Properties

Label 1.122.a.a.1.2
Level $1$
Weight $122$
Character 1.1
Self dual yes
Analytic conductor $92.717$
Analytic rank $1$
Dimension $9$
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] = [1,122,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, 122, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 122);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 122 \)
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: \(92.7173263878\)
Analytic rank: \(1\)
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: \( x^{9} - 2 x^{8} + \cdots + 32\!\cdots\!74 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{145}\cdot 3^{53}\cdot 5^{20}\cdot 7^{8}\cdot 11^{6}\cdot 13^{2}\cdot 17^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.77468e16\) 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)
 
\(f(q)\) \(=\) \(q-2.07240e18 q^{2} -2.64580e28 q^{3} +1.63640e36 q^{4} -5.16214e41 q^{5} +5.48316e46 q^{6} -2.39892e51 q^{7} +2.11812e54 q^{8} -4.69101e57 q^{9} +O(q^{10})\) \(q-2.07240e18 q^{2} -2.64580e28 q^{3} +1.63640e36 q^{4} -5.16214e41 q^{5} +5.48316e46 q^{6} -2.39892e51 q^{7} +2.11812e54 q^{8} -4.69101e57 q^{9} +1.06980e60 q^{10} +6.81449e62 q^{11} -4.32958e64 q^{12} +2.56774e66 q^{13} +4.97154e69 q^{14} +1.36580e70 q^{15} -8.73988e72 q^{16} -1.61699e74 q^{17} +9.72165e75 q^{18} -1.10211e76 q^{19} -8.44731e77 q^{20} +6.34707e79 q^{21} -1.41224e81 q^{22} -5.71167e81 q^{23} -5.60412e82 q^{24} -3.49510e84 q^{25} -5.32139e84 q^{26} +2.66751e86 q^{27} -3.92559e87 q^{28} +5.46451e88 q^{29} -2.83049e88 q^{30} -1.61112e90 q^{31} +1.24816e91 q^{32} -1.80298e91 q^{33} +3.35104e92 q^{34} +1.23836e93 q^{35} -7.67634e93 q^{36} -2.70619e94 q^{37} +2.28402e94 q^{38} -6.79372e94 q^{39} -1.09340e96 q^{40} +3.14111e97 q^{41} -1.31537e98 q^{42} +7.06608e98 q^{43} +1.11512e99 q^{44} +2.42156e99 q^{45} +1.18369e100 q^{46} +1.46831e101 q^{47} +2.31240e101 q^{48} +3.94823e102 q^{49} +7.24326e102 q^{50} +4.27822e102 q^{51} +4.20183e102 q^{52} +3.40087e104 q^{53} -5.52814e104 q^{54} -3.51774e104 q^{55} -5.08121e105 q^{56} +2.91597e104 q^{57} -1.13247e107 q^{58} +1.01537e107 q^{59} +2.23499e106 q^{60} -6.56814e107 q^{61} +3.33889e108 q^{62} +1.12534e109 q^{63} -2.63235e108 q^{64} -1.32550e108 q^{65} +3.73650e109 q^{66} -3.87624e110 q^{67} -2.64603e110 q^{68} +1.51119e110 q^{69} -2.56638e111 q^{70} +4.17247e110 q^{71} -9.93611e111 q^{72} +8.94067e112 q^{73} +5.60832e112 q^{74} +9.24735e112 q^{75} -1.80349e112 q^{76} -1.63474e114 q^{77} +1.40793e113 q^{78} -8.25811e114 q^{79} +4.51165e114 q^{80} +1.82317e115 q^{81} -6.50963e115 q^{82} +2.01103e116 q^{83} +1.03863e116 q^{84} +8.34711e115 q^{85} -1.46438e117 q^{86} -1.44580e117 q^{87} +1.44339e117 q^{88} -1.38039e118 q^{89} -5.01846e117 q^{90} -6.15981e117 q^{91} -9.34656e117 q^{92} +4.26270e118 q^{93} -3.04293e119 q^{94} +5.68926e117 q^{95} -3.30239e119 q^{96} -2.29519e120 q^{97} -8.18233e120 q^{98} -3.19668e120 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 23\!\cdots\!28 q^{2}+ \cdots + 79\!\cdots\!17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 23\!\cdots\!28 q^{2}+ \cdots - 44\!\cdots\!96 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.07240e18 −1.27104 −0.635520 0.772084i \(-0.719214\pi\)
−0.635520 + 0.772084i \(0.719214\pi\)
\(3\) −2.64580e28 −0.360347 −0.180174 0.983635i \(-0.557666\pi\)
−0.180174 + 0.983635i \(0.557666\pi\)
\(4\) 1.63640e36 0.615544
\(5\) −5.16214e41 −0.266161 −0.133081 0.991105i \(-0.542487\pi\)
−0.133081 + 0.991105i \(0.542487\pi\)
\(6\) 5.48316e46 0.458016
\(7\) −2.39892e51 −1.78478 −0.892391 0.451264i \(-0.850973\pi\)
−0.892391 + 0.451264i \(0.850973\pi\)
\(8\) 2.11812e54 0.488660
\(9\) −4.69101e57 −0.870150
\(10\) 1.06980e60 0.338302
\(11\) 6.81449e62 0.674802 0.337401 0.941361i \(-0.390452\pi\)
0.337401 + 0.941361i \(0.390452\pi\)
\(12\) −4.32958e64 −0.221809
\(13\) 2.56774e66 0.103748 0.0518738 0.998654i \(-0.483481\pi\)
0.0518738 + 0.998654i \(0.483481\pi\)
\(14\) 4.97154e69 2.26853
\(15\) 1.36580e70 0.0959105
\(16\) −8.73988e72 −1.23665
\(17\) −1.61699e74 −0.584180 −0.292090 0.956391i \(-0.594351\pi\)
−0.292090 + 0.956391i \(0.594351\pi\)
\(18\) 9.72165e75 1.10600
\(19\) −1.10211e76 −0.0476028 −0.0238014 0.999717i \(-0.507577\pi\)
−0.0238014 + 0.999717i \(0.507577\pi\)
\(20\) −8.44731e77 −0.163834
\(21\) 6.34707e79 0.643141
\(22\) −1.41224e81 −0.857700
\(23\) −5.71167e81 −0.235629 −0.117815 0.993036i \(-0.537589\pi\)
−0.117815 + 0.993036i \(0.537589\pi\)
\(24\) −5.60412e82 −0.176087
\(25\) −3.49510e84 −0.929158
\(26\) −5.32139e84 −0.131867
\(27\) 2.66751e86 0.673903
\(28\) −3.92559e87 −1.09861
\(29\) 5.46451e88 1.83009 0.915044 0.403354i \(-0.132156\pi\)
0.915044 + 0.403354i \(0.132156\pi\)
\(30\) −2.83049e88 −0.121906
\(31\) −1.61112e90 −0.954433 −0.477217 0.878786i \(-0.658354\pi\)
−0.477217 + 0.878786i \(0.658354\pi\)
\(32\) 1.24816e91 1.08317
\(33\) −1.80298e91 −0.243163
\(34\) 3.35104e92 0.742516
\(35\) 1.23836e93 0.475040
\(36\) −7.67634e93 −0.535615
\(37\) −2.70619e94 −0.359877 −0.179939 0.983678i \(-0.557590\pi\)
−0.179939 + 0.983678i \(0.557590\pi\)
\(38\) 2.28402e94 0.0605051
\(39\) −6.79372e94 −0.0373851
\(40\) −1.09340e96 −0.130062
\(41\) 3.14111e97 0.838801 0.419401 0.907801i \(-0.362240\pi\)
0.419401 + 0.907801i \(0.362240\pi\)
\(42\) −1.31537e98 −0.817458
\(43\) 7.06608e98 1.05764 0.528819 0.848734i \(-0.322635\pi\)
0.528819 + 0.848734i \(0.322635\pi\)
\(44\) 1.11512e99 0.415370
\(45\) 2.42156e99 0.231600
\(46\) 1.18369e100 0.299494
\(47\) 1.46831e101 1.01134 0.505671 0.862726i \(-0.331245\pi\)
0.505671 + 0.862726i \(0.331245\pi\)
\(48\) 2.31240e101 0.445623
\(49\) 3.94823e102 2.18544
\(50\) 7.24326e102 1.18100
\(51\) 4.27822e102 0.210507
\(52\) 4.20183e102 0.0638611
