Properties

Label 2200.2.a.x.1.2
Level $2200$
Weight $2$
Character 2200.1
Self dual yes
Analytic conductor $17.567$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(17.5670884447\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.655762\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-0.655762 q^{3} +0.415806 q^{7} -2.56998 q^{9} +O(q^{10})\) \(q-0.655762 q^{3} +0.415806 q^{7} -2.56998 q^{9} +1.00000 q^{11} -4.00000 q^{13} +6.51558 q^{17} +5.20406 q^{19} -0.272670 q^{21} -8.54830 q^{23} +3.65258 q^{27} +0.895717 q^{29} -6.73836 q^{31} -0.655762 q^{33} -8.96410 q^{37} +2.62305 q^{39} +10.0998 q^{41} +4.78825 q^{43} -5.61986 q^{47} -6.82711 q^{49} -4.27267 q^{51} -10.0357 q^{53} -3.41262 q^{57} -1.63408 q^{59} +7.10428 q^{61} -1.06861 q^{63} -10.6914 q^{67} +5.60565 q^{69} -6.19302 q^{71} -3.16839 q^{73} +0.415806 q^{77} -11.2682 q^{79} +5.31471 q^{81} -16.2429 q^{83} -0.587377 q^{87} +9.56998 q^{89} -1.66323 q^{91} +4.41876 q^{93} -0.591657 q^{97} -2.56998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - q^{7} + 7 q^{9} + 4 q^{11} - 16 q^{13} - 7 q^{17} - 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} - 15 q^{31} - q^{33} - 5 q^{37} + 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49} - 23 q^{51} - 5 q^{53} + 15 q^{57} + 6 q^{59} + 29 q^{61} - 40 q^{63} - 6 q^{67} - 30 q^{69} - q^{71} - 18 q^{73} - q^{77} - 20 q^{79} + 44 q^{81} - 26 q^{83} - 31 q^{87} + 21 q^{89} + 4 q^{91} - 25 q^{93} + 4 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.655762 −0.378604 −0.189302 0.981919i \(-0.560623\pi\)
−0.189302 + 0.981919i \(0.560623\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.415806 0.157160 0.0785800 0.996908i \(-0.474961\pi\)
0.0785800 + 0.996908i \(0.474961\pi\)
\(8\) 0 0
\(9\) −2.56998 −0.856659
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.51558 1.58026 0.790130 0.612939i \(-0.210013\pi\)
0.790130 + 0.612939i \(0.210013\pi\)
\(18\) 0 0
\(19\) 5.20406 1.19389 0.596946 0.802281i \(-0.296381\pi\)
0.596946 + 0.802281i \(0.296381\pi\)
\(20\) 0 0
\(21\) −0.272670 −0.0595015
\(22\) 0 0
\(23\) −8.54830 −1.78244 −0.891221 0.453568i \(-0.850151\pi\)
−0.891221 + 0.453568i \(0.850151\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.65258 0.702939
\(28\) 0 0
\(29\) 0.895717 0.166331 0.0831653 0.996536i \(-0.473497\pi\)
0.0831653 + 0.996536i \(0.473497\pi\)
\(30\) 0 0
\(31\) −6.73836 −1.21025 −0.605123 0.796132i \(-0.706876\pi\)
−0.605123 + 0.796132i \(0.706876\pi\)
\(32\) 0 0
\(33\) −0.655762 −0.114153
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.96410 −1.47369 −0.736845 0.676062i \(-0.763685\pi\)
−0.736845 + 0.676062i \(0.763685\pi\)
\(38\) 0 0
\(39\) 2.62305 0.420024
\(40\) 0 0
\(41\) 10.0998 1.57732 0.788660 0.614830i \(-0.210775\pi\)
0.788660 + 0.614830i \(0.210775\pi\)
\(42\) 0 0
\(43\) 4.78825 0.730201 0.365101 0.930968i \(-0.381035\pi\)
0.365101 + 0.930968i \(0.381035\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.61986 −0.819741 −0.409871 0.912144i \(-0.634426\pi\)
−0.409871 + 0.912144i \(0.634426\pi\)
\(48\) 0 0
\(49\) −6.82711 −0.975301
\(50\) 0 0
\(51\) −4.27267 −0.598293
\(52\) 0 0
\(53\) −10.0357 −1.37851 −0.689253 0.724521i \(-0.742061\pi\)
−0.689253 + 0.724521i \(0.742061\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.41262 −0.452013
\(58\) 0 0
\(59\) −1.63408 −0.212739 −0.106370 0.994327i \(-0.533923\pi\)
−0.106370 + 0.994327i \(0.533923\pi\)
\(60\) 0 0
\(61\) 7.10428 0.909610 0.454805 0.890591i \(-0.349709\pi\)
0.454805 + 0.890591i \(0.349709\pi\)
\(62\) 0 0
\(63\) −1.06861 −0.134633
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.6914 −1.30617 −0.653083 0.757286i \(-0.726525\pi\)
−0.653083 + 0.757286i \(0.726525\pi\)
\(68\) 0 0
\(69\) 5.60565 0.674841
\(70\) 0 0
\(71\) −6.19302 −0.734977 −0.367488 0.930028i \(-0.619782\pi\)
−0.367488 + 0.930028i \(0.619782\pi\)
\(72\) 0 0
\(73\) −3.16839 −0.370832 −0.185416 0.982660i \(-0.559363\pi\)
−0.185416 + 0.982660i \(0.559363\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.415806 0.0473855
\(78\) 0 0
