Properties

Label 225.8.a.h.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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{2} -103.000 q^{4} -930.000 q^{7} -1155.00 q^{8} +O(q^{10})\) \(q+5.00000 q^{2} -103.000 q^{4} -930.000 q^{7} -1155.00 q^{8} -8450.00 q^{11} -6220.00 q^{13} -4650.00 q^{14} +7409.00 q^{16} +9590.00 q^{17} -45884.0 q^{19} -42250.0 q^{22} +102120. q^{23} -31100.0 q^{26} +95790.0 q^{28} -87550.0 q^{29} -76212.0 q^{31} +184885. q^{32} +47950.0 q^{34} -264440. q^{37} -229420. q^{38} -103600. q^{41} +324680. q^{43} +870350. q^{44} +510600. q^{46} -855880. q^{47} +41357.0 q^{49} +640660. q^{52} +958190. q^{53} +1.07415e6 q^{56} -437750. q^{58} +1.23955e6 q^{59} +628522. q^{61} -381060. q^{62} -23927.0 q^{64} -310380. q^{67} -987770. q^{68} -3.93430e6 q^{71} +4.55609e6 q^{73} -1.32220e6 q^{74} +4.72605e6 q^{76} +7.85850e6 q^{77} +5.37164e6 q^{79} -518000. q^{82} +6.71106e6 q^{83} +1.62340e6 q^{86} +9.75975e6 q^{88} -3.34650e6 q^{89} +5.78460e6 q^{91} -1.05184e7 q^{92} -4.27940e6 q^{94} -1.58297e7 q^{97} +206785. q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.00000 0.441942 0.220971 0.975280i \(-0.429077\pi\)
0.220971 + 0.975280i \(0.429077\pi\)
\(3\) 0 0
\(4\) −103.000 −0.804688
\(5\) 0 0
\(6\) 0 0
\(7\) −930.000 −1.02480 −0.512401 0.858746i \(-0.671244\pi\)
−0.512401 + 0.858746i \(0.671244\pi\)
\(8\) −1155.00 −0.797567
\(9\) 0 0
\(10\) 0 0
\(11\) −8450.00 −1.91418 −0.957089 0.289794i \(-0.906413\pi\)
−0.957089 + 0.289794i \(0.906413\pi\)
\(12\) 0 0
\(13\) −6220.00 −0.785215 −0.392608 0.919706i \(-0.628427\pi\)
−0.392608 + 0.919706i \(0.628427\pi\)
\(14\) −4650.00 −0.452903
\(15\) 0 0
\(16\) 7409.00 0.452209
\(17\) 9590.00 0.473421 0.236710 0.971580i \(-0.423931\pi\)
0.236710 + 0.971580i \(0.423931\pi\)
\(18\) 0 0
\(19\) −45884.0 −1.53470 −0.767350 0.641228i \(-0.778425\pi\)
−0.767350 + 0.641228i \(0.778425\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −42250.0 −0.845955
\(23\) 102120. 1.75010 0.875051 0.484031i \(-0.160828\pi\)
0.875051 + 0.484031i \(0.160828\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −31100.0 −0.347019
\(27\) 0 0
\(28\) 95790.0 0.824645
\(29\) −87550.0 −0.666597 −0.333298 0.942821i \(-0.608162\pi\)
−0.333298 + 0.942821i \(0.608162\pi\)
\(30\) 0 0
\(31\) −76212.0 −0.459470 −0.229735 0.973253i \(-0.573786\pi\)
−0.229735 + 0.973253i \(0.573786\pi\)
\(32\) 184885. 0.997417
\(33\) 0 0
\(34\) 47950.0 0.209224
\(35\) 0 0
\(36\) 0 0
\(37\) −264440. −0.858264 −0.429132 0.903242i \(-0.641181\pi\)
−0.429132 + 0.903242i \(0.641181\pi\)
\(38\) −229420. −0.678248
\(39\) 0 0
\(40\) 0 0
\(41\) −103600. −0.234756 −0.117378 0.993087i \(-0.537449\pi\)
−0.117378 + 0.993087i \(0.537449\pi\)
\(42\) 0 0
\(43\) 324680. 0.622753 0.311377 0.950287i \(-0.399210\pi\)
0.311377 + 0.950287i \(0.399210\pi\)
\(44\) 870350. 1.54032
\(45\) 0 0
\(46\) 510600. 0.773443
\(47\) −855880. −1.20246 −0.601230 0.799076i \(-0.705322\pi\)
−0.601230 + 0.799076i \(0.705322\pi\)
\(48\) 0 0
\(49\) 41357.0 0.0502184
\(50\) 0 0
\(51\) 0 0
\(52\) 640660. 0.631853
\(53\) 958190. 0.884069 0.442034 0.896998i \(-0.354257\pi\)
0.442034 + 0.896998i \(0.354257\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.07415e6 0.817348
\(57\) 0 0
\(58\) −437750. −0.294597
\(59\) 1.23955e6 0.785746 0.392873 0.919593i \(-0.371481\pi\)
0.392873 + 0.919593i \(0.371481\pi\)
\(60\) 0 0
\(61\) 628522. 0.354541 0.177270 0.984162i \(-0.443273\pi\)
0.177270 + 0.984162i \(0.443273\pi\)
\(62\) −381060. −0.203059
\(63\) 0 0
\(64\) −23927.0 −0.0114093
\(65\) 0 0
\(66\) 0 0
\(67\) −310380. −0.126076 −0.0630379 0.998011i \(-0.520079\pi\)
−0.0630379 + 0.998011i \(0.520079\pi\)
\(68\) −987770. −0.380956
\(69\) 0 0
\(70\) 0 0
\(71\) −3.93430e6 −1.30456 −0.652279 0.757979i \(-0.726187\pi\)
−0.652279 + 0.757979i \(0.726187\pi\)
\(72\) 0 0
\(73\) 4.55609e6 1.37076 0.685381 0.728184i \(-0.259636\pi\)
0.685381 + 0.728184i \(0.259636\pi\)
\(74\) −1.32220e6 −0.379303
\(75\) 0 0
\(76\) 4.72605e6 1.23495
\(77\) 7.85850e6 1.96165
\(78\) 0 0
\(79\) 5.37164e6 1.22578 0.612890 0.790168i \(-0.290007\pi\)
0.612890 + 0.790168i \(0.290007\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −518000. −0.103748
\(83\) 6.71106e6 1.28830 0.644151 0.764898i \(-0.277211\pi\)
0.644151 + 0.764898i \(0.277211\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.62340e6 0.275221
\(87\) 0 0
\(88\) 9.75975e6 1.52668
