Properties

Label 192.6.a.o.1.1
Level $192$
Weight $6$
Character 192.1
Self dual yes
Analytic conductor $30.794$
Analytic rank $0$
Dimension $1$
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] = [192,6,Mod(1,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 192.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: \(30.7936934041\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 192.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.00000 q^{3} +66.0000 q^{5} +176.000 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+9.00000 q^{3} +66.0000 q^{5} +176.000 q^{7} +81.0000 q^{9} +60.0000 q^{11} +658.000 q^{13} +594.000 q^{15} -414.000 q^{17} -956.000 q^{19} +1584.00 q^{21} +600.000 q^{23} +1231.00 q^{25} +729.000 q^{27} -5574.00 q^{29} -3592.00 q^{31} +540.000 q^{33} +11616.0 q^{35} +8458.00 q^{37} +5922.00 q^{39} +19194.0 q^{41} -13316.0 q^{43} +5346.00 q^{45} -19680.0 q^{47} +14169.0 q^{49} -3726.00 q^{51} +31266.0 q^{53} +3960.00 q^{55} -8604.00 q^{57} -26340.0 q^{59} +31090.0 q^{61} +14256.0 q^{63} +43428.0 q^{65} +16804.0 q^{67} +5400.00 q^{69} +6120.00 q^{71} -25558.0 q^{73} +11079.0 q^{75} +10560.0 q^{77} +74408.0 q^{79} +6561.00 q^{81} +6468.00 q^{83} -27324.0 q^{85} -50166.0 q^{87} -32742.0 q^{89} +115808. q^{91} -32328.0 q^{93} -63096.0 q^{95} +166082. q^{97} +4860.00 q^{99} +O(q^{100})\)

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
\(3\) 9.00000 0.577350
\(4\) 0 0
\(5\) 66.0000 1.18064 0.590322 0.807168i \(-0.299001\pi\)
0.590322 + 0.807168i \(0.299001\pi\)
\(6\) 0 0
\(7\) 176.000 1.35759 0.678793 0.734329i \(-0.262503\pi\)
0.678793 + 0.734329i \(0.262503\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 60.0000 0.149510 0.0747549 0.997202i \(-0.476183\pi\)
0.0747549 + 0.997202i \(0.476183\pi\)
\(12\) 0 0
\(13\) 658.000 1.07986 0.539930 0.841710i \(-0.318451\pi\)
0.539930 + 0.841710i \(0.318451\pi\)
\(14\) 0 0
\(15\) 594.000 0.681645
\(16\) 0 0
\(17\) −414.000 −0.347439 −0.173719 0.984795i \(-0.555579\pi\)
−0.173719 + 0.984795i \(0.555579\pi\)
\(18\) 0 0
\(19\) −956.000 −0.607539 −0.303769 0.952746i \(-0.598245\pi\)
−0.303769 + 0.952746i \(0.598245\pi\)
\(20\) 0 0
\(21\) 1584.00 0.783803
\(22\) 0 0
\(23\) 600.000 0.236500 0.118250 0.992984i \(-0.462272\pi\)
0.118250 + 0.992984i \(0.462272\pi\)
\(24\) 0 0
\(25\) 1231.00 0.393920
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −5574.00 −1.23076 −0.615378 0.788232i \(-0.710997\pi\)
−0.615378 + 0.788232i \(0.710997\pi\)
\(30\) 0 0
\(31\) −3592.00 −0.671324 −0.335662 0.941983i \(-0.608960\pi\)
−0.335662 + 0.941983i \(0.608960\pi\)
\(32\) 0 0
\(33\) 540.000 0.0863195
\(34\) 0 0
\(35\) 11616.0 1.60283
\(36\) 0 0
\(37\) 8458.00 1.01570 0.507848 0.861447i \(-0.330441\pi\)
0.507848 + 0.861447i \(0.330441\pi\)
\(38\) 0 0
\(39\) 5922.00 0.623458
\(40\) 0 0
\(41\) 19194.0 1.78322 0.891612 0.452800i \(-0.149575\pi\)
0.891612 + 0.452800i \(0.149575\pi\)
\(42\) 0 0
\(43\) −13316.0 −1.09825 −0.549127 0.835739i \(-0.685040\pi\)
−0.549127 + 0.835739i \(0.685040\pi\)
\(44\) 0 0
\(45\) 5346.00 0.393548
\(46\) 0 0
\(47\) −19680.0 −1.29951 −0.649756 0.760143i \(-0.725129\pi\)
−0.649756 + 0.760143i \(0.725129\pi\)
\(48\) 0 0
\(49\) 14169.0 0.843042
\(50\) 0 0
\(51\) −3726.00 −0.200594
\(52\) 0 0
\(53\) 31266.0 1.52891 0.764456 0.644676i \(-0.223008\pi\)
0.764456 + 0.644676i \(0.223008\pi\)
\(54\) 0 0
\(55\) 3960.00 0.176518
\(56\) 0 0
\(57\) −8604.00 −0.350763
\(58\) 0 0
\(59\) −26340.0 −0.985112 −0.492556 0.870281i \(-0.663937\pi\)
−0.492556 + 0.870281i \(0.663937\pi\)
\(60\) 0 0
\(61\) 31090.0 1.06978 0.534892 0.844920i \(-0.320352\pi\)
0.534892 + 0.844920i \(0.320352\pi\)
\(62\) 0 0
\(63\) 14256.0 0.452529
\(64\) 0 0
\(65\) 43428.0 1.27493
\(66\) 0 0
\(67\) 16804.0 0.457326 0.228663 0.973506i \(-0.426565\pi\)
0.228663 + 0.973506i \(0.426565\pi\)
\(68\) 0 0
\(69\) 5400.00 0.136544
\(70\) 0 0
\(71\) 6120.00 0.144081 0.0720403 0.997402i \(-0.477049\pi\)
0.0720403 + 0.997402i \(0.477049\pi\)
\(72\) 0 0
\(73\) −25558.0 −0.561332 −0.280666 0.959806i \(-0.590555\pi\)
−0.280666 + 0.959806i \(0.590555\pi\)
\(74\) 0 0
\(75\) 11079.0 0.227430
\(76\) 0 0
\(77\) 10560.0 0.202972
\(78\) 0 0
\(79\) 74408.0 1.34138 0.670690 0.741738i \(-0.265998\pi\)
0.670690 + 0.741738i \(0.265998\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 6468.00 0.103056 0.0515282 0.998672i \(-0.483591\pi\)
0.0515282 + 0.998672i \(0.483591\pi\)
