Properties

Label 4140.2.i.b.1241.11
Level $4140$
Weight $2$
Character 4140.1241
Analytic conductor $33.058$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,2,Mod(1241,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.1241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.i (of order \(2\), degree \(1\), 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: \(33.0580664368\)
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: \( x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.11
Root \(-5.65286i\) of defining polynomial
Character \(\chi\) \(=\) 4140.1241
Dual form 4140.2.i.b.1241.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +0.826970i q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +0.826970i q^{7} +5.34857 q^{11} -4.20928 q^{13} -3.60504 q^{17} +5.22071i q^{19} +(4.78507 + 0.321128i) q^{23} +1.00000 q^{25} -5.68671i q^{29} +5.23310 q^{31} +0.826970i q^{35} -4.95459i q^{37} +11.4006i q^{41} +1.85889i q^{43} +1.79409i q^{47} +6.31612 q^{49} -6.66818 q^{53} +5.34857 q^{55} +4.65007i q^{59} +9.82671i q^{61} -4.20928 q^{65} -5.81889i q^{67} +11.1032i q^{71} +6.11519 q^{73} +4.42311i q^{77} +1.46423i q^{79} +8.97440 q^{83} -3.60504 q^{85} -0.0563452 q^{89} -3.48095i q^{91} +5.22071i q^{95} -15.0241i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{5} + 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} - 4 q^{53} + 8 q^{73} - 20 q^{83} - 32 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.826970i 0.312565i 0.987712 + 0.156283i \(0.0499511\pi\)
−0.987712 + 0.156283i \(0.950049\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.34857 1.61265 0.806327 0.591469i \(-0.201452\pi\)
0.806327 + 0.591469i \(0.201452\pi\)
\(12\) 0 0
\(13\) −4.20928 −1.16744 −0.583722 0.811953i \(-0.698404\pi\)
−0.583722 + 0.811953i \(0.698404\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.60504 −0.874352 −0.437176 0.899376i \(-0.644021\pi\)
−0.437176 + 0.899376i \(0.644021\pi\)
\(18\) 0 0
\(19\) 5.22071i 1.19771i 0.800857 + 0.598856i \(0.204378\pi\)
−0.800857 + 0.598856i \(0.795622\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.78507 + 0.321128i 0.997756 + 0.0669598i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.68671i 1.05600i −0.849246 0.527998i \(-0.822943\pi\)
0.849246 0.527998i \(-0.177057\pi\)
\(30\) 0 0
\(31\) 5.23310 0.939893 0.469946 0.882695i \(-0.344273\pi\)
0.469946 + 0.882695i \(0.344273\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.826970i 0.139783i
\(36\) 0 0
\(37\) 4.95459i 0.814530i −0.913310 0.407265i \(-0.866483\pi\)
0.913310 0.407265i \(-0.133517\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.4006i 1.78047i 0.455501 + 0.890235i \(0.349460\pi\)
−0.455501 + 0.890235i \(0.650540\pi\)
\(42\) 0 0
\(43\) 1.85889i 0.283478i 0.989904 + 0.141739i \(0.0452693\pi\)
−0.989904 + 0.141739i \(0.954731\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.79409i 0.261695i 0.991403 + 0.130847i \(0.0417698\pi\)
−0.991403 + 0.130847i \(0.958230\pi\)
\(48\) 0 0
\(49\) 6.31612 0.902303
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.66818 −0.915945 −0.457972 0.888966i \(-0.651424\pi\)
−0.457972 + 0.888966i \(0.651424\pi\)
\(54\) 0 0
\(55\) 5.34857 0.721201
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.65007i 0.605387i 0.953088 + 0.302694i \(0.0978859\pi\)
−0.953088 + 0.302694i \(0.902114\pi\)
\(60\) 0 0
\(61\) 9.82671i 1.25818i 0.777332 + 0.629090i \(0.216573\pi\)
−0.777332 + 0.629090i \(0.783427\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.20928 −0.522097
\(66\) 0 0
\(67\) 5.81889i 0.710891i −0.934697 0.355446i \(-0.884329\pi\)
0.934697 0.355446i \(-0.115671\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.1032i 1.31771i 0.752269 + 0.658856i \(0.228959\pi\)
−0.752269 + 0.658856i \(0.771041\pi\)
\(72\) 0 0
\(73\) 6.11519 0.715728 0.357864 0.933774i \(-0.383505\pi\)
0.357864 + 0.933774i \(0.383505\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.42311i 0.504060i
\(78\) 0 0
\(79\) 1.46423i 0.164739i 0.996602 + 0.0823694i \(0.0262487\pi\)
