Properties

Label 4840.2.a.m.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $2$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.56155 q^{3} -1.00000 q^{5} -4.56155 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q+2.56155 q^{3} -1.00000 q^{5} -4.56155 q^{7} +3.56155 q^{9} +1.12311 q^{13} -2.56155 q^{15} +7.68466 q^{17} -1.43845 q^{19} -11.6847 q^{21} +1.12311 q^{23} +1.00000 q^{25} +1.43845 q^{27} -8.56155 q^{29} -1.43845 q^{31} +4.56155 q^{35} +7.43845 q^{37} +2.87689 q^{39} +12.2462 q^{41} +3.12311 q^{43} -3.56155 q^{45} -11.3693 q^{47} +13.8078 q^{49} +19.6847 q^{51} +9.68466 q^{53} -3.68466 q^{57} +1.12311 q^{59} +12.5616 q^{61} -16.2462 q^{63} -1.12311 q^{65} +2.87689 q^{69} +3.68466 q^{71} -1.12311 q^{73} +2.56155 q^{75} +11.3693 q^{79} -7.00000 q^{81} +6.00000 q^{83} -7.68466 q^{85} -21.9309 q^{87} +9.68466 q^{89} -5.12311 q^{91} -3.68466 q^{93} +1.43845 q^{95} -4.87689 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{5} - 5 q^{7} + 3 q^{9} - 6 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 11 q^{21} - 6 q^{23} + 2 q^{25} + 7 q^{27} - 13 q^{29} - 7 q^{31} + 5 q^{35} + 19 q^{37} + 14 q^{39} + 8 q^{41} - 2 q^{43} - 3 q^{45} + 2 q^{47} + 7 q^{49} + 27 q^{51} + 7 q^{53} + 5 q^{57} - 6 q^{59} + 21 q^{61} - 16 q^{63} + 6 q^{65} + 14 q^{69} - 5 q^{71} + 6 q^{73} + q^{75} - 2 q^{79} - 14 q^{81} + 12 q^{83} - 3 q^{85} - 15 q^{87} + 7 q^{89} - 2 q^{91} + 5 q^{93} + 7 q^{95} - 18 q^{97}+O(q^{100}) \) 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
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.56155 −1.72410 −0.862052 0.506819i \(-0.830821\pi\)
−0.862052 + 0.506819i \(0.830821\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 1.12311 0.311493 0.155747 0.987797i \(-0.450222\pi\)
0.155747 + 0.987797i \(0.450222\pi\)
\(14\) 0 0
\(15\) −2.56155 −0.661390
\(16\) 0 0
\(17\) 7.68466 1.86380 0.931902 0.362711i \(-0.118149\pi\)
0.931902 + 0.362711i \(0.118149\pi\)
\(18\) 0 0
\(19\) −1.43845 −0.330002 −0.165001 0.986293i \(-0.552763\pi\)
−0.165001 + 0.986293i \(0.552763\pi\)
\(20\) 0 0
\(21\) −11.6847 −2.54980
\(22\) 0 0
\(23\) 1.12311 0.234184 0.117092 0.993121i \(-0.462643\pi\)
0.117092 + 0.993121i \(0.462643\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) −8.56155 −1.58984 −0.794920 0.606714i \(-0.792487\pi\)
−0.794920 + 0.606714i \(0.792487\pi\)
\(30\) 0 0
\(31\) −1.43845 −0.258353 −0.129176 0.991622i \(-0.541233\pi\)
−0.129176 + 0.991622i \(0.541233\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.56155 0.771043
\(36\) 0 0
\(37\) 7.43845 1.22287 0.611437 0.791293i \(-0.290592\pi\)
0.611437 + 0.791293i \(0.290592\pi\)
\(38\) 0 0
\(39\) 2.87689 0.460672
\(40\) 0 0
\(41\) 12.2462 1.91254 0.956268 0.292490i \(-0.0944840\pi\)
0.956268 + 0.292490i \(0.0944840\pi\)
\(42\) 0 0
\(43\) 3.12311 0.476269 0.238135 0.971232i \(-0.423464\pi\)
0.238135 + 0.971232i \(0.423464\pi\)
\(44\) 0 0
\(45\) −3.56155 −0.530925
\(46\) 0 0
\(47\) −11.3693 −1.65839 −0.829193 0.558963i \(-0.811199\pi\)
−0.829193 + 0.558963i \(0.811199\pi\)
\(48\) 0 0
\(49\) 13.8078 1.97254
\(50\) 0 0
\(51\) 19.6847 2.75640
\(52\) 0 0
\(53\) 9.68466 1.33029 0.665145 0.746714i \(-0.268370\pi\)
0.665145 + 0.746714i \(0.268370\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.68466 −0.488045
\(58\) 0 0
\(59\) 1.12311 0.146216 0.0731079 0.997324i \(-0.476708\pi\)
0.0731079 + 0.997324i \(0.476708\pi\)
\(60\) 0 0
\(61\) 12.5616 1.60834 0.804171 0.594398i \(-0.202610\pi\)
0.804171 + 0.594398i \(0.202610\pi\)
\(62\) 0 0
\(63\) −16.2462 −2.04683
\(64\) 0 0
\(65\) −1.12311 −0.139304
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 2.87689 0.346337
\(70\) 0 0
\(71\) 3.68466 0.437289 0.218644 0.975805i \(-0.429837\pi\)
0.218644 + 0.975805i \(0.429837\pi\)
\(72\) 0 0
\(73\) −1.12311 −0.131450 −0.0657248 0.997838i \(-0.520936\pi\)
−0.0657248 + 0.997838i \(0.520936\pi\)
\(74\) 0 0
\(75\) 2.56155 0.295783
