Properties

Label 6048.2.h.c.2591.1
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.1
Root \(-0.258819 - 0.965926i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.c.2591.8

$q$-expansion

\(f(q)\) \(=\) \(q-3.14626i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-3.14626i q^{5} -1.00000i q^{7} +1.41421 q^{11} +0.449490 q^{13} +2.51059i q^{17} +2.44949i q^{19} -5.65685 q^{23} -4.89898 q^{25} +8.34242i q^{29} +1.55051i q^{31} -3.14626 q^{35} +9.00000 q^{37} +10.0745i q^{41} +8.34847i q^{43} -0.460702 q^{47} -1.00000 q^{49} +5.02118i q^{53} -4.44949i q^{55} -11.4887 q^{59} -5.55051 q^{61} -1.41421i q^{65} +13.7980i q^{67} +5.65685 q^{71} +9.79796 q^{73} -1.41421i q^{77} +0.348469i q^{79} -2.36773 q^{83} +7.89898 q^{85} -1.55708i q^{89} -0.449490i q^{91} +7.70674 q^{95} +10.2474 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 16q^{13} + 72q^{37} - 8q^{49} - 64q^{61} + 24q^{85} - 16q^{97} + O(q^{100}) \)

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
\(5\) − 3.14626i − 1.40705i −0.710669 0.703526i \(-0.751608\pi\)
0.710669 0.703526i \(-0.248392\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) 0.449490 0.124666 0.0623330 0.998055i \(-0.480146\pi\)
0.0623330 + 0.998055i \(0.480146\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.51059i 0.608907i 0.952527 + 0.304454i \(0.0984739\pi\)
−0.952527 + 0.304454i \(0.901526\pi\)
\(18\) 0 0
\(19\) 2.44949i 0.561951i 0.959715 + 0.280976i \(0.0906580\pi\)
−0.959715 + 0.280976i \(0.909342\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.65685 −1.17954 −0.589768 0.807573i \(-0.700781\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) −4.89898 −0.979796
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.34242i 1.54915i 0.632483 + 0.774574i \(0.282036\pi\)
−0.632483 + 0.774574i \(0.717964\pi\)
\(30\) 0 0
\(31\) 1.55051i 0.278480i 0.990259 + 0.139240i \(0.0444659\pi\)
−0.990259 + 0.139240i \(0.955534\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.14626 −0.531816
\(36\) 0 0
\(37\) 9.00000 1.47959 0.739795 0.672832i \(-0.234922\pi\)
0.739795 + 0.672832i \(0.234922\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0745i 1.57337i 0.617356 + 0.786684i \(0.288204\pi\)
−0.617356 + 0.786684i \(0.711796\pi\)
\(42\) 0 0
\(43\) 8.34847i 1.27313i 0.771223 + 0.636565i \(0.219645\pi\)
−0.771223 + 0.636565i \(0.780355\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.460702 −0.0672003 −0.0336001 0.999435i \(-0.510697\pi\)
−0.0336001 + 0.999435i \(0.510697\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.02118i 0.689712i 0.938656 + 0.344856i \(0.112072\pi\)
−0.938656 + 0.344856i \(0.887928\pi\)
\(54\) 0 0
\(55\) − 4.44949i − 0.599969i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.4887 −1.49570 −0.747849 0.663868i \(-0.768914\pi\)
−0.747849 + 0.663868i \(0.768914\pi\)
\(60\) 0 0
\(61\) −5.55051 −0.710670 −0.355335 0.934739i \(-0.615633\pi\)
−0.355335 + 0.934739i \(0.615633\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1.41421i − 0.175412i
\(66\) 0 0
\(67\) 13.7980i 1.68569i 0.538157 + 0.842844i \(0.319121\pi\)
−0.538157 + 0.842844i \(0.680879\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) 9.79796 1.14676 0.573382 0.819288i \(-0.305631\pi\)
0.573382 + 0.819288i \(0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.41421i − 0.161165i
\(78\) 0 0
\(79\) 0.348469i 0.0392059i 0.999808 + 0.0196029i \(0.00624020\pi\)
−0.999808 + 0.0196029i \(0.993760\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.36773 −0.259892 −0.129946 0.991521i \(-0.541480\pi\)
−0.129946 + 0.991521i \(0.541480\pi\)
\(84\) 0 0
\(85\) 7.89898 0.856765
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.55708i − 0.165050i −0.996589 0.0825250i \(-0.973702\pi\)
0.996589 0.0825250i \(-0.0262984\pi\)
\(90\) 0 0