−0.996602 + 0.0823694i \(0.973751\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.97440 0.985069 0.492534 0.870293i \(-0.336071\pi\)
0.492534 + 0.870293i \(0.336071\pi\)
\(84\) 0 0
\(85\) −3.60504 −0.391022
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.0563452 −0.00597257 −0.00298629 0.999996i \(-0.500951\pi\)
−0.00298629 + 0.999996i \(0.500951\pi\)
\(90\) 0 0
\(91\) 3.48095i 0.364903i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.22071i 0.535633i
\(96\) 0 0
\(97\) 15.0241i 1.52547i −0.646714 0.762733i \(-0.723857\pi\)
0.646714 0.762733i \(-0.276143\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 19.4906i 1.93939i 0.244324 + 0.969694i \(0.421434\pi\)
−0.244324 + 0.969694i \(0.578566\pi\)
\(102\) 0 0
\(103\) 0.247690i 0.0244056i 0.999926 + 0.0122028i \(0.00388437\pi\)
−0.999926 + 0.0122028i \(0.996116\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.00052 −0.870113 −0.435057 0.900403i \(-0.643272\pi\)
−0.435057 + 0.900403i \(0.643272\pi\)
\(108\) 0 0
\(109\) 10.1556i 0.972732i −0.873755 0.486366i \(-0.838322\pi\)
0.873755 0.486366i \(-0.161678\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.63519 0.436042 0.218021 0.975944i \(-0.430040\pi\)
0.218021 + 0.975944i \(0.430040\pi\)
\(114\) 0 0
\(115\) 4.78507 + 0.321128i 0.446210 + 0.0299453i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.98126i 0.273292i
\(120\) 0 0
\(121\) 17.6072 1.60066
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.28911 0.203126 0.101563 0.994829i \(-0.467616\pi\)
0.101563 + 0.994829i \(0.467616\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.6358i 1.01662i −0.861173 0.508311i \(-0.830270\pi\)
0.861173 0.508311i \(-0.169730\pi\)
\(132\) 0 0
\(133\) −4.31737 −0.374363
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.80437 0.239593 0.119797 0.992798i \(-0.461776\pi\)
0.119797 + 0.992798i \(0.461776\pi\)
\(138\) 0 0
\(139\) −3.23304 −0.274223 −0.137111 0.990556i \(-0.543782\pi\)
−0.137111 + 0.990556i \(0.543782\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −22.5136 −1.88269
\(144\) 0 0
\(145\) 5.68671i 0.472255i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.32611 0.682101 0.341051 0.940045i \(-0.389217\pi\)
0.341051 + 0.940045i \(0.389217\pi\)
\(150\) 0 0
\(151\) −3.72932 −0.303488 −0.151744 0.988420i \(-0.548489\pi\)
−0.151744 + 0.988420i \(0.548489\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.23310 0.420333
\(156\) 0 0
\(157\) 14.9437i 1.19264i 0.802747 + 0.596320i \(0.203371\pi\)
−0.802747 + 0.596320i \(0.796629\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.265563 + 3.95711i −0.0209293 + 0.311864i
\(162\) 0 0
\(163\) 15.8043 1.23789 0.618945 0.785434i \(-0.287560\pi\)
0.618945 + 0.785434i \(0.287560\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.877759i 0.0679230i −0.999423 0.0339615i \(-0.989188\pi\)
0.999423 0.0339615i \(-0.0108124\pi\)
\(168\) 0 0
\(169\) 4.71806 0.362927
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.84136i 0.672196i 0.941827 + 0.336098i \(0.109107\pi\)
−0.941827 + 0.336098i \(0.890893\pi\)
\(174\) 0 0
\(175\) 0.826970i 0.0625130i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.7369i 0.952002i 0.879445 + 0.476001i \(0.157914\pi\)
−0.879445 + 0.476001i \(0.842086\pi\)
\(180\) 0 0
\(181\) 10.0635i 0.748017i 0.927425 + 0.374008i \(0.122017\pi\)
−0.927425 + 0.374008i \(0.877983\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.95459i 0.364269i
\(186\) 0 0
\(187\) −19.2818 −1.41003
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.9106 −0.934177 −0.467089 0.884210i \(-0.654697\pi\)
−0.467089 + 0.884210i \(0.654697\pi\)
\(192\) 0 0
\(193\) 26.4533 1.90415 0.952074 0.305869i \(-0.0989469\pi\)
0.952074 + 0.305869i \(0.0989469\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.30018i 0.591363i −0.955287 0.295682i \(-0.904453\pi\)
0.955287 0.295682i \(-0.0955467\pi\)
\(198\) 0 0
\(199\) 6.61245i 0.468744i −0.972147 0.234372i \(-0.924697\pi\)
