Properties

Label 1200.4.a.i.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 150)
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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -1.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -1.00000 q^{7} +9.00000 q^{9} -42.0000 q^{11} +67.0000 q^{13} -54.0000 q^{17} +115.000 q^{19} +3.00000 q^{21} -162.000 q^{23} -27.0000 q^{27} -210.000 q^{29} +193.000 q^{31} +126.000 q^{33} +286.000 q^{37} -201.000 q^{39} +12.0000 q^{41} +263.000 q^{43} +414.000 q^{47} -342.000 q^{49} +162.000 q^{51} +192.000 q^{53} -345.000 q^{57} -690.000 q^{59} -733.000 q^{61} -9.00000 q^{63} +299.000 q^{67} +486.000 q^{69} +228.000 q^{71} -938.000 q^{73} +42.0000 q^{77} +160.000 q^{79} +81.0000 q^{81} -462.000 q^{83} +630.000 q^{87} -240.000 q^{89} -67.0000 q^{91} -579.000 q^{93} +511.000 q^{97} -378.000 q^{99} +O(q^{100})\)

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
\(7\) −1.00000 −0.0539949 −0.0269975 0.999636i \(-0.508595\pi\)
−0.0269975 + 0.999636i \(0.508595\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −42.0000 −1.15123 −0.575613 0.817723i \(-0.695236\pi\)
−0.575613 + 0.817723i \(0.695236\pi\)
\(12\) 0 0
\(13\) 67.0000 1.42942 0.714710 0.699421i \(-0.246559\pi\)
0.714710 + 0.699421i \(0.246559\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) 0 0
\(19\) 115.000 1.38857 0.694284 0.719701i \(-0.255721\pi\)
0.694284 + 0.719701i \(0.255721\pi\)
\(20\) 0 0
\(21\) 3.00000 0.0311740
\(22\) 0 0
\(23\) −162.000 −1.46867 −0.734333 0.678789i \(-0.762505\pi\)
−0.734333 + 0.678789i \(0.762505\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −210.000 −1.34469 −0.672345 0.740238i \(-0.734713\pi\)
−0.672345 + 0.740238i \(0.734713\pi\)
\(30\) 0 0
\(31\) 193.000 1.11819 0.559094 0.829104i \(-0.311149\pi\)
0.559094 + 0.829104i \(0.311149\pi\)
\(32\) 0 0
\(33\) 126.000 0.664660
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 286.000 1.27076 0.635380 0.772200i \(-0.280844\pi\)
0.635380 + 0.772200i \(0.280844\pi\)
\(38\) 0 0
\(39\) −201.000 −0.825276
\(40\) 0 0
\(41\) 12.0000 0.0457094 0.0228547 0.999739i \(-0.492724\pi\)
0.0228547 + 0.999739i \(0.492724\pi\)
\(42\) 0 0
\(43\) 263.000 0.932724 0.466362 0.884594i \(-0.345564\pi\)
0.466362 + 0.884594i \(0.345564\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 414.000 1.28485 0.642427 0.766347i \(-0.277928\pi\)
0.642427 + 0.766347i \(0.277928\pi\)
\(48\) 0 0
\(49\) −342.000 −0.997085
\(50\) 0 0
\(51\) 162.000 0.444795
\(52\) 0 0
\(53\) 192.000 0.497608 0.248804 0.968554i \(-0.419962\pi\)
0.248804 + 0.968554i \(0.419962\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −345.000 −0.801691
\(58\) 0 0
\(59\) −690.000 −1.52255 −0.761274 0.648430i \(-0.775426\pi\)
−0.761274 + 0.648430i \(0.775426\pi\)
\(60\) 0 0
\(61\) −733.000 −1.53854 −0.769271 0.638923i \(-0.779380\pi\)
−0.769271 + 0.638923i \(0.779380\pi\)
\(62\) 0 0
\(63\) −9.00000 −0.0179983
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 299.000 0.545204 0.272602 0.962127i \(-0.412116\pi\)
0.272602 + 0.962127i \(0.412116\pi\)
\(68\) 0 0
\(69\) 486.000 0.847935
\(70\) 0 0
\(71\) 228.000 0.381107 0.190554 0.981677i \(-0.438972\pi\)
0.190554 + 0.981677i \(0.438972\pi\)
\(72\) 0 0
\(73\) −938.000 −1.50390 −0.751949 0.659221i \(-0.770886\pi\)
−0.751949 + 0.659221i \(0.770886\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 42.0000 0.0621603
\(78\) 0 0
\(79\) 160.000 0.227866 0.113933 0.993488i \(-0.463655\pi\)
0.113933 + 0.993488i \(0.463655\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −462.000 −0.610977 −0.305488 0.952196i \(-0.598820\pi\)
−0.305488 + 0.952196i \(0.598820\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 630.000 0.776357
\(88\) 0 0
\(89\) −240.000 −0.285842 −0.142921 0.989734i \(-0.545650\pi\)
−0.142921 + 0.989734i \(0.545650\pi\)
\(90\) 0 0
\(91\) −67.0000 −0.0771814
\(92\) 0 0
\(93\) −579.000 −0.645586
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 511.000 0.534889 0.267444 0.963573i \(-0.413821\pi\)
