Properties

Label 2312.4.a.o.1.2
Level $2312$
Weight $4$
Character 2312.1
Self dual yes
Analytic conductor $136.412$
Analytic rank $1$
Dimension $18$
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] = [2312,4,Mod(1,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, 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("2312.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2312.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: \(136.412415933\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 345 x^{16} - 182 x^{15} + 48165 x^{14} + 48078 x^{13} - 3485278 x^{12} - 4881882 x^{11} + 139131876 x^{10} + 239770606 x^{9} - 3013674039 x^{8} + \cdots - 119632152329 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2}\cdot 17^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.88256\) of defining polynomial
Character \(\chi\) \(=\) 2312.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.88256 q^{3} +2.51577 q^{5} +10.7880 q^{7} +51.8999 q^{9} +O(q^{10})\) \(q-8.88256 q^{3} +2.51577 q^{5} +10.7880 q^{7} +51.8999 q^{9} +4.61582 q^{11} +38.1572 q^{13} -22.3464 q^{15} +70.4236 q^{19} -95.8249 q^{21} -136.987 q^{23} -118.671 q^{25} -221.175 q^{27} -219.743 q^{29} +275.440 q^{31} -41.0003 q^{33} +27.1400 q^{35} +128.032 q^{37} -338.934 q^{39} -436.505 q^{41} -150.637 q^{43} +130.568 q^{45} +391.842 q^{47} -226.619 q^{49} +117.330 q^{53} +11.6123 q^{55} -625.542 q^{57} -31.7994 q^{59} -481.126 q^{61} +559.895 q^{63} +95.9947 q^{65} -722.761 q^{67} +1216.79 q^{69} +495.446 q^{71} -719.065 q^{73} +1054.10 q^{75} +49.7954 q^{77} +149.298 q^{79} +563.300 q^{81} +1214.22 q^{83} +1951.88 q^{87} +602.356 q^{89} +411.640 q^{91} -2446.62 q^{93} +177.169 q^{95} +1773.07 q^{97} +239.561 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 30 q^{5} - 33 q^{7} + 204 q^{9} + 66 q^{11} - 30 q^{13} - 102 q^{15} - 168 q^{19} - 510 q^{21} - 153 q^{23} + 594 q^{25} + 546 q^{27} - 447 q^{29} - 303 q^{31} + 153 q^{33} - 117 q^{35} - 939 q^{37} - 516 q^{39} - 1257 q^{41} + 306 q^{43} - 672 q^{45} + 633 q^{47} + 1239 q^{49} - 489 q^{53} + 1089 q^{55} - 1494 q^{57} + 696 q^{59} - 1686 q^{61} - 1908 q^{63} - 855 q^{65} + 513 q^{67} - 1329 q^{69} - 324 q^{71} - 1863 q^{73} + 3054 q^{75} + 1833 q^{77} - 3699 q^{79} + 2622 q^{81} + 1188 q^{83} - 3927 q^{87} + 1713 q^{89} - 252 q^{91} - 1470 q^{93} - 2109 q^{95} - 4611 q^{97} + 3918 q^{99}+O(q^{100}) \) 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
\(3\) −8.88256 −1.70945 −0.854725 0.519082i \(-0.826274\pi\)
−0.854725 + 0.519082i \(0.826274\pi\)
\(4\) 0 0
\(5\) 2.51577 0.225017 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(6\) 0 0
\(7\) 10.7880 0.582496 0.291248 0.956648i \(-0.405929\pi\)
0.291248 + 0.956648i \(0.405929\pi\)
\(8\) 0 0
\(9\) 51.8999 1.92222
\(10\) 0 0
\(11\) 4.61582 0.126520 0.0632602 0.997997i \(-0.479850\pi\)
0.0632602 + 0.997997i \(0.479850\pi\)
\(12\) 0 0
\(13\) 38.1572 0.814070 0.407035 0.913413i \(-0.366563\pi\)
0.407035 + 0.913413i \(0.366563\pi\)
\(14\) 0 0
\(15\) −22.3464 −0.384655
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 70.4236 0.850331 0.425166 0.905116i \(-0.360216\pi\)
0.425166 + 0.905116i \(0.360216\pi\)
\(20\) 0 0
\(21\) −95.8249 −0.995748
\(22\) 0 0
\(23\) −136.987 −1.24190 −0.620950 0.783850i \(-0.713253\pi\)
−0.620950 + 0.783850i \(0.713253\pi\)
\(24\) 0 0
\(25\) −118.671 −0.949367
\(26\) 0 0
\(27\) −221.175 −1.57648
\(28\) 0 0
\(29\) −219.743 −1.40708 −0.703540 0.710655i \(-0.748399\pi\)
−0.703540 + 0.710655i \(0.748399\pi\)
\(30\) 0 0
\(31\) 275.440 1.59582 0.797912 0.602774i \(-0.205938\pi\)
0.797912 + 0.602774i \(0.205938\pi\)
\(32\) 0 0
\(33\) −41.0003 −0.216280
\(34\) 0 0
\(35\) 27.1400 0.131072
\(36\) 0 0
\(37\) 128.032 0.568873 0.284436 0.958695i \(-0.408194\pi\)
0.284436 + 0.958695i \(0.408194\pi\)
\(38\) 0 0
\(39\) −338.934 −1.39161
\(40\) 0 0
\(41\) −436.505 −1.66270 −0.831348 0.555751i \(-0.812431\pi\)
−0.831348 + 0.555751i \(0.812431\pi\)
\(42\) 0 0
\(43\) −150.637 −0.534229 −0.267115 0.963665i \(-0.586070\pi\)
−0.267115 + 0.963665i \(0.586070\pi\)
\(44\) 0 0
\(45\) 130.568 0.432532
\(46\) 0 0
\(47\) 391.842 1.21609 0.608043 0.793904i \(-0.291955\pi\)
0.608043 + 0.793904i \(0.291955\pi\)
\(48\) 0 0
