Properties

Label 144.6.a.j.1.1
Level $144$
Weight $6$
Character 144.1
Self dual yes
Analytic conductor $23.095$
Analytic rank $1$
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] = [144,6,Mod(1,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, 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("144.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 144.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: \(23.0952700531\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 144.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+66.0000 q^{5} -176.000 q^{7} +O(q^{10})\) \(q+66.0000 q^{5} -176.000 q^{7} -60.0000 q^{11} -658.000 q^{13} +414.000 q^{17} -956.000 q^{19} +600.000 q^{23} +1231.00 q^{25} -5574.00 q^{29} +3592.00 q^{31} -11616.0 q^{35} -8458.00 q^{37} -19194.0 q^{41} -13316.0 q^{43} -19680.0 q^{47} +14169.0 q^{49} +31266.0 q^{53} -3960.00 q^{55} +26340.0 q^{59} -31090.0 q^{61} -43428.0 q^{65} +16804.0 q^{67} +6120.00 q^{71} -25558.0 q^{73} +10560.0 q^{77} -74408.0 q^{79} -6468.00 q^{83} +27324.0 q^{85} +32742.0 q^{89} +115808. q^{91} -63096.0 q^{95} +166082. q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 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 \(-0.737497\pi\)
−0.678793 + 0.734329i \(0.737497\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −60.0000 −0.149510 −0.0747549 0.997202i \(-0.523817\pi\)
−0.0747549 + 0.997202i \(0.523817\pi\)
\(12\) 0 0
\(13\) −658.000 −1.07986 −0.539930 0.841710i \(-0.681549\pi\)
−0.539930 + 0.841710i \(0.681549\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 414.000 0.347439 0.173719 0.984795i \(-0.444421\pi\)
0.173719 + 0.984795i \(0.444421\pi\)
\(18\) 0 0
\(19\) −956.000 −0.607539 −0.303769 0.952746i \(-0.598245\pi\)
−0.303769 + 0.952746i \(0.598245\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(27\) 0 0
\(28\) 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 \(-0.391040\pi\)
0.335662 + 0.941983i \(0.391040\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −11616.0 −1.60283
\(36\) 0 0
\(37\) −8458.00 −1.01570 −0.507848 0.861447i \(-0.669559\pi\)
−0.507848 + 0.861447i \(0.669559\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −19194.0 −1.78322 −0.891612 0.452800i \(-0.850425\pi\)
−0.891612 + 0.452800i \(0.850425\pi\)
\(42\) 0 0
\(43\) −13316.0 −1.09825 −0.549127 0.835739i \(-0.685040\pi\)
−0.549127 + 0.835739i \(0.685040\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 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
\(51\) 0 0
\(52\) 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
\(57\) 0 0
\(58\) 0 0
\(59\) 26340.0 0.985112 0.492556 0.870281i \(-0.336063\pi\)
0.492556 + 0.870281i \(0.336063\pi\)
\(60\) 0 0
\(61\) −31090.0 −1.06978 −0.534892 0.844920i \(-0.679648\pi\)
−0.534892 + 0.844920i \(0.679648\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 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
\(69\) 0 0
\(70\) 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
\(75\) 0 0
\(76\) 0 0
\(77\) 10560.0 0.202972
\(78\) 0 0
\(79\) −74408.0 −1.34138 −0.670690 0.741738i \(-0.734002\pi\)
−0.670690 + 0.741738i \(0.734002\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6468.00 −0.103056 −0.0515282 0.998672i \(-0.516409\pi\)
−0.0515282 + 0.998672i \(0.516409\pi\)
\(84\) 0 0
\(85\) 27324.0 0.410201
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 32742.0 0.438157 0.219079 0.975707i \(-0.429695\pi\)
0.219079 + 0.975707i \(0.429695\pi\)
\(90\) 0 0
\(91\) 115808. 1.46600
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(99\) 0 0
\(100\) 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 \(-0.380012\pi\)
