Properties

Label 1024.4.a.n.1.8
Level $1024$
Weight $4$
Character 1024.1
Self dual yes
Analytic conductor $60.418$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1024.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.4179558459\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 36 x^{8} + 405 x^{6} - 1380 x^{4} + 420 x^{2} - 32\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 16)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.446984\) of defining polynomial
Character \(\chi\) \(=\) 1024.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.62644 q^{3} -17.8826 q^{5} +13.8754 q^{7} -5.59607 q^{9} +O(q^{10})\) \(q+4.62644 q^{3} -17.8826 q^{5} +13.8754 q^{7} -5.59607 q^{9} -2.18771 q^{11} +46.3685 q^{13} -82.7326 q^{15} -18.6531 q^{17} -122.194 q^{19} +64.1936 q^{21} +134.006 q^{23} +194.786 q^{25} -150.804 q^{27} +84.5628 q^{29} -31.5391 q^{31} -10.1213 q^{33} -248.128 q^{35} +126.129 q^{37} +214.521 q^{39} +210.504 q^{41} -168.860 q^{43} +100.072 q^{45} +182.902 q^{47} -150.474 q^{49} -86.2972 q^{51} +37.0020 q^{53} +39.1219 q^{55} -565.323 q^{57} +624.494 q^{59} +246.759 q^{61} -77.6476 q^{63} -829.188 q^{65} -129.763 q^{67} +619.970 q^{69} +348.360 q^{71} +299.436 q^{73} +901.168 q^{75} -30.3553 q^{77} +943.487 q^{79} -546.590 q^{81} +443.034 q^{83} +333.565 q^{85} +391.224 q^{87} +1412.35 q^{89} +643.381 q^{91} -145.914 q^{93} +2185.14 q^{95} +1515.29 q^{97} +12.2426 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 28q^{7} + 54q^{9} + O(q^{10}) \) \( 10q + 28q^{7} + 54q^{9} + 124q^{15} + 4q^{17} + 276q^{23} + 50q^{25} + 368q^{31} - 4q^{33} + 732q^{39} + 944q^{47} - 94q^{49} + 1380q^{55} + 108q^{57} + 2628q^{63} - 492q^{65} + 3468q^{71} - 296q^{73} + 4416q^{79} - 482q^{81} + 6036q^{87} + 88q^{89} + 6900q^{95} - 4q^{97} + O(q^{100}) \)

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
\(3\) 4.62644 0.890358 0.445179 0.895441i \(-0.353140\pi\)
0.445179 + 0.895441i \(0.353140\pi\)
\(4\) 0 0
\(5\) −17.8826 −1.59947 −0.799733 0.600356i \(-0.795026\pi\)
−0.799733 + 0.600356i \(0.795026\pi\)
\(6\) 0 0
\(7\) 13.8754 0.749200 0.374600 0.927187i \(-0.377780\pi\)
0.374600 + 0.927187i \(0.377780\pi\)
\(8\) 0 0
\(9\) −5.59607 −0.207262
\(10\) 0 0
\(11\) −2.18771 −0.0599653 −0.0299827 0.999550i \(-0.509545\pi\)
−0.0299827 + 0.999550i \(0.509545\pi\)
\(12\) 0 0
\(13\) 46.3685 0.989255 0.494627 0.869105i \(-0.335305\pi\)
0.494627 + 0.869105i \(0.335305\pi\)
\(14\) 0 0
\(15\) −82.7326 −1.42410
\(16\) 0 0
\(17\) −18.6531 −0.266119 −0.133060 0.991108i \(-0.542480\pi\)
−0.133060 + 0.991108i \(0.542480\pi\)
\(18\) 0 0
\(19\) −122.194 −1.47543 −0.737716 0.675111i \(-0.764096\pi\)
−0.737716 + 0.675111i \(0.764096\pi\)
\(20\) 0 0
\(21\) 64.1936 0.667057
\(22\) 0 0
\(23\) 134.006 1.21488 0.607438 0.794367i \(-0.292197\pi\)
0.607438 + 0.794367i \(0.292197\pi\)
\(24\) 0 0
\(25\) 194.786 1.55829
\(26\) 0 0
\(27\) −150.804 −1.07490
\(28\) 0 0
\(29\) 84.5628 0.541480 0.270740 0.962653i \(-0.412732\pi\)
0.270740 + 0.962653i \(0.412732\pi\)
\(30\) 0 0
\(31\) −31.5391 −0.182729 −0.0913645 0.995818i \(-0.529123\pi\)
−0.0913645 + 0.995818i \(0.529123\pi\)
\(32\) 0 0
\(33\) −10.1213 −0.0533906
\(34\) 0 0
\(35\) −248.128 −1.19832
\(36\) 0 0
\(37\) 126.129 0.560418 0.280209 0.959939i \(-0.409596\pi\)
0.280209 + 0.959939i \(0.409596\pi\)
\(38\) 0 0
\(39\) 214.521 0.880791
\(40\) 0 0
\(41\) 210.504 0.801834 0.400917 0.916114i \(-0.368692\pi\)
0.400917 + 0.916114i \(0.368692\pi\)
\(42\) 0 0
\(43\) −168.860 −0.598857 −0.299429 0.954119i \(-0.596796\pi\)
−0.299429 + 0.954119i \(0.596796\pi\)
\(44\) 0 0
\(45\) 100.072 0.331508
\(46\) 0 0
\(47\) 182.902 0.567638 0.283819 0.958878i \(-0.408398\pi\)
0.283819 + 0.958878i \(0.408398\pi\)
\(48\) 0 0
\(49\) −150.474 −0.438699
\(50\) 0 0
\(51\) −86.2972 −0.236942
\(52\) 0 0
\(53\) 37.0020 0.0958984 0.0479492 0.998850i \(-0.484731\pi\)
0.0479492 + 0.998850i \(0.484731\pi\)
\(54\) 0 0
\(55\) 39.1219 0.0959125
\(56\) 0 0
\(57\) −565.323 −1.31366
\(58\) 0 0
\(59\) 624.494 1.37800 0.689001 0.724760i \(-0.258050\pi\)
0.689001 + 0.724760i \(0.258050\pi\)
\(60\) 0 0
\(61\) 246.759 0.517938 0.258969 0.965886i \(-0.416617\pi\)
0.258969 + 0.965886i \(0.416617\pi\)
\(62\) 0 0
\(63\) −77.6476 −0.155281
\(64\) 0 0
\(65\) −829.188 −1.58228
\(66\) 0 0
\(67\) −129.763 −0.236613 −0.118306 0.992977i \(-0.537747\pi\)
