Properties

Label 6048.2.c.e.3025.15
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 20
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.15
Root \(-0.725842 + 1.21374i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.e.3025.6

$q$-expansion

\(f(q)\) \(=\) \(q+3.06888i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+3.06888i q^{5} +1.00000 q^{7} +5.80820i q^{11} -3.52392i q^{13} -6.79736 q^{17} -5.28617i q^{19} +5.65085 q^{23} -4.41801 q^{25} +1.21394i q^{29} -0.107385 q^{31} +3.06888i q^{35} -4.90235i q^{37} -11.6855 q^{41} +1.85684i q^{43} -6.76459 q^{47} +1.00000 q^{49} +11.4861i q^{53} -17.8247 q^{55} +7.85494i q^{59} +12.0556i q^{61} +10.8145 q^{65} +6.66942i q^{67} -1.98478 q^{71} -6.52879 q^{73} +5.80820i q^{77} -2.30741 q^{79} -12.5795i q^{83} -20.8603i q^{85} -7.79151 q^{89} -3.52392i q^{91} +16.2226 q^{95} +4.05981 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 20q^{7} + O(q^{10}) \) \( 20q + 20q^{7} - 28q^{25} - 36q^{31} + 20q^{49} - 48q^{55} - 64q^{79} + 56q^{97} + O(q^{100}) \)

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
\(5\) 3.06888i 1.37244i 0.727392 + 0.686222i \(0.240732\pi\)
−0.727392 + 0.686222i \(0.759268\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.80820i 1.75124i 0.483001 + 0.875620i \(0.339547\pi\)
−0.483001 + 0.875620i \(0.660453\pi\)
\(12\) 0 0
\(13\) − 3.52392i − 0.977359i −0.872463 0.488680i \(-0.837479\pi\)
0.872463 0.488680i \(-0.162521\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.79736 −1.64860 −0.824301 0.566151i \(-0.808432\pi\)
−0.824301 + 0.566151i \(0.808432\pi\)
\(18\) 0 0
\(19\) − 5.28617i − 1.21273i −0.795186 0.606365i \(-0.792627\pi\)
0.795186 0.606365i \(-0.207373\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.65085 1.17828 0.589141 0.808030i \(-0.299466\pi\)
0.589141 + 0.808030i \(0.299466\pi\)
\(24\) 0 0
\(25\) −4.41801 −0.883601
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.21394i 0.225422i 0.993628 + 0.112711i \(0.0359534\pi\)
−0.993628 + 0.112711i \(0.964047\pi\)
\(30\) 0 0
\(31\) −0.107385 −0.0192868 −0.00964342 0.999954i \(-0.503070\pi\)
−0.00964342 + 0.999954i \(0.503070\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.06888i 0.518735i
\(36\) 0 0
\(37\) − 4.90235i − 0.805942i −0.915213 0.402971i \(-0.867978\pi\)
0.915213 0.402971i \(-0.132022\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.6855 −1.82497 −0.912485 0.409111i \(-0.865839\pi\)
−0.912485 + 0.409111i \(0.865839\pi\)
\(42\) 0 0
\(43\) 1.85684i 0.283165i 0.989926 + 0.141583i \(0.0452191\pi\)
−0.989926 + 0.141583i \(0.954781\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.76459 −0.986717 −0.493359 0.869826i \(-0.664231\pi\)
−0.493359 + 0.869826i \(0.664231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.4861i 1.57773i 0.614566 + 0.788865i \(0.289331\pi\)
−0.614566 + 0.788865i \(0.710669\pi\)
\(54\) 0 0
\(55\) −17.8247 −2.40348
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.85494i 1.02263i 0.859394 + 0.511313i \(0.170841\pi\)
−0.859394 + 0.511313i \(0.829159\pi\)
\(60\) 0 0
\(61\) 12.0556i 1.54356i 0.635890 + 0.771779i \(0.280633\pi\)
−0.635890 + 0.771779i \(0.719367\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.8145 1.34137
\(66\) 0 0
\(67\) 6.66942i 0.814800i 0.913250 + 0.407400i \(0.133564\pi\)
−0.913250 + 0.407400i \(0.866436\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.98478 −0.235550 −0.117775 0.993040i \(-0.537576\pi\)
−0.117775 + 0.993040i \(0.537576\pi\)
\(72\) 0 0
\(73\) −6.52879 −0.764137 −0.382068 0.924134i \(-0.624788\pi\)
−0.382068 + 0.924134i \(0.624788\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.80820i 0.661906i
\(78\) 0 0
\(79\) −2.30741 −0.259604 −0.129802 0.991540i \(-0.541434\pi\)
−0.129802 + 0.991540i \(0.541434\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.5795i − 1.38078i −0.723436 0.690391i \(-0.757438\pi\)
0.723436 0.690391i \(-0.242562\pi\)
\(84\) 0 0
\(85\) − 20.8603i − 2.26261i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.79151 −0.825898 −0.412949 0.910754i \(-0.635501\pi\)
