Properties

Label 3024.2.k.l.1889.16
Level $3024$
Weight $2$
Character 3024.1889
Analytic conductor $24.147$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(1889,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.k (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} - 45 x^{12} + 306 x^{11} - 378 x^{10} + 1704 x^{9} + 917 x^{8} - 12522 x^{7} + 27090 x^{6} - 35292 x^{5} + 30948 x^{4} - 18000 x^{3} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.16
Root \(0.121385 + 0.453016i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.l.1889.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.29876 q^{5} +(-0.832811 + 2.51126i) q^{7} +O(q^{10})\) \(q+4.29876 q^{5} +(-0.832811 + 2.51126i) q^{7} +2.51682i q^{11} +5.36755i q^{13} +3.29047 q^{17} +0.663261i q^{19} -6.44566i q^{23} +13.4793 q^{25} -4.26322i q^{29} +4.78838i q^{31} +(-3.58005 + 10.7953i) q^{35} -0.746404 q^{37} -7.99475 q^{41} -7.69926 q^{43} +9.32128 q^{47} +(-5.61285 - 4.18281i) q^{49} +11.8137i q^{53} +10.8192i q^{55} +1.15289 q^{59} -10.7083i q^{61} +23.0738i q^{65} -0.587975 q^{67} +9.77691i q^{71} +12.6408i q^{73} +(-6.32039 - 2.09603i) q^{77} -11.1113 q^{79} +7.07281 q^{83} +14.1449 q^{85} -3.25676 q^{89} +(-13.4793 - 4.47015i) q^{91} +2.85120i q^{95} -10.8192i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{7} + 36 q^{25} - 8 q^{37} - 20 q^{43} + 2 q^{49} - 44 q^{67} - 40 q^{79} + 16 q^{85} - 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) 4.29876 1.92246 0.961231 0.275744i \(-0.0889243\pi\)
0.961231 + 0.275744i \(0.0889243\pi\)
\(6\) 0 0
\(7\) −0.832811 + 2.51126i −0.314773 + 0.949167i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.51682i 0.758850i 0.925223 + 0.379425i \(0.123878\pi\)
−0.925223 + 0.379425i \(0.876122\pi\)
\(12\) 0 0
\(13\) 5.36755i 1.48869i 0.667796 + 0.744345i \(0.267238\pi\)
−0.667796 + 0.744345i \(0.732762\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.29047 0.798056 0.399028 0.916939i \(-0.369348\pi\)
0.399028 + 0.916939i \(0.369348\pi\)
\(18\) 0 0
\(19\) 0.663261i 0.152163i 0.997102 + 0.0760813i \(0.0242408\pi\)
−0.997102 + 0.0760813i \(0.975759\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.44566i 1.34401i −0.740545 0.672007i \(-0.765433\pi\)
0.740545 0.672007i \(-0.234567\pi\)
\(24\) 0 0
\(25\) 13.4793 2.69586
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.26322i 0.791661i −0.918324 0.395830i \(-0.870457\pi\)
0.918324 0.395830i \(-0.129543\pi\)
\(30\) 0 0
\(31\) 4.78838i 0.860019i 0.902824 + 0.430010i \(0.141490\pi\)
−0.902824 + 0.430010i \(0.858510\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.58005 + 10.7953i −0.605139 + 1.82474i
\(36\) 0 0
\(37\) −0.746404 −0.122708 −0.0613540 0.998116i \(-0.519542\pi\)
−0.0613540 + 0.998116i \(0.519542\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.99475 −1.24857 −0.624285 0.781196i \(-0.714610\pi\)
−0.624285 + 0.781196i \(0.714610\pi\)
\(42\) 0 0
\(43\) −7.69926 −1.17413 −0.587063 0.809541i \(-0.699716\pi\)
−0.587063 + 0.809541i \(0.699716\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.32128 1.35965 0.679824 0.733375i \(-0.262056\pi\)
0.679824 + 0.733375i \(0.262056\pi\)
\(48\) 0 0
\(49\) −5.61285 4.18281i −0.801836 0.597544i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.8137i 1.62273i 0.584537 + 0.811367i \(0.301276\pi\)
−0.584537 + 0.811367i \(0.698724\pi\)
\(54\) 0 0
\(55\) 10.8192i 1.45886i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.15289 0.150093 0.0750467 0.997180i \(-0.476089\pi\)
0.0750467 + 0.997180i \(0.476089\pi\)
\(60\) 0 0
\(61\) 10.7083i 1.37106i −0.728046 0.685529i \(-0.759571\pi\)
0.728046 0.685529i \(-0.240429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 23.0738i 2.86195i
\(66\) 0 0
\(67\) −0.587975 −0.0718326 −0.0359163 0.999355i \(-0.511435\pi\)
−0.0359163 + 0.999355i \(0.511435\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.77691i 1.16031i 0.814508 + 0.580153i \(0.197007\pi\)
−0.814508 + 0.580153i \(0.802993\pi\)
\(72\) 0 0
\(73\) 12.6408i 1.47949i 0.672887 + 0.739745i \(0.265054\pi\)
−0.672887 + 0.739745i \(0.734946\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.32039 2.09603i −0.720275 0.238865i
