Properties

Label 1280.4.d.v.641.3
Level $1280$
Weight $4$
Character 1280.641
Analytic conductor $75.522$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(75.5224448073\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.3
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.4.d.v.641.2

$q$-expansion

\(f(q)\) \(=\) \(q+4.47214i q^{3} -5.00000i q^{5} -31.3050 q^{7} +7.00000 q^{9} +O(q^{10})\) \(q+4.47214i q^{3} -5.00000i q^{5} -31.3050 q^{7} +7.00000 q^{9} +8.94427i q^{11} +62.0000i q^{13} +22.3607 q^{15} -46.0000 q^{17} +107.331i q^{19} -140.000i q^{21} +192.302 q^{23} -25.0000 q^{25} +152.053i q^{27} +90.0000i q^{29} +152.053 q^{31} -40.0000 q^{33} +156.525i q^{35} -214.000i q^{37} -277.272 q^{39} +10.0000 q^{41} +67.0820i q^{43} -35.0000i q^{45} -398.020 q^{47} +637.000 q^{49} -205.718i q^{51} -678.000i q^{53} +44.7214 q^{55} -480.000 q^{57} +411.437i q^{59} -250.000i q^{61} -219.135 q^{63} +310.000 q^{65} +49.1935i q^{67} +860.000i q^{69} -366.715 q^{71} -522.000 q^{73} -111.803i q^{75} -280.000i q^{77} -876.539 q^{79} -491.000 q^{81} +380.132i q^{83} +230.000i q^{85} -402.492 q^{87} -970.000 q^{89} -1940.91i q^{91} +680.000i q^{93} +536.656 q^{95} -934.000 q^{97} +62.6099i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{9} + O(q^{10}) \) \( 4 q + 28 q^{9} - 184 q^{17} - 100 q^{25} - 160 q^{33} + 40 q^{41} + 2548 q^{49} - 1920 q^{57} + 1240 q^{65} - 2088 q^{73} - 1964 q^{81} - 3880 q^{89} - 3736 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 4.47214i 0.860663i 0.902671 + 0.430331i \(0.141603\pi\)
−0.902671 + 0.430331i \(0.858397\pi\)
\(4\) 0 0
\(5\) − 5.00000i − 0.447214i
\(6\) 0 0
\(7\) −31.3050 −1.69031 −0.845154 0.534522i \(-0.820491\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 7.00000 0.259259
\(10\) 0 0
\(11\) 8.94427i 0.245164i 0.992458 + 0.122582i \(0.0391174\pi\)
−0.992458 + 0.122582i \(0.960883\pi\)
\(12\) 0 0
\(13\) 62.0000i 1.32275i 0.750057 + 0.661373i \(0.230026\pi\)
−0.750057 + 0.661373i \(0.769974\pi\)
\(14\) 0 0
\(15\) 22.3607 0.384900
\(16\) 0 0
\(17\) −46.0000 −0.656273 −0.328136 0.944630i \(-0.606421\pi\)
−0.328136 + 0.944630i \(0.606421\pi\)
\(18\) 0 0
\(19\) 107.331i 1.29597i 0.761652 + 0.647986i \(0.224389\pi\)
−0.761652 + 0.647986i \(0.775611\pi\)
\(20\) 0 0
\(21\) − 140.000i − 1.45479i
\(22\) 0 0
\(23\) 192.302 1.74338 0.871689 0.490059i \(-0.163025\pi\)
0.871689 + 0.490059i \(0.163025\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) 152.053i 1.08380i
\(28\) 0 0
\(29\) 90.0000i 0.576296i 0.957586 + 0.288148i \(0.0930395\pi\)
−0.957586 + 0.288148i \(0.906961\pi\)
\(30\) 0 0
\(31\) 152.053 0.880950 0.440475 0.897765i \(-0.354810\pi\)
0.440475 + 0.897765i \(0.354810\pi\)
\(32\) 0 0
\(33\) −40.0000 −0.211003
\(34\) 0 0
\(35\) 156.525i 0.755929i
\(36\) 0 0
\(37\) − 214.000i − 0.950848i −0.879757 0.475424i \(-0.842295\pi\)
0.879757 0.475424i \(-0.157705\pi\)
\(38\) 0 0
\(39\) −277.272 −1.13844
\(40\) 0 0
\(41\) 10.0000 0.0380912 0.0190456 0.999819i \(-0.493937\pi\)
0.0190456 + 0.999819i \(0.493937\pi\)
\(42\) 0 0
\(43\) 67.0820i 0.237905i 0.992900 + 0.118953i \(0.0379536\pi\)
−0.992900 + 0.118953i \(0.962046\pi\)
\(44\) 0 0
\(45\) − 35.0000i − 0.115944i
\(46\) 0 0
\(47\) −398.020 −1.23526 −0.617630 0.786469i \(-0.711907\pi\)
−0.617630 + 0.786469i \(0.711907\pi\)
\(48\) 0 0
\(49\) 637.000 1.85714
\(50\) 0 0
\(51\) − 205.718i − 0.564830i
\(52\) 0 0
\(53\) − 678.000i − 1.75718i −0.477578 0.878589i \(-0.658485\pi\)
0.477578 0.878589i \(-0.341515\pi\)
\(54\) 0 0
\(55\) 44.7214 0.109640
\(56\) 0 0
\(57\) −480.000 −1.11540
\(58\) 0 0
\(59\) 411.437i 0.907872i 0.891034 + 0.453936i \(0.149981\pi\)
−0.891034 + 0.453936i \(0.850019\pi\)
\(60\) 0 0
\(61\) − 250.000i − 0.524741i −0.964967 0.262371i \(-0.915496\pi\)
0.964967 0.262371i \(-0.0845043\pi\)
\(62\) 0 0
\(63\) −219.135 −0.438228
\(64\) 0 0
\(65\) 310.000 0.591550
\(66\) 0 0
\(67\) 49.1935i 0.0897006i 0.998994 + 0.0448503i \(0.0142811\pi\)
−0.998994 + 0.0448503i \(0.985719\pi\)
\(68\) 0 0
\(69\) 860.000i 1.50046i
\(70\) 0 0
\(71\) −366.715 −0.612973 −0.306486 0.951875i \(-0.599153\pi\)
−0.306486 + 0.951875i \(0.599153\pi\)
\(72\) 0 0
\(73\) −522.000 −0.836924 −0.418462 0.908234i \(-0.637431\pi\)
−0.418462 + 0.908234i \(0.637431\pi\)
\(74\) 0 0
\(75\) − 111.803i − 0.172133i
\(76\) 0 0
\(77\) − 280.000i − 0.414402i
\(78\) 0 0
