Properties

Label 4600.2.e.m.4049.4
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.m.4049.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155i q^{3} -5.12311i q^{7} -3.56155 q^{9} +O(q^{10})\) \(q+2.56155i q^{3} -5.12311i q^{7} -3.56155 q^{9} -4.00000 q^{11} -0.561553i q^{13} +3.12311i q^{17} -4.00000 q^{19} +13.1231 q^{21} +1.00000i q^{23} -1.43845i q^{27} +8.56155 q^{29} +1.43845 q^{31} -10.2462i q^{33} +7.12311i q^{37} +1.43845 q^{39} +0.561553 q^{41} -9.12311i q^{43} +3.68466i q^{47} -19.2462 q^{49} -8.00000 q^{51} -4.24621i q^{53} -10.2462i q^{57} +6.24621 q^{59} +11.1231 q^{61} +18.2462i q^{63} -6.24621i q^{67} -2.56155 q^{69} +3.68466 q^{71} +16.5616i q^{73} +20.4924i q^{77} +10.2462 q^{79} -7.00000 q^{81} +12.0000i q^{83} +21.9309i q^{87} -10.0000 q^{89} -2.87689 q^{91} +3.68466i q^{93} +16.2462i q^{97} +14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} + O(q^{10}) \) \( 4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 2.56155i 1.47891i 0.673204 + 0.739457i \(0.264917\pi\)
−0.673204 + 0.739457i \(0.735083\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 5.12311i − 1.93635i −0.250270 0.968176i \(-0.580520\pi\)
0.250270 0.968176i \(-0.419480\pi\)
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 0.561553i − 0.155747i −0.996963 0.0778734i \(-0.975187\pi\)
0.996963 0.0778734i \(-0.0248130\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.12311i 0.757464i 0.925506 + 0.378732i \(0.123640\pi\)
−0.925506 + 0.378732i \(0.876360\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 13.1231 2.86370
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.43845i − 0.276829i
\(28\) 0 0
\(29\) 8.56155 1.58984 0.794920 0.606714i \(-0.207513\pi\)
0.794920 + 0.606714i \(0.207513\pi\)
\(30\) 0 0
\(31\) 1.43845 0.258353 0.129176 0.991622i \(-0.458767\pi\)
0.129176 + 0.991622i \(0.458767\pi\)
\(32\) 0 0
\(33\) − 10.2462i − 1.78364i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.12311i 1.17103i 0.810661 + 0.585516i \(0.199108\pi\)
−0.810661 + 0.585516i \(0.800892\pi\)
\(38\) 0 0
\(39\) 1.43845 0.230336
\(40\) 0 0
\(41\) 0.561553 0.0876998 0.0438499 0.999038i \(-0.486038\pi\)
0.0438499 + 0.999038i \(0.486038\pi\)
\(42\) 0 0
\(43\) − 9.12311i − 1.39126i −0.718400 0.695630i \(-0.755125\pi\)
0.718400 0.695630i \(-0.244875\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.68466i 0.537463i 0.963215 + 0.268731i \(0.0866044\pi\)
−0.963215 + 0.268731i \(0.913396\pi\)
\(48\) 0 0
\(49\) −19.2462 −2.74946
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) − 4.24621i − 0.583262i −0.956531 0.291631i \(-0.905802\pi\)
0.956531 0.291631i \(-0.0941979\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 10.2462i − 1.35714i
\(58\) 0 0
\(59\) 6.24621 0.813187 0.406594 0.913609i \(-0.366716\pi\)
0.406594 + 0.913609i \(0.366716\pi\)
\(60\) 0 0
\(61\) 11.1231 1.42417 0.712084 0.702094i \(-0.247752\pi\)
0.712084 + 0.702094i \(0.247752\pi\)
\(62\) 0 0
\(63\) 18.2462i 2.29881i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.24621i − 0.763096i −0.924349 0.381548i \(-0.875391\pi\)
0.924349 0.381548i \(-0.124609\pi\)
\(68\) 0 0
\(69\) −2.56155 −0.308375
\(70\) 0 0
\(71\) 3.68466 0.437289 0.218644 0.975805i \(-0.429837\pi\)
0.218644 + 0.975805i \(0.429837\pi\)
\(72\) 0 0
\(73\) 16.5616i 1.93838i 0.246308 + 0.969192i \(0.420782\pi\)
−0.246308 + 0.969192i \(0.579218\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 20.4924i 2.33533i
\(78\) 0 0
\(79\) 10.2462 1.15279 0.576394 0.817172i \(-0.304459\pi\)
0.576394 + 0.817172i \(0.304459\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 21.9309i 2.35124i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −2.87689 −0.301580
\(92\) 0 0
\(93\) 3.68466i 0.382081i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.2462i 1.64955i 0.565459 + 0.824776i \(0.308699\pi\)
