↑

Properties

Label 432.8.a.c.1.1
Level $432$
Weight $8$
Character 432.1
Self dual yes
Analytic conductor $134.950$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,8,Mod(1,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, 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("432.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.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: \(134.950331009\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 432.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-105.000 q^{5} +937.000 q^{7} +O(q^{10})\) \(q-105.000 q^{5} +937.000 q^{7} +5943.00 q^{11} +68.0000 q^{13} +5400.00 q^{17} +48382.0 q^{19} -642.000 q^{23} -67100.0 q^{25} +125934. q^{29} +161275. q^{31} -98385.0 q^{35} -414286. q^{37} +627474. q^{41} -570590. q^{43} +538698. q^{47} +54426.0 q^{49} -356283. q^{53} -624015. q^{55} -2.91083e6 q^{59} +2.68417e6 q^{61} -7140.00 q^{65} -2.68108e6 q^{67} -3.70548e6 q^{71} -153151. q^{73} +5.56859e6 q^{77} +7.57929e6 q^{79} +9.34600e6 q^{83} -567000. q^{85} -4.03360e6 q^{89} +63716.0 q^{91} -5.08011e6 q^{95} -5.75410e6 q^{97} +O(q^{100})\)

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
\(5\) −105.000 −0.375659 −0.187830 − 0.982202i \(-0.560145\pi\)
−0.187830 + 0.982202i \(0.560145\pi\)
\(6\) 0 0
\(7\) 937.000 1.03252 0.516258 − 0.856433i \(-0.327325\pi\)
0.516258 + 0.856433i \(0.327325\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5943.00 1.34627 0.673134 − 0.739521i \(-0.264948\pi\)
0.673134 + 0.739521i \(0.264948\pi\)
\(12\) 0 0
\(13\) 68.0000 0.00858435 0.00429217 − 0.999991i \(-0.498634\pi\)
0.00429217 + 0.999991i \(0.498634\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5400.00 0.266577 0.133288 − 0.991077i \(-0.457446\pi\)
0.133288 + 0.991077i \(0.457446\pi\)
\(18\) 0 0
\(19\) 48382.0 1.61825 0.809126 − 0.587635i \(-0.199941\pi\)
0.809126 + 0.587635i \(0.199941\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −642.000 −0.0110024 −0.00550120 − 0.999985i \(-0.501751\pi\)
−0.00550120 + 0.999985i \(0.501751\pi\)
\(24\) 0 0
\(25\) −67100.0 −0.858880
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 125934. 0.958849 0.479424 − 0.877583i \(-0.340846\pi\)
0.479424 + 0.877583i \(0.340846\pi\)
\(30\) 0 0
\(31\) 161275. 0.972302 0.486151 − 0.873875i \(-0.338401\pi\)
0.486151 + 0.873875i \(0.338401\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −98385.0 −0.387874
\(36\) 0 0
\(37\) −414286. −1.34460 −0.672302 − 0.740277i \(-0.734694\pi\)
−0.672302 + 0.740277i \(0.734694\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 627474. 1.42184 0.710922 − 0.703270i \(-0.248278\pi\)
0.710922 + 0.703270i \(0.248278\pi\)
\(42\) 0 0
\(43\) −570590. −1.09442 −0.547211 − 0.836995i \(-0.684310\pi\)
−0.547211 + 0.836995i \(0.684310\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 538698. 0.756838 0.378419 − 0.925634i \(-0.376468\pi\)
0.378419 + 0.925634i \(0.376468\pi\)
\(48\) 0 0
\(49\) 54426.0 0.0660876
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −356283. −0.328723 −0.164361 − 0.986400i \(-0.552556\pi\)
−0.164361 + 0.986400i \(0.552556\pi\)
\(54\) 0 0
\(55\) −624015. −0.505738
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.91083e6 −1.84516 −0.922581 − 0.385803i \(-0.873924\pi\)
−0.922581 + 0.385803i \(0.873924\pi\)
\(60\) 0 0
\(61\) 2.68417e6 1.51410 0.757051 − 0.653356i \(-0.226639\pi\)
0.757051 + 0.653356i \(0.226639\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7140.00 −0.00322479
\(66\) 0 0
\(67\) −2.68108e6 −1.08905 −0.544525 − 0.838745i \(-0.683290\pi\)
−0.544525 + 0.838745i \(0.683290\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.70548e6 −1.22868 −0.614342 − 0.789040i \(-0.710578\pi\)
−0.614342 + 0.789040i \(0.710578\pi\)
\(72\) 0 0
\(73\) −153151. −0.0460776 −0.0230388 − 0.999735i \(-0.507334\pi\)
−0.0230388 + 0.999735i \(0.507334\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.56859e6 1.39004
\(78\) 0 0
\(79\) 7.57929e6 1.72955 0.864776 − 0.502158i \(-0.167460\pi\)
0.864776 + 0.502158i \(0.167460\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.34600e6 1.79412 0.897062 − 0.441905i \(-0.145697\pi\)
