Properties

Label 8624.2.a.bk.1.2
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
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] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, 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("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.64575 q^{3} +1.64575 q^{5} +4.00000 q^{9} +O(q^{10})\) \(q+2.64575 q^{3} +1.64575 q^{5} +4.00000 q^{9} -1.00000 q^{11} -5.00000 q^{13} +4.35425 q^{15} -6.00000 q^{17} -5.64575 q^{19} -1.64575 q^{23} -2.29150 q^{25} +2.64575 q^{27} +6.29150 q^{29} -4.00000 q^{31} -2.64575 q^{33} +3.64575 q^{37} -13.2288 q^{39} -10.9373 q^{41} +4.00000 q^{43} +6.58301 q^{45} +2.70850 q^{47} -15.8745 q^{51} +1.64575 q^{53} -1.64575 q^{55} -14.9373 q^{57} +4.64575 q^{59} -14.2915 q^{61} -8.22876 q^{65} +11.9373 q^{67} -4.35425 q^{69} -4.35425 q^{71} -0.354249 q^{73} -6.06275 q^{75} +2.64575 q^{79} -5.00000 q^{81} +2.70850 q^{83} -9.87451 q^{85} +16.6458 q^{87} -6.58301 q^{89} -10.5830 q^{93} -9.29150 q^{95} +16.2915 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 8 q^{9} - 2 q^{11} - 10 q^{13} + 14 q^{15} - 12 q^{17} - 6 q^{19} + 2 q^{23} + 6 q^{25} + 2 q^{29} - 8 q^{31} + 2 q^{37} - 6 q^{41} + 8 q^{43} - 8 q^{45} + 16 q^{47} - 2 q^{53} + 2 q^{55} - 14 q^{57} + 4 q^{59} - 18 q^{61} + 10 q^{65} + 8 q^{67} - 14 q^{69} - 14 q^{71} - 6 q^{73} - 28 q^{75} - 10 q^{81} + 16 q^{83} + 12 q^{85} + 28 q^{87} + 8 q^{89} - 8 q^{95} + 22 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.64575 1.52753 0.763763 0.645497i \(-0.223350\pi\)
0.763763 + 0.645497i \(0.223350\pi\)
\(4\) 0 0
\(5\) 1.64575 0.736002 0.368001 0.929825i \(-0.380042\pi\)
0.368001 + 0.929825i \(0.380042\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 4.00000 1.33333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 4.35425 1.12426
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −5.64575 −1.29522 −0.647612 0.761970i \(-0.724232\pi\)
−0.647612 + 0.761970i \(0.724232\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.64575 −0.343163 −0.171581 0.985170i \(-0.554888\pi\)
−0.171581 + 0.985170i \(0.554888\pi\)
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) 0 0
\(27\) 2.64575 0.509175
\(28\) 0 0
\(29\) 6.29150 1.16830 0.584151 0.811645i \(-0.301427\pi\)
0.584151 + 0.811645i \(0.301427\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −2.64575 −0.460566
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.64575 0.599358 0.299679 0.954040i \(-0.403120\pi\)
0.299679 + 0.954040i \(0.403120\pi\)
\(38\) 0 0
\(39\) −13.2288 −2.11830
\(40\) 0 0
\(41\) −10.9373 −1.70811 −0.854056 0.520181i \(-0.825864\pi\)
−0.854056 + 0.520181i \(0.825864\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 6.58301 0.981336
\(46\) 0 0
\(47\) 2.70850 0.395075 0.197537 0.980295i \(-0.436706\pi\)
0.197537 + 0.980295i \(0.436706\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −15.8745 −2.22288
\(52\) 0 0
\(53\) 1.64575 0.226061 0.113031 0.993592i \(-0.463944\pi\)
0.113031 + 0.993592i \(0.463944\pi\)
\(54\) 0 0
\(55\) −1.64575 −0.221913
\(56\) 0 0
\(57\) −14.9373 −1.97849
\(58\) 0 0
\(59\) 4.64575 0.604825 0.302413 0.953177i \(-0.402208\pi\)
0.302413 + 0.953177i \(0.402208\pi\)
\(60\) 0 0
\(61\) −14.2915 −1.82984 −0.914920 0.403636i \(-0.867746\pi\)
−0.914920 + 0.403636i \(0.867746\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.22876 −1.02065
\(66\) 0 0
\(67\) 11.9373 1.45837 0.729184 0.684318i \(-0.239900\pi\)
0.729184 + 0.684318i \(0.239900\pi\)
\(68\) 0 0
\(69\) −4.35425 −0.524190
\(70\) 0 0
\(71\) −4.35425 −0.516754 −0.258377 0.966044i \(-0.583188\pi\)
−0.258377 + 0.966044i \(0.583188\pi\)
\(72\) 0 0
\(73\) −0.354249 −0.0414617 −0.0207308 0.999785i \(-0.506599\pi\)
−0.0207308 + 0.999785i \(0.506599\pi\)
\(74\) 0 0
\(75\) −6.06275 −0.700066
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.64575 0.297670 0.148835 0.988862i \(-0.452448\pi\)
