Properties

Label 8008.2.a.q.1.6
Level $8008$
Weight $2$
Character 8008.1
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $9$
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] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 53x^{3} + 252x^{2} - 69x - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.829765\) of defining polynomial
Character \(\chi\) \(=\) 8008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.829765 q^{3} +4.12811 q^{5} +1.00000 q^{7} -2.31149 q^{9} +O(q^{10})\) \(q+0.829765 q^{3} +4.12811 q^{5} +1.00000 q^{7} -2.31149 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.42536 q^{15} -2.93739 q^{17} +6.03162 q^{19} +0.829765 q^{21} -4.47216 q^{23} +12.0413 q^{25} -4.40729 q^{27} -3.33589 q^{29} -5.08099 q^{31} +0.829765 q^{33} +4.12811 q^{35} +5.82151 q^{37} -0.829765 q^{39} +5.76715 q^{41} +9.45459 q^{43} -9.54208 q^{45} +4.47216 q^{47} +1.00000 q^{49} -2.43734 q^{51} +1.60150 q^{53} +4.12811 q^{55} +5.00483 q^{57} +9.48190 q^{59} +0.767842 q^{61} -2.31149 q^{63} -4.12811 q^{65} +0.951002 q^{67} -3.71084 q^{69} -5.32071 q^{71} -1.36820 q^{73} +9.99143 q^{75} +1.00000 q^{77} -1.68584 q^{79} +3.27746 q^{81} +8.97176 q^{83} -12.1259 q^{85} -2.76800 q^{87} +16.9049 q^{89} -1.00000 q^{91} -4.21603 q^{93} +24.8992 q^{95} +9.13254 q^{97} -2.31149 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.829765 0.479065 0.239532 0.970888i \(-0.423006\pi\)
0.239532 + 0.970888i \(0.423006\pi\)
\(4\) 0 0
\(5\) 4.12811 1.84615 0.923073 0.384625i \(-0.125669\pi\)
0.923073 + 0.384625i \(0.125669\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.31149 −0.770497
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.42536 0.884424
\(16\) 0 0
\(17\) −2.93739 −0.712421 −0.356211 0.934406i \(-0.615931\pi\)
−0.356211 + 0.934406i \(0.615931\pi\)
\(18\) 0 0
\(19\) 6.03162 1.38375 0.691874 0.722018i \(-0.256785\pi\)
0.691874 + 0.722018i \(0.256785\pi\)
\(20\) 0 0
\(21\) 0.829765 0.181070
\(22\) 0 0
\(23\) −4.47216 −0.932510 −0.466255 0.884650i \(-0.654397\pi\)
−0.466255 + 0.884650i \(0.654397\pi\)
\(24\) 0 0
\(25\) 12.0413 2.40826
\(26\) 0 0
\(27\) −4.40729 −0.848183
\(28\) 0 0
\(29\) −3.33589 −0.619459 −0.309729 0.950825i \(-0.600238\pi\)
−0.309729 + 0.950825i \(0.600238\pi\)
\(30\) 0 0
\(31\) −5.08099 −0.912572 −0.456286 0.889833i \(-0.650821\pi\)
−0.456286 + 0.889833i \(0.650821\pi\)
\(32\) 0 0
\(33\) 0.829765 0.144444
\(34\) 0 0
\(35\) 4.12811 0.697778
\(36\) 0 0
\(37\) 5.82151 0.957051 0.478525 0.878074i \(-0.341171\pi\)
0.478525 + 0.878074i \(0.341171\pi\)
\(38\) 0 0
\(39\) −0.829765 −0.132869
\(40\) 0 0
\(41\) 5.76715 0.900678 0.450339 0.892858i \(-0.351303\pi\)
0.450339 + 0.892858i \(0.351303\pi\)
\(42\) 0 0
\(43\) 9.45459 1.44181 0.720905 0.693033i \(-0.243726\pi\)
0.720905 + 0.693033i \(0.243726\pi\)
\(44\) 0 0
\(45\) −9.54208 −1.42245
\(46\) 0 0
\(47\) 4.47216 0.652331 0.326166 0.945313i \(-0.394243\pi\)
0.326166 + 0.945313i \(0.394243\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.43734 −0.341296
\(52\) 0 0
\(53\) 1.60150 0.219983 0.109991 0.993933i \(-0.464918\pi\)
0.109991 + 0.993933i \(0.464918\pi\)
\(54\) 0 0
\(55\) 4.12811 0.556634
\(56\) 0 0
\(57\) 5.00483 0.662905
\(58\) 0 0
\(59\) 9.48190 1.23444 0.617219 0.786791i \(-0.288259\pi\)
0.617219 + 0.786791i \(0.288259\pi\)
\(60\) 0 0
\(61\) 0.767842 0.0983121 0.0491561 0.998791i \(-0.484347\pi\)
0.0491561 + 0.998791i \(0.484347\pi\)
\(62\) 0 0
\(63\) −2.31149 −0.291220
\(64\) 0 0
\(65\) −4.12811 −0.512029
\(66\) 0 0
\(67\) 0.951002 0.116183 0.0580917 0.998311i \(-0.481498\pi\)
0.0580917 + 0.998311i \(0.481498\pi\)
\(68\) 0 0
\(69\) −3.71084 −0.446733
\(70\) 0 0
\(71\) −5.32071 −0.631452 −0.315726 0.948850i \(-0.602248\pi\)
−0.315726 + 0.948850i \(0.602248\pi\)
\(72\) 0 0
\(73\) −1.36820 −0.160135 −0.0800677 0.996789i \(-0.525514\pi\)
−0.0800677 + 0.996789i \(0.525514\pi\)
\(74\) 0 0
\(75\) 9.99143 1.15371
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −1.68584 −0.189672 −0.0948361 0.995493i \(-0.530233\pi\)
−0.0948361 + 0.995493i \(0.530233\pi\)
\(80\) 0 0
\(81\) 3.27746 0.364162
