Properties

Label 1800.4.f.k.649.1
Level $1800$
Weight $4$
Character 1800.649
Analytic conductor $106.203$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(106.203438010\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1800.649
Dual form 1800.4.f.k.649.2

$q$-expansion

\(f(q)\) \(=\) \(q-20.0000i q^{7} +O(q^{10})\) \(q-20.0000i q^{7} -16.0000 q^{11} +58.0000i q^{13} +38.0000i q^{17} -4.00000 q^{19} +80.0000i q^{23} +82.0000 q^{29} -8.00000 q^{31} -426.000i q^{37} +246.000 q^{41} -524.000i q^{43} -464.000i q^{47} -57.0000 q^{49} +702.000i q^{53} -592.000 q^{59} +574.000 q^{61} +172.000i q^{67} -768.000 q^{71} -558.000i q^{73} +320.000i q^{77} -408.000 q^{79} -164.000i q^{83} -510.000 q^{89} +1160.00 q^{91} -514.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{11} - 8 q^{19} + 164 q^{29} - 16 q^{31} + 492 q^{41} - 114 q^{49} - 1184 q^{59} + 1148 q^{61} - 1536 q^{71} - 816 q^{79} - 1020 q^{89} + 2320 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) − 20.0000i − 1.07990i −0.841698 0.539949i \(-0.818443\pi\)
0.841698 0.539949i \(-0.181557\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −16.0000 −0.438562 −0.219281 0.975662i \(-0.570371\pi\)
−0.219281 + 0.975662i \(0.570371\pi\)
\(12\) 0 0
\(13\) 58.0000i 1.23741i 0.785624 + 0.618704i \(0.212342\pi\)
−0.785624 + 0.618704i \(0.787658\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 38.0000i 0.542138i 0.962560 + 0.271069i \(0.0873772\pi\)
−0.962560 + 0.271069i \(0.912623\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.0482980 −0.0241490 0.999708i \(-0.507688\pi\)
−0.0241490 + 0.999708i \(0.507688\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 80.0000i 0.725268i 0.931932 + 0.362634i \(0.118122\pi\)
−0.931932 + 0.362634i \(0.881878\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 82.0000 0.525070 0.262535 0.964923i \(-0.415442\pi\)
0.262535 + 0.964923i \(0.415442\pi\)
\(30\) 0 0
\(31\) −8.00000 −0.0463498 −0.0231749 0.999731i \(-0.507377\pi\)
−0.0231749 + 0.999731i \(0.507377\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 426.000i − 1.89281i −0.322982 0.946405i \(-0.604685\pi\)
0.322982 0.946405i \(-0.395315\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 246.000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 524.000i − 1.85835i −0.369634 0.929177i \(-0.620517\pi\)
0.369634 0.929177i \(-0.379483\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 464.000i − 1.44003i −0.693959 0.720014i \(-0.744135\pi\)
0.693959 0.720014i \(-0.255865\pi\)
\(48\) 0 0
\(49\) −57.0000 −0.166181
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 702.000i 1.81938i 0.415288 + 0.909690i \(0.363681\pi\)
−0.415288 + 0.909690i \(0.636319\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −592.000 −1.30630 −0.653151 0.757228i \(-0.726553\pi\)
−0.653151 + 0.757228i \(0.726553\pi\)
\(60\) 0 0
\(61\) 574.000 1.20481 0.602403 0.798192i \(-0.294210\pi\)
0.602403 + 0.798192i \(0.294210\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 172.000i 0.313629i 0.987628 + 0.156815i \(0.0501225\pi\)
−0.987628 + 0.156815i \(0.949878\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −768.000 −1.28373 −0.641865 0.766818i \(-0.721839\pi\)
−0.641865 + 0.766818i \(0.721839\pi\)
\(72\) 0 0
\(73\) − 558.000i − 0.894643i −0.894373 0.447322i \(-0.852378\pi\)
0.894373 0.447322i \(-0.147622\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 320.000i 0.473602i
\(78\) 0 0
\(79\) −408.000 −0.581058 −0.290529 0.956866i \(-0.593831\pi\)
−0.290529 + 0.956866i \(0.593831\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 164.000i − 0.216884i −0.994103 0.108442i \(-0.965414\pi\)
