Properties

Label 400.4.c.f.49.1
Level $400$
Weight $4$
Character 400.49
Analytic conductor $23.601$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.4.c.f.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-6.00000i q^{3} +34.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-6.00000i q^{3} +34.0000i q^{7} -9.00000 q^{9} -16.0000 q^{11} -58.0000i q^{13} -70.0000i q^{17} +4.00000 q^{19} +204.000 q^{21} -134.000i q^{23} -108.000i q^{27} +242.000 q^{29} -100.000 q^{31} +96.0000i q^{33} -438.000i q^{37} -348.000 q^{39} -138.000 q^{41} +178.000i q^{43} -22.0000i q^{47} -813.000 q^{49} -420.000 q^{51} -162.000i q^{53} -24.0000i q^{57} -268.000 q^{59} +250.000 q^{61} -306.000i q^{63} -422.000i q^{67} -804.000 q^{69} +852.000 q^{71} -306.000i q^{73} -544.000i q^{77} -456.000 q^{79} -891.000 q^{81} +434.000i q^{83} -1452.00i q^{87} +726.000 q^{89} +1972.00 q^{91} +600.000i q^{93} +1378.00i q^{97} +144.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 32 q^{11} + 8 q^{19} + 408 q^{21} + 484 q^{29} - 200 q^{31} - 696 q^{39} - 276 q^{41} - 1626 q^{49} - 840 q^{51} - 536 q^{59} + 500 q^{61} - 1608 q^{69} + 1704 q^{71} - 912 q^{79} - 1782 q^{81} + 1452 q^{89} + 3944 q^{91} + 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 6.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 34.0000i 1.83583i 0.396780 + 0.917914i \(0.370128\pi\)
−0.396780 + 0.917914i \(0.629872\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 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 \(-0.787658\pi\)
0.785624 0.618704i \(-0.212342\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 70.0000i − 0.998676i −0.866407 0.499338i \(-0.833577\pi\)
0.866407 0.499338i \(-0.166423\pi\)
\(18\) 0 0
\(19\) 4.00000 0.0482980 0.0241490 0.999708i \(-0.492312\pi\)
0.0241490 + 0.999708i \(0.492312\pi\)
\(20\) 0 0
\(21\) 204.000 2.11983
\(22\) 0 0
\(23\) − 134.000i − 1.21482i −0.794387 0.607412i \(-0.792208\pi\)
0.794387 0.607412i \(-0.207792\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 108.000i − 0.769800i
\(28\) 0 0
\(29\) 242.000 1.54960 0.774798 0.632209i \(-0.217852\pi\)
0.774798 + 0.632209i \(0.217852\pi\)
\(30\) 0 0
\(31\) −100.000 −0.579372 −0.289686 0.957122i \(-0.593551\pi\)
−0.289686 + 0.957122i \(0.593551\pi\)
\(32\) 0 0
\(33\) 96.0000i 0.506408i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 438.000i − 1.94613i −0.230534 0.973064i \(-0.574047\pi\)
0.230534 0.973064i \(-0.425953\pi\)
\(38\) 0 0
\(39\) −348.000 −1.42884
\(40\) 0 0
\(41\) −138.000 −0.525658 −0.262829 0.964842i \(-0.584656\pi\)
−0.262829 + 0.964842i \(0.584656\pi\)
\(42\) 0 0
\(43\) 178.000i 0.631273i 0.948880 + 0.315637i \(0.102218\pi\)
−0.948880 + 0.315637i \(0.897782\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 22.0000i − 0.0682772i −0.999417 0.0341386i \(-0.989131\pi\)
0.999417 0.0341386i \(-0.0108688\pi\)
\(48\) 0 0
\(49\) −813.000 −2.37026
\(50\) 0 0
\(51\) −420.000 −1.15317
\(52\) 0 0
\(53\) − 162.000i − 0.419857i −0.977717 0.209928i \(-0.932677\pi\)
0.977717 0.209928i \(-0.0673231\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 24.0000i − 0.0557698i
\(58\) 0 0
\(59\) −268.000 −0.591367 −0.295683 0.955286i \(-0.595547\pi\)
−0.295683 + 0.955286i \(0.595547\pi\)
\(60\) 0 0
\(61\) 250.000 0.524741 0.262371 0.964967i \(-0.415496\pi\)
0.262371 + 0.964967i \(0.415496\pi\)
\(62\) 0 0
\(63\) − 306.000i − 0.611942i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 422.000i − 0.769485i −0.923024 0.384743i \(-0.874290\pi\)
0.923024 0.384743i \(-0.125710\pi\)
\(68\) 0 0
\(69\) −804.000 −1.40276
\(70\) 0 0
\(71\) 852.000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) − 306.000i − 0.490611i −0.969446 0.245305i \(-0.921112\pi\)
0.969446 0.245305i \(-0.0788882\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 544.000i − 0.805124i
\(78\) 0 0
\(79\) −456.000 −0.649418 −0.324709 0.945814i \(-0.605266\pi\)
−0.324709 + 0.945814i \(0.605266\pi\)
\(80\) 0 0
\(81\) −891.000 −1.22222
\(82\) 0 0
\(83\) 434.000i 0.573948i 0.957938 + 0.286974i \(0.0926493\pi\)
−0.957938 + 0.286974i \(0.907351\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 1452.00i − 1.78932i
\(88\) 0 0
\(89\) 726.000 0.864672 0.432336 0.901712i \(-0.357689\pi\)