\(82\) 0 0
\(83\) 8.97176 0.984779 0.492389 0.870375i \(-0.336124\pi\)
0.492389 + 0.870375i \(0.336124\pi\)
\(84\) 0 0
\(85\) −12.1259 −1.31523
\(86\) 0 0
\(87\) −2.76800 −0.296761
\(88\) 0 0
\(89\) 16.9049 1.79192 0.895959 0.444137i \(-0.146490\pi\)
0.895959 + 0.444137i \(0.146490\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −4.21603 −0.437181
\(94\) 0 0
\(95\) 24.8992 2.55460
\(96\) 0 0
\(97\) 9.13254 0.927269 0.463634 0.886027i \(-0.346545\pi\)
0.463634 + 0.886027i \(0.346545\pi\)
\(98\) 0 0
\(99\) −2.31149 −0.232314
\(100\) 0 0
\(101\) 10.3587 1.03073 0.515364 0.856971i \(-0.327657\pi\)
0.515364 + 0.856971i \(0.327657\pi\)
\(102\) 0 0
\(103\) −10.0074 −0.986061 −0.493031 0.870012i \(-0.664111\pi\)
−0.493031 + 0.870012i \(0.664111\pi\)
\(104\) 0 0
\(105\) 3.42536 0.334281
\(106\) 0 0
\(107\) −8.17531 −0.790337 −0.395168 0.918609i \(-0.629314\pi\)
−0.395168 + 0.918609i \(0.629314\pi\)
\(108\) 0 0
\(109\) −6.53533 −0.625971 −0.312985 0.949758i \(-0.601329\pi\)
−0.312985 + 0.949758i \(0.601329\pi\)
\(110\) 0 0
\(111\) 4.83049 0.458490
\(112\) 0 0
\(113\) 14.1221 1.32849 0.664247 0.747513i \(-0.268752\pi\)
0.664247 + 0.747513i \(0.268752\pi\)
\(114\) 0 0
\(115\) −18.4616 −1.72155
\(116\) 0 0
\(117\) 2.31149 0.213697
\(118\) 0 0
\(119\) −2.93739 −0.269270
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 4.78538 0.431483
\(124\) 0 0
\(125\) 29.0671 2.59984
\(126\) 0 0
\(127\) 17.4780 1.55092 0.775460 0.631396i \(-0.217518\pi\)
0.775460 + 0.631396i \(0.217518\pi\)
\(128\) 0 0
\(129\) 7.84508 0.690721
\(130\) 0 0
\(131\) 16.9632 1.48208 0.741040 0.671461i \(-0.234333\pi\)
0.741040 + 0.671461i \(0.234333\pi\)
\(132\) 0 0
\(133\) 6.03162 0.523008
\(134\) 0 0
\(135\) −18.1938 −1.56587
\(136\) 0 0
\(137\) −14.4110 −1.23122 −0.615609 0.788052i \(-0.711090\pi\)
−0.615609 + 0.788052i \(0.711090\pi\)
\(138\) 0 0
\(139\) −3.86031 −0.327427 −0.163714 0.986508i \(-0.552347\pi\)
−0.163714 + 0.986508i \(0.552347\pi\)
\(140\) 0 0
\(141\) 3.71084 0.312509
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −13.7709 −1.14361
\(146\) 0 0
\(147\) 0.829765 0.0684379
\(148\) 0 0
\(149\) −10.1712 −0.833257 −0.416628 0.909077i \(-0.636788\pi\)
−0.416628 + 0.909077i \(0.636788\pi\)
\(150\) 0 0
\(151\) 14.7023 1.19645 0.598227 0.801327i \(-0.295872\pi\)
0.598227 + 0.801327i \(0.295872\pi\)
\(152\) 0 0
\(153\) 6.78974 0.548918
\(154\) 0 0
\(155\) −20.9749 −1.68474
\(156\) 0 0
\(157\) −5.54984 −0.442926 −0.221463 0.975169i \(-0.571083\pi\)
−0.221463 + 0.975169i \(0.571083\pi\)
\(158\) 0 0
\(159\) 1.32887 0.105386
\(160\) 0 0
\(161\) −4.47216 −0.352455
\(162\) 0 0
\(163\) 1.11233 0.0871247 0.0435623 0.999051i \(-0.486129\pi\)
0.0435623 + 0.999051i \(0.486129\pi\)
\(164\) 0 0
\(165\) 3.42536 0.266664
\(166\) 0 0
\(167\) 9.80637 0.758840 0.379420 0.925225i \(-0.376124\pi\)
0.379420 + 0.925225i \(0.376124\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −13.9420 −1.06617
\(172\) 0 0
\(173\) −5.77299 −0.438913 −0.219456 0.975622i \(-0.570428\pi\)
−0.219456 + 0.975622i \(0.570428\pi\)
\(174\) 0 0
\(175\) 12.0413 0.910235
\(176\) 0 0
\(177\) 7.86775 0.591376
\(178\) 0 0
\(179\) 21.9988 1.64427 0.822133 0.569296i \(-0.192784\pi\)
0.822133 + 0.569296i \(0.192784\pi\)
\(180\) 0 0
\(181\) −25.8520 −1.92156 −0.960782 0.277304i \(-0.910559\pi\)
−0.960782 + 0.277304i \(0.910559\pi\)
\(182\) 0 0
\(183\) 0.637129 0.0470979
\(184\) 0 0
\(185\) 24.0318 1.76686
\(186\) 0 0
\(187\) −2.93739 −0.214803
\(188\) 0 0
\(189\) −4.40729 −0.320583
\(190\) 0 0
\(191\) 14.6902 1.06294 0.531472 0.847076i \(-0.321639\pi\)
0.531472 + 0.847076i \(0.321639\pi\)
\(192\) 0 0
\(193\) −23.3863 −1.68338 −0.841691 0.539959i \(-0.818440\pi\)
−0.841691 + 0.539959i \(0.818440\pi\)
\(194\) 0 0
\(195\) −3.42536 −0.245295
\(196\) 0 0
\(197\) 1.97258 0.140540 0.0702702 0.997528i \(-0.477614\pi\)
0.0702702 + 0.997528i \(0.477614\pi\)
\(198\) 0 0
\(199\) −1.39952 −0.0992093 −0.0496047 0.998769i \(-0.515796\pi\)
−0.0496047 + 0.998769i \(0.515796\pi\)
\(200\) 0 0
\(201\) 0.789108 0.0556594
\(202\) 0 0
\(203\) −3.33589 −0.234133
\(204\) 0 0
