Properties

Label 1.116.a.a.1.5
Level $1$
Weight $116$
Character 1.1
Self dual yes
Analytic conductor $83.750$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,116,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 116, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 116);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 116 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(83.7504016273\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2 x^{8} + \cdots + 17\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{142}\cdot 3^{52}\cdot 5^{17}\cdot 7^{8}\cdot 11^{3}\cdot 13^{3}\cdot 17\cdot 19^{3}\cdot 23^{3}\cdot 29^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-4.70183e14\) of defining polynomial
Character \(\chi\) \(=\) 1.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.62458e15 q^{2} -2.61982e27 q^{3} -4.15357e34 q^{4} +1.47212e40 q^{5} +4.25609e42 q^{6} +6.19961e48 q^{7} +1.34960e50 q^{8} -5.31663e53 q^{9} +O(q^{10})\) \(q-1.62458e15 q^{2} -2.61982e27 q^{3} -4.15357e34 q^{4} +1.47212e40 q^{5} +4.25609e42 q^{6} +6.19961e48 q^{7} +1.34960e50 q^{8} -5.31663e53 q^{9} -2.39156e55 q^{10} +2.18993e59 q^{11} +1.08816e62 q^{12} +1.46197e64 q^{13} -1.00717e64 q^{14} -3.85667e67 q^{15} +1.72511e69 q^{16} +4.52776e70 q^{17} +8.63727e68 q^{18} +3.61811e73 q^{19} -6.11454e74 q^{20} -1.62418e76 q^{21} -3.55771e74 q^{22} -1.94208e78 q^{23} -3.53571e77 q^{24} -2.40286e79 q^{25} -2.37507e79 q^{26} +2.07667e82 q^{27} -2.57505e83 q^{28} -2.21850e84 q^{29} +6.26546e82 q^{30} -2.45370e85 q^{31} -8.40859e84 q^{32} -5.73722e86 q^{33} -7.35569e85 q^{34} +9.12654e88 q^{35} +2.20830e88 q^{36} -7.78368e89 q^{37} -5.87790e88 q^{38} -3.83008e91 q^{39} +1.98677e90 q^{40} +8.43624e92 q^{41} +2.63861e91 q^{42} -4.15081e93 q^{43} -9.09604e93 q^{44} -7.82670e93 q^{45} +3.15505e93 q^{46} +5.23101e95 q^{47} -4.51947e96 q^{48} +2.30792e97 q^{49} +3.90363e94 q^{50} -1.18619e98 q^{51} -6.07239e98 q^{52} +2.18808e99 q^{53} -3.37370e97 q^{54} +3.22383e99 q^{55} +8.36700e98 q^{56} -9.47879e100 q^{57} +3.60412e99 q^{58} +1.47847e101 q^{59} +1.60190e102 q^{60} -2.05936e102 q^{61} +3.98621e100 q^{62} -3.29610e102 q^{63} -7.16445e103 q^{64} +2.15218e104 q^{65} +9.32055e101 q^{66} +9.97754e102 q^{67} -1.88064e105 q^{68} +5.08789e105 q^{69} -1.48267e104 q^{70} -2.71729e106 q^{71} -7.17533e103 q^{72} +1.83490e107 q^{73} +1.26452e105 q^{74} +6.29506e106 q^{75} -1.50281e108 q^{76} +1.35767e108 q^{77} +6.22226e106 q^{78} +1.76884e109 q^{79} +2.53956e109 q^{80} -5.04732e109 q^{81} -1.37053e108 q^{82} +1.20242e110 q^{83} +6.74616e110 q^{84} +6.66540e110 q^{85} +6.74330e108 q^{86} +5.81206e111 q^{87} +2.95554e109 q^{88} +1.26336e112 q^{89} +1.27151e109 q^{90} +9.06361e112 q^{91} +8.06657e112 q^{92} +6.42824e112 q^{93} -8.49817e110 q^{94} +5.32628e113 q^{95} +2.20290e112 q^{96} +6.14831e112 q^{97} -3.74939e112 q^{98} -1.16431e113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 11\!\cdots\!44 q^{2}+ \cdots + 26\!\cdots\!13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 11\!\cdots\!44 q^{2}+ \cdots - 81\!\cdots\!24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.62458e15 −0.00797104 −0.00398552 0.999992i \(-0.501269\pi\)
−0.00398552 + 0.999992i \(0.501269\pi\)
\(3\) −2.61982e27 −0.963383 −0.481691 0.876341i \(-0.659977\pi\)
−0.481691 + 0.876341i \(0.659977\pi\)
\(4\) −4.15357e34 −0.999936
\(5\) 1.47212e40 0.948783 0.474391 0.880314i \(-0.342668\pi\)
0.474391 + 0.880314i \(0.342668\pi\)
\(6\) 4.25609e42 0.00767916
\(7\) 6.19961e48 1.58207 0.791037 0.611768i \(-0.209542\pi\)
0.791037 + 0.611768i \(0.209542\pi\)
\(8\) 1.34960e50 0.0159416
\(9\) −5.31663e53 −0.0718939
\(10\) −2.39156e55 −0.00756279
\(11\) 2.18993e59 0.288636 0.144318 0.989531i \(-0.453901\pi\)
0.144318 + 0.989531i \(0.453901\pi\)
\(12\) 1.08816e62 0.963321
\(13\) 1.46197e64 1.29776 0.648880 0.760890i \(-0.275238\pi\)
0.648880 + 0.760890i \(0.275238\pi\)
\(14\) −1.00717e64 −0.0126108
\(15\) −3.85667e67 −0.914041
\(16\) 1.72511e69 0.999809
\(17\) 4.52776e70 0.803657 0.401829 0.915715i \(-0.368375\pi\)
0.401829 + 0.915715i \(0.368375\pi\)
\(18\) 8.63727e68 0.000573069 0
\(19\) 3.61811e73 1.07189 0.535945 0.844253i \(-0.319956\pi\)
0.535945 + 0.844253i \(0.319956\pi\)
\(20\) −6.11454e74 −0.948723
\(21\) −1.62418e76 −1.52414
\(22\) −3.55771e74 −0.00230073
\(23\) −1.94208e78 −0.974802 −0.487401 0.873178i \(-0.662055\pi\)
−0.487401 + 0.873178i \(0.662055\pi\)
\(24\) −3.53571e77 −0.0153578
\(25\) −2.40286e79 −0.0998111
\(26\) −2.37507e79 −0.0103445
\(27\) 2.07667e82 1.03264
\(28\) −2.57505e83 −1.58197
\(29\) −2.21850e84 −1.81207 −0.906034 0.423205i \(-0.860905\pi\)
−0.906034 + 0.423205i \(0.860905\pi\)
\(30\) 6.26546e82 0.00728586
\(31\) −2.45370e85 −0.433035 −0.216518 0.976279i \(-0.569470\pi\)
−0.216518 + 0.976279i \(0.569470\pi\)
\(32\) −8.40859e84 −0.0239111
\(33\) −5.73722e86 −0.278067
\(34\) −7.35569e85 −0.00640599
\(35\) 9.12654e88 1.50104
\(36\) 2.20830e88 0.0718894
\(37\) −7.78368e89 −0.524307 −0.262154 0.965026i \(-0.584433\pi\)
−0.262154 + 0.965026i \(0.584433\pi\)
\(38\) −5.87790e88 −0.00854408
\(39\) −3.83008e91 −1.25024
\(40\) 1.98677e90 0.0151251
\(41\) 8.43624e92 1.55266 0.776332 0.630324i \(-0.217078\pi\)
0.776332 + 0.630324i \(0.217078\pi\)
\(42\) 2.63861e91 0.0121490
\(43\) −4.15081e93 −0.493966 −0.246983 0.969020i \(-0.579439\pi\)
−0.246983 + 0.969020i \(0.579439\pi\)
\(44\) −9.09604e93 −0.288618
\(45\) −7.82670e93 −0.0682117
\(46\) 3.15505e93 0.00777018
\(47\) 5.23101e95 0.374075 0.187038 0.982353i \(-0.440111\pi\)
0.187038 + 0.982353i \(0.440111\pi\)
\(48\) −4.51947e96 −0.963199
\(49\) 2.30792e97 1.50296
\(50\) 3.90363e94 0.000795598 0
\(51\) −1.18619e98 −0.774229
\(52\) −6.07239e98 −1.29768
\(53\) 2.18808e99 1.56387 0.781936 0.623358i \(-0.214232\pi\)
0.781936 + 0.623358i \(0.214232\pi\)
