Properties

Label 784.4.a.ba.1.1
Level $784$
Weight $4$
Character 784.1
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
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)
 
\(f(q)\) \(=\) \(q-6.08276 q^{3} -5.16553 q^{5} +10.0000 q^{9} +O(q^{10})\) \(q-6.08276 q^{3} -5.16553 q^{5} +10.0000 q^{9} -58.5793 q^{11} +38.3311 q^{13} +31.4207 q^{15} +52.6689 q^{17} -130.248 q^{19} -76.5793 q^{23} -98.3174 q^{25} +103.407 q^{27} -288.317 q^{29} -140.579 q^{31} +356.324 q^{33} -82.4760 q^{37} -233.159 q^{39} +282.993 q^{41} +172.000 q^{43} -51.6553 q^{45} +133.241 q^{47} -320.373 q^{51} +36.4760 q^{53} +302.593 q^{55} +792.269 q^{57} -253.062 q^{59} -499.124 q^{61} -198.000 q^{65} -210.579 q^{67} +465.814 q^{69} -107.365 q^{71} -361.345 q^{73} +598.041 q^{75} +835.007 q^{79} -899.000 q^{81} +731.283 q^{83} -272.063 q^{85} +1753.77 q^{87} +217.966 q^{89} +855.111 q^{93} +672.801 q^{95} -252.290 q^{97} -585.793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{5} + 20 q^{9}+O(q^{10}) \) 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
\(5\) −5.16553 −0.462019 −0.231009 0.972952i \(-0.574203\pi\)
−0.231009 + 0.972952i \(0.574203\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 10.0000 0.370370
\(10\) 0 0
\(11\) −58.5793 −1.60567 −0.802833 0.596203i \(-0.796675\pi\)
−0.802833 + 0.596203i \(0.796675\pi\)
\(12\) 0 0
\(13\) 38.3311 0.817779 0.408889 0.912584i \(-0.365916\pi\)
0.408889 + 0.912584i \(0.365916\pi\)
\(14\) 0 0
\(15\) 31.4207 0.540852
\(16\) 0 0
\(17\) 52.6689 0.751417 0.375709 0.926738i \(-0.377399\pi\)
0.375709 + 0.926738i \(0.377399\pi\)
\(18\) 0 0
\(19\) −130.248 −1.57268 −0.786342 0.617791i \(-0.788028\pi\)
−0.786342 + 0.617791i \(0.788028\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −76.5793 −0.694256 −0.347128 0.937818i \(-0.612843\pi\)
−0.347128 + 0.937818i \(0.612843\pi\)
\(24\) 0 0
\(25\) −98.3174 −0.786539
\(26\) 0 0
\(27\) 103.407 0.737062
\(28\) 0 0
\(29\) −288.317 −1.84618 −0.923089 0.384585i \(-0.874344\pi\)
−0.923089 + 0.384585i \(0.874344\pi\)
\(30\) 0 0
\(31\) −140.579 −0.814477 −0.407239 0.913322i \(-0.633508\pi\)
−0.407239 + 0.913322i \(0.633508\pi\)
\(32\) 0 0
\(33\) 356.324 1.87964
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −82.4760 −0.366459 −0.183229 0.983070i \(-0.558655\pi\)
−0.183229 + 0.983070i \(0.558655\pi\)
\(38\) 0 0
\(39\) −233.159 −0.957315
\(40\) 0 0
\(41\) 282.993 1.07795 0.538977 0.842321i \(-0.318811\pi\)
0.538977 + 0.842321i \(0.318811\pi\)
\(42\) 0 0
\(43\) 172.000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −51.6553 −0.171118
\(46\) 0 0
\(47\) 133.241 0.413516 0.206758 0.978392i \(-0.433709\pi\)
0.206758 + 0.978392i \(0.433709\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −320.373 −0.879630
\(52\) 0 0
\(53\) 36.4760 0.0945352 0.0472676 0.998882i \(-0.484949\pi\)
0.0472676 + 0.998882i \(0.484949\pi\)
\(54\) 0 0
\(55\) 302.593 0.741848
\(56\) 0 0
\(57\) 792.269 1.84103
\(58\) 0 0
\(59\) −253.062 −0.558405 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(60\) 0 0
\(61\) −499.124 −1.04764 −0.523822 0.851827i \(-0.675494\pi\)
−0.523822 + 0.851827i \(0.675494\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −198.000 −0.377829
\(66\) 0 0
\(67\) −210.579 −0.383976 −0.191988 0.981397i \(-0.561493\pi\)
−0.191988 + 0.981397i \(0.561493\pi\)
\(68\) 0 0
\(69\) 465.814 0.812716
\(70\) 0 0
\(71\) −107.365 −0.179464 −0.0897318 0.995966i \(-0.528601\pi\)
−0.0897318 + 0.995966i \(0.528601\pi\)
\(72\) 0 0
\(73\) −361.345 −0.579345 −0.289673 0.957126i \(-0.593546\pi\)
−0.289673 + 0.957126i \(0.593546\pi\)
\(74\) 0 0
\(75\) 598.041 0.920745
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 835.007 1.18919 0.594593 0.804027i \(-0.297313\pi\)
0.594593 + 0.804027i \(0.297313\pi\)
\(80\) 0 0
\(81\) −899.000 −1.23320
\(82\) 0 0
\(83\) 731.283 0.967093 0.483547 0.875319i \(-0.339348\pi\)
0.483547 + 0.875319i \(0.339348\pi\)
\(84\) 0 0
\(85\) −272.063 −0.347169
\(86\) 0 0
\(87\) 1753.77 2.16119
\(88\) 0 0
\(89\) 217.966 0.259599 0.129800 0.991540i \(-0.458567\pi\)
0.129800 + 0.991540i \(0.458567\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 855.111 0.953450
\(94\) 0 0