\(205\) 23.8074 1.66278
\(206\) 0 0
\(207\) 10.3374 0.718496
\(208\) 0 0
\(209\) 6.03162 0.417216
\(210\) 0 0
\(211\) 16.3444 1.12519 0.562597 0.826731i \(-0.309802\pi\)
0.562597 + 0.826731i \(0.309802\pi\)
\(212\) 0 0
\(213\) −4.41494 −0.302506
\(214\) 0 0
\(215\) 39.0296 2.66179
\(216\) 0 0
\(217\) −5.08099 −0.344920
\(218\) 0 0
\(219\) −1.13528 −0.0767153
\(220\) 0 0
\(221\) 2.93739 0.197590
\(222\) 0 0
\(223\) 13.1235 0.878814 0.439407 0.898288i \(-0.355189\pi\)
0.439407 + 0.898288i \(0.355189\pi\)
\(224\) 0 0
\(225\) −27.8333 −1.85555
\(226\) 0 0
\(227\) −1.55421 −0.103156 −0.0515782 0.998669i \(-0.516425\pi\)
−0.0515782 + 0.998669i \(0.516425\pi\)
\(228\) 0 0
\(229\) −0.876231 −0.0579030 −0.0289515 0.999581i \(-0.509217\pi\)
−0.0289515 + 0.999581i \(0.509217\pi\)
\(230\) 0 0
\(231\) 0.829765 0.0545945
\(232\) 0 0
\(233\) −4.72345 −0.309444 −0.154722 0.987958i \(-0.549448\pi\)
−0.154722 + 0.987958i \(0.549448\pi\)
\(234\) 0 0
\(235\) 18.4616 1.20430
\(236\) 0 0
\(237\) −1.39885 −0.0908653
\(238\) 0 0
\(239\) −14.8265 −0.959048 −0.479524 0.877529i \(-0.659191\pi\)
−0.479524 + 0.877529i \(0.659191\pi\)
\(240\) 0 0
\(241\) −29.4642 −1.89796 −0.948979 0.315338i \(-0.897882\pi\)
−0.948979 + 0.315338i \(0.897882\pi\)
\(242\) 0 0
\(243\) 15.9414 1.02264
\(244\) 0 0
\(245\) 4.12811 0.263735
\(246\) 0 0
\(247\) −6.03162 −0.383783
\(248\) 0 0
\(249\) 7.44445 0.471773
\(250\) 0 0
\(251\) −21.7318 −1.37170 −0.685848 0.727745i \(-0.740569\pi\)
−0.685848 + 0.727745i \(0.740569\pi\)
\(252\) 0 0
\(253\) −4.47216 −0.281162
\(254\) 0 0
\(255\) −10.0616 −0.630082
\(256\) 0 0
\(257\) −25.3078 −1.57866 −0.789329 0.613971i \(-0.789571\pi\)
−0.789329 + 0.613971i \(0.789571\pi\)
\(258\) 0 0
\(259\) 5.82151 0.361731
\(260\) 0 0
\(261\) 7.71087 0.477291
\(262\) 0 0
\(263\) 19.9144 1.22797 0.613986 0.789317i \(-0.289565\pi\)
0.613986 + 0.789317i \(0.289565\pi\)
\(264\) 0 0
\(265\) 6.61117 0.406121
\(266\) 0 0
\(267\) 14.0271 0.858445
\(268\) 0 0
\(269\) −23.5469 −1.43568 −0.717840 0.696208i \(-0.754869\pi\)
−0.717840 + 0.696208i \(0.754869\pi\)
\(270\) 0 0
\(271\) −14.7577 −0.896469 −0.448235 0.893916i \(-0.647947\pi\)
−0.448235 + 0.893916i \(0.647947\pi\)
\(272\) 0 0
\(273\) −0.829765 −0.0502197
\(274\) 0 0
\(275\) 12.0413 0.726116
\(276\) 0 0
\(277\) 26.7616 1.60795 0.803975 0.594663i \(-0.202715\pi\)
0.803975 + 0.594663i \(0.202715\pi\)
\(278\) 0 0
\(279\) 11.7447 0.703134
\(280\) 0 0
\(281\) −7.83046 −0.467126 −0.233563 0.972342i \(-0.575038\pi\)
−0.233563 + 0.972342i \(0.575038\pi\)
\(282\) 0 0
\(283\) 13.6976 0.814238 0.407119 0.913375i \(-0.366533\pi\)
0.407119 + 0.913375i \(0.366533\pi\)
\(284\) 0 0
\(285\) 20.6605 1.22382
\(286\) 0 0
\(287\) 5.76715 0.340424
\(288\) 0 0
\(289\) −8.37176 −0.492456
\(290\) 0 0
\(291\) 7.57786 0.444222
\(292\) 0 0
\(293\) −1.22597 −0.0716217 −0.0358109 0.999359i \(-0.511401\pi\)
−0.0358109 + 0.999359i \(0.511401\pi\)
\(294\) 0 0
\(295\) 39.1423 2.27895
\(296\) 0 0
\(297\) −4.40729 −0.255737
\(298\) 0 0
\(299\) 4.47216 0.258632
\(300\) 0 0
\(301\) 9.45459 0.544953
\(302\) 0 0
\(303\) 8.59528 0.493786
\(304\) 0 0
\(305\) 3.16974 0.181499
\(306\) 0 0
\(307\) −14.7628 −0.842555 −0.421277 0.906932i \(-0.638418\pi\)
−0.421277 + 0.906932i \(0.638418\pi\)
\(308\) 0 0
\(309\) −8.30381 −0.472387
\(310\) 0 0
\(311\) −15.1137 −0.857017 −0.428508 0.903538i \(-0.640961\pi\)
−0.428508 + 0.903538i \(0.640961\pi\)
\(312\) 0 0
\(313\) −8.81459 −0.498230 −0.249115 0.968474i \(-0.580140\pi\)
−0.249115 + 0.968474i \(0.580140\pi\)
\(314\) 0 0
\(315\) −9.54208 −0.537635
\(316\) 0 0
\(317\) −20.1450 −1.13146 −0.565728 0.824592i \(-0.691405\pi\)
−0.565728 + 0.824592i \(0.691405\pi\)
\(318\) 0 0
\(319\) −3.33589 −0.186774
\(320\) 0 0
\(321\) −6.78358 −0.378623
\(322\) 0 0
\(323\) −17.7172 −0.985812
\(324\) 0 0
\(325\) −12.0413 −0.667930
\(326\) 0 0
\(327\) −5.42279 −0.299881
\(328\) 0 0
\(329\) 4.47216 0.246558
\(330\) 0 0
\(331\) −0.250156 −0.0137498 −0.00687492 0.999976i \(-0.502188\pi\)
−0.00687492 + 0.999976i \(0.502188\pi\)
\(332\) 0 0
\(333\) −13.4564 −0.737405
\(334\) 0 0