\(79\) −876.539 −1.24833 −0.624166 0.781291i \(-0.714561\pi\)
−0.624166 + 0.781291i \(0.714561\pi\)
\(80\) 0 0
\(81\) −491.000 −0.673525
\(82\) 0 0
\(83\) 380.132i 0.502709i 0.967895 + 0.251355i \(0.0808760\pi\)
−0.967895 + 0.251355i \(0.919124\pi\)
\(84\) 0 0
\(85\) 230.000i 0.293494i
\(86\) 0 0
\(87\) −402.492 −0.495997
\(88\) 0 0
\(89\) −970.000 −1.15528 −0.577639 0.816292i \(-0.696026\pi\)
−0.577639 + 0.816292i \(0.696026\pi\)
\(90\) 0 0
\(91\) − 1940.91i − 2.23585i
\(92\) 0 0
\(93\) 680.000i 0.758201i
\(94\) 0 0
\(95\) 536.656 0.579577
\(96\) 0 0
\(97\) −934.000 −0.977663 −0.488832 0.872378i \(-0.662577\pi\)
−0.488832 + 0.872378i \(0.662577\pi\)
\(98\) 0 0
\(99\) 62.6099i 0.0635609i
\(100\) 0 0
\(101\) − 602.000i − 0.593082i −0.955020 0.296541i \(-0.904167\pi\)
0.955020 0.296541i \(-0.0958331\pi\)
\(102\) 0 0
\(103\) −1829.10 −1.74978 −0.874888 0.484325i \(-0.839065\pi\)
−0.874888 + 0.484325i \(0.839065\pi\)
\(104\) 0 0
\(105\) −700.000 −0.650600
\(106\) 0 0
\(107\) 1525.00i 1.37782i 0.724845 + 0.688912i \(0.241911\pi\)
−0.724845 + 0.688912i \(0.758089\pi\)
\(108\) 0 0
\(109\) − 2154.00i − 1.89281i −0.322989 0.946403i \(-0.604688\pi\)
0.322989 0.946403i \(-0.395312\pi\)
\(110\) 0 0
\(111\) 957.037 0.818360
\(112\) 0 0
\(113\) −2182.00 −1.81651 −0.908254 0.418420i \(-0.862584\pi\)
−0.908254 + 0.418420i \(0.862584\pi\)
\(114\) 0 0
\(115\) − 961.509i − 0.779663i
\(116\) 0 0
\(117\) 434.000i 0.342934i
\(118\) 0 0
\(119\) 1440.03 1.10930
\(120\) 0 0
\(121\) 1251.00 0.939895
\(122\) 0 0
\(123\) 44.7214i 0.0327837i
\(124\) 0 0
\(125\) 125.000i 0.0894427i
\(126\) 0 0
\(127\) −1310.34 −0.915539 −0.457770 0.889071i \(-0.651352\pi\)
−0.457770 + 0.889071i \(0.651352\pi\)
\(128\) 0 0
\(129\) −300.000 −0.204756
\(130\) 0 0
\(131\) − 205.718i − 0.137204i −0.997644 0.0686019i \(-0.978146\pi\)
0.997644 0.0686019i \(-0.0218538\pi\)
\(132\) 0 0
\(133\) − 3360.00i − 2.19059i
\(134\) 0 0
\(135\) 760.263 0.484689
\(136\) 0 0
\(137\) 2094.00 1.30586 0.652929 0.757419i \(-0.273540\pi\)
0.652929 + 0.757419i \(0.273540\pi\)
\(138\) 0 0
\(139\) 1377.42i 0.840511i 0.907406 + 0.420256i \(0.138060\pi\)
−0.907406 + 0.420256i \(0.861940\pi\)
\(140\) 0 0
\(141\) − 1780.00i − 1.06314i
\(142\) 0 0
\(143\) −554.545 −0.324289
\(144\) 0 0
\(145\) 450.000 0.257727
\(146\) 0 0
\(147\) 2848.75i 1.59837i
\(148\) 0 0
\(149\) − 334.000i − 0.183640i −0.995776 0.0918200i \(-0.970732\pi\)
0.995776 0.0918200i \(-0.0292684\pi\)
\(150\) 0 0
\(151\) 3139.44 1.69195 0.845973 0.533225i \(-0.179020\pi\)
0.845973 + 0.533225i \(0.179020\pi\)
\(152\) 0 0
\(153\) −322.000 −0.170145
\(154\) 0 0
\(155\) − 760.263i − 0.393973i
\(156\) 0 0
\(157\) − 834.000i − 0.423952i −0.977275 0.211976i \(-0.932010\pi\)
0.977275 0.211976i \(-0.0679898\pi\)
\(158\) 0 0
\(159\) 3032.11 1.51234
\(160\) 0 0
\(161\) −6020.00 −2.94685
\(162\) 0 0
\(163\) − 3090.25i − 1.48495i −0.669874 0.742475i \(-0.733652\pi\)
0.669874 0.742475i \(-0.266348\pi\)
\(164\) 0 0
\(165\) 200.000i 0.0943635i
\(166\) 0 0
\(167\) 4.47214 0.00207224 0.00103612 0.999999i \(-0.499670\pi\)
0.00103612 + 0.999999i \(0.499670\pi\)
\(168\) 0 0
\(169\) −1647.00 −0.749659
\(170\) 0 0
\(171\) 751.319i 0.335993i
\(172\) 0 0
\(173\) 1838.00i 0.807749i 0.914814 + 0.403874i \(0.132337\pi\)
−0.914814 + 0.403874i \(0.867663\pi\)
\(174\) 0 0
\(175\) 782.624 0.338062
\(176\) 0 0
\(177\) −1840.00 −0.781372
\(178\) 0 0
\(179\) 1842.52i 0.769365i 0.923049 + 0.384683i \(0.125689\pi\)
−0.923049 + 0.384683i \(0.874311\pi\)
\(180\) 0 0
\(181\) 1862.00i 0.764648i 0.924028 + 0.382324i \(0.124876\pi\)
−0.924028 + 0.382324i \(0.875124\pi\)
\(182\) 0 0
\(183\) 1118.03 0.451625
\(184\) 0 0
\(185\) −1070.00 −0.425232
\(186\) 0 0
\(187\) − 411.437i − 0.160894i
\(188\) 0 0
\(189\) − 4760.00i − 1.83195i
\(190\) 0 0
\(191\) 2066.13 0.782721 0.391360 0.920237i \(-0.372005\pi\)
0.391360 + 0.920237i \(0.372005\pi\)
\(192\) 0 0
\(193\) 3378.00 1.25986 0.629932 0.776650i \(-0.283083\pi\)
0.629932 + 0.776650i \(0.283083\pi\)
\(194\) 0 0
\(195\) 1386.36i 0.509125i
\(196\) 0 0
\(197\) 66.0000i 0.0238696i 0.999929 + 0.0119348i \(0.00379905\pi\)
−0.999929 + 0.0119348i \(0.996201\pi\)
\(198\) 0 0
\(199\) −1216.42 −0.433316 −0.216658 0.976248i \(-0.569516\pi\)
−0.216658 + 0.976248i \(0.569516\pi\)
\(200\) 0 0
\(201\) −220.000 −0.0772020
\(202\) 0 0
\(203\) − 2817.45i − 0.974118i
\(204\) 0 0
