Properties

Label 225.8.a.j.1.1
Level $225$
Weight $8$
Character 225.1
Self dual yes
Analytic conductor $70.287$
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] = [225,8,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("225.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(70.2866307339\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.0000 q^{2} -28.0000 q^{4} +1170.00 q^{7} -1560.00 q^{8} +O(q^{10})\) \(q+10.0000 q^{2} -28.0000 q^{4} +1170.00 q^{7} -1560.00 q^{8} -2650.00 q^{11} +11180.0 q^{13} +11700.0 q^{14} -12016.0 q^{16} -31070.0 q^{17} +30316.0 q^{19} -26500.0 q^{22} -32760.0 q^{23} +111800. q^{26} -32760.0 q^{28} +163150. q^{29} +136188. q^{31} +79520.0 q^{32} -310700. q^{34} -16640.0 q^{37} +303160. q^{38} -483200. q^{41} +141080. q^{43} +74200.0 q^{44} -327600. q^{46} +103240. q^{47} +545357. q^{49} -313040. q^{52} +1.95013e6 q^{53} -1.82520e6 q^{56} +1.63150e6 q^{58} +2.64335e6 q^{59} +2.82092e6 q^{61} +1.36188e6 q^{62} +2.33325e6 q^{64} +506220. q^{67} +869960. q^{68} +2.89090e6 q^{71} +2.87729e6 q^{73} -166400. q^{74} -848848. q^{76} -3.10050e6 q^{77} -5.71756e6 q^{79} -4.83200e6 q^{82} -3.79038e6 q^{83} +1.41080e6 q^{86} +4.13400e6 q^{88} +1.05645e7 q^{89} +1.30806e7 q^{91} +917280. q^{92} +1.03240e6 q^{94} -2.15813e6 q^{97} +5.45357e6 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0000 0.883883 0.441942 0.897044i \(-0.354290\pi\)
0.441942 + 0.897044i \(0.354290\pi\)
\(3\) 0 0
\(4\) −28.0000 −0.218750
\(5\) 0 0
\(6\) 0 0
\(7\) 1170.00 1.28927 0.644633 0.764492i \(-0.277010\pi\)
0.644633 + 0.764492i \(0.277010\pi\)
\(8\) −1560.00 −1.07723
\(9\) 0 0
\(10\) 0 0
\(11\) −2650.00 −0.600304 −0.300152 0.953891i \(-0.597037\pi\)
−0.300152 + 0.953891i \(0.597037\pi\)
\(12\) 0 0
\(13\) 11180.0 1.41137 0.705684 0.708527i \(-0.250640\pi\)
0.705684 + 0.708527i \(0.250640\pi\)
\(14\) 11700.0 1.13956
\(15\) 0 0
\(16\) −12016.0 −0.733398
\(17\) −31070.0 −1.53380 −0.766902 0.641764i \(-0.778203\pi\)
−0.766902 + 0.641764i \(0.778203\pi\)
\(18\) 0 0
\(19\) 30316.0 1.01399 0.506996 0.861949i \(-0.330756\pi\)
0.506996 + 0.861949i \(0.330756\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −26500.0 −0.530599
\(23\) −32760.0 −0.561431 −0.280716 0.959791i \(-0.590572\pi\)
−0.280716 + 0.959791i \(0.590572\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 111800. 1.24748
\(27\) 0 0
\(28\) −32760.0 −0.282027
\(29\) 163150. 1.24221 0.621104 0.783728i \(-0.286685\pi\)
0.621104 + 0.783728i \(0.286685\pi\)
\(30\) 0 0
\(31\) 136188. 0.821056 0.410528 0.911848i \(-0.365344\pi\)
0.410528 + 0.911848i \(0.365344\pi\)
\(32\) 79520.0 0.428994
\(33\) 0 0
\(34\) −310700. −1.35570
\(35\) 0 0
\(36\) 0 0
\(37\) −16640.0 −0.0540067 −0.0270033 0.999635i \(-0.508596\pi\)
−0.0270033 + 0.999635i \(0.508596\pi\)
\(38\) 303160. 0.896250
\(39\) 0 0
\(40\) 0 0
\(41\) −483200. −1.09492 −0.547461 0.836831i \(-0.684406\pi\)
−0.547461 + 0.836831i \(0.684406\pi\)
\(42\) 0 0
\(43\) 141080. 0.270599 0.135299 0.990805i \(-0.456800\pi\)
0.135299 + 0.990805i \(0.456800\pi\)
\(44\) 74200.0 0.131317
\(45\) 0 0
\(46\) −327600. −0.496240
\(47\) 103240. 0.145046 0.0725230 0.997367i \(-0.476895\pi\)
0.0725230 + 0.997367i \(0.476895\pi\)
\(48\) 0 0
\(49\) 545357. 0.662208
\(50\) 0 0
\(51\) 0 0
\(52\) −313040. −0.308737
\(53\) 1.95013e6 1.79928 0.899638 0.436636i \(-0.143830\pi\)
0.899638 + 0.436636i \(0.143830\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.82520e6 −1.38884
\(57\) 0 0
\(58\) 1.63150e6 1.09797
\(59\) 2.64335e6 1.67561 0.837804 0.545971i \(-0.183839\pi\)
0.837804 + 0.545971i \(0.183839\pi\)
\(60\) 0 0
\(61\) 2.82092e6 1.59124 0.795622 0.605794i \(-0.207144\pi\)
0.795622 + 0.605794i \(0.207144\pi\)
\(62\) 1.36188e6 0.725718
\(63\) 0 0
\(64\) 2.33325e6 1.11258
\(65\) 0 0
\(66\) 0 0
\(67\) 506220. 0.205626 0.102813 0.994701i \(-0.467216\pi\)
0.102813 + 0.994701i \(0.467216\pi\)
\(68\) 869960. 0.335520
\(69\) 0 0
\(70\) 0 0
\(71\) 2.89090e6 0.958581 0.479291 0.877656i \(-0.340894\pi\)
0.479291 + 0.877656i \(0.340894\pi\)
\(72\) 0 0
\(73\) 2.87729e6 0.865673 0.432836 0.901473i \(-0.357513\pi\)
0.432836 + 0.901473i \(0.357513\pi\)
\(74\) −166400. −0.0477356
\(75\) 0 0
\(76\) −848848. −0.221811
\(77\) −3.10050e6 −0.773952
\(78\) 0 0
\(79\) −5.71756e6 −1.30471 −0.652357 0.757911i \(-0.726220\pi\)
−0.652357 + 0.757911i \(0.726220\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −4.83200e6 −0.967784