−0.412949 + 0.910754i \(0.635501\pi\)
\(90\) 0 0
\(91\) − 3.52392i − 0.369407i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.2226 1.66440
\(96\) 0 0
\(97\) 4.05981 0.412212 0.206106 0.978530i \(-0.433921\pi\)
0.206106 + 0.978530i \(0.433921\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 10.3336i − 1.02823i −0.857721 0.514116i \(-0.828120\pi\)
0.857721 0.514116i \(-0.171880\pi\)
\(102\) 0 0
\(103\) −8.94679 −0.881554 −0.440777 0.897617i \(-0.645297\pi\)
−0.440777 + 0.897617i \(0.645297\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 17.8845i − 1.72896i −0.502669 0.864479i \(-0.667649\pi\)
0.502669 0.864479i \(-0.332351\pi\)
\(108\) 0 0
\(109\) − 11.9610i − 1.14566i −0.819676 0.572828i \(-0.805846\pi\)
0.819676 0.572828i \(-0.194154\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −13.3192 −1.25297 −0.626484 0.779434i \(-0.715507\pi\)
−0.626484 + 0.779434i \(0.715507\pi\)
\(114\) 0 0
\(115\) 17.3417i 1.61713i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.79736 −0.623113
\(120\) 0 0
\(121\) −22.7352 −2.06684
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.78606i 0.159750i
\(126\) 0 0
\(127\) 7.42140 0.658543 0.329271 0.944235i \(-0.393197\pi\)
0.329271 + 0.944235i \(0.393197\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 4.82695i − 0.421733i −0.977515 0.210866i \(-0.932372\pi\)
0.977515 0.210866i \(-0.0676285\pi\)
\(132\) 0 0
\(133\) − 5.28617i − 0.458369i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.8496 −1.26868 −0.634342 0.773052i \(-0.718729\pi\)
−0.634342 + 0.773052i \(0.718729\pi\)
\(138\) 0 0
\(139\) 4.99694i 0.423835i 0.977288 + 0.211918i \(0.0679708\pi\)
−0.977288 + 0.211918i \(0.932029\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.4676 1.71159
\(144\) 0 0
\(145\) −3.72542 −0.309379
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 9.54688i − 0.782111i −0.920367 0.391055i \(-0.872110\pi\)
0.920367 0.391055i \(-0.127890\pi\)
\(150\) 0 0
\(151\) 11.4778 0.934052 0.467026 0.884244i \(-0.345325\pi\)
0.467026 + 0.884244i \(0.345325\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 0.329550i − 0.0264701i
\(156\) 0 0
\(157\) 1.38325i 0.110396i 0.998475 + 0.0551979i \(0.0175790\pi\)
−0.998475 + 0.0551979i \(0.982421\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65085 0.445349
\(162\) 0 0
\(163\) 10.0101i 0.784050i 0.919955 + 0.392025i \(0.128225\pi\)
−0.919955 + 0.392025i \(0.871775\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.63373 −0.126422 −0.0632110 0.998000i \(-0.520134\pi\)
−0.0632110 + 0.998000i \(0.520134\pi\)
\(168\) 0 0
\(169\) 0.581993 0.0447687
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.93513i 0.299182i 0.988748 + 0.149591i \(0.0477957\pi\)
−0.988748 + 0.149591i \(0.952204\pi\)
\(174\) 0 0
\(175\) −4.41801 −0.333970
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.199235i 0.0148915i 0.999972 + 0.00744577i \(0.00237008\pi\)
−0.999972 + 0.00744577i \(0.997630\pi\)
\(180\) 0 0
\(181\) 7.14783i 0.531294i 0.964070 + 0.265647i \(0.0855855\pi\)
−0.964070 + 0.265647i \(0.914414\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.0447 1.10611
\(186\) 0 0
\(187\) − 39.4805i − 2.88710i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.8761 1.29347 0.646733 0.762717i \(-0.276135\pi\)
0.646733 + 0.762717i \(0.276135\pi\)
\(192\) 0 0
\(193\) −14.2993 −1.02928 −0.514642 0.857405i \(-0.672075\pi\)
−0.514642 + 0.857405i \(0.672075\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 4.13436i − 0.294561i −0.989095 0.147280i \(-0.952948\pi\)
0.989095 0.147280i \(-0.0470520\pi\)
\(198\) 0 0
\(199\) −14.5173 −1.02910 −0.514550 0.857460i \(-0.672041\pi\)
−0.514550 + 0.857460i \(0.672041\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.21394i 0.0852015i
\(204\) 0 0
\(205\) − 35.8614i − 2.50467i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 30.7031 2.12378
\(210\) 0 0
\(211\) − 13.9076i − 0.957439i −0.877968 0.478719i \(-0.841101\pi\)
