Properties

Label 33.8.d.a.32.1
Level $33$
Weight $8$
Character 33.32
Analytic conductor $10.309$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,8,Mod(32,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.32");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3087058410\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 32.1
Root \(0.500000 + 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 33.32
Dual form 33.8.d.a.32.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-41.5000 - 21.5581i) q^{3} -128.000 q^{4} -235.480i q^{5} +(1257.50 + 1789.32i) q^{9} +O(q^{10})\) \(q+(-41.5000 - 21.5581i) q^{3} -128.000 q^{4} -235.480i q^{5} +(1257.50 + 1789.32i) q^{9} +4414.43i q^{11} +(5312.00 + 2759.43i) q^{12} +(-5076.50 + 9772.43i) q^{15} +16384.0 q^{16} +30141.5i q^{20} +72478.2i q^{23} +22674.0 q^{25} +(-13612.0 - 101366. i) q^{27} +39065.0 q^{31} +(95166.5 - 183199. i) q^{33} +(-160960. - 229033. i) q^{36} -562471. q^{37} -565047. i q^{44} +(421350. - 296117. i) q^{45} +545406. i q^{47} +(-679936. - 353207. i) q^{48} +823543. q^{49} +2.13701e6i q^{53} +1.03951e6 q^{55} +3.15153e6i q^{59} +(649792. - 1.25087e6i) q^{60} -2.09715e6 q^{64} +684671. q^{67} +(1.56249e6 - 3.00785e6i) q^{69} +1.88588e6i q^{71} +(-940971. - 488807. i) q^{75} -3.85811e6i q^{80} +(-1.62036e6 + 4.50014e6i) q^{81} -1.25440e7i q^{89} -9.27721e6i q^{92} +(-1.62120e6 - 842166. i) q^{93} -1.51825e7 q^{97} +(-7.89882e6 + 5.55114e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 83 q^{3} - 256 q^{4} + 2515 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 83 q^{3} - 256 q^{4} + 2515 q^{9} + 10624 q^{12} - 10153 q^{15} + 32768 q^{16} + 45348 q^{25} - 27224 q^{27} + 78130 q^{31} + 190333 q^{33} - 321920 q^{36} - 1124942 q^{37} + 842699 q^{45} - 1359872 q^{48} + 1647086 q^{49} + 2079022 q^{55} + 1299584 q^{60} - 4194304 q^{64} + 1369342 q^{67} + 3124979 q^{69} - 1881942 q^{75} - 3240713 q^{81} - 3242395 q^{93} - 30364958 q^{97} - 15797639 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −41.5000 21.5581i −0.887409 0.460983i
\(4\) −128.000 −1.00000
\(5\) 235.480i 0.842480i −0.906949 0.421240i \(-0.861595\pi\)
0.906949 0.421240i \(-0.138405\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1257.50 + 1789.32i 0.574989 + 0.818161i
\(10\) 0 0
\(11\) 4414.43i 1.00000i
\(12\) 5312.00 + 2759.43i 0.887409 + 0.460983i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −5076.50 + 9772.43i −0.388369 + 0.747624i
\(16\) 16384.0 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 30141.5i 0.842480i
\(21\) 0 0
\(22\) 0 0
\(23\) 72478.2i 1.24211i 0.783767 + 0.621055i \(0.213296\pi\)
−0.783767 + 0.621055i \(0.786704\pi\)
\(24\) 0 0
\(25\) 22674.0 0.290227
\(26\) 0 0
\(27\) −13612.0 101366.i −0.133091 0.991104i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 39065.0 0.235517 0.117758 0.993042i \(-0.462429\pi\)
0.117758 + 0.993042i \(0.462429\pi\)
\(32\) 0 0
\(33\) 95166.5 183199.i 0.460983 0.887409i
\(34\) 0 0
\(35\) 0 0
\(36\) −160960. 229033.i −0.574989 0.818161i
\(37\) −562471. −1.82555 −0.912776 0.408461i \(-0.866066\pi\)
−0.912776 + 0.408461i \(0.866066\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 565047.i 1.00000i
\(45\) 421350. 296117.i 0.689285 0.484416i
\(46\) 0 0
\(47\) 545406.i 0.766262i 0.923694 + 0.383131i \(0.125154\pi\)
