Properties

Label 1728.3.g.k.703.8
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 703.8
Root \(0.665665 + 1.24775i\) of defining polynomial
Character \(\chi\) \(=\) 1728.703
Dual form 1728.3.g.k.703.7

$q$-expansion

\(f(q)\) \(=\) \(q+4.18059 q^{5} +2.58429i q^{7} +O(q^{10})\) \(q+4.18059 q^{5} +2.58429i q^{7} +6.38876i q^{11} -11.5849 q^{13} -7.58491 q^{17} -0.0311239i q^{19} -7.52976i q^{23} -7.52266 q^{25} -13.5609 q^{29} +29.7852i q^{31} +10.8038i q^{35} -57.4290 q^{37} -27.1757 q^{41} -77.8454i q^{43} +43.8370i q^{47} +42.3215 q^{49} -87.7199 q^{53} +26.7088i q^{55} -67.8299i q^{59} -31.2943 q^{61} -48.4318 q^{65} -23.6302i q^{67} -50.8625i q^{71} +70.9868 q^{73} -16.5104 q^{77} +67.3835i q^{79} +8.55335i q^{83} -31.7094 q^{85} +5.22274 q^{89} -29.9387i q^{91} -0.130116i q^{95} +120.354 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) 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
\(5\) 4.18059 0.836118 0.418059 0.908420i \(-0.362711\pi\)
0.418059 + 0.908420i \(0.362711\pi\)
\(6\) 0 0
\(7\) 2.58429i 0.369184i 0.982815 + 0.184592i \(0.0590964\pi\)
−0.982815 + 0.184592i \(0.940904\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.38876i 0.580796i 0.956906 + 0.290398i \(0.0937877\pi\)
−0.956906 + 0.290398i \(0.906212\pi\)
\(12\) 0 0
\(13\) −11.5849 −0.891147 −0.445573 0.895245i \(-0.647000\pi\)
−0.445573 + 0.895245i \(0.647000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.58491 −0.446171 −0.223086 0.974799i \(-0.571613\pi\)
−0.223086 + 0.974799i \(0.571613\pi\)
\(18\) 0 0
\(19\) − 0.0311239i − 0.00163810i −1.00000 0.000819049i \(-0.999739\pi\)
1.00000 0.000819049i \(-0.000260711\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 7.52976i − 0.327381i −0.986512 0.163690i \(-0.947660\pi\)
0.986512 0.163690i \(-0.0523398\pi\)
\(24\) 0 0
\(25\) −7.52266 −0.300907
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −13.5609 −0.467617 −0.233808 0.972283i \(-0.575119\pi\)
−0.233808 + 0.972283i \(0.575119\pi\)
\(30\) 0 0
\(31\) 29.7852i 0.960814i 0.877046 + 0.480407i \(0.159511\pi\)
−0.877046 + 0.480407i \(0.840489\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.8038i 0.308681i
\(36\) 0 0
\(37\) −57.4290 −1.55214 −0.776068 0.630649i \(-0.782789\pi\)
−0.776068 + 0.630649i \(0.782789\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −27.1757 −0.662821 −0.331411 0.943487i \(-0.607525\pi\)
−0.331411 + 0.943487i \(0.607525\pi\)
\(42\) 0 0
\(43\) − 77.8454i − 1.81036i −0.425031 0.905179i \(-0.639737\pi\)
0.425031 0.905179i \(-0.360263\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 43.8370i 0.932703i 0.884600 + 0.466351i \(0.154432\pi\)
−0.884600 + 0.466351i \(0.845568\pi\)
\(48\) 0 0
\(49\) 42.3215 0.863703
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −87.7199 −1.65509 −0.827547 0.561397i \(-0.810264\pi\)
−0.827547 + 0.561397i \(0.810264\pi\)
\(54\) 0 0
\(55\) 26.7088i 0.485614i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 67.8299i − 1.14966i −0.818273 0.574830i \(-0.805068\pi\)
0.818273 0.574830i \(-0.194932\pi\)
\(60\) 0 0
\(61\) −31.2943 −0.513022 −0.256511 0.966541i \(-0.582573\pi\)
−0.256511 + 0.966541i \(0.582573\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −48.4318 −0.745104
\(66\) 0 0
\(67\) − 23.6302i − 0.352690i −0.984328 0.176345i \(-0.943573\pi\)
0.984328 0.176345i \(-0.0564274\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 50.8625i − 0.716374i −0.933650 0.358187i \(-0.883395\pi\)
0.933650 0.358187i \(-0.116605\pi\)
\(72\) 0 0
\(73\) 70.9868 0.972422 0.486211 0.873841i \(-0.338379\pi\)
0.486211 + 0.873841i \(0.338379\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −16.5104 −0.214421
\(78\) 0 0
\(79\) 67.3835i 0.852955i 0.904498 + 0.426478i \(0.140246\pi\)
−0.904498 + 0.426478i \(0.859754\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.55335i 0.103052i 0.998672 + 0.0515262i \(0.0164086\pi\)
−0.998672 + 0.0515262i \(0.983591\pi\)
\(84\) 0 0
\(85\) −31.7094 −0.373052
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.22274 0.0586825 0.0293412 0.999569i \(-0.490659\pi\)