0.877968 0.478719i \(-0.158899\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.69841 −0.388628
\(216\) 0 0
\(217\) −0.107385 −0.00728974
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 23.9534i 1.61128i
\(222\) 0 0
\(223\) 12.1591 0.814231 0.407115 0.913377i \(-0.366535\pi\)
0.407115 + 0.913377i \(0.366535\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 24.4059i − 1.61987i −0.586517 0.809937i \(-0.699501\pi\)
0.586517 0.809937i \(-0.300499\pi\)
\(228\) 0 0
\(229\) − 1.95033i − 0.128881i −0.997922 0.0644407i \(-0.979474\pi\)
0.997922 0.0644407i \(-0.0205263\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.3094 1.06847 0.534233 0.845337i \(-0.320600\pi\)
0.534233 + 0.845337i \(0.320600\pi\)
\(234\) 0 0
\(235\) − 20.7597i − 1.35421i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.65345 −0.106953 −0.0534764 0.998569i \(-0.517030\pi\)
−0.0534764 + 0.998569i \(0.517030\pi\)
\(240\) 0 0
\(241\) 3.63938 0.234433 0.117217 0.993106i \(-0.462603\pi\)
0.117217 + 0.993106i \(0.462603\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.06888i 0.196063i
\(246\) 0 0
\(247\) −18.6280 −1.18527
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.2386i 1.59305i 0.604607 + 0.796524i \(0.293330\pi\)
−0.604607 + 0.796524i \(0.706670\pi\)
\(252\) 0 0
\(253\) 32.8213i 2.06346i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.2051 −0.886087 −0.443044 0.896500i \(-0.646101\pi\)
−0.443044 + 0.896500i \(0.646101\pi\)
\(258\) 0 0
\(259\) − 4.90235i − 0.304617i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.3785 1.00994 0.504971 0.863137i \(-0.331503\pi\)
0.504971 + 0.863137i \(0.331503\pi\)
\(264\) 0 0
\(265\) −35.2493 −2.16535
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.73448i 0.288666i 0.989529 + 0.144333i \(0.0461037\pi\)
−0.989529 + 0.144333i \(0.953896\pi\)
\(270\) 0 0
\(271\) −20.3680 −1.23727 −0.618634 0.785679i \(-0.712314\pi\)
−0.618634 + 0.785679i \(0.712314\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 25.6607i − 1.54740i
\(276\) 0 0
\(277\) 29.0135i 1.74325i 0.490170 + 0.871627i \(0.336935\pi\)
−0.490170 + 0.871627i \(0.663065\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.5849 −0.989373 −0.494687 0.869071i \(-0.664717\pi\)
−0.494687 + 0.869071i \(0.664717\pi\)
\(282\) 0 0
\(283\) − 21.8603i − 1.29946i −0.760165 0.649730i \(-0.774882\pi\)
0.760165 0.649730i \(-0.225118\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −11.6855 −0.689774
\(288\) 0 0
\(289\) 29.2041 1.71789
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 3.59022i − 0.209743i −0.994486 0.104872i \(-0.966557\pi\)
0.994486 0.104872i \(-0.0334431\pi\)
\(294\) 0 0
\(295\) −24.1059 −1.40350
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 19.9131i − 1.15161i
\(300\) 0 0
\(301\) 1.85684i 0.107026i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −36.9971 −2.11845
\(306\) 0 0
\(307\) − 17.1418i − 0.978333i −0.872191 0.489166i \(-0.837301\pi\)
0.872191 0.489166i \(-0.162699\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −21.2814 −1.20676 −0.603379 0.797454i \(-0.706180\pi\)
−0.603379 + 0.797454i \(0.706180\pi\)
\(312\) 0 0
\(313\) −7.14361 −0.403781 −0.201890 0.979408i \(-0.564708\pi\)
−0.201890 + 0.979408i \(0.564708\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 31.4066i − 1.76397i −0.471277 0.881986i \(-0.656207\pi\)
0.471277 0.881986i \(-0.343793\pi\)
\(318\) 0 0
\(319\) −7.05078 −0.394768
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 35.9320i 1.99931i
\(324\) 0 0
\(325\) 15.5687i 0.863596i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.76459 −0.372944
\(330\) 0 0
\(331\) 11.6712i 0.641506i 0.947163 + 0.320753i \(0.103936\pi\)
−0.947163 + 0.320753i \(0.896064\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −20.4676 −1.11827
\(336\) 0 0
\(337\) 20.4008 1.11130 0.555652 0.831415i \(-0.312469\pi\)
0.555652 + 0.831415i \(0.312469\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 0.623712i − 0.0337759i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.5903i 0.675881i 0.941167 + 0.337940i \(0.109730\pi\)
