Properties

Label 400.9.h.b.399.3
Level $400$
Weight $9$
Character 400.399
Analytic conductor $162.951$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,9,Mod(399,400)]
 
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, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.399");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 400.h (of order \(2\), degree \(1\), not 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{35})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 17x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 399.3
Root \(-2.95804 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 400.399
Dual form 400.9.h.b.399.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+141.986 q^{3} +2555.75 q^{7} +13599.0 q^{9} +O(q^{10})\) \(q+141.986 q^{3} +2555.75 q^{7} +13599.0 q^{9} -19168.1i q^{11} -27710.0i q^{13} -50370.0i q^{17} +108619. i q^{19} +362880. q^{21} +176347. q^{23} +999297. q^{27} -54978.0 q^{29} -1.17564e6i q^{31} -2.72160e6i q^{33} -793730. i q^{37} -3.93443e6i q^{39} -75582.0 q^{41} +499648. q^{43} -2.86755e6 q^{47} +767039. q^{49} -7.15183e6i q^{51} +1.11662e7i q^{53} +1.54224e7i q^{57} -2.18325e7i q^{59} -2.38266e7 q^{61} +3.47556e7 q^{63} -7.49473e6 q^{67} +2.50387e7 q^{69} +1.00824e7i q^{71} +6.51661e6i q^{73} -4.89888e7i q^{77} -4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7 q^{83} -7.80610e6 q^{87} -8.67958e7 q^{89} -7.08197e7i q^{91} -1.66925e8i q^{93} +4.66703e7i q^{97} -2.60667e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 54396 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 54396 q^{9} + 1451520 q^{21} - 219912 q^{29} - 302328 q^{41} + 3068156 q^{49} - 95306488 q^{61} + 100154880 q^{69} + 210652164 q^{81} - 347183112 q^{89}+O(q^{100}) \) 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
\(3\) 141.986 1.75291 0.876456 0.481481i \(-0.159901\pi\)
0.876456 + 0.481481i \(0.159901\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2555.75 1.06445 0.532225 0.846603i \(-0.321356\pi\)
0.532225 + 0.846603i \(0.321356\pi\)
\(8\) 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
\(13\) − 27710.0i − 0.970204i −0.874458 0.485102i \(-0.838782\pi\)
0.874458 0.485102i \(-0.161218\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 50370.0i − 0.603082i −0.953453 0.301541i \(-0.902499\pi\)
0.953453 0.301541i \(-0.0975010\pi\)
\(18\) 0 0
\(19\) 108619.i 0.833474i 0.909027 + 0.416737i \(0.136826\pi\)
−0.909027 + 0.416737i \(0.863174\pi\)
\(20\) 0 0
\(21\) 362880. 1.86589
\(22\) 0 0
\(23\) 176347. 0.630167 0.315083 0.949064i \(-0.397968\pi\)
0.315083 + 0.949064i \(0.397968\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 999297. 1.88035
\(28\) 0 0
\(29\) −54978.0 −0.0777315 −0.0388657 0.999244i \(-0.512374\pi\)
−0.0388657 + 0.999244i \(0.512374\pi\)
\(30\) 0 0
\(31\) − 1.17564e6i − 1.27300i −0.771276 0.636501i \(-0.780381\pi\)
0.771276 0.636501i \(-0.219619\pi\)
\(32\) 0 0
\(33\) − 2.72160e6i − 2.29493i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 793730.i − 0.423512i −0.977323 0.211756i \(-0.932082\pi\)
0.977323 0.211756i \(-0.0679182\pi\)
\(38\) 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. 0.146147 0.0730736 0.997327i \(-0.476719\pi\)
0.0730736 + 0.997327i \(0.476719\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.86755e6 −0.587651 −0.293825 0.955859i \(-0.594928\pi\)
−0.293825 + 0.955859i \(0.594928\pi\)
\(48\) 0 0
\(49\) 767039. 0.133056
\(50\) 0 0
\(51\) − 7.15183e6i − 1.05715i
\(52\) 0 0
\(53\) 1.11662e7i 1.41515i 0.706639 + 0.707575i \(0.250211\pi\)
−0.706639 + 0.707575i \(0.749789\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.54224e7i 1.46101i
\(58\) 0 0
\(59\) − 2.18325e7i − 1.80175i −0.434078 0.900875i \(-0.642926\pi\)
0.434078 0.900875i \(-0.357074\pi\)
\(60\) 0 0
\(61\) −2.38266e7 −1.72085 −0.860425 0.509577i \(-0.829802\pi\)
−0.860425 + 0.509577i \(0.829802\pi\)
\(62\) 0 0
\(63\) 3.47556e7 2.20629
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.49473e6 −0.371926 −0.185963 0.982557i \(-0.559540\pi\)