\(95\) 672.801 0.726610
\(96\) 0 0
\(97\) −252.290 −0.264084 −0.132042 0.991244i \(-0.542153\pi\)
−0.132042 + 0.991244i \(0.542153\pi\)
\(98\) 0 0
\(99\) −585.793 −0.594691
\(100\) 0 0
\(101\) 1155.77 1.13865 0.569325 0.822112i \(-0.307205\pi\)
0.569325 + 0.822112i \(0.307205\pi\)
\(102\) 0 0
\(103\) 1051.26 1.00566 0.502831 0.864385i \(-0.332292\pi\)
0.502831 + 0.864385i \(0.332292\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2075.21 1.87494 0.937470 0.348067i \(-0.113162\pi\)
0.937470 + 0.348067i \(0.113162\pi\)
\(108\) 0 0
\(109\) −1216.16 −1.06869 −0.534343 0.845267i \(-0.679441\pi\)
−0.534343 + 0.845267i \(0.679441\pi\)
\(110\) 0 0
\(111\) 501.682 0.428987
\(112\) 0 0
\(113\) 1585.59 1.32000 0.659998 0.751268i \(-0.270557\pi\)
0.659998 + 0.751268i \(0.270557\pi\)
\(114\) 0 0
\(115\) 395.572 0.320759
\(116\) 0 0
\(117\) 383.311 0.302881
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2100.54 1.57817
\(122\) 0 0
\(123\) −1721.38 −1.26188
\(124\) 0 0
\(125\) 1153.55 0.825414
\(126\) 0 0
\(127\) 1107.37 0.773723 0.386861 0.922138i \(-0.373559\pi\)
0.386861 + 0.922138i \(0.373559\pi\)
\(128\) 0 0
\(129\) −1046.24 −0.714077
\(130\) 0 0
\(131\) 885.284 0.590440 0.295220 0.955429i \(-0.404607\pi\)
0.295220 + 0.955429i \(0.404607\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −534.151 −0.340536
\(136\) 0 0
\(137\) 2153.54 1.34299 0.671494 0.741010i \(-0.265653\pi\)
0.671494 + 0.741010i \(0.265653\pi\)
\(138\) 0 0
\(139\) −672.580 −0.410414 −0.205207 0.978719i \(-0.565787\pi\)
−0.205207 + 0.978719i \(0.565787\pi\)
\(140\) 0 0
\(141\) −810.476 −0.484074
\(142\) 0 0
\(143\) −2245.41 −1.31308
\(144\) 0 0
\(145\) 1489.31 0.852969
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1803.33 −0.991508 −0.495754 0.868463i \(-0.665108\pi\)
−0.495754 + 0.868463i \(0.665108\pi\)
\(150\) 0 0
\(151\) −1115.01 −0.600914 −0.300457 0.953795i \(-0.597139\pi\)
−0.300457 + 0.953795i \(0.597139\pi\)
\(152\) 0 0
\(153\) 526.689 0.278303
\(154\) 0 0
\(155\) 726.166 0.376304
\(156\) 0 0
\(157\) 1678.31 0.853146 0.426573 0.904453i \(-0.359721\pi\)
0.426573 + 0.904453i \(0.359721\pi\)
\(158\) 0 0
\(159\) −221.875 −0.110666
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1584.59 −0.761442 −0.380721 0.924690i \(-0.624324\pi\)
−0.380721 + 0.924690i \(0.624324\pi\)
\(164\) 0 0
\(165\) −1840.60 −0.868428
\(166\) 0 0
\(167\) 619.669 0.287134 0.143567 0.989641i \(-0.454143\pi\)
0.143567 + 0.989641i \(0.454143\pi\)
\(168\) 0 0
\(169\) −727.731 −0.331238
\(170\) 0 0
\(171\) −1302.48 −0.582476
\(172\) 0 0
\(173\) 3580.77 1.57365 0.786823 0.617179i \(-0.211725\pi\)
0.786823 + 0.617179i \(0.211725\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1539.32 0.653685
\(178\) 0 0
\(179\) 575.546 0.240326 0.120163 0.992754i \(-0.461658\pi\)
0.120163 + 0.992754i \(0.461658\pi\)
\(180\) 0 0
\(181\) −1556.51 −0.639197 −0.319598 0.947553i \(-0.603548\pi\)
−0.319598 + 0.947553i \(0.603548\pi\)
\(182\) 0 0
\(183\) 3036.06 1.22640
\(184\) 0 0
\(185\) 426.032 0.169311
\(186\) 0 0
\(187\) −3085.31 −1.20653
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −799.118 −0.302734 −0.151367 0.988478i \(-0.548367\pi\)
−0.151367 + 0.988478i \(0.548367\pi\)
\(192\) 0 0
\(193\) −558.048 −0.208130 −0.104065 0.994570i \(-0.533185\pi\)
−0.104065 + 0.994570i \(0.533185\pi\)
\(194\) 0 0
\(195\) 1204.39 0.442297
\(196\) 0 0
\(197\) −1729.59 −0.625523 −0.312761 0.949832i \(-0.601254\pi\)
−0.312761 + 0.949832i \(0.601254\pi\)
\(198\) 0 0
\(199\) 4285.55 1.52660 0.763302 0.646041i \(-0.223577\pi\)
0.763302 + 0.646041i \(0.223577\pi\)
\(200\) 0 0
\(201\) 1280.90 0.449493
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1461.81 −0.498035
\(206\) 0 0
\(207\) −765.793 −0.257132
\(208\) 0 0
\(209\) 7629.86 2.52521
\(210\) 0 0
\(211\) −849.904 −0.277298 −0.138649 0.990342i \(-0.544276\pi\)
−0.138649 + 0.990342i \(0.544276\pi\)
\(212\) 0 0
\(213\) 653.078 0.210085
\(214\) 0 0
\(215\) −888.470 −0.281829
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2197.97 0.678198
\(220\) 0 0
\(221\) 2018.86 0.614493
\(222\) 0 0
\(223\) 4377.66 1.31457 0.657286 0.753641i \(-0.271704\pi\)
0.657286 + 0.753641i \(0.271704\pi\)