0.148835 + 0.988862i \(0.452448\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 2.70850 0.297296 0.148648 0.988890i \(-0.452508\pi\)
0.148648 + 0.988890i \(0.452508\pi\)
\(84\) 0 0
\(85\) −9.87451 −1.07104
\(86\) 0 0
\(87\) 16.6458 1.78461
\(88\) 0 0
\(89\) −6.58301 −0.697797 −0.348899 0.937160i \(-0.613444\pi\)
−0.348899 + 0.937160i \(0.613444\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −10.5830 −1.09741
\(94\) 0 0
\(95\) −9.29150 −0.953288
\(96\) 0 0
\(97\) 16.2915 1.65415 0.827076 0.562090i \(-0.190003\pi\)
0.827076 + 0.562090i \(0.190003\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) −2.93725 −0.289416 −0.144708 0.989474i \(-0.546224\pi\)
−0.144708 + 0.989474i \(0.546224\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.9373 1.05734 0.528672 0.848826i \(-0.322690\pi\)
0.528672 + 0.848826i \(0.322690\pi\)
\(108\) 0 0
\(109\) −10.5830 −1.01367 −0.506834 0.862044i \(-0.669184\pi\)
−0.506834 + 0.862044i \(0.669184\pi\)
\(110\) 0 0
\(111\) 9.64575 0.915534
\(112\) 0 0
\(113\) −18.2915 −1.72072 −0.860360 0.509687i \(-0.829761\pi\)
−0.860360 + 0.509687i \(0.829761\pi\)
\(114\) 0 0
\(115\) −2.70850 −0.252569
\(116\) 0 0
\(117\) −20.0000 −1.84900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −28.9373 −2.60918
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −15.9373 −1.41420 −0.707101 0.707112i \(-0.749998\pi\)
−0.707101 + 0.707112i \(0.749998\pi\)
\(128\) 0 0
\(129\) 10.5830 0.931782
\(130\) 0 0
\(131\) 10.3542 0.904655 0.452327 0.891852i \(-0.350594\pi\)
0.452327 + 0.891852i \(0.350594\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.35425 0.374754
\(136\) 0 0
\(137\) −12.8745 −1.09994 −0.549972 0.835183i \(-0.685362\pi\)
−0.549972 + 0.835183i \(0.685362\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 7.16601 0.603487
\(142\) 0 0
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) 10.3542 0.859874
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −15.2915 −1.25273 −0.626364 0.779530i \(-0.715458\pi\)
−0.626364 + 0.779530i \(0.715458\pi\)
\(150\) 0 0
\(151\) 8.64575 0.703581 0.351791 0.936079i \(-0.385573\pi\)
0.351791 + 0.936079i \(0.385573\pi\)
\(152\) 0 0
\(153\) −24.0000 −1.94029
\(154\) 0 0
\(155\) −6.58301 −0.528760
\(156\) 0 0
\(157\) −21.1660 −1.68923 −0.844616 0.535373i \(-0.820171\pi\)
−0.844616 + 0.535373i \(0.820171\pi\)
\(158\) 0 0
\(159\) 4.35425 0.345314
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.645751 −0.0505791 −0.0252896 0.999680i \(-0.508051\pi\)
−0.0252896 + 0.999680i \(0.508051\pi\)
\(164\) 0 0
\(165\) −4.35425 −0.338978
\(166\) 0 0
\(167\) −11.2288 −0.868907 −0.434454 0.900694i \(-0.643059\pi\)
−0.434454 + 0.900694i \(0.643059\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −22.5830 −1.72697
\(172\) 0 0
\(173\) 0.291503 0.0221625 0.0110813 0.999939i \(-0.496473\pi\)
0.0110813 + 0.999939i \(0.496473\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.2915 0.923886
\(178\) 0 0
\(179\) 19.9373 1.49018 0.745090 0.666964i \(-0.232406\pi\)
0.745090 + 0.666964i \(0.232406\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −37.8118 −2.79513
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.70850 0.195980 0.0979900 0.995187i \(-0.468759\pi\)
0.0979900 + 0.995187i \(0.468759\pi\)
\(192\) 0 0
\(193\) 25.5203 1.83699 0.918494 0.395434i \(-0.129406\pi\)
0.918494 + 0.395434i \(0.129406\pi\)
\(194\) 0 0
\(195\) −21.7712 −1.55907
\(196\) 0 0
\(197\) −12.8745 −0.917271 −0.458635 0.888625i \(-0.651662\pi\)
−0.458635 + 0.888625i \(0.651662\pi\)
\(198\) 0 0
\(199\) 4.22876 0.299769 0.149884 0.988704i \(-0.452110\pi\)
0.149884 + 0.988704i \(0.452110\pi\)
\(200\) 0 0
\(201\) 31.5830 2.22769
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 0 0
\(207\) −6.58301 −0.457550
\(208\) 0 0
\(209\) 5.64575 0.390525
\(210\) 0 0
\(211\) −0.937254 −0.0645232 −0.0322616 0.999479i \(-0.510271\pi\)
−0.0322616 + 0.999479i \(0.510271\pi\)
\(212\) 0 0
\(213\) −11.5203 −0.789355
\(214\) 0 0
