Properties

Label 784.4.a.u.1.1
Level $784$
Weight $4$
Character 784.1
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,4,Mod(1,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(46.2574974445\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.54138\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.08276 q^{3} -19.1655 q^{5} +10.0000 q^{9} +O(q^{10})\) \(q-6.08276 q^{3} -19.1655 q^{5} +10.0000 q^{9} +26.5793 q^{11} +10.3311 q^{13} +116.579 q^{15} -101.331 q^{17} +93.7517 q^{19} +8.57934 q^{23} +242.317 q^{25} +103.407 q^{27} +52.3174 q^{29} +55.4207 q^{31} -161.676 q^{33} +428.476 q^{37} -62.8413 q^{39} -137.007 q^{41} +172.000 q^{43} -191.655 q^{45} +49.2414 q^{47} +616.373 q^{51} -474.476 q^{53} -509.407 q^{55} -570.269 q^{57} -197.062 q^{59} -401.124 q^{61} -198.000 q^{65} -125.421 q^{67} -52.1861 q^{69} -788.635 q^{71} +604.655 q^{73} -1473.96 q^{75} -783.007 q^{79} -899.000 q^{81} +339.283 q^{83} +1942.06 q^{85} -318.234 q^{87} +511.966 q^{89} -337.111 q^{93} -1796.80 q^{95} -672.290 q^{97} +265.793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{5} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{5} + 20 q^{9} - 32 q^{11} - 28 q^{13} + 148 q^{15} - 154 q^{17} + 224 q^{19} - 68 q^{23} + 144 q^{25} - 236 q^{29} + 196 q^{31} - 518 q^{33} + 346 q^{37} - 296 q^{39} - 420 q^{41} + 344 q^{43} - 140 q^{45} - 84 q^{47} + 296 q^{51} - 438 q^{53} - 812 q^{55} + 222 q^{57} + 56 q^{59} + 98 q^{61} - 396 q^{65} - 336 q^{67} - 518 q^{69} - 896 q^{71} + 966 q^{73} - 2072 q^{75} + 52 q^{79} - 1798 q^{81} - 392 q^{83} + 1670 q^{85} - 2072 q^{87} + 294 q^{89} + 518 q^{93} - 1124 q^{95} - 420 q^{97} - 320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −6.08276 −1.17063 −0.585314 0.810807i \(-0.699029\pi\)
−0.585314 + 0.810807i \(0.699029\pi\)
\(4\) 0 0
\(5\) −19.1655 −1.71422 −0.857108 0.515136i \(-0.827741\pi\)
−0.857108 + 0.515136i \(0.827741\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 10.0000 0.370370
\(10\) 0 0
\(11\) 26.5793 0.728543 0.364271 0.931293i \(-0.381318\pi\)
0.364271 + 0.931293i \(0.381318\pi\)
\(12\) 0 0
\(13\) 10.3311 0.220409 0.110205 0.993909i \(-0.464849\pi\)
0.110205 + 0.993909i \(0.464849\pi\)
\(14\) 0 0
\(15\) 116.579 2.00671
\(16\) 0 0
\(17\) −101.331 −1.44567 −0.722835 0.691021i \(-0.757161\pi\)
−0.722835 + 0.691021i \(0.757161\pi\)
\(18\) 0 0
\(19\) 93.7517 1.13201 0.566003 0.824403i \(-0.308489\pi\)
0.566003 + 0.824403i \(0.308489\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.57934 0.0777789 0.0388895 0.999244i \(-0.487618\pi\)
0.0388895 + 0.999244i \(0.487618\pi\)
\(24\) 0 0
\(25\) 242.317 1.93854
\(26\) 0 0
\(27\) 103.407 0.737062
\(28\) 0 0
\(29\) 52.3174 0.335003 0.167502 0.985872i \(-0.446430\pi\)
0.167502 + 0.985872i \(0.446430\pi\)
\(30\) 0 0
\(31\) 55.4207 0.321092 0.160546 0.987028i \(-0.448675\pi\)
0.160546 + 0.987028i \(0.448675\pi\)
\(32\) 0 0
\(33\) −161.676 −0.852853
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 428.476 1.90381 0.951906 0.306391i \(-0.0991215\pi\)
0.951906 + 0.306391i \(0.0991215\pi\)
\(38\) 0 0
\(39\) −62.8413 −0.258017
\(40\) 0 0
\(41\) −137.007 −0.521875 −0.260938 0.965356i \(-0.584032\pi\)
−0.260938 + 0.965356i \(0.584032\pi\)
\(42\) 0 0
\(43\) 172.000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −191.655 −0.634895
\(46\) 0 0
\(47\) 49.2414 0.152821 0.0764107 0.997076i \(-0.475654\pi\)
0.0764107 + 0.997076i \(0.475654\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 616.373 1.69234
\(52\) 0 0
\(53\) −474.476 −1.22970 −0.614852 0.788643i \(-0.710784\pi\)
−0.614852 + 0.788643i \(0.710784\pi\)
\(54\) 0 0
\(55\) −509.407 −1.24888
\(56\) 0 0
\(57\) −570.269 −1.32516
\(58\) 0 0
\(59\) −197.062 −0.434836 −0.217418 0.976079i \(-0.569763\pi\)
−0.217418 + 0.976079i \(0.569763\pi\)
\(60\) 0 0
\(61\) −401.124 −0.841946 −0.420973 0.907073i \(-0.638311\pi\)
−0.420973 + 0.907073i \(0.638311\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −198.000 −0.377829
\(66\) 0 0
\(67\) −125.421 −0.228695 −0.114348 0.993441i \(-0.536478\pi\)
−0.114348 + 0.993441i \(0.536478\pi\)
\(68\) 0 0
\(69\) −52.1861 −0.0910502
\(70\) 0 0
\(71\) −788.635 −1.31822 −0.659111 0.752046i \(-0.729067\pi\)
−0.659111 + 0.752046i \(0.729067\pi\)
\(72\) 0 0