0.994103 0.108442i \(-0.0345861\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −510.000 −0.607415 −0.303707 0.952765i \(-0.598224\pi\)
−0.303707 + 0.952765i \(0.598224\pi\)
\(90\) 0 0
\(91\) 1160.00 1.33628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 514.000i − 0.538029i −0.963136 0.269014i \(-0.913302\pi\)
0.963136 0.269014i \(-0.0866979\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −666.000 −0.656133 −0.328067 0.944655i \(-0.606397\pi\)
−0.328067 + 0.944655i \(0.606397\pi\)
\(102\) 0 0
\(103\) − 1100.00i − 1.05229i −0.850394 0.526147i \(-0.823636\pi\)
0.850394 0.526147i \(-0.176364\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1212.00i 1.09503i 0.836795 + 0.547516i \(0.184427\pi\)
−0.836795 + 0.547516i \(0.815573\pi\)
\(108\) 0 0
\(109\) −2078.00 −1.82602 −0.913011 0.407936i \(-0.866249\pi\)
−0.913011 + 0.407936i \(0.866249\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1458.00i 1.21378i 0.794786 + 0.606890i \(0.207583\pi\)
−0.794786 + 0.606890i \(0.792417\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 760.000 0.585455
\(120\) 0 0
\(121\) −1075.00 −0.807663
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2436.00i 1.70205i 0.525127 + 0.851024i \(0.324018\pi\)
−0.525127 + 0.851024i \(0.675982\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2544.00 −1.69672 −0.848360 0.529420i \(-0.822410\pi\)
−0.848360 + 0.529420i \(0.822410\pi\)
\(132\) 0 0
\(133\) 80.0000i 0.0521570i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 694.000i 0.432791i 0.976306 + 0.216396i \(0.0694301\pi\)
−0.976306 + 0.216396i \(0.930570\pi\)
\(138\) 0 0
\(139\) −516.000 −0.314867 −0.157434 0.987530i \(-0.550322\pi\)
−0.157434 + 0.987530i \(0.550322\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 928.000i − 0.542680i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 770.000 0.423361 0.211681 0.977339i \(-0.432106\pi\)
0.211681 + 0.977339i \(0.432106\pi\)
\(150\) 0 0
\(151\) −424.000 −0.228507 −0.114254 0.993452i \(-0.536448\pi\)
−0.114254 + 0.993452i \(0.536448\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 922.000i − 0.468685i −0.972154 0.234343i \(-0.924706\pi\)
0.972154 0.234343i \(-0.0752938\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1600.00 0.783215
\(162\) 0 0
\(163\) − 3788.00i − 1.82024i −0.414345 0.910120i \(-0.635989\pi\)
0.414345 0.910120i \(-0.364011\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 48.0000i − 0.0222416i −0.999938 0.0111208i \(-0.996460\pi\)
0.999938 0.0111208i \(-0.00353994\pi\)
\(168\) 0 0
\(169\) −1167.00 −0.531179
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 3242.00i − 1.42477i −0.701790 0.712384i \(-0.747616\pi\)
0.701790 0.712384i \(-0.252384\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2728.00 −1.13911 −0.569554 0.821954i \(-0.692884\pi\)
−0.569554 + 0.821954i \(0.692884\pi\)
\(180\) 0 0
\(181\) −4090.00 −1.67960 −0.839799 0.542897i \(-0.817327\pi\)
−0.839799 + 0.542897i \(0.817327\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 608.000i − 0.237761i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1480.00 0.560676 0.280338 0.959901i \(-0.409554\pi\)
0.280338 + 0.959901i \(0.409554\pi\)
\(192\) 0 0
\(193\) − 1622.00i − 0.604944i −0.953158 0.302472i \(-0.902188\pi\)
0.953158 0.302472i \(-0.0978118\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2530.00i 0.915000i 0.889210 + 0.457500i \(0.151255\pi\)
−0.889210 + 0.457500i \(0.848745\pi\)
\(198\) 0 0
\(199\) 2440.00 0.869181 0.434590 0.900628i \(-0.356893\pi\)
0.434590 + 0.900628i \(0.356893\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1640.00i − 0.567022i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 64.0000 0.0211817
\(210\) 0 0
\(211\) −148.000 −0.0482879 −0.0241439 0.999708i \(-0.507686\pi\)