0.897062 + 0.441905i \(0.145697\pi\)
\(84\) 0 0
\(85\) −567000. −0.100142
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.03360e6 −0.606496 −0.303248 − 0.952912i \(-0.598071\pi\)
−0.303248 + 0.952912i \(0.598071\pi\)
\(90\) 0 0
\(91\) 63716.0 0.00886347
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.08011e6 −0.607912
\(96\) 0 0
\(97\) −5.75410e6 −0.640142 −0.320071 − 0.947394i \(-0.603707\pi\)
−0.320071 + 0.947394i \(0.603707\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.87922e6 −0.857532 −0.428766 − 0.903416i \(-0.641051\pi\)
−0.428766 + 0.903416i \(0.641051\pi\)
\(102\) 0 0
\(103\) 2.65565e6 0.239464 0.119732 − 0.992806i \(-0.461796\pi\)
0.119732 + 0.992806i \(0.461796\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.11234e7 1.66694 0.833471 − 0.552564i \(-0.186350\pi\)
0.833471 + 0.552564i \(0.186350\pi\)
\(108\) 0 0
\(109\) −1.84213e7 −1.36247 −0.681237 − 0.732063i \(-0.738558\pi\)
−0.681237 + 0.732063i \(0.738558\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.86285e7 1.21452 0.607258 − 0.794505i \(-0.292270\pi\)
0.607258 + 0.794505i \(0.292270\pi\)
\(114\) 0 0
\(115\) 67410.0 0.00413316
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.05980e6 0.275245
\(120\) 0 0
\(121\) 1.58321e7 0.812436
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.52486e7 0.698306
\(126\) 0 0
\(127\) 2.58380e6 0.111930 0.0559649 − 0.998433i \(-0.482177\pi\)
0.0559649 + 0.998433i \(0.482177\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.19283e7 1.62951 0.814757 − 0.579803i \(-0.196871\pi\)
0.814757 + 0.579803i \(0.196871\pi\)
\(132\) 0 0
\(133\) 4.53339e7 1.67087
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.26662e6 −0.0420846 −0.0210423 − 0.999779i \(-0.506698\pi\)
−0.0210423 + 0.999779i \(0.506698\pi\)
\(138\) 0 0
\(139\) −2.30561e7 −0.728173 −0.364086 − 0.931365i \(-0.618619\pi\)
−0.364086 + 0.931365i \(0.618619\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 404124. 0.0115568
\(144\) 0 0
\(145\) −1.32231e7 −0.360200
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.38266e7 −0.590080 −0.295040 − 0.955485i \(-0.595333\pi\)
−0.295040 + 0.955485i \(0.595333\pi\)
\(150\) 0 0
\(151\) −4.50493e7 −1.06480 −0.532401 − 0.846492i \(-0.678710\pi\)
−0.532401 + 0.846492i \(0.678710\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.69339e7 −0.365254
\(156\) 0 0
\(157\) 3.52575e7 0.727115 0.363558 − 0.931572i \(-0.381562\pi\)
0.363558 + 0.931572i \(0.381562\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −601554. −0.0113601
\(162\) 0 0
\(163\) 6.55009e7 1.18465 0.592326 − 0.805699i \(-0.298210\pi\)
0.592326 + 0.805699i \(0.298210\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.28090e7 0.877405 0.438703 − 0.898632i \(-0.355438\pi\)
0.438703 + 0.898632i \(0.355438\pi\)
\(168\) 0 0
\(169\) −6.27439e7 −0.999926
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.10733e8 1.62598 0.812991 − 0.582277i \(-0.197838\pi\)
0.812991 + 0.582277i \(0.197838\pi\)
\(174\) 0 0
\(175\) −6.28727e7 −0.886807
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.19427e7 −0.155639 −0.0778195 − 0.996967i \(-0.524796\pi\)
−0.0778195 + 0.996967i \(0.524796\pi\)
\(180\) 0 0
\(181\) −7.95226e7 −0.996817 −0.498408 − 0.866942i \(-0.666082\pi\)
−0.498408 + 0.866942i \(0.666082\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.35000e7 0.505113
\(186\) 0 0
\(187\) 3.20922e7 0.358884
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.02315e6 −0.0625470 −0.0312735 − 0.999511i \(-0.509956\pi\)
−0.0312735 + 0.999511i \(0.509956\pi\)
\(192\) 0 0
\(193\) −5.85850e7 −0.586591 −0.293296 − 0.956022i \(-0.594752\pi\)
−0.293296 + 0.956022i \(0.594752\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.68414e8 1.56945 0.784725 − 0.619844i \(-0.212804\pi\)
0.784725 + 0.619844i \(0.212804\pi\)
\(198\) 0 0
\(199\) 1.01669e8 0.914543 0.457271 − 0.889327i \(-0.348827\pi\)
0.457271 + 0.889327i \(0.348827\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.18000e8 0.990026
\(204\) 0 0
\(205\) −6.58848e7 −0.534129
