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Properties

Label 1.118.a.a.1.5
Level $1$
Weight $118$
Character 1.1
Self dual yes
Analytic conductor $86.689$
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] = [1,118,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, 118, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 118);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 118 \)
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: \(86.6887159558\)
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: \( x^{9} - 4 x^{8} + \cdots - 93\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{151}\cdot 3^{56}\cdot 5^{18}\cdot 7^{7}\cdot 11^{4}\cdot 13^{4}\cdot 17^{2}\cdot 19\cdot 23\cdot 29^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(4.58648e14\) 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)
 
\(f(q)\) \(=\) \(q-1.27600e17 q^{2} +8.60094e27 q^{3} -1.49872e35 q^{4} +5.94769e40 q^{5} -1.09748e45 q^{6} +4.46500e49 q^{7} +4.03248e52 q^{8} +7.42032e54 q^{9} +O(q^{10})\) \(q-1.27600e17 q^{2} +8.60094e27 q^{3} -1.49872e35 q^{4} +5.94769e40 q^{5} -1.09748e45 q^{6} +4.46500e49 q^{7} +4.03248e52 q^{8} +7.42032e54 q^{9} -7.58924e57 q^{10} -4.86664e60 q^{11} -1.28904e63 q^{12} -1.65151e65 q^{13} -5.69734e66 q^{14} +5.11557e68 q^{15} +1.97563e70 q^{16} -1.52908e72 q^{17} -9.46831e71 q^{18} -2.16339e74 q^{19} -8.91391e75 q^{20} +3.84032e77 q^{21} +6.20982e77 q^{22} +3.41267e79 q^{23} +3.46831e80 q^{24} -2.48103e81 q^{25} +2.10733e82 q^{26} -5.08622e83 q^{27} -6.69178e84 q^{28} +2.34152e85 q^{29} -6.52746e85 q^{30} -2.97270e87 q^{31} -9.22100e87 q^{32} -4.18577e88 q^{33} +1.95110e89 q^{34} +2.65564e90 q^{35} -1.11210e90 q^{36} +2.20993e91 q^{37} +2.76048e91 q^{38} -1.42046e93 q^{39} +2.39839e93 q^{40} -1.03659e93 q^{41} -4.90025e94 q^{42} +5.21509e95 q^{43} +7.29372e95 q^{44} +4.41337e95 q^{45} -4.35457e96 q^{46} -6.36480e97 q^{47} +1.69923e98 q^{48} +1.24119e99 q^{49} +3.16579e98 q^{50} -1.31515e100 q^{51} +2.47515e100 q^{52} -9.00098e100 q^{53} +6.49001e100 q^{54} -2.89453e101 q^{55} +1.80050e102 q^{56} -1.86072e102 q^{57} -2.98778e102 q^{58} +2.04546e103 q^{59} -7.66680e103 q^{60} +1.30538e104 q^{61} +3.79316e104 q^{62} +3.31317e104 q^{63} -2.10598e105 q^{64} -9.82267e105 q^{65} +5.34104e105 q^{66} -5.10781e106 q^{67} +2.29166e107 q^{68} +2.93522e107 q^{69} -3.38860e107 q^{70} -2.64184e108 q^{71} +2.99223e107 q^{72} +8.88066e108 q^{73} -2.81986e108 q^{74} -2.13392e109 q^{75} +3.24231e109 q^{76} -2.17296e110 q^{77} +1.81250e110 q^{78} -5.52153e109 q^{79} +1.17504e111 q^{80} -4.86850e111 q^{81} +1.32269e110 q^{82} +1.11518e112 q^{83} -5.75556e112 q^{84} -9.09449e112 q^{85} -6.65445e112 q^{86} +2.01393e113 q^{87} -1.96246e113 q^{88} -1.64905e114 q^{89} -5.63146e112 q^{90} -7.37400e114 q^{91} -5.11463e114 q^{92} -2.55680e115 q^{93} +8.12147e114 q^{94} -1.28672e115 q^{95} -7.93093e115 q^{96} -2.61008e116 q^{97} -1.58375e116 q^{98} -3.61120e115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 40\!\cdots\!52 q^{2}+ \cdots + 22\!\cdots\!97 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 40\!\cdots\!52 q^{2}+ \cdots - 15\!\cdots\!16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27600e17 −0.313037 −0.156518 − 0.987675i \(-0.550027\pi\)
−0.156518 + 0.987675i \(0.550027\pi\)
\(3\) 8.60094e27 1.05427 0.527136 − 0.849781i \(-0.323266\pi\)
0.527136 + 0.849781i \(0.323266\pi\)
\(4\) −1.49872e35 −0.902008
\(5\) 5.94769e40 0.766660 0.383330 − 0.923611i \(-0.374777\pi\)
0.383330 + 0.923611i \(0.374777\pi\)
\(6\) −1.09748e45 −0.330026
\(7\) 4.46500e49 1.62775 0.813873 − 0.581043i \(-0.197355\pi\)
0.813873 + 0.581043i \(0.197355\pi\)
\(8\) 4.03248e52 0.595399
\(9\) 7.42032e54 0.111490
\(10\) −7.58924e57 −0.239993
\(11\) −4.86664e60 −0.583119 −0.291559 − 0.956553i \(-0.594174\pi\)
−0.291559 + 0.956553i \(0.594174\pi\)
\(12\) −1.28904e63 −0.950962
\(13\) −1.65151e65 −1.12770 −0.563852 − 0.825876i \(-0.690681\pi\)
−0.563852 + 0.825876i \(0.690681\pi\)
\(14\) −5.69734e66 −0.509544
\(15\) 5.11557e68 0.808269
\(16\) 1.97563e70 0.715626
\(17\) −1.52908e72 −1.59650 −0.798249 − 0.602328i \(-0.794240\pi\)
−0.798249 + 0.602328i \(0.794240\pi\)
\(18\) −9.46831e71 −0.0349005
\(19\) −2.16339e74 −0.337325 −0.168663 − 0.985674i \(-0.553945\pi\)
−0.168663 + 0.985674i \(0.553945\pi\)
\(20\) −8.91391e75 −0.691534
\(21\) 3.84032e77 1.71609
\(22\) 6.20982e77 0.182538
\(23\) 3.41267e79 0.744760 0.372380 − 0.928080i \(-0.378542\pi\)
0.372380 + 0.928080i \(0.378542\pi\)
\(24\) 3.46831e80 0.627712
\(25\) −2.48103e81 −0.412232
\(26\) 2.10733e82 0.353013
\(27\) −5.08622e83 −0.936731
\(28\) −6.69178e84 −1.46824
\(29\) 2.34152e85 0.659501 0.329751 − 0.944068i \(-0.393035\pi\)
0.329751 + 0.944068i \(0.393035\pi\)
\(30\) −6.52746e85 −0.253018
\(31\) −2.97270e87 −1.69236 −0.846178 − 0.532900i \(-0.821102\pi\)
−0.846178 + 0.532900i \(0.821102\pi\)
\(32\) −9.22100e87 −0.819416
\(33\) −4.18577e88 −0.614766
\(34\) 1.95110e89 0.499762
\(35\) 2.65564e90 1.24793
\(36\) −1.11210e90 −0.100565
\(37\) 2.20993e91 0.402325 0.201163 − 0.979558i \(-0.435528\pi\)
0.201163 + 0.979558i \(0.435528\pi\)
\(38\) 2.76048e91 0.105595
\(39\) −1.42046e93 −1.18891
\(40\) 2.39839e93 0.456468
\(41\) −1.03659e93 −0.0465321 −0.0232661 − 0.999729i \(-0.507406\pi\)
−0.0232661 + 0.999729i \(0.507406\pi\)
\(42\) −4.90025e94 −0.537198
\(43\) 5.21509e95 1.44330 0.721651 − 0.692257i \(-0.243383\pi\)
0.721651 + 0.692257i \(0.243383\pi\)
\(44\) 7.29372e95 0.525978
\(45\) 4.41337e95 0.0854749
\(46\) −4.35457e96 −0.233137
\(47\) −6.36480e97 −0.968412 −0.484206 − 0.874954i \(-0.660891\pi\)
−0.484206 + 0.874954i \(0.660891\pi\)
\(48\) 1.69923e98 0.754465
\(49\) 1.24119e99 1.64955
\(50\) 3.16579e98 0.129044
\(51\) −1.31515e100 −1.68314
\(52\) 2.47515e100 1.01720