\(84\) 0 0
\(85\) −27324.0 −0.410201
\(86\) 0 0
\(87\) −50166.0 −0.710577
\(88\) 0 0
\(89\) −32742.0 −0.438157 −0.219079 0.975707i \(-0.570305\pi\)
−0.219079 + 0.975707i \(0.570305\pi\)
\(90\) 0 0
\(91\) 115808. 1.46600
\(92\) 0 0
\(93\) −32328.0 −0.387589
\(94\) 0 0
\(95\) −63096.0 −0.717287
\(96\) 0 0
\(97\) 166082. 1.79223 0.896114 0.443824i \(-0.146378\pi\)
0.896114 + 0.443824i \(0.146378\pi\)
\(98\) 0 0
\(99\) 4860.00 0.0498366
\(100\) 0 0
\(101\) 22002.0 0.214614 0.107307 0.994226i \(-0.465777\pi\)
0.107307 + 0.994226i \(0.465777\pi\)
\(102\) 0 0
\(103\) −79264.0 −0.736178 −0.368089 0.929791i \(-0.619988\pi\)
−0.368089 + 0.929791i \(0.619988\pi\)
\(104\) 0 0
\(105\) 104544. 0.925392
\(106\) 0 0
\(107\) −227988. −1.92510 −0.962548 0.271110i \(-0.912609\pi\)
−0.962548 + 0.271110i \(0.912609\pi\)
\(108\) 0 0
\(109\) 8530.00 0.0687674 0.0343837 0.999409i \(-0.489053\pi\)
0.0343837 + 0.999409i \(0.489053\pi\)
\(110\) 0 0
\(111\) 76122.0 0.586412
\(112\) 0 0
\(113\) −195438. −1.43984 −0.719918 0.694059i \(-0.755821\pi\)
−0.719918 + 0.694059i \(0.755821\pi\)
\(114\) 0 0
\(115\) 39600.0 0.279223
\(116\) 0 0
\(117\) 53298.0 0.359953
\(118\) 0 0
\(119\) −72864.0 −0.471678
\(120\) 0 0
\(121\) −157451. −0.977647
\(122\) 0 0
\(123\) 172746. 1.02954
\(124\) 0 0
\(125\) −125004. −0.715565
\(126\) 0 0
\(127\) 173000. 0.951780 0.475890 0.879505i \(-0.342126\pi\)
0.475890 + 0.879505i \(0.342126\pi\)
\(128\) 0 0
\(129\) −119844. −0.634077
\(130\) 0 0
\(131\) −151260. −0.770098 −0.385049 0.922896i \(-0.625815\pi\)
−0.385049 + 0.922896i \(0.625815\pi\)
\(132\) 0 0
\(133\) −168256. −0.824786
\(134\) 0 0
\(135\) 48114.0 0.227215
\(136\) 0 0
\(137\) −128454. −0.584718 −0.292359 0.956309i \(-0.594440\pi\)
−0.292359 + 0.956309i \(0.594440\pi\)
\(138\) 0 0
\(139\) −154196. −0.676918 −0.338459 0.940981i \(-0.609906\pi\)
−0.338459 + 0.940981i \(0.609906\pi\)
\(140\) 0 0
\(141\) −177120. −0.750274
\(142\) 0 0
\(143\) 39480.0 0.161450
\(144\) 0 0
\(145\) −367884. −1.45308
\(146\) 0 0
\(147\) 127521. 0.486730
\(148\) 0 0
\(149\) −29454.0 −0.108687 −0.0543436 0.998522i \(-0.517307\pi\)
−0.0543436 + 0.998522i \(0.517307\pi\)
\(150\) 0 0
\(151\) −203872. −0.727638 −0.363819 0.931470i \(-0.618527\pi\)
−0.363819 + 0.931470i \(0.618527\pi\)
\(152\) 0 0
\(153\) −33534.0 −0.115813
\(154\) 0 0
\(155\) −237072. −0.792594
\(156\) 0 0
\(157\) −136142. −0.440801 −0.220401 0.975409i \(-0.570737\pi\)
−0.220401 + 0.975409i \(0.570737\pi\)
\(158\) 0 0
\(159\) 281394. 0.882718
\(160\) 0 0
\(161\) 105600. 0.321070
\(162\) 0 0
\(163\) 171124. 0.504478 0.252239 0.967665i \(-0.418833\pi\)
0.252239 + 0.967665i \(0.418833\pi\)
\(164\) 0 0
\(165\) 35640.0 0.101913
\(166\) 0 0
\(167\) −676200. −1.87622 −0.938110 0.346336i \(-0.887426\pi\)
−0.938110 + 0.346336i \(0.887426\pi\)
\(168\) 0 0
\(169\) 61671.0 0.166098
\(170\) 0 0
\(171\) −77436.0 −0.202513
\(172\) 0 0
\(173\) −133158. −0.338261 −0.169131 0.985594i \(-0.554096\pi\)
−0.169131 + 0.985594i \(0.554096\pi\)
\(174\) 0 0
\(175\) 216656. 0.534781
\(176\) 0 0
\(177\) −237060. −0.568755
\(178\) 0 0
\(179\) 693396. 1.61752 0.808758 0.588141i \(-0.200140\pi\)
0.808758 + 0.588141i \(0.200140\pi\)
\(180\) 0 0
\(181\) −377174. −0.855747 −0.427873 0.903839i \(-0.640737\pi\)
−0.427873 + 0.903839i \(0.640737\pi\)
\(182\) 0 0
\(183\) 279810. 0.617640
\(184\) 0 0
\(185\) 558228. 1.19917
\(186\) 0 0
\(187\) −24840.0 −0.0519455
\(188\) 0 0
\(189\) 128304. 0.261268
\(190\) 0 0
\(191\) −265344. −0.526291 −0.263145 0.964756i \(-0.584760\pi\)
−0.263145 + 0.964756i \(0.584760\pi\)
\(192\) 0 0
\(193\) 295298. 0.570647 0.285323 0.958431i \(-0.407899\pi\)
0.285323 + 0.958431i \(0.407899\pi\)
\(194\) 0 0
\(195\) 390852. 0.736081
\(196\) 0 0
\(197\) −201294. −0.369543 −0.184772 0.982781i \(-0.559155\pi\)
−0.184772 + 0.982781i \(0.559155\pi\)
\(198\) 0 0
\(199\) 652448. 1.16792 0.583960 0.811782i \(-0.301502\pi\)
0.583960 + 0.811782i \(0.301502\pi\)
\(200\) 0 0
\(201\) 151236. 0.264037
\(202\) 0 0
\(203\) −981024. −1.67086
\(204\) 0 0
\(205\) 1.26680e6 2.10535
\(206\) 0 0
\(207\) 48600.0 0.0788334
\(208\) 0 0
\(209\) −57360.0 −0.0908330
\(210\) 0 0
\(211\) 1.14706e6 1.77370 0.886850 0.462058i \(-0.152889\pi\)
0.886850 + 0.462058i \(0.152889\pi\)
\(212\) 0 0
\(213\) 55080.0 0.0831850
\(214\) 0 0
\(215\) −878856. −1.29665
\(216\) 0 0
\(217\) −632192. −0.911380
\(218\) 0 0
\(219\) −230022. −0.324085
