Properties

Label 48.28.a.l.1.2
Level $48$
Weight $28$
Character 48.1
Self dual yes
Analytic conductor $221.691$
Analytic rank $0$
Dimension $4$
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] = [48,28,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(221.690675922\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 360714331909x^{2} - 43287560841177118x + 8819337660421091919513 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{8}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(108627.\) of defining polynomial
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.59432e6 q^{3} -8.07135e8 q^{5} -1.51306e11 q^{7} +2.54187e12 q^{9} +O(q^{10})\) \(q-1.59432e6 q^{3} -8.07135e8 q^{5} -1.51306e11 q^{7} +2.54187e12 q^{9} -1.83635e14 q^{11} +1.01447e15 q^{13} +1.28683e15 q^{15} -4.47577e15 q^{17} -4.23813e16 q^{19} +2.41230e17 q^{21} -6.82342e17 q^{23} -6.79911e18 q^{25} -4.05256e18 q^{27} -4.66515e19 q^{29} -2.63899e19 q^{31} +2.92774e20 q^{33} +1.22124e20 q^{35} +7.09690e20 q^{37} -1.61739e21 q^{39} -2.45781e21 q^{41} -1.30361e22 q^{43} -2.05163e21 q^{45} -1.98925e22 q^{47} -4.28190e22 q^{49} +7.13583e21 q^{51} +2.98600e23 q^{53} +1.48219e23 q^{55} +6.75695e22 q^{57} +2.01063e23 q^{59} -1.80822e24 q^{61} -3.84599e23 q^{63} -8.18812e23 q^{65} -5.59396e24 q^{67} +1.08787e24 q^{69} +8.48444e24 q^{71} -1.17667e25 q^{73} +1.08400e25 q^{75} +2.77851e25 q^{77} -2.26494e25 q^{79} +6.46108e24 q^{81} -1.31501e26 q^{83} +3.61255e24 q^{85} +7.43775e25 q^{87} -1.37576e26 q^{89} -1.53495e26 q^{91} +4.20740e25 q^{93} +3.42074e25 q^{95} +1.54951e26 q^{97} -4.66777e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6377292 q^{3} + 4949050424 q^{5} + 88249731552 q^{7} + 10167463313316 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6377292 q^{3} + 4949050424 q^{5} + 88249731552 q^{7} + 10167463313316 q^{9} + 111189633716848 q^{11} - 7339657642664 q^{13} - 78\!\cdots\!52 q^{15}+ \cdots + 28\!\cdots\!92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.59432e6 −0.577350
\(4\) 0 0
\(5\) −8.07135e8 −0.295700 −0.147850 0.989010i \(-0.547235\pi\)
−0.147850 + 0.989010i \(0.547235\pi\)
\(6\) 0 0
\(7\) −1.51306e11 −0.590244 −0.295122 0.955460i \(-0.595360\pi\)
−0.295122 + 0.955460i \(0.595360\pi\)
\(8\) 0 0
\(9\) 2.54187e12 0.333333
\(10\) 0 0
\(11\) −1.83635e14 −1.60382 −0.801909 0.597446i \(-0.796182\pi\)
−0.801909 + 0.597446i \(0.796182\pi\)
\(12\) 0 0
\(13\) 1.01447e15 0.928972 0.464486 0.885581i \(-0.346239\pi\)
0.464486 + 0.885581i \(0.346239\pi\)
\(14\) 0 0
\(15\) 1.28683e15 0.170722
\(16\) 0 0
\(17\) −4.47577e15 −0.109599 −0.0547996 0.998497i \(-0.517452\pi\)
−0.0547996 + 0.998497i \(0.517452\pi\)
\(18\) 0 0
\(19\) −4.23813e16 −0.231207 −0.115603 0.993295i \(-0.536880\pi\)
−0.115603 + 0.993295i \(0.536880\pi\)
\(20\) 0 0
\(21\) 2.41230e17 0.340778
\(22\) 0 0
\(23\) −6.82342e17 −0.282278 −0.141139 0.989990i \(-0.545076\pi\)
−0.141139 + 0.989990i \(0.545076\pi\)
\(24\) 0 0
\(25\) −6.79911e18 −0.912562
\(26\) 0 0
\(27\) −4.05256e18 −0.192450
\(28\) 0 0
\(29\) −4.66515e19 −0.844291 −0.422146 0.906528i \(-0.638723\pi\)
−0.422146 + 0.906528i \(0.638723\pi\)
\(30\) 0 0
\(31\) −2.63899e19 −0.194113 −0.0970566 0.995279i \(-0.530943\pi\)
−0.0970566 + 0.995279i \(0.530943\pi\)
\(32\) 0 0
\(33\) 2.92774e20 0.925965
\(34\) 0 0
\(35\) 1.22124e20 0.174535
\(36\) 0 0
\(37\) 7.09690e20 0.479011 0.239505 0.970895i \(-0.423015\pi\)
0.239505 + 0.970895i \(0.423015\pi\)
\(38\) 0 0
\(39\) −1.61739e21 −0.536342
\(40\) 0 0
\(41\) −2.45781e21 −0.414921 −0.207461 0.978243i \(-0.566520\pi\)
−0.207461 + 0.978243i \(0.566520\pi\)
\(42\) 0 0
\(43\) −1.30361e22 −1.15697 −0.578486 0.815693i \(-0.696356\pi\)
−0.578486 + 0.815693i \(0.696356\pi\)
\(44\) 0 0
\(45\) −2.05163e21 −0.0985666
\(46\) 0 0
\(47\) −1.98925e22 −0.531335 −0.265667 0.964065i \(-0.585592\pi\)
−0.265667 + 0.964065i \(0.585592\pi\)
\(48\) 0 0
\(49\) −4.28190e22 −0.651612
\(50\) 0 0
\(51\) 7.13583e21 0.0632772
\(52\) 0 0
\(53\) 2.98600e23 1.57531 0.787655 0.616117i \(-0.211295\pi\)
0.787655 + 0.616117i \(0.211295\pi\)
\(54\) 0 0
\(55\) 1.48219e23 0.474249
\(56\) 0 0
\(57\) 6.75695e22 0.133487
\(58\) 0 0
\(59\) 2.01063e23 0.249361 0.124681 0.992197i \(-0.460209\pi\)
0.124681 + 0.992197i \(0.460209\pi\)
\(60\) 0 0
\(61\) −1.80822e24 −1.42988 −0.714939 0.699187i \(-0.753545\pi\)
−0.714939 + 0.699187i \(0.753545\pi\)
\(62\) 0 0
