Properties

Label 3332.2.a.u.1.5
Level $3332$
Weight $2$
Character 3332.1
Self dual yes
Analytic conductor $26.606$
Analytic rank $0$
Dimension $8$
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] = [3332,2,Mod(1,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.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: \(26.6061539535\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.26400\) of defining polynomial
Character \(\chi\) \(=\) 3332.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.26400 q^{3} -1.77431 q^{5} -1.40230 q^{9} +0.318204 q^{11} +0.205935 q^{13} -2.24272 q^{15} -1.00000 q^{17} +5.45902 q^{19} +9.43694 q^{23} -1.85184 q^{25} -5.56451 q^{27} -8.61859 q^{29} +4.94814 q^{31} +0.402209 q^{33} +7.64938 q^{37} +0.260301 q^{39} +1.44153 q^{41} -1.98578 q^{43} +2.48812 q^{45} -1.56955 q^{47} -1.26400 q^{51} -13.5386 q^{53} -0.564590 q^{55} +6.90020 q^{57} +13.2075 q^{59} +3.37365 q^{61} -0.365391 q^{65} +13.3478 q^{67} +11.9283 q^{69} -2.62870 q^{71} +3.86935 q^{73} -2.34072 q^{75} +12.2324 q^{79} -2.82663 q^{81} +10.4915 q^{83} +1.77431 q^{85} -10.8939 q^{87} +0.00620724 q^{89} +6.25445 q^{93} -9.68597 q^{95} +4.93669 q^{97} -0.446218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.26400 0.729771 0.364885 0.931052i \(-0.381108\pi\)
0.364885 + 0.931052i \(0.381108\pi\)
\(4\) 0 0
\(5\) −1.77431 −0.793494 −0.396747 0.917928i \(-0.629861\pi\)
−0.396747 + 0.917928i \(0.629861\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.40230 −0.467435
\(10\) 0 0
\(11\) 0.318204 0.0959420 0.0479710 0.998849i \(-0.484724\pi\)
0.0479710 + 0.998849i \(0.484724\pi\)
\(12\) 0 0
\(13\) 0.205935 0.0571160 0.0285580 0.999592i \(-0.490908\pi\)
0.0285580 + 0.999592i \(0.490908\pi\)
\(14\) 0 0
\(15\) −2.24272 −0.579069
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 5.45902 1.25238 0.626192 0.779669i \(-0.284612\pi\)
0.626192 + 0.779669i \(0.284612\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.43694 1.96774 0.983870 0.178888i \(-0.0572499\pi\)
0.983870 + 0.178888i \(0.0572499\pi\)
\(24\) 0 0
\(25\) −1.85184 −0.370368
\(26\) 0 0
\(27\) −5.56451 −1.07089
\(28\) 0 0
\(29\) −8.61859 −1.60043 −0.800216 0.599712i \(-0.795282\pi\)
−0.800216 + 0.599712i \(0.795282\pi\)
\(30\) 0 0
\(31\) 4.94814 0.888712 0.444356 0.895850i \(-0.353432\pi\)
0.444356 + 0.895850i \(0.353432\pi\)
\(32\) 0 0
\(33\) 0.402209 0.0700157
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.64938 1.25755 0.628775 0.777587i \(-0.283557\pi\)
0.628775 + 0.777587i \(0.283557\pi\)
\(38\) 0 0
\(39\) 0.260301 0.0416816
\(40\) 0 0
\(41\) 1.44153 0.225129 0.112565 0.993644i \(-0.464093\pi\)
0.112565 + 0.993644i \(0.464093\pi\)
\(42\) 0 0
\(43\) −1.98578 −0.302829 −0.151414 0.988470i \(-0.548383\pi\)
−0.151414 + 0.988470i \(0.548383\pi\)
\(44\) 0 0
\(45\) 2.48812 0.370906
\(46\) 0 0
\(47\) −1.56955 −0.228942 −0.114471 0.993427i \(-0.536517\pi\)
−0.114471 + 0.993427i \(0.536517\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.26400 −0.176995
\(52\) 0 0
\(53\) −13.5386 −1.85968 −0.929838 0.367969i \(-0.880053\pi\)
−0.929838 + 0.367969i \(0.880053\pi\)
\(54\) 0 0
\(55\) −0.564590 −0.0761293
\(56\) 0 0
\(57\) 6.90020 0.913954
\(58\) 0 0
\(59\) 13.2075 1.71947 0.859733 0.510743i \(-0.170630\pi\)
0.859733 + 0.510743i \(0.170630\pi\)
\(60\) 0 0
\(61\) 3.37365 0.431951 0.215976 0.976399i \(-0.430707\pi\)
0.215976 + 0.976399i \(0.430707\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.365391 −0.0453212
\(66\) 0 0
\(67\) 13.3478 1.63070 0.815348 0.578971i \(-0.196545\pi\)
0.815348 + 0.578971i \(0.196545\pi\)
\(68\) 0 0
\(69\) 11.9283 1.43600
\(70\) 0 0
\(71\) −2.62870 −0.311969 −0.155985 0.987759i \(-0.549855\pi\)
−0.155985 + 0.987759i \(0.549855\pi\)
\(72\) 0 0
\(73\) 3.86935 0.452873 0.226436 0.974026i \(-0.427292\pi\)
0.226436 + 0.974026i \(0.427292\pi\)
\(74\) 0 0
\(75\) −2.34072 −0.270284
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.2324 1.37625 0.688124 0.725594i \(-0.258435\pi\)
0.688124 + 0.725594i \(0.258435\pi\)
\(80\) 0 0
