Properties

Label 6048.2.a.bq.1.3
Level 6048
Weight 2
Character 6048.1
Self dual yes
Analytic conductor 48.294
Analytic rank 1
Dimension 4
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.707500\)
Character \(\chi\) = 6048.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.96666 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.96666 q^{5} -1.00000 q^{7} -3.38166 q^{11} +6.48057 q^{13} -5.06557 q^{17} +4.13225 q^{19} -7.09891 q^{23} -1.13225 q^{25} +6.48057 q^{29} -6.34832 q^{31} -1.96666 q^{35} -6.65057 q^{37} -8.44723 q^{41} -5.65057 q^{43} -3.71725 q^{47} +1.00000 q^{49} +1.17000 q^{53} -6.65057 q^{55} -10.5473 q^{59} +10.1311 q^{61} +12.7451 q^{65} -0.216067 q^{67} +8.16448 q^{71} -12.3472 q^{73} +3.38166 q^{77} +11.3106 q^{79} +2.48057 q^{83} -9.96225 q^{85} +12.5139 q^{89} -6.48057 q^{91} +8.12673 q^{95} +7.86664 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q + 2q^{5} - 4q^{7} - 2q^{11} - 4q^{13} + 4q^{17} + 4q^{19} - 10q^{23} + 8q^{25} - 4q^{29} - 8q^{31} - 2q^{35} - 8q^{37} + 2q^{41} - 4q^{43} - 8q^{47} + 4q^{49} + 16q^{53} - 8q^{55} - 24q^{59} - 8q^{61} - 4q^{65} + 4q^{67} - 10q^{71} + 4q^{73} + 2q^{77} + 4q^{79} - 20q^{83} - 16q^{85} + 26q^{89} + 4q^{91} - 34q^{95} + 8q^{97} + O(q^{100}) \)

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
\(5\) 1.96666 0.879517 0.439758 0.898116i \(-0.355064\pi\)
0.439758 + 0.898116i \(0.355064\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.38166 −1.01961 −0.509804 0.860290i \(-0.670282\pi\)
−0.509804 + 0.860290i \(0.670282\pi\)
\(12\) 0 0
\(13\) 6.48057 1.79739 0.898693 0.438578i \(-0.144518\pi\)
0.898693 + 0.438578i \(0.144518\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.06557 −1.22858 −0.614291 0.789080i \(-0.710558\pi\)
−0.614291 + 0.789080i \(0.710558\pi\)
\(18\) 0 0
\(19\) 4.13225 0.948003 0.474002 0.880524i \(-0.342809\pi\)
0.474002 + 0.880524i \(0.342809\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.09891 −1.48023 −0.740113 0.672483i \(-0.765228\pi\)
−0.740113 + 0.672483i \(0.765228\pi\)
\(24\) 0 0
\(25\) −1.13225 −0.226450
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.48057 1.20341 0.601706 0.798718i \(-0.294488\pi\)
0.601706 + 0.798718i \(0.294488\pi\)
\(30\) 0 0
\(31\) −6.34832 −1.14019 −0.570096 0.821578i \(-0.693094\pi\)
−0.570096 + 0.821578i \(0.693094\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.96666 −0.332426
\(36\) 0 0
\(37\) −6.65057 −1.09335 −0.546674 0.837346i \(-0.684106\pi\)
−0.546674 + 0.837346i \(0.684106\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.44723 −1.31924 −0.659618 0.751601i \(-0.729282\pi\)
−0.659618 + 0.751601i \(0.729282\pi\)
\(42\) 0 0
\(43\) −5.65057 −0.861704 −0.430852 0.902423i \(-0.641787\pi\)
−0.430852 + 0.902423i \(0.641787\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.71725 −0.542217 −0.271108 0.962549i \(-0.587390\pi\)
−0.271108 + 0.962549i \(0.587390\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.17000 0.160712 0.0803560 0.996766i \(-0.474394\pi\)
0.0803560 + 0.996766i \(0.474394\pi\)
\(54\) 0 0
\(55\) −6.65057 −0.896763
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.5473 −1.37314 −0.686568 0.727066i \(-0.740883\pi\)
−0.686568 + 0.727066i \(0.740883\pi\)
\(60\) 0 0
\(61\) 10.1311 1.29716 0.648580 0.761147i \(-0.275363\pi\)
0.648580 + 0.761147i \(0.275363\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.7451 1.58083
\(66\) 0 0
\(67\) −0.216067 −0.0263968 −0.0131984 0.999913i \(-0.504201\pi\)
−0.0131984 + 0.999913i \(0.504201\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.16448 0.968946 0.484473 0.874806i \(-0.339011\pi\)
0.484473 + 0.874806i \(0.339011\pi\)
\(72\) 0 0
\(73\) −12.3472 −1.44513 −0.722566 0.691302i \(-0.757037\pi\)
−0.722566 + 0.691302i \(0.757037\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.38166 0.385376
\(78\) 0 0
\(79\) 11.3106 1.27254 0.636269 0.771467i \(-0.280477\pi\)