\(63\) −3.84599e23 −0.196748
\(64\) 0 0
\(65\) −8.18812e23 −0.274697
\(66\) 0 0
\(67\) −5.59396e24 −1.24655 −0.623275 0.782003i \(-0.714198\pi\)
−0.623275 + 0.782003i \(0.714198\pi\)
\(68\) 0 0
\(69\) 1.08787e24 0.162973
\(70\) 0 0
\(71\) 8.48444e24 0.864242 0.432121 0.901816i \(-0.357765\pi\)
0.432121 + 0.901816i \(0.357765\pi\)
\(72\) 0 0
\(73\) −1.17667e25 −0.823753 −0.411876 0.911240i \(-0.635126\pi\)
−0.411876 + 0.911240i \(0.635126\pi\)
\(74\) 0 0
\(75\) 1.08400e25 0.526868
\(76\) 0 0
\(77\) 2.77851e25 0.946644
\(78\) 0 0
\(79\) −2.26494e25 −0.545871 −0.272936 0.962032i \(-0.587995\pi\)
−0.272936 + 0.962032i \(0.587995\pi\)
\(80\) 0 0
\(81\) 6.46108e24 0.111111
\(82\) 0 0
\(83\) −1.31501e26 −1.62695 −0.813474 0.581601i \(-0.802426\pi\)
−0.813474 + 0.581601i \(0.802426\pi\)
\(84\) 0 0
\(85\) 3.61255e24 0.0324085
\(86\) 0 0
\(87\) 7.43775e25 0.487452
\(88\) 0 0
\(89\) −1.37576e26 −0.663401 −0.331701 0.943385i \(-0.607622\pi\)
−0.331701 + 0.943385i \(0.607622\pi\)
\(90\) 0 0
\(91\) −1.53495e26 −0.548320
\(92\) 0 0
\(93\) 4.20740e25 0.112071
\(94\) 0 0
\(95\) 3.42074e25 0.0683678
\(96\) 0 0
\(97\) 1.54951e26 0.233763 0.116882 0.993146i \(-0.462710\pi\)
0.116882 + 0.993146i \(0.462710\pi\)
\(98\) 0 0
\(99\) −4.66777e26 −0.534606
\(100\) 0 0
\(101\) −8.25968e26 −0.722145 −0.361073 0.932538i \(-0.617589\pi\)
−0.361073 + 0.932538i \(0.617589\pi\)
\(102\) 0 0
\(103\) 4.67783e26 0.313864 0.156932 0.987609i \(-0.449840\pi\)
0.156932 + 0.987609i \(0.449840\pi\)
\(104\) 0 0
\(105\) −1.94705e26 −0.100768
\(106\) 0 0
\(107\) −4.41090e27 −1.76948 −0.884740 0.466084i \(-0.845664\pi\)
−0.884740 + 0.466084i \(0.845664\pi\)
\(108\) 0 0
\(109\) 1.86849e27 0.583758 0.291879 0.956455i \(-0.405720\pi\)
0.291879 + 0.956455i \(0.405720\pi\)
\(110\) 0 0
\(111\) −1.13147e27 −0.276557
\(112\) 0 0
\(113\) 5.28674e27 1.01538 0.507690 0.861540i \(-0.330499\pi\)
0.507690 + 0.861540i \(0.330499\pi\)
\(114\) 0 0
\(115\) 5.50743e26 0.0834695
\(116\) 0 0
\(117\) 2.57864e27 0.309657
\(118\) 0 0
\(119\) 6.77210e26 0.0646903
\(120\) 0 0
\(121\) 2.06120e28 1.57223
\(122\) 0 0
\(123\) 3.91854e27 0.239555
\(124\) 0 0
\(125\) 1.15014e28 0.565544
\(126\) 0 0
\(127\) −4.90297e28 −1.94585 −0.972925 0.231121i \(-0.925761\pi\)
−0.972925 + 0.231121i \(0.925761\pi\)
\(128\) 0 0
\(129\) 2.07837e28 0.667978
\(130\) 0 0
\(131\) −6.76726e27 −0.176706 −0.0883528 0.996089i \(-0.528160\pi\)
−0.0883528 + 0.996089i \(0.528160\pi\)
\(132\) 0 0
\(133\) 6.41253e27 0.136468
\(134\) 0 0
\(135\) 3.27096e27 0.0569075
\(136\) 0 0
\(137\) 3.73261e28 0.532458 0.266229 0.963910i \(-0.414222\pi\)
0.266229 + 0.963910i \(0.414222\pi\)
\(138\) 0 0
\(139\) 5.22819e27 0.0613270 0.0306635 0.999530i \(-0.490238\pi\)
0.0306635 + 0.999530i \(0.490238\pi\)
\(140\) 0 0
\(141\) 3.17151e28 0.306766
\(142\) 0 0
\(143\) −1.86292e29 −1.48990
\(144\) 0 0
\(145\) 3.76540e28 0.249657
\(146\) 0 0
\(147\) 6.82673e28 0.376208
\(148\) 0 0
\(149\) 2.78140e29 1.27717 0.638586 0.769551i \(-0.279520\pi\)
0.638586 + 0.769551i \(0.279520\pi\)
\(150\) 0 0
\(151\) −2.05376e29 −0.787698 −0.393849 0.919175i \(-0.628857\pi\)
−0.393849 + 0.919175i \(0.628857\pi\)
\(152\) 0 0
\(153\) −1.13768e28 −0.0365331
\(154\) 0 0
\(155\) 2.13002e28 0.0573993
\(156\) 0 0
\(157\) 1.70965e29 0.387492 0.193746 0.981052i \(-0.437936\pi\)
0.193746 + 0.981052i \(0.437936\pi\)
\(158\) 0 0
\(159\) −4.76065e29 −0.909506
\(160\) 0 0
\(161\) 1.03242e29 0.166613
\(162\) 0 0
\(163\) −2.52320e29 −0.344682 −0.172341 0.985037i \(-0.555133\pi\)
−0.172341 + 0.985037i \(0.555133\pi\)
\(164\) 0 0
\(165\) −2.36308e29 −0.273808
\(166\) 0 0
\(167\) 4.52090e29 0.445197 0.222598 0.974910i \(-0.428546\pi\)
0.222598 + 0.974910i \(0.428546\pi\)
\(168\) 0 0
\(169\) −1.63391e29 −0.137012
\(170\) 0 0
\(171\) −1.07728e29 −0.0770689
\(172\) 0 0
\(173\) −8.52083e29 −0.521026 −0.260513 0.965470i \(-0.583892\pi\)
−0.260513 + 0.965470i \(0.583892\pi\)
\(174\) 0 0
\(175\) 1.02874e30 0.538634
\(176\) 0 0
\(177\) −3.20559e29 −0.143969
\(178\) 0 0
\(179\) 1.37361e30 0.530088 0.265044 0.964236i \(-0.414613\pi\)
0.265044 + 0.964236i \(0.414613\pi\)
\(180\) 0 0
\(181\) −2.29003e30 −0.760644 −0.380322 0.924854i \(-0.624187\pi\)
−0.380322 + 0.924854i \(0.624187\pi\)
\(182\) 0 0
\(183\) 2.88289e30 0.825540
\(184\) 0 0
\(185\) −5.72816e29 −0.141643
\(186\) 0 0
\(187\) 8.21911e29 0.175777
\(188\) 0 0
\(189\) 6.13174e29 0.113593
\(190\) 0 0
