Properties

Label 48.22.a.h.1.1
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
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] = [48,22,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(134.149125258\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{537541}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 134385 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(367.086\) of defining polynomial
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-59049.0 q^{3} -3.80644e6 q^{5} +1.01851e9 q^{7} +3.48678e9 q^{9} +O(q^{10})\) \(q-59049.0 q^{3} -3.80644e6 q^{5} +1.01851e9 q^{7} +3.48678e9 q^{9} -1.06383e11 q^{11} +3.96922e11 q^{13} +2.24766e11 q^{15} -3.67718e12 q^{17} +7.39341e12 q^{19} -6.01419e13 q^{21} -2.08605e14 q^{23} -4.62348e14 q^{25} -2.05891e14 q^{27} -1.10405e15 q^{29} +7.87819e15 q^{31} +6.28179e15 q^{33} -3.87689e15 q^{35} +1.14668e16 q^{37} -2.34378e16 q^{39} +9.57394e15 q^{41} +1.54549e17 q^{43} -1.32722e16 q^{45} -2.63961e17 q^{47} +4.78814e17 q^{49} +2.17134e17 q^{51} -1.43334e18 q^{53} +4.04939e17 q^{55} -4.36574e17 q^{57} +5.89337e18 q^{59} +2.96513e18 q^{61} +3.55132e18 q^{63} -1.51086e18 q^{65} +1.75878e19 q^{67} +1.23179e19 q^{69} -3.35158e19 q^{71} -6.12183e19 q^{73} +2.73012e19 q^{75} -1.08352e20 q^{77} +5.83561e19 q^{79} +1.21577e19 q^{81} +1.25379e20 q^{83} +1.39970e19 q^{85} +6.51929e19 q^{87} -4.73528e20 q^{89} +4.04268e20 q^{91} -4.65200e20 q^{93} -2.81426e19 q^{95} -2.83797e20 q^{97} -3.70933e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 118098 q^{3} + 21948620 q^{5} + 659451408 q^{7} + 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 118098 q^{3} + 21948620 q^{5} + 659451408 q^{7} + 6973568802 q^{9} + 29910896872 q^{11} + 4468866812 q^{13} - 1296044062380 q^{15} - 17665404721820 q^{17} + 22467979297496 q^{19} - 38939946190992 q^{21} - 120995273049584 q^{23} - 275862509554850 q^{25} - 411782264189298 q^{27} - 33\!\cdots\!84 q^{29}+ \cdots + 10\!\cdots\!72 q^{99}+O(q^{100}) \) 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
\(3\) −59049.0 −0.577350
\(4\) 0 0
\(5\) −3.80644e6 −0.174314 −0.0871572 0.996195i \(-0.527778\pi\)
−0.0871572 + 0.996195i \(0.527778\pi\)
\(6\) 0 0
\(7\) 1.01851e9 1.36281 0.681405 0.731907i \(-0.261369\pi\)
0.681405 + 0.731907i \(0.261369\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −1.06383e11 −1.23665 −0.618326 0.785922i \(-0.712189\pi\)
−0.618326 + 0.785922i \(0.712189\pi\)
\(12\) 0 0
\(13\) 3.96922e11 0.798545 0.399273 0.916832i \(-0.369263\pi\)
0.399273 + 0.916832i \(0.369263\pi\)
\(14\) 0 0
\(15\) 2.24766e11 0.100640
\(16\) 0 0
\(17\) −3.67718e12 −0.442386 −0.221193 0.975230i \(-0.570995\pi\)
−0.221193 + 0.975230i \(0.570995\pi\)
\(18\) 0 0
\(19\) 7.39341e12 0.276651 0.138325 0.990387i \(-0.455828\pi\)
0.138325 + 0.990387i \(0.455828\pi\)
\(20\) 0 0
\(21\) −6.01419e13 −0.786819
\(22\) 0 0
\(23\) −2.08605e14 −1.04998 −0.524992 0.851107i \(-0.675932\pi\)
−0.524992 + 0.851107i \(0.675932\pi\)
\(24\) 0 0
\(25\) −4.62348e14 −0.969614
\(26\) 0 0
\(27\) −2.05891e14 −0.192450
\(28\) 0 0
\(29\) −1.10405e15 −0.487314 −0.243657 0.969862i \(-0.578347\pi\)
−0.243657 + 0.969862i \(0.578347\pi\)
\(30\) 0 0
\(31\) 7.87819e15 1.72635 0.863175 0.504905i \(-0.168472\pi\)
0.863175 + 0.504905i \(0.168472\pi\)
\(32\) 0 0
\(33\) 6.28179e15 0.713981
\(34\) 0 0
\(35\) −3.87689e15 −0.237557
\(36\) 0 0
\(37\) 1.14668e16 0.392034 0.196017 0.980601i \(-0.437199\pi\)
0.196017 + 0.980601i \(0.437199\pi\)
\(38\) 0 0
\(39\) −2.34378e16 −0.461040
\(40\) 0 0
\(41\) 9.57394e15 0.111394 0.0556968 0.998448i \(-0.482262\pi\)
0.0556968 + 0.998448i \(0.482262\pi\)
\(42\) 0 0
\(43\) 1.54549e17 1.09056 0.545278 0.838255i \(-0.316424\pi\)
0.545278 + 0.838255i \(0.316424\pi\)
\(44\) 0 0
\(45\) −1.32722e16 −0.0581048
\(46\) 0 0
\(47\) −2.63961e17 −0.732001 −0.366001 0.930615i \(-0.619273\pi\)
−0.366001 + 0.930615i \(0.619273\pi\)
\(48\) 0 0
\(49\) 4.78814e17 0.857250
\(50\) 0 0
\(51\) 2.17134e17 0.255412
\(52\) 0 0
\(53\) −1.43334e18 −1.12577 −0.562887 0.826534i \(-0.690309\pi\)
−0.562887 + 0.826534i \(0.690309\pi\)
\(54\) 0 0
\(55\) 4.04939e17 0.215566
\(56\) 0 0
\(57\) −4.36574e17 −0.159725
\(58\) 0 0
\(59\) 5.89337e18 1.50113 0.750564 0.660798i \(-0.229782\pi\)
0.750564 + 0.660798i \(0.229782\pi\)
\(60\) 0 0
\(61\) 2.96513e18 0.532207 0.266104 0.963944i \(-0.414264\pi\)
0.266104 + 0.963944i \(0.414264\pi\)