\(73\) 604.655 0.969446 0.484723 0.874668i \(-0.338920\pi\)
0.484723 + 0.874668i \(0.338920\pi\)
\(74\) 0 0
\(75\) −1473.96 −2.26931
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −783.007 −1.11513 −0.557565 0.830134i \(-0.688264\pi\)
−0.557565 + 0.830134i \(0.688264\pi\)
\(80\) 0 0
\(81\) −899.000 −1.23320
\(82\) 0 0
\(83\) 339.283 0.448689 0.224344 0.974510i \(-0.427976\pi\)
0.224344 + 0.974510i \(0.427976\pi\)
\(84\) 0 0
\(85\) 1942.06 2.47819
\(86\) 0 0
\(87\) −318.234 −0.392164
\(88\) 0 0
\(89\) 511.966 0.609756 0.304878 0.952391i \(-0.401384\pi\)
0.304878 + 0.952391i \(0.401384\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −337.111 −0.375879
\(94\) 0 0
\(95\) −1796.80 −1.94050
\(96\) 0 0
\(97\) −672.290 −0.703719 −0.351859 0.936053i \(-0.614450\pi\)
−0.351859 + 0.936053i \(0.614450\pi\)
\(98\) 0 0
\(99\) 265.793 0.269831
\(100\) 0 0
\(101\) 133.773 0.131791 0.0658955 0.997827i \(-0.479010\pi\)
0.0658955 + 0.997827i \(0.479010\pi\)
\(102\) 0 0
\(103\) −1160.74 −1.11040 −0.555202 0.831716i \(-0.687359\pi\)
−0.555202 + 0.831716i \(0.687359\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1308.79 1.18248 0.591239 0.806497i \(-0.298639\pi\)
0.591239 + 0.806497i \(0.298639\pi\)
\(108\) 0 0
\(109\) −1045.84 −0.919022 −0.459511 0.888172i \(-0.651975\pi\)
−0.459511 + 0.888172i \(0.651975\pi\)
\(110\) 0 0
\(111\) −2606.32 −2.22866
\(112\) 0 0
\(113\) −117.587 −0.0978905 −0.0489453 0.998801i \(-0.515586\pi\)
−0.0489453 + 0.998801i \(0.515586\pi\)
\(114\) 0 0
\(115\) −164.428 −0.133330
\(116\) 0 0
\(117\) 103.311 0.0816330
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −624.539 −0.469225
\(122\) 0 0
\(123\) 833.380 0.610922
\(124\) 0 0
\(125\) −2248.45 −1.60886
\(126\) 0 0
\(127\) 1788.63 1.24973 0.624865 0.780733i \(-0.285154\pi\)
0.624865 + 0.780733i \(0.285154\pi\)
\(128\) 0 0
\(129\) −1046.24 −0.714077
\(130\) 0 0
\(131\) 1949.28 1.30007 0.650037 0.759903i \(-0.274753\pi\)
0.650037 + 0.759903i \(0.274753\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1981.85 −1.26348
\(136\) 0 0
\(137\) −571.539 −0.356422 −0.178211 0.983992i \(-0.557031\pi\)
−0.178211 + 0.983992i \(0.557031\pi\)
\(138\) 0 0
\(139\) −1176.58 −0.717958 −0.358979 0.933346i \(-0.616875\pi\)
−0.358979 + 0.933346i \(0.616875\pi\)
\(140\) 0 0
\(141\) −299.524 −0.178897
\(142\) 0 0
\(143\) 274.592 0.160577
\(144\) 0 0
\(145\) −1002.69 −0.574268
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1773.33 0.975014 0.487507 0.873119i \(-0.337906\pi\)
0.487507 + 0.873119i \(0.337906\pi\)
\(150\) 0 0
\(151\) 503.007 0.271087 0.135544 0.990771i \(-0.456722\pi\)
0.135544 + 0.990771i \(0.456722\pi\)
\(152\) 0 0
\(153\) −1013.31 −0.535433
\(154\) 0 0
\(155\) −1062.17 −0.550421
\(156\) 0 0
\(157\) 2336.31 1.18763 0.593815 0.804601i \(-0.297621\pi\)
0.593815 + 0.804601i \(0.297621\pi\)
\(158\) 0 0
\(159\) 2886.12 1.43953
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1736.59 0.834482 0.417241 0.908796i \(-0.362997\pi\)
0.417241 + 0.908796i \(0.362997\pi\)
\(164\) 0 0
\(165\) 3098.60 1.46197
\(166\) 0 0
\(167\) −668.331 −0.309683 −0.154841 0.987939i \(-0.549487\pi\)
−0.154841 + 0.987939i \(0.549487\pi\)
\(168\) 0 0
\(169\) −2090.27 −0.951420
\(170\) 0 0
\(171\) 937.517 0.419262
\(172\) 0 0
\(173\) −2145.23 −0.942769 −0.471385 0.881928i \(-0.656246\pi\)
−0.471385 + 0.881928i \(0.656246\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1198.68 0.509031
\(178\) 0 0
\(179\) −3767.55 −1.57318 −0.786591 0.617474i \(-0.788156\pi\)
−0.786591 + 0.617474i \(0.788156\pi\)
\(180\) 0 0
\(181\) −1752.51 −0.719686 −0.359843 0.933013i \(-0.617170\pi\)
−0.359843 + 0.933013i \(0.617170\pi\)
\(182\) 0 0
\(183\) 2439.94 0.985606
\(184\) 0 0
\(185\) −8211.97 −3.26355
\(186\) 0 0
\(187\) −2693.31 −1.05323
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2011.12 0.761882 0.380941 0.924599i \(-0.375600\pi\)
0.380941 + 0.924599i \(0.375600\pi\)
\(192\) 0 0
\(193\) −1579.95 −0.589261 −0.294631 0.955611i \(-0.595197\pi\)
−0.294631 + 0.955611i \(0.595197\pi\)
\(194\) 0 0
\(195\) 1204.39 0.442297
\(196\) 0 0
\(197\) −26.4132 −0.00955262 −0.00477631 0.999989i \(-0.501520\pi\)
−0.00477631 + 0.999989i \(0.501520\pi\)
\(198\) 0 0
\(199\) 57.5462 0.0204992 0.0102496 0.999947i \(-0.496737\pi\)
0.0102496 + 0.999947i \(0.496737\pi\)