\(53\) −9.00098e100 −1.21381 −0.606906 − 0.794774i \(-0.707589\pi\)
−0.606906 + 0.794774i \(0.707589\pi\)
\(54\) 6.49001e100 0.293231
\(55\) −2.89453e101 −0.447054
\(56\) 1.80050e102 0.969157
\(57\) −1.86072e102 −0.355633
\(58\) −2.98778e102 −0.206448
\(59\) 2.04546e103 0.519934 0.259967 − 0.965617i \(-0.416288\pi\)
0.259967 + 0.965617i \(0.416288\pi\)
\(60\) −7.66680e103 −0.729065
\(61\) 1.30538e104 0.471998 0.235999 − 0.971753i \(-0.424164\pi\)
0.235999 + 0.971753i \(0.424164\pi\)
\(62\) 3.79316e104 0.529770
\(63\) 3.31317e104 0.181477
\(64\) −2.10598e105 −0.459119
\(65\) −9.82267e105 −0.864566
\(66\) 5.34104e105 0.192444
\(67\) −5.10781e106 −0.763588 −0.381794 − 0.924247i \(-0.624694\pi\)
−0.381794 + 0.924247i \(0.624694\pi\)
\(68\) 2.29166e107 1.44005
\(69\) 2.93522e107 0.785180
\(70\) −3.38860e107 −0.390647
\(71\) −2.64184e108 −1.32829 −0.664145 − 0.747604i \(-0.731204\pi\)
−0.664145 + 0.747604i \(0.731204\pi\)
\(72\) 2.99223e107 0.0663809
\(73\) 8.88066e108 0.879140 0.439570 − 0.898208i \(-0.355131\pi\)
0.439570 + 0.898208i \(0.355131\pi\)
\(74\) −2.81986e108 −0.125943
\(75\) −2.13392e109 −0.434605
\(76\) 3.24231e109 0.304270
\(77\) −2.17296e110 −0.949169
\(78\) 1.81250e110 0.372172
\(79\) −5.52153e109 −0.0538115 −0.0269057 − 0.999638i \(-0.508565\pi\)
−0.0269057 + 0.999638i \(0.508565\pi\)
\(80\) 1.17504e111 0.548642
\(81\) −4.86850e111 −1.09906
\(82\) 1.32269e110 0.0145663
\(83\) 1.11518e112 0.604335 0.302167 − 0.953255i \(-0.402290\pi\)
0.302167 + 0.953255i \(0.402290\pi\)
\(84\) −5.75556e112 −1.54792
\(85\) −9.09449e112 −1.22397
\(86\) −6.65445e112 −0.451807
\(87\) 2.01393e113 0.695294
\(88\) −1.96246e113 −0.347188
\(89\) −1.64905e114 −1.50632 −0.753162 − 0.657835i \(-0.771473\pi\)
−0.753162 + 0.657835i \(0.771473\pi\)
\(90\) −5.63146e112 −0.0267568
\(91\) −7.37400e114 −1.83562
\(92\) −5.11463e114 −0.671779
\(93\) −2.55680e115 −1.78420
\(94\) 8.12147e114 0.303148
\(95\) −1.28672e115 −0.258614
\(96\) −7.93093e115 −0.863887
\(97\) −2.61008e116 −1.55063 −0.775315 − 0.631575i \(-0.782409\pi\)
−0.775315 + 0.631575i \(0.782409\pi\)
\(98\) −1.58375e116 −0.516371
\(99\) −3.61120e115 −0.0650119
\(100\) 3.71837e116 0.371837
\(101\) −7.43804e115 −0.0415584 −0.0207792 − 0.999784i \(-0.506615\pi\)
−0.0207792 + 0.999784i \(0.506615\pi\)
\(102\) 1.67813e117 0.526886
\(103\) −1.30805e117 −0.232084 −0.116042 − 0.993244i \(-0.537021\pi\)
−0.116042 + 0.993244i \(0.537021\pi\)
\(104\) −6.65968e117 −0.671434
\(105\) 2.28411e118 1.31566
\(106\) 1.14852e118 0.379968
\(107\) 9.74210e118 1.86081 0.930404 − 0.366535i \(-0.119456\pi\)
0.930404 + 0.366535i \(0.119456\pi\)
\(108\) 7.62281e118 0.844939
\(109\) −3.52126e118 −0.227640 −0.113820 − 0.993501i \(-0.536309\pi\)
−0.113820 + 0.993501i \(0.536309\pi\)
\(110\) 3.69341e118 0.139944
\(111\) 1.90075e119 0.424160
\(112\) 8.82119e119 1.16486
\(113\) −2.32313e120 −1.82383 −0.911914 − 0.410382i \(-0.865395\pi\)
−0.911914 + 0.410382i \(0.865395\pi\)
\(114\) 2.37428e119 0.111326
\(115\) 2.02975e120 0.570978
\(116\) −3.50928e120 −0.594876
\(117\) −1.22547e120 −0.125728
\(118\) −2.61001e120 −0.162759
\(119\) −6.82734e121 −2.59869
\(120\) 2.06284e121 0.481242
\(121\) −4.59695e121 −0.659972
\(122\) −1.66566e121 −0.147753
\(123\) −8.91568e120 −0.0490575
\(124\) 4.45524e122 1.52652
\(125\) −5.05527e122 −1.08270
\(126\) −4.22760e121 −0.0568090
\(127\) 5.52156e122 0.467244 0.233622 − 0.972327i \(-0.424942\pi\)
0.233622 + 0.972327i \(0.424942\pi\)
\(128\) 1.80082e123 0.963137
\(129\) 4.48547e123 1.52163
\(130\) 1.25337e123 0.270641
\(131\) −2.60150e123 −0.358800 −0.179400 − 0.983776i \(-0.557416\pi\)
−0.179400 + 0.983776i \(0.557416\pi\)
\(132\) 6.27329e123 0.554524
\(133\) −9.65954e123 −0.549080
\(134\) 6.51756e123 0.239031
\(135\) −3.02513e124 −0.718155
\(136\) −6.16598e124 −0.950552
\(137\) 1.70684e125 1.71412 0.857059 − 0.515218i \(-0.172289\pi\)
0.857059 + 0.515218i \(0.172289\pi\)
\(138\) −3.74534e124 −0.245790
\(139\) −3.59581e125 −1.54679 −0.773394 − 0.633926i \(-0.781443\pi\)
−0.773394 + 0.633926i \(0.781443\pi\)
\(140\) −3.98006e125 −1.12564
\(141\) −5.47433e125 −1.02097
\(142\) 3.37098e125 0.415804
\(143\) 8.03731e125 0.657586
\(144\) 1.46598e125 0.0797851
\(145\) 1.39267e126 0.505614
\(146\) −1.13317e126 −0.275203
\(147\) 1.06754e127 1.73908
\(148\) −3.31206e126 −0.362901
\(149\) 5.42507e126 0.400875 0.200437 − 0.979707i \(-0.435764\pi\)
0.200437 + 0.979707i \(0.435764\pi\)
\(150\) 2.72288e126 0.136047
\(151\) 1.96760e127 0.666479 0.333239 − 0.942842i \(-0.391858\pi\)
0.333239 + 0.942842i \(0.391858\pi\)
\(152\) −8.72382e126 −0.200843
\(153\) −1.13463e127 −0.177993
\(154\) 2.77269e127 0.297125
\(155\) −1.76807e128 −1.29746
\(156\) 2.12886e128 1.07240
\(157\) −4.36847e128 −1.51426 −0.757128 − 0.653267i \(-0.773398\pi\)
−0.757128 + 0.653267i \(0.773398\pi\)
\(158\) 7.04547e126 0.0168450
\(159\) −7.74169e128 −1.27969
\(160\) −5.48436e128 −0.628214
\(161\) 1.52376e129 1.21228
\(162\) 6.21220e128 0.344046
\(163\) −3.54862e128 −0.137115 −0.0685574 − 0.997647i \(-0.521840\pi\)
−0.0685574 + 0.997647i \(0.521840\pi\)
\(164\) 1.55356e128 0.0419723
\(165\) −2.48957e129 −0.471317
\(166\) −1.42297e129 −0.189179
\(167\) 1.90004e130 1.77766 0.888831 − 0.458236i \(-0.151519\pi\)
0.888831 + 0.458236i \(0.151519\pi\)
\(168\) 1.54860e130 1.02176
\(169\) 5.82761e129 0.271718
\(170\) 1.16046e130 0.383148
\(171\) −1.60530e129 −0.0376084
\(172\) −7.81595e130 −1.30187
\(173\) −2.60257e130 −0.308819 −0.154410 − 0.988007i \(-0.549348\pi\)
−0.154410 + 0.988007i \(0.549348\pi\)
\(174\) −2.56977e130 −0.217653
\(175\) −1.10778e131 −0.671009
\(176\) −9.61467e130 −0.417295
\(177\) 1.75929e131 0.548152
\(178\) 2.10419e131 0.471535
\(179\) −6.85299e131 −1.10657 −0.553283 − 0.832993i \(-0.686625\pi\)
−0.553283 + 0.832993i \(0.686625\pi\)