\(62\) 0 0
\(63\) 3.55132e18 0.454270
\(64\) 0 0
\(65\) −1.51086e18 −0.139198
\(66\) 0 0
\(67\) 1.75878e19 1.17876 0.589380 0.807856i \(-0.299372\pi\)
0.589380 + 0.807856i \(0.299372\pi\)
\(68\) 0 0
\(69\) 1.23179e19 0.606208
\(70\) 0 0
\(71\) −3.35158e19 −1.22190 −0.610952 0.791668i \(-0.709213\pi\)
−0.610952 + 0.791668i \(0.709213\pi\)
\(72\) 0 0
\(73\) −6.12183e19 −1.66721 −0.833607 0.552358i \(-0.813728\pi\)
−0.833607 + 0.552358i \(0.813728\pi\)
\(74\) 0 0
\(75\) 2.73012e19 0.559807
\(76\) 0 0
\(77\) −1.08352e20 −1.68532
\(78\) 0 0
\(79\) 5.83561e19 0.693429 0.346714 0.937971i \(-0.387297\pi\)
0.346714 + 0.937971i \(0.387297\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) 1.25379e20 0.886958 0.443479 0.896285i \(-0.353744\pi\)
0.443479 + 0.896285i \(0.353744\pi\)
\(84\) 0 0
\(85\) 1.39970e19 0.0771143
\(86\) 0 0
\(87\) 6.51929e19 0.281351
\(88\) 0 0
\(89\) −4.73528e20 −1.60972 −0.804860 0.593465i \(-0.797760\pi\)
−0.804860 + 0.593465i \(0.797760\pi\)
\(90\) 0 0
\(91\) 4.04268e20 1.08827
\(92\) 0 0
\(93\) −4.65200e20 −0.996709
\(94\) 0 0
\(95\) −2.81426e19 −0.0482243
\(96\) 0 0
\(97\) −2.83797e20 −0.390756 −0.195378 0.980728i \(-0.562593\pi\)
−0.195378 + 0.980728i \(0.562593\pi\)
\(98\) 0 0
\(99\) −3.70933e20 −0.412217
\(100\) 0 0
\(101\) 4.60683e20 0.414980 0.207490 0.978237i \(-0.433470\pi\)
0.207490 + 0.978237i \(0.433470\pi\)
\(102\) 0 0
\(103\) −9.01928e20 −0.661273 −0.330637 0.943758i \(-0.607263\pi\)
−0.330637 + 0.943758i \(0.607263\pi\)
\(104\) 0 0
\(105\) 2.28926e20 0.137154
\(106\) 0 0
\(107\) 2.07173e21 1.01813 0.509065 0.860728i \(-0.329991\pi\)
0.509065 + 0.860728i \(0.329991\pi\)
\(108\) 0 0
\(109\) −1.82773e21 −0.739492 −0.369746 0.929133i \(-0.620555\pi\)
−0.369746 + 0.929133i \(0.620555\pi\)
\(110\) 0 0
\(111\) −6.77102e20 −0.226341
\(112\) 0 0
\(113\) 2.45088e21 0.679201 0.339601 0.940570i \(-0.389708\pi\)
0.339601 + 0.940570i \(0.389708\pi\)
\(114\) 0 0
\(115\) 7.94041e20 0.183027
\(116\) 0 0
\(117\) 1.38398e21 0.266182
\(118\) 0 0
\(119\) −3.74524e21 −0.602888
\(120\) 0 0
\(121\) 3.91702e21 0.529309
\(122\) 0 0
\(123\) −5.65332e20 −0.0643132
\(124\) 0 0
\(125\) 3.57495e21 0.343332
\(126\) 0 0
\(127\) −1.85246e22 −1.50595 −0.752976 0.658048i \(-0.771382\pi\)
−0.752976 + 0.658048i \(0.771382\pi\)
\(128\) 0 0
\(129\) −9.12598e21 −0.629633
\(130\) 0 0
\(131\) −2.23574e22 −1.31242 −0.656210 0.754579i \(-0.727841\pi\)
−0.656210 + 0.754579i \(0.727841\pi\)
\(132\) 0 0
\(133\) 7.53025e21 0.377023
\(134\) 0 0
\(135\) 7.83711e20 0.0335468
\(136\) 0 0
\(137\) −2.29591e22 −0.842151 −0.421075 0.907026i \(-0.638347\pi\)
−0.421075 + 0.907026i \(0.638347\pi\)
\(138\) 0 0
\(139\) −4.16594e22 −1.31237 −0.656187 0.754598i \(-0.727832\pi\)
−0.656187 + 0.754598i \(0.727832\pi\)
\(140\) 0 0
\(141\) 1.55866e22 0.422621
\(142\) 0 0
\(143\) −4.22256e22 −0.987523
\(144\) 0 0
\(145\) 4.20248e21 0.0849458
\(146\) 0 0
\(147\) −2.82735e22 −0.494934
\(148\) 0 0
\(149\) −4.33948e22 −0.659147 −0.329573 0.944130i \(-0.606905\pi\)
−0.329573 + 0.944130i \(0.606905\pi\)
\(150\) 0 0
\(151\) 8.01728e22 1.05869 0.529346 0.848406i \(-0.322437\pi\)
0.529346 + 0.848406i \(0.322437\pi\)
\(152\) 0 0
\(153\) −1.28215e22 −0.147462
\(154\) 0 0
\(155\) −2.99878e22 −0.300928
\(156\) 0 0
\(157\) −1.91718e23 −1.68157 −0.840787 0.541366i \(-0.817907\pi\)
−0.840787 + 0.541366i \(0.817907\pi\)
\(158\) 0 0
\(159\) 8.46370e22 0.649966
\(160\) 0 0
\(161\) −2.12466e23 −1.43093
\(162\) 0 0
\(163\) −1.98873e23 −1.17654 −0.588269 0.808665i \(-0.700191\pi\)
−0.588269 + 0.808665i \(0.700191\pi\)
\(164\) 0 0
\(165\) −2.39112e22 −0.124457
\(166\) 0 0
\(167\) −4.01098e22 −0.183962 −0.0919808 0.995761i \(-0.529320\pi\)
−0.0919808 + 0.995761i \(0.529320\pi\)
\(168\) 0 0
\(169\) −8.95177e22 −0.362325
\(170\) 0 0
\(171\) 2.57792e22 0.0922170
\(172\) 0 0
\(173\) −1.87093e23 −0.592344 −0.296172 0.955135i \(-0.595710\pi\)
−0.296172 + 0.955135i \(0.595710\pi\)
\(174\) 0 0
\(175\) −4.70906e23 −1.32140
\(176\) 0 0
\(177\) −3.47998e23 −0.866676
\(178\) 0 0
\(179\) 7.94050e23 1.75748 0.878742 0.477298i \(-0.158384\pi\)
0.878742 + 0.477298i \(0.158384\pi\)
\(180\) 0 0
\(181\) −1.89362e23 −0.372964 −0.186482 0.982458i \(-0.559709\pi\)
−0.186482 + 0.982458i \(0.559709\pi\)
\(182\) 0 0
\(183\) −1.75088e23 −0.307270
\(184\) 0 0
\(185\) −4.36476e22 −0.0683371
\(186\) 0 0
\(187\) 3.91189e23 0.547078
\(188\) 0 0
\(189\) −2.09702e23 −0.262273
