Properties

Label 18.26.a.e.1.1
Level $18$
Weight $26$
Character 18.1
Self dual yes
Analytic conductor $71.279$
Analytic rank $0$
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] = [18,26,Mod(1,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.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: \(71.2794203914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(163.829\) of defining polynomial
Character \(\chi\) \(=\) 18.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +1.67772e7 q^{4} -8.78994e8 q^{5} +3.00388e10 q^{7} -6.87195e10 q^{8} +3.60036e12 q^{10} -5.73599e12 q^{11} -1.07343e13 q^{13} -1.23039e14 q^{14} +2.81475e14 q^{16} -2.97473e15 q^{17} -5.42738e15 q^{19} -1.47471e16 q^{20} +2.34946e16 q^{22} +1.04540e17 q^{23} +4.74607e17 q^{25} +4.39677e16 q^{26} +5.03967e17 q^{28} -3.09182e18 q^{29} -4.26809e18 q^{31} -1.15292e18 q^{32} +1.21845e19 q^{34} -2.64039e19 q^{35} -4.51028e19 q^{37} +2.22305e19 q^{38} +6.04040e19 q^{40} +7.56724e19 q^{41} -1.39036e20 q^{43} -9.62339e19 q^{44} -4.28196e20 q^{46} -4.67328e20 q^{47} -4.38741e20 q^{49} -1.94399e21 q^{50} -1.80092e20 q^{52} +1.24315e21 q^{53} +5.04190e21 q^{55} -2.06425e21 q^{56} +1.26641e22 q^{58} +1.32468e22 q^{59} -2.36984e20 q^{61} +1.74821e22 q^{62} +4.72237e21 q^{64} +9.43539e21 q^{65} -5.36922e22 q^{67} -4.99076e22 q^{68} +1.08150e23 q^{70} +2.27032e23 q^{71} +2.78528e23 q^{73} +1.84741e23 q^{74} -9.10563e22 q^{76} -1.72302e23 q^{77} +6.36958e23 q^{79} -2.47415e23 q^{80} -3.09954e23 q^{82} -1.19662e24 q^{83} +2.61477e24 q^{85} +5.69491e23 q^{86} +3.94174e23 q^{88} +3.22134e24 q^{89} -3.22445e23 q^{91} +1.75389e24 q^{92} +1.91418e24 q^{94} +4.77063e24 q^{95} +3.58465e24 q^{97} +1.79708e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} + 33554432 q^{4} - 741953100 q^{5} - 376536944 q^{7} - 137438953472 q^{8} + 3039039897600 q^{10} - 8323034610264 q^{11} - 106467053152292 q^{13} + 1542295322624 q^{14} + 562949953421312 q^{16}+ \cdots + 35\!\cdots\!16 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) 0 0
\(4\) 1.67772e7 0.500000
\(5\) −8.78994e8 −1.61013 −0.805065 0.593187i \(-0.797870\pi\)
−0.805065 + 0.593187i \(0.797870\pi\)
\(6\) 0 0
\(7\) 3.00388e10 0.820270 0.410135 0.912025i \(-0.365482\pi\)
0.410135 + 0.912025i \(0.365482\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 0 0
\(10\) 3.60036e12 1.13853
\(11\) −5.73599e12 −0.551061 −0.275531 0.961292i \(-0.588854\pi\)
−0.275531 + 0.961292i \(0.588854\pi\)
\(12\) 0 0
\(13\) −1.07343e13 −0.127786 −0.0638928 0.997957i \(-0.520352\pi\)
−0.0638928 + 0.997957i \(0.520352\pi\)
\(14\) −1.23039e14 −0.580018
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −2.97473e15 −1.23833 −0.619164 0.785262i \(-0.712528\pi\)
−0.619164 + 0.785262i \(0.712528\pi\)
\(18\) 0 0
\(19\) −5.42738e15 −0.562561 −0.281281 0.959626i \(-0.590759\pi\)
−0.281281 + 0.959626i \(0.590759\pi\)
\(20\) −1.47471e16 −0.805065
\(21\) 0 0
\(22\) 2.34946e16 0.389659
\(23\) 1.04540e17 0.994683 0.497341 0.867555i \(-0.334310\pi\)
0.497341 + 0.867555i \(0.334310\pi\)
\(24\) 0 0
\(25\) 4.74607e17 1.59252
\(26\) 4.39677e16 0.0903581
\(27\) 0 0
\(28\) 5.03967e17 0.410135
\(29\) −3.09182e18 −1.62270 −0.811352 0.584558i \(-0.801268\pi\)
−0.811352 + 0.584558i \(0.801268\pi\)
\(30\) 0 0
\(31\) −4.26809e18 −0.973222 −0.486611 0.873619i \(-0.661767\pi\)
−0.486611 + 0.873619i \(0.661767\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 0 0
\(34\) 1.21845e19 0.875630
\(35\) −2.64039e19 −1.32074
\(36\) 0 0
\(37\) −4.51028e19 −1.12637 −0.563186 0.826330i \(-0.690424\pi\)
−0.563186 + 0.826330i \(0.690424\pi\)
\(38\) 2.22305e19 0.397791
\(39\) 0 0
\(40\) 6.04040e19 0.569267
\(41\) 7.56724e19 0.523769 0.261884 0.965099i \(-0.415656\pi\)
0.261884 + 0.965099i \(0.415656\pi\)
\(42\) 0 0
\(43\) −1.39036e20 −0.530605 −0.265302 0.964165i \(-0.585472\pi\)
−0.265302 + 0.964165i \(0.585472\pi\)
\(44\) −9.62339e19 −0.275531
\(45\) 0 0
\(46\) −4.28196e20 −0.703347
\(47\) −4.67328e20 −0.586676 −0.293338 0.956009i \(-0.594766\pi\)
−0.293338 + 0.956009i \(0.594766\pi\)
\(48\) 0 0
\(49\) −4.38741e20 −0.327158
\(50\) −1.94399e21 −1.12608
\(51\) 0 0
\(52\) −1.80092e20 −0.0638928
\(53\) 1.24315e21 0.347595 0.173797 0.984781i \(-0.444396\pi\)
0.173797 + 0.984781i \(0.444396\pi\)
\(54\) 0 0
\(55\) 5.04190e21 0.887280
\(56\) −2.06425e21 −0.290009
\(57\) 0 0
\(58\) 1.26641e22 1.14743
\(59\) 1.32468e22 0.969303 0.484652 0.874707i \(-0.338946\pi\)
0.484652 + 0.874707i \(0.338946\pi\)
\(60\) 0 0
\(61\) −2.36984e20 −0.0114313 −0.00571565 0.999984i \(-0.501819\pi\)
−0.00571565 + 0.999984i \(0.501819\pi\)
\(62\) 1.74821e22 0.688172
\(63\) 0 0
\(64\) 4.72237e21 0.125000
\(65\) 9.43539e21 0.205752
\(66\) 0 0
\(67\) −5.36922e22 −0.801634 −0.400817 0.916158i \(-0.631274\pi\)
