Properties

Label 6048.2.c.g.3025.5
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 24
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: \(24\)
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 3025.5
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.g.3025.20

$q$-expansion

\(f(q)\) \(=\) \(q-2.52623i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.52623i q^{5} -1.00000 q^{7} -5.70432i q^{11} -3.06046i q^{13} -5.49926 q^{17} -7.28486i q^{19} -0.539732 q^{23} -1.38184 q^{25} +8.35116i q^{29} -6.74848 q^{31} +2.52623i q^{35} -10.2711i q^{37} +5.58370 q^{41} -3.98332i q^{43} -4.83840 q^{47} +1.00000 q^{49} +11.1320i q^{53} -14.4104 q^{55} -10.1647i q^{59} -4.14075i q^{61} -7.73142 q^{65} +5.96348i q^{67} +4.93014 q^{71} +8.66798 q^{73} +5.70432i q^{77} -12.0532 q^{79} +5.83393i q^{83} +13.8924i q^{85} +9.28810 q^{89} +3.06046i q^{91} -18.4032 q^{95} +12.3500 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} + 16q^{31} + 24q^{49} + 8q^{55} - 32q^{79} + O(q^{100}) \)

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
\(5\) − 2.52623i − 1.12976i −0.825172 0.564882i \(-0.808922\pi\)
0.825172 0.564882i \(-0.191078\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.70432i − 1.71992i −0.510364 0.859958i \(-0.670489\pi\)
0.510364 0.859958i \(-0.329511\pi\)
\(12\) 0 0
\(13\) − 3.06046i − 0.848818i −0.905471 0.424409i \(-0.860482\pi\)
0.905471 0.424409i \(-0.139518\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.49926 −1.33377 −0.666883 0.745163i \(-0.732372\pi\)
−0.666883 + 0.745163i \(0.732372\pi\)
\(18\) 0 0
\(19\) − 7.28486i − 1.67126i −0.549291 0.835631i \(-0.685102\pi\)
0.549291 0.835631i \(-0.314898\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.539732 −0.112542 −0.0562710 0.998416i \(-0.517921\pi\)
−0.0562710 + 0.998416i \(0.517921\pi\)
\(24\) 0 0
\(25\) −1.38184 −0.276367
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.35116i 1.55077i 0.631488 + 0.775386i \(0.282445\pi\)
−0.631488 + 0.775386i \(0.717555\pi\)
\(30\) 0 0
\(31\) −6.74848 −1.21206 −0.606032 0.795441i \(-0.707239\pi\)
−0.606032 + 0.795441i \(0.707239\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.52623i 0.427011i
\(36\) 0 0
\(37\) − 10.2711i − 1.68855i −0.535908 0.844277i \(-0.680031\pi\)
0.535908 0.844277i \(-0.319969\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.58370 0.872028 0.436014 0.899940i \(-0.356390\pi\)
0.436014 + 0.899940i \(0.356390\pi\)
\(42\) 0 0
\(43\) − 3.98332i − 0.607451i −0.952760 0.303725i \(-0.901770\pi\)
0.952760 0.303725i \(-0.0982305\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.83840 −0.705754 −0.352877 0.935670i \(-0.614797\pi\)
−0.352877 + 0.935670i \(0.614797\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.1320i 1.52910i 0.644562 + 0.764552i \(0.277040\pi\)
−0.644562 + 0.764552i \(0.722960\pi\)
\(54\) 0 0
\(55\) −14.4104 −1.94310
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 10.1647i − 1.32333i −0.749798 0.661666i \(-0.769850\pi\)
0.749798 0.661666i \(-0.230150\pi\)
\(60\) 0 0
\(61\) − 4.14075i − 0.530169i −0.964225 0.265084i \(-0.914600\pi\)
0.964225 0.265084i \(-0.0853998\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.73142 −0.958964
\(66\) 0 0
\(67\) 5.96348i 0.728555i 0.931290 + 0.364278i \(0.118684\pi\)
−0.931290 + 0.364278i \(0.881316\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.93014 0.585100 0.292550 0.956250i \(-0.405496\pi\)
0.292550 + 0.956250i \(0.405496\pi\)
\(72\) 0 0
\(73\) 8.66798 1.01451 0.507255 0.861796i \(-0.330660\pi\)
0.507255 + 0.861796i \(0.330660\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.70432i 0.650067i
\(78\) 0 0
\(79\) −12.0532 −1.35609 −0.678044 0.735021i \(-0.737172\pi\)
−0.678044 + 0.735021i \(0.737172\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.83393i 0.640357i 0.947357 + 0.320178i \(0.103743\pi\)
−0.947357 + 0.320178i \(0.896257\pi\)
\(84\) 0 0
\(85\) 13.8924i 1.50684i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.28810 0.984537 0.492268 0.870444i \(-0.336168\pi\)