\(49\) −226.619 −0.660698
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 117.330 0.304084 0.152042 0.988374i \(-0.451415\pi\)
0.152042 + 0.988374i \(0.451415\pi\)
\(54\) 0 0
\(55\) 11.6123 0.0284692
\(56\) 0 0
\(57\) −625.542 −1.45360
\(58\) 0 0
\(59\) −31.7994 −0.0701683 −0.0350842 0.999384i \(-0.511170\pi\)
−0.0350842 + 0.999384i \(0.511170\pi\)
\(60\) 0 0
\(61\) −481.126 −1.00987 −0.504933 0.863158i \(-0.668483\pi\)
−0.504933 + 0.863158i \(0.668483\pi\)
\(62\) 0 0
\(63\) 559.895 1.11968
\(64\) 0 0
\(65\) 95.9947 0.183180
\(66\) 0 0
\(67\) −722.761 −1.31790 −0.658950 0.752187i \(-0.728999\pi\)
−0.658950 + 0.752187i \(0.728999\pi\)
\(68\) 0 0
\(69\) 1216.79 2.12297
\(70\) 0 0
\(71\) 495.446 0.828150 0.414075 0.910243i \(-0.364105\pi\)
0.414075 + 0.910243i \(0.364105\pi\)
\(72\) 0 0
\(73\) −719.065 −1.15288 −0.576440 0.817140i \(-0.695558\pi\)
−0.576440 + 0.817140i \(0.695558\pi\)
\(74\) 0 0
\(75\) 1054.10 1.62290
\(76\) 0 0
\(77\) 49.7954 0.0736976
\(78\) 0 0
\(79\) 149.298 0.212624 0.106312 0.994333i \(-0.466096\pi\)
0.106312 + 0.994333i \(0.466096\pi\)
\(80\) 0 0
\(81\) 563.300 0.772702
\(82\) 0 0
\(83\) 1214.22 1.60576 0.802879 0.596142i \(-0.203301\pi\)
0.802879 + 0.596142i \(0.203301\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1951.88 2.40533
\(88\) 0 0
\(89\) 602.356 0.717411 0.358706 0.933451i \(-0.383218\pi\)
0.358706 + 0.933451i \(0.383218\pi\)
\(90\) 0 0
\(91\) 411.640 0.474193
\(92\) 0 0
\(93\) −2446.62 −2.72798
\(94\) 0 0
\(95\) 177.169 0.191339
\(96\) 0 0
\(97\) 1773.07 1.85596 0.927979 0.372634i \(-0.121545\pi\)
0.927979 + 0.372634i \(0.121545\pi\)
\(98\) 0 0
\(99\) 239.561 0.243200
\(100\) 0 0
\(101\) 1480.23 1.45830 0.729148 0.684356i \(-0.239916\pi\)
0.729148 + 0.684356i \(0.239916\pi\)
\(102\) 0 0
\(103\) 948.709 0.907564 0.453782 0.891113i \(-0.350075\pi\)
0.453782 + 0.891113i \(0.350075\pi\)
\(104\) 0 0
\(105\) −241.073 −0.224060
\(106\) 0 0
\(107\) −1089.50 −0.984353 −0.492177 0.870495i \(-0.663799\pi\)
−0.492177 + 0.870495i \(0.663799\pi\)
\(108\) 0 0
\(109\) −1676.91 −1.47356 −0.736782 0.676131i \(-0.763656\pi\)
−0.736782 + 0.676131i \(0.763656\pi\)
\(110\) 0 0
\(111\) −1137.25 −0.972459
\(112\) 0 0
\(113\) −112.895 −0.0939846 −0.0469923 0.998895i \(-0.514964\pi\)
−0.0469923 + 0.998895i \(0.514964\pi\)
\(114\) 0 0
\(115\) −344.627 −0.279449
\(116\) 0 0
\(117\) 1980.36 1.56482
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1309.69 −0.983993
\(122\) 0 0
\(123\) 3877.28 2.84230
\(124\) 0 0
\(125\) −613.019 −0.438641
\(126\) 0 0
\(127\) −1023.09 −0.714842 −0.357421 0.933943i \(-0.616344\pi\)
−0.357421 + 0.933943i \(0.616344\pi\)
\(128\) 0 0
\(129\) 1338.04 0.913238
\(130\) 0 0
\(131\) 2415.30 1.61088 0.805441 0.592676i \(-0.201929\pi\)
0.805441 + 0.592676i \(0.201929\pi\)
\(132\) 0 0
\(133\) 759.729 0.495315
\(134\) 0 0
\(135\) −556.424 −0.354736
\(136\) 0 0
\(137\) 994.871 0.620420 0.310210 0.950668i \(-0.399601\pi\)
0.310210 + 0.950668i \(0.399601\pi\)
\(138\) 0 0
\(139\) −1363.31 −0.831904 −0.415952 0.909387i \(-0.636551\pi\)
−0.415952 + 0.909387i \(0.636551\pi\)
\(140\) 0 0
\(141\) −3480.56 −2.07884
\(142\) 0 0
\(143\) 176.127 0.102996
\(144\) 0 0
\(145\) −552.823 −0.316617
\(146\) 0 0
\(147\) 2012.96 1.12943
\(148\) 0 0
\(149\) 1720.70 0.946073 0.473036 0.881043i \(-0.343158\pi\)
0.473036 + 0.881043i \(0.343158\pi\)
\(150\) 0 0
\(151\) −128.263 −0.0691250 −0.0345625 0.999403i \(-0.511004\pi\)
−0.0345625 + 0.999403i \(0.511004\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 692.944 0.359088
\(156\) 0 0
\(157\) −726.645 −0.369380 −0.184690 0.982797i \(-0.559128\pi\)
−0.184690 + 0.982797i \(0.559128\pi\)
\(158\) 0 0
\(159\) −1042.19 −0.519817
\(160\) 0 0
\(161\) −1477.81 −0.723402
\(162\) 0 0
\(163\) −3372.62 −1.62064 −0.810320 0.585987i \(-0.800707\pi\)
−0.810320 + 0.585987i \(0.800707\pi\)
\(164\) 0 0
\(165\) −103.147 −0.0486667
\(166\) 0 0
\(167\) −3078.11 −1.42629 −0.713147 0.701015i \(-0.752731\pi\)
−0.713147 + 0.701015i \(0.752731\pi\)
\(168\) 0 0
\(169\) −741.026 −0.337290
\(170\) 0 0
\(171\) 3654.98 1.63452
\(172\) 0 0
\(173\) 1060.40 0.466014 0.233007 0.972475i \(-0.425144\pi\)
0.233007 + 0.972475i \(0.425144\pi\)
\(174\) 0 0
\(175\) −1280.22 −0.553003