\(180\) −6.61440e130 −0.0770990
\(181\) −1.03900e132 −0.875824 −0.437912 − 0.899018i \(-0.644282\pi\)
−0.437912 + 0.899018i \(0.644282\pi\)
\(182\) 9.40922e131 0.574615
\(183\) 1.12275e132 0.497614
\(184\) 1.37615e132 0.443429
\(185\) 1.31440e132 0.308447
\(186\) 3.26247e132 0.558521
\(187\) 7.44148e132 0.930948
\(188\) 9.53904e132 0.873515
\(189\) −2.27100e133 −1.52476
\(190\) 1.64185e132 0.0809557
\(191\) 7.05832e132 0.256005 0.128003 − 0.991774i \(-0.459143\pi\)
0.128003 + 0.991774i \(0.459143\pi\)
\(192\) −1.81134e133 −0.484036
\(193\) −1.08069e133 −0.213108 −0.106554 − 0.994307i \(-0.533982\pi\)
−0.106554 + 0.994307i \(0.533982\pi\)
\(194\) 3.33046e133 0.485404
\(195\) −8.44843e133 −0.911488
\(196\) −1.86019e134 −1.48791
\(197\) −3.89602e133 −0.231391 −0.115696 − 0.993285i \(-0.536910\pi\)
−0.115696 + 0.993285i \(0.536910\pi\)
\(198\) 4.60789e132 0.0203511
\(199\) 4.22902e134 1.39102 0.695512 − 0.718514i \(-0.255177\pi\)
0.695512 + 0.718514i \(0.255177\pi\)
\(200\) −1.00047e134 −0.245442
\(201\) −4.39320e134 −0.805029
\(202\) 9.49093e132 0.0130093
\(203\) 1.04549e135 1.07350
\(204\) 1.97104e135 1.51821
\(205\) −6.16533e133 −0.0356743
\(206\) 1.66907e134 0.0726509
\(207\) 2.53231e134 0.0830332
\(208\) −3.26277e135 −0.807015
\(209\) 1.05284e135 0.196701
\(210\) −2.91451e135 −0.411849
\(211\) 3.66071e134 0.0391779 0.0195890 − 0.999808i \(-0.493764\pi\)
0.0195890 + 0.999808i \(0.493764\pi\)
\(212\) 1.34899e136 1.09487
\(213\) −2.27223e136 −1.40038
\(214\) −1.24309e136 −0.582502
\(215\) 3.10177e136 1.10652
\(216\) −2.05101e136 −0.557729
\(217\) −1.32731e137 −2.75472
\(218\) 4.49312e135 0.0712596
\(219\) 7.63821e136 0.926853
\(220\) 4.33808e136 0.403246
\(221\) 2.52529e137 1.80038
\(222\) −2.42535e136 −0.132778
\(223\) 3.41526e137 1.43744 0.718720 − 0.695300i \(-0.244729\pi\)
0.718720 + 0.695300i \(0.244729\pi\)
\(224\) −4.11718e137 −1.33380
\(225\) −1.84100e136 −0.0459597
\(226\) 2.96430e137 0.570925
\(227\) −5.80709e136 −0.0863865 −0.0431933 − 0.999067i \(-0.513753\pi\)
−0.0431933 + 0.999067i \(0.513753\pi\)
\(228\) 2.78869e137 0.320784
\(229\) −5.68693e136 −0.0506410 −0.0253205 − 0.999679i \(-0.508061\pi\)
−0.0253205 + 0.999679i \(0.508061\pi\)
\(230\) −2.58996e137 −0.178737
\(231\) −1.86895e138 −1.00068
\(232\) 9.44214e137 0.392666
\(233\) −2.51348e138 −0.812745 −0.406373 − 0.913707i \(-0.633207\pi\)
−0.406373 + 0.913707i \(0.633207\pi\)
\(234\) 1.56370e137 0.0393574
\(235\) −3.78558e138 −0.742443
\(236\) −3.06557e138 −0.468985
\(237\) −4.74904e137 −0.0567319
\(238\) 8.71168e138 0.813486
\(239\) 6.15130e137 0.0449459 0.0224729 − 0.999747i \(-0.492846\pi\)
0.0224729 + 0.999747i \(0.492846\pi\)
\(240\) 1.01065e139 0.578418
\(241\) 1.22086e139 0.547861 0.273930 − 0.961750i \(-0.411676\pi\)
0.273930 + 0.961750i \(0.411676\pi\)
\(242\) 5.86570e138 0.206596
\(243\) −8.02185e138 −0.221977
\(244\) −1.95639e139 −0.425746
\(245\) 7.38220e139 1.26465
\(246\) 1.13764e138 0.0153568
\(247\) 3.57286e139 0.380404
\(248\) −1.19873e140 −1.00763
\(249\) 9.59162e139 0.637133
\(250\) 6.45052e139 0.338926
\(251\) −3.44309e140 −1.43231 −0.716153 − 0.697943i \(-0.754099\pi\)
−0.716153 + 0.697943i \(0.754099\pi\)
\(252\) −4.96551e139 −0.163694
\(253\) −1.66082e140 −0.434284
\(254\) −7.04550e139 −0.146265
\(255\) −7.82212e140 −1.29040
\(256\) 1.20131e140 0.157622
\(257\) 8.22624e140 0.859239 0.429619 − 0.903010i \(-0.358648\pi\)
0.429619 + 0.903010i \(0.358648\pi\)
\(258\) −5.72346e140 −0.476327
\(259\) 9.86733e140 0.654883
\(260\) 1.47214e141 0.779846
\(261\) 1.73748e140 0.0735278
\(262\) 3.31950e140 0.112318
\(263\) 6.97359e141 1.88819 0.944094 − 0.329678i \(-0.106940\pi\)
0.944094 + 0.329678i \(0.106940\pi\)
\(264\) −1.68790e141 −0.366031
\(265\) −5.35350e141 −0.930581
\(266\) 1.23256e141 0.171882
\(267\) −1.41834e142 −1.58808
\(268\) 7.65517e141 0.688762
\(269\) 8.44905e141 0.611363 0.305682 − 0.952134i \(-0.401116\pi\)
0.305682 + 0.952134i \(0.401116\pi\)
\(270\) 3.86006e141 0.224809
\(271\) 2.31110e142 1.08422 0.542108 − 0.840309i \(-0.317626\pi\)
0.542108 + 0.840309i \(0.317626\pi\)
\(272\) −3.02089e142 −1.14250
\(273\) −6.34234e142 −1.93524
\(274\) −2.17793e142 −0.536582
\(275\) 1.20743e142 0.240380
\(276\) −4.39907e142 −0.708238
\(277\) 7.27581e142 0.948014 0.474007 − 0.880521i \(-0.342807\pi\)
0.474007 + 0.880521i \(0.342807\pi\)
\(278\) 4.58825e142 0.484202
\(279\) −2.20584e142 −0.188681
\(280\) 1.07088e143 0.743014
\(281\) 1.14573e143 0.645303 0.322652 − 0.946518i \(-0.395426\pi\)
0.322652 + 0.946518i \(0.395426\pi\)
\(282\) 6.98523e142 0.319601
\(283\) −2.33024e143 −0.866752 −0.433376 − 0.901213i \(-0.642678\pi\)
−0.433376 + 0.901213i \(0.642678\pi\)
\(284\) 3.95937e143 1.19813
\(285\) −1.10670e143 −0.272650
\(286\) −1.02556e143 −0.205849
\(287\) −4.62839e142 −0.0757424
\(288\) −6.84227e142 −0.0913566
\(289\) 1.42076e144 1.54880
\(290\) −1.77704e143 −0.158276
\(291\) −2.24492e144 −1.63479
\(292\) −1.33096e144 −0.792991
\(293\) −1.53099e144 −0.746823 −0.373411 − 0.927666i \(-0.621812\pi\)
−0.373411 + 0.927666i \(0.621812\pi\)
\(294\) −1.36218e144 −0.544396
\(295\) 1.21658e144 0.398613
\(296\) 8.91148e143 0.239544
\(297\) 2.47528e144 0.546226
\(298\) −6.92238e143 −0.125489
\(299\) −5.63607e144 −0.839869
\(300\) 3.19815e144 0.392017
\(301\) 2.32854e145 2.34933
\(302\) −2.51066e144 −0.208632
\(303\) −6.39742e143 −0.0438138
\(304\) −4.27405e144 −0.241399
\(305\) 7.76396e144 0.361862
\(306\) 1.44778e144 0.0557185
\(307\) −3.81736e145 −1.21386 −0.606929 − 0.794756i \(-0.707599\pi\)
−0.606929 + 0.794756i \(0.707599\pi\)
\(308\) 3.25665e145 0.856158
\(309\) −1.12504e145 −0.244680
\(310\) 2.25605e145 0.406153
\(311\) 3.98834e145 0.594716 0.297358 − 0.954766i \(-0.403894\pi\)
0.297358 + 0.954766i \(0.403894\pi\)
\(312\) −5.72796e145 −0.707874