\(53\) 3.40087e104 1.63267 0.816337 0.577576i \(-0.196001\pi\)
0.816337 + 0.577576i \(0.196001\pi\)
\(54\) −5.52814e104 −0.856558
\(55\) −3.51774e104 −0.179606
\(56\) −5.08121e105 −0.872151
\(57\) 2.91597e104 0.0171535
\(58\) −1.13247e107 −2.32612
\(59\) 1.01537e107 0.741444 0.370722 0.928744i \(-0.379110\pi\)
0.370722 + 0.928744i \(0.379110\pi\)
\(60\) 2.23499e106 0.0590371
\(61\) −6.56814e107 −0.638245 −0.319122 0.947713i \(-0.603388\pi\)
−0.319122 + 0.947713i \(0.603388\pi\)
\(62\) 3.33889e108 1.21312
\(63\) 1.12534e109 1.55303
\(64\) −2.63235e108 −0.140106
\(65\) −1.32550e108 −0.0276136
\(66\) 3.73650e109 0.309070
\(67\) −3.87624e110 −1.29088 −0.645439 0.763812i \(-0.723325\pi\)
−0.645439 + 0.763812i \(0.723325\pi\)
\(68\) −2.64603e110 −0.359588
\(69\) 1.51119e110 0.0849084
\(70\) −2.56638e111 −0.603795
\(71\) 4.17247e110 0.0416163 0.0208081 0.999783i \(-0.493376\pi\)
0.0208081 + 0.999783i \(0.493376\pi\)
\(72\) −9.93611e111 −0.425207
\(73\) 8.94067e112 1.66088 0.830438 0.557111i \(-0.188090\pi\)
0.830438 + 0.557111i \(0.188090\pi\)
\(74\) 5.60832e112 0.457419
\(75\) 9.24735e112 0.334820
\(76\) −1.80349e112 −0.0293016
\(77\) −1.63474e114 −1.20437
\(78\) 1.40793e113 0.0475180
\(79\) −8.25811e114 −1.28956 −0.644780 0.764368i \(-0.723051\pi\)
−0.644780 + 0.764368i \(0.723051\pi\)
\(80\) 4.51165e114 0.329148
\(81\) 1.82317e115 0.627311
\(82\) −6.50963e115 −1.06615
\(83\) 2.01103e116 1.58196 0.790979 0.611843i \(-0.209572\pi\)
0.790979 + 0.611843i \(0.209572\pi\)
\(84\) 1.03863e116 0.395881
\(85\) 8.34711e115 0.155486
\(86\) −1.46438e117 −1.34430
\(87\) −1.44580e117 −0.659467
\(88\) 1.44339e117 0.329748
\(89\) −1.38039e118 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(90\) −5.01846e117 −0.294373
\(91\) −6.15981e117 −0.185167
\(92\) −9.34656e117 −0.145040
\(93\) 4.26270e118 0.343927
\(94\) −3.04293e119 −1.28546
\(95\) 5.68926e117 0.0126700
\(96\) −3.30239e119 −0.390318
\(97\) −2.29519e120 −1.44920 −0.724602 0.689168i \(-0.757976\pi\)
−0.724602 + 0.689168i \(0.757976\pi\)
\(98\) −8.18233e120 −2.77779
\(99\) −3.19668e120 −0.587179
\(100\) −5.71937e120 −0.571937
\(101\) −2.31541e120 −0.126819 −0.0634096 0.997988i \(-0.520197\pi\)
−0.0634096 + 0.997988i \(0.520197\pi\)
\(102\) −8.86619e120 −0.267563
\(103\) −3.55655e121 −0.594809 −0.297404 0.954752i \(-0.596121\pi\)
−0.297404 + 0.954752i \(0.596121\pi\)
\(104\) 5.43878e120 0.0506972
\(105\) −3.27645e121 −0.171179
\(106\) −7.04797e122 −2.07519
\(107\) 5.79788e121 0.0967277 0.0483639 0.998830i \(-0.484599\pi\)
0.0483639 + 0.998830i \(0.484599\pi\)
\(108\) 4.36509e122 0.414817
\(109\) −6.45454e122 −0.351206 −0.175603 0.984461i \(-0.556188\pi\)
−0.175603 + 0.984461i \(0.556188\pi\)
\(110\) 7.29017e122 0.228286
\(111\) 7.16005e122 0.129681
\(112\) 2.09663e124 2.20715
\(113\) 2.20120e124 1.35336 0.676680 0.736277i \(-0.263418\pi\)
0.676680 + 0.736277i \(0.263418\pi\)
\(114\) −6.04305e122 −0.0218028
\(115\) 2.94845e123 0.0627154
\(116\) 8.94209e124 1.12650
\(117\) −1.20453e124 −0.0902759
\(118\) −2.10426e125 −0.942406
\(119\) 3.87903e125 1.04263
\(120\) 2.89293e124 0.0468676
\(121\) −5.55427e125 −0.544643
\(122\) 1.36118e126 0.811235
\(123\) −8.31074e125 −0.302260
\(124\) −2.63643e126 −0.587495
\(125\) 3.74601e126 0.513467
\(126\) −2.33215e127 −1.97396
\(127\) −2.40916e127 −1.26398 −0.631991 0.774976i \(-0.717762\pi\)
−0.631991 + 0.774976i \(0.717762\pi\)
\(128\) −2.77266e127 −0.905092
\(129\) −1.86954e127 −0.381117
\(130\) 2.74698e126 0.0350980
\(131\) 1.88519e128 1.51510 0.757550 0.652778i \(-0.226396\pi\)
0.757550 + 0.652778i \(0.226396\pi\)
\(132\) −2.95039e127 −0.149677
\(133\) 2.64388e127 0.0849606
\(134\) 8.03313e128 1.64076
\(135\) −1.37700e128 −0.179367
\(136\) −3.42497e128 −0.285465
\(137\) 5.12965e128 0.274469 0.137235 0.990539i \(-0.456179\pi\)
0.137235 + 0.990539i \(0.456179\pi\)
\(138\) −3.13180e128 −0.107922
\(139\) 5.83869e129 1.29993 0.649965 0.759964i \(-0.274784\pi\)
0.649965 + 0.759964i \(0.274784\pi\)
\(140\) 2.02644e129 0.292408
\(141\) −3.88486e129 −0.364434
\(142\) −8.64704e128 −0.0528960
\(143\) 1.74978e129 0.0700090
\(144\) 4.09988e130 1.07607
\(145\) −2.82086e130 −0.487099
\(146\) −1.85287e131 −2.11104
\(147\) −1.04462e131 −0.787519
\(148\) −4.42840e130 −0.221520
\(149\) −2.54981e131 −0.848671 −0.424335 0.905505i \(-0.639492\pi\)
−0.424335 + 0.905505i \(0.639492\pi\)
\(150\) −1.91642e131 −0.425569
\(151\) 5.59487e131 0.831160 0.415580 0.909557i \(-0.363579\pi\)
0.415580 + 0.909557i \(0.363579\pi\)
\(152\) −2.33440e130 −0.0232616
\(153\) 7.58529e131 0.508324
\(154\) 3.38785e132 1.53081
\(155\) 8.31684e131 0.254033
\(156\) −1.11172e131 −0.0230122
\(157\) −5.66314e132 −0.796395 −0.398198 0.917300i \(-0.630364\pi\)
−0.398198 + 0.917300i \(0.630364\pi\)
\(158\) 1.71141e133 1.63908
\(159\) −8.99802e132 −0.588330
\(160\) −6.44319e132 −0.288298
\(161\) 1.37019e133 0.420547
\(162\) −3.77833e133 −0.797337
\(163\) 8.91375e133 1.29631 0.648157 0.761506i \(-0.275540\pi\)
0.648157 + 0.761506i \(0.275540\pi\)
\(164\) 5.14009e133 0.516319
\(165\) 9.30723e132 0.0647205
\(166\) −4.16767e134 −2.01073
\(167\) −3.85090e134 −1.29186 −0.645930 0.763397i \(-0.723530\pi\)
−0.645930 + 0.763397i \(0.723530\pi\)
\(168\) 1.34439e134 0.314277
\(169\) −6.05962e134 −0.989236
\(170\) −1.72986e134 −0.197629
\(171\) 5.17001e133 0.0414216
\(172\) 1.15629e135 0.651023
\(173\) 6.93703e134 0.275033 0.137516 0.990499i \(-0.456088\pi\)
0.137516 + 0.990499i \(0.456088\pi\)
\(174\) 2.99628e135 0.838209
\(175\) 8.38449e135 1.65834
\(176\) −5.95579e135 −0.834493
\(177\) −2.68647e135 −0.267177
\(178\) 2.86072e136 2.02332
\(179\) −1.40783e136 −0.709482 −0.354741 0.934965i \(-0.615431\pi\)
−0.354741 + 0.934965i \(0.615431\pi\)
\(180\) 3.96264e135 0.142560
