.486341i −0.969984 0.243171i \(-0.921812\pi\)
0.969984 0.243171i \(-0.0781875\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 15.4086i − 0.327844i −0.986473 0.163922i \(-0.947586\pi\)
0.986473 0.163922i \(-0.0524145\pi\)
\(48\) 0 0
\(49\) 5.80609 0.118492
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 46.8189 0.883376 0.441688 0.897169i \(-0.354380\pi\)
0.441688 + 0.897169i \(0.354380\pi\)
\(54\) 0 0
\(55\) 44.7980i 0.814508i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 40.4126i 0.684959i 0.939525 + 0.342480i \(0.111267\pi\)
−0.939525 + 0.342480i \(0.888733\pi\)
\(60\) 0 0
\(61\) 105.543 1.73022 0.865108 0.501586i \(-0.167250\pi\)
0.865108 + 0.501586i \(0.167250\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 61.5562 0.947018
\(66\) 0 0
\(67\) − 27.9369i − 0.416969i −0.978026 0.208485i \(-0.933147\pi\)
0.978026 0.208485i \(-0.0668531\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 24.0130i 0.338212i 0.985598 + 0.169106i \(0.0540880\pi\)
−0.985598 + 0.169106i \(0.945912\pi\)
\(72\) 0 0
\(73\) −120.117 −1.64545 −0.822723 0.568443i \(-0.807546\pi\)
−0.822723 + 0.568443i \(0.807546\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 91.2901 1.18559
\(78\) 0 0
\(79\) 95.6394i 1.21062i 0.795988 + 0.605312i \(0.206952\pi\)
−0.795988 + 0.605312i \(0.793048\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 115.446i − 1.39091i −0.718568 0.695457i \(-0.755202\pi\)
0.718568 0.695457i \(-0.244798\pi\)
\(84\) 0 0
\(85\) 74.4567 0.875961
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 169.589 1.90549 0.952745 0.303770i \(-0.0982455\pi\)
0.952745 + 0.303770i \(0.0982455\pi\)
\(90\) 0 0
\(91\) − 125.440i − 1.37846i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 60.7694i − 0.639678i
\(96\) 0 0
\(97\) −93.1142 −0.959940 −0.479970 0.877285i \(-0.659352\pi\)
−0.479970 + 0.877285i \(0.659352\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 159.090 1.57514 0.787572 0.616222i \(-0.211338\pi\)
0.787572 + 0.616222i \(0.211338\pi\)
\(102\) 0 0
\(103\) − 75.3674i − 0.731722i −0.930669 0.365861i \(-0.880774\pi\)
0.930669 0.365861i \(-0.119226\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 69.4279i − 0.648859i −0.945910 0.324429i \(-0.894828\pi\)
0.945910 0.324429i \(-0.105172\pi\)
\(108\) 0 0
\(109\) −60.4669 −0.554742 −0.277371 0.960763i \(-0.589463\pi\)
−0.277371 + 0.960763i \(0.589463\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 140.698 1.24512 0.622559 0.782573i \(-0.286093\pi\)
0.622559 + 0.782573i \(0.286093\pi\)
\(114\) 0 0
\(115\) 43.8444i 0.381255i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 151.729i − 1.27503i
\(120\) 0 0
\(121\) −71.9412 −0.594556
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −127.710 −1.02168
\(126\) 0 0
\(127\) − 66.9726i − 0.527343i −0.964612 0.263672i \(-0.915066\pi\)
0.964612 0.263672i \(-0.0849335\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 255.148i 1.94770i 0.227196 + 0.973849i \(0.427044\pi\)
−0.227196 + 0.973849i \(0.572956\pi\)
\(132\) 0 0
\(133\) −123.837 −0.931106
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −27.1946 −0.198501 −0.0992505 0.995062i \(-0.531645\pi\)
−0.0992505 + 0.995062i \(0.531645\pi\)
\(138\) 0 0
\(139\) − 203.956i − 1.46731i −0.679521 0.733656i \(-0.737812\pi\)
0.679521 0.733656i \(-0.262188\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 265.117i 1.85397i
\(144\) 0 0
\(145\) −2.42564 −0.0167286
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 251.507 1.68797 0.843984 0.536369i \(-0.180204\pi\)
0.843984 + 0.536369i \(0.180204\pi\)
\(150\) 0 0
\(151\) 250.251i 1.65729i 0.559774 + 0.828645i \(0.310888\pi\)
−0.559774 + 0.828645i \(0.689112\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 149.785i 0.966356i
