Properties

Label 1280.4.d.a.641.1
Level $1280$
Weight $4$
Character 1280.641
Analytic conductor $75.522$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.4.d.a.641.2

$q$-expansion

\(f(q)\) \(=\) \(q-6.00000i q^{3} -5.00000i q^{5} -34.0000 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-6.00000i q^{3} -5.00000i q^{5} -34.0000 q^{7} -9.00000 q^{9} -16.0000i q^{11} -58.0000i q^{13} -30.0000 q^{15} -70.0000 q^{17} +4.00000i q^{19} +204.000i q^{21} -134.000 q^{23} -25.0000 q^{25} -108.000i q^{27} +242.000i q^{29} -100.000 q^{31} -96.0000 q^{33} +170.000i q^{35} -438.000i q^{37} -348.000 q^{39} +138.000 q^{41} -178.000i q^{43} +45.0000i q^{45} -22.0000 q^{47} +813.000 q^{49} +420.000i q^{51} +162.000i q^{53} -80.0000 q^{55} +24.0000 q^{57} +268.000i q^{59} -250.000i q^{61} +306.000 q^{63} -290.000 q^{65} +422.000i q^{67} +804.000i q^{69} -852.000 q^{71} -306.000 q^{73} +150.000i q^{75} +544.000i q^{77} +456.000 q^{79} -891.000 q^{81} +434.000i q^{83} +350.000i q^{85} +1452.00 q^{87} +726.000 q^{89} +1972.00i q^{91} +600.000i q^{93} +20.0000 q^{95} +1378.00 q^{97} +144.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 68 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 68 q^{7} - 18 q^{9} - 60 q^{15} - 140 q^{17} - 268 q^{23} - 50 q^{25} - 200 q^{31} - 192 q^{33} - 696 q^{39} + 276 q^{41} - 44 q^{47} + 1626 q^{49} - 160 q^{55} + 48 q^{57} + 612 q^{63} - 580 q^{65} - 1704 q^{71} - 612 q^{73} + 912 q^{79} - 1782 q^{81} + 2904 q^{87} + 1452 q^{89} + 40 q^{95} + 2756 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) − 6.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) − 5.00000i − 0.447214i
\(6\) 0 0
\(7\) −34.0000 −1.83583 −0.917914 0.396780i \(-0.870128\pi\)
−0.917914 + 0.396780i \(0.870128\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) − 16.0000i − 0.438562i −0.975662 0.219281i \(-0.929629\pi\)
0.975662 0.219281i \(-0.0703711\pi\)
\(12\) 0 0
\(13\) − 58.0000i − 1.23741i −0.785624 0.618704i \(-0.787658\pi\)
0.785624 0.618704i \(-0.212342\pi\)
\(14\) 0 0
\(15\) −30.0000 −0.516398
\(16\) 0 0
\(17\) −70.0000 −0.998676 −0.499338 0.866407i \(-0.666423\pi\)
−0.499338 + 0.866407i \(0.666423\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.0482980i 0.999708 + 0.0241490i \(0.00768762\pi\)
−0.999708 + 0.0241490i \(0.992312\pi\)
\(20\) 0 0
\(21\) 204.000i 2.11983i
\(22\) 0 0
\(23\) −134.000 −1.21482 −0.607412 0.794387i \(-0.707792\pi\)
−0.607412 + 0.794387i \(0.707792\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) − 108.000i − 0.769800i
\(28\) 0 0
\(29\) 242.000i 1.54960i 0.632209 + 0.774798i \(0.282148\pi\)
−0.632209 + 0.774798i \(0.717852\pi\)
\(30\) 0 0
\(31\) −100.000 −0.579372 −0.289686 0.957122i \(-0.593551\pi\)
−0.289686 + 0.957122i \(0.593551\pi\)
\(32\) 0 0
\(33\) −96.0000 −0.506408
\(34\) 0 0
\(35\) 170.000i 0.821007i
\(36\) 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 \(-0.415344\pi\)
0.262829 + 0.964842i \(0.415344\pi\)
\(42\) 0 0
\(43\) − 178.000i − 0.631273i −0.948880 0.315637i \(-0.897782\pi\)
0.948880 0.315637i \(-0.102218\pi\)
\(44\) 0 0
\(45\) 45.0000i 0.149071i
\(46\) 0 0
\(47\) −22.0000 −0.0682772 −0.0341386 0.999417i \(-0.510869\pi\)
−0.0341386 + 0.999417i \(0.510869\pi\)
\(48\) 0 0
\(49\) 813.000 2.37026
\(50\) 0 0
\(51\) 420.000i 1.15317i
\(52\) 0 0
\(53\) 162.000i 0.419857i 0.977717 + 0.209928i \(0.0673231\pi\)
−0.977717 + 0.209928i \(0.932677\pi\)
\(54\) 0 0
\(55\) −80.0000 −0.196131
\(56\) 0 0
\(57\) 24.0000 0.0557698
\(58\) 0 0
\(59\) 268.000i 0.591367i 0.955286 + 0.295683i \(0.0955473\pi\)
−0.955286 + 0.295683i \(0.904453\pi\)
\(60\) 0 0
\(61\) − 250.000i − 0.524741i −0.964967 0.262371i \(-0.915496\pi\)
0.964967 0.262371i \(-0.0845043\pi\)
\(62\) 0 0
\(63\) 306.000 0.611942
\(64\) 0 0
\(65\) −290.000 −0.553386
\(66\) 0 0
\(67\) 422.000i 0.769485i 0.923024 + 0.384743i \(0.125710\pi\)
−0.923024 + 0.384743i \(0.874290\pi\)
\(68\) 0 0
\(69\) 804.000i 1.40276i
\(70\) 0 0