−0.118306 + 0.992977i \(0.537747\pi\)
\(68\) 0 0
\(69\) 619.970 1.08168
\(70\) 0 0
\(71\) 348.360 0.582291 0.291146 0.956679i \(-0.405964\pi\)
0.291146 + 0.956679i \(0.405964\pi\)
\(72\) 0 0
\(73\) 299.436 0.480087 0.240043 0.970762i \(-0.422838\pi\)
0.240043 + 0.970762i \(0.422838\pi\)
\(74\) 0 0
\(75\) 901.168 1.38744
\(76\) 0 0
\(77\) −30.3553 −0.0449260
\(78\) 0 0
\(79\) 943.487 1.34368 0.671839 0.740697i \(-0.265505\pi\)
0.671839 + 0.740697i \(0.265505\pi\)
\(80\) 0 0
\(81\) −546.590 −0.749781
\(82\) 0 0
\(83\) 443.034 0.585895 0.292947 0.956129i \(-0.405364\pi\)
0.292947 + 0.956129i \(0.405364\pi\)
\(84\) 0 0
\(85\) 333.565 0.425649
\(86\) 0 0
\(87\) 391.224 0.482111
\(88\) 0 0
\(89\) 1412.35 1.68212 0.841060 0.540942i \(-0.181932\pi\)
0.841060 + 0.540942i \(0.181932\pi\)
\(90\) 0 0
\(91\) 643.381 0.741150
\(92\) 0 0
\(93\) −145.914 −0.162694
\(94\) 0 0
\(95\) 2185.14 2.35990
\(96\) 0 0
\(97\) 1515.29 1.58613 0.793063 0.609140i \(-0.208485\pi\)
0.793063 + 0.609140i \(0.208485\pi\)
\(98\) 0 0
\(99\) 12.2426 0.0124285
\(100\) 0 0
\(101\) −810.631 −0.798621 −0.399311 0.916816i \(-0.630751\pi\)
−0.399311 + 0.916816i \(0.630751\pi\)
\(102\) 0 0
\(103\) −1021.00 −0.976717 −0.488359 0.872643i \(-0.662404\pi\)
−0.488359 + 0.872643i \(0.662404\pi\)
\(104\) 0 0
\(105\) −1147.95 −1.06693
\(106\) 0 0
\(107\) 1754.75 1.58540 0.792700 0.609612i \(-0.208675\pi\)
0.792700 + 0.609612i \(0.208675\pi\)
\(108\) 0 0
\(109\) −153.624 −0.134996 −0.0674980 0.997719i \(-0.521502\pi\)
−0.0674980 + 0.997719i \(0.521502\pi\)
\(110\) 0 0
\(111\) 583.528 0.498973
\(112\) 0 0
\(113\) 1722.22 1.43374 0.716870 0.697207i \(-0.245574\pi\)
0.716870 + 0.697207i \(0.245574\pi\)
\(114\) 0 0
\(115\) −2396.37 −1.94315
\(116\) 0 0
\(117\) −259.481 −0.205035
\(118\) 0 0
\(119\) −258.818 −0.199377
\(120\) 0 0
\(121\) −1326.21 −0.996404
\(122\) 0 0
\(123\) 973.883 0.713919
\(124\) 0 0
\(125\) −1247.96 −0.892969
\(126\) 0 0
\(127\) 699.127 0.488484 0.244242 0.969714i \(-0.421461\pi\)
0.244242 + 0.969714i \(0.421461\pi\)
\(128\) 0 0
\(129\) −781.219 −0.533198
\(130\) 0 0
\(131\) −279.972 −0.186727 −0.0933636 0.995632i \(-0.529762\pi\)
−0.0933636 + 0.995632i \(0.529762\pi\)
\(132\) 0 0
\(133\) −1695.49 −1.10539
\(134\) 0 0
\(135\) 2696.76 1.71926
\(136\) 0 0
\(137\) 271.386 0.169242 0.0846209 0.996413i \(-0.473032\pi\)
0.0846209 + 0.996413i \(0.473032\pi\)
\(138\) 0 0
\(139\) −650.449 −0.396909 −0.198455 0.980110i \(-0.563592\pi\)
−0.198455 + 0.980110i \(0.563592\pi\)
\(140\) 0 0
\(141\) 846.185 0.505402
\(142\) 0 0
\(143\) −101.441 −0.0593210
\(144\) 0 0
\(145\) −1512.20 −0.866079
\(146\) 0 0
\(147\) −696.158 −0.390600
\(148\) 0 0
\(149\) −856.692 −0.471026 −0.235513 0.971871i \(-0.575677\pi\)
−0.235513 + 0.971871i \(0.575677\pi\)
\(150\) 0 0
\(151\) 3534.47 1.90484 0.952421 0.304785i \(-0.0985848\pi\)
0.952421 + 0.304785i \(0.0985848\pi\)
\(152\) 0 0
\(153\) 104.384 0.0551564
\(154\) 0 0
\(155\) 564.001 0.292269
\(156\) 0 0
\(157\) 1744.48 0.886784 0.443392 0.896328i \(-0.353775\pi\)
0.443392 + 0.896328i \(0.353775\pi\)
\(158\) 0 0
\(159\) 171.187 0.0853839
\(160\) 0 0
\(161\) 1859.38 0.910186
\(162\) 0 0
\(163\) −3633.63 −1.74606 −0.873030 0.487666i \(-0.837848\pi\)
−0.873030 + 0.487666i \(0.837848\pi\)
\(164\) 0 0
\(165\) 180.995 0.0853965
\(166\) 0 0
\(167\) 3048.39 1.41252 0.706262 0.707950i \(-0.250380\pi\)
0.706262 + 0.707950i \(0.250380\pi\)
\(168\) 0 0
\(169\) −46.9618 −0.0213754
\(170\) 0 0
\(171\) 683.806 0.305801
\(172\) 0 0
\(173\) −2152.51 −0.945965 −0.472983 0.881072i \(-0.656823\pi\)
−0.472983 + 0.881072i \(0.656823\pi\)
\(174\) 0 0
\(175\) 2702.74 1.16747
\(176\) 0 0
\(177\) 2889.18 1.22692
\(178\) 0 0
\(179\) 427.457 0.178489 0.0892447 0.996010i \(-0.471555\pi\)
0.0892447 + 0.996010i \(0.471555\pi\)
\(180\) 0 0
\(181\) −2399.83 −0.985515 −0.492758 0.870167i \(-0.664011\pi\)
−0.492758 + 0.870167i \(0.664011\pi\)
\(182\) 0 0
\(183\) 1141.62 0.461151
\(184\) 0 0
\(185\) −2255.51 −0.896370
\(186\) 0 0
\(187\) 40.8074 0.0159579
\(188\) 0 0
\(189\) −2092.46 −0.805312
\(190\) 0 0
\(191\) −4035.31 −1.52872 −0.764358 0.644792i \(-0.776944\pi\)
−0.764358 + 0.644792i \(0.776944\pi\)
\(192\) 0 0
\(193\) −886.172 −0.330508 −0.165254 0.986251i \(-0.552844\pi\)
−0.165254 + 0.986251i \(0.552844\pi\)
\(194\) 0 0