−0.400817 + 0.916158i \(0.631274\pi\)
\(68\) −4.99076e22 −0.619164
\(69\) 0 0
\(70\) 1.08150e23 0.933905
\(71\) 2.27032e23 1.64194 0.820971 0.570970i \(-0.193433\pi\)
0.820971 + 0.570970i \(0.193433\pi\)
\(72\) 0 0
\(73\) 2.78528e23 1.42342 0.711710 0.702473i \(-0.247921\pi\)
0.711710 + 0.702473i \(0.247921\pi\)
\(74\) 1.84741e23 0.796465
\(75\) 0 0
\(76\) −9.10563e22 −0.281281
\(77\) −1.72302e23 −0.452019
\(78\) 0 0
\(79\) 6.36958e23 1.21275 0.606376 0.795178i \(-0.292622\pi\)
0.606376 + 0.795178i \(0.292622\pi\)
\(80\) −2.47415e23 −0.402532
\(81\) 0 0
\(82\) −3.09954e23 −0.370360
\(83\) −1.19662e24 −1.22880 −0.614398 0.788996i \(-0.710601\pi\)
−0.614398 + 0.788996i \(0.710601\pi\)
\(84\) 0 0
\(85\) 2.61477e24 1.99387
\(86\) 5.69491e23 0.375194
\(87\) 0 0
\(88\) 3.94174e23 0.194830
\(89\) 3.22134e24 1.38249 0.691244 0.722621i \(-0.257063\pi\)
0.691244 + 0.722621i \(0.257063\pi\)
\(90\) 0 0
\(91\) −3.22445e23 −0.104819
\(92\) 1.75389e24 0.497341
\(93\) 0 0
\(94\) 1.91418e24 0.414843
\(95\) 4.77063e24 0.905797
\(96\) 0 0
\(97\) 3.58465e24 0.524566 0.262283 0.964991i \(-0.415525\pi\)
0.262283 + 0.964991i \(0.415525\pi\)
\(98\) 1.79708e24 0.231335
\(99\) 0 0
\(100\) 7.96259e24 0.796259
\(101\) 3.40630e24 0.300792 0.150396 0.988626i \(-0.451945\pi\)
0.150396 + 0.988626i \(0.451945\pi\)
\(102\) 0 0
\(103\) 8.27075e24 0.571583 0.285792 0.958292i \(-0.407743\pi\)
0.285792 + 0.958292i \(0.407743\pi\)
\(104\) 7.37656e23 0.0451791
\(105\) 0 0
\(106\) −5.09192e24 −0.245787
\(107\) −1.57986e25 −0.678142 −0.339071 0.940761i \(-0.610113\pi\)
−0.339071 + 0.940761i \(0.610113\pi\)
\(108\) 0 0
\(109\) 3.00869e25 1.02458 0.512290 0.858813i \(-0.328797\pi\)
0.512290 + 0.858813i \(0.328797\pi\)
\(110\) −2.06516e25 −0.627402
\(111\) 0 0
\(112\) 8.45516e24 0.205067
\(113\) 8.06745e25 1.75088 0.875439 0.483329i \(-0.160572\pi\)
0.875439 + 0.483329i \(0.160572\pi\)
\(114\) 0 0
\(115\) −9.18901e25 −1.60157
\(116\) −5.18722e25 −0.811352
\(117\) 0 0
\(118\) −5.42587e25 −0.685401
\(119\) −8.93571e25 −1.01576
\(120\) 0 0
\(121\) −7.54455e25 −0.696332
\(122\) 9.70685e23 0.00808315
\(123\) 0 0
\(124\) −7.16066e25 −0.486611
\(125\) −1.55216e26 −0.954030
\(126\) 0 0
\(127\) −1.32973e26 −0.670221 −0.335111 0.942179i \(-0.608774\pi\)
−0.335111 + 0.942179i \(0.608774\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) 0 0
\(130\) −3.86474e25 −0.145488
\(131\) −6.73908e25 −0.230520 −0.115260 0.993335i \(-0.536770\pi\)
−0.115260 + 0.993335i \(0.536770\pi\)
\(132\) 0 0
\(133\) −1.63032e26 −0.461452
\(134\) 2.19923e26 0.566841
\(135\) 0 0
\(136\) 2.04422e26 0.437815
\(137\) 3.22963e26 0.631170 0.315585 0.948897i \(-0.397799\pi\)
0.315585 + 0.948897i \(0.397799\pi\)
\(138\) 0 0
\(139\) −9.21581e25 −0.150262 −0.0751309 0.997174i \(-0.523937\pi\)
−0.0751309 + 0.997174i \(0.523937\pi\)
\(140\) −4.42984e26 −0.660370
\(141\) 0 0
\(142\) −9.29923e26 −1.16103
\(143\) 6.15719e25 0.0704177
\(144\) 0 0
\(145\) 2.71769e27 2.61276
\(146\) −1.14085e27 −1.00651
\(147\) 0 0
\(148\) −7.56699e26 −0.563186
\(149\) 7.12667e26 0.487594 0.243797 0.969826i \(-0.421607\pi\)
0.243797 + 0.969826i \(0.421607\pi\)
\(150\) 0 0
\(151\) −3.06161e27 −1.77312 −0.886559 0.462616i \(-0.846911\pi\)
−0.886559 + 0.462616i \(0.846911\pi\)
\(152\) 3.72966e26 0.198895
\(153\) 0 0
\(154\) 7.05749e26 0.319626
\(155\) 3.75162e27 1.56701
\(156\) 0 0
\(157\) 2.38960e27 0.850315 0.425157 0.905119i \(-0.360219\pi\)
0.425157 + 0.905119i \(0.360219\pi\)
\(158\) −2.60898e27 −0.857546
\(159\) 0 0
\(160\) 1.01341e27 0.284633
\(161\) 3.14025e27 0.815908
\(162\) 0 0
\(163\) 6.63092e27 1.47648 0.738242 0.674536i \(-0.235656\pi\)
0.738242 + 0.674536i \(0.235656\pi\)
\(164\) 1.26957e27 0.261884
\(165\) 0 0
\(166\) 4.90135e27 0.868890
\(167\) −4.06997e27 −0.669323 −0.334662 0.942338i \(-0.608622\pi\)
−0.334662 + 0.942338i \(0.608622\pi\)
\(168\) 0 0
\(169\) −6.94118e27 −0.983671
\(170\) −1.07101e28 −1.40988
\(171\) 0 0
\(172\) −2.33263e27 −0.265302
\(173\) 8.68916e27 0.919182 0.459591 0.888131i \(-0.347996\pi\)
0.459591 + 0.888131i \(0.347996\pi\)
\(174\) 0 0
\(175\) 1.42566e28 1.30629
\(176\) −1.61454e27 −0.137765
\(177\) 0 0
\(178\) −1.31946e28 −0.977567
\(179\) 4.31406e27 0.298005 0.149002 0.988837i \(-0.452394\pi\)
0.149002 + 0.988837i \(0.452394\pi\)
\(180\) 0 0
\(181\) 2.22969e28 1.34049 0.670243 0.742142i \(-0.266190\pi\)
0.670243 + 0.742142i \(0.266190\pi\)
\(182\) 1.32074e27 0.0741180
\(183\) 0 0
\(184\) −7.18394e27 −0.351674
\(185\) 3.96451e28 1.81360
\(186\) 0 0
\(187\) 1.70630e28 0.682394
\(188\) −7.84046e27 −0.293338
\(189\) 0 0
\(190\) −1.95405e28 −0.640495
\(191\) 1.91337e28 0.587329 0.293665 0.955908i \(-0.405125\pi\)
0.293665 + 0.955908i \(0.405125\pi\)
\(192\) 0 0
\(193\) −5.96748e28 −1.60814 −0.804071 0.594533i \(-0.797337\pi\)
