Properties

Label 33.14.a.a.1.1
Level $33$
Weight $14$
Character 33.1
Self dual yes
Analytic conductor $35.386$
Analytic rank $0$
Dimension $1$
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] = [33,14,Mod(1,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 33.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: \(35.3862065541\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 33.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+140.000 q^{2} +729.000 q^{3} +11408.0 q^{4} +48740.0 q^{5} +102060. q^{6} +487486. q^{7} +450240. q^{8} +531441. q^{9} +O(q^{10})\) \(q+140.000 q^{2} +729.000 q^{3} +11408.0 q^{4} +48740.0 q^{5} +102060. q^{6} +487486. q^{7} +450240. q^{8} +531441. q^{9} +6.82360e6 q^{10} -1.77156e6 q^{11} +8.31643e6 q^{12} -1.83883e7 q^{13} +6.82480e7 q^{14} +3.55315e7 q^{15} -3.04207e7 q^{16} -9.62333e7 q^{17} +7.44017e7 q^{18} -1.49547e7 q^{19} +5.56026e8 q^{20} +3.55377e8 q^{21} -2.48019e8 q^{22} +1.53804e8 q^{23} +3.28225e8 q^{24} +1.15488e9 q^{25} -2.57436e9 q^{26} +3.87420e8 q^{27} +5.56124e9 q^{28} +5.21901e9 q^{29} +4.97440e9 q^{30} +1.18381e9 q^{31} -7.94727e9 q^{32} -1.29147e9 q^{33} -1.34727e10 q^{34} +2.37601e10 q^{35} +6.06268e9 q^{36} -1.76722e10 q^{37} -2.09365e9 q^{38} -1.34051e10 q^{39} +2.19447e10 q^{40} -1.94617e10 q^{41} +4.97528e10 q^{42} -3.41230e9 q^{43} -2.02100e10 q^{44} +2.59024e10 q^{45} +2.15326e10 q^{46} -1.00328e11 q^{47} -2.21767e10 q^{48} +1.40754e11 q^{49} +1.61684e11 q^{50} -7.01540e10 q^{51} -2.09774e11 q^{52} +2.75469e11 q^{53} +5.42389e10 q^{54} -8.63459e10 q^{55} +2.19486e11 q^{56} -1.09019e10 q^{57} +7.30661e11 q^{58} -2.67677e11 q^{59} +4.05343e11 q^{60} +5.63487e11 q^{61} +1.65734e11 q^{62} +2.59070e11 q^{63} -8.63411e11 q^{64} -8.96246e11 q^{65} -1.80806e11 q^{66} +1.08084e12 q^{67} -1.09783e12 q^{68} +1.12123e11 q^{69} +3.32641e12 q^{70} -1.15056e12 q^{71} +2.39276e11 q^{72} -3.45915e11 q^{73} -2.47411e12 q^{74} +8.41911e11 q^{75} -1.70603e11 q^{76} -8.63611e11 q^{77} -1.87671e12 q^{78} -2.00408e12 q^{79} -1.48271e12 q^{80} +2.82430e11 q^{81} -2.72464e12 q^{82} -3.33673e12 q^{83} +4.05414e12 q^{84} -4.69041e12 q^{85} -4.77723e11 q^{86} +3.80466e12 q^{87} -7.97628e11 q^{88} -5.69624e12 q^{89} +3.62634e12 q^{90} -8.96404e12 q^{91} +1.75460e12 q^{92} +8.62999e11 q^{93} -1.40459e13 q^{94} -7.28890e11 q^{95} -5.79356e12 q^{96} -6.55011e12 q^{97} +1.97055e13 q^{98} -9.41480e11 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 140.000 1.54680 0.773398 0.633921i \(-0.218555\pi\)
0.773398 + 0.633921i \(0.218555\pi\)
\(3\) 729.000 0.577350
\(4\) 11408.0 1.39258
\(5\) 48740.0 1.39502 0.697510 0.716575i \(-0.254291\pi\)
0.697510 + 0.716575i \(0.254291\pi\)
\(6\) 102060. 0.893043
\(7\) 487486. 1.56612 0.783060 0.621947i \(-0.213658\pi\)
0.783060 + 0.621947i \(0.213658\pi\)
\(8\) 450240. 0.607238
\(9\) 531441. 0.333333
\(10\) 6.82360e6 2.15781
\(11\) −1.77156e6 −0.301511
\(12\) 8.31643e6 0.804005
\(13\) −1.83883e7 −1.05660 −0.528299 0.849058i \(-0.677170\pi\)
−0.528299 + 0.849058i \(0.677170\pi\)
\(14\) 6.82480e7 2.42247
\(15\) 3.55315e7 0.805415
\(16\) −3.04207e7 −0.453304
\(17\) −9.62333e7 −0.966957 −0.483479 0.875356i \(-0.660627\pi\)
−0.483479 + 0.875356i \(0.660627\pi\)
\(18\) 7.44017e7 0.515599
\(19\) −1.49547e7 −0.0729252 −0.0364626 0.999335i \(-0.511609\pi\)
−0.0364626 + 0.999335i \(0.511609\pi\)
\(20\) 5.56026e8 1.94267
\(21\) 3.55377e8 0.904199
\(22\) −2.48019e8 −0.466377
\(23\) 1.53804e8 0.216640 0.108320 0.994116i \(-0.465453\pi\)
0.108320 + 0.994116i \(0.465453\pi\)
\(24\) 3.28225e8 0.350589
\(25\) 1.15488e9 0.946081
\(26\) −2.57436e9 −1.63434
\(27\) 3.87420e8 0.192450
\(28\) 5.56124e9 2.18094
\(29\) 5.21901e9 1.62930 0.814650 0.579953i \(-0.196929\pi\)
0.814650 + 0.579953i \(0.196929\pi\)
\(30\) 4.97440e9 1.24581
\(31\) 1.18381e9 0.239570 0.119785 0.992800i \(-0.461780\pi\)
0.119785 + 0.992800i \(0.461780\pi\)
\(32\) −7.94727e9 −1.30841
\(33\) −1.29147e9 −0.174078
\(34\) −1.34727e10 −1.49569
\(35\) 2.37601e10 2.18477
\(36\) 6.06268e9 0.464193
\(37\) −1.76722e10 −1.13235 −0.566173 0.824286i \(-0.691577\pi\)
−0.566173 + 0.824286i \(0.691577\pi\)
\(38\) −2.09365e9 −0.112800
\(39\) −1.34051e10 −0.610027
\(40\) 2.19447e10 0.847110
\(41\) −1.94617e10 −0.639862 −0.319931 0.947441i \(-0.603660\pi\)
−0.319931 + 0.947441i \(0.603660\pi\)
\(42\) 4.97528e10 1.39861
\(43\) −3.41230e9 −0.0823195 −0.0411598 0.999153i \(-0.513105\pi\)
−0.0411598 + 0.999153i \(0.513105\pi\)
\(44\) −2.02100e10 −0.419878
\(45\) 2.59024e10 0.465007
\(46\) 2.15326e10 0.335097
\(47\) −1.00328e11 −1.35764 −0.678820 0.734304i \(-0.737509\pi\)
−0.678820 + 0.734304i \(0.737509\pi\)
\(48\) −2.21767e10 −0.261715
\(49\) 1.40754e11 1.45273
\(50\) 1.61684e11 1.46339
\(51\) −7.01540e10 −0.558273
\(52\) −2.09774e11 −1.47140
\(53\) 2.75469e11 1.70718 0.853591 0.520944i \(-0.174420\pi\)
0.853591 + 0.520944i \(0.174420\pi\)
\(54\) 5.42389e10 0.297681
\(55\) −8.63459e10 −0.420614
