Properties

Label 16.9.c.a.15.1
Level $16$
Weight $9$
Character 16.15
Analytic conductor $6.518$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,9,Mod(15,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.15");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(6.51805776098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.1
Root \(0.500000 + 2.95804i\) of defining polynomial
Character \(\chi\) \(=\) 16.15
Dual form 16.9.c.a.15.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-141.986i q^{3} -510.000 q^{5} +2555.75i q^{7} -13599.0 q^{9} +O(q^{10})\) \(q-141.986i q^{3} -510.000 q^{5} +2555.75i q^{7} -13599.0 q^{9} -19168.1i q^{11} -27710.0 q^{13} +72412.8i q^{15} +50370.0 q^{17} -108619. i q^{19} +362880. q^{21} -176347. i q^{23} -130525. q^{25} +999297. i q^{27} +54978.0 q^{29} -1.17564e6i q^{31} -2.72160e6 q^{33} -1.30343e6i q^{35} +793730. q^{37} +3.93443e6i q^{39} -75582.0 q^{41} -499648. i q^{43} +6.93549e6 q^{45} -2.86755e6i q^{47} -767039. q^{49} -7.15183e6i q^{51} +1.11662e7 q^{53} +9.77573e6i q^{55} -1.54224e7 q^{57} +2.18325e7i q^{59} -2.38266e7 q^{61} -3.47556e7i q^{63} +1.41321e7 q^{65} -7.49473e6i q^{67} -2.50387e7 q^{69} +1.00824e7i q^{71} +6.51661e6 q^{73} +1.85327e7i q^{75} +4.89888e7 q^{77} +4.87892e7i q^{79} +5.26630e7 q^{81} -7.34483e7i q^{83} -2.56887e7 q^{85} -7.80610e6i q^{87} +8.67958e7 q^{89} -7.08197e7i q^{91} -1.66925e8 q^{93} +5.53958e7i q^{95} -4.66703e7 q^{97} +2.60667e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1020 q^{5} - 27198 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1020 q^{5} - 27198 q^{9} - 55420 q^{13} + 100740 q^{17} + 725760 q^{21} - 261050 q^{25} + 109956 q^{29} - 5443200 q^{33} + 1587460 q^{37} - 151164 q^{41} + 13870980 q^{45} - 1534078 q^{49} + 22332420 q^{53} - 30844800 q^{57} - 47653244 q^{61} + 28264200 q^{65} - 50077440 q^{69} + 13033220 q^{73} + 97977600 q^{77} + 105326082 q^{81} - 51377400 q^{85} + 173591556 q^{89} - 333849600 q^{93} - 93340540 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) − 141.986i − 1.75291i −0.481481 0.876456i \(-0.659901\pi\)
0.481481 0.876456i \(-0.340099\pi\)
\(4\) 0 0
\(5\) −510.000 −0.816000 −0.408000 0.912982i \(-0.633774\pi\)
−0.408000 + 0.912982i \(0.633774\pi\)
\(6\) 0 0
\(7\) 2555.75i 1.06445i 0.846603 + 0.532225i \(0.178644\pi\)
−0.846603 + 0.532225i \(0.821356\pi\)
\(8\) 0 0
\(9\) −13599.0 −2.07270
\(10\) 0 0
\(11\) − 19168.1i − 1.30921i −0.755972 0.654603i \(-0.772836\pi\)
0.755972 0.654603i \(-0.227164\pi\)
\(12\) 0 0
\(13\) −27710.0 −0.970204 −0.485102 0.874458i \(-0.661218\pi\)
−0.485102 + 0.874458i \(0.661218\pi\)
\(14\) 0 0
\(15\) 72412.8i 1.43038i
\(16\) 0 0
\(17\) 50370.0 0.603082 0.301541 0.953453i \(-0.402499\pi\)
0.301541 + 0.953453i \(0.402499\pi\)
\(18\) 0 0
\(19\) − 108619.i − 0.833474i −0.909027 0.416737i \(-0.863174\pi\)
0.909027 0.416737i \(-0.136826\pi\)
\(20\) 0 0
\(21\) 362880. 1.86589
\(22\) 0 0
\(23\) − 176347.i − 0.630167i −0.949064 0.315083i \(-0.897968\pi\)
0.949064 0.315083i \(-0.102032\pi\)
\(24\) 0 0
\(25\) −130525. −0.334144
\(26\) 0 0
\(27\) 999297.i 1.88035i
\(28\) 0 0
\(29\) 54978.0 0.0777315 0.0388657 0.999244i \(-0.487626\pi\)
0.0388657 + 0.999244i \(0.487626\pi\)
\(30\) 0 0
\(31\) − 1.17564e6i − 1.27300i −0.771276 0.636501i \(-0.780381\pi\)
0.771276 0.636501i \(-0.219619\pi\)
\(32\) 0 0
\(33\) −2.72160e6 −2.29493
\(34\) 0 0
\(35\) − 1.30343e6i − 0.868592i
\(36\) 0 0
\(37\) 793730. 0.423512 0.211756 0.977323i \(-0.432082\pi\)
0.211756 + 0.977323i \(0.432082\pi\)
\(38\) 0 0
\(39\) 3.93443e6i 1.70068i
\(40\) 0 0
\(41\) −75582.0 −0.0267475 −0.0133737 0.999911i \(-0.504257\pi\)
−0.0133737 + 0.999911i \(0.504257\pi\)
\(42\) 0 0
\(43\) − 499648.i − 0.146147i −0.997327 0.0730736i \(-0.976719\pi\)
0.997327 0.0730736i \(-0.0232808\pi\)
\(44\) 0 0
\(45\) 6.93549e6 1.69133
\(46\) 0 0
\(47\) − 2.86755e6i − 0.587651i −0.955859 0.293825i \(-0.905072\pi\)
0.955859 0.293825i \(-0.0949284\pi\)
\(48\) 0 0
\(49\) −767039. −0.133056
\(50\) 0 0
\(51\) − 7.15183e6i − 1.05715i
\(52\) 0 0
\(53\) 1.11662e7 1.41515 0.707575 0.706639i \(-0.249789\pi\)
0.707575 + 0.706639i \(0.249789\pi\)
\(54\) 0 0
\(55\) 9.77573e6i 1.06831i
\(56\) 0 0
\(57\) −1.54224e7 −1.46101
\(58\) 0 0
\(59\) 2.18325e7i 1.80175i 0.434078 + 0.900875i \(0.357074\pi\)
−0.434078 + 0.900875i \(0.642926\pi\)
\(60\) 0 0
\(61\) −2.38266e7 −1.72085 −0.860425 0.509577i \(-0.829802\pi\)
−0.860425 + 0.509577i \(0.829802\pi\)
\(62\) 0 0
\(63\) − 3.47556e7i − 2.20629i
\(64\) 0 0
\(65\) 1.41321e7 0.791687
\(66\) 0 0
