Properties

Label 2112.4.a.f.1.1
Level $2112$
Weight $4$
Character 2112.1
Self dual yes
Analytic conductor $124.612$
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] = [2112,4,Mod(1,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, 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("2112.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2112.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: \(124.612033932\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2112.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -2.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -2.00000 q^{7} +9.00000 q^{9} -11.0000 q^{11} +88.0000 q^{13} -66.0000 q^{17} -40.0000 q^{19} +6.00000 q^{21} -6.00000 q^{23} -125.000 q^{25} -27.0000 q^{27} +54.0000 q^{29} -8.00000 q^{31} +33.0000 q^{33} +106.000 q^{37} -264.000 q^{39} +354.000 q^{41} -124.000 q^{43} -546.000 q^{47} -339.000 q^{49} +198.000 q^{51} +408.000 q^{53} +120.000 q^{57} +552.000 q^{59} -404.000 q^{61} -18.0000 q^{63} -4.00000 q^{67} +18.0000 q^{69} -126.000 q^{71} -166.000 q^{73} +375.000 q^{75} +22.0000 q^{77} +874.000 q^{79} +81.0000 q^{81} +444.000 q^{83} -162.000 q^{87} +1002.00 q^{89} -176.000 q^{91} +24.0000 q^{93} -802.000 q^{97} -99.0000 q^{99} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.107990 −0.0539949 0.998541i \(-0.517195\pi\)
−0.0539949 + 0.998541i \(0.517195\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 88.0000 1.87745 0.938723 0.344671i \(-0.112010\pi\)
0.938723 + 0.344671i \(0.112010\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −66.0000 −0.941609 −0.470804 0.882238i \(-0.656036\pi\)
−0.470804 + 0.882238i \(0.656036\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 0 0
\(21\) 6.00000 0.0623480
\(22\) 0 0
\(23\) −6.00000 −0.0543951 −0.0271975 0.999630i \(-0.508658\pi\)
−0.0271975 + 0.999630i \(0.508658\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 54.0000 0.345778 0.172889 0.984941i \(-0.444690\pi\)
0.172889 + 0.984941i \(0.444690\pi\)
\(30\) 0 0
\(31\) −8.00000 −0.0463498 −0.0231749 0.999731i \(-0.507377\pi\)
−0.0231749 + 0.999731i \(0.507377\pi\)
\(32\) 0 0
\(33\) 33.0000 0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 106.000 0.470981 0.235490 0.971877i \(-0.424330\pi\)
0.235490 + 0.971877i \(0.424330\pi\)
\(38\) 0 0
\(39\) −264.000 −1.08394
\(40\) 0 0
\(41\) 354.000 1.34843 0.674214 0.738536i \(-0.264483\pi\)
0.674214 + 0.738536i \(0.264483\pi\)
\(42\) 0 0
\(43\) −124.000 −0.439763 −0.219882 0.975527i \(-0.570567\pi\)
−0.219882 + 0.975527i \(0.570567\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −546.000 −1.69452 −0.847258 0.531181i \(-0.821748\pi\)
−0.847258 + 0.531181i \(0.821748\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) 198.000 0.543638
\(52\) 0 0
\(53\) 408.000 1.05742 0.528709 0.848803i \(-0.322676\pi\)
0.528709 + 0.848803i \(0.322676\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 120.000 0.278849
\(58\) 0 0
\(59\) 552.000 1.21804 0.609019 0.793155i \(-0.291563\pi\)
0.609019 + 0.793155i \(0.291563\pi\)
\(60\) 0 0
\(61\) −404.000 −0.847982 −0.423991 0.905666i \(-0.639371\pi\)
−0.423991 + 0.905666i \(0.639371\pi\)
\(62\) 0 0
\(63\) −18.0000 −0.0359966
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.00729370 −0.00364685 0.999993i \(-0.501161\pi\)
−0.00364685 + 0.999993i \(0.501161\pi\)
\(68\) 0 0
\(69\) 18.0000 0.0314050
\(70\) 0 0
\(71\) −126.000 −0.210612 −0.105306 0.994440i \(-0.533582\pi\)
−0.105306 + 0.994440i \(0.533582\pi\)
\(72\) 0 0
\(73\) −166.000 −0.266148 −0.133074 0.991106i \(-0.542485\pi\)
−0.133074 + 0.991106i \(0.542485\pi\)
\(74\) 0 0
\(75\) 375.000 0.577350
\(76\) 0 0
\(77\) 22.0000 0.0325602
\(78\) 0 0
\(79\) 874.000 1.24472 0.622359 0.782732i \(-0.286175\pi\)
0.622359 + 0.782732i \(0.286175\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 444.000 0.587173 0.293586 0.955933i \(-0.405151\pi\)
0.293586 + 0.955933i \(0.405151\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −162.000 −0.199635
\(88\) 0 0
\(89\) 1002.00 1.19339 0.596695 0.802468i \(-0.296480\pi\)
