Properties

Label 8.22.a.a.1.2
Level $8$
Weight $22$
Character 8.1
Self dual yes
Analytic conductor $22.358$
Analytic rank $1$
Dimension $2$
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] = [8,22,Mod(1,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(22.3581875430\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{358549}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 89637 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-298.895\) of defining polynomial
Character \(\chi\) \(=\) 8.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+62251.6 q^{3} -1.24528e6 q^{5} -2.63007e8 q^{7} -6.58509e9 q^{9} +O(q^{10})\) \(q+62251.6 q^{3} -1.24528e6 q^{5} -2.63007e8 q^{7} -6.58509e9 q^{9} -3.37055e10 q^{11} +2.50845e11 q^{13} -7.75208e10 q^{15} +2.34664e11 q^{17} -3.24706e13 q^{19} -1.63726e13 q^{21} -9.63098e13 q^{23} -4.75286e14 q^{25} -1.06111e15 q^{27} -3.86879e15 q^{29} -2.26841e15 q^{31} -2.09822e15 q^{33} +3.27518e14 q^{35} +4.75366e15 q^{37} +1.56155e16 q^{39} +3.56442e16 q^{41} +2.20776e17 q^{43} +8.20030e15 q^{45} +4.79937e17 q^{47} -4.89373e17 q^{49} +1.46082e16 q^{51} -1.86349e18 q^{53} +4.19729e16 q^{55} -2.02135e18 q^{57} +1.23147e18 q^{59} -6.27167e18 q^{61} +1.73193e18 q^{63} -3.12373e17 q^{65} +1.72809e19 q^{67} -5.99544e18 q^{69} +3.13094e19 q^{71} +5.49389e19 q^{73} -2.95873e19 q^{75} +8.86480e18 q^{77} -8.34613e19 q^{79} +2.82681e18 q^{81} -1.22864e20 q^{83} -2.92223e17 q^{85} -2.40839e20 q^{87} +1.21225e20 q^{89} -6.59742e19 q^{91} -1.41212e20 q^{93} +4.04350e19 q^{95} +3.73762e20 q^{97} +2.21954e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 105432 q^{3} + 2108140 q^{5} + 444771792 q^{7} + 11072347578 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 105432 q^{3} + 2108140 q^{5} + 444771792 q^{7} + 11072347578 q^{9} + 53806403320 q^{11} - 490366676932 q^{13} - 639834721680 q^{15} - 6593864672092 q^{17} + 19302397925320 q^{19} - 135055584824256 q^{21} - 409737865776272 q^{23} - 940878149007650 q^{25} - 22\!\cdots\!56 q^{27}+ \cdots + 17\!\cdots\!04 q^{99}+O(q^{100}) \) 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
\(3\) 62251.6 0.608664 0.304332 0.952566i \(-0.401567\pi\)
0.304332 + 0.952566i \(0.401567\pi\)
\(4\) 0 0
\(5\) −1.24528e6 −0.0570273 −0.0285136 0.999593i \(-0.509077\pi\)
−0.0285136 + 0.999593i \(0.509077\pi\)
\(6\) 0 0
\(7\) −2.63007e8 −0.351916 −0.175958 0.984398i \(-0.556302\pi\)
−0.175958 + 0.984398i \(0.556302\pi\)
\(8\) 0 0
\(9\) −6.58509e9 −0.629529
\(10\) 0 0
\(11\) −3.37055e10 −0.391812 −0.195906 0.980623i \(-0.562765\pi\)
−0.195906 + 0.980623i \(0.562765\pi\)
\(12\) 0 0
\(13\) 2.50845e11 0.504662 0.252331 0.967641i \(-0.418803\pi\)
0.252331 + 0.967641i \(0.418803\pi\)
\(14\) 0 0
\(15\) −7.75208e10 −0.0347104
\(16\) 0 0
\(17\) 2.34664e11 0.0282314 0.0141157 0.999900i \(-0.495507\pi\)
0.0141157 + 0.999900i \(0.495507\pi\)
\(18\) 0 0
\(19\) −3.24706e13 −1.21500 −0.607501 0.794319i \(-0.707828\pi\)
−0.607501 + 0.794319i \(0.707828\pi\)
\(20\) 0 0
\(21\) −1.63726e13 −0.214198
\(22\) 0 0
\(23\) −9.63098e13 −0.484761 −0.242381 0.970181i \(-0.577928\pi\)
−0.242381 + 0.970181i \(0.577928\pi\)
\(24\) 0 0
\(25\) −4.75286e14 −0.996748
\(26\) 0 0
\(27\) −1.06111e15 −0.991835
\(28\) 0 0
\(29\) −3.86879e15 −1.70764 −0.853820 0.520568i \(-0.825720\pi\)
−0.853820 + 0.520568i \(0.825720\pi\)
\(30\) 0 0
\(31\) −2.26841e15 −0.497077 −0.248539 0.968622i \(-0.579950\pi\)
−0.248539 + 0.968622i \(0.579950\pi\)
\(32\) 0 0
\(33\) −2.09822e15 −0.238482
\(34\) 0 0
\(35\) 3.27518e14 0.0200688
\(36\) 0 0
\(37\) 4.75366e15 0.162521 0.0812606 0.996693i \(-0.474105\pi\)
0.0812606 + 0.996693i \(0.474105\pi\)
\(38\) 0 0
\(39\) 1.56155e16 0.307170
\(40\) 0 0
\(41\) 3.56442e16 0.414723 0.207361 0.978264i \(-0.433512\pi\)
0.207361 + 0.978264i \(0.433512\pi\)
\(42\) 0 0
\(43\) 2.20776e17 1.55788 0.778939 0.627100i \(-0.215758\pi\)
0.778939 + 0.627100i \(0.215758\pi\)
\(44\) 0 0
\(45\) 8.20030e15 0.0359003
\(46\) 0 0
\(47\) 4.79937e17 1.33093 0.665467 0.746427i \(-0.268232\pi\)
0.665467 + 0.746427i \(0.268232\pi\)
\(48\) 0 0
\(49\) −4.89373e17 −0.876155
\(50\) 0 0
\(51\) 1.46082e16 0.0171834
\(52\) 0 0
\(53\) −1.86349e18 −1.46362 −0.731812 0.681506i \(-0.761325\pi\)
−0.731812 + 0.681506i \(0.761325\pi\)
\(54\) 0 0
\(55\) 4.19729e16 0.0223440
\(56\) 0 0
\(57\) −2.02135e18 −0.739528
\(58\) 0 0
\(59\) 1.23147e18 0.313674 0.156837 0.987625i \(-0.449870\pi\)
0.156837 + 0.987625i \(0.449870\pi\)
