Properties

Label 1728.2.d.g.865.1
Level $1728$
Weight $2$
Character 1728.865
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 865.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.865
Dual form 1728.2.d.g.865.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{5} -1.73205 q^{7} +O(q^{10})\) \(q-1.73205i q^{5} -1.73205 q^{7} +3.00000i q^{11} +4.00000i q^{19} +6.92820 q^{23} +2.00000 q^{25} -6.92820i q^{29} +1.73205 q^{31} +3.00000i q^{35} +6.92820i q^{37} +12.0000 q^{41} -4.00000i q^{43} +6.92820 q^{47} -4.00000 q^{49} -8.66025i q^{53} +5.19615 q^{55} -13.8564i q^{61} -4.00000i q^{67} +13.8564 q^{71} -7.00000 q^{73} -5.19615i q^{77} -10.3923 q^{79} +9.00000i q^{83} +12.0000 q^{89} +6.92820 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 8 q^{25} + 48 q^{41} - 16 q^{49} - 28 q^{73} + 48 q^{89} + 28 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.73205i − 0.774597i −0.921954 0.387298i \(-0.873408\pi\)
0.921954 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) −1.73205 −0.654654 −0.327327 0.944911i \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.92820i − 1.28654i −0.765641 0.643268i \(-0.777578\pi\)
0.765641 0.643268i \(-0.222422\pi\)
\(30\) 0 0
\(31\) 1.73205 0.311086 0.155543 0.987829i \(-0.450287\pi\)
0.155543 + 0.987829i \(0.450287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000i 0.507093i
\(36\) 0 0
\(37\) 6.92820i 1.13899i 0.821995 + 0.569495i \(0.192861\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 8.66025i − 1.18958i −0.803882 0.594789i \(-0.797236\pi\)
0.803882 0.594789i \(-0.202764\pi\)
\(54\) 0 0
\(55\) 5.19615 0.700649
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 13.8564i − 1.77413i −0.461644 0.887066i \(-0.652740\pi\)
0.461644 0.887066i \(-0.347260\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.19615i − 0.592157i
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.92820 0.710819
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 8.66025i − 0.861727i −0.902417 0.430864i \(-0.858209\pi\)
0.902417 0.430864i \(-0.141791\pi\)
\(102\) 0 0
\(103\) 3.46410 0.341328 0.170664 0.985329i \(-0.445409\pi\)
0.170664 + 0.985329i \(0.445409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.00000i 0.870063i 0.900415 + 0.435031i \(0.143263\pi\)
−0.900415 + 0.435031i \(0.856737\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i 0.943349 + 0.331801i \(0.107656\pi\)
−0.943349 + 0.331801i \(0.892344\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) − 12.0000i − 1.11901i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 12.1244i − 1.08444i
\(126\) 0 0
\(127\) 19.0526 1.69064 0.845321 0.534259i \(-0.179409\pi\)
0.845321 + 0.534259i \(0.179409\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000i 0.262111i 0.991375 + 0.131056i \(0.0418366\pi\)
−0.991375 + 0.131056i \(0.958163\pi\)
\(132\) 0 0
\(133\) − 6.92820i − 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 5.19615i − 0.425685i −0.977086 0.212843i \(-0.931728\pi\)
0.977086 0.212843i \(-0.0682722\pi\)
\(150\) 0 0
\(151\) −22.5167 −1.83238 −0.916190 0.400744i \(-0.868752\pi\)
−0.916190 + 0.400744i \(0.868752\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 3.00000i − 0.240966i
\(156\) 0 0
\(157\) 13.8564i 1.10586i 0.833227 + 0.552931i \(0.186491\pi\)
−0.833227 + 0.552931i \(0.813509\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.92820 −0.536120 −0.268060 0.963402i \(-0.586383\pi\)
−0.268060 + 0.963402i \(0.586383\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.66025i 0.658427i 0.944256 + 0.329213i \(0.106784\pi\)
−0.944256 + 0.329213i \(0.893216\pi\)
\(174\) 0 0
\(175\) −3.46410 −0.261861
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 3.00000i − 0.224231i −0.993695 0.112115i \(-0.964237\pi\)
