Properties

Label 1152.2.k.d.289.2
Level $1152$
Weight $2$
Character 1152.289
Analytic conductor $9.199$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.629407744.1
Defining polynomial: \(x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 289.2
Root \(-1.38255 + 0.297594i\) of defining polynomial
Character \(\chi\) \(=\) 1152.289
Dual form 1152.2.k.d.865.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.595188 + 0.595188i) q^{5} +1.64575i q^{7} +O(q^{10})\) \(q+(-0.595188 + 0.595188i) q^{5} +1.64575i q^{7} +(3.36028 - 3.36028i) q^{11} +(-2.64575 - 2.64575i) q^{13} +5.53019 q^{17} +(-3.64575 - 3.64575i) q^{19} -4.33981i q^{23} +4.29150i q^{25} +(6.12538 + 6.12538i) q^{29} +5.64575 q^{31} +(-0.979531 - 0.979531i) q^{35} +(0.645751 - 0.645751i) q^{37} +7.91094i q^{41} +(-0.354249 + 0.354249i) q^{43} +9.10132 q^{47} +4.29150 q^{49} +(4.93500 - 4.93500i) q^{53} +4.00000i q^{55} +(4.33981 - 4.33981i) q^{59} +(0.645751 + 0.645751i) q^{61} +3.14944 q^{65} +(4.00000 + 4.00000i) q^{67} -13.4411i q^{71} -3.29150i q^{73} +(5.53019 + 5.53019i) q^{77} -9.64575 q^{79} +(-3.36028 - 3.36028i) q^{83} +(-3.29150 + 3.29150i) q^{85} +2.38075i q^{89} +(4.35425 - 4.35425i) q^{91} +4.33981 q^{95} -10.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 8 q^{19} + 24 q^{31} - 16 q^{37} - 24 q^{43} - 8 q^{49} - 16 q^{61} + 32 q^{67} - 56 q^{79} + 16 q^{85} + 56 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) −0.595188 + 0.595188i −0.266176 + 0.266176i −0.827557 0.561381i \(-0.810270\pi\)
0.561381 + 0.827557i \(0.310270\pi\)
\(6\) 0 0
\(7\) 1.64575i 0.622036i 0.950404 + 0.311018i \(0.100670\pi\)
−0.950404 + 0.311018i \(0.899330\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.36028 3.36028i 1.01316 1.01316i 0.0132513 0.999912i \(-0.495782\pi\)
0.999912 0.0132513i \(-0.00421814\pi\)
\(12\) 0 0
\(13\) −2.64575 2.64575i −0.733799 0.733799i 0.237571 0.971370i \(-0.423649\pi\)
−0.971370 + 0.237571i \(0.923649\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.53019 1.34127 0.670634 0.741788i \(-0.266022\pi\)
0.670634 + 0.741788i \(0.266022\pi\)
\(18\) 0 0
\(19\) −3.64575 3.64575i −0.836393 0.836393i 0.151989 0.988382i \(-0.451432\pi\)
−0.988382 + 0.151989i \(0.951432\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.33981i 0.904914i −0.891786 0.452457i \(-0.850548\pi\)
0.891786 0.452457i \(-0.149452\pi\)
\(24\) 0 0
\(25\) 4.29150i 0.858301i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.12538 + 6.12538i 1.13745 + 1.13745i 0.988905 + 0.148549i \(0.0474603\pi\)
0.148549 + 0.988905i \(0.452540\pi\)
\(30\) 0 0
\(31\) 5.64575 1.01401 0.507003 0.861944i \(-0.330753\pi\)
0.507003 + 0.861944i \(0.330753\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.979531 0.979531i −0.165571 0.165571i
\(36\) 0 0
\(37\) 0.645751 0.645751i 0.106161 0.106161i −0.652031 0.758192i \(-0.726083\pi\)
0.758192 + 0.652031i \(0.226083\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.91094i 1.23548i 0.786382 + 0.617741i \(0.211952\pi\)
−0.786382 + 0.617741i \(0.788048\pi\)
\(42\) 0 0
\(43\) −0.354249 + 0.354249i −0.0540224 + 0.0540224i −0.733602 0.679579i \(-0.762162\pi\)
0.679579 + 0.733602i \(0.262162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.10132 1.32756 0.663782 0.747926i \(-0.268950\pi\)
0.663782 + 0.747926i \(0.268950\pi\)
\(48\) 0 0
\(49\) 4.29150 0.613072
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.93500 4.93500i 0.677875 0.677875i −0.281644 0.959519i \(-0.590880\pi\)
0.959519 + 0.281644i \(0.0908798\pi\)
\(54\) 0 0
