Properties

Label 6048.2.a.bi.1.2
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.82843 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+3.82843 q^{5} +1.00000 q^{7} +0.828427 q^{11} -2.82843 q^{13} -5.82843 q^{17} +1.17157 q^{19} +4.00000 q^{23} +9.65685 q^{25} +2.82843 q^{29} -1.17157 q^{31} +3.82843 q^{35} -8.65685 q^{37} +1.82843 q^{41} +2.17157 q^{43} +2.65685 q^{47} +1.00000 q^{49} +2.00000 q^{53} +3.17157 q^{55} +10.6569 q^{59} +8.82843 q^{61} -10.8284 q^{65} +4.00000 q^{67} +9.65685 q^{71} +5.65685 q^{73} +0.828427 q^{77} +3.82843 q^{79} +16.6569 q^{83} -22.3137 q^{85} +6.00000 q^{89} -2.82843 q^{91} +4.48528 q^{95} +10.1421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{5} + 2q^{7} - 4q^{11} - 6q^{17} + 8q^{19} + 8q^{23} + 8q^{25} - 8q^{31} + 2q^{35} - 6q^{37} - 2q^{41} + 10q^{43} - 6q^{47} + 2q^{49} + 4q^{53} + 12q^{55} + 10q^{59} + 12q^{61} - 16q^{65} + 8q^{67} + 8q^{71} - 4q^{77} + 2q^{79} + 22q^{83} - 22q^{85} + 12q^{89} - 8q^{95} - 8q^{97} + O(q^{100}) \)

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
\(5\) 3.82843 1.71212 0.856062 0.516873i \(-0.172904\pi\)
0.856062 + 0.516873i \(0.172904\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) −2.82843 −0.784465 −0.392232 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.82843 −1.41360 −0.706801 0.707413i \(-0.749862\pi\)
−0.706801 + 0.707413i \(0.749862\pi\)
\(18\) 0 0
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 9.65685 1.93137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.82843 0.525226 0.262613 0.964901i \(-0.415416\pi\)
0.262613 + 0.964901i \(0.415416\pi\)
\(30\) 0 0
\(31\) −1.17157 −0.210421 −0.105210 0.994450i \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.82843 0.647122
\(36\) 0 0
\(37\) −8.65685 −1.42318 −0.711589 0.702596i \(-0.752024\pi\)
−0.711589 + 0.702596i \(0.752024\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.82843 0.285552 0.142776 0.989755i \(-0.454397\pi\)
0.142776 + 0.989755i \(0.454397\pi\)
\(42\) 0 0
\(43\) 2.17157 0.331162 0.165581 0.986196i \(-0.447050\pi\)
0.165581 + 0.986196i \(0.447050\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.65685 0.387542 0.193771 0.981047i \(-0.437928\pi\)
0.193771 + 0.981047i \(0.437928\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 3.17157 0.427655
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.6569 1.38740 0.693702 0.720262i \(-0.255978\pi\)
0.693702 + 0.720262i \(0.255978\pi\)
\(60\) 0 0
\(61\) 8.82843 1.13036 0.565182 0.824966i \(-0.308806\pi\)
0.565182 + 0.824966i \(0.308806\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.8284 −1.34310
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.65685 1.14606 0.573029 0.819535i \(-0.305768\pi\)
0.573029 + 0.819535i \(0.305768\pi\)
\(72\) 0 0
\(73\) 5.65685 0.662085 0.331042 0.943616i \(-0.392600\pi\)
0.331042 + 0.943616i \(0.392600\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.828427 0.0944080
\(78\) 0 0
\(79\) 3.82843 0.430732 0.215366 0.976533i \(-0.430906\pi\)
0.215366 + 0.976533i \(0.430906\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.6569 1.82833 0.914164 0.405345i \(-0.132849\pi\)
0.914164 + 0.405345i \(0.132849\pi\)
\(84\) 0 0
\(85\) −22.3137 −2.42026
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.48528 0.460180
\(96\) 0 0
\(97\) 10.1421 1.02978 0.514889 0.857257i \(-0.327833\pi\)
0.514889 + 0.857257i \(0.327833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.9706 −1.48963 −0.744813 0.667273i \(-0.767461\pi\)
−0.744813 + 0.667273i \(0.767461\pi\)
\(102\) 0 0