\(89\) −3.34650e6 −0.503183 −0.251591 0.967834i \(-0.580954\pi\)
−0.251591 + 0.967834i \(0.580954\pi\)
\(90\) 0 0
\(91\) 5.78460e6 0.804690
\(92\) −1.05184e7 −1.40829
\(93\) 0 0
\(94\) −4.27940e6 −0.531417
\(95\) 0 0
\(96\) 0 0
\(97\) −1.58297e7 −1.76105 −0.880527 0.473997i \(-0.842811\pi\)
−0.880527 + 0.473997i \(0.842811\pi\)
\(98\) 206785. 0.0221936
\(99\) 0 0
\(100\) 0 0
\(101\) −1.51995e6 −0.146793 −0.0733964 0.997303i \(-0.523384\pi\)
−0.0733964 + 0.997303i \(0.523384\pi\)
\(102\) 0 0
\(103\) 1.98367e7 1.78870 0.894352 0.447364i \(-0.147637\pi\)
0.894352 + 0.447364i \(0.147637\pi\)
\(104\) 7.18410e6 0.626261
\(105\) 0 0
\(106\) 4.79095e6 0.390707
\(107\) 413460. 0.0326280 0.0163140 0.999867i \(-0.494807\pi\)
0.0163140 + 0.999867i \(0.494807\pi\)
\(108\) 0 0
\(109\) −1.24127e7 −0.918067 −0.459034 0.888419i \(-0.651804\pi\)
−0.459034 + 0.888419i \(0.651804\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −6.89037e6 −0.463425
\(113\) −2.42499e7 −1.58101 −0.790507 0.612453i \(-0.790183\pi\)
−0.790507 + 0.612453i \(0.790183\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.01765e6 0.536402
\(117\) 0 0
\(118\) 6.19775e6 0.347254
\(119\) −8.91870e6 −0.485162
\(120\) 0 0
\(121\) 5.19153e7 2.66408
\(122\) 3.14261e6 0.156686
\(123\) 0 0
\(124\) 7.84984e6 0.369730
\(125\) 0 0
\(126\) 0 0
\(127\) −259570. −0.0112445 −0.00562227 0.999984i \(-0.501790\pi\)
−0.00562227 + 0.999984i \(0.501790\pi\)
\(128\) −2.37849e7 −1.00246
\(129\) 0 0
\(130\) 0 0
\(131\) 1.41784e7 0.551035 0.275518 0.961296i \(-0.411151\pi\)
0.275518 + 0.961296i \(0.411151\pi\)
\(132\) 0 0
\(133\) 4.26721e7 1.57276
\(134\) −1.55190e6 −0.0557182
\(135\) 0 0
\(136\) −1.10764e7 −0.377585
\(137\) −2.49710e7 −0.829685 −0.414842 0.909893i \(-0.636163\pi\)
−0.414842 + 0.909893i \(0.636163\pi\)
\(138\) 0 0
\(139\) 3.31920e7 1.04829 0.524145 0.851629i \(-0.324385\pi\)
0.524145 + 0.851629i \(0.324385\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.96715e7 −0.576538
\(143\) 5.25590e7 1.50304
\(144\) 0 0
\(145\) 0 0
\(146\) 2.27804e7 0.605797
\(147\) 0 0
\(148\) 2.72373e7 0.690635
\(149\) −3.85424e7 −0.954525 −0.477263 0.878761i \(-0.658371\pi\)
−0.477263 + 0.878761i \(0.658371\pi\)
\(150\) 0 0
\(151\) 2.34988e7 0.555427 0.277714 0.960664i \(-0.410423\pi\)
0.277714 + 0.960664i \(0.410423\pi\)
\(152\) 5.29960e7 1.22403
\(153\) 0 0
\(154\) 3.92925e7 0.866936
\(155\) 0 0
\(156\) 0 0
\(157\) 5.97169e7 1.23154 0.615770 0.787926i \(-0.288845\pi\)
0.615770 + 0.787926i \(0.288845\pi\)
\(158\) 2.68582e7 0.541723
\(159\) 0 0
\(160\) 0 0
\(161\) −9.49716e7 −1.79351
\(162\) 0 0
\(163\) 2.81491e7 0.509106 0.254553 0.967059i \(-0.418072\pi\)
0.254553 + 0.967059i \(0.418072\pi\)
\(164\) 1.06708e7 0.188905
\(165\) 0 0
\(166\) 3.35553e7 0.569355
\(167\) −2.22233e7 −0.369233 −0.184617 0.982811i \(-0.559104\pi\)
−0.184617 + 0.982811i \(0.559104\pi\)
\(168\) 0 0
\(169\) −2.40601e7 −0.383437
\(170\) 0 0
\(171\) 0 0
\(172\) −3.34420e7 −0.501122
\(173\) −1.00047e8 −1.46908 −0.734539 0.678567i \(-0.762601\pi\)
−0.734539 + 0.678567i \(0.762601\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.26060e7 −0.865609
\(177\) 0 0
\(178\) −1.67325e7 −0.222377
\(179\) 4.19711e7 0.546972 0.273486 0.961876i \(-0.411823\pi\)
0.273486 + 0.961876i \(0.411823\pi\)
\(180\) 0 0
\(181\) −1.22283e7 −0.153281 −0.0766407 0.997059i \(-0.524419\pi\)
−0.0766407 + 0.997059i \(0.524419\pi\)
\(182\) 2.89230e7 0.355626
\(183\) 0 0
\(184\) −1.17949e8 −1.39582
\(185\) 0 0
\(186\) 0 0
\(187\) −8.10355e7 −0.906212
\(188\) 8.81556e7 0.967604
\(189\) 0 0
\(190\) 0 0
\(191\) 1.56531e8 1.62549 0.812746 0.582618i \(-0.197972\pi\)
0.812746 + 0.582618i \(0.197972\pi\)
\(192\) 0 0
\(193\) −1.08056e8 −1.08193 −0.540965 0.841045i \(-0.681941\pi\)
−0.540965 + 0.841045i \(0.681941\pi\)
\(194\) −7.91486e7 −0.778283
\(195\) 0 0
\(196\) −4.25977e6 −0.0404101
\(197\) −1.95971e7 −0.182625 −0.0913125 0.995822i \(-0.529106\pi\)
−0.0913125 + 0.995822i \(0.529106\pi\)
\(198\) 0 0
\(199\) 1.22581e7 0.110265 0.0551323 0.998479i \(-0.482442\pi\)
0.0551323 + 0.998479i \(0.482442\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −7.59975e6 −0.0648738
\(203\) 8.14215e7 0.683129
\(204\) 0 0
\(205\) 0 0
\(206\) 9.91834e7 0.790503
\(207\) 0 0
\(208\) −4.60840e7 −0.355082
\(209\) 3.87720e8 2.93769
\(210\) 0 0
\(211\) −1.89778e8 −1.39077 −0.695387 0.718636i \(-0.744767\pi\)
−0.695387 + 0.718636i \(0.744767\pi\)