\(191\) 6.21927e30 0.999514 0.499757 0.866166i \(-0.333423\pi\)
0.499757 + 0.866166i \(0.333423\pi\)
\(192\) 0 0
\(193\) −2.56802e30 −0.358571 −0.179286 0.983797i \(-0.557379\pi\)
−0.179286 + 0.983797i \(0.557379\pi\)
\(194\) 0 0
\(195\) 1.30545e30 0.158596
\(196\) 0 0
\(197\) −2.45341e30 −0.259703 −0.129851 0.991533i \(-0.541450\pi\)
−0.129851 + 0.991533i \(0.541450\pi\)
\(198\) 0 0
\(199\) 7.88229e30 0.728007 0.364004 0.931398i \(-0.381410\pi\)
0.364004 + 0.931398i \(0.381410\pi\)
\(200\) 0 0
\(201\) 8.91858e30 0.719696
\(202\) 0 0
\(203\) 7.05863e30 0.498338
\(204\) 0 0
\(205\) 1.98378e30 0.122692
\(206\) 0 0
\(207\) −1.73442e30 −0.0940926
\(208\) 0 0
\(209\) 7.78271e30 0.370814
\(210\) 0 0
\(211\) 3.15286e31 1.32097 0.660484 0.750840i \(-0.270351\pi\)
0.660484 + 0.750840i \(0.270351\pi\)
\(212\) 0 0
\(213\) −1.35269e31 −0.498970
\(214\) 0 0
\(215\) 1.05219e31 0.342116
\(216\) 0 0
\(217\) 3.99294e30 0.114574
\(218\) 0 0
\(219\) 1.87600e31 0.475594
\(220\) 0 0
\(221\) −4.54052e30 −0.101815
\(222\) 0 0
\(223\) 8.82278e30 0.175182 0.0875910 0.996157i \(-0.472083\pi\)
0.0875910 + 0.996157i \(0.472083\pi\)
\(224\) 0 0
\(225\) −1.72824e31 −0.304187
\(226\) 0 0
\(227\) −5.18674e31 −0.810113 −0.405057 0.914292i \(-0.632748\pi\)
−0.405057 + 0.914292i \(0.632748\pi\)
\(228\) 0 0
\(229\) 1.08468e32 1.50495 0.752477 0.658619i \(-0.228859\pi\)
0.752477 + 0.658619i \(0.228859\pi\)
\(230\) 0 0
\(231\) −4.42984e31 −0.546545
\(232\) 0 0
\(233\) 8.48112e31 0.931426 0.465713 0.884936i \(-0.345798\pi\)
0.465713 + 0.884936i \(0.345798\pi\)
\(234\) 0 0
\(235\) 1.60559e31 0.157116
\(236\) 0 0
\(237\) 3.61104e31 0.315159
\(238\) 0 0
\(239\) 1.22307e32 0.952972 0.476486 0.879182i \(-0.341910\pi\)
0.476486 + 0.879182i \(0.341910\pi\)
\(240\) 0 0
\(241\) 1.16482e32 0.811014 0.405507 0.914092i \(-0.367095\pi\)
0.405507 + 0.914092i \(0.367095\pi\)
\(242\) 0 0
\(243\) −1.03011e31 −0.0641500
\(244\) 0 0
\(245\) 3.45607e31 0.192682
\(246\) 0 0
\(247\) −4.29944e31 −0.214785
\(248\) 0 0
\(249\) 2.09655e32 0.939319
\(250\) 0 0
\(251\) −2.61634e32 −1.05220 −0.526100 0.850422i \(-0.676346\pi\)
−0.526100 + 0.850422i \(0.676346\pi\)
\(252\) 0 0
\(253\) 1.25302e32 0.452722
\(254\) 0 0
\(255\) −5.75958e30 −0.0187110
\(256\) 0 0
\(257\) 5.84494e32 1.70877 0.854384 0.519643i \(-0.173935\pi\)
0.854384 + 0.519643i \(0.173935\pi\)
\(258\) 0 0
\(259\) −1.07380e32 −0.282733
\(260\) 0 0
\(261\) −1.18582e32 −0.281430
\(262\) 0 0
\(263\) 5.37994e32 1.15179 0.575897 0.817522i \(-0.304653\pi\)
0.575897 + 0.817522i \(0.304653\pi\)
\(264\) 0 0
\(265\) −2.41011e32 −0.465819
\(266\) 0 0
\(267\) 2.19340e32 0.383015
\(268\) 0 0
\(269\) 8.29944e32 1.31037 0.655183 0.755470i \(-0.272591\pi\)
0.655183 + 0.755470i \(0.272591\pi\)
\(270\) 0 0
\(271\) 1.01021e33 1.44320 0.721601 0.692309i \(-0.243406\pi\)
0.721601 + 0.692309i \(0.243406\pi\)
\(272\) 0 0
\(273\) 2.44720e32 0.316573
\(274\) 0 0
\(275\) 1.24856e33 1.46358
\(276\) 0 0
\(277\) 1.50312e33 1.59778 0.798890 0.601477i \(-0.205421\pi\)
0.798890 + 0.601477i \(0.205421\pi\)
\(278\) 0 0
\(279\) −6.70796e31 −0.0647044
\(280\) 0 0
\(281\) 3.76077e32 0.329413 0.164707 0.986343i \(-0.447332\pi\)
0.164707 + 0.986343i \(0.447332\pi\)
\(282\) 0 0
\(283\) 1.99259e33 1.58599 0.792997 0.609226i \(-0.208520\pi\)
0.792997 + 0.609226i \(0.208520\pi\)
\(284\) 0 0
\(285\) −5.45377e31 −0.0394722
\(286\) 0 0
\(287\) 3.71880e32 0.244905
\(288\) 0 0
\(289\) −1.64768e33 −0.987988
\(290\) 0 0
\(291\) −2.47042e32 −0.134963
\(292\) 0 0
\(293\) −2.14923e33 −1.07046 −0.535229 0.844707i \(-0.679775\pi\)
−0.535229 + 0.844707i \(0.679775\pi\)
\(294\) 0 0
\(295\) −1.62285e32 −0.0737361
\(296\) 0 0
\(297\) 7.44193e32 0.308655
\(298\) 0 0
\(299\) −6.92214e32 −0.262228
\(300\) 0 0
\(301\) 1.97243e33 0.682895
\(302\) 0 0
\(303\) 1.31686e33 0.416931
\(304\) 0 0
\(305\) 1.45948e33 0.422815
\(306\) 0 0
\(307\) 5.68598e33 1.50813 0.754063 0.656803i \(-0.228091\pi\)
0.754063 + 0.656803i \(0.228091\pi\)
\(308\) 0 0
\(309\) −7.45797e32 −0.181210
\(310\) 0 0
\(311\) 1.78610e33 0.397779 0.198889 0.980022i \(-0.436267\pi\)
0.198889 + 0.980022i \(0.436267\pi\)
\(312\) 0 0
\(313\) −6.22238e33 −1.27089 −0.635445 0.772146i \(-0.719183\pi\)
−0.635445 + 0.772146i \(0.719183\pi\)
\(314\) 0 0
\(315\) 3.10423e32 0.0581784
\(316\) 0 0
\(317\) 1.05915e33 0.182246 0.0911229 0.995840i \(-0.470954\pi\)
0.0911229 + 0.995840i \(0.470954\pi\)