0.432336 + 0.901712i \(0.357689\pi\)
\(90\) 0 0
\(91\) 1972.00 2.27167
\(92\) 0 0
\(93\) 600.000i 0.669001i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1378.00i 1.44242i 0.692717 + 0.721210i \(0.256414\pi\)
−0.692717 + 0.721210i \(0.743586\pi\)
\(98\) 0 0
\(99\) 144.000 0.146187
\(100\) 0 0
\(101\) 126.000 0.124133 0.0620667 0.998072i \(-0.480231\pi\)
0.0620667 + 0.998072i \(0.480231\pi\)
\(102\) 0 0
\(103\) − 1262.00i − 1.20727i −0.797262 0.603634i \(-0.793719\pi\)
0.797262 0.603634i \(-0.206281\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 510.000i − 0.460781i −0.973098 0.230390i \(-0.926000\pi\)
0.973098 0.230390i \(-0.0740003\pi\)
\(108\) 0 0
\(109\) −26.0000 −0.0228472 −0.0114236 0.999935i \(-0.503636\pi\)
−0.0114236 + 0.999935i \(0.503636\pi\)
\(110\) 0 0
\(111\) −2628.00 −2.24720
\(112\) 0 0
\(113\) − 1242.00i − 1.03396i −0.855997 0.516980i \(-0.827056\pi\)
0.855997 0.516980i \(-0.172944\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 522.000i 0.412469i
\(118\) 0 0
\(119\) 2380.00 1.83340
\(120\) 0 0
\(121\) −1075.00 −0.807663
\(122\) 0 0
\(123\) 828.000i 0.606978i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 978.000i 0.683334i 0.939821 + 0.341667i \(0.110992\pi\)
−0.939821 + 0.341667i \(0.889008\pi\)
\(128\) 0 0
\(129\) 1068.00 0.728931
\(130\) 0 0
\(131\) 912.000 0.608258 0.304129 0.952631i \(-0.401635\pi\)
0.304129 + 0.952631i \(0.401635\pi\)
\(132\) 0 0
\(133\) 136.000i 0.0886669i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 926.000i − 0.577471i −0.957409 0.288735i \(-0.906765\pi\)
0.957409 0.288735i \(-0.0932348\pi\)
\(138\) 0 0
\(139\) 516.000 0.314867 0.157434 0.987530i \(-0.449678\pi\)
0.157434 + 0.987530i \(0.449678\pi\)
\(140\) 0 0
\(141\) −132.000 −0.0788398
\(142\) 0 0
\(143\) 928.000i 0.542680i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4878.00i 2.73694i
\(148\) 0 0
\(149\) 958.000 0.526728 0.263364 0.964697i \(-0.415168\pi\)
0.263364 + 0.964697i \(0.415168\pi\)
\(150\) 0 0
\(151\) −332.000 −0.178926 −0.0894628 0.995990i \(-0.528515\pi\)
−0.0894628 + 0.995990i \(0.528515\pi\)
\(152\) 0 0
\(153\) 630.000i 0.332892i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 1022.00i − 0.519519i −0.965673 0.259759i \(-0.916357\pi\)
0.965673 0.259759i \(-0.0836433\pi\)
\(158\) 0 0
\(159\) −972.000 −0.484809
\(160\) 0 0
\(161\) 4556.00 2.23021
\(162\) 0 0
\(163\) − 926.000i − 0.444969i −0.974936 0.222484i \(-0.928583\pi\)
0.974936 0.222484i \(-0.0714166\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 654.000i − 0.303042i −0.988454 0.151521i \(-0.951583\pi\)
0.988454 0.151521i \(-0.0484171\pi\)
\(168\) 0 0
\(169\) −1167.00 −0.531179
\(170\) 0 0
\(171\) −36.0000 −0.0160993
\(172\) 0 0
\(173\) 1294.00i 0.568676i 0.958724 + 0.284338i \(0.0917738\pi\)
−0.958724 + 0.284338i \(0.908226\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1608.00i 0.682851i
\(178\) 0 0
\(179\) −2836.00 −1.18420 −0.592102 0.805863i \(-0.701702\pi\)
−0.592102 + 0.805863i \(0.701702\pi\)
\(180\) 0 0
\(181\) 1742.00 0.715369 0.357685 0.933842i \(-0.383566\pi\)
0.357685 + 0.933842i \(0.383566\pi\)
\(182\) 0 0
\(183\) − 1500.00i − 0.605919i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1120.00i 0.437981i
\(188\) 0 0
\(189\) 3672.00 1.41322
\(190\) 0 0
\(191\) −4460.00 −1.68960 −0.844802 0.535079i \(-0.820282\pi\)
−0.844802 + 0.535079i \(0.820282\pi\)
\(192\) 0 0
\(193\) 3782.00i 1.41054i 0.708939 + 0.705270i \(0.249174\pi\)
−0.708939 + 0.705270i \(0.750826\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4474.00i 1.61807i 0.587762 + 0.809034i \(0.300009\pi\)
−0.587762 + 0.809034i \(0.699991\pi\)
\(198\) 0 0
\(199\) 3608.00 1.28525 0.642624 0.766182i \(-0.277846\pi\)
0.642624 + 0.766182i \(0.277846\pi\)
\(200\) 0 0
\(201\) −2532.00 −0.888525
\(202\) 0 0
\(203\) 8228.00i 2.84479i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1206.00i 0.404941i
\(208\) 0 0
\(209\) −64.0000 −0.0211817
\(210\) 0 0
\(211\) 256.000 0.0835250 0.0417625 0.999128i \(-0.486703\pi\)
0.0417625 + 0.999128i \(0.486703\pi\)
\(212\) 0 0
\(213\) − 5112.00i − 1.64445i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 3400.00i − 1.06363i
\(218\) 0 0
\(219\) −1836.00 −0.566509