\(156\) 0 0
\(157\) −18.1730 −0.115751 −0.0578757 0.998324i \(-0.518433\pi\)
−0.0578757 + 0.998324i \(0.518433\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 89.3469 0.554950
\(162\) 0 0
\(163\) 50.0180i 0.306859i 0.988160 + 0.153429i \(0.0490318\pi\)
−0.988160 + 0.153429i \(0.950968\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 84.8389i − 0.508017i −0.967202 0.254009i \(-0.918251\pi\)
0.967202 0.254009i \(-0.0817492\pi\)
\(168\) 0 0
\(169\) 195.294 1.15558
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 144.703 0.836433 0.418216 0.908347i \(-0.362655\pi\)
0.418216 + 0.908347i \(0.362655\pi\)
\(174\) 0 0
\(175\) 95.9451i 0.548258i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 280.926i − 1.56942i −0.619864 0.784709i \(-0.712812\pi\)
0.619864 0.784709i \(-0.287188\pi\)
\(180\) 0 0
\(181\) 295.508 1.63264 0.816320 0.577600i \(-0.196011\pi\)
0.816320 + 0.577600i \(0.196011\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.64907 0.0467518
\(186\) 0 0
\(187\) 320.679i 1.71486i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 295.982i 1.54965i 0.632179 + 0.774823i \(0.282161\pi\)
−0.632179 + 0.774823i \(0.717839\pi\)
\(192\) 0 0
\(193\) −18.0202 −0.0933688 −0.0466844 0.998910i \(-0.514866\pi\)
−0.0466844 + 0.998910i \(0.514866\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −119.020 −0.604162 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(198\) 0 0
\(199\) − 304.912i − 1.53222i −0.642708 0.766111i \(-0.722189\pi\)
0.642708 0.766111i \(-0.277811\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.94302i 0.0243499i
\(204\) 0 0
\(205\) −110.107 −0.537108
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 261.729 1.25229
\(210\) 0 0
\(211\) 282.143i 1.33717i 0.743635 + 0.668585i \(0.233100\pi\)
−0.743635 + 0.668585i \(0.766900\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 67.4459i − 0.313702i
\(216\) 0 0
\(217\) 305.235 1.40661
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 440.640 1.99384
\(222\) 0 0
\(223\) − 91.9070i − 0.412139i −0.978537 0.206070i \(-0.933933\pi\)
0.978537 0.206070i \(-0.0660673\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 31.5125i 0.138822i 0.997588 + 0.0694108i \(0.0221119\pi\)
−0.997588 + 0.0694108i \(0.977888\pi\)
\(228\) 0 0
\(229\) −242.515 −1.05902 −0.529509 0.848304i \(-0.677624\pi\)
−0.529509 + 0.848304i \(0.677624\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 402.297 1.72660 0.863298 0.504695i \(-0.168395\pi\)
0.863298 + 0.504695i \(0.168395\pi\)
\(234\) 0 0
\(235\) − 49.6947i − 0.211467i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 167.034i − 0.698886i −0.936958 0.349443i \(-0.886371\pi\)
0.936958 0.349443i \(-0.113629\pi\)
\(240\) 0 0
\(241\) −21.4458 −0.0889868 −0.0444934 0.999010i \(-0.514167\pi\)
−0.0444934 + 0.999010i \(0.514167\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 18.7253 0.0764299
\(246\) 0 0
\(247\) − 359.638i − 1.45602i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.4825i 0.0895718i 0.998997 + 0.0447859i \(0.0142606\pi\)
−0.998997 + 0.0447859i \(0.985739\pi\)
\(252\) 0 0
\(253\) −188.834 −0.746380
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −437.652 −1.70293 −0.851463 0.524415i \(-0.824284\pi\)
−0.851463 + 0.524415i \(0.824284\pi\)
\(258\) 0 0
\(259\) − 17.6252i − 0.0680511i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 266.194i − 1.01214i −0.862491 0.506072i \(-0.831097\pi\)
0.862491 0.506072i \(-0.168903\pi\)
\(264\) 0 0
\(265\) 150.997 0.569799
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 220.671 0.820338 0.410169 0.912010i \(-0.365470\pi\)
0.410169 + 0.912010i \(0.365470\pi\)
\(270\) 0 0