\(71\) −852.000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −306.000 −0.490611 −0.245305 0.969446i \(-0.578888\pi\)
−0.245305 + 0.969446i \(0.578888\pi\)
\(74\) 0 0
\(75\) 150.000i 0.230940i
\(76\) 0 0
\(77\) 544.000i 0.805124i
\(78\) 0 0
\(79\) 456.000 0.649418 0.324709 0.945814i \(-0.394734\pi\)
0.324709 + 0.945814i \(0.394734\pi\)
\(80\) 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
\(85\) 350.000i 0.446622i
\(86\) 0 0
\(87\) 1452.00 1.78932
\(88\) 0 0
\(89\) 726.000 0.864672 0.432336 0.901712i \(-0.357689\pi\)
0.432336 + 0.901712i \(0.357689\pi\)
\(90\) 0 0
\(91\) 1972.00i 2.27167i
\(92\) 0 0
\(93\) 600.000i 0.669001i
\(94\) 0 0
\(95\) 20.0000 0.0215995
\(96\) 0 0
\(97\) 1378.00 1.44242 0.721210 0.692717i \(-0.243586\pi\)
0.721210 + 0.692717i \(0.243586\pi\)
\(98\) 0 0
\(99\) 144.000i 0.146187i
\(100\) 0 0
\(101\) 126.000i 0.124133i 0.998072 + 0.0620667i \(0.0197691\pi\)
−0.998072 + 0.0620667i \(0.980231\pi\)
\(102\) 0 0
\(103\) −1262.00 −1.20727 −0.603634 0.797262i \(-0.706281\pi\)
−0.603634 + 0.797262i \(0.706281\pi\)
\(104\) 0 0
\(105\) 1020.00 0.948017
\(106\) 0 0
\(107\) − 510.000i − 0.460781i −0.973098 0.230390i \(-0.926000\pi\)
0.973098 0.230390i \(-0.0740003\pi\)
\(108\) 0 0
\(109\) − 26.0000i − 0.0228472i −0.999935 0.0114236i \(-0.996364\pi\)
0.999935 0.0114236i \(-0.00363633\pi\)
\(110\) 0 0
\(111\) −2628.00 −2.24720
\(112\) 0 0
\(113\) 1242.00 1.03396 0.516980 0.855997i \(-0.327056\pi\)
0.516980 + 0.855997i \(0.327056\pi\)
\(114\) 0 0
\(115\) 670.000i 0.543285i
\(116\) 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
\(125\) 125.000i 0.0894427i
\(126\) 0 0
\(127\) 978.000 0.683334 0.341667 0.939821i \(-0.389008\pi\)
0.341667 + 0.939821i \(0.389008\pi\)
\(128\) 0 0
\(129\) −1068.00 −0.728931
\(130\) 0 0
\(131\) − 912.000i − 0.608258i −0.952631 0.304129i \(-0.901635\pi\)
0.952631 0.304129i \(-0.0983654\pi\)
\(132\) 0 0
\(133\) − 136.000i − 0.0886669i
\(134\) 0 0
\(135\) −540.000 −0.344265
\(136\) 0 0
\(137\) 926.000 0.577471 0.288735 0.957409i \(-0.406765\pi\)
0.288735 + 0.957409i \(0.406765\pi\)
\(138\) 0 0
\(139\) − 516.000i − 0.314867i −0.987530 0.157434i \(-0.949678\pi\)
0.987530 0.157434i \(-0.0503220\pi\)
\(140\) 0 0
\(141\) 132.000i 0.0788398i
\(142\) 0 0
\(143\) −928.000 −0.542680
\(144\) 0 0
\(145\) 1210.00 0.693000
\(146\) 0 0
\(147\) − 4878.00i − 2.73694i
\(148\) 0 0
\(149\) − 958.000i − 0.526728i −0.964697 0.263364i \(-0.915168\pi\)
0.964697 0.263364i \(-0.0848320\pi\)
\(150\) 0 0
\(151\) 332.000 0.178926 0.0894628 0.995990i \(-0.471485\pi\)
0.0894628 + 0.995990i \(0.471485\pi\)
\(152\) 0 0
\(153\) 630.000 0.332892
\(154\) 0 0
\(155\) 500.000i 0.259103i
\(156\) 0 0
\(157\) 1022.00i 0.519519i 0.965673 + 0.259759i \(0.0836433\pi\)
−0.965673 + 0.259759i \(0.916357\pi\)
\(158\) 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
\(165\) 480.000i 0.226472i
\(166\) 0 0
\(167\) 654.000 0.303042 0.151521 0.988454i \(-0.451583\pi\)
0.151521 + 0.988454i \(0.451583\pi\)
\(168\) 0 0
\(169\) −1167.00 −0.531179
\(170\) 0 0
\(171\) − 36.0000i − 0.0160993i
\(172\) 0 0
\(173\) 1294.00i 0.568676i 0.958724 + 0.284338i \(0.0917738\pi\)
−0.958724 + 0.284338i \(0.908226\pi\)
\(174\) 0 0
\(175\) 850.000 0.367165
\(176\) 0 0
\(177\) 1608.00 0.682851
\(178\) 0 0
\(179\) − 2836.00i − 1.18420i −0.805863 0.592102i \(-0.798298\pi\)
0.805863 0.592102i \(-0.201702\pi\)
\(180\) 0 0
\(181\) 1742.00i 0.715369i 0.933842 + 0.357685i \(0.116434\pi\)
−0.933842 + 0.357685i \(0.883566\pi\)
\(182\) 0 0
\(183\) −1500.00 −0.605919
\(184\) 0 0
\(185\) −2190.00 −0.870335
\(186\) 0 0
\(187\) 1120.00i 0.437981i
\(188\) 0 0
\(189\) 3672.00i 1.41322i
\(190\) 0 0
\(191\) −4460.00 −1.68960 −0.844802 0.535079i \(-0.820282\pi\)
−0.844802 + 0.535079i \(0.820282\pi\)
\(192\) 0 0
\(193\) −3782.00 −1.41054 −0.705270 0.708939i \(-0.749174\pi\)
−0.705270 + 0.708939i \(0.749174\pi\)
\(194\) 0 0
\(195\) 1740.00i 0.638995i
\(196\) 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
