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Properties

Label 252.8.a.b.1.1
Level $252$
Weight $8$
Character 252.1
Self dual yes
Analytic conductor $78.721$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(1,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.7210264220\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 252.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+240.000 q^{5} +343.000 q^{7} +O(q^{10})\) \(q+240.000 q^{5} +343.000 q^{7} -702.000 q^{11} -3958.00 q^{13} +3408.00 q^{17} -49036.0 q^{19} +11514.0 q^{23} -20525.0 q^{25} -49662.0 q^{29} -113320. q^{31} +82320.0 q^{35} -66886.0 q^{37} +360900. q^{41} -765292. q^{43} +1.34488e6 q^{47} +117649. q^{49} -358962. q^{53} -168480. q^{55} -930528. q^{59} -1.31883e6 q^{61} -949920. q^{65} +1.89346e6 q^{67} -227994. q^{71} +784934. q^{73} -240786. q^{77} -2.10089e6 q^{79} -8.62931e6 q^{83} +817920. q^{85} -5.90310e6 q^{89} -1.35759e6 q^{91} -1.17686e7 q^{95} +773846. q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) 0 0
\(4\) 0 0
\(5\) 240.000 0.858650 0.429325 − 0.903150i \(-0.358751\pi\)
0.429325 + 0.903150i \(0.358751\pi\)
\(6\) 0 0
\(7\) 343.000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −702.000 −0.159024 −0.0795120 − 0.996834i \(-0.525336\pi\)
−0.0795120 + 0.996834i \(0.525336\pi\)
\(12\) 0 0
\(13\) −3958.00 −0.499659 −0.249830 − 0.968290i \(-0.580375\pi\)
−0.249830 + 0.968290i \(0.580375\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3408.00 0.168240 0.0841198 − 0.996456i \(-0.473192\pi\)
0.0841198 + 0.996456i \(0.473192\pi\)
\(18\) 0 0
\(19\) −49036.0 −1.64013 −0.820063 − 0.572273i \(-0.806062\pi\)
−0.820063 + 0.572273i \(0.806062\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 11514.0 0.197323 0.0986617 − 0.995121i \(-0.468544\pi\)
0.0986617 + 0.995121i \(0.468544\pi\)
\(24\) 0 0
\(25\) −20525.0 −0.262720
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −49662.0 −0.378121 −0.189061 − 0.981965i \(-0.560544\pi\)
−0.189061 + 0.981965i \(0.560544\pi\)
\(30\) 0 0
\(31\) −113320. −0.683189 −0.341594 − 0.939847i \(-0.610967\pi\)
−0.341594 + 0.939847i \(0.610967\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 82320.0 0.324539
\(36\) 0 0
\(37\) −66886.0 −0.217085 −0.108542 − 0.994092i \(-0.534618\pi\)
−0.108542 + 0.994092i \(0.534618\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 360900. 0.817793 0.408896 − 0.912581i \(-0.365914\pi\)
0.408896 + 0.912581i \(0.365914\pi\)
\(42\) 0 0
\(43\) −765292. −1.46787 −0.733935 − 0.679220i \(-0.762318\pi\)
−0.733935 + 0.679220i \(0.762318\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.34488e6 1.88947 0.944734 − 0.327837i \(-0.106320\pi\)
0.944734 + 0.327837i \(0.106320\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −358962. −0.331194 −0.165597 − 0.986193i \(-0.552955\pi\)
−0.165597 + 0.986193i \(0.552955\pi\)
\(54\) 0 0
\(55\) −168480. −0.136546
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −930528. −0.589858 −0.294929 − 0.955519i \(-0.595296\pi\)
−0.294929 + 0.955519i \(0.595296\pi\)
\(60\) 0 0
\(61\) −1.31883e6 −0.743936 −0.371968 − 0.928246i \(-0.621317\pi\)
−0.371968 + 0.928246i \(0.621317\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −949920. −0.429033
\(66\) 0 0
\(67\) 1.89346e6 0.769122 0.384561 − 0.923100i \(-0.374353\pi\)
0.384561 + 0.923100i \(0.374353\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −227994. −0.0755995 −0.0377998 − 0.999285i \(-0.512035\pi\)
−0.0377998 + 0.999285i \(0.512035\pi\)
\(72\) 0 0
\(73\) 784934. 0.236158 0.118079 − 0.993004i \(-0.462326\pi\)
0.118079 + 0.993004i \(0.462326\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −240786. −0.0601054
\(78\) 0 0
\(79\) −2.10089e6 −0.479412 −0.239706 − 0.970846i \(-0.577051\pi\)
−0.239706 + 0.970846i \(0.577051\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.62931e6 −1.65654 −0.828271 − 0.560327i \(-0.810675\pi\)
−0.828271 + 0.560327i \(0.810675\pi\)
\(84\) 0 0
\(85\) 817920. 0.144459
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.90310e6 −0.887596 −0.443798 − 0.896127i \(-0.646369\pi\)
