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Properties

Label 432.8.a.h.1.1
Level $432$
Weight $8$
Character 432.1
Self dual yes
Analytic conductor $134.950$
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] = [432,8,Mod(1,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.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: \(134.950331009\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 432.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+312.000 q^{5} -323.000 q^{7} +O(q^{10})\) \(q+312.000 q^{5} -323.000 q^{7} -3720.00 q^{11} -14179.0 q^{13} +15912.0 q^{17} -22421.0 q^{19} -57768.0 q^{23} +19219.0 q^{25} +166656. q^{29} -94820.0 q^{31} -100776. q^{35} +453971. q^{37} +627072. q^{41} +42472.0 q^{43} +1.23526e6 q^{47} -719214. q^{49} +107280. q^{53} -1.16064e6 q^{55} +2.47922e6 q^{59} +2.87438e6 q^{61} -4.42385e6 q^{65} -1.50110e6 q^{67} -4.73314e6 q^{71} -85111.0 q^{73} +1.20156e6 q^{77} +1.18082e6 q^{79} +1.11653e6 q^{83} +4.96454e6 q^{85} +9.36814e6 q^{89} +4.57982e6 q^{91} -6.99535e6 q^{95} -2.03999e6 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 312.000 1.11625 0.558123 − 0.829759i \(-0.311522\pi\)
0.558123 + 0.829759i \(0.311522\pi\)
\(6\) 0 0
\(7\) −323.000 −0.355926 −0.177963 − 0.984037i \(-0.556951\pi\)
−0.177963 + 0.984037i \(0.556951\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3720.00 −0.842691 −0.421346 − 0.906900i \(-0.638442\pi\)
−0.421346 + 0.906900i \(0.638442\pi\)
\(12\) 0 0
\(13\) −14179.0 −1.78996 −0.894981 − 0.446104i \(-0.852811\pi\)
−0.894981 + 0.446104i \(0.852811\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15912.0 0.785513 0.392757 − 0.919642i \(-0.371521\pi\)
0.392757 + 0.919642i \(0.371521\pi\)
\(18\) 0 0
\(19\) −22421.0 −0.749924 −0.374962 − 0.927040i \(-0.622344\pi\)
−0.374962 + 0.927040i \(0.622344\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −57768.0 −0.990011 −0.495005 − 0.868890i \(-0.664834\pi\)
−0.495005 + 0.868890i \(0.664834\pi\)
\(24\) 0 0
\(25\) 19219.0 0.246003
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 166656. 1.26890 0.634451 − 0.772963i \(-0.281226\pi\)
0.634451 + 0.772963i \(0.281226\pi\)
\(30\) 0 0
\(31\) −94820.0 −0.571655 −0.285828 − 0.958281i \(-0.592268\pi\)
−0.285828 + 0.958281i \(0.592268\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −100776. −0.397300
\(36\) 0 0
\(37\) 453971. 1.47340 0.736702 − 0.676217i \(-0.236382\pi\)
0.736702 + 0.676217i \(0.236382\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 627072. 1.42093 0.710467 − 0.703731i \(-0.248484\pi\)
0.710467 + 0.703731i \(0.248484\pi\)
\(42\) 0 0
\(43\) 42472.0 0.0814635 0.0407318 − 0.999170i \(-0.487031\pi\)
0.0407318 + 0.999170i \(0.487031\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.23526e6 1.73546 0.867730 − 0.497036i \(-0.165578\pi\)
0.867730 + 0.497036i \(0.165578\pi\)
\(48\) 0 0
\(49\) −719214. −0.873317
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 107280. 0.0989813 0.0494907 − 0.998775i \(-0.484240\pi\)
0.0494907 + 0.998775i \(0.484240\pi\)
\(54\) 0 0
\(55\) −1.16064e6 −0.940650
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.47922e6 1.57157 0.785785 − 0.618500i \(-0.212259\pi\)
0.785785 + 0.618500i \(0.212259\pi\)
\(60\) 0 0
\(61\) 2.87438e6 1.62140 0.810700 − 0.585462i \(-0.199087\pi\)
0.810700 + 0.585462i \(0.199087\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.42385e6 −1.99804
\(66\) 0 0
\(67\) −1.50110e6 −0.609743 −0.304872 − 0.952393i \(-0.598614\pi\)
−0.304872 + 0.952393i \(0.598614\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.73314e6 −1.56944 −0.784720 − 0.619850i \(-0.787193\pi\)
−0.784720 + 0.619850i \(0.787193\pi\)
\(72\) 0 0
\(73\) −85111.0 −0.0256068 −0.0128034 − 0.999918i \(-0.504076\pi\)
−0.0128034 + 0.999918i \(0.504076\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.20156e6 0.299936
\(78\) 0 0
\(79\) 1.18082e6 0.269456 0.134728 − 0.990883i \(-0.456984\pi\)
0.134728 + 0.990883i \(0.456984\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.11653e6 0.214337 0.107168 − 0.994241i \(-0.465822\pi\)