\(200\) 0 0
\(201\) 762.904 0.267717
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2625.81 0.894607
\(206\) 0 0
\(207\) 85.7934 0.0288070
\(208\) 0 0
\(209\) 2491.86 0.824715
\(210\) 0 0
\(211\) 1193.90 0.389534 0.194767 0.980850i \(-0.437605\pi\)
0.194767 + 0.980850i \(0.437605\pi\)
\(212\) 0 0
\(213\) 4797.08 1.54315
\(214\) 0 0
\(215\) −3296.47 −1.04566
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3677.97 −1.13486
\(220\) 0 0
\(221\) −1046.86 −0.318639
\(222\) 0 0
\(223\) 2921.66 0.877348 0.438674 0.898646i \(-0.355448\pi\)
0.438674 + 0.898646i \(0.355448\pi\)
\(224\) 0 0
\(225\) 2423.17 0.717977
\(226\) 0 0
\(227\) 3112.34 0.910016 0.455008 0.890487i \(-0.349636\pi\)
0.455008 + 0.890487i \(0.349636\pi\)
\(228\) 0 0
\(229\) −5335.31 −1.53959 −0.769797 0.638289i \(-0.779643\pi\)
−0.769797 + 0.638289i \(0.779643\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5238.08 −1.47278 −0.736390 0.676557i \(-0.763471\pi\)
−0.736390 + 0.676557i \(0.763471\pi\)
\(234\) 0 0
\(235\) −943.738 −0.261969
\(236\) 0 0
\(237\) 4762.85 1.30540
\(238\) 0 0
\(239\) −4181.05 −1.13159 −0.565794 0.824547i \(-0.691430\pi\)
−0.565794 + 0.824547i \(0.691430\pi\)
\(240\) 0 0
\(241\) 5464.88 1.46068 0.730340 0.683084i \(-0.239362\pi\)
0.730340 + 0.683084i \(0.239362\pi\)
\(242\) 0 0
\(243\) 2676.42 0.706552
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 968.554 0.249504
\(248\) 0 0
\(249\) −2063.78 −0.525248
\(250\) 0 0
\(251\) 5718.95 1.43816 0.719078 0.694930i \(-0.244564\pi\)
0.719078 + 0.694930i \(0.244564\pi\)
\(252\) 0 0
\(253\) 228.033 0.0566653
\(254\) 0 0
\(255\) −11813.1 −2.90104
\(256\) 0 0
\(257\) 1018.28 0.247154 0.123577 0.992335i \(-0.460563\pi\)
0.123577 + 0.992335i \(0.460563\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 523.174 0.124075
\(262\) 0 0
\(263\) 8224.31 1.92826 0.964130 0.265430i \(-0.0855139\pi\)
0.964130 + 0.265430i \(0.0855139\pi\)
\(264\) 0 0
\(265\) 9093.58 2.10798
\(266\) 0 0
\(267\) −3114.17 −0.713797
\(268\) 0 0
\(269\) −5867.88 −1.33000 −0.665002 0.746841i \(-0.731569\pi\)
−0.665002 + 0.746841i \(0.731569\pi\)
\(270\) 0 0
\(271\) −6216.77 −1.39351 −0.696757 0.717307i \(-0.745374\pi\)
−0.696757 + 0.717307i \(0.745374\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6440.63 1.41231
\(276\) 0 0
\(277\) 1477.52 0.320490 0.160245 0.987077i \(-0.448772\pi\)
0.160245 + 0.987077i \(0.448772\pi\)
\(278\) 0 0
\(279\) 554.207 0.118923
\(280\) 0 0
\(281\) −3611.93 −0.766797 −0.383398 0.923583i \(-0.625246\pi\)
−0.383398 + 0.923583i \(0.625246\pi\)
\(282\) 0 0
\(283\) 7628.07 1.60227 0.801134 0.598485i \(-0.204230\pi\)
0.801134 + 0.598485i \(0.204230\pi\)
\(284\) 0 0
\(285\) 10929.5 2.27161
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5354.98 1.08996
\(290\) 0 0
\(291\) 4089.38 0.823793
\(292\) 0 0
\(293\) −5393.67 −1.07543 −0.537716 0.843126i \(-0.680713\pi\)
−0.537716 + 0.843126i \(0.680713\pi\)
\(294\) 0 0
\(295\) 3776.80 0.745403
\(296\) 0 0
\(297\) 2748.49 0.536981
\(298\) 0 0
\(299\) 88.6336 0.0171432
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −813.708 −0.154278
\(304\) 0 0
\(305\) 7687.76 1.44328
\(306\) 0 0
\(307\) −8054.16 −1.49731 −0.748656 0.662958i \(-0.769301\pi\)
−0.748656 + 0.662958i \(0.769301\pi\)
\(308\) 0 0
\(309\) 7060.54 1.29987
\(310\) 0 0
\(311\) −2284.61 −0.416554 −0.208277 0.978070i \(-0.566785\pi\)
−0.208277 + 0.978070i \(0.566785\pi\)
\(312\) 0 0
\(313\) 8336.13 1.50539 0.752694 0.658371i \(-0.228754\pi\)
0.752694 + 0.658371i \(0.228754\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10416.2 −1.84552 −0.922762 0.385369i \(-0.874074\pi\)
−0.922762 + 0.385369i \(0.874074\pi\)
\(318\) 0 0
\(319\) 1390.56 0.244064
\(320\) 0 0
\(321\) −7961.03 −1.38424
\(322\) 0 0
\(323\) −9499.96 −1.63651
\(324\) 0 0
\(325\) 2503.39 0.427272
\(326\) 0 0
\(327\) 6361.60 1.07583
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6943.88 −1.15308 −0.576541 0.817068i \(-0.695598\pi\)
−0.576541 + 0.817068i \(0.695598\pi\)
\(332\) 0 0
\(333\) 4284.76 0.705115
\(334\) 0 0
\(335\) 2403.75 0.392033
\(336\) 0 0
\(337\) 4951.93 0.800442 0.400221 0.916419i \(-0.368933\pi\)
0.400221 + 0.916419i \(0.368933\pi\)
\(338\) 0 0
\(339\) 715.252 0.114593
\(340\) 0 0
\(341\) 1473.04 0.233929
