Properties

Label 400.9.b.c.351.2
Level $400$
Weight $9$
Character 400.351
Analytic conductor $162.951$
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] = [400,9,Mod(351,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.351");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 400.b (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: \(162.951444024\)
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: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 351.2
Root \(0.500000 - 2.95804i\) of defining polynomial
Character \(\chi\) \(=\) 400.351
Dual form 400.9.b.c.351.1

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(0.340099\pi\)
−0.481481 + 0.876456i \(0.659901\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2555.75i − 1.06445i −0.846603 0.532225i \(-0.821356\pi\)
0.846603 0.532225i \(-0.178644\pi\)
\(8\) 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 \(-0.338782\pi\)
0.485102 + 0.874458i \(0.338782\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −50370.0 −0.603082 −0.301541 0.953453i \(-0.597501\pi\)
−0.301541 + 0.953453i \(0.597501\pi\)
\(18\) 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 \(0.102032\pi\)
−0.949064 + 0.315083i \(0.897968\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) −793730. −0.423512 −0.211756 0.977323i \(-0.567918\pi\)
−0.211756 + 0.977323i \(0.567918\pi\)
\(38\) 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 \(0.0232808\pi\)
−0.997327 + 0.0730736i \(0.976719\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.86755e6i 0.587651i 0.955859 + 0.293825i \(0.0949284\pi\)
−0.955859 + 0.293825i \(0.905072\pi\)
\(48\) 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 \(-0.750211\pi\)
−0.707575 + 0.706639i \(0.750211\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) 7.49473e6i 0.371926i 0.982557 + 0.185963i \(0.0595405\pi\)
−0.982557 + 0.185963i \(0.940460\pi\)
\(68\) 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 \(-0.536602\pi\)
−0.114736 + 0.993396i \(0.536602\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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 \(0.281656\pi\)
−0.633407 + 0.773819i \(0.718344\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) 4.66703e7 0.527173 0.263587 0.964636i \(-0.415095\pi\)
0.263587 + 0.964636i \(0.415095\pi\)
\(98\) 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 \(0.261640\pi\)
−0.680782 + 0.732486i \(0.738360\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.27326e8i 0.971364i 0.874136 + 0.485682i \(0.161429\pi\)
−0.874136 + 0.485682i \(0.838571\pi\)
\(108\) 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 \(-0.758057\pi\)
−0.724776 + 0.688984i \(0.758057\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) − 3.67741e8i − 1.41360i −0.707412 0.706802i \(-0.750137\pi\)
0.707412 0.706802i \(-0.249863\pi\)
\(128\) 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
\(135\) 0 0
\(136\) 0 0
\(137\) 3.86442e8 1.09699 0.548494 0.836155i \(-0.315201\pi\)
0.548494 + 0.836155i \(0.315201\pi\)
\(138\) 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
\(145\) 0 0
\(146\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) −4.35318e8 −0.716486 −0.358243 0.933628i \(-0.616624\pi\)
−0.358243 + 0.933628i \(0.616624\pi\)
\(158\) 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 \(0.0553047\pi\)
−0.984944 + 0.172872i \(0.944695\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.71351e8i 0.863145i 0.902078 + 0.431573i \(0.142041\pi\)
−0.902078 + 0.431573i \(0.857959\pi\)
\(168\) 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 \(-0.0520022\pi\)
0.986685 + 0.162644i \(0.0520022\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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
\(185\) 0 0
\(186\) 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 \(-0.496596\pi\)
0.0106940 + 0.999943i \(0.496596\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.12484e9 0.746837 0.373419 0.927663i \(-0.378186\pi\)
0.373419 + 0.927663i \(0.378186\pi\)
\(198\) 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
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 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 \(0.152374\pi\)
−0.887595 + 0.460624i \(0.847626\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.48863e9i − 0.560639i −0.959907 0.280319i \(-0.909560\pi\)
0.959907 0.280319i \(-0.0904404\pi\)
\(228\) 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 \(-0.528194\pi\)
−0.0884580 + 0.996080i \(0.528194\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 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
\(255\) 0 0
\(256\) 0 0
\(257\) −5.27789e9 −1.20984 −0.604920 0.796287i \(-0.706795\pi\)
−0.604920 + 0.796287i \(0.706795\pi\)
\(258\) 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 \(0.280765\pi\)
−0.635570 + 0.772044i \(0.719235\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) −2.15061e9 −0.365293 −0.182647 0.983179i \(-0.558466\pi\)
−0.182647 + 0.983179i \(0.558466\pi\)
\(278\) 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 \(0.0318238\pi\)
−0.995006 + 0.0998108i \(0.968176\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 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 \(-0.453670\pi\)
0.145037 + 0.989426i \(0.453670\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 5.45140e9i 0.613698i 0.951758 + 0.306849i \(0.0992747\pi\)
