Properties

Label 2200.4.a.d.1.1
Level $2200$
Weight $4$
Character 2200.1
Self dual yes
Analytic conductor $129.804$
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] = [2200,4,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, 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("2200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(129.804202013\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +8.00000 q^{7} -23.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +8.00000 q^{7} -23.0000 q^{9} -11.0000 q^{11} +25.0000 q^{13} -52.0000 q^{17} +19.0000 q^{19} -16.0000 q^{21} -75.0000 q^{23} +100.000 q^{27} +241.000 q^{29} +189.000 q^{31} +22.0000 q^{33} +68.0000 q^{37} -50.0000 q^{39} +240.000 q^{41} -183.000 q^{43} -56.0000 q^{47} -279.000 q^{49} +104.000 q^{51} -142.000 q^{53} -38.0000 q^{57} -584.000 q^{59} -622.000 q^{61} -184.000 q^{63} -142.000 q^{67} +150.000 q^{69} +435.000 q^{71} +628.000 q^{73} -88.0000 q^{77} -824.000 q^{79} +421.000 q^{81} +837.000 q^{83} -482.000 q^{87} -53.0000 q^{89} +200.000 q^{91} -378.000 q^{93} +1677.00 q^{97} +253.000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 8.00000 0.431959 0.215980 0.976398i \(-0.430705\pi\)
0.215980 + 0.976398i \(0.430705\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 25.0000 0.533366 0.266683 0.963784i \(-0.414072\pi\)
0.266683 + 0.963784i \(0.414072\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −52.0000 −0.741874 −0.370937 0.928658i \(-0.620963\pi\)
−0.370937 + 0.928658i \(0.620963\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −16.0000 −0.166261
\(22\) 0 0
\(23\) −75.0000 −0.679938 −0.339969 0.940437i \(-0.610417\pi\)
−0.339969 + 0.940437i \(0.610417\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 100.000 0.712778
\(28\) 0 0
\(29\) 241.000 1.54319 0.771596 0.636113i \(-0.219459\pi\)
0.771596 + 0.636113i \(0.219459\pi\)
\(30\) 0 0
\(31\) 189.000 1.09501 0.547506 0.836801i \(-0.315577\pi\)
0.547506 + 0.836801i \(0.315577\pi\)
\(32\) 0 0
\(33\) 22.0000 0.116052
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 68.0000 0.302139 0.151069 0.988523i \(-0.451728\pi\)
0.151069 + 0.988523i \(0.451728\pi\)
\(38\) 0 0
\(39\) −50.0000 −0.205293
\(40\) 0 0
\(41\) 240.000 0.914188 0.457094 0.889418i \(-0.348890\pi\)
0.457094 + 0.889418i \(0.348890\pi\)
\(42\) 0 0
\(43\) −183.000 −0.649006 −0.324503 0.945885i \(-0.605197\pi\)
−0.324503 + 0.945885i \(0.605197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −56.0000 −0.173797 −0.0868983 0.996217i \(-0.527696\pi\)
−0.0868983 + 0.996217i \(0.527696\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) 104.000 0.285547
\(52\) 0 0
\(53\) −142.000 −0.368023 −0.184011 0.982924i \(-0.558908\pi\)
−0.184011 + 0.982924i \(0.558908\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −38.0000 −0.0883022
\(58\) 0 0
\(59\) −584.000 −1.28865 −0.644325 0.764752i \(-0.722861\pi\)
−0.644325 + 0.764752i \(0.722861\pi\)
\(60\) 0 0
\(61\) −622.000 −1.30556 −0.652778 0.757549i \(-0.726397\pi\)
−0.652778 + 0.757549i \(0.726397\pi\)
\(62\) 0 0
\(63\) −184.000 −0.367965
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −142.000 −0.258926 −0.129463 0.991584i \(-0.541325\pi\)
−0.129463 + 0.991584i \(0.541325\pi\)
\(68\) 0 0
\(69\) 150.000 0.261708
\(70\) 0 0
\(71\) 435.000 0.727113 0.363556 0.931572i \(-0.381562\pi\)
0.363556 + 0.931572i \(0.381562\pi\)
\(72\) 0 0
\(73\) 628.000 1.00687 0.503437 0.864032i \(-0.332069\pi\)
0.503437 + 0.864032i \(0.332069\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −88.0000 −0.130241
\(78\) 0 0
\(79\) −824.000 −1.17351 −0.586755 0.809765i \(-0.699595\pi\)
−0.586755 + 0.809765i \(0.699595\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) 837.000 1.10690 0.553450 0.832882i \(-0.313311\pi\)
0.553450 + 0.832882i \(0.313311\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −482.000 −0.593975
\(88\) 0 0
\(89\) −53.0000 −0.0631235 −0.0315617 0.999502i \(-0.510048\pi\)
