Properties

Label 44.4.a.a.1.1
Level $44$
Weight $4$
Character 44.1
Self dual yes
Analytic conductor $2.596$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [44,4,Mod(1,44)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(44, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("44.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 44 = 2^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 44.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: \(2.59608404025\)
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 \(\chi\) \(=\) 44.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000 q^{3} -7.00000 q^{5} -26.0000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-5.00000 q^{3} -7.00000 q^{5} -26.0000 q^{7} -2.00000 q^{9} -11.0000 q^{11} +52.0000 q^{13} +35.0000 q^{15} +46.0000 q^{17} -96.0000 q^{19} +130.000 q^{21} +27.0000 q^{23} -76.0000 q^{25} +145.000 q^{27} +16.0000 q^{29} -293.000 q^{31} +55.0000 q^{33} +182.000 q^{35} -29.0000 q^{37} -260.000 q^{39} -472.000 q^{41} -110.000 q^{43} +14.0000 q^{45} -224.000 q^{47} +333.000 q^{49} -230.000 q^{51} +754.000 q^{53} +77.0000 q^{55} +480.000 q^{57} +825.000 q^{59} -548.000 q^{61} +52.0000 q^{63} -364.000 q^{65} -123.000 q^{67} -135.000 q^{69} +1001.00 q^{71} -1020.00 q^{73} +380.000 q^{75} +286.000 q^{77} +526.000 q^{79} -671.000 q^{81} -158.000 q^{83} -322.000 q^{85} -80.0000 q^{87} -1217.00 q^{89} -1352.00 q^{91} +1465.00 q^{93} +672.000 q^{95} -263.000 q^{97} +22.0000 q^{99} +O(q^{100})\)

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
\(3\) −5.00000 −0.962250 −0.481125 0.876652i \(-0.659772\pi\)
−0.481125 + 0.876652i \(0.659772\pi\)
\(4\) 0 0
\(5\) −7.00000 −0.626099 −0.313050 0.949737i \(-0.601351\pi\)
−0.313050 + 0.949737i \(0.601351\pi\)
\(6\) 0 0
\(7\) −26.0000 −1.40387 −0.701934 0.712242i \(-0.747680\pi\)
−0.701934 + 0.712242i \(0.747680\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.0740741
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 52.0000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 35.0000 0.602464
\(16\) 0 0
\(17\) 46.0000 0.656273 0.328136 0.944630i \(-0.393579\pi\)
0.328136 + 0.944630i \(0.393579\pi\)
\(18\) 0 0
\(19\) −96.0000 −1.15915 −0.579577 0.814918i \(-0.696782\pi\)
−0.579577 + 0.814918i \(0.696782\pi\)
\(20\) 0 0
\(21\) 130.000 1.35087
\(22\) 0 0
\(23\) 27.0000 0.244778 0.122389 0.992482i \(-0.460944\pi\)
0.122389 + 0.992482i \(0.460944\pi\)
\(24\) 0 0
\(25\) −76.0000 −0.608000
\(26\) 0 0
\(27\) 145.000 1.03353
\(28\) 0 0
\(29\) 16.0000 0.102453 0.0512263 0.998687i \(-0.483687\pi\)
0.0512263 + 0.998687i \(0.483687\pi\)
\(30\) 0 0
\(31\) −293.000 −1.69756 −0.848780 0.528746i \(-0.822662\pi\)
−0.848780 + 0.528746i \(0.822662\pi\)
\(32\) 0 0
\(33\) 55.0000 0.290129
\(34\) 0 0
\(35\) 182.000 0.878960
\(36\) 0 0
\(37\) −29.0000 −0.128853 −0.0644266 0.997922i \(-0.520522\pi\)
−0.0644266 + 0.997922i \(0.520522\pi\)
\(38\) 0 0
\(39\) −260.000 −1.06752
\(40\) 0 0
\(41\) −472.000 −1.79790 −0.898951 0.438048i \(-0.855670\pi\)
−0.898951 + 0.438048i \(0.855670\pi\)
\(42\) 0 0
\(43\) −110.000 −0.390113 −0.195056 0.980792i \(-0.562489\pi\)
−0.195056 + 0.980792i \(0.562489\pi\)
\(44\) 0 0
\(45\) 14.0000 0.0463777
\(46\) 0 0
\(47\) −224.000 −0.695186 −0.347593 0.937645i \(-0.613001\pi\)
−0.347593 + 0.937645i \(0.613001\pi\)
\(48\) 0 0
\(49\) 333.000 0.970845
\(50\) 0 0
\(51\) −230.000 −0.631499
\(52\) 0 0
\(53\) 754.000 1.95415 0.977074 0.212899i \(-0.0682905\pi\)
0.977074 + 0.212899i \(0.0682905\pi\)
\(54\) 0 0
\(55\) 77.0000 0.188776
\(56\) 0 0
\(57\) 480.000 1.11540
\(58\) 0 0
\(59\) 825.000 1.82044 0.910219 0.414127i \(-0.135913\pi\)
0.910219 + 0.414127i \(0.135913\pi\)
\(60\) 0 0
\(61\) −548.000 −1.15023 −0.575116 0.818072i \(-0.695043\pi\)
−0.575116 + 0.818072i \(0.695043\pi\)
\(62\) 0 0
\(63\) 52.0000 0.103990
\(64\) 0 0
\(65\) −364.000 −0.694595
\(66\) 0 0
\(67\) −123.000 −0.224281 −0.112141 0.993692i \(-0.535771\pi\)
−0.112141 + 0.993692i \(0.535771\pi\)
\(68\) 0 0
\(69\) −135.000 −0.235538
\(70\) 0 0
\(71\) 1001.00 1.67319 0.836597 0.547818i \(-0.184541\pi\)
0.836597 + 0.547818i \(0.184541\pi\)
\(72\) 0 0
\(73\) −1020.00 −1.63537 −0.817685 0.575666i \(-0.804743\pi\)
