Properties

Label 4608.2.a.r.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1536)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.41421 q^{5} +0.585786 q^{7} +O(q^{10})\) \(q+3.41421 q^{5} +0.585786 q^{7} +2.00000 q^{11} +2.82843 q^{13} +7.65685 q^{17} -5.65685 q^{19} -6.82843 q^{23} +6.65685 q^{25} +3.41421 q^{29} +7.41421 q^{31} +2.00000 q^{35} -1.65685 q^{37} +0.343146 q^{41} +9.65685 q^{43} +4.48528 q^{47} -6.65685 q^{49} -7.89949 q^{53} +6.82843 q^{55} -4.00000 q^{59} -1.65685 q^{61} +9.65685 q^{65} +8.00000 q^{67} -14.8284 q^{71} +9.65685 q^{73} +1.17157 q^{77} +14.2426 q^{79} -13.3137 q^{83} +26.1421 q^{85} -2.00000 q^{89} +1.65685 q^{91} -19.3137 q^{95} -9.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 4 q^{7} + 4 q^{11} + 4 q^{17} - 8 q^{23} + 2 q^{25} + 4 q^{29} + 12 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{41} + 8 q^{43} - 8 q^{47} - 2 q^{49} + 4 q^{53} + 8 q^{55} - 8 q^{59} + 8 q^{61} + 8 q^{65} + 16 q^{67} - 24 q^{71} + 8 q^{73} + 8 q^{77} + 20 q^{79} - 4 q^{83} + 24 q^{85} - 4 q^{89} - 8 q^{91} - 16 q^{95} + 4 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.41421 1.52688 0.763441 0.645877i \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) 0 0
\(7\) 0.585786 0.221406 0.110703 0.993854i \(-0.464690\pi\)
0.110703 + 0.993854i \(0.464690\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.82843 −1.42383 −0.711913 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.41421 0.634004 0.317002 0.948425i \(-0.397324\pi\)
0.317002 + 0.948425i \(0.397324\pi\)
\(30\) 0 0
\(31\) 7.41421 1.33163 0.665816 0.746116i \(-0.268084\pi\)
0.665816 + 0.746116i \(0.268084\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −1.65685 −0.272385 −0.136193 0.990682i \(-0.543487\pi\)
−0.136193 + 0.990682i \(0.543487\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.343146 0.0535904 0.0267952 0.999641i \(-0.491470\pi\)
0.0267952 + 0.999641i \(0.491470\pi\)
\(42\) 0 0
\(43\) 9.65685 1.47266 0.736328 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.48528 0.654246 0.327123 0.944982i \(-0.393921\pi\)
0.327123 + 0.944982i \(0.393921\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.89949 −1.08508 −0.542540 0.840030i \(-0.682537\pi\)
−0.542540 + 0.840030i \(0.682537\pi\)
\(54\) 0 0
\(55\) 6.82843 0.920745
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −1.65685 −0.212138 −0.106069 0.994359i \(-0.533827\pi\)
−0.106069 + 0.994359i \(0.533827\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.65685 1.19779
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.8284 −1.75981 −0.879905 0.475149i \(-0.842394\pi\)
−0.879905 + 0.475149i \(0.842394\pi\)
\(72\) 0 0
\(73\) 9.65685 1.13025 0.565125 0.825006i \(-0.308828\pi\)
0.565125 + 0.825006i \(0.308828\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.17157 0.133513
\(78\) 0 0
\(79\) 14.2426 1.60242 0.801211 0.598382i \(-0.204189\pi\)
0.801211 + 0.598382i \(0.204189\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −13.3137 −1.46137 −0.730685 0.682715i \(-0.760799\pi\)
−0.730685 + 0.682715i \(0.760799\pi\)
\(84\) 0 0
\(85\) 26.1421 2.83551
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 1.65685 0.173686
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −19.3137 −1.98154
\(96\) 0 0
\(97\) −9.31371 −0.945664 −0.472832 0.881153i \(-0.656768\pi\)
−0.472832 + 0.881153i \(0.656768\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.5563 1.34891 0.674454 0.738317i \(-0.264379\pi\)
0.674454 + 0.738317i \(0.264379\pi\)
\(102\) 0 0
\(103\) −5.07107 −0.499667 −0.249834 0.968289i \(-0.580376\pi\)
−0.249834 + 0.968289i \(0.580376\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.31371 0.707043 0.353521 0.935426i \(-0.384984\pi\)