\(67\) − 7.49473e6i − 0.371926i −0.982557 0.185963i \(-0.940460\pi\)
0.982557 0.185963i \(-0.0595405\pi\)
\(68\) 0 0
\(69\) −2.50387e7 −1.10463
\(70\) 0 0
\(71\) 1.00824e7i 0.396763i 0.980125 + 0.198382i \(0.0635685\pi\)
−0.980125 + 0.198382i \(0.936431\pi\)
\(72\) 0 0
\(73\) 6.51661e6 0.229472 0.114736 0.993396i \(-0.463398\pi\)
0.114736 + 0.993396i \(0.463398\pi\)
\(74\) 0 0
\(75\) 1.85327e7i 0.585725i
\(76\) 0 0
\(77\) 4.89888e7 1.39359
\(78\) 0 0
\(79\) 4.87892e7i 1.25261i 0.779579 + 0.626304i \(0.215433\pi\)
−0.779579 + 0.626304i \(0.784567\pi\)
\(80\) 0 0
\(81\) 5.26630e7 1.22339
\(82\) 0 0
\(83\) − 7.34483e7i − 1.54764i −0.633407 0.773819i \(-0.718344\pi\)
0.633407 0.773819i \(-0.281656\pi\)
\(84\) 0 0
\(85\) −2.56887e7 −0.492115
\(86\) 0 0
\(87\) − 7.80610e6i − 0.136256i
\(88\) 0 0
\(89\) 8.67958e7 1.38337 0.691685 0.722199i \(-0.256869\pi\)
0.691685 + 0.722199i \(0.256869\pi\)
\(90\) 0 0
\(91\) − 7.08197e7i − 1.03273i
\(92\) 0 0
\(93\) −1.66925e8 −2.23146
\(94\) 0 0
\(95\) 5.53958e7i 0.680115i
\(96\) 0 0
\(97\) −4.66703e7 −0.527173 −0.263587 0.964636i \(-0.584905\pi\)
−0.263587 + 0.964636i \(0.584905\pi\)
\(98\) 0 0
\(99\) 2.60667e8i 2.71360i
\(100\) 0 0
\(101\) 6.59910e7 0.634161 0.317080 0.948399i \(-0.397297\pi\)
0.317080 + 0.948399i \(0.397297\pi\)
\(102\) 0 0
\(103\) − 1.64884e8i − 1.46497i −0.680782 0.732486i \(-0.738360\pi\)
0.680782 0.732486i \(-0.261640\pi\)
\(104\) 0 0
\(105\) −1.85069e8 −1.52257
\(106\) 0 0
\(107\) − 1.27326e8i − 0.971364i −0.874136 0.485682i \(-0.838571\pi\)
0.874136 0.485682i \(-0.161429\pi\)
\(108\) 0 0
\(109\) −1.56119e8 −1.10598 −0.552992 0.833186i \(-0.686514\pi\)
−0.552992 + 0.833186i \(0.686514\pi\)
\(110\) 0 0
\(111\) − 1.12698e8i − 0.742380i
\(112\) 0 0
\(113\) 2.36346e8 1.44955 0.724776 0.688984i \(-0.241943\pi\)
0.724776 + 0.688984i \(0.241943\pi\)
\(114\) 0 0
\(115\) 8.99367e7i 0.514216i
\(116\) 0 0
\(117\) 3.76828e8 2.01094
\(118\) 0 0
\(119\) 1.28733e8i 0.641951i
\(120\) 0 0
\(121\) −1.53057e8 −0.714023
\(122\) 0 0
\(123\) 1.07316e7i 0.0468860i
\(124\) 0 0
\(125\) 2.65786e8 1.08866
\(126\) 0 0
\(127\) 3.67741e8i 1.41360i 0.707412 + 0.706802i \(0.249863\pi\)
−0.707412 + 0.706802i \(0.750137\pi\)
\(128\) 0 0
\(129\) −7.09430e7 −0.256183
\(130\) 0 0
\(131\) − 2.16350e8i − 0.734636i −0.930095 0.367318i \(-0.880276\pi\)
0.930095 0.367318i \(-0.119724\pi\)
\(132\) 0 0
\(133\) 2.77603e8 0.887193
\(134\) 0 0
\(135\) − 5.09641e8i − 1.53437i
\(136\) 0 0
\(137\) −3.86442e8 −1.09699 −0.548494 0.836155i \(-0.684799\pi\)
−0.548494 + 0.836155i \(0.684799\pi\)
\(138\) 0 0
\(139\) − 3.51077e8i − 0.940465i −0.882543 0.470232i \(-0.844170\pi\)
0.882543 0.470232i \(-0.155830\pi\)
\(140\) 0 0
\(141\) −4.07151e8 −1.03010
\(142\) 0 0
\(143\) 5.31148e8i 1.27020i
\(144\) 0 0
\(145\) −2.80388e7 −0.0634289
\(146\) 0 0
\(147\) 1.08909e8i 0.233235i
\(148\) 0 0
\(149\) −4.54099e8 −0.921308 −0.460654 0.887580i \(-0.652385\pi\)
−0.460654 + 0.887580i \(0.652385\pi\)
\(150\) 0 0
\(151\) 6.60188e8i 1.26987i 0.772565 + 0.634936i \(0.218973\pi\)
−0.772565 + 0.634936i \(0.781027\pi\)
\(152\) 0 0
\(153\) −6.84982e8 −1.25001
\(154\) 0 0
\(155\) 5.99578e8i 1.03877i
\(156\) 0 0
\(157\) 4.35318e8 0.716486 0.358243 0.933628i \(-0.383376\pi\)
0.358243 + 0.933628i \(0.383376\pi\)
\(158\) 0 0
\(159\) − 1.58544e9i − 2.48063i
\(160\) 0 0
\(161\) 4.50697e8 0.670782
\(162\) 0 0
\(163\) − 2.44065e8i − 0.345744i −0.984944 0.172872i \(-0.944695\pi\)
0.984944 0.172872i \(-0.0553047\pi\)
\(164\) 0 0
\(165\) 1.38802e9 1.87266
\(166\) 0 0
\(167\) − 6.71351e8i − 0.863145i −0.902078 0.431573i \(-0.857959\pi\)
0.902078 0.431573i \(-0.142041\pi\)
\(168\) 0 0
\(169\) −4.78866e7 −0.0587040
\(170\) 0 0
\(171\) 1.47711e9i 1.72754i
\(172\) 0 0
\(173\) −1.76764e9 −1.97337 −0.986685 0.162644i \(-0.947998\pi\)
−0.986685 + 0.162644i \(0.947998\pi\)
\(174\) 0 0
\(175\) − 3.33589e8i − 0.355680i
\(176\) 0 0
\(177\) 3.09990e9 3.15831
\(178\) 0 0
\(179\) − 7.56967e8i − 0.737335i −0.929561 0.368668i \(-0.879814\pi\)
0.929561 0.368668i \(-0.120186\pi\)
\(180\) 0 0
\(181\) 6.27094e8 0.584277 0.292138 0.956376i \(-0.405633\pi\)
0.292138 + 0.956376i \(0.405633\pi\)
\(182\) 0 0
\(183\) 3.38304e9i 3.01650i
\(184\) 0 0
\(185\) −4.04802e8 −0.345586
\(186\) 0 0
\(187\) − 9.65497e8i − 0.789559i
\(188\) 0 0
\(189\) −2.55395e9 −2.00154
\(190\) 0 0
\(191\) − 1.07924e9i − 0.810933i −0.914110 0.405466i \(-0.867109\pi\)
0.914110 0.405466i \(-0.132891\pi\)
\(192\) 0 0
\(193\) −2.96757e7 −0.0213881 −0.0106940 0.999943i \(-0.503404\pi\)
−0.0106940 + 0.999943i \(0.503404\pi\)
\(194\) 0 0
\(195\) − 2.00656e9i − 1.38776i