\(271\) − 224.295i − 0.827656i −0.910355 0.413828i \(-0.864192\pi\)
0.910355 0.413828i \(-0.135808\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 202.779i − 0.737380i
\(276\) 0 0
\(277\) 51.4458 0.185725 0.0928625 0.995679i \(-0.470398\pi\)
0.0928625 + 0.995679i \(0.470398\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 40.1606 0.142920 0.0714602 0.997443i \(-0.477234\pi\)
0.0714602 + 0.997443i \(0.477234\pi\)
\(282\) 0 0
\(283\) 190.774i 0.674112i 0.941485 + 0.337056i \(0.109431\pi\)
−0.941485 + 0.337056i \(0.890569\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 224.378i 0.781806i
\(288\) 0 0
\(289\) 243.986 0.844241
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −115.872 −0.395467 −0.197733 0.980256i \(-0.563358\pi\)
−0.197733 + 0.980256i \(0.563358\pi\)
\(294\) 0 0
\(295\) 130.335i 0.441815i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 259.474i 0.867806i
\(300\) 0 0
\(301\) −137.442 −0.456619
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 340.389 1.11603
\(306\) 0 0
\(307\) − 497.554i − 1.62070i −0.585949 0.810348i \(-0.699278\pi\)
0.585949 0.810348i \(-0.300722\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 358.447i − 1.15256i −0.817252 0.576281i \(-0.804503\pi\)
0.817252 0.576281i \(-0.195497\pi\)
\(312\) 0 0
\(313\) 32.9519 0.105278 0.0526388 0.998614i \(-0.483237\pi\)
0.0526388 + 0.998614i \(0.483237\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −216.101 −0.681707 −0.340853 0.940117i \(-0.610716\pi\)
−0.340853 + 0.940117i \(0.610716\pi\)
\(318\) 0 0
\(319\) − 10.4471i − 0.0327494i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 435.008i − 1.34677i
\(324\) 0 0
\(325\) −278.636 −0.857342
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −101.269 −0.307808
\(330\) 0 0
\(331\) 578.213i 1.74687i 0.486944 + 0.873433i \(0.338112\pi\)
−0.486944 + 0.873433i \(0.661888\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 90.0999i − 0.268955i
\(336\) 0 0
\(337\) 358.117 1.06266 0.531331 0.847164i \(-0.321692\pi\)
0.531331 + 0.847164i \(0.321692\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −645.113 −1.89183
\(342\) 0 0
\(343\) − 360.197i − 1.05014i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 483.855i 1.39440i 0.716879 + 0.697198i \(0.245570\pi\)
−0.716879 + 0.697198i \(0.754430\pi\)
\(348\) 0 0
\(349\) −22.1213 −0.0633849 −0.0316924 0.999498i \(-0.510090\pi\)
−0.0316924 + 0.999498i \(0.510090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −459.068 −1.30048 −0.650238 0.759731i \(-0.725331\pi\)
−0.650238 + 0.759731i \(0.725331\pi\)
\(354\) 0 0
\(355\) 77.4449i 0.218155i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 414.323i 1.15410i 0.816708 + 0.577052i \(0.195797\pi\)
−0.816708 + 0.577052i \(0.804203\pi\)
\(360\) 0 0
\(361\) 5.95856 0.0165057
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −387.393 −1.06135
\(366\) 0 0
\(367\) 90.5390i 0.246700i 0.992363 + 0.123350i \(0.0393638\pi\)
−0.992363 + 0.123350i \(0.960636\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 307.704i − 0.829390i
\(372\) 0 0
\(373\) −223.011 −0.597884 −0.298942 0.954271i \(-0.596634\pi\)
−0.298942 + 0.954271i \(0.596634\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −14.3551 −0.0380773
\(378\) 0 0
\(379\) 118.162i 0.311774i 0.987775 + 0.155887i \(0.0498236\pi\)
−0.987775 + 0.155887i \(0.950176\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 19.6418i − 0.0512840i −0.999671 0.0256420i \(-0.991837\pi\)
0.999671 0.0256420i \(-0.00816300\pi\)
\(384\) 0 0
\(385\) 294.422 0.764731
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 352.392 0.905892 0.452946 0.891538i \(-0.350373\pi\)
0.452946 + 0.891538i \(0.350373\pi\)
\(390\) 0 0
