Properties

Label 2400.2.d.e.49.6
Level 2400
Weight 2
Character 2400.49
Analytic conductor 19.164
Analytic rank 0
Dimension 6
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.6
Root \(-0.671462 + 1.24464i\) of defining polynomial
Character \(\chi\) \(=\) 2400.49
Dual form 2400.2.d.e.49.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.68585i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.68585i q^{7} +1.00000 q^{9} -2.29273i q^{11} +4.97858 q^{13} +2.97858i q^{17} +2.68585i q^{19} -4.68585i q^{21} -2.68585i q^{23} -1.00000 q^{27} -2.00000i q^{29} +6.97858 q^{31} +2.29273i q^{33} +4.39312 q^{37} -4.97858 q^{39} -11.3717 q^{41} +9.37169 q^{43} +7.27131i q^{47} -14.9572 q^{49} -2.97858i q^{51} +2.00000 q^{53} -2.68585i q^{57} +1.70727i q^{59} +4.58546i q^{61} +4.68585i q^{63} -4.00000 q^{67} +2.68585i q^{69} -0.585462 q^{71} -6.00000i q^{73} +10.7434 q^{77} +1.02142 q^{79} +1.00000 q^{81} -13.3717 q^{83} +2.00000i q^{87} -3.37169 q^{89} +23.3288i q^{91} -6.97858 q^{93} +3.95715i q^{97} -2.29273i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} + 6q^{9} - 6q^{27} + 12q^{31} + 8q^{37} - 20q^{41} + 8q^{43} - 30q^{49} + 12q^{53} - 24q^{67} + 8q^{71} - 32q^{77} + 36q^{79} + 6q^{81} - 32q^{83} + 28q^{89} - 12q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.68585i 1.77108i 0.464560 + 0.885542i \(0.346213\pi\)
−0.464560 + 0.885542i \(0.653787\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) − 2.29273i − 0.691284i −0.938366 0.345642i \(-0.887661\pi\)
0.938366 0.345642i \(-0.112339\pi\)
\(12\) 0 0
\(13\) 4.97858 1.38081 0.690404 0.723424i \(-0.257433\pi\)
0.690404 + 0.723424i \(0.257433\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.97858i 0.722411i 0.932486 + 0.361206i \(0.117635\pi\)
−0.932486 + 0.361206i \(0.882365\pi\)
\(18\) 0 0
\(19\) 2.68585i 0.616175i 0.951358 + 0.308088i \(0.0996890\pi\)
−0.951358 + 0.308088i \(0.900311\pi\)
\(20\) 0 0
\(21\) − 4.68585i − 1.02254i
\(22\) 0 0
\(23\) − 2.68585i − 0.560038i −0.959995 0.280019i \(-0.909659\pi\)
0.959995 0.280019i \(-0.0903407\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) − 2.00000i − 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) 6.97858 1.25339 0.626695 0.779265i \(-0.284407\pi\)
0.626695 + 0.779265i \(0.284407\pi\)
\(32\) 0 0
\(33\) 2.29273i 0.399113i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.39312 0.722224 0.361112 0.932523i \(-0.382397\pi\)
0.361112 + 0.932523i \(0.382397\pi\)
\(38\) 0 0
\(39\) −4.97858 −0.797210
\(40\) 0 0
\(41\) −11.3717 −1.77596 −0.887980 0.459882i \(-0.847892\pi\)
−0.887980 + 0.459882i \(0.847892\pi\)
\(42\) 0 0
\(43\) 9.37169 1.42917 0.714585 0.699549i \(-0.246616\pi\)
0.714585 + 0.699549i \(0.246616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.27131i 1.06063i 0.847801 + 0.530315i \(0.177926\pi\)
−0.847801 + 0.530315i \(0.822074\pi\)
\(48\) 0 0
\(49\) −14.9572 −2.13674
\(50\) 0 0
\(51\) − 2.97858i − 0.417084i
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.68585i − 0.355749i
\(58\) 0 0
\(59\) 1.70727i 0.222267i 0.993805 + 0.111134i \(0.0354482\pi\)
−0.993805 + 0.111134i \(0.964552\pi\)
\(60\) 0 0
\(61\) 4.58546i 0.587108i 0.955942 + 0.293554i \(0.0948381\pi\)
−0.955942 + 0.293554i \(0.905162\pi\)
\(62\) 0 0
\(63\) 4.68585i 0.590361i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 2.68585i 0.323338i
\(70\) 0 0
\(71\) −0.585462 −0.0694816 −0.0347408 0.999396i \(-0.511061\pi\)
−0.0347408 + 0.999396i \(0.511061\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.7434 1.22432
\(78\) 0 0
\(79\) 1.02142 0.114919 0.0574595 0.998348i \(-0.481700\pi\)
0.0574595 + 0.998348i \(0.481700\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.3717 −1.46773 −0.733867 0.679293i \(-0.762286\pi\)
−0.733867 + 0.679293i \(0.762286\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) −3.37169 −0.357399 −0.178699 0.983904i \(-0.557189\pi\)
−0.178699 + 0.983904i \(0.557189\pi\)
\(90\) 0 0
\(91\) 23.3288i 2.44553i
\(92\) 0 0
\(93\) −6.97858 −0.723645
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.95715i 0.401788i 0.979613 + 0.200894i \(0.0643847\pi\)
−0.979613 + 0.200894i \(0.935615\pi\)
\(98\) 0 0
\(99\) − 2.29273i − 0.230428i