\(103\) 13.6569 1.34565 0.672825 0.739802i \(-0.265081\pi\)
0.672825 + 0.739802i \(0.265081\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.65685 −0.546869 −0.273434 0.961891i \(-0.588160\pi\)
−0.273434 + 0.961891i \(0.588160\pi\)
\(108\) 0 0
\(109\) 0.656854 0.0629152 0.0314576 0.999505i \(-0.489985\pi\)
0.0314576 + 0.999505i \(0.489985\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.8284 1.20680 0.603398 0.797440i \(-0.293813\pi\)
0.603398 + 0.797440i \(0.293813\pi\)
\(114\) 0 0
\(115\) 15.3137 1.42801
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.82843 −0.534291
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 17.8284 1.59462
\(126\) 0 0
\(127\) 17.8284 1.58202 0.791009 0.611805i \(-0.209556\pi\)
0.791009 + 0.611805i \(0.209556\pi\)
\(128\) 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
\(133\) 1.17157 0.101588
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.17157 −0.783580 −0.391790 0.920055i \(-0.628144\pi\)
−0.391790 + 0.920055i \(0.628144\pi\)
\(138\) 0 0
\(139\) 16.8284 1.42737 0.713684 0.700468i \(-0.247025\pi\)
0.713684 + 0.700468i \(0.247025\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.34315 −0.195944
\(144\) 0 0
\(145\) 10.8284 0.899252
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.14214 −0.667030 −0.333515 0.942745i \(-0.608235\pi\)
−0.333515 + 0.942745i \(0.608235\pi\)
\(150\) 0 0
\(151\) 12.1716 0.990509 0.495254 0.868748i \(-0.335075\pi\)
0.495254 + 0.868748i \(0.335075\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.48528 −0.360266
\(156\) 0 0
\(157\) −2.82843 −0.225733 −0.112867 0.993610i \(-0.536003\pi\)
−0.112867 + 0.993610i \(0.536003\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 15.4853 1.21290 0.606450 0.795121i \(-0.292593\pi\)
0.606450 + 0.795121i \(0.292593\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.9706 −1.23584 −0.617920 0.786241i \(-0.712024\pi\)
−0.617920 + 0.786241i \(0.712024\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 9.65685 0.729990
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.1421 −1.80447 −0.902234 0.431247i \(-0.858074\pi\)
−0.902234 + 0.431247i \(0.858074\pi\)
\(180\) 0 0
\(181\) 8.34315 0.620141 0.310071 0.950714i \(-0.399647\pi\)
0.310071 + 0.950714i \(0.399647\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −33.1421 −2.43666
\(186\) 0 0
\(187\) −4.82843 −0.353090
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.6274 −1.49255 −0.746274 0.665639i \(-0.768159\pi\)
−0.746274 + 0.665639i \(0.768159\pi\)
\(192\) 0 0
\(193\) −12.3137 −0.886360 −0.443180 0.896433i \(-0.646150\pi\)
−0.443180 + 0.896433i \(0.646150\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 27.7990 1.98060 0.990298 0.138960i \(-0.0443759\pi\)
0.990298 + 0.138960i \(0.0443759\pi\)
\(198\) 0 0
\(199\) −14.9706 −1.06124 −0.530618 0.847611i \(-0.678040\pi\)
−0.530618 + 0.847611i \(0.678040\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.82843 0.198517
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.970563 0.0671352
\(210\) 0 0
\(211\) −16.9706 −1.16830 −0.584151 0.811645i \(-0.698572\pi\)
−0.584151 + 0.811645i \(0.698572\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.31371 0.566990
\(216\) 0 0
\(217\) −1.17157 −0.0795315
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 16.4853 1.10892
\(222\) 0 0
\(223\) −10.1421 −0.679168 −0.339584 0.940576i \(-0.610286\pi\)
−0.339584 + 0.940576i \(0.610286\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.34315 −0.421009 −0.210505 0.977593i \(-0.567511\pi\)
−0.210505 + 0.977593i \(0.567511\pi\)
\(228\) 0 0
