Properties

Label 256.9.d.f.127.4
Level $256$
Weight $9$
Character 256.127
Analytic conductor $104.289$
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] = [256,9,Mod(127,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 256.d (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: \(104.288924176\)
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: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.4
Root \(-2.95804 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 256.127
Dual form 256.9.d.f.127.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+141.986 q^{3} +510.000i q^{5} -2555.75i q^{7} +13599.0 q^{9} +O(q^{10})\) \(q+141.986 q^{3} +510.000i q^{5} -2555.75i q^{7} +13599.0 q^{9} -19168.1 q^{11} -27710.0i q^{13} +72412.8i q^{15} +50370.0 q^{17} +108619. q^{19} -362880. i q^{21} +176347. i q^{23} +130525. q^{25} +999297. q^{27} +54978.0i q^{29} -1.17564e6i q^{31} -2.72160e6 q^{33} +1.30343e6 q^{35} -793730. i q^{37} -3.93443e6i q^{39} +75582.0 q^{41} -499648. q^{43} +6.93549e6i q^{45} -2.86755e6i q^{47} -767039. q^{49} +7.15183e6 q^{51} -1.11662e7i q^{53} -9.77573e6i q^{55} +1.54224e7 q^{57} +2.18325e7 q^{59} -2.38266e7i q^{61} -3.47556e7i q^{63} +1.41321e7 q^{65} +7.49473e6 q^{67} +2.50387e7i q^{69} -1.00824e7i q^{71} -6.51661e6 q^{73} +1.85327e7 q^{75} +4.89888e7i q^{77} +4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7 q^{83} +2.56887e7i q^{85} +7.80610e6i q^{87} -8.67958e7 q^{89} -7.08197e7 q^{91} -1.66925e8i q^{93} +5.53958e7i q^{95} -4.66703e7 q^{97} -2.60667e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 54396 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 54396 q^{9} + 201480 q^{17} + 522100 q^{25} - 10886400 q^{33} + 302328 q^{41} - 3068156 q^{49} + 61689600 q^{57} + 56528400 q^{65} - 26066440 q^{73} + 210652164 q^{81} - 347183112 q^{89} - 186681080 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(3\) 141.986 1.75291 0.876456 0.481481i \(-0.159901\pi\)
0.876456 + 0.481481i \(0.159901\pi\)
\(4\) 0 0
\(5\) 510.000i 0.816000i 0.912982 + 0.408000i \(0.133774\pi\)
−0.912982 + 0.408000i \(0.866226\pi\)
\(6\) 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
\(11\) −19168.1 −1.30921 −0.654603 0.755972i \(-0.727164\pi\)
−0.654603 + 0.755972i \(0.727164\pi\)
\(12\) 0 0
\(13\) − 27710.0i − 0.970204i −0.874458 0.485102i \(-0.838782\pi\)
0.874458 0.485102i \(-0.161218\pi\)
\(14\) 0 0
\(15\) 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. 0.833474 0.416737 0.909027i \(-0.363174\pi\)
0.416737 + 0.909027i \(0.363174\pi\)
\(20\) 0 0
\(21\) − 362880.i − 1.86589i
\(22\) 0 0
\(23\) 176347.i 0.630167i 0.949064 + 0.315083i \(0.102032\pi\)
−0.949064 + 0.315083i \(0.897968\pi\)
\(24\) 0 0
\(25\) 130525. 0.334144
\(26\) 0 0
\(27\) 999297. 1.88035
\(28\) 0 0
\(29\) 54978.0i 0.0777315i 0.999244 + 0.0388657i \(0.0123745\pi\)
−0.999244 + 0.0388657i \(0.987626\pi\)
\(30\) 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
\(35\) 1.30343e6 0.868592
\(36\) 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 \(-0.495743\pi\)
0.0133737 + 0.999911i \(0.495743\pi\)
\(42\) 0 0
\(43\) −499648. −0.146147 −0.0730736 0.997327i \(-0.523281\pi\)
−0.0730736 + 0.997327i \(0.523281\pi\)
\(44\) 0 0
\(45\) 6.93549e6i 1.69133i
\(46\) 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
\(51\) 7.15183e6 1.05715
\(52\) 0 0
\(53\) − 1.11662e7i − 1.41515i −0.706639 0.707575i \(-0.749789\pi\)
0.706639 0.707575i \(-0.250211\pi\)
\(54\) 0 0
\(55\) − 9.77573e6i − 1.06831i
\(56\) 0 0
\(57\) 1.54224e7 1.46101
\(58\) 0 0