0.492268 + 0.870444i \(0.336168\pi\)
\(90\) 0 0
\(91\) 3.06046i 0.320823i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −18.4032 −1.88813
\(96\) 0 0
\(97\) 12.3500 1.25395 0.626974 0.779040i \(-0.284293\pi\)
0.626974 + 0.779040i \(0.284293\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.33719i 0.929085i 0.885551 + 0.464542i \(0.153781\pi\)
−0.885551 + 0.464542i \(0.846219\pi\)
\(102\) 0 0
\(103\) 7.28802 0.718110 0.359055 0.933316i \(-0.383099\pi\)
0.359055 + 0.933316i \(0.383099\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.31179i 0.706858i 0.935461 + 0.353429i \(0.114984\pi\)
−0.935461 + 0.353429i \(0.885016\pi\)
\(108\) 0 0
\(109\) 15.9330i 1.52611i 0.646336 + 0.763053i \(0.276300\pi\)
−0.646336 + 0.763053i \(0.723700\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.91459 −0.650470 −0.325235 0.945633i \(-0.605443\pi\)
−0.325235 + 0.945633i \(0.605443\pi\)
\(114\) 0 0
\(115\) 1.36349i 0.127146i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.49926 0.504116
\(120\) 0 0
\(121\) −21.5393 −1.95811
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 9.14031i − 0.817534i
\(126\) 0 0
\(127\) 17.6103 1.56267 0.781333 0.624115i \(-0.214540\pi\)
0.781333 + 0.624115i \(0.214540\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.7551i 0.939675i 0.882753 + 0.469838i \(0.155688\pi\)
−0.882753 + 0.469838i \(0.844312\pi\)
\(132\) 0 0
\(133\) 7.28486i 0.631678i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.56413 −0.133633 −0.0668164 0.997765i \(-0.521284\pi\)
−0.0668164 + 0.997765i \(0.521284\pi\)
\(138\) 0 0
\(139\) − 11.2728i − 0.956149i −0.878319 0.478074i \(-0.841335\pi\)
0.878319 0.478074i \(-0.158665\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −17.4578 −1.45990
\(144\) 0 0
\(145\) 21.0970 1.75201
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 6.48749i − 0.531476i −0.964045 0.265738i \(-0.914384\pi\)
0.964045 0.265738i \(-0.0856156\pi\)
\(150\) 0 0
\(151\) 0.383325 0.0311945 0.0155973 0.999878i \(-0.495035\pi\)
0.0155973 + 0.999878i \(0.495035\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 17.0482i 1.36935i
\(156\) 0 0
\(157\) 8.66658i 0.691669i 0.938296 + 0.345834i \(0.112404\pi\)
−0.938296 + 0.345834i \(0.887596\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.539732 0.0425369
\(162\) 0 0
\(163\) − 25.0771i − 1.96419i −0.188382 0.982096i \(-0.560324\pi\)
0.188382 0.982096i \(-0.439676\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.75411 −0.677414 −0.338707 0.940892i \(-0.609989\pi\)
−0.338707 + 0.940892i \(0.609989\pi\)
\(168\) 0 0
\(169\) 3.63361 0.279508
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 9.29760i − 0.706883i −0.935457 0.353442i \(-0.885011\pi\)
0.935457 0.353442i \(-0.114989\pi\)
\(174\) 0 0
\(175\) 1.38184 0.104457
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.65495i 0.572158i 0.958206 + 0.286079i \(0.0923520\pi\)
−0.958206 + 0.286079i \(0.907648\pi\)
\(180\) 0 0
\(181\) − 2.50424i − 0.186138i −0.995660 0.0930692i \(-0.970332\pi\)
0.995660 0.0930692i \(-0.0296678\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −25.9471 −1.90767
\(186\) 0 0
\(187\) 31.3695i 2.29397i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.20627 −0.231997 −0.115999 0.993249i \(-0.537007\pi\)
−0.115999 + 0.993249i \(0.537007\pi\)
\(192\) 0 0
\(193\) 10.9544 0.788518 0.394259 0.918999i \(-0.371001\pi\)
0.394259 + 0.918999i \(0.371001\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.87749i 0.561248i 0.959818 + 0.280624i \(0.0905414\pi\)
−0.959818 + 0.280624i \(0.909459\pi\)
\(198\) 0 0
\(199\) −1.00753 −0.0714217 −0.0357109 0.999362i \(-0.511370\pi\)
−0.0357109 + 0.999362i \(0.511370\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 8.35116i − 0.586137i
\(204\) 0 0
\(205\) − 14.1057i − 0.985186i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −41.5552 −2.87443
\(210\) 0 0
\(211\) − 16.2709i − 1.12014i −0.828446 0.560069i \(-0.810775\pi\)