−0.185963 + 0.982557i \(0.559540\pi\)
\(68\) 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
\(73\) 6.51661e6i 0.229472i 0.993396 + 0.114736i \(0.0366023\pi\)
−0.993396 + 0.114736i \(0.963398\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.89888e7i − 1.39359i
\(78\) 0 0
\(79\) − 4.87892e7i − 1.25261i −0.779579 0.626304i \(-0.784567\pi\)
0.779579 0.626304i \(-0.215433\pi\)
\(80\) 0 0
\(81\) 5.26630e7 1.22339
\(82\) 0 0
\(83\) 7.34483e7 1.54764 0.773819 0.633407i \(-0.218344\pi\)
0.773819 + 0.633407i \(0.218344\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.80610e6 −0.136256
\(88\) 0 0
\(89\) −8.67958e7 −1.38337 −0.691685 0.722199i \(-0.743131\pi\)
−0.691685 + 0.722199i \(0.743131\pi\)
\(90\) 0 0
\(91\) − 7.08197e7i − 1.03273i
\(92\) 0 0
\(93\) − 1.66925e8i − 2.23146i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.66703e7i 0.527173i 0.964636 + 0.263587i \(0.0849055\pi\)
−0.964636 + 0.263587i \(0.915095\pi\)
\(98\) 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
\(103\) 1.64884e8 1.46497 0.732486 0.680782i \(-0.238360\pi\)
0.732486 + 0.680782i \(0.238360\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.27326e8 −0.971364 −0.485682 0.874136i \(-0.661429\pi\)
−0.485682 + 0.874136i \(0.661429\pi\)
\(108\) 0 0
\(109\) 1.56119e8 1.10598 0.552992 0.833186i \(-0.313486\pi\)
0.552992 + 0.833186i \(0.313486\pi\)
\(110\) 0 0
\(111\) − 1.12698e8i − 0.742380i
\(112\) 0 0
\(113\) 2.36346e8i 1.44955i 0.688984 + 0.724776i \(0.258057\pi\)
−0.688984 + 0.724776i \(0.741943\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.76828e8i − 2.01094i
\(118\) 0 0
\(119\) − 1.28733e8i − 0.641951i
\(120\) 0 0
\(121\) −1.53057e8 −0.714023
\(122\) 0 0
\(123\) −1.07316e7 −0.0468860
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.67741e8 1.41360 0.706802 0.707412i \(-0.250137\pi\)
0.706802 + 0.707412i \(0.250137\pi\)
\(128\) 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
\(133\) 2.77603e8i 0.887193i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.86442e8i 1.09699i 0.836155 + 0.548494i \(0.184799\pi\)
−0.836155 + 0.548494i \(0.815201\pi\)
\(138\) 0 0
\(139\) 3.51077e8i 0.940465i 0.882543 + 0.470232i \(0.155830\pi\)
−0.882543 + 0.470232i \(0.844170\pi\)
\(140\) 0 0
\(141\) −4.07151e8 −1.03010
\(142\) 0 0
\(143\) −5.31148e8 −1.27020
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.08909e8 0.233235
\(148\) 0 0
\(149\) 4.54099e8 0.921308 0.460654 0.887580i \(-0.347615\pi\)
0.460654 + 0.887580i \(0.347615\pi\)
\(150\) 0 0
\(151\) 6.60188e8i 1.26987i 0.772565 + 0.634936i \(0.218973\pi\)
−0.772565 + 0.634936i \(0.781027\pi\)
\(152\) 0 0
\(153\) − 6.84982e8i − 1.25001i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.35318e8i − 0.716486i −0.933628 0.358243i \(-0.883376\pi\)
0.933628 0.358243i \(-0.116624\pi\)
\(158\) 0 0
\(159\) 1.58544e9i 2.48063i
\(160\) 0 0
\(161\) 4.50697e8 0.670782
\(162\) 0 0
\(163\) 2.44065e8 0.345744 0.172872 0.984944i \(-0.444695\pi\)
0.172872 + 0.984944i \(0.444695\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.71351e8 −0.863145 −0.431573 0.902078i \(-0.642041\pi\)
−0.431573 + 0.902078i \(0.642041\pi\)
\(168\) 0 0
\(169\) 4.78866e7 0.0587040
\(170\) 0 0
\(171\) 1.47711e9i 1.72754i
\(172\) 0 0
\(173\) − 1.76764e9i − 1.97337i −0.162644 0.986685i \(-0.552002\pi\)
0.162644 0.986685i \(-0.447998\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 3.09990e9i − 3.15831i
\(178\) 0 0
\(179\) 7.56967e8i 0.737335i 0.929561 + 0.368668i \(0.120186\pi\)
−0.929561 + 0.368668i \(0.879814\pi\)
\(180\) 0 0
\(181\) 6.27094e8 0.584277 0.292138 0.956376i \(-0.405633\pi\)
0.292138 + 0.956376i \(0.405633\pi\)
\(182\) 0 0
\(183\) −3.38304e9 −3.01650
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −9.65497e8 −0.789559
\(188\) 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
\(193\) − 2.96757e7i − 0.0213881i −0.999943 0.0106940i \(-0.996596\pi\)