\(176\) 0 0
\(177\) 282.460 0.119949
\(178\) 0 0
\(179\) 2972.05 1.24102 0.620508 0.784200i \(-0.286927\pi\)
0.620508 + 0.784200i \(0.286927\pi\)
\(180\) 0 0
\(181\) −170.006 −0.0698148 −0.0349074 0.999391i \(-0.511114\pi\)
−0.0349074 + 0.999391i \(0.511114\pi\)
\(182\) 0 0
\(183\) 4273.63 1.72632
\(184\) 0 0
\(185\) 322.098 0.128006
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2386.03 −0.918296
\(190\) 0 0
\(191\) 3586.29 1.35861 0.679306 0.733856i \(-0.262281\pi\)
0.679306 + 0.733856i \(0.262281\pi\)
\(192\) 0 0
\(193\) −3831.91 −1.42915 −0.714577 0.699557i \(-0.753381\pi\)
−0.714577 + 0.699557i \(0.753381\pi\)
\(194\) 0 0
\(195\) −852.679 −0.313136
\(196\) 0 0
\(197\) 1958.63 0.708360 0.354180 0.935177i \(-0.384760\pi\)
0.354180 + 0.935177i \(0.384760\pi\)
\(198\) 0 0
\(199\) −5257.05 −1.87267 −0.936337 0.351102i \(-0.885807\pi\)
−0.936337 + 0.351102i \(0.885807\pi\)
\(200\) 0 0
\(201\) 6419.96 2.25288
\(202\) 0 0
\(203\) −2370.59 −0.819619
\(204\) 0 0
\(205\) −1098.14 −0.374135
\(206\) 0 0
\(207\) −7109.59 −2.38720
\(208\) 0 0
\(209\) 325.063 0.107584
\(210\) 0 0
\(211\) 1549.01 0.505394 0.252697 0.967545i \(-0.418682\pi\)
0.252697 + 0.967545i \(0.418682\pi\)
\(212\) 0 0
\(213\) −4400.83 −1.41568
\(214\) 0 0
\(215\) −378.967 −0.120211
\(216\) 0 0
\(217\) 2971.45 0.929562
\(218\) 0 0
\(219\) 6387.14 1.97079
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2166.71 0.650643 0.325321 0.945603i \(-0.394527\pi\)
0.325321 + 0.945603i \(0.394527\pi\)
\(224\) 0 0
\(225\) −6159.00 −1.82489
\(226\) 0 0
\(227\) −2421.62 −0.708056 −0.354028 0.935235i \(-0.615188\pi\)
−0.354028 + 0.935235i \(0.615188\pi\)
\(228\) 0 0
\(229\) 4358.00 1.25757 0.628787 0.777578i \(-0.283552\pi\)
0.628787 + 0.777578i \(0.283552\pi\)
\(230\) 0 0
\(231\) −442.311 −0.125982
\(232\) 0 0
\(233\) −2416.05 −0.679316 −0.339658 0.940549i \(-0.610311\pi\)
−0.339658 + 0.940549i \(0.610311\pi\)
\(234\) 0 0
\(235\) 985.783 0.273640
\(236\) 0 0
\(237\) −1326.15 −0.363470
\(238\) 0 0
\(239\) −2512.23 −0.679928 −0.339964 0.940438i \(-0.610415\pi\)
−0.339964 + 0.940438i \(0.610415\pi\)
\(240\) 0 0
\(241\) −737.205 −0.197044 −0.0985219 0.995135i \(-0.531411\pi\)
−0.0985219 + 0.995135i \(0.531411\pi\)
\(242\) 0 0
\(243\) 968.169 0.255589
\(244\) 0 0
\(245\) −570.122 −0.148668
\(246\) 0 0
\(247\) 2687.17 0.692229
\(248\) 0 0
\(249\) −10785.4 −2.74496
\(250\) 0 0
\(251\) −4620.18 −1.16185 −0.580923 0.813958i \(-0.697308\pi\)
−0.580923 + 0.813958i \(0.697308\pi\)
\(252\) 0 0
\(253\) −632.307 −0.157126
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6573.88 −1.59559 −0.797797 0.602926i \(-0.794001\pi\)
−0.797797 + 0.602926i \(0.794001\pi\)
\(258\) 0 0
\(259\) 1381.20 0.331366
\(260\) 0 0
\(261\) −11404.7 −2.70471
\(262\) 0 0
\(263\) −1382.15 −0.324057 −0.162029 0.986786i \(-0.551804\pi\)
−0.162029 + 0.986786i \(0.551804\pi\)
\(264\) 0 0
\(265\) 295.174 0.0684241
\(266\) 0 0
\(267\) −5350.46 −1.22638
\(268\) 0 0
\(269\) 3193.92 0.723929 0.361965 0.932192i \(-0.382106\pi\)
0.361965 + 0.932192i \(0.382106\pi\)
\(270\) 0 0
\(271\) −1506.54 −0.337696 −0.168848 0.985642i \(-0.554005\pi\)
−0.168848 + 0.985642i \(0.554005\pi\)
\(272\) 0 0
\(273\) −3656.41 −0.810609
\(274\) 0 0
\(275\) −547.764 −0.120114
\(276\) 0 0
\(277\) −4911.09 −1.06527 −0.532633 0.846346i \(-0.678797\pi\)
−0.532633 + 0.846346i \(0.678797\pi\)
\(278\) 0 0
\(279\) 14295.3 3.06752
\(280\) 0 0
\(281\) −7198.28 −1.52816 −0.764081 0.645120i \(-0.776807\pi\)
−0.764081 + 0.645120i \(0.776807\pi\)
\(282\) 0 0
\(283\) 4315.91 0.906553 0.453276 0.891370i \(-0.350255\pi\)
0.453276 + 0.891370i \(0.350255\pi\)
\(284\) 0 0
\(285\) −1573.72 −0.327084
\(286\) 0 0
\(287\) −4709.00 −0.968515
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) −15749.4 −3.17266
\(292\) 0 0
\(293\) −3086.04 −0.615320 −0.307660 0.951496i \(-0.599546\pi\)
−0.307660 + 0.951496i \(0.599546\pi\)
\(294\) 0 0
\(295\) −79.9999 −0.0157891
\(296\) 0 0
\(297\) −1020.90 −0.199457
\(298\) 0 0
\(299\) −5227.03 −1.01099
\(300\) 0 0
\(301\) −1625.07 −0.311187
\(302\) 0 0
\(303\) −13148.2 −2.49288
\(304\) 0 0
\(305\) −1210.40 −0.227237
\(306\) 0 0
\(307\) −2132.90 −0.396518 −0.198259 0.980150i \(-0.563529\pi\)
−0.198259 + 0.980150i \(0.563529\pi\)