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.87534e8 2.17860
\(210\) 0 0
\(211\) 8.63820e7 0.633045 0.316523 − 0.948585i \(-0.397485\pi\)
0.316523 + 0.948585i \(0.397485\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.99120e7 0.411130
\(216\) 0 0
\(217\) 1.51115e8 1.00392
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 367200. 0.00228839
\(222\) 0 0
\(223\) −4.16173e7 −0.251308 −0.125654 − 0.992074i \(-0.540103\pi\)
−0.125654 + 0.992074i \(0.540103\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.09315e8 1.18771 0.593855 − 0.804572i \(-0.297605\pi\)
0.593855 + 0.804572i \(0.297605\pi\)
\(228\) 0 0
\(229\) 3.32919e8 1.83195 0.915976 − 0.401233i \(-0.131418\pi\)
0.915976 + 0.401233i \(0.131418\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.95428e7 −0.411960 −0.205980 − 0.978556i \(-0.566038\pi\)
−0.205980 + 0.978556i \(0.566038\pi\)
\(234\) 0 0
\(235\) −5.65633e7 −0.284313
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.46900e8 0.696034 0.348017 − 0.937488i \(-0.386855\pi\)
0.348017 + 0.937488i \(0.386855\pi\)
\(240\) 0 0
\(241\) −3.03055e7 −0.139464 −0.0697320 − 0.997566i \(-0.522214\pi\)
−0.0697320 + 0.997566i \(0.522214\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.71473e6 −0.0248264
\(246\) 0 0
\(247\) 3.28998e6 0.0138916
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.47027e8 0.586866 0.293433 − 0.955980i \(-0.405202\pi\)
0.293433 + 0.955980i \(0.405202\pi\)
\(252\) 0 0
\(253\) −3.81541e6 −0.0148122
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.67776e8 1.35150 0.675752 − 0.737129i \(-0.263819\pi\)
0.675752 + 0.737129i \(0.263819\pi\)
\(258\) 0 0
\(259\) −3.88186e8 −1.38832
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.25857e8 1.10454 0.552270 − 0.833665i \(-0.313762\pi\)
0.552270 + 0.833665i \(0.313762\pi\)
\(264\) 0 0
\(265\) 3.74097e7 0.123488
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.39029e8 1.06195 0.530976 − 0.847387i \(-0.321826\pi\)
0.530976 + 0.847387i \(0.321826\pi\)
\(270\) 0 0
\(271\) 2.50778e6 0.00765416 0.00382708 − 0.999993i \(-0.498782\pi\)
0.00382708 + 0.999993i \(0.498782\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.98775e8 −1.15628
\(276\) 0 0
\(277\) −5.28585e7 −0.149429 −0.0747146 − 0.997205i \(-0.523805\pi\)
−0.0747146 + 0.997205i \(0.523805\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.39552e8 −1.45064 −0.725322 − 0.688409i \(-0.758309\pi\)
−0.725322 + 0.688409i \(0.758309\pi\)
\(282\) 0 0
\(283\) −6.58565e7 −0.172721 −0.0863607 − 0.996264i \(-0.527524\pi\)
−0.0863607 + 0.996264i \(0.527524\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.87943e8 1.46808
\(288\) 0 0
\(289\) −3.81179e8 −0.928937
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.52583e7 0.0818889 0.0409444 − 0.999161i \(-0.486963\pi\)
0.0409444 + 0.999161i \(0.486963\pi\)
\(294\) 0 0
\(295\) 3.05637e8 0.693152
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −43656.0 −9.44484e−5 0
\(300\) 0 0
\(301\) −5.34643e8 −1.13001
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.81838e8 −0.568787
\(306\) 0 0
\(307\) −1.14259e8 −0.225376 −0.112688 − 0.993630i \(-0.535946\pi\)
−0.112688 + 0.993630i \(0.535946\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.55463e8 0.293066 0.146533 − 0.989206i \(-0.453189\pi\)
0.146533 + 0.989206i \(0.453189\pi\)
\(312\) 0 0
\(313\) 3.82299e8 0.704690 0.352345 − 0.935870i \(-0.385384\pi\)
0.352345 + 0.935870i \(0.385384\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.44887e8 −0.784408 −0.392204 − 0.919878i \(-0.628287\pi\)
−0.392204 + 0.919878i \(0.628287\pi\)
\(318\) 0 0
\(319\) 7.48426e8 1.29087
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.61263e8 0.431389
\(324\) 0 0
\(325\) −4.56280e6 −0.00737292
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.04760e8 0.781447
\(330\) 0 0
\(331\) 5.50888e8 0.834960 0.417480 − 0.908686i \(-0.362913\pi\)
0.417480 + 0.908686i \(0.362913\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.81513e8 0.409112
\(336\) 0 0
\(337\) −4.96793e8 −0.707084 −0.353542 − 0.935419i \(-0.615023\pi\)