0.368089 + 0.929791i \(0.380012\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 227988. 1.92510 0.962548 0.271110i \(-0.0873908\pi\)
0.962548 + 0.271110i \(0.0873908\pi\)
\(108\) 0 0
\(109\) −8530.00 −0.0687674 −0.0343837 0.999409i \(-0.510947\pi\)
−0.0343837 + 0.999409i \(0.510947\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 195438. 1.43984 0.719918 0.694059i \(-0.244179\pi\)
0.719918 + 0.694059i \(0.244179\pi\)
\(114\) 0 0
\(115\) 39600.0 0.279223
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −72864.0 −0.471678
\(120\) 0 0
\(121\) −157451. −0.977647
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125004. −0.715565
\(126\) 0 0
\(127\) −173000. −0.951780 −0.475890 0.879505i \(-0.657874\pi\)
−0.475890 + 0.879505i \(0.657874\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 151260. 0.770098 0.385049 0.922896i \(-0.374185\pi\)
0.385049 + 0.922896i \(0.374185\pi\)
\(132\) 0 0
\(133\) 168256. 0.824786
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 128454. 0.584718 0.292359 0.956309i \(-0.405560\pi\)
0.292359 + 0.956309i \(0.405560\pi\)
\(138\) 0 0
\(139\) −154196. −0.676918 −0.338459 0.940981i \(-0.609906\pi\)
−0.338459 + 0.940981i \(0.609906\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 39480.0 0.161450
\(144\) 0 0
\(145\) −367884. −1.45308
\(146\) 0 0
\(147\) 0 0
\(148\) 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 \(-0.381473\pi\)
0.363819 + 0.931470i \(0.381473\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 237072. 0.792594
\(156\) 0 0
\(157\) 136142. 0.440801 0.220401 0.975409i \(-0.429263\pi\)
0.220401 + 0.975409i \(0.429263\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 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
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 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
\(177\) 0 0
\(178\) 0 0
\(179\) −693396. −1.61752 −0.808758 0.588141i \(-0.799860\pi\)
−0.808758 + 0.588141i \(0.799860\pi\)
\(180\) 0 0
\(181\) 377174. 0.855747 0.427873 0.903839i \(-0.359263\pi\)
0.427873 + 0.903839i \(0.359263\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −558228. −1.19917
\(186\) 0 0
\(187\) −24840.0 −0.0519455
\(188\) 0 0
\(189\) 0 0
\(190\) 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
\(195\) 0 0
\(196\) 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 \(-0.698498\pi\)
−0.583960 + 0.811782i \(0.698498\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 981024. 1.67086
\(204\) 0 0
\(205\) −1.26680e6 −2.10535
\(206\) 0 0
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) −878856. −1.29665
\(216\) 0 0
\(217\) −632192. −0.911380
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −272412. −0.375185
\(222\) 0 0
\(223\) −701960. −0.945258 −0.472629 0.881262i \(-0.656695\pi\)
−0.472629 + 0.881262i \(0.656695\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.23611e6 1.59218 0.796089 0.605179i \(-0.206899\pi\)
0.796089 + 0.605179i \(0.206899\pi\)
\(228\) 0 0
\(229\) 105830. 0.133358 0.0666792 0.997774i \(-0.478760\pi\)
0.0666792 + 0.997774i \(0.478760\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 438678. 0.529366 0.264683 0.964335i \(-0.414733\pi\)
0.264683 + 0.964335i \(0.414733\pi\)
\(234\) 0 0
\(235\) −1.29888e6 −1.53426
\(236\) 0 0
\(237\) 0 0
\(238\) 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
\(243\) 0 0
\(244\) 0 0
\(245\) 935154. 0.995332
\(246\) 0 0
\(247\) 629048. 0.656057
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −110124. −0.110331 −0.0551655 0.998477i \(-0.517569\pi\)
−0.0551655 + 0.998477i \(0.517569\pi\)