\(308\) 0 0
\(309\) −8426.96 −1.55143
\(310\) 0 0
\(311\) 5896.43 1.07510 0.537550 0.843232i \(-0.319350\pi\)
0.537550 + 0.843232i \(0.319350\pi\)
\(312\) 0 0
\(313\) 2976.48 0.537510 0.268755 0.963209i \(-0.413388\pi\)
0.268755 + 0.963209i \(0.413388\pi\)
\(314\) 0 0
\(315\) 1408.56 0.251948
\(316\) 0 0
\(317\) 2742.33 0.485882 0.242941 0.970041i \(-0.421888\pi\)
0.242941 + 0.970041i \(0.421888\pi\)
\(318\) 0 0
\(319\) −1014.30 −0.178024
\(320\) 0 0
\(321\) 9677.54 1.68270
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4528.15 −0.772852
\(326\) 0 0
\(327\) 14895.2 2.51898
\(328\) 0 0
\(329\) 4227.19 0.708366
\(330\) 0 0
\(331\) 4210.57 0.699196 0.349598 0.936900i \(-0.386318\pi\)
0.349598 + 0.936900i \(0.386318\pi\)
\(332\) 0 0
\(333\) 6644.83 1.09350
\(334\) 0 0
\(335\) −1818.30 −0.296550
\(336\) 0 0
\(337\) −6629.30 −1.07158 −0.535788 0.844352i \(-0.679985\pi\)
−0.535788 + 0.844352i \(0.679985\pi\)
\(338\) 0 0
\(339\) 1002.80 0.160662
\(340\) 0 0
\(341\) 1271.38 0.201904
\(342\) 0 0
\(343\) −6145.04 −0.967350
\(344\) 0 0
\(345\) 3061.17 0.477703
\(346\) 0 0
\(347\) 10327.2 1.59768 0.798838 0.601546i \(-0.205448\pi\)
0.798838 + 0.601546i \(0.205448\pi\)
\(348\) 0 0
\(349\) −1403.59 −0.215279 −0.107640 0.994190i \(-0.534329\pi\)
−0.107640 + 0.994190i \(0.534329\pi\)
\(350\) 0 0
\(351\) −8439.41 −1.28337
\(352\) 0 0
\(353\) −3518.63 −0.530532 −0.265266 0.964175i \(-0.585460\pi\)
−0.265266 + 0.964175i \(0.585460\pi\)
\(354\) 0 0
\(355\) 1246.43 0.186348
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4829.92 −0.710066 −0.355033 0.934854i \(-0.615530\pi\)
−0.355033 + 0.934854i \(0.615530\pi\)
\(360\) 0 0
\(361\) −1899.51 −0.276937
\(362\) 0 0
\(363\) 11633.4 1.68209
\(364\) 0 0
\(365\) −1809.00 −0.259418
\(366\) 0 0
\(367\) 13511.1 1.92173 0.960863 0.277023i \(-0.0893477\pi\)
0.960863 + 0.277023i \(0.0893477\pi\)
\(368\) 0 0
\(369\) −22654.5 −3.19606
\(370\) 0 0
\(371\) 1265.75 0.177128
\(372\) 0 0
\(373\) 4797.04 0.665902 0.332951 0.942944i \(-0.391956\pi\)
0.332951 + 0.942944i \(0.391956\pi\)
\(374\) 0 0
\(375\) 5445.18 0.749834
\(376\) 0 0
\(377\) −8384.80 −1.14546
\(378\) 0 0
\(379\) 5901.75 0.799875 0.399937 0.916543i \(-0.369032\pi\)
0.399937 + 0.916543i \(0.369032\pi\)
\(380\) 0 0
\(381\) 9087.69 1.22199
\(382\) 0 0
\(383\) −13587.9 −1.81282 −0.906409 0.422401i \(-0.861188\pi\)
−0.906409 + 0.422401i \(0.861188\pi\)
\(384\) 0 0
\(385\) 125.274 0.0165832
\(386\) 0 0
\(387\) −7818.02 −1.02691
\(388\) 0 0
\(389\) 4100.95 0.534516 0.267258 0.963625i \(-0.413882\pi\)
0.267258 + 0.963625i \(0.413882\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −21454.0 −2.75372
\(394\) 0 0
\(395\) 375.598 0.0478440
\(396\) 0 0
\(397\) −12231.2 −1.54626 −0.773130 0.634248i \(-0.781310\pi\)
−0.773130 + 0.634248i \(0.781310\pi\)
\(398\) 0 0
\(399\) −6748.34 −0.846716
\(400\) 0 0
\(401\) 284.483 0.0354275 0.0177137 0.999843i \(-0.494361\pi\)
0.0177137 + 0.999843i \(0.494361\pi\)
\(402\) 0 0
\(403\) 10510.0 1.29911
\(404\) 0 0
\(405\) 1417.13 0.173871
\(406\) 0 0
\(407\) 590.972 0.0719739
\(408\) 0 0
\(409\) −10841.1 −1.31066 −0.655329 0.755343i \(-0.727470\pi\)
−0.655329 + 0.755343i \(0.727470\pi\)
\(410\) 0 0
\(411\) −8837.00 −1.06058
\(412\) 0 0
\(413\) −343.052 −0.0408728
\(414\) 0 0
\(415\) 3054.69 0.361323
\(416\) 0 0
\(417\) 12109.7 1.42210
\(418\) 0 0
\(419\) −2502.89 −0.291824 −0.145912 0.989298i \(-0.546612\pi\)
−0.145912 + 0.989298i \(0.546612\pi\)
\(420\) 0 0
\(421\) −7379.41 −0.854277 −0.427138 0.904186i \(-0.640478\pi\)
−0.427138 + 0.904186i \(0.640478\pi\)
\(422\) 0 0
\(423\) 20336.6 2.33758
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5190.38 −0.588243
\(428\) 0 0
\(429\) −1564.46 −0.176067
\(430\) 0 0
\(431\) −2269.80 −0.253671 −0.126836 0.991924i \(-0.540482\pi\)
−0.126836 + 0.991924i \(0.540482\pi\)
\(432\) 0 0
\(433\) −10132.3 −1.12454 −0.562272 0.826952i \(-0.690073\pi\)
−0.562272 + 0.826952i \(0.690073\pi\)
\(434\) 0 0
\(435\) 4910.49 0.541241
\(436\) 0 0
\(437\) −9647.10 −1.05603
\(438\) 0 0
\(439\) −12762.1 −1.38747 −0.693737 0.720228i \(-0.744037\pi\)
−0.693737 + 0.720228i \(0.744037\pi\)
\(440\) 0 0
\(441\) −11761.5 −1.27001
\(442\) 0 0
\(443\) −8883.68 −0.952768 −0.476384 0.879237i \(-0.658053\pi\)