\(220\) 0 0
\(221\) −272412. −0.375185
\(222\) 0 0
\(223\) 701960. 0.945258 0.472629 0.881262i \(-0.343305\pi\)
0.472629 + 0.881262i \(0.343305\pi\)
\(224\) 0 0
\(225\) 99711.0 0.131307
\(226\) 0 0
\(227\) −1.23611e6 −1.59218 −0.796089 0.605179i \(-0.793101\pi\)
−0.796089 + 0.605179i \(0.793101\pi\)
\(228\) 0 0
\(229\) −105830. −0.133358 −0.0666792 0.997774i \(-0.521240\pi\)
−0.0666792 + 0.997774i \(0.521240\pi\)
\(230\) 0 0
\(231\) 95040.0 0.117186
\(232\) 0 0
\(233\) −438678. −0.529366 −0.264683 0.964335i \(-0.585267\pi\)
−0.264683 + 0.964335i \(0.585267\pi\)
\(234\) 0 0
\(235\) −1.29888e6 −1.53426
\(236\) 0 0
\(237\) 669672. 0.774446
\(238\) 0 0
\(239\) 28464.0 0.0322330 0.0161165 0.999870i \(-0.494870\pi\)
0.0161165 + 0.999870i \(0.494870\pi\)
\(240\) 0 0
\(241\) 892562. 0.989910 0.494955 0.868919i \(-0.335185\pi\)
0.494955 + 0.868919i \(0.335185\pi\)
\(242\) 0 0
\(243\) 59049.0 0.0641500
\(244\) 0 0
\(245\) 935154. 0.995332
\(246\) 0 0
\(247\) −629048. −0.656057
\(248\) 0 0
\(249\) 58212.0 0.0594996
\(250\) 0 0
\(251\) 110124. 0.110331 0.0551655 0.998477i \(-0.482431\pi\)
0.0551655 + 0.998477i \(0.482431\pi\)
\(252\) 0 0
\(253\) 36000.0 0.0353591
\(254\) 0 0
\(255\) −245916. −0.236830
\(256\) 0 0
\(257\) 140802. 0.132977 0.0664884 0.997787i \(-0.478820\pi\)
0.0664884 + 0.997787i \(0.478820\pi\)
\(258\) 0 0
\(259\) 1.48861e6 1.37889
\(260\) 0 0
\(261\) −451494. −0.410252
\(262\) 0 0
\(263\) −938760. −0.836884 −0.418442 0.908244i \(-0.637424\pi\)
−0.418442 + 0.908244i \(0.637424\pi\)
\(264\) 0 0
\(265\) 2.06356e6 1.80510
\(266\) 0 0
\(267\) −294678. −0.252970
\(268\) 0 0
\(269\) 1.11451e6 0.939078 0.469539 0.882912i \(-0.344420\pi\)
0.469539 + 0.882912i \(0.344420\pi\)
\(270\) 0 0
\(271\) 567704. 0.469568 0.234784 0.972048i \(-0.424562\pi\)
0.234784 + 0.972048i \(0.424562\pi\)
\(272\) 0 0
\(273\) 1.04227e6 0.846398
\(274\) 0 0
\(275\) 73860.0 0.0588949
\(276\) 0 0
\(277\) 1.21326e6 0.950066 0.475033 0.879968i \(-0.342436\pi\)
0.475033 + 0.879968i \(0.342436\pi\)
\(278\) 0 0
\(279\) −290952. −0.223775
\(280\) 0 0
\(281\) 687738. 0.519586 0.259793 0.965664i \(-0.416346\pi\)
0.259793 + 0.965664i \(0.416346\pi\)
\(282\) 0 0
\(283\) 830908. 0.616718 0.308359 0.951270i \(-0.400220\pi\)
0.308359 + 0.951270i \(0.400220\pi\)
\(284\) 0 0
\(285\) −567864. −0.414126
\(286\) 0 0
\(287\) 3.37814e6 2.42088
\(288\) 0 0
\(289\) −1.24846e6 −0.879286
\(290\) 0 0
\(291\) 1.49474e6 1.03474
\(292\) 0 0
\(293\) 1.31263e6 0.893248 0.446624 0.894722i \(-0.352626\pi\)
0.446624 + 0.894722i \(0.352626\pi\)
\(294\) 0 0
\(295\) −1.73844e6 −1.16307
\(296\) 0 0
\(297\) 43740.0 0.0287732
\(298\) 0 0
\(299\) 394800. 0.255387
\(300\) 0 0
\(301\) −2.34362e6 −1.49097
\(302\) 0 0
\(303\) 198018. 0.123908
\(304\) 0 0
\(305\) 2.05194e6 1.26303
\(306\) 0 0
\(307\) −1.69022e6 −1.02352 −0.511761 0.859128i \(-0.671007\pi\)
−0.511761 + 0.859128i \(0.671007\pi\)
\(308\) 0 0
\(309\) −713376. −0.425033
\(310\) 0 0
\(311\) −1.50204e6 −0.880604 −0.440302 0.897850i \(-0.645129\pi\)
−0.440302 + 0.897850i \(0.645129\pi\)
\(312\) 0 0
\(313\) 810842. 0.467816 0.233908 0.972259i \(-0.424848\pi\)
0.233908 + 0.972259i \(0.424848\pi\)
\(314\) 0 0
\(315\) 940896. 0.534275
\(316\) 0 0
\(317\) −903558. −0.505019 −0.252510 0.967594i \(-0.581256\pi\)
−0.252510 + 0.967594i \(0.581256\pi\)
\(318\) 0 0
\(319\) −334440. −0.184010
\(320\) 0 0
\(321\) −2.05189e6 −1.11146
\(322\) 0 0
\(323\) 395784. 0.211082
\(324\) 0 0
\(325\) 809998. 0.425379
\(326\) 0 0
\(327\) 76770.0 0.0397029
\(328\) 0 0
\(329\) −3.46368e6 −1.76420
\(330\) 0 0
\(331\) −1.12197e6 −0.562875 −0.281438 0.959580i \(-0.590811\pi\)
−0.281438 + 0.959580i \(0.590811\pi\)
\(332\) 0 0
\(333\) 685098. 0.338565
\(334\) 0 0
\(335\) 1.10906e6 0.539939
\(336\) 0 0
\(337\) −2.75217e6 −1.32008 −0.660041 0.751229i \(-0.729461\pi\)
−0.660041 + 0.751229i \(0.729461\pi\)
\(338\) 0 0
\(339\) −1.75894e6 −0.831289
\(340\) 0 0
\(341\) −215520. −0.100369
\(342\) 0 0
\(343\) −464288. −0.213085
\(344\) 0 0
\(345\) 356400. 0.161209
\(346\) 0 0
\(347\) −1.91749e6 −0.854889 −0.427445 0.904042i \(-0.640586\pi\)
−0.427445 + 0.904042i \(0.640586\pi\)
\(348\) 0 0
\(349\) −1.83659e6 −0.807140 −0.403570 0.914949i \(-0.632231\pi\)
−0.403570 + 0.914949i \(0.632231\pi\)
\(350\) 0 0
\(351\) 479682. 0.207819
\(352\) 0 0
\(353\) −622014. −0.265683 −0.132841 0.991137i \(-0.542410\pi\)
−0.132841 + 0.991137i \(0.542410\pi\)
