Properties

Label 1764.3.c.h.197.3
Level $1764$
Weight $3$
Character 1764.197
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.224054542336.12
Defining polynomial: \(x^{8} + 199 x^{4} + 14641\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.3
Root \(-1.96485 - 2.67196i\) of defining polynomial
Character \(\chi\) \(=\) 1764.197
Dual form 1764.3.c.h.197.6

$q$-expansion

\(f(q)\) \(=\) \(q-3.55744i q^{5} +O(q^{10})\) \(q-3.55744i q^{5} -9.27362i q^{11} -4.24264 q^{13} +2.44256i q^{17} +20.7498 q^{19} +0.788337i q^{23} +12.3446 q^{25} -7.69694i q^{29} +23.5782 q^{31} -23.3446 q^{37} -51.5574i q^{41} -49.3446 q^{43} +15.7702i q^{47} -75.2782i q^{53} -32.9903 q^{55} +15.7702i q^{59} -8.02187 q^{61} +15.0929i q^{65} -87.3446 q^{67} -40.0614i q^{71} +34.4045 q^{73} +8.68926 q^{79} -115.115i q^{83} +8.68926 q^{85} +94.2467i q^{89} -73.8161i q^{95} -154.173 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 216 q^{25} + 128 q^{37} - 80 q^{43} - 384 q^{67} - 560 q^{79} - 560 q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 3.55744i − 0.711488i −0.934583 0.355744i \(-0.884228\pi\)
0.934583 0.355744i \(-0.115772\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 9.27362i − 0.843056i −0.906815 0.421528i \(-0.861494\pi\)
0.906815 0.421528i \(-0.138506\pi\)
\(12\) 0 0
\(13\) −4.24264 −0.326357 −0.163178 0.986597i \(-0.552175\pi\)
−0.163178 + 0.986597i \(0.552175\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.44256i 0.143680i 0.997416 + 0.0718400i \(0.0228871\pi\)
−0.997416 + 0.0718400i \(0.977113\pi\)
\(18\) 0 0
\(19\) 20.7498 1.09209 0.546047 0.837755i \(-0.316132\pi\)
0.546047 + 0.837755i \(0.316132\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.788337i 0.0342755i 0.999853 + 0.0171378i \(0.00545539\pi\)
−0.999853 + 0.0171378i \(0.994545\pi\)
\(24\) 0 0
\(25\) 12.3446 0.493785
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 7.69694i − 0.265412i −0.991155 0.132706i \(-0.957633\pi\)
0.991155 0.132706i \(-0.0423666\pi\)
\(30\) 0 0
\(31\) 23.5782 0.760588 0.380294 0.924866i \(-0.375823\pi\)
0.380294 + 0.924866i \(0.375823\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −23.3446 −0.630936 −0.315468 0.948936i \(-0.602161\pi\)
−0.315468 + 0.948936i \(0.602161\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 51.5574i − 1.25750i −0.777608 0.628749i \(-0.783567\pi\)
0.777608 0.628749i \(-0.216433\pi\)
\(42\) 0 0
\(43\) −49.3446 −1.14755 −0.573775 0.819013i \(-0.694522\pi\)
−0.573775 + 0.819013i \(0.694522\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 15.7702i 0.335537i 0.985826 + 0.167769i \(0.0536561\pi\)
−0.985826 + 0.167769i \(0.946344\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 75.2782i − 1.42034i −0.704028 0.710172i \(-0.748617\pi\)
0.704028 0.710172i \(-0.251383\pi\)
\(54\) 0 0
\(55\) −32.9903 −0.599824
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 15.7702i 0.267292i 0.991029 + 0.133646i \(0.0426686\pi\)
−0.991029 + 0.133646i \(0.957331\pi\)
\(60\) 0 0
\(61\) −8.02187 −0.131506 −0.0657530 0.997836i \(-0.520945\pi\)
−0.0657530 + 0.997836i \(0.520945\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.0929i 0.232199i
\(66\) 0 0
\(67\) −87.3446 −1.30365 −0.651826 0.758369i \(-0.725997\pi\)
−0.651826 + 0.758369i \(0.725997\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 40.0614i − 0.564245i −0.959378 0.282123i \(-0.908962\pi\)
0.959378 0.282123i \(-0.0910385\pi\)
\(72\) 0 0
\(73\) 34.4045 0.471295 0.235648 0.971839i \(-0.424279\pi\)
0.235648 + 0.971839i \(0.424279\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.68926 0.109991 0.0549953 0.998487i \(-0.482486\pi\)
0.0549953 + 0.998487i \(0.482486\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 115.115i − 1.38693i −0.720492 0.693463i \(-0.756084\pi\)
0.720492 0.693463i \(-0.243916\pi\)
\(84\) 0 0