\(78\) 0 0
\(79\) −11.1113 −1.25012 −0.625059 0.780578i \(-0.714925\pi\)
−0.625059 + 0.780578i \(0.714925\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.07281 0.776341 0.388171 0.921588i \(-0.373107\pi\)
0.388171 + 0.921588i \(0.373107\pi\)
\(84\) 0 0
\(85\) 14.1449 1.53423
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.25676 −0.345215 −0.172608 0.984991i \(-0.555219\pi\)
−0.172608 + 0.984991i \(0.555219\pi\)
\(90\) 0 0
\(91\) −13.4793 4.47015i −1.41301 0.468599i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.85120i 0.292527i
\(96\) 0 0
\(97\) 10.8192i 1.09852i −0.835651 0.549261i \(-0.814909\pi\)
0.835651 0.549261i \(-0.185091\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.16010 −0.712456 −0.356228 0.934399i \(-0.615937\pi\)
−0.356228 + 0.934399i \(0.615937\pi\)
\(102\) 0 0
\(103\) 2.22392i 0.219129i 0.993980 + 0.109565i \(0.0349457\pi\)
−0.993980 + 0.109565i \(0.965054\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.860825i 0.0832191i −0.999134 0.0416095i \(-0.986751\pi\)
0.999134 0.0416095i \(-0.0132486\pi\)
\(108\) 0 0
\(109\) 6.07765 0.582133 0.291066 0.956703i \(-0.405990\pi\)
0.291066 + 0.956703i \(0.405990\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.7425i 1.10464i −0.833631 0.552322i \(-0.813742\pi\)
0.833631 0.552322i \(-0.186258\pi\)
\(114\) 0 0
\(115\) 27.7083i 2.58382i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.74034 + 8.26322i −0.251206 + 0.757488i
\(120\) 0 0
\(121\) 4.66562 0.424147
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 36.4504 3.26023
\(126\) 0 0
\(127\) 7.44566 0.660696 0.330348 0.943859i \(-0.392834\pi\)
0.330348 + 0.943859i \(0.392834\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.81934 −0.158957 −0.0794784 0.996837i \(-0.525325\pi\)
−0.0794784 + 0.996837i \(0.525325\pi\)
\(132\) 0 0
\(133\) −1.66562 0.552371i −0.144428 0.0478966i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.51368i 0.812809i 0.913693 + 0.406404i \(0.133218\pi\)
−0.913693 + 0.406404i \(0.866782\pi\)
\(138\) 0 0
\(139\) 19.8749i 1.68576i −0.538099 0.842882i \(-0.680857\pi\)
0.538099 0.842882i \(-0.319143\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −13.5091 −1.12969
\(144\) 0 0
\(145\) 18.3266i 1.52194i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.29373i 0.679448i 0.940525 + 0.339724i \(0.110334\pi\)
−0.940525 + 0.339724i \(0.889666\pi\)
\(150\) 0 0
\(151\) 21.0057 1.70942 0.854712 0.519103i \(-0.173734\pi\)
0.854712 + 0.519103i \(0.173734\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 20.5841i 1.65335i
\(156\) 0 0
\(157\) 8.70616i 0.694827i −0.937712 0.347414i \(-0.887060\pi\)
0.937712 0.347414i \(-0.112940\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.1867 + 5.36802i 1.27569 + 0.423059i
\(162\) 0 0
\(163\) −21.2593 −1.66516 −0.832580 0.553905i \(-0.813137\pi\)
−0.832580 + 0.553905i \(0.813137\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.66854 0.748174 0.374087 0.927394i \(-0.377956\pi\)
0.374087 + 0.927394i \(0.377956\pi\)
\(168\) 0 0
\(169\) −15.8105 −1.21620
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.43741 0.109284 0.0546422 0.998506i \(-0.482598\pi\)
0.0546422 + 0.998506i \(0.482598\pi\)
\(174\) 0 0
\(175\) −11.2257 + 33.8500i −0.848584 + 2.55882i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.2129i 1.36130i 0.732609 + 0.680650i \(0.238302\pi\)
−0.732609 + 0.680650i \(0.761698\pi\)
\(180\) 0 0
\(181\) 1.11918i 0.0831878i −0.999135 0.0415939i \(-0.986756\pi\)
0.999135 0.0415939i \(-0.0132436\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.20861 −0.235902
\(186\) 0 0
\(187\) 8.28152i 0.605604i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.2976i 0.745108i −0.928010 0.372554i \(-0.878482\pi\)
0.928010 0.372554i \(-0.121518\pi\)
\(192\) 0 0
\(193\) 11.2226 0.807818 0.403909 0.914799i \(-0.367651\pi\)
0.403909 + 0.914799i \(0.367651\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.18558i 0.226963i 0.993540 + 0.113481i \(0.0362003\pi\)
−0.993540 + 0.113481i \(0.963800\pi\)
\(198\) 0 0
\(199\) 10.5895i 0.750672i −0.926889 0.375336i \(-0.877527\pi\)
