Properties

Label 4840.2.a.j.1.1
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
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: \(1\)
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 1.1
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)
 
\(f(q)\) \(=\) \(q-2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q-2.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} +3.56155 q^{9} -2.00000 q^{13} -2.56155 q^{15} +0.561553 q^{17} -2.56155 q^{19} -6.56155 q^{21} +5.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} -9.68466 q^{29} +6.56155 q^{31} +2.56155 q^{35} -5.68466 q^{37} +5.12311 q^{39} -2.00000 q^{41} -10.2462 q^{43} +3.56155 q^{45} -13.1231 q^{47} -0.438447 q^{49} -1.43845 q^{51} +4.56155 q^{53} +6.56155 q^{57} +1.12311 q^{59} -2.31534 q^{61} +9.12311 q^{63} -2.00000 q^{65} +6.24621 q^{67} -13.1231 q^{69} +3.68466 q^{71} +2.00000 q^{73} -2.56155 q^{75} +15.3693 q^{79} -7.00000 q^{81} -5.12311 q^{83} +0.561553 q^{85} +24.8078 q^{87} -12.5616 q^{89} -5.12311 q^{91} -16.8078 q^{93} -2.56155 q^{95} +7.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 4 q^{13} - q^{15} - 3 q^{17} - q^{19} - 9 q^{21} + 2 q^{23} + 2 q^{25} - 7 q^{27} - 7 q^{29} + 9 q^{31} + q^{35} + q^{37} + 2 q^{39} - 4 q^{41} - 4 q^{43} + 3 q^{45} - 18 q^{47} - 5 q^{49} - 7 q^{51} + 5 q^{53} + 9 q^{57} - 6 q^{59} - 17 q^{61} + 10 q^{63} - 4 q^{65} - 4 q^{67} - 18 q^{69} - 5 q^{71} + 4 q^{73} - q^{75} + 6 q^{79} - 14 q^{81} - 2 q^{83} - 3 q^{85} + 29 q^{87} - 21 q^{89} - 2 q^{91} - 13 q^{93} - q^{95} + 6 q^{97}+O(q^{100}) \) 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 \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.56155 0.968176 0.484088 0.875019i \(-0.339151\pi\)
0.484088 + 0.875019i \(0.339151\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.56155 −0.661390
\(16\) 0 0
\(17\) 0.561553 0.136197 0.0680983 0.997679i \(-0.478307\pi\)
0.0680983 + 0.997679i \(0.478307\pi\)
\(18\) 0 0
\(19\) −2.56155 −0.587661 −0.293830 0.955858i \(-0.594930\pi\)
−0.293830 + 0.955858i \(0.594930\pi\)
\(20\) 0 0
\(21\) −6.56155 −1.43185
\(22\) 0 0
\(23\) 5.12311 1.06824 0.534121 0.845408i \(-0.320643\pi\)
0.534121 + 0.845408i \(0.320643\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) −9.68466 −1.79840 −0.899198 0.437542i \(-0.855849\pi\)
−0.899198 + 0.437542i \(0.855849\pi\)
\(30\) 0 0
\(31\) 6.56155 1.17849 0.589245 0.807955i \(-0.299425\pi\)
0.589245 + 0.807955i \(0.299425\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.56155 0.432981
\(36\) 0 0
\(37\) −5.68466 −0.934552 −0.467276 0.884111i \(-0.654765\pi\)
−0.467276 + 0.884111i \(0.654765\pi\)
\(38\) 0 0
\(39\) 5.12311 0.820353
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −10.2462 −1.56253 −0.781266 0.624198i \(-0.785426\pi\)
−0.781266 + 0.624198i \(0.785426\pi\)
\(44\) 0 0
\(45\) 3.56155 0.530925
\(46\) 0 0
\(47\) −13.1231 −1.91420 −0.957101 0.289755i \(-0.906426\pi\)
−0.957101 + 0.289755i \(0.906426\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) −1.43845 −0.201423
\(52\) 0 0
\(53\) 4.56155 0.626577 0.313289 0.949658i \(-0.398569\pi\)
0.313289 + 0.949658i \(0.398569\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.56155 0.869099
\(58\) 0 0
\(59\) 1.12311 0.146216 0.0731079 0.997324i \(-0.476708\pi\)
0.0731079 + 0.997324i \(0.476708\pi\)
\(60\) 0 0
\(61\) −2.31534 −0.296449 −0.148225 0.988954i \(-0.547356\pi\)
−0.148225 + 0.988954i \(0.547356\pi\)
\(62\) 0 0
\(63\) 9.12311 1.14940
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 6.24621 0.763096 0.381548 0.924349i \(-0.375391\pi\)
0.381548 + 0.924349i \(0.375391\pi\)
\(68\) 0 0
\(69\) −13.1231 −1.57984
\(70\) 0 0
\(71\) 3.68466 0.437289 0.218644 0.975805i \(-0.429837\pi\)
0.218644 + 0.975805i \(0.429837\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 15.3693 1.72918 0.864592 0.502475i \(-0.167577\pi\)