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.3693 1.27915 0.639574 0.768729i \(-0.279111\pi\)
0.639574 + 0.768729i \(0.279111\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −7.68466 −0.833518
\(86\) 0 0
\(87\) −21.9309 −2.35124
\(88\) 0 0
\(89\) 9.68466 1.02657 0.513286 0.858218i \(-0.328428\pi\)
0.513286 + 0.858218i \(0.328428\pi\)
\(90\) 0 0
\(91\) −5.12311 −0.537047
\(92\) 0 0
\(93\) −3.68466 −0.382081
\(94\) 0 0
\(95\) 1.43845 0.147582
\(96\) 0 0
\(97\) −4.87689 −0.495174 −0.247587 0.968866i \(-0.579638\pi\)
−0.247587 + 0.968866i \(0.579638\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 11.6847 1.14031
\(106\) 0 0
\(107\) −7.12311 −0.688617 −0.344308 0.938857i \(-0.611887\pi\)
−0.344308 + 0.938857i \(0.611887\pi\)
\(108\) 0 0
\(109\) 4.24621 0.406713 0.203357 0.979105i \(-0.434815\pi\)
0.203357 + 0.979105i \(0.434815\pi\)
\(110\) 0 0
\(111\) 19.0540 1.80852
\(112\) 0 0
\(113\) 0.876894 0.0824913 0.0412456 0.999149i \(-0.486867\pi\)
0.0412456 + 0.999149i \(0.486867\pi\)
\(114\) 0 0
\(115\) −1.12311 −0.104730
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −35.0540 −3.21339
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 31.3693 2.82848
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.87689 −0.432754 −0.216377 0.976310i \(-0.569424\pi\)
−0.216377 + 0.976310i \(0.569424\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 16.8078 1.46850 0.734251 0.678879i \(-0.237534\pi\)
0.734251 + 0.678879i \(0.237534\pi\)
\(132\) 0 0
\(133\) 6.56155 0.568959
\(134\) 0 0
\(135\) −1.43845 −0.123802
\(136\) 0 0
\(137\) −7.12311 −0.608568 −0.304284 0.952581i \(-0.598417\pi\)
−0.304284 + 0.952581i \(0.598417\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −29.1231 −2.45261
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 8.56155 0.710998
\(146\) 0 0
\(147\) 35.3693 2.91721
\(148\) 0 0
\(149\) −1.68466 −0.138013 −0.0690063 0.997616i \(-0.521983\pi\)
−0.0690063 + 0.997616i \(0.521983\pi\)
\(150\) 0 0
\(151\) −6.87689 −0.559634 −0.279817 0.960053i \(-0.590274\pi\)
−0.279817 + 0.960053i \(0.590274\pi\)
\(152\) 0 0
\(153\) 27.3693 2.21268
\(154\) 0 0
\(155\) 1.43845 0.115539
\(156\) 0 0
\(157\) 21.6847 1.73062 0.865312 0.501233i \(-0.167120\pi\)
0.865312 + 0.501233i \(0.167120\pi\)
\(158\) 0 0
\(159\) 24.8078 1.96738
\(160\) 0 0
\(161\) −5.12311 −0.403757
\(162\) 0 0
\(163\) 20.8078 1.62979 0.814895 0.579609i \(-0.196795\pi\)
0.814895 + 0.579609i \(0.196795\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.68466 0.130363 0.0651814 0.997873i \(-0.479237\pi\)
0.0651814 + 0.997873i \(0.479237\pi\)
\(168\) 0 0
\(169\) −11.7386 −0.902972
\(170\) 0 0
\(171\) −5.12311 −0.391774
\(172\) 0 0
\(173\) −10.8769 −0.826955 −0.413477 0.910514i \(-0.635686\pi\)
−0.413477 + 0.910514i \(0.635686\pi\)
\(174\) 0 0
\(175\) −4.56155 −0.344821
\(176\) 0 0
\(177\) 2.87689 0.216241
\(178\) 0 0
\(179\) 14.2462 1.06481 0.532406 0.846489i \(-0.321288\pi\)
0.532406 + 0.846489i \(0.321288\pi\)
\(180\) 0 0
\(181\) −8.24621 −0.612936 −0.306468 0.951881i \(-0.599147\pi\)
−0.306468 + 0.951881i \(0.599147\pi\)
\(182\) 0 0
\(183\) 32.1771 2.37860
\(184\) 0 0
\(185\) −7.43845 −0.546886
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −6.56155 −0.477283
\(190\) 0 0
\(191\) 10.2462 0.741390 0.370695 0.928755i \(-0.379120\pi\)
0.370695 + 0.928755i \(0.379120\pi\)
\(192\) 0 0
\(193\) 21.4384 1.54317 0.771587 0.636124i \(-0.219463\pi\)
0.771587 + 0.636124i \(0.219463\pi\)
\(194\) 0 0
\(195\) −2.87689 −0.206019
\(196\) 0 0
\(197\) −22.7386 −1.62006 −0.810030 0.586388i \(-0.800549\pi\)
−0.810030 + 0.586388i \(0.800549\pi\)
\(198\) 0 0
\(199\) −4.31534 −0.305906 −0.152953 0.988233i \(-0.548878\pi\)
−0.152953 + 0.988233i \(0.548878\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 39.0540 2.74105
\(204\) 0 0
\(205\) −12.2462 −0.855312
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −19.6847 −1.35515 −0.677574 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(212\) 0 0