\(59\) 2.18325e7 1.80175 0.900875 0.434078i \(-0.142926\pi\)
0.900875 + 0.434078i \(0.142926\pi\)
\(60\) 0 0
\(61\) − 2.38266e7i − 1.72085i −0.509577 0.860425i \(-0.670198\pi\)
0.509577 0.860425i \(-0.329802\pi\)
\(62\) 0 0
\(63\) − 3.47556e7i − 2.20629i
\(64\) 0 0
\(65\) 1.41321e7 0.791687
\(66\) 0 0
\(67\) 7.49473e6 0.371926 0.185963 0.982557i \(-0.440460\pi\)
0.185963 + 0.982557i \(0.440460\pi\)
\(68\) 0 0
\(69\) 2.50387e7i 1.10463i
\(70\) 0 0
\(71\) − 1.00824e7i − 0.396763i −0.980125 0.198382i \(-0.936431\pi\)
0.980125 0.198382i \(-0.0635685\pi\)
\(72\) 0 0
\(73\) −6.51661e6 −0.229472 −0.114736 0.993396i \(-0.536602\pi\)
−0.114736 + 0.993396i \(0.536602\pi\)
\(74\) 0 0
\(75\) 1.85327e7 0.585725
\(76\) 0 0
\(77\) 4.89888e7i 1.39359i
\(78\) 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
\(83\) 7.34483e7 1.54764 0.773819 0.633407i \(-0.218344\pi\)
0.773819 + 0.633407i \(0.218344\pi\)
\(84\) 0 0
\(85\) 2.56887e7i 0.492115i
\(86\) 0 0
\(87\) 7.80610e6i 0.136256i
\(88\) 0 0
\(89\) −8.67958e7 −1.38337 −0.691685 0.722199i \(-0.743131\pi\)
−0.691685 + 0.722199i \(0.743131\pi\)
\(90\) 0 0
\(91\) −7.08197e7 −1.03273
\(92\) 0 0
\(93\) − 1.66925e8i − 2.23146i
\(94\) 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
\(99\) −2.60667e8 −2.71360
\(100\) 0 0
\(101\) − 6.59910e7i − 0.634161i −0.948399 0.317080i \(-0.897297\pi\)
0.948399 0.317080i \(-0.102703\pi\)
\(102\) 0 0
\(103\) 1.64884e8i 1.46497i 0.680782 + 0.732486i \(0.261640\pi\)
−0.680782 + 0.732486i \(0.738360\pi\)
\(104\) 0 0
\(105\) 1.85069e8 1.52257
\(106\) 0 0
\(107\) −1.27326e8 −0.971364 −0.485682 0.874136i \(-0.661429\pi\)
−0.485682 + 0.874136i \(0.661429\pi\)
\(108\) 0 0
\(109\) − 1.56119e8i − 1.10598i −0.833186 0.552992i \(-0.813486\pi\)
0.833186 0.552992i \(-0.186514\pi\)
\(110\) 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
\(115\) −8.99367e7 −0.514216
\(116\) 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
\(125\) 2.65786e8i 1.08866i
\(126\) 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
\(131\) 2.16350e8 0.734636 0.367318 0.930095i \(-0.380276\pi\)
0.367318 + 0.930095i \(0.380276\pi\)
\(132\) 0 0
\(133\) − 2.77603e8i − 0.887193i
\(134\) 0 0
\(135\) 5.09641e8i 1.53437i
\(136\) 0 0
\(137\) 3.86442e8 1.09699 0.548494 0.836155i \(-0.315201\pi\)
0.548494 + 0.836155i \(0.315201\pi\)
\(138\) 0 0
\(139\) −3.51077e8 −0.940465 −0.470232 0.882543i \(-0.655830\pi\)
−0.470232 + 0.882543i \(0.655830\pi\)
\(140\) 0 0
\(141\) − 4.07151e8i − 1.03010i
\(142\) 0 0
\(143\) 5.31148e8i 1.27020i
\(144\) 0 0
\(145\) −2.80388e7 −0.0634289
\(146\) 0 0
\(147\) −1.08909e8 −0.233235
\(148\) 0 0
\(149\) 4.54099e8i 0.921308i 0.887580 + 0.460654i \(0.152385\pi\)
−0.887580 + 0.460654i \(0.847615\pi\)
\(150\) 0 0
\(151\) − 6.60188e8i − 1.26987i −0.772565 0.634936i \(-0.781027\pi\)
0.772565 0.634936i \(-0.218973\pi\)
\(152\) 0 0
\(153\) 6.84982e8 1.25001
\(154\) 0 0
\(155\) 5.99578e8 1.03877
\(156\) 0 0
\(157\) 4.35318e8i 0.716486i 0.933628 + 0.358243i \(0.116624\pi\)
−0.933628 + 0.358243i \(0.883376\pi\)
\(158\) 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
\(165\) − 1.38802e9i − 1.87266i
\(166\) 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
\(171\) 1.47711e9 1.72754
\(172\) 0 0
\(173\) − 1.76764e9i − 1.97337i −0.162644 0.986685i \(-0.552002\pi\)
0.162644 0.986685i \(-0.447998\pi\)
\(174\) 0 0
\(175\) − 3.33589e8i − 0.355680i
\(176\) 0 0
\(177\) 3.09990e9 3.15831
\(178\) 0 0
\(179\) 7.56967e8 0.737335 0.368668 0.929561i \(-0.379814\pi\)
0.368668 + 0.929561i \(0.379814\pi\)