\(391\) 313.853i 0.802692i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 308.448i 0.780882i
\(396\) 0 0
\(397\) 5.66047 0.0142581 0.00712906 0.999975i \(-0.497731\pi\)
0.00712906 + 0.999975i \(0.497731\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −250.097 −0.623683 −0.311842 0.950134i \(-0.600946\pi\)
−0.311842 + 0.950134i \(0.600946\pi\)
\(402\) 0 0
\(403\) 886.439i 2.19960i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 37.2509i 0.0915255i
\(408\) 0 0
\(409\) −643.885 −1.57429 −0.787145 0.616767i \(-0.788442\pi\)
−0.787145 + 0.616767i \(0.788442\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 265.600 0.643099
\(414\) 0 0
\(415\) − 372.327i − 0.897173i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 439.563i − 1.04908i −0.851387 0.524539i \(-0.824238\pi\)
0.851387 0.524539i \(-0.175762\pi\)
\(420\) 0 0
\(421\) −65.4392 −0.155438 −0.0777188 0.996975i \(-0.524764\pi\)
−0.0777188 + 0.996975i \(0.524764\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −337.030 −0.793013
\(426\) 0 0
\(427\) − 693.651i − 1.62448i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 299.824i 0.695646i 0.937560 + 0.347823i \(0.113079\pi\)
−0.937560 + 0.347823i \(0.886921\pi\)
\(432\) 0 0
\(433\) −699.217 −1.61482 −0.807410 0.589991i \(-0.799131\pi\)
−0.807410 + 0.589991i \(0.799131\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 256.158 0.586174
\(438\) 0 0
\(439\) 744.832i 1.69666i 0.529472 + 0.848328i \(0.322390\pi\)
−0.529472 + 0.848328i \(0.677610\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 320.933i 0.724455i 0.932090 + 0.362227i \(0.117984\pi\)
−0.932090 + 0.362227i \(0.882016\pi\)
\(444\) 0 0
\(445\) 546.944 1.22909
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −439.651 −0.979179 −0.489589 0.871953i \(-0.662853\pi\)
−0.489589 + 0.871953i \(0.662853\pi\)
\(450\) 0 0
\(451\) − 474.222i − 1.05149i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 404.560i − 0.889143i
\(456\) 0 0
\(457\) −732.044 −1.60185 −0.800923 0.598767i \(-0.795658\pi\)
−0.800923 + 0.598767i \(0.795658\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −381.075 −0.826627 −0.413314 0.910589i \(-0.635629\pi\)
−0.413314 + 0.910589i \(0.635629\pi\)
\(462\) 0 0
\(463\) − 591.775i − 1.27813i −0.769152 0.639066i \(-0.779321\pi\)
0.769152 0.639066i \(-0.220679\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 50.8002i 0.108780i 0.998520 + 0.0543899i \(0.0173214\pi\)
−0.998520 + 0.0543899i \(0.982679\pi\)
\(468\) 0 0
\(469\) −183.607 −0.391487
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 290.484 0.614131
\(474\) 0 0
\(475\) 275.075i 0.579105i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 844.826i 1.76373i 0.471503 + 0.881865i \(0.343712\pi\)
−0.471503 + 0.881865i \(0.656288\pi\)
\(480\) 0 0
\(481\) 51.1858 0.106415
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −300.304 −0.619184
\(486\) 0 0
\(487\) 328.276i 0.674077i 0.941491 + 0.337039i \(0.109425\pi\)
−0.941491 + 0.337039i \(0.890575\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 558.319i 1.13711i 0.822647 + 0.568553i \(0.192496\pi\)
−0.822647 + 0.568553i \(0.807504\pi\)
\(492\) 0 0
\(493\) −17.3636 −0.0352202
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 157.819 0.317543
\(498\) 0 0
\(499\) − 507.059i − 1.01615i −0.861313 0.508075i \(-0.830357\pi\)
0.861313 0.508075i \(-0.169643\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 884.817i − 1.75908i −0.475825 0.879540i \(-0.657850\pi\)
0.475825 0.879540i \(-0.342150\pi\)
\(504\) 0 0
\(505\) 513.083 1.01601
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −398.001 −0.781928 −0.390964 0.920406i \(-0.627858\pi\)
−0.390964 + 0.920406i \(0.627858\pi\)
\(510\) 0 0