\(213\) 9.43845 0.646712
\(214\) 0 0
\(215\) −3.12311 −0.212994
\(216\) 0 0
\(217\) 6.56155 0.445427
\(218\) 0 0
\(219\) −2.87689 −0.194403
\(220\) 0 0
\(221\) 8.63068 0.580563
\(222\) 0 0
\(223\) 25.1231 1.68237 0.841184 0.540749i \(-0.181859\pi\)
0.841184 + 0.540749i \(0.181859\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) −8.24621 −0.547320 −0.273660 0.961826i \(-0.588234\pi\)
−0.273660 + 0.961826i \(0.588234\pi\)
\(228\) 0 0
\(229\) 16.2462 1.07358 0.536790 0.843716i \(-0.319637\pi\)
0.536790 + 0.843716i \(0.319637\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −27.0540 −1.77236 −0.886182 0.463336i \(-0.846652\pi\)
−0.886182 + 0.463336i \(0.846652\pi\)
\(234\) 0 0
\(235\) 11.3693 0.741652
\(236\) 0 0
\(237\) 29.1231 1.89175
\(238\) 0 0
\(239\) 21.6155 1.39819 0.699096 0.715028i \(-0.253586\pi\)
0.699096 + 0.715028i \(0.253586\pi\)
\(240\) 0 0
\(241\) −18.4924 −1.19120 −0.595601 0.803281i \(-0.703086\pi\)
−0.595601 + 0.803281i \(0.703086\pi\)
\(242\) 0 0
\(243\) −22.2462 −1.42710
\(244\) 0 0
\(245\) −13.8078 −0.882146
\(246\) 0 0
\(247\) −1.61553 −0.102794
\(248\) 0 0
\(249\) 15.3693 0.973991
\(250\) 0 0
\(251\) −27.3693 −1.72754 −0.863768 0.503890i \(-0.831902\pi\)
−0.863768 + 0.503890i \(0.831902\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −19.6847 −1.23270
\(256\) 0 0
\(257\) 6.49242 0.404986 0.202493 0.979284i \(-0.435096\pi\)
0.202493 + 0.979284i \(0.435096\pi\)
\(258\) 0 0
\(259\) −33.9309 −2.10836
\(260\) 0 0
\(261\) −30.4924 −1.88743
\(262\) 0 0
\(263\) 29.0540 1.79154 0.895772 0.444513i \(-0.146623\pi\)
0.895772 + 0.444513i \(0.146623\pi\)
\(264\) 0 0
\(265\) −9.68466 −0.594924
\(266\) 0 0
\(267\) 24.8078 1.51821
\(268\) 0 0
\(269\) 0.876894 0.0534652 0.0267326 0.999643i \(-0.491490\pi\)
0.0267326 + 0.999643i \(0.491490\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) 0 0
\(273\) −13.1231 −0.794246
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.2462 1.33665 0.668323 0.743872i \(-0.267013\pi\)
0.668323 + 0.743872i \(0.267013\pi\)
\(278\) 0 0
\(279\) −5.12311 −0.306712
\(280\) 0 0
\(281\) −8.24621 −0.491928 −0.245964 0.969279i \(-0.579104\pi\)
−0.245964 + 0.969279i \(0.579104\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 0 0
\(285\) 3.68466 0.218260
\(286\) 0 0
\(287\) −55.8617 −3.29741
\(288\) 0 0
\(289\) 42.0540 2.47376
\(290\) 0 0
\(291\) −12.4924 −0.732319
\(292\) 0 0
\(293\) −17.1231 −1.00034 −0.500171 0.865927i \(-0.666730\pi\)
−0.500171 + 0.865927i \(0.666730\pi\)
\(294\) 0 0
\(295\) −1.12311 −0.0653897
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.26137 0.0729467
\(300\) 0 0
\(301\) −14.2462 −0.821138
\(302\) 0 0
\(303\) 5.12311 0.294315
\(304\) 0 0
\(305\) −12.5616 −0.719272
\(306\) 0 0
\(307\) −18.0000 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.43845 −0.535205 −0.267603 0.963529i \(-0.586231\pi\)
−0.267603 + 0.963529i \(0.586231\pi\)
\(312\) 0 0
\(313\) 24.7386 1.39831 0.699155 0.714970i \(-0.253560\pi\)
0.699155 + 0.714970i \(0.253560\pi\)
\(314\) 0 0
\(315\) 16.2462 0.915370
\(316\) 0 0
\(317\) 6.31534 0.354705 0.177352 0.984147i \(-0.443247\pi\)
0.177352 + 0.984147i \(0.443247\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −18.2462 −1.01840
\(322\) 0 0
\(323\) −11.0540 −0.615060
\(324\) 0 0
\(325\) 1.12311 0.0622987
\(326\) 0 0
\(327\) 10.8769 0.601494
\(328\) 0 0
\(329\) 51.8617 2.85923
\(330\) 0 0
\(331\) −18.7386 −1.02997 −0.514984 0.857200i \(-0.672202\pi\)
−0.514984 + 0.857200i \(0.672202\pi\)
\(332\) 0 0
\(333\) 26.4924 1.45178
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.80776 0.261896 0.130948 0.991389i \(-0.458198\pi\)
0.130948 + 0.991389i \(0.458198\pi\)
\(338\) 0 0
\(339\) 2.24621 0.121997
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −31.0540 −1.67676
\(344\) 0 0
\(345\) −2.87689 −0.154887
\(346\) 0 0
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) 0 0