\(91\) − 0.449490i − 0.0471193i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.70674 0.790695
\(96\) 0 0
\(97\) 10.2474 1.04047 0.520235 0.854023i \(-0.325844\pi\)
0.520235 + 0.854023i \(0.325844\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 5.65685i − 0.562878i −0.959579 0.281439i \(-0.909188\pi\)
0.959579 0.281439i \(-0.0908117\pi\)
\(102\) 0 0
\(103\) 5.79796i 0.571290i 0.958336 + 0.285645i \(0.0922078\pi\)
−0.958336 + 0.285645i \(0.907792\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.8564 −1.33955 −0.669775 0.742564i \(-0.733609\pi\)
−0.669775 + 0.742564i \(0.733609\pi\)
\(108\) 0 0
\(109\) 0.797959 0.0764306 0.0382153 0.999270i \(-0.487833\pi\)
0.0382153 + 0.999270i \(0.487833\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.5352i 0.991065i 0.868589 + 0.495533i \(0.165027\pi\)
−0.868589 + 0.495533i \(0.834973\pi\)
\(114\) 0 0
\(115\) 17.7980i 1.65967i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.51059 0.230145
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 0.317837i − 0.0284282i
\(126\) 0 0
\(127\) − 12.5505i − 1.11368i −0.830621 0.556839i \(-0.812014\pi\)
0.830621 0.556839i \(-0.187986\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.84961 −0.685823 −0.342912 0.939368i \(-0.611413\pi\)
−0.342912 + 0.939368i \(0.611413\pi\)
\(132\) 0 0
\(133\) 2.44949 0.212398
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.4422i − 1.06301i −0.847056 0.531504i \(-0.821627\pi\)
0.847056 0.531504i \(-0.178373\pi\)
\(138\) 0 0
\(139\) − 0.449490i − 0.0381252i −0.999818 0.0190626i \(-0.993932\pi\)
0.999818 0.0190626i \(-0.00606819\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.635674 0.0531578
\(144\) 0 0
\(145\) 26.2474 2.17973
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 17.4634i − 1.43066i −0.698789 0.715328i \(-0.746277\pi\)
0.698789 0.715328i \(-0.253723\pi\)
\(150\) 0 0
\(151\) − 8.34847i − 0.679389i −0.940536 0.339694i \(-0.889676\pi\)
0.940536 0.339694i \(-0.110324\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.87832 0.391836
\(156\) 0 0
\(157\) −10.2474 −0.817835 −0.408918 0.912571i \(-0.634094\pi\)
−0.408918 + 0.912571i \(0.634094\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65685i 0.445823i
\(162\) 0 0
\(163\) − 0.550510i − 0.0431193i −0.999768 0.0215596i \(-0.993137\pi\)
0.999768 0.0215596i \(-0.00686318\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.58166 −0.741451 −0.370725 0.928742i \(-0.620891\pi\)
−0.370725 + 0.928742i \(0.620891\pi\)
\(168\) 0 0
\(169\) −12.7980 −0.984458
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.84961i 0.596795i 0.954442 + 0.298397i \(0.0964520\pi\)
−0.954442 + 0.298397i \(0.903548\pi\)
\(174\) 0 0
\(175\) 4.89898i 0.370328i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.1708 0.834948 0.417474 0.908689i \(-0.362915\pi\)
0.417474 + 0.908689i \(0.362915\pi\)
\(180\) 0 0
\(181\) 3.79796 0.282300 0.141150 0.989988i \(-0.454920\pi\)
0.141150 + 0.989988i \(0.454920\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 28.3164i − 2.08186i
\(186\) 0 0
\(187\) 3.55051i 0.259639i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.84961 0.567978 0.283989 0.958828i \(-0.408342\pi\)
0.283989 + 0.958828i \(0.408342\pi\)
\(192\) 0 0
\(193\) 20.5959 1.48253 0.741263 0.671214i \(-0.234227\pi\)
0.741263 + 0.671214i \(0.234227\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.89949i 0.705310i 0.935753 + 0.352655i \(0.114721\pi\)
−0.935753 + 0.352655i \(0.885279\pi\)
\(198\) 0 0
\(199\) 12.6969i 0.900062i 0.893013 + 0.450031i \(0.148587\pi\)
−0.893013 + 0.450031i \(0.851413\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.34242 0.585523
\(204\) 0 0
\(205\) 31.6969 2.21381
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.46410i 0.239617i
\(210\) 0 0
\(211\) 17.5959i 1.21135i 0.795711 + 0.605676i \(0.207097\pi\)