0.926889 0.375336i \(-0.122473\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.7061 + 3.55046i 0.751418 + 0.249193i
\(204\) 0 0
\(205\) −34.3675 −2.40033
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.66931 −0.115468
\(210\) 0 0
\(211\) 14.8105 1.01960 0.509800 0.860293i \(-0.329719\pi\)
0.509800 + 0.860293i \(0.329719\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −33.0972 −2.25721
\(216\) 0 0
\(217\) −12.0249 3.98782i −0.816302 0.270711i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 17.6617i 1.18806i
\(222\) 0 0
\(223\) 14.2934i 0.957157i −0.878045 0.478579i \(-0.841152\pi\)
0.878045 0.478579i \(-0.158848\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.5910 0.902066 0.451033 0.892507i \(-0.351056\pi\)
0.451033 + 0.892507i \(0.351056\pi\)
\(228\) 0 0
\(229\) 14.7102i 0.972076i 0.873938 + 0.486038i \(0.161558\pi\)
−0.873938 + 0.486038i \(0.838442\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.553594i 0.0362671i 0.999836 + 0.0181336i \(0.00577241\pi\)
−0.999836 + 0.0181336i \(0.994228\pi\)
\(234\) 0 0
\(235\) 40.0699 2.61387
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.2161i 0.984246i −0.870526 0.492123i \(-0.836221\pi\)
0.870526 0.492123i \(-0.163779\pi\)
\(240\) 0 0
\(241\) 5.80115i 0.373685i 0.982390 + 0.186842i \(0.0598254\pi\)
−0.982390 + 0.186842i \(0.940175\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −24.1283 17.9809i −1.54150 1.14876i
\(246\) 0 0
\(247\) −3.56008 −0.226523
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.5884 1.42577 0.712884 0.701282i \(-0.247389\pi\)
0.712884 + 0.701282i \(0.247389\pi\)
\(252\) 0 0
\(253\) 16.2226 1.01990
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.48068 0.341875 0.170938 0.985282i \(-0.445320\pi\)
0.170938 + 0.985282i \(0.445320\pi\)
\(258\) 0 0
\(259\) 0.621613 1.87441i 0.0386252 0.116470i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.6954i 1.64611i −0.567963 0.823054i \(-0.692268\pi\)
0.567963 0.823054i \(-0.307732\pi\)
\(264\) 0 0
\(265\) 50.7841i 3.11964i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0113 −0.610401 −0.305201 0.952288i \(-0.598724\pi\)
−0.305201 + 0.952288i \(0.598724\pi\)
\(270\) 0 0
\(271\) 11.5216i 0.699887i 0.936771 + 0.349944i \(0.113799\pi\)
−0.936771 + 0.349944i \(0.886201\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 33.9250i 2.04575i
\(276\) 0 0
\(277\) −3.91608 −0.235295 −0.117647 0.993055i \(-0.537535\pi\)
−0.117647 + 0.993055i \(0.537535\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 23.3235i 1.39136i 0.718351 + 0.695681i \(0.244897\pi\)
−0.718351 + 0.695681i \(0.755103\pi\)
\(282\) 0 0
\(283\) 2.67536i 0.159033i 0.996834 + 0.0795167i \(0.0253377\pi\)
−0.996834 + 0.0795167i \(0.974662\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.65812 20.0769i 0.393016 1.18510i
\(288\) 0 0
\(289\) −6.17281 −0.363107
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.67271 0.389824 0.194912 0.980821i \(-0.437558\pi\)
0.194912 + 0.980821i \(0.437558\pi\)
\(294\) 0 0
\(295\) 4.95599 0.288549
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 34.5974 2.00082
\(300\) 0 0
\(301\) 6.41203 19.3348i 0.369583 1.11444i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 46.0324i 2.63581i
\(306\) 0 0
\(307\) 26.6036i 1.51835i 0.650888 + 0.759174i \(0.274397\pi\)
−0.650888 + 0.759174i \(0.725603\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.18756 0.0673401 0.0336701 0.999433i \(-0.489280\pi\)
0.0336701 + 0.999433i \(0.489280\pi\)
\(312\) 0 0
\(313\) 3.70048i 0.209163i −0.994516 0.104582i \(-0.966650\pi\)
0.994516 0.104582i \(-0.0333504\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.1170i 0.568229i −0.958790 0.284115i \(-0.908300\pi\)
0.958790 0.284115i \(-0.0916996\pi\)
\(318\) 0 0
\(319\) 10.7298 0.600751
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.18244i 0.121434i
\(324\) 0 0
\(325\) 72.3508i 4.01330i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.76286 + 23.4081i −0.427980 + 1.29053i
\(330\) 0 0
\(331\) 5.03677 0.276846 0.138423 0.990373i \(-0.455797\pi\)
0.138423 + 0.990373i \(0.455797\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.52756 −0.138095
