Properties

Label 10.24.a.a.1.2
Level $10$
Weight $24$
Character 10.1
Self dual yes
Analytic conductor $33.520$
Analytic rank $1$
Dimension $2$
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] = [10,24,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(33.5204037345\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{219241}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 54810 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-233.616\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2048.00 q^{2} +229540. q^{3} +4.19430e6 q^{4} +4.88281e7 q^{5} -4.70098e8 q^{6} +1.11945e9 q^{7} -8.58993e9 q^{8} -4.14545e10 q^{9} -1.00000e11 q^{10} -9.54047e11 q^{11} +9.62761e11 q^{12} -2.42139e12 q^{13} -2.29263e12 q^{14} +1.12080e13 q^{15} +1.75922e13 q^{16} -1.35256e14 q^{17} +8.48988e13 q^{18} +8.28157e14 q^{19} +2.04800e14 q^{20} +2.56958e14 q^{21} +1.95389e15 q^{22} -7.88972e13 q^{23} -1.97174e15 q^{24} +2.38419e15 q^{25} +4.95901e15 q^{26} -3.11251e16 q^{27} +4.69531e15 q^{28} -3.21917e16 q^{29} -2.29540e16 q^{30} -1.85689e17 q^{31} -3.60288e16 q^{32} -2.18992e17 q^{33} +2.77004e17 q^{34} +5.46606e16 q^{35} -1.73873e17 q^{36} -3.74188e17 q^{37} -1.69607e18 q^{38} -5.55806e17 q^{39} -4.19430e17 q^{40} -3.22679e18 q^{41} -5.26251e17 q^{42} -2.74344e18 q^{43} -4.00156e18 q^{44} -2.02414e18 q^{45} +1.61582e17 q^{46} +5.25271e17 q^{47} +4.03811e18 q^{48} -2.61156e19 q^{49} -4.88281e18 q^{50} -3.10466e19 q^{51} -1.01560e19 q^{52} -4.40675e19 q^{53} +6.37442e19 q^{54} -4.65843e19 q^{55} -9.61599e18 q^{56} +1.90095e20 q^{57} +6.59286e19 q^{58} +1.56115e20 q^{59} +4.70098e19 q^{60} +5.13552e20 q^{61} +3.80291e20 q^{62} -4.64061e19 q^{63} +7.37870e19 q^{64} -1.18232e20 q^{65} +4.48496e20 q^{66} -7.59236e19 q^{67} -5.67304e20 q^{68} -1.81101e19 q^{69} -1.11945e20 q^{70} -4.54949e20 q^{71} +3.56091e20 q^{72} -4.50124e21 q^{73} +7.66337e20 q^{74} +5.47267e20 q^{75} +3.47354e21 q^{76} -1.06801e21 q^{77} +1.13829e21 q^{78} -1.03553e22 q^{79} +8.58993e20 q^{80} -3.24181e21 q^{81} +6.60847e21 q^{82} -5.91206e21 q^{83} +1.07776e21 q^{84} -6.60428e21 q^{85} +5.61857e21 q^{86} -7.38929e21 q^{87} +8.19520e21 q^{88} -2.39717e22 q^{89} +4.14545e21 q^{90} -2.71062e21 q^{91} -3.30919e20 q^{92} -4.26231e22 q^{93} -1.07575e21 q^{94} +4.04373e22 q^{95} -8.27006e21 q^{96} +1.24537e23 q^{97} +5.34847e22 q^{98} +3.95495e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4096 q^{2} - 18516 q^{3} + 8388608 q^{4} + 97656250 q^{5} + 37920768 q^{6} - 4415735108 q^{7} - 17179869184 q^{8} - 74065768326 q^{9} - 200000000000 q^{10} + 195017239824 q^{11} - 77661732864 q^{12}+ \cdots + 20\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2048.00 −0.707107
\(3\) 229540. 0.748108 0.374054 0.927407i \(-0.377967\pi\)
0.374054 + 0.927407i \(0.377967\pi\)
\(4\) 4.19430e6 0.500000
\(5\) 4.88281e7 0.447214
\(6\) −4.70098e8 −0.528992
\(7\) 1.11945e9 0.213982 0.106991 0.994260i \(-0.465878\pi\)
0.106991 + 0.994260i \(0.465878\pi\)
\(8\) −8.58993e9 −0.353553
\(9\) −4.14545e10 −0.440334
\(10\) −1.00000e11 −0.316228
\(11\) −9.54047e11 −1.00822 −0.504108 0.863640i \(-0.668179\pi\)
−0.504108 + 0.863640i \(0.668179\pi\)
\(12\) 9.62761e11 0.374054
\(13\) −2.42139e12 −0.374728 −0.187364 0.982291i \(-0.559994\pi\)
−0.187364 + 0.982291i \(0.559994\pi\)
\(14\) −2.29263e12 −0.151308
\(15\) 1.12080e13 0.334564
\(16\) 1.75922e13 0.250000
\(17\) −1.35256e14 −0.957179 −0.478589 0.878039i \(-0.658852\pi\)
−0.478589 + 0.878039i \(0.658852\pi\)
\(18\) 8.48988e13 0.311363
\(19\) 8.28157e14 1.63097 0.815485 0.578778i \(-0.196470\pi\)
0.815485 + 0.578778i \(0.196470\pi\)
\(20\) 2.04800e14 0.223607
\(21\) 2.56958e14 0.160081
\(22\) 1.95389e15 0.712917
\(23\) −7.88972e13 −0.0172660 −0.00863300 0.999963i \(-0.502748\pi\)
−0.00863300 + 0.999963i \(0.502748\pi\)
\(24\) −1.97174e15 −0.264496
\(25\) 2.38419e15 0.200000
\(26\) 4.95901e15 0.264973
\(27\) −3.11251e16 −1.07753
\(28\) 4.69531e15 0.106991
\(29\) −3.21917e16 −0.489967 −0.244983 0.969527i \(-0.578783\pi\)
−0.244983 + 0.969527i \(0.578783\pi\)
\(30\) −2.29540e16 −0.236573
\(31\) −1.85689e17 −1.31258 −0.656291 0.754508i \(-0.727876\pi\)
−0.656291 + 0.754508i \(0.727876\pi\)
\(32\) −3.60288e16 −0.176777
\(33\) −2.18992e17 −0.754255
\(34\) 2.77004e17 0.676827
\(35\) 5.46606e16 0.0956955
\(36\) −1.73873e17 −0.220167
\(37\) −3.74188e17 −0.345756 −0.172878 0.984943i \(-0.555307\pi\)
−0.172878 + 0.984943i \(0.555307\pi\)
\(38\) −1.69607e18 −1.15327
\(39\) −5.55806e17 −0.280337
\(40\) −4.19430e17 −0.158114
\(41\) −3.22679e18 −0.915707 −0.457853 0.889028i \(-0.651382\pi\)
−0.457853 + 0.889028i \(0.651382\pi\)
\(42\) −5.26251e17 −0.113195
\(43\) −2.74344e18 −0.450203 −0.225102 0.974335i \(-0.572271\pi\)
−0.225102 + 0.974335i \(0.572271\pi\)
\(44\) −4.00156e18 −0.504108
\(45\) −2.02414e18 −0.196923
\(46\) 1.61582e17 0.0122089
\(47\) 5.25271e17 0.0309926 0.0154963 0.999880i \(-0.495067\pi\)
0.0154963 + 0.999880i \(0.495067\pi\)
\(48\) 4.03811e18 0.187027
\(49\) −2.61156e19 −0.954212
\(50\) −4.88281e18 −0.141421
\(51\) −3.10466e19 −0.716073
\(52\) −1.01560e19 −0.187364
\(53\) −4.40675e19 −0.653050 −0.326525 0.945189i \(-0.605878\pi\)
−0.326525 + 0.945189i \(0.605878\pi\)
\(54\) 6.37442e19 0.761926
\(55\) −4.65843e19 −0.450888
\(56\) −9.61599e18 −0.0756539
\(57\) 1.90095e20 1.22014
\(58\) 6.59286e19 0.346459
\(59\) 1.56115e20 0.673979 0.336989 0.941508i \(-0.390591\pi\)
0.336989 + 0.941508i \(0.390591\pi\)
\(60\) 4.70098e19 0.167282
\(61\) 5.13552e20 1.51109 0.755546 0.655095i \(-0.227372\pi\)
0.755546 + 0.655095i \(0.227372\pi\)
\(62\) 3.80291e20 0.928136