\(100\) 0 0
\(101\) − 2.00000i − 0.199007i −0.995037 0.0995037i \(-0.968274\pi\)
0.995037 0.0995037i \(-0.0317255\pi\)
\(102\) 0 0
\(103\) 14.6430i 1.44282i 0.692509 + 0.721409i \(0.256505\pi\)
−0.692509 + 0.721409i \(0.743495\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3288 1.09520 0.547600 0.836740i \(-0.315541\pi\)
0.547600 + 0.836740i \(0.315541\pi\)
\(108\) 0 0
\(109\) 9.37169i 0.897645i 0.893621 + 0.448823i \(0.148157\pi\)
−0.893621 + 0.448823i \(0.851843\pi\)
\(110\) 0 0
\(111\) −4.39312 −0.416976
\(112\) 0 0
\(113\) 19.7648i 1.85932i 0.368423 + 0.929658i \(0.379898\pi\)
−0.368423 + 0.929658i \(0.620102\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.97858 0.460270
\(118\) 0 0
\(119\) −13.9572 −1.27945
\(120\) 0 0
\(121\) 5.74338 0.522126
\(122\) 0 0
\(123\) 11.3717 1.02535
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.64300i 0.589471i 0.955579 + 0.294735i \(0.0952315\pi\)
−0.955579 + 0.294735i \(0.904768\pi\)
\(128\) 0 0
\(129\) −9.37169 −0.825132
\(130\) 0 0
\(131\) − 7.07896i − 0.618492i −0.950982 0.309246i \(-0.899923\pi\)
0.950982 0.309246i \(-0.100077\pi\)
\(132\) 0 0
\(133\) −12.5855 −1.09130
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.9786i 1.27971i 0.768497 + 0.639853i \(0.221005\pi\)
−0.768497 + 0.639853i \(0.778995\pi\)
\(138\) 0 0
\(139\) 4.64300i 0.393814i 0.980422 + 0.196907i \(0.0630897\pi\)
−0.980422 + 0.196907i \(0.936910\pi\)
\(140\) 0 0
\(141\) − 7.27131i − 0.612355i
\(142\) 0 0
\(143\) − 11.4145i − 0.954532i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 14.9572 1.23365
\(148\) 0 0
\(149\) − 2.00000i − 0.163846i −0.996639 0.0819232i \(-0.973894\pi\)
0.996639 0.0819232i \(-0.0261062\pi\)
\(150\) 0 0
\(151\) 8.35027 0.679535 0.339768 0.940509i \(-0.389652\pi\)
0.339768 + 0.940509i \(0.389652\pi\)
\(152\) 0 0
\(153\) 2.97858i 0.240804i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −22.3503 −1.78375 −0.891873 0.452286i \(-0.850609\pi\)
−0.891873 + 0.452286i \(0.850609\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 12.5855 0.991873
\(162\) 0 0
\(163\) −1.37169 −0.107439 −0.0537196 0.998556i \(-0.517108\pi\)
−0.0537196 + 0.998556i \(0.517108\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.2713i 0.872200i 0.899898 + 0.436100i \(0.143641\pi\)
−0.899898 + 0.436100i \(0.856359\pi\)
\(168\) 0 0
\(169\) 11.7862 0.906633
\(170\) 0 0
\(171\) 2.68585i 0.205392i
\(172\) 0 0
\(173\) 10.7862 0.820062 0.410031 0.912072i \(-0.365518\pi\)
0.410031 + 0.912072i \(0.365518\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1.70727i − 0.128326i
\(178\) 0 0
\(179\) 3.66442i 0.273892i 0.990579 + 0.136946i \(0.0437287\pi\)
−0.990579 + 0.136946i \(0.956271\pi\)
\(180\) 0 0
\(181\) − 6.62831i − 0.492678i −0.969184 0.246339i \(-0.920772\pi\)
0.969184 0.246339i \(-0.0792277\pi\)
\(182\) 0 0
\(183\) − 4.58546i − 0.338967i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.82908 0.499392
\(188\) 0 0
\(189\) − 4.68585i − 0.340845i
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 1.21377i 0.0873690i 0.999045 + 0.0436845i \(0.0139096\pi\)
−0.999045 + 0.0436845i \(0.986090\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.9572 1.70688 0.853438 0.521194i \(-0.174513\pi\)
0.853438 + 0.521194i \(0.174513\pi\)
\(198\) 0 0
\(199\) 0.350269 0.0248299 0.0124150 0.999923i \(-0.496048\pi\)
0.0124150 + 0.999923i \(0.496048\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 9.37169 0.657764
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.68585i − 0.186679i
\(208\) 0 0
\(209\) 6.15792 0.425952
\(210\) 0 0
\(211\) − 14.1004i − 0.970710i −0.874317 0.485355i \(-0.838690\pi\)
0.874317 0.485355i \(-0.161310\pi\)
\(212\) 0 0
\(213\) 0.585462 0.0401152
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 32.7005i 2.21986i
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) 14.8291i 0.997512i
\(222\) 0 0
\(223\) 6.72869i 0.450587i 0.974291 + 0.225293i \(0.0723340\pi\)
−0.974291 + 0.225293i \(0.927666\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.95715 −0.660880 −0.330440 0.943827i \(-0.607197\pi\)
−0.330440 + 0.943827i \(0.607197\pi\)
\(228\) 0 0
\(229\) 11.3288i 0.748631i 0.927301 + 0.374316i \(0.122122\pi\)