0.636269 + 0.771467i \(0.280477\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.48057 0.272278 0.136139 0.990690i \(-0.456531\pi\)
0.136139 + 0.990690i \(0.456531\pi\)
\(84\) 0 0
\(85\) −9.96225 −1.08056
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.5139 1.32647 0.663236 0.748410i \(-0.269183\pi\)
0.663236 + 0.748410i \(0.269183\pi\)
\(90\) 0 0
\(91\) −6.48057 −0.679348
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.12673 0.833785
\(96\) 0 0
\(97\) 7.86664 0.798736 0.399368 0.916791i \(-0.369230\pi\)
0.399368 + 0.916791i \(0.369230\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.19893 −0.119298 −0.0596491 0.998219i \(-0.518998\pi\)
−0.0596491 + 0.998219i \(0.518998\pi\)
\(102\) 0 0
\(103\) −19.3095 −1.90262 −0.951309 0.308240i \(-0.900260\pi\)
−0.951309 + 0.308240i \(0.900260\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.86775 −0.663930 −0.331965 0.943292i \(-0.607712\pi\)
−0.331965 + 0.943292i \(0.607712\pi\)
\(108\) 0 0
\(109\) −6.34832 −0.608059 −0.304029 0.952663i \(-0.598332\pi\)
−0.304029 + 0.952663i \(0.598332\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15.9333 −1.49888 −0.749440 0.662072i \(-0.769677\pi\)
−0.749440 + 0.662072i \(0.769677\pi\)
\(114\) 0 0
\(115\) −13.9611 −1.30188
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.06557 0.464360
\(120\) 0 0
\(121\) 0.435615 0.0396014
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0600 −1.07868
\(126\) 0 0
\(127\) 3.65057 0.323936 0.161968 0.986796i \(-0.448216\pi\)
0.161968 + 0.986796i \(0.448216\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.41389 0.210902 0.105451 0.994424i \(-0.466371\pi\)
0.105451 + 0.994424i \(0.466371\pi\)
\(132\) 0 0
\(133\) −4.13225 −0.358312
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.9151 −1.18884 −0.594422 0.804153i \(-0.702619\pi\)
−0.594422 + 0.804153i \(0.702619\pi\)
\(138\) 0 0
\(139\) 6.69664 0.568001 0.284001 0.958824i \(-0.408338\pi\)
0.284001 + 0.958824i \(0.408338\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −21.9151 −1.83263
\(144\) 0 0
\(145\) 12.7451 1.05842
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.8945 1.05636 0.528178 0.849134i \(-0.322876\pi\)
0.528178 + 0.849134i \(0.322876\pi\)
\(150\) 0 0
\(151\) 16.1311 1.31273 0.656367 0.754442i \(-0.272092\pi\)
0.656367 + 0.754442i \(0.272092\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −12.4850 −1.00282
\(156\) 0 0
\(157\) −15.5172 −1.23841 −0.619204 0.785230i \(-0.712545\pi\)
−0.619204 + 0.785230i \(0.712545\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.09891 0.559473
\(162\) 0 0
\(163\) 7.04607 0.551890 0.275945 0.961173i \(-0.411009\pi\)
0.275945 + 0.961173i \(0.411009\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.6300 −1.75116 −0.875579 0.483075i \(-0.839520\pi\)
−0.875579 + 0.483075i \(0.839520\pi\)
\(168\) 0 0
\(169\) 28.9978 2.23060
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.09891 −0.539720 −0.269860 0.962900i \(-0.586977\pi\)
−0.269860 + 0.962900i \(0.586977\pi\)
\(174\) 0 0
\(175\) 1.13225 0.0855901
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −19.2634 −1.43981 −0.719907 0.694071i \(-0.755815\pi\)
−0.719907 + 0.694071i \(0.755815\pi\)
\(180\) 0 0
\(181\) −15.5266 −1.15409 −0.577043 0.816714i \(-0.695793\pi\)
−0.577043 + 0.816714i \(0.695793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −13.0794 −0.961617
\(186\) 0 0
\(187\) 17.1300 1.25267
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.99559 −0.433826 −0.216913 0.976191i \(-0.569599\pi\)
−0.216913 + 0.976191i \(0.569599\pi\)
\(192\) 0 0
\(193\) 16.6966 1.20185 0.600925 0.799305i \(-0.294799\pi\)
0.600925 + 0.799305i \(0.294799\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.8945 1.20368 0.601840 0.798617i \(-0.294434\pi\)
0.601840 + 0.798617i \(0.294434\pi\)
\(198\) 0 0
\(199\) 11.6117 0.823132 0.411566 0.911380i \(-0.364982\pi\)
0.411566 + 0.911380i \(0.364982\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.48057 −0.454847