\(349\) −14.4924 −0.775762 −0.387881 0.921710i \(-0.626793\pi\)
−0.387881 + 0.921710i \(0.626793\pi\)
\(350\) 0 0
\(351\) 1.61553 0.0862305
\(352\) 0 0
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) −3.68466 −0.195561
\(356\) 0 0
\(357\) −89.7926 −4.75233
\(358\) 0 0
\(359\) −18.7386 −0.988987 −0.494494 0.869181i \(-0.664646\pi\)
−0.494494 + 0.869181i \(0.664646\pi\)
\(360\) 0 0
\(361\) −16.9309 −0.891098
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.12311 0.0587860
\(366\) 0 0
\(367\) 16.4924 0.860897 0.430449 0.902615i \(-0.358355\pi\)
0.430449 + 0.902615i \(0.358355\pi\)
\(368\) 0 0
\(369\) 43.6155 2.27053
\(370\) 0 0
\(371\) −44.1771 −2.29356
\(372\) 0 0
\(373\) −23.3693 −1.21002 −0.605009 0.796219i \(-0.706830\pi\)
−0.605009 + 0.796219i \(0.706830\pi\)
\(374\) 0 0
\(375\) −2.56155 −0.132278
\(376\) 0 0
\(377\) −9.61553 −0.495225
\(378\) 0 0
\(379\) −14.8769 −0.764175 −0.382087 0.924126i \(-0.624795\pi\)
−0.382087 + 0.924126i \(0.624795\pi\)
\(380\) 0 0
\(381\) −12.4924 −0.640006
\(382\) 0 0
\(383\) −0.630683 −0.0322264 −0.0161132 0.999870i \(-0.505129\pi\)
−0.0161132 + 0.999870i \(0.505129\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.1231 0.565419
\(388\) 0 0
\(389\) −14.4924 −0.734795 −0.367397 0.930064i \(-0.619751\pi\)
−0.367397 + 0.930064i \(0.619751\pi\)
\(390\) 0 0
\(391\) 8.63068 0.436472
\(392\) 0 0
\(393\) 43.0540 2.17179
\(394\) 0 0
\(395\) −11.3693 −0.572052
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 16.8078 0.841441
\(400\) 0 0
\(401\) 30.8078 1.53847 0.769233 0.638968i \(-0.220638\pi\)
0.769233 + 0.638968i \(0.220638\pi\)
\(402\) 0 0
\(403\) −1.61553 −0.0804752
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −17.3693 −0.858857 −0.429429 0.903101i \(-0.641285\pi\)
−0.429429 + 0.903101i \(0.641285\pi\)
\(410\) 0 0
\(411\) −18.2462 −0.900019
\(412\) 0 0
\(413\) −5.12311 −0.252092
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 10.2462 0.501759
\(418\) 0 0
\(419\) 0.492423 0.0240564 0.0120282 0.999928i \(-0.496171\pi\)
0.0120282 + 0.999928i \(0.496171\pi\)
\(420\) 0 0
\(421\) 27.6155 1.34590 0.672949 0.739689i \(-0.265027\pi\)
0.672949 + 0.739689i \(0.265027\pi\)
\(422\) 0 0
\(423\) −40.4924 −1.96881
\(424\) 0 0
\(425\) 7.68466 0.372761
\(426\) 0 0
\(427\) −57.3002 −2.77295
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) −4.87689 −0.234369 −0.117184 0.993110i \(-0.537387\pi\)
−0.117184 + 0.993110i \(0.537387\pi\)
\(434\) 0 0
\(435\) 21.9309 1.05150
\(436\) 0 0
\(437\) −1.61553 −0.0772812
\(438\) 0 0
\(439\) −10.2462 −0.489025 −0.244512 0.969646i \(-0.578628\pi\)
−0.244512 + 0.969646i \(0.578628\pi\)
\(440\) 0 0
\(441\) 49.1771 2.34177
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −9.68466 −0.459097
\(446\) 0 0
\(447\) −4.31534 −0.204109
\(448\) 0 0
\(449\) 0.246211 0.0116194 0.00580971 0.999983i \(-0.498151\pi\)
0.00580971 + 0.999983i \(0.498151\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −17.6155 −0.827650
\(454\) 0 0
\(455\) 5.12311 0.240175
\(456\) 0 0
\(457\) −3.68466 −0.172361 −0.0861805 0.996280i \(-0.527466\pi\)
−0.0861805 + 0.996280i \(0.527466\pi\)
\(458\) 0 0
\(459\) 11.0540 0.515955
\(460\) 0 0
\(461\) 21.0540 0.980581 0.490291 0.871559i \(-0.336891\pi\)
0.490291 + 0.871559i \(0.336891\pi\)
\(462\) 0 0
\(463\) −27.3693 −1.27196 −0.635980 0.771706i \(-0.719404\pi\)
−0.635980 + 0.771706i \(0.719404\pi\)
\(464\) 0 0
\(465\) 3.68466 0.170872
\(466\) 0 0
\(467\) 32.8078 1.51816 0.759081 0.650996i \(-0.225649\pi\)
0.759081 + 0.650996i \(0.225649\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 55.5464 2.55944
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.43845 −0.0660005
\(476\) 0 0
\(477\) 34.4924 1.57930
\(478\) 0 0
\(479\) −14.2462 −0.650926 −0.325463 0.945555i \(-0.605520\pi\)
−0.325463 + 0.945555i \(0.605520\pi\)
\(480\) 0 0
\(481\) 8.35416 0.380917
\(482\) 0 0
\(483\) −13.1231 −0.597122
\(484\) 0 0
\(485\) 4.87689 0.221448
