Properties

Label 1.110.a.a.1.2
Level $1$
Weight $110$
Character 1.1
Self dual yes
Analytic conductor $75.239$
Analytic rank $1$
Dimension $8$
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] = [1,110,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, 110, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 110);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 110 \)
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: \(75.2394221917\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{118}\cdot 3^{40}\cdot 5^{14}\cdot 7^{6}\cdot 11^{3}\cdot 13 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.86979e14\) 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)
 
\(f(q)\) \(=\) \(q-3.56139e16 q^{2} +1.00458e26 q^{3} +6.19316e32 q^{4} -8.07110e36 q^{5} -3.57771e42 q^{6} -7.18940e45 q^{7} +1.05848e48 q^{8} -5.23267e49 q^{9} +O(q^{10})\) \(q-3.56139e16 q^{2} +1.00458e26 q^{3} +6.19316e32 q^{4} -8.07110e36 q^{5} -3.57771e42 q^{6} -7.18940e45 q^{7} +1.05848e48 q^{8} -5.23267e49 q^{9} +2.87444e53 q^{10} +3.15771e56 q^{11} +6.22154e58 q^{12} +3.96229e60 q^{13} +2.56043e62 q^{14} -8.10809e62 q^{15} -4.39656e65 q^{16} +7.90575e66 q^{17} +1.86356e66 q^{18} -8.36798e69 q^{19} -4.99857e69 q^{20} -7.22234e71 q^{21} -1.12459e73 q^{22} +1.60412e74 q^{23} +1.06333e74 q^{24} -1.53423e76 q^{25} -1.41113e77 q^{26} -1.02432e78 q^{27} -4.45251e78 q^{28} +2.30631e79 q^{29} +2.88761e79 q^{30} -6.52163e80 q^{31} +1.49709e82 q^{32} +3.17218e82 q^{33} -2.81555e83 q^{34} +5.80264e82 q^{35} -3.24067e82 q^{36} +2.83538e85 q^{37} +2.98017e86 q^{38} +3.98044e86 q^{39} -8.54311e84 q^{40} +1.40024e88 q^{41} +2.57216e88 q^{42} +4.04388e88 q^{43} +1.95562e89 q^{44} +4.22334e86 q^{45} -5.71289e90 q^{46} -1.37892e91 q^{47} -4.41670e91 q^{48} -7.88353e91 q^{49} +5.46400e92 q^{50} +7.94197e92 q^{51} +2.45391e93 q^{52} -9.81155e93 q^{53} +3.64802e94 q^{54} -2.54862e93 q^{55} -7.60984e93 q^{56} -8.40632e95 q^{57} -8.21369e95 q^{58} +4.69363e95 q^{59} -5.02147e95 q^{60} +3.61750e97 q^{61} +2.32261e97 q^{62} +3.76197e95 q^{63} -2.47819e98 q^{64} -3.19800e97 q^{65} -1.12974e99 q^{66} -4.15119e99 q^{67} +4.89616e99 q^{68} +1.61147e100 q^{69} -2.06655e99 q^{70} -6.22667e100 q^{71} -5.53868e97 q^{72} +1.10474e101 q^{73} -1.00979e102 q^{74} -1.54126e102 q^{75} -5.18242e102 q^{76} -2.27021e102 q^{77} -1.41759e103 q^{78} +2.39031e103 q^{79} +3.54851e102 q^{80} -1.02371e104 q^{81} -4.98682e104 q^{82} -6.41612e104 q^{83} -4.47291e104 q^{84} -6.38081e103 q^{85} -1.44018e105 q^{86} +2.31688e105 q^{87} +3.34238e104 q^{88} +1.35058e106 q^{89} -1.50410e103 q^{90} -2.84865e106 q^{91} +9.93455e106 q^{92} -6.55151e106 q^{93} +4.91087e107 q^{94} +6.75388e106 q^{95} +1.50395e108 q^{96} +1.03023e108 q^{97} +2.80764e108 q^{98} -1.65233e106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 22\!\cdots\!00 q^{2}+ \cdots + 24\!\cdots\!84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 22\!\cdots\!00 q^{2}+ \cdots + 71\!\cdots\!28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.56139e16 −1.39793 −0.698965 0.715156i \(-0.746356\pi\)
−0.698965 + 0.715156i \(0.746356\pi\)
\(3\) 1.00458e26 0.997418 0.498709 0.866770i \(-0.333808\pi\)
0.498709 + 0.866770i \(0.333808\pi\)
\(4\) 6.19316e32 0.954208
\(5\) −8.07110e36 −0.0650231 −0.0325116 0.999471i \(-0.510351\pi\)
−0.0325116 + 0.999471i \(0.510351\pi\)
\(6\) −3.57771e42 −1.39432
\(7\) −7.18940e45 −0.629288 −0.314644 0.949210i \(-0.601885\pi\)
−0.314644 + 0.949210i \(0.601885\pi\)
\(8\) 1.05848e48 0.0640145
\(9\) −5.23267e49 −0.00515830
\(10\) 2.87444e53 0.0908977
\(11\) 3.15771e56 0.553951 0.276976 0.960877i \(-0.410668\pi\)
0.276976 + 0.960877i \(0.410668\pi\)
\(12\) 6.22154e58 0.951743
\(13\) 3.96229e60 0.772740 0.386370 0.922344i \(-0.373729\pi\)
0.386370 + 0.922344i \(0.373729\pi\)
\(14\) 2.56043e62 0.879701
\(15\) −8.10809e62 −0.0648552
\(16\) −4.39656e65 −1.04370
\(17\) 7.90575e66 0.689409 0.344704 0.938711i \(-0.387979\pi\)
0.344704 + 0.938711i \(0.387979\pi\)
\(18\) 1.86356e66 0.00721094
\(19\) −8.36798e69 −1.70039 −0.850197 0.526465i \(-0.823517\pi\)
−0.850197 + 0.526465i \(0.823517\pi\)
\(20\) −4.99857e69 −0.0620455
\(21\) −7.22234e71 −0.627663
\(22\) −1.12459e73 −0.774385
\(23\) 1.60412e74 0.979645 0.489823 0.871822i \(-0.337062\pi\)
0.489823 + 0.871822i \(0.337062\pi\)
\(24\) 1.06333e74 0.0638492
\(25\) −1.53423e76 −0.995772
\(26\) −1.41113e77 −1.08024
\(27\) −1.02432e78 −1.00256
\(28\) −4.45251e78 −0.600472
\(29\) 2.30631e79 0.459438 0.229719 0.973257i \(-0.426219\pi\)
0.229719 + 0.973257i \(0.426219\pi\)
\(30\) 2.88761e79 0.0906630
\(31\) −6.52163e80 −0.342881 −0.171441 0.985194i \(-0.554842\pi\)
−0.171441 + 0.985194i \(0.554842\pi\)
\(32\) 1.49709e82 1.39500
\(33\) 3.17218e82 0.552520
\(34\) −2.81555e83 −0.963745
\(35\) 5.80264e82 0.0409183
\(36\) −3.24067e82 −0.00492209
\(37\) 2.83538e85 0.967425 0.483713 0.875227i \(-0.339288\pi\)
0.483713 + 0.875227i \(0.339288\pi\)
\(38\) 2.98017e86 2.37703
\(39\) 3.98044e86 0.770744
\(40\) −8.54311e84 −0.00416242
\(41\) 1.40024e88 1.77616 0.888082 0.459685i \(-0.152038\pi\)
0.888082 + 0.459685i \(0.152038\pi\)
\(42\) 2.57216e88 0.877429
\(43\) 4.04388e88 0.382619 0.191310 0.981530i \(-0.438727\pi\)
0.191310 + 0.981530i \(0.438727\pi\)
\(44\) 1.95562e89 0.528584
\(45\) 4.22334e86 0.000335408 0
\(46\) −5.71289e90 −1.36948
\(47\) −1.37892e91 −1.02378 −0.511888 0.859052i \(-0.671054\pi\)
−0.511888 + 0.859052i \(0.671054\pi\)
\(48\) −4.41670e91 −1.04100
\(49\) −7.88353e91 −0.603996
\(50\) 5.46400e92 1.39202
\(51\) 7.94197e92 0.687628
\(52\) 2.45391e93 0.737354
\(53\) −9.81155e93 −1.04401 −0.522003 0.852944i \(-0.674815\pi\)
−0.522003 + 0.852944i \(0.674815\pi\)
\(54\) 3.64802e94 1.40151
\(55\) −2.54862e93 −0.0360196