\(229\) −21.3137 −1.40845 −0.704225 0.709977i \(-0.748705\pi\)
−0.704225 + 0.709977i \(0.748705\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.1421 −1.18853 −0.594265 0.804269i \(-0.702557\pi\)
−0.594265 + 0.804269i \(0.702557\pi\)
\(234\) 0 0
\(235\) 10.1716 0.663520
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.1421 −1.04415 −0.522074 0.852900i \(-0.674842\pi\)
−0.522074 + 0.852900i \(0.674842\pi\)
\(240\) 0 0
\(241\) 10.1421 0.653312 0.326656 0.945143i \(-0.394078\pi\)
0.326656 + 0.945143i \(0.394078\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.82843 0.244589
\(246\) 0 0
\(247\) −3.31371 −0.210846
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −29.9706 −1.89173 −0.945863 0.324567i \(-0.894781\pi\)
−0.945863 + 0.324567i \(0.894781\pi\)
\(252\) 0 0
\(253\) 3.31371 0.208331
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −8.65685 −0.537911
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.17157 0.318893 0.159446 0.987207i \(-0.449029\pi\)
0.159446 + 0.987207i \(0.449029\pi\)
\(264\) 0 0
\(265\) 7.65685 0.470357
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −13.1421 −0.801290 −0.400645 0.916233i \(-0.631214\pi\)
−0.400645 + 0.916233i \(0.631214\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) −13.3431 −0.801712 −0.400856 0.916141i \(-0.631287\pi\)
−0.400856 + 0.916141i \(0.631287\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.970563 0.0578989 0.0289495 0.999581i \(-0.490784\pi\)
0.0289495 + 0.999581i \(0.490784\pi\)
\(282\) 0 0
\(283\) 28.8284 1.71367 0.856836 0.515589i \(-0.172427\pi\)
0.856836 + 0.515589i \(0.172427\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.82843 0.107929
\(288\) 0 0
\(289\) 16.9706 0.998268
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.8284 1.04155 0.520774 0.853695i \(-0.325643\pi\)
0.520774 + 0.853695i \(0.325643\pi\)
\(294\) 0 0
\(295\) 40.7990 2.37541
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.3137 −0.654289
\(300\) 0 0
\(301\) 2.17157 0.125167
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 33.7990 1.93532
\(306\) 0 0
\(307\) 32.2843 1.84256 0.921280 0.388899i \(-0.127145\pi\)
0.921280 + 0.388899i \(0.127145\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −23.9706 −1.35925 −0.679623 0.733562i \(-0.737857\pi\)
−0.679623 + 0.733562i \(0.737857\pi\)
\(312\) 0 0
\(313\) −20.6274 −1.16593 −0.582965 0.812497i \(-0.698108\pi\)
−0.582965 + 0.812497i \(0.698108\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.3431 −1.03025 −0.515127 0.857114i \(-0.672255\pi\)
−0.515127 + 0.857114i \(0.672255\pi\)
\(318\) 0 0
\(319\) 2.34315 0.131191
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.82843 −0.379944
\(324\) 0 0
\(325\) −27.3137 −1.51509
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.65685 0.146477
\(330\) 0 0
\(331\) −10.7990 −0.593566 −0.296783 0.954945i \(-0.595914\pi\)
−0.296783 + 0.954945i \(0.595914\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 15.3137 0.836677
\(336\) 0 0
\(337\) 2.31371 0.126036 0.0630179 0.998012i \(-0.479927\pi\)
0.0630179 + 0.998012i \(0.479927\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −0.970563 −0.0525589
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.1421 1.18865 0.594326 0.804224i \(-0.297419\pi\)
0.594326 + 0.804224i \(0.297419\pi\)
\(348\) 0 0
\(349\) 36.2843 1.94225 0.971126 0.238566i \(-0.0766774\pi\)
0.971126 + 0.238566i \(0.0766774\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −33.1421 −1.76398 −0.881989 0.471270i \(-0.843796\pi\)
−0.881989 + 0.471270i \(0.843796\pi\)
\(354\) 0 0
