Properties

Label 6048.2.c.f.3025.16
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 3025.16
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.f.3025.9

$q$-expansion

\(f(q)\) \(=\) \(q+1.25392i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.25392i q^{5} -1.00000 q^{7} -1.55766i q^{11} +1.07281i q^{13} -0.0158095 q^{17} -2.35077i q^{19} +5.95129 q^{23} +3.42768 q^{25} +0.469737i q^{29} -1.69031 q^{31} -1.25392i q^{35} +4.59800i q^{37} -12.2190 q^{41} +1.97482i q^{43} +7.12571 q^{47} +1.00000 q^{49} +1.86735i q^{53} +1.95319 q^{55} -8.54291i q^{59} +3.92502i q^{61} -1.34521 q^{65} -12.7403i q^{67} -4.22205 q^{71} +6.51561 q^{73} +1.55766i q^{77} +6.15343 q^{79} -8.88604i q^{83} -0.0198239i q^{85} +0.240788 q^{89} -1.07281i q^{91} +2.94767 q^{95} +5.86131 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} - 8q^{31} + 24q^{49} - 16q^{55} - 8q^{79} + O(q^{100}) \)

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
\(5\) 1.25392i 0.560770i 0.959888 + 0.280385i \(0.0904622\pi\)
−0.959888 + 0.280385i \(0.909538\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.55766i − 0.469653i −0.972037 0.234827i \(-0.924548\pi\)
0.972037 0.234827i \(-0.0754522\pi\)
\(12\) 0 0
\(13\) 1.07281i 0.297543i 0.988872 + 0.148771i \(0.0475319\pi\)
−0.988872 + 0.148771i \(0.952468\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.0158095 −0.00383438 −0.00191719 0.999998i \(-0.500610\pi\)
−0.00191719 + 0.999998i \(0.500610\pi\)
\(18\) 0 0
\(19\) − 2.35077i − 0.539303i −0.962958 0.269651i \(-0.913092\pi\)
0.962958 0.269651i \(-0.0869085\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.95129 1.24093 0.620465 0.784235i \(-0.286944\pi\)
0.620465 + 0.784235i \(0.286944\pi\)
\(24\) 0 0
\(25\) 3.42768 0.685537
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.469737i 0.0872279i 0.999048 + 0.0436139i \(0.0138872\pi\)
−0.999048 + 0.0436139i \(0.986113\pi\)
\(30\) 0 0
\(31\) −1.69031 −0.303589 −0.151795 0.988412i \(-0.548505\pi\)
−0.151795 + 0.988412i \(0.548505\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.25392i − 0.211951i
\(36\) 0 0
\(37\) 4.59800i 0.755906i 0.925825 + 0.377953i \(0.123372\pi\)
−0.925825 + 0.377953i \(0.876628\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.2190 −1.90829 −0.954143 0.299350i \(-0.903230\pi\)
−0.954143 + 0.299350i \(0.903230\pi\)
\(42\) 0 0
\(43\) 1.97482i 0.301158i 0.988598 + 0.150579i \(0.0481137\pi\)
−0.988598 + 0.150579i \(0.951886\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.12571 1.03939 0.519696 0.854351i \(-0.326045\pi\)
0.519696 + 0.854351i \(0.326045\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.86735i 0.256500i 0.991742 + 0.128250i \(0.0409360\pi\)
−0.991742 + 0.128250i \(0.959064\pi\)
\(54\) 0 0
\(55\) 1.95319 0.263368
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 8.54291i − 1.11219i −0.831118 0.556096i \(-0.812299\pi\)
0.831118 0.556096i \(-0.187701\pi\)
\(60\) 0 0
\(61\) 3.92502i 0.502547i 0.967916 + 0.251274i \(0.0808494\pi\)
−0.967916 + 0.251274i \(0.919151\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.34521 −0.166853
\(66\) 0 0
\(67\) − 12.7403i − 1.55647i −0.627972 0.778236i \(-0.716115\pi\)
0.627972 0.778236i \(-0.283885\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.22205 −0.501066 −0.250533 0.968108i \(-0.580606\pi\)
−0.250533 + 0.968108i \(0.580606\pi\)
\(72\) 0 0
\(73\) 6.51561 0.762595 0.381297 0.924452i \(-0.375477\pi\)
0.381297 + 0.924452i \(0.375477\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.55766i 0.177512i
\(78\) 0 0
\(79\) 6.15343 0.692315 0.346158 0.938176i \(-0.387486\pi\)
0.346158 + 0.938176i \(0.387486\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 8.88604i − 0.975370i −0.873020 0.487685i \(-0.837842\pi\)
0.873020 0.487685i \(-0.162158\pi\)
\(84\) 0 0
\(85\) − 0.0198239i − 0.00215021i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.240788 0.0255235 0.0127618 0.999919i \(-0.495938\pi\)
0.0127618 + 0.999919i \(0.495938\pi\)
\(90\) 0 0