\(195\) −3836.19 −1.40880
\(196\) 0 0
\(197\) 4624.65 1.67255 0.836276 0.548308i \(-0.184728\pi\)
0.836276 + 0.548308i \(0.184728\pi\)
\(198\) 0 0
\(199\) 222.513 0.0792639 0.0396319 0.999214i \(-0.487381\pi\)
0.0396319 + 0.999214i \(0.487381\pi\)
\(200\) 0 0
\(201\) −600.340 −0.210670
\(202\) 0 0
\(203\) 1173.34 0.405677
\(204\) 0 0
\(205\) −3764.35 −1.28251
\(206\) 0 0
\(207\) −749.907 −0.251798
\(208\) 0 0
\(209\) 267.325 0.0884748
\(210\) 0 0
\(211\) −4988.58 −1.62762 −0.813810 0.581131i \(-0.802610\pi\)
−0.813810 + 0.581131i \(0.802610\pi\)
\(212\) 0 0
\(213\) 1611.66 0.518448
\(214\) 0 0
\(215\) 3019.65 0.957852
\(216\) 0 0
\(217\) −437.618 −0.136901
\(218\) 0 0
\(219\) 1385.32 0.427449
\(220\) 0 0
\(221\) −864.914 −0.263260
\(222\) 0 0
\(223\) 5841.90 1.75427 0.877136 0.480242i \(-0.159451\pi\)
0.877136 + 0.480242i \(0.159451\pi\)
\(224\) 0 0
\(225\) −1090.04 −0.322975
\(226\) 0 0
\(227\) 1597.41 0.467065 0.233533 0.972349i \(-0.424971\pi\)
0.233533 + 0.972349i \(0.424971\pi\)
\(228\) 0 0
\(229\) −4108.86 −1.18568 −0.592840 0.805320i \(-0.701994\pi\)
−0.592840 + 0.805320i \(0.701994\pi\)
\(230\) 0 0
\(231\) −140.437 −0.0400003
\(232\) 0 0
\(233\) 734.054 0.206393 0.103196 0.994661i \(-0.467093\pi\)
0.103196 + 0.994661i \(0.467093\pi\)
\(234\) 0 0
\(235\) −3270.76 −0.907918
\(236\) 0 0
\(237\) 4364.98 1.19635
\(238\) 0 0
\(239\) 511.807 0.138519 0.0692595 0.997599i \(-0.477936\pi\)
0.0692595 + 0.997599i \(0.477936\pi\)
\(240\) 0 0
\(241\) −5920.31 −1.58241 −0.791204 0.611552i \(-0.790545\pi\)
−0.791204 + 0.611552i \(0.790545\pi\)
\(242\) 0 0
\(243\) 1542.93 0.407322
\(244\) 0 0
\(245\) 2690.86 0.701685
\(246\) 0 0
\(247\) −5665.95 −1.45958
\(248\) 0 0
\(249\) 2049.67 0.521656
\(250\) 0 0
\(251\) −437.462 −0.110009 −0.0550046 0.998486i \(-0.517517\pi\)
−0.0550046 + 0.998486i \(0.517517\pi\)
\(252\) 0 0
\(253\) −293.166 −0.0728505
\(254\) 0 0
\(255\) 1543.22 0.378980
\(256\) 0 0
\(257\) 323.723 0.0785730 0.0392865 0.999228i \(-0.487491\pi\)
0.0392865 + 0.999228i \(0.487491\pi\)
\(258\) 0 0
\(259\) 1750.09 0.419865
\(260\) 0 0
\(261\) −473.219 −0.112228
\(262\) 0 0
\(263\) 2689.15 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(264\) 0 0
\(265\) −661.691 −0.153386
\(266\) 0 0
\(267\) 6534.14 1.49769
\(268\) 0 0
\(269\) 6652.15 1.50776 0.753882 0.657009i \(-0.228179\pi\)
0.753882 + 0.657009i \(0.228179\pi\)
\(270\) 0 0
\(271\) 2018.97 0.452561 0.226280 0.974062i \(-0.427343\pi\)
0.226280 + 0.974062i \(0.427343\pi\)
\(272\) 0 0
\(273\) 2976.56 0.659889
\(274\) 0 0
\(275\) −426.136 −0.0934435
\(276\) 0 0
\(277\) 4356.63 0.944999 0.472499 0.881331i \(-0.343352\pi\)
0.472499 + 0.881331i \(0.343352\pi\)
\(278\) 0 0
\(279\) 176.495 0.0378728
\(280\) 0 0
\(281\) 3893.51 0.826575 0.413287 0.910601i \(-0.364381\pi\)
0.413287 + 0.910601i \(0.364381\pi\)
\(282\) 0 0
\(283\) 2865.74 0.601945 0.300972 0.953633i \(-0.402689\pi\)
0.300972 + 0.953633i \(0.402689\pi\)
\(284\) 0 0
\(285\) 10109.4 2.10116
\(286\) 0 0
\(287\) 2920.82 0.600734
\(288\) 0 0
\(289\) −4565.06 −0.929180
\(290\) 0 0
\(291\) 7010.39 1.41222
\(292\) 0 0
\(293\) −1416.60 −0.282452 −0.141226 0.989977i \(-0.545104\pi\)
−0.141226 + 0.989977i \(0.545104\pi\)
\(294\) 0 0
\(295\) −11167.6 −2.20407
\(296\) 0 0
\(297\) 329.914 0.0644565
\(298\) 0 0
\(299\) 6213.65 1.20182
\(300\) 0 0
\(301\) −2342.99 −0.448664
\(302\) 0 0
\(303\) −3750.33 −0.711059
\(304\) 0 0
\(305\) −4412.69 −0.828425
\(306\) 0 0
\(307\) 4195.32 0.779934 0.389967 0.920829i \(-0.372486\pi\)
0.389967 + 0.920829i \(0.372486\pi\)
\(308\) 0 0
\(309\) −4723.58 −0.869628
\(310\) 0 0
\(311\) 2911.18 0.530797 0.265399 0.964139i \(-0.414496\pi\)
0.265399 + 0.964139i \(0.414496\pi\)
\(312\) 0 0
\(313\) 8287.74 1.49665 0.748324 0.663333i \(-0.230859\pi\)
0.748324 + 0.663333i \(0.230859\pi\)
\(314\) 0 0
\(315\) 1388.54 0.248366
\(316\) 0 0
\(317\) −8120.67 −1.43881 −0.719404 0.694592i \(-0.755585\pi\)
−0.719404 + 0.694592i \(0.755585\pi\)
\(318\) 0 0
\(319\) −184.999 −0.0324700
\(320\) 0 0
\(321\) 8118.23 1.41157
\(322\) 0 0
\(323\) 2279.29 0.392641
\(324\) 0 0
\(325\) 9031.96 1.54155
\(326\) 0 0
\(327\) −710.734 −0.120195
\(328\) 0 0
\(329\) 2537.84 0.425275
\(330\) 0 0
\(331\) −6362.92 −1.05661 −0.528305 0.849055i \(-0.677172\pi\)
−0.528305 + 0.849055i \(0.677172\pi\)
\(332\) 0 0
\(333\) −705.827 −0.116153