−0.353542 + 0.935419i \(0.615023\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.58457e8 1.30898
\(342\) 0 0
\(343\) −7.20663e8 −0.964279
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.51162e8 −0.322701 −0.161350 − 0.986897i \(-0.551585\pi\)
−0.161350 + 0.986897i \(0.551585\pi\)
\(348\) 0 0
\(349\) −4.82914e8 −0.608108 −0.304054 − 0.952655i \(-0.598340\pi\)
−0.304054 + 0.952655i \(0.598340\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.86935e8 0.589195 0.294598 − 0.955621i \(-0.404814\pi\)
0.294598 + 0.955621i \(0.404814\pi\)
\(354\) 0 0
\(355\) 3.89075e8 0.461567
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.59969e8 −0.524684 −0.262342 − 0.964975i \(-0.584495\pi\)
−0.262342 + 0.964975i \(0.584495\pi\)
\(360\) 0 0
\(361\) 1.44695e9 1.61874
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.60809e7 0.0173095
\(366\) 0 0
\(367\) −6.52503e8 −0.689051 −0.344526 − 0.938777i \(-0.611960\pi\)
−0.344526 + 0.938777i \(0.611960\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.33837e8 −0.339411
\(372\) 0 0
\(373\) 1.91791e9 1.91358 0.956791 − 0.290775i \(-0.0939131\pi\)
0.956791 + 0.290775i \(0.0939131\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.56351e6 0.00823109
\(378\) 0 0
\(379\) −1.43151e9 −1.35070 −0.675348 − 0.737499i \(-0.736007\pi\)
−0.675348 + 0.737499i \(0.736007\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.39859e8 −0.127203 −0.0636013 − 0.997975i \(-0.520259\pi\)
−0.0636013 + 0.997975i \(0.520259\pi\)
\(384\) 0 0
\(385\) −5.84702e8 −0.522182
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.77592e9 1.52968 0.764838 − 0.644222i \(-0.222819\pi\)
0.764838 + 0.644222i \(0.222819\pi\)
\(390\) 0 0
\(391\) −3.46680e6 −0.00293299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.95825e8 −0.649722
\(396\) 0 0
\(397\) −1.05301e9 −0.844630 −0.422315 − 0.906449i \(-0.638782\pi\)
−0.422315 + 0.906449i \(0.638782\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.33012e9 −1.80457 −0.902284 − 0.431142i \(-0.858111\pi\)
−0.902284 + 0.431142i \(0.858111\pi\)
\(402\) 0 0
\(403\) 1.09667e7 0.00834658
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.46210e9 −1.81020
\(408\) 0 0
\(409\) −1.50422e8 −0.108712 −0.0543562 − 0.998522i \(-0.517311\pi\)
−0.0543562 + 0.998522i \(0.517311\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.72745e9 −1.90516
\(414\) 0 0
\(415\) −9.81330e8 −0.673980
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.11432e8 0.605306 0.302653 − 0.953101i \(-0.402128\pi\)
0.302653 + 0.953101i \(0.402128\pi\)
\(420\) 0 0
\(421\) −7.53482e8 −0.492137 −0.246068 − 0.969252i \(-0.579139\pi\)
−0.246068 + 0.969252i \(0.579139\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.62340e8 −0.228958
\(426\) 0 0
\(427\) 2.51507e9 1.56333
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.72892e9 −1.64180 −0.820900 − 0.571072i \(-0.806528\pi\)
−0.820900 + 0.571072i \(0.806528\pi\)
\(432\) 0 0
\(433\) 1.84699e9 1.09335 0.546673 − 0.837346i \(-0.315894\pi\)
0.546673 + 0.837346i \(0.315894\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.10612e7 −0.0178047
\(438\) 0 0
\(439\) −1.50066e9 −0.846557 −0.423279 − 0.906000i \(-0.639121\pi\)
−0.423279 + 0.906000i \(0.639121\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.31690e9 0.719680 0.359840 − 0.933014i \(-0.382831\pi\)
0.359840 + 0.933014i \(0.382831\pi\)
\(444\) 0 0
\(445\) 4.23528e8 0.227836
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.71216e8 −0.141401 −0.0707006 − 0.997498i \(-0.522523\pi\)
−0.0707006 + 0.997498i \(0.522523\pi\)
\(450\) 0 0
\(451\) 3.72908e9 1.91418
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.69018e6 −0.00332964
\(456\) 0 0
\(457\) −2.15061e9 −1.05404 −0.527018 − 0.849854i \(-0.676690\pi\)
−0.527018 + 0.849854i \(0.676690\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.84188e8 −0.467870 −0.233935 − 0.972252i \(-0.575160\pi\)
−0.233935 + 0.972252i \(0.575160\pi\)
\(462\) 0 0
\(463\) 2.56748e9 1.20219 0.601096 − 0.799177i \(-0.294731\pi\)
0.601096 + 0.799177i \(0.294731\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.97867e8 −0.317076 −0.158538 − 0.987353i \(-0.550678\pi\)