−0.476384 + 0.879237i \(0.658053\pi\)
\(444\) 0 0
\(445\) 1515.39 0.161430
\(446\) 0 0
\(447\) −15284.2 −1.61726
\(448\) 0 0
\(449\) −12729.0 −1.33790 −0.668949 0.743308i \(-0.733256\pi\)
−0.668949 + 0.743308i \(0.733256\pi\)
\(450\) 0 0
\(451\) −2014.83 −0.210365
\(452\) 0 0
\(453\) 1139.30 0.118166
\(454\) 0 0
\(455\) 1035.59 0.106701
\(456\) 0 0
\(457\) −15299.0 −1.56599 −0.782993 0.622030i \(-0.786308\pi\)
−0.782993 + 0.622030i \(0.786308\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2186.66 0.220917 0.110459 0.993881i \(-0.464768\pi\)
0.110459 + 0.993881i \(0.464768\pi\)
\(462\) 0 0
\(463\) −1074.21 −0.107825 −0.0539123 0.998546i \(-0.517169\pi\)
−0.0539123 + 0.998546i \(0.517169\pi\)
\(464\) 0 0
\(465\) −6155.11 −0.613842
\(466\) 0 0
\(467\) 3439.93 0.340859 0.170430 0.985370i \(-0.445484\pi\)
0.170430 + 0.985370i \(0.445484\pi\)
\(468\) 0 0
\(469\) −7797.13 −0.767671
\(470\) 0 0
\(471\) 6454.47 0.631436
\(472\) 0 0
\(473\) −695.312 −0.0675909
\(474\) 0 0
\(475\) −8357.24 −0.807277
\(476\) 0 0
\(477\) 6089.39 0.584516
\(478\) 0 0
\(479\) 2397.04 0.228650 0.114325 0.993443i \(-0.463529\pi\)
0.114325 + 0.993443i \(0.463529\pi\)
\(480\) 0 0
\(481\) 4885.34 0.463102
\(482\) 0 0
\(483\) 13126.7 1.23662
\(484\) 0 0
\(485\) 4460.63 0.417622
\(486\) 0 0
\(487\) −11064.2 −1.02950 −0.514751 0.857339i \(-0.672116\pi\)
−0.514751 + 0.857339i \(0.672116\pi\)
\(488\) 0 0
\(489\) 29957.5 2.77040
\(490\) 0 0
\(491\) 12258.8 1.12674 0.563371 0.826204i \(-0.309504\pi\)
0.563371 + 0.826204i \(0.309504\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 602.679 0.0547240
\(496\) 0 0
\(497\) 5344.86 0.482394
\(498\) 0 0
\(499\) −13076.1 −1.17308 −0.586541 0.809920i \(-0.699511\pi\)
−0.586541 + 0.809920i \(0.699511\pi\)
\(500\) 0 0
\(501\) 27341.5 2.43818
\(502\) 0 0
\(503\) 3204.51 0.284060 0.142030 0.989862i \(-0.454637\pi\)
0.142030 + 0.989862i \(0.454637\pi\)
\(504\) 0 0
\(505\) 3723.90 0.328142
\(506\) 0 0
\(507\) 6582.21 0.576580
\(508\) 0 0
\(509\) 5752.67 0.500948 0.250474 0.968123i \(-0.419413\pi\)
0.250474 + 0.968123i \(0.419413\pi\)
\(510\) 0 0
\(511\) −7757.26 −0.671548
\(512\) 0 0
\(513\) −15575.9 −1.34053
\(514\) 0 0
\(515\) 2386.73 0.204217
\(516\) 0 0
\(517\) 1808.67 0.153860
\(518\) 0 0
\(519\) −9419.02 −0.796627
\(520\) 0 0
\(521\) 10333.6 0.868954 0.434477 0.900683i \(-0.356933\pi\)
0.434477 + 0.900683i \(0.356933\pi\)
\(522\) 0 0
\(523\) 9797.30 0.819132 0.409566 0.912281i \(-0.365680\pi\)
0.409566 + 0.912281i \(0.365680\pi\)
\(524\) 0 0
\(525\) 11371.6 0.945331
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6598.35 0.542315
\(530\) 0 0
\(531\) −1650.39 −0.134879
\(532\) 0 0
\(533\) −16655.8 −1.35355
\(534\) 0 0
\(535\) −2740.92 −0.221496
\(536\) 0 0
\(537\) −26399.4 −2.12145
\(538\) 0 0
\(539\) −1046.04 −0.0835917
\(540\) 0 0
\(541\) 12700.1 1.00928 0.504638 0.863331i \(-0.331626\pi\)
0.504638 + 0.863331i \(0.331626\pi\)
\(542\) 0 0
\(543\) 1510.09 0.119345
\(544\) 0 0
\(545\) −4218.70 −0.331577
\(546\) 0 0
\(547\) −144.879 −0.0113246 −0.00566232 0.999984i \(-0.501802\pi\)
−0.00566232 + 0.999984i \(0.501802\pi\)
\(548\) 0 0
\(549\) −24970.4 −1.94118
\(550\) 0 0
\(551\) −15475.1 −1.19648
\(552\) 0 0
\(553\) 1610.62 0.123853
\(554\) 0 0
\(555\) −2861.05 −0.218820
\(556\) 0 0
\(557\) 22211.3 1.68963 0.844814 0.535061i \(-0.179711\pi\)
0.844814 + 0.535061i \(0.179711\pi\)
\(558\) 0 0
\(559\) −5747.88 −0.434900
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3615.69 −0.270663 −0.135331 0.990800i \(-0.543210\pi\)
−0.135331 + 0.990800i \(0.543210\pi\)
\(564\) 0 0
\(565\) −284.017 −0.0211481
\(566\) 0 0
\(567\) 6076.87 0.450096
\(568\) 0 0
\(569\) −428.770 −0.0315905 −0.0157952 0.999875i \(-0.505028\pi\)
−0.0157952 + 0.999875i \(0.505028\pi\)
\(570\) 0 0
\(571\) 1806.75 0.132417 0.0662086 0.997806i \(-0.478910\pi\)
0.0662086 + 0.997806i \(0.478910\pi\)
\(572\) 0 0
\(573\) −31855.4 −2.32248
\(574\) 0 0
\(575\) 16256.3 1.17902
\(576\) 0 0
\(577\) −22101.9 −1.59465 −0.797324 0.603551i \(-0.793752\pi\)
−0.797324 + 0.603551i \(0.793752\pi\)
\(578\) 0 0
\(579\) 34037.1 2.44307
\(580\) 0 0
\(581\) 13099.0 0.935348
\(582\) 0 0
\(583\) 541.573 0.0384728
\(584\) 0 0