\(486\) 0 0
\(487\) −30.2462 −1.37059 −0.685293 0.728267i \(-0.740326\pi\)
−0.685293 + 0.728267i \(0.740326\pi\)
\(488\) 0 0
\(489\) 53.3002 2.41032
\(490\) 0 0
\(491\) 16.3153 0.736301 0.368151 0.929766i \(-0.379991\pi\)
0.368151 + 0.929766i \(0.379991\pi\)
\(492\) 0 0
\(493\) −65.7926 −2.96315
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.8078 −0.753931
\(498\) 0 0
\(499\) −18.7386 −0.838856 −0.419428 0.907789i \(-0.637769\pi\)
−0.419428 + 0.907789i \(0.637769\pi\)
\(500\) 0 0
\(501\) 4.31534 0.192795
\(502\) 0 0
\(503\) 3.12311 0.139252 0.0696262 0.997573i \(-0.477819\pi\)
0.0696262 + 0.997573i \(0.477819\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) −30.0691 −1.33542
\(508\) 0 0
\(509\) −0.384472 −0.0170414 −0.00852071 0.999964i \(-0.502712\pi\)
−0.00852071 + 0.999964i \(0.502712\pi\)
\(510\) 0 0
\(511\) 5.12311 0.226633
\(512\) 0 0
\(513\) −2.06913 −0.0913543
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −27.8617 −1.22299
\(520\) 0 0
\(521\) −30.4924 −1.33590 −0.667949 0.744207i \(-0.732827\pi\)
−0.667949 + 0.744207i \(0.732827\pi\)
\(522\) 0 0
\(523\) −24.8769 −1.08779 −0.543895 0.839153i \(-0.683051\pi\)
−0.543895 + 0.839153i \(0.683051\pi\)
\(524\) 0 0
\(525\) −11.6847 −0.509960
\(526\) 0 0
\(527\) −11.0540 −0.481519
\(528\) 0 0
\(529\) −21.7386 −0.945158
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 13.7538 0.595743
\(534\) 0 0
\(535\) 7.12311 0.307959
\(536\) 0 0
\(537\) 36.4924 1.57476
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.93087 0.340975 0.170487 0.985360i \(-0.445466\pi\)
0.170487 + 0.985360i \(0.445466\pi\)
\(542\) 0 0
\(543\) −21.1231 −0.906479
\(544\) 0 0
\(545\) −4.24621 −0.181888
\(546\) 0 0
\(547\) 10.6307 0.454535 0.227268 0.973832i \(-0.427021\pi\)
0.227268 + 0.973832i \(0.427021\pi\)
\(548\) 0 0
\(549\) 44.7386 1.90940
\(550\) 0 0
\(551\) 12.3153 0.524651
\(552\) 0 0
\(553\) −51.8617 −2.20539
\(554\) 0 0
\(555\) −19.0540 −0.808796
\(556\) 0 0
\(557\) −16.6307 −0.704665 −0.352332 0.935875i \(-0.614611\pi\)
−0.352332 + 0.935875i \(0.614611\pi\)
\(558\) 0 0
\(559\) 3.50758 0.148355
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 42.9848 1.81160 0.905798 0.423711i \(-0.139273\pi\)
0.905798 + 0.423711i \(0.139273\pi\)
\(564\) 0 0
\(565\) −0.876894 −0.0368912
\(566\) 0 0
\(567\) 31.9309 1.34097
\(568\) 0 0
\(569\) −3.61553 −0.151571 −0.0757854 0.997124i \(-0.524146\pi\)
−0.0757854 + 0.997124i \(0.524146\pi\)
\(570\) 0 0
\(571\) −25.3002 −1.05878 −0.529390 0.848379i \(-0.677579\pi\)
−0.529390 + 0.848379i \(0.677579\pi\)
\(572\) 0 0
\(573\) 26.2462 1.09645
\(574\) 0 0
\(575\) 1.12311 0.0468367
\(576\) 0 0
\(577\) −15.7538 −0.655839 −0.327919 0.944706i \(-0.606347\pi\)
−0.327919 + 0.944706i \(0.606347\pi\)
\(578\) 0 0
\(579\) 54.9157 2.28222
\(580\) 0 0
\(581\) −27.3693 −1.13547
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) −3.05398 −0.126051 −0.0630255 0.998012i \(-0.520075\pi\)
−0.0630255 + 0.998012i \(0.520075\pi\)
\(588\) 0 0
\(589\) 2.06913 0.0852570
\(590\) 0 0
\(591\) −58.2462 −2.39593
\(592\) 0 0
\(593\) −21.6155 −0.887643 −0.443822 0.896115i \(-0.646378\pi\)
−0.443822 + 0.896115i \(0.646378\pi\)
\(594\) 0 0
\(595\) 35.0540 1.43707
\(596\) 0 0
\(597\) −11.0540 −0.452409
\(598\) 0 0
\(599\) −29.9309 −1.22294 −0.611471 0.791267i \(-0.709422\pi\)
−0.611471 + 0.791267i \(0.709422\pi\)
\(600\) 0 0
\(601\) 26.9848 1.10073 0.550367 0.834923i \(-0.314488\pi\)
0.550367 + 0.834923i \(0.314488\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 14.8078 0.601029 0.300514 0.953777i \(-0.402842\pi\)
0.300514 + 0.953777i \(0.402842\pi\)
\(608\) 0 0
\(609\) 100.039 4.05378
\(610\) 0 0
\(611\) −12.7689 −0.516576
\(612\) 0 0
\(613\) 2.87689 0.116197 0.0580983 0.998311i \(-0.481496\pi\)
0.0580983 + 0.998311i \(0.481496\pi\)
\(614\) 0 0
\(615\) −31.3693 −1.26493
\(616\) 0 0
\(617\) −27.6155 −1.11176 −0.555880 0.831263i \(-0.687618\pi\)
−0.555880 + 0.831263i \(0.687618\pi\)
