Properties

Label 1200.4.a.x.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(1,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -19.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -19.0000 q^{7} +9.00000 q^{9} -22.0000 q^{11} -1.00000 q^{13} +58.0000 q^{17} +53.0000 q^{19} -57.0000 q^{21} +58.0000 q^{23} +27.0000 q^{27} +22.0000 q^{29} +35.0000 q^{31} -66.0000 q^{33} +270.000 q^{37} -3.00000 q^{39} -468.000 q^{41} -431.000 q^{43} -230.000 q^{47} +18.0000 q^{49} +174.000 q^{51} +159.000 q^{57} -446.000 q^{59} +127.000 q^{61} -171.000 q^{63} -811.000 q^{67} +174.000 q^{69} -36.0000 q^{71} -522.000 q^{73} +418.000 q^{77} -1368.00 q^{79} +81.0000 q^{81} -1138.00 q^{83} +66.0000 q^{87} +144.000 q^{89} +19.0000 q^{91} +105.000 q^{93} +1079.00 q^{97} -198.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −19.0000 −1.02590 −0.512952 0.858417i \(-0.671448\pi\)
−0.512952 + 0.858417i \(0.671448\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −22.0000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.0213346 −0.0106673 0.999943i \(-0.503396\pi\)
−0.0106673 + 0.999943i \(0.503396\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 58.0000 0.827474 0.413737 0.910396i \(-0.364223\pi\)
0.413737 + 0.910396i \(0.364223\pi\)
\(18\) 0 0
\(19\) 53.0000 0.639949 0.319975 0.947426i \(-0.396326\pi\)
0.319975 + 0.947426i \(0.396326\pi\)
\(20\) 0 0
\(21\) −57.0000 −0.592306
\(22\) 0 0
\(23\) 58.0000 0.525819 0.262909 0.964821i \(-0.415318\pi\)
0.262909 + 0.964821i \(0.415318\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 22.0000 0.140872 0.0704362 0.997516i \(-0.477561\pi\)
0.0704362 + 0.997516i \(0.477561\pi\)
\(30\) 0 0
\(31\) 35.0000 0.202780 0.101390 0.994847i \(-0.467671\pi\)
0.101390 + 0.994847i \(0.467671\pi\)
\(32\) 0 0
\(33\) −66.0000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 270.000 1.19967 0.599834 0.800124i \(-0.295233\pi\)
0.599834 + 0.800124i \(0.295233\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.0123176
\(40\) 0 0
\(41\) −468.000 −1.78267 −0.891333 0.453349i \(-0.850229\pi\)
−0.891333 + 0.453349i \(0.850229\pi\)
\(42\) 0 0
\(43\) −431.000 −1.52853 −0.764266 0.644901i \(-0.776899\pi\)
−0.764266 + 0.644901i \(0.776899\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −230.000 −0.713807 −0.356904 0.934141i \(-0.616168\pi\)
−0.356904 + 0.934141i \(0.616168\pi\)
\(48\) 0 0
\(49\) 18.0000 0.0524781
\(50\) 0 0
\(51\) 174.000 0.477743
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 159.000 0.369475
\(58\) 0 0
\(59\) −446.000 −0.984140 −0.492070 0.870556i \(-0.663760\pi\)
−0.492070 + 0.870556i \(0.663760\pi\)
\(60\) 0 0
\(61\) 127.000 0.266569 0.133284 0.991078i \(-0.457448\pi\)
0.133284 + 0.991078i \(0.457448\pi\)
\(62\) 0 0
\(63\) −171.000 −0.341968
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −811.000 −1.47880 −0.739399 0.673268i \(-0.764890\pi\)
−0.739399 + 0.673268i \(0.764890\pi\)
\(68\) 0 0
\(69\) 174.000 0.303582
\(70\) 0 0
\(71\) −36.0000 −0.0601748 −0.0300874 0.999547i \(-0.509579\pi\)
−0.0300874 + 0.999547i \(0.509579\pi\)
\(72\) 0 0
\(73\) −522.000 −0.836924 −0.418462 0.908234i \(-0.637431\pi\)
−0.418462 + 0.908234i \(0.637431\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 418.000 0.618643
\(78\) 0 0
\(79\) −1368.00 −1.94825 −0.974127 0.226002i \(-0.927434\pi\)
−0.974127 + 0.226002i \(0.927434\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1138.00 −1.50496 −0.752480 0.658615i \(-0.771143\pi\)
−0.752480 + 0.658615i \(0.771143\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 66.0000 0.0813327
\(88\) 0 0
\(89\) 144.000 0.171505 0.0857526 0.996316i \(-0.472671\pi\)
0.0857526 + 0.996316i \(0.472671\pi\)
\(90\) 0 0
\(91\) 19.0000 0.0218873
\(92\) 0 0
\(93\) 105.000 0.117075
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1079.00 1.12944 0.564721 0.825282i \(-0.308984\pi\)
0.564721 + 0.825282i \(0.308984\pi\)
\(98\) 0 0
\(99\) −198.000 −0.201008
\(100\) 0 0
\(101\) −1440.00 −1.41867 −0.709333 0.704873i \(-0.751004\pi\)