\(190\) 0 0
\(191\) −1.17670e24 −1.31770 −0.658848 0.752276i \(-0.728956\pi\)
−0.658848 + 0.752276i \(0.728956\pi\)
\(192\) 0 0
\(193\) 1.30716e23 0.131213 0.0656064 0.997846i \(-0.479102\pi\)
0.0656064 + 0.997846i \(0.479102\pi\)
\(194\) 0 0
\(195\) 8.92146e22 0.0803660
\(196\) 0 0
\(197\) −7.43917e23 −0.602045 −0.301023 0.953617i \(-0.597328\pi\)
−0.301023 + 0.953617i \(0.597328\pi\)
\(198\) 0 0
\(199\) −2.02892e24 −1.47675 −0.738377 0.674388i \(-0.764408\pi\)
−0.738377 + 0.674388i \(0.764408\pi\)
\(200\) 0 0
\(201\) −1.03854e24 −0.680557
\(202\) 0 0
\(203\) −1.12448e24 −0.664116
\(204\) 0 0
\(205\) −3.64426e22 −0.0194175
\(206\) 0 0
\(207\) −7.27361e23 −0.349994
\(208\) 0 0
\(209\) −7.86531e23 −0.342121
\(210\) 0 0
\(211\) −2.73877e24 −1.07793 −0.538964 0.842329i \(-0.681184\pi\)
−0.538964 + 0.842329i \(0.681184\pi\)
\(212\) 0 0
\(213\) 1.97907e24 0.705466
\(214\) 0 0
\(215\) −5.88281e23 −0.190100
\(216\) 0 0
\(217\) 8.02401e24 2.35269
\(218\) 0 0
\(219\) 3.61488e24 0.962566
\(220\) 0 0
\(221\) −1.45955e24 −0.353266
\(222\) 0 0
\(223\) −4.35130e24 −0.958116 −0.479058 0.877783i \(-0.659022\pi\)
−0.479058 + 0.877783i \(0.659022\pi\)
\(224\) 0 0
\(225\) −1.61211e24 −0.323205
\(226\) 0 0
\(227\) 7.67094e23 0.140145 0.0700724 0.997542i \(-0.477677\pi\)
0.0700724 + 0.997542i \(0.477677\pi\)
\(228\) 0 0
\(229\) 7.54631e24 1.25737 0.628684 0.777661i \(-0.283594\pi\)
0.628684 + 0.777661i \(0.283594\pi\)
\(230\) 0 0
\(231\) 6.39805e24 0.973021
\(232\) 0 0
\(233\) 2.10251e24 0.292079 0.146039 0.989279i \(-0.453347\pi\)
0.146039 + 0.989279i \(0.453347\pi\)
\(234\) 0 0
\(235\) 1.00475e24 0.127598
\(236\) 0 0
\(237\) −3.44587e24 −0.400351
\(238\) 0 0
\(239\) 1.45098e25 1.54342 0.771709 0.635976i \(-0.219402\pi\)
0.771709 + 0.635976i \(0.219402\pi\)
\(240\) 0 0
\(241\) −3.56087e24 −0.347038 −0.173519 0.984831i \(-0.555514\pi\)
−0.173519 + 0.984831i \(0.555514\pi\)
\(242\) 0 0
\(243\) −7.17898e23 −0.0641500
\(244\) 0 0
\(245\) −1.82257e24 −0.149431
\(246\) 0 0
\(247\) 2.93461e24 0.220918
\(248\) 0 0
\(249\) −7.40348e24 −0.512086
\(250\) 0 0
\(251\) 2.01810e25 1.28342 0.641710 0.766948i \(-0.278225\pi\)
0.641710 + 0.766948i \(0.278225\pi\)
\(252\) 0 0
\(253\) 2.21920e25 1.29846
\(254\) 0 0
\(255\) −8.26507e23 −0.0445220
\(256\) 0 0
\(257\) 8.05388e24 0.399675 0.199838 0.979829i \(-0.435959\pi\)
0.199838 + 0.979829i \(0.435959\pi\)
\(258\) 0 0
\(259\) 1.16790e25 0.534267
\(260\) 0 0
\(261\) −3.84957e24 −0.162438
\(262\) 0 0
\(263\) 1.45231e25 0.565618 0.282809 0.959176i \(-0.408734\pi\)
0.282809 + 0.959176i \(0.408734\pi\)
\(264\) 0 0
\(265\) 5.45590e24 0.196239
\(266\) 0 0
\(267\) 2.79613e25 0.929372
\(268\) 0 0
\(269\) −5.59528e25 −1.71958 −0.859792 0.510645i \(-0.829407\pi\)
−0.859792 + 0.510645i \(0.829407\pi\)
\(270\) 0 0
\(271\) −3.65662e25 −1.03969 −0.519844 0.854261i \(-0.674010\pi\)
−0.519844 + 0.854261i \(0.674010\pi\)
\(272\) 0 0
\(273\) −2.38716e25 −0.628310
\(274\) 0 0
\(275\) 4.91858e25 1.19908
\(276\) 0 0
\(277\) 8.21467e25 1.85589 0.927946 0.372715i \(-0.121573\pi\)
0.927946 + 0.372715i \(0.121573\pi\)
\(278\) 0 0
\(279\) 2.74696e25 0.575450
\(280\) 0 0
\(281\) 9.16736e25 1.78167 0.890837 0.454324i \(-0.150119\pi\)
0.890837 + 0.454324i \(0.150119\pi\)
\(282\) 0 0
\(283\) −5.45902e25 −0.984821 −0.492411 0.870363i \(-0.663884\pi\)
−0.492411 + 0.870363i \(0.663884\pi\)
\(284\) 0 0
\(285\) 1.66179e24 0.0278423
\(286\) 0 0
\(287\) 9.75114e24 0.151808
\(288\) 0 0
\(289\) −5.55702e25 −0.804294
\(290\) 0 0
\(291\) 1.67580e25 0.225603
\(292\) 0 0
\(293\) −1.07858e26 −1.35127 −0.675635 0.737237i \(-0.736130\pi\)
−0.675635 + 0.737237i \(0.736130\pi\)
\(294\) 0 0
\(295\) −2.24327e25 −0.261668
\(296\) 0 0
\(297\) 2.19032e25 0.237994
\(298\) 0 0
\(299\) −8.27999e25 −0.838460
\(300\) 0 0
\(301\) 1.57410e26 1.48622
\(302\) 0 0
\(303\) −2.72029e25 −0.239589
\(304\) 0 0
\(305\) −1.12866e25 −0.0927714
\(306\) 0 0
\(307\) −1.57928e25 −0.121201 −0.0606007 0.998162i \(-0.519302\pi\)
−0.0606007 + 0.998162i \(0.519302\pi\)
\(308\) 0 0
\(309\) 5.32579e25 0.381786
\(310\) 0 0
\(311\) −2.16533e26 −1.45057 −0.725286 0.688447i \(-0.758293\pi\)
−0.725286 + 0.688447i \(0.758293\pi\)
\(312\) 0 0
\(313\) 2.08285e26 1.30450 0.652249 0.758005i \(-0.273826\pi\)
0.652249 + 0.758005i \(0.273826\pi\)
\(314\) 0 0
\(315\) −1.35179e25 −0.0791858
\(316\) 0 0
\(317\) 1.51043e26 0.827900 0.413950 0.910300i \(-0.364149\pi\)