\(60\) 0 0
\(61\) −6.27167e18 −1.12569 −0.562846 0.826562i \(-0.690294\pi\)
−0.562846 + 0.826562i \(0.690294\pi\)
\(62\) 0 0
\(63\) 1.73193e18 0.221541
\(64\) 0 0
\(65\) −3.12373e17 −0.0287795
\(66\) 0 0
\(67\) 1.72809e19 1.15820 0.579098 0.815258i \(-0.303405\pi\)
0.579098 + 0.815258i \(0.303405\pi\)
\(68\) 0 0
\(69\) −5.99544e18 −0.295057
\(70\) 0 0
\(71\) 3.13094e19 1.14146 0.570732 0.821137i \(-0.306660\pi\)
0.570732 + 0.821137i \(0.306660\pi\)
\(72\) 0 0
\(73\) 5.49389e19 1.49620 0.748100 0.663586i \(-0.230966\pi\)
0.748100 + 0.663586i \(0.230966\pi\)
\(74\) 0 0
\(75\) −2.95873e19 −0.606684
\(76\) 0 0
\(77\) 8.86480e18 0.137885
\(78\) 0 0
\(79\) −8.34613e19 −0.991747 −0.495873 0.868395i \(-0.665152\pi\)
−0.495873 + 0.868395i \(0.665152\pi\)
\(80\) 0 0
\(81\) 2.82681e18 0.0258347
\(82\) 0 0
\(83\) −1.22864e20 −0.869171 −0.434585 0.900631i \(-0.643105\pi\)
−0.434585 + 0.900631i \(0.643105\pi\)
\(84\) 0 0
\(85\) −2.92223e17 −0.00160996
\(86\) 0 0
\(87\) −2.40839e20 −1.03938
\(88\) 0 0
\(89\) 1.21225e20 0.412096 0.206048 0.978542i \(-0.433940\pi\)
0.206048 + 0.978542i \(0.433940\pi\)
\(90\) 0 0
\(91\) −6.59742e19 −0.177598
\(92\) 0 0
\(93\) −1.41212e20 −0.302553
\(94\) 0 0
\(95\) 4.04350e19 0.0692883
\(96\) 0 0
\(97\) 3.73762e20 0.514627 0.257313 0.966328i \(-0.417163\pi\)
0.257313 + 0.966328i \(0.417163\pi\)
\(98\) 0 0
\(99\) 2.21954e20 0.246657
\(100\) 0 0
\(101\) 1.12796e21 1.01606 0.508031 0.861339i \(-0.330374\pi\)
0.508031 + 0.861339i \(0.330374\pi\)
\(102\) 0 0
\(103\) 1.15466e21 0.846573 0.423287 0.905996i \(-0.360876\pi\)
0.423287 + 0.905996i \(0.360876\pi\)
\(104\) 0 0
\(105\) 2.03885e19 0.0122151
\(106\) 0 0
\(107\) 1.23814e21 0.608469 0.304235 0.952597i \(-0.401599\pi\)
0.304235 + 0.952597i \(0.401599\pi\)
\(108\) 0 0
\(109\) −1.33848e21 −0.541544 −0.270772 0.962643i \(-0.587279\pi\)
−0.270772 + 0.962643i \(0.587279\pi\)
\(110\) 0 0
\(111\) 2.95923e20 0.0989208
\(112\) 0 0
\(113\) −6.49910e21 −1.80107 −0.900533 0.434787i \(-0.856824\pi\)
−0.900533 + 0.434787i \(0.856824\pi\)
\(114\) 0 0
\(115\) 1.19933e20 0.0276446
\(116\) 0 0
\(117\) −1.65184e21 −0.317699
\(118\) 0 0
\(119\) −6.17183e19 −0.00993507
\(120\) 0 0
\(121\) −6.26419e21 −0.846483
\(122\) 0 0
\(123\) 2.21891e21 0.252427
\(124\) 0 0
\(125\) 1.18566e21 0.113869
\(126\) 0 0
\(127\) −5.02960e21 −0.408878 −0.204439 0.978879i \(-0.565537\pi\)
−0.204439 + 0.978879i \(0.565537\pi\)
\(128\) 0 0
\(129\) 1.37437e22 0.948223
\(130\) 0 0
\(131\) −2.11255e21 −0.124011 −0.0620053 0.998076i \(-0.519750\pi\)
−0.0620053 + 0.998076i \(0.519750\pi\)
\(132\) 0 0
\(133\) 8.54000e21 0.427578
\(134\) 0 0
\(135\) 1.32138e21 0.0565616
\(136\) 0 0
\(137\) 2.68746e22 0.985770 0.492885 0.870095i \(-0.335942\pi\)
0.492885 + 0.870095i \(0.335942\pi\)
\(138\) 0 0
\(139\) 9.03627e21 0.284665 0.142332 0.989819i \(-0.454540\pi\)
0.142332 + 0.989819i \(0.454540\pi\)
\(140\) 0 0
\(141\) 2.98768e22 0.810092
\(142\) 0 0
\(143\) −8.45488e21 −0.197733
\(144\) 0 0
\(145\) 4.81774e21 0.0973821
\(146\) 0 0
\(147\) −3.04643e22 −0.533284
\(148\) 0 0
\(149\) −5.86168e22 −0.890362 −0.445181 0.895441i \(-0.646861\pi\)
−0.445181 + 0.895441i \(0.646861\pi\)
\(150\) 0 0
\(151\) −8.08836e22 −1.06808 −0.534038 0.845460i \(-0.679326\pi\)
−0.534038 + 0.845460i \(0.679326\pi\)
\(152\) 0 0
\(153\) −1.54528e21 −0.0177725
\(154\) 0 0
\(155\) 2.82481e21 0.0283470
\(156\) 0 0
\(157\) −5.42012e22 −0.475404 −0.237702 0.971338i \(-0.576394\pi\)
−0.237702 + 0.971338i \(0.576394\pi\)
\(158\) 0 0
\(159\) −1.16005e23 −0.890855
\(160\) 0 0
\(161\) 2.53302e22 0.170595
\(162\) 0 0
\(163\) 1.61447e23 0.955126 0.477563 0.878597i \(-0.341520\pi\)
0.477563 + 0.878597i \(0.341520\pi\)
\(164\) 0 0
\(165\) 2.61288e21 0.0136000
\(166\) 0 0
\(167\) 2.19835e23 1.00826 0.504132 0.863627i \(-0.331812\pi\)
0.504132 + 0.863627i \(0.331812\pi\)
\(168\) 0 0
\(169\) −1.84141e23 −0.745316
\(170\) 0 0
\(171\) 2.13822e23 0.764879
\(172\) 0 0
\(173\) −3.84695e22 −0.121796 −0.0608979 0.998144i \(-0.519396\pi\)
−0.0608979 + 0.998144i \(0.519396\pi\)
\(174\) 0 0
\(175\) 1.25004e23 0.350771
\(176\) 0 0
\(177\) 7.66611e22 0.190922
\(178\) 0 0
\(179\) 6.03667e23 1.33610 0.668052 0.744115i \(-0.267128\pi\)
0.668052 + 0.744115i \(0.267128\pi\)
\(180\) 0 0
\(181\) −5.64803e21 −0.0111243 −0.00556214 0.999985i \(-0.501770\pi\)
−0.00556214 + 0.999985i \(0.501770\pi\)
\(182\) 0 0
\(183\) −3.90421e23 −0.685168
\(184\) 0 0
\(185\) −5.91965e21 −0.00926815
\(186\) 0 0