\(56\) 2.19486e11 0.951008
\(57\) −1.09019e10 −0.0421034
\(58\) 7.30661e11 2.52020
\(59\) −2.67677e11 −0.826176 −0.413088 0.910691i \(-0.635550\pi\)
−0.413088 + 0.910691i \(0.635550\pi\)
\(60\) 4.05343e11 1.12160
\(61\) 5.63487e11 1.40036 0.700180 0.713966i \(-0.253103\pi\)
0.700180 + 0.713966i \(0.253103\pi\)
\(62\) 1.65734e11 0.370565
\(63\) 2.59070e11 0.522040
\(64\) −8.63411e11 −1.57054
\(65\) −8.96246e11 −1.47398
\(66\) −1.80806e11 −0.269263
\(67\) 1.08084e12 1.45974 0.729871 0.683585i \(-0.239580\pi\)
0.729871 + 0.683585i \(0.239580\pi\)
\(68\) −1.09783e12 −1.34656
\(69\) 1.12123e11 0.125077
\(70\) 3.32641e12 3.37939
\(71\) −1.15056e12 −1.06594 −0.532968 0.846136i \(-0.678923\pi\)
−0.532968 + 0.846136i \(0.678923\pi\)
\(72\) 2.39276e11 0.202413
\(73\) −3.45915e11 −0.267529 −0.133764 0.991013i \(-0.542707\pi\)
−0.133764 + 0.991013i \(0.542707\pi\)
\(74\) −2.47411e12 −1.75151
\(75\) 8.41911e11 0.546220
\(76\) −1.70603e11 −0.101554
\(77\) −8.63611e11 −0.472203
\(78\) −1.87671e12 −0.943588
\(79\) −2.00408e12 −0.927554 −0.463777 0.885952i \(-0.653506\pi\)
−0.463777 + 0.885952i \(0.653506\pi\)
\(80\) −1.48271e12 −0.632369
\(81\) 2.82430e11 0.111111
\(82\) −2.72464e12 −0.989737
\(83\) −3.33673e12 −1.12025 −0.560123 0.828409i \(-0.689246\pi\)
−0.560123 + 0.828409i \(0.689246\pi\)
\(84\) 4.05414e12 1.25917
\(85\) −4.69041e12 −1.34892
\(86\) −4.77723e11 −0.127332
\(87\) 3.80466e12 0.940677
\(88\) −7.97628e11 −0.183089
\(89\) −5.69624e12 −1.21494 −0.607468 0.794344i \(-0.707815\pi\)
−0.607468 + 0.794344i \(0.707815\pi\)
\(90\) 3.62634e12 0.719271
\(91\) −8.96404e12 −1.65476
\(92\) 1.75460e12 0.301688
\(93\) 8.62999e11 0.138316
\(94\) −1.40459e13 −2.09999
\(95\) −7.28890e11 −0.101732
\(96\) −5.79356e12 −0.755409
\(97\) −6.55011e12 −0.798422 −0.399211 0.916859i \(-0.630716\pi\)
−0.399211 + 0.916859i \(0.630716\pi\)
\(98\) 1.97055e13 2.24708
\(99\) −9.41480e11 −0.100504
\(100\) 1.31749e13 1.31749
\(101\) 2.00225e13 1.87685 0.938424 0.345487i \(-0.112286\pi\)
0.938424 + 0.345487i \(0.112286\pi\)
\(102\) −9.82157e12 −0.863534
\(103\) 8.50230e12 0.701608 0.350804 0.936449i \(-0.385908\pi\)
0.350804 + 0.936449i \(0.385908\pi\)
\(104\) −8.27915e12 −0.641607
\(105\) 1.73211e13 1.26138
\(106\) 3.85657e13 2.64066
\(107\) 2.60740e13 1.67963 0.839814 0.542875i \(-0.182664\pi\)
0.839814 + 0.542875i \(0.182664\pi\)
\(108\) 4.41969e12 0.268002
\(109\) −2.65871e12 −0.151844 −0.0759222 0.997114i \(-0.524190\pi\)
−0.0759222 + 0.997114i \(0.524190\pi\)
\(110\) −1.20884e13 −0.650605
\(111\) −1.28830e13 −0.653761
\(112\) −1.48297e13 −0.709929
\(113\) −3.76049e13 −1.69916 −0.849580 0.527460i \(-0.823144\pi\)
−0.849580 + 0.527460i \(0.823144\pi\)
\(114\) −1.52627e12 −0.0651254
\(115\) 7.49643e12 0.302217
\(116\) 5.95385e13 2.26893
\(117\) −9.77230e12 −0.352199
\(118\) −3.74748e13 −1.27793
\(119\) −4.69124e13 −1.51437
\(120\) 1.59977e13 0.489079
\(121\) 3.13843e12 0.0909091
\(122\) 7.88881e13 2.16607
\(123\) −1.41876e13 −0.369425
\(124\) 1.35049e13 0.333619
\(125\) −3.20800e12 −0.0752176
\(126\) 3.62698e13 0.807489
\(127\) −6.93279e13 −1.46617 −0.733084 0.680138i \(-0.761920\pi\)
−0.733084 + 0.680138i \(0.761920\pi\)
\(128\) −5.57735e13 −1.12089
\(129\) −2.48757e12 −0.0475272
\(130\) −1.25474e14 −2.27994
\(131\) −5.31970e13 −0.919653 −0.459827 0.888009i \(-0.652088\pi\)
−0.459827 + 0.888009i \(0.652088\pi\)
\(132\) −1.47331e13 −0.242417
\(133\) −7.29018e12 −0.114210
\(134\) 1.51318e14 2.25792
\(135\) 1.88829e13 0.268472
\(136\) −4.33281e13 −0.587173
\(137\) 1.21037e14 1.56399 0.781994 0.623286i \(-0.214203\pi\)
0.781994 + 0.623286i \(0.214203\pi\)
\(138\) 1.56973e13 0.193468
\(139\) 1.03707e14 1.21958 0.609791 0.792562i \(-0.291253\pi\)
0.609791 + 0.792562i \(0.291253\pi\)
\(140\) 2.71055e14 3.04246
\(141\) −7.31389e13 −0.783834
\(142\) −1.61079e14 −1.64878
\(143\) 3.25760e13 0.318576
\(144\) −1.61668e13 −0.151101
\(145\) 2.54375e14 2.27291
\(146\) −4.84280e13 −0.413812
\(147\) 1.02609e14 0.838734
\(148\) −2.01604e14 −1.57688
\(149\) −2.94068e13 −0.220159 −0.110080 0.993923i \(-0.535111\pi\)
−0.110080 + 0.993923i \(0.535111\pi\)
\(150\) 1.17868e14 0.844891
\(151\) −4.86579e13 −0.334043 −0.167022 0.985953i \(-0.553415\pi\)
−0.167022 + 0.985953i \(0.553415\pi\)
\(152\) −6.73318e12 −0.0442830
\(153\) −5.11423e13 −0.322319
\(154\) −1.20906e14 −0.730401
\(155\) 5.76990e13 0.334204
\(156\) −1.52925e14 −0.849511
\(157\) −2.31548e14 −1.23394 −0.616969 0.786988i \(-0.711640\pi\)
−0.616969 + 0.786988i \(0.711640\pi\)
\(158\) −2.80571e14 −1.43474
\(159\) 2.00817e14 0.985642
\(160\) −3.87350e14 −1.82526
\(161\) 7.49775e13 0.339283
\(162\) 3.95401e13 0.171866
\(163\) 1.44315e14 0.602688 0.301344 0.953516i \(-0.402565\pi\)
0.301344 + 0.953516i \(0.402565\pi\)
\(164\) −2.22020e14 −0.891058
\(165\) −6.29461e13 −0.242842
\(166\) −4.67143e14 −1.73279
\(167\) 4.17920e14 1.49086 0.745429 0.666585i \(-0.232244\pi\)