\(63\) −4.64061e19 −0.0942235
\(64\) 7.37870e19 0.125000
\(65\) −1.18232e20 −0.167583
\(66\) 4.48496e20 0.533339
\(67\) −7.59236e19 −0.0759480 −0.0379740 0.999279i \(-0.512090\pi\)
−0.0379740 + 0.999279i \(0.512090\pi\)
\(68\) −5.67304e20 −0.478589
\(69\) −1.81101e19 −0.0129168
\(70\) −1.11945e20 −0.0676669
\(71\) −4.54949e20 −0.233610 −0.116805 0.993155i \(-0.537265\pi\)
−0.116805 + 0.993155i \(0.537265\pi\)
\(72\) 3.56091e20 0.155682
\(73\) −4.50124e21 −1.67927 −0.839633 0.543154i \(-0.817230\pi\)
−0.839633 + 0.543154i \(0.817230\pi\)
\(74\) 7.66337e20 0.244487
\(75\) 5.47267e20 0.149622
\(76\) 3.47354e21 0.815485
\(77\) −1.06801e21 −0.215740
\(78\) 1.13829e21 0.198228
\(79\) −1.03553e22 −1.55759 −0.778794 0.627280i \(-0.784168\pi\)
−0.778794 + 0.627280i \(0.784168\pi\)
\(80\) 8.58993e20 0.111803
\(81\) −3.24181e21 −0.365771
\(82\) 6.60847e21 0.647503
\(83\) −5.91206e21 −0.503895 −0.251948 0.967741i \(-0.581071\pi\)
−0.251948 + 0.967741i \(0.581071\pi\)
\(84\) 1.07776e21 0.0800407
\(85\) −6.60428e21 −0.428063
\(86\) 5.61857e21 0.318342
\(87\) −7.38929e21 −0.366548
\(88\) 8.19520e21 0.356458
\(89\) −2.39717e22 −0.915617 −0.457808 0.889051i \(-0.651365\pi\)
−0.457808 + 0.889051i \(0.651365\pi\)
\(90\) 4.14545e21 0.139246
\(91\) −2.71062e21 −0.0801849
\(92\) −3.30919e20 −0.00863300
\(93\) −4.26231e22 −0.981954
\(94\) −1.07575e21 −0.0219151
\(95\) 4.04373e22 0.729392
\(96\) −8.27006e21 −0.132248
\(97\) 1.24537e23 1.76776 0.883878 0.467717i \(-0.154923\pi\)
0.883878 + 0.467717i \(0.154923\pi\)
\(98\) 5.34847e22 0.674730
\(99\) 3.95495e22 0.443952
\(100\) 1.00000e22 0.100000
\(101\) −1.49773e23 −1.33579 −0.667893 0.744257i \(-0.732804\pi\)
−0.667893 + 0.744257i \(0.732804\pi\)
\(102\) 6.35835e22 0.506340
\(103\) 2.00999e23 1.43076 0.715380 0.698736i \(-0.246254\pi\)
0.715380 + 0.698736i \(0.246254\pi\)
\(104\) 2.07996e22 0.132486
\(105\) 1.25468e22 0.0715906
\(106\) 9.02503e22 0.461776
\(107\) 1.12988e23 0.518943 0.259471 0.965751i \(-0.416452\pi\)
0.259471 + 0.965751i \(0.416452\pi\)
\(108\) −1.30548e23 −0.538763
\(109\) −2.77607e23 −1.03045 −0.515224 0.857056i \(-0.672291\pi\)
−0.515224 + 0.857056i \(0.672291\pi\)
\(110\) 9.54047e22 0.318826
\(111\) −8.58912e22 −0.258663
\(112\) 1.96935e22 0.0534954
\(113\) 5.91159e22 0.144978 0.0724890 0.997369i \(-0.476906\pi\)
0.0724890 + 0.997369i \(0.476906\pi\)
\(114\) −3.89315e23 −0.862771
\(115\) −3.85240e21 −0.00772159
\(116\) −1.35022e23 −0.244983
\(117\) 1.00377e23 0.165006
\(118\) −3.19723e23 −0.476575
\(119\) −1.51412e23 −0.204819
\(120\) −9.62761e22 −0.118286
\(121\) 1.47751e22 0.0165006
\(122\) −1.05175e24 −1.06850
\(123\) −7.40678e23 −0.685048
\(124\) −7.78836e23 −0.656291
\(125\) 1.16415e23 0.0894427
\(126\) 9.50398e22 0.0666261
\(127\) 1.52582e24 0.976696 0.488348 0.872649i \(-0.337600\pi\)
0.488348 + 0.872649i \(0.337600\pi\)
\(128\) −1.51116e23 −0.0883883
\(129\) −6.29730e23 −0.336801
\(130\) 2.42139e23 0.118499
\(131\) 1.92350e24 0.861930 0.430965 0.902369i \(-0.358173\pi\)
0.430965 + 0.902369i \(0.358173\pi\)
\(132\) −9.18519e23 −0.377127
\(133\) 9.27079e23 0.348998
\(134\) 1.55492e23 0.0537034
\(135\) −1.51978e24 −0.481884
\(136\) 1.16184e24 0.338414
\(137\) 3.29993e24 0.883525 0.441762 0.897132i \(-0.354353\pi\)
0.441762 + 0.897132i \(0.354353\pi\)
\(138\) 3.70895e22 0.00913358
\(139\) 6.17390e24 1.39923 0.699616 0.714519i \(-0.253355\pi\)
0.699616 + 0.714519i \(0.253355\pi\)
\(140\) 2.29263e23 0.0478478
\(141\) 1.20571e23 0.0231858
\(142\) 9.31735e23 0.165187
\(143\) 2.31012e24 0.377807
\(144\) −7.29275e23 −0.110084
\(145\) −1.57186e24 −0.219120
\(146\) 9.21854e24 1.18742
\(147\) −5.99458e24 −0.713854
\(148\) −1.56946e24 −0.172878
\(149\) −4.79152e24 −0.488462 −0.244231 0.969717i \(-0.578536\pi\)
−0.244231 + 0.969717i \(0.578536\pi\)
\(150\) −1.12080e24 −0.105798
\(151\) −1.17176e25 −1.02472 −0.512358 0.858772i \(-0.671228\pi\)
−0.512358 + 0.858772i \(0.671228\pi\)
\(152\) −7.11381e24 −0.576635
\(153\) 5.60695e24 0.421479
\(154\) 2.18728e24 0.152551
\(155\) −9.06684e24 −0.587005
\(156\) −2.33122e24 −0.140169
\(157\) 1.45572e24 0.0813263 0.0406631 0.999173i \(-0.487053\pi\)
0.0406631 + 0.999173i \(0.487053\pi\)
\(158\) 2.12077e25 1.10138
\(159\) −1.01153e25 −0.488552
\(160\) −1.75922e24 −0.0790569
\(161\) −8.83214e22 −0.00369461
\(162\) 6.63923e24 0.258639
\(163\) 3.81476e25 1.38455 0.692277 0.721632i \(-0.256608\pi\)
0.692277 + 0.721632i \(0.256608\pi\)
\(164\) −1.35341e25 −0.457853
\(165\) −1.06930e25 −0.337313
\(166\) 1.21079e25 0.356308
\(167\) −6.46554e25 −1.77568 −0.887842 0.460149i \(-0.847796\pi\)
−0.887842 + 0.460149i \(0.847796\pi\)
\(168\) −2.20726e24 −0.0565973
\(169\) −3.58908e25 −0.859579
\(170\) 1.35256e25 0.302686
\(171\) −3.43308e25 −0.718172
\(172\) −1.15068e25 −0.225102
\(173\) −2.31249e25 −0.423203 −0.211602 0.977356i \(-0.567868\pi\)
−0.211602 + 0.977356i \(0.567868\pi\)
\(174\) 1.51333e25 0.259189
\(175\) 2.66897e24 0.0427963
\(176\) −1.67838e25 −0.252054
\(177\) 3.58347e25 0.504209
\(178\) 4.90941e25 0.647439
\(179\) −9.97940e25 −1.23394 −0.616971 0.786986i \(-0.711640\pi\)
−0.616971 + 0.786986i \(0.711640\pi\)
\(180\) −8.48988e24 −0.0984617
\(181\) 1.18405e26 1.28845 0.644224 0.764837i \(-0.277180\pi\)
0.644224 + 0.764837i \(0.277180\pi\)
\(182\) 5.55135e24 0.0566993
\(183\) 1.17881e26 1.13046
\(184\) 6.77722e23 0.00610445
\(185\) −1.82709e25 −0.154627
\(186\) 8.72921e25 0.694346
\(187\) 1.29040e26 0.965043
\(188\) 2.20315e24 0.0154963
\(189\) −3.48430e25 −0.230571
\(190\) −8.28157e25 −0.515758
\(191\) 2.47442e26 1.45074 0.725369 0.688360i \(-0.241669\pi\)