0.993695 0.112115i \(-0.0357626\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i 0.857209 + 0.514969i \(0.172197\pi\)
−0.857209 + 0.514969i \(0.827803\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8564 1.00261 0.501307 0.865269i \(-0.332853\pi\)
0.501307 + 0.865269i \(0.332853\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.5885i 1.11063i 0.831640 + 0.555316i \(0.187403\pi\)
−0.831640 + 0.555316i \(0.812597\pi\)
\(198\) 0 0
\(199\) 15.5885 1.10504 0.552518 0.833501i \(-0.313667\pi\)
0.552518 + 0.833501i \(0.313667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000i 0.842235i
\(204\) 0 0
\(205\) − 20.7846i − 1.45166i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) − 4.00000i − 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.92820 −0.472500
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −17.3205 −1.15987 −0.579934 0.814664i \(-0.696921\pi\)
−0.579934 + 0.814664i \(0.696921\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 0 0
\(229\) − 6.92820i − 0.457829i −0.973447 0.228914i \(-0.926482\pi\)
0.973447 0.228914i \(-0.0735176\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) − 12.0000i − 0.782794i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.92820 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.92820i 0.442627i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 24.0000i − 1.51487i −0.652913 0.757433i \(-0.726453\pi\)
0.652913 0.757433i \(-0.273547\pi\)
\(252\) 0 0
\(253\) 20.7846i 1.30672i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) − 12.0000i − 0.745644i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.7846 −1.28163 −0.640817 0.767694i \(-0.721404\pi\)
−0.640817 + 0.767694i \(0.721404\pi\)
\(264\) 0 0
\(265\) −15.0000 −0.921443
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.7846i 1.26726i 0.773636 + 0.633630i \(0.218436\pi\)
−0.773636 + 0.633630i \(0.781564\pi\)
\(270\) 0 0
\(271\) −1.73205 −0.105215 −0.0526073 0.998615i \(-0.516753\pi\)
−0.0526073 + 0.998615i \(0.516753\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000i 0.361814i
\(276\) 0 0
\(277\) − 27.7128i − 1.66510i −0.553949 0.832551i \(-0.686880\pi\)
0.553949 0.832551i \(-0.313120\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.7846 −1.22688
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.92820i 0.404750i 0.979308 + 0.202375i \(0.0648660\pi\)
−0.979308 + 0.202375i \(0.935134\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 6.92820i 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) 32.0000i 1.82634i 0.407583 + 0.913168i \(0.366372\pi\)
−0.407583 + 0.913168i \(0.633628\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.92820 −0.392862 −0.196431 0.980518i \(-0.562935\pi\)
−0.196431 + 0.980518i \(0.562935\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.5885i 0.875535i 0.899088 + 0.437767i \(0.144231\pi\)
−0.899088 + 0.437767i \(0.855769\pi\)
\(318\) 0 0
\(319\) 20.7846 1.16371
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) − 20.0000i − 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.92820 −0.378528
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.19615i 0.281387i
\(342\) 0 0
\(343\) 19.0526 1.02874
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 27.0000i − 1.44944i −0.689046 0.724718i \(-0.741970\pi\)
0.689046 0.724718i \(-0.258030\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i 0.982658 + 0.185429i \(0.0593675\pi\)
−0.982658 + 0.185429i \(0.940632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −36.0000 −1.91609 −0.958043 0.286623i \(-0.907467\pi\)
−0.958043 + 0.286623i \(0.907467\pi\)
\(354\) 0 0
\(355\) − 24.0000i − 1.27379i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.92820 −0.365657 −0.182828 0.983145i \(-0.558525\pi\)