−0.0315617 + 0.999502i \(0.510048\pi\)
\(90\) 0 0
\(91\) 200.000 0.230392
\(92\) 0 0
\(93\) −378.000 −0.421471
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1677.00 1.75540 0.877699 0.479213i \(-0.159078\pi\)
0.877699 + 0.479213i \(0.159078\pi\)
\(98\) 0 0
\(99\) 253.000 0.256843
\(100\) 0 0
\(101\) −1275.00 −1.25611 −0.628056 0.778168i \(-0.716149\pi\)
−0.628056 + 0.778168i \(0.716149\pi\)
\(102\) 0 0
\(103\) 391.000 0.374042 0.187021 0.982356i \(-0.440117\pi\)
0.187021 + 0.982356i \(0.440117\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1099.00 0.992938 0.496469 0.868055i \(-0.334630\pi\)
0.496469 + 0.868055i \(0.334630\pi\)
\(108\) 0 0
\(109\) −905.000 −0.795259 −0.397630 0.917546i \(-0.630167\pi\)
−0.397630 + 0.917546i \(0.630167\pi\)
\(110\) 0 0
\(111\) −136.000 −0.116293
\(112\) 0 0
\(113\) −226.000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −575.000 −0.454348
\(118\) 0 0
\(119\) −416.000 −0.320459
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −480.000 −0.351871
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1532.00 1.07042 0.535209 0.844720i \(-0.320233\pi\)
0.535209 + 0.844720i \(0.320233\pi\)
\(128\) 0 0
\(129\) 366.000 0.249802
\(130\) 0 0
\(131\) −2089.00 −1.39326 −0.696629 0.717432i \(-0.745318\pi\)
−0.696629 + 0.717432i \(0.745318\pi\)
\(132\) 0 0
\(133\) 152.000 0.0990983
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1979.00 −1.23414 −0.617071 0.786908i \(-0.711681\pi\)
−0.617071 + 0.786908i \(0.711681\pi\)
\(138\) 0 0
\(139\) −47.0000 −0.0286798 −0.0143399 0.999897i \(-0.504565\pi\)
−0.0143399 + 0.999897i \(0.504565\pi\)
\(140\) 0 0
\(141\) 112.000 0.0668943
\(142\) 0 0
\(143\) −275.000 −0.160816
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 558.000 0.313082
\(148\) 0 0
\(149\) −1602.00 −0.880812 −0.440406 0.897799i \(-0.645165\pi\)
−0.440406 + 0.897799i \(0.645165\pi\)
\(150\) 0 0
\(151\) −160.000 −0.0862292 −0.0431146 0.999070i \(-0.513728\pi\)
−0.0431146 + 0.999070i \(0.513728\pi\)
\(152\) 0 0
\(153\) 1196.00 0.631966
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1934.00 0.983121 0.491561 0.870843i \(-0.336427\pi\)
0.491561 + 0.870843i \(0.336427\pi\)
\(158\) 0 0
\(159\) 284.000 0.141652
\(160\) 0 0
\(161\) −600.000 −0.293706
\(162\) 0 0
\(163\) −2692.00 −1.29358 −0.646791 0.762668i \(-0.723889\pi\)
−0.646791 + 0.762668i \(0.723889\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −504.000 −0.233537 −0.116769 0.993159i \(-0.537254\pi\)
−0.116769 + 0.993159i \(0.537254\pi\)
\(168\) 0 0
\(169\) −1572.00 −0.715521
\(170\) 0 0
\(171\) −437.000 −0.195428
\(172\) 0 0
\(173\) −2759.00 −1.21250 −0.606251 0.795273i \(-0.707327\pi\)
−0.606251 + 0.795273i \(0.707327\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1168.00 0.496001
\(178\) 0 0
\(179\) −2400.00 −1.00215 −0.501074 0.865405i \(-0.667061\pi\)
−0.501074 + 0.865405i \(0.667061\pi\)
\(180\) 0 0
\(181\) −232.000 −0.0952731 −0.0476365 0.998865i \(-0.515169\pi\)
−0.0476365 + 0.998865i \(0.515169\pi\)
\(182\) 0 0
\(183\) 1244.00 0.502509
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 572.000 0.223683
\(188\) 0 0
\(189\) 800.000 0.307891
\(190\) 0 0
\(191\) 2039.00 0.772444 0.386222 0.922406i \(-0.373780\pi\)
0.386222 + 0.922406i \(0.373780\pi\)
\(192\) 0 0
\(193\) −484.000 −0.180513 −0.0902567 0.995919i \(-0.528769\pi\)
−0.0902567 + 0.995919i \(0.528769\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2081.00 −0.752615 −0.376307 0.926495i \(-0.622806\pi\)
−0.376307 + 0.926495i \(0.622806\pi\)
\(198\) 0 0
\(199\) −2437.00 −0.868112 −0.434056 0.900886i \(-0.642918\pi\)
−0.434056 + 0.900886i \(0.642918\pi\)
\(200\) 0 0
\(201\) 284.000 0.0996608
\(202\) 0 0
\(203\) 1928.00 0.666596
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1725.00 0.579207
\(208\) 0 0
\(209\) −209.000 −0.0691714
\(210\) 0 0
\(211\) 4332.00 1.41340 0.706699 0.707514i \(-0.250183\pi\)
0.706699 + 0.707514i \(0.250183\pi\)
\(212\) 0 0
\(213\) −870.000 −0.279866
