Properties

Label 6048.2.c.g.3025.13
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 3025.13
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.g.3025.12

$q$-expansion

\(f(q)\) \(=\) \(q+1.26947i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.26947i q^{5} -1.00000 q^{7} +2.02805i q^{11} -4.05339i q^{13} +1.33658 q^{17} +3.28346i q^{19} +2.25680 q^{23} +3.38845 q^{25} -3.58798i q^{29} +0.464312 q^{31} -1.26947i q^{35} -11.8743i q^{37} +1.73758 q^{41} +1.70380i q^{43} -8.45083 q^{47} +1.00000 q^{49} +11.7544i q^{53} -2.57454 q^{55} -5.49489i q^{59} -6.02150i q^{61} +5.14565 q^{65} +0.381481i q^{67} +9.47622 q^{71} -10.0336 q^{73} -2.02805i q^{77} -4.90442 q^{79} +6.95800i q^{83} +1.69674i q^{85} +17.4322 q^{89} +4.05339i q^{91} -4.16824 q^{95} +7.62794 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} + 16q^{31} + 24q^{49} + 8q^{55} - 32q^{79} + O(q^{100}) \)

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
\(5\) 1.26947i 0.567723i 0.958865 + 0.283862i \(0.0916156\pi\)
−0.958865 + 0.283862i \(0.908384\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.02805i 0.611480i 0.952115 + 0.305740i \(0.0989039\pi\)
−0.952115 + 0.305740i \(0.901096\pi\)
\(12\) 0 0
\(13\) − 4.05339i − 1.12421i −0.827066 0.562104i \(-0.809992\pi\)
0.827066 0.562104i \(-0.190008\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.33658 0.324167 0.162084 0.986777i \(-0.448179\pi\)
0.162084 + 0.986777i \(0.448179\pi\)
\(18\) 0 0
\(19\) 3.28346i 0.753277i 0.926360 + 0.376638i \(0.122920\pi\)
−0.926360 + 0.376638i \(0.877080\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.25680 0.470576 0.235288 0.971926i \(-0.424397\pi\)
0.235288 + 0.971926i \(0.424397\pi\)
\(24\) 0 0
\(25\) 3.38845 0.677690
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.58798i − 0.666272i −0.942879 0.333136i \(-0.891893\pi\)
0.942879 0.333136i \(-0.108107\pi\)
\(30\) 0 0
\(31\) 0.464312 0.0833930 0.0416965 0.999130i \(-0.486724\pi\)
0.0416965 + 0.999130i \(0.486724\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.26947i − 0.214579i
\(36\) 0 0
\(37\) − 11.8743i − 1.95212i −0.217507 0.976059i \(-0.569793\pi\)
0.217507 0.976059i \(-0.430207\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.73758 0.271365 0.135683 0.990752i \(-0.456677\pi\)
0.135683 + 0.990752i \(0.456677\pi\)
\(42\) 0 0
\(43\) 1.70380i 0.259827i 0.991525 + 0.129913i \(0.0414699\pi\)
−0.991525 + 0.129913i \(0.958530\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.45083 −1.23268 −0.616340 0.787480i \(-0.711385\pi\)
−0.616340 + 0.787480i \(0.711385\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.7544i 1.61459i 0.590146 + 0.807296i \(0.299070\pi\)
−0.590146 + 0.807296i \(0.700930\pi\)
\(54\) 0 0
\(55\) −2.57454 −0.347152
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 5.49489i − 0.715374i −0.933842 0.357687i \(-0.883565\pi\)
0.933842 0.357687i \(-0.116435\pi\)
\(60\) 0 0
\(61\) − 6.02150i − 0.770975i −0.922713 0.385487i \(-0.874033\pi\)
0.922713 0.385487i \(-0.125967\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.14565 0.638239
\(66\) 0 0
\(67\) 0.381481i 0.0466054i 0.999728 + 0.0233027i \(0.00741815\pi\)
−0.999728 + 0.0233027i \(0.992582\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.47622 1.12462 0.562310 0.826926i \(-0.309913\pi\)
0.562310 + 0.826926i \(0.309913\pi\)
\(72\) 0 0
\(73\) −10.0336 −1.17434 −0.587171 0.809463i \(-0.699759\pi\)
−0.587171 + 0.809463i \(0.699759\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.02805i − 0.231118i
\(78\) 0 0
\(79\) −4.90442 −0.551791 −0.275895 0.961188i \(-0.588974\pi\)
−0.275895 + 0.961188i \(0.588974\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.95800i 0.763740i 0.924216 + 0.381870i \(0.124720\pi\)
−0.924216 + 0.381870i \(0.875280\pi\)
\(84\) 0 0
\(85\) 1.69674i 0.184037i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 17.4322 1.84781 0.923903 0.382628i \(-0.124981\pi\)
0.923903 + 0.382628i \(0.124981\pi\)
\(90\) 0 0
