Properties

Label 6048.2.j.d.5615.4
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 5615.4
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.d.5615.3

$q$-expansion

\(f(q)\) \(=\) \(q-3.69622 q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-3.69622 q^{5} -1.00000i q^{7} -2.05843i q^{11} -4.65027i q^{13} +5.73497i q^{17} -3.27215 q^{19} +4.45085 q^{23} +8.66201 q^{25} +10.2594 q^{29} -1.26066i q^{31} +3.69622i q^{35} -3.25539i q^{37} +8.72747i q^{41} +2.56921 q^{43} -2.03048 q^{47} -1.00000 q^{49} +10.0794 q^{53} +7.60840i q^{55} +0.851213i q^{59} +2.31448i q^{61} +17.1884i q^{65} -15.4289 q^{67} +4.44544 q^{71} -6.97349 q^{73} -2.05843 q^{77} -11.6750i q^{79} -12.8349i q^{83} -21.1977i q^{85} +2.35419i q^{89} -4.65027 q^{91} +12.0946 q^{95} +4.02539 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q + O(q^{10}) \) \( 48q - 16q^{19} + 48q^{25} - 64q^{43} - 48q^{49} - 32q^{67} + O(q^{100}) \)

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
\(5\) −3.69622 −1.65300 −0.826499 0.562938i \(-0.809671\pi\)
−0.826499 + 0.562938i \(0.809671\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.05843i − 0.620640i −0.950632 0.310320i \(-0.899564\pi\)
0.950632 0.310320i \(-0.100436\pi\)
\(12\) 0 0
\(13\) − 4.65027i − 1.28975i −0.764287 0.644876i \(-0.776909\pi\)
0.764287 0.644876i \(-0.223091\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.73497i 1.39094i 0.718557 + 0.695468i \(0.244803\pi\)
−0.718557 + 0.695468i \(0.755197\pi\)
\(18\) 0 0
\(19\) −3.27215 −0.750682 −0.375341 0.926887i \(-0.622474\pi\)
−0.375341 + 0.926887i \(0.622474\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.45085 0.928067 0.464033 0.885818i \(-0.346402\pi\)
0.464033 + 0.885818i \(0.346402\pi\)
\(24\) 0 0
\(25\) 8.66201 1.73240
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.2594 1.90512 0.952561 0.304346i \(-0.0984380\pi\)
0.952561 + 0.304346i \(0.0984380\pi\)
\(30\) 0 0
\(31\) − 1.26066i − 0.226422i −0.993571 0.113211i \(-0.963886\pi\)
0.993571 0.113211i \(-0.0361136\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.69622i 0.624775i
\(36\) 0 0
\(37\) − 3.25539i − 0.535183i −0.963532 0.267592i \(-0.913772\pi\)
0.963532 0.267592i \(-0.0862278\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.72747i 1.36300i 0.731817 + 0.681501i \(0.238673\pi\)
−0.731817 + 0.681501i \(0.761327\pi\)
\(42\) 0 0
\(43\) 2.56921 0.391801 0.195901 0.980624i \(-0.437237\pi\)
0.195901 + 0.980624i \(0.437237\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.03048 −0.296176 −0.148088 0.988974i \(-0.547312\pi\)
−0.148088 + 0.988974i \(0.547312\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.0794 1.38451 0.692253 0.721655i \(-0.256618\pi\)
0.692253 + 0.721655i \(0.256618\pi\)
\(54\) 0 0
\(55\) 7.60840i 1.02592i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.851213i 0.110818i 0.998464 + 0.0554092i \(0.0176463\pi\)
−0.998464 + 0.0554092i \(0.982354\pi\)
\(60\) 0 0
\(61\) 2.31448i 0.296339i 0.988962 + 0.148170i \(0.0473382\pi\)
−0.988962 + 0.148170i \(0.952662\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 17.1884i 2.13196i
\(66\) 0 0
\(67\) −15.4289 −1.88494 −0.942468 0.334296i \(-0.891501\pi\)
−0.942468 + 0.334296i \(0.891501\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.44544 0.527576 0.263788 0.964581i \(-0.415028\pi\)
0.263788 + 0.964581i \(0.415028\pi\)
\(72\) 0 0
\(73\) −6.97349 −0.816186 −0.408093 0.912940i \(-0.633806\pi\)
−0.408093 + 0.912940i \(0.633806\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.05843 −0.234580
\(78\) 0 0
\(79\) − 11.6750i − 1.31354i −0.754092 0.656769i \(-0.771923\pi\)
0.754092 0.656769i \(-0.228077\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.8349i − 1.40882i −0.709796 0.704408i \(-0.751213\pi\)
0.709796 0.704408i \(-0.248787\pi\)
\(84\) 0 0
\(85\) − 21.1977i − 2.29921i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.35419i 0.249544i 0.992185 + 0.124772i \(0.0398200\pi\)
−0.992185 + 0.124772i \(0.960180\pi\)
\(90\) 0 0