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1512.00 0.473001
\(218\) 0 0
\(219\) −1256.00 −0.387546
\(220\) 0 0
\(221\) −1300.00 −0.395690
\(222\) 0 0
\(223\) −120.000 −0.0360350 −0.0180175 0.999838i \(-0.505735\pi\)
−0.0180175 + 0.999838i \(0.505735\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5001.00 1.46224 0.731119 0.682250i \(-0.238998\pi\)
0.731119 + 0.682250i \(0.238998\pi\)
\(228\) 0 0
\(229\) 278.000 0.0802216 0.0401108 0.999195i \(-0.487229\pi\)
0.0401108 + 0.999195i \(0.487229\pi\)
\(230\) 0 0
\(231\) 176.000 0.0501297
\(232\) 0 0
\(233\) −5358.00 −1.50650 −0.753249 0.657735i \(-0.771515\pi\)
−0.753249 + 0.657735i \(0.771515\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1648.00 0.451684
\(238\) 0 0
\(239\) 126.000 0.0341015 0.0170508 0.999855i \(-0.494572\pi\)
0.0170508 + 0.999855i \(0.494572\pi\)
\(240\) 0 0
\(241\) −510.000 −0.136315 −0.0681577 0.997675i \(-0.521712\pi\)
−0.0681577 + 0.997675i \(0.521712\pi\)
\(242\) 0 0
\(243\) −3542.00 −0.935059
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 475.000 0.122362
\(248\) 0 0
\(249\) −1674.00 −0.426046
\(250\) 0 0
\(251\) 2676.00 0.672939 0.336469 0.941694i \(-0.390767\pi\)
0.336469 + 0.941694i \(0.390767\pi\)
\(252\) 0 0
\(253\) 825.000 0.205009
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 617.000 0.149756 0.0748782 0.997193i \(-0.476143\pi\)
0.0748782 + 0.997193i \(0.476143\pi\)
\(258\) 0 0
\(259\) 544.000 0.130512
\(260\) 0 0
\(261\) −5543.00 −1.31457
\(262\) 0 0
\(263\) −3732.00 −0.875000 −0.437500 0.899218i \(-0.644136\pi\)
−0.437500 + 0.899218i \(0.644136\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 106.000 0.0242962
\(268\) 0 0
\(269\) −8418.00 −1.90801 −0.954005 0.299792i \(-0.903083\pi\)
−0.954005 + 0.299792i \(0.903083\pi\)
\(270\) 0 0
\(271\) −7786.00 −1.74526 −0.872631 0.488381i \(-0.837588\pi\)
−0.872631 + 0.488381i \(0.837588\pi\)
\(272\) 0 0
\(273\) −400.000 −0.0886780
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1974.00 −0.428181 −0.214091 0.976814i \(-0.568679\pi\)
−0.214091 + 0.976814i \(0.568679\pi\)
\(278\) 0 0
\(279\) −4347.00 −0.932789
\(280\) 0 0
\(281\) 428.000 0.0908624 0.0454312 0.998967i \(-0.485534\pi\)
0.0454312 + 0.998967i \(0.485534\pi\)
\(282\) 0 0
\(283\) 3220.00 0.676357 0.338179 0.941082i \(-0.390189\pi\)
0.338179 + 0.941082i \(0.390189\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1920.00 0.394892
\(288\) 0 0
\(289\) −2209.00 −0.449623
\(290\) 0 0
\(291\) −3354.00 −0.675653
\(292\) 0 0
\(293\) −8258.00 −1.64654 −0.823272 0.567647i \(-0.807854\pi\)
−0.823272 + 0.567647i \(0.807854\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1100.00 −0.214911
\(298\) 0 0
\(299\) −1875.00 −0.362656
\(300\) 0 0
\(301\) −1464.00 −0.280344
\(302\) 0 0
\(303\) 2550.00 0.483477
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3632.00 0.675209 0.337605 0.941288i \(-0.390383\pi\)
0.337605 + 0.941288i \(0.390383\pi\)
\(308\) 0 0
\(309\) −782.000 −0.143969
\(310\) 0 0
\(311\) −4413.00 −0.804625 −0.402312 0.915502i \(-0.631793\pi\)
−0.402312 + 0.915502i \(0.631793\pi\)
\(312\) 0 0
\(313\) −2266.00 −0.409207 −0.204604 0.978845i \(-0.565591\pi\)
−0.204604 + 0.978845i \(0.565591\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3632.00 0.643512 0.321756 0.946823i \(-0.395727\pi\)
0.321756 + 0.946823i \(0.395727\pi\)
\(318\) 0 0
\(319\) −2651.00 −0.465290
\(320\) 0 0
\(321\) −2198.00 −0.382182
\(322\) 0 0
\(323\) −988.000 −0.170197
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1810.00 0.306096
\(328\) 0 0
\(329\) −448.000 −0.0750731
\(330\) 0 0
\(331\) 8476.00 1.40750 0.703751 0.710447i \(-0.251507\pi\)
0.703751 + 0.710447i \(0.251507\pi\)
\(332\) 0 0
\(333\) −1564.00 −0.257377
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8834.00 −1.42795 −0.713974 0.700172i \(-0.753107\pi\)
−0.713974 + 0.700172i \(0.753107\pi\)
\(338\) 0 0
\(339\) 452.000 0.0724167
\(340\) 0 0
\(341\) −2079.00 −0.330159
\(342\) 0 0