\(355\) 36.9706 1.96219
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.6569 1.45967 0.729836 0.683622i \(-0.239596\pi\)
0.729836 + 0.683622i \(0.239596\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 21.6569 1.13357
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) 26.3137 1.36247 0.681236 0.732064i \(-0.261443\pi\)
0.681236 + 0.732064i \(0.261443\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −0.514719 −0.0264393 −0.0132197 0.999913i \(-0.504208\pi\)
−0.0132197 + 0.999913i \(0.504208\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.00000 0.0510976 0.0255488 0.999674i \(-0.491867\pi\)
0.0255488 + 0.999674i \(0.491867\pi\)
\(384\) 0 0
\(385\) 3.17157 0.161638
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 37.7990 1.91648 0.958242 0.285957i \(-0.0923115\pi\)
0.958242 + 0.285957i \(0.0923115\pi\)
\(390\) 0 0
\(391\) −23.3137 −1.17902
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.6569 0.737466
\(396\) 0 0
\(397\) 25.6569 1.28768 0.643840 0.765160i \(-0.277340\pi\)
0.643840 + 0.765160i \(0.277340\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.68629 0.134147 0.0670735 0.997748i \(-0.478634\pi\)
0.0670735 + 0.997748i \(0.478634\pi\)
\(402\) 0 0
\(403\) 3.31371 0.165068
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.17157 −0.355482
\(408\) 0 0
\(409\) −36.9706 −1.82808 −0.914038 0.405628i \(-0.867053\pi\)
−0.914038 + 0.405628i \(0.867053\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.6569 0.524390
\(414\) 0 0
\(415\) 63.7696 3.13032
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7.34315 −0.358736 −0.179368 0.983782i \(-0.557405\pi\)
−0.179368 + 0.983782i \(0.557405\pi\)
\(420\) 0 0
\(421\) −18.9706 −0.924569 −0.462284 0.886732i \(-0.652970\pi\)
−0.462284 + 0.886732i \(0.652970\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −56.2843 −2.73019
\(426\) 0 0
\(427\) 8.82843 0.427238
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.97056 −0.432097 −0.216048 0.976383i \(-0.569317\pi\)
−0.216048 + 0.976383i \(0.569317\pi\)
\(432\) 0 0
\(433\) 17.5147 0.841704 0.420852 0.907129i \(-0.361731\pi\)
0.420852 + 0.907129i \(0.361731\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.68629 0.224176
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −29.1716 −1.38598 −0.692992 0.720946i \(-0.743708\pi\)
−0.692992 + 0.720946i \(0.743708\pi\)
\(444\) 0 0
\(445\) 22.9706 1.08891
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.82843 0.322253 0.161127 0.986934i \(-0.448487\pi\)
0.161127 + 0.986934i \(0.448487\pi\)
\(450\) 0 0
\(451\) 1.51472 0.0713253
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.8284 −0.507644
\(456\) 0 0
\(457\) 18.6863 0.874108 0.437054 0.899435i \(-0.356022\pi\)
0.437054 + 0.899435i \(0.356022\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −17.4853 −0.814371 −0.407185 0.913346i \(-0.633490\pi\)
−0.407185 + 0.913346i \(0.633490\pi\)
\(462\) 0 0
\(463\) −23.8284 −1.10740 −0.553700 0.832716i \(-0.686785\pi\)
−0.553700 + 0.832716i \(0.686785\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.2843 −0.753546 −0.376773 0.926306i \(-0.622966\pi\)
−0.376773 + 0.926306i \(0.622966\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.79899 0.0827176
\(474\) 0 0
\(475\) 11.3137 0.519109
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.00000 −0.0456912 −0.0228456 0.999739i \(-0.507273\pi\)
−0.0228456 + 0.999739i \(0.507273\pi\)
\(480\) 0 0
\(481\) 24.4853 1.11643
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 38.8284 1.76311
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.85786 −0.0838442 −0.0419221 0.999121i \(-0.513348\pi\)