\(585\) 4982.11 0.352111
\(586\) 0 0
\(587\) −23557.8 −1.65645 −0.828225 0.560396i \(-0.810649\pi\)
−0.828225 + 0.560396i \(0.810649\pi\)
\(588\) 0 0
\(589\) 19397.5 1.35698
\(590\) 0 0
\(591\) −17397.7 −1.21091
\(592\) 0 0
\(593\) −13400.6 −0.927989 −0.463995 0.885838i \(-0.653584\pi\)
−0.463995 + 0.885838i \(0.653584\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 46696.0 3.20124
\(598\) 0 0
\(599\) −3641.89 −0.248420 −0.124210 0.992256i \(-0.539640\pi\)
−0.124210 + 0.992256i \(0.539640\pi\)
\(600\) 0 0
\(601\) 4853.99 0.329448 0.164724 0.986340i \(-0.447327\pi\)
0.164724 + 0.986340i \(0.447327\pi\)
\(602\) 0 0
\(603\) −37511.2 −2.53329
\(604\) 0 0
\(605\) −3294.89 −0.221415
\(606\) 0 0
\(607\) 15241.7 1.01918 0.509589 0.860418i \(-0.329798\pi\)
0.509589 + 0.860418i \(0.329798\pi\)
\(608\) 0 0
\(609\) 21056.9 1.40110
\(610\) 0 0
\(611\) 14951.6 0.989979
\(612\) 0 0
\(613\) 8519.26 0.561321 0.280660 0.959807i \(-0.409447\pi\)
0.280660 + 0.959807i \(0.409447\pi\)
\(614\) 0 0
\(615\) 9754.33 0.639565
\(616\) 0 0
\(617\) −13044.7 −0.851154 −0.425577 0.904922i \(-0.639929\pi\)
−0.425577 + 0.904922i \(0.639929\pi\)
\(618\) 0 0
\(619\) −11591.6 −0.752674 −0.376337 0.926483i \(-0.622816\pi\)
−0.376337 + 0.926483i \(0.622816\pi\)
\(620\) 0 0
\(621\) 30298.0 1.95784
\(622\) 0 0
\(623\) 6498.20 0.417889
\(624\) 0 0
\(625\) 13291.7 0.850666
\(626\) 0 0
\(627\) −2887.39 −0.183910
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −20451.8 −1.29029 −0.645145 0.764060i \(-0.723203\pi\)
−0.645145 + 0.764060i \(0.723203\pi\)
\(632\) 0 0
\(633\) −13759.2 −0.863946
\(634\) 0 0
\(635\) −2573.87 −0.160852
\(636\) 0 0
\(637\) −8647.17 −0.537855
\(638\) 0 0
\(639\) 25713.6 1.59188
\(640\) 0 0
\(641\) 29696.1 1.82984 0.914919 0.403637i \(-0.132254\pi\)
0.914919 + 0.403637i \(0.132254\pi\)
\(642\) 0 0
\(643\) 1806.65 0.110804 0.0554022 0.998464i \(-0.482356\pi\)
0.0554022 + 0.998464i \(0.482356\pi\)
\(644\) 0 0
\(645\) 3366.19 0.205494
\(646\) 0 0
\(647\) 14664.2 0.891052 0.445526 0.895269i \(-0.353017\pi\)
0.445526 + 0.895269i \(0.353017\pi\)
\(648\) 0 0
\(649\) −146.781 −0.00887772
\(650\) 0 0
\(651\) −26394.0 −1.58904
\(652\) 0 0
\(653\) 4054.25 0.242963 0.121482 0.992594i \(-0.461235\pi\)
0.121482 + 0.992594i \(0.461235\pi\)
\(654\) 0 0
\(655\) 6076.32 0.362476
\(656\) 0 0
\(657\) −37319.4 −2.21609
\(658\) 0 0
\(659\) −9241.66 −0.546288 −0.273144 0.961973i \(-0.588064\pi\)
−0.273144 + 0.961973i \(0.588064\pi\)
\(660\) 0 0
\(661\) 22727.1 1.33734 0.668671 0.743559i \(-0.266864\pi\)
0.668671 + 0.743559i \(0.266864\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1911.30 0.111454
\(666\) 0 0
\(667\) 30101.9 1.74745
\(668\) 0 0
\(669\) −19245.9 −1.11224
\(670\) 0 0
\(671\) −2220.79 −0.127769
\(672\) 0 0
\(673\) −25575.9 −1.46490 −0.732451 0.680819i \(-0.761624\pi\)
−0.732451 + 0.680819i \(0.761624\pi\)
\(674\) 0 0
\(675\) 26247.0 1.49666
\(676\) 0 0
\(677\) −25301.6 −1.43636 −0.718182 0.695855i \(-0.755026\pi\)
−0.718182 + 0.695855i \(0.755026\pi\)
\(678\) 0 0
\(679\) 19127.8 1.08109
\(680\) 0 0
\(681\) 21510.2 1.21039
\(682\) 0 0
\(683\) 6633.48 0.371630 0.185815 0.982585i \(-0.440508\pi\)
0.185815 + 0.982585i \(0.440508\pi\)
\(684\) 0 0
\(685\) 2502.86 0.139605
\(686\) 0 0
\(687\) −38710.2 −2.14976
\(688\) 0 0
\(689\) 4476.97 0.247546
\(690\) 0 0
\(691\) 21211.2 1.16775 0.583874 0.811844i \(-0.301536\pi\)
0.583874 + 0.811844i \(0.301536\pi\)
\(692\) 0 0
\(693\) 2584.38 0.141663
\(694\) 0 0
\(695\) −3429.77 −0.187192
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 21460.7 1.16126
\(700\) 0 0
\(701\) 31524.7 1.69853 0.849267 0.527963i \(-0.177044\pi\)
0.849267 + 0.527963i \(0.177044\pi\)
\(702\) 0 0
\(703\) 9016.46 0.483730
\(704\) 0 0
\(705\) −8756.28 −0.467774
\(706\) 0 0
\(707\) 15968.6 0.849452
\(708\) 0 0
\(709\) 23585.3 1.24931 0.624657 0.780899i \(-0.285239\pi\)
0.624657 + 0.780899i \(0.285239\pi\)
\(710\) 0 0
\(711\) 7748.53 0.408710
\(712\) 0 0
\(713\) −37731.7 −1.98185
\(714\) 0 0
\(715\) 443.095 0.0231759
\(716\) 0 0
\(717\) 22315.1 1.16230
\(718\) 0 0
\(719\) 5515.05 0.286060 0.143030 0.989718i \(-0.454316\pi\)
0.143030 + 0.989718i \(0.454316\pi\)
\(720\) 0 0
\(721\) 10234.7 0.528653
\(722\) 0 0
\(723\) 6548.27 0.336836