\(224\) 0 0
\(225\) −983.174 −0.291311
\(226\) 0 0
\(227\) −5119.66 −1.49693 −0.748466 0.663173i \(-0.769209\pi\)
−0.748466 + 0.663173i \(0.769209\pi\)
\(228\) 0 0
\(229\) −5881.31 −1.69715 −0.848576 0.529074i \(-0.822539\pi\)
−0.848576 + 0.529074i \(0.822539\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 212.078 0.0596295 0.0298147 0.999555i \(-0.490508\pi\)
0.0298147 + 0.999555i \(0.490508\pi\)
\(234\) 0 0
\(235\) −688.262 −0.191052
\(236\) 0 0
\(237\) −5079.15 −1.39209
\(238\) 0 0
\(239\) −5202.95 −1.40816 −0.704082 0.710119i \(-0.748641\pi\)
−0.704082 + 0.710119i \(0.748641\pi\)
\(240\) 0 0
\(241\) −2837.12 −0.758320 −0.379160 0.925331i \(-0.623787\pi\)
−0.379160 + 0.925331i \(0.623787\pi\)
\(242\) 0 0
\(243\) 2676.42 0.706552
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4992.55 −1.28611
\(248\) 0 0
\(249\) −4448.22 −1.13211
\(250\) 0 0
\(251\) −4697.05 −1.18118 −0.590588 0.806973i \(-0.701104\pi\)
−0.590588 + 0.806973i \(0.701104\pi\)
\(252\) 0 0
\(253\) 4485.97 1.11474
\(254\) 0 0
\(255\) 1654.89 0.406406
\(256\) 0 0
\(257\) −7003.72 −1.69992 −0.849961 0.526845i \(-0.823375\pi\)
−0.849961 + 0.526845i \(0.823375\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2883.17 −0.683770
\(262\) 0 0
\(263\) −1228.31 −0.287987 −0.143994 0.989579i \(-0.545994\pi\)
−0.143994 + 0.989579i \(0.545994\pi\)
\(264\) 0 0
\(265\) −188.418 −0.0436770
\(266\) 0 0
\(267\) −1325.83 −0.303894
\(268\) 0 0
\(269\) −141.885 −0.0321593 −0.0160797 0.999871i \(-0.505119\pi\)
−0.0160797 + 0.999871i \(0.505119\pi\)
\(270\) 0 0
\(271\) −364.775 −0.0817656 −0.0408828 0.999164i \(-0.513017\pi\)
−0.0408828 + 0.999164i \(0.513017\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5759.37 1.26292
\(276\) 0 0
\(277\) 1988.48 0.431321 0.215661 0.976468i \(-0.430810\pi\)
0.215661 + 0.976468i \(0.430810\pi\)
\(278\) 0 0
\(279\) −1405.79 −0.301658
\(280\) 0 0
\(281\) 4903.93 1.04108 0.520541 0.853837i \(-0.325730\pi\)
0.520541 + 0.853837i \(0.325730\pi\)
\(282\) 0 0
\(283\) 3260.07 0.684775 0.342387 0.939559i \(-0.388765\pi\)
0.342387 + 0.939559i \(0.388765\pi\)
\(284\) 0 0
\(285\) −4092.49 −0.850590
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2138.98 −0.435372
\(290\) 0 0
\(291\) 1534.62 0.309144
\(292\) 0 0
\(293\) −1613.67 −0.321746 −0.160873 0.986975i \(-0.551431\pi\)
−0.160873 + 0.986975i \(0.551431\pi\)
\(294\) 0 0
\(295\) 1307.20 0.257993
\(296\) 0 0
\(297\) −6057.51 −1.18348
\(298\) 0 0
\(299\) −2935.37 −0.567748
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −7030.29 −1.33294
\(304\) 0 0
\(305\) 2578.24 0.484031
\(306\) 0 0
\(307\) −2846.16 −0.529116 −0.264558 0.964370i \(-0.585226\pi\)
−0.264558 + 0.964370i \(0.585226\pi\)
\(308\) 0 0
\(309\) −6394.54 −1.17726
\(310\) 0 0
\(311\) 2783.39 0.507498 0.253749 0.967270i \(-0.418336\pi\)
0.253749 + 0.967270i \(0.418336\pi\)
\(312\) 0 0
\(313\) 3002.13 0.542143 0.271071 0.962559i \(-0.412622\pi\)
0.271071 + 0.962559i \(0.412622\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3773.81 −0.668638 −0.334319 0.942460i \(-0.608506\pi\)
−0.334319 + 0.942460i \(0.608506\pi\)
\(318\) 0 0
\(319\) 16889.4 2.96435
\(320\) 0 0
\(321\) −12623.0 −2.19486
\(322\) 0 0
\(323\) −6860.04 −1.18174
\(324\) 0 0
\(325\) −3768.61 −0.643215
\(326\) 0 0
\(327\) 7397.60 1.25104
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 975.878 0.162052 0.0810259 0.996712i \(-0.474180\pi\)
0.0810259 + 0.996712i \(0.474180\pi\)
\(332\) 0 0
\(333\) −824.760 −0.135725
\(334\) 0 0
\(335\) 1087.75 0.177404
\(336\) 0 0
\(337\) −3563.93 −0.576083 −0.288041 0.957618i \(-0.593004\pi\)
−0.288041 + 0.957618i \(0.593004\pi\)
\(338\) 0 0
\(339\) −9644.75 −1.54522
\(340\) 0 0
\(341\) 8235.04 1.30778
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2406.17 −0.375490
\(346\) 0 0
\(347\) 1155.35 0.178740 0.0893698 0.995999i \(-0.471515\pi\)
0.0893698 + 0.995999i \(0.471515\pi\)
\(348\) 0 0
\(349\) 4693.29 0.719845 0.359922 0.932982i \(-0.382803\pi\)
0.359922 + 0.932982i \(0.382803\pi\)
\(350\) 0 0
\(351\) 3963.70 0.602754
\(352\) 0 0
\(353\) 13257.1 1.99888 0.999439 0.0335035i \(-0.0106665\pi\)
0.999439 + 0.0335035i \(0.0106665\pi\)
\(354\) 0 0