\(335\) 3.92584 0.214491
\(336\) 0 0
\(337\) 22.0879 1.20321 0.601603 0.798795i \(-0.294529\pi\)
0.601603 + 0.798795i \(0.294529\pi\)
\(338\) 0 0
\(339\) 11.7180 0.636435
\(340\) 0 0
\(341\) −5.08099 −0.275151
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −15.3187 −0.824734
\(346\) 0 0
\(347\) −20.3480 −1.09234 −0.546169 0.837675i \(-0.683915\pi\)
−0.546169 + 0.837675i \(0.683915\pi\)
\(348\) 0 0
\(349\) −12.5565 −0.672135 −0.336067 0.941838i \(-0.609097\pi\)
−0.336067 + 0.941838i \(0.609097\pi\)
\(350\) 0 0
\(351\) 4.40729 0.235244
\(352\) 0 0
\(353\) −18.8811 −1.00494 −0.502469 0.864595i \(-0.667575\pi\)
−0.502469 + 0.864595i \(0.667575\pi\)
\(354\) 0 0
\(355\) −21.9645 −1.16575
\(356\) 0 0
\(357\) −2.43734 −0.128998
\(358\) 0 0
\(359\) 9.82369 0.518475 0.259237 0.965814i \(-0.416529\pi\)
0.259237 + 0.965814i \(0.416529\pi\)
\(360\) 0 0
\(361\) 17.3804 0.914760
\(362\) 0 0
\(363\) 0.829765 0.0435514
\(364\) 0 0
\(365\) −5.64807 −0.295633
\(366\) 0 0
\(367\) 0.451078 0.0235461 0.0117731 0.999931i \(-0.496252\pi\)
0.0117731 + 0.999931i \(0.496252\pi\)
\(368\) 0 0
\(369\) −13.3307 −0.693969
\(370\) 0 0
\(371\) 1.60150 0.0831457
\(372\) 0 0
\(373\) −35.8111 −1.85423 −0.927115 0.374778i \(-0.877719\pi\)
−0.927115 + 0.374778i \(0.877719\pi\)
\(374\) 0 0
\(375\) 24.1189 1.24549
\(376\) 0 0
\(377\) 3.33589 0.171807
\(378\) 0 0
\(379\) −14.2983 −0.734454 −0.367227 0.930131i \(-0.619693\pi\)
−0.367227 + 0.930131i \(0.619693\pi\)
\(380\) 0 0
\(381\) 14.5026 0.742992
\(382\) 0 0
\(383\) 7.25912 0.370924 0.185462 0.982651i \(-0.440622\pi\)
0.185462 + 0.982651i \(0.440622\pi\)
\(384\) 0 0
\(385\) 4.12811 0.210388
\(386\) 0 0
\(387\) −21.8542 −1.11091
\(388\) 0 0
\(389\) −37.5285 −1.90277 −0.951386 0.308001i \(-0.900340\pi\)
−0.951386 + 0.308001i \(0.900340\pi\)
\(390\) 0 0
\(391\) 13.1365 0.664339
\(392\) 0 0
\(393\) 14.0754 0.710012
\(394\) 0 0
\(395\) −6.95934 −0.350163
\(396\) 0 0
\(397\) −9.27221 −0.465359 −0.232680 0.972553i \(-0.574749\pi\)
−0.232680 + 0.972553i \(0.574749\pi\)
\(398\) 0 0
\(399\) 5.00483 0.250555
\(400\) 0 0
\(401\) 18.1124 0.904492 0.452246 0.891893i \(-0.350623\pi\)
0.452246 + 0.891893i \(0.350623\pi\)
\(402\) 0 0
\(403\) 5.08099 0.253102
\(404\) 0 0
\(405\) 13.5297 0.672296
\(406\) 0 0
\(407\) 5.82151 0.288562
\(408\) 0 0
\(409\) −3.97525 −0.196564 −0.0982818 0.995159i \(-0.531335\pi\)
−0.0982818 + 0.995159i \(0.531335\pi\)
\(410\) 0 0
\(411\) −11.9578 −0.589833
\(412\) 0 0
\(413\) 9.48190 0.466574
\(414\) 0 0
\(415\) 37.0364 1.81805
\(416\) 0 0
\(417\) −3.20315 −0.156859
\(418\) 0 0
\(419\) 7.61499 0.372017 0.186008 0.982548i \(-0.440445\pi\)
0.186008 + 0.982548i \(0.440445\pi\)
\(420\) 0 0
\(421\) −15.9412 −0.776926 −0.388463 0.921464i \(-0.626994\pi\)
−0.388463 + 0.921464i \(0.626994\pi\)
\(422\) 0 0
\(423\) −10.3374 −0.502619
\(424\) 0 0
\(425\) −35.3699 −1.71569
\(426\) 0 0
\(427\) 0.767842 0.0371585
\(428\) 0 0
\(429\) −0.829765 −0.0400614
\(430\) 0 0
\(431\) −5.41109 −0.260643 −0.130322 0.991472i \(-0.541601\pi\)
−0.130322 + 0.991472i \(0.541601\pi\)
\(432\) 0 0
\(433\) −26.0511 −1.25194 −0.625968 0.779849i \(-0.715296\pi\)
−0.625968 + 0.779849i \(0.715296\pi\)
\(434\) 0 0
\(435\) −11.4266 −0.547864
\(436\) 0 0
\(437\) −26.9744 −1.29036
\(438\) 0 0
\(439\) −26.3660 −1.25838 −0.629191 0.777250i \(-0.716614\pi\)
−0.629191 + 0.777250i \(0.716614\pi\)
\(440\) 0 0
\(441\) −2.31149 −0.110071
\(442\) 0 0
\(443\) −8.43273 −0.400651 −0.200325 0.979729i \(-0.564200\pi\)
−0.200325 + 0.979729i \(0.564200\pi\)
\(444\) 0 0
\(445\) 69.7853 3.30814
\(446\) 0 0
\(447\) −8.43970 −0.399184
\(448\) 0 0
\(449\) 11.9509 0.564000 0.282000 0.959414i \(-0.409002\pi\)
0.282000 + 0.959414i \(0.409002\pi\)
\(450\) 0 0
\(451\) 5.76715 0.271565
\(452\) 0 0
\(453\) 12.1994 0.573179
\(454\) 0 0
\(455\) −4.12811 −0.193529
\(456\) 0 0
\(457\) 36.2542 1.69590 0.847951 0.530074i \(-0.177836\pi\)
0.847951 + 0.530074i \(0.177836\pi\)
\(458\) 0 0
\(459\) 12.9459 0.604263
\(460\) 0 0
\(461\) −20.1950 −0.940573 −0.470287 0.882514i \(-0.655849\pi\)
−0.470287 + 0.882514i \(0.655849\pi\)
\(462\) 0 0