−0.565459 + 0.824776i \(0.691301\pi\)
\(98\) 0 0
\(99\) 14.2462 1.43180
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) − 2.24621i − 0.221326i −0.993858 0.110663i \(-0.964703\pi\)
0.993858 0.110663i \(-0.0352974\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) −8.24621 −0.789844 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(110\) 0 0
\(111\) −18.2462 −1.73185
\(112\) 0 0
\(113\) 20.2462i 1.90460i 0.305158 + 0.952302i \(0.401291\pi\)
−0.305158 + 0.952302i \(0.598709\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 1.43845i 0.129700i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.8078i 1.49145i 0.666255 + 0.745724i \(0.267896\pi\)
−0.666255 + 0.745724i \(0.732104\pi\)
\(128\) 0 0
\(129\) 23.3693 2.05755
\(130\) 0 0
\(131\) 9.93087 0.867664 0.433832 0.900994i \(-0.357161\pi\)
0.433832 + 0.900994i \(0.357161\pi\)
\(132\) 0 0
\(133\) 20.4924i 1.77692i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 15.1231i − 1.29205i −0.763315 0.646027i \(-0.776429\pi\)
0.763315 0.646027i \(-0.223571\pi\)
\(138\) 0 0
\(139\) −0.315342 −0.0267469 −0.0133735 0.999911i \(-0.504257\pi\)
−0.0133735 + 0.999911i \(0.504257\pi\)
\(140\) 0 0
\(141\) −9.43845 −0.794861
\(142\) 0 0
\(143\) 2.24621i 0.187838i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 49.3002i − 4.06621i
\(148\) 0 0
\(149\) 4.24621 0.347863 0.173932 0.984758i \(-0.444353\pi\)
0.173932 + 0.984758i \(0.444353\pi\)
\(150\) 0 0
\(151\) −16.8078 −1.36780 −0.683898 0.729577i \(-0.739717\pi\)
−0.683898 + 0.729577i \(0.739717\pi\)
\(152\) 0 0
\(153\) − 11.1231i − 0.899250i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 0 0
\(159\) 10.8769 0.862594
\(160\) 0 0
\(161\) 5.12311 0.403757
\(162\) 0 0
\(163\) − 0.315342i − 0.0246995i −0.999924 0.0123497i \(-0.996069\pi\)
0.999924 0.0123497i \(-0.00393114\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) 12.6847 0.975743
\(170\) 0 0
\(171\) 14.2462 1.08944
\(172\) 0 0
\(173\) 10.4924i 0.797724i 0.917011 + 0.398862i \(0.130595\pi\)
−0.917011 + 0.398862i \(0.869405\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 16.0000i 1.20263i
\(178\) 0 0
\(179\) 7.68466 0.574378 0.287189 0.957874i \(-0.407279\pi\)
0.287189 + 0.957874i \(0.407279\pi\)
\(180\) 0 0
\(181\) 3.12311 0.232139 0.116069 0.993241i \(-0.462971\pi\)
0.116069 + 0.993241i \(0.462971\pi\)
\(182\) 0 0
\(183\) 28.4924i 2.10622i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 12.4924i − 0.913536i
\(188\) 0 0
\(189\) −7.36932 −0.536039
\(190\) 0 0
\(191\) −2.87689 −0.208165 −0.104082 0.994569i \(-0.533191\pi\)
−0.104082 + 0.994569i \(0.533191\pi\)
\(192\) 0 0
\(193\) 24.5616i 1.76798i 0.467507 + 0.883990i \(0.345152\pi\)
−0.467507 + 0.883990i \(0.654848\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.4384i 0.814956i 0.913215 + 0.407478i \(0.133592\pi\)
−0.913215 + 0.407478i \(0.866408\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) − 43.8617i − 3.07849i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 3.56155i − 0.247545i
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −14.2462 −0.980750 −0.490375 0.871512i \(-0.663140\pi\)
−0.490375 + 0.871512i \(0.663140\pi\)
\(212\) 0 0
\(213\) 9.43845i 0.646712i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 7.36932i − 0.500262i
\(218\) 0 0
\(219\) −42.4233 −2.86670
\(220\) 0 0
\(221\) 1.75379 0.117973
\(222\) 0 0
\(223\) − 20.4924i − 1.37227i −0.727472 0.686137i \(-0.759305\pi\)
0.727472 0.686137i \(-0.240695\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 9.12311i − 0.605522i −0.953067 0.302761i \(-0.902092\pi\)
0.953067 0.302761i \(-0.0979084\pi\)
\(228\) 0 0
\(229\) −0.246211 −0.0162701 −0.00813505 0.999967i \(-0.502589\pi\)
−0.00813505 + 0.999967i \(0.502589\pi\)
\(230\) 0 0
\(231\) −52.4924 −3.45375
\(232\) 0 0
\(233\) − 12.5616i − 0.822935i −0.911425 0.411467i \(-0.865016\pi\)