\(336\) 0 0
\(337\) −26.2930 −1.43227 −0.716135 0.697962i \(-0.754091\pi\)
−0.716135 + 0.697962i \(0.754091\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0515 −0.652625
\(342\) 0 0
\(343\) 15.1786 10.6118i 0.819565 0.572986i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.8817i 0.798892i 0.916757 + 0.399446i \(0.130797\pi\)
−0.916757 + 0.399446i \(0.869203\pi\)
\(348\) 0 0
\(349\) 22.1413i 1.18520i −0.805498 0.592599i \(-0.798102\pi\)
0.805498 0.592599i \(-0.201898\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11.2024 −0.596244 −0.298122 0.954528i \(-0.596360\pi\)
−0.298122 + 0.954528i \(0.596360\pi\)
\(354\) 0 0
\(355\) 42.0285i 2.23064i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.90483i 0.469979i 0.971998 + 0.234990i \(0.0755056\pi\)
−0.971998 + 0.234990i \(0.924494\pi\)
\(360\) 0 0
\(361\) 18.5601 0.976847
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 54.3396i 2.84426i
\(366\) 0 0
\(367\) 38.2979i 1.99913i −0.0294506 0.999566i \(-0.509376\pi\)
0.0294506 0.999566i \(-0.490624\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −29.6672 9.83856i −1.54025 0.510793i
\(372\) 0 0
\(373\) −34.5963 −1.79133 −0.895665 0.444729i \(-0.853300\pi\)
−0.895665 + 0.444729i \(0.853300\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 22.8830 1.17854
\(378\) 0 0
\(379\) 13.6185 0.699534 0.349767 0.936837i \(-0.386261\pi\)
0.349767 + 0.936837i \(0.386261\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 37.0832 1.89486 0.947431 0.319959i \(-0.103669\pi\)
0.947431 + 0.319959i \(0.103669\pi\)
\(384\) 0 0
\(385\) −27.1698 9.01034i −1.38470 0.459209i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.9490i 0.656539i −0.944584 0.328270i \(-0.893535\pi\)
0.944584 0.328270i \(-0.106465\pi\)
\(390\) 0 0
\(391\) 21.2093i 1.07260i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −47.7647 −2.40330
\(396\) 0 0
\(397\) 10.9614i 0.550134i −0.961425 0.275067i \(-0.911300\pi\)
0.961425 0.275067i \(-0.0887001\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 37.2562i 1.86049i −0.366944 0.930243i \(-0.619596\pi\)
0.366944 0.930243i \(-0.380404\pi\)
\(402\) 0 0
\(403\) −25.7019 −1.28030
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.87856i 0.0931169i
\(408\) 0 0
\(409\) 26.2173i 1.29636i 0.761485 + 0.648182i \(0.224470\pi\)
−0.761485 + 0.648182i \(0.775530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.960139 + 2.89521i −0.0472453 + 0.142464i
\(414\) 0 0
\(415\) 30.4043 1.49249
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.87161 −0.286847 −0.143423 0.989661i \(-0.545811\pi\)
−0.143423 + 0.989661i \(0.545811\pi\)
\(420\) 0 0
\(421\) 33.1874 1.61746 0.808729 0.588182i \(-0.200156\pi\)
0.808729 + 0.588182i \(0.200156\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 44.3532 2.15145
\(426\) 0 0
\(427\) 26.8913 + 8.91799i 1.30136 + 0.431572i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.3985i 0.838057i −0.907973 0.419029i \(-0.862371\pi\)
0.907973 0.419029i \(-0.137629\pi\)
\(432\) 0 0
\(433\) 29.8045i 1.43231i −0.697939 0.716157i \(-0.745899\pi\)
0.697939 0.716157i \(-0.254101\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.27516 0.204509
\(438\) 0 0
\(439\) 0.950228i 0.0453519i −0.999743 0.0226759i \(-0.992781\pi\)
0.999743 0.0226759i \(-0.00721860\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.926232i 0.0440066i −0.999758 0.0220033i \(-0.992996\pi\)
0.999758 0.0220033i \(-0.00700444\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.31037i 0.0618401i 0.999522 + 0.0309200i \(0.00984372\pi\)
−0.999522 + 0.0309200i \(0.990156\pi\)
\(450\) 0 0
\(451\) 20.1213i 0.947477i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −57.9442 19.2161i −2.71647 0.900864i
\(456\) 0 0
\(457\) 37.5963 1.75868 0.879341 0.476192i \(-0.157983\pi\)
0.879341 + 0.476192i \(0.157983\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −21.6103 −1.00649 −0.503246 0.864143i \(-0.667861\pi\)
−0.503246 + 0.864143i \(0.667861\pi\)
\(462\) 0 0
\(463\) −5.29037 −0.245864 −0.122932 0.992415i \(-0.539230\pi\)
−0.122932 + 0.992415i \(0.539230\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −24.9242 −1.15335 −0.576676 0.816973i \(-0.695651\pi\)