0.0293412 + 0.999569i \(0.490659\pi\)
\(90\) 0 0
\(91\) − 29.9387i − 0.328997i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 0.130116i − 0.00136964i
\(96\) 0 0
\(97\) 120.354 1.24076 0.620379 0.784302i \(-0.286979\pi\)
0.620379 + 0.784302i \(0.286979\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −47.1221 −0.466555 −0.233278 0.972410i \(-0.574945\pi\)
−0.233278 + 0.972410i \(0.574945\pi\)
\(102\) 0 0
\(103\) 30.7664i 0.298703i 0.988784 + 0.149352i \(0.0477186\pi\)
−0.988784 + 0.149352i \(0.952281\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 149.818i 1.40017i 0.714061 + 0.700083i \(0.246854\pi\)
−0.714061 + 0.700083i \(0.753146\pi\)
\(108\) 0 0
\(109\) −12.0441 −0.110496 −0.0552480 0.998473i \(-0.517595\pi\)
−0.0552480 + 0.998473i \(0.517595\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −212.750 −1.88274 −0.941371 0.337373i \(-0.890462\pi\)
−0.941371 + 0.337373i \(0.890462\pi\)
\(114\) 0 0
\(115\) − 31.4788i − 0.273729i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 19.6016i − 0.164719i
\(120\) 0 0
\(121\) 80.1838 0.662676
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −135.964 −1.08771
\(126\) 0 0
\(127\) 192.982i 1.51954i 0.650190 + 0.759771i \(0.274689\pi\)
−0.650190 + 0.759771i \(0.725311\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 189.604i − 1.44736i −0.690135 0.723680i \(-0.742449\pi\)
0.690135 0.723680i \(-0.257551\pi\)
\(132\) 0 0
\(133\) 0.0804330 0.000604760 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −196.740 −1.43606 −0.718031 0.696011i \(-0.754956\pi\)
−0.718031 + 0.696011i \(0.754956\pi\)
\(138\) 0 0
\(139\) − 104.557i − 0.752211i −0.926577 0.376105i \(-0.877263\pi\)
0.926577 0.376105i \(-0.122737\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 74.0132i − 0.517575i
\(144\) 0 0
\(145\) −56.6925 −0.390983
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −51.2196 −0.343755 −0.171878 0.985118i \(-0.554983\pi\)
−0.171878 + 0.985118i \(0.554983\pi\)
\(150\) 0 0
\(151\) − 226.393i − 1.49929i −0.661839 0.749646i \(-0.730224\pi\)
0.661839 0.749646i \(-0.269776\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 124.520i 0.803354i
\(156\) 0 0
\(157\) 43.1698 0.274967 0.137484 0.990504i \(-0.456099\pi\)
0.137484 + 0.990504i \(0.456099\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 19.4591 0.120864
\(162\) 0 0
\(163\) 137.264i 0.842113i 0.907034 + 0.421057i \(0.138341\pi\)
−0.907034 + 0.421057i \(0.861659\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 143.149i 0.857177i 0.903500 + 0.428588i \(0.140989\pi\)
−0.903500 + 0.428588i \(0.859011\pi\)
\(168\) 0 0
\(169\) −34.7898 −0.205857
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −188.031 −1.08688 −0.543442 0.839446i \(-0.682879\pi\)
−0.543442 + 0.839446i \(0.682879\pi\)
\(174\) 0 0
\(175\) − 19.4407i − 0.111090i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 75.8284i 0.423622i 0.977311 + 0.211811i \(0.0679362\pi\)
−0.977311 + 0.211811i \(0.932064\pi\)
\(180\) 0 0
\(181\) 153.743 0.849406 0.424703 0.905333i \(-0.360378\pi\)
0.424703 + 0.905333i \(0.360378\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −240.087 −1.29777
\(186\) 0 0
\(187\) − 48.4582i − 0.259135i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 201.836i 1.05673i 0.849017 + 0.528366i \(0.177195\pi\)
−0.849017 + 0.528366i \(0.822805\pi\)
\(192\) 0 0
\(193\) −142.854 −0.740175 −0.370088 0.928997i \(-0.620672\pi\)
−0.370088 + 0.928997i \(0.620672\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −388.500 −1.97208 −0.986041 0.166504i \(-0.946752\pi\)
−0.986041 + 0.166504i \(0.946752\pi\)
\(198\) 0 0
\(199\) 19.9376i 0.100189i 0.998744 + 0.0500945i \(0.0159523\pi\)
−0.998744 + 0.0500945i \(0.984048\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 35.0452i − 0.172637i
\(204\) 0 0
\(205\) −113.610 −0.554197
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.198843 0.000951401 0
\(210\) 0 0
\(211\) 174.438i 0.826720i 0.910568 + 0.413360i \(0.135645\pi\)