0.911425 0.411467i \(-0.134984\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 26.2462i 1.70487i
\(238\) 0 0
\(239\) 8.80776 0.569727 0.284863 0.958568i \(-0.408052\pi\)
0.284863 + 0.958568i \(0.408052\pi\)
\(240\) 0 0
\(241\) −18.4924 −1.19120 −0.595601 0.803281i \(-0.703086\pi\)
−0.595601 + 0.803281i \(0.703086\pi\)
\(242\) 0 0
\(243\) − 22.2462i − 1.42710i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.24621i 0.142923i
\(248\) 0 0
\(249\) −30.7386 −1.94798
\(250\) 0 0
\(251\) 6.24621 0.394257 0.197129 0.980378i \(-0.436838\pi\)
0.197129 + 0.980378i \(0.436838\pi\)
\(252\) 0 0
\(253\) − 4.00000i − 0.251478i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.05398i 0.564771i 0.959301 + 0.282386i \(0.0911258\pi\)
−0.959301 + 0.282386i \(0.908874\pi\)
\(258\) 0 0
\(259\) 36.4924 2.26753
\(260\) 0 0
\(261\) −30.4924 −1.88743
\(262\) 0 0
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 25.6155i − 1.56764i
\(268\) 0 0
\(269\) 23.3002 1.42064 0.710319 0.703880i \(-0.248551\pi\)
0.710319 + 0.703880i \(0.248551\pi\)
\(270\) 0 0
\(271\) −2.24621 −0.136448 −0.0682238 0.997670i \(-0.521733\pi\)
−0.0682238 + 0.997670i \(0.521733\pi\)
\(272\) 0 0
\(273\) − 7.36932i − 0.446011i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.4384i 0.687270i 0.939103 + 0.343635i \(0.111658\pi\)
−0.939103 + 0.343635i \(0.888342\pi\)
\(278\) 0 0
\(279\) −5.12311 −0.306712
\(280\) 0 0
\(281\) 30.4924 1.81903 0.909513 0.415676i \(-0.136455\pi\)
0.909513 + 0.415676i \(0.136455\pi\)
\(282\) 0 0
\(283\) 22.8769i 1.35989i 0.733263 + 0.679945i \(0.237996\pi\)
−0.733263 + 0.679945i \(0.762004\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.87689i − 0.169818i
\(288\) 0 0
\(289\) 7.24621 0.426248
\(290\) 0 0
\(291\) −41.6155 −2.43955
\(292\) 0 0
\(293\) − 12.8769i − 0.752276i −0.926564 0.376138i \(-0.877252\pi\)
0.926564 0.376138i \(-0.122748\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.75379i 0.333869i
\(298\) 0 0
\(299\) 0.561553 0.0324754
\(300\) 0 0
\(301\) −46.7386 −2.69397
\(302\) 0 0
\(303\) 0.630683i 0.0362318i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 5.75379 0.327322
\(310\) 0 0
\(311\) 13.9309 0.789947 0.394974 0.918692i \(-0.370754\pi\)
0.394974 + 0.918692i \(0.370754\pi\)
\(312\) 0 0
\(313\) 15.1231i 0.854808i 0.904061 + 0.427404i \(0.140572\pi\)
−0.904061 + 0.427404i \(0.859428\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 20.7386i − 1.16480i −0.812903 0.582399i \(-0.802114\pi\)
0.812903 0.582399i \(-0.197886\pi\)
\(318\) 0 0
\(319\) −34.2462 −1.91742
\(320\) 0 0
\(321\) −30.7386 −1.71566
\(322\) 0 0
\(323\) − 12.4924i − 0.695097i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 21.1231i − 1.16811i
\(328\) 0 0
\(329\) 18.8769 1.04072
\(330\) 0 0
\(331\) 10.5616 0.580515 0.290258 0.956949i \(-0.406259\pi\)
0.290258 + 0.956949i \(0.406259\pi\)
\(332\) 0 0
\(333\) − 25.3693i − 1.39023i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 15.7538i − 0.858164i −0.903266 0.429082i \(-0.858837\pi\)
0.903266 0.429082i \(-0.141163\pi\)
\(338\) 0 0
\(339\) −51.8617 −2.81674
\(340\) 0 0
\(341\) −5.75379 −0.311585
\(342\) 0 0
\(343\) 62.7386i 3.38757i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.4924i 0.885360i 0.896680 + 0.442680i \(0.145972\pi\)
−0.896680 + 0.442680i \(0.854028\pi\)
\(348\) 0 0
\(349\) 2.80776 0.150296 0.0751481 0.997172i \(-0.476057\pi\)
0.0751481 + 0.997172i \(0.476057\pi\)
\(350\) 0 0
\(351\) −0.807764 −0.0431153
\(352\) 0 0
\(353\) 34.8078i 1.85263i 0.376750 + 0.926315i \(0.377042\pi\)
−0.376750 + 0.926315i \(0.622958\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 40.9848i 2.16915i
\(358\) 0 0
\(359\) 13.7538 0.725897 0.362949 0.931809i \(-0.381770\pi\)
0.362949 + 0.931809i \(0.381770\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 12.8078i 0.672233i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 20.4924i − 1.06970i −0.844948 0.534848i \(-0.820369\pi\)