\(85\) 8.68926 0.102227
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 94.2467i 1.05895i 0.848325 + 0.529476i \(0.177611\pi\)
−0.848325 + 0.529476i \(0.822389\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 73.8161i − 0.777012i
\(96\) 0 0
\(97\) −154.173 −1.58941 −0.794707 0.606993i \(-0.792376\pi\)
−0.794707 + 0.606993i \(0.792376\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 191.362i 1.89467i 0.320246 + 0.947335i \(0.396235\pi\)
−0.320246 + 0.947335i \(0.603765\pi\)
\(102\) 0 0
\(103\) 81.0736 0.787122 0.393561 0.919298i \(-0.371243\pi\)
0.393561 + 0.919298i \(0.371243\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 166.137i − 1.55268i −0.630314 0.776340i \(-0.717074\pi\)
0.630314 0.776340i \(-0.282926\pi\)
\(108\) 0 0
\(109\) −128.689 −1.18064 −0.590318 0.807171i \(-0.700998\pi\)
−0.590318 + 0.807171i \(0.700998\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 188.138i − 1.66494i −0.554069 0.832471i \(-0.686926\pi\)
0.554069 0.832471i \(-0.313074\pi\)
\(114\) 0 0
\(115\) 2.80446 0.0243866
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 35.0000 0.289256
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 132.851i − 1.06281i
\(126\) 0 0
\(127\) −40.6554 −0.320121 −0.160061 0.987107i \(-0.551169\pi\)
−0.160061 + 0.987107i \(0.551169\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 96.0000i 0.732824i 0.930453 + 0.366412i \(0.119414\pi\)
−0.930453 + 0.366412i \(0.880586\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 116.429i − 0.849846i −0.905229 0.424923i \(-0.860301\pi\)
0.905229 0.424923i \(-0.139699\pi\)
\(138\) 0 0
\(139\) −99.9218 −0.718862 −0.359431 0.933172i \(-0.617029\pi\)
−0.359431 + 0.933172i \(0.617029\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 39.3446i 0.275137i
\(144\) 0 0
\(145\) −27.3814 −0.188837
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 108.617i 0.728976i 0.931208 + 0.364488i \(0.118756\pi\)
−0.931208 + 0.364488i \(0.881244\pi\)
\(150\) 0 0
\(151\) −224.034 −1.48367 −0.741834 0.670583i \(-0.766044\pi\)
−0.741834 + 0.670583i \(0.766044\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 83.8780i − 0.541149i
\(156\) 0 0
\(157\) 105.115 0.669524 0.334762 0.942303i \(-0.391344\pi\)
0.334762 + 0.942303i \(0.391344\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −21.9661 −0.134761 −0.0673807 0.997727i \(-0.521464\pi\)
−0.0673807 + 0.997727i \(0.521464\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 311.838i − 1.86729i −0.358195 0.933647i \(-0.616608\pi\)
0.358195 0.933647i \(-0.383392\pi\)
\(168\) 0 0
\(169\) −151.000 −0.893491
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 64.6723i 0.373828i 0.982376 + 0.186914i \(0.0598486\pi\)
−0.982376 + 0.186914i \(0.940151\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 334.266i − 1.86741i −0.358048 0.933703i \(-0.616558\pi\)
0.358048 0.933703i \(-0.383442\pi\)
\(180\) 0 0
\(181\) −229.590 −1.26845 −0.634226 0.773147i \(-0.718681\pi\)
−0.634226 + 0.773147i \(0.718681\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 83.0471i 0.448903i
\(186\) 0 0
\(187\) 22.6514 0.121130
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 75.5792i − 0.395703i −0.980232 0.197851i \(-0.936604\pi\)
0.980232 0.197851i \(-0.0633963\pi\)
\(192\) 0 0
\(193\) 9.37852 0.0485934 0.0242967 0.999705i \(-0.492265\pi\)
0.0242967 + 0.999705i \(0.492265\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 38.5566i 0.195719i 0.995200 + 0.0978594i \(0.0311996\pi\)
−0.995200 + 0.0978594i \(0.968800\pi\)
\(198\) 0 0
\(199\) 65.1018 0.327145 0.163572 0.986531i \(-0.447698\pi\)
0.163572 + 0.986531i \(0.447698\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −183.412 −0.894695
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 192.426i − 0.920697i
\(210\) 0 0
\(211\) −176.068 −0.834444 −0.417222 0.908804i \(-0.636996\pi\)