0.107168 + 0.994241i \(0.465822\pi\)
\(84\) 0 0
\(85\) 4.96454e6 0.876825
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.36814e6 1.40860 0.704301 − 0.709902i \(-0.251261\pi\)
0.704301 + 0.709902i \(0.251261\pi\)
\(90\) 0 0
\(91\) 4.57982e6 0.637094
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.99535e6 −0.837099
\(96\) 0 0
\(97\) −2.03999e6 −0.226949 −0.113474 − 0.993541i \(-0.536198\pi\)
−0.113474 + 0.993541i \(0.536198\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.52575e7 −1.47353 −0.736763 − 0.676151i \(-0.763647\pi\)
−0.736763 + 0.676151i \(0.763647\pi\)
\(102\) 0 0
\(103\) 1.92433e7 1.73520 0.867601 − 0.497260i \(-0.165661\pi\)
0.867601 + 0.497260i \(0.165661\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.28571e7 −1.01461 −0.507306 − 0.861766i \(-0.669359\pi\)
−0.507306 + 0.861766i \(0.669359\pi\)
\(108\) 0 0
\(109\) −1.02835e7 −0.760589 −0.380294 − 0.924865i \(-0.624177\pi\)
−0.380294 + 0.924865i \(0.624177\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.86773e7 1.21769 0.608847 − 0.793287i \(-0.291632\pi\)
0.608847 + 0.793287i \(0.291632\pi\)
\(114\) 0 0
\(115\) −1.80236e7 −1.10509
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.13958e6 −0.279584
\(120\) 0 0
\(121\) −5.64877e6 −0.289871
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.83787e7 −0.841645
\(126\) 0 0
\(127\) −3.53659e6 −0.153204 −0.0766022 − 0.997062i \(-0.524407\pi\)
−0.0766022 + 0.997062i \(0.524407\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.01005e7 1.16983 0.584917 − 0.811093i \(-0.301127\pi\)
0.584917 + 0.811093i \(0.301127\pi\)
\(132\) 0 0
\(133\) 7.24198e6 0.266917
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.37326e6 −0.244984 −0.122492 − 0.992470i \(-0.539089\pi\)
−0.122492 + 0.992470i \(0.539089\pi\)
\(138\) 0 0
\(139\) 5.92522e7 1.87134 0.935670 − 0.352877i \(-0.114796\pi\)
0.935670 + 0.352877i \(0.114796\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.27459e7 1.50839
\(144\) 0 0
\(145\) 5.19967e7 1.41641
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.32632e7 −0.576127 −0.288064 − 0.957611i \(-0.593011\pi\)
−0.288064 + 0.957611i \(0.593011\pi\)
\(150\) 0 0
\(151\) 5.82460e7 1.37672 0.688361 − 0.725368i \(-0.258330\pi\)
0.688361 + 0.725368i \(0.258330\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.95838e7 −0.638107
\(156\) 0 0
\(157\) −2.93576e7 −0.605441 −0.302721 − 0.953079i \(-0.597895\pi\)
−0.302721 + 0.953079i \(0.597895\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.86591e7 0.352370
\(162\) 0 0
\(163\) 3.50196e7 0.633366 0.316683 − 0.948531i \(-0.397431\pi\)
0.316683 + 0.948531i \(0.397431\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.04548e7 1.00444 0.502219 − 0.864741i \(-0.332517\pi\)
0.502219 + 0.864741i \(0.332517\pi\)
\(168\) 0 0
\(169\) 1.38296e8 2.20396
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.43480e7 0.944874 0.472437 − 0.881364i \(-0.343374\pi\)
0.472437 + 0.881364i \(0.343374\pi\)
\(174\) 0 0
\(175\) −6.20774e6 −0.0875589
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.05862e7 −0.137960 −0.0689799 − 0.997618i \(-0.521974\pi\)
−0.0689799 + 0.997618i \(0.521974\pi\)
\(180\) 0 0
\(181\) 6.41578e7 0.804219 0.402109 − 0.915592i \(-0.368277\pi\)
0.402109 + 0.915592i \(0.368277\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.41639e8 1.64468
\(186\) 0 0
\(187\) −5.91926e7 −0.661945
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.56333e8 −1.62343 −0.811715 − 0.584054i \(-0.801466\pi\)
−0.811715 + 0.584054i \(0.801466\pi\)
\(192\) 0 0
\(193\) −3.53259e7 −0.353706 −0.176853 − 0.984237i \(-0.556592\pi\)
−0.176853 + 0.984237i \(0.556592\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.06306e8 −0.990666 −0.495333 − 0.868703i \(-0.664954\pi\)
−0.495333 + 0.868703i \(0.664954\pi\)
\(198\) 0 0
\(199\) −1.59628e7 −0.143590 −0.0717949 − 0.997419i \(-0.522873\pi\)
−0.0717949 + 0.997419i \(0.522873\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.38299e7 −0.451635
\(204\) 0 0
\(205\) 1.95646e8 1.58611
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.34061e7 0.631955