\(343\) −4976.00 −0.783320
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6724.00 −1.04024 −0.520120 0.854093i \(-0.674113\pi\)
−0.520120 + 0.854093i \(0.674113\pi\)
\(348\) 0 0
\(349\) 1825.00 0.279914 0.139957 0.990158i \(-0.455304\pi\)
0.139957 + 0.990158i \(0.455304\pi\)
\(350\) 0 0
\(351\) 2500.00 0.380171
\(352\) 0 0
\(353\) −3103.00 −0.467864 −0.233932 0.972253i \(-0.575159\pi\)
−0.233932 + 0.972253i \(0.575159\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 832.000 0.123345
\(358\) 0 0
\(359\) −9960.00 −1.46426 −0.732129 0.681166i \(-0.761473\pi\)
−0.732129 + 0.681166i \(0.761473\pi\)
\(360\) 0 0
\(361\) −6498.00 −0.947368
\(362\) 0 0
\(363\) −242.000 −0.0349909
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5695.00 0.810018 0.405009 0.914313i \(-0.367268\pi\)
0.405009 + 0.914313i \(0.367268\pi\)
\(368\) 0 0
\(369\) −5520.00 −0.778753
\(370\) 0 0
\(371\) −1136.00 −0.158971
\(372\) 0 0
\(373\) −9582.00 −1.33013 −0.665063 0.746787i \(-0.731595\pi\)
−0.665063 + 0.746787i \(0.731595\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6025.00 0.823086
\(378\) 0 0
\(379\) −3402.00 −0.461079 −0.230540 0.973063i \(-0.574049\pi\)
−0.230540 + 0.973063i \(0.574049\pi\)
\(380\) 0 0
\(381\) −3064.00 −0.412004
\(382\) 0 0
\(383\) 1651.00 0.220267 0.110133 0.993917i \(-0.464872\pi\)
0.110133 + 0.993917i \(0.464872\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4209.00 0.552857
\(388\) 0 0
\(389\) 8304.00 1.08234 0.541169 0.840914i \(-0.317982\pi\)
0.541169 + 0.840914i \(0.317982\pi\)
\(390\) 0 0
\(391\) 3900.00 0.504428
\(392\) 0 0
\(393\) 4178.00 0.536265
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4198.00 0.530709 0.265355 0.964151i \(-0.414511\pi\)
0.265355 + 0.964151i \(0.414511\pi\)
\(398\) 0 0
\(399\) −304.000 −0.0381429
\(400\) 0 0
\(401\) −12809.0 −1.59514 −0.797570 0.603227i \(-0.793881\pi\)
−0.797570 + 0.603227i \(0.793881\pi\)
\(402\) 0 0
\(403\) 4725.00 0.584042
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −748.000 −0.0910982
\(408\) 0 0
\(409\) −3608.00 −0.436196 −0.218098 0.975927i \(-0.569985\pi\)
−0.218098 + 0.975927i \(0.569985\pi\)
\(410\) 0 0
\(411\) 3958.00 0.475021
\(412\) 0 0
\(413\) −4672.00 −0.556644
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 94.0000 0.0110388
\(418\) 0 0
\(419\) 3462.00 0.403651 0.201825 0.979421i \(-0.435313\pi\)
0.201825 + 0.979421i \(0.435313\pi\)
\(420\) 0 0
\(421\) −7406.00 −0.857355 −0.428677 0.903458i \(-0.641020\pi\)
−0.428677 + 0.903458i \(0.641020\pi\)
\(422\) 0 0
\(423\) 1288.00 0.148049
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4976.00 −0.563947
\(428\) 0 0
\(429\) 550.000 0.0618980
\(430\) 0 0
\(431\) 3320.00 0.371041 0.185521 0.982640i \(-0.440603\pi\)
0.185521 + 0.982640i \(0.440603\pi\)
\(432\) 0 0
\(433\) −12043.0 −1.33660 −0.668302 0.743890i \(-0.732979\pi\)
−0.668302 + 0.743890i \(0.732979\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1425.00 −0.155989
\(438\) 0 0
\(439\) −9036.00 −0.982380 −0.491190 0.871053i \(-0.663438\pi\)
−0.491190 + 0.871053i \(0.663438\pi\)
\(440\) 0 0
\(441\) 6417.00 0.692906
\(442\) 0 0
\(443\) 360.000 0.0386097 0.0193049 0.999814i \(-0.493855\pi\)
0.0193049 + 0.999814i \(0.493855\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 3204.00 0.339025
\(448\) 0 0
\(449\) 3477.00 0.365456 0.182728 0.983163i \(-0.441507\pi\)
0.182728 + 0.983163i \(0.441507\pi\)
\(450\) 0 0
\(451\) −2640.00 −0.275638
\(452\) 0 0
\(453\) 320.000 0.0331897
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9212.00 0.942930 0.471465 0.881885i \(-0.343725\pi\)
0.471465 + 0.881885i \(0.343725\pi\)
\(458\) 0 0
\(459\) −5200.00 −0.528791
\(460\) 0 0
\(461\) −6106.00 −0.616887 −0.308443 0.951243i \(-0.599808\pi\)
−0.308443 + 0.951243i \(0.599808\pi\)
\(462\) 0 0
\(463\) 9133.00 0.916731 0.458366 0.888764i \(-0.348435\pi\)
0.458366 + 0.888764i \(0.348435\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9930.00 0.983952 0.491976 0.870609i \(-0.336275\pi\)