\(204\) 0 0
\(205\) −16.6128 −1.16029
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −13.9739 −0.966592
\(210\) 0 0
\(211\) 18.8278 1.29616 0.648079 0.761573i \(-0.275573\pi\)
0.648079 + 0.761573i \(0.275573\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.1127 −0.757883
\(216\) 0 0
\(217\) 6.34832 0.430952
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −32.8278 −2.20824
\(222\) 0 0
\(223\) 8.65057 0.579285 0.289643 0.957135i \(-0.406464\pi\)
0.289643 + 0.957135i \(0.406464\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.8945 0.855835 0.427918 0.903818i \(-0.359247\pi\)
0.427918 + 0.903818i \(0.359247\pi\)
\(228\) 0 0
\(229\) −26.1311 −1.72679 −0.863397 0.504525i \(-0.831668\pi\)
−0.863397 + 0.504525i \(0.831668\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.49881 0.556776 0.278388 0.960469i \(-0.410200\pi\)
0.278388 + 0.960469i \(0.410200\pi\)
\(234\) 0 0
\(235\) −7.31057 −0.476889
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −26.0934 −1.68784 −0.843921 0.536468i \(-0.819758\pi\)
−0.843921 + 0.536468i \(0.819758\pi\)
\(240\) 0 0
\(241\) −13.5172 −0.870720 −0.435360 0.900256i \(-0.643379\pi\)
−0.435360 + 0.900256i \(0.643379\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.96666 0.125645
\(246\) 0 0
\(247\) 26.7793 1.70393
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.39787 −0.151352 −0.0756760 0.997132i \(-0.524111\pi\)
−0.0756760 + 0.997132i \(0.524111\pi\)
\(252\) 0 0
\(253\) 24.0061 1.50925
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.3428 1.01944 0.509718 0.860342i \(-0.329750\pi\)
0.509718 + 0.860342i \(0.329750\pi\)
\(258\) 0 0
\(259\) 6.65057 0.413246
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −30.8900 −1.90476 −0.952381 0.304911i \(-0.901373\pi\)
−0.952381 + 0.304911i \(0.901373\pi\)
\(264\) 0 0
\(265\) 2.30099 0.141349
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.1934 1.23121 0.615607 0.788053i \(-0.288911\pi\)
0.615607 + 0.788053i \(0.288911\pi\)
\(270\) 0 0
\(271\) −23.2256 −1.41086 −0.705429 0.708781i \(-0.749245\pi\)
−0.705429 + 0.708781i \(0.749245\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.82889 0.230891
\(276\) 0 0
\(277\) 1.08270 0.0650534 0.0325267 0.999471i \(-0.489645\pi\)
0.0325267 + 0.999471i \(0.489645\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.58611 −0.0946195 −0.0473098 0.998880i \(-0.515065\pi\)
−0.0473098 + 0.998880i \(0.515065\pi\)
\(282\) 0 0
\(283\) −8.96114 −0.532684 −0.266342 0.963879i \(-0.585815\pi\)
−0.266342 + 0.963879i \(0.585815\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.44723 0.498624
\(288\) 0 0
\(289\) 8.66000 0.509412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.49770 0.554862 0.277431 0.960746i \(-0.410517\pi\)
0.277431 + 0.960746i \(0.410517\pi\)
\(294\) 0 0
\(295\) −20.7428 −1.20770
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −46.0050 −2.66054
\(300\) 0 0
\(301\) 5.65057 0.325693
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 19.9245 1.14087
\(306\) 0 0
\(307\) 5.86775 0.334890 0.167445 0.985881i \(-0.446448\pi\)
0.167445 + 0.985881i \(0.446448\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −22.2138 −1.25963 −0.629816 0.776745i \(-0.716870\pi\)
−0.629816 + 0.776745i \(0.716870\pi\)
\(312\) 0 0
\(313\) 13.5194 0.764163 0.382082 0.924129i \(-0.375207\pi\)
0.382082 + 0.924129i \(0.375207\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.11052 −0.511698 −0.255849 0.966717i \(-0.582355\pi\)
−0.255849 + 0.966717i \(0.582355\pi\)
\(318\) 0 0
\(319\) −21.9151 −1.22701
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −20.9322 −1.16470
\(324\) 0 0
\(325\) −7.33763 −0.407019
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.71725 0.204939
\(330\) 0 0
\(331\) −26.6966 −1.46738 −0.733690 0.679484i \(-0.762204\pi\)
−0.733690 + 0.679484i \(0.762204\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.424930 −0.0232164
