Properties

Label 2736.3.o.q.721.19
Level $2736$
Weight $3$
Character 2736.721
Analytic conductor $74.551$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,3,Mod(721,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 264 x^{18} + 28274 x^{16} - 1545308 x^{14} + 45358441 x^{12} - 637328868 x^{10} + 1825819356 x^{8} + 32794262368 x^{6} + 135580415344 x^{4} + \cdots + 194396337216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.19
Root \(8.89263 + 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2736.721
Dual form 2736.3.o.q.721.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.89263 q^{5} -4.08883 q^{7} +O(q^{10})\) \(q+8.89263 q^{5} -4.08883 q^{7} -4.37788 q^{11} -13.7670i q^{13} -14.0927 q^{17} +(-11.6112 + 15.0393i) q^{19} -6.99929 q^{23} +54.0788 q^{25} -31.8337i q^{29} +12.3543i q^{31} -36.3604 q^{35} -4.89820i q^{37} -73.1184i q^{41} +23.8862 q^{43} -43.3597 q^{47} -32.2815 q^{49} +65.4462i q^{53} -38.9309 q^{55} -62.3993i q^{59} -70.1235 q^{61} -122.425i q^{65} -126.526i q^{67} -44.9262i q^{71} -0.118545 q^{73} +17.9004 q^{77} +19.5093i q^{79} -97.8163 q^{83} -125.321 q^{85} -15.6943i q^{89} +56.2911i q^{91} +(-103.254 + 133.739i) q^{95} -6.98703i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{7} + 8 q^{19} + 68 q^{25} + 128 q^{43} + 116 q^{49} - 144 q^{55} - 104 q^{61} - 88 q^{73} - 280 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 8.89263 1.77853 0.889263 0.457397i \(-0.151218\pi\)
0.889263 + 0.457397i \(0.151218\pi\)
\(6\) 0 0
\(7\) −4.08883 −0.584118 −0.292059 0.956400i \(-0.594340\pi\)
−0.292059 + 0.956400i \(0.594340\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.37788 −0.397989 −0.198995 0.980001i \(-0.563768\pi\)
−0.198995 + 0.980001i \(0.563768\pi\)
\(12\) 0 0
\(13\) 13.7670i 1.05900i −0.848309 0.529502i \(-0.822379\pi\)
0.848309 0.529502i \(-0.177621\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −14.0927 −0.828981 −0.414490 0.910054i \(-0.636040\pi\)
−0.414490 + 0.910054i \(0.636040\pi\)
\(18\) 0 0
\(19\) −11.6112 + 15.0393i −0.611115 + 0.791542i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.99929 −0.304317 −0.152158 0.988356i \(-0.548622\pi\)
−0.152158 + 0.988356i \(0.548622\pi\)
\(24\) 0 0
\(25\) 54.0788 2.16315
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 31.8337i 1.09771i −0.835916 0.548857i \(-0.815063\pi\)
0.835916 0.548857i \(-0.184937\pi\)
\(30\) 0 0
\(31\) 12.3543i 0.398525i 0.979946 + 0.199262i \(0.0638546\pi\)
−0.979946 + 0.199262i \(0.936145\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −36.3604 −1.03887
\(36\) 0 0
\(37\) 4.89820i 0.132384i −0.997807 0.0661918i \(-0.978915\pi\)
0.997807 0.0661918i \(-0.0210849\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 73.1184i 1.78337i −0.452652 0.891687i \(-0.649522\pi\)
0.452652 0.891687i \(-0.350478\pi\)
\(42\) 0 0
\(43\) 23.8862 0.555492 0.277746 0.960655i \(-0.410413\pi\)
0.277746 + 0.960655i \(0.410413\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −43.3597 −0.922547 −0.461274 0.887258i \(-0.652607\pi\)
−0.461274 + 0.887258i \(0.652607\pi\)
\(48\) 0 0
\(49\) −32.2815 −0.658806
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 65.4462i 1.23483i 0.786636 + 0.617417i \(0.211821\pi\)
−0.786636 + 0.617417i \(0.788179\pi\)
\(54\) 0 0
\(55\) −38.9309 −0.707834
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 62.3993i 1.05761i −0.848742 0.528807i \(-0.822639\pi\)
0.848742 0.528807i \(-0.177361\pi\)
\(60\) 0 0
\(61\) −70.1235 −1.14957 −0.574783 0.818306i \(-0.694914\pi\)
−0.574783 + 0.818306i \(0.694914\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 122.425i 1.88346i
\(66\) 0 0
\(67\) 126.526i 1.88844i −0.329311 0.944222i \(-0.606816\pi\)
0.329311 0.944222i \(-0.393184\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 44.9262i 0.632763i −0.948632 0.316382i \(-0.897532\pi\)
0.948632 0.316382i \(-0.102468\pi\)
\(72\) 0 0
\(73\) −0.118545 −0.00162390 −0.000811950 1.00000i \(-0.500258\pi\)
−0.000811950 1.00000i \(0.500258\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 17.9004 0.232473
\(78\) 0 0