\(83\) −3.79038e6 −0.727628 −0.363814 0.931472i \(-0.618526\pi\)
−0.363814 + 0.931472i \(0.618526\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.41080e6 0.239178
\(87\) 0 0
\(88\) 4.13400e6 0.646668
\(89\) 1.05645e7 1.58849 0.794244 0.607599i \(-0.207867\pi\)
0.794244 + 0.607599i \(0.207867\pi\)
\(90\) 0 0
\(91\) 1.30806e7 1.81963
\(92\) 917280. 0.122813
\(93\) 0 0
\(94\) 1.03240e6 0.128204
\(95\) 0 0
\(96\) 0 0
\(97\) −2.15813e6 −0.240091 −0.120046 0.992768i \(-0.538304\pi\)
−0.120046 + 0.992768i \(0.538304\pi\)
\(98\) 5.45357e6 0.585315
\(99\) 0 0
\(100\) 0 0
\(101\) −5.00565e6 −0.483432 −0.241716 0.970347i \(-0.577710\pi\)
−0.241716 + 0.970347i \(0.577710\pi\)
\(102\) 0 0
\(103\) 2.19448e7 1.97879 0.989397 0.145234i \(-0.0463934\pi\)
0.989397 + 0.145234i \(0.0463934\pi\)
\(104\) −1.74408e7 −1.52037
\(105\) 0 0
\(106\) 1.95013e7 1.59035
\(107\) −8.47158e6 −0.668530 −0.334265 0.942479i \(-0.608488\pi\)
−0.334265 + 0.942479i \(0.608488\pi\)
\(108\) 0 0
\(109\) 1.68025e7 1.24274 0.621370 0.783517i \(-0.286576\pi\)
0.621370 + 0.783517i \(0.286576\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.40587e7 −0.945546
\(113\) −1.90436e7 −1.24158 −0.620789 0.783978i \(-0.713188\pi\)
−0.620789 + 0.783978i \(0.713188\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.56820e6 −0.271733
\(117\) 0 0
\(118\) 2.64335e7 1.48104
\(119\) −3.63519e7 −1.97748
\(120\) 0 0
\(121\) −1.24647e7 −0.639635
\(122\) 2.82092e7 1.40647
\(123\) 0 0
\(124\) −3.81326e6 −0.179606
\(125\) 0 0
\(126\) 0 0
\(127\) −1.18507e7 −0.513369 −0.256685 0.966495i \(-0.582630\pi\)
−0.256685 + 0.966495i \(0.582630\pi\)
\(128\) 1.31539e7 0.554396
\(129\) 0 0
\(130\) 0 0
\(131\) 2.07226e7 0.805371 0.402685 0.915338i \(-0.368077\pi\)
0.402685 + 0.915338i \(0.368077\pi\)
\(132\) 0 0
\(133\) 3.54697e7 1.30731
\(134\) 5.06220e6 0.181749
\(135\) 0 0
\(136\) 4.84692e7 1.65226
\(137\) 1.29435e6 0.0430061 0.0215030 0.999769i \(-0.493155\pi\)
0.0215030 + 0.999769i \(0.493155\pi\)
\(138\) 0 0
\(139\) −3.99420e6 −0.126147 −0.0630737 0.998009i \(-0.520090\pi\)
−0.0630737 + 0.998009i \(0.520090\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.89090e7 0.847274
\(143\) −2.96270e7 −0.847250
\(144\) 0 0
\(145\) 0 0
\(146\) 2.87729e7 0.765154
\(147\) 0 0
\(148\) 465920. 0.0118140
\(149\) −5.27315e6 −0.130592 −0.0652962 0.997866i \(-0.520799\pi\)
−0.0652962 + 0.997866i \(0.520799\pi\)
\(150\) 0 0
\(151\) 6.87634e7 1.62532 0.812659 0.582740i \(-0.198019\pi\)
0.812659 + 0.582740i \(0.198019\pi\)
\(152\) −4.72930e7 −1.09231
\(153\) 0 0
\(154\) −3.10050e7 −0.684084
\(155\) 0 0
\(156\) 0 0
\(157\) −6.43181e7 −1.32643 −0.663215 0.748429i \(-0.730809\pi\)
−0.663215 + 0.748429i \(0.730809\pi\)
\(158\) −5.71756e7 −1.15322
\(159\) 0 0
\(160\) 0 0
\(161\) −3.83292e7 −0.723834
\(162\) 0 0
\(163\) −3.10739e7 −0.562004 −0.281002 0.959707i \(-0.590667\pi\)
−0.281002 + 0.959707i \(0.590667\pi\)
\(164\) 1.35296e7 0.239514
\(165\) 0 0
\(166\) −3.79038e7 −0.643138
\(167\) −8.07466e7 −1.34158 −0.670790 0.741648i \(-0.734045\pi\)
−0.670790 + 0.741648i \(0.734045\pi\)
\(168\) 0 0
\(169\) 6.22439e7 0.991958
\(170\) 0 0
\(171\) 0 0
\(172\) −3.95024e6 −0.0591935
\(173\) 7.88638e7 1.15802 0.579011 0.815320i \(-0.303439\pi\)
0.579011 + 0.815320i \(0.303439\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.18424e7 0.440262
\(177\) 0 0
\(178\) 1.05645e8 1.40404
\(179\) 1.25273e8 1.63256 0.816282 0.577653i \(-0.196031\pi\)
0.816282 + 0.577653i \(0.196031\pi\)
\(180\) 0 0
\(181\) −5.95947e7 −0.747020 −0.373510 0.927626i \(-0.621846\pi\)
−0.373510 + 0.927626i \(0.621846\pi\)
\(182\) 1.30806e8 1.60834
\(183\) 0 0
\(184\) 5.11056e7 0.604792
\(185\) 0 0
\(186\) 0 0
\(187\) 8.23355e7 0.920750
\(188\) −2.89072e6 −0.0317288
\(189\) 0 0
\(190\) 0 0
\(191\) −1.40874e8 −1.46290 −0.731452 0.681893i \(-0.761157\pi\)
−0.731452 + 0.681893i \(0.761157\pi\)
\(192\) 0 0
\(193\) 5.43985e6 0.0544674 0.0272337 0.999629i \(-0.491330\pi\)
0.0272337 + 0.999629i \(0.491330\pi\)
\(194\) −2.15813e7 −0.212213
\(195\) 0 0
\(196\) −1.52700e7 −0.144858
\(197\) 9.05130e7 0.843489 0.421744 0.906715i \(-0.361418\pi\)
0.421744 + 0.906715i \(0.361418\pi\)
\(198\) 0 0
\(199\) −1.00226e8 −0.901558 −0.450779 0.892636i \(-0.648854\pi\)
−0.450779 + 0.892636i \(0.648854\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −5.00565e7 −0.427298
\(203\) 1.90886e8 1.60154
\(204\) 0 0
\(205\) 0 0
\(206\) 2.19448e8 1.74902
\(207\) 0 0
