Properties

Label 6048.2.c.f.3025.3
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 3025.3
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.f.3025.22

$q$-expansion

\(f(q)\) \(=\) \(q-3.04340i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-3.04340i q^{5} -1.00000 q^{7} +0.128573i q^{11} -6.30135i q^{13} -5.32168 q^{17} -6.68215i q^{19} +5.18212 q^{23} -4.26229 q^{25} -9.96821i q^{29} -3.27122 q^{31} +3.04340i q^{35} +0.796970i q^{37} +2.96782 q^{41} +6.99100i q^{43} +4.76595 q^{47} +1.00000 q^{49} +1.14264i q^{53} +0.391299 q^{55} -11.0999i q^{59} +14.6662i q^{61} -19.1775 q^{65} -1.05725i q^{67} +1.10582 q^{71} -12.1019 q^{73} -0.128573i q^{77} -2.62426 q^{79} +6.46403i q^{83} +16.1960i q^{85} +2.23283 q^{89} +6.30135i q^{91} -20.3365 q^{95} -7.88097 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} - 8q^{31} + 24q^{49} - 16q^{55} - 8q^{79} + O(q^{100}) \)

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
\(5\) − 3.04340i − 1.36105i −0.732725 0.680525i \(-0.761752\pi\)
0.732725 0.680525i \(-0.238248\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.128573i 0.0387662i 0.999812 + 0.0193831i \(0.00617022\pi\)
−0.999812 + 0.0193831i \(0.993830\pi\)
\(12\) 0 0
\(13\) − 6.30135i − 1.74768i −0.486214 0.873840i \(-0.661622\pi\)
0.486214 0.873840i \(-0.338378\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.32168 −1.29070 −0.645348 0.763889i \(-0.723288\pi\)
−0.645348 + 0.763889i \(0.723288\pi\)
\(18\) 0 0
\(19\) − 6.68215i − 1.53299i −0.642250 0.766495i \(-0.721999\pi\)
0.642250 0.766495i \(-0.278001\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.18212 1.08055 0.540273 0.841490i \(-0.318321\pi\)
0.540273 + 0.841490i \(0.318321\pi\)
\(24\) 0 0
\(25\) −4.26229 −0.852459
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 9.96821i − 1.85105i −0.378686 0.925525i \(-0.623624\pi\)
0.378686 0.925525i \(-0.376376\pi\)
\(30\) 0 0
\(31\) −3.27122 −0.587528 −0.293764 0.955878i \(-0.594908\pi\)
−0.293764 + 0.955878i \(0.594908\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.04340i 0.514429i
\(36\) 0 0
\(37\) 0.796970i 0.131021i 0.997852 + 0.0655106i \(0.0208676\pi\)
−0.997852 + 0.0655106i \(0.979132\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.96782 0.463496 0.231748 0.972776i \(-0.425556\pi\)
0.231748 + 0.972776i \(0.425556\pi\)
\(42\) 0 0
\(43\) 6.99100i 1.06612i 0.846078 + 0.533059i \(0.178958\pi\)
−0.846078 + 0.533059i \(0.821042\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.76595 0.695186 0.347593 0.937646i \(-0.386999\pi\)
0.347593 + 0.937646i \(0.386999\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.14264i 0.156954i 0.996916 + 0.0784772i \(0.0250058\pi\)
−0.996916 + 0.0784772i \(0.974994\pi\)
\(54\) 0 0
\(55\) 0.391299 0.0527627
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 11.0999i − 1.44508i −0.691329 0.722540i \(-0.742975\pi\)
0.691329 0.722540i \(-0.257025\pi\)
\(60\) 0 0
\(61\) 14.6662i 1.87782i 0.344165 + 0.938909i \(0.388162\pi\)
−0.344165 + 0.938909i \(0.611838\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −19.1775 −2.37868
\(66\) 0 0
\(67\) − 1.05725i − 0.129163i −0.997912 0.0645816i \(-0.979429\pi\)
0.997912 0.0645816i \(-0.0205713\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.10582 0.131237 0.0656185 0.997845i \(-0.479098\pi\)
0.0656185 + 0.997845i \(0.479098\pi\)
\(72\) 0 0
\(73\) −12.1019 −1.41642 −0.708212 0.706000i \(-0.750498\pi\)
−0.708212 + 0.706000i \(0.750498\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.128573i − 0.0146522i
\(78\) 0 0
\(79\) −2.62426 −0.295253 −0.147626 0.989043i \(-0.547163\pi\)
−0.147626 + 0.989043i \(0.547163\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.46403i 0.709520i 0.934957 + 0.354760i \(0.115437\pi\)
−0.934957 + 0.354760i \(0.884563\pi\)
\(84\) 0 0
\(85\) 16.1960i 1.75670i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.23283 0.236680 0.118340 0.992973i \(-0.462243\pi\)
0.118340 + 0.992973i \(0.462243\pi\)
\(90\) 0 0