−0.0241439 + 0.999708i \(0.507686\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 160.000i 0.0500530i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2204.00 −0.670847
\(222\) 0 0
\(223\) − 676.000i − 0.202997i −0.994836 0.101498i \(-0.967636\pi\)
0.994836 0.101498i \(-0.0323637\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 6276.00i − 1.83503i −0.397696 0.917517i \(-0.630190\pi\)
0.397696 0.917517i \(-0.369810\pi\)
\(228\) 0 0
\(229\) −6190.00 −1.78623 −0.893115 0.449828i \(-0.851485\pi\)
−0.893115 + 0.449828i \(0.851485\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 5406.00i − 1.52000i −0.649926 0.759998i \(-0.725200\pi\)
0.649926 0.759998i \(-0.274800\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −600.000 −0.162388 −0.0811941 0.996698i \(-0.525873\pi\)
−0.0811941 + 0.996698i \(0.525873\pi\)
\(240\) 0 0
\(241\) −1054.00 −0.281718 −0.140859 0.990030i \(-0.544986\pi\)
−0.140859 + 0.990030i \(0.544986\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 232.000i − 0.0597644i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2232.00 0.561285 0.280643 0.959812i \(-0.409452\pi\)
0.280643 + 0.959812i \(0.409452\pi\)
\(252\) 0 0
\(253\) − 1280.00i − 0.318075i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3630.00i 0.881063i 0.897737 + 0.440531i \(0.145210\pi\)
−0.897737 + 0.440531i \(0.854790\pi\)
\(258\) 0 0
\(259\) −8520.00 −2.04404
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 6960.00i − 1.63183i −0.578170 0.815916i \(-0.696233\pi\)
0.578170 0.815916i \(-0.303767\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2062.00 −0.467369 −0.233685 0.972312i \(-0.575078\pi\)
−0.233685 + 0.972312i \(0.575078\pi\)
\(270\) 0 0
\(271\) −2544.00 −0.570247 −0.285124 0.958491i \(-0.592035\pi\)
−0.285124 + 0.958491i \(0.592035\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 694.000i 0.150536i 0.997163 + 0.0752679i \(0.0239812\pi\)
−0.997163 + 0.0752679i \(0.976019\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1982.00 0.420769 0.210385 0.977619i \(-0.432528\pi\)
0.210385 + 0.977619i \(0.432528\pi\)
\(282\) 0 0
\(283\) 5228.00i 1.09814i 0.835778 + 0.549068i \(0.185017\pi\)
−0.835778 + 0.549068i \(0.814983\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 4920.00i − 1.01191i
\(288\) 0 0
\(289\) 3469.00 0.706086
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7454.00i 1.48624i 0.669160 + 0.743118i \(0.266654\pi\)
−0.669160 + 0.743118i \(0.733346\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4640.00 −0.897452
\(300\) 0 0
\(301\) −10480.0 −2.00683
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1316.00i 0.244652i 0.992490 + 0.122326i \(0.0390353\pi\)
−0.992490 + 0.122326i \(0.960965\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 832.000 0.151699 0.0758495 0.997119i \(-0.475833\pi\)
0.0758495 + 0.997119i \(0.475833\pi\)
\(312\) 0 0
\(313\) 6770.00i 1.22257i 0.791412 + 0.611283i \(0.209346\pi\)
−0.791412 + 0.611283i \(0.790654\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6582.00i − 1.16619i −0.812404 0.583095i \(-0.801842\pi\)
0.812404 0.583095i \(-0.198158\pi\)
\(318\) 0 0
\(319\) −1312.00 −0.230276
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 152.000i − 0.0261842i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9280.00 −1.55508
\(330\) 0 0
\(331\) 11292.0 1.87512 0.937560 0.347825i \(-0.113080\pi\)
0.937560 + 0.347825i \(0.113080\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8006.00i 1.29411i 0.762444 + 0.647054i \(0.223999\pi\)
−0.762444 + 0.647054i \(0.776001\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 128.000 0.0203272
\(342\) 0 0
\(343\) − 5720.00i − 0.900440i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 316.000i − 0.0488869i −0.999701 0.0244435i \(-0.992219\pi\)