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1000.17 0.156080
\(346\) 0 0
\(347\) −7275.35 −1.12554 −0.562769 0.826614i \(-0.690264\pi\)
−0.562769 + 0.826614i \(0.690264\pi\)
\(348\) 0 0
\(349\) 3433.29 0.526589 0.263294 0.964716i \(-0.415191\pi\)
0.263294 + 0.964716i \(0.415191\pi\)
\(350\) 0 0
\(351\) 1068.30 0.162455
\(352\) 0 0
\(353\) 6159.09 0.928655 0.464328 0.885664i \(-0.346296\pi\)
0.464328 + 0.885664i \(0.346296\pi\)
\(354\) 0 0
\(355\) 15114.6 2.25972
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8006.62 −1.17708 −0.588542 0.808466i \(-0.700298\pi\)
−0.588542 + 0.808466i \(0.700298\pi\)
\(360\) 0 0
\(361\) 1930.38 0.281438
\(362\) 0 0
\(363\) 3798.92 0.549288
\(364\) 0 0
\(365\) −11588.5 −1.66184
\(366\) 0 0
\(367\) −1184.49 −0.168474 −0.0842372 0.996446i \(-0.526845\pi\)
−0.0842372 + 0.996446i \(0.526845\pi\)
\(368\) 0 0
\(369\) −1370.07 −0.193287
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −944.218 −0.131072 −0.0655359 0.997850i \(-0.520876\pi\)
−0.0655359 + 0.997850i \(0.520876\pi\)
\(374\) 0 0
\(375\) 13676.8 1.88338
\(376\) 0 0
\(377\) 540.493 0.0738377
\(378\) 0 0
\(379\) 2906.32 0.393898 0.196949 0.980414i \(-0.436897\pi\)
0.196949 + 0.980414i \(0.436897\pi\)
\(380\) 0 0
\(381\) −10879.8 −1.46297
\(382\) 0 0
\(383\) 1146.84 0.153005 0.0765023 0.997069i \(-0.475625\pi\)
0.0765023 + 0.997069i \(0.475625\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1720.00 0.225924
\(388\) 0 0
\(389\) −5589.49 −0.728530 −0.364265 0.931295i \(-0.618680\pi\)
−0.364265 + 0.931295i \(0.618680\pi\)
\(390\) 0 0
\(391\) −869.353 −0.112443
\(392\) 0 0
\(393\) −11857.0 −1.52190
\(394\) 0 0
\(395\) 15006.7 1.91157
\(396\) 0 0
\(397\) 3154.10 0.398740 0.199370 0.979924i \(-0.436110\pi\)
0.199370 + 0.979924i \(0.436110\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6391.76 −0.795983 −0.397992 0.917389i \(-0.630293\pi\)
−0.397992 + 0.917389i \(0.630293\pi\)
\(402\) 0 0
\(403\) 572.554 0.0707715
\(404\) 0 0
\(405\) 17229.8 2.11397
\(406\) 0 0
\(407\) 11388.6 1.38701
\(408\) 0 0
\(409\) 196.630 0.0237720 0.0118860 0.999929i \(-0.496216\pi\)
0.0118860 + 0.999929i \(0.496216\pi\)
\(410\) 0 0
\(411\) 3476.53 0.417238
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6502.54 −0.769150
\(416\) 0 0
\(417\) 7156.86 0.840462
\(418\) 0 0
\(419\) 15699.9 1.83052 0.915262 0.402858i \(-0.131983\pi\)
0.915262 + 0.402858i \(0.131983\pi\)
\(420\) 0 0
\(421\) −8097.20 −0.937372 −0.468686 0.883365i \(-0.655272\pi\)
−0.468686 + 0.883365i \(0.655272\pi\)
\(422\) 0 0
\(423\) 492.414 0.0566005
\(424\) 0 0
\(425\) −24554.3 −2.80249
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1670.28 −0.187976
\(430\) 0 0
\(431\) −11578.4 −1.29399 −0.646995 0.762494i \(-0.723974\pi\)
−0.646995 + 0.762494i \(0.723974\pi\)
\(432\) 0 0
\(433\) −7270.25 −0.806896 −0.403448 0.915003i \(-0.632188\pi\)
−0.403448 + 0.915003i \(0.632188\pi\)
\(434\) 0 0
\(435\) 6099.12 0.672254
\(436\) 0 0
\(437\) 804.328 0.0880462
\(438\) 0 0
\(439\) 4127.39 0.448723 0.224362 0.974506i \(-0.427970\pi\)
0.224362 + 0.974506i \(0.427970\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 190.439 0.0204244 0.0102122 0.999948i \(-0.496749\pi\)
0.0102122 + 0.999948i \(0.496749\pi\)
\(444\) 0 0
\(445\) −9812.09 −1.04525
\(446\) 0 0
\(447\) −10786.8 −1.14138
\(448\) 0 0
\(449\) 17175.2 1.80523 0.902613 0.430454i \(-0.141646\pi\)
0.902613 + 0.430454i \(0.141646\pi\)
\(450\) 0 0
\(451\) −3641.55 −0.380208
\(452\) 0 0
\(453\) −3059.67 −0.317342
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3505.80 −0.358850 −0.179425 0.983772i \(-0.557424\pi\)
−0.179425 + 0.983772i \(0.557424\pi\)
\(458\) 0 0
\(459\) −10478.3 −1.06555
\(460\) 0 0
\(461\) 5983.80 0.604541 0.302270 0.953222i \(-0.402255\pi\)
0.302270 + 0.953222i \(0.402255\pi\)
\(462\) 0 0
\(463\) −14057.4 −1.41102 −0.705510 0.708700i \(-0.749282\pi\)
−0.705510 + 0.708700i \(0.749282\pi\)
\(464\) 0 0
\(465\) 6460.90 0.644338
\(466\) 0 0
\(467\) −18773.8 −1.86027 −0.930137 0.367211i \(-0.880313\pi\)
−0.930137 + 0.367211i \(0.880313\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −14211.2 −1.39027
\(472\) 0 0
\(473\) 4571.65 0.444407
\(474\) 0 0
\(475\) 22717.7 2.19444
\(476\) 0 0
\(477\) −4744.76 −0.455446
\(478\) 0 0
