Properties

Label 1024.4.b.k.513.6
Level $1024$
Weight $4$
Character 1024.513
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
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)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 513.6
Root \(-3.94652i\) of defining polynomial
Character \(\chi\) \(=\) 1024.513
Dual form 1024.4.b.k.513.5

$q$-expansion

\(f(q)\) \(=\) \(q+1.07024i q^{3} -11.6331i q^{5} +2.67171 q^{7} +25.8546 q^{9} +O(q^{10})\) \(q+1.07024i q^{3} -11.6331i q^{5} +2.67171 q^{7} +25.8546 q^{9} +63.9525i q^{11} +50.0586i q^{13} +12.4503 q^{15} +72.4991 q^{17} -27.4961i q^{19} +2.85937i q^{21} -139.462 q^{23} -10.3299 q^{25} +56.5672i q^{27} -93.3995i q^{29} -188.682 q^{31} -68.4447 q^{33} -31.0803i q^{35} +118.886i q^{37} -53.5748 q^{39} -104.629 q^{41} +44.5275i q^{43} -300.770i q^{45} -488.151 q^{47} -335.862 q^{49} +77.5916i q^{51} +211.510i q^{53} +743.968 q^{55} +29.4275 q^{57} +402.624i q^{59} +322.538i q^{61} +69.0758 q^{63} +582.338 q^{65} +196.789i q^{67} -149.258i q^{69} +453.655 q^{71} +259.747 q^{73} -11.0555i q^{75} +170.862i q^{77} +323.190 q^{79} +637.533 q^{81} +797.471i q^{83} -843.392i q^{85} +99.9602 q^{87} +866.853 q^{89} +133.742i q^{91} -201.935i q^{93} -319.866 q^{95} -936.077 q^{97} +1653.47i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 28q^{7} - 54q^{9} + O(q^{10}) \) \( 10q + 28q^{7} - 54q^{9} - 124q^{15} + 4q^{17} + 276q^{23} - 50q^{25} - 368q^{31} - 4q^{33} + 732q^{39} - 944q^{47} - 94q^{49} + 1380q^{55} - 108q^{57} - 2628q^{63} - 492q^{65} + 3468q^{71} + 296q^{73} - 4416q^{79} - 482q^{81} + 6036q^{87} - 88q^{89} - 6900q^{95} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 1.07024i 0.205968i 0.994683 + 0.102984i \(0.0328391\pi\)
−0.994683 + 0.102984i \(0.967161\pi\)
\(4\) 0 0
\(5\) − 11.6331i − 1.04050i −0.854014 0.520250i \(-0.825839\pi\)
0.854014 0.520250i \(-0.174161\pi\)
\(6\) 0 0
\(7\) 2.67171 0.144259 0.0721293 0.997395i \(-0.477021\pi\)
0.0721293 + 0.997395i \(0.477021\pi\)
\(8\) 0 0
\(9\) 25.8546 0.957577
\(10\) 0 0
\(11\) 63.9525i 1.75295i 0.481451 + 0.876473i \(0.340110\pi\)
−0.481451 + 0.876473i \(0.659890\pi\)
\(12\) 0 0
\(13\) 50.0586i 1.06798i 0.845491 + 0.533990i \(0.179308\pi\)
−0.845491 + 0.533990i \(0.820692\pi\)
\(14\) 0 0
\(15\) 12.4503 0.214310
\(16\) 0 0
\(17\) 72.4991 1.03433 0.517165 0.855886i \(-0.326987\pi\)
0.517165 + 0.855886i \(0.326987\pi\)
\(18\) 0 0
\(19\) − 27.4961i − 0.332002i −0.986126 0.166001i \(-0.946915\pi\)
0.986126 0.166001i \(-0.0530855\pi\)
\(20\) 0 0
\(21\) 2.85937i 0.0297127i
\(22\) 0 0
\(23\) −139.462 −1.26434 −0.632170 0.774830i \(-0.717835\pi\)
−0.632170 + 0.774830i \(0.717835\pi\)
\(24\) 0 0
\(25\) −10.3299 −0.0826390
\(26\) 0 0
\(27\) 56.5672i 0.403199i
\(28\) 0 0
\(29\) − 93.3995i − 0.598064i −0.954243 0.299032i \(-0.903336\pi\)
0.954243 0.299032i \(-0.0966637\pi\)
\(30\) 0 0
\(31\) −188.682 −1.09317 −0.546584 0.837404i \(-0.684072\pi\)
−0.546584 + 0.837404i \(0.684072\pi\)
\(32\) 0 0
\(33\) −68.4447 −0.361051
\(34\) 0 0
\(35\) − 31.0803i − 0.150101i
\(36\) 0 0
\(37\) 118.886i 0.528238i 0.964490 + 0.264119i \(0.0850811\pi\)
−0.964490 + 0.264119i \(0.914919\pi\)
\(38\) 0 0
\(39\) −53.5748 −0.219970
\(40\) 0 0
\(41\) −104.629 −0.398545 −0.199272 0.979944i \(-0.563858\pi\)
−0.199272 + 0.979944i \(0.563858\pi\)
\(42\) 0 0
\(43\) 44.5275i 0.157916i 0.996878 + 0.0789579i \(0.0251593\pi\)
−0.996878 + 0.0789579i \(0.974841\pi\)
\(44\) 0 0
\(45\) − 300.770i − 0.996358i
\(46\) 0 0
\(47\) −488.151 −1.51498 −0.757491 0.652846i \(-0.773575\pi\)
−0.757491 + 0.652846i \(0.773575\pi\)
\(48\) 0 0
\(49\) −335.862 −0.979189
\(50\) 0 0
\(51\) 77.5916i 0.213039i
\(52\) 0 0
\(53\) 211.510i 0.548173i 0.961705 + 0.274087i \(0.0883755\pi\)
−0.961705 + 0.274087i \(0.911624\pi\)
\(54\) 0 0
\(55\) 743.968 1.82394
\(56\) 0 0
\(57\) 29.4275 0.0683819
\(58\) 0 0
\(59\) 402.624i 0.888426i 0.895921 + 0.444213i \(0.146517\pi\)
−0.895921 + 0.444213i \(0.853483\pi\)
\(60\) 0 0
\(61\) 322.538i 0.676997i 0.940967 + 0.338498i \(0.109919\pi\)
−0.940967 + 0.338498i \(0.890081\pi\)
\(62\) 0 0
\(63\) 69.0758 0.138139
\(64\) 0 0
\(65\) 582.338 1.11123
\(66\) 0 0
\(67\) 196.789i 0.358829i 0.983774 + 0.179415i \(0.0574203\pi\)
−0.983774 + 0.179415i \(0.942580\pi\)
\(68\) 0 0
\(69\) − 149.258i − 0.260414i
\(70\) 0 0
\(71\) 453.655 0.758294 0.379147 0.925336i \(-0.376217\pi\)
0.379147 + 0.925336i \(0.376217\pi\)