\(79\) 19.5093i 0.246953i 0.992347 + 0.123477i \(0.0394044\pi\)
−0.992347 + 0.123477i \(0.960596\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −97.8163 −1.17851 −0.589255 0.807947i \(-0.700579\pi\)
−0.589255 + 0.807947i \(0.700579\pi\)
\(84\) 0 0
\(85\) −125.321 −1.47436
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.6943i 0.176340i −0.996105 0.0881702i \(-0.971898\pi\)
0.996105 0.0881702i \(-0.0281019\pi\)
\(90\) 0 0
\(91\) 56.2911i 0.618583i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −103.254 + 133.739i −1.08688 + 1.40778i
\(96\) 0 0
\(97\) 6.98703i 0.0720312i −0.999351 0.0360156i \(-0.988533\pi\)
0.999351 0.0360156i \(-0.0114666\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 75.8791 0.751279 0.375639 0.926766i \(-0.377423\pi\)
0.375639 + 0.926766i \(0.377423\pi\)
\(102\) 0 0
\(103\) 172.570i 1.67544i −0.546100 0.837720i \(-0.683888\pi\)
0.546100 0.837720i \(-0.316112\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 169.804i 1.58695i 0.608602 + 0.793475i \(0.291730\pi\)
−0.608602 + 0.793475i \(0.708270\pi\)
\(108\) 0 0
\(109\) 23.5902i 0.216423i 0.994128 + 0.108212i \(0.0345124\pi\)
−0.994128 + 0.108212i \(0.965488\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 68.2272i 0.603781i −0.953343 0.301890i \(-0.902382\pi\)
0.953343 0.301890i \(-0.0976177\pi\)
\(114\) 0 0
\(115\) −62.2421 −0.541235
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 57.6225 0.484223
\(120\) 0 0
\(121\) −101.834 −0.841605
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 258.587 2.06870
\(126\) 0 0
\(127\) 4.69781i 0.0369906i −0.999829 0.0184953i \(-0.994112\pi\)
0.999829 0.0184953i \(-0.00588758\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −24.4766 −0.186844 −0.0934220 0.995627i \(-0.529781\pi\)
−0.0934220 + 0.995627i \(0.529781\pi\)
\(132\) 0 0
\(133\) 47.4761 61.4931i 0.356963 0.462354i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −66.9886 −0.488968 −0.244484 0.969653i \(-0.578619\pi\)
−0.244484 + 0.969653i \(0.578619\pi\)
\(138\) 0 0
\(139\) 182.137 1.31034 0.655169 0.755482i \(-0.272597\pi\)
0.655169 + 0.755482i \(0.272597\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 60.2705i 0.421472i
\(144\) 0 0
\(145\) 283.086i 1.95231i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 203.480 1.36564 0.682818 0.730589i \(-0.260754\pi\)
0.682818 + 0.730589i \(0.260754\pi\)
\(150\) 0 0
\(151\) 225.098i 1.49071i −0.666666 0.745357i \(-0.732279\pi\)
0.666666 0.745357i \(-0.267721\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 109.862i 0.708786i
\(156\) 0 0
\(157\) 72.8209 0.463827 0.231914 0.972736i \(-0.425501\pi\)
0.231914 + 0.972736i \(0.425501\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 28.6189 0.177757
\(162\) 0 0
\(163\) −274.540 −1.68430 −0.842148 0.539246i \(-0.818709\pi\)
−0.842148 + 0.539246i \(0.818709\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 162.610i 0.973711i 0.873483 + 0.486855i \(0.161856\pi\)
−0.873483 + 0.486855i \(0.838144\pi\)
\(168\) 0 0
\(169\) −20.5314 −0.121488
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 180.593i 1.04389i 0.852979 + 0.521946i \(0.174794\pi\)
−0.852979 + 0.521946i \(0.825206\pi\)
\(174\) 0 0
\(175\) −221.119 −1.26354
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.9660i 0.139475i 0.997565 + 0.0697375i \(0.0222162\pi\)
−0.997565 + 0.0697375i \(0.977784\pi\)
\(180\) 0 0
\(181\) 73.8968i 0.408269i 0.978943 + 0.204135i \(0.0654380\pi\)
−0.978943 + 0.204135i \(0.934562\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 43.5578i 0.235448i
\(186\) 0 0
\(187\) 61.6961 0.329925
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −199.072 −1.04226 −0.521130 0.853477i \(-0.674489\pi\)
−0.521130 + 0.853477i \(0.674489\pi\)
\(192\) 0 0
\(193\) 225.689i 1.16937i −0.811259 0.584687i \(-0.801217\pi\)
0.811259 0.584687i \(-0.198783\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 184.669 0.937405 0.468703 0.883356i \(-0.344722\pi\)
0.468703 + 0.883356i \(0.344722\pi\)
\(198\) 0 0
\(199\) −341.471 −1.71594 −0.857968 0.513703i \(-0.828273\pi\)