\(355\) 554.598 0.0829155
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1786.62 0.262658 0.131329 0.991339i \(-0.458075\pi\)
0.131329 + 0.991339i \(0.458075\pi\)
\(360\) 0 0
\(361\) 10105.6 1.47334
\(362\) 0 0
\(363\) −12777.1 −1.84745
\(364\) 0 0
\(365\) 1866.54 0.267668
\(366\) 0 0
\(367\) 6403.51 0.910791 0.455395 0.890289i \(-0.349498\pi\)
0.455395 + 0.890289i \(0.349498\pi\)
\(368\) 0 0
\(369\) 2829.93 0.399242
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12170.2 1.68941 0.844705 0.535232i \(-0.179776\pi\)
0.844705 + 0.535232i \(0.179776\pi\)
\(374\) 0 0
\(375\) −7016.78 −0.966253
\(376\) 0 0
\(377\) −11051.5 −1.50977
\(378\) 0 0
\(379\) 2565.68 0.347732 0.173866 0.984769i \(-0.444374\pi\)
0.173866 + 0.984769i \(0.444374\pi\)
\(380\) 0 0
\(381\) −6735.84 −0.905742
\(382\) 0 0
\(383\) −6609.16 −0.881755 −0.440878 0.897567i \(-0.645333\pi\)
−0.440878 + 0.897567i \(0.645333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1720.00 0.225924
\(388\) 0 0
\(389\) 8887.49 1.15839 0.579195 0.815189i \(-0.303367\pi\)
0.579195 + 0.815189i \(0.303367\pi\)
\(390\) 0 0
\(391\) −4033.35 −0.521676
\(392\) 0 0
\(393\) −5384.97 −0.691185
\(394\) 0 0
\(395\) −4313.25 −0.549426
\(396\) 0 0
\(397\) −8871.90 −1.12158 −0.560791 0.827958i \(-0.689503\pi\)
−0.560791 + 0.827958i \(0.689503\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1282.24 −0.159681 −0.0798404 0.996808i \(-0.525441\pi\)
−0.0798404 + 0.996808i \(0.525441\pi\)
\(402\) 0 0
\(403\) −5388.55 −0.666062
\(404\) 0 0
\(405\) 4643.81 0.569760
\(406\) 0 0
\(407\) 4831.39 0.588411
\(408\) 0 0
\(409\) 7686.63 0.929289 0.464645 0.885497i \(-0.346182\pi\)
0.464645 + 0.885497i \(0.346182\pi\)
\(410\) 0 0
\(411\) −13099.5 −1.57214
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3777.46 −0.446815
\(416\) 0 0
\(417\) 4091.14 0.480442
\(418\) 0 0
\(419\) −13364.1 −1.55818 −0.779092 0.626909i \(-0.784320\pi\)
−0.779092 + 0.626909i \(0.784320\pi\)
\(420\) 0 0
\(421\) 1781.20 0.206201 0.103100 0.994671i \(-0.467124\pi\)
0.103100 + 0.994671i \(0.467124\pi\)
\(422\) 0 0
\(423\) 1332.41 0.153154
\(424\) 0 0
\(425\) −5178.27 −0.591019
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 13658.3 1.53713
\(430\) 0 0
\(431\) 7582.35 0.847399 0.423700 0.905803i \(-0.360731\pi\)
0.423700 + 0.905803i \(0.360731\pi\)
\(432\) 0 0
\(433\) −8642.25 −0.959169 −0.479585 0.877496i \(-0.659213\pi\)
−0.479585 + 0.877496i \(0.659213\pi\)
\(434\) 0 0
\(435\) −9059.12 −0.998510
\(436\) 0 0
\(437\) 9974.33 1.09185
\(438\) 0 0
\(439\) −14212.6 −1.54517 −0.772586 0.634910i \(-0.781037\pi\)
−0.772586 + 0.634910i \(0.781037\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7769.56 0.833280 0.416640 0.909072i \(-0.363208\pi\)
0.416640 + 0.909072i \(0.363208\pi\)
\(444\) 0 0
\(445\) −1125.91 −0.119940
\(446\) 0 0
\(447\) 10969.2 1.16069
\(448\) 0 0
\(449\) −7691.17 −0.808393 −0.404197 0.914672i \(-0.632449\pi\)
−0.404197 + 0.914672i \(0.632449\pi\)
\(450\) 0 0
\(451\) −16577.6 −1.73083
\(452\) 0 0
\(453\) 6782.33 0.703447
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13384.2 −1.36999 −0.684996 0.728547i \(-0.740196\pi\)
−0.684996 + 0.728547i \(0.740196\pi\)
\(458\) 0 0
\(459\) 5446.34 0.553841
\(460\) 0 0
\(461\) 5451.80 0.550793 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(462\) 0 0
\(463\) 13193.4 1.32430 0.662148 0.749373i \(-0.269645\pi\)
0.662148 + 0.749373i \(0.269645\pi\)
\(464\) 0 0
\(465\) −4417.10 −0.440512
\(466\) 0 0
\(467\) 2338.19 0.231688 0.115844 0.993267i \(-0.463043\pi\)
0.115844 + 0.993267i \(0.463043\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −10208.8 −0.998716
\(472\) 0 0
\(473\) −10075.6 −0.979448
\(474\) 0 0
\(475\) 12805.7 1.23698
\(476\) 0 0
\(477\) 364.760 0.0350131
\(478\) 0 0
\(479\) 18524.2 1.76700 0.883501 0.468430i \(-0.155180\pi\)
0.883501 + 0.468430i \(0.155180\pi\)
\(480\) 0 0
\(481\) −3161.39 −0.299682
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1303.21 0.122012
\(486\) 0 0
\(487\) −5757.07 −0.535684 −0.267842 0.963463i \(-0.586310\pi\)
−0.267842 + 0.963463i \(0.586310\pi\)
\(488\) 0 0
\(489\) 9638.71 0.891365
\(490\) 0 0
\(491\) 10593.6 0.973692 0.486846 0.873488i \(-0.338147\pi\)
