Properties

Label 21.30.a.a.1.1
Level $21$
Weight $30$
Character 21.1
Self dual yes
Analytic conductor $111.884$
Analytic rank $1$
Dimension $6$
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] = [21,30,Mod(1,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.1");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.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: \(111.883889004\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 110200839 x^{4} - 84515300136 x^{3} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 5^{2}\cdot 7^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-7908.19\) of defining polynomial
Character \(\chi\) \(=\) 21.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-30888.8 q^{2} +4.78297e6 q^{3} +4.17246e8 q^{4} -8.65686e9 q^{5} -1.47740e11 q^{6} +6.78223e11 q^{7} +3.69508e12 q^{8} +2.28768e13 q^{9} +O(q^{10})\) \(q-30888.8 q^{2} +4.78297e6 q^{3} +4.17246e8 q^{4} -8.65686e9 q^{5} -1.47740e11 q^{6} +6.78223e11 q^{7} +3.69508e12 q^{8} +2.28768e13 q^{9} +2.67400e14 q^{10} -2.24344e15 q^{11} +1.99567e15 q^{12} +1.22120e16 q^{13} -2.09495e16 q^{14} -4.14055e16 q^{15} -3.38144e17 q^{16} +9.55820e17 q^{17} -7.06636e17 q^{18} -1.62011e18 q^{19} -3.61204e18 q^{20} +3.24392e18 q^{21} +6.92971e19 q^{22} +3.65594e19 q^{23} +1.76734e19 q^{24} -1.11323e20 q^{25} -3.77213e20 q^{26} +1.09419e20 q^{27} +2.82986e20 q^{28} +1.08298e21 q^{29} +1.27897e21 q^{30} -6.95263e21 q^{31} +8.46106e21 q^{32} -1.07303e22 q^{33} -2.95241e22 q^{34} -5.87128e21 q^{35} +9.54524e21 q^{36} -9.83243e22 q^{37} +5.00431e22 q^{38} +5.84095e22 q^{39} -3.19878e22 q^{40} +3.16613e22 q^{41} -1.00201e23 q^{42} +8.33832e23 q^{43} -9.36065e23 q^{44} -1.98041e23 q^{45} -1.12928e24 q^{46} +1.90790e24 q^{47} -1.61733e24 q^{48} +4.59987e23 q^{49} +3.43864e24 q^{50} +4.57166e24 q^{51} +5.09539e24 q^{52} -4.34860e24 q^{53} -3.37982e24 q^{54} +1.94211e25 q^{55} +2.50609e24 q^{56} -7.74893e24 q^{57} -3.34520e25 q^{58} +2.04999e24 q^{59} -1.72763e25 q^{60} +1.44716e26 q^{61} +2.14758e26 q^{62} +1.55156e25 q^{63} -7.98123e25 q^{64} -1.05717e26 q^{65} +3.31446e26 q^{66} -1.63239e26 q^{67} +3.98812e26 q^{68} +1.74863e26 q^{69} +1.81357e26 q^{70} +5.57720e26 q^{71} +8.45316e25 q^{72} +1.06236e27 q^{73} +3.03712e27 q^{74} -5.32456e26 q^{75} -6.75983e26 q^{76} -1.52155e27 q^{77} -1.80420e27 q^{78} -3.19168e26 q^{79} +2.92726e27 q^{80} +5.23348e26 q^{81} -9.77979e26 q^{82} -1.02077e28 q^{83} +1.35351e27 q^{84} -8.27440e27 q^{85} -2.57561e28 q^{86} +5.17986e27 q^{87} -8.28969e27 q^{88} +3.36276e28 q^{89} +6.11725e27 q^{90} +8.28244e27 q^{91} +1.52543e28 q^{92} -3.32542e28 q^{93} -5.89326e28 q^{94} +1.40251e28 q^{95} +4.04690e28 q^{96} -8.94109e28 q^{97} -1.42084e28 q^{98} -5.13227e28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4464 q^{2} + 28697814 q^{3} + 308522592 q^{4} - 34057000980 q^{5} + 21351173616 q^{6} + 4069338437094 q^{7} + 19307209833216 q^{8} + 137260754729766 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4464 q^{2} + 28697814 q^{3} + 308522592 q^{4} - 34057000980 q^{5} + 21351173616 q^{6} + 4069338437094 q^{7} + 19307209833216 q^{8} + 137260754729766 q^{9} - 27416413415040 q^{10} - 19\!\cdots\!00 q^{11}+ \cdots - 44\!\cdots\!00 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30888.8 −1.33311 −0.666555 0.745456i \(-0.732232\pi\)
−0.666555 + 0.745456i \(0.732232\pi\)
\(3\) 4.78297e6 0.577350
\(4\) 4.17246e8 0.777181
\(5\) −8.65686e9 −0.634301 −0.317151 0.948375i \(-0.602726\pi\)
−0.317151 + 0.948375i \(0.602726\pi\)
\(6\) −1.47740e11 −0.769671
\(7\) 6.78223e11 0.377964
\(8\) 3.69508e12 0.297043
\(9\) 2.28768e13 0.333333
\(10\) 2.67400e14 0.845593
\(11\) −2.24344e15 −1.78123 −0.890616 0.454756i \(-0.849726\pi\)
−0.890616 + 0.454756i \(0.849726\pi\)
\(12\) 1.99567e15 0.448705
\(13\) 1.22120e16 0.860216 0.430108 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(14\) −2.09495e16 −0.503868
\(15\) −4.14055e16 −0.366214
\(16\) −3.38144e17 −1.17317
\(17\) 9.55820e17 1.37679 0.688394 0.725337i \(-0.258316\pi\)
0.688394 + 0.725337i \(0.258316\pi\)
\(18\) −7.06636e17 −0.444370
\(19\) −1.62011e18 −0.465175 −0.232588 0.972575i \(-0.574719\pi\)
−0.232588 + 0.972575i \(0.574719\pi\)
\(20\) −3.61204e18 −0.492966
\(21\) 3.24392e18 0.218218
\(22\) 6.92971e19 2.37458
\(23\) 3.65594e19 0.657576 0.328788 0.944404i \(-0.393360\pi\)
0.328788 + 0.944404i \(0.393360\pi\)
\(24\) 1.76734e19 0.171498
\(25\) −1.11323e20 −0.597662
\(26\) −3.77213e20 −1.14676
\(27\) 1.09419e20 0.192450
\(28\) 2.82986e20 0.293747
\(29\) 1.08298e21 0.675849 0.337925 0.941173i \(-0.390275\pi\)
0.337925 + 0.941173i \(0.390275\pi\)
\(30\) 1.27897e21 0.488203
\(31\) −6.95263e21 −1.64970 −0.824849 0.565353i \(-0.808740\pi\)
−0.824849 + 0.565353i \(0.808740\pi\)
\(32\) 8.46106e21 1.26692
\(33\) −1.07303e22 −1.02839
\(34\) −2.95241e22 −1.83541
\(35\) −5.87128e21 −0.239743
\(36\) 9.54524e21 0.259060
\(37\) −9.83243e22 −1.79364 −0.896821 0.442394i \(-0.854129\pi\)
−0.896821 + 0.442394i \(0.854129\pi\)
\(38\) 5.00431e22 0.620130
\(39\) 5.84095e22 0.496646
\(40\) −3.19878e22 −0.188415
\(41\) 3.16613e22 0.130366 0.0651828 0.997873i \(-0.479237\pi\)
0.0651828 + 0.997873i \(0.479237\pi\)
\(42\) −1.00201e23 −0.290908
\(43\) 8.33832e23 1.72102 0.860510 0.509433i \(-0.170145\pi\)
0.860510 + 0.509433i \(0.170145\pi\)
\(44\) −9.36065e23 −1.38434
\(45\) −1.98041e23 −0.211434
\(46\) −1.12928e24 −0.876620
\(47\) 1.90790e24 1.08427 0.542133 0.840293i \(-0.317617\pi\)
0.542133 + 0.840293i \(0.317617\pi\)
\(48\) −1.61733e24 −0.677331
\(49\) 4.59987e23 0.142857
\(50\) 3.43864e24 0.796749
\(51\) 4.57166e24 0.794888