−0.910568 + 0.413360i \(0.864355\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 325.440i − 1.51367i
\(216\) 0 0
\(217\) −76.9736 −0.354717
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 87.8705 0.397604
\(222\) 0 0
\(223\) − 354.898i − 1.59147i −0.605643 0.795736i \(-0.707084\pi\)
0.605643 0.795736i \(-0.292916\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 360.716i − 1.58906i −0.607227 0.794528i \(-0.707718\pi\)
0.607227 0.794528i \(-0.292282\pi\)
\(228\) 0 0
\(229\) −45.4706 −0.198562 −0.0992809 0.995059i \(-0.531654\pi\)
−0.0992809 + 0.995059i \(0.531654\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 294.123 1.26233 0.631166 0.775648i \(-0.282577\pi\)
0.631166 + 0.775648i \(0.282577\pi\)
\(234\) 0 0
\(235\) 183.265i 0.779850i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 295.342i 1.23574i 0.786279 + 0.617871i \(0.212005\pi\)
−0.786279 + 0.617871i \(0.787995\pi\)
\(240\) 0 0
\(241\) −200.546 −0.832142 −0.416071 0.909332i \(-0.636593\pi\)
−0.416071 + 0.909332i \(0.636593\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 176.929 0.722158
\(246\) 0 0
\(247\) 0.360567i 0.00145979i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 272.988i 1.08760i 0.839214 + 0.543801i \(0.183015\pi\)
−0.839214 + 0.543801i \(0.816985\pi\)
\(252\) 0 0
\(253\) 48.1058 0.190141
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −211.739 −0.823885 −0.411943 0.911210i \(-0.635150\pi\)
−0.411943 + 0.911210i \(0.635150\pi\)
\(258\) 0 0
\(259\) − 148.413i − 0.573024i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 189.044i − 0.718799i −0.933184 0.359400i \(-0.882981\pi\)
0.933184 0.359400i \(-0.117019\pi\)
\(264\) 0 0
\(265\) −366.721 −1.38385
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −139.727 −0.519432 −0.259716 0.965685i \(-0.583629\pi\)
−0.259716 + 0.965685i \(0.583629\pi\)
\(270\) 0 0
\(271\) 4.24913i 0.0156794i 0.999969 + 0.00783972i \(0.00249549\pi\)
−0.999969 + 0.00783972i \(0.997505\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 48.0605i − 0.174765i
\(276\) 0 0
\(277\) 230.546 0.832297 0.416149 0.909297i \(-0.363380\pi\)
0.416149 + 0.909297i \(0.363380\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −247.125 −0.879449 −0.439725 0.898133i \(-0.644924\pi\)
−0.439725 + 0.898133i \(0.644924\pi\)
\(282\) 0 0
\(283\) 542.670i 1.91756i 0.284149 + 0.958780i \(0.408289\pi\)
−0.284149 + 0.958780i \(0.591711\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 70.2297i − 0.244703i
\(288\) 0 0
\(289\) −231.469 −0.800931
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 245.662 0.838437 0.419218 0.907885i \(-0.362304\pi\)
0.419218 + 0.907885i \(0.362304\pi\)
\(294\) 0 0
\(295\) − 283.569i − 0.961251i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 87.2316i 0.291744i
\(300\) 0 0
\(301\) 201.175 0.668355
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −130.829 −0.428947
\(306\) 0 0
\(307\) 481.064i 1.56698i 0.621403 + 0.783491i \(0.286563\pi\)
−0.621403 + 0.783491i \(0.713437\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 71.8405i − 0.230998i −0.993308 0.115499i \(-0.963153\pi\)
0.993308 0.115499i \(-0.0368468\pi\)
\(312\) 0 0
\(313\) −152.089 −0.485907 −0.242953 0.970038i \(-0.578116\pi\)
−0.242953 + 0.970038i \(0.578116\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −183.035 −0.577396 −0.288698 0.957420i \(-0.593222\pi\)
−0.288698 + 0.957420i \(0.593222\pi\)
\(318\) 0 0
\(319\) − 86.6372i − 0.271590i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.236072i 0 0.000730872i
\(324\) 0 0
\(325\) 87.1494 0.268152
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −113.287 −0.344339
\(330\) 0 0
\(331\) − 546.449i − 1.65090i −0.564473 0.825452i \(-0.690921\pi\)
0.564473 0.825452i \(-0.309079\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 98.7883i − 0.294890i
\(336\) 0 0
\(337\) 176.360 0.523324 0.261662 0.965160i \(-0.415729\pi\)
0.261662 + 0.965160i \(0.415729\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −190.291 −0.558037
\(342\) 0 0
