Properties

Label 3200.2.a.bh.1.1
Level $3200$
Weight $2$
Character 3200.1
Self dual yes
Analytic conductor $25.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3200.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{3} -0.828427 q^{7} +2.82843 q^{9} +O(q^{10})\) \(q-2.41421 q^{3} -0.828427 q^{7} +2.82843 q^{9} +5.24264 q^{11} -5.65685 q^{13} +0.171573 q^{17} +1.58579 q^{19} +2.00000 q^{21} +4.82843 q^{23} +0.414214 q^{27} -8.00000 q^{29} -0.828427 q^{31} -12.6569 q^{33} -7.65685 q^{37} +13.6569 q^{39} +10.6569 q^{41} -10.0000 q^{43} +9.65685 q^{47} -6.31371 q^{49} -0.414214 q^{51} +7.65685 q^{53} -3.82843 q^{57} -3.65685 q^{59} -6.00000 q^{61} -2.34315 q^{63} +2.75736 q^{67} -11.6569 q^{69} -9.65685 q^{71} -5.82843 q^{73} -4.34315 q^{77} +12.8284 q^{79} -9.48528 q^{81} -1.24264 q^{83} +19.3137 q^{87} +6.17157 q^{89} +4.68629 q^{91} +2.00000 q^{93} +17.3137 q^{97} +14.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 4 q^{7} + 2 q^{11} + 6 q^{17} + 6 q^{19} + 4 q^{21} + 4 q^{23} - 2 q^{27} - 16 q^{29} + 4 q^{31} - 14 q^{33} - 4 q^{37} + 16 q^{39} + 10 q^{41} - 20 q^{43} + 8 q^{47} + 10 q^{49} + 2 q^{51} + 4 q^{53} - 2 q^{57} + 4 q^{59} - 12 q^{61} - 16 q^{63} + 14 q^{67} - 12 q^{69} - 8 q^{71} - 6 q^{73} - 20 q^{77} + 20 q^{79} - 2 q^{81} + 6 q^{83} + 16 q^{87} + 18 q^{89} + 32 q^{91} + 4 q^{93} + 12 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.828427 −0.313116 −0.156558 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) 5.24264 1.58072 0.790358 0.612646i \(-0.209895\pi\)
0.790358 + 0.612646i \(0.209895\pi\)
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.171573 0.0416125 0.0208063 0.999784i \(-0.493377\pi\)
0.0208063 + 0.999784i \(0.493377\pi\)
\(18\) 0 0
\(19\) 1.58579 0.363804 0.181902 0.983317i \(-0.441775\pi\)
0.181902 + 0.983317i \(0.441775\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −0.828427 −0.148790 −0.0743950 0.997229i \(-0.523703\pi\)
−0.0743950 + 0.997229i \(0.523703\pi\)
\(32\) 0 0
\(33\) −12.6569 −2.20328
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 13.6569 2.18685
\(40\) 0 0
\(41\) 10.6569 1.66432 0.832161 0.554535i \(-0.187104\pi\)
0.832161 + 0.554535i \(0.187104\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.65685 1.40860 0.704298 0.709904i \(-0.251262\pi\)
0.704298 + 0.709904i \(0.251262\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) −0.414214 −0.0580015
\(52\) 0 0
\(53\) 7.65685 1.05175 0.525875 0.850562i \(-0.323738\pi\)
0.525875 + 0.850562i \(0.323738\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.82843 −0.507088
\(58\) 0 0
\(59\) −3.65685 −0.476082 −0.238041 0.971255i \(-0.576505\pi\)
−0.238041 + 0.971255i \(0.576505\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −2.34315 −0.295209
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.75736 0.336865 0.168433 0.985713i \(-0.446129\pi\)
0.168433 + 0.985713i \(0.446129\pi\)
\(68\) 0 0
\(69\) −11.6569 −1.40332
\(70\) 0 0
\(71\) −9.65685 −1.14606 −0.573029 0.819535i \(-0.694232\pi\)
−0.573029 + 0.819535i \(0.694232\pi\)
\(72\) 0 0
\(73\) −5.82843 −0.682166 −0.341083 0.940033i \(-0.610794\pi\)
−0.341083 + 0.940033i \(0.610794\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.34315 −0.494947
\(78\) 0 0
\(79\) 12.8284 1.44331 0.721655 0.692252i \(-0.243382\pi\)
0.721655 + 0.692252i \(0.243382\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) −1.24264 −0.136398 −0.0681988 0.997672i \(-0.521725\pi\)
−0.0681988 + 0.997672i \(0.521725\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 19.3137 2.07065
\(88\) 0 0
\(89\) 6.17157 0.654185 0.327093 0.944992i \(-0.393931\pi\)
0.327093 + 0.944992i \(0.393931\pi\)
\(90\) 0 0
\(91\) 4.68629 0.491257
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.3137 1.75794 0.878970 0.476876i \(-0.158231\pi\)
0.878970 + 0.476876i \(0.158231\pi\)
\(98\) 0 0
\(99\) 14.8284 1.49031