−0.182828 + 0.983145i \(0.558525\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.1244i 0.634618i
\(366\) 0 0
\(367\) −12.1244 −0.632886 −0.316443 0.948611i \(-0.602489\pi\)
−0.316443 + 0.948611i \(0.602489\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 15.0000i 0.778761i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 16.0000i − 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −27.7128 −1.41606 −0.708029 0.706183i \(-0.750416\pi\)
−0.708029 + 0.706183i \(0.750416\pi\)
\(384\) 0 0
\(385\) −9.00000 −0.458682
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.5167i 1.14164i 0.821075 + 0.570820i \(0.193375\pi\)
−0.821075 + 0.570820i \(0.806625\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 18.0000i 0.905678i
\(396\) 0 0
\(397\) 13.8564i 0.695433i 0.937600 + 0.347717i \(0.113043\pi\)
−0.937600 + 0.347717i \(0.886957\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.7846 −1.03025
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 15.5885 0.765207
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 24.0000i − 1.17248i −0.810139 0.586238i \(-0.800608\pi\)
0.810139 0.586238i \(-0.199392\pi\)
\(420\) 0 0
\(421\) − 13.8564i − 0.675320i −0.941268 0.337660i \(-0.890365\pi\)
0.941268 0.337660i \(-0.109635\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 24.0000i 1.16144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) 25.0000 1.20142 0.600712 0.799466i \(-0.294884\pi\)
0.600712 + 0.799466i \(0.294884\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.7128i 1.32568i
\(438\) 0 0
\(439\) −25.9808 −1.23999 −0.619997 0.784604i \(-0.712866\pi\)
−0.619997 + 0.784604i \(0.712866\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) − 20.7846i − 0.985285i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 36.0000i 1.69517i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 15.5885i − 0.726027i −0.931784 0.363013i \(-0.881748\pi\)
0.931784 0.363013i \(-0.118252\pi\)
\(462\) 0 0
\(463\) −15.5885 −0.724457 −0.362229 0.932089i \(-0.617984\pi\)
−0.362229 + 0.932089i \(0.617984\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 15.0000i − 0.694117i −0.937843 0.347059i \(-0.887180\pi\)
0.937843 0.347059i \(-0.112820\pi\)
\(468\) 0 0
\(469\) 6.92820i 0.319915i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 8.00000i 0.367065i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.7846 −0.949673 −0.474837 0.880074i \(-0.657493\pi\)
−0.474837 + 0.880074i \(0.657493\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 12.1244i − 0.550539i
\(486\) 0 0
\(487\) −31.1769 −1.41276 −0.706380 0.707832i \(-0.749673\pi\)
−0.706380 + 0.707832i \(0.749673\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.0000i 0.676941i 0.940977 + 0.338470i \(0.109909\pi\)
−0.940977 + 0.338470i \(0.890091\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) − 28.0000i − 1.25345i −0.779240 0.626726i \(-0.784395\pi\)
0.779240 0.626726i \(-0.215605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.92820 0.308913 0.154457 0.988000i \(-0.450637\pi\)
0.154457 + 0.988000i \(0.450637\pi\)
\(504\) 0 0
\(505\) −15.0000 −0.667491
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 32.9090i 1.45866i 0.684160 + 0.729332i \(0.260169\pi\)
−0.684160 + 0.729332i \(0.739831\pi\)
\(510\) 0 0
\(511\) 12.1244 0.536350
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 6.00000i − 0.264392i
\(516\) 0 0
\(517\) 20.7846i 0.914106i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 15.5885 0.673948
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 12.0000i − 0.516877i
\(540\) 0 0
\(541\) − 13.8564i − 0.595733i −0.954607 0.297867i \(-0.903725\pi\)
0.954607 0.297867i \(-0.0962751\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.0000 0.514024