\(220\) 0 0
\(221\) −4060.00 −1.23577
\(222\) 0 0
\(223\) − 5158.00i − 1.54890i −0.632634 0.774451i \(-0.718026\pi\)
0.632634 0.774451i \(-0.281974\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2226.00i 0.650858i 0.945566 + 0.325429i \(0.105509\pi\)
−0.945566 + 0.325429i \(0.894491\pi\)
\(228\) 0 0
\(229\) −2086.00 −0.601951 −0.300975 0.953632i \(-0.597312\pi\)
−0.300975 + 0.953632i \(0.597312\pi\)
\(230\) 0 0
\(231\) −3264.00 −0.929677
\(232\) 0 0
\(233\) 5718.00i 1.60772i 0.594819 + 0.803860i \(0.297224\pi\)
−0.594819 + 0.803860i \(0.702776\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2736.00i 0.749883i
\(238\) 0 0
\(239\) −3624.00 −0.980825 −0.490412 0.871491i \(-0.663154\pi\)
−0.490412 + 0.871491i \(0.663154\pi\)
\(240\) 0 0
\(241\) −82.0000 −0.0219174 −0.0109587 0.999940i \(-0.503488\pi\)
−0.0109587 + 0.999940i \(0.503488\pi\)
\(242\) 0 0
\(243\) 2430.00i 0.641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 232.000i − 0.0597644i
\(248\) 0 0
\(249\) 2604.00 0.662738
\(250\) 0 0
\(251\) 5040.00 1.26742 0.633709 0.773571i \(-0.281532\pi\)
0.633709 + 0.773571i \(0.281532\pi\)
\(252\) 0 0
\(253\) 2144.00i 0.532775i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2310.00i − 0.560676i −0.959901 0.280338i \(-0.909553\pi\)
0.959901 0.280338i \(-0.0904466\pi\)
\(258\) 0 0
\(259\) 14892.0 3.57276
\(260\) 0 0
\(261\) −2178.00 −0.516532
\(262\) 0 0
\(263\) − 4110.00i − 0.963625i −0.876274 0.481813i \(-0.839979\pi\)
0.876274 0.481813i \(-0.160021\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4356.00i − 0.998438i
\(268\) 0 0
\(269\) −746.000 −0.169087 −0.0845435 0.996420i \(-0.526943\pi\)
−0.0845435 + 0.996420i \(0.526943\pi\)
\(270\) 0 0
\(271\) 4596.00 1.03021 0.515105 0.857127i \(-0.327753\pi\)
0.515105 + 0.857127i \(0.327753\pi\)
\(272\) 0 0
\(273\) − 11832.0i − 2.62310i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2206.00i − 0.478504i −0.970957 0.239252i \(-0.923098\pi\)
0.970957 0.239252i \(-0.0769023\pi\)
\(278\) 0 0
\(279\) 900.000 0.193124
\(280\) 0 0
\(281\) 8278.00 1.75738 0.878691 0.477392i \(-0.158418\pi\)
0.878691 + 0.477392i \(0.158418\pi\)
\(282\) 0 0
\(283\) 1178.00i 0.247438i 0.992317 + 0.123719i \(0.0394821\pi\)
−0.992317 + 0.123719i \(0.960518\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 4692.00i − 0.965017i
\(288\) 0 0
\(289\) 13.0000 0.00264604
\(290\) 0 0
\(291\) 8268.00 1.66556
\(292\) 0 0
\(293\) − 106.000i − 0.0211351i −0.999944 0.0105676i \(-0.996636\pi\)
0.999944 0.0105676i \(-0.00336382\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1728.00i 0.337605i
\(298\) 0 0
\(299\) −7772.00 −1.50323
\(300\) 0 0
\(301\) −6052.00 −1.15891
\(302\) 0 0
\(303\) − 756.000i − 0.143337i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 8134.00i − 1.51216i −0.654482 0.756078i \(-0.727113\pi\)
0.654482 0.756078i \(-0.272887\pi\)
\(308\) 0 0
\(309\) −7572.00 −1.39403
\(310\) 0 0
\(311\) 4396.00 0.801525 0.400763 0.916182i \(-0.368745\pi\)
0.400763 + 0.916182i \(0.368745\pi\)
\(312\) 0 0
\(313\) − 4826.00i − 0.871507i −0.900066 0.435753i \(-0.856482\pi\)
0.900066 0.435753i \(-0.143518\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7026.00i 1.24486i 0.782677 + 0.622428i \(0.213854\pi\)
−0.782677 + 0.622428i \(0.786146\pi\)
\(318\) 0 0
\(319\) −3872.00 −0.679594
\(320\) 0 0
\(321\) −3060.00 −0.532064
\(322\) 0 0
\(323\) − 280.000i − 0.0482341i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 156.000i 0.0263817i
\(328\) 0 0
\(329\) 748.000 0.125345
\(330\) 0 0
\(331\) −8808.00 −1.46263 −0.731316 0.682038i \(-0.761094\pi\)
−0.731316 + 0.682038i \(0.761094\pi\)
\(332\) 0 0
\(333\) 3942.00i 0.648710i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5602.00i 0.905520i 0.891632 + 0.452760i \(0.149561\pi\)
−0.891632 + 0.452760i \(0.850439\pi\)
\(338\) 0 0
\(339\) −7452.00 −1.19391
\(340\) 0 0
\(341\) 1600.00 0.254090
\(342\) 0 0
\(343\) − 15980.0i − 2.51557i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6634.00i 1.02632i 0.858294 + 0.513158i \(0.171525\pi\)
−0.858294 + 0.513158i \(0.828475\pi\)
\(348\) 0 0
\(349\) −3198.00 −0.490501 −0.245251 0.969460i \(-0.578870\pi\)
−0.245251 + 0.969460i \(0.578870\pi\)
\(350\) 0 0