\(100\) 0 0
\(101\) 18.9706 1.88764 0.943821 0.330458i \(-0.107203\pi\)
0.943821 + 0.330458i \(0.107203\pi\)
\(102\) 0 0
\(103\) −1.65685 −0.163255 −0.0816274 0.996663i \(-0.526012\pi\)
−0.0816274 + 0.996663i \(0.526012\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.92893 0.186477 0.0932385 0.995644i \(-0.470278\pi\)
0.0932385 + 0.995644i \(0.470278\pi\)
\(108\) 0 0
\(109\) −11.6569 −1.11652 −0.558262 0.829665i \(-0.688532\pi\)
−0.558262 + 0.829665i \(0.688532\pi\)
\(110\) 0 0
\(111\) 18.4853 1.75455
\(112\) 0 0
\(113\) −3.34315 −0.314497 −0.157248 0.987559i \(-0.550262\pi\)
−0.157248 + 0.987559i \(0.550262\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −16.0000 −1.47920
\(118\) 0 0
\(119\) −0.142136 −0.0130296
\(120\) 0 0
\(121\) 16.4853 1.49866
\(122\) 0 0
\(123\) −25.7279 −2.31981
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −20.1421 −1.78733 −0.893663 0.448739i \(-0.851873\pi\)
−0.893663 + 0.448739i \(0.851873\pi\)
\(128\) 0 0
\(129\) 24.1421 2.12560
\(130\) 0 0
\(131\) 5.31371 0.464261 0.232130 0.972685i \(-0.425430\pi\)
0.232130 + 0.972685i \(0.425430\pi\)
\(132\) 0 0
\(133\) −1.31371 −0.113913
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.00000 0.427179 0.213589 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 0 0
\(139\) 10.8995 0.924483 0.462242 0.886754i \(-0.347045\pi\)
0.462242 + 0.886754i \(0.347045\pi\)
\(140\) 0 0
\(141\) −23.3137 −1.96337
\(142\) 0 0
\(143\) −29.6569 −2.48003
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 15.2426 1.25719
\(148\) 0 0
\(149\) 3.31371 0.271470 0.135735 0.990745i \(-0.456660\pi\)
0.135735 + 0.990745i \(0.456660\pi\)
\(150\) 0 0
\(151\) 6.48528 0.527765 0.263882 0.964555i \(-0.414997\pi\)
0.263882 + 0.964555i \(0.414997\pi\)
\(152\) 0 0
\(153\) 0.485281 0.0392327
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.6569 1.56879 0.784394 0.620263i \(-0.212974\pi\)
0.784394 + 0.620263i \(0.212974\pi\)
\(158\) 0 0
\(159\) −18.4853 −1.46598
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 19.7279 1.54521 0.772605 0.634887i \(-0.218953\pi\)
0.772605 + 0.634887i \(0.218953\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.34315 0.181318 0.0906590 0.995882i \(-0.471103\pi\)
0.0906590 + 0.995882i \(0.471103\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 4.48528 0.342998
\(172\) 0 0
\(173\) 9.65685 0.734197 0.367099 0.930182i \(-0.380351\pi\)
0.367099 + 0.930182i \(0.380351\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.82843 0.663585
\(178\) 0 0
\(179\) 21.5858 1.61340 0.806699 0.590963i \(-0.201252\pi\)
0.806699 + 0.590963i \(0.201252\pi\)
\(180\) 0 0
\(181\) −15.6569 −1.16376 −0.581882 0.813273i \(-0.697684\pi\)
−0.581882 + 0.813273i \(0.697684\pi\)
\(182\) 0 0
\(183\) 14.4853 1.07078
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.899495 0.0657776
\(188\) 0 0
\(189\) −0.343146 −0.0249602
\(190\) 0 0
\(191\) 1.51472 0.109601 0.0548006 0.998497i \(-0.482548\pi\)
0.0548006 + 0.998497i \(0.482548\pi\)
\(192\) 0 0
\(193\) 8.17157 0.588203 0.294101 0.955774i \(-0.404980\pi\)
0.294101 + 0.955774i \(0.404980\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.65685 0.545528 0.272764 0.962081i \(-0.412062\pi\)
0.272764 + 0.962081i \(0.412062\pi\)
\(198\) 0 0
\(199\) 25.6569 1.81877 0.909383 0.415960i \(-0.136554\pi\)
0.909383 + 0.415960i \(0.136554\pi\)
\(200\) 0 0
\(201\) −6.65685 −0.469538
\(202\) 0 0
\(203\) 6.62742 0.465153
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 13.6569 0.949217
\(208\) 0 0
\(209\) 8.31371 0.575071
\(210\) 0 0
\(211\) −10.8995 −0.750352 −0.375176 0.926954i \(-0.622418\pi\)
−0.375176 + 0.926954i \(0.622418\pi\)
\(212\) 0 0
\(213\) 23.3137 1.59743
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.686292 0.0465885
\(218\) 0 0
\(219\) 14.0711 0.950835
\(220\) 0 0
\(221\) −0.970563 −0.0652871
\(222\) 0 0