0.725369 + 0.688360i \(0.241669\pi\)
\(192\) 1.69371e25 0.0935135
\(193\) 2.80515e26 1.45897 0.729486 0.683996i \(-0.239760\pi\)
0.729486 + 0.683996i \(0.239760\pi\)
\(194\) −2.55051e26 −1.24999
\(195\) −2.71390e25 −0.125371
\(196\) −1.09537e26 −0.477106
\(197\) 2.28927e26 0.940451 0.470226 0.882546i \(-0.344173\pi\)
0.470226 + 0.882546i \(0.344173\pi\)
\(198\) −8.09974e25 −0.313922
\(199\) −3.84077e26 −1.40478 −0.702390 0.711792i \(-0.747884\pi\)
−0.702390 + 0.711792i \(0.747884\pi\)
\(200\) −2.04800e25 −0.0707107
\(201\) −1.74275e25 −0.0568173
\(202\) 3.06735e26 0.944543
\(203\) −3.60369e25 −0.104844
\(204\) −1.30219e26 −0.358037
\(205\) −1.57558e26 −0.409517
\(206\) −4.11647e26 −1.01170
\(207\) 3.27064e24 0.00760281
\(208\) −4.25975e25 −0.0936820
\(209\) −7.90100e26 −1.64437
\(210\) −2.56958e25 −0.0506222
\(211\) 5.46973e26 1.02028 0.510138 0.860092i \(-0.329594\pi\)
0.510138 + 0.860092i \(0.329594\pi\)
\(212\) −1.84833e26 −0.326525
\(213\) −1.04429e26 −0.174766
\(214\) −2.31400e26 −0.366948
\(215\) −1.33957e26 −0.201337
\(216\) 2.67363e26 0.380963
\(217\) −2.07869e26 −0.280869
\(218\) 5.68539e26 0.728636
\(219\) −1.03322e27 −1.25627
\(220\) −1.95389e26 −0.225444
\(221\) 3.27507e26 0.358682
\(222\) 1.75905e26 0.182902
\(223\) 8.07759e26 0.797583 0.398792 0.917042i \(-0.369430\pi\)
0.398792 + 0.917042i \(0.369430\pi\)
\(224\) −4.03324e25 −0.0378270
\(225\) −9.88352e25 −0.0880669
\(226\) −1.21069e26 −0.102515
\(227\) 7.29025e26 0.586739 0.293370 0.955999i \(-0.405223\pi\)
0.293370 + 0.955999i \(0.405223\pi\)
\(228\) 7.97317e26 0.610071
\(229\) −1.53114e26 −0.111406 −0.0557028 0.998447i \(-0.517740\pi\)
−0.0557028 + 0.998447i \(0.517740\pi\)
\(230\) 7.88972e24 0.00545999
\(231\) −2.45150e26 −0.161397
\(232\) 2.76525e26 0.173229
\(233\) 2.78414e27 1.65996 0.829981 0.557791i \(-0.188351\pi\)
0.829981 + 0.557791i \(0.188351\pi\)
\(234\) −2.05573e26 −0.116677
\(235\) 2.56480e25 0.0138603
\(236\) 6.54794e26 0.336989
\(237\) −2.37696e27 −1.16524
\(238\) 3.10091e26 0.144829
\(239\) −1.57931e27 −0.702896 −0.351448 0.936207i \(-0.614311\pi\)
−0.351448 + 0.936207i \(0.614311\pi\)
\(240\) 1.97174e26 0.0836410
\(241\) 1.98820e27 0.804013 0.402006 0.915637i \(-0.368313\pi\)
0.402006 + 0.915637i \(0.368313\pi\)
\(242\) −3.02595e25 −0.0116677
\(243\) 2.18609e27 0.803889
\(244\) 2.15399e27 0.755546
\(245\) −1.27517e27 −0.426737
\(246\) 1.51691e27 0.484402
\(247\) −2.00529e27 −0.611170
\(248\) 1.59506e27 0.464068
\(249\) −1.35705e27 −0.376968
\(250\) −2.38419e26 −0.0632456
\(251\) −3.79632e27 −0.961866 −0.480933 0.876757i \(-0.659702\pi\)
−0.480933 + 0.876757i \(0.659702\pi\)
\(252\) −1.94641e26 −0.0471117
\(253\) 7.52716e25 0.0174079
\(254\) −3.12487e27 −0.690628
\(255\) −1.51595e27 −0.320238
\(256\) 3.09485e26 0.0625000
\(257\) 3.87160e27 0.747583 0.373791 0.927513i \(-0.378058\pi\)
0.373791 + 0.927513i \(0.378058\pi\)
\(258\) 1.28969e27 0.238154
\(259\) −4.18884e26 −0.0739855
\(260\) −4.95901e26 −0.0837917
\(261\) 1.33449e27 0.215749
\(262\) −3.93933e27 −0.609477
\(263\) −1.11317e28 −1.64842 −0.824212 0.566281i \(-0.808382\pi\)
−0.824212 + 0.566281i \(0.808382\pi\)
\(264\) 1.88113e27 0.266669
\(265\) −2.15173e27 −0.292053
\(266\) −1.89866e27 −0.246779
\(267\) −5.50247e27 −0.684980
\(268\) −3.18447e26 −0.0379740
\(269\) 1.52514e28 1.74244 0.871220 0.490893i \(-0.163329\pi\)
0.871220 + 0.490893i \(0.163329\pi\)
\(270\) 3.11251e27 0.340744
\(271\) −1.10913e27 −0.116369 −0.0581843 0.998306i \(-0.518531\pi\)
−0.0581843 + 0.998306i \(0.518531\pi\)
\(272\) −2.37944e27 −0.239295
\(273\) −6.22196e26 −0.0599870
\(274\) −6.75826e27 −0.624746
\(275\) −2.27462e27 −0.201643
\(276\) −7.59592e25 −0.00645841
\(277\) 1.90865e28 1.55671 0.778356 0.627824i \(-0.216054\pi\)
0.778356 + 0.627824i \(0.216054\pi\)
\(278\) −1.26442e28 −0.989406
\(279\) 7.69764e27 0.577975
\(280\) −4.69531e26 −0.0338335
\(281\) −5.75634e27 −0.398129 −0.199064 0.979986i \(-0.563790\pi\)
−0.199064 + 0.979986i \(0.563790\pi\)
\(282\) −2.46929e26 −0.0163948
\(283\) −9.14645e27 −0.583054 −0.291527 0.956563i \(-0.594163\pi\)
−0.291527 + 0.956563i \(0.594163\pi\)
\(284\) −1.90819e27 −0.116805
\(285\) 9.28200e27 0.545664
\(286\) −4.73112e27 −0.267150
\(287\) −3.61223e27 −0.195945
\(288\) 1.49355e27 0.0778408
\(289\) −1.67346e27 −0.0838091
\(290\) 3.21917e27 0.154941
\(291\) 2.85862e28 1.32247
\(292\) −1.88796e28 −0.839633
\(293\) 1.07789e28 0.460889 0.230445 0.973085i \(-0.425982\pi\)
0.230445 + 0.973085i \(0.425982\pi\)
\(294\) 1.22769e28 0.504771
\(295\) 7.62280e27 0.301413
\(296\) 3.21425e27 0.122243
\(297\) 2.96948e28 1.08638
\(298\) 9.81303e27 0.345395
\(299\) 1.91041e26 0.00647005
\(300\) 2.29540e27 0.0748108
\(301\) −3.07114e27 −0.0963353
\(302\) 2.39977e28 0.724584
\(303\) −3.43789e28 −0.999312
\(304\) 1.45691e28 0.407743
\(305\) 2.50758e28 0.675781
\(306\) −1.14830e28 −0.298030
\(307\) 1.06872e28 0.267161 0.133581 0.991038i \(-0.457352\pi\)
0.133581 + 0.991038i \(0.457352\pi\)
\(308\) −4.47954e27 −0.107870
\(309\) 4.61375e28 1.07036
\(310\) 1.85689e28 0.415075
\(311\) −8.23745e28 −1.77439 −0.887194 0.461397i \(-0.847348\pi\)
−0.887194 + 0.461397i \(0.847348\pi\)
\(312\) 4.77434e27 0.0991141
\(313\) −7.21679e28 −1.44406 −0.722028 0.691863i \(-0.756790\pi\)
−0.722028 + 0.691863i \(0.756790\pi\)
\(314\) −2.98131e27 −0.0575064
\(315\) −2.26592e27 −0.0421380
\(316\) −4.34334e28 −0.778794
\(317\) −8.73563e28 −1.51047 −0.755236 0.655453i \(-0.772478\pi\)
−0.755236 + 0.655453i \(0.772478\pi\)
\(318\) 2.07161e28 0.345458
\(319\) 3.07124e28 0.493993
\(320\) 3.60288e27 0.0559017