0.864592 + 0.502475i \(0.167577\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −5.12311 −0.562334 −0.281167 0.959659i \(-0.590721\pi\)
−0.281167 + 0.959659i \(0.590721\pi\)
\(84\) 0 0
\(85\) 0.561553 0.0609090
\(86\) 0 0
\(87\) 24.8078 2.65967
\(88\) 0 0
\(89\) −12.5616 −1.33152 −0.665761 0.746165i \(-0.731893\pi\)
−0.665761 + 0.746165i \(0.731893\pi\)
\(90\) 0 0
\(91\) −5.12311 −0.537047
\(92\) 0 0
\(93\) −16.8078 −1.74288
\(94\) 0 0
\(95\) −2.56155 −0.262810
\(96\) 0 0
\(97\) 7.12311 0.723242 0.361621 0.932325i \(-0.382223\pi\)
0.361621 + 0.932325i \(0.382223\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.24621 0.422514 0.211257 0.977431i \(-0.432244\pi\)
0.211257 + 0.977431i \(0.432244\pi\)
\(102\) 0 0
\(103\) 10.2462 1.00959 0.504795 0.863239i \(-0.331568\pi\)
0.504795 + 0.863239i \(0.331568\pi\)
\(104\) 0 0
\(105\) −6.56155 −0.640342
\(106\) 0 0
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) −18.4924 −1.77125 −0.885626 0.464398i \(-0.846271\pi\)
−0.885626 + 0.464398i \(0.846271\pi\)
\(110\) 0 0
\(111\) 14.5616 1.38212
\(112\) 0 0
\(113\) −19.1231 −1.79895 −0.899475 0.436972i \(-0.856051\pi\)
−0.899475 + 0.436972i \(0.856051\pi\)
\(114\) 0 0
\(115\) 5.12311 0.477732
\(116\) 0 0
\(117\) −7.12311 −0.658531
\(118\) 0 0
\(119\) 1.43845 0.131862
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 5.12311 0.461935
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −14.2462 −1.26415 −0.632073 0.774909i \(-0.717796\pi\)
−0.632073 + 0.774909i \(0.717796\pi\)
\(128\) 0 0
\(129\) 26.2462 2.31085
\(130\) 0 0
\(131\) 7.68466 0.671412 0.335706 0.941967i \(-0.391025\pi\)
0.335706 + 0.941967i \(0.391025\pi\)
\(132\) 0 0
\(133\) −6.56155 −0.568959
\(134\) 0 0
\(135\) −1.43845 −0.123802
\(136\) 0 0
\(137\) 19.6155 1.67587 0.837934 0.545772i \(-0.183763\pi\)
0.837934 + 0.545772i \(0.183763\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 33.6155 2.83094
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −9.68466 −0.804267
\(146\) 0 0
\(147\) 1.12311 0.0926322
\(148\) 0 0
\(149\) −2.31534 −0.189680 −0.0948401 0.995493i \(-0.530234\pi\)
−0.0948401 + 0.995493i \(0.530234\pi\)
\(150\) 0 0
\(151\) −13.1231 −1.06794 −0.533972 0.845502i \(-0.679301\pi\)
−0.533972 + 0.845502i \(0.679301\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 6.56155 0.527037
\(156\) 0 0
\(157\) 14.8078 1.18179 0.590894 0.806749i \(-0.298775\pi\)
0.590894 + 0.806749i \(0.298775\pi\)
\(158\) 0 0
\(159\) −11.6847 −0.926654
\(160\) 0 0
\(161\) 13.1231 1.03425
\(162\) 0 0
\(163\) 3.19224 0.250035 0.125018 0.992155i \(-0.460101\pi\)
0.125018 + 0.992155i \(0.460101\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.5616 0.817277 0.408639 0.912696i \(-0.366004\pi\)
0.408639 + 0.912696i \(0.366004\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −9.12311 −0.697661
\(172\) 0 0
\(173\) −4.24621 −0.322833 −0.161417 0.986886i \(-0.551606\pi\)
−0.161417 + 0.986886i \(0.551606\pi\)
\(174\) 0 0
\(175\) 2.56155 0.193635
\(176\) 0 0
\(177\) −2.87689 −0.216241
\(178\) 0 0
\(179\) −6.24621 −0.466864 −0.233432 0.972373i \(-0.574996\pi\)
−0.233432 + 0.972373i \(0.574996\pi\)
\(180\) 0 0
\(181\) −4.24621 −0.315618 −0.157809 0.987470i \(-0.550443\pi\)
−0.157809 + 0.987470i \(0.550443\pi\)
\(182\) 0 0
\(183\) 5.93087 0.438423
\(184\) 0 0
\(185\) −5.68466 −0.417944
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3.68466 −0.268019
\(190\) 0 0
\(191\) 10.2462 0.741390 0.370695 0.928755i \(-0.379120\pi\)
0.370695 + 0.928755i \(0.379120\pi\)
\(192\) 0 0
\(193\) −5.19224 −0.373745 −0.186873 0.982384i \(-0.559835\pi\)
−0.186873 + 0.982384i \(0.559835\pi\)
\(194\) 0 0
\(195\) 5.12311 0.366873
\(196\) 0 0
\(197\) 5.36932 0.382548 0.191274 0.981537i \(-0.438738\pi\)
0.191274 + 0.981537i \(0.438738\pi\)
\(198\) 0 0
\(199\) −0.807764 −0.0572609 −0.0286304 0.999590i \(-0.509115\pi\)
−0.0286304 + 0.999590i \(0.509115\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) −24.8078 −1.74116
