Properties

Label 1080.2.a.m.1.2
Level $1080$
Weight $2$
Character 1080.1
Self dual yes
Analytic conductor $8.624$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,2,Mod(1,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.77200\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +4.77200 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +4.77200 q^{7} -3.77200 q^{11} +3.00000 q^{13} +1.77200 q^{17} -2.77200 q^{19} +7.77200 q^{23} +1.00000 q^{25} +3.77200 q^{29} +2.22800 q^{31} -4.77200 q^{35} -0.772002 q^{37} -9.54400 q^{41} -1.77200 q^{43} +5.77200 q^{47} +15.7720 q^{49} -9.54400 q^{53} +3.77200 q^{55} +12.0000 q^{59} +0.772002 q^{61} -3.00000 q^{65} +14.7720 q^{67} +13.5440 q^{71} +4.77200 q^{73} -18.0000 q^{77} -5.00000 q^{79} +6.00000 q^{83} -1.77200 q^{85} +8.00000 q^{89} +14.3160 q^{91} +2.77200 q^{95} -18.3160 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + q^{7} + q^{11} + 6 q^{13} - 5 q^{17} + 3 q^{19} + 7 q^{23} + 2 q^{25} - q^{29} + 13 q^{31} - q^{35} + 7 q^{37} - 2 q^{41} + 5 q^{43} + 3 q^{47} + 23 q^{49} - 2 q^{53} - q^{55} + 24 q^{59} - 7 q^{61} - 6 q^{65} + 21 q^{67} + 10 q^{71} + q^{73} - 36 q^{77} - 10 q^{79} + 12 q^{83} + 5 q^{85} + 16 q^{89} + 3 q^{91} - 3 q^{95} - 11 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.77200 1.80365 0.901824 0.432104i \(-0.142229\pi\)
0.901824 + 0.432104i \(0.142229\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.77200 −1.13730 −0.568651 0.822579i \(-0.692534\pi\)
−0.568651 + 0.822579i \(0.692534\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.77200 0.429774 0.214887 0.976639i \(-0.431062\pi\)
0.214887 + 0.976639i \(0.431062\pi\)
\(18\) 0 0
\(19\) −2.77200 −0.635941 −0.317970 0.948101i \(-0.603001\pi\)
−0.317970 + 0.948101i \(0.603001\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.77200 1.62057 0.810287 0.586033i \(-0.199311\pi\)
0.810287 + 0.586033i \(0.199311\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.77200 0.700443 0.350222 0.936667i \(-0.386106\pi\)
0.350222 + 0.936667i \(0.386106\pi\)
\(30\) 0 0
\(31\) 2.22800 0.400160 0.200080 0.979780i \(-0.435880\pi\)
0.200080 + 0.979780i \(0.435880\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.77200 −0.806616
\(36\) 0 0
\(37\) −0.772002 −0.126916 −0.0634582 0.997984i \(-0.520213\pi\)
−0.0634582 + 0.997984i \(0.520213\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.54400 −1.49052 −0.745261 0.666772i \(-0.767675\pi\)
−0.745261 + 0.666772i \(0.767675\pi\)
\(42\) 0 0
\(43\) −1.77200 −0.270228 −0.135114 0.990830i \(-0.543140\pi\)
−0.135114 + 0.990830i \(0.543140\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.77200 0.841933 0.420967 0.907076i \(-0.361691\pi\)
0.420967 + 0.907076i \(0.361691\pi\)
\(48\) 0 0
\(49\) 15.7720 2.25314
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.54400 −1.31097 −0.655485 0.755208i \(-0.727536\pi\)
−0.655485 + 0.755208i \(0.727536\pi\)
\(54\) 0 0
\(55\) 3.77200 0.508617
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 0.772002 0.0988447 0.0494224 0.998778i \(-0.484262\pi\)
0.0494224 + 0.998778i \(0.484262\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 14.7720 1.80469 0.902344 0.431017i \(-0.141845\pi\)
0.902344 + 0.431017i \(0.141845\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.5440 1.60738 0.803689 0.595050i \(-0.202868\pi\)
0.803689 + 0.595050i \(0.202868\pi\)
\(72\) 0 0
\(73\) 4.77200 0.558521 0.279260 0.960215i \(-0.409911\pi\)
0.279260 + 0.960215i \(0.409911\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −18.0000 −2.05129
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −1.77200 −0.192201
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 14.3160 1.50073
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.77200 0.284401
\(96\) 0 0