−0.927301 + 0.374316i \(0.877878\pi\)
\(230\) 0 0
\(231\) −10.7434 −0.706863
\(232\) 0 0
\(233\) − 18.9786i − 1.24333i −0.783284 0.621664i \(-0.786457\pi\)
0.783284 0.621664i \(-0.213543\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.02142 −0.0663485
\(238\) 0 0
\(239\) 2.62831 0.170011 0.0850055 0.996380i \(-0.472909\pi\)
0.0850055 + 0.996380i \(0.472909\pi\)
\(240\) 0 0
\(241\) 10.7862 0.694802 0.347401 0.937717i \(-0.387064\pi\)
0.347401 + 0.937717i \(0.387064\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.3717i 0.850820i
\(248\) 0 0
\(249\) 13.3717 0.847397
\(250\) 0 0
\(251\) − 30.9933i − 1.95628i −0.207952 0.978139i \(-0.566680\pi\)
0.207952 0.978139i \(-0.433320\pi\)
\(252\) 0 0
\(253\) −6.15792 −0.387145
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 20.9357i − 1.30594i −0.757386 0.652968i \(-0.773524\pi\)
0.757386 0.652968i \(-0.226476\pi\)
\(258\) 0 0
\(259\) 20.5855i 1.27912i
\(260\) 0 0
\(261\) − 2.00000i − 0.123797i
\(262\) 0 0
\(263\) 19.2713i 1.18832i 0.804347 + 0.594160i \(0.202515\pi\)
−0.804347 + 0.594160i \(0.797485\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3.37169 0.206344
\(268\) 0 0
\(269\) 24.7434i 1.50863i 0.656512 + 0.754315i \(0.272031\pi\)
−0.656512 + 0.754315i \(0.727969\pi\)
\(270\) 0 0
\(271\) −27.5640 −1.67440 −0.837198 0.546900i \(-0.815808\pi\)
−0.837198 + 0.546900i \(0.815808\pi\)
\(272\) 0 0
\(273\) − 23.3288i − 1.41193i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −20.3074 −1.22015 −0.610077 0.792342i \(-0.708862\pi\)
−0.610077 + 0.792342i \(0.708862\pi\)
\(278\) 0 0
\(279\) 6.97858 0.417796
\(280\) 0 0
\(281\) −10.7862 −0.643453 −0.321726 0.946833i \(-0.604263\pi\)
−0.321726 + 0.946833i \(0.604263\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 53.2860i − 3.14537i
\(288\) 0 0
\(289\) 8.12808 0.478122
\(290\) 0 0
\(291\) − 3.95715i − 0.231972i
\(292\) 0 0
\(293\) 21.9143 1.28025 0.640124 0.768272i \(-0.278883\pi\)
0.640124 + 0.768272i \(0.278883\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.29273i 0.133038i
\(298\) 0 0
\(299\) − 13.3717i − 0.773305i
\(300\) 0 0
\(301\) 43.9143i 2.53118i
\(302\) 0 0
\(303\) 2.00000i 0.114897i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −26.5426 −1.51487 −0.757434 0.652912i \(-0.773547\pi\)
−0.757434 + 0.652912i \(0.773547\pi\)
\(308\) 0 0
\(309\) − 14.6430i − 0.833011i
\(310\) 0 0
\(311\) −12.2008 −0.691842 −0.345921 0.938264i \(-0.612434\pi\)
−0.345921 + 0.938264i \(0.612434\pi\)
\(312\) 0 0
\(313\) 15.9572i 0.901952i 0.892536 + 0.450976i \(0.148924\pi\)
−0.892536 + 0.450976i \(0.851076\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −33.5296 −1.88321 −0.941605 0.336718i \(-0.890683\pi\)
−0.941605 + 0.336718i \(0.890683\pi\)
\(318\) 0 0
\(319\) −4.58546 −0.256737
\(320\) 0 0
\(321\) −11.3288 −0.632315
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 9.37169i − 0.518256i
\(328\) 0 0
\(329\) −34.0722 −1.87846
\(330\) 0 0
\(331\) 19.8568i 1.09143i 0.837972 + 0.545713i \(0.183741\pi\)
−0.837972 + 0.545713i \(0.816259\pi\)
\(332\) 0 0
\(333\) 4.39312 0.240741
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 7.17092i − 0.390625i −0.980741 0.195313i \(-0.937428\pi\)
0.980741 0.195313i \(-0.0625721\pi\)
\(338\) 0 0
\(339\) − 19.7648i − 1.07348i
\(340\) 0 0
\(341\) − 16.0000i − 0.866449i
\(342\) 0 0
\(343\) − 37.2860i − 2.01325i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.786230 −0.0422071 −0.0211035 0.999777i \(-0.506718\pi\)
−0.0211035 + 0.999777i \(0.506718\pi\)
\(348\) 0 0
\(349\) − 6.15792i − 0.329626i −0.986325 0.164813i \(-0.947298\pi\)
0.986325 0.164813i \(-0.0527021\pi\)
\(350\) 0 0
\(351\) −4.97858 −0.265737
\(352\) 0 0
\(353\) 21.7220i 1.15614i 0.815986 + 0.578072i \(0.196195\pi\)
−0.815986 + 0.578072i \(0.803805\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 13.9572 0.738691
\(358\) 0 0
\(359\) 0.585462 0.0308995 0.0154498 0.999881i \(-0.495082\pi\)
0.0154498 + 0.999881i \(0.495082\pi\)
\(360\) 0 0
\(361\) 11.7862 0.620328
\(362\) 0 0
\(363\) −5.74338 −0.301450
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 0.485078i − 0.0253209i −0.999920 0.0126604i \(-0.995970\pi\)
0.999920 0.0126604i \(-0.00403005\pi\)