\(205\) − 690.000i − 0.235081i
\(206\) 0 0
\(207\) 1206.00 0.404941
\(208\) 0 0
\(209\) 64.0000 0.0211817
\(210\) 0 0
\(211\) − 256.000i − 0.0835250i −0.999128 0.0417625i \(-0.986703\pi\)
0.999128 0.0417625i \(-0.0132973\pi\)
\(212\) 0 0
\(213\) 5112.00i 1.64445i
\(214\) 0 0
\(215\) −890.000 −0.282314
\(216\) 0 0
\(217\) 3400.00 1.06363
\(218\) 0 0
\(219\) 1836.00i 0.566509i
\(220\) 0 0
\(221\) 4060.00i 1.23577i
\(222\) 0 0
\(223\) 5158.00 1.54890 0.774451 0.632634i \(-0.218026\pi\)
0.774451 + 0.632634i \(0.218026\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) − 2226.00i − 0.650858i −0.945566 0.325429i \(-0.894491\pi\)
0.945566 0.325429i \(-0.105509\pi\)
\(228\) 0 0
\(229\) 2086.00i 0.601951i 0.953632 + 0.300975i \(0.0973122\pi\)
−0.953632 + 0.300975i \(0.902688\pi\)
\(230\) 0 0
\(231\) 3264.00 0.929677
\(232\) 0 0
\(233\) 5718.00 1.60772 0.803860 0.594819i \(-0.202776\pi\)
0.803860 + 0.594819i \(0.202776\pi\)
\(234\) 0 0
\(235\) 110.000i 0.0305345i
\(236\) 0 0
\(237\) − 2736.00i − 0.749883i
\(238\) 0 0
\(239\) 3624.00 0.980825 0.490412 0.871491i \(-0.336846\pi\)
0.490412 + 0.871491i \(0.336846\pi\)
\(240\) 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
\(245\) − 4065.00i − 1.06001i
\(246\) 0 0
\(247\) 232.000 0.0597644
\(248\) 0 0
\(249\) 2604.00 0.662738
\(250\) 0 0
\(251\) 5040.00i 1.26742i 0.773571 + 0.633709i \(0.218468\pi\)
−0.773571 + 0.633709i \(0.781532\pi\)
\(252\) 0 0
\(253\) 2144.00i 0.532775i
\(254\) 0 0
\(255\) 2100.00 0.515714
\(256\) 0 0
\(257\) −2310.00 −0.560676 −0.280338 0.959901i \(-0.590447\pi\)
−0.280338 + 0.959901i \(0.590447\pi\)
\(258\) 0 0
\(259\) 14892.0i 3.57276i
\(260\) 0 0
\(261\) − 2178.00i − 0.516532i
\(262\) 0 0
\(263\) −4110.00 −0.963625 −0.481813 0.876274i \(-0.660021\pi\)
−0.481813 + 0.876274i \(0.660021\pi\)
\(264\) 0 0
\(265\) 810.000 0.187766
\(266\) 0 0
\(267\) − 4356.00i − 0.998438i
\(268\) 0 0
\(269\) − 746.000i − 0.169087i −0.996420 0.0845435i \(-0.973057\pi\)
0.996420 0.0845435i \(-0.0269432\pi\)
\(270\) 0 0
\(271\) 4596.00 1.03021 0.515105 0.857127i \(-0.327753\pi\)
0.515105 + 0.857127i \(0.327753\pi\)
\(272\) 0 0
\(273\) 11832.0 2.62310
\(274\) 0 0
\(275\) 400.000i 0.0877124i
\(276\) 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 \(-0.841582\pi\)
−0.878691 + 0.477392i \(0.841582\pi\)
\(282\) 0 0
\(283\) − 1178.00i − 0.247438i −0.992317 0.123719i \(-0.960518\pi\)
0.992317 0.123719i \(-0.0394821\pi\)
\(284\) 0 0
\(285\) − 120.000i − 0.0249410i
\(286\) 0 0
\(287\) −4692.00 −0.965017
\(288\) 0 0
\(289\) −13.0000 −0.00264604
\(290\) 0 0
\(291\) − 8268.00i − 1.66556i
\(292\) 0 0
\(293\) 106.000i 0.0211351i 0.999944 + 0.0105676i \(0.00336382\pi\)
−0.999944 + 0.0105676i \(0.996636\pi\)
\(294\) 0 0
\(295\) 1340.00 0.264467
\(296\) 0 0
\(297\) −1728.00 −0.337605
\(298\) 0 0
\(299\) 7772.00i 1.50323i
\(300\) 0 0
\(301\) 6052.00i 1.15891i
\(302\) 0 0
\(303\) 756.000 0.143337
\(304\) 0 0
\(305\) −1250.00 −0.234671
\(306\) 0 0
\(307\) 8134.00i 1.51216i 0.654482 + 0.756078i \(0.272887\pi\)
−0.654482 + 0.756078i \(0.727113\pi\)
\(308\) 0 0
\(309\) 7572.00i 1.39403i
\(310\) 0 0
\(311\) −4396.00 −0.801525 −0.400763 0.916182i \(-0.631255\pi\)
−0.400763 + 0.916182i \(0.631255\pi\)
\(312\) 0 0
\(313\) −4826.00 −0.871507 −0.435753 0.900066i \(-0.643518\pi\)
−0.435753 + 0.900066i \(0.643518\pi\)
\(314\) 0 0
\(315\) − 1530.00i − 0.273669i
\(316\) 0 0
\(317\) − 7026.00i − 1.24486i −0.782677 0.622428i \(-0.786146\pi\)
0.782677 0.622428i \(-0.213854\pi\)
\(318\) 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
\(325\) 1450.00i 0.247482i
\(326\) 0 0
\(327\) −156.000 −0.0263817
\(328\) 0 0
\(329\) 748.000 0.125345
\(330\) 0 0
\(331\) − 8808.00i − 1.46263i −0.682038 0.731316i \(-0.738906\pi\)
0.682038 0.731316i \(-0.261094\pi\)
\(332\) 0 0
\(333\) 3942.00i 0.648710i
\(334\) 0 0
\(335\) 2110.00 0.344124
\(336\) 0 0
\(337\) 5602.00 0.905520 0.452760 0.891632i \(-0.350439\pi\)
