Properties

Label 6048.2.a.bn.1.2
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 1.2
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 \(-0.644936\pi\)
−0.439758 + 0.898116i \(0.644936\pi\)
\(6\) 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 \(-0.289442\pi\)
0.614291 + 0.789080i \(0.289442\pi\)
\(18\) 0 0
\(19\) −4.13225 −0.948003 −0.474002 0.880524i \(-0.657191\pi\)
−0.474002 + 0.880524i \(0.657191\pi\)
\(20\) 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 \(-0.705512\pi\)
−0.601706 + 0.798718i \(0.705512\pi\)
\(30\) 0 0
\(31\) 6.34832 1.14019 0.570096 0.821578i \(-0.306906\pi\)
0.570096 + 0.821578i \(0.306906\pi\)
\(32\) 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 \(-0.270718\pi\)
0.659618 + 0.751601i \(0.270718\pi\)
\(42\) 0 0
\(43\) 5.65057 0.861704 0.430852 0.902423i \(-0.358213\pi\)
0.430852 + 0.902423i \(0.358213\pi\)
\(44\) 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 \(-0.525606\pi\)
−0.0803560 + 0.996766i \(0.525606\pi\)
\(54\) 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 \(-0.495799\pi\)
0.0131984 + 0.999913i \(0.495799\pi\)
\(68\) 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 \(-0.719523\pi\)
−0.636269 + 0.771467i \(0.719523\pi\)
\(80\) 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 \(-0.730817\pi\)
−0.663236 + 0.748410i \(0.730817\pi\)
\(90\) 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 \(-0.481002\pi\)
0.0596491 + 0.998219i \(0.481002\pi\)
\(102\) 0 0
\(103\) 19.3095 1.90262 0.951309 0.308240i \(-0.0997400\pi\)
0.951309 + 0.308240i \(0.0997400\pi\)
\(104\) 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 \(-0.230323\pi\)
0.749440 + 0.662072i \(0.230323\pi\)
\(114\) 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 \(-0.551784\pi\)
−0.161968 + 0.986796i \(0.551784\pi\)
\(128\) 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 \(-0.297381\pi\)
0.594422 + 0.804153i \(0.297381\pi\)
\(138\) 0 0
\(139\) −6.69664 −0.568001 −0.284001 0.958824i \(-0.591662\pi\)
−0.284001 + 0.958824i \(0.591662\pi\)
\(140\) 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 \(-0.677124\pi\)
−0.528178 + 0.849134i \(0.677124\pi\)
\(150\) 0 0
\(151\) −16.1311 −1.31273 −0.656367 0.754442i \(-0.727908\pi\)
−0.656367 + 0.754442i \(0.727908\pi\)
\(152\) 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 \(-0.588991\pi\)
−0.275945 + 0.961173i \(0.588991\pi\)
\(164\) 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 \(-0.413023\pi\)
0.269860 + 0.962900i \(0.413023\pi\)
\(174\) 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 \(-0.705566\pi\)
−0.601840 + 0.798617i \(0.705566\pi\)
\(198\) 0 0
\(199\) −11.6117 −0.823132 −0.411566 0.911380i \(-0.635018\pi\)
−0.411566 + 0.911380i \(0.635018\pi\)
\(200\) 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 \(-0.724427\pi\)
−0.648079 + 0.761573i \(0.724427\pi\)
\(212\) 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 \(-0.593536\pi\)
−0.289643 + 0.957135i \(0.593536\pi\)
\(224\) 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 \(-0.589800\pi\)
−0.278388 + 0.960469i \(0.589800\pi\)
\(234\) 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 \(-0.670250\pi\)
−0.509718 + 0.860342i \(0.670250\pi\)
\(258\) 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 \(-0.711089\pi\)
−0.615607 + 0.788053i \(0.711089\pi\)
\(270\) 0 0
\(271\) 23.2256 1.41086 0.705429 0.708781i \(-0.250755\pi\)
0.705429 + 0.708781i \(0.250755\pi\)
\(272\) 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 \(-0.484935\pi\)