\(55\) 4.00000i 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.33981 4.33981i 0.564996 0.564996i −0.365727 0.930722i \(-0.619179\pi\)
0.930722 + 0.365727i \(0.119179\pi\)
\(60\) 0 0
\(61\) 0.645751 + 0.645751i 0.0826800 + 0.0826800i 0.747237 0.664557i \(-0.231380\pi\)
−0.664557 + 0.747237i \(0.731380\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.14944 0.390640
\(66\) 0 0
\(67\) 4.00000 + 4.00000i 0.488678 + 0.488678i 0.907889 0.419211i \(-0.137693\pi\)
−0.419211 + 0.907889i \(0.637693\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4411i 1.59517i −0.603207 0.797584i \(-0.706111\pi\)
0.603207 0.797584i \(-0.293889\pi\)
\(72\) 0 0
\(73\) 3.29150i 0.385241i −0.981273 0.192621i \(-0.938301\pi\)
0.981273 0.192621i \(-0.0616987\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.53019 + 5.53019i 0.630224 + 0.630224i
\(78\) 0 0
\(79\) −9.64575 −1.08523 −0.542616 0.839981i \(-0.682566\pi\)
−0.542616 + 0.839981i \(0.682566\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.36028 3.36028i −0.368839 0.368839i 0.498215 0.867054i \(-0.333989\pi\)
−0.867054 + 0.498215i \(0.833989\pi\)
\(84\) 0 0
\(85\) −3.29150 + 3.29150i −0.357014 + 0.357014i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.38075i 0.252359i 0.992007 + 0.126180i \(0.0402716\pi\)
−0.992007 + 0.126180i \(0.959728\pi\)
\(90\) 0 0
\(91\) 4.35425 4.35425i 0.456449 0.456449i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.33981 0.445256
\(96\) 0 0
\(97\) −10.5830 −1.07454 −0.537271 0.843410i \(-0.680545\pi\)
−0.537271 + 0.843410i \(0.680545\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.595188 + 0.595188i −0.0592234 + 0.0592234i −0.736098 0.676875i \(-0.763334\pi\)
0.676875 + 0.736098i \(0.263334\pi\)
\(102\) 0 0
\(103\) 16.9373i 1.66888i −0.551101 0.834439i \(-0.685792\pi\)
0.551101 0.834439i \(-0.314208\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.38075 2.38075i 0.230156 0.230156i −0.582602 0.812758i \(-0.697965\pi\)
0.812758 + 0.582602i \(0.197965\pi\)
\(108\) 0 0
\(109\) 6.64575 + 6.64575i 0.636548 + 0.636548i 0.949702 0.313155i \(-0.101386\pi\)
−0.313155 + 0.949702i \(0.601386\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 2.58301 + 2.58301i 0.240866 + 0.240866i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.10132i 0.834316i
\(120\) 0 0
\(121\) 11.5830i 1.05300i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.53019 5.53019i −0.494635 0.494635i
\(126\) 0 0
\(127\) −0.937254 −0.0831678 −0.0415839 0.999135i \(-0.513240\pi\)
−0.0415839 + 0.999135i \(0.513240\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.72057 + 6.72057i 0.587179 + 0.587179i 0.936866 0.349688i \(-0.113712\pi\)
−0.349688 + 0.936866i \(0.613712\pi\)
\(132\) 0 0
\(133\) 6.00000 6.00000i 0.520266 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.2917i 0.879279i −0.898174 0.439639i \(-0.855106\pi\)
0.898174 0.439639i \(-0.144894\pi\)
\(138\) 0 0
\(139\) −14.5830 + 14.5830i −1.23691 + 1.23691i −0.275659 + 0.961256i \(0.588896\pi\)
−0.961256 + 0.275659i \(0.911104\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −17.7809 −1.48692
\(144\) 0 0
\(145\) −7.29150 −0.605526
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −11.6556 + 11.6556i −0.954861 + 0.954861i −0.999024 0.0441630i \(-0.985938\pi\)
0.0441630 + 0.999024i \(0.485938\pi\)
\(150\) 0 0
\(151\) 10.3542i 0.842617i 0.906917 + 0.421308i \(0.138429\pi\)
−0.906917 + 0.421308i \(0.861571\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.36028 + 3.36028i −0.269904 + 0.269904i
\(156\) 0 0