\(187\) −7.90947e21 −0.0110614
\(188\) 0 0
\(189\) 2.79079e23 0.349042
\(190\) 0 0
\(191\) −1.34983e24 −1.51157 −0.755783 0.654822i \(-0.772744\pi\)
−0.755783 + 0.654822i \(0.772744\pi\)
\(192\) 0 0
\(193\) −1.12919e24 −1.13348 −0.566741 0.823896i \(-0.691796\pi\)
−0.566741 + 0.823896i \(0.691796\pi\)
\(194\) 0 0
\(195\) −1.94457e22 −0.0175170
\(196\) 0 0
\(197\) 5.95420e23 0.481868 0.240934 0.970541i \(-0.422546\pi\)
0.240934 + 0.970541i \(0.422546\pi\)
\(198\) 0 0
\(199\) −2.68672e24 −1.95554 −0.977768 0.209690i \(-0.932755\pi\)
−0.977768 + 0.209690i \(0.932755\pi\)
\(200\) 0 0
\(201\) 1.07577e24 0.704952
\(202\) 0 0
\(203\) 1.01752e24 0.600945
\(204\) 0 0
\(205\) −4.43870e22 −0.0236505
\(206\) 0 0
\(207\) 6.34208e23 0.305171
\(208\) 0 0
\(209\) 1.09444e24 0.476053
\(210\) 0 0
\(211\) 1.31634e24 0.518085 0.259043 0.965866i \(-0.416593\pi\)
0.259043 + 0.965866i \(0.416593\pi\)
\(212\) 0 0
\(213\) 1.94906e24 0.694767
\(214\) 0 0
\(215\) −2.74928e23 −0.0888415
\(216\) 0 0
\(217\) 5.96609e23 0.174929
\(218\) 0 0
\(219\) 3.42003e24 0.910683
\(220\) 0 0
\(221\) 5.88643e22 0.0142473
\(222\) 0 0
\(223\) −7.81247e24 −1.72023 −0.860116 0.510098i \(-0.829609\pi\)
−0.860116 + 0.510098i \(0.829609\pi\)
\(224\) 0 0
\(225\) 3.12980e24 0.627481
\(226\) 0 0
\(227\) 9.76737e24 1.78446 0.892228 0.451584i \(-0.149141\pi\)
0.892228 + 0.451584i \(0.149141\pi\)
\(228\) 0 0
\(229\) −9.30771e24 −1.55085 −0.775426 0.631439i \(-0.782465\pi\)
−0.775426 + 0.631439i \(0.782465\pi\)
\(230\) 0 0
\(231\) 5.51848e23 0.0839255
\(232\) 0 0
\(233\) −1.31126e24 −0.182159 −0.0910797 0.995844i \(-0.529032\pi\)
−0.0910797 + 0.995844i \(0.529032\pi\)
\(234\) 0 0
\(235\) −5.97657e23 −0.0758996
\(236\) 0 0
\(237\) −5.19560e24 −0.603640
\(238\) 0 0
\(239\) −8.42576e24 −0.896254 −0.448127 0.893970i \(-0.647909\pi\)
−0.448127 + 0.893970i \(0.647909\pi\)
\(240\) 0 0
\(241\) −1.44027e25 −1.40367 −0.701834 0.712341i \(-0.747635\pi\)
−0.701834 + 0.712341i \(0.747635\pi\)
\(242\) 0 0
\(243\) 1.12755e25 1.00756
\(244\) 0 0
\(245\) 6.09407e23 0.0499648
\(246\) 0 0
\(247\) −8.14509e24 −0.613166
\(248\) 0 0
\(249\) −7.64849e24 −0.529033
\(250\) 0 0
\(251\) 1.42742e25 0.907772 0.453886 0.891060i \(-0.350037\pi\)
0.453886 + 0.891060i \(0.350037\pi\)
\(252\) 0 0
\(253\) 3.24617e24 0.189935
\(254\) 0 0
\(255\) −1.81913e22 −0.000979924 0
\(256\) 0 0
\(257\) 3.27132e25 1.62340 0.811700 0.584075i \(-0.198543\pi\)
0.811700 + 0.584075i \(0.198543\pi\)
\(258\) 0 0
\(259\) −1.25025e24 −0.0571938
\(260\) 0 0
\(261\) 2.54763e25 1.07501
\(262\) 0 0
\(263\) −1.57524e24 −0.0613494 −0.0306747 0.999529i \(-0.509766\pi\)
−0.0306747 + 0.999529i \(0.509766\pi\)
\(264\) 0 0
\(265\) 2.32057e24 0.0834665
\(266\) 0 0
\(267\) 7.54647e24 0.250828
\(268\) 0 0
\(269\) 2.47417e25 0.760378 0.380189 0.924909i \(-0.375859\pi\)
0.380189 + 0.924909i \(0.375859\pi\)
\(270\) 0 0
\(271\) −6.32727e25 −1.79903 −0.899515 0.436890i \(-0.856080\pi\)
−0.899515 + 0.436890i \(0.856080\pi\)
\(272\) 0 0
\(273\) −4.10700e24 −0.108098
\(274\) 0 0
\(275\) 1.60198e25 0.390538
\(276\) 0 0
\(277\) −1.52097e25 −0.343624 −0.171812 0.985130i \(-0.554962\pi\)
−0.171812 + 0.985130i \(0.554962\pi\)
\(278\) 0 0
\(279\) 1.49377e25 0.312924
\(280\) 0 0
\(281\) −4.87649e24 −0.0947743 −0.0473872 0.998877i \(-0.515089\pi\)
−0.0473872 + 0.998877i \(0.515089\pi\)
\(282\) 0 0
\(283\) −6.39728e25 −1.15408 −0.577042 0.816714i \(-0.695793\pi\)
−0.577042 + 0.816714i \(0.695793\pi\)
\(284\) 0 0
\(285\) 2.51714e24 0.0421733
\(286\) 0 0
\(287\) −9.37468e24 −0.145947
\(288\) 0 0
\(289\) −6.90369e25 −0.999203
\(290\) 0 0
\(291\) 2.32673e25 0.313234
\(292\) 0 0
\(293\) −2.68070e25 −0.335845 −0.167922 0.985800i \(-0.553706\pi\)
−0.167922 + 0.985800i \(0.553706\pi\)
\(294\) 0 0
\(295\) −1.53353e24 −0.0178880
\(296\) 0 0
\(297\) 3.57652e25 0.388613
\(298\) 0 0
\(299\) −2.41589e25 −0.244641
\(300\) 0 0
\(301\) −5.80657e25 −0.548241
\(302\) 0 0
\(303\) 7.02175e25 0.618440
\(304\) 0 0
\(305\) 7.81000e24 0.0641952
\(306\) 0 0
\(307\) 5.13119e25 0.393791 0.196895 0.980425i \(-0.436914\pi\)
0.196895 + 0.980425i \(0.436914\pi\)
\(308\) 0 0
\(309\) 7.18796e25 0.515278
\(310\) 0 0
\(311\) 2.73650e26 1.83321 0.916603 0.399798i \(-0.130920\pi\)
0.916603 + 0.399798i \(0.130920\pi\)
\(312\) 0 0
\(313\) 4.66082e25 0.291909 0.145954 0.989291i \(-0.453375\pi\)
0.145954 + 0.989291i \(0.453375\pi\)
\(314\) 0 0
\(315\) −2.15674e24 −0.0126339
\(316\) 0 0