0.972147 0.234372i \(-0.0753034\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.70274 0.330067
\(204\) 0 0
\(205\) 11.4006i 0.796251i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 27.9233i 1.93150i
\(210\) 0 0
\(211\) −2.04726 −0.140939 −0.0704695 0.997514i \(-0.522450\pi\)
−0.0704695 + 0.997514i \(0.522450\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.85889i 0.126775i
\(216\) 0 0
\(217\) 4.32762i 0.293778i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.1746 1.02076
\(222\) 0 0
\(223\) −19.3700 −1.29711 −0.648556 0.761167i \(-0.724627\pi\)
−0.648556 + 0.761167i \(0.724627\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.01201 −0.598148 −0.299074 0.954230i \(-0.596678\pi\)
−0.299074 + 0.954230i \(0.596678\pi\)
\(228\) 0 0
\(229\) 8.28944i 0.547781i 0.961761 + 0.273891i \(0.0883107\pi\)
−0.961761 + 0.273891i \(0.911689\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.2048i 1.52020i 0.649807 + 0.760099i \(0.274850\pi\)
−0.649807 + 0.760099i \(0.725150\pi\)
\(234\) 0 0
\(235\) 1.79409i 0.117034i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.5877i 0.814228i 0.913377 + 0.407114i \(0.133465\pi\)
−0.913377 + 0.407114i \(0.866535\pi\)
\(240\) 0 0
\(241\) 19.7065i 1.26941i 0.772756 + 0.634703i \(0.218878\pi\)
−0.772756 + 0.634703i \(0.781122\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.31612 0.403522
\(246\) 0 0
\(247\) 21.9754i 1.39826i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.4446 1.35357 0.676784 0.736181i \(-0.263373\pi\)
0.676784 + 0.736181i \(0.263373\pi\)
\(252\) 0 0
\(253\) 25.5933 + 1.71758i 1.60904 + 0.107983i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.5523i 0.658234i 0.944289 + 0.329117i \(0.106751\pi\)
−0.944289 + 0.329117i \(0.893249\pi\)
\(258\) 0 0
\(259\) 4.09730 0.254594
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.3888 0.825589 0.412794 0.910824i \(-0.364553\pi\)
0.412794 + 0.910824i \(0.364553\pi\)
\(264\) 0 0
\(265\) −6.66818 −0.409623
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.38744i 0.206536i −0.994654 0.103268i \(-0.967070\pi\)
0.994654 0.103268i \(-0.0329299\pi\)
\(270\) 0 0
\(271\) 14.4356 0.876902 0.438451 0.898755i \(-0.355527\pi\)
0.438451 + 0.898755i \(0.355527\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.34857 0.322531
\(276\) 0 0
\(277\) 19.7134 1.18446 0.592231 0.805768i \(-0.298247\pi\)
0.592231 + 0.805768i \(0.298247\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.73536 0.580763 0.290381 0.956911i \(-0.406218\pi\)
0.290381 + 0.956911i \(0.406218\pi\)
\(282\) 0 0
\(283\) 0.211034i 0.0125447i 0.999980 + 0.00627234i \(0.00199656\pi\)
−0.999980 + 0.00627234i \(0.998003\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.42793 −0.556513
\(288\) 0 0
\(289\) −4.00366 −0.235509
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −14.6156 −0.853851 −0.426925 0.904287i \(-0.640403\pi\)
−0.426925 + 0.904287i \(0.640403\pi\)
\(294\) 0 0
\(295\) 4.65007i 0.270737i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −20.1417 1.35172i −1.16482 0.0781719i
\(300\) 0 0
\(301\) −1.53724 −0.0886053
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.82671i 0.562676i
\(306\) 0 0
\(307\) 34.1867 1.95114 0.975569 0.219695i \(-0.0705064\pi\)
0.975569 + 0.219695i \(0.0705064\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 29.5570i 1.67602i −0.545651 0.838012i \(-0.683718\pi\)
0.545651 0.838012i \(-0.316282\pi\)
\(312\) 0 0
\(313\) 32.3791i 1.83018i −0.403255 0.915088i \(-0.632121\pi\)
0.403255 0.915088i \(-0.367879\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.62772i 0.0914216i 0.998955 + 0.0457108i \(0.0145553\pi\)
−0.998955 + 0.0457108i \(0.985445\pi\)
\(318\) 0 0
\(319\) 30.4158i 1.70296i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.8209i 1.04722i
\(324\) 0 0
\(325\) −4.20928 −0.233489
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.48366 −0.0817967
\(330\) 0 0
\(331\) −13.4951 −0.741759 −0.370880 0.928681i \(-0.620944\pi\)
−0.370880 + 0.928681i \(0.620944\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.81889i 0.317920i
