Properties

Label 6048.2.j.d.5615.2
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 48
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.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(48\)
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 5615.2
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.d.5615.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.18263 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-4.18263 q^{5} -1.00000i q^{7} -1.86730i q^{11} +3.07604i q^{13} -0.504885i q^{17} -3.05596 q^{19} -5.97229 q^{23} +12.4944 q^{25} -3.52884 q^{29} -10.8980i q^{31} +4.18263i q^{35} +11.3660i q^{37} +7.26960i q^{41} -9.02120 q^{43} +0.327367 q^{47} -1.00000 q^{49} -8.32940 q^{53} +7.81021i q^{55} -5.98008i q^{59} -13.2783i q^{61} -12.8659i q^{65} +9.21499 q^{67} -9.02460 q^{71} +0.416491 q^{73} -1.86730 q^{77} +6.00459i q^{79} +2.62368i q^{83} +2.11175i q^{85} -5.69486i q^{89} +3.07604 q^{91} +12.7819 q^{95} +6.21289 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q + O(q^{10}) \) \( 48q - 16q^{19} + 48q^{25} - 64q^{43} - 48q^{49} - 32q^{67} + O(q^{100}) \)

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
\(5\) −4.18263 −1.87053 −0.935264 0.353951i \(-0.884838\pi\)
−0.935264 + 0.353951i \(0.884838\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.86730i − 0.563011i −0.959560 0.281506i \(-0.909166\pi\)
0.959560 0.281506i \(-0.0908338\pi\)
\(12\) 0 0
\(13\) 3.07604i 0.853139i 0.904455 + 0.426569i \(0.140278\pi\)
−0.904455 + 0.426569i \(0.859722\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.504885i − 0.122453i −0.998124 0.0612263i \(-0.980499\pi\)
0.998124 0.0612263i \(-0.0195011\pi\)
\(18\) 0 0
\(19\) −3.05596 −0.701085 −0.350543 0.936547i \(-0.614003\pi\)
−0.350543 + 0.936547i \(0.614003\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.97229 −1.24531 −0.622654 0.782497i \(-0.713946\pi\)
−0.622654 + 0.782497i \(0.713946\pi\)
\(24\) 0 0
\(25\) 12.4944 2.49887
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.52884 −0.655289 −0.327644 0.944801i \(-0.606255\pi\)
−0.327644 + 0.944801i \(0.606255\pi\)
\(30\) 0 0
\(31\) − 10.8980i − 1.95734i −0.205435 0.978671i \(-0.565861\pi\)
0.205435 0.978671i \(-0.434139\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.18263i 0.706993i
\(36\) 0 0
\(37\) 11.3660i 1.86856i 0.356542 + 0.934279i \(0.383956\pi\)
−0.356542 + 0.934279i \(0.616044\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.26960i 1.13532i 0.823263 + 0.567660i \(0.192151\pi\)
−0.823263 + 0.567660i \(0.807849\pi\)
\(42\) 0 0
\(43\) −9.02120 −1.37572 −0.687860 0.725844i \(-0.741450\pi\)
−0.687860 + 0.725844i \(0.741450\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.327367 0.0477513 0.0238757 0.999715i \(-0.492399\pi\)
0.0238757 + 0.999715i \(0.492399\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.32940 −1.14413 −0.572066 0.820208i \(-0.693858\pi\)
−0.572066 + 0.820208i \(0.693858\pi\)
\(54\) 0 0
\(55\) 7.81021i 1.05313i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 5.98008i − 0.778540i −0.921124 0.389270i \(-0.872727\pi\)
0.921124 0.389270i \(-0.127273\pi\)
\(60\) 0 0
\(61\) − 13.2783i − 1.70011i −0.526691 0.850057i \(-0.676568\pi\)
0.526691 0.850057i \(-0.323432\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 12.8659i − 1.59582i
\(66\) 0 0
\(67\) 9.21499 1.12579 0.562895 0.826528i \(-0.309688\pi\)
0.562895 + 0.826528i \(0.309688\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.02460 −1.07102 −0.535512 0.844528i \(-0.679881\pi\)
−0.535512 + 0.844528i \(0.679881\pi\)
\(72\) 0 0
\(73\) 0.416491 0.0487466 0.0243733 0.999703i \(-0.492241\pi\)
0.0243733 + 0.999703i \(0.492241\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.86730 −0.212798
\(78\) 0 0
\(79\) 6.00459i 0.675569i 0.941224 + 0.337784i \(0.109677\pi\)
−0.941224 + 0.337784i \(0.890323\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.62368i 0.287987i 0.989579 + 0.143993i \(0.0459944\pi\)
−0.989579 + 0.143993i \(0.954006\pi\)
\(84\) 0 0