\(208\) −1.34339e8 −1.03509
\(209\) −8.03374e7 −0.608703
\(210\) 0 0
\(211\) −4.46103e6 −0.0326924 −0.0163462 0.999866i \(-0.505203\pi\)
−0.0163462 + 0.999866i \(0.505203\pi\)
\(212\) −5.46036e7 −0.393592
\(213\) 0 0
\(214\) −8.47158e7 −0.590903
\(215\) 0 0
\(216\) 0 0
\(217\) 1.59340e8 1.05856
\(218\) 1.68025e8 1.09844
\(219\) 0 0
\(220\) 0 0
\(221\) −3.47363e8 −2.16476
\(222\) 0 0
\(223\) −3.13973e8 −1.89594 −0.947972 0.318353i \(-0.896870\pi\)
−0.947972 + 0.318353i \(0.896870\pi\)
\(224\) 9.30384e7 0.553088
\(225\) 0 0
\(226\) −1.90436e8 −1.09741
\(227\) 6.07635e7 0.344788 0.172394 0.985028i \(-0.444850\pi\)
0.172394 + 0.985028i \(0.444850\pi\)
\(228\) 0 0
\(229\) 6.48864e7 0.357050 0.178525 0.983935i \(-0.442867\pi\)
0.178525 + 0.983935i \(0.442867\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.54514e8 −1.33815
\(233\) −1.22649e8 −0.635212 −0.317606 0.948223i \(-0.602879\pi\)
−0.317606 + 0.948223i \(0.602879\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.40138e7 −0.366539
\(237\) 0 0
\(238\) −3.63519e8 −1.74786
\(239\) 3.16220e8 1.49829 0.749146 0.662405i \(-0.230464\pi\)
0.749146 + 0.662405i \(0.230464\pi\)
\(240\) 0 0
\(241\) −1.37644e8 −0.633428 −0.316714 0.948521i \(-0.602579\pi\)
−0.316714 + 0.948521i \(0.602579\pi\)
\(242\) −1.24647e8 −0.565363
\(243\) 0 0
\(244\) −7.89858e7 −0.348085
\(245\) 0 0
\(246\) 0 0
\(247\) 3.38933e8 1.43111
\(248\) −2.12453e8 −0.884469
\(249\) 0 0
\(250\) 0 0
\(251\) 2.28522e8 0.912160 0.456080 0.889939i \(-0.349253\pi\)
0.456080 + 0.889939i \(0.349253\pi\)
\(252\) 0 0
\(253\) 8.68140e7 0.337030
\(254\) −1.18507e8 −0.453759
\(255\) 0 0
\(256\) −1.67117e8 −0.622558
\(257\) −2.83909e8 −1.04331 −0.521654 0.853157i \(-0.674685\pi\)
−0.521654 + 0.853157i \(0.674685\pi\)
\(258\) 0 0
\(259\) −1.94688e7 −0.0696290
\(260\) 0 0
\(261\) 0 0
\(262\) 2.07226e8 0.711854
\(263\) −2.59380e8 −0.879208 −0.439604 0.898192i \(-0.644881\pi\)
−0.439604 + 0.898192i \(0.644881\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.54697e8 1.15551
\(267\) 0 0
\(268\) −1.41742e7 −0.0449806
\(269\) −3.45707e8 −1.08287 −0.541434 0.840743i \(-0.682118\pi\)
−0.541434 + 0.840743i \(0.682118\pi\)
\(270\) 0 0
\(271\) −3.36373e8 −1.02666 −0.513332 0.858190i \(-0.671589\pi\)
−0.513332 + 0.858190i \(0.671589\pi\)
\(272\) 3.73337e8 1.12489
\(273\) 0 0
\(274\) 1.29435e7 0.0380124
\(275\) 0 0
\(276\) 0 0
\(277\) 3.09678e8 0.875448 0.437724 0.899109i \(-0.355785\pi\)
0.437724 + 0.899109i \(0.355785\pi\)
\(278\) −3.99420e7 −0.111500
\(279\) 0 0
\(280\) 0 0
\(281\) 1.79386e8 0.482300 0.241150 0.970488i \(-0.422475\pi\)
0.241150 + 0.970488i \(0.422475\pi\)
\(282\) 0 0
\(283\) −4.18903e8 −1.09865 −0.549327 0.835608i \(-0.685116\pi\)
−0.549327 + 0.835608i \(0.685116\pi\)
\(284\) −8.09452e7 −0.209690
\(285\) 0 0
\(286\) −2.96270e8 −0.748870
\(287\) −5.65344e8 −1.41165
\(288\) 0 0
\(289\) 5.55006e8 1.35256
\(290\) 0 0
\(291\) 0 0
\(292\) −8.05641e7 −0.189366
\(293\) 3.86471e7 0.0897595 0.0448797 0.998992i \(-0.485710\pi\)
0.0448797 + 0.998992i \(0.485710\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.59584e7 0.0581778
\(297\) 0 0
\(298\) −5.27315e7 −0.115429
\(299\) −3.66257e8 −0.792386
\(300\) 0 0
\(301\) 1.65064e8 0.348874
\(302\) 6.87634e8 1.43659
\(303\) 0 0
\(304\) −3.64277e8 −0.743660
\(305\) 0 0
\(306\) 0 0
\(307\) −6.28695e8 −1.24010 −0.620048 0.784564i \(-0.712887\pi\)
−0.620048 + 0.784564i \(0.712887\pi\)
\(308\) 8.68140e7 0.169302
\(309\) 0 0
\(310\) 0 0
\(311\) −4.04278e8 −0.762112 −0.381056 0.924552i \(-0.624439\pi\)
−0.381056 + 0.924552i \(0.624439\pi\)
\(312\) 0 0
\(313\) −6.49782e8 −1.19774 −0.598870 0.800846i \(-0.704383\pi\)
−0.598870 + 0.800846i \(0.704383\pi\)
\(314\) −6.43181e8 −1.17241
\(315\) 0 0
\(316\) 1.60092e8 0.285406
\(317\) 1.64523e8 0.290082 0.145041 0.989426i \(-0.453669\pi\)
0.145041 + 0.989426i \(0.453669\pi\)
\(318\) 0 0
\(319\) −4.32347e8 −0.745702
\(320\) 0 0
\(321\) 0 0
\(322\) −3.83292e8 −0.639785
\(323\) −9.41918e8 −1.55526
\(324\) 0 0
\(325\) 0 0
\(326\) −3.10739e8 −0.496746
\(327\) 0 0
\(328\) 7.53792e8 1.17949
\(329\) 1.20791e8 0.187003
\(330\) 0 0
\(331\) 1.52532e7 0.0231187 0.0115593 0.999933i \(-0.496320\pi\)
0.0115593 + 0.999933i \(0.496320\pi\)
\(332\) 1.06131e8 0.159169
\(333\) 0 0
\(334\) −8.07466e8 −1.18580
\(335\) 0 0
\(336\) 0 0
\(337\) 1.17071e9 1.66627 0.833136 0.553068i \(-0.186543\pi\)
0.833136 + 0.553068i \(0.186543\pi\)
\(338\) 6.22439e8 0.876775