\(334\) 0 0
\(335\) 2320.50 0.378454
\(336\) 0 0
\(337\) 5860.06 0.947234 0.473617 0.880731i \(-0.342948\pi\)
0.473617 + 0.880731i \(0.342948\pi\)
\(338\) 0 0
\(339\) 7967.73 1.27654
\(340\) 0 0
\(341\) 68.9984 0.0109574
\(342\) 0 0
\(343\) −6847.14 −1.07787
\(344\) 0 0
\(345\) −11086.7 −1.73010
\(346\) 0 0
\(347\) −2869.60 −0.443942 −0.221971 0.975053i \(-0.571249\pi\)
−0.221971 + 0.975053i \(0.571249\pi\)
\(348\) 0 0
\(349\) 2748.19 0.421511 0.210755 0.977539i \(-0.432408\pi\)
0.210755 + 0.977539i \(0.432408\pi\)
\(350\) 0 0
\(351\) −6992.54 −1.06335
\(352\) 0 0
\(353\) −7548.63 −1.13817 −0.569084 0.822280i \(-0.692702\pi\)
−0.569084 + 0.822280i \(0.692702\pi\)
\(354\) 0 0
\(355\) −6229.57 −0.931355
\(356\) 0 0
\(357\) −1197.41 −0.177517
\(358\) 0 0
\(359\) 5554.15 0.816537 0.408269 0.912862i \(-0.366133\pi\)
0.408269 + 0.912862i \(0.366133\pi\)
\(360\) 0 0
\(361\) 8072.37 1.17690
\(362\) 0 0
\(363\) −6135.65 −0.887157
\(364\) 0 0
\(365\) −5354.69 −0.767883
\(366\) 0 0
\(367\) 3610.98 0.513601 0.256800 0.966464i \(-0.417332\pi\)
0.256800 + 0.966464i \(0.417332\pi\)
\(368\) 0 0
\(369\) −1177.99 −0.166190
\(370\) 0 0
\(371\) 513.417 0.0718471
\(372\) 0 0
\(373\) −1718.96 −0.238618 −0.119309 0.992857i \(-0.538068\pi\)
−0.119309 + 0.992857i \(0.538068\pi\)
\(374\) 0 0
\(375\) −5773.62 −0.795062
\(376\) 0 0
\(377\) 3921.05 0.535661
\(378\) 0 0
\(379\) −10391.4 −1.40836 −0.704180 0.710021i \(-0.748685\pi\)
−0.704180 + 0.710021i \(0.748685\pi\)
\(380\) 0 0
\(381\) 3234.47 0.434926
\(382\) 0 0
\(383\) −7668.98 −1.02315 −0.511575 0.859238i \(-0.670938\pi\)
−0.511575 + 0.859238i \(0.670938\pi\)
\(384\) 0 0
\(385\) 542.831 0.0718577
\(386\) 0 0
\(387\) 944.951 0.124120
\(388\) 0 0
\(389\) −284.150 −0.0370359 −0.0185180 0.999829i \(-0.505895\pi\)
−0.0185180 + 0.999829i \(0.505895\pi\)
\(390\) 0 0
\(391\) −2499.62 −0.323302
\(392\) 0 0
\(393\) −1295.27 −0.166254
\(394\) 0 0
\(395\) −16872.0 −2.14917
\(396\) 0 0
\(397\) 9209.66 1.16428 0.582141 0.813088i \(-0.302215\pi\)
0.582141 + 0.813088i \(0.302215\pi\)
\(398\) 0 0
\(399\) −7844.07 −0.984197
\(400\) 0 0
\(401\) −5565.10 −0.693036 −0.346518 0.938043i \(-0.612636\pi\)
−0.346518 + 0.938043i \(0.612636\pi\)
\(402\) 0 0
\(403\) −1462.42 −0.180765
\(404\) 0 0
\(405\) 9774.44 1.19925
\(406\) 0 0
\(407\) −275.933 −0.0336057
\(408\) 0 0
\(409\) −12077.6 −1.46014 −0.730072 0.683370i \(-0.760514\pi\)
−0.730072 + 0.683370i \(0.760514\pi\)
\(410\) 0 0
\(411\) 1255.55 0.150686
\(412\) 0 0
\(413\) 8665.08 1.03240
\(414\) 0 0
\(415\) −7922.58 −0.937119
\(416\) 0 0
\(417\) −3009.26 −0.353391
\(418\) 0 0
\(419\) 2054.89 0.239590 0.119795 0.992799i \(-0.461776\pi\)
0.119795 + 0.992799i \(0.461776\pi\)
\(420\) 0 0
\(421\) −6819.69 −0.789480 −0.394740 0.918793i \(-0.629165\pi\)
−0.394740 + 0.918793i \(0.629165\pi\)
\(422\) 0 0
\(423\) −1023.53 −0.117650
\(424\) 0 0
\(425\) −3633.36 −0.414692
\(426\) 0 0
\(427\) 3423.87 0.388040
\(428\) 0 0
\(429\) −469.309 −0.0528169
\(430\) 0 0
\(431\) −12519.2 −1.39914 −0.699571 0.714563i \(-0.746626\pi\)
−0.699571 + 0.714563i \(0.746626\pi\)
\(432\) 0 0
\(433\) 2921.40 0.324235 0.162117 0.986771i \(-0.448168\pi\)
0.162117 + 0.986771i \(0.448168\pi\)
\(434\) 0 0
\(435\) −6996.10 −0.771120
\(436\) 0 0
\(437\) −16374.7 −1.79247
\(438\) 0 0
\(439\) −1140.50 −0.123993 −0.0619967 0.998076i \(-0.519747\pi\)
−0.0619967 + 0.998076i \(0.519747\pi\)
\(440\) 0 0
\(441\) 842.062 0.0909256
\(442\) 0 0
\(443\) −2606.46 −0.279541 −0.139771 0.990184i \(-0.544637\pi\)
−0.139771 + 0.990184i \(0.544637\pi\)
\(444\) 0 0
\(445\) −25256.4 −2.69049
\(446\) 0 0
\(447\) −3963.43 −0.419382
\(448\) 0 0
\(449\) 1752.13 0.184161 0.0920805 0.995752i \(-0.470648\pi\)
0.0920805 + 0.995752i \(0.470648\pi\)
\(450\) 0 0
\(451\) −460.521 −0.0480822
\(452\) 0 0
\(453\) 16352.0 1.69599
\(454\) 0 0
\(455\) −11505.3 −1.18544
\(456\) 0 0
\(457\) −12875.6 −1.31794 −0.658968 0.752171i \(-0.729007\pi\)
−0.658968 + 0.752171i \(0.729007\pi\)
\(458\) 0 0
\(459\) 2812.95 0.286051
\(460\) 0 0
\(461\) −19346.0 −1.95452 −0.977259 0.212050i \(-0.931986\pi\)
−0.977259 + 0.212050i \(0.931986\pi\)
\(462\) 0 0
\(463\) −15002.4 −1.50588 −0.752938 0.658091i \(-0.771364\pi\)
−0.752938 + 0.658091i \(0.771364\pi\)
\(464\) 0 0
\(465\) 2609.32 0.260224
\(466\) 0 0
\(467\) 13674.6 1.35500 0.677501 0.735522i \(-0.263063\pi\)