\(72\) 0 0
\(73\) 259.747 0.416454 0.208227 0.978081i \(-0.433231\pi\)
0.208227 + 0.978081i \(0.433231\pi\)
\(74\) 0 0
\(75\) − 11.0555i − 0.0170210i
\(76\) 0 0
\(77\) 170.862i 0.252877i
\(78\) 0 0
\(79\) 323.190 0.460275 0.230138 0.973158i \(-0.426082\pi\)
0.230138 + 0.973158i \(0.426082\pi\)
\(80\) 0 0
\(81\) 637.533 0.874531
\(82\) 0 0
\(83\) 797.471i 1.05462i 0.849672 + 0.527312i \(0.176800\pi\)
−0.849672 + 0.527312i \(0.823200\pi\)
\(84\) 0 0
\(85\) − 843.392i − 1.07622i
\(86\) 0 0
\(87\) 99.9602 0.123182
\(88\) 0 0
\(89\) 866.853 1.03243 0.516215 0.856459i \(-0.327341\pi\)
0.516215 + 0.856459i \(0.327341\pi\)
\(90\) 0 0
\(91\) 133.742i 0.154065i
\(92\) 0 0
\(93\) − 201.935i − 0.225158i
\(94\) 0 0
\(95\) −319.866 −0.345448
\(96\) 0 0
\(97\) −936.077 −0.979837 −0.489919 0.871768i \(-0.662974\pi\)
−0.489919 + 0.871768i \(0.662974\pi\)
\(98\) 0 0
\(99\) 1653.47i 1.67858i
\(100\) 0 0
\(101\) 2.24639i 0.00221311i 0.999999 + 0.00110656i \(0.000352228\pi\)
−0.999999 + 0.00110656i \(0.999648\pi\)
\(102\) 0 0
\(103\) 1388.28 1.32807 0.664036 0.747700i \(-0.268842\pi\)
0.664036 + 0.747700i \(0.268842\pi\)
\(104\) 0 0
\(105\) 33.2635 0.0309160
\(106\) 0 0
\(107\) 1161.81i 1.04969i 0.851198 + 0.524845i \(0.175877\pi\)
−0.851198 + 0.524845i \(0.824123\pi\)
\(108\) 0 0
\(109\) 753.489i 0.662121i 0.943610 + 0.331060i \(0.107406\pi\)
−0.943610 + 0.331060i \(0.892594\pi\)
\(110\) 0 0
\(111\) −127.237 −0.108800
\(112\) 0 0
\(113\) 67.2680 0.0560003 0.0280002 0.999608i \(-0.491086\pi\)
0.0280002 + 0.999608i \(0.491086\pi\)
\(114\) 0 0
\(115\) 1622.38i 1.31554i
\(116\) 0 0
\(117\) 1294.24i 1.02267i
\(118\) 0 0
\(119\) 193.696 0.149211
\(120\) 0 0
\(121\) −2758.92 −2.07282
\(122\) 0 0
\(123\) − 111.979i − 0.0820876i
\(124\) 0 0
\(125\) − 1333.97i − 0.954514i
\(126\) 0 0
\(127\) −1903.59 −1.33005 −0.665026 0.746820i \(-0.731579\pi\)
−0.665026 + 0.746820i \(0.731579\pi\)
\(128\) 0 0
\(129\) −47.6552 −0.0325257
\(130\) 0 0
\(131\) 1298.86i 0.866271i 0.901329 + 0.433136i \(0.142593\pi\)
−0.901329 + 0.433136i \(0.857407\pi\)
\(132\) 0 0
\(133\) − 73.4615i − 0.0478941i
\(134\) 0 0
\(135\) 658.054 0.419528
\(136\) 0 0
\(137\) −477.234 −0.297612 −0.148806 0.988866i \(-0.547543\pi\)
−0.148806 + 0.988866i \(0.547543\pi\)
\(138\) 0 0
\(139\) 2140.97i 1.30643i 0.757171 + 0.653217i \(0.226581\pi\)
−0.757171 + 0.653217i \(0.773419\pi\)
\(140\) 0 0
\(141\) − 522.440i − 0.312038i
\(142\) 0 0
\(143\) −3201.37 −1.87211
\(144\) 0 0
\(145\) −1086.53 −0.622285
\(146\) 0 0
\(147\) − 359.454i − 0.201682i
\(148\) 0 0
\(149\) 530.829i 0.291861i 0.989295 + 0.145930i \(0.0466175\pi\)
−0.989295 + 0.145930i \(0.953382\pi\)
\(150\) 0 0
\(151\) 2997.52 1.61546 0.807730 0.589553i \(-0.200696\pi\)
0.807730 + 0.589553i \(0.200696\pi\)
\(152\) 0 0
\(153\) 1874.43 0.990451
\(154\) 0 0
\(155\) 2194.96i 1.13744i
\(156\) 0 0
\(157\) 2134.06i 1.08482i 0.840115 + 0.542409i \(0.182488\pi\)
−0.840115 + 0.542409i \(0.817512\pi\)
\(158\) 0 0
\(159\) −226.368 −0.112906
\(160\) 0 0
\(161\) −372.601 −0.182392
\(162\) 0 0
\(163\) − 2015.52i − 0.968515i −0.874925 0.484258i \(-0.839090\pi\)
0.874925 0.484258i \(-0.160910\pi\)
\(164\) 0 0
\(165\) 796.227i 0.375674i
\(166\) 0 0
\(167\) 792.415 0.367179 0.183590 0.983003i \(-0.441228\pi\)
0.183590 + 0.983003i \(0.441228\pi\)
\(168\) 0 0
\(169\) −308.861 −0.140583
\(170\) 0 0
\(171\) − 710.900i − 0.317917i
\(172\) 0 0
\(173\) − 1094.03i − 0.480794i −0.970675 0.240397i \(-0.922722\pi\)
0.970675 0.240397i \(-0.0772776\pi\)
\(174\) 0 0
\(175\) −27.5984 −0.0119214
\(176\) 0 0
\(177\) −430.905 −0.182988
\(178\) 0 0
\(179\) − 602.526i − 0.251592i −0.992056 0.125796i \(-0.959852\pi\)
0.992056 0.125796i \(-0.0401484\pi\)
\(180\) 0 0
\(181\) − 3702.50i − 1.52047i −0.649649 0.760234i \(-0.725084\pi\)
0.649649 0.760234i \(-0.274916\pi\)
\(182\) 0 0
\(183\) −345.194 −0.139440
\(184\) 0 0
\(185\) 1383.02 0.549631
\(186\) 0 0
\(187\) 4636.50i 1.81312i
\(188\) 0 0
\(189\) 151.131i 0.0581649i
\(190\) 0 0
\(191\) −3216.39 −1.21848 −0.609240 0.792986i \(-0.708525\pi\)
−0.609240 + 0.792986i \(0.708525\pi\)
\(192\) 0 0
\(193\) 2852.57 1.06390 0.531950 0.846776i \(-0.321459\pi\)
0.531950 + 0.846776i \(0.321459\pi\)
\(194\) 0 0
\(195\) 623.243i 0.228879i
\(196\) 0 0
\(197\) − 2275.50i − 0.822956i −0.911420 0.411478i \(-0.865013\pi\)
0.911420 0.411478i \(-0.134987\pi\)
\(198\) 0 0
\(199\) −747.136 −0.266146 −0.133073 0.991106i \(-0.542484\pi\)