\(91\) −4.65027 −0.487480
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.0946 1.24088
\(96\) 0 0
\(97\) 4.02539 0.408717 0.204358 0.978896i \(-0.434489\pi\)
0.204358 + 0.978896i \(0.434489\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.2008 −1.41303 −0.706517 0.707696i \(-0.749735\pi\)
−0.706517 + 0.707696i \(0.749735\pi\)
\(102\) 0 0
\(103\) − 5.30631i − 0.522846i −0.965224 0.261423i \(-0.915808\pi\)
0.965224 0.261423i \(-0.0841918\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.3669i 1.29223i 0.763240 + 0.646115i \(0.223608\pi\)
−0.763240 + 0.646115i \(0.776392\pi\)
\(108\) 0 0
\(109\) 5.91832i 0.566872i 0.958991 + 0.283436i \(0.0914743\pi\)
−0.958991 + 0.283436i \(0.908526\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 18.5535i − 1.74537i −0.488287 0.872683i \(-0.662378\pi\)
0.488287 0.872683i \(-0.337622\pi\)
\(114\) 0 0
\(115\) −16.4513 −1.53409
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.73497 0.525724
\(120\) 0 0
\(121\) 6.76287 0.614806
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −13.5356 −1.21066
\(126\) 0 0
\(127\) − 9.76963i − 0.866915i −0.901174 0.433457i \(-0.857293\pi\)
0.901174 0.433457i \(-0.142707\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 15.5661i − 1.36002i −0.733204 0.680009i \(-0.761976\pi\)
0.733204 0.680009i \(-0.238024\pi\)
\(132\) 0 0
\(133\) 3.27215i 0.283731i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.54964i 0.730445i 0.930920 + 0.365223i \(0.119007\pi\)
−0.930920 + 0.365223i \(0.880993\pi\)
\(138\) 0 0
\(139\) 15.4607 1.31136 0.655679 0.755040i \(-0.272383\pi\)
0.655679 + 0.755040i \(0.272383\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.57225 −0.800472
\(144\) 0 0
\(145\) −37.9210 −3.14916
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.91181 −0.730084 −0.365042 0.930991i \(-0.618945\pi\)
−0.365042 + 0.930991i \(0.618945\pi\)
\(150\) 0 0
\(151\) − 14.8040i − 1.20473i −0.798219 0.602367i \(-0.794224\pi\)
0.798219 0.602367i \(-0.205776\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.65968i 0.374275i
\(156\) 0 0
\(157\) 14.3716i 1.14698i 0.819214 + 0.573488i \(0.194410\pi\)
−0.819214 + 0.573488i \(0.805590\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 4.45085i − 0.350776i
\(162\) 0 0
\(163\) −21.7846 −1.70630 −0.853149 0.521667i \(-0.825310\pi\)
−0.853149 + 0.521667i \(0.825310\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.4031 1.03717 0.518583 0.855027i \(-0.326460\pi\)
0.518583 + 0.855027i \(0.326460\pi\)
\(168\) 0 0
\(169\) −8.62498 −0.663460
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −13.3221 −1.01286 −0.506429 0.862282i \(-0.669035\pi\)
−0.506429 + 0.862282i \(0.669035\pi\)
\(174\) 0 0
\(175\) − 8.66201i − 0.654787i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 12.6787i − 0.947648i −0.880619 0.473824i \(-0.842873\pi\)
0.880619 0.473824i \(-0.157127\pi\)
\(180\) 0 0
\(181\) 11.1255i 0.826951i 0.910515 + 0.413476i \(0.135685\pi\)
−0.910515 + 0.413476i \(0.864315\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0326i 0.884657i
\(186\) 0 0
\(187\) 11.8050 0.863270
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.76387 −0.561774 −0.280887 0.959741i \(-0.590629\pi\)
−0.280887 + 0.959741i \(0.590629\pi\)
\(192\) 0 0
\(193\) 9.09582 0.654731 0.327366 0.944898i \(-0.393839\pi\)
0.327366 + 0.944898i \(0.393839\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.3722 −0.738988 −0.369494 0.929233i \(-0.620469\pi\)
−0.369494 + 0.929233i \(0.620469\pi\)
\(198\) 0 0
\(199\) − 9.92379i − 0.703479i −0.936098 0.351739i \(-0.885590\pi\)
0.936098 0.351739i \(-0.114410\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 10.2594i − 0.720069i
\(204\) 0 0
\(205\) − 32.2586i − 2.25304i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.73549i 0.465903i
\(210\) 0 0
\(211\) 10.9624 0.754680 0.377340 0.926075i \(-0.376839\pi\)
0.377340 + 0.926075i \(0.376839\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.49637 −0.647647
\(216\) 0 0