−0.417222 + 0.908804i \(0.636996\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 175.540i 0.816467i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 10.3629i − 0.0468910i
\(222\) 0 0
\(223\) 146.199 0.655602 0.327801 0.944747i \(-0.393692\pi\)
0.327801 + 0.944747i \(0.393692\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.6893i 0.108763i 0.998520 + 0.0543816i \(0.0173188\pi\)
−0.998520 + 0.0543816i \(0.982681\pi\)
\(228\) 0 0
\(229\) 209.839 0.916327 0.458164 0.888868i \(-0.348507\pi\)
0.458164 + 0.888868i \(0.348507\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 40.0614i 0.171937i 0.996298 + 0.0859687i \(0.0273985\pi\)
−0.996298 + 0.0859687i \(0.972602\pi\)
\(234\) 0 0
\(235\) 56.1017 0.238731
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 229.662i − 0.960928i −0.877014 0.480464i \(-0.840468\pi\)
0.877014 0.480464i \(-0.159532\pi\)
\(240\) 0 0
\(241\) 381.422 1.58266 0.791332 0.611386i \(-0.209388\pi\)
0.791332 + 0.611386i \(0.209388\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −88.0339 −0.356413
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 72.2636i 0.287903i 0.989585 + 0.143951i \(0.0459809\pi\)
−0.989585 + 0.143951i \(0.954019\pi\)
\(252\) 0 0
\(253\) 7.31074 0.0288962
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 222.740i − 0.866693i −0.901227 0.433347i \(-0.857333\pi\)
0.901227 0.433347i \(-0.142667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 322.025i − 1.22443i −0.790691 0.612215i \(-0.790279\pi\)
0.790691 0.612215i \(-0.209721\pi\)
\(264\) 0 0
\(265\) −267.798 −1.01056
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 307.591i 1.14346i 0.820441 + 0.571731i \(0.193728\pi\)
−0.820441 + 0.571731i \(0.806272\pi\)
\(270\) 0 0
\(271\) 362.111 1.33620 0.668101 0.744071i \(-0.267108\pi\)
0.668101 + 0.744071i \(0.267108\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 114.479i − 0.416289i
\(276\) 0 0
\(277\) 28.6893 0.103571 0.0517857 0.998658i \(-0.483509\pi\)
0.0517857 + 0.998658i \(0.483509\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 89.7693i 0.319464i 0.987160 + 0.159732i \(0.0510629\pi\)
−0.987160 + 0.159732i \(0.948937\pi\)
\(282\) 0 0
\(283\) −301.739 −1.06621 −0.533107 0.846048i \(-0.678976\pi\)
−0.533107 + 0.846048i \(0.678976\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 283.034 0.979356
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 235.855i − 0.804966i −0.915428 0.402483i \(-0.868147\pi\)
0.915428 0.402483i \(-0.131853\pi\)
\(294\) 0 0
\(295\) 56.1017 0.190175
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 3.34463i − 0.0111861i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 28.5373i 0.0935649i
\(306\) 0 0
\(307\) −68.8810 −0.224368 −0.112184 0.993687i \(-0.535785\pi\)
−0.112184 + 0.993687i \(0.535785\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 499.906i − 1.60741i −0.595025 0.803707i \(-0.702858\pi\)
0.595025 0.803707i \(-0.297142\pi\)
\(312\) 0 0
\(313\) −288.012 −0.920167 −0.460083 0.887876i \(-0.652180\pi\)
−0.460083 + 0.887876i \(0.652180\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 165.090i 0.520789i 0.965502 + 0.260395i \(0.0838526\pi\)
−0.965502 + 0.260395i \(0.916147\pi\)
\(318\) 0 0
\(319\) −71.3785 −0.223757
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 50.6826i 0.156912i
\(324\) 0 0
\(325\) −52.3738 −0.161150
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −265.311 −0.801543 −0.400772 0.916178i \(-0.631258\pi\)
−0.400772 + 0.916178i \(0.631258\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 310.723i 0.927532i
\(336\) 0 0
\(337\) 220.655 0.654764 0.327382 0.944892i \(-0.393834\pi\)
0.327382 + 0.944892i \(0.393834\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 218.655i − 0.641218i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 187.235i 0.539583i 0.962919 + 0.269792i \(0.0869549\pi\)