\(212\) −9.86936e7 −0.711399
\(213\) 0 0
\(214\) 2.06730e6 0.0144197
\(215\) 0 0
\(216\) 0 0
\(217\) 7.08772e7 0.470866
\(218\) −6.20636e7 −0.405732
\(219\) 0 0
\(220\) 0 0
\(221\) −5.96498e7 −0.371737
\(222\) 0 0
\(223\) −1.09802e7 −0.0663044 −0.0331522 0.999450i \(-0.510555\pi\)
−0.0331522 + 0.999450i \(0.510555\pi\)
\(224\) −1.71943e8 −1.02215
\(225\) 0 0
\(226\) −1.21250e8 −0.698716
\(227\) −5.04802e7 −0.286438 −0.143219 0.989691i \(-0.545745\pi\)
−0.143219 + 0.989691i \(0.545745\pi\)
\(228\) 0 0
\(229\) 6.18516e7 0.340351 0.170175 0.985414i \(-0.445567\pi\)
0.170175 + 0.985414i \(0.445567\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.01120e8 0.531655
\(233\) −1.37871e8 −0.714050 −0.357025 0.934095i \(-0.616209\pi\)
−0.357025 + 0.934095i \(0.616209\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.27674e8 −0.632280
\(237\) 0 0
\(238\) −4.45935e7 −0.214414
\(239\) −6.27719e7 −0.297422 −0.148711 0.988881i \(-0.547512\pi\)
−0.148711 + 0.988881i \(0.547512\pi\)
\(240\) 0 0
\(241\) −1.37149e8 −0.631153 −0.315576 0.948900i \(-0.602198\pi\)
−0.315576 + 0.948900i \(0.602198\pi\)
\(242\) 2.59577e8 1.17737
\(243\) 0 0
\(244\) −6.47378e7 −0.285294
\(245\) 0 0
\(246\) 0 0
\(247\) 2.85398e8 1.20507
\(248\) 8.80249e7 0.366458
\(249\) 0 0
\(250\) 0 0
\(251\) −4.05726e8 −1.61948 −0.809739 0.586790i \(-0.800391\pi\)
−0.809739 + 0.586790i \(0.800391\pi\)
\(252\) 0 0
\(253\) −8.62914e8 −3.35001
\(254\) −1.29785e6 −0.00496943
\(255\) 0 0
\(256\) −1.15862e8 −0.431619
\(257\) −9.00525e7 −0.330925 −0.165463 0.986216i \(-0.552912\pi\)
−0.165463 + 0.986216i \(0.552912\pi\)
\(258\) 0 0
\(259\) 2.45929e8 0.879551
\(260\) 0 0
\(261\) 0 0
\(262\) 7.08922e7 0.243525
\(263\) 2.06256e8 0.699135 0.349568 0.936911i \(-0.386328\pi\)
0.349568 + 0.936911i \(0.386328\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.13361e8 0.695070
\(267\) 0 0
\(268\) 3.19691e7 0.101452
\(269\) −4.91222e8 −1.53867 −0.769334 0.638847i \(-0.779412\pi\)
−0.769334 + 0.638847i \(0.779412\pi\)
\(270\) 0 0
\(271\) −3.69529e8 −1.12786 −0.563931 0.825822i \(-0.690712\pi\)
−0.563931 + 0.825822i \(0.690712\pi\)
\(272\) 7.10523e7 0.214085
\(273\) 0 0
\(274\) −1.24855e8 −0.366672
\(275\) 0 0
\(276\) 0 0
\(277\) 2.31059e8 0.653196 0.326598 0.945163i \(-0.394098\pi\)
0.326598 + 0.945163i \(0.394098\pi\)
\(278\) 1.65960e8 0.463283
\(279\) 0 0
\(280\) 0 0
\(281\) 5.73019e8 1.54063 0.770313 0.637666i \(-0.220100\pi\)
0.770313 + 0.637666i \(0.220100\pi\)
\(282\) 0 0
\(283\) −3.64618e8 −0.956281 −0.478140 0.878283i \(-0.658689\pi\)
−0.478140 + 0.878283i \(0.658689\pi\)
\(284\) 4.05233e8 1.04976
\(285\) 0 0
\(286\) 2.62795e8 0.664257
\(287\) 9.63480e7 0.240578
\(288\) 0 0
\(289\) −3.18371e8 −0.775873
\(290\) 0 0
\(291\) 0 0
\(292\) −4.69277e8 −1.10304
\(293\) −2.12858e8 −0.494372 −0.247186 0.968968i \(-0.579506\pi\)
−0.247186 + 0.968968i \(0.579506\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.05428e8 0.684523
\(297\) 0 0
\(298\) −1.92712e8 −0.421845
\(299\) −6.35186e8 −1.37421
\(300\) 0 0
\(301\) −3.01952e8 −0.638198
\(302\) 1.17494e8 0.245467
\(303\) 0 0
\(304\) −3.39955e8 −0.694006
\(305\) 0 0
\(306\) 0 0
\(307\) 6.55586e8 1.29314 0.646570 0.762855i \(-0.276203\pi\)
0.646570 + 0.762855i \(0.276203\pi\)
\(308\) −8.09426e8 −1.57852
\(309\) 0 0
\(310\) 0 0
\(311\) −9.17157e8 −1.72895 −0.864474 0.502677i \(-0.832349\pi\)
−0.864474 + 0.502677i \(0.832349\pi\)
\(312\) 0 0
\(313\) 6.52969e8 1.20361 0.601807 0.798641i \(-0.294447\pi\)
0.601807 + 0.798641i \(0.294447\pi\)
\(314\) 2.98584e8 0.544269
\(315\) 0 0
\(316\) −5.53279e8 −0.986370
\(317\) −7.34479e8 −1.29501 −0.647503 0.762063i \(-0.724187\pi\)
−0.647503 + 0.762063i \(0.724187\pi\)
\(318\) 0 0
\(319\) 7.39797e8 1.27598
\(320\) 0 0
\(321\) 0 0
\(322\) −4.74858e8 −0.792626
\(323\) −4.40028e8 −0.726559
\(324\) 0 0
\(325\) 0 0
\(326\) 1.40746e8 0.224995
\(327\) 0 0
\(328\) 1.19658e8 0.187233
\(329\) 7.95968e8 1.23228
\(330\) 0 0
\(331\) 8.79164e8 1.33251 0.666257 0.745722i \(-0.267895\pi\)
0.666257 + 0.745722i \(0.267895\pi\)
\(332\) −6.91239e8 −1.03668
\(333\) 0 0
\(334\) −1.11116e8 −0.163179
\(335\) 0 0
\(336\) 0 0
\(337\) −3.65182e8 −0.519762 −0.259881 0.965641i \(-0.583683\pi\)
−0.259881 + 0.965641i \(0.583683\pi\)
\(338\) −1.20301e8 −0.169457
\(339\) 0 0
\(340\) 0 0
\(341\) 6.43991e8 0.879508
\(342\) 0 0
\(343\) 7.27433e8 0.973338
\(344\) −3.75005e8 −0.496687