\(210\) 0 0
\(211\) 2.65780e7 0.194775 0.0973876 − 0.995247i \(-0.468951\pi\)
0.0973876 + 0.995247i \(0.468951\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.32513e7 0.0909332
\(216\) 0 0
\(217\) 3.06269e7 0.203467
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.25616e8 −1.40604
\(222\) 0 0
\(223\) 1.71223e8 1.03394 0.516969 − 0.856004i \(-0.327060\pi\)
0.516969 + 0.856004i \(0.327060\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.33897e7 0.246205 0.123102 − 0.992394i \(-0.460716\pi\)
0.123102 + 0.992394i \(0.460716\pi\)
\(228\) 0 0
\(229\) 2.64398e8 1.45490 0.727451 − 0.686159i \(-0.240705\pi\)
0.727451 + 0.686159i \(0.240705\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.69022e8 −0.875380 −0.437690 − 0.899126i \(-0.644203\pi\)
−0.437690 + 0.899126i \(0.644203\pi\)
\(234\) 0 0
\(235\) 3.85400e8 1.93720
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00906e7 −0.284717 −0.142359 − 0.989815i \(-0.545469\pi\)
−0.142359 + 0.989815i \(0.545469\pi\)
\(240\) 0 0
\(241\) −4.84056e7 −0.222759 −0.111380 − 0.993778i \(-0.535527\pi\)
−0.111380 + 0.993778i \(0.535527\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.24395e8 −0.974836
\(246\) 0 0
\(247\) 3.17907e8 1.34234
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.72664e7 −0.108835 −0.0544176 − 0.998518i \(-0.517330\pi\)
−0.0544176 + 0.998518i \(0.517330\pi\)
\(252\) 0 0
\(253\) 2.14897e8 0.834273
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.68776e7 −0.0987701 −0.0493850 − 0.998780i \(-0.515726\pi\)
−0.0493850 + 0.998780i \(0.515726\pi\)
\(258\) 0 0
\(259\) −1.46633e8 −0.524423
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.03708e8 1.36843 0.684214 − 0.729281i \(-0.260145\pi\)
0.684214 + 0.729281i \(0.260145\pi\)
\(264\) 0 0
\(265\) 3.34714e7 0.110487
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.44677e8 1.07964 0.539820 − 0.841780i \(-0.318492\pi\)
0.539820 + 0.841780i \(0.318492\pi\)
\(270\) 0 0
\(271\) −3.41243e8 −1.04153 −0.520765 − 0.853700i \(-0.674353\pi\)
−0.520765 + 0.853700i \(0.674353\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.14947e7 −0.207305
\(276\) 0 0
\(277\) −5.41619e7 −0.153114 −0.0765570 − 0.997065i \(-0.524393\pi\)
−0.0765570 + 0.997065i \(0.524393\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.49715e6 −0.0147797 −0.00738985 − 0.999973i \(-0.502352\pi\)
−0.00738985 + 0.999973i \(0.502352\pi\)
\(282\) 0 0
\(283\) 5.58773e8 1.46549 0.732745 − 0.680503i \(-0.238239\pi\)
0.732745 + 0.680503i \(0.238239\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.02544e8 −0.505747
\(288\) 0 0
\(289\) −1.57147e8 −0.382969
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.24737e7 −0.0289706 −0.0144853 − 0.999895i \(-0.504611\pi\)
−0.0144853 + 0.999895i \(0.504611\pi\)
\(294\) 0 0
\(295\) 7.73518e8 1.75426
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.19092e8 1.77208
\(300\) 0 0
\(301\) −1.37185e7 −0.0289950
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.96807e8 1.80988
\(306\) 0 0
\(307\) 1.80343e7 0.0355725 0.0177863 − 0.999842i \(-0.494338\pi\)
0.0177863 + 0.999842i \(0.494338\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.94654e8 −1.12099 −0.560497 − 0.828157i \(-0.689390\pi\)
−0.560497 + 0.828157i \(0.689390\pi\)
\(312\) 0 0
\(313\) 3.73871e8 0.689154 0.344577 − 0.938758i \(-0.388022\pi\)
0.344577 + 0.938758i \(0.388022\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.89363e8 1.03914 0.519572 − 0.854427i \(-0.326091\pi\)
0.519572 + 0.854427i \(0.326091\pi\)
\(318\) 0 0
\(319\) −6.19960e8 −1.06929
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.56763e8 −0.589075
\(324\) 0 0
\(325\) −2.72506e8 −0.440336
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.98988e8 −0.617695
\(330\) 0 0
\(331\) 1.15657e9 1.75297 0.876487 − 0.481425i \(-0.159881\pi\)
0.876487 + 0.481425i \(0.159881\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.68342e8 −0.680623
\(336\) 0 0
\(337\) 5.78818e8 0.823830 0.411915 − 0.911222i \(-0.364860\pi\)
0.411915 + 0.911222i \(0.364860\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.52730e8 0.481729