−0.962919 + 0.269792i \(0.913045\pi\)
\(348\) 0 0
\(349\) 318.270 0.911948 0.455974 0.889993i \(-0.349291\pi\)
0.455974 + 0.889993i \(0.349291\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 169.166i 0.479223i 0.970869 + 0.239611i \(0.0770201\pi\)
−0.970869 + 0.239611i \(0.922980\pi\)
\(354\) 0 0
\(355\) −142.516 −0.401453
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 111.097i − 0.309462i −0.987957 0.154731i \(-0.950549\pi\)
0.987957 0.154731i \(-0.0494511\pi\)
\(360\) 0 0
\(361\) 69.5537 0.192669
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 122.392i − 0.335321i
\(366\) 0 0
\(367\) −81.0496 −0.220844 −0.110422 0.993885i \(-0.535220\pi\)
−0.110422 + 0.993885i \(0.535220\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 439.446 1.17814 0.589070 0.808082i \(-0.299494\pi\)
0.589070 + 0.808082i \(0.299494\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 32.6554i 0.0866190i
\(378\) 0 0
\(379\) −81.3785 −0.214719 −0.107360 0.994220i \(-0.534240\pi\)
−0.107360 + 0.994220i \(0.534240\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 460.987i 1.20362i 0.798639 + 0.601810i \(0.205554\pi\)
−0.798639 + 0.601810i \(0.794446\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 555.256i 1.42739i 0.700455 + 0.713697i \(0.252980\pi\)
−0.700455 + 0.713697i \(0.747020\pi\)
\(390\) 0 0
\(391\) −1.92556 −0.00492471
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 30.9115i − 0.0782570i
\(396\) 0 0
\(397\) −304.128 −0.766065 −0.383033 0.923735i \(-0.625120\pi\)
−0.383033 + 0.923735i \(0.625120\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 246.489i 0.614685i 0.951599 + 0.307342i \(0.0994397\pi\)
−0.951599 + 0.307342i \(0.900560\pi\)
\(402\) 0 0
\(403\) −100.034 −0.248223
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 216.489i 0.531915i
\(408\) 0 0
\(409\) 14.6056 0.0357104 0.0178552 0.999841i \(-0.494316\pi\)
0.0178552 + 0.999841i \(0.494316\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −409.514 −0.986781
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 619.804i 1.47925i 0.673021 + 0.739623i \(0.264996\pi\)
−0.673021 + 0.739623i \(0.735004\pi\)
\(420\) 0 0
\(421\) −307.379 −0.730115 −0.365058 0.930985i \(-0.618951\pi\)
−0.365058 + 0.930985i \(0.618951\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 30.1525i 0.0709471i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 522.891i 1.21321i 0.795005 + 0.606603i \(0.207468\pi\)
−0.795005 + 0.606603i \(0.792532\pi\)
\(432\) 0 0
\(433\) 547.444 1.26431 0.632153 0.774844i \(-0.282171\pi\)
0.632153 + 0.774844i \(0.282171\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.3578i 0.0374321i
\(438\) 0 0
\(439\) −516.771 −1.17716 −0.588578 0.808441i \(-0.700312\pi\)
−0.588578 + 0.808441i \(0.700312\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 547.974i − 1.23696i −0.785799 0.618481i \(-0.787748\pi\)
0.785799 0.618481i \(-0.212252\pi\)
\(444\) 0 0
\(445\) 335.277 0.753431
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 575.681i 1.28214i 0.767482 + 0.641070i \(0.221509\pi\)
−0.767482 + 0.641070i \(0.778491\pi\)
\(450\) 0 0
\(451\) −478.124 −1.06014
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 406.825 0.890208 0.445104 0.895479i \(-0.353167\pi\)
0.445104 + 0.895479i \(0.353167\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 209.524i 0.454498i 0.973837 + 0.227249i \(0.0729731\pi\)
−0.973837 + 0.227249i \(0.927027\pi\)
\(462\) 0 0
\(463\) 870.169 1.87942 0.939708 0.341978i \(-0.111097\pi\)
0.939708 + 0.341978i \(0.111097\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 232.196i − 0.497207i −0.968605 0.248604i \(-0.920028\pi\)
0.968605 0.248604i \(-0.0799717\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 457.603i 0.967449i
\(474\) 0 0
\(475\) 256.148 0.539260