\(336\) 0 0
\(337\) 15.9611 0.869459 0.434729 0.900561i \(-0.356844\pi\)
0.434729 + 0.900561i \(0.356844\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 21.4678 1.16255
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 34.2361 1.83789 0.918946 0.394383i \(-0.129042\pi\)
0.918946 + 0.394383i \(0.129042\pi\)
\(348\) 0 0
\(349\) 14.9611 0.800851 0.400426 0.916329i \(-0.368862\pi\)
0.400426 + 0.916329i \(0.368862\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −30.6450 −1.63107 −0.815536 0.578707i \(-0.803558\pi\)
−0.815536 + 0.578707i \(0.803558\pi\)
\(354\) 0 0
\(355\) 16.0568 0.852204
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.3023 −0.543732 −0.271866 0.962335i \(-0.587641\pi\)
−0.271866 + 0.962335i \(0.587641\pi\)
\(360\) 0 0
\(361\) −1.92450 −0.101289
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −24.2827 −1.27102
\(366\) 0 0
\(367\) 4.38607 0.228951 0.114475 0.993426i \(-0.463481\pi\)
0.114475 + 0.993426i \(0.463481\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.17000 −0.0607434
\(372\) 0 0
\(373\) −12.0838 −0.625676 −0.312838 0.949806i \(-0.601280\pi\)
−0.312838 + 0.949806i \(0.601280\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 41.9978 2.16300
\(378\) 0 0
\(379\) 22.6117 1.16149 0.580743 0.814087i \(-0.302762\pi\)
0.580743 + 0.814087i \(0.302762\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.02782 0.256910 0.128455 0.991715i \(-0.458998\pi\)
0.128455 + 0.991715i \(0.458998\pi\)
\(384\) 0 0
\(385\) 6.65057 0.338944
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.39550 −0.374966 −0.187483 0.982268i \(-0.560033\pi\)
−0.187483 + 0.982268i \(0.560033\pi\)
\(390\) 0 0
\(391\) 35.9600 1.81858
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 22.2440 1.11922
\(396\) 0 0
\(397\) −28.5268 −1.43172 −0.715859 0.698245i \(-0.753965\pi\)
−0.715859 + 0.698245i \(0.753965\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.8762 −1.14238 −0.571192 0.820817i \(-0.693519\pi\)
−0.571192 + 0.820817i \(0.693519\pi\)
\(402\) 0 0
\(403\) −41.1407 −2.04936
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 22.4900 1.11479
\(408\) 0 0
\(409\) 37.0096 1.83001 0.915003 0.403447i \(-0.132188\pi\)
0.915003 + 0.403447i \(0.132188\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.5473 0.518996
\(414\) 0 0
\(415\) 4.87843 0.239473
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7.99118 −0.390395 −0.195197 0.980764i \(-0.562535\pi\)
−0.195197 + 0.980764i \(0.562535\pi\)
\(420\) 0 0
\(421\) −17.6139 −0.858451 −0.429225 0.903197i \(-0.641213\pi\)
−0.429225 + 0.903197i \(0.641213\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.73550 0.278212
\(426\) 0 0
\(427\) −10.1311 −0.490280
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.09891 0.341942 0.170971 0.985276i \(-0.445309\pi\)
0.170971 + 0.985276i \(0.445309\pi\)
\(432\) 0 0
\(433\) −5.87606 −0.282386 −0.141193 0.989982i \(-0.545094\pi\)
−0.141193 + 0.989982i \(0.545094\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −29.3345 −1.40326
\(438\) 0 0
\(439\) 2.34000 0.111682 0.0558411 0.998440i \(-0.482216\pi\)
0.0558411 + 0.998440i \(0.482216\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.3516 −0.586843 −0.293421 0.955983i \(-0.594794\pi\)
−0.293421 + 0.955983i \(0.594794\pi\)
\(444\) 0 0
\(445\) 24.6106 1.16665
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −29.9890 −1.41527 −0.707633 0.706580i \(-0.750237\pi\)
−0.707633 + 0.706580i \(0.750237\pi\)
\(450\) 0 0
\(451\) 28.5656 1.34510
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.7451 −0.597498
\(456\) 0 0
\(457\) −2.73327 −0.127857 −0.0639286 0.997954i \(-0.520363\pi\)
−0.0639286 + 0.997954i \(0.520363\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 29.7867 1.38731 0.693653 0.720309i \(-0.256000\pi\)
0.693653 + 0.720309i \(0.256000\pi\)
\(462\) 0 0
\(463\) 18.9611 0.881199 0.440599 0.897704i \(-0.354766\pi\)