−0.804071 + 0.594533i \(0.797337\pi\)
\(194\) −1.46827e28 −0.370924
\(195\) 0 0
\(196\) −7.36085e27 −0.163579
\(197\) −6.24204e28 −1.30166 −0.650832 0.759222i \(-0.725580\pi\)
−0.650832 + 0.759222i \(0.725580\pi\)
\(198\) 0 0
\(199\) −3.24229e28 −0.595921 −0.297960 0.954578i \(-0.596306\pi\)
−0.297960 + 0.954578i \(0.596306\pi\)
\(200\) −3.26148e28 −0.563040
\(201\) 0 0
\(202\) −1.39522e28 −0.212692
\(203\) −9.28745e28 −1.33106
\(204\) 0 0
\(205\) −6.65156e28 −0.843335
\(206\) −3.38770e28 −0.404171
\(207\) 0 0
\(208\) −3.02144e27 −0.0319464
\(209\) 3.11314e28 0.310006
\(210\) 0 0
\(211\) 3.82197e28 0.337875 0.168938 0.985627i \(-0.445966\pi\)
0.168938 + 0.985627i \(0.445966\pi\)
\(212\) 2.08565e28 0.173797
\(213\) 0 0
\(214\) 6.47109e28 0.479519
\(215\) 1.22212e29 0.854342
\(216\) 0 0
\(217\) −1.28208e29 −0.798305
\(218\) −1.23236e29 −0.724487
\(219\) 0 0
\(220\) 8.45891e28 0.443640
\(221\) 3.19316e28 0.158241
\(222\) 0 0
\(223\) 3.11100e29 1.37749 0.688746 0.725003i \(-0.258161\pi\)
0.688746 + 0.725003i \(0.258161\pi\)
\(224\) −3.46323e28 −0.145005
\(225\) 0 0
\(226\) −3.30443e29 −1.23806
\(227\) 1.05572e29 0.374304 0.187152 0.982331i \(-0.440074\pi\)
0.187152 + 0.982331i \(0.440074\pi\)
\(228\) 0 0
\(229\) −3.70322e29 −1.17662 −0.588310 0.808635i \(-0.700207\pi\)
−0.588310 + 0.808635i \(0.700207\pi\)
\(230\) 3.76382e29 1.13248
\(231\) 0 0
\(232\) 2.12468e29 0.573713
\(233\) 6.84781e28 0.175228 0.0876138 0.996155i \(-0.472076\pi\)
0.0876138 + 0.996155i \(0.472076\pi\)
\(234\) 0 0
\(235\) 4.10778e29 0.944625
\(236\) 2.22244e29 0.484652
\(237\) 0 0
\(238\) 3.66007e29 0.718253
\(239\) −3.87773e29 −0.722111 −0.361055 0.932544i \(-0.617583\pi\)
−0.361055 + 0.932544i \(0.617583\pi\)
\(240\) 0 0
\(241\) 1.03723e30 1.74046 0.870230 0.492646i \(-0.163970\pi\)
0.870230 + 0.492646i \(0.163970\pi\)
\(242\) 3.09025e29 0.492381
\(243\) 0 0
\(244\) −3.97593e27 −0.00571565
\(245\) 3.85651e29 0.526766
\(246\) 0 0
\(247\) 5.82591e28 0.0718873
\(248\) 2.93301e29 0.344086
\(249\) 0 0
\(250\) 6.35766e29 0.674601
\(251\) −1.47990e30 −1.49386 −0.746930 0.664903i \(-0.768473\pi\)
−0.746930 + 0.664903i \(0.768473\pi\)
\(252\) 0 0
\(253\) −5.99641e29 −0.548131
\(254\) 5.44659e29 0.473918
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −6.75384e29 −0.507442 −0.253721 0.967277i \(-0.581655\pi\)
−0.253721 + 0.967277i \(0.581655\pi\)
\(258\) 0 0
\(259\) −1.35483e30 −0.923929
\(260\) 1.58300e29 0.102876
\(261\) 0 0
\(262\) 2.76033e29 0.163003
\(263\) −1.42102e30 −0.800119 −0.400059 0.916489i \(-0.631011\pi\)
−0.400059 + 0.916489i \(0.631011\pi\)
\(264\) 0 0
\(265\) −1.09272e30 −0.559673
\(266\) 6.67778e29 0.326296
\(267\) 0 0
\(268\) −9.00805e29 −0.400817
\(269\) 1.90129e30 0.807504 0.403752 0.914868i \(-0.367706\pi\)
0.403752 + 0.914868i \(0.367706\pi\)
\(270\) 0 0
\(271\) −1.45800e30 −0.564471 −0.282236 0.959345i \(-0.591076\pi\)
−0.282236 + 0.959345i \(0.591076\pi\)
\(272\) −8.37311e29 −0.309582
\(273\) 0 0
\(274\) −1.32286e30 −0.446304
\(275\) −2.72234e30 −0.877575
\(276\) 0 0
\(277\) −5.31088e29 −0.156376 −0.0781879 0.996939i \(-0.524913\pi\)
−0.0781879 + 0.996939i \(0.524913\pi\)
\(278\) 3.77479e29 0.106251
\(279\) 0 0
\(280\) 1.81446e30 0.466952
\(281\) 3.64974e30 0.898323 0.449161 0.893451i \(-0.351723\pi\)
0.449161 + 0.893451i \(0.351723\pi\)
\(282\) 0 0
\(283\) −1.43965e30 −0.324285 −0.162143 0.986767i \(-0.551840\pi\)
−0.162143 + 0.986767i \(0.551840\pi\)
\(284\) 3.80897e30 0.820971
\(285\) 0 0
\(286\) −2.52199e29 −0.0497929
\(287\) 2.27311e30 0.429632
\(288\) 0 0
\(289\) 3.07837e30 0.533456
\(290\) −1.11317e31 −1.84750
\(291\) 0 0
\(292\) 4.67292e30 0.711710
\(293\) −1.24504e31 −1.81692 −0.908462 0.417967i \(-0.862743\pi\)
−0.908462 + 0.417967i \(0.862743\pi\)
\(294\) 0 0
\(295\) −1.16438e31 −1.56070
\(296\) 3.09944e30 0.398233
\(297\) 0 0
\(298\) −2.91909e30 −0.344781
\(299\) −1.12217e30 −0.127106
\(300\) 0 0
\(301\) −4.17647e30 −0.435239
\(302\) 1.25403e31 1.25378
\(303\) 0 0
\(304\) −1.52767e30 −0.140640
\(305\) 2.08307e29 0.0184059
\(306\) 0 0
\(307\) 2.27193e31 1.84997 0.924986 0.380001i \(-0.124076\pi\)
0.924986 + 0.380001i \(0.124076\pi\)
\(308\) −2.89075e30 −0.226009
\(309\) 0 0
\(310\) −1.53666e31 −1.10805
\(311\) 1.84336e31 1.27675 0.638374 0.769726i \(-0.279607\pi\)
0.638374 + 0.769726i \(0.279607\pi\)
\(312\) 0 0
\(313\) 4.98693e30 0.318808 0.159404 0.987213i \(-0.449043\pi\)
0.159404 + 0.987213i \(0.449043\pi\)
\(314\) −9.78780e30 −0.601263
\(315\) 0 0
\(316\) 1.06864e31 0.606376
\(317\) 8.79478e30 0.479716 0.239858 0.970808i \(-0.422899\pi\)
0.239858 + 0.970808i \(0.422899\pi\)
\(318\) 0 0
\(319\) 1.77347e31 0.894209
\(320\) −4.15093e30 −0.201266
\(321\) 0 0
\(322\) −1.28625e31 −0.576934
\(323\) 1.61450e31 0.696635
\(324\) 0 0
\(325\) −5.09458e30 −0.203501