\(336\) 0 0
\(337\) 2.59985i 0.141623i −0.997490 0.0708113i \(-0.977441\pi\)
0.997490 0.0708113i \(-0.0225588\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 27.9896 1.51572
\(342\) 0 0
\(343\) 11.0120i 0.594594i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 34.4803i 1.85100i 0.378748 + 0.925500i \(0.376355\pi\)
−0.378748 + 0.925500i \(0.623645\pi\)
\(348\) 0 0
\(349\) 3.09741 0.165800 0.0829002 0.996558i \(-0.473582\pi\)
0.0829002 + 0.996558i \(0.473582\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 33.1843i 1.76622i 0.469162 + 0.883112i \(0.344556\pi\)
−0.469162 + 0.883112i \(0.655444\pi\)
\(354\) 0 0
\(355\) 11.1032i 0.589299i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −27.1798 −1.43449 −0.717247 0.696819i \(-0.754598\pi\)
−0.717247 + 0.696819i \(0.754598\pi\)
\(360\) 0 0
\(361\) −8.25579 −0.434515
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.11519 0.320083
\(366\) 0 0
\(367\) 11.6520i 0.608229i −0.952635 0.304115i \(-0.901639\pi\)
0.952635 0.304115i \(-0.0983605\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5.51438i 0.286292i
\(372\) 0 0
\(373\) 27.0411i 1.40013i −0.714077 0.700067i \(-0.753153\pi\)
0.714077 0.700067i \(-0.246847\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 23.9370i 1.23282i
\(378\) 0 0
\(379\) 25.4729i 1.30845i −0.756298 0.654227i \(-0.772994\pi\)
0.756298 0.654227i \(-0.227006\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10.4529 0.534119 0.267059 0.963680i \(-0.413948\pi\)
0.267059 + 0.963680i \(0.413948\pi\)
\(384\) 0 0
\(385\) 4.42311i 0.225422i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −11.0966 −0.562621 −0.281310 0.959617i \(-0.590769\pi\)
−0.281310 + 0.959617i \(0.590769\pi\)
\(390\) 0 0
\(391\) −17.2504 1.15768i −0.872389 0.0585464i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.46423i 0.0736735i
\(396\) 0 0
\(397\) 10.5856 0.531274 0.265637 0.964073i \(-0.414418\pi\)
0.265637 + 0.964073i \(0.414418\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13.2109 −0.659720 −0.329860 0.944030i \(-0.607002\pi\)
−0.329860 + 0.944030i \(0.607002\pi\)
\(402\) 0 0
\(403\) −22.0276 −1.09727
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 26.5000i 1.31356i
\(408\) 0 0
\(409\) −6.12819 −0.303020 −0.151510 0.988456i \(-0.548414\pi\)
−0.151510 + 0.988456i \(0.548414\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.84547 −0.189223
\(414\) 0 0
\(415\) 8.97440 0.440536
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −18.1134 −0.884895 −0.442448 0.896794i \(-0.645890\pi\)
−0.442448 + 0.896794i \(0.645890\pi\)
\(420\) 0 0
\(421\) 4.12855i 0.201213i 0.994926 + 0.100607i \(0.0320784\pi\)
−0.994926 + 0.100607i \(0.967922\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.60504 −0.174870
\(426\) 0 0
\(427\) −8.12639 −0.393264
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.76534 −0.229539 −0.114769 0.993392i \(-0.536613\pi\)
−0.114769 + 0.993392i \(0.536613\pi\)
\(432\) 0 0
\(433\) 10.2357i 0.491897i 0.969283 + 0.245949i \(0.0790994\pi\)
−0.969283 + 0.245949i \(0.920901\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.67652 + 24.9814i −0.0801986 + 1.19502i
\(438\) 0 0
\(439\) 20.5730 0.981895 0.490948 0.871189i \(-0.336651\pi\)
0.490948 + 0.871189i \(0.336651\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.3996i 0.779169i −0.920991 0.389584i \(-0.872619\pi\)
0.920991 0.389584i \(-0.127381\pi\)
\(444\) 0 0
\(445\) −0.0563452 −0.00267102
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.82101i 0.463482i −0.972778 0.231741i \(-0.925558\pi\)
0.972778 0.231741i \(-0.0744422\pi\)
\(450\) 0 0
\(451\) 60.9768i 2.87128i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.48095i 0.163189i
\(456\) 0 0
\(457\) 30.5414i 1.42867i −0.699805 0.714334i \(-0.746730\pi\)
0.699805 0.714334i \(-0.253270\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.20057i 0.335364i −0.985841 0.167682i \(-0.946372\pi\)
0.985841 0.167682i \(-0.0536282\pi\)
\(462\) 0 0