\(342\) 0 0
\(343\) 4.98311e8 0.666762
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.25000e8 −0.417571 −0.208785 − 0.977961i \(-0.566951\pi\)
−0.208785 + 0.977961i \(0.566951\pi\)
\(348\) 0 0
\(349\) 2.50765e8 0.315776 0.157888 − 0.987457i \(-0.449532\pi\)
0.157888 + 0.987457i \(0.449532\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.04075e9 −1.25931 −0.629656 − 0.776874i \(-0.716804\pi\)
−0.629656 + 0.776874i \(0.716804\pi\)
\(354\) 0 0
\(355\) −1.47674e9 −1.75188
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.02411e9 1.16820 0.584101 − 0.811681i \(-0.301447\pi\)
0.584101 + 0.811681i \(0.301447\pi\)
\(360\) 0 0
\(361\) −3.91170e8 −0.437614
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.65546e7 −0.0285835
\(366\) 0 0
\(367\) −9.99933e8 −1.05594 −0.527971 − 0.849263i \(-0.677047\pi\)
−0.527971 + 0.849263i \(0.677047\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.46514e7 −0.0352300
\(372\) 0 0
\(373\) 1.92581e8 0.192146 0.0960732 − 0.995374i \(-0.469372\pi\)
0.0960732 + 0.995374i \(0.469372\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.36302e9 −2.27129
\(378\) 0 0
\(379\) −1.91512e9 −1.80700 −0.903501 − 0.428585i \(-0.859012\pi\)
−0.903501 + 0.428585i \(0.859012\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.51110e9 1.37435 0.687175 − 0.726492i \(-0.258851\pi\)
0.687175 + 0.726492i \(0.258851\pi\)
\(384\) 0 0
\(385\) 3.74887e8 0.334802
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.37168e8 0.634955 0.317477 − 0.948266i \(-0.397164\pi\)
0.317477 + 0.948266i \(0.397164\pi\)
\(390\) 0 0
\(391\) −9.19204e8 −0.777667
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.68416e8 0.300779
\(396\) 0 0
\(397\) −8.55916e8 −0.686538 −0.343269 − 0.939237i \(-0.611534\pi\)
−0.343269 + 0.939237i \(0.611534\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.38953e8 0.185058 0.0925288 − 0.995710i \(-0.470505\pi\)
0.0925288 + 0.995710i \(0.470505\pi\)
\(402\) 0 0
\(403\) 1.34445e9 1.02324
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.68877e9 −1.24163
\(408\) 0 0
\(409\) −9.53539e8 −0.689139 −0.344570 − 0.938761i \(-0.611975\pi\)
−0.344570 + 0.938761i \(0.611975\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00789e8 −0.559362
\(414\) 0 0
\(415\) 3.48357e8 0.239252
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.44668e8 −0.361729 −0.180864 − 0.983508i \(-0.557889\pi\)
−0.180864 + 0.983508i \(0.557889\pi\)
\(420\) 0 0
\(421\) −6.61270e8 −0.431909 −0.215954 − 0.976403i \(-0.569286\pi\)
−0.215954 + 0.976403i \(0.569286\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.05813e8 0.193239
\(426\) 0 0
\(427\) −9.28426e8 −0.577098
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.10789e8 −0.126817 −0.0634086 − 0.997988i \(-0.520197\pi\)
−0.0634086 + 0.997988i \(0.520197\pi\)
\(432\) 0 0
\(433\) −1.72888e9 −1.02343 −0.511714 − 0.859156i \(-0.670989\pi\)
−0.511714 + 0.859156i \(0.670989\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.29522e9 0.742433
\(438\) 0 0
\(439\) −1.39429e8 −0.0786554 −0.0393277 − 0.999226i \(-0.512522\pi\)
−0.0393277 + 0.999226i \(0.512522\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.40121e9 −0.765754 −0.382877 − 0.923799i \(-0.625067\pi\)
−0.382877 + 0.923799i \(0.625067\pi\)
\(444\) 0 0
\(445\) 2.92286e9 1.57234
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.78420e7 −0.00930209 −0.00465104 − 0.999989i \(-0.501480\pi\)
−0.00465104 + 0.999989i \(0.501480\pi\)
\(450\) 0 0
\(451\) −2.33271e9 −1.19741
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.42890e9 0.711153
\(456\) 0 0
\(457\) −1.74413e9 −0.854816 −0.427408 − 0.904059i \(-0.640573\pi\)
−0.427408 + 0.904059i \(0.640573\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.35991e9 0.646483 0.323242 − 0.946316i \(-0.395227\pi\)
0.323242 + 0.946316i \(0.395227\pi\)
\(462\) 0 0
\(463\) 1.97759e9 0.925981 0.462991 − 0.886363i \(-0.346776\pi\)
0.462991 + 0.886363i \(0.346776\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.46852e9 −1.57592 −0.787961 − 0.615726i \(-0.788863\pi\)