\(91\) 6.30135i 0.660561i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −20.3365 −2.08648
\(96\) 0 0
\(97\) −7.88097 −0.800192 −0.400096 0.916473i \(-0.631023\pi\)
−0.400096 + 0.916473i \(0.631023\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 16.9693i − 1.68851i −0.535945 0.844253i \(-0.680045\pi\)
0.535945 0.844253i \(-0.319955\pi\)
\(102\) 0 0
\(103\) 10.7049 1.05479 0.527393 0.849621i \(-0.323170\pi\)
0.527393 + 0.849621i \(0.323170\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.31462i 0.127089i 0.997979 + 0.0635445i \(0.0202405\pi\)
−0.997979 + 0.0635445i \(0.979760\pi\)
\(108\) 0 0
\(109\) 10.2677i 0.983465i 0.870746 + 0.491732i \(0.163636\pi\)
−0.870746 + 0.491732i \(0.836364\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.1796 1.23983 0.619916 0.784668i \(-0.287166\pi\)
0.619916 + 0.784668i \(0.287166\pi\)
\(114\) 0 0
\(115\) − 15.7713i − 1.47068i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.32168 0.487837
\(120\) 0 0
\(121\) 10.9835 0.998497
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 2.24514i − 0.200811i
\(126\) 0 0
\(127\) 0.840201 0.0745558 0.0372779 0.999305i \(-0.488131\pi\)
0.0372779 + 0.999305i \(0.488131\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.39979i 0.646523i 0.946310 + 0.323261i \(0.104779\pi\)
−0.946310 + 0.323261i \(0.895221\pi\)
\(132\) 0 0
\(133\) 6.68215i 0.579416i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.61682 −0.479877 −0.239939 0.970788i \(-0.577127\pi\)
−0.239939 + 0.970788i \(0.577127\pi\)
\(138\) 0 0
\(139\) − 5.19553i − 0.440679i −0.975423 0.220339i \(-0.929284\pi\)
0.975423 0.220339i \(-0.0707165\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.810183 0.0677509
\(144\) 0 0
\(145\) −30.3373 −2.51937
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 11.6207i − 0.952009i −0.879443 0.476004i \(-0.842085\pi\)
0.879443 0.476004i \(-0.157915\pi\)
\(150\) 0 0
\(151\) −21.5154 −1.75090 −0.875448 0.483313i \(-0.839433\pi\)
−0.875448 + 0.483313i \(0.839433\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.95563i 0.799655i
\(156\) 0 0
\(157\) − 18.3001i − 1.46050i −0.683178 0.730252i \(-0.739403\pi\)
0.683178 0.730252i \(-0.260597\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.18212 −0.408408
\(162\) 0 0
\(163\) 18.8550i 1.47684i 0.674344 + 0.738418i \(0.264427\pi\)
−0.674344 + 0.738418i \(0.735573\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.8283 1.61174 0.805872 0.592089i \(-0.201697\pi\)
0.805872 + 0.592089i \(0.201697\pi\)
\(168\) 0 0
\(169\) −26.7070 −2.05438
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 5.49174i − 0.417529i −0.977966 0.208765i \(-0.933056\pi\)
0.977966 0.208765i \(-0.0669443\pi\)
\(174\) 0 0
\(175\) 4.26229 0.322199
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.62941i 0.570249i 0.958491 + 0.285124i \(0.0920349\pi\)
−0.958491 + 0.285124i \(0.907965\pi\)
\(180\) 0 0
\(181\) − 10.0176i − 0.744600i −0.928113 0.372300i \(-0.878569\pi\)
0.928113 0.372300i \(-0.121431\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.42550 0.178326
\(186\) 0 0
\(187\) − 0.684223i − 0.0500354i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.25601 −0.452669 −0.226335 0.974050i \(-0.572674\pi\)
−0.226335 + 0.974050i \(0.572674\pi\)
\(192\) 0 0
\(193\) 2.46446 0.177396 0.0886980 0.996059i \(-0.471729\pi\)
0.0886980 + 0.996059i \(0.471729\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.8105i 1.69643i 0.529654 + 0.848214i \(0.322322\pi\)
−0.529654 + 0.848214i \(0.677678\pi\)
\(198\) 0 0
\(199\) −24.1296 −1.71050 −0.855252 0.518212i \(-0.826598\pi\)
−0.855252 + 0.518212i \(0.826598\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.96821i 0.699631i
\(204\) 0 0
\(205\) − 9.03227i − 0.630841i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.859144 0.0594282
\(210\) 0 0
\(211\) 14.9668i 1.03036i 0.857083 + 0.515179i \(0.172274\pi\)
−0.857083 + 0.515179i \(0.827726\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 21.2764 1.45104
\(216\) 0 0
\(217\) 3.27122 0.222065