0.999701 0.0244435i \(-0.00778137\pi\)
\(348\) 0 0
\(349\) −4926.00 −0.755538 −0.377769 0.925900i \(-0.623309\pi\)
−0.377769 + 0.925900i \(0.623309\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 2438.00i − 0.367597i −0.982964 0.183798i \(-0.941161\pi\)
0.982964 0.183798i \(-0.0588393\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3336.00 −0.490438 −0.245219 0.969468i \(-0.578860\pi\)
−0.245219 + 0.969468i \(0.578860\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 44.0000i − 0.00625826i −0.999995 0.00312913i \(-0.999004\pi\)
0.999995 0.00312913i \(-0.000996035\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14040.0 1.96475
\(372\) 0 0
\(373\) − 11966.0i − 1.66106i −0.556973 0.830531i \(-0.688037\pi\)
0.556973 0.830531i \(-0.311963\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4756.00i 0.649725i
\(378\) 0 0
\(379\) −12676.0 −1.71800 −0.859001 0.511975i \(-0.828914\pi\)
−0.859001 + 0.511975i \(0.828914\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 6672.00i − 0.890139i −0.895496 0.445070i \(-0.853179\pi\)
0.895496 0.445070i \(-0.146821\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 354.000 0.0461401 0.0230701 0.999734i \(-0.492656\pi\)
0.0230701 + 0.999734i \(0.492656\pi\)
\(390\) 0 0
\(391\) −3040.00 −0.393195
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5054.00i 0.638924i 0.947599 + 0.319462i \(0.103502\pi\)
−0.947599 + 0.319462i \(0.896498\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10266.0 −1.27845 −0.639226 0.769019i \(-0.720745\pi\)
−0.639226 + 0.769019i \(0.720745\pi\)
\(402\) 0 0
\(403\) − 464.000i − 0.0573536i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6816.00i 0.830114i
\(408\) 0 0
\(409\) 1526.00 0.184489 0.0922443 0.995736i \(-0.470596\pi\)
0.0922443 + 0.995736i \(0.470596\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11840.0i 1.41067i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2064.00 0.240652 0.120326 0.992734i \(-0.461606\pi\)
0.120326 + 0.992734i \(0.461606\pi\)
\(420\) 0 0
\(421\) 4590.00 0.531361 0.265680 0.964061i \(-0.414403\pi\)
0.265680 + 0.964061i \(0.414403\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 11480.0i − 1.30107i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5536.00 0.618700 0.309350 0.950948i \(-0.399889\pi\)
0.309350 + 0.950948i \(0.399889\pi\)
\(432\) 0 0
\(433\) 1850.00i 0.205324i 0.994716 + 0.102662i \(0.0327360\pi\)
−0.994716 + 0.102662i \(0.967264\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 320.000i − 0.0350290i
\(438\) 0 0
\(439\) −11704.0 −1.27244 −0.636220 0.771507i \(-0.719503\pi\)
−0.636220 + 0.771507i \(0.719503\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 6948.00i − 0.745168i −0.927998 0.372584i \(-0.878472\pi\)
0.927998 0.372584i \(-0.121528\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12090.0 1.27074 0.635370 0.772208i \(-0.280848\pi\)
0.635370 + 0.772208i \(0.280848\pi\)
\(450\) 0 0
\(451\) −3936.00 −0.410951
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 11626.0i − 1.19002i −0.803717 0.595012i \(-0.797147\pi\)
0.803717 0.595012i \(-0.202853\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16314.0 −1.64820 −0.824098 0.566447i \(-0.808318\pi\)
−0.824098 + 0.566447i \(0.808318\pi\)
\(462\) 0 0
\(463\) − 15756.0i − 1.58152i −0.612127 0.790760i \(-0.709686\pi\)
0.612127 0.790760i \(-0.290314\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5684.00i 0.563221i 0.959529 + 0.281610i \(0.0908686\pi\)
−0.959529 + 0.281610i \(0.909131\pi\)
\(468\) 0 0
\(469\) 3440.00 0.338688
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8384.00i 0.815004i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3368.00 −0.321269 −0.160634 0.987014i \(-0.551354\pi\)