−0.787961 + 0.615726i \(0.788863\pi\)
\(468\) 0 0
\(469\) 4.84854e8 0.217023
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.57996e8 −0.0686486
\(474\) 0 0
\(475\) −4.30909e8 −0.184484
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.18042e9 −0.906494 −0.453247 − 0.891385i \(-0.649735\pi\)
−0.453247 + 0.891385i \(0.649735\pi\)
\(480\) 0 0
\(481\) −6.43685e9 −2.63734
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.36478e8 −0.253331
\(486\) 0 0
\(487\) −3.00745e9 −1.17990 −0.589952 − 0.807439i \(-0.700853\pi\)
−0.589952 + 0.807439i \(0.700853\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.62970e9 1.00258 0.501292 − 0.865278i \(-0.332858\pi\)
0.501292 + 0.865278i \(0.332858\pi\)
\(492\) 0 0
\(493\) 2.65183e9 0.996739
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.52880e9 0.558604
\(498\) 0 0
\(499\) 2.76990e8 0.0997957 0.0498978 − 0.998754i \(-0.484110\pi\)
0.0498978 + 0.998754i \(0.484110\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.59216e9 1.60890 0.804451 − 0.594019i \(-0.202460\pi\)
0.804451 + 0.594019i \(0.202460\pi\)
\(504\) 0 0
\(505\) −4.76033e9 −1.64482
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.25466e9 1.09394 0.546970 − 0.837152i \(-0.315781\pi\)
0.546970 + 0.837152i \(0.315781\pi\)
\(510\) 0 0
\(511\) 2.74909e7 0.00911413
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.00392e9 1.93691
\(516\) 0 0
\(517\) −4.59515e9 −1.46246
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.57306e9 1.72648 0.863241 − 0.504793i \(-0.168431\pi\)
0.863241 + 0.504793i \(0.168431\pi\)
\(522\) 0 0
\(523\) −2.33783e9 −0.714590 −0.357295 − 0.933992i \(-0.616301\pi\)
−0.357295 + 0.933992i \(0.616301\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.50878e9 −0.449043
\(528\) 0 0
\(529\) −6.76836e7 −0.0198787
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −8.89125e9 −2.54342
\(534\) 0 0
\(535\) −4.01142e9 −1.13256
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.67548e9 0.735937
\(540\) 0 0
\(541\) −5.60874e9 −1.52291 −0.761456 − 0.648217i \(-0.775515\pi\)
−0.761456 + 0.648217i \(0.775515\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.20847e9 −0.849004
\(546\) 0 0
\(547\) 4.51033e9 1.17829 0.589146 − 0.808027i \(-0.299464\pi\)
0.589146 + 0.808027i \(0.299464\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.73659e9 −0.951580
\(552\) 0 0
\(553\) −3.81405e8 −0.0959065
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.30015e8 0.0809171 0.0404585 − 0.999181i \(-0.487118\pi\)
0.0404585 + 0.999181i \(0.487118\pi\)
\(558\) 0 0
\(559\) −6.02210e8 −0.145817
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.57572e9 −0.608303 −0.304151 − 0.952624i \(-0.598373\pi\)
−0.304151 + 0.952624i \(0.598373\pi\)
\(564\) 0 0
\(565\) 5.82730e9 1.35925
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.22891e9 −0.507225 −0.253612 − 0.967306i \(-0.581619\pi\)
−0.253612 + 0.967306i \(0.581619\pi\)
\(570\) 0 0
\(571\) −2.83433e9 −0.637123 −0.318562 − 0.947902i \(-0.603200\pi\)
−0.318562 + 0.947902i \(0.603200\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.11024e9 −0.243546
\(576\) 0 0
\(577\) 1.24427e7 0.00269649 0.00134825 − 0.999999i \(-0.499571\pi\)
0.00134825 + 0.999999i \(0.499571\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.60639e8 −0.0762879
\(582\) 0 0
\(583\) −3.99082e8 −0.0834107
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.61410e9 1.75783 0.878914 − 0.476980i \(-0.158269\pi\)
0.878914 + 0.476980i \(0.158269\pi\)
\(588\) 0 0
\(589\) 2.12596e9 0.428698
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.55490e9 −0.896989 −0.448495 − 0.893786i \(-0.648040\pi\)
−0.448495 + 0.893786i \(0.648040\pi\)
\(594\) 0 0
\(595\) −1.60355e9 −0.312085
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.57167e8 −0.124934 −0.0624672 − 0.998047i \(-0.519897\pi\)
−0.0624672 + 0.998047i \(0.519897\pi\)
\(600\) 0 0
\(601\) 8.39406e9 1.57729 0.788645 − 0.614849i \(-0.210783\pi\)
0.788645 + 0.614849i \(0.210783\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.76242e9 −0.323567
\(606\) 0 0