\(157\) 6.64575 + 6.64575i 0.530389 + 0.530389i 0.920688 0.390299i \(-0.127628\pi\)
−0.390299 + 0.920688i \(0.627628\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.14226 0.562889
\(162\) 0 0
\(163\) 11.6458 + 11.6458i 0.912166 + 0.912166i 0.996442 0.0842767i \(-0.0268580\pi\)
−0.0842767 + 0.996442i \(0.526858\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.7809i 1.37593i 0.725743 + 0.687965i \(0.241496\pi\)
−0.725743 + 0.687965i \(0.758504\pi\)
\(168\) 0 0
\(169\) 1.00000i 0.0769231i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.93500 4.93500i −0.375201 0.375201i 0.494166 0.869367i \(-0.335473\pi\)
−0.869367 + 0.494166i \(0.835473\pi\)
\(174\) 0 0
\(175\) −7.06275 −0.533893
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.38075 2.38075i −0.177946 0.177946i 0.612514 0.790460i \(-0.290158\pi\)
−0.790460 + 0.612514i \(0.790158\pi\)
\(180\) 0 0
\(181\) 0.645751 0.645751i 0.0479983 0.0479983i −0.682700 0.730699i \(-0.739195\pi\)
0.730699 + 0.682700i \(0.239195\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.768687i 0.0565150i
\(186\) 0 0
\(187\) 18.5830 18.5830i 1.35892 1.35892i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.10132 −0.658548 −0.329274 0.944234i \(-0.606804\pi\)
−0.329274 + 0.944234i \(0.606804\pi\)
\(192\) 0 0
\(193\) 11.8745 0.854746 0.427373 0.904075i \(-0.359439\pi\)
0.427373 + 0.904075i \(0.359439\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.50613 8.50613i 0.606037 0.606037i −0.335871 0.941908i \(-0.609031\pi\)
0.941908 + 0.335871i \(0.109031\pi\)
\(198\) 0 0
\(199\) 13.6458i 0.967322i −0.875256 0.483661i \(-0.839307\pi\)
0.875256 0.483661i \(-0.160693\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.0808 + 10.0808i −0.707537 + 0.707537i
\(204\) 0 0
\(205\) −4.70850 4.70850i −0.328856 0.328856i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −24.5015 −1.69481
\(210\) 0 0
\(211\) −14.5830 14.5830i −1.00394 1.00394i −0.999992 0.00394326i \(-0.998745\pi\)
−0.00394326 0.999992i \(-0.501255\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.421689i 0.0287590i
\(216\) 0 0
\(217\) 9.29150i 0.630748i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −14.6315 14.6315i −0.984222 0.984222i
\(222\) 0 0
\(223\) 14.3542 0.961232 0.480616 0.876931i \(-0.340413\pi\)
0.480616 + 0.876931i \(0.340413\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.40122 1.40122i −0.0930023 0.0930023i 0.659075 0.752077i \(-0.270948\pi\)
−0.752077 + 0.659075i \(0.770948\pi\)
\(228\) 0 0
\(229\) −17.9373 + 17.9373i −1.18533 + 1.18533i −0.206982 + 0.978345i \(0.566364\pi\)
−0.978345 + 0.206982i \(0.933636\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.7400i 1.29321i −0.762825 0.646606i \(-0.776188\pi\)
0.762825 0.646606i \(-0.223812\pi\)
\(234\) 0 0
\(235\) −5.41699 + 5.41699i −0.353366 + 0.353366i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.0194 0.842158 0.421079 0.907024i \(-0.361652\pi\)
0.421079 + 0.907024i \(0.361652\pi\)
\(240\) 0 0
\(241\) −1.29150 −0.0831930 −0.0415965 0.999134i \(-0.513244\pi\)
−0.0415965 + 0.999134i \(0.513244\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.55425 + 2.55425i −0.163185 + 0.163185i
\(246\) 0 0
\(247\) 19.2915i 1.22749i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.7605 + 18.7605i −1.18415 + 1.18415i −0.205492 + 0.978659i \(0.565879\pi\)
−0.978659 + 0.205492i \(0.934121\pi\)
\(252\) 0 0
\(253\) −14.5830 14.5830i −0.916826 0.916826i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −29.2630 −1.82538 −0.912688 0.408656i \(-0.865998\pi\)
−0.912688 + 0.408656i \(0.865998\pi\)