\(511\) 789.437i 1.54489i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 243.069i − 0.471978i
\(516\) 0 0
\(517\) 214.031 0.413987
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −716.672 −1.37557 −0.687785 0.725914i \(-0.741417\pi\)
−0.687785 + 0.725914i \(0.741417\pi\)
\(522\) 0 0
\(523\) − 706.115i − 1.35012i −0.737761 0.675062i \(-0.764117\pi\)
0.737761 0.675062i \(-0.235883\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1072.21i 2.03456i
\(528\) 0 0
\(529\) 344.185 0.650634
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −651.621 −1.22255
\(534\) 0 0
\(535\) − 223.913i − 0.418529i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 80.6485i 0.149626i
\(540\) 0 0
\(541\) 116.644 0.215609 0.107804 0.994172i \(-0.465618\pi\)
0.107804 + 0.994172i \(0.465618\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −195.013 −0.357822
\(546\) 0 0
\(547\) − 422.278i − 0.771989i −0.922501 0.385995i \(-0.873858\pi\)
0.922501 0.385995i \(-0.126142\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 14.1717i 0.0257199i
\(552\) 0 0
\(553\) 628.562 1.13664
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −686.288 −1.23212 −0.616058 0.787701i \(-0.711271\pi\)
−0.616058 + 0.787701i \(0.711271\pi\)
\(558\) 0 0
\(559\) − 399.149i − 0.714042i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 439.191i − 0.780090i −0.920796 0.390045i \(-0.872459\pi\)
0.920796 0.390045i \(-0.127541\pi\)
\(564\) 0 0
\(565\) 453.769 0.803130
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −406.032 −0.713589 −0.356794 0.934183i \(-0.616130\pi\)
−0.356794 + 0.934183i \(0.616130\pi\)
\(570\) 0 0
\(571\) 166.816i 0.292148i 0.989274 + 0.146074i \(0.0466637\pi\)
−0.989274 + 0.146074i \(0.953336\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 198.463i − 0.345153i
\(576\) 0 0
\(577\) −897.895 −1.55614 −0.778072 0.628175i \(-0.783802\pi\)
−0.778072 + 0.628175i \(0.783802\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −758.734 −1.30591
\(582\) 0 0
\(583\) 650.330i 1.11549i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 152.207i − 0.259296i −0.991560 0.129648i \(-0.958615\pi\)
0.991560 0.129648i \(-0.0413847\pi\)
\(588\) 0 0
\(589\) 875.110 1.48576
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −964.580 −1.62661 −0.813305 0.581838i \(-0.802334\pi\)
−0.813305 + 0.581838i \(0.802334\pi\)
\(594\) 0 0
\(595\) − 489.345i − 0.822428i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 983.578i − 1.64203i −0.570905 0.821016i \(-0.693407\pi\)
0.570905 0.821016i \(-0.306593\pi\)
\(600\) 0 0
\(601\) −443.392 −0.737757 −0.368878 0.929478i \(-0.620258\pi\)
−0.368878 + 0.929478i \(0.620258\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −232.019 −0.383503
\(606\) 0 0
\(607\) 487.520i 0.803163i 0.915823 + 0.401582i \(0.131539\pi\)
−0.915823 + 0.401582i \(0.868461\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 294.097i − 0.481337i
\(612\) 0 0
\(613\) 909.815 1.48420 0.742100 0.670289i \(-0.233830\pi\)
0.742100 + 0.670289i \(0.233830\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −481.557 −0.780481 −0.390241 0.920713i \(-0.627608\pi\)
−0.390241 + 0.920713i \(0.627608\pi\)
\(618\) 0 0
\(619\) − 350.103i − 0.565594i −0.959180 0.282797i \(-0.908738\pi\)
0.959180 0.282797i \(-0.0912623\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 1114.57i − 1.78904i
\(624\) 0 0
\(625\) −46.9156 −0.0750649
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 61.9130 0.0984308
\(630\) 0 0
\(631\) − 274.658i − 0.435274i −0.976030 0.217637i \(-0.930165\pi\)
0.976030 0.217637i \(-0.0698350\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 215.995i − 0.340149i
\(636\) 0 0
\(637\) 110.818 0.173968
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 383.085 0.597636 0.298818 0.954310i \(-0.403408\pi\)