\(313\) −1.88336e146 −1.93014 −0.965070 − 0.261990i \(-0.915621\pi\)
−0.965070 + 0.261990i \(0.915621\pi\)
\(314\) 5.57416e145 0.474018
\(315\) 1.97057e145 0.139131
\(316\) 8.27522e144 0.0485384
\(317\) −2.25718e146 −1.10052 −0.550262 − 0.834992i \(-0.685472\pi\)
−0.550262 + 0.834992i \(0.685472\pi\)
\(318\) 9.87838e145 0.400589
\(319\) −1.13954e146 −0.384568
\(320\) −1.25257e146 −0.351988
\(321\) 8.37913e146 1.96180
\(322\) −1.94432e146 −0.379488
\(323\) 3.30799e146 0.538539
\(324\) 7.29650e146 0.991361
\(325\) 4.09745e146 0.464876
\(326\) 4.52803e145 0.0429220
\(327\) −3.02861e146 −0.239994
\(328\) −4.18004e145 −0.0277051
\(329\) −2.84188e147 −1.57633
\(330\) 3.17668e146 0.147539
\(331\) 2.10008e147 0.817143 0.408572 − 0.912726i \(-0.366027\pi\)
0.408572 + 0.912726i \(0.366027\pi\)
\(332\) −1.67134e147 −0.545115
\(333\) 1.63984e146 0.0448552
\(334\) −2.42445e147 −0.556473
\(335\) −3.03797e147 −0.585412
\(336\) 7.58705e147 1.22808
\(337\) −9.50222e147 −1.29263 −0.646317 − 0.763069i \(-0.723692\pi\)
−0.646317 + 0.763069i \(0.723692\pi\)
\(338\) −7.43602e146 −0.0850577
\(339\) −1.99811e148 −1.92281
\(340\) 1.36301e148 1.10403
\(341\) 1.44670e148 0.986845
\(342\) 2.04837e146 0.0117728
\(343\) 2.18227e148 1.05731
\(344\) 2.10297e148 0.859340
\(345\) 1.74578e148 0.601966
\(346\) 3.32087e147 0.0966718
\(347\) −3.34873e148 −0.823391 −0.411696 − 0.911321i \(-0.635063\pi\)
−0.411696 + 0.911321i \(0.635063\pi\)
\(348\) −3.01832e148 −0.627161
\(349\) −4.09865e148 −0.720034 −0.360017 − 0.932946i \(-0.617229\pi\)
−0.360017 + 0.932946i \(0.617229\pi\)
\(350\) 1.41353e148 0.210050
\(351\) 8.39995e148 1.05636
\(352\) 4.48753e148 0.477817
\(353\) −5.46934e148 −0.493304 −0.246652 − 0.969104i \(-0.579331\pi\)
−0.246652 + 0.969104i \(0.579331\pi\)
\(354\) −2.24485e148 −0.171592
\(355\) −1.57128e149 −1.01835
\(356\) 2.47146e149 1.35872
\(357\) −5.87216e149 −2.73973
\(358\) 8.74440e148 0.346396
\(359\) 1.56908e149 0.527983 0.263991 − 0.964525i \(-0.414961\pi\)
0.263991 + 0.964525i \(0.414961\pi\)
\(360\) 1.77968e148 0.0508916
\(361\) −3.64509e149 −0.886212
\(362\) 1.32576e149 0.274165
\(363\) −3.95381e149 −0.695791
\(364\) 1.10515e150 1.65574
\(365\) 5.28194e149 0.674002
\(366\) −1.43262e149 −0.155772
\(367\) 1.17671e150 1.09070 0.545350 − 0.838208i \(-0.316397\pi\)
0.545350 + 0.838208i \(0.316397\pi\)
\(368\) 6.74217e149 0.532970
\(369\) −7.69185e147 −0.00518786
\(370\) −1.67717e149 −0.0965552
\(371\) −4.01894e150 −1.97578
\(372\) 3.83192e150 1.60937
\(373\) 5.78124e149 0.207517 0.103759 − 0.994603i \(-0.466913\pi\)
0.103759 + 0.994603i \(0.466913\pi\)
\(374\) −9.49531e149 −0.291421
\(375\) −4.34801e150 −1.14146
\(376\) −2.56659e150 −0.576591
\(377\) −3.86705e150 −0.743723
\(378\) 2.89779e150 0.477306
\(379\) −4.25346e149 −0.0600273 −0.0300136 − 0.999549i \(-0.509555\pi\)
−0.0300136 + 0.999549i \(0.509555\pi\)
\(380\) 1.92843e150 0.233272
\(381\) 4.74906e150 0.492603
\(382\) −9.00640e149 −0.0801391
\(383\) −1.72329e151 −1.31592 −0.657961 − 0.753052i \(-0.728581\pi\)
−0.657961 + 0.753052i \(0.728581\pi\)
\(384\) 1.54888e151 1.01541
\(385\) −1.29241e151 −0.727690
\(386\) 1.37896e150 0.0667106
\(387\) 3.86976e150 0.160914
\(388\) 3.91178e151 1.39868
\(389\) 3.05346e151 0.939162 0.469581 − 0.882889i \(-0.344405\pi\)
0.469581 + 0.882889i \(0.344405\pi\)
\(390\) 1.07802e151 0.285329
\(391\) −5.21825e151 −1.18901
\(392\) 5.00506e151 0.982142
\(393\) −2.23753e151 −0.378273
\(394\) 4.97132e150 0.0724340
\(395\) −3.28404e150 −0.0412551
\(396\) 5.41217e150 0.0586412
\(397\) −5.45839e151 −0.510294 −0.255147 − 0.966902i \(-0.582124\pi\)
−0.255147 + 0.966902i \(0.582124\pi\)
\(398\) −5.39622e151 −0.435442
\(399\) −8.30812e151 −0.578880
\(400\) −4.90160e151 −0.295004
\(401\) −2.17715e152 −1.13225 −0.566125 − 0.824319i \(-0.691558\pi\)
−0.566125 + 0.824319i \(0.691558\pi\)
\(402\) 5.60572e151 0.252004
\(403\) 4.90944e152 1.90848
\(404\) 1.11475e151 0.0374860
\(405\) −2.89563e152 −0.842606
\(406\) −1.33405e152 −0.336045
\(407\) −1.07549e152 −0.234603
\(408\) −5.30332e152 −1.00214
\(409\) 4.75051e152 0.777906 0.388953 − 0.921258i \(-0.372837\pi\)
0.388953 + 0.921258i \(0.372837\pi\)
\(410\) 7.86695e150 0.0111674
\(411\) 1.46805e153 1.80715
\(412\) 1.96039e152 0.209342
\(413\) 9.13299e152 0.846321
\(414\) −3.23123e151 −0.0259925
\(415\) 6.63276e152 0.463319
\(416\) 1.52286e153 0.924059
\(417\) −3.09274e153 −1.63074
\(418\) −1.34343e152 −0.0615746
\(419\) −3.20454e153 −1.27716 −0.638579 − 0.769556i \(-0.720478\pi\)
−0.638579 + 0.769556i \(0.720478\pi\)
\(420\) −3.42323e153 −1.18673
\(421\) 2.09715e153 0.632596 0.316298 − 0.948660i \(-0.397560\pi\)
0.316298 + 0.948660i \(0.397560\pi\)
\(422\) −4.67106e151 −0.0122641
\(423\) −4.72288e152 −0.107968
\(424\) −3.62962e153 −0.722701
\(425\) 3.79369e153 0.658127
\(426\) 2.89936e153 0.438370
\(427\) 5.82850e153 0.768292
\(428\) −1.46007e154 −1.67846
\(429\) 6.91285e153 0.693275
\(430\) −3.95786e153 −0.346382
\(431\) 1.19751e153 0.0914867 0.0457434 − 0.998953i \(-0.485434\pi\)
0.0457434 + 0.998953i \(0.485434\pi\)
\(432\) −1.00485e154 −0.670350
\(433\) 1.84399e154 1.07452 0.537262 − 0.843416i \(-0.319459\pi\)
0.537262 + 0.843416i \(0.319459\pi\)
\(434\) 1.69365e154 0.862330
\(435\) 1.19782e154 0.533054
\(436\) 5.27737e153 0.205333
\(437\) −7.38294e153 −0.251226
\(438\) −9.74634e153 −0.290139
\(439\) 2.12741e154 0.554214 0.277107 − 0.960839i \(-0.410624\pi\)
0.277107 + 0.960839i \(0.410624\pi\)
\(440\) −1.16721e154 −0.266175
\(441\) 9.21000e153 0.183909
\(442\) −3.22227e154 −0.563584
\(443\) 4.72853e154 0.724616 0.362308 − 0.932058i \(-0.381989\pi\)
0.362308 + 0.932058i \(0.381989\pi\)
\(444\) −2.84868e154 −0.382596
\(445\) −9.80804e154 −1.15484
\(446\) −4.35787e154 −0.449971
\(447\) 4.66607e154 0.422631