0.267444 + 0.963573i \(0.413821\pi\)
\(98\) 0 0
\(99\) −378.000 −0.383742
\(100\) 0 0
\(101\) 912.000 0.898489 0.449245 0.893409i \(-0.351693\pi\)
0.449245 + 0.893409i \(0.351693\pi\)
\(102\) 0 0
\(103\) 668.000 0.639029 0.319515 0.947581i \(-0.396480\pi\)
0.319515 + 0.947581i \(0.396480\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1296.00 −1.17093 −0.585463 0.810699i \(-0.699087\pi\)
−0.585463 + 0.810699i \(0.699087\pi\)
\(108\) 0 0
\(109\) −1735.00 −1.52461 −0.762307 0.647216i \(-0.775933\pi\)
−0.762307 + 0.647216i \(0.775933\pi\)
\(110\) 0 0
\(111\) −858.000 −0.733673
\(112\) 0 0
\(113\) 1092.00 0.909086 0.454543 0.890725i \(-0.349803\pi\)
0.454543 + 0.890725i \(0.349803\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 603.000 0.476473
\(118\) 0 0
\(119\) 54.0000 0.0415981
\(120\) 0 0
\(121\) 433.000 0.325319
\(122\) 0 0
\(123\) −36.0000 −0.0263903
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −0.0111793 −0.00558965 0.999984i \(-0.501779\pi\)
−0.00558965 + 0.999984i \(0.501779\pi\)
\(128\) 0 0
\(129\) −789.000 −0.538508
\(130\) 0 0
\(131\) −1992.00 −1.32856 −0.664282 0.747482i \(-0.731263\pi\)
−0.664282 + 0.747482i \(0.731263\pi\)
\(132\) 0 0
\(133\) −115.000 −0.0749757
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2346.00 1.46301 0.731505 0.681836i \(-0.238818\pi\)
0.731505 + 0.681836i \(0.238818\pi\)
\(138\) 0 0
\(139\) −2900.00 −1.76960 −0.884801 0.465968i \(-0.845706\pi\)
−0.884801 + 0.465968i \(0.845706\pi\)
\(140\) 0 0
\(141\) −1242.00 −0.741810
\(142\) 0 0
\(143\) −2814.00 −1.64558
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1026.00 0.575667
\(148\) 0 0
\(149\) −2070.00 −1.13813 −0.569064 0.822293i \(-0.692694\pi\)
−0.569064 + 0.822293i \(0.692694\pi\)
\(150\) 0 0
\(151\) −2237.00 −1.20559 −0.602796 0.797895i \(-0.705947\pi\)
−0.602796 + 0.797895i \(0.705947\pi\)
\(152\) 0 0
\(153\) −486.000 −0.256802
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 241.000 0.122509 0.0612544 0.998122i \(-0.480490\pi\)
0.0612544 + 0.998122i \(0.480490\pi\)
\(158\) 0 0
\(159\) −576.000 −0.287294
\(160\) 0 0
\(161\) 162.000 0.0793006
\(162\) 0 0
\(163\) −3547.00 −1.70443 −0.852216 0.523190i \(-0.824742\pi\)
−0.852216 + 0.523190i \(0.824742\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 984.000 0.455953 0.227977 0.973667i \(-0.426789\pi\)
0.227977 + 0.973667i \(0.426789\pi\)
\(168\) 0 0
\(169\) 2292.00 1.04324
\(170\) 0 0
\(171\) 1035.00 0.462856
\(172\) 0 0
\(173\) −3618.00 −1.59001 −0.795004 0.606604i \(-0.792531\pi\)
−0.795004 + 0.606604i \(0.792531\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2070.00 0.879044
\(178\) 0 0
\(179\) 150.000 0.0626342 0.0313171 0.999509i \(-0.490030\pi\)
0.0313171 + 0.999509i \(0.490030\pi\)
\(180\) 0 0
\(181\) 197.000 0.0809000 0.0404500 0.999182i \(-0.487121\pi\)
0.0404500 + 0.999182i \(0.487121\pi\)
\(182\) 0 0
\(183\) 2199.00 0.888277
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2268.00 0.886912
\(188\) 0 0
\(189\) 27.0000 0.0103913
\(190\) 0 0
\(191\) −1302.00 −0.493243 −0.246622 0.969112i \(-0.579320\pi\)
−0.246622 + 0.969112i \(0.579320\pi\)
\(192\) 0 0
\(193\) −4163.00 −1.55264 −0.776319 0.630340i \(-0.782916\pi\)
−0.776319 + 0.630340i \(0.782916\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3054.00 −1.10451 −0.552255 0.833675i \(-0.686233\pi\)
−0.552255 + 0.833675i \(0.686233\pi\)
\(198\) 0 0
\(199\) −3425.00 −1.22006 −0.610030 0.792379i \(-0.708842\pi\)
−0.610030 + 0.792379i \(0.708842\pi\)
\(200\) 0 0
\(201\) −897.000 −0.314774
\(202\) 0 0
\(203\) 210.000 0.0726065
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1458.00 −0.489556
\(208\) 0 0
\(209\) −4830.00 −1.59856
\(210\) 0 0
\(211\) 2443.00 0.797076 0.398538 0.917152i \(-0.369518\pi\)
0.398538 + 0.917152i \(0.369518\pi\)
\(212\) 0 0
\(213\) −684.000 −0.220032
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −193.000 −0.0603765
\(218\) 0 0
\(219\) 2814.00 0.868276
\(220\) 0 0
\(221\) −3618.00 −1.10124
\(222\) 0 0
\(223\) 23.0000 0.00690670 0.00345335 0.999994i \(-0.498901\pi\)