\(56\) −7.60984e93 −0.0402836
\(57\) −8.40632e95 −1.69600
\(58\) −8.21369e95 −0.642263
\(59\) 4.69363e95 0.144569 0.0722844 0.997384i \(-0.476971\pi\)
0.0722844 + 0.997384i \(0.476971\pi\)
\(60\) −5.02147e95 −0.0618853
\(61\) 3.61750e97 1.81106 0.905529 0.424284i \(-0.139474\pi\)
0.905529 + 0.424284i \(0.139474\pi\)
\(62\) 2.32261e97 0.479324
\(63\) 3.76197e95 0.00324606
\(64\) −2.47819e98 −0.906414
\(65\) −3.19800e97 −0.0502460
\(66\) −1.12974e99 −0.772385
\(67\) −4.15119e99 −1.25054 −0.625268 0.780410i \(-0.715010\pi\)
−0.625268 + 0.780410i \(0.715010\pi\)
\(68\) 4.89616e99 0.657839
\(69\) 1.61147e100 0.977115
\(70\) −2.06655e99 −0.0572009
\(71\) −6.22667e100 −0.795566 −0.397783 0.917480i \(-0.630220\pi\)
−0.397783 + 0.917480i \(0.630220\pi\)
\(72\) −5.53868e97 −0.000330206 0
\(73\) 1.10474e101 0.310575 0.155287 0.987869i \(-0.450370\pi\)
0.155287 + 0.987869i \(0.450370\pi\)
\(74\) −1.00979e102 −1.35239
\(75\) −1.54126e102 −0.993200
\(76\) −5.18242e102 −1.62253
\(77\) −2.27021e102 −0.348595
\(78\) −1.41759e103 −1.07745
\(79\) 2.39031e103 0.907357 0.453678 0.891165i \(-0.350111\pi\)
0.453678 + 0.891165i \(0.350111\pi\)
\(80\) 3.54851e102 0.0678643
\(81\) −1.02371e104 −0.994815
\(82\) −4.98682e104 −2.48295
\(83\) −6.41612e104 −1.65012 −0.825061 0.565043i \(-0.808860\pi\)
−0.825061 + 0.565043i \(0.808860\pi\)
\(84\) −4.47291e104 −0.598921
\(85\) −6.38081e103 −0.0448275
\(86\) −1.44018e105 −0.534875
\(87\) 2.31688e105 0.458252
\(88\) 3.34238e104 0.0354609
\(89\) 1.35058e106 0.774045 0.387022 0.922070i \(-0.373504\pi\)
0.387022 + 0.922070i \(0.373504\pi\)
\(90\) −1.50410e103 −0.000468877 0
\(91\) −2.84865e106 −0.486276
\(92\) 9.93455e106 0.934785
\(93\) −6.55151e106 −0.341996
\(94\) 4.91087e107 1.43117
\(95\) 6.75388e106 0.110565
\(96\) 1.50395e108 1.39140
\(97\) 1.03023e108 0.541847 0.270923 0.962601i \(-0.412671\pi\)
0.270923 + 0.962601i \(0.412671\pi\)
\(98\) 2.80764e108 0.844344
\(99\) −1.65233e106 −0.00285744
\(100\) −9.50173e108 −0.950173
\(101\) −3.45287e108 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(102\) −2.82845e109 −0.961256
\(103\) −9.34044e109 −1.86526 −0.932628 0.360840i \(-0.882490\pi\)
−0.932628 + 0.360840i \(0.882490\pi\)
\(104\) 4.19401e108 0.0494666
\(105\) 5.82923e108 0.0408126
\(106\) 3.49428e110 1.45945
\(107\) −1.30188e110 −0.325952 −0.162976 0.986630i \(-0.552109\pi\)
−0.162976 + 0.986630i \(0.552109\pi\)
\(108\) −6.34379e110 −0.956653
\(109\) −1.88543e111 −1.72055 −0.860274 0.509832i \(-0.829708\pi\)
−0.860274 + 0.509832i \(0.829708\pi\)
\(110\) 9.07665e109 0.0503529
\(111\) 2.84837e111 0.964927
\(112\) 3.16086e111 0.656785
\(113\) 1.04938e110 0.0134326 0.00671628 0.999977i \(-0.497862\pi\)
0.00671628 + 0.999977i \(0.497862\pi\)
\(114\) 2.99382e112 2.37089
\(115\) −1.29470e111 −0.0636996
\(116\) 1.42834e112 0.438400
\(117\) −2.07333e110 −0.00398602
\(118\) −1.67159e112 −0.202097
\(119\) −5.68376e112 −0.433837
\(120\) −8.58225e110 −0.00415167
\(121\) −2.25228e113 −0.693138
\(122\) −1.28833e114 −2.53173
\(123\) 1.40666e114 1.77158
\(124\) −4.03895e113 −0.327180
\(125\) 2.48184e113 0.129771
\(126\) −1.33979e112 −0.00453776
\(127\) −3.59232e114 −0.790810 −0.395405 0.918507i \(-0.629396\pi\)
−0.395405 + 0.918507i \(0.629396\pi\)
\(128\) −8.90833e113 −0.127895
\(129\) 4.06240e114 0.381631
\(130\) 1.13894e114 0.0702403
\(131\) −6.00882e113 −0.0244064 −0.0122032 0.999926i \(-0.503884\pi\)
−0.0122032 + 0.999926i \(0.503884\pi\)
\(132\) 1.96458e115 0.527219
\(133\) 6.01608e115 1.07004
\(134\) 1.47840e116 1.74816
\(135\) 8.26741e114 0.0651897
\(136\) 8.36808e114 0.0441322
\(137\) −2.53292e116 −0.896086 −0.448043 0.894012i \(-0.647879\pi\)
−0.448043 + 0.894012i \(0.647879\pi\)
\(138\) −5.73907e116 −1.36594
\(139\) −7.24815e116 −1.16391 −0.581956 0.813220i \(-0.697713\pi\)
−0.581956 + 0.813220i \(0.697713\pi\)
\(140\) 3.59367e115 0.0390445
\(141\) −1.38524e117 −1.02113
\(142\) 2.21756e117 1.11215
\(143\) 1.25118e117 0.428060
\(144\) 2.30057e115 0.00538369
\(145\) −1.86145e116 −0.0298741
\(146\) −3.93443e117 −0.434162
\(147\) −7.91965e117 −0.602436
\(148\) 1.75600e118 0.923124
\(149\) 7.84968e117 0.285891 0.142945 0.989731i \(-0.454343\pi\)
0.142945 + 0.989731i \(0.454343\pi\)
\(150\) 5.48903e118 1.38842
\(151\) −1.39761e118 −0.246117 −0.123058 0.992399i \(-0.539270\pi\)
−0.123058 + 0.992399i \(0.539270\pi\)
\(152\) −8.85734e117 −0.108850
\(153\) −4.13682e116 −0.00355618
\(154\) 8.08511e118 0.487311
\(155\) 5.26367e117 0.0222952
\(156\) 2.46515e119 0.735450
\(157\) −7.03358e119 −1.48131 −0.740654 0.671887i \(-0.765484\pi\)
−0.740654 + 0.671887i \(0.765484\pi\)
\(158\) −8.51284e119 −1.26842
\(159\) −9.85651e119 −1.04131
\(160\) −1.20832e119 −0.0907071
\(161\) −1.15326e120 −0.616479
\(162\) 3.64583e120 1.39068
\(163\) −2.09789e120 −0.572214 −0.286107 0.958198i \(-0.592361\pi\)
−0.286107 + 0.958198i \(0.592361\pi\)
\(164\) 8.67193e120 1.69483
\(165\) −2.56030e119 −0.0359266
\(166\) 2.28503e121 2.30676
\(167\) −2.23870e121 −1.62910 −0.814551 0.580092i \(-0.803017\pi\)
−0.814551 + 0.580092i \(0.803017\pi\)
\(168\) −7.64471e119 −0.0401796
\(169\) −1.05924e121 −0.402873
\(170\) 2.27246e120 0.0626657
\(171\) 4.37868e119 0.00877113
\(172\) 2.50444e121 0.365098
\(173\) 6.81871e120 0.0724753 0.0362376 0.999343i \(-0.488463\pi\)
0.0362376 + 0.999343i \(0.488463\pi\)
\(174\) −8.25132e121 −0.640604
\(175\) 1.10302e122 0.626628
\(176\) −1.38831e122 −0.578156
\(177\) 4.71514e121 0.144195
\(178\) −4.80996e122 −1.08206
\(179\) −1.16550e123 −1.93207 −0.966035 0.258411i \(-0.916801\pi\)
−0.966035 + 0.258411i \(0.916801\pi\)
\(180\) 2.61558e119 0.000320049 0
\(181\) 1.18514e123 1.07223 0.536115 0.844145i \(-0.319892\pi\)
0.536115 + 0.844145i \(0.319892\pi\)
\(182\) 1.01452e123 0.679780