\(368\) 0 0
\(369\) −11.3717 −0.591987
\(370\) 0 0
\(371\) 9.37169i 0.486554i
\(372\) 0 0
\(373\) 12.3931 0.641691 0.320846 0.947132i \(-0.396033\pi\)
0.320846 + 0.947132i \(0.396033\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 9.95715i − 0.512820i
\(378\) 0 0
\(379\) − 26.0147i − 1.33629i −0.744033 0.668143i \(-0.767090\pi\)
0.744033 0.668143i \(-0.232910\pi\)
\(380\) 0 0
\(381\) − 6.64300i − 0.340331i
\(382\) 0 0
\(383\) − 6.68585i − 0.341631i −0.985303 0.170815i \(-0.945360\pi\)
0.985303 0.170815i \(-0.0546402\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.37169 0.476390
\(388\) 0 0
\(389\) − 29.9143i − 1.51672i −0.651838 0.758358i \(-0.726002\pi\)
0.651838 0.758358i \(-0.273998\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 7.07896i 0.357086i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9.76481 0.490082 0.245041 0.969513i \(-0.421199\pi\)
0.245041 + 0.969513i \(0.421199\pi\)
\(398\) 0 0
\(399\) 12.5855 0.630061
\(400\) 0 0
\(401\) −6.58546 −0.328862 −0.164431 0.986389i \(-0.552579\pi\)
−0.164431 + 0.986389i \(0.552579\pi\)
\(402\) 0 0
\(403\) 34.7434 1.73069
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 10.0722i − 0.499262i
\(408\) 0 0
\(409\) 25.9143 1.28138 0.640690 0.767800i \(-0.278648\pi\)
0.640690 + 0.767800i \(0.278648\pi\)
\(410\) 0 0
\(411\) − 14.9786i − 0.738839i
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.64300i − 0.227369i
\(418\) 0 0
\(419\) − 12.2499i − 0.598446i −0.954183 0.299223i \(-0.903273\pi\)
0.954183 0.299223i \(-0.0967275\pi\)
\(420\) 0 0
\(421\) − 4.67115i − 0.227658i −0.993500 0.113829i \(-0.963688\pi\)
0.993500 0.113829i \(-0.0363116\pi\)
\(422\) 0 0
\(423\) 7.27131i 0.353543i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −21.4868 −1.03982
\(428\) 0 0
\(429\) 11.4145i 0.551099i
\(430\) 0 0
\(431\) −0.585462 −0.0282007 −0.0141004 0.999901i \(-0.504488\pi\)
−0.0141004 + 0.999901i \(0.504488\pi\)
\(432\) 0 0
\(433\) − 21.9143i − 1.05313i −0.850133 0.526567i \(-0.823479\pi\)
0.850133 0.526567i \(-0.176521\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.21377 0.345081
\(438\) 0 0
\(439\) −2.39312 −0.114217 −0.0571086 0.998368i \(-0.518188\pi\)
−0.0571086 + 0.998368i \(0.518188\pi\)
\(440\) 0 0
\(441\) −14.9572 −0.712245
\(442\) 0 0
\(443\) −20.7005 −0.983512 −0.491756 0.870733i \(-0.663645\pi\)
−0.491756 + 0.870733i \(0.663645\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.00000i 0.0945968i
\(448\) 0 0
\(449\) 37.9143 1.78929 0.894643 0.446781i \(-0.147430\pi\)
0.894643 + 0.446781i \(0.147430\pi\)
\(450\) 0 0
\(451\) 26.0722i 1.22769i
\(452\) 0 0
\(453\) −8.35027 −0.392330
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 38.7005i − 1.81033i −0.425055 0.905167i \(-0.639745\pi\)
0.425055 0.905167i \(-0.360255\pi\)
\(458\) 0 0
\(459\) − 2.97858i − 0.139028i
\(460\) 0 0
\(461\) − 4.74338i − 0.220921i −0.993880 0.110461i \(-0.964767\pi\)
0.993880 0.110461i \(-0.0352326\pi\)
\(462\) 0 0
\(463\) − 15.3142i − 0.711709i −0.934541 0.355855i \(-0.884190\pi\)
0.934541 0.355855i \(-0.115810\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.5426 1.41334 0.706672 0.707541i \(-0.250196\pi\)
0.706672 + 0.707541i \(0.250196\pi\)
\(468\) 0 0
\(469\) − 18.7434i − 0.865489i
\(470\) 0 0
\(471\) 22.3503 1.02985
\(472\) 0 0
\(473\) − 21.4868i − 0.987963i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) 3.32885 0.152099 0.0760494 0.997104i \(-0.475769\pi\)
0.0760494 + 0.997104i \(0.475769\pi\)
\(480\) 0 0
\(481\) 21.8715 0.997253
\(482\) 0 0
\(483\) −12.5855 −0.572658
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 12.1004i − 0.548321i −0.961684 0.274160i \(-0.911600\pi\)
0.961684 0.274160i \(-0.0883999\pi\)
\(488\) 0 0
\(489\) 1.37169 0.0620301
\(490\) 0 0
\(491\) − 14.2927i − 0.645022i −0.946566 0.322511i \(-0.895473\pi\)
0.946566 0.322511i \(-0.104527\pi\)
\(492\) 0 0
\(493\) 5.95715 0.268297
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2.74338i − 0.123058i
\(498\) 0 0
\(499\) − 9.22846i − 0.413123i −0.978434 0.206561i \(-0.933773\pi\)
0.978434 0.206561i \(-0.0662273\pi\)
\(500\) 0 0
\(501\) − 11.2713i − 0.503565i
\(502\) 0 0
\(503\) 14.1004i 0.628705i 0.949306 + 0.314353i \(0.101787\pi\)