0.486846 + 0.873488i \(0.338147\pi\)
\(492\) 0 0
\(493\) −15185.4 −1.38725
\(494\) 0 0
\(495\) 3025.93 0.274758
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3096.62 0.277803 0.138901 0.990306i \(-0.455643\pi\)
0.138901 + 0.990306i \(0.455643\pi\)
\(500\) 0 0
\(501\) −3769.30 −0.336127
\(502\) 0 0
\(503\) −3149.14 −0.279151 −0.139576 0.990211i \(-0.544574\pi\)
−0.139576 + 0.990211i \(0.544574\pi\)
\(504\) 0 0
\(505\) −5970.17 −0.526078
\(506\) 0 0
\(507\) 4426.61 0.387757
\(508\) 0 0
\(509\) 9133.85 0.795384 0.397692 0.917519i \(-0.369811\pi\)
0.397692 + 0.917519i \(0.369811\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −13468.6 −1.15917
\(514\) 0 0
\(515\) −5430.28 −0.464635
\(516\) 0 0
\(517\) −7805.20 −0.663969
\(518\) 0 0
\(519\) −21780.9 −1.84215
\(520\) 0 0
\(521\) 5117.77 0.430353 0.215176 0.976575i \(-0.430967\pi\)
0.215176 + 0.976575i \(0.430967\pi\)
\(522\) 0 0
\(523\) 5868.51 0.490654 0.245327 0.969440i \(-0.421105\pi\)
0.245327 + 0.969440i \(0.421105\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7404.17 −0.612012
\(528\) 0 0
\(529\) −6302.61 −0.518008
\(530\) 0 0
\(531\) −2530.62 −0.206817
\(532\) 0 0
\(533\) 10847.4 0.881527
\(534\) 0 0
\(535\) −10719.6 −0.866257
\(536\) 0 0
\(537\) −3500.91 −0.281332
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8280.32 −0.658038 −0.329019 0.944323i \(-0.606718\pi\)
−0.329019 + 0.944323i \(0.606718\pi\)
\(542\) 0 0
\(543\) 9467.89 0.748262
\(544\) 0 0
\(545\) 6282.10 0.493753
\(546\) 0 0
\(547\) −16832.7 −1.31575 −0.657875 0.753127i \(-0.728545\pi\)
−0.657875 + 0.753127i \(0.728545\pi\)
\(548\) 0 0
\(549\) −4991.24 −0.388017
\(550\) 0 0
\(551\) 37552.8 2.90346
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2591.45 −0.198200
\(556\) 0 0
\(557\) 10509.3 0.799450 0.399725 0.916635i \(-0.369106\pi\)
0.399725 + 0.916635i \(0.369106\pi\)
\(558\) 0 0
\(559\) 6592.94 0.498840
\(560\) 0 0
\(561\) 18767.2 1.41239
\(562\) 0 0
\(563\) 333.004 0.0249280 0.0124640 0.999922i \(-0.496032\pi\)
0.0124640 + 0.999922i \(0.496032\pi\)
\(564\) 0 0
\(565\) −8190.39 −0.609862
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15088.2 1.11166 0.555828 0.831298i \(-0.312401\pi\)
0.555828 + 0.831298i \(0.312401\pi\)
\(570\) 0 0
\(571\) 1834.44 0.134446 0.0672232 0.997738i \(-0.478586\pi\)
0.0672232 + 0.997738i \(0.478586\pi\)
\(572\) 0 0
\(573\) 4860.85 0.354389
\(574\) 0 0
\(575\) 7529.08 0.546060
\(576\) 0 0
\(577\) −8516.30 −0.614451 −0.307225 0.951637i \(-0.599401\pi\)
−0.307225 + 0.951637i \(0.599401\pi\)
\(578\) 0 0
\(579\) 3394.47 0.243643
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2136.74 −0.151792
\(584\) 0 0
\(585\) −1980.00 −0.139937
\(586\) 0 0
\(587\) 23957.0 1.68452 0.842258 0.539074i \(-0.181226\pi\)
0.842258 + 0.539074i \(0.181226\pi\)
\(588\) 0 0
\(589\) 18310.2 1.28092
\(590\) 0 0
\(591\) 10520.7 0.732254
\(592\) 0 0
\(593\) −11730.9 −0.812361 −0.406181 0.913793i \(-0.633140\pi\)
−0.406181 + 0.913793i \(0.633140\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −26068.0 −1.78709
\(598\) 0 0
\(599\) −4631.35 −0.315913 −0.157956 0.987446i \(-0.550491\pi\)
−0.157956 + 0.987446i \(0.550491\pi\)
\(600\) 0 0
\(601\) 25359.8 1.72121 0.860606 0.509272i \(-0.170085\pi\)
0.860606 + 0.509272i \(0.170085\pi\)
\(602\) 0 0
\(603\) −2105.79 −0.142213
\(604\) 0 0
\(605\) −10850.4 −0.729142
\(606\) 0 0
\(607\) 124.739 0.00834103 0.00417052 0.999991i \(-0.498672\pi\)
0.00417052 + 0.999991i \(0.498672\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5107.28 0.338165
\(612\) 0 0
\(613\) −17185.1 −1.13230 −0.566149 0.824303i \(-0.691567\pi\)
−0.566149 + 0.824303i \(0.691567\pi\)
\(614\) 0 0
\(615\) 8891.83 0.583013
\(616\) 0 0
\(617\) −25988.1 −1.69569 −0.847844 0.530246i \(-0.822100\pi\)
−0.847844 + 0.530246i \(0.822100\pi\)
\(618\) 0 0
\(619\) −18885.2 −1.22627 −0.613135 0.789978i \(-0.710092\pi\)
−0.613135 + 0.789978i \(0.710092\pi\)
\(620\) 0 0
\(621\) −7918.84 −0.511710
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6330.97 0.405182
\(626\) 0 0
\(627\) −46410.6 −2.95608
\(628\) 0 0
\(629\) −4343.93 −0.275364
\(630\) 0 0
\(631\) 6552.95 0.413421 0.206710 0.978402i \(-0.433724\pi\)