0.353521 + 0.935426i \(0.384984\pi\)
\(108\) 0 0
\(109\) 18.8284 1.80344 0.901718 0.432324i \(-0.142306\pi\)
0.901718 + 0.432324i \(0.142306\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −23.3137 −2.17401
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.48528 0.411165
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) −18.7279 −1.66183 −0.830917 0.556396i \(-0.812184\pi\)
−0.830917 + 0.556396i \(0.812184\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −3.31371 −0.287335
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.6569 −1.33766 −0.668828 0.743417i \(-0.733204\pi\)
−0.668828 + 0.743417i \(0.733204\pi\)
\(138\) 0 0
\(139\) 3.31371 0.281065 0.140533 0.990076i \(-0.455119\pi\)
0.140533 + 0.990076i \(0.455119\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) 11.6569 0.968049
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.75736 0.471661 0.235831 0.971794i \(-0.424219\pi\)
0.235831 + 0.971794i \(0.424219\pi\)
\(150\) 0 0
\(151\) 7.41421 0.603360 0.301680 0.953409i \(-0.402453\pi\)
0.301680 + 0.953409i \(0.402453\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 25.3137 2.03325
\(156\) 0 0
\(157\) 20.0000 1.59617 0.798087 0.602542i \(-0.205846\pi\)
0.798087 + 0.602542i \(0.205846\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −17.6569 −1.38299 −0.691496 0.722380i \(-0.743048\pi\)
−0.691496 + 0.722380i \(0.743048\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.3431 0.800377 0.400188 0.916433i \(-0.368945\pi\)
0.400188 + 0.916433i \(0.368945\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.72792 −0.511514 −0.255757 0.966741i \(-0.582325\pi\)
−0.255757 + 0.966741i \(0.582325\pi\)
\(174\) 0 0
\(175\) 3.89949 0.294774
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 8.48528 0.630706 0.315353 0.948974i \(-0.397877\pi\)
0.315353 + 0.948974i \(0.397877\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.65685 −0.415900
\(186\) 0 0
\(187\) 15.3137 1.11985
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.9706 −1.22795 −0.613973 0.789327i \(-0.710430\pi\)
−0.613973 + 0.789327i \(0.710430\pi\)
\(192\) 0 0
\(193\) −10.3431 −0.744516 −0.372258 0.928129i \(-0.621416\pi\)
−0.372258 + 0.928129i \(0.621416\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.0711 1.21626 0.608132 0.793836i \(-0.291919\pi\)
0.608132 + 0.793836i \(0.291919\pi\)
\(198\) 0 0
\(199\) 17.5563 1.24454 0.622268 0.782804i \(-0.286211\pi\)
0.622268 + 0.782804i \(0.286211\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 1.17157 0.0818262
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −11.3137 −0.782586
\(210\) 0 0
\(211\) −3.31371 −0.228125 −0.114063 0.993474i \(-0.536386\pi\)
−0.114063 + 0.993474i \(0.536386\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 32.9706 2.24857
\(216\) 0 0
\(217\) 4.34315 0.294832
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 21.6569 1.45680
\(222\) 0 0
\(223\) −11.8995 −0.796849 −0.398425 0.917201i \(-0.630443\pi\)
−0.398425 + 0.917201i \(0.630443\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.68629 0.443785 0.221892 0.975071i \(-0.428777\pi\)
0.221892 + 0.975071i \(0.428777\pi\)
\(228\) 0 0
\(229\) −0.485281 −0.0320683 −0.0160341 0.999871i \(-0.505104\pi\)
−0.0160341 + 0.999871i \(0.505104\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.3137 0.872210 0.436105 0.899896i \(-0.356358\pi\)
0.436105 + 0.899896i \(0.356358\pi\)
\(234\) 0 0
\(235\) 15.3137 0.998956
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.9706 1.09773 0.548867 0.835910i \(-0.315059\pi\)
0.548867 + 0.835910i \(0.315059\pi\)
\(240\) 0 0