\(258\) 0 0
\(259\) 1.06275 + 1.06275i 0.0660358 + 0.0660358i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.67963i 0.535209i 0.963529 + 0.267604i \(0.0862320\pi\)
−0.963529 + 0.267604i \(0.913768\pi\)
\(264\) 0 0
\(265\) 5.87451i 0.360868i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.4652 10.4652i −0.638074 0.638074i 0.312006 0.950080i \(-0.398999\pi\)
−0.950080 + 0.312006i \(0.898999\pi\)
\(270\) 0 0
\(271\) −6.35425 −0.385993 −0.192997 0.981199i \(-0.561821\pi\)
−0.192997 + 0.981199i \(0.561821\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.4207 + 14.4207i 0.869599 + 0.869599i
\(276\) 0 0
\(277\) 16.5203 16.5203i 0.992606 0.992606i −0.00736669 0.999973i \(-0.502345\pi\)
0.999973 + 0.00736669i \(0.00234491\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.38075i 0.142024i 0.997475 + 0.0710119i \(0.0226228\pi\)
−0.997475 + 0.0710119i \(0.977377\pi\)
\(282\) 0 0
\(283\) −2.58301 + 2.58301i −0.153544 + 0.153544i −0.779699 0.626155i \(-0.784628\pi\)
0.626155 + 0.779699i \(0.284628\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.0194 −0.768513
\(288\) 0 0
\(289\) 13.5830 0.799000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.55425 + 2.55425i −0.149221 + 0.149221i −0.777770 0.628549i \(-0.783649\pi\)
0.628549 + 0.777770i \(0.283649\pi\)
\(294\) 0 0
\(295\) 5.16601i 0.300777i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.4821 + 11.4821i −0.664025 + 0.664025i
\(300\) 0 0
\(301\) −0.583005 0.583005i −0.0336039 0.0336039i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.768687 −0.0440149
\(306\) 0 0
\(307\) −20.0000 20.0000i −1.14146 1.14146i −0.988183 0.153277i \(-0.951017\pi\)
−0.153277 0.988183i \(-0.548983\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.67963i 0.492177i 0.969247 + 0.246088i \(0.0791453\pi\)
−0.969247 + 0.246088i \(0.920855\pi\)
\(312\) 0 0
\(313\) 9.29150i 0.525187i 0.964907 + 0.262593i \(0.0845778\pi\)
−0.964907 + 0.262593i \(0.915422\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.595188 + 0.595188i 0.0334291 + 0.0334291i 0.723624 0.690195i \(-0.242475\pi\)
−0.690195 + 0.723624i \(0.742475\pi\)
\(318\) 0 0
\(319\) 41.1660 2.30485
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −20.1617 20.1617i −1.12183 1.12183i
\(324\) 0 0
\(325\) 11.3542 11.3542i 0.629820 0.629820i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.9785i 0.825792i
\(330\) 0 0
\(331\) −8.00000 + 8.00000i −0.439720 + 0.439720i −0.891918 0.452198i \(-0.850640\pi\)
0.452198 + 0.891918i \(0.350640\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.76150 −0.260149
\(336\) 0 0
\(337\) 4.70850 0.256488 0.128244 0.991743i \(-0.459066\pi\)
0.128244 + 0.991743i \(0.459066\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 18.9713 18.9713i 1.02735 1.02735i
\(342\) 0 0
\(343\) 18.5830i 1.00339i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.36028 3.36028i 0.180389 0.180389i −0.611136 0.791526i \(-0.709287\pi\)
0.791526 + 0.611136i \(0.209287\pi\)
\(348\) 0 0
\(349\) −3.22876 3.22876i −0.172831 0.172831i 0.615391 0.788222i \(-0.288998\pi\)
−0.788222 + 0.615391i \(0.788998\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.14226 0.380144 0.190072 0.981770i \(-0.439128\pi\)
0.190072 + 0.981770i \(0.439128\pi\)
\(354\) 0 0
\(355\) 8.00000 + 8.00000i 0.424596 + 0.424596i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.76150i 0.251303i 0.992074 + 0.125651i \(0.0401020\pi\)
−0.992074 + 0.125651i \(0.959898\pi\)
\(360\) 0 0