0.844948 0.534848i \(-0.179631\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −21.7538 −1.12940
\(372\) 0 0
\(373\) − 24.7386i − 1.28092i −0.767992 0.640459i \(-0.778744\pi\)
0.767992 0.640459i \(-0.221256\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.80776i − 0.247612i
\(378\) 0 0
\(379\) 14.2462 0.731779 0.365889 0.930658i \(-0.380765\pi\)
0.365889 + 0.930658i \(0.380765\pi\)
\(380\) 0 0
\(381\) −43.0540 −2.20572
\(382\) 0 0
\(383\) 33.6155i 1.71767i 0.512250 + 0.858837i \(0.328812\pi\)
−0.512250 + 0.858837i \(0.671188\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 32.4924i 1.65168i
\(388\) 0 0
\(389\) −7.61553 −0.386123 −0.193061 0.981187i \(-0.561842\pi\)
−0.193061 + 0.981187i \(0.561842\pi\)
\(390\) 0 0
\(391\) −3.12311 −0.157942
\(392\) 0 0
\(393\) 25.4384i 1.28320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 3.93087i − 0.197285i −0.995123 0.0986423i \(-0.968550\pi\)
0.995123 0.0986423i \(-0.0314500\pi\)
\(398\) 0 0
\(399\) −52.4924 −2.62791
\(400\) 0 0
\(401\) −28.7386 −1.43514 −0.717569 0.696487i \(-0.754745\pi\)
−0.717569 + 0.696487i \(0.754745\pi\)
\(402\) 0 0
\(403\) − 0.807764i − 0.0402376i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 28.4924i − 1.41232i
\(408\) 0 0
\(409\) 2.31534 0.114486 0.0572431 0.998360i \(-0.481769\pi\)
0.0572431 + 0.998360i \(0.481769\pi\)
\(410\) 0 0
\(411\) 38.7386 1.91084
\(412\) 0 0
\(413\) − 32.0000i − 1.57462i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 0.807764i − 0.0395564i
\(418\) 0 0
\(419\) 9.12311 0.445693 0.222846 0.974854i \(-0.428465\pi\)
0.222846 + 0.974854i \(0.428465\pi\)
\(420\) 0 0
\(421\) 26.4924 1.29116 0.645581 0.763692i \(-0.276615\pi\)
0.645581 + 0.763692i \(0.276615\pi\)
\(422\) 0 0
\(423\) − 13.1231i − 0.638067i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 56.9848i − 2.75769i
\(428\) 0 0
\(429\) −5.75379 −0.277796
\(430\) 0 0
\(431\) −33.6155 −1.61920 −0.809602 0.586980i \(-0.800317\pi\)
−0.809602 + 0.586980i \(0.800317\pi\)
\(432\) 0 0
\(433\) 24.7386i 1.18886i 0.804146 + 0.594431i \(0.202623\pi\)
−0.804146 + 0.594431i \(0.797377\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 4.00000i − 0.191346i
\(438\) 0 0
\(439\) −21.3002 −1.01660 −0.508301 0.861179i \(-0.669726\pi\)
−0.508301 + 0.861179i \(0.669726\pi\)
\(440\) 0 0
\(441\) 68.5464 3.26411
\(442\) 0 0
\(443\) − 15.6847i − 0.745201i −0.927992 0.372600i \(-0.878466\pi\)
0.927992 0.372600i \(-0.121534\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 10.8769i 0.514459i
\(448\) 0 0
\(449\) −16.7386 −0.789945 −0.394972 0.918693i \(-0.629246\pi\)
−0.394972 + 0.918693i \(0.629246\pi\)
\(450\) 0 0
\(451\) −2.24621 −0.105770
\(452\) 0 0
\(453\) − 43.0540i − 2.02285i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.876894i 0.0410194i 0.999790 + 0.0205097i \(0.00652890\pi\)
−0.999790 + 0.0205097i \(0.993471\pi\)
\(458\) 0 0
\(459\) 4.49242 0.209688
\(460\) 0 0
\(461\) 23.4384 1.09164 0.545819 0.837903i \(-0.316219\pi\)
0.545819 + 0.837903i \(0.316219\pi\)
\(462\) 0 0
\(463\) 10.2462i 0.476182i 0.971243 + 0.238091i \(0.0765216\pi\)
−0.971243 + 0.238091i \(0.923478\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 35.3693i − 1.63670i −0.574722 0.818348i \(-0.694890\pi\)
0.574722 0.818348i \(-0.305110\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) −5.12311 −0.236060
\(472\) 0 0
\(473\) 36.4924i 1.67792i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 15.1231i 0.692439i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 13.1231i 0.597122i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.05398i 0.138389i 0.997603 + 0.0691944i \(0.0220429\pi\)
−0.997603 + 0.0691944i \(0.977957\pi\)
\(488\) 0 0
\(489\) 0.807764 0.0365284
\(490\) 0 0
\(491\) 10.5616 0.476636 0.238318 0.971187i \(-0.423404\pi\)