\(241\) −6.34315 −0.408598 −0.204299 0.978909i \(-0.565491\pi\)
−0.204299 + 0.978909i \(0.565491\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −22.7279 −1.45203
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −22.0000 −1.38863 −0.694314 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) 0 0
\(253\) −13.6569 −0.858599
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −21.3137 −1.32951 −0.664756 0.747060i \(-0.731465\pi\)
−0.664756 + 0.747060i \(0.731465\pi\)
\(258\) 0 0
\(259\) −0.970563 −0.0603078
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.34315 0.144485 0.0722423 0.997387i \(-0.476985\pi\)
0.0722423 + 0.997387i \(0.476985\pi\)
\(264\) 0 0
\(265\) −26.9706 −1.65679
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.24264 0.136736 0.0683681 0.997660i \(-0.478221\pi\)
0.0683681 + 0.997660i \(0.478221\pi\)
\(270\) 0 0
\(271\) −16.5858 −1.00751 −0.503757 0.863845i \(-0.668049\pi\)
−0.503757 + 0.863845i \(0.668049\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.3137 0.802847
\(276\) 0 0
\(277\) 24.4853 1.47118 0.735589 0.677428i \(-0.236906\pi\)
0.735589 + 0.677428i \(0.236906\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 7.31371 0.434755 0.217377 0.976088i \(-0.430250\pi\)
0.217377 + 0.976088i \(0.430250\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.201010 0.0118653
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.10051 0.473237 0.236618 0.971603i \(-0.423961\pi\)
0.236618 + 0.971603i \(0.423961\pi\)
\(294\) 0 0
\(295\) −13.6569 −0.795133
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −19.3137 −1.11694
\(300\) 0 0
\(301\) 5.65685 0.326056
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.65685 −0.323911
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.3137 0.641542 0.320771 0.947157i \(-0.396058\pi\)
0.320771 + 0.947157i \(0.396058\pi\)
\(312\) 0 0
\(313\) −16.9706 −0.959233 −0.479616 0.877478i \(-0.659224\pi\)
−0.479616 + 0.877478i \(0.659224\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.07107 −0.509482 −0.254741 0.967009i \(-0.581990\pi\)
−0.254741 + 0.967009i \(0.581990\pi\)
\(318\) 0 0
\(319\) 6.82843 0.382319
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −43.3137 −2.41004
\(324\) 0 0
\(325\) 18.8284 1.04441
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.62742 0.144854
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 27.3137 1.49231
\(336\) 0 0
\(337\) −12.9706 −0.706552 −0.353276 0.935519i \(-0.614932\pi\)
−0.353276 + 0.935519i \(0.614932\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14.8284 0.803004
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.68629 0.144208 0.0721038 0.997397i \(-0.477029\pi\)
0.0721038 + 0.997397i \(0.477029\pi\)
\(348\) 0 0
\(349\) 23.3137 1.24795 0.623977 0.781443i \(-0.285516\pi\)
0.623977 + 0.781443i \(0.285516\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.31371 −0.495719 −0.247859 0.968796i \(-0.579727\pi\)
−0.247859 + 0.968796i \(0.579727\pi\)
\(354\) 0 0
\(355\) −50.6274 −2.68702
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −23.7990 −1.25606 −0.628031 0.778188i \(-0.716139\pi\)
−0.628031 + 0.778188i \(0.716139\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 32.9706 1.72576
\(366\) 0 0
\(367\) 2.72792 0.142396 0.0711982 0.997462i \(-0.477318\pi\)
0.0711982 + 0.997462i \(0.477318\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.62742 −0.240244
\(372\) 0 0
\(373\) −32.2843 −1.67162 −0.835808 0.549022i \(-0.815000\pi\)
−0.835808 + 0.549022i \(0.815000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.65685 0.497353
\(378\) 0 0
\(379\) 29.6569 1.52337 0.761685 0.647947i \(-0.224372\pi\)
