Properties

Label 6048.2.j.d.5615.31
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 48
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.31
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.d.5615.32

$q$-expansion

\(f(q)\) \(=\) \(q+1.27600 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+1.27600 q^{5} -1.00000i q^{7} +6.57401i q^{11} -4.05196i q^{13} +6.09788i q^{17} +2.08352 q^{19} -6.85760 q^{23} -3.37182 q^{25} -6.53304 q^{29} -3.26844i q^{31} -1.27600i q^{35} -2.95562i q^{37} -3.35381i q^{41} -10.9901 q^{43} -7.12041 q^{47} -1.00000 q^{49} +2.87278 q^{53} +8.38843i q^{55} -7.75213i q^{59} -12.0251i q^{61} -5.17030i q^{65} +3.01966 q^{67} -3.48158 q^{71} +2.76889 q^{73} +6.57401 q^{77} +0.849261i q^{79} -15.8068i q^{83} +7.78089i q^{85} +4.06349i q^{89} -4.05196 q^{91} +2.65858 q^{95} -4.90438 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q + O(q^{10}) \) \( 48q - 16q^{19} + 48q^{25} - 64q^{43} - 48q^{49} - 32q^{67} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(5\) 1.27600 0.570644 0.285322 0.958432i \(-0.407899\pi\)
0.285322 + 0.958432i \(0.407899\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.57401i 1.98214i 0.133353 + 0.991069i \(0.457426\pi\)
−0.133353 + 0.991069i \(0.542574\pi\)
\(12\) 0 0
\(13\) − 4.05196i − 1.12381i −0.827201 0.561906i \(-0.810068\pi\)
0.827201 0.561906i \(-0.189932\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.09788i 1.47895i 0.673182 + 0.739477i \(0.264927\pi\)
−0.673182 + 0.739477i \(0.735073\pi\)
\(18\) 0 0
\(19\) 2.08352 0.477993 0.238997 0.971020i \(-0.423181\pi\)
0.238997 + 0.971020i \(0.423181\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.85760 −1.42991 −0.714954 0.699171i \(-0.753552\pi\)
−0.714954 + 0.699171i \(0.753552\pi\)
\(24\) 0 0
\(25\) −3.37182 −0.674365
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.53304 −1.21316 −0.606578 0.795024i \(-0.707458\pi\)
−0.606578 + 0.795024i \(0.707458\pi\)
\(30\) 0 0
\(31\) − 3.26844i − 0.587028i −0.955955 0.293514i \(-0.905175\pi\)
0.955955 0.293514i \(-0.0948248\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.27600i − 0.215683i
\(36\) 0 0
\(37\) − 2.95562i − 0.485901i −0.970039 0.242950i \(-0.921885\pi\)
0.970039 0.242950i \(-0.0781152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.35381i − 0.523777i −0.965098 0.261888i \(-0.915655\pi\)
0.965098 0.261888i \(-0.0843452\pi\)
\(42\) 0 0
\(43\) −10.9901 −1.67598 −0.837991 0.545685i \(-0.816270\pi\)
−0.837991 + 0.545685i \(0.816270\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.12041 −1.03862 −0.519310 0.854586i \(-0.673811\pi\)
−0.519310 + 0.854586i \(0.673811\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.87278 0.394607 0.197304 0.980342i \(-0.436782\pi\)
0.197304 + 0.980342i \(0.436782\pi\)
\(54\) 0 0
\(55\) 8.38843i 1.13110i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 7.75213i − 1.00924i −0.863341 0.504621i \(-0.831632\pi\)
0.863341 0.504621i \(-0.168368\pi\)
\(60\) 0 0
\(61\) − 12.0251i − 1.53966i −0.638249 0.769830i \(-0.720341\pi\)
0.638249 0.769830i \(-0.279659\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 5.17030i − 0.641296i
\(66\) 0 0
\(67\) 3.01966 0.368910 0.184455 0.982841i \(-0.440948\pi\)
0.184455 + 0.982841i \(0.440948\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.48158 −0.413188 −0.206594 0.978427i \(-0.566238\pi\)
−0.206594 + 0.978427i \(0.566238\pi\)
\(72\) 0 0
\(73\) 2.76889 0.324075 0.162037 0.986785i \(-0.448194\pi\)
0.162037 + 0.986785i \(0.448194\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.57401 0.749177
\(78\) 0 0
\(79\) 0.849261i 0.0955494i 0.998858 + 0.0477747i \(0.0152129\pi\)
−0.998858 + 0.0477747i \(0.984787\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 15.8068i − 1.73503i −0.497414 0.867513i \(-0.665717\pi\)
0.497414 0.867513i \(-0.334283\pi\)
\(84\) 0 0
\(85\) 7.78089i 0.843956i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.06349i 0.430729i 0.976534 + 0.215365i \(0.0690940\pi\)
−0.976534 + 0.215365i \(0.930906\pi\)