\(479\) −263.773 −0.0251610 −0.0125805 0.999921i \(-0.504005\pi\)
−0.0125805 + 0.999921i \(0.504005\pi\)
\(480\) 0 0
\(481\) 4426.61 0.419617
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12884.8 1.20633
\(486\) 0 0
\(487\) −12654.9 −1.17751 −0.588757 0.808310i \(-0.700383\pi\)
−0.588757 + 0.808310i \(0.700383\pi\)
\(488\) 0 0
\(489\) −10563.3 −0.976868
\(490\) 0 0
\(491\) −19041.6 −1.75017 −0.875087 0.483965i \(-0.839196\pi\)
−0.875087 + 0.483965i \(0.839196\pi\)
\(492\) 0 0
\(493\) −5301.37 −0.484304
\(494\) 0 0
\(495\) −5094.07 −0.462548
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 17999.4 1.61476 0.807378 0.590035i \(-0.200886\pi\)
0.807378 + 0.590035i \(0.200886\pi\)
\(500\) 0 0
\(501\) 4065.30 0.362523
\(502\) 0 0
\(503\) −15245.1 −1.35139 −0.675693 0.737183i \(-0.736155\pi\)
−0.675693 + 0.737183i \(0.736155\pi\)
\(504\) 0 0
\(505\) −2563.83 −0.225918
\(506\) 0 0
\(507\) 12714.6 1.11376
\(508\) 0 0
\(509\) 4807.85 0.418672 0.209336 0.977844i \(-0.432870\pi\)
0.209336 + 0.977844i \(0.432870\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 9694.58 0.834359
\(514\) 0 0
\(515\) 22246.3 1.90347
\(516\) 0 0
\(517\) 1308.80 0.111337
\(518\) 0 0
\(519\) 13048.9 1.10363
\(520\) 0 0
\(521\) −300.226 −0.0252460 −0.0126230 0.999920i \(-0.504018\pi\)
−0.0126230 + 0.999920i \(0.504018\pi\)
\(522\) 0 0
\(523\) −795.488 −0.0665091 −0.0332546 0.999447i \(-0.510587\pi\)
−0.0332546 + 0.999447i \(0.510587\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5615.83 −0.464193
\(528\) 0 0
\(529\) −12093.4 −0.993950
\(530\) 0 0
\(531\) −1970.62 −0.161050
\(532\) 0 0
\(533\) −1415.42 −0.115026
\(534\) 0 0
\(535\) −25083.6 −2.02702
\(536\) 0 0
\(537\) 22917.1 1.84161
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18669.7 −1.48368 −0.741841 0.670576i \(-0.766047\pi\)
−0.741841 + 0.670576i \(0.766047\pi\)
\(542\) 0 0
\(543\) 10660.1 0.842485
\(544\) 0 0
\(545\) 20044.1 1.57540
\(546\) 0 0
\(547\) 3264.72 0.255191 0.127596 0.991826i \(-0.459274\pi\)
0.127596 + 0.991826i \(0.459274\pi\)
\(548\) 0 0
\(549\) −4011.24 −0.311832
\(550\) 0 0
\(551\) 4904.84 0.379226
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 49951.5 3.82040
\(556\) 0 0
\(557\) 13404.7 1.01970 0.509852 0.860262i \(-0.329700\pi\)
0.509852 + 0.860262i \(0.329700\pi\)
\(558\) 0 0
\(559\) 1776.94 0.134448
\(560\) 0 0
\(561\) 16382.8 1.23294
\(562\) 0 0
\(563\) −9299.00 −0.696103 −0.348051 0.937475i \(-0.613157\pi\)
−0.348051 + 0.937475i \(0.613157\pi\)
\(564\) 0 0
\(565\) 2253.61 0.167806
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20197.8 1.48811 0.744055 0.668119i \(-0.232900\pi\)
0.744055 + 0.668119i \(0.232900\pi\)
\(570\) 0 0
\(571\) 9413.56 0.689922 0.344961 0.938617i \(-0.387892\pi\)
0.344961 + 0.938617i \(0.387892\pi\)
\(572\) 0 0
\(573\) −12233.2 −0.891880
\(574\) 0 0
\(575\) 2078.92 0.150777
\(576\) 0 0
\(577\) 7737.70 0.558275 0.279138 0.960251i \(-0.409951\pi\)
0.279138 + 0.960251i \(0.409951\pi\)
\(578\) 0 0
\(579\) 9610.47 0.689806
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −12611.3 −0.895892
\(584\) 0 0
\(585\) −1980.00 −0.139937
\(586\) 0 0
\(587\) −16755.0 −1.17811 −0.589057 0.808091i \(-0.700501\pi\)
−0.589057 + 0.808091i \(0.700501\pi\)
\(588\) 0 0
\(589\) 5195.78 0.363478
\(590\) 0 0
\(591\) 160.666 0.0111826
\(592\) 0 0
\(593\) −11772.9 −0.815270 −0.407635 0.913145i \(-0.633646\pi\)
−0.407635 + 0.913145i \(0.633646\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −350.040 −0.0239970
\(598\) 0 0
\(599\) 25259.3 1.72299 0.861493 0.507769i \(-0.169530\pi\)
0.861493 + 0.507769i \(0.169530\pi\)
\(600\) 0 0
\(601\) 187.801 0.0127464 0.00637319 0.999980i \(-0.497971\pi\)
0.00637319 + 0.999980i \(0.497971\pi\)
\(602\) 0 0
\(603\) −1254.21 −0.0847019
\(604\) 0 0
\(605\) 11969.6 0.804354
\(606\) 0 0
\(607\) −17655.3 −1.18057 −0.590284 0.807196i \(-0.700984\pi\)
−0.590284 + 0.807196i \(0.700984\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 508.716 0.0336832
\(612\) 0 0
\(613\) −1004.93 −0.0662130 −0.0331065 0.999452i \(-0.510540\pi\)
−0.0331065 + 0.999452i \(0.510540\pi\)
\(614\) 0 0
\(615\) −15972.2 −1.04725
\(616\) 0 0
\(617\) 8416.05 0.549137 0.274568 0.961568i \(-0.411465\pi\)
0.274568 + 0.961568i \(0.411465\pi\)
\(618\) 0 0
\(619\) −6285.24 −0.408118 −0.204059 0.978959i \(-0.565413\pi\)