\(724\) 0 0
\(725\) 26077.2 1.33584
\(726\) 0 0
\(727\) −32407.1 −1.65325 −0.826625 0.562754i \(-0.809742\pi\)
−0.826625 + 0.562754i \(0.809742\pi\)
\(728\) 0 0
\(729\) −23808.9 −1.20962
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 32710.6 1.64828 0.824142 0.566383i \(-0.191658\pi\)
0.824142 + 0.566383i \(0.191658\pi\)
\(734\) 0 0
\(735\) 5064.14 0.254141
\(736\) 0 0
\(737\) −3336.14 −0.166741
\(738\) 0 0
\(739\) −33363.0 −1.66073 −0.830364 0.557221i \(-0.811868\pi\)
−0.830364 + 0.557221i \(0.811868\pi\)
\(740\) 0 0
\(741\) −23869.0 −1.18333
\(742\) 0 0
\(743\) 24474.4 1.20845 0.604226 0.796813i \(-0.293482\pi\)
0.604226 + 0.796813i \(0.293482\pi\)
\(744\) 0 0
\(745\) 4328.87 0.212882
\(746\) 0 0
\(747\) 63017.8 3.08661
\(748\) 0 0
\(749\) −11753.5 −0.573382
\(750\) 0 0
\(751\) −3172.55 −0.154152 −0.0770759 0.997025i \(-0.524558\pi\)
−0.0770759 + 0.997025i \(0.524558\pi\)
\(752\) 0 0
\(753\) 41039.1 1.98612
\(754\) 0 0
\(755\) −322.679 −0.0155543
\(756\) 0 0
\(757\) −36897.8 −1.77156 −0.885782 0.464102i \(-0.846377\pi\)
−0.885782 + 0.464102i \(0.846377\pi\)
\(758\) 0 0
\(759\) 5616.50 0.268598
\(760\) 0 0
\(761\) 15071.2 0.717913 0.358957 0.933354i \(-0.383133\pi\)
0.358957 + 0.933354i \(0.383133\pi\)
\(762\) 0 0
\(763\) −18090.4 −0.858345
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1213.38 −0.0571219
\(768\) 0 0
\(769\) 25033.3 1.17389 0.586947 0.809626i \(-0.300330\pi\)
0.586947 + 0.809626i \(0.300330\pi\)
\(770\) 0 0
\(771\) 58392.9 2.72759
\(772\) 0 0
\(773\) 26899.1 1.25161 0.625803 0.779981i \(-0.284771\pi\)
0.625803 + 0.779981i \(0.284771\pi\)
\(774\) 0 0
\(775\) −32686.8 −1.51502
\(776\) 0 0
\(777\) −12268.6 −0.566454
\(778\) 0 0
\(779\) −30740.3 −1.41384
\(780\) 0 0
\(781\) 2286.89 0.104778
\(782\) 0 0
\(783\) 48601.7 2.21824
\(784\) 0 0
\(785\) −1828.07 −0.0831167
\(786\) 0 0
\(787\) −15295.4 −0.692787 −0.346394 0.938089i \(-0.612594\pi\)
−0.346394 + 0.938089i \(0.612594\pi\)
\(788\) 0 0
\(789\) 12277.0 0.553960
\(790\) 0 0
\(791\) −1217.91 −0.0547457
\(792\) 0 0
\(793\) −18358.4 −0.822102
\(794\) 0 0
\(795\) −2621.90 −0.116968
\(796\) 0 0
\(797\) −4866.05 −0.216267 −0.108133 0.994136i \(-0.534487\pi\)
−0.108133 + 0.994136i \(0.534487\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 31262.2 1.37902
\(802\) 0 0
\(803\) −3319.08 −0.145863
\(804\) 0 0
\(805\) −3717.83 −0.162778
\(806\) 0 0
\(807\) −28370.2 −1.23752
\(808\) 0 0
\(809\) −11297.5 −0.490973 −0.245487 0.969400i \(-0.578948\pi\)
−0.245487 + 0.969400i \(0.578948\pi\)
\(810\) 0 0
\(811\) 22072.6 0.955700 0.477850 0.878441i \(-0.341416\pi\)
0.477850 + 0.878441i \(0.341416\pi\)
\(812\) 0 0
\(813\) 13381.9 0.577274
\(814\) 0 0
\(815\) −8484.74 −0.364672
\(816\) 0 0
\(817\) −10608.4 −0.454272
\(818\) 0 0
\(819\) 21364.0 0.911502
\(820\) 0 0
\(821\) −12329.0 −0.524097 −0.262048 0.965055i \(-0.584398\pi\)
−0.262048 + 0.965055i \(0.584398\pi\)
\(822\) 0 0
\(823\) −14234.2 −0.602885 −0.301442 0.953484i \(-0.597468\pi\)
−0.301442 + 0.953484i \(0.597468\pi\)
\(824\) 0 0
\(825\) 4865.55 0.205329
\(826\) 0 0
\(827\) −16569.6 −0.696713 −0.348357 0.937362i \(-0.613260\pi\)
−0.348357 + 0.937362i \(0.613260\pi\)
\(828\) 0 0
\(829\) −26164.3 −1.09617 −0.548085 0.836423i \(-0.684643\pi\)
−0.548085 + 0.836423i \(0.684643\pi\)
\(830\) 0 0
\(831\) 43623.0 1.82102
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −7743.80 −0.320940
\(836\) 0 0
\(837\) −60920.4 −2.51579
\(838\) 0 0
\(839\) −7683.47 −0.316165 −0.158083 0.987426i \(-0.550531\pi\)
−0.158083 + 0.987426i \(0.550531\pi\)
\(840\) 0 0
\(841\) 23898.2 0.979876
\(842\) 0 0
\(843\) 63939.2 2.61232
\(844\) 0 0
\(845\) −1864.25 −0.0758960
\(846\) 0 0
\(847\) −14129.0 −0.573172
\(848\) 0 0
\(849\) −38336.4 −1.54971
\(850\) 0 0
\(851\) −17538.6 −0.706483
\(852\) 0 0
\(853\) −5850.34 −0.234832 −0.117416 0.993083i \(-0.537461\pi\)
−0.117416 + 0.993083i \(0.537461\pi\)
\(854\) 0 0
\(855\) 9195.07 0.367795
\(856\) 0 0
\(857\) −33338.3 −1.32884 −0.664420 0.747360i \(-0.731321\pi\)
−0.664420 + 0.747360i \(0.731321\pi\)
\(858\) 0 0
\(859\) −27448.7 −1.09027 −0.545133 0.838350i \(-0.683521\pi\)
−0.545133 + 0.838350i \(0.683521\pi\)
\(860\) 0 0
\(861\) 41828.0 1.65563
\(862\) 0 0