\(345\) 0 0
\(346\) −5.00237e8 −0.649247
\(347\) 1.03136e9 1.32513 0.662563 0.749006i \(-0.269469\pi\)
0.662563 + 0.749006i \(0.269469\pi\)
\(348\) 0 0
\(349\) −1.24452e8 −0.156716 −0.0783579 0.996925i \(-0.524968\pi\)
−0.0783579 + 0.996925i \(0.524968\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.56228e9 −1.90923
\(353\) 1.32311e9 1.60097 0.800487 0.599350i \(-0.204574\pi\)
0.800487 + 0.599350i \(0.204574\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3.44690e8 0.404905
\(357\) 0 0
\(358\) 2.09856e8 0.241730
\(359\) −1.18844e8 −0.135564 −0.0677822 0.997700i \(-0.521592\pi\)
−0.0677822 + 0.997700i \(0.521592\pi\)
\(360\) 0 0
\(361\) 1.21147e9 1.35531
\(362\) −6.11413e7 −0.0677415
\(363\) 0 0
\(364\) −5.95814e8 −0.647524
\(365\) 0 0
\(366\) 0 0
\(367\) 1.10304e9 1.16482 0.582412 0.812893i \(-0.302109\pi\)
0.582412 + 0.812893i \(0.302109\pi\)
\(368\) 7.56607e8 0.791413
\(369\) 0 0
\(370\) 0 0
\(371\) −8.91117e8 −0.905995
\(372\) 0 0
\(373\) −4.53965e8 −0.452941 −0.226470 0.974018i \(-0.572719\pi\)
−0.226470 + 0.974018i \(0.572719\pi\)
\(374\) −4.05178e8 −0.400493
\(375\) 0 0
\(376\) 9.88541e8 0.959042
\(377\) 5.44561e8 0.523422
\(378\) 0 0
\(379\) 5.28520e8 0.498683 0.249341 0.968416i \(-0.419786\pi\)
0.249341 + 0.968416i \(0.419786\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 7.82658e8 0.718373
\(383\) 3.03673e8 0.276192 0.138096 0.990419i \(-0.455902\pi\)
0.138096 + 0.990419i \(0.455902\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.40281e8 −0.478150
\(387\) 0 0
\(388\) 1.63046e9 1.41710
\(389\) 2.38311e8 0.205267 0.102634 0.994719i \(-0.467273\pi\)
0.102634 + 0.994719i \(0.467273\pi\)
\(390\) 0 0
\(391\) 9.79331e8 0.828535
\(392\) −4.77673e7 −0.0400525
\(393\) 0 0
\(394\) −9.79856e7 −0.0807096
\(395\) 0 0
\(396\) 0 0
\(397\) 7.12127e8 0.571203 0.285601 0.958348i \(-0.407807\pi\)
0.285601 + 0.958348i \(0.407807\pi\)
\(398\) 6.12903e7 0.0487305
\(399\) 0 0
\(400\) 0 0
\(401\) −1.42308e8 −0.110211 −0.0551053 0.998481i \(-0.517549\pi\)
−0.0551053 + 0.998481i \(0.517549\pi\)
\(402\) 0 0
\(403\) 4.74039e8 0.360783
\(404\) 1.56555e8 0.118122
\(405\) 0 0
\(406\) 4.07108e8 0.301903
\(407\) 2.23452e9 1.64287
\(408\) 0 0
\(409\) 1.81943e9 1.31493 0.657466 0.753484i \(-0.271628\pi\)
0.657466 + 0.753484i \(0.271628\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.04318e9 −1.43935
\(413\) −1.15278e9 −0.805233
\(414\) 0 0
\(415\) 0 0
\(416\) −1.14998e9 −0.783187
\(417\) 0 0
\(418\) 1.93860e9 1.29829
\(419\) 2.51122e8 0.166777 0.0833883 0.996517i \(-0.473426\pi\)
0.0833883 + 0.996517i \(0.473426\pi\)
\(420\) 0 0
\(421\) 6.49600e7 0.0424286 0.0212143 0.999775i \(-0.493247\pi\)
0.0212143 + 0.999775i \(0.493247\pi\)
\(422\) −9.48888e8 −0.614641
\(423\) 0 0
\(424\) −1.10671e9 −0.705104
\(425\) 0 0
\(426\) 0 0
\(427\) −5.84525e8 −0.363334
\(428\) −4.25864e7 −0.0262553
\(429\) 0 0
\(430\) 0 0
\(431\) 7.64855e8 0.460160 0.230080 0.973172i \(-0.426101\pi\)
0.230080 + 0.973172i \(0.426101\pi\)
\(432\) 0 0
\(433\) 1.64928e9 0.976308 0.488154 0.872758i \(-0.337670\pi\)
0.488154 + 0.872758i \(0.337670\pi\)
\(434\) 3.54386e8 0.208095
\(435\) 0 0
\(436\) 1.27851e9 0.738757
\(437\) −4.68567e9 −2.68588
\(438\) 0 0
\(439\) 2.27102e9 1.28113 0.640567 0.767902i \(-0.278699\pi\)
0.640567 + 0.767902i \(0.278699\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.98249e8 −0.164286
\(443\) 1.47221e9 0.804559 0.402279 0.915517i \(-0.368218\pi\)
0.402279 + 0.915517i \(0.368218\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −5.49008e7 −0.0293027
\(447\) 0 0
\(448\) 2.22521e7 0.0116923
\(449\) 3.41486e9 1.78037 0.890185 0.455600i \(-0.150575\pi\)
0.890185 + 0.455600i \(0.150575\pi\)
\(450\) 0 0
\(451\) 8.75420e8 0.449364
\(452\) 2.49774e9 1.27222
\(453\) 0 0
\(454\) −2.52401e8 −0.126589
\(455\) 0 0
\(456\) 0 0
\(457\) 1.10215e9 0.540174 0.270087 0.962836i \(-0.412948\pi\)
0.270087 + 0.962836i \(0.412948\pi\)
\(458\) 3.09258e8 0.150415
\(459\) 0 0
\(460\) 0 0
\(461\) 3.76947e9 1.79195 0.895976 0.444102i \(-0.146477\pi\)
0.895976 + 0.444102i \(0.146477\pi\)
\(462\) 0 0
\(463\) 1.10706e9 0.518366 0.259183 0.965828i \(-0.416547\pi\)
0.259183 + 0.965828i \(0.416547\pi\)
\(464\) −6.48658e8 −0.301441
\(465\) 0 0
\(466\) −6.89357e8 −0.315568
\(467\) −1.50674e9 −0.684586 −0.342293 0.939593i \(-0.611204\pi\)
−0.342293 + 0.939593i \(0.611204\pi\)
\(468\) 0 0
\(469\) 2.88653e8 0.129203