0.413950 + 0.910300i \(0.364149\pi\)
\(318\) 0 0
\(319\) 1.17451e26 0.602638
\(320\) 0 0
\(321\) −1.22333e26 −0.587817
\(322\) 0 0
\(323\) −2.71869e25 −0.122387
\(324\) 0 0
\(325\) −1.83516e26 −0.774281
\(326\) 0 0
\(327\) 1.07925e26 0.426946
\(328\) 0 0
\(329\) −2.68846e26 −0.997579
\(330\) 0 0
\(331\) −3.02767e26 −1.05418 −0.527090 0.849809i \(-0.676717\pi\)
−0.527090 + 0.849809i \(0.676717\pi\)
\(332\) 0 0
\(333\) 3.99822e25 0.130678
\(334\) 0 0
\(335\) −6.69467e25 −0.205475
\(336\) 0 0
\(337\) −5.07259e26 −1.46257 −0.731284 0.682073i \(-0.761079\pi\)
−0.731284 + 0.682073i \(0.761079\pi\)
\(338\) 0 0
\(339\) −1.44722e26 −0.392137
\(340\) 0 0
\(341\) −8.38103e26 −2.13489
\(342\) 0 0
\(343\) −8.12080e25 −0.194541
\(344\) 0 0
\(345\) −4.68873e25 −0.105671
\(346\) 0 0
\(347\) −1.92246e26 −0.407753 −0.203876 0.978997i \(-0.565354\pi\)
−0.203876 + 0.978997i \(0.565354\pi\)
\(348\) 0 0
\(349\) −4.69006e26 −0.936507 −0.468254 0.883594i \(-0.655117\pi\)
−0.468254 + 0.883594i \(0.655117\pi\)
\(350\) 0 0
\(351\) −8.17227e25 −0.153680
\(352\) 0 0
\(353\) 5.16954e26 0.915836 0.457918 0.888994i \(-0.348595\pi\)
0.457918 + 0.888994i \(0.348595\pi\)
\(354\) 0 0
\(355\) 1.27576e26 0.212995
\(356\) 0 0
\(357\) 2.21153e26 0.348078
\(358\) 0 0
\(359\) 8.26652e26 1.22696 0.613481 0.789710i \(-0.289769\pi\)
0.613481 + 0.789710i \(0.289769\pi\)
\(360\) 0 0
\(361\) −6.59547e26 −0.923464
\(362\) 0 0
\(363\) −2.31296e26 −0.305597
\(364\) 0 0
\(365\) 2.33023e26 0.290619
\(366\) 0 0
\(367\) 1.35866e27 1.59999 0.799995 0.600007i \(-0.204835\pi\)
0.799995 + 0.600007i \(0.204835\pi\)
\(368\) 0 0
\(369\) 3.33823e25 0.0371312
\(370\) 0 0
\(371\) −1.45986e27 −1.53422
\(372\) 0 0
\(373\) −1.22128e27 −1.21303 −0.606517 0.795071i \(-0.707434\pi\)
−0.606517 + 0.795071i \(0.707434\pi\)
\(374\) 0 0
\(375\) −2.11097e26 −0.198223
\(376\) 0 0
\(377\) −4.38220e26 −0.389142
\(378\) 0 0
\(379\) 6.57479e26 0.552294 0.276147 0.961115i \(-0.410942\pi\)
0.276147 + 0.961115i \(0.410942\pi\)
\(380\) 0 0
\(381\) 1.09386e27 0.869461
\(382\) 0 0
\(383\) −1.90325e27 −1.43188 −0.715942 0.698159i \(-0.754003\pi\)
−0.715942 + 0.698159i \(0.754003\pi\)
\(384\) 0 0
\(385\) 4.12433e26 0.293776
\(386\) 0 0
\(387\) 5.38880e26 0.363519
\(388\) 0 0
\(389\) 5.12948e26 0.327795 0.163897 0.986477i \(-0.447593\pi\)
0.163897 + 0.986477i \(0.447593\pi\)
\(390\) 0 0
\(391\) 7.67079e26 0.464498
\(392\) 0 0
\(393\) 1.32018e27 0.757726
\(394\) 0 0
\(395\) −2.22129e26 −0.120875
\(396\) 0 0
\(397\) 2.36102e27 1.21843 0.609213 0.793007i \(-0.291485\pi\)
0.609213 + 0.793007i \(0.291485\pi\)
\(398\) 0 0
\(399\) −4.44654e26 −0.217674
\(400\) 0 0
\(401\) −3.27592e27 −1.52166 −0.760829 0.648952i \(-0.775208\pi\)
−0.760829 + 0.648952i \(0.775208\pi\)
\(402\) 0 0
\(403\) 3.12703e27 1.37857
\(404\) 0 0
\(405\) −4.62774e25 −0.0193683
\(406\) 0 0
\(407\) −1.21987e27 −0.484809
\(408\) 0 0
\(409\) 4.16749e27 1.57318 0.786592 0.617473i \(-0.211844\pi\)
0.786592 + 0.617473i \(0.211844\pi\)
\(410\) 0 0
\(411\) 1.35571e27 0.486216
\(412\) 0 0
\(413\) 6.00245e27 2.04575
\(414\) 0 0
\(415\) −4.77245e26 −0.154610
\(416\) 0 0
\(417\) 2.45995e27 0.757700
\(418\) 0 0
\(419\) −2.60990e27 −0.764497 −0.382249 0.924060i \(-0.624850\pi\)
−0.382249 + 0.924060i \(0.624850\pi\)
\(420\) 0 0
\(421\) −2.10137e27 −0.585518 −0.292759 0.956186i \(-0.594573\pi\)
−0.292759 + 0.956186i \(0.594573\pi\)
\(422\) 0 0
\(423\) −9.20374e26 −0.244000
\(424\) 0 0
\(425\) 1.70014e27 0.428944
\(426\) 0 0
\(427\) 3.02001e27 0.725297
\(428\) 0 0
\(429\) 2.49338e27 0.570147
\(430\) 0 0
\(431\) −3.31265e27 −0.721380 −0.360690 0.932686i \(-0.617459\pi\)
−0.360690 + 0.932686i \(0.617459\pi\)
\(432\) 0 0
\(433\) −4.60260e27 −0.954730 −0.477365 0.878705i \(-0.658408\pi\)
−0.477365 + 0.878705i \(0.658408\pi\)
\(434\) 0 0
\(435\) −2.48152e26 −0.0490435
\(436\) 0 0
\(437\) −1.54230e27 −0.290479
\(438\) 0 0
\(439\) 2.40231e27 0.431273 0.215637 0.976474i \(-0.430817\pi\)
0.215637 + 0.976474i \(0.430817\pi\)
\(440\) 0 0
\(441\) 1.66952e27 0.285750
\(442\) 0 0
\(443\) 9.24244e27 1.50851 0.754254 0.656583i \(-0.227999\pi\)
0.754254 + 0.656583i \(0.227999\pi\)
\(444\) 0 0
\(445\) 1.80245e27 0.280597
\(446\) 0 0
\(447\) 2.56242e27 0.380559
\(448\) 0 0
\(449\) −6.68141e27 −0.946851 −0.473426 0.880834i \(-0.656983\pi\)
−0.473426 + 0.880834i \(0.656983\pi\)
\(450\) 0 0
\(451\) −1.01850e27 −0.137755
\(452\) 0 0