\(863\) 4907.37 0.193568 0.0967838 0.995305i \(-0.469144\pi\)
0.0967838 + 0.995305i \(0.469144\pi\)
\(864\) 0 0
\(865\) 2667.71 0.104861
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 689.132 0.0269013
\(870\) 0 0
\(871\) −27578.5 −1.07286
\(872\) 0 0
\(873\) 92022.0 3.56755
\(874\) 0 0
\(875\) −6613.24 −0.255507
\(876\) 0 0
\(877\) 27586.7 1.06219 0.531093 0.847313i \(-0.321781\pi\)
0.531093 + 0.847313i \(0.321781\pi\)
\(878\) 0 0
\(879\) 27412.0 1.05186
\(880\) 0 0
\(881\) 45088.4 1.72425 0.862126 0.506693i \(-0.169132\pi\)
0.862126 + 0.506693i \(0.169132\pi\)
\(882\) 0 0
\(883\) 15927.0 0.607005 0.303502 0.952831i \(-0.401844\pi\)
0.303502 + 0.952831i \(0.401844\pi\)
\(884\) 0 0
\(885\) 710.604 0.0269906
\(886\) 0 0
\(887\) 37913.5 1.43519 0.717593 0.696463i \(-0.245244\pi\)
0.717593 + 0.696463i \(0.245244\pi\)
\(888\) 0 0
\(889\) −11037.1 −0.416393
\(890\) 0 0
\(891\) 2600.09 0.0977625
\(892\) 0 0
\(893\) 27595.0 1.03408
\(894\) 0 0
\(895\) 7477.00 0.279250
\(896\) 0 0
\(897\) 46429.4 1.72824
\(898\) 0 0
\(899\) −60526.2 −2.24545
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 14434.7 0.531958
\(904\) 0 0
\(905\) −427.697 −0.0157095
\(906\) 0 0
\(907\) −25256.3 −0.924611 −0.462305 0.886721i \(-0.652978\pi\)
−0.462305 + 0.886721i \(0.652978\pi\)
\(908\) 0 0
\(909\) 76823.5 2.80316
\(910\) 0 0
\(911\) −11135.6 −0.404983 −0.202491 0.979284i \(-0.564904\pi\)
−0.202491 + 0.979284i \(0.564904\pi\)
\(912\) 0 0
\(913\) 5604.62 0.203161
\(914\) 0 0
\(915\) 10751.5 0.388450
\(916\) 0 0
\(917\) 26056.2 0.938333
\(918\) 0 0
\(919\) −45609.6 −1.63713 −0.818565 0.574414i \(-0.805230\pi\)
−0.818565 + 0.574414i \(0.805230\pi\)
\(920\) 0 0
\(921\) 18945.6 0.677827
\(922\) 0 0
\(923\) 18904.8 0.674172
\(924\) 0 0
\(925\) −15193.6 −0.540069
\(926\) 0 0
\(927\) 49237.9 1.74453
\(928\) 0 0
\(929\) −23419.6 −0.827097 −0.413548 0.910482i \(-0.635711\pi\)
−0.413548 + 0.910482i \(0.635711\pi\)
\(930\) 0 0
\(931\) −15959.4 −0.561812
\(932\) 0 0
\(933\) −52375.4 −1.83783
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 14729.0 0.513527 0.256764 0.966474i \(-0.417344\pi\)
0.256764 + 0.966474i \(0.417344\pi\)
\(938\) 0 0
\(939\) −26438.7 −0.918845
\(940\) 0 0
\(941\) −53609.0 −1.85718 −0.928589 0.371110i \(-0.878977\pi\)
−0.928589 + 0.371110i \(0.878977\pi\)
\(942\) 0 0
\(943\) 59795.3 2.06490
\(944\) 0 0
\(945\) −6002.69 −0.206632
\(946\) 0 0
\(947\) −44343.7 −1.52162 −0.760811 0.648973i \(-0.775199\pi\)
−0.760811 + 0.648973i \(0.775199\pi\)
\(948\) 0 0
\(949\) −27437.5 −0.938525
\(950\) 0 0
\(951\) −24358.9 −0.830591
\(952\) 0 0
\(953\) −32953.0 −1.12010 −0.560049 0.828460i \(-0.689218\pi\)
−0.560049 + 0.828460i \(0.689218\pi\)
\(954\) 0 0
\(955\) 9022.26 0.305711
\(956\) 0 0
\(957\) 9009.56 0.304324
\(958\) 0 0
\(959\) 10732.7 0.361393
\(960\) 0 0
\(961\) 46076.4 1.54666
\(962\) 0 0
\(963\) −56544.8 −1.89214
\(964\) 0 0
\(965\) −9640.19 −0.321584
\(966\) 0 0
\(967\) 15193.2 0.505254 0.252627 0.967564i \(-0.418705\pi\)
0.252627 + 0.967564i \(0.418705\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27985.5 −0.924920 −0.462460 0.886640i \(-0.653033\pi\)
−0.462460 + 0.886640i \(0.653033\pi\)
\(972\) 0 0
\(973\) −14707.4 −0.484581
\(974\) 0 0
\(975\) 40221.6 1.32115
\(976\) 0 0
\(977\) −54353.9 −1.77987 −0.889936 0.456086i \(-0.849251\pi\)
−0.889936 + 0.456086i \(0.849251\pi\)
\(978\) 0 0
\(979\) 2780.37 0.0907671
\(980\) 0 0
\(981\) −87031.2 −2.83251
\(982\) 0 0
\(983\) 2179.36 0.0707130 0.0353565 0.999375i \(-0.488743\pi\)
0.0353565 + 0.999375i \(0.488743\pi\)
\(984\) 0 0
\(985\) 4927.47 0.159393
\(986\) 0 0
\(987\) −37548.2 −1.21092
\(988\) 0 0
\(989\) 20635.2 0.663460
\(990\) 0 0
\(991\) 60236.3 1.93085 0.965423 0.260688i \(-0.0839493\pi\)
0.965423 + 0.260688i \(0.0839493\pi\)
\(992\) 0 0
\(993\) −37400.7 −1.19524
\(994\) 0 0
\(995\) −13225.5 −0.421384
\(996\) 0 0
\(997\) 38415.9 1.22031 0.610153 0.792284i \(-0.291108\pi\)
0.610153 + 0.792284i \(0.291108\pi\)
\(998\) 0 0
\(999\) −28317.4 −0.896819
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 2312.4.a.o.1.2 18
17.16 even 2 2312.4.a.p.1.17 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2312.4.a.o.1.2 18 1.1 even 1 trivial
2312.4.a.p.1.17 yes 18 17.16 even 2