−0.158538 + 0.987353i \(0.550678\pi\)
\(468\) 0 0
\(469\) −2.51217e9 −1.12446
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.39102e9 −1.47338
\(474\) 0 0
\(475\) −3.24643e9 −1.38988
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.07066e9 −1.27661 −0.638304 − 0.769785i \(-0.720364\pi\)
−0.638304 + 0.769785i \(0.720364\pi\)
\(480\) 0 0
\(481\) −2.81714e7 −0.0115425
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.04180e8 0.240475
\(486\) 0 0
\(487\) 1.74254e9 0.683646 0.341823 − 0.939764i \(-0.388956\pi\)
0.341823 + 0.939764i \(0.388956\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.01054e9 −0.385272 −0.192636 − 0.981270i \(-0.561704\pi\)
−0.192636 + 0.981270i \(0.561704\pi\)
\(492\) 0 0
\(493\) 6.80044e8 0.255607
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.47203e9 −1.26864
\(498\) 0 0
\(499\) −1.56087e9 −0.562361 −0.281181 − 0.959655i \(-0.590726\pi\)
−0.281181 + 0.959655i \(0.590726\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.63663e9 0.573407 0.286704 − 0.958019i \(-0.407441\pi\)
0.286704 + 0.958019i \(0.407441\pi\)
\(504\) 0 0
\(505\) 9.32318e8 0.322140
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.28526e8 0.177645 0.0888227 − 0.996047i \(-0.471690\pi\)
0.0888227 + 0.996047i \(0.471690\pi\)
\(510\) 0 0
\(511\) −1.43502e8 −0.0475758
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.78843e8 −0.0899570
\(516\) 0 0
\(517\) 3.20148e9 1.01891
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.50707e9 −1.39625 −0.698124 − 0.715977i \(-0.745982\pi\)
−0.698124 + 0.715977i \(0.745982\pi\)
\(522\) 0 0
\(523\) −3.60454e9 −1.10178 −0.550888 − 0.834579i \(-0.685711\pi\)
−0.550888 + 0.834579i \(0.685711\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.70885e8 0.259193
\(528\) 0 0
\(529\) −3.40441e9 −0.999879
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.26682e7 0.0122056
\(534\) 0 0
\(535\) −2.21796e9 −0.626202
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.23454e8 0.0889716
\(540\) 0 0
\(541\) −2.24714e9 −0.610154 −0.305077 − 0.952328i \(-0.598682\pi\)
−0.305077 + 0.952328i \(0.598682\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.93424e9 0.511827
\(546\) 0 0
\(547\) −4.26741e9 −1.11483 −0.557415 − 0.830234i \(-0.688207\pi\)
−0.557415 + 0.830234i \(0.688207\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.09294e9 1.55166
\(552\) 0 0
\(553\) 7.10179e9 1.78579
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.18311e8 −0.176124 −0.0880622 − 0.996115i \(-0.528067\pi\)
−0.0880622 + 0.996115i \(0.528067\pi\)
\(558\) 0 0
\(559\) −3.88001e7 −0.00939489
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.71155e9 −1.34888 −0.674442 − 0.738328i \(-0.735616\pi\)
−0.674442 + 0.738328i \(0.735616\pi\)
\(564\) 0 0
\(565\) −1.95599e9 −0.456244
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6.36000e9 −1.44732 −0.723660 − 0.690157i \(-0.757542\pi\)
−0.723660 + 0.690157i \(0.757542\pi\)
\(570\) 0 0
\(571\) 4.74950e9 1.06763 0.533816 − 0.845600i \(-0.320757\pi\)
0.533816 + 0.845600i \(0.320757\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.30782e7 0.00944975
\(576\) 0 0
\(577\) −8.95938e8 −0.194161 −0.0970807 − 0.995277i \(-0.530950\pi\)
−0.0970807 + 0.995277i \(0.530950\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.75720e9 1.85246
\(582\) 0 0
\(583\) −2.11739e9 −0.442549
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.42764e9 −1.31165 −0.655826 − 0.754912i \(-0.727679\pi\)
−0.655826 + 0.754912i \(0.727679\pi\)
\(588\) 0 0
\(589\) 7.80281e9 1.57343
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.89741e9 1.35830 0.679148 − 0.734001i \(-0.262349\pi\)
0.679148 + 0.734001i \(0.262349\pi\)
\(594\) 0 0
\(595\) −5.31279e8 −0.103398
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.50564e9 −1.61701 −0.808506 − 0.588488i \(-0.799723\pi\)
−0.808506 + 0.588488i \(0.799723\pi\)
\(600\) 0 0
\(601\) −7.03680e9 −1.32225 −0.661126 − 0.750275i \(-0.729921\pi\)
−0.661126 + 0.750275i \(0.729921\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.66237e9 −0.305199
\(606\) 0 0