−0.949306 + 0.314353i \(0.898213\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −11.7862 −0.523445
\(508\) 0 0
\(509\) − 43.4011i − 1.92372i −0.273544 0.961859i \(-0.588196\pi\)
0.273544 0.961859i \(-0.411804\pi\)
\(510\) 0 0
\(511\) 28.1151 1.24374
\(512\) 0 0
\(513\) − 2.68585i − 0.118583i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.6712 0.733196
\(518\) 0 0
\(519\) −10.7862 −0.473463
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) −13.5725 −0.593482 −0.296741 0.954958i \(-0.595900\pi\)
−0.296741 + 0.954958i \(0.595900\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.7862i 0.905462i
\(528\) 0 0
\(529\) 15.7862 0.686358
\(530\) 0 0
\(531\) 1.70727i 0.0740892i
\(532\) 0 0
\(533\) −56.6148 −2.45226
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 3.66442i − 0.158132i
\(538\) 0 0
\(539\) 34.2927i 1.47709i
\(540\) 0 0
\(541\) 37.2860i 1.60305i 0.597961 + 0.801525i \(0.295978\pi\)
−0.597961 + 0.801525i \(0.704022\pi\)
\(542\) 0 0
\(543\) 6.62831i 0.284448i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.200768 −0.00858424 −0.00429212 0.999991i \(-0.501366\pi\)
−0.00429212 + 0.999991i \(0.501366\pi\)
\(548\) 0 0
\(549\) 4.58546i 0.195703i
\(550\) 0 0
\(551\) 5.37169 0.228842
\(552\) 0 0
\(553\) 4.78623i 0.203531i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.21377 −0.390400 −0.195200 0.980763i \(-0.562536\pi\)
−0.195200 + 0.980763i \(0.562536\pi\)
\(558\) 0 0
\(559\) 46.6577 1.97341
\(560\) 0 0
\(561\) −6.82908 −0.288324
\(562\) 0 0
\(563\) 36.7005 1.54674 0.773372 0.633953i \(-0.218569\pi\)
0.773372 + 0.633953i \(0.218569\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.68585i 0.196787i
\(568\) 0 0
\(569\) −13.4145 −0.562367 −0.281183 0.959654i \(-0.590727\pi\)
−0.281183 + 0.959654i \(0.590727\pi\)
\(570\) 0 0
\(571\) 18.6858i 0.781978i 0.920395 + 0.390989i \(0.127867\pi\)
−0.920395 + 0.390989i \(0.872133\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.78623i 0.115992i 0.998317 + 0.0579961i \(0.0184711\pi\)
−0.998317 + 0.0579961i \(0.981529\pi\)
\(578\) 0 0
\(579\) − 1.21377i − 0.0504425i
\(580\) 0 0
\(581\) − 62.6577i − 2.59948i
\(582\) 0 0
\(583\) − 4.58546i − 0.189910i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27.3288 −1.12798 −0.563991 0.825781i \(-0.690735\pi\)
−0.563991 + 0.825781i \(0.690735\pi\)
\(588\) 0 0
\(589\) 18.7434i 0.772308i
\(590\) 0 0
\(591\) −23.9572 −0.985466
\(592\) 0 0
\(593\) − 6.97858i − 0.286576i −0.989681 0.143288i \(-0.954233\pi\)
0.989681 0.143288i \(-0.0457675\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.350269 −0.0143356
\(598\) 0 0
\(599\) 36.4998 1.49134 0.745670 0.666315i \(-0.232130\pi\)
0.745670 + 0.666315i \(0.232130\pi\)
\(600\) 0 0
\(601\) −15.5725 −0.635214 −0.317607 0.948222i \(-0.602879\pi\)
−0.317607 + 0.948222i \(0.602879\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 31.2285i − 1.26752i −0.773528 0.633762i \(-0.781510\pi\)
0.773528 0.633762i \(-0.218490\pi\)
\(608\) 0 0
\(609\) −9.37169 −0.379760
\(610\) 0 0
\(611\) 36.2008i 1.46453i
\(612\) 0 0
\(613\) −0.978577 −0.0395244 −0.0197622 0.999805i \(-0.506291\pi\)
−0.0197622 + 0.999805i \(0.506291\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 32.9357i − 1.32594i −0.748645 0.662971i \(-0.769295\pi\)
0.748645 0.662971i \(-0.230705\pi\)
\(618\) 0 0
\(619\) − 3.35700i − 0.134929i −0.997722 0.0674646i \(-0.978509\pi\)
0.997722 0.0674646i \(-0.0214910\pi\)
\(620\) 0 0
\(621\) 2.68585i 0.107779i
\(622\) 0 0
\(623\) − 15.7992i − 0.632983i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −6.15792 −0.245924
\(628\) 0 0
\(629\) 13.0852i 0.521742i
\(630\) 0 0
\(631\) 27.7648 1.10530 0.552650 0.833414i \(-0.313617\pi\)
0.552650 + 0.833414i \(0.313617\pi\)
\(632\) 0 0
\(633\) 14.1004i 0.560440i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −74.4653 −2.95042
\(638\) 0 0
\(639\) −0.585462 −0.0231605
\(640\) 0 0
\(641\) −21.1281 −0.834509 −0.417254 0.908790i \(-0.637008\pi\)
−0.417254 + 0.908790i \(0.637008\pi\)
\(642\) 0 0
\(643\) 29.2860 1.15493 0.577464 0.816416i \(-0.304043\pi\)
0.577464 + 0.816416i \(0.304043\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.6728i 0.616163i 0.951360 + 0.308082i \(0.0996870\pi\)