0.761685 + 0.647947i \(0.224372\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.31371 −0.169323 −0.0846613 0.996410i \(-0.526981\pi\)
−0.0846613 + 0.996410i \(0.526981\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13.7574 −0.697526 −0.348763 0.937211i \(-0.613398\pi\)
−0.348763 + 0.937211i \(0.613398\pi\)
\(390\) 0 0
\(391\) −52.2843 −2.64413
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 48.6274 2.44671
\(396\) 0 0
\(397\) 4.00000 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.3431 1.01589 0.507944 0.861390i \(-0.330406\pi\)
0.507944 + 0.861390i \(0.330406\pi\)
\(402\) 0 0
\(403\) 20.9706 1.04462
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.31371 −0.164254
\(408\) 0 0
\(409\) −21.3137 −1.05390 −0.526948 0.849898i \(-0.676664\pi\)
−0.526948 + 0.849898i \(0.676664\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.34315 −0.115299
\(414\) 0 0
\(415\) −45.4558 −2.23134
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.3137 1.04124 0.520621 0.853788i \(-0.325700\pi\)
0.520621 + 0.853788i \(0.325700\pi\)
\(420\) 0 0
\(421\) 6.14214 0.299349 0.149675 0.988735i \(-0.452177\pi\)
0.149675 + 0.988735i \(0.452177\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 50.9706 2.47244
\(426\) 0 0
\(427\) −0.970563 −0.0469688
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.1127 1.30597 0.652986 0.757370i \(-0.273516\pi\)
0.652986 + 0.757370i \(0.273516\pi\)
\(432\) 0 0
\(433\) 28.6274 1.37575 0.687873 0.725831i \(-0.258545\pi\)
0.687873 + 0.725831i \(0.258545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 38.6274 1.84780
\(438\) 0 0
\(439\) 3.89949 0.186113 0.0930564 0.995661i \(-0.470336\pi\)
0.0930564 + 0.995661i \(0.470336\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 39.9411 1.89766 0.948830 0.315787i \(-0.102269\pi\)
0.948830 + 0.315787i \(0.102269\pi\)
\(444\) 0 0
\(445\) −6.82843 −0.323698
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.65685 −0.172578 −0.0862888 0.996270i \(-0.527501\pi\)
−0.0862888 + 0.996270i \(0.527501\pi\)
\(450\) 0 0
\(451\) 0.686292 0.0323162
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.65685 0.265197
\(456\) 0 0
\(457\) 5.31371 0.248565 0.124282 0.992247i \(-0.460337\pi\)
0.124282 + 0.992247i \(0.460337\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −25.0711 −1.16768 −0.583838 0.811870i \(-0.698450\pi\)
−0.583838 + 0.811870i \(0.698450\pi\)
\(462\) 0 0
\(463\) 33.5563 1.55950 0.779748 0.626094i \(-0.215347\pi\)
0.779748 + 0.626094i \(0.215347\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.3137 0.616085 0.308042 0.951373i \(-0.400326\pi\)
0.308042 + 0.951373i \(0.400326\pi\)
\(468\) 0 0
\(469\) 4.68629 0.216393
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 19.3137 0.888045
\(474\) 0 0
\(475\) −37.6569 −1.72781
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.4853 −0.570467 −0.285234 0.958458i \(-0.592071\pi\)
−0.285234 + 0.958458i \(0.592071\pi\)
\(480\) 0 0
\(481\) −4.68629 −0.213676
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −31.7990 −1.44392
\(486\) 0 0
\(487\) 25.5563 1.15807 0.579034 0.815303i \(-0.303430\pi\)
0.579034 + 0.815303i \(0.303430\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −23.3137 −1.05213 −0.526066 0.850443i \(-0.676334\pi\)
−0.526066 + 0.850443i \(0.676334\pi\)
\(492\) 0 0
\(493\) 26.1421 1.17738
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.68629 −0.389633
\(498\) 0 0
\(499\) −0.686292 −0.0307226 −0.0153613 0.999882i \(-0.504890\pi\)
−0.0153613 + 0.999882i \(0.504890\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.4853 −0.556691 −0.278346 0.960481i \(-0.589786\pi\)
−0.278346 + 0.960481i \(0.589786\pi\)
\(504\) 0 0