\(448\) −9.40320e154 −0.747329
\(449\) −1.47156e155 −1.02652 −0.513261 − 0.858232i \(-0.671563\pi\)
−0.513261 + 0.858232i \(0.671563\pi\)
\(450\) 2.34912e153 0.0143871
\(451\) 5.04472e153 0.0271337
\(452\) 3.48171e155 1.64511
\(453\) 1.69233e155 0.702650
\(454\) 7.40984e153 0.0270422
\(455\) −4.38583e155 −1.40729
\(456\) −7.50331e154 −0.211743
\(457\) 5.49700e155 1.36467 0.682335 − 0.731039i \(-0.260964\pi\)
0.682335 + 0.731039i \(0.260964\pi\)
\(458\) 7.25652e153 0.0158525
\(459\) 7.77724e155 1.49549
\(460\) −3.04202e155 −0.515026
\(461\) −4.61551e155 −0.688202 −0.344101 − 0.938933i \(-0.611816\pi\)
−0.344101 + 0.938933i \(0.611816\pi\)
\(462\) 2.38477e155 0.313250
\(463\) 6.31519e155 0.730969 0.365484 − 0.930818i \(-0.380903\pi\)
0.365484 + 0.930818i \(0.380903\pi\)
\(464\) 4.62598e155 0.471957
\(465\) −1.52071e156 −1.36788
\(466\) 3.20720e155 0.254419
\(467\) −5.93527e154 −0.0415339 −0.0207670 − 0.999784i \(-0.506611\pi\)
−0.0207670 + 0.999784i \(0.506611\pi\)
\(468\) 1.83664e155 0.113407
\(469\) −2.28064e156 −1.24293
\(470\) 4.83040e155 0.232412
\(471\) −3.75729e156 −1.59644
\(472\) 8.24828e155 0.309568
\(473\) −2.53800e156 −0.841617
\(474\) 6.05977e154 0.0177592
\(475\) 5.36744e155 0.139056
\(476\) 1.02323e157 2.34404
\(477\) −6.67901e155 −0.135328
\(478\) −7.84905e154 −0.0140697
\(479\) 4.59363e156 0.728666 0.364333 − 0.931269i \(-0.381297\pi\)
0.364333 + 0.931269i \(0.381297\pi\)
\(480\) −4.71707e156 −0.662308
\(481\) −3.64972e156 −0.453704
\(482\) −1.55782e156 −0.171501
\(483\) 1.31058e157 1.27807
\(484\) 6.88953e156 0.595300
\(485\) −1.55240e157 −1.18881
\(486\) 1.02359e156 0.0694869
\(487\) 4.23910e156 0.255170 0.127585 − 0.991828i \(-0.459277\pi\)
0.127585 + 0.991828i \(0.459277\pi\)
\(488\) 5.26390e156 0.281027
\(489\) −3.05214e156 −0.144556
\(490\) −9.41967e156 −0.395881
\(491\) 3.51234e157 1.31018 0.655088 − 0.755552i \(-0.272631\pi\)
0.655088 + 0.755552i \(0.272631\pi\)
\(492\) 1.33621e156 0.0442503
\(493\) −3.58038e157 −1.05289
\(494\) −4.55897e156 −0.119080
\(495\) −2.14783e156 −0.0498420
\(496\) −5.87295e157 −1.21109
\(497\) −1.17958e158 −2.16212
\(498\) −1.22389e157 −0.199446
\(499\) 1.55906e157 0.225933 0.112967 − 0.993599i \(-0.463965\pi\)
0.112967 + 0.993599i \(0.463965\pi\)
\(500\) 7.57643e157 0.976606
\(501\) 1.63421e158 1.87414
\(502\) 4.39338e157 0.448365
\(503\) −1.97687e157 −0.179577 −0.0897886 − 0.995961i \(-0.528619\pi\)
−0.0897886 + 0.995961i \(0.528619\pi\)
\(504\) 1.33603e157 0.108051
\(505\) −4.42392e156 −0.0318611
\(506\) 2.11921e157 0.135947
\(507\) 5.01229e157 0.286465
\(508\) −8.27526e157 −0.421458
\(509\) −5.96996e156 −0.0271007 −0.0135504 − 0.999908i \(-0.504313\pi\)
−0.0135504 + 0.999908i \(0.504313\pi\)
\(510\) 9.98101e157 0.403942
\(511\) 3.96522e158 1.43102
\(512\) −3.14542e158 −1.01248
\(513\) 1.10035e158 0.315983
\(514\) −1.04967e158 −0.268973
\(515\) −7.77986e157 −0.177930
\(516\) −6.72246e158 −1.37253
\(517\) 3.09752e158 0.564699
\(518\) −1.25907e158 −0.205002
\(519\) −2.23845e158 −0.325580
\(520\) −3.96097e158 −0.514762
\(521\) −2.37291e158 −0.275597 −0.137799 − 0.990460i \(-0.544003\pi\)
−0.137799 + 0.990460i \(0.544003\pi\)
\(522\) −2.21703e157 −0.0230169
\(523\) 8.75144e156 0.00812327 0.00406164 − 0.999992i \(-0.498707\pi\)
0.00406164 + 0.999992i \(0.498707\pi\)
\(524\) 3.89891e158 0.323640
\(525\) −9.52797e158 −0.707426
\(526\) −8.89829e158 −0.591072
\(527\) 4.54549e159 2.70184
\(528\) −8.26953e158 −0.439943
\(529\) −9.35064e158 −0.445333
\(530\) 6.83106e158 0.291306
\(531\) 1.51780e158 0.0579674
\(532\) 1.44769e159 0.495274
\(533\) 1.71194e158 0.0524745
\(534\) 1.80980e159 0.497126
\(535\) 5.79430e159 1.42661
\(536\) −2.05971e159 −0.454639
\(537\) −5.89422e159 −1.16662
\(538\) −1.07810e159 −0.191379
\(539\) −6.04041e159 −0.961886
\(540\) 4.53381e159 0.647781
\(541\) −1.01393e159 −0.130008 −0.0650038 − 0.997885i \(-0.520706\pi\)
−0.0650038 + 0.997885i \(0.520706\pi\)
\(542\) −2.94897e159 −0.339400
\(543\) −8.93634e159 −0.923357
\(544\) 1.40996e160 1.30820
\(545\) −2.09433e159 −0.174522
\(546\) 8.09281e159 0.605801
\(547\) −8.73812e159 −0.587705 −0.293853 − 0.955851i \(-0.594937\pi\)
−0.293853 + 0.955851i \(0.594937\pi\)
\(548\) −2.55808e160 −1.54615
\(549\) 9.68630e158 0.0526230
\(550\) −1.54068e159 −0.0752479
\(551\) −5.06563e159 −0.222467
\(552\) 1.18362e160 0.467495
\(553\) −2.46537e159 −0.0875914
\(554\) −9.28392e159 −0.296763
\(555\) 1.13050e160 0.325187
\(556\) 5.38910e160 1.39522
\(557\) 7.87194e160 1.83465 0.917324 − 0.398142i \(-0.130345\pi\)
0.917324 + 0.398142i \(0.130345\pi\)
\(558\) 2.81464e159 0.0590640
\(559\) −8.61278e160 −1.62762
\(560\) 5.24657e160 0.893050
\(561\) 6.40037e160 0.981472
\(562\) −1.46195e160 −0.202004
\(563\) −7.96041e160 −0.991275 −0.495638 − 0.868529i \(-0.665066\pi\)
−0.495638 + 0.868529i \(0.665066\pi\)
\(564\) 8.20447e160 0.920922
\(565\) −1.38172e161 −1.39826
\(566\) 2.97339e160 0.271325
\(567\) −2.17379e161 −1.78899
\(568\) −1.06532e161 −0.790862
\(569\) 5.16908e160 0.346215 0.173107 − 0.984903i \(-0.444619\pi\)
0.173107 + 0.984903i \(0.444619\pi\)
\(570\) 1.41215e160 0.0853494
\(571\) 2.31075e161 1.26050 0.630248 − 0.776394i \(-0.282953\pi\)
0.630248 + 0.776394i \(0.282953\pi\)
\(572\) −1.20457e161 −0.593148
\(573\) 6.07082e160 0.269899
\(574\) 5.90582e159 0.0237102
\(575\) −8.46695e160 −0.307014
\(576\) −1.56270e160 −0.0511871
\(577\) 8.53587e160 0.252617 0.126309 − 0.991991i \(-0.459687\pi\)
0.126309 + 0.991991i \(0.459687\pi\)
\(578\) −1.81288e161 −0.484832
\(579\) −9.29497e160 −0.224674
\(580\) −2.08721e161 −0.456067
\(581\) 4.97929e161 0.983703
\(582\) 2.86451e161 0.511748
\(583\) 4.38045e161 0.707796
\(584\) 3.58111e161 0.523439
\(585\) −7.28873e160 −0.0963904
\(586\) 1.95355e161 0.233783
\(587\) 9.60172e161 1.03997 0.519984 − 0.854176i \(-0.325938\pi\)