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 33.5337i 2.25572i
\(222\) 0 0
\(223\) −11.6108 −0.777518 −0.388759 0.921340i \(-0.627096\pi\)
−0.388759 + 0.921340i \(0.627096\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.939445i 0.0623531i 0.999514 + 0.0311766i \(0.00992542\pi\)
−0.999514 + 0.0311766i \(0.990075\pi\)
\(228\) 0 0
\(229\) − 0.0137124i 0 0.000906143i −1.00000 0.000453071i \(-0.999856\pi\)
1.00000 0.000453071i \(-0.000144217\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.2195 1.19360 0.596800 0.802390i \(-0.296439\pi\)
0.596800 + 0.802390i \(0.296439\pi\)
\(234\) 0 0
\(235\) − 14.5047i − 0.946183i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.37153 0.0887169 0.0443585 0.999016i \(-0.485876\pi\)
0.0443585 + 0.999016i \(0.485876\pi\)
\(240\) 0 0
\(241\) −10.5054 −0.676711 −0.338355 0.941018i \(-0.609871\pi\)
−0.338355 + 0.941018i \(0.609871\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 3.04340i − 0.194436i
\(246\) 0 0
\(247\) −42.1066 −2.67918
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0556i 1.26590i 0.774192 + 0.632950i \(0.218156\pi\)
−0.774192 + 0.632950i \(0.781844\pi\)
\(252\) 0 0
\(253\) 0.666280i 0.0418887i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.6631 −1.53844 −0.769220 0.638984i \(-0.779355\pi\)
−0.769220 + 0.638984i \(0.779355\pi\)
\(258\) 0 0
\(259\) − 0.796970i − 0.0495213i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.0542 0.866616 0.433308 0.901246i \(-0.357346\pi\)
0.433308 + 0.901246i \(0.357346\pi\)
\(264\) 0 0
\(265\) 3.47753 0.213623
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 13.1276i − 0.800404i −0.916427 0.400202i \(-0.868940\pi\)
0.916427 0.400202i \(-0.131060\pi\)
\(270\) 0 0
\(271\) 15.4626 0.939285 0.469642 0.882857i \(-0.344383\pi\)
0.469642 + 0.882857i \(0.344383\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 0.548015i − 0.0330466i
\(276\) 0 0
\(277\) − 11.1880i − 0.672223i −0.941822 0.336112i \(-0.890888\pi\)
0.941822 0.336112i \(-0.109112\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.179766 0.0107240 0.00536198 0.999986i \(-0.498293\pi\)
0.00536198 + 0.999986i \(0.498293\pi\)
\(282\) 0 0
\(283\) 29.0480i 1.72673i 0.504583 + 0.863363i \(0.331646\pi\)
−0.504583 + 0.863363i \(0.668354\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.96782 −0.175185
\(288\) 0 0
\(289\) 11.3202 0.665897
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 4.42498i − 0.258510i −0.991611 0.129255i \(-0.958741\pi\)
0.991611 0.129255i \(-0.0412586\pi\)
\(294\) 0 0
\(295\) −33.7813 −1.96683
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 32.6543i − 1.88845i
\(300\) 0 0
\(301\) − 6.99100i − 0.402955i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 44.6352 2.55581
\(306\) 0 0
\(307\) − 8.36831i − 0.477605i −0.971068 0.238802i \(-0.923245\pi\)
0.971068 0.238802i \(-0.0767548\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −28.1701 −1.59738 −0.798690 0.601743i \(-0.794473\pi\)
−0.798690 + 0.601743i \(0.794473\pi\)
\(312\) 0 0
\(313\) 14.0081 0.791784 0.395892 0.918297i \(-0.370435\pi\)
0.395892 + 0.918297i \(0.370435\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 14.4390i − 0.810974i −0.914101 0.405487i \(-0.867102\pi\)
0.914101 0.405487i \(-0.132898\pi\)
\(318\) 0 0
\(319\) 1.28164 0.0717582
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 35.5603i 1.97863i
\(324\) 0 0
\(325\) 26.8582i 1.48982i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.76595 −0.262756
\(330\) 0 0
\(331\) 32.5738i 1.79042i 0.445648 + 0.895208i \(0.352973\pi\)
−0.445648 + 0.895208i \(0.647027\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.21762 −0.175798
\(336\) 0 0
\(337\) −2.17181 −0.118306 −0.0591529 0.998249i \(-0.518840\pi\)
−0.0591529 + 0.998249i \(0.518840\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 0.420590i − 0.0227762i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.79477i − 0.150031i −0.997182 0.0750157i \(-0.976099\pi\)
0.997182 0.0750157i \(-0.0239007\pi\)