\(607\) 9.16881e9 1.66400 0.831999 − 0.554778i \(-0.187197\pi\)
0.831999 + 0.554778i \(0.187197\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.66315e7 0.00649696
\(612\) 0 0
\(613\) 1.89814e9 0.332826 0.166413 − 0.986056i \(-0.446781\pi\)
0.166413 + 0.986056i \(0.446781\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.00801e10 −1.72769 −0.863847 − 0.503755i \(-0.831951\pi\)
−0.863847 + 0.503755i \(0.831951\pi\)
\(618\) 0 0
\(619\) 2.07797e9 0.352145 0.176073 − 0.984377i \(-0.443661\pi\)
0.176073 + 0.984377i \(0.443661\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.77949e9 −0.626216
\(624\) 0 0
\(625\) 3.64108e9 0.596555
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.23714e9 −0.358440
\(630\) 0 0
\(631\) 8.91619e9 1.41279 0.706393 − 0.707820i \(-0.250321\pi\)
0.706393 + 0.707820i \(0.250321\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.71299e8 −0.0420475
\(636\) 0 0
\(637\) 3.70097e6 0.000567319 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.87905e9 0.281796 0.140898 − 0.990024i \(-0.455001\pi\)
0.140898 + 0.990024i \(0.455001\pi\)
\(642\) 0 0
\(643\) 2.30520e9 0.341956 0.170978 − 0.985275i \(-0.445307\pi\)
0.170978 + 0.985275i \(0.445307\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.73867e9 −0.833002 −0.416501 − 0.909135i \(-0.636744\pi\)
−0.416501 + 0.909135i \(0.636744\pi\)
\(648\) 0 0
\(649\) −1.72991e10 −2.48408
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.36875e9 1.03561 0.517807 − 0.855497i \(-0.326749\pi\)
0.517807 + 0.855497i \(0.326749\pi\)
\(654\) 0 0
\(655\) −4.40247e9 −0.612142
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.24540e10 1.69515 0.847577 − 0.530672i \(-0.178060\pi\)
0.847577 + 0.530672i \(0.178060\pi\)
\(660\) 0 0
\(661\) −5.53207e9 −0.745045 −0.372523 − 0.928023i \(-0.621507\pi\)
−0.372523 + 0.928023i \(0.621507\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.76006e9 −0.627678
\(666\) 0 0
\(667\) −8.08496e7 −0.0105496
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.59520e10 2.03839
\(672\) 0 0
\(673\) 1.11788e10 1.41366 0.706829 − 0.707385i \(-0.250125\pi\)
0.706829 + 0.707385i \(0.250125\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.60037e10 1.98226 0.991130 − 0.132893i \(-0.0424267\pi\)
0.991130 + 0.132893i \(0.0424267\pi\)
\(678\) 0 0
\(679\) −5.39159e9 −0.660956
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.70052e8 0.116499 0.0582496 − 0.998302i \(-0.481448\pi\)
0.0582496 + 0.998302i \(0.481448\pi\)
\(684\) 0 0
\(685\) 1.32995e8 0.0158095
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.42272e7 −0.00282187
\(690\) 0 0
\(691\) −2.09761e8 −0.0241853 −0.0120927 − 0.999927i \(-0.503849\pi\)
−0.0120927 + 0.999927i \(0.503849\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.42089e9 0.273545
\(696\) 0 0
\(697\) 3.38836e9 0.379031
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.65792e10 1.81782 0.908910 − 0.416993i \(-0.136916\pi\)
0.908910 + 0.416993i \(0.136916\pi\)
\(702\) 0 0
\(703\) −2.00440e10 −2.17591
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.31983e9 −0.885414
\(708\) 0 0
\(709\) −9.78901e9 −1.03152 −0.515759 − 0.856734i \(-0.672490\pi\)
−0.515759 + 0.856734i \(0.672490\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.03539e8 −0.0106977
\(714\) 0 0
\(715\) −4.24330e7 −0.00434143
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.08605e10 1.08968 0.544838 − 0.838541i \(-0.316591\pi\)
0.544838 + 0.838541i \(0.316591\pi\)
\(720\) 0 0
\(721\) 2.48835e9 0.247251
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.45017e9 −0.823536
\(726\) 0 0
\(727\) −8.01058e9 −0.773203 −0.386602 − 0.922247i \(-0.626351\pi\)
−0.386602 + 0.922247i \(0.626351\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.08119e9 −0.291747
\(732\) 0 0
\(733\) 1.67173e10 1.56784 0.783921 − 0.620860i \(-0.213217\pi\)
0.783921 + 0.620860i \(0.213217\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.59336e10 −1.46615
\(738\) 0 0
\(739\) 9.35702e9 0.852869 0.426434 − 0.904519i \(-0.359770\pi\)
