Properties

Label 3724.2.a.i.1.2
Level $3724$
Weight $2$
Character 3724.1
Self dual yes
Analytic conductor $29.736$
Analytic rank $0$
Dimension $3$
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] = [3724,2,Mod(1,3724)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3724, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3724.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3724.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: \(29.7362897127\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.17819\) of defining polynomial
Character \(\chi\) \(=\) 3724.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.17819 q^{3} -3.43366 q^{5} -1.61186 q^{9} +O(q^{10})\) \(q+1.17819 q^{3} -3.43366 q^{5} -1.61186 q^{9} +2.61186 q^{11} -5.43366 q^{13} -4.04552 q^{15} +0.611859 q^{17} +1.00000 q^{19} -1.43366 q^{23} +6.79005 q^{25} -5.43366 q^{27} +1.74453 q^{29} -0.255471 q^{31} +3.07728 q^{33} +5.79005 q^{37} -6.40191 q^{39} -8.96825 q^{41} +7.58011 q^{43} +5.53458 q^{45} +10.6574 q^{47} +0.720888 q^{51} +5.53458 q^{53} -8.96825 q^{55} +1.17819 q^{57} -4.30099 q^{59} +1.27911 q^{61} +18.6574 q^{65} +0.611859 q^{67} -1.68914 q^{69} -1.07728 q^{71} +11.6891 q^{73} +8.00000 q^{75} -12.4474 q^{79} -1.56634 q^{81} +10.4019 q^{83} -2.10092 q^{85} +2.05539 q^{87} +13.2237 q^{89} -0.300994 q^{93} -3.43366 q^{95} +7.88110 q^{97} -4.20995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 2 q^{5} + 6 q^{9} - 3 q^{11} - 8 q^{13} + 7 q^{15} - 9 q^{17} + 3 q^{19} + 4 q^{23} + 7 q^{25} - 8 q^{27} + 11 q^{29} + 5 q^{31} + 6 q^{33} + 4 q^{37} + 5 q^{39} - 11 q^{41} - 4 q^{43} + 9 q^{45} + 2 q^{47} + 4 q^{51} + 9 q^{53} - 11 q^{55} + q^{57} + 12 q^{59} + 2 q^{61} + 26 q^{65} - 9 q^{67} + 9 q^{69} + 21 q^{73} + 24 q^{75} + 6 q^{79} - 13 q^{81} + 7 q^{83} - 7 q^{85} + 26 q^{87} + 18 q^{89} + 24 q^{93} - 2 q^{95} - 28 q^{97} - 26 q^{99}+O(q^{100}) \) 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
\(3\) 1.17819 0.680231 0.340115 0.940384i \(-0.389534\pi\)
0.340115 + 0.940384i \(0.389534\pi\)
\(4\) 0 0
\(5\) −3.43366 −1.53558 −0.767791 0.640701i \(-0.778644\pi\)
−0.767791 + 0.640701i \(0.778644\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.61186 −0.537286
\(10\) 0 0
\(11\) 2.61186 0.787505 0.393753 0.919216i \(-0.371177\pi\)
0.393753 + 0.919216i \(0.371177\pi\)
\(12\) 0 0
\(13\) −5.43366 −1.50703 −0.753514 0.657432i \(-0.771643\pi\)
−0.753514 + 0.657432i \(0.771643\pi\)
\(14\) 0 0
\(15\) −4.04552 −1.04455
\(16\) 0 0
\(17\) 0.611859 0.148398 0.0741988 0.997243i \(-0.476360\pi\)
0.0741988 + 0.997243i \(0.476360\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.43366 −0.298940 −0.149470 0.988766i \(-0.547757\pi\)
−0.149470 + 0.988766i \(0.547757\pi\)
\(24\) 0 0
\(25\) 6.79005 1.35801
\(26\) 0 0
\(27\) −5.43366 −1.04571
\(28\) 0 0
\(29\) 1.74453 0.323951 0.161975 0.986795i \(-0.448213\pi\)
0.161975 + 0.986795i \(0.448213\pi\)
\(30\) 0 0
\(31\) −0.255471 −0.0458839 −0.0229419 0.999737i \(-0.507303\pi\)
−0.0229419 + 0.999737i \(0.507303\pi\)
\(32\) 0 0
\(33\) 3.07728 0.535685
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.79005 0.951879 0.475939 0.879478i \(-0.342108\pi\)
0.475939 + 0.879478i \(0.342108\pi\)
\(38\) 0 0
\(39\) −6.40191 −1.02513
\(40\) 0 0
\(41\) −8.96825 −1.40060 −0.700302 0.713846i \(-0.746952\pi\)
−0.700302 + 0.713846i \(0.746952\pi\)
\(42\) 0 0
\(43\) 7.58011 1.15596 0.577978 0.816053i \(-0.303842\pi\)
0.577978 + 0.816053i \(0.303842\pi\)
\(44\) 0 0
\(45\) 5.53458 0.825047
\(46\) 0 0
\(47\) 10.6574 1.55454 0.777269 0.629168i \(-0.216604\pi\)
0.777269 + 0.629168i \(0.216604\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.720888 0.100945
\(52\) 0 0
\(53\) 5.53458 0.760233 0.380117 0.924939i \(-0.375884\pi\)
0.380117 + 0.924939i \(0.375884\pi\)
\(54\) 0 0
\(55\) −8.96825 −1.20928
\(56\) 0 0
\(57\) 1.17819 0.156056
\(58\) 0 0
\(59\) −4.30099 −0.559942 −0.279971 0.960009i \(-0.590325\pi\)
−0.279971 + 0.960009i \(0.590325\pi\)
\(60\) 0 0
\(61\) 1.27911 0.163773 0.0818867 0.996642i \(-0.473905\pi\)
0.0818867 + 0.996642i \(0.473905\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 18.6574 2.31416