−0.817685 + 0.575666i \(0.804743\pi\)
\(74\) 0 0
\(75\) 380.000 0.585048
\(76\) 0 0
\(77\) 286.000 0.423282
\(78\) 0 0
\(79\) 526.000 0.749109 0.374555 0.927205i \(-0.377796\pi\)
0.374555 + 0.927205i \(0.377796\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) 0 0
\(83\) −158.000 −0.208949 −0.104474 0.994528i \(-0.533316\pi\)
−0.104474 + 0.994528i \(0.533316\pi\)
\(84\) 0 0
\(85\) −322.000 −0.410892
\(86\) 0 0
\(87\) −80.0000 −0.0985851
\(88\) 0 0
\(89\) −1217.00 −1.44946 −0.724729 0.689034i \(-0.758035\pi\)
−0.724729 + 0.689034i \(0.758035\pi\)
\(90\) 0 0
\(91\) −1352.00 −1.55745
\(92\) 0 0
\(93\) 1465.00 1.63348
\(94\) 0 0
\(95\) 672.000 0.725745
\(96\) 0 0
\(97\) −263.000 −0.275295 −0.137647 0.990481i \(-0.543954\pi\)
−0.137647 + 0.990481i \(0.543954\pi\)
\(98\) 0 0
\(99\) 22.0000 0.0223342
\(100\) 0 0
\(101\) −814.000 −0.801941 −0.400970 0.916091i \(-0.631327\pi\)
−0.400970 + 0.916091i \(0.631327\pi\)
\(102\) 0 0
\(103\) 376.000 0.359693 0.179847 0.983695i \(-0.442440\pi\)
0.179847 + 0.983695i \(0.442440\pi\)
\(104\) 0 0
\(105\) −910.000 −0.845780
\(106\) 0 0
\(107\) −1494.00 −1.34982 −0.674909 0.737901i \(-0.735817\pi\)
−0.674909 + 0.737901i \(0.735817\pi\)
\(108\) 0 0
\(109\) 842.000 0.739899 0.369949 0.929052i \(-0.379375\pi\)
0.369949 + 0.929052i \(0.379375\pi\)
\(110\) 0 0
\(111\) 145.000 0.123989
\(112\) 0 0
\(113\) 1281.00 1.06643 0.533214 0.845980i \(-0.320984\pi\)
0.533214 + 0.845980i \(0.320984\pi\)
\(114\) 0 0
\(115\) −189.000 −0.153255
\(116\) 0 0
\(117\) −104.000 −0.0821778
\(118\) 0 0
\(119\) −1196.00 −0.921321
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) 2360.00 1.73003
\(124\) 0 0
\(125\) 1407.00 1.00677
\(126\) 0 0
\(127\) 1736.00 1.21295 0.606477 0.795101i \(-0.292582\pi\)
0.606477 + 0.795101i \(0.292582\pi\)
\(128\) 0 0
\(129\) 550.000 0.375386
\(130\) 0 0
\(131\) 6.00000 0.00400170 0.00200085 0.999998i \(-0.499363\pi\)
0.00200085 + 0.999998i \(0.499363\pi\)
\(132\) 0 0
\(133\) 2496.00 1.62730
\(134\) 0 0
\(135\) −1015.00 −0.647091
\(136\) 0 0
\(137\) −1439.00 −0.897387 −0.448694 0.893686i \(-0.648111\pi\)
−0.448694 + 0.893686i \(0.648111\pi\)
\(138\) 0 0
\(139\) −318.000 −0.194046 −0.0970231 0.995282i \(-0.530932\pi\)
−0.0970231 + 0.995282i \(0.530932\pi\)
\(140\) 0 0
\(141\) 1120.00 0.668943
\(142\) 0 0
\(143\) −572.000 −0.334497
\(144\) 0 0
\(145\) −112.000 −0.0641455
\(146\) 0 0
\(147\) −1665.00 −0.934196
\(148\) 0 0
\(149\) −922.000 −0.506934 −0.253467 0.967344i \(-0.581571\pi\)
−0.253467 + 0.967344i \(0.581571\pi\)
\(150\) 0 0
\(151\) −1030.00 −0.555101 −0.277550 0.960711i \(-0.589523\pi\)
−0.277550 + 0.960711i \(0.589523\pi\)
\(152\) 0 0
\(153\) −92.0000 −0.0486128
\(154\) 0 0
\(155\) 2051.00 1.06284
\(156\) 0 0
\(157\) 1017.00 0.516977 0.258489 0.966014i \(-0.416776\pi\)
0.258489 + 0.966014i \(0.416776\pi\)
\(158\) 0 0
\(159\) −3770.00 −1.88038
\(160\) 0 0
\(161\) −702.000 −0.343636
\(162\) 0 0
\(163\) −2444.00 −1.17441 −0.587205 0.809438i \(-0.699772\pi\)
−0.587205 + 0.809438i \(0.699772\pi\)
\(164\) 0 0
\(165\) −385.000 −0.181650
\(166\) 0 0
\(167\) −1452.00 −0.672809 −0.336405 0.941718i \(-0.609211\pi\)
−0.336405 + 0.941718i \(0.609211\pi\)
\(168\) 0 0
\(169\) 507.000 0.230769
\(170\) 0 0
\(171\) 192.000 0.0858632
\(172\) 0 0
\(173\) 1914.00 0.841149 0.420574 0.907258i \(-0.361829\pi\)
0.420574 + 0.907258i \(0.361829\pi\)
\(174\) 0 0
\(175\) 1976.00 0.853552
\(176\) 0 0
\(177\) −4125.00 −1.75172
\(178\) 0 0
\(179\) 1293.00 0.539907 0.269954 0.962873i \(-0.412992\pi\)
0.269954 + 0.962873i \(0.412992\pi\)
\(180\) 0 0
\(181\) 455.000 0.186850 0.0934251 0.995626i \(-0.470218\pi\)
0.0934251 + 0.995626i \(0.470218\pi\)
\(182\) 0 0
\(183\) 2740.00 1.10681
\(184\) 0 0
\(185\) 203.000 0.0806749
\(186\) 0 0
\(187\) −506.000 −0.197874
\(188\) 0 0
\(189\) −3770.00 −1.45094
\(190\) 0 0
\(191\) −1115.00 −0.422401 −0.211200 0.977443i \(-0.567737\pi\)
−0.211200 + 0.977443i \(0.567737\pi\)
\(192\) 0 0
\(193\) 5012.00 1.86928 0.934642 0.355591i \(-0.115720\pi\)
0.934642 + 0.355591i \(0.115720\pi\)
\(194\) 0 0
\(195\) 1820.00 0.668374
\(196\) 0 0
\(197\) −4146.00 −1.49944 −0.749721 0.661753i \(-0.769813\pi\)
−0.749721 + 0.661753i \(0.769813\pi\)
\(198\) 0 0