−0.133073 + 0.991106i \(0.542484\pi\)
\(200\) 0 0
\(201\) −210.612 −0.0739074
\(202\) 0 0
\(203\) − 249.536i − 0.0862758i
\(204\) 0 0
\(205\) 1217.17i 0.414685i
\(206\) 0 0
\(207\) −3605.73 −1.21070
\(208\) 0 0
\(209\) 1758.44 0.581981
\(210\) 0 0
\(211\) 3149.64i 1.02763i 0.857901 + 0.513816i \(0.171769\pi\)
−0.857901 + 0.513816i \(0.828231\pi\)
\(212\) 0 0
\(213\) 485.520i 0.156185i
\(214\) 0 0
\(215\) 517.995 0.164311
\(216\) 0 0
\(217\) −504.102 −0.157699
\(218\) 0 0
\(219\) 277.993i 0.0857763i
\(220\) 0 0
\(221\) 3629.20i 1.10464i
\(222\) 0 0
\(223\) 358.053 0.107520 0.0537601 0.998554i \(-0.482879\pi\)
0.0537601 + 0.998554i \(0.482879\pi\)
\(224\) 0 0
\(225\) −267.075 −0.0791332
\(226\) 0 0
\(227\) 4886.67i 1.42881i 0.699733 + 0.714404i \(0.253302\pi\)
−0.699733 + 0.714404i \(0.746698\pi\)
\(228\) 0 0
\(229\) 2022.37i 0.583589i 0.956481 + 0.291794i \(0.0942523\pi\)
−0.956481 + 0.291794i \(0.905748\pi\)
\(230\) 0 0
\(231\) −182.864 −0.0520847
\(232\) 0 0
\(233\) 926.479 0.260496 0.130248 0.991481i \(-0.458423\pi\)
0.130248 + 0.991481i \(0.458423\pi\)
\(234\) 0 0
\(235\) 5678.73i 1.57634i
\(236\) 0 0
\(237\) 345.892i 0.0948021i
\(238\) 0 0
\(239\) −792.472 −0.214480 −0.107240 0.994233i \(-0.534201\pi\)
−0.107240 + 0.994233i \(0.534201\pi\)
\(240\) 0 0
\(241\) −1449.01 −0.387299 −0.193650 0.981071i \(-0.562033\pi\)
−0.193650 + 0.981071i \(0.562033\pi\)
\(242\) 0 0
\(243\) 2209.63i 0.583325i
\(244\) 0 0
\(245\) 3907.13i 1.01885i
\(246\) 0 0
\(247\) 1376.42 0.354572
\(248\) 0 0
\(249\) −853.487 −0.217219
\(250\) 0 0
\(251\) − 5062.94i − 1.27319i −0.771199 0.636594i \(-0.780343\pi\)
0.771199 0.636594i \(-0.219657\pi\)
\(252\) 0 0
\(253\) − 8918.93i − 2.21632i
\(254\) 0 0
\(255\) 902.634 0.221667
\(256\) 0 0
\(257\) −4708.87 −1.14292 −0.571461 0.820629i \(-0.693623\pi\)
−0.571461 + 0.820629i \(0.693623\pi\)
\(258\) 0 0
\(259\) 317.629i 0.0762028i
\(260\) 0 0
\(261\) − 2414.81i − 0.572692i
\(262\) 0 0
\(263\) 2967.82 0.695830 0.347915 0.937526i \(-0.386890\pi\)
0.347915 + 0.937526i \(0.386890\pi\)
\(264\) 0 0
\(265\) 2460.53 0.570374
\(266\) 0 0
\(267\) 927.743i 0.212648i
\(268\) 0 0
\(269\) − 938.519i − 0.212723i −0.994328 0.106362i \(-0.966080\pi\)
0.994328 0.106362i \(-0.0339201\pi\)
\(270\) 0 0
\(271\) 8058.74 1.80640 0.903199 0.429223i \(-0.141212\pi\)
0.903199 + 0.429223i \(0.141212\pi\)
\(272\) 0 0
\(273\) −143.136 −0.0317326
\(274\) 0 0
\(275\) − 660.622i − 0.144862i
\(276\) 0 0
\(277\) 682.325i 0.148003i 0.997258 + 0.0740017i \(0.0235770\pi\)
−0.997258 + 0.0740017i \(0.976423\pi\)
\(278\) 0 0
\(279\) −4878.29 −1.04679
\(280\) 0 0
\(281\) −5899.10 −1.25235 −0.626175 0.779682i \(-0.715381\pi\)
−0.626175 + 0.779682i \(0.715381\pi\)
\(282\) 0 0
\(283\) − 961.519i − 0.201966i −0.994888 0.100983i \(-0.967801\pi\)
0.994888 0.100983i \(-0.0321988\pi\)
\(284\) 0 0
\(285\) − 342.334i − 0.0711513i
\(286\) 0 0
\(287\) −279.538 −0.0574935
\(288\) 0 0
\(289\) 343.118 0.0698388
\(290\) 0 0
\(291\) − 1001.83i − 0.201815i
\(292\) 0 0
\(293\) 5024.51i 1.00183i 0.865498 + 0.500913i \(0.167002\pi\)
−0.865498 + 0.500913i \(0.832998\pi\)
\(294\) 0 0
\(295\) 4683.78 0.924407
\(296\) 0 0
\(297\) −3617.62 −0.706786
\(298\) 0 0
\(299\) − 6981.26i − 1.35029i
\(300\) 0 0
\(301\) 118.964i 0.0227807i
\(302\) 0 0
\(303\) −2.40419 −0.000455831 0
\(304\) 0 0
\(305\) 3752.13 0.704415
\(306\) 0 0
\(307\) − 3868.67i − 0.719207i −0.933105 0.359603i \(-0.882912\pi\)
0.933105 0.359603i \(-0.117088\pi\)
\(308\) 0 0
\(309\) 1485.80i 0.273541i
\(310\) 0 0
\(311\) 5796.70 1.05692 0.528458 0.848960i \(-0.322771\pi\)
0.528458 + 0.848960i \(0.322771\pi\)
\(312\) 0 0
\(313\) 8362.62 1.51017 0.755085 0.655627i \(-0.227596\pi\)
0.755085 + 0.655627i \(0.227596\pi\)
\(314\) 0 0
\(315\) − 803.569i − 0.143733i
\(316\) 0 0
\(317\) 487.064i 0.0862973i 0.999069 + 0.0431487i \(0.0137389\pi\)
−0.999069 + 0.0431487i \(0.986261\pi\)
\(318\) 0 0
\(319\) 5973.13 1.04837
\(320\) 0 0
\(321\) −1243.42 −0.216203
\(322\) 0 0
\(323\) − 1993.44i − 0.343400i
\(324\) 0 0
\(325\) − 517.099i − 0.0882569i
\(326\) 0 0
\(327\) −806.416 −0.136376
\(328\) 0 0
\(329\) −1304.20 −0.218549
\(330\) 0 0
\(331\) − 3801.21i − 0.631218i −0.948889 0.315609i \(-0.897791\pi\)
0.948889 0.315609i \(-0.102209\pi\)
\(332\) 0 0
\(333\) 3073.76i 0.505828i
\(334\) 0 0
\(335\) 2289.27 0.373361
\(336\) 0 0
\(337\) 1795.31 0.290199 0.145099 0.989417i \(-0.453650\pi\)