0.440599 + 0.897704i \(0.354766\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.3384 0.478404 0.239202 0.970970i \(-0.423114\pi\)
0.239202 + 0.970970i \(0.423114\pi\)
\(468\) 0 0
\(469\) 0.216067 0.00997704
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 19.1083 0.878601
\(474\) 0 0
\(475\) −4.67875 −0.214676
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 28.3956 1.29743 0.648715 0.761031i \(-0.275307\pi\)
0.648715 + 0.761031i \(0.275307\pi\)
\(480\) 0 0
\(481\) −43.0995 −1.96517
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 15.4710 0.702502
\(486\) 0 0
\(487\) 5.44171 0.246587 0.123294 0.992370i \(-0.460654\pi\)
0.123294 + 0.992370i \(0.460654\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.4750 −0.788638 −0.394319 0.918974i \(-0.629019\pi\)
−0.394319 + 0.918974i \(0.629019\pi\)
\(492\) 0 0
\(493\) −32.8278 −1.47849
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.16448 −0.366227
\(498\) 0 0
\(499\) −17.7084 −0.792738 −0.396369 0.918091i \(-0.629730\pi\)
−0.396369 + 0.918091i \(0.629730\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −11.5172 −0.513527 −0.256763 0.966474i \(-0.582656\pi\)
−0.256763 + 0.966474i \(0.582656\pi\)
\(504\) 0 0
\(505\) −2.35789 −0.104925
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.6288 −0.870033 −0.435017 0.900422i \(-0.643257\pi\)
−0.435017 + 0.900422i \(0.643257\pi\)
\(510\) 0 0
\(511\) 12.3472 0.546208
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −37.9751 −1.67338
\(516\) 0 0
\(517\) 12.5705 0.552849
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0506 1.31654 0.658271 0.752781i \(-0.271288\pi\)
0.658271 + 0.752781i \(0.271288\pi\)
\(522\) 0 0
\(523\) −6.56564 −0.287096 −0.143548 0.989643i \(-0.545851\pi\)
−0.143548 + 0.989643i \(0.545851\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 32.1578 1.40082
\(528\) 0 0
\(529\) 27.3945 1.19107
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −54.7428 −2.37118
\(534\) 0 0
\(535\) −13.5065 −0.583938
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.38166 −0.145658
\(540\) 0 0
\(541\) 2.91507 0.125329 0.0626644 0.998035i \(-0.480040\pi\)
0.0626644 + 0.998035i \(0.480040\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −12.4850 −0.534798
\(546\) 0 0
\(547\) −16.4321 −0.702587 −0.351294 0.936265i \(-0.614258\pi\)
−0.351294 + 0.936265i \(0.614258\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 26.7793 1.14084
\(552\) 0 0
\(553\) −11.3106 −0.480974
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.9333 0.505631 0.252815 0.967515i \(-0.418643\pi\)
0.252815 + 0.967515i \(0.418643\pi\)
\(558\) 0 0
\(559\) −36.6189 −1.54881
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.5751 −0.656411 −0.328205 0.944606i \(-0.606444\pi\)
−0.328205 + 0.944606i \(0.606444\pi\)
\(564\) 0 0
\(565\) −31.3354 −1.31829
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 42.8922 1.79814 0.899068 0.437809i \(-0.144245\pi\)
0.899068 + 0.437809i \(0.144245\pi\)
\(570\) 0 0
\(571\) 23.0945 0.966475 0.483237 0.875489i \(-0.339461\pi\)
0.483237 + 0.875489i \(0.339461\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.03775 0.335197
\(576\) 0 0
\(577\) −17.2256 −0.717113 −0.358556 0.933508i \(-0.616731\pi\)
−0.358556 + 0.933508i \(0.616731\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.48057 −0.102911
\(582\) 0 0
\(583\) −3.95654 −0.163863
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.200044 −0.00825671 −0.00412835 0.999991i \(-0.501314\pi\)
−0.00412835 + 0.999991i \(0.501314\pi\)
\(588\) 0 0
\(589\) −26.2328 −1.08091
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.4872 0.759178 0.379589 0.925155i \(-0.376065\pi\)
0.379589 + 0.925155i \(0.376065\pi\)
\(594\) 0 0
\(595\) 9.96225 0.408412
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.37223 −0.219503 −0.109752 0.993959i \(-0.535006\pi\)
−0.109752 + 0.993959i \(0.535006\pi\)
\(600\) 0 0