\(52\) 5.09539e24 0.668543
\(53\) −4.34860e24 −0.432861 −0.216431 0.976298i \(-0.569442\pi\)
−0.216431 + 0.976298i \(0.569442\pi\)
\(54\) −3.37982e24 −0.256557
\(55\) 1.94211e25 1.12984
\(56\) 2.50609e24 0.112272
\(57\) −7.74893e24 −0.268569
\(58\) −3.34520e25 −0.900981
\(59\) 2.04999e24 0.0430922 0.0215461 0.999768i \(-0.493141\pi\)
0.0215461 + 0.999768i \(0.493141\pi\)
\(60\) −1.72763e25 −0.284614
\(61\) 1.44716e26 1.87600 0.937999 0.346637i \(-0.112676\pi\)
0.937999 + 0.346637i \(0.112676\pi\)
\(62\) 2.14758e26 2.19923
\(63\) 1.55156e25 0.125988
\(64\) −7.98123e25 −0.515775
\(65\) −1.05717e26 −0.545636
\(66\) 3.31446e26 1.37096
\(67\) −1.63239e26 −0.542923 −0.271461 0.962449i \(-0.587507\pi\)
−0.271461 + 0.962449i \(0.587507\pi\)
\(68\) 3.98812e26 1.07001
\(69\) 1.74863e26 0.379651
\(70\) 1.81357e26 0.319604
\(71\) 5.57720e26 0.800148 0.400074 0.916483i \(-0.368984\pi\)
0.400074 + 0.916483i \(0.368984\pi\)
\(72\) 8.45316e25 0.0990142
\(73\) 1.06236e27 1.01880 0.509400 0.860530i \(-0.329867\pi\)
0.509400 + 0.860530i \(0.329867\pi\)
\(74\) 3.03712e27 2.39112
\(75\) −5.32456e26 −0.345060
\(76\) −6.75983e26 −0.361525
\(77\) −1.52155e27 −0.673242
\(78\) −1.80420e27 −0.662083
\(79\) −3.19168e26 −0.0973702 −0.0486851 0.998814i \(-0.515503\pi\)
−0.0486851 + 0.998814i \(0.515503\pi\)
\(80\) 2.92726e27 0.744144
\(81\) 5.23348e26 0.111111
\(82\) −9.77979e26 −0.173792
\(83\) −1.02077e28 −1.52158 −0.760789 0.648999i \(-0.775188\pi\)
−0.760789 + 0.648999i \(0.775188\pi\)
\(84\) 1.35351e27 0.169595
\(85\) −8.27440e27 −0.873298
\(86\) −2.57561e28 −2.29431
\(87\) 5.17986e27 0.390202
\(88\) −8.28969e27 −0.529102
\(89\) 3.36276e28 1.82197 0.910985 0.412439i \(-0.135323\pi\)
0.910985 + 0.412439i \(0.135323\pi\)
\(90\) 6.11725e27 0.281864
\(91\) 8.28244e27 0.325131
\(92\) 1.52543e28 0.511055
\(93\) −3.32542e28 −0.952453
\(94\) −5.89326e28 −1.44545
\(95\) 1.40251e28 0.295061
\(96\) 4.04690e28 0.731458
\(97\) −8.94109e28 −1.39059 −0.695297 0.718723i \(-0.744727\pi\)
−0.695297 + 0.718723i \(0.744727\pi\)
\(98\) −1.42084e28 −0.190444
\(99\) −5.13227e28 −0.593744
\(100\) −4.64491e28 −0.464491
\(101\) 1.50865e28 0.130596 0.0652979 0.997866i \(-0.479200\pi\)
0.0652979 + 0.997866i \(0.479200\pi\)
\(102\) −1.41213e29 −1.05967
\(103\) 5.98748e28 0.390036 0.195018 0.980800i \(-0.437523\pi\)
0.195018 + 0.980800i \(0.437523\pi\)
\(104\) 4.51242e28 0.255521
\(105\) −2.80822e28 −0.138416
\(106\) 1.34323e29 0.577052
\(107\) −1.38251e29 −0.518328 −0.259164 0.965833i \(-0.583447\pi\)
−0.259164 + 0.965833i \(0.583447\pi\)
\(108\) 4.56546e28 0.149568
\(109\) −3.00269e29 −0.860649 −0.430324 0.902674i \(-0.641601\pi\)
−0.430324 + 0.902674i \(0.641601\pi\)
\(110\) −5.99896e29 −1.50620
\(111\) −4.70282e29 −1.03556
\(112\) −2.29337e29 −0.443417
\(113\) 6.59917e29 1.12164 0.560818 0.827939i \(-0.310487\pi\)
0.560818 + 0.827939i \(0.310487\pi\)
\(114\) 2.39355e29 0.358032
\(115\) −3.16490e29 −0.417101
\(116\) 4.51869e29 0.525257
\(117\) 2.79371e29 0.286739
\(118\) −6.33218e28 −0.0574466
\(119\) 6.48259e29 0.520377
\(120\) −1.52997e29 −0.108781
\(121\) 3.44671e30 2.17279
\(122\) −4.47009e30 −2.50091
\(123\) 1.51435e29 0.0752666
\(124\) −2.90095e30 −1.28211
\(125\) 2.57618e30 1.01340
\(126\) −4.79257e29 −0.167956
\(127\) −5.55715e30 −1.73659 −0.868297 0.496044i \(-0.834785\pi\)
−0.868297 + 0.496044i \(0.834785\pi\)
\(128\) −2.07719e30 −0.579338
\(129\) 3.98819e30 0.993632
\(130\) 3.26548e30 0.727392
\(131\) −1.14915e30 −0.229056 −0.114528 0.993420i \(-0.536536\pi\)
−0.114528 + 0.993420i \(0.536536\pi\)
\(132\) −4.47717e30 −0.799248
\(133\) −1.09879e30 −0.175820
\(134\) 5.04224e30 0.723776
\(135\) −9.47225e29 −0.122071
\(136\) 3.53183e30 0.408965
\(137\) −1.26350e31 −1.31561 −0.657804 0.753189i \(-0.728514\pi\)
−0.657804 + 0.753189i \(0.728514\pi\)
\(138\) −5.40129e30 −0.506117
\(139\) −1.81277e31 −1.52977 −0.764887 0.644164i \(-0.777205\pi\)
−0.764887 + 0.644164i \(0.777205\pi\)
\(140\) −2.44977e30 −0.186324
\(141\) 9.12541e30 0.626002
\(142\) −1.72273e31 −1.06668
\(143\) −2.73968e31 −1.53224
\(144\) −7.73564e30 −0.391057
\(145\) −9.37522e30 −0.428692
\(146\) −3.28149e31 −1.35817
\(147\) 2.20010e30 0.0824786
\(148\) −4.10254e31 −1.39398
\(149\) −3.04363e31 −0.937975 −0.468988 0.883205i \(-0.655381\pi\)
−0.468988 + 0.883205i \(0.655381\pi\)
\(150\) 1.64469e31 0.460003
\(151\) −6.86856e31 −1.74462 −0.872308 0.488956i \(-0.837378\pi\)
−0.872308 + 0.488956i \(0.837378\pi\)
\(152\) −5.98643e30 −0.138177
\(153\) 2.18661e31 0.458929
\(154\) 4.69989e31 0.897506
\(155\) 6.01879e31 1.04641
\(156\) 2.43711e31 0.385983
\(157\) −6.33549e31 −0.914609 −0.457304 0.889310i \(-0.651185\pi\)
−0.457304 + 0.889310i \(0.651185\pi\)
\(158\) 9.85869e30 0.129805
\(159\) −2.07992e31 −0.249913
\(160\) −7.32462e31 −0.803610
\(161\) 2.47954e31 0.248540
\(162\) −1.61656e31 −0.148123
\(163\) 3.82410e30 0.0320486 0.0160243 0.999872i \(-0.494899\pi\)
0.0160243 + 0.999872i \(0.494899\pi\)
\(164\) 1.32105e31 0.101318
\(165\) 9.28908e31 0.652312
\(166\) 3.15302e32 2.02843
\(167\) 4.74464e31 0.279778 0.139889 0.990167i \(-0.455325\pi\)
0.139889 + 0.990167i \(0.455325\pi\)
\(168\) 1.19865e31 0.0648200
\(169\) −5.24058e31 −0.260029
\(170\) 2.55586e32 1.16420
\(171\) −3.70629e31 −0.155058
\(172\) 3.47913e32 1.33754
\(173\) 1.42061e32 0.502119 0.251060 0.967972i \(-0.419221\pi\)
0.251060 + 0.967972i \(0.419221\pi\)
\(174\) −1.60000e32 −0.520182
\(175\) −7.55020e31 −0.225895
\(176\) 7.58605e32 2.08969
\(177\) 9.80506e30 0.0248793
\(178\) −1.03872e33 −2.42889
\(179\) −6.38161e32 −1.37582 −0.687909 0.725797i \(-0.741471\pi\)