0.206710 + 0.978402i \(0.433724\pi\)
\(632\) 0 0
\(633\) 5169.76 0.324612
\(634\) 0 0
\(635\) −5720.12 −0.357474
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1073.65 −0.0664680
\(640\) 0 0
\(641\) −27671.0 −1.70505 −0.852525 0.522686i \(-0.824930\pi\)
−0.852525 + 0.522686i \(0.824930\pi\)
\(642\) 0 0
\(643\) −27154.8 −1.66544 −0.832722 0.553691i \(-0.813219\pi\)
−0.832722 + 0.553691i \(0.813219\pi\)
\(644\) 0 0
\(645\) 5404.35 0.329917
\(646\) 0 0
\(647\) 27322.6 1.66022 0.830110 0.557600i \(-0.188278\pi\)
0.830110 + 0.557600i \(0.188278\pi\)
\(648\) 0 0
\(649\) 14824.2 0.896612
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1432.86 0.0858685 0.0429342 0.999078i \(-0.486329\pi\)
0.0429342 + 0.999078i \(0.486329\pi\)
\(654\) 0 0
\(655\) −4572.95 −0.272794
\(656\) 0 0
\(657\) −3613.45 −0.214572
\(658\) 0 0
\(659\) −12914.0 −0.763367 −0.381684 0.924293i \(-0.624656\pi\)
−0.381684 + 0.924293i \(0.624656\pi\)
\(660\) 0 0
\(661\) 10068.8 0.592485 0.296243 0.955113i \(-0.404266\pi\)
0.296243 + 0.955113i \(0.404266\pi\)
\(662\) 0 0
\(663\) −12280.2 −0.719343
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 22079.2 1.28172
\(668\) 0 0
\(669\) −26628.3 −1.53888
\(670\) 0 0
\(671\) 29238.4 1.68217
\(672\) 0 0
\(673\) −1686.86 −0.0966174 −0.0483087 0.998832i \(-0.515383\pi\)
−0.0483087 + 0.998832i \(0.515383\pi\)
\(674\) 0 0
\(675\) −10166.7 −0.579728
\(676\) 0 0
\(677\) 22535.0 1.27931 0.639654 0.768663i \(-0.279078\pi\)
0.639654 + 0.768663i \(0.279078\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 31141.7 1.75235
\(682\) 0 0
\(683\) 29406.1 1.64743 0.823714 0.567006i \(-0.191898\pi\)
0.823714 + 0.567006i \(0.191898\pi\)
\(684\) 0 0
\(685\) −11124.2 −0.620485
\(686\) 0 0
\(687\) 35774.6 1.98673
\(688\) 0 0
\(689\) 1398.16 0.0773089
\(690\) 0 0
\(691\) 251.526 0.0138473 0.00692365 0.999976i \(-0.497796\pi\)
0.00692365 + 0.999976i \(0.497796\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3474.23 0.189619
\(696\) 0 0
\(697\) 14905.0 0.809993
\(698\) 0 0
\(699\) −1290.02 −0.0698039
\(700\) 0 0
\(701\) −6188.42 −0.333429 −0.166714 0.986005i \(-0.553316\pi\)
−0.166714 + 0.986005i \(0.553316\pi\)
\(702\) 0 0
\(703\) 10742.4 0.576324
\(704\) 0 0
\(705\) 4186.53 0.223651
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 27439.4 1.45347 0.726734 0.686919i \(-0.241037\pi\)
0.726734 + 0.686919i \(0.241037\pi\)
\(710\) 0 0
\(711\) 8350.07 0.440439
\(712\) 0 0
\(713\) 10765.5 0.565456
\(714\) 0 0
\(715\) 11598.7 0.606667
\(716\) 0 0
\(717\) 31648.3 1.64844
\(718\) 0 0
\(719\) −12278.5 −0.636871 −0.318436 0.947944i \(-0.603157\pi\)
−0.318436 + 0.947944i \(0.603157\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 17257.5 0.887711
\(724\) 0 0
\(725\) 28346.6 1.45209
\(726\) 0 0
\(727\) 8576.11 0.437510 0.218755 0.975780i \(-0.429800\pi\)
0.218755 + 0.975780i \(0.429800\pi\)
\(728\) 0 0
\(729\) 7993.00 0.406086
\(730\) 0 0
\(731\) 9059.06 0.458360
\(732\) 0 0
\(733\) 12874.8 0.648762 0.324381 0.945926i \(-0.394844\pi\)
0.324381 + 0.945926i \(0.394844\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12335.6 0.616537
\(738\) 0 0
\(739\) −19917.3 −0.991432 −0.495716 0.868485i \(-0.665094\pi\)
−0.495716 + 0.868485i \(0.665094\pi\)
\(740\) 0 0
\(741\) 30368.5 1.50555
\(742\) 0 0
\(743\) −22672.6 −1.11948 −0.559741 0.828667i \(-0.689100\pi\)
−0.559741 + 0.828667i \(0.689100\pi\)
\(744\) 0 0
\(745\) 9315.16 0.458095
\(746\) 0 0
\(747\) 7312.83 0.358183
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20953.6 1.01812 0.509059 0.860732i \(-0.329994\pi\)
0.509059 + 0.860732i \(0.329994\pi\)
\(752\) 0 0
\(753\) 28571.0 1.38272
\(754\) 0 0
\(755\) 5759.60 0.277633
\(756\) 0 0
\(757\) −12684.5 −0.609018 −0.304509 0.952510i \(-0.598492\pi\)
−0.304509 + 0.952510i \(0.598492\pi\)
\(758\) 0 0
\(759\) −27287.1 −1.30495
\(760\) 0 0
\(761\) 33663.7 1.60356 0.801779 0.597621i \(-0.203887\pi\)
0.801779 + 0.597621i \(0.203887\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2720.63 −0.128581
\(766\) 0 0
\(767\) −9700.14 −0.456652
\(768\) 0 0
\(769\) 20358.4 0.954672 0.477336 0.878721i \(-0.341602\pi\)
0.477336 + 0.878721i \(0.341602\pi\)
\(770\) 0 0