\(90\) 0 0
\(91\) −4.05196 −0.424761
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.65858 0.272764
\(96\) 0 0
\(97\) −4.90438 −0.497964 −0.248982 0.968508i \(-0.580096\pi\)
−0.248982 + 0.968508i \(0.580096\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.5134 1.64314 0.821572 0.570105i \(-0.193098\pi\)
0.821572 + 0.570105i \(0.193098\pi\)
\(102\) 0 0
\(103\) 10.7097i 1.05526i 0.849475 + 0.527629i \(0.176919\pi\)
−0.849475 + 0.527629i \(0.823081\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.41994i − 0.910660i −0.890323 0.455330i \(-0.849521\pi\)
0.890323 0.455330i \(-0.150479\pi\)
\(108\) 0 0
\(109\) − 7.95370i − 0.761826i −0.924611 0.380913i \(-0.875610\pi\)
0.924611 0.380913i \(-0.124390\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 1.56398i − 0.147127i −0.997291 0.0735636i \(-0.976563\pi\)
0.997291 0.0735636i \(-0.0234372\pi\)
\(114\) 0 0
\(115\) −8.75029 −0.815969
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.09788 0.558992
\(120\) 0 0
\(121\) −32.2175 −2.92887
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.6824 −0.955467
\(126\) 0 0
\(127\) − 19.7895i − 1.75603i −0.478629 0.878017i \(-0.658866\pi\)
0.478629 0.878017i \(-0.341134\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 13.6907i − 1.19616i −0.801436 0.598080i \(-0.795931\pi\)
0.801436 0.598080i \(-0.204069\pi\)
\(132\) 0 0
\(133\) − 2.08352i − 0.180664i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.0099i 1.70956i 0.518990 + 0.854780i \(0.326308\pi\)
−0.518990 + 0.854780i \(0.673692\pi\)
\(138\) 0 0
\(139\) 11.0306 0.935607 0.467804 0.883832i \(-0.345045\pi\)
0.467804 + 0.883832i \(0.345045\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 26.6376 2.22755
\(144\) 0 0
\(145\) −8.33616 −0.692280
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.391867 −0.0321030 −0.0160515 0.999871i \(-0.505110\pi\)
−0.0160515 + 0.999871i \(0.505110\pi\)
\(150\) 0 0
\(151\) 10.3302i 0.840662i 0.907371 + 0.420331i \(0.138086\pi\)
−0.907371 + 0.420331i \(0.861914\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 4.17052i − 0.334984i
\(156\) 0 0
\(157\) − 9.78139i − 0.780640i −0.920679 0.390320i \(-0.872364\pi\)
0.920679 0.390320i \(-0.127636\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.85760i 0.540455i
\(162\) 0 0
\(163\) −1.11525 −0.0873533 −0.0436766 0.999046i \(-0.513907\pi\)
−0.0436766 + 0.999046i \(0.513907\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.7973 −1.29982 −0.649908 0.760013i \(-0.725193\pi\)
−0.649908 + 0.760013i \(0.725193\pi\)
\(168\) 0 0
\(169\) −3.41837 −0.262951
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.2063 1.38420 0.692100 0.721802i \(-0.256686\pi\)
0.692100 + 0.721802i \(0.256686\pi\)
\(174\) 0 0
\(175\) 3.37182i 0.254886i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.03488i 0.600555i 0.953852 + 0.300278i \(0.0970793\pi\)
−0.953852 + 0.300278i \(0.902921\pi\)
\(180\) 0 0
\(181\) 18.5478i 1.37865i 0.724454 + 0.689323i \(0.242092\pi\)
−0.724454 + 0.689323i \(0.757908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 3.77137i − 0.277276i
\(186\) 0 0
\(187\) −40.0875 −2.93149
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.8458 −1.65306 −0.826532 0.562890i \(-0.809689\pi\)
−0.826532 + 0.562890i \(0.809689\pi\)
\(192\) 0 0
\(193\) 3.49311 0.251439 0.125720 0.992066i \(-0.459876\pi\)
0.125720 + 0.992066i \(0.459876\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.9432 −1.56339 −0.781693 0.623663i \(-0.785644\pi\)
−0.781693 + 0.623663i \(0.785644\pi\)
\(198\) 0 0
\(199\) − 23.0870i − 1.63660i −0.574793 0.818299i \(-0.694918\pi\)
0.574793 0.818299i \(-0.305082\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.53304i 0.458529i
\(204\) 0 0
\(205\) − 4.27946i − 0.298890i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.6971i 0.947448i
\(210\) 0 0
\(211\) 1.71426 0.118014 0.0590072 0.998258i \(-0.481207\pi\)
0.0590072 + 0.998258i \(0.481207\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −14.0234 −0.956389