−0.709333 + 0.704873i \(0.751004\pi\)
\(102\) 0 0
\(103\) 124.000 0.118622 0.0593111 0.998240i \(-0.481110\pi\)
0.0593111 + 0.998240i \(0.481110\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −432.000 −0.390309 −0.195154 0.980773i \(-0.562521\pi\)
−0.195154 + 0.980773i \(0.562521\pi\)
\(108\) 0 0
\(109\) 701.000 0.615997 0.307998 0.951387i \(-0.400341\pi\)
0.307998 + 0.951387i \(0.400341\pi\)
\(110\) 0 0
\(111\) 810.000 0.692629
\(112\) 0 0
\(113\) 1044.00 0.869126 0.434563 0.900641i \(-0.356903\pi\)
0.434563 + 0.900641i \(0.356903\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −9.00000 −0.00711154
\(118\) 0 0
\(119\) −1102.00 −0.848909
\(120\) 0 0
\(121\) −847.000 −0.636364
\(122\) 0 0
\(123\) −1404.00 −1.02922
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 504.000 0.352148 0.176074 0.984377i \(-0.443660\pi\)
0.176074 + 0.984377i \(0.443660\pi\)
\(128\) 0 0
\(129\) −1293.00 −0.882498
\(130\) 0 0
\(131\) −72.0000 −0.0480204 −0.0240102 0.999712i \(-0.507643\pi\)
−0.0240102 + 0.999712i \(0.507643\pi\)
\(132\) 0 0
\(133\) −1007.00 −0.656526
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1850.00 1.15369 0.576847 0.816852i \(-0.304283\pi\)
0.576847 + 0.816852i \(0.304283\pi\)
\(138\) 0 0
\(139\) −1836.00 −1.12034 −0.560171 0.828377i \(-0.689265\pi\)
−0.560171 + 0.828377i \(0.689265\pi\)
\(140\) 0 0
\(141\) −690.000 −0.412117
\(142\) 0 0
\(143\) 22.0000 0.0128653
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 54.0000 0.0302983
\(148\) 0 0
\(149\) −2398.00 −1.31847 −0.659234 0.751938i \(-0.729119\pi\)
−0.659234 + 0.751938i \(0.729119\pi\)
\(150\) 0 0
\(151\) −1871.00 −1.00834 −0.504172 0.863604i \(-0.668202\pi\)
−0.504172 + 0.863604i \(0.668202\pi\)
\(152\) 0 0
\(153\) 522.000 0.275825
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3293.00 1.67395 0.836975 0.547242i \(-0.184322\pi\)
0.836975 + 0.547242i \(0.184322\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1102.00 −0.539440
\(162\) 0 0
\(163\) 883.000 0.424306 0.212153 0.977236i \(-0.431952\pi\)
0.212153 + 0.977236i \(0.431952\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4104.00 1.90166 0.950830 0.309715i \(-0.100234\pi\)
0.950830 + 0.309715i \(0.100234\pi\)
\(168\) 0 0
\(169\) −2196.00 −0.999545
\(170\) 0 0
\(171\) 477.000 0.213316
\(172\) 0 0
\(173\) −1706.00 −0.749739 −0.374869 0.927078i \(-0.622312\pi\)
−0.374869 + 0.927078i \(0.622312\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1338.00 −0.568193
\(178\) 0 0
\(179\) −662.000 −0.276426 −0.138213 0.990403i \(-0.544136\pi\)
−0.138213 + 0.990403i \(0.544136\pi\)
\(180\) 0 0
\(181\) 4121.00 1.69233 0.846164 0.532922i \(-0.178906\pi\)
0.846164 + 0.532922i \(0.178906\pi\)
\(182\) 0 0
\(183\) 381.000 0.153903
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1276.00 −0.498986
\(188\) 0 0
\(189\) −513.000 −0.197435
\(190\) 0 0
\(191\) 958.000 0.362924 0.181462 0.983398i \(-0.441917\pi\)
0.181462 + 0.983398i \(0.441917\pi\)
\(192\) 0 0
\(193\) −3187.00 −1.18863 −0.594314 0.804233i \(-0.702576\pi\)
−0.594314 + 0.804233i \(0.702576\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2282.00 0.825308 0.412654 0.910888i \(-0.364602\pi\)
0.412654 + 0.910888i \(0.364602\pi\)
\(198\) 0 0
\(199\) −1043.00 −0.371539 −0.185770 0.982593i \(-0.559478\pi\)
−0.185770 + 0.982593i \(0.559478\pi\)
\(200\) 0 0
\(201\) −2433.00 −0.853784
\(202\) 0 0
\(203\) −418.000 −0.144521
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 522.000 0.175273
\(208\) 0 0
\(209\) −1166.00 −0.385904
\(210\) 0 0
\(211\) −4139.00 −1.35043 −0.675214 0.737621i \(-0.735949\pi\)
−0.675214 + 0.737621i \(0.735949\pi\)
\(212\) 0 0
\(213\) −108.000 −0.0347420
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −665.000 −0.208033
\(218\) 0 0
\(219\) −1566.00 −0.483199
\(220\) 0 0
\(221\) −58.0000 −0.0176539
\(222\) 0 0
\(223\) 413.000 0.124020 0.0620101 0.998076i \(-0.480249\pi\)