\(180\) 0 0
\(181\) − 6.27094e8i − 0.584277i −0.956376 0.292138i \(-0.905633\pi\)
0.956376 0.292138i \(-0.0943668\pi\)
\(182\) 0 0
\(183\) − 3.38304e9i − 3.01650i
\(184\) 0 0
\(185\) 4.04802e8 0.345586
\(186\) 0 0
\(187\) −9.65497e8 −0.789559
\(188\) 0 0
\(189\) − 2.55395e9i − 2.00154i
\(190\) 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
\(195\) 2.00656e9 1.38776
\(196\) 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
\(205\) 3.85468e7i 0.0218259i
\(206\) 0 0
\(207\) 2.39814e9i 1.30615i
\(208\) 0 0
\(209\) −2.08202e9 −1.09119
\(210\) 0 0
\(211\) −2.77676e9 −1.40090 −0.700452 0.713699i \(-0.747018\pi\)
−0.700452 + 0.713699i \(0.747018\pi\)
\(212\) 0 0
\(213\) − 1.43156e9i − 0.695491i
\(214\) 0 0
\(215\) − 2.54821e8i − 0.119256i
\(216\) 0 0
\(217\) −3.00465e9 −1.35505
\(218\) 0 0
\(219\) −9.25267e8 −0.402245
\(220\) 0 0
\(221\) − 1.39575e9i − 0.585113i
\(222\) 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
\(227\) −1.48863e9 −0.560639 −0.280319 0.959907i \(-0.590440\pi\)
−0.280319 + 0.959907i \(0.590440\pi\)
\(228\) 0 0
\(229\) − 1.69447e9i − 0.616157i −0.951361 0.308079i \(-0.900314\pi\)
0.951361 0.308079i \(-0.0996860\pi\)
\(230\) 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
\(235\) 1.46245e9 0.479523
\(236\) 0 0
\(237\) 6.92738e9i 2.19571i
\(238\) 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
\(243\) 9.21023e8 0.264147
\(244\) 0 0
\(245\) − 3.91190e8i − 0.108573i
\(246\) 0 0
\(247\) − 3.00984e9i − 0.808640i
\(248\) 0 0
\(249\) 1.04286e10 2.71287
\(250\) 0 0
\(251\) 1.31321e8 0.0330855 0.0165428 0.999863i \(-0.494734\pi\)
0.0165428 + 0.999863i \(0.494734\pi\)
\(252\) 0 0
\(253\) − 3.38023e9i − 0.825019i
\(254\) 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
\(259\) −2.02857e9 −0.450808
\(260\) 0 0
\(261\) 7.47646e8i 0.161114i
\(262\) 0 0
\(263\) 7.38745e9i 1.54409i 0.635570 + 0.772044i \(0.280765\pi\)
−0.635570 + 0.772044i \(0.719235\pi\)
\(264\) 0 0
\(265\) 5.69477e9 1.15476
\(266\) 0 0
\(267\) −1.23238e10 −2.42493
\(268\) 0 0
\(269\) 3.46450e8i 0.0661655i 0.999453 + 0.0330828i \(0.0105325\pi\)
−0.999453 + 0.0330828i \(0.989468\pi\)
\(270\) 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
\(275\) −2.50192e9 −0.437464
\(276\) 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 \(-0.815155\pi\)
−0.836074 + 0.548616i \(0.815155\pi\)
\(282\) 0 0
\(283\) −1.28042e9 −0.199622 −0.0998108 0.995006i \(-0.531824\pi\)
−0.0998108 + 0.995006i \(0.531824\pi\)
\(284\) 0 0
\(285\) 7.86542e9i 1.19218i
\(286\) 0 0
\(287\) − 1.93168e8i − 0.0284714i
\(288\) 0 0
\(289\) −4.43862e9 −0.636292
\(290\) 0 0
\(291\) −6.62652e9 −0.924089
\(292\) 0 0
\(293\) 2.13786e9i 0.290074i 0.989426 + 0.145037i \(0.0463301\pi\)
−0.989426 + 0.145037i \(0.953670\pi\)
\(294\) 0 0
\(295\) 1.11346e10i 1.47023i
\(296\) 0 0
\(297\) −1.91546e10 −2.46177
\(298\) 0 0
\(299\) 4.88656e9 0.611390
\(300\) 0 0
\(301\) 1.27697e9i 0.155567i
\(302\) 0 0
\(303\) − 9.36980e9i − 1.11163i
\(304\) 0 0
\(305\) 1.21516e10 1.40421
\(306\) 0 0
\(307\) 5.45140e9 0.613698 0.306849 0.951758i \(-0.400725\pi\)
0.306849 + 0.951758i \(0.400725\pi\)
\(308\) 0 0
\(309\) 2.34112e10i 2.56797i
\(310\) 0 0
\(311\) 1.07550e10i 1.14965i 0.818275 + 0.574827i \(0.194931\pi\)
−0.818275 + 0.574827i \(0.805069\pi\)
\(312\) 0 0
\(313\) 2.99804e8 0.0312364 0.0156182 0.999878i \(-0.495028\pi\)
0.0156182 + 0.999878i \(0.495028\pi\)
\(314\) 0 0
\(315\) 1.77254e10 1.80033
\(316\) 0 0
\(317\) 5.31172e9i 0.526015i 0.964794 + 0.263007i \(0.0847144\pi\)
−0.964794 + 0.263007i \(0.915286\pi\)