−0.795711 + 0.605676i \(0.792903\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 26.2665 1.79136
\(216\) 0 0
\(217\) 1.55051 0.105255
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.12848i 0.0759101i
\(222\) 0 0
\(223\) 28.2474i 1.89159i 0.324766 + 0.945795i \(0.394715\pi\)
−0.324766 + 0.945795i \(0.605285\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.6848 −1.10741 −0.553706 0.832712i \(-0.686787\pi\)
−0.553706 + 0.832712i \(0.686787\pi\)
\(228\) 0 0
\(229\) 23.7980 1.57261 0.786307 0.617836i \(-0.211991\pi\)
0.786307 + 0.617836i \(0.211991\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.07107i 0.463241i 0.972806 + 0.231621i \(0.0744028\pi\)
−0.972806 + 0.231621i \(0.925597\pi\)
\(234\) 0 0
\(235\) 1.44949i 0.0945543i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.3704 1.25297 0.626483 0.779435i \(-0.284494\pi\)
0.626483 + 0.779435i \(0.284494\pi\)
\(240\) 0 0
\(241\) 9.34847 0.602188 0.301094 0.953594i \(-0.402648\pi\)
0.301094 + 0.953594i \(0.402648\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.14626i 0.201007i
\(246\) 0 0
\(247\) 1.10102i 0.0700563i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.6814 0.863564 0.431782 0.901978i \(-0.357885\pi\)
0.431782 + 0.901978i \(0.357885\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 14.1421i − 0.882162i −0.897467 0.441081i \(-0.854595\pi\)
0.897467 0.441081i \(-0.145405\pi\)
\(258\) 0 0
\(259\) − 9.00000i − 0.559233i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.7347 1.15523 0.577616 0.816308i \(-0.303983\pi\)
0.577616 + 0.816308i \(0.303983\pi\)
\(264\) 0 0
\(265\) 15.7980 0.970461
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 22.9453i − 1.39900i −0.714634 0.699498i \(-0.753407\pi\)
0.714634 0.699498i \(-0.246593\pi\)
\(270\) 0 0
\(271\) − 8.69694i − 0.528301i −0.964481 0.264151i \(-0.914908\pi\)
0.964481 0.264151i \(-0.0850916\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.92820 −0.417786
\(276\) 0 0
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 4.38551i − 0.261617i −0.991408 0.130809i \(-0.958243\pi\)
0.991408 0.130809i \(-0.0417574\pi\)
\(282\) 0 0
\(283\) − 2.65153i − 0.157617i −0.996890 0.0788086i \(-0.974888\pi\)
0.996890 0.0788086i \(-0.0251116\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0745 0.594677
\(288\) 0 0
\(289\) 10.6969 0.629232
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 11.6315i − 0.679522i −0.940512 0.339761i \(-0.889654\pi\)
0.940512 0.339761i \(-0.110346\pi\)
\(294\) 0 0
\(295\) 36.1464i 2.10453i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.54270 −0.147048
\(300\) 0 0
\(301\) 8.34847 0.481198
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 17.4634i 0.999950i
\(306\) 0 0
\(307\) 2.20204i 0.125677i 0.998024 + 0.0628386i \(0.0200153\pi\)
−0.998024 + 0.0628386i \(0.979985\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.8954 −1.18487 −0.592434 0.805619i \(-0.701833\pi\)
−0.592434 + 0.805619i \(0.701833\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.5851i 0.706847i 0.935463 + 0.353424i \(0.114983\pi\)
−0.935463 + 0.353424i \(0.885017\pi\)
\(318\) 0 0
\(319\) 11.7980i 0.660559i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.14966 −0.342176
\(324\) 0 0
\(325\) −2.20204 −0.122147
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.460702i 0.0253993i
\(330\) 0 0
\(331\) 28.1464i 1.54707i 0.633755 + 0.773534i \(0.281513\pi\)
−0.633755 + 0.773534i \(0.718487\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 43.4120 2.37185
\(336\) 0 0
\(337\) −20.5959 −1.12193 −0.560966 0.827839i \(-0.689570\pi\)
−0.560966 + 0.827839i \(0.689570\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.19275i 0.118744i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.5911 −1.74958 −0.874792 0.484499i \(-0.839002\pi\)