−0.923694 + 0.383131i \(0.874846\pi\)
\(48\) −679936. 353207.i −0.887409 0.460983i
\(49\) 823543. 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.13701e6i 1.97170i 0.167634 + 0.985849i \(0.446387\pi\)
−0.167634 + 0.985849i \(0.553613\pi\)
\(54\) 0 0
\(55\) 1.03951e6 0.842480
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.15153e6i 1.99774i 0.0475279 + 0.998870i \(0.484866\pi\)
−0.0475279 + 0.998870i \(0.515134\pi\)
\(60\) 649792. 1.25087e6i 0.388369 0.747624i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −2.09715e6 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 684671. 0.278112 0.139056 0.990285i \(-0.455593\pi\)
0.139056 + 0.990285i \(0.455593\pi\)
\(68\) 0 0
\(69\) 1.56249e6 3.00785e6i 0.572592 1.10226i
\(70\) 0 0
\(71\) 1.88588e6i 0.625332i 0.949863 + 0.312666i \(0.101222\pi\)
−0.949863 + 0.312666i \(0.898778\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −940971. 488807.i −0.257550 0.133790i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 3.85811e6i 0.842480i
\(81\) −1.62036e6 + 4.50014e6i −0.338776 + 0.940867i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.25440e7i 1.88613i −0.332607 0.943065i \(-0.607928\pi\)
0.332607 0.943065i \(-0.392072\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.27721e6i 1.24211i
\(93\) −1.62120e6 842166.i −0.209000 0.108569i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.51825e7 −1.68905 −0.844523 0.535519i \(-0.820116\pi\)
−0.844523 + 0.535519i \(0.820116\pi\)
\(98\) 0 0
\(99\) −7.89882e6 + 5.55114e6i −0.818161 + 0.574989i
\(100\) −2.90227e6 −0.290227
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 2.18126e7 1.96688 0.983439 0.181240i \(-0.0580110\pi\)
0.983439 + 0.181240i \(0.0580110\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.74234e6 + 1.29748e7i 0.133091 + 0.991104i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 2.33425e7 + 1.21258e7i 1.62001 + 0.841549i
\(112\) 0 0
\(113\) 2.72900e7i 1.77921i 0.456726 + 0.889607i \(0.349022\pi\)
−0.456726 + 0.889607i \(0.650978\pi\)
\(114\) 0 0
\(115\) 1.70672e7 1.04645
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.94872e7 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −5.00032e6 −0.235517
\(125\) 2.37362e7i 1.08699i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −1.21813e7 + 2.34494e7i −0.460983 + 0.887409i
\(133\) 0 0
\(134\) 0 0
\(135\) −2.38697e7 + 3.20536e6i −0.834985 + 0.112127i
\(136\) 0 0
\(137\) 3.74383e7i 1.24393i −0.783047 0.621963i \(-0.786335\pi\)
0.783047 0.621963i \(-0.213665\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 1.17579e7 2.26343e7i 0.353234 0.679987i
\(142\) 0 0
\(143\) 0 0
\(144\) 2.06029e7 + 2.93162e7i 0.574989 + 0.818161i
\(145\) 0 0
\(146\) 0 0
\(147\) −3.41770e7 1.77540e7i −0.887409 0.460983i
\(148\) 7.19963e7 1.82555
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.19904e6i 0.198418i
\(156\) 0 0
\(157\) −9.31069e7 −1.92014 −0.960071 0.279758i \(-0.909746\pi\)
−0.960071 + 0.279758i \(0.909746\pi\)
\(158\) 0 0
\(159\) 4.60697e7 8.86858e7i 0.908920 1.74970i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.10539e8 −1.99922 −0.999609 0.0279625i \(-0.991098\pi\)
−0.999609 + 0.0279625i \(0.991098\pi\)
\(164\) 0 0
\(165\) −4.31397e7 2.24098e7i −0.747624 0.388369i