\(318\) 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
\(325\) − 3.61685e9i − 0.324188i
\(326\) 0 0
\(327\) − 2.21667e10i − 1.93869i
\(328\) 0 0
\(329\) −7.32872e9 −0.625525
\(330\) 0 0
\(331\) 1.01004e10 0.841446 0.420723 0.907189i \(-0.361776\pi\)
0.420723 + 0.907189i \(0.361776\pi\)
\(332\) 0 0
\(333\) − 1.07939e10i − 0.877815i
\(334\) 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
\(339\) 3.35578e10 2.54094
\(340\) 0 0
\(341\) 2.25348e10i 1.66662i
\(342\) 0 0
\(343\) − 1.27730e10i − 0.922820i
\(344\) 0 0
\(345\) −1.27697e10 −0.901376
\(346\) 0 0
\(347\) −1.27822e10 −0.881636 −0.440818 0.897597i \(-0.645312\pi\)
−0.440818 + 0.897597i \(0.645312\pi\)
\(348\) 0 0
\(349\) − 6.39381e8i − 0.0430981i −0.999768 0.0215490i \(-0.993140\pi\)
0.999768 0.0215490i \(-0.00685980\pi\)
\(350\) 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
\(355\) 5.14203e9 0.323759
\(356\) 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
\(365\) − 3.32347e9i − 0.187249i
\(366\) 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
\(371\) −2.85380e10 −1.50636
\(372\) 0 0
\(373\) − 1.19680e10i − 0.618283i −0.951016 0.309141i \(-0.899958\pi\)
0.951016 0.309141i \(-0.100042\pi\)
\(374\) 0 0
\(375\) 3.77379e10i 1.90833i
\(376\) 0 0
\(377\) 1.52344e9 0.0754154
\(378\) 0 0
\(379\) −2.30787e10 −1.11855 −0.559275 0.828982i \(-0.688920\pi\)
−0.559275 + 0.828982i \(0.688920\pi\)
\(380\) 0 0
\(381\) 5.22141e10i 2.47792i
\(382\) 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
\(387\) −6.79472e9 −0.302920
\(388\) 0 0
\(389\) 2.73457e10i 1.19424i 0.802152 + 0.597119i \(0.203688\pi\)
−0.802152 + 0.597119i \(0.796312\pi\)
\(390\) 0 0
\(391\) 8.88257e9i 0.380042i
\(392\) 0 0
\(393\) 3.07187e10 1.28775
\(394\) 0 0
\(395\) −2.48825e10 −1.02213
\(396\) 0 0
\(397\) 3.99456e10i 1.60808i 0.594576 + 0.804039i \(0.297320\pi\)
−0.594576 + 0.804039i \(0.702680\pi\)
\(398\) 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
\(405\) 2.68582e10i 0.998288i
\(406\) 0 0
\(407\) 1.52143e10i 0.554465i
\(408\) 0 0
\(409\) −1.14283e10 −0.408404 −0.204202 0.978929i \(-0.565460\pi\)
−0.204202 + 0.978929i \(0.565460\pi\)
\(410\) 0 0
\(411\) 5.48693e10 1.92292
\(412\) 0 0
\(413\) − 5.57982e10i − 1.91788i
\(414\) 0 0
\(415\) 3.74586e10i 1.26287i
\(416\) 0 0
\(417\) −4.98479e10 −1.64855
\(418\) 0 0
\(419\) 1.10009e10 0.356922 0.178461 0.983947i \(-0.442888\pi\)
0.178461 + 0.983947i \(0.442888\pi\)
\(420\) 0 0
\(421\) − 2.28766e10i − 0.728220i −0.931356 0.364110i \(-0.881373\pi\)
0.931356 0.364110i \(-0.118627\pi\)
\(422\) 0 0
\(423\) − 3.89958e10i − 1.21802i
\(424\) 0 0
\(425\) 6.57454e9 0.201516
\(426\) 0 0
\(427\) −6.08948e10 −1.83176
\(428\) 0 0
\(429\) 7.54155e10i 2.22655i
\(430\) 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
\(435\) −3.98111e9 −0.111185
\(436\) 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 \(-0.922678\pi\)
−0.970641 + 0.240533i \(0.922678\pi\)
\(444\) 0 0
\(445\) − 4.42658e10i − 1.12883i
\(446\) 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
\(451\) −1.44876e9 −0.0350180
\(452\) 0 0
\(453\) − 9.37373e10i − 2.22597i
\(454\) 0 0
\(455\) − 3.61181e10i − 0.842711i
\(456\) 0 0
\(457\) −4.66828e9 −0.107027 −0.0535133 0.998567i \(-0.517042\pi\)
−0.0535133 + 0.998567i \(0.517042\pi\)
\(458\) 0 0
\(459\) 5.03346e10 1.13401
\(460\) 0 0
\(461\) − 3.88096e10i − 0.859281i −0.903000 0.429641i \(-0.858640\pi\)
0.903000 0.429641i \(-0.141360\pi\)
\(462\) 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
\(467\) −2.31902e10 −0.487570 −0.243785 0.969829i \(-0.578389\pi\)