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) 18.2462 1.26820
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.31534 −0.572452 −0.286226 0.958162i \(-0.592401\pi\)
−0.286226 + 0.958162i \(0.592401\pi\)
\(212\) 0 0
\(213\) −9.43845 −0.646712
\(214\) 0 0
\(215\) −10.2462 −0.698786
\(216\) 0 0
\(217\) 16.8078 1.14099
\(218\) 0 0
\(219\) −5.12311 −0.346187
\(220\) 0 0
\(221\) −1.12311 −0.0755483
\(222\) 0 0
\(223\) 13.1231 0.878788 0.439394 0.898294i \(-0.355193\pi\)
0.439394 + 0.898294i \(0.355193\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) −2.87689 −0.190946 −0.0954731 0.995432i \(-0.530436\pi\)
−0.0954731 + 0.995432i \(0.530436\pi\)
\(228\) 0 0
\(229\) −24.7386 −1.63477 −0.817387 0.576088i \(-0.804578\pi\)
−0.817387 + 0.576088i \(0.804578\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19.9309 −1.30571 −0.652857 0.757481i \(-0.726430\pi\)
−0.652857 + 0.757481i \(0.726430\pi\)
\(234\) 0 0
\(235\) −13.1231 −0.856057
\(236\) 0 0
\(237\) −39.3693 −2.55731
\(238\) 0 0
\(239\) −25.6155 −1.65693 −0.828465 0.560040i \(-0.810786\pi\)
−0.828465 + 0.560040i \(0.810786\pi\)
\(240\) 0 0
\(241\) 16.2462 1.04651 0.523255 0.852176i \(-0.324717\pi\)
0.523255 + 0.852176i \(0.324717\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) −0.438447 −0.0280114
\(246\) 0 0
\(247\) 5.12311 0.325975
\(248\) 0 0
\(249\) 13.1231 0.831643
\(250\) 0 0
\(251\) −29.6155 −1.86932 −0.934658 0.355549i \(-0.884294\pi\)
−0.934658 + 0.355549i \(0.884294\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −1.43845 −0.0900791
\(256\) 0 0
\(257\) −28.7386 −1.79267 −0.896333 0.443381i \(-0.853779\pi\)
−0.896333 + 0.443381i \(0.853779\pi\)
\(258\) 0 0
\(259\) −14.5616 −0.904811
\(260\) 0 0
\(261\) −34.4924 −2.13503
\(262\) 0 0
\(263\) −4.80776 −0.296459 −0.148230 0.988953i \(-0.547357\pi\)
−0.148230 + 0.988953i \(0.547357\pi\)
\(264\) 0 0
\(265\) 4.56155 0.280214
\(266\) 0 0
\(267\) 32.1771 1.96921
\(268\) 0 0
\(269\) −19.6155 −1.19598 −0.597990 0.801504i \(-0.704034\pi\)
−0.597990 + 0.801504i \(0.704034\pi\)
\(270\) 0 0
\(271\) −28.4924 −1.73079 −0.865396 0.501089i \(-0.832933\pi\)
−0.865396 + 0.501089i \(0.832933\pi\)
\(272\) 0 0
\(273\) 13.1231 0.794246
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.12311 −0.427986 −0.213993 0.976835i \(-0.568647\pi\)
−0.213993 + 0.976835i \(0.568647\pi\)
\(278\) 0 0
\(279\) 23.3693 1.39908
\(280\) 0 0
\(281\) 8.24621 0.491928 0.245964 0.969279i \(-0.420896\pi\)
0.245964 + 0.969279i \(0.420896\pi\)
\(282\) 0 0
\(283\) 23.3693 1.38916 0.694581 0.719415i \(-0.255590\pi\)
0.694581 + 0.719415i \(0.255590\pi\)
\(284\) 0 0
\(285\) 6.56155 0.388673
\(286\) 0 0
\(287\) −5.12311 −0.302407
\(288\) 0 0
\(289\) −16.6847 −0.981450
\(290\) 0 0
\(291\) −18.2462 −1.06961
\(292\) 0 0
\(293\) −14.4924 −0.846656 −0.423328 0.905976i \(-0.639138\pi\)
−0.423328 + 0.905976i \(0.639138\pi\)
\(294\) 0 0
\(295\) 1.12311 0.0653897
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.2462 −0.592554
\(300\) 0 0
\(301\) −26.2462 −1.51281
\(302\) 0 0
\(303\) −10.8769 −0.624861
\(304\) 0 0
\(305\) −2.31534 −0.132576
\(306\) 0 0
\(307\) 25.6155 1.46196 0.730978 0.682401i \(-0.239064\pi\)
0.730978 + 0.682401i \(0.239064\pi\)
\(308\) 0 0
\(309\) −26.2462 −1.49309
\(310\) 0 0
\(311\) −25.4384 −1.44248 −0.721241 0.692684i \(-0.756428\pi\)
−0.721241 + 0.692684i \(0.756428\pi\)
\(312\) 0 0
\(313\) 30.4924 1.72353 0.861767 0.507305i \(-0.169358\pi\)
0.861767 + 0.507305i \(0.169358\pi\)
\(314\) 0 0
\(315\) 9.12311 0.514029
\(316\) 0 0
\(317\) 13.1922 0.740950 0.370475 0.928842i \(-0.379195\pi\)
0.370475 + 0.928842i \(0.379195\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −40.9848 −2.28755
\(322\) 0 0
\(323\) −1.43845 −0.0800373
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 47.3693 2.61953
\(328\) 0 0
\(329\) −33.6155 −1.85328
\(330\) 0 0
\(331\) −8.49242 −0.466786 −0.233393 0.972383i \(-0.574983\pi\)