\(317\) −1.24217e26 −0.680860 −0.340430 0.940270i \(-0.610573\pi\)
−0.340430 + 0.940270i \(0.610573\pi\)
\(318\) 0 0
\(319\) 1.30400e26 0.669074
\(320\) 0 0
\(321\) 7.70760e25 0.370353
\(322\) 0 0
\(323\) −7.61966e24 −0.0343012
\(324\) 0 0
\(325\) −1.19223e26 −0.503021
\(326\) 0 0
\(327\) −8.33226e25 −0.329618
\(328\) 0 0
\(329\) −1.26227e26 −0.468377
\(330\) 0 0
\(331\) −3.57554e26 −1.24494 −0.622469 0.782644i \(-0.713870\pi\)
−0.622469 + 0.782644i \(0.713870\pi\)
\(332\) 0 0
\(333\) −3.13033e25 −0.102312
\(334\) 0 0
\(335\) −2.15196e25 −0.0660488
\(336\) 0 0
\(337\) 5.04047e25 0.145331 0.0726653 0.997356i \(-0.476850\pi\)
0.0726653 + 0.997356i \(0.476850\pi\)
\(338\) 0 0
\(339\) −4.04580e26 −1.09624
\(340\) 0 0
\(341\) 7.64580e25 0.194761
\(342\) 0 0
\(343\) 2.75610e26 0.660248
\(344\) 0 0
\(345\) 7.46601e24 0.0168263
\(346\) 0 0
\(347\) −7.30981e26 −1.55041 −0.775205 0.631709i \(-0.782354\pi\)
−0.775205 + 0.631709i \(0.782354\pi\)
\(348\) 0 0
\(349\) 6.72114e26 1.34207 0.671036 0.741425i \(-0.265850\pi\)
0.671036 + 0.741425i \(0.265850\pi\)
\(350\) 0 0
\(351\) −2.66174e26 −0.500542
\(352\) 0 0
\(353\) 5.30749e25 0.0940275 0.0470138 0.998894i \(-0.485030\pi\)
0.0470138 + 0.998894i \(0.485030\pi\)
\(354\) 0 0
\(355\) −3.89890e25 −0.0650945
\(356\) 0 0
\(357\) −3.84206e24 −0.00604711
\(358\) 0 0
\(359\) 8.31857e26 1.23469 0.617343 0.786694i \(-0.288209\pi\)
0.617343 + 0.786694i \(0.288209\pi\)
\(360\) 0 0
\(361\) 3.40128e26 0.476231
\(362\) 0 0
\(363\) −3.89956e26 −0.515224
\(364\) 0 0
\(365\) −6.84144e25 −0.0853243
\(366\) 0 0
\(367\) 1.02517e27 1.20727 0.603635 0.797261i \(-0.293719\pi\)
0.603635 + 0.797261i \(0.293719\pi\)
\(368\) 0 0
\(369\) −2.34720e26 −0.261080
\(370\) 0 0
\(371\) 4.90111e26 0.515072
\(372\) 0 0
\(373\) 1.01508e27 1.00822 0.504112 0.863638i \(-0.331820\pi\)
0.504112 + 0.863638i \(0.331820\pi\)
\(374\) 0 0
\(375\) 7.38094e25 0.0693080
\(376\) 0 0
\(377\) −9.70468e26 −0.861782
\(378\) 0 0
\(379\) 9.76777e26 0.820510 0.410255 0.911971i \(-0.365440\pi\)
0.410255 + 0.911971i \(0.365440\pi\)
\(380\) 0 0
\(381\) −3.13100e26 −0.248869
\(382\) 0 0
\(383\) −3.96291e26 −0.298145 −0.149072 0.988826i \(-0.547629\pi\)
−0.149072 + 0.988826i \(0.547629\pi\)
\(384\) 0 0
\(385\) −1.10392e25 −0.00786320
\(386\) 0 0
\(387\) −1.45383e27 −0.980728
\(388\) 0 0
\(389\) −2.89702e27 −1.85132 −0.925658 0.378361i \(-0.876488\pi\)
−0.925658 + 0.378361i \(0.876488\pi\)
\(390\) 0 0
\(391\) −2.26004e25 −0.0136855
\(392\) 0 0
\(393\) −1.31510e26 −0.0754807
\(394\) 0 0
\(395\) 1.03933e26 0.0565566
\(396\) 0 0
\(397\) −5.79178e26 −0.298890 −0.149445 0.988770i \(-0.547749\pi\)
−0.149445 + 0.988770i \(0.547749\pi\)
\(398\) 0 0
\(399\) 5.31629e26 0.260251
\(400\) 0 0
\(401\) −1.05664e27 −0.490807 −0.245403 0.969421i \(-0.578920\pi\)
−0.245403 + 0.969421i \(0.578920\pi\)
\(402\) 0 0
\(403\) −5.69020e26 −0.250856
\(404\) 0 0
\(405\) −3.52017e24 −0.00147328
\(406\) 0 0
\(407\) −1.60225e26 −0.0636778
\(408\) 0 0
\(409\) 2.30402e27 0.869746 0.434873 0.900492i \(-0.356793\pi\)
0.434873 + 0.900492i \(0.356793\pi\)
\(410\) 0 0
\(411\) 1.67299e27 0.600002
\(412\) 0 0
\(413\) −3.23886e26 −0.110387
\(414\) 0 0
\(415\) 1.53000e26 0.0495664
\(416\) 0 0
\(417\) 5.62522e26 0.173265
\(418\) 0 0
\(419\) 2.38736e27 0.699311 0.349655 0.936878i \(-0.386299\pi\)
0.349655 + 0.936878i \(0.386299\pi\)
\(420\) 0 0
\(421\) 4.44803e27 1.23938 0.619691 0.784846i \(-0.287258\pi\)
0.619691 + 0.784846i \(0.287258\pi\)
\(422\) 0 0
\(423\) −3.16043e27 −0.837861
\(424\) 0 0
\(425\) −1.11532e26 −0.0281396
\(426\) 0 0
\(427\) 1.64949e27 0.396149
\(428\) 0 0
\(429\) −5.26330e26 −0.120353
\(430\) 0 0
\(431\) −2.79963e27 −0.609662 −0.304831 0.952406i \(-0.598600\pi\)
−0.304831 + 0.952406i \(0.598600\pi\)
\(432\) 0 0
\(433\) 6.23965e27 1.29431 0.647154 0.762360i \(-0.275959\pi\)
0.647154 + 0.762360i \(0.275959\pi\)
\(434\) 0 0
\(435\) 2.99912e26 0.0592729
\(436\) 0 0
\(437\) 3.12723e27 0.588986
\(438\) 0 0
\(439\) −7.65214e27 −1.37374 −0.686871 0.726779i \(-0.741016\pi\)
−0.686871 + 0.726779i \(0.741016\pi\)
\(440\) 0 0
\(441\) 3.22257e27 0.551565
\(442\) 0 0
\(443\) 2.50862e27 0.409446 0.204723 0.978820i \(-0.434371\pi\)
0.204723 + 0.978820i \(0.434371\pi\)
\(444\) 0 0
\(445\) −1.50960e26 −0.0235007
\(446\) 0 0
\(447\) −3.64899e27 −0.541931
\(448\) 0 0
\(449\) −9.77491e27 −1.38524 −0.692622 0.721301i \(-0.743544\pi\)
−0.692622 + 0.721301i \(0.743544\pi\)
\(450\) 0 0
\(451\) −1.20141e27 −0.162493
\(452\) 0 0