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 6.27485e7 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 7.23260e7i 1.00000i
\(177\) 6.79408e7 1.30788e8i 0.920925 1.77281i
\(178\) 0 0
\(179\) 1.53381e8i 1.99888i 0.0334992 + 0.999439i \(0.489335\pi\)
−0.0334992 + 0.999439i \(0.510665\pi\)
\(180\) −5.39327e7 + 3.79029e7i −0.689285 + 0.484416i
\(181\) 1.40767e8 1.76452 0.882260 0.470763i \(-0.156021\pi\)
0.882260 + 0.470763i \(0.156021\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.32451e8i 1.53799i
\(186\) 0 0
\(187\) 0 0
\(188\) 6.98119e7i 0.766262i
\(189\) 0 0
\(190\) 0 0
\(191\) 1.23633e8i 1.28385i 0.766765 + 0.641927i \(0.221865\pi\)
−0.766765 + 0.641927i \(0.778135\pi\)
\(192\) 8.70318e7 + 4.52105e7i 0.887409 + 0.460983i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.05414e8 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 1.54428e8 1.38912 0.694561 0.719433i \(-0.255598\pi\)
0.694561 + 0.719433i \(0.255598\pi\)
\(200\) 0 0
\(201\) −2.84138e7 1.47602e7i −0.246799 0.128205i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.29687e8 + 9.11413e7i −1.01625 + 0.714199i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 2.73537e8i 1.97170i
\(213\) 4.06560e7 7.82641e7i 0.288268 0.554925i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −1.33057e8 −0.842480
\(221\) 0 0
\(222\) 0 0
\(223\) 7.69323e7 0.464560 0.232280 0.972649i \(-0.425381\pi\)
0.232280 + 0.972649i \(0.425381\pi\)
\(224\) 0 0
\(225\) 2.85126e7 + 4.05710e7i 0.166877 + 0.237453i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −3.33474e8 −1.83501 −0.917504 0.397727i \(-0.869799\pi\)
−0.917504 + 0.397727i \(0.869799\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 1.28432e8 0.645560
\(236\) 4.03395e8i 1.99774i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −8.31734e7 + 1.60112e8i −0.388369 + 0.747624i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 1.64259e8 1.51824e8i 0.734357 0.678763i
\(244\) 0 0
\(245\) 1.93928e8i 0.842480i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.28095e8i 1.30961i 0.755799 + 0.654803i \(0.227249\pi\)
−0.755799 + 0.654803i \(0.772751\pi\)
\(252\) 0 0
\(253\) −3.19950e8 −1.24211
\(254\) 0 0
\(255\) 0 0
\(256\) 2.68435e8 1.00000
\(257\) 2.79730e8i 1.02795i −0.857804 0.513977i \(-0.828172\pi\)
0.857804 0.513977i \(-0.171828\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 5.03223e8 1.66112
\(266\) 0 0
\(267\) −2.70425e8 + 5.20577e8i −0.869475 + 1.67377i
\(268\) −8.76379e7 −0.278112
\(269\) 1.43733e8i 0.450219i 0.974333 + 0.225110i \(0.0722740\pi\)
−0.974333 + 0.225110i \(0.927726\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00093e8i 0.290227i
\(276\) −1.99999e8 + 3.85004e8i −0.572592 + 1.10226i
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 4.91242e7 + 6.98997e7i 0.135419 + 0.192691i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 2.41393e8i 0.625332i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.10339e8 −1.00000
\(290\) 0 0
\(291\) 6.30073e8 + 3.27305e8i 1.49887 + 0.778622i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 7.42123e8 1.68306
\(296\) 0 0
\(297\) 4.47473e8 6.00892e7i 0.991104 0.133091i
\(298\) 0 0
\(299\) 0 0
\(300\) 1.20444e8 + 6.25674e7i 0.257550 + 0.133790i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −9.05223e8 4.70238e8i −1.74542 0.906698i
\(310\) 0 0