−0.857968 + 0.513703i \(0.828273\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 130.163i 0.641196i
\(204\) 0 0
\(205\) 650.214i 3.17178i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 50.8324 65.8403i 0.243217 0.315025i
\(210\) 0 0
\(211\) 11.7087i 0.0554916i 0.999615 + 0.0277458i \(0.00883290\pi\)
−0.999615 + 0.0277458i \(0.991167\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 212.411 0.987957
\(216\) 0 0
\(217\) 50.5145i 0.232786i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 194.014i 0.877893i
\(222\) 0 0
\(223\) 107.679i 0.482866i −0.970418 0.241433i \(-0.922383\pi\)
0.970418 0.241433i \(-0.0776174\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 231.845i 1.02134i −0.859776 0.510671i \(-0.829397\pi\)
0.859776 0.510671i \(-0.170603\pi\)
\(228\) 0 0
\(229\) −94.0527 −0.410710 −0.205355 0.978688i \(-0.565835\pi\)
−0.205355 + 0.978688i \(0.565835\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 64.2586 0.275788 0.137894 0.990447i \(-0.455967\pi\)
0.137894 + 0.990447i \(0.455967\pi\)
\(234\) 0 0
\(235\) −385.582 −1.64077
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 254.410 1.06448 0.532239 0.846594i \(-0.321351\pi\)
0.532239 + 0.846594i \(0.321351\pi\)
\(240\) 0 0
\(241\) 360.169i 1.49448i 0.664557 + 0.747238i \(0.268620\pi\)
−0.664557 + 0.747238i \(0.731380\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −287.067 −1.17170
\(246\) 0 0
\(247\) 207.047 + 159.852i 0.838246 + 0.647172i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 190.633 0.759494 0.379747 0.925090i \(-0.376011\pi\)
0.379747 + 0.925090i \(0.376011\pi\)
\(252\) 0 0
\(253\) 30.6421 0.121115
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.0030i 0.101179i −0.998720 0.0505895i \(-0.983890\pi\)
0.998720 0.0505895i \(-0.0161100\pi\)
\(258\) 0 0
\(259\) 20.0279i 0.0773278i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −187.438 −0.712693 −0.356346 0.934354i \(-0.615978\pi\)
−0.356346 + 0.934354i \(0.615978\pi\)
\(264\) 0 0
\(265\) 581.989i 2.19618i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 345.243i 1.28343i 0.766943 + 0.641715i \(0.221777\pi\)
−0.766943 + 0.641715i \(0.778223\pi\)
\(270\) 0 0
\(271\) 196.509 0.725124 0.362562 0.931960i \(-0.381902\pi\)
0.362562 + 0.931960i \(0.381902\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −236.751 −0.860911
\(276\) 0 0
\(277\) −501.779 −1.81148 −0.905738 0.423839i \(-0.860682\pi\)
−0.905738 + 0.423839i \(0.860682\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 448.171i 1.59492i −0.603375 0.797458i \(-0.706178\pi\)
0.603375 0.797458i \(-0.293822\pi\)
\(282\) 0 0
\(283\) 186.965 0.660655 0.330328 0.943866i \(-0.392841\pi\)
0.330328 + 0.943866i \(0.392841\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 298.969i 1.04170i
\(288\) 0 0
\(289\) −90.3966 −0.312791
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 152.099i 0.519111i 0.965728 + 0.259555i \(0.0835760\pi\)
−0.965728 + 0.259555i \(0.916424\pi\)
\(294\) 0 0
\(295\) 554.893i 1.88099i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 96.3595i 0.322273i
\(300\) 0 0
\(301\) −97.6665 −0.324473
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −623.582 −2.04453
\(306\) 0 0
\(307\) 223.721i 0.728733i 0.931256 + 0.364367i \(0.118714\pi\)
−0.931256 + 0.364367i \(0.881286\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −107.817 −0.346678 −0.173339 0.984862i \(-0.555456\pi\)
−0.173339 + 0.984862i \(0.555456\pi\)
\(312\) 0 0
\(313\) −18.9501 −0.0605435 −0.0302718 0.999542i \(-0.509637\pi\)
−0.0302718 + 0.999542i \(0.509637\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 316.377i 0.998034i −0.866592 0.499017i \(-0.833694\pi\)
0.866592 0.499017i \(-0.166306\pi\)
\(318\) 0 0
\(319\) 139.364i 0.436879i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 163.633 211.944i 0.506602 0.656173i
\(324\) 0 0
\(325\) 744.505i 2.29079i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 177.291 0.538877
\(330\) 0 0
\(331\) 562.873i 1.70052i −0.526362 0.850261i \(-0.676444\pi\)
0.526362 0.850261i \(-0.323556\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1125.15i 3.35864i