0.745429 + 0.666585i \(0.232244\pi\)
\(168\) 1.60005e14 0.549065
\(169\) 3.52546e13 0.116400
\(170\) −6.56657e14 −2.08651
\(171\) −7.94752e12 −0.0243084
\(172\) −3.89276e13 −0.114636
\(173\) 4.17435e14 1.18383 0.591914 0.806001i \(-0.298372\pi\)
0.591914 + 0.806001i \(0.298372\pi\)
\(174\) 5.32652e14 1.45504
\(175\) 5.62990e14 1.48168
\(176\) 5.38922e13 0.136676
\(177\) −1.95136e14 −0.476993
\(178\) −7.97473e14 −1.87926
\(179\) 4.89549e14 1.11238 0.556188 0.831056i \(-0.312263\pi\)
0.556188 + 0.831056i \(0.312263\pi\)
\(180\) 2.95495e14 0.647558
\(181\) 3.82069e14 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(182\) −1.25497e15 −2.55957
\(183\) 4.10782e14 0.808498
\(184\) 6.92489e13 0.131552
\(185\) −8.61343e14 −1.57965
\(186\) 1.20820e14 0.213946
\(187\) 1.70483e14 0.291549
\(188\) −1.14454e15 −1.89062
\(189\) 1.88862e14 0.301400
\(190\) −1.02045e14 −0.157359
\(191\) 2.81803e14 0.419980 0.209990 0.977704i \(-0.432657\pi\)
0.209990 + 0.977704i \(0.432657\pi\)
\(192\) −6.29427e14 −0.906749
\(193\) 6.26551e14 0.872638 0.436319 0.899792i \(-0.356282\pi\)
0.436319 + 0.899792i \(0.356282\pi\)
\(194\) −9.17016e14 −1.23500
\(195\) −6.53363e14 −0.851000
\(196\) 1.60572e15 2.02304
\(197\) −6.60082e14 −0.804576 −0.402288 0.915513i \(-0.631785\pi\)
−0.402288 + 0.915513i \(0.631785\pi\)
\(198\) −1.31807e14 −0.155459
\(199\) −1.45047e15 −1.65564 −0.827818 0.560996i \(-0.810418\pi\)
−0.827818 + 0.560996i \(0.810418\pi\)
\(200\) 5.19975e14 0.574497
\(201\) 7.87934e14 0.842783
\(202\) 2.80315e15 2.90310
\(203\) 2.54419e15 2.55168
\(204\) −8.00317e14 −0.777439
\(205\) −9.48565e14 −0.892621
\(206\) 1.19032e15 1.08524
\(207\) 8.17380e13 0.0722132
\(208\) 5.59386e14 0.478961
\(209\) 2.64931e13 0.0219878
\(210\) 2.42495e15 1.95109
\(211\) −1.86139e15 −1.45212 −0.726059 0.687633i \(-0.758650\pi\)
−0.726059 + 0.687633i \(0.758650\pi\)
\(212\) 3.14255e15 2.37738
\(213\) −8.38760e14 −0.615418
\(214\) 3.65036e15 2.59804
\(215\) −1.66316e14 −0.114837
\(216\) 1.74432e14 0.116863
\(217\) 5.77092e14 0.375195
\(218\) −3.72219e14 −0.234872
\(219\) −2.52172e14 −0.154458
\(220\) −9.85034e14 −0.585738
\(221\) 1.76957e15 1.02169
\(222\) −1.80362e15 −1.01123
\(223\) 2.64472e15 1.44012 0.720059 0.693913i \(-0.244115\pi\)
0.720059 + 0.693913i \(0.244115\pi\)
\(224\) −3.87418e15 −2.04912
\(225\) 6.13753e14 0.315360
\(226\) −5.26468e15 −2.62825
\(227\) 3.07427e15 1.49133 0.745666 0.666319i \(-0.232131\pi\)
0.745666 + 0.666319i \(0.232131\pi\)
\(228\) −1.24369e14 −0.0586323
\(229\) −2.98623e15 −1.36834 −0.684168 0.729324i \(-0.739835\pi\)
−0.684168 + 0.729324i \(0.739835\pi\)
\(230\) 1.04950e15 0.467467
\(231\) −6.29573e14 −0.272626
\(232\) 2.34981e15 0.989374
\(233\) −1.86637e15 −0.764159 −0.382079 0.924130i \(-0.624792\pi\)
−0.382079 + 0.924130i \(0.624792\pi\)
\(234\) −1.36812e15 −0.544781
\(235\) −4.88997e15 −1.89394
\(236\) −3.05366e15 −1.15052
\(237\) −1.46098e15 −0.535524
\(238\) −6.56773e15 −2.34242
\(239\) −3.11970e14 −0.108275 −0.0541373 0.998534i \(-0.517241\pi\)
−0.0541373 + 0.998534i \(0.517241\pi\)
\(240\) −1.08089e15 −0.365098
\(241\) 5.25176e15 1.72661 0.863305 0.504683i \(-0.168391\pi\)
0.863305 + 0.504683i \(0.168391\pi\)
\(242\) 4.39380e14 0.140618
\(243\) 2.05891e14 0.0641500
\(244\) 6.42826e15 1.95011
\(245\) 6.86033e15 2.02659
\(246\) −1.98627e15 −0.571425
\(247\) 2.74991e14 0.0770527
\(248\) 5.32999e14 0.145476
\(249\) −2.43248e15 −0.646775
\(250\) −4.49120e14 −0.116346
\(251\) −2.46350e15 −0.621833 −0.310916 0.950437i \(-0.600636\pi\)
−0.310916 + 0.950437i \(0.600636\pi\)
\(252\) 2.95547e15 0.726981
\(253\) −2.72474e14 −0.0653193
\(254\) −9.70590e15 −2.26786
\(255\) −3.41931e15 −0.778802
\(256\) −7.35229e14 −0.163254
\(257\) 2.41514e15 0.522851 0.261425 0.965224i \(-0.415808\pi\)
0.261425 + 0.965224i \(0.415808\pi\)
\(258\) −3.48260e14 −0.0735149
\(259\) −8.61495e15 −1.77339
\(260\) −1.02244e16 −2.05263
\(261\) 2.77360e15 0.543100
\(262\) −7.44759e15 −1.42252
\(263\) −1.68025e15 −0.313084 −0.156542 0.987671i \(-0.550035\pi\)
−0.156542 + 0.987671i \(0.550035\pi\)
\(264\) −5.81471e14 −0.105707
\(265\) 1.34264e16 2.38155
\(266\) −1.02063e15 −0.176659
\(267\) −4.15256e15 −0.701443
\(268\) 1.23303e16 2.03281
\(269\) −5.87926e14 −0.0946091 −0.0473046 0.998881i \(-0.515063\pi\)
−0.0473046 + 0.998881i \(0.515063\pi\)
\(270\) 2.64360e15 0.415271
\(271\) 3.69154e14 0.0566119 0.0283059 0.999599i \(-0.490989\pi\)
0.0283059 + 0.999599i \(0.490989\pi\)
\(272\) 2.92749e15 0.438326
\(273\) −6.53479e15 −0.955376
\(274\) 1.69451e16 2.41917
\(275\) −2.04595e15 −0.285254
\(276\) 1.27910e15 0.174179
\(277\) 3.44319e15 0.457976 0.228988 0.973429i \(-0.426458\pi\)
0.228988 + 0.973429i \(0.426458\pi\)
\(278\) 1.45190e16 1.88645
\(279\) 6.29126e14 0.0798565
\(280\) 1.06977e16 1.32667
\(281\) 1.73559e15 0.210309 0.105154 0.994456i \(-0.466466\pi\)
0.105154 + 0.994456i \(0.466466\pi\)
\(282\) −1.02394e16 −1.21243