\(81\) −2.82663 −0.314070
\(82\) 0 0
\(83\) 10.4915 1.15159 0.575797 0.817592i \(-0.304692\pi\)
0.575797 + 0.817592i \(0.304692\pi\)
\(84\) 0 0
\(85\) 1.77431 0.192450
\(86\) 0 0
\(87\) −10.8939 −1.16795
\(88\) 0 0
\(89\) 0.00620724 0.000657966 0 0.000328983 1.00000i \(-0.499895\pi\)
0.000328983 1.00000i \(0.499895\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.25445 0.648556
\(94\) 0 0
\(95\) −9.68597 −0.993759
\(96\) 0 0
\(97\) 4.93669 0.501245 0.250622 0.968085i \(-0.419365\pi\)
0.250622 + 0.968085i \(0.419365\pi\)
\(98\) 0 0
\(99\) −0.446218 −0.0448466
\(100\) 0 0
\(101\) 8.69105 0.864792 0.432396 0.901684i \(-0.357668\pi\)
0.432396 + 0.901684i \(0.357668\pi\)
\(102\) 0 0
\(103\) 4.08166 0.402178 0.201089 0.979573i \(-0.435552\pi\)
0.201089 + 0.979573i \(0.435552\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.4600 1.39790 0.698950 0.715171i \(-0.253651\pi\)
0.698950 + 0.715171i \(0.253651\pi\)
\(108\) 0 0
\(109\) 1.57368 0.150731 0.0753655 0.997156i \(-0.475988\pi\)
0.0753655 + 0.997156i \(0.475988\pi\)
\(110\) 0 0
\(111\) 9.66882 0.917723
\(112\) 0 0
\(113\) −0.830486 −0.0781256 −0.0390628 0.999237i \(-0.512437\pi\)
−0.0390628 + 0.999237i \(0.512437\pi\)
\(114\) 0 0
\(115\) −16.7440 −1.56139
\(116\) 0 0
\(117\) −0.288783 −0.0266980
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.8987 −0.990795
\(122\) 0 0
\(123\) 1.82210 0.164293
\(124\) 0 0
\(125\) 12.1573 1.08738
\(126\) 0 0
\(127\) −4.93809 −0.438185 −0.219092 0.975704i \(-0.570310\pi\)
−0.219092 + 0.975704i \(0.570310\pi\)
\(128\) 0 0
\(129\) −2.51003 −0.220995
\(130\) 0 0
\(131\) 18.2925 1.59823 0.799113 0.601181i \(-0.205303\pi\)
0.799113 + 0.601181i \(0.205303\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 9.87315 0.849745
\(136\) 0 0
\(137\) 9.89826 0.845665 0.422833 0.906208i \(-0.361036\pi\)
0.422833 + 0.906208i \(0.361036\pi\)
\(138\) 0 0
\(139\) −12.6946 −1.07675 −0.538373 0.842707i \(-0.680961\pi\)
−0.538373 + 0.842707i \(0.680961\pi\)
\(140\) 0 0
\(141\) −1.98391 −0.167075
\(142\) 0 0
\(143\) 0.0655291 0.00547982
\(144\) 0 0
\(145\) 15.2920 1.26993
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.30758 −0.107121 −0.0535607 0.998565i \(-0.517057\pi\)
−0.0535607 + 0.998565i \(0.517057\pi\)
\(150\) 0 0
\(151\) 3.68720 0.300060 0.150030 0.988681i \(-0.452063\pi\)
0.150030 + 0.988681i \(0.452063\pi\)
\(152\) 0 0
\(153\) 1.40230 0.113370
\(154\) 0 0
\(155\) −8.77952 −0.705188
\(156\) 0 0
\(157\) 15.0157 1.19838 0.599190 0.800607i \(-0.295489\pi\)
0.599190 + 0.800607i \(0.295489\pi\)
\(158\) 0 0
\(159\) −17.1128 −1.35714
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.11421 0.713880 0.356940 0.934127i \(-0.383820\pi\)
0.356940 + 0.934127i \(0.383820\pi\)
\(164\) 0 0
\(165\) −0.713642 −0.0555570
\(166\) 0 0
\(167\) 9.18362 0.710650 0.355325 0.934743i \(-0.384370\pi\)
0.355325 + 0.934743i \(0.384370\pi\)
\(168\) 0 0
\(169\) −12.9576 −0.996738
\(170\) 0 0
\(171\) −7.65520 −0.585408
\(172\) 0 0
\(173\) −15.5531 −1.18248 −0.591241 0.806495i \(-0.701362\pi\)
−0.591241 + 0.806495i \(0.701362\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 16.6942 1.25482
\(178\) 0 0
\(179\) −8.10572 −0.605850 −0.302925 0.953014i \(-0.597963\pi\)
−0.302925 + 0.953014i \(0.597963\pi\)
\(180\) 0 0
\(181\) 14.8172 1.10135 0.550677 0.834718i \(-0.314370\pi\)
0.550677 + 0.834718i \(0.314370\pi\)
\(182\) 0 0
\(183\) 4.26429 0.315226
\(184\) 0 0
\(185\) −13.5723 −0.997858
\(186\) 0 0
\(187\) −0.318204 −0.0232693
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.5824 −0.765719 −0.382860 0.923807i \(-0.625061\pi\)
−0.382860 + 0.923807i \(0.625061\pi\)
\(192\) 0 0
\(193\) −20.4950 −1.47526 −0.737631 0.675204i \(-0.764055\pi\)
−0.737631 + 0.675204i \(0.764055\pi\)
\(194\) 0 0
\(195\) −0.461854 −0.0330741
\(196\) 0 0
\(197\) −12.3238 −0.878037 −0.439018 0.898478i \(-0.644674\pi\)
−0.439018 + 0.898478i \(0.644674\pi\)
\(198\) 0 0
\(199\) −1.24045 −0.0879335 −0.0439668 0.999033i \(-0.514000\pi\)
−0.0439668 + 0.999033i \(0.514000\pi\)
\(200\) 0 0