−0.687909 + 0.725797i \(0.741471\pi\)
\(180\) −8.26318e31 −0.164322
\(181\) 9.60436e32 1.76250 0.881250 0.472650i \(-0.156703\pi\)
0.881250 + 0.472650i \(0.156703\pi\)
\(182\) −2.55835e32 −0.433435
\(183\) 6.92170e32 1.08311
\(184\) 1.35090e32 0.195328
\(185\) 8.51180e32 1.13771
\(186\) 1.02718e33 1.26972
\(187\) −2.14432e33 −2.45238
\(188\) 7.96062e32 0.842671
\(189\) 7.42105e31 0.0727393
\(190\) −4.33217e32 −0.393349
\(191\) 8.79105e32 0.739702 0.369851 0.929091i \(-0.379409\pi\)
0.369851 + 0.929091i \(0.379409\pi\)
\(192\) −3.81740e32 −0.297783
\(193\) −8.01268e32 −0.579691 −0.289846 0.957073i \(-0.593604\pi\)
−0.289846 + 0.957073i \(0.593604\pi\)
\(194\) 2.76179e33 1.85381
\(195\) −5.05643e32 −0.315023
\(196\) 1.91927e32 0.111026
\(197\) −7.50182e32 −0.403094 −0.201547 0.979479i \(-0.564597\pi\)
−0.201547 + 0.979479i \(0.564597\pi\)
\(198\) 1.58530e33 0.791526
\(199\) −1.54091e33 −0.715168 −0.357584 0.933881i \(-0.616399\pi\)
−0.357584 + 0.933881i \(0.616399\pi\)
\(200\) −4.11348e32 −0.177531
\(201\) −7.80765e32 −0.313457
\(202\) −4.66004e32 −0.174098
\(203\) 7.34503e32 0.255447
\(204\) 1.90750e33 0.617772
\(205\) −2.74088e32 −0.0826910
\(206\) −1.84946e33 −0.519960
\(207\) 8.36362e32 0.219192
\(208\) −4.12940e33 −1.00918
\(209\) 3.63461e33 0.828585
\(210\) 8.67424e32 0.184523
\(211\) −2.96174e33 −0.588100 −0.294050 0.955790i \(-0.595003\pi\)
−0.294050 + 0.955790i \(0.595003\pi\)
\(212\) −1.81443e33 −0.336412
\(213\) 2.66756e33 0.461966
\(214\) 4.27042e33 0.690988
\(215\) −7.21837e33 −1.09165
\(216\) 4.04312e32 0.0571659
\(217\) −4.71543e33 −0.623527
\(218\) 9.27493e33 1.14734
\(219\) 5.08122e33 0.588205
\(220\) 8.10339e33 0.878088
\(221\) 1.16725e34 1.18433
\(222\) 1.45264e34 1.38051
\(223\) 1.43711e33 0.127958 0.0639791 0.997951i \(-0.479621\pi\)
0.0639791 + 0.997951i \(0.479621\pi\)
\(224\) 5.73849e33 0.478852
\(225\) −2.54672e33 −0.199221
\(226\) −2.03840e34 −1.49526
\(227\) −1.46215e34 −1.00604 −0.503021 0.864274i \(-0.667778\pi\)
−0.503021 + 0.864274i \(0.667778\pi\)
\(228\) −3.23321e33 −0.208727
\(229\) −4.58587e33 −0.277848 −0.138924 0.990303i \(-0.544364\pi\)
−0.138924 + 0.990303i \(0.544364\pi\)
\(230\) 9.77598e33 0.556041
\(231\) −7.27754e33 −0.388697
\(232\) 4.00170e33 0.200756
\(233\) 1.29997e33 0.0612735 0.0306368 0.999531i \(-0.490246\pi\)
0.0306368 + 0.999531i \(0.490246\pi\)
\(234\) −8.62942e33 −0.382254
\(235\) −1.65164e34 −0.687751
\(236\) 8.55351e32 0.0334904
\(237\) −1.52657e33 −0.0562167
\(238\) −2.00239e34 −0.693719
\(239\) −4.10278e33 −0.133755 −0.0668773 0.997761i \(-0.521304\pi\)
−0.0668773 + 0.997761i \(0.521304\pi\)
\(240\) 1.40010e34 0.429632
\(241\) −3.78274e34 −1.09285 −0.546424 0.837509i \(-0.684011\pi\)
−0.546424 + 0.837509i \(0.684011\pi\)
\(242\) −1.06465e35 −2.89656
\(243\) 2.50316e33 0.0641500
\(244\) 6.03819e34 1.45799
\(245\) −3.98204e33 −0.0906144
\(246\) −4.67764e33 −0.100339
\(247\) −1.97847e34 −0.400151
\(248\) −2.56905e34 −0.490031
\(249\) −4.88230e34 −0.878484
\(250\) −7.95749e34 −1.35097
\(251\) 3.98771e34 0.638933 0.319466 0.947598i \(-0.396496\pi\)
0.319466 + 0.947598i \(0.396496\pi\)
\(252\) 6.47380e33 0.0979155
\(253\) −8.20188e34 −1.17129
\(254\) 1.71654e35 2.31507
\(255\) −3.95762e34 −0.504199
\(256\) 1.07011e35 1.28810
\(257\) −4.60277e34 −0.523587 −0.261794 0.965124i \(-0.584314\pi\)
−0.261794 + 0.965124i \(0.584314\pi\)
\(258\) −1.23190e35 −1.32462
\(259\) −6.66858e34 −0.677933
\(260\) −4.41101e34 −0.424057
\(261\) 2.47751e34 0.225283
\(262\) 3.54957e34 0.305357
\(263\) −6.06941e34 −0.494069 −0.247035 0.969007i \(-0.579456\pi\)
−0.247035 + 0.969007i \(0.579456\pi\)
\(264\) −3.96493e34 −0.305477
\(265\) 3.76452e34 0.274564
\(266\) 3.39404e34 0.234387
\(267\) 1.60840e35 1.05192
\(268\) −6.81106e34 −0.421949
\(269\) 4.09618e34 0.240420 0.120210 0.992748i \(-0.461643\pi\)
0.120210 + 0.992748i \(0.461643\pi\)
\(270\) 2.92586e34 0.162734
\(271\) 1.78750e35 0.942302 0.471151 0.882052i \(-0.343839\pi\)
0.471151 + 0.882052i \(0.343839\pi\)
\(272\) −3.23204e35 −1.61521
\(273\) 3.96147e34 0.187714
\(274\) 3.90279e35 1.75385
\(275\) 2.49747e35 1.06457
\(276\) 7.29606e34 0.295058
\(277\) −2.52704e35 −0.969741 −0.484870 0.874586i \(-0.661133\pi\)
−0.484870 + 0.874586i \(0.661133\pi\)
\(278\) 5.59942e35 2.03936
\(279\) −1.59054e35 −0.549899
\(280\) −2.16949e34 −0.0712140
\(281\) −5.43269e35 −1.69345 −0.846727 0.532028i \(-0.821430\pi\)
−0.846727 + 0.532028i \(0.821430\pi\)
\(282\) −2.81873e35 −0.834529
\(283\) 2.46362e35 0.692899 0.346449 0.938069i \(-0.387387\pi\)
0.346449 + 0.938069i \(0.387387\pi\)
\(284\) 2.32706e35 0.621860
\(285\) 6.70814e34 0.170354
\(286\) 8.46255e35 2.04265
\(287\) 2.14734e34 0.0492735
\(288\) 1.93562e35 0.422307
\(289\) 4.31623e35 0.895543
\(290\) 2.89589e35 0.571493
\(291\) −4.27649e35 −0.802859
\(292\) 4.43264e35 0.791792
\(293\) −1.66570e35 −0.283149 −0.141575 0.989928i \(-0.545217\pi\)
−0.141575 + 0.989928i \(0.545217\pi\)
\(294\) −6.79584e34 −0.109953
\(295\) −1.77465e34 −0.0273334
\(296\) −3.63316e35 −0.532788
\(297\) −2.45475e35 −0.342798
\(298\) 9.40141e35 1.25042
\(299\) 4.46463e35 0.565657
\(300\) −2.22165e35 −0.268174
\(301\) 5.65524e35 0.650485
\(302\) 2.12161e36 2.32576
\(303\) 7.21583e34 0.0753995
\(304\) 5.47829e35 0.545730
\(305\) −1.25278e36 −1.18995
\(306\) −6.75417e35 −0.611803
\(307\) 5.24663e34 0.0453287 0.0226644 0.999743i \(-0.492785\pi\)
0.0226644 + 0.999743i \(0.492785\pi\)
\(308\) −6.34861e35 −0.523231
\(309\) 2.86379e35 0.225187
\(310\) −1.85913e36 −1.39497
\(311\) −9.88783e35 −0.708068 −0.354034 0.935232i \(-0.615190\pi\)
−0.354034 + 0.935232i \(0.615190\pi\)