0.828446 0.560069i \(-0.189225\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.0628 −0.686276
\(216\) 0 0
\(217\) 6.74848 0.458117
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 16.8302i 1.13212i
\(222\) 0 0
\(223\) −2.78339 −0.186390 −0.0931949 0.995648i \(-0.529708\pi\)
−0.0931949 + 0.995648i \(0.529708\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 11.4440i − 0.759565i −0.925076 0.379782i \(-0.875999\pi\)
0.925076 0.379782i \(-0.124001\pi\)
\(228\) 0 0
\(229\) 4.54391i 0.300270i 0.988666 + 0.150135i \(0.0479708\pi\)
−0.988666 + 0.150135i \(0.952029\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0188 1.04943 0.524715 0.851278i \(-0.324172\pi\)
0.524715 + 0.851278i \(0.324172\pi\)
\(234\) 0 0
\(235\) 12.2229i 0.797335i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −19.2968 −1.24821 −0.624104 0.781341i \(-0.714536\pi\)
−0.624104 + 0.781341i \(0.714536\pi\)
\(240\) 0 0
\(241\) −22.1358 −1.42589 −0.712947 0.701218i \(-0.752640\pi\)
−0.712947 + 0.701218i \(0.752640\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.52623i − 0.161395i
\(246\) 0 0
\(247\) −22.2950 −1.41860
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.94172i 0.501277i 0.968081 + 0.250638i \(0.0806405\pi\)
−0.968081 + 0.250638i \(0.919360\pi\)
\(252\) 0 0
\(253\) 3.07881i 0.193563i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.02501 −0.438208 −0.219104 0.975701i \(-0.570313\pi\)
−0.219104 + 0.975701i \(0.570313\pi\)
\(258\) 0 0
\(259\) 10.2711i 0.638213i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.4695 1.20054 0.600270 0.799798i \(-0.295060\pi\)
0.600270 + 0.799798i \(0.295060\pi\)
\(264\) 0 0
\(265\) 28.1221 1.72753
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.4258i 0.940526i 0.882526 + 0.470263i \(0.155841\pi\)
−0.882526 + 0.470263i \(0.844159\pi\)
\(270\) 0 0
\(271\) −7.28198 −0.442349 −0.221174 0.975234i \(-0.570989\pi\)
−0.221174 + 0.975234i \(0.570989\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.88244i 0.475329i
\(276\) 0 0
\(277\) 2.43449i 0.146275i 0.997322 + 0.0731373i \(0.0233011\pi\)
−0.997322 + 0.0731373i \(0.976699\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.1555 −1.08307 −0.541534 0.840679i \(-0.682156\pi\)
−0.541534 + 0.840679i \(0.682156\pi\)
\(282\) 0 0
\(283\) − 31.2909i − 1.86005i −0.367496 0.930025i \(-0.619785\pi\)
0.367496 0.930025i \(-0.380215\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.58370 −0.329596
\(288\) 0 0
\(289\) 13.2418 0.778931
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 18.3419i − 1.07155i −0.844362 0.535773i \(-0.820020\pi\)
0.844362 0.535773i \(-0.179980\pi\)
\(294\) 0 0
\(295\) −25.6784 −1.49505
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.65183i 0.0955276i
\(300\) 0 0
\(301\) 3.98332i 0.229595i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −10.4605 −0.598966
\(306\) 0 0
\(307\) 24.5811i 1.40292i 0.712709 + 0.701460i \(0.247468\pi\)
−0.712709 + 0.701460i \(0.752532\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.95251 0.394240 0.197120 0.980379i \(-0.436841\pi\)
0.197120 + 0.980379i \(0.436841\pi\)
\(312\) 0 0
\(313\) −9.81498 −0.554775 −0.277388 0.960758i \(-0.589469\pi\)
−0.277388 + 0.960758i \(0.589469\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.69289i 0.375910i 0.982178 + 0.187955i \(0.0601860\pi\)
−0.982178 + 0.187955i \(0.939814\pi\)
\(318\) 0 0
\(319\) 47.6377 2.66720
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 40.0613i 2.22907i
\(324\) 0 0
\(325\) 4.22905i 0.234586i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.83840 0.266750
\(330\) 0 0
\(331\) 2.10916i 0.115930i 0.998319 + 0.0579651i \(0.0184612\pi\)
−0.998319 + 0.0579651i \(0.981539\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 15.0651 0.823096
\(336\) 0 0
\(337\) −5.53954 −0.301758 −0.150879 0.988552i \(-0.548210\pi\)
−0.150879 + 0.988552i \(0.548210\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 38.4955i 2.08465i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 9.63790i − 0.517389i −0.965959 0.258695i \(-0.916708\pi\)