0.238318 + 0.971187i \(0.423404\pi\)
\(492\) 0 0
\(493\) 26.7386i 1.20425i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 18.8769i − 0.846744i
\(498\) 0 0
\(499\) −0.946025 −0.0423499 −0.0211749 0.999776i \(-0.506741\pi\)
−0.0211749 + 0.999776i \(0.506741\pi\)
\(500\) 0 0
\(501\) −20.4924 −0.915534
\(502\) 0 0
\(503\) 25.6155i 1.14214i 0.820901 + 0.571070i \(0.193472\pi\)
−0.820901 + 0.571070i \(0.806528\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 32.4924i 1.44304i
\(508\) 0 0
\(509\) 8.56155 0.379484 0.189742 0.981834i \(-0.439235\pi\)
0.189742 + 0.981834i \(0.439235\pi\)
\(510\) 0 0
\(511\) 84.8466 3.75339
\(512\) 0 0
\(513\) 5.75379i 0.254036i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 14.7386i − 0.648204i
\(518\) 0 0
\(519\) −26.8769 −1.17976
\(520\) 0 0
\(521\) −17.8617 −0.782537 −0.391269 0.920277i \(-0.627964\pi\)
−0.391269 + 0.920277i \(0.627964\pi\)
\(522\) 0 0
\(523\) − 23.8617i − 1.04340i −0.853129 0.521701i \(-0.825298\pi\)
0.853129 0.521701i \(-0.174702\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.49242i 0.195693i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −22.2462 −0.965403
\(532\) 0 0
\(533\) − 0.315342i − 0.0136590i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 19.6847i 0.849456i
\(538\) 0 0
\(539\) 76.9848 3.31597
\(540\) 0 0
\(541\) 33.6847 1.44822 0.724108 0.689686i \(-0.242252\pi\)
0.724108 + 0.689686i \(0.242252\pi\)
\(542\) 0 0
\(543\) 8.00000i 0.343313i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 0.946025i − 0.0404491i −0.999795 0.0202245i \(-0.993562\pi\)
0.999795 0.0202245i \(-0.00643811\pi\)
\(548\) 0 0
\(549\) −39.6155 −1.69075
\(550\) 0 0
\(551\) −34.2462 −1.45894
\(552\) 0 0
\(553\) − 52.4924i − 2.23220i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.75379i 0.328539i 0.986416 + 0.164269i \(0.0525266\pi\)
−0.986416 + 0.164269i \(0.947473\pi\)
\(558\) 0 0
\(559\) −5.12311 −0.216684
\(560\) 0 0
\(561\) 32.0000 1.35104
\(562\) 0 0
\(563\) − 39.2311i − 1.65339i −0.562649 0.826696i \(-0.690218\pi\)
0.562649 0.826696i \(-0.309782\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 35.8617i 1.50605i
\(568\) 0 0
\(569\) 20.7386 0.869409 0.434704 0.900573i \(-0.356853\pi\)
0.434704 + 0.900573i \(0.356853\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) − 7.36932i − 0.307858i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 32.4233i 1.34980i 0.737910 + 0.674900i \(0.235813\pi\)
−0.737910 + 0.674900i \(0.764187\pi\)
\(578\) 0 0
\(579\) −62.9157 −2.61469
\(580\) 0 0
\(581\) 61.4773 2.55051
\(582\) 0 0
\(583\) 16.9848i 0.703440i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.1771i 1.49319i 0.665280 + 0.746594i \(0.268312\pi\)
−0.665280 + 0.746594i \(0.731688\pi\)
\(588\) 0 0
\(589\) −5.75379 −0.237081
\(590\) 0 0
\(591\) −29.3002 −1.20525
\(592\) 0 0
\(593\) 26.9848i 1.10813i 0.832472 + 0.554067i \(0.186925\pi\)
−0.832472 + 0.554067i \(0.813075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 61.4773i − 2.51610i
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) 18.1771 0.741459 0.370729 0.928741i \(-0.379108\pi\)
0.370729 + 0.928741i \(0.379108\pi\)
\(602\) 0 0
\(603\) 22.2462i 0.905936i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 112.354 4.55282
\(610\) 0 0
\(611\) 2.06913 0.0837081
\(612\) 0 0
\(613\) 3.12311i 0.126141i 0.998009 + 0.0630705i \(0.0200893\pi\)
−0.998009 + 0.0630705i \(0.979911\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 35.6155i − 1.43383i −0.697162 0.716914i \(-0.745554\pi\)
0.697162 0.716914i \(-0.254446\pi\)
\(618\) 0 0
\(619\) 28.9848 1.16500 0.582500 0.812831i \(-0.302075\pi\)
0.582500 + 0.812831i \(0.302075\pi\)
\(620\) 0 0
\(621\) 1.43845 0.0577229
\(622\) 0 0
\(623\) 51.2311i 2.05253i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 40.9848i 1.63678i
\(628\) 0 0
\(629\) −22.2462 −0.887015