−0.443798 + 0.896127i \(0.646369\pi\)
\(90\) 0 0
\(91\) −1.35759e6 −0.188853
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.17686e7 −1.40830
\(96\) 0 0
\(97\) 773846. 0.0860902 0.0430451 − 0.999073i \(-0.486294\pi\)
0.0430451 + 0.999073i \(0.486294\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.33340e6 −0.321930 −0.160965 − 0.986960i \(-0.551461\pi\)
−0.160965 + 0.986960i \(0.551461\pi\)
\(102\) 0 0
\(103\) −5.65905e6 −0.510285 −0.255143 − 0.966903i \(-0.582122\pi\)
−0.255143 + 0.966903i \(0.582122\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.53822e7 −1.21388 −0.606940 − 0.794748i \(-0.707603\pi\)
−0.606940 + 0.794748i \(0.707603\pi\)
\(108\) 0 0
\(109\) 4.47690e6 0.331120 0.165560 − 0.986200i \(-0.447057\pi\)
0.165560 + 0.986200i \(0.447057\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.72279e6 0.633894 0.316947 − 0.948443i \(-0.397342\pi\)
0.316947 + 0.948443i \(0.397342\pi\)
\(114\) 0 0
\(115\) 2.76336e6 0.169432
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.16894e6 0.0635886
\(120\) 0 0
\(121\) −1.89944e7 −0.974711
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.36760e7 −1.08423
\(126\) 0 0
\(127\) 1.52949e7 0.662572 0.331286 − 0.943530i \(-0.392517\pi\)
0.331286 + 0.943530i \(0.392517\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.36966e7 −1.69824 −0.849119 − 0.528202i \(-0.822867\pi\)
−0.849119 + 0.528202i \(0.822867\pi\)
\(132\) 0 0
\(133\) −1.68193e7 −0.619910
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00071e6 0.0997017 0.0498509 − 0.998757i \(-0.484125\pi\)
0.0498509 + 0.998757i \(0.484125\pi\)
\(138\) 0 0
\(139\) −1.73192e7 −0.546987 −0.273494 − 0.961874i \(-0.588179\pi\)
−0.273494 + 0.961874i \(0.588179\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.77852e6 0.0794578
\(144\) 0 0
\(145\) −1.19189e7 −0.324674
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.90458e7 0.966990 0.483495 − 0.875347i \(-0.339367\pi\)
0.483495 + 0.875347i \(0.339367\pi\)
\(150\) 0 0
\(151\) −1.06602e7 −0.251968 −0.125984 − 0.992032i \(-0.540209\pi\)
−0.125984 + 0.992032i \(0.540209\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.71968e7 −0.586620
\(156\) 0 0
\(157\) −5.52292e7 −1.13899 −0.569496 − 0.821994i \(-0.692861\pi\)
−0.569496 + 0.821994i \(0.692861\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.94930e6 0.0745813
\(162\) 0 0
\(163\) −1.65741e7 −0.299760 −0.149880 − 0.988704i \(-0.547889\pi\)
−0.149880 + 0.988704i \(0.547889\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.70004e7 0.947044 0.473522 − 0.880782i \(-0.342982\pi\)
0.473522 + 0.880782i \(0.342982\pi\)
\(168\) 0 0
\(169\) −4.70828e7 −0.750340
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.84256e7 0.417396 0.208698 − 0.977980i \(-0.433077\pi\)
0.208698 + 0.977980i \(0.433077\pi\)
\(174\) 0 0
\(175\) −7.04008e6 −0.0992988
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.44314e8 −1.88071 −0.940357 − 0.340190i \(-0.889509\pi\)
−0.940357 + 0.340190i \(0.889509\pi\)
\(180\) 0 0
\(181\) −1.17741e8 −1.47588 −0.737942 − 0.674865i \(-0.764202\pi\)
−0.737942 + 0.674865i \(0.764202\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.60526e7 −0.186400
\(186\) 0 0
\(187\) −2.39242e6 −0.0267541
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.47184e8 −1.52842 −0.764211 − 0.644966i \(-0.776872\pi\)
−0.764211 + 0.644966i \(0.776872\pi\)
\(192\) 0 0
\(193\) 1.20940e8 1.21093 0.605465 − 0.795872i \(-0.292987\pi\)
0.605465 + 0.795872i \(0.292987\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.80341e7 0.540818 0.270409 − 0.962745i \(-0.412841\pi\)
0.270409 + 0.962745i \(0.412841\pi\)
\(198\) 0 0
\(199\) 7.80160e7 0.701775 0.350888 − 0.936418i \(-0.385880\pi\)
0.350888 + 0.936418i \(0.385880\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.70341e7 −0.142916
\(204\) 0 0
\(205\) 8.66160e7 0.702198
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.44233e7 0.260820
\(210\) 0 0
\(211\) 2.11854e8 1.55256 0.776279 − 0.630390i \(-0.217105\pi\)