−0.243785 + 0.969829i \(0.578389\pi\)
\(468\) 0 0
\(469\) − 1.91546e10i − 0.395897i
\(470\) 0 0
\(471\) 6.18090e10i 1.25594i
\(472\) 0 0
\(473\) 9.57731e9 0.191337
\(474\) 0 0
\(475\) 1.41775e10 0.278500
\(476\) 0 0
\(477\) − 1.51849e11i − 2.93318i
\(478\) 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
\(483\) 6.39926e10 1.17582
\(484\) 0 0
\(485\) − 2.38018e10i − 0.430173i
\(486\) 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
\(491\) 5.56483e10 0.957472 0.478736 0.877959i \(-0.341095\pi\)
0.478736 + 0.877959i \(0.341095\pi\)
\(492\) 0 0
\(493\) 2.76924e9i 0.0468784i
\(494\) 0 0
\(495\) − 1.32940e11i − 2.21429i
\(496\) 0 0
\(497\) −2.57681e10 −0.422335
\(498\) 0 0
\(499\) −7.88458e10 −1.27168 −0.635838 0.771822i \(-0.719345\pi\)
−0.635838 + 0.771822i \(0.719345\pi\)
\(500\) 0 0
\(501\) 9.53224e10i 1.51302i
\(502\) 0 0
\(503\) 4.41092e10i 0.689061i 0.938775 + 0.344530i \(0.111962\pi\)
−0.938775 + 0.344530i \(0.888038\pi\)
\(504\) 0 0
\(505\) 3.36554e10 0.517475
\(506\) 0 0
\(507\) 6.79923e9 0.102903
\(508\) 0 0
\(509\) 1.05927e10i 0.157811i 0.996882 + 0.0789055i \(0.0251425\pi\)
−0.996882 + 0.0789055i \(0.974857\pi\)
\(510\) 0 0
\(511\) 1.66548e10i 0.244262i
\(512\) 0 0
\(513\) 1.08543e11 1.56723
\(514\) 0 0
\(515\) −8.40908e10 −1.19542
\(516\) 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 \(-0.822177\pi\)
−0.847973 + 0.530039i \(0.822177\pi\)
\(522\) 0 0
\(523\) −2.80408e10 −0.374786 −0.187393 0.982285i \(-0.560004\pi\)
−0.187393 + 0.982285i \(0.560004\pi\)
\(524\) 0 0
\(525\) − 4.73649e10i − 0.623476i
\(526\) 0 0
\(527\) − 5.92172e10i − 0.767724i
\(528\) 0 0
\(529\) 4.72129e10 0.602890
\(530\) 0 0
\(531\) 2.96900e11 3.73449
\(532\) 0 0
\(533\) − 2.09438e9i − 0.0259505i
\(534\) 0 0
\(535\) − 6.49363e10i − 0.792633i
\(536\) 0 0
\(537\) 1.07479e11 1.29248
\(538\) 0 0
\(539\) 1.47027e10 0.174197
\(540\) 0 0
\(541\) 1.44659e11i 1.68871i 0.535782 + 0.844356i \(0.320017\pi\)
−0.535782 + 0.844356i \(0.679983\pi\)
\(542\) 0 0
\(543\) − 8.90386e10i − 1.02419i
\(544\) 0 0
\(545\) 7.96205e10 0.902483
\(546\) 0 0
\(547\) 1.03774e10 0.115915 0.0579573 0.998319i \(-0.481541\pi\)
0.0579573 + 0.998319i \(0.481541\pi\)
\(548\) 0 0
\(549\) − 3.24018e11i − 3.56681i
\(550\) 0 0
\(551\) 5.97167e9i 0.0647872i
\(552\) 0 0
\(553\) 1.24693e11 1.33334
\(554\) 0 0
\(555\) 5.74762e10 0.605782
\(556\) 0 0
\(557\) − 5.47312e9i − 0.0568610i −0.999596 0.0284305i \(-0.990949\pi\)
0.999596 0.0284305i \(-0.00905093\pi\)
\(558\) 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
\(565\) 1.20536e11i 1.18283i
\(566\) 0 0
\(567\) − 1.34593e11i − 1.30224i
\(568\) 0 0
\(569\) −1.27822e10 −0.121943 −0.0609716 0.998140i \(-0.519420\pi\)
−0.0609716 + 0.998140i \(0.519420\pi\)
\(570\) 0 0
\(571\) 7.59455e10 0.714427 0.357213 0.934023i \(-0.383727\pi\)
0.357213 + 0.934023i \(0.383727\pi\)
\(572\) 0 0
\(573\) − 1.53237e11i − 1.42149i
\(574\) 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
\(579\) −4.21353e9 −0.0374914
\(580\) 0 0
\(581\) − 1.87715e11i − 1.64738i
\(582\) 0 0
\(583\) 2.14035e11i 1.85272i
\(584\) 0 0
\(585\) 1.92182e11 1.64093
\(586\) 0 0
\(587\) −6.07298e10 −0.511504 −0.255752 0.966742i \(-0.582323\pi\)
−0.255752 + 0.966742i \(0.582323\pi\)
\(588\) 0 0
\(589\) − 1.27697e11i − 1.06101i
\(590\) 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
\(595\) 6.56538e10 0.523832
\(596\) 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 \(-0.235093\pi\)
0.739434 + 0.673229i \(0.235093\pi\)
\(602\) 0 0
\(603\) 1.01921e11 0.770892