\(66\) 0 0
\(67\) 0.611859 0.0747504 0.0373752 0.999301i \(-0.488100\pi\)
0.0373752 + 0.999301i \(0.488100\pi\)
\(68\) 0 0
\(69\) −1.68914 −0.203348
\(70\) 0 0
\(71\) −1.07728 −0.127849 −0.0639246 0.997955i \(-0.520362\pi\)
−0.0639246 + 0.997955i \(0.520362\pi\)
\(72\) 0 0
\(73\) 11.6891 1.36811 0.684055 0.729431i \(-0.260215\pi\)
0.684055 + 0.729431i \(0.260215\pi\)
\(74\) 0 0
\(75\) 8.00000 0.923760
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.4474 −1.40045 −0.700223 0.713924i \(-0.746916\pi\)
−0.700223 + 0.713924i \(0.746916\pi\)
\(80\) 0 0
\(81\) −1.56634 −0.174037
\(82\) 0 0
\(83\) 10.4019 1.14176 0.570879 0.821034i \(-0.306603\pi\)
0.570879 + 0.821034i \(0.306603\pi\)
\(84\) 0 0
\(85\) −2.10092 −0.227877
\(86\) 0 0
\(87\) 2.05539 0.220361
\(88\) 0 0
\(89\) 13.2237 1.40171 0.700856 0.713303i \(-0.252802\pi\)
0.700856 + 0.713303i \(0.252802\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −0.300994 −0.0312116
\(94\) 0 0
\(95\) −3.43366 −0.352287
\(96\) 0 0
\(97\) 7.88110 0.800204 0.400102 0.916471i \(-0.368975\pi\)
0.400102 + 0.916471i \(0.368975\pi\)
\(98\) 0 0
\(99\) −4.20995 −0.423116
\(100\) 0 0
\(101\) 16.2375 1.61569 0.807845 0.589395i \(-0.200634\pi\)
0.807845 + 0.589395i \(0.200634\pi\)
\(102\) 0 0
\(103\) −6.14644 −0.605627 −0.302813 0.953050i \(-0.597926\pi\)
−0.302813 + 0.953050i \(0.597926\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.07728 0.490839 0.245419 0.969417i \(-0.421074\pi\)
0.245419 + 0.969417i \(0.421074\pi\)
\(108\) 0 0
\(109\) −18.3010 −1.75292 −0.876459 0.481477i \(-0.840100\pi\)
−0.876459 + 0.481477i \(0.840100\pi\)
\(110\) 0 0
\(111\) 6.82181 0.647497
\(112\) 0 0
\(113\) 3.17819 0.298979 0.149490 0.988763i \(-0.452237\pi\)
0.149490 + 0.988763i \(0.452237\pi\)
\(114\) 0 0
\(115\) 4.92272 0.459046
\(116\) 0 0
\(117\) 8.75830 0.809705
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.17819 −0.379836
\(122\) 0 0
\(123\) −10.5663 −0.952734
\(124\) 0 0
\(125\) −6.14644 −0.549754
\(126\) 0 0
\(127\) 10.0910 0.895436 0.447718 0.894175i \(-0.352237\pi\)
0.447718 + 0.894175i \(0.352237\pi\)
\(128\) 0 0
\(129\) 8.93084 0.786316
\(130\) 0 0
\(131\) −10.1920 −0.890476 −0.445238 0.895412i \(-0.646881\pi\)
−0.445238 + 0.895412i \(0.646881\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 18.6574 1.60577
\(136\) 0 0
\(137\) 11.7347 1.00256 0.501280 0.865285i \(-0.332863\pi\)
0.501280 + 0.865285i \(0.332863\pi\)
\(138\) 0 0
\(139\) 9.94461 0.843490 0.421745 0.906714i \(-0.361418\pi\)
0.421745 + 0.906714i \(0.361418\pi\)
\(140\) 0 0
\(141\) 12.5565 1.05745
\(142\) 0 0
\(143\) −14.1920 −1.18679
\(144\) 0 0
\(145\) −5.99013 −0.497453
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.0773 1.07133 0.535666 0.844430i \(-0.320061\pi\)
0.535666 + 0.844430i \(0.320061\pi\)
\(150\) 0 0
\(151\) −18.3465 −1.49302 −0.746509 0.665375i \(-0.768272\pi\)
−0.746509 + 0.665375i \(0.768272\pi\)
\(152\) 0 0
\(153\) −0.986230 −0.0797320
\(154\) 0 0
\(155\) 0.877200 0.0704584
\(156\) 0 0
\(157\) 15.9901 1.27615 0.638076 0.769974i \(-0.279731\pi\)
0.638076 + 0.769974i \(0.279731\pi\)
\(158\) 0 0
\(159\) 6.52081 0.517134
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.33275 −0.261041 −0.130520 0.991446i \(-0.541665\pi\)
−0.130520 + 0.991446i \(0.541665\pi\)
\(164\) 0 0
\(165\) −10.5663 −0.822588
\(166\) 0 0
\(167\) −12.5028 −0.967498 −0.483749 0.875207i \(-0.660725\pi\)
−0.483749 + 0.875207i \(0.660725\pi\)
\(168\) 0 0
\(169\) 16.5247 1.27113
\(170\) 0 0
\(171\) −1.61186 −0.123262
\(172\) 0 0
\(173\) −10.7763 −0.819306 −0.409653 0.912242i \(-0.634350\pi\)
−0.409653 + 0.912242i \(0.634350\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5.06741 −0.380890
\(178\) 0 0
\(179\) −3.47919 −0.260047 −0.130023 0.991511i \(-0.541505\pi\)
−0.130023 + 0.991511i \(0.541505\pi\)
\(180\) 0 0
\(181\) 7.83558 0.582414 0.291207 0.956660i \(-0.405943\pi\)
0.291207 + 0.956660i \(0.405943\pi\)
\(182\) 0 0
\(183\) 1.50704 0.111404
\(184\) 0 0
\(185\) −19.8811 −1.46169
\(186\) 0 0
\(187\) 1.59809 0.116864
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 23.4238 1.69489 0.847443 0.530886i \(-0.178141\pi\)