\(470\) 0 0
\(471\) 0 0
\(472\) −1.43168e9 −0.626685
\(473\) −2.74355e9 −1.19206
\(474\) 0 0
\(475\) 0 0
\(476\) 9.18626e8 0.390404
\(477\) 0 0
\(478\) −3.13860e8 −0.131443
\(479\) −3.55245e9 −1.47691 −0.738454 0.674304i \(-0.764444\pi\)
−0.738454 + 0.674304i \(0.764444\pi\)
\(480\) 0 0
\(481\) 1.64482e9 0.673922
\(482\) −6.85747e8 −0.278933
\(483\) 0 0
\(484\) −5.34728e9 −2.14375
\(485\) 0 0
\(486\) 0 0
\(487\) −2.62227e9 −1.02879 −0.514395 0.857553i \(-0.671983\pi\)
−0.514395 + 0.857553i \(0.671983\pi\)
\(488\) −7.25943e8 −0.282770
\(489\) 0 0
\(490\) 0 0
\(491\) −1.76846e9 −0.674232 −0.337116 0.941463i \(-0.609451\pi\)
−0.337116 + 0.941463i \(0.609451\pi\)
\(492\) 0 0
\(493\) −8.39604e8 −0.315581
\(494\) 1.42699e9 0.532571
\(495\) 0 0
\(496\) −5.64655e8 −0.207777
\(497\) 3.65890e9 1.33691
\(498\) 0 0
\(499\) −2.46316e9 −0.887442 −0.443721 0.896165i \(-0.646342\pi\)
−0.443721 + 0.896165i \(0.646342\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.02863e9 −0.715715
\(503\) −1.50060e9 −0.525748 −0.262874 0.964830i \(-0.584670\pi\)
−0.262874 + 0.964830i \(0.584670\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −4.31457e9 −1.48051
\(507\) 0 0
\(508\) 2.67357e7 0.00904834
\(509\) −4.31485e9 −1.45029 −0.725143 0.688598i \(-0.758226\pi\)
−0.725143 + 0.688598i \(0.758226\pi\)
\(510\) 0 0
\(511\) −4.23716e9 −1.40476
\(512\) 2.46516e9 0.811709
\(513\) 0 0
\(514\) −4.50262e8 −0.146250
\(515\) 0 0
\(516\) 0 0
\(517\) 7.23219e9 2.30172
\(518\) 1.22965e9 0.388710
\(519\) 0 0
\(520\) 0 0
\(521\) 9.53559e8 0.295404 0.147702 0.989032i \(-0.452812\pi\)
0.147702 + 0.989032i \(0.452812\pi\)
\(522\) 0 0
\(523\) −4.25294e9 −1.29997 −0.649986 0.759946i \(-0.725225\pi\)
−0.649986 + 0.759946i \(0.725225\pi\)
\(524\) −1.46038e9 −0.443411
\(525\) 0 0
\(526\) 1.03128e9 0.308977
\(527\) −7.30873e8 −0.217523
\(528\) 0 0
\(529\) 7.02367e9 2.06286
\(530\) 0 0
\(531\) 0 0
\(532\) −4.39523e9 −1.26558
\(533\) 6.44392e8 0.184334
\(534\) 0 0
\(535\) 0 0
\(536\) 3.58489e8 0.100554
\(537\) 0 0
\(538\) −2.45611e9 −0.680002
\(539\) −3.49467e8 −0.0961269
\(540\) 0 0
\(541\) −2.25815e9 −0.613144 −0.306572 0.951848i \(-0.599182\pi\)
−0.306572 + 0.951848i \(0.599182\pi\)
\(542\) −1.84765e9 −0.498450
\(543\) 0 0
\(544\) 1.77305e9 0.472198
\(545\) 0 0
\(546\) 0 0
\(547\) 6.43075e9 1.67999 0.839994 0.542595i \(-0.182558\pi\)
0.839994 + 0.542595i \(0.182558\pi\)
\(548\) 2.57201e9 0.667637
\(549\) 0 0
\(550\) 0 0
\(551\) 4.01714e9 1.02303
\(552\) 0 0
\(553\) −4.99563e9 −1.25618
\(554\) 1.15529e9 0.288675
\(555\) 0 0
\(556\) −3.41878e9 −0.843546
\(557\) 7.37598e8 0.180853 0.0904266 0.995903i \(-0.471177\pi\)
0.0904266 + 0.995903i \(0.471177\pi\)
\(558\) 0 0
\(559\) −2.01951e9 −0.488995
\(560\) 0 0
\(561\) 0 0
\(562\) 2.86510e9 0.680867
\(563\) 3.56268e9 0.841390 0.420695 0.907202i \(-0.361786\pi\)
0.420695 + 0.907202i \(0.361786\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.82309e9 −0.422620
\(567\) 0 0
\(568\) 4.54412e9 1.04047
\(569\) 2.20501e9 0.501785 0.250892 0.968015i \(-0.419276\pi\)
0.250892 + 0.968015i \(0.419276\pi\)
\(570\) 0 0
\(571\) −5.41646e8 −0.121756 −0.0608778 0.998145i \(-0.519390\pi\)
−0.0608778 + 0.998145i \(0.519390\pi\)
\(572\) −5.41358e9 −1.20948
\(573\) 0 0
\(574\) 4.81740e8 0.106321
\(575\) 0 0
\(576\) 0 0
\(577\) 1.71040e9 0.370665 0.185333 0.982676i \(-0.440664\pi\)
0.185333 + 0.982676i \(0.440664\pi\)
\(578\) −1.59185e9 −0.342891
\(579\) 0 0
\(580\) 0 0
\(581\) −6.24129e9 −1.32025
\(582\) 0 0
\(583\) −8.09671e9 −1.69227
\(584\) −5.26228e9 −1.09327
\(585\) 0 0
\(586\) −1.06429e9 −0.218483
\(587\) 1.60774e9 0.328083 0.164041 0.986453i \(-0.447547\pi\)
0.164041 + 0.986453i \(0.447547\pi\)
\(588\) 0 0
\(589\) 3.49691e9 0.705149
\(590\) 0 0
\(591\) 0 0
\(592\) −1.95924e9 −0.388115
\(593\) −6.40212e9 −1.26076 −0.630380 0.776286i \(-0.717101\pi\)
−0.630380 + 0.776286i \(0.717101\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.96987e9 0.768094
\(597\) 0 0
\(598\) −3.17593e9 −0.607319
\(599\) 7.65106e9 1.45455 0.727274 0.686347i \(-0.240787\pi\)
0.727274 + 0.686347i \(0.240787\pi\)
\(600\) 0 0
\(601\) −6.49677e9 −1.22078 −0.610389 0.792102i \(-0.708987\pi\)
−0.610389 + 0.792102i \(0.708987\pi\)
\(602\) −1.50976e9 −0.282047
\(603\) 0 0
\(604\) −2.42038e9 −0.446945
\(605\) 0 0
\(606\) 0 0
\(607\) 8.27081e9 1.50102 0.750512 0.660857i \(-0.229807\pi\)
0.750512 + 0.660857i \(0.229807\pi\)