\(181\) −6.92545e136 −1.78195 −0.890973 0.454057i \(-0.849976\pi\)
−0.890973 + 0.454057i \(0.849976\pi\)
\(182\) 1.27656e136 0.235354
\(183\) 1.73780e136 0.229990
\(184\) −1.20980e136 −0.115143
\(185\) 1.39698e136 0.0957854
\(186\) −8.83404e136 −0.437146
\(187\) −1.10189e137 −0.394205
\(188\) 2.40274e137 0.622525
\(189\) −6.39914e137 −1.20277
\(190\) −1.17904e136 −0.0161041
\(191\) −9.75491e137 −0.969850 −0.484925 0.874556i \(-0.661153\pi\)
−0.484925 + 0.874556i \(0.661153\pi\)
\(192\) 6.96468e136 0.0504867
\(193\) 1.27458e137 0.0674762 0.0337381 0.999431i \(-0.489259\pi\)
0.0337381 + 0.999431i \(0.489259\pi\)
\(194\) 4.75657e138 1.84200
\(195\) 3.50702e136 0.00995047
\(196\) 6.46087e138 1.34524
\(197\) 8.72607e137 0.133540 0.0667701 0.997768i \(-0.478731\pi\)
0.0667701 + 0.997768i \(0.478731\pi\)
\(198\) 6.62481e138 0.746328
\(199\) −5.08881e138 −0.422674 −0.211337 0.977413i \(-0.567782\pi\)
−0.211337 + 0.977413i \(0.567782\pi\)
\(200\) −7.40305e138 −0.454042
\(201\) 1.02558e139 0.465164
\(202\) 4.79846e138 0.161192
\(203\) −1.31089e140 −3.26631
\(204\) 7.00086e138 0.129577
\(205\) −1.62148e139 −0.223256
\(206\) 7.37061e139 0.756026
\(207\) 2.67935e139 0.205033
\(208\) −2.24417e139 −0.128299
\(209\) −7.51033e138 −0.0321224
\(210\) 6.79012e139 0.217576
\(211\) −3.22420e140 −0.775056 −0.387528 0.921858i \(-0.626671\pi\)
−0.387528 + 0.921858i \(0.626671\pi\)
\(212\) 5.56516e140 1.00498
\(213\) −1.10395e139 −0.0149963
\(214\) −1.20155e140 −0.122945
\(215\) −3.64761e140 −0.281502
\(216\) 5.65010e140 0.329309
\(217\) 3.86496e141 1.70346
\(218\) 1.33764e141 0.446397
\(219\) −2.36552e141 −0.598492
\(220\) −5.75641e140 −0.110555
\(221\) −4.15199e140 −0.0606072
\(222\) −1.48385e141 −0.164830
\(223\) 8.38698e141 0.709842 0.354921 0.934896i \(-0.384508\pi\)
0.354921 + 0.934896i \(0.384508\pi\)
\(224\) −2.99425e142 −1.93323
\(225\) 1.63956e142 0.808507
\(226\) −4.56177e142 −1.72018
\(227\) −5.86880e142 −1.69428 −0.847140 0.531369i \(-0.821678\pi\)
−0.847140 + 0.531369i \(0.821678\pi\)
\(228\) 4.77167e140 0.0105587
\(229\) 1.01653e143 1.72613 0.863066 0.505091i \(-0.168541\pi\)
0.863066 + 0.505091i \(0.168541\pi\)
\(230\) −6.11037e141 −0.0797138
\(231\) 4.32521e142 0.433993
\(232\) 1.15745e143 0.894290
\(233\) 1.52230e143 0.906704 0.453352 0.891331i \(-0.350228\pi\)
0.453352 + 0.891331i \(0.350228\pi\)
\(234\) 2.49626e142 0.114744
\(235\) −7.57964e142 −0.269180
\(236\) 1.66155e143 0.456391
\(237\) 2.18493e143 0.464689
\(238\) −8.03890e143 −1.32523
\(239\) −4.56142e143 −0.583481 −0.291741 0.956497i \(-0.594234\pi\)
−0.291741 + 0.956497i \(0.594234\pi\)
\(240\) −1.19369e143 −0.118608
\(241\) 1.10935e144 0.857110 0.428555 0.903516i \(-0.359023\pi\)
0.428555 + 0.903516i \(0.359023\pi\)
\(242\) 1.15107e144 0.692263
\(243\) −1.92043e144 −0.899953
\(244\) −1.07481e144 −0.392868
\(245\) −2.03813e144 −0.581681
\(246\) 1.72232e144 0.384184
\(247\) −2.82993e142 −0.00493867
\(248\) −3.41255e144 −0.466393
\(249\) −5.32080e144 −0.570054
\(250\) −7.76323e144 −0.652637
\(251\) 1.21642e144 0.0803200 0.0401600 0.999193i \(-0.487213\pi\)
0.0401600 + 0.999193i \(0.487213\pi\)
\(252\) 1.84150e145 0.955956
\(253\) −3.89222e144 −0.159003
\(254\) 4.99275e145 1.60657
\(255\) −2.20848e144 −0.0560289
\(256\) 6.44586e145 1.29051
\(257\) −3.59155e145 −0.567973 −0.283987 0.958828i \(-0.591657\pi\)
−0.283987 + 0.958828i \(0.591657\pi\)
\(258\) 3.87445e145 0.484415
\(259\) 6.49195e145 0.642302
\(260\) −2.16905e144 −0.0169974
\(261\) −2.56340e146 −1.59245
\(262\) −3.90687e146 −1.92575
\(263\) 1.64079e146 0.642289 0.321145 0.947030i \(-0.395932\pi\)
0.321145 + 0.947030i \(0.395932\pi\)
\(264\) −3.81893e145 −0.118824
\(265\) −1.75558e146 −0.434555
\(266\) −5.47918e145 −0.107988
\(267\) 3.65223e146 0.573624
\(268\) −6.34306e146 −0.794592
\(269\) −1.43687e146 −0.143683 −0.0718414 0.997416i \(-0.522888\pi\)
−0.0718414 + 0.997416i \(0.522888\pi\)
\(270\) 2.85371e146 0.227983
\(271\) 2.23153e147 1.42548 0.712740 0.701428i \(-0.247454\pi\)
0.712740 + 0.701428i \(0.247454\pi\)
\(272\) 1.41323e147 0.722426
\(273\) 1.62976e146 0.0667243
\(274\) −1.06307e147 −0.348861
\(275\) −2.38174e147 −0.626997
\(276\) 2.47291e146 0.0522648
\(277\) 7.37038e147 1.25160 0.625798 0.779985i \(-0.284774\pi\)
0.625798 + 0.779985i \(0.284774\pi\)
\(278\) −1.21001e148 −1.65226
\(279\) 7.55778e147 0.830500
\(280\) 2.62299e147 0.232133
\(281\) 1.22628e148 0.874696 0.437348 0.899292i \(-0.355918\pi\)
0.437348 + 0.899292i \(0.355918\pi\)
\(282\) 8.05099e147 0.463211
\(283\) −3.36601e148 −1.56328 −0.781638 0.623733i \(-0.785615\pi\)
−0.781638 + 0.623733i \(0.785615\pi\)
\(284\) 6.82781e146 0.0256166
\(285\) −1.50526e146 −0.00456561
\(286\) −3.62625e147 −0.0889843
\(287\) −7.53527e148 −1.49708
\(288\) −5.85514e148 −0.942522
\(289\) −5.04696e148 −0.658734
\(290\) 5.84595e148 0.619122
\(291\) 6.07263e148 0.522217
\(292\) 1.46305e149 1.02234
\(293\) 9.89552e148 0.562273 0.281137 0.959668i \(-0.409289\pi\)
0.281137 + 0.959668i \(0.409289\pi\)
\(294\) 2.16488e149 1.00097
\(295\) −5.24150e148 −0.197344
\(296\) −5.73204e148 −0.175858
\(297\) 1.81777e149 0.454751
\(298\) 5.28424e149 1.07869
\(299\) −1.46661e148 −0.0244460
\(300\) 1.51323e149 0.206096
\(301\) −1.69510e150 −1.88765
\(302\) −1.15948e150 −1.05644
\(303\) 6.12611e148 0.0456989
\(304\) 9.63232e148 0.0588680
\(305\) 3.39057e149 0.169876
\(306\) −1.57198e150 −0.646100
\(307\) 2.35422e150 0.794284 0.397142 0.917757i \(-0.370002\pi\)
0.397142 + 0.917757i \(0.370002\pi\)
\(308\) −2.67509e150 −0.741344
\(309\) 9.40993e149 0.214338
\(310\) −1.72358e150 −0.322886
\(311\) 6.35294e150 0.979424 0.489712 0.871884i \(-0.337102\pi\)
0.489712 + 0.871884i \(0.337102\pi\)
\(312\) −1.43899e149 −0.0182686
\(313\) −6.10266e150 −0.638391 −0.319195 0.947689i \(-0.603413\pi\)