−0.233393 + 0.972383i \(0.574983\pi\)
\(332\) 0 0
\(333\) −20.2462 −1.10949
\(334\) 0 0
\(335\) 6.24621 0.341267
\(336\) 0 0
\(337\) −2.31534 −0.126125 −0.0630623 0.998010i \(-0.520087\pi\)
−0.0630623 + 0.998010i \(0.520087\pi\)
\(338\) 0 0
\(339\) 48.9848 2.66049
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −19.0540 −1.02882
\(344\) 0 0
\(345\) −13.1231 −0.706524
\(346\) 0 0
\(347\) 18.8769 1.01336 0.506682 0.862133i \(-0.330872\pi\)
0.506682 + 0.862133i \(0.330872\pi\)
\(348\) 0 0
\(349\) −10.4924 −0.561646 −0.280823 0.959760i \(-0.590607\pi\)
−0.280823 + 0.959760i \(0.590607\pi\)
\(350\) 0 0
\(351\) 2.87689 0.153557
\(352\) 0 0
\(353\) −34.4924 −1.83585 −0.917923 0.396758i \(-0.870135\pi\)
−0.917923 + 0.396758i \(0.870135\pi\)
\(354\) 0 0
\(355\) 3.68466 0.195561
\(356\) 0 0
\(357\) −3.68466 −0.195013
\(358\) 0 0
\(359\) −26.2462 −1.38522 −0.692611 0.721311i \(-0.743540\pi\)
−0.692611 + 0.721311i \(0.743540\pi\)
\(360\) 0 0
\(361\) −12.4384 −0.654655
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) −14.7386 −0.769350 −0.384675 0.923052i \(-0.625687\pi\)
−0.384675 + 0.923052i \(0.625687\pi\)
\(368\) 0 0
\(369\) −7.12311 −0.370814
\(370\) 0 0
\(371\) 11.6847 0.606637
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) −2.56155 −0.132278
\(376\) 0 0
\(377\) 19.3693 0.997571
\(378\) 0 0
\(379\) −25.1231 −1.29049 −0.645244 0.763977i \(-0.723244\pi\)
−0.645244 + 0.763977i \(0.723244\pi\)
\(380\) 0 0
\(381\) 36.4924 1.86956
\(382\) 0 0
\(383\) −31.3693 −1.60290 −0.801449 0.598064i \(-0.795937\pi\)
−0.801449 + 0.598064i \(0.795937\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −36.4924 −1.85501
\(388\) 0 0
\(389\) 32.2462 1.63495 0.817474 0.575966i \(-0.195374\pi\)
0.817474 + 0.575966i \(0.195374\pi\)
\(390\) 0 0
\(391\) 2.87689 0.145491
\(392\) 0 0
\(393\) −19.6847 −0.992960
\(394\) 0 0
\(395\) 15.3693 0.773314
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 16.8078 0.841441
\(400\) 0 0
\(401\) −7.43845 −0.371458 −0.185729 0.982601i \(-0.559465\pi\)
−0.185729 + 0.982601i \(0.559465\pi\)
\(402\) 0 0
\(403\) −13.1231 −0.653708
\(404\) 0 0
\(405\) −7.00000 −0.347833
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −7.12311 −0.352215 −0.176107 0.984371i \(-0.556351\pi\)
−0.176107 + 0.984371i \(0.556351\pi\)
\(410\) 0 0
\(411\) −50.2462 −2.47846
\(412\) 0 0
\(413\) 2.87689 0.141563
\(414\) 0 0
\(415\) −5.12311 −0.251483
\(416\) 0 0
\(417\) 30.7386 1.50528
\(418\) 0 0
\(419\) −26.7386 −1.30627 −0.653134 0.757242i \(-0.726546\pi\)
−0.653134 + 0.757242i \(0.726546\pi\)
\(420\) 0 0
\(421\) 7.61553 0.371158 0.185579 0.982629i \(-0.440584\pi\)
0.185579 + 0.982629i \(0.440584\pi\)
\(422\) 0 0
\(423\) −46.7386 −2.27251
\(424\) 0 0
\(425\) 0.561553 0.0272393
\(426\) 0 0
\(427\) −5.93087 −0.287015
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 33.3693 1.60363 0.801814 0.597574i \(-0.203869\pi\)
0.801814 + 0.597574i \(0.203869\pi\)
\(434\) 0 0
\(435\) 24.8078 1.18944
\(436\) 0 0
\(437\) −13.1231 −0.627763
\(438\) 0 0
\(439\) 34.2462 1.63448 0.817241 0.576296i \(-0.195502\pi\)
0.817241 + 0.576296i \(0.195502\pi\)
\(440\) 0 0
\(441\) −1.56155 −0.0743597
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −12.5616 −0.595475
\(446\) 0 0
\(447\) 5.93087 0.280521
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 33.6155 1.57940
\(454\) 0 0
\(455\) −5.12311 −0.240175
\(456\) 0 0
\(457\) −28.5616 −1.33605 −0.668027 0.744137i \(-0.732861\pi\)
−0.668027 + 0.744137i \(0.732861\pi\)
\(458\) 0 0
\(459\) −0.807764 −0.0377032
\(460\) 0 0
\(461\) −20.5616 −0.957647 −0.478823 0.877911i \(-0.658937\pi\)
−0.478823 + 0.877911i \(0.658937\pi\)
\(462\) 0 0
\(463\) −7.36932 −0.342481 −0.171241 0.985229i \(-0.554778\pi\)
−0.171241 + 0.985229i \(0.554778\pi\)
\(464\) 0 0
\(465\) −16.8078 −0.779441
\(466\) 0 0
\(467\) 9.93087 0.459546 0.229773 0.973244i \(-0.426202\pi\)
0.229773 + 0.973244i \(0.426202\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −37.9309 −1.74776