\(216\) 0 0
\(217\) −3.26844 −0.221876
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 24.7084 1.66206
\(222\) 0 0
\(223\) − 0.545891i − 0.0365556i −0.999833 0.0182778i \(-0.994182\pi\)
0.999833 0.0182778i \(-0.00581832\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.60514i 0.106537i 0.998580 + 0.0532685i \(0.0169639\pi\)
−0.998580 + 0.0532685i \(0.983036\pi\)
\(228\) 0 0
\(229\) 1.39412i 0.0921257i 0.998939 + 0.0460629i \(0.0146675\pi\)
−0.998939 + 0.0460629i \(0.985333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0130i 0.655971i 0.944683 + 0.327985i \(0.106370\pi\)
−0.944683 + 0.327985i \(0.893630\pi\)
\(234\) 0 0
\(235\) −9.08565 −0.592682
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.62073 0.622313 0.311157 0.950359i \(-0.399284\pi\)
0.311157 + 0.950359i \(0.399284\pi\)
\(240\) 0 0
\(241\) 13.9788 0.900456 0.450228 0.892914i \(-0.351343\pi\)
0.450228 + 0.892914i \(0.351343\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.27600 −0.0815206
\(246\) 0 0
\(247\) − 8.44235i − 0.537174i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.99814i 0.441719i 0.975306 + 0.220859i \(0.0708862\pi\)
−0.975306 + 0.220859i \(0.929114\pi\)
\(252\) 0 0
\(253\) − 45.0819i − 2.83427i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.39346i 0.274057i 0.990567 + 0.137028i \(0.0437551\pi\)
−0.990567 + 0.137028i \(0.956245\pi\)
\(258\) 0 0
\(259\) −2.95562 −0.183653
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.54094 −0.526657 −0.263328 0.964706i \(-0.584820\pi\)
−0.263328 + 0.964706i \(0.584820\pi\)
\(264\) 0 0
\(265\) 3.66567 0.225180
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.7389 −0.898647 −0.449324 0.893369i \(-0.648335\pi\)
−0.449324 + 0.893369i \(0.648335\pi\)
\(270\) 0 0
\(271\) − 4.21696i − 0.256162i −0.991764 0.128081i \(-0.959118\pi\)
0.991764 0.128081i \(-0.0408817\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 22.1664i − 1.33668i
\(276\) 0 0
\(277\) 2.73096i 0.164088i 0.996629 + 0.0820438i \(0.0261447\pi\)
−0.996629 + 0.0820438i \(0.973855\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.32789i 0.556455i 0.960515 + 0.278228i \(0.0897469\pi\)
−0.960515 + 0.278228i \(0.910253\pi\)
\(282\) 0 0
\(283\) −20.2355 −1.20288 −0.601438 0.798919i \(-0.705405\pi\)
−0.601438 + 0.798919i \(0.705405\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.35381 −0.197969
\(288\) 0 0
\(289\) −20.1841 −1.18730
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.259973 −0.0151878 −0.00759391 0.999971i \(-0.502417\pi\)
−0.00759391 + 0.999971i \(0.502417\pi\)
\(294\) 0 0
\(295\) − 9.89172i − 0.575918i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 27.7867i 1.60695i
\(300\) 0 0
\(301\) 10.9901i 0.633461i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 15.3441i − 0.878598i
\(306\) 0 0
\(307\) −25.8255 −1.47394 −0.736969 0.675927i \(-0.763743\pi\)
−0.736969 + 0.675927i \(0.763743\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.8253 1.12419 0.562094 0.827074i \(-0.309996\pi\)
0.562094 + 0.827074i \(0.309996\pi\)
\(312\) 0 0
\(313\) 10.2201 0.577673 0.288837 0.957378i \(-0.406732\pi\)
0.288837 + 0.957378i \(0.406732\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.21191 −0.180399 −0.0901995 0.995924i \(-0.528750\pi\)
−0.0901995 + 0.995924i \(0.528750\pi\)
\(318\) 0 0
\(319\) − 42.9482i − 2.40464i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.7051i 0.706930i
\(324\) 0 0
\(325\) 13.6625i 0.757859i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 7.12041i 0.392561i
\(330\) 0 0
\(331\) 29.8260 1.63939 0.819694 0.572802i \(-0.194144\pi\)
0.819694 + 0.572802i \(0.194144\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.85308 0.210516
\(336\) 0 0
\(337\) −26.9588 −1.46854 −0.734271 0.678857i \(-0.762476\pi\)
−0.734271 + 0.678857i \(0.762476\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 21.4867 1.16357
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.73769i 0.0932843i 0.998912 + 0.0466421i \(0.0148520\pi\)