0.596695 + 0.802468i \(0.296480\pi\)
\(90\) 0 0
\(91\) −176.000 −0.202745
\(92\) 0 0
\(93\) 24.0000 0.0267600
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −802.000 −0.839492 −0.419746 0.907642i \(-0.637881\pi\)
−0.419746 + 0.907642i \(0.637881\pi\)
\(98\) 0 0
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 1710.00 1.68467 0.842333 0.538957i \(-0.181181\pi\)
0.842333 + 0.538957i \(0.181181\pi\)
\(102\) 0 0
\(103\) −572.000 −0.547193 −0.273596 0.961845i \(-0.588213\pi\)
−0.273596 + 0.961845i \(0.588213\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −108.000 −0.0975771 −0.0487886 0.998809i \(-0.515536\pi\)
−0.0487886 + 0.998809i \(0.515536\pi\)
\(108\) 0 0
\(109\) 712.000 0.625663 0.312831 0.949809i \(-0.398723\pi\)
0.312831 + 0.949809i \(0.398723\pi\)
\(110\) 0 0
\(111\) −318.000 −0.271921
\(112\) 0 0
\(113\) −1302.00 −1.08391 −0.541955 0.840407i \(-0.682316\pi\)
−0.541955 + 0.840407i \(0.682316\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 792.000 0.625816
\(118\) 0 0
\(119\) 132.000 0.101684
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −1062.00 −0.778515
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −854.000 −0.596695 −0.298347 0.954457i \(-0.596435\pi\)
−0.298347 + 0.954457i \(0.596435\pi\)
\(128\) 0 0
\(129\) 372.000 0.253897
\(130\) 0 0
\(131\) −1548.00 −1.03244 −0.516219 0.856457i \(-0.672661\pi\)
−0.516219 + 0.856457i \(0.672661\pi\)
\(132\) 0 0
\(133\) 80.0000 0.0521570
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1194.00 −0.744601 −0.372300 0.928112i \(-0.621431\pi\)
−0.372300 + 0.928112i \(0.621431\pi\)
\(138\) 0 0
\(139\) −1816.00 −1.10814 −0.554069 0.832471i \(-0.686926\pi\)
−0.554069 + 0.832471i \(0.686926\pi\)
\(140\) 0 0
\(141\) 1638.00 0.978330
\(142\) 0 0
\(143\) −968.000 −0.566072
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1017.00 0.570617
\(148\) 0 0
\(149\) −2046.00 −1.12493 −0.562466 0.826820i \(-0.690147\pi\)
−0.562466 + 0.826820i \(0.690147\pi\)
\(150\) 0 0
\(151\) −1406.00 −0.757739 −0.378870 0.925450i \(-0.623687\pi\)
−0.378870 + 0.925450i \(0.623687\pi\)
\(152\) 0 0
\(153\) −594.000 −0.313870
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2354.00 −1.19662 −0.598311 0.801264i \(-0.704161\pi\)
−0.598311 + 0.801264i \(0.704161\pi\)
\(158\) 0 0
\(159\) −1224.00 −0.610500
\(160\) 0 0
\(161\) 12.0000 0.00587411
\(162\) 0 0
\(163\) −988.000 −0.474762 −0.237381 0.971417i \(-0.576289\pi\)
−0.237381 + 0.971417i \(0.576289\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −732.000 −0.339185 −0.169592 0.985514i \(-0.554245\pi\)
−0.169592 + 0.985514i \(0.554245\pi\)
\(168\) 0 0
\(169\) 5547.00 2.52481
\(170\) 0 0
\(171\) −360.000 −0.160993
\(172\) 0 0
\(173\) 2790.00 1.22613 0.613063 0.790034i \(-0.289937\pi\)
0.613063 + 0.790034i \(0.289937\pi\)
\(174\) 0 0
\(175\) 250.000 0.107990
\(176\) 0 0
\(177\) −1656.00 −0.703235
\(178\) 0 0
\(179\) −408.000 −0.170365 −0.0851825 0.996365i \(-0.527147\pi\)
−0.0851825 + 0.996365i \(0.527147\pi\)
\(180\) 0 0
\(181\) −3458.00 −1.42006 −0.710031 0.704171i \(-0.751319\pi\)
−0.710031 + 0.704171i \(0.751319\pi\)
\(182\) 0 0
\(183\) 1212.00 0.489583
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 726.000 0.283906
\(188\) 0 0
\(189\) 54.0000 0.0207827
\(190\) 0 0
\(191\) −3498.00 −1.32516 −0.662582 0.748989i \(-0.730539\pi\)
−0.662582 + 0.748989i \(0.730539\pi\)
\(192\) 0 0
\(193\) 3878.00 1.44634 0.723172 0.690668i \(-0.242683\pi\)
0.723172 + 0.690668i \(0.242683\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1710.00 0.618439 0.309219 0.950991i \(-0.399932\pi\)
0.309219 + 0.950991i \(0.399932\pi\)
\(198\) 0 0
\(199\) −2876.00 −1.02449 −0.512247 0.858838i \(-0.671187\pi\)
−0.512247 + 0.858838i \(0.671187\pi\)
\(200\) 0 0
\(201\) 12.0000 0.00421102
\(202\) 0 0
\(203\) −108.000 −0.0373405
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −54.0000 −0.0181317
\(208\) 0 0
\(209\) 440.000 0.145624
\(210\) 0 0
\(211\) −3604.00 −1.17587 −0.587937 0.808906i \(-0.700060\pi\)