\(91\) 4.05339i 0.424911i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.16824 −0.427653
\(96\) 0 0
\(97\) 7.62794 0.774500 0.387250 0.921975i \(-0.373425\pi\)
0.387250 + 0.921975i \(0.373425\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 5.27674i − 0.525055i −0.964924 0.262528i \(-0.915444\pi\)
0.964924 0.262528i \(-0.0845561\pi\)
\(102\) 0 0
\(103\) 5.07253 0.499812 0.249906 0.968270i \(-0.419600\pi\)
0.249906 + 0.968270i \(0.419600\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 5.22269i − 0.504896i −0.967610 0.252448i \(-0.918764\pi\)
0.967610 0.252448i \(-0.0812357\pi\)
\(108\) 0 0
\(109\) 10.8354i 1.03784i 0.854822 + 0.518922i \(0.173667\pi\)
−0.854822 + 0.518922i \(0.826333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.8783 1.02334 0.511672 0.859181i \(-0.329026\pi\)
0.511672 + 0.859181i \(0.329026\pi\)
\(114\) 0 0
\(115\) 2.86494i 0.267157i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.33658 −0.122524
\(120\) 0 0
\(121\) 6.88701 0.626092
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.6489i 0.952464i
\(126\) 0 0
\(127\) −1.57175 −0.139471 −0.0697353 0.997566i \(-0.522215\pi\)
−0.0697353 + 0.997566i \(0.522215\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 11.7947i − 1.03051i −0.857037 0.515256i \(-0.827697\pi\)
0.857037 0.515256i \(-0.172303\pi\)
\(132\) 0 0
\(133\) − 3.28346i − 0.284712i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.4006 1.57207 0.786033 0.618184i \(-0.212131\pi\)
0.786033 + 0.618184i \(0.212131\pi\)
\(138\) 0 0
\(139\) 11.3851i 0.965674i 0.875710 + 0.482837i \(0.160394\pi\)
−0.875710 + 0.482837i \(0.839606\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.22048 0.687431
\(144\) 0 0
\(145\) 4.55483 0.378258
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 14.6966i − 1.20400i −0.798498 0.601998i \(-0.794372\pi\)
0.798498 0.601998i \(-0.205628\pi\)
\(150\) 0 0
\(151\) 0.220342 0.0179312 0.00896559 0.999960i \(-0.497146\pi\)
0.00896559 + 0.999960i \(0.497146\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.589430i 0.0473441i
\(156\) 0 0
\(157\) − 10.4367i − 0.832941i −0.909149 0.416471i \(-0.863267\pi\)
0.909149 0.416471i \(-0.136733\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.25680 −0.177861
\(162\) 0 0
\(163\) 4.46955i 0.350082i 0.984561 + 0.175041i \(0.0560058\pi\)
−0.984561 + 0.175041i \(0.943994\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.0617 1.08812 0.544062 0.839045i \(-0.316886\pi\)
0.544062 + 0.839045i \(0.316886\pi\)
\(168\) 0 0
\(169\) −3.42998 −0.263844
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.334570i 0.0254369i 0.999919 + 0.0127184i \(0.00404851\pi\)
−0.999919 + 0.0127184i \(0.995951\pi\)
\(174\) 0 0
\(175\) −3.38845 −0.256143
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.85419i 0.138589i 0.997596 + 0.0692944i \(0.0220748\pi\)
−0.997596 + 0.0692944i \(0.977925\pi\)
\(180\) 0 0
\(181\) − 11.8864i − 0.883512i −0.897135 0.441756i \(-0.854356\pi\)
0.897135 0.441756i \(-0.145644\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.0740 1.10826
\(186\) 0 0
\(187\) 2.71064i 0.198222i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.4777 1.91586 0.957928 0.287008i \(-0.0926607\pi\)
0.957928 + 0.287008i \(0.0926607\pi\)
\(192\) 0 0
\(193\) −17.6831 −1.27285 −0.636427 0.771337i \(-0.719588\pi\)
−0.636427 + 0.771337i \(0.719588\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.22153i 0.657007i 0.944503 + 0.328504i \(0.106544\pi\)
−0.944503 + 0.328504i \(0.893456\pi\)
\(198\) 0 0
\(199\) 8.46365 0.599973 0.299986 0.953944i \(-0.403018\pi\)
0.299986 + 0.953944i \(0.403018\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.58798i 0.251827i
\(204\) 0 0
\(205\) 2.20581i 0.154060i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.65902 −0.460614
\(210\) 0 0
\(211\) 4.66745i 0.321321i 0.987010 + 0.160660i \(0.0513624\pi\)
−0.987010 + 0.160660i \(0.948638\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.16292 −0.147510
\(216\) 0 0
\(217\) −0.464312 −0.0315196