\(54\) −3.37370e97 −0.00823125
\(55\) 3.22383e99 0.273853
\(56\) 8.36700e98 0.0252208
\(57\) −9.47879e100 −1.03264
\(58\) 3.60412e99 0.0144441
\(59\) 1.47847e101 0.221728 0.110864 0.993836i \(-0.464638\pi\)
0.110864 + 0.993836i \(0.464638\pi\)
\(60\) 1.60190e102 0.913983
\(61\) −2.05936e102 −0.454220 −0.227110 0.973869i \(-0.572928\pi\)
−0.227110 + 0.973869i \(0.572928\pi\)
\(62\) 3.98621e100 0.00345174
\(63\) −3.29610e102 −0.113741
\(64\) −7.16445e103 −0.999619
\(65\) 2.15218e104 1.23129
\(66\) 9.32055e101 0.00221649
\(67\) 9.97754e102 0.00999361 0.00499680 0.999988i \(-0.498409\pi\)
0.00499680 + 0.999988i \(0.498409\pi\)
\(68\) −1.88064e105 −0.803606
\(69\) 5.08789e105 0.939107
\(70\) −1.48267e104 −0.0119649
\(71\) −2.71729e106 −0.970021 −0.485010 0.874508i \(-0.661184\pi\)
−0.485010 + 0.874508i \(0.661184\pi\)
\(72\) −7.17533e103 −0.00114610
\(73\) 1.83490e107 1.32601 0.663005 0.748615i \(-0.269281\pi\)
0.663005 + 0.748615i \(0.269281\pi\)
\(74\) 1.26452e105 0.00417927
\(75\) 6.29506e106 0.0961562
\(76\) −1.50281e108 −1.07182
\(77\) 1.35767e108 0.456644
\(78\) 6.22226e106 0.00996572
\(79\) 1.76884e109 1.36186 0.680929 0.732350i \(-0.261576\pi\)
0.680929 + 0.732350i \(0.261576\pi\)
\(80\) 2.53956e109 0.948602
\(81\) −5.04732e109 −0.922937
\(82\) −1.37053e108 −0.0123763
\(83\) 1.20242e110 0.540836 0.270418 0.962743i \(-0.412838\pi\)
0.270418 + 0.962743i \(0.412838\pi\)
\(84\) 6.74616e110 1.52405
\(85\) 6.66540e110 0.762496
\(86\) 6.74330e108 0.00393742
\(87\) 5.81206e111 1.74572
\(88\) 2.95554e109 0.00460132
\(89\) 1.26336e112 1.02708 0.513539 0.858066i \(-0.328334\pi\)
0.513539 + 0.858066i \(0.328334\pi\)
\(90\) 1.27151e109 0.000543718 0
\(91\) 9.06361e112 2.05315
\(92\) 8.06657e112 0.974740
\(93\) 6.42824e112 0.417179
\(94\) −8.49817e110 −0.00298177
\(95\) 5.32628e113 1.01699
\(96\) 2.20290e112 0.0230355
\(97\) 6.14831e112 0.0354308 0.0177154 0.999843i \(-0.494361\pi\)
0.0177154 + 0.999843i \(0.494361\pi\)
\(98\) −3.74939e112 −0.0119801
\(99\) −1.16431e113 −0.0207512
\(100\) 9.98047e113 0.0998047
\(101\) −2.04862e115 −1.15606 −0.578032 0.816014i \(-0.696179\pi\)
−0.578032 + 0.816014i \(0.696179\pi\)
\(102\) 1.92706e113 0.00617142
\(103\) −9.39193e115 −1.71638 −0.858192 0.513329i \(-0.828412\pi\)
−0.858192 + 0.513329i \(0.828412\pi\)
\(104\) 1.97307e114 0.0206884
\(105\) −2.39099e116 −1.44608
\(106\) −3.55471e114 −0.0124657
\(107\) 6.02443e116 1.23126 0.615628 0.788037i \(-0.288902\pi\)
0.615628 + 0.788037i \(0.288902\pi\)
\(108\) −8.62559e116 −1.03258
\(109\) −2.07798e117 −1.46426 −0.732130 0.681165i \(-0.761474\pi\)
−0.732130 + 0.681165i \(0.761474\pi\)
\(110\) −5.23736e114 −0.00218290
\(111\) 2.03918e117 0.505108
\(112\) 1.06950e118 1.58177
\(113\) 1.21712e118 1.07974 0.539872 0.841747i \(-0.318472\pi\)
0.539872 + 0.841747i \(0.318472\pi\)
\(114\) 1.53990e116 0.00823121
\(115\) −2.85897e118 −0.924875
\(116\) 9.21470e118 1.81195
\(117\) −7.77274e117 −0.0933011
\(118\) −2.40188e116 −0.00176740
\(119\) 2.80704e119 1.27145
\(120\) −5.20497e117 −0.0145713
\(121\) −5.27692e119 −0.916689
\(122\) 3.34558e117 0.00362061
\(123\) −2.21014e120 −1.49581
\(124\) 1.01916e120 0.433008
\(125\) −3.89772e120 −1.04348
\(126\) 5.35476e117 0.000906638 0
\(127\) −8.61025e120 −0.925341 −0.462670 0.886530i \(-0.653109\pi\)
−0.462670 + 0.886530i \(0.653109\pi\)
\(128\) 4.65671e119 0.0318791
\(129\) 1.08744e121 0.475878
\(130\) −3.49639e119 −0.00981469
\(131\) −4.62276e121 −0.835221 −0.417611 0.908626i \(-0.637132\pi\)
−0.417611 + 0.908626i \(0.637132\pi\)
\(132\) 2.38300e121 0.278050
\(133\) 2.24309e122 1.69581
\(134\) −1.62093e118 −7.96595e−5 0
\(135\) 3.05710e122 0.979755
\(136\) 6.11068e120 0.0128116
\(137\) 2.86128e122 0.393665 0.196833 0.980437i \(-0.436934\pi\)
0.196833 + 0.980437i \(0.436934\pi\)
\(138\) −8.26566e120 −0.00748566
\(139\) 1.27509e123 0.762410 0.381205 0.924491i \(-0.375509\pi\)
0.381205 + 0.924491i \(0.375509\pi\)
\(140\) −3.79078e123 −1.50095
\(141\) −1.37043e123 −0.360377
\(142\) 4.41444e121 0.00773208
\(143\) 3.20161e123 0.374581
\(144\) −9.17176e122 −0.0718802
\(145\) −3.26589e124 −1.71926
\(146\) −2.98093e122 −0.0105697
\(147\) −6.04634e124 −1.44792
\(148\) 3.23301e124 0.524274
\(149\) 9.52826e124 1.04907 0.524534 0.851390i \(-0.324240\pi\)
0.524534 + 0.851390i \(0.324240\pi\)
\(150\) −1.02268e122 −0.000766465 0
\(151\) 2.48757e125 1.27234 0.636168 0.771551i \(-0.280519\pi\)
0.636168 + 0.771551i \(0.280519\pi\)
\(152\) 4.88301e123 0.0170876
\(153\) −2.40725e124 −0.0577781
\(154\) −2.20564e123 −0.00363993
\(155\) −3.61213e125 −0.410857
\(156\) 1.59085e126 1.25016
\(157\) 2.34288e126 1.27503 0.637514 0.770438i \(-0.279963\pi\)
0.637514 + 0.770438i \(0.279963\pi\)
\(158\) −2.87362e124 −0.0108554
\(159\) −5.73238e126 −1.50661
\(160\) −1.23784e125 −0.0226864
\(161\) −1.20401e127 −1.54221
\(162\) 8.19975e124 0.00735677
\(163\) 1.61392e127 1.01647 0.508234 0.861219i \(-0.330298\pi\)
0.508234 + 0.861219i \(0.330298\pi\)
\(164\) −3.50406e127 −1.55257
\(165\) −8.44586e126 −0.263826
\(166\) −1.95342e125 −0.00431103
\(167\) 3.44692e127 0.538560 0.269280 0.963062i \(-0.413214\pi\)
0.269280 + 0.963062i \(0.413214\pi\)
\(168\) −2.19200e126 −0.0242972
\(169\) 8.68276e127 0.684183
\(170\) −1.08284e126 −0.00607789
\(171\) −1.92362e127 −0.0770623
\(172\) 1.72407e128 0.493934
\(173\) 6.31383e128 1.29611 0.648055 0.761594i \(-0.275583\pi\)
0.648055 + 0.761594i \(0.275583\pi\)
\(174\) −9.44214e126 −0.0139152
\(175\) −1.48968e128 −0.157908
\(176\) 3.77787e128 0.288581
\(177\) −3.87331e128 −0.213609
\(178\) −2.05243e127 −0.00818688
\(179\) −5.59755e129 −1.61789 −0.808944 0.587885i \(-0.799961\pi\)
−0.808944 + 0.587885i \(0.799961\pi\)
\(180\) 3.25088e128 0.0682074
\(181\) −5.19054e129 −0.791943 −0.395972 0.918263i \(-0.629592\pi\)