\(217\) −1.26066 −0.0855794
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 26.6692 1.79396
\(222\) 0 0
\(223\) − 1.18980i − 0.0796749i −0.999206 0.0398375i \(-0.987316\pi\)
0.999206 0.0398375i \(-0.0126840\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.62724i 0.638983i 0.947589 + 0.319491i \(0.103512\pi\)
−0.947589 + 0.319491i \(0.896488\pi\)
\(228\) 0 0
\(229\) − 21.5143i − 1.42170i −0.703343 0.710851i \(-0.748310\pi\)
0.703343 0.710851i \(-0.251690\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 6.82082i − 0.446847i −0.974722 0.223423i \(-0.928277\pi\)
0.974722 0.223423i \(-0.0717232\pi\)
\(234\) 0 0
\(235\) 7.50510 0.489579
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.24506 0.403959 0.201980 0.979390i \(-0.435263\pi\)
0.201980 + 0.979390i \(0.435263\pi\)
\(240\) 0 0
\(241\) 25.5573 1.64629 0.823145 0.567831i \(-0.192217\pi\)
0.823145 + 0.567831i \(0.192217\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.69622 0.236143
\(246\) 0 0
\(247\) 15.2164i 0.968194i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 0.373698i − 0.0235876i −0.999930 0.0117938i \(-0.996246\pi\)
0.999930 0.0117938i \(-0.00375417\pi\)
\(252\) 0 0
\(253\) − 9.16177i − 0.575995i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 11.0514i − 0.689369i −0.938719 0.344684i \(-0.887986\pi\)
0.938719 0.344684i \(-0.112014\pi\)
\(258\) 0 0
\(259\) −3.25539 −0.202280
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −27.6685 −1.70612 −0.853058 0.521817i \(-0.825255\pi\)
−0.853058 + 0.521817i \(0.825255\pi\)
\(264\) 0 0
\(265\) −37.2555 −2.28859
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −29.3247 −1.78796 −0.893981 0.448105i \(-0.852099\pi\)
−0.893981 + 0.448105i \(0.852099\pi\)
\(270\) 0 0
\(271\) − 5.13414i − 0.311877i −0.987767 0.155938i \(-0.950160\pi\)
0.987767 0.155938i \(-0.0498401\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 17.8301i − 1.07520i
\(276\) 0 0
\(277\) 5.93089i 0.356353i 0.983999 + 0.178176i \(0.0570197\pi\)
−0.983999 + 0.178176i \(0.942980\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 10.7827i − 0.643242i −0.946868 0.321621i \(-0.895772\pi\)
0.946868 0.321621i \(-0.104228\pi\)
\(282\) 0 0
\(283\) 18.1556 1.07924 0.539619 0.841910i \(-0.318568\pi\)
0.539619 + 0.841910i \(0.318568\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.72747 0.515166
\(288\) 0 0
\(289\) −15.8899 −0.934701
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.8863 1.27861 0.639306 0.768953i \(-0.279222\pi\)
0.639306 + 0.768953i \(0.279222\pi\)
\(294\) 0 0
\(295\) − 3.14627i − 0.183183i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 20.6976i − 1.19698i
\(300\) 0 0
\(301\) − 2.56921i − 0.148087i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 8.55484i − 0.489848i
\(306\) 0 0
\(307\) −16.8301 −0.960546 −0.480273 0.877119i \(-0.659462\pi\)
−0.480273 + 0.877119i \(0.659462\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −33.1293 −1.87859 −0.939294 0.343113i \(-0.888519\pi\)
−0.939294 + 0.343113i \(0.888519\pi\)
\(312\) 0 0
\(313\) 13.4187 0.758469 0.379235 0.925301i \(-0.376187\pi\)
0.379235 + 0.925301i \(0.376187\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.764835 −0.0429574 −0.0214787 0.999769i \(-0.506837\pi\)
−0.0214787 + 0.999769i \(0.506837\pi\)
\(318\) 0 0
\(319\) − 21.1183i − 1.18240i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 18.7657i − 1.04415i
\(324\) 0 0
\(325\) − 40.2807i − 2.23437i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.03048i 0.111944i
\(330\) 0 0
\(331\) −29.7299 −1.63410 −0.817051 0.576566i \(-0.804392\pi\)
−0.817051 + 0.576566i \(0.804392\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 57.0284 3.11580
\(336\) 0 0
\(337\) −19.0487 −1.03765 −0.518824 0.854881i \(-0.673630\pi\)
−0.518824 + 0.854881i \(0.673630\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.59499 −0.140526
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 30.0749i − 1.61451i −0.590205 0.807253i \(-0.700953\pi\)