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 889.642i − 1.85729i −0.370969 0.928645i \(-0.620974\pi\)
0.370969 0.928645i \(-0.379026\pi\)
\(480\) 0 0
\(481\) 99.0429 0.205910
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 548.462i 1.13085i
\(486\) 0 0
\(487\) −65.4124 −0.134317 −0.0671585 0.997742i \(-0.521393\pi\)
−0.0671585 + 0.997742i \(0.521393\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 82.4878i 0.168000i 0.996466 + 0.0839998i \(0.0267695\pi\)
−0.996466 + 0.0839998i \(0.973230\pi\)
\(492\) 0 0
\(493\) 18.8003 0.0381344
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −66.6554 −0.133578 −0.0667889 0.997767i \(-0.521275\pi\)
−0.0667889 + 0.997767i \(0.521275\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 592.885i 1.17870i 0.807879 + 0.589349i \(0.200616\pi\)
−0.807879 + 0.589349i \(0.799384\pi\)
\(504\) 0 0
\(505\) 680.757 1.34803
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 777.923i − 1.52834i −0.645018 0.764168i \(-0.723150\pi\)
0.645018 0.764168i \(-0.276850\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 288.414i − 0.560028i
\(516\) 0 0
\(517\) 146.247 0.282877
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 433.166i 0.831412i 0.909499 + 0.415706i \(0.136465\pi\)
−0.909499 + 0.415706i \(0.863535\pi\)
\(522\) 0 0
\(523\) 277.234 0.530084 0.265042 0.964237i \(-0.414614\pi\)
0.265042 + 0.964237i \(0.414614\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 57.5912i 0.109281i
\(528\) 0 0
\(529\) 528.379 0.998825
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 218.740i 0.410393i
\(534\) 0 0
\(535\) −591.021 −1.10471
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 720.757 1.33227 0.666134 0.745832i \(-0.267948\pi\)
0.666134 + 0.745832i \(0.267948\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 457.804i 0.840008i
\(546\) 0 0
\(547\) −744.825 −1.36165 −0.680827 0.732444i \(-0.738379\pi\)
−0.680827 + 0.732444i \(0.738379\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 159.710i − 0.289855i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 716.032i 1.28551i 0.766070 + 0.642757i \(0.222210\pi\)
−0.766070 + 0.642757i \(0.777790\pi\)
\(558\) 0 0
\(559\) 209.352 0.374511
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 750.851i − 1.33366i −0.745209 0.666831i \(-0.767650\pi\)
0.745209 0.666831i \(-0.232350\pi\)
\(564\) 0 0
\(565\) −669.291 −1.18459
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 169.748i 0.298327i 0.988813 + 0.149164i \(0.0476581\pi\)
−0.988813 + 0.149164i \(0.952342\pi\)
\(570\) 0 0
\(571\) 627.514 1.09897 0.549487 0.835502i \(-0.314823\pi\)
0.549487 + 0.835502i \(0.314823\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.73173i 0.0169247i
\(576\) 0 0
\(577\) 164.632 0.285324 0.142662 0.989771i \(-0.454434\pi\)
0.142662 + 0.989771i \(0.454434\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −698.102 −1.19743
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1040.76i 1.77301i 0.462719 + 0.886505i \(0.346874\pi\)
−0.462719 + 0.886505i \(0.653126\pi\)
\(588\) 0 0
\(589\) 489.243 0.830633
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 627.923i − 1.05889i −0.848344 0.529446i \(-0.822400\pi\)
0.848344 0.529446i \(-0.177600\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 263.832i 0.440454i 0.975449 + 0.220227i \(0.0706798\pi\)
−0.975449 + 0.220227i \(0.929320\pi\)
\(600\) 0 0
\(601\) 502.094 0.835431 0.417715 0.908578i \(-0.362831\pi\)
0.417715 + 0.908578i \(0.362831\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 124.510i − 0.205802i
\(606\) 0 0
\(607\) 999.601 1.64679 0.823395 0.567469i \(-0.192077\pi\)
0.823395 + 0.567469i \(0.192077\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 66.9075i − 0.109505i
\(612\) 0 0
\(613\) 920.169 1.50109 0.750546 0.660818i \(-0.229791\pi\)