\(348\) 0 0
\(349\) − 3.19085i − 0.170802i −0.996347 0.0854011i \(-0.972783\pi\)
0.996347 0.0854011i \(-0.0272172\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.10245 −0.431250 −0.215625 0.976476i \(-0.569179\pi\)
−0.215625 + 0.976476i \(0.569179\pi\)
\(354\) 0 0
\(355\) − 3.36546i − 0.178620i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.2635 −0.911134 −0.455567 0.890202i \(-0.650563\pi\)
−0.455567 + 0.890202i \(0.650563\pi\)
\(360\) 0 0
\(361\) −25.6511 −1.35006
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 36.8310i 1.92782i
\(366\) 0 0
\(367\) 13.5508 0.707348 0.353674 0.935369i \(-0.384932\pi\)
0.353674 + 0.935369i \(0.384932\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 1.14264i − 0.0593232i
\(372\) 0 0
\(373\) − 16.3399i − 0.846048i −0.906118 0.423024i \(-0.860969\pi\)
0.906118 0.423024i \(-0.139031\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −62.8132 −3.23504
\(378\) 0 0
\(379\) − 2.37754i − 0.122126i −0.998134 0.0610630i \(-0.980551\pi\)
0.998134 0.0610630i \(-0.0194491\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.6210 −0.900389 −0.450194 0.892931i \(-0.648645\pi\)
−0.450194 + 0.892931i \(0.648645\pi\)
\(384\) 0 0
\(385\) −0.391299 −0.0199424
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.6831i 0.643058i 0.946900 + 0.321529i \(0.104197\pi\)
−0.946900 + 0.321529i \(0.895803\pi\)
\(390\) 0 0
\(391\) −27.5776 −1.39466
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.98668i 0.401854i
\(396\) 0 0
\(397\) 34.3856i 1.72576i 0.505405 + 0.862882i \(0.331343\pi\)
−0.505405 + 0.862882i \(0.668657\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.75947 0.187739 0.0938696 0.995585i \(-0.470076\pi\)
0.0938696 + 0.995585i \(0.470076\pi\)
\(402\) 0 0
\(403\) 20.6131i 1.02681i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.102469 −0.00507919
\(408\) 0 0
\(409\) 13.0996 0.647731 0.323866 0.946103i \(-0.395017\pi\)
0.323866 + 0.946103i \(0.395017\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.0999i 0.546189i
\(414\) 0 0
\(415\) 19.6726 0.965692
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.4344i 1.19370i 0.802353 + 0.596849i \(0.203581\pi\)
−0.802353 + 0.596849i \(0.796419\pi\)
\(420\) 0 0
\(421\) − 26.3907i − 1.28620i −0.765781 0.643101i \(-0.777647\pi\)
0.765781 0.643101i \(-0.222353\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.6825 1.10027
\(426\) 0 0
\(427\) − 14.6662i − 0.709749i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 32.7161 1.57588 0.787940 0.615751i \(-0.211147\pi\)
0.787940 + 0.615751i \(0.211147\pi\)
\(432\) 0 0
\(433\) 12.3235 0.592228 0.296114 0.955153i \(-0.404309\pi\)
0.296114 + 0.955153i \(0.404309\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 34.6277i − 1.65647i
\(438\) 0 0
\(439\) −4.67878 −0.223306 −0.111653 0.993747i \(-0.535615\pi\)
−0.111653 + 0.993747i \(0.535615\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 35.6471i − 1.69364i −0.531877 0.846821i \(-0.678513\pi\)
0.531877 0.846821i \(-0.321487\pi\)
\(444\) 0 0
\(445\) − 6.79540i − 0.322133i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.02786 −0.284472 −0.142236 0.989833i \(-0.545429\pi\)
−0.142236 + 0.989833i \(0.545429\pi\)
\(450\) 0 0
\(451\) 0.381581i 0.0179680i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 19.1775 0.899057
\(456\) 0 0
\(457\) 4.88761 0.228633 0.114316 0.993444i \(-0.463532\pi\)
0.114316 + 0.993444i \(0.463532\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 9.16306i − 0.426766i −0.976969 0.213383i \(-0.931552\pi\)
0.976969 0.213383i \(-0.0684483\pi\)
\(462\) 0 0
\(463\) −32.8412 −1.52626 −0.763130 0.646245i \(-0.776338\pi\)
−0.763130 + 0.646245i \(0.776338\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 6.40559i − 0.296415i −0.988956 0.148208i \(-0.952650\pi\)
0.988956 0.148208i \(-0.0473504\pi\)
\(468\) 0 0
\(469\) 1.05725i 0.0488191i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.898854 −0.0413293
\(474\) 0 0