−0.395972 + 0.918263i \(0.629592\pi\)
\(182\) −1.47245e128 −0.0163658
\(183\) 5.39514e129 0.437588
\(184\) −2.62103e128 −0.0155399
\(185\) −1.14585e130 −0.497454
\(186\) −1.04432e128 −0.00332535
\(187\) 9.91550e129 0.231965
\(188\) −2.17274e130 −0.374051
\(189\) 1.28745e131 1.63372
\(190\) −8.65295e128 −0.00810647
\(191\) −1.41019e130 −0.0976920 −0.0488460 0.998806i \(-0.515554\pi\)
−0.0488460 + 0.998806i \(0.515554\pi\)
\(192\) 1.87696e131 0.963015
\(193\) 2.91112e131 1.10794 0.553968 0.832538i \(-0.313113\pi\)
0.553968 + 0.832538i \(0.313113\pi\)
\(194\) −9.98839e127 −0.000282421 0
\(195\) −5.63833e131 −1.18621
\(196\) −9.58613e131 −1.50286
\(197\) 2.52276e130 0.0295167 0.0147583 0.999891i \(-0.495302\pi\)
0.0147583 + 0.999891i \(0.495302\pi\)
\(198\) 1.89150e128 0.000165409 0
\(199\) 4.98876e131 0.326543 0.163272 0.986581i \(-0.447795\pi\)
0.163272 + 0.986581i \(0.447795\pi\)
\(200\) −3.24291e129 −0.00159115
\(201\) −2.61393e130 −0.00962767
\(202\) 3.32813e130 0.00921503
\(203\) −1.37538e133 −2.86683
\(204\) 4.92693e132 0.774180
\(205\) 1.24191e133 1.47314
\(206\) 1.52579e131 0.0136814
\(207\) 1.03253e132 0.0700823
\(208\) 2.52205e133 1.29751
\(209\) 7.92342e132 0.309386
\(210\) 3.88434e131 0.0115268
\(211\) 2.97918e133 0.672753 0.336377 0.941728i \(-0.390798\pi\)
0.336377 + 0.941728i \(0.390798\pi\)
\(212\) −9.08836e133 −1.56377
\(213\) 7.11880e133 0.934501
\(214\) −9.78714e131 −0.00981440
\(215\) −6.11047e133 −0.468666
\(216\) 2.80267e132 0.0164620
\(217\) −1.52119e134 −0.685094
\(218\) 3.37584e132 0.0116717
\(219\) −4.80709e134 −1.27746
\(220\) −1.33904e134 −0.273836
\(221\) 6.61944e134 1.04295
\(222\) −3.31280e132 −0.00402624
\(223\) −6.35559e134 −0.596522 −0.298261 0.954484i \(-0.596407\pi\)
−0.298261 + 0.954484i \(0.596407\pi\)
\(224\) −5.21300e133 −0.0378291
\(225\) 1.27751e133 0.00717581
\(226\) −1.97730e133 −0.00860669
\(227\) 5.60799e135 1.89374 0.946869 0.321618i \(-0.104227\pi\)
0.946869 + 0.321618i \(0.104227\pi\)
\(228\) 3.93709e135 1.03257
\(229\) −1.66776e135 −0.340089 −0.170045 0.985436i \(-0.554391\pi\)
−0.170045 + 0.985436i \(0.554391\pi\)
\(230\) 4.64460e133 0.00737222
\(231\) −3.55685e135 −0.439923
\(232\) −2.99409e134 −0.0288872
\(233\) −8.65932e135 −0.652407 −0.326203 0.945300i \(-0.605769\pi\)
−0.326203 + 0.945300i \(0.605769\pi\)
\(234\) 1.26274e133 0.000743707 0
\(235\) 7.70066e135 0.354916
\(236\) −6.14092e135 −0.221714
\(237\) −4.63405e136 −1.31199
\(238\) −4.56024e134 −0.0101347
\(239\) 5.88042e136 1.02690 0.513451 0.858119i \(-0.328367\pi\)
0.513451 + 0.858119i \(0.328367\pi\)
\(240\) −6.65318e136 −0.913867
\(241\) −6.96996e136 −0.753789 −0.376895 0.926256i \(-0.623008\pi\)
−0.376895 + 0.926256i \(0.623008\pi\)
\(242\) 8.57276e134 0.00730697
\(243\) −2.13412e136 −0.143502
\(244\) 8.55370e136 0.454192
\(245\) 3.39753e137 1.42598
\(246\) 3.59054e135 0.0119232
\(247\) 5.28956e137 1.39106
\(248\) −3.31151e135 −0.00690327
\(249\) −3.15012e137 −0.521032
\(250\) 6.33214e135 0.00831764
\(251\) −5.22050e137 −0.545096 −0.272548 0.962142i \(-0.587866\pi\)
−0.272548 + 0.962142i \(0.587866\pi\)
\(252\) 1.36906e137 0.113734
\(253\) −4.25302e137 −0.281363
\(254\) 1.39880e136 0.00737593
\(255\) −1.74621e138 −0.734576
\(256\) 2.97524e138 0.999365
\(257\) −2.81349e138 −0.755249 −0.377624 0.925959i \(-0.623259\pi\)
−0.377624 + 0.925959i \(0.623259\pi\)
\(258\) −1.76662e136 −0.00379324
\(259\) −4.82557e138 −0.829492
\(260\) −8.93926e138 −1.23121
\(261\) 1.17949e138 0.130277
\(262\) 7.51002e136 0.00665758
\(263\) 2.55665e139 1.82061 0.910303 0.413942i \(-0.135848\pi\)
0.910303 + 0.413942i \(0.135848\pi\)
\(264\) −7.74296e136 −0.00443283
\(265\) 3.22111e139 1.48378
\(266\) −3.64406e137 −0.0135174
\(267\) −3.30978e139 −0.989468
\(268\) −4.14424e137 −0.00999297
\(269\) 3.45524e139 0.672545 0.336273 0.941765i \(-0.390834\pi\)
0.336273 + 0.941765i \(0.390834\pi\)
\(270\) −4.96648e137 −0.00780967
\(271\) 2.56173e139 0.325686 0.162843 0.986652i \(-0.447934\pi\)
0.162843 + 0.986652i \(0.447934\pi\)
\(272\) 7.81088e139 0.803504
\(273\) −2.37450e140 −1.97797
\(274\) −4.64836e137 −0.00313792
\(275\) −5.26211e138 −0.0288091
\(276\) −2.11329e140 −0.939047
\(277\) 7.50119e139 0.270734 0.135367 0.990796i \(-0.456779\pi\)
0.135367 + 0.990796i \(0.456779\pi\)
\(278\) −2.07147e138 −0.00607720
\(279\) 1.30454e139 0.0311326
\(280\) 1.23172e139 0.0239290
\(281\) −7.56779e140 −1.19772 −0.598861 0.800853i \(-0.704380\pi\)
−0.598861 + 0.800853i \(0.704380\pi\)
\(282\) 2.22637e138 0.00287258
\(283\) 1.09452e141 1.15213 0.576066 0.817403i \(-0.304587\pi\)
0.576066 + 0.817403i \(0.304587\pi\)
\(284\) 1.12865e141 0.969959
\(285\) −1.39539e141 −0.979751
\(286\) −5.20125e138 −0.00298580
\(287\) 5.23014e141 2.45643
\(288\) 4.47054e138 0.00171906
\(289\) −1.12407e141 −0.354135
\(290\) 5.30568e139 0.0137043
\(291\) −1.61075e140 −0.0341334
\(292\) −7.62137e141 −1.32593
\(293\) −3.01490e141 −0.430909 −0.215454 0.976514i \(-0.569123\pi\)
−0.215454 + 0.976514i \(0.569123\pi\)
\(294\) 9.82273e139 0.0115415
\(295\) 2.17647e141 0.210372
\(296\) −1.05049e140 −0.00835828
\(297\) 4.54776e141 0.298059
\(298\) −1.54794e140 −0.00836216
\(299\) −2.83925e142 −1.26506
\(300\) −2.61470e141 −0.0961501
\(301\) −2.57334e142 −0.781490
\(302\) −4.04125e140 −0.0101418
\(303\) 5.36700e142 1.11373
\(304\) 6.24163e142 1.07169
\(305\) −3.03162e142 −0.430957
\(306\) 3.91075e139 0.000460551 0
\(307\) 1.31713e143 1.28580 0.642898 0.765952i \(-0.277732\pi\)
0.642898 + 0.765952i \(0.277732\pi\)
\(308\) −5.63919e142 −0.456615
\(309\) 2.46051e143 1.65353
\(310\) 5.86817e140 0.00327496
\(311\) −3.57939e143 −1.65992 −0.829959 0.557824i \(-0.811636\pi\)
−0.829959 + 0.557824i \(0.811636\pi\)
\(312\) −5.16909e141 −0.0199308
\(313\) −1.61081e143 −0.516706 −0.258353 0.966051i \(-0.583180\pi\)
−0.258353 + 0.966051i \(0.583180\pi\)