−0.874792 + 0.484499i \(0.839002\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 22.6595i − 1.20604i −0.797724 0.603022i \(-0.793963\pi\)
0.797724 0.603022i \(-0.206037\pi\)
\(354\) 0 0
\(355\) − 17.7980i − 0.944618i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 30.8270i − 1.61356i
\(366\) 0 0
\(367\) − 17.1010i − 0.892666i −0.894867 0.446333i \(-0.852730\pi\)
0.894867 0.446333i \(-0.147270\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.02118 0.260687
\(372\) 0 0
\(373\) −12.1010 −0.626567 −0.313284 0.949660i \(-0.601429\pi\)
−0.313284 + 0.949660i \(0.601429\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.74983i 0.193126i
\(378\) 0 0
\(379\) 11.0454i 0.567364i 0.958918 + 0.283682i \(0.0915561\pi\)
−0.958918 + 0.283682i \(0.908444\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.02458 −0.410037 −0.205018 0.978758i \(-0.565725\pi\)
−0.205018 + 0.978758i \(0.565725\pi\)
\(384\) 0 0
\(385\) −4.44949 −0.226767
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 25.9487i 1.31565i 0.753171 + 0.657824i \(0.228523\pi\)
−0.753171 + 0.657824i \(0.771477\pi\)
\(390\) 0 0
\(391\) − 14.2020i − 0.718228i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.09638 0.0551647
\(396\) 0 0
\(397\) 21.7980 1.09401 0.547004 0.837130i \(-0.315768\pi\)
0.547004 + 0.837130i \(0.315768\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 2.47848i − 0.123769i −0.998083 0.0618847i \(-0.980289\pi\)
0.998083 0.0618847i \(-0.0197111\pi\)
\(402\) 0 0
\(403\) 0.696938i 0.0347170i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.7279 0.630900
\(408\) 0 0
\(409\) 5.79796 0.286691 0.143345 0.989673i \(-0.454214\pi\)
0.143345 + 0.989673i \(0.454214\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.4887i 0.565321i
\(414\) 0 0
\(415\) 7.44949i 0.365681i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.36773 −0.115671 −0.0578355 0.998326i \(-0.518420\pi\)
−0.0578355 + 0.998326i \(0.518420\pi\)
\(420\) 0 0
\(421\) 32.8990 1.60340 0.801699 0.597728i \(-0.203930\pi\)
0.801699 + 0.597728i \(0.203930\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 12.2993i − 0.596605i
\(426\) 0 0
\(427\) 5.55051i 0.268608i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) 26.4495 1.27108 0.635541 0.772067i \(-0.280777\pi\)
0.635541 + 0.772067i \(0.280777\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 13.8564i − 0.662842i
\(438\) 0 0
\(439\) − 5.10102i − 0.243458i −0.992563 0.121729i \(-0.961156\pi\)
0.992563 0.121729i \(-0.0388439\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −31.8912 −1.51520 −0.757599 0.652720i \(-0.773628\pi\)
−0.757599 + 0.652720i \(0.773628\pi\)
\(444\) 0 0
\(445\) −4.89898 −0.232234
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 26.5843i 1.25459i 0.778781 + 0.627296i \(0.215838\pi\)
−0.778781 + 0.627296i \(0.784162\pi\)
\(450\) 0 0
\(451\) 14.2474i 0.670886i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.41421 −0.0662994
\(456\) 0 0
\(457\) 11.5959 0.542434 0.271217 0.962518i \(-0.412574\pi\)
0.271217 + 0.962518i \(0.412574\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.5091i 1.42095i 0.703721 + 0.710476i \(0.251520\pi\)
−0.703721 + 0.710476i \(0.748480\pi\)
\(462\) 0 0
\(463\) 31.2474i 1.45219i 0.687593 + 0.726096i \(0.258667\pi\)
−0.687593 + 0.726096i \(0.741333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.2275 −0.889744 −0.444872 0.895594i \(-0.646751\pi\)
−0.444872 + 0.895594i \(0.646751\pi\)
\(468\) 0 0
\(469\) 13.7980 0.637131
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.8065i 0.542864i
\(474\) 0 0
\(475\) − 12.0000i − 0.550598i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.6028 −0.667221 −0.333610 0.942711i \(-0.608267\pi\)
−0.333610 + 0.942711i \(0.608267\pi\)
\(480\) 0 0
\(481\) 4.04541 0.184455
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 32.2412i − 1.46400i
\(486\) 0 0