0.776279 + 0.630390i \(0.217105\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.83670e8 −1.26039
\(216\) 0 0
\(217\) −3.88688e7 −0.258221
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.34889e7 −0.0840625
\(222\) 0 0
\(223\) −1.71323e8 −1.03454 −0.517271 − 0.855822i \(-0.673052\pi\)
−0.517271 + 0.855822i \(0.673052\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.64284e8 1.49962 0.749808 − 0.661656i \(-0.230146\pi\)
0.749808 + 0.661656i \(0.230146\pi\)
\(228\) 0 0
\(229\) 5.30633e6 0.0291991 0.0145996 − 0.999893i \(-0.495353\pi\)
0.0145996 + 0.999893i \(0.495353\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.38215e8 −0.715829 −0.357914 − 0.933754i \(-0.616512\pi\)
−0.357914 + 0.933754i \(0.616512\pi\)
\(234\) 0 0
\(235\) 3.22770e8 1.62239
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.71430e8 1.28607 0.643037 − 0.765836i \(-0.277674\pi\)
0.643037 + 0.765836i \(0.277674\pi\)
\(240\) 0 0
\(241\) 2.46061e7 0.113236 0.0566179 − 0.998396i \(-0.481968\pi\)
0.0566179 + 0.998396i \(0.481968\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.82358e7 0.122664
\(246\) 0 0
\(247\) 1.94084e8 0.819505
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.08916e8 0.833900 0.416950 − 0.908929i \(-0.363099\pi\)
0.416950 + 0.908929i \(0.363099\pi\)
\(252\) 0 0
\(253\) −8.08283e6 −0.0313792
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.12952e8 0.415075 0.207537 − 0.978227i \(-0.433455\pi\)
0.207537 + 0.978227i \(0.433455\pi\)
\(258\) 0 0
\(259\) −2.29419e7 −0.0820503
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.61527e8 −0.547520 −0.273760 − 0.961798i \(-0.588267\pi\)
−0.273760 + 0.961798i \(0.588267\pi\)
\(264\) 0 0
\(265\) −8.61509e7 −0.284380
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.84015e8 0.889628 0.444814 − 0.895623i \(-0.353270\pi\)
0.444814 + 0.895623i \(0.353270\pi\)
\(270\) 0 0
\(271\) 2.88374e8 0.880165 0.440083 − 0.897957i \(-0.354949\pi\)
0.440083 + 0.897957i \(0.354949\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.44086e7 0.0417788
\(276\) 0 0
\(277\) 3.07325e8 0.868797 0.434398 − 0.900721i \(-0.356961\pi\)
0.434398 + 0.900721i \(0.356961\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.81803e8 1.02652 0.513260 − 0.858233i \(-0.328438\pi\)
0.513260 + 0.858233i \(0.328438\pi\)
\(282\) 0 0
\(283\) −6.05387e8 −1.58775 −0.793873 − 0.608084i \(-0.791938\pi\)
−0.793873 + 0.608084i \(0.791938\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.23789e8 0.309097
\(288\) 0 0
\(289\) −3.98724e8 −0.971695
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.33619e8 1.47161 0.735804 − 0.677195i \(-0.236805\pi\)
0.735804 + 0.677195i \(0.236805\pi\)
\(294\) 0 0
\(295\) −2.23327e8 −0.506482
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.55724e7 −0.0985945
\(300\) 0 0
\(301\) −2.62495e8 −0.554803
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.16520e8 −0.638781
\(306\) 0 0
\(307\) −3.79251e8 −0.748070 −0.374035 − 0.927415i \(-0.622026\pi\)
−0.374035 + 0.927415i \(0.622026\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.80859e8 0.906476 0.453238 − 0.891390i \(-0.350269\pi\)
0.453238 + 0.891390i \(0.350269\pi\)
\(312\) 0 0
\(313\) 8.04070e8 1.48214 0.741069 − 0.671429i \(-0.234319\pi\)
0.741069 + 0.671429i \(0.234319\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.14707e8 1.26015 0.630073 − 0.776536i \(-0.283025\pi\)
0.630073 + 0.776536i \(0.283025\pi\)
\(318\) 0 0
\(319\) 3.48627e7 0.0601304
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.67115e8 −0.275934
\(324\) 0 0
\(325\) 8.12380e7 0.131271
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.61292e8 0.714152
\(330\) 0 0
\(331\) 2.00218e8 0.303462 0.151731 − 0.988422i \(-0.451515\pi\)
0.151731 + 0.988422i \(0.451515\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.54431e8 0.660407
\(336\) 0 0
\(337\) −6.36870e8 −0.906455 −0.453228 − 0.891395i \(-0.649728\pi\)
−0.453228 + 0.891395i \(0.649728\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.95506e7 0.108643
\(342\) 0 0
\(343\) 4.03536e7 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.00147e9 −1.28672 −0.643361 − 0.765563i \(-0.722460\pi\)