0.452760 + 0.891632i \(0.350439\pi\)
\(338\) 0 0
\(339\) − 7452.00i − 1.19391i
\(340\) 0 0
\(341\) 1600.00i 0.254090i
\(342\) 0 0
\(343\) −15980.0 −2.51557
\(344\) 0 0
\(345\) 4020.00 0.627332
\(346\) 0 0
\(347\) 6634.00i 1.02632i 0.858294 + 0.513158i \(0.171525\pi\)
−0.858294 + 0.513158i \(0.828475\pi\)
\(348\) 0 0
\(349\) − 3198.00i − 0.490501i −0.969460 0.245251i \(-0.921130\pi\)
0.969460 0.245251i \(-0.0788703\pi\)
\(350\) 0 0
\(351\) −6264.00 −0.952557
\(352\) 0 0
\(353\) −5230.00 −0.788569 −0.394284 0.918988i \(-0.629008\pi\)
−0.394284 + 0.918988i \(0.629008\pi\)
\(354\) 0 0
\(355\) 4260.00i 0.636894i
\(356\) 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
\(365\) 1530.00i 0.219408i
\(366\) 0 0
\(367\) −10790.0 −1.53470 −0.767348 0.641231i \(-0.778424\pi\)
−0.767348 + 0.641231i \(0.778424\pi\)
\(368\) 0 0
\(369\) −1242.00 −0.175219
\(370\) 0 0
\(371\) − 5508.00i − 0.770785i
\(372\) 0 0
\(373\) − 4190.00i − 0.581635i −0.956778 0.290818i \(-0.906073\pi\)
0.956778 0.290818i \(-0.0939273\pi\)
\(374\) 0 0
\(375\) 750.000 0.103280
\(376\) 0 0
\(377\) 14036.0 1.91748
\(378\) 0 0
\(379\) 6980.00i 0.946012i 0.881059 + 0.473006i \(0.156831\pi\)
−0.881059 + 0.473006i \(0.843169\pi\)
\(380\) 0 0
\(381\) − 5868.00i − 0.789047i
\(382\) 0 0
\(383\) −13962.0 −1.86273 −0.931364 0.364089i \(-0.881380\pi\)
−0.931364 + 0.364089i \(0.881380\pi\)
\(384\) 0 0
\(385\) 2720.00 0.360062
\(386\) 0 0
\(387\) 1602.00i 0.210424i
\(388\) 0 0
\(389\) 3810.00i 0.496593i 0.968684 + 0.248296i \(0.0798707\pi\)
−0.968684 + 0.248296i \(0.920129\pi\)
\(390\) 0 0
\(391\) 9380.00 1.21321
\(392\) 0 0
\(393\) −5472.00 −0.702356
\(394\) 0 0
\(395\) − 2280.00i − 0.290428i
\(396\) 0 0
\(397\) 9158.00i 1.15775i 0.815416 + 0.578875i \(0.196508\pi\)
−0.815416 + 0.578875i \(0.803492\pi\)
\(398\) 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
\(405\) 4455.00i 0.546594i
\(406\) 0 0
\(407\) −7008.00 −0.853498
\(408\) 0 0
\(409\) −13486.0 −1.63042 −0.815208 0.579169i \(-0.803377\pi\)
−0.815208 + 0.579169i \(0.803377\pi\)
\(410\) 0 0
\(411\) − 5556.00i − 0.666806i
\(412\) 0 0
\(413\) − 9112.00i − 1.08565i
\(414\) 0 0
\(415\) 2170.00 0.256677
\(416\) 0 0
\(417\) −3096.00 −0.363577
\(418\) 0 0
\(419\) 5628.00i 0.656195i 0.944644 + 0.328098i \(0.106407\pi\)
−0.944644 + 0.328098i \(0.893593\pi\)
\(420\) 0 0
\(421\) 7938.00i 0.918942i 0.888193 + 0.459471i \(0.151961\pi\)
−0.888193 + 0.459471i \(0.848039\pi\)
\(422\) 0 0
\(423\) 198.000 0.0227591
\(424\) 0 0
\(425\) 1750.00 0.199735
\(426\) 0 0
\(427\) 8500.00i 0.963334i
\(428\) 0 0
\(429\) 5568.00i 0.626633i
\(430\) 0 0
\(431\) −1916.00 −0.214131 −0.107066 0.994252i \(-0.534145\pi\)
−0.107066 + 0.994252i \(0.534145\pi\)
\(432\) 0 0
\(433\) −16510.0 −1.83238 −0.916189 0.400746i \(-0.868751\pi\)
−0.916189 + 0.400746i \(0.868751\pi\)
\(434\) 0 0
\(435\) − 7260.00i − 0.800208i
\(436\) 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 \(0.227500\pi\)
−0.755282 + 0.655400i \(0.772500\pi\)
\(444\) 0 0
\(445\) − 3630.00i − 0.386693i
\(446\) 0 0
\(447\) −5748.00 −0.608213
\(448\) 0 0
\(449\) −5946.00 −0.624965 −0.312482 0.949924i \(-0.601160\pi\)
−0.312482 + 0.949924i \(0.601160\pi\)
\(450\) 0 0
\(451\) − 2208.00i − 0.230534i
\(452\) 0 0
\(453\) − 1992.00i − 0.206606i
\(454\) 0 0
\(455\) 9860.00 1.01592
\(456\) 0 0
\(457\) −1258.00 −0.128768 −0.0643838 0.997925i \(-0.520508\pi\)
−0.0643838 + 0.997925i \(0.520508\pi\)
\(458\) 0 0
\(459\) 7560.00i 0.768781i
\(460\) 0 0
\(461\) − 16422.0i − 1.65911i −0.558426 0.829554i \(-0.688595\pi\)
0.558426 0.829554i \(-0.311405\pi\)
\(462\) 0 0
\(463\) −2658.00 −0.266799 −0.133399 0.991062i \(-0.542589\pi\)
−0.133399 + 0.991062i \(0.542589\pi\)
\(464\) 0 0
\(465\) 3000.00 0.299186
\(466\) 0 0
\(467\) 3686.00i 0.365241i 0.983183 + 0.182621i \(0.0584580\pi\)
−0.983183 + 0.182621i \(0.941542\pi\)
\(468\) 0 0
\(469\) − 14348.0i − 1.41264i
\(470\) 0 0
\(471\) 6132.00 0.599889
\(472\) 0 0
\(473\) −2848.00 −0.276852