0.298818 + 0.954310i \(0.403408\pi\)
\(642\) 0 0
\(643\) − 177.134i − 0.275480i −0.990468 0.137740i \(-0.956016\pi\)
0.990468 0.137740i \(-0.0439838\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 877.305i − 1.35596i −0.735081 0.677979i \(-0.762856\pi\)
0.735081 0.677979i \(-0.237144\pi\)
\(648\) 0 0
\(649\) −561.344 −0.864937
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −35.3934 −0.0542012 −0.0271006 0.999633i \(-0.508627\pi\)
−0.0271006 + 0.999633i \(0.508627\pi\)
\(654\) 0 0
\(655\) 822.884i 1.25631i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 556.337i 0.844214i 0.906546 + 0.422107i \(0.138709\pi\)
−0.906546 + 0.422107i \(0.861291\pi\)
\(660\) 0 0
\(661\) −318.875 −0.482413 −0.241206 0.970474i \(-0.577543\pi\)
−0.241206 + 0.970474i \(0.577543\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −399.389 −0.600586
\(666\) 0 0
\(667\) − 10.2247i − 0.0153293i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1466.03i 2.18484i
\(672\) 0 0
\(673\) 135.295 0.201033 0.100517 0.994935i \(-0.467950\pi\)
0.100517 + 0.994935i \(0.467950\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 123.989 0.183145 0.0915723 0.995798i \(-0.470811\pi\)
0.0915723 + 0.995798i \(0.470811\pi\)
\(678\) 0 0
\(679\) 611.966i 0.901275i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 609.347i 0.892163i 0.894992 + 0.446081i \(0.147181\pi\)
−0.894992 + 0.446081i \(0.852819\pi\)
\(684\) 0 0
\(685\) −87.7059 −0.128038
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 893.609 1.29696
\(690\) 0 0
\(691\) 353.682i 0.511840i 0.966698 + 0.255920i \(0.0823784\pi\)
−0.966698 + 0.255920i \(0.917622\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 657.784i − 0.946452i
\(696\) 0 0
\(697\) −788.183 −1.13082
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 85.2186 0.121567 0.0607836 0.998151i \(-0.480640\pi\)
0.0607836 + 0.998151i \(0.480640\pi\)
\(702\) 0 0
\(703\) − 50.5316i − 0.0718800i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1045.57i − 1.47888i
\(708\) 0 0
\(709\) 701.943 0.990046 0.495023 0.868880i \(-0.335160\pi\)
0.495023 + 0.868880i \(0.335160\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −631.381 −0.885527
\(714\) 0 0
\(715\) 855.036i 1.19585i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 993.471i 1.38174i 0.722979 + 0.690870i \(0.242772\pi\)
−0.722979 + 0.690870i \(0.757228\pi\)
\(720\) 0 0
\(721\) −495.330 −0.687004
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10.9798 0.0151445
\(726\) 0 0
\(727\) − 485.145i − 0.667325i −0.942693 0.333662i \(-0.891715\pi\)
0.942693 0.333662i \(-0.108285\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 482.800i − 0.660465i
\(732\) 0 0
\(733\) −706.167 −0.963394 −0.481697 0.876338i \(-0.659979\pi\)
−0.481697 + 0.876338i \(0.659979\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 388.053 0.526531
\(738\) 0 0
\(739\) 414.250i 0.560554i 0.959919 + 0.280277i \(0.0904264\pi\)
−0.959919 + 0.280277i \(0.909574\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1386.38i 1.86592i 0.359978 + 0.932961i \(0.382784\pi\)
−0.359978 + 0.932961i \(0.617216\pi\)
\(744\) 0 0
\(745\) 811.141 1.08878
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −456.294 −0.609205
\(750\) 0 0
\(751\) 601.365i 0.800752i 0.916351 + 0.400376i \(0.131121\pi\)
−0.916351 + 0.400376i \(0.868879\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 807.089i 1.06899i
\(756\) 0 0
\(757\) 254.294 0.335924 0.167962 0.985794i \(-0.446281\pi\)
0.167962 + 0.985794i \(0.446281\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −466.017 −0.612375 −0.306187 0.951971i \(-0.599053\pi\)
−0.306187 + 0.951971i \(0.599053\pi\)
\(762\) 0 0