−0.643361 + 0.765563i \(0.722460\pi\)
\(348\) 0 0
\(349\) 9.60061e8 1.20895 0.604477 − 0.796623i \(-0.293382\pi\)
0.604477 + 0.796623i \(0.293382\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.88199e8 −0.469724 −0.234862 − 0.972029i \(-0.575464\pi\)
−0.234862 + 0.972029i \(0.575464\pi\)
\(354\) 0 0
\(355\) −5.47186e7 −0.0649136
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.18153e7 −0.0933263 −0.0466632 − 0.998911i \(-0.514859\pi\)
−0.0466632 + 0.998911i \(0.514859\pi\)
\(360\) 0 0
\(361\) 1.51066e9 1.69002
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.88384e8 0.202777
\(366\) 0 0
\(367\) −1.09898e9 −1.16053 −0.580266 − 0.814427i \(-0.697051\pi\)
−0.580266 + 0.814427i \(0.697051\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.23124e8 −0.125180
\(372\) 0 0
\(373\) 1.90918e9 1.90488 0.952438 − 0.304733i \(-0.0985671\pi\)
0.952438 + 0.304733i \(0.0985671\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.96562e8 0.188932
\(378\) 0 0
\(379\) −1.07421e9 −1.01356 −0.506781 − 0.862075i \(-0.669165\pi\)
−0.506781 + 0.862075i \(0.669165\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.16642e8 0.197037 0.0985184 − 0.995135i \(-0.468590\pi\)
0.0985184 + 0.995135i \(0.468590\pi\)
\(384\) 0 0
\(385\) −5.77886e7 −0.0516095
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.04803e9 −1.76406 −0.882028 − 0.471197i \(-0.843822\pi\)
−0.882028 + 0.471197i \(0.843822\pi\)
\(390\) 0 0
\(391\) 3.92397e7 0.0331976
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.04214e8 −0.411647
\(396\) 0 0
\(397\) 1.73429e9 1.39109 0.695544 − 0.718483i \(-0.255163\pi\)
0.695544 + 0.718483i \(0.255163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.64740e9 −1.27583 −0.637917 − 0.770105i \(-0.720204\pi\)
−0.637917 + 0.770105i \(0.720204\pi\)
\(402\) 0 0
\(403\) 4.48521e8 0.341362
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.69540e7 0.0345217
\(408\) 0 0
\(409\) −1.95391e8 −0.141213 −0.0706063 − 0.997504i \(-0.522493\pi\)
−0.0706063 + 0.997504i \(0.522493\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.19171e8 −0.222945
\(414\) 0 0
\(415\) −2.07103e9 −1.42239
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.63659e9 −1.08690 −0.543451 − 0.839441i \(-0.682883\pi\)
−0.543451 + 0.839441i \(0.682883\pi\)
\(420\) 0 0
\(421\) −1.40514e8 −0.0917765 −0.0458883 − 0.998947i \(-0.514612\pi\)
−0.0458883 + 0.998947i \(0.514612\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.99492e7 −0.0441999
\(426\) 0 0
\(427\) −4.52360e8 −0.281181
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.14319e9 −0.687780 −0.343890 − 0.939010i \(-0.611745\pi\)
−0.343890 + 0.939010i \(0.611745\pi\)
\(432\) 0 0
\(433\) −2.24052e9 −1.32630 −0.663148 − 0.748488i \(-0.730780\pi\)
−0.663148 + 0.748488i \(0.730780\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.64601e8 −0.323636
\(438\) 0 0
\(439\) −1.69630e9 −0.956921 −0.478461 − 0.878109i \(-0.658805\pi\)
−0.478461 + 0.878109i \(0.658805\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.04940e9 −1.11999 −0.559994 − 0.828497i \(-0.689197\pi\)
−0.559994 + 0.828497i \(0.689197\pi\)
\(444\) 0 0
\(445\) −1.41674e9 −0.762134
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.62831e9 0.848937 0.424469 − 0.905443i \(-0.360461\pi\)
0.424469 + 0.905443i \(0.360461\pi\)
\(450\) 0 0
\(451\) −2.53352e8 −0.130049
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.25823e8 −0.162159
\(456\) 0 0
\(457\) −1.17242e8 −0.0574612 −0.0287306 − 0.999587i \(-0.509146\pi\)
−0.0287306 + 0.999587i \(0.509146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.50354e9 1.19015 0.595075 − 0.803670i \(-0.297122\pi\)
0.595075 + 0.803670i \(0.297122\pi\)
\(462\) 0 0
\(463\) −1.01316e9 −0.474398 −0.237199 − 0.971461i \(-0.576229\pi\)
−0.237199 + 0.971461i \(0.576229\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.82198e8 0.355392 0.177696 − 0.984085i \(-0.443136\pi\)
0.177696 + 0.984085i \(0.443136\pi\)
\(468\) 0 0
\(469\) 6.49458e8 0.290701