\(336\) 0 0
\(337\) 335.863i 0.996628i 0.866997 + 0.498314i \(0.166047\pi\)
−0.866997 + 0.498314i \(0.833953\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 54.0855i 0.158609i
\(342\) 0 0
\(343\) 332.346 0.968939
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 288.891 0.832538 0.416269 0.909241i \(-0.363337\pi\)
0.416269 + 0.909241i \(0.363337\pi\)
\(348\) 0 0
\(349\) −332.076 −0.951508 −0.475754 0.879578i \(-0.657825\pi\)
−0.475754 + 0.879578i \(0.657825\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 100.942 0.285954 0.142977 0.989726i \(-0.454333\pi\)
0.142977 + 0.989726i \(0.454333\pi\)
\(354\) 0 0
\(355\) 399.512i 1.12539i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −126.478 −0.352306 −0.176153 0.984363i \(-0.556365\pi\)
−0.176153 + 0.984363i \(0.556365\pi\)
\(360\) 0 0
\(361\) −91.3610 349.248i −0.253078 0.967446i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.05417 −0.00288815
\(366\) 0 0
\(367\) −385.079 −1.04926 −0.524631 0.851330i \(-0.675797\pi\)
−0.524631 + 0.851330i \(0.675797\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 267.598i 0.721290i
\(372\) 0 0
\(373\) 188.228i 0.504632i −0.967645 0.252316i \(-0.918808\pi\)
0.967645 0.252316i \(-0.0811922\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −438.256 −1.16248
\(378\) 0 0
\(379\) 694.899i 1.83351i −0.399453 0.916754i \(-0.630800\pi\)
0.399453 0.916754i \(-0.369200\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 496.495i 1.29633i −0.761500 0.648165i \(-0.775537\pi\)
0.761500 0.648165i \(-0.224463\pi\)
\(384\) 0 0
\(385\) 159.182 0.413459
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −454.909 −1.16943 −0.584715 0.811239i \(-0.698794\pi\)
−0.584715 + 0.811239i \(0.698794\pi\)
\(390\) 0 0
\(391\) 98.6387 0.252273
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 173.489i 0.439213i
\(396\) 0 0
\(397\) 56.0875 0.141278 0.0706392 0.997502i \(-0.477496\pi\)
0.0706392 + 0.997502i \(0.477496\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.3922i 0.0533470i 0.999644 + 0.0266735i \(0.00849145\pi\)
−0.999644 + 0.0266735i \(0.991509\pi\)
\(402\) 0 0
\(403\) 170.082 0.422039
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.4437i 0.0526873i
\(408\) 0 0
\(409\) 259.092i 0.633476i 0.948513 + 0.316738i \(0.102588\pi\)
−0.948513 + 0.316738i \(0.897412\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 255.140i 0.617772i
\(414\) 0 0
\(415\) −869.844 −2.09601
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −822.708 −1.96350 −0.981752 0.190166i \(-0.939097\pi\)
−0.981752 + 0.190166i \(0.939097\pi\)
\(420\) 0 0
\(421\) 512.963i 1.21844i −0.793002 0.609219i \(-0.791483\pi\)
0.793002 0.609219i \(-0.208517\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −762.115 −1.79321
\(426\) 0 0
\(427\) 286.723 0.671483
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 217.995i 0.505789i −0.967494 0.252894i \(-0.918617\pi\)
0.967494 0.252894i \(-0.0813826\pi\)
\(432\) 0 0
\(433\) 474.721i 1.09635i −0.836363 0.548177i \(-0.815322\pi\)
0.836363 0.548177i \(-0.184678\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 81.2700 105.264i 0.185973 0.240880i
\(438\) 0 0
\(439\) 610.886i 1.39154i 0.718264 + 0.695770i \(0.244937\pi\)
−0.718264 + 0.695770i \(0.755063\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 743.578 1.67851 0.839253 0.543741i \(-0.182992\pi\)
0.839253 + 0.543741i \(0.182992\pi\)
\(444\) 0 0
\(445\) 139.564i 0.313626i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 244.636i 0.544846i 0.962178 + 0.272423i \(0.0878249\pi\)
−0.962178 + 0.272423i \(0.912175\pi\)
\(450\) 0 0
\(451\) 320.104i 0.709764i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 500.576i 1.10017i
\(456\) 0 0
\(457\) 456.513 0.998934 0.499467 0.866333i \(-0.333529\pi\)
0.499467 + 0.866333i \(0.333529\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −230.954 −0.500986 −0.250493 0.968118i \(-0.580593\pi\)
−0.250493 + 0.968118i \(0.580593\pi\)
\(462\) 0 0
\(463\) 438.242 0.946527 0.473264 0.880921i \(-0.343076\pi\)