\(199\) −1240.00 −0.441715 −0.220857 0.975306i \(-0.570886\pi\)
−0.220857 + 0.975306i \(0.570886\pi\)
\(200\) 0 0
\(201\) 615.000 0.215815
\(202\) 0 0
\(203\) −416.000 −0.143830
\(204\) 0 0
\(205\) 3304.00 1.12567
\(206\) 0 0
\(207\) −54.0000 −0.0181317
\(208\) 0 0
\(209\) 1056.00 0.349498
\(210\) 0 0
\(211\) −2820.00 −0.920080 −0.460040 0.887898i \(-0.652165\pi\)
−0.460040 + 0.887898i \(0.652165\pi\)
\(212\) 0 0
\(213\) −5005.00 −1.61003
\(214\) 0 0
\(215\) 770.000 0.244249
\(216\) 0 0
\(217\) 7618.00 2.38315
\(218\) 0 0
\(219\) 5100.00 1.57363
\(220\) 0 0
\(221\) 2392.00 0.728069
\(222\) 0 0
\(223\) 3695.00 1.10958 0.554788 0.831992i \(-0.312799\pi\)
0.554788 + 0.831992i \(0.312799\pi\)
\(224\) 0 0
\(225\) 152.000 0.0450370
\(226\) 0 0
\(227\) −486.000 −0.142101 −0.0710506 0.997473i \(-0.522635\pi\)
−0.0710506 + 0.997473i \(0.522635\pi\)
\(228\) 0 0
\(229\) 4231.00 1.22093 0.610464 0.792044i \(-0.290983\pi\)
0.610464 + 0.792044i \(0.290983\pi\)
\(230\) 0 0
\(231\) −1430.00 −0.407303
\(232\) 0 0
\(233\) −3336.00 −0.937977 −0.468988 0.883204i \(-0.655381\pi\)
−0.468988 + 0.883204i \(0.655381\pi\)
\(234\) 0 0
\(235\) 1568.00 0.435255
\(236\) 0 0
\(237\) −2630.00 −0.720831
\(238\) 0 0
\(239\) 3610.00 0.977036 0.488518 0.872554i \(-0.337538\pi\)
0.488518 + 0.872554i \(0.337538\pi\)
\(240\) 0 0
\(241\) −6408.00 −1.71276 −0.856381 0.516345i \(-0.827292\pi\)
−0.856381 + 0.516345i \(0.827292\pi\)
\(242\) 0 0
\(243\) −560.000 −0.147835
\(244\) 0 0
\(245\) −2331.00 −0.607845
\(246\) 0 0
\(247\) −4992.00 −1.28596
\(248\) 0 0
\(249\) 790.000 0.201061
\(250\) 0 0
\(251\) −1787.00 −0.449380 −0.224690 0.974430i \(-0.572137\pi\)
−0.224690 + 0.974430i \(0.572137\pi\)
\(252\) 0 0
\(253\) −297.000 −0.0738033
\(254\) 0 0
\(255\) 1610.00 0.395381
\(256\) 0 0
\(257\) −354.000 −0.0859218 −0.0429609 0.999077i \(-0.513679\pi\)
−0.0429609 + 0.999077i \(0.513679\pi\)
\(258\) 0 0
\(259\) 754.000 0.180893
\(260\) 0 0
\(261\) −32.0000 −0.00758908
\(262\) 0 0
\(263\) −2026.00 −0.475013 −0.237507 0.971386i \(-0.576330\pi\)
−0.237507 + 0.971386i \(0.576330\pi\)
\(264\) 0 0
\(265\) −5278.00 −1.22349
\(266\) 0 0
\(267\) 6085.00 1.39474
\(268\) 0 0
\(269\) −8750.00 −1.98326 −0.991630 0.129112i \(-0.958787\pi\)
−0.991630 + 0.129112i \(0.958787\pi\)
\(270\) 0 0
\(271\) −5036.00 −1.12884 −0.564419 0.825488i \(-0.690900\pi\)
−0.564419 + 0.825488i \(0.690900\pi\)
\(272\) 0 0
\(273\) 6760.00 1.49866
\(274\) 0 0
\(275\) 836.000 0.183319
\(276\) 0 0
\(277\) −4306.00 −0.934016 −0.467008 0.884253i \(-0.654668\pi\)
−0.467008 + 0.884253i \(0.654668\pi\)
\(278\) 0 0
\(279\) 586.000 0.125745
\(280\) 0 0
\(281\) −1202.00 −0.255179 −0.127590 0.991827i \(-0.540724\pi\)
−0.127590 + 0.991827i \(0.540724\pi\)
\(282\) 0 0
\(283\) −6620.00 −1.39052 −0.695262 0.718757i \(-0.744712\pi\)
−0.695262 + 0.718757i \(0.744712\pi\)
\(284\) 0 0
\(285\) −3360.00 −0.698348
\(286\) 0 0
\(287\) 12272.0 2.52402
\(288\) 0 0
\(289\) −2797.00 −0.569306
\(290\) 0 0
\(291\) 1315.00 0.264903
\(292\) 0 0
\(293\) 6968.00 1.38933 0.694667 0.719331i \(-0.255552\pi\)
0.694667 + 0.719331i \(0.255552\pi\)
\(294\) 0 0
\(295\) −5775.00 −1.13977
\(296\) 0 0
\(297\) −1595.00 −0.311620
\(298\) 0 0
\(299\) 1404.00 0.271557
\(300\) 0 0
\(301\) 2860.00 0.547667
\(302\) 0 0
\(303\) 4070.00 0.771668
\(304\) 0 0
\(305\) 3836.00 0.720160
\(306\) 0 0
\(307\) 7640.00 1.42032 0.710159 0.704041i \(-0.248623\pi\)
0.710159 + 0.704041i \(0.248623\pi\)
\(308\) 0 0
\(309\) −1880.00 −0.346115
\(310\) 0 0
\(311\) −652.000 −0.118880 −0.0594398 0.998232i \(-0.518931\pi\)
−0.0594398 + 0.998232i \(0.518931\pi\)
\(312\) 0 0
\(313\) 8055.00 1.45462 0.727309 0.686310i \(-0.240771\pi\)
0.727309 + 0.686310i \(0.240771\pi\)
\(314\) 0 0
\(315\) −364.000 −0.0651082
\(316\) 0 0
\(317\) −5675.00 −1.00549 −0.502744 0.864435i \(-0.667676\pi\)
−0.502744 + 0.864435i \(0.667676\pi\)
\(318\) 0 0
\(319\) −176.000 −0.0308906
\(320\) 0 0
\(321\) 7470.00 1.29886
\(322\) 0 0
\(323\) −4416.00 −0.760721
\(324\) 0 0
\(325\) −3952.00 −0.674515
\(326\) 0 0
\(327\) −4210.00 −0.711968
\(328\) 0 0
\(329\) 5824.00 0.975950
\(330\) 0 0
\(331\) −2245.00 −0.372799 −0.186399 0.982474i \(-0.559682\pi\)
−0.186399 + 0.982474i \(0.559682\pi\)
\(332\) 0 0