0.750546 + 0.660818i \(0.229791\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 520.713i 0.843943i 0.906609 + 0.421972i \(0.138662\pi\)
−0.906609 + 0.421972i \(0.861338\pi\)
\(618\) 0 0
\(619\) −238.611 −0.385478 −0.192739 0.981250i \(-0.561737\pi\)
−0.192739 + 0.981250i \(0.561737\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −163.994 −0.262391
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 57.0207i − 0.0906529i
\(630\) 0 0
\(631\) 534.034 0.846329 0.423165 0.906053i \(-0.360919\pi\)
0.423165 + 0.906053i \(0.360919\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 144.629i 0.227762i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1003.03i 1.56478i 0.622787 + 0.782392i \(0.286001\pi\)
−0.622787 + 0.782392i \(0.713999\pi\)
\(642\) 0 0
\(643\) 744.923 1.15851 0.579256 0.815146i \(-0.303343\pi\)
0.579256 + 0.815146i \(0.303343\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 891.446i − 1.37782i −0.724849 0.688908i \(-0.758091\pi\)
0.724849 0.688908i \(-0.241909\pi\)
\(648\) 0 0
\(649\) 146.247 0.225342
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 81.5130i 0.124829i 0.998050 + 0.0624143i \(0.0198800\pi\)
−0.998050 + 0.0624143i \(0.980120\pi\)
\(654\) 0 0
\(655\) 341.514 0.521396
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 185.430i 0.281380i 0.990054 + 0.140690i \(0.0449322\pi\)
−0.990054 + 0.140690i \(0.955068\pi\)
\(660\) 0 0
\(661\) 545.567 0.825366 0.412683 0.910875i \(-0.364592\pi\)
0.412683 + 0.910875i \(0.364592\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.06779 0.00909713
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 74.3917i 0.110867i
\(672\) 0 0
\(673\) −994.136 −1.47717 −0.738585 0.674160i \(-0.764506\pi\)
−0.738585 + 0.674160i \(0.764506\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 209.949i − 0.310117i −0.987905 0.155058i \(-0.950443\pi\)
0.987905 0.155058i \(-0.0495566\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 410.633i − 0.601220i −0.953747 0.300610i \(-0.902810\pi\)
0.953747 0.300610i \(-0.0971903\pi\)
\(684\) 0 0
\(685\) −414.189 −0.604655
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 319.379i 0.463539i
\(690\) 0 0
\(691\) −1324.68 −1.91705 −0.958523 0.285016i \(-0.908001\pi\)
−0.958523 + 0.285016i \(0.908001\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 355.466i 0.511461i
\(696\) 0 0
\(697\) 125.932 0.180677
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 580.339i 0.827873i 0.910306 + 0.413936i \(0.135846\pi\)
−0.910306 + 0.413936i \(0.864154\pi\)
\(702\) 0 0
\(703\) −484.396 −0.689041
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −327.379 −0.461747 −0.230873 0.972984i \(-0.574158\pi\)
−0.230873 + 0.972984i \(0.574158\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.5876i 0.0260695i
\(714\) 0 0
\(715\) 139.966 0.195757
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 392.392i 0.545746i 0.962050 + 0.272873i \(0.0879740\pi\)
−0.962050 + 0.272873i \(0.912026\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 95.0159i − 0.131056i
\(726\) 0 0
\(727\) 1340.99 1.84455 0.922274 0.386537i \(-0.126329\pi\)
0.922274 + 0.386537i \(0.126329\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 120.527i − 0.164880i
\(732\) 0 0
\(733\) 130.403 0.177904 0.0889518 0.996036i \(-0.471648\pi\)
0.0889518 + 0.996036i \(0.471648\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 810.001i 1.09905i
\(738\) 0 0
\(739\) −202.034 −0.273388 −0.136694 0.990613i \(-0.543648\pi\)
−0.136694 + 0.990613i \(0.543648\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 821.453i 1.10559i 0.833317 + 0.552795i \(0.186439\pi\)
−0.833317 + 0.552795i \(0.813561\pi\)
\(744\) 0 0
\(745\) 386.400 0.518658
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 804.136 1.07075 0.535377 0.844613i \(-0.320170\pi\)