\(354\) 0 0
\(355\) 403920. 0.170108
\(356\) 0 0
\(357\) −655776. −0.272323
\(358\) 0 0
\(359\) 3.74062e6 1.53182 0.765909 0.642949i \(-0.222289\pi\)
0.765909 + 0.642949i \(0.222289\pi\)
\(360\) 0 0
\(361\) −1.56216e6 −0.630897
\(362\) 0 0
\(363\) −1.41706e6 −0.564445
\(364\) 0 0
\(365\) −1.68683e6 −0.662733
\(366\) 0 0
\(367\) 16232.0 0.00629081 0.00314541 0.999995i \(-0.498999\pi\)
0.00314541 + 0.999995i \(0.498999\pi\)
\(368\) 0 0
\(369\) 1.55471e6 0.594408
\(370\) 0 0
\(371\) 5.50282e6 2.07563
\(372\) 0 0
\(373\) −293606. −0.109268 −0.0546340 0.998506i \(-0.517399\pi\)
−0.0546340 + 0.998506i \(0.517399\pi\)
\(374\) 0 0
\(375\) −1.12504e6 −0.413131
\(376\) 0 0
\(377\) −3.66769e6 −1.32904
\(378\) 0 0
\(379\) −3.18012e6 −1.13722 −0.568611 0.822607i \(-0.692519\pi\)
−0.568611 + 0.822607i \(0.692519\pi\)
\(380\) 0 0
\(381\) 1.55700e6 0.549511
\(382\) 0 0
\(383\) −2.97984e6 −1.03800 −0.518998 0.854775i \(-0.673695\pi\)
−0.518998 + 0.854775i \(0.673695\pi\)
\(384\) 0 0
\(385\) 696960. 0.239638
\(386\) 0 0
\(387\) −1.07860e6 −0.366085
\(388\) 0 0
\(389\) −3.45977e6 −1.15924 −0.579620 0.814887i \(-0.696799\pi\)
−0.579620 + 0.814887i \(0.696799\pi\)
\(390\) 0 0
\(391\) −248400. −0.0821693
\(392\) 0 0
\(393\) −1.36134e6 −0.444616
\(394\) 0 0
\(395\) 4.91093e6 1.58369
\(396\) 0 0
\(397\) 3.90416e6 1.24323 0.621615 0.783323i \(-0.286477\pi\)
0.621615 + 0.783323i \(0.286477\pi\)
\(398\) 0 0
\(399\) −1.51430e6 −0.476191
\(400\) 0 0
\(401\) 5.44115e6 1.68978 0.844890 0.534940i \(-0.179666\pi\)
0.844890 + 0.534940i \(0.179666\pi\)
\(402\) 0 0
\(403\) −2.36354e6 −0.724936
\(404\) 0 0
\(405\) 433026. 0.131183
\(406\) 0 0
\(407\) 507480. 0.151856
\(408\) 0 0
\(409\) 1.96995e6 0.582299 0.291150 0.956678i \(-0.405962\pi\)
0.291150 + 0.956678i \(0.405962\pi\)
\(410\) 0 0
\(411\) −1.15609e6 −0.337587
\(412\) 0 0
\(413\) −4.63584e6 −1.33738
\(414\) 0 0
\(415\) 426888. 0.121673
\(416\) 0 0
\(417\) −1.38776e6 −0.390819
\(418\) 0 0
\(419\) −139020. −0.0386850 −0.0193425 0.999813i \(-0.506157\pi\)
−0.0193425 + 0.999813i \(0.506157\pi\)
\(420\) 0 0
\(421\) −4.32743e6 −1.18994 −0.594970 0.803748i \(-0.702836\pi\)
−0.594970 + 0.803748i \(0.702836\pi\)
\(422\) 0 0
\(423\) −1.59408e6 −0.433171
\(424\) 0 0
\(425\) −509634. −0.136863
\(426\) 0 0
\(427\) 5.47184e6 1.45232
\(428\) 0 0
\(429\) 355320. 0.0932130
\(430\) 0 0
\(431\) −2.79936e6 −0.725881 −0.362941 0.931812i \(-0.618227\pi\)
−0.362941 + 0.931812i \(0.618227\pi\)
\(432\) 0 0
\(433\) −5.90241e6 −1.51290 −0.756449 0.654052i \(-0.773068\pi\)
−0.756449 + 0.654052i \(0.773068\pi\)
\(434\) 0 0
\(435\) −3.31096e6 −0.838939
\(436\) 0 0
\(437\) −573600. −0.143683
\(438\) 0 0
\(439\) −446512. −0.110579 −0.0552894 0.998470i \(-0.517608\pi\)
−0.0552894 + 0.998470i \(0.517608\pi\)
\(440\) 0 0
\(441\) 1.14769e6 0.281014
\(442\) 0 0
\(443\) −3.49525e6 −0.846193 −0.423096 0.906085i \(-0.639057\pi\)
−0.423096 + 0.906085i \(0.639057\pi\)
\(444\) 0 0
\(445\) −2.16097e6 −0.517308
\(446\) 0 0
\(447\) −265086. −0.0627506
\(448\) 0 0
\(449\) −1.20613e6 −0.282343 −0.141171 0.989985i \(-0.545087\pi\)
−0.141171 + 0.989985i \(0.545087\pi\)
\(450\) 0 0
\(451\) 1.15164e6 0.266609
\(452\) 0 0
\(453\) −1.83485e6 −0.420102
\(454\) 0 0
\(455\) 7.64333e6 1.73083
\(456\) 0 0
\(457\) 233546. 0.0523097 0.0261548 0.999658i \(-0.491674\pi\)
0.0261548 + 0.999658i \(0.491674\pi\)
\(458\) 0 0
\(459\) −301806. −0.0668646
\(460\) 0 0
\(461\) 1.74489e6 0.382398 0.191199 0.981551i \(-0.438762\pi\)
0.191199 + 0.981551i \(0.438762\pi\)
\(462\) 0 0
\(463\) −2.91786e6 −0.632576 −0.316288 0.948663i \(-0.602437\pi\)
−0.316288 + 0.948663i \(0.602437\pi\)
\(464\) 0 0
\(465\) −2.13365e6 −0.457605
\(466\) 0 0
\(467\) 5.31076e6 1.12684 0.563422 0.826169i \(-0.309484\pi\)
0.563422 + 0.826169i \(0.309484\pi\)
\(468\) 0 0
\(469\) 2.95750e6 0.620859
\(470\) 0 0
\(471\) −1.22528e6 −0.254497
\(472\) 0 0
\(473\) −798960. −0.164200
\(474\) 0 0
\(475\) −1.17684e6 −0.239322
\(476\) 0 0
\(477\) 2.53255e6 0.509638
\(478\) 0 0
\(479\) 2.34466e6 0.466918 0.233459 0.972367i \(-0.424996\pi\)
0.233459 + 0.972367i \(0.424996\pi\)
\(480\) 0 0
\(481\) 5.56536e6 1.09681
\(482\) 0 0
\(483\) 950400. 0.185370
\(484\) 0 0
\(485\) 1.09614e7 2.11598
\(486\) 0 0
\(487\) 9.81531e6 1.87535 0.937674 0.347517i \(-0.112975\pi\)
0.937674 + 0.347517i \(0.112975\pi\)
\(488\) 0 0
\(489\) 1.54012e6 0.291260
\(490\) 0 0