\(333\) 58.0000 0.00954469
\(334\) 0 0
\(335\) 861.000 0.140422
\(336\) 0 0
\(337\) −694.000 −0.112180 −0.0560899 0.998426i \(-0.517863\pi\)
−0.0560899 + 0.998426i \(0.517863\pi\)
\(338\) 0 0
\(339\) −6405.00 −1.02617
\(340\) 0 0
\(341\) 3223.00 0.511834
\(342\) 0 0
\(343\) 260.000 0.0409291
\(344\) 0 0
\(345\) 945.000 0.147470
\(346\) 0 0
\(347\) 4092.00 0.633055 0.316527 0.948583i \(-0.397483\pi\)
0.316527 + 0.948583i \(0.397483\pi\)
\(348\) 0 0
\(349\) 334.000 0.0512281 0.0256141 0.999672i \(-0.491846\pi\)
0.0256141 + 0.999672i \(0.491846\pi\)
\(350\) 0 0
\(351\) 7540.00 1.14660
\(352\) 0 0
\(353\) 891.000 0.134343 0.0671716 0.997741i \(-0.478603\pi\)
0.0671716 + 0.997741i \(0.478603\pi\)
\(354\) 0 0
\(355\) −7007.00 −1.04759
\(356\) 0 0
\(357\) 5980.00 0.886541
\(358\) 0 0
\(359\) 2476.00 0.364006 0.182003 0.983298i \(-0.441742\pi\)
0.182003 + 0.983298i \(0.441742\pi\)
\(360\) 0 0
\(361\) 2357.00 0.343636
\(362\) 0 0
\(363\) −605.000 −0.0874773
\(364\) 0 0
\(365\) 7140.00 1.02390
\(366\) 0 0
\(367\) 1379.00 0.196140 0.0980698 0.995180i \(-0.468733\pi\)
0.0980698 + 0.995180i \(0.468733\pi\)
\(368\) 0 0
\(369\) 944.000 0.133178
\(370\) 0 0
\(371\) −19604.0 −2.74337
\(372\) 0 0
\(373\) −6266.00 −0.869816 −0.434908 0.900475i \(-0.643219\pi\)
−0.434908 + 0.900475i \(0.643219\pi\)
\(374\) 0 0
\(375\) −7035.00 −0.968762
\(376\) 0 0
\(377\) 832.000 0.113661
\(378\) 0 0
\(379\) 151.000 0.0204653 0.0102327 0.999948i \(-0.496743\pi\)
0.0102327 + 0.999948i \(0.496743\pi\)
\(380\) 0 0
\(381\) −8680.00 −1.16717
\(382\) 0 0
\(383\) −1989.00 −0.265361 −0.132680 0.991159i \(-0.542358\pi\)
−0.132680 + 0.991159i \(0.542358\pi\)
\(384\) 0 0
\(385\) −2002.00 −0.265017
\(386\) 0 0
\(387\) 220.000 0.0288972
\(388\) 0 0
\(389\) 6817.00 0.888523 0.444262 0.895897i \(-0.353466\pi\)
0.444262 + 0.895897i \(0.353466\pi\)
\(390\) 0 0
\(391\) 1242.00 0.160641
\(392\) 0 0
\(393\) −30.0000 −0.00385064
\(394\) 0 0
\(395\) −3682.00 −0.469017
\(396\) 0 0
\(397\) −4170.00 −0.527170 −0.263585 0.964636i \(-0.584905\pi\)
−0.263585 + 0.964636i \(0.584905\pi\)
\(398\) 0 0
\(399\) −12480.0 −1.56587
\(400\) 0 0
\(401\) 10914.0 1.35915 0.679575 0.733606i \(-0.262164\pi\)
0.679575 + 0.733606i \(0.262164\pi\)
\(402\) 0 0
\(403\) −15236.0 −1.88327
\(404\) 0 0
\(405\) 4697.00 0.576286
\(406\) 0 0
\(407\) 319.000 0.0388507
\(408\) 0 0
\(409\) −1102.00 −0.133228 −0.0666142 0.997779i \(-0.521220\pi\)
−0.0666142 + 0.997779i \(0.521220\pi\)
\(410\) 0 0
\(411\) 7195.00 0.863511
\(412\) 0 0
\(413\) −21450.0 −2.55565
\(414\) 0 0
\(415\) 1106.00 0.130823
\(416\) 0 0
\(417\) 1590.00 0.186721
\(418\) 0 0
\(419\) 11028.0 1.28581 0.642903 0.765947i \(-0.277730\pi\)
0.642903 + 0.765947i \(0.277730\pi\)
\(420\) 0 0
\(421\) 2622.00 0.303536 0.151768 0.988416i \(-0.451503\pi\)
0.151768 + 0.988416i \(0.451503\pi\)
\(422\) 0 0
\(423\) 448.000 0.0514953
\(424\) 0 0
\(425\) −3496.00 −0.399014
\(426\) 0 0
\(427\) 14248.0 1.61478
\(428\) 0 0
\(429\) 2860.00 0.321870
\(430\) 0 0
\(431\) 16598.0 1.85498 0.927491 0.373845i \(-0.121961\pi\)
0.927491 + 0.373845i \(0.121961\pi\)
\(432\) 0 0
\(433\) −5763.00 −0.639612 −0.319806 0.947483i \(-0.603618\pi\)
−0.319806 + 0.947483i \(0.603618\pi\)
\(434\) 0 0
\(435\) 560.000 0.0617240
\(436\) 0 0
\(437\) −2592.00 −0.283735
\(438\) 0 0
\(439\) −3128.00 −0.340071 −0.170036 0.985438i \(-0.554388\pi\)
−0.170036 + 0.985438i \(0.554388\pi\)
\(440\) 0 0
\(441\) −666.000 −0.0719145
\(442\) 0 0
\(443\) 6369.00 0.683071 0.341535 0.939869i \(-0.389053\pi\)
0.341535 + 0.939869i \(0.389053\pi\)
\(444\) 0 0
\(445\) 8519.00 0.907504
\(446\) 0 0
\(447\) 4610.00 0.487798
\(448\) 0 0
\(449\) 8691.00 0.913483 0.456741 0.889600i \(-0.349017\pi\)
0.456741 + 0.889600i \(0.349017\pi\)
\(450\) 0 0
\(451\) 5192.00 0.542088
\(452\) 0 0
\(453\) 5150.00 0.534146
\(454\) 0 0
\(455\) 9464.00 0.975119
\(456\) 0 0
\(457\) 2260.00 0.231331 0.115666 0.993288i \(-0.463100\pi\)
0.115666 + 0.993288i \(0.463100\pi\)
\(458\) 0 0
\(459\) 6670.00 0.678277
\(460\) 0 0
\(461\) −12756.0 −1.28873 −0.644367 0.764717i \(-0.722879\pi\)
−0.644367 + 0.764717i \(0.722879\pi\)
\(462\) 0 0
\(463\) −6887.00 −0.691287 −0.345644 0.938366i \(-0.612339\pi\)
−0.345644 + 0.938366i \(0.612339\pi\)