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.37235e8 0.233427
\(474\) 0 0
\(475\) 1.00646e9 0.430894
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.85704e8 −0.0772055 −0.0386027 − 0.999255i \(-0.512291\pi\)
−0.0386027 + 0.999255i \(0.512291\pi\)
\(480\) 0 0
\(481\) 2.64735e8 0.108468
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.85723e8 0.0739213
\(486\) 0 0
\(487\) −1.68521e9 −0.661155 −0.330577 − 0.943779i \(-0.607243\pi\)
−0.330577 + 0.943779i \(0.607243\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.19408e9 −1.21776 −0.608879 − 0.793263i \(-0.708380\pi\)
−0.608879 + 0.793263i \(0.708380\pi\)
\(492\) 0 0
\(493\) −1.69248e8 −0.0636150
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.82019e7 −0.0285739
\(498\) 0 0
\(499\) 4.03224e8 0.145276 0.0726380 − 0.997358i \(-0.476858\pi\)
0.0726380 + 0.997358i \(0.476858\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.05407e8 −0.107002 −0.0535009 − 0.998568i \(-0.517038\pi\)
−0.0535009 + 0.998568i \(0.517038\pi\)
\(504\) 0 0
\(505\) −8.00015e8 −0.276426
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.73298e9 0.582479 0.291240 − 0.956650i \(-0.405932\pi\)
0.291240 + 0.956650i \(0.405932\pi\)
\(510\) 0 0
\(511\) 2.69232e8 0.0892594
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.35817e9 −0.438157
\(516\) 0 0
\(517\) −9.44103e8 −0.300471
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.29373e8 0.163995 0.0819974 − 0.996633i \(-0.473870\pi\)
0.0819974 + 0.996633i \(0.473870\pi\)
\(522\) 0 0
\(523\) 3.60373e9 1.10153 0.550765 − 0.834661i \(-0.314336\pi\)
0.550765 + 0.834661i \(0.314336\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.86195e8 −0.114939
\(528\) 0 0
\(529\) −3.27225e9 −0.961063
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.42844e9 −0.408618
\(534\) 0 0
\(535\) −3.69173e9 −1.04230
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8.25896e7 −0.0227177
\(540\) 0 0
\(541\) −2.75672e9 −0.748517 −0.374259 − 0.927324i \(-0.622103\pi\)
−0.374259 + 0.927324i \(0.622103\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.07446e9 0.284316
\(546\) 0 0
\(547\) 6.28471e9 1.64183 0.820917 − 0.571047i \(-0.193463\pi\)
0.820917 + 0.571047i \(0.193463\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.43523e9 0.620167
\(552\) 0 0
\(553\) −7.20606e8 −0.181201
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.01153e9 1.47398 0.736991 − 0.675903i \(-0.236246\pi\)
0.736991 + 0.675903i \(0.236246\pi\)
\(558\) 0 0
\(559\) 3.02903e9 0.733435
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.21783e9 −1.23228 −0.616142 − 0.787635i \(-0.711305\pi\)
−0.616142 + 0.787635i \(0.711305\pi\)
\(564\) 0 0
\(565\) 2.33347e9 0.544293
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.35896e8 −0.121952 −0.0609758 − 0.998139i \(-0.519421\pi\)
−0.0609758 + 0.998139i \(0.519421\pi\)
\(570\) 0 0
\(571\) −5.46646e9 −1.22880 −0.614398 − 0.788997i \(-0.710601\pi\)
−0.614398 + 0.788997i \(0.710601\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.36325e8 −0.0518408
\(576\) 0 0
\(577\) 3.69396e9 0.800528 0.400264 − 0.916400i \(-0.368918\pi\)
0.400264 + 0.916400i \(0.368918\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.95985e9 −0.626114
\(582\) 0 0
\(583\) 2.51991e8 0.0526679
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.08755e9 −1.03819 −0.519093 − 0.854718i \(-0.673730\pi\)
−0.519093 + 0.854718i \(0.673730\pi\)
\(588\) 0 0
\(589\) 5.55676e9 1.12052
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.88519e9 1.15896 0.579481 − 0.814986i \(-0.303255\pi\)
0.579481 + 0.814986i \(0.303255\pi\)
\(594\) 0 0
\(595\) 2.80547e8 0.0546004
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4.10449e9 −0.780306 −0.390153 − 0.920750i \(-0.627578\pi\)
−0.390153 + 0.920750i \(0.627578\pi\)
\(600\) 0 0
\(601\) 6.08667e8 0.114372 0.0571859 − 0.998364i \(-0.481787\pi\)
0.0571859 + 0.998364i \(0.481787\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.55865e9 −0.836936
\(606\) 0 0
\(607\) −6.46963e9 −1.17414 −0.587069 − 0.809537i \(-0.699718\pi\)