\(453\) −5.03513e27 −0.650099
\(454\) 0 0
\(455\) 8.21564e25 0.0101280
\(456\) 0 0
\(457\) −6.63390e27 −0.780996 −0.390498 0.920604i \(-0.627697\pi\)
−0.390498 + 0.920604i \(0.627697\pi\)
\(458\) 0 0
\(459\) −2.49003e26 −0.0280009
\(460\) 0 0
\(461\) −1.50903e27 −0.162121 −0.0810605 0.996709i \(-0.525831\pi\)
−0.0810605 + 0.996709i \(0.525831\pi\)
\(462\) 0 0
\(463\) −5.89041e27 −0.604707 −0.302354 0.953196i \(-0.597772\pi\)
−0.302354 + 0.953196i \(0.597772\pi\)
\(464\) 0 0
\(465\) 1.75849e26 0.0172538
\(466\) 0 0
\(467\) 1.38550e28 1.29951 0.649756 0.760142i \(-0.274871\pi\)
0.649756 + 0.760142i \(0.274871\pi\)
\(468\) 0 0
\(469\) −4.54501e27 −0.407587
\(470\) 0 0
\(471\) −3.37411e27 −0.289361
\(472\) 0 0
\(473\) −7.44138e27 −0.610395
\(474\) 0 0
\(475\) 1.54328e28 1.21105
\(476\) 0 0
\(477\) 1.22712e28 0.921393
\(478\) 0 0
\(479\) 2.10471e28 1.51241 0.756204 0.654336i \(-0.227052\pi\)
0.756204 + 0.654336i \(0.227052\pi\)
\(480\) 0 0
\(481\) 1.19243e27 0.0820183
\(482\) 0 0
\(483\) 1.57684e27 0.103835
\(484\) 0 0
\(485\) −4.65439e26 −0.0293478
\(486\) 0 0
\(487\) −1.93093e28 −1.16604 −0.583018 0.812459i \(-0.698128\pi\)
−0.583018 + 0.812459i \(0.698128\pi\)
\(488\) 0 0
\(489\) 1.00504e28 0.581351
\(490\) 0 0
\(491\) 2.38236e28 1.32023 0.660117 0.751163i \(-0.270507\pi\)
0.660117 + 0.751163i \(0.270507\pi\)
\(492\) 0 0
\(493\) −9.07865e26 −0.0482091
\(494\) 0 0
\(495\) −2.76395e26 −0.0140662
\(496\) 0 0
\(497\) −8.23460e27 −0.401699
\(498\) 0 0
\(499\) −2.22172e28 −1.03905 −0.519523 0.854457i \(-0.673890\pi\)
−0.519523 + 0.854457i \(0.673890\pi\)
\(500\) 0 0
\(501\) 1.36851e28 0.613694
\(502\) 0 0
\(503\) 8.37368e27 0.360124 0.180062 0.983655i \(-0.442370\pi\)
0.180062 + 0.983655i \(0.442370\pi\)
\(504\) 0 0
\(505\) −1.40463e27 −0.0579433
\(506\) 0 0
\(507\) −1.14631e28 −0.453647
\(508\) 0 0
\(509\) −3.38122e28 −1.28392 −0.641958 0.766740i \(-0.721878\pi\)
−0.641958 + 0.766740i \(0.721878\pi\)
\(510\) 0 0
\(511\) −1.44493e28 −0.526536
\(512\) 0 0
\(513\) 3.44547e28 1.20508
\(514\) 0 0
\(515\) −1.43788e27 −0.0482778
\(516\) 0 0
\(517\) −1.61765e28 −0.521477
\(518\) 0 0
\(519\) −2.39479e27 −0.0741327
\(520\) 0 0
\(521\) 8.19578e27 0.243665 0.121833 0.992551i \(-0.461123\pi\)
0.121833 + 0.992551i \(0.461123\pi\)
\(522\) 0 0
\(523\) −6.28168e28 −1.79394 −0.896970 0.442092i \(-0.854236\pi\)
−0.896970 + 0.442092i \(0.854236\pi\)
\(524\) 0 0
\(525\) 7.78169e27 0.213502
\(526\) 0 0
\(527\) −5.32314e26 −0.0140332
\(528\) 0 0
\(529\) −3.01960e28 −0.765006
\(530\) 0 0
\(531\) −8.10935e27 −0.197467
\(532\) 0 0
\(533\) 8.94117e27 0.209295
\(534\) 0 0
\(535\) −1.54183e27 −0.0346994
\(536\) 0 0
\(537\) 3.75792e28 0.813238
\(538\) 0 0
\(539\) 1.64946e28 0.343288
\(540\) 0 0
\(541\) 4.40803e28 0.882416 0.441208 0.897405i \(-0.354550\pi\)
0.441208 + 0.897405i \(0.354550\pi\)
\(542\) 0 0
\(543\) −3.51599e26 −0.00677094
\(544\) 0 0
\(545\) 1.66679e27 0.0308828
\(546\) 0 0
\(547\) −3.93832e28 −0.702174 −0.351087 0.936343i \(-0.614188\pi\)
−0.351087 + 0.936343i \(0.614188\pi\)
\(548\) 0 0
\(549\) 4.12995e28 0.708656
\(550\) 0 0
\(551\) 1.25622e29 2.07479
\(552\) 0 0
\(553\) 2.19509e28 0.349011
\(554\) 0 0
\(555\) −3.68508e26 −0.00564118
\(556\) 0 0
\(557\) −5.79761e28 −0.854612 −0.427306 0.904107i \(-0.640537\pi\)
−0.427306 + 0.904107i \(0.640537\pi\)
\(558\) 0 0
\(559\) 5.53806e28 0.786202
\(560\) 0 0
\(561\) −4.92377e26 −0.00673268
\(562\) 0 0
\(563\) 3.32225e28 0.437618 0.218809 0.975768i \(-0.429783\pi\)
0.218809 + 0.975768i \(0.429783\pi\)
\(564\) 0 0
\(565\) 8.09322e27 0.102710
\(566\) 0 0
\(567\) −7.43471e26 −0.00909163
\(568\) 0 0
\(569\) −1.08751e29 −1.28160 −0.640801 0.767707i \(-0.721398\pi\)
−0.640801 + 0.767707i \(0.721398\pi\)
\(570\) 0 0
\(571\) −1.31209e28 −0.149033 −0.0745166 0.997220i \(-0.523741\pi\)
−0.0745166 + 0.997220i \(0.523741\pi\)
\(572\) 0 0
\(573\) −8.40288e28 −0.920036
\(574\) 0 0
\(575\) 4.57747e28 0.483185
\(576\) 0 0
\(577\) −1.17664e29 −1.19756 −0.598778 0.800915i \(-0.704347\pi\)
−0.598778 + 0.800915i \(0.704347\pi\)
\(578\) 0 0
\(579\) −7.02938e28 −0.689910
\(580\) 0 0
\(581\) 3.23142e28 0.305875
\(582\) 0 0
\(583\) 6.28098e28 0.573466
\(584\) 0 0
\(585\) 2.05701e27 0.0181175
\(586\) 0 0
\(587\) −1.41639e29 −1.20360 −0.601799 0.798647i \(-0.705549\pi\)
−0.601799 + 0.798647i \(0.705549\pi\)
\(588\) 0 0
\(589\) 7.36566e28 0.603950
\(590\) 0 0
\(591\) 3.70659e28 0.293296
\(592\) 0 0
\(593\) −6.70941e28 −0.512401 −0.256201 0.966624i \(-0.582471\pi\)