0.519984 + 0.854176i \(0.325938\pi\)
\(588\) −1.59994e162 −1.56866
\(589\) 6.43110e161 0.570875
\(590\) −1.55235e161 −0.124781
\(591\) −3.35095e161 −0.243949
\(592\) 4.36600e161 0.287914
\(593\) −1.45887e162 −0.871600 −0.435800 − 0.900043i \(-0.643535\pi\)
−0.435800 + 0.900043i \(0.643535\pi\)
\(594\) −3.15845e161 −0.170989
\(595\) −4.06069e162 −1.99231
\(596\) −8.13065e161 −0.361592
\(597\) 3.63736e162 1.46652
\(598\) 7.19161e161 0.262910
\(599\) 8.78836e161 0.291366 0.145683 − 0.989331i \(-0.453462\pi\)
0.145683 + 0.989331i \(0.453462\pi\)
\(600\) −8.60499e161 −0.258763
\(601\) 1.69839e162 0.463319 0.231659 − 0.972797i \(-0.425585\pi\)
0.231659 + 0.972797i \(0.425585\pi\)
\(602\) −2.97121e162 −0.735426
\(603\) −3.79016e161 −0.0851324
\(604\) −2.94888e162 −0.601169
\(605\) −2.73412e162 −0.505975
\(606\) 8.16310e160 0.0137153
\(607\) 8.24586e162 1.25805 0.629024 − 0.777386i \(-0.283455\pi\)
0.629024 + 0.777386i \(0.283455\pi\)
\(608\) 1.99486e162 0.276410
\(609\) 8.99221e162 1.13176
\(610\) −9.90681e161 −0.113276
\(611\) 1.05115e163 1.09208
\(612\) 1.70048e162 0.160551
\(613\) −1.11618e163 −0.957844 −0.478922 − 0.877858i \(-0.658972\pi\)
−0.478922 + 0.877858i \(0.658972\pi\)
\(614\) 4.87095e162 0.379983
\(615\) −5.30277e161 −0.0376104
\(616\) −8.76240e162 −0.565134
\(617\) −8.04710e162 −0.472016 −0.236008 − 0.971751i \(-0.575839\pi\)
−0.236008 + 0.971751i \(0.575839\pi\)
\(618\) 1.43555e162 0.0765938
\(619\) 7.00303e162 0.339925 0.169963 − 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(620\) 2.64983e163 1.17032
\(621\) −1.73576e163 −0.697640
\(622\) −5.08912e162 −0.186168
\(623\) −7.36302e163 −2.45191
\(624\) −2.80629e163 −0.850814
\(625\) −1.51350e163 −0.417833
\(626\) 2.40316e163 0.604205
\(627\) 9.05545e162 0.207376
\(628\) 6.54710e163 1.36587
\(629\) −3.37915e163 −0.642311
\(630\) −2.51445e162 −0.0435532
\(631\) 8.31298e163 1.31232 0.656159 − 0.754623i \(-0.272180\pi\)
0.656159 + 0.754623i \(0.272180\pi\)
\(632\) −2.22655e162 −0.0320393
\(633\) 3.14855e162 0.0413042
\(634\) 2.88016e163 0.344504
\(635\) 3.28405e163 0.358218
\(636\) 1.16026e164 1.15429
\(637\) −2.04983e164 −1.86021
\(638\) 1.45405e163 0.120384
\(639\) −1.96033e163 −0.148091
\(640\) 1.07107e164 0.738399
\(641\) 2.18768e163 0.137654 0.0688272 − 0.997629i \(-0.478074\pi\)
0.0688272 + 0.997629i \(0.478074\pi\)
\(642\) −1.06918e164 −0.614115
\(643\) 9.67157e163 0.507172 0.253586 − 0.967313i \(-0.418390\pi\)
0.253586 + 0.967313i \(0.418390\pi\)
\(644\) −2.28369e164 −1.09349
\(645\) 2.66782e164 1.16658
\(646\) −4.22100e163 −0.168583
\(647\) 3.42590e164 1.24990 0.624949 − 0.780665i \(-0.285120\pi\)
0.624949 + 0.780665i \(0.285120\pi\)
\(648\) −1.96321e164 −0.654379
\(649\) −9.95452e163 −0.303184
\(650\) −5.22834e163 −0.145523
\(651\) −1.14161e165 −2.90423
\(652\) 5.31837e163 0.123679
\(653\) 1.28519e164 0.273241 0.136620 − 0.990623i \(-0.456376\pi\)
0.136620 + 0.990623i \(0.456376\pi\)
\(654\) 3.86451e163 0.0751270
\(655\) −1.54729e164 −0.275078
\(656\) −2.04792e163 −0.0332996
\(657\) 6.58973e163 0.0980152
\(658\) 3.62624e164 0.493448
\(659\) 1.27240e165 1.58426 0.792132 − 0.610350i \(-0.208971\pi\)
0.792132 + 0.610350i \(0.208971\pi\)
\(660\) 3.73116e164 0.425131
\(661\) −1.26749e165 −1.32178 −0.660890 − 0.750483i \(-0.729821\pi\)
−0.660890 + 0.750483i \(0.729821\pi\)
\(662\) −2.67970e164 −0.255796
\(663\) 2.17199e165 1.89809
\(664\) 4.49695e164 0.359820
\(665\) −5.74519e164 −0.420958
\(666\) −2.09243e163 −0.0140413
\(667\) 7.99086e164 0.491170
\(668\) −2.84762e165 −1.60346
\(669\) 2.93745e165 1.51545
\(670\) 3.87644e164 0.183256
\(671\) −6.35279e164 −0.275231
\(672\) −3.54116e165 −1.40619
\(673\) −6.15688e163 −0.0224119 −0.0112060 − 0.999937i \(-0.503567\pi\)
−0.0112060 + 0.999937i \(0.503567\pi\)
\(674\) 1.21248e165 0.404642
\(675\) 1.26191e165 0.386151
\(676\) −8.73394e164 −0.245092
\(677\) 1.92091e165 0.494391 0.247195 − 0.968966i \(-0.420491\pi\)
0.247195 + 0.968966i \(0.420491\pi\)
\(678\) 2.54958e165 0.601910
\(679\) −1.16540e166 −2.52403
\(680\) −3.66733e165 −0.728750
\(681\) −4.99465e164 −0.0910749
\(682\) −1.84599e165 −0.308919
\(683\) −6.81285e164 −0.104645 −0.0523224 − 0.998630i \(-0.516662\pi\)
−0.0523224 + 0.998630i \(0.516662\pi\)
\(684\) 2.40590e164 0.0339231
\(685\) 1.01518e166 1.31415
\(686\) −2.78457e165 −0.330977
\(687\) −4.89130e164 −0.0533894
\(688\) 1.03031e166 1.03287
\(689\) 1.48652e166 1.36882
\(690\) −2.22761e165 −0.188438
\(691\) 2.31551e165 0.179962 0.0899808 − 0.995943i \(-0.471319\pi\)
0.0899808 + 0.995943i \(0.471319\pi\)
\(692\) 3.90051e165 0.278558
\(693\) −1.61240e165 −0.105823
\(694\) 4.27298e165 0.257752
\(695\) −2.13868e166 −1.18586
\(696\) 8.12114e165 0.413977
\(697\) 1.58503e165 0.0742884
\(698\) 5.22987e165 0.225397
\(699\) −2.16183e166 −0.856855
\(700\) 1.66025e166 0.605255
\(701\) −9.90224e165 −0.332070 −0.166035 − 0.986120i \(-0.553096\pi\)
−0.166035 + 0.986120i \(0.553096\pi\)
\(702\) −1.07183e166 −0.330678
\(703\) −4.78094e165 −0.135715
\(704\) 1.02490e166 0.267721
\(705\) −3.25596e166 −0.782737
\(706\) 6.97887e165 0.154422
\(707\) −3.32109e165 −0.0676464
\(708\) −2.63668e166 −0.494438
\(709\) 1.57713e166 0.272310 0.136155 − 0.990688i \(-0.456526\pi\)
0.136155 + 0.990688i \(0.456526\pi\)
\(710\) 2.00495e166 0.318780
\(711\) −4.09715e164 −0.00599944
\(712\) −6.64976e166 −0.896864
\(713\) −1.01448e167 −1.26040
\(714\) 7.49287e166 0.857636
\(715\) 4.78034e166 0.504145
\(716\) 1.02707e167 0.998132
\(717\) 5.29070e165 0.0473852
\(718\) −2.00214e166 −0.165278
\(719\) 6.65333e166 0.506289 0.253145 − 0.967428i \(-0.418535\pi\)
0.253145 + 0.967428i \(0.418535\pi\)
\(720\) 8.71918e165 0.0611681
\(721\) −5.84043e166 −0.377774
\(722\) 4.65113e166 0.277417
\(723\) 1.05006e167 0.577594
\(724\) 1.55716e167 0.790001