\(546\) 0 0
\(547\) − 8.00000i − 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 27.7128 1.18061
\(552\) 0 0
\(553\) 18.0000 0.765438
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 25.9808i 1.10084i 0.834888 + 0.550420i \(0.185532\pi\)
−0.834888 + 0.550420i \(0.814468\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 9.00000i − 0.379305i −0.981851 0.189652i \(-0.939264\pi\)
0.981851 0.189652i \(-0.0607361\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 40.0000i 1.67395i 0.547243 + 0.836974i \(0.315677\pi\)
−0.547243 + 0.836974i \(0.684323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.8564 0.577852
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 15.5885i − 0.646718i
\(582\) 0 0
\(583\) 25.9808 1.07601
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.0000i 1.36206i 0.732257 + 0.681028i \(0.238467\pi\)
−0.732257 + 0.681028i \(0.761533\pi\)
\(588\) 0 0
\(589\) 6.92820i 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.92820 0.283079 0.141539 0.989933i \(-0.454795\pi\)
0.141539 + 0.989933i \(0.454795\pi\)
\(600\) 0 0
\(601\) −1.00000 −0.0407909 −0.0203954 0.999792i \(-0.506493\pi\)
−0.0203954 + 0.999792i \(0.506493\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 3.46410i − 0.140836i
\(606\) 0 0
\(607\) −17.3205 −0.703018 −0.351509 0.936185i \(-0.614331\pi\)
−0.351509 + 0.936185i \(0.614331\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 6.92820i 0.279827i 0.990164 + 0.139914i \(0.0446825\pi\)
−0.990164 + 0.139914i \(0.955317\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 40.0000i 1.60774i 0.594808 + 0.803868i \(0.297228\pi\)
−0.594808 + 0.803868i \(0.702772\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −20.7846 −0.832718
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 12.1244 0.482663 0.241331 0.970443i \(-0.422416\pi\)
0.241331 + 0.970443i \(0.422416\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 33.0000i − 1.30957i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) − 8.00000i − 0.315489i −0.987480 0.157745i \(-0.949578\pi\)
0.987480 0.157745i \(-0.0504223\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27.7128 −1.08950 −0.544752 0.838597i \(-0.683376\pi\)
−0.544752 + 0.838597i \(0.683376\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 46.7654i − 1.83007i −0.403374 0.915035i \(-0.632163\pi\)
0.403374 0.915035i \(-0.367837\pi\)
\(654\) 0 0
\(655\) 5.19615 0.203030
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 21.0000i 0.818044i 0.912525 + 0.409022i \(0.134130\pi\)
−0.912525 + 0.409022i \(0.865870\pi\)
\(660\) 0 0
\(661\) 6.92820i 0.269476i 0.990881 + 0.134738i \(0.0430193\pi\)
−0.990881 + 0.134738i \(0.956981\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) − 48.0000i − 1.85857i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 41.5692 1.60476
\(672\) 0 0
\(673\) −47.0000 −1.81172 −0.905858 0.423581i \(-0.860773\pi\)
−0.905858 + 0.423581i \(0.860773\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 6.92820i − 0.266272i −0.991098 0.133136i \(-0.957495\pi\)
0.991098 0.133136i \(-0.0425048\pi\)
\(678\) 0 0
\(679\) −12.1244 −0.465290
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) − 20.7846i − 0.794139i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 27.7128 1.05121
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 36.3731i − 1.37379i −0.726756 0.686896i \(-0.758973\pi\)
0.726756 0.686896i \(-0.241027\pi\)
\(702\) 0 0
\(703\) −27.7128 −1.04521
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.0000i 0.564133i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 13.8564i − 0.514614i
\(726\) 0 0
\(727\) 25.9808 0.963573 0.481787 0.876289i \(-0.339988\pi\)