−0.587069 + 0.809537i \(0.699718\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.32302e9 −0.944091
\(612\) 0 0
\(613\) 4.74144e9 0.831378 0.415689 − 0.909507i \(-0.363541\pi\)
0.415689 + 0.909507i \(0.363541\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.22535e9 1.40980 0.704898 − 0.709308i \(-0.250993\pi\)
0.704898 + 0.709308i \(0.250993\pi\)
\(618\) 0 0
\(619\) 1.09006e10 1.84728 0.923640 − 0.383262i \(-0.125199\pi\)
0.923640 + 0.383262i \(0.125199\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.02476e9 −0.335480
\(624\) 0 0
\(625\) −4.07872e9 −0.668258
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.27947e8 −0.0365223
\(630\) 0 0
\(631\) 5.18857e9 0.822138 0.411069 − 0.911604i \(-0.365156\pi\)
0.411069 + 0.911604i \(0.365156\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.67077e9 0.568918
\(636\) 0 0
\(637\) −4.65655e8 −0.0713799
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.12275e10 1.68375 0.841877 − 0.539669i \(-0.181451\pi\)
0.841877 + 0.539669i \(0.181451\pi\)
\(642\) 0 0
\(643\) 8.49079e9 1.25953 0.629767 − 0.776784i \(-0.283151\pi\)
0.629767 + 0.776784i \(0.283151\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.96743e8 0.115652 0.0578260 − 0.998327i \(-0.481583\pi\)
0.0578260 + 0.998327i \(0.481583\pi\)
\(648\) 0 0
\(649\) 6.53231e8 0.0938016
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.80739e9 1.37834 0.689171 − 0.724598i \(-0.257975\pi\)
0.689171 + 0.724598i \(0.257975\pi\)
\(654\) 0 0
\(655\) −1.04872e10 −1.45819
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.00172e10 1.36348 0.681741 − 0.731594i \(-0.261223\pi\)
0.681741 + 0.731594i \(0.261223\pi\)
\(660\) 0 0
\(661\) −1.49593e9 −0.201469 −0.100734 − 0.994913i \(-0.532119\pi\)
−0.100734 + 0.994913i \(0.532119\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.03664e9 −0.532286
\(666\) 0 0
\(667\) −5.71808e8 −0.0746122
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.25821e8 0.118304
\(672\) 0 0
\(673\) 7.25457e9 0.917401 0.458700 − 0.888591i \(-0.348315\pi\)
0.458700 + 0.888591i \(0.348315\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.47562e10 1.82774 0.913870 − 0.406006i \(-0.133079\pi\)
0.913870 + 0.406006i \(0.133079\pi\)
\(678\) 0 0
\(679\) 2.65429e8 0.0325390
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5.53531e9 −0.664767 −0.332383 − 0.943144i \(-0.607853\pi\)
−0.332383 + 0.943144i \(0.607853\pi\)
\(684\) 0 0
\(685\) 7.20171e8 0.0856089
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.42077e9 0.165484
\(690\) 0 0
\(691\) −5.55281e9 −0.640236 −0.320118 − 0.947378i \(-0.603723\pi\)
−0.320118 + 0.947378i \(0.603723\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.15662e9 −0.469671
\(696\) 0 0
\(697\) 1.22995e9 0.137585
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.06044e10 1.16271 0.581357 − 0.813648i \(-0.302522\pi\)
0.581357 + 0.813648i \(0.302522\pi\)
\(702\) 0 0
\(703\) 3.27982e9 0.356046
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.14335e9 −0.121678
\(708\) 0 0
\(709\) −5.44638e9 −0.573913 −0.286957 − 0.957944i \(-0.592644\pi\)
−0.286957 + 0.957944i \(0.592644\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.30477e9 −0.134809
\(714\) 0 0
\(715\) 6.66844e8 0.0682265
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.90216e10 1.90851 0.954257 − 0.298986i \(-0.0966484\pi\)
0.954257 + 0.298986i \(0.0966484\pi\)
\(720\) 0 0
\(721\) −1.94105e9 −0.192870
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.01931e9 0.0993400
\(726\) 0 0
\(727\) 7.80283e9 0.753150 0.376575 − 0.926386i \(-0.377102\pi\)
0.376575 + 0.926386i \(0.377102\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.60812e9 −0.246954
\(732\) 0 0
\(733\) −1.20743e9 −0.113239 −0.0566197 − 0.998396i \(-0.518032\pi\)
−0.0566197 + 0.998396i \(0.518032\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.32921e9 −0.122309
\(738\) 0 0
\(739\) −1.64892e10 −1.50294 −0.751472 − 0.659765i \(-0.770656\pi\)
−0.751472 + 0.659765i \(0.770656\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.43983e9 0.218222 0.109111 − 0.994030i \(-0.465200\pi\)