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 5.41767i − 0.364432i
\(222\) 0 0
\(223\) −7.85009 −0.525681 −0.262841 0.964839i \(-0.584659\pi\)
−0.262841 + 0.964839i \(0.584659\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19.2857i 1.28004i 0.768360 + 0.640018i \(0.221073\pi\)
−0.768360 + 0.640018i \(0.778927\pi\)
\(228\) 0 0
\(229\) 1.71588i 0.113389i 0.998392 + 0.0566944i \(0.0180561\pi\)
−0.998392 + 0.0566944i \(0.981944\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −23.8734 −1.56400 −0.781999 0.623280i \(-0.785800\pi\)
−0.781999 + 0.623280i \(0.785800\pi\)
\(234\) 0 0
\(235\) − 10.7281i − 0.699821i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.6697 1.40170 0.700848 0.713310i \(-0.252805\pi\)
0.700848 + 0.713310i \(0.252805\pi\)
\(240\) 0 0
\(241\) 27.8432 1.79354 0.896770 0.442497i \(-0.145907\pi\)
0.896770 + 0.442497i \(0.145907\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.26947i 0.0811033i
\(246\) 0 0
\(247\) 13.3091 0.846840
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 27.7106i 1.74908i 0.484957 + 0.874538i \(0.338835\pi\)
−0.484957 + 0.874538i \(0.661165\pi\)
\(252\) 0 0
\(253\) 4.57691i 0.287748i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.90462 0.118807 0.0594035 0.998234i \(-0.481080\pi\)
0.0594035 + 0.998234i \(0.481080\pi\)
\(258\) 0 0
\(259\) 11.8743i 0.737831i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.2186 1.18507 0.592534 0.805545i \(-0.298128\pi\)
0.592534 + 0.805545i \(0.298128\pi\)
\(264\) 0 0
\(265\) −14.9218 −0.916642
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.2931i 1.11535i 0.830059 + 0.557676i \(0.188307\pi\)
−0.830059 + 0.557676i \(0.811693\pi\)
\(270\) 0 0
\(271\) −19.1450 −1.16297 −0.581487 0.813555i \(-0.697529\pi\)
−0.581487 + 0.813555i \(0.697529\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.87195i 0.414394i
\(276\) 0 0
\(277\) 17.6288i 1.05921i 0.848244 + 0.529606i \(0.177660\pi\)
−0.848244 + 0.529606i \(0.822340\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.3772 0.738362 0.369181 0.929358i \(-0.379638\pi\)
0.369181 + 0.929358i \(0.379638\pi\)
\(282\) 0 0
\(283\) − 2.56225i − 0.152310i −0.997096 0.0761550i \(-0.975736\pi\)
0.997096 0.0761550i \(-0.0242644\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.73758 −0.102566
\(288\) 0 0
\(289\) −15.2136 −0.894916
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.12608i 0.0657866i 0.999459 + 0.0328933i \(0.0104721\pi\)
−0.999459 + 0.0328933i \(0.989528\pi\)
\(294\) 0 0
\(295\) 6.97559 0.406135
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 9.14770i − 0.529025i
\(300\) 0 0
\(301\) − 1.70380i − 0.0982052i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.64411 0.437700
\(306\) 0 0
\(307\) 24.9543i 1.42422i 0.702070 + 0.712108i \(0.252259\pi\)
−0.702070 + 0.712108i \(0.747741\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.4110 1.15740 0.578701 0.815540i \(-0.303560\pi\)
0.578701 + 0.815540i \(0.303560\pi\)
\(312\) 0 0
\(313\) 18.5902 1.05078 0.525389 0.850862i \(-0.323920\pi\)
0.525389 + 0.850862i \(0.323920\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.68959i − 0.0948970i −0.998874 0.0474485i \(-0.984891\pi\)
0.998874 0.0474485i \(-0.0151090\pi\)
\(318\) 0 0
\(319\) 7.27661 0.407412
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.38859i 0.244188i
\(324\) 0 0
\(325\) − 13.7347i − 0.761865i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.45083 0.465909
\(330\) 0 0
\(331\) − 25.4187i − 1.39714i −0.715542 0.698570i \(-0.753820\pi\)
0.715542 0.698570i \(-0.246180\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.484278 −0.0264590
\(336\) 0 0
\(337\) −10.5368 −0.573979 −0.286989 0.957934i \(-0.592654\pi\)
−0.286989 + 0.957934i \(0.592654\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.941649i 0.0509932i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.6098i − 0.676930i −0.940979 0.338465i \(-0.890092\pi\)
0.940979 0.338465i \(-0.109908\pi\)
\(348\) 0 0