−0.319195 + 0.947689i \(0.603413\pi\)
\(314\) 1.17363e151 1.01225
\(315\) −5.80915e150 −0.413356
\(316\) −1.35135e151 −0.793780
\(317\) 8.99646e150 0.436502 0.218251 0.975893i \(-0.429965\pi\)
0.218251 + 0.975893i \(0.429965\pi\)
\(318\) 1.86475e151 0.747791
\(319\) 3.72378e151 1.23495
\(320\) 1.35886e150 0.0372907
\(321\) −1.53400e150 −0.0348556
\(322\) −2.83958e151 −0.534532
\(323\) 1.78210e150 0.0278086
\(324\) 2.98342e151 0.386137
\(325\) −8.97451e150 −0.0963979
\(326\) −1.84729e152 −1.64767
\(327\) 1.70774e151 0.126556
\(328\) 6.65324e151 0.409888
\(329\) −3.52237e152 −1.80502
\(330\) −1.92883e151 −0.0822624
\(331\) −3.69989e152 −1.31400 −0.657000 0.753890i \(-0.728175\pi\)
−0.657000 + 0.753890i \(0.728175\pi\)
\(332\) 3.29085e152 0.973764
\(333\) 1.26948e152 0.313147
\(334\) 7.98061e152 1.64201
\(335\) 2.00097e152 0.343582
\(336\) −5.54727e152 −0.795340
\(337\) 1.38638e153 1.66063 0.830316 0.557292i \(-0.188160\pi\)
0.830316 + 0.557292i \(0.188160\pi\)
\(338\) 1.25580e153 1.25736
\(339\) −5.82393e152 −0.487680
\(340\) 1.36592e152 0.0957084
\(341\) −1.09790e153 −0.644053
\(342\) −1.07143e152 −0.0526485
\(343\) −5.13760e153 −2.11576
\(344\) 1.49668e153 0.516825
\(345\) −7.80100e151 −0.0225993
\(346\) −1.43763e153 −0.349578
\(347\) 7.57101e153 1.54604 0.773020 0.634382i \(-0.218745\pi\)
0.773020 + 0.634382i \(0.218745\pi\)
\(348\) −2.36590e153 −0.405931
\(349\) 1.46318e153 0.211037 0.105519 0.994417i \(-0.466350\pi\)
0.105519 + 0.994417i \(0.466350\pi\)
\(350\) −1.73760e154 −2.10782
\(351\) 6.84945e152 0.0699158
\(352\) 8.50559e153 0.730926
\(353\) 2.50797e153 0.181532 0.0907660 0.995872i \(-0.471068\pi\)
0.0907660 + 0.995872i \(0.471068\pi\)
\(354\) 5.56745e153 0.339593
\(355\) −2.15389e152 −0.0110766
\(356\) −2.25886e154 −0.979861
\(357\) −1.02631e154 −0.375710
\(358\) 2.91759e154 0.901780
\(359\) 3.19117e154 0.833175 0.416587 0.909096i \(-0.363226\pi\)
0.416587 + 0.909096i \(0.363226\pi\)
\(360\) 5.12916e153 0.113174
\(361\) −5.34811e154 −0.997734
\(362\) 1.43523e155 2.26492
\(363\) 1.46955e154 0.196261
\(364\) −1.00799e154 −0.113978
\(365\) −4.61530e154 −0.442061
\(366\) −3.60142e154 −0.292326
\(367\) 4.61606e154 0.317669 0.158835 0.987305i \(-0.449226\pi\)
0.158835 + 0.987305i \(0.449226\pi\)
\(368\) 4.99194e154 0.291391
\(369\) −1.47349e155 −0.729883
\(370\) −2.89510e154 −0.121747
\(371\) −8.15842e155 −2.91397
\(372\) 6.97547e154 0.211702
\(373\) −6.56950e154 −0.169492 −0.0847458 0.996403i \(-0.527008\pi\)
−0.0847458 + 0.996403i \(0.527008\pi\)
\(374\) 2.28357e155 0.501051
\(375\) −9.91118e154 −0.185026
\(376\) 3.11006e155 0.494202
\(377\) 1.40314e155 0.189867
\(378\) 1.32616e156 1.52877
\(379\) −2.34599e155 −0.230491 −0.115245 0.993337i \(-0.536765\pi\)
−0.115245 + 0.993337i \(0.536765\pi\)
\(380\) 9.30987e153 0.00779895
\(381\) 6.37416e155 0.455472
\(382\) 2.02161e156 1.23272
\(383\) −1.63444e156 −0.850830 −0.425415 0.904998i \(-0.639872\pi\)
−0.425415 + 0.904998i \(0.639872\pi\)
\(384\) 7.33589e155 0.326147
\(385\) 8.43879e155 0.320557
\(386\) −2.64145e155 −0.0857650
\(387\) −3.31470e156 −0.920304
\(388\) −3.75585e156 −0.892048
\(389\) −1.85102e156 −0.376237 −0.188118 0.982146i \(-0.560239\pi\)
−0.188118 + 0.982146i \(0.560239\pi\)
\(390\) −7.26795e154 −0.0126475
\(391\) 9.23569e155 0.137650
\(392\) 8.36283e156 1.06794
\(393\) −4.98783e156 −0.545962
\(394\) −1.80839e156 −0.169735
\(395\) 4.26295e156 0.343231
\(396\) −5.23104e156 −0.361434
\(397\) −2.01690e157 −1.19635 −0.598177 0.801364i \(-0.704108\pi\)
−0.598177 + 0.801364i \(0.704108\pi\)
\(398\) 1.05461e157 0.537236
\(399\) −6.99518e155 −0.0306153
\(400\) 3.05468e157 1.14904
\(401\) −3.04168e157 −0.983738 −0.491869 0.870669i \(-0.663686\pi\)
−0.491869 + 0.870669i \(0.663686\pi\)
\(402\) −2.12541e157 −0.591242
\(403\) −4.13694e156 −0.0990201
\(404\) −3.78893e156 −0.0780627
\(405\) −9.41145e156 −0.166966
\(406\) 2.71670e158 4.15161
\(407\) −1.84413e157 −0.242846
\(408\) 9.06179e156 0.102867
\(409\) −8.33715e157 −0.816126 −0.408063 0.912954i \(-0.633796\pi\)
−0.408063 + 0.912954i \(0.633796\pi\)
\(410\) 3.36037e157 0.283768
\(411\) −1.35720e157 −0.0989042
\(412\) −5.81993e157 −0.366131
\(413\) −2.43580e158 −1.32332
\(414\) −5.55269e157 −0.260605
\(415\) −1.03812e158 −0.421056
\(416\) 3.20495e157 0.112376
\(417\) −1.54480e158 −0.468426
\(418\) 1.55644e157 0.0408289
\(419\) −6.42807e158 −1.45926 −0.729630 0.683842i \(-0.760308\pi\)
−0.729630 + 0.683842i \(0.760308\pi\)
\(420\) −5.36157e157 −0.105368
\(421\) 7.49095e158 1.27488 0.637441 0.770499i \(-0.279993\pi\)
0.637441 + 0.770499i \(0.279993\pi\)
\(422\) 6.68183e158 0.985128
\(423\) −6.88786e158 −0.880019
\(424\) 7.20345e158 0.797822
\(425\) 5.65153e158 0.542795
\(426\) 2.28783e157 0.0190609
\(427\) 1.57565e159 1.13913
\(428\) 9.48763e157 0.0595401
\(429\) −4.62958e157 −0.0252275
\(430\) 7.55932e158 0.357801
\(431\) −1.69880e159 −0.698664 −0.349332 0.936999i \(-0.613591\pi\)
−0.349332 + 0.936999i \(0.613591\pi\)
\(432\) −2.33137e159 −0.833382
\(433\) 4.37631e159 1.36016 0.680080 0.733138i \(-0.261945\pi\)
0.680080 + 0.733138i \(0.261945\pi\)
\(434\) −8.00974e159 −2.16516
\(435\) 7.46342e158 0.175525
\(436\) −1.05622e159 −0.216183
\(437\) 6.29490e157 0.0112166
\(438\) 4.90232e159 0.760707
\(439\) 6.94989e159 0.939452 0.469726 0.882812i \(-0.344353\pi\)
0.469726 + 0.882812i \(0.344353\pi\)
\(440\) −7.45099e158 −0.0877662
\(441\) −1.85212e160 −1.90166
\(442\) 8.60460e158 0.0770342
\(443\) −8.71532e158 −0.0680547 −0.0340273 0.999421i \(-0.510833\pi\)
−0.0340273 + 0.999421i \(0.510833\pi\)
\(444\) 1.17167e159 0.0798242
\(445\) 7.12576e159 0.423692
\(446\) −1.73812e160 −0.902238
\(447\) 6.74630e159 0.305816
\(448\) 6.31481e159 0.250058