−0.256201 + 0.966624i \(0.582471\pi\)
\(594\) 0 0
\(595\) 7.68567e25 0.000566570 0
\(596\) 0 0
\(597\) −1.67253e29 −1.19026
\(598\) 0 0
\(599\) 2.42035e28 0.166302 0.0831510 0.996537i \(-0.473502\pi\)
0.0831510 + 0.996537i \(0.473502\pi\)
\(600\) 0 0
\(601\) 2.08024e29 1.38016 0.690082 0.723731i \(-0.257575\pi\)
0.690082 + 0.723731i \(0.257575\pi\)
\(602\) 0 0
\(603\) −1.13797e29 −0.729117
\(604\) 0 0
\(605\) 7.80068e27 0.0482726
\(606\) 0 0
\(607\) −1.15837e29 −0.692417 −0.346208 0.938158i \(-0.612531\pi\)
−0.346208 + 0.938158i \(0.612531\pi\)
\(608\) 0 0
\(609\) 6.33423e28 0.365773
\(610\) 0 0
\(611\) 1.20390e29 0.671672
\(612\) 0 0
\(613\) 2.36853e29 1.27686 0.638430 0.769680i \(-0.279584\pi\)
0.638430 + 0.769680i \(0.279584\pi\)
\(614\) 0 0
\(615\) −2.76316e27 −0.0143952
\(616\) 0 0
\(617\) −4.18061e28 −0.210497 −0.105248 0.994446i \(-0.533564\pi\)
−0.105248 + 0.994446i \(0.533564\pi\)
\(618\) 0 0
\(619\) −4.89038e27 −0.0238008 −0.0119004 0.999929i \(-0.503788\pi\)
−0.0119004 + 0.999929i \(0.503788\pi\)
\(620\) 0 0
\(621\) 1.02195e29 0.480803
\(622\) 0 0
\(623\) −3.18831e28 −0.145023
\(624\) 0 0
\(625\) 2.25158e29 0.990254
\(626\) 0 0
\(627\) 6.81305e28 0.289756
\(628\) 0 0
\(629\) 1.11551e27 0.00458820
\(630\) 0 0
\(631\) 2.10752e29 0.838424 0.419212 0.907888i \(-0.362306\pi\)
0.419212 + 0.907888i \(0.362306\pi\)
\(632\) 0 0
\(633\) 8.19441e28 0.315340
\(634\) 0 0
\(635\) 6.26327e27 0.0233172
\(636\) 0 0
\(637\) −1.22757e29 −0.442163
\(638\) 0 0
\(639\) −2.06175e29 −0.718584
\(640\) 0 0
\(641\) 6.89760e28 0.232642 0.116321 0.993212i \(-0.462890\pi\)
0.116321 + 0.993212i \(0.462890\pi\)
\(642\) 0 0
\(643\) 8.28400e28 0.270411 0.135206 0.990818i \(-0.456830\pi\)
0.135206 + 0.990818i \(0.456830\pi\)
\(644\) 0 0
\(645\) −1.71147e28 −0.0540746
\(646\) 0 0
\(647\) −5.69599e29 −1.74211 −0.871053 0.491189i \(-0.836562\pi\)
−0.871053 + 0.491189i \(0.836562\pi\)
\(648\) 0 0
\(649\) −4.15074e28 −0.122901
\(650\) 0 0
\(651\) 3.71398e28 0.106473
\(652\) 0 0
\(653\) −3.86586e29 −1.07314 −0.536571 0.843855i \(-0.680281\pi\)
−0.536571 + 0.843855i \(0.680281\pi\)
\(654\) 0 0
\(655\) 2.63072e27 0.00707199
\(656\) 0 0
\(657\) −3.61778e29 −0.941901
\(658\) 0 0
\(659\) −3.00496e29 −0.757777 −0.378889 0.925442i \(-0.623694\pi\)
−0.378889 + 0.925442i \(0.623694\pi\)
\(660\) 0 0
\(661\) 2.82911e29 0.691089 0.345545 0.938402i \(-0.387694\pi\)
0.345545 + 0.938402i \(0.387694\pi\)
\(662\) 0 0
\(663\) 3.66440e27 0.00867182
\(664\) 0 0
\(665\) −1.06347e28 −0.0243836
\(666\) 0 0
\(667\) 3.72602e29 0.827798
\(668\) 0 0
\(669\) −4.86339e29 −1.04704
\(670\) 0 0
\(671\) 2.11390e29 0.441060
\(672\) 0 0
\(673\) −4.94836e29 −1.00070 −0.500349 0.865824i \(-0.666795\pi\)
−0.500349 + 0.865824i \(0.666795\pi\)
\(674\) 0 0
\(675\) 5.04329e29 0.988609
\(676\) 0 0
\(677\) 9.03975e28 0.171781 0.0858906 0.996305i \(-0.472626\pi\)
0.0858906 + 0.996305i \(0.472626\pi\)
\(678\) 0 0
\(679\) −9.83022e28 −0.181105
\(680\) 0 0
\(681\) 6.08034e29 1.08613
\(682\) 0 0
\(683\) 6.74953e29 1.16911 0.584556 0.811354i \(-0.301269\pi\)
0.584556 + 0.811354i \(0.301269\pi\)
\(684\) 0 0
\(685\) −3.34664e28 −0.0562158
\(686\) 0 0
\(687\) −5.79420e29 −0.943947
\(688\) 0 0
\(689\) −4.67447e29 −0.738636
\(690\) 0 0
\(691\) 1.18142e30 1.81087 0.905433 0.424489i \(-0.139546\pi\)
0.905433 + 0.424489i \(0.139546\pi\)
\(692\) 0 0
\(693\) −5.83755e28 −0.0868024
\(694\) 0 0
\(695\) −1.12527e28 −0.0162337
\(696\) 0 0
\(697\) 8.36439e27 0.0117082
\(698\) 0 0
\(699\) −8.16280e28 −0.110874
\(700\) 0 0
\(701\) 8.90277e29 1.17351 0.586754 0.809766i \(-0.300406\pi\)
0.586754 + 0.809766i \(0.300406\pi\)
\(702\) 0 0
\(703\) −1.54354e29 −0.197464
\(704\) 0 0
\(705\) −3.72051e28 −0.0461973
\(706\) 0 0
\(707\) −2.96662e29 −0.357568
\(708\) 0 0
\(709\) 4.82848e29 0.564970 0.282485 0.959272i \(-0.408841\pi\)
0.282485 + 0.959272i \(0.408841\pi\)
\(710\) 0 0
\(711\) 5.49600e29 0.624333
\(712\) 0 0
\(713\) 2.18470e29 0.240964
\(714\) 0 0
\(715\) 1.05287e28 0.0112762
\(716\) 0 0
\(717\) −5.24517e29 −0.545517
\(718\) 0 0
\(719\) −9.70706e29 −0.980470 −0.490235 0.871590i \(-0.663089\pi\)
−0.490235 + 0.871590i \(0.663089\pi\)
\(720\) 0 0
\(721\) −3.03685e29 −0.297922
\(722\) 0 0
\(723\) −8.96589e29 −0.854361
\(724\) 0 0
\(725\) 1.83878e30 1.70209
\(726\) 0 0
\(727\) −1.65557e30 −1.48880 −0.744401 0.667733i \(-0.767265\pi\)
−0.744401 + 0.667733i \(0.767265\pi\)
\(728\) 0 0
\(729\) 6.72350e29 0.587430