0.0620101 + 0.998076i \(0.480249\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5652.00 1.65258 0.826292 0.563242i \(-0.190446\pi\)
0.826292 + 0.563242i \(0.190446\pi\)
\(228\) 0 0
\(229\) −4391.00 −1.26710 −0.633549 0.773703i \(-0.718403\pi\)
−0.633549 + 0.773703i \(0.718403\pi\)
\(230\) 0 0
\(231\) 1254.00 0.357174
\(232\) 0 0
\(233\) 2052.00 0.576957 0.288479 0.957486i \(-0.406851\pi\)
0.288479 + 0.957486i \(0.406851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4104.00 −1.12482
\(238\) 0 0
\(239\) 4320.00 1.16919 0.584597 0.811324i \(-0.301252\pi\)
0.584597 + 0.811324i \(0.301252\pi\)
\(240\) 0 0
\(241\) 4265.00 1.13997 0.569985 0.821655i \(-0.306949\pi\)
0.569985 + 0.821655i \(0.306949\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −53.0000 −0.0136531
\(248\) 0 0
\(249\) −3414.00 −0.868889
\(250\) 0 0
\(251\) 2412.00 0.606550 0.303275 0.952903i \(-0.401920\pi\)
0.303275 + 0.952903i \(0.401920\pi\)
\(252\) 0 0
\(253\) −1276.00 −0.317081
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6948.00 −1.68640 −0.843199 0.537601i \(-0.819331\pi\)
−0.843199 + 0.537601i \(0.819331\pi\)
\(258\) 0 0
\(259\) −5130.00 −1.23074
\(260\) 0 0
\(261\) 198.000 0.0469574
\(262\) 0 0
\(263\) −3564.00 −0.835611 −0.417805 0.908537i \(-0.637201\pi\)
−0.417805 + 0.908537i \(0.637201\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 432.000 0.0990186
\(268\) 0 0
\(269\) −1534.00 −0.347694 −0.173847 0.984773i \(-0.555620\pi\)
−0.173847 + 0.984773i \(0.555620\pi\)
\(270\) 0 0
\(271\) −7704.00 −1.72688 −0.863440 0.504451i \(-0.831695\pi\)
−0.863440 + 0.504451i \(0.831695\pi\)
\(272\) 0 0
\(273\) 57.0000 0.0126366
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4877.00 −1.05787 −0.528936 0.848662i \(-0.677409\pi\)
−0.528936 + 0.848662i \(0.677409\pi\)
\(278\) 0 0
\(279\) 315.000 0.0675934
\(280\) 0 0
\(281\) −3758.00 −0.797806 −0.398903 0.916993i \(-0.630609\pi\)
−0.398903 + 0.916993i \(0.630609\pi\)
\(282\) 0 0
\(283\) 935.000 0.196396 0.0981978 0.995167i \(-0.468692\pi\)
0.0981978 + 0.995167i \(0.468692\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8892.00 1.82884
\(288\) 0 0
\(289\) −1549.00 −0.315286
\(290\) 0 0
\(291\) 3237.00 0.652084
\(292\) 0 0
\(293\) −6214.00 −1.23900 −0.619498 0.784998i \(-0.712664\pi\)
−0.619498 + 0.784998i \(0.712664\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −594.000 −0.116052
\(298\) 0 0
\(299\) −58.0000 −0.0112181
\(300\) 0 0
\(301\) 8189.00 1.56813
\(302\) 0 0
\(303\) −4320.00 −0.819068
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3905.00 −0.725961 −0.362981 0.931797i \(-0.618241\pi\)
−0.362981 + 0.931797i \(0.618241\pi\)
\(308\) 0 0
\(309\) 372.000 0.0684865
\(310\) 0 0
\(311\) −9662.00 −1.76168 −0.880839 0.473416i \(-0.843021\pi\)
−0.880839 + 0.473416i \(0.843021\pi\)
\(312\) 0 0
\(313\) −5147.00 −0.929475 −0.464737 0.885449i \(-0.653851\pi\)
−0.464737 + 0.885449i \(0.653851\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9216.00 1.63288 0.816439 0.577432i \(-0.195945\pi\)
0.816439 + 0.577432i \(0.195945\pi\)
\(318\) 0 0
\(319\) −484.000 −0.0849492
\(320\) 0 0
\(321\) −1296.00 −0.225345
\(322\) 0 0
\(323\) 3074.00 0.529542
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2103.00 0.355646
\(328\) 0 0
\(329\) 4370.00 0.732298
\(330\) 0 0
\(331\) −2196.00 −0.364662 −0.182331 0.983237i \(-0.558364\pi\)
−0.182331 + 0.983237i \(0.558364\pi\)
\(332\) 0 0
\(333\) 2430.00 0.399889
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2521.00 −0.407500 −0.203750 0.979023i \(-0.565313\pi\)
−0.203750 + 0.979023i \(0.565313\pi\)
\(338\) 0 0
\(339\) 3132.00 0.501790
\(340\) 0 0
\(341\) −770.000 −0.122281
\(342\) 0 0
\(343\) 6175.00 0.972066
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7034.00 1.08820 0.544099 0.839021i \(-0.316871\pi\)
0.544099 + 0.839021i \(0.316871\pi\)
\(348\) 0 0
\(349\) 7362.00 1.12917 0.564583 0.825376i \(-0.309037\pi\)
0.564583 + 0.825376i \(0.309037\pi\)
\(350\) 0 0
\(351\) −27.0000 −0.00410585
\(352\) 0 0
\(353\) 9382.00 1.41460 0.707300 0.706914i \(-0.249913\pi\)