\(349\) − 10.3681i − 0.554990i −0.960727 0.277495i \(-0.910496\pi\)
0.960727 0.277495i \(-0.0895043\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.4319 1.14071 0.570353 0.821399i \(-0.306806\pi\)
0.570353 + 0.821399i \(0.306806\pi\)
\(354\) 0 0
\(355\) 12.0298i 0.638473i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.5414 0.925798 0.462899 0.886411i \(-0.346809\pi\)
0.462899 + 0.886411i \(0.346809\pi\)
\(360\) 0 0
\(361\) 8.21890 0.432574
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 12.7373i − 0.666702i
\(366\) 0 0
\(367\) −28.4730 −1.48628 −0.743140 0.669136i \(-0.766665\pi\)
−0.743140 + 0.669136i \(0.766665\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 11.7544i − 0.610259i
\(372\) 0 0
\(373\) − 11.7775i − 0.609818i −0.952382 0.304909i \(-0.901374\pi\)
0.952382 0.304909i \(-0.0986260\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −14.5435 −0.749028
\(378\) 0 0
\(379\) − 14.4058i − 0.739975i −0.929037 0.369988i \(-0.879362\pi\)
0.929037 0.369988i \(-0.120638\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.8612 −0.606078 −0.303039 0.952978i \(-0.598001\pi\)
−0.303039 + 0.952978i \(0.598001\pi\)
\(384\) 0 0
\(385\) 2.57454 0.131211
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 16.3633i − 0.829655i −0.909900 0.414827i \(-0.863842\pi\)
0.909900 0.414827i \(-0.136158\pi\)
\(390\) 0 0
\(391\) 3.01639 0.152545
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 6.22601i − 0.313265i
\(396\) 0 0
\(397\) 10.7133i 0.537685i 0.963184 + 0.268843i \(0.0866411\pi\)
−0.963184 + 0.268843i \(0.913359\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −21.7286 −1.08507 −0.542536 0.840032i \(-0.682536\pi\)
−0.542536 + 0.840032i \(0.682536\pi\)
\(402\) 0 0
\(403\) − 1.88204i − 0.0937511i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0816 1.19368
\(408\) 0 0
\(409\) −14.3607 −0.710092 −0.355046 0.934849i \(-0.615535\pi\)
−0.355046 + 0.934849i \(0.615535\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.49489i 0.270386i
\(414\) 0 0
\(415\) −8.83296 −0.433593
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 33.7826i − 1.65039i −0.564849 0.825194i \(-0.691065\pi\)
0.564849 0.825194i \(-0.308935\pi\)
\(420\) 0 0
\(421\) 10.6036i 0.516788i 0.966040 + 0.258394i \(0.0831933\pi\)
−0.966040 + 0.258394i \(0.916807\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.52892 0.219685
\(426\) 0 0
\(427\) 6.02150i 0.291401i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.2155 0.925577 0.462788 0.886469i \(-0.346849\pi\)
0.462788 + 0.886469i \(0.346849\pi\)
\(432\) 0 0
\(433\) 39.8723 1.91614 0.958070 0.286534i \(-0.0925030\pi\)
0.958070 + 0.286534i \(0.0925030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.41011i 0.354474i
\(438\) 0 0
\(439\) −2.96048 −0.141296 −0.0706481 0.997501i \(-0.522507\pi\)
−0.0706481 + 0.997501i \(0.522507\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.7091i 0.603827i 0.953335 + 0.301914i \(0.0976254\pi\)
−0.953335 + 0.301914i \(0.902375\pi\)
\(444\) 0 0
\(445\) 22.1296i 1.04904i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 27.4353 1.29475 0.647375 0.762171i \(-0.275867\pi\)
0.647375 + 0.762171i \(0.275867\pi\)
\(450\) 0 0
\(451\) 3.52391i 0.165934i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.14565 −0.241232
\(456\) 0 0
\(457\) 18.9049 0.884335 0.442168 0.896932i \(-0.354210\pi\)
0.442168 + 0.896932i \(0.354210\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 30.3182i − 1.41206i −0.708181 0.706030i \(-0.750484\pi\)
0.708181 0.706030i \(-0.249516\pi\)
\(462\) 0 0
\(463\) 29.2518 1.35944 0.679722 0.733470i \(-0.262100\pi\)
0.679722 + 0.733470i \(0.262100\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 32.7692i 1.51638i 0.652036 + 0.758188i \(0.273915\pi\)
−0.652036 + 0.758188i \(0.726085\pi\)
\(468\) 0 0
\(469\) − 0.381481i − 0.0176152i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.45539 −0.158879
\(474\) 0 0
\(475\) 11.1258i 0.510488i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −25.8900 −1.18294 −0.591472 0.806326i \(-0.701453\pi\)