0.473264 + 0.880921i \(0.343076\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 443.535 0.949753 0.474877 0.880052i \(-0.342493\pi\)
0.474877 + 0.880052i \(0.342493\pi\)
\(468\) 0 0
\(469\) 517.342i 1.10307i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −104.571 −0.221080
\(474\) 0 0
\(475\) −627.919 + 813.307i −1.32193 + 1.71223i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 263.159 0.549393 0.274696 0.961531i \(-0.411423\pi\)
0.274696 + 0.961531i \(0.411423\pi\)
\(480\) 0 0
\(481\) −67.4337 −0.140195
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 62.1331i 0.128109i
\(486\) 0 0
\(487\) 270.643i 0.555735i −0.960619 0.277868i \(-0.910372\pi\)
0.960619 0.277868i \(-0.0896276\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 331.509 0.675172 0.337586 0.941295i \(-0.390390\pi\)
0.337586 + 0.941295i \(0.390390\pi\)
\(492\) 0 0
\(493\) 448.622i 0.909985i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 183.696i 0.369609i
\(498\) 0 0
\(499\) −846.591 −1.69658 −0.848288 0.529536i \(-0.822366\pi\)
−0.848288 + 0.529536i \(0.822366\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 714.432 1.42034 0.710171 0.704030i \(-0.248618\pi\)
0.710171 + 0.704030i \(0.248618\pi\)
\(504\) 0 0
\(505\) 674.765 1.33617
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 171.981i 0.337880i −0.985626 0.168940i \(-0.945966\pi\)
0.985626 0.168940i \(-0.0540345\pi\)
\(510\) 0 0
\(511\) 0.484709 0.000948550
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1534.60i 2.97981i
\(516\) 0 0
\(517\) 189.824 0.367164
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 624.308i 1.19829i 0.800641 + 0.599144i \(0.204492\pi\)
−0.800641 + 0.599144i \(0.795508\pi\)
\(522\) 0 0
\(523\) 874.036i 1.67120i −0.549340 0.835599i \(-0.685121\pi\)
0.549340 0.835599i \(-0.314879\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 174.105i 0.330369i
\(528\) 0 0
\(529\) −480.010 −0.907391
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1006.62 −1.88860
\(534\) 0 0
\(535\) 1510.00i 2.82243i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 141.324 0.262198
\(540\) 0 0
\(541\) −382.026 −0.706149 −0.353074 0.935595i \(-0.614864\pi\)
−0.353074 + 0.935595i \(0.614864\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 209.778i 0.384915i
\(546\) 0 0
\(547\) 494.309i 0.903672i 0.892101 + 0.451836i \(0.149231\pi\)
−0.892101 + 0.451836i \(0.850769\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 478.757 + 369.627i 0.868888 + 0.670830i
\(552\) 0 0
\(553\) 79.7703i 0.144250i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 987.867 1.77355 0.886775 0.462201i \(-0.152940\pi\)
0.886775 + 0.462201i \(0.152940\pi\)
\(558\) 0 0
\(559\) 328.842i 0.588268i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 444.761i 0.789984i 0.918685 + 0.394992i \(0.129253\pi\)
−0.918685 + 0.394992i \(0.870747\pi\)
\(564\) 0 0
\(565\) 606.719i 1.07384i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 760.247i 1.33611i 0.744111 + 0.668056i \(0.232873\pi\)
−0.744111 + 0.668056i \(0.767127\pi\)
\(570\) 0 0
\(571\) −587.441 −1.02879 −0.514397 0.857552i \(-0.671984\pi\)
−0.514397 + 0.857552i \(0.671984\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −378.513 −0.658284
\(576\) 0 0
\(577\) 470.148 0.814814 0.407407 0.913247i \(-0.366433\pi\)
0.407407 + 0.913247i \(0.366433\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 399.954 0.688389
\(582\) 0 0
\(583\) 286.516i 0.491451i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −559.569 −0.953269 −0.476634 0.879102i \(-0.658143\pi\)
−0.476634 + 0.879102i \(0.658143\pi\)
\(588\) 0 0
\(589\) −185.799 143.448i −0.315449 0.243544i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 923.095 1.55665 0.778326 0.627860i \(-0.216069\pi\)
0.778326 + 0.627860i \(0.216069\pi\)
\(594\) 0 0
\(595\) 512.416 0.861203
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 569.367i 0.950529i 0.879843 + 0.475264i \(0.157648\pi\)
−0.879843 + 0.475264i \(0.842352\pi\)
\(600\) 0 0
\(601\) 575.057i 0.956833i 0.878133 + 0.478417i \(0.158789\pi\)