\(311\) 8.03559e8i 1.51480i 0.652949 + 0.757402i \(0.273532\pi\)
−0.652949 + 0.757402i \(0.726468\pi\)
\(312\) 0 0
\(313\) −7.97959e8 −1.47087 −0.735436 0.677594i \(-0.763023\pi\)
−0.735436 + 0.677594i \(0.763023\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.09650e9i 1.93331i −0.256083 0.966655i \(-0.582432\pi\)
0.256083 0.966655i \(-0.417568\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 4.93838e8i 0.842480i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 2.07406e8 5.76018e8i 0.338776 0.940867i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.74747e8 −0.719555 −0.359777 0.933038i \(-0.617147\pi\)
−0.359777 + 0.933038i \(0.617147\pi\)
\(332\) 0 0
\(333\) −7.07307e8 1.00644e9i −1.04967 1.49360i
\(334\) 0 0
\(335\) 1.61227e8i 0.234304i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 5.88319e8 1.13253e9i 0.820189 1.57889i
\(340\) 0 0
\(341\) 1.72450e8i 0.235517i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −7.08289e8 3.67936e8i −0.928631 0.482397i
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.01541e8i 0.243867i −0.992538 0.121933i \(-0.961091\pi\)
0.992538 0.121933i \(-0.0389094\pi\)
\(354\) 0 0
\(355\) 4.44088e8 0.526830
\(356\) 1.60563e9i 1.88613i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 8.93872e8 1.00000
\(362\) 0 0
\(363\) 8.08718e8 + 4.20106e8i 0.887409 + 0.460983i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.01715e8 0.529817 0.264909 0.964274i \(-0.414658\pi\)
0.264909 + 0.964274i \(0.414658\pi\)
\(368\) 1.18748e9i 1.24211i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 2.07513e8 + 1.07797e8i 0.209000 + 0.108569i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −5.11706e8 + 9.85052e8i −0.501085 + 0.964605i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.08979e9 −1.97181 −0.985907 0.167296i \(-0.946496\pi\)
−0.985907 + 0.167296i \(0.946496\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.23049e9i 1.11913i 0.828785 + 0.559567i \(0.189032\pi\)
−0.828785 + 0.559567i \(0.810968\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 1.94336e9 1.68905
\(389\) 2.12857e9i 1.83343i −0.399538 0.916717i \(-0.630829\pi\)
0.399538 0.916717i \(-0.369171\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 1.01105e9 7.10546e8i 0.818161 0.574989i
\(397\) −8.58427e8 −0.688552 −0.344276 0.938868i \(-0.611876\pi\)
−0.344276 + 0.938868i \(0.611876\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 3.71491e8 0.290227
\(401\) 2.41996e9i 1.87415i 0.349133 + 0.937073i \(0.386476\pi\)
−0.349133 + 0.937073i \(0.613524\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.05969e9 + 3.81562e8i 0.792662 + 0.285412i
\(406\) 0 0
\(407\) 2.48299e9i 1.82555i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −8.07098e8 + 1.55369e9i −0.573429 + 1.10387i
\(412\) −2.79201e9 −1.96688
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.68820e8i 0.643419i 0.946838 + 0.321709i \(0.104257\pi\)
−0.946838 + 0.321709i \(0.895743\pi\)
\(420\) 0 0
\(421\) 3.02662e9 1.97683 0.988417 0.151761i \(-0.0484943\pi\)
0.988417 + 0.151761i \(0.0484943\pi\)
\(422\) 0 0
\(423\) −9.75905e8 + 6.85848e8i −0.626926 + 0.440592i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −2.23019e8 1.66078e9i −0.133091 0.991104i
\(433\) 2.61786e9 1.54967 0.774833 0.632166i \(-0.217834\pi\)