0.590205 0.807253i \(-0.299047\pi\)
\(348\) 0 0
\(349\) − 28.7995i − 1.54160i −0.637078 0.770800i \(-0.719857\pi\)
0.637078 0.770800i \(-0.280143\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 26.2377i − 1.39649i −0.715857 0.698247i \(-0.753964\pi\)
0.715857 0.698247i \(-0.246036\pi\)
\(354\) 0 0
\(355\) −16.4313 −0.872083
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.4178 −0.972053 −0.486027 0.873944i \(-0.661554\pi\)
−0.486027 + 0.873944i \(0.661554\pi\)
\(360\) 0 0
\(361\) −8.29305 −0.436476
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 25.7755 1.34915
\(366\) 0 0
\(367\) − 37.2862i − 1.94632i −0.230127 0.973161i \(-0.573914\pi\)
0.230127 0.973161i \(-0.426086\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 10.0794i − 0.523294i
\(372\) 0 0
\(373\) 4.59095i 0.237711i 0.992912 + 0.118855i \(0.0379224\pi\)
−0.992912 + 0.118855i \(0.962078\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 47.7090i − 2.45714i
\(378\) 0 0
\(379\) 21.0149 1.07947 0.539733 0.841836i \(-0.318525\pi\)
0.539733 + 0.841836i \(0.318525\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.09047 0.464501 0.232251 0.972656i \(-0.425391\pi\)
0.232251 + 0.972656i \(0.425391\pi\)
\(384\) 0 0
\(385\) 7.60840 0.387760
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.06330 0.104614 0.0523068 0.998631i \(-0.483343\pi\)
0.0523068 + 0.998631i \(0.483343\pi\)
\(390\) 0 0
\(391\) 25.5255i 1.29088i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 43.1533i 2.17128i
\(396\) 0 0
\(397\) − 0.0369436i − 0.00185415i −1.00000 0.000927073i \(-0.999705\pi\)
1.00000 0.000927073i \(-0.000295096\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.6814i 0.932906i 0.884546 + 0.466453i \(0.154468\pi\)
−0.884546 + 0.466453i \(0.845532\pi\)
\(402\) 0 0
\(403\) −5.86242 −0.292028
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.70100 −0.332156
\(408\) 0 0
\(409\) −11.5630 −0.571755 −0.285878 0.958266i \(-0.592285\pi\)
−0.285878 + 0.958266i \(0.592285\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.851213 0.0418854
\(414\) 0 0
\(415\) 47.4406i 2.32877i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.5911i 0.712822i 0.934329 + 0.356411i \(0.116000\pi\)
−0.934329 + 0.356411i \(0.884000\pi\)
\(420\) 0 0
\(421\) − 17.6204i − 0.858767i −0.903122 0.429384i \(-0.858731\pi\)
0.903122 0.429384i \(-0.141269\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 49.6764i 2.40966i
\(426\) 0 0
\(427\) 2.31448 0.112006
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.6500 −0.850170 −0.425085 0.905153i \(-0.639756\pi\)
−0.425085 + 0.905153i \(0.639756\pi\)
\(432\) 0 0
\(433\) −6.31965 −0.303703 −0.151852 0.988403i \(-0.548524\pi\)
−0.151852 + 0.988403i \(0.548524\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −14.5638 −0.696683
\(438\) 0 0
\(439\) − 10.1993i − 0.486786i −0.969928 0.243393i \(-0.921739\pi\)
0.969928 0.243393i \(-0.0782605\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.46579i 0.449733i 0.974390 + 0.224867i \(0.0721947\pi\)
−0.974390 + 0.224867i \(0.927805\pi\)
\(444\) 0 0
\(445\) − 8.70161i − 0.412496i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.1482i 1.32840i 0.747556 + 0.664198i \(0.231227\pi\)
−0.747556 + 0.664198i \(0.768773\pi\)
\(450\) 0 0
\(451\) 17.9649 0.845934
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 17.1884 0.805804
\(456\) 0 0
\(457\) −34.9936 −1.63693 −0.818465 0.574556i \(-0.805175\pi\)
−0.818465 + 0.574556i \(0.805175\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.2558 −1.40915 −0.704577 0.709628i \(-0.748863\pi\)
−0.704577 + 0.709628i \(0.748863\pi\)
\(462\) 0 0
\(463\) − 2.87077i − 0.133416i −0.997773 0.0667080i \(-0.978750\pi\)
0.997773 0.0667080i \(-0.0212496\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.1957i 1.67494i 0.546484 + 0.837469i \(0.315966\pi\)
−0.546484 + 0.837469i \(0.684034\pi\)
\(468\) 0 0
\(469\) 15.4289i 0.712439i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 5.28855i − 0.243168i