0.677501 + 0.735522i \(0.263063\pi\)
\(468\) 0 0
\(469\) −1800.51 −0.177270
\(470\) 0 0
\(471\) 8070.75 0.789555
\(472\) 0 0
\(473\) 369.416 0.0359107
\(474\) 0 0
\(475\) −23801.7 −2.29915
\(476\) 0 0
\(477\) −207.066 −0.0198761
\(478\) 0 0
\(479\) −3072.68 −0.293099 −0.146550 0.989203i \(-0.546817\pi\)
−0.146550 + 0.989203i \(0.546817\pi\)
\(480\) 0 0
\(481\) 5848.41 0.554396
\(482\) 0 0
\(483\) 8602.32 0.810392
\(484\) 0 0
\(485\) −27097.3 −2.53695
\(486\) 0 0
\(487\) 8689.64 0.808553 0.404276 0.914637i \(-0.367523\pi\)
0.404276 + 0.914637i \(0.367523\pi\)
\(488\) 0 0
\(489\) −16810.8 −1.55462
\(490\) 0 0
\(491\) −15831.2 −1.45509 −0.727547 0.686057i \(-0.759340\pi\)
−0.727547 + 0.686057i \(0.759340\pi\)
\(492\) 0 0
\(493\) −1577.35 −0.144098
\(494\) 0 0
\(495\) −218.929 −0.0198790
\(496\) 0 0
\(497\) 4833.62 0.436253
\(498\) 0 0
\(499\) −2309.01 −0.207146 −0.103573 0.994622i \(-0.533027\pi\)
−0.103573 + 0.994622i \(0.533027\pi\)
\(500\) 0 0
\(501\) 14103.2 1.25765
\(502\) 0 0
\(503\) 6901.81 0.611802 0.305901 0.952063i \(-0.401042\pi\)
0.305901 + 0.952063i \(0.401042\pi\)
\(504\) 0 0
\(505\) 14496.2 1.27737
\(506\) 0 0
\(507\) −217.266 −0.0190318
\(508\) 0 0
\(509\) −131.481 −0.0114495 −0.00572475 0.999984i \(-0.501822\pi\)
−0.00572475 + 0.999984i \(0.501822\pi\)
\(510\) 0 0
\(511\) 4154.79 0.359681
\(512\) 0 0
\(513\) 18427.3 1.58594
\(514\) 0 0
\(515\) 18258.1 1.56223
\(516\) 0 0
\(517\) −400.136 −0.0340386
\(518\) 0 0
\(519\) −9958.43 −0.842248
\(520\) 0 0
\(521\) 11931.3 1.00330 0.501649 0.865071i \(-0.332727\pi\)
0.501649 + 0.865071i \(0.332727\pi\)
\(522\) 0 0
\(523\) 13721.4 1.14721 0.573607 0.819131i \(-0.305544\pi\)
0.573607 + 0.819131i \(0.305544\pi\)
\(524\) 0 0
\(525\) 12504.0 1.03947
\(526\) 0 0
\(527\) 588.301 0.0486277
\(528\) 0 0
\(529\) 5790.59 0.475926
\(530\) 0 0
\(531\) −3494.71 −0.285607
\(532\) 0 0
\(533\) 9760.75 0.793218
\(534\) 0 0
\(535\) −31379.4 −2.53579
\(536\) 0 0
\(537\) 1977.60 0.158920
\(538\) 0 0
\(539\) 329.193 0.0263067
\(540\) 0 0
\(541\) 12101.0 0.961665 0.480833 0.876812i \(-0.340334\pi\)
0.480833 + 0.876812i \(0.340334\pi\)
\(542\) 0 0
\(543\) −11102.7 −0.877462
\(544\) 0 0
\(545\) 2747.20 0.215921
\(546\) 0 0
\(547\) 63.9157 0.00499605 0.00249803 0.999997i \(-0.499205\pi\)
0.00249803 + 0.999997i \(0.499205\pi\)
\(548\) 0 0
\(549\) −1380.88 −0.107349
\(550\) 0 0
\(551\) −10333.1 −0.798917
\(552\) 0 0
\(553\) 13091.2 1.00668
\(554\) 0 0
\(555\) −10435.0 −0.798090
\(556\) 0 0
\(557\) 6052.26 0.460400 0.230200 0.973143i \(-0.426062\pi\)
0.230200 + 0.973143i \(0.426062\pi\)
\(558\) 0 0
\(559\) −7829.77 −0.592422
\(560\) 0 0
\(561\) 188.793 0.0142083
\(562\) 0 0
\(563\) 20638.9 1.54498 0.772491 0.635026i \(-0.219011\pi\)
0.772491 + 0.635026i \(0.219011\pi\)
\(564\) 0 0
\(565\) −30797.7 −2.29322
\(566\) 0 0
\(567\) −7584.14 −0.561736
\(568\) 0 0
\(569\) −21728.1 −1.60086 −0.800431 0.599425i \(-0.795396\pi\)
−0.800431 + 0.599425i \(0.795396\pi\)
\(570\) 0 0
\(571\) 22737.8 1.66646 0.833229 0.552929i \(-0.186490\pi\)
0.833229 + 0.552929i \(0.186490\pi\)
\(572\) 0 0
\(573\) −18669.1 −1.36110
\(574\) 0 0
\(575\) 26102.5 1.89313
\(576\) 0 0
\(577\) −26648.2 −1.92267 −0.961335 0.275383i \(-0.911195\pi\)
−0.961335 + 0.275383i \(0.911195\pi\)
\(578\) 0 0
\(579\) −4099.82 −0.294270
\(580\) 0 0
\(581\) 6147.26 0.438952
\(582\) 0 0
\(583\) −80.9495 −0.00575058
\(584\) 0 0
\(585\) 4640.20 0.327946
\(586\) 0 0
\(587\) −1898.75 −0.133509 −0.0667544 0.997769i \(-0.521264\pi\)
−0.0667544 + 0.997769i \(0.521264\pi\)
\(588\) 0 0
\(589\) 3853.89 0.269604
\(590\) 0 0
\(591\) 21395.7 1.48917
\(592\) 0 0
\(593\) −4474.79 −0.309878 −0.154939 0.987924i \(-0.549518\pi\)
−0.154939 + 0.987924i \(0.549518\pi\)
\(594\) 0 0
\(595\) 4628.34 0.318896
\(596\) 0 0
\(597\) 1029.44 0.0705733
\(598\) 0 0
\(599\) −12603.8 −0.859725 −0.429863 0.902894i \(-0.641438\pi\)
−0.429863 + 0.902894i \(0.641438\pi\)
\(600\) 0 0
\(601\) 7220.64 0.490077 0.245038 0.969513i \(-0.421199\pi\)
0.245038 + 0.969513i \(0.421199\pi\)
\(602\) 0 0
\(603\) 726.163 0.0490408
\(604\) 0 0
\(605\) 23716.1 1.59371
\(606\) 0 0
\(607\) −13695.6 −0.915796 −0.457898 0.889005i \(-0.651397\pi\)
−0.457898 + 0.889005i \(0.651397\pi\)
\(608\) 0 0
\(609\) 5428.39 0.361198
\(610\) 0 0
\(611\) 8480.89 0.561539
\(612\) 0 0