0.426434 + 0.904519i \(0.359770\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.58318e10 1.41602 0.708012 − 0.706201i \(-0.249592\pi\)
0.708012 + 0.706201i \(0.249592\pi\)
\(744\) 0 0
\(745\) 2.50180e9 0.221669
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.97926e10 1.72114
\(750\) 0 0
\(751\) 1.50540e10 1.29692 0.648458 − 0.761250i \(-0.275414\pi\)
0.648458 + 0.761250i \(0.275414\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.73018e9 0.400003
\(756\) 0 0
\(757\) 6.81448e9 0.570949 0.285474 − 0.958386i \(-0.407849\pi\)
0.285474 + 0.958386i \(0.407849\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.42269e9 −0.281527 −0.140764 − 0.990043i \(-0.544956\pi\)
−0.140764 + 0.990043i \(0.544956\pi\)
\(762\) 0 0
\(763\) −1.72608e10 −1.40678
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.97936e8 −0.0158395
\(768\) 0 0
\(769\) 9.62370e9 0.763133 0.381566 − 0.924341i \(-0.375385\pi\)
0.381566 + 0.924341i \(0.375385\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.46796e10 1.92181 0.960905 − 0.276880i \(-0.0893004\pi\)
0.960905 + 0.276880i \(0.0893004\pi\)
\(774\) 0 0
\(775\) −1.08216e10 −0.835091
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.03584e10 2.30090
\(780\) 0 0
\(781\) −2.20217e10 −1.65414
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.70204e9 −0.273148
\(786\) 0 0
\(787\) 2.46260e10 1.80087 0.900436 − 0.434988i \(-0.143247\pi\)
0.900436 + 0.434988i \(0.143247\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.74549e10 1.25401
\(792\) 0 0
\(793\) 1.82523e8 0.0129976
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.46329e9 −0.312285 −0.156142 − 0.987735i \(-0.549906\pi\)
−0.156142 + 0.987735i \(0.549906\pi\)
\(798\) 0 0
\(799\) 2.90897e9 0.201756
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −9.10176e8 −0.0620328
\(804\) 0 0
\(805\) 6.31632e7 0.00426755
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.56496e9 0.170318 0.0851590 − 0.996367i \(-0.472860\pi\)
0.0851590 + 0.996367i \(0.472860\pi\)
\(810\) 0 0
\(811\) −3.95820e9 −0.260570 −0.130285 − 0.991477i \(-0.541589\pi\)
−0.130285 + 0.991477i \(0.541589\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.87759e9 −0.445026
\(816\) 0 0
\(817\) −2.76063e10 −1.77105
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.58510e10 −0.999669 −0.499835 − 0.866121i \(-0.666606\pi\)
−0.499835 + 0.866121i \(0.666606\pi\)
\(822\) 0 0
\(823\) 1.57453e10 0.984583 0.492292 − 0.870430i \(-0.336159\pi\)
0.492292 + 0.870430i \(0.336159\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.15640e9 −0.132574 −0.0662872 − 0.997801i \(-0.521115\pi\)
−0.0662872 + 0.997801i \(0.521115\pi\)
\(828\) 0 0
\(829\) 2.95703e10 1.80267 0.901333 − 0.433126i \(-0.142589\pi\)
0.901333 + 0.433126i \(0.142589\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.93900e8 0.0176174
\(834\) 0 0
\(835\) −5.54494e9 −0.329605
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.11982e9 −0.182374 −0.0911869 − 0.995834i \(-0.529066\pi\)
−0.0911869 + 0.995834i \(0.529066\pi\)
\(840\) 0 0
\(841\) −1.39050e9 −0.0806095
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.58811e9 0.375632
\(846\) 0 0
\(847\) 1.48347e10 0.838852
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.65972e8 0.0147939
\(852\) 0 0
\(853\) −1.43942e10 −0.794082 −0.397041 − 0.917801i \(-0.629963\pi\)
−0.397041 + 0.917801i \(0.629963\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.73201e9 −0.256810 −0.128405 − 0.991722i \(-0.540986\pi\)
−0.128405 + 0.991722i \(0.540986\pi\)
\(858\) 0 0
\(859\) 4.83735e9 0.260394 0.130197 − 0.991488i \(-0.458439\pi\)
0.130197 + 0.991488i \(0.458439\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.90890e10 −1.01099 −0.505493 − 0.862831i \(-0.668689\pi\)
−0.505493 + 0.862831i \(0.668689\pi\)
\(864\) 0 0
\(865\) −1.16270e10 −0.610815
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.50437e10 2.32844
\(870\) 0 0
\(871\) −1.82313e8 −0.00934878
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.42880e10 0.721011
\(876\) 0 0
\(877\) −3.36363e10 −1.68388 −0.841938 − 0.539575i \(-0.818585\pi\)