\(79\) −11.2682 −1.26777 −0.633884 0.773428i \(-0.718540\pi\)
−0.633884 + 0.773428i \(0.718540\pi\)
\(80\) 0 0
\(81\) 5.31471 0.590523
\(82\) 0 0
\(83\) −16.2429 −1.78289 −0.891446 0.453128i \(-0.850308\pi\)
−0.891446 + 0.453128i \(0.850308\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.587377 −0.0629735
\(88\) 0 0
\(89\) 9.56998 1.01442 0.507208 0.861824i \(-0.330678\pi\)
0.507208 + 0.861824i \(0.330678\pi\)
\(90\) 0 0
\(91\) −1.66323 −0.174353
\(92\) 0 0
\(93\) 4.41876 0.458204
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.591657 −0.0600737 −0.0300368 0.999549i \(-0.509562\pi\)
−0.0300368 + 0.999549i \(0.509562\pi\)
\(98\) 0 0
\(99\) −2.56998 −0.258292
\(100\) 0 0
\(101\) −4.62305 −0.460010 −0.230005 0.973189i \(-0.573874\pi\)
−0.230005 + 0.973189i \(0.573874\pi\)
\(102\) 0 0
\(103\) 8.24291 0.812198 0.406099 0.913829i \(-0.366889\pi\)
0.406099 + 0.913829i \(0.366889\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.6199 −1.70338 −0.851688 0.524049i \(-0.824421\pi\)
−0.851688 + 0.524049i \(0.824421\pi\)
\(108\) 0 0
\(109\) 3.66323 0.350873 0.175437 0.984491i \(-0.443866\pi\)
0.175437 + 0.984491i \(0.443866\pi\)
\(110\) 0 0
\(111\) 5.87832 0.557945
\(112\) 0 0
\(113\) 11.4797 1.07992 0.539959 0.841691i \(-0.318440\pi\)
0.539959 + 0.841691i \(0.318440\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 10.2799 0.950378
\(118\) 0 0
\(119\) 2.70922 0.248354
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.62305 −0.597180
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.13995 0.811040 0.405520 0.914086i \(-0.367091\pi\)
0.405520 + 0.914086i \(0.367091\pi\)
\(128\) 0 0
\(129\) −3.13995 −0.276457
\(130\) 0 0
\(131\) −2.48123 −0.216787 −0.108393 0.994108i \(-0.534571\pi\)
−0.108393 + 0.994108i \(0.534571\pi\)
\(132\) 0 0
\(133\) 2.16388 0.187632
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.40834 −0.632937 −0.316469 0.948603i \(-0.602497\pi\)
−0.316469 + 0.948603i \(0.602497\pi\)
\(138\) 0 0
\(139\) −1.69166 −0.143485 −0.0717424 0.997423i \(-0.522856\pi\)
−0.0717424 + 0.997423i \(0.522856\pi\)
\(140\) 0 0
\(141\) 3.68529 0.310358
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.47696 0.369253
\(148\) 0 0
\(149\) 3.74940 0.307163 0.153581 0.988136i \(-0.450919\pi\)
0.153581 + 0.988136i \(0.450919\pi\)
\(150\) 0 0
\(151\) −15.1400 −1.23207 −0.616036 0.787718i \(-0.711262\pi\)
−0.616036 + 0.787718i \(0.711262\pi\)
\(152\) 0 0
\(153\) −16.7449 −1.35374
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.5871 −0.924755 −0.462378 0.886683i \(-0.653004\pi\)
−0.462378 + 0.886683i \(0.653004\pi\)
\(158\) 0 0
\(159\) 6.58101 0.521908
\(160\) 0 0
\(161\) −3.55444 −0.280129
\(162\) 0 0
\(163\) −6.82392 −0.534491 −0.267245 0.963629i \(-0.586113\pi\)
−0.267245 + 0.963629i \(0.586113\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.10428 −0.549746 −0.274873 0.961481i \(-0.588636\pi\)
−0.274873 + 0.961481i \(0.588636\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −13.3743 −1.02276
\(172\) 0 0
\(173\) −9.31152 −0.707942 −0.353971 0.935256i \(-0.615169\pi\)
−0.353971 + 0.935256i \(0.615169\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.07157 0.0805440
\(178\) 0 0
\(179\) −26.6368 −1.99093 −0.995464 0.0951359i \(-0.969671\pi\)
−0.995464 + 0.0951359i \(0.969671\pi\)
\(180\) 0 0
\(181\) 1.01103 0.0751495 0.0375748 0.999294i \(-0.488037\pi\)
0.0375748 + 0.999294i \(0.488037\pi\)
\(182\) 0 0
\(183\) −4.65872 −0.344382
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.51558 0.476466
\(188\) 0 0
\(189\) 1.51877 0.110474
\(190\) 0 0
\(191\) 10.5655 0.764490 0.382245 0.924061i \(-0.375151\pi\)
0.382245 + 0.924061i \(0.375151\pi\)
\(192\) 0 0
\(193\) −11.8271 −0.851334 −0.425667 0.904880i \(-0.639960\pi\)
−0.425667 + 0.904880i \(0.639960\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.31152 0.0934422 0.0467211 0.998908i \(-0.485123\pi\)
0.0467211 + 0.998908i \(0.485123\pi\)
\(198\) 0 0
\(199\) −11.8271 −0.838401 −0.419201 0.907894i \(-0.637690\pi\)
−0.419201 + 0.907894i \(0.637690\pi\)
\(200\) 0 0
\(201\) 7.01103 0.494520
\(202\) 0 0
\(203\) 0.372445 0.0261405