\(85\) 2.11175i 0.229051i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 5.69486i − 0.603654i −0.953363 0.301827i \(-0.902403\pi\)
0.953363 0.301827i \(-0.0975966\pi\)
\(90\) 0 0
\(91\) 3.07604 0.322456
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.7819 1.31140
\(96\) 0 0
\(97\) 6.21289 0.630824 0.315412 0.948955i \(-0.397857\pi\)
0.315412 + 0.948955i \(0.397857\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.85216 0.184296 0.0921482 0.995745i \(-0.470627\pi\)
0.0921482 + 0.995745i \(0.470627\pi\)
\(102\) 0 0
\(103\) 2.74001i 0.269982i 0.990847 + 0.134991i \(0.0431005\pi\)
−0.990847 + 0.134991i \(0.956900\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 4.22506i − 0.408452i −0.978924 0.204226i \(-0.934532\pi\)
0.978924 0.204226i \(-0.0654677\pi\)
\(108\) 0 0
\(109\) 13.9420i 1.33540i 0.744432 + 0.667698i \(0.232720\pi\)
−0.744432 + 0.667698i \(0.767280\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 7.49447i − 0.705020i −0.935808 0.352510i \(-0.885328\pi\)
0.935808 0.352510i \(-0.114672\pi\)
\(114\) 0 0
\(115\) 24.9799 2.32938
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.504885 −0.0462827
\(120\) 0 0
\(121\) 7.51320 0.683018
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −31.3462 −2.80369
\(126\) 0 0
\(127\) 18.0939i 1.60557i 0.596267 + 0.802787i \(0.296650\pi\)
−0.596267 + 0.802787i \(0.703350\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 9.32625i − 0.814839i −0.913241 0.407419i \(-0.866429\pi\)
0.913241 0.407419i \(-0.133571\pi\)
\(132\) 0 0
\(133\) 3.05596i 0.264985i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.53069i 0.557955i 0.960298 + 0.278977i \(0.0899954\pi\)
−0.960298 + 0.278977i \(0.910005\pi\)
\(138\) 0 0
\(139\) 10.1589 0.861664 0.430832 0.902432i \(-0.358220\pi\)
0.430832 + 0.902432i \(0.358220\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.74387 0.480327
\(144\) 0 0
\(145\) 14.7598 1.22574
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.72516 −0.796716 −0.398358 0.917230i \(-0.630420\pi\)
−0.398358 + 0.917230i \(0.630420\pi\)
\(150\) 0 0
\(151\) − 12.4664i − 1.01450i −0.861800 0.507249i \(-0.830662\pi\)
0.861800 0.507249i \(-0.169338\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 45.5823i 3.66126i
\(156\) 0 0
\(157\) 11.3627i 0.906838i 0.891297 + 0.453419i \(0.149796\pi\)
−0.891297 + 0.453419i \(0.850204\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.97229i 0.470682i
\(162\) 0 0
\(163\) 3.59449 0.281542 0.140771 0.990042i \(-0.455042\pi\)
0.140771 + 0.990042i \(0.455042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.3801 1.57706 0.788531 0.614995i \(-0.210842\pi\)
0.788531 + 0.614995i \(0.210842\pi\)
\(168\) 0 0
\(169\) 3.53800 0.272154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.49761 0.341947 0.170974 0.985276i \(-0.445309\pi\)
0.170974 + 0.985276i \(0.445309\pi\)
\(174\) 0 0
\(175\) − 12.4944i − 0.944486i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.63430i 0.196897i 0.995142 + 0.0984485i \(0.0313880\pi\)
−0.995142 + 0.0984485i \(0.968612\pi\)
\(180\) 0 0
\(181\) − 10.3748i − 0.771151i −0.922676 0.385576i \(-0.874003\pi\)
0.922676 0.385576i \(-0.125997\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 47.5397i − 3.49519i
\(186\) 0 0
\(187\) −0.942770 −0.0689422
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.1707 1.31479 0.657395 0.753547i \(-0.271659\pi\)
0.657395 + 0.753547i \(0.271659\pi\)
\(192\) 0 0
\(193\) −16.7704 −1.20716 −0.603578 0.797304i \(-0.706259\pi\)
−0.603578 + 0.797304i \(0.706259\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.2240 1.36965 0.684826 0.728706i \(-0.259878\pi\)
0.684826 + 0.728706i \(0.259878\pi\)
\(198\) 0 0
\(199\) − 1.68931i − 0.119752i −0.998206 0.0598761i \(-0.980929\pi\)
0.998206 0.0598761i \(-0.0190706\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.52884i 0.247676i
\(204\) 0 0
\(205\) − 30.4060i − 2.12365i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.70638i 0.394719i
\(210\) 0 0
\(211\) 14.1544 0.974427 0.487213 0.873283i \(-0.338013\pi\)