\(326\) −2.71602e31 −1.04403
\(327\) 0 0
\(328\) −5.20017e30 −0.185180
\(329\) −1.40380e31 −0.481233
\(330\) 0 0
\(331\) 5.97122e31 1.89764 0.948818 0.315822i \(-0.102280\pi\)
0.948818 + 0.315822i \(0.102280\pi\)
\(332\) −2.00759e31 −0.614398
\(333\) 0 0
\(334\) 1.66706e31 0.473283
\(335\) 4.71951e31 1.29073
\(336\) 0 0
\(337\) 1.08428e31 0.275275 0.137638 0.990483i \(-0.456049\pi\)
0.137638 + 0.990483i \(0.456049\pi\)
\(338\) 2.84311e31 0.695560
\(339\) 0 0
\(340\) 4.38685e31 0.996934
\(341\) 2.44817e31 0.536305
\(342\) 0 0
\(343\) −5.34633e31 −1.08863
\(344\) 9.55447e30 0.187597
\(345\) 0 0
\(346\) −3.55908e31 −0.649960
\(347\) −7.20193e30 −0.126862 −0.0634308 0.997986i \(-0.520204\pi\)
−0.0634308 + 0.997986i \(0.520204\pi\)
\(348\) 0 0
\(349\) 5.27779e31 0.865235 0.432618 0.901577i \(-0.357590\pi\)
0.432618 + 0.901577i \(0.357590\pi\)
\(350\) −5.83951e31 −0.923689
\(351\) 0 0
\(352\) 6.61315e30 0.0974148
\(353\) 5.67715e31 0.807135 0.403567 0.914950i \(-0.367770\pi\)
0.403567 + 0.914950i \(0.367770\pi\)
\(354\) 0 0
\(355\) −1.99560e32 −2.64374
\(356\) 5.40451e31 0.691244
\(357\) 0 0
\(358\) −1.76704e31 −0.210721
\(359\) −2.18925e31 −0.252125 −0.126062 0.992022i \(-0.540234\pi\)
−0.126062 + 0.992022i \(0.540234\pi\)
\(360\) 0 0
\(361\) −6.36201e31 −0.683525
\(362\) −9.13280e31 −0.947866
\(363\) 0 0
\(364\) −5.40974e30 −0.0524094
\(365\) −2.44824e32 −2.29189
\(366\) 0 0
\(367\) 1.00500e32 0.878699 0.439349 0.898316i \(-0.355209\pi\)
0.439349 + 0.898316i \(0.355209\pi\)
\(368\) 2.94254e31 0.248671
\(369\) 0 0
\(370\) −1.62386e32 −1.28241
\(371\) 3.73425e31 0.285122
\(372\) 0 0
\(373\) 6.31052e31 0.450510 0.225255 0.974300i \(-0.427678\pi\)
0.225255 + 0.974300i \(0.427678\pi\)
\(374\) −6.98901e31 −0.482526
\(375\) 0 0
\(376\) 3.21145e31 0.207421
\(377\) 3.31886e31 0.207358
\(378\) 0 0
\(379\) −8.85755e31 −0.517993 −0.258996 0.965878i \(-0.583392\pi\)
−0.258996 + 0.965878i \(0.583392\pi\)
\(380\) 8.00379e31 0.452898
\(381\) 0 0
\(382\) −7.83716e31 −0.415305
\(383\) −9.08315e31 −0.465856 −0.232928 0.972494i \(-0.574831\pi\)
−0.232928 + 0.972494i \(0.574831\pi\)
\(384\) 0 0
\(385\) 1.51453e32 0.727809
\(386\) 2.44428e32 1.13713
\(387\) 0 0
\(388\) 6.01405e31 0.262283
\(389\) −7.13043e31 −0.301124 −0.150562 0.988601i \(-0.548108\pi\)
−0.150562 + 0.988601i \(0.548108\pi\)
\(390\) 0 0
\(391\) −3.10978e32 −1.23174
\(392\) 3.01500e31 0.115668
\(393\) 0 0
\(394\) 2.55674e32 0.920416
\(395\) −5.59883e32 −1.95269
\(396\) 0 0
\(397\) 2.99427e31 0.0980413 0.0490206 0.998798i \(-0.484390\pi\)
0.0490206 + 0.998798i \(0.484390\pi\)
\(398\) 1.32804e32 0.421380
\(399\) 0 0
\(400\) 1.33590e32 0.398129
\(401\) 5.00588e32 1.44603 0.723013 0.690835i \(-0.242757\pi\)
0.723013 + 0.690835i \(0.242757\pi\)
\(402\) 0 0
\(403\) 4.58150e31 0.124364
\(404\) 5.71482e31 0.150396
\(405\) 0 0
\(406\) 3.80414e32 0.941198
\(407\) 2.58709e32 0.620700
\(408\) 0 0
\(409\) −2.78379e32 −0.628195 −0.314097 0.949391i \(-0.601702\pi\)
−0.314097 + 0.949391i \(0.601702\pi\)
\(410\) 2.72448e32 0.596328
\(411\) 0 0
\(412\) 1.38760e32 0.285792
\(413\) 3.97916e32 0.795090
\(414\) 0 0
\(415\) 1.05182e33 1.97852
\(416\) 1.23758e31 0.0225895
\(417\) 0 0
\(418\) −1.27514e32 −0.219207
\(419\) 6.36828e32 1.06254 0.531272 0.847201i \(-0.321714\pi\)
0.531272 + 0.847201i \(0.321714\pi\)
\(420\) 0 0
\(421\) −3.92275e32 −0.616686 −0.308343 0.951275i \(-0.599775\pi\)
−0.308343 + 0.951275i \(0.599775\pi\)
\(422\) −1.56548e32 −0.238914
\(423\) 0 0
\(424\) −8.54283e31 −0.122893
\(425\) −1.41183e33 −1.97206
\(426\) 0 0
\(427\) −7.11870e30 −0.00937675
\(428\) −2.65056e32 −0.339071
\(429\) 0 0
\(430\) −5.00579e32 −0.604111
\(431\) −2.44178e32 −0.286247 −0.143124 0.989705i \(-0.545715\pi\)
−0.143124 + 0.989705i \(0.545715\pi\)
\(432\) 0 0
\(433\) 9.12872e32 1.00998 0.504988 0.863126i \(-0.331497\pi\)
0.504988 + 0.863126i \(0.331497\pi\)
\(434\) 5.25140e32 0.564487
\(435\) 0 0
\(436\) 5.04774e32 0.512290
\(437\) −5.67378e32 −0.559570
\(438\) 0 0
\(439\) 6.07601e32 0.565995 0.282997 0.959121i \(-0.408671\pi\)
0.282997 + 0.959121i \(0.408671\pi\)
\(440\) −3.46477e32 −0.313701
\(441\) 0 0
\(442\) −1.30792e32 −0.111893
\(443\) −9.06289e32 −0.753737 −0.376869 0.926267i \(-0.622999\pi\)
−0.376869 + 0.926267i \(0.622999\pi\)
\(444\) 0 0
\(445\) −2.83154e33 −2.22598
\(446\) −1.27427e33 −0.974034
\(447\) 0 0
\(448\) 1.41854e32 0.102534
\(449\) 1.21579e33 0.854630 0.427315 0.904103i \(-0.359460\pi\)
0.427315 + 0.904103i \(0.359460\pi\)
\(450\) 0 0
\(451\) −4.34056e32 −0.288629
\(452\) 1.35349e33 0.875439
\(453\) 0 0
\(454\) −4.32422e32 −0.264673
\(455\) 2.83428e32 0.168772
\(456\) 0 0
\(457\) −5.59530e32 −0.315407 −0.157703 0.987487i \(-0.550409\pi\)
−0.157703 + 0.987487i \(0.550409\pi\)
\(458\) 1.51684e33 0.831997