\(601\) 8.34943 0.340580 0.170290 0.985394i \(-0.445529\pi\)
0.170290 + 0.985394i \(0.445529\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.856707 0.0348301
\(606\) 0 0
\(607\) −25.0366 −1.01621 −0.508103 0.861296i \(-0.669653\pi\)
−0.508103 + 0.861296i \(0.669653\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0899 −0.974573
\(612\) 0 0
\(613\) 10.6106 0.428558 0.214279 0.976772i \(-0.431260\pi\)
0.214279 + 0.976772i \(0.431260\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.5473 −1.22979 −0.614893 0.788611i \(-0.710801\pi\)
−0.614893 + 0.788611i \(0.710801\pi\)
\(618\) 0 0
\(619\) 41.6189 1.67281 0.836403 0.548115i \(-0.184655\pi\)
0.836403 + 0.548115i \(0.184655\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.5139 −0.501359
\(624\) 0 0
\(625\) −18.0568 −0.722270
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.6889 1.34327
\(630\) 0 0
\(631\) −24.4806 −0.974556 −0.487278 0.873247i \(-0.662010\pi\)
−0.487278 + 0.873247i \(0.662010\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.17943 0.284907
\(636\) 0 0
\(637\) 6.48057 0.256769
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 31.0256 1.22544 0.612719 0.790301i \(-0.290076\pi\)
0.612719 + 0.790301i \(0.290076\pi\)
\(642\) 0 0
\(643\) −41.7145 −1.64506 −0.822530 0.568722i \(-0.807438\pi\)
−0.822530 + 0.568722i \(0.807438\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.3194 −0.838152 −0.419076 0.907951i \(-0.637646\pi\)
−0.419076 + 0.907951i \(0.637646\pi\)
\(648\) 0 0
\(649\) 35.6672 1.40006
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −32.5060 −1.27206 −0.636029 0.771665i \(-0.719424\pi\)
−0.636029 + 0.771665i \(0.719424\pi\)
\(654\) 0 0
\(655\) 4.74729 0.185492
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −24.9083 −0.970289 −0.485145 0.874434i \(-0.661233\pi\)
−0.485145 + 0.874434i \(0.661233\pi\)
\(660\) 0 0
\(661\) −34.3472 −1.33595 −0.667976 0.744183i \(-0.732839\pi\)
−0.667976 + 0.744183i \(0.732839\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.12673 −0.315141
\(666\) 0 0
\(667\) −46.0050 −1.78132
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −34.2601 −1.32259
\(672\) 0 0
\(673\) −28.9589 −1.11628 −0.558142 0.829745i \(-0.688486\pi\)
−0.558142 + 0.829745i \(0.688486\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.5045 0.711185 0.355593 0.934641i \(-0.384279\pi\)
0.355593 + 0.934641i \(0.384279\pi\)
\(678\) 0 0
\(679\) −7.86664 −0.301894
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.92830 0.303368 0.151684 0.988429i \(-0.451530\pi\)
0.151684 + 0.988429i \(0.451530\pi\)
\(684\) 0 0
\(685\) −27.3662 −1.04561
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.58227 0.288862
\(690\) 0 0
\(691\) 7.12877 0.271191 0.135596 0.990764i \(-0.456705\pi\)
0.135596 + 0.990764i \(0.456705\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.1700 0.499567
\(696\) 0 0
\(697\) 42.7900 1.62079
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −43.7795 −1.65353 −0.826764 0.562549i \(-0.809821\pi\)
−0.826764 + 0.562549i \(0.809821\pi\)
\(702\) 0 0
\(703\) −27.4818 −1.03650
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.19893 0.0450905
\(708\) 0 0
\(709\) 0.877323 0.0329486 0.0164743 0.999864i \(-0.494756\pi\)
0.0164743 + 0.999864i \(0.494756\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 45.0661 1.68774
\(714\) 0 0
\(715\) −43.0995 −1.61183
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −31.9040 −1.18982 −0.594910 0.803792i \(-0.702812\pi\)
−0.594910 + 0.803792i \(0.702812\pi\)
\(720\) 0 0
\(721\) 19.3095 0.719122
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.33763 −0.272513
\(726\) 0 0
\(727\) 8.69886 0.322623 0.161311 0.986904i \(-0.448428\pi\)
0.161311 + 0.986904i \(0.448428\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 28.6234 1.05867
\(732\) 0 0
\(733\) 43.5728 1.60940 0.804700 0.593682i \(-0.202326\pi\)
0.804700 + 0.593682i \(0.202326\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.730664 0.0269144