\(97\) −18.3160 −1.85971 −0.929854 0.367928i \(-0.880067\pi\)
−0.929854 + 0.367928i \(0.880067\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.2280 −1.01772 −0.508862 0.860848i \(-0.669934\pi\)
−0.508862 + 0.860848i \(0.669934\pi\)
\(102\) 0 0
\(103\) 16.7720 1.65259 0.826297 0.563234i \(-0.190443\pi\)
0.826297 + 0.563234i \(0.190443\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.0000 −1.35343 −0.676716 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 0 0
\(109\) 9.54400 0.914150 0.457075 0.889428i \(-0.348897\pi\)
0.457075 + 0.889428i \(0.348897\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.3160 1.06452 0.532260 0.846581i \(-0.321343\pi\)
0.532260 + 0.846581i \(0.321343\pi\)
\(114\) 0 0
\(115\) −7.77200 −0.724743
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.45600 0.775160
\(120\) 0 0
\(121\) 3.22800 0.293454
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −11.5440 −1.02436 −0.512182 0.858877i \(-0.671163\pi\)
−0.512182 + 0.858877i \(0.671163\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.22800 0.544143 0.272071 0.962277i \(-0.412291\pi\)
0.272071 + 0.962277i \(0.412291\pi\)
\(132\) 0 0
\(133\) −13.2280 −1.14701
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.0880 −1.80167 −0.900835 0.434162i \(-0.857045\pi\)
−0.900835 + 0.434162i \(0.857045\pi\)
\(138\) 0 0
\(139\) −12.7720 −1.08331 −0.541654 0.840602i \(-0.682202\pi\)
−0.541654 + 0.840602i \(0.682202\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −11.3160 −0.946292
\(144\) 0 0
\(145\) −3.77200 −0.313248
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.77200 −0.800554 −0.400277 0.916394i \(-0.631086\pi\)
−0.400277 + 0.916394i \(0.631086\pi\)
\(150\) 0 0
\(151\) −24.0880 −1.96025 −0.980127 0.198370i \(-0.936435\pi\)
−0.980127 + 0.198370i \(0.936435\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.22800 −0.178957
\(156\) 0 0
\(157\) 7.77200 0.620273 0.310137 0.950692i \(-0.399625\pi\)
0.310137 + 0.950692i \(0.399625\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 37.0880 2.92294
\(162\) 0 0
\(163\) −8.54400 −0.669218 −0.334609 0.942357i \(-0.608604\pi\)
−0.334609 + 0.942357i \(0.608604\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −23.0880 −1.78660 −0.893302 0.449457i \(-0.851617\pi\)
−0.893302 + 0.449457i \(0.851617\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.54400 0.573560 0.286780 0.957996i \(-0.407415\pi\)
0.286780 + 0.957996i \(0.407415\pi\)
\(174\) 0 0
\(175\) 4.77200 0.360729
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.54400 0.713352 0.356676 0.934228i \(-0.383910\pi\)
0.356676 + 0.934228i \(0.383910\pi\)
\(180\) 0 0
\(181\) −24.3160 −1.80739 −0.903697 0.428172i \(-0.859158\pi\)
−0.903697 + 0.428172i \(0.859158\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.772002 0.0567587
\(186\) 0 0
\(187\) −6.68399 −0.488782
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.54400 0.256435 0.128218 0.991746i \(-0.459074\pi\)
0.128218 + 0.991746i \(0.459074\pi\)
\(192\) 0 0
\(193\) −12.7720 −0.919349 −0.459674 0.888088i \(-0.652034\pi\)
−0.459674 + 0.888088i \(0.652034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.0000 −1.42494 −0.712470 0.701702i \(-0.752424\pi\)
−0.712470 + 0.701702i \(0.752424\pi\)
\(198\) 0 0
\(199\) −15.0000 −1.06332 −0.531661 0.846957i \(-0.678432\pi\)
−0.531661 + 0.846957i \(0.678432\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 18.0000 1.26335
\(204\) 0 0
\(205\) 9.54400 0.666582
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.4560 0.723256
\(210\) 0 0
\(211\) 14.3160 0.985554 0.492777 0.870156i \(-0.335982\pi\)
0.492777 + 0.870156i \(0.335982\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.77200 0.120850
\(216\) 0 0
\(217\) 10.6320 0.721748
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.31601 0.357593