\(463\) −5.93798 −0.275961 −0.137981 0.990435i \(-0.544061\pi\)
−0.137981 + 0.990435i \(0.544061\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.4933 0.948317 0.474158 0.880440i \(-0.342752\pi\)
0.474158 + 0.880440i \(0.342752\pi\)
\(468\) 0 0
\(469\) 4.81205 0.222200
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.94239i 0.457152i
\(474\) 0 0
\(475\) 5.22071i 0.239543i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.9538 1.27724 0.638622 0.769521i \(-0.279505\pi\)
0.638622 + 0.769521i \(0.279505\pi\)
\(480\) 0 0
\(481\) 20.8553i 0.950919i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 15.0241i 0.682209i
\(486\) 0 0
\(487\) −27.2625 −1.23538 −0.617690 0.786422i \(-0.711931\pi\)
−0.617690 + 0.786422i \(0.711931\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 37.4663i 1.69083i −0.534110 0.845415i \(-0.679353\pi\)
0.534110 0.845415i \(-0.320647\pi\)
\(492\) 0 0
\(493\) 20.5008i 0.923311i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.18204 −0.411871
\(498\) 0 0
\(499\) −30.3473 −1.35853 −0.679266 0.733892i \(-0.737702\pi\)
−0.679266 + 0.733892i \(0.737702\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.08858 0.316064 0.158032 0.987434i \(-0.449485\pi\)
0.158032 + 0.987434i \(0.449485\pi\)
\(504\) 0 0
\(505\) 19.4906i 0.867320i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 23.4017i 1.03726i −0.854999 0.518630i \(-0.826442\pi\)
0.854999 0.518630i \(-0.173558\pi\)
\(510\) 0 0
\(511\) 5.05707i 0.223712i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.247690i 0.0109145i
\(516\) 0 0
\(517\) 9.59582i 0.422024i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.74404 0.339273 0.169636 0.985507i \(-0.445741\pi\)
0.169636 + 0.985507i \(0.445741\pi\)
\(522\) 0 0
\(523\) 21.9918i 0.961634i 0.876821 + 0.480817i \(0.159660\pi\)
−0.876821 + 0.480817i \(0.840340\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.8656 −0.821797
\(528\) 0 0
\(529\) 22.7938 + 3.07324i 0.991033 + 0.133619i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 47.9882i 2.07860i
\(534\) 0 0
\(535\) −9.00052 −0.389127
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 33.7822 1.45510
\(540\) 0 0
\(541\) −10.6968 −0.459892 −0.229946 0.973203i \(-0.573855\pi\)
−0.229946 + 0.973203i \(0.573855\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.1556i 0.435019i
\(546\) 0 0
\(547\) 27.4632 1.17424 0.587122 0.809499i \(-0.300261\pi\)
0.587122 + 0.809499i \(0.300261\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 29.6886 1.26478
\(552\) 0 0
\(553\) −1.21088 −0.0514916
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.1611 0.642398 0.321199 0.947012i \(-0.395914\pi\)
0.321199 + 0.947012i \(0.395914\pi\)
\(558\) 0 0
\(559\) 7.82458i 0.330945i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −26.3602 −1.11095 −0.555475 0.831533i \(-0.687464\pi\)
−0.555475 + 0.831533i \(0.687464\pi\)
\(564\) 0 0
\(565\) 4.63519 0.195004
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.2551 −1.26836 −0.634179 0.773186i \(-0.718662\pi\)
−0.634179 + 0.773186i \(0.718662\pi\)
\(570\) 0 0
\(571\) 42.0330i 1.75903i 0.475874 + 0.879513i \(0.342132\pi\)
−0.475874 + 0.879513i \(0.657868\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.78507 + 0.321128i 0.199551 + 0.0133920i
\(576\) 0 0
\(577\) 21.2284 0.883750 0.441875 0.897077i \(-0.354314\pi\)
0.441875 + 0.897077i \(0.354314\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.42156i 0.307898i
\(582\) 0 0
\(583\) −35.6652 −1.47710
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0478i 0.744911i 0.928050 + 0.372455i \(0.121484\pi\)
−0.928050 + 0.372455i \(0.878516\pi\)
\(588\) 0 0
\(589\) 27.3205i 1.12572i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.6934i 0.972973i 0.873688 + 0.486486i \(0.161722\pi\)
−0.873688 + 0.486486i \(0.838278\pi\)
\(594\) 0 0
\(595\) 2.98126i 0.122220i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.1037i 0.576261i −0.957591 0.288131i \(-0.906966\pi\)
0.957591 0.288131i \(-0.0930337\pi\)
\(600\) 0 0
\(601\) 32.8218 1.33883 0.669415 0.742889i \(-0.266545\pi\)