\(464\) 0 0
\(465\) −10255.0 −1.02272
\(466\) 0 0
\(467\) −1535.00 −0.152101 −0.0760507 0.997104i \(-0.524231\pi\)
−0.0760507 + 0.997104i \(0.524231\pi\)
\(468\) 0 0
\(469\) 3198.00 0.314861
\(470\) 0 0
\(471\) −5085.00 −0.497462
\(472\) 0 0
\(473\) 1210.00 0.117623
\(474\) 0 0
\(475\) 7296.00 0.704765
\(476\) 0 0
\(477\) −1508.00 −0.144752
\(478\) 0 0
\(479\) −17564.0 −1.67541 −0.837703 0.546126i \(-0.816102\pi\)
−0.837703 + 0.546126i \(0.816102\pi\)
\(480\) 0 0
\(481\) −1508.00 −0.142950
\(482\) 0 0
\(483\) 3510.00 0.330664
\(484\) 0 0
\(485\) 1841.00 0.172362
\(486\) 0 0
\(487\) −7541.00 −0.701674 −0.350837 0.936437i \(-0.614103\pi\)
−0.350837 + 0.936437i \(0.614103\pi\)
\(488\) 0 0
\(489\) 12220.0 1.13008
\(490\) 0 0
\(491\) 12552.0 1.15369 0.576847 0.816852i \(-0.304283\pi\)
0.576847 + 0.816852i \(0.304283\pi\)
\(492\) 0 0
\(493\) 736.000 0.0672369
\(494\) 0 0
\(495\) −154.000 −0.0139834
\(496\) 0 0
\(497\) −26026.0 −2.34894
\(498\) 0 0
\(499\) −8396.00 −0.753220 −0.376610 0.926372i \(-0.622910\pi\)
−0.376610 + 0.926372i \(0.622910\pi\)
\(500\) 0 0
\(501\) 7260.00 0.647411
\(502\) 0 0
\(503\) −12194.0 −1.08092 −0.540461 0.841369i \(-0.681750\pi\)
−0.540461 + 0.841369i \(0.681750\pi\)
\(504\) 0 0
\(505\) 5698.00 0.502094
\(506\) 0 0
\(507\) −2535.00 −0.222058
\(508\) 0 0
\(509\) 18295.0 1.59315 0.796573 0.604542i \(-0.206644\pi\)
0.796573 + 0.604542i \(0.206644\pi\)
\(510\) 0 0
\(511\) 26520.0 2.29584
\(512\) 0 0
\(513\) −13920.0 −1.19802
\(514\) 0 0
\(515\) −2632.00 −0.225203
\(516\) 0 0
\(517\) 2464.00 0.209607
\(518\) 0 0
\(519\) −9570.00 −0.809396
\(520\) 0 0
\(521\) 7101.00 0.597122 0.298561 0.954391i \(-0.403493\pi\)
0.298561 + 0.954391i \(0.403493\pi\)
\(522\) 0 0
\(523\) −4912.00 −0.410682 −0.205341 0.978690i \(-0.565830\pi\)
−0.205341 + 0.978690i \(0.565830\pi\)
\(524\) 0 0
\(525\) −9880.00 −0.821331
\(526\) 0 0
\(527\) −13478.0 −1.11406
\(528\) 0 0
\(529\) −11438.0 −0.940084
\(530\) 0 0
\(531\) −1650.00 −0.134847
\(532\) 0 0
\(533\) −24544.0 −1.99459
\(534\) 0 0
\(535\) 10458.0 0.845119
\(536\) 0 0
\(537\) −6465.00 −0.519526
\(538\) 0 0
\(539\) −3663.00 −0.292721
\(540\) 0 0
\(541\) 11496.0 0.913589 0.456794 0.889572i \(-0.348997\pi\)
0.456794 + 0.889572i \(0.348997\pi\)
\(542\) 0 0
\(543\) −2275.00 −0.179797
\(544\) 0 0
\(545\) −5894.00 −0.463250
\(546\) 0 0
\(547\) 19048.0 1.48891 0.744455 0.667673i \(-0.232709\pi\)
0.744455 + 0.667673i \(0.232709\pi\)
\(548\) 0 0
\(549\) 1096.00 0.0852024
\(550\) 0 0
\(551\) −1536.00 −0.118758
\(552\) 0 0
\(553\) −13676.0 −1.05165
\(554\) 0 0
\(555\) −1015.00 −0.0776295
\(556\) 0 0
\(557\) 10718.0 0.815325 0.407663 0.913133i \(-0.366344\pi\)
0.407663 + 0.913133i \(0.366344\pi\)
\(558\) 0 0
\(559\) −5720.00 −0.432791
\(560\) 0 0
\(561\) 2530.00 0.190404
\(562\) 0 0
\(563\) 6660.00 0.498553 0.249277 0.968432i \(-0.419807\pi\)
0.249277 + 0.968432i \(0.419807\pi\)
\(564\) 0 0
\(565\) −8967.00 −0.667689
\(566\) 0 0
\(567\) 17446.0 1.29217
\(568\) 0 0
\(569\) −10960.0 −0.807499 −0.403750 0.914870i \(-0.632293\pi\)
−0.403750 + 0.914870i \(0.632293\pi\)
\(570\) 0 0
\(571\) 18596.0 1.36290 0.681452 0.731863i \(-0.261349\pi\)
0.681452 + 0.731863i \(0.261349\pi\)
\(572\) 0 0
\(573\) 5575.00 0.406455
\(574\) 0 0
\(575\) −2052.00 −0.148825
\(576\) 0 0
\(577\) −21119.0 −1.52374 −0.761868 0.647733i \(-0.775717\pi\)
−0.761868 + 0.647733i \(0.775717\pi\)
\(578\) 0 0
\(579\) −25060.0 −1.79872
\(580\) 0 0
\(581\) 4108.00 0.293337
\(582\) 0 0
\(583\) −8294.00 −0.589198
\(584\) 0 0
\(585\) 728.000 0.0514514
\(586\) 0 0
\(587\) −2836.00 −0.199411 −0.0997055 0.995017i \(-0.531790\pi\)
−0.0997055 + 0.995017i \(0.531790\pi\)
\(588\) 0 0
\(589\) 28128.0 1.96773
\(590\) 0 0
\(591\) 20730.0 1.44284
\(592\) 0 0
\(593\) 18044.0 1.24954 0.624771 0.780808i \(-0.285192\pi\)
0.624771 + 0.780808i \(0.285192\pi\)
\(594\) 0 0
\(595\) 8372.00 0.576838
\(596\) 0 0
\(597\) 6200.00 0.425040
\(598\) 0 0
\(599\) −9264.00 −0.631914 −0.315957 0.948773i \(-0.602326\pi\)
−0.315957 + 0.948773i \(0.602326\pi\)
\(600\) 0 0
\(601\) −19326.0 −1.31169 −0.655844 0.754897i \(-0.727687\pi\)
−0.655844 + 0.754897i \(0.727687\pi\)
\(602\) 0 0
\(603\) 246.000 0.0166134
\(604\) 0 0
\(605\) −847.000 −0.0569181