\(318\) 0 0
\(319\) 8.56686e33 1.35409
\(320\) 0 0
\(321\) 7.03239e33 1.02161
\(322\) 0 0
\(323\) 1.89689e32 0.0253401
\(324\) 0 0
\(325\) −6.89747e33 −0.847744
\(326\) 0 0
\(327\) −2.97897e33 −0.337033
\(328\) 0 0
\(329\) 3.00985e33 0.313617
\(330\) 0 0
\(331\) −1.92231e34 −1.84563 −0.922815 0.385244i \(-0.874117\pi\)
−0.922815 + 0.385244i \(0.874117\pi\)
\(332\) 0 0
\(333\) 1.80394e33 0.159670
\(334\) 0 0
\(335\) 4.51508e33 0.368605
\(336\) 0 0
\(337\) −4.81862e33 −0.363010 −0.181505 0.983390i \(-0.558097\pi\)
−0.181505 + 0.983390i \(0.558097\pi\)
\(338\) 0 0
\(339\) −8.42877e33 −0.586230
\(340\) 0 0
\(341\) 4.84612e33 0.311322
\(342\) 0 0
\(343\) 1.64214e34 0.974854
\(344\) 0 0
\(345\) −8.78061e32 −0.0481911
\(346\) 0 0
\(347\) −2.68701e34 −1.36402 −0.682011 0.731342i \(-0.738895\pi\)
−0.682011 + 0.731342i \(0.738895\pi\)
\(348\) 0 0
\(349\) 2.42556e34 1.13938 0.569689 0.821860i \(-0.307064\pi\)
0.569689 + 0.821860i \(0.307064\pi\)
\(350\) 0 0
\(351\) −4.11118e33 −0.178781
\(352\) 0 0
\(353\) 3.46426e34 1.39525 0.697624 0.716464i \(-0.254240\pi\)
0.697624 + 0.716464i \(0.254240\pi\)
\(354\) 0 0
\(355\) −6.84809e33 −0.255556
\(356\) 0 0
\(357\) −1.07969e33 −0.0373490
\(358\) 0 0
\(359\) −3.55859e34 −1.14157 −0.570785 0.821100i \(-0.693361\pi\)
−0.570785 + 0.821100i \(0.693361\pi\)
\(360\) 0 0
\(361\) −3.18044e34 −0.946543
\(362\) 0 0
\(363\) −3.28622e34 −0.907730
\(364\) 0 0
\(365\) 9.49733e33 0.243584
\(366\) 0 0
\(367\) 2.46193e34 0.586521 0.293260 0.956033i \(-0.405260\pi\)
0.293260 + 0.956033i \(0.405260\pi\)
\(368\) 0 0
\(369\) −6.24741e33 −0.138307
\(370\) 0 0
\(371\) −4.51799e34 −0.929817
\(372\) 0 0
\(373\) −7.61316e34 −1.45712 −0.728560 0.684982i \(-0.759810\pi\)
−0.728560 + 0.684982i \(0.759810\pi\)
\(374\) 0 0
\(375\) −1.83370e34 −0.326517
\(376\) 0 0
\(377\) −4.73263e34 −0.784323
\(378\) 0 0
\(379\) 5.82910e34 0.899439 0.449720 0.893170i \(-0.351524\pi\)
0.449720 + 0.893170i \(0.351524\pi\)
\(380\) 0 0
\(381\) 7.81692e34 1.12344
\(382\) 0 0
\(383\) −5.12399e34 −0.686160 −0.343080 0.939306i \(-0.611470\pi\)
−0.343080 + 0.939306i \(0.611470\pi\)
\(384\) 0 0
\(385\) −2.24263e34 −0.279923
\(386\) 0 0
\(387\) −3.31359e34 −0.385657
\(388\) 0 0
\(389\) −7.44474e34 −0.808220 −0.404110 0.914710i \(-0.632419\pi\)
−0.404110 + 0.914710i \(0.632419\pi\)
\(390\) 0 0
\(391\) 3.05401e33 0.0309374
\(392\) 0 0
\(393\) 1.07892e34 0.102021
\(394\) 0 0
\(395\) 1.82811e34 0.161414
\(396\) 0 0
\(397\) −2.30723e35 −1.90291 −0.951455 0.307787i \(-0.900411\pi\)
−0.951455 + 0.307787i \(0.900411\pi\)
\(398\) 0 0
\(399\) −1.02236e34 −0.0787901
\(400\) 0 0
\(401\) 1.64034e34 0.118164 0.0590821 0.998253i \(-0.481183\pi\)
0.0590821 + 0.998253i \(0.481183\pi\)
\(402\) 0 0
\(403\) −2.67717e34 −0.180326
\(404\) 0 0
\(405\) −5.21497e33 −0.0328555
\(406\) 0 0
\(407\) −1.30324e35 −0.768247
\(408\) 0 0
\(409\) −3.15758e35 −1.74217 −0.871083 0.491135i \(-0.836582\pi\)
−0.871083 + 0.491135i \(0.836582\pi\)
\(410\) 0 0
\(411\) −5.95099e34 −0.307414
\(412\) 0 0
\(413\) −3.04219e34 −0.147184
\(414\) 0 0
\(415\) 1.06139e35 0.481088
\(416\) 0 0
\(417\) −8.33542e33 −0.0354071
\(418\) 0 0
\(419\) −1.05931e35 −0.421827 −0.210914 0.977505i \(-0.567644\pi\)
−0.210914 + 0.977505i \(0.567644\pi\)
\(420\) 0 0
\(421\) −5.20613e35 −1.94405 −0.972023 0.234886i \(-0.924528\pi\)
−0.972023 + 0.234886i \(0.924528\pi\)
\(422\) 0 0
\(423\) −5.05641e34 −0.177112
\(424\) 0 0
\(425\) 3.04313e34 0.100016
\(426\) 0 0
\(427\) 2.73594e35 0.843976
\(428\) 0 0
\(429\) 2.97010e35 0.860195
\(430\) 0 0
\(431\) 3.56552e35 0.969795 0.484897 0.874571i \(-0.338857\pi\)
0.484897 + 0.874571i \(0.338857\pi\)
\(432\) 0 0
\(433\) 3.96319e35 1.01265 0.506323 0.862344i \(-0.331004\pi\)
0.506323 + 0.862344i \(0.331004\pi\)
\(434\) 0 0
\(435\) −6.00327e34 −0.144139
\(436\) 0 0
\(437\) 2.89186e34 0.0652645
\(438\) 0 0
\(439\) 6.91014e35 1.46628 0.733138 0.680080i \(-0.238055\pi\)
0.733138 + 0.680080i \(0.238055\pi\)
\(440\) 0 0
\(441\) −1.08840e35 −0.217204
\(442\) 0 0
\(443\) −6.24863e35 −1.17310 −0.586550 0.809913i \(-0.699514\pi\)
−0.586550 + 0.809913i \(0.699514\pi\)
\(444\) 0 0
\(445\) 1.11042e35 0.196168
\(446\) 0 0
\(447\) −4.43445e35 −0.737375
\(448\) 0 0
\(449\) 1.73334e35 0.271367 0.135684 0.990752i \(-0.456677\pi\)
0.135684 + 0.990752i \(0.456677\pi\)
\(450\) 0 0
\(451\) 4.51340e35 0.665458
\(452\) 0 0
\(453\) 3.27435e35 0.454778