\(618\) 0 0
\(619\) −34.7386 −1.39626 −0.698132 0.715969i \(-0.745985\pi\)
−0.698132 + 0.715969i \(0.745985\pi\)
\(620\) 0 0
\(621\) 1.61553 0.0648289
\(622\) 0 0
\(623\) −44.1771 −1.76992
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 57.1619 2.27920
\(630\) 0 0
\(631\) −32.8078 −1.30606 −0.653028 0.757334i \(-0.726502\pi\)
−0.653028 + 0.757334i \(0.726502\pi\)
\(632\) 0 0
\(633\) −50.4233 −2.00415
\(634\) 0 0
\(635\) 4.87689 0.193534
\(636\) 0 0
\(637\) 15.5076 0.614433
\(638\) 0 0
\(639\) 13.1231 0.519142
\(640\) 0 0
\(641\) 20.5616 0.812133 0.406066 0.913844i \(-0.366900\pi\)
0.406066 + 0.913844i \(0.366900\pi\)
\(642\) 0 0
\(643\) 25.3002 0.997742 0.498871 0.866676i \(-0.333748\pi\)
0.498871 + 0.866676i \(0.333748\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 45.1231 1.77397 0.886986 0.461796i \(-0.152795\pi\)
0.886986 + 0.461796i \(0.152795\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 16.8078 0.658748
\(652\) 0 0
\(653\) 28.5616 1.11770 0.558850 0.829269i \(-0.311243\pi\)
0.558850 + 0.829269i \(0.311243\pi\)
\(654\) 0 0
\(655\) −16.8078 −0.656734
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 33.9309 1.32176 0.660880 0.750492i \(-0.270183\pi\)
0.660880 + 0.750492i \(0.270183\pi\)
\(660\) 0 0
\(661\) 2.49242 0.0969440 0.0484720 0.998825i \(-0.484565\pi\)
0.0484720 + 0.998825i \(0.484565\pi\)
\(662\) 0 0
\(663\) 22.1080 0.858602
\(664\) 0 0
\(665\) −6.56155 −0.254446
\(666\) 0 0
\(667\) −9.61553 −0.372315
\(668\) 0 0
\(669\) 64.3542 2.48808
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 41.9309 1.61632 0.808158 0.588966i \(-0.200465\pi\)
0.808158 + 0.588966i \(0.200465\pi\)
\(674\) 0 0
\(675\) 1.43845 0.0553659
\(676\) 0 0
\(677\) −46.2462 −1.77739 −0.888693 0.458502i \(-0.848386\pi\)
−0.888693 + 0.458502i \(0.848386\pi\)
\(678\) 0 0
\(679\) 22.2462 0.853731
\(680\) 0 0
\(681\) −21.1231 −0.809439
\(682\) 0 0
\(683\) 40.1771 1.53733 0.768667 0.639650i \(-0.220921\pi\)
0.768667 + 0.639650i \(0.220921\pi\)
\(684\) 0 0
\(685\) 7.12311 0.272160
\(686\) 0 0
\(687\) 41.6155 1.58773
\(688\) 0 0
\(689\) 10.8769 0.414377
\(690\) 0 0
\(691\) 5.61553 0.213625 0.106812 0.994279i \(-0.465936\pi\)
0.106812 + 0.994279i \(0.465936\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 94.1080 3.56459
\(698\) 0 0
\(699\) −69.3002 −2.62117
\(700\) 0 0
\(701\) 20.5616 0.776599 0.388300 0.921533i \(-0.373063\pi\)
0.388300 + 0.921533i \(0.373063\pi\)
\(702\) 0 0
\(703\) −10.6998 −0.403551
\(704\) 0 0
\(705\) 29.1231 1.09684
\(706\) 0 0
\(707\) −9.12311 −0.343110
\(708\) 0 0
\(709\) −32.2462 −1.21103 −0.605516 0.795833i \(-0.707033\pi\)
−0.605516 + 0.795833i \(0.707033\pi\)
\(710\) 0 0
\(711\) 40.4924 1.51858
\(712\) 0 0
\(713\) −1.61553 −0.0605020
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 55.3693 2.06781
\(718\) 0 0
\(719\) −48.1771 −1.79670 −0.898351 0.439278i \(-0.855234\pi\)
−0.898351 + 0.439278i \(0.855234\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −47.3693 −1.76168
\(724\) 0 0
\(725\) −8.56155 −0.317968
\(726\) 0 0
\(727\) −10.8769 −0.403402 −0.201701 0.979447i \(-0.564647\pi\)
−0.201701 + 0.979447i \(0.564647\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 13.7538 0.508008 0.254004 0.967203i \(-0.418252\pi\)
0.254004 + 0.967203i \(0.418252\pi\)
\(734\) 0 0
\(735\) −35.3693 −1.30462
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −22.2462 −0.818340 −0.409170 0.912458i \(-0.634182\pi\)
−0.409170 + 0.912458i \(0.634182\pi\)
\(740\) 0 0
\(741\) −4.13826 −0.152023
\(742\) 0 0
\(743\) −20.5616 −0.754330 −0.377165 0.926146i \(-0.623101\pi\)
−0.377165 + 0.926146i \(0.623101\pi\)
\(744\) 0 0
\(745\) 1.68466 0.0617211
\(746\) 0 0
\(747\) 21.3693 0.781862
\(748\) 0 0
\(749\) 32.4924 1.18725
\(750\) 0 0
\(751\) −31.1922 −1.13822 −0.569110 0.822261i \(-0.692712\pi\)
−0.569110 + 0.822261i \(0.692712\pi\)
\(752\) 0 0
\(753\) −70.1080 −2.55488
\(754\) 0 0