0.487213 + 0.873283i \(0.338013\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 37.7323 2.57332
\(216\) 0 0
\(217\) −10.8980 −0.739806
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.55304 0.104469
\(222\) 0 0
\(223\) − 1.33032i − 0.0890851i −0.999007 0.0445425i \(-0.985817\pi\)
0.999007 0.0445425i \(-0.0141830\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 6.34646i − 0.421229i −0.977569 0.210615i \(-0.932453\pi\)
0.977569 0.210615i \(-0.0675465\pi\)
\(228\) 0 0
\(229\) − 13.9920i − 0.924619i −0.886719 0.462309i \(-0.847021\pi\)
0.886719 0.462309i \(-0.152979\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0797i 0.922392i 0.887298 + 0.461196i \(0.152580\pi\)
−0.887298 + 0.461196i \(0.847420\pi\)
\(234\) 0 0
\(235\) −1.36925 −0.0893202
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −13.1737 −0.852134 −0.426067 0.904691i \(-0.640101\pi\)
−0.426067 + 0.904691i \(0.640101\pi\)
\(240\) 0 0
\(241\) 7.46846 0.481086 0.240543 0.970639i \(-0.422675\pi\)
0.240543 + 0.970639i \(0.422675\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.18263 0.267218
\(246\) 0 0
\(247\) − 9.40024i − 0.598123i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.3203i 1.03013i 0.857151 + 0.515065i \(0.172232\pi\)
−0.857151 + 0.515065i \(0.827768\pi\)
\(252\) 0 0
\(253\) 11.1520i 0.701123i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.6939i 0.854205i 0.904203 + 0.427102i \(0.140466\pi\)
−0.904203 + 0.427102i \(0.859534\pi\)
\(258\) 0 0
\(259\) 11.3660 0.706249
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.63480 −0.285794 −0.142897 0.989738i \(-0.545642\pi\)
−0.142897 + 0.989738i \(0.545642\pi\)
\(264\) 0 0
\(265\) 34.8388 2.14013
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 28.6544 1.74709 0.873545 0.486743i \(-0.161815\pi\)
0.873545 + 0.486743i \(0.161815\pi\)
\(270\) 0 0
\(271\) 8.27458i 0.502645i 0.967903 + 0.251322i \(0.0808654\pi\)
−0.967903 + 0.251322i \(0.919135\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 23.3307i − 1.40689i
\(276\) 0 0
\(277\) 12.5278i 0.752724i 0.926473 + 0.376362i \(0.122825\pi\)
−0.926473 + 0.376362i \(0.877175\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.63112i 0.514889i 0.966293 + 0.257445i \(0.0828805\pi\)
−0.966293 + 0.257445i \(0.917119\pi\)
\(282\) 0 0
\(283\) −8.73498 −0.519241 −0.259620 0.965711i \(-0.583597\pi\)
−0.259620 + 0.965711i \(0.583597\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.26960 0.429111
\(288\) 0 0
\(289\) 16.7451 0.985005
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 30.3572 1.77348 0.886742 0.462265i \(-0.152963\pi\)
0.886742 + 0.462265i \(0.152963\pi\)
\(294\) 0 0
\(295\) 25.0124i 1.45628i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 18.3710i − 1.06242i
\(300\) 0 0
\(301\) 9.02120i 0.519973i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 55.5382i 3.18011i
\(306\) 0 0
\(307\) 29.1705 1.66485 0.832425 0.554138i \(-0.186952\pi\)
0.832425 + 0.554138i \(0.186952\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.4778 0.764253 0.382127 0.924110i \(-0.375192\pi\)
0.382127 + 0.924110i \(0.375192\pi\)
\(312\) 0 0
\(313\) 20.7291 1.17168 0.585838 0.810428i \(-0.300765\pi\)
0.585838 + 0.810428i \(0.300765\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.453146 0.0254512 0.0127256 0.999919i \(-0.495949\pi\)
0.0127256 + 0.999919i \(0.495949\pi\)
\(318\) 0 0
\(319\) 6.58939i 0.368935i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.54291i 0.0858497i
\(324\) 0 0
\(325\) 38.4331i 2.13189i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 0.327367i − 0.0180483i
\(330\) 0 0
\(331\) −24.8798 −1.36752 −0.683759 0.729708i \(-0.739656\pi\)
−0.683759 + 0.729708i \(0.739656\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −38.5429 −2.10582
\(336\) 0 0
\(337\) 21.8036 1.18772 0.593860 0.804568i \(-0.297603\pi\)
0.593860 + 0.804568i \(0.297603\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −20.3498 −1.10201