\(312\) 2.15828e35 0.147525
\(313\) −3.78881e35 −0.247235 −0.123617 0.992330i \(-0.539450\pi\)
−0.123617 + 0.992330i \(0.539450\pi\)
\(314\) 1.95696e36 1.21927
\(315\) −1.34316e35 −0.0799144
\(316\) −1.33171e35 −0.0756742
\(317\) −2.72717e36 −1.48031 −0.740155 0.672437i \(-0.765248\pi\)
−0.740155 + 0.672437i \(0.765248\pi\)
\(318\) 6.42462e35 0.333161
\(319\) −2.42960e36 −1.20384
\(320\) 6.90925e35 0.327157
\(321\) −6.61252e35 −0.299257
\(322\) −7.65901e35 −0.331331
\(323\) −1.54853e36 −0.640447
\(324\) 2.18365e35 0.0863534
\(325\) −1.35948e36 −0.514118
\(326\) −1.18122e35 −0.0427243
\(327\) −1.43618e36 −0.496896
\(328\) 1.16991e35 0.0387241
\(329\) 1.29398e36 0.409814
\(330\) −2.86928e36 −0.869603
\(331\) −3.93676e35 −0.114191 −0.0570956 0.998369i \(-0.518184\pi\)
−0.0570956 + 0.998369i \(0.518184\pi\)
\(332\) −4.25911e36 −1.18254
\(333\) −2.24934e36 −0.597881
\(334\) −1.46556e36 −0.372975
\(335\) 1.41313e36 0.344377
\(336\) −1.09691e36 −0.256007
\(337\) −1.56266e36 −0.349325 −0.174662 0.984628i \(-0.555883\pi\)
−0.174662 + 0.984628i \(0.555883\pi\)
\(338\) 1.61875e36 0.346647
\(339\) 3.15636e36 0.647577
\(340\) −3.45246e36 −0.678710
\(341\) 1.55978e37 2.93849
\(342\) 1.14483e36 0.206710
\(343\) 3.11973e35 0.0539949
\(344\) 3.08108e36 0.511217
\(345\) −1.51376e36 −0.240813
\(346\) −4.38810e36 −0.669380
\(347\) 1.22019e37 1.78504 0.892521 0.451005i \(-0.148935\pi\)
0.892521 + 0.451005i \(0.148935\pi\)
\(348\) 2.16128e36 0.303257
\(349\) −9.92389e36 −1.33571 −0.667857 0.744290i \(-0.732788\pi\)
−0.667857 + 0.744290i \(0.732788\pi\)
\(350\) 2.33216e36 0.301143
\(351\) 1.33622e36 0.165549
\(352\) −1.89819e37 −2.25668
\(353\) 1.07725e37 1.22909 0.614543 0.788883i \(-0.289340\pi\)
0.614543 + 0.788883i \(0.289340\pi\)
\(354\) −3.02866e35 −0.0331668
\(355\) −4.82811e36 −0.507535
\(356\) 1.40310e37 1.41600
\(357\) 3.10060e36 0.300440
\(358\) 1.97120e37 1.83412
\(359\) −5.90053e36 −0.527256 −0.263628 0.964624i \(-0.584919\pi\)
−0.263628 + 0.964624i \(0.584919\pi\)
\(360\) −7.31778e35 −0.0628048
\(361\) −9.50507e36 −0.783612
\(362\) −2.96667e37 −2.34961
\(363\) 1.64855e37 1.25446
\(364\) 3.45581e36 0.252685
\(365\) −9.19668e36 −0.646226
\(366\) −2.13803e37 −1.44390
\(367\) −8.73597e36 −0.567091 −0.283546 0.958959i \(-0.591511\pi\)
−0.283546 + 0.958959i \(0.591511\pi\)
\(368\) −1.23623e37 −0.771449
\(369\) 7.24309e35 0.0434552
\(370\) −2.62919e37 −1.51669
\(371\) −2.94932e36 −0.163606
\(372\) −1.38752e37 −0.740228
\(373\) 1.49457e37 0.766900 0.383450 0.923562i \(-0.374736\pi\)
0.383450 + 0.923562i \(0.374736\pi\)
\(374\) 6.62356e37 3.26929
\(375\) 1.23218e37 0.585086
\(376\) 7.04983e36 0.322073
\(377\) 1.32253e37 0.581376
\(378\) −2.29227e36 −0.0969694
\(379\) −1.52013e37 −0.618889 −0.309445 0.950917i \(-0.600143\pi\)
−0.309445 + 0.950917i \(0.600143\pi\)
\(380\) 5.85189e36 0.229316
\(381\) −2.65797e37 −1.00262
\(382\) −2.71545e37 −0.986104
\(383\) 5.24927e36 0.183534 0.0917670 0.995781i \(-0.470749\pi\)
0.0917670 + 0.995781i \(0.470749\pi\)
\(384\) −9.93514e36 −0.334481
\(385\) 1.31719e37 0.427038
\(386\) 2.47502e37 0.772792
\(387\) 1.90754e37 0.573673
\(388\) −3.73063e37 −1.08074
\(389\) −4.27696e37 −1.19362 −0.596810 0.802382i \(-0.703566\pi\)
−0.596810 + 0.802382i \(0.703566\pi\)
\(390\) 1.56187e37 0.419960
\(391\) 3.49442e37 0.905341
\(392\) 1.69969e36 0.0424347
\(393\) −5.49633e36 −0.132245
\(394\) 2.31722e37 0.537368
\(395\) 2.76299e36 0.0617620
\(396\) −2.14142e37 −0.461446
\(397\) 1.50576e37 0.312819 0.156409 0.987692i \(-0.450008\pi\)
0.156409 + 0.987692i \(0.450008\pi\)
\(398\) 4.75969e37 0.953397
\(399\) −5.25550e36 −0.101510
\(400\) 3.76432e37 0.701160
\(401\) 7.67293e37 1.37838 0.689188 0.724583i \(-0.257967\pi\)
0.689188 + 0.724583i \(0.257967\pi\)
\(402\) 2.41169e37 0.417872
\(403\) −8.49053e37 −1.41910
\(404\) 6.29478e36 0.101496
\(405\) −4.53055e36 −0.0704779
\(406\) −2.26879e37 −0.340539
\(407\) 2.20585e38 3.19489
\(408\) 1.68926e37 0.236116
\(409\) 4.94517e37 0.667103 0.333552 0.942732i \(-0.391753\pi\)
0.333552 + 0.942732i \(0.391753\pi\)
\(410\) 8.46623e36 0.110236
\(411\) −6.04327e37 −0.759567
\(412\) 2.49825e37 0.303128
\(413\) 1.39035e36 0.0162873
\(414\) −2.58342e37 −0.292207
\(415\) 8.83664e37 0.965139
\(416\) 1.03326e38 1.08983
\(417\) −8.67041e37 −0.883216
\(418\) −1.12269e38 −1.10459
\(419\) 8.54276e36 0.0811886 0.0405943 0.999176i \(-0.487075\pi\)
0.0405943 + 0.999176i \(0.487075\pi\)
\(420\) −1.17172e37 −0.107574
\(421\) −4.56419e37 −0.404830 −0.202415 0.979300i \(-0.564879\pi\)
−0.202415 + 0.979300i \(0.564879\pi\)
\(422\) 9.14845e37 0.784002
\(423\) 4.36466e37 0.361422
\(424\) −1.60684e37 −0.128578
\(425\) −1.06405e38 −0.822853
\(426\) −8.23976e37 −0.615851
\(427\) 9.81494e37 0.709061
\(428\) −5.76848e37 −0.402834
\(429\) −1.31038e38 −0.884641
\(430\) 2.22967e38 1.45528
\(431\) −1.01193e38 −0.638601 −0.319301 0.947654i \(-0.603448\pi\)
−0.319301 + 0.947654i \(0.603448\pi\)
\(432\) −3.69993e37 −0.225777
\(433\) 2.91333e38 1.71915 0.859577 0.511006i \(-0.170727\pi\)
0.859577 + 0.511006i \(0.170727\pi\)
\(434\) 1.45654e38 0.831230
\(435\) −4.48414e37 −0.247505
\(436\) −1.25286e38 −0.668879
\(437\) −5.92302e37 −0.305888
\(438\) −1.56953e38 −0.784141
\(439\) −3.78609e38 −1.83002 −0.915009 0.403434i \(-0.867817\pi\)
−0.915009 + 0.403434i \(0.867817\pi\)
\(440\) 7.17627e37 0.335610
\(441\) 1.05230e37 0.0476190
\(442\) −3.60548e38 −1.57885
\(443\) −2.91221e38 −1.23415 −0.617076 0.786904i \(-0.711683\pi\)
−0.617076 + 0.786904i \(0.711683\pi\)
\(444\) −1.96223e38 −0.804817
\(445\) −2.91110e38 −1.15568
\(446\) −4.43905e37 −0.170582
\(447\) −1.45576e38 −0.541540