\(463\) −15.5119 −0.720901 −0.360451 0.932778i \(-0.617377\pi\)
−0.360451 + 0.932778i \(0.617377\pi\)
\(464\) 0 0
\(465\) −17.4042 −0.807101
\(466\) 0 0
\(467\) −36.2692 −1.67834 −0.839169 0.543871i \(-0.816958\pi\)
−0.839169 + 0.543871i \(0.816958\pi\)
\(468\) 0 0
\(469\) 0.951002 0.0439132
\(470\) 0 0
\(471\) −4.60506 −0.212190
\(472\) 0 0
\(473\) 9.45459 0.434722
\(474\) 0 0
\(475\) 72.6284 3.33242
\(476\) 0 0
\(477\) −3.70185 −0.169496
\(478\) 0 0
\(479\) 10.0795 0.460545 0.230273 0.973126i \(-0.426038\pi\)
0.230273 + 0.973126i \(0.426038\pi\)
\(480\) 0 0
\(481\) −5.82151 −0.265438
\(482\) 0 0
\(483\) −3.71084 −0.168849
\(484\) 0 0
\(485\) 37.7001 1.71187
\(486\) 0 0
\(487\) −32.5017 −1.47279 −0.736396 0.676550i \(-0.763474\pi\)
−0.736396 + 0.676550i \(0.763474\pi\)
\(488\) 0 0
\(489\) 0.922975 0.0417384
\(490\) 0 0
\(491\) −4.34874 −0.196256 −0.0981280 0.995174i \(-0.531285\pi\)
−0.0981280 + 0.995174i \(0.531285\pi\)
\(492\) 0 0
\(493\) 9.79879 0.441315
\(494\) 0 0
\(495\) −9.54208 −0.428885
\(496\) 0 0
\(497\) −5.32071 −0.238666
\(498\) 0 0
\(499\) −5.57133 −0.249407 −0.124704 0.992194i \(-0.539798\pi\)
−0.124704 + 0.992194i \(0.539798\pi\)
\(500\) 0 0
\(501\) 8.13698 0.363534
\(502\) 0 0
\(503\) 43.6064 1.94431 0.972157 0.234330i \(-0.0752897\pi\)
0.972157 + 0.234330i \(0.0752897\pi\)
\(504\) 0 0
\(505\) 42.7618 1.90287
\(506\) 0 0
\(507\) 0.829765 0.0368512
\(508\) 0 0
\(509\) −21.5763 −0.956354 −0.478177 0.878263i \(-0.658702\pi\)
−0.478177 + 0.878263i \(0.658702\pi\)
\(510\) 0 0
\(511\) −1.36820 −0.0605255
\(512\) 0 0
\(513\) −26.5831 −1.17367
\(514\) 0 0
\(515\) −41.3117 −1.82041
\(516\) 0 0
\(517\) 4.47216 0.196685
\(518\) 0 0
\(519\) −4.79023 −0.210268
\(520\) 0 0
\(521\) −0.0561274 −0.00245899 −0.00122949 0.999999i \(-0.500391\pi\)
−0.00122949 + 0.999999i \(0.500391\pi\)
\(522\) 0 0
\(523\) 32.0264 1.40042 0.700209 0.713938i \(-0.253090\pi\)
0.700209 + 0.713938i \(0.253090\pi\)
\(524\) 0 0
\(525\) 9.99143 0.436062
\(526\) 0 0
\(527\) 14.9248 0.650136
\(528\) 0 0
\(529\) −2.99980 −0.130426
\(530\) 0 0
\(531\) −21.9173 −0.951131
\(532\) 0 0
\(533\) −5.76715 −0.249803
\(534\) 0 0
\(535\) −33.7486 −1.45908
\(536\) 0 0
\(537\) 18.2538 0.787710
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 26.3027 1.13084 0.565420 0.824803i \(-0.308714\pi\)
0.565420 + 0.824803i \(0.308714\pi\)
\(542\) 0 0
\(543\) −21.4511 −0.920554
\(544\) 0 0
\(545\) −26.9785 −1.15563
\(546\) 0 0
\(547\) −7.46262 −0.319079 −0.159539 0.987192i \(-0.551001\pi\)
−0.159539 + 0.987192i \(0.551001\pi\)
\(548\) 0 0
\(549\) −1.77486 −0.0757492
\(550\) 0 0
\(551\) −20.1208 −0.857175
\(552\) 0 0
\(553\) −1.68584 −0.0716894
\(554\) 0 0
\(555\) 19.9408 0.846439
\(556\) 0 0
\(557\) 13.8144 0.585335 0.292667 0.956214i \(-0.405457\pi\)
0.292667 + 0.956214i \(0.405457\pi\)
\(558\) 0 0
\(559\) −9.45459 −0.399886
\(560\) 0 0
\(561\) −2.43734 −0.102905
\(562\) 0 0
\(563\) −34.1499 −1.43925 −0.719624 0.694364i \(-0.755686\pi\)
−0.719624 + 0.694364i \(0.755686\pi\)
\(564\) 0 0
\(565\) 58.2975 2.45259
\(566\) 0 0
\(567\) 3.27746 0.137640
\(568\) 0 0
\(569\) 33.1747 1.39076 0.695379 0.718643i \(-0.255237\pi\)
0.695379 + 0.718643i \(0.255237\pi\)
\(570\) 0 0
\(571\) −26.7096 −1.11776 −0.558881 0.829248i \(-0.688769\pi\)
−0.558881 + 0.829248i \(0.688769\pi\)
\(572\) 0 0
\(573\) 12.1894 0.509219
\(574\) 0 0
\(575\) −53.8505 −2.24572
\(576\) 0 0
\(577\) 5.01972 0.208974 0.104487 0.994526i \(-0.466680\pi\)
0.104487 + 0.994526i \(0.466680\pi\)
\(578\) 0 0
\(579\) −19.4051 −0.806449
\(580\) 0 0
\(581\) 8.97176 0.372211
\(582\) 0 0
\(583\) 1.60150 0.0663274
\(584\) 0 0
\(585\) 9.54208 0.394517
\(586\) 0 0
\(587\) −4.99583 −0.206200 −0.103100 0.994671i \(-0.532876\pi\)
−0.103100 + 0.994671i \(0.532876\pi\)
\(588\) 0 0
\(589\) −30.6466 −1.26277
\(590\) 0 0
\(591\) 1.63678 0.0673280
\(592\) 0 0
\(593\) −29.2892 −1.20276 −0.601382 0.798961i \(-0.705383\pi\)
−0.601382 + 0.798961i \(0.705383\pi\)
\(594\) 0 0
\(595\) −12.1259 −0.497111
\(596\) 0 0
\(597\) −1.16127 −0.0475277
\(598\) 0 0
\(599\) 42.0732 1.71907 0.859533 0.511081i \(-0.170755\pi\)