0.491976 + 0.870609i \(0.336275\pi\)
\(468\) 0 0
\(469\) −1136.00 −0.111846
\(470\) 0 0
\(471\) −3868.00 −0.378403
\(472\) 0 0
\(473\) 2013.00 0.195683
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3266.00 0.313501
\(478\) 0 0
\(479\) −15906.0 −1.51725 −0.758626 0.651526i \(-0.774129\pi\)
−0.758626 + 0.651526i \(0.774129\pi\)
\(480\) 0 0
\(481\) 1700.00 0.161150
\(482\) 0 0
\(483\) 1200.00 0.113047
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3461.00 −0.322039 −0.161019 0.986951i \(-0.551478\pi\)
−0.161019 + 0.986951i \(0.551478\pi\)
\(488\) 0 0
\(489\) 5384.00 0.497900
\(490\) 0 0
\(491\) −12717.0 −1.16886 −0.584430 0.811444i \(-0.698682\pi\)
−0.584430 + 0.811444i \(0.698682\pi\)
\(492\) 0 0
\(493\) −12532.0 −1.14485
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3480.00 0.314083
\(498\) 0 0
\(499\) −8178.00 −0.733662 −0.366831 0.930288i \(-0.619557\pi\)
−0.366831 + 0.930288i \(0.619557\pi\)
\(500\) 0 0
\(501\) 1008.00 0.0898885
\(502\) 0 0
\(503\) −1278.00 −0.113287 −0.0566433 0.998394i \(-0.518040\pi\)
−0.0566433 + 0.998394i \(0.518040\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3144.00 0.275404
\(508\) 0 0
\(509\) 8022.00 0.698564 0.349282 0.937018i \(-0.386426\pi\)
0.349282 + 0.937018i \(0.386426\pi\)
\(510\) 0 0
\(511\) 5024.00 0.434929
\(512\) 0 0
\(513\) 1900.00 0.163523
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 616.000 0.0524016
\(518\) 0 0
\(519\) 5518.00 0.466692
\(520\) 0 0
\(521\) 5503.00 0.462746 0.231373 0.972865i \(-0.425678\pi\)
0.231373 + 0.972865i \(0.425678\pi\)
\(522\) 0 0
\(523\) −5521.00 −0.461599 −0.230800 0.973001i \(-0.574134\pi\)
−0.230800 + 0.973001i \(0.574134\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9828.00 −0.812361
\(528\) 0 0
\(529\) −6542.00 −0.537684
\(530\) 0 0
\(531\) 13432.0 1.09774
\(532\) 0 0
\(533\) 6000.00 0.487596
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4800.00 0.385727
\(538\) 0 0
\(539\) 3069.00 0.245253
\(540\) 0 0
\(541\) 13763.0 1.09375 0.546874 0.837215i \(-0.315818\pi\)
0.546874 + 0.837215i \(0.315818\pi\)
\(542\) 0 0
\(543\) 464.000 0.0366706
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17939.0 1.40222 0.701112 0.713051i \(-0.252687\pi\)
0.701112 + 0.713051i \(0.252687\pi\)
\(548\) 0 0
\(549\) 14306.0 1.11214
\(550\) 0 0
\(551\) 4579.00 0.354033
\(552\) 0 0
\(553\) −6592.00 −0.506908
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 119.000 0.00905241 0.00452620 0.999990i \(-0.498559\pi\)
0.00452620 + 0.999990i \(0.498559\pi\)
\(558\) 0 0
\(559\) −4575.00 −0.346157
\(560\) 0 0
\(561\) −1144.00 −0.0860958
\(562\) 0 0
\(563\) 9876.00 0.739296 0.369648 0.929172i \(-0.379478\pi\)
0.369648 + 0.929172i \(0.379478\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3368.00 0.249458
\(568\) 0 0
\(569\) −2504.00 −0.184487 −0.0922435 0.995736i \(-0.529404\pi\)
−0.0922435 + 0.995736i \(0.529404\pi\)
\(570\) 0 0
\(571\) 3947.00 0.289276 0.144638 0.989485i \(-0.453798\pi\)
0.144638 + 0.989485i \(0.453798\pi\)
\(572\) 0 0
\(573\) −4078.00 −0.297314
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2938.00 −0.211977 −0.105988 0.994367i \(-0.533801\pi\)
−0.105988 + 0.994367i \(0.533801\pi\)
\(578\) 0 0
\(579\) 968.000 0.0694796
\(580\) 0 0
\(581\) 6696.00 0.478136
\(582\) 0 0
\(583\) 1562.00 0.110963
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1484.00 0.104346 0.0521731 0.998638i \(-0.483385\pi\)
0.0521731 + 0.998638i \(0.483385\pi\)
\(588\) 0 0
\(589\) 3591.00 0.251213
\(590\) 0 0
\(591\) 4162.00 0.289682
\(592\) 0 0
\(593\) 534.000 0.0369793 0.0184897 0.999829i \(-0.494114\pi\)
0.0184897 + 0.999829i \(0.494114\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4874.00 0.334137
\(598\) 0 0
\(599\) −5460.00 −0.372437 −0.186218 0.982508i \(-0.559623\pi\)
−0.186218 + 0.982508i \(0.559623\pi\)
\(600\) 0 0
\(601\) 4844.00 0.328770 0.164385 0.986396i \(-0.447436\pi\)
0.164385 + 0.986396i \(0.447436\pi\)
\(602\) 0 0