\(321\) 2.59353e28 0.388225
\(322\) 1.80882e26 0.00261248
\(323\) −1.12013e29 −1.56113
\(324\) −1.35971e28 −0.182886
\(325\) −5.77304e27 −0.0749456
\(326\) −7.81263e28 −0.979027
\(327\) −6.37220e28 −0.770886
\(328\) 2.77179e28 0.323751
\(329\) 5.88014e26 0.00663185
\(330\) 2.18992e28 0.238516
\(331\) −4.91673e28 −0.517195 −0.258597 0.965985i \(-0.583260\pi\)
−0.258597 + 0.965985i \(0.583260\pi\)
\(332\) −2.47970e28 −0.251948
\(333\) 1.55118e28 0.152248
\(334\) 1.32414e29 1.25560
\(335\) −3.70721e27 −0.0339650
\(336\) 4.52046e27 0.0400204
\(337\) 1.56219e29 1.33656 0.668280 0.743910i \(-0.267031\pi\)
0.668280 + 0.743910i \(0.267031\pi\)
\(338\) 7.35043e28 0.607814
\(339\) 1.35695e28 0.108459
\(340\) −2.77004e28 −0.214032
\(341\) 1.77156e29 1.32337
\(342\) 7.03095e28 0.507825
\(343\) −5.98729e28 −0.418166
\(344\) 2.35660e28 0.159171
\(345\) −8.84281e26 −0.00577658
\(346\) 4.73597e28 0.299250
\(347\) −1.50078e29 −0.917335 −0.458668 0.888608i \(-0.651673\pi\)
−0.458668 + 0.888608i \(0.651673\pi\)
\(348\) −3.09929e28 −0.183274
\(349\) 2.61518e29 1.49627 0.748134 0.663548i \(-0.230950\pi\)
0.748134 + 0.663548i \(0.230950\pi\)
\(350\) −5.46606e27 −0.0302616
\(351\) 7.53660e28 0.403779
\(352\) 3.43732e28 0.178229
\(353\) −2.25089e29 −1.12965 −0.564827 0.825209i \(-0.691057\pi\)
−0.564827 + 0.825209i \(0.691057\pi\)
\(354\) −7.33894e28 −0.356530
\(355\) −2.22143e28 −0.104474
\(356\) −1.00545e29 −0.457808
\(357\) −3.47551e28 −0.153227
\(358\) 2.04378e29 0.872528
\(359\) −4.75724e28 −0.196684 −0.0983420 0.995153i \(-0.531354\pi\)
−0.0983420 + 0.995153i \(0.531354\pi\)
\(360\) 1.73873e28 0.0696230
\(361\) 4.28014e29 1.66007
\(362\) −2.42494e29 −0.911070
\(363\) 3.39149e27 0.0123442
\(364\) −1.13692e28 −0.0400925
\(365\) −2.19787e29 −0.750991
\(366\) −2.41420e29 −0.799356
\(367\) 1.94572e29 0.624339 0.312170 0.950026i \(-0.398944\pi\)
0.312170 + 0.950026i \(0.398944\pi\)
\(368\) −1.38797e27 −0.00431650
\(369\) 1.33765e29 0.403217
\(370\) 3.74188e28 0.109338
\(371\) −4.93313e28 −0.139741
\(372\) −1.78774e29 −0.490977
\(373\) 5.99622e29 1.59671 0.798355 0.602187i \(-0.205704\pi\)
0.798355 + 0.602187i \(0.205704\pi\)
\(374\) −2.64274e29 −0.682389
\(375\) 2.67220e28 0.0669128
\(376\) −4.51204e27 −0.0109575
\(377\) 7.79486e28 0.183604
\(378\) 7.13584e28 0.163038
\(379\) −6.46441e29 −1.43277 −0.716387 0.697703i \(-0.754206\pi\)
−0.716387 + 0.697703i \(0.754206\pi\)
\(380\) 1.69607e29 0.364696
\(381\) 3.50236e29 0.730674
\(382\) −5.06761e29 −1.02583
\(383\) −8.91098e29 −1.75041 −0.875205 0.483752i \(-0.839274\pi\)
−0.875205 + 0.483752i \(0.839274\pi\)
\(384\) −3.46871e28 −0.0661240
\(385\) −5.21487e28 −0.0964818
\(386\) −5.74494e29 −1.03165
\(387\) 1.13728e29 0.198240
\(388\) 5.22345e29 0.883878
\(389\) −3.75725e29 −0.617233 −0.308617 0.951187i \(-0.599866\pi\)
−0.308617 + 0.951187i \(0.599866\pi\)
\(390\) 5.55806e28 0.0886504
\(391\) 1.06713e28 0.0165266
\(392\) 2.24331e29 0.337365
\(393\) 4.41520e29 0.644817
\(394\) −4.68843e29 −0.664999
\(395\) −5.05631e29 −0.696574
\(396\) 1.65883e29 0.221976
\(397\) 7.49782e29 0.974640 0.487320 0.873223i \(-0.337975\pi\)
0.487320 + 0.873223i \(0.337975\pi\)
\(398\) 7.86590e29 0.993329
\(399\) 2.12802e29 0.261088
\(400\) 4.19430e28 0.0500000
\(401\) −8.25665e29 −0.956409 −0.478205 0.878248i \(-0.658712\pi\)
−0.478205 + 0.878248i \(0.658712\pi\)
\(402\) 3.56916e28 0.0401759
\(403\) 4.49625e29 0.491861
\(404\) −6.28192e29 −0.667893
\(405\) −1.58291e29 −0.163578
\(406\) 7.38036e28 0.0741359
\(407\) 3.56993e29 0.348597
\(408\) 2.66688e29 0.253170
\(409\) −2.03509e30 −1.87830 −0.939149 0.343509i \(-0.888384\pi\)
−0.939149 + 0.343509i \(0.888384\pi\)
\(410\) 3.22679e29 0.289572
\(411\) 7.57467e29 0.660972
\(412\) 8.43053e29 0.715380
\(413\) 1.74763e29 0.144219
\(414\) −6.69828e27 −0.00537600
\(415\) −2.88675e29 −0.225349
\(416\) 8.72398e28 0.0662432
\(417\) 1.41716e30 1.04678
\(418\) 1.61813e30 1.16275
\(419\) −1.17686e30 −0.822742 −0.411371 0.911468i \(-0.634950\pi\)
−0.411371 + 0.911468i \(0.634950\pi\)
\(420\) 5.26251e28 0.0357953
\(421\) 1.99473e30 1.32020 0.660100 0.751178i \(-0.270514\pi\)
0.660100 + 0.751178i \(0.270514\pi\)
\(422\) −1.12020e30 −0.721444
\(423\) −2.17748e28 −0.0136471
\(424\) 3.78537e29 0.230888
\(425\) −3.22475e29 −0.191436
\(426\) 2.13871e29 0.123578
\(427\) 5.74894e29 0.323346
\(428\) 4.73907e29 0.259471
\(429\) 5.30265e29 0.282640
\(430\) 2.74344e29 0.142367
\(431\) 9.01027e29 0.455249 0.227625 0.973749i \(-0.426904\pi\)
0.227625 + 0.973749i \(0.426904\pi\)
\(432\) −5.47559e29 −0.269381
\(433\) −3.28011e30 −1.57137 −0.785685 0.618627i \(-0.787689\pi\)
−0.785685 + 0.618627i \(0.787689\pi\)
\(434\) 4.25716e29 0.198604
\(435\) −3.60805e29 −0.163925
\(436\) −1.16437e30 −0.515224
\(437\) −6.53393e28 −0.0281603
\(438\) 2.11603e30 0.888319
\(439\) 1.34681e30 0.550763 0.275382 0.961335i \(-0.411196\pi\)
0.275382 + 0.961335i \(0.411196\pi\)
\(440\) 4.00156e29 0.159413
\(441\) 1.08261e30 0.420172
\(442\) −6.70734e29 −0.253626
\(443\) −2.42126e30 −0.892068 −0.446034 0.895016i \(-0.647164\pi\)
−0.446034 + 0.895016i \(0.647164\pi\)
\(444\) −3.60254e29 −0.129331
\(445\) −1.17049e30 −0.409476
\(446\) −1.65429e30 −0.563977
\(447\) −1.09985e30 −0.365423
\(448\) 8.26007e28 0.0267477
\(449\) −3.10296e30 −0.979361 −0.489681 0.871902i \(-0.662887\pi\)
−0.489681 + 0.871902i \(0.662887\pi\)
\(450\) 2.02414e29 0.0622727
\(451\) 3.07851e30 0.923231
\(452\) 2.47950e29 0.0724890
\(453\) −2.68966e30 −0.766599
\(454\) −1.49304e30 −0.414887
\(455\) −1.32355e29 −0.0358598
\(456\) −1.63291e30 −0.431385