−0.587937 + 0.808906i \(0.700060\pi\)
\(212\) 0 0
\(213\) 378.000 0.121597
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000 0.00500530
\(218\) 0 0
\(219\) 498.000 0.153661
\(220\) 0 0
\(221\) −5808.00 −1.76782
\(222\) 0 0
\(223\) 6112.00 1.83538 0.917690 0.397297i \(-0.130052\pi\)
0.917690 + 0.397297i \(0.130052\pi\)
\(224\) 0 0
\(225\) −1125.00 −0.333333
\(226\) 0 0
\(227\) −2796.00 −0.817520 −0.408760 0.912642i \(-0.634039\pi\)
−0.408760 + 0.912642i \(0.634039\pi\)
\(228\) 0 0
\(229\) 214.000 0.0617534 0.0308767 0.999523i \(-0.490170\pi\)
0.0308767 + 0.999523i \(0.490170\pi\)
\(230\) 0 0
\(231\) −66.0000 −0.0187986
\(232\) 0 0
\(233\) −5142.00 −1.44577 −0.722883 0.690970i \(-0.757184\pi\)
−0.722883 + 0.690970i \(0.757184\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2622.00 −0.718638
\(238\) 0 0
\(239\) 156.000 0.0422209 0.0211105 0.999777i \(-0.493280\pi\)
0.0211105 + 0.999777i \(0.493280\pi\)
\(240\) 0 0
\(241\) 3350.00 0.895404 0.447702 0.894183i \(-0.352242\pi\)
0.447702 + 0.894183i \(0.352242\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3520.00 −0.906770
\(248\) 0 0
\(249\) −1332.00 −0.339004
\(250\) 0 0
\(251\) −5616.00 −1.41227 −0.706133 0.708079i \(-0.749562\pi\)
−0.706133 + 0.708079i \(0.749562\pi\)
\(252\) 0 0
\(253\) 66.0000 0.0164007
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7122.00 1.72863 0.864315 0.502950i \(-0.167752\pi\)
0.864315 + 0.502950i \(0.167752\pi\)
\(258\) 0 0
\(259\) −212.000 −0.0508612
\(260\) 0 0
\(261\) 486.000 0.115259
\(262\) 0 0
\(263\) 5568.00 1.30547 0.652733 0.757588i \(-0.273622\pi\)
0.652733 + 0.757588i \(0.273622\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3006.00 −0.689005
\(268\) 0 0
\(269\) −2016.00 −0.456943 −0.228472 0.973551i \(-0.573373\pi\)
−0.228472 + 0.973551i \(0.573373\pi\)
\(270\) 0 0
\(271\) 5974.00 1.33909 0.669547 0.742769i \(-0.266488\pi\)
0.669547 + 0.742769i \(0.266488\pi\)
\(272\) 0 0
\(273\) 528.000 0.117055
\(274\) 0 0
\(275\) 1375.00 0.301511
\(276\) 0 0
\(277\) 1324.00 0.287189 0.143595 0.989637i \(-0.454134\pi\)
0.143595 + 0.989637i \(0.454134\pi\)
\(278\) 0 0
\(279\) −72.0000 −0.0154499
\(280\) 0 0
\(281\) −5178.00 −1.09927 −0.549633 0.835406i \(-0.685232\pi\)
−0.549633 + 0.835406i \(0.685232\pi\)
\(282\) 0 0
\(283\) −4492.00 −0.943540 −0.471770 0.881722i \(-0.656385\pi\)
−0.471770 + 0.881722i \(0.656385\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −708.000 −0.145616
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) 0 0
\(291\) 2406.00 0.484681
\(292\) 0 0
\(293\) −486.000 −0.0969025 −0.0484512 0.998826i \(-0.515429\pi\)
−0.0484512 + 0.998826i \(0.515429\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 297.000 0.0580259
\(298\) 0 0
\(299\) −528.000 −0.102124
\(300\) 0 0
\(301\) 248.000 0.0474900
\(302\) 0 0
\(303\) −5130.00 −0.972643
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9664.00 −1.79659 −0.898296 0.439391i \(-0.855194\pi\)
−0.898296 + 0.439391i \(0.855194\pi\)
\(308\) 0 0
\(309\) 1716.00 0.315922
\(310\) 0 0
\(311\) −9246.00 −1.68583 −0.842914 0.538048i \(-0.819162\pi\)
−0.842914 + 0.538048i \(0.819162\pi\)
\(312\) 0 0
\(313\) −9718.00 −1.75493 −0.877466 0.479638i \(-0.840768\pi\)
−0.877466 + 0.479638i \(0.840768\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7236.00 1.28206 0.641032 0.767514i \(-0.278507\pi\)
0.641032 + 0.767514i \(0.278507\pi\)
\(318\) 0 0
\(319\) −594.000 −0.104256
\(320\) 0 0
\(321\) 324.000 0.0563362
\(322\) 0 0
\(323\) 2640.00 0.454779
\(324\) 0 0
\(325\) −11000.0 −1.87745
\(326\) 0 0
\(327\) −2136.00 −0.361227
\(328\) 0 0
\(329\) 1092.00 0.182991
\(330\) 0 0
\(331\) −10540.0 −1.75024 −0.875122 0.483902i \(-0.839219\pi\)
−0.875122 + 0.483902i \(0.839219\pi\)
\(332\) 0 0
\(333\) 954.000 0.156994
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3382.00 −0.546674 −0.273337 0.961918i \(-0.588127\pi\)
−0.273337 + 0.961918i \(0.588127\pi\)
\(338\) 0 0
\(339\) 3906.00 0.625796
\(340\) 0 0
\(341\) 88.0000 0.0139750
\(342\) 0 0
\(343\) 1364.00 0.214720