0.145099 + 0.989417i \(0.453650\pi\)
\(338\) 0 0
\(339\) 71.9931i 0.0115343i
\(340\) 0 0
\(341\) − 12066.7i − 1.91627i
\(342\) 0 0
\(343\) −1813.72 −0.285515
\(344\) 0 0
\(345\) −1736.34 −0.270960
\(346\) 0 0
\(347\) − 2782.23i − 0.430426i −0.976567 0.215213i \(-0.930955\pi\)
0.976567 0.215213i \(-0.0690445\pi\)
\(348\) 0 0
\(349\) 10413.4i 1.59718i 0.601876 + 0.798589i \(0.294420\pi\)
−0.601876 + 0.798589i \(0.705580\pi\)
\(350\) 0 0
\(351\) −2831.68 −0.430609
\(352\) 0 0
\(353\) 10644.3 1.60493 0.802466 0.596698i \(-0.203521\pi\)
0.802466 + 0.596698i \(0.203521\pi\)
\(354\) 0 0
\(355\) − 5277.43i − 0.789005i
\(356\) 0 0
\(357\) 207.302i 0.0307327i
\(358\) 0 0
\(359\) −7459.42 −1.09664 −0.548319 0.836269i \(-0.684732\pi\)
−0.548319 + 0.836269i \(0.684732\pi\)
\(360\) 0 0
\(361\) 6102.96 0.889775
\(362\) 0 0
\(363\) − 2952.72i − 0.426935i
\(364\) 0 0
\(365\) − 3021.68i − 0.433320i
\(366\) 0 0
\(367\) −6251.35 −0.889149 −0.444574 0.895742i \(-0.646645\pi\)
−0.444574 + 0.895742i \(0.646645\pi\)
\(368\) 0 0
\(369\) −2705.14 −0.381637
\(370\) 0 0
\(371\) 565.094i 0.0790787i
\(372\) 0 0
\(373\) 12603.3i 1.74953i 0.484552 + 0.874763i \(0.338983\pi\)
−0.484552 + 0.874763i \(0.661017\pi\)
\(374\) 0 0
\(375\) 1427.68 0.196600
\(376\) 0 0
\(377\) 4675.45 0.638721
\(378\) 0 0
\(379\) 1674.47i 0.226943i 0.993541 + 0.113472i \(0.0361971\pi\)
−0.993541 + 0.113472i \(0.963803\pi\)
\(380\) 0 0
\(381\) − 2037.31i − 0.273949i
\(382\) 0 0
\(383\) 2880.38 0.384283 0.192142 0.981367i \(-0.438457\pi\)
0.192142 + 0.981367i \(0.438457\pi\)
\(384\) 0 0
\(385\) 1987.66 0.263119
\(386\) 0 0
\(387\) 1151.24i 0.151217i
\(388\) 0 0
\(389\) − 13073.3i − 1.70397i −0.523567 0.851984i \(-0.675399\pi\)
0.523567 0.851984i \(-0.324601\pi\)
\(390\) 0 0
\(391\) −10110.9 −1.30774
\(392\) 0 0
\(393\) −1390.09 −0.178424
\(394\) 0 0
\(395\) − 3759.72i − 0.478916i
\(396\) 0 0
\(397\) − 6021.44i − 0.761228i −0.924734 0.380614i \(-0.875713\pi\)
0.924734 0.380614i \(-0.124287\pi\)
\(398\) 0 0
\(399\) 78.6216 0.00986467
\(400\) 0 0
\(401\) −12722.6 −1.58437 −0.792187 0.610278i \(-0.791058\pi\)
−0.792187 + 0.610278i \(0.791058\pi\)
\(402\) 0 0
\(403\) − 9445.14i − 1.16748i
\(404\) 0 0
\(405\) − 7416.51i − 0.909949i
\(406\) 0 0
\(407\) −7603.08 −0.925972
\(408\) 0 0
\(409\) 232.991 0.0281678 0.0140839 0.999901i \(-0.495517\pi\)
0.0140839 + 0.999901i \(0.495517\pi\)
\(410\) 0 0
\(411\) − 510.756i − 0.0612987i
\(412\) 0 0
\(413\) 1075.69i 0.128163i
\(414\) 0 0
\(415\) 9277.08 1.09734
\(416\) 0 0
\(417\) −2291.35 −0.269084
\(418\) 0 0
\(419\) − 8663.03i − 1.01006i −0.863101 0.505032i \(-0.831481\pi\)
0.863101 0.505032i \(-0.168519\pi\)
\(420\) 0 0
\(421\) 11749.9i 1.36023i 0.733107 + 0.680113i \(0.238069\pi\)
−0.733107 + 0.680113i \(0.761931\pi\)
\(422\) 0 0
\(423\) −12620.9 −1.45071
\(424\) 0 0
\(425\) −748.907 −0.0854760
\(426\) 0 0
\(427\) 861.727i 0.0976626i
\(428\) 0 0
\(429\) − 3426.24i − 0.385596i
\(430\) 0 0
\(431\) 8737.57 0.976506 0.488253 0.872702i \(-0.337634\pi\)
0.488253 + 0.872702i \(0.337634\pi\)
\(432\) 0 0
\(433\) 11627.5 1.29049 0.645247 0.763974i \(-0.276755\pi\)
0.645247 + 0.763974i \(0.276755\pi\)
\(434\) 0 0
\(435\) − 1162.85i − 0.128171i
\(436\) 0 0
\(437\) 3834.66i 0.419763i
\(438\) 0 0
\(439\) −17631.8 −1.91690 −0.958450 0.285261i \(-0.907920\pi\)
−0.958450 + 0.285261i \(0.907920\pi\)
\(440\) 0 0
\(441\) −8683.57 −0.937649
\(442\) 0 0
\(443\) − 6434.41i − 0.690085i −0.938587 0.345043i \(-0.887864\pi\)
0.938587 0.345043i \(-0.112136\pi\)
\(444\) 0 0
\(445\) − 10084.2i − 1.07424i
\(446\) 0 0
\(447\) −568.116 −0.0601140
\(448\) 0 0
\(449\) −12926.5 −1.35867 −0.679334 0.733830i \(-0.737731\pi\)
−0.679334 + 0.733830i \(0.737731\pi\)
\(450\) 0 0
\(451\) − 6691.30i − 0.698627i
\(452\) 0 0
\(453\) 3208.07i 0.332734i
\(454\) 0 0
\(455\) 1555.84 0.160305
\(456\) 0 0
\(457\) −9320.32 −0.954018 −0.477009 0.878898i \(-0.658279\pi\)
−0.477009 + 0.878898i \(0.658279\pi\)
\(458\) 0 0
\(459\) 4101.07i 0.417041i
\(460\) 0 0
\(461\) 18222.1i 1.84098i 0.390771 + 0.920488i \(0.372208\pi\)
−0.390771 + 0.920488i \(0.627792\pi\)
\(462\) 0 0
\(463\) 7038.37 0.706482 0.353241 0.935532i \(-0.385080\pi\)
0.353241 + 0.935532i \(0.385080\pi\)
\(464\) 0 0
\(465\) −2349.14 −0.234277
\(466\) 0 0
\(467\) − 8487.77i − 0.841043i −0.907283 0.420522i \(-0.861847\pi\)
0.907283 0.420522i \(-0.138153\pi\)
\(468\) 0 0