\(457\) −5.07608e30 −1.30765 −0.653826 0.756645i \(-0.726837\pi\)
−0.653826 + 0.756645i \(0.726837\pi\)
\(458\) 3.13578e29 0.0787757
\(459\) 4.20985e30 1.03138
\(460\) −1.61582e28 −0.00386079
\(461\) 4.69993e29 0.109530 0.0547648 0.998499i \(-0.482559\pi\)
0.0547648 + 0.998499i \(0.482559\pi\)
\(462\) 5.02068e29 0.114125
\(463\) −3.39682e29 −0.0753166 −0.0376583 0.999291i \(-0.511990\pi\)
−0.0376583 + 0.999291i \(0.511990\pi\)
\(464\) −5.66322e29 −0.122492
\(465\) −2.08121e30 −0.439143
\(466\) −5.70192e30 −1.17377
\(467\) 1.07622e30 0.216150 0.108075 0.994143i \(-0.465531\pi\)
0.108075 + 0.994143i \(0.465531\pi\)
\(468\) 4.21013e29 0.0825028
\(469\) −8.49926e28 −0.0162515
\(470\) −5.25271e28 −0.00980072
\(471\) 3.34145e29 0.0608408
\(472\) −1.34102e30 −0.238288
\(473\) 2.61737e30 0.453902
\(474\) 4.86802e30 0.823952
\(475\) 1.97448e30 0.326194
\(476\) −6.35067e29 −0.102409
\(477\) 1.82680e30 0.287560
\(478\) 3.23442e30 0.497022
\(479\) 1.27284e31 1.90949 0.954744 0.297429i \(-0.0961293\pi\)
0.954744 + 0.297429i \(0.0961293\pi\)
\(480\) −4.03811e29 −0.0591431
\(481\) 9.06055e29 0.129565
\(482\) −4.07182e30 −0.568523
\(483\) −2.02733e28 −0.00276397
\(484\) 6.19714e28 0.00825030
\(485\) 6.08089e30 0.790565
\(486\) −4.47712e30 −0.568436
\(487\) 4.08137e30 0.506084 0.253042 0.967455i \(-0.418569\pi\)
0.253042 + 0.967455i \(0.418569\pi\)
\(488\) −4.41137e30 −0.534252
\(489\) 8.75641e30 1.03580
\(490\) 2.61156e30 0.301748
\(491\) 6.22475e30 0.702562 0.351281 0.936270i \(-0.385746\pi\)
0.351281 + 0.936270i \(0.385746\pi\)
\(492\) −3.10663e30 −0.342524
\(493\) 4.35411e30 0.468986
\(494\) 4.10683e30 0.432163
\(495\) 1.93113e30 0.198542
\(496\) −3.26667e30 −0.328146
\(497\) −5.09292e29 −0.0499883
\(498\) 2.77925e30 0.266557
\(499\) 6.71277e30 0.629137 0.314569 0.949235i \(-0.398140\pi\)
0.314569 + 0.949235i \(0.398140\pi\)
\(500\) 4.88281e29 0.0447214
\(501\) −1.48410e31 −1.32840
\(502\) 7.77485e30 0.680142
\(503\) −2.13145e31 −1.82240 −0.911200 0.411965i \(-0.864843\pi\)
−0.911200 + 0.411965i \(0.864843\pi\)
\(504\) 3.98626e29 0.0333130
\(505\) −7.31312e30 −0.597382
\(506\) −1.54156e29 −0.0123092
\(507\) −8.23838e30 −0.643058
\(508\) 6.39974e30 0.488348
\(509\) −3.48709e30 −0.260141 −0.130070 0.991505i \(-0.541520\pi\)
−0.130070 + 0.991505i \(0.541520\pi\)
\(510\) 3.10466e30 0.226442
\(511\) −5.03891e30 −0.359332
\(512\) −6.33825e29 −0.0441942
\(513\) −2.57765e31 −1.75741
\(514\) −7.92903e30 −0.528621
\(515\) 9.81443e30 0.639855
\(516\) −2.64128e30 −0.168400
\(517\) −5.01133e29 −0.0312472
\(518\) 8.57875e29 0.0523156
\(519\) −5.30809e30 −0.316602
\(520\) 1.01560e30 0.0592497
\(521\) 6.35728e30 0.362775 0.181388 0.983412i \(-0.441941\pi\)
0.181388 + 0.983412i \(0.441941\pi\)
\(522\) −2.73303e30 −0.152558
\(523\) −2.07982e31 −1.13568 −0.567842 0.823138i \(-0.692221\pi\)
−0.567842 + 0.823138i \(0.692221\pi\)
\(524\) 8.06774e30 0.430965
\(525\) 6.12637e29 0.0320163
\(526\) 2.27977e31 1.16561
\(527\) 2.51155e31 1.25638
\(528\) −3.85255e30 −0.188564
\(529\) −2.08742e31 −0.999702
\(530\) 4.40675e30 0.206512
\(531\) −6.47166e30 −0.296776
\(532\) 3.88845e30 0.174499
\(533\) 7.81332e30 0.343141
\(534\) 1.12691e31 0.484354
\(535\) 5.51700e30 0.232078
\(536\) 6.52179e29 0.0268517
\(537\) −2.29067e31 −0.923122
\(538\) −3.12348e31 −1.23209
\(539\) 2.49155e31 0.962052
\(540\) −6.37442e30 −0.240942
\(541\) −5.19131e30 −0.192092 −0.0960458 0.995377i \(-0.530620\pi\)
−0.0960458 + 0.995377i \(0.530620\pi\)
\(542\) 2.27150e30 0.0822850
\(543\) 2.71787e31 0.963899
\(544\) 4.87310e30 0.169207
\(545\) −1.35550e31 −0.460830
\(546\) 1.27426e30 0.0424172
\(547\) 1.22758e31 0.400125 0.200063 0.979783i \(-0.435885\pi\)
0.200063 + 0.979783i \(0.435885\pi\)
\(548\) 1.38409e31 0.441762
\(549\) −2.12890e31 −0.665386
\(550\) 4.65843e30 0.142583
\(551\) −2.66598e31 −0.799122
\(552\) 1.55564e29 0.00456679
\(553\) −1.15923e31 −0.333295
\(554\) −3.90891e31 −1.10076
\(555\) −4.19390e30 −0.115678
\(556\) 2.58952e31 0.699616
\(557\) 3.14791e31 0.833081 0.416540 0.909117i \(-0.363242\pi\)
0.416540 + 0.909117i \(0.363242\pi\)
\(558\) −1.57648e31 −0.408690
\(559\) 6.64294e30 0.168704
\(560\) 9.61599e29 0.0239239
\(561\) 2.96199e31 0.721957
\(562\) 1.17890e31 0.281520
\(563\) 4.85033e31 1.13481 0.567407 0.823437i \(-0.307947\pi\)
0.567407 + 0.823437i \(0.307947\pi\)
\(564\) 5.05711e29 0.0115929
\(565\) 2.88652e30 0.0648362
\(566\) 1.87319e31 0.412281
\(567\) −3.62904e30 −0.0782684
\(568\) 3.90798e30 0.0825936
\(569\) 1.58999e31 0.329309 0.164654 0.986351i \(-0.447349\pi\)
0.164654 + 0.986351i \(0.447349\pi\)
\(570\) −1.90095e31 −0.385843
\(571\) −3.92722e31 −0.781214 −0.390607 0.920557i \(-0.627735\pi\)
−0.390607 + 0.920557i \(0.627735\pi\)
\(572\) 9.68934e30 0.188903
\(573\) 5.67978e31 1.08531
\(574\) 7.39784e30 0.138554
\(575\) −1.88106e29 −0.00345320
\(576\) −3.05880e30 −0.0550418
\(577\) 5.05036e30 0.0890841 0.0445420 0.999008i \(-0.485817\pi\)
0.0445420 + 0.999008i \(0.485817\pi\)
\(578\) 3.42725e30 0.0592620
\(579\) 6.43894e31 1.09147
\(580\) −6.59286e30 −0.109560
\(581\) −6.61824e30 −0.107824
\(582\) −5.85445e31 −0.935130
\(583\) 4.20425e31 0.658415
\(584\) 3.86654e31 0.593710
\(585\) 4.90124e30 0.0737927
\(586\) −2.20752e31 −0.325898
\(587\) −3.04905e31 −0.441394 −0.220697 0.975342i \(-0.570833\pi\)
−0.220697 + 0.975342i \(0.570833\pi\)
\(588\) −2.51431e31 −0.356927
\(589\) −1.53780e32 −2.14078
\(590\) −1.56115e31 −0.213131
\(591\) 5.25481e31 0.703559
\(592\) −6.58278e30 −0.0864390
\(593\) 1.36625e32 1.75955 0.879777 0.475387i \(-0.157692\pi\)
0.879777 + 0.475387i \(0.157692\pi\)