\(613\) −3335.20 −0.219751 −0.109876 0.993945i \(-0.535045\pi\)
−0.109876 + 0.993945i \(0.535045\pi\)
\(614\) 0 0
\(615\) −17415.5 −1.14189
\(616\) 0 0
\(617\) −4186.39 −0.273157 −0.136579 0.990629i \(-0.543611\pi\)
−0.136579 + 0.990629i \(0.543611\pi\)
\(618\) 0 0
\(619\) 6788.94 0.440825 0.220412 0.975407i \(-0.429260\pi\)
0.220412 + 0.975407i \(0.429260\pi\)
\(620\) 0 0
\(621\) −20208.6 −1.30587
\(622\) 0 0
\(623\) 19596.9 1.26024
\(624\) 0 0
\(625\) −2031.54 −0.130018
\(626\) 0 0
\(627\) 1236.76 0.0787743
\(628\) 0 0
\(629\) −2352.69 −0.149138
\(630\) 0 0
\(631\) 16106.3 1.01614 0.508069 0.861316i \(-0.330360\pi\)
0.508069 + 0.861316i \(0.330360\pi\)
\(632\) 0 0
\(633\) −23079.3 −1.44917
\(634\) 0 0
\(635\) −12502.2 −0.781314
\(636\) 0 0
\(637\) −6977.25 −0.433985
\(638\) 0 0
\(639\) −1949.45 −0.120687
\(640\) 0 0
\(641\) −6682.21 −0.411749 −0.205875 0.978578i \(-0.566004\pi\)
−0.205875 + 0.978578i \(0.566004\pi\)
\(642\) 0 0
\(643\) 7047.69 0.432245 0.216123 0.976366i \(-0.430659\pi\)
0.216123 + 0.976366i \(0.430659\pi\)
\(644\) 0 0
\(645\) 13970.2 0.852832
\(646\) 0 0
\(647\) 5078.45 0.308585 0.154292 0.988025i \(-0.450690\pi\)
0.154292 + 0.988025i \(0.450690\pi\)
\(648\) 0 0
\(649\) −1366.21 −0.0826324
\(650\) 0 0
\(651\) −2024.61 −0.121891
\(652\) 0 0
\(653\) 8753.86 0.524602 0.262301 0.964986i \(-0.415519\pi\)
0.262301 + 0.964986i \(0.415519\pi\)
\(654\) 0 0
\(655\) 5006.62 0.298664
\(656\) 0 0
\(657\) −1675.67 −0.0995037
\(658\) 0 0
\(659\) 8133.42 0.480778 0.240389 0.970677i \(-0.422725\pi\)
0.240389 + 0.970677i \(0.422725\pi\)
\(660\) 0 0
\(661\) 8917.15 0.524716 0.262358 0.964971i \(-0.415500\pi\)
0.262358 + 0.964971i \(0.415500\pi\)
\(662\) 0 0
\(663\) −4001.47 −0.234396
\(664\) 0 0
\(665\) 30319.7 1.76804
\(666\) 0 0
\(667\) 11331.9 0.657831
\(668\) 0 0
\(669\) 27027.2 1.56193
\(670\) 0 0
\(671\) −539.837 −0.0310584
\(672\) 0 0
\(673\) 14664.4 0.839925 0.419963 0.907541i \(-0.362043\pi\)
0.419963 + 0.907541i \(0.362043\pi\)
\(674\) 0 0
\(675\) −29374.5 −1.67500
\(676\) 0 0
\(677\) 8195.60 0.465262 0.232631 0.972565i \(-0.425267\pi\)
0.232631 + 0.972565i \(0.425267\pi\)
\(678\) 0 0
\(679\) 21025.2 1.18833
\(680\) 0 0
\(681\) 7390.32 0.415855
\(682\) 0 0
\(683\) 18574.4 1.04060 0.520300 0.853983i \(-0.325820\pi\)
0.520300 + 0.853983i \(0.325820\pi\)
\(684\) 0 0
\(685\) −4853.09 −0.270696
\(686\) 0 0
\(687\) −19009.4 −1.05568
\(688\) 0 0
\(689\) 1715.73 0.0948679
\(690\) 0 0
\(691\) 20257.2 1.11523 0.557613 0.830101i \(-0.311718\pi\)
0.557613 + 0.830101i \(0.311718\pi\)
\(692\) 0 0
\(693\) 169.870 0.00931146
\(694\) 0 0
\(695\) 11631.7 0.634843
\(696\) 0 0
\(697\) −3926.54 −0.213383
\(698\) 0 0
\(699\) 3396.06 0.183763
\(700\) 0 0
\(701\) −22609.3 −1.21818 −0.609089 0.793102i \(-0.708465\pi\)
−0.609089 + 0.793102i \(0.708465\pi\)
\(702\) 0 0
\(703\) −15412.2 −0.826859
\(704\) 0 0
\(705\) −15132.0 −0.808373
\(706\) 0 0
\(707\) −11247.8 −0.598327
\(708\) 0 0
\(709\) 27690.9 1.46679 0.733395 0.679802i \(-0.237934\pi\)
0.733395 + 0.679802i \(0.237934\pi\)
\(710\) 0 0
\(711\) −5279.82 −0.278493
\(712\) 0 0
\(713\) −4226.43 −0.221993
\(714\) 0 0
\(715\) 1814.02 0.0948819
\(716\) 0 0
\(717\) 2367.84 0.123331
\(718\) 0 0
\(719\) 2111.24 0.109507 0.0547537 0.998500i \(-0.482563\pi\)
0.0547537 + 0.998500i \(0.482563\pi\)
\(720\) 0 0
\(721\) −14166.7 −0.731757
\(722\) 0 0
\(723\) −27389.9 −1.40891
\(724\) 0 0
\(725\) 16471.7 0.843784
\(726\) 0 0
\(727\) 14763.6 0.753164 0.376582 0.926383i \(-0.377099\pi\)
0.376582 + 0.926383i \(0.377099\pi\)
\(728\) 0 0
\(729\) 21896.2 1.11244
\(730\) 0 0
\(731\) 3149.75 0.159368
\(732\) 0 0
\(733\) −4836.29 −0.243700 −0.121850 0.992549i \(-0.538883\pi\)
−0.121850 + 0.992549i \(0.538883\pi\)
\(734\) 0 0
\(735\) 12449.1 0.624751
\(736\) 0 0
\(737\) 283.883 0.0141886
\(738\) 0 0
\(739\) 16015.7 0.797224 0.398612 0.917120i \(-0.369492\pi\)
0.398612 + 0.917120i \(0.369492\pi\)
\(740\) 0 0
\(741\) −26213.2 −1.29955
\(742\) 0 0
\(743\) 20313.3 1.00299 0.501495 0.865160i \(-0.332783\pi\)
0.501495 + 0.865160i \(0.332783\pi\)
\(744\) 0 0
\(745\) 15319.9 0.753391
\(746\) 0 0
\(747\) −2479.25 −0.121434
\(748\) 0 0
\(749\) 24347.8 1.18778
\(750\) 0 0
\(751\) −28755.3 −1.39720 −0.698598 0.715514i \(-0.746192\pi\)
−0.698598 + 0.715514i \(0.746192\pi\)
\(752\) 0 0
\(753\) −2023.89 −0.0979477