0.999943 0.0106940i \(-0.00340408\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.12484e9i 0.746837i 0.927663 + 0.373419i \(0.121814\pi\)
−0.927663 + 0.373419i \(0.878186\pi\)
\(198\) 0 0
\(199\) 1.04718e9i 0.667742i 0.942619 + 0.333871i \(0.108355\pi\)
−0.942619 + 0.333871i \(0.891645\pi\)
\(200\) 0 0
\(201\) −1.06415e9 −0.651954
\(202\) 0 0
\(203\) −1.40510e8 −0.0827413
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.39814e9 1.30615
\(208\) 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
\(213\) 1.43156e9i 0.695491i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 3.00465e9i − 1.35505i
\(218\) 0 0
\(219\) 9.25267e8i 0.402245i
\(220\) 0 0
\(221\) −1.39575e9 −0.585113
\(222\) 0 0
\(223\) 2.27822e9 0.921248 0.460624 0.887595i \(-0.347626\pi\)
0.460624 + 0.887595i \(0.347626\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.48863e9 0.560639 0.280319 0.959907i \(-0.409560\pi\)
0.280319 + 0.959907i \(0.409560\pi\)
\(228\) 0 0
\(229\) −1.69447e9 −0.616157 −0.308079 0.951361i \(-0.599686\pi\)
−0.308079 + 0.951361i \(0.599686\pi\)
\(230\) 0 0
\(231\) − 6.95572e9i − 2.44284i
\(232\) 0 0
\(233\) 5.21423e8i 0.176916i 0.996080 + 0.0884580i \(0.0281939\pi\)
−0.996080 + 0.0884580i \(0.971806\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 6.92738e9i − 2.19571i
\(238\) 0 0
\(239\) − 4.40690e9i − 1.35065i −0.737522 0.675323i \(-0.764004\pi\)
0.737522 0.675323i \(-0.235996\pi\)
\(240\) 0 0
\(241\) −1.62148e9 −0.480666 −0.240333 0.970691i \(-0.577257\pi\)
−0.240333 + 0.970691i \(0.577257\pi\)
\(242\) 0 0
\(243\) 9.21023e8 0.264147
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.00984e9 0.808640
\(248\) 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
\(253\) − 3.38023e9i − 0.825019i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 5.27789e9i − 1.20984i −0.796287 0.604920i \(-0.793205\pi\)
0.796287 0.604920i \(-0.206795\pi\)
\(258\) 0 0
\(259\) − 2.02857e9i − 0.450808i
\(260\) 0 0
\(261\) −7.47646e8 −0.161114
\(262\) 0 0
\(263\) 7.38745e9 1.54409 0.772044 0.635570i \(-0.219235\pi\)
0.772044 + 0.635570i \(0.219235\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.23238e10 −2.42493
\(268\) 0 0
\(269\) −3.46450e8 −0.0661655 −0.0330828 0.999453i \(-0.510532\pi\)
−0.0330828 + 0.999453i \(0.510532\pi\)
\(270\) 0 0
\(271\) − 6.51715e6i − 0.00120832i −1.00000 0.000604158i \(-0.999808\pi\)
1.00000 0.000604158i \(-0.000192310\pi\)
\(272\) 0 0
\(273\) − 1.00554e10i − 1.81029i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.15061e9i − 0.365293i −0.983179 0.182647i \(-0.941534\pi\)
0.983179 0.182647i \(-0.0584664\pi\)
\(278\) 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
\(283\) 1.28042e9 0.199622 0.0998108 0.995006i \(-0.468176\pi\)
0.0998108 + 0.995006i \(0.468176\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.93168e8 −0.0284714
\(288\) 0 0
\(289\) 4.43862e9 0.636292
\(290\) 0 0
\(291\) 6.62652e9i 0.924089i
\(292\) 0 0
\(293\) − 2.13786e9i − 0.290074i −0.989426 0.145037i \(-0.953670\pi\)
0.989426 0.145037i \(-0.0463301\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.91546e10i − 2.46177i
\(298\) 0 0
\(299\) − 4.88656e9i − 0.611390i
\(300\) 0 0
\(301\) 1.27697e9 0.155567
\(302\) 0 0
\(303\) 9.36980e9 1.11163
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5.45140e9 −0.613698 −0.306849 0.951758i \(-0.599275\pi\)
−0.306849 + 0.951758i \(0.599275\pi\)
\(308\) 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
\(313\) − 2.99804e8i − 0.0312364i −0.999878 0.0156182i \(-0.995028\pi\)
0.999878 0.0156182i \(-0.00497163\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 5.31172e9i − 0.526015i −0.964794 0.263007i \(-0.915286\pi\)
0.964794 0.263007i \(-0.0847144\pi\)
\(318\) 0 0
\(319\) 1.05382e9i 0.101767i
\(320\) 0 0
\(321\) −1.80785e10 −1.70272
\(322\) 0 0
\(323\) 5.47115e9 0.502653
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.21667e10 1.93869
\(328\) 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