\(361\) 7.58301i 0.399106i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.95906 + 1.95906i 0.102542 + 0.102542i
\(366\) 0 0
\(367\) −34.8118 −1.81716 −0.908580 0.417712i \(-0.862832\pi\)
−0.908580 + 0.417712i \(0.862832\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.12179 + 8.12179i 0.421662 + 0.421662i
\(372\) 0 0
\(373\) −11.9373 + 11.9373i −0.618088 + 0.618088i −0.945041 0.326953i \(-0.893978\pi\)
0.326953 + 0.945041i \(0.393978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 32.4125i 1.66933i
\(378\) 0 0
\(379\) 8.35425 8.35425i 0.429129 0.429129i −0.459203 0.888332i \(-0.651865\pi\)
0.888332 + 0.459203i \(0.151865\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.10132 0.465056 0.232528 0.972590i \(-0.425300\pi\)
0.232528 + 0.972590i \(0.425300\pi\)
\(384\) 0 0
\(385\) −6.58301 −0.335501
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −20.7569 + 20.7569i −1.05242 + 1.05242i −0.0538679 + 0.998548i \(0.517155\pi\)
−0.998548 + 0.0538679i \(0.982845\pi\)
\(390\) 0 0
\(391\) 24.0000i 1.21373i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.74103 5.74103i 0.288863 0.288863i
\(396\) 0 0
\(397\) −8.06275 8.06275i −0.404658 0.404658i 0.475213 0.879871i \(-0.342371\pi\)
−0.879871 + 0.475213i \(0.842371\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.5906 0.828494 0.414247 0.910165i \(-0.364045\pi\)
0.414247 + 0.910165i \(0.364045\pi\)
\(402\) 0 0
\(403\) −14.9373 14.9373i −0.744078 0.744078i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.33981i 0.215117i
\(408\) 0 0
\(409\) 25.1660i 1.24438i −0.782867 0.622190i \(-0.786243\pi\)
0.782867 0.622190i \(-0.213757\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.14226 + 7.14226i 0.351447 + 0.351447i
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.7605 + 18.7605i 0.916509 + 0.916509i 0.996774 0.0802643i \(-0.0255764\pi\)
−0.0802643 + 0.996774i \(0.525576\pi\)
\(420\) 0 0
\(421\) −11.3542 + 11.3542i −0.553372 + 0.553372i −0.927412 0.374040i \(-0.877972\pi\)
0.374040 + 0.927412i \(0.377972\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 23.7328i 1.15121i
\(426\) 0 0
\(427\) −1.06275 + 1.06275i −0.0514299 + 0.0514299i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.2221 −1.50391 −0.751957 0.659212i \(-0.770890\pi\)
−0.751957 + 0.659212i \(0.770890\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −15.8219 + 15.8219i −0.756863 + 0.756863i
\(438\) 0 0
\(439\) 0.479741i 0.0228968i 0.999934 + 0.0114484i \(0.00364422\pi\)
−0.999934 + 0.0114484i \(0.996356\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.5629 21.5629i 1.02449 1.02449i 0.0247926 0.999693i \(-0.492107\pi\)
0.999693 0.0247926i \(-0.00789253\pi\)
\(444\) 0 0
\(445\) −1.41699 1.41699i −0.0671720 0.0671720i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.44832 0.445894 0.222947 0.974831i \(-0.428432\pi\)
0.222947 + 0.974831i \(0.428432\pi\)
\(450\) 0 0
\(451\) 26.5830 + 26.5830i 1.25174 + 1.25174i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.18319i 0.242992i
\(456\) 0 0
\(457\) 5.41699i 0.253396i 0.991941 + 0.126698i \(0.0404380\pi\)
−0.991941 + 0.126698i \(0.959562\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.50613 8.50613i −0.396170 0.396170i 0.480710 0.876880i \(-0.340379\pi\)
−0.876880 + 0.480710i \(0.840379\pi\)
\(462\) 0 0
\(463\) −15.0627 −0.700025 −0.350013 0.936745i \(-0.613823\pi\)
−0.350013 + 0.936745i \(0.613823\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −25.4810 25.4810i −1.17912 1.17912i −0.979968 0.199154i \(-0.936181\pi\)
−0.199154 0.979968i \(-0.563819\pi\)
\(468\) 0 0