0.00345335 + 0.999994i \(0.498901\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1956.00 −0.571913 −0.285957 0.958243i \(-0.592311\pi\)
−0.285957 + 0.958243i \(0.592311\pi\)
\(228\) 0 0
\(229\) 1805.00 0.520864 0.260432 0.965492i \(-0.416135\pi\)
0.260432 + 0.965492i \(0.416135\pi\)
\(230\) 0 0
\(231\) −126.000 −0.0358883
\(232\) 0 0
\(233\) −3468.00 −0.975091 −0.487546 0.873098i \(-0.662108\pi\)
−0.487546 + 0.873098i \(0.662108\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −480.000 −0.131558
\(238\) 0 0
\(239\) −2640.00 −0.714508 −0.357254 0.934007i \(-0.616287\pi\)
−0.357254 + 0.934007i \(0.616287\pi\)
\(240\) 0 0
\(241\) −5383.00 −1.43879 −0.719397 0.694599i \(-0.755582\pi\)
−0.719397 + 0.694599i \(0.755582\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7705.00 1.98485
\(248\) 0 0
\(249\) 1386.00 0.352748
\(250\) 0 0
\(251\) 5028.00 1.26440 0.632200 0.774805i \(-0.282152\pi\)
0.632200 + 0.774805i \(0.282152\pi\)
\(252\) 0 0
\(253\) 6804.00 1.69077
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −564.000 −0.136892 −0.0684462 0.997655i \(-0.521804\pi\)
−0.0684462 + 0.997655i \(0.521804\pi\)
\(258\) 0 0
\(259\) −286.000 −0.0686146
\(260\) 0 0
\(261\) −1890.00 −0.448230
\(262\) 0 0
\(263\) −1812.00 −0.424839 −0.212420 0.977179i \(-0.568134\pi\)
−0.212420 + 0.977179i \(0.568134\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 720.000 0.165031
\(268\) 0 0
\(269\) −5190.00 −1.17636 −0.588178 0.808731i \(-0.700155\pi\)
−0.588178 + 0.808731i \(0.700155\pi\)
\(270\) 0 0
\(271\) −4592.00 −1.02931 −0.514657 0.857396i \(-0.672081\pi\)
−0.514657 + 0.857396i \(0.672081\pi\)
\(272\) 0 0
\(273\) 201.000 0.0445607
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2191.00 0.475251 0.237625 0.971357i \(-0.423631\pi\)
0.237625 + 0.971357i \(0.423631\pi\)
\(278\) 0 0
\(279\) 1737.00 0.372729
\(280\) 0 0
\(281\) 7842.00 1.66482 0.832410 0.554160i \(-0.186960\pi\)
0.832410 + 0.554160i \(0.186960\pi\)
\(282\) 0 0
\(283\) −247.000 −0.0518821 −0.0259410 0.999663i \(-0.508258\pi\)
−0.0259410 + 0.999663i \(0.508258\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.00246808
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) 0 0
\(291\) −1533.00 −0.308818
\(292\) 0 0
\(293\) 5442.00 1.08507 0.542534 0.840034i \(-0.317465\pi\)
0.542534 + 0.840034i \(0.317465\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1134.00 0.221553
\(298\) 0 0
\(299\) −10854.0 −2.09934
\(300\) 0 0
\(301\) −263.000 −0.0503624
\(302\) 0 0
\(303\) −2736.00 −0.518743
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3871.00 −0.719641 −0.359820 0.933022i \(-0.617162\pi\)
−0.359820 + 0.933022i \(0.617162\pi\)
\(308\) 0 0
\(309\) −2004.00 −0.368944
\(310\) 0 0
\(311\) 5718.00 1.04257 0.521283 0.853384i \(-0.325454\pi\)
0.521283 + 0.853384i \(0.325454\pi\)
\(312\) 0 0
\(313\) 3637.00 0.656790 0.328395 0.944540i \(-0.393492\pi\)
0.328395 + 0.944540i \(0.393492\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1296.00 0.229623 0.114812 0.993387i \(-0.463374\pi\)
0.114812 + 0.993387i \(0.463374\pi\)
\(318\) 0 0
\(319\) 8820.00 1.54804
\(320\) 0 0
\(321\) 3888.00 0.676034
\(322\) 0 0
\(323\) −6210.00 −1.06976
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5205.00 0.880236
\(328\) 0 0
\(329\) −414.000 −0.0693756
\(330\) 0 0
\(331\) −5132.00 −0.852206 −0.426103 0.904675i \(-0.640114\pi\)
−0.426103 + 0.904675i \(0.640114\pi\)
\(332\) 0 0
\(333\) 2574.00 0.423587
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6751.00 1.09125 0.545624 0.838030i \(-0.316293\pi\)
0.545624 + 0.838030i \(0.316293\pi\)
\(338\) 0 0
\(339\) −3276.00 −0.524861
\(340\) 0 0
\(341\) −8106.00 −1.28729
\(342\) 0 0
\(343\) 685.000 0.107832
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5226.00 −0.808491 −0.404246 0.914651i \(-0.632466\pi\)
−0.404246 + 0.914651i \(0.632466\pi\)
\(348\) 0 0
\(349\) −6190.00 −0.949407 −0.474704 0.880146i \(-0.657445\pi\)
−0.474704 + 0.880146i \(0.657445\pi\)
\(350\) 0 0
\(351\) −1809.00 −0.275092