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.56155 −0.117532
\(476\) 0 0
\(477\) 16.2462 0.743863
\(478\) 0 0
\(479\) 32.9848 1.50712 0.753558 0.657381i \(-0.228336\pi\)
0.753558 + 0.657381i \(0.228336\pi\)
\(480\) 0 0
\(481\) 11.3693 0.518396
\(482\) 0 0
\(483\) −33.6155 −1.52956
\(484\) 0 0
\(485\) 7.12311 0.323444
\(486\) 0 0
\(487\) −2.24621 −0.101786 −0.0508928 0.998704i \(-0.516207\pi\)
−0.0508928 + 0.998704i \(0.516207\pi\)
\(488\) 0 0
\(489\) −8.17708 −0.369780
\(490\) 0 0
\(491\) −7.05398 −0.318341 −0.159171 0.987251i \(-0.550882\pi\)
−0.159171 + 0.987251i \(0.550882\pi\)
\(492\) 0 0
\(493\) −5.43845 −0.244935
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.43845 0.423372
\(498\) 0 0
\(499\) 16.4924 0.738302 0.369151 0.929369i \(-0.379648\pi\)
0.369151 + 0.929369i \(0.379648\pi\)
\(500\) 0 0
\(501\) −27.0540 −1.20868
\(502\) 0 0
\(503\) 14.2462 0.635207 0.317604 0.948224i \(-0.397122\pi\)
0.317604 + 0.948224i \(0.397122\pi\)
\(504\) 0 0
\(505\) 4.24621 0.188954
\(506\) 0 0
\(507\) 23.0540 1.02386
\(508\) 0 0
\(509\) 21.3693 0.947178 0.473589 0.880746i \(-0.342958\pi\)
0.473589 + 0.880746i \(0.342958\pi\)
\(510\) 0 0
\(511\) 5.12311 0.226633
\(512\) 0 0
\(513\) 3.68466 0.162682
\(514\) 0 0
\(515\) 10.2462 0.451502
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 10.8769 0.477443
\(520\) 0 0
\(521\) 8.73863 0.382846 0.191423 0.981508i \(-0.438690\pi\)
0.191423 + 0.981508i \(0.438690\pi\)
\(522\) 0 0
\(523\) 3.50758 0.153376 0.0766878 0.997055i \(-0.475566\pi\)
0.0766878 + 0.997055i \(0.475566\pi\)
\(524\) 0 0
\(525\) −6.56155 −0.286370
\(526\) 0 0
\(527\) 3.68466 0.160506
\(528\) 0 0
\(529\) 3.24621 0.141140
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 33.5464 1.44227 0.721136 0.692793i \(-0.243620\pi\)
0.721136 + 0.692793i \(0.243620\pi\)
\(542\) 0 0
\(543\) 10.8769 0.466772
\(544\) 0 0
\(545\) −18.4924 −0.792128
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) −8.24621 −0.351940
\(550\) 0 0
\(551\) 24.8078 1.05685
\(552\) 0 0
\(553\) 39.3693 1.67415
\(554\) 0 0
\(555\) 14.5616 0.618103
\(556\) 0 0
\(557\) −46.4924 −1.96995 −0.984974 0.172705i \(-0.944749\pi\)
−0.984974 + 0.172705i \(0.944749\pi\)
\(558\) 0 0
\(559\) 20.4924 0.866737
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.87689 0.121247 0.0606233 0.998161i \(-0.480691\pi\)
0.0606233 + 0.998161i \(0.480691\pi\)
\(564\) 0 0
\(565\) −19.1231 −0.804515
\(566\) 0 0
\(567\) −17.9309 −0.753026
\(568\) 0 0
\(569\) 28.1080 1.17835 0.589173 0.808007i \(-0.299454\pi\)
0.589173 + 0.808007i \(0.299454\pi\)
\(570\) 0 0
\(571\) −30.4233 −1.27318 −0.636588 0.771204i \(-0.719655\pi\)
−0.636588 + 0.771204i \(0.719655\pi\)
\(572\) 0 0
\(573\) −26.2462 −1.09645
\(574\) 0 0
\(575\) 5.12311 0.213648
\(576\) 0 0
\(577\) 24.7386 1.02988 0.514941 0.857225i \(-0.327814\pi\)
0.514941 + 0.857225i \(0.327814\pi\)
\(578\) 0 0
\(579\) 13.3002 0.552737
\(580\) 0 0
\(581\) −13.1231 −0.544438
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −7.12311 −0.294504
\(586\) 0 0
\(587\) 32.3153 1.33380 0.666898 0.745149i \(-0.267621\pi\)
0.666898 + 0.745149i \(0.267621\pi\)
\(588\) 0 0
\(589\) −16.8078 −0.692552
\(590\) 0 0
\(591\) −13.7538 −0.565755
\(592\) 0 0
\(593\) −1.50758 −0.0619088 −0.0309544 0.999521i \(-0.509855\pi\)
−0.0309544 + 0.999521i \(0.509855\pi\)
\(594\) 0 0
\(595\) 1.43845 0.0589706
\(596\) 0 0
\(597\) 2.06913 0.0846839
\(598\) 0 0
\(599\) −17.4384 −0.712516 −0.356258 0.934388i \(-0.615948\pi\)
−0.356258 + 0.934388i \(0.615948\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 22.2462 0.905936
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −20.8078 −0.844561 −0.422281 0.906465i \(-0.638770\pi\)
−0.422281 + 0.906465i \(0.638770\pi\)
\(608\) 0 0
\(609\) 63.5464 2.57503
\(610\) 0 0
\(611\) 26.2462 1.06181
\(612\) 0 0
\(613\) −16.7386 −0.676067 −0.338034 0.941134i \(-0.609762\pi\)