\(469\) −6.58301 + 6.58301i −0.303975 + 0.303975i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.38075i 0.109467i
\(474\) 0 0
\(475\) 15.6458 15.6458i 0.717876 0.717876i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 22.1208 1.01072 0.505362 0.862908i \(-0.331359\pi\)
0.505362 + 0.862908i \(0.331359\pi\)
\(480\) 0 0
\(481\) −3.41699 −0.155802
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.29888 6.29888i 0.286017 0.286017i
\(486\) 0 0
\(487\) 8.22876i 0.372881i −0.982466 0.186440i \(-0.940305\pi\)
0.982466 0.186440i \(-0.0596951\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.4002 15.4002i 0.695001 0.695001i −0.268327 0.963328i \(-0.586471\pi\)
0.963328 + 0.268327i \(0.0864707\pi\)
\(492\) 0 0
\(493\) 33.8745 + 33.8745i 1.52563 + 1.52563i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.1208 0.992252
\(498\) 0 0
\(499\) 10.5830 + 10.5830i 0.473760 + 0.473760i 0.903129 0.429369i \(-0.141264\pi\)
−0.429369 + 0.903129i \(0.641264\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.6438i 1.41093i −0.708747 0.705463i \(-0.750739\pi\)
0.708747 0.705463i \(-0.249261\pi\)
\(504\) 0 0
\(505\) 0.708497i 0.0315277i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 24.6750 + 24.6750i 1.09370 + 1.09370i 0.995130 + 0.0985706i \(0.0314270\pi\)
0.0985706 + 0.995130i \(0.468573\pi\)
\(510\) 0 0
\(511\) 5.41699 0.239634
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10.0808 + 10.0808i 0.444215 + 0.444215i
\(516\) 0 0
\(517\) 30.5830 30.5830i 1.34504 1.34504i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.2098i 0.622543i −0.950321 0.311272i \(-0.899245\pi\)
0.950321 0.311272i \(-0.100755\pi\)
\(522\) 0 0
\(523\) −18.9373 + 18.9373i −0.828068 + 0.828068i −0.987249 0.159181i \(-0.949115\pi\)
0.159181 + 0.987249i \(0.449115\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31.2221 1.36006
\(528\) 0 0
\(529\) 4.16601 0.181131
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20.9304 20.9304i 0.906596 0.906596i
\(534\) 0 0
\(535\) 2.83399i 0.122524i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14.4207 14.4207i 0.621142 0.621142i
\(540\) 0 0
\(541\) 28.5203 + 28.5203i 1.22618 + 1.22618i 0.965397 + 0.260785i \(0.0839813\pi\)
0.260785 + 0.965397i \(0.416019\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.91094 −0.338868
\(546\) 0 0
\(547\) 14.9373 + 14.9373i 0.638671 + 0.638671i 0.950228 0.311557i \(-0.100850\pi\)
−0.311557 + 0.950228i \(0.600850\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 44.6632i 1.90272i
\(552\) 0 0
\(553\) 15.8745i 0.675053i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.7978 + 18.7978i 0.796489 + 0.796489i 0.982540 0.186051i \(-0.0595689\pi\)
−0.186051 + 0.982540i \(0.559569\pi\)
\(558\) 0 0
\(559\) 1.87451 0.0792832
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.31935 5.31935i −0.224184 0.224184i 0.586074 0.810258i \(-0.300673\pi\)
−0.810258 + 0.586074i \(0.800673\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 41.0921i 1.72267i 0.508038 + 0.861335i \(0.330371\pi\)
−0.508038 + 0.861335i \(0.669629\pi\)
\(570\) 0 0
\(571\) 4.00000 4.00000i 0.167395 0.167395i −0.618438 0.785833i \(-0.712234\pi\)
0.785833 + 0.618438i \(0.212234\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 18.6243 0.776688
\(576\) 0 0
\(577\) −39.0405 −1.62528 −0.812639 0.582767i \(-0.801970\pi\)
−0.812639 + 0.582767i \(0.801970\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.53019 5.53019i 0.229431 0.229431i
\(582\) 0 0
\(583\) 33.1660i 1.37360i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.76150 + 4.76150i −0.196528 + 0.196528i −0.798510 0.601982i \(-0.794378\pi\)