\(222\) 0 0
\(223\) 15.0880 1.01037 0.505184 0.863012i \(-0.331425\pi\)
0.505184 + 0.863012i \(0.331425\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.54400 0.102479 0.0512396 0.998686i \(-0.483683\pi\)
0.0512396 + 0.998686i \(0.483683\pi\)
\(228\) 0 0
\(229\) 10.4560 0.690952 0.345476 0.938428i \(-0.387718\pi\)
0.345476 + 0.938428i \(0.387718\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.4560 0.816019 0.408010 0.912978i \(-0.366223\pi\)
0.408010 + 0.912978i \(0.366223\pi\)
\(234\) 0 0
\(235\) −5.77200 −0.376524
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 8.08801 0.520994 0.260497 0.965475i \(-0.416114\pi\)
0.260497 + 0.965475i \(0.416114\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −15.7720 −1.00764
\(246\) 0 0
\(247\) −8.31601 −0.529135
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.227998 0.0143911 0.00719556 0.999974i \(-0.497710\pi\)
0.00719556 + 0.999974i \(0.497710\pi\)
\(252\) 0 0
\(253\) −29.3160 −1.84308
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.22800 −0.138979 −0.0694894 0.997583i \(-0.522137\pi\)
−0.0694894 + 0.997583i \(0.522137\pi\)
\(258\) 0 0
\(259\) −3.68399 −0.228912
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 9.54400 0.586283
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.31601 −0.324123 −0.162061 0.986781i \(-0.551814\pi\)
−0.162061 + 0.986781i \(0.551814\pi\)
\(270\) 0 0
\(271\) −10.7720 −0.654353 −0.327176 0.944963i \(-0.606097\pi\)
−0.327176 + 0.944963i \(0.606097\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.77200 −0.227460
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.5440 −0.688658 −0.344329 0.938849i \(-0.611893\pi\)
−0.344329 + 0.938849i \(0.611893\pi\)
\(282\) 0 0
\(283\) −11.0880 −0.659114 −0.329557 0.944136i \(-0.606899\pi\)
−0.329557 + 0.944136i \(0.606899\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −45.5440 −2.68838
\(288\) 0 0
\(289\) −13.8600 −0.815295
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −29.0880 −1.69934 −0.849670 0.527315i \(-0.823199\pi\)
−0.849670 + 0.527315i \(0.823199\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 23.3160 1.34840
\(300\) 0 0
\(301\) −8.45600 −0.487396
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.772002 −0.0442047
\(306\) 0 0
\(307\) −5.77200 −0.329426 −0.164713 0.986342i \(-0.552670\pi\)
−0.164713 + 0.986342i \(0.552670\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −28.3160 −1.60052 −0.800258 0.599656i \(-0.795304\pi\)
−0.800258 + 0.599656i \(0.795304\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.0880 0.622765 0.311382 0.950285i \(-0.399208\pi\)
0.311382 + 0.950285i \(0.399208\pi\)
\(318\) 0 0
\(319\) −14.2280 −0.796615
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.91199 −0.273311
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 27.5440 1.51855
\(330\) 0 0
\(331\) 3.22800 0.177427 0.0887134 0.996057i \(-0.471724\pi\)
0.0887134 + 0.996057i \(0.471724\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −14.7720 −0.807081
\(336\) 0 0
\(337\) −26.7720 −1.45836 −0.729182 0.684320i \(-0.760099\pi\)
−0.729182 + 0.684320i \(0.760099\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.40401 −0.455103
\(342\) 0 0
\(343\) 41.8600 2.26023
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.6320 1.53705 0.768523 0.639822i \(-0.220992\pi\)
0.768523 + 0.639822i \(0.220992\pi\)
\(348\) 0 0
\(349\) −25.4040 −1.35985 −0.679923 0.733284i \(-0.737987\pi\)
−0.679923 + 0.733284i \(0.737987\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.2280 0.757280 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(354\) 0 0
\(355\) −13.5440 −0.718841
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.45600 −0.129623 −0.0648113 0.997898i \(-0.520645\pi\)
−0.0648113 + 0.997898i \(0.520645\pi\)
\(360\) 0 0