\(607\) 1.04953e9 0.190473 0.0952363 − 0.995455i \(-0.469639\pi\)
0.0952363 + 0.995455i \(0.469639\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.75147e10 −3.10641
\(612\) 0 0
\(613\) 4.22848e9 0.741433 0.370717 − 0.928746i \(-0.379112\pi\)
0.370717 + 0.928746i \(0.379112\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.85460e9 0.832061 0.416031 − 0.909351i \(-0.363421\pi\)
0.416031 + 0.909351i \(0.363421\pi\)
\(618\) 0 0
\(619\) −2.32275e9 −0.393627 −0.196813 − 0.980441i \(-0.563059\pi\)
−0.196813 + 0.980441i \(0.563059\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.02591e9 −0.501358
\(624\) 0 0
\(625\) −7.23563e9 −1.18549
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.22359e9 1.15738
\(630\) 0 0
\(631\) −5.84987e9 −0.926923 −0.463461 − 0.886117i \(-0.653393\pi\)
−0.463461 + 0.886117i \(0.653393\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.10342e9 −0.171014
\(636\) 0 0
\(637\) 1.01977e10 1.56320
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.81719e9 1.47226 0.736129 − 0.676841i \(-0.236652\pi\)
0.736129 + 0.676841i \(0.236652\pi\)
\(642\) 0 0
\(643\) 4.73044e9 0.701719 0.350859 − 0.936428i \(-0.385890\pi\)
0.350859 + 0.936428i \(0.385890\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.28999e9 0.622718 0.311359 − 0.950292i \(-0.399216\pi\)
0.311359 + 0.950292i \(0.399216\pi\)
\(648\) 0 0
\(649\) −9.22271e9 −1.32435
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.32108e10 −1.85666 −0.928328 − 0.371762i \(-0.878754\pi\)
−0.928328 + 0.371762i \(0.878754\pi\)
\(654\) 0 0
\(655\) 9.39136e9 1.30582
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.60547e9 −0.762980 −0.381490 − 0.924373i \(-0.624589\pi\)
−0.381490 + 0.924373i \(0.624589\pi\)
\(660\) 0 0
\(661\) −1.77070e9 −0.238474 −0.119237 − 0.992866i \(-0.538045\pi\)
−0.119237 + 0.992866i \(0.538045\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.25950e9 0.297945
\(666\) 0 0
\(667\) −9.62738e9 −1.25623
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.06927e10 −1.36634
\(672\) 0 0
\(673\) −7.36036e8 −0.0930778 −0.0465389 − 0.998916i \(-0.514819\pi\)
−0.0465389 + 0.998916i \(0.514819\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.22562e9 0.771120 0.385560 − 0.922683i \(-0.374008\pi\)
0.385560 + 0.922683i \(0.374008\pi\)
\(678\) 0 0
\(679\) 6.58918e8 0.0807769
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.78088e9 0.814355 0.407178 − 0.913349i \(-0.366513\pi\)
0.407178 + 0.913349i \(0.366513\pi\)
\(684\) 0 0
\(685\) −2.30046e9 −0.273462
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.52112e9 −0.177173
\(690\) 0 0
\(691\) 4.14997e9 0.478489 0.239244 − 0.970959i \(-0.423100\pi\)
0.239244 + 0.970959i \(0.423100\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.84867e10 2.08887
\(696\) 0 0
\(697\) 9.97797e9 1.11616
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.61210e10 −1.76758 −0.883791 − 0.467881i \(-0.845017\pi\)
−0.883791 + 0.467881i \(0.845017\pi\)
\(702\) 0 0
\(703\) −1.01785e10 −1.10494
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.92816e9 0.524466
\(708\) 0 0
\(709\) 1.13288e10 1.19378 0.596889 − 0.802324i \(-0.296403\pi\)
0.596889 + 0.802324i \(0.296403\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.47756e9 0.565945
\(714\) 0 0
\(715\) 1.64567e10 1.68373
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.97759e9 0.900760 0.450380 − 0.892837i \(-0.351289\pi\)
0.450380 + 0.892837i \(0.351289\pi\)
\(720\) 0 0
\(721\) −6.21560e9 −0.617603
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.20296e9 0.312154
\(726\) 0 0
\(727\) 1.98049e10 1.91163 0.955813 − 0.293976i \(-0.0949787\pi\)
0.955813 + 0.293976i \(0.0949787\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.75814e8 0.0639907
\(732\) 0 0
\(733\) 1.88605e10 1.76884 0.884420 − 0.466692i \(-0.154554\pi\)
0.884420 + 0.466692i \(0.154554\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.58408e9 0.513825
\(738\) 0 0
\(739\) 1.38397e10 1.26145 0.630725 − 0.776006i \(-0.282757\pi\)
0.630725 + 0.776006i \(0.282757\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.80884e10 1.61785 0.808926 − 0.587911i \(-0.200049\pi\)