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.7794i 1.11550i 0.830010 + 0.557748i \(0.188335\pi\)
−0.830010 + 0.557748i \(0.811665\pi\)
\(348\) 0 0
\(349\) − 30.5928i − 1.63760i −0.574081 0.818799i \(-0.694640\pi\)
0.574081 0.818799i \(-0.305360\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 19.5174i − 1.03881i −0.854529 0.519403i \(-0.826154\pi\)
0.854529 0.519403i \(-0.173846\pi\)
\(354\) 0 0
\(355\) 37.7466 2.00338
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.3969 1.07651 0.538254 0.842783i \(-0.319084\pi\)
0.538254 + 0.842783i \(0.319084\pi\)
\(360\) 0 0
\(361\) −9.66111 −0.508479
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.74203 −0.0911818
\(366\) 0 0
\(367\) 25.6785i 1.34041i 0.742177 + 0.670203i \(0.233793\pi\)
−0.742177 + 0.670203i \(0.766207\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.32940i 0.432441i
\(372\) 0 0
\(373\) − 12.4313i − 0.643669i −0.946796 0.321835i \(-0.895700\pi\)
0.946796 0.321835i \(-0.104300\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 10.8548i − 0.559053i
\(378\) 0 0
\(379\) 9.90048 0.508553 0.254277 0.967132i \(-0.418163\pi\)
0.254277 + 0.967132i \(0.418163\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −23.6536 −1.20864 −0.604320 0.796741i \(-0.706555\pi\)
−0.604320 + 0.796741i \(0.706555\pi\)
\(384\) 0 0
\(385\) 7.81021 0.398045
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.14513 −0.159465 −0.0797323 0.996816i \(-0.525407\pi\)
−0.0797323 + 0.996816i \(0.525407\pi\)
\(390\) 0 0
\(391\) 3.01532i 0.152491i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 25.1150i − 1.26367i
\(396\) 0 0
\(397\) 12.2176i 0.613183i 0.951841 + 0.306592i \(0.0991886\pi\)
−0.951841 + 0.306592i \(0.900811\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 35.0797i − 1.75180i −0.482494 0.875899i \(-0.660269\pi\)
0.482494 0.875899i \(-0.339731\pi\)
\(402\) 0 0
\(403\) 33.5227 1.66988
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.2237 1.05202
\(408\) 0 0
\(409\) −27.3489 −1.35231 −0.676157 0.736757i \(-0.736356\pi\)
−0.676157 + 0.736757i \(0.736356\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.98008 −0.294260
\(414\) 0 0
\(415\) − 10.9739i − 0.538687i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.6442i 0.715417i 0.933833 + 0.357709i \(0.116442\pi\)
−0.933833 + 0.357709i \(0.883558\pi\)
\(420\) 0 0
\(421\) − 20.2168i − 0.985305i −0.870226 0.492653i \(-0.836027\pi\)
0.870226 0.492653i \(-0.163973\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 6.30822i − 0.305994i
\(426\) 0 0
\(427\) −13.2783 −0.642583
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.6656 −0.947260 −0.473630 0.880724i \(-0.657057\pi\)
−0.473630 + 0.880724i \(0.657057\pi\)
\(432\) 0 0
\(433\) 11.9996 0.576666 0.288333 0.957530i \(-0.406899\pi\)
0.288333 + 0.957530i \(0.406899\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 18.2511 0.873068
\(438\) 0 0
\(439\) 7.82113i 0.373282i 0.982428 + 0.186641i \(0.0597601\pi\)
−0.982428 + 0.186641i \(0.940240\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 29.9854i − 1.42465i −0.701850 0.712325i \(-0.747642\pi\)
0.701850 0.712325i \(-0.252358\pi\)
\(444\) 0 0
\(445\) 23.8195i 1.12915i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 42.0892i 1.98631i 0.116797 + 0.993156i \(0.462737\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(450\) 0 0
\(451\) 13.5745 0.639198
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.8659 −0.603163
\(456\) 0 0
\(457\) −12.1655 −0.569079 −0.284540 0.958664i \(-0.591841\pi\)
−0.284540 + 0.958664i \(0.591841\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 29.0054 1.35091 0.675457 0.737399i \(-0.263946\pi\)
0.675457 + 0.737399i \(0.263946\pi\)
\(462\) 0 0
\(463\) 5.21317i 0.242277i 0.992636 + 0.121138i \(0.0386545\pi\)
−0.992636 + 0.121138i \(0.961346\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.3061i 0.569460i 0.958608 + 0.284730i \(0.0919039\pi\)
−0.958608 + 0.284730i \(0.908096\pi\)
\(468\) 0 0
\(469\) − 9.21499i − 0.425509i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.8452i 0.774545i