\(361\) −11.3160 −0.595579
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.77200 −0.249778
\(366\) 0 0
\(367\) −10.3160 −0.538491 −0.269246 0.963072i \(-0.586774\pi\)
−0.269246 + 0.963072i \(0.586774\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −45.5440 −2.36453
\(372\) 0 0
\(373\) 20.0880 1.04012 0.520059 0.854130i \(-0.325910\pi\)
0.520059 + 0.854130i \(0.325910\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.3160 0.582804
\(378\) 0 0
\(379\) 0.316006 0.0162321 0.00811606 0.999967i \(-0.497417\pi\)
0.00811606 + 0.999967i \(0.497417\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −13.7720 −0.703716 −0.351858 0.936053i \(-0.614450\pi\)
−0.351858 + 0.936053i \(0.614450\pi\)
\(384\) 0 0
\(385\) 18.0000 0.917365
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 29.3160 1.48638 0.743190 0.669080i \(-0.233312\pi\)
0.743190 + 0.669080i \(0.233312\pi\)
\(390\) 0 0
\(391\) 13.7720 0.696480
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.00000 0.251577
\(396\) 0 0
\(397\) 31.3160 1.57171 0.785853 0.618414i \(-0.212224\pi\)
0.785853 + 0.618414i \(0.212224\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.54400 −0.0771039 −0.0385519 0.999257i \(-0.512275\pi\)
−0.0385519 + 0.999257i \(0.512275\pi\)
\(402\) 0 0
\(403\) 6.68399 0.332953
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.91199 0.144342
\(408\) 0 0
\(409\) 20.5440 1.01584 0.507918 0.861406i \(-0.330415\pi\)
0.507918 + 0.861406i \(0.330415\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 57.2640 2.81778
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.77200 −0.0865680 −0.0432840 0.999063i \(-0.513782\pi\)
−0.0432840 + 0.999063i \(0.513782\pi\)
\(420\) 0 0
\(421\) −5.22800 −0.254797 −0.127399 0.991852i \(-0.540663\pi\)
−0.127399 + 0.991852i \(0.540663\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.77200 0.0859547
\(426\) 0 0
\(427\) 3.68399 0.178281
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.6320 −1.18648 −0.593241 0.805025i \(-0.702152\pi\)
−0.593241 + 0.805025i \(0.702152\pi\)
\(432\) 0 0
\(433\) 1.08801 0.0522863 0.0261432 0.999658i \(-0.491677\pi\)
0.0261432 + 0.999658i \(0.491677\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −21.5440 −1.03059
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −27.0880 −1.28699 −0.643495 0.765450i \(-0.722516\pi\)
−0.643495 + 0.765450i \(0.722516\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 27.5440 1.29988 0.649941 0.759985i \(-0.274794\pi\)
0.649941 + 0.759985i \(0.274794\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −14.3160 −0.671145
\(456\) 0 0
\(457\) 17.5440 0.820674 0.410337 0.911934i \(-0.365411\pi\)
0.410337 + 0.911934i \(0.365411\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −35.5440 −1.65545 −0.827725 0.561134i \(-0.810365\pi\)
−0.827725 + 0.561134i \(0.810365\pi\)
\(462\) 0 0
\(463\) 7.86001 0.365286 0.182643 0.983179i \(-0.441535\pi\)
0.182643 + 0.983179i \(0.441535\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −32.0000 −1.48078 −0.740392 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 0 0
\(469\) 70.4920 3.25502
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.68399 0.307330
\(474\) 0 0
\(475\) −2.77200 −0.127188
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.4560 0.477747 0.238873 0.971051i \(-0.423222\pi\)
0.238873 + 0.971051i \(0.423222\pi\)
\(480\) 0 0
\(481\) −2.31601 −0.105601
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18.3160 0.831687
\(486\) 0 0
\(487\) −6.77200 −0.306869 −0.153434 0.988159i \(-0.549033\pi\)
−0.153434 + 0.988159i \(0.549033\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.5440 1.51382 0.756910 0.653519i \(-0.226708\pi\)
0.756910 + 0.653519i \(0.226708\pi\)
\(492\) 0 0
\(493\) 6.68399 0.301032
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 64.6320 2.89914