\(469\) 525.761i 0.0517642i
\(470\) 0 0
\(471\) −2283.96 −0.223438
\(472\) 0 0
\(473\) −2847.65 −0.276818
\(474\) 0 0
\(475\) 284.031i 0.0274363i
\(476\) 0 0
\(477\) 5468.51i 0.524918i
\(478\) 0 0
\(479\) 587.317 0.0560234 0.0280117 0.999608i \(-0.491082\pi\)
0.0280117 + 0.999608i \(0.491082\pi\)
\(480\) 0 0
\(481\) −5951.28 −0.564148
\(482\) 0 0
\(483\) − 398.773i − 0.0375669i
\(484\) 0 0
\(485\) 10889.5i 1.01952i
\(486\) 0 0
\(487\) 8366.45 0.778481 0.389240 0.921136i \(-0.372738\pi\)
0.389240 + 0.921136i \(0.372738\pi\)
\(488\) 0 0
\(489\) 2157.10 0.199483
\(490\) 0 0
\(491\) − 2162.76i − 0.198786i −0.995048 0.0993929i \(-0.968310\pi\)
0.995048 0.0993929i \(-0.0316901\pi\)
\(492\) 0 0
\(493\) − 6771.38i − 0.618596i
\(494\) 0 0
\(495\) 19235.0 1.74656
\(496\) 0 0
\(497\) 1212.03 0.109390
\(498\) 0 0
\(499\) − 16071.8i − 1.44183i −0.693026 0.720913i \(-0.743723\pi\)
0.693026 0.720913i \(-0.256277\pi\)
\(500\) 0 0
\(501\) 848.077i 0.0756273i
\(502\) 0 0
\(503\) −12570.2 −1.11427 −0.557137 0.830421i \(-0.688100\pi\)
−0.557137 + 0.830421i \(0.688100\pi\)
\(504\) 0 0
\(505\) 26.1326 0.00230274
\(506\) 0 0
\(507\) − 330.556i − 0.0289556i
\(508\) 0 0
\(509\) − 16801.4i − 1.46308i −0.681797 0.731542i \(-0.738801\pi\)
0.681797 0.731542i \(-0.261199\pi\)
\(510\) 0 0
\(511\) 693.968 0.0600770
\(512\) 0 0
\(513\) 1555.38 0.133863
\(514\) 0 0
\(515\) − 16150.1i − 1.38186i
\(516\) 0 0
\(517\) − 31218.5i − 2.65568i
\(518\) 0 0
\(519\) 1170.87 0.0990283
\(520\) 0 0
\(521\) 6612.98 0.556085 0.278042 0.960569i \(-0.410314\pi\)
0.278042 + 0.960569i \(0.410314\pi\)
\(522\) 0 0
\(523\) − 7253.92i − 0.606485i −0.952913 0.303243i \(-0.901931\pi\)
0.952913 0.303243i \(-0.0980693\pi\)
\(524\) 0 0
\(525\) − 29.5370i − 0.00245543i
\(526\) 0 0
\(527\) −13679.3 −1.13070
\(528\) 0 0
\(529\) 7282.60 0.598553
\(530\) 0 0
\(531\) 10409.7i 0.850737i
\(532\) 0 0
\(533\) − 5237.59i − 0.425638i
\(534\) 0 0
\(535\) 13515.5 1.09220
\(536\) 0 0
\(537\) 644.849 0.0518199
\(538\) 0 0
\(539\) − 21479.2i − 1.71647i
\(540\) 0 0
\(541\) 15511.9i 1.23273i 0.787461 + 0.616365i \(0.211395\pi\)
−0.787461 + 0.616365i \(0.788605\pi\)
\(542\) 0 0
\(543\) 3962.57 0.313168
\(544\) 0 0
\(545\) 8765.44 0.688936
\(546\) 0 0
\(547\) 18510.4i 1.44689i 0.690382 + 0.723445i \(0.257442\pi\)
−0.690382 + 0.723445i \(0.742558\pi\)
\(548\) 0 0
\(549\) 8339.09i 0.648277i
\(550\) 0 0
\(551\) −2568.12 −0.198558
\(552\) 0 0
\(553\) 863.469 0.0663986
\(554\) 0 0
\(555\) 1480.17i 0.113207i
\(556\) 0 0
\(557\) 7141.59i 0.543266i 0.962401 + 0.271633i \(0.0875637\pi\)
−0.962401 + 0.271633i \(0.912436\pi\)
\(558\) 0 0
\(559\) −2228.98 −0.168651
\(560\) 0 0
\(561\) −4962.18 −0.373446
\(562\) 0 0
\(563\) 4594.86i 0.343961i 0.985100 + 0.171981i \(0.0550167\pi\)
−0.985100 + 0.171981i \(0.944983\pi\)
\(564\) 0 0
\(565\) − 782.538i − 0.0582683i
\(566\) 0 0
\(567\) 1703.30 0.126159
\(568\) 0 0
\(569\) 2806.05 0.206741 0.103371 0.994643i \(-0.467037\pi\)
0.103371 + 0.994643i \(0.467037\pi\)
\(570\) 0 0
\(571\) − 17025.4i − 1.24779i −0.781506 0.623897i \(-0.785548\pi\)
0.781506 0.623897i \(-0.214452\pi\)
\(572\) 0 0
\(573\) − 3442.31i − 0.250968i
\(574\) 0 0
\(575\) 1440.62 0.104484
\(576\) 0 0
\(577\) 7206.84 0.519973 0.259987 0.965612i \(-0.416282\pi\)
0.259987 + 0.965612i \(0.416282\pi\)
\(578\) 0 0
\(579\) 3052.95i 0.219130i
\(580\) 0 0
\(581\) 2130.61i 0.152138i
\(582\) 0 0
\(583\) −13526.6 −0.960918
\(584\) 0 0
\(585\) 15056.1 1.06409
\(586\) 0 0
\(587\) − 14675.2i − 1.03188i −0.856626 0.515938i \(-0.827443\pi\)
0.856626 0.515938i \(-0.172557\pi\)
\(588\) 0 0
\(589\) 5188.01i 0.362934i
\(590\) 0 0
\(591\) 2435.33 0.169503
\(592\) 0 0
\(593\) 4758.60 0.329531 0.164766 0.986333i \(-0.447313\pi\)
0.164766 + 0.986333i \(0.447313\pi\)
\(594\) 0 0
\(595\) − 2253.29i − 0.155254i
\(596\) 0 0
\(597\) − 799.617i − 0.0548176i
\(598\) 0 0
\(599\) 14256.4 0.972455 0.486227 0.873832i \(-0.338373\pi\)
0.486227 + 0.873832i \(0.338373\pi\)
\(600\) 0 0
\(601\) 10385.2 0.704862 0.352431 0.935838i \(-0.385355\pi\)
0.352431 + 0.935838i \(0.385355\pi\)
\(602\) 0 0
\(603\) 5087.89i 0.343606i
\(604\) 0 0
\(605\) 32094.9i 2.15677i
\(606\) 0 0
\(607\) 16243.6 1.08618 0.543088 0.839676i \(-0.317255\pi\)
0.543088 + 0.839676i \(0.317255\pi\)
\(608\) 0 0
\(609\) 267.064 0.0177701
\(610\) 0 0
\(611\) − 24436.1i − 1.61797i
\(612\) 0 0
\(613\) 707.817i 0.0466369i 0.999728 + 0.0233185i \(0.00742317\pi\)