−0.0419221 + 0.999121i \(0.513348\pi\)
\(492\) 0 0
\(493\) −16.4853 −0.742460
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.65685 0.433169
\(498\) 0 0
\(499\) −29.7696 −1.33267 −0.666334 0.745653i \(-0.732138\pi\)
−0.666334 + 0.745653i \(0.732138\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 35.6274 1.58855 0.794274 0.607560i \(-0.207851\pi\)
0.794274 + 0.607560i \(0.207851\pi\)
\(504\) 0 0
\(505\) −57.3137 −2.55043
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11.4853 −0.509076 −0.254538 0.967063i \(-0.581923\pi\)
−0.254538 + 0.967063i \(0.581923\pi\)
\(510\) 0 0
\(511\) 5.65685 0.250244
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 52.2843 2.30392
\(516\) 0 0
\(517\) 2.20101 0.0968003
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.4558 −0.545700 −0.272850 0.962057i \(-0.587966\pi\)
−0.272850 + 0.962057i \(0.587966\pi\)
\(522\) 0 0
\(523\) −44.4264 −1.94263 −0.971316 0.237794i \(-0.923576\pi\)
−0.971316 + 0.237794i \(0.923576\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.82843 0.297451
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.17157 −0.224006
\(534\) 0 0
\(535\) −21.6569 −0.936307
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.828427 0.0356829
\(540\) 0 0
\(541\) 29.2843 1.25903 0.629515 0.776989i \(-0.283254\pi\)
0.629515 + 0.776989i \(0.283254\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.51472 0.107719
\(546\) 0 0
\(547\) −11.4853 −0.491075 −0.245538 0.969387i \(-0.578964\pi\)
−0.245538 + 0.969387i \(0.578964\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.31371 0.141169
\(552\) 0 0
\(553\) 3.82843 0.162801
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.6863 0.537535 0.268768 0.963205i \(-0.413384\pi\)
0.268768 + 0.963205i \(0.413384\pi\)
\(558\) 0 0
\(559\) −6.14214 −0.259785
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.6863 0.703243 0.351622 0.936142i \(-0.385630\pi\)
0.351622 + 0.936142i \(0.385630\pi\)
\(564\) 0 0
\(565\) 49.1127 2.06619
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −37.6569 −1.57866 −0.789329 0.613971i \(-0.789571\pi\)
−0.789329 + 0.613971i \(0.789571\pi\)
\(570\) 0 0
\(571\) 41.7696 1.74800 0.874001 0.485925i \(-0.161517\pi\)
0.874001 + 0.485925i \(0.161517\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 38.6274 1.61087
\(576\) 0 0
\(577\) −22.9706 −0.956277 −0.478139 0.878284i \(-0.658688\pi\)
−0.478139 + 0.878284i \(0.658688\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 16.6569 0.691043
\(582\) 0 0
\(583\) 1.65685 0.0686199
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.9706 −1.19574 −0.597872 0.801592i \(-0.703987\pi\)
−0.597872 + 0.801592i \(0.703987\pi\)
\(588\) 0 0
\(589\) −1.37258 −0.0565563
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.48528 0.225254 0.112627 0.993637i \(-0.464074\pi\)
0.112627 + 0.993637i \(0.464074\pi\)
\(594\) 0 0
\(595\) −22.3137 −0.914773
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −32.8284 −1.34133 −0.670667 0.741759i \(-0.733992\pi\)
−0.670667 + 0.741759i \(0.733992\pi\)
\(600\) 0 0
\(601\) −43.1127 −1.75860 −0.879302 0.476265i \(-0.841990\pi\)
−0.879302 + 0.476265i \(0.841990\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −39.4853 −1.60530
\(606\) 0 0
\(607\) 24.1421 0.979899 0.489950 0.871751i \(-0.337015\pi\)
0.489950 + 0.871751i \(0.337015\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.51472 −0.304013
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.6863 −0.510731 −0.255365 0.966845i \(-0.582196\pi\)