\(339\) 0 0
\(340\) 0 0
\(341\) −3.60898e8 −0.492884
\(342\) 0 0
\(343\) −3.25478e8 −0.435504
\(344\) −2.20085e8 −0.291498
\(345\) 0 0
\(346\) 7.88639e8 1.02356
\(347\) −8.86549e8 −1.13907 −0.569534 0.821968i \(-0.692876\pi\)
−0.569534 + 0.821968i \(0.692876\pi\)
\(348\) 0 0
\(349\) 9.30443e8 1.17166 0.585829 0.810435i \(-0.300769\pi\)
0.585829 + 0.810435i \(0.300769\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.10728e8 −0.257527
\(353\) 9.09069e8 1.09998 0.549991 0.835171i \(-0.314631\pi\)
0.549991 + 0.835171i \(0.314631\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.95806e8 −0.347482
\(357\) 0 0
\(358\) 1.25273e9 1.44300
\(359\) 1.07136e9 1.22209 0.611047 0.791594i \(-0.290749\pi\)
0.611047 + 0.791594i \(0.290749\pi\)
\(360\) 0 0
\(361\) 2.51881e7 0.0281787
\(362\) −5.95947e8 −0.660279
\(363\) 0 0
\(364\) −3.66257e8 −0.398044
\(365\) 0 0
\(366\) 0 0
\(367\) −4.38881e8 −0.463463 −0.231732 0.972780i \(-0.574439\pi\)
−0.231732 + 0.972780i \(0.574439\pi\)
\(368\) 3.93644e8 0.411753
\(369\) 0 0
\(370\) 0 0
\(371\) 2.28165e9 2.31975
\(372\) 0 0
\(373\) 1.86773e9 1.86352 0.931759 0.363077i \(-0.118274\pi\)
0.931759 + 0.363077i \(0.118274\pi\)
\(374\) 8.23355e8 0.813835
\(375\) 0 0
\(376\) −1.61054e8 −0.156248
\(377\) 1.82402e9 1.75321
\(378\) 0 0
\(379\) 1.59566e9 1.50558 0.752788 0.658263i \(-0.228708\pi\)
0.752788 + 0.658263i \(0.228708\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.40874e9 −1.29304
\(383\) −1.19991e9 −1.09132 −0.545661 0.838006i \(-0.683721\pi\)
−0.545661 + 0.838006i \(0.683721\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.43985e7 0.0481428
\(387\) 0 0
\(388\) 6.04276e7 0.0525200
\(389\) 9.25788e8 0.797421 0.398711 0.917077i \(-0.369458\pi\)
0.398711 + 0.917077i \(0.369458\pi\)
\(390\) 0 0
\(391\) 1.01785e9 0.861126
\(392\) −8.50757e8 −0.713353
\(393\) 0 0
\(394\) 9.05130e8 0.745546
\(395\) 0 0
\(396\) 0 0
\(397\) −9.87392e7 −0.0791995 −0.0395998 0.999216i \(-0.512608\pi\)
−0.0395998 + 0.999216i \(0.512608\pi\)
\(398\) −1.00226e9 −0.796872
\(399\) 0 0
\(400\) 0 0
\(401\) −1.88875e9 −1.46275 −0.731373 0.681977i \(-0.761120\pi\)
−0.731373 + 0.681977i \(0.761120\pi\)
\(402\) 0 0
\(403\) 1.52258e9 1.15881
\(404\) 1.40158e8 0.105751
\(405\) 0 0
\(406\) 1.90886e9 1.41557
\(407\) 4.40960e7 0.0324204
\(408\) 0 0
\(409\) 4.19515e8 0.303191 0.151595 0.988443i \(-0.451559\pi\)
0.151595 + 0.988443i \(0.451559\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6.14454e8 −0.432861
\(413\) 3.09272e9 2.16031
\(414\) 0 0
\(415\) 0 0
\(416\) 8.89034e8 0.605468
\(417\) 0 0
\(418\) −8.03374e8 −0.538023
\(419\) 8.54891e8 0.567756 0.283878 0.958860i \(-0.408379\pi\)
0.283878 + 0.958860i \(0.408379\pi\)
\(420\) 0 0
\(421\) 1.65667e9 1.08205 0.541026 0.841006i \(-0.318036\pi\)
0.541026 + 0.841006i \(0.318036\pi\)
\(422\) −4.46103e7 −0.0288962
\(423\) 0 0
\(424\) −3.04220e9 −1.93824
\(425\) 0 0
\(426\) 0 0
\(427\) 3.30048e9 2.05154
\(428\) 2.37204e8 0.146241
\(429\) 0 0
\(430\) 0 0
\(431\) −1.57827e9 −0.949536 −0.474768 0.880111i \(-0.657468\pi\)
−0.474768 + 0.880111i \(0.657468\pi\)
\(432\) 0 0
\(433\) 6.85271e8 0.405653 0.202827 0.979215i \(-0.434987\pi\)
0.202827 + 0.979215i \(0.434987\pi\)
\(434\) 1.59340e9 0.935644
\(435\) 0 0
\(436\) −4.70469e8 −0.271850
\(437\) −9.93152e8 −0.569286
\(438\) 0 0
\(439\) −7.80243e8 −0.440154 −0.220077 0.975483i \(-0.570631\pi\)
−0.220077 + 0.975483i \(0.570631\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.47363e9 −1.91340
\(443\) 7.41931e8 0.405462 0.202731 0.979234i \(-0.435018\pi\)
0.202731 + 0.979234i \(0.435018\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.13973e9 −1.67579
\(447\) 0 0
\(448\) 2.72990e9 1.43441
\(449\) −2.86391e9 −1.49313 −0.746565 0.665312i \(-0.768298\pi\)
−0.746565 + 0.665312i \(0.768298\pi\)
\(450\) 0 0
\(451\) 1.28048e9 0.657287
\(452\) 5.33220e8 0.271595
\(453\) 0 0
\(454\) 6.07635e8 0.304753
\(455\) 0 0
\(456\) 0 0
\(457\) 6.37820e8 0.312602 0.156301 0.987709i \(-0.450043\pi\)
0.156301 + 0.987709i \(0.450043\pi\)
\(458\) 6.48864e8 0.315591
\(459\) 0 0
\(460\) 0 0
\(461\) 2.31053e9 1.09839 0.549196 0.835693i \(-0.314934\pi\)
0.549196 + 0.835693i \(0.314934\pi\)
\(462\) 0 0
\(463\) −8.74079e8 −0.409277 −0.204638 0.978838i \(-0.565602\pi\)
−0.204638 + 0.978838i \(0.565602\pi\)
\(464\) −1.96041e9 −0.911033
\(465\) 0 0
\(466\) −1.22649e9 −0.561454
\(467\) 2.81303e9 1.27810 0.639051 0.769164i \(-0.279327\pi\)