\(604\) 0 0
\(605\) 7.80591e10i 0.582643i
\(606\) 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
\(611\) −7.94597e10 −0.570141
\(612\) 0 0
\(613\) − 1.76424e11i − 1.24944i −0.780847 0.624722i \(-0.785212\pi\)
0.780847 0.624722i \(-0.214788\pi\)
\(614\) 0 0
\(615\) 5.47311e9i 0.0382590i
\(616\) 0 0
\(617\) 9.84986e10 0.679656 0.339828 0.940488i \(-0.389631\pi\)
0.339828 + 0.940488i \(0.389631\pi\)
\(618\) 0 0
\(619\) 1.28596e10 0.0875923 0.0437961 0.999040i \(-0.486055\pi\)
0.0437961 + 0.999040i \(0.486055\pi\)
\(620\) 0 0
\(621\) 1.76223e11i 1.18494i
\(622\) 0 0
\(623\) 2.21828e11i 1.47253i
\(624\) 0 0
\(625\) −8.45648e10 −0.554204
\(626\) 0 0
\(627\) −2.95618e11 −1.91276
\(628\) 0 0
\(629\) − 3.99802e10i − 0.255413i
\(630\) 0 0
\(631\) − 1.73463e11i − 1.09418i −0.837073 0.547091i \(-0.815735\pi\)
0.837073 0.547091i \(-0.184265\pi\)
\(632\) 0 0
\(633\) −3.94261e11 −2.45566
\(634\) 0 0
\(635\) −1.87548e11 −1.15350
\(636\) 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
\(645\) − 3.61810e10i − 0.209046i
\(646\) 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
\(651\) −4.26617e11 −2.37528
\(652\) 0 0
\(653\) 6.49972e9i 0.0357472i 0.999840 + 0.0178736i \(0.00568964\pi\)
−0.999840 + 0.0178736i \(0.994310\pi\)
\(654\) 0 0
\(655\) 1.10339e11i 0.599463i
\(656\) 0 0
\(657\) −8.86194e10 −0.475628
\(658\) 0 0
\(659\) 2.20982e11 1.17170 0.585848 0.810421i \(-0.300761\pi\)
0.585848 + 0.810421i \(0.300761\pi\)
\(660\) 0 0
\(661\) 2.69549e11i 1.41199i 0.708217 + 0.705995i \(0.249500\pi\)
−0.708217 + 0.705995i \(0.750500\pi\)
\(662\) 0 0
\(663\) − 1.98177e11i − 1.02565i
\(664\) 0 0
\(665\) 1.41578e11 0.723949
\(666\) 0 0
\(667\) −9.69518e9 −0.0489838
\(668\) 0 0
\(669\) − 3.23476e11i − 1.61487i
\(670\) 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
\(675\) 1.30433e11 0.628309
\(676\) 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 \(-0.257127\pi\)
0.691099 + 0.722760i \(0.257127\pi\)
\(684\) 0 0
\(685\) 1.97085e11i 0.895142i
\(686\) 0 0
\(687\) − 2.40591e11i − 1.08007i
\(688\) 0 0
\(689\) −3.09416e11 −1.37298
\(690\) 0 0
\(691\) −3.06208e11 −1.34309 −0.671544 0.740964i \(-0.734369\pi\)
−0.671544 + 0.740964i \(0.734369\pi\)
\(692\) 0 0
\(693\) 6.66199e11i 2.88849i
\(694\) 0 0
\(695\) − 1.79049e11i − 0.767419i
\(696\) 0 0
\(697\) 3.80707e9 0.0161309
\(698\) 0 0
\(699\) −7.40348e10 −0.310118
\(700\) 0 0
\(701\) − 2.73603e11i − 1.13305i −0.824045 0.566524i \(-0.808288\pi\)
0.824045 0.566524i \(-0.191712\pi\)
\(702\) 0 0
\(703\) − 8.62143e10i − 0.352987i
\(704\) 0 0
\(705\) 2.07647e11 0.840562
\(706\) 0 0
\(707\) −1.68656e11 −0.675033
\(708\) 0 0
\(709\) 1.76662e11i 0.699129i 0.936912 + 0.349564i \(0.113670\pi\)
−0.936912 + 0.349564i \(0.886330\pi\)
\(710\) 0 0
\(711\) 6.63484e11i 2.59628i
\(712\) 0 0
\(713\) 2.07321e11 0.802203
\(714\) 0 0
\(715\) −2.70885e11 −1.03648
\(716\) 0 0
\(717\) 6.25718e11i 2.36756i
\(718\) 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
\(723\) −2.30227e11 −0.842565
\(724\) 0 0
\(725\) 7.17600e9i 0.0259735i
\(726\) 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
\(731\) −2.51673e10 −0.0881388
\(732\) 0 0
\(733\) − 2.94176e11i − 1.01904i −0.860459 0.509520i \(-0.829823\pi\)
0.860459 0.509520i \(-0.170177\pi\)
\(734\) 0 0
\(735\) − 5.55435e10i − 0.190320i
\(736\) 0 0
\(737\) −1.43660e11 −0.486928
\(738\) 0 0
\(739\) 9.69888e10 0.325195 0.162598 0.986692i \(-0.448013\pi\)
0.162598 + 0.986692i \(0.448013\pi\)
\(740\) 0 0
\(741\) − 4.27355e11i − 1.41748i
\(742\) 0 0