−0.951758 + 0.306849i \(0.900725\pi\)
\(308\) 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 \(-0.495028\pi\)
0.0156182 + 0.999878i \(0.495028\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.31172e9 −0.526015 −0.263007 0.964794i \(-0.584714\pi\)
−0.263007 + 0.964794i \(0.584714\pi\)
\(318\) 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
\(325\) 0 0
\(326\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) 1.84359e10 1.42937 0.714684 0.699448i \(-0.246571\pi\)
0.714684 + 0.699448i \(0.246571\pi\)
\(338\) 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
\(345\) 0 0
\(346\) 0 0
\(347\) 1.27822e10i 0.881636i 0.897597 + 0.440818i \(0.145312\pi\)
−0.897597 + 0.440818i \(0.854688\pi\)
\(348\) 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 \(-0.815521\pi\)
−0.836705 + 0.547653i \(0.815521\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) − 1.86229e10i − 1.02656i −0.858221 0.513280i \(-0.828430\pi\)
0.858221 0.513280i \(-0.171570\pi\)
\(368\) 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 \(-0.600042\pi\)
−0.309141 + 0.951016i \(0.600042\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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 \(-0.891852\pi\)
0.942835 0.333259i \(-0.108148\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 0 0
\(397\) −3.99456e10 −1.60808 −0.804039 0.594576i \(-0.797320\pi\)
−0.804039 + 0.594576i \(0.797320\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 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
\(415\) 0 0
\(416\) 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
\(425\) 0 0
\(426\) 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 \(-0.682967\pi\)
−0.543673 + 0.839297i \(0.682967\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 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 \(0.422678\pi\)
−0.240533 + 0.970641i \(0.577322\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 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
\(455\) 0 0
\(456\) 0 0
\(457\) −4.66828e9 −0.107027 −0.0535133 0.998567i \(-0.517042\pi\)
−0.0535133 + 0.998567i \(0.517042\pi\)
\(458\) 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 \(-0.885533\pi\)
0.936034 0.351908i \(-0.114467\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 2.31902e10i − 0.487570i −0.969829 0.243785i \(-0.921611\pi\)
0.969829 0.243785i \(-0.0783891\pi\)
\(468\) 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
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 3.03878e10i 0.540236i 0.962827 + 0.270118i \(0.0870628\pi\)
−0.962827 + 0.270118i \(0.912937\pi\)
\(488\) 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
\(495\) 0 0
\(496\) 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 \(0.111962\pi\)
−0.938775 + 0.344530i \(0.888038\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 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 \(0.0600038\pi\)
−0.982285 + 0.187393i \(0.939996\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 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
\(535\) 0 0
\(536\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) 1.03774e10i 0.115915i 0.998319 + 0.0579573i \(0.0184587\pi\)
−0.998319 + 0.0579573i \(0.981541\pi\)
\(548\) 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
\(555\) 0 0
\(556\) 0 0
\(557\) 5.47312e9 0.0568610 0.0284305 0.999596i \(-0.490949\pi\)
0.0284305 + 0.999596i \(0.490949\pi\)
\(558\) 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 \(-0.930360\pi\)
0.976163 0.217040i \(-0.0696403\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) −2.13827e10 −0.192912 −0.0964560 0.995337i \(-0.530751\pi\)
−0.0964560 + 0.995337i \(0.530751\pi\)
\(578\) 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
\(585\) 0 0
\(586\) 0 0
\(587\) 6.07298e10i 0.511504i 0.966742 + 0.255752i \(0.0823231\pi\)
−0.966742 + 0.255752i \(0.917677\pi\)
\(588\) 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 \(-0.655367\pi\)
−0.468949 + 0.883225i \(0.655367\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 1.62042e11i 1.19364i 0.802376 + 0.596819i \(0.203569\pi\)
−0.802376 + 0.596819i \(0.796431\pi\)
\(608\) 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 \(-0.714788\pi\)
−0.624722 + 0.780847i \(0.714788\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.84986e10 0.679656 0.339828 0.940488i \(-0.389631\pi\)
0.339828 + 0.940488i \(0.389631\pi\)
\(618\) 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
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 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 \(-0.933975\pi\)
0.978565 0.205940i \(-0.0660250\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.99175e11i 1.13663i 0.822812 + 0.568314i \(0.192404\pi\)
−0.822812 + 0.568314i \(0.807596\pi\)
\(648\) 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 \(-0.505690\pi\)
−0.0178736 + 0.999840i \(0.505690\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 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
\(665\) 0 0
\(666\) 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 \(-0.573937\pi\)
−0.230196 + 0.973144i \(0.573937\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.02735e10 −0.382136 −0.191068 0.981577i \(-0.561195\pi\)