\(459\) 0 0
\(460\) −1.54166e33 −0.800784
\(461\) 3.78592e32 0.191386 0.0956930 0.995411i \(-0.469493\pi\)
0.0956930 + 0.995411i \(0.469493\pi\)
\(462\) 0 0
\(463\) 2.77039e33 1.32672 0.663359 0.748301i \(-0.269130\pi\)
0.663359 + 0.748301i \(0.269130\pi\)
\(464\) −8.70270e32 −0.405676
\(465\) 0 0
\(466\) −2.80486e32 −0.123905
\(467\) 2.31922e33 0.997428 0.498714 0.866767i \(-0.333806\pi\)
0.498714 + 0.866767i \(0.333806\pi\)
\(468\) 0 0
\(469\) −1.61285e33 −0.657556
\(470\) −1.68255e33 −0.667951
\(471\) 0 0
\(472\) −9.10311e32 −0.342700
\(473\) 7.97508e32 0.292396
\(474\) 0 0
\(475\) −2.57587e33 −0.895889
\(476\) −1.49916e33 −0.507881
\(477\) 0 0
\(478\) 1.58832e33 0.510609
\(479\) −2.24376e33 −0.702720 −0.351360 0.936240i \(-0.614281\pi\)
−0.351360 + 0.936240i \(0.614281\pi\)
\(480\) 0 0
\(481\) 4.84147e32 0.143934
\(482\) −4.24851e33 −1.23069
\(483\) 0 0
\(484\) −1.26577e33 −0.348166
\(485\) −3.15089e33 −0.844620
\(486\) 0 0
\(487\) 3.71393e32 0.0945632 0.0472816 0.998882i \(-0.484944\pi\)
0.0472816 + 0.998882i \(0.484944\pi\)
\(488\) 1.62854e31 0.00404157
\(489\) 0 0
\(490\) −1.57963e33 −0.372480
\(491\) 2.65508e33 0.610323 0.305161 0.952301i \(-0.401290\pi\)
0.305161 + 0.952301i \(0.401290\pi\)
\(492\) 0 0
\(493\) 9.19732e33 2.00944
\(494\) −2.38629e32 −0.0508320
\(495\) 0 0
\(496\) −1.20136e33 −0.243306
\(497\) 6.81976e33 1.34684
\(498\) 0 0
\(499\) −2.39117e33 −0.449111 −0.224556 0.974461i \(-0.572093\pi\)
−0.224556 + 0.974461i \(0.572093\pi\)
\(500\) −2.60410e33 −0.477015
\(501\) 0 0
\(502\) 6.06166e33 1.05632
\(503\) 6.34146e33 1.07793 0.538964 0.842329i \(-0.318816\pi\)
0.538964 + 0.842329i \(0.318816\pi\)
\(504\) 0 0
\(505\) −2.99412e33 −0.484314
\(506\) 2.45613e33 0.387587
\(507\) 0 0
\(508\) −2.23092e33 −0.335111
\(509\) 3.62528e32 0.0531337 0.0265668 0.999647i \(-0.491543\pi\)
0.0265668 + 0.999647i \(0.491543\pi\)
\(510\) 0 0
\(511\) 8.36664e33 1.16759
\(512\) −3.24519e32 −0.0441942
\(513\) 0 0
\(514\) 2.76637e33 0.358816
\(515\) −7.26994e33 −0.920323
\(516\) 0 0
\(517\) 2.68059e33 0.323295
\(518\) 5.54939e33 0.653316
\(519\) 0 0
\(520\) −6.48395e32 −0.0727442
\(521\) −8.23573e33 −0.902050 −0.451025 0.892511i \(-0.648942\pi\)
−0.451025 + 0.892511i \(0.648942\pi\)
\(522\) 0 0
\(523\) −1.20443e34 −1.25750 −0.628752 0.777606i \(-0.716434\pi\)
−0.628752 + 0.777606i \(0.716434\pi\)
\(524\) −1.13063e33 −0.115260
\(525\) 0 0
\(526\) 5.82051e33 0.565770
\(527\) 1.26964e34 1.20517
\(528\) 0 0
\(529\) −1.17149e32 −0.0106057
\(530\) 4.47577e33 0.395749
\(531\) 0 0
\(532\) −2.73522e33 −0.230726
\(533\) −8.12291e32 −0.0669301
\(534\) 0 0
\(535\) 1.38868e34 1.09190
\(536\) 3.68970e33 0.283420
\(537\) 0 0
\(538\) −7.78769e33 −0.570992
\(539\) 2.51661e33 0.180284
\(540\) 0 0
\(541\) 1.30366e34 0.891657 0.445829 0.895118i \(-0.352909\pi\)
0.445829 + 0.895118i \(0.352909\pi\)
\(542\) 5.97197e33 0.399141
\(543\) 0 0
\(544\) 3.42963e33 0.218907
\(545\) −2.64462e34 −1.64971
\(546\) 0 0
\(547\) −1.78334e34 −1.06266 −0.531328 0.847166i \(-0.678307\pi\)
−0.531328 + 0.847166i \(0.678307\pi\)
\(548\) 5.41843e33 0.315585
\(549\) 0 0
\(550\) 1.11507e34 0.620539
\(551\) 1.67805e34 0.912871
\(552\) 0 0
\(553\) 1.91334e34 0.994784
\(554\) 2.17534e33 0.110574
\(555\) 0 0
\(556\) −1.54616e33 −0.0751309
\(557\) −8.73339e33 −0.414947 −0.207474 0.978241i \(-0.566524\pi\)
−0.207474 + 0.978241i \(0.566524\pi\)
\(558\) 0 0
\(559\) 1.49245e33 0.0678037
\(560\) −7.43204e33 −0.330185
\(561\) 0 0
\(562\) −1.49493e34 −0.635210
\(563\) −3.82416e34 −1.58921 −0.794606 0.607126i \(-0.792322\pi\)
−0.794606 + 0.607126i \(0.792322\pi\)
\(564\) 0 0
\(565\) −7.09124e34 −2.81914
\(566\) 5.89682e33 0.229304
\(567\) 0 0
\(568\) −1.56015e34 −0.580514
\(569\) −3.49591e34 −1.27250 −0.636248 0.771484i \(-0.719515\pi\)
−0.636248 + 0.771484i \(0.719515\pi\)
\(570\) 0 0
\(571\) −3.60524e34 −1.25598 −0.627989 0.778222i \(-0.716122\pi\)
−0.627989 + 0.778222i \(0.716122\pi\)
\(572\) 1.03301e33 0.0352089
\(573\) 0 0
\(574\) −9.31065e33 −0.303795
\(575\) 4.96155e34 1.58405
\(576\) 0 0
\(577\) −4.45926e34 −1.36322 −0.681609 0.731717i \(-0.738719\pi\)
−0.681609 + 0.731717i \(0.738719\pi\)
\(578\) −1.26090e34 −0.377210
\(579\) 0 0
\(580\) 4.55953e34 1.30638
\(581\) −3.59450e34 −1.00794
\(582\) 0 0
\(583\) −7.13067e33 −0.191546
\(584\) −1.91403e34 −0.503255
\(585\) 0 0
\(586\) 5.09967e34 1.28476
\(587\) 2.50958e34 0.618905 0.309452 0.950915i \(-0.399854\pi\)
0.309452 + 0.950915i \(0.399854\pi\)
\(588\) 0 0
\(589\) 2.31645e34 0.547497
\(590\) 4.76931e34 1.10358
\(591\) 0 0
\(592\) −1.26953e34 −0.281593
\(593\) 4.88078e33 0.106000 0.0530000 0.998595i \(-0.483122\pi\)
0.0530000 + 0.998595i \(0.483122\pi\)
\(594\) 0 0
\(595\) 7.85444e34 1.63551
\(596\) 1.19566e34 0.243797
\(597\) 0 0
\(598\) 4.59639e33 0.0898777