0.535377 + 0.844613i \(0.320170\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 796.987i 1.05561i
\(756\) 0 0
\(757\) 518.000 0.684280 0.342140 0.939649i \(-0.388848\pi\)
0.342140 + 0.939649i \(0.388848\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 673.855i − 0.885486i −0.896649 0.442743i \(-0.854005\pi\)
0.896649 0.442743i \(-0.145995\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 66.9075i − 0.0872327i
\(768\) 0 0
\(769\) −653.023 −0.849185 −0.424592 0.905385i \(-0.639583\pi\)
−0.424592 + 0.905385i \(0.639583\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 461.301i − 0.596768i −0.954446 0.298384i \(-0.903552\pi\)
0.954446 0.298384i \(-0.0964475\pi\)
\(774\) 0 0
\(775\) 291.064 0.375567
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 1069.81i − 1.37331i
\(780\) 0 0
\(781\) −371.514 −0.475690
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 373.941i − 0.476358i
\(786\) 0 0
\(787\) −793.126 −1.00778 −0.503892 0.863767i \(-0.668099\pi\)
−0.503892 + 0.863767i \(0.668099\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 34.0339 0.0429179
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 435.599i 0.546548i 0.961936 + 0.273274i \(0.0881066\pi\)
−0.961936 + 0.273274i \(0.911893\pi\)
\(798\) 0 0
\(799\) −38.5198 −0.0482100
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 319.055i − 0.397328i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 577.028i 0.713261i 0.934245 + 0.356631i \(0.116075\pi\)
−0.934245 + 0.356631i \(0.883925\pi\)
\(810\) 0 0
\(811\) −38.5274 −0.0475060 −0.0237530 0.999718i \(-0.507562\pi\)
−0.0237530 + 0.999718i \(0.507562\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 78.1431i 0.0958811i
\(816\) 0 0
\(817\) −1023.89 −1.25323
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 455.768i 0.555138i 0.960706 + 0.277569i \(0.0895287\pi\)
−0.960706 + 0.277569i \(0.910471\pi\)
\(822\) 0 0
\(823\) −399.582 −0.485519 −0.242759 0.970087i \(-0.578053\pi\)
−0.242759 + 0.970087i \(0.578053\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1182.94i − 1.43040i −0.698922 0.715198i \(-0.746336\pi\)
0.698922 0.715198i \(-0.253664\pi\)
\(828\) 0 0
\(829\) 163.561 0.197300 0.0986498 0.995122i \(-0.468548\pi\)
0.0986498 + 0.995122i \(0.468548\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1109.34 −1.32856
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 980.535i − 1.16869i −0.811504 0.584347i \(-0.801351\pi\)
0.811504 0.584347i \(-0.198649\pi\)
\(840\) 0 0
\(841\) 781.757 0.929557
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 537.173i 0.635708i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 18.4034i − 0.0216257i
\(852\) 0 0
\(853\) −184.455 −0.216243 −0.108121 0.994138i \(-0.534483\pi\)
−0.108121 + 0.994138i \(0.534483\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1146.42i 1.33771i 0.743394 + 0.668854i \(0.233215\pi\)
−0.743394 + 0.668854i \(0.766785\pi\)
\(858\) 0 0
\(859\) 891.074 1.03734 0.518670 0.854975i \(-0.326427\pi\)
0.518670 + 0.854975i \(0.326427\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1430.73i − 1.65786i −0.559354 0.828929i \(-0.688951\pi\)
0.559354 0.828929i \(-0.311049\pi\)
\(864\) 0 0
\(865\) 230.068 0.265974
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 80.5809i − 0.0927283i
\(870\) 0 0
\(871\) 370.572 0.425456
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1414.86 1.61329 0.806647 0.591034i \(-0.201280\pi\)
0.806647 + 0.591034i \(0.201280\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 407.260i − 0.462270i −0.972922 0.231135i \(-0.925756\pi\)
0.972922 0.231135i \(-0.0742439\pi\)
\(882\) 0 0
\(883\) −386.723 −0.437965 −0.218983 0.975729i \(-0.570274\pi\)
−0.218983 + 0.975729i \(0.570274\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1153.74i 1.30073i 0.759624 + 0.650363i \(0.225383\pi\)