\(475\) 28.4813i 1.30681i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.3218 1.24837 0.624184 0.781278i \(-0.285432\pi\)
0.624184 + 0.781278i \(0.285432\pi\)
\(480\) 0 0
\(481\) 5.02199 0.228983
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 23.9850i 1.08910i
\(486\) 0 0
\(487\) −9.05264 −0.410214 −0.205107 0.978740i \(-0.565754\pi\)
−0.205107 + 0.978740i \(0.565754\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 10.1183i − 0.456632i −0.973587 0.228316i \(-0.926678\pi\)
0.973587 0.228316i \(-0.0733219\pi\)
\(492\) 0 0
\(493\) 53.0476i 2.38914i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.10582 −0.0496029
\(498\) 0 0
\(499\) 19.9396i 0.892621i 0.894878 + 0.446310i \(0.147262\pi\)
−0.894878 + 0.446310i \(0.852738\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 24.3936 1.08766 0.543828 0.839197i \(-0.316974\pi\)
0.543828 + 0.839197i \(0.316974\pi\)
\(504\) 0 0
\(505\) −51.6443 −2.29814
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.20394i 0.319309i 0.987173 + 0.159655i \(0.0510381\pi\)
−0.987173 + 0.159655i \(0.948962\pi\)
\(510\) 0 0
\(511\) 12.1019 0.535358
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 32.5793i − 1.43562i
\(516\) 0 0
\(517\) 0.612773i 0.0269497i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.8648 −0.651240 −0.325620 0.945501i \(-0.605573\pi\)
−0.325620 + 0.945501i \(0.605573\pi\)
\(522\) 0 0
\(523\) − 32.1286i − 1.40489i −0.711740 0.702443i \(-0.752093\pi\)
0.711740 0.702443i \(-0.247907\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.4084 0.758320
\(528\) 0 0
\(529\) 3.85438 0.167582
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 18.7013i − 0.810042i
\(534\) 0 0
\(535\) 4.00091 0.172975
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.128573i 0.00553803i
\(540\) 0 0
\(541\) − 30.6123i − 1.31612i −0.752964 0.658062i \(-0.771376\pi\)
0.752964 0.658062i \(-0.228624\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 31.2487 1.33855
\(546\) 0 0
\(547\) − 7.64518i − 0.326884i −0.986553 0.163442i \(-0.947740\pi\)
0.986553 0.163442i \(-0.0522597\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −66.6091 −2.83764
\(552\) 0 0
\(553\) 2.62426 0.111595
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.9682i 1.05794i 0.848641 + 0.528969i \(0.177421\pi\)
−0.848641 + 0.528969i \(0.822579\pi\)
\(558\) 0 0
\(559\) 44.0527 1.86323
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 33.1836i − 1.39852i −0.714865 0.699262i \(-0.753512\pi\)
0.714865 0.699262i \(-0.246488\pi\)
\(564\) 0 0
\(565\) − 40.1108i − 1.68748i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.0679 0.715523 0.357761 0.933813i \(-0.383540\pi\)
0.357761 + 0.933813i \(0.383540\pi\)
\(570\) 0 0
\(571\) − 5.13058i − 0.214708i −0.994221 0.107354i \(-0.965762\pi\)
0.994221 0.107354i \(-0.0342378\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −22.0877 −0.921121
\(576\) 0 0
\(577\) −47.5819 −1.98086 −0.990429 0.138021i \(-0.955926\pi\)
−0.990429 + 0.138021i \(0.955926\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 6.46403i − 0.268173i
\(582\) 0 0
\(583\) −0.146913 −0.00608452
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 23.6803i − 0.977390i −0.872455 0.488695i \(-0.837473\pi\)
0.872455 0.488695i \(-0.162527\pi\)
\(588\) 0 0
\(589\) 21.8588i 0.900675i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.8919 0.734734 0.367367 0.930076i \(-0.380259\pi\)
0.367367 + 0.930076i \(0.380259\pi\)
\(594\) 0 0
\(595\) − 16.1960i − 0.663971i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −32.0167 −1.30817 −0.654083 0.756423i \(-0.726945\pi\)
−0.654083 + 0.756423i \(0.726945\pi\)
\(600\) 0 0
\(601\) 23.7187 0.967507 0.483754 0.875204i \(-0.339273\pi\)
0.483754 + 0.875204i \(0.339273\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 33.4271i − 1.35901i
\(606\) 0 0
\(607\) 14.4444 0.586281 0.293141 0.956069i \(-0.405300\pi\)
0.293141 + 0.956069i \(0.405300\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 30.0319i − 1.21496i
\(612\) 0 0