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 21.9689 1.52695
\(208\) 0 0
\(209\) 5.20406 0.359972
\(210\) 0 0
\(211\) −8.37244 −0.576383 −0.288191 0.957573i \(-0.593054\pi\)
−0.288191 + 0.957573i \(0.593054\pi\)
\(212\) 0 0
\(213\) 4.06115 0.278265
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.80185 −0.190202
\(218\) 0 0
\(219\) 2.07771 0.140398
\(220\) 0 0
\(221\) −26.0623 −1.75314
\(222\) 0 0
\(223\) 22.1461 1.48301 0.741506 0.670946i \(-0.234112\pi\)
0.741506 + 0.670946i \(0.234112\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.8194 1.44821 0.724103 0.689692i \(-0.242254\pi\)
0.724103 + 0.689692i \(0.242254\pi\)
\(228\) 0 0
\(229\) 2.80247 0.185192 0.0925962 0.995704i \(-0.470483\pi\)
0.0925962 + 0.995704i \(0.470483\pi\)
\(230\) 0 0
\(231\) −0.272670 −0.0179404
\(232\) 0 0
\(233\) −8.24424 −0.540098 −0.270049 0.962847i \(-0.587040\pi\)
−0.270049 + 0.962847i \(0.587040\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.38923 0.479982
\(238\) 0 0
\(239\) −11.2682 −0.728877 −0.364438 0.931227i \(-0.618739\pi\)
−0.364438 + 0.931227i \(0.618739\pi\)
\(240\) 0 0
\(241\) 22.3860 1.44201 0.721006 0.692929i \(-0.243680\pi\)
0.721006 + 0.692929i \(0.243680\pi\)
\(242\) 0 0
\(243\) −14.4429 −0.926514
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −20.8162 −1.32451
\(248\) 0 0
\(249\) 10.6515 0.675010
\(250\) 0 0
\(251\) 3.84265 0.242546 0.121273 0.992619i \(-0.461302\pi\)
0.121273 + 0.992619i \(0.461302\pi\)
\(252\) 0 0
\(253\) −8.54830 −0.537427
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.6230 −0.662648 −0.331324 0.943517i \(-0.607495\pi\)
−0.331324 + 0.943517i \(0.607495\pi\)
\(258\) 0 0
\(259\) −3.72733 −0.231605
\(260\) 0 0
\(261\) −2.30197 −0.142489
\(262\) 0 0
\(263\) −5.79276 −0.357197 −0.178598 0.983922i \(-0.557156\pi\)
−0.178598 + 0.983922i \(0.557156\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.27563 −0.384062
\(268\) 0 0
\(269\) −8.20857 −0.500485 −0.250243 0.968183i \(-0.580510\pi\)
−0.250243 + 0.968183i \(0.580510\pi\)
\(270\) 0 0
\(271\) 27.3111 1.65903 0.829515 0.558485i \(-0.188617\pi\)
0.829515 + 0.558485i \(0.188617\pi\)
\(272\) 0 0
\(273\) 1.09068 0.0660109
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.4735 0.869631 0.434815 0.900520i \(-0.356814\pi\)
0.434815 + 0.900520i \(0.356814\pi\)
\(278\) 0 0
\(279\) 17.3174 1.03677
\(280\) 0 0
\(281\) 16.2799 0.971178 0.485589 0.874187i \(-0.338605\pi\)
0.485589 + 0.874187i \(0.338605\pi\)
\(282\) 0 0
\(283\) −10.5169 −0.625165 −0.312583 0.949891i \(-0.601194\pi\)
−0.312583 + 0.949891i \(0.601194\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.19955 0.247892
\(288\) 0 0
\(289\) 25.4528 1.49722
\(290\) 0 0
\(291\) 0.387986 0.0227441
\(292\) 0 0
\(293\) 9.37695 0.547807 0.273904 0.961757i \(-0.411685\pi\)
0.273904 + 0.961757i \(0.411685\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.65258 0.211944
\(298\) 0 0
\(299\) 34.1932 1.97744
\(300\) 0 0
\(301\) 1.99099 0.114758
\(302\) 0 0
\(303\) 3.03162 0.174162
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.09978 −0.119840 −0.0599202 0.998203i \(-0.519085\pi\)
−0.0599202 + 0.998203i \(0.519085\pi\)
\(308\) 0 0
\(309\) −5.40539 −0.307502
\(310\) 0 0
\(311\) −20.3724 −1.15522 −0.577608 0.816314i \(-0.696014\pi\)
−0.577608 + 0.816314i \(0.696014\pi\)
\(312\) 0 0
\(313\) 23.8968 1.35073 0.675364 0.737485i \(-0.263987\pi\)
0.675364 + 0.737485i \(0.263987\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.86114 0.329195 0.164597 0.986361i \(-0.447368\pi\)
0.164597 + 0.986361i \(0.447368\pi\)
\(318\) 0 0
\(319\) 0.895717 0.0501506
\(320\) 0 0
\(321\) 11.5544 0.644906
\(322\) 0 0
\(323\) 33.9075 1.88666
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.40220 −0.132842
\(328\) 0 0
\(329\) −2.33677 −0.128831
\(330\) 0 0
\(331\) −0.574484 −0.0315765 −0.0157882 0.999875i \(-0.505026\pi\)
−0.0157882 + 0.999875i \(0.505026\pi\)
\(332\) 0 0
\(333\) 23.0375 1.26245
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.10747 0.223748 0.111874 0.993722i \(-0.464315\pi\)