−0.941167 + 0.337940i \(0.890270\pi\)
\(348\) 0 0
\(349\) − 12.8022i − 0.685286i −0.939466 0.342643i \(-0.888678\pi\)
0.939466 0.342643i \(-0.111322\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 17.2495 0.918099 0.459050 0.888411i \(-0.348190\pi\)
0.459050 + 0.888411i \(0.348190\pi\)
\(354\) 0 0
\(355\) − 6.09103i − 0.323278i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.52892 −0.0806935 −0.0403468 0.999186i \(-0.512846\pi\)
−0.0403468 + 0.999186i \(0.512846\pi\)
\(360\) 0 0
\(361\) −8.94358 −0.470715
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 20.0360i − 1.04873i
\(366\) 0 0
\(367\) 4.04525 0.211160 0.105580 0.994411i \(-0.466330\pi\)
0.105580 + 0.994411i \(0.466330\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.4861i 0.596326i
\(372\) 0 0
\(373\) − 25.1416i − 1.30178i −0.759171 0.650891i \(-0.774395\pi\)
0.759171 0.650891i \(-0.225605\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.27781 0.220318
\(378\) 0 0
\(379\) 10.1885i 0.523349i 0.965156 + 0.261675i \(0.0842747\pi\)
−0.965156 + 0.261675i \(0.915725\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 35.9423 1.83657 0.918284 0.395923i \(-0.129575\pi\)
0.918284 + 0.395923i \(0.129575\pi\)
\(384\) 0 0
\(385\) −17.8247 −0.908429
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 6.81471i − 0.345519i −0.984964 0.172760i \(-0.944732\pi\)
0.984964 0.172760i \(-0.0552684\pi\)
\(390\) 0 0
\(391\) −38.4108 −1.94252
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 7.08116i − 0.356292i
\(396\) 0 0
\(397\) 11.1982i 0.562021i 0.959705 + 0.281010i \(0.0906695\pi\)
−0.959705 + 0.281010i \(0.909330\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −29.9404 −1.49515 −0.747577 0.664175i \(-0.768783\pi\)
−0.747577 + 0.664175i \(0.768783\pi\)
\(402\) 0 0
\(403\) 0.378415i 0.0188502i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 28.4739 1.41140
\(408\) 0 0
\(409\) −11.0213 −0.544968 −0.272484 0.962160i \(-0.587845\pi\)
−0.272484 + 0.962160i \(0.587845\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.85494i 0.386516i
\(414\) 0 0
\(415\) 38.6050 1.89505
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 14.0133i − 0.684594i −0.939592 0.342297i \(-0.888795\pi\)
0.939592 0.342297i \(-0.111205\pi\)
\(420\) 0 0
\(421\) 19.4716i 0.948988i 0.880259 + 0.474494i \(0.157369\pi\)
−0.880259 + 0.474494i \(0.842631\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 30.0308 1.45671
\(426\) 0 0
\(427\) 12.0556i 0.583410i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −37.2120 −1.79244 −0.896219 0.443612i \(-0.853697\pi\)
−0.896219 + 0.443612i \(0.853697\pi\)
\(432\) 0 0
\(433\) −23.4042 −1.12474 −0.562368 0.826887i \(-0.690109\pi\)
−0.562368 + 0.826887i \(0.690109\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 29.8713i − 1.42894i
\(438\) 0 0
\(439\) −21.7533 −1.03823 −0.519115 0.854705i \(-0.673738\pi\)
−0.519115 + 0.854705i \(0.673738\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 12.8369i − 0.609901i −0.952368 0.304951i \(-0.901360\pi\)
0.952368 0.304951i \(-0.0986400\pi\)
\(444\) 0 0
\(445\) − 23.9112i − 1.13350i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.7889 0.556353 0.278176 0.960530i \(-0.410270\pi\)
0.278176 + 0.960530i \(0.410270\pi\)
\(450\) 0 0
\(451\) − 67.8718i − 3.19596i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 10.8145 0.506990
\(456\) 0 0
\(457\) −33.5157 −1.56780 −0.783900 0.620888i \(-0.786772\pi\)
−0.783900 + 0.620888i \(0.786772\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 10.0902i − 0.469946i −0.972002 0.234973i \(-0.924500\pi\)
0.972002 0.234973i \(-0.0755002\pi\)
\(462\) 0 0
\(463\) 23.6131 1.09739 0.548697 0.836021i \(-0.315124\pi\)
0.548697 + 0.836021i \(0.315124\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.52944i 0.117048i 0.998286 + 0.0585242i \(0.0186395\pi\)
−0.998286 + 0.0585242i \(0.981361\pi\)
\(468\) 0 0
\(469\) 6.66942i 0.307965i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.7849 −0.495890