0.669415 + 0.742889i \(0.266545\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 17.6072 0.715835
\(606\) 0 0
\(607\) −23.9134 −0.970616 −0.485308 0.874343i \(-0.661293\pi\)
−0.485308 + 0.874343i \(0.661293\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.55183i 0.305514i
\(612\) 0 0
\(613\) 37.1500i 1.50047i −0.661170 0.750237i \(-0.729940\pi\)
0.661170 0.750237i \(-0.270060\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.1226 −0.488036 −0.244018 0.969771i \(-0.578466\pi\)
−0.244018 + 0.969771i \(0.578466\pi\)
\(618\) 0 0
\(619\) 13.1429i 0.528256i −0.964488 0.264128i \(-0.914916\pi\)
0.964488 0.264128i \(-0.0850842\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0.0465957i 0.00186682i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.8615i 0.712186i
\(630\) 0 0
\(631\) 2.82487i 0.112456i −0.998418 0.0562282i \(-0.982093\pi\)
0.998418 0.0562282i \(-0.0179075\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.28911 0.0908405
\(636\) 0 0
\(637\) −26.5863 −1.05339
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.81398 −0.0716478 −0.0358239 0.999358i \(-0.511406\pi\)
−0.0358239 + 0.999358i \(0.511406\pi\)
\(642\) 0 0
\(643\) 24.7415i 0.975711i 0.872924 + 0.487855i \(0.162221\pi\)
−0.872924 + 0.487855i \(0.837779\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 41.3075i 1.62397i −0.583681 0.811983i \(-0.698388\pi\)
0.583681 0.811983i \(-0.301612\pi\)
\(648\) 0 0
\(649\) 24.8712i 0.976281i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.5923i 0.492774i 0.969172 + 0.246387i \(0.0792434\pi\)
−0.969172 + 0.246387i \(0.920757\pi\)
\(654\) 0 0
\(655\) 11.6358i 0.454647i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.4875 0.798080 0.399040 0.916934i \(-0.369344\pi\)
0.399040 + 0.916934i \(0.369344\pi\)
\(660\) 0 0
\(661\) 40.3876i 1.57090i −0.618927 0.785449i \(-0.712432\pi\)
0.618927 0.785449i \(-0.287568\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.31737 −0.167420
\(666\) 0 0
\(667\) 1.82616 27.2113i 0.0707092 1.05363i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 52.5588i 2.02901i
\(672\) 0 0
\(673\) −47.1930 −1.81915 −0.909577 0.415535i \(-0.863594\pi\)
−0.909577 + 0.415535i \(0.863594\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −47.4443 −1.82343 −0.911716 0.410821i \(-0.865242\pi\)
−0.911716 + 0.410821i \(0.865242\pi\)
\(678\) 0 0
\(679\) 12.4245 0.476807
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.1260i 1.15274i −0.817188 0.576371i \(-0.804469\pi\)
0.817188 0.576371i \(-0.195531\pi\)
\(684\) 0 0
\(685\) 2.80437 0.107149
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 28.0682 1.06931
\(690\) 0 0
\(691\) −41.3816 −1.57423 −0.787115 0.616806i \(-0.788426\pi\)
−0.787115 + 0.616806i \(0.788426\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.23304 −0.122636
\(696\) 0 0
\(697\) 41.0996i 1.55676i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 24.1046 0.910418 0.455209 0.890385i \(-0.349565\pi\)
0.455209 + 0.890385i \(0.349565\pi\)
\(702\) 0 0
\(703\) 25.8665 0.975573
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.1181 −0.606185
\(708\) 0 0
\(709\) 41.1005i 1.54356i −0.635890 0.771780i \(-0.719367\pi\)
0.635890 0.771780i \(-0.280633\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 25.0407 + 1.68050i 0.937783 + 0.0629350i
\(714\) 0 0
\(715\) −22.5136 −0.841963
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 19.0130i 0.709065i −0.935044 0.354532i \(-0.884640\pi\)
0.935044 0.354532i \(-0.115360\pi\)
\(720\) 0 0
\(721\) −0.204832 −0.00762835
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.68671i 0.211199i
\(726\) 0 0
\(727\) 0.332437i 0.0123294i −0.999981 0.00616470i \(-0.998038\pi\)
0.999981 0.00616470i \(-0.00196230\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.70137i 0.247859i
\(732\) 0 0
\(733\) 4.37626i 0.161641i 0.996729 + 0.0808203i \(0.0257540\pi\)
−0.996729 + 0.0808203i \(0.974246\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.1228i 1.14642i
\(738\) 0 0