\(755\) 6.87689 0.250276
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.75379 −0.281075 −0.140537 0.990075i \(-0.544883\pi\)
−0.140537 + 0.990075i \(0.544883\pi\)
\(762\) 0 0
\(763\) −19.3693 −0.701216
\(764\) 0 0
\(765\) −27.3693 −0.989540
\(766\) 0 0
\(767\) 1.26137 0.0455453
\(768\) 0 0
\(769\) −3.75379 −0.135365 −0.0676825 0.997707i \(-0.521561\pi\)
−0.0676825 + 0.997707i \(0.521561\pi\)
\(770\) 0 0
\(771\) 16.6307 0.598939
\(772\) 0 0
\(773\) 43.9309 1.58008 0.790042 0.613053i \(-0.210059\pi\)
0.790042 + 0.613053i \(0.210059\pi\)
\(774\) 0 0
\(775\) −1.43845 −0.0516705
\(776\) 0 0
\(777\) −86.9157 −3.11808
\(778\) 0 0
\(779\) −17.6155 −0.631142
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −12.3153 −0.440114
\(784\) 0 0
\(785\) −21.6847 −0.773959
\(786\) 0 0
\(787\) 17.8617 0.636702 0.318351 0.947973i \(-0.396871\pi\)
0.318351 + 0.947973i \(0.396871\pi\)
\(788\) 0 0
\(789\) 74.4233 2.64954
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 14.1080 0.500988
\(794\) 0 0
\(795\) −24.8078 −0.879841
\(796\) 0 0
\(797\) 20.2462 0.717158 0.358579 0.933499i \(-0.383261\pi\)
0.358579 + 0.933499i \(0.383261\pi\)
\(798\) 0 0
\(799\) −87.3693 −3.09090
\(800\) 0 0
\(801\) 34.4924 1.21873
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 5.12311 0.180566
\(806\) 0 0
\(807\) 2.24621 0.0790704
\(808\) 0 0
\(809\) 40.7386 1.43229 0.716147 0.697949i \(-0.245904\pi\)
0.716147 + 0.697949i \(0.245904\pi\)
\(810\) 0 0
\(811\) 1.93087 0.0678020 0.0339010 0.999425i \(-0.489207\pi\)
0.0339010 + 0.999425i \(0.489207\pi\)
\(812\) 0 0
\(813\) 10.2462 0.359350
\(814\) 0 0
\(815\) −20.8078 −0.728864
\(816\) 0 0
\(817\) −4.49242 −0.157170
\(818\) 0 0
\(819\) −18.2462 −0.637574
\(820\) 0 0
\(821\) −14.4924 −0.505789 −0.252895 0.967494i \(-0.581383\pi\)
−0.252895 + 0.967494i \(0.581383\pi\)
\(822\) 0 0
\(823\) 12.4924 0.435458 0.217729 0.976009i \(-0.430135\pi\)
0.217729 + 0.976009i \(0.430135\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −29.8617 −1.03839 −0.519197 0.854654i \(-0.673769\pi\)
−0.519197 + 0.854654i \(0.673769\pi\)
\(828\) 0 0
\(829\) −32.8769 −1.14186 −0.570931 0.820998i \(-0.693418\pi\)
−0.570931 + 0.820998i \(0.693418\pi\)
\(830\) 0 0
\(831\) 56.9848 1.97678
\(832\) 0 0
\(833\) 106.108 3.67642
\(834\) 0 0
\(835\) −1.68466 −0.0583000
\(836\) 0 0
\(837\) −2.06913 −0.0715196
\(838\) 0 0
\(839\) 19.5076 0.673476 0.336738 0.941598i \(-0.390676\pi\)
0.336738 + 0.941598i \(0.390676\pi\)
\(840\) 0 0
\(841\) 44.3002 1.52759
\(842\) 0 0
\(843\) −21.1231 −0.727518
\(844\) 0 0
\(845\) 11.7386 0.403821
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 15.3693 0.527474
\(850\) 0 0
\(851\) 8.35416 0.286377
\(852\) 0 0
\(853\) 37.1231 1.27107 0.635535 0.772072i \(-0.280779\pi\)
0.635535 + 0.772072i \(0.280779\pi\)
\(854\) 0 0
\(855\) 5.12311 0.175207
\(856\) 0 0
\(857\) −12.8078 −0.437505 −0.218752 0.975780i \(-0.570199\pi\)
−0.218752 + 0.975780i \(0.570199\pi\)
\(858\) 0 0
\(859\) 23.8617 0.814152 0.407076 0.913394i \(-0.366548\pi\)
0.407076 + 0.913394i \(0.366548\pi\)
\(860\) 0 0
\(861\) −143.093 −4.87659
\(862\) 0 0
\(863\) 29.1231 0.991362 0.495681 0.868505i \(-0.334919\pi\)
0.495681 + 0.868505i \(0.334919\pi\)
\(864\) 0 0
\(865\) 10.8769 0.369826
\(866\) 0 0
\(867\) 107.723 3.65848
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −17.3693 −0.587862
\(874\) 0 0
\(875\) 4.56155 0.154209
\(876\) 0 0
\(877\) 25.6155 0.864975 0.432487 0.901640i \(-0.357636\pi\)
0.432487 + 0.901640i \(0.357636\pi\)
\(878\) 0 0
\(879\) −43.8617 −1.47942
\(880\) 0 0
\(881\) 5.50758 0.185555 0.0927775 0.995687i \(-0.470425\pi\)
0.0927775 + 0.995687i \(0.470425\pi\)
\(882\) 0 0
\(883\) −27.5464 −0.927010 −0.463505 0.886094i \(-0.653408\pi\)
−0.463505 + 0.886094i \(0.653408\pi\)
\(884\) 0 0
\(885\) −2.87689 −0.0967057
\(886\) 0 0
\(887\) 27.1231 0.910705 0.455352 0.890311i \(-0.349513\pi\)
0.455352 + 0.890311i \(0.349513\pi\)
\(888\) 0 0