0.965959 0.258695i \(-0.0832924\pi\)
\(348\) 0 0
\(349\) 8.04565i 0.430674i 0.976540 + 0.215337i \(0.0690850\pi\)
−0.976540 + 0.215337i \(0.930915\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.2775 1.45184 0.725918 0.687782i \(-0.241415\pi\)
0.725918 + 0.687782i \(0.241415\pi\)
\(354\) 0 0
\(355\) − 12.4547i − 0.661025i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.36413 0.388664 0.194332 0.980936i \(-0.437746\pi\)
0.194332 + 0.980936i \(0.437746\pi\)
\(360\) 0 0
\(361\) −34.0692 −1.79312
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 21.8973i − 1.14616i
\(366\) 0 0
\(367\) −21.8746 −1.14185 −0.570923 0.821003i \(-0.693415\pi\)
−0.570923 + 0.821003i \(0.693415\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 11.1320i − 0.577947i
\(372\) 0 0
\(373\) 8.99106i 0.465539i 0.972532 + 0.232770i \(0.0747788\pi\)
−0.972532 + 0.232770i \(0.925221\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.5584 1.31632
\(378\) 0 0
\(379\) 28.9165i 1.48534i 0.669656 + 0.742671i \(0.266442\pi\)
−0.669656 + 0.742671i \(0.733558\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.5404 −0.640786 −0.320393 0.947285i \(-0.603815\pi\)
−0.320393 + 0.947285i \(0.603815\pi\)
\(384\) 0 0
\(385\) 14.4104 0.734423
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 7.50374i − 0.380455i −0.981740 0.190227i \(-0.939077\pi\)
0.981740 0.190227i \(-0.0609225\pi\)
\(390\) 0 0
\(391\) 2.96813 0.150105
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 30.4491i 1.53206i
\(396\) 0 0
\(397\) − 27.6639i − 1.38841i −0.719778 0.694205i \(-0.755756\pi\)
0.719778 0.694205i \(-0.244244\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −35.8414 −1.78983 −0.894916 0.446234i \(-0.852765\pi\)
−0.894916 + 0.446234i \(0.852765\pi\)
\(402\) 0 0
\(403\) 20.6534i 1.02882i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −58.5894 −2.90417
\(408\) 0 0
\(409\) 18.4056 0.910097 0.455048 0.890467i \(-0.349622\pi\)
0.455048 + 0.890467i \(0.349622\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.1647i 0.500173i
\(414\) 0 0
\(415\) 14.7378 0.723452
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.0173724i 0 0.000848699i 1.00000 0.000424349i \(0.000135075\pi\)
−1.00000 0.000424349i \(0.999865\pi\)
\(420\) 0 0
\(421\) 1.03437i 0.0504121i 0.999682 + 0.0252060i \(0.00802418\pi\)
−0.999682 + 0.0252060i \(0.991976\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.59908 0.368609
\(426\) 0 0
\(427\) 4.14075i 0.200385i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.1392 −0.488390 −0.244195 0.969726i \(-0.578524\pi\)
−0.244195 + 0.969726i \(0.578524\pi\)
\(432\) 0 0
\(433\) −33.5806 −1.61378 −0.806890 0.590701i \(-0.798851\pi\)
−0.806890 + 0.590701i \(0.798851\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.93188i 0.188087i
\(438\) 0 0
\(439\) 3.92098 0.187138 0.0935692 0.995613i \(-0.470172\pi\)
0.0935692 + 0.995613i \(0.470172\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 34.6306i 1.64535i 0.568514 + 0.822674i \(0.307519\pi\)
−0.568514 + 0.822674i \(0.692481\pi\)
\(444\) 0 0
\(445\) − 23.4639i − 1.11229i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17.7862 −0.839383 −0.419691 0.907667i \(-0.637862\pi\)
−0.419691 + 0.907667i \(0.637862\pi\)
\(450\) 0 0
\(451\) − 31.8512i − 1.49982i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.73142 0.362454
\(456\) 0 0
\(457\) 11.5852 0.541933 0.270966 0.962589i \(-0.412657\pi\)
0.270966 + 0.962589i \(0.412657\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 12.5948i − 0.586596i −0.956021 0.293298i \(-0.905247\pi\)
0.956021 0.293298i \(-0.0947529\pi\)
\(462\) 0 0
\(463\) 3.16107 0.146907 0.0734537 0.997299i \(-0.476598\pi\)
0.0734537 + 0.997299i \(0.476598\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 20.8974i − 0.967015i −0.875340 0.483507i \(-0.839363\pi\)
0.875340 0.483507i \(-0.160637\pi\)
\(468\) 0 0
\(469\) − 5.96348i − 0.275368i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −22.7221 −1.04476
\(474\) 0 0