0.774833 + 0.632166i \(0.217834\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.03561e9 + 1.47358e9i 0.574989 + 0.818161i
\(442\) 0 0
\(443\) 3.18595e9i 1.74111i −0.492073 0.870554i \(-0.663761\pi\)
0.492073 0.870554i \(-0.336239\pi\)
\(444\) −2.98785e9 1.55210e9i −1.62001 0.841549i
\(445\) −2.95387e9 −1.58903
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.23313e9i 0.642905i −0.946926 0.321453i \(-0.895829\pi\)
0.946926 0.321453i \(-0.104171\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 3.49312e9i 1.77921i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −2.18460e9 −1.04645
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −2.59540e9 −1.21527 −0.607633 0.794218i \(-0.707881\pi\)
−0.607633 + 0.794218i \(0.707881\pi\)
\(464\) 0 0
\(465\) −1.98313e8 + 3.81760e8i −0.0914675 + 0.176078i
\(466\) 0 0
\(467\) 3.89163e9i 1.76816i −0.467334 0.884081i \(-0.654786\pi\)
0.467334 0.884081i \(-0.345214\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3.86394e9 + 2.00720e9i 1.70395 + 0.885153i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.82379e9 + 2.68729e9i −1.61317 + 1.13370i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.49436e9 1.00000
\(485\) 3.57518e9i 1.42299i
\(486\) 0 0
\(487\) 1.06724e8 0.0418706 0.0209353 0.999781i \(-0.493336\pi\)
0.0209353 + 0.999781i \(0.493336\pi\)
\(488\) 0 0
\(489\) 4.58738e9 + 2.38301e9i 1.77412 + 0.921606i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.30719e9 + 1.86002e9i 0.484416 + 0.689285i
\(496\) 6.40041e8 0.235517
\(497\) 0 0
\(498\) 0 0
\(499\) 5.52872e9 1.99192 0.995962 0.0897704i \(-0.0286133\pi\)
0.995962 + 0.0897704i \(0.0286133\pi\)
\(500\) 3.03823e9i 1.08699i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.60406e9 1.35274e9i −0.887409 0.460983i
\(508\) 0 0
\(509\) 2.93069e9i 0.985049i 0.870298 + 0.492525i \(0.163926\pi\)
−0.870298 + 0.492525i \(0.836074\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.13644e9i 1.65706i
\(516\) 0 0
\(517\) −2.40765e9 −0.766262
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.50723e9i 1.39630i 0.715953 + 0.698149i \(0.245993\pi\)
−0.715953 + 0.698149i \(0.754007\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.55921e9 3.00153e9i 0.460983 0.887409i
\(529\) −1.84826e9 −0.542837
\(530\) 0 0
\(531\) −5.63909e9 + 3.96304e9i −1.63447 + 1.14868i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 3.30660e9 6.36531e9i 0.921449 1.77382i
\(538\) 0 0
\(539\) 3.63547e9i 1.00000i
\(540\) 3.05532e9 4.10286e8i 0.834985 0.112127i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −5.84184e9 3.03467e9i −1.56585 0.813414i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 4.79211e9i 1.24393i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.85538e9 5.49671e9i 0.708988 1.36483i
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −1.50501e9 + 2.89719e9i −0.353234 + 0.679987i
\(565\) 6.42625e9 1.49895
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 2.66528e9 5.13075e9i 0.591836 1.13930i
\(574\) 0 0
\(575\) 1.64337e9i 0.360494i
\(576\) −2.63717e9 3.75247e9i −0.574989 0.818161i
\(577\) 1.01735e9 0.220473 0.110237 0.993905i \(-0.464839\pi\)
0.110237 + 0.993905i \(0.464839\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −9.43366e9 −1.97170
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.03585e9i 1.63983i −0.572487 0.819914i \(-0.694021\pi\)