\(252\) 0 0
\(253\) −36000.0 −0.0353591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −140802. −0.132977 −0.0664884 0.997787i \(-0.521180\pi\)
−0.0664884 + 0.997787i \(0.521180\pi\)
\(258\) 0 0
\(259\) 1.48861e6 1.37889
\(260\) 0 0
\(261\) 0 0
\(262\) 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
\(267\) 0 0
\(268\) 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 \(-0.575438\pi\)
−0.234784 + 0.972048i \(0.575438\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −73860.0 −0.0588949
\(276\) 0 0
\(277\) −1.21326e6 −0.950066 −0.475033 0.879968i \(-0.657564\pi\)
−0.475033 + 0.879968i \(0.657564\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −687738. −0.519586 −0.259793 0.965664i \(-0.583654\pi\)
−0.259793 + 0.965664i \(0.583654\pi\)
\(282\) 0 0
\(283\) 830908. 0.616718 0.308359 0.951270i \(-0.400220\pi\)
0.308359 + 0.951270i \(0.400220\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.37814e6 2.42088
\(288\) 0 0
\(289\) −1.24846e6 −0.879286
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 0 0
\(299\) −394800. −0.255387
\(300\) 0 0
\(301\) 2.34362e6 1.49097
\(302\) 0 0
\(303\) 0 0
\(304\) 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
\(309\) 0 0
\(310\) 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
\(315\) 0 0
\(316\) 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
\(321\) 0 0
\(322\) 0 0
\(323\) −395784. −0.211082
\(324\) 0 0
\(325\) −809998. −0.425379
\(326\) 0 0
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 0 0
\(341\) −215520. −0.100369
\(342\) 0 0
\(343\) 464288. 0.213085
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.91749e6 0.854889 0.427445 0.904042i \(-0.359414\pi\)
0.427445 + 0.904042i \(0.359414\pi\)
\(348\) 0 0
\(349\) 1.83659e6 0.807140 0.403570 0.914949i \(-0.367769\pi\)
0.403570 + 0.914949i \(0.367769\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 622014. 0.265683 0.132841 0.991137i \(-0.457590\pi\)
0.132841 + 0.991137i \(0.457590\pi\)
\(354\) 0 0
\(355\) 403920. 0.170108
\(356\) 0 0
\(357\) 0 0
\(358\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) −1.68683e6 −0.662733
\(366\) 0 0
\(367\) −16232.0 −0.00629081 −0.00314541 0.999995i \(-0.501001\pi\)
−0.00314541 + 0.999995i \(0.501001\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.50282e6 −2.07563
\(372\) 0 0
\(373\) 293606. 0.109268 0.0546340 0.998506i \(-0.482601\pi\)
0.0546340 + 0.998506i \(0.482601\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) 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
\(387\) 0 0
\(388\) 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
\(393\) 0 0
\(394\) 0 0
\(395\) −4.91093e6 −1.58369
\(396\) 0 0
\(397\) −3.90416e6 −1.24323 −0.621615 0.783323i \(-0.713523\pi\)
−0.621615 + 0.783323i \(0.713523\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.44115e6 −1.68978 −0.844890 0.534940i \(-0.820334\pi\)
−0.844890 + 0.534940i \(0.820334\pi\)
\(402\) 0 0
\(403\) −2.36354e6 −0.724936
\(404\) 0 0
\(405\) 0 0
\(406\) 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
\(411\) 0 0
\(412\) 0 0
\(413\) −4.63584e6 −1.33738
\(414\) 0 0
\(415\) −426888. −0.121673
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 139020. 0.0386850 0.0193425 0.999813i \(-0.493843\pi\)
0.0193425 + 0.999813i \(0.493843\pi\)
\(420\) 0 0
\(421\) 4.32743e6 1.18994 0.594970 0.803748i \(-0.297164\pi\)
0.594970 + 0.803748i \(0.297164\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 509634. 0.136863
\(426\) 0 0
\(427\) 5.47184e6 1.45232
\(428\) 0 0
\(429\) 0 0
\(430\) 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
\(435\) 0 0