\(223\) −13.6569 −0.914531 −0.457265 0.889330i \(-0.651171\pi\)
−0.457265 + 0.889330i \(0.651171\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.3137 1.41464 0.707320 0.706893i \(-0.249904\pi\)
0.707320 + 0.706893i \(0.249904\pi\)
\(228\) 0 0
\(229\) −0.686292 −0.0453514 −0.0226757 0.999743i \(-0.507219\pi\)
−0.0226757 + 0.999743i \(0.507219\pi\)
\(230\) 0 0
\(231\) 10.4853 0.689881
\(232\) 0 0
\(233\) 13.3137 0.872210 0.436105 0.899896i \(-0.356358\pi\)
0.436105 + 0.899896i \(0.356358\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −30.9706 −2.01175
\(238\) 0 0
\(239\) 18.6274 1.20491 0.602454 0.798154i \(-0.294190\pi\)
0.602454 + 0.798154i \(0.294190\pi\)
\(240\) 0 0
\(241\) 9.82843 0.633105 0.316552 0.948575i \(-0.397475\pi\)
0.316552 + 0.948575i \(0.397475\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.97056 −0.570783
\(248\) 0 0
\(249\) 3.00000 0.190117
\(250\) 0 0
\(251\) 8.75736 0.552760 0.276380 0.961048i \(-0.410865\pi\)
0.276380 + 0.961048i \(0.410865\pi\)
\(252\) 0 0
\(253\) 25.3137 1.59146
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.31371 −0.331460 −0.165730 0.986171i \(-0.552998\pi\)
−0.165730 + 0.986171i \(0.552998\pi\)
\(258\) 0 0
\(259\) 6.34315 0.394144
\(260\) 0 0
\(261\) −22.6274 −1.40060
\(262\) 0 0
\(263\) −10.4853 −0.646550 −0.323275 0.946305i \(-0.604784\pi\)
−0.323275 + 0.946305i \(0.604784\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −14.8995 −0.911834
\(268\) 0 0
\(269\) 6.34315 0.386748 0.193374 0.981125i \(-0.438057\pi\)
0.193374 + 0.981125i \(0.438057\pi\)
\(270\) 0 0
\(271\) 19.1716 1.16459 0.582295 0.812978i \(-0.302155\pi\)
0.582295 + 0.812978i \(0.302155\pi\)
\(272\) 0 0
\(273\) −11.3137 −0.684737
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 9.65685 0.580224 0.290112 0.956993i \(-0.406307\pi\)
0.290112 + 0.956993i \(0.406307\pi\)
\(278\) 0 0
\(279\) −2.34315 −0.140280
\(280\) 0 0
\(281\) −20.6274 −1.23053 −0.615264 0.788321i \(-0.710951\pi\)
−0.615264 + 0.788321i \(0.710951\pi\)
\(282\) 0 0
\(283\) 9.72792 0.578265 0.289132 0.957289i \(-0.406633\pi\)
0.289132 + 0.957289i \(0.406633\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.82843 −0.521126
\(288\) 0 0
\(289\) −16.9706 −0.998268
\(290\) 0 0
\(291\) −41.7990 −2.45030
\(292\) 0 0
\(293\) 17.3137 1.01148 0.505739 0.862687i \(-0.331220\pi\)
0.505739 + 0.862687i \(0.331220\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.17157 0.126007
\(298\) 0 0
\(299\) −27.3137 −1.57959
\(300\) 0 0
\(301\) 8.28427 0.477497
\(302\) 0 0
\(303\) −45.7990 −2.63108
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.24264 0.185067 0.0925336 0.995710i \(-0.470503\pi\)
0.0925336 + 0.995710i \(0.470503\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −22.4853 −1.27502 −0.637512 0.770441i \(-0.720036\pi\)
−0.637512 + 0.770441i \(0.720036\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.34315 0.131604 0.0658021 0.997833i \(-0.479039\pi\)
0.0658021 + 0.997833i \(0.479039\pi\)
\(318\) 0 0
\(319\) −41.9411 −2.34825
\(320\) 0 0
\(321\) −4.65685 −0.259920
\(322\) 0 0
\(323\) 0.272078 0.0151388
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 28.1421 1.55626
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 22.8995 1.25867 0.629335 0.777134i \(-0.283327\pi\)
0.629335 + 0.777134i \(0.283327\pi\)
\(332\) 0 0
\(333\) −21.6569 −1.18679
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.34315 −0.400007 −0.200003 0.979795i \(-0.564095\pi\)
−0.200003 + 0.979795i \(0.564095\pi\)
\(338\) 0 0
\(339\) 8.07107 0.438360
\(340\) 0 0
\(341\) −4.34315 −0.235195
\(342\) 0 0
\(343\) 11.0294 0.595534
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.8995 1.01458 0.507289 0.861776i \(-0.330648\pi\)
0.507289 + 0.861776i \(0.330648\pi\)
\(348\) 0 0
\(349\) −29.3137 −1.56913 −0.784563 0.620049i \(-0.787113\pi\)
−0.784563 + 0.620049i \(0.787113\pi\)
\(350\) 0 0