0.111874 + 0.993722i \(0.464315\pi\)
\(338\) 0 0
\(339\) −7.52794 −0.408862
\(340\) 0 0
\(341\) −6.73836 −0.364903
\(342\) 0 0
\(343\) −5.74940 −0.310438
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.30834 0.446015 0.223008 0.974817i \(-0.428413\pi\)
0.223008 + 0.974817i \(0.428413\pi\)
\(348\) 0 0
\(349\) −18.4858 −0.989523 −0.494762 0.869029i \(-0.664745\pi\)
−0.494762 + 0.869029i \(0.664745\pi\)
\(350\) 0 0
\(351\) −14.6103 −0.779841
\(352\) 0 0
\(353\) −26.5199 −1.41151 −0.705755 0.708456i \(-0.749392\pi\)
−0.705755 + 0.708456i \(0.749392\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.77660 −0.0940278
\(358\) 0 0
\(359\) −18.8226 −0.993419 −0.496709 0.867917i \(-0.665459\pi\)
−0.496709 + 0.867917i \(0.665459\pi\)
\(360\) 0 0
\(361\) 8.08222 0.425380
\(362\) 0 0
\(363\) −0.655762 −0.0344186
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.2115 0.533037 0.266519 0.963830i \(-0.414127\pi\)
0.266519 + 0.963830i \(0.414127\pi\)
\(368\) 0 0
\(369\) −25.9562 −1.35122
\(370\) 0 0
\(371\) −4.17289 −0.216646
\(372\) 0 0
\(373\) 32.3363 1.67431 0.837156 0.546965i \(-0.184217\pi\)
0.837156 + 0.546965i \(0.184217\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.58287 −0.184527
\(378\) 0 0
\(379\) 15.3686 0.789434 0.394717 0.918803i \(-0.370843\pi\)
0.394717 + 0.918803i \(0.370843\pi\)
\(380\) 0 0
\(381\) −5.99363 −0.307063
\(382\) 0 0
\(383\) 13.6662 0.698309 0.349155 0.937065i \(-0.386469\pi\)
0.349155 + 0.937065i \(0.386469\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.3057 −0.625534
\(388\) 0 0
\(389\) −9.01103 −0.456878 −0.228439 0.973558i \(-0.573362\pi\)
−0.228439 + 0.973558i \(0.573362\pi\)
\(390\) 0 0
\(391\) −55.6971 −2.81672
\(392\) 0 0
\(393\) 1.62710 0.0820763
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.45466 0.173384 0.0866922 0.996235i \(-0.472370\pi\)
0.0866922 + 0.996235i \(0.472370\pi\)
\(398\) 0 0
\(399\) −1.41899 −0.0710384
\(400\) 0 0
\(401\) −1.62756 −0.0812762 −0.0406381 0.999174i \(-0.512939\pi\)
−0.0406381 + 0.999174i \(0.512939\pi\)
\(402\) 0 0
\(403\) 26.9535 1.34265
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.96410 −0.444334
\(408\) 0 0
\(409\) 6.54534 0.323646 0.161823 0.986820i \(-0.448263\pi\)
0.161823 + 0.986820i \(0.448263\pi\)
\(410\) 0 0
\(411\) 4.85811 0.239633
\(412\) 0 0
\(413\) −0.679461 −0.0334341
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.10933 0.0543239
\(418\) 0 0
\(419\) −34.4795 −1.68443 −0.842216 0.539141i \(-0.818749\pi\)
−0.842216 + 0.539141i \(0.818749\pi\)
\(420\) 0 0
\(421\) 6.54534 0.319000 0.159500 0.987198i \(-0.449012\pi\)
0.159500 + 0.987198i \(0.449012\pi\)
\(422\) 0 0
\(423\) 14.4429 0.702239
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.95401 0.142954
\(428\) 0 0
\(429\) 2.62305 0.126642
\(430\) 0 0
\(431\) 36.1711 1.74230 0.871151 0.491016i \(-0.163374\pi\)
0.871151 + 0.491016i \(0.163374\pi\)
\(432\) 0 0
\(433\) 38.1027 1.83110 0.915550 0.402204i \(-0.131756\pi\)
0.915550 + 0.402204i \(0.131756\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −44.4858 −2.12805
\(438\) 0 0
\(439\) 12.1282 0.578848 0.289424 0.957201i \(-0.406536\pi\)
0.289424 + 0.957201i \(0.406536\pi\)
\(440\) 0 0
\(441\) 17.5455 0.835500
\(442\) 0 0
\(443\) 16.5483 0.786233 0.393117 0.919489i \(-0.371397\pi\)
0.393117 + 0.919489i \(0.371397\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2.45871 −0.116293
\(448\) 0 0
\(449\) 15.6056 0.736476 0.368238 0.929732i \(-0.379961\pi\)
0.368238 + 0.929732i \(0.379961\pi\)
\(450\) 0 0
\(451\) 10.0998 0.475580
\(452\) 0 0
\(453\) 9.92820 0.466468
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.87761 0.274943 0.137471 0.990506i \(-0.456102\pi\)
0.137471 + 0.990506i \(0.456102\pi\)
\(458\) 0 0
\(459\) 23.7987 1.11083
\(460\) 0 0
\(461\) 36.7495 1.71159 0.855797 0.517312i \(-0.173067\pi\)
0.855797 + 0.517312i \(0.173067\pi\)
\(462\) 0 0
\(463\) 19.3081 0.897324 0.448662 0.893702i \(-0.351901\pi\)
0.448662 + 0.893702i \(0.351901\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.4129 0.528127 0.264064 0.964505i \(-0.414937\pi\)