\(730\) 0 0
\(731\) 5.18081e28 0.0439810
\(732\) 0 0
\(733\) −1.70639e30 −1.40762 −0.703811 0.710387i \(-0.748520\pi\)
−0.703811 + 0.710387i \(0.748520\pi\)
\(734\) 0 0
\(735\) 3.79366e28 0.0304117
\(736\) 0 0
\(737\) −5.82463e29 −0.453795
\(738\) 0 0
\(739\) −1.51635e30 −1.14824 −0.574118 0.818772i \(-0.694655\pi\)
−0.574118 + 0.818772i \(0.694655\pi\)
\(740\) 0 0
\(741\) −5.07045e29 −0.373212
\(742\) 0 0
\(743\) −1.01870e30 −0.728890 −0.364445 0.931225i \(-0.618741\pi\)
−0.364445 + 0.931225i \(0.618741\pi\)
\(744\) 0 0
\(745\) 7.29945e28 0.0507749
\(746\) 0 0
\(747\) 8.09071e29 0.547168
\(748\) 0 0
\(749\) −3.25639e29 −0.214130
\(750\) 0 0
\(751\) 2.00722e30 1.28344 0.641721 0.766938i \(-0.278221\pi\)
0.641721 + 0.766938i \(0.278221\pi\)
\(752\) 0 0
\(753\) 8.88590e29 0.552528
\(754\) 0 0
\(755\) 1.00723e29 0.0609095
\(756\) 0 0
\(757\) −6.30787e29 −0.371002 −0.185501 0.982644i \(-0.559391\pi\)
−0.185501 + 0.982644i \(0.559391\pi\)
\(758\) 0 0
\(759\) 2.02079e29 0.115607
\(760\) 0 0
\(761\) 2.91644e30 1.62299 0.811493 0.584363i \(-0.198655\pi\)
0.811493 + 0.584363i \(0.198655\pi\)
\(762\) 0 0
\(763\) 3.52030e29 0.190578
\(764\) 0 0
\(765\) 1.92431e27 0.00101352
\(766\) 0 0
\(767\) 3.08909e29 0.158299
\(768\) 0 0
\(769\) −2.22144e30 −1.10766 −0.553831 0.832629i \(-0.686835\pi\)
−0.553831 + 0.832629i \(0.686835\pi\)
\(770\) 0 0
\(771\) 2.03645e30 0.988104
\(772\) 0 0
\(773\) 1.99610e30 0.942535 0.471267 0.881990i \(-0.343797\pi\)
0.471267 + 0.881990i \(0.343797\pi\)
\(774\) 0 0
\(775\) 1.07814e30 0.495461
\(776\) 0 0
\(777\) −7.78299e28 −0.0348118
\(778\) 0 0
\(779\) −1.15739e30 −0.503889
\(780\) 0 0
\(781\) −1.05530e30 −0.447239
\(782\) 0 0
\(783\) 4.10520e30 1.69370
\(784\) 0 0
\(785\) 6.74958e28 0.0271110
\(786\) 0 0
\(787\) −3.43376e30 −1.34287 −0.671437 0.741062i \(-0.734323\pi\)
−0.671437 + 0.741062i \(0.734323\pi\)
\(788\) 0 0
\(789\) −9.80610e28 −0.0373412
\(790\) 0 0
\(791\) 1.70931e30 0.633823
\(792\) 0 0
\(793\) −1.57322e30 −0.568095
\(794\) 0 0
\(795\) 1.44459e29 0.0508030
\(796\) 0 0
\(797\) −2.08927e30 −0.715620 −0.357810 0.933794i \(-0.616476\pi\)
−0.357810 + 0.933794i \(0.616476\pi\)
\(798\) 0 0
\(799\) 1.12624e29 0.0375741
\(800\) 0 0
\(801\) −7.98280e29 −0.259426
\(802\) 0 0
\(803\) −1.85174e30 −0.586230
\(804\) 0 0
\(805\) −3.15432e28 −0.00972857
\(806\) 0 0
\(807\) 1.54021e30 0.462815
\(808\) 0 0
\(809\) 1.63457e30 0.478567 0.239284 0.970950i \(-0.423087\pi\)
0.239284 + 0.970950i \(0.423087\pi\)
\(810\) 0 0
\(811\) −6.21829e28 −0.0177399 −0.00886997 0.999961i \(-0.502823\pi\)
−0.00886997 + 0.999961i \(0.502823\pi\)
\(812\) 0 0
\(813\) −3.93882e30 −1.09500
\(814\) 0 0
\(815\) −2.01047e29 −0.0544683
\(816\) 0 0
\(817\) −7.16872e30 −1.89282
\(818\) 0 0
\(819\) 4.34446e29 0.111803
\(820\) 0 0
\(821\) 5.54995e30 1.39215 0.696074 0.717970i \(-0.254928\pi\)
0.696074 + 0.717970i \(0.254928\pi\)
\(822\) 0 0
\(823\) −8.10790e29 −0.198249 −0.0991243 0.995075i \(-0.531604\pi\)
−0.0991243 + 0.995075i \(0.531604\pi\)
\(824\) 0 0
\(825\) 9.97257e29 0.237706
\(826\) 0 0
\(827\) −8.17222e30 −1.89903 −0.949516 0.313719i \(-0.898425\pi\)
−0.949516 + 0.313719i \(0.898425\pi\)
\(828\) 0 0
\(829\) −3.53326e29 −0.0800484 −0.0400242 0.999199i \(-0.512744\pi\)
−0.0400242 + 0.999199i \(0.512744\pi\)
\(830\) 0 0
\(831\) −9.46829e29 −0.209151
\(832\) 0 0
\(833\) −1.14838e29 −0.0247351
\(834\) 0 0
\(835\) −2.73757e29 −0.0574986
\(836\) 0 0
\(837\) 2.40702e30 0.493019
\(838\) 0 0
\(839\) 2.12371e30 0.424223 0.212112 0.977245i \(-0.431966\pi\)
0.212112 + 0.977245i \(0.431966\pi\)
\(840\) 0 0
\(841\) 9.83471e30 1.91604
\(842\) 0 0
\(843\) −3.03569e29 −0.0576857
\(844\) 0 0
\(845\) 2.29308e29 0.0425034
\(846\) 0 0
\(847\) 1.64753e30 0.297891
\(848\) 0 0
\(849\) −3.98241e30 −0.702450
\(850\) 0 0
\(851\) −4.57824e29 −0.0787840
\(852\) 0 0
\(853\) 9.31841e30 1.56451 0.782253 0.622961i \(-0.214071\pi\)
0.782253 + 0.622961i \(0.214071\pi\)
\(854\) 0 0
\(855\) −2.66268e29 −0.0436190
\(856\) 0 0
\(857\) 8.07467e30 1.29070 0.645352 0.763886i \(-0.276711\pi\)
0.645352 + 0.763886i \(0.276711\pi\)
\(858\) 0 0
\(859\) 2.78850e30 0.434953 0.217476 0.976066i \(-0.430218\pi\)
0.217476 + 0.976066i \(0.430218\pi\)
\(860\) 0 0
\(861\) −5.83589e29 −0.0888329
\(862\) 0 0
\(863\) −7.81319e29 −0.116069 −0.0580344 0.998315i \(-0.518483\pi\)
−0.0580344 + 0.998315i \(0.518483\pi\)
\(864\) 0 0
\(865\) 4.79054e28 0.00694569
\(866\) 0 0
\(867\) −4.29766e30 −0.608179