\(738\) 0 0
\(739\) 28.8300 1.06053 0.530264 0.847832i \(-0.322093\pi\)
0.530264 + 0.847832i \(0.322093\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.8611 1.27893 0.639465 0.768820i \(-0.279156\pi\)
0.639465 + 0.768820i \(0.279156\pi\)
\(744\) 0 0
\(745\) 25.3590 0.929082
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6.86775 0.250942
\(750\) 0 0
\(751\) 38.9495 1.42129 0.710644 0.703552i \(-0.248404\pi\)
0.710644 + 0.703552i \(0.248404\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 31.7245 1.15457
\(756\) 0 0
\(757\) 20.3544 0.739794 0.369897 0.929073i \(-0.379393\pi\)
0.369897 + 0.929073i \(0.379393\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.9534 −0.578311 −0.289156 0.957282i \(-0.593375\pi\)
−0.289156 + 0.957282i \(0.593375\pi\)
\(762\) 0 0
\(763\) 6.34832 0.229825
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −68.3522 −2.46805
\(768\) 0 0
\(769\) 48.5362 1.75026 0.875130 0.483887i \(-0.160776\pi\)
0.875130 + 0.483887i \(0.160776\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −28.0223 −1.00789 −0.503946 0.863735i \(-0.668119\pi\)
−0.503946 + 0.863735i \(0.668119\pi\)
\(774\) 0 0
\(775\) 7.18789 0.258197
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −34.9061 −1.25064
\(780\) 0 0
\(781\) −27.6095 −0.987945
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −30.5171 −1.08920
\(786\) 0 0
\(787\) −12.3378 −0.439794 −0.219897 0.975523i \(-0.570572\pi\)
−0.219897 + 0.975523i \(0.570572\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 15.9333 0.566524
\(792\) 0 0
\(793\) 65.6556 2.33150
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.1258 1.35048 0.675242 0.737596i \(-0.264039\pi\)
0.675242 + 0.737596i \(0.264039\pi\)
\(798\) 0 0
\(799\) 18.8300 0.666157
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 41.7540 1.47347
\(804\) 0 0
\(805\) 13.9611 0.492065
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −37.4395 −1.31630 −0.658151 0.752886i \(-0.728661\pi\)
−0.658151 + 0.752886i \(0.728661\pi\)
\(810\) 0 0
\(811\) 18.4901 0.649277 0.324638 0.945838i \(-0.394757\pi\)
0.324638 + 0.945838i \(0.394757\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 13.8572 0.485397
\(816\) 0 0
\(817\) −23.3496 −0.816898
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −14.8850 −0.519491 −0.259746 0.965677i \(-0.583639\pi\)
−0.259746 + 0.965677i \(0.583639\pi\)
\(822\) 0 0
\(823\) −1.47100 −0.0512757 −0.0256378 0.999671i \(-0.508162\pi\)
−0.0256378 + 0.999671i \(0.508162\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.3027 1.29714 0.648571 0.761155i \(-0.275367\pi\)
0.648571 + 0.761155i \(0.275367\pi\)
\(828\) 0 0
\(829\) −8.04844 −0.279534 −0.139767 0.990184i \(-0.544635\pi\)
−0.139767 + 0.990184i \(0.544635\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.06557 −0.175512
\(834\) 0 0
\(835\) −44.5054 −1.54017
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10.4806 0.361829 0.180915 0.983499i \(-0.442094\pi\)
0.180915 + 0.983499i \(0.442094\pi\)
\(840\) 0 0
\(841\) 12.9978 0.448199
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 57.0287 1.96185
\(846\) 0 0
\(847\) −0.435615 −0.0149679
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 47.2118 1.61840
\(852\) 0 0
\(853\) −28.3956 −0.972248 −0.486124 0.873890i \(-0.661590\pi\)
−0.486124 + 0.873890i \(0.661590\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −26.5784 −0.907900 −0.453950 0.891027i \(-0.649985\pi\)
−0.453950 + 0.891027i \(0.649985\pi\)
\(858\) 0 0
\(859\) 4.47448 0.152667 0.0763336 0.997082i \(-0.475679\pi\)
0.0763336 + 0.997082i \(0.475679\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31.2954 1.06531 0.532655 0.846333i \(-0.321195\pi\)
0.532655 + 0.846333i \(0.321195\pi\)
\(864\) 0 0
\(865\) −13.9611 −0.474693
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −38.2485 −1.29749
\(870\) 0 0
\(871\) −1.40024 −0.0474452