0.859533 + 0.511081i \(0.170755\pi\)
\(600\) 0 0
\(601\) 33.3963 1.36227 0.681133 0.732160i \(-0.261488\pi\)
0.681133 + 0.732160i \(0.261488\pi\)
\(602\) 0 0
\(603\) −2.19823 −0.0895189
\(604\) 0 0
\(605\) 4.12811 0.167831
\(606\) 0 0
\(607\) 20.6117 0.836604 0.418302 0.908308i \(-0.362625\pi\)
0.418302 + 0.908308i \(0.362625\pi\)
\(608\) 0 0
\(609\) −2.76800 −0.112165
\(610\) 0 0
\(611\) −4.47216 −0.180924
\(612\) 0 0
\(613\) −34.2550 −1.38354 −0.691772 0.722116i \(-0.743170\pi\)
−0.691772 + 0.722116i \(0.743170\pi\)
\(614\) 0 0
\(615\) 19.7546 0.796581
\(616\) 0 0
\(617\) 8.50685 0.342473 0.171236 0.985230i \(-0.445224\pi\)
0.171236 + 0.985230i \(0.445224\pi\)
\(618\) 0 0
\(619\) −3.16839 −0.127348 −0.0636742 0.997971i \(-0.520282\pi\)
−0.0636742 + 0.997971i \(0.520282\pi\)
\(620\) 0 0
\(621\) 19.7101 0.790939
\(622\) 0 0
\(623\) 16.9049 0.677281
\(624\) 0 0
\(625\) 59.7859 2.39144
\(626\) 0 0
\(627\) 5.00483 0.199874
\(628\) 0 0
\(629\) −17.1000 −0.681823
\(630\) 0 0
\(631\) −31.6741 −1.26092 −0.630462 0.776220i \(-0.717135\pi\)
−0.630462 + 0.776220i \(0.717135\pi\)
\(632\) 0 0
\(633\) 13.5620 0.539041
\(634\) 0 0
\(635\) 72.1510 2.86323
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 12.2988 0.486532
\(640\) 0 0
\(641\) −28.0040 −1.10609 −0.553046 0.833150i \(-0.686535\pi\)
−0.553046 + 0.833150i \(0.686535\pi\)
\(642\) 0 0
\(643\) 38.5194 1.51906 0.759528 0.650474i \(-0.225430\pi\)
0.759528 + 0.650474i \(0.225430\pi\)
\(644\) 0 0
\(645\) 32.3853 1.27517
\(646\) 0 0
\(647\) 41.2557 1.62193 0.810964 0.585097i \(-0.198943\pi\)
0.810964 + 0.585097i \(0.198943\pi\)
\(648\) 0 0
\(649\) 9.48190 0.372197
\(650\) 0 0
\(651\) −4.21603 −0.165239
\(652\) 0 0
\(653\) −6.51770 −0.255057 −0.127529 0.991835i \(-0.540704\pi\)
−0.127529 + 0.991835i \(0.540704\pi\)
\(654\) 0 0
\(655\) 70.0258 2.73613
\(656\) 0 0
\(657\) 3.16257 0.123384
\(658\) 0 0
\(659\) 25.8407 1.00661 0.503305 0.864109i \(-0.332117\pi\)
0.503305 + 0.864109i \(0.332117\pi\)
\(660\) 0 0
\(661\) 9.40006 0.365620 0.182810 0.983148i \(-0.441481\pi\)
0.182810 + 0.983148i \(0.441481\pi\)
\(662\) 0 0
\(663\) 2.43734 0.0946585
\(664\) 0 0
\(665\) 24.8992 0.965549
\(666\) 0 0
\(667\) 14.9186 0.577651
\(668\) 0 0
\(669\) 10.8894 0.421009
\(670\) 0 0
\(671\) 0.767842 0.0296422
\(672\) 0 0
\(673\) 36.8867 1.42188 0.710938 0.703255i \(-0.248271\pi\)
0.710938 + 0.703255i \(0.248271\pi\)
\(674\) 0 0
\(675\) −53.0694 −2.04264
\(676\) 0 0
\(677\) 42.5835 1.63662 0.818308 0.574780i \(-0.194912\pi\)
0.818308 + 0.574780i \(0.194912\pi\)
\(678\) 0 0
\(679\) 9.13254 0.350475
\(680\) 0 0
\(681\) −1.28963 −0.0494186
\(682\) 0 0
\(683\) −46.1158 −1.76457 −0.882286 0.470713i \(-0.843997\pi\)
−0.882286 + 0.470713i \(0.843997\pi\)
\(684\) 0 0
\(685\) −59.4903 −2.27301
\(686\) 0 0
\(687\) −0.727066 −0.0277393
\(688\) 0 0
\(689\) −1.60150 −0.0610123
\(690\) 0 0
\(691\) −34.7814 −1.32315 −0.661574 0.749880i \(-0.730111\pi\)
−0.661574 + 0.749880i \(0.730111\pi\)
\(692\) 0 0
\(693\) −2.31149 −0.0878063
\(694\) 0 0
\(695\) −15.9358 −0.604478
\(696\) 0 0
\(697\) −16.9404 −0.641662
\(698\) 0 0
\(699\) −3.91935 −0.148244
\(700\) 0 0
\(701\) 22.3275 0.843298 0.421649 0.906759i \(-0.361451\pi\)
0.421649 + 0.906759i \(0.361451\pi\)
\(702\) 0 0
\(703\) 35.1132 1.32432
\(704\) 0 0
\(705\) 15.3187 0.576937
\(706\) 0 0
\(707\) 10.3587 0.389579
\(708\) 0 0
\(709\) −15.8408 −0.594915 −0.297458 0.954735i \(-0.596139\pi\)
−0.297458 + 0.954735i \(0.596139\pi\)
\(710\) 0 0
\(711\) 3.89681 0.146142
\(712\) 0 0
\(713\) 22.7230 0.850982
\(714\) 0 0
\(715\) −4.12811 −0.154382
\(716\) 0 0
\(717\) −12.3025 −0.459446
\(718\) 0 0
\(719\) −18.2766 −0.681602 −0.340801 0.940135i \(-0.610698\pi\)
−0.340801 + 0.940135i \(0.610698\pi\)
\(720\) 0 0
\(721\) −10.0074 −0.372696
\(722\) 0 0
\(723\) −24.4484 −0.909246
\(724\) 0 0
\(725\) −40.1683 −1.49181
\(726\) 0 0
\(727\) 11.8528 0.439597 0.219798 0.975545i \(-0.429460\pi\)
0.219798 + 0.975545i \(0.429460\pi\)
\(728\) 0 0
\(729\) 3.39523 0.125749
\(730\) 0 0
\(731\) −27.7718 −1.02718
\(732\) 0 0