\(505\) 46.2843 2.05962
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.2132 −0.851610 −0.425805 0.904815i \(-0.640009\pi\)
−0.425805 + 0.904815i \(0.640009\pi\)
\(510\) 0 0
\(511\) 5.65685 0.250244
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.3137 −0.762933
\(516\) 0 0
\(517\) 8.97056 0.394525
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −22.9706 −1.00636 −0.503179 0.864182i \(-0.667836\pi\)
−0.503179 + 0.864182i \(0.667836\pi\)
\(522\) 0 0
\(523\) −37.6569 −1.64662 −0.823310 0.567593i \(-0.807875\pi\)
−0.823310 + 0.567593i \(0.807875\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 56.7696 2.47292
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.970563 0.0420397
\(534\) 0 0
\(535\) 24.9706 1.07957
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −13.3137 −0.573462
\(540\) 0 0
\(541\) 10.8284 0.465550 0.232775 0.972531i \(-0.425219\pi\)
0.232775 + 0.972531i \(0.425219\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 64.2843 2.75364
\(546\) 0 0
\(547\) 4.97056 0.212526 0.106263 0.994338i \(-0.466111\pi\)
0.106263 + 0.994338i \(0.466111\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −19.3137 −0.822792
\(552\) 0 0
\(553\) 8.34315 0.354787
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.07107 0.0453826 0.0226913 0.999743i \(-0.492777\pi\)
0.0226913 + 0.999743i \(0.492777\pi\)
\(558\) 0 0
\(559\) 27.3137 1.15525
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.6274 0.700762 0.350381 0.936607i \(-0.386052\pi\)
0.350381 + 0.936607i \(0.386052\pi\)
\(564\) 0 0
\(565\) −20.4853 −0.861822
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.9706 0.627599 0.313799 0.949489i \(-0.398398\pi\)
0.313799 + 0.949489i \(0.398398\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −45.4558 −1.89564
\(576\) 0 0
\(577\) −28.2843 −1.17749 −0.588745 0.808319i \(-0.700378\pi\)
−0.588745 + 0.808319i \(0.700378\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.79899 −0.323557
\(582\) 0 0
\(583\) −15.7990 −0.654327
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −31.3137 −1.29246 −0.646228 0.763145i \(-0.723654\pi\)
−0.646228 + 0.763145i \(0.723654\pi\)
\(588\) 0 0
\(589\) −41.9411 −1.72815
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) 15.3137 0.627801
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.5147 −0.470479 −0.235239 0.971937i \(-0.575587\pi\)
−0.235239 + 0.971937i \(0.575587\pi\)
\(600\) 0 0
\(601\) −5.65685 −0.230748 −0.115374 0.993322i \(-0.536807\pi\)
−0.115374 + 0.993322i \(0.536807\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −23.8995 −0.971653
\(606\) 0 0
\(607\) 3.89949 0.158276 0.0791378 0.996864i \(-0.474783\pi\)
0.0791378 + 0.996864i \(0.474783\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.6863 0.513232
\(612\) 0 0
\(613\) 10.6274 0.429237 0.214619 0.976698i \(-0.431149\pi\)
0.214619 + 0.976698i \(0.431149\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.9411 −0.963833 −0.481917 0.876217i \(-0.660059\pi\)
−0.481917 + 0.876217i \(0.660059\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.17157 −0.0469381
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.6863 −0.505836
\(630\) 0 0
\(631\) 18.9289 0.753549 0.376774 0.926305i \(-0.377033\pi\)
0.376774 + 0.926305i \(0.377033\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −63.9411 −2.53743
\(636\) 0 0
\(637\) −18.8284 −0.746009
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.02944 −0.356641 −0.178321 0.983972i \(-0.557066\pi\)
−0.178321 + 0.983972i \(0.557066\pi\)
\(642\) 0 0
\(643\) 32.2843 1.27317 0.636584 0.771208i \(-0.280347\pi\)