\(343\) 236.001i 0.688049i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 113.338i 0.326622i 0.986575 + 0.163311i \(0.0522174\pi\)
−0.986575 + 0.163311i \(0.947783\pi\)
\(348\) 0 0
\(349\) 408.318 1.16996 0.584982 0.811046i \(-0.301102\pi\)
0.584982 + 0.811046i \(0.301102\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 374.570 1.06111 0.530553 0.847652i \(-0.321984\pi\)
0.530553 + 0.847652i \(0.321984\pi\)
\(354\) 0 0
\(355\) − 212.635i − 0.598973i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 336.934i − 0.938534i −0.883056 0.469267i \(-0.844518\pi\)
0.883056 0.469267i \(-0.155482\pi\)
\(360\) 0 0
\(361\) 360.999 0.999997
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 296.767 0.813060
\(366\) 0 0
\(367\) − 430.222i − 1.17227i −0.810214 0.586134i \(-0.800649\pi\)
0.810214 0.586134i \(-0.199351\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 226.694i − 0.611034i
\(372\) 0 0
\(373\) 62.2557 0.166905 0.0834527 0.996512i \(-0.473405\pi\)
0.0834527 + 0.996512i \(0.473405\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 157.102 0.416715
\(378\) 0 0
\(379\) 657.970i 1.73607i 0.496505 + 0.868034i \(0.334616\pi\)
−0.496505 + 0.868034i \(0.665384\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 726.694i 1.89737i 0.316219 + 0.948686i \(0.397587\pi\)
−0.316219 + 0.948686i \(0.602413\pi\)
\(384\) 0 0
\(385\) −69.0232 −0.179281
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 363.139 0.933521 0.466760 0.884384i \(-0.345421\pi\)
0.466760 + 0.884384i \(0.345421\pi\)
\(390\) 0 0
\(391\) 57.1125i 0.146068i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 281.703i 0.713171i
\(396\) 0 0
\(397\) −69.9305 −0.176147 −0.0880737 0.996114i \(-0.528071\pi\)
−0.0880737 + 0.996114i \(0.528071\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 371.449 0.926307 0.463153 0.886278i \(-0.346718\pi\)
0.463153 + 0.886278i \(0.346718\pi\)
\(402\) 0 0
\(403\) − 345.059i − 0.856226i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 366.900i − 0.901475i
\(408\) 0 0
\(409\) 33.3496 0.0815394 0.0407697 0.999169i \(-0.487019\pi\)
0.0407697 + 0.999169i \(0.487019\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 175.292 0.424436
\(414\) 0 0
\(415\) 35.7581i 0.0861640i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 221.058i − 0.527585i −0.964579 0.263792i \(-0.915027\pi\)
0.964579 0.263792i \(-0.0849734\pi\)
\(420\) 0 0
\(421\) −199.813 −0.474615 −0.237307 0.971435i \(-0.576265\pi\)
−0.237307 + 0.971435i \(0.576265\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 57.0587 0.134256
\(426\) 0 0
\(427\) − 80.8735i − 0.189399i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 575.274i 1.33474i 0.744725 + 0.667371i \(0.232581\pi\)
−0.744725 + 0.667371i \(0.767419\pi\)
\(432\) 0 0
\(433\) 90.1509 0.208201 0.104100 0.994567i \(-0.466804\pi\)
0.104100 + 0.994567i \(0.466804\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.234355 −0.000536282 0
\(438\) 0 0
\(439\) − 153.475i − 0.349602i −0.984604 0.174801i \(-0.944072\pi\)
0.984604 0.174801i \(-0.0559282\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 89.5187i − 0.202074i −0.994883 0.101037i \(-0.967784\pi\)
0.994883 0.101037i \(-0.0322160\pi\)
\(444\) 0 0
\(445\) 21.8341 0.0490655
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 487.678 1.08614 0.543071 0.839687i \(-0.317261\pi\)
0.543071 + 0.839687i \(0.317261\pi\)
\(450\) 0 0
\(451\) − 173.619i − 0.384964i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 125.162i − 0.275080i
\(456\) 0 0
\(457\) 77.2928 0.169131 0.0845654 0.996418i \(-0.473050\pi\)
0.0845654 + 0.996418i \(0.473050\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 201.504 0.437101 0.218551 0.975826i \(-0.429867\pi\)
0.218551 + 0.975826i \(0.429867\pi\)
\(462\) 0 0
\(463\) − 82.5940i − 0.178389i −0.996014 0.0891944i \(-0.971571\pi\)
0.996014 0.0891944i \(-0.0284292\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 444.443i − 0.951698i −0.879527 0.475849i \(-0.842141\pi\)
0.879527 0.475849i \(-0.157859\pi\)
\(468\) 0 0