−0.255365 + 0.966845i \(0.582196\pi\)
\(618\) 0 0
\(619\) −6.82843 −0.274458 −0.137229 0.990539i \(-0.543820\pi\)
−0.137229 + 0.990539i \(0.543820\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 50.4558 2.01181
\(630\) 0 0
\(631\) 34.1127 1.35801 0.679003 0.734136i \(-0.262413\pi\)
0.679003 + 0.734136i \(0.262413\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 68.2548 2.70861
\(636\) 0 0
\(637\) −2.82843 −0.112066
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −39.4558 −1.55841 −0.779206 0.626768i \(-0.784377\pi\)
−0.779206 + 0.626768i \(0.784377\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.3431 −0.406631 −0.203316 0.979113i \(-0.565172\pi\)
−0.203316 + 0.979113i \(0.565172\pi\)
\(648\) 0 0
\(649\) 8.82843 0.346546
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.4853 −0.410321 −0.205160 0.978728i \(-0.565772\pi\)
−0.205160 + 0.978728i \(0.565772\pi\)
\(654\) 0 0
\(655\) −15.3137 −0.598356
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 23.6569 0.921540 0.460770 0.887520i \(-0.347573\pi\)
0.460770 + 0.887520i \(0.347573\pi\)
\(660\) 0 0
\(661\) 45.1127 1.75468 0.877340 0.479869i \(-0.159316\pi\)
0.877340 + 0.479869i \(0.159316\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.48528 0.173932
\(666\) 0 0
\(667\) 11.3137 0.438069
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.31371 0.282343
\(672\) 0 0
\(673\) −24.3431 −0.938359 −0.469180 0.883103i \(-0.655450\pi\)
−0.469180 + 0.883103i \(0.655450\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.68629 −0.103243 −0.0516213 0.998667i \(-0.516439\pi\)
−0.0516213 + 0.998667i \(0.516439\pi\)
\(678\) 0 0
\(679\) 10.1421 0.389219
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.8284 1.02656 0.513281 0.858221i \(-0.328430\pi\)
0.513281 + 0.858221i \(0.328430\pi\)
\(684\) 0 0
\(685\) −35.1127 −1.34159
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.65685 −0.215509
\(690\) 0 0
\(691\) −15.1716 −0.577154 −0.288577 0.957457i \(-0.593182\pi\)
−0.288577 + 0.957457i \(0.593182\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 64.4264 2.44383
\(696\) 0 0
\(697\) −10.6569 −0.403657
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.5147 1.11476 0.557378 0.830259i \(-0.311808\pi\)
0.557378 + 0.830259i \(0.311808\pi\)
\(702\) 0 0
\(703\) −10.1421 −0.382518
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.9706 −0.563026
\(708\) 0 0
\(709\) −25.2843 −0.949571 −0.474785 0.880102i \(-0.657474\pi\)
−0.474785 + 0.880102i \(0.657474\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.68629 −0.175503
\(714\) 0 0
\(715\) −8.97056 −0.335480
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.6569 −0.472021 −0.236011 0.971751i \(-0.575840\pi\)
−0.236011 + 0.971751i \(0.575840\pi\)
\(720\) 0 0
\(721\) 13.6569 0.508608
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 27.3137 1.01441
\(726\) 0 0
\(727\) −18.1421 −0.672855 −0.336427 0.941709i \(-0.609219\pi\)
−0.336427 + 0.941709i \(0.609219\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.6569 −0.468131
\(732\) 0 0
\(733\) 25.3137 0.934983 0.467492 0.883998i \(-0.345158\pi\)
0.467492 + 0.883998i \(0.345158\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.31371 0.122062
\(738\) 0 0
\(739\) −29.9411 −1.10140 −0.550701 0.834703i \(-0.685640\pi\)
−0.550701 + 0.834703i \(0.685640\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.82843 0.250511 0.125255 0.992125i \(-0.460025\pi\)
0.125255 + 0.992125i \(0.460025\pi\)
\(744\) 0 0
\(745\) −31.1716 −1.14204
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.65685 −0.206697