\(333\) − 1.07939e10i − 0.877815i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.84359e10i 1.42937i 0.699448 + 0.714684i \(0.253429\pi\)
−0.699448 + 0.714684i \(0.746571\pi\)
\(338\) 0 0
\(339\) 3.35578e10i 2.54094i
\(340\) 0 0
\(341\) −2.25348e10 −1.66662
\(342\) 0 0
\(343\) −1.27730e10 −0.922820
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.27822e10 −0.881636 −0.440818 0.897597i \(-0.645312\pi\)
−0.440818 + 0.897597i \(0.645312\pi\)
\(348\) 0 0
\(349\) 6.39381e8 0.0430981 0.0215490 0.999768i \(-0.493140\pi\)
0.0215490 + 0.999768i \(0.493140\pi\)
\(350\) 0 0
\(351\) − 2.76905e10i − 1.82433i
\(352\) 0 0
\(353\) 2.59837e10i 1.67341i 0.547653 + 0.836705i \(0.315521\pi\)
−0.547653 + 0.836705i \(0.684479\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 1.82783e10i − 1.12528i
\(358\) 0 0
\(359\) − 2.22541e10i − 1.33978i −0.742461 0.669889i \(-0.766342\pi\)
0.742461 0.669889i \(-0.233658\pi\)
\(360\) 0 0
\(361\) 5.18543e9 0.305320
\(362\) 0 0
\(363\) −2.17320e10 −1.25162
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.86229e10 1.02656 0.513280 0.858221i \(-0.328430\pi\)
0.513280 + 0.858221i \(0.328430\pi\)
\(368\) 0 0
\(369\) −1.02784e9 −0.0554396
\(370\) 0 0
\(371\) 2.85380e10i 1.50636i
\(372\) 0 0
\(373\) 1.19680e10i 0.618283i 0.951016 + 0.309141i \(0.100042\pi\)
−0.951016 + 0.309141i \(0.899958\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.52344e9i 0.0754154i
\(378\) 0 0
\(379\) 2.30787e10i 1.11855i 0.828982 + 0.559275i \(0.188920\pi\)
−0.828982 + 0.559275i \(0.811080\pi\)
\(380\) 0 0
\(381\) 5.22141e10 2.47792
\(382\) 0 0
\(383\) −1.43419e10 −0.666518 −0.333259 0.942835i \(-0.608148\pi\)
−0.333259 + 0.942835i \(0.608148\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.79472e9 0.302920
\(388\) 0 0
\(389\) 2.73457e10 1.19424 0.597119 0.802152i \(-0.296312\pi\)
0.597119 + 0.802152i \(0.296312\pi\)
\(390\) 0 0
\(391\) − 8.88257e9i − 0.380042i
\(392\) 0 0
\(393\) − 3.07187e10i − 1.28775i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 3.99456e10i − 1.60808i −0.594576 0.804039i \(-0.702680\pi\)
0.594576 0.804039i \(-0.297320\pi\)
\(398\) 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
\(403\) −3.25771e10 −1.23507
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.52143e10 −0.554465
\(408\) 0 0
\(409\) −1.14283e10 −0.408404 −0.204202 0.978929i \(-0.565460\pi\)
−0.204202 + 0.978929i \(0.565460\pi\)
\(410\) 0 0
\(411\) 5.48693e10i 1.92292i
\(412\) 0 0
\(413\) − 5.57982e10i − 1.91788i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.98479e10i 1.64855i
\(418\) 0 0
\(419\) 1.10009e10i 0.356922i 0.983947 + 0.178461i \(0.0571119\pi\)
−0.983947 + 0.178461i \(0.942888\pi\)
\(420\) 0 0
\(421\) 2.28766e10 0.728220 0.364110 0.931356i \(-0.381373\pi\)
0.364110 + 0.931356i \(0.381373\pi\)
\(422\) 0 0
\(423\) −3.89958e10 −1.21802
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.08948e10 −1.83176
\(428\) 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
\(433\) 3.82225e10i 1.08735i 0.839297 + 0.543673i \(0.182967\pi\)
−0.839297 + 0.543673i \(0.817033\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.91546e10i 0.525228i
\(438\) 0 0
\(439\) 6.40288e10i 1.72392i 0.506976 + 0.861960i \(0.330763\pi\)
−0.506976 + 0.861960i \(0.669237\pi\)
\(440\) 0 0
\(441\) 1.04310e10 0.275785
\(442\) 0 0
\(443\) 7.47659e10 1.94128 0.970641 0.240533i \(-0.0773222\pi\)
0.970641 + 0.240533i \(0.0773222\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.44756e10 1.61497
\(448\) 0 0
\(449\) −2.51987e10 −0.620001 −0.310000 0.950736i \(-0.600329\pi\)
−0.310000 + 0.950736i \(0.600329\pi\)
\(450\) 0 0
\(451\) 1.44876e9i 0.0350180i
\(452\) 0 0
\(453\) 9.37373e10i 2.22597i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 4.66828e9i − 0.107027i −0.998567 0.0535133i \(-0.982958\pi\)
0.998567 0.0535133i \(-0.0170420\pi\)
\(458\) 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
\(463\) −3.23432e10 −0.703817 −0.351908 0.936034i \(-0.614467\pi\)
−0.351908 + 0.936034i \(0.614467\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.31902e10 0.487570 0.243785 0.969829i \(-0.421611\pi\)