\(608\) −8.48326e9 −1.53074
\(609\) 0 0
\(610\) 0 0
\(611\) 5.32357e9 0.944189
\(612\) 0 0
\(613\) −1.95155e9 −0.342190 −0.171095 0.985255i \(-0.554731\pi\)
−0.171095 + 0.985255i \(0.554731\pi\)
\(614\) 3.27793e9 0.571492
\(615\) 0 0
\(616\) −9.07657e9 −1.56455
\(617\) 1.00173e10 1.71693 0.858464 0.512874i \(-0.171419\pi\)
0.858464 + 0.512874i \(0.171419\pi\)
\(618\) 0 0
\(619\) −1.54387e9 −0.261634 −0.130817 0.991407i \(-0.541760\pi\)
−0.130817 + 0.991407i \(0.541760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4.58578e9 −0.764095
\(623\) 3.11224e9 0.515663
\(624\) 0 0
\(625\) 0 0
\(626\) 3.26485e9 0.531928
\(627\) 0 0
\(628\) −6.15084e9 −0.991005
\(629\) −2.53598e9 −0.406320
\(630\) 0 0
\(631\) 7.16453e9 1.13523 0.567616 0.823293i \(-0.307866\pi\)
0.567616 + 0.823293i \(0.307866\pi\)
\(632\) −6.20425e9 −0.977641
\(633\) 0 0
\(634\) −3.67239e9 −0.572317
\(635\) 0 0
\(636\) 0 0
\(637\) −2.57241e8 −0.0394322
\(638\) 3.69899e9 0.563911
\(639\) 0 0
\(640\) 0 0
\(641\) −3.32101e9 −0.498043 −0.249021 0.968498i \(-0.580109\pi\)
−0.249021 + 0.968498i \(0.580109\pi\)
\(642\) 0 0
\(643\) −6.42196e9 −0.952641 −0.476320 0.879272i \(-0.658030\pi\)
−0.476320 + 0.879272i \(0.658030\pi\)
\(644\) 9.78207e9 1.44321
\(645\) 0 0
\(646\) −2.20014e9 −0.321097
\(647\) −4.97907e9 −0.722743 −0.361371 0.932422i \(-0.617691\pi\)
−0.361371 + 0.932422i \(0.617691\pi\)
\(648\) 0 0
\(649\) −1.04742e10 −1.50406
\(650\) 0 0
\(651\) 0 0
\(652\) −2.89936e9 −0.409671
\(653\) 1.08435e10 1.52396 0.761978 0.647603i \(-0.224229\pi\)
0.761978 + 0.647603i \(0.224229\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −7.67572e8 −0.106159
\(657\) 0 0
\(658\) 3.97984e9 0.544597
\(659\) −3.26221e9 −0.444030 −0.222015 0.975043i \(-0.571263\pi\)
−0.222015 + 0.975043i \(0.571263\pi\)
\(660\) 0 0
\(661\) 9.36712e8 0.126154 0.0630770 0.998009i \(-0.479909\pi\)
0.0630770 + 0.998009i \(0.479909\pi\)
\(662\) 4.39582e9 0.588894
\(663\) 0 0
\(664\) −7.75127e9 −1.02751
\(665\) 0 0
\(666\) 0 0
\(667\) −8.94061e9 −1.16661
\(668\) 2.28900e9 0.297117
\(669\) 0 0
\(670\) 0 0
\(671\) −5.31101e9 −0.678654
\(672\) 0 0
\(673\) 3.64203e9 0.460565 0.230282 0.973124i \(-0.426035\pi\)
0.230282 + 0.973124i \(0.426035\pi\)
\(674\) −1.82591e9 −0.229705
\(675\) 0 0
\(676\) 2.47819e9 0.308547
\(677\) −7.13369e9 −0.883597 −0.441798 0.897114i \(-0.645659\pi\)
−0.441798 + 0.897114i \(0.645659\pi\)
\(678\) 0 0
\(679\) 1.47216e10 1.80473
\(680\) 0 0
\(681\) 0 0
\(682\) 3.21996e9 0.388691
\(683\) 7.57192e9 0.909355 0.454678 0.890656i \(-0.349754\pi\)
0.454678 + 0.890656i \(0.349754\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 3.63716e9 0.430159
\(687\) 0 0
\(688\) 2.40555e9 0.281615
\(689\) −5.95994e9 −0.694184
\(690\) 0 0
\(691\) −9.66534e8 −0.111441 −0.0557204 0.998446i \(-0.517746\pi\)
−0.0557204 + 0.998446i \(0.517746\pi\)
\(692\) 1.03049e10 1.18215
\(693\) 0 0
\(694\) 5.15680e9 0.585629
\(695\) 0 0
\(696\) 0 0
\(697\) −9.93524e8 −0.111138
\(698\) −6.22260e8 −0.0692593
\(699\) 0 0
\(700\) 0 0
\(701\) −1.35138e10 −1.48172 −0.740859 0.671660i \(-0.765582\pi\)
−0.740859 + 0.671660i \(0.765582\pi\)
\(702\) 0 0
\(703\) 1.21336e10 1.31718
\(704\) 2.02183e8 0.0218394
\(705\) 0 0
\(706\) 6.61555e9 0.707537
\(707\) 1.41355e9 0.150433
\(708\) 0 0
\(709\) 5.99067e9 0.631268 0.315634 0.948881i \(-0.397783\pi\)
0.315634 + 0.948881i \(0.397783\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3.86521e9 0.401322
\(713\) −7.78277e9 −0.804120
\(714\) 0 0
\(715\) 0 0
\(716\) −4.32303e9 −0.440142
\(717\) 0 0
\(718\) −5.94219e8 −0.0599116
\(719\) −1.52251e9 −0.152760 −0.0763800 0.997079i \(-0.524336\pi\)
−0.0763800 + 0.997079i \(0.524336\pi\)
\(720\) 0 0
\(721\) −1.84481e10 −1.83307
\(722\) 6.05735e9 0.598966
\(723\) 0 0
\(724\) 1.25951e9 0.123344
\(725\) 0 0
\(726\) 0 0
\(727\) 5.91664e9 0.571091 0.285545 0.958365i \(-0.407825\pi\)
0.285545 + 0.958365i \(0.407825\pi\)
\(728\) −6.68121e9 −0.641794
\(729\) 0 0
\(730\) 0 0
\(731\) 3.11368e9 0.294824
\(732\) 0 0
\(733\) 2.26254e9 0.212194 0.106097 0.994356i \(-0.466165\pi\)
0.106097 + 0.994356i \(0.466165\pi\)
\(734\) 5.51521e9 0.514785
\(735\) 0 0
\(736\) 1.88805e10 1.74558
\(737\) 2.62271e9 0.241332
\(738\) 0 0
\(739\) −9.50501e9 −0.866358 −0.433179 0.901308i \(-0.642608\pi\)
−0.433179 + 0.901308i \(0.642608\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.45558e9 −0.400397