\(491\) 5.94520e6 1.11292 0.556458 0.830876i \(-0.312160\pi\)
0.556458 + 0.830876i \(0.312160\pi\)
\(492\) 0 0
\(493\) 2.30764e6 0.427612
\(494\) 0 0
\(495\) 320760. 0.0588393
\(496\) 0 0
\(497\) 1.07712e6 0.195602
\(498\) 0 0
\(499\) −6.47832e6 −1.16469 −0.582346 0.812941i \(-0.697865\pi\)
−0.582346 + 0.812941i \(0.697865\pi\)
\(500\) 0 0
\(501\) −6.08580e6 −1.08324
\(502\) 0 0
\(503\) 4.71794e6 0.831444 0.415722 0.909492i \(-0.363529\pi\)
0.415722 + 0.909492i \(0.363529\pi\)
\(504\) 0 0
\(505\) 1.45213e6 0.253383
\(506\) 0 0
\(507\) 555039. 0.0958967
\(508\) 0 0
\(509\) 1.90771e6 0.326375 0.163188 0.986595i \(-0.447822\pi\)
0.163188 + 0.986595i \(0.447822\pi\)
\(510\) 0 0
\(511\) −4.49821e6 −0.762057
\(512\) 0 0
\(513\) −696924. −0.116921
\(514\) 0 0
\(515\) −5.23142e6 −0.869164
\(516\) 0 0
\(517\) −1.18080e6 −0.194290
\(518\) 0 0
\(519\) −1.19842e6 −0.195295
\(520\) 0 0
\(521\) 8.01974e6 1.29439 0.647196 0.762324i \(-0.275941\pi\)
0.647196 + 0.762324i \(0.275941\pi\)
\(522\) 0 0
\(523\) −1.91162e6 −0.305596 −0.152798 0.988257i \(-0.548828\pi\)
−0.152798 + 0.988257i \(0.548828\pi\)
\(524\) 0 0
\(525\) 1.94990e6 0.308756
\(526\) 0 0
\(527\) 1.48709e6 0.233244
\(528\) 0 0
\(529\) −6.07634e6 −0.944068
\(530\) 0 0
\(531\) −2.13354e6 −0.328371
\(532\) 0 0
\(533\) 1.26297e7 1.92563
\(534\) 0 0
\(535\) −1.50472e7 −2.27285
\(536\) 0 0
\(537\) 6.24056e6 0.933874
\(538\) 0 0
\(539\) 850140. 0.126043
\(540\) 0 0
\(541\) 1.19900e7 1.76128 0.880639 0.473788i \(-0.157114\pi\)
0.880639 + 0.473788i \(0.157114\pi\)
\(542\) 0 0
\(543\) −3.39457e6 −0.494066
\(544\) 0 0
\(545\) 562980. 0.0811898
\(546\) 0 0
\(547\) −4.45809e6 −0.637061 −0.318530 0.947913i \(-0.603189\pi\)
−0.318530 + 0.947913i \(0.603189\pi\)
\(548\) 0 0
\(549\) 2.51829e6 0.356595
\(550\) 0 0
\(551\) 5.32874e6 0.747732
\(552\) 0 0
\(553\) 1.30958e7 1.82104
\(554\) 0 0
\(555\) 5.02405e6 0.692344
\(556\) 0 0
\(557\) −9.02612e6 −1.23272 −0.616358 0.787466i \(-0.711393\pi\)
−0.616358 + 0.787466i \(0.711393\pi\)
\(558\) 0 0
\(559\) −8.76193e6 −1.18596
\(560\) 0 0
\(561\) −223560. −0.0299907
\(562\) 0 0
\(563\) −6.84899e6 −0.910658 −0.455329 0.890323i \(-0.650478\pi\)
−0.455329 + 0.890323i \(0.650478\pi\)
\(564\) 0 0
\(565\) −1.28989e7 −1.69993
\(566\) 0 0
\(567\) 1.15474e6 0.150843
\(568\) 0 0
\(569\) −5.46322e6 −0.707405 −0.353703 0.935358i \(-0.615077\pi\)
−0.353703 + 0.935358i \(0.615077\pi\)
\(570\) 0 0
\(571\) 1.02324e7 1.31337 0.656684 0.754166i \(-0.271959\pi\)
0.656684 + 0.754166i \(0.271959\pi\)
\(572\) 0 0
\(573\) −2.38810e6 −0.303854
\(574\) 0 0
\(575\) 738600. 0.0931622
\(576\) 0 0
\(577\) 1.59437e7 1.99365 0.996825 0.0796186i \(-0.0253702\pi\)
0.996825 + 0.0796186i \(0.0253702\pi\)
\(578\) 0 0
\(579\) 2.65768e6 0.329463
\(580\) 0 0
\(581\) 1.13837e6 0.139908
\(582\) 0 0
\(583\) 1.87596e6 0.228587
\(584\) 0 0
\(585\) 3.51767e6 0.424977
\(586\) 0 0
\(587\) 9.47713e6 1.13522 0.567612 0.823296i \(-0.307867\pi\)
0.567612 + 0.823296i \(0.307867\pi\)
\(588\) 0 0
\(589\) 3.43395e6 0.407855
\(590\) 0 0
\(591\) −1.81165e6 −0.213356
\(592\) 0 0
\(593\) 2.45349e6 0.286515 0.143258 0.989685i \(-0.454242\pi\)
0.143258 + 0.989685i \(0.454242\pi\)
\(594\) 0 0
\(595\) −4.80902e6 −0.556884
\(596\) 0 0
\(597\) 5.87203e6 0.674299
\(598\) 0 0
\(599\) −9.29978e6 −1.05902 −0.529512 0.848302i \(-0.677625\pi\)
−0.529512 + 0.848302i \(0.677625\pi\)
\(600\) 0 0
\(601\) −1.14617e7 −1.29438 −0.647192 0.762327i \(-0.724057\pi\)
−0.647192 + 0.762327i \(0.724057\pi\)
\(602\) 0 0
\(603\) 1.36112e6 0.152442
\(604\) 0 0
\(605\) −1.03918e7 −1.15425
\(606\) 0 0
\(607\) 1.12784e7 1.24244 0.621219 0.783637i \(-0.286638\pi\)
0.621219 + 0.783637i \(0.286638\pi\)
\(608\) 0 0
\(609\) −8.82922e6 −0.964670
\(610\) 0 0
\(611\) −1.29494e7 −1.40329
\(612\) 0 0
\(613\) −93782.0 −0.0100802 −0.00504009 0.999987i \(-0.501604\pi\)
−0.00504009 + 0.999987i \(0.501604\pi\)
\(614\) 0 0
\(615\) 1.14012e7 1.21553
\(616\) 0 0
\(617\) −1.49642e7 −1.58248 −0.791242 0.611504i \(-0.790565\pi\)
−0.791242 + 0.611504i \(0.790565\pi\)
\(618\) 0 0
\(619\) 5.06888e6 0.531723 0.265861 0.964011i \(-0.414344\pi\)
0.265861 + 0.964011i \(0.414344\pi\)
\(620\) 0 0
\(621\) 437400. 0.0455145
\(622\) 0 0
\(623\) −5.76259e6 −0.594837
\(624\) 0 0
\(625\) −1.20971e7 −1.23875
\(626\) 0 0
\(627\) −516240. −0.0524424
\(628\) 0 0
\(629\) −3.50161e6 −0.352892
\(630\) 0 0
\(631\) 1.55919e7 1.55892 0.779462 0.626450i \(-0.215493\pi\)