\(436\) 0 0
\(437\) −573600. −0.143683
\(438\) 0 0
\(439\) 446512. 0.110579 0.0552894 0.998470i \(-0.482392\pi\)
0.0552894 + 0.998470i \(0.482392\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.49525e6 0.846193 0.423096 0.906085i \(-0.360943\pi\)
0.423096 + 0.906085i \(0.360943\pi\)
\(444\) 0 0
\(445\) 2.16097e6 0.517308
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.20613e6 0.282343 0.141171 0.989985i \(-0.454913\pi\)
0.141171 + 0.989985i \(0.454913\pi\)
\(450\) 0 0
\(451\) 1.15164e6 0.266609
\(452\) 0 0
\(453\) 0 0
\(454\) 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
\(459\) 0 0
\(460\) 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 \(-0.397563\pi\)
0.316288 + 0.948663i \(0.397563\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.31076e6 −1.12684 −0.563422 0.826169i \(-0.690516\pi\)
−0.563422 + 0.826169i \(0.690516\pi\)
\(468\) 0 0
\(469\) −2.95750e6 −0.620859
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 798960. 0.164200
\(474\) 0 0
\(475\) −1.17684e6 −0.239322
\(476\) 0 0
\(477\) 0 0
\(478\) 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
\(483\) 0 0
\(484\) 0 0
\(485\) 1.09614e7 2.11598
\(486\) 0 0
\(487\) −9.81531e6 −1.87535 −0.937674 0.347517i \(-0.887025\pi\)
−0.937674 + 0.347517i \(0.887025\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.94520e6 −1.11292 −0.556458 0.830876i \(-0.687840\pi\)
−0.556458 + 0.830876i \(0.687840\pi\)
\(492\) 0 0
\(493\) −2.30764e6 −0.427612
\(494\) 0 0
\(495\) 0 0
\(496\) 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
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 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
\(513\) 0 0
\(514\) 0 0
\(515\) 5.23142e6 0.869164
\(516\) 0 0
\(517\) 1.18080e6 0.194290
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −8.01974e6 −1.29439 −0.647196 0.762324i \(-0.724059\pi\)
−0.647196 + 0.762324i \(0.724059\pi\)
\(522\) 0 0
\(523\) −1.91162e6 −0.305596 −0.152798 0.988257i \(-0.548828\pi\)
−0.152798 + 0.988257i \(0.548828\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.48709e6 0.233244
\(528\) 0 0
\(529\) −6.07634e6 −0.944068
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.26297e7 1.92563
\(534\) 0 0
\(535\) 1.50472e7 2.27285
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −850140. −0.126043
\(540\) 0 0
\(541\) −1.19900e7 −1.76128 −0.880639 0.473788i \(-0.842886\pi\)
−0.880639 + 0.473788i \(0.842886\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 0 0
\(551\) 5.32874e6 0.747732
\(552\) 0 0
\(553\) 1.30958e7 1.82104
\(554\) 0 0
\(555\) 0 0
\(556\) 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
\(561\) 0 0
\(562\) 0 0
\(563\) 6.84899e6 0.910658 0.455329 0.890323i \(-0.349522\pi\)
0.455329 + 0.890323i \(0.349522\pi\)
\(564\) 0 0
\(565\) 1.28989e7 1.69993
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.46322e6 0.707405 0.353703 0.935358i \(-0.384923\pi\)
0.353703 + 0.935358i \(0.384923\pi\)
\(570\) 0 0
\(571\) 1.02324e7 1.31337 0.656684 0.754166i \(-0.271959\pi\)
0.656684 + 0.754166i \(0.271959\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 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
\(579\) 0 0
\(580\) 0 0
\(581\) 1.13837e6 0.139908
\(582\) 0 0
\(583\) −1.87596e6 −0.228587
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.47713e6 −1.13522 −0.567612 0.823296i \(-0.692133\pi\)
−0.567612 + 0.823296i \(0.692133\pi\)
\(588\) 0 0
\(589\) −3.43395e6 −0.407855
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.45349e6 −0.286515 −0.143258 0.989685i \(-0.545758\pi\)
−0.143258 + 0.989685i \(0.545758\pi\)
\(594\) 0 0
\(595\) −4.80902e6 −0.556884