0.243785 + 0.969829i \(0.421611\pi\)
\(468\) 0 0
\(469\) −1.91546e10 −0.395897
\(470\) 0 0
\(471\) − 6.18090e10i − 1.25594i
\(472\) 0 0
\(473\) − 9.57731e9i − 0.191337i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.51849e11i 2.93318i
\(478\) 0 0
\(479\) − 4.83542e9i − 0.0918528i −0.998945 0.0459264i \(-0.985376\pi\)
0.998945 0.0459264i \(-0.0146240\pi\)
\(480\) 0 0
\(481\) −2.19943e10 −0.410893
\(482\) 0 0
\(483\) 6.39926e10 1.17582
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3.03878e10 −0.540236 −0.270118 0.962827i \(-0.587063\pi\)
−0.270118 + 0.962827i \(0.587063\pi\)
\(488\) 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
\(493\) 2.76924e9i 0.0468784i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.57681e10i 0.422335i
\(498\) 0 0
\(499\) − 7.88458e10i − 1.27168i −0.771822 0.635838i \(-0.780655\pi\)
0.771822 0.635838i \(-0.219345\pi\)
\(500\) 0 0
\(501\) −9.53224e10 −1.51302
\(502\) 0 0
\(503\) 4.41092e10 0.689061 0.344530 0.938775i \(-0.388038\pi\)
0.344530 + 0.938775i \(0.388038\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.79923e9 0.102903
\(508\) 0 0
\(509\) −1.05927e10 −0.157811 −0.0789055 0.996882i \(-0.525143\pi\)
−0.0789055 + 0.996882i \(0.525143\pi\)
\(510\) 0 0
\(511\) 1.66548e10i 0.244262i
\(512\) 0 0
\(513\) 1.08543e11i 1.56723i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.49654e10i 0.769356i
\(518\) 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
\(523\) 2.80408e10 0.374786 0.187393 0.982285i \(-0.439996\pi\)
0.187393 + 0.982285i \(0.439996\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.92172e10 −0.767724
\(528\) 0 0
\(529\) −4.72129e10 −0.602890
\(530\) 0 0
\(531\) − 2.96900e11i − 3.73449i
\(532\) 0 0
\(533\) 2.09438e9i 0.0259505i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.07479e11i 1.29248i
\(538\) 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
\(543\) 8.90386e10 1.02419
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.03774e10 −0.115915 −0.0579573 0.998319i \(-0.518459\pi\)
−0.0579573 + 0.998319i \(0.518459\pi\)
\(548\) 0 0
\(549\) −3.24018e11 −3.56681
\(550\) 0 0
\(551\) − 5.97167e9i − 0.0647872i
\(552\) 0 0
\(553\) − 1.24693e11i − 1.33334i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.47312e9i 0.0568610i 0.999596 + 0.0284305i \(0.00905093\pi\)
−0.999596 + 0.0284305i \(0.990949\pi\)
\(558\) 0 0
\(559\) − 1.38453e10i − 0.141793i
\(560\) 0 0
\(561\) −1.37087e11 −1.38403
\(562\) 0 0
\(563\) −4.36118e10 −0.434081 −0.217040 0.976163i \(-0.569640\pi\)
−0.217040 + 0.976163i \(0.569640\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.34593e11 1.30224
\(568\) 0 0
\(569\) −1.27822e10 −0.121943 −0.0609716 0.998140i \(-0.519420\pi\)
−0.0609716 + 0.998140i \(0.519420\pi\)
\(570\) 0 0
\(571\) 7.59455e10i 0.714427i 0.934023 + 0.357213i \(0.116273\pi\)
−0.934023 + 0.357213i \(0.883727\pi\)
\(572\) 0 0
\(573\) − 1.53237e11i − 1.42149i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.13827e10i − 0.192912i −0.995337 0.0964560i \(-0.969249\pi\)
0.995337 0.0964560i \(-0.0307507\pi\)
\(578\) 0 0
\(579\) − 4.21353e9i − 0.0374914i
\(580\) 0 0
\(581\) 1.87715e11 1.64738
\(582\) 0 0
\(583\) 2.14035e11 1.85272
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.07298e10 −0.511504 −0.255752 0.966742i \(-0.582323\pi\)
−0.255752 + 0.966742i \(0.582323\pi\)
\(588\) 0 0
\(589\) 1.27697e11 1.06101
\(590\) 0 0
\(591\) 1.59711e11i 1.30914i
\(592\) 0 0
\(593\) 1.15978e11i 0.937899i 0.883225 + 0.468949i \(0.155367\pi\)
−0.883225 + 0.468949i \(0.844633\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.48685e11i 1.17049i
\(598\) 0 0
\(599\) 2.40647e11i 1.86927i 0.355607 + 0.934636i \(0.384274\pi\)
−0.355607 + 0.934636i \(0.615726\pi\)
\(600\) 0 0
\(601\) −1.92942e11 −1.47887 −0.739434 0.673229i \(-0.764907\pi\)
−0.739434 + 0.673229i \(0.764907\pi\)
\(602\) 0 0
\(603\) −1.01921e11 −0.770892