\(739\) 8.49427 0.312467 0.156233 0.987720i \(-0.450065\pi\)
0.156233 + 0.987720i \(0.450065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −19.8899 −0.729689 −0.364844 0.931068i \(-0.618878\pi\)
−0.364844 + 0.931068i \(0.618878\pi\)
\(744\) 0 0
\(745\) 8.32611 0.305045
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.44316i 0.271967i
\(750\) 0 0
\(751\) 4.99053i 0.182107i −0.995846 0.0910535i \(-0.970977\pi\)
0.995846 0.0910535i \(-0.0290234\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3.72932 −0.135724
\(756\) 0 0
\(757\) 38.7722i 1.40920i −0.709606 0.704599i \(-0.751127\pi\)
0.709606 0.704599i \(-0.248873\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.7993i 0.500224i −0.968217 0.250112i \(-0.919533\pi\)
0.968217 0.250112i \(-0.0804674\pi\)
\(762\) 0 0
\(763\) 8.39839 0.304042
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 19.5734i 0.706756i
\(768\) 0 0
\(769\) 12.0874i 0.435883i −0.975962 0.217942i \(-0.930066\pi\)
0.975962 0.217942i \(-0.0699342\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −35.7194 −1.28474 −0.642368 0.766396i \(-0.722048\pi\)
−0.642368 + 0.766396i \(0.722048\pi\)
\(774\) 0 0
\(775\) 5.23310 0.187979
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −59.5191 −2.13249
\(780\) 0 0
\(781\) 59.3865i 2.12501i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.9437i 0.533365i
\(786\) 0 0
\(787\) 21.0986i 0.752082i −0.926603 0.376041i \(-0.877285\pi\)
0.926603 0.376041i \(-0.122715\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.83316i 0.136292i
\(792\) 0 0
\(793\) 41.3634i 1.46886i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −41.8022 −1.48071 −0.740355 0.672216i \(-0.765343\pi\)
−0.740355 + 0.672216i \(0.765343\pi\)
\(798\) 0 0
\(799\) 6.46777i 0.228813i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 32.7075 1.15422
\(804\) 0 0
\(805\) −0.265563 + 3.95711i −0.00935987 + 0.139470i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 27.2362i 0.957573i −0.877931 0.478786i \(-0.841077\pi\)
0.877931 0.478786i \(-0.158923\pi\)
\(810\) 0 0
\(811\) 41.3116 1.45065 0.725323 0.688408i \(-0.241690\pi\)
0.725323 + 0.688408i \(0.241690\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 15.8043 0.553602
\(816\) 0 0
\(817\) −9.70471 −0.339525
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.54365i 0.0887741i −0.999014 0.0443870i \(-0.985867\pi\)
0.999014 0.0443870i \(-0.0141335\pi\)
\(822\) 0 0
\(823\) −30.4594 −1.06175 −0.530873 0.847451i \(-0.678136\pi\)
−0.530873 + 0.847451i \(0.678136\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −51.1175 −1.77753 −0.888764 0.458365i \(-0.848435\pi\)
−0.888764 + 0.458365i \(0.848435\pi\)
\(828\) 0 0
\(829\) −47.5460 −1.65134 −0.825670 0.564153i \(-0.809203\pi\)
−0.825670 + 0.564153i \(0.809203\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −22.7699 −0.788930
\(834\) 0 0
\(835\) 0.877759i 0.0303761i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.8450 −0.926792 −0.463396 0.886151i \(-0.653369\pi\)
−0.463396 + 0.886151i \(0.653369\pi\)
\(840\) 0 0
\(841\) −3.33865 −0.115126
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.71806 0.162306
\(846\) 0 0
\(847\) 14.5606i 0.500309i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.59106 23.7081i 0.0545408 0.812702i
\(852\) 0 0
\(853\) −32.4644 −1.11156 −0.555780 0.831330i \(-0.687580\pi\)
−0.555780 + 0.831330i \(0.687580\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.4155i 0.355787i −0.984050 0.177893i \(-0.943072\pi\)
0.984050 0.177893i \(-0.0569282\pi\)
\(858\) 0 0
\(859\) −34.1733 −1.16598 −0.582989 0.812480i \(-0.698117\pi\)
−0.582989 + 0.812480i \(0.698117\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.6417i 0.634572i −0.948330 0.317286i \(-0.897229\pi\)
0.948330 0.317286i \(-0.102771\pi\)
\(864\) 0 0
\(865\) 8.84136i 0.300615i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 7.83154i 0.265667i
\(870\) 0 0
\(871\) 24.4934i 0.829926i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.826970i 0.0279567i