\(469\) 61.0673 0.130207
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 497.335 1.05145
\(474\) 0 0
\(475\) 0.234134i 0 0.000492914i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 556.519i 1.16184i 0.813962 + 0.580918i \(0.197306\pi\)
−0.813962 + 0.580918i \(0.802694\pi\)
\(480\) 0 0
\(481\) 665.310 1.38318
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 503.149 1.03742
\(486\) 0 0
\(487\) 705.234i 1.44812i 0.689737 + 0.724060i \(0.257726\pi\)
−0.689737 + 0.724060i \(0.742274\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 605.015i 1.23221i 0.787664 + 0.616105i \(0.211290\pi\)
−0.787664 + 0.616105i \(0.788710\pi\)
\(492\) 0 0
\(493\) 102.858 0.208637
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 131.443 0.264474
\(498\) 0 0
\(499\) − 192.748i − 0.386268i −0.981172 0.193134i \(-0.938135\pi\)
0.981172 0.193134i \(-0.0618652\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 558.416i − 1.11017i −0.831793 0.555085i \(-0.812686\pi\)
0.831793 0.555085i \(-0.187314\pi\)
\(504\) 0 0
\(505\) −196.998 −0.390095
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 126.597 0.248718 0.124359 0.992237i \(-0.460313\pi\)
0.124359 + 0.992237i \(0.460313\pi\)
\(510\) 0 0
\(511\) 183.450i 0.359003i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 128.622i 0.249751i
\(516\) 0 0
\(517\) −280.064 −0.541710
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 590.984 1.13433 0.567163 0.823605i \(-0.308041\pi\)
0.567163 + 0.823605i \(0.308041\pi\)
\(522\) 0 0
\(523\) 50.5498i 0.0966535i 0.998832 + 0.0483268i \(0.0153889\pi\)
−0.998832 + 0.0483268i \(0.984611\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 225.918i − 0.428688i
\(528\) 0 0
\(529\) 472.303 0.892822
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 314.828 0.590671
\(534\) 0 0
\(535\) 626.327i 1.17070i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 270.382i 0.501636i
\(540\) 0 0
\(541\) −909.128 −1.68046 −0.840229 0.542231i \(-0.817580\pi\)
−0.840229 + 0.542231i \(0.817580\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −50.3513 −0.0923877
\(546\) 0 0
\(547\) − 731.866i − 1.33796i −0.743279 0.668981i \(-0.766731\pi\)
0.743279 0.668981i \(-0.233269\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.422067i 0 0.000766002i
\(552\) 0 0
\(553\) −174.138 −0.314897
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 617.363 1.10837 0.554186 0.832393i \(-0.313030\pi\)
0.554186 + 0.832393i \(0.313030\pi\)
\(558\) 0 0
\(559\) 901.832i 1.61329i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 829.105i − 1.47266i −0.676625 0.736328i \(-0.736558\pi\)
0.676625 0.736328i \(-0.263442\pi\)
\(564\) 0 0
\(565\) −889.420 −1.57419
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 287.157 0.504670 0.252335 0.967640i \(-0.418802\pi\)
0.252335 + 0.967640i \(0.418802\pi\)
\(570\) 0 0
\(571\) − 161.987i − 0.283691i −0.989889 0.141845i \(-0.954696\pi\)
0.989889 0.141845i \(-0.0453036\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 56.6438i 0.0985110i
\(576\) 0 0
\(577\) −197.092 −0.341581 −0.170790 0.985307i \(-0.554632\pi\)
−0.170790 + 0.985307i \(0.554632\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −22.1043 −0.0380453
\(582\) 0 0
\(583\) − 560.421i − 0.961272i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 757.516i 1.29049i 0.763977 + 0.645244i \(0.223244\pi\)
−0.763977 + 0.645244i \(0.776756\pi\)
\(588\) 0 0
\(589\) 0.927032 0.00157391
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −767.673 −1.29456 −0.647279 0.762253i \(-0.724093\pi\)
−0.647279 + 0.762253i \(0.724093\pi\)
\(594\) 0 0
\(595\) − 81.9462i − 0.137725i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1043.77i − 1.74252i −0.490821 0.871260i \(-0.663303\pi\)
0.490821 0.871260i \(-0.336697\pi\)
\(600\) 0 0
\(601\) 697.990 1.16138 0.580691 0.814124i \(-0.302783\pi\)
0.580691 + 0.814124i \(0.302783\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 335.215 0.554075
\(606\) 0 0
\(607\) 67.5726i 0.111322i 0.998450 + 0.0556611i \(0.0177267\pi\)