\(196\) 0 0
\(197\) −1.12484e9 −0.746837 −0.373419 0.927663i \(-0.621814\pi\)
−0.373419 + 0.927663i \(0.621814\pi\)
\(198\) 0 0
\(199\) − 1.04718e9i − 0.667742i −0.942619 0.333871i \(-0.891645\pi\)
0.942619 0.333871i \(-0.108355\pi\)
\(200\) 0 0
\(201\) −1.06415e9 −0.651954
\(202\) 0 0
\(203\) 1.40510e8i 0.0827413i
\(204\) 0 0
\(205\) 3.85468e7 0.0218259
\(206\) 0 0
\(207\) 2.39814e9i 1.30615i
\(208\) 0 0
\(209\) −2.08202e9 −1.09119
\(210\) 0 0
\(211\) 2.77676e9i 1.40090i 0.713699 + 0.700452i \(0.247018\pi\)
−0.713699 + 0.700452i \(0.752982\pi\)
\(212\) 0 0
\(213\) 1.43156e9 0.695491
\(214\) 0 0
\(215\) 2.54821e8i 0.119256i
\(216\) 0 0
\(217\) 3.00465e9 1.35505
\(218\) 0 0
\(219\) − 9.25267e8i − 0.402245i
\(220\) 0 0
\(221\) −1.39575e9 −0.585113
\(222\) 0 0
\(223\) − 2.27822e9i − 0.921248i −0.887595 0.460624i \(-0.847626\pi\)
0.887595 0.460624i \(-0.152374\pi\)
\(224\) 0 0
\(225\) 1.77501e9 0.692581
\(226\) 0 0
\(227\) 1.48863e9i 0.560639i 0.959907 + 0.280319i \(0.0904404\pi\)
−0.959907 + 0.280319i \(0.909560\pi\)
\(228\) 0 0
\(229\) 1.69447e9 0.616157 0.308079 0.951361i \(-0.400314\pi\)
0.308079 + 0.951361i \(0.400314\pi\)
\(230\) 0 0
\(231\) − 6.95572e9i − 2.44284i
\(232\) 0 0
\(233\) 5.21423e8 0.176916 0.0884580 0.996080i \(-0.471806\pi\)
0.0884580 + 0.996080i \(0.471806\pi\)
\(234\) 0 0
\(235\) 1.46245e9i 0.479523i
\(236\) 0 0
\(237\) 6.92738e9 2.19571
\(238\) 0 0
\(239\) 4.40690e9i 1.35065i 0.737522 + 0.675323i \(0.235996\pi\)
−0.737522 + 0.675323i \(0.764004\pi\)
\(240\) 0 0
\(241\) −1.62148e9 −0.480666 −0.240333 0.970691i \(-0.577257\pi\)
−0.240333 + 0.970691i \(0.577257\pi\)
\(242\) 0 0
\(243\) − 9.21023e8i − 0.264147i
\(244\) 0 0
\(245\) 3.91190e8 0.108573
\(246\) 0 0
\(247\) 3.00984e9i 0.808640i
\(248\) 0 0
\(249\) −1.04286e10 −2.71287
\(250\) 0 0
\(251\) 1.31321e8i 0.0330855i 0.999863 + 0.0165428i \(0.00526597\pi\)
−0.999863 + 0.0165428i \(0.994734\pi\)
\(252\) 0 0
\(253\) −3.38023e9 −0.825019
\(254\) 0 0
\(255\) 3.64743e9i 0.862634i
\(256\) 0 0
\(257\) 5.27789e9 1.20984 0.604920 0.796287i \(-0.293205\pi\)
0.604920 + 0.796287i \(0.293205\pi\)
\(258\) 0 0
\(259\) 2.02857e9i 0.450808i
\(260\) 0 0
\(261\) −7.47646e8 −0.161114
\(262\) 0 0
\(263\) − 7.38745e9i − 1.54409i −0.635570 0.772044i \(-0.719235\pi\)
0.635570 0.772044i \(-0.280765\pi\)
\(264\) 0 0
\(265\) −5.69477e9 −1.15476
\(266\) 0 0
\(267\) − 1.23238e10i − 2.42493i
\(268\) 0 0
\(269\) 3.46450e8 0.0661655 0.0330828 0.999453i \(-0.489468\pi\)
0.0330828 + 0.999453i \(0.489468\pi\)
\(270\) 0 0
\(271\) − 6.51715e6i − 0.00120832i −1.00000 0.000604158i \(-0.999808\pi\)
1.00000 0.000604158i \(-0.000192310\pi\)
\(272\) 0 0
\(273\) −1.00554e10 −1.81029
\(274\) 0 0
\(275\) 2.50192e9i 0.437464i
\(276\) 0 0
\(277\) 2.15061e9 0.365293 0.182647 0.983179i \(-0.441534\pi\)
0.182647 + 0.983179i \(0.441534\pi\)
\(278\) 0 0
\(279\) 1.59876e10i 2.63855i
\(280\) 0 0
\(281\) 1.04256e10 1.67215 0.836074 0.548616i \(-0.184845\pi\)
0.836074 + 0.548616i \(0.184845\pi\)
\(282\) 0 0
\(283\) − 1.28042e9i − 0.199622i −0.995006 0.0998108i \(-0.968176\pi\)
0.995006 0.0998108i \(-0.0318238\pi\)
\(284\) 0 0
\(285\) 7.86542e9 1.19218
\(286\) 0 0
\(287\) − 1.93168e8i − 0.0284714i
\(288\) 0 0
\(289\) −4.43862e9 −0.636292
\(290\) 0 0
\(291\) 6.62652e9i 0.924089i
\(292\) 0 0
\(293\) −2.13786e9 −0.290074 −0.145037 0.989426i \(-0.546330\pi\)
−0.145037 + 0.989426i \(0.546330\pi\)
\(294\) 0 0
\(295\) − 1.11346e10i − 1.47023i
\(296\) 0 0
\(297\) 1.91546e10 2.46177
\(298\) 0 0
\(299\) 4.88656e9i 0.611390i
\(300\) 0 0
\(301\) 1.27697e9 0.155567
\(302\) 0 0
\(303\) − 9.36980e9i − 1.11163i
\(304\) 0 0
\(305\) 1.21516e10 1.40421
\(306\) 0 0
\(307\) − 5.45140e9i − 0.613698i −0.951758 0.306849i \(-0.900725\pi\)
0.951758 0.306849i \(-0.0992747\pi\)
\(308\) 0 0
\(309\) −2.34112e10 −2.56797
\(310\) 0 0
\(311\) − 1.07550e10i − 1.14965i −0.818275 0.574827i \(-0.805069\pi\)
0.818275 0.574827i \(-0.194931\pi\)
\(312\) 0 0
\(313\) −2.99804e8 −0.0312364 −0.0156182 0.999878i \(-0.504972\pi\)
−0.0156182 + 0.999878i \(0.504972\pi\)
\(314\) 0 0
\(315\) 1.77254e10i 1.80033i
\(316\) 0 0
\(317\) 5.31172e9 0.526015 0.263007 0.964794i \(-0.415286\pi\)
0.263007 + 0.964794i \(0.415286\pi\)
\(318\) 0 0
\(319\) − 1.05382e9i − 0.101767i
\(320\) 0 0
\(321\) −1.80785e10 −1.70272
\(322\) 0 0
\(323\) − 5.47115e9i − 0.502653i
\(324\) 0 0
\(325\) 3.61685e9 0.324188
\(326\) 0 0
\(327\) 2.21667e10i 1.93869i
\(328\) 0 0
\(329\) 7.32872e9 0.625525
\(330\) 0 0
\(331\) 1.01004e10i 0.841446i 0.907189 + 0.420723i \(0.138224\pi\)