\(351\) −6264.00 −0.952557
\(352\) 0 0
\(353\) 5230.00i 0.788569i 0.918988 + 0.394284i \(0.129008\pi\)
−0.918988 + 0.394284i \(0.870992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 14280.0i − 2.11702i
\(358\) 0 0
\(359\) −312.000 −0.0458683 −0.0229342 0.999737i \(-0.507301\pi\)
−0.0229342 + 0.999737i \(0.507301\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) 6450.00i 0.932609i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 10790.0i − 1.53470i −0.641231 0.767348i \(-0.721576\pi\)
0.641231 0.767348i \(-0.278424\pi\)
\(368\) 0 0
\(369\) 1242.00 0.175219
\(370\) 0 0
\(371\) 5508.00 0.770785
\(372\) 0 0
\(373\) 4190.00i 0.581635i 0.956778 + 0.290818i \(0.0939273\pi\)
−0.956778 + 0.290818i \(0.906073\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 14036.0i − 1.91748i
\(378\) 0 0
\(379\) −6980.00 −0.946012 −0.473006 0.881059i \(-0.656831\pi\)
−0.473006 + 0.881059i \(0.656831\pi\)
\(380\) 0 0
\(381\) 5868.00 0.789047
\(382\) 0 0
\(383\) 13962.0i 1.86273i 0.364089 + 0.931364i \(0.381380\pi\)
−0.364089 + 0.931364i \(0.618620\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1602.00i − 0.210424i
\(388\) 0 0
\(389\) −3810.00 −0.496593 −0.248296 0.968684i \(-0.579871\pi\)
−0.248296 + 0.968684i \(0.579871\pi\)
\(390\) 0 0
\(391\) −9380.00 −1.21321
\(392\) 0 0
\(393\) − 5472.00i − 0.702356i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 9158.00i − 1.15775i −0.815416 0.578875i \(-0.803492\pi\)
0.815416 0.578875i \(-0.196508\pi\)
\(398\) 0 0
\(399\) 816.000 0.102384
\(400\) 0 0
\(401\) 4866.00 0.605976 0.302988 0.952994i \(-0.402016\pi\)
0.302988 + 0.952994i \(0.402016\pi\)
\(402\) 0 0
\(403\) 5800.00i 0.716920i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7008.00i 0.853498i
\(408\) 0 0
\(409\) −13486.0 −1.63042 −0.815208 0.579169i \(-0.803377\pi\)
−0.815208 + 0.579169i \(0.803377\pi\)
\(410\) 0 0
\(411\) −5556.00 −0.666806
\(412\) 0 0
\(413\) − 9112.00i − 1.08565i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 3096.00i − 0.363577i
\(418\) 0 0
\(419\) 5628.00 0.656195 0.328098 0.944644i \(-0.393593\pi\)
0.328098 + 0.944644i \(0.393593\pi\)
\(420\) 0 0
\(421\) 7938.00 0.918942 0.459471 0.888193i \(-0.348039\pi\)
0.459471 + 0.888193i \(0.348039\pi\)
\(422\) 0 0
\(423\) 198.000i 0.0227591i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8500.00i 0.963334i
\(428\) 0 0
\(429\) 5568.00 0.626633
\(430\) 0 0
\(431\) −1916.00 −0.214131 −0.107066 0.994252i \(-0.534145\pi\)
−0.107066 + 0.994252i \(0.534145\pi\)
\(432\) 0 0
\(433\) 16510.0i 1.83238i 0.400746 + 0.916189i \(0.368751\pi\)
−0.400746 + 0.916189i \(0.631249\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 536.000i − 0.0586736i
\(438\) 0 0
\(439\) −1256.00 −0.136550 −0.0682752 0.997667i \(-0.521750\pi\)
−0.0682752 + 0.997667i \(0.521750\pi\)
\(440\) 0 0
\(441\) 7317.00 0.790087
\(442\) 0 0
\(443\) − 12222.0i − 1.31080i −0.755282 0.655400i \(-0.772500\pi\)
0.755282 0.655400i \(-0.227500\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 5748.00i − 0.608213i
\(448\) 0 0
\(449\) 5946.00 0.624965 0.312482 0.949924i \(-0.398840\pi\)
0.312482 + 0.949924i \(0.398840\pi\)
\(450\) 0 0
\(451\) 2208.00 0.230534
\(452\) 0 0
\(453\) 1992.00i 0.206606i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1258.00i 0.128768i 0.997925 + 0.0643838i \(0.0205082\pi\)
−0.997925 + 0.0643838i \(0.979492\pi\)
\(458\) 0 0
\(459\) −7560.00 −0.768781
\(460\) 0 0
\(461\) 16422.0 1.65911 0.829554 0.558426i \(-0.188595\pi\)
0.829554 + 0.558426i \(0.188595\pi\)
\(462\) 0 0
\(463\) 2658.00i 0.266799i 0.991062 + 0.133399i \(0.0425893\pi\)
−0.991062 + 0.133399i \(0.957411\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 3686.00i − 0.365241i −0.983183 0.182621i \(-0.941542\pi\)
0.983183 0.182621i \(-0.0584580\pi\)
\(468\) 0 0
\(469\) 14348.0 1.41264
\(470\) 0 0
\(471\) −6132.00 −0.599889
\(472\) 0 0
\(473\) − 2848.00i − 0.276852i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1458.00i 0.139952i
\(478\) 0 0
\(479\) 88.0000 0.00839420 0.00419710 0.999991i \(-0.498664\pi\)
0.00419710 + 0.999991i \(0.498664\pi\)
\(480\) 0 0
\(481\) −25404.0 −2.40816
\(482\) 0 0