0.847443 + 0.530886i \(0.178141\pi\)
\(192\) 0 0
\(193\) 21.4157 1.54153 0.770767 0.637117i \(-0.219873\pi\)
0.770767 + 0.637117i \(0.219873\pi\)
\(194\) 0 0
\(195\) 21.9820 1.57416
\(196\) 0 0
\(197\) −21.8275 −1.55514 −0.777571 0.628795i \(-0.783549\pi\)
−0.777571 + 0.628795i \(0.783549\pi\)
\(198\) 0 0
\(199\) 24.3920 1.72911 0.864553 0.502542i \(-0.167602\pi\)
0.864553 + 0.502542i \(0.167602\pi\)
\(200\) 0 0
\(201\) 0.720888 0.0508475
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 30.7940 2.15074
\(206\) 0 0
\(207\) 2.31086 0.160616
\(208\) 0 0
\(209\) 2.61186 0.180666
\(210\) 0 0
\(211\) −5.62563 −0.387284 −0.193642 0.981072i \(-0.562030\pi\)
−0.193642 + 0.981072i \(0.562030\pi\)
\(212\) 0 0
\(213\) −1.26924 −0.0869669
\(214\) 0 0
\(215\) −26.0275 −1.77506
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 13.7721 0.930630
\(220\) 0 0
\(221\) −3.32464 −0.223639
\(222\) 0 0
\(223\) −19.3702 −1.29712 −0.648561 0.761163i \(-0.724629\pi\)
−0.648561 + 0.761163i \(0.724629\pi\)
\(224\) 0 0
\(225\) −10.9446 −0.729640
\(226\) 0 0
\(227\) 15.2138 1.00978 0.504889 0.863184i \(-0.331533\pi\)
0.504889 + 0.863184i \(0.331533\pi\)
\(228\) 0 0
\(229\) 0.265341 0.0175343 0.00876713 0.999962i \(-0.497209\pi\)
0.00876713 + 0.999962i \(0.497209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.6119 1.35033 0.675164 0.737668i \(-0.264073\pi\)
0.675164 + 0.737668i \(0.264073\pi\)
\(234\) 0 0
\(235\) −36.5939 −2.38712
\(236\) 0 0
\(237\) −14.6655 −0.952626
\(238\) 0 0
\(239\) 0.00811199 0.000524721 0 0.000262360 1.00000i \(-0.499916\pi\)
0.000262360 1.00000i \(0.499916\pi\)
\(240\) 0 0
\(241\) 20.0910 1.29418 0.647089 0.762414i \(-0.275986\pi\)
0.647089 + 0.762414i \(0.275986\pi\)
\(242\) 0 0
\(243\) 14.4555 0.927324
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.43366 −0.345736
\(248\) 0 0
\(249\) 12.2555 0.776659
\(250\) 0 0
\(251\) 24.3465 1.53674 0.768369 0.640007i \(-0.221068\pi\)
0.768369 + 0.640007i \(0.221068\pi\)
\(252\) 0 0
\(253\) −3.74453 −0.235417
\(254\) 0 0
\(255\) −2.47529 −0.155009
\(256\) 0 0
\(257\) 23.2057 1.44753 0.723767 0.690044i \(-0.242409\pi\)
0.723767 + 0.690044i \(0.242409\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.81193 −0.174054
\(262\) 0 0
\(263\) 23.3327 1.43876 0.719379 0.694617i \(-0.244426\pi\)
0.719379 + 0.694617i \(0.244426\pi\)
\(264\) 0 0
\(265\) −19.0039 −1.16740
\(266\) 0 0
\(267\) 15.5801 0.953487
\(268\) 0 0
\(269\) −25.9820 −1.58415 −0.792076 0.610423i \(-0.791000\pi\)
−0.792076 + 0.610423i \(0.791000\pi\)
\(270\) 0 0
\(271\) 25.1503 1.52777 0.763887 0.645350i \(-0.223288\pi\)
0.763887 + 0.645350i \(0.223288\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.7347 1.06944
\(276\) 0 0
\(277\) −19.2237 −1.15504 −0.577521 0.816376i \(-0.695980\pi\)
−0.577521 + 0.816376i \(0.695980\pi\)
\(278\) 0 0
\(279\) 0.411782 0.0246528
\(280\) 0 0
\(281\) −1.27911 −0.0763054 −0.0381527 0.999272i \(-0.512147\pi\)
−0.0381527 + 0.999272i \(0.512147\pi\)
\(282\) 0 0
\(283\) −23.1147 −1.37403 −0.687013 0.726645i \(-0.741078\pi\)
−0.687013 + 0.726645i \(0.741078\pi\)
\(284\) 0 0
\(285\) −4.04552 −0.239636
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.6256 −0.977978
\(290\) 0 0
\(291\) 9.28546 0.544324
\(292\) 0 0
\(293\) −23.7901 −1.38983 −0.694915 0.719092i \(-0.744558\pi\)
−0.694915 + 0.719092i \(0.744558\pi\)
\(294\) 0 0
\(295\) 14.7682 0.859836
\(296\) 0 0
\(297\) −14.1920 −0.823501
\(298\) 0 0
\(299\) 7.79005 0.450510
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 19.1309 1.09904
\(304\) 0 0
\(305\) −4.39204 −0.251488
\(306\) 0 0
\(307\) −18.5011 −1.05591 −0.527956 0.849272i \(-0.677041\pi\)
−0.527956 + 0.849272i \(0.677041\pi\)
\(308\) 0 0
\(309\) −7.24170 −0.411966
\(310\) 0 0
\(311\) −12.3465 −0.700107 −0.350053 0.936730i \(-0.613837\pi\)
−0.350053 + 0.936730i \(0.613837\pi\)
\(312\) 0 0
\(313\) 5.67115 0.320552 0.160276 0.987072i \(-0.448761\pi\)
0.160276 + 0.987072i \(0.448761\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.0219 −0.843713 −0.421856 0.906663i \(-0.638621\pi\)
−0.421856 + 0.906663i \(0.638621\pi\)
\(318\) 0 0
\(319\) 4.55646 0.255113
\(320\) 0 0