−0.907189 + 0.420723i \(0.861776\pi\)
\(332\) 0 0
\(333\) −1.07939e10 −0.877815
\(334\) 0 0
\(335\) 3.82231e9i 0.303492i
\(336\) 0 0
\(337\) −1.84359e10 −1.42937 −0.714684 0.699448i \(-0.753429\pi\)
−0.714684 + 0.699448i \(0.753429\pi\)
\(338\) 0 0
\(339\) − 3.35578e10i − 2.54094i
\(340\) 0 0
\(341\) −2.25348e10 −1.66662
\(342\) 0 0
\(343\) 1.27730e10i 0.922820i
\(344\) 0 0
\(345\) 1.27697e10 0.901376
\(346\) 0 0
\(347\) − 1.27822e10i − 0.881636i −0.897597 0.440818i \(-0.854688\pi\)
0.897597 0.440818i \(-0.145312\pi\)
\(348\) 0 0
\(349\) −6.39381e8 −0.0430981 −0.0215490 0.999768i \(-0.506860\pi\)
−0.0215490 + 0.999768i \(0.506860\pi\)
\(350\) 0 0
\(351\) − 2.76905e10i − 1.82433i
\(352\) 0 0
\(353\) 2.59837e10 1.67341 0.836705 0.547653i \(-0.184479\pi\)
0.836705 + 0.547653i \(0.184479\pi\)
\(354\) 0 0
\(355\) − 5.14203e9i − 0.323759i
\(356\) 0 0
\(357\) 1.82783e10 1.12528
\(358\) 0 0
\(359\) 2.22541e10i 1.33978i 0.742461 + 0.669889i \(0.233658\pi\)
−0.742461 + 0.669889i \(0.766342\pi\)
\(360\) 0 0
\(361\) 5.18543e9 0.305320
\(362\) 0 0
\(363\) 2.17320e10i 1.25162i
\(364\) 0 0
\(365\) −3.32347e9 −0.187249
\(366\) 0 0
\(367\) 1.86229e10i 1.02656i 0.858221 + 0.513280i \(0.171570\pi\)
−0.858221 + 0.513280i \(0.828430\pi\)
\(368\) 0 0
\(369\) 1.02784e9 0.0554396
\(370\) 0 0
\(371\) 2.85380e10i 1.50636i
\(372\) 0 0
\(373\) 1.19680e10 0.618283 0.309141 0.951016i \(-0.399958\pi\)
0.309141 + 0.951016i \(0.399958\pi\)
\(374\) 0 0
\(375\) − 3.77379e10i − 1.90833i
\(376\) 0 0
\(377\) −1.52344e9 −0.0754154
\(378\) 0 0
\(379\) − 2.30787e10i − 1.11855i −0.828982 0.559275i \(-0.811080\pi\)
0.828982 0.559275i \(-0.188920\pi\)
\(380\) 0 0
\(381\) 5.22141e10 2.47792
\(382\) 0 0
\(383\) 1.43419e10i 0.666518i 0.942835 + 0.333259i \(0.108148\pi\)
−0.942835 + 0.333259i \(0.891852\pi\)
\(384\) 0 0
\(385\) −2.49843e10 −1.13717
\(386\) 0 0
\(387\) 6.79472e9i 0.302920i
\(388\) 0 0
\(389\) −2.73457e10 −1.19424 −0.597119 0.802152i \(-0.703688\pi\)
−0.597119 + 0.802152i \(0.703688\pi\)
\(390\) 0 0
\(391\) − 8.88257e9i − 0.380042i
\(392\) 0 0
\(393\) −3.07187e10 −1.28775
\(394\) 0 0
\(395\) − 2.48825e10i − 1.02213i
\(396\) 0 0
\(397\) 3.99456e10 1.60808 0.804039 0.594576i \(-0.202680\pi\)
0.804039 + 0.594576i \(0.202680\pi\)
\(398\) 0 0
\(399\) − 3.94157e10i − 1.55517i
\(400\) 0 0
\(401\) −2.12767e10 −0.822863 −0.411431 0.911441i \(-0.634971\pi\)
−0.411431 + 0.911441i \(0.634971\pi\)
\(402\) 0 0
\(403\) 3.25771e10i 1.23507i
\(404\) 0 0
\(405\) −2.68582e10 −0.998288
\(406\) 0 0
\(407\) − 1.52143e10i − 0.554465i
\(408\) 0 0
\(409\) 1.14283e10 0.408404 0.204202 0.978929i \(-0.434540\pi\)
0.204202 + 0.978929i \(0.434540\pi\)
\(410\) 0 0
\(411\) 5.48693e10i 1.92292i
\(412\) 0 0
\(413\) −5.57982e10 −1.91788
\(414\) 0 0
\(415\) 3.74586e10i 1.26287i
\(416\) 0 0
\(417\) −4.98479e10 −1.64855
\(418\) 0 0
\(419\) − 1.10009e10i − 0.356922i −0.983947 0.178461i \(-0.942888\pi\)
0.983947 0.178461i \(-0.0571119\pi\)
\(420\) 0 0
\(421\) 2.28766e10 0.728220 0.364110 0.931356i \(-0.381373\pi\)
0.364110 + 0.931356i \(0.381373\pi\)
\(422\) 0 0
\(423\) 3.89958e10i 1.21802i
\(424\) 0 0
\(425\) −6.57454e9 −0.201516
\(426\) 0 0
\(427\) − 6.08948e10i − 1.83176i
\(428\) 0 0
\(429\) 7.54155e10 2.22655
\(430\) 0 0
\(431\) 9.55108e8i 0.0276786i 0.999904 + 0.0138393i \(0.00440532\pi\)
−0.999904 + 0.0138393i \(0.995595\pi\)
\(432\) 0 0
\(433\) 3.82225e10 1.08735 0.543673 0.839297i \(-0.317033\pi\)
0.543673 + 0.839297i \(0.317033\pi\)
\(434\) 0 0
\(435\) 3.98111e9i 0.111185i
\(436\) 0 0
\(437\) −1.91546e10 −0.525228
\(438\) 0 0
\(439\) − 6.40288e10i − 1.72392i −0.506976 0.861960i \(-0.669237\pi\)
0.506976 0.861960i \(-0.330763\pi\)
\(440\) 0 0
\(441\) 1.04310e10 0.275785
\(442\) 0 0
\(443\) − 7.47659e10i − 1.94128i −0.240533 0.970641i \(-0.577322\pi\)
0.240533 0.970641i \(-0.422678\pi\)
\(444\) 0 0
\(445\) −4.42658e10 −1.12883
\(446\) 0 0
\(447\) 6.44756e10i 1.61497i
\(448\) 0 0
\(449\) 2.51987e10 0.620001 0.310000 0.950736i \(-0.399671\pi\)
0.310000 + 0.950736i \(0.399671\pi\)
\(450\) 0 0
\(451\) 1.44876e9i 0.0350180i
\(452\) 0 0
\(453\) 9.37373e10 2.22597
\(454\) 0 0
\(455\) 3.61181e10i 0.842711i
\(456\) 0 0
\(457\) 4.66828e9 0.107027 0.0535133 0.998567i \(-0.482958\pi\)
0.0535133 + 0.998567i \(0.482958\pi\)
\(458\) 0 0
\(459\) 5.03346e10i 1.13401i
\(460\) 0 0
\(461\) −3.88096e10 −0.859281 −0.429641 0.903000i \(-0.641360\pi\)
−0.429641 + 0.903000i \(0.641360\pi\)
\(462\) 0 0
\(463\) 3.23432e10i 0.703817i 0.936034 + 0.351908i \(0.114467\pi\)
−0.936034 + 0.351908i \(0.885533\pi\)
\(464\) 0 0
\(465\) 8.51316e10 1.82087