\(351\) −2.34315 −0.125068
\(352\) 0 0
\(353\) 12.6274 0.672090 0.336045 0.941846i \(-0.390911\pi\)
0.336045 + 0.941846i \(0.390911\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.343146 0.0181612
\(358\) 0 0
\(359\) 8.82843 0.465947 0.232973 0.972483i \(-0.425155\pi\)
0.232973 + 0.972483i \(0.425155\pi\)
\(360\) 0 0
\(361\) −16.4853 −0.867646
\(362\) 0 0
\(363\) −39.7990 −2.08891
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.970563 0.0506630 0.0253315 0.999679i \(-0.491936\pi\)
0.0253315 + 0.999679i \(0.491936\pi\)
\(368\) 0 0
\(369\) 30.1421 1.56914
\(370\) 0 0
\(371\) −6.34315 −0.329320
\(372\) 0 0
\(373\) −33.3137 −1.72492 −0.862459 0.506127i \(-0.831077\pi\)
−0.862459 + 0.506127i \(0.831077\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 45.2548 2.33074
\(378\) 0 0
\(379\) −16.8995 −0.868069 −0.434034 0.900896i \(-0.642910\pi\)
−0.434034 + 0.900896i \(0.642910\pi\)
\(380\) 0 0
\(381\) 48.6274 2.49126
\(382\) 0 0
\(383\) 27.4558 1.40293 0.701464 0.712705i \(-0.252530\pi\)
0.701464 + 0.712705i \(0.252530\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −28.2843 −1.43777
\(388\) 0 0
\(389\) −5.31371 −0.269416 −0.134708 0.990885i \(-0.543010\pi\)
−0.134708 + 0.990885i \(0.543010\pi\)
\(390\) 0 0
\(391\) 0.828427 0.0418954
\(392\) 0 0
\(393\) −12.8284 −0.647109
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.68629 −0.435952 −0.217976 0.975954i \(-0.569946\pi\)
−0.217976 + 0.975954i \(0.569946\pi\)
\(398\) 0 0
\(399\) 3.17157 0.158777
\(400\) 0 0
\(401\) 20.4558 1.02152 0.510758 0.859724i \(-0.329365\pi\)
0.510758 + 0.859724i \(0.329365\pi\)
\(402\) 0 0
\(403\) 4.68629 0.233441
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −40.1421 −1.98977
\(408\) 0 0
\(409\) 5.97056 0.295225 0.147613 0.989045i \(-0.452841\pi\)
0.147613 + 0.989045i \(0.452841\pi\)
\(410\) 0 0
\(411\) −12.0711 −0.595422
\(412\) 0 0
\(413\) 3.02944 0.149069
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −26.3137 −1.28859
\(418\) 0 0
\(419\) 6.55635 0.320299 0.160149 0.987093i \(-0.448802\pi\)
0.160149 + 0.987093i \(0.448802\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) 27.3137 1.32804
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.97056 0.240542
\(428\) 0 0
\(429\) 71.5980 3.45678
\(430\) 0 0
\(431\) 12.1421 0.584866 0.292433 0.956286i \(-0.405535\pi\)
0.292433 + 0.956286i \(0.405535\pi\)
\(432\) 0 0
\(433\) 21.8284 1.04901 0.524504 0.851408i \(-0.324251\pi\)
0.524504 + 0.851408i \(0.324251\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.65685 0.366277
\(438\) 0 0
\(439\) 0.686292 0.0327549 0.0163775 0.999866i \(-0.494787\pi\)
0.0163775 + 0.999866i \(0.494787\pi\)
\(440\) 0 0
\(441\) −17.8579 −0.850374
\(442\) 0 0
\(443\) −22.5563 −1.07168 −0.535842 0.844318i \(-0.680006\pi\)
−0.535842 + 0.844318i \(0.680006\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −8.00000 −0.378387
\(448\) 0 0
\(449\) −34.4558 −1.62607 −0.813036 0.582214i \(-0.802187\pi\)
−0.813036 + 0.582214i \(0.802187\pi\)
\(450\) 0 0
\(451\) 55.8701 2.63082
\(452\) 0 0
\(453\) −15.6569 −0.735623
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.3431 −0.811278 −0.405639 0.914033i \(-0.632951\pi\)
−0.405639 + 0.914033i \(0.632951\pi\)
\(458\) 0 0
\(459\) 0.0710678 0.00331716
\(460\) 0 0
\(461\) −5.65685 −0.263466 −0.131733 0.991285i \(-0.542054\pi\)
−0.131733 + 0.991285i \(0.542054\pi\)
\(462\) 0 0
\(463\) 12.9706 0.602793 0.301397 0.953499i \(-0.402547\pi\)
0.301397 + 0.953499i \(0.402547\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9.31371 −0.430987 −0.215494 0.976505i \(-0.569136\pi\)
−0.215494 + 0.976505i \(0.569136\pi\)
\(468\) 0 0
\(469\) −2.28427 −0.105478
\(470\) 0 0
\(471\) −47.4558 −2.18665
\(472\) 0 0
\(473\) −52.4264 −2.41057
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 21.6569 0.991599
\(478\) 0 0