\(448\) −5.41306e37 −0.194945
\(449\) −6.73368e37 −0.234791 −0.117395 0.993085i \(-0.537454\pi\)
−0.117395 + 0.993085i \(0.537454\pi\)
\(450\) 7.86650e37 0.265583
\(451\) −7.10302e37 −0.232211
\(452\) 2.75348e38 0.871714
\(453\) −3.28521e38 −1.00725
\(454\) 4.51639e38 1.34116
\(455\) −7.17000e37 −0.206231
\(456\) −2.86329e37 −0.0797765
\(457\) −4.64507e37 −0.125374 −0.0626870 0.998033i \(-0.519967\pi\)
−0.0626870 + 0.998033i \(0.519967\pi\)
\(458\) 1.41652e38 0.370402
\(459\) 1.04585e38 0.264963
\(460\) −1.32054e38 −0.324163
\(461\) −9.77508e37 −0.232518 −0.116259 0.993219i \(-0.537090\pi\)
−0.116259 + 0.993219i \(0.537090\pi\)
\(462\) 2.24794e38 0.518175
\(463\) 2.31687e38 0.517581 0.258790 0.965934i \(-0.416676\pi\)
0.258790 + 0.965934i \(0.416676\pi\)
\(464\) −3.66203e38 −0.792887
\(465\) 2.87877e38 0.604142
\(466\) −4.01545e37 −0.0816843
\(467\) −1.90269e38 −0.375209 −0.187605 0.982245i \(-0.560072\pi\)
−0.187605 + 0.982245i \(0.560072\pi\)
\(468\) 1.16566e38 0.222848
\(469\) −1.10712e38 −0.205206
\(470\) 5.10172e38 0.916848
\(471\) −3.03025e38 −0.528050
\(472\) 7.57489e36 0.0128002
\(473\) −1.87065e39 −3.06554
\(474\) 4.71538e37 0.0749430
\(475\) 1.80356e38 0.278018
\(476\) 2.70483e38 0.404427
\(477\) −9.94819e37 −0.144287
\(478\) 1.26730e38 0.178309
\(479\) 5.73704e38 0.783110 0.391555 0.920155i \(-0.371937\pi\)
0.391555 + 0.920155i \(0.371937\pi\)
\(480\) −3.50334e38 −0.463965
\(481\) −1.20073e39 −1.54292
\(482\) 1.16844e39 1.45689
\(483\) 1.18596e38 0.143495
\(484\) 1.43813e39 1.68865
\(485\) 7.74018e38 0.882055
\(486\) −7.73194e37 −0.0855190
\(487\) −5.36038e38 −0.575473 −0.287737 0.957710i \(-0.592903\pi\)
−0.287737 + 0.957710i \(0.592903\pi\)
\(488\) 5.34735e38 0.557252
\(489\) 1.82905e37 0.0185033
\(490\) 1.23000e38 0.120799
\(491\) 5.02527e38 0.479157 0.239579 0.970877i \(-0.422991\pi\)
0.239579 + 0.970877i \(0.422991\pi\)
\(492\) 6.31856e37 0.0584957
\(493\) 1.03513e39 0.930501
\(494\) 6.11126e38 0.533445
\(495\) 4.44294e38 0.376612
\(496\) 2.35099e39 1.93538
\(497\) 3.78259e38 0.302428
\(498\) 1.50808e39 1.17111
\(499\) −2.60379e39 −1.96403 −0.982015 0.188802i \(-0.939540\pi\)
−0.982015 + 0.188802i \(0.939540\pi\)
\(500\) 1.07490e39 0.787594
\(501\) 2.26934e38 0.161530
\(502\) −1.23176e39 −0.851767
\(503\) −2.11973e38 −0.142411 −0.0712057 0.997462i \(-0.522685\pi\)
−0.0712057 + 0.997462i \(0.522685\pi\)
\(504\) 5.73312e37 0.0374239
\(505\) −1.30602e38 −0.0828370
\(506\) 2.53346e39 1.56146
\(507\) −2.50655e38 −0.150128
\(508\) −2.31870e39 −1.34965
\(509\) 5.11976e38 0.289629 0.144814 0.989459i \(-0.453741\pi\)
0.144814 + 0.989459i \(0.453741\pi\)
\(510\) 1.22246e39 0.672152
\(511\) 7.20515e38 0.385071
\(512\) −2.19025e39 −1.13783
\(513\) −1.77271e38 −0.0895230
\(514\) 1.42174e39 0.697999
\(515\) −5.18328e38 −0.247400
\(516\) 1.66406e39 0.772231
\(517\) −4.28025e39 −1.93133
\(518\) 2.05984e39 0.903759
\(519\) 6.79474e38 0.289899
\(520\) −3.90634e38 −0.162077
\(521\) 1.71985e39 0.693975 0.346988 0.937870i \(-0.387205\pi\)
0.346988 + 0.937870i \(0.387205\pi\)
\(522\) −7.65274e38 −0.300327
\(523\) −8.39812e37 −0.0320559 −0.0160279 0.999872i \(-0.505102\pi\)
−0.0160279 + 0.999872i \(0.505102\pi\)
\(524\) −4.79476e38 −0.178018
\(525\) −3.61124e38 −0.130421
\(526\) 1.87477e39 0.658648
\(527\) −6.64546e39 −2.27128
\(528\) 3.62838e39 1.20648
\(529\) −1.75447e39 −0.567594
\(530\) −1.16281e39 −0.366025
\(531\) 4.68973e37 0.0143641
\(532\) −4.58467e38 −0.136644
\(533\) 3.86647e38 0.112142
\(534\) −4.96815e39 −1.40232
\(535\) 1.19682e39 0.328776
\(536\) −6.03179e38 −0.161271
\(537\) −3.05230e39 −0.794329
\(538\) −1.26526e39 −0.320507
\(539\) −1.03195e39 −0.254462
\(540\) −3.95226e38 −0.0948714
\(541\) 5.59912e39 1.30846 0.654229 0.756297i \(-0.272993\pi\)
0.654229 + 0.756297i \(0.272993\pi\)
\(542\) −5.52136e39 −1.25619
\(543\) 4.59374e39 1.01758
\(544\) 8.08725e39 1.74428
\(545\) 2.59939e39 0.545910
\(546\) −1.22365e39 −0.250244
\(547\) −2.35539e39 −0.469080 −0.234540 0.972107i \(-0.575358\pi\)
−0.234540 + 0.972107i \(0.575358\pi\)
\(548\) −5.27189e39 −1.02246
\(549\) 3.31063e39 0.625333
\(550\) −7.71438e39 −1.41919
\(551\) −1.75455e39 −0.314388
\(552\) 6.46131e38 0.112773
\(553\) −2.16467e38 −0.0368025
\(554\) 7.80572e39 1.29277
\(555\) 4.07117e39 0.656857
\(556\) −7.56369e39 −1.18891
\(557\) −1.15628e40 −1.77077 −0.885387 0.464855i \(-0.846106\pi\)
−0.885387 + 0.464855i \(0.846106\pi\)
\(558\) 4.91298e39 0.733076
\(559\) 1.01827e40 1.48045
\(560\) 1.98534e39 0.281260
\(561\) −1.02562e40 −1.41588
\(562\) 1.67809e40 2.25756
\(563\) 4.36317e39 0.572046 0.286023 0.958223i \(-0.407667\pi\)
0.286023 + 0.958223i \(0.407667\pi\)
\(564\) 3.80754e39 0.486516
\(565\) −5.71281e39 −0.711455
\(566\) −7.60981e39 −0.923710
\(567\) 3.54946e38 0.0419961
\(568\) 2.06082e39 0.237678
\(569\) −9.83480e39 −1.10570 −0.552850 0.833281i \(-0.686460\pi\)
−0.552850 + 0.833281i \(0.686460\pi\)
\(570\) −2.07206e39 −0.227100
\(571\) −8.52295e39 −0.910681 −0.455341 0.890317i \(-0.650483\pi\)
−0.455341 + 0.890317i \(0.650483\pi\)
\(572\) −1.14312e40 −1.19083
\(573\) 4.20473e39 0.427067
\(574\) −6.63288e38 −0.0656870
\(575\) −4.06991e39 −0.393008
\(576\) −1.82585e39 −0.171925
\(577\) 4.58587e39 0.421087 0.210544 0.977584i \(-0.432477\pi\)
0.210544 + 0.977584i \(0.432477\pi\)
\(578\) −1.33323e40 −1.19386
\(579\) −3.83244e39 −0.334685
\(580\) −3.91177e39 −0.333171
\(581\) −6.92308e39 −0.575103
\(582\) 1.32096e40 1.07030
\(583\) 9.75581e39 0.771027
\(584\) 3.92550e39 0.302627
\(585\) −2.41847e39 −0.181879
\(586\) 5.14514e39 0.377469
\(587\) −6.07209e39 −0.434596 −0.217298 0.976105i \(-0.569724\pi\)