\(321\) 5.98202 0.333884
\(322\) 0 0
\(323\) 0.611859 0.0340447
\(324\) 0 0
\(325\) −36.8949 −2.04656
\(326\) 0 0
\(327\) −21.5621 −1.19239
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.73076 0.479886 0.239943 0.970787i \(-0.422871\pi\)
0.239943 + 0.970787i \(0.422871\pi\)
\(332\) 0 0
\(333\) −9.33275 −0.511431
\(334\) 0 0
\(335\) −2.10092 −0.114785
\(336\) 0 0
\(337\) −22.7502 −1.23928 −0.619641 0.784885i \(-0.712722\pi\)
−0.619641 + 0.784885i \(0.712722\pi\)
\(338\) 0 0
\(339\) 3.74453 0.203375
\(340\) 0 0
\(341\) −0.667253 −0.0361338
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5.79992 0.312257
\(346\) 0 0
\(347\) 18.0731 0.970213 0.485106 0.874455i \(-0.338781\pi\)
0.485106 + 0.874455i \(0.338781\pi\)
\(348\) 0 0
\(349\) −10.3384 −0.553402 −0.276701 0.960956i \(-0.589241\pi\)
−0.276701 + 0.960956i \(0.589241\pi\)
\(350\) 0 0
\(351\) 29.5247 1.57591
\(352\) 0 0
\(353\) 17.0318 0.906509 0.453254 0.891381i \(-0.350263\pi\)
0.453254 + 0.891381i \(0.350263\pi\)
\(354\) 0 0
\(355\) 3.69901 0.196323
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.8575 −0.889703 −0.444851 0.895604i \(-0.646743\pi\)
−0.444851 + 0.895604i \(0.646743\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −4.92272 −0.258376
\(364\) 0 0
\(365\) −40.1366 −2.10084
\(366\) 0 0
\(367\) −34.7484 −1.81385 −0.906927 0.421289i \(-0.861578\pi\)
−0.906927 + 0.421289i \(0.861578\pi\)
\(368\) 0 0
\(369\) 14.4555 0.752526
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −29.0039 −1.50176 −0.750882 0.660436i \(-0.770372\pi\)
−0.750882 + 0.660436i \(0.770372\pi\)
\(374\) 0 0
\(375\) −7.24170 −0.373960
\(376\) 0 0
\(377\) −9.47919 −0.488203
\(378\) 0 0
\(379\) 9.88110 0.507558 0.253779 0.967262i \(-0.418326\pi\)
0.253779 + 0.967262i \(0.418326\pi\)
\(380\) 0 0
\(381\) 11.8892 0.609103
\(382\) 0 0
\(383\) −19.5882 −1.00091 −0.500456 0.865762i \(-0.666834\pi\)
−0.500456 + 0.865762i \(0.666834\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.2181 −0.621079
\(388\) 0 0
\(389\) 16.6119 0.842255 0.421127 0.907002i \(-0.361635\pi\)
0.421127 + 0.907002i \(0.361635\pi\)
\(390\) 0 0
\(391\) −0.877200 −0.0443619
\(392\) 0 0
\(393\) −12.0081 −0.605729
\(394\) 0 0
\(395\) 42.7403 2.15050
\(396\) 0 0
\(397\) −19.1129 −0.959250 −0.479625 0.877473i \(-0.659227\pi\)
−0.479625 + 0.877473i \(0.659227\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.96825 0.347978 0.173989 0.984748i \(-0.444334\pi\)
0.173989 + 0.984748i \(0.444334\pi\)
\(402\) 0 0
\(403\) 1.38814 0.0691482
\(404\) 0 0
\(405\) 5.37827 0.267248
\(406\) 0 0
\(407\) 15.1228 0.749609
\(408\) 0 0
\(409\) 25.6891 1.27025 0.635123 0.772411i \(-0.280949\pi\)
0.635123 + 0.772411i \(0.280949\pi\)
\(410\) 0 0
\(411\) 13.8257 0.681972
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −35.7167 −1.75326
\(416\) 0 0
\(417\) 11.7167 0.573768
\(418\) 0 0
\(419\) −0.519053 −0.0253574 −0.0126787 0.999920i \(-0.504036\pi\)
−0.0126787 + 0.999920i \(0.504036\pi\)
\(420\) 0 0
\(421\) 36.2375 1.76611 0.883054 0.469272i \(-0.155484\pi\)
0.883054 + 0.469272i \(0.155484\pi\)
\(422\) 0 0
\(423\) −17.1782 −0.835232
\(424\) 0 0
\(425\) 4.15455 0.201525
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −16.7209 −0.807292
\(430\) 0 0
\(431\) −3.73466 −0.179892 −0.0899461 0.995947i \(-0.528669\pi\)
−0.0899461 + 0.995947i \(0.528669\pi\)
\(432\) 0 0
\(433\) 3.79005 0.182138 0.0910692 0.995845i \(-0.470972\pi\)
0.0910692 + 0.995845i \(0.470972\pi\)
\(434\) 0 0
\(435\) −7.05753 −0.338383
\(436\) 0 0
\(437\) −1.43366 −0.0685815
\(438\) 0 0
\(439\) −6.57445 −0.313781 −0.156891 0.987616i \(-0.550147\pi\)
−0.156891 + 0.987616i \(0.550147\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −23.4238 −1.11290 −0.556449 0.830882i \(-0.687837\pi\)
−0.556449 + 0.830882i \(0.687837\pi\)
\(444\) 0 0
\(445\) −45.4058 −2.15244
\(446\) 0 0
\(447\) 15.4076 0.728753
\(448\) 0 0
\(449\) −32.2001 −1.51962 −0.759808 0.650148i \(-0.774707\pi\)
−0.759808 + 0.650148i \(0.774707\pi\)
\(450\) 0 0
\(451\) −23.4238 −1.10298
\(452\) 0 0
\(453\) −21.6158 −1.01560
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.0674 0.891936 0.445968 0.895049i \(-0.352860\pi\)