\(91\) − 1.07281i − 0.112461i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.94767 0.302425
\(96\) 0 0
\(97\) 5.86131 0.595126 0.297563 0.954702i \(-0.403826\pi\)
0.297563 + 0.954702i \(0.403826\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 3.12870i − 0.311317i −0.987811 0.155659i \(-0.950250\pi\)
0.987811 0.155659i \(-0.0497500\pi\)
\(102\) 0 0
\(103\) −2.72832 −0.268830 −0.134415 0.990925i \(-0.542915\pi\)
−0.134415 + 0.990925i \(0.542915\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.05577i 0.198738i 0.995051 + 0.0993692i \(0.0316825\pi\)
−0.995051 + 0.0993692i \(0.968318\pi\)
\(108\) 0 0
\(109\) 13.8179i 1.32351i 0.749719 + 0.661756i \(0.230189\pi\)
−0.749719 + 0.661756i \(0.769811\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.72061 0.350005 0.175003 0.984568i \(-0.444007\pi\)
0.175003 + 0.984568i \(0.444007\pi\)
\(114\) 0 0
\(115\) 7.46244i 0.695876i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.0158095 0.00144926
\(120\) 0 0
\(121\) 8.57368 0.779426
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.5676i 0.945199i
\(126\) 0 0
\(127\) 19.8627 1.76253 0.881264 0.472625i \(-0.156693\pi\)
0.881264 + 0.472625i \(0.156693\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 7.24798i − 0.633259i −0.948549 0.316629i \(-0.897449\pi\)
0.948549 0.316629i \(-0.102551\pi\)
\(132\) 0 0
\(133\) 2.35077i 0.203837i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.1526 1.12371 0.561853 0.827237i \(-0.310089\pi\)
0.561853 + 0.827237i \(0.310089\pi\)
\(138\) 0 0
\(139\) 5.29486i 0.449104i 0.974462 + 0.224552i \(0.0720919\pi\)
−0.974462 + 0.224552i \(0.927908\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.67107 0.139742
\(144\) 0 0
\(145\) −0.589012 −0.0489148
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.59368i 0.540175i 0.962836 + 0.270088i \(0.0870526\pi\)
−0.962836 + 0.270088i \(0.912947\pi\)
\(150\) 0 0
\(151\) 9.51585 0.774389 0.387195 0.921998i \(-0.373444\pi\)
0.387195 + 0.921998i \(0.373444\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 2.11952i − 0.170244i
\(156\) 0 0
\(157\) 16.0109i 1.27781i 0.769285 + 0.638905i \(0.220612\pi\)
−0.769285 + 0.638905i \(0.779388\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.95129 −0.469027
\(162\) 0 0
\(163\) 12.5751i 0.984959i 0.870324 + 0.492480i \(0.163909\pi\)
−0.870324 + 0.492480i \(0.836091\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.2841 1.33748 0.668741 0.743495i \(-0.266834\pi\)
0.668741 + 0.743495i \(0.266834\pi\)
\(168\) 0 0
\(169\) 11.8491 0.911468
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 8.78719i − 0.668078i −0.942559 0.334039i \(-0.891588\pi\)
0.942559 0.334039i \(-0.108412\pi\)
\(174\) 0 0
\(175\) −3.42768 −0.259109
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.6964i 0.874233i 0.899405 + 0.437117i \(0.144000\pi\)
−0.899405 + 0.437117i \(0.856000\pi\)
\(180\) 0 0
\(181\) − 24.2624i − 1.80341i −0.432351 0.901705i \(-0.642316\pi\)
0.432351 0.901705i \(-0.357684\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.76552 −0.423890
\(186\) 0 0
\(187\) 0.0246260i 0.00180083i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.4850 −0.758665 −0.379333 0.925260i \(-0.623846\pi\)
−0.379333 + 0.925260i \(0.623846\pi\)
\(192\) 0 0
\(193\) 12.7092 0.914832 0.457416 0.889253i \(-0.348775\pi\)
0.457416 + 0.889253i \(0.348775\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.0117i 1.14079i 0.821372 + 0.570394i \(0.193209\pi\)
−0.821372 + 0.570394i \(0.806791\pi\)
\(198\) 0 0
\(199\) 11.1888 0.793156 0.396578 0.918001i \(-0.370198\pi\)
0.396578 + 0.918001i \(0.370198\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 0.469737i − 0.0329690i
\(204\) 0 0
\(205\) − 15.3217i − 1.07011i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.66170 −0.253285
\(210\) 0 0
\(211\) 20.1657i 1.38827i 0.719847 + 0.694133i \(0.244212\pi\)
−0.719847 + 0.694133i \(0.755788\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.47627 −0.168880
\(216\) 0 0
\(217\) 1.69031 0.114746