−0.217298 + 0.976105i \(0.569724\pi\)
\(588\) 9.17983e38 0.0641008
\(589\) 1.12640e40 0.767399
\(590\) 5.48168e38 0.0364384
\(591\) −3.58809e39 −0.232726
\(592\) 3.32477e40 2.10425
\(593\) 1.41593e40 0.874479 0.437240 0.899345i \(-0.355956\pi\)
0.437240 + 0.899345i \(0.355956\pi\)
\(594\) 7.58242e39 0.456988
\(595\) −5.61189e39 −0.330075
\(596\) −1.26994e40 −0.728976
\(597\) −7.37013e39 −0.412903
\(598\) −1.37907e40 −0.754082
\(599\) −1.06418e39 −0.0567969 −0.0283984 0.999597i \(-0.509041\pi\)
−0.0283984 + 0.999597i \(0.509041\pi\)
\(600\) −1.96747e39 −0.102498
\(601\) 3.01640e40 1.53394 0.766970 0.641682i \(-0.221763\pi\)
0.766970 + 0.641682i \(0.221763\pi\)
\(602\) −1.74683e40 −0.867167
\(603\) −3.73437e39 −0.180974
\(604\) −2.86588e40 −1.35588
\(605\) −2.98377e40 −1.37820
\(606\) −2.22888e39 −0.100516
\(607\) −4.98397e39 −0.219452 −0.109726 0.993962i \(-0.534997\pi\)
−0.109726 + 0.993962i \(0.534997\pi\)
\(608\) −1.37078e40 −0.589341
\(609\) 3.51310e39 0.147482
\(610\) 3.86969e40 1.58633
\(611\) 2.32992e40 0.932703
\(612\) 9.12353e39 0.356671
\(613\) 9.73763e39 0.371772 0.185886 0.982571i \(-0.440485\pi\)
0.185886 + 0.982571i \(0.440485\pi\)
\(614\) −1.62062e39 −0.0604282
\(615\) −1.31095e39 −0.0477417
\(616\) −5.62226e39 −0.199982
\(617\) −1.41936e40 −0.493125 −0.246562 0.969127i \(-0.579301\pi\)
−0.246562 + 0.969127i \(0.579301\pi\)
\(618\) −8.84591e39 −0.300199
\(619\) −1.78871e40 −0.592959 −0.296480 0.955039i \(-0.595813\pi\)
−0.296480 + 0.955039i \(0.595813\pi\)
\(620\) 2.51132e40 0.813246
\(621\) 4.00029e39 0.126550
\(622\) 3.05423e40 0.943933
\(623\) 2.28070e40 0.688640
\(624\) −1.97508e40 −0.582650
\(625\) −1.56603e39 −0.0451379
\(626\) 1.17032e40 0.329591
\(627\) 1.73842e40 0.478384
\(628\) −2.64346e40 −0.710816
\(629\) −9.39803e40 −2.46946
\(630\) 4.14886e39 0.106535
\(631\) −6.50199e40 −1.63163 −0.815813 0.578316i \(-0.803710\pi\)
−0.815813 + 0.578316i \(0.803710\pi\)
\(632\) −1.17935e39 −0.0289231
\(633\) −1.41659e40 −0.339540
\(634\) 8.42389e40 1.97341
\(635\) 4.81075e40 1.10152
\(636\) −8.67837e39 −0.194227
\(637\) 5.61734e39 0.122888
\(638\) 7.50474e40 1.60486
\(639\) 1.27589e40 0.266716
\(640\) 1.79820e40 0.367475
\(641\) 9.19232e40 1.83647 0.918235 0.396035i \(-0.129614\pi\)
0.918235 + 0.396035i \(0.129614\pi\)
\(642\) 2.04253e40 0.398942
\(643\) −5.16413e40 −0.986139 −0.493070 0.869990i \(-0.664125\pi\)
−0.493070 + 0.869990i \(0.664125\pi\)
\(644\) 1.03458e40 0.193161
\(645\) −3.45252e40 −0.630262
\(646\) 4.78322e40 0.853786
\(647\) −6.55912e40 −1.14481 −0.572406 0.819971i \(-0.693990\pi\)
−0.572406 + 0.819971i \(0.693990\pi\)
\(648\) 1.93381e39 0.0330047
\(649\) −4.59904e39 −0.0767572
\(650\) 4.19926e40 0.685376
\(651\) −2.25538e40 −0.359994
\(652\) 1.59559e39 0.0249075
\(653\) 7.10534e40 1.08479 0.542393 0.840125i \(-0.317519\pi\)
0.542393 + 0.840125i \(0.317519\pi\)
\(654\) 4.43617e40 0.662416
\(655\) 9.94800e39 0.145290
\(656\) −1.07061e40 −0.152941
\(657\) 2.43033e40 0.339600
\(658\) −3.99695e40 −0.546327
\(659\) 8.95221e40 1.19699 0.598497 0.801125i \(-0.295765\pi\)
0.598497 + 0.801125i \(0.295765\pi\)
\(660\) 3.87583e40 0.506964
\(661\) 1.26579e41 1.61972 0.809862 0.586620i \(-0.199542\pi\)
0.809862 + 0.586620i \(0.199542\pi\)
\(662\) 1.21602e40 0.152229
\(663\) 5.58290e40 0.683775
\(664\) −3.77181e40 −0.451974
\(665\) 9.51211e39 0.111523
\(666\) 6.94795e40 0.797040
\(667\) 3.95932e40 0.444422
\(668\) 1.97968e40 0.217438
\(669\) 6.87364e39 0.0738768
\(670\) −4.36500e40 −0.459092
\(671\) −3.24661e41 −3.34159
\(672\) 2.74470e40 0.276465
\(673\) −3.05450e40 −0.301108 −0.150554 0.988602i \(-0.548106\pi\)
−0.150554 + 0.988602i \(0.548106\pi\)
\(674\) 4.82685e40 0.465688
\(675\) −1.21809e40 −0.115020
\(676\) −2.18661e40 −0.202090
\(677\) −4.39630e40 −0.397696 −0.198848 0.980030i \(-0.563720\pi\)
−0.198848 + 0.980030i \(0.563720\pi\)
\(678\) −9.74962e40 −0.863291
\(679\) −6.06405e40 −0.525595
\(680\) −3.05746e40 −0.259407
\(681\) −6.99340e40 −0.580838
\(682\) −4.81797e41 −3.91733
\(683\) 1.17734e41 0.937131 0.468565 0.883429i \(-0.344771\pi\)
0.468565 + 0.883429i \(0.344771\pi\)
\(684\) −1.54643e40 −0.120508
\(685\) 1.09379e41 0.834491
\(686\) −9.63648e39 −0.0719811
\(687\) −2.19341e40 −0.160416
\(688\) −2.81955e41 −2.01905
\(689\) −5.31049e40 −0.372354
\(690\) 4.67582e40 0.321030
\(691\) −5.40635e40 −0.363473 −0.181737 0.983347i \(-0.558172\pi\)
−0.181737 + 0.983347i \(0.558172\pi\)
\(692\) 5.92744e40 0.390237
\(693\) −3.48082e40 −0.224414
\(694\) −3.76900e41 −2.37966
\(695\) 1.56929e41 0.970338
\(696\) 1.91400e40 0.115907
\(697\) 3.02625e40 0.179486
\(698\) 3.06537e41 1.78065
\(699\) 6.21772e39 0.0353763
\(700\) −3.15029e40 −0.175561
\(701\) −3.18652e41 −1.73942 −0.869712 0.493559i \(-0.835696\pi\)
−0.869712 + 0.493559i \(0.835696\pi\)
\(702\) −4.12743e40 −0.220694
\(703\) 1.59296e41 0.834358
\(704\) 1.79054e41 0.918715
\(705\) −7.89974e40 −0.397073
\(706\) −3.32749e41 −1.63851
\(707\) 1.02320e40 0.0493606
\(708\) 4.09112e39 0.0193357
\(709\) −2.38651e41 −1.10508 −0.552538 0.833488i \(-0.686341\pi\)
−0.552538 + 0.833488i \(0.686341\pi\)
\(710\) 1.49134e41 0.676599
\(711\) −7.30153e39 −0.0324567
\(712\) 1.24257e41 0.541203
\(713\) −2.54184e41 −1.08480
\(714\) −9.57739e40 −0.400519
\(715\) 2.37171e41 0.971904
\(716\) −2.66270e41 −1.06926
\(717\) −1.96235e40 −0.0772232
\(718\) 1.82260e41 0.702890
\(719\) −1.96842e41 −0.743959 −0.371980 0.928241i \(-0.621321\pi\)
−0.371980 + 0.928241i \(0.621321\pi\)
\(720\) 6.69664e40 0.248048
\(721\) 4.06085e40 0.147420
\(722\) 2.93600e41 1.04464
\(723\) −1.80927e41 −0.630956