\(183\) 3.63408e123 1.80638
\(184\) 1.69793e122 0.0627115
\(185\) −2.28847e122 −0.0629050
\(186\) 2.33325e123 0.478086
\(187\) 2.49641e123 0.381899
\(188\) −8.53986e123 −0.976895
\(189\) 7.36427e123 0.630901
\(190\) −2.40532e123 −0.154562
\(191\) −2.68673e124 −1.29690 −0.648449 0.761258i \(-0.724582\pi\)
−0.648449 + 0.761258i \(0.724582\pi\)
\(192\) −2.48955e124 −0.904074
\(193\) 2.04325e124 0.559048 0.279524 0.960139i \(-0.409823\pi\)
0.279524 + 0.960139i \(0.409823\pi\)
\(194\) −3.66907e124 −0.757463
\(195\) −3.21266e123 −0.0501162
\(196\) −4.88240e124 −0.576338
\(197\) 1.67102e125 1.49476 0.747381 0.664396i \(-0.231311\pi\)
0.747381 + 0.664396i \(0.231311\pi\)
\(198\) 5.88459e122 0.00399451
\(199\) 1.79326e125 0.925019 0.462509 0.886614i \(-0.346949\pi\)
0.462509 + 0.886614i \(0.346949\pi\)
\(200\) −1.62395e124 −0.0637439
\(201\) −4.17022e125 −1.24731
\(202\) 1.22970e125 0.280641
\(203\) −1.65810e125 −0.289119
\(204\) 4.91859e125 0.656140
\(205\) −1.13015e125 −0.115492
\(206\) 3.32650e126 2.60750
\(207\) −8.39381e123 −0.00505330
\(208\) −1.74204e126 −0.806505
\(209\) −2.64237e126 −0.941935
\(210\) −2.07602e125 −0.0570532
\(211\) −4.55403e126 −0.966055 −0.483027 0.875605i \(-0.660463\pi\)
−0.483027 + 0.875605i \(0.660463\pi\)
\(212\) −6.07645e126 −0.996198
\(213\) −6.25520e126 −0.793511
\(214\) 4.63650e126 0.455658
\(215\) −3.26385e125 −0.0248791
\(216\) −1.08423e126 −0.0641786
\(217\) 4.68866e126 0.215771
\(218\) 6.71477e127 2.40521
\(219\) 1.10981e127 0.309773
\(220\) −1.57840e126 −0.0343702
\(221\) 3.13249e127 0.532734
\(222\) −1.01442e128 −1.34890
\(223\) 7.56923e127 0.787838 0.393919 0.919145i \(-0.371119\pi\)
0.393919 + 0.919145i \(0.371119\pi\)
\(224\) −1.07632e128 −0.877856
\(225\) 8.02811e125 0.00513649
\(226\) −3.73727e126 −0.0187778
\(227\) 1.83524e128 0.724910 0.362455 0.932001i \(-0.381939\pi\)
0.362455 + 0.932001i \(0.381939\pi\)
\(228\) −5.20617e128 −1.61834
\(229\) 2.38042e128 0.582934 0.291467 0.956581i \(-0.405857\pi\)
0.291467 + 0.956581i \(0.405857\pi\)
\(230\) 4.61093e127 0.0890475
\(231\) −2.28061e128 −0.347695
\(232\) 2.44119e127 0.0294107
\(233\) −9.64479e128 −0.919168 −0.459584 0.888134i \(-0.652001\pi\)
−0.459584 + 0.888134i \(0.652001\pi\)
\(234\) 7.38396e126 0.00557218
\(235\) 1.11294e128 0.0665691
\(236\) 2.90684e128 0.137949
\(237\) 2.40126e129 0.905014
\(238\) 2.02421e129 0.606474
\(239\) −2.75183e129 −0.656049 −0.328025 0.944669i \(-0.606383\pi\)
−0.328025 + 0.944669i \(0.606383\pi\)
\(240\) 3.56477e128 0.0676891
\(241\) 2.31148e129 0.349913 0.174957 0.984576i \(-0.444021\pi\)
0.174957 + 0.984576i \(0.444021\pi\)
\(242\) 8.02126e129 0.968959
\(243\) 1.06924e128 0.0103165
\(244\) 2.24038e130 1.72813
\(245\) 6.36288e128 0.0392737
\(246\) −5.00967e130 −2.47654
\(247\) −3.31563e130 −1.31396
\(248\) −6.90302e128 −0.0219494
\(249\) −6.44552e130 −1.64586
\(250\) −8.83882e129 −0.181411
\(251\) 7.59986e130 1.25484 0.627419 0.778682i \(-0.284111\pi\)
0.627419 + 0.778682i \(0.284111\pi\)
\(252\) 2.32985e128 0.00309741
\(253\) 5.06534e130 0.542675
\(254\) 1.27937e131 1.10550
\(255\) −6.41005e129 −0.0447117
\(256\) 1.92570e131 1.08520
\(257\) −2.73816e130 −0.124768 −0.0623840 0.998052i \(-0.519870\pi\)
−0.0623840 + 0.998052i \(0.519870\pi\)
\(258\) −1.44678e131 −0.533494
\(259\) −2.03847e131 −0.608789
\(260\) −1.98058e130 −0.0479451
\(261\) −1.20682e129 −0.00236992
\(262\) 2.13998e130 0.0341184
\(263\) 8.10061e131 1.04937 0.524686 0.851296i \(-0.324183\pi\)
0.524686 + 0.851296i \(0.324183\pi\)
\(264\) 3.35769e130 0.0353693
\(265\) 7.91901e130 0.0678845
\(266\) −2.14256e132 −1.49584
\(267\) 1.35677e132 0.772046
\(268\) −2.57090e132 −1.19327
\(269\) 1.10436e132 0.418420 0.209210 0.977871i \(-0.432911\pi\)
0.209210 + 0.977871i \(0.432911\pi\)
\(270\) −2.94435e131 −0.0911307
\(271\) −1.79737e132 −0.454790 −0.227395 0.973803i \(-0.573021\pi\)
−0.227395 + 0.973803i \(0.573021\pi\)
\(272\) −3.47581e132 −0.719533
\(273\) −2.86170e132 −0.485020
\(274\) 9.02071e132 1.25267
\(275\) −4.84466e132 −0.551609
\(276\) 9.98007e132 0.932371
\(277\) 9.56140e132 0.733456 0.366728 0.930328i \(-0.380478\pi\)
0.366728 + 0.930328i \(0.380478\pi\)
\(278\) 2.58135e133 1.62707
\(279\) 3.41255e130 0.00176868
\(280\) 6.14198e130 0.00261936
\(281\) −4.46085e133 −1.56647 −0.783236 0.621724i \(-0.786433\pi\)
−0.783236 + 0.621724i \(0.786433\pi\)
\(282\) 4.93337e133 1.42747
\(283\) −3.18805e133 −0.760615 −0.380308 0.924860i \(-0.624182\pi\)
−0.380308 + 0.924860i \(0.624182\pi\)
\(284\) −3.85628e133 −0.759135
\(285\) 6.78483e132 0.110279
\(286\) −4.45594e133 −0.598398
\(287\) −1.00669e134 −1.11772
\(288\) −7.83377e131 −0.00719582
\(289\) −6.90011e133 −0.524715
\(290\) 6.62935e132 0.0417619
\(291\) 1.03495e134 0.540447
\(292\) 6.84186e133 0.296353
\(293\) 3.11001e134 1.11809 0.559045 0.829137i \(-0.311168\pi\)
0.559045 + 0.829137i \(0.311168\pi\)
\(294\) 2.82050e134 0.842164
\(295\) −3.78828e132 −0.00940031
\(296\) 3.00120e133 0.0619293
\(297\) −3.23452e134 −0.555371
\(298\) −2.79558e134 −0.399655
\(299\) 6.35597e134 0.757011
\(300\) −9.54527e134 −0.947719
\(301\) −2.90731e134 −0.240778
\(302\) 4.97743e134 0.344054
\(303\) −3.46869e134 −0.200236
\(304\) 3.67903e135 1.77469
\(305\) −2.91972e134 −0.117761
\(306\) 1.47328e133 0.00497128
\(307\) −2.51941e135 −0.711636 −0.355818 0.934555i \(-0.615798\pi\)
−0.355818 + 0.934555i \(0.615798\pi\)
\(308\) −1.40598e135 −0.332632
\(309\) −9.38323e135 −1.86044
\(310\) −1.87460e134 −0.0311671
\(311\) −1.03475e136 −1.44342 −0.721709 0.692196i \(-0.756643\pi\)
−0.721709 + 0.692196i \(0.756643\pi\)
\(312\) 4.21322e134 0.0493388
\(313\) 1.02153e136 1.00481 0.502405 0.864632i \(-0.332449\pi\)
0.502405 + 0.864632i \(0.332449\pi\)
\(314\) 2.50493e136 2.07076
\(315\) −3.03633e132 −0.000211069 0