\(352\) 0 0
\(353\) −6618.00 −0.997849 −0.498924 0.866646i \(-0.666271\pi\)
−0.498924 + 0.866646i \(0.666271\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −162.000 −0.0240167
\(358\) 0 0
\(359\) 3420.00 0.502787 0.251394 0.967885i \(-0.419111\pi\)
0.251394 + 0.967885i \(0.419111\pi\)
\(360\) 0 0
\(361\) 6366.00 0.928124
\(362\) 0 0
\(363\) −1299.00 −0.187823
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −871.000 −0.123885 −0.0619425 0.998080i \(-0.519730\pi\)
−0.0619425 + 0.998080i \(0.519730\pi\)
\(368\) 0 0
\(369\) 108.000 0.0152365
\(370\) 0 0
\(371\) −192.000 −0.0268683
\(372\) 0 0
\(373\) −6383.00 −0.886057 −0.443028 0.896508i \(-0.646096\pi\)
−0.443028 + 0.896508i \(0.646096\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −14070.0 −1.92213
\(378\) 0 0
\(379\) 9865.00 1.33702 0.668511 0.743703i \(-0.266932\pi\)
0.668511 + 0.743703i \(0.266932\pi\)
\(380\) 0 0
\(381\) 48.0000 0.00645437
\(382\) 0 0
\(383\) 9828.00 1.31119 0.655597 0.755111i \(-0.272417\pi\)
0.655597 + 0.755111i \(0.272417\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2367.00 0.310908
\(388\) 0 0
\(389\) 12540.0 1.63446 0.817228 0.576315i \(-0.195510\pi\)
0.817228 + 0.576315i \(0.195510\pi\)
\(390\) 0 0
\(391\) 8748.00 1.13147
\(392\) 0 0
\(393\) 5976.00 0.767047
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1381.00 0.174585 0.0872927 0.996183i \(-0.472178\pi\)
0.0872927 + 0.996183i \(0.472178\pi\)
\(398\) 0 0
\(399\) 345.000 0.0432872
\(400\) 0 0
\(401\) 14232.0 1.77235 0.886175 0.463351i \(-0.153353\pi\)
0.886175 + 0.463351i \(0.153353\pi\)
\(402\) 0 0
\(403\) 12931.0 1.59836
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12012.0 −1.46293
\(408\) 0 0
\(409\) 2645.00 0.319772 0.159886 0.987135i \(-0.448887\pi\)
0.159886 + 0.987135i \(0.448887\pi\)
\(410\) 0 0
\(411\) −7038.00 −0.844669
\(412\) 0 0
\(413\) 690.000 0.0822099
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8700.00 1.02168
\(418\) 0 0
\(419\) −3000.00 −0.349784 −0.174892 0.984588i \(-0.555958\pi\)
−0.174892 + 0.984588i \(0.555958\pi\)
\(420\) 0 0
\(421\) −11338.0 −1.31254 −0.656271 0.754525i \(-0.727867\pi\)
−0.656271 + 0.754525i \(0.727867\pi\)
\(422\) 0 0
\(423\) 3726.00 0.428284
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 733.000 0.0830734
\(428\) 0 0
\(429\) 8442.00 0.950078
\(430\) 0 0
\(431\) 3258.00 0.364112 0.182056 0.983288i \(-0.441725\pi\)
0.182056 + 0.983288i \(0.441725\pi\)
\(432\) 0 0
\(433\) −1163.00 −0.129077 −0.0645384 0.997915i \(-0.520557\pi\)
−0.0645384 + 0.997915i \(0.520557\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −18630.0 −2.03934
\(438\) 0 0
\(439\) −6695.00 −0.727870 −0.363935 0.931424i \(-0.618567\pi\)
−0.363935 + 0.931424i \(0.618567\pi\)
\(440\) 0 0
\(441\) −3078.00 −0.332362
\(442\) 0 0
\(443\) 16368.0 1.75546 0.877728 0.479159i \(-0.159058\pi\)
0.877728 + 0.479159i \(0.159058\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6210.00 0.657098
\(448\) 0 0
\(449\) 16380.0 1.72165 0.860824 0.508903i \(-0.169949\pi\)
0.860824 + 0.508903i \(0.169949\pi\)
\(450\) 0 0
\(451\) −504.000 −0.0526218
\(452\) 0 0
\(453\) 6711.00 0.696049
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13786.0 1.41112 0.705560 0.708650i \(-0.250696\pi\)
0.705560 + 0.708650i \(0.250696\pi\)
\(458\) 0 0
\(459\) 1458.00 0.148265
\(460\) 0 0
\(461\) 11832.0 1.19538 0.597691 0.801726i \(-0.296085\pi\)
0.597691 + 0.801726i \(0.296085\pi\)
\(462\) 0 0
\(463\) 3008.00 0.301930 0.150965 0.988539i \(-0.451762\pi\)
0.150965 + 0.988539i \(0.451762\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4434.00 0.439360 0.219680 0.975572i \(-0.429499\pi\)
0.219680 + 0.975572i \(0.429499\pi\)
\(468\) 0 0
\(469\) −299.000 −0.0294382
\(470\) 0 0
\(471\) −723.000 −0.0707305
\(472\) 0 0
\(473\) −11046.0 −1.07378
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1728.00 0.165869
\(478\) 0 0
\(479\) −7410.00 −0.706830 −0.353415 0.935467i \(-0.614980\pi\)
−0.353415 + 0.935467i \(0.614980\pi\)
\(480\) 0 0
\(481\) 19162.0 1.81645