−0.160634 + 0.987014i \(0.551354\pi\)
\(480\) 0 0
\(481\) 24708.0 2.34218
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5588.00i 0.519952i 0.965615 + 0.259976i \(0.0837146\pi\)
−0.965615 + 0.259976i \(0.916285\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10584.0 −0.972809 −0.486405 0.873734i \(-0.661692\pi\)
−0.486405 + 0.873734i \(0.661692\pi\)
\(492\) 0 0
\(493\) 3116.00i 0.284660i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15360.0i 1.38630i
\(498\) 0 0
\(499\) 12220.0 1.09628 0.548139 0.836388i \(-0.315337\pi\)
0.548139 + 0.836388i \(0.315337\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 16152.0i − 1.43177i −0.698216 0.715887i \(-0.746023\pi\)
0.698216 0.715887i \(-0.253977\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10642.0 0.926716 0.463358 0.886171i \(-0.346644\pi\)
0.463358 + 0.886171i \(0.346644\pi\)
\(510\) 0 0
\(511\) −11160.0 −0.966124
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 7424.00i 0.631542i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −22882.0 −1.92414 −0.962072 0.272797i \(-0.912051\pi\)
−0.962072 + 0.272797i \(0.912051\pi\)
\(522\) 0 0
\(523\) − 10052.0i − 0.840427i −0.907425 0.420213i \(-0.861955\pi\)
0.907425 0.420213i \(-0.138045\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 304.000i − 0.0251280i
\(528\) 0 0
\(529\) 5767.00 0.473987
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14268.0i 1.15950i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 912.000 0.0728806
\(540\) 0 0
\(541\) −6530.00 −0.518940 −0.259470 0.965751i \(-0.583548\pi\)
−0.259470 + 0.965751i \(0.583548\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 16652.0i − 1.30162i −0.759239 0.650812i \(-0.774429\pi\)
0.759239 0.650812i \(-0.225571\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −328.000 −0.0253598
\(552\) 0 0
\(553\) 8160.00i 0.627484i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 12886.0i − 0.980247i −0.871653 0.490123i \(-0.836952\pi\)
0.871653 0.490123i \(-0.163048\pi\)
\(558\) 0 0
\(559\) 30392.0 2.29954
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11108.0i 0.831521i 0.909474 + 0.415761i \(0.136485\pi\)
−0.909474 + 0.415761i \(0.863515\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9214.00 −0.678859 −0.339430 0.940631i \(-0.610234\pi\)
−0.339430 + 0.940631i \(0.610234\pi\)
\(570\) 0 0
\(571\) −4052.00 −0.296972 −0.148486 0.988915i \(-0.547440\pi\)
−0.148486 + 0.988915i \(0.547440\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8446.00i 0.609379i 0.952452 + 0.304689i \(0.0985527\pi\)
−0.952452 + 0.304689i \(0.901447\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3280.00 −0.234212
\(582\) 0 0
\(583\) − 11232.0i − 0.797911i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2172.00i 0.152722i 0.997080 + 0.0763612i \(0.0243302\pi\)
−0.997080 + 0.0763612i \(0.975670\pi\)
\(588\) 0 0
\(589\) 32.0000 0.00223860
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1218.00i 0.0843461i 0.999110 + 0.0421731i \(0.0134281\pi\)
−0.999110 + 0.0421731i \(0.986572\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21240.0 1.44882 0.724410 0.689370i \(-0.242112\pi\)
0.724410 + 0.689370i \(0.242112\pi\)
\(600\) 0 0
\(601\) 17626.0 1.19631 0.598153 0.801382i \(-0.295902\pi\)
0.598153 + 0.801382i \(0.295902\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 2580.00i − 0.172519i −0.996273 0.0862594i \(-0.972509\pi\)
0.996273 0.0862594i \(-0.0274914\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 26912.0 1.78190
\(612\) 0 0
\(613\) − 14166.0i − 0.933376i −0.884422 0.466688i \(-0.845447\pi\)