\(316\) 1.48036e136 0.865807
\(317\) −2.45989e136 −1.21111 −0.605557 0.795802i \(-0.707050\pi\)
−0.605557 + 0.795802i \(0.707050\pi\)
\(318\) 3.51029e136 1.45568
\(319\) 7.28267e135 0.254506
\(320\) 2.00018e135 0.0589379
\(321\) −1.30784e136 −0.325111
\(322\) 4.10723e136 0.861795
\(323\) −6.61552e136 −1.17227
\(324\) −6.33999e136 −0.949260
\(325\) −6.07906e136 −0.769473
\(326\) 7.47142e136 0.799915
\(327\) −1.89407e137 −1.71610
\(328\) 1.48213e136 0.113700
\(329\) 9.91359e136 0.644250
\(330\) 9.11824e135 0.0502229
\(331\) 1.66846e137 0.779274 0.389637 0.920968i \(-0.372600\pi\)
0.389637 + 0.920968i \(0.372600\pi\)
\(332\) −3.97361e137 −1.57456
\(333\) −1.48366e135 −0.00499027
\(334\) 7.97289e137 2.27737
\(335\) 3.35047e136 0.0813138
\(336\) 3.17535e137 0.655089
\(337\) −9.18402e137 −1.61140 −0.805700 0.592324i \(-0.798211\pi\)
−0.805700 + 0.592324i \(0.798211\pi\)
\(338\) 3.77236e137 0.563188
\(339\) 1.05419e136 0.0133979
\(340\) −3.95174e136 −0.0427747
\(341\) −2.05934e137 −0.189939
\(342\) −1.55942e136 −0.0122614
\(343\) 1.50516e138 1.00938
\(344\) 4.28036e136 0.0244932
\(345\) −1.30063e137 −0.0635351
\(346\) −2.42841e137 −0.101315
\(347\) −4.53577e138 −1.61694 −0.808472 0.588535i \(-0.799705\pi\)
−0.808472 + 0.588535i \(0.799705\pi\)
\(348\) 1.43488e138 0.437267
\(349\) 3.55561e138 0.926678 0.463339 0.886181i \(-0.346651\pi\)
0.463339 + 0.886181i \(0.346651\pi\)
\(350\) −3.92829e138 −0.875981
\(351\) −4.05866e138 −0.774720
\(352\) 4.72738e138 0.772761
\(353\) −1.37878e139 −1.93096 −0.965480 0.260476i \(-0.916120\pi\)
−0.965480 + 0.260476i \(0.916120\pi\)
\(354\) −1.67925e138 −0.201575
\(355\) 5.02561e137 0.0517302
\(356\) 8.36438e138 0.738600
\(357\) −5.70980e138 −0.432717
\(358\) 4.15081e139 2.70090
\(359\) 1.90732e139 1.06605 0.533023 0.846101i \(-0.321056\pi\)
0.533023 + 0.846101i \(0.321056\pi\)
\(360\) 4.47032e134 2.14710e−5 0
\(361\) 4.58048e139 1.89134
\(362\) −4.22075e139 −1.49890
\(363\) −2.26260e139 −0.691348
\(364\) −1.76421e139 −0.464008
\(365\) −8.91651e137 −0.0201945
\(366\) −1.29424e140 −2.52519
\(367\) 7.10976e139 1.19551 0.597757 0.801678i \(-0.296059\pi\)
0.597757 + 0.801678i \(0.296059\pi\)
\(368\) −7.05259e139 −1.02245
\(369\) −7.32700e137 −0.00916198
\(370\) 8.15013e138 0.0879368
\(371\) 7.05392e139 0.656981
\(372\) −4.05746e139 −0.326335
\(373\) 1.48848e140 1.03421 0.517107 0.855921i \(-0.327009\pi\)
0.517107 + 0.855921i \(0.327009\pi\)
\(374\) −8.89070e139 −0.533868
\(375\) 2.49322e139 0.129436
\(376\) −1.45956e139 −0.0655366
\(377\) 9.13828e139 0.355026
\(378\) −2.62271e140 −0.881955
\(379\) 3.23786e140 0.942803 0.471402 0.881919i \(-0.343748\pi\)
0.471402 + 0.881919i \(0.343748\pi\)
\(380\) 4.18279e139 0.105502
\(381\) −3.60878e140 −0.788768
\(382\) 9.56852e140 1.81297
\(383\) −2.90202e138 −0.00476834 −0.00238417 0.999997i \(-0.500759\pi\)
−0.00238417 + 0.999997i \(0.500759\pi\)
\(384\) −8.94915e139 −0.127565
\(385\) 1.83231e139 0.0226667
\(386\) −7.27683e140 −0.781509
\(387\) −2.11603e138 −0.00197366
\(388\) 6.38040e140 0.517034
\(389\) −1.30244e141 −0.917287 −0.458644 0.888620i \(-0.651665\pi\)
−0.458644 + 0.888620i \(0.651665\pi\)
\(390\) 1.14415e140 0.0700589
\(391\) 1.26817e141 0.675376
\(392\) −8.34456e139 −0.0386645
\(393\) −6.03636e139 −0.0243434
\(394\) −5.95116e141 −2.08957
\(395\) −1.92925e140 −0.0589992
\(396\) −1.02331e139 −0.00272659
\(397\) 4.65632e140 0.108134 0.0540668 0.998537i \(-0.482782\pi\)
0.0540668 + 0.998537i \(0.482782\pi\)
\(398\) −6.38651e141 −1.29311
\(399\) 6.04364e141 1.06727
\(400\) 6.74533e141 1.03928
\(401\) 4.10237e141 0.551654 0.275827 0.961207i \(-0.411048\pi\)
0.275827 + 0.961207i \(0.411048\pi\)
\(402\) 1.48518e142 1.74365
\(403\) −2.58406e141 −0.264958
\(404\) −2.13842e141 −0.191561
\(405\) 8.26245e140 0.0646860
\(406\) 5.90515e141 0.404168
\(407\) 8.95332e141 0.535906
\(408\) 8.40643e140 0.0440182
\(409\) −3.31649e142 −1.51970 −0.759850 0.650098i \(-0.774728\pi\)
−0.759850 + 0.650098i \(0.774728\pi\)
\(410\) 4.02491e141 0.161449
\(411\) −2.54452e142 −0.893772
\(412\) −5.78468e142 −1.77984
\(413\) −3.37444e141 −0.0909754
\(414\) 2.98937e140 0.00706416
\(415\) 5.17852e141 0.107296
\(416\) 5.93190e142 1.07797
\(417\) −7.28136e142 −1.16091
\(418\) 9.41052e142 1.31676
\(419\) 1.21164e143 1.48837 0.744185 0.667973i \(-0.232838\pi\)
0.744185 + 0.667973i \(0.232838\pi\)
\(420\) 3.61014e141 0.0389437
\(421\) −1.96025e143 −1.85754 −0.928768 0.370661i \(-0.879131\pi\)
−0.928768 + 0.370661i \(0.879131\pi\)
\(422\) 1.62187e143 1.35048
\(423\) 7.21542e140 0.00528094
\(424\) −1.03853e142 −0.0668315
\(425\) −1.21292e143 −0.686494
\(426\) 2.22773e143 1.10927
\(427\) −2.60077e143 −1.13968
\(428\) −8.06273e142 −0.311026
\(429\) 1.25691e143 0.426955
\(430\) 1.16239e142 0.0347792
\(431\) 3.86754e143 1.01959 0.509793 0.860297i \(-0.329722\pi\)
0.509793 + 0.860297i \(0.329722\pi\)
\(432\) 4.50349e143 1.04637
\(433\) 7.82836e142 0.160354 0.0801771 0.996781i \(-0.474451\pi\)
0.0801771 + 0.996781i \(0.474451\pi\)
\(434\) −1.66982e143 −0.301633
\(435\) −1.86998e142 −0.0297970
\(436\) −1.16768e144 −1.64176
\(437\) −1.34232e144 −1.66578
\(438\) −3.95246e143 −0.433040
\(439\) 3.98186e143 0.385274 0.192637 0.981270i \(-0.438296\pi\)
0.192637 + 0.981270i \(0.438296\pi\)
\(440\) −2.69767e141 −0.00230578
\(441\) 4.12519e141 0.00311559
\(442\) −1.11560e144 −0.744724
\(443\) 2.91682e144 1.72150 0.860750 0.509028i \(-0.169995\pi\)
0.860750 + 0.509028i \(0.169995\pi\)
\(444\) 1.76404e144 0.920741
\(445\) −1.09007e143 −0.0503308
\(446\) −2.69570e144 −1.10134
\(447\) 7.88564e143 0.285152
\(448\) 1.78167e144 0.570396
\(449\) 4.18120e144 1.18543 0.592715 0.805413i \(-0.298056\pi\)
0.592715 + 0.805413i \(0.298056\pi\)
\(450\) −2.85913e142 −0.00718045