\(733\) 21.8421 0.806757 0.403379 0.915033i \(-0.367836\pi\)
0.403379 + 0.915033i \(0.367836\pi\)
\(734\) 0 0
\(735\) 3.42536 0.126346
\(736\) 0 0
\(737\) 0.951002 0.0350306
\(738\) 0 0
\(739\) 10.5532 0.388207 0.194103 0.980981i \(-0.437820\pi\)
0.194103 + 0.980981i \(0.437820\pi\)
\(740\) 0 0
\(741\) −5.00483 −0.183857
\(742\) 0 0
\(743\) 53.1325 1.94924 0.974620 0.223864i \(-0.0718672\pi\)
0.974620 + 0.223864i \(0.0718672\pi\)
\(744\) 0 0
\(745\) −41.9878 −1.53831
\(746\) 0 0
\(747\) −20.7381 −0.758769
\(748\) 0 0
\(749\) −8.17531 −0.298719
\(750\) 0 0
\(751\) 25.2243 0.920447 0.460224 0.887803i \(-0.347769\pi\)
0.460224 + 0.887803i \(0.347769\pi\)
\(752\) 0 0
\(753\) −18.0323 −0.657132
\(754\) 0 0
\(755\) 60.6926 2.20883
\(756\) 0 0
\(757\) −15.9861 −0.581025 −0.290512 0.956871i \(-0.593826\pi\)
−0.290512 + 0.956871i \(0.593826\pi\)
\(758\) 0 0
\(759\) −3.71084 −0.134695
\(760\) 0 0
\(761\) −31.5280 −1.14289 −0.571444 0.820641i \(-0.693617\pi\)
−0.571444 + 0.820641i \(0.693617\pi\)
\(762\) 0 0
\(763\) −6.53533 −0.236595
\(764\) 0 0
\(765\) 28.0288 1.01338
\(766\) 0 0
\(767\) −9.48190 −0.342372
\(768\) 0 0
\(769\) −17.2315 −0.621384 −0.310692 0.950511i \(-0.600561\pi\)
−0.310692 + 0.950511i \(0.600561\pi\)
\(770\) 0 0
\(771\) −20.9995 −0.756280
\(772\) 0 0
\(773\) 12.9833 0.466978 0.233489 0.972359i \(-0.424986\pi\)
0.233489 + 0.972359i \(0.424986\pi\)
\(774\) 0 0
\(775\) −61.1816 −2.19771
\(776\) 0 0
\(777\) 4.83049 0.173293
\(778\) 0 0
\(779\) 34.7853 1.24631
\(780\) 0 0
\(781\) −5.32071 −0.190390
\(782\) 0 0
\(783\) 14.7022 0.525414
\(784\) 0 0
\(785\) −22.9103 −0.817705
\(786\) 0 0
\(787\) −2.36830 −0.0844209 −0.0422105 0.999109i \(-0.513440\pi\)
−0.0422105 + 0.999109i \(0.513440\pi\)
\(788\) 0 0
\(789\) 16.5242 0.588278
\(790\) 0 0
\(791\) 14.1221 0.502123
\(792\) 0 0
\(793\) −0.767842 −0.0272669
\(794\) 0 0
\(795\) 5.48571 0.194558
\(796\) 0 0
\(797\) −24.0580 −0.852176 −0.426088 0.904682i \(-0.640109\pi\)
−0.426088 + 0.904682i \(0.640109\pi\)
\(798\) 0 0
\(799\) −13.1365 −0.464735
\(800\) 0 0
\(801\) −39.0755 −1.38067
\(802\) 0 0
\(803\) −1.36820 −0.0482826
\(804\) 0 0
\(805\) −18.4616 −0.650684
\(806\) 0 0
\(807\) −19.5384 −0.687784
\(808\) 0 0
\(809\) 31.9465 1.12318 0.561589 0.827416i \(-0.310190\pi\)
0.561589 + 0.827416i \(0.310190\pi\)
\(810\) 0 0
\(811\) −24.8926 −0.874097 −0.437049 0.899438i \(-0.643976\pi\)
−0.437049 + 0.899438i \(0.643976\pi\)
\(812\) 0 0
\(813\) −12.2455 −0.429467
\(814\) 0 0
\(815\) 4.59183 0.160845
\(816\) 0 0
\(817\) 57.0265 1.99510
\(818\) 0 0
\(819\) 2.31149 0.0807700
\(820\) 0 0
\(821\) −27.4488 −0.957969 −0.478985 0.877823i \(-0.658995\pi\)
−0.478985 + 0.877823i \(0.658995\pi\)
\(822\) 0 0
\(823\) −3.68587 −0.128481 −0.0642407 0.997934i \(-0.520463\pi\)
−0.0642407 + 0.997934i \(0.520463\pi\)
\(824\) 0 0
\(825\) 9.99143 0.347857
\(826\) 0 0
\(827\) 15.1209 0.525807 0.262903 0.964822i \(-0.415320\pi\)
0.262903 + 0.964822i \(0.415320\pi\)
\(828\) 0 0
\(829\) 4.78178 0.166078 0.0830391 0.996546i \(-0.473537\pi\)
0.0830391 + 0.996546i \(0.473537\pi\)
\(830\) 0 0
\(831\) 22.2058 0.770312
\(832\) 0 0
\(833\) −2.93739 −0.101774
\(834\) 0 0
\(835\) 40.4818 1.40093
\(836\) 0 0
\(837\) 22.3934 0.774028
\(838\) 0 0
\(839\) 41.3658 1.42811 0.714053 0.700092i \(-0.246858\pi\)
0.714053 + 0.700092i \(0.246858\pi\)
\(840\) 0 0
\(841\) −17.8719 −0.616271
\(842\) 0 0
\(843\) −6.49744 −0.223784
\(844\) 0 0
\(845\) 4.12811 0.142011
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 11.3658 0.390073
\(850\) 0 0
\(851\) −26.0347 −0.892459
\(852\) 0 0
\(853\) −21.8152 −0.746937 −0.373469 0.927643i \(-0.621832\pi\)
−0.373469 + 0.927643i \(0.621832\pi\)
\(854\) 0 0
\(855\) −57.5542 −1.96831
\(856\) 0 0
\(857\) 18.8757 0.644783 0.322391 0.946606i \(-0.395513\pi\)
0.322391 + 0.946606i \(0.395513\pi\)
\(858\) 0 0
\(859\) 16.3038 0.556279 0.278140 0.960541i \(-0.410282\pi\)
0.278140 + 0.960541i \(0.410282\pi\)
\(860\) 0 0
\(861\) 4.78538 0.163085
\(862\) 0 0
\(863\) 15.4059 0.524421 0.262211 0.965011i \(-0.415548\pi\)
0.262211 + 0.965011i \(0.415548\pi\)
\(864\) 0 0