0.808926 + 0.587911i \(0.200049\pi\)
\(744\) 0 0
\(745\) −7.25813e9 −0.643099
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.15285e9 0.361127
\(750\) 0 0
\(751\) 2.22027e9 0.191278 0.0956390 − 0.995416i \(-0.469511\pi\)
0.0956390 + 0.995416i \(0.469511\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.81727e10 1.53676
\(756\) 0 0
\(757\) 2.21883e10 1.85904 0.929521 − 0.368770i \(-0.120221\pi\)
0.929521 + 0.368770i \(0.120221\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.03821e9 0.249903 0.124952 − 0.992163i \(-0.460122\pi\)
0.124952 + 0.992163i \(0.460122\pi\)
\(762\) 0 0
\(763\) 3.32158e9 0.270713
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3.51529e10 −2.81305
\(768\) 0 0
\(769\) −1.82409e10 −1.44645 −0.723226 − 0.690611i \(-0.757342\pi\)
−0.723226 + 0.690611i \(0.757342\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.20659e10 −0.939576 −0.469788 − 0.882779i \(-0.655670\pi\)
−0.469788 + 0.882779i \(0.655670\pi\)
\(774\) 0 0
\(775\) −1.82235e9 −0.140629
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.40596e10 −1.06559
\(780\) 0 0
\(781\) 1.76073e10 1.32255
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.15958e9 −0.675821
\(786\) 0 0
\(787\) −1.26799e10 −0.927269 −0.463634 − 0.886027i \(-0.653455\pi\)
−0.463634 + 0.886027i \(0.653455\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.03275e9 −0.433409
\(792\) 0 0
\(793\) −4.07559e10 −2.90225
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.86451e9 0.340357 0.170179 − 0.985413i \(-0.445565\pi\)
0.170179 + 0.985413i \(0.445565\pi\)
\(798\) 0 0
\(799\) 1.96554e10 1.36323
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.16613e8 0.0215786
\(804\) 0 0
\(805\) 5.82163e9 0.393332
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.07879e10 −0.716338 −0.358169 − 0.933657i \(-0.616599\pi\)
−0.358169 + 0.933657i \(0.616599\pi\)
\(810\) 0 0
\(811\) 2.41592e10 1.59041 0.795204 − 0.606341i \(-0.207364\pi\)
0.795204 + 0.606341i \(0.207364\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.09261e10 0.706992
\(816\) 0 0
\(817\) −9.52265e8 −0.0610915
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.71882e10 1.71467 0.857333 − 0.514763i \(-0.172120\pi\)
0.857333 + 0.514763i \(0.172120\pi\)
\(822\) 0 0
\(823\) −7.45342e9 −0.466075 −0.233038 − 0.972468i \(-0.574867\pi\)
−0.233038 + 0.972468i \(0.574867\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −7.48871e9 −0.460402 −0.230201 − 0.973143i \(-0.573938\pi\)
−0.230201 + 0.973143i \(0.573938\pi\)
\(828\) 0 0
\(829\) 5.23995e9 0.319437 0.159719 − 0.987163i \(-0.448941\pi\)
0.159719 + 0.987163i \(0.448941\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.14441e10 −0.686002
\(834\) 0 0
\(835\) 1.88619e10 1.12120
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.92901e9 0.288133 0.144066 − 0.989568i \(-0.453982\pi\)
0.144066 + 0.989568i \(0.453982\pi\)
\(840\) 0 0
\(841\) 1.05243e10 0.610111
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.31482e10 2.46017
\(846\) 0 0
\(847\) 1.82455e9 0.103173
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.62250e10 −1.45869
\(852\) 0 0
\(853\) 3.17690e10 1.75260 0.876300 − 0.481767i \(-0.160005\pi\)
0.876300 + 0.481767i \(0.160005\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.40083e10 −1.30295 −0.651475 − 0.758670i \(-0.725850\pi\)
−0.651475 + 0.758670i \(0.725850\pi\)
\(858\) 0 0
\(859\) −2.89979e10 −1.56096 −0.780478 − 0.625183i \(-0.785024\pi\)
−0.780478 + 0.625183i \(0.785024\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9.49509e9 −0.502877 −0.251438 − 0.967873i \(-0.580904\pi\)
−0.251438 + 0.967873i \(0.580904\pi\)
\(864\) 0 0
\(865\) 2.00766e10 1.05471
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.39265e9 −0.227069
\(870\) 0 0
\(871\) 2.12841e10 1.09142
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5.93631e9 0.299563
\(876\) 0 0
\(877\) 1.79366e10 0.897930 0.448965 − 0.893549i \(-0.351793\pi\)
0.448965 + 0.893549i \(0.351793\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 5.83458e8 0.0287471 0.0143735 − 0.999897i \(-0.495425\pi\)