−0.591472 + 0.806326i \(0.701453\pi\)
\(480\) 0 0
\(481\) −48.1310 −2.19459
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.68342i 0.439701i
\(486\) 0 0
\(487\) 5.94127 0.269225 0.134612 0.990898i \(-0.457021\pi\)
0.134612 + 0.990898i \(0.457021\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.77413i 0.170324i 0.996367 + 0.0851621i \(0.0271408\pi\)
−0.996367 + 0.0851621i \(0.972859\pi\)
\(492\) 0 0
\(493\) − 4.79561i − 0.215984i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.47622 −0.425067
\(498\) 0 0
\(499\) − 7.66921i − 0.343321i −0.985156 0.171661i \(-0.945087\pi\)
0.985156 0.171661i \(-0.0549132\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.29637 −0.325329 −0.162664 0.986681i \(-0.552009\pi\)
−0.162664 + 0.986681i \(0.552009\pi\)
\(504\) 0 0
\(505\) 6.69865 0.298086
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 7.32231i − 0.324556i −0.986745 0.162278i \(-0.948116\pi\)
0.986745 0.162278i \(-0.0518841\pi\)
\(510\) 0 0
\(511\) 10.0336 0.443860
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.43942i 0.283755i
\(516\) 0 0
\(517\) − 17.1387i − 0.753759i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.2300 1.28059 0.640295 0.768129i \(-0.278812\pi\)
0.640295 + 0.768129i \(0.278812\pi\)
\(522\) 0 0
\(523\) − 37.4676i − 1.63835i −0.573547 0.819173i \(-0.694433\pi\)
0.573547 0.819173i \(-0.305567\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.620589 0.0270333
\(528\) 0 0
\(529\) −17.9068 −0.778559
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 7.04311i − 0.305071i
\(534\) 0 0
\(535\) 6.63003 0.286641
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.02805i 0.0873543i
\(540\) 0 0
\(541\) 22.9143i 0.985164i 0.870266 + 0.492582i \(0.163947\pi\)
−0.870266 + 0.492582i \(0.836053\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −13.7552 −0.589208
\(546\) 0 0
\(547\) 14.0319i 0.599959i 0.953946 + 0.299979i \(0.0969798\pi\)
−0.953946 + 0.299979i \(0.903020\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.7810 0.501887
\(552\) 0 0
\(553\) 4.90442 0.208557
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.1958i 1.23706i 0.785760 + 0.618532i \(0.212272\pi\)
−0.785760 + 0.618532i \(0.787728\pi\)
\(558\) 0 0
\(559\) 6.90615 0.292099
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 29.5633i − 1.24594i −0.782245 0.622971i \(-0.785925\pi\)
0.782245 0.622971i \(-0.214075\pi\)
\(564\) 0 0
\(565\) 13.8096i 0.580976i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.2748 −1.18534 −0.592672 0.805444i \(-0.701927\pi\)
−0.592672 + 0.805444i \(0.701927\pi\)
\(570\) 0 0
\(571\) − 40.3483i − 1.68852i −0.535932 0.844261i \(-0.680040\pi\)
0.535932 0.844261i \(-0.319960\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.64706 0.318904
\(576\) 0 0
\(577\) −0.0318593 −0.00132632 −0.000663161 1.00000i \(-0.500211\pi\)
−0.000663161 1.00000i \(0.500211\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 6.95800i − 0.288667i
\(582\) 0 0
\(583\) −23.8385 −0.987291
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 22.9245i − 0.946195i −0.881010 0.473098i \(-0.843136\pi\)
0.881010 0.473098i \(-0.156864\pi\)
\(588\) 0 0
\(589\) 1.52455i 0.0628180i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −19.8895 −0.816764 −0.408382 0.912811i \(-0.633907\pi\)
−0.408382 + 0.912811i \(0.633907\pi\)
\(594\) 0 0
\(595\) − 1.69674i − 0.0695596i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.01493 −0.368340 −0.184170 0.982894i \(-0.558960\pi\)
−0.184170 + 0.982894i \(0.558960\pi\)
\(600\) 0 0
\(601\) −30.3934 −1.23977 −0.619886 0.784692i \(-0.712821\pi\)
−0.619886 + 0.784692i \(0.712821\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.74284i 0.355447i
\(606\) 0 0
\(607\) 35.5668 1.44361 0.721807 0.692095i \(-0.243312\pi\)
0.721807 + 0.692095i \(0.243312\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 34.2545i 1.38579i
\(612\) 0 0
\(613\) 36.1874i 1.46159i 0.682595 + 0.730797i \(0.260851\pi\)