\(868\) 0 0
\(869\) 2.81311e30 0.388579
\(870\) 0 0
\(871\) 4.33484e30 0.584498
\(872\) 0 0
\(873\) −2.46126e30 −0.323972
\(874\) 0 0
\(875\) −3.11838e29 −0.0400723
\(876\) 0 0
\(877\) −7.95009e30 −0.997416 −0.498708 0.866770i \(-0.666192\pi\)
−0.498708 + 0.866770i \(0.666192\pi\)
\(878\) 0 0
\(879\) −1.66878e30 −0.204417
\(880\) 0 0
\(881\) −9.87210e30 −1.18076 −0.590381 0.807125i \(-0.701023\pi\)
−0.590381 + 0.807125i \(0.701023\pi\)
\(882\) 0 0
\(883\) 2.79595e30 0.326544 0.163272 0.986581i \(-0.447795\pi\)
0.163272 + 0.986581i \(0.447795\pi\)
\(884\) 0 0
\(885\) −9.54647e28 −0.0108878
\(886\) 0 0
\(887\) −2.90087e30 −0.323095 −0.161547 0.986865i \(-0.551648\pi\)
−0.161547 + 0.986865i \(0.551648\pi\)
\(888\) 0 0
\(889\) 1.32282e30 0.143891
\(890\) 0 0
\(891\) −9.52791e28 −0.0101224
\(892\) 0 0
\(893\) −1.55838e31 −1.61709
\(894\) 0 0
\(895\) −7.51735e29 −0.0761944
\(896\) 0 0
\(897\) −1.50393e30 −0.148904
\(898\) 0 0
\(899\) 8.77601e30 0.848829
\(900\) 0 0
\(901\) −4.37293e29 −0.0413202
\(902\) 0 0
\(903\) −3.61468e30 −0.333694
\(904\) 0 0
\(905\) 7.03339e27 0.000634387 0
\(906\) 0 0
\(907\) 5.28748e30 0.465985 0.232993 0.972479i \(-0.425148\pi\)
0.232993 + 0.972479i \(0.425148\pi\)
\(908\) 0 0
\(909\) −7.42773e30 −0.639640
\(910\) 0 0
\(911\) −1.11142e31 −0.935269 −0.467634 0.883922i \(-0.654894\pi\)
−0.467634 + 0.883922i \(0.654894\pi\)
\(912\) 0 0
\(913\) 4.14120e30 0.340552
\(914\) 0 0
\(915\) 4.86185e29 0.0390733
\(916\) 0 0
\(917\) 5.55617e29 0.0436413
\(918\) 0 0
\(919\) 6.90233e30 0.529886 0.264943 0.964264i \(-0.414647\pi\)
0.264943 + 0.964264i \(0.414647\pi\)
\(920\) 0 0
\(921\) 3.19425e30 0.239686
\(922\) 0 0
\(923\) 7.85381e30 0.576053
\(924\) 0 0
\(925\) −2.25935e30 −0.161993
\(926\) 0 0
\(927\) −7.60356e30 −0.532942
\(928\) 0 0
\(929\) −2.04813e31 −1.40344 −0.701718 0.712455i \(-0.747583\pi\)
−0.701718 + 0.712455i \(0.747583\pi\)
\(930\) 0 0
\(931\) 1.58902e31 1.06453
\(932\) 0 0
\(933\) 1.70352e31 1.11581
\(934\) 0 0
\(935\) 9.84952e27 0.000630802 0
\(936\) 0 0
\(937\) 2.29135e31 1.43492 0.717458 0.696601i \(-0.245305\pi\)
0.717458 + 0.696601i \(0.245305\pi\)
\(938\) 0 0
\(939\) 2.90144e30 0.177674
\(940\) 0 0
\(941\) −7.14878e30 −0.428096 −0.214048 0.976823i \(-0.568665\pi\)
−0.214048 + 0.976823i \(0.568665\pi\)
\(942\) 0 0
\(943\) −3.43288e30 −0.201042
\(944\) 0 0
\(945\) −3.47532e29 −0.0199049
\(946\) 0 0
\(947\) 2.01049e31 1.12623 0.563116 0.826378i \(-0.309603\pi\)
0.563116 + 0.826378i \(0.309603\pi\)
\(948\) 0 0
\(949\) 1.37812e31 0.755076
\(950\) 0 0
\(951\) −7.73269e30 −0.414415
\(952\) 0 0
\(953\) 1.77316e31 0.929550 0.464775 0.885429i \(-0.346135\pi\)
0.464775 + 0.885429i \(0.346135\pi\)
\(954\) 0 0
\(955\) 1.68091e30 0.0862005
\(956\) 0 0
\(957\) 8.11759e30 0.407241
\(958\) 0 0
\(959\) −7.06821e30 −0.346908
\(960\) 0 0
\(961\) −1.56798e31 −0.752914
\(962\) 0 0
\(963\) −8.15324e30 −0.383049
\(964\) 0 0
\(965\) 1.40616e30 0.0646394
\(966\) 0 0
\(967\) −1.43980e31 −0.647624 −0.323812 0.946121i \(-0.604965\pi\)
−0.323812 + 0.946121i \(0.604965\pi\)
\(968\) 0 0
\(969\) −4.74336e29 −0.0208779
\(970\) 0 0
\(971\) −1.69355e31 −0.729452 −0.364726 0.931115i \(-0.618837\pi\)
−0.364726 + 0.931115i \(0.618837\pi\)
\(972\) 0 0
\(973\) −2.37660e30 −0.100178
\(974\) 0 0
\(975\) −7.42185e30 −0.306171
\(976\) 0 0
\(977\) −6.75065e30 −0.272554 −0.136277 0.990671i \(-0.543514\pi\)
−0.136277 + 0.990671i \(0.543514\pi\)
\(978\) 0 0
\(979\) −4.08596e30 −0.161464
\(980\) 0 0
\(981\) 8.81402e30 0.340918
\(982\) 0 0
\(983\) −1.83872e31 −0.696153 −0.348076 0.937466i \(-0.613165\pi\)
−0.348076 + 0.937466i \(0.613165\pi\)
\(984\) 0 0
\(985\) −7.41466e29 −0.0274796
\(986\) 0 0
\(987\) −7.85783e30 −0.285084
\(988\) 0 0
\(989\) −2.12629e31 −0.755199
\(990\) 0 0
\(991\) −4.23711e31 −1.47332 −0.736659 0.676264i \(-0.763598\pi\)
−0.736659 + 0.676264i \(0.763598\pi\)
\(992\) 0 0
\(993\) −2.22583e31 −0.757749
\(994\) 0 0
\(995\) 3.34573e30 0.111519
\(996\) 0 0
\(997\) 5.70834e31 1.86299 0.931495 0.363753i \(-0.118505\pi\)
0.931495 + 0.363753i \(0.118505\pi\)
\(998\) 0 0
\(999\) −5.04414e30 −0.161194
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 8.22.a.a.1.2 2
3.2 odd 2 72.22.a.b.1.2 2
4.3 odd 2 16.22.a.e.1.1 2
8.3 odd 2 64.22.a.h.1.2 2
8.5 even 2 64.22.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.a.1.2 2 1.1 even 1 trivial
16.22.a.e.1.1 2 4.3 odd 2
64.22.a.h.1.2 2 8.3 odd 2
64.22.a.k.1.1 2 8.5 even 2
72.22.a.b.1.2 2 3.2 odd 2