\(449\) 1.45705e160 0.504163 0.252082 0.967706i \(-0.418885\pi\)
0.252082 + 0.967706i \(0.418885\pi\)
\(450\) −3.39782e160 −1.02764
\(451\) 2.14050e160 0.566024
\(452\) 3.60203e160 0.833052
\(453\) −1.48029e160 −0.299506
\(454\) 1.21625e161 2.15350
\(455\) 3.17978e159 0.0492842
\(456\) 6.17637e158 0.00838224
\(457\) 9.72851e160 1.15642 0.578211 0.815887i \(-0.303751\pi\)
0.578211 + 0.815887i \(0.303751\pi\)
\(458\) −2.10666e161 −2.19398
\(459\) −4.31332e160 −0.393681
\(460\) 4.82483e159 0.0386041
\(461\) −2.24952e160 −0.157828 −0.0789141 0.996881i \(-0.525145\pi\)
−0.0789141 + 0.996881i \(0.525145\pi\)
\(462\) −8.96357e160 −0.551622
\(463\) 1.06518e161 0.575141 0.287571 0.957759i \(-0.407152\pi\)
0.287571 + 0.957759i \(0.407152\pi\)
\(464\) −4.77591e161 −2.26318
\(465\) −2.20047e160 −0.0915401
\(466\) −3.15481e161 −1.15246
\(467\) 9.57907e160 0.307363 0.153682 0.988120i \(-0.450887\pi\)
0.153682 + 0.988120i \(0.450887\pi\)
\(468\) −1.97108e160 −0.0555688
\(469\) 9.29881e161 2.30393
\(470\) 1.57081e161 0.342139
\(471\) 1.49835e161 0.286979
\(472\) 2.15068e161 0.362314
\(473\) 4.81518e161 0.713696
\(474\) −4.52806e161 −0.590639
\(475\) 3.85199e160 0.0442305
\(476\) 6.34762e161 0.641786
\(477\) −1.59535e162 −1.42067
\(478\) 9.45310e161 0.741629
\(479\) 5.22132e161 0.360979 0.180489 0.983577i \(-0.442232\pi\)
0.180489 + 0.983577i \(0.442232\pi\)
\(480\) 1.70474e161 0.103888
\(481\) −6.94880e160 −0.0373364
\(482\) −2.29901e162 −1.08942
\(483\) −3.62524e161 −0.151543
\(484\) −9.08898e161 −0.335251
\(485\) 1.18481e162 0.385722
\(486\) 3.97991e162 1.14388
\(487\) −4.57656e162 −1.16155 −0.580774 0.814065i \(-0.697250\pi\)
−0.580774 + 0.814065i \(0.697250\pi\)
\(488\) −1.39121e162 −0.311885
\(489\) −2.35840e162 −0.467123
\(490\) 4.22383e162 0.739340
\(491\) −8.60617e161 −0.133162 −0.0665809 0.997781i \(-0.521209\pi\)
−0.0665809 + 0.997781i \(0.521209\pi\)
\(492\) −1.35997e162 −0.186054
\(493\) −8.83603e162 −1.06910
\(494\) 5.86476e160 0.00627725
\(495\) 1.65017e162 0.156284
\(496\) 1.40810e163 1.18030
\(497\) −1.00094e162 −0.0742760
\(498\) 1.10268e163 0.724562
\(499\) −2.43499e163 −1.41715 −0.708573 0.705637i \(-0.750661\pi\)
−0.708573 + 0.705637i \(0.750661\pi\)
\(500\) 6.12995e162 0.316061
\(501\) 1.01887e163 0.465518
\(502\) −2.52091e162 −0.102090
\(503\) 4.06905e163 1.46093 0.730466 0.682949i \(-0.239303\pi\)
0.730466 + 0.682949i \(0.239303\pi\)
\(504\) 2.38360e163 0.758902
\(505\) 1.19525e162 0.0337543
\(506\) 8.06624e162 0.202099
\(507\) 1.60326e163 0.356469
\(508\) −3.94234e163 −0.778036
\(509\) −1.84603e163 −0.323455 −0.161728 0.986835i \(-0.551707\pi\)
−0.161728 + 0.986835i \(0.551707\pi\)
\(510\) 4.57686e162 0.0712150
\(511\) −2.14480e164 −2.96430
\(512\) −5.98743e163 −0.735204
\(513\) −2.93989e162 −0.0320797
\(514\) 7.44313e163 0.721917
\(515\) 1.83594e163 0.158315
\(516\) −3.05931e163 −0.234594
\(517\) 1.00058e164 0.682455
\(518\) −1.34539e164 −0.816392
\(519\) −1.83540e163 −0.0991073
\(520\) −2.80758e162 −0.0134936
\(521\) 3.64619e164 1.56012 0.780058 0.625708i \(-0.215190\pi\)
0.780058 + 0.625708i \(0.215190\pi\)
\(522\) 5.31240e164 2.02407
\(523\) −2.38094e164 −0.807971 −0.403986 0.914765i \(-0.632375\pi\)
−0.403986 + 0.914765i \(0.632375\pi\)
\(524\) 3.08491e164 0.932610
\(525\) −2.21837e164 −0.597580
\(526\) −3.40038e164 −0.816376
\(527\) 2.60516e164 0.557561
\(528\) 1.57578e164 0.300707
\(529\) −5.54958e164 −0.944479
\(530\) 3.63826e164 0.552336
\(531\) −4.76312e164 −0.645168
\(532\) 4.32643e163 0.0522970
\(533\) 8.06553e163 0.0870236
\(534\) −7.56889e164 −0.729099
\(535\) −2.99295e163 −0.0257452
\(536\) −8.21034e164 −0.630800
\(537\) 3.72483e164 0.255660
\(538\) 2.97778e164 0.182627
\(539\) 2.69052e165 1.47474
\(540\) −2.25332e164 −0.110408
\(541\) −9.91470e164 −0.434355 −0.217178 0.976132i \(-0.569685\pi\)
−0.217178 + 0.976132i \(0.569685\pi\)
\(542\) −4.62464e165 −1.81184
\(543\) 1.83234e165 0.642119
\(544\) −2.01826e165 −0.632767
\(545\) 3.33193e164 0.0934775
\(546\) −3.37752e164 −0.0848093
\(547\) 5.30962e164 0.119352 0.0596761 0.998218i \(-0.480993\pi\)
0.0596761 + 0.998218i \(0.480993\pi\)
\(548\) 8.39414e164 0.168948
\(549\) 3.08112e165 0.555369
\(550\) 4.93592e165 0.796939
\(551\) −6.02249e164 −0.0871173
\(552\) 3.20089e164 0.0414913
\(553\) 1.98106e166 2.30158
\(554\) −1.52744e166 −1.59083
\(555\) −3.69612e164 −0.0345160
\(556\) 9.55440e165 0.800163
\(557\) 9.51657e165 0.714893 0.357447 0.933934i \(-0.383647\pi\)
0.357447 + 0.933934i \(0.383647\pi\)
\(558\) −1.56628e166 −1.05560
\(559\) 1.81439e165 0.109727
\(560\) −1.08231e166 −0.587458
\(561\) 2.91539e165 0.142051
\(562\) −2.54134e166 −1.11177
\(563\) −3.86845e166 −1.51977 −0.759887 0.650055i \(-0.774746\pi\)
−0.759887 + 0.650055i \(0.774746\pi\)
\(564\) −6.35717e165 −0.224325
\(565\) −1.13629e166 −0.360212
\(566\) 6.97572e166 1.98699
\(567\) −4.37364e166 −1.11961
\(568\) 8.83780e164 0.0203362
\(569\) −3.35645e166 −0.694366 −0.347183 0.937797i \(-0.612862\pi\)
−0.347183 + 0.937797i \(0.612862\pi\)
\(570\) 3.11951e164 0.00580307
\(571\) 4.30635e166 0.720484 0.360242 0.932859i \(-0.382694\pi\)
0.360242 + 0.932859i \(0.382694\pi\)
\(572\) 2.86334e165 0.0430936
\(573\) 2.58095e166 0.349483
\(574\) 1.56161e167 1.90285
\(575\) 1.99629e166 0.218937
\(576\) 1.23484e166 0.121913
\(577\) 4.03725e166 0.358879 0.179440 0.983769i \(-0.442572\pi\)
0.179440 + 0.983769i \(0.442572\pi\)
\(578\) 1.04593e167 0.837278
\(579\) −3.37229e165 −0.0243149
\(580\) −4.61604e166 −0.299830
\(581\) −4.82432e167 −2.82345
\(582\) −1.25849e167 −0.663758
\(583\) 2.31752e167 1.10173
\(584\) 1.89374e167 0.811603
\(585\) 6.21794e165 0.0240280
\(586\) −2.05075e167 −0.714672
\(587\) −8.96368e166 −0.281761 −0.140881 0.990027i \(-0.544993\pi\)
−0.140881 + 0.990027i \(0.544993\pi\)