\(474\) 0 0
\(475\) − 100.000i − 0.00965961i
\(476\) 0 0
\(477\) − 1458.00i − 0.139952i
\(478\) 0 0
\(479\) −88.0000 −0.00839420 −0.00419710 0.999991i \(-0.501336\pi\)
−0.00419710 + 0.999991i \(0.501336\pi\)
\(480\) 0 0
\(481\) −25404.0 −2.40816
\(482\) 0 0
\(483\) − 27336.0i − 2.57522i
\(484\) 0 0
\(485\) − 6890.00i − 0.645070i
\(486\) 0 0
\(487\) −14714.0 −1.36911 −0.684553 0.728963i \(-0.740003\pi\)
−0.684553 + 0.728963i \(0.740003\pi\)
\(488\) 0 0
\(489\) −5556.00 −0.513806
\(490\) 0 0
\(491\) 7344.00i 0.675010i 0.941324 + 0.337505i \(0.109583\pi\)
−0.941324 + 0.337505i \(0.890417\pi\)
\(492\) 0 0
\(493\) − 16940.0i − 1.54754i
\(494\) 0 0
\(495\) 720.000 0.0653770
\(496\) 0 0
\(497\) 28968.0 2.61447
\(498\) 0 0
\(499\) 1604.00i 0.143898i 0.997408 + 0.0719488i \(0.0229218\pi\)
−0.997408 + 0.0719488i \(0.977078\pi\)
\(500\) 0 0
\(501\) − 3924.00i − 0.349923i
\(502\) 0 0
\(503\) 14802.0 1.31210 0.656052 0.754715i \(-0.272225\pi\)
0.656052 + 0.754715i \(0.272225\pi\)
\(504\) 0 0
\(505\) 630.000 0.0555141
\(506\) 0 0
\(507\) 7002.00i 0.613353i
\(508\) 0 0
\(509\) 22514.0i 1.96054i 0.197660 + 0.980271i \(0.436666\pi\)
−0.197660 + 0.980271i \(0.563334\pi\)
\(510\) 0 0
\(511\) 10404.0 0.900677
\(512\) 0 0
\(513\) 432.000 0.0371799
\(514\) 0 0
\(515\) 6310.00i 0.539906i
\(516\) 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 \(-0.408962\pi\)
0.282121 + 0.959379i \(0.408962\pi\)
\(522\) 0 0
\(523\) − 7930.00i − 0.663011i −0.943453 0.331505i \(-0.892443\pi\)
0.943453 0.331505i \(-0.107557\pi\)
\(524\) 0 0
\(525\) − 5100.00i − 0.423966i
\(526\) 0 0
\(527\) 7000.00 0.578605
\(528\) 0 0
\(529\) 5789.00 0.475795
\(530\) 0 0
\(531\) − 2412.00i − 0.197122i
\(532\) 0 0
\(533\) − 8004.00i − 0.650454i
\(534\) 0 0
\(535\) −2550.00 −0.206068
\(536\) 0 0
\(537\) −17016.0 −1.36740
\(538\) 0 0
\(539\) − 13008.0i − 1.03951i
\(540\) 0 0
\(541\) − 4918.00i − 0.390834i −0.980720 0.195417i \(-0.937394\pi\)
0.980720 0.195417i \(-0.0626061\pi\)
\(542\) 0 0
\(543\) 10452.0 0.826037
\(544\) 0 0
\(545\) −130.000 −0.0102176
\(546\) 0 0
\(547\) − 3922.00i − 0.306568i −0.988182 0.153284i \(-0.951015\pi\)
0.988182 0.153284i \(-0.0489849\pi\)
\(548\) 0 0
\(549\) 2250.00i 0.174914i
\(550\) 0 0
\(551\) −968.000 −0.0748424
\(552\) 0 0
\(553\) −15504.0 −1.19222
\(554\) 0 0
\(555\) 13140.0i 1.00498i
\(556\) 0 0
\(557\) − 17786.0i − 1.35299i −0.736446 0.676496i \(-0.763497\pi\)
0.736446 0.676496i \(-0.236503\pi\)
\(558\) 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
\(565\) − 6210.00i − 0.462401i
\(566\) 0 0
\(567\) 30294.0 2.24379
\(568\) 0 0
\(569\) −13358.0 −0.984177 −0.492088 0.870545i \(-0.663766\pi\)
−0.492088 + 0.870545i \(0.663766\pi\)
\(570\) 0 0
\(571\) − 16360.0i − 1.19903i −0.800364 0.599514i \(-0.795361\pi\)
0.800364 0.599514i \(-0.204639\pi\)
\(572\) 0 0
\(573\) 26760.0i 1.95099i
\(574\) 0 0
\(575\) 3350.00 0.242965
\(576\) 0 0
\(577\) −15574.0 −1.12366 −0.561832 0.827251i \(-0.689903\pi\)
−0.561832 + 0.827251i \(0.689903\pi\)
\(578\) 0 0
\(579\) 22692.0i 1.62875i
\(580\) 0 0
\(581\) − 14756.0i − 1.05367i
\(582\) 0 0
\(583\) 2592.00 0.184133
\(584\) 0 0
\(585\) 2610.00 0.184462
\(586\) 0 0
\(587\) − 6654.00i − 0.467870i −0.972252 0.233935i \(-0.924840\pi\)
0.972252 0.233935i \(-0.0751604\pi\)
\(588\) 0 0
\(589\) − 400.000i − 0.0279825i
\(590\) 0 0
\(591\) 26844.0 1.86838
\(592\) 0 0
\(593\) −17742.0 −1.22863 −0.614314 0.789062i \(-0.710567\pi\)
−0.614314 + 0.789062i \(0.710567\pi\)
\(594\) 0 0
\(595\) − 11900.0i − 0.819920i
\(596\) 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 \(-0.467516\pi\)
0.101875 + 0.994797i \(0.467516\pi\)
\(602\) 0 0
\(603\) − 3798.00i − 0.256495i
\(604\) 0 0
\(605\) − 5375.00i − 0.361198i
\(606\) 0 0
\(607\) 23610.0 1.57875 0.789374 0.613912i \(-0.210405\pi\)
0.789374 + 0.613912i \(0.210405\pi\)
\(608\) 0 0
\(609\) −49368.0 −3.28488
\(610\) 0 0
\(611\) 1276.00i 0.0844868i
\(612\) 0 0
\(613\) 23850.0i 1.57144i 0.618583 + 0.785720i \(0.287707\pi\)