−0.576676 + 0.816973i \(0.695651\pi\)
\(468\) 0 0
\(469\) 0.489672 1.47656i 0.0226109 0.0681811i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 19.3776i 0.890985i
\(474\) 0 0
\(475\) 8.94030i 0.410209i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −21.4503 −0.980090 −0.490045 0.871697i \(-0.663020\pi\)
−0.490045 + 0.871697i \(0.663020\pi\)
\(480\) 0 0
\(481\) 4.00636i 0.182674i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 46.5091i 2.11187i
\(486\) 0 0
\(487\) −23.6946 −1.07371 −0.536853 0.843676i \(-0.680387\pi\)
−0.536853 + 0.843676i \(0.680387\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.91534i 0.176697i 0.996090 + 0.0883484i \(0.0281589\pi\)
−0.996090 + 0.0883484i \(0.971841\pi\)
\(492\) 0 0
\(493\) 14.0280i 0.631790i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.5524 8.14231i −1.10132 0.365233i
\(498\) 0 0
\(499\) 28.5098 1.27627 0.638137 0.769923i \(-0.279705\pi\)
0.638137 + 0.769923i \(0.279705\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 23.6995 1.05671 0.528355 0.849024i \(-0.322809\pi\)
0.528355 + 0.849024i \(0.322809\pi\)
\(504\) 0 0
\(505\) −30.7795 −1.36967
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.18192 −0.0523877 −0.0261938 0.999657i \(-0.508339\pi\)
−0.0261938 + 0.999657i \(0.508339\pi\)
\(510\) 0 0
\(511\) −31.7443 10.5274i −1.40428 0.465703i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.56008i 0.421268i
\(516\) 0 0
\(517\) 23.4600i 1.03177i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −31.0033 −1.35828 −0.679139 0.734009i \(-0.737647\pi\)
−0.679139 + 0.734009i \(0.737647\pi\)
\(522\) 0 0
\(523\) 13.8296i 0.604727i 0.953193 + 0.302363i \(0.0977756\pi\)
−0.953193 + 0.302363i \(0.902224\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.7560i 0.686343i
\(528\) 0 0
\(529\) −18.5466 −0.806373
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 42.9122i 1.85873i
\(534\) 0 0
\(535\) 3.70048i 0.159986i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.5274 14.1265i 0.453446 0.608473i
\(540\) 0 0
\(541\) −38.6026 −1.65966 −0.829828 0.558019i \(-0.811561\pi\)
−0.829828 + 0.558019i \(0.811561\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 26.1263 1.11913
\(546\) 0 0
\(547\) 12.0089 0.513463 0.256731 0.966483i \(-0.417354\pi\)
0.256731 + 0.966483i \(0.417354\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.82763 0.120461
\(552\) 0 0
\(553\) 9.25360 27.9033i 0.393503 1.18657i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.8059i 1.34766i 0.738886 + 0.673830i \(0.235352\pi\)
−0.738886 + 0.673830i \(0.764648\pi\)
\(558\) 0 0
\(559\) 41.3261i 1.74791i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.38005 0.353177 0.176589 0.984285i \(-0.443494\pi\)
0.176589 + 0.984285i \(0.443494\pi\)
\(564\) 0 0
\(565\) 50.4782i 2.12364i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.1087i 0.633389i 0.948528 + 0.316694i \(0.102573\pi\)
−0.948528 + 0.316694i \(0.897427\pi\)
\(570\) 0 0
\(571\) −14.3007 −0.598467 −0.299234 0.954180i \(-0.596731\pi\)
−0.299234 + 0.954180i \(0.596731\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 86.8830i 3.62327i
\(576\) 0 0
\(577\) 11.3671i 0.473218i −0.971605 0.236609i \(-0.923964\pi\)
0.971605 0.236609i \(-0.0760360\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.89031 + 17.7617i −0.244371 + 0.736878i
\(582\) 0 0
\(583\) −29.7329 −1.23141
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.8647 0.613532 0.306766 0.951785i \(-0.400753\pi\)
0.306766 + 0.951785i \(0.400753\pi\)
\(588\) 0 0
\(589\) −3.17595 −0.130863
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −24.3207 −0.998730 −0.499365 0.866392i \(-0.666433\pi\)
−0.499365 + 0.866392i \(0.666433\pi\)
\(594\) 0 0
\(595\) −11.7800 + 35.5216i −0.482935 + 1.45624i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13.8041i 0.564018i −0.959412 0.282009i \(-0.908999\pi\)
0.959412 0.282009i \(-0.0910009\pi\)
\(600\) 0 0
\(601\) 1.70621i 0.0695979i 0.999394 + 0.0347990i \(0.0110791\pi\)
−0.999394 + 0.0347990i \(0.988921\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 20.0564 0.815407
\(606\) 0 0