\(771\) 42602.0 1.98998
\(772\) 0 0
\(773\) 330.841 0.0153940 0.00769698 0.999970i \(-0.497550\pi\)
0.00769698 + 0.999970i \(0.497550\pi\)
\(774\) 0 0
\(775\) 13821.4 0.640618
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −36859.4 −1.69528
\(780\) 0 0
\(781\) 6289.39 0.288159
\(782\) 0 0
\(783\) −29814.0 −1.36075
\(784\) 0 0
\(785\) −8669.36 −0.394169
\(786\) 0 0
\(787\) 6762.20 0.306285 0.153143 0.988204i \(-0.451061\pi\)
0.153143 + 0.988204i \(0.451061\pi\)
\(788\) 0 0
\(789\) 7471.50 0.337126
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −19132.0 −0.856741
\(794\) 0 0
\(795\) 1146.10 0.0511296
\(796\) 0 0
\(797\) −23848.2 −1.05991 −0.529954 0.848027i \(-0.677791\pi\)
−0.529954 + 0.848027i \(0.677791\pi\)
\(798\) 0 0
\(799\) 7017.69 0.310723
\(800\) 0 0
\(801\) 2179.66 0.0961478
\(802\) 0 0
\(803\) 21167.3 0.930235
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 863.051 0.0376466
\(808\) 0 0
\(809\) −927.139 −0.0402923 −0.0201462 0.999797i \(-0.506413\pi\)
−0.0201462 + 0.999797i \(0.506413\pi\)
\(810\) 0 0
\(811\) −4288.65 −0.185690 −0.0928451 0.995681i \(-0.529596\pi\)
−0.0928451 + 0.995681i \(0.529596\pi\)
\(812\) 0 0
\(813\) 2218.84 0.0957171
\(814\) 0 0
\(815\) 8185.26 0.351800
\(816\) 0 0
\(817\) −22402.7 −0.959329
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 27752.0 1.17972 0.589862 0.807504i \(-0.299182\pi\)
0.589862 + 0.807504i \(0.299182\pi\)
\(822\) 0 0
\(823\) −8690.15 −0.368068 −0.184034 0.982920i \(-0.558916\pi\)
−0.184034 + 0.982920i \(0.558916\pi\)
\(824\) 0 0
\(825\) −35032.9 −1.47841
\(826\) 0 0
\(827\) −29503.0 −1.24053 −0.620267 0.784391i \(-0.712976\pi\)
−0.620267 + 0.784391i \(0.712976\pi\)
\(828\) 0 0
\(829\) 20276.3 0.849488 0.424744 0.905314i \(-0.360364\pi\)
0.424744 + 0.905314i \(0.360364\pi\)
\(830\) 0 0
\(831\) −12095.4 −0.504917
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −3200.92 −0.132661
\(836\) 0 0
\(837\) −14536.9 −0.600320
\(838\) 0 0
\(839\) 5229.63 0.215193 0.107596 0.994195i \(-0.465685\pi\)
0.107596 + 0.994195i \(0.465685\pi\)
\(840\) 0 0
\(841\) 58737.9 2.40838
\(842\) 0 0
\(843\) −29829.5 −1.21872
\(844\) 0 0
\(845\) 3759.11 0.153038
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −19830.2 −0.801616
\(850\) 0 0
\(851\) 6315.96 0.254416
\(852\) 0 0
\(853\) 16423.9 0.659255 0.329628 0.944111i \(-0.393077\pi\)
0.329628 + 0.944111i \(0.393077\pi\)
\(854\) 0 0
\(855\) 6728.01 0.269115
\(856\) 0 0
\(857\) −30991.8 −1.23531 −0.617655 0.786449i \(-0.711917\pi\)
−0.617655 + 0.786449i \(0.711917\pi\)
\(858\) 0 0
\(859\) −32514.3 −1.29147 −0.645735 0.763561i \(-0.723449\pi\)
−0.645735 + 0.763561i \(0.723449\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17365.6 −0.684971 −0.342486 0.939523i \(-0.611269\pi\)
−0.342486 + 0.939523i \(0.611269\pi\)
\(864\) 0 0
\(865\) −18496.5 −0.727053
\(866\) 0 0
\(867\) 13010.9 0.509659
\(868\) 0 0
\(869\) −48914.2 −1.90944
\(870\) 0 0
\(871\) −8071.73 −0.314007
\(872\) 0 0
\(873\) −2522.90 −0.0978089
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20606.3 −0.793417 −0.396709 0.917945i \(-0.629848\pi\)
−0.396709 + 0.917945i \(0.629848\pi\)
\(878\) 0 0
\(879\) 9815.58 0.376645
\(880\) 0 0
\(881\) 4704.88 0.179922 0.0899610 0.995945i \(-0.471326\pi\)
0.0899610 + 0.995945i \(0.471326\pi\)
\(882\) 0 0
\(883\) 13075.1 0.498313 0.249157 0.968463i \(-0.419847\pi\)
0.249157 + 0.968463i \(0.419847\pi\)
\(884\) 0 0
\(885\) −7951.38 −0.302014
\(886\) 0 0
\(887\) 44569.2 1.68714 0.843568 0.537023i \(-0.180451\pi\)
0.843568 + 0.537023i \(0.180451\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 52662.8 1.98010
\(892\) 0 0
\(893\) −17354.5 −0.650331
\(894\) 0 0
\(895\) −2973.00 −0.111035
\(896\) 0 0
\(897\) 17855.1 0.664622
\(898\) 0 0
\(899\) 40531.5 1.50367
\(900\) 0 0
\(901\) 1921.15 0.0710354
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8040.20 0.295321
\(906\) 0 0
\(907\) 16003.3 0.585866 0.292933 0.956133i \(-0.405369\pi\)
0.292933 + 0.956133i \(0.405369\pi\)
\(908\) 0 0
\(909\) 11557.7 0.421722
\(910\) 0 0
\(911\) −10072.4 −0.366315 −0.183157 0.983084i \(-0.558632\pi\)
−0.183157 + 0.983084i \(0.558632\pi\)
\(912\) 0 0
\(913\) −42838.1 −1.55283