0.264064 + 0.964505i \(0.414937\pi\)
\(468\) 0 0
\(469\) −4.44556 −0.205277
\(470\) 0 0
\(471\) 7.59841 0.350116
\(472\) 0 0
\(473\) 4.78825 0.220164
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 25.7914 1.18091
\(478\) 0 0
\(479\) 29.2307 1.33559 0.667793 0.744347i \(-0.267239\pi\)
0.667793 + 0.744347i \(0.267239\pi\)
\(480\) 0 0
\(481\) 35.8564 1.63491
\(482\) 0 0
\(483\) 2.33086 0.106058
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 33.2932 1.50866 0.754329 0.656496i \(-0.227962\pi\)
0.754329 + 0.656496i \(0.227962\pi\)
\(488\) 0 0
\(489\) 4.47487 0.202361
\(490\) 0 0
\(491\) −24.5151 −1.10635 −0.553176 0.833064i \(-0.686584\pi\)
−0.553176 + 0.833064i \(0.686584\pi\)
\(492\) 0 0
\(493\) 5.83612 0.262846
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.57510 −0.115509
\(498\) 0 0
\(499\) −18.9093 −0.846497 −0.423249 0.906014i \(-0.639110\pi\)
−0.423249 + 0.906014i \(0.639110\pi\)
\(500\) 0 0
\(501\) 4.65872 0.208136
\(502\) 0 0
\(503\) −9.42031 −0.420031 −0.210016 0.977698i \(-0.567351\pi\)
−0.210016 + 0.977698i \(0.567351\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.96729 −0.0873702
\(508\) 0 0
\(509\) −16.7804 −0.743778 −0.371889 0.928277i \(-0.621290\pi\)
−0.371889 + 0.928277i \(0.621290\pi\)
\(510\) 0 0
\(511\) −1.31744 −0.0582799
\(512\) 0 0
\(513\) 19.0082 0.839234
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5.61986 −0.247161
\(518\) 0 0
\(519\) 6.10614 0.268030
\(520\) 0 0
\(521\) −37.8273 −1.65724 −0.828621 0.559810i \(-0.810874\pi\)
−0.828621 + 0.559810i \(0.810874\pi\)
\(522\) 0 0
\(523\) 10.0998 0.441632 0.220816 0.975315i \(-0.429128\pi\)
0.220816 + 0.975315i \(0.429128\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −43.9044 −1.91250
\(528\) 0 0
\(529\) 50.0734 2.17710
\(530\) 0 0
\(531\) 4.19955 0.182245
\(532\) 0 0
\(533\) −40.3991 −1.74988
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 17.4674 0.753774
\(538\) 0 0
\(539\) −6.82711 −0.294064
\(540\) 0 0
\(541\) −29.9269 −1.28666 −0.643329 0.765590i \(-0.722447\pi\)
−0.643329 + 0.765590i \(0.722447\pi\)
\(542\) 0 0
\(543\) −0.662997 −0.0284519
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 23.7630 1.01603 0.508016 0.861347i \(-0.330379\pi\)
0.508016 + 0.861347i \(0.330379\pi\)
\(548\) 0 0
\(549\) −18.2578 −0.779226
\(550\) 0 0
\(551\) 4.66137 0.198581
\(552\) 0 0
\(553\) −4.68537 −0.199242
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.3363 −1.03116 −0.515581 0.856841i \(-0.672424\pi\)
−0.515581 + 0.856841i \(0.672424\pi\)
\(558\) 0 0
\(559\) −19.1530 −0.810086
\(560\) 0 0
\(561\) −4.27267 −0.180392
\(562\) 0 0
\(563\) −8.85368 −0.373138 −0.186569 0.982442i \(-0.559737\pi\)
−0.186569 + 0.982442i \(0.559737\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.20989 0.0928066
\(568\) 0 0
\(569\) 26.3860 1.10616 0.553080 0.833128i \(-0.313452\pi\)
0.553080 + 0.833128i \(0.313452\pi\)
\(570\) 0 0
\(571\) 6.45280 0.270041 0.135021 0.990843i \(-0.456890\pi\)
0.135021 + 0.990843i \(0.456890\pi\)
\(572\) 0 0
\(573\) −6.92843 −0.289439
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 12.1745 0.506832 0.253416 0.967357i \(-0.418446\pi\)
0.253416 + 0.967357i \(0.418446\pi\)
\(578\) 0 0
\(579\) 7.75576 0.322319
\(580\) 0 0
\(581\) −6.75390 −0.280199
\(582\) 0 0
\(583\) −10.0357 −0.415635
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −22.9521 −0.947336 −0.473668 0.880704i \(-0.657070\pi\)
−0.473668 + 0.880704i \(0.657070\pi\)
\(588\) 0 0
\(589\) −35.0668 −1.44490
\(590\) 0 0
\(591\) −0.860047 −0.0353776
\(592\) 0 0
\(593\) 10.4081 0.427410 0.213705 0.976898i \(-0.431447\pi\)
0.213705 + 0.976898i \(0.431447\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.75576 0.317422
\(598\) 0 0
\(599\) −8.98913 −0.367286 −0.183643 0.982993i \(-0.558789\pi\)
−0.183643 + 0.982993i \(0.558789\pi\)
\(600\) 0 0
\(601\) −3.57650 −0.145889 −0.0729443 0.997336i \(-0.523240\pi\)
−0.0729443 + 0.997336i \(0.523240\pi\)
\(602\) 0 0
\(603\) 27.4767 1.11894
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 27.0235 1.09685 0.548424 0.836200i \(-0.315228\pi\)