−0.998912 + 0.0466421i \(0.985148\pi\)
\(348\) 0 0
\(349\) 16.8981i 0.904536i 0.891882 + 0.452268i \(0.149385\pi\)
−0.891882 + 0.452268i \(0.850615\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 12.2774i − 0.653459i −0.945118 0.326730i \(-0.894053\pi\)
0.945118 0.326730i \(-0.105947\pi\)
\(354\) 0 0
\(355\) −4.44249 −0.235783
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.77380 −0.199173 −0.0995867 0.995029i \(-0.531752\pi\)
−0.0995867 + 0.995029i \(0.531752\pi\)
\(360\) 0 0
\(361\) −14.6589 −0.771522
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.53311 0.184931
\(366\) 0 0
\(367\) 28.6703i 1.49658i 0.663372 + 0.748289i \(0.269125\pi\)
−0.663372 + 0.748289i \(0.730875\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 2.87278i − 0.149147i
\(372\) 0 0
\(373\) 27.3540i 1.41634i 0.706043 + 0.708169i \(0.250478\pi\)
−0.706043 + 0.708169i \(0.749522\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 26.4716i 1.36336i
\(378\) 0 0
\(379\) 18.6336 0.957146 0.478573 0.878048i \(-0.341154\pi\)
0.478573 + 0.878048i \(0.341154\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 27.8368 1.42240 0.711198 0.702992i \(-0.248153\pi\)
0.711198 + 0.702992i \(0.248153\pi\)
\(384\) 0 0
\(385\) 8.38843 0.427514
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.10860 0.360420 0.180210 0.983628i \(-0.442322\pi\)
0.180210 + 0.983628i \(0.442322\pi\)
\(390\) 0 0
\(391\) − 41.8168i − 2.11477i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.08366i 0.0545247i
\(396\) 0 0
\(397\) 32.8312i 1.64775i 0.566771 + 0.823875i \(0.308192\pi\)
−0.566771 + 0.823875i \(0.691808\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 29.0585i − 1.45111i −0.688164 0.725555i \(-0.741583\pi\)
0.688164 0.725555i \(-0.258417\pi\)
\(402\) 0 0
\(403\) −13.2436 −0.659709
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 19.4303 0.963122
\(408\) 0 0
\(409\) −5.68368 −0.281040 −0.140520 0.990078i \(-0.544877\pi\)
−0.140520 + 0.990078i \(0.544877\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.75213 −0.381458
\(414\) 0 0
\(415\) − 20.1695i − 0.990083i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.00749i 0.0980722i 0.998797 + 0.0490361i \(0.0156149\pi\)
−0.998797 + 0.0490361i \(0.984385\pi\)
\(420\) 0 0
\(421\) − 16.4735i − 0.802871i −0.915887 0.401435i \(-0.868511\pi\)
0.915887 0.401435i \(-0.131489\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 20.5610i − 0.997354i
\(426\) 0 0
\(427\) −12.0251 −0.581936
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.2399 −0.830417 −0.415209 0.909726i \(-0.636291\pi\)
−0.415209 + 0.909726i \(0.636291\pi\)
\(432\) 0 0
\(433\) −38.1465 −1.83321 −0.916603 0.399799i \(-0.869080\pi\)
−0.916603 + 0.399799i \(0.869080\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −14.2880 −0.683487
\(438\) 0 0
\(439\) − 7.30831i − 0.348806i −0.984674 0.174403i \(-0.944200\pi\)
0.984674 0.174403i \(-0.0557996\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.7787i 1.03474i 0.855763 + 0.517368i \(0.173088\pi\)
−0.855763 + 0.517368i \(0.826912\pi\)
\(444\) 0 0
\(445\) 5.18501i 0.245793i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.30200i 0.0614451i 0.999528 + 0.0307225i \(0.00978083\pi\)
−0.999528 + 0.0307225i \(0.990219\pi\)
\(450\) 0 0
\(451\) 22.0479 1.03820
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.17030 −0.242387
\(456\) 0 0
\(457\) 3.17657 0.148594 0.0742968 0.997236i \(-0.476329\pi\)
0.0742968 + 0.997236i \(0.476329\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.56764 −0.399035 −0.199517 0.979894i \(-0.563937\pi\)
−0.199517 + 0.979894i \(0.563937\pi\)
\(462\) 0 0
\(463\) 19.9440i 0.926877i 0.886129 + 0.463438i \(0.153385\pi\)
−0.886129 + 0.463438i \(0.846615\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.5091i 0.532579i 0.963893 + 0.266290i \(0.0857978\pi\)
−0.963893 + 0.266290i \(0.914202\pi\)
\(468\) 0 0
\(469\) − 3.01966i − 0.139435i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 72.2493i − 3.32203i
\(474\) 0 0