\(876\) 0 0
\(877\) −42.2978 −1.42830 −0.714148 0.699995i \(-0.753186\pi\)
−0.714148 + 0.699995i \(0.753186\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 44.8656 1.51156 0.755780 0.654826i \(-0.227258\pi\)
0.755780 + 0.654826i \(0.227258\pi\)
\(882\) 0 0
\(883\) −10.4176 −0.350579 −0.175289 0.984517i \(-0.556086\pi\)
−0.175289 + 0.984517i \(0.556086\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.0868i 1.61460i −0.590145 0.807298i \(-0.700929\pi\)
0.590145 0.807298i \(-0.299071\pi\)
\(888\) 0 0
\(889\) 1.89302i 0.0634900i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.36642 −0.313435
\(894\) 0 0
\(895\) 12.7369i 0.425748i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 29.7591i 0.992522i
\(900\) 0 0
\(901\) 24.0391 0.800858
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0635i 0.334523i
\(906\) 0 0
\(907\) 18.0597i 0.599661i −0.953993 0.299830i \(-0.903070\pi\)
0.953993 0.299830i \(-0.0969301\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 46.3139 1.53445 0.767225 0.641379i \(-0.221637\pi\)
0.767225 + 0.641379i \(0.221637\pi\)
\(912\) 0 0
\(913\) 48.0002 1.58858
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.62243 0.317761
\(918\) 0 0
\(919\) 0.917053i 0.0302508i 0.999886 + 0.0151254i \(0.00481474\pi\)
−0.999886 + 0.0151254i \(0.995185\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 46.7367i 1.53836i
\(924\) 0 0
\(925\) 4.95459i 0.162906i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.70815i 0.0888516i 0.999013 + 0.0444258i \(0.0141458\pi\)
−0.999013 + 0.0444258i \(0.985854\pi\)
\(930\) 0 0
\(931\) 32.9746i 1.08070i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −19.2818 −0.630583
\(936\) 0 0
\(937\) 58.5686i 1.91335i −0.291154 0.956676i \(-0.594039\pi\)
0.291154 0.956676i \(-0.405961\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −15.0376 −0.490213 −0.245107 0.969496i \(-0.578823\pi\)
−0.245107 + 0.969496i \(0.578823\pi\)
\(942\) 0 0
\(943\) −3.66104 + 54.5525i −0.119220 + 1.77647i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.78805i 0.0581037i 0.999578 + 0.0290518i \(0.00924879\pi\)
−0.999578 + 0.0290518i \(0.990751\pi\)
\(948\) 0 0
\(949\) −25.7405 −0.835573
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 21.9100 0.709736 0.354868 0.934916i \(-0.384526\pi\)
0.354868 + 0.934916i \(0.384526\pi\)
\(954\) 0 0
\(955\) −12.9106 −0.417777
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.31913i 0.0748885i
\(960\) 0 0
\(961\) −3.61464 −0.116601
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 26.4533 0.851561
\(966\) 0 0
\(967\) −47.3740 −1.52344 −0.761722 0.647904i \(-0.775646\pi\)
−0.761722 + 0.647904i \(0.775646\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 19.8635 0.637449 0.318725 0.947847i \(-0.396745\pi\)
0.318725 + 0.947847i \(0.396745\pi\)
\(972\) 0 0
\(973\) 2.67362i 0.0857125i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −29.0169 −0.928332 −0.464166 0.885748i \(-0.653646\pi\)
−0.464166 + 0.885748i \(0.653646\pi\)
\(978\) 0 0
\(979\) −0.301366 −0.00963170
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 40.9587 1.30638 0.653190 0.757194i \(-0.273430\pi\)
0.653190 + 0.757194i \(0.273430\pi\)
\(984\) 0 0
\(985\) 8.30018i 0.264466i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.596941 + 8.89490i −0.0189816 + 0.282842i
\(990\) 0 0
\(991\) 37.6334 1.19546 0.597732 0.801696i \(-0.296069\pi\)
0.597732 + 0.801696i \(0.296069\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.61245i 0.209629i
\(996\) 0 0
\(997\) −27.7183 −0.877849 −0.438924 0.898524i \(-0.644640\pi\)
−0.438924 + 0.898524i \(0.644640\pi\)
\(998\) 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 4140.2.i.b.1241.11 yes 16
3.2 odd 2 4140.2.i.a.1241.11 yes 16
23.22 odd 2 4140.2.i.a.1241.6 16
69.68 even 2 inner 4140.2.i.b.1241.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.i.a.1241.6 16 23.22 odd 2
4140.2.i.a.1241.11 yes 16 3.2 odd 2
4140.2.i.b.1241.6 yes 16 69.68 even 2 inner
4140.2.i.b.1241.11 yes 16 1.1 even 1 trivial