\(451\) 4.42157e144 0.983908
\(452\) 6.49900e142 0.0128175
\(453\) −1.40401e144 −0.245481
\(454\) −6.53602e144 −1.01337
\(455\) 2.29917e143 0.0316192
\(456\) −8.89793e143 −0.108569
\(457\) −5.85132e144 −0.633609 −0.316804 0.948491i \(-0.602610\pi\)
−0.316804 + 0.948491i \(0.602610\pi\)
\(458\) −8.47761e144 −0.814901
\(459\) −8.09804e144 −0.691175
\(460\) −8.01828e143 −0.0607826
\(461\) −1.72477e145 −1.16153 −0.580766 0.814071i \(-0.697247\pi\)
−0.580766 + 0.814071i \(0.697247\pi\)
\(462\) 8.12215e144 0.486053
\(463\) 2.23810e145 1.19046 0.595230 0.803555i \(-0.297061\pi\)
0.595230 + 0.803555i \(0.297061\pi\)
\(464\) −1.01398e145 −0.479514
\(465\) 5.28779e143 0.0222376
\(466\) 3.43489e145 1.28493
\(467\) 2.29371e145 0.763428 0.381714 0.924280i \(-0.375334\pi\)
0.381714 + 0.924280i \(0.375334\pi\)
\(468\) −1.28405e143 −0.00380349
\(469\) 2.98446e145 0.786948
\(470\) −3.96361e144 −0.0930589
\(471\) −7.06580e145 −1.47748
\(472\) 4.96812e143 0.00925450
\(473\) 1.27694e145 0.211952
\(474\) −8.55185e145 −1.26515
\(475\) 1.28384e146 1.69320
\(476\) −3.52005e145 −0.413970
\(477\) 5.13406e143 0.00538529
\(478\) 9.80036e145 0.917111
\(479\) −4.89548e145 −0.408799 −0.204400 0.978888i \(-0.565524\pi\)
−0.204400 + 0.978888i \(0.565524\pi\)
\(480\) −1.21385e145 −0.0904729
\(481\) 1.12346e146 0.747568
\(482\) −8.23208e145 −0.489154
\(483\) −1.15855e146 −0.614887
\(484\) −1.39487e146 −0.661398
\(485\) −8.31512e144 −0.0352325
\(486\) −3.80797e144 −0.0144217
\(487\) −1.96256e146 −0.664500 −0.332250 0.943191i \(-0.607808\pi\)
−0.332250 + 0.943191i \(0.607808\pi\)
\(488\) 3.82905e145 0.115934
\(489\) −2.10750e146 −0.570736
\(490\) −2.26607e145 −0.0549019
\(491\) −7.63740e146 −1.65579 −0.827893 0.560886i \(-0.810461\pi\)
−0.827893 + 0.560886i \(0.810461\pi\)
\(492\) 8.71166e146 1.69045
\(493\) 1.82331e146 0.316741
\(494\) 1.18083e147 1.83683
\(495\) 1.33361e143 0.000185800 0
\(496\) 2.86727e146 0.357864
\(497\) 4.47661e146 0.500640
\(498\) 2.29550e147 2.30080
\(499\) −1.09773e147 −0.986314 −0.493157 0.869940i \(-0.664157\pi\)
−0.493157 + 0.869940i \(0.664157\pi\)
\(500\) 1.53705e146 0.123829
\(501\) −2.24896e147 −1.62489
\(502\) −2.70661e147 −1.75418
\(503\) 7.26062e146 0.422200 0.211100 0.977464i \(-0.432295\pi\)
0.211100 + 0.977464i \(0.432295\pi\)
\(504\) 3.98198e143 0.000207795 0
\(505\) 2.78685e145 0.0130537
\(506\) −1.80397e147 −0.758622
\(507\) −1.06409e147 −0.401832
\(508\) −2.22478e147 −0.754597
\(509\) 4.39329e147 1.33866 0.669331 0.742964i \(-0.266581\pi\)
0.669331 + 0.742964i \(0.266581\pi\)
\(510\) 2.28287e146 0.0625039
\(511\) −7.94246e146 −0.195441
\(512\) −6.28000e147 −1.38914
\(513\) 8.57151e147 1.70475
\(514\) 9.75166e146 0.174417
\(515\) 7.53876e146 0.121285
\(516\) 2.51591e147 0.364155
\(517\) −4.35423e147 −0.567122
\(518\) 7.25979e147 0.851045
\(519\) 6.84996e146 0.0722881
\(520\) −3.38503e145 −0.00321647
\(521\) 3.17006e147 0.271276 0.135638 0.990758i \(-0.456692\pi\)
0.135638 + 0.990758i \(0.456692\pi\)
\(522\) 4.29795e145 0.00331298
\(523\) 4.11162e147 0.285543 0.142771 0.989756i \(-0.454399\pi\)
0.142771 + 0.989756i \(0.454399\pi\)
\(524\) −3.72136e146 −0.0232888
\(525\) 1.10807e148 0.625009
\(526\) −2.88495e148 −1.46695
\(527\) −5.15584e147 −0.236385
\(528\) −1.39467e148 −0.576663
\(529\) −1.08041e147 −0.0402954
\(530\) −2.82027e147 −0.0948978
\(531\) −2.45602e145 −0.000745729 0
\(532\) 3.72585e148 1.02104
\(533\) 5.54817e148 1.37251
\(534\) −4.83200e148 −1.07927
\(535\) 1.05076e147 0.0211944
\(536\) −4.39396e147 −0.0800525
\(537\) −1.17084e149 −1.92708
\(538\) −3.93307e148 −0.584922
\(539\) −2.48939e148 −0.334584
\(540\) 5.12014e147 0.0622045
\(541\) −8.28104e146 −0.00909565 −0.00454783 0.999990i \(-0.501448\pi\)
−0.00454783 + 0.999990i \(0.501448\pi\)
\(542\) 6.40114e148 0.635764
\(543\) 1.19057e149 1.06946
\(544\) 1.18356e149 0.961724
\(545\) 1.52175e148 0.111875
\(546\) 1.01916e149 0.678024
\(547\) 2.33897e149 1.40837 0.704183 0.710019i \(-0.251313\pi\)
0.704183 + 0.710019i \(0.251313\pi\)
\(548\) −1.56868e149 −0.855052
\(549\) −1.89292e147 −0.00934198
\(550\) 1.72537e149 0.771111
\(551\) −1.92992e149 −0.781226
\(552\) 1.70571e148 0.0625496
\(553\) −1.71849e149 −0.570989
\(554\) −3.40519e149 −1.02532
\(555\) −2.29895e148 −0.0627425
\(556\) −4.48889e149 −1.11061
\(557\) −6.72972e149 −1.50969 −0.754847 0.655901i \(-0.772289\pi\)
−0.754847 + 0.655901i \(0.772289\pi\)
\(558\) −1.21534e147 −0.00247249
\(559\) 1.60230e149 0.295665
\(560\) −2.55117e148 −0.0427062
\(561\) 2.50785e149 0.380913
\(562\) 1.58868e150 2.18982
\(563\) −1.56865e149 −0.196254 −0.0981270 0.995174i \(-0.531285\pi\)
−0.0981270 + 0.995174i \(0.531285\pi\)
\(564\) −8.57899e149 −0.974372
\(565\) −8.46968e146 −0.000873427 0
\(566\) 1.13539e150 1.06329
\(567\) 7.35985e149 0.626025
\(568\) −6.59081e148 −0.0509278
\(569\) 2.40146e149 0.168600 0.0842998 0.996440i \(-0.473135\pi\)
0.0842998 + 0.996440i \(0.473135\pi\)
\(570\) −2.41635e149 −0.154163
\(571\) −2.97299e149 −0.172395 −0.0861975 0.996278i \(-0.527472\pi\)
−0.0861975 + 0.996278i \(0.527472\pi\)
\(572\) 7.74874e149 0.408458
\(573\) −2.69904e150 −1.29355
\(574\) 3.58522e150 1.56249
\(575\) −2.46108e150 −0.975503
\(576\) 1.29676e148 0.00467555
\(577\) −3.73020e150 −1.22363 −0.611816 0.791000i \(-0.709561\pi\)
−0.611816 + 0.791000i \(0.709561\pi\)
\(578\) 2.45740e150 0.733515
\(579\) 2.05261e150 0.557604
\(580\) −1.15283e149 −0.0285061
\(581\) 4.61281e150 1.03840
\(582\) −3.68588e150 −0.755507
\(583\) −3.09821e150 −0.578328
\(584\) 1.16935e149 0.0198813
\(585\) 1.67341e147 0.000259184 0
\(586\) −1.10760e151 −1.56301
\(587\) −2.27443e149 −0.0292480 −0.0146240 0.999893i \(-0.504655\pi\)
−0.0146240 + 0.999893i \(0.504655\pi\)
\(588\) −4.90477e150 −0.574849