\(599\) −4.61838e34 −0.884412 −0.442206 0.896914i \(-0.645804\pi\)
−0.442206 + 0.896914i \(0.645804\pi\)
\(600\) 0 0
\(601\) −3.07405e34 −0.564651 −0.282325 0.959319i \(-0.591106\pi\)
−0.282325 + 0.959319i \(0.591106\pi\)
\(602\) 1.71068e34 0.307760
\(603\) 0 0
\(604\) −5.13652e34 −0.886559
\(605\) 6.63161e34 1.12118
\(606\) 0 0
\(607\) 1.03737e35 1.68297 0.841484 0.540282i \(-0.181682\pi\)
0.841484 + 0.540282i \(0.181682\pi\)
\(608\) 6.25734e33 0.0994477
\(609\) 0 0
\(610\) −8.53226e32 −0.0130149
\(611\) 5.01644e33 0.0749688
\(612\) 0 0
\(613\) 2.85744e34 0.409941 0.204970 0.978768i \(-0.434290\pi\)
0.204970 + 0.978768i \(0.434290\pi\)
\(614\) −9.30582e34 −1.30813
\(615\) 0 0
\(616\) 1.18405e34 0.159813
\(617\) −5.36336e34 −0.709370 −0.354685 0.934986i \(-0.615412\pi\)
−0.354685 + 0.934986i \(0.615412\pi\)
\(618\) 0 0
\(619\) 6.60909e34 0.839477 0.419738 0.907645i \(-0.362122\pi\)
0.419738 + 0.907645i \(0.362122\pi\)
\(620\) 6.29418e34 0.783507
\(621\) 0 0
\(622\) −7.55041e34 −0.902798
\(623\) 9.67651e34 1.13401
\(624\) 0 0
\(625\) −5.00979e33 −0.0564053
\(626\) −2.04264e34 −0.225431
\(627\) 0 0
\(628\) 4.00908e34 0.425157
\(629\) 1.34168e35 1.39482
\(630\) 0 0
\(631\) 8.62816e34 0.862087 0.431043 0.902331i \(-0.358146\pi\)
0.431043 + 0.902331i \(0.358146\pi\)
\(632\) −4.37714e34 −0.428773
\(633\) 0 0
\(634\) −3.60234e34 −0.339210
\(635\) 1.16883e35 1.07914
\(636\) 0 0
\(637\) 4.70958e33 0.0418061
\(638\) −7.26412e34 −0.632302
\(639\) 0 0
\(640\) 1.70022e34 0.142317
\(641\) −1.77093e35 −1.45370 −0.726850 0.686796i \(-0.759017\pi\)
−0.726850 + 0.686796i \(0.759017\pi\)
\(642\) 0 0
\(643\) −7.98617e34 −0.630524 −0.315262 0.949005i \(-0.602092\pi\)
−0.315262 + 0.949005i \(0.602092\pi\)
\(644\) 5.26847e34 0.407954
\(645\) 0 0
\(646\) −6.61298e34 −0.492596
\(647\) 2.16671e35 1.58306 0.791530 0.611130i \(-0.209285\pi\)
0.791530 + 0.611130i \(0.209285\pi\)
\(648\) 0 0
\(649\) −7.59833e34 −0.534145
\(650\) 2.08674e34 0.143897
\(651\) 0 0
\(652\) 1.11248e35 0.738242
\(653\) −2.41184e35 −1.57013 −0.785063 0.619416i \(-0.787369\pi\)
−0.785063 + 0.619416i \(0.787369\pi\)
\(654\) 0 0
\(655\) 5.92361e34 0.371168
\(656\) 2.12999e34 0.130942
\(657\) 0 0
\(658\) 5.74995e34 0.340283
\(659\) −2.18788e35 −1.27045 −0.635223 0.772329i \(-0.719092\pi\)
−0.635223 + 0.772329i \(0.719092\pi\)
\(660\) 0 0
\(661\) −8.90411e34 −0.497820 −0.248910 0.968527i \(-0.580072\pi\)
−0.248910 + 0.968527i \(0.580072\pi\)
\(662\) −2.44581e35 −1.34183
\(663\) 0 0
\(664\) 8.22310e34 0.434445
\(665\) 1.43304e35 0.742998
\(666\) 0 0
\(667\) −3.23219e35 −1.61408
\(668\) −6.82828e34 −0.334662
\(669\) 0 0
\(670\) −1.93311e35 −0.912687
\(671\) 1.35934e33 0.00629934
\(672\) 0 0
\(673\) 2.39665e35 1.07008 0.535040 0.844827i \(-0.320297\pi\)
0.535040 + 0.844827i \(0.320297\pi\)
\(674\) −4.44120e34 −0.194649
\(675\) 0 0
\(676\) −1.16454e35 −0.491835
\(677\) 3.79216e34 0.157227 0.0786137 0.996905i \(-0.474951\pi\)
0.0786137 + 0.996905i \(0.474951\pi\)
\(678\) 0 0
\(679\) 1.07679e35 0.430286
\(680\) −1.79685e35 −0.704939
\(681\) 0 0
\(682\) −1.00277e35 −0.379225
\(683\) 3.21208e35 1.19269 0.596345 0.802729i \(-0.296619\pi\)
0.596345 + 0.802729i \(0.296619\pi\)
\(684\) 0 0
\(685\) −2.83883e35 −1.01627
\(686\) 2.18986e35 0.769776
\(687\) 0 0
\(688\) −3.91351e34 −0.132651
\(689\) −1.33443e34 −0.0444177
\(690\) 0 0
\(691\) 8.34540e34 0.267899 0.133950 0.990988i \(-0.457234\pi\)
0.133950 + 0.990988i \(0.457234\pi\)
\(692\) 1.45780e35 0.459591
\(693\) 0 0
\(694\) 2.94991e34 0.0897047
\(695\) 8.10064e34 0.241941
\(696\) 0 0
\(697\) −2.25105e35 −0.648597
\(698\) −2.16178e35 −0.611814
\(699\) 0 0
\(700\) 2.39186e35 0.653147
\(701\) −1.50008e35 −0.402383 −0.201191 0.979552i \(-0.564481\pi\)
−0.201191 + 0.979552i \(0.564481\pi\)
\(702\) 0 0
\(703\) 2.44790e35 0.633653
\(704\) −2.70874e34 −0.0688827
\(705\) 0 0
\(706\) −2.32536e35 −0.570731
\(707\) 1.02321e35 0.246730
\(708\) 0 0
\(709\) −2.12042e35 −0.493564 −0.246782 0.969071i \(-0.579373\pi\)
−0.246782 + 0.969071i \(0.579373\pi\)
\(710\) 8.17397e35 1.86941
\(711\) 0 0
\(712\) −2.21369e35 −0.488783
\(713\) −4.46186e35 −0.968048
\(714\) 0 0
\(715\) −5.41213e34 −0.113382
\(716\) 7.23778e34 0.149002
\(717\) 0 0
\(718\) 8.96718e34 0.178279
\(719\) −3.27911e35 −0.640686 −0.320343 0.947302i \(-0.603798\pi\)
−0.320343 + 0.947302i \(0.603798\pi\)
\(720\) 0 0
\(721\) 2.48443e35 0.468853
\(722\) 2.60588e35 0.483325
\(723\) 0 0
\(724\) 3.74080e35 0.670243
\(725\) −1.46740e36 −2.58419
\(726\) 0 0
\(727\) 2.40795e35 0.409701 0.204850 0.978793i \(-0.434329\pi\)
0.204850 + 0.978793i \(0.434329\pi\)
\(728\) 2.21583e34 0.0370590
\(729\) 0 0
\(730\) 1.00280e36 1.62061
\(731\) 4.13594e35 0.657063
\(732\) 0 0
\(733\) −3.39196e35 −0.520776 −0.260388 0.965504i \(-0.583850\pi\)