−0.951360 + 0.308082i \(0.900313\pi\)
\(648\) 0 0
\(649\) 3.91431 0.153650
\(650\) 0 0
\(651\) − 32.7005i − 1.28164i
\(652\) 0 0
\(653\) −17.5296 −0.685987 −0.342993 0.939338i \(-0.611441\pi\)
−0.342993 + 0.939338i \(0.611441\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 6.00000i − 0.234082i
\(658\) 0 0
\(659\) − 23.8652i − 0.929656i −0.885401 0.464828i \(-0.846116\pi\)
0.885401 0.464828i \(-0.153884\pi\)
\(660\) 0 0
\(661\) − 30.1579i − 1.17301i −0.809947 0.586504i \(-0.800504\pi\)
0.809947 0.586504i \(-0.199496\pi\)
\(662\) 0 0
\(663\) − 14.8291i − 0.575914i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.37169 −0.207993
\(668\) 0 0
\(669\) − 6.72869i − 0.260146i
\(670\) 0 0
\(671\) 10.5132 0.405859
\(672\) 0 0
\(673\) − 18.0000i − 0.693849i −0.937893 0.346925i \(-0.887226\pi\)
0.937893 0.346925i \(-0.112774\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.61531 0.369546 0.184773 0.982781i \(-0.440845\pi\)
0.184773 + 0.982781i \(0.440845\pi\)
\(678\) 0 0
\(679\) −18.5426 −0.711600
\(680\) 0 0
\(681\) 9.95715 0.381559
\(682\) 0 0
\(683\) 18.6283 0.712792 0.356396 0.934335i \(-0.384005\pi\)
0.356396 + 0.934335i \(0.384005\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 11.3288i − 0.432222i
\(688\) 0 0
\(689\) 9.95715 0.379337
\(690\) 0 0
\(691\) 13.4292i 0.510872i 0.966826 + 0.255436i \(0.0822190\pi\)
−0.966826 + 0.255436i \(0.917781\pi\)
\(692\) 0 0
\(693\) 10.7434 0.408107
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 33.8715i − 1.28297i
\(698\) 0 0
\(699\) 18.9786i 0.717836i
\(700\) 0 0
\(701\) 19.1709i 0.724076i 0.932163 + 0.362038i \(0.117919\pi\)
−0.932163 + 0.362038i \(0.882081\pi\)
\(702\) 0 0
\(703\) 11.7992i 0.445016i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.37169 0.352459
\(708\) 0 0
\(709\) − 15.4145i − 0.578905i −0.957192 0.289453i \(-0.906527\pi\)
0.957192 0.289453i \(-0.0934732\pi\)
\(710\) 0 0
\(711\) 1.02142 0.0383063
\(712\) 0 0
\(713\) − 18.7434i − 0.701945i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −2.62831 −0.0981559
\(718\) 0 0
\(719\) 20.7862 0.775196 0.387598 0.921829i \(-0.373305\pi\)
0.387598 + 0.921829i \(0.373305\pi\)
\(720\) 0 0
\(721\) −68.6148 −2.55535
\(722\) 0 0
\(723\) −10.7862 −0.401144
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12.3012i 0.456224i 0.973635 + 0.228112i \(0.0732553\pi\)
−0.973635 + 0.228112i \(0.926745\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 27.9143i 1.03245i
\(732\) 0 0
\(733\) 35.9227 1.32684 0.663418 0.748249i \(-0.269105\pi\)
0.663418 + 0.748249i \(0.269105\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.17092i 0.337815i
\(738\) 0 0
\(739\) 29.0277i 1.06780i 0.845547 + 0.533900i \(0.179274\pi\)
−0.845547 + 0.533900i \(0.820726\pi\)
\(740\) 0 0
\(741\) − 13.3717i − 0.491221i
\(742\) 0 0
\(743\) − 2.60015i − 0.0953904i −0.998862 0.0476952i \(-0.984812\pi\)
0.998862 0.0476952i \(-0.0151876\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −13.3717 −0.489245
\(748\) 0 0
\(749\) 53.0852i 1.93969i
\(750\) 0 0
\(751\) 10.8929 0.397487 0.198744 0.980052i \(-0.436314\pi\)
0.198744 + 0.980052i \(0.436314\pi\)
\(752\) 0 0
\(753\) 30.9933i 1.12946i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34.3503 1.24848 0.624241 0.781232i \(-0.285408\pi\)
0.624241 + 0.781232i \(0.285408\pi\)
\(758\) 0 0
\(759\) 6.15792 0.223518
\(760\) 0 0
\(761\) −19.0852 −0.691839 −0.345920 0.938264i \(-0.612433\pi\)
−0.345920 + 0.938264i \(0.612433\pi\)
\(762\) 0 0
\(763\) −43.9143 −1.58980
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.49977i 0.306909i
\(768\) 0 0
\(769\) −31.8715 −1.14931 −0.574657 0.818394i \(-0.694865\pi\)
−0.574657 + 0.818394i \(0.694865\pi\)
\(770\) 0 0
\(771\) 20.9357i 0.753982i
\(772\) 0 0
\(773\) 11.9572 0.430069 0.215034 0.976606i \(-0.431014\pi\)
0.215034 + 0.976606i \(0.431014\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 20.5855i − 0.738499i
\(778\) 0 0
\(779\) − 30.5426i − 1.09430i
\(780\) 0 0
\(781\) 1.34231i 0.0480315i
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −33.0852 −1.17936 −0.589681 0.807637i \(-0.700746\pi\)
−0.589681 + 0.807637i \(0.700746\pi\)