\(589\) 5.45728e150 0.583033
\(590\) 1.34916e149 0.0131410
\(591\) 1.67868e151 1.49090
\(592\) −1.24659e151 −1.00970
\(593\) 2.75411e150 0.203470 0.101735 0.994812i \(-0.467561\pi\)
0.101735 + 0.994812i \(0.467561\pi\)
\(594\) 1.15194e151 0.776369
\(595\) 4.58742e149 0.0282094
\(596\) 4.86143e150 0.272799
\(597\) 1.80148e151 0.922630
\(598\) −2.26361e151 −1.05825
\(599\) 2.69438e150 0.115000 0.0575000 0.998346i \(-0.481687\pi\)
0.0575000 + 0.998346i \(0.481687\pi\)
\(600\) −1.63139e150 −0.0635793
\(601\) 3.99687e151 1.42253 0.711265 0.702924i \(-0.248123\pi\)
0.711265 + 0.702924i \(0.248123\pi\)
\(602\) 1.03541e151 0.336591
\(603\) 2.17218e149 0.00645064
\(604\) −8.65560e150 −0.234847
\(605\) 1.81784e150 0.0450700
\(606\) 1.23534e151 0.279916
\(607\) −4.71998e151 −0.977590 −0.488795 0.872399i \(-0.662563\pi\)
−0.488795 + 0.872399i \(0.662563\pi\)
\(608\) −1.25276e152 −2.37205
\(609\) −1.66570e151 −0.288373
\(610\) 1.03983e151 0.164621
\(611\) −5.46367e151 −0.791113
\(612\) −2.56200e149 −0.00339333
\(613\) −6.97271e151 −0.844901 −0.422451 0.906386i \(-0.638830\pi\)
−0.422451 + 0.906386i \(0.638830\pi\)
\(614\) 8.97263e151 0.994817
\(615\) −1.13533e151 −0.115193
\(616\) −2.40297e150 −0.0223151
\(617\) 1.38408e152 1.17658 0.588289 0.808650i \(-0.299802\pi\)
0.588289 + 0.808650i \(0.299802\pi\)
\(618\) 3.34174e152 2.60076
\(619\) −2.77559e151 −0.197794 −0.0988972 0.995098i \(-0.531532\pi\)
−0.0988972 + 0.995098i \(0.531532\pi\)
\(620\) 3.25988e150 0.0212743
\(621\) −1.64313e152 −0.982155
\(622\) 3.68514e152 2.01780
\(623\) −9.70989e151 −0.487097
\(624\) −1.75003e152 −0.804422
\(625\) 2.34382e152 0.987334
\(626\) −3.63807e152 −1.40465
\(627\) −2.65448e152 −0.939502
\(628\) −4.35601e152 −1.41347
\(629\) 2.24158e152 0.666951
\(630\) 1.08136e149 0.000295059 0
\(631\) 5.40222e152 1.35199 0.675993 0.736908i \(-0.263715\pi\)
0.675993 + 0.736908i \(0.263715\pi\)
\(632\) 2.53010e151 0.0580840
\(633\) −4.57489e152 −0.963560
\(634\) 8.76063e152 1.69305
\(635\) 2.89940e151 0.0514209
\(636\) −6.10430e152 −0.993625
\(637\) −3.12368e152 −0.466732
\(638\) −2.59365e152 −0.355782
\(639\) 3.25821e150 0.00410376
\(640\) 7.19001e150 0.00831612
\(641\) −1.28863e153 −1.36888 −0.684441 0.729069i \(-0.739954\pi\)
−0.684441 + 0.729069i \(0.739954\pi\)
\(642\) 4.65774e152 0.454482
\(643\) 1.11168e153 0.996513 0.498256 0.867030i \(-0.333974\pi\)
0.498256 + 0.867030i \(0.333974\pi\)
\(644\) −7.14235e152 −0.588249
\(645\) −3.27881e151 −0.0248149
\(646\) 2.35605e153 1.63875
\(647\) −2.23161e153 −1.42671 −0.713354 0.700804i \(-0.752825\pi\)
−0.713354 + 0.700804i \(0.752825\pi\)
\(648\) −1.08357e152 −0.0636826
\(649\) 1.48211e152 0.0800840
\(650\) 2.16499e153 1.07567
\(651\) 4.71014e152 0.215214
\(652\) −1.29926e153 −0.546011
\(653\) 2.84275e153 1.09893 0.549466 0.835516i \(-0.314831\pi\)
0.549466 + 0.835516i \(0.314831\pi\)
\(654\) 6.74554e153 2.39899
\(655\) 4.84978e150 0.00158698
\(656\) −6.15625e153 −1.85377
\(657\) −5.78076e150 −0.00160204
\(658\) −3.53062e153 −0.900617
\(659\) 2.20008e153 0.516635 0.258317 0.966060i \(-0.416832\pi\)
0.258317 + 0.966060i \(0.416832\pi\)
\(660\) −1.58564e152 −0.0342814
\(661\) −8.61845e153 −1.71573 −0.857867 0.513872i \(-0.828210\pi\)
−0.857867 + 0.513872i \(0.828210\pi\)
\(662\) −5.94205e153 −1.08937
\(663\) 3.14684e153 0.531358
\(664\) −6.79134e152 −0.105632
\(665\) −4.85564e152 −0.0695772
\(666\) 5.28390e151 0.00697604
\(667\) 3.69959e153 0.450087
\(668\) −1.38646e154 −1.55450
\(669\) 7.60392e153 0.785803
\(670\) −1.19324e153 −0.113671
\(671\) 1.14230e154 1.00324
\(672\) −1.08125e154 −0.875589
\(673\) 6.72013e153 0.501830 0.250915 0.968009i \(-0.419269\pi\)
0.250915 + 0.968009i \(0.419269\pi\)
\(674\) 3.27079e154 2.25262
\(675\) 1.57155e154 0.998324
\(676\) −6.56003e153 −0.384424
\(677\) −8.09368e153 −0.437586 −0.218793 0.975771i \(-0.570212\pi\)
−0.218793 + 0.975771i \(0.570212\pi\)
\(678\) −3.75439e152 −0.0187293
\(679\) −7.40677e153 −0.340978
\(680\) −6.75397e151 −0.00286961
\(681\) 1.84365e154 0.723038
\(682\) 7.33413e153 0.265522
\(683\) −3.88180e154 −1.29749 −0.648746 0.761005i \(-0.724706\pi\)
−0.648746 + 0.761005i \(0.724706\pi\)
\(684\) 2.71179e152 0.00836948
\(685\) 2.04434e153 0.0582663
\(686\) −5.36047e154 −1.41104
\(687\) 2.39133e154 0.581429
\(688\) −1.77791e154 −0.399338
\(689\) −3.88762e154 −0.806745
\(690\) 4.63206e153 0.0888176
\(691\) −3.61696e154 −0.640900 −0.320450 0.947265i \(-0.603834\pi\)
−0.320450 + 0.947265i \(0.603834\pi\)
\(692\) 4.22294e153 0.0691565
\(693\) 1.18792e152 0.00179816
\(694\) 1.61537e155 2.26037
\(695\) 5.85006e153 0.0756812
\(696\) 2.45237e153 0.0293348
\(697\) 1.10700e155 1.22450
\(698\) −1.26629e155 −1.29543
\(699\) −9.68898e154 −0.916794
\(700\) 6.83118e154 0.597933
\(701\) −1.75193e155 −1.41868 −0.709340 0.704867i \(-0.751007\pi\)
−0.709340 + 0.704867i \(0.751007\pi\)
\(702\) 1.44545e155 1.08300
\(703\) −2.37264e155 −1.64500
\(704\) −7.82543e154 −0.502109
\(705\) 1.11804e154 0.0663972
\(706\) 4.91038e155 2.69935
\(707\) 2.48241e154 0.126332
\(708\) 2.92016e154 0.137592
\(709\) 3.25375e155 1.41959 0.709796 0.704407i \(-0.248787\pi\)
0.709796 + 0.704407i \(0.248787\pi\)
\(710\) −1.78982e154 −0.0723151
\(711\) −1.25077e153 −0.00468042
\(712\) 1.42957e154 0.0495501
\(713\) −1.04614e155 −0.335902
\(714\) 2.03349e155 0.604907
\(715\) −1.00984e154 −0.0278338
\(716\) −7.21814e155 −1.84360
\(717\) −2.76444e155 −0.654355
\(718\) −6.79272e155 −1.49026
\(719\) 2.80951e154 0.0571354 0.0285677 0.999592i \(-0.490905\pi\)
0.0285677 + 0.999592i \(0.490905\pi\)
\(720\) −1.85682e152 −0.000350064 0
\(721\) 6.71522e155 1.17378
\(722\) −1.63129e156 −2.64396
\(723\) 2.32207e155 0.349010
\(724\) 7.33977e155 1.02313
\(725\) −3.53841e155 −0.457496
\(726\) 8.05801e155 0.966456