\(479\) −35.1716 −1.60703 −0.803515 0.595284i \(-0.797039\pi\)
−0.803515 + 0.595284i \(0.797039\pi\)
\(480\) 0 0
\(481\) 43.3137 1.97494
\(482\) 0 0
\(483\) 9.65685 0.439402
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 25.1127 1.13796 0.568982 0.822350i \(-0.307337\pi\)
0.568982 + 0.822350i \(0.307337\pi\)
\(488\) 0 0
\(489\) −47.6274 −2.15379
\(490\) 0 0
\(491\) −16.6274 −0.750385 −0.375192 0.926947i \(-0.622423\pi\)
−0.375192 + 0.926947i \(0.622423\pi\)
\(492\) 0 0
\(493\) −1.37258 −0.0618180
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) −5.65685 −0.252730
\(502\) 0 0
\(503\) 16.9706 0.756680 0.378340 0.925667i \(-0.376495\pi\)
0.378340 + 0.925667i \(0.376495\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −45.8701 −2.03716
\(508\) 0 0
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 4.82843 0.213597
\(512\) 0 0
\(513\) 0.656854 0.0290008
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 50.6274 2.22659
\(518\) 0 0
\(519\) −23.3137 −1.02336
\(520\) 0 0
\(521\) 16.3137 0.714717 0.357358 0.933967i \(-0.383678\pi\)
0.357358 + 0.933967i \(0.383678\pi\)
\(522\) 0 0
\(523\) 19.5858 0.856427 0.428213 0.903678i \(-0.359143\pi\)
0.428213 + 0.903678i \(0.359143\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.142136 −0.00619153
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) −10.3431 −0.448854
\(532\) 0 0
\(533\) −60.2843 −2.61120
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −52.1127 −2.24883
\(538\) 0 0
\(539\) −33.1005 −1.42574
\(540\) 0 0
\(541\) 17.6569 0.759127 0.379564 0.925166i \(-0.376074\pi\)
0.379564 + 0.925166i \(0.376074\pi\)
\(542\) 0 0
\(543\) 37.7990 1.62211
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.5563 0.622385 0.311192 0.950347i \(-0.399272\pi\)
0.311192 + 0.950347i \(0.399272\pi\)
\(548\) 0 0
\(549\) −16.9706 −0.724286
\(550\) 0 0
\(551\) −12.6863 −0.540454
\(552\) 0 0
\(553\) −10.6274 −0.451924
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −26.3431 −1.11619 −0.558097 0.829775i \(-0.688468\pi\)
−0.558097 + 0.829775i \(0.688468\pi\)
\(558\) 0 0
\(559\) 56.5685 2.39259
\(560\) 0 0
\(561\) −2.17157 −0.0916839
\(562\) 0 0
\(563\) 6.97056 0.293774 0.146887 0.989153i \(-0.453075\pi\)
0.146887 + 0.989153i \(0.453075\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.85786 0.329999
\(568\) 0 0
\(569\) −26.3137 −1.10313 −0.551564 0.834133i \(-0.685969\pi\)
−0.551564 + 0.834133i \(0.685969\pi\)
\(570\) 0 0
\(571\) 31.9411 1.33669 0.668347 0.743849i \(-0.267002\pi\)
0.668347 + 0.743849i \(0.267002\pi\)
\(572\) 0 0
\(573\) −3.65685 −0.152767
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −41.2843 −1.71869 −0.859343 0.511399i \(-0.829127\pi\)
−0.859343 + 0.511399i \(0.829127\pi\)
\(578\) 0 0
\(579\) −19.7279 −0.819864
\(580\) 0 0
\(581\) 1.02944 0.0427083
\(582\) 0 0
\(583\) 40.1421 1.66252
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.0710678 0.00293328 0.00146664 0.999999i \(-0.499533\pi\)
0.00146664 + 0.999999i \(0.499533\pi\)
\(588\) 0 0
\(589\) −1.31371 −0.0541304
\(590\) 0 0
\(591\) −18.4853 −0.760383
\(592\) 0 0
\(593\) 7.00000 0.287456 0.143728 0.989617i \(-0.454091\pi\)
0.143728 + 0.989617i \(0.454091\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −61.9411 −2.53508
\(598\) 0 0
\(599\) 26.7696 1.09377 0.546887 0.837206i \(-0.315813\pi\)
0.546887 + 0.837206i \(0.315813\pi\)
\(600\) 0 0
\(601\) 35.1421 1.43348 0.716739 0.697342i \(-0.245634\pi\)
0.716739 + 0.697342i \(0.245634\pi\)
\(602\) 0 0
\(603\) 7.79899 0.317599
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −36.0000 −1.46119 −0.730597 0.682808i \(-0.760758\pi\)
−0.730597 + 0.682808i \(0.760758\pi\)
\(608\) 0 0
\(609\) −16.0000 −0.648353
\(610\) 0 0
\(611\) −54.6274 −2.20999
\(612\) 0 0
\(613\) −6.68629 −0.270057 −0.135028 0.990842i \(-0.543113\pi\)
−0.135028 + 0.990842i \(0.543113\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −13.3137 −0.535123 −0.267562 0.963541i \(-0.586218\pi\)