\(201\) 16.8717 1.19003
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.55772 −0.178639
\(206\) 0 0
\(207\) −13.2335 −0.919789
\(208\) 0 0
\(209\) 1.73708 0.120156
\(210\) 0 0
\(211\) 6.88477 0.473967 0.236984 0.971514i \(-0.423841\pi\)
0.236984 + 0.971514i \(0.423841\pi\)
\(212\) 0 0
\(213\) −3.32268 −0.227666
\(214\) 0 0
\(215\) 3.52338 0.240293
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4.89086 0.330493
\(220\) 0 0
\(221\) −0.205935 −0.0138527
\(222\) 0 0
\(223\) 13.5034 0.904258 0.452129 0.891953i \(-0.350665\pi\)
0.452129 + 0.891953i \(0.350665\pi\)
\(224\) 0 0
\(225\) 2.59684 0.173123
\(226\) 0 0
\(227\) −0.645775 −0.0428616 −0.0214308 0.999770i \(-0.506822\pi\)
−0.0214308 + 0.999770i \(0.506822\pi\)
\(228\) 0 0
\(229\) 17.8239 1.17784 0.588919 0.808192i \(-0.299554\pi\)
0.588919 + 0.808192i \(0.299554\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.09800 −0.530518 −0.265259 0.964177i \(-0.585457\pi\)
−0.265259 + 0.964177i \(0.585457\pi\)
\(234\) 0 0
\(235\) 2.78485 0.181664
\(236\) 0 0
\(237\) 15.4617 1.00434
\(238\) 0 0
\(239\) −3.12106 −0.201885 −0.100942 0.994892i \(-0.532186\pi\)
−0.100942 + 0.994892i \(0.532186\pi\)
\(240\) 0 0
\(241\) 11.3366 0.730252 0.365126 0.930958i \(-0.381026\pi\)
0.365126 + 0.930958i \(0.381026\pi\)
\(242\) 0 0
\(243\) 13.1207 0.841692
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.12420 0.0715312
\(248\) 0 0
\(249\) 13.2613 0.840400
\(250\) 0 0
\(251\) 3.57825 0.225857 0.112929 0.993603i \(-0.463977\pi\)
0.112929 + 0.993603i \(0.463977\pi\)
\(252\) 0 0
\(253\) 3.00287 0.188789
\(254\) 0 0
\(255\) 2.24272 0.140445
\(256\) 0 0
\(257\) −9.06411 −0.565404 −0.282702 0.959208i \(-0.591231\pi\)
−0.282702 + 0.959208i \(0.591231\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0859 0.748097
\(262\) 0 0
\(263\) 4.93268 0.304162 0.152081 0.988368i \(-0.451403\pi\)
0.152081 + 0.988368i \(0.451403\pi\)
\(264\) 0 0
\(265\) 24.0217 1.47564
\(266\) 0 0
\(267\) 0.00784595 0.000480165 0
\(268\) 0 0
\(269\) −17.2672 −1.05280 −0.526400 0.850237i \(-0.676458\pi\)
−0.526400 + 0.850237i \(0.676458\pi\)
\(270\) 0 0
\(271\) 29.4257 1.78748 0.893742 0.448581i \(-0.148070\pi\)
0.893742 + 0.448581i \(0.148070\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.589262 −0.0355338
\(276\) 0 0
\(277\) −29.4835 −1.77149 −0.885745 0.464172i \(-0.846352\pi\)
−0.885745 + 0.464172i \(0.846352\pi\)
\(278\) 0 0
\(279\) −6.93880 −0.415415
\(280\) 0 0
\(281\) 5.51539 0.329021 0.164510 0.986375i \(-0.447396\pi\)
0.164510 + 0.986375i \(0.447396\pi\)
\(282\) 0 0
\(283\) −7.46379 −0.443676 −0.221838 0.975084i \(-0.571206\pi\)
−0.221838 + 0.975084i \(0.571206\pi\)
\(284\) 0 0
\(285\) −12.2431 −0.725217
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 6.23997 0.365794
\(292\) 0 0
\(293\) −20.6687 −1.20748 −0.603740 0.797181i \(-0.706323\pi\)
−0.603740 + 0.797181i \(0.706323\pi\)
\(294\) 0 0
\(295\) −23.4341 −1.36439
\(296\) 0 0
\(297\) −1.77065 −0.102743
\(298\) 0 0
\(299\) 1.94339 0.112389
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 10.9855 0.631100
\(304\) 0 0
\(305\) −5.98588 −0.342751
\(306\) 0 0
\(307\) −14.0562 −0.802228 −0.401114 0.916028i \(-0.631377\pi\)
−0.401114 + 0.916028i \(0.631377\pi\)
\(308\) 0 0
\(309\) 5.15922 0.293498
\(310\) 0 0
\(311\) −4.76105 −0.269975 −0.134987 0.990847i \(-0.543099\pi\)
−0.134987 + 0.990847i \(0.543099\pi\)
\(312\) 0 0
\(313\) −26.9097 −1.52103 −0.760513 0.649323i \(-0.775052\pi\)
−0.760513 + 0.649323i \(0.775052\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.3180 −1.14117 −0.570587 0.821237i \(-0.693284\pi\)
−0.570587 + 0.821237i \(0.693284\pi\)
\(318\) 0 0
\(319\) −2.74246 −0.153549
\(320\) 0 0
\(321\) 18.2774 1.02015
\(322\) 0 0
\(323\) −5.45902 −0.303748
\(324\) 0 0
\(325\) −0.381358 −0.0211539
\(326\) 0 0
\(327\) 1.98913 0.109999
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.29216 0.0710238 0.0355119 0.999369i \(-0.488694\pi\)
0.0355119 + 0.999369i \(0.488694\pi\)
\(332\) 0 0
\(333\) −10.7268 −0.587822
\(334\) 0 0
\(335\) −23.6831 −1.29395