\(889\) 22.2462 0.746114
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 16.3542 0.547271
\(894\) 0 0
\(895\) −14.2462 −0.476198
\(896\) 0 0
\(897\) 3.23106 0.107882
\(898\) 0 0
\(899\) 12.3153 0.410740
\(900\) 0 0
\(901\) 74.4233 2.47940
\(902\) 0 0
\(903\) −36.4924 −1.21439
\(904\) 0 0
\(905\) 8.24621 0.274113
\(906\) 0 0
\(907\) −43.6847 −1.45053 −0.725263 0.688472i \(-0.758282\pi\)
−0.725263 + 0.688472i \(0.758282\pi\)
\(908\) 0 0
\(909\) 7.12311 0.236259
\(910\) 0 0
\(911\) 41.4384 1.37292 0.686459 0.727169i \(-0.259164\pi\)
0.686459 + 0.727169i \(0.259164\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −32.1771 −1.06374
\(916\) 0 0
\(917\) −76.6695 −2.53185
\(918\) 0 0
\(919\) 35.2311 1.16217 0.581083 0.813845i \(-0.302629\pi\)
0.581083 + 0.813845i \(0.302629\pi\)
\(920\) 0 0
\(921\) −46.1080 −1.51931
\(922\) 0 0
\(923\) 4.13826 0.136213
\(924\) 0 0
\(925\) 7.43845 0.244575
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 33.5464 1.10062 0.550311 0.834960i \(-0.314509\pi\)
0.550311 + 0.834960i \(0.314509\pi\)
\(930\) 0 0
\(931\) −19.8617 −0.650942
\(932\) 0 0
\(933\) −24.1771 −0.791522
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 49.6155 1.62087 0.810434 0.585829i \(-0.199231\pi\)
0.810434 + 0.585829i \(0.199231\pi\)
\(938\) 0 0
\(939\) 63.3693 2.06798
\(940\) 0 0
\(941\) −9.05398 −0.295151 −0.147576 0.989051i \(-0.547147\pi\)
−0.147576 + 0.989051i \(0.547147\pi\)
\(942\) 0 0
\(943\) 13.7538 0.447885
\(944\) 0 0
\(945\) 6.56155 0.213447
\(946\) 0 0
\(947\) 12.9460 0.420689 0.210345 0.977627i \(-0.432541\pi\)
0.210345 + 0.977627i \(0.432541\pi\)
\(948\) 0 0
\(949\) −1.26137 −0.0409457
\(950\) 0 0
\(951\) 16.1771 0.524578
\(952\) 0 0
\(953\) 6.56155 0.212550 0.106275 0.994337i \(-0.466108\pi\)
0.106275 + 0.994337i \(0.466108\pi\)
\(954\) 0 0
\(955\) −10.2462 −0.331560
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 32.4924 1.04924
\(960\) 0 0
\(961\) −28.9309 −0.933254
\(962\) 0 0
\(963\) −25.3693 −0.817515
\(964\) 0 0
\(965\) −21.4384 −0.690128
\(966\) 0 0
\(967\) 12.5616 0.403952 0.201976 0.979390i \(-0.435264\pi\)
0.201976 + 0.979390i \(0.435264\pi\)
\(968\) 0 0
\(969\) −28.3153 −0.909620
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) −18.2462 −0.584947
\(974\) 0 0
\(975\) 2.87689 0.0921344
\(976\) 0 0
\(977\) −21.8617 −0.699419 −0.349710 0.936858i \(-0.613720\pi\)
−0.349710 + 0.936858i \(0.613720\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 15.1231 0.482844
\(982\) 0 0
\(983\) −16.6307 −0.530436 −0.265218 0.964188i \(-0.585444\pi\)
−0.265218 + 0.964188i \(0.585444\pi\)
\(984\) 0 0
\(985\) 22.7386 0.724513
\(986\) 0 0
\(987\) 132.847 4.22855
\(988\) 0 0
\(989\) 3.50758 0.111534
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) −48.0000 −1.52323
\(994\) 0 0
\(995\) 4.31534 0.136806
\(996\) 0 0
\(997\) −1.26137 −0.0399479 −0.0199739 0.999801i \(-0.506358\pi\)
−0.0199739 + 0.999801i \(0.506358\pi\)
\(998\) 0 0
\(999\) 10.6998 0.338527
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 4840.2.a.m.1.2 2
4.3 odd 2 9680.2.a.bm.1.1 2
11.10 odd 2 440.2.a.g.1.2 2
33.32 even 2 3960.2.a.bf.1.2 2
44.43 even 2 880.2.a.k.1.1 2
55.32 even 4 2200.2.b.f.1849.1 4
55.43 even 4 2200.2.b.f.1849.4 4
55.54 odd 2 2200.2.a.l.1.1 2
88.21 odd 2 3520.2.a.bm.1.1 2
88.43 even 2 3520.2.a.br.1.2 2
132.131 odd 2 7920.2.a.by.1.1 2
220.43 odd 4 4400.2.b.w.4049.1 4
220.87 odd 4 4400.2.b.w.4049.4 4
220.219 even 2 4400.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.g.1.2 2 11.10 odd 2
880.2.a.k.1.1 2 44.43 even 2
2200.2.a.l.1.1 2 55.54 odd 2
2200.2.b.f.1849.1 4 55.32 even 4
2200.2.b.f.1849.4 4 55.43 even 4
3520.2.a.bm.1.1 2 88.21 odd 2
3520.2.a.br.1.2 2 88.43 even 2
3960.2.a.bf.1.2 2 33.32 even 2
4400.2.a.bt.1.2 2 220.219 even 2
4400.2.b.w.4049.1 4 220.43 odd 4
4400.2.b.w.4049.4 4 220.87 odd 4
4840.2.a.m.1.2 2 1.1 even 1 trivial
7920.2.a.by.1.1 2 132.131 odd 2
9680.2.a.bm.1.1 2 4.3 odd 2