0.548424 + 0.836200i \(0.315228\pi\)
\(608\) 0 0
\(609\) −0.244235 −0.00989691
\(610\) 0 0
\(611\) 22.4795 0.909421
\(612\) 0 0
\(613\) −44.5572 −1.79965 −0.899823 0.436254i \(-0.856305\pi\)
−0.899823 + 0.436254i \(0.856305\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.2966 −0.615818 −0.307909 0.951416i \(-0.599629\pi\)
−0.307909 + 0.951416i \(0.599629\pi\)
\(618\) 0 0
\(619\) 14.5655 0.585436 0.292718 0.956199i \(-0.405440\pi\)
0.292718 + 0.956199i \(0.405440\pi\)
\(620\) 0 0
\(621\) −31.2233 −1.25295
\(622\) 0 0
\(623\) 3.97926 0.159426
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3.41262 −0.136287
\(628\) 0 0
\(629\) −58.4063 −2.32881
\(630\) 0 0
\(631\) −22.0779 −0.878906 −0.439453 0.898266i \(-0.644828\pi\)
−0.439453 + 0.898266i \(0.644828\pi\)
\(632\) 0 0
\(633\) 5.49033 0.218221
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 27.3084 1.08200
\(638\) 0 0
\(639\) 15.9159 0.629624
\(640\) 0 0
\(641\) −31.2152 −1.23293 −0.616463 0.787384i \(-0.711435\pi\)
−0.616463 + 0.787384i \(0.711435\pi\)
\(642\) 0 0
\(643\) −10.4498 −0.412102 −0.206051 0.978541i \(-0.566061\pi\)
−0.206051 + 0.978541i \(0.566061\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26.1307 1.02730 0.513652 0.857999i \(-0.328292\pi\)
0.513652 + 0.857999i \(0.328292\pi\)
\(648\) 0 0
\(649\) −1.63408 −0.0641433
\(650\) 0 0
\(651\) 1.83735 0.0720114
\(652\) 0 0
\(653\) 44.2097 1.73006 0.865030 0.501719i \(-0.167299\pi\)
0.865030 + 0.501719i \(0.167299\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.14268 0.317676
\(658\) 0 0
\(659\) 6.96706 0.271398 0.135699 0.990750i \(-0.456672\pi\)
0.135699 + 0.990750i \(0.456672\pi\)
\(660\) 0 0
\(661\) −9.19753 −0.357743 −0.178871 0.983872i \(-0.557245\pi\)
−0.178871 + 0.983872i \(0.557245\pi\)
\(662\) 0 0
\(663\) 17.0907 0.663747
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.65686 −0.296475
\(668\) 0 0
\(669\) −14.5226 −0.561475
\(670\) 0 0
\(671\) 7.10428 0.274258
\(672\) 0 0
\(673\) −8.72415 −0.336291 −0.168146 0.985762i \(-0.553778\pi\)
−0.168146 + 0.985762i \(0.553778\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.76618 0.336912 0.168456 0.985709i \(-0.446122\pi\)
0.168456 + 0.985709i \(0.446122\pi\)
\(678\) 0 0
\(679\) −0.246015 −0.00944118
\(680\) 0 0
\(681\) −14.3083 −0.548297
\(682\) 0 0
\(683\) −49.2320 −1.88381 −0.941906 0.335877i \(-0.890967\pi\)
−0.941906 + 0.335877i \(0.890967\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.83775 −0.0701146
\(688\) 0 0
\(689\) 40.1427 1.52931
\(690\) 0 0
\(691\) −18.1483 −0.690395 −0.345198 0.938530i \(-0.612188\pi\)
−0.345198 + 0.938530i \(0.612188\pi\)
\(692\) 0 0
\(693\) −1.06861 −0.0405932
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 65.8059 2.49258
\(698\) 0 0
\(699\) 5.40626 0.204483
\(700\) 0 0
\(701\) 22.6587 0.855808 0.427904 0.903824i \(-0.359252\pi\)
0.427904 + 0.903824i \(0.359252\pi\)
\(702\) 0 0
\(703\) −46.6497 −1.75943
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.92229 −0.0722952
\(708\) 0 0
\(709\) −36.7366 −1.37967 −0.689836 0.723966i \(-0.742317\pi\)
−0.689836 + 0.723966i \(0.742317\pi\)
\(710\) 0 0
\(711\) 28.9589 1.08604
\(712\) 0 0
\(713\) 57.6015 2.15719
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.38923 0.275956
\(718\) 0 0
\(719\) 8.74738 0.326222 0.163111 0.986608i \(-0.447847\pi\)
0.163111 + 0.986608i \(0.447847\pi\)
\(720\) 0 0
\(721\) 3.42745 0.127645
\(722\) 0 0
\(723\) −14.6799 −0.545952
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.20887 −0.0819226 −0.0409613 0.999161i \(-0.513042\pi\)
−0.0409613 + 0.999161i \(0.513042\pi\)
\(728\) 0 0
\(729\) −6.47301 −0.239741
\(730\) 0 0
\(731\) 31.1982 1.15391
\(732\) 0 0
\(733\) 3.58552 0.132434 0.0662171 0.997805i \(-0.478907\pi\)
0.0662171 + 0.997805i \(0.478907\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.6914 −0.393824
\(738\) 0 0
\(739\) −24.9314 −0.917116 −0.458558 0.888665i \(-0.651634\pi\)
−0.458558 + 0.888665i \(0.651634\pi\)
\(740\) 0 0
\(741\) 13.6505 0.501463
\(742\) 0 0
\(743\) 21.4257 0.786032 0.393016 0.919532i \(-0.371432\pi\)