\(474\) 0 0
\(475\) 23.3543i 1.07157i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15.1106 0.690421 0.345210 0.938525i \(-0.387808\pi\)
0.345210 + 0.938525i \(0.387808\pi\)
\(480\) 0 0
\(481\) −17.2755 −0.787695
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.4591i 0.565737i
\(486\) 0 0
\(487\) −41.1541 −1.86487 −0.932436 0.361335i \(-0.882321\pi\)
−0.932436 + 0.361335i \(0.882321\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.6868i 1.65565i 0.560984 + 0.827827i \(0.310423\pi\)
−0.560984 + 0.827827i \(0.689577\pi\)
\(492\) 0 0
\(493\) − 8.25156i − 0.371631i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.98478 −0.0890294
\(498\) 0 0
\(499\) 20.1824i 0.903489i 0.892147 + 0.451744i \(0.149198\pi\)
−0.892147 + 0.451744i \(0.850802\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.5011 −0.468222 −0.234111 0.972210i \(-0.575218\pi\)
−0.234111 + 0.972210i \(0.575218\pi\)
\(504\) 0 0
\(505\) 31.7125 1.41119
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.5910i 1.26727i 0.773630 + 0.633637i \(0.218439\pi\)
−0.773630 + 0.633637i \(0.781561\pi\)
\(510\) 0 0
\(511\) −6.52879 −0.288816
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 27.4566i − 1.20988i
\(516\) 0 0
\(517\) − 39.2901i − 1.72798i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.8325 0.474578 0.237289 0.971439i \(-0.423741\pi\)
0.237289 + 0.971439i \(0.423741\pi\)
\(522\) 0 0
\(523\) 1.77247i 0.0775047i 0.999249 + 0.0387523i \(0.0123383\pi\)
−0.999249 + 0.0387523i \(0.987662\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.729932 0.0317963
\(528\) 0 0
\(529\) 8.93205 0.388350
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 41.1788i 1.78365i
\(534\) 0 0
\(535\) 54.8853 2.37290
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.80820i 0.250177i
\(540\) 0 0
\(541\) 11.0556i 0.475316i 0.971349 + 0.237658i \(0.0763797\pi\)
−0.971349 + 0.237658i \(0.923620\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 36.7068 1.57235
\(546\) 0 0
\(547\) 5.55978i 0.237719i 0.992911 + 0.118859i \(0.0379238\pi\)
−0.992911 + 0.118859i \(0.962076\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.41707 0.273376
\(552\) 0 0
\(553\) −2.30741 −0.0981211
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.30560i 0.182434i 0.995831 + 0.0912171i \(0.0290757\pi\)
−0.995831 + 0.0912171i \(0.970924\pi\)
\(558\) 0 0
\(559\) 6.54335 0.276754
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.43447i 0.102601i 0.998683 + 0.0513003i \(0.0163366\pi\)
−0.998683 + 0.0513003i \(0.983663\pi\)
\(564\) 0 0
\(565\) − 40.8751i − 1.71963i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.9163 0.457635 0.228818 0.973469i \(-0.426514\pi\)
0.228818 + 0.973469i \(0.426514\pi\)
\(570\) 0 0
\(571\) 11.3442i 0.474740i 0.971419 + 0.237370i \(0.0762854\pi\)
−0.971419 + 0.237370i \(0.923715\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.9655 −1.04113
\(576\) 0 0
\(577\) 7.97962 0.332196 0.166098 0.986109i \(-0.446883\pi\)
0.166098 + 0.986109i \(0.446883\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 12.5795i − 0.521887i
\(582\) 0 0
\(583\) −66.7133 −2.76298
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 41.7410i 1.72283i 0.507898 + 0.861417i \(0.330423\pi\)
−0.507898 + 0.861417i \(0.669577\pi\)
\(588\) 0 0
\(589\) 0.567653i 0.0233897i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.43527 0.182135 0.0910673 0.995845i \(-0.470972\pi\)
0.0910673 + 0.995845i \(0.470972\pi\)
\(594\) 0 0
\(595\) − 20.8603i − 0.855188i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.4875 −0.510224 −0.255112 0.966911i \(-0.582112\pi\)
−0.255112 + 0.966911i \(0.582112\pi\)
\(600\) 0 0
\(601\) 37.1871 1.51690 0.758448 0.651734i \(-0.225958\pi\)
0.758448 + 0.651734i \(0.225958\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 69.7716i − 2.83662i
\(606\) 0 0
\(607\) 21.8403 0.886470 0.443235 0.896405i \(-0.353831\pi\)
0.443235 + 0.896405i \(0.353831\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 23.8379i 0.964377i