\(725\) −5.80940e166 −0.271868
\(726\) 5.04506e166 0.217808
\(727\) 2.89239e167 1.15211 0.576055 − 0.817411i \(-0.304591\pi\)
0.576055 + 0.817411i \(0.304591\pi\)
\(728\) −2.97355e167 −1.09292
\(729\) 2.55032e167 0.865036
\(730\) −6.73975e166 −0.210987
\(731\) −7.97429e167 −2.30423
\(732\) −1.68268e167 −0.448852
\(733\) −6.97887e167 −1.71871 −0.859355 − 0.511379i \(-0.829135\pi\)
−0.859355 + 0.511379i \(0.829135\pi\)
\(734\) −1.50148e167 −0.341429
\(735\) 6.34939e167 1.33328
\(736\) −3.14683e167 −0.610268
\(737\) 2.48579e167 0.445262
\(738\) 9.81479e164 0.00162399
\(739\) 8.57490e167 1.31078 0.655390 − 0.755291i \(-0.272504\pi\)
0.655390 + 0.755291i \(0.272504\pi\)
\(740\) −1.96991e167 −0.278221
\(741\) 3.07300e167 0.401049
\(742\) 5.12816e167 0.618491
\(743\) −8.33130e167 −0.928682 −0.464341 − 0.885656i \(-0.653709\pi\)
−0.464341 + 0.885656i \(0.653709\pi\)
\(744\) −1.03102e168 −1.06231
\(745\) 3.22666e167 0.307335
\(746\) −7.37685e166 −0.0649606
\(747\) 8.27501e166 0.0673772
\(748\) −1.11527e168 −0.839722
\(749\) 4.34985e168 3.02892
\(750\) 5.54806e167 0.357320
\(751\) 1.60530e168 0.956359 0.478179 − 0.878262i \(-0.341297\pi\)
0.478179 + 0.878262i \(0.341297\pi\)
\(752\) −1.25745e168 −0.693021
\(753\) −2.96139e168 −1.51004
\(754\) 4.93435e167 0.232813
\(755\) 1.17027e168 0.510963
\(756\) 3.40359e168 1.37535
\(757\) −2.55720e168 −0.956437 −0.478219 − 0.878241i \(-0.658717\pi\)
−0.478219 + 0.878241i \(0.658717\pi\)
\(758\) 5.42741e166 0.0187907
\(759\) −1.42847e168 −0.457853
\(760\) −5.18866e167 −0.153978
\(761\) −5.53493e168 −1.52093 −0.760466 − 0.649378i \(-0.775029\pi\)
−0.760466 + 0.649378i \(0.775029\pi\)
\(762\) −6.05980e167 −0.154203
\(763\) −1.57224e168 −0.370539
\(764\) −1.05784e168 −0.230919
\(765\) −6.74840e167 −0.136460
\(766\) 2.19892e168 0.411932
\(767\) −3.37810e168 −0.586332
\(768\) 1.03324e168 0.166176
\(769\) 5.90428e168 0.879986 0.439993 − 0.898001i \(-0.354981\pi\)
0.439993 + 0.898001i \(0.354981\pi\)
\(770\) 1.64911e168 0.227794
\(771\) 7.07535e168 0.905872
\(772\) 1.61965e168 0.192225
\(773\) −1.13100e169 −1.24440 −0.622201 − 0.782858i \(-0.713761\pi\)
−0.622201 + 0.782858i \(0.713761\pi\)
\(774\) −4.93781e167 −0.0503719
\(775\) 7.37536e168 0.697643
\(776\) −1.05251e169 −0.923243
\(777\) 8.48684e168 0.690425
\(778\) −3.89621e168 −0.293992
\(779\) 2.24255e167 0.0156965
\(780\) 1.26618e169 0.822170
\(781\) 1.28569e169 0.774551
\(782\) 6.65848e168 0.372203
\(783\) −1.19095e169 −0.617776
\(784\) 2.45213e169 1.18046
\(785\) −2.59823e169 −1.16092
\(786\) 2.85509e168 0.118413
\(787\) −1.31905e169 −0.507853 −0.253927 − 0.967224i \(-0.581722\pi\)
−0.253927 + 0.967224i \(0.581722\pi\)
\(788\) 5.83903e168 0.208717
\(789\) 5.99794e169 1.99066
\(790\) 4.19043e167 0.0129144
\(791\) −1.03728e170 −2.96873
\(792\) −1.45621e168 −0.0387080
\(793\) −2.15584e169 −0.532274
\(794\) 6.96490e168 0.159741
\(795\) −4.60452e169 −0.981086
\(796\) −6.33811e169 −1.25472
\(797\) 1.16898e169 0.215029 0.107514 − 0.994204i \(-0.465711\pi\)
0.107514 + 0.994204i \(0.465711\pi\)
\(798\) 1.06011e169 0.181211
\(799\) 9.73228e169 1.54607
\(800\) 2.28776e169 0.337790
\(801\) −1.22365e169 −0.167940
\(802\) 2.77804e169 0.354436
\(803\) −4.32190e169 −0.512643
\(804\) 6.58417e169 0.726143
\(805\) 9.06285e169 0.929406
\(806\) −6.26444e169 −0.597424
\(807\) 7.26698e169 0.644544
\(808\) −2.99937e168 −0.0247438
\(809\) −7.44687e169 −0.571459 −0.285730 − 0.958310i \(-0.592236\pi\)
−0.285730 + 0.958310i \(0.592236\pi\)
\(810\) 3.69482e169 0.263767
\(811\) 1.92852e170 1.28086 0.640432 − 0.768015i \(-0.278755\pi\)
0.640432 + 0.768015i \(0.278755\pi\)
\(812\) −1.56690e170 −0.968306
\(813\) 1.98777e170 1.14306
\(814\) 1.37233e169 0.0734395
\(815\) −2.11061e169 −0.105120
\(816\) −2.59825e170 −1.20450
\(817\) −1.12823e170 −0.486863
\(818\) −6.06165e169 −0.243513
\(819\) −5.47174e169 −0.204653
\(820\) 9.24009e168 0.0321785
\(821\) 1.88040e169 0.0609784 0.0304892 − 0.999535i \(-0.490293\pi\)
0.0304892 + 0.999535i \(0.490293\pi\)
\(822\) −1.87323e170 −0.565704
\(823\) 3.36615e170 0.946767 0.473383 − 0.880857i \(-0.343033\pi\)
0.473383 + 0.880857i \(0.343033\pi\)
\(824\) −5.27467e169 −0.138183
\(825\) 1.03850e170 0.253426
\(826\) −1.16537e170 −0.264930
\(827\) −8.49206e170 −1.79863 −0.899313 − 0.437306i \(-0.855933\pi\)
−0.899313 + 0.437306i \(0.855933\pi\)
\(828\) −3.79522e169 −0.0748966
\(829\) 3.03563e170 0.558224 0.279112 − 0.960259i \(-0.409960\pi\)
0.279112 + 0.960259i \(0.409960\pi\)
\(830\) −8.46339e169 −0.145036
\(831\) 6.25788e170 0.999465
\(832\) 3.47804e170 0.517751
\(833\) −1.89787e171 −2.63351
\(834\) 3.94633e170 0.510480
\(835\) 1.13008e171 1.36286
\(836\) −1.57792e170 −0.177426
\(837\) 1.51198e171 1.58528
\(838\) 4.08898e170 0.399798
\(839\) −1.86948e170 −0.170470 −0.0852348 − 0.996361i \(-0.527164\pi\)
−0.0852348 + 0.996361i \(0.527164\pi\)
\(840\) 9.21060e170 0.783339
\(841\) −7.12293e170 −0.565058
\(842\) −2.67596e170 −0.198026
\(843\) 9.85439e170 0.680325
\(844\) −5.48637e169 −0.0353388
\(845\) 3.46608e170 0.208315
\(846\) 6.02639e169 0.0337980
\(847\) −2.05254e171 −1.07427
\(848\) −1.77826e171 −0.868635
\(849\) −2.00423e171 −0.913792
\(850\) −4.84075e170 −0.206018
\(851\) 7.54176e170 0.299636
\(852\) 3.40543e171 1.26315
\(853\) −6.78167e170 −0.234865 −0.117432 − 0.993081i \(-0.537466\pi\)
−0.117432 + 0.993081i \(0.537466\pi\)
\(854\) −7.43716e170 −0.240504
\(855\) −9.54785e169 −0.0288329
\(856\) 3.92848e171 1.10792
\(857\) −4.50432e171 −1.18646 −0.593228 − 0.805035i \(-0.702147\pi\)
−0.593228 + 0.805035i \(0.702147\pi\)
\(858\) −8.82078e170 −0.217020
\(859\) −9.15613e170 −0.210432 −0.105216 − 0.994449i \(-0.533553\pi\)
−0.105216 + 0.994449i \(0.533553\pi\)
\(860\) −4.64868e171 −0.998092
\(861\) −3.98085e170 −0.0798531