\(498\) 0 0
\(499\) −0.911993 −0.0408264 −0.0204132 0.999792i \(-0.506498\pi\)
−0.0204132 + 0.999792i \(0.506498\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.8600 0.840926 0.420463 0.907310i \(-0.361868\pi\)
0.420463 + 0.907310i \(0.361868\pi\)
\(504\) 0 0
\(505\) 10.2280 0.455140
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8.22800 −0.364700 −0.182350 0.983234i \(-0.558370\pi\)
−0.182350 + 0.983234i \(0.558370\pi\)
\(510\) 0 0
\(511\) 22.7720 1.00737
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.7720 −0.739063
\(516\) 0 0
\(517\) −21.7720 −0.957532
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 26.4560 1.15906 0.579529 0.814952i \(-0.303236\pi\)
0.579529 + 0.814952i \(0.303236\pi\)
\(522\) 0 0
\(523\) 1.45600 0.0636663 0.0318331 0.999493i \(-0.489865\pi\)
0.0318331 + 0.999493i \(0.489865\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.94802 0.171978
\(528\) 0 0
\(529\) 37.4040 1.62626
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −28.6320 −1.24019
\(534\) 0 0
\(535\) 14.0000 0.605273
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −59.4920 −2.56250
\(540\) 0 0
\(541\) −16.7720 −0.721085 −0.360542 0.932743i \(-0.617408\pi\)
−0.360542 + 0.932743i \(0.617408\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.54400 −0.408820
\(546\) 0 0
\(547\) −0.544004 −0.0232599 −0.0116300 0.999932i \(-0.503702\pi\)
−0.0116300 + 0.999932i \(0.503702\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10.4560 −0.445440
\(552\) 0 0
\(553\) −23.8600 −1.01463
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −38.1760 −1.61757 −0.808785 0.588105i \(-0.799874\pi\)
−0.808785 + 0.588105i \(0.799874\pi\)
\(558\) 0 0
\(559\) −5.31601 −0.224843
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7.54400 −0.317942 −0.158971 0.987283i \(-0.550818\pi\)
−0.158971 + 0.987283i \(0.550818\pi\)
\(564\) 0 0
\(565\) −11.3160 −0.476068
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19.0880 0.800211 0.400105 0.916469i \(-0.368973\pi\)
0.400105 + 0.916469i \(0.368973\pi\)
\(570\) 0 0
\(571\) −19.2280 −0.804667 −0.402333 0.915493i \(-0.631801\pi\)
−0.402333 + 0.915493i \(0.631801\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.77200 0.324115
\(576\) 0 0
\(577\) −19.4040 −0.807800 −0.403900 0.914803i \(-0.632346\pi\)
−0.403900 + 0.914803i \(0.632346\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 28.6320 1.18786
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.91199 0.120191 0.0600954 0.998193i \(-0.480860\pi\)
0.0600954 + 0.998193i \(0.480860\pi\)
\(588\) 0 0
\(589\) −6.17601 −0.254478
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.8600 −0.610227 −0.305114 0.952316i \(-0.598694\pi\)
−0.305114 + 0.952316i \(0.598694\pi\)
\(594\) 0 0
\(595\) −8.45600 −0.346662
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.08801 0.207890 0.103945 0.994583i \(-0.466853\pi\)
0.103945 + 0.994583i \(0.466853\pi\)
\(600\) 0 0
\(601\) 20.6840 0.843718 0.421859 0.906662i \(-0.361378\pi\)
0.421859 + 0.906662i \(0.361378\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.22800 −0.131237
\(606\) 0 0
\(607\) 39.8600 1.61787 0.808934 0.587900i \(-0.200045\pi\)
0.808934 + 0.587900i \(0.200045\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 17.3160 0.700531
\(612\) 0 0
\(613\) −2.54400 −0.102751 −0.0513757 0.998679i \(-0.516361\pi\)
−0.0513757 + 0.998679i \(0.516361\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.31601 −0.294531 −0.147266 0.989097i \(-0.547047\pi\)
−0.147266 + 0.989097i \(0.547047\pi\)
\(618\) 0 0
\(619\) 5.68399 0.228459 0.114230 0.993454i \(-0.463560\pi\)
0.114230 + 0.993454i \(0.463560\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 38.1760 1.52949
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.36799 −0.0545453