−0.267562 + 0.963541i \(0.586218\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) 0 0
\(623\) −5.11270 −0.204836
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −20.0711 −0.801561
\(628\) 0 0
\(629\) −1.31371 −0.0523810
\(630\) 0 0
\(631\) 15.8579 0.631292 0.315646 0.948877i \(-0.397779\pi\)
0.315646 + 0.948877i \(0.397779\pi\)
\(632\) 0 0
\(633\) 26.3137 1.04588
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 35.7157 1.41511
\(638\) 0 0
\(639\) −27.3137 −1.08051
\(640\) 0 0
\(641\) −45.3137 −1.78978 −0.894892 0.446283i \(-0.852748\pi\)
−0.894892 + 0.446283i \(0.852748\pi\)
\(642\) 0 0
\(643\) 14.6863 0.579171 0.289585 0.957152i \(-0.406483\pi\)
0.289585 + 0.957152i \(0.406483\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.65685 0.222394 0.111197 0.993798i \(-0.464532\pi\)
0.111197 + 0.993798i \(0.464532\pi\)
\(648\) 0 0
\(649\) −19.1716 −0.752550
\(650\) 0 0
\(651\) −1.65685 −0.0649372
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −16.4853 −0.643152
\(658\) 0 0
\(659\) −51.1838 −1.99384 −0.996918 0.0784478i \(-0.975004\pi\)
−0.996918 + 0.0784478i \(0.975004\pi\)
\(660\) 0 0
\(661\) −6.34315 −0.246720 −0.123360 0.992362i \(-0.539367\pi\)
−0.123360 + 0.992362i \(0.539367\pi\)
\(662\) 0 0
\(663\) 2.34315 0.0910002
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −38.6274 −1.49566
\(668\) 0 0
\(669\) 32.9706 1.27472
\(670\) 0 0
\(671\) −31.4558 −1.21434
\(672\) 0 0
\(673\) 25.3137 0.975772 0.487886 0.872907i \(-0.337768\pi\)
0.487886 + 0.872907i \(0.337768\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.6863 1.10250 0.551252 0.834339i \(-0.314150\pi\)
0.551252 + 0.834339i \(0.314150\pi\)
\(678\) 0 0
\(679\) −14.3431 −0.550439
\(680\) 0 0
\(681\) −51.4558 −1.97179
\(682\) 0 0
\(683\) −16.6985 −0.638950 −0.319475 0.947595i \(-0.603507\pi\)
−0.319475 + 0.947595i \(0.603507\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.65685 0.0632129
\(688\) 0 0
\(689\) −43.3137 −1.65012
\(690\) 0 0
\(691\) 36.2132 1.37762 0.688808 0.724944i \(-0.258134\pi\)
0.688808 + 0.724944i \(0.258134\pi\)
\(692\) 0 0
\(693\) −12.2843 −0.466641
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.82843 0.0692566
\(698\) 0 0
\(699\) −32.1421 −1.21573
\(700\) 0 0
\(701\) 47.3137 1.78701 0.893507 0.449049i \(-0.148237\pi\)
0.893507 + 0.449049i \(0.148237\pi\)
\(702\) 0 0
\(703\) −12.1421 −0.457949
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −15.7157 −0.591051
\(708\) 0 0
\(709\) 21.9411 0.824016 0.412008 0.911180i \(-0.364828\pi\)
0.412008 + 0.911180i \(0.364828\pi\)
\(710\) 0 0
\(711\) 36.2843 1.36077
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −44.9706 −1.67946
\(718\) 0 0
\(719\) −23.4558 −0.874755 −0.437378 0.899278i \(-0.644093\pi\)
−0.437378 + 0.899278i \(0.644093\pi\)
\(720\) 0 0
\(721\) 1.37258 0.0511177
\(722\) 0 0
\(723\) −23.7279 −0.882451
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) −1.71573 −0.0634585
\(732\) 0 0
\(733\) −26.3431 −0.973006 −0.486503 0.873679i \(-0.661728\pi\)
−0.486503 + 0.873679i \(0.661728\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14.4558 0.532488
\(738\) 0 0
\(739\) −28.6274 −1.05308 −0.526538 0.850151i \(-0.676510\pi\)
−0.526538 + 0.850151i \(0.676510\pi\)
\(740\) 0 0
\(741\) 21.6569 0.795584
\(742\) 0 0
\(743\) −25.7990 −0.946473 −0.473237 0.880935i \(-0.656914\pi\)
−0.473237 + 0.880935i \(0.656914\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.51472 −0.128597
\(748\) 0 0
\(749\) −1.59798 −0.0583889
\(750\) 0 0
\(751\) −34.6274 −1.26357 −0.631786 0.775143i \(-0.717678\pi\)
−0.631786 + 0.775143i \(0.717678\pi\)
\(752\) 0 0
\(753\) −21.1421 −0.770462
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 25.6569 0.932514 0.466257 0.884649i \(-0.345602\pi\)
0.466257 + 0.884649i \(0.345602\pi\)
\(758\) 0 0