\(594\) −6.08150e31 −0.768186
\(595\) −7.39315e30 −0.0915977
\(596\) −2.00971e31 −0.244231
\(597\) −8.81612e31 −1.05093
\(598\) −3.91252e29 −0.00457502
\(599\) −9.92624e31 −1.13861 −0.569307 0.822125i \(-0.692788\pi\)
−0.569307 + 0.822125i \(0.692788\pi\)
\(600\) −4.70098e30 −0.0528992
\(601\) −1.76086e31 −0.194387 −0.0971937 0.995265i \(-0.530987\pi\)
−0.0971937 + 0.995265i \(0.530987\pi\)
\(602\) 6.28970e30 0.0681193
\(603\) 3.14737e30 0.0334425
\(604\) −4.91472e31 −0.512358
\(605\) 7.21442e29 0.00737929
\(606\) 7.04079e31 0.706621
\(607\) −1.39464e32 −1.37338 −0.686691 0.726949i \(-0.740937\pi\)
−0.686691 + 0.726949i \(0.740937\pi\)
\(608\) −2.98375e31 −0.288318
\(609\) −8.27193e30 −0.0784346
\(610\) −5.13552e31 −0.477849
\(611\) −1.27189e30 −0.0116138
\(612\) 2.35173e31 0.210739
\(613\) 1.82366e32 1.60380 0.801898 0.597461i \(-0.203824\pi\)
0.801898 + 0.597461i \(0.203824\pi\)
\(614\) −2.18874e31 −0.188912
\(615\) −3.61659e31 −0.306363
\(616\) 9.17410e30 0.0762756
\(617\) 1.13184e32 0.923650 0.461825 0.886971i \(-0.347195\pi\)
0.461825 + 0.886971i \(0.347195\pi\)
\(618\) −9.44895e31 −0.756861
\(619\) −8.35127e31 −0.656614 −0.328307 0.944571i \(-0.606478\pi\)
−0.328307 + 0.944571i \(0.606478\pi\)
\(620\) −3.80291e31 −0.293502
\(621\) 2.45568e30 0.0186046
\(622\) 1.68703e32 1.25468
\(623\) −2.68351e31 −0.195925
\(624\) −9.77785e30 −0.0700843
\(625\) 5.68434e30 0.0400000
\(626\) 1.47800e32 1.02110
\(627\) −1.81360e32 −1.23017
\(628\) 6.10572e30 0.0406631
\(629\) 5.06110e31 0.330950
\(630\) 4.64061e30 0.0297961
\(631\) −7.94078e31 −0.500640 −0.250320 0.968163i \(-0.580536\pi\)
−0.250320 + 0.968163i \(0.580536\pi\)
\(632\) 8.89516e31 0.550690
\(633\) 1.25552e32 0.763277
\(634\) 1.78906e32 1.06807
\(635\) 7.45027e31 0.436792
\(636\) −4.24265e31 −0.244276
\(637\) 6.32360e31 0.357570
\(638\) −6.28990e31 −0.349306
\(639\) 1.88597e31 0.102867
\(640\) −7.37870e30 −0.0395285
\(641\) 1.60774e32 0.845955 0.422978 0.906140i \(-0.360985\pi\)
0.422978 + 0.906140i \(0.360985\pi\)
\(642\) −5.31155e31 −0.274517
\(643\) −3.25183e32 −1.65083 −0.825413 0.564529i \(-0.809058\pi\)
−0.825413 + 0.564529i \(0.809058\pi\)
\(644\) −3.70447e29 −0.00184730
\(645\) −3.07486e31 −0.150622
\(646\) 2.29402e32 1.10389
\(647\) 2.35559e32 1.11353 0.556763 0.830672i \(-0.312043\pi\)
0.556763 + 0.830672i \(0.312043\pi\)
\(648\) 2.78469e31 0.129320
\(649\) −1.48941e32 −0.679517
\(650\) 1.18232e31 0.0529945
\(651\) −4.77143e31 −0.210120
\(652\) 1.60003e32 0.692277
\(653\) 1.72242e32 0.732215 0.366108 0.930573i \(-0.380690\pi\)
0.366108 + 0.930573i \(0.380690\pi\)
\(654\) 1.30503e32 0.545099
\(655\) 9.39208e31 0.385467
\(656\) −5.67663e31 −0.228927
\(657\) 1.86597e32 0.739438
\(658\) −1.20425e30 −0.00468942
\(659\) −2.10755e32 −0.806483 −0.403242 0.915094i \(-0.632117\pi\)
−0.403242 + 0.915094i \(0.632117\pi\)
\(660\) −4.48496e31 −0.168657
\(661\) −1.56345e32 −0.577786 −0.288893 0.957361i \(-0.593287\pi\)
−0.288893 + 0.957361i \(0.593287\pi\)
\(662\) 1.00695e32 0.365712
\(663\) 7.51760e31 0.268333
\(664\) 5.07842e31 0.178154
\(665\) 4.52675e31 0.156077
\(666\) −3.17681e31 −0.107656
\(667\) 2.53983e30 0.00845977
\(668\) −2.71184e32 −0.887842
\(669\) 1.85413e32 0.596678
\(670\) 7.59236e30 0.0240169
\(671\) −4.89952e32 −1.52351
\(672\) −9.25790e30 −0.0282987
\(673\) −3.51032e32 −1.05481 −0.527405 0.849614i \(-0.676835\pi\)
−0.527405 + 0.849614i \(0.676835\pi\)
\(674\) −3.19936e32 −0.945091
\(675\) −7.42081e31 −0.215505
\(676\) −1.50537e32 −0.429789
\(677\) −5.04787e32 −1.41690 −0.708448 0.705763i \(-0.750604\pi\)
−0.708448 + 0.705763i \(0.750604\pi\)
\(678\) −2.77903e31 −0.0766923
\(679\) 1.39412e32 0.378268
\(680\) 5.67304e31 0.151343
\(681\) 1.67341e32 0.438944
\(682\) −3.62815e32 −0.935762
\(683\) 2.13585e32 0.541668 0.270834 0.962626i \(-0.412701\pi\)
0.270834 + 0.962626i \(0.412701\pi\)
\(684\) −1.43994e32 −0.359086
\(685\) 1.61130e32 0.395124
\(686\) 1.22620e32 0.295688
\(687\) −3.51458e31 −0.0833435
\(688\) −4.82632e31 −0.112551
\(689\) 1.06705e32 0.244716
\(690\) 1.81101e30 0.00408466
\(691\) −4.74070e32 −1.05159 −0.525793 0.850613i \(-0.676231\pi\)
−0.525793 + 0.850613i \(0.676231\pi\)
\(692\) −9.69927e31 −0.211602
\(693\) 4.42736e31 0.0949977
\(694\) 3.07360e32 0.648654
\(695\) 3.01460e32 0.625755
\(696\) 6.34735e31 0.129594
\(697\) 4.36442e32 0.876495
\(698\) −5.35589e32 −1.05802
\(699\) 6.39073e32 1.24183
\(700\) 1.11945e31 0.0213982
\(701\) −4.92407e32 −0.925906 −0.462953 0.886383i \(-0.653210\pi\)
−0.462953 + 0.886383i \(0.653210\pi\)
\(702\) −1.54350e32 −0.285515
\(703\) −3.09886e32 −0.563918
\(704\) −7.03962e31 −0.126027
\(705\) 5.88725e30 0.0103690
\(706\) 4.60983e32 0.798786
\(707\) −1.67663e32 −0.285834
\(708\) 1.50301e32 0.252105
\(709\) 8.64776e32 1.42716 0.713578 0.700575i \(-0.247073\pi\)
0.713578 + 0.700575i \(0.247073\pi\)
\(710\) 4.54949e31 0.0738740
\(711\) 4.29275e32 0.685859
\(712\) 2.05915e32 0.323719
\(713\) 1.46503e31 0.0226630
\(714\) 7.11784e31 0.108348
\(715\) 1.12799e32 0.168960
\(716\) −4.18566e32 −0.616971
\(717\) −3.62515e32 −0.525842
\(718\) 9.74283e31 0.139077
\(719\) −6.11269e31 −0.0858717 −0.0429359 0.999078i \(-0.513671\pi\)
−0.0429359 + 0.999078i \(0.513671\pi\)
\(720\) −3.56091e31 −0.0492309
\(721\) 2.25009e32 0.306156
\(722\) −8.76573e32 −1.17384
\(723\) 4.56371e32 0.601489
\(724\) 4.96627e32 0.644224
\(725\) −7.67510e31 −0.0979934
\(726\) −6.94577e30 −0.00872869
\(727\) 3.03244e30 0.00375099 0.00187550 0.999998i \(-0.499403\pi\)
0.00187550 + 0.999998i \(0.499403\pi\)
\(728\) 2.32841e31 0.0283497
\(729\) 8.06990e32 0.967167