0.601982 + 0.798510i \(0.294378\pi\)
\(588\) 0 0
\(589\) −20.5830 20.5830i −0.848108 0.848108i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.1208 0.908391 0.454195 0.890902i \(-0.349927\pi\)
0.454195 + 0.890902i \(0.349927\pi\)
\(594\) 0 0
\(595\) −5.41699 5.41699i −0.222075 0.222075i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13.8628i 0.566420i 0.959058 + 0.283210i \(0.0913993\pi\)
−0.959058 + 0.283210i \(0.908601\pi\)
\(600\) 0 0
\(601\) 3.29150i 0.134263i −0.997744 0.0671316i \(-0.978615\pi\)
0.997744 0.0671316i \(-0.0213847\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.89407 + 6.89407i 0.280284 + 0.280284i
\(606\) 0 0
\(607\) −14.1033 −0.572434 −0.286217 0.958165i \(-0.592398\pi\)
−0.286217 + 0.958165i \(0.592398\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0798 24.0798i −0.974165 0.974165i
\(612\) 0 0
\(613\) −26.6458 + 26.6458i −1.07621 + 1.07621i −0.0793662 + 0.996846i \(0.525290\pi\)
−0.996846 + 0.0793662i \(0.974710\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.6438i 1.27393i 0.770893 + 0.636965i \(0.219810\pi\)
−0.770893 + 0.636965i \(0.780190\pi\)
\(618\) 0 0
\(619\) −21.1660 + 21.1660i −0.850734 + 0.850734i −0.990224 0.139490i \(-0.955454\pi\)
0.139490 + 0.990224i \(0.455454\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.91813 −0.156976
\(624\) 0 0
\(625\) −14.8745 −0.594980
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.57113 3.57113i 0.142390 0.142390i
\(630\) 0 0
\(631\) 8.22876i 0.327582i 0.986495 + 0.163791i \(0.0523722\pi\)
−0.986495 + 0.163791i \(0.947628\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.557842 0.557842i 0.0221373 0.0221373i
\(636\) 0 0
\(637\) −11.3542 11.3542i −0.449872 0.449872i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5906 0.655288 0.327644 0.944801i \(-0.393745\pi\)
0.327644 + 0.944801i \(0.393745\pi\)
\(642\) 0 0
\(643\) −3.64575 3.64575i −0.143774 0.143774i 0.631556 0.775330i \(-0.282417\pi\)
−0.775330 + 0.631556i \(0.782417\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.33981i 0.170616i −0.996355 0.0853079i \(-0.972813\pi\)
0.996355 0.0853079i \(-0.0271874\pi\)
\(648\) 0 0
\(649\) 29.1660i 1.14487i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.7569 + 20.7569i 0.812280 + 0.812280i 0.984975 0.172696i \(-0.0552477\pi\)
−0.172696 + 0.984975i \(0.555248\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.9232 + 24.9232i 0.970870 + 0.970870i 0.999588 0.0287175i \(-0.00914231\pi\)
−0.0287175 + 0.999588i \(0.509142\pi\)
\(660\) 0 0
\(661\) 2.77124 2.77124i 0.107789 0.107789i −0.651155 0.758944i \(-0.725715\pi\)
0.758944 + 0.651155i \(0.225715\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.14226i 0.276965i
\(666\) 0 0
\(667\) 26.5830 26.5830i 1.02930 1.02930i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.33981 0.167537
\(672\) 0 0
\(673\) 20.0000 0.770943 0.385472 0.922720i \(-0.374039\pi\)
0.385472 + 0.922720i \(0.374039\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.0363 14.0363i 0.539460 0.539460i −0.383911 0.923370i \(-0.625423\pi\)
0.923370 + 0.383911i \(0.125423\pi\)
\(678\) 0 0
\(679\) 17.4170i 0.668403i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.557842 + 0.557842i −0.0213452 + 0.0213452i −0.717699 0.696354i \(-0.754805\pi\)
0.696354 + 0.717699i \(0.254805\pi\)
\(684\) 0 0
\(685\) 6.12549 + 6.12549i 0.234043 + 0.234043i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −26.1136 −0.994848
\(690\) 0 0