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4308.00 −0.666471 −0.333236 0.942844i \(-0.608140\pi\)
−0.333236 + 0.942844i \(0.608140\pi\)
\(348\) 0 0
\(349\) −7040.00 −1.07978 −0.539889 0.841736i \(-0.681534\pi\)
−0.539889 + 0.841736i \(0.681534\pi\)
\(350\) 0 0
\(351\) −2376.00 −0.361315
\(352\) 0 0
\(353\) 8010.00 1.20773 0.603866 0.797086i \(-0.293626\pi\)
0.603866 + 0.797086i \(0.293626\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −396.000 −0.0587074
\(358\) 0 0
\(359\) −10932.0 −1.60716 −0.803578 0.595200i \(-0.797073\pi\)
−0.803578 + 0.595200i \(0.797073\pi\)
\(360\) 0 0
\(361\) −5259.00 −0.766730
\(362\) 0 0
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6104.00 −0.868191 −0.434096 0.900867i \(-0.642932\pi\)
−0.434096 + 0.900867i \(0.642932\pi\)
\(368\) 0 0
\(369\) 3186.00 0.449476
\(370\) 0 0
\(371\) −816.000 −0.114190
\(372\) 0 0
\(373\) −440.000 −0.0610786 −0.0305393 0.999534i \(-0.509722\pi\)
−0.0305393 + 0.999534i \(0.509722\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4752.00 0.649179
\(378\) 0 0
\(379\) 7004.00 0.949265 0.474632 0.880184i \(-0.342581\pi\)
0.474632 + 0.880184i \(0.342581\pi\)
\(380\) 0 0
\(381\) 2562.00 0.344502
\(382\) 0 0
\(383\) −12774.0 −1.70423 −0.852116 0.523353i \(-0.824681\pi\)
−0.852116 + 0.523353i \(0.824681\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1116.00 −0.146588
\(388\) 0 0
\(389\) 4992.00 0.650654 0.325327 0.945602i \(-0.394526\pi\)
0.325327 + 0.945602i \(0.394526\pi\)
\(390\) 0 0
\(391\) 396.000 0.0512189
\(392\) 0 0
\(393\) 4644.00 0.596078
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3362.00 −0.425023 −0.212511 0.977159i \(-0.568164\pi\)
−0.212511 + 0.977159i \(0.568164\pi\)
\(398\) 0 0
\(399\) −240.000 −0.0301129
\(400\) 0 0
\(401\) −10170.0 −1.26650 −0.633249 0.773948i \(-0.718279\pi\)
−0.633249 + 0.773948i \(0.718279\pi\)
\(402\) 0 0
\(403\) −704.000 −0.0870192
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1166.00 −0.142006
\(408\) 0 0
\(409\) −4210.00 −0.508976 −0.254488 0.967076i \(-0.581907\pi\)
−0.254488 + 0.967076i \(0.581907\pi\)
\(410\) 0 0
\(411\) 3582.00 0.429895
\(412\) 0 0
\(413\) −1104.00 −0.131536
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5448.00 0.639783
\(418\) 0 0
\(419\) 15480.0 1.80489 0.902443 0.430809i \(-0.141772\pi\)
0.902443 + 0.430809i \(0.141772\pi\)
\(420\) 0 0
\(421\) 2698.00 0.312334 0.156167 0.987731i \(-0.450086\pi\)
0.156167 + 0.987731i \(0.450086\pi\)
\(422\) 0 0
\(423\) −4914.00 −0.564839
\(424\) 0 0
\(425\) 8250.00 0.941609
\(426\) 0 0
\(427\) 808.000 0.0915734
\(428\) 0 0
\(429\) 2904.00 0.326822
\(430\) 0 0
\(431\) 720.000 0.0804668 0.0402334 0.999190i \(-0.487190\pi\)
0.0402334 + 0.999190i \(0.487190\pi\)
\(432\) 0 0
\(433\) −16438.0 −1.82439 −0.912194 0.409759i \(-0.865613\pi\)
−0.912194 + 0.409759i \(0.865613\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 240.000 0.0262718
\(438\) 0 0
\(439\) −16598.0 −1.80451 −0.902254 0.431204i \(-0.858089\pi\)
−0.902254 + 0.431204i \(0.858089\pi\)
\(440\) 0 0
\(441\) −3051.00 −0.329446
\(442\) 0 0
\(443\) −13392.0 −1.43628 −0.718141 0.695897i \(-0.755007\pi\)
−0.718141 + 0.695897i \(0.755007\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6138.00 0.649480
\(448\) 0 0
\(449\) −906.000 −0.0952267 −0.0476133 0.998866i \(-0.515162\pi\)
−0.0476133 + 0.998866i \(0.515162\pi\)
\(450\) 0 0
\(451\) −3894.00 −0.406566
\(452\) 0 0
\(453\) 4218.00 0.437481
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8170.00 −0.836272 −0.418136 0.908384i \(-0.637317\pi\)
−0.418136 + 0.908384i \(0.637317\pi\)
\(458\) 0 0
\(459\) 1782.00 0.181213
\(460\) 0 0
\(461\) 12378.0 1.25054 0.625272 0.780407i \(-0.284988\pi\)
0.625272 + 0.780407i \(0.284988\pi\)
\(462\) 0 0
\(463\) 6964.00 0.699016 0.349508 0.936933i \(-0.386349\pi\)
0.349508 + 0.936933i \(0.386349\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8052.00 −0.797863 −0.398932 0.916981i \(-0.630619\pi\)
−0.398932 + 0.916981i \(0.630619\pi\)
\(468\) 0 0
\(469\) 8.00000 0.000787645 0
\(470\) 0 0