0.707300 + 0.706914i \(0.249913\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3306.00 −0.490118
\(358\) 0 0
\(359\) 10116.0 1.48719 0.743596 0.668629i \(-0.233119\pi\)
0.743596 + 0.668629i \(0.233119\pi\)
\(360\) 0 0
\(361\) −4050.00 −0.590465
\(362\) 0 0
\(363\) −2541.00 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3277.00 −0.466098 −0.233049 0.972465i \(-0.574870\pi\)
−0.233049 + 0.972465i \(0.574870\pi\)
\(368\) 0 0
\(369\) −4212.00 −0.594222
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10891.0 −1.51184 −0.755918 0.654667i \(-0.772809\pi\)
−0.755918 + 0.654667i \(0.772809\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −22.0000 −0.00300546
\(378\) 0 0
\(379\) 2591.00 0.351163 0.175581 0.984465i \(-0.443819\pi\)
0.175581 + 0.984465i \(0.443819\pi\)
\(380\) 0 0
\(381\) 1512.00 0.203313
\(382\) 0 0
\(383\) −612.000 −0.0816494 −0.0408247 0.999166i \(-0.512999\pi\)
−0.0408247 + 0.999166i \(0.512999\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3879.00 −0.509511
\(388\) 0 0
\(389\) 3708.00 0.483298 0.241649 0.970364i \(-0.422312\pi\)
0.241649 + 0.970364i \(0.422312\pi\)
\(390\) 0 0
\(391\) 3364.00 0.435102
\(392\) 0 0
\(393\) −216.000 −0.0277246
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3833.00 0.484566 0.242283 0.970206i \(-0.422104\pi\)
0.242283 + 0.970206i \(0.422104\pi\)
\(398\) 0 0
\(399\) −3021.00 −0.379046
\(400\) 0 0
\(401\) −9288.00 −1.15666 −0.578330 0.815803i \(-0.696295\pi\)
−0.578330 + 0.815803i \(0.696295\pi\)
\(402\) 0 0
\(403\) −35.0000 −0.00432624
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5940.00 −0.723427
\(408\) 0 0
\(409\) −755.000 −0.0912771 −0.0456386 0.998958i \(-0.514532\pi\)
−0.0456386 + 0.998958i \(0.514532\pi\)
\(410\) 0 0
\(411\) 5550.00 0.666086
\(412\) 0 0
\(413\) 8474.00 1.00963
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5508.00 −0.646830
\(418\) 0 0
\(419\) 9576.00 1.11651 0.558256 0.829669i \(-0.311471\pi\)
0.558256 + 0.829669i \(0.311471\pi\)
\(420\) 0 0
\(421\) 9414.00 1.08981 0.544905 0.838498i \(-0.316566\pi\)
0.544905 + 0.838498i \(0.316566\pi\)
\(422\) 0 0
\(423\) −2070.00 −0.237936
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2413.00 −0.273474
\(428\) 0 0
\(429\) 66.0000 0.00742776
\(430\) 0 0
\(431\) 2254.00 0.251906 0.125953 0.992036i \(-0.459801\pi\)
0.125953 + 0.992036i \(0.459801\pi\)
\(432\) 0 0
\(433\) 6301.00 0.699323 0.349661 0.936876i \(-0.386297\pi\)
0.349661 + 0.936876i \(0.386297\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3074.00 0.336497
\(438\) 0 0
\(439\) 3779.00 0.410847 0.205423 0.978673i \(-0.434143\pi\)
0.205423 + 0.978673i \(0.434143\pi\)
\(440\) 0 0
\(441\) 162.000 0.0174927
\(442\) 0 0
\(443\) 7632.00 0.818527 0.409263 0.912416i \(-0.365786\pi\)
0.409263 + 0.912416i \(0.365786\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −7194.00 −0.761218
\(448\) 0 0
\(449\) 2988.00 0.314059 0.157029 0.987594i \(-0.449808\pi\)
0.157029 + 0.987594i \(0.449808\pi\)
\(450\) 0 0
\(451\) 10296.0 1.07499
\(452\) 0 0
\(453\) −5613.00 −0.582167
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1370.00 0.140232 0.0701159 0.997539i \(-0.477663\pi\)
0.0701159 + 0.997539i \(0.477663\pi\)
\(458\) 0 0
\(459\) 1566.00 0.159248
\(460\) 0 0
\(461\) −7992.00 −0.807429 −0.403714 0.914885i \(-0.632281\pi\)
−0.403714 + 0.914885i \(0.632281\pi\)
\(462\) 0 0
\(463\) −3096.00 −0.310763 −0.155382 0.987855i \(-0.549661\pi\)
−0.155382 + 0.987855i \(0.549661\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8338.00 −0.826203 −0.413101 0.910685i \(-0.635554\pi\)
−0.413101 + 0.910685i \(0.635554\pi\)
\(468\) 0 0
\(469\) 15409.0 1.51710
\(470\) 0 0
\(471\) 9879.00 0.966455
\(472\) 0 0
\(473\) 9482.00 0.921740
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4918.00 −0.469121 −0.234561 0.972101i \(-0.575365\pi\)
−0.234561 + 0.972101i \(0.575365\pi\)
\(480\) 0 0
\(481\) −270.000 −0.0255945
\(482\) 0 0
\(483\) −3306.00 −0.311446