\(691\) −3.64575 3.64575i −0.138691 0.138691i 0.634353 0.773044i \(-0.281267\pi\)
−0.773044 + 0.634353i \(0.781267\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.3593i 0.658474i
\(696\) 0 0
\(697\) 43.7490i 1.65711i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.16632 + 4.16632i 0.157360 + 0.157360i 0.781396 0.624036i \(-0.214508\pi\)
−0.624036 + 0.781396i \(0.714508\pi\)
\(702\) 0 0
\(703\) −4.70850 −0.177584
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.979531 0.979531i −0.0368391 0.0368391i
\(708\) 0 0
\(709\) 8.77124 8.77124i 0.329411 0.329411i −0.522951 0.852362i \(-0.675169\pi\)
0.852362 + 0.522951i \(0.175169\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.5015i 0.917589i
\(714\) 0 0
\(715\) 10.5830 10.5830i 0.395782 0.395782i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 40.3234 1.50381 0.751904 0.659272i \(-0.229135\pi\)
0.751904 + 0.659272i \(0.229135\pi\)
\(720\) 0 0
\(721\) 27.8745 1.03810
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −26.2871 + 26.2871i −0.976278 + 0.976278i
\(726\) 0 0
\(727\) 33.3948i 1.23854i 0.785177 + 0.619272i \(0.212572\pi\)
−0.785177 + 0.619272i \(0.787428\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.95906 + 1.95906i −0.0724586 + 0.0724586i
\(732\) 0 0
\(733\) 12.6458 + 12.6458i 0.467081 + 0.467081i 0.900968 0.433886i \(-0.142858\pi\)
−0.433886 + 0.900968i \(0.642858\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26.8823 0.990221
\(738\) 0 0
\(739\) −9.16601 9.16601i −0.337177 0.337177i 0.518127 0.855304i \(-0.326630\pi\)
−0.855304 + 0.518127i \(0.826630\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.5425i 0.827002i −0.910504 0.413501i \(-0.864306\pi\)
0.910504 0.413501i \(-0.135694\pi\)
\(744\) 0 0
\(745\) 13.8745i 0.508323i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.91813 + 3.91813i 0.143165 + 0.143165i
\(750\) 0 0
\(751\) −12.9373 −0.472087 −0.236044 0.971742i \(-0.575851\pi\)
−0.236044 + 0.971742i \(0.575851\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.16272 6.16272i −0.224284 0.224284i
\(756\) 0 0
\(757\) 7.22876 7.22876i 0.262734 0.262734i −0.563430 0.826164i \(-0.690519\pi\)
0.826164 + 0.563430i \(0.190519\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.91094i 0.286771i 0.989667 + 0.143386i \(0.0457989\pi\)
−0.989667 + 0.143386i \(0.954201\pi\)
\(762\) 0 0
\(763\) −10.9373 + 10.9373i −0.395955 + 0.395955i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.9641 −0.829187
\(768\) 0 0
\(769\) 17.2915 0.623548 0.311774 0.950156i \(-0.399077\pi\)
0.311774 + 0.950156i \(0.399077\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 21.5256 21.5256i 0.774221 0.774221i −0.204620 0.978841i \(-0.565596\pi\)
0.978841 + 0.204620i \(0.0655960\pi\)
\(774\) 0 0
\(775\) 24.2288i 0.870323i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 28.8413 28.8413i 1.03335 1.03335i
\(780\) 0 0
\(781\) −45.1660 45.1660i −1.61617 1.61617i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.91094 −0.282354
\(786\) 0 0
\(787\) 6.22876 + 6.22876i 0.222031 + 0.222031i 0.809353 0.587322i \(-0.199818\pi\)
−0.587322 + 0.809353i \(0.699818\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.41699i 0.121341i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.5256 21.5256i −0.762475 0.762475i 0.214294 0.976769i \(-0.431255\pi\)
−0.976769 + 0.214294i \(0.931255\pi\)
\(798\) 0 0
\(799\) 50.3320 1.78062
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.0604 11.0604i −0.390312 0.390312i
\(804\) 0 0
\(805\) −4.25098 + 4.25098i −0.149828 + 0.149828i
\(806\) 0