\(483\) − 27336.0i − 2.57522i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 14714.0i 1.36911i 0.728963 + 0.684553i \(0.240003\pi\)
−0.728963 + 0.684553i \(0.759997\pi\)
\(488\) 0 0
\(489\) −5556.00 −0.513806
\(490\) 0 0
\(491\) 7344.00 0.675010 0.337505 0.941324i \(-0.390417\pi\)
0.337505 + 0.941324i \(0.390417\pi\)
\(492\) 0 0
\(493\) − 16940.0i − 1.54754i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 28968.0i 2.61447i
\(498\) 0 0
\(499\) 1604.00 0.143898 0.0719488 0.997408i \(-0.477078\pi\)
0.0719488 + 0.997408i \(0.477078\pi\)
\(500\) 0 0
\(501\) −3924.00 −0.349923
\(502\) 0 0
\(503\) 14802.0i 1.31210i 0.754715 + 0.656052i \(0.227775\pi\)
−0.754715 + 0.656052i \(0.772225\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7002.00i 0.613353i
\(508\) 0 0
\(509\) 22514.0 1.96054 0.980271 0.197660i \(-0.0633342\pi\)
0.980271 + 0.197660i \(0.0633342\pi\)
\(510\) 0 0
\(511\) 10404.0 0.900677
\(512\) 0 0
\(513\) − 432.000i − 0.0371799i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 352.000i 0.0299438i
\(518\) 0 0
\(519\) 7764.00 0.656651
\(520\) 0 0
\(521\) −6710.00 −0.564243 −0.282121 0.959379i \(-0.591038\pi\)
−0.282121 + 0.959379i \(0.591038\pi\)
\(522\) 0 0
\(523\) 7930.00i 0.663011i 0.943453 + 0.331505i \(0.107557\pi\)
−0.943453 + 0.331505i \(0.892443\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7000.00i 0.578605i
\(528\) 0 0
\(529\) −5789.00 −0.475795
\(530\) 0 0
\(531\) 2412.00 0.197122
\(532\) 0 0
\(533\) 8004.00i 0.650454i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 17016.0i 1.36740i
\(538\) 0 0
\(539\) 13008.0 1.03951
\(540\) 0 0
\(541\) 4918.00 0.390834 0.195417 0.980720i \(-0.437394\pi\)
0.195417 + 0.980720i \(0.437394\pi\)
\(542\) 0 0
\(543\) − 10452.0i − 0.826037i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3922.00i 0.306568i 0.988182 + 0.153284i \(0.0489849\pi\)
−0.988182 + 0.153284i \(0.951015\pi\)
\(548\) 0 0
\(549\) −2250.00 −0.174914
\(550\) 0 0
\(551\) 968.000 0.0748424
\(552\) 0 0
\(553\) − 15504.0i − 1.19222i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17786.0i 1.35299i 0.736446 + 0.676496i \(0.236503\pi\)
−0.736446 + 0.676496i \(0.763497\pi\)
\(558\) 0 0
\(559\) 10324.0 0.781143
\(560\) 0 0
\(561\) 6720.00 0.505737
\(562\) 0 0
\(563\) 20266.0i 1.51707i 0.651633 + 0.758535i \(0.274084\pi\)
−0.651633 + 0.758535i \(0.725916\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 30294.0i − 2.24379i
\(568\) 0 0
\(569\) −13358.0 −0.984177 −0.492088 0.870545i \(-0.663766\pi\)
−0.492088 + 0.870545i \(0.663766\pi\)
\(570\) 0 0
\(571\) −16360.0 −1.19903 −0.599514 0.800364i \(-0.704639\pi\)
−0.599514 + 0.800364i \(0.704639\pi\)
\(572\) 0 0
\(573\) 26760.0i 1.95099i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 15574.0i − 1.12366i −0.827251 0.561832i \(-0.810097\pi\)
0.827251 0.561832i \(-0.189903\pi\)
\(578\) 0 0
\(579\) 22692.0 1.62875
\(580\) 0 0
\(581\) −14756.0 −1.05367
\(582\) 0 0
\(583\) 2592.00i 0.184133i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 6654.00i − 0.467870i −0.972252 0.233935i \(-0.924840\pi\)
0.972252 0.233935i \(-0.0751604\pi\)
\(588\) 0 0
\(589\) −400.000 −0.0279825
\(590\) 0 0
\(591\) 26844.0 1.86838
\(592\) 0 0
\(593\) 17742.0i 1.22863i 0.789062 + 0.614314i \(0.210567\pi\)
−0.789062 + 0.614314i \(0.789433\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 21648.0i − 1.48408i
\(598\) 0 0
\(599\) 15840.0 1.08048 0.540238 0.841512i \(-0.318334\pi\)
0.540238 + 0.841512i \(0.318334\pi\)
\(600\) 0 0
\(601\) −3002.00 −0.203751 −0.101875 0.994797i \(-0.532484\pi\)
−0.101875 + 0.994797i \(0.532484\pi\)
\(602\) 0 0
\(603\) 3798.00i 0.256495i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 23610.0i 1.57875i 0.613912 + 0.789374i \(0.289595\pi\)
−0.613912 + 0.789374i \(0.710405\pi\)
\(608\) 0 0
\(609\) 49368.0 3.28488
\(610\) 0 0
\(611\) −1276.00 −0.0844868
\(612\) 0 0
\(613\) − 23850.0i − 1.57144i −0.618583 0.785720i \(-0.712293\pi\)
0.618583 0.785720i \(-0.287707\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 5334.00i − 0.348037i −0.984742 0.174018i \(-0.944325\pi\)
0.984742 0.174018i \(-0.0556752\pi\)
\(618\) 0 0
\(619\) −2164.00 −0.140515 −0.0702573 0.997529i \(-0.522382\pi\)
−0.0702573 + 0.997529i \(0.522382\pi\)
\(620\) 0 0
\(621\) −14472.0 −0.935171