\(336\) 0 0
\(337\) 4.35840 0.237417 0.118708 0.992929i \(-0.462125\pi\)
0.118708 + 0.992929i \(0.462125\pi\)
\(338\) 0 0
\(339\) −1.04973 −0.0570138
\(340\) 0 0
\(341\) 1.57452 0.0852648
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −21.1645 −1.13946
\(346\) 0 0
\(347\) 5.83404 0.313188 0.156594 0.987663i \(-0.449949\pi\)
0.156594 + 0.987663i \(0.449949\pi\)
\(348\) 0 0
\(349\) −35.2609 −1.88747 −0.943736 0.330700i \(-0.892715\pi\)
−0.943736 + 0.330700i \(0.892715\pi\)
\(350\) 0 0
\(351\) −1.14593 −0.0611650
\(352\) 0 0
\(353\) 21.7586 1.15809 0.579046 0.815295i \(-0.303425\pi\)
0.579046 + 0.815295i \(0.303425\pi\)
\(354\) 0 0
\(355\) 4.66412 0.247546
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.68306 0.194385 0.0971923 0.995266i \(-0.469014\pi\)
0.0971923 + 0.995266i \(0.469014\pi\)
\(360\) 0 0
\(361\) 10.8009 0.568468
\(362\) 0 0
\(363\) −13.7760 −0.723053
\(364\) 0 0
\(365\) −6.86541 −0.359352
\(366\) 0 0
\(367\) 8.43369 0.440235 0.220117 0.975473i \(-0.429356\pi\)
0.220117 + 0.975473i \(0.429356\pi\)
\(368\) 0 0
\(369\) −2.02146 −0.105233
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −31.3060 −1.62097 −0.810483 0.585762i \(-0.800795\pi\)
−0.810483 + 0.585762i \(0.800795\pi\)
\(374\) 0 0
\(375\) 15.3668 0.793537
\(376\) 0 0
\(377\) −1.77486 −0.0914102
\(378\) 0 0
\(379\) −27.3855 −1.40670 −0.703349 0.710844i \(-0.748313\pi\)
−0.703349 + 0.710844i \(0.748313\pi\)
\(380\) 0 0
\(381\) −6.24175 −0.319774
\(382\) 0 0
\(383\) −22.2937 −1.13916 −0.569579 0.821937i \(-0.692894\pi\)
−0.569579 + 0.821937i \(0.692894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.78467 0.141553
\(388\) 0 0
\(389\) −26.3008 −1.33351 −0.666753 0.745279i \(-0.732316\pi\)
−0.666753 + 0.745279i \(0.732316\pi\)
\(390\) 0 0
\(391\) −9.43694 −0.477247
\(392\) 0 0
\(393\) 23.1218 1.16634
\(394\) 0 0
\(395\) −21.7039 −1.09204
\(396\) 0 0
\(397\) 38.8034 1.94749 0.973745 0.227644i \(-0.0731021\pi\)
0.973745 + 0.227644i \(0.0731021\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.9758 0.747855 0.373928 0.927458i \(-0.378011\pi\)
0.373928 + 0.927458i \(0.378011\pi\)
\(402\) 0 0
\(403\) 1.01899 0.0507597
\(404\) 0 0
\(405\) 5.01531 0.249213
\(406\) 0 0
\(407\) 2.43406 0.120652
\(408\) 0 0
\(409\) 38.3051 1.89407 0.947034 0.321134i \(-0.104064\pi\)
0.947034 + 0.321134i \(0.104064\pi\)
\(410\) 0 0
\(411\) 12.5114 0.617142
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −18.6152 −0.913783
\(416\) 0 0
\(417\) −16.0460 −0.785778
\(418\) 0 0
\(419\) 23.3744 1.14192 0.570958 0.820979i \(-0.306572\pi\)
0.570958 + 0.820979i \(0.306572\pi\)
\(420\) 0 0
\(421\) 8.17711 0.398528 0.199264 0.979946i \(-0.436145\pi\)
0.199264 + 0.979946i \(0.436145\pi\)
\(422\) 0 0
\(423\) 2.20098 0.107015
\(424\) 0 0
\(425\) 1.85184 0.0898274
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.0828288 0.00399901
\(430\) 0 0
\(431\) −20.1422 −0.970217 −0.485108 0.874454i \(-0.661220\pi\)
−0.485108 + 0.874454i \(0.661220\pi\)
\(432\) 0 0
\(433\) 30.8004 1.48017 0.740085 0.672513i \(-0.234785\pi\)
0.740085 + 0.672513i \(0.234785\pi\)
\(434\) 0 0
\(435\) 19.3291 0.926759
\(436\) 0 0
\(437\) 51.5165 2.46437
\(438\) 0 0
\(439\) −37.7595 −1.80216 −0.901082 0.433649i \(-0.857226\pi\)
−0.901082 + 0.433649i \(0.857226\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.877302 0.0416819 0.0208409 0.999783i \(-0.493366\pi\)
0.0208409 + 0.999783i \(0.493366\pi\)
\(444\) 0 0
\(445\) −0.0110135 −0.000522092 0
\(446\) 0 0
\(447\) −1.65279 −0.0781741
\(448\) 0 0
\(449\) 33.0313 1.55884 0.779421 0.626500i \(-0.215513\pi\)
0.779421 + 0.626500i \(0.215513\pi\)
\(450\) 0 0
\(451\) 0.458700 0.0215993
\(452\) 0 0
\(453\) 4.66062 0.218975
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 31.6690 1.48141 0.740707 0.671828i \(-0.234491\pi\)
0.740707 + 0.671828i \(0.234491\pi\)
\(458\) 0 0
\(459\) 5.56451 0.259729
\(460\) 0 0
\(461\) −29.9124 −1.39316 −0.696580 0.717479i \(-0.745296\pi\)
−0.696580 + 0.717479i \(0.745296\pi\)
\(462\) 0 0