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −19637.0 −1.82718 −0.913591 0.406635i \(-0.866702\pi\)
−0.913591 + 0.406635i \(0.866702\pi\)
\(488\) 0 0
\(489\) 2649.00 0.244973
\(490\) 0 0
\(491\) −3096.00 −0.284563 −0.142282 0.989826i \(-0.545444\pi\)
−0.142282 + 0.989826i \(0.545444\pi\)
\(492\) 0 0
\(493\) 1276.00 0.116568
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 684.000 0.0617336
\(498\) 0 0
\(499\) −6875.00 −0.616768 −0.308384 0.951262i \(-0.599788\pi\)
−0.308384 + 0.951262i \(0.599788\pi\)
\(500\) 0 0
\(501\) 12312.0 1.09792
\(502\) 0 0
\(503\) 11268.0 0.998838 0.499419 0.866361i \(-0.333547\pi\)
0.499419 + 0.866361i \(0.333547\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6588.00 −0.577087
\(508\) 0 0
\(509\) 16078.0 1.40009 0.700044 0.714100i \(-0.253164\pi\)
0.700044 + 0.714100i \(0.253164\pi\)
\(510\) 0 0
\(511\) 9918.00 0.858604
\(512\) 0 0
\(513\) 1431.00 0.123158
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5060.00 0.430442
\(518\) 0 0
\(519\) −5118.00 −0.432862
\(520\) 0 0
\(521\) −17366.0 −1.46030 −0.730152 0.683285i \(-0.760551\pi\)
−0.730152 + 0.683285i \(0.760551\pi\)
\(522\) 0 0
\(523\) −4913.00 −0.410766 −0.205383 0.978682i \(-0.565844\pi\)
−0.205383 + 0.978682i \(0.565844\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2030.00 0.167795
\(528\) 0 0
\(529\) −8803.00 −0.723514
\(530\) 0 0
\(531\) −4014.00 −0.328047
\(532\) 0 0
\(533\) 468.000 0.0380325
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1986.00 −0.159594
\(538\) 0 0
\(539\) −396.000 −0.0316455
\(540\) 0 0
\(541\) 17605.0 1.39907 0.699536 0.714597i \(-0.253390\pi\)
0.699536 + 0.714597i \(0.253390\pi\)
\(542\) 0 0
\(543\) 12363.0 0.977067
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −14560.0 −1.13810 −0.569050 0.822303i \(-0.692689\pi\)
−0.569050 + 0.822303i \(0.692689\pi\)
\(548\) 0 0
\(549\) 1143.00 0.0888562
\(550\) 0 0
\(551\) 1166.00 0.0901511
\(552\) 0 0
\(553\) 25992.0 1.99872
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20614.0 1.56812 0.784060 0.620685i \(-0.213145\pi\)
0.784060 + 0.620685i \(0.213145\pi\)
\(558\) 0 0
\(559\) 431.000 0.0326107
\(560\) 0 0
\(561\) −3828.00 −0.288090
\(562\) 0 0
\(563\) 3442.00 0.257661 0.128830 0.991667i \(-0.458878\pi\)
0.128830 + 0.991667i \(0.458878\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1539.00 −0.113989
\(568\) 0 0
\(569\) 22082.0 1.62693 0.813467 0.581611i \(-0.197577\pi\)
0.813467 + 0.581611i \(0.197577\pi\)
\(570\) 0 0
\(571\) −451.000 −0.0330539 −0.0165269 0.999863i \(-0.505261\pi\)
−0.0165269 + 0.999863i \(0.505261\pi\)
\(572\) 0 0
\(573\) 2874.00 0.209534
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −20987.0 −1.51421 −0.757106 0.653292i \(-0.773387\pi\)
−0.757106 + 0.653292i \(0.773387\pi\)
\(578\) 0 0
\(579\) −9561.00 −0.686255
\(580\) 0 0
\(581\) 21622.0 1.54394
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3888.00 0.273381 0.136691 0.990614i \(-0.456353\pi\)
0.136691 + 0.990614i \(0.456353\pi\)
\(588\) 0 0
\(589\) 1855.00 0.129769
\(590\) 0 0
\(591\) 6846.00 0.476492
\(592\) 0 0
\(593\) 11268.0 0.780306 0.390153 0.920750i \(-0.372422\pi\)
0.390153 + 0.920750i \(0.372422\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3129.00 −0.214508
\(598\) 0 0
\(599\) 3996.00 0.272575 0.136287 0.990669i \(-0.456483\pi\)
0.136287 + 0.990669i \(0.456483\pi\)
\(600\) 0 0
\(601\) −24965.0 −1.69442 −0.847208 0.531262i \(-0.821718\pi\)
−0.847208 + 0.531262i \(0.821718\pi\)
\(602\) 0 0
\(603\) −7299.00 −0.492932
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4176.00 0.279240 0.139620 0.990205i \(-0.455412\pi\)
0.139620 + 0.990205i \(0.455412\pi\)
\(608\) 0 0
\(609\) −1254.00 −0.0834395
\(610\) 0 0
\(611\) 230.000 0.0152288
\(612\) 0 0
\(613\) 9558.00 0.629762 0.314881 0.949131i \(-0.398035\pi\)
0.314881 + 0.949131i \(0.398035\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9000.00 0.587239 0.293619 0.955922i \(-0.405140\pi\)
0.293619 + 0.955922i \(0.405140\pi\)
\(618\) 0 0