\(482\) 0 0
\(483\) −486.000 −0.0457842
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −8671.00 −0.806818 −0.403409 0.915020i \(-0.632175\pi\)
−0.403409 + 0.915020i \(0.632175\pi\)
\(488\) 0 0
\(489\) 10641.0 0.984055
\(490\) 0 0
\(491\) 19368.0 1.78017 0.890087 0.455790i \(-0.150643\pi\)
0.890087 + 0.455790i \(0.150643\pi\)
\(492\) 0 0
\(493\) 11340.0 1.03596
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −228.000 −0.0205779
\(498\) 0 0
\(499\) 8875.00 0.796192 0.398096 0.917344i \(-0.369671\pi\)
0.398096 + 0.917344i \(0.369671\pi\)
\(500\) 0 0
\(501\) −2952.00 −0.263245
\(502\) 0 0
\(503\) −10452.0 −0.926504 −0.463252 0.886227i \(-0.653318\pi\)
−0.463252 + 0.886227i \(0.653318\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6876.00 −0.602315
\(508\) 0 0
\(509\) −19770.0 −1.72159 −0.860796 0.508951i \(-0.830033\pi\)
−0.860796 + 0.508951i \(0.830033\pi\)
\(510\) 0 0
\(511\) 938.000 0.0812029
\(512\) 0 0
\(513\) −3105.00 −0.267230
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −17388.0 −1.47916
\(518\) 0 0
\(519\) 10854.0 0.917992
\(520\) 0 0
\(521\) −11238.0 −0.945001 −0.472501 0.881330i \(-0.656649\pi\)
−0.472501 + 0.881330i \(0.656649\pi\)
\(522\) 0 0
\(523\) −7447.00 −0.622628 −0.311314 0.950307i \(-0.600769\pi\)
−0.311314 + 0.950307i \(0.600769\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10422.0 −0.861460
\(528\) 0 0
\(529\) 14077.0 1.15698
\(530\) 0 0
\(531\) −6210.00 −0.507516
\(532\) 0 0
\(533\) 804.000 0.0653379
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −450.000 −0.0361619
\(538\) 0 0
\(539\) 14364.0 1.14787
\(540\) 0 0
\(541\) −17623.0 −1.40050 −0.700251 0.713896i \(-0.746929\pi\)
−0.700251 + 0.713896i \(0.746929\pi\)
\(542\) 0 0
\(543\) −591.000 −0.0467076
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −10096.0 −0.789166 −0.394583 0.918860i \(-0.629111\pi\)
−0.394583 + 0.918860i \(0.629111\pi\)
\(548\) 0 0
\(549\) −6597.00 −0.512847
\(550\) 0 0
\(551\) −24150.0 −1.86720
\(552\) 0 0
\(553\) −160.000 −0.0123036
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14514.0 −1.10409 −0.552045 0.833814i \(-0.686152\pi\)
−0.552045 + 0.833814i \(0.686152\pi\)
\(558\) 0 0
\(559\) 17621.0 1.33325
\(560\) 0 0
\(561\) −6804.00 −0.512059
\(562\) 0 0
\(563\) −10242.0 −0.766694 −0.383347 0.923604i \(-0.625229\pi\)
−0.383347 + 0.923604i \(0.625229\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −81.0000 −0.00599944
\(568\) 0 0
\(569\) −6750.00 −0.497319 −0.248660 0.968591i \(-0.579990\pi\)
−0.248660 + 0.968591i \(0.579990\pi\)
\(570\) 0 0
\(571\) −17117.0 −1.25451 −0.627254 0.778815i \(-0.715821\pi\)
−0.627254 + 0.778815i \(0.715821\pi\)
\(572\) 0 0
\(573\) 3906.00 0.284774
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 301.000 0.0217171 0.0108586 0.999941i \(-0.496544\pi\)
0.0108586 + 0.999941i \(0.496544\pi\)
\(578\) 0 0
\(579\) 12489.0 0.896416
\(580\) 0 0
\(581\) 462.000 0.0329897
\(582\) 0 0
\(583\) −8064.00 −0.572859
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15456.0 −1.08678 −0.543388 0.839482i \(-0.682859\pi\)
−0.543388 + 0.839482i \(0.682859\pi\)
\(588\) 0 0
\(589\) 22195.0 1.55268
\(590\) 0 0
\(591\) 9162.00 0.637689
\(592\) 0 0
\(593\) 9492.00 0.657318 0.328659 0.944449i \(-0.393403\pi\)
0.328659 + 0.944449i \(0.393403\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10275.0 0.704402
\(598\) 0 0
\(599\) −1500.00 −0.102318 −0.0511589 0.998691i \(-0.516291\pi\)
−0.0511589 + 0.998691i \(0.516291\pi\)
\(600\) 0 0
\(601\) 14627.0 0.992758 0.496379 0.868106i \(-0.334663\pi\)
0.496379 + 0.868106i \(0.334663\pi\)
\(602\) 0 0
\(603\) 2691.00 0.181735
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16184.0 1.08219 0.541094 0.840962i \(-0.318010\pi\)
0.541094 + 0.840962i \(0.318010\pi\)
\(608\) 0 0
\(609\) −630.000 −0.0419194
\(610\) 0 0
\(611\) 27738.0 1.83659
\(612\) 0 0
\(613\) 18502.0 1.21907 0.609534 0.792760i \(-0.291357\pi\)
0.609534 + 0.792760i \(0.291357\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13896.0 0.906697 0.453348 0.891333i \(-0.350229\pi\)