\(474\) 0 0
\(475\) −38.1823 −1.75192
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.9609 −0.912038 −0.456019 0.889970i \(-0.650725\pi\)
−0.456019 + 0.889970i \(0.650725\pi\)
\(480\) 0 0
\(481\) −34.9622 −1.59414
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −25.9862 −1.17997
\(486\) 0 0
\(487\) − 5.61420i − 0.254404i −0.991877 0.127202i \(-0.959400\pi\)
0.991877 0.127202i \(-0.0405996\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 9.07131i − 0.409382i −0.978827 0.204691i \(-0.934381\pi\)
0.978827 0.204691i \(-0.0656190\pi\)
\(492\) 0 0
\(493\) 1.78166i 0.0802418i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.02460i 0.404809i
\(498\) 0 0
\(499\) 36.4346 1.63103 0.815517 0.578732i \(-0.196452\pi\)
0.815517 + 0.578732i \(0.196452\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.10995 −0.183253 −0.0916267 0.995793i \(-0.529207\pi\)
−0.0916267 + 0.995793i \(0.529207\pi\)
\(504\) 0 0
\(505\) −7.74688 −0.344732
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.5175 0.776450 0.388225 0.921565i \(-0.373088\pi\)
0.388225 + 0.921565i \(0.373088\pi\)
\(510\) 0 0
\(511\) − 0.416491i − 0.0184245i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 11.4605i − 0.505008i
\(516\) 0 0
\(517\) − 0.611291i − 0.0268845i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 40.7567i − 1.78558i −0.450472 0.892791i \(-0.648744\pi\)
0.450472 0.892791i \(-0.351256\pi\)
\(522\) 0 0
\(523\) −7.03412 −0.307581 −0.153790 0.988103i \(-0.549148\pi\)
−0.153790 + 0.988103i \(0.549148\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.50225 −0.239682
\(528\) 0 0
\(529\) 12.6683 0.550794
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −22.3615 −0.968586
\(534\) 0 0
\(535\) 17.6718i 0.764020i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.86730i 0.0804302i
\(540\) 0 0
\(541\) − 19.6764i − 0.845953i −0.906141 0.422976i \(-0.860985\pi\)
0.906141 0.422976i \(-0.139015\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 58.3140i − 2.49790i
\(546\) 0 0
\(547\) −17.8078 −0.761408 −0.380704 0.924697i \(-0.624318\pi\)
−0.380704 + 0.924697i \(0.624318\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.7840 0.459413
\(552\) 0 0
\(553\) 6.00459 0.255341
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.99646 −0.254078 −0.127039 0.991898i \(-0.540547\pi\)
−0.127039 + 0.991898i \(0.540547\pi\)
\(558\) 0 0
\(559\) − 27.7495i − 1.17368i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 20.4848i − 0.863333i −0.902033 0.431667i \(-0.857926\pi\)
0.902033 0.431667i \(-0.142074\pi\)
\(564\) 0 0
\(565\) 31.3466i 1.31876i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 38.1889i 1.60096i 0.599359 + 0.800480i \(0.295422\pi\)
−0.599359 + 0.800480i \(0.704578\pi\)
\(570\) 0 0
\(571\) −11.3551 −0.475198 −0.237599 0.971363i \(-0.576360\pi\)
−0.237599 + 0.971363i \(0.576360\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −74.6200 −3.11187
\(576\) 0 0
\(577\) −20.7869 −0.865371 −0.432685 0.901545i \(-0.642434\pi\)
−0.432685 + 0.901545i \(0.642434\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.62368 0.108849
\(582\) 0 0
\(583\) 15.5535i 0.644159i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.3849i 1.00647i 0.864148 + 0.503237i \(0.167858\pi\)
−0.864148 + 0.503237i \(0.832142\pi\)
\(588\) 0 0
\(589\) 33.3039i 1.37226i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 32.5053i 1.33483i 0.744684 + 0.667417i \(0.232600\pi\)
−0.744684 + 0.667417i \(0.767400\pi\)
\(594\) 0 0
\(595\) 2.11175 0.0865731
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.4800 0.428200 0.214100 0.976812i \(-0.431318\pi\)
0.214100 + 0.976812i \(0.431318\pi\)
\(600\) 0 0
\(601\) −22.4946 −0.917575 −0.458787 0.888546i \(-0.651716\pi\)
−0.458787 + 0.888546i \(0.651716\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −31.4249 −1.27761
\(606\) 0 0
\(607\) − 32.3672i − 1.31374i −0.754002 0.656872i \(-0.771879\pi\)
0.754002 0.656872i \(-0.228121\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.00699i 0.0407385i