\(487\) 19.5959i 0.887976i 0.896033 + 0.443988i \(0.146437\pi\)
−0.896033 + 0.443988i \(0.853563\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.4634 −0.788111 −0.394055 0.919087i \(-0.628928\pi\)
−0.394055 + 0.919087i \(0.628928\pi\)
\(492\) 0 0
\(493\) −20.9444 −0.943288
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 5.65685i − 0.253745i
\(498\) 0 0
\(499\) − 25.2474i − 1.13023i −0.825012 0.565116i \(-0.808832\pi\)
0.825012 0.565116i \(-0.191168\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.4169 0.821168 0.410584 0.911823i \(-0.365325\pi\)
0.410584 + 0.911823i \(0.365325\pi\)
\(504\) 0 0
\(505\) −17.7980 −0.791999
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 1.87492i − 0.0831042i −0.999136 0.0415521i \(-0.986770\pi\)
0.999136 0.0415521i \(-0.0132302\pi\)
\(510\) 0 0
\(511\) − 9.79796i − 0.433436i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.2419 0.803835
\(516\) 0 0
\(517\) −0.651531 −0.0286543
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 5.68896i − 0.249238i −0.992205 0.124619i \(-0.960229\pi\)
0.992205 0.124619i \(-0.0397709\pi\)
\(522\) 0 0
\(523\) 12.0454i 0.526709i 0.964699 + 0.263354i \(0.0848289\pi\)
−0.964699 + 0.263354i \(0.915171\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.89270 −0.169568
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.52837i 0.196145i
\(534\) 0 0
\(535\) 43.5959i 1.88482i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.41421 −0.0609145
\(540\) 0 0
\(541\) −29.0000 −1.24681 −0.623404 0.781900i \(-0.714251\pi\)
−0.623404 + 0.781900i \(0.714251\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 2.51059i − 0.107542i
\(546\) 0 0
\(547\) − 0.146428i − 0.00626082i −0.999995 0.00313041i \(-0.999004\pi\)
0.999995 0.00313041i \(-0.000996442\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.4347 −0.870546
\(552\) 0 0
\(553\) 0.348469 0.0148184
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.7128i 1.17423i 0.809504 + 0.587115i \(0.199736\pi\)
−0.809504 + 0.587115i \(0.800264\pi\)
\(558\) 0 0
\(559\) 3.75255i 0.158716i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −32.6698 −1.37687 −0.688433 0.725300i \(-0.741701\pi\)
−0.688433 + 0.725300i \(0.741701\pi\)
\(564\) 0 0
\(565\) 33.1464 1.39448
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.38551i 0.183850i 0.995766 + 0.0919250i \(0.0293020\pi\)
−0.995766 + 0.0919250i \(0.970698\pi\)
\(570\) 0 0
\(571\) 14.3485i 0.600465i 0.953866 + 0.300232i \(0.0970642\pi\)
−0.953866 + 0.300232i \(0.902936\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 27.7128 1.15570
\(576\) 0 0
\(577\) −1.59592 −0.0664389 −0.0332195 0.999448i \(-0.510576\pi\)
−0.0332195 + 0.999448i \(0.510576\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.36773i 0.0982298i
\(582\) 0 0
\(583\) 7.10102i 0.294094i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.8844 −1.02709 −0.513544 0.858063i \(-0.671668\pi\)
−0.513544 + 0.858063i \(0.671668\pi\)
\(588\) 0 0
\(589\) −3.79796 −0.156492
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.603566i 0.0247855i 0.999923 + 0.0123928i \(0.00394484\pi\)
−0.999923 + 0.0123928i \(0.996055\pi\)
\(594\) 0 0
\(595\) − 7.89898i − 0.323827i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 29.1270 1.19010 0.595049 0.803689i \(-0.297133\pi\)
0.595049 + 0.803689i \(0.297133\pi\)
\(600\) 0 0
\(601\) 46.7423 1.90666 0.953330 0.301930i \(-0.0976310\pi\)
0.953330 + 0.301930i \(0.0976310\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 28.3164i 1.15122i
\(606\) 0 0
\(607\) 1.75255i 0.0711339i 0.999367 + 0.0355669i \(0.0113237\pi\)
−0.999367 + 0.0355669i \(0.988676\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.207081 −0.00837759
\(612\) 0 0
\(613\) −41.3939 −1.67188 −0.835941 0.548819i \(-0.815078\pi\)