\(730\) 4.50124e32 0.531031
\(731\) 3.71066e32 0.430925
\(732\) 4.94428e32 0.565230
\(733\) −5.01686e32 −0.564595 −0.282297 0.959327i \(-0.591096\pi\)
−0.282297 + 0.959327i \(0.591096\pi\)
\(734\) −3.98484e32 −0.441475
\(735\) −2.92704e32 −0.319245
\(736\) 2.84257e30 0.00305223
\(737\) 7.24347e31 0.0765721
\(738\) −2.73951e32 −0.285118
\(739\) 7.88978e32 0.808451 0.404226 0.914659i \(-0.367541\pi\)
0.404226 + 0.914659i \(0.367541\pi\)
\(740\) −7.66337e31 −0.0773134
\(741\) −4.60295e32 −0.457221
\(742\) 1.01031e32 0.0988116
\(743\) −1.01205e33 −0.974609 −0.487304 0.873232i \(-0.662020\pi\)
−0.487304 + 0.873232i \(0.662020\pi\)
\(744\) 3.66129e32 0.347173
\(745\) −2.33961e32 −0.218447
\(746\) −1.22803e33 −1.12904
\(747\) 2.45081e32 0.221882
\(748\) 5.41234e32 0.482522
\(749\) 1.26484e32 0.111044
\(750\) −5.47267e31 −0.0473145
\(751\) 9.01052e32 0.767169 0.383584 0.923506i \(-0.374690\pi\)
0.383584 + 0.923506i \(0.374690\pi\)
\(752\) 9.24066e30 0.00774815
\(753\) −8.71407e32 −0.719580
\(754\) −1.59639e32 −0.129828
\(755\) −5.72149e32 −0.458267
\(756\) −1.46142e32 −0.115285
\(757\) 3.32781e32 0.258557 0.129279 0.991608i \(-0.458734\pi\)
0.129279 + 0.991608i \(0.458734\pi\)
\(758\) 1.32391e33 1.01312
\(759\) 1.72779e31 0.0130230
\(760\) −3.47354e32 −0.257879
\(761\) −3.01945e32 −0.220803 −0.110401 0.993887i \(-0.535214\pi\)
−0.110401 + 0.993887i \(0.535214\pi\)
\(762\) −7.17284e32 −0.516665
\(763\) −3.10767e32 −0.220497
\(764\) 1.03785e33 0.725369
\(765\) 2.73777e32 0.188491
\(766\) 1.82497e33 1.23773
\(767\) −3.78015e32 −0.252559
\(768\) 7.10393e31 0.0467568
\(769\) −2.38000e33 −1.54320 −0.771602 0.636106i \(-0.780544\pi\)
−0.771602 + 0.636106i \(0.780544\pi\)
\(770\) 1.06801e32 0.0682229
\(771\) 8.88687e32 0.559273
\(772\) 1.17656e33 0.729486
\(773\) −1.65561e33 −1.01133 −0.505666 0.862729i \(-0.668753\pi\)
−0.505666 + 0.862729i \(0.668753\pi\)
\(774\) −2.32915e32 −0.140177
\(775\) −4.42717e32 −0.262516
\(776\) −1.06976e33 −0.624996
\(777\) −9.61507e31 −0.0553491
\(778\) 7.69484e32 0.436450
\(779\) −2.67229e33 −1.49349
\(780\) −1.13829e32 −0.0626853
\(781\) 4.34043e32 0.235530
\(782\) −2.18548e31 −0.0116861
\(783\) 1.00197e33 0.527952
\(784\) −4.59430e32 −0.238553
\(785\) 7.10799e31 0.0363702
\(786\) −9.04234e32 −0.455954
\(787\) −1.04035e33 −0.516974 −0.258487 0.966015i \(-0.583224\pi\)
−0.258487 + 0.966015i \(0.583224\pi\)
\(788\) 9.60191e32 0.470226
\(789\) −2.55517e33 −1.23320
\(790\) 1.03553e33 0.492552
\(791\) 6.61772e31 0.0310227
\(792\) −3.39728e32 −0.156961
\(793\) −1.24351e33 −0.566249
\(794\) −1.53555e33 −0.689175
\(795\) −4.93910e32 −0.218487
\(796\) −1.61094e33 −0.702390
\(797\) 2.16453e32 0.0930238 0.0465119 0.998918i \(-0.485189\pi\)
0.0465119 + 0.998918i \(0.485189\pi\)
\(798\) −4.35818e32 −0.184617
\(799\) −7.10459e31 −0.0296655
\(800\) −8.58993e31 −0.0353553
\(801\) 9.93734e32 0.403178
\(802\) 1.69096e33 0.676283
\(803\) 4.29440e33 1.69306
\(804\) −7.30963e31 −0.0284087
\(805\) −4.31257e30 −0.00165228
\(806\) −9.20833e32 −0.347799
\(807\) 3.50080e33 1.30353
\(808\) 1.28654e33 0.472272
\(809\) 2.79297e33 1.01078 0.505392 0.862890i \(-0.331348\pi\)
0.505392 + 0.862890i \(0.331348\pi\)
\(810\) 3.24181e32 0.115667
\(811\) 2.01537e33 0.708948 0.354474 0.935066i \(-0.384660\pi\)
0.354474 + 0.935066i \(0.384660\pi\)
\(812\) −1.51150e32 −0.0524220
\(813\) −2.54590e32 −0.0870562
\(814\) −7.31121e32 −0.246495
\(815\) 1.86268e33 0.619191
\(816\) −5.46178e32 −0.179018
\(817\) −2.27200e33 −0.734268
\(818\) 4.16786e33 1.32816
\(819\) 1.12367e32 0.0353082
\(820\) −6.60847e32 −0.204758
\(821\) 4.42385e33 1.35162 0.675808 0.737077i \(-0.263795\pi\)
0.675808 + 0.737077i \(0.263795\pi\)
\(822\) −1.55129e33 −0.467378
\(823\) −6.10731e31 −0.0181448 −0.00907239 0.999959i \(-0.502888\pi\)
−0.00907239 + 0.999959i \(0.502888\pi\)
\(824\) −1.72657e33 −0.505850
\(825\) −5.22118e32 −0.150851
\(826\) −3.57914e32 −0.101978
\(827\) 4.78632e33 1.34489 0.672447 0.740145i \(-0.265243\pi\)
0.672447 + 0.740145i \(0.265243\pi\)
\(828\) 1.37181e31 0.00380140
\(829\) −3.13389e33 −0.856459 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(830\) 5.91206e32 0.159346
\(831\) 4.38111e33 1.16459
\(832\) −1.78667e32 −0.0468410
\(833\) 3.53228e33 0.913351
\(834\) −2.90234e33 −0.740183
\(835\) −3.15700e33 −0.794110
\(836\) −3.31392e33 −0.822186
\(837\) 5.77959e33 1.41434
\(838\) 2.41021e33 0.581767
\(839\) −4.05251e33 −0.964855 −0.482427 0.875936i \(-0.660245\pi\)
−0.482427 + 0.875936i \(0.660245\pi\)
\(840\) −1.07776e32 −0.0253111
\(841\) −3.28042e33 −0.759932
\(842\) −4.08520e33 −0.933522
\(843\) −1.32131e33 −0.297843
\(844\) 2.29417e33 0.510138
\(845\) −1.75248e33 −0.384415
\(846\) 4.45948e31 0.00964996
\(847\) 1.65400e31 0.00353083
\(848\) −7.75244e32 −0.163262
\(849\) −2.09948e33 −0.436187
\(850\) 6.60428e32 0.135365
\(851\) 2.95224e31 0.00596982
\(852\) −4.38007e32 −0.0873828
\(853\) −4.75268e33 −0.935459 −0.467729 0.883872i \(-0.654928\pi\)
−0.467729 + 0.883872i \(0.654928\pi\)
\(854\) −1.17738e33 −0.228640
\(855\) −1.67631e33 −0.321176
\(856\) −9.70561e32 −0.183474
\(857\) −9.71517e33 −1.81205 −0.906027 0.423221i \(-0.860900\pi\)
−0.906027 + 0.423221i \(0.860900\pi\)
\(858\) −1.08598e33 −0.199857
\(859\) −8.89769e32 −0.161568 −0.0807841 0.996732i \(-0.525742\pi\)
−0.0807841 + 0.996732i \(0.525742\pi\)
\(860\) −5.61857e32 −0.100669
\(861\) −8.29151e32 −0.146588
\(862\) −1.84530e33 −0.321910
\(863\) 4.93774e33 0.849972 0.424986 0.905200i \(-0.360279\pi\)
0.424986 + 0.905200i \(0.360279\pi\)
\(864\) 1.12140e33 0.190481
\(865\) −1.12914e33 −0.189262