−0.204059 + 0.978959i \(0.565413\pi\)
\(620\) 0 0
\(621\) 887.163 0.0573279
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 12803.0 0.819394
\(626\) 0 0
\(627\) −15157.4 −0.965435
\(628\) 0 0
\(629\) −43417.9 −2.75228
\(630\) 0 0
\(631\) −15928.9 −1.00495 −0.502473 0.864593i \(-0.667576\pi\)
−0.502473 + 0.864593i \(0.667576\pi\)
\(632\) 0 0
\(633\) −7262.24 −0.456000
\(634\) 0 0
\(635\) −34280.1 −2.14231
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −7886.35 −0.488230
\(640\) 0 0
\(641\) −12683.0 −0.781513 −0.390756 0.920494i \(-0.627787\pi\)
−0.390756 + 0.920494i \(0.627787\pi\)
\(642\) 0 0
\(643\) 1461.21 0.0896179 0.0448090 0.998996i \(-0.485732\pi\)
0.0448090 + 0.998996i \(0.485732\pi\)
\(644\) 0 0
\(645\) 20051.6 1.22408
\(646\) 0 0
\(647\) −24001.4 −1.45841 −0.729206 0.684294i \(-0.760110\pi\)
−0.729206 + 0.684294i \(0.760110\pi\)
\(648\) 0 0
\(649\) −5237.78 −0.316797
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9097.14 0.545174 0.272587 0.962131i \(-0.412121\pi\)
0.272587 + 0.962131i \(0.412121\pi\)
\(654\) 0 0
\(655\) −37359.0 −2.22861
\(656\) 0 0
\(657\) 6046.55 0.359054
\(658\) 0 0
\(659\) 15018.0 0.887738 0.443869 0.896092i \(-0.353606\pi\)
0.443869 + 0.896092i \(0.353606\pi\)
\(660\) 0 0
\(661\) −24497.2 −1.44150 −0.720748 0.693197i \(-0.756202\pi\)
−0.720748 + 0.693197i \(0.756202\pi\)
\(662\) 0 0
\(663\) 6367.78 0.373008
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 448.848 0.0260562
\(668\) 0 0
\(669\) −17771.7 −1.02705
\(670\) 0 0
\(671\) −10661.6 −0.613394
\(672\) 0 0
\(673\) 1378.86 0.0789762 0.0394881 0.999220i \(-0.487427\pi\)
0.0394881 + 0.999220i \(0.487427\pi\)
\(674\) 0 0
\(675\) 25057.3 1.42882
\(676\) 0 0
\(677\) −12535.0 −0.711607 −0.355804 0.934561i \(-0.615793\pi\)
−0.355804 + 0.934561i \(0.615793\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −18931.7 −1.06529
\(682\) 0 0
\(683\) −27054.1 −1.51566 −0.757830 0.652452i \(-0.773741\pi\)
−0.757830 + 0.652452i \(0.773741\pi\)
\(684\) 0 0
\(685\) 10953.8 0.610985
\(686\) 0 0
\(687\) 32453.4 1.80229
\(688\) 0 0
\(689\) −4901.84 −0.271038
\(690\) 0 0
\(691\) 25843.5 1.42277 0.711385 0.702803i \(-0.248068\pi\)
0.711385 + 0.702803i \(0.248068\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 22549.8 1.23074
\(696\) 0 0
\(697\) 13883.0 0.754459
\(698\) 0 0
\(699\) 31862.0 1.72408
\(700\) 0 0
\(701\) −29351.6 −1.58145 −0.790723 0.612174i \(-0.790295\pi\)
−0.790723 + 0.612174i \(0.790295\pi\)
\(702\) 0 0
\(703\) 40170.4 2.15513
\(704\) 0 0
\(705\) 5740.53 0.306668
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17013.4 −0.901202 −0.450601 0.892725i \(-0.648790\pi\)
−0.450601 + 0.892725i \(0.648790\pi\)
\(710\) 0 0
\(711\) −7830.07 −0.413011
\(712\) 0 0
\(713\) 475.473 0.0249742
\(714\) 0 0
\(715\) −5262.71 −0.275265
\(716\) 0 0
\(717\) 25432.3 1.32467
\(718\) 0 0
\(719\) −14546.5 −0.754510 −0.377255 0.926109i \(-0.623132\pi\)
−0.377255 + 0.926109i \(0.623132\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −33241.5 −1.70991
\(724\) 0 0
\(725\) 12677.4 0.649416
\(726\) 0 0
\(727\) −25023.9 −1.27660 −0.638298 0.769790i \(-0.720361\pi\)
−0.638298 + 0.769790i \(0.720361\pi\)
\(728\) 0 0
\(729\) 7993.00 0.406086
\(730\) 0 0
\(731\) −17428.9 −0.881850
\(732\) 0 0
\(733\) −12899.2 −0.649988 −0.324994 0.945716i \(-0.605362\pi\)
−0.324994 + 0.945716i \(0.605362\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3333.60 −0.166614
\(738\) 0 0
\(739\) 15253.3 0.759270 0.379635 0.925136i \(-0.376050\pi\)
0.379635 + 0.925136i \(0.376050\pi\)
\(740\) 0 0
\(741\) −5891.48 −0.292077
\(742\) 0 0
\(743\) 7984.56 0.394247 0.197123 0.980379i \(-0.436840\pi\)
0.197123 + 0.980379i \(0.436840\pi\)
\(744\) 0 0
\(745\) −33986.8 −1.67138
\(746\) 0 0
\(747\) 3392.83 0.166181
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −32781.6 −1.59283 −0.796416 0.604750i \(-0.793273\pi\)
−0.796416 + 0.604750i \(0.793273\pi\)
\(752\) 0 0
\(753\) −34787.0 −1.68355
\(754\) 0 0
\(755\) −9640.40 −0.464702
\(756\) 0 0
\(757\) −16431.5 −0.788920 −0.394460 0.918913i \(-0.629068\pi\)
−0.394460 + 0.918913i \(0.629068\pi\)
\(758\) 0 0
\(759\) −1387.07 −0.0663340
\(760\) 0 0
\(761\) −33858.3 −1.61283 −0.806415 0.591350i \(-0.798595\pi\)