\(606\) 0 0
\(607\) 10082.0 0.674161 0.337081 0.941476i \(-0.390561\pi\)
0.337081 + 0.941476i \(0.390561\pi\)
\(608\) 0 0
\(609\) 2080.00 0.138400
\(610\) 0 0
\(611\) −11648.0 −0.771240
\(612\) 0 0
\(613\) 13088.0 0.862348 0.431174 0.902269i \(-0.358100\pi\)
0.431174 + 0.902269i \(0.358100\pi\)
\(614\) 0 0
\(615\) −16520.0 −1.08317
\(616\) 0 0
\(617\) 23426.0 1.52852 0.764259 0.644910i \(-0.223105\pi\)
0.764259 + 0.644910i \(0.223105\pi\)
\(618\) 0 0
\(619\) 23587.0 1.53157 0.765785 0.643097i \(-0.222351\pi\)
0.765785 + 0.643097i \(0.222351\pi\)
\(620\) 0 0
\(621\) 3915.00 0.252985
\(622\) 0 0
\(623\) 31642.0 2.03485
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) 0 0
\(627\) −5280.00 −0.336304
\(628\) 0 0
\(629\) −1334.00 −0.0845629
\(630\) 0 0
\(631\) 19683.0 1.24179 0.620894 0.783895i \(-0.286770\pi\)
0.620894 + 0.783895i \(0.286770\pi\)
\(632\) 0 0
\(633\) 14100.0 0.885347
\(634\) 0 0
\(635\) −12152.0 −0.759429
\(636\) 0 0
\(637\) 17316.0 1.07706
\(638\) 0 0
\(639\) −2002.00 −0.123940
\(640\) 0 0
\(641\) 375.000 0.0231070 0.0115535 0.999933i \(-0.496322\pi\)
0.0115535 + 0.999933i \(0.496322\pi\)
\(642\) 0 0
\(643\) −21055.0 −1.29133 −0.645667 0.763619i \(-0.723421\pi\)
−0.645667 + 0.763619i \(0.723421\pi\)
\(644\) 0 0
\(645\) −3850.00 −0.235029
\(646\) 0 0
\(647\) −6427.00 −0.390528 −0.195264 0.980751i \(-0.562556\pi\)
−0.195264 + 0.980751i \(0.562556\pi\)
\(648\) 0 0
\(649\) −9075.00 −0.548883
\(650\) 0 0
\(651\) −38090.0 −2.29319
\(652\) 0 0
\(653\) −7617.00 −0.456472 −0.228236 0.973606i \(-0.573296\pi\)
−0.228236 + 0.973606i \(0.573296\pi\)
\(654\) 0 0
\(655\) −42.0000 −0.00250546
\(656\) 0 0
\(657\) 2040.00 0.121138
\(658\) 0 0
\(659\) −17630.0 −1.04214 −0.521068 0.853515i \(-0.674466\pi\)
−0.521068 + 0.853515i \(0.674466\pi\)
\(660\) 0 0
\(661\) 4605.00 0.270974 0.135487 0.990779i \(-0.456740\pi\)
0.135487 + 0.990779i \(0.456740\pi\)
\(662\) 0 0
\(663\) −11960.0 −0.700585
\(664\) 0 0
\(665\) −17472.0 −1.01885
\(666\) 0 0
\(667\) 432.000 0.0250781
\(668\) 0 0
\(669\) −18475.0 −1.06769
\(670\) 0 0
\(671\) 6028.00 0.346808
\(672\) 0 0
\(673\) −2818.00 −0.161406 −0.0807028 0.996738i \(-0.525716\pi\)
−0.0807028 + 0.996738i \(0.525716\pi\)
\(674\) 0 0
\(675\) −11020.0 −0.628385
\(676\) 0 0
\(677\) 8438.00 0.479023 0.239512 0.970894i \(-0.423013\pi\)
0.239512 + 0.970894i \(0.423013\pi\)
\(678\) 0 0
\(679\) 6838.00 0.386478
\(680\) 0 0
\(681\) 2430.00 0.136737
\(682\) 0 0
\(683\) −17344.0 −0.971669 −0.485834 0.874051i \(-0.661484\pi\)
−0.485834 + 0.874051i \(0.661484\pi\)
\(684\) 0 0
\(685\) 10073.0 0.561853
\(686\) 0 0
\(687\) −21155.0 −1.17484
\(688\) 0 0
\(689\) 39208.0 2.16793
\(690\) 0 0
\(691\) −3947.00 −0.217295 −0.108648 0.994080i \(-0.534652\pi\)
−0.108648 + 0.994080i \(0.534652\pi\)
\(692\) 0 0
\(693\) −572.000 −0.0313542
\(694\) 0 0
\(695\) 2226.00 0.121492
\(696\) 0 0
\(697\) −21712.0 −1.17991
\(698\) 0 0
\(699\) 16680.0 0.902569
\(700\) 0 0
\(701\) −7998.00 −0.430928 −0.215464 0.976512i \(-0.569126\pi\)
−0.215464 + 0.976512i \(0.569126\pi\)
\(702\) 0 0
\(703\) 2784.00 0.149361
\(704\) 0 0
\(705\) −7840.00 −0.418825
\(706\) 0 0
\(707\) 21164.0 1.12582
\(708\) 0 0
\(709\) −881.000 −0.0466666 −0.0233333 0.999728i \(-0.507428\pi\)
−0.0233333 + 0.999728i \(0.507428\pi\)
\(710\) 0 0
\(711\) −1052.00 −0.0554896
\(712\) 0 0
\(713\) −7911.00 −0.415525
\(714\) 0 0
\(715\) 4004.00 0.209428
\(716\) 0 0
\(717\) −18050.0 −0.940153
\(718\) 0 0
\(719\) −26093.0 −1.35341 −0.676707 0.736252i \(-0.736594\pi\)
−0.676707 + 0.736252i \(0.736594\pi\)
\(720\) 0 0
\(721\) −9776.00 −0.504962
\(722\) 0 0
\(723\) 32040.0 1.64811
\(724\) 0 0
\(725\) −1216.00 −0.0622912
\(726\) 0 0
\(727\) −7481.00 −0.381644 −0.190822 0.981625i \(-0.561115\pi\)
−0.190822 + 0.981625i \(0.561115\pi\)
\(728\) 0 0
\(729\) 20917.0 1.06269
\(730\) 0 0
\(731\) −5060.00 −0.256020
\(732\) 0 0
\(733\) 17788.0 0.896337 0.448168 0.893949i \(-0.352077\pi\)
0.448168 + 0.893949i \(0.352077\pi\)
\(734\) 0 0
\(735\) 11655.0 0.584900
\(736\) 0 0
\(737\) 1353.00 0.0676233
\(738\) 0 0
\(739\) −32182.0 −1.60194 −0.800970 0.598704i \(-0.795683\pi\)
−0.800970 + 0.598704i \(0.795683\pi\)
\(740\) 0 0
\(741\) 24960.0 1.23742
\(742\) 0 0