\(619\) 15625.0 1.01457 0.507287 0.861777i \(-0.330648\pi\)
0.507287 + 0.861777i \(0.330648\pi\)
\(620\) 0 0
\(621\) 1566.00 0.101194
\(622\) 0 0
\(623\) −2736.00 −0.175948
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3498.00 −0.222802
\(628\) 0 0
\(629\) 15660.0 0.992695
\(630\) 0 0
\(631\) 31175.0 1.96681 0.983405 0.181424i \(-0.0580705\pi\)
0.983405 + 0.181424i \(0.0580705\pi\)
\(632\) 0 0
\(633\) −12417.0 −0.779671
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −18.0000 −0.00111960
\(638\) 0 0
\(639\) −324.000 −0.0200583
\(640\) 0 0
\(641\) −6732.00 −0.414817 −0.207409 0.978254i \(-0.566503\pi\)
−0.207409 + 0.978254i \(0.566503\pi\)
\(642\) 0 0
\(643\) −6228.00 −0.381973 −0.190986 0.981593i \(-0.561169\pi\)
−0.190986 + 0.981593i \(0.561169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −396.000 −0.0240624 −0.0120312 0.999928i \(-0.503830\pi\)
−0.0120312 + 0.999928i \(0.503830\pi\)
\(648\) 0 0
\(649\) 9812.00 0.593459
\(650\) 0 0
\(651\) −1995.00 −0.120108
\(652\) 0 0
\(653\) 4702.00 0.281782 0.140891 0.990025i \(-0.455003\pi\)
0.140891 + 0.990025i \(0.455003\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4698.00 −0.278975
\(658\) 0 0
\(659\) 22140.0 1.30873 0.654364 0.756180i \(-0.272936\pi\)
0.654364 + 0.756180i \(0.272936\pi\)
\(660\) 0 0
\(661\) −13518.0 −0.795445 −0.397723 0.917506i \(-0.630199\pi\)
−0.397723 + 0.917506i \(0.630199\pi\)
\(662\) 0 0
\(663\) −174.000 −0.0101925
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1276.00 0.0740733
\(668\) 0 0
\(669\) 1239.00 0.0716032
\(670\) 0 0
\(671\) −2794.00 −0.160747
\(672\) 0 0
\(673\) 11250.0 0.644362 0.322181 0.946678i \(-0.395584\pi\)
0.322181 + 0.946678i \(0.395584\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15170.0 0.861197 0.430599 0.902544i \(-0.358302\pi\)
0.430599 + 0.902544i \(0.358302\pi\)
\(678\) 0 0
\(679\) −20501.0 −1.15870
\(680\) 0 0
\(681\) 16956.0 0.954119
\(682\) 0 0
\(683\) 22680.0 1.27061 0.635305 0.772262i \(-0.280875\pi\)
0.635305 + 0.772262i \(0.280875\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −13173.0 −0.731559
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −128.000 −0.00704682 −0.00352341 0.999994i \(-0.501122\pi\)
−0.00352341 + 0.999994i \(0.501122\pi\)
\(692\) 0 0
\(693\) 3762.00 0.206214
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −27144.0 −1.47511
\(698\) 0 0
\(699\) 6156.00 0.333106
\(700\) 0 0
\(701\) 25682.0 1.38373 0.691866 0.722026i \(-0.256789\pi\)
0.691866 + 0.722026i \(0.256789\pi\)
\(702\) 0 0
\(703\) 14310.0 0.767727
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27360.0 1.45542
\(708\) 0 0
\(709\) −4951.00 −0.262255 −0.131127 0.991366i \(-0.541860\pi\)
−0.131127 + 0.991366i \(0.541860\pi\)
\(710\) 0 0
\(711\) −12312.0 −0.649418
\(712\) 0 0
\(713\) 2030.00 0.106626
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12960.0 0.675035
\(718\) 0 0
\(719\) 22190.0 1.15097 0.575485 0.817812i \(-0.304813\pi\)
0.575485 + 0.817812i \(0.304813\pi\)
\(720\) 0 0
\(721\) −2356.00 −0.121695
\(722\) 0 0
\(723\) 12795.0 0.658162
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7685.00 0.392051 0.196025 0.980599i \(-0.437197\pi\)
0.196025 + 0.980599i \(0.437197\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −24998.0 −1.26482
\(732\) 0 0
\(733\) −29574.0 −1.49023 −0.745116 0.666934i \(-0.767606\pi\)
−0.745116 + 0.666934i \(0.767606\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17842.0 0.891748
\(738\) 0 0
\(739\) 32580.0 1.62175 0.810876 0.585218i \(-0.198991\pi\)
0.810876 + 0.585218i \(0.198991\pi\)
\(740\) 0 0
\(741\) −159.000 −0.00788261
\(742\) 0 0
\(743\) −3060.00 −0.151091 −0.0755454 0.997142i \(-0.524070\pi\)
−0.0755454 + 0.997142i \(0.524070\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −10242.0 −0.501654
\(748\) 0 0
\(749\) 8208.00 0.400419
\(750\) 0 0
\(751\) 7992.00 0.388325 0.194163 0.980969i \(-0.437801\pi\)
0.194163 + 0.980969i \(0.437801\pi\)
\(752\) 0 0
\(753\) 7236.00 0.350192