−0.835941 + 0.548819i \(0.815078\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 4.38551i − 0.176554i −0.996096 0.0882769i \(-0.971864\pi\)
0.996096 0.0882769i \(-0.0281360\pi\)
\(618\) 0 0
\(619\) − 47.8434i − 1.92299i −0.274827 0.961494i \(-0.588621\pi\)
0.274827 0.961494i \(-0.411379\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.55708 −0.0623830
\(624\) 0 0
\(625\) −25.4949 −1.01980
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 22.5953i 0.900934i
\(630\) 0 0
\(631\) 33.7423i 1.34326i 0.740886 + 0.671631i \(0.234406\pi\)
−0.740886 + 0.671631i \(0.765594\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −39.4872 −1.56700
\(636\) 0 0
\(637\) −0.449490 −0.0178094
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 25.9487i 1.02491i 0.858714 + 0.512455i \(0.171264\pi\)
−0.858714 + 0.512455i \(0.828736\pi\)
\(642\) 0 0
\(643\) − 38.4949i − 1.51809i −0.651038 0.759045i \(-0.725666\pi\)
0.651038 0.759045i \(-0.274334\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 35.4982 1.39558 0.697789 0.716303i \(-0.254167\pi\)
0.697789 + 0.716303i \(0.254167\pi\)
\(648\) 0 0
\(649\) −16.2474 −0.637768
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 6.43539i − 0.251836i −0.992041 0.125918i \(-0.959812\pi\)
0.992041 0.125918i \(-0.0401877\pi\)
\(654\) 0 0
\(655\) 24.6969i 0.964989i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.2351 0.476612 0.238306 0.971190i \(-0.423408\pi\)
0.238306 + 0.971190i \(0.423408\pi\)
\(660\) 0 0
\(661\) 42.0454 1.63538 0.817688 0.575661i \(-0.195255\pi\)
0.817688 + 0.575661i \(0.195255\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 7.70674i − 0.298855i
\(666\) 0 0
\(667\) − 47.1918i − 1.82728i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.84961 −0.303031
\(672\) 0 0
\(673\) −12.8990 −0.497219 −0.248610 0.968604i \(-0.579974\pi\)
−0.248610 + 0.968604i \(0.579974\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 0.285729i − 0.0109815i −0.999985 0.00549073i \(-0.998252\pi\)
0.999985 0.00549073i \(-0.00174776\pi\)
\(678\) 0 0
\(679\) − 10.2474i − 0.393261i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33.1626 1.26893 0.634466 0.772951i \(-0.281220\pi\)
0.634466 + 0.772951i \(0.281220\pi\)
\(684\) 0 0
\(685\) −39.1464 −1.49571
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.25697i 0.0859837i
\(690\) 0 0
\(691\) − 40.9444i − 1.55760i −0.627274 0.778799i \(-0.715829\pi\)
0.627274 0.778799i \(-0.284171\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.41421 −0.0536442
\(696\) 0 0
\(697\) −25.2929 −0.958035
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 43.2049i 1.63183i 0.578173 + 0.815914i \(0.303766\pi\)
−0.578173 + 0.815914i \(0.696234\pi\)
\(702\) 0 0
\(703\) 22.0454i 0.831458i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.65685 −0.212748
\(708\) 0 0
\(709\) 52.3939 1.96769 0.983847 0.179013i \(-0.0572905\pi\)
0.983847 + 0.179013i \(0.0572905\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 8.77101i − 0.328477i
\(714\) 0 0
\(715\) − 2.00000i − 0.0747958i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32.5590 −1.21425 −0.607123 0.794608i \(-0.707677\pi\)
−0.607123 + 0.794608i \(0.707677\pi\)
\(720\) 0 0
\(721\) 5.79796 0.215927
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 40.8693i − 1.51785i
\(726\) 0 0
\(727\) − 26.0454i − 0.965971i −0.875628 0.482985i \(-0.839552\pi\)
0.875628 0.482985i \(-0.160448\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −20.9596 −0.775218
\(732\) 0 0
\(733\) 17.5959 0.649920 0.324960 0.945728i \(-0.394649\pi\)
0.324960 + 0.945728i \(0.394649\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.5133i 0.718780i
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32.8769 −1.20613 −0.603067 0.797690i \(-0.706055\pi\)
−0.603067 + 0.797690i \(0.706055\pi\)
\(744\) 0 0
\(745\) −54.9444 −2.01301