−0.999728 + 0.0233185i \(0.992577\pi\)
\(614\) 0 0
\(615\) −1302.66 −0.0854121
\(616\) 0 0
\(617\) −11575.9 −0.755316 −0.377658 0.925945i \(-0.623271\pi\)
−0.377658 + 0.925945i \(0.623271\pi\)
\(618\) 0 0
\(619\) − 25959.4i − 1.68562i −0.538214 0.842808i \(-0.680901\pi\)
0.538214 0.842808i \(-0.319099\pi\)
\(620\) 0 0
\(621\) − 7888.97i − 0.509780i
\(622\) 0 0
\(623\) 2315.98 0.148937
\(624\) 0 0
\(625\) −16809.5 −1.07581
\(626\) 0 0
\(627\) 1881.96i 0.119870i
\(628\) 0 0
\(629\) 8619.15i 0.546372i
\(630\) 0 0
\(631\) −10224.8 −0.645079 −0.322539 0.946556i \(-0.604537\pi\)
−0.322539 + 0.946556i \(0.604537\pi\)
\(632\) 0 0
\(633\) −3370.88 −0.211660
\(634\) 0 0
\(635\) 22144.8i 1.38392i
\(636\) 0 0
\(637\) − 16812.8i − 1.04576i
\(638\) 0 0
\(639\) 11729.0 0.726125
\(640\) 0 0
\(641\) 19804.4 1.22032 0.610162 0.792277i \(-0.291104\pi\)
0.610162 + 0.792277i \(0.291104\pi\)
\(642\) 0 0
\(643\) 22175.9i 1.36008i 0.733174 + 0.680041i \(0.238038\pi\)
−0.733174 + 0.680041i \(0.761962\pi\)
\(644\) 0 0
\(645\) 554.380i 0.0338429i
\(646\) 0 0
\(647\) 9232.26 0.560985 0.280493 0.959856i \(-0.409502\pi\)
0.280493 + 0.959856i \(0.409502\pi\)
\(648\) 0 0
\(649\) −25748.8 −1.55736
\(650\) 0 0
\(651\) − 539.511i − 0.0324810i
\(652\) 0 0
\(653\) − 27857.1i − 1.66942i −0.550690 0.834710i \(-0.685635\pi\)
0.550690 0.834710i \(-0.314365\pi\)
\(654\) 0 0
\(655\) 15109.8 0.901355
\(656\) 0 0
\(657\) 6715.66 0.398786
\(658\) 0 0
\(659\) − 5498.55i − 0.325028i −0.986706 0.162514i \(-0.948040\pi\)
0.986706 0.162514i \(-0.0519602\pi\)
\(660\) 0 0
\(661\) − 11469.6i − 0.674908i −0.941342 0.337454i \(-0.890434\pi\)
0.941342 0.337454i \(-0.109566\pi\)
\(662\) 0 0
\(663\) −3884.13 −0.227522
\(664\) 0 0
\(665\) −854.587 −0.0498338
\(666\) 0 0
\(667\) 13025.7i 0.756156i
\(668\) 0 0
\(669\) 383.204i 0.0221458i
\(670\) 0 0
\(671\) −20627.1 −1.18674
\(672\) 0 0
\(673\) −28428.2 −1.62827 −0.814135 0.580676i \(-0.802788\pi\)
−0.814135 + 0.580676i \(0.802788\pi\)
\(674\) 0 0
\(675\) − 584.333i − 0.0333200i
\(676\) 0 0
\(677\) 23995.5i 1.36222i 0.732181 + 0.681110i \(0.238502\pi\)
−0.732181 + 0.681110i \(0.761498\pi\)
\(678\) 0 0
\(679\) −2500.92 −0.141350
\(680\) 0 0
\(681\) −5229.92 −0.294289
\(682\) 0 0
\(683\) 13506.0i 0.756650i 0.925673 + 0.378325i \(0.123500\pi\)
−0.925673 + 0.378325i \(0.876500\pi\)
\(684\) 0 0
\(685\) 5551.73i 0.309665i
\(686\) 0 0
\(687\) −2164.42 −0.120201
\(688\) 0 0
\(689\) −10587.9 −0.585439
\(690\) 0 0
\(691\) 29500.1i 1.62408i 0.583605 + 0.812038i \(0.301642\pi\)
−0.583605 + 0.812038i \(0.698358\pi\)
\(692\) 0 0
\(693\) 4417.57i 0.242150i
\(694\) 0 0
\(695\) 24906.1 1.35934
\(696\) 0 0
\(697\) −7585.52 −0.412227
\(698\) 0 0
\(699\) 991.557i 0.0536540i
\(700\) 0 0
\(701\) − 33227.5i − 1.79028i −0.445787 0.895139i \(-0.647076\pi\)
0.445787 0.895139i \(-0.352924\pi\)
\(702\) 0 0
\(703\) 3268.91 0.175376
\(704\) 0 0
\(705\) −6077.62 −0.324676
\(706\) 0 0
\(707\) 6.00170i 0 0.000319261i
\(708\) 0 0
\(709\) − 6448.04i − 0.341553i −0.985310 0.170777i \(-0.945372\pi\)
0.985310 0.170777i \(-0.0546277\pi\)
\(710\) 0 0
\(711\) 8355.95 0.440749
\(712\) 0 0
\(713\) 26313.9 1.38214
\(714\) 0 0
\(715\) 37242.0i 1.94793i
\(716\) 0 0
\(717\) − 848.138i − 0.0441761i
\(718\) 0 0
\(719\) −6494.67 −0.336871 −0.168436 0.985713i \(-0.553872\pi\)
−0.168436 + 0.985713i \(0.553872\pi\)
\(720\) 0 0
\(721\) 3709.08 0.191586
\(722\) 0 0
\(723\) − 1550.80i − 0.0797714i
\(724\) 0 0
\(725\) 964.806i 0.0494234i
\(726\) 0 0
\(727\) −24866.4 −1.26856 −0.634280 0.773103i \(-0.718704\pi\)
−0.634280 + 0.773103i \(0.718704\pi\)
\(728\) 0 0
\(729\) 14848.5 0.754384
\(730\) 0 0
\(731\) 3228.20i 0.163337i
\(732\) 0 0
\(733\) 21092.1i 1.06283i 0.847112 + 0.531414i \(0.178339\pi\)
−0.847112 + 0.531414i \(0.821661\pi\)
\(734\) 0 0
\(735\) −4181.58 −0.209850
\(736\) 0 0
\(737\) −12585.1 −0.629008
\(738\) 0 0
\(739\) 11952.7i 0.594977i 0.954725 + 0.297489i \(0.0961491\pi\)
−0.954725 + 0.297489i \(0.903851\pi\)
\(740\) 0 0
\(741\) 1473.10i 0.0730305i
\(742\) 0 0
\(743\) 5622.43 0.277614 0.138807 0.990319i \(-0.455673\pi\)
0.138807 + 0.990319i \(0.455673\pi\)
\(744\) 0 0
\(745\) 6175.21 0.303681
\(746\) 0 0
\(747\) 20618.3i 1.00988i
\(748\) 0 0
\(749\) 3104.02i 0.151427i
\(750\) 0 0
\(751\) −32314.9 −1.57016 −0.785079 0.619396i \(-0.787378\pi\)
−0.785079 + 0.619396i \(0.787378\pi\)
\(752\) 0 0
\(753\) 5418.58 0.262236
\(754\) 0 0