\(622\) 0 0
\(623\) 24684.0i 1.58739i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 384.000i 0.0244585i
\(628\) 0 0
\(629\) −30660.0 −1.94355
\(630\) 0 0
\(631\) 25220.0 1.59111 0.795557 0.605879i \(-0.207179\pi\)
0.795557 + 0.605879i \(0.207179\pi\)
\(632\) 0 0
\(633\) − 1536.00i − 0.0964463i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 47154.0i 2.93298i
\(638\) 0 0
\(639\) −7668.00 −0.474713
\(640\) 0 0
\(641\) −12306.0 −0.758280 −0.379140 0.925339i \(-0.623780\pi\)
−0.379140 + 0.925339i \(0.623780\pi\)
\(642\) 0 0
\(643\) − 27414.0i − 1.68134i −0.541547 0.840671i \(-0.682161\pi\)
0.541547 0.840671i \(-0.317839\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21834.0i 1.32671i 0.748304 + 0.663356i \(0.230869\pi\)
−0.748304 + 0.663356i \(0.769131\pi\)
\(648\) 0 0
\(649\) 4288.00 0.259351
\(650\) 0 0
\(651\) −20400.0 −1.22817
\(652\) 0 0
\(653\) 23998.0i 1.43815i 0.694931 + 0.719077i \(0.255435\pi\)
−0.694931 + 0.719077i \(0.744565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2754.00i 0.163537i
\(658\) 0 0
\(659\) −32004.0 −1.89180 −0.945902 0.324452i \(-0.894820\pi\)
−0.945902 + 0.324452i \(0.894820\pi\)
\(660\) 0 0
\(661\) −8526.00 −0.501699 −0.250849 0.968026i \(-0.580710\pi\)
−0.250849 + 0.968026i \(0.580710\pi\)
\(662\) 0 0
\(663\) 24360.0i 1.42694i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 32428.0i − 1.88248i
\(668\) 0 0
\(669\) −30948.0 −1.78852
\(670\) 0 0
\(671\) −4000.00 −0.230132
\(672\) 0 0
\(673\) − 8178.00i − 0.468408i −0.972187 0.234204i \(-0.924752\pi\)
0.972187 0.234204i \(-0.0752484\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 16646.0i − 0.944989i −0.881334 0.472495i \(-0.843354\pi\)
0.881334 0.472495i \(-0.156646\pi\)
\(678\) 0 0
\(679\) −46852.0 −2.64803
\(680\) 0 0
\(681\) 13356.0 0.751546
\(682\) 0 0
\(683\) − 22446.0i − 1.25750i −0.777608 0.628750i \(-0.783567\pi\)
0.777608 0.628750i \(-0.216433\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 12516.0i 0.695073i
\(688\) 0 0
\(689\) −9396.00 −0.519534
\(690\) 0 0
\(691\) −35336.0 −1.94536 −0.972681 0.232147i \(-0.925425\pi\)
−0.972681 + 0.232147i \(0.925425\pi\)
\(692\) 0 0
\(693\) 4896.00i 0.268375i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 9660.00i 0.524962i
\(698\) 0 0
\(699\) 34308.0 1.85643
\(700\) 0 0
\(701\) 3482.00 0.187608 0.0938041 0.995591i \(-0.470097\pi\)
0.0938041 + 0.995591i \(0.470097\pi\)
\(702\) 0 0
\(703\) − 1752.00i − 0.0939942i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4284.00i 0.227887i
\(708\) 0 0
\(709\) 19402.0 1.02773 0.513863 0.857872i \(-0.328214\pi\)
0.513863 + 0.857872i \(0.328214\pi\)
\(710\) 0 0
\(711\) 4104.00 0.216473
\(712\) 0 0
\(713\) 13400.0i 0.703834i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 21744.0i 1.13256i
\(718\) 0 0
\(719\) −9896.00 −0.513294 −0.256647 0.966505i \(-0.582618\pi\)
−0.256647 + 0.966505i \(0.582618\pi\)
\(720\) 0 0
\(721\) 42908.0 2.21633
\(722\) 0 0
\(723\) 492.000i 0.0253080i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 494.000i − 0.0252014i −0.999921 0.0126007i \(-0.995989\pi\)
0.999921 0.0126007i \(-0.00401104\pi\)
\(728\) 0 0
\(729\) −9477.00 −0.481481
\(730\) 0 0
\(731\) 12460.0 0.630437
\(732\) 0 0
\(733\) − 9282.00i − 0.467720i −0.972270 0.233860i \(-0.924864\pi\)
0.972270 0.233860i \(-0.0751357\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6752.00i 0.337467i
\(738\) 0 0
\(739\) −3252.00 −0.161877 −0.0809383 0.996719i \(-0.525792\pi\)
−0.0809383 + 0.996719i \(0.525792\pi\)
\(740\) 0 0
\(741\) −1392.00 −0.0690100
\(742\) 0 0
\(743\) − 4710.00i − 0.232561i −0.993216 0.116281i \(-0.962903\pi\)
0.993216 0.116281i \(-0.0370972\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 3906.00i − 0.191316i
\(748\) 0 0
\(749\) 17340.0 0.845914
\(750\) 0 0
\(751\) −25764.0 −1.25185 −0.625927 0.779882i \(-0.715279\pi\)
−0.625927 + 0.779882i \(0.715279\pi\)
\(752\) 0 0
\(753\) − 30240.0i − 1.46349i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 30094.0i − 1.44489i −0.691426 0.722447i \(-0.743017\pi\)
0.691426 0.722447i \(-0.256983\pi\)
\(758\) 0 0
\(759\) 12864.0 0.615196
\(760\) 0 0
\(761\) 22362.0 1.06521 0.532603 0.846365i \(-0.321214\pi\)
0.532603 + 0.846365i \(0.321214\pi\)