0.481787 + 0.876289i \(0.339988\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 34.6410i − 1.27950i −0.768585 0.639748i \(-0.779039\pi\)
0.768585 0.639748i \(-0.220961\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) 8.00000i 0.294285i 0.989115 + 0.147142i \(0.0470076\pi\)
−0.989115 + 0.147142i \(0.952992\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.8564 0.508342 0.254171 0.967159i \(-0.418197\pi\)
0.254171 + 0.967159i \(0.418197\pi\)
\(744\) 0 0
\(745\) −9.00000 −0.329734
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 15.5885i − 0.569590i
\(750\) 0 0
\(751\) 29.4449 1.07446 0.537229 0.843436i \(-0.319471\pi\)
0.537229 + 0.843436i \(0.319471\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 39.0000i 1.41936i
\(756\) 0 0
\(757\) − 48.4974i − 1.76267i −0.472493 0.881334i \(-0.656646\pi\)
0.472493 0.881334i \(-0.343354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) − 12.0000i − 0.434429i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.92820i 0.249190i 0.992208 + 0.124595i \(0.0397632\pi\)
−0.992208 + 0.124595i \(0.960237\pi\)
\(774\) 0 0
\(775\) 3.46410 0.124434
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 48.0000i 1.71978i
\(780\) 0 0
\(781\) 41.5692i 1.48746i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.0000 0.856597
\(786\) 0 0
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 20.7846 0.739016
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.66025i 0.306762i 0.988167 + 0.153381i \(0.0490162\pi\)
−0.988167 + 0.153381i \(0.950984\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 21.0000i − 0.741074i
\(804\) 0 0
\(805\) 20.7846i 0.732561i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) − 52.0000i − 1.82597i −0.407997 0.912983i \(-0.633772\pi\)
0.407997 0.912983i \(-0.366228\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 27.7128 0.970737
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 48.4974i − 1.69257i −0.532729 0.846286i \(-0.678834\pi\)
0.532729 0.846286i \(-0.321166\pi\)
\(822\) 0 0
\(823\) −36.3731 −1.26789 −0.633943 0.773380i \(-0.718565\pi\)
−0.633943 + 0.773380i \(0.718565\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 48.0000i − 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) 0 0
\(829\) − 20.7846i − 0.721879i −0.932589 0.360940i \(-0.882456\pi\)
0.932589 0.360940i \(-0.117544\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 12.0000i 0.415277i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 27.7128 0.956753 0.478376 0.878155i \(-0.341226\pi\)
0.478376 + 0.878155i \(0.341226\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 22.5167i − 0.774597i
\(846\) 0 0
\(847\) −3.46410 −0.119028
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 48.0000i 1.64542i
\(852\) 0 0
\(853\) 34.6410i 1.18609i 0.805171 + 0.593043i \(0.202074\pi\)
−0.805171 + 0.593043i \(0.797926\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) 32.0000i 1.09183i 0.837842 + 0.545913i \(0.183817\pi\)
−0.837842 + 0.545913i \(0.816183\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 55.4256 1.88671 0.943355 0.331785i \(-0.107651\pi\)
0.943355 + 0.331785i \(0.107651\pi\)
\(864\) 0 0
\(865\) 15.0000 0.510015
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 31.1769i − 1.05760i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 21.0000i 0.709930i
\(876\) 0 0
\(877\) − 34.6410i − 1.16974i −0.811126 0.584872i \(-0.801145\pi\)
0.811126 0.584872i \(-0.198855\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) 52.0000i 1.74994i 0.484178 + 0.874970i \(0.339119\pi\)
−0.484178 + 0.874970i \(0.660881\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −41.5692 −1.39576 −0.697879 0.716216i \(-0.745873\pi\)
−0.697879 + 0.716216i \(0.745873\pi\)
\(888\) 0 0
\(889\) −33.0000 −1.10678