\(588\) −1.70942e167 −0.484752
\(589\) 1.77563e166 0.0454337
\(590\) 1.08625e167 0.250832
\(591\) −2.30874e166 −0.0481208
\(592\) 2.36518e167 0.445042
\(593\) 4.04604e167 0.687418 0.343709 0.939076i \(-0.388317\pi\)
0.343709 + 0.939076i \(0.388317\pi\)
\(594\) −3.76715e167 −0.578007
\(595\) −2.00241e167 −0.277508
\(596\) −4.17250e167 −0.522394
\(597\) 1.34640e167 0.152309
\(598\) 3.03940e166 0.0310718
\(599\) −1.16804e168 −1.07928 −0.539640 0.841896i \(-0.681440\pi\)
−0.539640 + 0.841896i \(0.681440\pi\)
\(600\) 1.95870e167 0.163613
\(601\) −1.51560e168 −1.14466 −0.572332 0.820022i \(-0.693961\pi\)
−0.572332 + 0.820022i \(0.693961\pi\)
\(602\) 3.51293e168 2.39928
\(603\) 1.81835e168 1.12326
\(604\) 9.15542e167 0.511615
\(605\) 2.86719e167 0.144963
\(606\) −1.26958e167 −0.0580852
\(607\) −2.05167e168 −0.849554 −0.424777 0.905298i \(-0.639648\pi\)
−0.424777 + 0.905298i \(0.639648\pi\)
\(608\) −1.37561e167 −0.0515620
\(609\) 3.46836e168 1.17700
\(610\) −7.02662e167 −0.215919
\(611\) 3.77024e167 0.104924
\(612\) 1.24125e168 0.312895
\(613\) −2.50276e168 −0.571558 −0.285779 0.958295i \(-0.592252\pi\)
−0.285779 + 0.958295i \(0.592252\pi\)
\(614\) −4.87890e168 −1.00957
\(615\) 4.29012e167 0.0804498
\(616\) −3.46259e168 −0.588529
\(617\) −6.17889e168 −0.952046 −0.476023 0.879433i \(-0.657922\pi\)
−0.476023 + 0.879433i \(0.657922\pi\)
\(618\) −1.95012e168 −0.272432
\(619\) −8.34218e168 −1.05681 −0.528403 0.848994i \(-0.677209\pi\)
−0.528403 + 0.848994i \(0.677209\pi\)
\(620\) 1.36096e168 0.156368
\(621\) −1.52359e168 −0.158791
\(622\) −1.31658e169 −1.24489
\(623\) 3.31145e169 2.84113
\(624\) 5.93763e167 0.0462323
\(625\) 1.12134e169 0.792493
\(626\) 1.26472e169 0.811420
\(627\) 1.98708e167 0.0115752
\(628\) −9.26714e168 −0.490216
\(629\) 4.37588e168 0.210233
\(630\) 1.20389e169 0.525392
\(631\) 2.07517e169 0.822768 0.411384 0.911462i \(-0.365046\pi\)
0.411384 + 0.911462i \(0.365046\pi\)
\(632\) −1.74917e169 −0.630156
\(633\) 8.53058e168 0.279289
\(634\) −1.86443e169 −0.554812
\(635\) 1.24364e169 0.336423
\(636\) −1.47243e169 −0.362143
\(637\) 1.01380e169 0.226735
\(638\) −7.71718e169 −1.56967
\(639\) −1.95731e168 −0.0362124
\(640\) 1.43128e169 0.240900
\(641\) −2.59447e169 −0.397318 −0.198659 0.980069i \(-0.563659\pi\)
−0.198659 + 0.980069i \(0.563659\pi\)
\(642\) 3.17907e168 0.0443028
\(643\) −5.49700e168 −0.0697207 −0.0348603 0.999392i \(-0.511099\pi\)
−0.0348603 + 0.999392i \(0.511099\pi\)
\(644\) 2.24217e169 0.258865
\(645\) 9.65086e168 0.101439
\(646\) −3.69322e168 −0.0353458
\(647\) −1.23674e170 −1.07788 −0.538941 0.842344i \(-0.681175\pi\)
−0.538941 + 0.842344i \(0.681175\pi\)
\(648\) 3.86169e169 0.306541
\(649\) 6.91925e169 0.500328
\(650\) 1.85988e169 0.122526
\(651\) −1.02259e170 −0.613835
\(652\) 1.45864e170 0.797938
\(653\) 1.99178e170 0.993102 0.496551 0.868007i \(-0.334600\pi\)
0.496551 + 0.868007i \(0.334600\pi\)
\(654\) −3.53913e169 −0.160858
\(655\) −9.73161e169 −0.403261
\(656\) −2.74529e170 −1.03730
\(657\) −4.19407e170 −1.44521
\(658\) 7.29977e170 2.29426
\(659\) 6.47582e169 0.185664 0.0928319 0.995682i \(-0.470408\pi\)
0.0928319 + 0.995682i \(0.470408\pi\)
\(660\) 1.52303e169 0.0398383
\(661\) −1.46234e170 −0.349028 −0.174514 0.984655i \(-0.555835\pi\)
−0.174514 + 0.984655i \(0.555835\pi\)
\(662\) 7.66767e170 1.67015
\(663\) 1.09853e169 0.0218396
\(664\) 4.25961e170 0.773039
\(665\) −1.36481e169 −0.0226132
\(666\) −2.63087e170 −0.398023
\(667\) −3.12115e170 −0.431222
\(668\) −6.30159e170 −0.795196
\(669\) −2.21903e170 −0.255790
\(670\) −4.14682e170 −0.436706
\(671\) −4.47585e170 −0.430689
\(672\) 7.92218e170 0.696632
\(673\) −4.46853e170 −0.359131 −0.179566 0.983746i \(-0.557469\pi\)
−0.179566 + 0.983746i \(0.557469\pi\)
\(674\) −2.87314e171 −2.11073
\(675\) −9.32321e170 −0.626163
\(676\) −9.91594e170 −0.608918
\(677\) −8.54958e169 −0.0480098 −0.0240049 0.999712i \(-0.507642\pi\)
−0.0240049 + 0.999712i \(0.507642\pi\)
\(678\) 1.20695e171 0.619860
\(679\) 5.50600e171 2.58651
\(680\) 1.76802e170 0.0759797
\(681\) 1.55277e171 0.610529
\(682\) 2.27528e171 0.818618
\(683\) −2.95078e171 −0.971592 −0.485796 0.874072i \(-0.661470\pi\)
−0.485796 + 0.874072i \(0.661470\pi\)
\(684\) 8.46018e169 0.0254968
\(685\) −2.64800e170 −0.0730530
\(686\) 1.06472e172 2.68922
\(687\) −2.68954e171 −0.622007
\(688\) −6.17568e171 −1.30793
\(689\) 8.73253e170 0.169386
\(690\) 1.61668e170 0.0287246
\(691\) −1.96563e171 −0.319948 −0.159974 0.987121i \(-0.551141\pi\)
−0.159974 + 0.987121i \(0.551141\pi\)
\(692\) 1.13517e171 0.169295
\(693\) 7.66860e171 1.04799
\(694\) −1.56902e172 −1.96508
\(695\) −3.01401e171 −0.345991
\(696\) −3.06238e171 −0.322255
\(697\) −5.07912e171 −0.490011
\(698\) −3.03229e171 −0.268237
\(699\) −4.02769e171 −0.326728
\(700\) 1.37203e172 1.02078
\(701\) 4.98212e171 0.339996 0.169998 0.985444i \(-0.445624\pi\)
0.169998 + 0.985444i \(0.445624\pi\)
\(702\) −1.41948e171 −0.0888658
\(703\) 2.98253e170 0.0171312
\(704\) −1.79382e171 −0.0945435
\(705\) 2.00542e171 0.0969983
\(706\) −5.19753e171 −0.230734
\(707\) 5.55449e171 0.226344
\(708\) −4.39613e171 −0.164459
\(709\) −2.82066e171 −0.0968844 −0.0484422 0.998826i \(-0.515426\pi\)
−0.0484422 + 0.998826i \(0.515426\pi\)
\(710\) 4.46373e170 0.0140789
\(711\) 3.87388e172 1.12211
\(712\) −2.92383e172 −0.777879
\(713\) 9.20220e171 0.224892
\(714\) 2.12693e172 0.477542
\(715\) −9.03263e170 −0.0186337
\(716\) −2.30376e172 −0.436717
\(717\) 1.20686e172 0.210256
\(718\) −6.61339e172 −1.05900
\(719\) 1.34032e173 1.97293 0.986466 0.163965i \(-0.0524285\pi\)
0.986466 + 0.163965i \(0.0524285\pi\)
\(720\) −2.11642e172 −0.286408
\(721\) 8.53190e172 1.06160
\(722\) 1.10834e173 1.26816
\(723\) −2.93511e172 −0.308857
\(724\) −1.13328e173 −1.09686