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.62042e11 −1.19364 −0.596819 0.802376i \(-0.703569\pi\)
−0.596819 + 0.802376i \(0.703569\pi\)
\(608\) 0 0
\(609\) −1.99504e10 −0.145038
\(610\) 0 0
\(611\) 7.94597e10i 0.570141i
\(612\) 0 0
\(613\) 1.76424e11i 1.24944i 0.780847 + 0.624722i \(0.214788\pi\)
−0.780847 + 0.624722i \(0.785212\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.84986e10i 0.679656i 0.940488 + 0.339828i \(0.110369\pi\)
−0.940488 + 0.339828i \(0.889631\pi\)
\(618\) 0 0
\(619\) − 1.28596e10i − 0.0875923i −0.999040 0.0437961i \(-0.986055\pi\)
0.999040 0.0437961i \(-0.0139452\pi\)
\(620\) 0 0
\(621\) 1.76223e11 1.18494
\(622\) 0 0
\(623\) −2.21828e11 −1.47253
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.95618e11 1.91276
\(628\) 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
\(633\) 3.94261e11i 2.45566i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 2.12547e10i − 0.129091i
\(638\) 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
\(643\) −7.04067e10 −0.411879 −0.205940 0.978565i \(-0.566025\pi\)
−0.205940 + 0.978565i \(0.566025\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.99175e11 −1.13663 −0.568314 0.822812i \(-0.692404\pi\)
−0.568314 + 0.822812i \(0.692404\pi\)
\(648\) 0 0
\(649\) −4.18487e11 −2.35886
\(650\) 0 0
\(651\) − 4.26617e11i − 2.37528i
\(652\) 0 0
\(653\) 6.49972e9i 0.0357472i 0.999840 + 0.0178736i \(0.00568964\pi\)
−0.999840 + 0.0178736i \(0.994310\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.86194e10i 0.475628i
\(658\) 0 0
\(659\) 2.20982e11i 1.17170i 0.810421 + 0.585848i \(0.199239\pi\)
−0.810421 + 0.585848i \(0.800761\pi\)
\(660\) 0 0
\(661\) −2.69549e11 −1.41199 −0.705995 0.708217i \(-0.749500\pi\)
−0.705995 + 0.708217i \(0.749500\pi\)
\(662\) 0 0
\(663\) −1.98177e11 −1.02565
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.69518e9 −0.0489838
\(668\) 0 0
\(669\) 3.23476e11 1.61487
\(670\) 0 0
\(671\) 4.56711e11i 2.25295i
\(672\) 0 0
\(673\) 9.44470e10i 0.460392i 0.973144 + 0.230196i \(0.0739368\pi\)
−0.973144 + 0.230196i \(0.926063\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 8.02735e10i − 0.382136i −0.981577 0.191068i \(-0.938805\pi\)
0.981577 0.191068i \(-0.0611950\pi\)
\(678\) 0 0
\(679\) 1.19277e11i 0.561150i
\(680\) 0 0
\(681\) 2.11364e11 0.982751
\(682\) 0 0
\(683\) −3.00783e11 −1.38220 −0.691099 0.722760i \(-0.742873\pi\)
−0.691099 + 0.722760i \(0.742873\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −2.40591e11 −1.08007
\(688\) 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
\(693\) − 6.66199e11i − 2.88849i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.80707e9i 0.0161309i
\(698\) 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
\(703\) 8.62143e10 0.352987
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.68656e11 0.675033
\(708\) 0 0
\(709\) 1.76662e11 0.699129 0.349564 0.936912i \(-0.386330\pi\)
0.349564 + 0.936912i \(0.386330\pi\)
\(710\) 0 0
\(711\) − 6.63484e11i − 2.59628i
\(712\) 0 0
\(713\) − 2.07321e11i − 0.802203i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 6.25718e11i − 2.36756i
\(718\) 0 0
\(719\) 2.25510e11i 0.843821i 0.906637 + 0.421911i \(0.138640\pi\)
−0.906637 + 0.421911i \(0.861360\pi\)
\(720\) 0 0
\(721\) 4.21402e11 1.55939
\(722\) 0 0
\(723\) −2.30227e11 −0.842565
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.87080e11 1.02770 0.513849 0.857881i \(-0.328219\pi\)
0.513849 + 0.857881i \(0.328219\pi\)
\(728\) 0 0
\(729\) −2.14750e11 −0.760366
\(730\) 0 0
\(731\) − 2.51673e10i − 0.0881388i
\(732\) 0 0
\(733\) − 2.94176e11i − 1.01904i −0.860459 0.509520i \(-0.829823\pi\)
0.860459 0.509520i \(-0.170177\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.43660e11i 0.486928i
\(738\) 0 0
\(739\) 9.69888e10i 0.325195i 0.986692 + 0.162598i \(0.0519872\pi\)
−0.986692 + 0.162598i \(0.948013\pi\)
\(740\) 0 0
\(741\) 4.27355e11 1.41748
\(742\) 0 0
\(743\) 1.01567e11 0.333271 0.166635 0.986019i \(-0.446710\pi\)
0.166635 + 0.986019i \(0.446710\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 9.98824e11 3.20779