0.445968 + 0.895049i \(0.352860\pi\)
\(458\) 0 0
\(459\) −3.32464 −0.155181
\(460\) 0 0
\(461\) −19.8712 −0.925495 −0.462748 0.886490i \(-0.653136\pi\)
−0.462748 + 0.886490i \(0.653136\pi\)
\(462\) 0 0
\(463\) −24.5028 −1.13874 −0.569372 0.822080i \(-0.692813\pi\)
−0.569372 + 0.822080i \(0.692813\pi\)
\(464\) 0 0
\(465\) 1.03351 0.0479280
\(466\) 0 0
\(467\) 29.7802 1.37806 0.689031 0.724732i \(-0.258036\pi\)
0.689031 + 0.724732i \(0.258036\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 18.8395 0.868077
\(472\) 0 0
\(473\) 19.7982 0.910321
\(474\) 0 0
\(475\) 6.79005 0.311549
\(476\) 0 0
\(477\) −8.92096 −0.408463
\(478\) 0 0
\(479\) 17.4792 0.798644 0.399322 0.916811i \(-0.369245\pi\)
0.399322 + 0.916811i \(0.369245\pi\)
\(480\) 0 0
\(481\) −31.4612 −1.43451
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −27.0611 −1.22878
\(486\) 0 0
\(487\) 34.0829 1.54445 0.772223 0.635352i \(-0.219145\pi\)
0.772223 + 0.635352i \(0.219145\pi\)
\(488\) 0 0
\(489\) −3.92662 −0.177568
\(490\) 0 0
\(491\) 32.8949 1.48452 0.742262 0.670109i \(-0.233753\pi\)
0.742262 + 0.670109i \(0.233753\pi\)
\(492\) 0 0
\(493\) 1.06741 0.0480735
\(494\) 0 0
\(495\) 14.4555 0.649729
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −35.5346 −1.59075 −0.795373 0.606120i \(-0.792725\pi\)
−0.795373 + 0.606120i \(0.792725\pi\)
\(500\) 0 0
\(501\) −14.7308 −0.658122
\(502\) 0 0
\(503\) −32.5858 −1.45293 −0.726464 0.687205i \(-0.758837\pi\)
−0.726464 + 0.687205i \(0.758837\pi\)
\(504\) 0 0
\(505\) −55.7541 −2.48102
\(506\) 0 0
\(507\) 19.4693 0.864663
\(508\) 0 0
\(509\) 19.7622 0.875944 0.437972 0.898989i \(-0.355697\pi\)
0.437972 + 0.898989i \(0.355697\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −5.43366 −0.239902
\(514\) 0 0
\(515\) 21.1048 0.929989
\(516\) 0 0
\(517\) 27.8356 1.22421
\(518\) 0 0
\(519\) −12.6966 −0.557317
\(520\) 0 0
\(521\) 26.9503 1.18071 0.590356 0.807143i \(-0.298987\pi\)
0.590356 + 0.807143i \(0.298987\pi\)
\(522\) 0 0
\(523\) −15.6793 −0.685606 −0.342803 0.939407i \(-0.611376\pi\)
−0.342803 + 0.939407i \(0.611376\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.156312 −0.00680905
\(528\) 0 0
\(529\) −20.9446 −0.910635
\(530\) 0 0
\(531\) 6.93259 0.300849
\(532\) 0 0
\(533\) 48.7304 2.11075
\(534\) 0 0
\(535\) −17.4337 −0.753723
\(536\) 0 0
\(537\) −4.09916 −0.176892
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 24.6295 1.05891 0.529453 0.848339i \(-0.322397\pi\)
0.529453 + 0.848339i \(0.322397\pi\)
\(542\) 0 0
\(543\) 9.23183 0.396176
\(544\) 0 0
\(545\) 62.8395 2.69175
\(546\) 0 0
\(547\) −0.603747 −0.0258143 −0.0129072 0.999917i \(-0.504109\pi\)
−0.0129072 + 0.999917i \(0.504109\pi\)
\(548\) 0 0
\(549\) −2.06175 −0.0879932
\(550\) 0 0
\(551\) 1.74453 0.0743195
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −23.4238 −0.994285
\(556\) 0 0
\(557\) 27.6810 1.17288 0.586441 0.809992i \(-0.300528\pi\)
0.586441 + 0.809992i \(0.300528\pi\)
\(558\) 0 0
\(559\) −41.1878 −1.74206
\(560\) 0 0
\(561\) 1.88286 0.0794943
\(562\) 0 0
\(563\) 0.831677 0.0350510 0.0175255 0.999846i \(-0.494421\pi\)
0.0175255 + 0.999846i \(0.494421\pi\)
\(564\) 0 0
\(565\) −10.9129 −0.459107
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 39.4139 1.65232 0.826159 0.563437i \(-0.190521\pi\)
0.826159 + 0.563437i \(0.190521\pi\)
\(570\) 0 0
\(571\) 9.63550 0.403233 0.201617 0.979465i \(-0.435381\pi\)
0.201617 + 0.979465i \(0.435381\pi\)
\(572\) 0 0
\(573\) 27.5978 1.15291
\(574\) 0 0
\(575\) −9.73466 −0.405963
\(576\) 0 0
\(577\) −7.48730 −0.311700 −0.155850 0.987781i \(-0.549812\pi\)
−0.155850 + 0.987781i \(0.549812\pi\)
\(578\) 0 0
\(579\) 25.2318 1.04860
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 14.4555 0.598688
\(584\) 0 0
\(585\) −30.0731 −1.24337
\(586\) 0 0
\(587\) 28.6020 1.18053 0.590265 0.807209i \(-0.299023\pi\)
0.590265 + 0.807209i \(0.299023\pi\)
\(588\) 0 0
\(589\) −0.255471 −0.0105265
\(590\) 0 0
\(591\) −25.7170 −1.05786
\(592\) 0 0
\(593\) 18.0829 0.742577 0.371289 0.928518i \(-0.378916\pi\)
0.371289 + 0.928518i \(0.378916\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 28.7386 1.17619