\(743\) −7.55711e9 −0.675919 −0.337960 0.941161i \(-0.609737\pi\)
−0.337960 + 0.941161i \(0.609737\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.26982e9 −0.200173
\(747\) 0 0
\(748\) 8.34666e9 0.729217
\(749\) −3.84518e8 −0.0334372
\(750\) 0 0
\(751\) 7.06008e9 0.608233 0.304117 0.952635i \(-0.401639\pi\)
0.304117 + 0.952635i \(0.401639\pi\)
\(752\) −6.34121e9 −0.543763
\(753\) 0 0
\(754\) 2.72281e9 0.231322
\(755\) 0 0
\(756\) 0 0
\(757\) −1.38354e10 −1.15919 −0.579595 0.814905i \(-0.696789\pi\)
−0.579595 + 0.814905i \(0.696789\pi\)
\(758\) 2.64260e9 0.220389
\(759\) 0 0
\(760\) 0 0
\(761\) 9.16838e9 0.754131 0.377065 0.926187i \(-0.376933\pi\)
0.377065 + 0.926187i \(0.376933\pi\)
\(762\) 0 0
\(763\) 1.15438e10 0.940837
\(764\) −1.61227e10 −1.30801
\(765\) 0 0
\(766\) 1.51837e9 0.122061
\(767\) −7.71000e9 −0.616979
\(768\) 0 0
\(769\) 1.09798e10 0.870664 0.435332 0.900270i \(-0.356631\pi\)
0.435332 + 0.900270i \(0.356631\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.11298e10 0.870615
\(773\) 1.33749e10 1.04151 0.520753 0.853707i \(-0.325651\pi\)
0.520753 + 0.853707i \(0.325651\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.82833e10 1.40456
\(777\) 0 0
\(778\) 1.19155e9 0.0907162
\(779\) 4.75358e9 0.360280
\(780\) 0 0
\(781\) 3.32448e10 2.49716
\(782\) 4.89665e9 0.366164
\(783\) 0 0
\(784\) 3.06414e8 0.0227092
\(785\) 0 0
\(786\) 0 0
\(787\) 6.82502e9 0.499105 0.249553 0.968361i \(-0.419716\pi\)
0.249553 + 0.968361i \(0.419716\pi\)
\(788\) 2.01850e9 0.146956
\(789\) 0 0
\(790\) 0 0
\(791\) 2.25524e10 1.62023
\(792\) 0 0
\(793\) −3.90941e9 −0.278391
\(794\) 3.56063e9 0.252438
\(795\) 0 0
\(796\) −1.26258e9 −0.0887285
\(797\) −1.56291e10 −1.09353 −0.546765 0.837286i \(-0.684141\pi\)
−0.546765 + 0.837286i \(0.684141\pi\)
\(798\) 0 0
\(799\) −8.20789e9 −0.569269
\(800\) 0 0
\(801\) 0 0
\(802\) −7.11538e8 −0.0487066
\(803\) −3.84990e10 −2.62388
\(804\) 0 0
\(805\) 0 0
\(806\) 2.37019e9 0.159445
\(807\) 0 0
\(808\) 1.75554e9 0.117077
\(809\) 1.52410e10 1.01203 0.506016 0.862524i \(-0.331118\pi\)
0.506016 + 0.862524i \(0.331118\pi\)
\(810\) 0 0
\(811\) 1.52930e10 1.00675 0.503373 0.864069i \(-0.332092\pi\)
0.503373 + 0.864069i \(0.332092\pi\)
\(812\) −8.38641e9 −0.549706
\(813\) 0 0
\(814\) 1.11726e10 0.726053
\(815\) 0 0
\(816\) 0 0
\(817\) −1.48976e10 −0.955740
\(818\) 9.09714e9 0.581123
\(819\) 0 0
\(820\) 0 0
\(821\) −3.84785e9 −0.242671 −0.121335 0.992612i \(-0.538718\pi\)
−0.121335 + 0.992612i \(0.538718\pi\)
\(822\) 0 0
\(823\) 2.44305e10 1.52768 0.763840 0.645405i \(-0.223312\pi\)
0.763840 + 0.645405i \(0.223312\pi\)
\(824\) −2.29114e10 −1.42661
\(825\) 0 0
\(826\) −5.76391e9 −0.355866
\(827\) −2.44311e9 −0.150201 −0.0751007 0.997176i \(-0.523928\pi\)
−0.0751007 + 0.997176i \(0.523928\pi\)
\(828\) 0 0
\(829\) −2.35220e10 −1.43395 −0.716974 0.697100i \(-0.754474\pi\)
−0.716974 + 0.697100i \(0.754474\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.48826e8 0.00895874
\(833\) 3.96614e8 0.0237744
\(834\) 0 0
\(835\) 0 0
\(836\) −3.99351e10 −2.36392
\(837\) 0 0
\(838\) 1.25561e9 0.0737055
\(839\) 2.27195e10 1.32810 0.664050 0.747688i \(-0.268836\pi\)
0.664050 + 0.747688i \(0.268836\pi\)
\(840\) 0 0
\(841\) −9.58487e9 −0.555649
\(842\) 3.24800e8 0.0187510
\(843\) 0 0
\(844\) 1.95471e10 1.11914
\(845\) 0 0
\(846\) 0 0
\(847\) −4.82813e10 −2.73015
\(848\) 7.09923e9 0.399784
\(849\) 0 0
\(850\) 0 0
\(851\) −2.70046e10 −1.50205
\(852\) 0 0
\(853\) 3.13423e10 1.72906 0.864528 0.502584i \(-0.167617\pi\)
0.864528 + 0.502584i \(0.167617\pi\)
\(854\) −2.92263e9 −0.160572
\(855\) 0 0
\(856\) −4.77546e8 −0.0260230
\(857\) 2.95845e10 1.60558 0.802790 0.596262i \(-0.203348\pi\)
0.802790 + 0.596262i \(0.203348\pi\)
\(858\) 0 0
\(859\) 7.18615e9 0.386830 0.193415 0.981117i \(-0.438044\pi\)
0.193415 + 0.981117i \(0.438044\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.82427e9 0.203364
\(863\) −2.82174e10 −1.49444 −0.747221 0.664575i \(-0.768613\pi\)
−0.747221 + 0.664575i \(0.768613\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8.24640e9 0.431471
\(867\) 0 0
\(868\) −7.30035e9 −0.378900
\(869\) −4.53904e10 −2.34636
\(870\) 0 0
\(871\) 1.93056e9 0.0989967
\(872\) 1.43367e10 0.732220
\(873\) 0 0
\(874\) −2.34284e10 −1.18700
\(875\) 0 0
\(876\) 0 0
\(877\) 3.44481e10 1.72451 0.862256 0.506472i \(-0.169051\pi\)
0.862256 + 0.506472i \(0.169051\pi\)
\(878\) 1.13551e10 0.566187