\(314\) −3.80619e141 −0.0101633
\(315\) −4.85224e142 −0.107916
\(316\) −7.34702e143 −1.36177
\(317\) 1.23920e144 1.91529 0.957645 0.287950i \(-0.0929738\pi\)
0.957645 + 0.287950i \(0.0929738\pi\)
\(318\) 9.31268e141 0.0120092
\(319\) −4.85836e143 −0.523029
\(320\) −1.05469e144 −0.948421
\(321\) −1.57829e144 −1.18617
\(322\) 1.95601e142 0.0122930
\(323\) 1.63820e144 0.861432
\(324\) 2.09644e144 0.922879
\(325\) −3.51291e143 −0.129531
\(326\) −2.62193e142 −0.00810232
\(327\) 5.44393e144 1.41064
\(328\) 1.13856e143 0.0247519
\(329\) 3.24302e144 0.591814
\(330\) 1.37209e142 0.00210296
\(331\) −3.05925e144 −0.394009 −0.197004 0.980403i \(-0.563121\pi\)
−0.197004 + 0.980403i \(0.563121\pi\)
\(332\) −4.99434e144 −0.540802
\(333\) 4.13829e143 0.0376945
\(334\) −5.59979e142 −0.00429289
\(335\) 1.46881e143 0.00948176
\(336\) −2.80189e145 −1.52385
\(337\) 2.65995e145 1.21942 0.609712 0.792623i \(-0.291285\pi\)
0.609712 + 0.792623i \(0.291285\pi\)
\(338\) −1.41058e143 −0.00545365
\(339\) −3.18862e145 −1.04021
\(340\) −2.76852e145 −0.762448
\(341\) −5.37343e144 −0.124990
\(342\) 3.12506e142 0.000614267 0
\(343\) 4.78817e145 0.795716
\(344\) −5.60194e143 −0.00787459
\(345\) 7.48997e145 0.891009
\(346\) −1.02573e144 −0.0103313
\(347\) 1.27354e145 0.108660 0.0543299 0.998523i \(-0.482698\pi\)
0.0543299 + 0.998523i \(0.482698\pi\)
\(348\) −2.41408e146 −1.74560
\(349\) 2.91182e146 1.78526 0.892631 0.450789i \(-0.148857\pi\)
0.892631 + 0.450789i \(0.148857\pi\)
\(350\) 2.42010e143 0.00125870
\(351\) 3.03602e146 1.34012
\(352\) −1.84142e144 −0.00690162
\(353\) 1.89065e146 0.601956 0.300978 0.953631i \(-0.402687\pi\)
0.300978 + 0.953631i \(0.402687\pi\)
\(354\) 6.29249e143 0.00170269
\(355\) −4.00017e146 −0.920339
\(356\) −5.24748e146 −1.02701
\(357\) −7.35392e146 −1.22489
\(358\) 9.09364e144 0.0128963
\(359\) 1.20462e146 0.145519 0.0727596 0.997350i \(-0.476819\pi\)
0.0727596 + 0.997350i \(0.476819\pi\)
\(360\) −1.05629e144 −0.00108740
\(361\) 1.69705e146 0.148947
\(362\) 8.43242e144 0.00631261
\(363\) 1.38246e147 0.883122
\(364\) −3.76464e147 −2.05302
\(365\) 2.70118e147 1.25810
\(366\) −8.76482e144 −0.00348803
\(367\) 1.97259e147 0.671022 0.335511 0.942036i \(-0.391091\pi\)
0.335511 + 0.942036i \(0.391091\pi\)
\(368\) −3.35030e147 −0.974616
\(369\) −4.48524e146 −0.111627
\(370\) 1.86152e145 0.00396522
\(371\) 1.35653e148 2.47416
\(372\) −2.67001e147 −0.417152
\(373\) 2.30431e147 0.308519 0.154260 0.988030i \(-0.450701\pi\)
0.154260 + 0.988030i \(0.450701\pi\)
\(374\) −1.61085e145 −0.00184900
\(375\) 1.02113e148 1.00527
\(376\) 7.05978e145 0.00596335
\(377\) −3.24337e148 −2.35163
\(378\) −2.09156e146 −0.0130224
\(379\) −4.02043e147 −0.215039 −0.107520 0.994203i \(-0.534291\pi\)
−0.107520 + 0.994203i \(0.534291\pi\)
\(380\) −2.21231e148 −1.01693
\(381\) 2.25573e148 0.891457
\(382\) 2.29096e145 0.000778707 0
\(383\) −2.98479e148 −0.872941 −0.436471 0.899719i \(-0.643772\pi\)
−0.436471 + 0.899719i \(0.643772\pi\)
\(384\) −1.21997e147 −0.0307118
\(385\) 1.99865e148 0.433256
\(386\) −4.72933e146 −0.00883140
\(387\) 2.20683e147 0.0355131
\(388\) −2.55375e147 −0.0354286
\(389\) 7.83898e148 0.937903 0.468951 0.883224i \(-0.344632\pi\)
0.468951 + 0.883224i \(0.344632\pi\)
\(390\) 9.15989e146 0.00945530
\(391\) −8.79328e148 −0.783406
\(392\) 3.11478e147 0.0239595
\(393\) 1.21108e149 0.804638
\(394\) −4.09841e145 −0.000235279 0
\(395\) 2.60394e149 1.29211
\(396\) 4.83603e147 0.0207499
\(397\) 2.24295e149 0.832464 0.416232 0.909258i \(-0.363350\pi\)
0.416232 + 0.909258i \(0.363350\pi\)
\(398\) −8.10462e146 −0.00260289
\(399\) −5.87648e149 −1.63371
\(400\) −4.14520e148 −0.0997920
\(401\) −5.56701e149 −1.16097 −0.580485 0.814271i \(-0.697137\pi\)
−0.580485 + 0.814271i \(0.697137\pi\)
\(402\) 4.24653e145 7.67425e−5 0
\(403\) −3.58722e149 −0.561976
\(404\) 8.50908e149 1.15599
\(405\) −7.43024e149 −0.875667
\(406\) 2.23441e148 0.0228516
\(407\) −1.70457e149 −0.151334
\(408\) −1.60089e148 −0.0123424
\(409\) −3.09893e149 −0.207549 −0.103775 0.994601i \(-0.533092\pi\)
−0.103775 + 0.994601i \(0.533092\pi\)
\(410\) −2.01758e148 −0.0117425
\(411\) −7.49602e149 −0.379250
\(412\) 3.90101e150 1.71627
\(413\) 9.16591e149 0.350790
\(414\) −1.67743e147 −0.000558629 0
\(415\) 1.77010e150 0.513136
\(416\) −1.22931e149 −0.0310309
\(417\) −3.34049e150 −0.734492
\(418\) −1.28722e148 −0.00246613
\(419\) 3.68395e150 0.615187 0.307593 0.951518i \(-0.400476\pi\)
0.307593 + 0.951518i \(0.400476\pi\)
\(420\) 9.93114e150 1.44599
\(421\) −1.33076e151 −1.68997 −0.844987 0.534786i \(-0.820392\pi\)
−0.844987 + 0.534786i \(0.820392\pi\)
\(422\) −4.83991e148 −0.00536254
\(423\) −2.78114e149 −0.0268937
\(424\) 2.95304e149 0.0249306
\(425\) −1.08796e150 −0.0802139
\(426\) −1.15650e149 −0.00744895
\(427\) −1.27672e151 −0.718610
\(428\) −2.50229e151 −1.23118
\(429\) −8.38762e150 −0.360865
\(430\) 9.92693e148 0.00373576
\(431\) −1.47034e151 −0.484145 −0.242073 0.970258i \(-0.577827\pi\)
−0.242073 + 0.970258i \(0.577827\pi\)
\(432\) 3.58248e151 1.03245
\(433\) 8.18435e150 0.206505 0.103252 0.994655i \(-0.467075\pi\)
0.103252 + 0.994655i \(0.467075\pi\)
\(434\) 2.47130e149 0.00546091
\(435\) 8.55603e151 1.65630
\(436\) 8.63105e151 1.46417
\(437\) −7.02666e151 −1.04488
\(438\) 7.80948e149 0.0101827
\(439\) −9.86539e151 −1.12825 −0.564124 0.825690i \(-0.690786\pi\)
−0.564124 + 0.825690i \(0.690786\pi\)
\(440\) 4.35089e149 0.00436565
\(441\) −1.22704e151 −0.108054
\(442\) −1.07538e150 −0.00831344
\(443\) 2.53820e152 1.72311 0.861553 0.507667i \(-0.169492\pi\)
0.861553 + 0.507667i \(0.169492\pi\)
\(444\) −8.46989e151 −0.505076
\(445\) 1.85982e152 0.974473
\(446\) 1.03251e150 0.00475490
\(447\) −2.49623e152 −1.01065
\(448\) −4.44168e152 −1.58147
\(449\) 4.51737e150 0.0141488 0.00707442 0.999975i \(-0.497748\pi\)