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.8564i 0.506302i
\(750\) 0 0
\(751\) − 25.7980i − 0.941381i −0.882298 0.470690i \(-0.844005\pi\)
0.882298 0.470690i \(-0.155995\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −26.2665 −0.955935
\(756\) 0 0
\(757\) −33.6969 −1.22474 −0.612368 0.790573i \(-0.709783\pi\)
−0.612368 + 0.790573i \(0.709783\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 19.1954i − 0.695834i −0.937525 0.347917i \(-0.886889\pi\)
0.937525 0.347917i \(-0.113111\pi\)
\(762\) 0 0
\(763\) − 0.797959i − 0.0288881i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.16404 −0.186463
\(768\) 0 0
\(769\) 35.7980 1.29091 0.645454 0.763799i \(-0.276668\pi\)
0.645454 + 0.763799i \(0.276668\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 12.5529i − 0.451498i −0.974185 0.225749i \(-0.927517\pi\)
0.974185 0.225749i \(-0.0724830\pi\)
\(774\) 0 0
\(775\) − 7.59592i − 0.272853i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.6773 −0.884156
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 32.2412i 1.15074i
\(786\) 0 0
\(787\) 37.3939i 1.33295i 0.745528 + 0.666474i \(0.232197\pi\)
−0.745528 + 0.666474i \(0.767803\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.5352 0.374588
\(792\) 0 0
\(793\) −2.49490 −0.0885964
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 40.5836i − 1.43754i −0.695245 0.718772i \(-0.744704\pi\)
0.695245 0.718772i \(-0.255296\pi\)
\(798\) 0 0
\(799\) − 1.15663i − 0.0409187i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.8564 0.488982
\(804\) 0 0
\(805\) 17.7980 0.627296
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.13534i 0.286023i 0.989721 + 0.143012i \(0.0456786\pi\)
−0.989721 + 0.143012i \(0.954321\pi\)
\(810\) 0 0
\(811\) 16.0454i 0.563430i 0.959498 + 0.281715i \(0.0909033\pi\)
−0.959498 + 0.281715i \(0.909097\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.73205 −0.0606711
\(816\) 0 0
\(817\) −20.4495 −0.715437
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 35.9124i − 1.25335i −0.779281 0.626675i \(-0.784415\pi\)
0.779281 0.626675i \(-0.215585\pi\)
\(822\) 0 0
\(823\) − 42.8434i − 1.49343i −0.665146 0.746713i \(-0.731631\pi\)
0.665146 0.746713i \(-0.268369\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.1708 −0.388448 −0.194224 0.980957i \(-0.562219\pi\)
−0.194224 + 0.980957i \(0.562219\pi\)
\(828\) 0 0
\(829\) −52.0454 −1.80761 −0.903806 0.427943i \(-0.859238\pi\)
−0.903806 + 0.427943i \(0.859238\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2.51059i − 0.0869868i
\(834\) 0 0
\(835\) 30.1464i 1.04326i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.56048 0.157445 0.0787226 0.996897i \(-0.474916\pi\)
0.0787226 + 0.996897i \(0.474916\pi\)
\(840\) 0 0
\(841\) −40.5959 −1.39986
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 40.2658i 1.38518i
\(846\) 0 0
\(847\) 9.00000i 0.309244i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −50.9117 −1.74523
\(852\) 0 0
\(853\) 24.2020 0.828662 0.414331 0.910126i \(-0.364016\pi\)
0.414331 + 0.910126i \(0.364016\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.1380i 0.858698i 0.903139 + 0.429349i \(0.141257\pi\)
−0.903139 + 0.429349i \(0.858743\pi\)
\(858\) 0 0
\(859\) 29.1010i 0.992914i 0.868061 + 0.496457i \(0.165366\pi\)
−0.868061 + 0.496457i \(0.834634\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.4347 0.695604 0.347802 0.937568i \(-0.386928\pi\)
0.347802 + 0.937568i \(0.386928\pi\)
\(864\) 0 0
\(865\) 24.6969 0.839721
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.492810i 0.0167174i
\(870\) 0 0
\(871\) 6.20204i 0.210148i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.317837 −0.0107449
\(876\) 0 0
\(877\) 38.5959 1.30329 0.651646 0.758523i \(-0.274079\pi\)
0.651646 + 0.758523i \(0.274079\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.4974i 1.63392i 0.576695 + 0.816960i \(0.304342\pi\)