\(466\) 0 0
\(467\) 2.31902e10i 0.487570i 0.969829 + 0.243785i \(0.0783891\pi\)
−0.969829 + 0.243785i \(0.921611\pi\)
\(468\) 0 0
\(469\) 1.91546e10 0.395897
\(470\) 0 0
\(471\) − 6.18090e10i − 1.25594i
\(472\) 0 0
\(473\) −9.57731e9 −0.191337
\(474\) 0 0
\(475\) 1.41775e10i 0.278500i
\(476\) 0 0
\(477\) −1.51849e11 −2.93318
\(478\) 0 0
\(479\) 4.83542e9i 0.0918528i 0.998945 + 0.0459264i \(0.0146240\pi\)
−0.998945 + 0.0459264i \(0.985376\pi\)
\(480\) 0 0
\(481\) −2.19943e10 −0.410893
\(482\) 0 0
\(483\) − 6.39926e10i − 1.17582i
\(484\) 0 0
\(485\) 2.38018e10 0.430173
\(486\) 0 0
\(487\) − 3.03878e10i − 0.540236i −0.962827 0.270118i \(-0.912937\pi\)
0.962827 0.270118i \(-0.0870628\pi\)
\(488\) 0 0
\(489\) −3.46538e10 −0.606059
\(490\) 0 0
\(491\) 5.56483e10i 0.957472i 0.877959 + 0.478736i \(0.158905\pi\)
−0.877959 + 0.478736i \(0.841095\pi\)
\(492\) 0 0
\(493\) 2.76924e9 0.0468784
\(494\) 0 0
\(495\) − 1.32940e11i − 2.21429i
\(496\) 0 0
\(497\) −2.57681e10 −0.422335
\(498\) 0 0
\(499\) 7.88458e10i 1.27168i 0.771822 + 0.635838i \(0.219345\pi\)
−0.771822 + 0.635838i \(0.780655\pi\)
\(500\) 0 0
\(501\) −9.53224e10 −1.51302
\(502\) 0 0
\(503\) − 4.41092e10i − 0.689061i −0.938775 0.344530i \(-0.888038\pi\)
0.938775 0.344530i \(-0.111962\pi\)
\(504\) 0 0
\(505\) −3.36554e10 −0.517475
\(506\) 0 0
\(507\) 6.79923e9i 0.102903i
\(508\) 0 0
\(509\) 1.05927e10 0.157811 0.0789055 0.996882i \(-0.474857\pi\)
0.0789055 + 0.996882i \(0.474857\pi\)
\(510\) 0 0
\(511\) 1.66548e10i 0.244262i
\(512\) 0 0
\(513\) 1.08543e11 1.56723
\(514\) 0 0
\(515\) 8.40908e10i 1.19542i
\(516\) 0 0
\(517\) −5.49654e10 −0.769356
\(518\) 0 0
\(519\) 2.50979e11i 3.45914i
\(520\) 0 0
\(521\) 1.24958e11 1.69595 0.847973 0.530039i \(-0.177823\pi\)
0.847973 + 0.530039i \(0.177823\pi\)
\(522\) 0 0
\(523\) − 2.80408e10i − 0.374786i −0.982285 0.187393i \(-0.939996\pi\)
0.982285 0.187393i \(-0.0600038\pi\)
\(524\) 0 0
\(525\) −4.73649e10 −0.623476
\(526\) 0 0
\(527\) − 5.92172e10i − 0.767724i
\(528\) 0 0
\(529\) 4.72129e10 0.602890
\(530\) 0 0
\(531\) − 2.96900e11i − 3.73449i
\(532\) 0 0
\(533\) 2.09438e9 0.0259505
\(534\) 0 0
\(535\) 6.49363e10i 0.792633i
\(536\) 0 0
\(537\) −1.07479e11 −1.29248
\(538\) 0 0
\(539\) 1.47027e10i 0.174197i
\(540\) 0 0
\(541\) 1.44659e11 1.68871 0.844356 0.535782i \(-0.179983\pi\)
0.844356 + 0.535782i \(0.179983\pi\)
\(542\) 0 0
\(543\) − 8.90386e10i − 1.02419i
\(544\) 0 0
\(545\) 7.96205e10 0.902483
\(546\) 0 0
\(547\) − 1.03774e10i − 0.115915i −0.998319 0.0579573i \(-0.981541\pi\)
0.998319 0.0579573i \(-0.0184587\pi\)
\(548\) 0 0
\(549\) 3.24018e11 3.56681
\(550\) 0 0
\(551\) − 5.97167e9i − 0.0647872i
\(552\) 0 0
\(553\) −1.24693e11 −1.33334
\(554\) 0 0
\(555\) 5.74762e10i 0.605782i
\(556\) 0 0
\(557\) −5.47312e9 −0.0568610 −0.0284305 0.999596i \(-0.509051\pi\)
−0.0284305 + 0.999596i \(0.509051\pi\)
\(558\) 0 0
\(559\) 1.38453e10i 0.141793i
\(560\) 0 0
\(561\) −1.37087e11 −1.38403
\(562\) 0 0
\(563\) 4.36118e10i 0.434081i 0.976163 + 0.217040i \(0.0696403\pi\)
−0.976163 + 0.217040i \(0.930360\pi\)
\(564\) 0 0
\(565\) −1.20536e11 −1.18283
\(566\) 0 0
\(567\) 1.34593e11i 1.30224i
\(568\) 0 0
\(569\) 1.27822e10 0.121943 0.0609716 0.998140i \(-0.480580\pi\)
0.0609716 + 0.998140i \(0.480580\pi\)
\(570\) 0 0
\(571\) 7.59455e10i 0.714427i 0.934023 + 0.357213i \(0.116273\pi\)
−0.934023 + 0.357213i \(0.883727\pi\)
\(572\) 0 0
\(573\) −1.53237e11 −1.42149
\(574\) 0 0
\(575\) 2.30176e10i 0.210566i
\(576\) 0 0
\(577\) 2.13827e10 0.192912 0.0964560 0.995337i \(-0.469249\pi\)
0.0964560 + 0.995337i \(0.469249\pi\)
\(578\) 0 0
\(579\) 4.21353e9i 0.0374914i
\(580\) 0 0
\(581\) 1.87715e11 1.64738
\(582\) 0 0
\(583\) − 2.14035e11i − 1.85272i
\(584\) 0 0
\(585\) −1.92182e11 −1.64093
\(586\) 0 0
\(587\) − 6.07298e10i − 0.511504i −0.966742 0.255752i \(-0.917677\pi\)
0.966742 0.255752i \(-0.0823231\pi\)
\(588\) 0 0
\(589\) −1.27697e11 −1.06101
\(590\) 0 0
\(591\) 1.59711e11i 1.30914i
\(592\) 0 0
\(593\) 1.15978e11 0.937899 0.468949 0.883225i \(-0.344633\pi\)
0.468949 + 0.883225i \(0.344633\pi\)
\(594\) 0 0
\(595\) − 6.56538e10i − 0.523832i
\(596\) 0 0
\(597\) −1.48685e11 −1.17049
\(598\) 0 0
\(599\) − 2.40647e11i − 1.86927i −0.355607 0.934636i \(-0.615726\pi\)
0.355607 0.934636i \(-0.384274\pi\)
\(600\) 0 0
\(601\) −1.92942e11 −1.47887 −0.739434 0.673229i \(-0.764907\pi\)
−0.739434 + 0.673229i \(0.764907\pi\)
\(602\) 0 0
\(603\) 1.01921e11i 0.770892i
\(604\) 0 0
\(605\) 7.80591e10 0.582643
\(606\) 0 0
\(607\) − 1.62042e11i − 1.19364i −0.802376 0.596819i \(-0.796431\pi\)