\(471\) 7062.00 0.690870
\(472\) 0 0
\(473\) 1364.00 0.132594
\(474\) 0 0
\(475\) 5000.00 0.482980
\(476\) 0 0
\(477\) 3672.00 0.352472
\(478\) 0 0
\(479\) 10872.0 1.03707 0.518533 0.855058i \(-0.326478\pi\)
0.518533 + 0.855058i \(0.326478\pi\)
\(480\) 0 0
\(481\) 9328.00 0.884242
\(482\) 0 0
\(483\) −36.0000 −0.00339142
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −8804.00 −0.819194 −0.409597 0.912267i \(-0.634331\pi\)
−0.409597 + 0.912267i \(0.634331\pi\)
\(488\) 0 0
\(489\) 2964.00 0.274104
\(490\) 0 0
\(491\) 8940.00 0.821704 0.410852 0.911702i \(-0.365231\pi\)
0.410852 + 0.911702i \(0.365231\pi\)
\(492\) 0 0
\(493\) −3564.00 −0.325587
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 252.000 0.0227440
\(498\) 0 0
\(499\) 1964.00 0.176194 0.0880969 0.996112i \(-0.471921\pi\)
0.0880969 + 0.996112i \(0.471921\pi\)
\(500\) 0 0
\(501\) 2196.00 0.195828
\(502\) 0 0
\(503\) 21252.0 1.88386 0.941928 0.335814i \(-0.109011\pi\)
0.941928 + 0.335814i \(0.109011\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −16641.0 −1.45770
\(508\) 0 0
\(509\) −12912.0 −1.12439 −0.562195 0.827005i \(-0.690043\pi\)
−0.562195 + 0.827005i \(0.690043\pi\)
\(510\) 0 0
\(511\) 332.000 0.0287413
\(512\) 0 0
\(513\) 1080.00 0.0929496
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6006.00 0.510916
\(518\) 0 0
\(519\) −8370.00 −0.707904
\(520\) 0 0
\(521\) −10602.0 −0.891520 −0.445760 0.895152i \(-0.647067\pi\)
−0.445760 + 0.895152i \(0.647067\pi\)
\(522\) 0 0
\(523\) 5084.00 0.425063 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(524\) 0 0
\(525\) −750.000 −0.0623480
\(526\) 0 0
\(527\) 528.000 0.0436433
\(528\) 0 0
\(529\) −12131.0 −0.997041
\(530\) 0 0
\(531\) 4968.00 0.406013
\(532\) 0 0
\(533\) 31152.0 2.53160
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1224.00 0.0983603
\(538\) 0 0
\(539\) 3729.00 0.297995
\(540\) 0 0
\(541\) −4160.00 −0.330596 −0.165298 0.986244i \(-0.552859\pi\)
−0.165298 + 0.986244i \(0.552859\pi\)
\(542\) 0 0
\(543\) 10374.0 0.819873
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3368.00 0.263264 0.131632 0.991299i \(-0.457978\pi\)
0.131632 + 0.991299i \(0.457978\pi\)
\(548\) 0 0
\(549\) −3636.00 −0.282661
\(550\) 0 0
\(551\) −2160.00 −0.167004
\(552\) 0 0
\(553\) −1748.00 −0.134417
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15534.0 −1.18168 −0.590841 0.806788i \(-0.701204\pi\)
−0.590841 + 0.806788i \(0.701204\pi\)
\(558\) 0 0
\(559\) −10912.0 −0.825632
\(560\) 0 0
\(561\) −2178.00 −0.163913
\(562\) 0 0
\(563\) 23628.0 1.76874 0.884371 0.466785i \(-0.154588\pi\)
0.884371 + 0.466785i \(0.154588\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −162.000 −0.0119989
\(568\) 0 0
\(569\) −2490.00 −0.183456 −0.0917278 0.995784i \(-0.529239\pi\)
−0.0917278 + 0.995784i \(0.529239\pi\)
\(570\) 0 0
\(571\) 4232.00 0.310164 0.155082 0.987902i \(-0.450436\pi\)
0.155082 + 0.987902i \(0.450436\pi\)
\(572\) 0 0
\(573\) 10494.0 0.765084
\(574\) 0 0
\(575\) 750.000 0.0543951
\(576\) 0 0
\(577\) 12446.0 0.897979 0.448989 0.893537i \(-0.351784\pi\)
0.448989 + 0.893537i \(0.351784\pi\)
\(578\) 0 0
\(579\) −11634.0 −0.835048
\(580\) 0 0
\(581\) −888.000 −0.0634087
\(582\) 0 0
\(583\) −4488.00 −0.318823
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20112.0 −1.41416 −0.707079 0.707134i \(-0.749988\pi\)
−0.707079 + 0.707134i \(0.749988\pi\)
\(588\) 0 0
\(589\) 320.000 0.0223860
\(590\) 0 0
\(591\) −5130.00 −0.357056
\(592\) 0 0
\(593\) −10866.0 −0.752467 −0.376234 0.926525i \(-0.622781\pi\)
−0.376234 + 0.926525i \(0.622781\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8628.00 0.591492
\(598\) 0 0
\(599\) −19422.0 −1.32481 −0.662405 0.749146i \(-0.730464\pi\)
−0.662405 + 0.749146i \(0.730464\pi\)
\(600\) 0 0
\(601\) 6602.00 0.448089 0.224044 0.974579i \(-0.428074\pi\)
0.224044 + 0.974579i \(0.428074\pi\)
\(602\) 0 0
\(603\) −36.0000 −0.00243123
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −26966.0 −1.80316 −0.901578 0.432616i \(-0.857591\pi\)
−0.901578 + 0.432616i \(0.857591\pi\)