\(205\) − 50.0000i − 0.0170349i
\(206\) 0 0
\(207\) 1346.11 0.451987
\(208\) 0 0
\(209\) −960.000 −0.317725
\(210\) 0 0
\(211\) − 5286.06i − 1.72468i −0.506329 0.862341i \(-0.668998\pi\)
0.506329 0.862341i \(-0.331002\pi\)
\(212\) 0 0
\(213\) − 1640.00i − 0.527563i
\(214\) 0 0
\(215\) 335.410 0.106394
\(216\) 0 0
\(217\) −4760.00 −1.48908
\(218\) 0 0
\(219\) − 2334.45i − 0.720310i
\(220\) 0 0
\(221\) − 2852.00i − 0.868083i
\(222\) 0 0
\(223\) −2965.03 −0.890371 −0.445186 0.895438i \(-0.646862\pi\)
−0.445186 + 0.895438i \(0.646862\pi\)
\(224\) 0 0
\(225\) −175.000 −0.0518519
\(226\) 0 0
\(227\) − 4369.28i − 1.27753i −0.769402 0.638765i \(-0.779446\pi\)
0.769402 0.638765i \(-0.220554\pi\)
\(228\) 0 0
\(229\) − 3250.00i − 0.937843i −0.883240 0.468921i \(-0.844643\pi\)
0.883240 0.468921i \(-0.155357\pi\)
\(230\) 0 0
\(231\) 1252.20 0.356661
\(232\) 0 0
\(233\) −3298.00 −0.927293 −0.463646 0.886020i \(-0.653459\pi\)
−0.463646 + 0.886020i \(0.653459\pi\)
\(234\) 0 0
\(235\) 1990.10i 0.552425i
\(236\) 0 0
\(237\) − 3920.00i − 1.07439i
\(238\) 0 0
\(239\) 554.545 0.150086 0.0750429 0.997180i \(-0.476091\pi\)
0.0750429 + 0.997180i \(0.476091\pi\)
\(240\) 0 0
\(241\) 5150.00 1.37652 0.688259 0.725465i \(-0.258375\pi\)
0.688259 + 0.725465i \(0.258375\pi\)
\(242\) 0 0
\(243\) 1909.60i 0.504119i
\(244\) 0 0
\(245\) − 3185.00i − 0.830540i
\(246\) 0 0
\(247\) −6654.54 −1.71424
\(248\) 0 0
\(249\) −1700.00 −0.432663
\(250\) 0 0
\(251\) − 1386.36i − 0.348631i −0.984690 0.174316i \(-0.944229\pi\)
0.984690 0.174316i \(-0.0557713\pi\)
\(252\) 0 0
\(253\) 1720.00i 0.427413i
\(254\) 0 0
\(255\) −1028.59 −0.252600
\(256\) 0 0
\(257\) −4166.00 −1.01116 −0.505580 0.862780i \(-0.668721\pi\)
−0.505580 + 0.862780i \(0.668721\pi\)
\(258\) 0 0
\(259\) 6699.26i 1.60723i
\(260\) 0 0
\(261\) 630.000i 0.149410i
\(262\) 0 0
\(263\) 961.509 0.225434 0.112717 0.993627i \(-0.464045\pi\)
0.112717 + 0.993627i \(0.464045\pi\)
\(264\) 0 0
\(265\) −3390.00 −0.785834
\(266\) 0 0
\(267\) − 4337.97i − 0.994305i
\(268\) 0 0
\(269\) 1494.00i 0.338627i 0.985562 + 0.169314i \(0.0541551\pi\)
−0.985562 + 0.169314i \(0.945845\pi\)
\(270\) 0 0
\(271\) −5017.74 −1.12474 −0.562372 0.826884i \(-0.690111\pi\)
−0.562372 + 0.826884i \(0.690111\pi\)
\(272\) 0 0
\(273\) 8680.00 1.92431
\(274\) 0 0
\(275\) − 223.607i − 0.0490327i
\(276\) 0 0
\(277\) − 1006.00i − 0.218212i −0.994030 0.109106i \(-0.965201\pi\)
0.994030 0.109106i \(-0.0347988\pi\)
\(278\) 0 0
\(279\) 1064.37 0.228395
\(280\) 0 0
\(281\) 3210.00 0.681468 0.340734 0.940160i \(-0.389324\pi\)
0.340734 + 0.940160i \(0.389324\pi\)
\(282\) 0 0
\(283\) − 3635.85i − 0.763705i −0.924223 0.381853i \(-0.875286\pi\)
0.924223 0.381853i \(-0.124714\pi\)
\(284\) 0 0
\(285\) 2400.00i 0.498820i
\(286\) 0 0
\(287\) −313.050 −0.0643858
\(288\) 0 0
\(289\) −2797.00 −0.569306
\(290\) 0 0
\(291\) − 4176.97i − 0.841439i
\(292\) 0 0
\(293\) − 3622.00i − 0.722183i −0.932530 0.361091i \(-0.882404\pi\)
0.932530 0.361091i \(-0.117596\pi\)
\(294\) 0 0
\(295\) 2057.18 0.406013
\(296\) 0 0
\(297\) −1360.00 −0.265708
\(298\) 0 0
\(299\) 11922.7i 2.30605i
\(300\) 0 0
\(301\) − 2100.00i − 0.402133i
\(302\) 0 0
\(303\) 2692.23 0.510443
\(304\) 0 0
\(305\) −1250.00 −0.234671
\(306\) 0 0
\(307\) 2088.49i 0.388261i 0.980976 + 0.194131i \(0.0621886\pi\)
−0.980976 + 0.194131i \(0.937811\pi\)
\(308\) 0 0
\(309\) − 8180.00i − 1.50597i
\(310\) 0 0
\(311\) −8899.55 −1.62266 −0.811330 0.584589i \(-0.801256\pi\)
−0.811330 + 0.584589i \(0.801256\pi\)
\(312\) 0 0
\(313\) −8778.00 −1.58518 −0.792591 0.609754i \(-0.791268\pi\)
−0.792591 + 0.609754i \(0.791268\pi\)
\(314\) 0 0
\(315\) 1095.67i 0.195982i
\(316\) 0 0
\(317\) 5046.00i 0.894043i 0.894523 + 0.447021i \(0.147515\pi\)
−0.894523 + 0.447021i \(0.852485\pi\)
\(318\) 0 0
\(319\) −804.984 −0.141287
\(320\) 0 0
\(321\) −6820.00 −1.18584
\(322\) 0 0
\(323\) − 4937.24i − 0.850512i
\(324\) 0 0
\(325\) − 1550.00i − 0.264549i
\(326\) 0 0
\(327\) 9632.98 1.62907
\(328\) 0 0
\(329\) 12460.0 2.08797
\(330\) 0 0
\(331\) 313.050i 0.0519842i 0.999662 + 0.0259921i \(0.00827447\pi\)
−0.999662 + 0.0259921i \(0.991726\pi\)
\(332\) 0 0
\(333\) − 1498.00i − 0.246516i
\(334\) 0 0
\(335\) 245.967 0.0401153
\(336\) 0 0
\(337\) −2574.00 −0.416067 −0.208034 0.978122i \(-0.566706\pi\)
−0.208034 + 0.978122i \(0.566706\pi\)
\(338\) 0 0
\(339\) − 9758.20i − 1.56340i