\(755\) − 34870.5i − 1.68089i
\(756\) 0 0
\(757\) 17950.4i 0.861847i 0.902389 + 0.430923i \(0.141812\pi\)
−0.902389 + 0.430923i \(0.858188\pi\)
\(758\) 0 0
\(759\) 9545.42 0.456491
\(760\) 0 0
\(761\) −13108.2 −0.624404 −0.312202 0.950016i \(-0.601067\pi\)
−0.312202 + 0.950016i \(0.601067\pi\)
\(762\) 0 0
\(763\) 2013.10i 0.0955166i
\(764\) 0 0
\(765\) − 21805.5i − 1.03056i
\(766\) 0 0
\(767\) −20154.8 −0.948822
\(768\) 0 0
\(769\) 23661.2 1.10955 0.554776 0.832000i \(-0.312804\pi\)
0.554776 + 0.832000i \(0.312804\pi\)
\(770\) 0 0
\(771\) − 5039.63i − 0.235406i
\(772\) 0 0
\(773\) − 30222.4i − 1.40624i −0.711071 0.703120i \(-0.751790\pi\)
0.711071 0.703120i \(-0.248210\pi\)
\(774\) 0 0
\(775\) 1949.06 0.0903384
\(776\) 0 0
\(777\) −339.940 −0.0156954
\(778\) 0 0
\(779\) 2876.89i 0.132318i
\(780\) 0 0
\(781\) 29012.3i 1.32925i
\(782\) 0 0
\(783\) 5283.35 0.241139
\(784\) 0 0
\(785\) 24825.8 1.12875
\(786\) 0 0
\(787\) 29544.2i 1.33817i 0.743188 + 0.669083i \(0.233313\pi\)
−0.743188 + 0.669083i \(0.766687\pi\)
\(788\) 0 0
\(789\) 3176.28i 0.143319i
\(790\) 0 0
\(791\) 179.720 0.00807853
\(792\) 0 0
\(793\) −16145.8 −0.723020
\(794\) 0 0
\(795\) 2633.36i 0.117479i
\(796\) 0 0
\(797\) 9665.91i 0.429591i 0.976659 + 0.214795i \(0.0689085\pi\)
−0.976659 + 0.214795i \(0.931092\pi\)
\(798\) 0 0
\(799\) −35390.5 −1.56699
\(800\) 0 0
\(801\) 22412.1 0.988631
\(802\) 0 0
\(803\) 16611.5i 0.730021i
\(804\) 0 0
\(805\) 4334.52i 0.189778i
\(806\) 0 0
\(807\) 1004.44 0.0438142
\(808\) 0 0
\(809\) 15807.0 0.686952 0.343476 0.939162i \(-0.388396\pi\)
0.343476 + 0.939162i \(0.388396\pi\)
\(810\) 0 0
\(811\) 16295.5i 0.705565i 0.935705 + 0.352783i \(0.114764\pi\)
−0.935705 + 0.352783i \(0.885236\pi\)
\(812\) 0 0
\(813\) 8624.81i 0.372061i
\(814\) 0 0
\(815\) −23446.9 −1.00774
\(816\) 0 0
\(817\) 1224.33 0.0524284
\(818\) 0 0
\(819\) 3457.84i 0.147529i
\(820\) 0 0
\(821\) 436.743i 0.0185657i 0.999957 + 0.00928286i \(0.00295487\pi\)
−0.999957 + 0.00928286i \(0.997045\pi\)
\(822\) 0 0
\(823\) −5633.49 −0.238604 −0.119302 0.992858i \(-0.538066\pi\)
−0.119302 + 0.992858i \(0.538066\pi\)
\(824\) 0 0
\(825\) 707.026 0.0298369
\(826\) 0 0
\(827\) − 22394.7i − 0.941645i −0.882228 0.470822i \(-0.843957\pi\)
0.882228 0.470822i \(-0.156043\pi\)
\(828\) 0 0
\(829\) 8768.39i 0.367357i 0.982986 + 0.183678i \(0.0588005\pi\)
−0.982986 + 0.183678i \(0.941199\pi\)
\(830\) 0 0
\(831\) −730.253 −0.0304840
\(832\) 0 0
\(833\) −24349.7 −1.01281
\(834\) 0 0
\(835\) − 9218.27i − 0.382050i
\(836\) 0 0
\(837\) − 10673.2i − 0.440764i
\(838\) 0 0
\(839\) 30644.3 1.26098 0.630488 0.776199i \(-0.282855\pi\)
0.630488 + 0.776199i \(0.282855\pi\)
\(840\) 0 0
\(841\) 15665.5 0.642319
\(842\) 0 0
\(843\) − 6313.47i − 0.257945i
\(844\) 0 0
\(845\) 3593.02i 0.146277i
\(846\) 0 0
\(847\) −7371.03 −0.299022
\(848\) 0 0
\(849\) 1029.06 0.0415986
\(850\) 0 0
\(851\) − 16580.1i − 0.667871i
\(852\) 0 0
\(853\) − 43559.2i − 1.74846i −0.485508 0.874232i \(-0.661365\pi\)
0.485508 0.874232i \(-0.338635\pi\)
\(854\) 0 0
\(855\) −8270.00 −0.330793
\(856\) 0 0
\(857\) 41788.5 1.66566 0.832828 0.553533i \(-0.186721\pi\)
0.832828 + 0.553533i \(0.186721\pi\)
\(858\) 0 0
\(859\) 16849.6i 0.669267i 0.942348 + 0.334633i \(0.108612\pi\)
−0.942348 + 0.334633i \(0.891388\pi\)
\(860\) 0 0
\(861\) − 299.174i − 0.0118418i
\(862\) 0 0
\(863\) −27636.7 −1.09011 −0.545054 0.838401i \(-0.683491\pi\)
−0.545054 + 0.838401i \(0.683491\pi\)
\(864\) 0 0
\(865\) −12727.0 −0.500266
\(866\) 0 0
\(867\) 367.220i 0.0143846i
\(868\) 0 0
\(869\) 20668.8i 0.806837i
\(870\) 0 0
\(871\) −9850.95 −0.383223
\(872\) 0 0
\(873\) −24201.9 −0.938270
\(874\) 0 0
\(875\) − 3563.98i − 0.137697i
\(876\) 0 0
\(877\) − 16855.0i − 0.648977i −0.945890 0.324488i \(-0.894808\pi\)
0.945890 0.324488i \(-0.105192\pi\)
\(878\) 0 0
\(879\) −5377.45 −0.206344
\(880\) 0 0
\(881\) 13330.0 0.509759 0.254880 0.966973i \(-0.417964\pi\)
0.254880 + 0.966973i \(0.417964\pi\)
\(882\) 0 0
\(883\) 35598.7i 1.35673i 0.734725 + 0.678365i \(0.237311\pi\)
−0.734725 + 0.678365i \(0.762689\pi\)
\(884\) 0 0
\(885\) 5012.78i 0.190399i
\(886\) 0 0
\(887\) 48821.3 1.84810 0.924048 0.382278i \(-0.124860\pi\)
0.924048 + 0.382278i \(0.124860\pi\)
\(888\) 0 0
\(889\) −5085.84 −0.191871
\(890\) 0 0
\(891\) 40771.8i 1.53301i
\(892\) 0 0
\(893\) 13422.2i 0.502977i
\(894\) 0 0
\(895\) −7009.27 −0.261781
\(896\) 0 0
\(897\) 7471.64 0.278117