−0.806415 + 0.591350i \(0.798595\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 19420.6 0.917849
\(766\) 0 0
\(767\) −2035.86 −0.0958418
\(768\) 0 0
\(769\) −36173.6 −1.69630 −0.848149 0.529758i \(-0.822283\pi\)
−0.848149 + 0.529758i \(0.822283\pi\)
\(770\) 0 0
\(771\) −6193.96 −0.289326
\(772\) 0 0
\(773\) 20812.8 0.968416 0.484208 0.874953i \(-0.339108\pi\)
0.484208 + 0.874953i \(0.339108\pi\)
\(774\) 0 0
\(775\) 13429.4 0.622449
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12844.6 −0.590766
\(780\) 0 0
\(781\) −20961.4 −0.960381
\(782\) 0 0
\(783\) 5409.98 0.246918
\(784\) 0 0
\(785\) −44776.6 −2.03586
\(786\) 0 0
\(787\) 29722.2 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(788\) 0 0
\(789\) −50026.5 −2.25728
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4144.04 −0.185573
\(794\) 0 0
\(795\) −55314.1 −2.46766
\(796\) 0 0
\(797\) 6475.82 0.287811 0.143905 0.989591i \(-0.454034\pi\)
0.143905 + 0.989591i \(0.454034\pi\)
\(798\) 0 0
\(799\) −4989.69 −0.220929
\(800\) 0 0
\(801\) 5119.66 0.225835
\(802\) 0 0
\(803\) 16071.3 0.706283
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 35692.9 1.55694
\(808\) 0 0
\(809\) −39418.9 −1.71309 −0.856547 0.516069i \(-0.827395\pi\)
−0.856547 + 0.516069i \(0.827395\pi\)
\(810\) 0 0
\(811\) −38728.6 −1.67688 −0.838438 0.544997i \(-0.816531\pi\)
−0.838438 + 0.544997i \(0.816531\pi\)
\(812\) 0 0
\(813\) 37815.2 1.63129
\(814\) 0 0
\(815\) −33282.7 −1.43048
\(816\) 0 0
\(817\) 16125.3 0.690517
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 32010.0 1.36073 0.680363 0.732876i \(-0.261822\pi\)
0.680363 + 0.732876i \(0.261822\pi\)
\(822\) 0 0
\(823\) −10137.8 −0.429384 −0.214692 0.976682i \(-0.568875\pi\)
−0.214692 + 0.976682i \(0.568875\pi\)
\(824\) 0 0
\(825\) −39176.9 −1.65329
\(826\) 0 0
\(827\) −9064.96 −0.381160 −0.190580 0.981672i \(-0.561037\pi\)
−0.190580 + 0.981672i \(0.561037\pi\)
\(828\) 0 0
\(829\) 19310.3 0.809017 0.404508 0.914534i \(-0.367443\pi\)
0.404508 + 0.914534i \(0.367443\pi\)
\(830\) 0 0
\(831\) −8987.43 −0.375175
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 12808.9 0.530863
\(836\) 0 0
\(837\) 5730.88 0.236665
\(838\) 0 0
\(839\) −11458.4 −0.471498 −0.235749 0.971814i \(-0.575754\pi\)
−0.235749 + 0.971814i \(0.575754\pi\)
\(840\) 0 0
\(841\) −21651.9 −0.887773
\(842\) 0 0
\(843\) 21970.5 0.897634
\(844\) 0 0
\(845\) 40061.1 1.63094
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −46399.8 −1.87566
\(850\) 0 0
\(851\) 3676.04 0.148076
\(852\) 0 0
\(853\) 6203.94 0.249026 0.124513 0.992218i \(-0.460263\pi\)
0.124513 + 0.992218i \(0.460263\pi\)
\(854\) 0 0
\(855\) −17968.0 −0.718705
\(856\) 0 0
\(857\) −15577.8 −0.620920 −0.310460 0.950587i \(-0.600483\pi\)
−0.310460 + 0.950587i \(0.600483\pi\)
\(858\) 0 0
\(859\) 11893.7 0.472420 0.236210 0.971702i \(-0.424095\pi\)
0.236210 + 0.971702i \(0.424095\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11673.6 0.460455 0.230227 0.973137i \(-0.426053\pi\)
0.230227 + 0.973137i \(0.426053\pi\)
\(864\) 0 0
\(865\) 41114.5 1.61611
\(866\) 0 0
\(867\) −32573.1 −1.27594
\(868\) 0 0
\(869\) −20811.8 −0.812420
\(870\) 0 0
\(871\) −1295.73 −0.0504065
\(872\) 0 0
\(873\) −6722.90 −0.260637
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −3063.66 −0.117962 −0.0589808 0.998259i \(-0.518785\pi\)
−0.0589808 + 0.998259i \(0.518785\pi\)
\(878\) 0 0
\(879\) 32808.4 1.25893
\(880\) 0 0
\(881\) 19236.9 0.735649 0.367825 0.929895i \(-0.380103\pi\)
0.367825 + 0.929895i \(0.380103\pi\)
\(882\) 0 0
\(883\) 35556.9 1.35514 0.677569 0.735459i \(-0.263034\pi\)
0.677569 + 0.735459i \(0.263034\pi\)
\(884\) 0 0
\(885\) −22973.4 −0.872590
\(886\) 0 0
\(887\) −23218.8 −0.878928 −0.439464 0.898260i \(-0.644832\pi\)
−0.439464 + 0.898260i \(0.644832\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −23894.8 −0.898436
\(892\) 0 0
\(893\) 4616.47 0.172995
\(894\) 0 0
\(895\) 72207.0 2.69678
\(896\) 0 0
\(897\) −539.137 −0.0200683
\(898\) 0 0
\(899\) 2899.46 0.107567
\(900\) 0 0
\(901\) 48079.2 1.77775
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 33587.8 1.23370
\(906\) 0 0
\(907\) 20516.7 0.751098 0.375549 0.926803i \(-0.377454\pi\)
0.375549 + 0.926803i \(0.377454\pi\)
\(908\) 0 0
\(909\) 1337.73 0.0488115
\(910\) 0 0
\(911\) −18247.6 −0.663634 −0.331817 0.943344i \(-0.607662\pi\)