\(759\) −61.1127 −2.21825
\(760\) 0 0
\(761\) −39.2843 −1.42405 −0.712027 0.702152i \(-0.752223\pi\)
−0.712027 + 0.702152i \(0.752223\pi\)
\(762\) 0 0
\(763\) 9.65685 0.349602
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.6863 0.746939
\(768\) 0 0
\(769\) 34.1127 1.23014 0.615068 0.788474i \(-0.289129\pi\)
0.615068 + 0.788474i \(0.289129\pi\)
\(770\) 0 0
\(771\) 12.8284 0.462005
\(772\) 0 0
\(773\) 19.3137 0.694666 0.347333 0.937742i \(-0.387087\pi\)
0.347333 + 0.937742i \(0.387087\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −15.3137 −0.549376
\(778\) 0 0
\(779\) 16.8995 0.605487
\(780\) 0 0
\(781\) −50.6274 −1.81159
\(782\) 0 0
\(783\) −3.31371 −0.118422
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.6274 0.592703 0.296352 0.955079i \(-0.404230\pi\)
0.296352 + 0.955079i \(0.404230\pi\)
\(788\) 0 0
\(789\) 25.3137 0.901192
\(790\) 0 0
\(791\) 2.76955 0.0984740
\(792\) 0 0
\(793\) 33.9411 1.20528
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 1.65685 0.0586153
\(800\) 0 0
\(801\) 17.4558 0.616772
\(802\) 0 0
\(803\) −30.5563 −1.07831
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −15.3137 −0.539068
\(808\) 0 0
\(809\) 40.6274 1.42838 0.714192 0.699950i \(-0.246794\pi\)
0.714192 + 0.699950i \(0.246794\pi\)
\(810\) 0 0
\(811\) −35.9411 −1.26206 −0.631032 0.775757i \(-0.717368\pi\)
−0.631032 + 0.775757i \(0.717368\pi\)
\(812\) 0 0
\(813\) −46.2843 −1.62326
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −15.8579 −0.554796
\(818\) 0 0
\(819\) 13.2548 0.463161
\(820\) 0 0
\(821\) 40.6274 1.41791 0.708953 0.705255i \(-0.249168\pi\)
0.708953 + 0.705255i \(0.249168\pi\)
\(822\) 0 0
\(823\) 12.9706 0.452125 0.226063 0.974113i \(-0.427415\pi\)
0.226063 + 0.974113i \(0.427415\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0711 1.67159 0.835797 0.549038i \(-0.185006\pi\)
0.835797 + 0.549038i \(0.185006\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) −23.3137 −0.808744
\(832\) 0 0
\(833\) −1.08326 −0.0375328
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.343146 −0.0118609
\(838\) 0 0
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 49.7990 1.71517
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −13.6569 −0.469255
\(848\) 0 0
\(849\) −23.4853 −0.806013
\(850\) 0 0
\(851\) −36.9706 −1.26733
\(852\) 0 0
\(853\) 16.6274 0.569312 0.284656 0.958630i \(-0.408121\pi\)
0.284656 + 0.958630i \(0.408121\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −11.0000 −0.375753 −0.187876 0.982193i \(-0.560160\pi\)
−0.187876 + 0.982193i \(0.560160\pi\)
\(858\) 0 0
\(859\) 54.2132 1.84973 0.924865 0.380295i \(-0.124177\pi\)
0.924865 + 0.380295i \(0.124177\pi\)
\(860\) 0 0
\(861\) 21.3137 0.726369
\(862\) 0 0
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 40.9706 1.39143
\(868\) 0 0
\(869\) 67.2548 2.28146
\(870\) 0 0
\(871\) −15.5980 −0.528517
\(872\) 0 0
\(873\) 48.9706 1.65740
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −26.3431 −0.889545 −0.444772 0.895644i \(-0.646715\pi\)
−0.444772 + 0.895644i \(0.646715\pi\)
\(878\) 0 0
\(879\) −41.7990 −1.40984
\(880\) 0 0
\(881\) −14.6863 −0.494794 −0.247397 0.968914i \(-0.579575\pi\)
−0.247397 + 0.968914i \(0.579575\pi\)
\(882\) 0 0
\(883\) 43.2426 1.45523 0.727615 0.685985i \(-0.240629\pi\)
0.727615 + 0.685985i \(0.240629\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.3431 −0.750209 −0.375105 0.926982i \(-0.622393\pi\)
−0.375105 + 0.926982i \(0.622393\pi\)
\(888\) 0 0
\(889\) 16.6863 0.559640
\(890\) 0 0
\(891\) −49.7279 −1.66595
\(892\) 0 0
\(893\) 15.3137 0.512454
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 65.9411 2.20171
\(898\) 0 0
\(899\) 6.62742 0.221037
\(900\) 0 0
\(901\) 1.31371 0.0437660
\(902\) 0 0
\(903\) −20.0000 −0.665558
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −45.3137 −1.50462 −0.752308 0.658811i \(-0.771060\pi\)