\(453\) −4.73413e27 −0.611236
\(454\) 0 0
\(455\) −1.53882e27 −0.189700
\(456\) 0 0
\(457\) 1.53406e27 0.180602 0.0903011 0.995915i \(-0.471217\pi\)
0.0903011 + 0.995915i \(0.471217\pi\)
\(458\) 0 0
\(459\) 7.57100e26 0.0851373
\(460\) 0 0
\(461\) −8.04683e27 −0.864500 −0.432250 0.901754i \(-0.642280\pi\)
−0.432250 + 0.901754i \(0.642280\pi\)
\(462\) 0 0
\(463\) 1.43789e28 1.47613 0.738067 0.674727i \(-0.235739\pi\)
0.738067 + 0.674727i \(0.235739\pi\)
\(464\) 0 0
\(465\) 1.77075e27 0.173741
\(466\) 0 0
\(467\) −3.22594e27 −0.302573 −0.151286 0.988490i \(-0.548342\pi\)
−0.151286 + 0.988490i \(0.548342\pi\)
\(468\) 0 0
\(469\) 1.79133e28 1.60642
\(470\) 0 0
\(471\) 1.13207e28 0.970857
\(472\) 0 0
\(473\) −1.64414e28 −1.34864
\(474\) 0 0
\(475\) −3.41833e27 −0.268245
\(476\) 0 0
\(477\) −4.99773e27 −0.375258
\(478\) 0 0
\(479\) 3.52220e27 0.253099 0.126550 0.991960i \(-0.459610\pi\)
0.126550 + 0.991960i \(0.459610\pi\)
\(480\) 0 0
\(481\) 4.55141e27 0.313057
\(482\) 0 0
\(483\) 1.25459e28 0.826146
\(484\) 0 0
\(485\) 1.08026e27 0.0681143
\(486\) 0 0
\(487\) −5.18350e27 −0.313018 −0.156509 0.987677i \(-0.550024\pi\)
−0.156509 + 0.987677i \(0.550024\pi\)
\(488\) 0 0
\(489\) 1.17433e28 0.679275
\(490\) 0 0
\(491\) −2.74020e28 −1.51854 −0.759270 0.650776i \(-0.774444\pi\)
−0.759270 + 0.650776i \(0.774444\pi\)
\(492\) 0 0
\(493\) 4.05978e27 0.215581
\(494\) 0 0
\(495\) 1.41193e27 0.0718554
\(496\) 0 0
\(497\) −3.41361e28 −1.66522
\(498\) 0 0
\(499\) −1.08162e28 −0.505848 −0.252924 0.967486i \(-0.581392\pi\)
−0.252924 + 0.967486i \(0.581392\pi\)
\(500\) 0 0
\(501\) 2.36844e27 0.106210
\(502\) 0 0
\(503\) −3.89398e28 −1.67467 −0.837337 0.546687i \(-0.815889\pi\)
−0.837337 + 0.546687i \(0.815889\pi\)
\(504\) 0 0
\(505\) −1.75356e27 −0.0723371
\(506\) 0 0
\(507\) 5.28593e27 0.209189
\(508\) 0 0
\(509\) −2.89861e28 −1.10066 −0.550329 0.834948i \(-0.685498\pi\)
−0.550329 + 0.834948i \(0.685498\pi\)
\(510\) 0 0
\(511\) −6.23514e28 −2.27209
\(512\) 0 0
\(513\) −1.52224e27 −0.0532415
\(514\) 0 0
\(515\) 3.43313e27 0.115269
\(516\) 0 0
\(517\) 2.80808e28 0.905231
\(518\) 0 0
\(519\) 1.10477e28 0.341990
\(520\) 0 0
\(521\) −3.41212e28 −1.01444 −0.507222 0.861816i \(-0.669327\pi\)
−0.507222 + 0.861816i \(0.669327\pi\)
\(522\) 0 0
\(523\) 1.37991e28 0.394078 0.197039 0.980396i \(-0.436867\pi\)
0.197039 + 0.980396i \(0.436867\pi\)
\(524\) 0 0
\(525\) 2.78065e28 0.762911
\(526\) 0 0
\(527\) −2.89696e28 −0.763714
\(528\) 0 0
\(529\) 4.04446e27 0.102465
\(530\) 0 0
\(531\) 2.05489e28 0.500376
\(532\) 0 0
\(533\) 3.80011e27 0.0889529
\(534\) 0 0
\(535\) −7.88589e27 −0.177475
\(536\) 0 0
\(537\) −4.68879e28 −1.01468
\(538\) 0 0
\(539\) −5.09375e28 −1.06012
\(540\) 0 0
\(541\) 5.75791e28 1.15264 0.576320 0.817224i \(-0.304488\pi\)
0.576320 + 0.817224i \(0.304488\pi\)
\(542\) 0 0
\(543\) 1.11816e28 0.215331
\(544\) 0 0
\(545\) 6.95713e27 0.128904
\(546\) 0 0
\(547\) −7.41040e28 −1.32122 −0.660609 0.750730i \(-0.729702\pi\)
−0.660609 + 0.750730i \(0.729702\pi\)
\(548\) 0 0
\(549\) 1.03388e28 0.177402
\(550\) 0 0
\(551\) −8.16268e27 −0.134816
\(552\) 0 0
\(553\) 5.94362e28 0.945012
\(554\) 0 0
\(555\) 2.57734e27 0.0394545
\(556\) 0 0
\(557\) −7.21534e28 −1.06360 −0.531799 0.846871i \(-0.678484\pi\)
−0.531799 + 0.846871i \(0.678484\pi\)
\(558\) 0 0
\(559\) 6.13439e28 0.870859
\(560\) 0 0
\(561\) −2.30993e28 −0.315856
\(562\) 0 0
\(563\) 7.75204e27 0.102112 0.0510561 0.998696i \(-0.483741\pi\)
0.0510561 + 0.998696i \(0.483741\pi\)
\(564\) 0 0
\(565\) −9.32911e27 −0.118395
\(566\) 0 0
\(567\) 1.23827e28 0.151423
\(568\) 0 0
\(569\) −1.42718e29 −1.68190 −0.840952 0.541109i \(-0.818005\pi\)
−0.840952 + 0.541109i \(0.818005\pi\)
\(570\) 0 0
\(571\) 1.19587e29 1.35833 0.679165 0.733986i \(-0.262342\pi\)
0.679165 + 0.733986i \(0.262342\pi\)
\(572\) 0 0
\(573\) 6.94829e28 0.760772
\(574\) 0 0
\(575\) 9.64481e28 1.01808
\(576\) 0 0
\(577\) −1.02439e29 −1.04261 −0.521304 0.853371i \(-0.674554\pi\)
−0.521304 + 0.853371i \(0.674554\pi\)
\(578\) 0 0
\(579\) −7.71864e27 −0.0757557
\(580\) 0 0
\(581\) 1.27699e29 1.20876
\(582\) 0 0
\(583\) 1.52482e29 1.39219
\(584\) 0 0
\(585\) −5.26803e27 −0.0463993
\(586\) 0 0
\(587\) −8.48071e28 −0.720663 −0.360332 0.932824i \(-0.617336\pi\)
−0.360332 + 0.932824i \(0.617336\pi\)
\(588\) 0 0
\(589\) 5.82468e28 0.477597
\(590\) 0 0
\(591\) 4.39276e28 0.347591
\(592\) 0 0