0.639051 + 0.769164i \(0.279327\pi\)
\(468\) 0 0
\(469\) 5.92277e8 0.265106
\(470\) 0 0
\(471\) 0 0
\(472\) −4.12363e9 −1.80502
\(473\) −3.73862e8 −0.162442
\(474\) 0 0
\(475\) 0 0
\(476\) 1.01785e9 0.432574
\(477\) 0 0
\(478\) 3.16220e9 1.32431
\(479\) −1.85406e9 −0.770812 −0.385406 0.922747i \(-0.625939\pi\)
−0.385406 + 0.922747i \(0.625939\pi\)
\(480\) 0 0
\(481\) −1.86035e8 −0.0762232
\(482\) −1.37644e9 −0.559876
\(483\) 0 0
\(484\) 3.49011e8 0.139920
\(485\) 0 0
\(486\) 0 0
\(487\) 2.78028e9 1.09078 0.545390 0.838183i \(-0.316382\pi\)
0.545390 + 0.838183i \(0.316382\pi\)
\(488\) −4.40064e9 −1.71414
\(489\) 0 0
\(490\) 0 0
\(491\) 1.73954e9 0.663206 0.331603 0.943419i \(-0.392411\pi\)
0.331603 + 0.943419i \(0.392411\pi\)
\(492\) 0 0
\(493\) −5.06907e9 −1.90530
\(494\) 3.38933e9 1.26494
\(495\) 0 0
\(496\) −1.63644e9 −0.602161
\(497\) 3.38235e9 1.23587
\(498\) 0 0
\(499\) 5.39056e9 1.94215 0.971074 0.238778i \(-0.0767470\pi\)
0.971074 + 0.238778i \(0.0767470\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.28522e9 0.806243
\(503\) −6.05020e6 −0.00211974 −0.00105987 0.999999i \(-0.500337\pi\)
−0.00105987 + 0.999999i \(0.500337\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.68140e8 0.297895
\(507\) 0 0
\(508\) 3.31819e8 0.112300
\(509\) −4.04539e9 −1.35972 −0.679858 0.733344i \(-0.737959\pi\)
−0.679858 + 0.733344i \(0.737959\pi\)
\(510\) 0 0
\(511\) 3.36643e9 1.11608
\(512\) −3.35487e9 −1.10466
\(513\) 0 0
\(514\) −2.83909e9 −0.922163
\(515\) 0 0
\(516\) 0 0
\(517\) −2.73586e8 −0.0870717
\(518\) −1.94688e8 −0.0615439
\(519\) 0 0
\(520\) 0 0
\(521\) 1.47330e9 0.456413 0.228207 0.973613i \(-0.426714\pi\)
0.228207 + 0.973613i \(0.426714\pi\)
\(522\) 0 0
\(523\) 2.56401e9 0.783725 0.391863 0.920024i \(-0.371831\pi\)
0.391863 + 0.920024i \(0.371831\pi\)
\(524\) −5.80234e8 −0.176175
\(525\) 0 0
\(526\) −2.59380e9 −0.777117
\(527\) −4.23136e9 −1.25934
\(528\) 0 0
\(529\) −2.33161e9 −0.684795
\(530\) 0 0
\(531\) 0 0
\(532\) −9.93152e8 −0.285973
\(533\) −5.40218e9 −1.54534
\(534\) 0 0
\(535\) 0 0
\(536\) −7.89703e8 −0.221507
\(537\) 0 0
\(538\) −3.45707e9 −0.957129
\(539\) −1.44520e9 −0.397527
\(540\) 0 0
\(541\) −5.08476e9 −1.38064 −0.690320 0.723504i \(-0.742530\pi\)
−0.690320 + 0.723504i \(0.742530\pi\)
\(542\) −3.36373e9 −0.907451
\(543\) 0 0
\(544\) −2.47069e9 −0.657993
\(545\) 0 0
\(546\) 0 0
\(547\) 3.06578e9 0.800912 0.400456 0.916316i \(-0.368852\pi\)
0.400456 + 0.916316i \(0.368852\pi\)
\(548\) −3.62418e7 −0.00940758
\(549\) 0 0
\(550\) 0 0
\(551\) 4.94606e9 1.25959
\(552\) 0 0
\(553\) −6.68954e9 −1.68213
\(554\) 3.09678e9 0.773794
\(555\) 0 0
\(556\) 1.11838e8 0.0275948
\(557\) 2.13766e9 0.524137 0.262069 0.965049i \(-0.415595\pi\)
0.262069 + 0.965049i \(0.415595\pi\)
\(558\) 0 0
\(559\) 1.57727e9 0.381914
\(560\) 0 0
\(561\) 0 0
\(562\) 1.79386e9 0.426297
\(563\) −2.21110e9 −0.522190 −0.261095 0.965313i \(-0.584084\pi\)
−0.261095 + 0.965313i \(0.584084\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.18903e9 −0.971082
\(567\) 0 0
\(568\) −4.50980e9 −1.03262
\(569\) 5.70449e8 0.129815 0.0649074 0.997891i \(-0.479325\pi\)
0.0649074 + 0.997891i \(0.479325\pi\)
\(570\) 0 0
\(571\) −7.58930e9 −1.70599 −0.852993 0.521922i \(-0.825215\pi\)
−0.852993 + 0.521922i \(0.825215\pi\)
\(572\) 8.29556e8 0.185336
\(573\) 0 0
\(574\) −5.65344e9 −1.24773
\(575\) 0 0
\(576\) 0 0
\(577\) 1.22091e9 0.264587 0.132293 0.991211i \(-0.457766\pi\)
0.132293 + 0.991211i \(0.457766\pi\)
\(578\) 5.55006e9 1.19550
\(579\) 0 0
\(580\) 0 0
\(581\) −4.43474e9 −0.938107
\(582\) 0 0
\(583\) −5.16784e9 −1.08011
\(584\) −4.48857e9 −0.932531
\(585\) 0 0
\(586\) 3.86471e8 0.0793369
\(587\) 4.95204e9 1.01053 0.505267 0.862963i \(-0.331394\pi\)
0.505267 + 0.862963i \(0.331394\pi\)
\(588\) 0 0
\(589\) 4.12868e9 0.832544
\(590\) 0 0
\(591\) 0 0
\(592\) 1.99946e8 0.0396084
\(593\) 7.15564e8 0.140915 0.0704575 0.997515i \(-0.477554\pi\)
0.0704575 + 0.997515i \(0.477554\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.47648e8 0.0285671
\(597\) 0 0
\(598\) −3.66257e9 −0.700376
\(599\) 7.09937e9 1.34966 0.674832 0.737971i \(-0.264216\pi\)
0.674832 + 0.737971i \(0.264216\pi\)
\(600\) 0 0
\(601\) −1.34874e9 −0.253435 −0.126718 0.991939i \(-0.540444\pi\)
−0.126718 + 0.991939i \(0.540444\pi\)
\(602\) 1.65064e9 0.308364
\(603\) 0 0
\(604\) −1.92538e9 −0.355538
\(605\) 0 0
\(606\) 0 0
\(607\) 3.16618e9 0.574613 0.287307 0.957839i \(-0.407240\pi\)