0.393016 + 0.919532i \(0.371432\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 41.7439 1.52733
\(748\) 0 0
\(749\) −7.32645 −0.267703
\(750\) 0 0
\(751\) 28.1582 1.02751 0.513754 0.857938i \(-0.328254\pi\)
0.513754 + 0.857938i \(0.328254\pi\)
\(752\) 0 0
\(753\) −2.51986 −0.0918288
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.7513 −0.390761 −0.195381 0.980728i \(-0.562594\pi\)
−0.195381 + 0.980728i \(0.562594\pi\)
\(758\) 0 0
\(759\) 5.60565 0.203472
\(760\) 0 0
\(761\) −38.0429 −1.37905 −0.689527 0.724260i \(-0.742182\pi\)
−0.689527 + 0.724260i \(0.742182\pi\)
\(762\) 0 0
\(763\) 1.52319 0.0551433
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.53632 0.236013
\(768\) 0 0
\(769\) 26.2280 0.945805 0.472903 0.881115i \(-0.343206\pi\)
0.472903 + 0.881115i \(0.343206\pi\)
\(770\) 0 0
\(771\) 6.96619 0.250881
\(772\) 0 0
\(773\) −8.94499 −0.321729 −0.160864 0.986977i \(-0.551428\pi\)
−0.160864 + 0.986977i \(0.551428\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.44424 0.0876867
\(778\) 0 0
\(779\) 52.5598 1.88315
\(780\) 0 0
\(781\) −6.19302 −0.221604
\(782\) 0 0
\(783\) 3.27168 0.116920
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −29.9716 −1.06837 −0.534185 0.845367i \(-0.679382\pi\)
−0.534185 + 0.845367i \(0.679382\pi\)
\(788\) 0 0
\(789\) 3.79867 0.135236
\(790\) 0 0
\(791\) 4.77332 0.169720
\(792\) 0 0
\(793\) −28.4171 −1.00912
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.80022 −0.240876 −0.120438 0.992721i \(-0.538430\pi\)
−0.120438 + 0.992721i \(0.538430\pi\)
\(798\) 0 0
\(799\) −36.6167 −1.29541
\(800\) 0 0
\(801\) −24.5946 −0.869008
\(802\) 0 0
\(803\) −3.16839 −0.111810
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.38286 0.189486
\(808\) 0 0
\(809\) 53.1246 1.86776 0.933880 0.357586i \(-0.116400\pi\)
0.933880 + 0.357586i \(0.116400\pi\)
\(810\) 0 0
\(811\) −30.0487 −1.05515 −0.527577 0.849507i \(-0.676899\pi\)
−0.527577 + 0.849507i \(0.676899\pi\)
\(812\) 0 0
\(813\) −17.9096 −0.628116
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 24.9183 0.871782
\(818\) 0 0
\(819\) 4.27445 0.149361
\(820\) 0 0
\(821\) 19.9069 0.694756 0.347378 0.937725i \(-0.387072\pi\)
0.347378 + 0.937725i \(0.387072\pi\)
\(822\) 0 0
\(823\) 33.2491 1.15899 0.579495 0.814976i \(-0.303250\pi\)
0.579495 + 0.814976i \(0.303250\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.3770 −0.986766 −0.493383 0.869812i \(-0.664240\pi\)
−0.493383 + 0.869812i \(0.664240\pi\)
\(828\) 0 0
\(829\) 3.63143 0.126125 0.0630625 0.998010i \(-0.479913\pi\)
0.0630625 + 0.998010i \(0.479913\pi\)
\(830\) 0 0
\(831\) −9.49120 −0.329246
\(832\) 0 0
\(833\) −44.4826 −1.54123
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −24.6124 −0.850729
\(838\) 0 0
\(839\) −3.45660 −0.119335 −0.0596675 0.998218i \(-0.519004\pi\)
−0.0596675 + 0.998218i \(0.519004\pi\)
\(840\) 0 0
\(841\) −28.1977 −0.972334
\(842\) 0 0
\(843\) −10.6757 −0.367692
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0.415806 0.0142873
\(848\) 0 0
\(849\) 6.89659 0.236690
\(850\) 0 0
\(851\) 76.6278 2.62677
\(852\) 0 0
\(853\) 47.4452 1.62449 0.812246 0.583315i \(-0.198245\pi\)
0.812246 + 0.583315i \(0.198245\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −35.8271 −1.22383 −0.611915 0.790923i \(-0.709601\pi\)
−0.611915 + 0.790923i \(0.709601\pi\)
\(858\) 0 0
\(859\) −39.0539 −1.33250 −0.666252 0.745727i \(-0.732102\pi\)
−0.666252 + 0.745727i \(0.732102\pi\)
\(860\) 0 0
\(861\) −2.75390 −0.0938528
\(862\) 0 0
\(863\) 5.75072 0.195757 0.0978784 0.995198i \(-0.468794\pi\)
0.0978784 + 0.995198i \(0.468794\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −16.6910 −0.566855
\(868\) 0 0
\(869\) −11.2682 −0.382246
\(870\) 0 0
\(871\) 42.7657 1.44906
\(872\) 0 0
\(873\) 1.52054 0.0514626
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.32776 0.281208 0.140604 0.990066i \(-0.455095\pi\)
0.140604 + 0.990066i \(0.455095\pi\)
\(878\) 0 0
\(879\) −6.14905 −0.207402
\(880\) 0 0
\(881\) −49.0954 −1.65407 −0.827033 0.562153i \(-0.809973\pi\)
−0.827033 + 0.562153i \(0.809973\pi\)
\(882\) 0 0