\(283\) 1.72244e15 0.199311 0.0996556 0.995022i \(-0.468226\pi\)
0.0996556 + 0.995022i \(0.468226\pi\)
\(284\) −1.31256e16 −1.48440
\(285\) −5.31361e14 −0.0587351
\(286\) 4.56064e15 0.492773
\(287\) −9.48733e15 −1.00210
\(288\) −4.22350e15 −0.436136
\(289\) −6.43739e14 −0.0649941
\(290\) 3.56124e16 3.51572
\(291\) −4.77503e15 −0.460969
\(292\) −3.94619e15 −0.372554
\(293\) −9.32911e15 −0.861392 −0.430696 0.902497i \(-0.641732\pi\)
−0.430696 + 0.902497i \(0.641732\pi\)
\(294\) 1.43653e16 1.29735
\(295\) −1.30466e16 −1.15253
\(296\) −7.95673e15 −0.687604
\(297\) −6.86339e14 −0.0580259
\(298\) −4.11695e15 −0.340541
\(299\) −2.82820e15 −0.228901
\(300\) 9.60452e15 0.760654
\(301\) −1.66345e15 −0.128922
\(302\) −6.81210e15 −0.516697
\(303\) 1.45964e16 1.08360
\(304\) 4.54932e14 0.0330573
\(305\) 2.74643e16 1.95353
\(306\) −7.15992e15 −0.498562
\(307\) −1.85795e16 −1.26658 −0.633292 0.773913i \(-0.718297\pi\)
−0.633292 + 0.773913i \(0.718297\pi\)
\(308\) −9.85208e15 −0.657579
\(309\) 6.19818e15 0.405074
\(310\) 8.07786e15 0.516946
\(311\) 1.10219e16 0.690739 0.345369 0.938467i \(-0.387754\pi\)
0.345369 + 0.938467i \(0.387754\pi\)
\(312\) −6.03550e15 −0.370432
\(313\) 2.44634e16 1.47055 0.735273 0.677771i \(-0.237054\pi\)
0.735273 + 0.677771i \(0.237054\pi\)
\(314\) −3.24167e16 −1.90865
\(315\) 1.26271e16 0.728256
\(316\) −2.28626e16 −1.29169
\(317\) 7.28014e15 0.402953 0.201477 0.979493i \(-0.435426\pi\)
0.201477 + 0.979493i \(0.435426\pi\)
\(318\) 2.81144e16 1.52459
\(319\) −9.24580e15 −0.491253
\(320\) −4.20827e16 −2.19093
\(321\) 1.90079e16 0.969733
\(322\) 1.04968e16 0.524802
\(323\) 1.43913e15 0.0705156
\(324\) 3.22196e15 0.154731
\(325\) −2.12364e16 −0.999628
\(326\) 2.02041e16 0.932235
\(327\) −1.93820e15 −0.0876674
\(328\) −8.76245e15 −0.388549
\(329\) −4.89084e16 −2.12623
\(330\) −8.81246e15 −0.375627
\(331\) 4.91941e15 0.205604 0.102802 0.994702i \(-0.467219\pi\)
0.102802 + 0.994702i \(0.467219\pi\)
\(332\) −3.80654e16 −1.56003
\(333\) −9.39173e15 −0.377449
\(334\) 5.85089e16 2.30605
\(335\) 5.26803e16 2.03637
\(336\) −1.08108e16 −0.409878
\(337\) −3.61707e15 −0.134512 −0.0672562 0.997736i \(-0.521424\pi\)
−0.0672562 + 0.997736i \(0.521424\pi\)
\(338\) 4.93565e15 0.180047
\(339\) −2.74139e16 −0.981010
\(340\) −5.35082e16 −1.87848
\(341\) −2.09719e15 −0.0722329
\(342\) −1.11265e15 −0.0376002
\(343\) 2.13834e16 0.709030
\(344\) −1.53636e15 −0.0499876
\(345\) 5.46489e15 0.174485
\(346\) 5.84408e16 1.83114
\(347\) 2.03135e16 0.624660 0.312330 0.949974i \(-0.398890\pi\)
0.312330 + 0.949974i \(0.398890\pi\)
\(348\) 4.34035e16 1.30997
\(349\) −1.16091e16 −0.343902 −0.171951 0.985106i \(-0.555007\pi\)
−0.171951 + 0.985106i \(0.555007\pi\)
\(350\) 7.88186e16 2.29185
\(351\) −7.12401e15 −0.203342
\(352\) 1.40791e16 0.394500
\(353\) 1.93894e16 0.533370 0.266685 0.963784i \(-0.414072\pi\)
0.266685 + 0.963784i \(0.414072\pi\)
\(354\) −2.73191e16 −0.737811
\(355\) −5.60784e16 −1.48700
\(356\) −6.49827e16 −1.69189
\(357\) −3.41991e16 −0.874322
\(358\) 6.85369e16 1.72062
\(359\) −1.45895e16 −0.359689 −0.179844 0.983695i \(-0.557559\pi\)
−0.179844 + 0.983695i \(0.557559\pi\)
\(360\) 1.16623e16 0.282370
\(361\) −4.18293e16 −0.994682
\(362\) 5.34896e16 1.24929
\(363\) 2.28791e15 0.0524864
\(364\) −1.02262e17 −2.30438
\(365\) −1.68599e16 −0.373208
\(366\) 5.75094e16 1.25058
\(367\) −5.04342e16 −1.07745 −0.538724 0.842482i \(-0.681093\pi\)
−0.538724 + 0.842482i \(0.681093\pi\)
\(368\) −4.67884e15 −0.0982037
\(369\) −1.03428e16 −0.213287
\(370\) −1.20588e17 −2.44339
\(371\) 1.34287e17 2.67365
\(372\) 9.84509e15 0.192615
\(373\) 6.41403e15 0.123317 0.0616586 0.998097i \(-0.480361\pi\)
0.0616586 + 0.998097i \(0.480361\pi\)
\(374\) 2.38676e16 0.450966
\(375\) −2.33863e15 −0.0434269
\(376\) −4.51716e16 −0.824411
\(377\) −9.59688e16 −1.72152
\(378\) 2.64407e16 0.466204
\(379\) 6.85830e15 0.118867 0.0594335 0.998232i \(-0.481071\pi\)
0.0594335 + 0.998232i \(0.481071\pi\)
\(380\) −8.31517e15 −0.141670
\(381\) −5.05400e16 −0.846492
\(382\) 3.94524e16 0.649623
\(383\) 5.27865e16 0.854537 0.427268 0.904125i \(-0.359476\pi\)
0.427268 + 0.904125i \(0.359476\pi\)
\(384\) −4.06589e16 −0.647146
\(385\) −4.20924e16 −0.658732
\(386\) 8.77171e16 1.34979
\(387\) −1.81344e15 −0.0274398
\(388\) −7.47237e16 −1.11187
\(389\) 6.57216e16 0.961692 0.480846 0.876805i \(-0.340330\pi\)
0.480846 + 0.876805i \(0.340330\pi\)
\(390\) −9.14709e16 −1.31632
\(391\) −1.48011e16 −0.209481
\(392\) 6.33729e16 0.882153
\(393\) −3.87806e16 −0.530962
\(394\) −9.24114e16 −1.24452
\(395\) −9.76789e16 −1.29396
\(396\) −1.07404e16 −0.139959
\(397\) −1.04942e17 −1.34528 −0.672639 0.739971i \(-0.734839\pi\)
−0.672639 + 0.739971i \(0.734839\pi\)
\(398\) −2.03066e17 −2.56093
\(399\) −5.31454e15 −0.0659390
\(400\) −3.51324e16 −0.428863
\(401\) 2.14759e16 0.257936 0.128968 0.991649i \(-0.458834\pi\)
0.128968 + 0.991649i \(0.458834\pi\)
\(402\) 1.10311e17 1.30361
\(403\) −2.17683e16 −0.253129