0.884422 0.466688i \(-0.154553\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 21426.0i − 1.39802i −0.715112 0.699010i \(-0.753624\pi\)
0.715112 0.699010i \(-0.246376\pi\)
\(618\) 0 0
\(619\) −3668.00 −0.238173 −0.119087 0.992884i \(-0.537997\pi\)
−0.119087 + 0.992884i \(0.537997\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10200.0i 0.655946i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16188.0 1.02617
\(630\) 0 0
\(631\) 20032.0 1.26381 0.631903 0.775048i \(-0.282274\pi\)
0.631903 + 0.775048i \(0.282274\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3306.00i − 0.205633i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7458.00 −0.459553 −0.229776 0.973243i \(-0.573799\pi\)
−0.229776 + 0.973243i \(0.573799\pi\)
\(642\) 0 0
\(643\) 7092.00i 0.434963i 0.976064 + 0.217481i \(0.0697842\pi\)
−0.976064 + 0.217481i \(0.930216\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3384.00i 0.205624i 0.994701 + 0.102812i \(0.0327840\pi\)
−0.994701 + 0.102812i \(0.967216\pi\)
\(648\) 0 0
\(649\) 9472.00 0.572894
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 29398.0i 1.76177i 0.473335 + 0.880883i \(0.343050\pi\)
−0.473335 + 0.880883i \(0.656950\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6624.00 −0.391554 −0.195777 0.980648i \(-0.562723\pi\)
−0.195777 + 0.980648i \(0.562723\pi\)
\(660\) 0 0
\(661\) 8646.00 0.508760 0.254380 0.967104i \(-0.418129\pi\)
0.254380 + 0.967104i \(0.418129\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6560.00i 0.380816i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9184.00 −0.528382
\(672\) 0 0
\(673\) 28698.0i 1.64372i 0.569686 + 0.821862i \(0.307065\pi\)
−0.569686 + 0.821862i \(0.692935\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19426.0i 1.10281i 0.834238 + 0.551405i \(0.185908\pi\)
−0.834238 + 0.551405i \(0.814092\pi\)
\(678\) 0 0
\(679\) −10280.0 −0.581016
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 8604.00i − 0.482025i −0.970522 0.241012i \(-0.922521\pi\)
0.970522 0.241012i \(-0.0774794\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −40716.0 −2.25132
\(690\) 0 0
\(691\) 12980.0 0.714591 0.357296 0.933991i \(-0.383699\pi\)
0.357296 + 0.933991i \(0.383699\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 9348.00i 0.508007i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19630.0 1.05765 0.528827 0.848730i \(-0.322632\pi\)
0.528827 + 0.848730i \(0.322632\pi\)
\(702\) 0 0
\(703\) 1704.00i 0.0914190i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13320.0i 0.708558i
\(708\) 0 0
\(709\) −8030.00 −0.425350 −0.212675 0.977123i \(-0.568218\pi\)
−0.212675 + 0.977123i \(0.568218\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 640.000i − 0.0336160i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22720.0 1.17846 0.589230 0.807965i \(-0.299431\pi\)
0.589230 + 0.807965i \(0.299431\pi\)
\(720\) 0 0
\(721\) −22000.0 −1.13637
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 27116.0i − 1.38332i −0.722221 0.691662i \(-0.756879\pi\)
0.722221 0.691662i \(-0.243121\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 19912.0 1.00749
\(732\) 0 0
\(733\) 30882.0i 1.55614i 0.628176 + 0.778071i \(0.283802\pi\)
−0.628176 + 0.778071i \(0.716198\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 2752.00i − 0.137546i
\(738\) 0 0
\(739\) 13836.0 0.688722 0.344361 0.938837i \(-0.388096\pi\)
0.344361 + 0.938837i \(0.388096\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 32712.0i − 1.61519i −0.589737 0.807595i \(-0.700769\pi\)
0.589737 0.807595i \(-0.299231\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24240.0 1.18252
\(750\) 0 0
\(751\) −8472.00 −0.411648 −0.205824 0.978589i \(-0.565987\pi\)