0.00707442 + 0.999975i \(0.497748\pi\)
\(450\) −2.07542e148 −5.71987e−5 0
\(451\) 1.84748e152 0.448155
\(452\) −5.05538e152 −1.07968
\(453\) −6.51699e152 −1.22575
\(454\) −9.11059e150 −0.0150951
\(455\) 1.33427e153 1.94800
\(456\) −1.27926e151 −0.0164619
\(457\) −4.05863e152 −0.460467 −0.230233 0.973135i \(-0.573949\pi\)
−0.230233 + 0.973135i \(0.573949\pi\)
\(458\) 2.70940e150 0.00271087
\(459\) 9.40266e152 0.829892
\(460\) 1.18749e153 0.924816
\(461\) 7.16459e152 0.492479 0.246240 0.969209i \(-0.420805\pi\)
0.246240 + 0.969209i \(0.420805\pi\)
\(462\) 5.77837e150 0.00350665
\(463\) 7.38254e152 0.395639 0.197819 0.980238i \(-0.436614\pi\)
0.197819 + 0.980238i \(0.436614\pi\)
\(464\) −3.82715e153 −1.81172
\(465\) 9.46311e152 0.395812
\(466\) 1.40677e151 0.00520036
\(467\) 2.49693e153 0.815992 0.407996 0.912984i \(-0.366228\pi\)
0.407996 + 0.912984i \(0.366228\pi\)
\(468\) 3.22846e152 0.0932952
\(469\) 6.18568e151 0.0158106
\(470\) −1.25103e151 −0.00282905
\(471\) −6.13792e153 −1.22834
\(472\) 1.99534e151 0.00353470
\(473\) −9.08999e152 −0.142576
\(474\) 7.52836e151 0.0104579
\(475\) −8.69383e152 −0.106986
\(476\) −1.16592e154 −1.27136
\(477\) −1.16332e153 −0.112433
\(478\) −9.55319e151 −0.00818548
\(479\) 1.11081e154 0.844011 0.422006 0.906593i \(-0.361326\pi\)
0.422006 + 0.906593i \(0.361326\pi\)
\(480\) 3.24292e152 0.0218557
\(481\) −1.13795e154 −0.680425
\(482\) 1.13232e152 0.00600849
\(483\) 3.15429e154 1.48574
\(484\) 2.19181e154 0.916631
\(485\) 9.05103e152 0.0336161
\(486\) 3.46704e151 0.00114386
\(487\) 1.39646e154 0.409366 0.204683 0.978828i \(-0.434384\pi\)
0.204683 + 0.978828i \(0.434384\pi\)
\(488\) −2.77931e152 −0.00724099
\(489\) −4.22817e154 −0.979248
\(490\) −5.51954e152 −0.0113665
\(491\) 1.61060e154 0.294986 0.147493 0.989063i \(-0.452880\pi\)
0.147493 + 0.989063i \(0.452880\pi\)
\(492\) 9.17999e154 1.49571
\(493\) −1.00448e155 −1.45628
\(494\) −8.59329e152 −0.0110882
\(495\) −1.71399e153 −0.0196884
\(496\) −4.23289e154 −0.432953
\(497\) −1.68461e155 −1.53464
\(498\) 5.11761e152 0.00415317
\(499\) −1.15267e154 −0.0833532 −0.0416766 0.999131i \(-0.513270\pi\)
−0.0416766 + 0.999131i \(0.513270\pi\)
\(500\) 1.61895e155 1.04342
\(501\) −9.03031e154 −0.518840
\(502\) 8.48110e152 0.00434498
\(503\) 1.73196e155 0.791370 0.395685 0.918386i \(-0.370507\pi\)
0.395685 + 0.918386i \(0.370507\pi\)
\(504\) −4.44842e152 −0.00181322
\(505\) −3.01580e155 −1.09685
\(506\) 6.90935e152 0.00224276
\(507\) −2.27472e155 −0.659130
\(508\) 3.57633e155 0.925282
\(509\) 2.00777e155 0.463917 0.231959 0.972726i \(-0.425487\pi\)
0.231959 + 0.972726i \(0.425487\pi\)
\(510\) 2.83685e153 0.00585533
\(511\) 1.13756e156 2.09785
\(512\) −2.41767e154 −0.0398451
\(513\) 7.51362e155 1.10688
\(514\) 4.57072e153 0.00602012
\(515\) −1.38260e156 −1.62847
\(516\) −4.51675e155 −0.475848
\(517\) 1.14556e155 0.107972
\(518\) 7.83951e153 0.00661192
\(519\) −1.65411e156 −1.24865
\(520\) 2.90459e154 0.0196288
\(521\) −7.35144e155 −0.444839 −0.222420 0.974951i \(-0.571395\pi\)
−0.222420 + 0.974951i \(0.571395\pi\)
\(522\) −1.91618e153 −0.00103844
\(523\) −4.27171e155 −0.207374 −0.103687 0.994610i \(-0.533064\pi\)
−0.103687 + 0.994610i \(0.533064\pi\)
\(524\) 1.92010e156 0.835168
\(525\) 3.90269e155 0.152126
\(526\) −4.15347e154 −0.0145121
\(527\) −1.11098e156 −0.348012
\(528\) −9.89732e155 −0.278014
\(529\) −1.97513e155 −0.0497617
\(530\) −5.23294e154 −0.0118272
\(531\) −7.86046e154 −0.0159409
\(532\) −9.31683e156 −1.69570
\(533\) 1.23335e157 2.01499
\(534\) 5.37699e154 0.00788709
\(535\) 8.86866e156 1.16820
\(536\) 1.34657e153 0.000159314 0
\(537\) 1.46646e157 1.55865
\(538\) −5.61329e154 −0.00536089
\(539\) 5.05419e156 0.433808
\(540\) −1.26979e157 −0.979693
\(541\) −2.22349e157 −1.54238 −0.771191 0.636604i \(-0.780339\pi\)
−0.771191 + 0.636604i \(0.780339\pi\)
\(542\) −4.16172e154 −0.00259605
\(543\) 1.35983e157 0.762944
\(544\) −3.80721e155 −0.0192163
\(545\) −3.05903e157 −1.38926
\(546\) 3.85756e155 0.0157665
\(547\) 1.07884e157 0.396905 0.198453 0.980110i \(-0.436408\pi\)
0.198453 + 0.980110i \(0.436408\pi\)
\(548\) −1.18845e157 −0.393640
\(549\) 1.09489e156 0.0326557
\(550\) 8.54869e153 0.000229639 0
\(551\) −8.02678e157 −1.94234
\(552\) 6.86663e155 0.0149708
\(553\) 1.09661e158 2.15456
\(554\) −1.21862e155 −0.00215803
\(555\) 3.00191e157 0.479238
\(556\) −5.29616e157 −0.762361
\(557\) 1.33887e157 0.173806 0.0869028 0.996217i \(-0.472303\pi\)
0.0869028 + 0.996217i \(0.472303\pi\)
\(558\) −2.11932e154 −0.000248159 0
\(559\) −6.06834e157 −0.641049
\(560\) 1.57443e158 1.50076
\(561\) −2.59768e157 −0.223471
\(562\) 1.22945e156 0.00954709
\(563\) 8.65513e157 0.606793 0.303397 0.952864i \(-0.401879\pi\)
0.303397 + 0.952864i \(0.401879\pi\)
\(564\) 5.69218e157 0.360355
\(565\) 1.79173e158 1.02444
\(566\) −1.77813e156 −0.00918370
\(567\) −3.12914e158 −1.46015
\(568\) −3.66726e156 −0.0154637
\(569\) 1.11867e157 0.0426330 0.0213165 0.999773i \(-0.493214\pi\)
0.0213165 + 0.999773i \(0.493214\pi\)
\(570\) 2.26691e156 0.00780963
\(571\) −3.78486e158 −1.17889 −0.589446 0.807808i \(-0.700654\pi\)
−0.589446 + 0.807808i \(0.700654\pi\)
\(572\) −1.32981e158 −0.374557
\(573\) 3.69444e157 0.0941147
\(574\) −8.49675e156 −0.0195803
\(575\) 4.66655e157 0.0972960
\(576\) 3.80907e157 0.0718665
\(577\) −6.89298e158 −1.17706 −0.588529 0.808476i \(-0.700293\pi\)
−0.588529 + 0.808476i \(0.700293\pi\)
\(578\) 1.82614e156 0.00282283
\(579\) −7.62660e158 −1.06737
\(580\) 1.35651e159 1.71915
\(581\) 7.45453e158 0.855643
\(582\) 2.61678e155 0.000272079 0
\(583\) 4.79175e158 0.451391
\(584\) 2.47638e157 0.0211387
\(585\) −1.14424e158 −0.0885225
\(586\) 4.89794e156 0.00343479
\(587\) −2.75880e158 −0.175400 −0.0877000 0.996147i \(-0.527952\pi\)
−0.0877000 + 0.996147i \(0.527952\pi\)
\(588\) 2.51139e159 1.44783