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 0.0169606i − 0.00114089i
\(222\) 0 0
\(223\) 11.1501 0.746663 0.373332 0.927698i \(-0.378215\pi\)
0.373332 + 0.927698i \(0.378215\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 18.7414i − 1.24391i −0.783053 0.621955i \(-0.786338\pi\)
0.783053 0.621955i \(-0.213662\pi\)
\(228\) 0 0
\(229\) 13.5444i 0.895038i 0.894274 + 0.447519i \(0.147692\pi\)
−0.894274 + 0.447519i \(0.852308\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.0070 −0.721094 −0.360547 0.932741i \(-0.617410\pi\)
−0.360547 + 0.932741i \(0.617410\pi\)
\(234\) 0 0
\(235\) 8.93508i 0.582860i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.4366 −0.998509 −0.499255 0.866455i \(-0.666393\pi\)
−0.499255 + 0.866455i \(0.666393\pi\)
\(240\) 0 0
\(241\) 16.0354 1.03293 0.516466 0.856308i \(-0.327247\pi\)
0.516466 + 0.856308i \(0.327247\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.25392i 0.0801100i
\(246\) 0 0
\(247\) 2.52192 0.160466
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.4441i 0.722343i 0.932499 + 0.361172i \(0.117623\pi\)
−0.932499 + 0.361172i \(0.882377\pi\)
\(252\) 0 0
\(253\) − 9.27010i − 0.582806i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.7447 −0.670236 −0.335118 0.942176i \(-0.608776\pi\)
−0.335118 + 0.942176i \(0.608776\pi\)
\(258\) 0 0
\(259\) − 4.59800i − 0.285706i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −14.6747 −0.904878 −0.452439 0.891795i \(-0.649446\pi\)
−0.452439 + 0.891795i \(0.649446\pi\)
\(264\) 0 0
\(265\) −2.34151 −0.143838
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.93110i 0.300655i 0.988636 + 0.150327i \(0.0480328\pi\)
−0.988636 + 0.150327i \(0.951967\pi\)
\(270\) 0 0
\(271\) 22.6605 1.37652 0.688262 0.725462i \(-0.258374\pi\)
0.688262 + 0.725462i \(0.258374\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 5.33918i − 0.321965i
\(276\) 0 0
\(277\) 5.99172i 0.360008i 0.983666 + 0.180004i \(0.0576110\pi\)
−0.983666 + 0.180004i \(0.942389\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.92544 0.114862 0.0574309 0.998349i \(-0.481709\pi\)
0.0574309 + 0.998349i \(0.481709\pi\)
\(282\) 0 0
\(283\) − 28.2195i − 1.67747i −0.544537 0.838737i \(-0.683295\pi\)
0.544537 0.838737i \(-0.316705\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.2190 0.721265
\(288\) 0 0
\(289\) −16.9998 −0.999985
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 29.0957i 1.69979i 0.526955 + 0.849893i \(0.323334\pi\)
−0.526955 + 0.849893i \(0.676666\pi\)
\(294\) 0 0
\(295\) 10.7121 0.623685
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.38458i 0.369230i
\(300\) 0 0
\(301\) − 1.97482i − 0.113827i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.92166 −0.281814
\(306\) 0 0
\(307\) 15.8372i 0.903879i 0.892049 + 0.451940i \(0.149268\pi\)
−0.892049 + 0.451940i \(0.850732\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 14.5620 0.825737 0.412869 0.910791i \(-0.364527\pi\)
0.412869 + 0.910791i \(0.364527\pi\)
\(312\) 0 0
\(313\) 6.47077 0.365749 0.182875 0.983136i \(-0.441460\pi\)
0.182875 + 0.983136i \(0.441460\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 11.9823i − 0.672994i −0.941685 0.336497i \(-0.890758\pi\)
0.941685 0.336497i \(-0.109242\pi\)
\(318\) 0 0
\(319\) 0.731691 0.0409669
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.0371646i 0.00206789i
\(324\) 0 0
\(325\) 3.67724i 0.203977i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.12571 −0.392853
\(330\) 0 0
\(331\) 21.9422i 1.20605i 0.797720 + 0.603027i \(0.206039\pi\)
−0.797720 + 0.603027i \(0.793961\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 15.9753 0.872823
\(336\) 0 0
\(337\) 22.3150 1.21558 0.607788 0.794100i \(-0.292057\pi\)
0.607788 + 0.794100i \(0.292057\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.63294i 0.142582i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.5806i 1.31956i 0.751460 + 0.659779i \(0.229350\pi\)
−0.751460 + 0.659779i \(0.770650\pi\)
\(348\) 0 0