\(613\) 16.2817i 0.657610i 0.944398 + 0.328805i \(0.106646\pi\)
−0.944398 + 0.328805i \(0.893354\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.7753 1.15845 0.579225 0.815167i \(-0.303355\pi\)
0.579225 + 0.815167i \(0.303355\pi\)
\(618\) 0 0
\(619\) − 36.0871i − 1.45046i −0.688506 0.725231i \(-0.741733\pi\)
0.688506 0.725231i \(-0.258267\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.23283 −0.0894564
\(624\) 0 0
\(625\) −28.1443 −1.12577
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 4.24122i − 0.169108i
\(630\) 0 0
\(631\) −18.0509 −0.718595 −0.359297 0.933223i \(-0.616984\pi\)
−0.359297 + 0.933223i \(0.616984\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 2.55707i − 0.101474i
\(636\) 0 0
\(637\) − 6.30135i − 0.249668i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0253 0.474971 0.237486 0.971391i \(-0.423677\pi\)
0.237486 + 0.971391i \(0.423677\pi\)
\(642\) 0 0
\(643\) 9.54316i 0.376345i 0.982136 + 0.188173i \(0.0602565\pi\)
−0.982136 + 0.188173i \(0.939744\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.2201 −1.26670 −0.633351 0.773865i \(-0.718321\pi\)
−0.633351 + 0.773865i \(0.718321\pi\)
\(648\) 0 0
\(649\) 1.42714 0.0560202
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 14.5245i − 0.568388i −0.958767 0.284194i \(-0.908274\pi\)
0.958767 0.284194i \(-0.0917259\pi\)
\(654\) 0 0
\(655\) 22.5205 0.879950
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 10.5726i − 0.411849i −0.978568 0.205924i \(-0.933980\pi\)
0.978568 0.205924i \(-0.0660201\pi\)
\(660\) 0 0
\(661\) 27.0965i 1.05393i 0.849887 + 0.526965i \(0.176670\pi\)
−0.849887 + 0.526965i \(0.823330\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 20.3365 0.788614
\(666\) 0 0
\(667\) − 51.6565i − 2.00015i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.88568 −0.0727959
\(672\) 0 0
\(673\) 38.2417 1.47411 0.737055 0.675833i \(-0.236216\pi\)
0.737055 + 0.675833i \(0.236216\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.2289i 0.546860i 0.961892 + 0.273430i \(0.0881582\pi\)
−0.961892 + 0.273430i \(0.911842\pi\)
\(678\) 0 0
\(679\) 7.88097 0.302444
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 39.0552i − 1.49441i −0.664596 0.747203i \(-0.731397\pi\)
0.664596 0.747203i \(-0.268603\pi\)
\(684\) 0 0
\(685\) 17.0942i 0.653137i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.20020 0.274306
\(690\) 0 0
\(691\) 20.1602i 0.766929i 0.923556 + 0.383464i \(0.125269\pi\)
−0.923556 + 0.383464i \(0.874731\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.8121 −0.599786
\(696\) 0 0
\(697\) −15.7938 −0.598232
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 22.4760i − 0.848908i −0.905449 0.424454i \(-0.860466\pi\)
0.905449 0.424454i \(-0.139534\pi\)
\(702\) 0 0
\(703\) 5.32548 0.200854
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.9693i 0.638195i
\(708\) 0 0
\(709\) 20.2609i 0.760915i 0.924798 + 0.380458i \(0.124233\pi\)
−0.924798 + 0.380458i \(0.875767\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16.9518 −0.634852
\(714\) 0 0
\(715\) − 2.46571i − 0.0922124i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −21.2440 −0.792269 −0.396134 0.918193i \(-0.629649\pi\)
−0.396134 + 0.918193i \(0.629649\pi\)
\(720\) 0 0
\(721\) −10.7049 −0.398672
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 42.4874i 1.57794i
\(726\) 0 0
\(727\) 32.9048 1.22037 0.610185 0.792259i \(-0.291095\pi\)
0.610185 + 0.792259i \(0.291095\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 37.2039i − 1.37603i
\(732\) 0 0
\(733\) − 32.0598i − 1.18415i −0.805881 0.592077i \(-0.798308\pi\)
0.805881 0.592077i \(-0.201692\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.135933 0.00500716
\(738\) 0 0
\(739\) 46.3826i 1.70621i 0.521738 + 0.853106i \(0.325284\pi\)
−0.521738 + 0.853106i \(0.674716\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.1782 0.446777 0.223388 0.974730i \(-0.428288\pi\)
0.223388 + 0.974730i \(0.428288\pi\)
\(744\) 0 0
\(745\) −35.3666 −1.29573