\(475\) −7.02528 −0.322342
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.91391 −0.270213 −0.135107 0.990831i \(-0.543138\pi\)
−0.135107 + 0.990831i \(0.543138\pi\)
\(480\) 0 0
\(481\) −11.9760 −0.546061
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.25798 −0.284160
\(486\) 0 0
\(487\) 20.4470i 0.926541i 0.886217 + 0.463271i \(0.153324\pi\)
−0.886217 + 0.463271i \(0.846676\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 24.0391i − 1.08487i −0.840098 0.542434i \(-0.817503\pi\)
0.840098 0.542434i \(-0.182497\pi\)
\(492\) 0 0
\(493\) − 39.8377i − 1.79420i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.48158i 0.156170i
\(498\) 0 0
\(499\) −29.1375 −1.30438 −0.652188 0.758057i \(-0.726149\pi\)
−0.652188 + 0.758057i \(0.726149\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9.70604 −0.432771 −0.216386 0.976308i \(-0.569427\pi\)
−0.216386 + 0.976308i \(0.569427\pi\)
\(504\) 0 0
\(505\) 21.0711 0.937651
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.5042 0.687213 0.343606 0.939114i \(-0.388351\pi\)
0.343606 + 0.939114i \(0.388351\pi\)
\(510\) 0 0
\(511\) − 2.76889i − 0.122489i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13.6656i 0.602177i
\(516\) 0 0
\(517\) − 46.8096i − 2.05869i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.22420i 0.272687i 0.990662 + 0.136344i \(0.0435351\pi\)
−0.990662 + 0.136344i \(0.956465\pi\)
\(522\) 0 0
\(523\) −33.3752 −1.45939 −0.729697 0.683770i \(-0.760339\pi\)
−0.729697 + 0.683770i \(0.760339\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 19.9305 0.868188
\(528\) 0 0
\(529\) 24.0267 1.04464
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −13.5895 −0.588626
\(534\) 0 0
\(535\) − 12.0198i − 0.519663i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 6.57401i − 0.283162i
\(540\) 0 0
\(541\) 4.04207i 0.173782i 0.996218 + 0.0868911i \(0.0276932\pi\)
−0.996218 + 0.0868911i \(0.972307\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 10.1489i − 0.434732i
\(546\) 0 0
\(547\) 19.7497 0.844436 0.422218 0.906494i \(-0.361252\pi\)
0.422218 + 0.906494i \(0.361252\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −13.6117 −0.579880
\(552\) 0 0
\(553\) 0.849261 0.0361143
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.2192 −0.475373 −0.237687 0.971342i \(-0.576389\pi\)
−0.237687 + 0.971342i \(0.576389\pi\)
\(558\) 0 0
\(559\) 44.5316i 1.88349i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 8.66666i − 0.365256i −0.983182 0.182628i \(-0.941540\pi\)
0.983182 0.182628i \(-0.0584604\pi\)
\(564\) 0 0
\(565\) − 1.99564i − 0.0839573i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 27.0778i − 1.13516i −0.823317 0.567581i \(-0.807879\pi\)
0.823317 0.567581i \(-0.192121\pi\)
\(570\) 0 0
\(571\) −19.5104 −0.816485 −0.408243 0.912873i \(-0.633858\pi\)
−0.408243 + 0.912873i \(0.633858\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.1226 0.964280
\(576\) 0 0
\(577\) −23.4544 −0.976421 −0.488211 0.872726i \(-0.662350\pi\)
−0.488211 + 0.872726i \(0.662350\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −15.8068 −0.655778
\(582\) 0 0
\(583\) 18.8857i 0.782165i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 8.29331i − 0.342302i −0.985245 0.171151i \(-0.945251\pi\)
0.985245 0.171151i \(-0.0547485\pi\)
\(588\) 0 0
\(589\) − 6.80987i − 0.280596i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 5.99359i − 0.246127i −0.992399 0.123064i \(-0.960728\pi\)
0.992399 0.123064i \(-0.0392719\pi\)
\(594\) 0 0
\(595\) 7.78089 0.318986
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −41.0912 −1.67894 −0.839470 0.543405i \(-0.817135\pi\)
−0.839470 + 0.543405i \(0.817135\pi\)
\(600\) 0 0
\(601\) −14.9084 −0.608128 −0.304064 0.952652i \(-0.598344\pi\)
−0.304064 + 0.952652i \(0.598344\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −41.1096 −1.67134
\(606\) 0 0
\(607\) 21.1672i 0.859152i 0.903031 + 0.429576i \(0.141337\pi\)
−0.903031 + 0.429576i \(0.858663\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 28.8516i 1.16721i