\(349\) 12.5930i 0.674089i 0.941489 + 0.337045i \(0.109427\pi\)
−0.941489 + 0.337045i \(0.890573\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.1946 0.861953 0.430977 0.902363i \(-0.358169\pi\)
0.430977 + 0.902363i \(0.358169\pi\)
\(354\) 0 0
\(355\) − 5.29412i − 0.280983i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.1809 0.537329 0.268665 0.963234i \(-0.413418\pi\)
0.268665 + 0.963234i \(0.413418\pi\)
\(360\) 0 0
\(361\) 13.4739 0.709152
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.17006i 0.427640i
\(366\) 0 0
\(367\) 12.3939 0.646955 0.323478 0.946236i \(-0.395148\pi\)
0.323478 + 0.946236i \(0.395148\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 1.86735i − 0.0969480i
\(372\) 0 0
\(373\) − 7.57921i − 0.392437i −0.980560 0.196218i \(-0.937134\pi\)
0.980560 0.196218i \(-0.0628661\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.503936 −0.0259540
\(378\) 0 0
\(379\) 14.1179i 0.725187i 0.931947 + 0.362594i \(0.118109\pi\)
−0.931947 + 0.362594i \(0.881891\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.28678 0.0657512 0.0328756 0.999459i \(-0.489533\pi\)
0.0328756 + 0.999459i \(0.489533\pi\)
\(384\) 0 0
\(385\) −1.95319 −0.0995436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 33.3310i − 1.68995i −0.534806 0.844975i \(-0.679615\pi\)
0.534806 0.844975i \(-0.320385\pi\)
\(390\) 0 0
\(391\) −0.0940872 −0.00475819
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.71591i 0.388230i
\(396\) 0 0
\(397\) − 20.7770i − 1.04277i −0.853322 0.521385i \(-0.825416\pi\)
0.853322 0.521385i \(-0.174584\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.38157 0.368618 0.184309 0.982868i \(-0.440995\pi\)
0.184309 + 0.982868i \(0.440995\pi\)
\(402\) 0 0
\(403\) − 1.81338i − 0.0903308i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.16213 0.355014
\(408\) 0 0
\(409\) 31.9846 1.58154 0.790769 0.612114i \(-0.209681\pi\)
0.790769 + 0.612114i \(0.209681\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.54291i 0.420369i
\(414\) 0 0
\(415\) 11.1424 0.546958
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 14.6646i − 0.716413i −0.933642 0.358207i \(-0.883388\pi\)
0.933642 0.358207i \(-0.116612\pi\)
\(420\) 0 0
\(421\) 28.4958i 1.38880i 0.719589 + 0.694400i \(0.244330\pi\)
−0.719589 + 0.694400i \(0.755670\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.0541901 −0.00262861
\(426\) 0 0
\(427\) − 3.92502i − 0.189945i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −35.4443 −1.70729 −0.853645 0.520855i \(-0.825613\pi\)
−0.853645 + 0.520855i \(0.825613\pi\)
\(432\) 0 0
\(433\) −31.5907 −1.51815 −0.759077 0.651001i \(-0.774349\pi\)
−0.759077 + 0.651001i \(0.774349\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 13.9901i − 0.669237i
\(438\) 0 0
\(439\) −23.3732 −1.11554 −0.557771 0.829995i \(-0.688343\pi\)
−0.557771 + 0.829995i \(0.688343\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 3.52932i − 0.167683i −0.996479 0.0838416i \(-0.973281\pi\)
0.996479 0.0838416i \(-0.0267190\pi\)
\(444\) 0 0
\(445\) 0.301929i 0.0143128i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.94601 −0.139031 −0.0695154 0.997581i \(-0.522145\pi\)
−0.0695154 + 0.997581i \(0.522145\pi\)
\(450\) 0 0
\(451\) 19.0331i 0.896233i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.34521 0.0630646
\(456\) 0 0
\(457\) −22.1796 −1.03752 −0.518758 0.854921i \(-0.673605\pi\)
−0.518758 + 0.854921i \(0.673605\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 23.4545i − 1.09238i −0.837660 0.546192i \(-0.816077\pi\)
0.837660 0.546192i \(-0.183923\pi\)
\(462\) 0 0
\(463\) 29.2354 1.35868 0.679342 0.733822i \(-0.262265\pi\)
0.679342 + 0.733822i \(0.262265\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.43057i 0.251297i 0.992075 + 0.125648i \(0.0401011\pi\)
−0.992075 + 0.125648i \(0.959899\pi\)
\(468\) 0 0
\(469\) 12.7403i 0.588291i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.07611 0.141440
\(474\) 0 0
\(475\) − 8.05768i − 0.369712i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.07525 0.231894 0.115947 0.993255i \(-0.463010\pi\)