\(593\) 1.14783e29 0.876601 0.438300 0.898829i \(-0.355581\pi\)
0.438300 + 0.898829i \(0.355581\pi\)
\(594\) 0 0
\(595\) 1.42560e28 0.105092
\(596\) 0 0
\(597\) 1.19806e29 0.852605
\(598\) 0 0
\(599\) 1.72380e29 1.18442 0.592208 0.805785i \(-0.298256\pi\)
0.592208 + 0.805785i \(0.298256\pi\)
\(600\) 0 0
\(601\) −2.73184e29 −1.81248 −0.906241 0.422761i \(-0.861061\pi\)
−0.906241 + 0.422761i \(0.861061\pi\)
\(602\) 0 0
\(603\) 6.13247e28 0.392920
\(604\) 0 0
\(605\) −1.49099e28 −0.0922662
\(606\) 0 0
\(607\) 2.31744e29 1.38525 0.692625 0.721298i \(-0.256454\pi\)
0.692625 + 0.721298i \(0.256454\pi\)
\(608\) 0 0
\(609\) 6.63995e28 0.383427
\(610\) 0 0
\(611\) −1.04772e29 −0.584536
\(612\) 0 0
\(613\) 2.48641e29 1.34041 0.670205 0.742176i \(-0.266206\pi\)
0.670205 + 0.742176i \(0.266206\pi\)
\(614\) 0 0
\(615\) 2.15190e27 0.0112107
\(616\) 0 0
\(617\) −2.47882e29 −1.24810 −0.624052 0.781383i \(-0.714514\pi\)
−0.624052 + 0.781383i \(0.714514\pi\)
\(618\) 0 0
\(619\) 5.18780e28 0.252483 0.126241 0.992000i \(-0.459709\pi\)
0.126241 + 0.992000i \(0.459709\pi\)
\(620\) 0 0
\(621\) 4.29499e28 0.202069
\(622\) 0 0
\(623\) −4.82292e29 −2.19374
\(624\) 0 0
\(625\) 2.06857e29 0.909767
\(626\) 0 0
\(627\) 4.64439e28 0.197524
\(628\) 0 0
\(629\) −4.21655e28 −0.173430
\(630\) 0 0
\(631\) −2.91890e29 −1.16121 −0.580605 0.814186i \(-0.697184\pi\)
−0.580605 + 0.814186i \(0.697184\pi\)
\(632\) 0 0
\(633\) 1.61722e29 0.622342
\(634\) 0 0
\(635\) 7.05129e28 0.262509
\(636\) 0 0
\(637\) 1.90051e29 0.684553
\(638\) 0 0
\(639\) −1.16862e29 −0.407301
\(640\) 0 0
\(641\) 3.18026e29 1.07264 0.536319 0.844015i \(-0.319814\pi\)
0.536319 + 0.844015i \(0.319814\pi\)
\(642\) 0 0
\(643\) 4.41840e29 1.44228 0.721140 0.692790i \(-0.243619\pi\)
0.721140 + 0.692790i \(0.243619\pi\)
\(644\) 0 0
\(645\) 3.47374e28 0.109754
\(646\) 0 0
\(647\) 5.85233e28 0.178992 0.0894960 0.995987i \(-0.471474\pi\)
0.0894960 + 0.995987i \(0.471474\pi\)
\(648\) 0 0
\(649\) −6.26952e29 −1.85637
\(650\) 0 0
\(651\) −4.73810e29 −1.35832
\(652\) 0 0
\(653\) 6.75541e29 1.87527 0.937634 0.347623i \(-0.113011\pi\)
0.937634 + 0.347623i \(0.113011\pi\)
\(654\) 0 0
\(655\) 8.51020e28 0.228774
\(656\) 0 0
\(657\) −2.13455e29 −0.555738
\(658\) 0 0
\(659\) −5.16262e28 −0.130189 −0.0650943 0.997879i \(-0.520735\pi\)
−0.0650943 + 0.997879i \(0.520735\pi\)
\(660\) 0 0
\(661\) 1.39834e29 0.341583 0.170792 0.985307i \(-0.445367\pi\)
0.170792 + 0.985307i \(0.445367\pi\)
\(662\) 0 0
\(663\) 8.61852e28 0.203958
\(664\) 0 0
\(665\) −2.86634e28 −0.0657205
\(666\) 0 0
\(667\) 2.30310e29 0.511671
\(668\) 0 0
\(669\) 2.56940e29 0.553169
\(670\) 0 0
\(671\) −3.15439e29 −0.658155
\(672\) 0 0
\(673\) −8.25080e29 −1.66854 −0.834272 0.551353i \(-0.814112\pi\)
−0.834272 + 0.551353i \(0.814112\pi\)
\(674\) 0 0
\(675\) 9.51934e28 0.186602
\(676\) 0 0
\(677\) 5.97987e29 1.13635 0.568173 0.822909i \(-0.307650\pi\)
0.568173 + 0.822909i \(0.307650\pi\)
\(678\) 0 0
\(679\) −2.89050e29 −0.532525
\(680\) 0 0
\(681\) −4.52962e28 −0.0809127
\(682\) 0 0
\(683\) 5.71862e29 0.990545 0.495272 0.868738i \(-0.335068\pi\)
0.495272 + 0.868738i \(0.335068\pi\)
\(684\) 0 0
\(685\) 8.73925e28 0.146799
\(686\) 0 0
\(687\) −4.45602e29 −0.725942
\(688\) 0 0
\(689\) −5.68922e29 −0.898982
\(690\) 0 0
\(691\) −9.55681e29 −1.46485 −0.732426 0.680847i \(-0.761612\pi\)
−0.732426 + 0.680847i \(0.761612\pi\)
\(692\) 0 0
\(693\) −3.77799e29 −0.561774
\(694\) 0 0
\(695\) 1.58574e29 0.228766
\(696\) 0 0
\(697\) −3.52052e28 −0.0492790
\(698\) 0 0
\(699\) −1.24151e29 −0.168632
\(700\) 0 0
\(701\) 6.42010e27 0.00846258 0.00423129 0.999991i \(-0.498653\pi\)
0.00423129 + 0.999991i \(0.498653\pi\)
\(702\) 0 0
\(703\) 8.47787e28 0.108457
\(704\) 0 0
\(705\) −5.93295e28 −0.0736690
\(706\) 0 0
\(707\) 4.69209e29 0.565539
\(708\) 0 0
\(709\) −6.52437e29 −0.763402 −0.381701 0.924286i \(-0.624662\pi\)
−0.381701 + 0.924286i \(0.624662\pi\)
\(710\) 0 0
\(711\) 2.03475e29 0.231143
\(712\) 0 0
\(713\) −1.64343e30 −1.81264
\(714\) 0 0
\(715\) 1.60729e29 0.172140
\(716\) 0 0
\(717\) −8.56789e29 −0.891093
\(718\) 0 0
\(719\) 1.22078e29 0.123306 0.0616530 0.998098i \(-0.480363\pi\)
0.0616530 + 0.998098i \(0.480363\pi\)
\(720\) 0 0
\(721\) −9.18621e29 −0.901189
\(722\) 0 0
\(723\) 2.10266e29 0.200362
\(724\) 0 0
\(725\) 5.10454e29 0.472506
\(726\) 0 0
\(727\) −1.04525e30 −0.939959 −0.469980 0.882677i \(-0.655739\pi\)
−0.469980 + 0.882677i \(0.655739\pi\)