0.287307 + 0.957839i \(0.407240\pi\)
\(608\) 2.41073e9 0.434996
\(609\) 0 0
\(610\) 0 0
\(611\) 1.15422e9 0.204713
\(612\) 0 0
\(613\) −1.83727e9 −0.322153 −0.161076 0.986942i \(-0.551497\pi\)
−0.161076 + 0.986942i \(0.551497\pi\)
\(614\) −6.28695e9 −1.09610
\(615\) 0 0
\(616\) 4.83678e9 0.833727
\(617\) −1.90555e9 −0.326605 −0.163302 0.986576i \(-0.552215\pi\)
−0.163302 + 0.986576i \(0.552215\pi\)
\(618\) 0 0
\(619\) 6.93437e9 1.17514 0.587570 0.809173i \(-0.300085\pi\)
0.587570 + 0.809173i \(0.300085\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4.04278e9 −0.673618
\(623\) 1.23605e10 2.04798
\(624\) 0 0
\(625\) 0 0
\(626\) −6.49782e9 −1.05866
\(627\) 0 0
\(628\) 1.80091e9 0.290157
\(629\) 5.17005e8 0.0828357
\(630\) 0 0
\(631\) −5.23914e9 −0.830152 −0.415076 0.909787i \(-0.636245\pi\)
−0.415076 + 0.909787i \(0.636245\pi\)
\(632\) 8.91939e9 1.40548
\(633\) 0 0
\(634\) 1.64523e9 0.256398
\(635\) 0 0
\(636\) 0 0
\(637\) 6.09709e9 0.934619
\(638\) −4.32348e9 −0.659114
\(639\) 0 0
\(640\) 0 0
\(641\) 3.00504e9 0.450657 0.225329 0.974283i \(-0.427654\pi\)
0.225329 + 0.974283i \(0.427654\pi\)
\(642\) 0 0
\(643\) 3.72219e9 0.552154 0.276077 0.961136i \(-0.410966\pi\)
0.276077 + 0.961136i \(0.410966\pi\)
\(644\) 1.07322e9 0.158339
\(645\) 0 0
\(646\) −9.41918e9 −1.37467
\(647\) −3.81203e9 −0.553340 −0.276670 0.960965i \(-0.589231\pi\)
−0.276670 + 0.960965i \(0.589231\pi\)
\(648\) 0 0
\(649\) −7.00488e9 −1.00588
\(650\) 0 0
\(651\) 0 0
\(652\) 8.70069e8 0.122938
\(653\) −4.15124e8 −0.0583420 −0.0291710 0.999574i \(-0.509287\pi\)
−0.0291710 + 0.999574i \(0.509287\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.80613e9 0.803015
\(657\) 0 0
\(658\) 1.20791e9 0.165289
\(659\) 5.54774e9 0.755122 0.377561 0.925985i \(-0.376763\pi\)
0.377561 + 0.925985i \(0.376763\pi\)
\(660\) 0 0
\(661\) −2.32765e9 −0.313482 −0.156741 0.987640i \(-0.550099\pi\)
−0.156741 + 0.987640i \(0.550099\pi\)
\(662\) 1.52532e8 0.0204342
\(663\) 0 0
\(664\) 5.91299e9 0.783825
\(665\) 0 0
\(666\) 0 0
\(667\) −5.34479e9 −0.697414
\(668\) 2.26090e9 0.293470
\(669\) 0 0
\(670\) 0 0
\(671\) −7.47544e9 −0.955230
\(672\) 0 0
\(673\) −6.28697e9 −0.795039 −0.397520 0.917594i \(-0.630129\pi\)
−0.397520 + 0.917594i \(0.630129\pi\)
\(674\) 1.17071e10 1.47279
\(675\) 0 0
\(676\) −1.74283e9 −0.216991
\(677\) −8.01998e9 −0.993374 −0.496687 0.867930i \(-0.665450\pi\)
−0.496687 + 0.867930i \(0.665450\pi\)
\(678\) 0 0
\(679\) −2.52501e9 −0.309542
\(680\) 0 0
\(681\) 0 0
\(682\) −3.60898e9 −0.435652
\(683\) −2.94863e9 −0.354118 −0.177059 0.984200i \(-0.556658\pi\)
−0.177059 + 0.984200i \(0.556658\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −3.25478e9 −0.384934
\(687\) 0 0
\(688\) −1.69522e9 −0.198457
\(689\) 2.18025e10 2.53944
\(690\) 0 0
\(691\) −5.55069e7 −0.00639991 −0.00319996 0.999995i \(-0.501019\pi\)
−0.00319996 + 0.999995i \(0.501019\pi\)
\(692\) −2.20819e9 −0.253317
\(693\) 0 0
\(694\) −8.86549e9 −1.00680
\(695\) 0 0
\(696\) 0 0
\(697\) 1.50130e10 1.67940
\(698\) 9.30443e9 1.03561
\(699\) 0 0
\(700\) 0 0
\(701\) 6.99424e9 0.766880 0.383440 0.923566i \(-0.374739\pi\)
0.383440 + 0.923566i \(0.374739\pi\)
\(702\) 0 0
\(703\) −5.04458e8 −0.0547623
\(704\) −6.18311e9 −0.667886
\(705\) 0 0
\(706\) 9.09069e9 0.972256
\(707\) −5.85661e9 −0.623273
\(708\) 0 0
\(709\) −6.04433e9 −0.636922 −0.318461 0.947936i \(-0.603166\pi\)
−0.318461 + 0.947936i \(0.603166\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.64806e10 −1.71117
\(713\) −4.46152e9 −0.460967
\(714\) 0 0
\(715\) 0 0
\(716\) −3.50763e9 −0.357123
\(717\) 0 0
\(718\) 1.07136e10 1.08019
\(719\) −1.82101e10 −1.82710 −0.913548 0.406731i \(-0.866669\pi\)
−0.913548 + 0.406731i \(0.866669\pi\)
\(720\) 0 0
\(721\) 2.56754e10 2.55119
\(722\) 2.51881e8 0.0249067
\(723\) 0 0
\(724\) 1.66865e9 0.163411
\(725\) 0 0
\(726\) 0 0
\(727\) −1.18107e10 −1.14000 −0.570001 0.821644i \(-0.693057\pi\)
−0.570001 + 0.821644i \(0.693057\pi\)
\(728\) −2.04057e10 −1.96016
\(729\) 0 0
\(730\) 0 0
\(731\) −4.38336e9 −0.415046
\(732\) 0 0
\(733\) 2.95438e7 0.00277078 0.00138539 0.999999i \(-0.499559\pi\)
0.00138539 + 0.999999i \(0.499559\pi\)
\(734\) −4.38881e9 −0.409647
\(735\) 0 0
\(736\) −2.60508e9 −0.240851
\(737\) −1.34148e9 −0.123438
\(738\) 0 0
\(739\) 3.04219e8 0.0277288 0.0138644 0.999904i \(-0.495587\pi\)
0.0138644 + 0.999904i \(0.495587\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.28165e10 2.05039