−0.618583 + 0.785720i \(0.712293\pi\)
\(614\) 0 0
\(615\) −4140.00 −0.271449
\(616\) 0 0
\(617\) 5334.00 0.348037 0.174018 0.984742i \(-0.444325\pi\)
0.174018 + 0.984742i \(0.444325\pi\)
\(618\) 0 0
\(619\) 2164.00i 0.140515i 0.997529 + 0.0702573i \(0.0223820\pi\)
−0.997529 + 0.0702573i \(0.977618\pi\)
\(620\) 0 0
\(621\) 14472.0i 0.935171i
\(622\) 0 0
\(623\) −24684.0 −1.58739
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) − 384.000i − 0.0244585i
\(628\) 0 0
\(629\) 30660.0i 1.94355i
\(630\) 0 0
\(631\) −25220.0 −1.59111 −0.795557 0.605879i \(-0.792821\pi\)
−0.795557 + 0.605879i \(0.792821\pi\)
\(632\) 0 0
\(633\) −1536.00 −0.0964463
\(634\) 0 0
\(635\) − 4890.00i − 0.305596i
\(636\) 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
\(645\) 5340.00i 0.325988i
\(646\) 0 0
\(647\) −21834.0 −1.32671 −0.663356 0.748304i \(-0.730869\pi\)
−0.663356 + 0.748304i \(0.730869\pi\)
\(648\) 0 0
\(649\) 4288.00 0.259351
\(650\) 0 0
\(651\) − 20400.0i − 1.22817i
\(652\) 0 0
\(653\) 23998.0i 1.43815i 0.694931 + 0.719077i \(0.255435\pi\)
−0.694931 + 0.719077i \(0.744565\pi\)
\(654\) 0 0
\(655\) −4560.00 −0.272021
\(656\) 0 0
\(657\) 2754.00 0.163537
\(658\) 0 0
\(659\) − 32004.0i − 1.89180i −0.324452 0.945902i \(-0.605180\pi\)
0.324452 0.945902i \(-0.394820\pi\)
\(660\) 0 0
\(661\) − 8526.00i − 0.501699i −0.968026 0.250849i \(-0.919290\pi\)
0.968026 0.250849i \(-0.0807099\pi\)
\(662\) 0 0
\(663\) 24360.0 1.42694
\(664\) 0 0
\(665\) −680.000 −0.0396530
\(666\) 0 0
\(667\) − 32428.0i − 1.88248i
\(668\) 0 0
\(669\) − 30948.0i − 1.78852i
\(670\) 0 0
\(671\) −4000.00 −0.230132
\(672\) 0 0
\(673\) 8178.00 0.468408 0.234204 0.972187i \(-0.424752\pi\)
0.234204 + 0.972187i \(0.424752\pi\)
\(674\) 0 0
\(675\) 2700.00i 0.153960i
\(676\) 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 \(0.216433\pi\)
−0.777608 + 0.628750i \(0.783567\pi\)
\(684\) 0 0
\(685\) − 4630.00i − 0.258253i
\(686\) 0 0
\(687\) 12516.0 0.695073
\(688\) 0 0
\(689\) 9396.00 0.519534
\(690\) 0 0
\(691\) 35336.0i 1.94536i 0.232147 + 0.972681i \(0.425425\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(692\) 0 0
\(693\) − 4896.00i − 0.268375i
\(694\) 0 0
\(695\) −2580.00 −0.140813
\(696\) 0 0
\(697\) −9660.00 −0.524962
\(698\) 0 0
\(699\) − 34308.0i − 1.85643i
\(700\) 0 0
\(701\) − 3482.00i − 0.187608i −0.995591 0.0938041i \(-0.970097\pi\)
0.995591 0.0938041i \(-0.0299027\pi\)
\(702\) 0 0
\(703\) 1752.00 0.0939942
\(704\) 0 0
\(705\) 660.000 0.0352582
\(706\) 0 0
\(707\) − 4284.00i − 0.227887i
\(708\) 0 0
\(709\) − 19402.0i − 1.02773i −0.857872 0.513863i \(-0.828214\pi\)
0.857872 0.513863i \(-0.171786\pi\)
\(710\) 0 0
\(711\) −4104.00 −0.216473
\(712\) 0 0
\(713\) 13400.0 0.703834
\(714\) 0 0
\(715\) 4640.00i 0.242694i
\(716\) 0 0
\(717\) − 21744.0i − 1.13256i
\(718\) 0 0
\(719\) 9896.00 0.513294 0.256647 0.966505i \(-0.417382\pi\)
0.256647 + 0.966505i \(0.417382\pi\)
\(720\) 0 0
\(721\) 42908.0 2.21633
\(722\) 0 0
\(723\) 492.000i 0.0253080i
\(724\) 0 0
\(725\) − 6050.00i − 0.309919i
\(726\) 0 0
\(727\) 494.000 0.0252014 0.0126007 0.999921i \(-0.495989\pi\)
0.0126007 + 0.999921i \(0.495989\pi\)
\(728\) 0 0
\(729\) −9477.00 −0.481481
\(730\) 0 0
\(731\) 12460.0i 0.630437i
\(732\) 0 0
\(733\) − 9282.00i − 0.467720i −0.972270 0.233860i \(-0.924864\pi\)
0.972270 0.233860i \(-0.0751357\pi\)
\(734\) 0 0
\(735\) −24390.0 −1.22400
\(736\) 0 0
\(737\) 6752.00 0.337467
\(738\) 0 0
\(739\) − 3252.00i − 0.161877i −0.996719 0.0809383i \(-0.974208\pi\)
0.996719 0.0809383i \(-0.0257917\pi\)
\(740\) 0 0
\(741\) − 1392.00i − 0.0690100i
\(742\) 0 0
\(743\) −4710.00 −0.232561 −0.116281 0.993216i \(-0.537097\pi\)
−0.116281 + 0.993216i \(0.537097\pi\)
\(744\) 0 0
\(745\) −4790.00 −0.235560
\(746\) 0 0
\(747\) − 3906.00i − 0.191316i
\(748\) 0 0
\(749\) 17340.0i 0.845914i
\(750\) 0 0
\(751\) −25764.0 −1.25185 −0.625927 0.779882i \(-0.715279\pi\)
−0.625927 + 0.779882i \(0.715279\pi\)