\(474\) 0 0
\(475\) −28.3434 −1.30048
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −17.1156 −0.782030 −0.391015 0.920384i \(-0.627876\pi\)
−0.391015 + 0.920384i \(0.627876\pi\)
\(480\) 0 0
\(481\) −15.1384 −0.690253
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.8787 −0.675608
\(486\) 0 0
\(487\) − 26.8336i − 1.21595i −0.793957 0.607974i \(-0.791982\pi\)
0.793957 0.607974i \(-0.208018\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.17336i 0.323729i 0.986813 + 0.161865i \(0.0517508\pi\)
−0.986813 + 0.161865i \(0.948249\pi\)
\(492\) 0 0
\(493\) 58.8374i 2.64990i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 4.44544i − 0.199405i
\(498\) 0 0
\(499\) 36.8731 1.65067 0.825334 0.564646i \(-0.190987\pi\)
0.825334 + 0.564646i \(0.190987\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17.4855 −0.779642 −0.389821 0.920891i \(-0.627463\pi\)
−0.389821 + 0.920891i \(0.627463\pi\)
\(504\) 0 0
\(505\) 52.4893 2.33574
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 38.6034 1.71106 0.855532 0.517750i \(-0.173230\pi\)
0.855532 + 0.517750i \(0.173230\pi\)
\(510\) 0 0
\(511\) 6.97349i 0.308489i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 19.6133i 0.864264i
\(516\) 0 0
\(517\) 4.17960i 0.183819i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.3406i 1.28543i 0.766104 + 0.642716i \(0.222193\pi\)
−0.766104 + 0.642716i \(0.777807\pi\)
\(522\) 0 0
\(523\) 7.90612 0.345711 0.172855 0.984947i \(-0.444701\pi\)
0.172855 + 0.984947i \(0.444701\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.22987 0.314938
\(528\) 0 0
\(529\) −3.18992 −0.138692
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 40.5851 1.75793
\(534\) 0 0
\(535\) − 49.4071i − 2.13606i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.05843i 0.0886629i
\(540\) 0 0
\(541\) − 6.25573i − 0.268955i −0.990917 0.134477i \(-0.957064\pi\)
0.990917 0.134477i \(-0.0429356\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 21.8754i − 0.937038i
\(546\) 0 0
\(547\) −18.4597 −0.789279 −0.394640 0.918836i \(-0.629131\pi\)
−0.394640 + 0.918836i \(0.629131\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −33.5703 −1.43014
\(552\) 0 0
\(553\) −11.6750 −0.496471
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.6497 0.451243 0.225622 0.974215i \(-0.427559\pi\)
0.225622 + 0.974215i \(0.427559\pi\)
\(558\) 0 0
\(559\) − 11.9475i − 0.505327i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.9117i 0.459874i 0.973206 + 0.229937i \(0.0738520\pi\)
−0.973206 + 0.229937i \(0.926148\pi\)
\(564\) 0 0
\(565\) 68.5778i 2.88509i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 10.7145i − 0.449175i −0.974454 0.224588i \(-0.927896\pi\)
0.974454 0.224588i \(-0.0721035\pi\)
\(570\) 0 0
\(571\) 21.6134 0.904493 0.452247 0.891893i \(-0.350623\pi\)
0.452247 + 0.891893i \(0.350623\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 38.5533 1.60779
\(576\) 0 0
\(577\) 17.2449 0.717914 0.358957 0.933354i \(-0.383133\pi\)
0.358957 + 0.933354i \(0.383133\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.8349 −0.532482
\(582\) 0 0
\(583\) − 20.7476i − 0.859279i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.83868i 0.282263i 0.989991 + 0.141131i \(0.0450740\pi\)
−0.989991 + 0.141131i \(0.954926\pi\)
\(588\) 0 0
\(589\) 4.12508i 0.169971i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.2262i 0.953787i 0.878961 + 0.476893i \(0.158237\pi\)
−0.878961 + 0.476893i \(0.841763\pi\)
\(594\) 0 0
\(595\) −21.1977 −0.869021
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.71919 0.192821 0.0964105 0.995342i \(-0.469264\pi\)
0.0964105 + 0.995342i \(0.469264\pi\)
\(600\) 0 0
\(601\) −0.230392 −0.00939787 −0.00469893 0.999989i \(-0.501496\pi\)
−0.00469893 + 0.999989i \(0.501496\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −24.9970 −1.01627
\(606\) 0 0
\(607\) 5.43638i 0.220656i 0.993895 + 0.110328i \(0.0351901\pi\)
−0.993895 + 0.110328i \(0.964810\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.44228i 0.381994i