\(862\) −1.52802e170 −0.0286387
\(863\) −4.83835e171 −0.847356 −0.423678 − 0.905813i \(-0.639261\pi\)
−0.423678 + 0.905813i \(0.639261\pi\)
\(864\) 4.69001e171 0.767573
\(865\) −1.54792e171 −0.236760
\(866\) −2.35292e171 −0.336365
\(867\) 1.22199e172 1.63286
\(868\) 1.98926e172 2.48478
\(869\) 2.68713e170 0.0313785
\(870\) −1.52842e171 −0.166866
\(871\) 8.43561e171 0.861102
\(872\) −1.41994e171 −0.135536
\(873\) −1.93677e171 −0.172880
\(874\) 9.42062e170 0.0786431
\(875\) −2.25718e172 −1.76236
\(876\) −1.14475e172 −0.836029
\(877\) 2.84933e172 1.94656 0.973278 − 0.229631i \(-0.0737518\pi\)
0.973278 + 0.229631i \(0.0737518\pi\)
\(878\) −2.71458e171 −0.173489
\(879\) −1.31680e172 −0.787355
\(880\) −5.71851e171 −0.319924
\(881\) 1.82712e171 0.0956481 0.0478241 − 0.998856i \(-0.484771\pi\)
0.0478241 + 0.998856i \(0.484771\pi\)
\(882\) −1.17519e171 −0.0575702
\(883\) −1.38803e172 −0.636353 −0.318177 − 0.948031i \(-0.603071\pi\)
−0.318177 + 0.948031i \(0.603071\pi\)
\(884\) −3.78470e172 −1.62395
\(885\) 1.04637e172 0.420247
\(886\) −6.03359e171 −0.226831
\(887\) 7.60436e171 0.267628 0.133814 − 0.991006i \(-0.457278\pi\)
0.133814 + 0.991006i \(0.457278\pi\)
\(888\) 7.66472e171 0.252544
\(889\) 2.46538e172 0.760555
\(890\) 1.25150e172 0.361507
\(891\) 2.36932e172 0.640883
\(892\) −5.11851e172 −1.29658
\(893\) 1.37695e172 0.326670
\(894\) −5.95390e171 −0.132299
\(895\) −4.07594e172 −0.848361
\(896\) 8.04069e172 1.56774
\(897\) −4.84755e172 −0.885451
\(898\) 1.87771e172 0.321339
\(899\) −6.96064e172 −1.11611
\(900\) 2.75915e171 0.0414560
\(901\) 1.37632e173 1.93785
\(902\) −6.43706e170 −0.00849386
\(903\) 2.00276e173 2.47683
\(904\) −9.36795e172 −1.08590
\(905\) −6.17962e172 −0.671460
\(906\) −2.15941e172 −0.219955
\(907\) 1.67599e172 0.160046 0.0800230 − 0.996793i \(-0.474501\pi\)
0.0800230 + 0.996793i \(0.474501\pi\)
\(908\) 8.70319e171 0.0779213
\(909\) −5.51926e170 −0.00463334
\(910\) 5.59631e172 0.440535
\(911\) −1.06661e173 −0.787370 −0.393685 − 0.919245i \(-0.628800\pi\)
−0.393685 + 0.919245i \(0.628800\pi\)
\(912\) −3.67609e172 −0.254500
\(913\) −5.42719e172 −0.352399
\(914\) −7.01416e172 −0.427192
\(915\) 6.67774e172 0.381501
\(916\) 8.52311e171 0.0456786
\(917\) −1.16157e173 −0.584035
\(918\) −9.92374e172 −0.468143
\(919\) −2.18625e173 −0.967704 −0.483852 − 0.875150i \(-0.660763\pi\)
−0.483852 + 0.875150i \(0.660763\pi\)
\(920\) 8.18493e172 0.339959
\(921\) −3.28329e173 −1.27974
\(922\) 5.88939e172 0.215433
\(923\) 4.36303e173 1.49792
\(924\) 2.80102e173 0.902624
\(925\) −5.48290e172 −0.165851
\(926\) −8.05818e172 −0.228820
\(927\) −9.70612e171 −0.0258751
\(928\) −2.15912e173 −0.540406
\(929\) 2.82656e173 0.664262 0.332131 − 0.943233i \(-0.392232\pi\)
0.332131 + 0.943233i \(0.392232\pi\)
\(930\) 1.94042e173 0.428196
\(931\) −2.68517e173 −0.556437
\(932\) 3.76700e173 0.733103
\(933\) 3.43035e173 0.626993
\(934\) 7.57339e171 0.0130016
\(935\) 4.42596e173 0.713720
\(936\) −4.94169e172 −0.0748581
\(937\) 3.27037e173 0.465404 0.232702 − 0.972548i \(-0.425243\pi\)
0.232702 + 0.972548i \(0.425243\pi\)
\(938\) 2.91009e173 0.389082
\(939\) −1.61987e174 −2.03489
\(940\) 5.67352e173 0.669689
\(941\) −7.02396e173 −0.779093 −0.389546 − 0.921007i \(-0.627368\pi\)
−0.389546 + 0.921007i \(0.627368\pi\)
\(942\) 4.79430e173 0.499744
\(943\) −3.53755e172 −0.0346552
\(944\) 4.04107e173 0.372079
\(945\) −1.35072e174 −1.16897
\(946\) 3.23848e173 0.263457
\(947\) 1.01339e174 0.775004 0.387502 − 0.921869i \(-0.373338\pi\)
0.387502 + 0.921869i \(0.373338\pi\)
\(948\) 7.11747e172 0.0511727
\(949\) −1.46665e174 −0.991410
\(950\) −6.84884e172 −0.0435298
\(951\) −1.94139e174 −1.16025
\(952\) −2.75311e174 −1.54726
\(953\) 1.11657e174 0.590133 0.295066 − 0.955477i \(-0.404658\pi\)
0.295066 + 0.955477i \(0.404658\pi\)
\(954\) 8.52240e172 0.0423626
\(955\) 4.19807e173 0.196269
\(956\) −9.21906e172 −0.0405415
\(957\) −9.80108e173 −0.405439
\(958\) −5.86146e173 −0.228099
\(959\) 7.62107e174 2.79015
\(960\) −1.07733e174 −0.371091
\(961\) 5.75149e174 1.86407
\(962\) 4.65704e173 0.142026
\(963\) 7.22895e173 0.207461
\(964\) −1.82973e174 −0.494175
\(965\) −6.42762e173 −0.163381
\(966\) −1.67229e174 −0.400084
\(967\) −3.47900e174 −0.783440 −0.391720 − 0.920084i \(-0.628120\pi\)
−0.391720 + 0.920084i \(0.628120\pi\)
\(968\) −1.85371e174 −0.392947
\(969\) 2.84519e174 0.567767
\(970\) 1.98086e174 0.372140
\(971\) 5.00660e174 0.885560 0.442780 − 0.896630i \(-0.353992\pi\)
0.442780 + 0.896630i \(0.353992\pi\)
\(972\) 1.20225e174 0.200225
\(973\) −1.60553e175 −2.51778
\(974\) −5.40909e173 −0.0798776
\(975\) 3.52420e174 0.490106
\(976\) 2.57894e174 0.337774
\(977\) 8.60745e174 1.06180 0.530900 − 0.847435i \(-0.321854\pi\)
0.530900 + 0.847435i \(0.321854\pi\)
\(978\) 3.89453e173 0.0452514
\(979\) 8.02534e174 0.878366
\(980\) −1.10638e175 −1.14072
\(981\) −2.61288e173 −0.0253795
\(982\) −4.48175e174 −0.410133
\(983\) −1.59423e175 −1.37458 −0.687292 − 0.726381i \(-0.741201\pi\)
−0.687292 + 0.726381i \(0.741201\pi\)
\(984\) −3.59523e173 −0.0292088
\(985\) −2.31723e174 −0.177398
\(986\) 4.56855e174 0.329594
\(987\) −2.44429e175 −1.66188
\(988\) −5.35471e174 −0.343127
\(989\) 1.77974e175 1.07491
\(990\) 2.74063e173 0.0156024
\(991\) −2.15140e175 −1.15455 −0.577275 − 0.816550i \(-0.695884\pi\)
−0.577275 + 0.816550i \(0.695884\pi\)
\(992\) 2.74113e175 1.38674
\(993\) 1.80627e175 0.861491
\(994\) 1.50514e175 0.676822
\(995\) 2.51529e175 1.06644
\(996\) −1.43751e175 −0.574699
\(997\) 3.16088e175 1.19163 0.595814 − 0.803122i \(-0.296829\pi\)
0.595814 + 0.803122i \(0.296829\pi\)
\(998\) −1.98935e174 −0.0707254
\(999\) −1.12402e175 −0.376871
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 1.118.a.a.1.5 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.118.a.a.1.5 ✓ 9 1.1 even 1 trivial