\(463\) −15.8809 −0.738050 −0.369025 0.929419i \(-0.620308\pi\)
−0.369025 + 0.929419i \(0.620308\pi\)
\(464\) 0 0
\(465\) −11.0973 −0.514625
\(466\) 0 0
\(467\) −13.0111 −0.602081 −0.301041 0.953611i \(-0.597334\pi\)
−0.301041 + 0.953611i \(0.597334\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 18.9798 0.874543
\(472\) 0 0
\(473\) −0.631882 −0.0290540
\(474\) 0 0
\(475\) −10.1092 −0.463843
\(476\) 0 0
\(477\) 18.9853 0.869277
\(478\) 0 0
\(479\) −33.6377 −1.53695 −0.768473 0.639882i \(-0.778983\pi\)
−0.768473 + 0.639882i \(0.778983\pi\)
\(480\) 0 0
\(481\) 1.57527 0.0718262
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.75919 −0.397734
\(486\) 0 0
\(487\) 33.3793 1.51256 0.756281 0.654247i \(-0.227014\pi\)
0.756281 + 0.654247i \(0.227014\pi\)
\(488\) 0 0
\(489\) 11.5204 0.520969
\(490\) 0 0
\(491\) 1.61369 0.0728249 0.0364125 0.999337i \(-0.488407\pi\)
0.0364125 + 0.999337i \(0.488407\pi\)
\(492\) 0 0
\(493\) 8.61859 0.388162
\(494\) 0 0
\(495\) 0.791727 0.0355855
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 7.08060 0.316971 0.158486 0.987361i \(-0.449339\pi\)
0.158486 + 0.987361i \(0.449339\pi\)
\(500\) 0 0
\(501\) 11.6081 0.518611
\(502\) 0 0
\(503\) −17.2097 −0.767342 −0.383671 0.923470i \(-0.625340\pi\)
−0.383671 + 0.923470i \(0.625340\pi\)
\(504\) 0 0
\(505\) −15.4206 −0.686207
\(506\) 0 0
\(507\) −16.3784 −0.727390
\(508\) 0 0
\(509\) 0.569329 0.0252351 0.0126175 0.999920i \(-0.495984\pi\)
0.0126175 + 0.999920i \(0.495984\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −30.3768 −1.34117
\(514\) 0 0
\(515\) −7.24211 −0.319125
\(516\) 0 0
\(517\) −0.499435 −0.0219651
\(518\) 0 0
\(519\) −19.6591 −0.862941
\(520\) 0 0
\(521\) −4.96590 −0.217560 −0.108780 0.994066i \(-0.534694\pi\)
−0.108780 + 0.994066i \(0.534694\pi\)
\(522\) 0 0
\(523\) 10.5968 0.463364 0.231682 0.972792i \(-0.425577\pi\)
0.231682 + 0.972792i \(0.425577\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.94814 −0.215544
\(528\) 0 0
\(529\) 66.0559 2.87200
\(530\) 0 0
\(531\) −18.5209 −0.803738
\(532\) 0 0
\(533\) 0.296861 0.0128585
\(534\) 0 0
\(535\) −25.6564 −1.10922
\(536\) 0 0
\(537\) −10.2456 −0.442132
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.64750 −0.371785 −0.185892 0.982570i \(-0.559518\pi\)
−0.185892 + 0.982570i \(0.559518\pi\)
\(542\) 0 0
\(543\) 18.7289 0.803736
\(544\) 0 0
\(545\) −2.79219 −0.119604
\(546\) 0 0
\(547\) 27.9601 1.19549 0.597744 0.801687i \(-0.296064\pi\)
0.597744 + 0.801687i \(0.296064\pi\)
\(548\) 0 0
\(549\) −4.73088 −0.201909
\(550\) 0 0
\(551\) −47.0490 −2.00436
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −17.1554 −0.728208
\(556\) 0 0
\(557\) 23.4737 0.994613 0.497306 0.867575i \(-0.334322\pi\)
0.497306 + 0.867575i \(0.334322\pi\)
\(558\) 0 0
\(559\) −0.408941 −0.0172963
\(560\) 0 0
\(561\) −0.402209 −0.0169813
\(562\) 0 0
\(563\) 26.6059 1.12130 0.560652 0.828052i \(-0.310551\pi\)
0.560652 + 0.828052i \(0.310551\pi\)
\(564\) 0 0
\(565\) 1.47354 0.0619921
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.0713 1.17681 0.588405 0.808566i \(-0.299756\pi\)
0.588405 + 0.808566i \(0.299756\pi\)
\(570\) 0 0
\(571\) 39.1974 1.64036 0.820180 0.572105i \(-0.193873\pi\)
0.820180 + 0.572105i \(0.193873\pi\)
\(572\) 0 0
\(573\) −13.3762 −0.558799
\(574\) 0 0
\(575\) −17.4757 −0.728787
\(576\) 0 0
\(577\) 6.80895 0.283460 0.141730 0.989905i \(-0.454733\pi\)
0.141730 + 0.989905i \(0.454733\pi\)
\(578\) 0 0
\(579\) −25.9057 −1.07660
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.30804 −0.178421
\(584\) 0 0
\(585\) 0.512389 0.0211847
\(586\) 0 0
\(587\) −23.8016 −0.982397 −0.491198 0.871048i \(-0.663441\pi\)
−0.491198 + 0.871048i \(0.663441\pi\)
\(588\) 0 0
\(589\) 27.0120 1.11301
\(590\) 0 0
\(591\) −15.5773 −0.640766
\(592\) 0 0
\(593\) −30.2650 −1.24284 −0.621418 0.783479i \(-0.713443\pi\)
−0.621418 + 0.783479i \(0.713443\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.56793 −0.0641713
\(598\) 0 0
\(599\) −20.3977 −0.833427 −0.416714 0.909038i \(-0.636818\pi\)
−0.416714 + 0.909038i \(0.636818\pi\)