−0.191068 + 0.981577i \(0.561195\pi\)
\(678\) 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 \(-0.757127\pi\)
0.722760 0.691099i \(-0.242873\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 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
\(705\) 0 0
\(706\) 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
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) − 2.87080e11i − 1.02770i −0.857881 0.513849i \(-0.828219\pi\)
0.857881 0.513849i \(-0.171781\pi\)
\(728\) 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 \(-0.329823\pi\)
0.509520 + 0.860459i \(0.329823\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 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 \(0.0532902\pi\)
−0.986019 + 0.166635i \(0.946710\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) −1.82006e11 −0.554244 −0.277122 0.960835i \(-0.589381\pi\)
−0.277122 + 0.960835i \(0.589381\pi\)
\(758\) 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
\(765\) 0 0
\(766\) 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 \(-0.432904\pi\)
0.209231 + 0.977866i \(0.432904\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) 7.33252e11i 1.91141i 0.294323 + 0.955706i \(0.404906\pi\)
−0.294323 + 0.955706i \(0.595094\pi\)
\(788\) 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
\(795\) 0 0
\(796\) 0 0
\(797\) −3.02703e11 −0.750212 −0.375106 0.926982i \(-0.622394\pi\)
−0.375106 + 0.926982i \(0.622394\pi\)
\(798\) 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
\(805\) 0 0
\(806\) 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
\(815\) 0 0
\(816\) 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 \(0.108488\pi\)
−0.942479 + 0.334264i \(0.891512\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.93276e11i 0.626982i 0.949591 + 0.313491i \(0.101499\pi\)
−0.949591 + 0.313491i \(0.898501\pi\)
\(828\) 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
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 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 \(-0.340491\pi\)
0.480400 + 0.877049i \(0.340491\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.06764e11 −1.12486 −0.562428 0.826846i \(-0.690133\pi\)
−0.562428 + 0.826846i \(0.690133\pi\)
\(858\) 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 \(-0.991396\pi\)
0.999635 0.0270278i \(-0.00860426\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) 8.80195e11 1.48792 0.743961 0.668223i \(-0.232945\pi\)
0.743961 + 0.668223i \(0.232945\pi\)
\(878\) 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 \(-0.784914\pi\)
0.780261 0.625454i \(-0.215086\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.34097e11i 0.539732i 0.962898 + 0.269866i \(0.0869794\pi\)
−0.962898 + 0.269866i \(0.913021\pi\)
\(888\) 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
\(895\) 0 0
\(896\) 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
\(905\) 0 0
\(906\) 0 0
\(907\) 7.65213e11i 1.13071i 0.824846 + 0.565357i \(0.191262\pi\)
−0.824846 + 0.565357i \(0.808738\pi\)
\(908\) 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
\(915\) 0 0
\(916\) 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
\(925\) 0 0
\(926\) 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
\(935\) 0 0
\(936\) 0 0
\(937\) −1.03941e12 −1.34843 −0.674217 0.738533i \(-0.735519\pi\)
−0.674217 + 0.738533i \(0.735519\pi\)
\(938\) 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
\(945\) 0 0
\(946\) 0 0
\(947\) − 6.13064e11i − 0.762265i −0.924520 0.381133i \(-0.875534\pi\)
0.924520 0.381133i \(-0.124466\pi\)
\(948\) 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 \(-0.630643\pi\)
−0.399001 + 0.916951i \(0.630643\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) − 5.41485e11i − 0.619271i −0.950855 0.309635i \(-0.899793\pi\)
0.950855 0.309635i \(-0.100207\pi\)
\(968\) 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
\(975\) 0 0
\(976\) 0 0
\(977\) −6.02238e11 −0.660982 −0.330491 0.943809i \(-0.607214\pi\)
−0.330491 + 0.943809i \(0.607214\pi\)
\(978\) 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 \(0.263343\pi\)
−0.676854 + 0.736117i \(0.736657\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) 3.28556e11 0.332528 0.166264 0.986081i \(-0.446830\pi\)
0.166264 + 0.986081i \(0.446830\pi\)
\(998\) 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 400.9.b.c.351.2 2
4.3 odd 2 inner 400.9.b.c.351.1 2
5.2 odd 4 400.9.h.b.399.3 4
5.3 odd 4 400.9.h.b.399.1 4
5.4 even 2 16.9.c.a.15.1 2
15.14 odd 2 144.9.g.g.127.2 2
20.3 even 4 400.9.h.b.399.4 4
20.7 even 4 400.9.h.b.399.2 4
20.19 odd 2 16.9.c.a.15.2 yes 2
40.19 odd 2 64.9.c.d.63.1 2
40.29 even 2 64.9.c.d.63.2 2
60.59 even 2 144.9.g.g.127.1 2
80.19 odd 4 256.9.d.f.127.2 4
80.29 even 4 256.9.d.f.127.4 4
80.59 odd 4 256.9.d.f.127.3 4
80.69 even 4 256.9.d.f.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 5.4 even 2
16.9.c.a.15.2 yes 2 20.19 odd 2
64.9.c.d.63.1 2 40.19 odd 2
64.9.c.d.63.2 2 40.29 even 2
144.9.g.g.127.1 2 60.59 even 2
144.9.g.g.127.2 2 15.14 odd 2
256.9.d.f.127.1 4 80.69 even 4
256.9.d.f.127.2 4 80.19 odd 4
256.9.d.f.127.3 4 80.59 odd 4
256.9.d.f.127.4 4 80.29 even 4
400.9.b.c.351.1 2 4.3 odd 2 inner
400.9.b.c.351.2 2 1.1 even 1 trivial
400.9.h.b.399.1 4 5.3 odd 4
400.9.h.b.399.2 4 20.7 even 4
400.9.h.b.399.3 4 5.2 odd 4
400.9.h.b.399.4 4 20.3 even 4