0.109111 + 0.994030i \(0.465200\pi\)
\(744\) 0 0
\(745\) 9.37099e9 0.830306
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.27610e9 −0.458803
\(750\) 0 0
\(751\) −1.51959e10 −1.30914 −0.654571 − 0.756000i \(-0.727151\pi\)
−0.654571 + 0.756000i \(0.727151\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.55844e9 −0.216352
\(756\) 0 0
\(757\) −1.52456e9 −0.127735 −0.0638673 − 0.997958i \(-0.520343\pi\)
−0.0638673 + 0.997958i \(0.520343\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.33134e10 1.91761 0.958803 − 0.284072i \(-0.0916854\pi\)
0.958803 + 0.284072i \(0.0916854\pi\)
\(762\) 0 0
\(763\) 1.53558e9 0.125151
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.68303e9 0.294728
\(768\) 0 0
\(769\) −5.50489e9 −0.436523 −0.218261 − 0.975890i \(-0.570039\pi\)
−0.218261 + 0.975890i \(0.570039\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.72465e10 −1.34299 −0.671494 − 0.741010i \(-0.734347\pi\)
−0.671494 + 0.741010i \(0.734347\pi\)
\(774\) 0 0
\(775\) 2.32589e9 0.179487
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.76971e10 −1.34128
\(780\) 0 0
\(781\) 1.60052e8 0.0120221
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.32550e10 −0.977995
\(786\) 0 0
\(787\) −8.41133e9 −0.615111 −0.307555 − 0.951530i \(-0.599511\pi\)
−0.307555 + 0.951530i \(0.599511\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.33492e9 0.239589
\(792\) 0 0
\(793\) 5.21994e9 0.371715
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7.89074e9 −0.552095 −0.276047 − 0.961144i \(-0.589025\pi\)
−0.276047 + 0.961144i \(0.589025\pi\)
\(798\) 0 0
\(799\) 4.58334e9 0.317884
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −5.51024e8 −0.0375548
\(804\) 0 0
\(805\) 9.47832e8 0.0640392
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.56202e10 −1.03721 −0.518606 − 0.855013i \(-0.673549\pi\)
−0.518606 + 0.855013i \(0.673549\pi\)
\(810\) 0 0
\(811\) 8.33712e9 0.548836 0.274418 − 0.961610i \(-0.411515\pi\)
0.274418 + 0.961610i \(0.411515\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.97779e9 −0.257389
\(816\) 0 0
\(817\) 3.75269e10 2.40749
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.17639e10 0.741907 0.370954 − 0.928651i \(-0.379031\pi\)
0.370954 + 0.928651i \(0.379031\pi\)
\(822\) 0 0
\(823\) −2.54212e10 −1.58963 −0.794816 − 0.606851i \(-0.792433\pi\)
−0.794816 + 0.606851i \(0.792433\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.92118e10 −1.18113 −0.590566 − 0.806990i \(-0.701095\pi\)
−0.590566 + 0.806990i \(0.701095\pi\)
\(828\) 0 0
\(829\) −2.08097e10 −1.26860 −0.634299 − 0.773088i \(-0.718711\pi\)
−0.634299 + 0.773088i \(0.718711\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.00948e8 0.0240342
\(834\) 0 0
\(835\) 1.36801e10 0.813179
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.17898e10 1.27376 0.636878 − 0.770964i \(-0.280225\pi\)
0.636878 + 0.770964i \(0.280225\pi\)
\(840\) 0 0
\(841\) −1.47836e10 −0.857024
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.12999e10 −0.644280
\(846\) 0 0
\(847\) −6.51507e9 −0.368406
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −7.70125e8 −0.0428359
\(852\) 0 0
\(853\) 2.42027e10 1.33519 0.667594 − 0.744526i \(-0.267324\pi\)
0.667594 + 0.744526i \(0.267324\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.21583e10 −1.20255 −0.601276 − 0.799042i \(-0.705341\pi\)
−0.601276 + 0.799042i \(0.705341\pi\)
\(858\) 0 0
\(859\) 2.12882e10 1.14594 0.572970 − 0.819576i \(-0.305791\pi\)
0.572970 + 0.819576i \(0.305791\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.55835e10 −1.88456 −0.942281 − 0.334823i \(-0.891323\pi\)
−0.942281 + 0.334823i \(0.891323\pi\)
\(864\) 0 0
\(865\) 6.82215e9 0.358397
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.47483e9 0.0762380
\(870\) 0 0
\(871\) −7.49433e9 −0.384299
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −8.12087e9 −0.409802
\(876\) 0 0
\(877\) −3.86385e10 −1.93429 −0.967146 − 0.254223i \(-0.918180\pi\)
−0.967146 + 0.254223i \(0.918180\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.74469e9 0.0859615 0.0429807 − 0.999076i \(-0.486315\pi\)