\(748\) 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
\(753\) 1.86457e10i 0.0579960i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 1.82006e11i − 0.554244i −0.960835 0.277122i \(-0.910619\pi\)
0.960835 0.277122i \(-0.0893806\pi\)
\(758\) 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
\(763\) 3.99000e11 1.17727
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.04978e11 −1.74807
\(768\) 0 0
\(769\) 5.09969e11 1.45827 0.729136 0.684368i \(-0.239922\pi\)
0.729136 + 0.684368i \(0.239922\pi\)
\(770\) 0 0
\(771\) − 7.49386e11i − 2.12074i
\(772\) 0 0
\(773\) − 1.49408e11i − 0.418462i −0.977866 0.209231i \(-0.932904\pi\)
0.977866 0.209231i \(-0.0670961\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.88029e11i − 0.790227i
\(778\) 0 0
\(779\) − 8.20966e9i − 0.0222933i
\(780\) 0 0
\(781\) 1.93261e11 0.519445
\(782\) 0 0
\(783\) −5.49393e10 −0.146163
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −7.33252e11 −1.91141 −0.955706 0.294323i \(-0.904906\pi\)
−0.955706 + 0.294323i \(0.904906\pi\)
\(788\) 0 0
\(789\) 1.04891e12 2.70665
\(790\) 0 0
\(791\) 6.04040e11i 1.54298i
\(792\) 0 0
\(793\) 6.60236e11i 1.66958i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 3.02703e11i − 0.750212i −0.926982 0.375106i \(-0.877606\pi\)
0.926982 0.375106i \(-0.122394\pi\)
\(798\) 0 0
\(799\) 1.44438e11i 0.354401i
\(800\) 0 0
\(801\) −1.18034e12 −2.86732
\(802\) 0 0
\(803\) 1.24911e11 0.300427
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.91911e10 −0.115982
\(808\) 0 0
\(809\) 5.84316e11 1.36412 0.682062 0.731295i \(-0.261084\pi\)
0.682062 + 0.731295i \(0.261084\pi\)
\(810\) 0 0
\(811\) 1.21470e11i 0.280793i 0.990095 + 0.140396i \(0.0448377\pi\)
−0.990095 + 0.140396i \(0.955162\pi\)
\(812\) 0 0
\(813\) − 9.25344e8i − 0.00211807i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5.42714e10i 0.121810i
\(818\) 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
\(823\) 3.06704e11 0.668528 0.334264 0.942479i \(-0.391512\pi\)
0.334264 + 0.942479i \(0.391512\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.93276e11 −0.626982 −0.313491 0.949591i \(-0.601499\pi\)
−0.313491 + 0.949591i \(0.601499\pi\)
\(828\) 0 0
\(829\) −3.35532e11 −0.710421 −0.355210 0.934786i \(-0.615591\pi\)
−0.355210 + 0.934786i \(0.615591\pi\)
\(830\) 0 0
\(831\) − 3.05356e11i − 0.640327i
\(832\) 0 0
\(833\) − 3.86358e10i − 0.0802434i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 1.17482e12i − 2.39369i
\(838\) 0 0
\(839\) − 3.42844e11i − 0.691908i −0.938252 0.345954i \(-0.887555\pi\)
0.938252 0.345954i \(-0.112445\pi\)
\(840\) 0 0
\(841\) −4.97224e11 −0.993958
\(842\) 0 0
\(843\) 1.48029e12 2.93113
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.91175e11 −0.760042
\(848\) 0 0
\(849\) 1.81802e11 0.349919
\(850\) 0 0
\(851\) − 1.39972e11i − 0.266883i
\(852\) 0 0
\(853\) − 5.08662e11i − 0.960801i −0.877049 0.480400i \(-0.840491\pi\)
0.877049 0.480400i \(-0.159509\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 6.06764e11i − 1.12486i −0.826846 0.562428i \(-0.809867\pi\)
0.826846 0.562428i \(-0.190133\pi\)
\(858\) 0 0
\(859\) − 9.49431e11i − 1.74378i −0.489705 0.871888i \(-0.662895\pi\)
0.489705 0.871888i \(-0.337105\pi\)
\(860\) 0 0
\(861\) −2.74272e10 −0.0499078
\(862\) 0 0
\(863\) −2.99836e10 −0.0540556 −0.0270278 0.999635i \(-0.508604\pi\)
−0.0270278 + 0.999635i \(0.508604\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 6.30222e11 1.11536
\(868\) 0 0
\(869\) −9.35196e11 −1.63992
\(870\) 0 0
\(871\) 2.07679e11i 0.360844i
\(872\) 0 0
\(873\) 6.34669e11i 1.09267i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.80195e11i 1.48792i 0.668223 + 0.743961i \(0.267055\pi\)
−0.668223 + 0.743961i \(0.732945\pi\)
\(878\) 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
\(883\) −7.60446e11 −1.25091 −0.625454 0.780261i \(-0.715086\pi\)
−0.625454 + 0.780261i \(0.715086\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.34097e11 −0.539732 −0.269866 0.962898i \(-0.586979\pi\)