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −22841.0 −1.09666 −0.548329 0.836263i \(-0.684736\pi\)
−0.548329 + 0.836263i \(0.684736\pi\)
\(758\) 0 0
\(759\) −3828.00 −0.183067
\(760\) 0 0
\(761\) −17172.0 −0.817982 −0.408991 0.912538i \(-0.634119\pi\)
−0.408991 + 0.912538i \(0.634119\pi\)
\(762\) 0 0
\(763\) −13319.0 −0.631953
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 446.000 0.0209963
\(768\) 0 0
\(769\) −30869.0 −1.44755 −0.723774 0.690037i \(-0.757594\pi\)
−0.723774 + 0.690037i \(0.757594\pi\)
\(770\) 0 0
\(771\) −20844.0 −0.973642
\(772\) 0 0
\(773\) −34884.0 −1.62314 −0.811572 0.584252i \(-0.801388\pi\)
−0.811572 + 0.584252i \(0.801388\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −15390.0 −0.710570
\(778\) 0 0
\(779\) −24804.0 −1.14082
\(780\) 0 0
\(781\) 792.000 0.0362868
\(782\) 0 0
\(783\) 594.000 0.0271109
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5039.00 0.228235 0.114118 0.993467i \(-0.463596\pi\)
0.114118 + 0.993467i \(0.463596\pi\)
\(788\) 0 0
\(789\) −10692.0 −0.482440
\(790\) 0 0
\(791\) −19836.0 −0.891640
\(792\) 0 0
\(793\) −127.000 −0.00568714
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −540.000 −0.0239997 −0.0119999 0.999928i \(-0.503820\pi\)
−0.0119999 + 0.999928i \(0.503820\pi\)
\(798\) 0 0
\(799\) −13340.0 −0.590657
\(800\) 0 0
\(801\) 1296.00 0.0571684
\(802\) 0 0
\(803\) 11484.0 0.504684
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4602.00 −0.200741
\(808\) 0 0
\(809\) −41328.0 −1.79606 −0.898032 0.439931i \(-0.855003\pi\)
−0.898032 + 0.439931i \(0.855003\pi\)
\(810\) 0 0
\(811\) 12853.0 0.556510 0.278255 0.960507i \(-0.410244\pi\)
0.278255 + 0.960507i \(0.410244\pi\)
\(812\) 0 0
\(813\) −23112.0 −0.997015
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −22843.0 −0.978183
\(818\) 0 0
\(819\) 171.000 0.00729576
\(820\) 0 0
\(821\) 29470.0 1.25275 0.626376 0.779521i \(-0.284537\pi\)
0.626376 + 0.779521i \(0.284537\pi\)
\(822\) 0 0
\(823\) −24407.0 −1.03375 −0.516874 0.856062i \(-0.672904\pi\)
−0.516874 + 0.856062i \(0.672904\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15048.0 0.632733 0.316367 0.948637i \(-0.397537\pi\)
0.316367 + 0.948637i \(0.397537\pi\)
\(828\) 0 0
\(829\) −28406.0 −1.19009 −0.595043 0.803694i \(-0.702865\pi\)
−0.595043 + 0.803694i \(0.702865\pi\)
\(830\) 0 0
\(831\) −14631.0 −0.610763
\(832\) 0 0
\(833\) 1044.00 0.0434243
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 945.000 0.0390251
\(838\) 0 0
\(839\) 26914.0 1.10748 0.553739 0.832690i \(-0.313200\pi\)
0.553739 + 0.832690i \(0.313200\pi\)
\(840\) 0 0
\(841\) −23905.0 −0.980155
\(842\) 0 0
\(843\) −11274.0 −0.460614
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 16093.0 0.652848
\(848\) 0 0
\(849\) 2805.00 0.113389
\(850\) 0 0
\(851\) 15660.0 0.630808
\(852\) 0 0
\(853\) −21275.0 −0.853977 −0.426988 0.904257i \(-0.640425\pi\)
−0.426988 + 0.904257i \(0.640425\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39132.0 −1.55977 −0.779885 0.625922i \(-0.784723\pi\)
−0.779885 + 0.625922i \(0.784723\pi\)
\(858\) 0 0
\(859\) 448.000 0.0177946 0.00889730 0.999960i \(-0.497168\pi\)
0.00889730 + 0.999960i \(0.497168\pi\)
\(860\) 0 0
\(861\) 26676.0 1.05588
\(862\) 0 0
\(863\) −26856.0 −1.05932 −0.529658 0.848212i \(-0.677680\pi\)
−0.529658 + 0.848212i \(0.677680\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4647.00 −0.182030
\(868\) 0 0
\(869\) 30096.0 1.17484
\(870\) 0 0
\(871\) 811.000 0.0315496
\(872\) 0 0
\(873\) 9711.00 0.376481
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3653.00 0.140653 0.0703267 0.997524i \(-0.477596\pi\)
0.0703267 + 0.997524i \(0.477596\pi\)
\(878\) 0 0
\(879\) −18642.0 −0.715335
\(880\) 0 0
\(881\) −6552.00 −0.250559 −0.125280 0.992121i \(-0.539983\pi\)
−0.125280 + 0.992121i \(0.539983\pi\)
\(882\) 0 0
\(883\) 4481.00 0.170779 0.0853894 0.996348i \(-0.472787\pi\)
0.0853894 + 0.996348i \(0.472787\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −14666.0 −0.555170 −0.277585 0.960701i \(-0.589534\pi\)