0.115947 + 0.993255i \(0.463010\pi\)
\(480\) 0 0
\(481\) −4.93276 −0.224914
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.34962i 0.333729i
\(486\) 0 0
\(487\) 3.23366 0.146531 0.0732656 0.997312i \(-0.476658\pi\)
0.0732656 + 0.997312i \(0.476658\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.6437i 1.47319i 0.676335 + 0.736594i \(0.263567\pi\)
−0.676335 + 0.736594i \(0.736433\pi\)
\(492\) 0 0
\(493\) − 0.00742632i 0 0.000334465i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.22205 0.189385
\(498\) 0 0
\(499\) − 1.77339i − 0.0793877i −0.999212 0.0396939i \(-0.987362\pi\)
0.999212 0.0396939i \(-0.0126383\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −25.3666 −1.13104 −0.565519 0.824735i \(-0.691324\pi\)
−0.565519 + 0.824735i \(0.691324\pi\)
\(504\) 0 0
\(505\) 3.92314 0.174578
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 14.3204i − 0.634739i −0.948302 0.317369i \(-0.897201\pi\)
0.948302 0.317369i \(-0.102799\pi\)
\(510\) 0 0
\(511\) −6.51561 −0.288234
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 3.42110i − 0.150752i
\(516\) 0 0
\(517\) − 11.0995i − 0.488154i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.72827 0.250960 0.125480 0.992096i \(-0.459953\pi\)
0.125480 + 0.992096i \(0.459953\pi\)
\(522\) 0 0
\(523\) 11.3779i 0.497519i 0.968565 + 0.248759i \(0.0800229\pi\)
−0.968565 + 0.248759i \(0.919977\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.0267231 0.00116408
\(528\) 0 0
\(529\) 12.4178 0.539905
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 13.1086i − 0.567797i
\(534\) 0 0
\(535\) −2.57777 −0.111447
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 1.55766i − 0.0670933i
\(540\) 0 0
\(541\) − 43.3953i − 1.86571i −0.360252 0.932855i \(-0.617309\pi\)
0.360252 0.932855i \(-0.382691\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −17.3265 −0.742187
\(546\) 0 0
\(547\) − 18.5677i − 0.793898i −0.917841 0.396949i \(-0.870069\pi\)
0.917841 0.396949i \(-0.129931\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.10424 0.0470422
\(552\) 0 0
\(553\) −6.15343 −0.261671
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 17.4689i − 0.740182i −0.928996 0.370091i \(-0.879327\pi\)
0.928996 0.370091i \(-0.120673\pi\)
\(558\) 0 0
\(559\) −2.11860 −0.0896073
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 26.8370i − 1.13105i −0.824733 0.565523i \(-0.808675\pi\)
0.824733 0.565523i \(-0.191325\pi\)
\(564\) 0 0
\(565\) 4.66534i 0.196272i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.4375 1.35985 0.679926 0.733281i \(-0.262012\pi\)
0.679926 + 0.733281i \(0.262012\pi\)
\(570\) 0 0
\(571\) 34.0452i 1.42475i 0.701800 + 0.712374i \(0.252380\pi\)
−0.701800 + 0.712374i \(0.747620\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 20.3991 0.850702
\(576\) 0 0
\(577\) 6.09534 0.253752 0.126876 0.991919i \(-0.459505\pi\)
0.126876 + 0.991919i \(0.459505\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.88604i 0.368655i
\(582\) 0 0
\(583\) 2.90870 0.120466
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 10.7735i − 0.444669i −0.974970 0.222334i \(-0.928632\pi\)
0.974970 0.222334i \(-0.0713677\pi\)
\(588\) 0 0
\(589\) 3.97353i 0.163727i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.4720 0.553230 0.276615 0.960981i \(-0.410787\pi\)
0.276615 + 0.960981i \(0.410787\pi\)
\(594\) 0 0
\(595\) 0.0198239i 0 0.000812701i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −37.4766 −1.53125 −0.765626 0.643285i \(-0.777571\pi\)
−0.765626 + 0.643285i \(0.777571\pi\)
\(600\) 0 0
\(601\) −3.99803 −0.163083 −0.0815415 0.996670i \(-0.525984\pi\)
−0.0815415 + 0.996670i \(0.525984\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.7507i 0.437079i
\(606\) 0 0
\(607\) −36.8295 −1.49486 −0.747432 0.664338i \(-0.768713\pi\)
−0.747432 + 0.664338i \(0.768713\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.64451i 0.309264i
\(612\) 0 0
\(613\) − 9.76469i − 0.394392i −0.980364 0.197196i \(-0.936816\pi\)