−0.682595 + 0.730797i \(0.739149\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 27.7558 1.11741 0.558704 0.829367i \(-0.311299\pi\)
0.558704 + 0.829367i \(0.311299\pi\)
\(618\) 0 0
\(619\) − 30.0187i − 1.20656i −0.797531 0.603278i \(-0.793861\pi\)
0.797531 0.603278i \(-0.206139\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −17.4322 −0.698405
\(624\) 0 0
\(625\) 3.42386 0.136954
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 15.8709i − 0.632813i
\(630\) 0 0
\(631\) −46.3724 −1.84606 −0.923028 0.384732i \(-0.874294\pi\)
−0.923028 + 0.384732i \(0.874294\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 1.99529i − 0.0791807i
\(636\) 0 0
\(637\) − 4.05339i − 0.160601i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.0590 0.634294 0.317147 0.948376i \(-0.397275\pi\)
0.317147 + 0.948376i \(0.397275\pi\)
\(642\) 0 0
\(643\) 11.8356i 0.466752i 0.972387 + 0.233376i \(0.0749774\pi\)
−0.972387 + 0.233376i \(0.925023\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.05582 −0.198765 −0.0993824 0.995049i \(-0.531687\pi\)
−0.0993824 + 0.995049i \(0.531687\pi\)
\(648\) 0 0
\(649\) 11.1439 0.437437
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 21.1804i − 0.828851i −0.910083 0.414426i \(-0.863982\pi\)
0.910083 0.414426i \(-0.136018\pi\)
\(654\) 0 0
\(655\) 14.9730 0.585045
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 18.6255i − 0.725548i −0.931877 0.362774i \(-0.881830\pi\)
0.931877 0.362774i \(-0.118170\pi\)
\(660\) 0 0
\(661\) − 8.43660i − 0.328146i −0.986448 0.164073i \(-0.947537\pi\)
0.986448 0.164073i \(-0.0524632\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.16824 0.161638
\(666\) 0 0
\(667\) − 8.09736i − 0.313531i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.2119 0.471436
\(672\) 0 0
\(673\) −11.6372 −0.448581 −0.224291 0.974522i \(-0.572006\pi\)
−0.224291 + 0.974522i \(0.572006\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.4251i 1.63053i 0.579089 + 0.815264i \(0.303408\pi\)
−0.579089 + 0.815264i \(0.696592\pi\)
\(678\) 0 0
\(679\) −7.62794 −0.292733
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.8253i 1.02644i 0.858257 + 0.513221i \(0.171548\pi\)
−0.858257 + 0.513221i \(0.828452\pi\)
\(684\) 0 0
\(685\) 23.3589i 0.892499i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 47.6452 1.81514
\(690\) 0 0
\(691\) − 49.9065i − 1.89853i −0.314476 0.949265i \(-0.601829\pi\)
0.314476 0.949265i \(-0.398171\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −14.4530 −0.548235
\(696\) 0 0
\(697\) 2.32241 0.0879677
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 47.6375i 1.79924i 0.436671 + 0.899621i \(0.356157\pi\)
−0.436671 + 0.899621i \(0.643843\pi\)
\(702\) 0 0
\(703\) 38.9887 1.47049
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.27674i 0.198452i
\(708\) 0 0
\(709\) 24.0726i 0.904064i 0.892002 + 0.452032i \(0.149301\pi\)
−0.892002 + 0.452032i \(0.850699\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.04786 0.0392427
\(714\) 0 0
\(715\) 10.4356i 0.390271i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −49.0237 −1.82827 −0.914137 0.405406i \(-0.867130\pi\)
−0.914137 + 0.405406i \(0.867130\pi\)
\(720\) 0 0
\(721\) −5.07253 −0.188911
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 12.1577i − 0.451526i
\(726\) 0 0
\(727\) −18.0843 −0.670710 −0.335355 0.942092i \(-0.608856\pi\)
−0.335355 + 0.942092i \(0.608856\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.27725i 0.0842273i
\(732\) 0 0
\(733\) 2.54526i 0.0940114i 0.998895 + 0.0470057i \(0.0149679\pi\)
−0.998895 + 0.0470057i \(0.985032\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.773664 −0.0284983
\(738\) 0 0
\(739\) − 36.7876i − 1.35326i −0.736325 0.676628i \(-0.763441\pi\)
0.736325 0.676628i \(-0.236559\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.1051 0.994391 0.497196 0.867638i \(-0.334363\pi\)
0.497196 + 0.867638i \(0.334363\pi\)
\(744\) 0 0
\(745\) 18.6569 0.683537