−0.998450 + 0.0556611i \(0.982273\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 507.848i − 0.831175i
\(612\) 0 0
\(613\) 627.420 1.02352 0.511762 0.859127i \(-0.328993\pi\)
0.511762 + 0.859127i \(0.328993\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 98.1234 0.159033 0.0795165 0.996834i \(-0.474662\pi\)
0.0795165 + 0.996834i \(0.474662\pi\)
\(618\) 0 0
\(619\) − 622.064i − 1.00495i −0.864592 0.502475i \(-0.832423\pi\)
0.864592 0.502475i \(-0.167577\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.4971i 0.0216646i
\(624\) 0 0
\(625\) −380.343 −0.608549
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 435.594 0.692518
\(630\) 0 0
\(631\) 583.009i 0.923944i 0.886895 + 0.461972i \(0.152858\pi\)
−0.886895 + 0.461972i \(0.847142\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 806.779i 1.27052i
\(636\) 0 0
\(637\) −490.290 −0.769687
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 642.626 1.00254 0.501268 0.865292i \(-0.332867\pi\)
0.501268 + 0.865292i \(0.332867\pi\)
\(642\) 0 0
\(643\) 224.989i 0.349904i 0.984577 + 0.174952i \(0.0559771\pi\)
−0.984577 + 0.174952i \(0.944023\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1234.87i 1.90861i 0.298840 + 0.954303i \(0.403400\pi\)
−0.298840 + 0.954303i \(0.596600\pi\)
\(648\) 0 0
\(649\) 433.349 0.667718
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 54.3250 0.0831929 0.0415965 0.999134i \(-0.486756\pi\)
0.0415965 + 0.999134i \(0.486756\pi\)
\(654\) 0 0
\(655\) − 792.658i − 1.21016i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 592.696i − 0.899387i −0.893183 0.449694i \(-0.851533\pi\)
0.893183 0.449694i \(-0.148467\pi\)
\(660\) 0 0
\(661\) −817.609 −1.23693 −0.618464 0.785813i \(-0.712245\pi\)
−0.618464 + 0.785813i \(0.712245\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.336258 0.000505650 0
\(666\) 0 0
\(667\) 102.110i 0.153089i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 199.932i − 0.297961i
\(672\) 0 0
\(673\) −294.801 −0.438040 −0.219020 0.975720i \(-0.570286\pi\)
−0.219020 + 0.975720i \(0.570286\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −238.903 −0.352884 −0.176442 0.984311i \(-0.556459\pi\)
−0.176442 + 0.984311i \(0.556459\pi\)
\(678\) 0 0
\(679\) 311.028i 0.458068i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 358.601i − 0.525038i −0.964927 0.262519i \(-0.915447\pi\)
0.964927 0.262519i \(-0.0845532\pi\)
\(684\) 0 0
\(685\) −822.491 −1.20072
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1016.23 1.47493
\(690\) 0 0
\(691\) − 590.750i − 0.854921i −0.904034 0.427460i \(-0.859408\pi\)
0.904034 0.427460i \(-0.140592\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 437.111i − 0.628937i
\(696\) 0 0
\(697\) 206.125 0.295732
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1122.36 1.60108 0.800539 0.599281i \(-0.204547\pi\)
0.800539 + 0.599281i \(0.204547\pi\)
\(702\) 0 0
\(703\) 1.78741i 0.00254255i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 121.777i − 0.172245i
\(708\) 0 0
\(709\) −209.168 −0.295018 −0.147509 0.989061i \(-0.547126\pi\)
−0.147509 + 0.989061i \(0.547126\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 224.276 0.314552
\(714\) 0 0
\(715\) − 309.419i − 0.432754i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 601.883i 0.837111i 0.908191 + 0.418555i \(0.137463\pi\)
−0.908191 + 0.418555i \(0.862537\pi\)
\(720\) 0 0
\(721\) −79.5093 −0.110276
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 102.014 0.140709
\(726\) 0 0
\(727\) 131.548i 0.180947i 0.995899 + 0.0904734i \(0.0288380\pi\)
−0.995899 + 0.0904734i \(0.971162\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 590.450i 0.807729i
\(732\) 0 0
\(733\) 401.891 0.548283 0.274141 0.961689i \(-0.411606\pi\)
0.274141 + 0.961689i \(0.411606\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 150.968 0.204841
\(738\) 0 0
\(739\) 214.088i 0.289699i 0.989454 + 0.144850i \(0.0462698\pi\)
−0.989454 + 0.144850i \(0.953730\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 891.968i − 1.20050i −0.799814 0.600248i \(-0.795069\pi\)