\(608\) 0 0
\(609\) 324.000 0.0215585
\(610\) 0 0
\(611\) −48048.0 −3.18137
\(612\) 0 0
\(613\) 436.000 0.0287274 0.0143637 0.999897i \(-0.495428\pi\)
0.0143637 + 0.999897i \(0.495428\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10938.0 0.713691 0.356845 0.934163i \(-0.383852\pi\)
0.356845 + 0.934163i \(0.383852\pi\)
\(618\) 0 0
\(619\) −8620.00 −0.559721 −0.279860 0.960041i \(-0.590288\pi\)
−0.279860 + 0.960041i \(0.590288\pi\)
\(620\) 0 0
\(621\) 162.000 0.0104683
\(622\) 0 0
\(623\) −2004.00 −0.128874
\(624\) 0 0
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) −1320.00 −0.0840761
\(628\) 0 0
\(629\) −6996.00 −0.443480
\(630\) 0 0
\(631\) 13840.0 0.873156 0.436578 0.899666i \(-0.356190\pi\)
0.436578 + 0.899666i \(0.356190\pi\)
\(632\) 0 0
\(633\) 10812.0 0.678892
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −29832.0 −1.85555
\(638\) 0 0
\(639\) −1134.00 −0.0702040
\(640\) 0 0
\(641\) 23622.0 1.45556 0.727779 0.685812i \(-0.240553\pi\)
0.727779 + 0.685812i \(0.240553\pi\)
\(642\) 0 0
\(643\) 22772.0 1.39664 0.698320 0.715785i \(-0.253931\pi\)
0.698320 + 0.715785i \(0.253931\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5718.00 −0.347446 −0.173723 0.984795i \(-0.555580\pi\)
−0.173723 + 0.984795i \(0.555580\pi\)
\(648\) 0 0
\(649\) −6072.00 −0.367252
\(650\) 0 0
\(651\) −48.0000 −0.00288981
\(652\) 0 0
\(653\) −6852.00 −0.410627 −0.205314 0.978696i \(-0.565821\pi\)
−0.205314 + 0.978696i \(0.565821\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1494.00 −0.0887161
\(658\) 0 0
\(659\) 10188.0 0.602228 0.301114 0.953588i \(-0.402642\pi\)
0.301114 + 0.953588i \(0.402642\pi\)
\(660\) 0 0
\(661\) 9094.00 0.535122 0.267561 0.963541i \(-0.413782\pi\)
0.267561 + 0.963541i \(0.413782\pi\)
\(662\) 0 0
\(663\) 17424.0 1.02065
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −324.000 −0.0188086
\(668\) 0 0
\(669\) −18336.0 −1.05966
\(670\) 0 0
\(671\) 4444.00 0.255676
\(672\) 0 0
\(673\) −8362.00 −0.478947 −0.239474 0.970903i \(-0.576975\pi\)
−0.239474 + 0.970903i \(0.576975\pi\)
\(674\) 0 0
\(675\) 3375.00 0.192450
\(676\) 0 0
\(677\) 31686.0 1.79881 0.899403 0.437120i \(-0.144002\pi\)
0.899403 + 0.437120i \(0.144002\pi\)
\(678\) 0 0
\(679\) 1604.00 0.0906567
\(680\) 0 0
\(681\) 8388.00 0.471995
\(682\) 0 0
\(683\) 22248.0 1.24641 0.623204 0.782060i \(-0.285831\pi\)
0.623204 + 0.782060i \(0.285831\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −642.000 −0.0356533
\(688\) 0 0
\(689\) 35904.0 1.98524
\(690\) 0 0
\(691\) 12860.0 0.707985 0.353992 0.935248i \(-0.384824\pi\)
0.353992 + 0.935248i \(0.384824\pi\)
\(692\) 0 0
\(693\) 198.000 0.0108534
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −23364.0 −1.26969
\(698\) 0 0
\(699\) 15426.0 0.834714
\(700\) 0 0
\(701\) −7878.00 −0.424462 −0.212231 0.977220i \(-0.568073\pi\)
−0.212231 + 0.977220i \(0.568073\pi\)
\(702\) 0 0
\(703\) −4240.00 −0.227475
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3420.00 −0.181927
\(708\) 0 0
\(709\) 22966.0 1.21651 0.608255 0.793741i \(-0.291870\pi\)
0.608255 + 0.793741i \(0.291870\pi\)
\(710\) 0 0
\(711\) 7866.00 0.414906
\(712\) 0 0
\(713\) 48.0000 0.00252120
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −468.000 −0.0243763
\(718\) 0 0
\(719\) 1194.00 0.0619314 0.0309657 0.999520i \(-0.490142\pi\)
0.0309657 + 0.999520i \(0.490142\pi\)
\(720\) 0 0
\(721\) 1144.00 0.0590912
\(722\) 0 0
\(723\) −10050.0 −0.516962
\(724\) 0 0
\(725\) −6750.00 −0.345778
\(726\) 0 0
\(727\) 16252.0 0.829097 0.414548 0.910027i \(-0.363940\pi\)
0.414548 + 0.910027i \(0.363940\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 8184.00 0.414085
\(732\) 0 0
\(733\) −3368.00 −0.169713 −0.0848567 0.996393i \(-0.527043\pi\)
−0.0848567 + 0.996393i \(0.527043\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 44.0000 0.00219913
\(738\) 0 0
\(739\) 33044.0 1.64485 0.822424 0.568874i \(-0.192621\pi\)
0.822424 + 0.568874i \(0.192621\pi\)
\(740\) 0 0
\(741\) 10560.0 0.523524
\(742\) 0 0