\(879\) 0 0
\(880\) 0 0
\(881\) 1.96916e10 0.970211 0.485105 0.874456i \(-0.338781\pi\)
0.485105 + 0.874456i \(0.338781\pi\)
\(882\) 0 0
\(883\) 2.79896e10 1.36815 0.684075 0.729412i \(-0.260206\pi\)
0.684075 + 0.729412i \(0.260206\pi\)
\(884\) 6.14393e9 0.299132
\(885\) 0 0
\(886\) 7.36107e9 0.355568
\(887\) −1.03338e10 −0.497198 −0.248599 0.968606i \(-0.579970\pi\)
−0.248599 + 0.968606i \(0.579970\pi\)
\(888\) 0 0
\(889\) 2.41400e8 0.0115234
\(890\) 0 0
\(891\) 0 0
\(892\) 1.13096e9 0.0533543
\(893\) 3.92712e10 1.84541
\(894\) 0 0
\(895\) 0 0
\(896\) 2.21200e10 1.02732
\(897\) 0 0
\(898\) 1.70743e10 0.786820
\(899\) 6.67236e9 0.306281
\(900\) 0 0
\(901\) 9.18904e9 0.418537
\(902\) 4.37710e9 0.198593
\(903\) 0 0
\(904\) 2.80086e10 1.26096
\(905\) 0 0
\(906\) 0 0
\(907\) −2.24060e10 −0.997099 −0.498550 0.866861i \(-0.666134\pi\)
−0.498550 + 0.866861i \(0.666134\pi\)
\(908\) 5.19946e9 0.230493
\(909\) 0 0
\(910\) 0 0
\(911\) −2.53010e10 −1.10872 −0.554361 0.832276i \(-0.687037\pi\)
−0.554361 + 0.832276i \(0.687037\pi\)
\(912\) 0 0
\(913\) −5.67085e10 −2.46604
\(914\) 5.51075e9 0.238725
\(915\) 0 0
\(916\) −6.37072e9 −0.273876
\(917\) −1.31860e10 −0.564702
\(918\) 0 0
\(919\) −1.88277e10 −0.800191 −0.400095 0.916474i \(-0.631023\pi\)
−0.400095 + 0.916474i \(0.631023\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.88473e10 0.791939
\(923\) 2.44713e10 1.02436
\(924\) 0 0
\(925\) 0 0
\(926\) 5.53528e9 0.229087
\(927\) 0 0
\(928\) −1.61867e10 −0.664875
\(929\) 9.06966e9 0.371138 0.185569 0.982631i \(-0.440587\pi\)
0.185569 + 0.982631i \(0.440587\pi\)
\(930\) 0 0
\(931\) −1.89762e9 −0.0770702
\(932\) 1.42008e10 0.574587
\(933\) 0 0
\(934\) −7.53368e9 −0.302547
\(935\) 0 0
\(936\) 0 0
\(937\) −2.38178e10 −0.945828 −0.472914 0.881109i \(-0.656798\pi\)
−0.472914 + 0.881109i \(0.656798\pi\)
\(938\) 1.44327e9 0.0571001
\(939\) 0 0
\(940\) 0 0
\(941\) 6.64810e9 0.260096 0.130048 0.991508i \(-0.458487\pi\)
0.130048 + 0.991508i \(0.458487\pi\)
\(942\) 0 0
\(943\) −1.05796e10 −0.410847
\(944\) 9.18383e9 0.355322
\(945\) 0 0
\(946\) −1.37177e10 −0.526821
\(947\) −1.09645e10 −0.419532 −0.209766 0.977752i \(-0.567270\pi\)
−0.209766 + 0.977752i \(0.567270\pi\)
\(948\) 0 0
\(949\) −2.83389e10 −1.07634
\(950\) 0 0
\(951\) 0 0
\(952\) 1.03011e10 0.386949
\(953\) 1.79028e10 0.670032 0.335016 0.942212i \(-0.391258\pi\)
0.335016 + 0.942212i \(0.391258\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.46551e9 0.239332
\(957\) 0 0
\(958\) −1.77622e10 −0.652707
\(959\) 2.32230e10 0.850262
\(960\) 0 0
\(961\) −2.17043e10 −0.788887
\(962\) 8.22408e9 0.297834
\(963\) 0 0
\(964\) 1.41264e10 0.507881
\(965\) 0 0
\(966\) 0 0
\(967\) 3.39231e10 1.20643 0.603216 0.797578i \(-0.293886\pi\)
0.603216 + 0.797578i \(0.293886\pi\)
\(968\) −5.99622e10 −2.12478
\(969\) 0 0
\(970\) 0 0
\(971\) −1.00724e7 −0.000353072 0 −0.000176536 1.00000i \(-0.500056\pi\)
−0.000176536 1.00000i \(0.500056\pi\)
\(972\) 0 0
\(973\) −3.08686e10 −1.07429
\(974\) −1.31114e10 −0.454665
\(975\) 0 0
\(976\) 4.65672e9 0.160327
\(977\) 7.44894e9 0.255543 0.127771 0.991804i \(-0.459218\pi\)
0.127771 + 0.991804i \(0.459218\pi\)
\(978\) 0 0
\(979\) 2.82779e10 0.963181
\(980\) 0 0
\(981\) 0 0
\(982\) −8.84229e9 −0.297971
\(983\) −1.43718e10 −0.482584 −0.241292 0.970453i \(-0.577571\pi\)
−0.241292 + 0.970453i \(0.577571\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −4.19802e9 −0.139468
\(987\) 0 0
\(988\) −2.93960e10 −0.969705
\(989\) 3.31563e10 1.08988
\(990\) 0 0
\(991\) −3.76560e10 −1.22907 −0.614535 0.788890i \(-0.710656\pi\)
−0.614535 + 0.788890i \(0.710656\pi\)
\(992\) −1.40905e10 −0.458283
\(993\) 0 0
\(994\) 1.82945e10 0.590838
\(995\) 0 0
\(996\) 0 0
\(997\) 2.47986e10 0.792490 0.396245 0.918145i \(-0.370313\pi\)
0.396245 + 0.918145i \(0.370313\pi\)
\(998\) −1.23158e10 −0.392198
\(999\) 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.h.1.1 1
3.2 odd 2 225.8.a.e.1.1 1
5.2 odd 4 225.8.b.g.199.2 2
5.3 odd 4 225.8.b.g.199.1 2
5.4 even 2 45.8.a.b.1.1 1
15.2 even 4 225.8.b.h.199.1 2
15.8 even 4 225.8.b.h.199.2 2
15.14 odd 2 45.8.a.c.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.8.a.b.1.1 1 5.4 even 2
45.8.a.c.1.1 yes 1 15.14 odd 2
225.8.a.e.1.1 1 3.2 odd 2
225.8.a.h.1.1 1 1.1 even 1 trivial
225.8.b.g.199.1 2 5.3 odd 4
225.8.b.g.199.2 2 5.2 odd 4
225.8.b.h.199.1 2 15.2 even 4
225.8.b.h.199.2 2 15.8 even 4