−0.752308 + 0.658811i \(0.771060\pi\)
\(908\) 0 0
\(909\) 53.6569 1.77969
\(910\) 0 0
\(911\) 36.2843 1.20215 0.601076 0.799192i \(-0.294739\pi\)
0.601076 + 0.799192i \(0.294739\pi\)
\(912\) 0 0
\(913\) −6.51472 −0.215606
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.40202 −0.145368
\(918\) 0 0
\(919\) −39.1716 −1.29215 −0.646075 0.763274i \(-0.723591\pi\)
−0.646075 + 0.763274i \(0.723591\pi\)
\(920\) 0 0
\(921\) −7.82843 −0.257955
\(922\) 0 0
\(923\) 54.6274 1.79808
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.68629 −0.153918
\(928\) 0 0
\(929\) 0.627417 0.0205849 0.0102924 0.999947i \(-0.496724\pi\)
0.0102924 + 0.999947i \(0.496724\pi\)
\(930\) 0 0
\(931\) −10.0122 −0.328136
\(932\) 0 0
\(933\) 54.2843 1.77719
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20.1127 0.657053 0.328527 0.944495i \(-0.393448\pi\)
0.328527 + 0.944495i \(0.393448\pi\)
\(938\) 0 0
\(939\) −24.1421 −0.787849
\(940\) 0 0
\(941\) −40.9706 −1.33560 −0.667801 0.744340i \(-0.732764\pi\)
−0.667801 + 0.744340i \(0.732764\pi\)
\(942\) 0 0
\(943\) 51.4558 1.67563
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −33.5980 −1.09179 −0.545894 0.837854i \(-0.683810\pi\)
−0.545894 + 0.837854i \(0.683810\pi\)
\(948\) 0 0
\(949\) 32.9706 1.07027
\(950\) 0 0
\(951\) −5.65685 −0.183436
\(952\) 0 0
\(953\) 12.3137 0.398880 0.199440 0.979910i \(-0.436088\pi\)
0.199440 + 0.979910i \(0.436088\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 101.255 3.27310
\(958\) 0 0
\(959\) −4.14214 −0.133757
\(960\) 0 0
\(961\) −30.3137 −0.977862
\(962\) 0 0
\(963\) 5.45584 0.175812
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −42.6274 −1.37081 −0.685403 0.728164i \(-0.740374\pi\)
−0.685403 + 0.728164i \(0.740374\pi\)
\(968\) 0 0
\(969\) −0.656854 −0.0211012
\(970\) 0 0
\(971\) 42.8995 1.37671 0.688355 0.725374i \(-0.258333\pi\)
0.688355 + 0.725374i \(0.258333\pi\)
\(972\) 0 0
\(973\) −9.02944 −0.289470
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −24.1127 −0.771434 −0.385717 0.922617i \(-0.626046\pi\)
−0.385717 + 0.922617i \(0.626046\pi\)
\(978\) 0 0
\(979\) 32.3553 1.03408
\(980\) 0 0
\(981\) −32.9706 −1.05267
\(982\) 0 0
\(983\) −33.7990 −1.07802 −0.539010 0.842299i \(-0.681202\pi\)
−0.539010 + 0.842299i \(0.681202\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 19.3137 0.614762
\(988\) 0 0
\(989\) −48.2843 −1.53535
\(990\) 0 0
\(991\) 62.0833 1.97214 0.986070 0.166331i \(-0.0531922\pi\)
0.986070 + 0.166331i \(0.0531922\pi\)
\(992\) 0 0
\(993\) −55.2843 −1.75439
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 16.9706 0.537463 0.268732 0.963215i \(-0.413396\pi\)
0.268732 + 0.963215i \(0.413396\pi\)
\(998\) 0 0
\(999\) −3.17157 −0.100344
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 3200.2.a.bh.1.1 yes 2
4.3 odd 2 3200.2.a.bi.1.2 yes 2
5.2 odd 4 3200.2.c.bb.2049.4 4
5.3 odd 4 3200.2.c.bb.2049.1 4
5.4 even 2 3200.2.a.bj.1.2 yes 2
8.3 odd 2 3200.2.a.bd.1.1 yes 2
8.5 even 2 3200.2.a.bm.1.2 yes 2
20.3 even 4 3200.2.c.z.2049.4 4
20.7 even 4 3200.2.c.z.2049.1 4
20.19 odd 2 3200.2.a.bg.1.1 yes 2
40.3 even 4 3200.2.c.ba.2049.1 4
40.13 odd 4 3200.2.c.y.2049.4 4
40.19 odd 2 3200.2.a.bn.1.2 yes 2
40.27 even 4 3200.2.c.ba.2049.4 4
40.29 even 2 3200.2.a.bc.1.1 2
40.37 odd 4 3200.2.c.y.2049.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.a.bc.1.1 2 40.29 even 2
3200.2.a.bd.1.1 yes 2 8.3 odd 2
3200.2.a.bg.1.1 yes 2 20.19 odd 2
3200.2.a.bh.1.1 yes 2 1.1 even 1 trivial
3200.2.a.bi.1.2 yes 2 4.3 odd 2
3200.2.a.bj.1.2 yes 2 5.4 even 2
3200.2.a.bm.1.2 yes 2 8.5 even 2
3200.2.a.bn.1.2 yes 2 40.19 odd 2
3200.2.c.y.2049.1 4 40.37 odd 4
3200.2.c.y.2049.4 4 40.13 odd 4
3200.2.c.z.2049.1 4 20.7 even 4
3200.2.c.z.2049.4 4 20.3 even 4
3200.2.c.ba.2049.1 4 40.3 even 4
3200.2.c.ba.2049.4 4 40.27 even 4
3200.2.c.bb.2049.1 4 5.3 odd 4
3200.2.c.bb.2049.4 4 5.2 odd 4