\(762\) 0 0
\(763\) − 884.000i − 0.0419436i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15544.0i 0.731762i
\(768\) 0 0
\(769\) 30398.0 1.42546 0.712731 0.701438i \(-0.247458\pi\)
0.712731 + 0.701438i \(0.247458\pi\)
\(770\) 0 0
\(771\) −13860.0 −0.647413
\(772\) 0 0
\(773\) − 1290.00i − 0.0600234i −0.999550 0.0300117i \(-0.990446\pi\)
0.999550 0.0300117i \(-0.00955445\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 89352.0i − 4.12546i
\(778\) 0 0
\(779\) −552.000 −0.0253883
\(780\) 0 0
\(781\) −13632.0 −0.624573
\(782\) 0 0
\(783\) − 26136.0i − 1.19288i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 14.0000i 0 0.000634112i −1.00000 0.000317056i \(-0.999899\pi\)
1.00000 0.000317056i \(-0.000100922\pi\)
\(788\) 0 0
\(789\) −24660.0 −1.11270
\(790\) 0 0
\(791\) 42228.0 1.89817
\(792\) 0 0
\(793\) − 14500.0i − 0.649319i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 38814.0i − 1.72505i −0.506017 0.862523i \(-0.668883\pi\)
0.506017 0.862523i \(-0.331117\pi\)
\(798\) 0 0
\(799\) −1540.00 −0.0681868
\(800\) 0 0
\(801\) −6534.00 −0.288224
\(802\) 0 0
\(803\) 4896.00i 0.215163i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4476.00i 0.195245i
\(808\) 0 0
\(809\) −27402.0 −1.19086 −0.595428 0.803408i \(-0.703018\pi\)
−0.595428 + 0.803408i \(0.703018\pi\)
\(810\) 0 0
\(811\) 28576.0 1.23729 0.618643 0.785672i \(-0.287683\pi\)
0.618643 + 0.785672i \(0.287683\pi\)
\(812\) 0 0
\(813\) − 27576.0i − 1.18958i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 712.000i 0.0304893i
\(818\) 0 0
\(819\) −17748.0 −0.757223
\(820\) 0 0
\(821\) 31762.0 1.35018 0.675092 0.737733i \(-0.264104\pi\)
0.675092 + 0.737733i \(0.264104\pi\)
\(822\) 0 0
\(823\) 20506.0i 0.868523i 0.900787 + 0.434261i \(0.142991\pi\)
−0.900787 + 0.434261i \(0.857009\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 13014.0i − 0.547208i −0.961842 0.273604i \(-0.911784\pi\)
0.961842 0.273604i \(-0.0882158\pi\)
\(828\) 0 0
\(829\) 22790.0 0.954800 0.477400 0.878686i \(-0.341579\pi\)
0.477400 + 0.878686i \(0.341579\pi\)
\(830\) 0 0
\(831\) −13236.0 −0.552529
\(832\) 0 0
\(833\) 56910.0i 2.36712i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 10800.0i 0.446001i
\(838\) 0 0
\(839\) 23696.0 0.975062 0.487531 0.873106i \(-0.337898\pi\)
0.487531 + 0.873106i \(0.337898\pi\)
\(840\) 0 0
\(841\) 34175.0 1.40125
\(842\) 0 0
\(843\) − 49668.0i − 2.02925i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 36550.0i − 1.48273i
\(848\) 0 0
\(849\) 7068.00 0.285716
\(850\) 0 0
\(851\) −58692.0 −2.36420
\(852\) 0 0
\(853\) − 5306.00i − 0.212982i −0.994314 0.106491i \(-0.966038\pi\)
0.994314 0.106491i \(-0.0339616\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 21054.0i − 0.839196i −0.907710 0.419598i \(-0.862171\pi\)
0.907710 0.419598i \(-0.137829\pi\)
\(858\) 0 0
\(859\) 7364.00 0.292499 0.146249 0.989248i \(-0.453280\pi\)
0.146249 + 0.989248i \(0.453280\pi\)
\(860\) 0 0
\(861\) −28152.0 −1.11431
\(862\) 0 0
\(863\) 17226.0i 0.679467i 0.940522 + 0.339733i \(0.110337\pi\)
−0.940522 + 0.339733i \(0.889663\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 78.0000i − 0.00305539i
\(868\) 0 0
\(869\) 7296.00 0.284810
\(870\) 0 0
\(871\) −24476.0 −0.952167
\(872\) 0 0
\(873\) − 12402.0i − 0.480807i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 21202.0i 0.816352i 0.912903 + 0.408176i \(0.133835\pi\)
−0.912903 + 0.408176i \(0.866165\pi\)
\(878\) 0 0
\(879\) −636.000 −0.0244047
\(880\) 0 0
\(881\) −29490.0 −1.12774 −0.563872 0.825862i \(-0.690689\pi\)
−0.563872 + 0.825862i \(0.690689\pi\)
\(882\) 0 0
\(883\) 2570.00i 0.0979472i 0.998800 + 0.0489736i \(0.0155950\pi\)
−0.998800 + 0.0489736i \(0.984405\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 36334.0i − 1.37540i −0.725997 0.687698i \(-0.758621\pi\)
0.725997 0.687698i \(-0.241379\pi\)
\(888\) 0 0
\(889\) −33252.0 −1.25448
\(890\) 0 0
\(891\) 14256.0 0.536020
\(892\) 0 0
\(893\) − 88.0000i − 0.00329766i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 46632.0i 1.73578i
\(898\) 0 0
\(899\) −24200.0 −0.897792
\(900\) 0 0
\(901\) −11340.0 −0.419301
\(902\) 0 0
\(903\) 36312.0i 1.33819i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 12474.0i 0.456662i 0.973584 + 0.228331i \(0.0733268\pi\)