\(727\) 3.43575e155 0.382293 0.191147 0.981561i \(-0.438779\pi\)
0.191147 + 0.981561i \(0.438779\pi\)
\(728\) −3.01524e154 −0.0311287
\(729\) 1.04921e156 1.00510
\(730\) 3.17552e154 0.0282305
\(731\) 3.19699e155 0.263781
\(732\) 2.25064e156 1.72366
\(733\) −5.00657e154 −0.0355938 −0.0177969 0.999842i \(-0.505665\pi\)
−0.0177969 + 0.999842i \(0.505665\pi\)
\(734\) −2.53207e156 −1.67124
\(735\) 6.39203e154 0.0391723
\(736\) 2.40150e156 1.36660
\(737\) −1.31083e156 −0.692736
\(738\) 2.60944e154 0.0128078
\(739\) 1.34210e156 0.611875 0.305937 0.952052i \(-0.401030\pi\)
0.305937 + 0.952052i \(0.401030\pi\)
\(740\) −1.41728e155 −0.0600244
\(741\) −3.33083e156 −1.31057
\(742\) −2.51218e156 −0.918413
\(743\) 9.29385e155 0.315722 0.157861 0.987461i \(-0.449540\pi\)
0.157861 + 0.987461i \(0.449540\pi\)
\(744\) −6.93465e154 −0.0218927
\(745\) −6.33555e154 −0.0185895
\(746\) −5.30105e156 −1.44576
\(747\) 3.35734e154 0.00851182
\(748\) 1.54607e156 0.364411
\(749\) 9.35972e155 0.205118
\(750\) −8.87932e155 −0.180943
\(751\) −2.43545e156 −0.461533 −0.230766 0.973009i \(-0.574123\pi\)
−0.230766 + 0.973009i \(0.574123\pi\)
\(752\) 6.06249e156 1.06851
\(753\) 7.63468e156 1.25160
\(754\) −3.25450e156 −0.496302
\(755\) 1.12802e155 0.0160033
\(756\) 4.56081e156 0.602010
\(757\) −5.10211e156 −0.626649 −0.313325 0.949646i \(-0.601443\pi\)
−0.313325 + 0.949646i \(0.601443\pi\)
\(758\) −1.15313e157 −1.31797
\(759\) 5.08855e156 0.541274
\(760\) 7.14885e154 0.00707776
\(761\) 1.33251e156 0.122802 0.0614011 0.998113i \(-0.480443\pi\)
0.0614011 + 0.998113i \(0.480443\pi\)
\(762\) 1.28523e157 1.10264
\(763\) 1.35551e157 1.08272
\(764\) −1.66394e157 −1.23751
\(765\) 3.33887e153 0.000231234 0
\(766\) 1.03352e155 0.00666580
\(767\) 1.85975e156 0.111714
\(768\) 1.93452e157 1.08240
\(769\) −2.29704e157 −1.19725 −0.598623 0.801031i \(-0.704285\pi\)
−0.598623 + 0.801031i \(0.704285\pi\)
\(770\) −6.52557e155 −0.0316865
\(771\) −2.75070e156 −0.124446
\(772\) 1.26542e157 0.533447
\(773\) −3.91504e156 −0.153799 −0.0768995 0.997039i \(-0.524502\pi\)
−0.0768995 + 0.997039i \(0.524502\pi\)
\(774\) 7.53600e154 0.00275904
\(775\) 1.00057e157 0.341431
\(776\) 1.09048e156 0.0346861
\(777\) −2.04781e157 −0.607217
\(778\) 4.63851e157 1.28230
\(779\) −1.17172e158 −3.02018
\(780\) −1.98965e156 −0.0478213
\(781\) −1.96621e157 −0.440705
\(782\) −4.51647e157 −0.944128
\(783\) −2.36241e157 −0.460616
\(784\) 3.46604e157 0.630388
\(785\) 5.67687e156 0.0963192
\(786\) 2.14978e156 0.0340303
\(787\) 7.06698e157 1.04378 0.521892 0.853011i \(-0.325226\pi\)
0.521892 + 0.853011i \(0.325226\pi\)
\(788\) 1.03489e158 1.42631
\(789\) 8.13773e157 1.04666
\(790\) 6.87081e156 0.0824767
\(791\) −7.54444e155 −0.00845296
\(792\) −1.74896e154 −0.000182918 0
\(793\) 1.43336e158 1.39948
\(794\) −1.65830e157 −0.151163
\(795\) 7.95529e156 0.0677092
\(796\) 1.11060e158 0.882660
\(797\) −3.65107e157 −0.270982 −0.135491 0.990779i \(-0.543261\pi\)
−0.135491 + 0.990779i \(0.543261\pi\)
\(798\) −2.15238e158 −1.49197
\(799\) −1.09014e158 −0.705800
\(800\) −2.29688e158 −1.38910
\(801\) −7.06716e155 −0.00399275
\(802\) −1.46102e158 −0.771173
\(803\) 3.48847e157 0.172043
\(804\) −2.58268e158 −1.19019
\(805\) 9.30811e156 0.0400854
\(806\) 9.20285e157 0.370393
\(807\) 1.10942e158 0.417340
\(808\) −3.65479e156 −0.0128512
\(809\) 5.80414e158 1.90785 0.953923 0.300053i \(-0.0970043\pi\)
0.953923 + 0.300053i \(0.0970043\pi\)
\(810\) −2.94258e157 −0.0904264
\(811\) 1.75473e158 0.504165 0.252083 0.967706i \(-0.418885\pi\)
0.252083 + 0.967706i \(0.418885\pi\)
\(812\) −1.02689e158 −0.275880
\(813\) −1.80560e158 −0.453615
\(814\) −3.18863e158 −0.749159
\(815\) 1.69323e157 0.0372071
\(816\) −3.49174e158 −0.717675
\(817\) −3.38391e158 −0.650603
\(818\) 1.18113e159 2.12443
\(819\) 1.49060e156 0.00250836
\(820\) −6.99921e157 −0.110203
\(821\) 4.54576e158 0.669737 0.334869 0.942265i \(-0.391308\pi\)
0.334869 + 0.942265i \(0.391308\pi\)
\(822\) 9.06204e158 1.24943
\(823\) −6.60073e158 −0.851728 −0.425864 0.904787i \(-0.640030\pi\)
−0.425864 + 0.904787i \(0.640030\pi\)
\(824\) −9.88667e157 −0.119403
\(825\) −4.86686e158 −0.550184
\(826\) 1.20177e158 0.127177
\(827\) 3.88758e158 0.385149 0.192575 0.981282i \(-0.438316\pi\)
0.192575 + 0.981282i \(0.438316\pi\)
\(828\) −5.19842e156 −0.00482190
\(829\) −9.56043e158 −0.830339 −0.415170 0.909744i \(-0.636278\pi\)
−0.415170 + 0.909744i \(0.636278\pi\)
\(830\) −1.84427e158 −0.149992
\(831\) 9.60521e158 0.731562
\(832\) −9.81932e158 −0.700423
\(833\) −6.23252e158 −0.416400
\(834\) 2.59318e159 1.62287
\(835\) 1.80688e158 0.105929
\(836\) −1.63646e159 −0.898801
\(837\) 6.68025e158 0.343760
\(838\) −4.31515e159 −2.08064
\(839\) 1.42210e158 0.0642545 0.0321272 0.999484i \(-0.489772\pi\)
0.0321272 + 0.999484i \(0.489772\pi\)
\(840\) 6.17013e156 0.00261260
\(841\) −1.98798e159 −0.788916
\(842\) 6.98123e159 2.59671
\(843\) −4.48129e159 −1.56243
\(844\) −2.82038e159 −0.921817
\(845\) 8.54922e157 0.0261960
\(846\) −2.56969e157 −0.00738238
\(847\) 1.61926e159 0.436184
\(848\) 4.31371e159 1.08962
\(849\) −3.20266e159 −0.758651
\(850\) 4.31970e159 0.959670
\(851\) 4.54828e159 0.947733
\(852\) −3.87395e159 −0.757175
\(853\) 4.56241e159 0.836512 0.418256 0.908329i \(-0.362642\pi\)
0.418256 + 0.908329i \(0.362642\pi\)
\(854\) 9.26236e159 1.59319
\(855\) −3.53408e156 −0.000570326 0
\(856\) −1.37801e158 −0.0208657
\(857\) −4.25587e157 −0.00604692 −0.00302346 0.999995i \(-0.500962\pi\)
−0.00302346 + 0.999995i \(0.500962\pi\)
\(858\) −4.47635e159 −0.596853
\(859\) 5.71553e159 0.715203 0.357601 0.933874i \(-0.383595\pi\)
0.357601 + 0.933874i \(0.383595\pi\)
\(860\) −2.02136e158 −0.0237398
\(861\) −1.01130e160 −1.11483
\(862\) −1.37738e160 −1.42531