\(612\) 0 0
\(613\) − 21.0293i − 0.849367i −0.905342 0.424683i \(-0.860385\pi\)
0.905342 0.424683i \(-0.139615\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 29.2848i 1.17896i 0.807783 + 0.589480i \(0.200667\pi\)
−0.807783 + 0.589480i \(0.799333\pi\)
\(618\) 0 0
\(619\) −43.6407 −1.75407 −0.877034 0.480428i \(-0.840481\pi\)
−0.877034 + 0.480428i \(0.840481\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.06349 0.162800
\(624\) 0 0
\(625\) 3.22833 0.129133
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18.0230 0.718624
\(630\) 0 0
\(631\) − 33.7501i − 1.34357i −0.740747 0.671784i \(-0.765528\pi\)
0.740747 0.671784i \(-0.234472\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 25.2514i − 1.00207i
\(636\) 0 0
\(637\) 4.05196i 0.160544i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.1478i 0.519306i 0.965702 + 0.259653i \(0.0836083\pi\)
−0.965702 + 0.259653i \(0.916392\pi\)
\(642\) 0 0
\(643\) 26.7796 1.05608 0.528042 0.849218i \(-0.322926\pi\)
0.528042 + 0.849218i \(0.322926\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13.7850 −0.541944 −0.270972 0.962587i \(-0.587345\pi\)
−0.270972 + 0.962587i \(0.587345\pi\)
\(648\) 0 0
\(649\) 50.9626 2.00046
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.5245 1.35105 0.675524 0.737338i \(-0.263917\pi\)
0.675524 + 0.737338i \(0.263917\pi\)
\(654\) 0 0
\(655\) − 17.4693i − 0.682582i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 31.8035i 1.23889i 0.785040 + 0.619445i \(0.212642\pi\)
−0.785040 + 0.619445i \(0.787358\pi\)
\(660\) 0 0
\(661\) − 14.7156i − 0.572370i −0.958174 0.286185i \(-0.907613\pi\)
0.958174 0.286185i \(-0.0923873\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 2.65858i − 0.103095i
\(666\) 0 0
\(667\) 44.8010 1.73470
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 79.0532 3.05182
\(672\) 0 0
\(673\) 14.1705 0.546233 0.273116 0.961981i \(-0.411946\pi\)
0.273116 + 0.961981i \(0.411946\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −33.9994 −1.30670 −0.653351 0.757055i \(-0.726638\pi\)
−0.653351 + 0.757055i \(0.726638\pi\)
\(678\) 0 0
\(679\) 4.90438i 0.188213i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.7265i 0.946132i 0.881027 + 0.473066i \(0.156853\pi\)
−0.881027 + 0.473066i \(0.843147\pi\)
\(684\) 0 0
\(685\) 25.5326i 0.975551i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 11.6404i − 0.443464i
\(690\) 0 0
\(691\) 19.2490 0.732267 0.366134 0.930562i \(-0.380681\pi\)
0.366134 + 0.930562i \(0.380681\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.0751 0.533899
\(696\) 0 0
\(697\) 20.4511 0.774641
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13.4914 −0.509563 −0.254781 0.966999i \(-0.582004\pi\)
−0.254781 + 0.966999i \(0.582004\pi\)
\(702\) 0 0
\(703\) − 6.15810i − 0.232257i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 16.5134i − 0.621050i
\(708\) 0 0
\(709\) − 12.1839i − 0.457575i −0.973476 0.228788i \(-0.926524\pi\)
0.973476 0.228788i \(-0.0734761\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 22.4136i 0.839397i
\(714\) 0 0
\(715\) 33.9896 1.27114
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.14139 −0.0798603 −0.0399301 0.999202i \(-0.512714\pi\)
−0.0399301 + 0.999202i \(0.512714\pi\)
\(720\) 0 0
\(721\) 10.7097 0.398850
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 22.0283 0.818109
\(726\) 0 0
\(727\) 36.8865i 1.36804i 0.729462 + 0.684022i \(0.239771\pi\)
−0.729462 + 0.684022i \(0.760229\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 67.0166i − 2.47870i
\(732\) 0 0
\(733\) 28.2007i 1.04162i 0.853674 + 0.520808i \(0.174369\pi\)
−0.853674 + 0.520808i \(0.825631\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.8513i 0.731230i
\(738\) 0 0
\(739\) 13.9006 0.511341 0.255670 0.966764i \(-0.417704\pi\)
0.255670 + 0.966764i \(0.417704\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.81892 −0.213476 −0.106738 0.994287i \(-0.534041\pi\)
−0.106738 + 0.994287i \(0.534041\pi\)
\(744\) 0 0