−0.878133 + 0.478417i \(0.841211\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −905.573 −1.49682
\(606\) 0 0
\(607\) 941.747i 1.55148i −0.631054 0.775739i \(-0.717377\pi\)
0.631054 0.775739i \(-0.282623\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 596.935i 0.976981i
\(612\) 0 0
\(613\) −835.577 −1.36309 −0.681547 0.731774i \(-0.738693\pi\)
−0.681547 + 0.731774i \(0.738693\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 725.675 1.17613 0.588067 0.808812i \(-0.299889\pi\)
0.588067 + 0.808812i \(0.299889\pi\)
\(618\) 0 0
\(619\) 539.760 0.871988 0.435994 0.899950i \(-0.356397\pi\)
0.435994 + 0.899950i \(0.356397\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 64.1713i 0.103004i
\(624\) 0 0
\(625\) 947.548 1.51608
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 69.0287i 0.109744i
\(630\) 0 0
\(631\) 1166.75 1.84905 0.924524 0.381123i \(-0.124463\pi\)
0.924524 + 0.381123i \(0.124463\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 41.7759i 0.0657888i
\(636\) 0 0
\(637\) 444.420i 0.697677i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1000.68i 1.56112i 0.625083 + 0.780558i \(0.285065\pi\)
−0.625083 + 0.780558i \(0.714935\pi\)
\(642\) 0 0
\(643\) 857.488 1.33357 0.666787 0.745248i \(-0.267669\pi\)
0.666787 + 0.745248i \(0.267669\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −83.5434 −0.129124 −0.0645621 0.997914i \(-0.520565\pi\)
−0.0645621 + 0.997914i \(0.520565\pi\)
\(648\) 0 0
\(649\) 273.177i 0.420919i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −482.731 −0.739252 −0.369626 0.929181i \(-0.620514\pi\)
−0.369626 + 0.929181i \(0.620514\pi\)
\(654\) 0 0
\(655\) −217.661 −0.332307
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 292.175i 0.443360i 0.975119 + 0.221680i \(0.0711541\pi\)
−0.975119 + 0.221680i \(0.928846\pi\)
\(660\) 0 0
\(661\) 524.668i 0.793748i −0.917873 0.396874i \(-0.870095\pi\)
0.917873 0.396874i \(-0.129905\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 422.188 546.835i 0.634868 0.822309i
\(666\) 0 0
\(667\) 222.813i 0.334053i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 306.992 0.457515
\(672\) 0 0
\(673\) 933.628i 1.38726i 0.720330 + 0.693632i \(0.243990\pi\)
−0.720330 + 0.693632i \(0.756010\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 221.809i 0.327635i 0.986491 + 0.163818i \(0.0523809\pi\)
−0.986491 + 0.163818i \(0.947619\pi\)
\(678\) 0 0
\(679\) 28.5688i 0.0420748i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1042.38i 1.52618i 0.646293 + 0.763090i \(0.276319\pi\)
−0.646293 + 0.763090i \(0.723681\pi\)
\(684\) 0 0
\(685\) −595.705 −0.869642
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 901.001 1.30769
\(690\) 0 0
\(691\) −296.453 −0.429020 −0.214510 0.976722i \(-0.568815\pi\)
−0.214510 + 0.976722i \(0.568815\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1619.68 2.33047
\(696\) 0 0
\(697\) 1030.43i 1.47838i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1000.32 −1.42700 −0.713499 0.700657i \(-0.752890\pi\)
−0.713499 + 0.700657i \(0.752890\pi\)
\(702\) 0 0
\(703\) 73.6654 + 56.8738i 0.104787 + 0.0809016i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −310.257 −0.438836
\(708\) 0 0
\(709\) −198.456 −0.279909 −0.139955 0.990158i \(-0.544696\pi\)
−0.139955 + 0.990158i \(0.544696\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 86.4711i 0.121278i
\(714\) 0 0
\(715\) 535.963i 0.749598i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −648.586 −0.902067 −0.451033 0.892507i \(-0.648944\pi\)
−0.451033 + 0.892507i \(0.648944\pi\)
\(720\) 0 0
\(721\) 705.611i 0.978656i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1721.53i 2.37452i
\(726\) 0 0
\(727\) 1282.65 1.76431 0.882155 0.470960i \(-0.156092\pi\)
0.882155 + 0.470960i \(0.156092\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −336.620 −0.460492
\(732\) 0 0
\(733\) 892.888 1.21813 0.609064 0.793121i \(-0.291545\pi\)
0.609064 + 0.793121i \(0.291545\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 553.915i 0.751580i
\(738\) 0 0
\(739\) 1150.31 1.55658 0.778290 0.627905i \(-0.216087\pi\)