0.980364 0.197196i \(-0.0631836\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.4832 0.583070 0.291535 0.956560i \(-0.405834\pi\)
0.291535 + 0.956560i \(0.405834\pi\)
\(618\) 0 0
\(619\) − 6.72500i − 0.270301i −0.990825 0.135150i \(-0.956848\pi\)
0.990825 0.135150i \(-0.0431518\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.240788 −0.00964698
\(624\) 0 0
\(625\) 3.88743 0.155497
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 0.0726923i − 0.00289843i
\(630\) 0 0
\(631\) 23.2251 0.924577 0.462288 0.886730i \(-0.347029\pi\)
0.462288 + 0.886730i \(0.347029\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.9062i 0.988373i
\(636\) 0 0
\(637\) 1.07281i 0.0425061i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30.3809 1.19998 0.599988 0.800009i \(-0.295172\pi\)
0.599988 + 0.800009i \(0.295172\pi\)
\(642\) 0 0
\(643\) 5.74408i 0.226524i 0.993565 + 0.113262i \(0.0361300\pi\)
−0.993565 + 0.113262i \(0.963870\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.19948 −0.165099 −0.0825494 0.996587i \(-0.526306\pi\)
−0.0825494 + 0.996587i \(0.526306\pi\)
\(648\) 0 0
\(649\) −13.3070 −0.522345
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 45.2069i − 1.76908i −0.466460 0.884542i \(-0.654471\pi\)
0.466460 0.884542i \(-0.345529\pi\)
\(654\) 0 0
\(655\) 9.08839 0.355113
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 47.0951i − 1.83456i −0.398239 0.917282i \(-0.630379\pi\)
0.398239 0.917282i \(-0.369621\pi\)
\(660\) 0 0
\(661\) 41.9805i 1.63285i 0.577449 + 0.816427i \(0.304048\pi\)
−0.577449 + 0.816427i \(0.695952\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.94767 −0.114306
\(666\) 0 0
\(667\) 2.79554i 0.108244i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.11386 0.236023
\(672\) 0 0
\(673\) 30.0898 1.15987 0.579937 0.814661i \(-0.303077\pi\)
0.579937 + 0.814661i \(0.303077\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 40.5494i − 1.55844i −0.626749 0.779221i \(-0.715615\pi\)
0.626749 0.779221i \(-0.284385\pi\)
\(678\) 0 0
\(679\) −5.86131 −0.224936
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 50.1042i 1.91718i 0.284786 + 0.958591i \(0.408077\pi\)
−0.284786 + 0.958591i \(0.591923\pi\)
\(684\) 0 0
\(685\) 16.4924i 0.630141i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.00330 −0.0763198
\(690\) 0 0
\(691\) − 43.2174i − 1.64407i −0.569438 0.822034i \(-0.692839\pi\)
0.569438 0.822034i \(-0.307161\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.63933 −0.251844
\(696\) 0 0
\(697\) 0.193177 0.00731709
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19.4337i 0.734002i 0.930221 + 0.367001i \(0.119615\pi\)
−0.930221 + 0.367001i \(0.880385\pi\)
\(702\) 0 0
\(703\) 10.8088 0.407662
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.12870i 0.117667i
\(708\) 0 0
\(709\) − 24.0279i − 0.902387i −0.892426 0.451194i \(-0.850998\pi\)
0.892426 0.451194i \(-0.149002\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10.0595 −0.376733
\(714\) 0 0
\(715\) 2.09539i 0.0783631i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −36.0898 −1.34592 −0.672961 0.739678i \(-0.734978\pi\)
−0.672961 + 0.739678i \(0.734978\pi\)
\(720\) 0 0
\(721\) 2.72832 0.101608
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.61011i 0.0597979i
\(726\) 0 0
\(727\) −36.3230 −1.34714 −0.673572 0.739122i \(-0.735241\pi\)
−0.673572 + 0.739122i \(0.735241\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 0.0312211i − 0.00115475i
\(732\) 0 0
\(733\) 11.9814i 0.442544i 0.975212 + 0.221272i \(0.0710209\pi\)
−0.975212 + 0.221272i \(0.928979\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19.8451 −0.731002
\(738\) 0 0
\(739\) − 36.7377i − 1.35142i −0.737169 0.675708i \(-0.763838\pi\)
0.737169 0.675708i \(-0.236162\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13.2966 −0.487805 −0.243903 0.969800i \(-0.578428\pi\)
−0.243903 + 0.969800i \(0.578428\pi\)
\(744\) 0 0
\(745\) −8.26795 −0.302914