\(598\) 0 0
\(599\) 22.9601 0.938126 0.469063 0.883165i \(-0.344592\pi\)
0.469063 + 0.883165i \(0.344592\pi\)
\(600\) 0 0
\(601\) 11.8437 0.483114 0.241557 0.970387i \(-0.422342\pi\)
0.241557 + 0.970387i \(0.422342\pi\)
\(602\) 0 0
\(603\) −0.986230 −0.0401624
\(604\) 0 0
\(605\) 14.3465 0.583269
\(606\) 0 0
\(607\) 40.5939 1.64765 0.823827 0.566841i \(-0.191835\pi\)
0.823827 + 0.566841i \(0.191835\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −57.9086 −2.34273
\(612\) 0 0
\(613\) −36.8021 −1.48642 −0.743211 0.669057i \(-0.766698\pi\)
−0.743211 + 0.669057i \(0.766698\pi\)
\(614\) 0 0
\(615\) 36.2813 1.46300
\(616\) 0 0
\(617\) −5.41568 −0.218027 −0.109014 0.994040i \(-0.534769\pi\)
−0.109014 + 0.994040i \(0.534769\pi\)
\(618\) 0 0
\(619\) 36.4295 1.46422 0.732112 0.681185i \(-0.238535\pi\)
0.732112 + 0.681185i \(0.238535\pi\)
\(620\) 0 0
\(621\) 7.79005 0.312604
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.8454 −0.513818
\(626\) 0 0
\(627\) 3.07728 0.122895
\(628\) 0 0
\(629\) 3.54269 0.141256
\(630\) 0 0
\(631\) 40.7957 1.62405 0.812026 0.583622i \(-0.198365\pi\)
0.812026 + 0.583622i \(0.198365\pi\)
\(632\) 0 0
\(633\) −6.62808 −0.263443
\(634\) 0 0
\(635\) −34.6493 −1.37501
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1.73642 0.0686916
\(640\) 0 0
\(641\) −4.39380 −0.173545 −0.0867723 0.996228i \(-0.527655\pi\)
−0.0867723 + 0.996228i \(0.527655\pi\)
\(642\) 0 0
\(643\) −20.4831 −0.807774 −0.403887 0.914809i \(-0.632341\pi\)
−0.403887 + 0.914809i \(0.632341\pi\)
\(644\) 0 0
\(645\) −30.6655 −1.20745
\(646\) 0 0
\(647\) 11.6793 0.459159 0.229580 0.973290i \(-0.426265\pi\)
0.229580 + 0.973290i \(0.426265\pi\)
\(648\) 0 0
\(649\) −11.2336 −0.440957
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14.1902 −0.555306 −0.277653 0.960681i \(-0.589556\pi\)
−0.277653 + 0.960681i \(0.589556\pi\)
\(654\) 0 0
\(655\) 34.9958 1.36740
\(656\) 0 0
\(657\) −18.8412 −0.735067
\(658\) 0 0
\(659\) 14.2392 0.554682 0.277341 0.960772i \(-0.410547\pi\)
0.277341 + 0.960772i \(0.410547\pi\)
\(660\) 0 0
\(661\) 28.2929 1.10047 0.550233 0.835011i \(-0.314539\pi\)
0.550233 + 0.835011i \(0.314539\pi\)
\(662\) 0 0
\(663\) −3.91707 −0.152126
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.50107 −0.0968418
\(668\) 0 0
\(669\) −22.8218 −0.882342
\(670\) 0 0
\(671\) 3.34086 0.128972
\(672\) 0 0
\(673\) −38.1203 −1.46943 −0.734716 0.678375i \(-0.762684\pi\)
−0.734716 + 0.678375i \(0.762684\pi\)
\(674\) 0 0
\(675\) −36.8949 −1.42008
\(676\) 0 0
\(677\) −30.2911 −1.16418 −0.582091 0.813123i \(-0.697765\pi\)
−0.582091 + 0.813123i \(0.697765\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 17.9249 0.686882
\(682\) 0 0
\(683\) −13.9721 −0.534629 −0.267315 0.963609i \(-0.586136\pi\)
−0.267315 + 0.963609i \(0.586136\pi\)
\(684\) 0 0
\(685\) −40.2929 −1.53951
\(686\) 0 0
\(687\) 0.312624 0.0119273
\(688\) 0 0
\(689\) −30.0731 −1.14569
\(690\) 0 0
\(691\) −46.9422 −1.78576 −0.892882 0.450291i \(-0.851320\pi\)
−0.892882 + 0.450291i \(0.851320\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −34.1464 −1.29525
\(696\) 0 0
\(697\) −5.48730 −0.207846
\(698\) 0 0
\(699\) 24.2848 0.918534
\(700\) 0 0
\(701\) 19.3148 0.729509 0.364754 0.931104i \(-0.381153\pi\)
0.364754 + 0.931104i \(0.381153\pi\)
\(702\) 0 0
\(703\) 5.79005 0.218376
\(704\) 0 0
\(705\) −43.1147 −1.62379
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −18.1902 −0.683148 −0.341574 0.939855i \(-0.610960\pi\)
−0.341574 + 0.939855i \(0.610960\pi\)
\(710\) 0 0
\(711\) 20.0635 0.752440
\(712\) 0 0
\(713\) 0.366259 0.0137165
\(714\) 0 0
\(715\) 48.7304 1.82242
\(716\) 0 0
\(717\) 0.00955750 0.000356931 0
\(718\) 0 0
\(719\) 30.1383 1.12397 0.561985 0.827147i \(-0.310038\pi\)
0.561985 + 0.827147i \(0.310038\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 23.6712 0.880340
\(724\) 0 0
\(725\) 11.8454 0.439929
\(726\) 0 0
\(727\) 20.1902 0.748813 0.374407 0.927265i \(-0.377846\pi\)
0.374407 + 0.927265i \(0.377846\pi\)
\(728\) 0 0
\(729\) 21.7304 0.804831
\(730\) 0 0
\(731\) 4.63795 0.171541
\(732\) 0 0
\(733\) −39.1878 −1.44743 −0.723716 0.690098i \(-0.757568\pi\)