\(454\) 0 0
\(455\) 1.23891e35 0.162138
\(456\) 0 0
\(457\) −7.72120e35 −0.952393 −0.476197 0.879339i \(-0.657985\pi\)
−0.476197 + 0.879339i \(0.657985\pi\)
\(458\) 0 0
\(459\) 1.81383e34 0.0210924
\(460\) 0 0
\(461\) 1.41111e36 1.54738 0.773692 0.633562i \(-0.218408\pi\)
0.773692 + 0.633562i \(0.218408\pi\)
\(462\) 0 0
\(463\) −4.08110e35 −0.422118 −0.211059 0.977473i \(-0.567691\pi\)
−0.211059 + 0.977473i \(0.567691\pi\)
\(464\) 0 0
\(465\) −3.39594e34 −0.0331395
\(466\) 0 0
\(467\) −5.33107e35 −0.490949 −0.245474 0.969403i \(-0.578944\pi\)
−0.245474 + 0.969403i \(0.578944\pi\)
\(468\) 0 0
\(469\) 8.46398e35 0.735769
\(470\) 0 0
\(471\) −2.72574e35 −0.223718
\(472\) 0 0
\(473\) 2.39388e36 1.85557
\(474\) 0 0
\(475\) 2.88155e35 0.210990
\(476\) 0 0
\(477\) 7.59002e35 0.525103
\(478\) 0 0
\(479\) −2.89621e35 −0.189365 −0.0946826 0.995508i \(-0.530184\pi\)
−0.0946826 + 0.995508i \(0.530184\pi\)
\(480\) 0 0
\(481\) 7.19957e35 0.444987
\(482\) 0 0
\(483\) −1.64602e35 −0.0961939
\(484\) 0 0
\(485\) −1.25067e35 −0.0691238
\(486\) 0 0
\(487\) −1.59425e36 −0.833517 −0.416758 0.909017i \(-0.636834\pi\)
−0.416758 + 0.909017i \(0.636834\pi\)
\(488\) 0 0
\(489\) 4.02279e35 0.199002
\(490\) 0 0
\(491\) 3.38076e36 1.58276 0.791378 0.611327i \(-0.209364\pi\)
0.791378 + 0.611327i \(0.209364\pi\)
\(492\) 0 0
\(493\) 2.08801e35 0.0925337
\(494\) 0 0
\(495\) 3.76752e35 0.158083
\(496\) 0 0
\(497\) −1.28374e36 −0.510114
\(498\) 0 0
\(499\) 1.44120e36 0.542458 0.271229 0.962515i \(-0.412570\pi\)
0.271229 + 0.962515i \(0.412570\pi\)
\(500\) 0 0
\(501\) −7.20777e35 −0.257035
\(502\) 0 0
\(503\) 2.62386e35 0.0886693 0.0443346 0.999017i \(-0.485883\pi\)
0.0443346 + 0.999017i \(0.485883\pi\)
\(504\) 0 0
\(505\) 6.66668e35 0.213538
\(506\) 0 0
\(507\) 2.60498e35 0.0791037
\(508\) 0 0
\(509\) −2.40955e36 −0.693820 −0.346910 0.937898i \(-0.612769\pi\)
−0.346910 + 0.937898i \(0.612769\pi\)
\(510\) 0 0
\(511\) 1.78037e36 0.486215
\(512\) 0 0
\(513\) 1.71753e35 0.0444958
\(514\) 0 0
\(515\) −3.77564e35 −0.0928097
\(516\) 0 0
\(517\) 3.65297e36 0.852165
\(518\) 0 0
\(519\) 1.35850e36 0.300815
\(520\) 0 0
\(521\) 4.29398e36 0.902715 0.451357 0.892343i \(-0.350940\pi\)
0.451357 + 0.892343i \(0.350940\pi\)
\(522\) 0 0
\(523\) 6.68903e36 1.33533 0.667667 0.744460i \(-0.267293\pi\)
0.667667 + 0.744460i \(0.267293\pi\)
\(524\) 0 0
\(525\) −1.64015e36 −0.310980
\(526\) 0 0
\(527\) 1.18115e35 0.0212747
\(528\) 0 0
\(529\) −5.37762e36 −0.920319
\(530\) 0 0
\(531\) 5.11074e35 0.0831205
\(532\) 0 0
\(533\) −2.49336e36 −0.385450
\(534\) 0 0
\(535\) 3.56019e36 0.523235
\(536\) 0 0
\(537\) −2.18998e36 −0.306047
\(538\) 0 0
\(539\) 7.86308e36 1.04507
\(540\) 0 0
\(541\) 9.50157e36 1.20125 0.600624 0.799532i \(-0.294919\pi\)
0.600624 + 0.799532i \(0.294919\pi\)
\(542\) 0 0
\(543\) 3.65105e36 0.439158
\(544\) 0 0
\(545\) −1.50812e36 −0.172617
\(546\) 0 0
\(547\) −1.21980e37 −1.32880 −0.664401 0.747376i \(-0.731313\pi\)
−0.664401 + 0.747376i \(0.731313\pi\)
\(548\) 0 0
\(549\) −4.59626e36 −0.476626
\(550\) 0 0
\(551\) 1.97715e36 0.195206
\(552\) 0 0
\(553\) 3.42698e36 0.322197
\(554\) 0 0
\(555\) 9.13253e35 0.0817779
\(556\) 0 0
\(557\) −1.48409e37 −1.26594 −0.632972 0.774175i \(-0.718165\pi\)
−0.632972 + 0.774175i \(0.718165\pi\)
\(558\) 0 0
\(559\) −1.32247e37 −1.07479
\(560\) 0 0
\(561\) −1.31039e36 −0.101485
\(562\) 0 0
\(563\) 2.32350e37 1.71506 0.857530 0.514434i \(-0.171998\pi\)
0.857530 + 0.514434i \(0.171998\pi\)
\(564\) 0 0
\(565\) −4.26711e36 −0.300248
\(566\) 0 0
\(567\) −9.77598e35 −0.0655827
\(568\) 0 0
\(569\) 8.24546e36 0.527472 0.263736 0.964595i \(-0.415045\pi\)
0.263736 + 0.964595i \(0.415045\pi\)
\(570\) 0 0
\(571\) −1.85314e37 −1.13063 −0.565315 0.824875i \(-0.691245\pi\)
−0.565315 + 0.824875i \(0.691245\pi\)
\(572\) 0 0
\(573\) −9.91552e36 −0.577069
\(574\) 0 0
\(575\) 4.63932e36 0.257596
\(576\) 0 0
\(577\) −1.73606e37 −0.919794 −0.459897 0.887972i \(-0.652114\pi\)
−0.459897 + 0.887972i \(0.652114\pi\)
\(578\) 0 0
\(579\) 4.09426e36 0.207021
\(580\) 0 0
\(581\) 1.98968e37 0.960296
\(582\) 0 0
\(583\) −5.48336e37 −2.52651
\(584\) 0 0
\(585\) −2.08131e36 −0.0915656
\(586\) 0 0
\(587\) −4.04428e37 −1.69913 −0.849564 0.527485i \(-0.823135\pi\)
−0.849564 + 0.527485i \(0.823135\pi\)
\(588\) 0 0
\(589\) 1.11844e36 0.0448803
\(590\) 0 0
\(591\) 3.91153e36 0.149940
\(592\) 0 0