−0.260388 + 0.965504i \(0.583850\pi\)
\(734\) −4.11648e35 −0.621334
\(735\) 0 0
\(736\) −1.20526e35 −0.175837
\(737\) 3.07978e35 0.441749
\(738\) 0 0
\(739\) 1.23366e36 1.71057 0.855284 0.518159i \(-0.173383\pi\)
0.855284 + 0.518159i \(0.173383\pi\)
\(740\) 6.65134e35 0.906802
\(741\) 0 0
\(742\) −1.52955e35 −0.201611
\(743\) 6.24478e35 0.809389 0.404694 0.914452i \(-0.367378\pi\)
0.404694 + 0.914452i \(0.367378\pi\)
\(744\) 0 0
\(745\) −6.26430e35 −0.785090
\(746\) −2.58479e35 −0.318559
\(747\) 0 0
\(748\) 2.86270e35 0.341197
\(749\) −4.74570e35 −0.556259
\(750\) 0 0
\(751\) −4.41291e35 −0.500294 −0.250147 0.968208i \(-0.580479\pi\)
−0.250147 + 0.968208i \(0.580479\pi\)
\(752\) −1.31541e35 −0.146669
\(753\) 0 0
\(754\) −1.35940e35 −0.146625
\(755\) 2.69113e36 2.85495
\(756\) 0 0
\(757\) 1.83266e35 0.188098 0.0940489 0.995568i \(-0.470019\pi\)
0.0940489 + 0.995568i \(0.470019\pi\)
\(758\) 3.62805e35 0.366276
\(759\) 0 0
\(760\) −3.27835e35 −0.320247
\(761\) −1.88027e36 −1.80681 −0.903406 0.428787i \(-0.858941\pi\)
−0.903406 + 0.428787i \(0.858941\pi\)
\(762\) 0 0
\(763\) 9.03772e35 0.840431
\(764\) 3.21010e35 0.293665
\(765\) 0 0
\(766\) 3.72046e35 0.329410
\(767\) −1.42195e35 −0.123863
\(768\) 0 0
\(769\) 1.28453e36 1.08309 0.541546 0.840671i \(-0.317839\pi\)
0.541546 + 0.840671i \(0.317839\pi\)
\(770\) −6.20349e35 −0.514639
\(771\) 0 0
\(772\) −1.00118e36 −0.804071
\(773\) 1.04808e36 0.828226 0.414113 0.910225i \(-0.364092\pi\)
0.414113 + 0.910225i \(0.364092\pi\)
\(774\) 0 0
\(775\) −2.02566e36 −1.54987
\(776\) −2.46335e35 −0.185462
\(777\) 0 0
\(778\) 2.92063e35 0.212927
\(779\) −4.10703e35 −0.294652
\(780\) 0 0
\(781\) −1.30225e36 −0.904811
\(782\) 1.27377e36 0.870974
\(783\) 0 0
\(784\) −1.23495e35 −0.0817894
\(785\) −2.10044e36 −1.36912
\(786\) 0 0
\(787\) −8.93146e35 −0.563947 −0.281974 0.959422i \(-0.590989\pi\)
−0.281974 + 0.959422i \(0.590989\pi\)
\(788\) −1.04724e36 −0.650832
\(789\) 0 0
\(790\) 2.29328e36 1.38076
\(791\) 2.42336e36 1.43619
\(792\) 0 0
\(793\) 2.54386e33 0.00146076
\(794\) −1.22645e35 −0.0693257
\(795\) 0 0
\(796\) −5.43966e35 −0.297960
\(797\) 1.02840e36 0.554540 0.277270 0.960792i \(-0.410570\pi\)
0.277270 + 0.960792i \(0.410570\pi\)
\(798\) 0 0
\(799\) 1.39017e36 0.726498
\(800\) −5.47185e35 −0.281520
\(801\) 0 0
\(802\) −2.05041e36 −1.02249
\(803\) −1.59763e36 −0.784392
\(804\) 0 0
\(805\) −2.76026e36 −1.31372
\(806\) −1.87658e35 −0.0879385
\(807\) 0 0
\(808\) −2.34079e35 −0.106346
\(809\) −1.33897e36 −0.598981 −0.299490 0.954099i \(-0.596817\pi\)
−0.299490 + 0.954099i \(0.596817\pi\)
\(810\) 0 0
\(811\) 1.29485e36 0.561643 0.280821 0.959760i \(-0.409393\pi\)
0.280821 + 0.959760i \(0.409393\pi\)
\(812\) −1.55818e36 −0.665528
\(813\) 0 0
\(814\) −1.05967e36 −0.438901
\(815\) −5.82854e36 −2.37733
\(816\) 0 0
\(817\) 7.54600e35 0.298498
\(818\) 1.14024e36 0.444201
\(819\) 0 0
\(820\) −1.11595e36 −0.421668
\(821\) −3.91664e34 −0.0145755 −0.00728777 0.999973i \(-0.502320\pi\)
−0.00728777 + 0.999973i \(0.502320\pi\)
\(822\) 0 0
\(823\) −3.40400e36 −1.22883 −0.614414 0.788984i \(-0.710608\pi\)
−0.614414 + 0.788984i \(0.710608\pi\)
\(824\) −5.68362e35 −0.202085
\(825\) 0 0
\(826\) −1.62987e36 −0.562213
\(827\) 5.02186e36 1.70626 0.853132 0.521696i \(-0.174700\pi\)
0.853132 + 0.521696i \(0.174700\pi\)
\(828\) 0 0
\(829\) 8.20485e35 0.270482 0.135241 0.990813i \(-0.456819\pi\)
0.135241 + 0.990813i \(0.456819\pi\)
\(830\) −4.30826e36 −1.39902
\(831\) 0 0
\(832\) −5.06913e34 −0.0159732
\(833\) 1.30513e36 0.405128
\(834\) 0 0
\(835\) 3.57748e36 1.07770
\(836\) 5.22298e35 0.155003
\(837\) 0 0
\(838\) −2.60845e36 −0.751332
\(839\) 5.38664e36 1.52860 0.764300 0.644861i \(-0.223085\pi\)
0.764300 + 0.644861i \(0.223085\pi\)
\(840\) 0 0
\(841\) 5.92900e36 1.63317
\(842\) 1.60676e36 0.436063
\(843\) 0 0
\(844\) 6.41220e35 0.168938
\(845\) 6.10126e36 1.58384
\(846\) 0 0
\(847\) −2.26629e36 −0.571180
\(848\) 3.49914e35 0.0868987
\(849\) 0 0
\(850\) 5.78284e36 1.39446
\(851\) −4.71505e36 −1.12038
\(852\) 0 0
\(853\) 9.55369e35 0.220449 0.110225 0.993907i \(-0.464843\pi\)
0.110225 + 0.993907i \(0.464843\pi\)
\(854\) 2.91582e34 0.00663036
\(855\) 0 0
\(856\) 1.08567e36 0.239759
\(857\) 5.39118e36 1.17334 0.586669 0.809827i \(-0.300439\pi\)
0.586669 + 0.809827i \(0.300439\pi\)
\(858\) 0 0
\(859\) 5.48992e36 1.16052 0.580258 0.814432i \(-0.302952\pi\)
0.580258 + 0.814432i \(0.302952\pi\)
\(860\) 2.05037e36 0.427171
\(861\) 0 0
\(862\) 1.00016e36 0.202407
\(863\) 2.89510e36 0.577469 0.288735 0.957409i \(-0.406766\pi\)
0.288735 + 0.957409i \(0.406766\pi\)
\(864\) 0 0
\(865\) −7.63772e36 −1.48000
\(866\) −3.73912e36 −0.714161
\(867\) 0 0
\(868\) −2.15097e36 −0.399152
\(869\) −3.65359e36 −0.668301
\(870\) 0 0
\(871\) 5.76349e35 0.102437