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 2.05577i − 0.0751161i
\(750\) 0 0
\(751\) 13.5953 0.496101 0.248051 0.968747i \(-0.420210\pi\)
0.248051 + 0.968747i \(0.420210\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.9321i 0.434254i
\(756\) 0 0
\(757\) 8.04991i 0.292579i 0.989242 + 0.146290i \(0.0467331\pi\)
−0.989242 + 0.146290i \(0.953267\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 53.5813 1.94232 0.971160 0.238429i \(-0.0766325\pi\)
0.971160 + 0.238429i \(0.0766325\pi\)
\(762\) 0 0
\(763\) − 13.8179i − 0.500241i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.16489 0.330925
\(768\) 0 0
\(769\) 3.40107 0.122646 0.0613229 0.998118i \(-0.480468\pi\)
0.0613229 + 0.998118i \(0.480468\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.6537i 0.455123i 0.973764 + 0.227561i \(0.0730752\pi\)
−0.973764 + 0.227561i \(0.926925\pi\)
\(774\) 0 0
\(775\) −5.79386 −0.208122
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 28.7240i 1.02914i
\(780\) 0 0
\(781\) 6.57654i 0.235327i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −20.0764 −0.716558
\(786\) 0 0
\(787\) − 37.3539i − 1.33152i −0.746165 0.665761i \(-0.768107\pi\)
0.746165 0.665761i \(-0.231893\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.72061 −0.132290
\(792\) 0 0
\(793\) −4.21079 −0.149529
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.3333i 0.543134i 0.962420 + 0.271567i \(0.0875418\pi\)
−0.962420 + 0.271567i \(0.912458\pi\)
\(798\) 0 0
\(799\) −0.112654 −0.00398542
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 10.1491i − 0.358155i
\(804\) 0 0
\(805\) − 7.46244i − 0.263016i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 48.1700 1.69357 0.846784 0.531938i \(-0.178536\pi\)
0.846784 + 0.531938i \(0.178536\pi\)
\(810\) 0 0
\(811\) − 21.4720i − 0.753984i −0.926216 0.376992i \(-0.876958\pi\)
0.926216 0.376992i \(-0.123042\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −15.7682 −0.552336
\(816\) 0 0
\(817\) 4.64235 0.162415
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 24.2783i − 0.847318i −0.905822 0.423659i \(-0.860746\pi\)
0.905822 0.423659i \(-0.139254\pi\)
\(822\) 0 0
\(823\) −15.7002 −0.547275 −0.273637 0.961833i \(-0.588227\pi\)
−0.273637 + 0.961833i \(0.588227\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.47290i 0.0512177i 0.999672 + 0.0256089i \(0.00815244\pi\)
−0.999672 + 0.0256089i \(0.991848\pi\)
\(828\) 0 0
\(829\) 25.1759i 0.874395i 0.899365 + 0.437198i \(0.144029\pi\)
−0.899365 + 0.437198i \(0.855971\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.0158095 −0.000547768 0
\(834\) 0 0
\(835\) 21.6729i 0.750020i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 14.4703 0.499570 0.249785 0.968301i \(-0.419640\pi\)
0.249785 + 0.968301i \(0.419640\pi\)
\(840\) 0 0
\(841\) 28.7793 0.992391
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 14.8578i 0.511124i
\(846\) 0 0
\(847\) −8.57368 −0.294595
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 27.3640i 0.938026i
\(852\) 0 0
\(853\) 0.552410i 0.0189142i 0.999955 + 0.00945708i \(0.00301033\pi\)
−0.999955 + 0.00945708i \(0.996990\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −47.3043 −1.61588 −0.807942 0.589262i \(-0.799419\pi\)
−0.807942 + 0.589262i \(0.799419\pi\)
\(858\) 0 0
\(859\) − 42.9031i − 1.46384i −0.681393 0.731918i \(-0.738625\pi\)
0.681393 0.731918i \(-0.261375\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.39600 −0.0815609 −0.0407805 0.999168i \(-0.512984\pi\)
−0.0407805 + 0.999168i \(0.512984\pi\)
\(864\) 0 0
\(865\) 11.0184 0.374638
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 9.58498i − 0.325148i
\(870\) 0 0
\(871\) 13.6678 0.463117
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 10.5676i − 0.357252i
\(876\) 0 0
\(877\) 36.8060i 1.24285i 0.783473 + 0.621426i \(0.213446\pi\)
−0.783473 + 0.621426i \(0.786554\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 45.1322 1.52054 0.760271 0.649606i \(-0.225066\pi\)
0.760271 + 0.649606i \(0.225066\pi\)
\(882\) 0 0