\(596\) 0 0
\(597\) 0 0
\(598\) 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
\(603\) 0 0
\(604\) 0 0
\(605\) −1.03918e7 −1.15425
\(606\) 0 0
\(607\) −1.12784e7 −1.24244 −0.621219 0.783637i \(-0.713362\pi\)
−0.621219 + 0.783637i \(0.713362\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.29494e7 1.40329
\(612\) 0 0
\(613\) 93782.0 0.0100802 0.00504009 0.999987i \(-0.498396\pi\)
0.00504009 + 0.999987i \(0.498396\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.49642e7 1.58248 0.791242 0.611504i \(-0.209435\pi\)
0.791242 + 0.611504i \(0.209435\pi\)
\(618\) 0 0
\(619\) 5.06888e6 0.531723 0.265861 0.964011i \(-0.414344\pi\)
0.265861 + 0.964011i \(0.414344\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.76259e6 −0.594837
\(624\) 0 0
\(625\) −1.20971e7 −1.23875
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.50161e6 −0.352892
\(630\) 0 0
\(631\) −1.55919e7 −1.55892 −0.779462 0.626450i \(-0.784507\pi\)
−0.779462 + 0.626450i \(0.784507\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.14180e7 −1.12371
\(636\) 0 0
\(637\) −9.32320e6 −0.910367
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.09701e7 −1.05455 −0.527274 0.849695i \(-0.676786\pi\)
−0.527274 + 0.849695i \(0.676786\pi\)
\(642\) 0 0
\(643\) 2.83704e6 0.270607 0.135303 0.990804i \(-0.456799\pi\)
0.135303 + 0.990804i \(0.456799\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 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
\(651\) 0 0
\(652\) 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
\(657\) 0 0
\(658\) 0 0
\(659\) 7.41803e6 0.665388 0.332694 0.943035i \(-0.392042\pi\)
0.332694 + 0.943035i \(0.392042\pi\)
\(660\) 0 0
\(661\) 767654. 0.0683379 0.0341690 0.999416i \(-0.489122\pi\)
0.0341690 + 0.999416i \(0.489122\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.11049e7 0.973779
\(666\) 0 0
\(667\) −3.34440e6 −0.291074
\(668\) 0 0
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) 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
\(681\) 0 0
\(682\) 0 0
\(683\) 1.50621e7 1.23548 0.617739 0.786383i \(-0.288049\pi\)
0.617739 + 0.786383i \(0.288049\pi\)
\(684\) 0 0
\(685\) 8.47796e6 0.690343
\(686\) 0 0
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) 0 0
\(695\) −1.01769e7 −0.799199
\(696\) 0 0
\(697\) −7.94632e6 −0.619561
\(698\) 0 0
\(699\) 0 0
\(700\) 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
\(705\) 0 0
\(706\) 0 0
\(707\) −3.87235e6 −0.291358
\(708\) 0 0
\(709\) 9.22516e6 0.689221 0.344610 0.938746i \(-0.388011\pi\)
0.344610 + 0.938746i \(0.388011\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.15520e6 0.158768
\(714\) 0 0
\(715\) 2.60568e6 0.190615
\(716\) 0 0
\(717\) 0 0
\(718\) 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
\(723\) 0 0
\(724\) 0 0
\(725\) −6.86159e6 −0.484819
\(726\) 0 0
\(727\) 9.79485e6 0.687324 0.343662 0.939093i \(-0.388333\pi\)
0.343662 + 0.939093i \(0.388333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −5.51282e6 −0.381576
\(732\) 0 0
\(733\) 4.07584e6 0.280193 0.140096 0.990138i \(-0.455259\pi\)
0.140096 + 0.990138i \(0.455259\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 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
\(741\) 0 0
\(742\) 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
\(747\) 0 0
\(748\) 0 0
\(749\) −4.01259e7 −2.61349
\(750\) 0 0
\(751\) −1.67944e7 −1.08659 −0.543295 0.839542i \(-0.682823\pi\)
−0.543295 + 0.839542i \(0.682823\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.34556e7 0.859081
\(756\) 0 0
\(757\) 1.32943e7 0.843188 0.421594 0.906785i \(-0.361471\pi\)