−0.841938 + 0.539575i \(0.818585\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.81256e10 −1.38575 −0.692876 − 0.721057i \(-0.743657\pi\)
−0.692876 + 0.721057i \(0.743657\pi\)
\(882\) 0 0
\(883\) −1.99944e10 −0.977341 −0.488671 − 0.872468i \(-0.662518\pi\)
−0.488671 + 0.872468i \(0.662518\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.22676e10 −1.55251 −0.776255 − 0.630419i \(-0.782883\pi\)
−0.776255 + 0.630419i \(0.782883\pi\)
\(888\) 0 0
\(889\) 2.42102e9 0.115569
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.60633e10 1.22475
\(894\) 0 0
\(895\) 1.25399e9 0.0584672
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.03100e10 0.932290
\(900\) 0 0
\(901\) −1.92393e9 −0.0876299
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.34987e9 0.374464
\(906\) 0 0
\(907\) −1.15232e10 −0.512801 −0.256400 − 0.966571i \(-0.582537\pi\)
−0.256400 + 0.966571i \(0.582537\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.89709e9 −0.258418 −0.129209 − 0.991617i \(-0.541244\pi\)
−0.129209 + 0.991617i \(0.541244\pi\)
\(912\) 0 0
\(913\) 5.55433e10 2.41537
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.92868e10 1.68250
\(918\) 0 0
\(919\) −4.37679e9 −0.186017 −0.0930083 − 0.995665i \(-0.529648\pi\)
−0.0930083 + 0.995665i \(0.529648\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.51973e8 −0.0105474
\(924\) 0 0
\(925\) 2.77986e10 1.15485
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.03024e10 −1.24000 −0.620000 − 0.784602i \(-0.712867\pi\)
−0.620000 + 0.784602i \(0.712867\pi\)
\(930\) 0 0
\(931\) 2.63324e9 0.106946
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.36968e9 −0.134818
\(936\) 0 0
\(937\) 2.61397e10 1.03804 0.519018 − 0.854763i \(-0.326298\pi\)
0.519018 + 0.854763i \(0.326298\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.42439e10 0.948505 0.474253 − 0.880389i \(-0.342718\pi\)
0.474253 + 0.880389i \(0.342718\pi\)
\(942\) 0 0
\(943\) −4.02838e8 −0.0156437
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.57755e10 0.986237 0.493119 − 0.869962i \(-0.335857\pi\)
0.493119 + 0.869962i \(0.335857\pi\)
\(948\) 0 0
\(949\) −1.04143e7 −0.000395546 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.22288e10 −0.831937 −0.415968 − 0.909379i \(-0.636557\pi\)
−0.415968 + 0.909379i \(0.636557\pi\)
\(954\) 0 0
\(955\) 6.32431e8 0.0234964
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.18682e9 −0.0434530
\(960\) 0 0
\(961\) −1.50299e9 −0.0546291
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.15142e9 0.220358
\(966\) 0 0
\(967\) 3.09410e9 0.110038 0.0550189 − 0.998485i \(-0.482478\pi\)
0.0550189 + 0.998485i \(0.482478\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.62621e10 −0.570044 −0.285022 − 0.958521i \(-0.592001\pi\)
−0.285022 + 0.958521i \(0.592001\pi\)
\(972\) 0 0
\(973\) −2.16036e10 −0.751850
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.42256e9 −0.151720 −0.0758601 − 0.997118i \(-0.524170\pi\)
−0.0758601 + 0.997118i \(0.524170\pi\)
\(978\) 0 0
\(979\) −2.39717e10 −0.816506
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.73847e10 1.25533 0.627663 − 0.778485i \(-0.284012\pi\)
0.627663 + 0.778485i \(0.284012\pi\)
\(984\) 0 0
\(985\) −1.76835e10 −0.589579
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.66319e8 0.0120413
\(990\) 0 0
\(991\) 1.24473e10 0.406273 0.203136 − 0.979150i \(-0.434887\pi\)
0.203136 + 0.979150i \(0.434887\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.06753e10 −0.343557
\(996\) 0 0
\(997\) 2.84335e10 0.908652 0.454326 − 0.890836i \(-0.349880\pi\)
0.454326 + 0.890836i \(0.349880\pi\)
\(998\) 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 432.8.a.c.1.1 1
3.2 odd 2 432.8.a.f.1.1 1
4.3 odd 2 54.8.a.e.1.1 yes 1
12.11 even 2 54.8.a.b.1.1 ✓ 1
36.7 odd 6 162.8.c.c.109.1 2
36.11 even 6 162.8.c.j.109.1 2
36.23 even 6 162.8.c.j.55.1 2
36.31 odd 6 162.8.c.c.55.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.8.a.b.1.1 ✓ 1 12.11 even 2
54.8.a.e.1.1 yes 1 4.3 odd 2
162.8.c.c.55.1 2 36.31 odd 6
162.8.c.c.109.1 2 36.7 odd 6
162.8.c.j.55.1 2 36.23 even 6
162.8.c.j.109.1 2 36.11 even 6
432.8.a.c.1.1 1 1.1 even 1 trivial
432.8.a.f.1.1 1 3.2 odd 2