\(475\) 10.0665i 0.461882i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −28.2766 −1.29199 −0.645996 0.763341i \(-0.723558\pi\)
−0.645996 + 0.763341i \(0.723558\pi\)
\(480\) 0 0
\(481\) −31.4341 −1.43327
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 31.1988i − 1.41667i
\(486\) 0 0
\(487\) 26.8879 1.21841 0.609203 0.793014i \(-0.291489\pi\)
0.609203 + 0.793014i \(0.291489\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.00142i 0.270840i 0.990788 + 0.135420i \(0.0432384\pi\)
−0.990788 + 0.135420i \(0.956762\pi\)
\(492\) 0 0
\(493\) − 45.9252i − 2.06837i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.93014 −0.221147
\(498\) 0 0
\(499\) 18.3124i 0.819777i 0.912136 + 0.409888i \(0.134432\pi\)
−0.912136 + 0.409888i \(0.865568\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.95874 0.310275 0.155137 0.987893i \(-0.450418\pi\)
0.155137 + 0.987893i \(0.450418\pi\)
\(504\) 0 0
\(505\) 23.5879 1.04965
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 15.2426i − 0.675618i −0.941215 0.337809i \(-0.890314\pi\)
0.941215 0.337809i \(-0.109686\pi\)
\(510\) 0 0
\(511\) −8.66798 −0.383449
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 18.4112i − 0.811295i
\(516\) 0 0
\(517\) 27.5998i 1.21384i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −13.0764 −0.572887 −0.286444 0.958097i \(-0.592473\pi\)
−0.286444 + 0.958097i \(0.592473\pi\)
\(522\) 0 0
\(523\) 23.9123i 1.04561i 0.852452 + 0.522806i \(0.175115\pi\)
−0.852452 + 0.522806i \(0.824885\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 37.1116 1.61661
\(528\) 0 0
\(529\) −22.7087 −0.987334
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 17.0887i − 0.740193i
\(534\) 0 0
\(535\) 18.4713 0.798583
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 5.70432i − 0.245702i
\(540\) 0 0
\(541\) − 13.9717i − 0.600689i −0.953831 0.300345i \(-0.902898\pi\)
0.953831 0.300345i \(-0.0971017\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 40.2504 1.72414
\(546\) 0 0
\(547\) − 21.5760i − 0.922522i −0.887264 0.461261i \(-0.847397\pi\)
0.887264 0.461261i \(-0.152603\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 60.8371 2.59175
\(552\) 0 0
\(553\) 12.0532 0.512553
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 0.679074i − 0.0287733i −0.999897 0.0143866i \(-0.995420\pi\)
0.999897 0.0143866i \(-0.00457957\pi\)
\(558\) 0 0
\(559\) −12.1908 −0.515615
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.8429i 0.667698i 0.942626 + 0.333849i \(0.108348\pi\)
−0.942626 + 0.333849i \(0.891652\pi\)
\(564\) 0 0
\(565\) 17.4678i 0.734877i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.24267 −0.135940 −0.0679698 0.997687i \(-0.521652\pi\)
−0.0679698 + 0.997687i \(0.521652\pi\)
\(570\) 0 0
\(571\) − 1.38251i − 0.0578563i −0.999581 0.0289281i \(-0.990791\pi\)
0.999581 0.0289281i \(-0.00920939\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.745822 0.0311029
\(576\) 0 0
\(577\) −7.57598 −0.315392 −0.157696 0.987488i \(-0.550407\pi\)
−0.157696 + 0.987488i \(0.550407\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 5.83393i − 0.242032i
\(582\) 0 0
\(583\) 63.5007 2.62993
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 5.03970i − 0.208011i −0.994577 0.104005i \(-0.966834\pi\)
0.994577 0.104005i \(-0.0331659\pi\)
\(588\) 0 0
\(589\) 49.1618i 2.02568i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.94121 0.326106 0.163053 0.986617i \(-0.447866\pi\)
0.163053 + 0.986617i \(0.447866\pi\)
\(594\) 0 0
\(595\) − 13.8924i − 0.569532i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −33.0131 −1.34888 −0.674439 0.738331i \(-0.735614\pi\)
−0.674439 + 0.738331i \(0.735614\pi\)
\(600\) 0 0
\(601\) 40.4038 1.64810 0.824052 0.566513i \(-0.191708\pi\)
0.824052 + 0.566513i \(0.191708\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 54.4131i 2.21221i
\(606\) 0 0
\(607\) −0.544975 −0.0221199 −0.0110599 0.999939i \(-0.503521\pi\)
−0.0110599 + 0.999939i \(0.503521\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 14.8077i 0.599056i