\(630\) 0 0
\(631\) −2.77200 −0.110352 −0.0551758 0.998477i \(-0.517572\pi\)
−0.0551758 + 0.998477i \(0.517572\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 11.5440 0.458110
\(636\) 0 0
\(637\) 47.3160 1.87473
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.6320 0.419939 0.209970 0.977708i \(-0.432663\pi\)
0.209970 + 0.977708i \(0.432663\pi\)
\(642\) 0 0
\(643\) −9.77200 −0.385370 −0.192685 0.981261i \(-0.561720\pi\)
−0.192685 + 0.981261i \(0.561720\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28.6320 1.12564 0.562820 0.826579i \(-0.309716\pi\)
0.562820 + 0.826579i \(0.309716\pi\)
\(648\) 0 0
\(649\) −45.2640 −1.77677
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27.0880 −1.06004 −0.530018 0.847987i \(-0.677815\pi\)
−0.530018 + 0.847987i \(0.677815\pi\)
\(654\) 0 0
\(655\) −6.22800 −0.243348
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) −7.22800 −0.281137 −0.140568 0.990071i \(-0.544893\pi\)
−0.140568 + 0.990071i \(0.544893\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 13.2280 0.512960
\(666\) 0 0
\(667\) 29.3160 1.13512
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.91199 −0.112416
\(672\) 0 0
\(673\) 13.4040 0.516687 0.258343 0.966053i \(-0.416823\pi\)
0.258343 + 0.966053i \(0.416823\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.91199 0.342516 0.171258 0.985226i \(-0.445217\pi\)
0.171258 + 0.985226i \(0.445217\pi\)
\(678\) 0 0
\(679\) −87.4040 −3.35426
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.4560 0.400088 0.200044 0.979787i \(-0.435892\pi\)
0.200044 + 0.979787i \(0.435892\pi\)
\(684\) 0 0
\(685\) 21.0880 0.805731
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −28.6320 −1.09079
\(690\) 0 0
\(691\) 38.6320 1.46963 0.734815 0.678267i \(-0.237269\pi\)
0.734815 + 0.678267i \(0.237269\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.7720 0.484470
\(696\) 0 0
\(697\) −16.9120 −0.640587
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19.7720 0.746778 0.373389 0.927675i \(-0.378196\pi\)
0.373389 + 0.927675i \(0.378196\pi\)
\(702\) 0 0
\(703\) 2.13999 0.0807113
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −48.8080 −1.83561
\(708\) 0 0
\(709\) 49.8600 1.87253 0.936266 0.351292i \(-0.114258\pi\)
0.936266 + 0.351292i \(0.114258\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 17.3160 0.648490
\(714\) 0 0
\(715\) 11.3160 0.423195
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 80.0360 2.98070
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.77200 0.140089
\(726\) 0 0
\(727\) −0.911993 −0.0338239 −0.0169120 0.999857i \(-0.505384\pi\)
−0.0169120 + 0.999857i \(0.505384\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.13999 −0.116137
\(732\) 0 0
\(733\) −6.91199 −0.255300 −0.127650 0.991819i \(-0.540743\pi\)
−0.127650 + 0.991819i \(0.540743\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −55.7200 −2.05247
\(738\) 0 0
\(739\) −35.0880 −1.29073 −0.645367 0.763873i \(-0.723295\pi\)
−0.645367 + 0.763873i \(0.723295\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −44.4040 −1.62903 −0.814513 0.580146i \(-0.802996\pi\)
−0.814513 + 0.580146i \(0.802996\pi\)
\(744\) 0 0
\(745\) 9.77200 0.358018
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −66.8080 −2.44111
\(750\) 0 0
\(751\) −39.1760 −1.42955 −0.714777 0.699353i \(-0.753472\pi\)
−0.714777 + 0.699353i \(0.753472\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 24.0880 0.876652
\(756\) 0 0
\(757\) −46.7200 −1.69807 −0.849034 0.528338i \(-0.822815\pi\)
−0.849034 + 0.528338i \(0.822815\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 29.0880 1.05444 0.527220 0.849729i \(-0.323234\pi\)
0.527220 + 0.849729i \(0.323234\pi\)
\(762\) 0 0
\(763\) 45.5440 1.64880
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36.0000 1.29988