−0.331817 + 0.943344i \(0.607662\pi\)
\(912\) 0 0
\(913\) 9017.92 0.326889
\(914\) 0 0
\(915\) −46762.8 −1.68954
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 27362.2 0.982149 0.491074 0.871118i \(-0.336604\pi\)
0.491074 + 0.871118i \(0.336604\pi\)
\(920\) 0 0
\(921\) 48991.5 1.75280
\(922\) 0 0
\(923\) −8147.42 −0.290548
\(924\) 0 0
\(925\) 103827. 3.69061
\(926\) 0 0
\(927\) −11607.4 −0.411261
\(928\) 0 0
\(929\) 43542.6 1.53777 0.768884 0.639389i \(-0.220813\pi\)
0.768884 + 0.639389i \(0.220813\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 13896.7 0.487629
\(934\) 0 0
\(935\) 51618.7 1.80547
\(936\) 0 0
\(937\) 32990.1 1.15020 0.575101 0.818083i \(-0.304963\pi\)
0.575101 + 0.818083i \(0.304963\pi\)
\(938\) 0 0
\(939\) −50706.7 −1.76225
\(940\) 0 0
\(941\) −17311.6 −0.599726 −0.299863 0.953982i \(-0.596941\pi\)
−0.299863 + 0.953982i \(0.596941\pi\)
\(942\) 0 0
\(943\) −1175.43 −0.0405909
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1335.71 0.0458339 0.0229169 0.999737i \(-0.492705\pi\)
0.0229169 + 0.999737i \(0.492705\pi\)
\(948\) 0 0
\(949\) 6246.72 0.213675
\(950\) 0 0
\(951\) 63359.2 2.16042
\(952\) 0 0
\(953\) 18877.6 0.641663 0.320831 0.947136i \(-0.396038\pi\)
0.320831 + 0.947136i \(0.396038\pi\)
\(954\) 0 0
\(955\) −38544.1 −1.30603
\(956\) 0 0
\(957\) −8458.45 −0.285708
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −26719.6 −0.896900
\(962\) 0 0
\(963\) 13087.9 0.437955
\(964\) 0 0
\(965\) 30280.6 1.01012
\(966\) 0 0
\(967\) 39814.6 1.32405 0.662023 0.749483i \(-0.269698\pi\)
0.662023 + 0.749483i \(0.269698\pi\)
\(968\) 0 0
\(969\) 57786.0 1.91574
\(970\) 0 0
\(971\) 22844.8 0.755021 0.377511 0.926005i \(-0.376780\pi\)
0.377511 + 0.926005i \(0.376780\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −15227.5 −0.500176
\(976\) 0 0
\(977\) 24396.6 0.798891 0.399446 0.916757i \(-0.369203\pi\)
0.399446 + 0.916757i \(0.369203\pi\)
\(978\) 0 0
\(979\) 13607.7 0.444233
\(980\) 0 0
\(981\) −10458.4 −0.340379
\(982\) 0 0
\(983\) −32975.5 −1.06994 −0.534972 0.844870i \(-0.679678\pi\)
−0.534972 + 0.844870i \(0.679678\pi\)
\(984\) 0 0
\(985\) 506.224 0.0163753
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1475.65 0.0474447
\(990\) 0 0
\(991\) 11877.1 0.380717 0.190358 0.981715i \(-0.439035\pi\)
0.190358 + 0.981715i \(0.439035\pi\)
\(992\) 0 0
\(993\) 42238.0 1.34983
\(994\) 0 0
\(995\) −1102.90 −0.0351401
\(996\) 0 0
\(997\) 16386.3 0.520522 0.260261 0.965538i \(-0.416191\pi\)
0.260261 + 0.965538i \(0.416191\pi\)
\(998\) 0 0
\(999\) 44307.4 1.40323
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 784.4.a.u.1.1 2
4.3 odd 2 196.4.a.e.1.2 2
7.2 even 3 112.4.i.d.81.2 4
7.4 even 3 112.4.i.d.65.2 4
7.6 odd 2 784.4.a.ba.1.2 2
12.11 even 2 1764.4.a.z.1.2 2
28.3 even 6 196.4.e.g.177.2 4
28.11 odd 6 28.4.e.a.9.1 4
28.19 even 6 196.4.e.g.165.2 4
28.23 odd 6 28.4.e.a.25.1 yes 4
28.27 even 2 196.4.a.g.1.1 2
56.11 odd 6 448.4.i.h.65.2 4
56.37 even 6 448.4.i.g.193.1 4
56.51 odd 6 448.4.i.h.193.2 4
56.53 even 6 448.4.i.g.65.1 4
84.11 even 6 252.4.k.c.37.1 4
84.23 even 6 252.4.k.c.109.1 4
84.47 odd 6 1764.4.k.ba.361.2 4
84.59 odd 6 1764.4.k.ba.1549.2 4
84.83 odd 2 1764.4.a.n.1.1 2
140.23 even 12 700.4.r.d.249.1 8
140.39 odd 6 700.4.i.g.401.2 4
140.67 even 12 700.4.r.d.149.1 8
140.79 odd 6 700.4.i.g.501.2 4
140.107 even 12 700.4.r.d.249.4 8
140.123 even 12 700.4.r.d.149.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.4.e.a.9.1 4 28.11 odd 6
28.4.e.a.25.1 yes 4 28.23 odd 6
112.4.i.d.65.2 4 7.4 even 3
112.4.i.d.81.2 4 7.2 even 3
196.4.a.e.1.2 2 4.3 odd 2
196.4.a.g.1.1 2 28.27 even 2
196.4.e.g.165.2 4 28.19 even 6
196.4.e.g.177.2 4 28.3 even 6
252.4.k.c.37.1 4 84.11 even 6
252.4.k.c.109.1 4 84.23 even 6
448.4.i.g.65.1 4 56.53 even 6
448.4.i.g.193.1 4 56.37 even 6
448.4.i.h.65.2 4 56.11 odd 6
448.4.i.h.193.2 4 56.51 odd 6
700.4.i.g.401.2 4 140.39 odd 6
700.4.i.g.501.2 4 140.79 odd 6
700.4.r.d.149.1 8 140.67 even 12
700.4.r.d.149.4 8 140.123 even 12
700.4.r.d.249.1 8 140.23 even 12
700.4.r.d.249.4 8 140.107 even 12
784.4.a.u.1.1 2 1.1 even 1 trivial
784.4.a.ba.1.2 2 7.6 odd 2
1764.4.a.n.1.1 2 84.83 odd 2
1764.4.a.z.1.2 2 12.11 even 2
1764.4.k.ba.361.2 4 84.47 odd 6
1764.4.k.ba.1549.2 4 84.59 odd 6