\(728\) 0 0
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) −5.68306e29 −0.482447
\(732\) 0 0
\(733\) 8.69967e29 0.717647 0.358824 0.933405i \(-0.383178\pi\)
0.358824 + 0.933405i \(0.383178\pi\)
\(734\) 0 0
\(735\) 1.07621e29 0.0862741
\(736\) 0 0
\(737\) −1.87103e30 −1.45772
\(738\) 0 0
\(739\) −1.11587e29 −0.0844980 −0.0422490 0.999107i \(-0.513452\pi\)
−0.0422490 + 0.999107i \(0.513452\pi\)
\(740\) 0 0
\(741\) −1.73286e29 −0.127547
\(742\) 0 0
\(743\) 9.69344e29 0.693578 0.346789 0.937943i \(-0.387272\pi\)
0.346789 + 0.937943i \(0.387272\pi\)
\(744\) 0 0
\(745\) 1.65180e29 0.114899
\(746\) 0 0
\(747\) 4.37168e29 0.295653
\(748\) 0 0
\(749\) 2.11007e30 1.38752
\(750\) 0 0
\(751\) 7.68531e29 0.491408 0.245704 0.969345i \(-0.420981\pi\)
0.245704 + 0.969345i \(0.420981\pi\)
\(752\) 0 0
\(753\) −1.19167e30 −0.740982
\(754\) 0 0
\(755\) −3.05173e29 −0.184545
\(756\) 0 0
\(757\) −2.16553e29 −0.127367 −0.0636836 0.997970i \(-0.520285\pi\)
−0.0636836 + 0.997970i \(0.520285\pi\)
\(758\) 0 0
\(759\) −1.31041e30 −0.749669
\(760\) 0 0
\(761\) −9.88897e28 −0.0550316 −0.0275158 0.999621i \(-0.508760\pi\)
−0.0275158 + 0.999621i \(0.508760\pi\)
\(762\) 0 0
\(763\) −1.86156e30 −1.00779
\(764\) 0 0
\(765\) 4.88044e28 0.0257048
\(766\) 0 0
\(767\) 2.33921e30 1.19872
\(768\) 0 0
\(769\) 6.21082e29 0.309686 0.154843 0.987939i \(-0.450513\pi\)
0.154843 + 0.987939i \(0.450513\pi\)
\(770\) 0 0
\(771\) −4.75574e29 −0.230753
\(772\) 0 0
\(773\) 3.43242e30 1.62075 0.810373 0.585914i \(-0.199264\pi\)
0.810373 + 0.585914i \(0.199264\pi\)
\(774\) 0 0
\(775\) −3.64247e30 −1.67389
\(776\) 0 0
\(777\) −6.89634e29 −0.308459
\(778\) 0 0
\(779\) 7.07841e28 0.0308172
\(780\) 0 0
\(781\) 3.56550e30 1.51107
\(782\) 0 0
\(783\) 2.27313e29 0.0937836
\(784\) 0 0
\(785\) 7.29761e29 0.293123
\(786\) 0 0
\(787\) −3.39619e30 −1.32818 −0.664091 0.747652i \(-0.731181\pi\)
−0.664091 + 0.747652i \(0.731181\pi\)
\(788\) 0 0
\(789\) −8.57573e29 −0.326560
\(790\) 0 0
\(791\) 2.49624e30 0.925622
\(792\) 0 0
\(793\) 1.17693e30 0.424992
\(794\) 0 0
\(795\) −3.22165e29 −0.113298
\(796\) 0 0
\(797\) 3.15218e30 1.07969 0.539844 0.841765i \(-0.318483\pi\)
0.539844 + 0.841765i \(0.318483\pi\)
\(798\) 0 0
\(799\) 9.70633e29 0.323827
\(800\) 0 0
\(801\) −1.65109e30 −0.536573
\(802\) 0 0
\(803\) 6.51256e30 2.06176
\(804\) 0 0
\(805\) 8.08738e29 0.249431
\(806\) 0 0
\(807\) 3.30396e30 0.992802
\(808\) 0 0
\(809\) 5.77781e30 1.69162 0.845811 0.533482i \(-0.179117\pi\)
0.845811 + 0.533482i \(0.179117\pi\)
\(810\) 0 0
\(811\) 7.71903e29 0.220213 0.110107 0.993920i \(-0.464881\pi\)
0.110107 + 0.993920i \(0.464881\pi\)
\(812\) 0 0
\(813\) 2.15920e30 0.600264
\(814\) 0 0
\(815\) 7.56998e29 0.205088
\(816\) 0 0
\(817\) 1.14265e30 0.301703
\(818\) 0 0
\(819\) 1.40960e30 0.362755
\(820\) 0 0
\(821\) −1.63389e30 −0.409845 −0.204923 0.978778i \(-0.565694\pi\)
−0.204923 + 0.978778i \(0.565694\pi\)
\(822\) 0 0
\(823\) −4.56986e29 −0.111739 −0.0558695 0.998438i \(-0.517793\pi\)
−0.0558695 + 0.998438i \(0.517793\pi\)
\(824\) 0 0
\(825\) −2.90437e30 −0.692287
\(826\) 0 0
\(827\) 2.30833e30 0.536401 0.268201 0.963363i \(-0.413571\pi\)
0.268201 + 0.963363i \(0.413571\pi\)
\(828\) 0 0
\(829\) −3.52497e29 −0.0798607 −0.0399303 0.999202i \(-0.512714\pi\)
−0.0399303 + 0.999202i \(0.512714\pi\)
\(830\) 0 0
\(831\) −4.85068e30 −1.07150
\(832\) 0 0
\(833\) −1.76069e30 −0.379236
\(834\) 0 0
\(835\) 1.52675e29 0.0320672
\(836\) 0 0
\(837\) −1.62205e30 −0.332236
\(838\) 0 0
\(839\) 7.70025e29 0.153817 0.0769084 0.997038i \(-0.475495\pi\)
0.0769084 + 0.997038i \(0.475495\pi\)
\(840\) 0 0
\(841\) −3.91392e30 −0.762525
\(842\) 0 0
\(843\) −5.41324e30 −1.02865
\(844\) 0 0
\(845\) 3.40743e29 0.0631585
\(846\) 0 0
\(847\) 3.98951e30 0.721347
\(848\) 0 0
\(849\) 3.22350e30 0.568587
\(850\) 0 0
\(851\) −2.39203e30 −0.411629
\(852\) 0 0
\(853\) −6.31709e30 −1.06060 −0.530301 0.847810i \(-0.677921\pi\)
−0.530301 + 0.847810i \(0.677921\pi\)
\(854\) 0 0
\(855\) −9.81270e28 −0.0160748
\(856\) 0 0
\(857\) −3.16894e30 −0.506541 −0.253271 0.967395i \(-0.581506\pi\)
−0.253271 + 0.967395i \(0.581506\pi\)
\(858\) 0 0
\(859\) 8.21661e30 1.28164 0.640818 0.767693i \(-0.278595\pi\)
0.640818 + 0.767693i \(0.278595\pi\)
\(860\) 0 0
\(861\) −5.75795e29 −0.0876466
\(862\) 0 0
\(863\) −4.74681e30 −0.705162 −0.352581 0.935781i \(-0.614696\pi\)
−0.352581 + 0.935781i \(0.614696\pi\)