\(752\) 0 0
\(753\) 30240.0 1.46349
\(754\) 0 0
\(755\) − 1660.00i − 0.0800180i
\(756\) 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 \(-0.678786\pi\)
−0.532603 + 0.846365i \(0.678786\pi\)
\(762\) 0 0
\(763\) 884.000i 0.0419436i
\(764\) 0 0
\(765\) − 3150.00i − 0.148874i
\(766\) 0 0
\(767\) 15544.0 0.731762
\(768\) 0 0
\(769\) −30398.0 −1.42546 −0.712731 0.701438i \(-0.752542\pi\)
−0.712731 + 0.701438i \(0.752542\pi\)
\(770\) 0 0
\(771\) 13860.0i 0.647413i
\(772\) 0 0
\(773\) 1290.00i 0.0600234i 0.999550 + 0.0300117i \(0.00955445\pi\)
−0.999550 + 0.0300117i \(0.990446\pi\)
\(774\) 0 0
\(775\) 2500.00 0.115874
\(776\) 0 0
\(777\) 89352.0 4.12546
\(778\) 0 0
\(779\) 552.000i 0.0253883i
\(780\) 0 0
\(781\) 13632.0i 0.624573i
\(782\) 0 0
\(783\) 26136.0 1.19288
\(784\) 0 0
\(785\) 5110.00 0.232336
\(786\) 0 0
\(787\) 14.0000i 0 0.000634112i 1.00000 0.000317056i \(0.000100922\pi\)
−1.00000 0.000317056i \(0.999899\pi\)
\(788\) 0 0
\(789\) 24660.0i 1.11270i
\(790\) 0 0
\(791\) −42228.0 −1.89817
\(792\) 0 0
\(793\) −14500.0 −0.649319
\(794\) 0 0
\(795\) − 4860.00i − 0.216813i
\(796\) 0 0
\(797\) 38814.0i 1.72505i 0.506017 + 0.862523i \(0.331117\pi\)
−0.506017 + 0.862523i \(0.668883\pi\)
\(798\) 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
\(805\) − 22780.0i − 0.997378i
\(806\) 0 0
\(807\) −4476.00 −0.195245
\(808\) 0 0
\(809\) −27402.0 −1.19086 −0.595428 0.803408i \(-0.703018\pi\)
−0.595428 + 0.803408i \(0.703018\pi\)
\(810\) 0 0
\(811\) 28576.0i 1.23729i 0.785672 + 0.618643i \(0.212317\pi\)
−0.785672 + 0.618643i \(0.787683\pi\)
\(812\) 0 0
\(813\) − 27576.0i − 1.18958i
\(814\) 0 0
\(815\) −4630.00 −0.198996
\(816\) 0 0
\(817\) 712.000 0.0304893
\(818\) 0 0
\(819\) − 17748.0i − 0.757223i
\(820\) 0 0
\(821\) 31762.0i 1.35018i 0.737733 + 0.675092i \(0.235896\pi\)
−0.737733 + 0.675092i \(0.764104\pi\)
\(822\) 0 0
\(823\) 20506.0 0.868523 0.434261 0.900787i \(-0.357009\pi\)
0.434261 + 0.900787i \(0.357009\pi\)
\(824\) 0 0
\(825\) 2400.00 0.101282
\(826\) 0 0
\(827\) − 13014.0i − 0.547208i −0.961842 0.273604i \(-0.911784\pi\)
0.961842 0.273604i \(-0.0882158\pi\)
\(828\) 0 0
\(829\) 22790.0i 0.954800i 0.878686 + 0.477400i \(0.158421\pi\)
−0.878686 + 0.477400i \(0.841579\pi\)
\(830\) 0 0
\(831\) −13236.0 −0.552529
\(832\) 0 0
\(833\) −56910.0 −2.36712
\(834\) 0 0
\(835\) − 3270.00i − 0.135525i
\(836\) 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
\(845\) 5835.00i 0.237550i
\(846\) 0 0
\(847\) −36550.0 −1.48273
\(848\) 0 0
\(849\) −7068.00 −0.285716
\(850\) 0 0
\(851\) 58692.0i 2.36420i
\(852\) 0 0
\(853\) 5306.00i 0.212982i 0.994314 + 0.106491i \(0.0339616\pi\)
−0.994314 + 0.106491i \(0.966038\pi\)
\(854\) 0 0
\(855\) −180.000 −0.00719985
\(856\) 0 0
\(857\) 21054.0 0.839196 0.419598 0.907710i \(-0.362171\pi\)
0.419598 + 0.907710i \(0.362171\pi\)
\(858\) 0 0
\(859\) − 7364.00i − 0.292499i −0.989248 0.146249i \(-0.953280\pi\)
0.989248 0.146249i \(-0.0467202\pi\)
\(860\) 0 0
\(861\) 28152.0i 1.11431i
\(862\) 0 0
\(863\) −17226.0 −0.679467 −0.339733 0.940522i \(-0.610337\pi\)
−0.339733 + 0.940522i \(0.610337\pi\)
\(864\) 0 0
\(865\) 6470.00 0.254320
\(866\) 0 0
\(867\) 78.0000i 0.00305539i
\(868\) 0 0
\(869\) − 7296.00i − 0.284810i
\(870\) 0 0
\(871\) 24476.0 0.952167
\(872\) 0 0
\(873\) −12402.0 −0.480807
\(874\) 0 0
\(875\) − 4250.00i − 0.164201i
\(876\) 0 0
\(877\) − 21202.0i − 0.816352i −0.912903 0.408176i \(-0.866165\pi\)
0.912903 0.408176i \(-0.133835\pi\)
\(878\) 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
\(885\) − 8040.00i − 0.305380i
\(886\) 0 0
\(887\) 36334.0 1.37540 0.687698 0.725997i \(-0.258621\pi\)
0.687698 + 0.725997i \(0.258621\pi\)
\(888\) 0 0
\(889\) −33252.0 −1.25448
\(890\) 0 0
\(891\) 14256.0i 0.536020i
\(892\) 0 0
\(893\) − 88.0000i − 0.00329766i
\(894\) 0 0
\(895\) −14180.0 −0.529592
\(896\) 0 0
\(897\) 46632.0 1.73578
\(898\) 0 0
\(899\) − 24200.0i − 0.897792i
\(900\) 0 0
\(901\) − 11340.0i − 0.419301i