\(743\) −7.28471e9 −0.651555 −0.325778 0.945446i \(-0.605626\pi\)
−0.325778 + 0.945446i \(0.605626\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.86773e10 1.64713
\(747\) 0 0
\(748\) −2.30539e9 −0.201414
\(749\) −9.91175e9 −0.861914
\(750\) 0 0
\(751\) 1.35327e10 1.16586 0.582929 0.812523i \(-0.301907\pi\)
0.582929 + 0.812523i \(0.301907\pi\)
\(752\) −1.24053e9 −0.106376
\(753\) 0 0
\(754\) 1.82402e10 1.54963
\(755\) 0 0
\(756\) 0 0
\(757\) −1.47276e10 −1.23395 −0.616974 0.786983i \(-0.711642\pi\)
−0.616974 + 0.786983i \(0.711642\pi\)
\(758\) 1.59566e10 1.33075
\(759\) 0 0
\(760\) 0 0
\(761\) 2.37523e9 0.195370 0.0976852 0.995217i \(-0.468856\pi\)
0.0976852 + 0.995217i \(0.468856\pi\)
\(762\) 0 0
\(763\) 1.96589e10 1.60222
\(764\) 3.94449e9 0.320010
\(765\) 0 0
\(766\) −1.19991e10 −0.964602
\(767\) 2.95527e10 2.36490
\(768\) 0 0
\(769\) 1.41939e10 1.12554 0.562768 0.826615i \(-0.309737\pi\)
0.562768 + 0.826615i \(0.309737\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.52316e8 −0.0119147
\(773\) 7.82792e9 0.609562 0.304781 0.952422i \(-0.401417\pi\)
0.304781 + 0.952422i \(0.401417\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3.36668e9 0.258634
\(777\) 0 0
\(778\) 9.25788e9 0.704828
\(779\) −1.46487e10 −1.11024
\(780\) 0 0
\(781\) −7.66088e9 −0.575440
\(782\) 1.01785e10 0.761135
\(783\) 0 0
\(784\) −6.55301e9 −0.485663
\(785\) 0 0
\(786\) 0 0
\(787\) −2.14353e8 −0.0156754 −0.00783769 0.999969i \(-0.502495\pi\)
−0.00783769 + 0.999969i \(0.502495\pi\)
\(788\) −2.53436e9 −0.184513
\(789\) 0 0
\(790\) 0 0
\(791\) −2.22810e10 −1.60072
\(792\) 0 0
\(793\) 3.15379e10 2.24583
\(794\) −9.87392e8 −0.0700032
\(795\) 0 0
\(796\) 2.80632e9 0.197216
\(797\) −1.67153e10 −1.16952 −0.584762 0.811205i \(-0.698812\pi\)
−0.584762 + 0.811205i \(0.698812\pi\)
\(798\) 0 0
\(799\) −3.20767e9 −0.222472
\(800\) 0 0
\(801\) 0 0
\(802\) −1.88875e10 −1.29290
\(803\) −7.62482e9 −0.519667
\(804\) 0 0
\(805\) 0 0
\(806\) 1.52258e10 1.02425
\(807\) 0 0
\(808\) 7.80881e9 0.520769
\(809\) 1.13448e9 0.0753313 0.0376657 0.999290i \(-0.488008\pi\)
0.0376657 + 0.999290i \(0.488008\pi\)
\(810\) 0 0
\(811\) 7.87794e9 0.518609 0.259304 0.965796i \(-0.416507\pi\)
0.259304 + 0.965796i \(0.416507\pi\)
\(812\) −5.34479e9 −0.350336
\(813\) 0 0
\(814\) 4.40960e8 0.0286559
\(815\) 0 0
\(816\) 0 0
\(817\) 4.27698e9 0.274385
\(818\) 4.19515e9 0.267985
\(819\) 0 0
\(820\) 0 0
\(821\) −1.67119e10 −1.05396 −0.526981 0.849877i \(-0.676676\pi\)
−0.526981 + 0.849877i \(0.676676\pi\)
\(822\) 0 0
\(823\) −4.60135e8 −0.0287730 −0.0143865 0.999897i \(-0.504580\pi\)
−0.0143865 + 0.999897i \(0.504580\pi\)
\(824\) −3.42338e10 −2.13162
\(825\) 0 0
\(826\) 3.09272e10 1.90946
\(827\) −1.80809e10 −1.11160 −0.555802 0.831315i \(-0.687589\pi\)
−0.555802 + 0.831315i \(0.687589\pi\)
\(828\) 0 0
\(829\) −4.31759e9 −0.263209 −0.131604 0.991302i \(-0.542013\pi\)
−0.131604 + 0.991302i \(0.542013\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.60857e10 1.57026
\(833\) −1.69442e10 −1.01570
\(834\) 0 0
\(835\) 0 0
\(836\) 2.24945e9 0.133154
\(837\) 0 0
\(838\) 8.54891e9 0.501830
\(839\) −1.47269e10 −0.860882 −0.430441 0.902619i \(-0.641642\pi\)
−0.430441 + 0.902619i \(0.641642\pi\)
\(840\) 0 0
\(841\) 9.36805e9 0.543079
\(842\) 1.65667e10 0.956408
\(843\) 0 0
\(844\) 1.24909e8 0.00715146
\(845\) 0 0
\(846\) 0 0
\(847\) −1.45837e10 −0.824660
\(848\) −2.34328e10 −1.31959
\(849\) 0 0
\(850\) 0 0
\(851\) 5.45126e8 0.0303210
\(852\) 0 0
\(853\) −1.33675e10 −0.737444 −0.368722 0.929540i \(-0.620204\pi\)
−0.368722 + 0.929540i \(0.620204\pi\)
\(854\) 3.30048e10 1.81332
\(855\) 0 0
\(856\) 1.32157e10 0.720163
\(857\) −9.72498e9 −0.527784 −0.263892 0.964552i \(-0.585006\pi\)
−0.263892 + 0.964552i \(0.585006\pi\)
\(858\) 0 0
\(859\) −7.19642e8 −0.0387383 −0.0193691 0.999812i \(-0.506166\pi\)
−0.0193691 + 0.999812i \(0.506166\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.57827e10 −0.839279
\(863\) 1.73662e10 0.919743 0.459871 0.887985i \(-0.347895\pi\)
0.459871 + 0.887985i \(0.347895\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 6.85271e9 0.358550
\(867\) 0 0
\(868\) −4.46152e9 −0.231560
\(869\) 1.51515e10 0.783226
\(870\) 0 0
\(871\) 5.65954e9 0.290213
\(872\) −2.62119e10 −1.33872
\(873\) 0 0
\(874\) −9.93152e9 −0.503183
\(875\) 0 0
\(876\) 0 0
\(877\) −2.97495e10 −1.48930 −0.744649 0.667456i \(-0.767383\pi\)