\(215\) 6.58301 0.448957
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −0.937254 −0.0633338
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) −17.6458 −1.18165 −0.590823 0.806801i \(-0.701197\pi\)
−0.590823 + 0.806801i \(0.701197\pi\)
\(224\) 0 0
\(225\) −9.16601 −0.611067
\(226\) 0 0
\(227\) −2.70850 −0.179769 −0.0898846 0.995952i \(-0.528650\pi\)
−0.0898846 + 0.995952i \(0.528650\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.06275 0.0696228 0.0348114 0.999394i \(-0.488917\pi\)
0.0348114 + 0.999394i \(0.488917\pi\)
\(234\) 0 0
\(235\) 4.45751 0.290776
\(236\) 0 0
\(237\) 7.00000 0.454699
\(238\) 0 0
\(239\) 17.2288 1.11444 0.557218 0.830366i \(-0.311869\pi\)
0.557218 + 0.830366i \(0.311869\pi\)
\(240\) 0 0
\(241\) 24.8118 1.59827 0.799133 0.601154i \(-0.205292\pi\)
0.799133 + 0.601154i \(0.205292\pi\)
\(242\) 0 0
\(243\) −21.1660 −1.35780
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 28.2288 1.79615
\(248\) 0 0
\(249\) 7.16601 0.454127
\(250\) 0 0
\(251\) −3.29150 −0.207758 −0.103879 0.994590i \(-0.533125\pi\)
−0.103879 + 0.994590i \(0.533125\pi\)
\(252\) 0 0
\(253\) 1.64575 0.103467
\(254\) 0 0
\(255\) −26.1255 −1.63604
\(256\) 0 0
\(257\) 21.5830 1.34631 0.673155 0.739501i \(-0.264938\pi\)
0.673155 + 0.739501i \(0.264938\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 25.1660 1.55774
\(262\) 0 0
\(263\) −19.9373 −1.22938 −0.614692 0.788767i \(-0.710720\pi\)
−0.614692 + 0.788767i \(0.710720\pi\)
\(264\) 0 0
\(265\) 2.70850 0.166382
\(266\) 0 0
\(267\) −17.4170 −1.06590
\(268\) 0 0
\(269\) −5.41699 −0.330280 −0.165140 0.986270i \(-0.552808\pi\)
−0.165140 + 0.986270i \(0.552808\pi\)
\(270\) 0 0
\(271\) −2.06275 −0.125303 −0.0626514 0.998035i \(-0.519956\pi\)
−0.0626514 + 0.998035i \(0.519956\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.29150 0.138183
\(276\) 0 0
\(277\) −22.2915 −1.33937 −0.669683 0.742647i \(-0.733570\pi\)
−0.669683 + 0.742647i \(0.733570\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) −22.9373 −1.36832 −0.684161 0.729331i \(-0.739831\pi\)
−0.684161 + 0.729331i \(0.739831\pi\)
\(282\) 0 0
\(283\) −2.35425 −0.139946 −0.0699728 0.997549i \(-0.522291\pi\)
−0.0699728 + 0.997549i \(0.522291\pi\)
\(284\) 0 0
\(285\) −24.5830 −1.45617
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 43.1033 2.52676
\(292\) 0 0
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 7.64575 0.445153
\(296\) 0 0
\(297\) −2.64575 −0.153522
\(298\) 0 0
\(299\) 8.22876 0.475881
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −7.93725 −0.455983
\(304\) 0 0
\(305\) −23.5203 −1.34677
\(306\) 0 0
\(307\) 22.2288 1.26866 0.634331 0.773062i \(-0.281276\pi\)
0.634331 + 0.773062i \(0.281276\pi\)
\(308\) 0 0
\(309\) −7.77124 −0.442091
\(310\) 0 0
\(311\) 1.06275 0.0602628 0.0301314 0.999546i \(-0.490407\pi\)
0.0301314 + 0.999546i \(0.490407\pi\)
\(312\) 0 0
\(313\) −23.5830 −1.33299 −0.666495 0.745509i \(-0.732206\pi\)
−0.666495 + 0.745509i \(0.732206\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) −6.29150 −0.352257
\(320\) 0 0
\(321\) 28.9373 1.61512
\(322\) 0 0
\(323\) 33.8745 1.88483
\(324\) 0 0
\(325\) 11.4575 0.635548
\(326\) 0 0
\(327\) −28.0000 −1.54840
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.6458 1.13479 0.567397 0.823445i \(-0.307951\pi\)
0.567397 + 0.823445i \(0.307951\pi\)
\(332\) 0 0
\(333\) 14.5830 0.799144
\(334\) 0 0
\(335\) 19.6458 1.07336
\(336\) 0 0
\(337\) 9.06275 0.493679 0.246840 0.969056i \(-0.420608\pi\)
0.246840 + 0.969056i \(0.420608\pi\)
\(338\) 0 0
\(339\) −48.3948 −2.62844
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −7.16601 −0.385805
\(346\) 0 0
\(347\) 20.8118 1.11723 0.558617 0.829426i \(-0.311332\pi\)
0.558617 + 0.829426i \(0.311332\pi\)
\(348\) 0 0
\(349\) 1.87451 0.100340 0.0501701 0.998741i \(-0.484024\pi\)
0.0501701 + 0.998741i \(0.484024\pi\)
\(350\) 0 0
\(351\) −13.2288 −0.706099
\(352\) 0 0
\(353\) −7.16601 −0.381408 −0.190704 0.981648i \(-0.561077\pi\)