\(603\) 3266.00 0.220567
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4994.00 0.333938 0.166969 0.985962i \(-0.446602\pi\)
0.166969 + 0.985962i \(0.446602\pi\)
\(608\) 0 0
\(609\) −3856.00 −0.256573
\(610\) 0 0
\(611\) −1400.00 −0.0926971
\(612\) 0 0
\(613\) −9502.00 −0.626072 −0.313036 0.949741i \(-0.601346\pi\)
−0.313036 + 0.949741i \(0.601346\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25029.0 −1.63311 −0.816556 0.577267i \(-0.804119\pi\)
−0.816556 + 0.577267i \(0.804119\pi\)
\(618\) 0 0
\(619\) −15180.0 −0.985680 −0.492840 0.870120i \(-0.664041\pi\)
−0.492840 + 0.870120i \(0.664041\pi\)
\(620\) 0 0
\(621\) −7500.00 −0.484645
\(622\) 0 0
\(623\) −424.000 −0.0272668
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 418.000 0.0266241
\(628\) 0 0
\(629\) −3536.00 −0.224149
\(630\) 0 0
\(631\) −26624.0 −1.67969 −0.839845 0.542826i \(-0.817354\pi\)
−0.839845 + 0.542826i \(0.817354\pi\)
\(632\) 0 0
\(633\) −8664.00 −0.544018
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6975.00 −0.433845
\(638\) 0 0
\(639\) −10005.0 −0.619392
\(640\) 0 0
\(641\) −2691.00 −0.165816 −0.0829080 0.996557i \(-0.526421\pi\)
−0.0829080 + 0.996557i \(0.526421\pi\)
\(642\) 0 0
\(643\) 28748.0 1.76316 0.881579 0.472037i \(-0.156481\pi\)
0.881579 + 0.472037i \(0.156481\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26864.0 1.63235 0.816177 0.577802i \(-0.196089\pi\)
0.816177 + 0.577802i \(0.196089\pi\)
\(648\) 0 0
\(649\) 6424.00 0.388542
\(650\) 0 0
\(651\) −3024.00 −0.182058
\(652\) 0 0
\(653\) −4286.00 −0.256852 −0.128426 0.991719i \(-0.540992\pi\)
−0.128426 + 0.991719i \(0.540992\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14444.0 −0.857708
\(658\) 0 0
\(659\) −23167.0 −1.36944 −0.684718 0.728808i \(-0.740075\pi\)
−0.684718 + 0.728808i \(0.740075\pi\)
\(660\) 0 0
\(661\) −12472.0 −0.733895 −0.366947 0.930242i \(-0.619597\pi\)
−0.366947 + 0.930242i \(0.619597\pi\)
\(662\) 0 0
\(663\) 2600.00 0.152301
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −18075.0 −1.04928
\(668\) 0 0
\(669\) 240.000 0.0138699
\(670\) 0 0
\(671\) 6842.00 0.393640
\(672\) 0 0
\(673\) −6940.00 −0.397500 −0.198750 0.980050i \(-0.563688\pi\)
−0.198750 + 0.980050i \(0.563688\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −27187.0 −1.54340 −0.771700 0.635987i \(-0.780593\pi\)
−0.771700 + 0.635987i \(0.780593\pi\)
\(678\) 0 0
\(679\) 13416.0 0.758260
\(680\) 0 0
\(681\) −10002.0 −0.562816
\(682\) 0 0
\(683\) 19880.0 1.11374 0.556872 0.830598i \(-0.312001\pi\)
0.556872 + 0.830598i \(0.312001\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −556.000 −0.0308773
\(688\) 0 0
\(689\) −3550.00 −0.196291
\(690\) 0 0
\(691\) 13040.0 0.717894 0.358947 0.933358i \(-0.383136\pi\)
0.358947 + 0.933358i \(0.383136\pi\)
\(692\) 0 0
\(693\) 2024.00 0.110946
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −12480.0 −0.678212
\(698\) 0 0
\(699\) 10716.0 0.579852
\(700\) 0 0
\(701\) 5463.00 0.294343 0.147172 0.989111i \(-0.452983\pi\)
0.147172 + 0.989111i \(0.452983\pi\)
\(702\) 0 0
\(703\) 1292.00 0.0693154
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10200.0 −0.542589
\(708\) 0 0
\(709\) 14732.0 0.780355 0.390178 0.920740i \(-0.372414\pi\)
0.390178 + 0.920740i \(0.372414\pi\)
\(710\) 0 0
\(711\) 18952.0 0.999656
\(712\) 0 0
\(713\) −14175.0 −0.744541
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −252.000 −0.0131257
\(718\) 0 0
\(719\) 23616.0 1.22493 0.612467 0.790496i \(-0.290177\pi\)
0.612467 + 0.790496i \(0.290177\pi\)
\(720\) 0 0
\(721\) 3128.00 0.161571
\(722\) 0 0
\(723\) 1020.00 0.0524678
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −3869.00 −0.197377 −0.0986886 0.995118i \(-0.531465\pi\)
−0.0986886 + 0.995118i \(0.531465\pi\)
\(728\) 0 0
\(729\) −4283.00 −0.217599
\(730\) 0 0
\(731\) 9516.00 0.481480
\(732\) 0 0
\(733\) 13993.0 0.705107 0.352553 0.935792i \(-0.385313\pi\)
0.352553 + 0.935792i \(0.385313\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1562.00 0.0780692