0.799814 0.600248i \(-0.204931\pi\)
\(744\) 0 0
\(745\) −214.128 −0.287420
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −387.172 −0.516919
\(750\) 0 0
\(751\) 879.020i 1.17047i 0.810865 + 0.585233i \(0.198997\pi\)
−0.810865 + 0.585233i \(0.801003\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 946.457i − 1.25358i
\(756\) 0 0
\(757\) 510.218 0.673999 0.337000 0.941505i \(-0.390588\pi\)
0.337000 + 0.941505i \(0.390588\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 926.923 1.21803 0.609016 0.793158i \(-0.291564\pi\)
0.609016 + 0.793158i \(0.291564\pi\)
\(762\) 0 0
\(763\) − 31.1253i − 0.0407933i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 785.804i 1.02452i
\(768\) 0 0
\(769\) 890.596 1.15812 0.579061 0.815284i \(-0.303419\pi\)
0.579061 + 0.815284i \(0.303419\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −175.401 −0.226910 −0.113455 0.993543i \(-0.536192\pi\)
−0.113455 + 0.993543i \(0.536192\pi\)
\(774\) 0 0
\(775\) − 224.064i − 0.289115i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.845812i 0.00108577i
\(780\) 0 0
\(781\) 324.949 0.416067
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 180.475 0.229905
\(786\) 0 0
\(787\) 461.631i 0.586571i 0.956025 + 0.293286i \(0.0947487\pi\)
−0.956025 + 0.293286i \(0.905251\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 549.807i − 0.695078i
\(792\) 0 0
\(793\) 362.542 0.457178
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −792.157 −0.993923 −0.496961 0.867773i \(-0.665551\pi\)
−0.496961 + 0.867773i \(0.665551\pi\)
\(798\) 0 0
\(799\) − 332.500i − 0.416145i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 453.518i 0.564779i
\(804\) 0 0
\(805\) 81.3503 0.101056
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1078.34 −1.33292 −0.666462 0.745539i \(-0.732192\pi\)
−0.666462 + 0.745539i \(0.732192\pi\)
\(810\) 0 0
\(811\) − 606.040i − 0.747275i −0.927575 0.373638i \(-0.878110\pi\)
0.927575 0.373638i \(-0.121890\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 573.846i 0.704106i
\(816\) 0 0
\(817\) −2.42285 −0.00296554
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 326.498 0.397683 0.198842 0.980032i \(-0.436282\pi\)
0.198842 + 0.980032i \(0.436282\pi\)
\(822\) 0 0
\(823\) − 1114.44i − 1.35412i −0.735928 0.677060i \(-0.763254\pi\)
0.735928 0.677060i \(-0.236746\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 381.985i 0.461892i 0.972967 + 0.230946i \(0.0741821\pi\)
−0.972967 + 0.230946i \(0.925818\pi\)
\(828\) 0 0
\(829\) 238.484 0.287677 0.143838 0.989601i \(-0.454055\pi\)
0.143838 + 0.989601i \(0.454055\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −321.004 −0.385360
\(834\) 0 0
\(835\) 598.445i 0.716701i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 478.703i 0.570563i 0.958444 + 0.285282i \(0.0920872\pi\)
−0.958444 + 0.285282i \(0.907913\pi\)
\(840\) 0 0
\(841\) −657.103 −0.781335
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −145.442 −0.172121
\(846\) 0 0
\(847\) 207.218i 0.244649i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 432.427i 0.508139i
\(852\) 0 0
\(853\) 1425.70 1.67140 0.835700 0.549186i \(-0.185062\pi\)
0.835700 + 0.549186i \(0.185062\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −11.4474 −0.0133575 −0.00667876 0.999978i \(-0.502126\pi\)
−0.00667876 + 0.999978i \(0.502126\pi\)
\(858\) 0 0
\(859\) 221.720i 0.258114i 0.991637 + 0.129057i \(0.0411951\pi\)
−0.991637 + 0.129057i \(0.958805\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1675.77i 1.94180i 0.239487 + 0.970900i \(0.423021\pi\)
−0.239487 + 0.970900i \(0.576979\pi\)
\(864\) 0 0
\(865\) −786.081 −0.908764
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −430.497 −0.495393
\(870\) 0 0
\(871\) 273.754i 0.314299i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 351.370i − 0.401566i
\(876\) 0 0
\(877\) 748.650 0.853649 0.426825 0.904334i \(-0.359632\pi\)
0.426825 + 0.904334i \(0.359632\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −705.252 −0.800513 −0.400256 0.916403i \(-0.631079\pi\)