0.453348 + 0.891333i \(0.350229\pi\)
\(618\) 0 0
\(619\) 9895.00 0.642510 0.321255 0.946993i \(-0.395895\pi\)
0.321255 + 0.946993i \(0.395895\pi\)
\(620\) 0 0
\(621\) 4374.00 0.282645
\(622\) 0 0
\(623\) 240.000 0.0154340
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 14490.0 0.922926
\(628\) 0 0
\(629\) −15444.0 −0.979003
\(630\) 0 0
\(631\) −467.000 −0.0294627 −0.0147314 0.999891i \(-0.504689\pi\)
−0.0147314 + 0.999891i \(0.504689\pi\)
\(632\) 0 0
\(633\) −7329.00 −0.460192
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −22914.0 −1.42525
\(638\) 0 0
\(639\) 2052.00 0.127036
\(640\) 0 0
\(641\) 30612.0 1.88627 0.943137 0.332405i \(-0.107860\pi\)
0.943137 + 0.332405i \(0.107860\pi\)
\(642\) 0 0
\(643\) −1852.00 −0.113586 −0.0567930 0.998386i \(-0.518087\pi\)
−0.0567930 + 0.998386i \(0.518087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21156.0 −1.28551 −0.642757 0.766070i \(-0.722210\pi\)
−0.642757 + 0.766070i \(0.722210\pi\)
\(648\) 0 0
\(649\) 28980.0 1.75280
\(650\) 0 0
\(651\) 579.000 0.0348584
\(652\) 0 0
\(653\) 9702.00 0.581422 0.290711 0.956811i \(-0.406108\pi\)
0.290711 + 0.956811i \(0.406108\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −8442.00 −0.501300
\(658\) 0 0
\(659\) −1980.00 −0.117041 −0.0585204 0.998286i \(-0.518638\pi\)
−0.0585204 + 0.998286i \(0.518638\pi\)
\(660\) 0 0
\(661\) −20158.0 −1.18617 −0.593083 0.805142i \(-0.702089\pi\)
−0.593083 + 0.805142i \(0.702089\pi\)
\(662\) 0 0
\(663\) 10854.0 0.635799
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 34020.0 1.97490
\(668\) 0 0
\(669\) −69.0000 −0.00398758
\(670\) 0 0
\(671\) 30786.0 1.77121
\(672\) 0 0
\(673\) 16882.0 0.966944 0.483472 0.875360i \(-0.339376\pi\)
0.483472 + 0.875360i \(0.339376\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20934.0 −1.18842 −0.594209 0.804311i \(-0.702535\pi\)
−0.594209 + 0.804311i \(0.702535\pi\)
\(678\) 0 0
\(679\) −511.000 −0.0288813
\(680\) 0 0
\(681\) 5868.00 0.330194
\(682\) 0 0
\(683\) −8712.00 −0.488075 −0.244038 0.969766i \(-0.578472\pi\)
−0.244038 + 0.969766i \(0.578472\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5415.00 −0.300721
\(688\) 0 0
\(689\) 12864.0 0.711291
\(690\) 0 0
\(691\) 14128.0 0.777792 0.388896 0.921282i \(-0.372856\pi\)
0.388896 + 0.921282i \(0.372856\pi\)
\(692\) 0 0
\(693\) 378.000 0.0207201
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −648.000 −0.0352148
\(698\) 0 0
\(699\) 10404.0 0.562969
\(700\) 0 0
\(701\) −28278.0 −1.52360 −0.761801 0.647811i \(-0.775685\pi\)
−0.761801 + 0.647811i \(0.775685\pi\)
\(702\) 0 0
\(703\) 32890.0 1.76454
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −912.000 −0.0485138
\(708\) 0 0
\(709\) 8885.00 0.470639 0.235320 0.971918i \(-0.424386\pi\)
0.235320 + 0.971918i \(0.424386\pi\)
\(710\) 0 0
\(711\) 1440.00 0.0759553
\(712\) 0 0
\(713\) −31266.0 −1.64225
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7920.00 0.412521
\(718\) 0 0
\(719\) 7530.00 0.390572 0.195286 0.980746i \(-0.437436\pi\)
0.195286 + 0.980746i \(0.437436\pi\)
\(720\) 0 0
\(721\) −668.000 −0.0345043
\(722\) 0 0
\(723\) 16149.0 0.830688
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1801.00 −0.0918781 −0.0459391 0.998944i \(-0.514628\pi\)
−0.0459391 + 0.998944i \(0.514628\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −14202.0 −0.718577
\(732\) 0 0
\(733\) 7882.00 0.397174 0.198587 0.980083i \(-0.436365\pi\)
0.198587 + 0.980083i \(0.436365\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12558.0 −0.627652
\(738\) 0 0
\(739\) −33860.0 −1.68547 −0.842734 0.538331i \(-0.819055\pi\)
−0.842734 + 0.538331i \(0.819055\pi\)
\(740\) 0 0
\(741\) −23115.0 −1.14595
\(742\) 0 0
\(743\) −20652.0 −1.01972 −0.509858 0.860259i \(-0.670302\pi\)
−0.509858 + 0.860259i \(0.670302\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −4158.00 −0.203659
\(748\) 0 0
\(749\) 1296.00 0.0632240
\(750\) 0 0
\(751\) −7472.00 −0.363059 −0.181529 0.983386i \(-0.558105\pi\)