\(883\) 38.0204i 1.27949i 0.768588 + 0.639744i \(0.220960\pi\)
−0.768588 + 0.639744i \(0.779040\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −30.4326 −1.02183 −0.510913 0.859633i \(-0.670692\pi\)
−0.510913 + 0.859633i \(0.670692\pi\)
\(888\) 0 0
\(889\) −19.8627 −0.666173
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 16.7509i − 0.560547i
\(894\) 0 0
\(895\) −14.6664 −0.490244
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 0.794002i − 0.0264814i
\(900\) 0 0
\(901\) − 0.0295220i 0 0.000983519i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 30.4231 1.01130
\(906\) 0 0
\(907\) 8.80211i 0.292269i 0.989265 + 0.146135i \(0.0466833\pi\)
−0.989265 + 0.146135i \(0.953317\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 31.9516 1.05860 0.529302 0.848434i \(-0.322454\pi\)
0.529302 + 0.848434i \(0.322454\pi\)
\(912\) 0 0
\(913\) −13.8415 −0.458086
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.24798i 0.239349i
\(918\) 0 0
\(919\) −55.5377 −1.83202 −0.916009 0.401157i \(-0.868608\pi\)
−0.916009 + 0.401157i \(0.868608\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 4.52944i − 0.149088i
\(924\) 0 0
\(925\) 15.7605i 0.518201i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 22.4706 0.737235 0.368617 0.929581i \(-0.379831\pi\)
0.368617 + 0.929581i \(0.379831\pi\)
\(930\) 0 0
\(931\) − 2.35077i − 0.0770433i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.0308790 −0.00100985
\(936\) 0 0
\(937\) −47.5975 −1.55494 −0.777471 0.628919i \(-0.783498\pi\)
−0.777471 + 0.628919i \(0.783498\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 5.03276i − 0.164063i −0.996630 0.0820317i \(-0.973859\pi\)
0.996630 0.0820317i \(-0.0261409\pi\)
\(942\) 0 0
\(943\) −72.7188 −2.36805
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 18.9034i − 0.614277i −0.951665 0.307138i \(-0.900629\pi\)
0.951665 0.307138i \(-0.0993714\pi\)
\(948\) 0 0
\(949\) 6.98999i 0.226905i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −59.5108 −1.92774 −0.963872 0.266365i \(-0.914177\pi\)
−0.963872 + 0.266365i \(0.914177\pi\)
\(954\) 0 0
\(955\) − 13.1473i − 0.425437i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13.1526 −0.424721
\(960\) 0 0
\(961\) −28.1428 −0.907834
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 15.9364i 0.513010i
\(966\) 0 0
\(967\) 26.3394 0.847018 0.423509 0.905892i \(-0.360798\pi\)
0.423509 + 0.905892i \(0.360798\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 50.9352i 1.63459i 0.576222 + 0.817293i \(0.304526\pi\)
−0.576222 + 0.817293i \(0.695474\pi\)
\(972\) 0 0
\(973\) − 5.29486i − 0.169745i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.5022 0.751903 0.375951 0.926639i \(-0.377316\pi\)
0.375951 + 0.926639i \(0.377316\pi\)
\(978\) 0 0
\(979\) − 0.375067i − 0.0119872i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 41.3639 1.31930 0.659651 0.751572i \(-0.270704\pi\)
0.659651 + 0.751572i \(0.270704\pi\)
\(984\) 0 0
\(985\) −20.0774 −0.639720
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.7527i 0.373715i
\(990\) 0 0
\(991\) 8.06637 0.256237 0.128118 0.991759i \(-0.459106\pi\)
0.128118 + 0.991759i \(0.459106\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.0299i 0.444778i
\(996\) 0 0
\(997\) − 38.8503i − 1.23040i −0.788371 0.615200i \(-0.789075\pi\)
0.788371 0.615200i \(-0.210925\pi\)
\(998\) 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.3025.16 24
3.2 odd 2 inner 6048.2.c.f.3025.10 24
4.3 odd 2 1512.2.c.g.757.9 24
8.3 odd 2 1512.2.c.g.757.10 yes 24
8.5 even 2 inner 6048.2.c.f.3025.9 24
12.11 even 2 1512.2.c.g.757.16 yes 24
24.5 odd 2 inner 6048.2.c.f.3025.15 24
24.11 even 2 1512.2.c.g.757.15 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.g.757.9 24 4.3 odd 2
1512.2.c.g.757.10 yes 24 8.3 odd 2
1512.2.c.g.757.15 yes 24 24.11 even 2
1512.2.c.g.757.16 yes 24 12.11 even 2
6048.2.c.f.3025.9 24 8.5 even 2 inner
6048.2.c.f.3025.10 24 3.2 odd 2 inner
6048.2.c.f.3025.15 24 24.5 odd 2 inner
6048.2.c.f.3025.16 24 1.1 even 1 trivial