\(630\) 0 0
\(631\) −31.3693 −1.24879 −0.624396 0.781108i \(-0.714655\pi\)
−0.624396 + 0.781108i \(0.714655\pi\)
\(632\) 0 0
\(633\) − 36.4924i − 1.45044i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 10.8078i 0.428219i
\(638\) 0 0
\(639\) −13.1231 −0.519142
\(640\) 0 0
\(641\) 23.7538 0.938218 0.469109 0.883140i \(-0.344575\pi\)
0.469109 + 0.883140i \(0.344575\pi\)
\(642\) 0 0
\(643\) 44.3542i 1.74916i 0.484884 + 0.874579i \(0.338862\pi\)
−0.484884 + 0.874579i \(0.661138\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.43845i 0.0565512i 0.999600 + 0.0282756i \(0.00900160\pi\)
−0.999600 + 0.0282756i \(0.990998\pi\)
\(648\) 0 0
\(649\) −24.9848 −0.980741
\(650\) 0 0
\(651\) 18.8769 0.739844
\(652\) 0 0
\(653\) − 35.7926i − 1.40067i −0.713813 0.700337i \(-0.753033\pi\)
0.713813 0.700337i \(-0.246967\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 58.9848i − 2.30122i
\(658\) 0 0
\(659\) −14.2462 −0.554954 −0.277477 0.960732i \(-0.589498\pi\)
−0.277477 + 0.960732i \(0.589498\pi\)
\(660\) 0 0
\(661\) 10.4924 0.408108 0.204054 0.978960i \(-0.434588\pi\)
0.204054 + 0.978960i \(0.434588\pi\)
\(662\) 0 0
\(663\) 4.49242i 0.174471i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.56155i 0.331505i
\(668\) 0 0
\(669\) 52.4924 2.02947
\(670\) 0 0
\(671\) −44.4924 −1.71761
\(672\) 0 0
\(673\) − 4.56155i − 0.175835i −0.996128 0.0879175i \(-0.971979\pi\)
0.996128 0.0879175i \(-0.0280212\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 20.7386i − 0.797050i −0.917157 0.398525i \(-0.869522\pi\)
0.917157 0.398525i \(-0.130478\pi\)
\(678\) 0 0
\(679\) 83.2311 3.19411
\(680\) 0 0
\(681\) 23.3693 0.895514
\(682\) 0 0
\(683\) − 18.5616i − 0.710238i −0.934821 0.355119i \(-0.884440\pi\)
0.934821 0.355119i \(-0.115560\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 0.630683i − 0.0240621i
\(688\) 0 0
\(689\) −2.38447 −0.0908411
\(690\) 0 0
\(691\) 6.24621 0.237617 0.118809 0.992917i \(-0.462093\pi\)
0.118809 + 0.992917i \(0.462093\pi\)
\(692\) 0 0
\(693\) − 72.9848i − 2.77247i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.75379i 0.0664295i
\(698\) 0 0
\(699\) 32.1771 1.21705
\(700\) 0 0
\(701\) 38.9848 1.47244 0.736219 0.676744i \(-0.236610\pi\)
0.736219 + 0.676744i \(0.236610\pi\)
\(702\) 0 0
\(703\) − 28.4924i − 1.07461i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.26137i − 0.0474386i
\(708\) 0 0
\(709\) 40.7386 1.52997 0.764986 0.644047i \(-0.222746\pi\)
0.764986 + 0.644047i \(0.222746\pi\)
\(710\) 0 0
\(711\) −36.4924 −1.36857
\(712\) 0 0
\(713\) 1.43845i 0.0538703i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 22.5616i 0.842577i
\(718\) 0 0
\(719\) −38.7386 −1.44471 −0.722354 0.691524i \(-0.756940\pi\)
−0.722354 + 0.691524i \(0.756940\pi\)
\(720\) 0 0
\(721\) −11.5076 −0.428565
\(722\) 0 0
\(723\) − 47.3693i − 1.76168i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 37.1231i − 1.37682i −0.725322 0.688410i \(-0.758309\pi\)
0.725322 0.688410i \(-0.241691\pi\)
\(728\) 0 0
\(729\) 35.9848 1.33277
\(730\) 0 0
\(731\) 28.4924 1.05383
\(732\) 0 0
\(733\) 11.1231i 0.410841i 0.978674 + 0.205421i \(0.0658562\pi\)
−0.978674 + 0.205421i \(0.934144\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.9848i 0.920329i
\(738\) 0 0
\(739\) −27.5464 −1.01331 −0.506655 0.862149i \(-0.669118\pi\)
−0.506655 + 0.862149i \(0.669118\pi\)
\(740\) 0 0
\(741\) −5.75379 −0.211371
\(742\) 0 0
\(743\) − 13.7538i − 0.504578i −0.967652 0.252289i \(-0.918817\pi\)
0.967652 0.252289i \(-0.0811833\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 42.7386i − 1.56372i
\(748\) 0 0
\(749\) 61.4773 2.24633
\(750\) 0 0
\(751\) 49.6155 1.81050 0.905248 0.424883i \(-0.139685\pi\)
0.905248 + 0.424883i \(0.139685\pi\)
\(752\) 0 0
\(753\) 16.0000i 0.583072i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 24.8769i − 0.904166i −0.891976 0.452083i \(-0.850681\pi\)