\(745\) −0.500022 −0.0183194
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.41994 −0.344197
\(750\) 0 0
\(751\) 36.7860i 1.34234i 0.741304 + 0.671170i \(0.234208\pi\)
−0.741304 + 0.671170i \(0.765792\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 13.1814i 0.479719i
\(756\) 0 0
\(757\) − 16.6555i − 0.605356i −0.953093 0.302678i \(-0.902119\pi\)
0.953093 0.302678i \(-0.0978806\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 38.2510i − 1.38660i −0.720651 0.693298i \(-0.756157\pi\)
0.720651 0.693298i \(-0.243843\pi\)
\(762\) 0 0
\(763\) −7.95370 −0.287943
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −31.4113 −1.13420
\(768\) 0 0
\(769\) −40.1021 −1.44612 −0.723059 0.690787i \(-0.757264\pi\)
−0.723059 + 0.690787i \(0.757264\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.84516 −0.0663659 −0.0331830 0.999449i \(-0.510564\pi\)
−0.0331830 + 0.999449i \(0.510564\pi\)
\(774\) 0 0
\(775\) 11.0206i 0.395871i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 6.98774i − 0.250362i
\(780\) 0 0
\(781\) − 22.8879i − 0.818994i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 12.4810i − 0.445468i
\(786\) 0 0
\(787\) 40.4160 1.44068 0.720338 0.693623i \(-0.243987\pi\)
0.720338 + 0.693623i \(0.243987\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.56398 −0.0556088
\(792\) 0 0
\(793\) −48.7253 −1.73029
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −47.9867 −1.69978 −0.849889 0.526962i \(-0.823331\pi\)
−0.849889 + 0.526962i \(0.823331\pi\)
\(798\) 0 0
\(799\) − 43.4194i − 1.53607i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 18.2027i 0.642360i
\(804\) 0 0
\(805\) 8.75029i 0.308407i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.1754i 0.498379i 0.968455 + 0.249190i \(0.0801643\pi\)
−0.968455 + 0.249190i \(0.919836\pi\)
\(810\) 0 0
\(811\) 5.80289 0.203767 0.101884 0.994796i \(-0.467513\pi\)
0.101884 + 0.994796i \(0.467513\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.42306 −0.0498477
\(816\) 0 0
\(817\) −22.8982 −0.801108
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.67601 −0.302795 −0.151397 0.988473i \(-0.548377\pi\)
−0.151397 + 0.988473i \(0.548377\pi\)
\(822\) 0 0
\(823\) − 0.199240i − 0.00694506i −0.999994 0.00347253i \(-0.998895\pi\)
0.999994 0.00347253i \(-0.00110534\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.2411i 1.46887i 0.678681 + 0.734433i \(0.262552\pi\)
−0.678681 + 0.734433i \(0.737448\pi\)
\(828\) 0 0
\(829\) − 10.9282i − 0.379554i −0.981827 0.189777i \(-0.939224\pi\)
0.981827 0.189777i \(-0.0607765\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 6.09788i − 0.211279i
\(834\) 0 0
\(835\) −21.4334 −0.741733
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 41.6038 1.43632 0.718161 0.695877i \(-0.244984\pi\)
0.718161 + 0.695877i \(0.244984\pi\)
\(840\) 0 0
\(841\) 13.6806 0.471745
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.36184 −0.150052
\(846\) 0 0
\(847\) 32.2175i 1.10701i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 20.2684i 0.694793i
\(852\) 0 0
\(853\) 10.8964i 0.373084i 0.982447 + 0.186542i \(0.0597281\pi\)
−0.982447 + 0.186542i \(0.940272\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.9467i 0.408093i 0.978961 + 0.204046i \(0.0654093\pi\)
−0.978961 + 0.204046i \(0.934591\pi\)
\(858\) 0 0
\(859\) 15.8684 0.541423 0.270711 0.962661i \(-0.412741\pi\)
0.270711 + 0.962661i \(0.412741\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 36.7712 1.25171 0.625853 0.779941i \(-0.284751\pi\)
0.625853 + 0.779941i \(0.284751\pi\)
\(864\) 0 0
\(865\) 23.2312 0.789886
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −5.58305 −0.189392
\(870\) 0 0
\(871\) − 12.2355i − 0.414585i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.6824i 0.361133i
\(876\) 0 0
\(877\) 48.2439i 1.62908i 0.580106 + 0.814541i \(0.303011\pi\)
−0.580106 + 0.814541i \(0.696989\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 21.4780i − 0.723611i −0.932254 0.361805i \(-0.882160\pi\)
0.932254 0.361805i \(-0.117840\pi\)