\(754\) 0 0
\(755\) −63205.4 −3.04673
\(756\) 0 0
\(757\) 32535.4 1.56211 0.781057 0.624460i \(-0.214681\pi\)
0.781057 + 0.624460i \(0.214681\pi\)
\(758\) 0 0
\(759\) −1356.31 −0.0648631
\(760\) 0 0
\(761\) −9298.53 −0.442932 −0.221466 0.975168i \(-0.571084\pi\)
−0.221466 + 0.975168i \(0.571084\pi\)
\(762\) 0 0
\(763\) −2131.60 −0.101139
\(764\) 0 0
\(765\) −1866.65 −0.0882208
\(766\) 0 0
\(767\) 28956.8 1.36319
\(768\) 0 0
\(769\) −20402.0 −0.956717 −0.478358 0.878165i \(-0.658768\pi\)
−0.478358 + 0.878165i \(0.658768\pi\)
\(770\) 0 0
\(771\) 1497.68 0.0699582
\(772\) 0 0
\(773\) 10376.1 0.482798 0.241399 0.970426i \(-0.422394\pi\)
0.241399 + 0.970426i \(0.422394\pi\)
\(774\) 0 0
\(775\) −6143.40 −0.284745
\(776\) 0 0
\(777\) 8096.67 0.373831
\(778\) 0 0
\(779\) −25722.3 −1.18305
\(780\) 0 0
\(781\) −762.109 −0.0349173
\(782\) 0 0
\(783\) −12752.4 −0.582034
\(784\) 0 0
\(785\) −31195.9 −1.41838
\(786\) 0 0
\(787\) −16869.6 −0.764088 −0.382044 0.924144i \(-0.624780\pi\)
−0.382044 + 0.924144i \(0.624780\pi\)
\(788\) 0 0
\(789\) 12441.2 0.561367
\(790\) 0 0
\(791\) 23896.4 1.07416
\(792\) 0 0
\(793\) 11441.8 0.512373
\(794\) 0 0
\(795\) −3061.27 −0.136569
\(796\) 0 0
\(797\) 9300.12 0.413334 0.206667 0.978411i \(-0.433738\pi\)
0.206667 + 0.978411i \(0.433738\pi\)
\(798\) 0 0
\(799\) −3411.68 −0.151060
\(800\) 0 0
\(801\) −7903.61 −0.348639
\(802\) 0 0
\(803\) −655.079 −0.0287886
\(804\) 0 0
\(805\) −33250.6 −1.45581
\(806\) 0 0
\(807\) 30775.8 1.34245
\(808\) 0 0
\(809\) −29320.9 −1.27425 −0.637126 0.770760i \(-0.719877\pi\)
−0.637126 + 0.770760i \(0.719877\pi\)
\(810\) 0 0
\(811\) 20488.9 0.887132 0.443566 0.896242i \(-0.353713\pi\)
0.443566 + 0.896242i \(0.353713\pi\)
\(812\) 0 0
\(813\) 9340.66 0.402941
\(814\) 0 0
\(815\) 64978.7 2.79276
\(816\) 0 0
\(817\) 20633.6 0.883574
\(818\) 0 0
\(819\) −3600.40 −0.153612
\(820\) 0 0
\(821\) −28650.9 −1.21793 −0.608966 0.793196i \(-0.708415\pi\)
−0.608966 + 0.793196i \(0.708415\pi\)
\(822\) 0 0
\(823\) 24605.9 1.04217 0.521086 0.853504i \(-0.325527\pi\)
0.521086 + 0.853504i \(0.325527\pi\)
\(824\) 0 0
\(825\) −1971.49 −0.0831982
\(826\) 0 0
\(827\) −34076.6 −1.43284 −0.716421 0.697668i \(-0.754221\pi\)
−0.716421 + 0.697668i \(0.754221\pi\)
\(828\) 0 0
\(829\) 1293.46 0.0541904 0.0270952 0.999633i \(-0.491374\pi\)
0.0270952 + 0.999633i \(0.491374\pi\)
\(830\) 0 0
\(831\) 20155.7 0.841388
\(832\) 0 0
\(833\) 2806.80 0.116746
\(834\) 0 0
\(835\) −54513.1 −2.25929
\(836\) 0 0
\(837\) 4756.22 0.196415
\(838\) 0 0
\(839\) 28847.5 1.18704 0.593521 0.804819i \(-0.297737\pi\)
0.593521 + 0.804819i \(0.297737\pi\)
\(840\) 0 0
\(841\) −17238.1 −0.706800
\(842\) 0 0
\(843\) 18013.1 0.735948
\(844\) 0 0
\(845\) 839.798 0.0341893
\(846\) 0 0
\(847\) −18401.7 −0.746506
\(848\) 0 0
\(849\) 13258.2 0.535947
\(850\) 0 0
\(851\) 16902.0 0.680839
\(852\) 0 0
\(853\) −58.7693 −0.00235900 −0.00117950 0.999999i \(-0.500375\pi\)
−0.00117950 + 0.999999i \(0.500375\pi\)
\(854\) 0 0
\(855\) −12228.2 −0.489118
\(856\) 0 0
\(857\) −20953.6 −0.835194 −0.417597 0.908632i \(-0.637128\pi\)
−0.417597 + 0.908632i \(0.637128\pi\)
\(858\) 0 0
\(859\) 41459.5 1.64677 0.823387 0.567481i \(-0.192082\pi\)
0.823387 + 0.567481i \(0.192082\pi\)
\(860\) 0 0
\(861\) 13513.0 0.534868
\(862\) 0 0
\(863\) −3389.59 −0.133700 −0.0668499 0.997763i \(-0.521295\pi\)
−0.0668499 + 0.997763i \(0.521295\pi\)
\(864\) 0 0
\(865\) 38492.3 1.51304
\(866\) 0 0
\(867\) −21120.0 −0.827304
\(868\) 0 0
\(869\) −2064.07 −0.0805741
\(870\) 0 0
\(871\) −6016.91 −0.234070
\(872\) 0 0
\(873\) −8479.66 −0.328743
\(874\) 0 0
\(875\) −17315.9 −0.669012
\(876\) 0 0
\(877\) −19675.3 −0.757568 −0.378784 0.925485i \(-0.623658\pi\)
−0.378784 + 0.925485i \(0.623658\pi\)
\(878\) 0 0
\(879\) −6553.79 −0.251484
\(880\) 0 0
\(881\) 1497.48 0.0572660 0.0286330 0.999590i \(-0.490885\pi\)
0.0286330 + 0.999590i \(0.490885\pi\)
\(882\) 0 0
\(883\) −5860.23 −0.223344 −0.111672 0.993745i \(-0.535621\pi\)
−0.111672 + 0.993745i \(0.535621\pi\)
\(884\) 0 0
\(885\) −51666.0 −1.96241
\(886\) 0 0
\(887\) 18058.0 0.683573 0.341787 0.939778i \(-0.388968\pi\)
0.341787 + 0.939778i \(0.388968\pi\)
\(888\) 0 0
\(889\) 9700.66 0.365973
\(890\) 0 0
\(891\) 1195.78 0.0449609
\(892\) 0 0
\(893\) −22349.5 −0.837512
\(894\) 0 0
\(895\) −7644.03 −0.285488
\(896\) 0 0