\(743\) 39660.0 1.95826 0.979128 0.203244i \(-0.0651486\pi\)
0.979128 + 0.203244i \(0.0651486\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3996.00 0.195724
\(748\) 0 0
\(749\) 216.000 0.0105373
\(750\) 0 0
\(751\) 19240.0 0.934857 0.467428 0.884031i \(-0.345181\pi\)
0.467428 + 0.884031i \(0.345181\pi\)
\(752\) 0 0
\(753\) 16848.0 0.815372
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1310.00 −0.0628966 −0.0314483 0.999505i \(-0.510012\pi\)
−0.0314483 + 0.999505i \(0.510012\pi\)
\(758\) 0 0
\(759\) −198.000 −0.00946897
\(760\) 0 0
\(761\) 6918.00 0.329537 0.164768 0.986332i \(-0.447312\pi\)
0.164768 + 0.986332i \(0.447312\pi\)
\(762\) 0 0
\(763\) −1424.00 −0.0675652
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 48576.0 2.28680
\(768\) 0 0
\(769\) −25498.0 −1.19568 −0.597842 0.801614i \(-0.703975\pi\)
−0.597842 + 0.801614i \(0.703975\pi\)
\(770\) 0 0
\(771\) −21366.0 −0.998026
\(772\) 0 0
\(773\) −17688.0 −0.823018 −0.411509 0.911406i \(-0.634998\pi\)
−0.411509 + 0.911406i \(0.634998\pi\)
\(774\) 0 0
\(775\) 1000.00 0.0463498
\(776\) 0 0
\(777\) 636.000 0.0293647
\(778\) 0 0
\(779\) −14160.0 −0.651264
\(780\) 0 0
\(781\) 1386.00 0.0635019
\(782\) 0 0
\(783\) −1458.00 −0.0665449
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3364.00 −0.152368 −0.0761840 0.997094i \(-0.524274\pi\)
−0.0761840 + 0.997094i \(0.524274\pi\)
\(788\) 0 0
\(789\) −16704.0 −0.753711
\(790\) 0 0
\(791\) 2604.00 0.117051
\(792\) 0 0
\(793\) −35552.0 −1.59204
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15456.0 0.686925 0.343463 0.939166i \(-0.388400\pi\)
0.343463 + 0.939166i \(0.388400\pi\)
\(798\) 0 0
\(799\) 36036.0 1.59557
\(800\) 0 0
\(801\) 9018.00 0.397797
\(802\) 0 0
\(803\) 1826.00 0.0802468
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6048.00 0.263816
\(808\) 0 0
\(809\) 8154.00 0.354363 0.177181 0.984178i \(-0.443302\pi\)
0.177181 + 0.984178i \(0.443302\pi\)
\(810\) 0 0
\(811\) −28660.0 −1.24092 −0.620462 0.784237i \(-0.713055\pi\)
−0.620462 + 0.784237i \(0.713055\pi\)
\(812\) 0 0
\(813\) −17922.0 −0.773127
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4960.00 0.212397
\(818\) 0 0
\(819\) −1584.00 −0.0675817
\(820\) 0 0
\(821\) −6570.00 −0.279287 −0.139643 0.990202i \(-0.544596\pi\)
−0.139643 + 0.990202i \(0.544596\pi\)
\(822\) 0 0
\(823\) 26824.0 1.13612 0.568059 0.822988i \(-0.307694\pi\)
0.568059 + 0.822988i \(0.307694\pi\)
\(824\) 0 0
\(825\) −4125.00 −0.174078
\(826\) 0 0
\(827\) 31452.0 1.32248 0.661241 0.750173i \(-0.270030\pi\)
0.661241 + 0.750173i \(0.270030\pi\)
\(828\) 0 0
\(829\) 15178.0 0.635891 0.317946 0.948109i \(-0.397007\pi\)
0.317946 + 0.948109i \(0.397007\pi\)
\(830\) 0 0
\(831\) −3972.00 −0.165809
\(832\) 0 0
\(833\) 22374.0 0.930628
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 216.000 0.00892001
\(838\) 0 0
\(839\) −12066.0 −0.496501 −0.248251 0.968696i \(-0.579856\pi\)
−0.248251 + 0.968696i \(0.579856\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 0 0
\(843\) 15534.0 0.634661
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −242.000 −0.00981726
\(848\) 0 0
\(849\) 13476.0 0.544753
\(850\) 0 0
\(851\) −636.000 −0.0256190
\(852\) 0 0
\(853\) −68.0000 −0.00272951 −0.00136476 0.999999i \(-0.500434\pi\)
−0.00136476 + 0.999999i \(0.500434\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −25158.0 −1.00278 −0.501389 0.865222i \(-0.667177\pi\)
−0.501389 + 0.865222i \(0.667177\pi\)
\(858\) 0 0
\(859\) −13732.0 −0.545436 −0.272718 0.962094i \(-0.587923\pi\)
−0.272718 + 0.962094i \(0.587923\pi\)
\(860\) 0 0
\(861\) 2124.00 0.0840717
\(862\) 0 0
\(863\) 32694.0 1.28959 0.644795 0.764355i \(-0.276943\pi\)
0.644795 + 0.764355i \(0.276943\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1671.00 0.0654558
\(868\) 0 0
\(869\) −9614.00 −0.375296
\(870\) 0 0
\(871\) −352.000 −0.0136935
\(872\) 0 0
\(873\) −7218.00 −0.279831
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −34976.0 −1.34670 −0.673350 0.739324i \(-0.735145\pi\)
−0.673350 + 0.739324i \(0.735145\pi\)
\(878\) 0 0