\(404\) 2.28416e17 2.61366
\(405\) 1.37656e16 0.155002
\(406\) 3.56187e17 3.94693
\(407\) 3.13074e16 0.341415
\(408\) −3.15862e16 −0.339005
\(409\) 1.57661e17 1.66542 0.832708 0.553713i \(-0.186790\pi\)
0.832708 + 0.553713i \(0.186790\pi\)
\(410\) −1.32799e17 −1.38070
\(411\) 8.82358e16 0.902969
\(412\) 9.69943e16 0.977044
\(413\) −1.30489e17 −1.29389
\(414\) 1.14433e16 0.111699
\(415\) −1.62632e17 −1.56277
\(416\) 1.46137e17 1.38246
\(417\) 7.56023e16 0.704126
\(418\) 3.70903e15 0.0340106
\(419\) 8.20257e16 0.740558 0.370279 0.928921i \(-0.379262\pi\)
0.370279 + 0.928921i \(0.379262\pi\)
\(420\) 1.97599e17 1.75657
\(421\) −5.59695e16 −0.489912 −0.244956 0.969534i \(-0.578774\pi\)
−0.244956 + 0.969534i \(0.578774\pi\)
\(422\) −2.60595e17 −2.24613
\(423\) −5.33183e16 −0.452547
\(424\) 1.24027e17 1.03667
\(425\) −1.11138e17 −0.914820
\(426\) −1.17426e17 −0.951926
\(427\) 2.74692e17 2.19313
\(428\) 2.97452e17 2.33901
\(429\) 2.37479e16 0.183930
\(430\) −2.32842e16 −0.177630
\(431\) 1.77267e17 1.33207 0.666034 0.745921i \(-0.267991\pi\)
0.666034 + 0.745921i \(0.267991\pi\)
\(432\) −1.17856e16 −0.0872385
\(433\) −5.06598e16 −0.369396 −0.184698 0.982795i \(-0.559131\pi\)
−0.184698 + 0.982795i \(0.559131\pi\)
\(434\) 8.07928e16 0.580349
\(435\) 1.85439e17 1.31226
\(436\) −3.03306e16 −0.211455
\(437\) −2.30009e15 −0.0157985
\(438\) −3.53040e16 −0.238915
\(439\) −6.76669e16 −0.451187 −0.225594 0.974221i \(-0.572432\pi\)
−0.225594 + 0.974221i \(0.572432\pi\)
\(440\) −3.88764e16 −0.255413
\(441\) 7.48022e16 0.484243
\(442\) 2.47739e17 1.58034
\(443\) 1.70873e17 1.07411 0.537057 0.843546i \(-0.319536\pi\)
0.537057 + 0.843546i \(0.319536\pi\)
\(444\) −1.46970e17 −0.910413
\(445\) −2.77635e17 −1.69486
\(446\) 3.70261e17 2.22757
\(447\) −2.14376e16 −0.127109
\(448\) −4.20901e17 −2.45965
\(449\) 2.13667e14 0.00123066 0.000615329 1.00000i \(-0.499804\pi\)
0.000615329 1.00000i \(0.499804\pi\)
\(450\) 8.59254e16 0.487798
\(451\) 3.44777e16 0.192926
\(452\) −4.28996e17 −2.36621
\(453\) −3.54716e16 −0.192860
\(454\) 4.30398e17 2.30679
\(455\) −4.36907e17 −2.30842
\(456\) −4.90849e15 −0.0255668
\(457\) 3.31395e17 1.70173 0.850864 0.525386i \(-0.176079\pi\)
0.850864 + 0.525386i \(0.176079\pi\)
\(458\) −4.18073e17 −2.11654
\(459\) −3.72827e16 −0.186091
\(460\) 8.55192e16 0.420860
\(461\) −2.65757e17 −1.28952 −0.644761 0.764384i \(-0.723043\pi\)
−0.644761 + 0.764384i \(0.723043\pi\)
\(462\) −8.81402e16 −0.421697
\(463\) −4.27497e15 −0.0201677 −0.0100839 0.999949i \(-0.503210\pi\)
−0.0100839 + 0.999949i \(0.503210\pi\)
\(464\) −1.58766e17 −0.738569
\(465\) 4.20626e16 0.192953
\(466\) −2.61291e17 −1.18200
\(467\) −2.56398e16 −0.114381 −0.0571907 0.998363i \(-0.518214\pi\)
−0.0571907 + 0.998363i \(0.518214\pi\)
\(468\) −1.11482e17 −0.490465
\(469\) 5.26896e17 2.28613
\(470\) −6.84596e17 −2.92953
\(471\) −1.68798e17 −0.712414
\(472\) −1.20519e17 −0.501686
\(473\) 6.04511e15 0.0248203
\(474\) −2.04537e17 −0.828346
\(475\) −1.72709e16 −0.0689932
\(476\) −5.35176e17 −2.10888
\(477\) 1.46396e17 0.569061
\(478\) −4.36758e16 −0.167479
\(479\) 3.69707e17 1.39855 0.699274 0.714854i \(-0.253507\pi\)
0.699274 + 0.714854i \(0.253507\pi\)
\(480\) −2.82378e17 −1.05381
\(481\) 3.24962e17 1.19644
\(482\) 7.35246e17 2.67071
\(483\) 5.46586e16 0.195885
\(484\) 3.58032e16 0.126598
\(485\) −3.19253e17 −1.11382
\(486\) 2.88248e16 0.0992270
\(487\) −5.17795e17 −1.75881 −0.879406 0.476073i \(-0.842060\pi\)
−0.879406 + 0.476073i \(0.842060\pi\)
\(488\) 2.53704e17 0.850352
\(489\) 1.05206e17 0.347962
\(490\) 9.60446e17 3.13472
\(491\) 1.72056e17 0.554165 0.277082 0.960846i \(-0.410632\pi\)
0.277082 + 0.960846i \(0.410632\pi\)
\(492\) −1.61852e17 −0.514453
\(493\) −5.02242e17 −1.57546
\(494\) 3.84987e16 0.119185
\(495\) −4.58877e16 −0.140205
\(496\) −3.60124e16 −0.108598
\(497\) −5.60883e17 −1.66938
\(498\) −3.40547e17 −1.00043
\(499\) 1.31941e17 0.382584 0.191292 0.981533i \(-0.438732\pi\)
0.191292 + 0.981533i \(0.438732\pi\)
\(500\) −3.65969e16 −0.104746
\(501\) 3.04664e17 0.860748
\(502\) −3.44890e17 −0.961848
\(503\) 2.26740e17 0.624218 0.312109 0.950046i \(-0.398965\pi\)
0.312109 + 0.950046i \(0.398965\pi\)
\(504\) 1.16644e17 0.317003
\(505\) 9.75896e17 2.61824
\(506\) −3.81463e16 −0.101036
\(507\) 2.57006e16 0.0672035
\(508\) −7.90892e17 −2.04175
\(509\) −8.67070e16 −0.220998 −0.110499 0.993876i \(-0.535245\pi\)
−0.110499 + 0.993876i \(0.535245\pi\)
\(510\) −4.78703e17 −1.20465
\(511\) −1.68628e17 −0.418982
\(512\) 3.53965e17 0.868371
\(513\) −5.79374e15 −0.0140345
\(514\) 3.38120e17 0.808743
\(515\) 4.14402e17 0.978758
\(516\) −2.83782e16 −0.0661853
\(517\) 1.77737e17 0.409344
\(518\) −1.20609e18 −2.74307
\(519\) 3.04310e17 0.683484
\(520\) −4.03526e17 −0.895055
\(521\) 5.57568e17 1.22138 0.610692 0.791868i \(-0.290891\pi\)
0.610692 + 0.791868i \(0.290891\pi\)
\(522\) 3.88303e17 0.840065
\(523\) −4.80021e17 −1.02565 −0.512826 0.858493i \(-0.671401\pi\)