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.7128i 0.927374i
\(894\) 0 0
\(895\) −5.19615 −0.173688
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 12.0000i − 0.400222i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) 20.0000i 0.664089i 0.943264 + 0.332045i \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48.4974 1.60679 0.803396 0.595446i \(-0.203024\pi\)
0.803396 + 0.595446i \(0.203024\pi\)
\(912\) 0 0
\(913\) −27.0000 −0.893570
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 5.19615i − 0.171592i
\(918\) 0 0
\(919\) −36.3731 −1.19984 −0.599918 0.800061i \(-0.704800\pi\)
−0.599918 + 0.800061i \(0.704800\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 13.8564i 0.455596i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 48.0000 1.57483 0.787414 0.616424i \(-0.211419\pi\)
0.787414 + 0.616424i \(0.211419\pi\)
\(930\) 0 0
\(931\) − 16.0000i − 0.524379i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 37.0000 1.20874 0.604369 0.796705i \(-0.293425\pi\)
0.604369 + 0.796705i \(0.293425\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 43.3013i − 1.41158i −0.708421 0.705791i \(-0.750592\pi\)
0.708421 0.705791i \(-0.249408\pi\)
\(942\) 0 0
\(943\) 83.1384 2.70736
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.00000i − 0.0974869i −0.998811 0.0487435i \(-0.984478\pi\)
0.998811 0.0487435i \(-0.0155217\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) − 24.0000i − 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −20.7846 −0.671170
\(960\) 0 0
\(961\) −28.0000 −0.903226
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 19.0526i − 0.613324i
\(966\) 0 0
\(967\) 22.5167 0.724087 0.362043 0.932161i \(-0.382079\pi\)
0.362043 + 0.932161i \(0.382079\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 51.0000i 1.63667i 0.574743 + 0.818334i \(0.305102\pi\)
−0.574743 + 0.818334i \(0.694898\pi\)
\(972\) 0 0
\(973\) − 27.7128i − 0.888432i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) 36.0000i 1.15056i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20.7846 0.662926 0.331463 0.943468i \(-0.392458\pi\)
0.331463 + 0.943468i \(0.392458\pi\)
\(984\) 0 0
\(985\) 27.0000 0.860292
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 27.7128i − 0.881216i
\(990\) 0 0
\(991\) −32.9090 −1.04539 −0.522694 0.852520i \(-0.675073\pi\)
−0.522694 + 0.852520i \(0.675073\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 27.0000i − 0.855958i
\(996\) 0 0
\(997\) 41.5692i 1.31651i 0.752795 + 0.658255i \(0.228705\pi\)
−0.752795 + 0.658255i \(0.771295\pi\)
\(998\) 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 1728.2.d.g.865.1 yes 4
3.2 odd 2 1728.2.d.f.865.3 yes 4
4.3 odd 2 inner 1728.2.d.g.865.2 yes 4
8.3 odd 2 inner 1728.2.d.g.865.4 yes 4
8.5 even 2 inner 1728.2.d.g.865.3 yes 4
12.11 even 2 1728.2.d.f.865.4 yes 4
16.3 odd 4 6912.2.a.bj.1.1 2
16.5 even 4 6912.2.a.bj.1.2 2
16.11 odd 4 6912.2.a.bo.1.2 2
16.13 even 4 6912.2.a.bo.1.1 2
24.5 odd 2 1728.2.d.f.865.1 4
24.11 even 2 1728.2.d.f.865.2 yes 4
48.5 odd 4 6912.2.a.bp.1.1 2
48.11 even 4 6912.2.a.bi.1.1 2
48.29 odd 4 6912.2.a.bi.1.2 2
48.35 even 4 6912.2.a.bp.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.2.d.f.865.1 4 24.5 odd 2
1728.2.d.f.865.2 yes 4 24.11 even 2
1728.2.d.f.865.3 yes 4 3.2 odd 2
1728.2.d.f.865.4 yes 4 12.11 even 2
1728.2.d.g.865.1 yes 4 1.1 even 1 trivial
1728.2.d.g.865.2 yes 4 4.3 odd 2 inner
1728.2.d.g.865.3 yes 4 8.5 even 2 inner
1728.2.d.g.865.4 yes 4 8.3 odd 2 inner
6912.2.a.bi.1.1 2 48.11 even 4
6912.2.a.bi.1.2 2 48.29 odd 4
6912.2.a.bj.1.1 2 16.3 odd 4
6912.2.a.bj.1.2 2 16.5 even 4
6912.2.a.bo.1.1 2 16.13 even 4
6912.2.a.bo.1.2 2 16.11 odd 4
6912.2.a.bp.1.1 2 48.5 odd 4
6912.2.a.bp.1.2 2 48.35 even 4