0.572487 0.819914i \(-0.305979\pi\)
\(588\) 4.37466e9 + 2.27251e9i 0.887409 + 0.460983i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −9.21552e9 −1.82555
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.40877e9 3.32917e9i −1.23272 0.640363i
\(598\) 0 0
\(599\) 9.23855e9i 1.75635i −0.478343 0.878173i \(-0.658763\pi\)
0.478343 0.878173i \(-0.341237\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 8.60974e8 + 1.22509e9i 0.159911 + 0.227541i
\(604\) 0 0
\(605\) 4.58885e9i 0.842480i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.39035e9i 1.43808i 0.694971 + 0.719038i \(0.255417\pi\)
−0.694971 + 0.719038i \(0.744583\pi\)
\(618\) 0 0
\(619\) 1.65488e9 0.280445 0.140223 0.990120i \(-0.455218\pi\)
0.140223 + 0.990120i \(0.455218\pi\)
\(620\) 1.17748e9i 0.198418i
\(621\) 7.34683e9 9.86573e8i 1.23106 0.165314i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −3.81800e9 −0.625541
\(626\) 0 0
\(627\) 0 0
\(628\) 1.19177e10 1.92014
\(629\) 0 0
\(630\) 0 0
\(631\) 1.04722e10 1.65934 0.829668 0.558257i \(-0.188530\pi\)
0.829668 + 0.558257i \(0.188530\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −5.89693e9 + 1.13518e10i −0.908920 + 1.74970i
\(637\) 0 0
\(638\) 0 0
\(639\) −3.37445e9 + 2.37150e9i −0.511622 + 0.359559i
\(640\) 0 0
\(641\) 2.55167e9i 0.382667i −0.981525 0.191333i \(-0.938719\pi\)
0.981525 0.191333i \(-0.0612812\pi\)
\(642\) 0 0
\(643\) −4.07832e9 −0.604983 −0.302491 0.953152i \(-0.597818\pi\)
−0.302491 + 0.953152i \(0.597818\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.84620e9i 1.42923i 0.699516 + 0.714617i \(0.253399\pi\)
−0.699516 + 0.714617i \(0.746601\pi\)
\(648\) 0 0
\(649\) −1.39122e10 −1.99774
\(650\) 0 0
\(651\) 0 0
\(652\) 1.41490e10 1.99922
\(653\) 6.24852e9i 0.878175i 0.898444 + 0.439088i \(0.144698\pi\)
−0.898444 + 0.439088i \(0.855302\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 5.52188e9 + 2.86846e9i 0.747624 + 0.388369i
\(661\) 1.44976e10 1.95251 0.976253 0.216634i \(-0.0695079\pi\)
0.976253 + 0.216634i \(0.0695079\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.19269e9 1.65851e9i −0.412255 0.214154i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −3.08638e8 2.29837e9i −0.0386266 0.287645i
\(676\) −8.03181e9 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.63286e10i 1.96100i −0.196523 0.980499i \(-0.562965\pi\)
0.196523 0.980499i \(-0.437035\pi\)
\(684\) 0 0
\(685\) −8.81599e9 −1.04798
\(686\) 0 0
\(687\) 1.38392e10 + 7.18906e9i 1.62840 + 0.845908i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.28782e10 −1.48485 −0.742424 0.669931i \(-0.766324\pi\)
−0.742424 + 0.669931i \(0.766324\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 9.25773e9i 1.00000i
\(705\) −5.32994e9 2.76875e9i −0.572876 0.297593i
\(706\) 0 0
\(707\) 0 0
\(708\) −8.69642e9 + 1.67409e10i −0.920925 + 1.77281i
\(709\) 1.05259e10 1.10917 0.554583 0.832129i \(-0.312878\pi\)
0.554583 + 0.832129i \(0.312878\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.83136e9i 0.292538i
\(714\) 0 0
\(715\) 0 0
\(716\) 1.96328e10i 1.99888i
\(717\) 0 0
\(718\) 0 0
\(719\) 1.61087e10i 1.61625i −0.589012 0.808124i \(-0.700483\pi\)
0.589012 0.808124i \(-0.299517\pi\)
\(720\) 6.90339e9 4.85157e9i 0.689285 0.484416i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −1.80182e10 −1.76452