0.778290 + 0.627905i \(0.216087\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 436.802i 0.587889i 0.955822 + 0.293945i \(0.0949681\pi\)
−0.955822 + 0.293945i \(0.905032\pi\)
\(744\) 0 0
\(745\) 1809.47 2.42882
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 694.298i 0.926967i
\(750\) 0 0
\(751\) 491.952i 0.655063i 0.944840 + 0.327531i \(0.106217\pi\)
−0.944840 + 0.327531i \(0.893783\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2001.71i 2.65127i
\(756\) 0 0
\(757\) −1120.04 −1.47958 −0.739789 0.672839i \(-0.765075\pi\)
−0.739789 + 0.672839i \(0.765075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 235.547 0.309523 0.154761 0.987952i \(-0.450539\pi\)
0.154761 + 0.987952i \(0.450539\pi\)
\(762\) 0 0
\(763\) 96.4561i 0.126417i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −859.053 −1.12002
\(768\) 0 0
\(769\) 558.287 0.725991 0.362996 0.931791i \(-0.381754\pi\)
0.362996 + 0.931791i \(0.381754\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 964.213i 1.24737i 0.781678 + 0.623683i \(0.214364\pi\)
−0.781678 + 0.623683i \(0.785636\pi\)
\(774\) 0 0
\(775\) 668.104i 0.862070i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1099.65 + 848.991i 1.41162 + 1.08985i
\(780\) 0 0
\(781\) 196.682i 0.251833i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 647.569 0.824928
\(786\) 0 0
\(787\) 734.077i 0.932754i 0.884586 + 0.466377i \(0.154441\pi\)
−0.884586 + 0.466377i \(0.845559\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 278.969i 0.352680i
\(792\) 0 0
\(793\) 965.393i 1.21739i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1208.84i 1.51673i −0.651827 0.758367i \(-0.725997\pi\)
0.651827 0.758367i \(-0.274003\pi\)
\(798\) 0 0
\(799\) 611.054 0.764774
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.518975 0.000646295
\(804\) 0 0
\(805\) 254.497 0.316146
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1294.97 1.60070 0.800349 0.599534i \(-0.204647\pi\)
0.800349 + 0.599534i \(0.204647\pi\)
\(810\) 0 0
\(811\) 698.570i 0.861369i −0.902503 0.430685i \(-0.858272\pi\)
0.902503 0.430685i \(-0.141728\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2441.38 −2.99556
\(816\) 0 0
\(817\) −277.347 + 359.231i −0.339470 + 0.439695i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 642.039 0.782021 0.391011 0.920386i \(-0.372126\pi\)
0.391011 + 0.920386i \(0.372126\pi\)
\(822\) 0 0
\(823\) −808.191 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1092.17i 1.32064i 0.750983 + 0.660321i \(0.229580\pi\)
−0.750983 + 0.660321i \(0.770420\pi\)
\(828\) 0 0
\(829\) 552.397i 0.666341i −0.942867 0.333170i \(-0.891882\pi\)
0.942867 0.333170i \(-0.108118\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 454.932 0.546137
\(834\) 0 0
\(835\) 1446.03i 1.73177i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 308.616i 0.367838i 0.982941 + 0.183919i \(0.0588784\pi\)
−0.982941 + 0.183919i \(0.941122\pi\)
\(840\) 0 0
\(841\) −172.387 −0.204978
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −182.578 −0.216069
\(846\) 0 0
\(847\) 416.382 0.491597
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 34.2839i 0.0402866i
\(852\) 0 0
\(853\) 513.281 0.601737 0.300868 0.953666i \(-0.402724\pi\)
0.300868 + 0.953666i \(0.402724\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 902.528i 1.05312i 0.850137 + 0.526562i \(0.176519\pi\)
−0.850137 + 0.526562i \(0.823481\pi\)
\(858\) 0 0
\(859\) −279.345 −0.325197 −0.162599 0.986692i \(-0.551988\pi\)
−0.162599 + 0.986692i \(0.551988\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 974.538i 1.12924i −0.825349 0.564622i \(-0.809022\pi\)
0.825349 0.564622i \(-0.190978\pi\)
\(864\) 0 0
\(865\) 1605.95i 1.85659i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 85.4095i 0.0982848i
\(870\) 0 0
\(871\) −1741.88 −1.99987
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1057.32 −1.20836
\(876\) 0 0
\(877\) 1355.21i 1.54527i −0.634848 0.772637i \(-0.718937\pi\)
0.634848 0.772637i \(-0.281063\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1598.89 −1.81486 −0.907431 0.420200i \(-0.861960\pi\)