−0.512826 + 0.858493i \(0.671401\pi\)
\(524\) −6.06872e17 −1.28069
\(525\) 4.10420e17 0.855446
\(526\) −2.35235e17 −0.484277
\(527\) −1.13922e17 −0.231653
\(528\) 3.92874e16 0.0789101
\(529\) −4.80381e17 −0.953067
\(530\) 1.87969e18 3.68378
\(531\) −1.42254e17 −0.275392
\(532\) −8.31664e16 −0.159046
\(533\) 3.57868e17 0.676077
\(534\) −5.81358e17 −1.08499
\(535\) 1.27085e18 2.34311
\(536\) 4.86639e17 0.886412
\(537\) 3.56881e17 0.642231
\(538\) −8.23097e16 −0.146341
\(539\) −2.49354e17 −0.438015
\(540\) 2.15416e17 0.373868
\(541\) −3.07367e17 −0.527078 −0.263539 0.964649i \(-0.584890\pi\)
−0.263539 + 0.964649i \(0.584890\pi\)
\(542\) 5.16816e16 0.0875670
\(543\) 2.78528e17 0.466305
\(544\) 7.64792e17 1.26517
\(545\) −1.29585e17 −0.211826
\(546\) −9.14870e17 −1.47777
\(547\) 6.25426e17 0.998294 0.499147 0.866517i \(-0.333647\pi\)
0.499147 + 0.866517i \(0.333647\pi\)
\(548\) 1.38079e18 2.17798
\(549\) 2.99460e17 0.466787
\(550\) −2.86433e17 −0.441230
\(551\) −7.80485e16 −0.118817
\(552\) 5.04824e16 0.0759515
\(553\) −9.76961e17 −1.45266
\(554\) 4.82047e17 0.708396
\(555\) −6.27919e17 −0.912009
\(556\) 1.18309e18 1.69836
\(557\) 5.97923e17 0.848372 0.424186 0.905575i \(-0.360560\pi\)
0.424186 + 0.905575i \(0.360560\pi\)
\(558\) 8.80777e16 0.123522
\(559\) 6.27465e16 0.0869787
\(560\) −7.22799e17 −0.990365
\(561\) 1.24282e17 0.168326
\(562\) 2.42983e17 0.325305
\(563\) −9.48274e16 −0.125496 −0.0627479 0.998029i \(-0.519986\pi\)
−0.0627479 + 0.998029i \(0.519986\pi\)
\(564\) −8.34369e17 −1.09155
\(565\) −1.83286e18 −2.37036
\(566\) 2.41141e17 0.308294
\(567\) 1.37680e17 0.174013
\(568\) −5.18029e17 −0.647277
\(569\) 7.42364e17 0.917038 0.458519 0.888685i \(-0.348380\pi\)
0.458519 + 0.888685i \(0.348380\pi\)
\(570\) −7.43905e16 −0.0908512
\(571\) 9.72970e17 1.17480 0.587401 0.809296i \(-0.300151\pi\)
0.587401 + 0.809296i \(0.300151\pi\)
\(572\) 3.71627e17 0.443642
\(573\) 2.05434e17 0.242475
\(574\) −1.32823e18 −1.55005
\(575\) 1.77626e17 0.204959
\(576\) −4.58852e17 −0.523512
\(577\) −4.88742e17 −0.551362 −0.275681 0.961249i \(-0.588903\pi\)
−0.275681 + 0.961249i \(0.588903\pi\)
\(578\) −9.01234e16 −0.100533
\(579\) 4.56756e17 0.503817
\(580\) 2.90191e18 3.16520
\(581\) −1.62661e18 −1.75444
\(582\) −6.68505e17 −0.713025
\(583\) −4.88010e17 −0.514735
\(584\) −1.55745e17 −0.162454
\(585\) −4.76302e17 −0.491325
\(586\) −1.30608e18 −1.33240
\(587\) 1.09582e18 1.10558 0.552792 0.833320i \(-0.313563\pi\)
0.552792 + 0.833320i \(0.313563\pi\)
\(588\) 1.17057e18 1.16800
\(589\) −1.77035e16 −0.0174707
\(590\) −1.82652e18 −1.78273
\(591\) −4.81199e17 −0.464522
\(592\) 5.37601e17 0.513298
\(593\) −1.82087e18 −1.71959 −0.859794 0.510641i \(-0.829408\pi\)
−0.859794 + 0.510641i \(0.829408\pi\)
\(594\) −9.60875e16 −0.0897542
\(595\) −2.28651e18 −2.11258
\(596\) −3.35473e17 −0.306589
\(597\) −1.05740e18 −0.955882
\(598\) −3.95948e17 −0.354063
\(599\) 5.79052e17 0.512204 0.256102 0.966650i \(-0.417562\pi\)
0.256102 + 0.966650i \(0.417562\pi\)
\(600\) 3.79062e17 0.331686
\(601\) −1.68040e18 −1.45455 −0.727273 0.686348i \(-0.759213\pi\)
−0.727273 + 0.686348i \(0.759213\pi\)
\(602\) −2.32883e17 −0.199416
\(603\) 5.74404e17 0.486581
\(604\) −5.55089e17 −0.465182
\(605\) 1.52967e17 0.126820
\(606\) 2.04349e18 1.67611
\(607\) −2.18412e17 −0.177235 −0.0886176 0.996066i \(-0.528245\pi\)
−0.0886176 + 0.996066i \(0.528245\pi\)
\(608\) 1.18849e17 0.0954159
\(609\) 1.85472e18 1.47321
\(610\) 3.84501e18 3.02171
\(611\) 1.84486e18 1.43448
\(612\) −5.83431e17 −0.448854
\(613\) −8.91819e16 −0.0678865 −0.0339433 0.999424i \(-0.510807\pi\)
−0.0339433 + 0.999424i \(0.510807\pi\)
\(614\) −2.60113e18 −1.95915
\(615\) −6.91504e17 −0.515355
\(616\) −3.88832e17 −0.286740
\(617\) 1.19832e18 0.874419 0.437210 0.899360i \(-0.355967\pi\)
0.437210 + 0.899360i \(0.355967\pi\)
\(618\) 8.67745e17 0.626566
\(619\) 2.57248e18 1.83807 0.919037 0.394170i \(-0.128968\pi\)
0.919037 + 0.394170i \(0.128968\pi\)
\(620\) 6.58230e17 0.465406
\(621\) 5.95870e16 0.0416923
\(622\) 1.54307e18 1.06843
\(623\) −2.77684e18 −1.90273
\(624\) 4.07792e17 0.276528
\(625\) −1.56613e18 −1.05101
\(626\) 3.42488e18 2.27463
\(627\) 1.93135e16 0.0126947
\(628\) −2.64150e18 −1.71835
\(629\) 1.70065e18 1.09493
\(630\) 1.76779e18 1.12646
\(631\) 1.23798e18 0.780767 0.390383 0.920652i \(-0.372342\pi\)
0.390383 + 0.920652i \(0.372342\pi\)
\(632\) −9.02317e17 −0.563246
\(633\) −1.35695e18 −0.838380
\(634\) 1.01922e18 0.623286
\(635\) −3.37904e18 −2.04533
\(636\) 2.29092e18 1.37258
\(637\) −2.58822e18 −1.53495
\(638\) −1.29441e18 −0.759868
\(639\) −6.11456e17 −0.355312
\(640\) −2.71840e18 −1.56366
\(641\) −3.15860e17 −0.179853 −0.0899264 0.995948i \(-0.528663\pi\)
−0.0899264 + 0.995948i \(0.528663\pi\)
\(642\) 2.66111e18 1.49998
\(643\) 1.62866e18 0.908782 0.454391 0.890802i \(-0.349857\pi\)
0.454391 + 0.890802i \(0.349857\pi\)
\(644\) 8.55343e17 0.472479