\(788\) 0 0
\(789\) − 19.2713i − 0.686077i
\(790\) 0 0
\(791\) −92.6148 −3.29300
\(792\) 0 0
\(793\) 22.8291i 0.810684i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) −21.6582 −0.766210
\(800\) 0 0
\(801\) −3.37169 −0.119133
\(802\) 0 0
\(803\) −13.7564 −0.485452
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 24.7434i − 0.871008i
\(808\) 0 0
\(809\) 30.6148 1.07636 0.538180 0.842830i \(-0.319112\pi\)
0.538180 + 0.842830i \(0.319112\pi\)
\(810\) 0 0
\(811\) − 53.9290i − 1.89370i −0.321670 0.946852i \(-0.604244\pi\)
0.321670 0.946852i \(-0.395756\pi\)
\(812\) 0 0
\(813\) 27.5640 0.966713
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 25.1709i 0.880619i
\(818\) 0 0
\(819\) 23.3288i 0.815176i
\(820\) 0 0
\(821\) 12.6577i 0.441757i 0.975301 + 0.220878i \(0.0708924\pi\)
−0.975301 + 0.220878i \(0.929108\pi\)
\(822\) 0 0
\(823\) 19.8139i 0.690670i 0.938479 + 0.345335i \(0.112235\pi\)
−0.938479 + 0.345335i \(0.887765\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) 41.3717i 1.43690i 0.695580 + 0.718449i \(0.255148\pi\)
−0.695580 + 0.718449i \(0.744852\pi\)
\(830\) 0 0
\(831\) 20.3074 0.704457
\(832\) 0 0
\(833\) − 44.5510i − 1.54360i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −6.97858 −0.241215
\(838\) 0 0
\(839\) 20.9013 0.721593 0.360797 0.932645i \(-0.382505\pi\)
0.360797 + 0.932645i \(0.382505\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 10.7862 0.371498
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 26.9126i 0.924728i
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) − 11.7992i − 0.404472i
\(852\) 0 0
\(853\) −6.63673 −0.227237 −0.113619 0.993524i \(-0.536244\pi\)
−0.113619 + 0.993524i \(0.536244\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 9.80765i − 0.335023i −0.985870 0.167512i \(-0.946427\pi\)
0.985870 0.167512i \(-0.0535731\pi\)
\(858\) 0 0
\(859\) 39.3864i 1.34385i 0.740621 + 0.671923i \(0.234531\pi\)
−0.740621 + 0.671923i \(0.765469\pi\)
\(860\) 0 0
\(861\) 53.2860i 1.81598i
\(862\) 0 0
\(863\) − 7.07054i − 0.240684i −0.992732 0.120342i \(-0.961601\pi\)
0.992732 0.120342i \(-0.0383991\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.12808 −0.276044
\(868\) 0 0
\(869\) − 2.34185i − 0.0794417i
\(870\) 0 0
\(871\) −19.9143 −0.674771
\(872\) 0 0
\(873\) 3.95715i 0.133929i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 23.1365 0.781264 0.390632 0.920547i \(-0.372256\pi\)
0.390632 + 0.920547i \(0.372256\pi\)
\(878\) 0 0
\(879\) −21.9143 −0.739151
\(880\) 0 0
\(881\) 28.4569 0.958738 0.479369 0.877613i \(-0.340866\pi\)
0.479369 + 0.877613i \(0.340866\pi\)
\(882\) 0 0
\(883\) −41.2003 −1.38650 −0.693250 0.720697i \(-0.743822\pi\)
−0.693250 + 0.720697i \(0.743822\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 3.55777i − 0.119458i −0.998215 0.0597291i \(-0.980976\pi\)
0.998215 0.0597291i \(-0.0190237\pi\)
\(888\) 0 0
\(889\) −31.1281 −1.04400
\(890\) 0 0
\(891\) − 2.29273i − 0.0768094i
\(892\) 0 0
\(893\) −19.5296 −0.653534
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 13.3717i 0.446468i
\(898\) 0 0
\(899\) − 13.9572i − 0.465497i
\(900\) 0 0
\(901\) 5.95715i 0.198462i
\(902\) 0 0
\(903\) − 43.9143i − 1.46138i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 50.6577 1.68206 0.841031 0.540988i \(-0.181949\pi\)
0.841031 + 0.540988i \(0.181949\pi\)
\(908\) 0 0
\(909\) − 2.00000i − 0.0663358i
\(910\) 0 0
\(911\) 26.4569 0.876557 0.438279 0.898839i \(-0.355588\pi\)
0.438279 + 0.898839i \(0.355588\pi\)
\(912\) 0 0
\(913\) 30.6577i 1.01462i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 33.1709 1.09540
\(918\) 0 0
\(919\) 29.8077 0.983264 0.491632 0.870803i \(-0.336401\pi\)
0.491632 + 0.870803i \(0.336401\pi\)
\(920\) 0 0
\(921\) 26.5426 0.874609
\(922\) 0 0
\(923\) −2.91477 −0.0959407
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 14.6430i 0.480939i
\(928\) 0 0
\(929\) 8.82908 0.289673 0.144836 0.989456i \(-0.453734\pi\)
0.144836 + 0.989456i \(0.453734\pi\)
\(930\) 0 0
\(931\) − 40.1726i − 1.31660i
\(932\) 0 0
\(933\) 12.2008 0.399435
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 42.2302i 1.37960i 0.724000 + 0.689799i \(0.242301\pi\)
−0.724000 + 0.689799i \(0.757699\pi\)