0.0473098 + 0.998880i \(0.484935\pi\)
\(282\) 0 0
\(283\) 8.96114 0.532684 0.266342 0.963879i \(-0.414185\pi\)
0.266342 + 0.963879i \(0.414185\pi\)
\(284\) 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 \(-0.589483\pi\)
−0.277431 + 0.960746i \(0.589483\pi\)
\(294\) 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 \(-0.553552\pi\)
−0.167445 + 0.985881i \(0.553552\pi\)
\(308\) 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 \(-0.417645\pi\)
0.255849 + 0.966717i \(0.417645\pi\)
\(318\) 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 \(-0.237796\pi\)
0.733690 + 0.679484i \(0.237796\pi\)
\(332\) 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 \(-0.196442\pi\)
0.815536 + 0.578707i \(0.196442\pi\)
\(354\) 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 \(-0.536519\pi\)
−0.114475 + 0.993426i \(0.536519\pi\)
\(368\) 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 \(-0.697238\pi\)
−0.580743 + 0.814087i \(0.697238\pi\)
\(380\) 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 \(-0.439967\pi\)
0.187483 + 0.982268i \(0.439967\pi\)
\(390\) 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 \(-0.306481\pi\)
0.571192 + 0.820817i \(0.306481\pi\)
\(402\) 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 \(-0.517784\pi\)
−0.0558411 + 0.998440i \(0.517784\pi\)
\(440\) 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 \(-0.249763\pi\)
0.707633 + 0.706580i \(0.249763\pi\)
\(450\) 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 \(-0.744000\pi\)
−0.693653 + 0.720309i \(0.744000\pi\)
\(462\) 0 0
\(463\) −18.9611 −0.881199 −0.440599 0.897704i \(-0.645234\pi\)
−0.440599 + 0.897704i \(0.645234\pi\)
\(464\) 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 \(-0.539346\pi\)
−0.123294 + 0.992370i \(0.539346\pi\)
\(488\) 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 \(-0.370270\pi\)
0.396369 + 0.918091i \(0.370270\pi\)
\(500\) 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 \(-0.356743\pi\)
0.435017 + 0.900422i \(0.356743\pi\)
\(510\) 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 \(-0.728712\pi\)
−0.658271 + 0.752781i \(0.728712\pi\)
\(522\) 0 0
\(523\) 6.56564 0.287096 0.143548 0.989643i \(-0.454149\pi\)
0.143548 + 0.989643i \(0.454149\pi\)
\(524\) 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 \(-0.385742\pi\)
0.351294 + 0.936265i \(0.385742\pi\)
\(548\) 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 \(-0.581357\pi\)
−0.252815 + 0.967515i \(0.581357\pi\)
\(558\) 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 \(-0.855755\pi\)
−0.899068 + 0.437809i \(0.855755\pi\)
\(570\) 0 0
\(571\) −23.0945 −0.966475 −0.483237 0.875489i \(-0.660539\pi\)
−0.483237 + 0.875489i \(0.660539\pi\)
\(572\) 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 \(-0.623935\pi\)
−0.379589 + 0.925155i \(0.623935\pi\)
\(594\) 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 \(-0.330347\pi\)
0.508103 + 0.861296i \(0.330347\pi\)
\(608\) 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 \(-0.289199\pi\)
0.614893 + 0.788611i \(0.289199\pi\)
\(618\) 0 0
\(619\) −41.6189 −1.67281 −0.836403 0.548115i \(-0.815345\pi\)
−0.836403 + 0.548115i \(0.815345\pi\)
\(620\) 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 \(-0.337990\pi\)
0.487278 + 0.873247i \(0.337990\pi\)
\(632\) 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 \(-0.709924\pi\)
−0.612719 + 0.790301i \(0.709924\pi\)
\(642\) 0 0
\(643\) 41.7145 1.64506 0.822530 0.568722i \(-0.192562\pi\)
0.822530 + 0.568722i \(0.192562\pi\)