\(763\) 397.401i 0.520840i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 771.334i 1.00565i
\(768\) 0 0
\(769\) −365.822 −0.475712 −0.237856 0.971300i \(-0.576445\pi\)
−0.237856 + 0.971300i \(0.576445\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1426.70 1.84566 0.922830 0.385207i \(-0.125870\pi\)
0.922830 + 0.385207i \(0.125870\pi\)
\(774\) 0 0
\(775\) − 678.007i − 0.874848i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 643.293i 0.825794i
\(780\) 0 0
\(781\) −333.549 −0.427079
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −58.6100 −0.0746624
\(786\) 0 0
\(787\) − 410.337i − 0.521394i −0.965421 0.260697i \(-0.916048\pi\)
0.965421 0.260697i \(-0.0839523\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 924.698i − 1.16902i
\(792\) 0 0
\(793\) 2014.45 2.54029
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −738.968 −0.927187 −0.463594 0.886048i \(-0.653440\pi\)
−0.463594 + 0.886048i \(0.653440\pi\)
\(798\) 0 0
\(799\) − 355.731i − 0.445221i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 1668.47i − 2.07780i
\(804\) 0 0
\(805\) 288.154 0.357956
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −130.444 −0.161241 −0.0806204 0.996745i \(-0.525690\pi\)
−0.0806204 + 0.996745i \(0.525690\pi\)
\(810\) 0 0
\(811\) − 1558.44i − 1.92163i −0.277187 0.960816i \(-0.589402\pi\)
0.277187 0.960816i \(-0.410598\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 161.314i 0.197931i
\(816\) 0 0
\(817\) −394.048 −0.482311
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −55.0345 −0.0670335 −0.0335168 0.999438i \(-0.510671\pi\)
−0.0335168 + 0.999438i \(0.510671\pi\)
\(822\) 0 0
\(823\) − 293.058i − 0.356085i −0.984023 0.178043i \(-0.943023\pi\)
0.984023 0.178043i \(-0.0569765\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1118.55i 1.35253i 0.736657 + 0.676267i \(0.236403\pi\)
−0.736657 + 0.676267i \(0.763597\pi\)
\(828\) 0 0
\(829\) −1069.30 −1.28986 −0.644932 0.764240i \(-0.723115\pi\)
−0.644932 + 0.764240i \(0.723115\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 134.042 0.160915
\(834\) 0 0
\(835\) − 273.616i − 0.327683i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 41.6050i 0.0495888i 0.999693 + 0.0247944i \(0.00789311\pi\)
−0.999693 + 0.0247944i \(0.992107\pi\)
\(840\) 0 0
\(841\) −840.434 −0.999327
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 629.846 0.745379
\(846\) 0 0
\(847\) 472.813i 0.558220i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 36.4579i 0.0428413i
\(852\) 0 0
\(853\) −1302.63 −1.52711 −0.763555 0.645743i \(-0.776548\pi\)
−0.763555 + 0.645743i \(0.776548\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 734.683 0.857273 0.428636 0.903477i \(-0.358994\pi\)
0.428636 + 0.903477i \(0.358994\pi\)
\(858\) 0 0
\(859\) − 445.296i − 0.518388i −0.965825 0.259194i \(-0.916543\pi\)
0.965825 0.259194i \(-0.0834570\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1338.23i − 1.55067i −0.631550 0.775335i \(-0.717581\pi\)
0.631550 0.775335i \(-0.282419\pi\)
\(864\) 0 0
\(865\) 466.684 0.539519
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1328.46 −1.52873
\(870\) 0 0
\(871\) − 533.218i − 0.612190i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 839.338i 0.959244i
\(876\) 0 0
\(877\) 329.803 0.376058 0.188029 0.982163i \(-0.439790\pi\)
0.188029 + 0.982163i \(0.439790\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1173.32 −1.33181 −0.665903 0.746038i \(-0.731954\pi\)
−0.665903 + 0.746038i \(0.731954\pi\)
\(882\) 0 0
\(883\) 410.459i 0.464846i 0.972615 + 0.232423i \(0.0746654\pi\)
−0.972615 + 0.232423i \(0.925335\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 604.280i − 0.681263i −0.940197 0.340631i \(-0.889359\pi\)
0.940197 0.340631i \(-0.110641\pi\)