\(612\) 0 0
\(613\) − 34.8714i − 1.40844i −0.709981 0.704220i \(-0.751297\pi\)
0.709981 0.704220i \(-0.248703\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.47324i − 0.0995688i −0.998760 0.0497844i \(-0.984147\pi\)
0.998760 0.0497844i \(-0.0158534\pi\)
\(618\) 0 0
\(619\) 15.2375 0.612447 0.306223 0.951960i \(-0.400935\pi\)
0.306223 + 0.951960i \(0.400935\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.69486 −0.228160
\(624\) 0 0
\(625\) 68.6375 2.74550
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.73852 0.228810
\(630\) 0 0
\(631\) 23.2064i 0.923832i 0.886924 + 0.461916i \(0.152838\pi\)
−0.886924 + 0.461916i \(0.847162\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 75.6800i − 3.00327i
\(636\) 0 0
\(637\) − 3.07604i − 0.121877i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 29.5988i 1.16908i 0.811364 + 0.584541i \(0.198725\pi\)
−0.811364 + 0.584541i \(0.801275\pi\)
\(642\) 0 0
\(643\) −28.8060 −1.13600 −0.567999 0.823029i \(-0.692282\pi\)
−0.567999 + 0.823029i \(0.692282\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27.3622 −1.07572 −0.537860 0.843034i \(-0.680767\pi\)
−0.537860 + 0.843034i \(0.680767\pi\)
\(648\) 0 0
\(649\) −11.1666 −0.438327
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.9292 1.56255 0.781275 0.624187i \(-0.214570\pi\)
0.781275 + 0.624187i \(0.214570\pi\)
\(654\) 0 0
\(655\) 39.0082i 1.52418i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 14.9809i 0.583572i 0.956484 + 0.291786i \(0.0942495\pi\)
−0.956484 + 0.291786i \(0.905751\pi\)
\(660\) 0 0
\(661\) 33.0963i 1.28730i 0.765322 + 0.643648i \(0.222580\pi\)
−0.765322 + 0.643648i \(0.777420\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 12.7819i − 0.495662i
\(666\) 0 0
\(667\) 21.0753 0.816037
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −24.7945 −0.957183
\(672\) 0 0
\(673\) −12.1189 −0.467150 −0.233575 0.972339i \(-0.575043\pi\)
−0.233575 + 0.972339i \(0.575043\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −27.7231 −1.06549 −0.532743 0.846277i \(-0.678839\pi\)
−0.532743 + 0.846277i \(0.678839\pi\)
\(678\) 0 0
\(679\) − 6.21289i − 0.238429i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 9.79813i − 0.374915i −0.982273 0.187457i \(-0.939975\pi\)
0.982273 0.187457i \(-0.0600247\pi\)
\(684\) 0 0
\(685\) − 27.3155i − 1.04367i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 25.6215i − 0.976103i
\(690\) 0 0
\(691\) 37.4765 1.42567 0.712837 0.701330i \(-0.247410\pi\)
0.712837 + 0.701330i \(0.247410\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −42.4908 −1.61177
\(696\) 0 0
\(697\) 3.67031 0.139023
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 22.6525 0.855575 0.427787 0.903879i \(-0.359293\pi\)
0.427787 + 0.903879i \(0.359293\pi\)
\(702\) 0 0
\(703\) − 34.7340i − 1.31002i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.85216i − 0.0696575i
\(708\) 0 0
\(709\) − 1.58535i − 0.0595393i −0.999557 0.0297696i \(-0.990523\pi\)
0.999557 0.0297696i \(-0.00947737\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 65.0861i 2.43749i
\(714\) 0 0
\(715\) −24.0245 −0.898465
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17.7228 −0.660948 −0.330474 0.943815i \(-0.607209\pi\)
−0.330474 + 0.943815i \(0.607209\pi\)
\(720\) 0 0
\(721\) 2.74001 0.102043
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −44.0906 −1.63749
\(726\) 0 0
\(727\) − 3.31812i − 0.123062i −0.998105 0.0615312i \(-0.980402\pi\)
0.998105 0.0615312i \(-0.0195984\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.55467i 0.168460i
\(732\) 0 0
\(733\) − 18.8547i − 0.696414i −0.937418 0.348207i \(-0.886791\pi\)
0.937418 0.348207i \(-0.113209\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 17.2071i − 0.633832i
\(738\) 0 0
\(739\) 33.3560 1.22702 0.613510 0.789687i \(-0.289757\pi\)
0.613510 + 0.789687i \(0.289757\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.0826 0.993565 0.496783 0.867875i \(-0.334515\pi\)
0.496783 + 0.867875i \(0.334515\pi\)