\(879\) 1458.00 0.0559467
\(880\) 0 0
\(881\) 39354.0 1.50496 0.752480 0.658615i \(-0.228857\pi\)
0.752480 + 0.658615i \(0.228857\pi\)
\(882\) 0 0
\(883\) −9820.00 −0.374257 −0.187129 0.982335i \(-0.559918\pi\)
−0.187129 + 0.982335i \(0.559918\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −25356.0 −0.959832 −0.479916 0.877314i \(-0.659333\pi\)
−0.479916 + 0.877314i \(0.659333\pi\)
\(888\) 0 0
\(889\) 1708.00 0.0644370
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 0 0
\(893\) 21840.0 0.818419
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1584.00 0.0589612
\(898\) 0 0
\(899\) −432.000 −0.0160267
\(900\) 0 0
\(901\) −26928.0 −0.995673
\(902\) 0 0
\(903\) −744.000 −0.0274183
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 40892.0 1.49702 0.748510 0.663124i \(-0.230770\pi\)
0.748510 + 0.663124i \(0.230770\pi\)
\(908\) 0 0
\(909\) 15390.0 0.561556
\(910\) 0 0
\(911\) −27966.0 −1.01707 −0.508537 0.861040i \(-0.669814\pi\)
−0.508537 + 0.861040i \(0.669814\pi\)
\(912\) 0 0
\(913\) −4884.00 −0.177039
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3096.00 0.111493
\(918\) 0 0
\(919\) 14038.0 0.503886 0.251943 0.967742i \(-0.418931\pi\)
0.251943 + 0.967742i \(0.418931\pi\)
\(920\) 0 0
\(921\) 28992.0 1.03726
\(922\) 0 0
\(923\) −11088.0 −0.395413
\(924\) 0 0
\(925\) −13250.0 −0.470981
\(926\) 0 0
\(927\) −5148.00 −0.182398
\(928\) 0 0
\(929\) −4170.00 −0.147269 −0.0736347 0.997285i \(-0.523460\pi\)
−0.0736347 + 0.997285i \(0.523460\pi\)
\(930\) 0 0
\(931\) 13560.0 0.477348
\(932\) 0 0
\(933\) 27738.0 0.973313
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −41830.0 −1.45841 −0.729203 0.684297i \(-0.760109\pi\)
−0.729203 + 0.684297i \(0.760109\pi\)
\(938\) 0 0
\(939\) 29154.0 1.01321
\(940\) 0 0
\(941\) −28458.0 −0.985871 −0.492935 0.870066i \(-0.664076\pi\)
−0.492935 + 0.870066i \(0.664076\pi\)
\(942\) 0 0
\(943\) −2124.00 −0.0733478
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9420.00 0.323241 0.161620 0.986853i \(-0.448328\pi\)
0.161620 + 0.986853i \(0.448328\pi\)
\(948\) 0 0
\(949\) −14608.0 −0.499679
\(950\) 0 0
\(951\) −21708.0 −0.740200
\(952\) 0 0
\(953\) −7734.00 −0.262884 −0.131442 0.991324i \(-0.541961\pi\)
−0.131442 + 0.991324i \(0.541961\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1782.00 0.0601921
\(958\) 0 0
\(959\) 2388.00 0.0804093
\(960\) 0 0
\(961\) −29727.0 −0.997852
\(962\) 0 0
\(963\) −972.000 −0.0325257
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −31670.0 −1.05319 −0.526597 0.850115i \(-0.676532\pi\)
−0.526597 + 0.850115i \(0.676532\pi\)
\(968\) 0 0
\(969\) −7920.00 −0.262567
\(970\) 0 0
\(971\) −16680.0 −0.551274 −0.275637 0.961262i \(-0.588889\pi\)
−0.275637 + 0.961262i \(0.588889\pi\)
\(972\) 0 0
\(973\) 3632.00 0.119668
\(974\) 0 0
\(975\) 33000.0 1.08394
\(976\) 0 0
\(977\) 26718.0 0.874907 0.437454 0.899241i \(-0.355880\pi\)
0.437454 + 0.899241i \(0.355880\pi\)
\(978\) 0 0
\(979\) −11022.0 −0.359821
\(980\) 0 0
\(981\) 6408.00 0.208554
\(982\) 0 0
\(983\) −20238.0 −0.656655 −0.328328 0.944564i \(-0.606485\pi\)
−0.328328 + 0.944564i \(0.606485\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −3276.00 −0.105650
\(988\) 0 0
\(989\) 744.000 0.0239210
\(990\) 0 0
\(991\) −25544.0 −0.818801 −0.409401 0.912355i \(-0.634262\pi\)
−0.409401 + 0.912355i \(0.634262\pi\)
\(992\) 0 0
\(993\) 31620.0 1.01050
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −15968.0 −0.507233 −0.253617 0.967305i \(-0.581620\pi\)
−0.253617 + 0.967305i \(0.581620\pi\)
\(998\) 0 0
\(999\) −2862.00 −0.0906403
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 2112.4.a.f.1.1 1
4.3 odd 2 2112.4.a.t.1.1 1
8.3 odd 2 132.4.a.b.1.1 1
8.5 even 2 528.4.a.i.1.1 1
24.5 odd 2 1584.4.a.j.1.1 1
24.11 even 2 396.4.a.d.1.1 1
88.43 even 2 1452.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.a.b.1.1 1 8.3 odd 2
396.4.a.d.1.1 1 24.11 even 2
528.4.a.i.1.1 1 8.5 even 2
1452.4.a.b.1.1 1 88.43 even 2
1584.4.a.j.1.1 1 24.5 odd 2
2112.4.a.f.1.1 1 1.1 even 1 trivial
2112.4.a.t.1.1 1 4.3 odd 2