\(340\) 0 0
\(341\) 1360.00i 0.215977i
\(342\) 0 0
\(343\) −9203.66 −1.44884
\(344\) 0 0
\(345\) 4300.00 0.671027
\(346\) 0 0
\(347\) − 2643.03i − 0.408892i −0.978878 0.204446i \(-0.934461\pi\)
0.978878 0.204446i \(-0.0655392\pi\)
\(348\) 0 0
\(349\) 10170.0i 1.55985i 0.625873 + 0.779925i \(0.284743\pi\)
−0.625873 + 0.779925i \(0.715257\pi\)
\(350\) 0 0
\(351\) −9427.26 −1.43359
\(352\) 0 0
\(353\) −318.000 −0.0479474 −0.0239737 0.999713i \(-0.507632\pi\)
−0.0239737 + 0.999713i \(0.507632\pi\)
\(354\) 0 0
\(355\) 1833.58i 0.274130i
\(356\) 0 0
\(357\) 6440.00i 0.954737i
\(358\) 0 0
\(359\) 12378.9 1.81987 0.909933 0.414755i \(-0.136133\pi\)
0.909933 + 0.414755i \(0.136133\pi\)
\(360\) 0 0
\(361\) −4661.00 −0.679545
\(362\) 0 0
\(363\) 5594.64i 0.808933i
\(364\) 0 0
\(365\) 2610.00i 0.374284i
\(366\) 0 0
\(367\) 3072.36 0.436991 0.218496 0.975838i \(-0.429885\pi\)
0.218496 + 0.975838i \(0.429885\pi\)
\(368\) 0 0
\(369\) 70.0000 0.00987549
\(370\) 0 0
\(371\) 21224.8i 2.97017i
\(372\) 0 0
\(373\) − 3278.00i − 0.455036i −0.973774 0.227518i \(-0.926939\pi\)
0.973774 0.227518i \(-0.0730610\pi\)
\(374\) 0 0
\(375\) −559.017 −0.0769800
\(376\) 0 0
\(377\) −5580.00 −0.762293
\(378\) 0 0
\(379\) − 5116.12i − 0.693397i −0.937977 0.346699i \(-0.887303\pi\)
0.937977 0.346699i \(-0.112697\pi\)
\(380\) 0 0
\(381\) − 5860.00i − 0.787971i
\(382\) 0 0
\(383\) 1149.34 0.153338 0.0766690 0.997057i \(-0.475572\pi\)
0.0766690 + 0.997057i \(0.475572\pi\)
\(384\) 0 0
\(385\) −1400.00 −0.185326
\(386\) 0 0
\(387\) 469.574i 0.0616791i
\(388\) 0 0
\(389\) 834.000i 0.108703i 0.998522 + 0.0543515i \(0.0173092\pi\)
−0.998522 + 0.0543515i \(0.982691\pi\)
\(390\) 0 0
\(391\) −8845.88 −1.14413
\(392\) 0 0
\(393\) 920.000 0.118086
\(394\) 0 0
\(395\) 4382.69i 0.558271i
\(396\) 0 0
\(397\) 8734.00i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) 0 0
\(399\) 15026.4 1.88536
\(400\) 0 0
\(401\) 242.000 0.0301369 0.0150685 0.999886i \(-0.495203\pi\)
0.0150685 + 0.999886i \(0.495203\pi\)
\(402\) 0 0
\(403\) 9427.26i 1.16527i
\(404\) 0 0
\(405\) 2455.00i 0.301210i
\(406\) 0 0
\(407\) 1914.07 0.233113
\(408\) 0 0
\(409\) 6514.00 0.787522 0.393761 0.919213i \(-0.371174\pi\)
0.393761 + 0.919213i \(0.371174\pi\)
\(410\) 0 0
\(411\) 9364.65i 1.12390i
\(412\) 0 0
\(413\) − 12880.0i − 1.53458i
\(414\) 0 0
\(415\) 1900.66 0.224818
\(416\) 0 0
\(417\) −6160.00 −0.723397
\(418\) 0 0
\(419\) 16081.8i 1.87505i 0.347913 + 0.937527i \(0.386890\pi\)
−0.347913 + 0.937527i \(0.613110\pi\)
\(420\) 0 0
\(421\) 7250.00i 0.839295i 0.907687 + 0.419648i \(0.137846\pi\)
−0.907687 + 0.419648i \(0.862154\pi\)
\(422\) 0 0
\(423\) −2786.14 −0.320252
\(424\) 0 0
\(425\) 1150.00 0.131255
\(426\) 0 0
\(427\) 7826.24i 0.886975i
\(428\) 0 0
\(429\) − 2480.00i − 0.279104i
\(430\) 0 0
\(431\) −4981.96 −0.556781 −0.278390 0.960468i \(-0.589801\pi\)
−0.278390 + 0.960468i \(0.589801\pi\)
\(432\) 0 0
\(433\) 11482.0 1.27434 0.637171 0.770723i \(-0.280105\pi\)
0.637171 + 0.770723i \(0.280105\pi\)
\(434\) 0 0
\(435\) 2012.46i 0.221816i
\(436\) 0 0
\(437\) 20640.0i 2.25937i
\(438\) 0 0
\(439\) 3792.37 0.412301 0.206150 0.978520i \(-0.433906\pi\)
0.206150 + 0.978520i \(0.433906\pi\)
\(440\) 0 0
\(441\) 4459.00 0.481481
\(442\) 0 0
\(443\) 746.847i 0.0800988i 0.999198 + 0.0400494i \(0.0127515\pi\)
−0.999198 + 0.0400494i \(0.987248\pi\)
\(444\) 0 0
\(445\) 4850.00i 0.516656i
\(446\) 0 0
\(447\) 1493.69 0.158052
\(448\) 0 0
\(449\) −1306.00 −0.137269 −0.0686347 0.997642i \(-0.521864\pi\)
−0.0686347 + 0.997642i \(0.521864\pi\)
\(450\) 0 0
\(451\) 89.4427i 0.00933857i
\(452\) 0 0
\(453\) 14040.0i 1.45620i
\(454\) 0 0
\(455\) −9704.54 −0.999902
\(456\) 0 0
\(457\) 9526.00 0.975071 0.487536 0.873103i \(-0.337896\pi\)
0.487536 + 0.873103i \(0.337896\pi\)
\(458\) 0 0
\(459\) − 6994.42i − 0.711267i
\(460\) 0 0
\(461\) − 1518.00i − 0.153363i −0.997056 0.0766815i \(-0.975568\pi\)
0.997056 0.0766815i \(-0.0244325\pi\)
\(462\) 0 0
\(463\) −17293.7 −1.73587 −0.867936 0.496676i \(-0.834554\pi\)
−0.867936 + 0.496676i \(0.834554\pi\)
\(464\) 0 0
\(465\) 3400.00 0.339078
\(466\) 0 0
\(467\) − 16980.7i − 1.68260i −0.540570 0.841299i \(-0.681792\pi\)
0.540570 0.841299i \(-0.318208\pi\)
\(468\) 0 0
\(469\) − 1540.00i − 0.151622i
\(470\) 0 0
\(471\) 3729.76 0.364880
\(472\) 0 0
\(473\) −600.000 −0.0583256
\(474\) 0 0