0.0143735 + 0.999897i \(0.495425\pi\)
\(882\) 0 0
\(883\) −3.60031e10 −1.75986 −0.879929 − 0.475106i \(-0.842410\pi\)
−0.879929 + 0.475106i \(0.842410\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.65301e10 −0.795320 −0.397660 − 0.917533i \(-0.630178\pi\)
−0.397660 + 0.917533i \(0.630178\pi\)
\(888\) 0 0
\(889\) 1.14232e9 0.0545294
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.76957e10 −1.30146
\(894\) 0 0
\(895\) −3.30288e9 −0.153997
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.58023e10 −0.725374
\(900\) 0 0
\(901\) 1.70704e9 0.0777511
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.00172e10 0.897705
\(906\) 0 0
\(907\) 3.20080e10 1.42440 0.712201 − 0.701975i \(-0.247698\pi\)
0.712201 + 0.701975i \(0.247698\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.23494e10 0.541168 0.270584 − 0.962696i \(-0.412783\pi\)
0.270584 + 0.962696i \(0.412783\pi\)
\(912\) 0 0
\(913\) −4.15348e9 −0.180620
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.72247e9 −0.416374
\(918\) 0 0
\(919\) 1.97181e10 0.838032 0.419016 − 0.907979i \(-0.362375\pi\)
0.419016 + 0.907979i \(0.362375\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.71111e10 2.80924
\(924\) 0 0
\(925\) 8.72487e9 0.362462
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.86505e10 −1.17240 −0.586201 − 0.810166i \(-0.699377\pi\)
−0.586201 + 0.810166i \(0.699377\pi\)
\(930\) 0 0
\(931\) 1.61255e10 0.654921
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.84681e10 −0.738893
\(936\) 0 0
\(937\) −4.85335e8 −0.0192732 −0.00963659 − 0.999954i \(-0.503067\pi\)
−0.00963659 + 0.999954i \(0.503067\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.90015e9 0.113464 0.0567319 − 0.998389i \(-0.481932\pi\)
0.0567319 + 0.998389i \(0.481932\pi\)
\(942\) 0 0
\(943\) −3.62247e10 −1.40674
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.38658e9 0.0530544 0.0265272 − 0.999648i \(-0.491555\pi\)
0.0265272 + 0.999648i \(0.491555\pi\)
\(948\) 0 0
\(949\) 1.20679e9 0.0458352
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.45920e10 −0.546123 −0.273062 − 0.961997i \(-0.588036\pi\)
−0.273062 + 0.961997i \(0.588036\pi\)
\(954\) 0 0
\(955\) −4.87759e10 −1.81215
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.38156e9 0.0871960
\(960\) 0 0
\(961\) −1.85218e10 −0.673211
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.10217e10 −0.394823
\(966\) 0 0
\(967\) −3.50961e9 −0.124815 −0.0624074 − 0.998051i \(-0.519878\pi\)
−0.0624074 + 0.998051i \(0.519878\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.19076e10 0.417403 0.208701 − 0.977979i \(-0.433076\pi\)
0.208701 + 0.977979i \(0.433076\pi\)
\(972\) 0 0
\(973\) −1.91385e10 −0.666058
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.27872e10 0.438676 0.219338 − 0.975649i \(-0.429610\pi\)
0.219338 + 0.975649i \(0.429610\pi\)
\(978\) 0 0
\(979\) −3.48495e10 −1.18702
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.75522e10 0.925163 0.462582 − 0.886577i \(-0.346923\pi\)
0.462582 + 0.886577i \(0.346923\pi\)
\(984\) 0 0
\(985\) −3.31676e10 −1.10583
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.45352e9 −0.0806497
\(990\) 0 0
\(991\) −1.90312e9 −0.0621167 −0.0310584 − 0.999518i \(-0.509888\pi\)
−0.0310584 + 0.999518i \(0.509888\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.98040e9 −0.160281
\(996\) 0 0
\(997\) −1.52301e10 −0.486711 −0.243355 − 0.969937i \(-0.578248\pi\)
−0.243355 + 0.969937i \(0.578248\pi\)
\(998\) 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 432.8.a.h.1.1 1
3.2 odd 2 432.8.a.a.1.1 1
4.3 odd 2 54.8.a.c.1.1 ✓ 1
12.11 even 2 54.8.a.d.1.1 yes 1
36.7 odd 6 162.8.c.g.109.1 2
36.11 even 6 162.8.c.f.109.1 2
36.23 even 6 162.8.c.f.55.1 2
36.31 odd 6 162.8.c.g.55.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.8.a.c.1.1 ✓ 1 4.3 odd 2
54.8.a.d.1.1 yes 1 12.11 even 2
162.8.c.f.55.1 2 36.23 even 6
162.8.c.f.109.1 2 36.11 even 6
162.8.c.g.55.1 2 36.31 odd 6
162.8.c.g.109.1 2 36.7 odd 6
432.8.a.a.1.1 1 3.2 odd 2
432.8.a.h.1.1 1 1.1 even 1 trivial