−0.338034 + 0.941134i \(0.609762\pi\)
\(614\) 0 0
\(615\) 5.12311 0.206584
\(616\) 0 0
\(617\) −6.63068 −0.266941 −0.133471 0.991053i \(-0.542612\pi\)
−0.133471 + 0.991053i \(0.542612\pi\)
\(618\) 0 0
\(619\) −8.49242 −0.341339 −0.170670 0.985328i \(-0.554593\pi\)
−0.170670 + 0.985328i \(0.554593\pi\)
\(620\) 0 0
\(621\) −7.36932 −0.295720
\(622\) 0 0
\(623\) −32.1771 −1.28915
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.19224 −0.127283
\(630\) 0 0
\(631\) −16.8078 −0.669107 −0.334553 0.942377i \(-0.608585\pi\)
−0.334553 + 0.942377i \(0.608585\pi\)
\(632\) 0 0
\(633\) 21.3002 0.846606
\(634\) 0 0
\(635\) −14.2462 −0.565344
\(636\) 0 0
\(637\) 0.876894 0.0347438
\(638\) 0 0
\(639\) 13.1231 0.519142
\(640\) 0 0
\(641\) −38.1771 −1.50790 −0.753952 0.656929i \(-0.771855\pi\)
−0.753952 + 0.656929i \(0.771855\pi\)
\(642\) 0 0
\(643\) 23.6847 0.934032 0.467016 0.884249i \(-0.345329\pi\)
0.467016 + 0.884249i \(0.345329\pi\)
\(644\) 0 0
\(645\) 26.2462 1.03344
\(646\) 0 0
\(647\) −13.1231 −0.515923 −0.257961 0.966155i \(-0.583051\pi\)
−0.257961 + 0.966155i \(0.583051\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −43.0540 −1.68742
\(652\) 0 0
\(653\) −10.8078 −0.422940 −0.211470 0.977384i \(-0.567825\pi\)
−0.211470 + 0.977384i \(0.567825\pi\)
\(654\) 0 0
\(655\) 7.68466 0.300264
\(656\) 0 0
\(657\) 7.12311 0.277899
\(658\) 0 0
\(659\) −19.1922 −0.747623 −0.373812 0.927505i \(-0.621949\pi\)
−0.373812 + 0.927505i \(0.621949\pi\)
\(660\) 0 0
\(661\) 16.2462 0.631904 0.315952 0.948775i \(-0.397676\pi\)
0.315952 + 0.948775i \(0.397676\pi\)
\(662\) 0 0
\(663\) 2.87689 0.111729
\(664\) 0 0
\(665\) −6.56155 −0.254446
\(666\) 0 0
\(667\) −49.6155 −1.92112
\(668\) 0 0
\(669\) −33.6155 −1.29965
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 35.7926 1.37970 0.689852 0.723951i \(-0.257676\pi\)
0.689852 + 0.723951i \(0.257676\pi\)
\(674\) 0 0
\(675\) −1.43845 −0.0553659
\(676\) 0 0
\(677\) −27.6155 −1.06135 −0.530675 0.847575i \(-0.678062\pi\)
−0.530675 + 0.847575i \(0.678062\pi\)
\(678\) 0 0
\(679\) 18.2462 0.700225
\(680\) 0 0
\(681\) 7.36932 0.282393
\(682\) 0 0
\(683\) −25.9309 −0.992217 −0.496109 0.868260i \(-0.665238\pi\)
−0.496109 + 0.868260i \(0.665238\pi\)
\(684\) 0 0
\(685\) 19.6155 0.749471
\(686\) 0 0
\(687\) 63.3693 2.41769
\(688\) 0 0
\(689\) −9.12311 −0.347563
\(690\) 0 0
\(691\) 3.36932 0.128175 0.0640874 0.997944i \(-0.479586\pi\)
0.0640874 + 0.997944i \(0.479586\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −1.12311 −0.0425407
\(698\) 0 0
\(699\) 51.0540 1.93104
\(700\) 0 0
\(701\) −34.3153 −1.29607 −0.648036 0.761609i \(-0.724409\pi\)
−0.648036 + 0.761609i \(0.724409\pi\)
\(702\) 0 0
\(703\) 14.5616 0.549199
\(704\) 0 0
\(705\) 33.6155 1.26603
\(706\) 0 0
\(707\) 10.8769 0.409068
\(708\) 0 0
\(709\) 1.50758 0.0566183 0.0283091 0.999599i \(-0.490988\pi\)
0.0283091 + 0.999599i \(0.490988\pi\)
\(710\) 0 0
\(711\) 54.7386 2.05286
\(712\) 0 0
\(713\) 33.6155 1.25891
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 65.6155 2.45046
\(718\) 0 0
\(719\) 7.82292 0.291746 0.145873 0.989303i \(-0.453401\pi\)
0.145873 + 0.989303i \(0.453401\pi\)
\(720\) 0 0
\(721\) 26.2462 0.977460
\(722\) 0 0
\(723\) −41.6155 −1.54770
\(724\) 0 0
\(725\) −9.68466 −0.359679
\(726\) 0 0
\(727\) 17.6155 0.653324 0.326662 0.945141i \(-0.394076\pi\)
0.326662 + 0.945141i \(0.394076\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) −5.75379 −0.212812
\(732\) 0 0
\(733\) 19.1231 0.706328 0.353164 0.935561i \(-0.385106\pi\)
0.353164 + 0.935561i \(0.385106\pi\)
\(734\) 0 0
\(735\) 1.12311 0.0414264
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 42.7386 1.57217 0.786083 0.618121i \(-0.212106\pi\)
0.786083 + 0.618121i \(0.212106\pi\)
\(740\) 0 0
\(741\) −13.1231 −0.482089
\(742\) 0 0
\(743\) 2.56155 0.0939743 0.0469871 0.998895i \(-0.485038\pi\)
0.0469871 + 0.998895i \(0.485038\pi\)
\(744\) 0 0