−0.744649 + 0.667456i \(0.767383\pi\)
\(878\) −7.80243e9 −0.389044
\(879\) 0 0
\(880\) 0 0
\(881\) 6.89920e9 0.339925 0.169962 0.985451i \(-0.445635\pi\)
0.169962 + 0.985451i \(0.445635\pi\)
\(882\) 0 0
\(883\) 8.12764e9 0.397285 0.198643 0.980072i \(-0.436347\pi\)
0.198643 + 0.980072i \(0.436347\pi\)
\(884\) 9.72615e9 0.473542
\(885\) 0 0
\(886\) 7.41931e9 0.358382
\(887\) 1.19176e10 0.573397 0.286698 0.958021i \(-0.407442\pi\)
0.286698 + 0.958021i \(0.407442\pi\)
\(888\) 0 0
\(889\) −1.38653e10 −0.661870
\(890\) 0 0
\(891\) 0 0
\(892\) 8.79125e9 0.414738
\(893\) 3.12982e9 0.147075
\(894\) 0 0
\(895\) 0 0
\(896\) 1.53901e10 0.714765
\(897\) 0 0
\(898\) −2.86391e10 −1.31975
\(899\) 2.22191e10 1.01992
\(900\) 0 0
\(901\) −6.05905e10 −2.75974
\(902\) 1.28048e10 0.580965
\(903\) 0 0
\(904\) 2.97080e10 1.33747
\(905\) 0 0
\(906\) 0 0
\(907\) 3.67453e9 0.163522 0.0817611 0.996652i \(-0.473946\pi\)
0.0817611 + 0.996652i \(0.473946\pi\)
\(908\) −1.70138e9 −0.0754224
\(909\) 0 0
\(910\) 0 0
\(911\) 2.49328e9 0.109259 0.0546294 0.998507i \(-0.482602\pi\)
0.0546294 + 0.998507i \(0.482602\pi\)
\(912\) 0 0
\(913\) 1.00445e10 0.436798
\(914\) 6.37820e9 0.276304
\(915\) 0 0
\(916\) −1.81682e9 −0.0781048
\(917\) 2.42455e10 1.03834
\(918\) 0 0
\(919\) −4.51548e10 −1.91911 −0.959554 0.281526i \(-0.909160\pi\)
−0.959554 + 0.281526i \(0.909160\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 2.31053e10 0.970851
\(923\) 3.23203e10 1.35291
\(924\) 0 0
\(925\) 0 0
\(926\) −8.74079e9 −0.361753
\(927\) 0 0
\(928\) 1.29737e10 0.532900
\(929\) 1.26182e10 0.516346 0.258173 0.966099i \(-0.416880\pi\)
0.258173 + 0.966099i \(0.416880\pi\)
\(930\) 0 0
\(931\) 1.65330e10 0.671474
\(932\) 3.43418e9 0.138953
\(933\) 0 0
\(934\) 2.81303e10 1.12969
\(935\) 0 0
\(936\) 0 0
\(937\) −2.30706e10 −0.916157 −0.458079 0.888912i \(-0.651462\pi\)
−0.458079 + 0.888912i \(0.651462\pi\)
\(938\) 5.92277e9 0.234323
\(939\) 0 0
\(940\) 0 0
\(941\) 2.95344e10 1.15549 0.577743 0.816218i \(-0.303933\pi\)
0.577743 + 0.816218i \(0.303933\pi\)
\(942\) 0 0
\(943\) 1.58296e10 0.614724
\(944\) −3.17625e10 −1.22889
\(945\) 0 0
\(946\) −3.73862e9 −0.143579
\(947\) −3.33791e10 −1.27717 −0.638586 0.769551i \(-0.720480\pi\)
−0.638586 + 0.769551i \(0.720480\pi\)
\(948\) 0 0
\(949\) 3.21681e10 1.22178
\(950\) 0 0
\(951\) 0 0
\(952\) 5.67090e10 2.13021
\(953\) 5.59328e9 0.209335 0.104667 0.994507i \(-0.466622\pi\)
0.104667 + 0.994507i \(0.466622\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.85415e9 −0.327751
\(957\) 0 0
\(958\) −1.85406e10 −0.681308
\(959\) 1.51439e9 0.0554463
\(960\) 0 0
\(961\) −8.96544e9 −0.325867
\(962\) −1.86035e9 −0.0673725
\(963\) 0 0
\(964\) 3.85403e9 0.138562
\(965\) 0 0
\(966\) 0 0
\(967\) 6.61612e9 0.235294 0.117647 0.993055i \(-0.462465\pi\)
0.117647 + 0.993055i \(0.462465\pi\)
\(968\) 1.94449e10 0.689036
\(969\) 0 0
\(970\) 0 0
\(971\) −5.48042e10 −1.92108 −0.960542 0.278136i \(-0.910284\pi\)
−0.960542 + 0.278136i \(0.910284\pi\)
\(972\) 0 0
\(973\) −4.67322e9 −0.162638
\(974\) 2.78028e10 0.964122
\(975\) 0 0
\(976\) −3.38962e10 −1.16702
\(977\) −2.89446e9 −0.0992971 −0.0496485 0.998767i \(-0.515810\pi\)
−0.0496485 + 0.998767i \(0.515810\pi\)
\(978\) 0 0
\(979\) −2.79959e10 −0.953576
\(980\) 0 0
\(981\) 0 0
\(982\) 1.73954e10 0.586196
\(983\) −7.15494e9 −0.240253 −0.120127 0.992759i \(-0.538330\pi\)
−0.120127 + 0.992759i \(0.538330\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.06907e10 −1.68407
\(987\) 0 0
\(988\) −9.49012e9 −0.313056
\(989\) −4.62178e9 −0.151923
\(990\) 0 0
\(991\) −2.07992e9 −0.0678872 −0.0339436 0.999424i \(-0.510807\pi\)
−0.0339436 + 0.999424i \(0.510807\pi\)
\(992\) 1.08297e10 0.352228
\(993\) 0 0
\(994\) 3.38235e10 1.09236
\(995\) 0 0
\(996\) 0 0
\(997\) −9.10829e9 −0.291074 −0.145537 0.989353i \(-0.546491\pi\)
−0.145537 + 0.989353i \(0.546491\pi\)
\(998\) 5.39056e10 1.71663
\(999\) 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 225.8.a.j.1.1 1
3.2 odd 2 225.8.a.d.1.1 1
5.2 odd 4 225.8.b.d.199.2 2
5.3 odd 4 225.8.b.d.199.1 2
5.4 even 2 45.8.a.a.1.1 1
15.2 even 4 225.8.b.e.199.1 2
15.8 even 4 225.8.b.e.199.2 2
15.14 odd 2 45.8.a.d.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.8.a.a.1.1 1 5.4 even 2
45.8.a.d.1.1 yes 1 15.14 odd 2
225.8.a.d.1.1 1 3.2 odd 2
225.8.a.j.1.1 1 1.1 even 1 trivial
225.8.b.d.199.1 2 5.3 odd 4
225.8.b.d.199.2 2 5.2 odd 4
225.8.b.e.199.1 2 15.2 even 4
225.8.b.e.199.2 2 15.8 even 4