\(768\) 0 0
\(769\) 48.2640 1.74045 0.870223 0.492659i \(-0.163975\pi\)
0.870223 + 0.492659i \(0.163975\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −10.9120 −0.392477 −0.196239 0.980556i \(-0.562873\pi\)
−0.196239 + 0.980556i \(0.562873\pi\)
\(774\) 0 0
\(775\) 2.22800 0.0800321
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 26.4560 0.947884
\(780\) 0 0
\(781\) −51.0880 −1.82807
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.77200 −0.277395
\(786\) 0 0
\(787\) 39.1760 1.39647 0.698237 0.715867i \(-0.253968\pi\)
0.698237 + 0.715867i \(0.253968\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 54.0000 1.92002
\(792\) 0 0
\(793\) 2.31601 0.0822438
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 46.1760 1.63564 0.817819 0.575475i \(-0.195183\pi\)
0.817819 + 0.575475i \(0.195183\pi\)
\(798\) 0 0
\(799\) 10.2280 0.361841
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −18.0000 −0.635206
\(804\) 0 0
\(805\) −37.0880 −1.30718
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) −31.5440 −1.10766 −0.553830 0.832630i \(-0.686834\pi\)
−0.553830 + 0.832630i \(0.686834\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.54400 0.299283
\(816\) 0 0
\(817\) 4.91199 0.171849
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23.5440 −0.821691 −0.410846 0.911705i \(-0.634767\pi\)
−0.410846 + 0.911705i \(0.634767\pi\)
\(822\) 0 0
\(823\) −39.4040 −1.37354 −0.686769 0.726876i \(-0.740972\pi\)
−0.686769 + 0.726876i \(0.740972\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.54400 −0.123237 −0.0616185 0.998100i \(-0.519626\pi\)
−0.0616185 + 0.998100i \(0.519626\pi\)
\(828\) 0 0
\(829\) 38.7720 1.34661 0.673304 0.739366i \(-0.264875\pi\)
0.673304 + 0.739366i \(0.264875\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 27.9480 0.968341
\(834\) 0 0
\(835\) 23.0880 0.798993
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 54.6320 1.88611 0.943053 0.332642i \(-0.107940\pi\)
0.943053 + 0.332642i \(0.107940\pi\)
\(840\) 0 0
\(841\) −14.7720 −0.509379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 15.4040 0.529288
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −15.4560 −0.529203 −0.264602 0.964358i \(-0.585240\pi\)
−0.264602 + 0.964358i \(0.585240\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 41.0880 1.40354 0.701770 0.712404i \(-0.252394\pi\)
0.701770 + 0.712404i \(0.252394\pi\)
\(858\) 0 0
\(859\) −24.7720 −0.845210 −0.422605 0.906314i \(-0.638884\pi\)
−0.422605 + 0.906314i \(0.638884\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.3160 −0.861767 −0.430883 0.902408i \(-0.641798\pi\)
−0.430883 + 0.902408i \(0.641798\pi\)
\(864\) 0 0
\(865\) −7.54400 −0.256504
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 18.8600 0.639782
\(870\) 0 0
\(871\) 44.3160 1.50159
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.77200 −0.161323
\(876\) 0 0
\(877\) −23.0000 −0.776655 −0.388327 0.921521i \(-0.626947\pi\)
−0.388327 + 0.921521i \(0.626947\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.91199 −0.232871 −0.116435 0.993198i \(-0.537147\pi\)
−0.116435 + 0.993198i \(0.537147\pi\)
\(882\) 0 0
\(883\) 12.3160 0.414467 0.207233 0.978292i \(-0.433554\pi\)
0.207233 + 0.978292i \(0.433554\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.68399 0.224427 0.112213 0.993684i \(-0.464206\pi\)
0.112213 + 0.993684i \(0.464206\pi\)
\(888\) 0 0
\(889\) −55.0880 −1.84759
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) −9.54400 −0.319021
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.40401 0.280290
\(900\) 0 0
\(901\) −16.9120 −0.563420
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.3160 0.808291
\(906\) 0 0
\(907\) −35.0000 −1.16216 −0.581078 0.813848i \(-0.697369\pi\)