−0.723716 + 0.690098i \(0.757568\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.59809 0.0588663
\(738\) 0 0
\(739\) 21.4337 0.788450 0.394225 0.919014i \(-0.371013\pi\)
0.394225 + 0.919014i \(0.371013\pi\)
\(740\) 0 0
\(741\) −6.40191 −0.235180
\(742\) 0 0
\(743\) 11.5409 0.423396 0.211698 0.977335i \(-0.432101\pi\)
0.211698 + 0.977335i \(0.432101\pi\)
\(744\) 0 0
\(745\) −44.9030 −1.64512
\(746\) 0 0
\(747\) −16.7664 −0.613451
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −50.5205 −1.84352 −0.921760 0.387762i \(-0.873248\pi\)
−0.921760 + 0.387762i \(0.873248\pi\)
\(752\) 0 0
\(753\) 28.6849 1.04534
\(754\) 0 0
\(755\) 62.9958 2.29265
\(756\) 0 0
\(757\) 4.97812 0.180933 0.0904664 0.995900i \(-0.471164\pi\)
0.0904664 + 0.995900i \(0.471164\pi\)
\(758\) 0 0
\(759\) −4.41178 −0.160138
\(760\) 0 0
\(761\) −28.8592 −1.04615 −0.523073 0.852288i \(-0.675214\pi\)
−0.523073 + 0.852288i \(0.675214\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.38638 0.122435
\(766\) 0 0
\(767\) 23.3702 0.843848
\(768\) 0 0
\(769\) 15.6239 0.563411 0.281706 0.959501i \(-0.409100\pi\)
0.281706 + 0.959501i \(0.409100\pi\)
\(770\) 0 0
\(771\) 27.3409 0.984657
\(772\) 0 0
\(773\) −27.9640 −1.00580 −0.502898 0.864346i \(-0.667733\pi\)
−0.502898 + 0.864346i \(0.667733\pi\)
\(774\) 0 0
\(775\) −1.73466 −0.0623108
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.96825 −0.321321
\(780\) 0 0
\(781\) −2.81369 −0.100682
\(782\) 0 0
\(783\) −9.47919 −0.338759
\(784\) 0 0
\(785\) −54.9047 −1.95963
\(786\) 0 0
\(787\) −18.5744 −0.662108 −0.331054 0.943612i \(-0.607404\pi\)
−0.331054 + 0.943612i \(0.607404\pi\)
\(788\) 0 0
\(789\) 27.4905 0.978688
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.95026 −0.246811
\(794\) 0 0
\(795\) −22.3903 −0.794101
\(796\) 0 0
\(797\) 21.8631 0.774431 0.387216 0.921989i \(-0.373437\pi\)
0.387216 + 0.921989i \(0.373437\pi\)
\(798\) 0 0
\(799\) 6.52081 0.230690
\(800\) 0 0
\(801\) −21.3148 −0.753120
\(802\) 0 0
\(803\) 30.5304 1.07739
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −30.6119 −1.07759
\(808\) 0 0
\(809\) 35.0138 1.23102 0.615509 0.788130i \(-0.288950\pi\)
0.615509 + 0.788130i \(0.288950\pi\)
\(810\) 0 0
\(811\) −36.6020 −1.28527 −0.642635 0.766173i \(-0.722159\pi\)
−0.642635 + 0.766173i \(0.722159\pi\)
\(812\) 0 0
\(813\) 29.6320 1.03924
\(814\) 0 0
\(815\) 11.4435 0.400850
\(816\) 0 0
\(817\) 7.58011 0.265194
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.4199 −0.642859 −0.321429 0.946934i \(-0.604163\pi\)
−0.321429 + 0.946934i \(0.604163\pi\)
\(822\) 0 0
\(823\) −33.8257 −1.17909 −0.589545 0.807736i \(-0.700693\pi\)
−0.589545 + 0.807736i \(0.700693\pi\)
\(824\) 0 0
\(825\) 20.8949 0.727466
\(826\) 0 0
\(827\) 32.5304 1.13119 0.565596 0.824683i \(-0.308646\pi\)
0.565596 + 0.824683i \(0.308646\pi\)
\(828\) 0 0
\(829\) 13.9168 0.483349 0.241674 0.970357i \(-0.422303\pi\)
0.241674 + 0.970357i \(0.422303\pi\)
\(830\) 0 0
\(831\) −22.6493 −0.785694
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 42.9305 1.48567
\(836\) 0 0
\(837\) 1.38814 0.0479812
\(838\) 0 0
\(839\) 27.2872 0.942060 0.471030 0.882117i \(-0.343882\pi\)
0.471030 + 0.882117i \(0.343882\pi\)
\(840\) 0 0
\(841\) −25.9566 −0.895056
\(842\) 0 0
\(843\) −1.50704 −0.0519053
\(844\) 0 0
\(845\) −56.7403 −1.95193
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −27.2336 −0.934654
\(850\) 0 0
\(851\) −8.30099 −0.284554
\(852\) 0 0
\(853\) −9.95448 −0.340835 −0.170417 0.985372i \(-0.554512\pi\)
−0.170417 + 0.985372i \(0.554512\pi\)
\(854\) 0 0
\(855\) 5.53458 0.189279
\(856\) 0 0
\(857\) −20.1644 −0.688804 −0.344402 0.938822i \(-0.611918\pi\)
−0.344402 + 0.938822i \(0.611918\pi\)
\(858\) 0 0
\(859\) −9.78829 −0.333972 −0.166986 0.985959i \(-0.553404\pi\)
−0.166986 + 0.985959i \(0.553404\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 41.0039 1.39579 0.697894 0.716201i \(-0.254120\pi\)
0.697894 + 0.716201i \(0.254120\pi\)
\(864\) 0 0
\(865\) 37.0021 1.25811
\(866\) 0 0
\(867\) −19.5882 −0.665251
\(868\) 0 0
\(869\) −32.5109 −1.10286
\(870\) 0 0
\(871\) −3.32464 −0.112651
\(872\) 0 0
\(873\) −12.7032 −0.429939