\(600\) 0 0
\(601\) −19.6591 −0.801912 −0.400956 0.916097i \(-0.631322\pi\)
−0.400956 + 0.916097i \(0.631322\pi\)
\(602\) 0 0
\(603\) −18.7177 −0.762244
\(604\) 0 0
\(605\) 19.3377 0.786190
\(606\) 0 0
\(607\) 0.670435 0.0272121 0.0136061 0.999907i \(-0.495669\pi\)
0.0136061 + 0.999907i \(0.495669\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.323224 −0.0130762
\(612\) 0 0
\(613\) 20.1900 0.815466 0.407733 0.913101i \(-0.366319\pi\)
0.407733 + 0.913101i \(0.366319\pi\)
\(614\) 0 0
\(615\) −3.23295 −0.130365
\(616\) 0 0
\(617\) 15.1299 0.609107 0.304554 0.952495i \(-0.401493\pi\)
0.304554 + 0.952495i \(0.401493\pi\)
\(618\) 0 0
\(619\) −23.6956 −0.952407 −0.476204 0.879335i \(-0.657987\pi\)
−0.476204 + 0.879335i \(0.657987\pi\)
\(620\) 0 0
\(621\) −52.5120 −2.10723
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.3115 −0.492460
\(626\) 0 0
\(627\) 2.19567 0.0876865
\(628\) 0 0
\(629\) −7.64938 −0.305001
\(630\) 0 0
\(631\) −0.879472 −0.0350112 −0.0175056 0.999847i \(-0.505572\pi\)
−0.0175056 + 0.999847i \(0.505572\pi\)
\(632\) 0 0
\(633\) 8.70235 0.345887
\(634\) 0 0
\(635\) 8.76168 0.347697
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.68624 0.145825
\(640\) 0 0
\(641\) −32.1852 −1.27124 −0.635620 0.772002i \(-0.719255\pi\)
−0.635620 + 0.772002i \(0.719255\pi\)
\(642\) 0 0
\(643\) 18.7957 0.741231 0.370615 0.928786i \(-0.379147\pi\)
0.370615 + 0.928786i \(0.379147\pi\)
\(644\) 0 0
\(645\) 4.45355 0.175358
\(646\) 0 0
\(647\) −44.7477 −1.75921 −0.879607 0.475701i \(-0.842194\pi\)
−0.879607 + 0.475701i \(0.842194\pi\)
\(648\) 0 0
\(649\) 4.20266 0.164969
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13.6090 −0.532560 −0.266280 0.963896i \(-0.585795\pi\)
−0.266280 + 0.963896i \(0.585795\pi\)
\(654\) 0 0
\(655\) −32.4565 −1.26818
\(656\) 0 0
\(657\) −5.42600 −0.211688
\(658\) 0 0
\(659\) 31.7616 1.23725 0.618627 0.785684i \(-0.287689\pi\)
0.618627 + 0.785684i \(0.287689\pi\)
\(660\) 0 0
\(661\) 20.4414 0.795078 0.397539 0.917585i \(-0.369864\pi\)
0.397539 + 0.917585i \(0.369864\pi\)
\(662\) 0 0
\(663\) −0.260301 −0.0101093
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −81.3331 −3.14923
\(668\) 0 0
\(669\) 17.0684 0.659901
\(670\) 0 0
\(671\) 1.07351 0.0414423
\(672\) 0 0
\(673\) −8.40756 −0.324087 −0.162044 0.986784i \(-0.551809\pi\)
−0.162044 + 0.986784i \(0.551809\pi\)
\(674\) 0 0
\(675\) 10.3046 0.396624
\(676\) 0 0
\(677\) 10.9126 0.419405 0.209703 0.977765i \(-0.432750\pi\)
0.209703 + 0.977765i \(0.432750\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −0.816259 −0.0312791
\(682\) 0 0
\(683\) 0.402969 0.0154192 0.00770959 0.999970i \(-0.497546\pi\)
0.00770959 + 0.999970i \(0.497546\pi\)
\(684\) 0 0
\(685\) −17.5625 −0.671030
\(686\) 0 0
\(687\) 22.5294 0.859552
\(688\) 0 0
\(689\) −2.78807 −0.106217
\(690\) 0 0
\(691\) 28.7597 1.09407 0.547036 0.837109i \(-0.315756\pi\)
0.547036 + 0.837109i \(0.315756\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 22.5242 0.854391
\(696\) 0 0
\(697\) −1.44153 −0.0546019
\(698\) 0 0
\(699\) −10.2359 −0.387157
\(700\) 0 0
\(701\) 30.4661 1.15069 0.575345 0.817911i \(-0.304868\pi\)
0.575345 + 0.817911i \(0.304868\pi\)
\(702\) 0 0
\(703\) 41.7581 1.57494
\(704\) 0 0
\(705\) 3.52005 0.132573
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −13.1020 −0.492057 −0.246029 0.969263i \(-0.579126\pi\)
−0.246029 + 0.969263i \(0.579126\pi\)
\(710\) 0 0
\(711\) −17.1535 −0.643305
\(712\) 0 0
\(713\) 46.6953 1.74875
\(714\) 0 0
\(715\) −0.116269 −0.00434820
\(716\) 0 0
\(717\) −3.94502 −0.147330
\(718\) 0 0
\(719\) 15.4948 0.577858 0.288929 0.957350i \(-0.406701\pi\)
0.288929 + 0.957350i \(0.406701\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 14.3294 0.532917
\(724\) 0 0
\(725\) 15.9602 0.592748
\(726\) 0 0
\(727\) −35.6680 −1.32285 −0.661427 0.750010i \(-0.730049\pi\)
−0.661427 + 0.750010i \(0.730049\pi\)
\(728\) 0 0
\(729\) 25.0644 0.928312
\(730\) 0 0
\(731\) 1.98578 0.0734467
\(732\) 0 0
\(733\) −12.5978 −0.465311 −0.232656 0.972559i \(-0.574742\pi\)