\(743\) 1.01567e11i 0.333271i 0.986019 + 0.166635i \(0.0532902\pi\)
−0.986019 + 0.166635i \(0.946710\pi\)
\(744\) 0 0
\(745\) −2.31590e11 −0.751788
\(746\) 0 0
\(747\) 9.98824e11 3.20779
\(748\) 0 0
\(749\) 3.25413e11i 1.03397i
\(750\) 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
\(755\) 3.36696e11 1.03621
\(756\) 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 \(-0.280087\pi\)
0.637213 + 0.770687i \(0.280087\pi\)
\(762\) 0 0
\(763\) −3.99000e11 −1.17727
\(764\) 0 0
\(765\) 3.49341e11i 1.02001i
\(766\) 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
\(771\) 7.49386e11 2.12074
\(772\) 0 0
\(773\) 1.49408e11i 0.418462i 0.977866 + 0.209231i \(0.0670961\pi\)
−0.977866 + 0.209231i \(0.932904\pi\)
\(774\) 0 0
\(775\) − 1.53451e11i − 0.425366i
\(776\) 0 0
\(777\) −2.88029e11 −0.790227
\(778\) 0 0
\(779\) 8.20966e9 0.0222933
\(780\) 0 0
\(781\) 1.93261e11i 0.519445i
\(782\) 0 0
\(783\) 5.49393e10i 0.146163i
\(784\) 0 0
\(785\) −2.22012e11 −0.584652
\(786\) 0 0
\(787\) 7.33252e11 1.91141 0.955706 0.294323i \(-0.0950943\pi\)
0.955706 + 0.294323i \(0.0950943\pi\)
\(788\) 0 0
\(789\) 1.04891e12i 2.70665i
\(790\) 0 0
\(791\) − 6.04040e11i − 1.54298i
\(792\) 0 0
\(793\) −6.60236e11 −1.66958
\(794\) 0 0
\(795\) 8.08577e11 2.02420
\(796\) 0 0
\(797\) 3.02703e11i 0.750212i 0.926982 + 0.375106i \(0.122394\pi\)
−0.926982 + 0.375106i \(0.877606\pi\)
\(798\) 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
\(805\) 2.29855e11i 0.547358i
\(806\) 0 0
\(807\) 4.91911e10i 0.115982i
\(808\) 0 0
\(809\) 5.84316e11 1.36412 0.682062 0.731295i \(-0.261084\pi\)
0.682062 + 0.731295i \(0.261084\pi\)
\(810\) 0 0
\(811\) 1.21470e11 0.280793 0.140396 0.990095i \(-0.455162\pi\)
0.140396 + 0.990095i \(0.455162\pi\)
\(812\) 0 0
\(813\) − 9.25344e8i − 0.00211807i
\(814\) 0 0
\(815\) 1.24473e11i 0.282127i
\(816\) 0 0
\(817\) −5.42714e10 −0.121810
\(818\) 0 0
\(819\) −9.63078e11 −2.14055
\(820\) 0 0
\(821\) − 4.52470e11i − 0.995903i −0.867205 0.497952i \(-0.834086\pi\)
0.867205 0.497952i \(-0.165914\pi\)
\(822\) 0 0
\(823\) 3.06704e11i 0.668528i 0.942479 + 0.334264i \(0.108488\pi\)
−0.942479 + 0.334264i \(0.891512\pi\)
\(824\) 0 0
\(825\) −3.55237e11 −0.766835
\(826\) 0 0
\(827\) −2.93276e11 −0.626982 −0.313491 0.949591i \(-0.601499\pi\)
−0.313491 + 0.949591i \(0.601499\pi\)
\(828\) 0 0
\(829\) 3.35532e11i 0.710421i 0.934786 + 0.355210i \(0.115591\pi\)
−0.934786 + 0.355210i \(0.884409\pi\)
\(830\) 0 0
\(831\) − 3.05356e11i − 0.640327i
\(832\) 0 0
\(833\) −3.86358e10 −0.0802434
\(834\) 0 0
\(835\) −3.42389e11 −0.704326
\(836\) 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
\(845\) 2.44222e10i 0.0479024i
\(846\) 0 0
\(847\) − 3.91175e11i − 0.760042i
\(848\) 0 0
\(849\) −1.81802e11 −0.349919
\(850\) 0 0
\(851\) 1.39972e11 0.266883
\(852\) 0 0
\(853\) 5.08662e11i 0.960801i 0.877049 + 0.480400i \(0.159509\pi\)
−0.877049 + 0.480400i \(0.840491\pi\)
\(854\) 0 0
\(855\) 7.53328e11i 1.40968i
\(856\) 0 0
\(857\) −6.06764e11 −1.12486 −0.562428 0.826846i \(-0.690133\pi\)
−0.562428 + 0.826846i \(0.690133\pi\)
\(858\) 0 0
\(859\) 9.49431e11 1.74378 0.871888 0.489705i \(-0.162895\pi\)
0.871888 + 0.489705i \(0.162895\pi\)
\(860\) 0 0
\(861\) − 2.74272e10i − 0.0499078i
\(862\) 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
\(867\) −6.30222e11 −1.11536
\(868\) 0 0
\(869\) − 9.35196e11i − 1.63992i
\(870\) 0 0
\(871\) − 2.07679e11i − 0.360844i
\(872\) 0 0
\(873\) −6.34669e11 −1.09267
\(874\) 0 0
\(875\) 6.79283e11 1.15883
\(876\) 0 0
\(877\) − 8.80195e11i − 1.48792i −0.668223 0.743961i \(-0.732945\pi\)