0.779462 + 0.626450i \(0.215493\pi\)
\(632\) 0 0
\(633\) 1.03235e7 1.02405
\(634\) 0 0
\(635\) 1.14180e7 1.12371
\(636\) 0 0
\(637\) 9.32320e6 0.910367
\(638\) 0 0
\(639\) 495720. 0.0480269
\(640\) 0 0
\(641\) 1.09701e7 1.05455 0.527274 0.849695i \(-0.323214\pi\)
0.527274 + 0.849695i \(0.323214\pi\)
\(642\) 0 0
\(643\) 2.83704e6 0.270607 0.135303 0.990804i \(-0.456799\pi\)
0.135303 + 0.990804i \(0.456799\pi\)
\(644\) 0 0
\(645\) −7.90970e6 −0.748619
\(646\) 0 0
\(647\) −6.05686e6 −0.568835 −0.284418 0.958700i \(-0.591800\pi\)
−0.284418 + 0.958700i \(0.591800\pi\)
\(648\) 0 0
\(649\) −1.58040e6 −0.147284
\(650\) 0 0
\(651\) −5.68973e6 −0.526186
\(652\) 0 0
\(653\) 1.08892e6 0.0999341 0.0499671 0.998751i \(-0.484088\pi\)
0.0499671 + 0.998751i \(0.484088\pi\)
\(654\) 0 0
\(655\) −9.98316e6 −0.909211
\(656\) 0 0
\(657\) −2.07020e6 −0.187111
\(658\) 0 0
\(659\) −7.41803e6 −0.665388 −0.332694 0.943035i \(-0.607958\pi\)
−0.332694 + 0.943035i \(0.607958\pi\)
\(660\) 0 0
\(661\) −767654. −0.0683379 −0.0341690 0.999416i \(-0.510878\pi\)
−0.0341690 + 0.999416i \(0.510878\pi\)
\(662\) 0 0
\(663\) −2.45171e6 −0.216613
\(664\) 0 0
\(665\) −1.11049e7 −0.973779
\(666\) 0 0
\(667\) −3.34440e6 −0.291074
\(668\) 0 0
\(669\) 6.31764e6 0.545745
\(670\) 0 0
\(671\) 1.86540e6 0.159943
\(672\) 0 0
\(673\) 1.42263e6 0.121075 0.0605373 0.998166i \(-0.480719\pi\)
0.0605373 + 0.998166i \(0.480719\pi\)
\(674\) 0 0
\(675\) 897399. 0.0758099
\(676\) 0 0
\(677\) 6.16231e6 0.516739 0.258370 0.966046i \(-0.416815\pi\)
0.258370 + 0.966046i \(0.416815\pi\)
\(678\) 0 0
\(679\) 2.92304e7 2.43310
\(680\) 0 0
\(681\) −1.11250e7 −0.919245
\(682\) 0 0
\(683\) −1.50621e7 −1.23548 −0.617739 0.786383i \(-0.711951\pi\)
−0.617739 + 0.786383i \(0.711951\pi\)
\(684\) 0 0
\(685\) −8.47796e6 −0.690343
\(686\) 0 0
\(687\) −952470. −0.0769945
\(688\) 0 0
\(689\) 2.05730e7 1.65101
\(690\) 0 0
\(691\) 5.87636e6 0.468180 0.234090 0.972215i \(-0.424789\pi\)
0.234090 + 0.972215i \(0.424789\pi\)
\(692\) 0 0
\(693\) 855360. 0.0676575
\(694\) 0 0
\(695\) −1.01769e7 −0.799199
\(696\) 0 0
\(697\) −7.94632e6 −0.619561
\(698\) 0 0
\(699\) −3.94810e6 −0.305630
\(700\) 0 0
\(701\) −3.60077e6 −0.276758 −0.138379 0.990379i \(-0.544189\pi\)
−0.138379 + 0.990379i \(0.544189\pi\)
\(702\) 0 0
\(703\) −8.08585e6 −0.617074
\(704\) 0 0
\(705\) −1.16899e7 −0.885806
\(706\) 0 0
\(707\) 3.87235e6 0.291358
\(708\) 0 0
\(709\) −9.22516e6 −0.689221 −0.344610 0.938746i \(-0.611989\pi\)
−0.344610 + 0.938746i \(0.611989\pi\)
\(710\) 0 0
\(711\) 6.02705e6 0.447127
\(712\) 0 0
\(713\) −2.15520e6 −0.158768
\(714\) 0 0
\(715\) 2.60568e6 0.190615
\(716\) 0 0
\(717\) 256176. 0.0186098
\(718\) 0 0
\(719\) −2.63923e7 −1.90395 −0.951975 0.306177i \(-0.900950\pi\)
−0.951975 + 0.306177i \(0.900950\pi\)
\(720\) 0 0
\(721\) −1.39505e7 −0.999426
\(722\) 0 0
\(723\) 8.03306e6 0.571525
\(724\) 0 0
\(725\) −6.86159e6 −0.484819
\(726\) 0 0
\(727\) −9.79485e6 −0.687324 −0.343662 0.939093i \(-0.611667\pi\)
−0.343662 + 0.939093i \(0.611667\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 5.51282e6 0.381576
\(732\) 0 0
\(733\) −4.07584e6 −0.280193 −0.140096 0.990138i \(-0.544741\pi\)
−0.140096 + 0.990138i \(0.544741\pi\)
\(734\) 0 0
\(735\) 8.41639e6 0.574655
\(736\) 0 0
\(737\) 1.00824e6 0.0683747
\(738\) 0 0
\(739\) 1.65709e7 1.11618 0.558089 0.829781i \(-0.311535\pi\)
0.558089 + 0.829781i \(0.311535\pi\)
\(740\) 0 0
\(741\) −5.66143e6 −0.378775
\(742\) 0 0
\(743\) 1.44141e7 0.957892 0.478946 0.877844i \(-0.341019\pi\)
0.478946 + 0.877844i \(0.341019\pi\)
\(744\) 0 0
\(745\) −1.94396e6 −0.128321
\(746\) 0 0
\(747\) 523908. 0.0343521
\(748\) 0 0
\(749\) −4.01259e7 −2.61349
\(750\) 0 0
\(751\) 1.67944e7 1.08659 0.543295 0.839542i \(-0.317177\pi\)
0.543295 + 0.839542i \(0.317177\pi\)
\(752\) 0 0
\(753\) 991116. 0.0636997
\(754\) 0 0
\(755\) −1.34556e7 −0.859081
\(756\) 0 0
\(757\) −1.32943e7 −0.843188 −0.421594 0.906785i \(-0.638529\pi\)
−0.421594 + 0.906785i \(0.638529\pi\)
\(758\) 0 0
\(759\) 324000. 0.0204146
\(760\) 0 0
\(761\) −2.14786e6 −0.134445 −0.0672225 0.997738i \(-0.521414\pi\)
−0.0672225 + 0.997738i \(0.521414\pi\)
\(762\) 0 0
\(763\) 1.50128e6 0.0933577
\(764\) 0 0
\(765\) −2.21324e6 −0.136734
\(766\) 0 0
\(767\) −1.73317e7 −1.06378
\(768\) 0 0
\(769\) −1.31059e7 −0.799193 −0.399596 0.916691i \(-0.630850\pi\)
−0.399596 + 0.916691i \(0.630850\pi\)
\(770\) 0 0