−0.973584 + 0.228331i \(0.926673\pi\)
\(908\) 0 0
\(909\) −1134.00 −0.0413778
\(910\) 0 0
\(911\) 41132.0 1.49590 0.747949 0.663756i \(-0.231039\pi\)
0.747949 + 0.663756i \(0.231039\pi\)
\(912\) 0 0
\(913\) − 6944.00i − 0.251712i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 31008.0i 1.11666i
\(918\) 0 0
\(919\) −38416.0 −1.37892 −0.689460 0.724324i \(-0.742152\pi\)
−0.689460 + 0.724324i \(0.742152\pi\)
\(920\) 0 0
\(921\) −48804.0 −1.74609
\(922\) 0 0
\(923\) − 49416.0i − 1.76224i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 11358.0i 0.402423i
\(928\) 0 0
\(929\) −41302.0 −1.45864 −0.729319 0.684174i \(-0.760163\pi\)
−0.729319 + 0.684174i \(0.760163\pi\)
\(930\) 0 0
\(931\) −3252.00 −0.114479
\(932\) 0 0
\(933\) − 26376.0i − 0.925521i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 26150.0i − 0.911722i −0.890051 0.455861i \(-0.849331\pi\)
0.890051 0.455861i \(-0.150669\pi\)
\(938\) 0 0
\(939\) −28956.0 −1.00633
\(940\) 0 0
\(941\) 35254.0 1.22130 0.610652 0.791899i \(-0.290907\pi\)
0.610652 + 0.791899i \(0.290907\pi\)
\(942\) 0 0
\(943\) 18492.0i 0.638582i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 18550.0i − 0.636530i −0.948002 0.318265i \(-0.896900\pi\)
0.948002 0.318265i \(-0.103100\pi\)
\(948\) 0 0
\(949\) −17748.0 −0.607086
\(950\) 0 0
\(951\) 42156.0 1.43744
\(952\) 0 0
\(953\) − 17322.0i − 0.588788i −0.955684 0.294394i \(-0.904882\pi\)
0.955684 0.294394i \(-0.0951177\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 23232.0i 0.784727i
\(958\) 0 0
\(959\) 31484.0 1.06014
\(960\) 0 0
\(961\) −19791.0 −0.664328
\(962\) 0 0
\(963\) 4590.00i 0.153594i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 35190.0i − 1.17025i −0.810942 0.585126i \(-0.801045\pi\)
0.810942 0.585126i \(-0.198955\pi\)
\(968\) 0 0
\(969\) −1680.00 −0.0556960
\(970\) 0 0
\(971\) 40696.0 1.34500 0.672501 0.740096i \(-0.265220\pi\)
0.672501 + 0.740096i \(0.265220\pi\)
\(972\) 0 0
\(973\) 17544.0i 0.578042i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 44306.0i 1.45084i 0.688304 + 0.725422i \(0.258355\pi\)
−0.688304 + 0.725422i \(0.741645\pi\)
\(978\) 0 0
\(979\) −11616.0 −0.379212
\(980\) 0 0
\(981\) 234.000 0.00761574
\(982\) 0 0
\(983\) − 18798.0i − 0.609932i −0.952363 0.304966i \(-0.901355\pi\)
0.952363 0.304966i \(-0.0986451\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 4488.00i − 0.144736i
\(988\) 0 0
\(989\) 23852.0 0.766885
\(990\) 0 0
\(991\) −2468.00 −0.0791106 −0.0395553 0.999217i \(-0.512594\pi\)
−0.0395553 + 0.999217i \(0.512594\pi\)
\(992\) 0 0
\(993\) 52848.0i 1.68890i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 61086.0i − 1.94043i −0.242237 0.970217i \(-0.577881\pi\)
0.242237 0.970217i \(-0.422119\pi\)
\(998\) 0 0
\(999\) −47304.0 −1.49813
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 400.4.c.f.49.1 2
4.3 odd 2 200.4.c.c.49.2 2
5.2 odd 4 400.4.a.e.1.1 1
5.3 odd 4 80.4.a.e.1.1 1
5.4 even 2 inner 400.4.c.f.49.2 2
12.11 even 2 1800.4.f.j.649.1 2
15.8 even 4 720.4.a.bd.1.1 1
20.3 even 4 40.4.a.a.1.1 1
20.7 even 4 200.4.a.i.1.1 1
20.19 odd 2 200.4.c.c.49.1 2
40.3 even 4 320.4.a.l.1.1 1
40.13 odd 4 320.4.a.c.1.1 1
40.27 even 4 1600.4.a.j.1.1 1
40.37 odd 4 1600.4.a.br.1.1 1
60.23 odd 4 360.4.a.h.1.1 1
60.47 odd 4 1800.4.a.bi.1.1 1
60.59 even 2 1800.4.f.j.649.2 2
80.3 even 4 1280.4.d.p.641.1 2
80.13 odd 4 1280.4.d.a.641.2 2
80.43 even 4 1280.4.d.p.641.2 2
80.53 odd 4 1280.4.d.a.641.1 2
140.83 odd 4 1960.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.a.1.1 1 20.3 even 4
80.4.a.e.1.1 1 5.3 odd 4
200.4.a.i.1.1 1 20.7 even 4
200.4.c.c.49.1 2 20.19 odd 2
200.4.c.c.49.2 2 4.3 odd 2
320.4.a.c.1.1 1 40.13 odd 4
320.4.a.l.1.1 1 40.3 even 4
360.4.a.h.1.1 1 60.23 odd 4
400.4.a.e.1.1 1 5.2 odd 4
400.4.c.f.49.1 2 1.1 even 1 trivial
400.4.c.f.49.2 2 5.4 even 2 inner
720.4.a.bd.1.1 1 15.8 even 4
1280.4.d.a.641.1 2 80.53 odd 4
1280.4.d.a.641.2 2 80.13 odd 4
1280.4.d.p.641.1 2 80.3 even 4
1280.4.d.p.641.2 2 80.43 even 4
1600.4.a.j.1.1 1 40.27 even 4
1600.4.a.br.1.1 1 40.37 odd 4
1800.4.a.bi.1.1 1 60.47 odd 4
1800.4.f.j.649.1 2 12.11 even 2
1800.4.f.j.649.2 2 60.59 even 2
1960.4.a.h.1.1 1 140.83 odd 4