\(607\) 27.0595i 1.09831i −0.835720 0.549156i \(-0.814949\pi\)
0.835720 0.549156i \(-0.185051\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 50.0324i 2.02409i
\(612\) 0 0
\(613\) 12.3456 0.498635 0.249318 0.968422i \(-0.419794\pi\)
0.249318 + 0.968422i \(0.419794\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.5875i 0.506752i −0.967368 0.253376i \(-0.918459\pi\)
0.967368 0.253376i \(-0.0815409\pi\)
\(618\) 0 0
\(619\) 12.5031i 0.502542i −0.967917 0.251271i \(-0.919152\pi\)
0.967917 0.251271i \(-0.0808485\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.71226 8.17856i 0.108664 0.327667i
\(624\) 0 0
\(625\) 89.2951 3.57180
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.45602 −0.0979279
\(630\) 0 0
\(631\) −46.4715 −1.85000 −0.925001 0.379964i \(-0.875937\pi\)
−0.925001 + 0.379964i \(0.875937\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 32.0071 1.27016
\(636\) 0 0
\(637\) 22.4514 30.1272i 0.889557 1.19368i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.81129i 0.150537i −0.997163 0.0752684i \(-0.976019\pi\)
0.997163 0.0752684i \(-0.0239814\pi\)
\(642\) 0 0
\(643\) 1.84837i 0.0728927i −0.999336 0.0364464i \(-0.988396\pi\)
0.999336 0.0364464i \(-0.0116038\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.9174 0.783034 0.391517 0.920171i \(-0.371950\pi\)
0.391517 + 0.920171i \(0.371950\pi\)
\(648\) 0 0
\(649\) 2.90162i 0.113898i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.2937i 1.42028i −0.704059 0.710142i \(-0.748631\pi\)
0.704059 0.710142i \(-0.251369\pi\)
\(654\) 0 0
\(655\) −7.82091 −0.305588
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.43678i 0.367605i 0.982963 + 0.183802i \(0.0588406\pi\)
−0.982963 + 0.183802i \(0.941159\pi\)
\(660\) 0 0
\(661\) 21.5015i 0.836310i 0.908376 + 0.418155i \(0.137323\pi\)
−0.908376 + 0.418155i \(0.862677\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7.16010 2.37451i −0.277657 0.0920795i
\(666\) 0 0
\(667\) −27.4793 −1.06400
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 26.9509 1.04043
\(672\) 0 0
\(673\) −7.30649 −0.281644 −0.140822 0.990035i \(-0.544975\pi\)
−0.140822 + 0.990035i \(0.544975\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.03284 0.0396952 0.0198476 0.999803i \(-0.493682\pi\)
0.0198476 + 0.999803i \(0.493682\pi\)
\(678\) 0 0
\(679\) 27.1698 + 9.01034i 1.04268 + 0.345785i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.7363i 0.984770i −0.870377 0.492385i \(-0.836125\pi\)
0.870377 0.492385i \(-0.163875\pi\)
\(684\) 0 0
\(685\) 40.8970i 1.56259i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −63.4105 −2.41575
\(690\) 0 0
\(691\) 21.4970i 0.817783i −0.912583 0.408892i \(-0.865915\pi\)
0.912583 0.408892i \(-0.134085\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 85.4372i 3.24082i
\(696\) 0 0
\(697\) −26.3065 −0.996429
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.27211i 0.350203i −0.984550 0.175101i \(-0.943975\pi\)
0.984550 0.175101i \(-0.0560253\pi\)
\(702\) 0 0
\(703\) 0.495061i 0.0186716i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.96301 17.9809i 0.224262 0.676240i
\(708\) 0 0
\(709\) −4.67285 −0.175493 −0.0877464 0.996143i \(-0.527967\pi\)
−0.0877464 + 0.996143i \(0.527967\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 30.8643 1.15588
\(714\) 0 0
\(715\) −58.0725 −2.17179
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −35.6248 −1.32858 −0.664290 0.747475i \(-0.731266\pi\)
−0.664290 + 0.747475i \(0.731266\pi\)
\(720\) 0 0
\(721\) −5.58484 1.85210i −0.207990 0.0689759i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 57.4653i 2.13421i
\(726\) 0 0
\(727\) 33.8768i 1.25642i −0.778043 0.628211i \(-0.783788\pi\)
0.778043 0.628211i \(-0.216212\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −25.3342 −0.937018
\(732\) 0 0
\(733\) 29.0002i 1.07115i −0.844489 0.535573i \(-0.820096\pi\)
0.844489 0.535573i \(-0.179904\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.47983i 0.0545101i
\(738\) 0 0
\(739\) −2.22257 −0.0817585 −0.0408793 0.999164i \(-0.513016\pi\)
−0.0408793 + 0.999164i \(0.513016\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 37.5002i 1.37575i −0.725830 0.687874i \(-0.758544\pi\)