−0.269866 + 0.962898i \(0.586979\pi\)
\(888\) 0 0
\(889\) 9.39853e11 1.50471
\(890\) 0 0
\(891\) − 1.00945e12i − 1.60167i
\(892\) 0 0
\(893\) − 3.11471e11i − 0.489792i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 6.93823e11i − 1.07171i
\(898\) 0 0
\(899\) 6.46345e10i 0.0989523i
\(900\) 0 0
\(901\) 5.62442e11 0.853451
\(902\) 0 0
\(903\) 1.81312e11 0.272695
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −7.65213e11 −1.13071 −0.565357 0.824846i \(-0.691262\pi\)
−0.565357 + 0.824846i \(0.691262\pi\)
\(908\) 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
\(913\) − 1.40786e12i − 2.02618i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 5.52937e11i − 0.781984i
\(918\) 0 0
\(919\) 6.82775e11i 0.957229i 0.878025 + 0.478615i \(0.158861\pi\)
−0.878025 + 0.478615i \(0.841139\pi\)
\(920\) 0 0
\(921\) −7.74022e11 −1.07576
\(922\) 0 0
\(923\) 2.79384e11 0.384941
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.24226e12 3.03645
\(928\) 0 0
\(929\) −2.94973e11 −0.396021 −0.198011 0.980200i \(-0.563448\pi\)
−0.198011 + 0.980200i \(0.563448\pi\)
\(930\) 0 0
\(931\) 8.33152e10i 0.110898i
\(932\) 0 0
\(933\) − 1.52705e12i − 2.01524i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1.03941e12i − 1.34843i −0.738533 0.674217i \(-0.764481\pi\)
0.738533 0.674217i \(-0.235519\pi\)
\(938\) 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
\(943\) −1.33286e10 −0.0168554
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.13064e11 0.762265 0.381133 0.924520i \(-0.375534\pi\)
0.381133 + 0.924520i \(0.375534\pi\)
\(948\) 0 0
\(949\) 1.80575e11 0.222635
\(950\) 0 0
\(951\) − 7.54189e11i − 0.922058i
\(952\) 0 0
\(953\) 6.58227e11i 0.798002i 0.916951 + 0.399001i \(0.130643\pi\)
−0.916951 + 0.399001i \(0.869357\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.49628e11i 0.178388i
\(958\) 0 0
\(959\) 9.87647e11i 1.16769i
\(960\) 0 0
\(961\) −5.29246e11 −0.620532
\(962\) 0 0
\(963\) −1.73151e12 −2.01335
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 5.41485e11 0.619271 0.309635 0.950855i \(-0.399793\pi\)
0.309635 + 0.950855i \(0.399793\pi\)
\(968\) 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
\(973\) 8.97263e11i 1.00108i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 6.02238e11i − 0.660982i −0.943809 0.330491i \(-0.892786\pi\)
0.943809 0.330491i \(-0.107214\pi\)
\(978\) 0 0
\(979\) 1.66371e12i 1.81112i
\(980\) 0 0
\(981\) 2.12306e12 2.29238
\(982\) 0 0
\(983\) 1.37465e12 1.47223 0.736117 0.676854i \(-0.236657\pi\)
0.736117 + 0.676854i \(0.236657\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.04058e12 −1.09649
\(988\) 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
\(993\) 1.43411e12i 1.47498i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.28556e11i 0.332528i 0.986081 + 0.166264i \(0.0531704\pi\)
−0.986081 + 0.166264i \(0.946830\pi\)
\(998\) 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.h.b.399.3 4
4.3 odd 2 inner 400.9.h.b.399.2 4
5.2 odd 4 16.9.c.a.15.1 2
5.3 odd 4 400.9.b.c.351.2 2
5.4 even 2 inner 400.9.h.b.399.1 4
15.2 even 4 144.9.g.g.127.2 2
20.3 even 4 400.9.b.c.351.1 2
20.7 even 4 16.9.c.a.15.2 yes 2
20.19 odd 2 inner 400.9.h.b.399.4 4
40.27 even 4 64.9.c.d.63.1 2
40.37 odd 4 64.9.c.d.63.2 2
60.47 odd 4 144.9.g.g.127.1 2
80.27 even 4 256.9.d.f.127.3 4
80.37 odd 4 256.9.d.f.127.1 4
80.67 even 4 256.9.d.f.127.2 4
80.77 odd 4 256.9.d.f.127.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 5.2 odd 4
16.9.c.a.15.2 yes 2 20.7 even 4
64.9.c.d.63.1 2 40.27 even 4
64.9.c.d.63.2 2 40.37 odd 4
144.9.g.g.127.1 2 60.47 odd 4
144.9.g.g.127.2 2 15.2 even 4
256.9.d.f.127.1 4 80.37 odd 4
256.9.d.f.127.2 4 80.67 even 4
256.9.d.f.127.3 4 80.27 even 4
256.9.d.f.127.4 4 80.77 odd 4
400.9.b.c.351.1 2 20.3 even 4
400.9.b.c.351.2 2 5.3 odd 4
400.9.h.b.399.1 4 5.4 even 2 inner
400.9.h.b.399.2 4 4.3 odd 2 inner
400.9.h.b.399.3 4 1.1 even 1 trivial
400.9.h.b.399.4 4 20.19 odd 2 inner