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0600 0.407704
\(876\) 0 0
\(877\) −3.99778 −0.134995 −0.0674977 0.997719i \(-0.521502\pi\)
−0.0674977 + 0.997719i \(0.521502\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 35.1439 1.18403 0.592013 0.805928i \(-0.298333\pi\)
0.592013 + 0.805928i \(0.298333\pi\)
\(882\) 0 0
\(883\) 19.7911 0.666025 0.333012 0.942922i \(-0.391935\pi\)
0.333012 + 0.942922i \(0.391935\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.9128 0.937222 0.468611 0.883405i \(-0.344755\pi\)
0.468611 + 0.883405i \(0.344755\pi\)
\(888\) 0 0
\(889\) −3.65057 −0.122436
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −15.3606 −0.514023
\(894\) 0 0
\(895\) −37.8845 −1.26634
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −41.1407 −1.37212
\(900\) 0 0
\(901\) −5.92672 −0.197448
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −30.5356 −1.01504
\(906\) 0 0
\(907\) 48.6922 1.61680 0.808399 0.588635i \(-0.200335\pi\)
0.808399 + 0.588635i \(0.200335\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −20.1667 −0.668151 −0.334076 0.942546i \(-0.608424\pi\)
−0.334076 + 0.942546i \(0.608424\pi\)
\(912\) 0 0
\(913\) −8.38844 −0.277617
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.41389 −0.0797136
\(918\) 0 0
\(919\) −51.9779 −1.71459 −0.857297 0.514823i \(-0.827858\pi\)
−0.857297 + 0.514823i \(0.827858\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 52.9105 1.74157
\(924\) 0 0
\(925\) 7.53012 0.247589
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.15992 0.300527 0.150264 0.988646i \(-0.451988\pi\)
0.150264 + 0.988646i \(0.451988\pi\)
\(930\) 0 0
\(931\) 4.13225 0.135429
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 33.6889 1.10175
\(936\) 0 0
\(937\) 50.9224 1.66356 0.831782 0.555103i \(-0.187321\pi\)
0.831782 + 0.555103i \(0.187321\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −31.6568 −1.03198 −0.515992 0.856593i \(-0.672577\pi\)
−0.515992 + 0.856593i \(0.672577\pi\)
\(942\) 0 0
\(943\) 59.9661 1.95277
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.3804 0.369813 0.184907 0.982756i \(-0.440802\pi\)
0.184907 + 0.982756i \(0.440802\pi\)
\(948\) 0 0
\(949\) −80.0169 −2.59746
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22.7428 0.736713 0.368357 0.929685i \(-0.379921\pi\)
0.368357 + 0.929685i \(0.379921\pi\)
\(954\) 0 0
\(955\) −11.7913 −0.381557
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 13.9151 0.449341
\(960\) 0 0
\(961\) 9.30114 0.300037
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 32.8366 1.05705
\(966\) 0 0
\(967\) −15.5845 −0.501164 −0.250582 0.968095i \(-0.580622\pi\)
−0.250582 + 0.968095i \(0.580622\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12.6507 −0.405981 −0.202990 0.979181i \(-0.565066\pi\)
−0.202990 + 0.979181i \(0.565066\pi\)
\(972\) 0 0
\(973\) −6.69664 −0.214684
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −15.5679 −0.498060 −0.249030 0.968496i \(-0.580112\pi\)
−0.249030 + 0.968496i \(0.580112\pi\)
\(978\) 0 0
\(979\) −42.3178 −1.35248
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.45497 0.110197 0.0550983 0.998481i \(-0.482453\pi\)
0.0550983 + 0.998481i \(0.482453\pi\)
\(984\) 0 0
\(985\) 33.2256 1.05866
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 40.1129 1.27552
\(990\) 0 0
\(991\) −9.44393 −0.299996 −0.149998 0.988686i \(-0.547927\pi\)
−0.149998 + 0.988686i \(0.547927\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.8363 0.723959
\(996\) 0 0
\(997\) 18.5290 0.586819 0.293410 0.955987i \(-0.405210\pi\)
0.293410 + 0.955987i \(0.405210\pi\)
\(998\) 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.bq.1.3 yes 4
3.2 odd 2 6048.2.a.bl.1.2 4
4.3 odd 2 6048.2.a.bu.1.3 yes 4
12.11 even 2 6048.2.a.bn.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bl.1.2 4 3.2 odd 2
6048.2.a.bn.1.2 yes 4 12.11 even 2
6048.2.a.bq.1.3 yes 4 1.1 even 1 trivial
6048.2.a.bu.1.3 yes 4 4.3 odd 2