\(738\) 0 0
\(739\) 8736.00 0.434857 0.217428 0.976076i \(-0.430233\pi\)
0.217428 + 0.976076i \(0.430233\pi\)
\(740\) 0 0
\(741\) −950.000 −0.0470973
\(742\) 0 0
\(743\) 32842.0 1.62161 0.810805 0.585316i \(-0.199030\pi\)
0.810805 + 0.585316i \(0.199030\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −19251.0 −0.942915
\(748\) 0 0
\(749\) 8792.00 0.428909
\(750\) 0 0
\(751\) 7431.00 0.361067 0.180533 0.983569i \(-0.442218\pi\)
0.180533 + 0.983569i \(0.442218\pi\)
\(752\) 0 0
\(753\) −5352.00 −0.259014
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22460.0 1.07837 0.539183 0.842189i \(-0.318733\pi\)
0.539183 + 0.842189i \(0.318733\pi\)
\(758\) 0 0
\(759\) −1650.00 −0.0789080
\(760\) 0 0
\(761\) −21940.0 −1.04510 −0.522552 0.852607i \(-0.675020\pi\)
−0.522552 + 0.852607i \(0.675020\pi\)
\(762\) 0 0
\(763\) −7240.00 −0.343520
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14600.0 −0.687321
\(768\) 0 0
\(769\) 26002.0 1.21932 0.609659 0.792664i \(-0.291306\pi\)
0.609659 + 0.792664i \(0.291306\pi\)
\(770\) 0 0
\(771\) −1234.00 −0.0576413
\(772\) 0 0
\(773\) 5060.00 0.235441 0.117720 0.993047i \(-0.462441\pi\)
0.117720 + 0.993047i \(0.462441\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1088.00 −0.0502340
\(778\) 0 0
\(779\) 4560.00 0.209729
\(780\) 0 0
\(781\) −4785.00 −0.219233
\(782\) 0 0
\(783\) 24100.0 1.09995
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5460.00 0.247304 0.123652 0.992326i \(-0.460539\pi\)
0.123652 + 0.992326i \(0.460539\pi\)
\(788\) 0 0
\(789\) 7464.00 0.336788
\(790\) 0 0
\(791\) −1808.00 −0.0812706
\(792\) 0 0
\(793\) −15550.0 −0.696339
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10108.0 0.449239 0.224620 0.974447i \(-0.427886\pi\)
0.224620 + 0.974447i \(0.427886\pi\)
\(798\) 0 0
\(799\) 2912.00 0.128935
\(800\) 0 0
\(801\) 1219.00 0.0537718
\(802\) 0 0
\(803\) −6908.00 −0.303584
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16836.0 0.734393
\(808\) 0 0
\(809\) 17466.0 0.759051 0.379525 0.925181i \(-0.376087\pi\)
0.379525 + 0.925181i \(0.376087\pi\)
\(810\) 0 0
\(811\) 25388.0 1.09925 0.549626 0.835411i \(-0.314770\pi\)
0.549626 + 0.835411i \(0.314770\pi\)
\(812\) 0 0
\(813\) 15572.0 0.671751
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3477.00 −0.148892
\(818\) 0 0
\(819\) −4600.00 −0.196260
\(820\) 0 0
\(821\) −4523.00 −0.192270 −0.0961351 0.995368i \(-0.530648\pi\)
−0.0961351 + 0.995368i \(0.530648\pi\)
\(822\) 0 0
\(823\) 31824.0 1.34789 0.673946 0.738781i \(-0.264598\pi\)
0.673946 + 0.738781i \(0.264598\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3207.00 −0.134847 −0.0674234 0.997724i \(-0.521478\pi\)
−0.0674234 + 0.997724i \(0.521478\pi\)
\(828\) 0 0
\(829\) −28174.0 −1.18037 −0.590183 0.807269i \(-0.700944\pi\)
−0.590183 + 0.807269i \(0.700944\pi\)
\(830\) 0 0
\(831\) 3948.00 0.164807
\(832\) 0 0
\(833\) 14508.0 0.603448
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 18900.0 0.780501
\(838\) 0 0
\(839\) −20904.0 −0.860174 −0.430087 0.902787i \(-0.641517\pi\)
−0.430087 + 0.902787i \(0.641517\pi\)
\(840\) 0 0
\(841\) 33692.0 1.38144
\(842\) 0 0
\(843\) −856.000 −0.0349730
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 968.000 0.0392690
\(848\) 0 0
\(849\) −6440.00 −0.260330
\(850\) 0 0
\(851\) −5100.00 −0.205436
\(852\) 0 0
\(853\) 13398.0 0.537795 0.268897 0.963169i \(-0.413341\pi\)
0.268897 + 0.963169i \(0.413341\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1284.00 0.0511792 0.0255896 0.999673i \(-0.491854\pi\)
0.0255896 + 0.999673i \(0.491854\pi\)
\(858\) 0 0
\(859\) 27178.0 1.07951 0.539756 0.841821i \(-0.318516\pi\)
0.539756 + 0.841821i \(0.318516\pi\)
\(860\) 0 0
\(861\) −3840.00 −0.151994
\(862\) 0 0
\(863\) −15321.0 −0.604325 −0.302163 0.953256i \(-0.597709\pi\)
−0.302163 + 0.953256i \(0.597709\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4418.00 0.173060
\(868\) 0 0
\(869\) 9064.00 0.353826