\(475\) − 2683.28i − 0.259195i
\(476\) 0 0
\(477\) − 4746.00i − 0.455565i
\(478\) 0 0
\(479\) −3810.26 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(480\) 0 0
\(481\) 13268.0 1.25773
\(482\) 0 0
\(483\) − 26922.3i − 2.53624i
\(484\) 0 0
\(485\) 4670.00i 0.437224i
\(486\) 0 0
\(487\) 1310.34 0.121924 0.0609620 0.998140i \(-0.480583\pi\)
0.0609620 + 0.998140i \(0.480583\pi\)
\(488\) 0 0
\(489\) 13820.0 1.27804
\(490\) 0 0
\(491\) − 2960.55i − 0.272114i −0.990701 0.136057i \(-0.956557\pi\)
0.990701 0.136057i \(-0.0434430\pi\)
\(492\) 0 0
\(493\) − 4140.00i − 0.378207i
\(494\) 0 0
\(495\) 313.050 0.0284253
\(496\) 0 0
\(497\) 11480.0 1.03611
\(498\) 0 0
\(499\) − 19319.6i − 1.73320i −0.499006 0.866598i \(-0.666301\pi\)
0.499006 0.866598i \(-0.333699\pi\)
\(500\) 0 0
\(501\) 20.0000i 0.00178350i
\(502\) 0 0
\(503\) 3072.36 0.272345 0.136173 0.990685i \(-0.456520\pi\)
0.136173 + 0.990685i \(0.456520\pi\)
\(504\) 0 0
\(505\) −3010.00 −0.265234
\(506\) 0 0
\(507\) − 7365.61i − 0.645203i
\(508\) 0 0
\(509\) − 18550.0i − 1.61535i −0.589626 0.807676i \(-0.700725\pi\)
0.589626 0.807676i \(-0.299275\pi\)
\(510\) 0 0
\(511\) 16341.2 1.41466
\(512\) 0 0
\(513\) −16320.0 −1.40457
\(514\) 0 0
\(515\) 9145.52i 0.782524i
\(516\) 0 0
\(517\) − 3560.00i − 0.302841i
\(518\) 0 0
\(519\) −8219.79 −0.695200
\(520\) 0 0
\(521\) 2102.00 0.176757 0.0883784 0.996087i \(-0.471832\pi\)
0.0883784 + 0.996087i \(0.471832\pi\)
\(522\) 0 0
\(523\) − 17696.2i − 1.47955i −0.672856 0.739773i \(-0.734933\pi\)
0.672856 0.739773i \(-0.265067\pi\)
\(524\) 0 0
\(525\) 3500.00i 0.290957i
\(526\) 0 0
\(527\) −6994.42 −0.578144
\(528\) 0 0
\(529\) 24813.0 2.03937
\(530\) 0 0
\(531\) 2880.06i 0.235374i
\(532\) 0 0
\(533\) 620.000i 0.0503850i
\(534\) 0 0
\(535\) 7624.99 0.616182
\(536\) 0 0
\(537\) −8240.00 −0.662164
\(538\) 0 0
\(539\) 5697.50i 0.455304i
\(540\) 0 0
\(541\) 9922.00i 0.788503i 0.919003 + 0.394251i \(0.128996\pi\)
−0.919003 + 0.394251i \(0.871004\pi\)
\(542\) 0 0
\(543\) −8327.12 −0.658105
\(544\) 0 0
\(545\) −10770.0 −0.846488
\(546\) 0 0
\(547\) 3716.34i 0.290493i 0.989396 + 0.145246i \(0.0463975\pi\)
−0.989396 + 0.145246i \(0.953603\pi\)
\(548\) 0 0
\(549\) − 1750.00i − 0.136044i
\(550\) 0 0
\(551\) −9659.81 −0.746864
\(552\) 0 0
\(553\) 27440.0 2.11007
\(554\) 0 0
\(555\) − 4785.19i − 0.365982i
\(556\) 0 0
\(557\) 15094.0i 1.14821i 0.818781 + 0.574105i \(0.194650\pi\)
−0.818781 + 0.574105i \(0.805350\pi\)
\(558\) 0 0
\(559\) −4159.09 −0.314688
\(560\) 0 0
\(561\) 1840.00 0.138476
\(562\) 0 0
\(563\) 5657.25i 0.423490i 0.977325 + 0.211745i \(0.0679146\pi\)
−0.977325 + 0.211745i \(0.932085\pi\)
\(564\) 0 0
\(565\) 10910.0i 0.812367i
\(566\) 0 0
\(567\) 15370.7 1.13847
\(568\) 0 0
\(569\) 5906.00 0.435136 0.217568 0.976045i \(-0.430188\pi\)
0.217568 + 0.976045i \(0.430188\pi\)
\(570\) 0 0
\(571\) 4892.52i 0.358573i 0.983797 + 0.179287i \(0.0573790\pi\)
−0.983797 + 0.179287i \(0.942621\pi\)
\(572\) 0 0
\(573\) 9240.00i 0.673659i
\(574\) 0 0
\(575\) −4807.55 −0.348676
\(576\) 0 0
\(577\) −13286.0 −0.958585 −0.479292 0.877655i \(-0.659107\pi\)
−0.479292 + 0.877655i \(0.659107\pi\)
\(578\) 0 0
\(579\) 15106.9i 1.08432i
\(580\) 0 0
\(581\) − 11900.0i − 0.849734i
\(582\) 0 0
\(583\) 6064.22 0.430796
\(584\) 0 0
\(585\) 2170.00 0.153365
\(586\) 0 0
\(587\) − 9029.24i − 0.634884i −0.948278 0.317442i \(-0.897176\pi\)
0.948278 0.317442i \(-0.102824\pi\)
\(588\) 0 0
\(589\) 16320.0i 1.14169i
\(590\) 0 0
\(591\) −295.161 −0.0205437
\(592\) 0 0
\(593\) 11442.0 0.792355 0.396178 0.918174i \(-0.370336\pi\)
0.396178 + 0.918174i \(0.370336\pi\)
\(594\) 0 0
\(595\) − 7200.14i − 0.496096i
\(596\) 0 0
\(597\) − 5440.00i − 0.372939i
\(598\) 0 0
\(599\) −14149.8 −0.965187 −0.482593 0.875845i \(-0.660305\pi\)
−0.482593 + 0.875845i \(0.660305\pi\)
\(600\) 0 0
\(601\) −3110.00 −0.211081 −0.105540 0.994415i \(-0.533657\pi\)
−0.105540 + 0.994415i \(0.533657\pi\)
\(602\) 0 0
\(603\) 344.354i 0.0232557i
\(604\) 0 0
\(605\) − 6255.00i − 0.420334i
\(606\) 0 0
\(607\) 11193.8 0.748502 0.374251 0.927327i \(-0.377900\pi\)
0.374251 + 0.927327i \(0.377900\pi\)
\(608\) 0 0
\(609\) 12600.0 0.838387
\(610\) 0 0
\(611\) − 24677.2i − 1.63394i
\(612\) 0 0
\(613\) − 5342.00i − 0.351976i −0.984392 0.175988i \(-0.943688\pi\)
0.984392 0.175988i \(-0.0563120\pi\)
\(614\) 0 0
\(615\) 223.607 0.0146613
\(616\) 0 0