\(612\) 0 0
\(613\) 30.7687i 1.24274i 0.783519 + 0.621368i \(0.213423\pi\)
−0.783519 + 0.621368i \(0.786577\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0402 1.53144 0.765721 0.643173i \(-0.222383\pi\)
0.765721 + 0.643173i \(0.222383\pi\)
\(618\) 0 0
\(619\) 14.9005i 0.598903i 0.954111 + 0.299452i \(0.0968037\pi\)
−0.954111 + 0.299452i \(0.903196\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −7.79151 −0.312160
\(624\) 0 0
\(625\) −27.5712 −1.10285
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.3231i 1.32868i
\(630\) 0 0
\(631\) 26.8787 1.07002 0.535011 0.844845i \(-0.320307\pi\)
0.535011 + 0.844845i \(0.320307\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.7754i 0.903813i
\(636\) 0 0
\(637\) − 3.52392i − 0.139623i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12.4910 −0.493365 −0.246682 0.969096i \(-0.579340\pi\)
−0.246682 + 0.969096i \(0.579340\pi\)
\(642\) 0 0
\(643\) 3.53414i 0.139373i 0.997569 + 0.0696864i \(0.0221999\pi\)
−0.997569 + 0.0696864i \(0.977800\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −28.3282 −1.11370 −0.556848 0.830614i \(-0.687990\pi\)
−0.556848 + 0.830614i \(0.687990\pi\)
\(648\) 0 0
\(649\) −45.6231 −1.79086
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 29.9962i 1.17384i 0.809644 + 0.586921i \(0.199660\pi\)
−0.809644 + 0.586921i \(0.800340\pi\)
\(654\) 0 0
\(655\) 14.8133 0.578804
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 16.0647i − 0.625793i −0.949787 0.312896i \(-0.898701\pi\)
0.949787 0.312896i \(-0.101299\pi\)
\(660\) 0 0
\(661\) 44.3984i 1.72690i 0.504435 + 0.863450i \(0.331701\pi\)
−0.504435 + 0.863450i \(0.668299\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.2226 0.629086
\(666\) 0 0
\(667\) 6.85976i 0.265611i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −70.0213 −2.70314
\(672\) 0 0
\(673\) −50.0649 −1.92986 −0.964930 0.262508i \(-0.915450\pi\)
−0.964930 + 0.262508i \(0.915450\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 30.1370i − 1.15826i −0.815236 0.579129i \(-0.803393\pi\)
0.815236 0.579129i \(-0.196607\pi\)
\(678\) 0 0
\(679\) 4.05981 0.155801
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.49520i 0.0572124i 0.999591 + 0.0286062i \(0.00910688\pi\)
−0.999591 + 0.0286062i \(0.990893\pi\)
\(684\) 0 0
\(685\) − 45.5715i − 1.74120i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 40.4759 1.54201
\(690\) 0 0
\(691\) 0.438347i 0.0166755i 0.999965 + 0.00833774i \(0.00265402\pi\)
−0.999965 + 0.00833774i \(0.997346\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.3350 −0.581690
\(696\) 0 0
\(697\) 79.4306 3.00865
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 43.1330i 1.62911i 0.580084 + 0.814556i \(0.303020\pi\)
−0.580084 + 0.814556i \(0.696980\pi\)
\(702\) 0 0
\(703\) −25.9147 −0.977390
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 10.3336i − 0.388635i
\(708\) 0 0
\(709\) − 30.8542i − 1.15875i −0.815060 0.579376i \(-0.803296\pi\)
0.815060 0.579376i \(-0.196704\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −0.606814 −0.0227254
\(714\) 0 0
\(715\) 62.8127i 2.34906i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.24406 −0.0836895 −0.0418447 0.999124i \(-0.513323\pi\)
−0.0418447 + 0.999124i \(0.513323\pi\)
\(720\) 0 0
\(721\) −8.94679 −0.333196
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 5.36317i − 0.199183i
\(726\) 0 0
\(727\) 16.9794 0.629733 0.314866 0.949136i \(-0.398040\pi\)
0.314866 + 0.949136i \(0.398040\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 12.6216i − 0.466827i
\(732\) 0 0
\(733\) 34.0255i 1.25676i 0.777906 + 0.628380i \(0.216282\pi\)
−0.777906 + 0.628380i \(0.783718\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −38.7374 −1.42691
\(738\) 0 0
\(739\) − 0.0107994i 0 0.000397263i −1.00000 0.000198632i \(-0.999937\pi\)
1.00000 0.000198632i \(-6.32264e-5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −45.6360 −1.67422 −0.837112 0.547032i \(-0.815758\pi\)
−0.837112 + 0.547032i \(0.815758\pi\)
\(744\) 0 0