0.636584 + 0.771208i \(0.280347\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.8284 0.897478 0.448739 0.893663i \(-0.351873\pi\)
0.448739 + 0.893663i \(0.351873\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.5858 0.805584 0.402792 0.915292i \(-0.368040\pi\)
0.402792 + 0.915292i \(0.368040\pi\)
\(654\) 0 0
\(655\) −13.6569 −0.533617
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −39.3137 −1.53144 −0.765722 0.643171i \(-0.777618\pi\)
−0.765722 + 0.643171i \(0.777618\pi\)
\(660\) 0 0
\(661\) −39.3137 −1.52913 −0.764563 0.644549i \(-0.777045\pi\)
−0.764563 + 0.644549i \(0.777045\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.3137 −0.438727
\(666\) 0 0
\(667\) −23.3137 −0.902710
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3.31371 −0.127924
\(672\) 0 0
\(673\) −8.62742 −0.332562 −0.166281 0.986078i \(-0.553176\pi\)
−0.166281 + 0.986078i \(0.553176\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −35.2132 −1.35335 −0.676677 0.736280i \(-0.736580\pi\)
−0.676677 + 0.736280i \(0.736580\pi\)
\(678\) 0 0
\(679\) −5.45584 −0.209376
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.68629 −0.102788 −0.0513940 0.998678i \(-0.516366\pi\)
−0.0513940 + 0.998678i \(0.516366\pi\)
\(684\) 0 0
\(685\) −53.4558 −2.04244
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −22.3431 −0.851206
\(690\) 0 0
\(691\) 6.34315 0.241305 0.120652 0.992695i \(-0.461501\pi\)
0.120652 + 0.992695i \(0.461501\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11.3137 0.429153
\(696\) 0 0
\(697\) 2.62742 0.0995205
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.58579 0.173203 0.0866014 0.996243i \(-0.472399\pi\)
0.0866014 + 0.996243i \(0.472399\pi\)
\(702\) 0 0
\(703\) 9.37258 0.353494
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.94113 0.298657
\(708\) 0 0
\(709\) 31.1127 1.16846 0.584231 0.811587i \(-0.301396\pi\)
0.584231 + 0.811587i \(0.301396\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −50.6274 −1.89601
\(714\) 0 0
\(715\) 19.3137 0.722292
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.8284 −1.14971 −0.574853 0.818257i \(-0.694941\pi\)
−0.574853 + 0.818257i \(0.694941\pi\)
\(720\) 0 0
\(721\) −2.97056 −0.110630
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 22.7279 0.844094
\(726\) 0 0
\(727\) −22.2426 −0.824934 −0.412467 0.910973i \(-0.635333\pi\)
−0.412467 + 0.910973i \(0.635333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 73.9411 2.73481
\(732\) 0 0
\(733\) −23.5147 −0.868536 −0.434268 0.900784i \(-0.642993\pi\)
−0.434268 + 0.900784i \(0.642993\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 10.6274 0.390936 0.195468 0.980710i \(-0.437377\pi\)
0.195468 + 0.980710i \(0.437377\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 19.6569 0.720171
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.28427 0.156544
\(750\) 0 0
\(751\) 5.27208 0.192381 0.0961904 0.995363i \(-0.469334\pi\)
0.0961904 + 0.995363i \(0.469334\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 25.3137 0.921260
\(756\) 0 0
\(757\) −32.4853 −1.18070 −0.590349 0.807148i \(-0.701010\pi\)
−0.590349 + 0.807148i \(0.701010\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −35.6569 −1.29256 −0.646280 0.763100i \(-0.723676\pi\)
−0.646280 + 0.763100i \(0.723676\pi\)
\(762\) 0 0
\(763\) 11.0294 0.399292
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −11.3137 −0.408514
\(768\) 0 0
\(769\) −2.34315 −0.0844960 −0.0422480 0.999107i \(-0.513452\pi\)
−0.0422480 + 0.999107i \(0.513452\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.72792 −0.241987 −0.120993 0.992653i \(-0.538608\pi\)
−0.120993 + 0.992653i \(0.538608\pi\)
\(774\) 0 0
\(775\) 49.3553 1.77290