\(864\) 0 0
\(865\) 7.12158e29 0.103254
\(866\) 0 0
\(867\) 3.28137e30 0.464360
\(868\) 0 0
\(869\) −6.20808e30 −0.857530
\(870\) 0 0
\(871\) 6.98097e30 0.941293
\(872\) 0 0
\(873\) −9.89540e29 −0.130252
\(874\) 0 0
\(875\) 3.64111e30 0.467897
\(876\) 0 0
\(877\) 6.72283e30 0.843444 0.421722 0.906725i \(-0.361426\pi\)
0.421722 + 0.906725i \(0.361426\pi\)
\(878\) 0 0
\(879\) 6.36890e30 0.780156
\(880\) 0 0
\(881\) 1.20529e31 1.44160 0.720800 0.693143i \(-0.243775\pi\)
0.720800 + 0.693143i \(0.243775\pi\)
\(882\) 0 0
\(883\) −2.79063e30 −0.325923 −0.162962 0.986632i \(-0.552105\pi\)
−0.162962 + 0.986632i \(0.552105\pi\)
\(884\) 0 0
\(885\) 1.32463e30 0.151074
\(886\) 0 0
\(887\) 4.22230e30 0.470274 0.235137 0.971962i \(-0.424446\pi\)
0.235137 + 0.971962i \(0.424446\pi\)
\(888\) 0 0
\(889\) −1.88675e31 −2.05232
\(890\) 0 0
\(891\) −1.29336e30 −0.137406
\(892\) 0 0
\(893\) −1.95157e30 −0.202509
\(894\) 0 0
\(895\) −3.02250e30 −0.306355
\(896\) 0 0
\(897\) 4.88925e30 0.484085
\(898\) 0 0
\(899\) −8.69790e30 −0.841274
\(900\) 0 0
\(901\) 5.27064e30 0.498027
\(902\) 0 0
\(903\) −9.29488e30 −0.858070
\(904\) 0 0
\(905\) 7.20793e29 0.0650130
\(906\) 0 0
\(907\) −8.17027e30 −0.720045 −0.360023 0.932944i \(-0.617231\pi\)
−0.360023 + 0.932944i \(0.617231\pi\)
\(908\) 0 0
\(909\) 1.60630e30 0.138327
\(910\) 0 0
\(911\) −8.02621e30 −0.675410 −0.337705 0.941252i \(-0.609651\pi\)
−0.337705 + 0.941252i \(0.609651\pi\)
\(912\) 0 0
\(913\) −1.33381e31 −1.09686
\(914\) 0 0
\(915\) 6.66462e29 0.0535616
\(916\) 0 0
\(917\) −2.27712e31 −1.78858
\(918\) 0 0
\(919\) 3.31927e30 0.254818 0.127409 0.991850i \(-0.459334\pi\)
0.127409 + 0.991850i \(0.459334\pi\)
\(920\) 0 0
\(921\) 9.32552e29 0.0699756
\(922\) 0 0
\(923\) −1.33031e31 −0.975746
\(924\) 0 0
\(925\) −5.30165e30 −0.380122
\(926\) 0 0
\(927\) −3.14483e30 −0.220424
\(928\) 0 0
\(929\) 2.23059e31 1.52846 0.764232 0.644941i \(-0.223118\pi\)
0.764232 + 0.644941i \(0.223118\pi\)
\(930\) 0 0
\(931\) 3.54007e30 0.237159
\(932\) 0 0
\(933\) 1.27860e31 0.837489
\(934\) 0 0
\(935\) −1.48903e30 −0.0953636
\(936\) 0 0
\(937\) 8.24036e30 0.516037 0.258018 0.966140i \(-0.416931\pi\)
0.258018 + 0.966140i \(0.416931\pi\)
\(938\) 0 0
\(939\) −1.22990e31 −0.753152
\(940\) 0 0
\(941\) 1.13254e31 0.678208 0.339104 0.940749i \(-0.389876\pi\)
0.339104 + 0.940749i \(0.389876\pi\)
\(942\) 0 0
\(943\) −1.99717e30 −0.116961
\(944\) 0 0
\(945\) 7.98216e29 0.0457179
\(946\) 0 0
\(947\) −2.85285e31 −1.59810 −0.799050 0.601264i \(-0.794664\pi\)
−0.799050 + 0.601264i \(0.794664\pi\)
\(948\) 0 0
\(949\) −2.42989e31 −1.33135
\(950\) 0 0
\(951\) −8.91893e30 −0.477988
\(952\) 0 0
\(953\) −2.28550e30 −0.119814 −0.0599068 0.998204i \(-0.519080\pi\)
−0.0599068 + 0.998204i \(0.519080\pi\)
\(954\) 0 0
\(955\) 4.47903e30 0.229693
\(956\) 0 0
\(957\) −6.93539e30 −0.347933
\(958\) 0 0
\(959\) −2.33841e31 −1.14769
\(960\) 0 0
\(961\) 4.12404e31 1.98029
\(962\) 0 0
\(963\) 7.22366e30 0.339376
\(964\) 0 0
\(965\) −4.97561e29 −0.0228723
\(966\) 0 0
\(967\) −2.86814e31 −1.29010 −0.645049 0.764141i \(-0.723163\pi\)
−0.645049 + 0.764141i \(0.723163\pi\)
\(968\) 0 0
\(969\) 1.60536e30 0.0706600
\(970\) 0 0
\(971\) 3.70387e31 1.59534 0.797670 0.603094i \(-0.206066\pi\)
0.797670 + 0.603094i \(0.206066\pi\)
\(972\) 0 0
\(973\) −4.24305e31 −1.78852
\(974\) 0 0
\(975\) 1.08364e31 0.447031
\(976\) 0 0
\(977\) −1.87685e31 −0.757770 −0.378885 0.925444i \(-0.623692\pi\)
−0.378885 + 0.925444i \(0.623692\pi\)
\(978\) 0 0
\(979\) 5.03751e31 1.99066
\(980\) 0 0
\(981\) −6.37289e30 −0.246497
\(982\) 0 0
\(983\) −2.00084e31 −0.757531 −0.378766 0.925493i \(-0.623651\pi\)
−0.378766 + 0.925493i \(0.623651\pi\)
\(984\) 0 0
\(985\) 2.83167e30 0.104945
\(986\) 0 0
\(987\) 1.58751e31 0.575952
\(988\) 0 0
\(989\) −3.22397e31 −1.14507
\(990\) 0 0
\(991\) 1.44667e31 0.503034 0.251517 0.967853i \(-0.419071\pi\)
0.251517 + 0.967853i \(0.419071\pi\)
\(992\) 0 0
\(993\) 1.78781e31 0.608631
\(994\) 0 0
\(995\) 7.72297e30 0.257420
\(996\) 0 0
\(997\) 3.92902e31 1.28229 0.641144 0.767421i \(-0.278460\pi\)
0.641144 + 0.767421i \(0.278460\pi\)
\(998\) 0 0
\(999\) −2.36091e30 −0.0754469
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 48.22.a.h.1.1 2
4.3 odd 2 24.22.a.a.1.1 2
12.11 even 2 72.22.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.a.1.1 2 4.3 odd 2
48.22.a.h.1.1 2 1.1 even 1 trivial
72.22.a.a.1.2 2 12.11 even 2