−0.576695 + 0.816960i \(0.695658\pi\)
\(882\) 0 0
\(883\) 7.65153i 0.257495i 0.991677 + 0.128747i \(0.0410956\pi\)
−0.991677 + 0.128747i \(0.958904\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33.8304 1.13591 0.567956 0.823059i \(-0.307734\pi\)
0.567956 + 0.823059i \(0.307734\pi\)
\(888\) 0 0
\(889\) −12.5505 −0.420931
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 1.12848i − 0.0377633i
\(894\) 0 0
\(895\) − 35.1464i − 1.17482i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.9350 −0.431406
\(900\) 0 0
\(901\) −12.6061 −0.419971
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 11.9494i − 0.397211i
\(906\) 0 0
\(907\) − 23.2474i − 0.771919i −0.922516 0.385959i \(-0.873870\pi\)
0.922516 0.385959i \(-0.126130\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 49.2117 1.63046 0.815229 0.579139i \(-0.196611\pi\)
0.815229 + 0.579139i \(0.196611\pi\)
\(912\) 0 0
\(913\) −3.34847 −0.110818
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.84961i 0.259217i
\(918\) 0 0
\(919\) − 28.8434i − 0.951455i −0.879593 0.475727i \(-0.842185\pi\)
0.879593 0.475727i \(-0.157815\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.54270 0.0836939
\(924\) 0 0
\(925\) −44.0908 −1.44970
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.14626i 0.103226i 0.998667 + 0.0516128i \(0.0164362\pi\)
−0.998667 + 0.0516128i \(0.983564\pi\)
\(930\) 0 0
\(931\) − 2.44949i − 0.0802788i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 11.1708 0.365326
\(936\) 0 0
\(937\) 54.7423 1.78835 0.894177 0.447713i \(-0.147761\pi\)
0.894177 + 0.447713i \(0.147761\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 58.8576i 1.91870i 0.282213 + 0.959352i \(0.408932\pi\)
−0.282213 + 0.959352i \(0.591068\pi\)
\(942\) 0 0
\(943\) − 56.9898i − 1.85584i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 43.4120 1.41070 0.705351 0.708859i \(-0.250790\pi\)
0.705351 + 0.708859i \(0.250790\pi\)
\(948\) 0 0
\(949\) 4.40408 0.142963
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 3.81405i − 0.123549i −0.998090 0.0617745i \(-0.980324\pi\)
0.998090 0.0617745i \(-0.0196760\pi\)
\(954\) 0 0
\(955\) − 24.6969i − 0.799174i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.4422 −0.401779
\(960\) 0 0
\(961\) 28.5959 0.922449
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 64.8002i − 2.08599i
\(966\) 0 0
\(967\) − 12.8990i − 0.414803i −0.978256 0.207402i \(-0.933499\pi\)
0.978256 0.207402i \(-0.0665007\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 39.2015 1.25804 0.629018 0.777391i \(-0.283457\pi\)
0.629018 + 0.777391i \(0.283457\pi\)
\(972\) 0 0
\(973\) −0.449490 −0.0144100
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 31.4626i − 1.00658i −0.864118 0.503290i \(-0.832123\pi\)
0.864118 0.503290i \(-0.167877\pi\)
\(978\) 0 0
\(979\) − 2.20204i − 0.0703775i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9.58166 −0.305607 −0.152804 0.988257i \(-0.548830\pi\)
−0.152804 + 0.988257i \(0.548830\pi\)
\(984\) 0 0
\(985\) 31.1464 0.992408
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 47.2261i − 1.50170i
\(990\) 0 0
\(991\) − 7.65153i − 0.243059i −0.992588 0.121529i \(-0.961220\pi\)
0.992588 0.121529i \(-0.0387799\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 39.9479 1.26643
\(996\) 0 0
\(997\) −41.8434 −1.32519 −0.662596 0.748977i \(-0.730545\pi\)
−0.662596 + 0.748977i \(0.730545\pi\)
\(998\) 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.h.c.2591.1 8
3.2 odd 2 inner 6048.2.h.c.2591.7 yes 8
4.3 odd 2 inner 6048.2.h.c.2591.2 yes 8
12.11 even 2 inner 6048.2.h.c.2591.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.c.2591.1 8 1.1 even 1 trivial
6048.2.h.c.2591.2 yes 8 4.3 odd 2 inner
6048.2.h.c.2591.7 yes 8 3.2 odd 2 inner
6048.2.h.c.2591.8 yes 8 12.11 even 2 inner