−0.759624 + 0.650363i \(0.774617\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 327.229i 0.366438i
\(894\) 0 0
\(895\) −1189.13 −1.32864
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 181.480i − 0.201869i
\(900\) 0 0
\(901\) 183.872 0.204075
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 816.752i 0.902489i
\(906\) 0 0
\(907\) −568.169 −0.626427 −0.313214 0.949683i \(-0.601406\pi\)
−0.313214 + 0.949683i \(0.601406\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 78.5035i − 0.0861729i −0.999071 0.0430864i \(-0.986281\pi\)
0.999071 0.0430864i \(-0.0137191\pi\)
\(912\) 0 0
\(913\) −1067.53 −1.16926
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 542.927 0.590780 0.295390 0.955377i \(-0.404550\pi\)
0.295390 + 0.955377i \(0.404550\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 169.966i 0.184145i
\(924\) 0 0
\(925\) −288.181 −0.311547
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 183.335i 0.197347i 0.995120 + 0.0986734i \(0.0314599\pi\)
−0.995120 + 0.0986734i \(0.968540\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 80.5809i − 0.0861828i
\(936\) 0 0
\(937\) −532.184 −0.567966 −0.283983 0.958829i \(-0.591656\pi\)
−0.283983 + 0.958829i \(0.591656\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1666.13i 1.77059i 0.465029 + 0.885295i \(0.346044\pi\)
−0.465029 + 0.885295i \(0.653956\pi\)
\(942\) 0 0
\(943\) 40.6446 0.0431014
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 538.887i − 0.569047i −0.958669 0.284523i \(-0.908165\pi\)
0.958669 0.284523i \(-0.0918353\pi\)
\(948\) 0 0
\(949\) −145.966 −0.153810
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1047.62i − 1.09928i −0.835400 0.549642i \(-0.814764\pi\)
0.835400 0.549642i \(-0.185236\pi\)
\(954\) 0 0
\(955\) −268.868 −0.281538
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −405.068 −0.421507
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 33.3635i − 0.0345736i
\(966\) 0 0
\(967\) 347.209 0.359058 0.179529 0.983753i \(-0.442543\pi\)
0.179529 + 0.983753i \(0.442543\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 448.885i − 0.462292i −0.972919 0.231146i \(-0.925753\pi\)
0.972919 0.231146i \(-0.0742474\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 945.494i − 0.967752i −0.875137 0.483876i \(-0.839229\pi\)
0.875137 0.483876i \(-0.160771\pi\)
\(978\) 0 0
\(979\) 874.008 0.892756
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1079.25i − 1.09792i −0.835850 0.548958i \(-0.815025\pi\)
0.835850 0.548958i \(-0.184975\pi\)
\(984\) 0 0
\(985\) 137.163 0.139252
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 38.9002i − 0.0393329i
\(990\) 0 0
\(991\) −786.859 −0.794005 −0.397002 0.917818i \(-0.629950\pi\)
−0.397002 + 0.917818i \(0.629950\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 231.595i − 0.232759i
\(996\) 0 0
\(997\) 1876.00 1.88164 0.940822 0.338902i \(-0.110056\pi\)
0.940822 + 0.338902i \(0.110056\pi\)
\(998\) 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 1764.3.c.h.197.3 8
3.2 odd 2 inner 1764.3.c.h.197.6 yes 8
7.2 even 3 1764.3.bk.g.557.4 16
7.3 odd 6 1764.3.bk.g.1745.3 16
7.4 even 3 1764.3.bk.g.1745.5 16
7.5 odd 6 1764.3.bk.g.557.6 16
7.6 odd 2 inner 1764.3.c.h.197.5 yes 8
21.2 odd 6 1764.3.bk.g.557.5 16
21.5 even 6 1764.3.bk.g.557.3 16
21.11 odd 6 1764.3.bk.g.1745.4 16
21.17 even 6 1764.3.bk.g.1745.6 16
21.20 even 2 inner 1764.3.c.h.197.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.3.c.h.197.3 8 1.1 even 1 trivial
1764.3.c.h.197.4 yes 8 21.20 even 2 inner
1764.3.c.h.197.5 yes 8 7.6 odd 2 inner
1764.3.c.h.197.6 yes 8 3.2 odd 2 inner
1764.3.bk.g.557.3 16 21.5 even 6
1764.3.bk.g.557.4 16 7.2 even 3
1764.3.bk.g.557.5 16 21.2 odd 6
1764.3.bk.g.557.6 16 7.5 odd 6
1764.3.bk.g.1745.3 16 7.3 odd 6
1764.3.bk.g.1745.4 16 21.11 odd 6
1764.3.bk.g.1745.5 16 7.4 even 3
1764.3.bk.g.1745.6 16 21.17 even 6