\(872\) −2.06755e36 −0.362244
\(873\) 0 0
\(874\) 2.32398e36 0.395676
\(875\) −4.66251e36 −0.782562
\(876\) 0 0
\(877\) 2.04429e36 0.333463 0.166731 0.986002i \(-0.446679\pi\)
0.166731 + 0.986002i \(0.446679\pi\)
\(878\) −2.48874e36 −0.400219
\(879\) 0 0
\(880\) 1.41917e36 0.221820
\(881\) 1.76163e36 0.271466 0.135733 0.990745i \(-0.456661\pi\)
0.135733 + 0.990745i \(0.456661\pi\)
\(882\) 0 0
\(883\) 1.25672e37 1.88248 0.941240 0.337738i \(-0.109662\pi\)
0.941240 + 0.337738i \(0.109662\pi\)
\(884\) 5.35724e35 0.0791203
\(885\) 0 0
\(886\) 3.71216e36 0.532973
\(887\) −8.81003e36 −1.24719 −0.623595 0.781748i \(-0.714328\pi\)
−0.623595 + 0.781748i \(0.714328\pi\)
\(888\) 0 0
\(889\) −3.99436e36 −0.549762
\(890\) 1.15980e37 1.57401
\(891\) 0 0
\(892\) 5.21939e36 0.688746
\(893\) 2.53636e36 0.330041
\(894\) 0 0
\(895\) −3.79203e36 −0.479826
\(896\) −5.81034e35 −0.0725023
\(897\) 0 0
\(898\) −4.97986e36 −0.604315
\(899\) 1.31962e37 1.57925
\(900\) 0 0
\(901\) −3.69802e36 −0.430436
\(902\) 1.77789e36 0.204091
\(903\) 0 0
\(904\) −5.54391e36 −0.619029
\(905\) −1.95988e37 −2.15835
\(906\) 0 0
\(907\) −1.59523e37 −1.70896 −0.854482 0.519481i \(-0.826125\pi\)
−0.854482 + 0.519481i \(0.826125\pi\)
\(908\) 1.77120e36 0.187152
\(909\) 0 0
\(910\) −1.16092e36 −0.119340
\(911\) 1.71775e36 0.174172 0.0870862 0.996201i \(-0.472244\pi\)
0.0870862 + 0.996201i \(0.472244\pi\)
\(912\) 0 0
\(913\) 6.86379e36 0.677142
\(914\) 2.29183e36 0.223026
\(915\) 0 0
\(916\) −6.21298e36 −0.588310
\(917\) −2.02434e36 −0.189089
\(918\) 0 0
\(919\) −7.70373e35 −0.0700256 −0.0350128 0.999387i \(-0.511147\pi\)
−0.0350128 + 0.999387i \(0.511147\pi\)
\(920\) 6.31464e36 0.566240
\(921\) 0 0
\(922\) −1.55071e36 −0.135330
\(923\) −2.43703e36 −0.209817
\(924\) 0 0
\(925\) −2.14061e37 −1.79377
\(926\) −1.13475e37 −0.938132
\(927\) 0 0
\(928\) 3.56463e36 0.286856
\(929\) −1.66420e37 −1.32132 −0.660662 0.750684i \(-0.729724\pi\)
−0.660662 + 0.750684i \(0.729724\pi\)
\(930\) 0 0
\(931\) 2.38121e36 0.184046
\(932\) 1.14887e36 0.0876138
\(933\) 0 0
\(934\) −9.49954e36 −0.705288
\(935\) −1.49983e37 −1.09874
\(936\) 0 0
\(937\) 9.66427e36 0.689325 0.344663 0.938727i \(-0.387993\pi\)
0.344663 + 0.938727i \(0.387993\pi\)
\(938\) 6.60622e36 0.464962
\(939\) 0 0
\(940\) 6.89172e36 0.472312
\(941\) 1.33808e37 0.904923 0.452461 0.891784i \(-0.350546\pi\)
0.452461 + 0.891784i \(0.350546\pi\)
\(942\) 0 0
\(943\) 7.91080e36 0.520984
\(944\) 3.72863e36 0.242326
\(945\) 0 0
\(946\) −3.26659e36 −0.206755
\(947\) −2.63078e37 −1.64328 −0.821638 0.570010i \(-0.806939\pi\)
−0.821638 + 0.570010i \(0.806939\pi\)
\(948\) 0 0
\(949\) −2.98981e36 −0.181893
\(950\) 1.05508e37 0.633489
\(951\) 0 0
\(952\) 6.14058e36 0.359126
\(953\) 9.10378e36 0.525485 0.262742 0.964866i \(-0.415373\pi\)
0.262742 + 0.964866i \(0.415373\pi\)
\(954\) 0 0
\(955\) −1.68184e37 −0.945677
\(956\) −6.50575e36 −0.361055
\(957\) 0 0
\(958\) 9.19044e36 0.496898
\(959\) 9.70142e36 0.517729
\(960\) 0 0
\(961\) −1.01623e36 −0.0528386
\(962\) −1.98307e36 −0.101777
\(963\) 0 0
\(964\) 1.74019e37 0.870230
\(965\) 5.24538e37 2.58932
\(966\) 0 0
\(967\) −1.34794e36 −0.0648394 −0.0324197 0.999474i \(-0.510321\pi\)
−0.0324197 + 0.999474i \(0.510321\pi\)
\(968\) 5.18457e36 0.246190
\(969\) 0 0
\(970\) 1.29060e37 0.597236
\(971\) −3.82504e36 −0.174741 −0.0873707 0.996176i \(-0.527846\pi\)
−0.0873707 + 0.996176i \(0.527846\pi\)
\(972\) 0 0
\(973\) −2.76831e36 −0.123255
\(974\) −1.52123e36 −0.0668663
\(975\) 0 0
\(976\) −6.67050e34 −0.00285782
\(977\) −1.16914e37 −0.494522 −0.247261 0.968949i \(-0.579530\pi\)
−0.247261 + 0.968949i \(0.579530\pi\)
\(978\) 0 0
\(979\) −1.84776e37 −0.761835
\(980\) 6.47014e36 0.263383
\(981\) 0 0
\(982\) −1.08752e37 −0.431563
\(983\) −1.38138e37 −0.541244 −0.270622 0.962686i \(-0.587229\pi\)
−0.270622 + 0.962686i \(0.587229\pi\)
\(984\) 0 0
\(985\) 5.48672e37 2.09585
\(986\) −3.76722e37 −1.42089
\(987\) 0 0
\(988\) 9.77426e35 0.0359436
\(989\) −1.45348e37 −0.527784
\(990\) 0 0
\(991\) −3.79186e37 −1.34255 −0.671276 0.741207i \(-0.734254\pi\)
−0.671276 + 0.741207i \(0.734254\pi\)
\(992\) 4.92077e36 0.172043
\(993\) 0 0
\(994\) −2.79338e37 −0.952356
\(995\) 2.84995e37 0.959510
\(996\) 0 0
\(997\) 5.34619e37 1.75531 0.877657 0.479289i \(-0.159106\pi\)
0.877657 + 0.479289i \(0.159106\pi\)
\(998\) 9.79424e36 0.317570
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 18.26.a.e.1.1 2
3.2 odd 2 2.26.a.b.1.1 2
12.11 even 2 16.26.a.c.1.2 2
15.2 even 4 50.26.b.e.49.4 4
15.8 even 4 50.26.b.e.49.1 4
15.14 odd 2 50.26.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.1 2 3.2 odd 2
16.26.a.c.1.2 2 12.11 even 2
18.26.a.e.1.1 2 1.1 even 1 trivial
50.26.a.c.1.2 2 15.14 odd 2
50.26.b.e.49.1 4 15.8 even 4
50.26.b.e.49.4 4 15.2 even 4