\(883\) −10.7526 −0.361853 −0.180927 0.983497i \(-0.557910\pi\)
−0.180927 + 0.983497i \(0.557910\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −53.0591 −1.78155 −0.890776 0.454443i \(-0.849838\pi\)
−0.890776 + 0.454443i \(0.849838\pi\)
\(888\) 0 0
\(889\) 3.80045 0.127463
\(890\) 0 0
\(891\) 5.31471 0.178049
\(892\) 0 0
\(893\) −29.2461 −0.978683
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −22.4226 −0.748668
\(898\) 0 0
\(899\) −6.03567 −0.201301
\(900\) 0 0
\(901\) −65.3882 −2.17840
\(902\) 0 0
\(903\) −1.30561 −0.0434481
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 7.03885 0.233721 0.116861 0.993148i \(-0.462717\pi\)
0.116861 + 0.993148i \(0.462717\pi\)
\(908\) 0 0
\(909\) 11.8811 0.394072
\(910\) 0 0
\(911\) 32.0980 1.06345 0.531727 0.846916i \(-0.321543\pi\)
0.531727 + 0.846916i \(0.321543\pi\)
\(912\) 0 0
\(913\) −16.2429 −0.537562
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.03171 −0.0340702
\(918\) 0 0
\(919\) −1.05960 −0.0349529 −0.0174764 0.999847i \(-0.505563\pi\)
−0.0174764 + 0.999847i \(0.505563\pi\)
\(920\) 0 0
\(921\) 1.37695 0.0453721
\(922\) 0 0
\(923\) 24.7721 0.815383
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −21.1841 −0.695777
\(928\) 0 0
\(929\) −12.9981 −0.426455 −0.213228 0.977003i \(-0.568398\pi\)
−0.213228 + 0.977003i \(0.568398\pi\)
\(930\) 0 0
\(931\) −35.5286 −1.16440
\(932\) 0 0
\(933\) 13.3595 0.437370
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −39.9846 −1.30624 −0.653120 0.757254i \(-0.726540\pi\)
−0.653120 + 0.757254i \(0.726540\pi\)
\(938\) 0 0
\(939\) −15.6706 −0.511391
\(940\) 0 0
\(941\) 34.0267 1.10924 0.554619 0.832105i \(-0.312864\pi\)
0.554619 + 0.832105i \(0.312864\pi\)
\(942\) 0 0
\(943\) −86.3359 −2.81148
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −27.1506 −0.882276 −0.441138 0.897439i \(-0.645425\pi\)
−0.441138 + 0.897439i \(0.645425\pi\)
\(948\) 0 0
\(949\) 12.6735 0.411401
\(950\) 0 0
\(951\) −3.84351 −0.124634
\(952\) 0 0
\(953\) 43.1169 1.39669 0.698346 0.715760i \(-0.253920\pi\)
0.698346 + 0.715760i \(0.253920\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −0.587377 −0.0189872
\(958\) 0 0
\(959\) −3.08044 −0.0994725
\(960\) 0 0
\(961\) 14.4055 0.464695
\(962\) 0 0
\(963\) 45.2826 1.45921
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 12.0641 0.387955 0.193978 0.981006i \(-0.437861\pi\)
0.193978 + 0.981006i \(0.437861\pi\)
\(968\) 0 0
\(969\) −22.2352 −0.714298
\(970\) 0 0
\(971\) −3.81156 −0.122319 −0.0611595 0.998128i \(-0.519480\pi\)
−0.0611595 + 0.998128i \(0.519480\pi\)
\(972\) 0 0
\(973\) −0.703403 −0.0225501
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −39.4202 −1.26116 −0.630581 0.776123i \(-0.717184\pi\)
−0.630581 + 0.776123i \(0.717184\pi\)
\(978\) 0 0
\(979\) 9.56998 0.305858
\(980\) 0 0
\(981\) −9.41440 −0.300579
\(982\) 0 0
\(983\) 8.67651 0.276738 0.138369 0.990381i \(-0.455814\pi\)
0.138369 + 0.990381i \(0.455814\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1.53237 0.0487758
\(988\) 0 0
\(989\) −40.9314 −1.30154
\(990\) 0 0
\(991\) 35.7256 1.13486 0.567430 0.823422i \(-0.307938\pi\)
0.567430 + 0.823422i \(0.307938\pi\)
\(992\) 0 0
\(993\) 0.376725 0.0119550
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 26.3453 0.834365 0.417183 0.908823i \(-0.363018\pi\)
0.417183 + 0.908823i \(0.363018\pi\)
\(998\) 0 0
\(999\) −32.7421 −1.03591
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 2200.2.a.x.1.2 4
4.3 odd 2 4400.2.a.ce.1.3 4
5.2 odd 4 440.2.b.d.89.6 yes 8
5.3 odd 4 440.2.b.d.89.3 8
5.4 even 2 2200.2.a.y.1.3 4
15.2 even 4 3960.2.d.f.3169.7 8
15.8 even 4 3960.2.d.f.3169.8 8
20.3 even 4 880.2.b.j.529.6 8
20.7 even 4 880.2.b.j.529.3 8
20.19 odd 2 4400.2.a.cb.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.b.d.89.3 8 5.3 odd 4
440.2.b.d.89.6 yes 8 5.2 odd 4
880.2.b.j.529.3 8 20.7 even 4
880.2.b.j.529.6 8 20.3 even 4
2200.2.a.x.1.2 4 1.1 even 1 trivial
2200.2.a.y.1.3 4 5.4 even 2
3960.2.d.f.3169.7 8 15.2 even 4
3960.2.d.f.3169.8 8 15.8 even 4
4400.2.a.cb.1.2 4 20.19 odd 2
4400.2.a.ce.1.3 4 4.3 odd 2