\(914\) 0 0
\(915\) −15682.8 −0.566621
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −45278.2 −1.62523 −0.812617 0.582799i \(-0.801958\pi\)
−0.812617 + 0.582799i \(0.801958\pi\)
\(920\) 0 0
\(921\) 17312.5 0.619398
\(922\) 0 0
\(923\) −4115.42 −0.146761
\(924\) 0 0
\(925\) 8108.82 0.288234
\(926\) 0 0
\(927\) 10512.6 0.372468
\(928\) 0 0
\(929\) 8136.58 0.287355 0.143677 0.989625i \(-0.454107\pi\)
0.143677 + 0.989625i \(0.454107\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −16930.7 −0.594091
\(934\) 0 0
\(935\) 15937.3 0.557437
\(936\) 0 0
\(937\) −34741.9 −1.21128 −0.605640 0.795739i \(-0.707083\pi\)
−0.605640 + 0.795739i \(0.707083\pi\)
\(938\) 0 0
\(939\) −18261.3 −0.634648
\(940\) 0 0
\(941\) −37457.6 −1.29764 −0.648822 0.760940i \(-0.724738\pi\)
−0.648822 + 0.760940i \(0.724738\pi\)
\(942\) 0 0
\(943\) −21671.4 −0.748376
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7552.29 0.259152 0.129576 0.991570i \(-0.458638\pi\)
0.129576 + 0.991570i \(0.458638\pi\)
\(948\) 0 0
\(949\) −13850.7 −0.473776
\(950\) 0 0
\(951\) 22955.2 0.782727
\(952\) 0 0
\(953\) −25745.6 −0.875111 −0.437556 0.899191i \(-0.644156\pi\)
−0.437556 + 0.899191i \(0.644156\pi\)
\(954\) 0 0
\(955\) 4127.86 0.139869
\(956\) 0 0
\(957\) −102734. −3.47015
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −10028.4 −0.336627
\(962\) 0 0
\(963\) 20752.1 0.694422
\(964\) 0 0
\(965\) 2882.61 0.0961601
\(966\) 0 0
\(967\) −31718.6 −1.05481 −0.527406 0.849614i \(-0.676835\pi\)
−0.527406 + 0.849614i \(0.676835\pi\)
\(968\) 0 0
\(969\) 41728.0 1.38338
\(970\) 0 0
\(971\) 39308.8 1.29916 0.649578 0.760295i \(-0.274946\pi\)
0.649578 + 0.760295i \(0.274946\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 22923.5 0.752965
\(976\) 0 0
\(977\) −5238.61 −0.171543 −0.0857717 0.996315i \(-0.527336\pi\)
−0.0857717 + 0.996315i \(0.527336\pi\)
\(978\) 0 0
\(979\) −12768.3 −0.416830
\(980\) 0 0
\(981\) −12161.6 −0.395810
\(982\) 0 0
\(983\) −25163.5 −0.816472 −0.408236 0.912876i \(-0.633856\pi\)
−0.408236 + 0.912876i \(0.633856\pi\)
\(984\) 0 0
\(985\) 8934.22 0.289003
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13171.6 −0.423492
\(990\) 0 0
\(991\) −43561.1 −1.39633 −0.698166 0.715936i \(-0.746000\pi\)
−0.698166 + 0.715936i \(0.746000\pi\)
\(992\) 0 0
\(993\) −5936.04 −0.189702
\(994\) 0 0
\(995\) −22137.1 −0.705320
\(996\) 0 0
\(997\) 50840.3 1.61497 0.807487 0.589885i \(-0.200827\pi\)
0.807487 + 0.589885i \(0.200827\pi\)
\(998\) 0 0
\(999\) −8528.60 −0.270103
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.ba.1.1 2
4.3 odd 2 196.4.a.g.1.2 2
7.3 odd 6 112.4.i.d.65.1 4
7.5 odd 6 112.4.i.d.81.1 4
7.6 odd 2 784.4.a.u.1.2 2
12.11 even 2 1764.4.a.n.1.2 2
28.3 even 6 28.4.e.a.9.2 4
28.11 odd 6 196.4.e.g.177.1 4
28.19 even 6 28.4.e.a.25.2 yes 4
28.23 odd 6 196.4.e.g.165.1 4
28.27 even 2 196.4.a.e.1.1 2
56.3 even 6 448.4.i.h.65.1 4
56.5 odd 6 448.4.i.g.193.2 4
56.19 even 6 448.4.i.h.193.1 4
56.45 odd 6 448.4.i.g.65.2 4
84.11 even 6 1764.4.k.ba.1549.1 4
84.23 even 6 1764.4.k.ba.361.1 4
84.47 odd 6 252.4.k.c.109.2 4
84.59 odd 6 252.4.k.c.37.2 4
84.83 odd 2 1764.4.a.z.1.1 2
140.3 odd 12 700.4.r.d.149.2 8
140.19 even 6 700.4.i.g.501.1 4
140.47 odd 12 700.4.r.d.249.2 8
140.59 even 6 700.4.i.g.401.1 4
140.87 odd 12 700.4.r.d.149.3 8
140.103 odd 12 700.4.r.d.249.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.4.e.a.9.2 4 28.3 even 6
28.4.e.a.25.2 yes 4 28.19 even 6
112.4.i.d.65.1 4 7.3 odd 6
112.4.i.d.81.1 4 7.5 odd 6
196.4.a.e.1.1 2 28.27 even 2
196.4.a.g.1.2 2 4.3 odd 2
196.4.e.g.165.1 4 28.23 odd 6
196.4.e.g.177.1 4 28.11 odd 6
252.4.k.c.37.2 4 84.59 odd 6
252.4.k.c.109.2 4 84.47 odd 6
448.4.i.g.65.2 4 56.45 odd 6
448.4.i.g.193.2 4 56.5 odd 6
448.4.i.h.65.1 4 56.3 even 6
448.4.i.h.193.1 4 56.19 even 6
700.4.i.g.401.1 4 140.59 even 6
700.4.i.g.501.1 4 140.19 even 6
700.4.r.d.149.2 8 140.3 odd 12
700.4.r.d.149.3 8 140.87 odd 12
700.4.r.d.249.2 8 140.47 odd 12
700.4.r.d.249.3 8 140.103 odd 12
784.4.a.u.1.2 2 7.6 odd 2
784.4.a.ba.1.1 2 1.1 even 1 trivial
1764.4.a.n.1.2 2 12.11 even 2
1764.4.a.z.1.1 2 84.83 odd 2
1764.4.k.ba.361.1 4 84.23 even 6
1764.4.k.ba.1549.1 4 84.11 even 6