0.421594 + 0.906785i \(0.361471\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.14786e6 0.134445 0.0672225 0.997738i \(-0.478586\pi\)
0.0672225 + 0.997738i \(0.478586\pi\)
\(762\) 0 0
\(763\) 1.50128e6 0.0933577
\(764\) 0 0
\(765\) 0 0
\(766\) 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
\(771\) 0 0
\(772\) 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
\(777\) 0 0
\(778\) 0 0
\(779\) 1.83495e7 1.08338
\(780\) 0 0
\(781\) −367200. −0.0215415
\(782\) 0 0
\(783\) 0 0
\(784\) 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
\(789\) 0 0
\(790\) 0 0
\(791\) −3.43971e7 −1.95470
\(792\) 0 0
\(793\) 2.04572e7 1.15522
\(794\) 0 0
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) 0 0
\(803\) 1.53348e6 0.0839246
\(804\) 0 0
\(805\) −6.96960e6 −0.379069
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.60777e7 1.40087 0.700436 0.713715i \(-0.252989\pi\)
0.700436 + 0.713715i \(0.252989\pi\)
\(810\) 0 0
\(811\) −1.90021e7 −1.01449 −0.507247 0.861800i \(-0.669337\pi\)
−0.507247 + 0.861800i \(0.669337\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.12942e7 0.595608
\(816\) 0 0
\(817\) 1.27301e7 0.667231
\(818\) 0 0
\(819\) 0 0
\(820\) 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 \(-0.368259\pi\)
0.402162 + 0.915569i \(0.368259\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.58421e7 0.805467 0.402733 0.915317i \(-0.368060\pi\)
0.402733 + 0.915317i \(0.368060\pi\)
\(828\) 0 0
\(829\) 2.06176e6 0.104196 0.0520980 0.998642i \(-0.483409\pi\)
0.0520980 + 0.998642i \(0.483409\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.86597e6 0.292905
\(834\) 0 0
\(835\) −4.46292e7 −2.21515
\(836\) 0 0
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 0 0
\(845\) 4.07029e6 0.196103
\(846\) 0 0
\(847\) 2.77114e7 1.32724
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.07480e6 −0.240212
\(852\) 0 0
\(853\) −2.97738e7 −1.40108 −0.700538 0.713615i \(-0.747056\pi\)
−0.700538 + 0.713615i \(0.747056\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8.64100e6 −0.401894 −0.200947 0.979602i \(-0.564402\pi\)
−0.200947 + 0.979602i \(0.564402\pi\)
\(858\) 0 0
\(859\) 3.35663e7 1.55210 0.776051 0.630670i \(-0.217220\pi\)
0.776051 + 0.630670i \(0.217220\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 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
\(867\) 0 0
\(868\) 0 0
\(869\) 4.46448e6 0.200549
\(870\) 0 0
\(871\) −1.10570e7 −0.493848
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.20007e7 0.971441
\(876\) 0 0
\(877\) −1.81382e7 −0.796333 −0.398166 0.917313i \(-0.630353\pi\)
−0.398166 + 0.917313i \(0.630353\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −3.05312e7 −1.32527 −0.662634 0.748943i \(-0.730562\pi\)
−0.662634 + 0.748943i \(0.730562\pi\)
\(882\) 0 0
\(883\) 4.35533e7 1.87983 0.939916 0.341405i \(-0.110903\pi\)
0.939916 + 0.341405i \(0.110903\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 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
\(891\) 0 0
\(892\) 0 0
\(893\) 1.88141e7 0.789504
\(894\) 0 0
\(895\) −4.57641e7 −1.90971
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.00218e7 −0.826236
\(900\) 0 0
\(901\) 1.29441e7 0.531203
\(902\) 0 0
\(903\) 0 0
\(904\) 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
\(909\) 0 0
\(910\) 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
\(915\) 0 0
\(916\) 0 0
\(917\) −2.66218e7 −1.04547
\(918\) 0 0
\(919\) 4.71996e7 1.84353 0.921764 0.387752i \(-0.126748\pi\)
0.921764 + 0.387752i \(0.126748\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.02696e6 −0.155587
\(924\) 0 0
\(925\) −1.04118e7 −0.400103