\(888\) 0 0
\(889\) −440.158 −0.495116
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −290.338 −0.325127
\(894\) 0 0
\(895\) − 906.020i − 1.01231i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 34.9305i − 0.0388548i
\(900\) 0 0
\(901\) 1080.88 1.19965
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 953.048 1.05309
\(906\) 0 0
\(907\) 735.914i 0.811371i 0.914013 + 0.405686i \(0.132967\pi\)
−0.914013 + 0.405686i \(0.867033\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 586.390i 0.643677i 0.946795 + 0.321838i \(0.104301\pi\)
−0.946795 + 0.321838i \(0.895699\pi\)
\(912\) 0 0
\(913\) 1603.58 1.75639
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1676.89 1.82867
\(918\) 0 0
\(919\) 158.999i 0.173013i 0.996251 + 0.0865063i \(0.0275703\pi\)
−0.996251 + 0.0865063i \(0.972430\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 458.324i 0.496560i
\(924\) 0 0
\(925\) −39.1503 −0.0423247
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −540.670 −0.581992 −0.290996 0.956724i \(-0.593987\pi\)
−0.290996 + 0.956724i \(0.593987\pi\)
\(930\) 0 0
\(931\) − 109.402i − 0.117510i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1034.23i 1.10613i
\(936\) 0 0
\(937\) 384.947 0.410829 0.205415 0.978675i \(-0.434146\pi\)
0.205415 + 0.978675i \(0.434146\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −39.6287 −0.0421134 −0.0210567 0.999778i \(-0.506703\pi\)
−0.0210567 + 0.999778i \(0.506703\pi\)
\(942\) 0 0
\(943\) − 464.128i − 0.492182i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 449.648i − 0.474813i −0.971410 0.237407i \(-0.923703\pi\)
0.971410 0.237407i \(-0.0762974\pi\)
\(948\) 0 0
\(949\) −2292.62 −2.41583
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −21.5070 −0.0225676 −0.0112838 0.999936i \(-0.503592\pi\)
−0.0112838 + 0.999936i \(0.503592\pi\)
\(954\) 0 0
\(955\) 954.578i 0.999558i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 178.729i 0.186370i
\(960\) 0 0
\(961\) −1195.98 −1.24452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −58.1172 −0.0602251
\(966\) 0 0
\(967\) − 1355.49i − 1.40175i −0.713283 0.700876i \(-0.752793\pi\)
0.713283 0.700876i \(-0.247207\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 938.488i 0.966517i 0.875478 + 0.483258i \(0.160547\pi\)
−0.875478 + 0.483258i \(0.839453\pi\)
\(972\) 0 0
\(973\) −1340.44 −1.37764
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1090.81 1.11649 0.558243 0.829678i \(-0.311476\pi\)
0.558243 + 0.829678i \(0.311476\pi\)
\(978\) 0 0
\(979\) 2355.64i 2.40617i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 27.7492i 0.0282291i 0.999900 + 0.0141146i \(0.00449296\pi\)
−0.999900 + 0.0141146i \(0.995507\pi\)
\(984\) 0 0
\(985\) −383.853 −0.389699
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 284.301 0.287463
\(990\) 0 0
\(991\) − 272.858i − 0.275336i −0.990478 0.137668i \(-0.956039\pi\)
0.990478 0.137668i \(-0.0439607\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 983.378i − 0.988320i
\(996\) 0 0
\(997\) −1190.35 −1.19393 −0.596964 0.802268i \(-0.703626\pi\)
−0.596964 + 0.802268i \(0.703626\pi\)
\(998\) 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.703.5 8
3.2 odd 2 1728.3.g.n.703.3 8
4.3 odd 2 inner 1728.3.g.k.703.6 8
8.3 odd 2 864.3.g.c.703.4 yes 8
8.5 even 2 864.3.g.c.703.3 yes 8
12.11 even 2 1728.3.g.n.703.4 8
24.5 odd 2 864.3.g.a.703.5 8
24.11 even 2 864.3.g.a.703.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.a.703.5 8 24.5 odd 2
864.3.g.a.703.6 yes 8 24.11 even 2
864.3.g.c.703.3 yes 8 8.5 even 2
864.3.g.c.703.4 yes 8 8.3 odd 2
1728.3.g.k.703.5 8 1.1 even 1 trivial
1728.3.g.k.703.6 8 4.3 odd 2 inner
1728.3.g.n.703.3 8 3.2 odd 2
1728.3.g.n.703.4 8 12.11 even 2