−0.277585 + 0.960701i \(0.589534\pi\)
\(888\) 0 0
\(889\) −9576.00 −0.361270
\(890\) 0 0
\(891\) −1782.00 −0.0670025
\(892\) 0 0
\(893\) −12190.0 −0.456800
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −174.000 −0.00647680
\(898\) 0 0
\(899\) 770.000 0.0285661
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 24567.0 0.905358
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 51804.0 1.89650 0.948249 0.317528i \(-0.102853\pi\)
0.948249 + 0.317528i \(0.102853\pi\)
\(908\) 0 0
\(909\) −12960.0 −0.472889
\(910\) 0 0
\(911\) −31198.0 −1.13462 −0.567308 0.823505i \(-0.692015\pi\)
−0.567308 + 0.823505i \(0.692015\pi\)
\(912\) 0 0
\(913\) 25036.0 0.907525
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1368.00 0.0492643
\(918\) 0 0
\(919\) 27001.0 0.969185 0.484592 0.874740i \(-0.338968\pi\)
0.484592 + 0.874740i \(0.338968\pi\)
\(920\) 0 0
\(921\) −11715.0 −0.419134
\(922\) 0 0
\(923\) 36.0000 0.00128381
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1116.00 0.0395407
\(928\) 0 0
\(929\) −22694.0 −0.801470 −0.400735 0.916194i \(-0.631245\pi\)
−0.400735 + 0.916194i \(0.631245\pi\)
\(930\) 0 0
\(931\) 954.000 0.0335833
\(932\) 0 0
\(933\) −28986.0 −1.01711
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −29503.0 −1.02862 −0.514312 0.857603i \(-0.671953\pi\)
−0.514312 + 0.857603i \(0.671953\pi\)
\(938\) 0 0
\(939\) −15441.0 −0.536633
\(940\) 0 0
\(941\) 14566.0 0.504610 0.252305 0.967648i \(-0.418811\pi\)
0.252305 + 0.967648i \(0.418811\pi\)
\(942\) 0 0
\(943\) −27144.0 −0.937360
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42322.0 1.45225 0.726125 0.687563i \(-0.241319\pi\)
0.726125 + 0.687563i \(0.241319\pi\)
\(948\) 0 0
\(949\) 522.000 0.0178555
\(950\) 0 0
\(951\) 27648.0 0.942742
\(952\) 0 0
\(953\) −43416.0 −1.47574 −0.737871 0.674942i \(-0.764169\pi\)
−0.737871 + 0.674942i \(0.764169\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1452.00 −0.0490454
\(958\) 0 0
\(959\) −35150.0 −1.18358
\(960\) 0 0
\(961\) −28566.0 −0.958880
\(962\) 0 0
\(963\) −3888.00 −0.130103
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 21528.0 0.715919 0.357960 0.933737i \(-0.383473\pi\)
0.357960 + 0.933737i \(0.383473\pi\)
\(968\) 0 0
\(969\) 9222.00 0.305731
\(970\) 0 0
\(971\) −27050.0 −0.894002 −0.447001 0.894533i \(-0.647508\pi\)
−0.447001 + 0.894533i \(0.647508\pi\)
\(972\) 0 0
\(973\) 34884.0 1.14936
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24934.0 0.816489 0.408244 0.912873i \(-0.366141\pi\)
0.408244 + 0.912873i \(0.366141\pi\)
\(978\) 0 0
\(979\) −3168.00 −0.103422
\(980\) 0 0
\(981\) 6309.00 0.205332
\(982\) 0 0
\(983\) 8388.00 0.272162 0.136081 0.990698i \(-0.456549\pi\)
0.136081 + 0.990698i \(0.456549\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 13110.0 0.422792
\(988\) 0 0
\(989\) −24998.0 −0.803731
\(990\) 0 0
\(991\) −29033.0 −0.930639 −0.465320 0.885143i \(-0.654061\pi\)
−0.465320 + 0.885143i \(0.654061\pi\)
\(992\) 0 0
\(993\) −6588.00 −0.210538
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25326.0 0.804496 0.402248 0.915531i \(-0.368229\pi\)
0.402248 + 0.915531i \(0.368229\pi\)
\(998\) 0 0
\(999\) 7290.00 0.230876
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 1200.4.a.x.1.1 1
4.3 odd 2 600.4.a.g.1.1 1
5.2 odd 4 1200.4.f.f.49.1 2
5.3 odd 4 1200.4.f.f.49.2 2
5.4 even 2 1200.4.a.n.1.1 1
12.11 even 2 1800.4.a.bf.1.1 1
20.3 even 4 600.4.f.h.49.1 2
20.7 even 4 600.4.f.h.49.2 2
20.19 odd 2 600.4.a.j.1.1 yes 1
60.23 odd 4 1800.4.f.g.649.1 2
60.47 odd 4 1800.4.f.g.649.2 2
60.59 even 2 1800.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.g.1.1 1 4.3 odd 2
600.4.a.j.1.1 yes 1 20.19 odd 2
600.4.f.h.49.1 2 20.3 even 4
600.4.f.h.49.2 2 20.7 even 4
1200.4.a.n.1.1 1 5.4 even 2
1200.4.a.x.1.1 1 1.1 even 1 trivial
1200.4.f.f.49.1 2 5.2 odd 4
1200.4.f.f.49.2 2 5.3 odd 4
1800.4.a.g.1.1 1 60.59 even 2
1800.4.a.bf.1.1 1 12.11 even 2
1800.4.f.g.649.1 2 60.23 odd 4
1800.4.f.g.649.2 2 60.47 odd 4