\(645\) −1.21244e17 −0.0663014
\(646\) 2.01479e17 0.109073
\(647\) −1.20409e18 −0.645327 −0.322664 0.946514i \(-0.604578\pi\)
−0.322664 + 0.946514i \(0.604578\pi\)
\(648\) 1.27161e17 0.0674709
\(649\) 4.74206e17 0.249102
\(650\) −2.97309e18 −1.54622
\(651\) 4.20700e17 0.216619
\(652\) 1.64635e18 0.839290
\(653\) −2.58301e18 −1.30374 −0.651869 0.758332i \(-0.726015\pi\)
−0.651869 + 0.758332i \(0.726015\pi\)
\(654\) −2.71348e17 −0.135604
\(655\) −2.59282e18 −1.28293
\(656\) 5.92040e17 0.290052
\(657\) −1.83833e17 −0.0891762
\(658\) −6.84717e18 −3.28884
\(659\) 2.90720e18 1.38267 0.691337 0.722532i \(-0.257022\pi\)
0.691337 + 0.722532i \(0.257022\pi\)
\(660\) −7.18090e17 −0.338176
\(661\) −3.29270e18 −1.53547 −0.767737 0.640765i \(-0.778617\pi\)
−0.767737 + 0.640765i \(0.778617\pi\)
\(662\) 6.88717e17 0.318027
\(663\) 1.29001e18 0.589870
\(664\) −1.50233e18 −0.680257
\(665\) −3.55324e17 −0.159325
\(666\) −1.31484e18 −0.583836
\(667\) 8.02707e17 0.352971
\(668\) 4.76764e18 2.07614
\(669\) 1.92800e18 0.831453
\(670\) 7.37524e18 3.14985
\(671\) −9.98251e17 −0.422224
\(672\) −2.82428e18 −1.18306
\(673\) −2.98149e18 −1.23690 −0.618451 0.785823i \(-0.712240\pi\)
−0.618451 + 0.785823i \(0.712240\pi\)
\(674\) −5.06390e17 −0.208063
\(675\) 4.47426e17 0.182073
\(676\) 4.02185e17 0.162096
\(677\) −6.19017e17 −0.247102 −0.123551 0.992338i \(-0.539428\pi\)
−0.123551 + 0.992338i \(0.539428\pi\)
\(678\) −3.83795e18 −1.51742
\(679\) −3.19309e18 −1.25042
\(680\) −2.11181e18 −0.819119
\(681\) 2.24115e18 0.861021
\(682\) −2.93607e17 −0.111730
\(683\) 1.07700e18 0.405960 0.202980 0.979183i \(-0.434937\pi\)
0.202980 + 0.979183i \(0.434937\pi\)
\(684\) −9.06653e16 −0.0338514
\(685\) 5.89933e18 2.18180
\(686\) 2.99367e18 1.09672
\(687\) −2.17696e18 −0.790010
\(688\) 1.03805e17 0.0373158
\(689\) −5.06541e18 −1.80381
\(690\) 7.65085e17 0.269892
\(691\) −4.77336e17 −0.166808 −0.0834039 0.996516i \(-0.526579\pi\)
−0.0834039 + 0.996516i \(0.526579\pi\)
\(692\) 4.76209e18 1.64857
\(693\) −4.58958e17 −0.157401
\(694\) 2.84390e18 0.966222
\(695\) 5.05467e18 1.70134
\(696\) 1.71301e18 0.571215
\(697\) 1.87287e18 0.618719
\(698\) −1.62528e18 −0.531946
\(699\) −1.36058e18 −0.441187
\(700\) 6.42259e18 2.06335
\(701\) 4.37219e18 1.39166 0.695828 0.718208i \(-0.255038\pi\)
0.695828 + 0.718208i \(0.255038\pi\)
\(702\) −9.97361e17 −0.314529
\(703\) 2.64282e17 0.0825767
\(704\) 1.52959e18 0.473534
\(705\) −3.56479e18 −1.09346
\(706\) 2.71451e18 0.825014
\(707\) 9.76068e18 2.93937
\(708\) −2.22612e18 −0.664250
\(709\) −2.41723e18 −0.714689 −0.357344 0.933973i \(-0.616318\pi\)
−0.357344 + 0.933973i \(0.616318\pi\)
\(710\) −7.85098e18 −2.30009
\(711\) −1.06505e18 −0.309185
\(712\) −2.56467e18 −0.737755
\(713\) 1.82075e17 0.0519002
\(714\) −4.78788e18 −1.35240
\(715\) 1.58775e18 0.444420
\(716\) 5.58478e18 1.54907
\(717\) −2.27426e17 −0.0625123
\(718\) −2.04253e18 −0.556365
\(719\) −1.88829e18 −0.509719 −0.254860 0.966978i \(-0.582029\pi\)
−0.254860 + 0.966978i \(0.582029\pi\)
\(720\) −7.87971e17 −0.210790
\(721\) 4.14475e18 1.09880
\(722\) −5.85611e18 −1.53857
\(723\) 3.82853e18 0.996858
\(724\) 4.35864e18 1.12474
\(725\) 6.02735e18 1.54145
\(726\) 3.20308e17 0.0811857
\(727\) −5.58551e17 −0.140310 −0.0701551 0.997536i \(-0.522349\pi\)
−0.0701551 + 0.997536i \(0.522349\pi\)
\(728\) −4.03597e18 −1.00483
\(729\) 1.50095e17 0.0370370
\(730\) −2.36038e18 −0.577276
\(731\) 3.28377e17 0.0795994
\(732\) 4.68620e18 1.12590
\(733\) −1.54585e18 −0.368122 −0.184061 0.982915i \(-0.558924\pi\)
−0.184061 + 0.982915i \(0.558924\pi\)
\(734\) −7.06079e18 −1.66659
\(735\) 5.00118e18 1.17005
\(736\) −1.22232e18 −0.283453
\(737\) −1.91478e18 −0.440129
\(738\) −1.44799e18 −0.329912
\(739\) −6.42538e18 −1.45114 −0.725571 0.688147i \(-0.758424\pi\)
−0.725571 + 0.688147i \(0.758424\pi\)
\(740\) −9.82620e18 −2.19978
\(741\) 2.00468e17 0.0444864
\(742\) 1.88002e19 4.13559
\(743\) 4.38566e18 0.956330 0.478165 0.878270i \(-0.341302\pi\)
0.478165 + 0.878270i \(0.341302\pi\)
\(744\) 3.88557e17 0.0839905
\(745\) −1.43329e18 −0.307127
\(746\) 8.97965e17 0.190747
\(747\) −1.77328e18 −0.373416
\(748\) 1.94487e18 0.406004
\(749\) 1.27107e19 2.63050
\(750\) −3.27409e17 −0.0671726
\(751\) 1.75685e18 0.357334 0.178667 0.983910i \(-0.442821\pi\)
0.178667 + 0.983910i \(0.442821\pi\)
\(752\) 3.05204e18 0.615424
\(753\) −1.79589e18 −0.359015
\(754\) −1.34356e19 −2.66283
\(755\) −2.37158e18 −0.465997
\(756\) 2.15454e18 0.419723
\(757\) −1.97367e18 −0.381199 −0.190600 0.981668i \(-0.561043\pi\)
−0.190600 + 0.981668i \(0.561043\pi\)
\(758\) 9.60161e17 0.183863
\(759\) −1.98633e17 −0.0377121
\(760\) −3.28175e17 −0.0617757
\(761\) 1.91433e18 0.357286 0.178643 0.983914i \(-0.442829\pi\)
0.178643 + 0.983914i \(0.442829\pi\)
\(762\) −7.07560e18 −1.30935
\(763\) −1.29608e18 −0.237806
\(764\) 3.21481e18 0.584855
\(765\) −2.49268e18 −0.449642
\(766\) 7.39011e18 1.32179
\(767\) 4.92212e18 0.872936