0.725830 0.687874i \(-0.241456\pi\)
\(744\) 0 0
\(745\) 35.6527i 1.30621i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.16176 + 0.716904i 0.0789888 + 0.0261951i
\(750\) 0 0
\(751\) 11.0912 0.404722 0.202361 0.979311i \(-0.435139\pi\)
0.202361 + 0.979311i \(0.435139\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 90.2986 3.28630
\(756\) 0 0
\(757\) 26.0714 0.947580 0.473790 0.880638i \(-0.342886\pi\)
0.473790 + 0.880638i \(0.342886\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26.3809 0.956305 0.478153 0.878277i \(-0.341307\pi\)
0.478153 + 0.878277i \(0.341307\pi\)
\(762\) 0 0
\(763\) −5.06153 + 15.2626i −0.183240 + 0.552541i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.18819i 0.223443i
\(768\) 0 0
\(769\) 43.9095i 1.58342i −0.610899 0.791709i \(-0.709192\pi\)
0.610899 0.791709i \(-0.290808\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.5616 0.667616 0.333808 0.942641i \(-0.391666\pi\)
0.333808 + 0.942641i \(0.391666\pi\)
\(774\) 0 0
\(775\) 64.5441i 2.31849i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.30261i 0.189986i
\(780\) 0 0
\(781\) −24.6067 −0.880497
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 37.4257i 1.33578i
\(786\) 0 0
\(787\) 2.42679i 0.0865056i 0.999064 + 0.0432528i \(0.0137721\pi\)
−0.999064 + 0.0432528i \(0.986228\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 29.4885 + 9.77930i 1.04849 + 0.347712i
\(792\) 0 0
\(793\) 57.4773 2.04108
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.6530 0.766988 0.383494 0.923543i \(-0.374721\pi\)
0.383494 + 0.923543i \(0.374721\pi\)
\(798\) 0 0
\(799\) 30.6714 1.08508
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −31.8145 −1.12271
\(804\) 0 0
\(805\) 69.5828 + 23.0758i 2.45247 + 0.813315i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 25.3442i 0.891055i 0.895268 + 0.445528i \(0.146984\pi\)
−0.895268 + 0.445528i \(0.853016\pi\)
\(810\) 0 0
\(811\) 39.0486i 1.37118i 0.727987 + 0.685591i \(0.240456\pi\)
−0.727987 + 0.685591i \(0.759544\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −91.3887 −3.20121
\(816\) 0 0
\(817\) 5.10662i 0.178658i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 40.6745i 1.41955i 0.704428 + 0.709775i \(0.251203\pi\)
−0.704428 + 0.709775i \(0.748797\pi\)
\(822\) 0 0
\(823\) −36.6153 −1.27633 −0.638165 0.769899i \(-0.720306\pi\)
−0.638165 + 0.769899i \(0.720306\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23.6002i 0.820660i 0.911937 + 0.410330i \(0.134586\pi\)
−0.911937 + 0.410330i \(0.865414\pi\)
\(828\) 0 0
\(829\) 24.1455i 0.838609i −0.907846 0.419305i \(-0.862274\pi\)
0.907846 0.419305i \(-0.137726\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18.4689 13.7634i −0.639910 0.476874i
\(834\) 0 0
\(835\) 41.5627 1.43834
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.0720 −1.17630 −0.588148 0.808753i \(-0.700143\pi\)
−0.588148 + 0.808753i \(0.700143\pi\)
\(840\) 0 0
\(841\) 10.8249 0.373273
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −67.9657 −2.33809
\(846\) 0 0
\(847\) −3.88558 + 11.7166i −0.133510 + 0.402587i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.81107i 0.164921i
\(852\) 0 0
\(853\) 9.31584i 0.318968i −0.987200 0.159484i \(-0.949017\pi\)
0.987200 0.159484i \(-0.0509831\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −41.7920 −1.42759 −0.713793 0.700357i \(-0.753024\pi\)
−0.713793 + 0.700357i \(0.753024\pi\)
\(858\) 0 0
\(859\) 37.5649i 1.28170i 0.767666 + 0.640850i \(0.221418\pi\)
−0.767666 + 0.640850i \(0.778582\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.8043i 0.435862i −0.975964 0.217931i \(-0.930069\pi\)
0.975964 0.217931i \(-0.0699309\pi\)
\(864\) 0 0
\(865\) 6.17909 0.210095
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 27.9651i 0.948651i
\(870\) 0 0
\(871\) 3.15598i 0.106936i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −30.3563 + 91.5366i −1.02623 + 3.09450i
\(876\) 0 0
\(877\) 39.5497 1.33550 0.667749 0.744386i \(-0.267258\pi\)
0.667749 + 0.744386i \(0.267258\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.6284 1.63833 0.819166 0.573557i \(-0.194437\pi\)