\(725\) −1.90990e173 −1.70044
\(726\) −3.04549e172 −0.249455
\(727\) −5.97341e172 −0.450185 −0.225092 0.974337i \(-0.572268\pi\)
−0.225092 + 0.974337i \(0.572268\pi\)
\(728\) −1.30472e172 −0.0904835
\(729\) −4.74766e172 −0.303015
\(730\) 9.56476e172 0.561877
\(731\) −1.14258e173 −0.617851
\(732\) 2.84373e172 0.141569
\(733\) −5.60300e172 −0.256821 −0.128410 0.991721i \(-0.540987\pi\)
−0.128410 + 0.991721i \(0.540987\pi\)
\(734\) −9.56634e172 −0.403770
\(735\) 5.39250e172 0.209607
\(736\) −7.12910e172 −0.255227
\(737\) −2.64146e173 −0.871086
\(738\) 3.05367e173 0.927711
\(739\) 1.59736e173 0.447109 0.223555 0.974691i \(-0.428234\pi\)
0.223555 + 0.974691i \(0.428234\pi\)
\(740\) 2.28601e172 0.0589601
\(741\) 7.48744e170 0.00177964
\(742\) 1.69075e174 3.70377
\(743\) −1.32421e173 −0.267382 −0.133691 0.991023i \(-0.542683\pi\)
−0.133691 + 0.991023i \(0.542683\pi\)
\(744\) 9.02892e172 0.168063
\(745\) 1.31625e173 0.225883
\(746\) 1.36147e173 0.215431
\(747\) −9.43377e173 −1.37654
\(748\) −1.80313e173 −0.242651
\(749\) −1.39087e173 −0.172638
\(750\) 2.05400e173 0.235176
\(751\) −4.83823e173 −0.511058 −0.255529 0.966801i \(-0.582250\pi\)
−0.255529 + 0.966801i \(0.582250\pi\)
\(752\) −1.28329e174 −1.25068
\(753\) −3.21841e172 −0.0289431
\(754\) −2.90787e173 −0.241329
\(755\) −2.88815e173 −0.221223
\(756\) −1.04715e174 −0.740357
\(757\) −1.03035e173 −0.0672486 −0.0336243 0.999435i \(-0.510705\pi\)
−0.0336243 + 0.999435i \(0.510705\pi\)
\(758\) 4.86183e173 0.292963
\(759\) 1.02980e173 0.0572963
\(760\) 1.20505e172 0.00619133
\(761\) 3.07415e174 1.45866 0.729328 0.684164i \(-0.239833\pi\)
0.729328 + 0.684164i \(0.239833\pi\)
\(762\) −1.32098e174 −0.578924
\(763\) 1.54839e174 0.626826
\(764\) −1.59629e174 −0.596985
\(765\) −3.91563e173 −0.135296
\(766\) 3.38722e174 1.08144
\(767\) 2.60721e173 0.0769230
\(768\) −1.70545e174 −0.465033
\(769\) −2.09468e174 −0.527927 −0.263963 0.964533i \(-0.585030\pi\)
−0.263963 + 0.964533i \(0.585030\pi\)
\(770\) −1.74886e174 −0.407442
\(771\) 9.50252e173 0.204668
\(772\) 2.08572e173 0.0415346
\(773\) 3.16270e173 0.0582370 0.0291185 0.999576i \(-0.490730\pi\)
0.0291185 + 0.999576i \(0.490730\pi\)
\(774\) 6.86940e174 1.16974
\(775\) 5.63104e174 0.886820
\(776\) −4.86150e174 −0.708168
\(777\) −1.71764e174 −0.231452
\(778\) 3.83607e174 0.478212
\(779\) −3.46185e173 −0.0399293
\(780\) 5.73887e172 0.00612495
\(781\) 2.84333e173 0.0280827
\(782\) −1.91401e174 −0.174958
\(783\) 1.45766e175 1.23330
\(784\) −3.45071e175 −2.70263
\(785\) 2.92340e174 0.211969
\(786\) 1.03368e175 0.693939
\(787\) 5.09657e173 0.0316816 0.0158408 0.999875i \(-0.494958\pi\)
0.0158408 + 0.999875i \(0.494958\pi\)
\(788\) 1.42793e174 0.0821998
\(789\) −4.34121e174 −0.231447
\(790\) −8.83456e174 −0.436260
\(791\) −5.28051e175 −2.41545
\(792\) −6.77096e174 −0.286930
\(793\) −1.68653e174 −0.0662163
\(794\) 4.17984e175 1.52061
\(795\) 4.64490e174 0.156591
\(796\) −8.32731e174 −0.260174
\(797\) 4.71982e175 1.36677 0.683387 0.730056i \(-0.260506\pi\)
0.683387 + 0.730056i \(0.260506\pi\)
\(798\) 1.44968e174 0.0389133
\(799\) −2.37424e175 −0.590805
\(800\) −4.36246e175 −1.00644
\(801\) 6.47541e175 1.38516
\(802\) 6.30359e175 1.25037
\(803\) 6.09261e175 1.12076
\(804\) 1.67825e175 0.286329
\(805\) −7.07310e174 −0.111933
\(806\) 8.57340e174 0.125859
\(807\) 3.80167e174 0.0517757
\(808\) −4.90432e174 −0.0619714
\(809\) −4.53797e175 −0.532079 −0.266039 0.963962i \(-0.585715\pi\)
−0.266039 + 0.963962i \(0.585715\pi\)
\(810\) 1.95043e175 0.212220
\(811\) −6.54588e175 −0.661007 −0.330504 0.943805i \(-0.607219\pi\)
−0.330504 + 0.943805i \(0.607219\pi\)
\(812\) −2.14514e176 −2.01055
\(813\) −5.90419e175 −0.513668
\(814\) 3.82179e175 0.308667
\(815\) −4.60141e175 −0.345029
\(816\) −3.73911e175 −0.260324
\(817\) −7.78761e174 −0.0503466
\(818\) 1.72779e176 1.03733
\(819\) 2.88957e175 0.161123
\(820\) −2.65339e175 −0.137424
\(821\) −3.47295e176 −1.67085 −0.835425 0.549604i \(-0.814779\pi\)
−0.835425 + 0.549604i \(0.814779\pi\)
\(822\) 2.81267e175 0.125711
\(823\) 1.97889e176 0.821736 0.410868 0.911695i \(-0.365226\pi\)
0.410868 + 0.911695i \(0.365226\pi\)
\(824\) −7.53321e175 −0.290659
\(825\) 6.30160e175 0.225937
\(826\) 5.04796e176 1.68199
\(827\) 1.70660e176 0.528503 0.264252 0.964454i \(-0.414875\pi\)
0.264252 + 0.964454i \(0.414875\pi\)
\(828\) 4.38447e175 0.126207
\(829\) −1.27004e176 −0.339834 −0.169917 0.985458i \(-0.554350\pi\)
−0.169917 + 0.985458i \(0.554350\pi\)
\(830\) 2.15141e176 0.535179
\(831\) −1.95006e176 −0.451009
\(832\) −6.75919e174 −0.0145356
\(833\) −6.38423e176 −1.27669
\(834\) 3.20145e176 0.595388
\(835\) 1.98789e176 0.343843
\(836\) −1.22899e175 −0.0197728
\(837\) −4.29767e176 −0.643196
\(838\) 1.33216e177 1.85478
\(839\) −1.39765e177 −1.81051 −0.905254 0.424870i \(-0.860320\pi\)
−0.905254 + 0.424870i \(0.860320\pi\)
\(840\) −6.93992e175 −0.0836484
\(841\) 2.09451e177 2.34922
\(842\) −1.55243e177 −1.62043
\(843\) −3.24449e176 −0.315194
\(844\) −5.27606e176 −0.477081
\(845\) 3.12807e176 0.263296
\(846\) 1.42744e177 1.11854
\(847\) 1.33243e177 0.972068
\(848\) −2.97232e177 −2.01905
\(849\) 8.90578e176 0.563322
\(850\) −1.17123e177 −0.689915
\(851\) 1.54569e176 0.0847977
\(852\) −1.80650e175 −0.00923088
\(853\) −3.49520e177 −1.66363 −0.831813 0.555057i \(-0.812697\pi\)
−0.831813 + 0.555057i \(0.812697\pi\)
\(854\) −3.26537e177 −1.44788
\(855\) −2.66883e175 −0.0110248
\(856\) 1.22806e176 0.0472669
\(857\) 4.28472e177 1.53668 0.768339 0.640043i \(-0.221083\pi\)
0.768339 + 0.640043i \(0.221083\pi\)
\(858\) 9.59434e175 0.0320652
\(859\) 1.98589e177 0.618542 0.309271 0.950974i \(-0.399915\pi\)
0.309271 + 0.950974i \(0.399915\pi\)
\(860\) −5.96894e176 −0.173277
\(861\) 1.99368e177 0.539468