\(882\) 0 0
\(883\) 2.78771 0.0938139 0.0469070 0.998899i \(-0.485064\pi\)
0.0469070 + 0.998899i \(0.485064\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −37.3399 −1.25375 −0.626875 0.779120i \(-0.715666\pi\)
−0.626875 + 0.779120i \(0.715666\pi\)
\(888\) 0 0
\(889\) −19.7895 −0.663719
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −14.8356 −0.496453
\(894\) 0 0
\(895\) 10.2525i 0.342703i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 21.3528i 0.712156i
\(900\) 0 0
\(901\) 17.5179i 0.583605i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 23.6670i 0.786717i
\(906\) 0 0
\(907\) −38.2433 −1.26985 −0.634924 0.772574i \(-0.718969\pi\)
−0.634924 + 0.772574i \(0.718969\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.40878 0.0798063 0.0399032 0.999204i \(-0.487295\pi\)
0.0399032 + 0.999204i \(0.487295\pi\)
\(912\) 0 0
\(913\) 103.914 3.43906
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −13.6907 −0.452106
\(918\) 0 0
\(919\) 21.5153i 0.709723i 0.934919 + 0.354862i \(0.115472\pi\)
−0.934919 + 0.354862i \(0.884528\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 14.1072i 0.464345i
\(924\) 0 0
\(925\) 9.96583i 0.327674i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21.9659i 0.720677i 0.932822 + 0.360339i \(0.117339\pi\)
−0.932822 + 0.360339i \(0.882661\pi\)
\(930\) 0 0
\(931\) −2.08352 −0.0682848
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −51.1516 −1.67284
\(936\) 0 0
\(937\) −23.9371 −0.781991 −0.390995 0.920393i \(-0.627869\pi\)
−0.390995 + 0.920393i \(0.627869\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.56414 0.311782 0.155891 0.987774i \(-0.450175\pi\)
0.155891 + 0.987774i \(0.450175\pi\)
\(942\) 0 0
\(943\) 22.9991i 0.748953i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 35.0824i 1.14003i 0.821636 + 0.570013i \(0.193062\pi\)
−0.821636 + 0.570013i \(0.806938\pi\)
\(948\) 0 0
\(949\) − 11.2194i − 0.364199i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 14.3190i 0.463837i 0.972735 + 0.231919i \(0.0745003\pi\)
−0.972735 + 0.231919i \(0.925500\pi\)
\(954\) 0 0
\(955\) −29.1512 −0.943311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 20.0099 0.646153
\(960\) 0 0
\(961\) 20.3173 0.655398
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.45720 0.143482
\(966\) 0 0
\(967\) − 48.7056i − 1.56627i −0.621855 0.783133i \(-0.713621\pi\)
0.621855 0.783133i \(-0.286379\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 2.39723i − 0.0769308i −0.999260 0.0384654i \(-0.987753\pi\)
0.999260 0.0384654i \(-0.0122469\pi\)
\(972\) 0 0
\(973\) − 11.0306i − 0.353626i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 6.33844i − 0.202785i −0.994847 0.101392i \(-0.967670\pi\)
0.994847 0.101392i \(-0.0323297\pi\)
\(978\) 0 0
\(979\) −26.7134 −0.853764
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −13.6385 −0.434999 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(984\) 0 0
\(985\) −27.9995 −0.892138
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 75.3660 2.39650
\(990\) 0 0
\(991\) 24.8997i 0.790964i 0.918474 + 0.395482i \(0.129422\pi\)
−0.918474 + 0.395482i \(0.870578\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 29.4591i − 0.933915i
\(996\) 0 0
\(997\) − 25.3869i − 0.804011i −0.915637 0.402005i \(-0.868313\pi\)
0.915637 0.402005i \(-0.131687\pi\)
\(998\) 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.j.d.5615.31 48
3.2 odd 2 inner 6048.2.j.d.5615.18 48
4.3 odd 2 1512.2.j.d.323.41 yes 48
8.3 odd 2 inner 6048.2.j.d.5615.17 48
8.5 even 2 1512.2.j.d.323.7 48
12.11 even 2 1512.2.j.d.323.8 yes 48
24.5 odd 2 1512.2.j.d.323.42 yes 48
24.11 even 2 inner 6048.2.j.d.5615.32 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.7 48 8.5 even 2
1512.2.j.d.323.8 yes 48 12.11 even 2
1512.2.j.d.323.41 yes 48 4.3 odd 2
1512.2.j.d.323.42 yes 48 24.5 odd 2
6048.2.j.d.5615.17 48 8.3 odd 2 inner
6048.2.j.d.5615.18 48 3.2 odd 2 inner
6048.2.j.d.5615.31 48 1.1 even 1 trivial
6048.2.j.d.5615.32 48 24.11 even 2 inner