\(725\) 0 0
\(726\) 0 0
\(727\) 5.56239e9 0.536897 0.268448 0.963294i \(-0.413489\pi\)
0.268448 + 0.963294i \(0.413489\pi\)
\(728\) 0 0
\(729\) −1.00898e10 + 2.75959e9i −0.964574 + 0.263814i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −4.18072e9 + 8.04802e9i −0.388369 + 0.747624i
\(736\) 0 0
\(737\) 3.02243e9i 0.278112i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 1.69537e10i 1.53799i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.55415e9 −0.220043 −0.110021 0.993929i \(-0.535092\pi\)
−0.110021 + 0.993929i \(0.535092\pi\)
\(752\) 8.93593e9i 0.766262i
\(753\) 7.07308e9 1.36159e10i 0.603707 1.16216i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.93790e10 1.62366 0.811830 0.583894i \(-0.198472\pi\)
0.811830 + 0.583894i \(0.198472\pi\)
\(758\) 0 0
\(759\) 1.32779e10 + 6.89750e9i 1.10226 + 0.572592i
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.58250e10i 1.28385i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.11401e10 5.78695e9i −0.887409 0.460983i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −6.03044e9 + 1.16088e10i −0.473869 + 0.912215i
\(772\) 0 0
\(773\) 2.55125e10i 1.98667i 0.115280 + 0.993333i \(0.463223\pi\)
−0.115280 + 0.993333i \(0.536777\pi\)
\(774\) 0 0
\(775\) 8.85760e8 0.0683534
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −8.32509e9 −0.625332
\(782\) 0 0
\(783\) 0 0
\(784\) 1.34929e10 1.00000
\(785\) 2.19249e10i 1.61768i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −2.08838e10 1.08485e10i −1.47409 0.765747i
\(796\) −1.97668e10 −1.38912
\(797\) 2.66285e10i 1.86313i 0.363577 + 0.931564i \(0.381556\pi\)
−0.363577 + 0.931564i \(0.618444\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 2.24453e10 1.57741e10i 1.54316 1.08450i
\(802\) 0 0
\(803\) 0 0
\(804\) 3.63697e9 + 1.88930e9i 0.246799 + 0.128205i
\(805\) 0 0
\(806\) 0 0
\(807\) 3.09861e9 5.96492e9i 0.207544 0.399528i
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.60298e10i 1.68430i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 2.47002e10 1.54455 0.772275 0.635289i \(-0.219119\pi\)
0.772275 + 0.635289i \(0.219119\pi\)
\(824\) 0 0
\(825\) 2.15781e9 4.15385e9i 0.133790 0.257550i
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 1.65999e10 1.16661e10i 1.01625 0.714199i
\(829\) 1.72256e10 1.05011 0.525055 0.851068i \(-0.324045\pi\)
0.525055 + 0.851068i \(0.324045\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5.31753e8 3.95986e9i −0.0313452 0.233422i
\(838\) 0 0
\(839\) 3.41591e10i 1.99682i 0.0563296 + 0.998412i \(0.482060\pi\)
−0.0563296 + 0.998412i \(0.517940\pi\)
\(840\) 0 0
\(841\) −1.72499e10 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.47760e10i 0.842480i
\(846\) 0 0
\(847\) 0 0
\(848\) 3.50127e10i 1.97170i
\(849\) 0 0
\(850\) 0 0
\(851\) 4.07669e10i 2.26754i
\(852\) −5.20396e9 + 1.00178e10i −0.288268 + 0.554925i
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −2.55733e10 −1.37661 −0.688304 0.725422i \(-0.741644\pi\)
−0.688304 + 0.725422i \(0.741644\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.61374e9i 0.191390i −0.995411 0.0956950i \(-0.969493\pi\)
0.995411 0.0956950i \(-0.0305073\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.70291e10 + 8.84611e9i 0.887409 + 0.460983i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.90920e10 2.71663e10i −0.971182 1.38191i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 1.70313e10 0.842480