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1.31462i − 0.0480351i
\(750\) 0 0
\(751\) −24.4371 −0.891723 −0.445862 0.895102i \(-0.647103\pi\)
−0.445862 + 0.895102i \(0.647103\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 65.4799i 2.38306i
\(756\) 0 0
\(757\) − 7.15224i − 0.259953i −0.991517 0.129976i \(-0.958510\pi\)
0.991517 0.129976i \(-0.0414902\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.7809 −0.535806 −0.267903 0.963446i \(-0.586331\pi\)
−0.267903 + 0.963446i \(0.586331\pi\)
\(762\) 0 0
\(763\) − 10.2677i − 0.371715i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −69.9441 −2.52554
\(768\) 0 0
\(769\) −1.90576 −0.0687235 −0.0343618 0.999409i \(-0.510940\pi\)
−0.0343618 + 0.999409i \(0.510940\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.63435i 0.238621i 0.992857 + 0.119310i \(0.0380684\pi\)
−0.992857 + 0.119310i \(0.961932\pi\)
\(774\) 0 0
\(775\) 13.9429 0.500843
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 19.8314i − 0.710534i
\(780\) 0 0
\(781\) 0.142179i 0.00508756i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −55.6944 −1.98782
\(786\) 0 0
\(787\) − 2.91096i − 0.103765i −0.998653 0.0518823i \(-0.983478\pi\)
0.998653 0.0518823i \(-0.0165221\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −13.1796 −0.468613
\(792\) 0 0
\(793\) 92.4170 3.28182
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.96910i 0.246858i 0.992353 + 0.123429i \(0.0393891\pi\)
−0.992353 + 0.123429i \(0.960611\pi\)
\(798\) 0 0
\(799\) −25.3629 −0.897274
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 1.55598i − 0.0549094i
\(804\) 0 0
\(805\) 15.7713i 0.555864i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21.3012 0.748909 0.374454 0.927245i \(-0.377830\pi\)
0.374454 + 0.927245i \(0.377830\pi\)
\(810\) 0 0
\(811\) 18.1357i 0.636832i 0.947951 + 0.318416i \(0.103151\pi\)
−0.947951 + 0.318416i \(0.896849\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 57.3832 2.01005
\(816\) 0 0
\(817\) 46.7149 1.63435
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24.3030i 0.848180i 0.905620 + 0.424090i \(0.139406\pi\)
−0.905620 + 0.424090i \(0.860594\pi\)
\(822\) 0 0
\(823\) −1.29306 −0.0450734 −0.0225367 0.999746i \(-0.507174\pi\)
−0.0225367 + 0.999746i \(0.507174\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 23.6268i − 0.821585i −0.911729 0.410792i \(-0.865252\pi\)
0.911729 0.410792i \(-0.134748\pi\)
\(828\) 0 0
\(829\) − 5.66492i − 0.196751i −0.995149 0.0983754i \(-0.968635\pi\)
0.995149 0.0983754i \(-0.0313646\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.32168 −0.184385
\(834\) 0 0
\(835\) − 63.3890i − 2.19367i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.1554 −0.730366 −0.365183 0.930936i \(-0.618994\pi\)
−0.365183 + 0.930936i \(0.618994\pi\)
\(840\) 0 0
\(841\) −70.3652 −2.42639
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 81.2801i 2.79612i
\(846\) 0 0
\(847\) −10.9835 −0.377396
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.13000i 0.141574i
\(852\) 0 0
\(853\) − 4.41255i − 0.151083i −0.997143 0.0755413i \(-0.975932\pi\)
0.997143 0.0755413i \(-0.0240685\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −20.9783 −0.716606 −0.358303 0.933605i \(-0.616644\pi\)
−0.358303 + 0.933605i \(0.616644\pi\)
\(858\) 0 0
\(859\) − 15.6318i − 0.533351i −0.963786 0.266676i \(-0.914075\pi\)
0.963786 0.266676i \(-0.0859253\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.1880 0.551047 0.275523 0.961294i \(-0.411149\pi\)
0.275523 + 0.961294i \(0.411149\pi\)
\(864\) 0 0
\(865\) −16.7136 −0.568278
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 0.337409i − 0.0114458i
\(870\) 0 0
\(871\) −6.66208 −0.225736
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.24514i 0.0758995i
\(876\) 0 0
\(877\) − 38.1001i − 1.28655i −0.765636 0.643274i \(-0.777576\pi\)
0.765636 0.643274i \(-0.222424\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 46.4289 1.56423 0.782114 0.623135i \(-0.214141\pi\)
0.782114 + 0.623135i \(0.214141\pi\)