0.819166 + 0.573557i \(0.194437\pi\)
\(882\) 0 0
\(883\) 9.35363 0.314775 0.157387 0.987537i \(-0.449693\pi\)
0.157387 + 0.987537i \(0.449693\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −43.7316 −1.46836 −0.734181 0.678954i \(-0.762434\pi\)
−0.734181 + 0.678954i \(0.762434\pi\)
\(888\) 0 0
\(889\) −6.20083 + 18.6980i −0.207969 + 0.627111i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.18244i 0.206887i
\(894\) 0 0
\(895\) 78.2930i 2.61705i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 20.4140 0.680843
\(900\) 0 0
\(901\) 38.8726i 1.29503i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.81107i 0.159925i
\(906\) 0 0
\(907\) 4.21996 0.140121 0.0700607 0.997543i \(-0.477681\pi\)
0.0700607 + 0.997543i \(0.477681\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 47.5267i 1.57463i 0.616552 + 0.787314i \(0.288529\pi\)
−0.616552 + 0.787314i \(0.711471\pi\)
\(912\) 0 0
\(913\) 17.8010i 0.589126i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.51517 4.56884i 0.0500353 0.150877i
\(918\) 0 0
\(919\) 3.28410 0.108332 0.0541662 0.998532i \(-0.482750\pi\)
0.0541662 + 0.998532i \(0.482750\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −52.4780 −1.72733
\(924\) 0 0
\(925\) −10.0610 −0.330804
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 21.6294 0.709637 0.354818 0.934935i \(-0.384543\pi\)
0.354818 + 0.934935i \(0.384543\pi\)
\(930\) 0 0
\(931\) 2.77429 3.72279i 0.0909238 0.122009i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 35.6002i 1.16425i
\(936\) 0 0
\(937\) 3.01251i 0.0984143i 0.998789 + 0.0492071i \(0.0156694\pi\)
−0.998789 + 0.0492071i \(0.984331\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 29.9167 0.975257 0.487628 0.873051i \(-0.337862\pi\)
0.487628 + 0.873051i \(0.337862\pi\)
\(942\) 0 0
\(943\) 51.5315i 1.67810i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.8209i 1.48898i −0.667634 0.744490i \(-0.732693\pi\)
0.667634 0.744490i \(-0.267307\pi\)
\(948\) 0 0
\(949\) −67.8499 −2.20250
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 14.5009i 0.469731i 0.972028 + 0.234865i \(0.0754649\pi\)
−0.972028 + 0.234865i \(0.924535\pi\)
\(954\) 0 0
\(955\) 44.2669i 1.43244i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −23.8913 7.92310i −0.771491 0.255850i
\(960\) 0 0
\(961\) 8.07137 0.260367
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 48.2431 1.55300
\(966\) 0 0
\(967\) −14.4224 −0.463793 −0.231896 0.972740i \(-0.574493\pi\)
−0.231896 + 0.972740i \(0.574493\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 31.4034 1.00778 0.503891 0.863767i \(-0.331901\pi\)
0.503891 + 0.863767i \(0.331901\pi\)
\(972\) 0 0
\(973\) 49.9109 + 16.5520i 1.60007 + 0.530633i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45.2067i 1.44629i 0.690696 + 0.723145i \(0.257304\pi\)
−0.690696 + 0.723145i \(0.742696\pi\)
\(978\) 0 0
\(979\) 8.19667i 0.261967i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −11.1835 −0.356698 −0.178349 0.983967i \(-0.557076\pi\)
−0.178349 + 0.983967i \(0.557076\pi\)
\(984\) 0 0
\(985\) 13.6940i 0.436328i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 49.6268i 1.57804i
\(990\) 0 0
\(991\) −39.4509 −1.25320 −0.626599 0.779342i \(-0.715554\pi\)
−0.626599 + 0.779342i \(0.715554\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 45.5218i 1.44314i
\(996\) 0 0
\(997\) 15.8764i 0.502810i 0.967882 + 0.251405i \(0.0808926\pi\)
−0.967882 + 0.251405i \(0.919107\pi\)
\(998\) 0 0
\(999\) 0 0
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 3024.2.k.l.1889.16 16
3.2 odd 2 inner 3024.2.k.l.1889.2 16
4.3 odd 2 1512.2.k.b.377.15 yes 16
7.6 odd 2 inner 3024.2.k.l.1889.1 16
12.11 even 2 1512.2.k.b.377.1 16
21.20 even 2 inner 3024.2.k.l.1889.15 16
28.27 even 2 1512.2.k.b.377.2 yes 16
84.83 odd 2 1512.2.k.b.377.16 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.k.b.377.1 16 12.11 even 2
1512.2.k.b.377.2 yes 16 28.27 even 2
1512.2.k.b.377.15 yes 16 4.3 odd 2
1512.2.k.b.377.16 yes 16 84.83 odd 2
3024.2.k.l.1889.1 16 7.6 odd 2 inner
3024.2.k.l.1889.2 16 3.2 odd 2 inner
3024.2.k.l.1889.15 16 21.20 even 2 inner
3024.2.k.l.1889.16 16 1.1 even 1 trivial