\(612\) 0 0
\(613\) − 24.7976i − 1.00156i −0.865573 0.500782i \(-0.833046\pi\)
0.865573 0.500782i \(-0.166954\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 13.4024i − 0.539561i −0.962922 0.269781i \(-0.913049\pi\)
0.962922 0.269781i \(-0.0869512\pi\)
\(618\) 0 0
\(619\) 12.8992 0.518465 0.259232 0.965815i \(-0.416530\pi\)
0.259232 + 0.965815i \(0.416530\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.35419 0.0943188
\(624\) 0 0
\(625\) 6.72041 0.268817
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18.6696 0.744405
\(630\) 0 0
\(631\) − 24.0217i − 0.956288i −0.878281 0.478144i \(-0.841310\pi\)
0.878281 0.478144i \(-0.158690\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 36.1107i 1.43301i
\(636\) 0 0
\(637\) 4.65027i 0.184250i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 10.4176i − 0.411469i −0.978608 0.205735i \(-0.934042\pi\)
0.978608 0.205735i \(-0.0659583\pi\)
\(642\) 0 0
\(643\) −42.2723 −1.66706 −0.833528 0.552477i \(-0.813683\pi\)
−0.833528 + 0.552477i \(0.813683\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.75700 0.304959 0.152480 0.988307i \(-0.451274\pi\)
0.152480 + 0.988307i \(0.451274\pi\)
\(648\) 0 0
\(649\) 1.75216 0.0687784
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.9159 0.779370 0.389685 0.920948i \(-0.372584\pi\)
0.389685 + 0.920948i \(0.372584\pi\)
\(654\) 0 0
\(655\) 57.5357i 2.24811i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 39.4456i 1.53658i 0.640101 + 0.768291i \(0.278892\pi\)
−0.640101 + 0.768291i \(0.721108\pi\)
\(660\) 0 0
\(661\) − 35.8650i − 1.39499i −0.716591 0.697494i \(-0.754298\pi\)
0.716591 0.697494i \(-0.245702\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 12.0946i − 0.469007i
\(666\) 0 0
\(667\) 45.6631 1.76808
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.76420 0.183920
\(672\) 0 0
\(673\) 30.5668 1.17827 0.589133 0.808036i \(-0.299470\pi\)
0.589133 + 0.808036i \(0.299470\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.14804 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(678\) 0 0
\(679\) − 4.02539i − 0.154480i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.07293i 0.0793186i 0.999213 + 0.0396593i \(0.0126273\pi\)
−0.999213 + 0.0396593i \(0.987373\pi\)
\(684\) 0 0
\(685\) − 31.6013i − 1.20742i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 46.8717i − 1.78567i
\(690\) 0 0
\(691\) −34.7325 −1.32129 −0.660644 0.750700i \(-0.729717\pi\)
−0.660644 + 0.750700i \(0.729717\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −57.1460 −2.16767
\(696\) 0 0
\(697\) −50.0518 −1.89585
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.1113 −1.13729 −0.568643 0.822584i \(-0.692531\pi\)
−0.568643 + 0.822584i \(0.692531\pi\)
\(702\) 0 0
\(703\) 10.6521i 0.401752i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14.2008i 0.534077i
\(708\) 0 0
\(709\) − 30.7194i − 1.15369i −0.816853 0.576846i \(-0.804283\pi\)
0.816853 0.576846i \(-0.195717\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 5.61102i − 0.210135i
\(714\) 0 0
\(715\) 35.3811 1.32318
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.93357 −0.0721099 −0.0360549 0.999350i \(-0.511479\pi\)
−0.0360549 + 0.999350i \(0.511479\pi\)
\(720\) 0 0
\(721\) −5.30631 −0.197617
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 88.8671 3.30044
\(726\) 0 0
\(727\) − 21.4593i − 0.795881i −0.917411 0.397940i \(-0.869725\pi\)
0.917411 0.397940i \(-0.130275\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14.7344i 0.544970i
\(732\) 0 0
\(733\) − 7.43101i − 0.274471i −0.990538 0.137235i \(-0.956178\pi\)
0.990538 0.137235i \(-0.0438217\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.7592i 1.16987i
\(738\) 0 0
\(739\) 6.48141 0.238423 0.119211 0.992869i \(-0.461963\pi\)
0.119211 + 0.992869i \(0.461963\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 50.4752 1.85176 0.925879 0.377821i \(-0.123327\pi\)
0.925879 + 0.377821i \(0.123327\pi\)
\(744\) 0 0
\(745\) 32.9400 1.20683