\(593\) −2.72712e37 −0.998771 −0.499386 0.866380i \(-0.666441\pi\)
−0.499386 + 0.866380i \(0.666441\pi\)
\(594\) 0 0
\(595\) −5.46600e35 −0.0191289
\(596\) 0 0
\(597\) −1.25669e37 −0.420315
\(598\) 0 0
\(599\) −7.42066e36 −0.237236 −0.118618 0.992940i \(-0.537846\pi\)
−0.118618 + 0.992940i \(0.537846\pi\)
\(600\) 0 0
\(601\) 5.26594e37 1.60942 0.804712 0.593665i \(-0.202320\pi\)
0.804712 + 0.593665i \(0.202320\pi\)
\(602\) 0 0
\(603\) −1.42191e37 −0.415517
\(604\) 0 0
\(605\) −1.66367e37 −0.464910
\(606\) 0 0
\(607\) −2.96287e37 −0.791890 −0.395945 0.918274i \(-0.629583\pi\)
−0.395945 + 0.918274i \(0.629583\pi\)
\(608\) 0 0
\(609\) −1.12537e37 −0.287715
\(610\) 0 0
\(611\) −2.01803e37 −0.493595
\(612\) 0 0
\(613\) 6.79040e36 0.158920 0.0794601 0.996838i \(-0.474680\pi\)
0.0794601 + 0.996838i \(0.474680\pi\)
\(614\) 0 0
\(615\) −3.16279e36 −0.0708363
\(616\) 0 0
\(617\) 1.07350e37 0.230118 0.115059 0.993359i \(-0.463294\pi\)
0.115059 + 0.993359i \(0.463294\pi\)
\(618\) 0 0
\(619\) −5.51172e37 −1.13100 −0.565501 0.824748i \(-0.691317\pi\)
−0.565501 + 0.824748i \(0.691317\pi\)
\(620\) 0 0
\(621\) 2.76523e36 0.0543244
\(622\) 0 0
\(623\) 2.08160e37 0.391568
\(624\) 0 0
\(625\) 4.13741e37 0.745330
\(626\) 0 0
\(627\) −1.24082e37 −0.214089
\(628\) 0 0
\(629\) −3.17641e36 −0.0524992
\(630\) 0 0
\(631\) 6.29297e37 0.996459 0.498230 0.867045i \(-0.333984\pi\)
0.498230 + 0.867045i \(0.333984\pi\)
\(632\) 0 0
\(633\) −5.02668e37 −0.762661
\(634\) 0 0
\(635\) 3.95736e37 0.575388
\(636\) 0 0
\(637\) −4.34384e37 −0.605329
\(638\) 0 0
\(639\) 2.15663e37 0.288081
\(640\) 0 0
\(641\) −7.82906e37 −1.00260 −0.501298 0.865274i \(-0.667144\pi\)
−0.501298 + 0.865274i \(0.667144\pi\)
\(642\) 0 0
\(643\) −2.29842e37 −0.282216 −0.141108 0.989994i \(-0.545066\pi\)
−0.141108 + 0.989994i \(0.545066\pi\)
\(644\) 0 0
\(645\) −1.67753e37 −0.197521
\(646\) 0 0
\(647\) 6.00638e37 0.678274 0.339137 0.940737i \(-0.389865\pi\)
0.339137 + 0.940737i \(0.389865\pi\)
\(648\) 0 0
\(649\) −3.69222e37 −0.399930
\(650\) 0 0
\(651\) −6.36604e36 −0.0661494
\(652\) 0 0
\(653\) 1.68189e38 1.67676 0.838378 0.545090i \(-0.183504\pi\)
0.838378 + 0.545090i \(0.183504\pi\)
\(654\) 0 0
\(655\) 5.46209e36 0.0522518
\(656\) 0 0
\(657\) −2.99094e37 −0.274584
\(658\) 0 0
\(659\) −3.71730e37 −0.327548 −0.163774 0.986498i \(-0.552367\pi\)
−0.163774 + 0.986498i \(0.552367\pi\)
\(660\) 0 0
\(661\) −1.47751e38 −1.24971 −0.624857 0.780739i \(-0.714843\pi\)
−0.624857 + 0.780739i \(0.714843\pi\)
\(662\) 0 0
\(663\) 7.23906e36 0.0587827
\(664\) 0 0
\(665\) −5.17578e36 −0.0403537
\(666\) 0 0
\(667\) 3.18323e37 0.238325
\(668\) 0 0
\(669\) −1.40664e37 −0.101141
\(670\) 0 0
\(671\) 3.32053e38 2.29326
\(672\) 0 0
\(673\) 1.40235e37 0.0930362 0.0465181 0.998917i \(-0.485187\pi\)
0.0465181 + 0.998917i \(0.485187\pi\)
\(674\) 0 0
\(675\) 2.75538e37 0.175623
\(676\) 0 0
\(677\) −5.70098e37 −0.349142 −0.174571 0.984645i \(-0.555854\pi\)
−0.174571 + 0.984645i \(0.555854\pi\)
\(678\) 0 0
\(679\) −2.34450e37 −0.137977
\(680\) 0 0
\(681\) 8.26935e37 0.467719
\(682\) 0 0
\(683\) 1.52052e38 0.826630 0.413315 0.910588i \(-0.364371\pi\)
0.413315 + 0.910588i \(0.364371\pi\)
\(684\) 0 0
\(685\) −3.01272e37 −0.157448
\(686\) 0 0
\(687\) −1.72933e38 −0.868885
\(688\) 0 0
\(689\) 3.02920e38 1.46342
\(690\) 0 0
\(691\) 2.89989e38 1.34719 0.673594 0.739101i \(-0.264750\pi\)
0.673594 + 0.739101i \(0.264750\pi\)
\(692\) 0 0
\(693\) 7.06259e37 0.315548
\(694\) 0 0
\(695\) −4.21985e36 −0.0181344
\(696\) 0 0
\(697\) 1.10006e37 0.0454750
\(698\) 0 0
\(699\) −1.35216e38 −0.537759
\(700\) 0 0
\(701\) 3.62729e38 1.38800 0.694000 0.719975i \(-0.255847\pi\)
0.694000 + 0.719975i \(0.255847\pi\)
\(702\) 0 0
\(703\) −3.00776e37 −0.110751
\(704\) 0 0
\(705\) −2.55983e37 −0.0907107
\(706\) 0 0
\(707\) 1.24974e38 0.426242
\(708\) 0 0
\(709\) −3.77266e38 −1.23858 −0.619289 0.785163i \(-0.712579\pi\)
−0.619289 + 0.785163i \(0.712579\pi\)
\(710\) 0 0
\(711\) −5.75716e37 −0.181957
\(712\) 0 0
\(713\) 1.80070e37 0.0547938
\(714\) 0 0
\(715\) 1.50363e38 0.440564
\(716\) 0 0
\(717\) −1.94997e38 −0.550199
\(718\) 0 0
\(719\) −3.37121e38 −0.916107 −0.458053 0.888925i \(-0.651453\pi\)
−0.458053 + 0.888925i \(0.651453\pi\)
\(720\) 0 0
\(721\) −7.07782e37 −0.185257
\(722\) 0 0
\(723\) −1.85710e38 −0.468239
\(724\) 0 0
\(725\) 3.17189e38 0.770468
\(726\) 0 0
\(727\) 2.14930e38 0.503016 0.251508 0.967855i \(-0.419073\pi\)