\(743\) −32044.0 −1.58221 −0.791104 0.611682i \(-0.790493\pi\)
−0.791104 + 0.611682i \(0.790493\pi\)
\(744\) 0 0
\(745\) 6454.00 0.317391
\(746\) 0 0
\(747\) 316.000 0.0154777
\(748\) 0 0
\(749\) 38844.0 1.89497
\(750\) 0 0
\(751\) −33779.0 −1.64130 −0.820648 0.571434i \(-0.806387\pi\)
−0.820648 + 0.571434i \(0.806387\pi\)
\(752\) 0 0
\(753\) 8935.00 0.432416
\(754\) 0 0
\(755\) 7210.00 0.347548
\(756\) 0 0
\(757\) −15630.0 −0.750439 −0.375219 0.926936i \(-0.622433\pi\)
−0.375219 + 0.926936i \(0.622433\pi\)
\(758\) 0 0
\(759\) 1485.00 0.0710172
\(760\) 0 0
\(761\) 1948.00 0.0927923 0.0463962 0.998923i \(-0.485226\pi\)
0.0463962 + 0.998923i \(0.485226\pi\)
\(762\) 0 0
\(763\) −21892.0 −1.03872
\(764\) 0 0
\(765\) 644.000 0.0304364
\(766\) 0 0
\(767\) 42900.0 2.01959
\(768\) 0 0
\(769\) −17420.0 −0.816881 −0.408440 0.912785i \(-0.633927\pi\)
−0.408440 + 0.912785i \(0.633927\pi\)
\(770\) 0 0
\(771\) 1770.00 0.0826783
\(772\) 0 0
\(773\) 11122.0 0.517504 0.258752 0.965944i \(-0.416689\pi\)
0.258752 + 0.965944i \(0.416689\pi\)
\(774\) 0 0
\(775\) 22268.0 1.03212
\(776\) 0 0
\(777\) −3770.00 −0.174064
\(778\) 0 0
\(779\) 45312.0 2.08404
\(780\) 0 0
\(781\) −11011.0 −0.504487
\(782\) 0 0
\(783\) 2320.00 0.105888
\(784\) 0 0
\(785\) −7119.00 −0.323679
\(786\) 0 0
\(787\) 29648.0 1.34287 0.671434 0.741064i \(-0.265679\pi\)
0.671434 + 0.741064i \(0.265679\pi\)
\(788\) 0 0
\(789\) 10130.0 0.457082
\(790\) 0 0
\(791\) −33306.0 −1.49712
\(792\) 0 0
\(793\) −28496.0 −1.27607
\(794\) 0 0
\(795\) 26390.0 1.17730
\(796\) 0 0
\(797\) −23219.0 −1.03194 −0.515972 0.856606i \(-0.672569\pi\)
−0.515972 + 0.856606i \(0.672569\pi\)
\(798\) 0 0
\(799\) −10304.0 −0.456232
\(800\) 0 0
\(801\) 2434.00 0.107367
\(802\) 0 0
\(803\) 11220.0 0.493082
\(804\) 0 0
\(805\) 4914.00 0.215150
\(806\) 0 0
\(807\) 43750.0 1.90839
\(808\) 0 0
\(809\) 9232.00 0.401211 0.200606 0.979672i \(-0.435709\pi\)
0.200606 + 0.979672i \(0.435709\pi\)
\(810\) 0 0
\(811\) −32286.0 −1.39792 −0.698961 0.715160i \(-0.746354\pi\)
−0.698961 + 0.715160i \(0.746354\pi\)
\(812\) 0 0
\(813\) 25180.0 1.08623
\(814\) 0 0
\(815\) 17108.0 0.735297
\(816\) 0 0
\(817\) 10560.0 0.452200
\(818\) 0 0
\(819\) 2704.00 0.115367
\(820\) 0 0
\(821\) −31706.0 −1.34780 −0.673902 0.738821i \(-0.735383\pi\)
−0.673902 + 0.738821i \(0.735383\pi\)
\(822\) 0 0
\(823\) 41139.0 1.74242 0.871212 0.490906i \(-0.163334\pi\)
0.871212 + 0.490906i \(0.163334\pi\)
\(824\) 0 0
\(825\) −4180.00 −0.176399
\(826\) 0 0
\(827\) −15252.0 −0.641311 −0.320655 0.947196i \(-0.603903\pi\)
−0.320655 + 0.947196i \(0.603903\pi\)
\(828\) 0 0
\(829\) 369.000 0.0154595 0.00772973 0.999970i \(-0.497540\pi\)
0.00772973 + 0.999970i \(0.497540\pi\)
\(830\) 0 0
\(831\) 21530.0 0.898757
\(832\) 0 0
\(833\) 15318.0 0.637140
\(834\) 0 0
\(835\) 10164.0 0.421245
\(836\) 0 0
\(837\) −42485.0 −1.75448
\(838\) 0 0
\(839\) −10257.0 −0.422063 −0.211032 0.977479i \(-0.567682\pi\)
−0.211032 + 0.977479i \(0.567682\pi\)
\(840\) 0 0
\(841\) −24133.0 −0.989503
\(842\) 0 0
\(843\) 6010.00 0.245546
\(844\) 0 0
\(845\) −3549.00 −0.144484
\(846\) 0 0
\(847\) −3146.00 −0.127624
\(848\) 0 0
\(849\) 33100.0 1.33803
\(850\) 0 0
\(851\) −783.000 −0.0315404
\(852\) 0 0
\(853\) −34386.0 −1.38025 −0.690126 0.723690i \(-0.742445\pi\)
−0.690126 + 0.723690i \(0.742445\pi\)
\(854\) 0 0
\(855\) −1344.00 −0.0537589
\(856\) 0 0
\(857\) −23464.0 −0.935257 −0.467628 0.883925i \(-0.654891\pi\)
−0.467628 + 0.883925i \(0.654891\pi\)
\(858\) 0 0
\(859\) −22475.0 −0.892709 −0.446355 0.894856i \(-0.647278\pi\)
−0.446355 + 0.894856i \(0.647278\pi\)
\(860\) 0 0
\(861\) −61360.0 −2.42874
\(862\) 0 0
\(863\) 2880.00 0.113599 0.0567997 0.998386i \(-0.481910\pi\)
0.0567997 + 0.998386i \(0.481910\pi\)
\(864\) 0 0
\(865\) −13398.0 −0.526642
\(866\) 0 0
\(867\) 13985.0 0.547815
\(868\) 0 0
\(869\) −5786.00 −0.225865
\(870\) 0 0
\(871\) −6396.00 −0.248818
\(872\) 0 0
\(873\) 526.000 0.0203922
\(874\) 0 0
\(875\) −36582.0 −1.41337
\(876\) 0 0
\(877\) −11084.0 −0.426773 −0.213387 0.976968i \(-0.568449\pi\)
−0.213387 + 0.976968i \(0.568449\pi\)
\(878\) 0 0
\(879\) −34840.0 −1.33689
\(880\) 0 0
\(881\) 41797.0 1.59838 0.799192 0.601076i \(-0.205261\pi\)
0.799192 + 0.601076i \(0.205261\pi\)