0.891976 0.452083i \(-0.149319\pi\)
\(758\) 0 0
\(759\) 10.2462 0.371914
\(760\) 0 0
\(761\) −11.3002 −0.409631 −0.204816 0.978801i \(-0.565660\pi\)
−0.204816 + 0.978801i \(0.565660\pi\)
\(762\) 0 0
\(763\) 42.2462i 1.52942i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 3.50758i − 0.126651i
\(768\) 0 0
\(769\) −38.4924 −1.38807 −0.694036 0.719940i \(-0.744169\pi\)
−0.694036 + 0.719940i \(0.744169\pi\)
\(770\) 0 0
\(771\) −23.1922 −0.835248
\(772\) 0 0
\(773\) − 4.24621i − 0.152726i −0.997080 0.0763628i \(-0.975669\pi\)
0.997080 0.0763628i \(-0.0243307\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 93.4773i 3.35348i
\(778\) 0 0
\(779\) −2.24621 −0.0804789
\(780\) 0 0
\(781\) −14.7386 −0.527390
\(782\) 0 0
\(783\) − 12.3153i − 0.440114i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 46.2462i 1.64850i 0.566226 + 0.824250i \(0.308403\pi\)
−0.566226 + 0.824250i \(0.691597\pi\)
\(788\) 0 0
\(789\) 20.4924 0.729550
\(790\) 0 0
\(791\) 103.723 3.68798
\(792\) 0 0
\(793\) − 6.24621i − 0.221809i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 27.1231i − 0.960750i −0.877063 0.480375i \(-0.840501\pi\)
0.877063 0.480375i \(-0.159499\pi\)
\(798\) 0 0
\(799\) −11.5076 −0.407109
\(800\) 0 0
\(801\) 35.6155 1.25841
\(802\) 0 0
\(803\) − 66.2462i − 2.33778i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 59.6847i 2.10100i
\(808\) 0 0
\(809\) 10.4924 0.368894 0.184447 0.982842i \(-0.440951\pi\)
0.184447 + 0.982842i \(0.440951\pi\)
\(810\) 0 0
\(811\) −8.31534 −0.291991 −0.145996 0.989285i \(-0.546639\pi\)
−0.145996 + 0.989285i \(0.546639\pi\)
\(812\) 0 0
\(813\) − 5.75379i − 0.201794i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 36.4924i 1.27671i
\(818\) 0 0
\(819\) 10.2462 0.358032
\(820\) 0 0
\(821\) −31.7538 −1.10821 −0.554107 0.832445i \(-0.686940\pi\)
−0.554107 + 0.832445i \(0.686940\pi\)
\(822\) 0 0
\(823\) − 17.4384i − 0.607866i −0.952693 0.303933i \(-0.901700\pi\)
0.952693 0.303933i \(-0.0982999\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) −50.4924 −1.75367 −0.876837 0.480787i \(-0.840351\pi\)
−0.876837 + 0.480787i \(0.840351\pi\)
\(830\) 0 0
\(831\) −29.3002 −1.01641
\(832\) 0 0
\(833\) − 60.1080i − 2.08262i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 2.06913i − 0.0715196i
\(838\) 0 0
\(839\) −9.61553 −0.331965 −0.165982 0.986129i \(-0.553080\pi\)
−0.165982 + 0.986129i \(0.553080\pi\)
\(840\) 0 0
\(841\) 44.3002 1.52759
\(842\) 0 0
\(843\) 78.1080i 2.69018i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 25.6155i − 0.880160i
\(848\) 0 0
\(849\) −58.6004 −2.01116
\(850\) 0 0
\(851\) −7.12311 −0.244177
\(852\) 0 0
\(853\) 22.9848i 0.786986i 0.919328 + 0.393493i \(0.128733\pi\)
−0.919328 + 0.393493i \(0.871267\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 34.1771i − 1.16747i −0.811945 0.583733i \(-0.801591\pi\)
0.811945 0.583733i \(-0.198409\pi\)
\(858\) 0 0
\(859\) −38.4233 −1.31099 −0.655493 0.755201i \(-0.727539\pi\)
−0.655493 + 0.755201i \(0.727539\pi\)
\(860\) 0 0
\(861\) 7.36932 0.251146
\(862\) 0 0
\(863\) 18.4233i 0.627136i 0.949566 + 0.313568i \(0.101524\pi\)
−0.949566 + 0.313568i \(0.898476\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 18.5616i 0.630383i
\(868\) 0 0
\(869\) −40.9848 −1.39032
\(870\) 0 0
\(871\) −3.50758 −0.118850
\(872\) 0 0
\(873\) − 57.8617i − 1.95832i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 40.7386i 1.37565i 0.725878 + 0.687823i \(0.241433\pi\)
−0.725878 + 0.687823i \(0.758567\pi\)
\(878\) 0 0
\(879\) 32.9848 1.11255
\(880\) 0 0
\(881\) 24.1080 0.812217 0.406109 0.913825i \(-0.366885\pi\)
0.406109 + 0.913825i \(0.366885\pi\)
\(882\) 0 0
\(883\) − 40.4924i − 1.36268i −0.731968 0.681339i \(-0.761398\pi\)
0.731968 0.681339i \(-0.238602\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33.4384i 1.12275i 0.827560 + 0.561377i \(0.189728\pi\)