\(744\) 0 0
\(745\) 40.6767 1.49028
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.22506 −0.154380
\(750\) 0 0
\(751\) 44.3284i 1.61757i 0.588107 + 0.808783i \(0.299874\pi\)
−0.588107 + 0.808783i \(0.700126\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 52.1421i 1.89765i
\(756\) 0 0
\(757\) 12.3093i 0.447391i 0.974659 + 0.223695i \(0.0718121\pi\)
−0.974659 + 0.223695i \(0.928188\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 1.85054i − 0.0670820i −0.999437 0.0335410i \(-0.989322\pi\)
0.999437 0.0335410i \(-0.0106784\pi\)
\(762\) 0 0
\(763\) 13.9420 0.504733
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.3949 0.664203
\(768\) 0 0
\(769\) −31.2463 −1.12677 −0.563386 0.826194i \(-0.690502\pi\)
−0.563386 + 0.826194i \(0.690502\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 44.4521 1.59883 0.799415 0.600779i \(-0.205143\pi\)
0.799415 + 0.600779i \(0.205143\pi\)
\(774\) 0 0
\(775\) − 136.164i − 4.89115i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 22.2156i − 0.795956i
\(780\) 0 0
\(781\) 16.8516i 0.602998i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 47.5257i − 1.69627i
\(786\) 0 0
\(787\) −28.0995 −1.00164 −0.500820 0.865551i \(-0.666968\pi\)
−0.500820 + 0.865551i \(0.666968\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7.49447 −0.266473
\(792\) 0 0
\(793\) 40.8446 1.45043
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.9492 −1.52134 −0.760670 0.649139i \(-0.775129\pi\)
−0.760670 + 0.649139i \(0.775129\pi\)
\(798\) 0 0
\(799\) − 0.165282i − 0.00584727i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 0.777712i − 0.0274449i
\(804\) 0 0
\(805\) − 24.9799i − 0.880425i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 8.54301i − 0.300356i −0.988659 0.150178i \(-0.952015\pi\)
0.988659 0.150178i \(-0.0479847\pi\)
\(810\) 0 0
\(811\) −34.6880 −1.21806 −0.609030 0.793147i \(-0.708441\pi\)
−0.609030 + 0.793147i \(0.708441\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −15.0344 −0.526632
\(816\) 0 0
\(817\) 27.5684 0.964497
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.46951 0.295588 0.147794 0.989018i \(-0.452783\pi\)
0.147794 + 0.989018i \(0.452783\pi\)
\(822\) 0 0
\(823\) − 44.5109i − 1.55155i −0.631008 0.775777i \(-0.717358\pi\)
0.631008 0.775777i \(-0.282642\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.3895i 1.26539i 0.774403 + 0.632693i \(0.218051\pi\)
−0.774403 + 0.632693i \(0.781949\pi\)
\(828\) 0 0
\(829\) − 39.2607i − 1.36358i −0.731548 0.681790i \(-0.761202\pi\)
0.731548 0.681790i \(-0.238798\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.504885i 0.0174932i
\(834\) 0 0
\(835\) −85.2425 −2.94994
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −15.1101 −0.521659 −0.260830 0.965385i \(-0.583996\pi\)
−0.260830 + 0.965385i \(0.583996\pi\)
\(840\) 0 0
\(841\) −16.5473 −0.570596
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −14.7981 −0.509072
\(846\) 0 0
\(847\) − 7.51320i − 0.258157i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 67.8810i − 2.32693i
\(852\) 0 0
\(853\) − 23.4153i − 0.801724i −0.916138 0.400862i \(-0.868711\pi\)
0.916138 0.400862i \(-0.131289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 20.7800i − 0.709833i −0.934898 0.354916i \(-0.884509\pi\)
0.934898 0.354916i \(-0.115491\pi\)
\(858\) 0 0
\(859\) −12.0088 −0.409735 −0.204868 0.978790i \(-0.565676\pi\)
−0.204868 + 0.978790i \(0.565676\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.5254 −0.392328 −0.196164 0.980571i \(-0.562849\pi\)
−0.196164 + 0.980571i \(0.562849\pi\)
\(864\) 0 0
\(865\) −18.8118 −0.639622
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.2123 0.380353
\(870\) 0 0
\(871\) 28.3456i 0.960455i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 31.3462i 1.05969i
\(876\) 0 0
\(877\) − 7.25986i − 0.245148i −0.992459 0.122574i \(-0.960885\pi\)
0.992459 0.122574i \(-0.0391149\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 10.7411i − 0.361876i −0.983495 0.180938i \(-0.942087\pi\)