−0.181529 + 0.983386i \(0.558105\pi\)
\(752\) 0 0
\(753\) −15084.0 −0.730002
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 32251.0 1.54846 0.774229 0.632906i \(-0.218138\pi\)
0.774229 + 0.632906i \(0.218138\pi\)
\(758\) 0 0
\(759\) −20412.0 −0.976164
\(760\) 0 0
\(761\) 16812.0 0.800834 0.400417 0.916333i \(-0.368865\pi\)
0.400417 + 0.916333i \(0.368865\pi\)
\(762\) 0 0
\(763\) 1735.00 0.0823214
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −46230.0 −2.17636
\(768\) 0 0
\(769\) −34645.0 −1.62462 −0.812309 0.583228i \(-0.801789\pi\)
−0.812309 + 0.583228i \(0.801789\pi\)
\(770\) 0 0
\(771\) 1692.00 0.0790349
\(772\) 0 0
\(773\) 8412.00 0.391408 0.195704 0.980663i \(-0.437301\pi\)
0.195704 + 0.980663i \(0.437301\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 858.000 0.0396146
\(778\) 0 0
\(779\) 1380.00 0.0634706
\(780\) 0 0
\(781\) −9576.00 −0.438740
\(782\) 0 0
\(783\) 5670.00 0.258786
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18329.0 0.830188 0.415094 0.909778i \(-0.363749\pi\)
0.415094 + 0.909778i \(0.363749\pi\)
\(788\) 0 0
\(789\) 5436.00 0.245281
\(790\) 0 0
\(791\) −1092.00 −0.0490860
\(792\) 0 0
\(793\) −49111.0 −2.19922
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16044.0 −0.713059 −0.356529 0.934284i \(-0.616040\pi\)
−0.356529 + 0.934284i \(0.616040\pi\)
\(798\) 0 0
\(799\) −22356.0 −0.989860
\(800\) 0 0
\(801\) −2160.00 −0.0952807
\(802\) 0 0
\(803\) 39396.0 1.73133
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15570.0 0.679170
\(808\) 0 0
\(809\) −24000.0 −1.04301 −0.521505 0.853248i \(-0.674629\pi\)
−0.521505 + 0.853248i \(0.674629\pi\)
\(810\) 0 0
\(811\) −5117.00 −0.221556 −0.110778 0.993845i \(-0.535334\pi\)
−0.110778 + 0.993845i \(0.535334\pi\)
\(812\) 0 0
\(813\) 13776.0 0.594275
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 30245.0 1.29515
\(818\) 0 0
\(819\) −603.000 −0.0257271
\(820\) 0 0
\(821\) 13542.0 0.575663 0.287831 0.957681i \(-0.407066\pi\)
0.287831 + 0.957681i \(0.407066\pi\)
\(822\) 0 0
\(823\) 1283.00 0.0543409 0.0271705 0.999631i \(-0.491350\pi\)
0.0271705 + 0.999631i \(0.491350\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16344.0 0.687227 0.343613 0.939111i \(-0.388349\pi\)
0.343613 + 0.939111i \(0.388349\pi\)
\(828\) 0 0
\(829\) −790.000 −0.0330975 −0.0165488 0.999863i \(-0.505268\pi\)
−0.0165488 + 0.999863i \(0.505268\pi\)
\(830\) 0 0
\(831\) −6573.00 −0.274386
\(832\) 0 0
\(833\) 18468.0 0.768161
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5211.00 −0.215195
\(838\) 0 0
\(839\) 9990.00 0.411076 0.205538 0.978649i \(-0.434106\pi\)
0.205538 + 0.978649i \(0.434106\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) 0 0
\(843\) −23526.0 −0.961184
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −433.000 −0.0175656
\(848\) 0 0
\(849\) 741.000 0.0299541
\(850\) 0 0
\(851\) −46332.0 −1.86632
\(852\) 0 0
\(853\) −24743.0 −0.993182 −0.496591 0.867985i \(-0.665415\pi\)
−0.496591 + 0.867985i \(0.665415\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23556.0 0.938924 0.469462 0.882953i \(-0.344448\pi\)
0.469462 + 0.882953i \(0.344448\pi\)
\(858\) 0 0
\(859\) 34000.0 1.35048 0.675242 0.737597i \(-0.264039\pi\)
0.675242 + 0.737597i \(0.264039\pi\)
\(860\) 0 0
\(861\) 36.0000 0.00142494
\(862\) 0 0
\(863\) −37032.0 −1.46070 −0.730350 0.683073i \(-0.760643\pi\)
−0.730350 + 0.683073i \(0.760643\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5991.00 0.234677
\(868\) 0 0
\(869\) −6720.00 −0.262325
\(870\) 0 0
\(871\) 20033.0 0.779325
\(872\) 0 0
\(873\) 4599.00 0.178296
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2519.00 −0.0969904 −0.0484952 0.998823i \(-0.515443\pi\)
−0.0484952 + 0.998823i \(0.515443\pi\)
\(878\) 0 0
\(879\) −16326.0 −0.626465
\(880\) 0 0
\(881\) 43992.0 1.68232 0.841162 0.540783i \(-0.181872\pi\)
0.841162 + 0.540783i \(0.181872\pi\)
\(882\) 0 0
\(883\) −19177.0 −0.730869 −0.365435 0.930837i \(-0.619080\pi\)
−0.365435 + 0.930837i \(0.619080\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 44994.0 1.70321 0.851607 0.524181i \(-0.175628\pi\)