\(938\) 0 0
\(939\) − 15.9572i − 0.520742i
\(940\) 0 0
\(941\) 32.7434i 1.06740i 0.845673 + 0.533702i \(0.179200\pi\)
−0.845673 + 0.533702i \(0.820800\pi\)
\(942\) 0 0
\(943\) 30.5426i 0.994604i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.9143 0.517146 0.258573 0.965992i \(-0.416748\pi\)
0.258573 + 0.965992i \(0.416748\pi\)
\(948\) 0 0
\(949\) − 29.8715i − 0.969669i
\(950\) 0 0
\(951\) 33.5296 1.08727
\(952\) 0 0
\(953\) 55.6791i 1.80362i 0.432129 + 0.901812i \(0.357762\pi\)
−0.432129 + 0.901812i \(0.642238\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.58546 0.148227
\(958\) 0 0
\(959\) −70.1873 −2.26647
\(960\) 0 0
\(961\) 17.7005 0.570985
\(962\) 0 0
\(963\) 11.3288 0.365067
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 54.7581i − 1.76090i −0.474138 0.880451i \(-0.657240\pi\)
0.474138 0.880451i \(-0.342760\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 30.3221i 0.973083i 0.873657 + 0.486542i \(0.161742\pi\)
−0.873657 + 0.486542i \(0.838258\pi\)
\(972\) 0 0
\(973\) −21.7564 −0.697478
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.7220i 0.950890i 0.879746 + 0.475445i \(0.157713\pi\)
−0.879746 + 0.475445i \(0.842287\pi\)
\(978\) 0 0
\(979\) 7.73038i 0.247064i
\(980\) 0 0
\(981\) 9.37169i 0.299215i
\(982\) 0 0
\(983\) 49.5443i 1.58022i 0.612966 + 0.790109i \(0.289976\pi\)
−0.612966 + 0.790109i \(0.710024\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 34.0722 1.08453
\(988\) 0 0
\(989\) − 25.1709i − 0.800389i
\(990\) 0 0
\(991\) −19.0937 −0.606530 −0.303265 0.952906i \(-0.598077\pi\)
−0.303265 + 0.952906i \(0.598077\pi\)
\(992\) 0 0
\(993\) − 19.8568i − 0.630136i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.8500 1.23039 0.615197 0.788374i \(-0.289077\pi\)
0.615197 + 0.788374i \(0.289077\pi\)
\(998\) 0 0
\(999\) −4.39312 −0.138992
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 2400.2.d.e.49.6 6
3.2 odd 2 7200.2.d.r.2449.6 6
4.3 odd 2 600.2.d.f.349.6 6
5.2 odd 4 480.2.k.b.241.4 6
5.3 odd 4 2400.2.k.c.1201.3 6
5.4 even 2 2400.2.d.f.49.1 6
8.3 odd 2 600.2.d.e.349.2 6
8.5 even 2 2400.2.d.f.49.6 6
12.11 even 2 1800.2.d.r.1549.1 6
15.2 even 4 1440.2.k.f.721.4 6
15.8 even 4 7200.2.k.p.3601.6 6
15.14 odd 2 7200.2.d.q.2449.1 6
20.3 even 4 600.2.k.c.301.5 6
20.7 even 4 120.2.k.b.61.2 yes 6
20.19 odd 2 600.2.d.e.349.1 6
24.5 odd 2 7200.2.d.q.2449.6 6
24.11 even 2 1800.2.d.q.1549.5 6
40.3 even 4 600.2.k.c.301.6 6
40.13 odd 4 2400.2.k.c.1201.6 6
40.19 odd 2 600.2.d.f.349.5 6
40.27 even 4 120.2.k.b.61.1 6
40.29 even 2 inner 2400.2.d.e.49.1 6
40.37 odd 4 480.2.k.b.241.1 6
60.23 odd 4 1800.2.k.p.901.2 6
60.47 odd 4 360.2.k.f.181.5 6
60.59 even 2 1800.2.d.q.1549.6 6
80.27 even 4 3840.2.a.bp.1.1 3
80.37 odd 4 3840.2.a.br.1.3 3
80.67 even 4 3840.2.a.bq.1.1 3
80.77 odd 4 3840.2.a.bo.1.3 3
120.29 odd 2 7200.2.d.r.2449.1 6
120.53 even 4 7200.2.k.p.3601.5 6
120.59 even 2 1800.2.d.r.1549.2 6
120.77 even 4 1440.2.k.f.721.1 6
120.83 odd 4 1800.2.k.p.901.1 6
120.107 odd 4 360.2.k.f.181.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.k.b.61.1 6 40.27 even 4
120.2.k.b.61.2 yes 6 20.7 even 4
360.2.k.f.181.5 6 60.47 odd 4
360.2.k.f.181.6 6 120.107 odd 4
480.2.k.b.241.1 6 40.37 odd 4
480.2.k.b.241.4 6 5.2 odd 4
600.2.d.e.349.1 6 20.19 odd 2
600.2.d.e.349.2 6 8.3 odd 2
600.2.d.f.349.5 6 40.19 odd 2
600.2.d.f.349.6 6 4.3 odd 2
600.2.k.c.301.5 6 20.3 even 4
600.2.k.c.301.6 6 40.3 even 4
1440.2.k.f.721.1 6 120.77 even 4
1440.2.k.f.721.4 6 15.2 even 4
1800.2.d.q.1549.5 6 24.11 even 2
1800.2.d.q.1549.6 6 60.59 even 2
1800.2.d.r.1549.1 6 12.11 even 2
1800.2.d.r.1549.2 6 120.59 even 2
1800.2.k.p.901.1 6 120.83 odd 4
1800.2.k.p.901.2 6 60.23 odd 4
2400.2.d.e.49.1 6 40.29 even 2 inner
2400.2.d.e.49.6 6 1.1 even 1 trivial
2400.2.d.f.49.1 6 5.4 even 2
2400.2.d.f.49.6 6 8.5 even 2
2400.2.k.c.1201.3 6 5.3 odd 4
2400.2.k.c.1201.6 6 40.13 odd 4
3840.2.a.bo.1.3 3 80.77 odd 4
3840.2.a.bp.1.1 3 80.27 even 4
3840.2.a.bq.1.1 3 80.67 even 4
3840.2.a.br.1.3 3 80.37 odd 4
7200.2.d.q.2449.1 6 15.14 odd 2
7200.2.d.q.2449.6 6 24.5 odd 2
7200.2.d.r.2449.1 6 120.29 odd 2
7200.2.d.r.2449.6 6 3.2 odd 2
7200.2.k.p.3601.5 6 120.53 even 4
7200.2.k.p.3601.6 6 15.8 even 4