−0.190704 + 0.981648i \(0.561077\pi\)
\(354\) 0 0
\(355\) −7.16601 −0.380332
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.9373 1.36892 0.684458 0.729052i \(-0.260039\pi\)
0.684458 + 0.729052i \(0.260039\pi\)
\(360\) 0 0
\(361\) 12.8745 0.677606
\(362\) 0 0
\(363\) 2.64575 0.138866
\(364\) 0 0
\(365\) −0.583005 −0.0305159
\(366\) 0 0
\(367\) −36.2288 −1.89113 −0.945563 0.325440i \(-0.894488\pi\)
−0.945563 + 0.325440i \(0.894488\pi\)
\(368\) 0 0
\(369\) −43.7490 −2.27748
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 20.8745 1.08084 0.540421 0.841395i \(-0.318265\pi\)
0.540421 + 0.841395i \(0.318265\pi\)
\(374\) 0 0
\(375\) −31.7490 −1.63951
\(376\) 0 0
\(377\) −31.4575 −1.62014
\(378\) 0 0
\(379\) −6.06275 −0.311422 −0.155711 0.987803i \(-0.549767\pi\)
−0.155711 + 0.987803i \(0.549767\pi\)
\(380\) 0 0
\(381\) −42.1660 −2.16023
\(382\) 0 0
\(383\) 24.1033 1.23162 0.615810 0.787895i \(-0.288829\pi\)
0.615810 + 0.787895i \(0.288829\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.0000 0.813326
\(388\) 0 0
\(389\) 26.8118 1.35941 0.679705 0.733485i \(-0.262108\pi\)
0.679705 + 0.733485i \(0.262108\pi\)
\(390\) 0 0
\(391\) 9.87451 0.499375
\(392\) 0 0
\(393\) 27.3948 1.38188
\(394\) 0 0
\(395\) 4.35425 0.219086
\(396\) 0 0
\(397\) 11.1660 0.560406 0.280203 0.959941i \(-0.409598\pi\)
0.280203 + 0.959941i \(0.409598\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.5830 1.07780 0.538902 0.842369i \(-0.318839\pi\)
0.538902 + 0.842369i \(0.318839\pi\)
\(402\) 0 0
\(403\) 20.0000 0.996271
\(404\) 0 0
\(405\) −8.22876 −0.408890
\(406\) 0 0
\(407\) −3.64575 −0.180713
\(408\) 0 0
\(409\) −3.06275 −0.151443 −0.0757215 0.997129i \(-0.524126\pi\)
−0.0757215 + 0.997129i \(0.524126\pi\)
\(410\) 0 0
\(411\) −34.0627 −1.68019
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.45751 0.218811
\(416\) 0 0
\(417\) −10.5830 −0.518252
\(418\) 0 0
\(419\) −9.87451 −0.482401 −0.241201 0.970475i \(-0.577541\pi\)
−0.241201 + 0.970475i \(0.577541\pi\)
\(420\) 0 0
\(421\) 9.16601 0.446724 0.223362 0.974736i \(-0.428297\pi\)
0.223362 + 0.974736i \(0.428297\pi\)
\(422\) 0 0
\(423\) 10.8340 0.526767
\(424\) 0 0
\(425\) 13.7490 0.666925
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 13.2288 0.638690
\(430\) 0 0
\(431\) −29.2288 −1.40790 −0.703950 0.710250i \(-0.748582\pi\)
−0.703950 + 0.710250i \(0.748582\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 27.3948 1.31348
\(436\) 0 0
\(437\) 9.29150 0.444473
\(438\) 0 0
\(439\) 3.93725 0.187915 0.0939574 0.995576i \(-0.470048\pi\)
0.0939574 + 0.995576i \(0.470048\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −34.4575 −1.63713 −0.818563 0.574417i \(-0.805229\pi\)
−0.818563 + 0.574417i \(0.805229\pi\)
\(444\) 0 0
\(445\) −10.8340 −0.513580
\(446\) 0 0
\(447\) −40.4575 −1.91357
\(448\) 0 0
\(449\) −21.8745 −1.03232 −0.516161 0.856492i \(-0.672639\pi\)
−0.516161 + 0.856492i \(0.672639\pi\)
\(450\) 0 0
\(451\) 10.9373 0.515015
\(452\) 0 0
\(453\) 22.8745 1.07474
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.16601 0.148100 0.0740499 0.997255i \(-0.476408\pi\)
0.0740499 + 0.997255i \(0.476408\pi\)
\(458\) 0 0
\(459\) −15.8745 −0.740959
\(460\) 0 0
\(461\) −10.1660 −0.473478 −0.236739 0.971573i \(-0.576079\pi\)
−0.236739 + 0.971573i \(0.576079\pi\)
\(462\) 0 0
\(463\) −30.4575 −1.41548 −0.707740 0.706473i \(-0.750285\pi\)
−0.707740 + 0.706473i \(0.750285\pi\)
\(464\) 0 0
\(465\) −17.4170 −0.807694
\(466\) 0 0
\(467\) 21.2915 0.985253 0.492627 0.870241i \(-0.336037\pi\)
0.492627 + 0.870241i \(0.336037\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −56.0000 −2.58034
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 12.9373 0.593602
\(476\) 0 0
\(477\) 6.58301 0.301415
\(478\) 0 0
\(479\) −10.0627 −0.459779 −0.229889 0.973217i \(-0.573836\pi\)
−0.229889 + 0.973217i \(0.573836\pi\)
\(480\) 0 0
\(481\) −18.2288 −0.831160