\(612\) 0 0
\(613\) 2.75425i 0.111243i 0.998452 + 0.0556215i \(0.0177140\pi\)
−0.998452 + 0.0556215i \(0.982286\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −32.6720 −1.31533 −0.657664 0.753312i \(-0.728455\pi\)
−0.657664 + 0.753312i \(0.728455\pi\)
\(618\) 0 0
\(619\) 10.4789i 0.421183i 0.977574 + 0.210591i \(0.0675389\pi\)
−0.977574 + 0.210591i \(0.932461\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.28810 −0.372120
\(624\) 0 0
\(625\) −29.9997 −1.19999
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 56.4832i 2.25213i
\(630\) 0 0
\(631\) 7.94128 0.316138 0.158069 0.987428i \(-0.449473\pi\)
0.158069 + 0.987428i \(0.449473\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 44.4878i − 1.76544i
\(636\) 0 0
\(637\) − 3.06046i − 0.121260i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −17.5222 −0.692084 −0.346042 0.938219i \(-0.612475\pi\)
−0.346042 + 0.938219i \(0.612475\pi\)
\(642\) 0 0
\(643\) − 34.7998i − 1.37237i −0.727426 0.686186i \(-0.759284\pi\)
0.727426 0.686186i \(-0.240716\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.4054 0.959474 0.479737 0.877412i \(-0.340732\pi\)
0.479737 + 0.877412i \(0.340732\pi\)
\(648\) 0 0
\(649\) −57.9827 −2.27602
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 20.9374i − 0.819345i −0.912233 0.409673i \(-0.865643\pi\)
0.912233 0.409673i \(-0.134357\pi\)
\(654\) 0 0
\(655\) 27.1698 1.06161
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 27.6061i − 1.07538i −0.843143 0.537690i \(-0.819297\pi\)
0.843143 0.537690i \(-0.180703\pi\)
\(660\) 0 0
\(661\) − 31.2679i − 1.21618i −0.793868 0.608090i \(-0.791936\pi\)
0.793868 0.608090i \(-0.208064\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 18.4032 0.713647
\(666\) 0 0
\(667\) − 4.50739i − 0.174527i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −23.6201 −0.911846
\(672\) 0 0
\(673\) 18.7023 0.720923 0.360461 0.932774i \(-0.382619\pi\)
0.360461 + 0.932774i \(0.382619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.58757i 0.0610152i 0.999535 + 0.0305076i \(0.00971238\pi\)
−0.999535 + 0.0305076i \(0.990288\pi\)
\(678\) 0 0
\(679\) −12.3500 −0.473948
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.1798i 1.30785i 0.756557 + 0.653927i \(0.226880\pi\)
−0.756557 + 0.653927i \(0.773120\pi\)
\(684\) 0 0
\(685\) 3.95135i 0.150974i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 34.0691 1.29793
\(690\) 0 0
\(691\) − 39.5718i − 1.50538i −0.658373 0.752692i \(-0.728755\pi\)
0.658373 0.752692i \(-0.271245\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −28.4778 −1.08022
\(696\) 0 0
\(697\) −30.7062 −1.16308
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.2500i 1.14253i 0.820767 + 0.571263i \(0.193546\pi\)
−0.820767 + 0.571263i \(0.806454\pi\)
\(702\) 0 0
\(703\) −74.8233 −2.82202
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 9.33719i − 0.351161i
\(708\) 0 0
\(709\) − 51.5735i − 1.93688i −0.249244 0.968441i \(-0.580182\pi\)
0.249244 0.968441i \(-0.419818\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.64237 0.136408
\(714\) 0 0
\(715\) 44.1025i 1.64934i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 11.2028 0.417795 0.208898 0.977938i \(-0.433012\pi\)
0.208898 + 0.977938i \(0.433012\pi\)
\(720\) 0 0
\(721\) −7.28802 −0.271420
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 11.5399i − 0.428583i
\(726\) 0 0
\(727\) 0.814724 0.0302164 0.0151082 0.999886i \(-0.495191\pi\)
0.0151082 + 0.999886i \(0.495191\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 21.9053i 0.810197i
\(732\) 0 0
\(733\) 12.7283i 0.470131i 0.971980 + 0.235066i \(0.0755305\pi\)
−0.971980 + 0.235066i \(0.924470\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 34.0176 1.25305
\(738\) 0 0
\(739\) − 9.13395i − 0.335998i −0.985787 0.167999i \(-0.946269\pi\)
0.985787 0.167999i \(-0.0537305\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.4112 0.565383 0.282692 0.959211i \(-0.408773\pi\)
0.282692 + 0.959211i \(0.408773\pi\)
\(744\) 0 0