−0.827560 + 0.561377i \(0.810272\pi\)
\(888\) 0 0
\(889\) 86.1080 2.88797
\(890\) 0 0
\(891\) 28.0000 0.938035
\(892\) 0 0
\(893\) − 14.7386i − 0.493210i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.43845i 0.0480284i
\(898\) 0 0
\(899\) 12.3153 0.410740
\(900\) 0 0
\(901\) 13.2614 0.441800
\(902\) 0 0
\(903\) − 119.723i − 3.98415i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 20.0000i − 0.664089i −0.943264 0.332045i \(-0.892262\pi\)
0.943264 0.332045i \(-0.107738\pi\)
\(908\) 0 0
\(909\) −0.876894 −0.0290848
\(910\) 0 0
\(911\) −20.4924 −0.678944 −0.339472 0.940616i \(-0.610248\pi\)
−0.339472 + 0.940616i \(0.610248\pi\)
\(912\) 0 0
\(913\) − 48.0000i − 1.58857i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 50.8769i − 1.68010i
\(918\) 0 0
\(919\) 35.8617 1.18297 0.591485 0.806316i \(-0.298542\pi\)
0.591485 + 0.806316i \(0.298542\pi\)
\(920\) 0 0
\(921\) 30.7386 1.01287
\(922\) 0 0
\(923\) − 2.06913i − 0.0681063i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) 0 0
\(929\) −13.0540 −0.428287 −0.214144 0.976802i \(-0.568696\pi\)
−0.214144 + 0.976802i \(0.568696\pi\)
\(930\) 0 0
\(931\) 76.9848 2.52308
\(932\) 0 0
\(933\) 35.6847i 1.16826i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 41.3693i − 1.35148i −0.737142 0.675738i \(-0.763825\pi\)
0.737142 0.675738i \(-0.236175\pi\)
\(938\) 0 0
\(939\) −38.7386 −1.26419
\(940\) 0 0
\(941\) −60.6004 −1.97552 −0.987758 0.155995i \(-0.950142\pi\)
−0.987758 + 0.155995i \(0.950142\pi\)
\(942\) 0 0
\(943\) 0.561553i 0.0182867i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 35.1922i 1.14359i 0.820395 + 0.571797i \(0.193754\pi\)
−0.820395 + 0.571797i \(0.806246\pi\)
\(948\) 0 0
\(949\) 9.30019 0.301897
\(950\) 0 0
\(951\) 53.1231 1.72263
\(952\) 0 0
\(953\) 13.5076i 0.437553i 0.975775 + 0.218777i \(0.0702066\pi\)
−0.975775 + 0.218777i \(0.929793\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 87.7235i − 2.83570i
\(958\) 0 0
\(959\) −77.4773 −2.50187
\(960\) 0 0
\(961\) −28.9309 −0.933254
\(962\) 0 0
\(963\) − 42.7386i − 1.37723i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 0.177081i − 0.00569454i −0.999996 0.00284727i \(-0.999094\pi\)
0.999996 0.00284727i \(-0.000906315\pi\)
\(968\) 0 0
\(969\) 32.0000 1.02799
\(970\) 0 0
\(971\) 3.36932 0.108127 0.0540633 0.998538i \(-0.482783\pi\)
0.0540633 + 0.998538i \(0.482783\pi\)
\(972\) 0 0
\(973\) 1.61553i 0.0517915i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 48.1080i − 1.53911i −0.638581 0.769555i \(-0.720478\pi\)
0.638581 0.769555i \(-0.279522\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 29.3693 0.937690
\(982\) 0 0
\(983\) 37.1231i 1.18404i 0.805922 + 0.592022i \(0.201670\pi\)
−0.805922 + 0.592022i \(0.798330\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 48.3542i 1.53913i
\(988\) 0 0
\(989\) 9.12311 0.290098
\(990\) 0 0
\(991\) −12.4924 −0.396835 −0.198417 0.980118i \(-0.563580\pi\)
−0.198417 + 0.980118i \(0.563580\pi\)
\(992\) 0 0
\(993\) 27.0540i 0.858532i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 48.7386i 1.54357i 0.635885 + 0.771784i \(0.280635\pi\)
−0.635885 + 0.771784i \(0.719365\pi\)
\(998\) 0 0
\(999\) 10.2462 0.324176
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 4600.2.e.m.4049.4 4
5.2 odd 4 920.2.a.f.1.2 2
5.3 odd 4 4600.2.a.r.1.1 2
5.4 even 2 inner 4600.2.e.m.4049.1 4
15.2 even 4 8280.2.a.bb.1.2 2
20.3 even 4 9200.2.a.bx.1.2 2
20.7 even 4 1840.2.a.k.1.1 2
40.27 even 4 7360.2.a.bm.1.2 2
40.37 odd 4 7360.2.a.bj.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.f.1.2 2 5.2 odd 4
1840.2.a.k.1.1 2 20.7 even 4
4600.2.a.r.1.1 2 5.3 odd 4
4600.2.e.m.4049.1 4 5.4 even 2 inner
4600.2.e.m.4049.4 4 1.1 even 1 trivial
7360.2.a.bj.1.1 2 40.37 odd 4
7360.2.a.bm.1.2 2 40.27 even 4
8280.2.a.bb.1.2 2 15.2 even 4
9200.2.a.bx.1.2 2 20.3 even 4