0.983495 0.180938i \(-0.0579133\pi\)
\(882\) 0 0
\(883\) −9.09009 −0.305906 −0.152953 0.988233i \(-0.548878\pi\)
−0.152953 + 0.988233i \(0.548878\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 29.0988 0.977041 0.488521 0.872552i \(-0.337537\pi\)
0.488521 + 0.872552i \(0.337537\pi\)
\(888\) 0 0
\(889\) 18.0939 0.606850
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.00042 −0.0334778
\(894\) 0 0
\(895\) − 11.0183i − 0.368301i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 38.4574i 1.28262i
\(900\) 0 0
\(901\) 4.20539i 0.140102i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 43.3938i 1.44246i
\(906\) 0 0
\(907\) −29.9962 −0.996008 −0.498004 0.867175i \(-0.665933\pi\)
−0.498004 + 0.867175i \(0.665933\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 43.3253 1.43543 0.717716 0.696336i \(-0.245188\pi\)
0.717716 + 0.696336i \(0.245188\pi\)
\(912\) 0 0
\(913\) 4.89919 0.162140
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.32625 −0.307980
\(918\) 0 0
\(919\) 44.6079i 1.47148i 0.677264 + 0.735740i \(0.263165\pi\)
−0.677264 + 0.735740i \(0.736835\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 27.7600i − 0.913732i
\(924\) 0 0
\(925\) 142.011i 4.66929i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.73251i 0.253695i 0.991922 + 0.126848i \(0.0404860\pi\)
−0.991922 + 0.126848i \(0.959514\pi\)
\(930\) 0 0
\(931\) 3.05596 0.100155
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.94326 0.128958
\(936\) 0 0
\(937\) 25.6210 0.837002 0.418501 0.908216i \(-0.362556\pi\)
0.418501 + 0.908216i \(0.362556\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −30.8529 −1.00578 −0.502888 0.864352i \(-0.667729\pi\)
−0.502888 + 0.864352i \(0.667729\pi\)
\(942\) 0 0
\(943\) − 43.4161i − 1.41382i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.5682i 1.28579i 0.765953 + 0.642897i \(0.222268\pi\)
−0.765953 + 0.642897i \(0.777732\pi\)
\(948\) 0 0
\(949\) 1.28114i 0.0415876i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 4.36408i − 0.141366i −0.997499 0.0706832i \(-0.977482\pi\)
0.997499 0.0706832i \(-0.0225179\pi\)
\(954\) 0 0
\(955\) −76.0015 −2.45935
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.53069 0.210887
\(960\) 0 0
\(961\) −87.7668 −2.83119
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 70.1442 2.25802
\(966\) 0 0
\(967\) 51.1938i 1.64628i 0.567837 + 0.823141i \(0.307780\pi\)
−0.567837 + 0.823141i \(0.692220\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 47.6773i 1.53004i 0.644008 + 0.765019i \(0.277270\pi\)
−0.644008 + 0.765019i \(0.722730\pi\)
\(972\) 0 0
\(973\) − 10.1589i − 0.325678i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.1656i 0.325225i 0.986690 + 0.162613i \(0.0519921\pi\)
−0.986690 + 0.162613i \(0.948008\pi\)
\(978\) 0 0
\(979\) −10.6340 −0.339864
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 7.35716 0.234657 0.117328 0.993093i \(-0.462567\pi\)
0.117328 + 0.993093i \(0.462567\pi\)
\(984\) 0 0
\(985\) −80.4068 −2.56197
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 53.8772 1.71320
\(990\) 0 0
\(991\) 18.5907i 0.590552i 0.955412 + 0.295276i \(0.0954116\pi\)
−0.955412 + 0.295276i \(0.904588\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.06577i 0.224000i
\(996\) 0 0
\(997\) 4.19288i 0.132790i 0.997793 + 0.0663950i \(0.0211497\pi\)
−0.997793 + 0.0663950i \(0.978850\pi\)
\(998\) 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.j.d.5615.2 48
3.2 odd 2 inner 6048.2.j.d.5615.47 48
4.3 odd 2 1512.2.j.d.323.45 yes 48
8.3 odd 2 inner 6048.2.j.d.5615.48 48
8.5 even 2 1512.2.j.d.323.3 48
12.11 even 2 1512.2.j.d.323.4 yes 48
24.5 odd 2 1512.2.j.d.323.46 yes 48
24.11 even 2 inner 6048.2.j.d.5615.1 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.3 48 8.5 even 2
1512.2.j.d.323.4 yes 48 12.11 even 2
1512.2.j.d.323.45 yes 48 4.3 odd 2
1512.2.j.d.323.46 yes 48 24.5 odd 2
6048.2.j.d.5615.1 48 24.11 even 2 inner
6048.2.j.d.5615.2 48 1.1 even 1 trivial
6048.2.j.d.5615.47 48 3.2 odd 2 inner
6048.2.j.d.5615.48 48 8.3 odd 2 inner