\(865\) −23.8315 −0.810297
\(866\) 0 0
\(867\) −6.94659 −0.235919
\(868\) 0 0
\(869\) −1.68584 −0.0571883
\(870\) 0 0
\(871\) −0.951002 −0.0322235
\(872\) 0 0
\(873\) −21.1098 −0.714458
\(874\) 0 0
\(875\) 29.0671 0.982649
\(876\) 0 0
\(877\) −0.787645 −0.0265969 −0.0132984 0.999912i \(-0.504233\pi\)
−0.0132984 + 0.999912i \(0.504233\pi\)
\(878\) 0 0
\(879\) −1.01726 −0.0343115
\(880\) 0 0
\(881\) 19.2246 0.647694 0.323847 0.946109i \(-0.395024\pi\)
0.323847 + 0.946109i \(0.395024\pi\)
\(882\) 0 0
\(883\) 40.7815 1.37241 0.686204 0.727409i \(-0.259276\pi\)
0.686204 + 0.727409i \(0.259276\pi\)
\(884\) 0 0
\(885\) 32.4789 1.09177
\(886\) 0 0
\(887\) −44.1173 −1.48131 −0.740657 0.671884i \(-0.765485\pi\)
−0.740657 + 0.671884i \(0.765485\pi\)
\(888\) 0 0
\(889\) 17.4780 0.586193
\(890\) 0 0
\(891\) 3.27746 0.109799
\(892\) 0 0
\(893\) 26.9744 0.902663
\(894\) 0 0
\(895\) 90.8133 3.03555
\(896\) 0 0
\(897\) 3.71084 0.123901
\(898\) 0 0
\(899\) 16.9496 0.565301
\(900\) 0 0
\(901\) −4.70423 −0.156720
\(902\) 0 0
\(903\) 7.84508 0.261068
\(904\) 0 0
\(905\) −106.720 −3.54749
\(906\) 0 0
\(907\) 11.1654 0.370741 0.185371 0.982669i \(-0.440651\pi\)
0.185371 + 0.982669i \(0.440651\pi\)
\(908\) 0 0
\(909\) −23.9440 −0.794173
\(910\) 0 0
\(911\) 19.8784 0.658602 0.329301 0.944225i \(-0.393187\pi\)
0.329301 + 0.944225i \(0.393187\pi\)
\(912\) 0 0
\(913\) 8.97176 0.296922
\(914\) 0 0
\(915\) 2.63014 0.0869496
\(916\) 0 0
\(917\) 16.9632 0.560173
\(918\) 0 0
\(919\) −28.7842 −0.949504 −0.474752 0.880120i \(-0.657462\pi\)
−0.474752 + 0.880120i \(0.657462\pi\)
\(920\) 0 0
\(921\) −12.2496 −0.403639
\(922\) 0 0
\(923\) 5.32071 0.175133
\(924\) 0 0
\(925\) 70.0984 2.30482
\(926\) 0 0
\(927\) 23.1321 0.759757
\(928\) 0 0
\(929\) 6.61071 0.216890 0.108445 0.994102i \(-0.465413\pi\)
0.108445 + 0.994102i \(0.465413\pi\)
\(930\) 0 0
\(931\) 6.03162 0.197678
\(932\) 0 0
\(933\) −12.5408 −0.410567
\(934\) 0 0
\(935\) −12.1259 −0.396558
\(936\) 0 0
\(937\) 16.7109 0.545921 0.272961 0.962025i \(-0.411997\pi\)
0.272961 + 0.962025i \(0.411997\pi\)
\(938\) 0 0
\(939\) −7.31404 −0.238685
\(940\) 0 0
\(941\) 24.9377 0.812946 0.406473 0.913663i \(-0.366759\pi\)
0.406473 + 0.913663i \(0.366759\pi\)
\(942\) 0 0
\(943\) −25.7916 −0.839891
\(944\) 0 0
\(945\) −18.1938 −0.591843
\(946\) 0 0
\(947\) 11.8721 0.385790 0.192895 0.981219i \(-0.438212\pi\)
0.192895 + 0.981219i \(0.438212\pi\)
\(948\) 0 0
\(949\) 1.36820 0.0444136
\(950\) 0 0
\(951\) −16.7156 −0.542041
\(952\) 0 0
\(953\) −22.5839 −0.731564 −0.365782 0.930700i \(-0.619199\pi\)
−0.365782 + 0.930700i \(0.619199\pi\)
\(954\) 0 0
\(955\) 60.6427 1.96235
\(956\) 0 0
\(957\) −2.76800 −0.0894768
\(958\) 0 0
\(959\) −14.4110 −0.465356
\(960\) 0 0
\(961\) −5.18356 −0.167212
\(962\) 0 0
\(963\) 18.8971 0.608952
\(964\) 0 0
\(965\) −96.5411 −3.10777
\(966\) 0 0
\(967\) −10.3246 −0.332017 −0.166008 0.986124i \(-0.553088\pi\)
−0.166008 + 0.986124i \(0.553088\pi\)
\(968\) 0 0
\(969\) −14.7011 −0.472268
\(970\) 0 0
\(971\) −4.82234 −0.154756 −0.0773781 0.997002i \(-0.524655\pi\)
−0.0773781 + 0.997002i \(0.524655\pi\)
\(972\) 0 0
\(973\) −3.86031 −0.123756
\(974\) 0 0
\(975\) −9.99143 −0.319982
\(976\) 0 0
\(977\) 24.0395 0.769090 0.384545 0.923106i \(-0.374358\pi\)
0.384545 + 0.923106i \(0.374358\pi\)
\(978\) 0 0
\(979\) 16.9049 0.540283
\(980\) 0 0
\(981\) 15.1063 0.482308
\(982\) 0 0
\(983\) −33.9072 −1.08147 −0.540736 0.841193i \(-0.681854\pi\)
−0.540736 + 0.841193i \(0.681854\pi\)
\(984\) 0 0
\(985\) 8.14302 0.259458
\(986\) 0 0
\(987\) 3.71084 0.118117
\(988\) 0 0
\(989\) −42.2824 −1.34450
\(990\) 0 0
\(991\) −16.0541 −0.509974 −0.254987 0.966944i \(-0.582071\pi\)
−0.254987 + 0.966944i \(0.582071\pi\)
\(992\) 0 0
\(993\) −0.207571 −0.00658707
\(994\) 0 0
\(995\) −5.77737 −0.183155
\(996\) 0 0
\(997\) 55.1878 1.74781 0.873907 0.486093i \(-0.161578\pi\)
0.873907 + 0.486093i \(0.161578\pi\)
\(998\) 0 0
\(999\) −25.6571 −0.811754
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 8008.2.a.q.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8008.2.a.q.1.6 9 1.1 even 1 trivial