\(644\) 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 \(-0.280576\pi\)
0.636029 + 0.771665i \(0.280576\pi\)
\(654\) 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 \(-0.615721\pi\)
−0.355593 + 0.934641i \(0.615721\pi\)
\(678\) 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 \(-0.543295\pi\)
−0.135596 + 0.990764i \(0.543295\pi\)
\(692\) 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 \(-0.190179\pi\)
0.826764 + 0.562549i \(0.190179\pi\)
\(702\) 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 \(-0.551572\pi\)
−0.161311 + 0.986904i \(0.551572\pi\)
\(728\) 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 \(-0.677907\pi\)
−0.530264 + 0.847832i \(0.677907\pi\)
\(740\) 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 \(-0.751596\pi\)
−0.710644 + 0.703552i \(0.751596\pi\)
\(752\) 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 \(-0.406625\pi\)
0.289156 + 0.957282i \(0.406625\pi\)
\(762\) 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 \(-0.331881\pi\)
0.503946 + 0.863735i \(0.331881\pi\)
\(774\) 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 \(-0.429428\pi\)
0.219897 + 0.975523i \(0.429428\pi\)
\(788\) 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 \(-0.735961\pi\)
−0.675242 + 0.737596i \(0.735961\pi\)
\(798\) 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 \(-0.271339\pi\)
0.658151 + 0.752886i \(0.271339\pi\)
\(810\) 0 0
\(811\) −18.4901 −0.649277 −0.324638 0.945838i \(-0.605243\pi\)
−0.324638 + 0.945838i \(0.605243\pi\)
\(812\) 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 \(-0.416361\pi\)
0.259746 + 0.965677i \(0.416361\pi\)
\(822\) 0 0
\(823\) 1.47100 0.0512757 0.0256378 0.999671i \(-0.491838\pi\)
0.0256378 + 0.999671i \(0.491838\pi\)
\(824\) 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 \(-0.350015\pi\)
0.453950 + 0.891027i \(0.350015\pi\)
\(858\) 0 0
\(859\) −4.47448 −0.152667 −0.0763336 0.997082i \(-0.524321\pi\)
−0.0763336 + 0.997082i \(0.524321\pi\)
\(860\) 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 \(-0.701667\pi\)
−0.592013 + 0.805928i \(0.701667\pi\)
\(882\) 0 0
\(883\) −19.7911 −0.666025 −0.333012 0.942922i \(-0.608065\pi\)
−0.333012 + 0.942922i \(0.608065\pi\)
\(884\) 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 \(-0.799665\pi\)
−0.808399 + 0.588635i \(0.799665\pi\)
\(908\) 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 \(-0.172142\pi\)
0.857297 + 0.514823i \(0.172142\pi\)
\(920\) 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 \(-0.548012\pi\)
−0.150264 + 0.988646i \(0.548012\pi\)
\(930\) 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 \(-0.327423\pi\)
0.515992 + 0.856593i \(0.327423\pi\)
\(942\) 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 \(-0.620079\pi\)
−0.368357 + 0.929685i \(0.620079\pi\)
\(954\) 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 \(-0.419378\pi\)
0.250582 + 0.968095i \(0.419378\pi\)
\(968\) 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 \(-0.419888\pi\)
0.249030 + 0.968496i \(0.419888\pi\)
\(978\) 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 \(-0.452073\pi\)
0.149998 + 0.988686i \(0.452073\pi\)
\(992\) 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.bn.1.2 yes 4
3.2 odd 2 6048.2.a.bu.1.3 yes 4
4.3 odd 2 6048.2.a.bl.1.2 4
12.11 even 2 6048.2.a.bq.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.a.bl.1.2 4 4.3 odd 2
6048.2.a.bn.1.2 yes 4 1.1 even 1 trivial
6048.2.a.bq.1.3 yes 4 12.11 even 2
6048.2.a.bu.1.3 yes 4 3.2 odd 2