\(771\) 1.26722e6 0.0767742
\(772\) 0 0
\(773\) 2.37154e7 1.42752 0.713759 0.700392i \(-0.246991\pi\)
0.713759 + 0.700392i \(0.246991\pi\)
\(774\) 0 0
\(775\) −4.42175e6 −0.264448
\(776\) 0 0
\(777\) 1.33975e7 0.796105
\(778\) 0 0
\(779\) −1.83495e7 −1.08338
\(780\) 0 0
\(781\) 367200. 0.0215415
\(782\) 0 0
\(783\) −4.06345e6 −0.236859
\(784\) 0 0
\(785\) −8.98537e6 −0.520430
\(786\) 0 0
\(787\) 8.40048e6 0.483468 0.241734 0.970343i \(-0.422284\pi\)
0.241734 + 0.970343i \(0.422284\pi\)
\(788\) 0 0
\(789\) −8.44884e6 −0.483175
\(790\) 0 0
\(791\) −3.43971e7 −1.95470
\(792\) 0 0
\(793\) 2.04572e7 1.15522
\(794\) 0 0
\(795\) 1.85720e7 1.04218
\(796\) 0 0
\(797\) −5.41023e6 −0.301696 −0.150848 0.988557i \(-0.548200\pi\)
−0.150848 + 0.988557i \(0.548200\pi\)
\(798\) 0 0
\(799\) 8.14752e6 0.451501
\(800\) 0 0
\(801\) −2.65210e6 −0.146052
\(802\) 0 0
\(803\) −1.53348e6 −0.0839246
\(804\) 0 0
\(805\) 6.96960e6 0.379069
\(806\) 0 0
\(807\) 1.00306e7 0.542177
\(808\) 0 0
\(809\) −2.60777e7 −1.40087 −0.700436 0.713715i \(-0.747011\pi\)
−0.700436 + 0.713715i \(0.747011\pi\)
\(810\) 0 0
\(811\) −1.90021e7 −1.01449 −0.507247 0.861800i \(-0.669337\pi\)
−0.507247 + 0.861800i \(0.669337\pi\)
\(812\) 0 0
\(813\) 5.10934e6 0.271105
\(814\) 0 0
\(815\) 1.12942e7 0.595608
\(816\) 0 0
\(817\) 1.27301e7 0.667231
\(818\) 0 0
\(819\) 9.38045e6 0.488668
\(820\) 0 0
\(821\) 3.10173e7 1.60600 0.803001 0.595978i \(-0.203236\pi\)
0.803001 + 0.595978i \(0.203236\pi\)
\(822\) 0 0
\(823\) −1.56290e7 −0.804323 −0.402162 0.915569i \(-0.631741\pi\)
−0.402162 + 0.915569i \(0.631741\pi\)
\(824\) 0 0
\(825\) 664740. 0.0340030
\(826\) 0 0
\(827\) −1.58421e7 −0.805467 −0.402733 0.915317i \(-0.631940\pi\)
−0.402733 + 0.915317i \(0.631940\pi\)
\(828\) 0 0
\(829\) −2.06176e6 −0.104196 −0.0520980 0.998642i \(-0.516591\pi\)
−0.0520980 + 0.998642i \(0.516591\pi\)
\(830\) 0 0
\(831\) 1.09193e7 0.548521
\(832\) 0 0
\(833\) −5.86597e6 −0.292905
\(834\) 0 0
\(835\) −4.46292e7 −2.21515
\(836\) 0 0
\(837\) −2.61857e6 −0.129196
\(838\) 0 0
\(839\) 3.03900e7 1.49048 0.745240 0.666796i \(-0.232335\pi\)
0.745240 + 0.666796i \(0.232335\pi\)
\(840\) 0 0
\(841\) 1.05583e7 0.514760
\(842\) 0 0
\(843\) 6.18964e6 0.299983
\(844\) 0 0
\(845\) 4.07029e6 0.196103
\(846\) 0 0
\(847\) −2.77114e7 −1.32724
\(848\) 0 0
\(849\) 7.47817e6 0.356062
\(850\) 0 0
\(851\) 5.07480e6 0.240212
\(852\) 0 0
\(853\) 2.97738e7 1.40108 0.700538 0.713615i \(-0.252944\pi\)
0.700538 + 0.713615i \(0.252944\pi\)
\(854\) 0 0
\(855\) −5.11078e6 −0.239096
\(856\) 0 0
\(857\) 8.64100e6 0.401894 0.200947 0.979602i \(-0.435598\pi\)
0.200947 + 0.979602i \(0.435598\pi\)
\(858\) 0 0
\(859\) 3.35663e7 1.55210 0.776051 0.630670i \(-0.217220\pi\)
0.776051 + 0.630670i \(0.217220\pi\)
\(860\) 0 0
\(861\) 3.04033e7 1.39770
\(862\) 0 0
\(863\) 3.90191e7 1.78341 0.891703 0.452621i \(-0.149511\pi\)
0.891703 + 0.452621i \(0.149511\pi\)
\(864\) 0 0
\(865\) −8.78843e6 −0.399366
\(866\) 0 0
\(867\) −1.12361e7 −0.507656
\(868\) 0 0
\(869\) 4.46448e6 0.200549
\(870\) 0 0
\(871\) 1.10570e7 0.493848
\(872\) 0 0
\(873\) 1.34526e7 0.597409
\(874\) 0 0
\(875\) −2.20007e7 −0.971441
\(876\) 0 0
\(877\) 1.81382e7 0.796333 0.398166 0.917313i \(-0.369647\pi\)
0.398166 + 0.917313i \(0.369647\pi\)
\(878\) 0 0
\(879\) 1.18136e7 0.515717
\(880\) 0 0
\(881\) 3.05312e7 1.32527 0.662634 0.748943i \(-0.269438\pi\)
0.662634 + 0.748943i \(0.269438\pi\)
\(882\) 0 0
\(883\) 4.35533e7 1.87983 0.939916 0.341405i \(-0.110903\pi\)
0.939916 + 0.341405i \(0.110903\pi\)
\(884\) 0 0
\(885\) −1.56460e7 −0.671497
\(886\) 0 0
\(887\) −1.34152e7 −0.572515 −0.286257 0.958153i \(-0.592411\pi\)
−0.286257 + 0.958153i \(0.592411\pi\)
\(888\) 0 0
\(889\) 3.04480e7 1.29212
\(890\) 0 0
\(891\) 393660. 0.0166122
\(892\) 0 0
\(893\) 1.88141e7 0.789504
\(894\) 0 0
\(895\) 4.57641e7 1.90971
\(896\) 0 0
\(897\) 3.55320e6 0.147448
\(898\) 0 0
\(899\) 2.00218e7 0.826236
\(900\) 0 0
\(901\) −1.29441e7 −0.531203
\(902\) 0 0
\(903\) −2.10925e7 −0.860815
\(904\) 0 0
\(905\) −2.48935e7 −1.01033
\(906\) 0 0
\(907\) −3.10816e6 −0.125454 −0.0627272 0.998031i \(-0.519980\pi\)
−0.0627272 + 0.998031i \(0.519980\pi\)
\(908\) 0 0
\(909\) 1.78216e6 0.0715381
\(910\) 0 0
\(911\) 1.19035e6 0.0475203 0.0237602 0.999718i \(-0.492436\pi\)
0.0237602 + 0.999718i \(0.492436\pi\)
\(912\) 0 0
\(913\) 388080. 0.0154079
\(914\) 0 0
\(915\) 1.84675e7 0.729213
\(916\) 0 0