\(866\) 6.71768e33 1.11113
\(867\) −3.84127e32 −0.0626983
\(868\) −8.71867e32 −0.140434
\(869\) 9.87947e33 1.57039
\(870\) 7.38929e32 0.115913
\(871\) 1.83841e32 0.0284599
\(872\) 2.38463e33 0.364318
\(873\) −5.16260e33 −0.778404
\(874\) 1.33815e32 0.0199124
\(875\) 1.30321e32 0.0191391
\(876\) −4.33362e33 −0.628136
\(877\) 8.97071e32 0.128331 0.0641655 0.997939i \(-0.479561\pi\)
0.0641655 + 0.997939i \(0.479561\pi\)
\(878\) −2.75827e33 −0.389448
\(879\) 2.47419e33 0.344795
\(880\) −8.19520e32 −0.112722
\(881\) −1.00868e34 −1.36940 −0.684699 0.728826i \(-0.740066\pi\)
−0.684699 + 0.728826i \(0.740066\pi\)
\(882\) −2.21718e33 −0.297107
\(883\) −9.65735e33 −1.27735 −0.638675 0.769476i \(-0.720517\pi\)
−0.638675 + 0.769476i \(0.720517\pi\)
\(884\) 1.37366e33 0.179341
\(885\) 1.74974e33 0.225489
\(886\) 4.95873e33 0.630787
\(887\) 5.68517e33 0.713874 0.356937 0.934128i \(-0.383821\pi\)
0.356937 + 0.934128i \(0.383821\pi\)
\(888\) 7.37800e32 0.0914512
\(889\) 1.70807e33 0.208995
\(890\) 2.39717e33 0.289544
\(891\) 3.09284e33 0.368777
\(892\) 3.38799e33 0.398792
\(893\) 4.35007e32 0.0505480
\(894\) 2.25249e33 0.258393
\(895\) −4.87275e33 −0.551835
\(896\) −1.69166e32 −0.0189135
\(897\) 4.38516e31 0.00484030
\(898\) 6.35486e33 0.692513
\(899\) 5.97764e33 0.643122
\(900\) −4.14545e32 −0.0440334
\(901\) 5.96038e33 0.625085
\(902\) −6.30479e33 −0.652823
\(903\) −7.04951e32 −0.0720692
\(904\) −5.07802e32 −0.0512575
\(905\) 5.78150e33 0.576212
\(906\) 5.50843e33 0.542067
\(907\) −1.49169e34 −1.44942 −0.724712 0.689052i \(-0.758027\pi\)
−0.724712 + 0.689052i \(0.758027\pi\)
\(908\) 3.05775e33 0.293370
\(909\) 6.20875e33 0.588192
\(910\) 2.71062e32 0.0253567
\(911\) 1.98559e33 0.183412 0.0917062 0.995786i \(-0.470768\pi\)
0.0917062 + 0.995786i \(0.470768\pi\)
\(912\) 3.34419e33 0.305036
\(913\) 5.64038e33 0.508036
\(914\) 1.03958e34 0.924650
\(915\) 5.75590e33 0.505557
\(916\) −6.42207e32 −0.0557028
\(917\) 2.15326e33 0.184437
\(918\) −8.62177e33 −0.729299
\(919\) −2.72785e32 −0.0227873 −0.0113936 0.999935i \(-0.503627\pi\)
−0.0113936 + 0.999935i \(0.503627\pi\)
\(920\) 3.30919e31 0.00272999
\(921\) 2.45315e33 0.199866
\(922\) −9.62547e32 −0.0774491
\(923\) 1.10161e33 0.0875402
\(924\) −1.02824e33 −0.0806984
\(925\) −8.92133e32 −0.0691512
\(926\) 6.95668e32 0.0532569
\(927\) −8.33233e33 −0.630013
\(928\) 1.15983e33 0.0866147
\(929\) 1.45553e34 1.07360 0.536798 0.843711i \(-0.319634\pi\)
0.536798 + 0.843711i \(0.319634\pi\)
\(930\) 4.26231e33 0.310521
\(931\) −2.16278e34 −1.55629
\(932\) 1.16775e34 0.829981
\(933\) −1.89083e34 −1.32743
\(934\) −2.20409e33 −0.152841
\(935\) 6.30079e33 0.431580
\(936\) −8.62236e32 −0.0583383
\(937\) 1.67847e34 1.12178 0.560889 0.827891i \(-0.310459\pi\)
0.560889 + 0.827891i \(0.310459\pi\)
\(938\) 1.74065e32 0.0114915
\(939\) −1.65654e34 −1.08031
\(940\) 1.07575e32 0.00693016
\(941\) −1.26349e34 −0.804064 −0.402032 0.915626i \(-0.631696\pi\)
−0.402032 + 0.915626i \(0.631696\pi\)
\(942\) −6.84330e32 −0.0430210
\(943\) 2.54585e32 0.0158106
\(944\) 2.74640e33 0.168495
\(945\) −1.70132e33 −0.103114
\(946\) −5.36038e33 −0.320957
\(947\) −1.44322e34 −0.853703 −0.426851 0.904322i \(-0.640377\pi\)
−0.426851 + 0.904322i \(0.640377\pi\)
\(948\) −9.96971e33 −0.582622
\(949\) 1.08993e34 0.629268
\(950\) −4.04373e33 −0.230654
\(951\) −2.00518e34 −1.13000
\(952\) 1.30062e33 0.0724143
\(953\) 1.65674e34 0.911350 0.455675 0.890146i \(-0.349398\pi\)
0.455675 + 0.890146i \(0.349398\pi\)
\(954\) −3.74128e33 −0.203336
\(955\) 1.20821e34 0.648790
\(956\) −6.62409e33 −0.351448
\(957\) 7.04973e33 0.369560
\(958\) −2.60678e34 −1.35021
\(959\) 3.69411e33 0.189058
\(960\) 8.27006e32 0.0418205
\(961\) 1.44671e34 0.722873
\(962\) −1.85560e33 −0.0916159
\(963\) −4.68386e33 −0.228508
\(964\) 8.33909e33 0.402006
\(965\) 1.36970e34 0.652472
\(966\) 4.15197e31 0.00195442
\(967\) 1.67020e33 0.0776896 0.0388448 0.999245i \(-0.487632\pi\)
0.0388448 + 0.999245i \(0.487632\pi\)
\(968\) −1.26917e32 −0.00583384
\(969\) −2.57115e34 −1.16789
\(970\) −1.24537e34 −0.559014
\(971\) −1.89959e34 −0.842633 −0.421317 0.906914i \(-0.638432\pi\)
−0.421317 + 0.906914i \(0.638432\pi\)
\(972\) 9.16913e33 0.401945
\(973\) 6.91137e33 0.299410
\(974\) −8.35865e33 −0.357856
\(975\) −1.32515e33 −0.0560674
\(976\) 9.03450e33 0.377773
\(977\) 1.78605e34 0.738086 0.369043 0.929412i \(-0.379686\pi\)
0.369043 + 0.929412i \(0.379686\pi\)
\(978\) −1.79331e34 −0.732418
\(979\) 2.28701e34 0.923140
\(980\) −5.34847e33 −0.213368
\(981\) 1.15081e34 0.453741
\(982\) −1.27483e34 −0.496786
\(983\) 9.74109e33 0.375182 0.187591 0.982247i \(-0.439932\pi\)
0.187591 + 0.982247i \(0.439932\pi\)
\(984\) 6.36238e33 0.242201
\(985\) 1.11781e34 0.420583
\(986\) −8.91722e33 −0.331623
\(987\) 1.34973e32 0.00496134
\(988\) −8.41080e33 −0.305585
\(989\) 2.16450e32 0.00777321
\(990\) −3.95495e33 −0.140390
\(991\) 1.52559e34 0.535292 0.267646 0.963517i \(-0.413754\pi\)
0.267646 + 0.963517i \(0.413754\pi\)
\(992\) 6.69015e33 0.232034
\(993\) −1.12859e34 −0.386918
\(994\) 1.04303e33 0.0353471
\(995\) −1.87538e34 −0.628237
\(996\) −5.69190e33 −0.188484
\(997\) −9.21743e33 −0.301728 −0.150864 0.988555i \(-0.548206\pi\)
−0.150864 + 0.988555i \(0.548206\pi\)
\(998\) −1.37477e34 −0.444867
\(999\) 1.16466e34 0.372561
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 10.24.a.a.1.2 2
5.2 odd 4 50.24.b.d.49.1 4
5.3 odd 4 50.24.b.d.49.4 4
5.4 even 2 50.24.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.24.a.a.1.2 2 1.1 even 1 trivial
50.24.a.d.1.1 2 5.4 even 2
50.24.b.d.49.1 4 5.2 odd 4
50.24.b.d.49.4 4 5.3 odd 4