0.251508 + 0.967855i \(0.419073\pi\)
\(728\) 0 0
\(729\) 1.64232e37 0.0370370
\(730\) 0 0
\(731\) 5.83465e37 0.126803
\(732\) 0 0
\(733\) 4.15923e38 0.871183 0.435592 0.900144i \(-0.356539\pi\)
0.435592 + 0.900144i \(0.356539\pi\)
\(734\) 0 0
\(735\) −5.51009e37 −0.111245
\(736\) 0 0
\(737\) 1.02725e39 1.99924
\(738\) 0 0
\(739\) −1.98293e38 −0.372054 −0.186027 0.982545i \(-0.559561\pi\)
−0.186027 + 0.982545i \(0.559561\pi\)
\(740\) 0 0
\(741\) 6.85470e37 0.124006
\(742\) 0 0
\(743\) 6.16838e38 1.07602 0.538012 0.842937i \(-0.319176\pi\)
0.538012 + 0.842937i \(0.319176\pi\)
\(744\) 0 0
\(745\) −2.24497e38 −0.377659
\(746\) 0 0
\(747\) −3.34257e38 −0.542316
\(748\) 0 0
\(749\) 6.67393e38 1.04443
\(750\) 0 0
\(751\) −3.81388e37 −0.0575742 −0.0287871 0.999586i \(-0.509164\pi\)
−0.0287871 + 0.999586i \(0.509164\pi\)
\(752\) 0 0
\(753\) 4.17129e38 0.607489
\(754\) 0 0
\(755\) 1.65766e38 0.232922
\(756\) 0 0
\(757\) −1.63378e38 −0.221513 −0.110757 0.993848i \(-0.535327\pi\)
−0.110757 + 0.993848i \(0.535327\pi\)
\(758\) 0 0
\(759\) −1.99772e38 −0.261379
\(760\) 0 0
\(761\) 1.09698e39 1.38517 0.692587 0.721335i \(-0.256471\pi\)
0.692587 + 0.721335i \(0.256471\pi\)
\(762\) 0 0
\(763\) −2.82713e38 −0.344560
\(764\) 0 0
\(765\) 9.18263e36 0.0108028
\(766\) 0 0
\(767\) 2.03971e38 0.231650
\(768\) 0 0
\(769\) −7.55174e38 −0.828021 −0.414010 0.910272i \(-0.635872\pi\)
−0.414010 + 0.910272i \(0.635872\pi\)
\(770\) 0 0
\(771\) −9.31872e38 −0.986557
\(772\) 0 0
\(773\) −4.48700e38 −0.458704 −0.229352 0.973344i \(-0.573661\pi\)
−0.229352 + 0.973344i \(0.573661\pi\)
\(774\) 0 0
\(775\) 1.79428e38 0.177140
\(776\) 0 0
\(777\) 1.71199e38 0.163236
\(778\) 0 0
\(779\) 1.04165e38 0.0959326
\(780\) 0 0
\(781\) −1.55804e39 −1.38609
\(782\) 0 0
\(783\) 1.89058e38 0.162484
\(784\) 0 0
\(785\) −1.37992e38 −0.114581
\(786\) 0 0
\(787\) 6.54790e38 0.525344 0.262672 0.964885i \(-0.415396\pi\)
0.262672 + 0.964885i \(0.415396\pi\)
\(788\) 0 0
\(789\) −8.57737e38 −0.664989
\(790\) 0 0
\(791\) −7.99913e38 −0.599322
\(792\) 0 0
\(793\) −1.83438e39 −1.32832
\(794\) 0 0
\(795\) 3.84249e38 0.268941
\(796\) 0 0
\(797\) −8.33821e38 −0.564138 −0.282069 0.959394i \(-0.591021\pi\)
−0.282069 + 0.959394i \(0.591021\pi\)
\(798\) 0 0
\(799\) 8.90343e37 0.0582339
\(800\) 0 0
\(801\) −3.49699e38 −0.221134
\(802\) 0 0
\(803\) 2.16079e39 1.32115
\(804\) 0 0
\(805\) −8.33305e37 −0.0492674
\(806\) 0 0
\(807\) −1.32320e39 −0.756541
\(808\) 0 0
\(809\) 1.15396e39 0.638097 0.319048 0.947738i \(-0.396637\pi\)
0.319048 + 0.947738i \(0.396637\pi\)
\(810\) 0 0
\(811\) −7.44170e38 −0.398007 −0.199004 0.979999i \(-0.563771\pi\)
−0.199004 + 0.979999i \(0.563771\pi\)
\(812\) 0 0
\(813\) −1.61061e39 −0.833234
\(814\) 0 0
\(815\) 2.03656e38 0.101922
\(816\) 0 0
\(817\) 5.52486e38 0.267500
\(818\) 0 0
\(819\) −3.90162e38 −0.182773
\(820\) 0 0
\(821\) −2.30112e39 −1.04305 −0.521527 0.853235i \(-0.674637\pi\)
−0.521527 + 0.853235i \(0.674637\pi\)
\(822\) 0 0
\(823\) −2.70878e39 −1.18816 −0.594082 0.804404i \(-0.702485\pi\)
−0.594082 + 0.804404i \(0.702485\pi\)
\(824\) 0 0
\(825\) −1.99061e39 −0.845000
\(826\) 0 0
\(827\) 2.41109e39 0.990580 0.495290 0.868728i \(-0.335062\pi\)
0.495290 + 0.868728i \(0.335062\pi\)
\(828\) 0 0
\(829\) −3.96338e39 −1.57608 −0.788042 0.615621i \(-0.788905\pi\)
−0.788042 + 0.615621i \(0.788905\pi\)
\(830\) 0 0
\(831\) −2.39646e39 −0.922479
\(832\) 0 0
\(833\) 1.91648e38 0.0714162
\(834\) 0 0
\(835\) −3.64897e38 −0.131645
\(836\) 0 0
\(837\) 1.06947e38 0.0373571
\(838\) 0 0
\(839\) 3.82794e38 0.129473 0.0647364 0.997902i \(-0.479379\pi\)
0.0647364 + 0.997902i \(0.479379\pi\)
\(840\) 0 0
\(841\) −8.76776e38 −0.287173
\(842\) 0 0
\(843\) −5.99587e38 −0.190187
\(844\) 0 0
\(845\) 1.31879e38 0.0405143
\(846\) 0 0
\(847\) −3.11871e39 −0.928002
\(848\) 0 0
\(849\) −3.17683e39 −0.915674
\(850\) 0 0
\(851\) −4.84252e38 −0.135214
\(852\) 0 0
\(853\) 3.96671e39 1.07305 0.536523 0.843885i \(-0.319737\pi\)
0.536523 + 0.843885i \(0.319737\pi\)
\(854\) 0 0
\(855\) 8.69507e37 0.0227893
\(856\) 0 0
\(857\) 7.42146e39 1.88473 0.942363 0.334593i \(-0.108599\pi\)
0.942363 + 0.334593i \(0.108599\pi\)
\(858\) 0 0
\(859\) −2.37083e39 −0.583436 −0.291718 0.956504i \(-0.594227\pi\)
−0.291718 + 0.956504i \(0.594227\pi\)
\(860\) 0 0
\(861\) −5.92897e38 −0.141396
\(862\) 0 0
\(863\) 6.45487e39 1.49191 0.745954 0.665998i \(-0.231994\pi\)