\(902\) 0 0
\(903\) 36312.0 1.33819
\(904\) 0 0
\(905\) 8710.00 0.319923
\(906\) 0 0
\(907\) 12474.0i 0.456662i 0.973584 + 0.228331i \(0.0733268\pi\)
−0.973584 + 0.228331i \(0.926673\pi\)
\(908\) 0 0
\(909\) − 1134.00i − 0.0413778i
\(910\) 0 0
\(911\) 41132.0 1.49590 0.747949 0.663756i \(-0.231039\pi\)
0.747949 + 0.663756i \(0.231039\pi\)
\(912\) 0 0
\(913\) 6944.00 0.251712
\(914\) 0 0
\(915\) 7500.00i 0.270975i
\(916\) 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
\(925\) 10950.0i 0.389226i
\(926\) 0 0
\(927\) 11358.0 0.402423
\(928\) 0 0
\(929\) 41302.0 1.45864 0.729319 0.684174i \(-0.239837\pi\)
0.729319 + 0.684174i \(0.239837\pi\)
\(930\) 0 0
\(931\) 3252.00i 0.114479i
\(932\) 0 0
\(933\) 26376.0i 0.925521i
\(934\) 0 0
\(935\) 5600.00 0.195871
\(936\) 0 0
\(937\) 26150.0 0.911722 0.455861 0.890051i \(-0.349331\pi\)
0.455861 + 0.890051i \(0.349331\pi\)
\(938\) 0 0
\(939\) 28956.0i 1.00633i
\(940\) 0 0
\(941\) − 35254.0i − 1.22130i −0.791899 0.610652i \(-0.790907\pi\)
0.791899 0.610652i \(-0.209093\pi\)
\(942\) 0 0
\(943\) −18492.0 −0.638582
\(944\) 0 0
\(945\) 18360.0 0.632011
\(946\) 0 0
\(947\) 18550.0i 0.636530i 0.948002 + 0.318265i \(0.103100\pi\)
−0.948002 + 0.318265i \(0.896900\pi\)
\(948\) 0 0
\(949\) 17748.0i 0.607086i
\(950\) 0 0
\(951\) −42156.0 −1.43744
\(952\) 0 0
\(953\) −17322.0 −0.588788 −0.294394 0.955684i \(-0.595118\pi\)
−0.294394 + 0.955684i \(0.595118\pi\)
\(954\) 0 0
\(955\) 22300.0i 0.755614i
\(956\) 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
\(965\) 18910.0i 0.630813i
\(966\) 0 0
\(967\) 35190.0 1.17025 0.585126 0.810942i \(-0.301045\pi\)
0.585126 + 0.810942i \(0.301045\pi\)
\(968\) 0 0
\(969\) −1680.00 −0.0556960
\(970\) 0 0
\(971\) 40696.0i 1.34500i 0.740096 + 0.672501i \(0.234780\pi\)
−0.740096 + 0.672501i \(0.765220\pi\)
\(972\) 0 0
\(973\) 17544.0i 0.578042i
\(974\) 0 0
\(975\) 8700.00 0.285767
\(976\) 0 0
\(977\) 44306.0 1.45084 0.725422 0.688304i \(-0.241645\pi\)
0.725422 + 0.688304i \(0.241645\pi\)
\(978\) 0 0
\(979\) − 11616.0i − 0.379212i
\(980\) 0 0
\(981\) 234.000i 0.00761574i
\(982\) 0 0
\(983\) −18798.0 −0.609932 −0.304966 0.952363i \(-0.598645\pi\)
−0.304966 + 0.952363i \(0.598645\pi\)
\(984\) 0 0
\(985\) 22370.0 0.723622
\(986\) 0 0
\(987\) − 4488.00i − 0.144736i
\(988\) 0 0
\(989\) 23852.0i 0.766885i
\(990\) 0 0
\(991\) −2468.00 −0.0791106 −0.0395553 0.999217i \(-0.512594\pi\)
−0.0395553 + 0.999217i \(0.512594\pi\)
\(992\) 0 0
\(993\) −52848.0 −1.68890
\(994\) 0 0
\(995\) − 18040.0i − 0.574780i
\(996\) 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 1280.4.d.a.641.1 2
4.3 odd 2 1280.4.d.p.641.2 2
8.3 odd 2 1280.4.d.p.641.1 2
8.5 even 2 inner 1280.4.d.a.641.2 2
16.3 odd 4 40.4.a.a.1.1 1
16.5 even 4 320.4.a.c.1.1 1
16.11 odd 4 320.4.a.l.1.1 1
16.13 even 4 80.4.a.e.1.1 1
48.29 odd 4 720.4.a.bd.1.1 1
48.35 even 4 360.4.a.h.1.1 1
80.3 even 4 200.4.c.c.49.1 2
80.13 odd 4 400.4.c.f.49.2 2
80.19 odd 4 200.4.a.i.1.1 1
80.29 even 4 400.4.a.e.1.1 1
80.59 odd 4 1600.4.a.j.1.1 1
80.67 even 4 200.4.c.c.49.2 2
80.69 even 4 1600.4.a.br.1.1 1
80.77 odd 4 400.4.c.f.49.1 2
112.83 even 4 1960.4.a.h.1.1 1
240.83 odd 4 1800.4.f.j.649.2 2
240.179 even 4 1800.4.a.bi.1.1 1
240.227 odd 4 1800.4.f.j.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.a.1.1 1 16.3 odd 4
80.4.a.e.1.1 1 16.13 even 4
200.4.a.i.1.1 1 80.19 odd 4
200.4.c.c.49.1 2 80.3 even 4
200.4.c.c.49.2 2 80.67 even 4
320.4.a.c.1.1 1 16.5 even 4
320.4.a.l.1.1 1 16.11 odd 4
360.4.a.h.1.1 1 48.35 even 4
400.4.a.e.1.1 1 80.29 even 4
400.4.c.f.49.1 2 80.77 odd 4
400.4.c.f.49.2 2 80.13 odd 4
720.4.a.bd.1.1 1 48.29 odd 4
1280.4.d.a.641.1 2 1.1 even 1 trivial
1280.4.d.a.641.2 2 8.5 even 2 inner
1280.4.d.p.641.1 2 8.3 odd 2
1280.4.d.p.641.2 2 4.3 odd 2
1600.4.a.j.1.1 1 80.59 odd 4
1600.4.a.br.1.1 1 80.69 even 4
1800.4.a.bi.1.1 1 240.179 even 4
1800.4.f.j.649.1 2 240.227 odd 4
1800.4.f.j.649.2 2 240.83 odd 4
1960.4.a.h.1.1 1 112.83 even 4