\(881\) 3.72117e10i 1.83343i −0.399546 0.916713i \(-0.630832\pi\)
0.399546 0.916713i \(-0.369168\pi\)
\(882\) 0 0
\(883\) −2.98946e10 −1.46127 −0.730635 0.682769i \(-0.760776\pi\)
−0.730635 + 0.682769i \(0.760776\pi\)
\(884\) 0 0
\(885\) −3.07981e10 1.59987e10i −1.49356 0.775861i
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.98655e10 7.15295e9i −0.940867 0.338776i
\(892\) −9.84734e9 −0.464560
\(893\) 0 0
\(894\) 0 0
\(895\) 3.61182e10 1.68401
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −3.64961e9 5.19309e9i −0.166877 0.237453i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.31479e10i 1.48657i
\(906\) 0 0
\(907\) −3.60932e10 −1.60620 −0.803100 0.595844i \(-0.796818\pi\)
−0.803100 + 0.595844i \(0.796818\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.17961e10i 1.39335i −0.717389 0.696673i \(-0.754663\pi\)
0.717389 0.696673i \(-0.245337\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 4.26847e10 1.83501
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.27535e10 −0.529825
\(926\) 0 0
\(927\) 2.74294e10 + 3.90297e10i 1.13093 + 1.60922i
\(928\) 0 0
\(929\) 4.37904e10i 1.79194i 0.444113 + 0.895971i \(0.353519\pi\)
−0.444113 + 0.895971i \(0.646481\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.73232e10 3.33477e10i 0.698300 1.34425i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 3.31153e10 + 1.72024e10i 1.30527 + 0.678048i
\(940\) −1.64393e10 −0.645560
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 5.16346e10i 1.99774i
\(945\) 0 0
\(946\) 0 0
\(947\) 2.19836e10i 0.841149i 0.907258 + 0.420575i \(0.138172\pi\)
−0.907258 + 0.420575i \(0.861828\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −2.36384e10 + 4.55048e10i −0.891224 + 1.71564i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 2.91130e10 1.08162
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 1.06462e10 2.04943e10i 0.388369 0.747624i
\(961\) −2.59865e10 −0.944532
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.59079e10i 1.60924i −0.593793 0.804618i \(-0.702370\pi\)
0.593793 0.804618i \(-0.297630\pi\)
\(972\) −2.10252e10 + 1.94335e10i −0.734357 + 0.678763i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.82921e10i 1.99976i −0.0153784 0.999882i \(-0.504895\pi\)
0.0153784 0.999882i \(-0.495105\pi\)
\(978\) 0 0
\(979\) 5.53747e10 1.88613
\(980\) 2.48228e10i 0.842480i
\(981\) 0 0
\(982\) 0 0
\(983\) 5.56229e10i 1.86774i 0.357612 + 0.933870i \(0.383591\pi\)
−0.357612 + 0.933870i \(0.616409\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −4.71792e10 −1.53990 −0.769951 0.638103i \(-0.779719\pi\)
−0.769951 + 0.638103i \(0.779719\pi\)
\(992\) 0 0
\(993\) 1.97020e10 + 1.02346e10i 0.638539 + 0.331703i
\(994\) 0 0
\(995\) 3.63648e10i 1.17031i
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 7.65636e9 + 5.70154e10i 0.242965 + 1.80931i
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 33.8.d.a.32.1 2
3.2 odd 2 inner 33.8.d.a.32.2 yes 2
11.10 odd 2 CM 33.8.d.a.32.1 2
33.32 even 2 inner 33.8.d.a.32.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.8.d.a.32.1 2 1.1 even 1 trivial
33.8.d.a.32.1 2 11.10 odd 2 CM
33.8.d.a.32.2 yes 2 3.2 odd 2 inner
33.8.d.a.32.2 yes 2 33.32 even 2 inner