\(897\) 28747.1 1.07005
\(898\) 0 0
\(899\) −2667.04 −0.0989441
\(900\) 0 0
\(901\) −690.200 −0.0255204
\(902\) 0 0
\(903\) −10839.7 −0.399472
\(904\) 0 0
\(905\) 42915.2 1.57630
\(906\) 0 0
\(907\) 30297.4 1.10916 0.554580 0.832130i \(-0.312879\pi\)
0.554580 + 0.832130i \(0.312879\pi\)
\(908\) 0 0
\(909\) 4536.35 0.165524
\(910\) 0 0
\(911\) 31977.7 1.16297 0.581487 0.813556i \(-0.302471\pi\)
0.581487 + 0.813556i \(0.302471\pi\)
\(912\) 0 0
\(913\) −969.228 −0.0351334
\(914\) 0 0
\(915\) −20415.0 −0.737595
\(916\) 0 0
\(917\) −3884.72 −0.139896
\(918\) 0 0
\(919\) 40696.7 1.46078 0.730391 0.683029i \(-0.239338\pi\)
0.730391 + 0.683029i \(0.239338\pi\)
\(920\) 0 0
\(921\) 19409.4 0.694421
\(922\) 0 0
\(923\) 16152.9 0.576034
\(924\) 0 0
\(925\) 24568.2 0.873295
\(926\) 0 0
\(927\) 5713.57 0.202436
\(928\) 0 0
\(929\) −11467.5 −0.404989 −0.202495 0.979283i \(-0.564905\pi\)
−0.202495 + 0.979283i \(0.564905\pi\)
\(930\) 0 0
\(931\) 18387.0 0.647271
\(932\) 0 0
\(933\) 13468.4 0.472600
\(934\) 0 0
\(935\) −729.742 −0.0255242
\(936\) 0 0
\(937\) 14100.2 0.491603 0.245802 0.969320i \(-0.420949\pi\)
0.245802 + 0.969320i \(0.420949\pi\)
\(938\) 0 0
\(939\) 38342.7 1.33255
\(940\) 0 0
\(941\) −29781.6 −1.03172 −0.515862 0.856672i \(-0.672528\pi\)
−0.515862 + 0.856672i \(0.672528\pi\)
\(942\) 0 0
\(943\) 28208.8 0.974129
\(944\) 0 0
\(945\) 37418.5 1.28807
\(946\) 0 0
\(947\) 35353.8 1.21314 0.606571 0.795030i \(-0.292545\pi\)
0.606571 + 0.795030i \(0.292545\pi\)
\(948\) 0 0
\(949\) 13884.4 0.474928
\(950\) 0 0
\(951\) −37569.8 −1.28106
\(952\) 0 0
\(953\) 6456.01 0.219445 0.109722 0.993962i \(-0.465004\pi\)
0.109722 + 0.993962i \(0.465004\pi\)
\(954\) 0 0
\(955\) 72161.7 2.44513
\(956\) 0 0
\(957\) −855.885 −0.0289100
\(958\) 0 0
\(959\) 3765.59 0.126796
\(960\) 0 0
\(961\) −28796.3 −0.966610
\(962\) 0 0
\(963\) −9819.69 −0.328593
\(964\) 0 0
\(965\) 15847.0 0.528636
\(966\) 0 0
\(967\) 15099.9 0.502153 0.251076 0.967967i \(-0.419215\pi\)
0.251076 + 0.967967i \(0.419215\pi\)
\(968\) 0 0
\(969\) 10545.0 0.349591
\(970\) 0 0
\(971\) 8800.82 0.290867 0.145433 0.989368i \(-0.453542\pi\)
0.145433 + 0.989368i \(0.453542\pi\)
\(972\) 0 0
\(973\) −9025.23 −0.297364
\(974\) 0 0
\(975\) 41785.8 1.37253
\(976\) 0 0
\(977\) 34900.0 1.14284 0.571418 0.820659i \(-0.306393\pi\)
0.571418 + 0.820659i \(0.306393\pi\)
\(978\) 0 0
\(979\) −3089.81 −0.100869
\(980\) 0 0
\(981\) 859.693 0.0279795
\(982\) 0 0
\(983\) −21221.5 −0.688567 −0.344283 0.938866i \(-0.611878\pi\)
−0.344283 + 0.938866i \(0.611878\pi\)
\(984\) 0 0
\(985\) −82700.7 −2.67519
\(986\) 0 0
\(987\) 11741.1 0.378647
\(988\) 0 0
\(989\) −22628.2 −0.727538
\(990\) 0 0
\(991\) 23985.3 0.768838 0.384419 0.923159i \(-0.374402\pi\)
0.384419 + 0.923159i \(0.374402\pi\)
\(992\) 0 0
\(993\) −29437.7 −0.940761
\(994\) 0 0
\(995\) −3979.10 −0.126780
\(996\) 0 0
\(997\) 15222.3 0.483547 0.241773 0.970333i \(-0.422271\pi\)
0.241773 + 0.970333i \(0.422271\pi\)
\(998\) 0 0
\(999\) −19020.7 −0.602391
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 1024.4.a.n.1.8 10
4.3 odd 2 1024.4.a.m.1.3 10
8.3 odd 2 1024.4.a.m.1.8 10
8.5 even 2 inner 1024.4.a.n.1.3 10
16.3 odd 4 1024.4.b.k.513.3 10
16.5 even 4 1024.4.b.j.513.3 10
16.11 odd 4 1024.4.b.k.513.8 10
16.13 even 4 1024.4.b.j.513.8 10
32.3 odd 8 128.4.e.a.33.2 10
32.5 even 8 16.4.e.a.5.5 10
32.11 odd 8 128.4.e.a.97.2 10
32.13 even 8 16.4.e.a.13.5 yes 10
32.19 odd 8 64.4.e.a.17.4 10
32.21 even 8 128.4.e.b.97.4 10
32.27 odd 8 64.4.e.a.49.4 10
32.29 even 8 128.4.e.b.33.4 10
96.5 odd 8 144.4.k.a.37.1 10
96.59 even 8 576.4.k.a.433.5 10
96.77 odd 8 144.4.k.a.109.1 10
96.83 even 8 576.4.k.a.145.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.5 10 32.5 even 8
16.4.e.a.13.5 yes 10 32.13 even 8
64.4.e.a.17.4 10 32.19 odd 8
64.4.e.a.49.4 10 32.27 odd 8
128.4.e.a.33.2 10 32.3 odd 8
128.4.e.a.97.2 10 32.11 odd 8
128.4.e.b.33.4 10 32.29 even 8
128.4.e.b.97.4 10 32.21 even 8
144.4.k.a.37.1 10 96.5 odd 8
144.4.k.a.109.1 10 96.77 odd 8
576.4.k.a.145.5 10 96.83 even 8
576.4.k.a.433.5 10 96.59 even 8
1024.4.a.m.1.3 10 4.3 odd 2
1024.4.a.m.1.8 10 8.3 odd 2
1024.4.a.n.1.3 10 8.5 even 2 inner
1024.4.a.n.1.8 10 1.1 even 1 trivial
1024.4.b.j.513.3 10 16.5 even 4
1024.4.b.j.513.8 10 16.13 even 4
1024.4.b.k.513.3 10 16.3 odd 4
1024.4.b.k.513.8 10 16.11 odd 4