\(617\) −19714.0 −1.28631 −0.643157 0.765734i \(-0.722376\pi\)
−0.643157 + 0.765734i \(0.722376\pi\)
\(618\) 0 0
\(619\) 13166.0i 0.854903i 0.904038 + 0.427451i \(0.140589\pi\)
−0.904038 + 0.427451i \(0.859411\pi\)
\(620\) 0 0
\(621\) 29240.0i 1.88947i
\(622\) 0 0
\(623\) 30365.8 1.95278
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) − 4293.25i − 0.273454i
\(628\) 0 0
\(629\) 9844.00i 0.624016i
\(630\) 0 0
\(631\) 12262.6 0.773639 0.386820 0.922155i \(-0.373574\pi\)
0.386820 + 0.922155i \(0.373574\pi\)
\(632\) 0 0
\(633\) 23640.0 1.48437
\(634\) 0 0
\(635\) 6551.68i 0.409442i
\(636\) 0 0
\(637\) 39494.0i 2.45653i
\(638\) 0 0
\(639\) −2567.01 −0.158919
\(640\) 0 0
\(641\) −2690.00 −0.165754 −0.0828772 0.996560i \(-0.526411\pi\)
−0.0828772 + 0.996560i \(0.526411\pi\)
\(642\) 0 0
\(643\) 12240.2i 0.750712i 0.926881 + 0.375356i \(0.122480\pi\)
−0.926881 + 0.375356i \(0.877520\pi\)
\(644\) 0 0
\(645\) 1500.00i 0.0915697i
\(646\) 0 0
\(647\) −17973.5 −1.09214 −0.546068 0.837741i \(-0.683876\pi\)
−0.546068 + 0.837741i \(0.683876\pi\)
\(648\) 0 0
\(649\) −3680.00 −0.222577
\(650\) 0 0
\(651\) − 21287.4i − 1.28159i
\(652\) 0 0
\(653\) 3478.00i 0.208430i 0.994555 + 0.104215i \(0.0332330\pi\)
−0.994555 + 0.104215i \(0.966767\pi\)
\(654\) 0 0
\(655\) −1028.59 −0.0613594
\(656\) 0 0
\(657\) −3654.00 −0.216980
\(658\) 0 0
\(659\) − 10572.1i − 0.624934i −0.949929 0.312467i \(-0.898845\pi\)
0.949929 0.312467i \(-0.101155\pi\)
\(660\) 0 0
\(661\) − 110.000i − 0.00647277i −0.999995 0.00323639i \(-0.998970\pi\)
0.999995 0.00323639i \(-0.00103018\pi\)
\(662\) 0 0
\(663\) 12754.5 0.747127
\(664\) 0 0
\(665\) −16800.0 −0.979663
\(666\) 0 0
\(667\) 17307.2i 1.00470i
\(668\) 0 0
\(669\) − 13260.0i − 0.766310i
\(670\) 0 0
\(671\) 2236.07 0.128647
\(672\) 0 0
\(673\) −14278.0 −0.817796 −0.408898 0.912580i \(-0.634087\pi\)
−0.408898 + 0.912580i \(0.634087\pi\)
\(674\) 0 0
\(675\) − 3801.32i − 0.216760i
\(676\) 0 0
\(677\) 18386.0i 1.04377i 0.853016 + 0.521884i \(0.174771\pi\)
−0.853016 + 0.521884i \(0.825229\pi\)
\(678\) 0 0
\(679\) 29238.8 1.65255
\(680\) 0 0
\(681\) 19540.0 1.09952
\(682\) 0 0
\(683\) − 15317.1i − 0.858113i −0.903278 0.429057i \(-0.858846\pi\)
0.903278 0.429057i \(-0.141154\pi\)
\(684\) 0 0
\(685\) − 10470.0i − 0.583997i
\(686\) 0 0
\(687\) 14534.4 0.807167
\(688\) 0 0
\(689\) 42036.0 2.32430
\(690\) 0 0
\(691\) 9507.76i 0.523433i 0.965145 + 0.261717i \(0.0842886\pi\)
−0.965145 + 0.261717i \(0.915711\pi\)
\(692\) 0 0
\(693\) − 1960.00i − 0.107438i
\(694\) 0 0
\(695\) 6887.09 0.375888
\(696\) 0 0
\(697\) −460.000 −0.0249982
\(698\) 0 0
\(699\) − 14749.1i − 0.798086i
\(700\) 0 0
\(701\) 15830.0i 0.852911i 0.904508 + 0.426456i \(0.140238\pi\)
−0.904508 + 0.426456i \(0.859762\pi\)
\(702\) 0 0
\(703\) 22968.9 1.23227
\(704\) 0 0
\(705\) −8900.00 −0.475452
\(706\) 0 0
\(707\) 18845.6i 1.00249i
\(708\) 0 0
\(709\) − 20050.0i − 1.06205i −0.847356 0.531025i \(-0.821807\pi\)
0.847356 0.531025i \(-0.178193\pi\)
\(710\) 0 0
\(711\) −6135.77 −0.323642
\(712\) 0 0
\(713\) 29240.0 1.53583
\(714\) 0 0
\(715\) 2772.72i 0.145027i
\(716\) 0 0
\(717\) 2480.00i 0.129173i
\(718\) 0 0
\(719\) −21126.4 −1.09580 −0.547900 0.836544i \(-0.684573\pi\)
−0.547900 + 0.836544i \(0.684573\pi\)
\(720\) 0 0
\(721\) 57260.0 2.95766
\(722\) 0 0
\(723\) 23031.5i 1.18472i
\(724\) 0 0
\(725\) − 2250.00i − 0.115259i
\(726\) 0 0
\(727\) −11336.9 −0.578351 −0.289175 0.957276i \(-0.593381\pi\)
−0.289175 + 0.957276i \(0.593381\pi\)
\(728\) 0 0
\(729\) −21797.0 −1.10740
\(730\) 0 0
\(731\) − 3085.77i − 0.156131i
\(732\) 0 0
\(733\) 17198.0i 0.866607i 0.901248 + 0.433303i \(0.142652\pi\)
−0.901248 + 0.433303i \(0.857348\pi\)
\(734\) 0 0
\(735\) 14243.8 0.714815
\(736\) 0 0
\(737\) −440.000 −0.0219913
\(738\) 0 0
\(739\) 4597.36i 0.228845i 0.993432 + 0.114423i \(0.0365018\pi\)
−0.993432 + 0.114423i \(0.963498\pi\)
\(740\) 0 0
\(741\) − 29760.0i − 1.47539i
\(742\) 0 0
\(743\) −2419.43 −0.119462 −0.0597309 0.998215i \(-0.519024\pi\)
−0.0597309 + 0.998215i \(0.519024\pi\)
\(744\) 0 0
\(745\) −1670.00 −0.0821263
\(746\) 0 0
\(747\) 2660.92i 0.130332i
\(748\) 0 0
\(749\) − 47740.0i − 2.32895i
\(750\) 0 0
\(751\) −7432.69 −0.361149 −0.180574 0.983561i \(-0.557796\pi\)
−0.180574 + 0.983561i \(0.557796\pi\)
\(752\) 0 0
\(753\) 6200.00 0.300054
\(754\) 0 0
\(755\) − 15697.2i − 0.756662i