0.668223 0.743961i \(-0.267055\pi\)
\(878\) 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
\(885\) 1.58095e12i 2.57718i
\(886\) 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
\(891\) −1.00945e12 −1.60167
\(892\) 0 0
\(893\) − 3.11471e11i − 0.489792i
\(894\) 0 0
\(895\) 3.86053e11i 0.601666i
\(896\) 0 0
\(897\) 6.93823e11 1.07171
\(898\) 0 0
\(899\) 6.46345e10 0.0989523
\(900\) 0 0
\(901\) − 5.62442e11i − 0.853451i
\(902\) 0 0
\(903\) 1.81312e11i 0.272695i
\(904\) 0 0
\(905\) 3.19818e11 0.476770
\(906\) 0 0
\(907\) −7.65213e11 −1.13071 −0.565357 0.824846i \(-0.691262\pi\)
−0.565357 + 0.824846i \(0.691262\pi\)
\(908\) 0 0
\(909\) − 8.97412e11i − 1.31443i
\(910\) 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
\(915\) 1.72535e12 2.46146
\(916\) 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
\(925\) − 1.03602e11i − 0.141514i
\(926\) 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
\(931\) −8.33152e10 −0.110898
\(932\) 0 0
\(933\) 1.52705e12i 2.01524i
\(934\) 0 0
\(935\) − 4.92404e11i − 0.644280i
\(936\) 0 0
\(937\) −1.03941e12 −1.34843 −0.674217 0.738533i \(-0.735519\pi\)
−0.674217 + 0.738533i \(0.735519\pi\)
\(938\) 0 0
\(939\) 4.25680e10 0.0547546
\(940\) 0 0
\(941\) 1.26891e11i 0.161836i 0.996721 + 0.0809178i \(0.0257851\pi\)
−0.996721 + 0.0809178i \(0.974215\pi\)
\(942\) 0 0
\(943\) 1.33286e10i 0.0168554i
\(944\) 0 0
\(945\) 1.30251e12 1.63326
\(946\) 0 0
\(947\) −6.13064e11 −0.762265 −0.381133 0.924520i \(-0.624466\pi\)
−0.381133 + 0.924520i \(0.624466\pi\)
\(948\) 0 0
\(949\) 1.80575e11i 0.222635i
\(950\) 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
\(955\) 5.50413e11 0.661721
\(956\) 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
\(965\) − 1.51346e10i − 0.0174527i
\(966\) 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
\(971\) 3.54981e11 0.399327 0.199663 0.979865i \(-0.436015\pi\)
0.199663 + 0.979865i \(0.436015\pi\)
\(972\) 0 0
\(973\) 8.97263e11i 1.00108i
\(974\) 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
\(979\) 1.66371e12 1.81112
\(980\) 0 0
\(981\) − 2.12306e12i − 2.29238i
\(982\) 0 0
\(983\) 1.37465e12i 1.47223i 0.676854 + 0.736117i \(0.263343\pi\)
−0.676854 + 0.736117i \(0.736657\pi\)
\(984\) 0 0
\(985\) −5.73668e11 −0.609419
\(986\) 0 0
\(987\) −1.04058e12 −1.09649
\(988\) 0 0
\(989\) − 8.81113e10i − 0.0920972i
\(990\) 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
\(995\) −5.34061e11 −0.544877
\(996\) 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 256.9.d.f.127.4 4
4.3 odd 2 inner 256.9.d.f.127.2 4
8.3 odd 2 inner 256.9.d.f.127.3 4
8.5 even 2 inner 256.9.d.f.127.1 4
16.3 odd 4 64.9.c.d.63.1 2
16.5 even 4 16.9.c.a.15.1 2
16.11 odd 4 16.9.c.a.15.2 yes 2
16.13 even 4 64.9.c.d.63.2 2
48.5 odd 4 144.9.g.g.127.2 2
48.11 even 4 144.9.g.g.127.1 2
80.27 even 4 400.9.h.b.399.4 4
80.37 odd 4 400.9.h.b.399.1 4
80.43 even 4 400.9.h.b.399.2 4
80.53 odd 4 400.9.h.b.399.3 4
80.59 odd 4 400.9.b.c.351.1 2
80.69 even 4 400.9.b.c.351.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 16.5 even 4
16.9.c.a.15.2 yes 2 16.11 odd 4
64.9.c.d.63.1 2 16.3 odd 4
64.9.c.d.63.2 2 16.13 even 4
144.9.g.g.127.1 2 48.11 even 4
144.9.g.g.127.2 2 48.5 odd 4
256.9.d.f.127.1 4 8.5 even 2 inner
256.9.d.f.127.2 4 4.3 odd 2 inner
256.9.d.f.127.3 4 8.3 odd 2 inner
256.9.d.f.127.4 4 1.1 even 1 trivial
400.9.b.c.351.1 2 80.59 odd 4
400.9.b.c.351.2 2 80.69 even 4
400.9.h.b.399.1 4 80.37 odd 4
400.9.h.b.399.2 4 80.43 even 4
400.9.h.b.399.3 4 80.53 odd 4
400.9.h.b.399.4 4 80.27 even 4