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 46.5980 1.69587
\(756\) 0 0
\(757\) 9.62742 0.349914 0.174957 0.984576i \(-0.444021\pi\)
0.174957 + 0.984576i \(0.444021\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.82843 −0.283780 −0.141890 0.989882i \(-0.545318\pi\)
−0.141890 + 0.989882i \(0.545318\pi\)
\(762\) 0 0
\(763\) 0.656854 0.0237797
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −30.1421 −1.08837
\(768\) 0 0
\(769\) 15.6569 0.564601 0.282300 0.959326i \(-0.408903\pi\)
0.282300 + 0.959326i \(0.408903\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 32.1716 1.15713 0.578566 0.815636i \(-0.303613\pi\)
0.578566 + 0.815636i \(0.303613\pi\)
\(774\) 0 0
\(775\) −11.3137 −0.406400
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.14214 0.0767500
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10.8284 −0.386483
\(786\) 0 0
\(787\) 41.9411 1.49504 0.747520 0.664239i \(-0.231244\pi\)
0.747520 + 0.664239i \(0.231244\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.8284 0.456126
\(792\) 0 0
\(793\) −24.9706 −0.886731
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.31371 0.329908 0.164954 0.986301i \(-0.447252\pi\)
0.164954 + 0.986301i \(0.447252\pi\)
\(798\) 0 0
\(799\) −15.4853 −0.547830
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.68629 0.165376
\(804\) 0 0
\(805\) 15.3137 0.539737
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 35.9411 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(810\) 0 0
\(811\) 30.7696 1.08047 0.540233 0.841516i \(-0.318336\pi\)
0.540233 + 0.841516i \(0.318336\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 59.2843 2.07664
\(816\) 0 0
\(817\) 2.54416 0.0890087
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.3431 0.360978 0.180489 0.983577i \(-0.442232\pi\)
0.180489 + 0.983577i \(0.442232\pi\)
\(822\) 0 0
\(823\) −31.8284 −1.10947 −0.554735 0.832027i \(-0.687180\pi\)
−0.554735 + 0.832027i \(0.687180\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.82843 −0.306994 −0.153497 0.988149i \(-0.549054\pi\)
−0.153497 + 0.988149i \(0.549054\pi\)
\(828\) 0 0
\(829\) 14.8284 0.515013 0.257506 0.966277i \(-0.417099\pi\)
0.257506 + 0.966277i \(0.417099\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.82843 −0.201943
\(834\) 0 0
\(835\) −61.1421 −2.11591
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.68629 0.0582172 0.0291086 0.999576i \(-0.490733\pi\)
0.0291086 + 0.999576i \(0.490733\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −19.1421 −0.658509
\(846\) 0 0
\(847\) −10.3137 −0.354383
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −34.6274 −1.18701
\(852\) 0 0
\(853\) −18.9706 −0.649540 −0.324770 0.945793i \(-0.605287\pi\)
−0.324770 + 0.945793i \(0.605287\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.8579 −0.507535 −0.253767 0.967265i \(-0.581670\pi\)
−0.253767 + 0.967265i \(0.581670\pi\)
\(858\) 0 0
\(859\) −0.343146 −0.0117080 −0.00585399 0.999983i \(-0.501863\pi\)
−0.00585399 + 0.999983i \(0.501863\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −41.3137 −1.40633 −0.703167 0.711024i \(-0.748232\pi\)
−0.703167 + 0.711024i \(0.748232\pi\)
\(864\) 0 0
\(865\) 7.65685 0.260341
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.17157 0.107588
\(870\) 0 0
\(871\) −11.3137 −0.383350
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 17.8284 0.602711
\(876\) 0 0
\(877\) 1.97056 0.0665412 0.0332706 0.999446i \(-0.489408\pi\)
0.0332706 + 0.999446i \(0.489408\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.97056 −0.234844 −0.117422 0.993082i \(-0.537463\pi\)