−0.907431 + 0.420200i \(0.861960\pi\)
\(882\) 0 0
\(883\) 298.404 0.337944 0.168972 0.985621i \(-0.445955\pi\)
0.168972 + 0.985621i \(0.445955\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 277.416i 0.312757i 0.987697 + 0.156379i \(0.0499820\pi\)
−0.987697 + 0.156379i \(0.950018\pi\)
\(888\) 0 0
\(889\) 19.2085i 0.0216069i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 503.458 652.100i 0.563782 0.730235i
\(894\) 0 0
\(895\) 222.014i 0.248060i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 393.282 0.437467
\(900\) 0 0
\(901\) 922.312i 1.02365i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 657.136i 0.726117i
\(906\) 0 0
\(907\) 588.225i 0.648539i 0.945965 + 0.324270i \(0.105119\pi\)
−0.945965 + 0.324270i \(0.894881\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 456.868i 0.501502i −0.968052 0.250751i \(-0.919323\pi\)
0.968052 0.250751i \(-0.0806775\pi\)
\(912\) 0 0
\(913\) 428.228 0.469034
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 100.080 0.109139
\(918\) 0 0
\(919\) 3.98423 0.00433540 0.00216770 0.999998i \(-0.499310\pi\)
0.00216770 + 0.999998i \(0.499310\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −618.501 −0.670099
\(924\) 0 0
\(925\) 264.889i 0.286366i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −742.264 −0.798993 −0.399496 0.916735i \(-0.630815\pi\)
−0.399496 + 0.916735i \(0.630815\pi\)
\(930\) 0 0
\(931\) 374.826 485.491i 0.402606 0.521472i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 548.640 0.586781
\(936\) 0 0
\(937\) 415.097 0.443006 0.221503 0.975160i \(-0.428904\pi\)
0.221503 + 0.975160i \(0.428904\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 883.401i 0.938790i −0.882988 0.469395i \(-0.844472\pi\)
0.882988 0.469395i \(-0.155528\pi\)
\(942\) 0 0
\(943\) 511.777i 0.542711i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 465.843 0.491915 0.245957 0.969281i \(-0.420898\pi\)
0.245957 + 0.969281i \(0.420898\pi\)
\(948\) 0 0
\(949\) 1.63201i 0.00171972i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 479.320i 0.502959i −0.967863 0.251479i \(-0.919083\pi\)
0.967863 0.251479i \(-0.0809171\pi\)
\(954\) 0 0
\(955\) −1770.27 −1.85369
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 273.905 0.285615
\(960\) 0 0
\(961\) 808.372 0.841178
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2006.97i 2.07976i
\(966\) 0 0
\(967\) −48.0426 −0.0496821 −0.0248411 0.999691i \(-0.507908\pi\)
−0.0248411 + 0.999691i \(0.507908\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 954.276i 0.982776i 0.870941 + 0.491388i \(0.163510\pi\)
−0.870941 + 0.491388i \(0.836490\pi\)
\(972\) 0 0
\(973\) −744.727 −0.765393
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 230.509i 0.235935i 0.993017 + 0.117968i \(0.0376379\pi\)
−0.993017 + 0.117968i \(0.962362\pi\)
\(978\) 0 0
\(979\) 68.7078i 0.0701816i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 525.483i 0.534571i 0.963617 + 0.267285i \(0.0861266\pi\)
−0.963617 + 0.267285i \(0.913873\pi\)
\(984\) 0 0
\(985\) 1642.19 1.66720
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −167.186 −0.169046
\(990\) 0 0
\(991\) 988.597i 0.997575i −0.866724 0.498787i \(-0.833779\pi\)
0.866724 0.498787i \(-0.166221\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3036.58 −3.05184
\(996\) 0 0
\(997\) 307.985 0.308912 0.154456 0.988000i \(-0.450638\pi\)
0.154456 + 0.988000i \(0.450638\pi\)
\(998\) 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 2736.3.o.q.721.19 20
3.2 odd 2 inner 2736.3.o.q.721.1 20
4.3 odd 2 1368.3.o.d.721.19 yes 20
12.11 even 2 1368.3.o.d.721.1 20
19.18 odd 2 inner 2736.3.o.q.721.20 20
57.56 even 2 inner 2736.3.o.q.721.2 20
76.75 even 2 1368.3.o.d.721.20 yes 20
228.227 odd 2 1368.3.o.d.721.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.3.o.d.721.1 20 12.11 even 2
1368.3.o.d.721.2 yes 20 228.227 odd 2
1368.3.o.d.721.19 yes 20 4.3 odd 2
1368.3.o.d.721.20 yes 20 76.75 even 2
2736.3.o.q.721.1 20 3.2 odd 2 inner
2736.3.o.q.721.2 20 57.56 even 2 inner
2736.3.o.q.721.19 20 1.1 even 1 trivial
2736.3.o.q.721.20 20 19.18 odd 2 inner