\(862\) 3.52061e177 0.888030
\(863\) 6.91411e177 1.62586 0.812930 0.582361i \(-0.197871\pi\)
0.812930 + 0.582361i \(0.197871\pi\)
\(864\) 3.32948e177 0.729953
\(865\) −3.58100e176 −0.0732031
\(866\) −9.06947e177 −1.72882
\(867\) 1.33532e177 0.237373
\(868\) 6.32460e177 1.04855
\(869\) −5.62748e177 −0.870197
\(870\) −1.54672e177 −0.223099
\(871\) −9.95317e176 −0.133925
\(872\) −1.36715e177 −0.171620
\(873\) 1.07668e178 1.26102
\(874\) −1.30456e176 −0.0142568
\(875\) −8.98638e177 −0.916427
\(876\) −3.87093e177 −0.368398
\(877\) −2.22632e178 −1.97748 −0.988740 0.149645i \(-0.952187\pi\)
−0.988740 + 0.149645i \(0.952187\pi\)
\(878\) −1.44030e178 −1.19408
\(879\) −2.61816e177 −0.202614
\(880\) 3.07446e177 0.222110
\(881\) −8.76614e177 −0.591243 −0.295621 0.955305i \(-0.595527\pi\)
−0.295621 + 0.955305i \(0.595527\pi\)
\(882\) 3.83833e178 2.41709
\(883\) −3.56005e177 −0.209331 −0.104666 0.994507i \(-0.533377\pi\)
−0.104666 + 0.994507i \(0.533377\pi\)
\(884\) −6.79431e176 −0.0373064
\(885\) 1.38680e177 0.0711123
\(886\) 1.80617e177 0.0865002
\(887\) −2.22560e178 −0.995562 −0.497781 0.867303i \(-0.665852\pi\)
−0.497781 + 0.867303i \(0.665852\pi\)
\(888\) 1.51658e177 0.0633698
\(889\) 5.77940e178 2.25593
\(890\) −1.47674e178 −0.538530
\(891\) 1.24240e178 0.423310
\(892\) 1.37244e178 0.436939
\(893\) −1.61824e177 −0.0481427
\(894\) −1.39810e178 −0.388705
\(895\) 7.26741e177 0.188837
\(896\) 6.65139e178 1.61539
\(897\) 3.88035e176 0.00880903
\(898\) −3.01960e178 −0.640812
\(899\) −8.80398e178 −1.74670
\(900\) 2.68296e178 0.497671
\(901\) −5.49915e178 −0.953775
\(902\) −4.43599e178 −0.719440
\(903\) 4.48490e178 0.680211
\(904\) 4.66240e178 0.661333
\(905\) 3.57502e178 0.474285
\(906\) 3.06776e178 0.380685
\(907\) 1.67040e178 0.193900 0.0969501 0.995289i \(-0.469091\pi\)
0.0969501 + 0.995289i \(0.469091\pi\)
\(908\) −9.60369e178 −1.04290
\(909\) 1.08616e178 0.110352
\(910\) −6.58979e177 −0.0626422
\(911\) −5.90627e178 −0.525354 −0.262677 0.964884i \(-0.584605\pi\)
−0.262677 + 0.964884i \(0.584605\pi\)
\(912\) −2.54852e177 −0.0212129
\(913\) 1.37042e179 1.06751
\(914\) −2.01614e179 −1.46986
\(915\) −8.97077e177 −0.0612144
\(916\) 1.66345e179 1.06251
\(917\) −4.52242e179 −2.70412
\(918\) 8.93893e178 0.500384
\(919\) 1.92508e179 1.00893 0.504464 0.863433i \(-0.331690\pi\)
0.504464 + 0.863433i \(0.331690\pi\)
\(920\) 6.24517e177 0.0306465
\(921\) −6.22881e178 −0.286218
\(922\) 4.66192e178 0.200606
\(923\) 1.07138e177 0.00431759
\(924\) 7.07775e178 0.267141
\(925\) 9.45843e178 0.334383
\(926\) −2.20749e179 −0.731028
\(927\) 1.66838e179 0.517573
\(928\) 6.82059e179 1.98230
\(929\) 1.82038e179 0.495692 0.247846 0.968799i \(-0.420277\pi\)
0.247846 + 0.968799i \(0.420277\pi\)
\(930\) 4.56026e178 0.116351
\(931\) −4.35139e178 −0.104033
\(932\) 2.49108e179 0.558116
\(933\) −1.68086e179 −0.352933
\(934\) −1.98517e179 −0.390671
\(935\) 5.68813e178 0.104922
\(936\) −2.55133e178 −0.0441142
\(937\) 8.01956e179 1.29989 0.649943 0.759983i \(-0.274793\pi\)
0.649943 + 0.759983i \(0.274793\pi\)
\(938\) −1.92709e180 −2.92839
\(939\) 1.61464e179 0.230042
\(940\) −1.24033e179 −0.165692
\(941\) −7.73694e179 −0.969163 −0.484582 0.874746i \(-0.661028\pi\)
−0.484582 + 0.874746i \(0.661028\pi\)
\(942\) −3.10519e179 −0.364762
\(943\) −1.79410e179 −0.197646
\(944\) −8.87424e179 −0.916907
\(945\) 3.30333e179 0.320131
\(946\) −9.97899e179 −0.907137
\(947\) 7.33813e179 0.625766 0.312883 0.949792i \(-0.398705\pi\)
0.312883 + 0.949792i \(0.398705\pi\)
\(948\) 3.57541e179 0.286037
\(949\) 2.29573e179 0.172312
\(950\) −7.98288e178 −0.0562188
\(951\) −2.38028e179 −0.157292
\(952\) 8.21624e179 0.509493
\(953\) 2.98138e180 1.73499 0.867494 0.497447i \(-0.165729\pi\)
0.867494 + 0.497447i \(0.165729\pi\)
\(954\) 3.30620e180 1.80573
\(955\) 5.03562e179 0.258136
\(956\) −7.46429e179 −0.359158
\(957\) −9.85239e179 −0.445009
\(958\) −1.08207e180 −0.458818
\(959\) −1.23056e180 −0.489867
\(960\) −3.59527e178 −0.0134376
\(961\) −2.53765e179 −0.0890568
\(962\) 1.44007e179 0.0474561
\(963\) −2.71979e179 −0.0841676
\(964\) 1.81533e180 0.527589
\(965\) −6.57958e178 −0.0179596
\(966\) 7.51296e179 0.192617
\(967\) −2.21131e180 −0.532536 −0.266268 0.963899i \(-0.585791\pi\)
−0.266268 + 0.963899i \(0.585791\pi\)
\(968\) −1.17646e180 −0.266145
\(969\) −4.71507e178 −0.0100207
\(970\) −2.45541e180 −0.490268
\(971\) 5.37812e180 1.00894 0.504472 0.863428i \(-0.331687\pi\)
0.504472 + 0.863428i \(0.331687\pi\)
\(972\) −3.14259e180 −0.553960
\(973\) −1.40066e181 −2.32009
\(974\) 9.48448e180 1.47638
\(975\) 2.37448e179 0.0347367
\(976\) 5.74048e180 0.789285
\(977\) −3.49022e180 −0.451058 −0.225529 0.974236i \(-0.572411\pi\)
−0.225529 + 0.974236i \(0.572411\pi\)
\(978\) 4.88756e180 0.593733
\(979\) −9.40664e180 −1.07419
\(980\) −3.33519e180 −0.358050
\(981\) 3.02783e180 0.305602
\(982\) 1.78354e180 0.169254
\(983\) −1.07879e181 −0.962610 −0.481305 0.876553i \(-0.659837\pi\)
−0.481305 + 0.876553i \(0.659837\pi\)
\(984\) −1.76031e180 −0.147702
\(985\) −4.50452e179 −0.0355432
\(986\) 1.83118e181 1.35887
\(987\) 9.31949e180 0.650435
\(988\) −4.63089e178 −0.00303997
\(989\) −4.03592e180 −0.249211
\(990\) −3.41982e180 −0.198643
\(991\) 7.09661e180 0.387789 0.193894 0.981022i \(-0.437888\pi\)
0.193894 + 0.981022i \(0.437888\pi\)
\(992\) −2.01094e181 −1.03382
\(993\) 9.78918e180 0.473496
\(994\) 2.07436e180 0.0944077
\(995\) 2.62692e180 0.112499
\(996\) −8.70693e180 −0.350893
\(997\) 3.75837e181 1.42542 0.712709 0.701460i \(-0.247468\pi\)
0.712709 + 0.701460i \(0.247468\pi\)
\(998\) 5.04629e181 1.80125
\(999\) −7.21879e180 −0.242523
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 1.122.a.a.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.122.a.a.1.2 9 1.1 even 1 trivial