\(724\) 4.00738e41 1.36978
\(725\) −1.20561e41 −0.403930
\(726\) −5.09217e41 −1.67233
\(727\) −6.04250e40 −0.194522 −0.0972608 0.995259i \(-0.531008\pi\)
−0.0972608 + 0.995259i \(0.531008\pi\)
\(728\) 3.06043e40 0.0965778
\(729\) 1.19725e40 0.0370370
\(730\) 2.84074e41 0.861491
\(731\) 7.96993e41 2.36948
\(732\) 2.88805e41 0.841771
\(733\) −3.82878e40 −0.109409 −0.0547045 0.998503i \(-0.517422\pi\)
−0.0547045 + 0.998503i \(0.517422\pi\)
\(734\) 2.69843e41 0.755995
\(735\) −1.90460e40 −0.0523163
\(736\) 3.09331e41 0.833097
\(737\) 3.66216e41 0.967071
\(738\) −2.23730e40 −0.0579305
\(739\) 7.59644e41 1.92870 0.964352 0.264623i \(-0.0852476\pi\)
0.964352 + 0.264623i \(0.0852476\pi\)
\(740\) 3.55151e41 0.884205
\(741\) −9.46297e40 −0.231027
\(742\) 9.11008e40 0.218105
\(743\) 7.01191e41 1.64626 0.823130 0.567853i \(-0.192226\pi\)
0.823130 + 0.567853i \(0.192226\pi\)
\(744\) −1.22877e41 −0.282919
\(745\) 2.63483e41 0.594959
\(746\) −4.61655e41 −1.02236
\(747\) −2.33519e41 −0.507193
\(748\) −8.94710e41 −1.90594
\(749\) −9.37653e40 −0.195910
\(750\) −3.80604e41 −0.779984
\(751\) 1.74160e41 0.350082 0.175041 0.984561i \(-0.443994\pi\)
0.175041 + 0.984561i \(0.443994\pi\)
\(752\) −6.45143e41 −1.27203
\(753\) 1.90731e41 0.368888
\(754\) −4.08515e41 −0.775038
\(755\) 5.94602e41 1.10661
\(756\) 3.09640e40 0.0565316
\(757\) 1.20658e40 0.0216106 0.0108053 0.999942i \(-0.496561\pi\)
0.0108053 + 0.999942i \(0.496561\pi\)
\(758\) 4.69550e41 0.825047
\(759\) −3.92294e41 −0.676247
\(760\) 5.18237e40 0.0876458
\(761\) 3.98373e41 0.661017 0.330508 0.943803i \(-0.392780\pi\)
0.330508 + 0.943803i \(0.392780\pi\)
\(762\) 8.21014e41 1.33661
\(763\) −2.03649e41 −0.325295
\(764\) 3.66803e41 0.574882
\(765\) −1.89292e41 −0.291099
\(766\) −1.62143e41 −0.244671
\(767\) 2.50345e40 0.0370686
\(768\) 5.11829e41 0.743682
\(769\) 8.20510e41 1.16991 0.584954 0.811066i \(-0.301113\pi\)
0.584954 + 0.811066i \(0.301113\pi\)
\(770\) −4.06863e41 −0.569289
\(771\) −2.20149e41 −0.302293
\(772\) −3.34326e41 −0.450525
\(773\) −8.53207e41 −1.12837 −0.564185 0.825648i \(-0.690810\pi\)
−0.564185 + 0.825648i \(0.690810\pi\)
\(774\) −5.89216e41 −0.764769
\(775\) 7.73989e41 0.985962
\(776\) −3.30380e41 −0.413066
\(777\) −3.18956e41 −0.391405
\(778\) 1.32110e42 1.59123
\(779\) −5.12947e40 −0.0606428
\(780\) −2.10977e41 −0.244830
\(781\) −1.25121e42 −1.42525
\(782\) −1.07938e42 −1.20692
\(783\) 1.18499e41 0.130067
\(784\) −1.55541e41 −0.167596
\(785\) 5.48455e41 0.580137
\(786\) 1.69775e41 0.176298
\(787\) 8.22773e41 0.838777 0.419389 0.907807i \(-0.362244\pi\)
0.419389 + 0.907807i \(0.362244\pi\)
\(788\) −3.13010e41 −0.313277
\(789\) −2.90298e41 −0.285251
\(790\) −8.53454e40 −0.0823355
\(791\) 4.47571e41 0.423939
\(792\) −1.89641e41 −0.176367
\(793\) 1.76726e42 1.61376
\(794\) −4.65110e41 −0.417022
\(795\) 1.80056e41 0.158520
\(796\) −6.42939e41 −0.555815
\(797\) −1.03982e42 −0.882701 −0.441350 0.897335i \(-0.645500\pi\)
−0.441350 + 0.897335i \(0.645500\pi\)
\(798\) 1.62336e41 0.135323
\(799\) 1.82361e42 1.49280
\(800\) −9.41913e41 −0.757192
\(801\) 7.69293e41 0.607323
\(802\) −2.37007e42 −1.83753
\(803\) −2.38334e42 −1.81472
\(804\) −3.25771e41 −0.243612
\(805\) −2.14651e41 −0.157649
\(806\) 2.62262e42 1.89181
\(807\) 1.95919e41 0.138807
\(808\) 5.57459e40 0.0387925
\(809\) 1.08854e42 0.744031 0.372015 0.928227i \(-0.378667\pi\)
0.372015 + 0.928227i \(0.378667\pi\)
\(810\) 1.39943e41 0.0939547
\(811\) 1.94986e42 1.28588 0.642941 0.765916i \(-0.277714\pi\)
0.642941 + 0.765916i \(0.277714\pi\)
\(812\) 3.06468e41 0.198528
\(813\) 8.54954e41 0.544039
\(814\) −6.81359e42 −4.25914
\(815\) −3.31047e40 −0.0203285
\(816\) −1.54588e42 −0.932540
\(817\) −1.35090e42 −0.800576
\(818\) −1.52750e42 −0.889321
\(819\) 1.89476e41 0.108377
\(820\) −1.14362e41 −0.0642658
\(821\) −3.00016e41 −0.165641 −0.0828205 0.996564i \(-0.526393\pi\)
−0.0828205 + 0.996564i \(0.526393\pi\)
\(822\) 1.86669e42 1.01259
\(823\) 1.07893e42 0.575037 0.287519 0.957775i \(-0.407170\pi\)
0.287519 + 0.957775i \(0.407170\pi\)
\(824\) 2.21242e41 0.115857
\(825\) 1.19453e42 0.614633
\(826\) −4.29463e40 −0.0217128
\(827\) 1.02449e42 0.508952 0.254476 0.967079i \(-0.418097\pi\)
0.254476 + 0.967079i \(0.418097\pi\)
\(828\) 3.48968e41 0.170352
\(829\) 1.75052e42 0.839703 0.419852 0.907593i \(-0.362082\pi\)
0.419852 + 0.907593i \(0.362082\pi\)
\(830\) −2.72953e42 −1.28664
\(831\) −1.20868e42 −0.559880
\(832\) −9.74666e41 −0.443678
\(833\) 4.39664e41 0.196684
\(834\) 2.67818e42 1.17742
\(835\) −4.10737e41 −0.177464
\(836\) 1.51653e42 0.643960
\(837\) −7.60749e41 −0.317484
\(838\) −2.63875e41 −0.108233
\(839\) −3.66014e41 −0.147553 −0.0737766 0.997275i \(-0.523505\pi\)
−0.0737766 + 0.997275i \(0.523505\pi\)
\(840\) −1.03766e41 −0.0411154
\(841\) −1.39484e42 −0.543228
\(842\) 1.40982e42 0.539683
\(843\) −2.59844e42 −0.977716
\(844\) −1.23577e42 −0.457060
\(845\) 4.53670e41 0.164937
\(846\) −1.34819e42 −0.481815
\(847\) 2.33764e42 0.821236
\(848\) 1.47045e42 0.507820
\(849\) 1.17834e42 0.400045
\(850\) 3.28672e42 1.09695
\(851\) −3.59468e42 −1.17945
\(852\) 1.11303e42 0.359031
\(853\) −6.63603e41 −0.210449 −0.105225 0.994448i \(-0.533556\pi\)
−0.105225 + 0.994448i \(0.533556\pi\)
\(854\) −3.03172e42 −0.945256
\(855\) 3.20848e41 0.0983537
\(856\) −5.10850e41 −0.153966
\(857\) 5.53619e42 1.64055 0.820274 0.571971i \(-0.193821\pi\)
0.820274 + 0.571971i \(0.193821\pi\)
\(858\) 4.04761e42 1.17932
\(859\) 1.48226e41 0.0424643 0.0212322 0.999775i \(-0.493241\pi\)
0.0212322 + 0.999775i \(0.493241\pi\)
\(860\) −3.01183e42 −0.848405
\(861\) 1.02707e41 0.0284481