\(882\) 0 0
\(883\) 23780.0 0.906298 0.453149 0.891435i \(-0.350301\pi\)
0.453149 + 0.891435i \(0.350301\pi\)
\(884\) 0 0
\(885\) 28875.0 1.09675
\(886\) 0 0
\(887\) −14334.0 −0.542603 −0.271301 0.962494i \(-0.587454\pi\)
−0.271301 + 0.962494i \(0.587454\pi\)
\(888\) 0 0
\(889\) −45136.0 −1.70283
\(890\) 0 0
\(891\) 7381.00 0.277523
\(892\) 0 0
\(893\) 21504.0 0.805827
\(894\) 0 0
\(895\) −9051.00 −0.338035
\(896\) 0 0
\(897\) −7020.00 −0.261305
\(898\) 0 0
\(899\) −4688.00 −0.173919
\(900\) 0 0
\(901\) 34684.0 1.28245
\(902\) 0 0
\(903\) −14300.0 −0.526992
\(904\) 0 0
\(905\) −3185.00 −0.116987
\(906\) 0 0
\(907\) 25716.0 0.941440 0.470720 0.882283i \(-0.343994\pi\)
0.470720 + 0.882283i \(0.343994\pi\)
\(908\) 0 0
\(909\) 1628.00 0.0594030
\(910\) 0 0
\(911\) −28300.0 −1.02922 −0.514611 0.857424i \(-0.672064\pi\)
−0.514611 + 0.857424i \(0.672064\pi\)
\(912\) 0 0
\(913\) 1738.00 0.0630004
\(914\) 0 0
\(915\) −19180.0 −0.692974
\(916\) 0 0
\(917\) −156.000 −0.00561786
\(918\) 0 0
\(919\) 21410.0 0.768499 0.384250 0.923229i \(-0.374460\pi\)
0.384250 + 0.923229i \(0.374460\pi\)
\(920\) 0 0
\(921\) −38200.0 −1.36670
\(922\) 0 0
\(923\) 52052.0 1.85624
\(924\) 0 0
\(925\) 2204.00 0.0783428
\(926\) 0 0
\(927\) −752.000 −0.0266439
\(928\) 0 0
\(929\) 19938.0 0.704138 0.352069 0.935974i \(-0.385478\pi\)
0.352069 + 0.935974i \(0.385478\pi\)
\(930\) 0 0
\(931\) −31968.0 −1.12536
\(932\) 0 0
\(933\) 3260.00 0.114392
\(934\) 0 0
\(935\) 3542.00 0.123889
\(936\) 0 0
\(937\) 26728.0 0.931874 0.465937 0.884818i \(-0.345717\pi\)
0.465937 + 0.884818i \(0.345717\pi\)
\(938\) 0 0
\(939\) −40275.0 −1.39971
\(940\) 0 0
\(941\) 8634.00 0.299108 0.149554 0.988754i \(-0.452216\pi\)
0.149554 + 0.988754i \(0.452216\pi\)
\(942\) 0 0
\(943\) −12744.0 −0.440087
\(944\) 0 0
\(945\) 26390.0 0.908430
\(946\) 0 0
\(947\) 24841.0 0.852401 0.426201 0.904629i \(-0.359852\pi\)
0.426201 + 0.904629i \(0.359852\pi\)
\(948\) 0 0
\(949\) −53040.0 −1.81428
\(950\) 0 0
\(951\) 28375.0 0.967531
\(952\) 0 0
\(953\) 51234.0 1.74148 0.870741 0.491742i \(-0.163640\pi\)
0.870741 + 0.491742i \(0.163640\pi\)
\(954\) 0 0
\(955\) 7805.00 0.264465
\(956\) 0 0
\(957\) 880.000 0.0297245
\(958\) 0 0
\(959\) 37414.0 1.25981
\(960\) 0 0
\(961\) 56058.0 1.88171
\(962\) 0 0
\(963\) 2988.00 0.0999865
\(964\) 0 0
\(965\) −35084.0 −1.17036
\(966\) 0 0
\(967\) −26752.0 −0.889645 −0.444822 0.895619i \(-0.646733\pi\)
−0.444822 + 0.895619i \(0.646733\pi\)
\(968\) 0 0
\(969\) 22080.0 0.732004
\(970\) 0 0
\(971\) 21155.0 0.699172 0.349586 0.936904i \(-0.386322\pi\)
0.349586 + 0.936904i \(0.386322\pi\)
\(972\) 0 0
\(973\) 8268.00 0.272415
\(974\) 0 0
\(975\) 19760.0 0.649053
\(976\) 0 0
\(977\) −56323.0 −1.84435 −0.922176 0.386770i \(-0.873591\pi\)
−0.922176 + 0.386770i \(0.873591\pi\)
\(978\) 0 0
\(979\) 13387.0 0.437028
\(980\) 0 0
\(981\) −1684.00 −0.0548073
\(982\) 0 0
\(983\) 24683.0 0.800880 0.400440 0.916323i \(-0.368857\pi\)
0.400440 + 0.916323i \(0.368857\pi\)
\(984\) 0 0
\(985\) 29022.0 0.938800
\(986\) 0 0
\(987\) −29120.0 −0.939108
\(988\) 0 0
\(989\) −2970.00 −0.0954909
\(990\) 0 0
\(991\) −26816.0 −0.859574 −0.429787 0.902930i \(-0.641411\pi\)
−0.429787 + 0.902930i \(0.641411\pi\)
\(992\) 0 0
\(993\) 11225.0 0.358726
\(994\) 0 0
\(995\) 8680.00 0.276557
\(996\) 0 0
\(997\) 18614.0 0.591285 0.295643 0.955299i \(-0.404466\pi\)
0.295643 + 0.955299i \(0.404466\pi\)
\(998\) 0 0
\(999\) −4205.00 −0.133173
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 44.4.a.a.1.1 1
3.2 odd 2 396.4.a.e.1.1 1
4.3 odd 2 176.4.a.e.1.1 1
5.2 odd 4 1100.4.b.c.749.2 2
5.3 odd 4 1100.4.b.c.749.1 2
5.4 even 2 1100.4.a.d.1.1 1
7.6 odd 2 2156.4.a.b.1.1 1
8.3 odd 2 704.4.a.c.1.1 1
8.5 even 2 704.4.a.j.1.1 1
11.10 odd 2 484.4.a.a.1.1 1
12.11 even 2 1584.4.a.p.1.1 1
44.43 even 2 1936.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.4.a.a.1.1 1 1.1 even 1 trivial
176.4.a.e.1.1 1 4.3 odd 2
396.4.a.e.1.1 1 3.2 odd 2
484.4.a.a.1.1 1 11.10 odd 2
704.4.a.c.1.1 1 8.3 odd 2
704.4.a.j.1.1 1 8.5 even 2
1100.4.a.d.1.1 1 5.4 even 2
1100.4.b.c.749.1 2 5.3 odd 4
1100.4.b.c.749.2 2 5.2 odd 4
1584.4.a.p.1.1 1 12.11 even 2
1936.4.a.m.1.1 1 44.43 even 2
2156.4.a.b.1.1 1 7.6 odd 2