\(589\) −8.87775e158 −0.464166
\(590\) −3.53585e156 −0.00167688
\(591\) −6.60916e157 −0.0284358
\(592\) −1.34277e159 −0.524207
\(593\) −1.62267e158 −0.0574891 −0.0287445 0.999587i \(-0.509151\pi\)
−0.0287445 + 0.999587i \(0.509151\pi\)
\(594\) −7.38818e156 −0.00237584
\(595\) 4.13228e159 1.20633
\(596\) −3.95763e159 −1.04900
\(597\) −1.30696e159 −0.314586
\(598\) 4.61258e157 0.0100838
\(599\) 3.63775e156 0.000722421 0 0.000361210 1.00000i \(-0.499885\pi\)
0.000361210 1.00000i \(0.499885\pi\)
\(600\) 8.49583e156 0.00153288
\(601\) −7.08617e159 −1.16179 −0.580897 0.813977i \(-0.697298\pi\)
−0.580897 + 0.813977i \(0.697298\pi\)
\(602\) 4.18058e157 0.00622929
\(603\) −5.30469e156 −0.000718480 0
\(604\) −1.03323e160 −1.27225
\(605\) −7.76824e159 −0.869739
\(606\) −8.71910e157 −0.00887760
\(607\) 1.57783e160 1.46120 0.730601 0.682805i \(-0.239240\pi\)
0.730601 + 0.682805i \(0.239240\pi\)
\(608\) −3.04232e158 −0.0256301
\(609\) 3.60325e160 2.76185
\(610\) 4.92509e157 0.00343517
\(611\) 7.64756e159 0.485460
\(612\) 9.99867e158 0.0577744
\(613\) −8.70148e159 −0.457737 −0.228868 0.973457i \(-0.573503\pi\)
−0.228868 + 0.973457i \(0.573503\pi\)
\(614\) −2.13978e158 −0.0102491
\(615\) −3.25359e160 −1.41920
\(616\) 1.83232e158 0.00727963
\(617\) 4.79825e159 0.173655 0.0868273 0.996223i \(-0.472327\pi\)
0.0868273 + 0.996223i \(0.472327\pi\)
\(618\) −3.99729e158 −0.0131804
\(619\) 3.52731e160 1.05982 0.529909 0.848054i \(-0.322226\pi\)
0.529909 + 0.848054i \(0.322226\pi\)
\(620\) 1.50032e160 0.410830
\(621\) −4.03305e160 −1.00662
\(622\) 5.81499e158 0.0132313
\(623\) 7.83236e160 1.62491
\(624\) −6.60731e160 −1.25000
\(625\) −5.15943e160 −0.890227
\(626\) 2.61687e158 0.00411869
\(627\) −2.07579e160 −0.298057
\(628\) −9.73132e160 −1.27495
\(629\) −3.52427e160 −0.421363
\(630\) 7.88283e157 0.000860203 0
\(631\) 1.23110e161 1.22632 0.613161 0.789958i \(-0.289898\pi\)
0.613161 + 0.789958i \(0.289898\pi\)
\(632\) 2.38723e159 0.0217102
\(633\) −7.80491e160 −0.648119
\(634\) −2.01318e159 −0.0152669
\(635\) −1.26753e161 −0.877947
\(636\) 2.38099e161 1.50651
\(637\) 3.37411e161 1.95048
\(638\) 7.89278e158 0.00416909
\(639\) 1.44468e160 0.0697386
\(640\) 6.85522e159 0.0302463
\(641\) −1.05174e161 −0.424201 −0.212101 0.977248i \(-0.568030\pi\)
−0.212101 + 0.977248i \(0.568030\pi\)
\(642\) 2.56405e159 0.00945502
\(643\) −3.21803e161 −1.08507 −0.542537 0.840032i \(-0.682536\pi\)
−0.542537 + 0.840032i \(0.682536\pi\)
\(644\) 5.00095e161 1.54211
\(645\) 1.60083e161 0.451505
\(646\) −2.66137e159 −0.00686651
\(647\) −5.92463e161 −1.39851 −0.699255 0.714873i \(-0.746485\pi\)
−0.699255 + 0.714873i \(0.746485\pi\)
\(648\) −6.81187e159 −0.0147131
\(649\) 3.23774e160 0.0639988
\(650\) 5.70698e158 0.00103250
\(651\) 3.98525e161 0.660008
\(652\) −6.70353e161 −1.01640
\(653\) −8.21189e161 −1.14008 −0.570040 0.821617i \(-0.693072\pi\)
−0.570040 + 0.821617i \(0.693072\pi\)
\(654\) −8.84408e159 −0.0112443
\(655\) −6.80523e161 −0.792443
\(656\) 1.45534e162 1.55237
\(657\) −9.75546e160 −0.0953321
\(658\) −5.26853e159 −0.00471738
\(659\) −7.69425e161 −0.631328 −0.315664 0.948871i \(-0.602227\pi\)
−0.315664 + 0.948871i \(0.602227\pi\)
\(660\) 3.50805e161 0.263809
\(661\) 4.39883e160 0.0303217 0.0151608 0.999885i \(-0.495174\pi\)
0.0151608 + 0.999885i \(0.495174\pi\)
\(662\) 4.96999e159 0.00314066
\(663\) −1.73417e162 −1.00476
\(664\) 1.62279e160 0.00862179
\(665\) 3.30208e162 1.60895
\(666\) −6.72297e158 −0.000300464 0
\(667\) 4.30850e162 1.76641
\(668\) −1.43170e162 −0.538526
\(669\) 1.66505e162 0.574678
\(670\) −2.38619e158 −7.55795e−5 0
\(671\) −4.50986e161 −0.131105
\(672\) 1.36571e161 0.0364439
\(673\) −1.88994e162 −0.463002 −0.231501 0.972835i \(-0.574364\pi\)
−0.231501 + 0.972835i \(0.574364\pi\)
\(674\) −4.32129e160 −0.00972007
\(675\) −4.98995e161 −0.103069
\(676\) −3.60645e162 −0.684140
\(677\) 1.09963e161 0.0191601 0.00958006 0.999954i \(-0.496951\pi\)
0.00958006 + 0.999954i \(0.496951\pi\)
\(678\) 5.18015e160 0.00829154
\(679\) 3.81171e161 0.0560542
\(680\) 8.99563e160 0.0121554
\(681\) −1.46919e163 −1.82440
\(682\) 8.72954e159 0.000996299 0
\(683\) −1.01812e162 −0.106809 −0.0534045 0.998573i \(-0.517007\pi\)
−0.0534045 + 0.998573i \(0.517007\pi\)
\(684\) 7.98988e161 0.0770574
\(685\) 4.21213e162 0.373503
\(686\) −7.77875e160 −0.00634268
\(687\) 4.36922e162 0.327636
\(688\) −7.16059e162 −0.493871
\(689\) 3.19890e163 2.02953
\(690\) −1.21680e161 −0.00710227
\(691\) −7.27763e162 −0.390843 −0.195421 0.980719i \(-0.562607\pi\)
−0.195421 + 0.980719i \(0.562607\pi\)
\(692\) −2.62249e163 −1.29603
\(693\) −7.21824e161 −0.0328299
\(694\) −2.06897e160 −0.000866132 0
\(695\) 1.87707e163 0.723361
\(696\) 7.84397e161 0.0278295
\(697\) 3.81973e163 1.24781
\(698\) −4.73046e161 −0.0142304
\(699\) 2.26858e163 0.628517
\(700\) 6.18750e162 0.157898
\(701\) 1.33169e163 0.313053 0.156527 0.987674i \(-0.449970\pi\)
0.156527 + 0.987674i \(0.449970\pi\)
\(702\) −4.93224e161 −0.0106822
\(703\) −2.81622e163 −0.561999
\(704\) −1.56897e163 −0.288526
\(705\) −2.01743e163 −0.341920
\(706\) −3.07150e161 −0.00479822
\(707\) −1.27006e164 −1.82898
\(708\) 1.60881e163 0.213595
\(709\) −7.79346e163 −0.954052 −0.477026 0.878889i \(-0.658285\pi\)
−0.477026 + 0.878889i \(0.658285\pi\)
\(710\) 6.49857e161 0.00733606
\(711\) −9.40429e162 −0.0979093
\(712\) 1.70504e162 0.0163732
\(713\) 4.76527e163 0.422124
\(714\) 1.19470e162 0.00976363
\(715\) 4.71314e163 0.355396
\(716\) 2.32498e164 1.61779
\(717\) −1.54056e164 −0.989300
\(718\) −1.95700e161 −0.00115994
\(719\) 2.98307e164 1.63212 0.816060 0.577967i \(-0.196154\pi\)
0.816060 + 0.577967i \(0.196154\pi\)
\(720\) −1.35019e163 −0.0681987
\(721\) −5.82262e164 −2.71544
\(722\) −2.75699e161 −0.00118726
\(723\) 1.82600e164 0.726188
\(724\) 2.15593e164 0.791893