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.94113 −0.0695480
\(780\) 0 0
\(781\) −29.6569 −1.06121
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 68.2843 2.43717
\(786\) 0 0
\(787\) −21.6569 −0.771983 −0.385992 0.922502i \(-0.626141\pi\)
−0.385992 + 0.922502i \(0.626141\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.51472 −0.124969
\(792\) 0 0
\(793\) −4.68629 −0.166415
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.4142 1.25444 0.627218 0.778844i \(-0.284194\pi\)
0.627218 + 0.778844i \(0.284194\pi\)
\(798\) 0 0
\(799\) 34.3431 1.21497
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.3137 0.681566
\(804\) 0 0
\(805\) −13.6569 −0.481341
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.02944 0.176826 0.0884128 0.996084i \(-0.471821\pi\)
0.0884128 + 0.996084i \(0.471821\pi\)
\(810\) 0 0
\(811\) 18.3431 0.644115 0.322057 0.946720i \(-0.395626\pi\)
0.322057 + 0.946720i \(0.395626\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −60.2843 −2.11167
\(816\) 0 0
\(817\) −54.6274 −1.91117
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.87006 0.309567 0.154784 0.987948i \(-0.450532\pi\)
0.154784 + 0.987948i \(0.450532\pi\)
\(822\) 0 0
\(823\) 39.2132 1.36689 0.683443 0.730004i \(-0.260482\pi\)
0.683443 + 0.730004i \(0.260482\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −29.9411 −1.04115 −0.520577 0.853814i \(-0.674283\pi\)
−0.520577 + 0.853814i \(0.674283\pi\)
\(828\) 0 0
\(829\) 52.7696 1.83276 0.916381 0.400307i \(-0.131096\pi\)
0.916381 + 0.400307i \(0.131096\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −50.9706 −1.76603
\(834\) 0 0
\(835\) 35.3137 1.22208
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.4853 −0.983421 −0.491711 0.870759i \(-0.663628\pi\)
−0.491711 + 0.870759i \(0.663628\pi\)
\(840\) 0 0
\(841\) −17.3431 −0.598040
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −17.0711 −0.587263
\(846\) 0 0
\(847\) −4.10051 −0.140895
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.3137 0.387829
\(852\) 0 0
\(853\) −34.6274 −1.18562 −0.592810 0.805342i \(-0.701982\pi\)
−0.592810 + 0.805342i \(0.701982\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −41.5980 −1.42096 −0.710480 0.703717i \(-0.751522\pi\)
−0.710480 + 0.703717i \(0.751522\pi\)
\(858\) 0 0
\(859\) −44.9706 −1.53438 −0.767188 0.641422i \(-0.778345\pi\)
−0.767188 + 0.641422i \(0.778345\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −15.0294 −0.511608 −0.255804 0.966729i \(-0.582340\pi\)
−0.255804 + 0.966729i \(0.582340\pi\)
\(864\) 0 0
\(865\) −22.9706 −0.781023
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 28.4853 0.966297
\(870\) 0 0
\(871\) 22.6274 0.766701
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.31371 0.112024
\(876\) 0 0
\(877\) 23.3137 0.787248 0.393624 0.919272i \(-0.371221\pi\)
0.393624 + 0.919272i \(0.371221\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −45.3137 −1.52666 −0.763329 0.646010i \(-0.776436\pi\)
−0.763329 + 0.646010i \(0.776436\pi\)
\(882\) 0 0
\(883\) −30.3431 −1.02113 −0.510564 0.859840i \(-0.670563\pi\)
−0.510564 + 0.859840i \(0.670563\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 29.6569 0.995780 0.497890 0.867240i \(-0.334108\pi\)
0.497890 + 0.867240i \(0.334108\pi\)
\(888\) 0 0
\(889\) −10.9706 −0.367941
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −25.3726 −0.849061
\(894\) 0 0
\(895\) −13.6569 −0.456498
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 25.3137 0.844259
\(900\) 0 0
\(901\) −60.4853 −2.01506
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 28.9706 0.963014
\(906\) 0 0
\(907\) −51.5980 −1.71328 −0.856641 0.515912i \(-0.827453\pi\)