−0.205824 + 0.978589i \(0.565987\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 9866.00i − 0.473693i −0.971547 0.236847i \(-0.923886\pi\)
0.971547 0.236847i \(-0.0761139\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3774.00 0.179773 0.0898866 0.995952i \(-0.471350\pi\)
0.0898866 + 0.995952i \(0.471350\pi\)
\(762\) 0 0
\(763\) 41560.0i 1.97192i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 34336.0i − 1.61643i
\(768\) 0 0
\(769\) 28670.0 1.34443 0.672215 0.740356i \(-0.265343\pi\)
0.672215 + 0.740356i \(0.265343\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3246.00i 0.151036i 0.997144 + 0.0755178i \(0.0240610\pi\)
−0.997144 + 0.0755178i \(0.975939\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −984.000 −0.0452573
\(780\) 0 0
\(781\) 12288.0 0.562995
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 19372.0i 0.877430i 0.898626 + 0.438715i \(0.144566\pi\)
−0.898626 + 0.438715i \(0.855434\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 29160.0 1.31076
\(792\) 0 0
\(793\) 33292.0i 1.49084i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 11814.0i − 0.525061i −0.964924 0.262530i \(-0.915443\pi\)
0.964924 0.262530i \(-0.0845570\pi\)
\(798\) 0 0
\(799\) 17632.0 0.780695
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8928.00i 0.392357i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30054.0 −1.30611 −0.653055 0.757311i \(-0.726513\pi\)
−0.653055 + 0.757311i \(0.726513\pi\)
\(810\) 0 0
\(811\) 2852.00 0.123486 0.0617431 0.998092i \(-0.480334\pi\)
0.0617431 + 0.998092i \(0.480334\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2096.00i 0.0897549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2170.00 −0.0922455 −0.0461227 0.998936i \(-0.514687\pi\)
−0.0461227 + 0.998936i \(0.514687\pi\)
\(822\) 0 0
\(823\) 19804.0i 0.838790i 0.907804 + 0.419395i \(0.137758\pi\)
−0.907804 + 0.419395i \(0.862242\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5508.00i 0.231598i 0.993273 + 0.115799i \(0.0369429\pi\)
−0.993273 + 0.115799i \(0.963057\pi\)
\(828\) 0 0
\(829\) −33262.0 −1.39353 −0.696765 0.717299i \(-0.745378\pi\)
−0.696765 + 0.717299i \(0.745378\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2166.00i − 0.0900930i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4600.00 −0.189284 −0.0946422 0.995511i \(-0.530171\pi\)
−0.0946422 + 0.995511i \(0.530171\pi\)
\(840\) 0 0
\(841\) −17665.0 −0.724302
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 21500.0i 0.872195i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 34080.0 1.37279
\(852\) 0 0
\(853\) − 4198.00i − 0.168507i −0.996444 0.0842537i \(-0.973149\pi\)
0.996444 0.0842537i \(-0.0268506\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 5826.00i − 0.232220i −0.993236 0.116110i \(-0.962958\pi\)
0.993236 0.116110i \(-0.0370425\pi\)
\(858\) 0 0
\(859\) 3004.00 0.119319 0.0596596 0.998219i \(-0.480998\pi\)
0.0596596 + 0.998219i \(0.480998\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 36936.0i 1.45691i 0.685092 + 0.728457i \(0.259762\pi\)
−0.685092 + 0.728457i \(0.740238\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6528.00 0.254830
\(870\) 0 0
\(871\) −9976.00 −0.388087
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 5434.00i − 0.209228i −0.994513 0.104614i \(-0.966639\pi\)
0.994513 0.104614i \(-0.0333607\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4758.00 0.181954 0.0909768 0.995853i \(-0.471001\pi\)
0.0909768 + 0.995853i \(0.471001\pi\)
\(882\) 0 0
\(883\) 15476.0i 0.589818i 0.955525 + 0.294909i \(0.0952893\pi\)
−0.955525 + 0.294909i \(0.904711\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 27440.0i − 1.03872i −0.854555 0.519360i \(-0.826170\pi\)
0.854555 0.519360i \(-0.173830\pi\)
\(888\) 0 0