−0.232656 + 0.972559i \(0.574742\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.24733 0.156452
\(738\) 0 0
\(739\) 32.5853 1.19867 0.599334 0.800499i \(-0.295432\pi\)
0.599334 + 0.800499i \(0.295432\pi\)
\(740\) 0 0
\(741\) 1.42099 0.0522014
\(742\) 0 0
\(743\) −1.53311 −0.0562444 −0.0281222 0.999604i \(-0.508953\pi\)
−0.0281222 + 0.999604i \(0.508953\pi\)
\(744\) 0 0
\(745\) 2.32005 0.0850001
\(746\) 0 0
\(747\) −14.7123 −0.538295
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −24.5508 −0.895871 −0.447935 0.894066i \(-0.647841\pi\)
−0.447935 + 0.894066i \(0.647841\pi\)
\(752\) 0 0
\(753\) 4.52291 0.164824
\(754\) 0 0
\(755\) −6.54221 −0.238095
\(756\) 0 0
\(757\) 30.8744 1.12215 0.561075 0.827765i \(-0.310388\pi\)
0.561075 + 0.827765i \(0.310388\pi\)
\(758\) 0 0
\(759\) 3.79563 0.137773
\(760\) 0 0
\(761\) −26.8426 −0.973045 −0.486522 0.873668i \(-0.661735\pi\)
−0.486522 + 0.873668i \(0.661735\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.48812 −0.0899580
\(766\) 0 0
\(767\) 2.71988 0.0982090
\(768\) 0 0
\(769\) −52.2087 −1.88269 −0.941347 0.337440i \(-0.890439\pi\)
−0.941347 + 0.337440i \(0.890439\pi\)
\(770\) 0 0
\(771\) −11.4570 −0.412615
\(772\) 0 0
\(773\) 15.3808 0.553207 0.276604 0.960984i \(-0.410791\pi\)
0.276604 + 0.960984i \(0.410791\pi\)
\(774\) 0 0
\(775\) −9.16316 −0.329150
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.86934 0.281949
\(780\) 0 0
\(781\) −0.836462 −0.0299309
\(782\) 0 0
\(783\) 47.9582 1.71389
\(784\) 0 0
\(785\) −26.6424 −0.950907
\(786\) 0 0
\(787\) 25.6237 0.913387 0.456693 0.889624i \(-0.349034\pi\)
0.456693 + 0.889624i \(0.349034\pi\)
\(788\) 0 0
\(789\) 6.23491 0.221969
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.694751 0.0246713
\(794\) 0 0
\(795\) 30.3634 1.07688
\(796\) 0 0
\(797\) −15.4806 −0.548350 −0.274175 0.961680i \(-0.588405\pi\)
−0.274175 + 0.961680i \(0.588405\pi\)
\(798\) 0 0
\(799\) 1.56955 0.0555265
\(800\) 0 0
\(801\) −0.00870444 −0.000307556 0
\(802\) 0 0
\(803\) 1.23124 0.0434495
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −21.8257 −0.768302
\(808\) 0 0
\(809\) −38.5138 −1.35407 −0.677036 0.735950i \(-0.736736\pi\)
−0.677036 + 0.735950i \(0.736736\pi\)
\(810\) 0 0
\(811\) −41.6065 −1.46100 −0.730502 0.682911i \(-0.760714\pi\)
−0.730502 + 0.682911i \(0.760714\pi\)
\(812\) 0 0
\(813\) 37.1941 1.30445
\(814\) 0 0
\(815\) −16.1714 −0.566459
\(816\) 0 0
\(817\) −10.8404 −0.379258
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 20.4853 0.714943 0.357472 0.933924i \(-0.383639\pi\)
0.357472 + 0.933924i \(0.383639\pi\)
\(822\) 0 0
\(823\) 17.1271 0.597014 0.298507 0.954407i \(-0.403511\pi\)
0.298507 + 0.954407i \(0.403511\pi\)
\(824\) 0 0
\(825\) −0.744827 −0.0259315
\(826\) 0 0
\(827\) −50.9089 −1.77028 −0.885138 0.465329i \(-0.845936\pi\)
−0.885138 + 0.465329i \(0.845936\pi\)
\(828\) 0 0
\(829\) −37.5712 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(830\) 0 0
\(831\) −37.2671 −1.29278
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −16.2945 −0.563896
\(836\) 0 0
\(837\) −27.5340 −0.951714
\(838\) 0 0
\(839\) 33.2696 1.14860 0.574298 0.818646i \(-0.305275\pi\)
0.574298 + 0.818646i \(0.305275\pi\)
\(840\) 0 0
\(841\) 45.2800 1.56138
\(842\) 0 0
\(843\) 6.97146 0.240110
\(844\) 0 0
\(845\) 22.9907 0.790905
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9.43423 −0.323782
\(850\) 0 0
\(851\) 72.1868 2.47453
\(852\) 0 0
\(853\) −30.3159 −1.03800 −0.518998 0.854776i \(-0.673695\pi\)
−0.518998 + 0.854776i \(0.673695\pi\)
\(854\) 0 0
\(855\) 13.5827 0.464517
\(856\) 0 0
\(857\) −38.7778 −1.32463 −0.662313 0.749228i \(-0.730425\pi\)
−0.662313 + 0.749228i \(0.730425\pi\)
\(858\) 0 0
\(859\) 36.4346 1.24313 0.621566 0.783362i \(-0.286497\pi\)
0.621566 + 0.783362i \(0.286497\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −41.0825 −1.39847 −0.699233 0.714894i \(-0.746475\pi\)
−0.699233 + 0.714894i \(0.746475\pi\)
\(864\) 0 0
\(865\) 27.5960 0.938292
\(866\) 0 0
\(867\) 1.26400 0.0429277
\(868\) 0 0