0.0429807 + 0.999076i \(0.486315\pi\)
\(882\) 0 0
\(883\) 2.42096e10 1.18338 0.591691 − 0.806165i \(-0.298461\pi\)
0.591691 + 0.806165i \(0.298461\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.46025e10 1.18371 0.591857 − 0.806043i \(-0.298395\pi\)
0.591857 + 0.806043i \(0.298395\pi\)
\(888\) 0 0
\(889\) 5.24615e9 0.250429
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.59473e10 −3.09897
\(894\) 0 0
\(895\) −3.46353e10 −1.61487
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.62770e9 0.258328
\(900\) 0 0
\(901\) −1.22334e9 −0.0557200
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.82578e10 −1.26727
\(906\) 0 0
\(907\) 8.41700e9 0.374569 0.187284 − 0.982306i \(-0.440031\pi\)
0.187284 + 0.982306i \(0.440031\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.62674e10 1.58929 0.794644 − 0.607076i \(-0.207658\pi\)
0.794644 + 0.607076i \(0.207658\pi\)
\(912\) 0 0
\(913\) 6.05777e9 0.263430
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.49879e10 −0.641874
\(918\) 0 0
\(919\) 3.50569e10 1.48994 0.744971 − 0.667097i \(-0.232463\pi\)
0.744971 + 0.667097i \(0.232463\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.02400e8 0.0377740
\(924\) 0 0
\(925\) 1.37284e9 0.0570325
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −5.98739e9 −0.245009 −0.122505 − 0.992468i \(-0.539093\pi\)
−0.122505 + 0.992468i \(0.539093\pi\)
\(930\) 0 0
\(931\) −5.76904e9 −0.234304
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.74180e8 −0.0229724
\(936\) 0 0
\(937\) −1.27744e10 −0.507283 −0.253642 − 0.967298i \(-0.581628\pi\)
−0.253642 + 0.967298i \(0.581628\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.08847e9 −0.0817080 −0.0408540 − 0.999165i \(-0.513008\pi\)
−0.0408540 + 0.999165i \(0.513008\pi\)
\(942\) 0 0
\(943\) 4.15540e9 0.161370
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.26786e9 −0.201562 −0.100781 − 0.994909i \(-0.532134\pi\)
−0.100781 + 0.994909i \(0.532134\pi\)
\(948\) 0 0
\(949\) −3.10677e9 −0.117999
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.40878e10 0.527252 0.263626 − 0.964625i \(-0.415082\pi\)
0.263626 + 0.964625i \(0.415082\pi\)
\(954\) 0 0
\(955\) −3.53241e10 −1.31238
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.02924e9 0.0376837
\(960\) 0 0
\(961\) −1.46712e10 −0.533253
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.90256e10 1.03976
\(966\) 0 0
\(967\) −4.95778e10 −1.76317 −0.881587 − 0.472022i \(-0.843524\pi\)
−0.881587 + 0.472022i \(0.843524\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.72967e10 1.65792 0.828960 − 0.559308i \(-0.188933\pi\)
0.828960 + 0.559308i \(0.188933\pi\)
\(972\) 0 0
\(973\) −5.94050e9 −0.206742
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.99103e10 −0.683041 −0.341520 − 0.939874i \(-0.610942\pi\)
−0.341520 + 0.939874i \(0.610942\pi\)
\(978\) 0 0
\(979\) 4.14398e9 0.141149
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.84811e9 −0.0620568 −0.0310284 − 0.999519i \(-0.509878\pi\)
−0.0310284 + 0.999519i \(0.509878\pi\)
\(984\) 0 0
\(985\) 1.39282e10 0.464374
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.81157e9 −0.289645
\(990\) 0 0
\(991\) −1.39246e10 −0.454489 −0.227245 − 0.973838i \(-0.572972\pi\)
−0.227245 + 0.973838i \(0.572972\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.87238e10 0.602579
\(996\) 0 0
\(997\) −3.58707e10 −1.14632 −0.573161 − 0.819443i \(-0.694283\pi\)
−0.573161 + 0.819443i \(0.694283\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 252.8.a.b.1.1 1
3.2 odd 2 84.8.a.b.1.1 ✓ 1
12.11 even 2 336.8.a.c.1.1 1
21.2 odd 6 588.8.i.d.361.1 2
21.5 even 6 588.8.i.e.361.1 2
21.11 odd 6 588.8.i.d.373.1 2
21.17 even 6 588.8.i.e.373.1 2
21.20 even 2 588.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.b.1.1 ✓ 1 3.2 odd 2
252.8.a.b.1.1 1 1.1 even 1 trivial
336.8.a.c.1.1 1 12.11 even 2
588.8.a.b.1.1 1 21.20 even 2
588.8.i.d.361.1 2 21.2 odd 6
588.8.i.d.373.1 2 21.11 odd 6
588.8.i.e.361.1 2 21.5 even 6
588.8.i.e.373.1 2 21.17 even 6