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.3669 0.488417
\(750\) 0 0
\(751\) 34.8254i 1.27080i 0.772185 + 0.635398i \(0.219164\pi\)
−0.772185 + 0.635398i \(0.780836\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 54.7189i 1.99142i
\(756\) 0 0
\(757\) 15.6422i 0.568524i 0.958747 + 0.284262i \(0.0917485\pi\)
−0.958747 + 0.284262i \(0.908251\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 8.77808i − 0.318205i −0.987262 0.159103i \(-0.949140\pi\)
0.987262 0.159103i \(-0.0508600\pi\)
\(762\) 0 0
\(763\) 5.91832 0.214257
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.95837 0.142928
\(768\) 0 0
\(769\) 20.4092 0.735975 0.367988 0.929831i \(-0.380047\pi\)
0.367988 + 0.929831i \(0.380047\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.204759 0.00736468 0.00368234 0.999993i \(-0.498828\pi\)
0.00368234 + 0.999993i \(0.498828\pi\)
\(774\) 0 0
\(775\) − 10.9199i − 0.392254i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 28.5576i − 1.02318i
\(780\) 0 0
\(781\) − 9.15062i − 0.327435i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 53.1205i − 1.89595i
\(786\) 0 0
\(787\) −51.9705 −1.85255 −0.926274 0.376850i \(-0.877007\pi\)
−0.926274 + 0.376850i \(0.877007\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.5535 −0.659687
\(792\) 0 0
\(793\) 10.7630 0.382204
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.9252 −0.599522 −0.299761 0.954014i \(-0.596907\pi\)
−0.299761 + 0.954014i \(0.596907\pi\)
\(798\) 0 0
\(799\) − 11.6448i − 0.411962i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.3544i 0.506557i
\(804\) 0 0
\(805\) 16.4513i 0.579832i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10.4440i 0.367191i 0.983002 + 0.183596i \(0.0587737\pi\)
−0.983002 + 0.183596i \(0.941226\pi\)
\(810\) 0 0
\(811\) −15.2976 −0.537170 −0.268585 0.963256i \(-0.586556\pi\)
−0.268585 + 0.963256i \(0.586556\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 80.5204 2.82051
\(816\) 0 0
\(817\) −8.40685 −0.294118
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.7719 −0.724946 −0.362473 0.931994i \(-0.618067\pi\)
−0.362473 + 0.931994i \(0.618067\pi\)
\(822\) 0 0
\(823\) − 6.01697i − 0.209738i −0.994486 0.104869i \(-0.966558\pi\)
0.994486 0.104869i \(-0.0334424\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 53.8903i − 1.87395i −0.349399 0.936974i \(-0.613614\pi\)
0.349399 0.936974i \(-0.386386\pi\)
\(828\) 0 0
\(829\) − 5.69863i − 0.197922i −0.995091 0.0989608i \(-0.968448\pi\)
0.995091 0.0989608i \(-0.0315519\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 5.73497i − 0.198705i
\(834\) 0 0
\(835\) −49.5409 −1.71443
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.43315 0.325668 0.162834 0.986653i \(-0.447936\pi\)
0.162834 + 0.986653i \(0.447936\pi\)
\(840\) 0 0
\(841\) 76.2553 2.62949
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 31.8798 1.09670
\(846\) 0 0
\(847\) − 6.76287i − 0.232375i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 14.4893i − 0.496686i
\(852\) 0 0
\(853\) 3.49254i 0.119582i 0.998211 + 0.0597912i \(0.0190435\pi\)
−0.998211 + 0.0597912i \(0.980957\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.2047i 0.382745i 0.981517 + 0.191372i \(0.0612938\pi\)
−0.981517 + 0.191372i \(0.938706\pi\)
\(858\) 0 0
\(859\) −43.4564 −1.48271 −0.741357 0.671111i \(-0.765817\pi\)
−0.741357 + 0.671111i \(0.765817\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19.8312 −0.675063 −0.337532 0.941314i \(-0.609592\pi\)
−0.337532 + 0.941314i \(0.609592\pi\)
\(864\) 0 0
\(865\) 49.2412 1.67425
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −24.0321 −0.815235
\(870\) 0 0
\(871\) 71.7483i 2.43110i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 13.5356i 0.457587i
\(876\) 0 0
\(877\) 40.6652i 1.37317i 0.727051 + 0.686584i \(0.240890\pi\)
−0.727051 + 0.686584i \(0.759110\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 46.8298i 1.57774i 0.614562 + 0.788868i \(0.289333\pi\)
−0.614562 + 0.788868i \(0.710667\pi\)