\(745\) 29.2982 1.07340
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 17.8845i − 0.653485i
\(750\) 0 0
\(751\) 3.95000 0.144138 0.0720688 0.997400i \(-0.477040\pi\)
0.0720688 + 0.997400i \(0.477040\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 35.2240i 1.28193i
\(756\) 0 0
\(757\) − 18.6412i − 0.677524i −0.940872 0.338762i \(-0.889992\pi\)
0.940872 0.338762i \(-0.110008\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.86842 0.176480 0.0882401 0.996099i \(-0.471876\pi\)
0.0882401 + 0.996099i \(0.471876\pi\)
\(762\) 0 0
\(763\) − 11.9610i − 0.433017i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27.6802 0.999473
\(768\) 0 0
\(769\) −1.48442 −0.0535297 −0.0267649 0.999642i \(-0.508521\pi\)
−0.0267649 + 0.999642i \(0.508521\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 14.2868i − 0.513861i −0.966430 0.256931i \(-0.917289\pi\)
0.966430 0.256931i \(-0.0827112\pi\)
\(774\) 0 0
\(775\) 0.474426 0.0170419
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 61.7716i 2.21320i
\(780\) 0 0
\(781\) − 11.5280i − 0.412504i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.24504 −0.151512
\(786\) 0 0
\(787\) 22.3799i 0.797758i 0.917004 + 0.398879i \(0.130601\pi\)
−0.917004 + 0.398879i \(0.869399\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −13.3192 −0.473578
\(792\) 0 0
\(793\) 42.4829 1.50861
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23.1872i 0.821334i 0.911785 + 0.410667i \(0.134704\pi\)
−0.911785 + 0.410667i \(0.865296\pi\)
\(798\) 0 0
\(799\) 45.9814 1.62670
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 37.9205i − 1.33819i
\(804\) 0 0
\(805\) 17.3417i 0.611216i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −44.8263 −1.57601 −0.788004 0.615670i \(-0.788886\pi\)
−0.788004 + 0.615670i \(0.788886\pi\)
\(810\) 0 0
\(811\) − 18.4358i − 0.647370i −0.946165 0.323685i \(-0.895078\pi\)
0.946165 0.323685i \(-0.104922\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −30.7197 −1.07606
\(816\) 0 0
\(817\) 9.81556 0.343403
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21.0533i 0.734764i 0.930070 + 0.367382i \(0.119746\pi\)
−0.930070 + 0.367382i \(0.880254\pi\)
\(822\) 0 0
\(823\) 32.3206 1.12663 0.563313 0.826244i \(-0.309527\pi\)
0.563313 + 0.826244i \(0.309527\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.4599i 0.433272i 0.976252 + 0.216636i \(0.0695085\pi\)
−0.976252 + 0.216636i \(0.930492\pi\)
\(828\) 0 0
\(829\) − 6.78073i − 0.235504i −0.993043 0.117752i \(-0.962431\pi\)
0.993043 0.117752i \(-0.0375689\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.79736 −0.235515
\(834\) 0 0
\(835\) − 5.01372i − 0.173507i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −23.6268 −0.815687 −0.407843 0.913052i \(-0.633719\pi\)
−0.407843 + 0.913052i \(0.633719\pi\)
\(840\) 0 0
\(841\) 27.5264 0.949185
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.78606i 0.0614425i
\(846\) 0 0
\(847\) −22.7352 −0.781192
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 27.7024i − 0.949628i
\(852\) 0 0
\(853\) 17.7202i 0.606727i 0.952875 + 0.303363i \(0.0981096\pi\)
−0.952875 + 0.303363i \(0.901890\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.5262 0.462046 0.231023 0.972948i \(-0.425793\pi\)
0.231023 + 0.972948i \(0.425793\pi\)
\(858\) 0 0
\(859\) 27.6249i 0.942548i 0.881987 + 0.471274i \(0.156206\pi\)
−0.881987 + 0.471274i \(0.843794\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −39.9545 −1.36007 −0.680034 0.733181i \(-0.738035\pi\)
−0.680034 + 0.733181i \(0.738035\pi\)
\(864\) 0 0
\(865\) −12.0764 −0.410610
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 13.4019i − 0.454629i
\(870\) 0 0
\(871\) 23.5025 0.796352
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.78606i 0.0603800i
\(876\) 0 0
\(877\) − 11.3922i − 0.384687i −0.981328 0.192343i \(-0.938391\pi\)
0.981328 0.192343i \(-0.0616087\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −36.3753 −1.22551 −0.612757 0.790271i \(-0.709940\pi\)
−0.612757 + 0.790271i \(0.709940\pi\)