−0.581078 + 0.813848i \(0.697369\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −38.1760 −1.26483 −0.632414 0.774631i \(-0.717936\pi\)
−0.632414 + 0.774631i \(0.717936\pi\)
\(912\) 0 0
\(913\) −22.6320 −0.749010
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 29.7200 0.981441
\(918\) 0 0
\(919\) 4.86001 0.160317 0.0801585 0.996782i \(-0.474457\pi\)
0.0801585 + 0.996782i \(0.474457\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 40.6320 1.33742
\(924\) 0 0
\(925\) −0.772002 −0.0253833
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −16.6320 −0.545679 −0.272839 0.962060i \(-0.587963\pi\)
−0.272839 + 0.962060i \(0.587963\pi\)
\(930\) 0 0
\(931\) −43.7200 −1.43287
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.68399 0.218590
\(936\) 0 0
\(937\) −16.7720 −0.547917 −0.273959 0.961741i \(-0.588333\pi\)
−0.273959 + 0.961741i \(0.588333\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −22.4040 −0.730350 −0.365175 0.930939i \(-0.618991\pi\)
−0.365175 + 0.930939i \(0.618991\pi\)
\(942\) 0 0
\(943\) −74.1760 −2.41550
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.45600 0.274783 0.137391 0.990517i \(-0.456128\pi\)
0.137391 + 0.990517i \(0.456128\pi\)
\(948\) 0 0
\(949\) 14.3160 0.464717
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −48.8600 −1.58273 −0.791365 0.611343i \(-0.790629\pi\)
−0.791365 + 0.611343i \(0.790629\pi\)
\(954\) 0 0
\(955\) −3.54400 −0.114681
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −100.632 −3.24958
\(960\) 0 0
\(961\) −26.0360 −0.839872
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 12.7720 0.411145
\(966\) 0 0
\(967\) −4.77200 −0.153457 −0.0767286 0.997052i \(-0.524448\pi\)
−0.0767286 + 0.997052i \(0.524448\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 37.7720 1.21216 0.606081 0.795403i \(-0.292741\pi\)
0.606081 + 0.795403i \(0.292741\pi\)
\(972\) 0 0
\(973\) −60.9480 −1.95390
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.227998 0.00729431 0.00364715 0.999993i \(-0.498839\pi\)
0.00364715 + 0.999993i \(0.498839\pi\)
\(978\) 0 0
\(979\) −30.1760 −0.964430
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −14.4040 −0.459417 −0.229708 0.973260i \(-0.573777\pi\)
−0.229708 + 0.973260i \(0.573777\pi\)
\(984\) 0 0
\(985\) 20.0000 0.637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13.7720 −0.437924
\(990\) 0 0
\(991\) 8.08801 0.256924 0.128462 0.991714i \(-0.458996\pi\)
0.128462 + 0.991714i \(0.458996\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 15.0000 0.475532
\(996\) 0 0
\(997\) 12.6840 0.401706 0.200853 0.979621i \(-0.435629\pi\)
0.200853 + 0.979621i \(0.435629\pi\)
\(998\) 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 1080.2.a.m.1.2 2
3.2 odd 2 1080.2.a.n.1.2 yes 2
4.3 odd 2 2160.2.a.z.1.1 2
5.2 odd 4 5400.2.f.be.649.4 4
5.3 odd 4 5400.2.f.be.649.1 4
5.4 even 2 5400.2.a.cb.1.1 2
8.3 odd 2 8640.2.a.db.1.1 2
8.5 even 2 8640.2.a.de.1.2 2
9.2 odd 6 3240.2.q.z.1081.1 4
9.4 even 3 3240.2.q.bc.2161.1 4
9.5 odd 6 3240.2.q.z.2161.1 4
9.7 even 3 3240.2.q.bc.1081.1 4
12.11 even 2 2160.2.a.bb.1.1 2
15.2 even 4 5400.2.f.bd.649.4 4
15.8 even 4 5400.2.f.bd.649.1 4
15.14 odd 2 5400.2.a.ca.1.1 2
24.5 odd 2 8640.2.a.cq.1.2 2
24.11 even 2 8640.2.a.cn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.a.m.1.2 2 1.1 even 1 trivial
1080.2.a.n.1.2 yes 2 3.2 odd 2
2160.2.a.z.1.1 2 4.3 odd 2
2160.2.a.bb.1.1 2 12.11 even 2
3240.2.q.z.1081.1 4 9.2 odd 6
3240.2.q.z.2161.1 4 9.5 odd 6
3240.2.q.bc.1081.1 4 9.7 even 3
3240.2.q.bc.2161.1 4 9.4 even 3
5400.2.a.ca.1.1 2 15.14 odd 2
5400.2.a.cb.1.1 2 5.4 even 2
5400.2.f.bd.649.1 4 15.8 even 4
5400.2.f.bd.649.4 4 15.2 even 4
5400.2.f.be.649.1 4 5.3 odd 4
5400.2.f.be.649.4 4 5.2 odd 4
8640.2.a.cn.1.1 2 24.11 even 2
8640.2.a.cq.1.2 2 24.5 odd 2
8640.2.a.db.1.1 2 8.3 odd 2
8640.2.a.de.1.2 2 8.5 even 2