0.851607 + 0.524181i \(0.175628\pi\)
\(888\) 0 0
\(889\) 16.0000 0.000603625 0
\(890\) 0 0
\(891\) −3402.00 −0.127914
\(892\) 0 0
\(893\) 47610.0 1.78411
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 32562.0 1.21206
\(898\) 0 0
\(899\) −40530.0 −1.50362
\(900\) 0 0
\(901\) −10368.0 −0.383361
\(902\) 0 0
\(903\) 789.000 0.0290767
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −52396.0 −1.91817 −0.959085 0.283117i \(-0.908631\pi\)
−0.959085 + 0.283117i \(0.908631\pi\)
\(908\) 0 0
\(909\) 8208.00 0.299496
\(910\) 0 0
\(911\) −7242.00 −0.263379 −0.131689 0.991291i \(-0.542040\pi\)
−0.131689 + 0.991291i \(0.542040\pi\)
\(912\) 0 0
\(913\) 19404.0 0.703372
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1992.00 0.0717357
\(918\) 0 0
\(919\) −4085.00 −0.146629 −0.0733143 0.997309i \(-0.523358\pi\)
−0.0733143 + 0.997309i \(0.523358\pi\)
\(920\) 0 0
\(921\) 11613.0 0.415485
\(922\) 0 0
\(923\) 15276.0 0.544762
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6012.00 0.213010
\(928\) 0 0
\(929\) −3030.00 −0.107009 −0.0535043 0.998568i \(-0.517039\pi\)
−0.0535043 + 0.998568i \(0.517039\pi\)
\(930\) 0 0
\(931\) −39330.0 −1.38452
\(932\) 0 0
\(933\) −17154.0 −0.601926
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5759.00 −0.200788 −0.100394 0.994948i \(-0.532010\pi\)
−0.100394 + 0.994948i \(0.532010\pi\)
\(938\) 0 0
\(939\) −10911.0 −0.379198
\(940\) 0 0
\(941\) −258.000 −0.00893790 −0.00446895 0.999990i \(-0.501423\pi\)
−0.00446895 + 0.999990i \(0.501423\pi\)
\(942\) 0 0
\(943\) −1944.00 −0.0671319
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1374.00 0.0471478 0.0235739 0.999722i \(-0.492495\pi\)
0.0235739 + 0.999722i \(0.492495\pi\)
\(948\) 0 0
\(949\) −62846.0 −2.14970
\(950\) 0 0
\(951\) −3888.00 −0.132573
\(952\) 0 0
\(953\) −9288.00 −0.315706 −0.157853 0.987463i \(-0.550457\pi\)
−0.157853 + 0.987463i \(0.550457\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −26460.0 −0.893762
\(958\) 0 0
\(959\) −2346.00 −0.0789951
\(960\) 0 0
\(961\) 7458.00 0.250344
\(962\) 0 0
\(963\) −11664.0 −0.390309
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −21616.0 −0.718846 −0.359423 0.933175i \(-0.617026\pi\)
−0.359423 + 0.933175i \(0.617026\pi\)
\(968\) 0 0
\(969\) 18630.0 0.617628
\(970\) 0 0
\(971\) 19098.0 0.631188 0.315594 0.948894i \(-0.397796\pi\)
0.315594 + 0.948894i \(0.397796\pi\)
\(972\) 0 0
\(973\) 2900.00 0.0955496
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18246.0 0.597483 0.298742 0.954334i \(-0.403433\pi\)
0.298742 + 0.954334i \(0.403433\pi\)
\(978\) 0 0
\(979\) 10080.0 0.329069
\(980\) 0 0
\(981\) −15615.0 −0.508204
\(982\) 0 0
\(983\) −38772.0 −1.25802 −0.629011 0.777397i \(-0.716540\pi\)
−0.629011 + 0.777397i \(0.716540\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1242.00 0.0400540
\(988\) 0 0
\(989\) −42606.0 −1.36986
\(990\) 0 0
\(991\) 23053.0 0.738953 0.369477 0.929240i \(-0.379537\pi\)
0.369477 + 0.929240i \(0.379537\pi\)
\(992\) 0 0
\(993\) 15396.0 0.492021
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10366.0 0.329282 0.164641 0.986354i \(-0.447353\pi\)
0.164641 + 0.986354i \(0.447353\pi\)
\(998\) 0 0
\(999\) −7722.00 −0.244558
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.i.1.1 1
4.3 odd 2 150.4.a.h.1.1 yes 1
5.2 odd 4 1200.4.f.c.49.2 2
5.3 odd 4 1200.4.f.c.49.1 2
5.4 even 2 1200.4.a.bb.1.1 1
12.11 even 2 450.4.a.f.1.1 1
20.3 even 4 150.4.c.e.49.1 2
20.7 even 4 150.4.c.e.49.2 2
20.19 odd 2 150.4.a.a.1.1 1
60.23 odd 4 450.4.c.a.199.2 2
60.47 odd 4 450.4.c.a.199.1 2
60.59 even 2 450.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.4.a.a.1.1 1 20.19 odd 2
150.4.a.h.1.1 yes 1 4.3 odd 2
150.4.c.e.49.1 2 20.3 even 4
150.4.c.e.49.2 2 20.7 even 4
450.4.a.f.1.1 1 12.11 even 2
450.4.a.o.1.1 1 60.59 even 2
450.4.c.a.199.1 2 60.47 odd 4
450.4.c.a.199.2 2 60.23 odd 4
1200.4.a.i.1.1 1 1.1 even 1 trivial
1200.4.a.bb.1.1 1 5.4 even 2
1200.4.f.c.49.1 2 5.3 odd 4
1200.4.f.c.49.2 2 5.2 odd 4