−0.117422 + 0.993082i \(0.537463\pi\)
\(882\) 0 0
\(883\) −8.79899 −0.296110 −0.148055 0.988979i \(-0.547301\pi\)
−0.148055 + 0.988979i \(0.547301\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −26.5980 −0.893073 −0.446536 0.894765i \(-0.647343\pi\)
−0.446536 + 0.894765i \(0.647343\pi\)
\(888\) 0 0
\(889\) 17.8284 0.597946
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.11270 0.104162
\(894\) 0 0
\(895\) −92.4264 −3.08947
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.31371 −0.110518
\(900\) 0 0
\(901\) −11.6569 −0.388346
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 31.9411 1.06176
\(906\) 0 0
\(907\) 43.7696 1.45334 0.726672 0.686984i \(-0.241066\pi\)
0.726672 + 0.686984i \(0.241066\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 13.5147 0.447763 0.223881 0.974616i \(-0.428127\pi\)
0.223881 + 0.974616i \(0.428127\pi\)
\(912\) 0 0
\(913\) 13.7990 0.456680
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) 51.0833 1.68508 0.842541 0.538633i \(-0.181059\pi\)
0.842541 + 0.538633i \(0.181059\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −27.3137 −0.899042
\(924\) 0 0
\(925\) −83.5980 −2.74868
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −10.1716 −0.333718 −0.166859 0.985981i \(-0.553363\pi\)
−0.166859 + 0.985981i \(0.553363\pi\)
\(930\) 0 0
\(931\) 1.17157 0.0383968
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −18.4853 −0.604533
\(936\) 0 0
\(937\) 38.8284 1.26847 0.634235 0.773141i \(-0.281315\pi\)
0.634235 + 0.773141i \(0.281315\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.20101 −0.104350 −0.0521750 0.998638i \(-0.516615\pi\)
−0.0521750 + 0.998638i \(0.516615\pi\)
\(942\) 0 0
\(943\) 7.31371 0.238167
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.94113 0.323043 0.161522 0.986869i \(-0.448360\pi\)
0.161522 + 0.986869i \(0.448360\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.3137 −0.625632 −0.312816 0.949814i \(-0.601272\pi\)
−0.312816 + 0.949814i \(0.601272\pi\)
\(954\) 0 0
\(955\) −78.9706 −2.55543
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9.17157 −0.296166
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −47.1421 −1.51756
\(966\) 0 0
\(967\) 5.65685 0.181912 0.0909561 0.995855i \(-0.471008\pi\)
0.0909561 + 0.995855i \(0.471008\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.656854 0.0210795 0.0105397 0.999944i \(-0.496645\pi\)
0.0105397 + 0.999944i \(0.496645\pi\)
\(972\) 0 0
\(973\) 16.8284 0.539495
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.02944 0.288877 0.144439 0.989514i \(-0.453862\pi\)
0.144439 + 0.989514i \(0.453862\pi\)
\(978\) 0 0
\(979\) 4.97056 0.158860
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −59.2843 −1.89087 −0.945437 0.325804i \(-0.894365\pi\)
−0.945437 + 0.325804i \(0.894365\pi\)
\(984\) 0 0
\(985\) 106.426 3.39103
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.68629 0.276208
\(990\) 0 0
\(991\) −57.4264 −1.82421 −0.912105 0.409957i \(-0.865544\pi\)
−0.912105 + 0.409957i \(0.865544\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −57.3137 −1.81697
\(996\) 0 0
\(997\) −46.1421 −1.46134 −0.730668 0.682733i \(-0.760791\pi\)
−0.730668 + 0.682733i \(0.760791\pi\)
\(998\) 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 6048.2.a.bi.1.2 yes 2
3.2 odd 2 6048.2.a.bb.1.1 yes 2
4.3 odd 2 6048.2.a.bh.1.2 yes 2
12.11 even 2 6048.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.y.1.1 2 12.11 even 2
6048.2.a.bb.1.1 yes 2 3.2 odd 2
6048.2.a.bh.1.2 yes 2 4.3 odd 2
6048.2.a.bi.1.2 yes 2 1.1 even 1 trivial