\(725\) 5.33075e163 0.180864
\(726\) −2.24591e162 −0.00703940
\(727\) 4.23599e164 1.22667 0.613333 0.789824i \(-0.289828\pi\)
0.613333 + 0.789824i \(0.289828\pi\)
\(728\) 1.22323e163 0.0327305
\(729\) 4.29165e164 1.06118
\(730\) −4.38827e162 −0.0100283
\(731\) −1.87939e164 −0.396979
\(732\) −2.24091e164 −0.437560
\(733\) −7.22331e164 −1.30394 −0.651970 0.758245i \(-0.726057\pi\)
−0.651970 + 0.758245i \(0.726057\pi\)
\(734\) −3.20461e162 −0.00534875
\(735\) −8.90091e164 −1.37376
\(736\) 1.63301e163 0.0233086
\(737\) 2.18501e162 0.00288452
\(738\) 7.28661e161 0.000889784 0
\(739\) 5.23614e164 0.591501 0.295751 0.955265i \(-0.404430\pi\)
0.295751 + 0.955265i \(0.404430\pi\)
\(740\) 4.75936e164 0.497422
\(741\) −1.38577e165 −1.34012
\(742\) −2.20378e163 −0.0197216
\(743\) −7.62830e164 −0.631788 −0.315894 0.948795i \(-0.602304\pi\)
−0.315894 + 0.948795i \(0.602304\pi\)
\(744\) 8.67556e162 0.00665049
\(745\) 1.40267e165 0.995337
\(746\) −3.74352e162 −0.00245922
\(747\) −6.39282e163 −0.0388829
\(748\) −4.11847e164 −0.231950
\(749\) 3.73491e165 1.94794
\(750\) −1.65890e163 −0.00801307
\(751\) −1.43926e165 −0.643934 −0.321967 0.946751i \(-0.604344\pi\)
−0.321967 + 0.946751i \(0.604344\pi\)
\(752\) 9.02406e164 0.374004
\(753\) 1.36768e165 0.525136
\(754\) 5.26910e163 0.0187450
\(755\) 3.66200e165 1.20717
\(756\) −5.34753e165 −1.63362
\(757\) −1.10275e165 −0.312224 −0.156112 0.987739i \(-0.549896\pi\)
−0.156112 + 0.987739i \(0.549896\pi\)
\(758\) 6.53149e162 0.00171409
\(759\) 1.11421e165 0.271061
\(760\) 7.18836e163 0.0162124
\(761\) 6.71001e165 1.40315 0.701577 0.712594i \(-0.252480\pi\)
0.701577 + 0.712594i \(0.252480\pi\)
\(762\) −3.66460e163 −0.00710584
\(763\) −1.28827e166 −2.31657
\(764\) 5.85732e164 0.0976857
\(765\) −3.54374e164 −0.0548188
\(766\) 4.84902e163 0.00695825
\(767\) 2.16147e165 0.287750
\(768\) −7.79458e165 −0.962771
\(769\) 1.13066e165 0.129588 0.0647942 0.997899i \(-0.479361\pi\)
0.0647942 + 0.997899i \(0.479361\pi\)
\(770\) −3.24696e163 −0.00345350
\(771\) 7.37082e165 0.727594
\(772\) −1.20915e166 −1.10787
\(773\) −8.71638e165 −0.741336 −0.370668 0.928765i \(-0.620871\pi\)
−0.370668 + 0.928765i \(0.620871\pi\)
\(774\) −3.58517e162 −0.000283077 0
\(775\) 5.89590e164 0.0432217
\(776\) 8.29777e162 0.000564823 0
\(777\) 1.26421e166 0.799119
\(778\) −1.27350e164 −0.00747606
\(779\) 3.05233e166 1.66428
\(780\) 2.34192e166 1.18613
\(781\) −5.95068e165 −0.279983
\(782\) 1.42853e164 0.00624457
\(783\) −4.60709e166 −1.87122
\(784\) 3.98142e166 1.50267
\(785\) 3.44899e166 1.20973
\(786\) −1.96749e164 −0.00641380
\(787\) −3.70181e166 −1.12167 −0.560837 0.827926i \(-0.689521\pi\)
−0.560837 + 0.827926i \(0.689521\pi\)
\(788\) −1.04785e165 −0.0295148
\(789\) −6.69796e166 −1.75394
\(790\) −4.23030e164 −0.0102994
\(791\) 7.54563e166 1.70824
\(792\) −1.57135e163 −0.000330807 0
\(793\) −3.01071e166 −0.589469
\(794\) −3.64384e164 −0.00663561
\(795\) −8.43873e166 −1.42944
\(796\) −2.07212e166 −0.326523
\(797\) 2.81578e166 0.412806 0.206403 0.978467i \(-0.433824\pi\)
0.206403 + 0.978467i \(0.433824\pi\)
\(798\) 9.54678e164 0.0130224
\(799\) 2.36848e166 0.300628
\(800\) 2.02047e164 0.00238659
\(801\) −6.71684e165 −0.0738406
\(802\) 9.04403e164 0.00925414
\(803\) 4.01830e166 0.382735
\(804\) 1.08572e165 0.00962706
\(805\) −1.77245e167 −1.46322
\(806\) 5.82771e164 0.00447954
\(807\) −9.05209e166 −0.647919
\(808\) −2.76482e165 −0.0184295
\(809\) 3.11134e167 1.93156 0.965779 0.259367i \(-0.0835140\pi\)
0.965779 + 0.259367i \(0.0835140\pi\)
\(810\) 1.20710e165 0.00697998
\(811\) −2.71844e166 −0.146427 −0.0732133 0.997316i \(-0.523325\pi\)
−0.0732133 + 0.997316i \(0.523325\pi\)
\(812\) 5.71275e167 2.86664
\(813\) −6.71126e166 −0.313760
\(814\) 2.76921e164 0.00120629
\(815\) 2.37587e167 0.964408
\(816\) −2.04631e167 −0.774082
\(817\) −1.50181e167 −0.529476
\(818\) 5.03445e164 0.00165439
\(819\) −4.81879e166 −0.147609
\(820\) −5.15838e167 −1.47305
\(821\) −4.32991e167 −1.15278 −0.576391 0.817174i \(-0.695539\pi\)
−0.576391 + 0.817174i \(0.695539\pi\)
\(822\) 1.21778e165 0.00302302
\(823\) 2.56133e167 0.592891 0.296446 0.955050i \(-0.404199\pi\)
0.296446 + 0.955050i \(0.404199\pi\)
\(824\) −1.26754e166 −0.0273619
\(825\) 1.37858e166 0.0277542
\(826\) −1.48907e165 −0.00279616
\(827\) 4.60673e167 0.806912 0.403456 0.914999i \(-0.367809\pi\)
0.403456 + 0.914999i \(0.367809\pi\)
\(828\) −4.28870e166 −0.0700779
\(829\) −5.24129e167 −0.799010 −0.399505 0.916731i \(-0.630818\pi\)
−0.399505 + 0.916731i \(0.630818\pi\)
\(830\) −2.87566e165 −0.00409023
\(831\) −1.96517e167 −0.260821
\(832\) −1.04742e168 −1.29727
\(833\) 1.04497e168 1.20786
\(834\) 5.42688e165 0.00585467
\(835\) 5.07427e167 0.510977
\(836\) −3.29105e167 −0.309367
\(837\) −5.09551e167 −0.447171
\(838\) −5.98485e165 −0.00490368
\(839\) −3.73650e167 −0.285859 −0.142930 0.989733i \(-0.545652\pi\)
−0.142930 + 0.989733i \(0.545652\pi\)
\(840\) −3.22688e166 −0.0230528
\(841\) 3.42285e168 2.28359
\(842\) 2.16192e166 0.0134709
\(843\) 1.98262e168 1.15386
\(844\) −1.23743e168 −0.672711
\(845\) 1.27820e168 0.649141
\(846\) 4.51816e164 0.000214371 0
\(847\) −3.27148e168 −1.45027
\(848\) 3.77468e168 1.56357
\(849\) −2.86744e168 −1.10994
\(850\) 1.76747e165 0.000639388 0
\(851\) 1.51165e168 0.511095
\(852\) −2.95685e168 −0.934442
\(853\) −3.79346e168 −1.12064 −0.560321 0.828276i \(-0.689322\pi\)
−0.560321 + 0.828276i \(0.689322\pi\)
\(854\) 2.07413e166 0.00572807
\(855\) −2.83179e167 −0.0731154
\(856\) 8.13058e166 0.0196282
\(857\) 6.21573e168 1.40312 0.701560 0.712610i \(-0.252487\pi\)
0.701560 + 0.712610i \(0.252487\pi\)
\(858\) 1.36263e166 0.00287647
\(859\) −8.56154e168 −1.69023 −0.845113 0.534587i \(-0.820467\pi\)
−0.845113 + 0.534587i \(0.820467\pi\)
\(860\) 2.53803e168 0.468636
\(861\) −1.37020e169 −2.36648