\(745\) −2.31534 −0.0848276
\(746\) 0 0
\(747\) −18.2462 −0.667594
\(748\) 0 0
\(749\) 40.9848 1.49755
\(750\) 0 0
\(751\) −16.1771 −0.590310 −0.295155 0.955449i \(-0.595371\pi\)
−0.295155 + 0.955449i \(0.595371\pi\)
\(752\) 0 0
\(753\) 75.8617 2.76456
\(754\) 0 0
\(755\) −13.1231 −0.477599
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36.7386 1.33177 0.665887 0.746052i \(-0.268053\pi\)
0.665887 + 0.746052i \(0.268053\pi\)
\(762\) 0 0
\(763\) −47.3693 −1.71488
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) −2.24621 −0.0811060
\(768\) 0 0
\(769\) −24.7386 −0.892098 −0.446049 0.895009i \(-0.647169\pi\)
−0.446049 + 0.895009i \(0.647169\pi\)
\(770\) 0 0
\(771\) 73.6155 2.65120
\(772\) 0 0
\(773\) −37.6847 −1.35542 −0.677711 0.735328i \(-0.737028\pi\)
−0.677711 + 0.735328i \(0.737028\pi\)
\(774\) 0 0
\(775\) 6.56155 0.235698
\(776\) 0 0
\(777\) 37.3002 1.33814
\(778\) 0 0
\(779\) 5.12311 0.183554
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 13.9309 0.497849
\(784\) 0 0
\(785\) 14.8078 0.528512
\(786\) 0 0
\(787\) 54.7386 1.95122 0.975611 0.219508i \(-0.0704451\pi\)
0.975611 + 0.219508i \(0.0704451\pi\)
\(788\) 0 0
\(789\) 12.3153 0.438438
\(790\) 0 0
\(791\) −48.9848 −1.74170
\(792\) 0 0
\(793\) 4.63068 0.164440
\(794\) 0 0
\(795\) −11.6847 −0.414412
\(796\) 0 0
\(797\) −23.7538 −0.841402 −0.420701 0.907199i \(-0.638216\pi\)
−0.420701 + 0.907199i \(0.638216\pi\)
\(798\) 0 0
\(799\) −7.36932 −0.260708
\(800\) 0 0
\(801\) −44.7386 −1.58076
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 13.1231 0.462529
\(806\) 0 0
\(807\) 50.2462 1.76875
\(808\) 0 0
\(809\) −6.49242 −0.228261 −0.114131 0.993466i \(-0.536408\pi\)
−0.114131 + 0.993466i \(0.536408\pi\)
\(810\) 0 0
\(811\) −27.1922 −0.954849 −0.477424 0.878673i \(-0.658430\pi\)
−0.477424 + 0.878673i \(0.658430\pi\)
\(812\) 0 0
\(813\) 72.9848 2.55969
\(814\) 0 0
\(815\) 3.19224 0.111819
\(816\) 0 0
\(817\) 26.2462 0.918239
\(818\) 0 0
\(819\) −18.2462 −0.637574
\(820\) 0 0
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) −52.4924 −1.82977 −0.914885 0.403714i \(-0.867719\pi\)
−0.914885 + 0.403714i \(0.867719\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.49242 0.156217 0.0781084 0.996945i \(-0.475112\pi\)
0.0781084 + 0.996945i \(0.475112\pi\)
\(828\) 0 0
\(829\) 5.36932 0.186484 0.0932420 0.995643i \(-0.470277\pi\)
0.0932420 + 0.995643i \(0.470277\pi\)
\(830\) 0 0
\(831\) 18.2462 0.632954
\(832\) 0 0
\(833\) −0.246211 −0.00853071
\(834\) 0 0
\(835\) 10.5616 0.365498
\(836\) 0 0
\(837\) −9.43845 −0.326240
\(838\) 0 0
\(839\) 11.5076 0.397286 0.198643 0.980072i \(-0.436347\pi\)
0.198643 + 0.980072i \(0.436347\pi\)
\(840\) 0 0
\(841\) 64.7926 2.23423
\(842\) 0 0
\(843\) −21.1231 −0.727518
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −59.8617 −2.05445
\(850\) 0 0
\(851\) −29.1231 −0.998327
\(852\) 0 0
\(853\) 3.75379 0.128527 0.0642636 0.997933i \(-0.479530\pi\)
0.0642636 + 0.997933i \(0.479530\pi\)
\(854\) 0 0
\(855\) −9.12311 −0.312004
\(856\) 0 0
\(857\) −32.4233 −1.10756 −0.553779 0.832664i \(-0.686815\pi\)
−0.553779 + 0.832664i \(0.686815\pi\)
\(858\) 0 0
\(859\) −38.8769 −1.32646 −0.663231 0.748415i \(-0.730815\pi\)
−0.663231 + 0.748415i \(0.730815\pi\)
\(860\) 0 0
\(861\) 13.1231 0.447234
\(862\) 0 0
\(863\) 9.61553 0.327316 0.163658 0.986517i \(-0.447671\pi\)
0.163658 + 0.986517i \(0.447671\pi\)
\(864\) 0 0
\(865\) −4.24621 −0.144376
\(866\) 0 0
\(867\) 42.7386 1.45148
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −12.4924 −0.423290
\(872\) 0 0
\(873\) 25.3693 0.858621
\(874\) 0 0
\(875\) 2.56155 0.0865963
\(876\) 0 0
\(877\) 32.2462 1.08888 0.544439 0.838801i \(-0.316743\pi\)
0.544439 + 0.838801i \(0.316743\pi\)
\(878\) 0 0
\(879\) 37.1231 1.25213
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) 6.06913 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(884\) 0 0
\(885\) −2.87689 −0.0967057
\(886\) 0 0