−0.856641 + 0.515912i \(0.827453\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28.6863 −0.950419 −0.475210 0.879873i \(-0.657628\pi\)
−0.475210 + 0.879873i \(0.657628\pi\)
\(912\) 0 0
\(913\) −26.6274 −0.881239
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.34315 −0.0773775
\(918\) 0 0
\(919\) −17.3553 −0.572500 −0.286250 0.958155i \(-0.592409\pi\)
−0.286250 + 0.958155i \(0.592409\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −41.9411 −1.38051
\(924\) 0 0
\(925\) −11.0294 −0.362646
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −29.5980 −0.971078 −0.485539 0.874215i \(-0.661377\pi\)
−0.485539 + 0.874215i \(0.661377\pi\)
\(930\) 0 0
\(931\) 37.6569 1.23415
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 52.2843 1.70988
\(936\) 0 0
\(937\) 36.6274 1.19657 0.598283 0.801285i \(-0.295850\pi\)
0.598283 + 0.801285i \(0.295850\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −20.3848 −0.664525 −0.332262 0.943187i \(-0.607812\pi\)
−0.332262 + 0.943187i \(0.607812\pi\)
\(942\) 0 0
\(943\) −2.34315 −0.0763033
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.6274 −1.12524 −0.562620 0.826716i \(-0.690207\pi\)
−0.562620 + 0.826716i \(0.690207\pi\)
\(948\) 0 0
\(949\) 27.3137 0.886640
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −30.9706 −1.00323 −0.501617 0.865090i \(-0.667261\pi\)
−0.501617 + 0.865090i \(0.667261\pi\)
\(954\) 0 0
\(955\) −57.9411 −1.87493
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9.17157 −0.296166
\(960\) 0 0
\(961\) 23.9706 0.773244
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −35.3137 −1.13679
\(966\) 0 0
\(967\) 1.75736 0.0565129 0.0282564 0.999601i \(-0.491005\pi\)
0.0282564 + 0.999601i \(0.491005\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 29.3137 0.940722 0.470361 0.882474i \(-0.344124\pi\)
0.470361 + 0.882474i \(0.344124\pi\)
\(972\) 0 0
\(973\) 1.94113 0.0622296
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.2843 0.968880 0.484440 0.874825i \(-0.339023\pi\)
0.484440 + 0.874825i \(0.339023\pi\)
\(978\) 0 0
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.28427 0.136647 0.0683235 0.997663i \(-0.478235\pi\)
0.0683235 + 0.997663i \(0.478235\pi\)
\(984\) 0 0
\(985\) 58.2843 1.85709
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −65.9411 −2.09681
\(990\) 0 0
\(991\) 11.8995 0.378000 0.189000 0.981977i \(-0.439475\pi\)
0.189000 + 0.981977i \(0.439475\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 59.9411 1.90026
\(996\) 0 0
\(997\) −12.9706 −0.410782 −0.205391 0.978680i \(-0.565847\pi\)
−0.205391 + 0.978680i \(0.565847\pi\)
\(998\) 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 4608.2.a.r.1.2 2
3.2 odd 2 1536.2.a.b.1.1 2
4.3 odd 2 4608.2.a.n.1.2 2
8.3 odd 2 4608.2.a.a.1.1 2
8.5 even 2 4608.2.a.e.1.1 2
12.11 even 2 1536.2.a.g.1.1 yes 2
16.3 odd 4 4608.2.d.o.2305.1 4
16.5 even 4 4608.2.d.c.2305.4 4
16.11 odd 4 4608.2.d.o.2305.4 4
16.13 even 4 4608.2.d.c.2305.1 4
24.5 odd 2 1536.2.a.l.1.2 yes 2
24.11 even 2 1536.2.a.e.1.2 yes 2
48.5 odd 4 1536.2.d.a.769.3 4
48.11 even 4 1536.2.d.f.769.1 4
48.29 odd 4 1536.2.d.a.769.2 4
48.35 even 4 1536.2.d.f.769.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.b.1.1 2 3.2 odd 2
1536.2.a.e.1.2 yes 2 24.11 even 2
1536.2.a.g.1.1 yes 2 12.11 even 2
1536.2.a.l.1.2 yes 2 24.5 odd 2
1536.2.d.a.769.2 4 48.29 odd 4
1536.2.d.a.769.3 4 48.5 odd 4
1536.2.d.f.769.1 4 48.11 even 4
1536.2.d.f.769.4 4 48.35 even 4
4608.2.a.a.1.1 2 8.3 odd 2
4608.2.a.e.1.1 2 8.5 even 2
4608.2.a.n.1.2 2 4.3 odd 2
4608.2.a.r.1.2 2 1.1 even 1 trivial
4608.2.d.c.2305.1 4 16.13 even 4
4608.2.d.c.2305.4 4 16.5 even 4
4608.2.d.o.2305.1 4 16.3 odd 4
4608.2.d.o.2305.4 4 16.11 odd 4