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 26.8118 1.21746
\(486\) 0 0
\(487\) 9.41699 0.426725 0.213362 0.976973i \(-0.431559\pi\)
0.213362 + 0.976973i \(0.431559\pi\)
\(488\) 0 0
\(489\) −1.70850 −0.0772609
\(490\) 0 0
\(491\) −21.2915 −0.960872 −0.480436 0.877030i \(-0.659522\pi\)
−0.480436 + 0.877030i \(0.659522\pi\)
\(492\) 0 0
\(493\) −37.7490 −1.70013
\(494\) 0 0
\(495\) −6.58301 −0.295884
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 13.8745 0.621108 0.310554 0.950556i \(-0.399485\pi\)
0.310554 + 0.950556i \(0.399485\pi\)
\(500\) 0 0
\(501\) −29.7085 −1.32728
\(502\) 0 0
\(503\) −4.06275 −0.181149 −0.0905744 0.995890i \(-0.528870\pi\)
−0.0905744 + 0.995890i \(0.528870\pi\)
\(504\) 0 0
\(505\) −4.93725 −0.219705
\(506\) 0 0
\(507\) 31.7490 1.41002
\(508\) 0 0
\(509\) 0.583005 0.0258413 0.0129206 0.999917i \(-0.495887\pi\)
0.0129206 + 0.999917i \(0.495887\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −14.9373 −0.659496
\(514\) 0 0
\(515\) −4.83399 −0.213011
\(516\) 0 0
\(517\) −2.70850 −0.119120
\(518\) 0 0
\(519\) 0.771243 0.0338538
\(520\) 0 0
\(521\) 33.8745 1.48407 0.742035 0.670362i \(-0.233861\pi\)
0.742035 + 0.670362i \(0.233861\pi\)
\(522\) 0 0
\(523\) −21.5203 −0.941015 −0.470508 0.882396i \(-0.655929\pi\)
−0.470508 + 0.882396i \(0.655929\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −20.2915 −0.882239
\(530\) 0 0
\(531\) 18.5830 0.806434
\(532\) 0 0
\(533\) 54.6863 2.36873
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 52.7490 2.27629
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.29150 0.0985194 0.0492597 0.998786i \(-0.484314\pi\)
0.0492597 + 0.998786i \(0.484314\pi\)
\(542\) 0 0
\(543\) 26.4575 1.13540
\(544\) 0 0
\(545\) −17.4170 −0.746062
\(546\) 0 0
\(547\) 9.52026 0.407057 0.203528 0.979069i \(-0.434759\pi\)
0.203528 + 0.979069i \(0.434759\pi\)
\(548\) 0 0
\(549\) −57.1660 −2.43979
\(550\) 0 0
\(551\) −35.5203 −1.51321
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 15.8745 0.673835
\(556\) 0 0
\(557\) −31.7490 −1.34525 −0.672624 0.739984i \(-0.734833\pi\)
−0.672624 + 0.739984i \(0.734833\pi\)
\(558\) 0 0
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 15.8745 0.670222
\(562\) 0 0
\(563\) 28.9373 1.21956 0.609780 0.792571i \(-0.291258\pi\)
0.609780 + 0.792571i \(0.291258\pi\)
\(564\) 0 0
\(565\) −30.1033 −1.26645
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.16601 0.0488817 0.0244409 0.999701i \(-0.492219\pi\)
0.0244409 + 0.999701i \(0.492219\pi\)
\(570\) 0 0
\(571\) 29.0627 1.21624 0.608119 0.793846i \(-0.291924\pi\)
0.608119 + 0.793846i \(0.291924\pi\)
\(572\) 0 0
\(573\) 7.16601 0.299364
\(574\) 0 0
\(575\) 3.77124 0.157272
\(576\) 0 0
\(577\) −27.4575 −1.14307 −0.571536 0.820577i \(-0.693652\pi\)
−0.571536 + 0.820577i \(0.693652\pi\)
\(578\) 0 0
\(579\) 67.5203 2.80605
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.64575 −0.0681601
\(584\) 0 0
\(585\) −32.9150 −1.36087
\(586\) 0 0
\(587\) 7.93725 0.327606 0.163803 0.986493i \(-0.447624\pi\)
0.163803 + 0.986493i \(0.447624\pi\)
\(588\) 0 0
\(589\) 22.5830 0.930517
\(590\) 0 0
\(591\) −34.0627 −1.40115
\(592\) 0 0
\(593\) −7.06275 −0.290032 −0.145016 0.989429i \(-0.546323\pi\)
−0.145016 + 0.989429i \(0.546323\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.1882 0.457904
\(598\) 0 0
\(599\) −43.7490 −1.78754 −0.893768 0.448529i \(-0.851948\pi\)
−0.893768 + 0.448529i \(0.851948\pi\)
\(600\) 0 0
\(601\) 3.41699 0.139382 0.0696911 0.997569i \(-0.477799\pi\)
0.0696911 + 0.997569i \(0.477799\pi\)
\(602\) 0 0
\(603\) 47.7490 1.94449
\(604\) 0 0
\(605\) 1.64575 0.0669093
\(606\) 0 0
\(607\) 10.7085 0.434645 0.217322 0.976100i \(-0.430268\pi\)
0.217322 + 0.976100i \(0.430268\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13.5425 −0.547870
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) −47.6235 −1.92037
\(616\) 0 0
\(617\) −5.70850 −0.229815 −0.114908 0.993376i \(-0.536657\pi\)