\(887\) 30.2462 1.01557 0.507784 0.861484i \(-0.330465\pi\)
0.507784 + 0.861484i \(0.330465\pi\)
\(888\) 0 0
\(889\) −36.4924 −1.22392
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 33.6155 1.12490
\(894\) 0 0
\(895\) −6.24621 −0.208788
\(896\) 0 0
\(897\) 26.2462 0.876335
\(898\) 0 0
\(899\) −63.5464 −2.11939
\(900\) 0 0
\(901\) 2.56155 0.0853377
\(902\) 0 0
\(903\) 67.2311 2.23731
\(904\) 0 0
\(905\) −4.24621 −0.141149
\(906\) 0 0
\(907\) 8.94602 0.297048 0.148524 0.988909i \(-0.452548\pi\)
0.148524 + 0.988909i \(0.452548\pi\)
\(908\) 0 0
\(909\) 15.1231 0.501602
\(910\) 0 0
\(911\) −2.06913 −0.0685533 −0.0342767 0.999412i \(-0.510913\pi\)
−0.0342767 + 0.999412i \(0.510913\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 5.93087 0.196069
\(916\) 0 0
\(917\) 19.6847 0.650045
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −65.6155 −2.16211
\(922\) 0 0
\(923\) −7.36932 −0.242564
\(924\) 0 0
\(925\) −5.68466 −0.186910
\(926\) 0 0
\(927\) 36.4924 1.19857
\(928\) 0 0
\(929\) 45.0540 1.47817 0.739086 0.673611i \(-0.235257\pi\)
0.739086 + 0.673611i \(0.235257\pi\)
\(930\) 0 0
\(931\) 1.12311 0.0368083
\(932\) 0 0
\(933\) 65.1619 2.13331
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) −78.1080 −2.54896
\(940\) 0 0
\(941\) 4.06913 0.132650 0.0663249 0.997798i \(-0.478873\pi\)
0.0663249 + 0.997798i \(0.478873\pi\)
\(942\) 0 0
\(943\) −10.2462 −0.333663
\(944\) 0 0
\(945\) −3.68466 −0.119862
\(946\) 0 0
\(947\) −39.6847 −1.28958 −0.644789 0.764361i \(-0.723055\pi\)
−0.644789 + 0.764361i \(0.723055\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) −33.7926 −1.09580
\(952\) 0 0
\(953\) 34.1771 1.10710 0.553552 0.832815i \(-0.313272\pi\)
0.553552 + 0.832815i \(0.313272\pi\)
\(954\) 0 0
\(955\) 10.2462 0.331560
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 50.2462 1.62253
\(960\) 0 0
\(961\) 12.0540 0.388838
\(962\) 0 0
\(963\) 56.9848 1.83631
\(964\) 0 0
\(965\) −5.19224 −0.167144
\(966\) 0 0
\(967\) −11.5464 −0.371307 −0.185654 0.982615i \(-0.559440\pi\)
−0.185654 + 0.982615i \(0.559440\pi\)
\(968\) 0 0
\(969\) 3.68466 0.118368
\(970\) 0 0
\(971\) 6.24621 0.200450 0.100225 0.994965i \(-0.468044\pi\)
0.100225 + 0.994965i \(0.468044\pi\)
\(972\) 0 0
\(973\) −30.7386 −0.985435
\(974\) 0 0
\(975\) 5.12311 0.164071
\(976\) 0 0
\(977\) −24.8769 −0.795882 −0.397941 0.917411i \(-0.630275\pi\)
−0.397941 + 0.917411i \(0.630275\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −65.8617 −2.10280
\(982\) 0 0
\(983\) 18.8769 0.602079 0.301040 0.953612i \(-0.402666\pi\)
0.301040 + 0.953612i \(0.402666\pi\)
\(984\) 0 0
\(985\) 5.36932 0.171081
\(986\) 0 0
\(987\) 86.1080 2.74085
\(988\) 0 0
\(989\) −52.4924 −1.66916
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 21.7538 0.690336
\(994\) 0 0
\(995\) −0.807764 −0.0256078
\(996\) 0 0
\(997\) −13.8617 −0.439006 −0.219503 0.975612i \(-0.570444\pi\)
−0.219503 + 0.975612i \(0.570444\pi\)
\(998\) 0 0
\(999\) 8.17708 0.258711
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.j.1.1 2
4.3 odd 2 9680.2.a.bs.1.2 2
11.10 odd 2 440.2.a.e.1.1 2
33.32 even 2 3960.2.a.w.1.1 2
44.43 even 2 880.2.a.o.1.2 2
55.32 even 4 2200.2.b.i.1849.4 4
55.43 even 4 2200.2.b.i.1849.1 4
55.54 odd 2 2200.2.a.s.1.2 2
88.21 odd 2 3520.2.a.bp.1.2 2
88.43 even 2 3520.2.a.bk.1.1 2
132.131 odd 2 7920.2.a.bu.1.2 2
220.43 odd 4 4400.2.b.t.4049.4 4
220.87 odd 4 4400.2.b.t.4049.1 4
220.219 even 2 4400.2.a.bj.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.e.1.1 2 11.10 odd 2
880.2.a.o.1.2 2 44.43 even 2
2200.2.a.s.1.2 2 55.54 odd 2
2200.2.b.i.1849.1 4 55.43 even 4
2200.2.b.i.1849.4 4 55.32 even 4
3520.2.a.bk.1.1 2 88.43 even 2
3520.2.a.bp.1.2 2 88.21 odd 2
3960.2.a.w.1.1 2 33.32 even 2
4400.2.a.bj.1.1 2 220.219 even 2
4400.2.b.t.4049.1 4 220.87 odd 4
4400.2.b.t.4049.4 4 220.43 odd 4
4840.2.a.j.1.1 2 1.1 even 1 trivial
7920.2.a.bu.1.2 2 132.131 odd 2
9680.2.a.bs.1.2 2 4.3 odd 2