\(869\) 3.89238 0.132040
\(870\) 0 0
\(871\) 2.74878 0.0931388
\(872\) 0 0
\(873\) −6.92273 −0.234299
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.12907 0.206964 0.103482 0.994631i \(-0.467002\pi\)
0.103482 + 0.994631i \(0.467002\pi\)
\(878\) 0 0
\(879\) −26.1253 −0.881183
\(880\) 0 0
\(881\) −32.4381 −1.09287 −0.546434 0.837502i \(-0.684015\pi\)
−0.546434 + 0.837502i \(0.684015\pi\)
\(882\) 0 0
\(883\) −35.1411 −1.18259 −0.591296 0.806454i \(-0.701384\pi\)
−0.591296 + 0.806454i \(0.701384\pi\)
\(884\) 0 0
\(885\) −29.6207 −0.995689
\(886\) 0 0
\(887\) −26.9435 −0.904673 −0.452336 0.891847i \(-0.649409\pi\)
−0.452336 + 0.891847i \(0.649409\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.899445 −0.0301325
\(892\) 0 0
\(893\) −8.56818 −0.286723
\(894\) 0 0
\(895\) 14.3820 0.480738
\(896\) 0 0
\(897\) 2.45645 0.0820184
\(898\) 0 0
\(899\) −42.6460 −1.42232
\(900\) 0 0
\(901\) 13.5386 0.451038
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −26.2903 −0.873918
\(906\) 0 0
\(907\) −23.1647 −0.769170 −0.384585 0.923090i \(-0.625655\pi\)
−0.384585 + 0.923090i \(0.625655\pi\)
\(908\) 0 0
\(909\) −12.1875 −0.404234
\(910\) 0 0
\(911\) −28.5642 −0.946376 −0.473188 0.880962i \(-0.656897\pi\)
−0.473188 + 0.880962i \(0.656897\pi\)
\(912\) 0 0
\(913\) 3.33844 0.110486
\(914\) 0 0
\(915\) −7.56616 −0.250129
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −29.4563 −0.971673 −0.485836 0.874050i \(-0.661485\pi\)
−0.485836 + 0.874050i \(0.661485\pi\)
\(920\) 0 0
\(921\) −17.7670 −0.585442
\(922\) 0 0
\(923\) −0.541340 −0.0178184
\(924\) 0 0
\(925\) −14.1654 −0.465756
\(926\) 0 0
\(927\) −5.72372 −0.187992
\(928\) 0 0
\(929\) −6.91683 −0.226934 −0.113467 0.993542i \(-0.536196\pi\)
−0.113467 + 0.993542i \(0.536196\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −6.01797 −0.197020
\(934\) 0 0
\(935\) 0.564590 0.0184641
\(936\) 0 0
\(937\) 39.1000 1.27734 0.638671 0.769480i \(-0.279484\pi\)
0.638671 + 0.769480i \(0.279484\pi\)
\(938\) 0 0
\(939\) −34.0138 −1.11000
\(940\) 0 0
\(941\) 19.8018 0.645522 0.322761 0.946481i \(-0.395389\pi\)
0.322761 + 0.946481i \(0.395389\pi\)
\(942\) 0 0
\(943\) 13.6036 0.442996
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6.95286 −0.225938 −0.112969 0.993599i \(-0.536036\pi\)
−0.112969 + 0.993599i \(0.536036\pi\)
\(948\) 0 0
\(949\) 0.796833 0.0258663
\(950\) 0 0
\(951\) −25.6820 −0.832795
\(952\) 0 0
\(953\) 44.2489 1.43336 0.716681 0.697402i \(-0.245661\pi\)
0.716681 + 0.697402i \(0.245661\pi\)
\(954\) 0 0
\(955\) 18.7765 0.607593
\(956\) 0 0
\(957\) −3.46648 −0.112055
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.51590 −0.210190
\(962\) 0 0
\(963\) −20.2773 −0.653427
\(964\) 0 0
\(965\) 36.3644 1.17061
\(966\) 0 0
\(967\) 3.70560 0.119164 0.0595821 0.998223i \(-0.481023\pi\)
0.0595821 + 0.998223i \(0.481023\pi\)
\(968\) 0 0
\(969\) −6.90020 −0.221666
\(970\) 0 0
\(971\) 7.59954 0.243881 0.121940 0.992537i \(-0.461088\pi\)
0.121940 + 0.992537i \(0.461088\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −0.482036 −0.0154375
\(976\) 0 0
\(977\) −37.4738 −1.19889 −0.599446 0.800415i \(-0.704612\pi\)
−0.599446 + 0.800415i \(0.704612\pi\)
\(978\) 0 0
\(979\) 0.00197517 6.31266e−5 0
\(980\) 0 0
\(981\) −2.20678 −0.0704569
\(982\) 0 0
\(983\) 20.6504 0.658647 0.329323 0.944217i \(-0.393179\pi\)
0.329323 + 0.944217i \(0.393179\pi\)
\(984\) 0 0
\(985\) 21.8663 0.696717
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −18.7397 −0.595888
\(990\) 0 0
\(991\) −17.2648 −0.548435 −0.274218 0.961668i \(-0.588419\pi\)
−0.274218 + 0.961668i \(0.588419\pi\)
\(992\) 0 0
\(993\) 1.63330 0.0518311
\(994\) 0 0
\(995\) 2.20095 0.0697747
\(996\) 0 0
\(997\) 13.6573 0.432530 0.216265 0.976335i \(-0.430613\pi\)
0.216265 + 0.976335i \(0.430613\pi\)
\(998\) 0 0
\(999\) −42.5651 −1.34670
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 3332.2.a.u.1.5 yes 8
7.6 odd 2 3332.2.a.t.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.2.a.t.1.4 8 7.6 odd 2
3332.2.a.u.1.5 yes 8 1.1 even 1 trivial