\(882\) 0 0
\(883\) 18.8388 0.633975 0.316988 0.948430i \(-0.397329\pi\)
0.316988 + 0.948430i \(0.397329\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.52873 −0.0513297 −0.0256648 0.999671i \(-0.508170\pi\)
−0.0256648 + 0.999671i \(0.508170\pi\)
\(888\) 0 0
\(889\) −9.76963 −0.327663
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.64403 0.222334
\(894\) 0 0
\(895\) 46.8631i 1.56646i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 12.9336i − 0.431361i
\(900\) 0 0
\(901\) 57.8048i 1.92576i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 41.1222i − 1.36695i
\(906\) 0 0
\(907\) 24.5682 0.815773 0.407886 0.913033i \(-0.366266\pi\)
0.407886 + 0.913033i \(0.366266\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 14.8864 0.493209 0.246604 0.969116i \(-0.420685\pi\)
0.246604 + 0.969116i \(0.420685\pi\)
\(912\) 0 0
\(913\) −26.4198 −0.874367
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.5661 −0.514038
\(918\) 0 0
\(919\) 55.6464i 1.83560i 0.397038 + 0.917802i \(0.370038\pi\)
−0.397038 + 0.917802i \(0.629962\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 20.6725i − 0.680443i
\(924\) 0 0
\(925\) − 28.1982i − 0.927153i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 29.3098i − 0.961623i −0.876824 0.480812i \(-0.840342\pi\)
0.876824 0.480812i \(-0.159658\pi\)
\(930\) 0 0
\(931\) 3.27215 0.107240
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −43.6340 −1.42698
\(936\) 0 0
\(937\) 12.6626 0.413669 0.206834 0.978376i \(-0.433684\pi\)
0.206834 + 0.978376i \(0.433684\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −26.6135 −0.867575 −0.433788 0.901015i \(-0.642823\pi\)
−0.433788 + 0.901015i \(0.642823\pi\)
\(942\) 0 0
\(943\) 38.8447i 1.26496i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.05029i − 0.131617i −0.997832 0.0658084i \(-0.979037\pi\)
0.997832 0.0658084i \(-0.0209626\pi\)
\(948\) 0 0
\(949\) 32.4286i 1.05268i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 7.19155i − 0.232957i −0.993193 0.116479i \(-0.962839\pi\)
0.993193 0.116479i \(-0.0371606\pi\)
\(954\) 0 0
\(955\) 28.6969 0.928611
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.54964 0.276082
\(960\) 0 0
\(961\) 29.4107 0.948733
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −33.6201 −1.08227
\(966\) 0 0
\(967\) 46.3071i 1.48914i 0.667547 + 0.744568i \(0.267344\pi\)
−0.667547 + 0.744568i \(0.732656\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 22.1689i − 0.711432i −0.934594 0.355716i \(-0.884237\pi\)
0.934594 0.355716i \(-0.115763\pi\)
\(972\) 0 0
\(973\) − 15.4607i − 0.495647i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 13.0854i − 0.418638i −0.977847 0.209319i \(-0.932875\pi\)
0.977847 0.209319i \(-0.0671248\pi\)
\(978\) 0 0
\(979\) 4.84594 0.154877
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.9454 0.476684 0.238342 0.971181i \(-0.423396\pi\)
0.238342 + 0.971181i \(0.423396\pi\)
\(984\) 0 0
\(985\) 38.3379 1.22155
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.4352 0.363618
\(990\) 0 0
\(991\) − 25.6587i − 0.815075i −0.913188 0.407537i \(-0.866388\pi\)
0.913188 0.407537i \(-0.133612\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 36.6805i 1.16285i
\(996\) 0 0
\(997\) − 10.1507i − 0.321476i −0.986997 0.160738i \(-0.948613\pi\)
0.986997 0.160738i \(-0.0513874\pi\)
\(998\) 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.5615.4 48
3.2 odd 2 inner 6048.2.j.d.5615.45 48
4.3 odd 2 1512.2.j.d.323.33 yes 48
8.3 odd 2 inner 6048.2.j.d.5615.46 48
8.5 even 2 1512.2.j.d.323.15 48
12.11 even 2 1512.2.j.d.323.16 yes 48
24.5 odd 2 1512.2.j.d.323.34 yes 48
24.11 even 2 inner 6048.2.j.d.5615.3 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.15 48 8.5 even 2
1512.2.j.d.323.16 yes 48 12.11 even 2
1512.2.j.d.323.33 yes 48 4.3 odd 2
1512.2.j.d.323.34 yes 48 24.5 odd 2
6048.2.j.d.5615.3 48 24.11 even 2 inner
6048.2.j.d.5615.4 48 1.1 even 1 trivial
6048.2.j.d.5615.45 48 3.2 odd 2 inner
6048.2.j.d.5615.46 48 8.3 odd 2 inner