0.802376 0.596819i \(-0.203569\pi\)
\(608\) 0 0
\(609\) 1.99504e10 0.145038
\(610\) 0 0
\(611\) 7.94597e10i 0.570141i
\(612\) 0 0
\(613\) 1.76424e11 1.24944 0.624722 0.780847i \(-0.285212\pi\)
0.624722 + 0.780847i \(0.285212\pi\)
\(614\) 0 0
\(615\) − 5.47311e9i − 0.0382590i
\(616\) 0 0
\(617\) −9.84986e10 −0.679656 −0.339828 0.940488i \(-0.610369\pi\)
−0.339828 + 0.940488i \(0.610369\pi\)
\(618\) 0 0
\(619\) 1.28596e10i 0.0875923i 0.999040 + 0.0437961i \(0.0139452\pi\)
−0.999040 + 0.0437961i \(0.986055\pi\)
\(620\) 0 0
\(621\) 1.76223e11 1.18494
\(622\) 0 0
\(623\) 2.21828e11i 1.47253i
\(624\) 0 0
\(625\) −8.45648e10 −0.554204
\(626\) 0 0
\(627\) 2.95618e11i 1.91276i
\(628\) 0 0
\(629\) 3.99802e10 0.255413
\(630\) 0 0
\(631\) 1.73463e11i 1.09418i 0.837073 + 0.547091i \(0.184265\pi\)
−0.837073 + 0.547091i \(0.815735\pi\)
\(632\) 0 0
\(633\) 3.94261e11 2.45566
\(634\) 0 0
\(635\) − 1.87548e11i − 1.15350i
\(636\) 0 0
\(637\) 2.12547e10 0.129091
\(638\) 0 0
\(639\) − 1.37111e11i − 0.822372i
\(640\) 0 0
\(641\) −1.13903e11 −0.674690 −0.337345 0.941381i \(-0.609529\pi\)
−0.337345 + 0.941381i \(0.609529\pi\)
\(642\) 0 0
\(643\) 7.04067e10i 0.411879i 0.978565 + 0.205940i \(0.0660250\pi\)
−0.978565 + 0.205940i \(0.933975\pi\)
\(644\) 0 0
\(645\) 3.61810e10 0.209046
\(646\) 0 0
\(647\) − 1.99175e11i − 1.13663i −0.822812 0.568314i \(-0.807596\pi\)
0.822812 0.568314i \(-0.192404\pi\)
\(648\) 0 0
\(649\) 4.18487e11 2.35886
\(650\) 0 0
\(651\) − 4.26617e11i − 2.37528i
\(652\) 0 0
\(653\) 6.49972e9 0.0357472 0.0178736 0.999840i \(-0.494310\pi\)
0.0178736 + 0.999840i \(0.494310\pi\)
\(654\) 0 0
\(655\) 1.10339e11i 0.599463i
\(656\) 0 0
\(657\) −8.86194e10 −0.475628
\(658\) 0 0
\(659\) − 2.20982e11i − 1.17170i −0.810421 0.585848i \(-0.800761\pi\)
0.810421 0.585848i \(-0.199239\pi\)
\(660\) 0 0
\(661\) −2.69549e11 −1.41199 −0.705995 0.708217i \(-0.749500\pi\)
−0.705995 + 0.708217i \(0.749500\pi\)
\(662\) 0 0
\(663\) 1.98177e11i 1.02565i
\(664\) 0 0
\(665\) −1.41578e11 −0.723949
\(666\) 0 0
\(667\) − 9.69518e9i − 0.0489838i
\(668\) 0 0
\(669\) −3.23476e11 −1.61487
\(670\) 0 0
\(671\) 4.56711e11i 2.25295i
\(672\) 0 0
\(673\) 9.44470e10 0.460392 0.230196 0.973144i \(-0.426063\pi\)
0.230196 + 0.973144i \(0.426063\pi\)
\(674\) 0 0
\(675\) − 1.30433e11i − 0.628309i
\(676\) 0 0
\(677\) 8.02735e10 0.382136 0.191068 0.981577i \(-0.438805\pi\)
0.191068 + 0.981577i \(0.438805\pi\)
\(678\) 0 0
\(679\) − 1.19277e11i − 0.561150i
\(680\) 0 0
\(681\) 2.11364e11 0.982751
\(682\) 0 0
\(683\) 3.00783e11i 1.38220i 0.722760 + 0.691099i \(0.242873\pi\)
−0.722760 + 0.691099i \(0.757127\pi\)
\(684\) 0 0
\(685\) 1.97085e11 0.895142
\(686\) 0 0
\(687\) − 2.40591e11i − 1.08007i
\(688\) 0 0
\(689\) −3.09416e11 −1.37298
\(690\) 0 0
\(691\) 3.06208e11i 1.34309i 0.740964 + 0.671544i \(0.234369\pi\)
−0.740964 + 0.671544i \(0.765631\pi\)
\(692\) 0 0
\(693\) −6.66199e11 −2.88849
\(694\) 0 0
\(695\) 1.79049e11i 0.767419i
\(696\) 0 0
\(697\) −3.80707e9 −0.0161309
\(698\) 0 0
\(699\) − 7.40348e10i − 0.310118i
\(700\) 0 0
\(701\) −2.73603e11 −1.13305 −0.566524 0.824045i \(-0.691712\pi\)
−0.566524 + 0.824045i \(0.691712\pi\)
\(702\) 0 0
\(703\) − 8.62143e10i − 0.352987i
\(704\) 0 0
\(705\) 2.07647e11 0.840562
\(706\) 0 0
\(707\) 1.68656e11i 0.675033i
\(708\) 0 0
\(709\) −1.76662e11 −0.699129 −0.349564 0.936912i \(-0.613670\pi\)
−0.349564 + 0.936912i \(0.613670\pi\)
\(710\) 0 0
\(711\) − 6.63484e11i − 2.59628i
\(712\) 0 0
\(713\) −2.07321e11 −0.802203
\(714\) 0 0
\(715\) − 2.70885e11i − 1.03648i
\(716\) 0 0
\(717\) 6.25718e11 2.36756
\(718\) 0 0
\(719\) − 2.25510e11i − 0.843821i −0.906637 0.421911i \(-0.861360\pi\)
0.906637 0.421911i \(-0.138640\pi\)
\(720\) 0 0
\(721\) 4.21402e11 1.55939
\(722\) 0 0
\(723\) 2.30227e11i 0.842565i
\(724\) 0 0
\(725\) −7.17600e9 −0.0259735
\(726\) 0 0
\(727\) 2.87080e11i 1.02770i 0.857881 + 0.513849i \(0.171781\pi\)
−0.857881 + 0.513849i \(0.828219\pi\)
\(728\) 0 0
\(729\) 2.14750e11 0.760366
\(730\) 0 0
\(731\) − 2.51673e10i − 0.0881388i
\(732\) 0 0
\(733\) −2.94176e11 −1.01904 −0.509520 0.860459i \(-0.670177\pi\)
−0.509520 + 0.860459i \(0.670177\pi\)
\(734\) 0 0
\(735\) − 5.55435e10i − 0.190320i
\(736\) 0 0
\(737\) −1.43660e11 −0.486928
\(738\) 0 0
\(739\) − 9.69888e10i − 0.325195i −0.986692 0.162598i \(-0.948013\pi\)
0.986692 0.162598i \(-0.0519872\pi\)
\(740\) 0 0
\(741\) 4.27355e11 1.41748
\(742\) 0 0
\(743\) − 1.01567e11i − 0.333271i −0.986019 0.166635i \(-0.946710\pi\)
0.986019 0.166635i \(-0.0532902\pi\)
\(744\) 0 0
\(745\) 2.31590e11 0.751788