\(882\) 0 0
\(883\) − 17.8873i − 0.601956i −0.953631 0.300978i \(-0.902687\pi\)
0.953631 0.300978i \(-0.0973131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 35.5228 1.19274 0.596369 0.802711i \(-0.296610\pi\)
0.596369 + 0.802711i \(0.296610\pi\)
\(888\) 0 0
\(889\) −0.840201 −0.0281795
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 31.8468i − 1.06571i
\(894\) 0 0
\(895\) 23.2194 0.776137
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 32.6082i 1.08754i
\(900\) 0 0
\(901\) − 6.08079i − 0.202580i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −30.4875 −1.01344
\(906\) 0 0
\(907\) − 41.8996i − 1.39125i −0.718404 0.695627i \(-0.755127\pi\)
0.718404 0.695627i \(-0.244873\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.35427 0.177395 0.0886974 0.996059i \(-0.471730\pi\)
0.0886974 + 0.996059i \(0.471730\pi\)
\(912\) 0 0
\(913\) −0.831099 −0.0275054
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 7.39979i − 0.244363i
\(918\) 0 0
\(919\) 54.8843 1.81047 0.905234 0.424914i \(-0.139696\pi\)
0.905234 + 0.424914i \(0.139696\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 6.96817i − 0.229360i
\(924\) 0 0
\(925\) − 3.39692i − 0.111690i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.4091 0.341512 0.170756 0.985313i \(-0.445379\pi\)
0.170756 + 0.985313i \(0.445379\pi\)
\(930\) 0 0
\(931\) − 6.68215i − 0.218999i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.08237 −0.0681007
\(936\) 0 0
\(937\) 49.8177 1.62747 0.813737 0.581233i \(-0.197430\pi\)
0.813737 + 0.581233i \(0.197430\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.61503i 0.183045i 0.995803 + 0.0915224i \(0.0291733\pi\)
−0.995803 + 0.0915224i \(0.970827\pi\)
\(942\) 0 0
\(943\) 15.3796 0.500829
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.3495i 0.726262i 0.931738 + 0.363131i \(0.118292\pi\)
−0.931738 + 0.363131i \(0.881708\pi\)
\(948\) 0 0
\(949\) 76.2585i 2.47545i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 19.9449 0.646080 0.323040 0.946385i \(-0.395295\pi\)
0.323040 + 0.946385i \(0.395295\pi\)
\(954\) 0 0
\(955\) 19.0396i 0.616106i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.61682 0.181376
\(960\) 0 0
\(961\) −20.2991 −0.654811
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 7.50035i − 0.241445i
\(966\) 0 0
\(967\) −32.6951 −1.05140 −0.525702 0.850669i \(-0.676197\pi\)
−0.525702 + 0.850669i \(0.676197\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 33.7391i − 1.08274i −0.840785 0.541369i \(-0.817906\pi\)
0.840785 0.541369i \(-0.182094\pi\)
\(972\) 0 0
\(973\) 5.19553i 0.166561i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45.8374 1.46647 0.733235 0.679976i \(-0.238010\pi\)
0.733235 + 0.679976i \(0.238010\pi\)
\(978\) 0 0
\(979\) 0.287081i 0.00917516i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.0557 0.448307 0.224154 0.974554i \(-0.428038\pi\)
0.224154 + 0.974554i \(0.428038\pi\)
\(984\) 0 0
\(985\) 72.4649 2.30892
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 36.2282i 1.15199i
\(990\) 0 0
\(991\) −18.3875 −0.584099 −0.292050 0.956403i \(-0.594337\pi\)
−0.292050 + 0.956403i \(0.594337\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 73.4362i 2.32808i
\(996\) 0 0
\(997\) − 20.8011i − 0.658776i −0.944195 0.329388i \(-0.893158\pi\)
0.944195 0.329388i \(-0.106842\pi\)
\(998\) 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.f.3025.3 24
3.2 odd 2 inner 6048.2.c.f.3025.21 24
4.3 odd 2 1512.2.c.g.757.13 yes 24
8.3 odd 2 1512.2.c.g.757.14 yes 24
8.5 even 2 inner 6048.2.c.f.3025.22 24
12.11 even 2 1512.2.c.g.757.12 yes 24
24.5 odd 2 inner 6048.2.c.f.3025.4 24
24.11 even 2 1512.2.c.g.757.11 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.g.757.11 24 24.11 even 2
1512.2.c.g.757.12 yes 24 12.11 even 2
1512.2.c.g.757.13 yes 24 4.3 odd 2
1512.2.c.g.757.14 yes 24 8.3 odd 2
6048.2.c.f.3025.3 24 1.1 even 1 trivial
6048.2.c.f.3025.4 24 24.5 odd 2 inner
6048.2.c.f.3025.21 24 3.2 odd 2 inner
6048.2.c.f.3025.22 24 8.5 even 2 inner