\(862\) 3.12572e42 0.851325
\(863\) −6.02545e42 −1.61374 −0.806869 0.590730i \(-0.798840\pi\)
−0.806869 + 0.590730i \(0.798840\pi\)
\(864\) 9.25801e41 0.243819
\(865\) −1.22980e42 −0.318495
\(866\) −8.99891e42 −2.29182
\(867\) 2.06444e42 0.517042
\(868\) −1.96749e42 −0.484593
\(869\) 7.16033e41 0.173439
\(870\) 1.38510e42 0.329952
\(871\) −1.99347e42 −0.467031
\(872\) −1.10952e42 −0.255649
\(873\) −2.04543e42 −0.463531
\(874\) 1.82955e42 0.407782
\(875\) 1.74722e42 0.383029
\(876\) 2.12012e42 0.457141
\(877\) −7.08427e42 −1.50245 −0.751227 0.660044i \(-0.770538\pi\)
−0.751227 + 0.660044i \(0.770538\pi\)
\(878\) 1.16948e43 2.43961
\(879\) −7.96698e41 −0.163476
\(880\) −6.56714e42 −1.32549
\(881\) −4.20382e42 −0.834630 −0.417315 0.908762i \(-0.637029\pi\)
−0.417315 + 0.908762i \(0.637029\pi\)
\(882\) −3.25043e41 −0.0634814
\(883\) 7.79972e41 0.149847 0.0749236 0.997189i \(-0.476129\pi\)
0.0749236 + 0.997189i \(0.476129\pi\)
\(884\) 4.87028e42 0.920441
\(885\) −8.48810e40 −0.0157810
\(886\) 8.99545e42 1.64526
\(887\) −2.79977e42 −0.503768 −0.251884 0.967757i \(-0.581050\pi\)
−0.251884 + 0.967757i \(0.581050\pi\)
\(888\) −1.73773e42 −0.307605
\(889\) −3.76899e42 −0.656371
\(890\) 8.99203e42 1.54064
\(891\) −1.17410e42 −0.197915
\(892\) 5.99626e41 0.0994467
\(893\) −3.09100e42 −0.504374
\(894\) 4.49666e42 0.721932
\(895\) 5.52447e42 0.872683
\(896\) −1.40880e42 −0.218969
\(897\) 2.13542e42 0.326582
\(898\) 2.07995e42 0.313002
\(899\) −7.52956e42 −1.11495
\(900\) −1.06261e42 −0.154830
\(901\) −4.15647e42 −0.595958
\(902\) 2.19404e42 0.309563
\(903\) 2.70488e42 0.375557
\(904\) 2.43845e42 0.333174
\(905\) −8.31436e42 −1.11796
\(906\) 1.01476e43 1.34278
\(907\) −4.53883e42 −0.591069 −0.295534 0.955332i \(-0.595498\pi\)
−0.295534 + 0.955332i \(0.595498\pi\)
\(908\) −6.10074e42 −0.781876
\(909\) 3.45131e41 0.0435319
\(910\) 2.21472e42 0.274928
\(911\) −1.15803e42 −0.141483 −0.0707414 0.997495i \(-0.522536\pi\)
−0.0707414 + 0.997495i \(0.522536\pi\)
\(912\) 2.62025e42 0.315077
\(913\) 2.29003e43 2.71028
\(914\) 1.43481e42 0.167137
\(915\) −5.99202e42 −0.687017
\(916\) −1.91343e42 −0.215938
\(917\) −7.79377e41 −0.0865750
\(918\) −3.23050e42 −0.353224
\(919\) 6.69251e42 0.720301 0.360150 0.932894i \(-0.382725\pi\)
0.360150 + 0.932894i \(0.382725\pi\)
\(920\) −1.16945e42 −0.123897
\(921\) 2.50944e41 0.0261706
\(922\) 3.01940e42 0.309972
\(923\) 6.81087e42 0.688300
\(924\) −3.03652e42 −0.302087
\(925\) 1.09458e43 1.07199
\(926\) −7.15654e42 −0.689992
\(927\) 1.36974e42 0.130012
\(928\) 9.16317e42 0.856249
\(929\) 7.56474e42 0.695931 0.347965 0.937507i \(-0.386873\pi\)
0.347965 + 0.937507i \(0.386873\pi\)
\(930\) −8.89217e42 −0.805388
\(931\) −7.45228e41 −0.0664536
\(932\) 5.42407e41 0.0476206
\(933\) −4.72932e42 −0.408804
\(934\) 5.87718e42 0.500195
\(935\) 1.85631e43 1.55555
\(936\) 1.03230e42 0.0851736
\(937\) −2.01813e43 −1.63955 −0.819774 0.572687i \(-0.805901\pi\)
−0.819774 + 0.572687i \(0.805901\pi\)
\(938\) 3.41976e42 0.273561
\(939\) −1.81218e42 −0.142741
\(940\) −6.89140e42 −0.534507
\(941\) −2.21689e43 −1.69315 −0.846575 0.532270i \(-0.821339\pi\)
−0.846575 + 0.532270i \(0.821339\pi\)
\(942\) 9.36006e42 0.703948
\(943\) 1.15752e42 0.0857252
\(944\) −6.93192e41 −0.0505545
\(945\) −6.42430e41 −0.0461386
\(946\) 5.77821e43 4.08669
\(947\) −2.42844e43 −1.69142 −0.845711 0.533641i \(-0.820823\pi\)
−0.845711 + 0.533641i \(0.820823\pi\)
\(948\) −6.36954e41 −0.0436905
\(949\) 1.29735e43 0.876388
\(950\) −5.57097e42 −0.370628
\(951\) −1.30440e43 −0.854657
\(952\) 2.39537e42 0.154574
\(953\) −1.45517e43 −0.924839 −0.462420 0.886661i \(-0.653019\pi\)
−0.462420 + 0.886661i \(0.653019\pi\)
\(954\) 3.07287e42 0.192351
\(955\) −7.61029e42 −0.469194
\(956\) −1.71187e42 −0.103951
\(957\) −1.16207e43 −0.695040
\(958\) −1.77210e43 −1.04397
\(959\) −8.56934e42 −0.497253
\(960\) 3.30467e42 0.188884
\(961\) 3.05771e43 1.72150
\(962\) 3.70892e43 2.05688
\(963\) −3.16275e42 −0.172776
\(964\) −1.57833e43 −0.849340
\(965\) 6.93647e42 0.367699
\(966\) −3.66328e42 −0.191294
\(967\) 8.30591e42 0.427271 0.213635 0.976913i \(-0.431470\pi\)
0.213635 + 0.976913i \(0.431470\pi\)
\(968\) 1.27359e43 0.645411
\(969\) −7.40658e42 −0.369762
\(970\) −2.39085e43 −1.17588
\(971\) −1.55969e43 −0.755718 −0.377859 0.925863i \(-0.623340\pi\)
−0.377859 + 0.925863i \(0.623340\pi\)
\(972\) 1.04443e42 0.0498562
\(973\) −1.22946e43 −0.578201
\(974\) 1.65575e43 0.767169
\(975\) −6.50234e42 −0.296826
\(976\) −4.89346e43 −2.20087
\(977\) 8.29683e42 0.367656 0.183828 0.982958i \(-0.441151\pi\)
0.183828 + 0.982958i \(0.441151\pi\)
\(978\) −5.64973e41 −0.0246669
\(979\) −7.54416e43 −3.24535
\(980\) −1.66149e42 −0.0704238
\(981\) −6.86919e42 −0.286883
\(982\) −1.55225e43 −0.638769
\(983\) 4.23448e42 0.171701 0.0858507 0.996308i \(-0.472639\pi\)
0.0858507 + 0.996308i \(0.472639\pi\)
\(984\) 5.59564e41 0.0223574
\(985\) 6.49422e42 0.255683
\(986\) −3.19741e43 −1.24046
\(987\) 6.18907e42 0.236606
\(988\) −8.25509e42 −0.310990
\(989\) 3.04844e43 1.13170
\(990\) −1.37237e43 −0.502066
\(991\) 1.25102e43 0.451019 0.225510 0.974241i \(-0.427595\pi\)
0.225510 + 0.974241i \(0.427595\pi\)
\(992\) −5.88266e43 −2.09004
\(993\) −1.88294e42 −0.0659283
\(994\) −1.16840e43 −0.403169
\(995\) 1.33395e43 0.453632
\(996\) −2.03712e43 −0.682740
\(997\) −1.78459e43 −0.589467 −0.294734 0.955579i \(-0.595231\pi\)
−0.294734 + 0.955579i \(0.595231\pi\)
\(998\) 8.04279e43 2.61827
\(999\) −1.07585e43 −0.345186
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 21.30.a.a.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.30.a.a.1.1 6 1.1 even 1 trivial