\(746\) 0 0
\(747\) 9.98824e11i 3.20779i
\(748\) 0 0
\(749\) 3.25413e11 1.03397
\(750\) 0 0
\(751\) 4.17899e11i 1.31375i 0.754001 + 0.656873i \(0.228121\pi\)
−0.754001 + 0.656873i \(0.771879\pi\)
\(752\) 0 0
\(753\) 1.86457e10 0.0579960
\(754\) 0 0
\(755\) − 3.36696e11i − 1.03621i
\(756\) 0 0
\(757\) 1.82006e11 0.554244 0.277122 0.960835i \(-0.410619\pi\)
0.277122 + 0.960835i \(0.410619\pi\)
\(758\) 0 0
\(759\) 4.79945e11i 1.44619i
\(760\) 0 0
\(761\) −4.27419e11 −1.27443 −0.637213 0.770687i \(-0.719913\pi\)
−0.637213 + 0.770687i \(0.719913\pi\)
\(762\) 0 0
\(763\) − 3.99000e11i − 1.17727i
\(764\) 0 0
\(765\) 3.49341e11 1.02001
\(766\) 0 0
\(767\) − 6.04978e11i − 1.74807i
\(768\) 0 0
\(769\) −5.09969e11 −1.45827 −0.729136 0.684368i \(-0.760078\pi\)
−0.729136 + 0.684368i \(0.760078\pi\)
\(770\) 0 0
\(771\) − 7.49386e11i − 2.12074i
\(772\) 0 0
\(773\) −1.49408e11 −0.418462 −0.209231 0.977866i \(-0.567096\pi\)
−0.209231 + 0.977866i \(0.567096\pi\)
\(774\) 0 0
\(775\) 1.53451e11i 0.425366i
\(776\) 0 0
\(777\) 2.88029e11 0.790227
\(778\) 0 0
\(779\) 8.20966e9i 0.0222933i
\(780\) 0 0
\(781\) 1.93261e11 0.519445
\(782\) 0 0
\(783\) 5.49393e10i 0.146163i
\(784\) 0 0
\(785\) −2.22012e11 −0.584652
\(786\) 0 0
\(787\) − 7.33252e11i − 1.91141i −0.294323 0.955706i \(-0.595094\pi\)
0.294323 0.955706i \(-0.404906\pi\)
\(788\) 0 0
\(789\) −1.04891e12 −2.70665
\(790\) 0 0
\(791\) 6.04040e11i 1.54298i
\(792\) 0 0
\(793\) 6.60236e11 1.66958
\(794\) 0 0
\(795\) 8.08577e11i 2.02420i
\(796\) 0 0
\(797\) 3.02703e11 0.750212 0.375106 0.926982i \(-0.377606\pi\)
0.375106 + 0.926982i \(0.377606\pi\)
\(798\) 0 0
\(799\) − 1.44438e11i − 0.354401i
\(800\) 0 0
\(801\) −1.18034e12 −2.86732
\(802\) 0 0
\(803\) − 1.24911e11i − 0.300427i
\(804\) 0 0
\(805\) −2.29855e11 −0.547358
\(806\) 0 0
\(807\) − 4.91911e10i − 0.115982i
\(808\) 0 0
\(809\) −5.84316e11 −1.36412 −0.682062 0.731295i \(-0.738916\pi\)
−0.682062 + 0.731295i \(0.738916\pi\)
\(810\) 0 0
\(811\) 1.21470e11i 0.280793i 0.990095 + 0.140396i \(0.0448377\pi\)
−0.990095 + 0.140396i \(0.955162\pi\)
\(812\) 0 0
\(813\) −9.25344e8 −0.00211807
\(814\) 0 0
\(815\) 1.24473e11i 0.282127i
\(816\) 0 0
\(817\) −5.42714e10 −0.121810
\(818\) 0 0
\(819\) 9.63078e11i 2.14055i
\(820\) 0 0
\(821\) 4.52470e11 0.995903 0.497952 0.867205i \(-0.334086\pi\)
0.497952 + 0.867205i \(0.334086\pi\)
\(822\) 0 0
\(823\) − 3.06704e11i − 0.668528i −0.942479 0.334264i \(-0.891512\pi\)
0.942479 0.334264i \(-0.108488\pi\)
\(824\) 0 0
\(825\) 3.55237e11 0.766835
\(826\) 0 0
\(827\) − 2.93276e11i − 0.626982i −0.949591 0.313491i \(-0.898501\pi\)
0.949591 0.313491i \(-0.101499\pi\)
\(828\) 0 0
\(829\) 3.35532e11 0.710421 0.355210 0.934786i \(-0.384409\pi\)
0.355210 + 0.934786i \(0.384409\pi\)
\(830\) 0 0
\(831\) − 3.05356e11i − 0.640327i
\(832\) 0 0
\(833\) −3.86358e10 −0.0802434
\(834\) 0 0
\(835\) 3.42389e11i 0.704326i
\(836\) 0 0
\(837\) 1.17482e12 2.39369
\(838\) 0 0
\(839\) 3.42844e11i 0.691908i 0.938252 + 0.345954i \(0.112445\pi\)
−0.938252 + 0.345954i \(0.887555\pi\)
\(840\) 0 0
\(841\) −4.97224e11 −0.993958
\(842\) 0 0
\(843\) − 1.48029e12i − 2.93113i
\(844\) 0 0
\(845\) 2.44222e10 0.0479024
\(846\) 0 0
\(847\) − 3.91175e11i − 0.760042i
\(848\) 0 0
\(849\) −1.81802e11 −0.349919
\(850\) 0 0
\(851\) − 1.39972e11i − 0.266883i
\(852\) 0 0
\(853\) −5.08662e11 −0.960801 −0.480400 0.877049i \(-0.659509\pi\)
−0.480400 + 0.877049i \(0.659509\pi\)
\(854\) 0 0
\(855\) − 7.53328e11i − 1.40968i
\(856\) 0 0
\(857\) 6.06764e11 1.12486 0.562428 0.826846i \(-0.309867\pi\)
0.562428 + 0.826846i \(0.309867\pi\)
\(858\) 0 0
\(859\) 9.49431e11i 1.74378i 0.489705 + 0.871888i \(0.337105\pi\)
−0.489705 + 0.871888i \(0.662895\pi\)
\(860\) 0 0
\(861\) −2.74272e10 −0.0499078
\(862\) 0 0
\(863\) 2.99836e10i 0.0540556i 0.999635 + 0.0270278i \(0.00860426\pi\)
−0.999635 + 0.0270278i \(0.991396\pi\)
\(864\) 0 0
\(865\) 9.01494e11 1.61027
\(866\) 0 0
\(867\) 6.30222e11i 1.11536i
\(868\) 0 0
\(869\) 9.35196e11 1.63992
\(870\) 0 0
\(871\) 2.07679e11i 0.360844i
\(872\) 0 0
\(873\) 6.34669e11 1.09267
\(874\) 0 0
\(875\) 6.79283e11i 1.15883i
\(876\) 0 0
\(877\) −8.80195e11 −1.48792 −0.743961 0.668223i \(-0.767055\pi\)
−0.743961 + 0.668223i \(0.767055\pi\)
\(878\) 0 0
\(879\) 3.03546e11i 0.508474i
\(880\) 0 0
\(881\) 1.04085e12 1.72776 0.863879 0.503699i \(-0.168028\pi\)
0.863879 + 0.503699i \(0.168028\pi\)
\(882\) 0 0
\(883\) 7.60446e11i 1.25091i 0.780261 + 0.625454i \(0.215086\pi\)
−0.780261 + 0.625454i \(0.784914\pi\)
\(884\) 0 0
\(885\) −1.58095e12 −2.57718
\(886\) 0 0