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.22269i 0.190833i
\(750\) 0 0
\(751\) 35.3786 1.29098 0.645491 0.763768i \(-0.276653\pi\)
0.645491 + 0.763768i \(0.276653\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.279717i 0.0101799i
\(756\) 0 0
\(757\) 0.340328i 0.0123694i 0.999981 + 0.00618471i \(0.00196867\pi\)
−0.999981 + 0.00618471i \(0.998031\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.2601 −1.09693 −0.548464 0.836174i \(-0.684787\pi\)
−0.548464 + 0.836174i \(0.684787\pi\)
\(762\) 0 0
\(763\) − 10.8354i − 0.392268i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.2729 −0.804230
\(768\) 0 0
\(769\) 18.0636 0.651389 0.325695 0.945475i \(-0.394402\pi\)
0.325695 + 0.945475i \(0.394402\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 3.29847i − 0.118638i −0.998239 0.0593188i \(-0.981107\pi\)
0.998239 0.0593188i \(-0.0188929\pi\)
\(774\) 0 0
\(775\) 1.57330 0.0565146
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.70529i 0.204413i
\(780\) 0 0
\(781\) 19.2183i 0.687683i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 13.2491 0.472880
\(786\) 0 0
\(787\) 14.6607i 0.522597i 0.965258 + 0.261298i \(0.0841506\pi\)
−0.965258 + 0.261298i \(0.915849\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −10.8783 −0.386787
\(792\) 0 0
\(793\) −24.4075 −0.866736
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.7001i 0.379016i 0.981879 + 0.189508i \(0.0606892\pi\)
−0.981879 + 0.189508i \(0.939311\pi\)
\(798\) 0 0
\(799\) −11.2952 −0.399595
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 20.3486i − 0.718087i
\(804\) 0 0
\(805\) − 2.86494i − 0.100976i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.7701 0.695078 0.347539 0.937666i \(-0.387017\pi\)
0.347539 + 0.937666i \(0.387017\pi\)
\(810\) 0 0
\(811\) − 29.8088i − 1.04673i −0.852109 0.523364i \(-0.824677\pi\)
0.852109 0.523364i \(-0.175323\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.67395 −0.198750
\(816\) 0 0
\(817\) −5.59435 −0.195721
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 38.3014i 1.33673i 0.743834 + 0.668364i \(0.233005\pi\)
−0.743834 + 0.668364i \(0.766995\pi\)
\(822\) 0 0
\(823\) −26.3640 −0.918990 −0.459495 0.888180i \(-0.651970\pi\)
−0.459495 + 0.888180i \(0.651970\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.5542i 1.54930i 0.632390 + 0.774650i \(0.282074\pi\)
−0.632390 + 0.774650i \(0.717926\pi\)
\(828\) 0 0
\(829\) − 48.8761i − 1.69754i −0.528765 0.848768i \(-0.677345\pi\)
0.528765 0.848768i \(-0.322655\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.33658 0.0463096
\(834\) 0 0
\(835\) 17.8508i 0.617753i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −39.1944 −1.35314 −0.676571 0.736377i \(-0.736535\pi\)
−0.676571 + 0.736377i \(0.736535\pi\)
\(840\) 0 0
\(841\) 16.1264 0.556082
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 4.35425i − 0.149791i
\(846\) 0 0
\(847\) −6.88701 −0.236641
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 26.7979i − 0.918619i
\(852\) 0 0
\(853\) − 22.6315i − 0.774887i −0.921893 0.387444i \(-0.873358\pi\)
0.921893 0.387444i \(-0.126642\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −25.6866 −0.877436 −0.438718 0.898625i \(-0.644567\pi\)
−0.438718 + 0.898625i \(0.644567\pi\)
\(858\) 0 0
\(859\) − 45.4628i − 1.55117i −0.631244 0.775585i \(-0.717455\pi\)
0.631244 0.775585i \(-0.282545\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.81159 −0.231869 −0.115935 0.993257i \(-0.536986\pi\)
−0.115935 + 0.993257i \(0.536986\pi\)
\(864\) 0 0
\(865\) −0.424725 −0.0144411
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 9.94642i − 0.337409i
\(870\) 0 0
\(871\) 1.54629 0.0523942
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 10.6489i − 0.359997i
\(876\) 0 0
\(877\) 34.9922i 1.18160i 0.806817 + 0.590802i \(0.201188\pi\)
−0.806817 + 0.590802i \(0.798812\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −16.1155 −0.542946 −0.271473 0.962446i \(-0.587511\pi\)
−0.271473 + 0.962446i \(0.587511\pi\)
\(882\) 0 0