−0.114908 + 0.993376i \(0.536657\pi\)
\(618\) 0 0
\(619\) 13.4170 0.539275 0.269637 0.962962i \(-0.413096\pi\)
0.269637 + 0.962962i \(0.413096\pi\)
\(620\) 0 0
\(621\) −4.35425 −0.174730
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −8.29150 −0.331660
\(626\) 0 0
\(627\) 14.9373 0.596536
\(628\) 0 0
\(629\) −21.8745 −0.872194
\(630\) 0 0
\(631\) 12.8118 0.510028 0.255014 0.966937i \(-0.417920\pi\)
0.255014 + 0.966937i \(0.417920\pi\)
\(632\) 0 0
\(633\) −2.47974 −0.0985608
\(634\) 0 0
\(635\) −26.2288 −1.04086
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −17.4170 −0.689006
\(640\) 0 0
\(641\) 12.8745 0.508512 0.254256 0.967137i \(-0.418169\pi\)
0.254256 + 0.967137i \(0.418169\pi\)
\(642\) 0 0
\(643\) −30.5203 −1.20360 −0.601801 0.798646i \(-0.705550\pi\)
−0.601801 + 0.798646i \(0.705550\pi\)
\(644\) 0 0
\(645\) 17.4170 0.685793
\(646\) 0 0
\(647\) −8.81176 −0.346426 −0.173213 0.984884i \(-0.555415\pi\)
−0.173213 + 0.984884i \(0.555415\pi\)
\(648\) 0 0
\(649\) −4.64575 −0.182362
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.64575 0.299201 0.149601 0.988746i \(-0.452201\pi\)
0.149601 + 0.988746i \(0.452201\pi\)
\(654\) 0 0
\(655\) 17.0405 0.665828
\(656\) 0 0
\(657\) −1.41699 −0.0552822
\(658\) 0 0
\(659\) 6.58301 0.256437 0.128219 0.991746i \(-0.459074\pi\)
0.128219 + 0.991746i \(0.459074\pi\)
\(660\) 0 0
\(661\) −7.41699 −0.288488 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(662\) 0 0
\(663\) 79.3725 3.08257
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10.3542 −0.400918
\(668\) 0 0
\(669\) −46.6863 −1.80500
\(670\) 0 0
\(671\) 14.2915 0.551717
\(672\) 0 0
\(673\) 0.937254 0.0361285 0.0180642 0.999837i \(-0.494250\pi\)
0.0180642 + 0.999837i \(0.494250\pi\)
\(674\) 0 0
\(675\) −6.06275 −0.233355
\(676\) 0 0
\(677\) 33.8745 1.30190 0.650952 0.759119i \(-0.274370\pi\)
0.650952 + 0.759119i \(0.274370\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −7.16601 −0.274602
\(682\) 0 0
\(683\) −1.93725 −0.0741270 −0.0370635 0.999313i \(-0.511800\pi\)
−0.0370635 + 0.999313i \(0.511800\pi\)
\(684\) 0 0
\(685\) −21.1882 −0.809561
\(686\) 0 0
\(687\) 42.3320 1.61507
\(688\) 0 0
\(689\) −8.22876 −0.313491
\(690\) 0 0
\(691\) −45.2288 −1.72058 −0.860291 0.509802i \(-0.829718\pi\)
−0.860291 + 0.509802i \(0.829718\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.58301 −0.249708
\(696\) 0 0
\(697\) 65.6235 2.48567
\(698\) 0 0
\(699\) 2.81176 0.106351
\(700\) 0 0
\(701\) 24.8745 0.939497 0.469749 0.882800i \(-0.344345\pi\)
0.469749 + 0.882800i \(0.344345\pi\)
\(702\) 0 0
\(703\) −20.5830 −0.776303
\(704\) 0 0
\(705\) 11.7935 0.444168
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 40.8118 1.53272 0.766359 0.642413i \(-0.222066\pi\)
0.766359 + 0.642413i \(0.222066\pi\)
\(710\) 0 0
\(711\) 10.5830 0.396894
\(712\) 0 0
\(713\) 6.58301 0.246535
\(714\) 0 0
\(715\) 8.22876 0.307738
\(716\) 0 0
\(717\) 45.5830 1.70233
\(718\) 0 0
\(719\) 27.8745 1.03954 0.519772 0.854305i \(-0.326017\pi\)
0.519772 + 0.854305i \(0.326017\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 65.6458 2.44139
\(724\) 0 0
\(725\) −14.4170 −0.535434
\(726\) 0 0
\(727\) −6.70850 −0.248804 −0.124402 0.992232i \(-0.539701\pi\)
−0.124402 + 0.992232i \(0.539701\pi\)
\(728\) 0 0
\(729\) −41.0000 −1.51852
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 11.4575 0.423193 0.211596 0.977357i \(-0.432134\pi\)
0.211596 + 0.977357i \(0.432134\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.9373 −0.439714
\(738\) 0 0
\(739\) −23.8745 −0.878238 −0.439119 0.898429i \(-0.644709\pi\)
−0.439119 + 0.898429i \(0.644709\pi\)
\(740\) 0 0
\(741\) 74.6863 2.74367
\(742\) 0 0
\(743\) 45.2915 1.66158 0.830792 0.556583i \(-0.187888\pi\)
0.830792 + 0.556583i \(0.187888\pi\)
\(744\) 0 0
\(745\) −25.1660 −0.922011
\(746\) 0 0
\(747\) 10.8340 0.396395
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) −8.70850 −0.317355
\(754\) 0 0
\(755\) 14.2288 0.517837