\(862\) 2.38868e166 0.00385914
\(863\) 5.82180e168 0.879907 0.439953 0.898021i \(-0.354995\pi\)
0.439953 + 0.898021i \(0.354995\pi\)
\(864\) −1.74619e167 −0.0246917
\(865\) 9.29469e168 1.22973
\(866\) −1.32961e166 −0.00164606
\(867\) 2.94487e168 0.341168
\(868\) 6.31839e168 0.685050
\(869\) 3.87365e168 0.393082
\(870\) −1.38999e167 −0.0132025
\(871\) 1.45868e167 0.0129693
\(872\) −2.80445e167 −0.0233426
\(873\) −3.26883e166 −0.00254726
\(874\) 1.14153e167 0.00832878
\(875\) −2.41643e169 −1.65087
\(876\) 1.99666e169 1.27737
\(877\) 7.89824e168 0.473210 0.236605 0.971606i \(-0.423965\pi\)
0.236605 + 0.971606i \(0.423965\pi\)
\(878\) 1.60271e167 0.00899331
\(879\) 7.89850e168 0.415130
\(880\) 5.56146e168 0.273801
\(881\) 4.01189e168 0.185027 0.0925133 0.995711i \(-0.470510\pi\)
0.0925133 + 0.995711i \(0.470510\pi\)
\(882\) 1.99341e166 0.000861299 0
\(883\) −7.29233e167 −0.0295206 −0.0147603 0.999891i \(-0.504699\pi\)
−0.0147603 + 0.999891i \(0.504699\pi\)
\(884\) −2.74943e169 −1.04289
\(885\) −5.70196e168 −0.202669
\(886\) −4.12350e167 −0.0137350
\(887\) −1.65482e169 −0.516585 −0.258292 0.966067i \(-0.583160\pi\)
−0.258292 + 0.966067i \(0.583160\pi\)
\(888\) 2.75208e167 0.00805222
\(889\) −5.33801e169 −1.46396
\(890\) −3.02142e167 −0.00776757
\(891\) −1.10533e169 −0.266393
\(892\) 2.63984e169 0.596484
\(893\) 1.89264e169 0.400967
\(894\) 4.05531e167 0.00805596
\(895\) −8.24024e169 −1.53502
\(896\) 2.88698e168 0.0504351
\(897\) 7.43833e169 1.21874
\(898\) −7.33880e165 −0.000112781 0
\(899\) 5.44353e169 0.784690
\(900\) −5.30625e167 −0.00717535
\(901\) 9.90713e169 1.25682
\(902\) −3.00137e167 −0.00357227
\(903\) 6.74168e169 0.752874
\(904\) 1.64262e168 0.0172128
\(905\) −7.64107e169 −0.751382
\(906\) 1.05873e168 0.00977047
\(907\) 9.78926e169 0.847873 0.423937 0.905692i \(-0.360648\pi\)
0.423937 + 0.905692i \(0.360648\pi\)
\(908\) −2.32932e170 −1.89362
\(909\) 1.08917e169 0.0831139
\(910\) −2.16762e168 −0.0155276
\(911\) 2.27135e170 1.52749 0.763744 0.645519i \(-0.223359\pi\)
0.763744 + 0.645519i \(0.223359\pi\)
\(912\) −1.63519e170 −1.03244
\(913\) 2.63322e169 0.156105
\(914\) 6.59356e167 0.00367040
\(915\) 7.94228e169 0.415176
\(916\) 6.92716e169 0.340068
\(917\) −2.86593e170 −1.32138
\(918\) −1.52753e168 −0.00661510
\(919\) 1.31904e170 0.536558 0.268279 0.963341i \(-0.413545\pi\)
0.268279 + 0.963341i \(0.413545\pi\)
\(920\) −3.85846e168 −0.0147440
\(921\) −3.45064e170 −1.23871
\(922\) −1.16394e168 −0.00392557
\(923\) −3.97259e170 −1.25885
\(924\) 1.47736e170 0.439895
\(925\) 1.87031e169 0.0523316
\(926\) −1.19935e168 −0.00315365
\(927\) 4.99334e169 0.123398
\(928\) 1.86545e169 0.0433285
\(929\) −6.52268e170 −1.42404 −0.712021 0.702158i \(-0.752220\pi\)
−0.712021 + 0.702158i \(0.752220\pi\)
\(930\) −1.53735e168 −0.00315503
\(931\) 8.35032e170 1.61100
\(932\) 3.59671e170 0.652365
\(933\) 9.37735e170 1.59914
\(934\) −4.05645e168 −0.00650431
\(935\) 1.45968e170 0.220084
\(936\) −1.04901e168 −0.00148737
\(937\) 4.54488e170 0.606033 0.303016 0.952985i \(-0.402006\pi\)
0.303016 + 0.952985i \(0.402006\pi\)
\(938\) −1.00491e167 −0.000126027 0
\(939\) 4.22002e170 0.497786
\(940\) −3.19852e170 −0.354893
\(941\) −1.60984e171 −1.68028 −0.840138 0.542373i \(-0.817526\pi\)
−0.840138 + 0.542373i \(0.817526\pi\)
\(942\) 9.97151e168 0.00979116
\(943\) −1.63839e171 −1.51354
\(944\) 2.55051e170 0.221686
\(945\) 1.89528e171 1.55004
\(946\) 1.47674e168 0.00113648
\(947\) −2.25031e171 −1.62974 −0.814870 0.579644i \(-0.803191\pi\)
−0.814870 + 0.579644i \(0.803191\pi\)
\(948\) 1.92479e171 1.31191
\(949\) 2.68256e171 1.72084
\(950\) 1.41238e168 0.000852793 0
\(951\) −3.24648e171 −1.84516
\(952\) 3.78838e169 0.0202688
\(953\) −3.01429e171 −1.51825 −0.759126 0.650944i \(-0.774373\pi\)
−0.759126 + 0.650944i \(0.774373\pi\)
\(954\) 1.88991e168 0.000896207 0
\(955\) −2.07596e170 −0.0926884
\(956\) −2.44248e171 −1.02684
\(957\) 1.27280e171 0.503877
\(958\) −1.80460e169 −0.00672765
\(959\) 1.77388e171 0.622808
\(960\) 2.76310e171 0.913692
\(961\) −2.60860e171 −0.812480
\(962\) 1.84868e169 0.00542370
\(963\) −3.20297e170 −0.0885199
\(964\) 2.89502e171 0.753742
\(965\) 4.28551e171 1.05119
\(966\) −5.12438e169 −0.0118429
\(967\) 1.35320e171 0.294673 0.147337 0.989086i \(-0.452930\pi\)
0.147337 + 0.989086i \(0.452930\pi\)
\(968\) −7.12174e169 −0.0146135
\(969\) −4.29177e171 −0.829888
\(970\) −1.47041e168 −0.000267956 0
\(971\) 4.11549e171 0.706832 0.353416 0.935466i \(-0.385020\pi\)
0.353416 + 0.935466i \(0.385020\pi\)
\(972\) 8.86423e170 0.143493
\(973\) 7.90503e171 1.20619
\(974\) −2.26865e169 −0.00326307
\(975\) 9.20317e170 0.124788
\(976\) −3.55262e171 −0.454134
\(977\) −7.47661e171 −0.901088 −0.450544 0.892754i \(-0.648770\pi\)
−0.450544 + 0.892754i \(0.648770\pi\)
\(978\) 6.86898e169 0.00780563
\(979\) 2.76668e171 0.296452
\(980\) −1.41119e172 −1.42589
\(981\) 1.10479e171 0.105271
\(982\) −2.61654e169 −0.00235134
\(983\) −5.23743e171 −0.443906 −0.221953 0.975057i \(-0.571243\pi\)
−0.221953 + 0.975057i \(0.571243\pi\)
\(984\) −2.98281e170 −0.0238456
\(985\) 3.71379e170 0.0280049
\(986\) 1.63186e170 0.0116081
\(987\) −8.49612e171 −0.570144
\(988\) −2.19706e172 −1.39097
\(989\) 8.06120e171 0.481518
\(990\) 2.78451e168 0.000156937 0
\(991\) 2.83687e172 1.50870 0.754351 0.656471i \(-0.227952\pi\)
0.754351 + 0.656471i \(0.227952\pi\)
\(992\) 2.06321e170 0.0103544
\(993\) 8.01468e171 0.379581
\(994\) 2.73678e170 0.0122327
\(995\) 7.34404e171 0.309819
\(996\) 1.30843e172 0.520999
\(997\) −2.05510e172 −0.772436 −0.386218 0.922408i \(-0.626219\pi\)
−0.386218 + 0.922408i \(0.626219\pi\)
\(998\) 1.87259e169 0.000664412 0
\(999\) −1.61641e172 −0.541423
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 1.116.a.a.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.116.a.a.1.5 9 1.1 even 1 trivial