\(756\) 0 0
\(757\) 11474.0i 0.550898i 0.961316 + 0.275449i \(0.0888265\pi\)
−0.961316 + 0.275449i \(0.911174\pi\)
\(758\) 0 0
\(759\) −7692.07 −0.367858
\(760\) 0 0
\(761\) −31802.0 −1.51488 −0.757439 0.652906i \(-0.773550\pi\)
−0.757439 + 0.652906i \(0.773550\pi\)
\(762\) 0 0
\(763\) 67430.9i 3.19942i
\(764\) 0 0
\(765\) 1610.00i 0.0760911i
\(766\) 0 0
\(767\) −25509.1 −1.20089
\(768\) 0 0
\(769\) −5310.00 −0.249003 −0.124502 0.992219i \(-0.539733\pi\)
−0.124502 + 0.992219i \(0.539733\pi\)
\(770\) 0 0
\(771\) − 18630.9i − 0.870267i
\(772\) 0 0
\(773\) 37938.0i 1.76525i 0.470082 + 0.882623i \(0.344224\pi\)
−0.470082 + 0.882623i \(0.655776\pi\)
\(774\) 0 0
\(775\) −3801.32 −0.176190
\(776\) 0 0
\(777\) −29960.0 −1.38328
\(778\) 0 0
\(779\) 1073.31i 0.0493651i
\(780\) 0 0
\(781\) − 3280.00i − 0.150279i
\(782\) 0 0
\(783\) −13684.7 −0.624588
\(784\) 0 0
\(785\) −4170.00 −0.189597
\(786\) 0 0
\(787\) 37633.0i 1.70454i 0.523103 + 0.852270i \(0.324774\pi\)
−0.523103 + 0.852270i \(0.675226\pi\)
\(788\) 0 0
\(789\) 4300.00i 0.194023i
\(790\) 0 0
\(791\) 68307.4 3.07046
\(792\) 0 0
\(793\) 15500.0 0.694100
\(794\) 0 0
\(795\) − 15160.5i − 0.676338i
\(796\) 0 0
\(797\) 17526.0i 0.778924i 0.921042 + 0.389462i \(0.127339\pi\)
−0.921042 + 0.389462i \(0.872661\pi\)
\(798\) 0 0
\(799\) 18308.9 0.810667
\(800\) 0 0
\(801\) −6790.00 −0.299517
\(802\) 0 0
\(803\) − 4668.91i − 0.205183i
\(804\) 0 0
\(805\) 30100.0i 1.31787i
\(806\) 0 0
\(807\) −6681.37 −0.291444
\(808\) 0 0
\(809\) −8970.00 −0.389825 −0.194912 0.980821i \(-0.562442\pi\)
−0.194912 + 0.980821i \(0.562442\pi\)
\(810\) 0 0
\(811\) 3550.88i 0.153746i 0.997041 + 0.0768731i \(0.0244936\pi\)
−0.997041 + 0.0768731i \(0.975506\pi\)
\(812\) 0 0
\(813\) − 22440.0i − 0.968026i
\(814\) 0 0
\(815\) −15451.2 −0.664090
\(816\) 0 0
\(817\) −7200.00 −0.308318
\(818\) 0 0
\(819\) − 13586.3i − 0.579665i
\(820\) 0 0
\(821\) − 15550.0i − 0.661022i −0.943802 0.330511i \(-0.892779\pi\)
0.943802 0.330511i \(-0.107221\pi\)
\(822\) 0 0
\(823\) −26712.1 −1.13138 −0.565689 0.824619i \(-0.691390\pi\)
−0.565689 + 0.824619i \(0.691390\pi\)
\(824\) 0 0
\(825\) 1000.00 0.0422006
\(826\) 0 0
\(827\) 863.122i 0.0362923i 0.999835 + 0.0181461i \(0.00577641\pi\)
−0.999835 + 0.0181461i \(0.994224\pi\)
\(828\) 0 0
\(829\) − 19066.0i − 0.798781i −0.916781 0.399391i \(-0.869222\pi\)
0.916781 0.399391i \(-0.130778\pi\)
\(830\) 0 0
\(831\) 4498.97 0.187807
\(832\) 0 0
\(833\) −29302.0 −1.21879
\(834\) 0 0
\(835\) − 22.3607i 0 0.000926734i
\(836\) 0 0
\(837\) 23120.0i 0.954772i
\(838\) 0 0
\(839\) −47744.5 −1.96463 −0.982315 0.187238i \(-0.940047\pi\)
−0.982315 + 0.187238i \(0.940047\pi\)
\(840\) 0 0
\(841\) 16289.0 0.667883
\(842\) 0 0
\(843\) 14355.6i 0.586514i
\(844\) 0 0
\(845\) 8235.00i 0.335258i
\(846\) 0 0
\(847\) −39162.5 −1.58871
\(848\) 0 0
\(849\) 16260.0 0.657293
\(850\) 0 0
\(851\) − 41152.6i − 1.65769i
\(852\) 0 0
\(853\) − 14462.0i − 0.580503i −0.956950 0.290252i \(-0.906261\pi\)
0.956950 0.290252i \(-0.0937390\pi\)
\(854\) 0 0
\(855\) 3756.59 0.150261
\(856\) 0 0
\(857\) −29346.0 −1.16971 −0.584854 0.811138i \(-0.698848\pi\)
−0.584854 + 0.811138i \(0.698848\pi\)
\(858\) 0 0
\(859\) − 22807.9i − 0.905932i −0.891528 0.452966i \(-0.850366\pi\)
0.891528 0.452966i \(-0.149634\pi\)
\(860\) 0 0
\(861\) − 1400.00i − 0.0554145i
\(862\) 0 0
\(863\) −24753.3 −0.976375 −0.488187 0.872739i \(-0.662342\pi\)
−0.488187 + 0.872739i \(0.662342\pi\)
\(864\) 0 0
\(865\) 9190.00 0.361236
\(866\) 0 0
\(867\) − 12508.6i − 0.489981i
\(868\) 0 0
\(869\) − 7840.00i − 0.306046i
\(870\) 0 0
\(871\) −3050.00 −0.118651
\(872\) 0 0
\(873\) −6538.00 −0.253468
\(874\) 0 0
\(875\) − 3913.12i − 0.151186i
\(876\) 0 0
\(877\) 32126.0i 1.23696i 0.785799 + 0.618482i \(0.212252\pi\)
−0.785799 + 0.618482i \(0.787748\pi\)
\(878\) 0 0
\(879\) 16198.1 0.621556
\(880\) 0 0
\(881\) −33570.0 −1.28377 −0.641885 0.766801i \(-0.721848\pi\)
−0.641885 + 0.766801i \(0.721848\pi\)
\(882\) 0 0
\(883\) − 6435.40i − 0.245265i −0.992452 0.122632i \(-0.960866\pi\)
0.992452 0.122632i \(-0.0391336\pi\)
\(884\) 0 0
\(885\) 9200.00i 0.349440i
\(886\) 0 0
\(887\) −46827.7 −1.77263 −0.886314 0.463084i \(-0.846743\pi\)
−0.886314 + 0.463084i \(0.846743\pi\)
\(888\) 0 0
\(889\) 41020.0 1.54754
\(890\) 0 0
\(891\) − 4391.64i − 0.165124i
\(892\) 0 0
\(893\) − 42720.0i − 1.60086i
\(894\) 0 0