\(745\) −16.3889 −0.600442
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 7.31179i − 0.267167i
\(750\) 0 0
\(751\) −40.7526 −1.48708 −0.743542 0.668690i \(-0.766856\pi\)
−0.743542 + 0.668690i \(0.766856\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 0.968366i − 0.0352424i
\(756\) 0 0
\(757\) − 26.0085i − 0.945296i −0.881251 0.472648i \(-0.843298\pi\)
0.881251 0.472648i \(-0.156702\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −10.3686 −0.375861 −0.187931 0.982182i \(-0.560178\pi\)
−0.187931 + 0.982182i \(0.560178\pi\)
\(762\) 0 0
\(763\) − 15.9330i − 0.576814i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −31.1086 −1.12327
\(768\) 0 0
\(769\) −31.7525 −1.14502 −0.572512 0.819896i \(-0.694031\pi\)
−0.572512 + 0.819896i \(0.694031\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16.3614i 0.588478i 0.955732 + 0.294239i \(0.0950662\pi\)
−0.955732 + 0.294239i \(0.904934\pi\)
\(774\) 0 0
\(775\) 9.32530 0.334975
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 40.6765i − 1.45739i
\(780\) 0 0
\(781\) − 28.1231i − 1.00632i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 21.8938 0.781422
\(786\) 0 0
\(787\) − 25.4664i − 0.907781i −0.891058 0.453890i \(-0.850036\pi\)
0.891058 0.453890i \(-0.149964\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.91459 0.245854
\(792\) 0 0
\(793\) −12.6726 −0.450016
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 27.4312i − 0.971664i −0.874052 0.485832i \(-0.838517\pi\)
0.874052 0.485832i \(-0.161483\pi\)
\(798\) 0 0
\(799\) 26.6076 0.941310
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 49.4449i − 1.74487i
\(804\) 0 0
\(805\) − 1.36349i − 0.0480566i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 23.8507 0.838546 0.419273 0.907860i \(-0.362285\pi\)
0.419273 + 0.907860i \(0.362285\pi\)
\(810\) 0 0
\(811\) 27.8123i 0.976622i 0.872670 + 0.488311i \(0.162387\pi\)
−0.872670 + 0.488311i \(0.837613\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −63.3506 −2.21907
\(816\) 0 0
\(817\) −29.0179 −1.01521
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 53.9215i − 1.88187i −0.338583 0.940936i \(-0.609948\pi\)
0.338583 0.940936i \(-0.390052\pi\)
\(822\) 0 0
\(823\) 11.4932 0.400628 0.200314 0.979732i \(-0.435804\pi\)
0.200314 + 0.979732i \(0.435804\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.946387i 0.0329091i 0.999865 + 0.0164546i \(0.00523788\pi\)
−0.999865 + 0.0164546i \(0.994762\pi\)
\(828\) 0 0
\(829\) 42.9151i 1.49050i 0.666783 + 0.745252i \(0.267671\pi\)
−0.666783 + 0.745252i \(0.732329\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.49926 −0.190538
\(834\) 0 0
\(835\) 22.1149i 0.765318i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 31.2098 1.07748 0.538741 0.842471i \(-0.318900\pi\)
0.538741 + 0.842471i \(0.318900\pi\)
\(840\) 0 0
\(841\) −40.7419 −1.40489
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 9.17933i − 0.315779i
\(846\) 0 0
\(847\) 21.5393 0.740097
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5.54363i 0.190033i
\(852\) 0 0
\(853\) − 8.84389i − 0.302809i −0.988472 0.151404i \(-0.951620\pi\)
0.988472 0.151404i \(-0.0483796\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −49.1414 −1.67864 −0.839319 0.543640i \(-0.817046\pi\)
−0.839319 + 0.543640i \(0.817046\pi\)
\(858\) 0 0
\(859\) − 43.8201i − 1.49512i −0.664193 0.747561i \(-0.731225\pi\)
0.664193 0.747561i \(-0.268775\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31.5811 1.07503 0.537516 0.843253i \(-0.319363\pi\)
0.537516 + 0.843253i \(0.319363\pi\)
\(864\) 0 0
\(865\) −23.4879 −0.798612
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 68.7552i 2.33236i
\(870\) 0 0
\(871\) 18.2510 0.618411
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.14031i 0.308999i
\(876\) 0 0
\(877\) 33.7451i 1.13949i 0.821821 + 0.569746i \(0.192959\pi\)
−0.821821 + 0.569746i \(0.807041\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8.58523 0.289244 0.144622 0.989487i \(-0.453803\pi\)
0.144622 + 0.989487i \(0.453803\pi\)