\(756\) 0 0
\(757\) −23.1660 −0.841983 −0.420991 0.907065i \(-0.638318\pi\)
−0.420991 + 0.907065i \(0.638318\pi\)
\(758\) 0 0
\(759\) 4.35425 0.158049
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −39.4980 −1.42805
\(766\) 0 0
\(767\) −23.2288 −0.838742
\(768\) 0 0
\(769\) −27.1660 −0.979631 −0.489816 0.871826i \(-0.662936\pi\)
−0.489816 + 0.871826i \(0.662936\pi\)
\(770\) 0 0
\(771\) 57.1033 2.05652
\(772\) 0 0
\(773\) 18.5830 0.668384 0.334192 0.942505i \(-0.391537\pi\)
0.334192 + 0.942505i \(0.391537\pi\)
\(774\) 0 0
\(775\) 9.16601 0.329253
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 61.7490 2.21239
\(780\) 0 0
\(781\) 4.35425 0.155807
\(782\) 0 0
\(783\) 16.6458 0.594871
\(784\) 0 0
\(785\) −34.8340 −1.24328
\(786\) 0 0
\(787\) 46.8118 1.66866 0.834330 0.551266i \(-0.185855\pi\)
0.834330 + 0.551266i \(0.185855\pi\)
\(788\) 0 0
\(789\) −52.7490 −1.87791
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 71.4575 2.53753
\(794\) 0 0
\(795\) 7.16601 0.254152
\(796\) 0 0
\(797\) −7.16601 −0.253833 −0.126917 0.991913i \(-0.540508\pi\)
−0.126917 + 0.991913i \(0.540508\pi\)
\(798\) 0 0
\(799\) −16.2510 −0.574918
\(800\) 0 0
\(801\) −26.3320 −0.930396
\(802\) 0 0
\(803\) 0.354249 0.0125012
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.3320 −0.504511
\(808\) 0 0
\(809\) 29.4170 1.03425 0.517123 0.855911i \(-0.327003\pi\)
0.517123 + 0.855911i \(0.327003\pi\)
\(810\) 0 0
\(811\) −35.7490 −1.25532 −0.627659 0.778489i \(-0.715987\pi\)
−0.627659 + 0.778489i \(0.715987\pi\)
\(812\) 0 0
\(813\) −5.45751 −0.191403
\(814\) 0 0
\(815\) −1.06275 −0.0372264
\(816\) 0 0
\(817\) −22.5830 −0.790079
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.2915 −0.638378 −0.319189 0.947691i \(-0.603410\pi\)
−0.319189 + 0.947691i \(0.603410\pi\)
\(822\) 0 0
\(823\) 12.1255 0.422668 0.211334 0.977414i \(-0.432219\pi\)
0.211334 + 0.977414i \(0.432219\pi\)
\(824\) 0 0
\(825\) 6.06275 0.211078
\(826\) 0 0
\(827\) −33.3948 −1.16125 −0.580625 0.814171i \(-0.697192\pi\)
−0.580625 + 0.814171i \(0.697192\pi\)
\(828\) 0 0
\(829\) 25.3948 0.881997 0.440998 0.897508i \(-0.354624\pi\)
0.440998 + 0.897508i \(0.354624\pi\)
\(830\) 0 0
\(831\) −58.9778 −2.04592
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −18.4797 −0.639518
\(836\) 0 0
\(837\) −10.5830 −0.365802
\(838\) 0 0
\(839\) −3.87451 −0.133763 −0.0668814 0.997761i \(-0.521305\pi\)
−0.0668814 + 0.997761i \(0.521305\pi\)
\(840\) 0 0
\(841\) 10.5830 0.364931
\(842\) 0 0
\(843\) −60.6863 −2.09015
\(844\) 0 0
\(845\) 19.7490 0.679387
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −6.22876 −0.213770
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −39.1660 −1.34102 −0.670509 0.741901i \(-0.733924\pi\)
−0.670509 + 0.741901i \(0.733924\pi\)
\(854\) 0 0
\(855\) −37.1660 −1.27105
\(856\) 0 0
\(857\) −36.0000 −1.22974 −0.614868 0.788630i \(-0.710791\pi\)
−0.614868 + 0.788630i \(0.710791\pi\)
\(858\) 0 0
\(859\) −9.81176 −0.334773 −0.167386 0.985891i \(-0.553533\pi\)
−0.167386 + 0.985891i \(0.553533\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.4797 0.424815 0.212408 0.977181i \(-0.431870\pi\)
0.212408 + 0.977181i \(0.431870\pi\)
\(864\) 0 0
\(865\) 0.479741 0.0163117
\(866\) 0 0
\(867\) 50.2693 1.70723
\(868\) 0 0
\(869\) −2.64575 −0.0897510
\(870\) 0 0
\(871\) −59.6863 −2.02239
\(872\) 0 0
\(873\) 65.1660 2.20554
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.8745 −0.772417 −0.386209 0.922411i \(-0.626216\pi\)
−0.386209 + 0.922411i \(0.626216\pi\)
\(878\) 0 0
\(879\) −31.7490 −1.07087
\(880\) 0 0
\(881\) 24.8745 0.838043 0.419022 0.907976i \(-0.362373\pi\)
0.419022 + 0.907976i \(0.362373\pi\)
\(882\) 0 0
\(883\) −21.9373 −0.738247 −0.369124 0.929380i \(-0.620342\pi\)
−0.369124 + 0.929380i \(0.620342\pi\)
\(884\) 0 0
\(885\) 20.2288 0.679982
\(886\) 0 0
\(887\) 45.1033 1.51442 0.757210 0.653172i \(-0.226562\pi\)