−0.400256 + 0.916403i \(0.631079\pi\)
\(882\) 0 0
\(883\) 1657.97i 1.87765i 0.344392 + 0.938826i \(0.388085\pi\)
−0.344392 + 0.938826i \(0.611915\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1407.84i − 1.58719i −0.608444 0.793597i \(-0.708206\pi\)
0.608444 0.793597i \(-0.291794\pi\)
\(888\) 0 0
\(889\) −498.721 −0.560991
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.36438 0.00152786
\(894\) 0 0
\(895\) 317.007i 0.354198i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 403.914i − 0.449293i
\(900\) 0 0
\(901\) 665.348 0.738455
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 642.735 0.710204
\(906\) 0 0
\(907\) − 238.955i − 0.263457i −0.991286 0.131728i \(-0.957947\pi\)
0.991286 0.131728i \(-0.0420527\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 659.103i − 0.723494i −0.932276 0.361747i \(-0.882180\pi\)
0.932276 0.361747i \(-0.117820\pi\)
\(912\) 0 0
\(913\) −54.6453 −0.0598525
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 489.992 0.534342
\(918\) 0 0
\(919\) − 1171.93i − 1.27522i −0.770357 0.637612i \(-0.779922\pi\)
0.770357 0.637612i \(-0.220078\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 589.238i 0.638394i
\(924\) 0 0
\(925\) 432.019 0.467048
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1676.14 −1.80424 −0.902121 0.431482i \(-0.857991\pi\)
−0.902121 + 0.431482i \(0.857991\pi\)
\(930\) 0 0
\(931\) − 1.31721i − 0.00141483i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 202.584i − 0.216667i
\(936\) 0 0
\(937\) 1791.65 1.91212 0.956058 0.293176i \(-0.0947123\pi\)
0.956058 + 0.293176i \(0.0947123\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −269.577 −0.286479 −0.143239 0.989688i \(-0.545752\pi\)
−0.143239 + 0.989688i \(0.545752\pi\)
\(942\) 0 0
\(943\) 204.626i 0.216995i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.8401i 0.0399579i 0.999800 + 0.0199789i \(0.00635992\pi\)
−0.999800 + 0.0199789i \(0.993640\pi\)
\(948\) 0 0
\(949\) −822.376 −0.866571
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1574.25 −1.65188 −0.825942 0.563755i \(-0.809356\pi\)
−0.825942 + 0.563755i \(0.809356\pi\)
\(954\) 0 0
\(955\) 843.793i 0.883553i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 508.434i − 0.530171i
\(960\) 0 0
\(961\) 73.8399 0.0768365
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −597.213 −0.618874
\(966\) 0 0
\(967\) − 1105.96i − 1.14371i −0.820356 0.571853i \(-0.806225\pi\)
0.820356 0.571853i \(-0.193775\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 390.576i − 0.402241i −0.979567 0.201120i \(-0.935542\pi\)
0.979567 0.201120i \(-0.0644583\pi\)
\(972\) 0 0
\(973\) 270.206 0.277704
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −125.557 −0.128513 −0.0642564 0.997933i \(-0.520468\pi\)
−0.0642564 + 0.997933i \(0.520468\pi\)
\(978\) 0 0
\(979\) 33.3668i 0.0340826i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1264.46i − 1.28633i −0.765729 0.643164i \(-0.777621\pi\)
0.765729 0.643164i \(-0.222379\pi\)
\(984\) 0 0
\(985\) −1624.16 −1.64889
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −586.157 −0.592676
\(990\) 0 0
\(991\) − 264.800i − 0.267205i −0.991035 0.133602i \(-0.957346\pi\)
0.991035 0.133602i \(-0.0426545\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 83.3511i 0.0837699i
\(996\) 0 0
\(997\) 166.052 0.166552 0.0832758 0.996527i \(-0.473462\pi\)
0.0832758 + 0.996527i \(0.473462\pi\)
\(998\) 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.703.8 8
3.2 odd 2 1728.3.g.n.703.2 8
4.3 odd 2 inner 1728.3.g.k.703.7 8
8.3 odd 2 864.3.g.c.703.1 yes 8
8.5 even 2 864.3.g.c.703.2 yes 8
12.11 even 2 1728.3.g.n.703.1 8
24.5 odd 2 864.3.g.a.703.8 yes 8
24.11 even 2 864.3.g.a.703.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.a.703.7 8 24.11 even 2
864.3.g.a.703.8 yes 8 24.5 odd 2
864.3.g.c.703.1 yes 8 8.3 odd 2
864.3.g.c.703.2 yes 8 8.5 even 2
1728.3.g.k.703.7 8 4.3 odd 2 inner
1728.3.g.k.703.8 8 1.1 even 1 trivial
1728.3.g.n.703.1 8 12.11 even 2
1728.3.g.n.703.2 8 3.2 odd 2