0.745954 + 0.665998i \(0.231994\pi\)
\(864\) 0 0
\(865\) 6.87746e38 0.154067
\(866\) 0 0
\(867\) 2.62693e39 0.570415
\(868\) 0 0
\(869\) 4.15923e39 0.875479
\(870\) 0 0
\(871\) −5.67489e39 −1.15801
\(872\) 0 0
\(873\) 3.93865e38 0.0779211
\(874\) 0 0
\(875\) −1.74023e39 −0.333809
\(876\) 0 0
\(877\) −3.78716e39 −0.704400 −0.352200 0.935925i \(-0.614566\pi\)
−0.352200 + 0.935925i \(0.614566\pi\)
\(878\) 0 0
\(879\) 3.42656e39 0.618029
\(880\) 0 0
\(881\) −2.28239e39 −0.399222 −0.199611 0.979875i \(-0.563968\pi\)
−0.199611 + 0.979875i \(0.563968\pi\)
\(882\) 0 0
\(883\) −3.77953e39 −0.641163 −0.320582 0.947221i \(-0.603878\pi\)
−0.320582 + 0.947221i \(0.603878\pi\)
\(884\) 0 0
\(885\) 2.58734e38 0.0425716
\(886\) 0 0
\(887\) 9.78168e38 0.156115 0.0780575 0.996949i \(-0.475128\pi\)
0.0780575 + 0.996949i \(0.475128\pi\)
\(888\) 0 0
\(889\) 7.41848e39 1.14853
\(890\) 0 0
\(891\) −1.18648e39 −0.178202
\(892\) 0 0
\(893\) 8.43070e38 0.122848
\(894\) 0 0
\(895\) −1.10869e39 −0.156747
\(896\) 0 0
\(897\) 1.10361e39 0.151397
\(898\) 0 0
\(899\) 1.23113e39 0.163888
\(900\) 0 0
\(901\) −1.33647e39 −0.172653
\(902\) 0 0
\(903\) −3.14469e39 −0.394270
\(904\) 0 0
\(905\) 1.84837e39 0.224922
\(906\) 0 0
\(907\) −1.50937e40 −1.78278 −0.891390 0.453237i \(-0.850269\pi\)
−0.891390 + 0.453237i \(0.850269\pi\)
\(908\) 0 0
\(909\) −2.09950e39 −0.240715
\(910\) 0 0
\(911\) 1.32834e40 1.47847 0.739233 0.673450i \(-0.235188\pi\)
0.739233 + 0.673450i \(0.235188\pi\)
\(912\) 0 0
\(913\) 2.41482e40 2.60933
\(914\) 0 0
\(915\) −2.32688e39 −0.244112
\(916\) 0 0
\(917\) 1.02392e39 0.104299
\(918\) 0 0
\(919\) 8.25375e39 0.816379 0.408190 0.912897i \(-0.366160\pi\)
0.408190 + 0.912897i \(0.366160\pi\)
\(920\) 0 0
\(921\) −9.06529e39 −0.870717
\(922\) 0 0
\(923\) 8.60718e39 0.802856
\(924\) 0 0
\(925\) −4.82526e39 −0.437127
\(926\) 0 0
\(927\) 1.18904e39 0.104621
\(928\) 0 0
\(929\) 1.76611e40 1.50940 0.754702 0.656068i \(-0.227781\pi\)
0.754702 + 0.656068i \(0.227781\pi\)
\(930\) 0 0
\(931\) 1.81472e39 0.150657
\(932\) 0 0
\(933\) −2.84763e39 −0.229658
\(934\) 0 0
\(935\) −6.63393e38 −0.0519773
\(936\) 0 0
\(937\) 1.45298e40 1.10605 0.553024 0.833165i \(-0.313474\pi\)
0.553024 + 0.833165i \(0.313474\pi\)
\(938\) 0 0
\(939\) 9.92048e39 0.733749
\(940\) 0 0
\(941\) −1.91467e40 −1.37605 −0.688025 0.725687i \(-0.741522\pi\)
−0.688025 + 0.725687i \(0.741522\pi\)
\(942\) 0 0
\(943\) 1.67707e39 0.117123
\(944\) 0 0
\(945\) −4.94915e38 −0.0335893
\(946\) 0 0
\(947\) 9.97334e39 0.657833 0.328917 0.944359i \(-0.393317\pi\)
0.328917 + 0.944359i \(0.393317\pi\)
\(948\) 0 0
\(949\) −1.19369e40 −0.765243
\(950\) 0 0
\(951\) −1.68863e39 −0.105220
\(952\) 0 0
\(953\) 1.27709e40 0.773512 0.386756 0.922182i \(-0.373596\pi\)
0.386756 + 0.922182i \(0.373596\pi\)
\(954\) 0 0
\(955\) −5.01979e39 −0.295556
\(956\) 0 0
\(957\) −1.36583e40 −0.781784
\(958\) 0 0
\(959\) −5.64765e39 −0.314280
\(960\) 0 0
\(961\) −1.77863e40 −0.962320
\(962\) 0 0
\(963\) −1.12119e40 −0.589827
\(964\) 0 0
\(965\) 2.07274e39 0.106029
\(966\) 0 0
\(967\) 2.34693e40 1.16746 0.583730 0.811948i \(-0.301593\pi\)
0.583730 + 0.811948i \(0.301593\pi\)
\(968\) 0 0
\(969\) −3.02426e38 −0.0146301
\(970\) 0 0
\(971\) 2.98139e40 1.40268 0.701341 0.712826i \(-0.252585\pi\)
0.701341 + 0.712826i \(0.252585\pi\)
\(972\) 0 0
\(973\) −7.91054e38 −0.0361979
\(974\) 0 0
\(975\) 1.09968e40 0.489445
\(976\) 0 0
\(977\) −7.06988e39 −0.306080 −0.153040 0.988220i \(-0.548906\pi\)
−0.153040 + 0.988220i \(0.548906\pi\)
\(978\) 0 0
\(979\) 2.52637e40 1.06397
\(980\) 0 0
\(981\) 4.74945e39 0.194586
\(982\) 0 0
\(983\) 3.06712e40 1.22252 0.611262 0.791428i \(-0.290662\pi\)
0.611262 + 0.791428i \(0.290662\pi\)
\(984\) 0 0
\(985\) 1.98024e39 0.0767941
\(986\) 0 0
\(987\) −4.79867e39 −0.181067
\(988\) 0 0
\(989\) 8.89506e39 0.326587
\(990\) 0 0
\(991\) −7.77866e39 −0.277914 −0.138957 0.990298i \(-0.544375\pi\)
−0.138957 + 0.990298i \(0.544375\pi\)
\(992\) 0 0
\(993\) 3.06478e40 1.06557
\(994\) 0 0
\(995\) −6.36207e39 −0.215272
\(996\) 0 0
\(997\) −3.19922e40 −1.05356 −0.526780 0.850002i \(-0.676601\pi\)
−0.526780 + 0.850002i \(0.676601\pi\)
\(998\) 0 0
\(999\) −2.87606e39 −0.0921857
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 48.28.a.l.1.2 4
4.3 odd 2 24.28.a.d.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.28.a.d.1.2 4 4.3 odd 2
48.28.a.l.1.2 4 1.1 even 1 trivial