\(917\) −2.66218e7 −1.04547
\(918\) 0 0
\(919\) −4.71996e7 −1.84353 −0.921764 0.387752i \(-0.873252\pi\)
−0.921764 + 0.387752i \(0.873252\pi\)
\(920\) 0 0
\(921\) −1.52120e7 −0.590931
\(922\) 0 0
\(923\) 4.02696e6 0.155587
\(924\) 0 0
\(925\) 1.04118e7 0.400103
\(926\) 0 0
\(927\) −6.42038e6 −0.245393
\(928\) 0 0
\(929\) 1.33595e6 0.0507870 0.0253935 0.999678i \(-0.491916\pi\)
0.0253935 + 0.999678i \(0.491916\pi\)
\(930\) 0 0
\(931\) −1.35456e7 −0.512180
\(932\) 0 0
\(933\) −1.35184e7 −0.508417
\(934\) 0 0
\(935\) −1.63944e6 −0.0613291
\(936\) 0 0
\(937\) 1.47238e7 0.547861 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(938\) 0 0
\(939\) 7.29758e6 0.270094
\(940\) 0 0
\(941\) 2.69196e7 0.991049 0.495525 0.868594i \(-0.334976\pi\)
0.495525 + 0.868594i \(0.334976\pi\)
\(942\) 0 0
\(943\) 1.15164e7 0.421733
\(944\) 0 0
\(945\) 8.46806e6 0.308464
\(946\) 0 0
\(947\) 3.73160e6 0.135214 0.0676068 0.997712i \(-0.478464\pi\)
0.0676068 + 0.997712i \(0.478464\pi\)
\(948\) 0 0
\(949\) −1.68172e7 −0.606160
\(950\) 0 0
\(951\) −8.13202e6 −0.291573
\(952\) 0 0
\(953\) 2.18735e7 0.780166 0.390083 0.920780i \(-0.372446\pi\)
0.390083 + 0.920780i \(0.372446\pi\)
\(954\) 0 0
\(955\) −1.75127e7 −0.621362
\(956\) 0 0
\(957\) −3.00996e6 −0.106238
\(958\) 0 0
\(959\) −2.26079e7 −0.793805
\(960\) 0 0
\(961\) −1.57267e7 −0.549324
\(962\) 0 0
\(963\) −1.84670e7 −0.641699
\(964\) 0 0
\(965\) 1.94897e7 0.673730
\(966\) 0 0
\(967\) 1.76025e7 0.605352 0.302676 0.953093i \(-0.402120\pi\)
0.302676 + 0.953093i \(0.402120\pi\)
\(968\) 0 0
\(969\) 3.56206e6 0.121868
\(970\) 0 0
\(971\) −1.67317e7 −0.569497 −0.284749 0.958602i \(-0.591910\pi\)
−0.284749 + 0.958602i \(0.591910\pi\)
\(972\) 0 0
\(973\) −2.71385e7 −0.918975
\(974\) 0 0
\(975\) 7.28998e6 0.245592
\(976\) 0 0
\(977\) 5.55382e7 1.86147 0.930733 0.365699i \(-0.119170\pi\)
0.930733 + 0.365699i \(0.119170\pi\)
\(978\) 0 0
\(979\) −1.96452e6 −0.0655088
\(980\) 0 0
\(981\) 690930. 0.0229225
\(982\) 0 0
\(983\) −3.86784e7 −1.27669 −0.638344 0.769751i \(-0.720380\pi\)
−0.638344 + 0.769751i \(0.720380\pi\)
\(984\) 0 0
\(985\) −1.32854e7 −0.436299
\(986\) 0 0
\(987\) −3.11731e7 −1.01856
\(988\) 0 0
\(989\) −7.98960e6 −0.259737
\(990\) 0 0
\(991\) 9.58498e6 0.310033 0.155016 0.987912i \(-0.450457\pi\)
0.155016 + 0.987912i \(0.450457\pi\)
\(992\) 0 0
\(993\) −1.00977e7 −0.324976
\(994\) 0 0
\(995\) 4.30616e7 1.37890
\(996\) 0 0
\(997\) 1.03650e7 0.330242 0.165121 0.986273i \(-0.447198\pi\)
0.165121 + 0.986273i \(0.447198\pi\)
\(998\) 0 0
\(999\) 6.16588e6 0.195471
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 192.6.a.o.1.1 1
3.2 odd 2 576.6.a.j.1.1 1
4.3 odd 2 192.6.a.g.1.1 1
8.3 odd 2 48.6.a.c.1.1 1
8.5 even 2 6.6.a.a.1.1 1
12.11 even 2 576.6.a.i.1.1 1
16.3 odd 4 768.6.d.p.385.1 2
16.5 even 4 768.6.d.c.385.1 2
16.11 odd 4 768.6.d.p.385.2 2
16.13 even 4 768.6.d.c.385.2 2
24.5 odd 2 18.6.a.b.1.1 1
24.11 even 2 144.6.a.j.1.1 1
40.13 odd 4 150.6.c.b.49.1 2
40.29 even 2 150.6.a.d.1.1 1
40.37 odd 4 150.6.c.b.49.2 2
56.5 odd 6 294.6.e.a.67.1 2
56.13 odd 2 294.6.a.m.1.1 1
56.37 even 6 294.6.e.g.67.1 2
56.45 odd 6 294.6.e.a.79.1 2
56.53 even 6 294.6.e.g.79.1 2
72.5 odd 6 162.6.c.h.55.1 2
72.13 even 6 162.6.c.e.55.1 2
72.29 odd 6 162.6.c.h.109.1 2
72.61 even 6 162.6.c.e.109.1 2
88.21 odd 2 726.6.a.a.1.1 1
104.77 even 2 1014.6.a.c.1.1 1
120.29 odd 2 450.6.a.m.1.1 1
120.53 even 4 450.6.c.j.199.2 2
120.77 even 4 450.6.c.j.199.1 2
168.125 even 2 882.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.6.a.a.1.1 1 8.5 even 2
18.6.a.b.1.1 1 24.5 odd 2
48.6.a.c.1.1 1 8.3 odd 2
144.6.a.j.1.1 1 24.11 even 2
150.6.a.d.1.1 1 40.29 even 2
150.6.c.b.49.1 2 40.13 odd 4
150.6.c.b.49.2 2 40.37 odd 4
162.6.c.e.55.1 2 72.13 even 6
162.6.c.e.109.1 2 72.61 even 6
162.6.c.h.55.1 2 72.5 odd 6
162.6.c.h.109.1 2 72.29 odd 6
192.6.a.g.1.1 1 4.3 odd 2
192.6.a.o.1.1 1 1.1 even 1 trivial
294.6.a.m.1.1 1 56.13 odd 2
294.6.e.a.67.1 2 56.5 odd 6
294.6.e.a.79.1 2 56.45 odd 6
294.6.e.g.67.1 2 56.37 even 6
294.6.e.g.79.1 2 56.53 even 6
450.6.a.m.1.1 1 120.29 odd 2
450.6.c.j.199.1 2 120.77 even 4
450.6.c.j.199.2 2 120.53 even 4
576.6.a.i.1.1 1 12.11 even 2
576.6.a.j.1.1 1 3.2 odd 2
726.6.a.a.1.1 1 88.21 odd 2
768.6.d.c.385.1 2 16.5 even 4
768.6.d.c.385.2 2 16.13 even 4
768.6.d.p.385.1 2 16.3 odd 4
768.6.d.p.385.2 2 16.11 odd 4
882.6.a.a.1.1 1 168.125 even 2
1014.6.a.c.1.1 1 104.77 even 2