\(895\) 9212.60 0.344071
\(896\) 0 0
\(897\) −53320.0 −1.98473
\(898\) 0 0
\(899\) 13684.7i 0.507688i
\(900\) 0 0
\(901\) 31188.0i 1.15319i
\(902\) 0 0
\(903\) 9391.49 0.346101
\(904\) 0 0
\(905\) 9310.00 0.341961
\(906\) 0 0
\(907\) − 11980.9i − 0.438608i −0.975657 0.219304i \(-0.929621\pi\)
0.975657 0.219304i \(-0.0703787\pi\)
\(908\) 0 0
\(909\) − 4214.00i − 0.153762i
\(910\) 0 0
\(911\) 24194.3 0.879903 0.439951 0.898022i \(-0.354996\pi\)
0.439951 + 0.898022i \(0.354996\pi\)
\(912\) 0 0
\(913\) −3400.00 −0.123246
\(914\) 0 0
\(915\) − 5590.17i − 0.201973i
\(916\) 0 0
\(917\) 6440.00i 0.231917i
\(918\) 0 0
\(919\) −37512.3 −1.34648 −0.673240 0.739424i \(-0.735098\pi\)
−0.673240 + 0.739424i \(0.735098\pi\)
\(920\) 0 0
\(921\) −9340.00 −0.334162
\(922\) 0 0
\(923\) − 22736.3i − 0.810808i
\(924\) 0 0
\(925\) 5350.00i 0.190170i
\(926\) 0 0
\(927\) −12803.7 −0.453646
\(928\) 0 0
\(929\) −21994.0 −0.776749 −0.388374 0.921502i \(-0.626963\pi\)
−0.388374 + 0.921502i \(0.626963\pi\)
\(930\) 0 0
\(931\) 68370.0i 2.40681i
\(932\) 0 0
\(933\) − 39800.0i − 1.39656i
\(934\) 0 0
\(935\) −2057.18 −0.0719541
\(936\) 0 0
\(937\) 16286.0 0.567813 0.283906 0.958852i \(-0.408370\pi\)
0.283906 + 0.958852i \(0.408370\pi\)
\(938\) 0 0
\(939\) − 39256.4i − 1.36431i
\(940\) 0 0
\(941\) − 24302.0i − 0.841894i −0.907085 0.420947i \(-0.861698\pi\)
0.907085 0.420947i \(-0.138302\pi\)
\(942\) 0 0
\(943\) 1923.02 0.0664073
\(944\) 0 0
\(945\) −23800.0 −0.819274
\(946\) 0 0
\(947\) 19869.7i 0.681815i 0.940097 + 0.340907i \(0.110734\pi\)
−0.940097 + 0.340907i \(0.889266\pi\)
\(948\) 0 0
\(949\) − 32364.0i − 1.10704i
\(950\) 0 0
\(951\) −22566.4 −0.769470
\(952\) 0 0
\(953\) 22422.0 0.762140 0.381070 0.924546i \(-0.375556\pi\)
0.381070 + 0.924546i \(0.375556\pi\)
\(954\) 0 0
\(955\) − 10330.6i − 0.350043i
\(956\) 0 0
\(957\) − 3600.00i − 0.121600i
\(958\) 0 0
\(959\) −65552.6 −2.20730
\(960\) 0 0
\(961\) −6671.00 −0.223927
\(962\) 0 0
\(963\) 10675.0i 0.357214i
\(964\) 0 0
\(965\) − 16890.0i − 0.563428i
\(966\) 0 0
\(967\) 43777.7 1.45584 0.727920 0.685662i \(-0.240487\pi\)
0.727920 + 0.685662i \(0.240487\pi\)
\(968\) 0 0
\(969\) 22080.0 0.732004
\(970\) 0 0
\(971\) 25714.8i 0.849873i 0.905223 + 0.424936i \(0.139704\pi\)
−0.905223 + 0.424936i \(0.860296\pi\)
\(972\) 0 0
\(973\) − 43120.0i − 1.42072i
\(974\) 0 0
\(975\) 6931.81 0.227688
\(976\) 0 0
\(977\) 28986.0 0.949175 0.474588 0.880208i \(-0.342597\pi\)
0.474588 + 0.880208i \(0.342597\pi\)
\(978\) 0 0
\(979\) − 8675.94i − 0.283232i
\(980\) 0 0
\(981\) − 15078.0i − 0.490727i
\(982\) 0 0
\(983\) 32123.4 1.04229 0.521147 0.853467i \(-0.325504\pi\)
0.521147 + 0.853467i \(0.325504\pi\)
\(984\) 0 0
\(985\) 330.000 0.0106748
\(986\) 0 0
\(987\) 55722.8i 1.79704i
\(988\) 0 0
\(989\) 12900.0i 0.414758i
\(990\) 0 0
\(991\) 11994.3 0.384471 0.192235 0.981349i \(-0.438426\pi\)
0.192235 + 0.981349i \(0.438426\pi\)
\(992\) 0 0
\(993\) −1400.00 −0.0447408
\(994\) 0 0
\(995\) 6082.10i 0.193785i
\(996\) 0 0
\(997\) − 406.000i − 0.0128968i −0.999979 0.00644842i \(-0.997947\pi\)
0.999979 0.00644842i \(-0.00205261\pi\)
\(998\) 0 0
\(999\) 32539.3 1.03053
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 1280.4.d.v.641.3 4
4.3 odd 2 inner 1280.4.d.v.641.1 4
8.3 odd 2 inner 1280.4.d.v.641.4 4
8.5 even 2 inner 1280.4.d.v.641.2 4
16.3 odd 4 160.4.a.d.1.2 yes 2
16.5 even 4 320.4.a.q.1.2 2
16.11 odd 4 320.4.a.q.1.1 2
16.13 even 4 160.4.a.d.1.1 2
48.29 odd 4 1440.4.a.bb.1.2 2
48.35 even 4 1440.4.a.bb.1.1 2
80.3 even 4 800.4.c.l.449.3 4
80.13 odd 4 800.4.c.l.449.2 4
80.19 odd 4 800.4.a.o.1.1 2
80.29 even 4 800.4.a.o.1.2 2
80.59 odd 4 1600.4.a.cg.1.2 2
80.67 even 4 800.4.c.l.449.1 4
80.69 even 4 1600.4.a.cg.1.1 2
80.77 odd 4 800.4.c.l.449.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.d.1.1 2 16.13 even 4
160.4.a.d.1.2 yes 2 16.3 odd 4
320.4.a.q.1.1 2 16.11 odd 4
320.4.a.q.1.2 2 16.5 even 4
800.4.a.o.1.1 2 80.19 odd 4
800.4.a.o.1.2 2 80.29 even 4
800.4.c.l.449.1 4 80.67 even 4
800.4.c.l.449.2 4 80.13 odd 4
800.4.c.l.449.3 4 80.3 even 4
800.4.c.l.449.4 4 80.77 odd 4
1280.4.d.v.641.1 4 4.3 odd 2 inner
1280.4.d.v.641.2 4 8.5 even 2 inner
1280.4.d.v.641.3 4 1.1 even 1 trivial
1280.4.d.v.641.4 4 8.3 odd 2 inner
1440.4.a.bb.1.1 2 48.35 even 4
1440.4.a.bb.1.2 2 48.29 odd 4
1600.4.a.cg.1.1 2 80.69 even 4
1600.4.a.cg.1.2 2 80.59 odd 4