\(882\) 0 0
\(883\) 52.0983i 1.75325i 0.481177 + 0.876624i \(0.340210\pi\)
−0.481177 + 0.876624i \(0.659790\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 45.0095 1.51127 0.755635 0.654992i \(-0.227328\pi\)
0.755635 + 0.654992i \(0.227328\pi\)
\(888\) 0 0
\(889\) 7.42140 0.248906
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 35.7588i 1.19662i
\(894\) 0 0
\(895\) −0.611428 −0.0204378
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 0.130358i − 0.00434768i
\(900\) 0 0
\(901\) − 78.0748i − 2.60105i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −21.9358 −0.729171
\(906\) 0 0
\(907\) − 7.96941i − 0.264620i −0.991208 0.132310i \(-0.957761\pi\)
0.991208 0.132310i \(-0.0422394\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −47.8457 −1.58520 −0.792600 0.609742i \(-0.791273\pi\)
−0.792600 + 0.609742i \(0.791273\pi\)
\(912\) 0 0
\(913\) 73.0645 2.41808
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.82695i − 0.159400i
\(918\) 0 0
\(919\) −36.2279 −1.19505 −0.597525 0.801851i \(-0.703849\pi\)
−0.597525 + 0.801851i \(0.703849\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.99419i 0.230217i
\(924\) 0 0
\(925\) 21.6586i 0.712132i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.44479 −0.309874 −0.154937 0.987924i \(-0.549517\pi\)
−0.154937 + 0.987924i \(0.549517\pi\)
\(930\) 0 0
\(931\) − 5.28617i − 0.173247i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 121.161 3.96238
\(936\) 0 0
\(937\) −15.2959 −0.499695 −0.249847 0.968285i \(-0.580380\pi\)
−0.249847 + 0.968285i \(0.580380\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 29.2847i 0.954654i 0.878726 + 0.477327i \(0.158394\pi\)
−0.878726 + 0.477327i \(0.841606\pi\)
\(942\) 0 0
\(943\) −66.0330 −2.15033
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 23.0133i − 0.747832i −0.927463 0.373916i \(-0.878015\pi\)
0.927463 0.373916i \(-0.121985\pi\)
\(948\) 0 0
\(949\) 23.0069i 0.746836i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.48602 0.145316 0.0726582 0.997357i \(-0.476852\pi\)
0.0726582 + 0.997357i \(0.476852\pi\)
\(954\) 0 0
\(955\) 54.8594i 1.77521i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.8496 −0.479518
\(960\) 0 0
\(961\) −30.9885 −0.999628
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 43.8827i − 1.41263i
\(966\) 0 0
\(967\) 18.4520 0.593376 0.296688 0.954974i \(-0.404118\pi\)
0.296688 + 0.954974i \(0.404118\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 21.1370i 0.678317i 0.940729 + 0.339159i \(0.110142\pi\)
−0.940729 + 0.339159i \(0.889858\pi\)
\(972\) 0 0
\(973\) 4.99694i 0.160195i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 52.4783 1.67893 0.839465 0.543414i \(-0.182869\pi\)
0.839465 + 0.543414i \(0.182869\pi\)
\(978\) 0 0
\(979\) − 45.2547i − 1.44635i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 29.7453 0.948727 0.474364 0.880329i \(-0.342678\pi\)
0.474364 + 0.880329i \(0.342678\pi\)
\(984\) 0 0
\(985\) 12.6878 0.404268
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 10.4927i 0.333649i
\(990\) 0 0
\(991\) −9.11149 −0.289436 −0.144718 0.989473i \(-0.546227\pi\)
−0.144718 + 0.989473i \(0.546227\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 44.5517i − 1.41238i
\(996\) 0 0
\(997\) 4.52824i 0.143411i 0.997426 + 0.0717054i \(0.0228442\pi\)
−0.997426 + 0.0717054i \(0.977156\pi\)
\(998\) 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.c.e.3025.15 20
3.2 odd 2 inner 6048.2.c.e.3025.5 20
4.3 odd 2 1512.2.c.e.757.14 yes 20
8.3 odd 2 1512.2.c.e.757.13 yes 20
8.5 even 2 inner 6048.2.c.e.3025.6 20
12.11 even 2 1512.2.c.e.757.7 20
24.5 odd 2 inner 6048.2.c.e.3025.16 20
24.11 even 2 1512.2.c.e.757.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.e.757.7 20 12.11 even 2
1512.2.c.e.757.8 yes 20 24.11 even 2
1512.2.c.e.757.13 yes 20 8.3 odd 2
1512.2.c.e.757.14 yes 20 4.3 odd 2
6048.2.c.e.3025.5 20 3.2 odd 2 inner
6048.2.c.e.3025.6 20 8.5 even 2 inner
6048.2.c.e.3025.15 20 1.1 even 1 trivial
6048.2.c.e.3025.16 20 24.5 odd 2 inner