\(898\) 0 0
\(899\) 17622.8i 0.653785i
\(900\) 0 0
\(901\) 15334.3i 0.566992i
\(902\) 0 0
\(903\) −127.321 −0.00469210
\(904\) 0 0
\(905\) −43071.7 −1.58205
\(906\) 0 0
\(907\) − 7523.90i − 0.275443i −0.990471 0.137722i \(-0.956022\pi\)
0.990471 0.137722i \(-0.0439779\pi\)
\(908\) 0 0
\(909\) 58.0796i 0.00211923i
\(910\) 0 0
\(911\) −26016.0 −0.946155 −0.473077 0.881021i \(-0.656857\pi\)
−0.473077 + 0.881021i \(0.656857\pi\)
\(912\) 0 0
\(913\) −51000.2 −1.84870
\(914\) 0 0
\(915\) 4015.69i 0.145087i
\(916\) 0 0
\(917\) 3470.16i 0.124967i
\(918\) 0 0
\(919\) 24082.1 0.864413 0.432206 0.901775i \(-0.357735\pi\)
0.432206 + 0.901775i \(0.357735\pi\)
\(920\) 0 0
\(921\) 4140.41 0.148134
\(922\) 0 0
\(923\) 22709.3i 0.809844i
\(924\) 0 0
\(925\) − 1228.08i − 0.0436530i
\(926\) 0 0
\(927\) 35893.5 1.27173
\(928\) 0 0
\(929\) 9324.93 0.329323 0.164661 0.986350i \(-0.447347\pi\)
0.164661 + 0.986350i \(0.447347\pi\)
\(930\) 0 0
\(931\) 9234.89i 0.325093i
\(932\) 0 0
\(933\) 6203.87i 0.217691i
\(934\) 0 0
\(935\) 53937.0 1.88656
\(936\) 0 0
\(937\) 15535.2 0.541636 0.270818 0.962631i \(-0.412706\pi\)
0.270818 + 0.962631i \(0.412706\pi\)
\(938\) 0 0
\(939\) 8950.03i 0.311047i
\(940\) 0 0
\(941\) − 48118.3i − 1.66696i −0.552547 0.833482i \(-0.686344\pi\)
0.552547 0.833482i \(-0.313656\pi\)
\(942\) 0 0
\(943\) 14591.8 0.503896
\(944\) 0 0
\(945\) 1758.13 0.0605205
\(946\) 0 0
\(947\) 5396.67i 0.185183i 0.995704 + 0.0925915i \(0.0295151\pi\)
−0.995704 + 0.0925915i \(0.970485\pi\)
\(948\) 0 0
\(949\) 13002.6i 0.444765i
\(950\) 0 0
\(951\) −521.277 −0.0177745
\(952\) 0 0
\(953\) −21759.7 −0.739629 −0.369815 0.929106i \(-0.620579\pi\)
−0.369815 + 0.929106i \(0.620579\pi\)
\(954\) 0 0
\(955\) 37416.7i 1.26783i
\(956\) 0 0
\(957\) 6392.70i 0.215932i
\(958\) 0 0
\(959\) −1275.03 −0.0429331
\(960\) 0 0
\(961\) 5809.78 0.195018
\(962\) 0 0
\(963\) 30038.2i 1.00516i
\(964\) 0 0
\(965\) − 33184.4i − 1.10699i
\(966\) 0 0
\(967\) −19338.5 −0.643108 −0.321554 0.946891i \(-0.604205\pi\)
−0.321554 + 0.946891i \(0.604205\pi\)
\(968\) 0 0
\(969\) 2133.47 0.0707294
\(970\) 0 0
\(971\) − 5918.37i − 0.195602i −0.995206 0.0978009i \(-0.968819\pi\)
0.995206 0.0978009i \(-0.0311809\pi\)
\(972\) 0 0
\(973\) 5720.03i 0.188464i
\(974\) 0 0
\(975\) 553.421 0.0181781
\(976\) 0 0
\(977\) −17841.9 −0.584249 −0.292125 0.956380i \(-0.594362\pi\)
−0.292125 + 0.956380i \(0.594362\pi\)
\(978\) 0 0
\(979\) 55437.4i 1.80979i
\(980\) 0 0
\(981\) 19481.1i 0.634032i
\(982\) 0 0
\(983\) 61512.0 1.99586 0.997929 0.0643304i \(-0.0204912\pi\)
0.997929 + 0.0643304i \(0.0204912\pi\)
\(984\) 0 0
\(985\) −26471.2 −0.856286
\(986\) 0 0
\(987\) − 1395.81i − 0.0450142i
\(988\) 0 0
\(989\) − 6209.89i − 0.199659i
\(990\) 0 0
\(991\) −1827.73 −0.0585870 −0.0292935 0.999571i \(-0.509326\pi\)
−0.0292935 + 0.999571i \(0.509326\pi\)
\(992\) 0 0
\(993\) 4068.21 0.130011
\(994\) 0 0
\(995\) 8691.53i 0.276925i
\(996\) 0 0
\(997\) 35379.4i 1.12385i 0.827188 + 0.561925i \(0.189939\pi\)
−0.827188 + 0.561925i \(0.810061\pi\)
\(998\) 0 0
\(999\) −6725.07 −0.212985
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.b.k.513.6 10
4.3 odd 2 1024.4.b.j.513.5 10
8.3 odd 2 1024.4.b.j.513.6 10
8.5 even 2 inner 1024.4.b.k.513.5 10
16.3 odd 4 1024.4.a.n.1.6 10
16.5 even 4 1024.4.a.m.1.6 10
16.11 odd 4 1024.4.a.n.1.5 10
16.13 even 4 1024.4.a.m.1.5 10
32.3 odd 8 16.4.e.a.5.3 10
32.5 even 8 64.4.e.a.17.3 10
32.11 odd 8 128.4.e.b.33.3 10
32.13 even 8 128.4.e.a.97.3 10
32.19 odd 8 128.4.e.b.97.3 10
32.21 even 8 128.4.e.a.33.3 10
32.27 odd 8 16.4.e.a.13.3 yes 10
32.29 even 8 64.4.e.a.49.3 10
96.5 odd 8 576.4.k.a.145.2 10
96.29 odd 8 576.4.k.a.433.2 10
96.35 even 8 144.4.k.a.37.3 10
96.59 even 8 144.4.k.a.109.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.3 10 32.3 odd 8
16.4.e.a.13.3 yes 10 32.27 odd 8
64.4.e.a.17.3 10 32.5 even 8
64.4.e.a.49.3 10 32.29 even 8
128.4.e.a.33.3 10 32.21 even 8
128.4.e.a.97.3 10 32.13 even 8
128.4.e.b.33.3 10 32.11 odd 8
128.4.e.b.97.3 10 32.19 odd 8
144.4.k.a.37.3 10 96.35 even 8
144.4.k.a.109.3 10 96.59 even 8
576.4.k.a.145.2 10 96.5 odd 8
576.4.k.a.433.2 10 96.29 odd 8
1024.4.a.m.1.5 10 16.13 even 4
1024.4.a.m.1.6 10 16.5 even 4
1024.4.a.n.1.5 10 16.11 odd 4
1024.4.a.n.1.6 10 16.3 odd 4
1024.4.b.j.513.5 10 4.3 odd 2
1024.4.b.j.513.6 10 8.3 odd 2
1024.4.b.k.513.5 10 8.5 even 2 inner
1024.4.b.k.513.6 10 1.1 even 1 trivial