\(889\) 48720.0 1.83804
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1856.00i 0.0695506i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −656.000 −0.0243368
\(900\) 0 0
\(901\) −26676.0 −0.986356
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 48924.0i 1.79106i 0.444997 + 0.895532i \(0.353205\pi\)
−0.444997 + 0.895532i \(0.646795\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3440.00 0.125107 0.0625534 0.998042i \(-0.480076\pi\)
0.0625534 + 0.998042i \(0.480076\pi\)
\(912\) 0 0
\(913\) 2624.00i 0.0951169i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 50880.0i 1.83229i
\(918\) 0 0
\(919\) 27184.0 0.975753 0.487877 0.872913i \(-0.337772\pi\)
0.487877 + 0.872913i \(0.337772\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 44544.0i − 1.58850i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 42490.0 1.50059 0.750297 0.661101i \(-0.229911\pi\)
0.750297 + 0.661101i \(0.229911\pi\)
\(930\) 0 0
\(931\) 228.000 0.00802621
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 37354.0i − 1.30235i −0.758928 0.651175i \(-0.774276\pi\)
0.758928 0.651175i \(-0.225724\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 24470.0 0.847714 0.423857 0.905729i \(-0.360676\pi\)
0.423857 + 0.905729i \(0.360676\pi\)
\(942\) 0 0
\(943\) 19680.0i 0.679607i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 34100.0i − 1.17012i −0.810991 0.585059i \(-0.801071\pi\)
0.810991 0.585059i \(-0.198929\pi\)
\(948\) 0 0
\(949\) 32364.0 1.10704
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1878.00i − 0.0638346i −0.999491 0.0319173i \(-0.989839\pi\)
0.999491 0.0319173i \(-0.0101613\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 13880.0 0.467371
\(960\) 0 0
\(961\) −29727.0 −0.997852
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 38484.0i − 1.27980i −0.768460 0.639898i \(-0.778977\pi\)
0.768460 0.639898i \(-0.221023\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −45272.0 −1.49624 −0.748119 0.663564i \(-0.769043\pi\)
−0.748119 + 0.663564i \(0.769043\pi\)
\(972\) 0 0
\(973\) 10320.0i 0.340025i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 25354.0i − 0.830242i −0.909766 0.415121i \(-0.863739\pi\)
0.909766 0.415121i \(-0.136261\pi\)
\(978\) 0 0
\(979\) 8160.00 0.266389
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18744.0i 0.608180i 0.952643 + 0.304090i \(0.0983523\pi\)
−0.952643 + 0.304090i \(0.901648\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 41920.0 1.34780
\(990\) 0 0
\(991\) 59600.0 1.91045 0.955225 0.295880i \(-0.0956127\pi\)
0.955225 + 0.295880i \(0.0956127\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 17886.0i 0.568160i 0.958801 + 0.284080i \(0.0916881\pi\)
−0.958801 + 0.284080i \(0.908312\pi\)
\(998\) 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 1800.4.f.k.649.1 2
3.2 odd 2 600.4.f.f.49.2 2
5.2 odd 4 360.4.a.m.1.1 1
5.3 odd 4 1800.4.a.e.1.1 1
5.4 even 2 inner 1800.4.f.k.649.2 2
12.11 even 2 1200.4.f.h.49.1 2
15.2 even 4 120.4.a.e.1.1 1
15.8 even 4 600.4.a.a.1.1 1
15.14 odd 2 600.4.f.f.49.1 2
20.7 even 4 720.4.a.s.1.1 1
60.23 odd 4 1200.4.a.bj.1.1 1
60.47 odd 4 240.4.a.a.1.1 1
60.59 even 2 1200.4.f.h.49.2 2
120.77 even 4 960.4.a.q.1.1 1
120.107 odd 4 960.4.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.e.1.1 1 15.2 even 4
240.4.a.a.1.1 1 60.47 odd 4
360.4.a.m.1.1 1 5.2 odd 4
600.4.a.a.1.1 1 15.8 even 4
600.4.f.f.49.1 2 15.14 odd 2
600.4.f.f.49.2 2 3.2 odd 2
720.4.a.s.1.1 1 20.7 even 4
960.4.a.q.1.1 1 120.77 even 4
960.4.a.bd.1.1 1 120.107 odd 4
1200.4.a.bj.1.1 1 60.23 odd 4
1200.4.f.h.49.1 2 12.11 even 2
1200.4.f.h.49.2 2 60.59 even 2
1800.4.a.e.1.1 1 5.3 odd 4
1800.4.f.k.649.1 2 1.1 even 1 trivial
1800.4.f.k.649.2 2 5.4 even 2 inner