\(882\) 0 0
\(883\) 8.07348i 0.271694i 0.990730 + 0.135847i \(0.0433756\pi\)
−0.990730 + 0.135847i \(0.956624\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 31.4397 1.05564 0.527821 0.849356i \(-0.323009\pi\)
0.527821 + 0.849356i \(0.323009\pi\)
\(888\) 0 0
\(889\) −17.6103 −0.590632
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 35.2471i 1.17950i
\(894\) 0 0
\(895\) 19.3382 0.646404
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 56.3577i − 1.87963i
\(900\) 0 0
\(901\) − 61.2180i − 2.03947i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.32628 −0.210293
\(906\) 0 0
\(907\) 10.4183i 0.345932i 0.984928 + 0.172966i \(0.0553352\pi\)
−0.984928 + 0.172966i \(0.944665\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −47.9123 −1.58741 −0.793703 0.608305i \(-0.791850\pi\)
−0.793703 + 0.608305i \(0.791850\pi\)
\(912\) 0 0
\(913\) 33.2786 1.10136
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 10.7551i − 0.355164i
\(918\) 0 0
\(919\) 37.6472 1.24187 0.620933 0.783863i \(-0.286754\pi\)
0.620933 + 0.783863i \(0.286754\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 15.0885i − 0.496644i
\(924\) 0 0
\(925\) 14.1929i 0.466661i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 44.8135 1.47028 0.735141 0.677914i \(-0.237116\pi\)
0.735141 + 0.677914i \(0.237116\pi\)
\(930\) 0 0
\(931\) − 7.28486i − 0.238752i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 79.2466 2.59164
\(936\) 0 0
\(937\) 29.2431 0.955330 0.477665 0.878542i \(-0.341483\pi\)
0.477665 + 0.878542i \(0.341483\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 11.7936i − 0.384460i −0.981350 0.192230i \(-0.938428\pi\)
0.981350 0.192230i \(-0.0615719\pi\)
\(942\) 0 0
\(943\) −3.01371 −0.0981398
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.5710i 0.473493i 0.971572 + 0.236746i \(0.0760810\pi\)
−0.971572 + 0.236746i \(0.923919\pi\)
\(948\) 0 0
\(949\) − 26.5280i − 0.861134i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −54.0151 −1.74972 −0.874860 0.484375i \(-0.839047\pi\)
−0.874860 + 0.484375i \(0.839047\pi\)
\(954\) 0 0
\(955\) 8.09977i 0.262102i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.56413 0.0505084
\(960\) 0 0
\(961\) 14.5420 0.469097
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 27.6735i − 0.890840i
\(966\) 0 0
\(967\) −39.1750 −1.25978 −0.629892 0.776683i \(-0.716901\pi\)
−0.629892 + 0.776683i \(0.716901\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 2.39896i − 0.0769864i −0.999259 0.0384932i \(-0.987744\pi\)
0.999259 0.0384932i \(-0.0122558\pi\)
\(972\) 0 0
\(973\) 11.2728i 0.361390i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.7447 0.503717 0.251858 0.967764i \(-0.418958\pi\)
0.251858 + 0.967764i \(0.418958\pi\)
\(978\) 0 0
\(979\) − 52.9823i − 1.69332i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.5566 0.464285 0.232142 0.972682i \(-0.425426\pi\)
0.232142 + 0.972682i \(0.425426\pi\)
\(984\) 0 0
\(985\) 19.9004 0.634078
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.14993i 0.0683637i
\(990\) 0 0
\(991\) 23.2853 0.739681 0.369841 0.929095i \(-0.379412\pi\)
0.369841 + 0.929095i \(0.379412\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.54525i 0.0806897i
\(996\) 0 0
\(997\) − 1.38107i − 0.0437390i −0.999761 0.0218695i \(-0.993038\pi\)
0.999761 0.0218695i \(-0.00696184\pi\)
\(998\) 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.g.3025.5 24
3.2 odd 2 inner 6048.2.c.g.3025.19 24
4.3 odd 2 1512.2.c.f.757.8 yes 24
8.3 odd 2 1512.2.c.f.757.7 24
8.5 even 2 inner 6048.2.c.g.3025.20 24
12.11 even 2 1512.2.c.f.757.17 yes 24
24.5 odd 2 inner 6048.2.c.g.3025.6 24
24.11 even 2 1512.2.c.f.757.18 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.f.757.7 24 8.3 odd 2
1512.2.c.f.757.8 yes 24 4.3 odd 2
1512.2.c.f.757.17 yes 24 12.11 even 2
1512.2.c.f.757.18 yes 24 24.11 even 2
6048.2.c.g.3025.5 24 1.1 even 1 trivial
6048.2.c.g.3025.6 24 24.5 odd 2 inner
6048.2.c.g.3025.19 24 3.2 odd 2 inner
6048.2.c.g.3025.20 24 8.5 even 2 inner