0.757210 + 0.653172i \(0.226562\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) −15.2915 −0.511711
\(894\) 0 0
\(895\) 32.8118 1.09678
\(896\) 0 0
\(897\) 21.7712 0.726921
\(898\) 0 0
\(899\) −25.1660 −0.839333
\(900\) 0 0
\(901\) −9.87451 −0.328968
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.4575 0.547066
\(906\) 0 0
\(907\) −42.4575 −1.40978 −0.704889 0.709317i \(-0.749003\pi\)
−0.704889 + 0.709317i \(0.749003\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −9.29150 −0.307841 −0.153921 0.988083i \(-0.549190\pi\)
−0.153921 + 0.988083i \(0.549190\pi\)
\(912\) 0 0
\(913\) −2.70850 −0.0896382
\(914\) 0 0
\(915\) −62.2288 −2.05722
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 24.7085 0.815058 0.407529 0.913192i \(-0.366391\pi\)
0.407529 + 0.913192i \(0.366391\pi\)
\(920\) 0 0
\(921\) 58.8118 1.93791
\(922\) 0 0
\(923\) 21.7712 0.716609
\(924\) 0 0
\(925\) −8.35425 −0.274686
\(926\) 0 0
\(927\) −11.7490 −0.385888
\(928\) 0 0
\(929\) −14.4170 −0.473006 −0.236503 0.971631i \(-0.576001\pi\)
−0.236503 + 0.971631i \(0.576001\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.81176 0.0920529
\(934\) 0 0
\(935\) 9.87451 0.322931
\(936\) 0 0
\(937\) −32.6863 −1.06781 −0.533907 0.845543i \(-0.679277\pi\)
−0.533907 + 0.845543i \(0.679277\pi\)
\(938\) 0 0
\(939\) −62.3948 −2.03618
\(940\) 0 0
\(941\) −50.6235 −1.65028 −0.825140 0.564929i \(-0.808904\pi\)
−0.825140 + 0.564929i \(0.808904\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.12549 0.0690692 0.0345346 0.999404i \(-0.489005\pi\)
0.0345346 + 0.999404i \(0.489005\pi\)
\(948\) 0 0
\(949\) 1.77124 0.0574970
\(950\) 0 0
\(951\) 31.7490 1.02953
\(952\) 0 0
\(953\) 11.5203 0.373178 0.186589 0.982438i \(-0.440257\pi\)
0.186589 + 0.982438i \(0.440257\pi\)
\(954\) 0 0
\(955\) 4.45751 0.144242
\(956\) 0 0
\(957\) −16.6458 −0.538081
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 43.7490 1.40979
\(964\) 0 0
\(965\) 42.0000 1.35203
\(966\) 0 0
\(967\) −58.3320 −1.87583 −0.937916 0.346863i \(-0.887247\pi\)
−0.937916 + 0.346863i \(0.887247\pi\)
\(968\) 0 0
\(969\) 89.6235 2.87912
\(970\) 0 0
\(971\) −10.0627 −0.322929 −0.161464 0.986879i \(-0.551622\pi\)
−0.161464 + 0.986879i \(0.551622\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 30.3137 0.970816
\(976\) 0 0
\(977\) −16.4575 −0.526522 −0.263261 0.964725i \(-0.584798\pi\)
−0.263261 + 0.964725i \(0.584798\pi\)
\(978\) 0 0
\(979\) 6.58301 0.210394
\(980\) 0 0
\(981\) −42.3320 −1.35156
\(982\) 0 0
\(983\) −26.7085 −0.851869 −0.425934 0.904754i \(-0.640055\pi\)
−0.425934 + 0.904754i \(0.640055\pi\)
\(984\) 0 0
\(985\) −21.1882 −0.675113
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.58301 −0.209327
\(990\) 0 0
\(991\) 22.6863 0.720653 0.360327 0.932826i \(-0.382665\pi\)
0.360327 + 0.932826i \(0.382665\pi\)
\(992\) 0 0
\(993\) 54.6235 1.73343
\(994\) 0 0
\(995\) 6.95948 0.220630
\(996\) 0 0
\(997\) 21.4170 0.678283 0.339142 0.940735i \(-0.389863\pi\)
0.339142 + 0.940735i \(0.389863\pi\)
\(998\) 0 0
\(999\) 9.64575 0.305178
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 8624.2.a.bk.1.2 2
4.3 odd 2 1078.2.a.n.1.1 2
7.3 odd 6 1232.2.q.g.177.2 4
7.5 odd 6 1232.2.q.g.529.2 4
7.6 odd 2 8624.2.a.ca.1.1 2
12.11 even 2 9702.2.a.dr.1.1 2
28.3 even 6 154.2.e.f.23.1 4
28.11 odd 6 1078.2.e.v.177.2 4
28.19 even 6 154.2.e.f.67.1 yes 4
28.23 odd 6 1078.2.e.v.67.2 4
28.27 even 2 1078.2.a.s.1.2 2
84.47 odd 6 1386.2.k.s.991.1 4
84.59 odd 6 1386.2.k.s.793.1 4
84.83 odd 2 9702.2.a.cz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.f.23.1 4 28.3 even 6
154.2.e.f.67.1 yes 4 28.19 even 6
1078.2.a.n.1.1 2 4.3 odd 2
1078.2.a.s.1.2 2 28.27 even 2
1078.2.e.v.67.2 4 28.23 odd 6
1078.2.e.v.177.2 4 28.11 odd 6
1232.2.q.g.177.2 4 7.3 odd 6
1232.2.q.g.529.2 4 7.5 odd 6
1386.2.k.s.793.1 4 84.59 odd 6
1386.2.k.s.991.1 4 84.47 odd 6
8624.2.a.bk.1.2 2 1.1 even 1 trivial
8624.2.a.ca.1.1 2 7.6 odd 2
9702.2.a.cz.1.2 2 84.83 odd 2
9702.2.a.dr.1.1 2 12.11 even 2