\(870\) 0 0
\(871\) −3550.00 −0.138102
\(872\) 0 0
\(873\) −38571.0 −1.49534
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −27389.0 −1.05457 −0.527287 0.849687i \(-0.676791\pi\)
−0.527287 + 0.849687i \(0.676791\pi\)
\(878\) 0 0
\(879\) 16516.0 0.633755
\(880\) 0 0
\(881\) 2725.00 0.104208 0.0521042 0.998642i \(-0.483407\pi\)
0.0521042 + 0.998642i \(0.483407\pi\)
\(882\) 0 0
\(883\) 45272.0 1.72540 0.862698 0.505720i \(-0.168773\pi\)
0.862698 + 0.505720i \(0.168773\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23894.0 0.904489 0.452245 0.891894i \(-0.350623\pi\)
0.452245 + 0.891894i \(0.350623\pi\)
\(888\) 0 0
\(889\) 12256.0 0.462377
\(890\) 0 0
\(891\) −4631.00 −0.174124
\(892\) 0 0
\(893\) −1064.00 −0.0398717
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3750.00 0.139586
\(898\) 0 0
\(899\) 45549.0 1.68982
\(900\) 0 0
\(901\) 7384.00 0.273026
\(902\) 0 0
\(903\) 2928.00 0.107904
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −8134.00 −0.297778 −0.148889 0.988854i \(-0.547570\pi\)
−0.148889 + 0.988854i \(0.547570\pi\)
\(908\) 0 0
\(909\) 29325.0 1.07002
\(910\) 0 0
\(911\) 11648.0 0.423617 0.211809 0.977311i \(-0.432065\pi\)
0.211809 + 0.977311i \(0.432065\pi\)
\(912\) 0 0
\(913\) −9207.00 −0.333743
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −16712.0 −0.601831
\(918\) 0 0
\(919\) −11424.0 −0.410058 −0.205029 0.978756i \(-0.565729\pi\)
−0.205029 + 0.978756i \(0.565729\pi\)
\(920\) 0 0
\(921\) −7264.00 −0.259888
\(922\) 0 0
\(923\) 10875.0 0.387817
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −8993.00 −0.318629
\(928\) 0 0
\(929\) 32935.0 1.16315 0.581573 0.813494i \(-0.302438\pi\)
0.581573 + 0.813494i \(0.302438\pi\)
\(930\) 0 0
\(931\) −5301.00 −0.186609
\(932\) 0 0
\(933\) 8826.00 0.309700
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −42670.0 −1.48769 −0.743846 0.668351i \(-0.767001\pi\)
−0.743846 + 0.668351i \(0.767001\pi\)
\(938\) 0 0
\(939\) 4532.00 0.157504
\(940\) 0 0
\(941\) 2.00000 6.92860e−5 0 3.46430e−5 1.00000i \(-0.499989\pi\)
3.46430e−5 1.00000i \(0.499989\pi\)
\(942\) 0 0
\(943\) −18000.0 −0.621591
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −31390.0 −1.07713 −0.538563 0.842585i \(-0.681033\pi\)
−0.538563 + 0.842585i \(0.681033\pi\)
\(948\) 0 0
\(949\) 15700.0 0.537032
\(950\) 0 0
\(951\) −7264.00 −0.247688
\(952\) 0 0
\(953\) −20902.0 −0.710474 −0.355237 0.934776i \(-0.615600\pi\)
−0.355237 + 0.934776i \(0.615600\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5302.00 0.179090
\(958\) 0 0
\(959\) −15832.0 −0.533099
\(960\) 0 0
\(961\) 5930.00 0.199053
\(962\) 0 0
\(963\) −25277.0 −0.845836
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −3630.00 −0.120717 −0.0603583 0.998177i \(-0.519224\pi\)
−0.0603583 + 0.998177i \(0.519224\pi\)
\(968\) 0 0
\(969\) 1976.00 0.0655090
\(970\) 0 0
\(971\) 23282.0 0.769470 0.384735 0.923027i \(-0.374293\pi\)
0.384735 + 0.923027i \(0.374293\pi\)
\(972\) 0 0
\(973\) −376.000 −0.0123885
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −42910.0 −1.40513 −0.702565 0.711619i \(-0.747962\pi\)
−0.702565 + 0.711619i \(0.747962\pi\)
\(978\) 0 0
\(979\) 583.000 0.0190324
\(980\) 0 0
\(981\) 20815.0 0.677443
\(982\) 0 0
\(983\) 26849.0 0.871160 0.435580 0.900150i \(-0.356543\pi\)
0.435580 + 0.900150i \(0.356543\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 896.000 0.0288956
\(988\) 0 0
\(989\) 13725.0 0.441284
\(990\) 0 0
\(991\) 30024.0 0.962405 0.481203 0.876609i \(-0.340200\pi\)
0.481203 + 0.876609i \(0.340200\pi\)
\(992\) 0 0
\(993\) −16952.0 −0.541748
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22810.0 0.724574 0.362287 0.932067i \(-0.381996\pi\)
0.362287 + 0.932067i \(0.381996\pi\)
\(998\) 0 0
\(999\) 6800.00 0.215358
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 2200.4.a.d.1.1 1
5.4 even 2 2200.4.a.g.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.4.a.d.1.1 1 1.1 even 1 trivial
2200.4.a.g.1.1 yes 1 5.4 even 2