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 21.2513 0.717604 0.358802 0.933414i \(-0.383185\pi\)
0.358802 + 0.933414i \(0.383185\pi\)
\(878\) 0 0
\(879\) −28.0293 −0.945405
\(880\) 0 0
\(881\) 9.66159 0.325507 0.162754 0.986667i \(-0.447962\pi\)
0.162754 + 0.986667i \(0.447962\pi\)
\(882\) 0 0
\(883\) −27.7703 −0.934545 −0.467273 0.884113i \(-0.654763\pi\)
−0.467273 + 0.884113i \(0.654763\pi\)
\(884\) 0 0
\(885\) 17.3998 0.584887
\(886\) 0 0
\(887\) 10.3483 0.347461 0.173731 0.984793i \(-0.444418\pi\)
0.173731 + 0.984793i \(0.444418\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.09105 −0.137055
\(892\) 0 0
\(893\) 10.6574 0.356636
\(894\) 0 0
\(895\) 11.9464 0.399323
\(896\) 0 0
\(897\) 9.17819 0.306451
\(898\) 0 0
\(899\) −0.445676 −0.0148641
\(900\) 0 0
\(901\) 3.38638 0.112817
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −26.9047 −0.894344
\(906\) 0 0
\(907\) −13.2791 −0.440926 −0.220463 0.975395i \(-0.570757\pi\)
−0.220463 + 0.975395i \(0.570757\pi\)
\(908\) 0 0
\(909\) −26.1725 −0.868088
\(910\) 0 0
\(911\) 10.1821 0.337348 0.168674 0.985672i \(-0.446052\pi\)
0.168674 + 0.985672i \(0.446052\pi\)
\(912\) 0 0
\(913\) 27.1683 0.899140
\(914\) 0 0
\(915\) −5.17468 −0.171070
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −11.0773 −0.365406 −0.182703 0.983168i \(-0.558485\pi\)
−0.182703 + 0.983168i \(0.558485\pi\)
\(920\) 0 0
\(921\) −21.7979 −0.718264
\(922\) 0 0
\(923\) 5.85356 0.192672
\(924\) 0 0
\(925\) 39.3148 1.29266
\(926\) 0 0
\(927\) 9.90719 0.325395
\(928\) 0 0
\(929\) −23.0512 −0.756285 −0.378142 0.925747i \(-0.623437\pi\)
−0.378142 + 0.925747i \(0.623437\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −14.5466 −0.476234
\(934\) 0 0
\(935\) −5.48730 −0.179454
\(936\) 0 0
\(937\) −6.36626 −0.207977 −0.103988 0.994579i \(-0.533160\pi\)
−0.103988 + 0.994579i \(0.533160\pi\)
\(938\) 0 0
\(939\) 6.68172 0.218050
\(940\) 0 0
\(941\) 2.15455 0.0702364 0.0351182 0.999383i \(-0.488819\pi\)
0.0351182 + 0.999383i \(0.488819\pi\)
\(942\) 0 0
\(943\) 12.8575 0.418696
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −24.7745 −0.805064 −0.402532 0.915406i \(-0.631870\pi\)
−0.402532 + 0.915406i \(0.631870\pi\)
\(948\) 0 0
\(949\) −63.5148 −2.06178
\(950\) 0 0
\(951\) −17.6987 −0.573919
\(952\) 0 0
\(953\) −4.11714 −0.133367 −0.0666836 0.997774i \(-0.521242\pi\)
−0.0666836 + 0.997774i \(0.521242\pi\)
\(954\) 0 0
\(955\) −80.4295 −2.60264
\(956\) 0 0
\(957\) 5.36840 0.173536
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.9347 −0.997895
\(962\) 0 0
\(963\) −8.18385 −0.263721
\(964\) 0 0
\(965\) −73.5343 −2.36715
\(966\) 0 0
\(967\) −18.9047 −0.607935 −0.303968 0.952682i \(-0.598312\pi\)
−0.303968 + 0.952682i \(0.598312\pi\)
\(968\) 0 0
\(969\) 0.720888 0.0231583
\(970\) 0 0
\(971\) −7.57199 −0.242997 −0.121498 0.992592i \(-0.538770\pi\)
−0.121498 + 0.992592i \(0.538770\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −43.4693 −1.39213
\(976\) 0 0
\(977\) 44.4277 1.42137 0.710684 0.703511i \(-0.248386\pi\)
0.710684 + 0.703511i \(0.248386\pi\)
\(978\) 0 0
\(979\) 34.5385 1.10385
\(980\) 0 0
\(981\) 29.4986 0.941818
\(982\) 0 0
\(983\) 49.3780 1.57491 0.787456 0.616371i \(-0.211398\pi\)
0.787456 + 0.616371i \(0.211398\pi\)
\(984\) 0 0
\(985\) 74.9482 2.38805
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −10.8673 −0.345561
\(990\) 0 0
\(991\) 53.1405 1.68806 0.844031 0.536294i \(-0.180176\pi\)
0.844031 + 0.536294i \(0.180176\pi\)
\(992\) 0 0
\(993\) 10.2865 0.326433
\(994\) 0 0
\(995\) −83.7541 −2.65518
\(996\) 0 0
\(997\) 22.3546 0.707978 0.353989 0.935250i \(-0.384825\pi\)
0.353989 + 0.935250i \(0.384825\pi\)
\(998\) 0 0
\(999\) −31.4612 −0.995388
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 3724.2.a.i.1.2 3
7.6 odd 2 532.2.a.e.1.2 3
21.20 even 2 4788.2.a.o.1.1 3
28.27 even 2 2128.2.a.r.1.2 3
56.13 odd 2 8512.2.a.bn.1.2 3
56.27 even 2 8512.2.a.bl.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.e.1.2 3 7.6 odd 2
2128.2.a.r.1.2 3 28.27 even 2
3724.2.a.i.1.2 3 1.1 even 1 trivial
4788.2.a.o.1.1 3 21.20 even 2
8512.2.a.bl.1.2 3 56.27 even 2
8512.2.a.bn.1.2 3 56.13 odd 2