\(887\) − 3.34097e11i − 0.539732i −0.962898 0.269866i \(-0.913021\pi\)
0.962898 0.269866i \(-0.0869794\pi\)
\(888\) 0 0
\(889\) −9.39853e11 −1.50471
\(890\) 0 0
\(891\) − 1.00945e12i − 1.60167i
\(892\) 0 0
\(893\) −3.11471e11 −0.489792
\(894\) 0 0
\(895\) 3.86053e11i 0.601666i
\(896\) 0 0
\(897\) 6.93823e11 1.07171
\(898\) 0 0
\(899\) − 6.46345e10i − 0.0989523i
\(900\) 0 0
\(901\) 5.62442e11 0.853451
\(902\) 0 0
\(903\) − 1.81312e11i − 0.272695i
\(904\) 0 0
\(905\) −3.19818e11 −0.476770
\(906\) 0 0
\(907\) − 7.65213e11i − 1.13071i −0.824846 0.565357i \(-0.808738\pi\)
0.824846 0.565357i \(-0.191262\pi\)
\(908\) 0 0
\(909\) −8.97412e11 −1.31443
\(910\) 0 0
\(911\) 3.83541e11i 0.556851i 0.960458 + 0.278425i \(0.0898125\pi\)
−0.960458 + 0.278425i \(0.910188\pi\)
\(912\) 0 0
\(913\) −1.40786e12 −2.02618
\(914\) 0 0
\(915\) − 1.72535e12i − 2.46146i
\(916\) 0 0
\(917\) 5.52937e11 0.781984
\(918\) 0 0
\(919\) − 6.82775e11i − 0.957229i −0.878025 0.478615i \(-0.841139\pi\)
0.878025 0.478615i \(-0.158861\pi\)
\(920\) 0 0
\(921\) −7.74022e11 −1.07576
\(922\) 0 0
\(923\) − 2.79384e11i − 0.384941i
\(924\) 0 0
\(925\) −1.03602e11 −0.141514
\(926\) 0 0
\(927\) 2.24226e12i 3.03645i
\(928\) 0 0
\(929\) 2.94973e11 0.396021 0.198011 0.980200i \(-0.436552\pi\)
0.198011 + 0.980200i \(0.436552\pi\)
\(930\) 0 0
\(931\) 8.33152e10i 0.110898i
\(932\) 0 0
\(933\) −1.52705e12 −2.01524
\(934\) 0 0
\(935\) 4.92404e11i 0.644280i
\(936\) 0 0
\(937\) 1.03941e12 1.34843 0.674217 0.738533i \(-0.264481\pi\)
0.674217 + 0.738533i \(0.264481\pi\)
\(938\) 0 0
\(939\) 4.25680e10i 0.0547546i
\(940\) 0 0
\(941\) 1.26891e11 0.161836 0.0809178 0.996721i \(-0.474215\pi\)
0.0809178 + 0.996721i \(0.474215\pi\)
\(942\) 0 0
\(943\) 1.33286e10i 0.0168554i
\(944\) 0 0
\(945\) 1.30251e12 1.63326
\(946\) 0 0
\(947\) 6.13064e11i 0.762265i 0.924520 + 0.381133i \(0.124466\pi\)
−0.924520 + 0.381133i \(0.875534\pi\)
\(948\) 0 0
\(949\) −1.80575e11 −0.222635
\(950\) 0 0
\(951\) − 7.54189e11i − 0.922058i
\(952\) 0 0
\(953\) 6.58227e11 0.798002 0.399001 0.916951i \(-0.369357\pi\)
0.399001 + 0.916951i \(0.369357\pi\)
\(954\) 0 0
\(955\) 5.50413e11i 0.661721i
\(956\) 0 0
\(957\) −1.49628e11 −0.178388
\(958\) 0 0
\(959\) − 9.87647e11i − 1.16769i
\(960\) 0 0
\(961\) −5.29246e11 −0.620532
\(962\) 0 0
\(963\) 1.73151e12i 2.01335i
\(964\) 0 0
\(965\) 1.51346e10 0.0174527
\(966\) 0 0
\(967\) 5.41485e11i 0.619271i 0.950855 + 0.309635i \(0.100207\pi\)
−0.950855 + 0.309635i \(0.899793\pi\)
\(968\) 0 0
\(969\) −7.76826e11 −0.881107
\(970\) 0 0
\(971\) 3.54981e11i 0.399327i 0.979865 + 0.199663i \(0.0639849\pi\)
−0.979865 + 0.199663i \(0.936015\pi\)
\(972\) 0 0
\(973\) 8.97263e11 1.00108
\(974\) 0 0
\(975\) − 5.13541e11i − 0.568273i
\(976\) 0 0
\(977\) 6.02238e11 0.660982 0.330491 0.943809i \(-0.392786\pi\)
0.330491 + 0.943809i \(0.392786\pi\)
\(978\) 0 0
\(979\) − 1.66371e12i − 1.81112i
\(980\) 0 0
\(981\) 2.12306e12 2.29238
\(982\) 0 0
\(983\) − 1.37465e12i − 1.47223i −0.676854 0.736117i \(-0.736657\pi\)
0.676854 0.736117i \(-0.263343\pi\)
\(984\) 0 0
\(985\) 5.73668e11 0.609419
\(986\) 0 0
\(987\) − 1.04058e12i − 1.09649i
\(988\) 0 0
\(989\) −8.81113e10 −0.0920972
\(990\) 0 0
\(991\) − 1.01081e12i − 1.04803i −0.851709 0.524015i \(-0.824433\pi\)
0.851709 0.524015i \(-0.175567\pi\)
\(992\) 0 0
\(993\) 1.43411e12 1.47498
\(994\) 0 0
\(995\) 5.34061e11i 0.544877i
\(996\) 0 0
\(997\) −3.28556e11 −0.332528 −0.166264 0.986081i \(-0.553170\pi\)
−0.166264 + 0.986081i \(0.553170\pi\)
\(998\) 0 0
\(999\) 7.93172e11i 0.796353i
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 16.9.c.a.15.1 2
3.2 odd 2 144.9.g.g.127.2 2
4.3 odd 2 inner 16.9.c.a.15.2 yes 2
5.2 odd 4 400.9.h.b.399.1 4
5.3 odd 4 400.9.h.b.399.3 4
5.4 even 2 400.9.b.c.351.2 2
8.3 odd 2 64.9.c.d.63.1 2
8.5 even 2 64.9.c.d.63.2 2
12.11 even 2 144.9.g.g.127.1 2
16.3 odd 4 256.9.d.f.127.2 4
16.5 even 4 256.9.d.f.127.1 4
16.11 odd 4 256.9.d.f.127.3 4
16.13 even 4 256.9.d.f.127.4 4
20.3 even 4 400.9.h.b.399.2 4
20.7 even 4 400.9.h.b.399.4 4
20.19 odd 2 400.9.b.c.351.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 1.1 even 1 trivial
16.9.c.a.15.2 yes 2 4.3 odd 2 inner
64.9.c.d.63.1 2 8.3 odd 2
64.9.c.d.63.2 2 8.5 even 2
144.9.g.g.127.1 2 12.11 even 2
144.9.g.g.127.2 2 3.2 odd 2
256.9.d.f.127.1 4 16.5 even 4
256.9.d.f.127.2 4 16.3 odd 4
256.9.d.f.127.3 4 16.11 odd 4
256.9.d.f.127.4 4 16.13 even 4
400.9.b.c.351.1 2 20.19 odd 2
400.9.b.c.351.2 2 5.4 even 2
400.9.h.b.399.1 4 5.2 odd 4
400.9.h.b.399.2 4 20.3 even 4
400.9.h.b.399.3 4 5.3 odd 4
400.9.h.b.399.4 4 20.7 even 4