\(863\) 1.02491e159 0.0995632 0.0497816 0.998760i \(-0.484147\pi\)
0.0497816 + 0.998760i \(0.484147\pi\)
\(864\) −1.53350e160 −1.39857
\(865\) −5.50345e157 −0.00471257
\(866\) −2.78799e159 −0.224164
\(867\) −6.93173e159 −0.523360
\(868\) 2.90376e159 0.205890
\(869\) 7.54792e159 0.502631
\(870\) 6.65973e158 0.0416541
\(871\) −1.64482e160 −0.966340
\(872\) −1.99569e159 −0.110140
\(873\) −5.39087e157 −0.00279501
\(874\) 4.78054e160 2.32865
\(875\) −1.78430e159 −0.0816636
\(876\) 6.87321e159 0.295587
\(877\) 2.15553e160 0.871118 0.435559 0.900160i \(-0.356551\pi\)
0.435559 + 0.900160i \(0.356551\pi\)
\(878\) −1.41810e160 −0.538586
\(879\) 3.12426e160 1.11520
\(880\) 1.12052e159 0.0375935
\(881\) −1.34416e160 −0.423901 −0.211950 0.977280i \(-0.567981\pi\)
−0.211950 + 0.977280i \(0.567981\pi\)
\(882\) −1.46914e158 −0.00435538
\(883\) 7.25500e159 0.202199 0.101100 0.994876i \(-0.467764\pi\)
0.101100 + 0.994876i \(0.467764\pi\)
\(884\) 1.94000e160 0.508339
\(885\) −3.80564e158 −0.00937603
\(886\) −1.03880e161 −2.40654
\(887\) −2.66568e160 −0.580725 −0.290362 0.956917i \(-0.593776\pi\)
−0.290362 + 0.956917i \(0.593776\pi\)
\(888\) 3.01495e159 0.0617693
\(889\) 2.58266e160 0.497647
\(890\) 3.88217e159 0.0703589
\(891\) −3.23258e160 −0.551079
\(892\) 4.68775e160 0.751761
\(893\) 1.15388e161 1.74082
\(894\) −2.80839e160 −0.398623
\(895\) 9.40689e159 0.125629
\(896\) 6.40456e159 0.0804827
\(897\) 6.38509e160 0.755056
\(898\) −1.48909e161 −1.65715
\(899\) −1.50409e160 −0.157533
\(900\) 4.97194e158 0.00490128
\(901\) −7.75677e160 −0.719747
\(902\) −1.57469e161 −1.37543
\(903\) −2.92063e160 −0.240156
\(904\) 1.11075e158 0.000859879 0
\(905\) −9.56539e159 −0.0697197
\(906\) 5.00024e160 0.343166
\(907\) 4.81256e160 0.311013 0.155507 0.987835i \(-0.450299\pi\)
0.155507 + 0.987835i \(0.450299\pi\)
\(908\) 1.13660e161 0.691715
\(909\) 1.80677e158 0.00103555
\(910\) −8.18827e159 −0.0442014
\(911\) 3.02239e160 0.153673 0.0768366 0.997044i \(-0.475518\pi\)
0.0768366 + 0.997044i \(0.475518\pi\)
\(912\) 3.69589e161 1.77011
\(913\) −2.02603e161 −0.914087
\(914\) 2.08389e161 0.885740
\(915\) −2.93310e160 −0.117457
\(916\) 1.47423e161 0.556240
\(917\) 4.31999e159 0.0153587
\(918\) 2.88403e161 0.966215
\(919\) −2.95847e161 −0.934053 −0.467027 0.884243i \(-0.654675\pi\)
−0.467027 + 0.884243i \(0.654675\pi\)
\(920\) −1.37041e159 −0.00407770
\(921\) −2.53096e161 −0.709798
\(922\) 6.14259e161 1.62374
\(923\) −2.46719e161 −0.614766
\(924\) −1.41242e161 −0.331773
\(925\) −4.35013e161 −0.963335
\(926\) −7.97075e161 −1.66418
\(927\) 4.88754e159 0.00962154
\(928\) 3.45275e161 0.640916
\(929\) −4.06546e160 −0.0711629 −0.0355815 0.999367i \(-0.511328\pi\)
−0.0355815 + 0.999367i \(0.511328\pi\)
\(930\) −1.88319e160 −0.0310866
\(931\) 6.59692e161 1.02703
\(932\) −5.97317e161 −0.877077
\(933\) −1.03949e162 −1.43969
\(934\) −8.16880e161 −1.06722
\(935\) −2.01488e160 −0.0248322
\(936\) −2.19458e158 −0.000255163 0
\(937\) 1.78882e162 1.96228 0.981138 0.193311i \(-0.0619226\pi\)
0.981138 + 0.193311i \(0.0619226\pi\)
\(938\) −1.06288e162 −1.10010
\(939\) 1.02621e162 1.00222
\(940\) 6.89261e160 0.0635207
\(941\) 6.89283e160 0.0599464 0.0299732 0.999551i \(-0.490458\pi\)
0.0299732 + 0.999551i \(0.490458\pi\)
\(942\) 2.51641e162 2.06542
\(943\) 2.24615e162 1.74001
\(944\) −2.06358e161 −0.150886
\(945\) −5.94378e160 −0.0410231
\(946\) −4.54769e161 −0.296295
\(947\) −1.13965e162 −0.700965 −0.350482 0.936569i \(-0.613982\pi\)
−0.350482 + 0.936569i \(0.613982\pi\)
\(948\) 1.48714e162 0.863571
\(949\) 4.37732e161 0.239993
\(950\) −4.57226e162 −2.36698
\(951\) −2.47116e162 −1.20799
\(952\) −6.01615e160 −0.0277719
\(953\) 1.50922e162 0.657944 0.328972 0.944340i \(-0.393298\pi\)
0.328972 + 0.944340i \(0.393298\pi\)
\(954\) −1.82844e160 −0.00752826
\(955\) 2.16849e161 0.0843284
\(956\) −1.70425e162 −0.626007
\(957\) 7.31604e161 0.253849
\(958\) 1.74347e162 0.571473
\(959\) 1.82102e162 0.563897
\(960\) 2.00934e161 0.0587857
\(961\) −3.19232e162 −0.882432
\(962\) −4.00108e162 −1.04505
\(963\) 6.81229e159 0.00168136
\(964\) 1.43153e162 0.333890
\(965\) −1.64913e161 −0.0363510
\(966\) 4.12605e162 0.859569
\(967\) 5.66771e162 1.11600 0.558000 0.829841i \(-0.311569\pi\)
0.558000 + 0.829841i \(0.311569\pi\)
\(968\) −2.38400e161 −0.0443709
\(969\) −6.64583e162 −1.16924
\(970\) 2.96134e161 0.0492526
\(971\) −8.55654e162 −1.34540 −0.672698 0.739917i \(-0.734865\pi\)
−0.672698 + 0.739917i \(0.734865\pi\)
\(972\) 6.62196e160 0.00984407
\(973\) 5.21099e162 0.732437
\(974\) 6.98945e162 0.928924
\(975\) −6.10691e162 −0.767486
\(976\) −1.59046e163 −1.89019
\(977\) 1.12692e163 1.26660 0.633300 0.773906i \(-0.281700\pi\)
0.633300 + 0.773906i \(0.281700\pi\)
\(978\) 7.50565e162 0.797849
\(979\) 4.26476e162 0.428783
\(980\) 3.94063e161 0.0374753
\(981\) 9.86584e160 0.00887510
\(982\) 2.71998e163 2.31467
\(983\) 2.66629e160 0.00214655 0.00107327 0.999999i \(-0.499658\pi\)
0.00107327 + 0.999999i \(0.499658\pi\)
\(984\) 1.48892e162 0.113407
\(985\) −1.34870e162 −0.0971940
\(986\) −6.49354e162 −0.442781
\(987\) 9.95902e162 0.642587
\(988\) −2.05343e163 −1.25379
\(989\) 6.48685e162 0.374831
\(990\) −4.74951e159 −0.000259735 0
\(991\) −3.68903e163 −1.90940 −0.954702 0.297562i \(-0.903826\pi\)
−0.954702 + 0.297562i \(0.903826\pi\)
\(992\) −9.76345e162 −0.478319
\(993\) 1.67611e163 0.777262
\(994\) −1.59430e163 −0.699860
\(995\) −1.44736e162 −0.0601476
\(996\) −3.99181e163 −1.57049
\(997\) −3.31436e163 −1.23456 −0.617282 0.786742i \(-0.711766\pi\)
−0.617282 + 0.786742i \(0.711766\pi\)
\(998\) 3.90944e163 1.37880
\(999\) −2.90434e163 −0.969904
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 1.110.a.a.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.110.a.a.1.2 8 1.1 even 1 trivial