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.33595e6 −0.0507870 −0.0253935 0.999678i \(-0.508084\pi\)
−0.0253935 + 0.999678i \(0.508084\pi\)
\(930\) 0 0
\(931\) −1.35456e7 −0.512180
\(932\) 0 0
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 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
\(945\) 0 0
\(946\) 0 0
\(947\) −3.73160e6 −0.135214 −0.0676068 0.997712i \(-0.521536\pi\)
−0.0676068 + 0.997712i \(0.521536\pi\)
\(948\) 0 0
\(949\) 1.68172e7 0.606160
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.18735e7 −0.780166 −0.390083 0.920780i \(-0.627554\pi\)
−0.390083 + 0.920780i \(0.627554\pi\)
\(954\) 0 0
\(955\) −1.75127e7 −0.621362
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.26079e7 −0.793805
\(960\) 0 0
\(961\) −1.57267e7 −0.549324
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.94897e7 0.673730
\(966\) 0 0
\(967\) −1.76025e7 −0.605352 −0.302676 0.953093i \(-0.597880\pi\)
−0.302676 + 0.953093i \(0.597880\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.67317e7 0.569497 0.284749 0.958602i \(-0.408090\pi\)
0.284749 + 0.958602i \(0.408090\pi\)
\(972\) 0 0
\(973\) 2.71385e7 0.918975
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.55382e7 −1.86147 −0.930733 0.365699i \(-0.880830\pi\)
−0.930733 + 0.365699i \(0.880830\pi\)
\(978\) 0 0
\(979\) −1.96452e6 −0.0655088
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(987\) 0 0
\(988\) 0 0
\(989\) −7.98960e6 −0.259737
\(990\) 0 0
\(991\) −9.58498e6 −0.310033 −0.155016 0.987912i \(-0.549543\pi\)
−0.155016 + 0.987912i \(0.549543\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.30616e7 −1.37890
\(996\) 0 0
\(997\) −1.03650e7 −0.330242 −0.165121 0.986273i \(-0.552802\pi\)
−0.165121 + 0.986273i \(0.552802\pi\)
\(998\) 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 144.6.a.j.1.1 1
3.2 odd 2 48.6.a.c.1.1 1
4.3 odd 2 18.6.a.b.1.1 1
8.3 odd 2 576.6.a.j.1.1 1
8.5 even 2 576.6.a.i.1.1 1
12.11 even 2 6.6.a.a.1.1 1
20.3 even 4 450.6.c.j.199.2 2
20.7 even 4 450.6.c.j.199.1 2
20.19 odd 2 450.6.a.m.1.1 1
24.5 odd 2 192.6.a.g.1.1 1
24.11 even 2 192.6.a.o.1.1 1
28.27 even 2 882.6.a.a.1.1 1
36.7 odd 6 162.6.c.h.109.1 2
36.11 even 6 162.6.c.e.109.1 2
36.23 even 6 162.6.c.e.55.1 2
36.31 odd 6 162.6.c.h.55.1 2
48.5 odd 4 768.6.d.p.385.1 2
48.11 even 4 768.6.d.c.385.2 2
48.29 odd 4 768.6.d.p.385.2 2
48.35 even 4 768.6.d.c.385.1 2
60.23 odd 4 150.6.c.b.49.1 2
60.47 odd 4 150.6.c.b.49.2 2
60.59 even 2 150.6.a.d.1.1 1
84.11 even 6 294.6.e.g.79.1 2
84.23 even 6 294.6.e.g.67.1 2
84.47 odd 6 294.6.e.a.67.1 2
84.59 odd 6 294.6.e.a.79.1 2
84.83 odd 2 294.6.a.m.1.1 1
132.131 odd 2 726.6.a.a.1.1 1
156.155 even 2 1014.6.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.6.a.a.1.1 1 12.11 even 2
18.6.a.b.1.1 1 4.3 odd 2
48.6.a.c.1.1 1 3.2 odd 2
144.6.a.j.1.1 1 1.1 even 1 trivial
150.6.a.d.1.1 1 60.59 even 2
150.6.c.b.49.1 2 60.23 odd 4
150.6.c.b.49.2 2 60.47 odd 4
162.6.c.e.55.1 2 36.23 even 6
162.6.c.e.109.1 2 36.11 even 6
162.6.c.h.55.1 2 36.31 odd 6
162.6.c.h.109.1 2 36.7 odd 6
192.6.a.g.1.1 1 24.5 odd 2
192.6.a.o.1.1 1 24.11 even 2
294.6.a.m.1.1 1 84.83 odd 2
294.6.e.a.67.1 2 84.47 odd 6
294.6.e.a.79.1 2 84.59 odd 6
294.6.e.g.67.1 2 84.23 even 6
294.6.e.g.79.1 2 84.11 even 6
450.6.a.m.1.1 1 20.19 odd 2
450.6.c.j.199.1 2 20.7 even 4
450.6.c.j.199.2 2 20.3 even 4
576.6.a.i.1.1 1 8.5 even 2
576.6.a.j.1.1 1 8.3 odd 2
726.6.a.a.1.1 1 132.131 odd 2
768.6.d.c.385.1 2 48.35 even 4
768.6.d.c.385.2 2 48.11 even 4
768.6.d.p.385.1 2 48.5 odd 4
768.6.d.p.385.2 2 48.29 odd 4
882.6.a.a.1.1 1 28.27 even 2
1014.6.a.c.1.1 1 156.155 even 2