\(883\) 46.5075i 1.56510i 0.622586 + 0.782551i \(0.286082\pi\)
−0.622586 + 0.782551i \(0.713918\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −43.0256 −1.44466 −0.722328 0.691550i \(-0.756928\pi\)
−0.722328 + 0.691550i \(0.756928\pi\)
\(888\) 0 0
\(889\) 1.57175 0.0527149
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 27.7479i − 0.928550i
\(894\) 0 0
\(895\) −2.35384 −0.0786801
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1.66595i − 0.0555624i
\(900\) 0 0
\(901\) 15.7107i 0.523398i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.0895 0.501590
\(906\) 0 0
\(907\) − 49.6912i − 1.64997i −0.565156 0.824984i \(-0.691184\pi\)
0.565156 0.824984i \(-0.308816\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 44.6723 1.48006 0.740029 0.672575i \(-0.234812\pi\)
0.740029 + 0.672575i \(0.234812\pi\)
\(912\) 0 0
\(913\) −14.1112 −0.467012
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11.7947i 0.389497i
\(918\) 0 0
\(919\) 51.7662 1.70761 0.853804 0.520594i \(-0.174290\pi\)
0.853804 + 0.520594i \(0.174290\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 38.4108i − 1.26431i
\(924\) 0 0
\(925\) − 40.2354i − 1.32293i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −16.4271 −0.538956 −0.269478 0.963006i \(-0.586851\pi\)
−0.269478 + 0.963006i \(0.586851\pi\)
\(930\) 0 0
\(931\) 3.28346i 0.107611i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.44108 −0.112535
\(936\) 0 0
\(937\) −24.9154 −0.813951 −0.406976 0.913439i \(-0.633417\pi\)
−0.406976 + 0.913439i \(0.633417\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 35.3088i − 1.15103i −0.817790 0.575517i \(-0.804801\pi\)
0.817790 0.575517i \(-0.195199\pi\)
\(942\) 0 0
\(943\) 3.92138 0.127698
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 41.3026i − 1.34215i −0.741388 0.671077i \(-0.765832\pi\)
0.741388 0.671077i \(-0.234168\pi\)
\(948\) 0 0
\(949\) 40.6701i 1.32021i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −11.2564 −0.364630 −0.182315 0.983240i \(-0.558359\pi\)
−0.182315 + 0.983240i \(0.558359\pi\)
\(954\) 0 0
\(955\) 33.6125i 1.08768i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.4006 −0.594185
\(960\) 0 0
\(961\) −30.7844 −0.993046
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 22.4481i − 0.722629i
\(966\) 0 0
\(967\) 1.51158 0.0486093 0.0243046 0.999705i \(-0.492263\pi\)
0.0243046 + 0.999705i \(0.492263\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 9.03959i − 0.290094i −0.989425 0.145047i \(-0.953667\pi\)
0.989425 0.145047i \(-0.0463334\pi\)
\(972\) 0 0
\(973\) − 11.3851i − 0.364990i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.32341 −0.0743325 −0.0371662 0.999309i \(-0.511833\pi\)
−0.0371662 + 0.999309i \(0.511833\pi\)
\(978\) 0 0
\(979\) 35.3533i 1.12990i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.8206 0.600284 0.300142 0.953894i \(-0.402966\pi\)
0.300142 + 0.953894i \(0.402966\pi\)
\(984\) 0 0
\(985\) −11.7064 −0.372998
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.84513i 0.122268i
\(990\) 0 0
\(991\) 37.9699 1.20615 0.603077 0.797683i \(-0.293941\pi\)
0.603077 + 0.797683i \(0.293941\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.7443i 0.340618i
\(996\) 0 0
\(997\) 9.55023i 0.302459i 0.988499 + 0.151229i \(0.0483232\pi\)
−0.988499 + 0.151229i \(0.951677\pi\)
\(998\) 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.3025.13 24
3.2 odd 2 inner 6048.2.c.g.3025.11 24
4.3 odd 2 1512.2.c.f.757.14 yes 24
8.3 odd 2 1512.2.c.f.757.13 yes 24
8.5 even 2 inner 6048.2.c.g.3025.12 24
12.11 even 2 1512.2.c.f.757.11 24
24.5 odd 2 inner 6048.2.c.g.3025.14 24
24.11 even 2 1512.2.c.f.757.12 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.f.757.11 24 12.11 even 2
1512.2.c.f.757.12 yes 24 24.11 even 2
1512.2.c.f.757.13 yes 24 8.3 odd 2
1512.2.c.f.757.14 yes 24 4.3 odd 2
6048.2.c.g.3025.11 24 3.2 odd 2 inner
6048.2.c.g.3025.12 24 8.5 even 2 inner
6048.2.c.g.3025.13 24 1.1 even 1 trivial
6048.2.c.g.3025.14 24 24.5 odd 2 inner