\(768\) −5.35982e17 −0.0942545
\(769\) −9.69949e18 −1.69133 −0.845664 0.533716i \(-0.820795\pi\)
−0.845664 + 0.533716i \(0.820795\pi\)
\(770\) −5.89294e18 −1.01892
\(771\) 1.76064e18 0.301868
\(772\) 7.14769e18 1.21522
\(773\) −1.83115e18 −0.308714 −0.154357 0.988015i \(-0.549331\pi\)
−0.154357 + 0.988015i \(0.549331\pi\)
\(774\) −2.53881e17 −0.0424438
\(775\) 1.36717e18 0.226652
\(776\) −2.94912e18 −0.484833
\(777\) −6.28030e18 −1.02387
\(778\) 9.20103e18 1.48754
\(779\) 2.91044e17 0.0466621
\(780\) −7.45357e18 −1.18508
\(781\) 2.03829e18 0.321392
\(782\) −2.07215e18 −0.324025
\(783\) 2.02195e18 0.313559
\(784\) −4.28183e18 −0.658529
\(785\) −1.12856e19 −1.72137
\(786\) −5.42929e18 −0.821290
\(787\) 4.83945e17 0.0726039 0.0363019 0.999341i \(-0.488442\pi\)
0.0363019 + 0.999341i \(0.488442\pi\)
\(788\) −7.53021e18 −1.12044
\(789\) −1.22490e18 −0.180759
\(790\) −1.36750e19 −2.00149
\(791\) −1.83318e19 −2.66109
\(792\) −4.23892e17 −0.0610297
\(793\) −1.03616e19 −1.47962
\(794\) −1.46919e19 −2.08087
\(795\) 9.78782e18 1.37499
\(796\) −1.65470e19 −2.30560
\(797\) −1.28612e19 −1.77747 −0.888734 0.458423i \(-0.848415\pi\)
−0.888734 + 0.458423i \(0.848415\pi\)
\(798\) −7.44036e17 −0.101994
\(799\) 9.65486e18 1.31278
\(800\) −9.17818e18 −1.23786
\(801\) −3.02721e18 −0.404978
\(802\) 3.00662e18 0.398975
\(803\) 6.12809e17 0.0806629
\(804\) 8.98876e18 1.17364
\(805\) 3.65440e18 0.473307
\(806\) −3.04756e18 −0.391539
\(807\) −4.28598e17 −0.0546226
\(808\) 9.01492e18 1.13969
\(809\) 6.97044e18 0.874168 0.437084 0.899421i \(-0.356011\pi\)
0.437084 + 0.899421i \(0.356011\pi\)
\(810\) 1.92719e18 0.239757
\(811\) 1.21111e19 1.49468 0.747338 0.664444i \(-0.231332\pi\)
0.747338 + 0.664444i \(0.231332\pi\)
\(812\) 2.90242e19 3.55341
\(813\) 2.69113e17 0.0326849
\(814\) 4.38303e18 0.528100
\(815\) 7.03392e18 0.840762
\(816\) 2.13414e18 0.253068
\(817\) 5.10298e16 0.00600317
\(818\) 2.20725e19 2.57606
\(819\) −4.76386e18 −0.551586
\(820\) −1.08212e19 −1.24304
\(821\) 1.13387e19 1.29220 0.646102 0.763251i \(-0.276398\pi\)
0.646102 + 0.763251i \(0.276398\pi\)
\(822\) 1.23530e19 1.39671
\(823\) −4.21754e17 −0.0473108 −0.0236554 0.999720i \(-0.507530\pi\)
−0.0236554 + 0.999720i \(0.507530\pi\)
\(824\) 3.82808e18 0.426043
\(825\) −1.49150e18 −0.164692
\(826\) −1.82684e19 −2.00139
\(827\) 9.62297e18 1.04598 0.522990 0.852339i \(-0.324817\pi\)
0.522990 + 0.852339i \(0.324817\pi\)
\(828\) 9.32467e17 0.100563
\(829\) −1.01074e19 −1.08152 −0.540759 0.841178i \(-0.681863\pi\)
−0.540759 + 0.841178i \(0.681863\pi\)
\(830\) −2.27685e19 −2.41728
\(831\) 2.51009e18 0.264413
\(832\) 1.58767e19 1.65943
\(833\) −1.35452e19 −1.40473
\(834\) 1.05843e19 1.08914
\(835\) 2.03694e19 2.07978
\(836\) 3.02233e17 0.0306197
\(837\) 4.58633e17 0.0461052
\(838\) 1.14836e19 1.14549
\(839\) −9.83113e17 −0.0973085 −0.0486543 0.998816i \(-0.515493\pi\)
−0.0486543 + 0.998816i \(0.515493\pi\)
\(840\) 7.79865e18 0.765956
\(841\) 1.69774e19 1.65462
\(842\) −7.83573e18 −0.757794
\(843\) 1.26525e18 0.121422
\(844\) −2.12348e19 −2.02219
\(845\) 1.71831e18 0.162380
\(846\) −7.46456e18 −0.699998
\(847\) 1.52994e18 0.142374
\(848\) −8.37997e18 −0.773873
\(849\) 1.25566e18 0.115072
\(850\) −1.55594e19 −1.41504
\(851\) −2.71806e18 −0.245311
\(852\) −9.56857e18 −0.857018
\(853\) 1.20347e19 1.06971 0.534854 0.844944i \(-0.320367\pi\)
0.534854 + 0.844944i \(0.320367\pi\)
\(854\) 3.84569e19 3.39233
\(855\) −3.87362e17 −0.0339107
\(856\) 1.17396e19 1.01993
\(857\) −5.60320e18 −0.483127 −0.241563 0.970385i \(-0.577660\pi\)
−0.241563 + 0.970385i \(0.577660\pi\)
\(858\) 3.32471e18 0.284502
\(859\) 1.16060e17 0.00985656 0.00492828 0.999988i \(-0.498431\pi\)
0.00492828 + 0.999988i \(0.498431\pi\)
\(860\) −1.89733e18 −0.159920
\(861\) −6.91626e18 −0.578563
\(862\) 2.48174e19 2.06044
\(863\) −1.14438e18 −0.0942978 −0.0471489 0.998888i \(-0.515014\pi\)
−0.0471489 + 0.998888i \(0.515014\pi\)
\(864\) −3.07893e18 −0.251803
\(865\) 2.03458e19 1.65146
\(866\) −7.09237e18 −0.571380
\(867\) −4.69286e17 −0.0375243
\(868\) 6.58346e18 0.522488
\(869\) 3.55035e18 0.279668
\(870\) 2.59615e19 2.02980
\(871\) −1.98749e19 −1.54236
\(872\) −1.19706e18 −0.0922057
\(873\) −3.48100e18 −0.266141
\(874\) −3.22013e17 −0.0244371
\(875\) −1.56386e18 −0.117800
\(876\) −2.87677e18 −0.215094
\(877\) 5.25918e17 0.0390320 0.0195160 0.999810i \(-0.493787\pi\)
0.0195160 + 0.999810i \(0.493787\pi\)
\(878\) −9.47337e18 −0.697895
\(879\) −6.80092e18 −0.497325
\(880\) 2.62671e18 0.190666
\(881\) −1.02154e19 −0.736056 −0.368028 0.929815i \(-0.619967\pi\)
−0.368028 + 0.929815i \(0.619967\pi\)
\(882\) 1.04723e19 0.749026
\(883\) −1.67904e19 −1.19211 −0.596055 0.802943i \(-0.703266\pi\)
−0.596055 + 0.802943i \(0.703266\pi\)
\(884\) 2.01872e19 1.42278
\(885\) −9.51095e18 −0.665415
\(886\) 2.39223e19 1.66143
\(887\) −9.98165e18 −0.688175 −0.344087 0.938938i \(-0.611812\pi\)
−0.344087 + 0.938938i \(0.611812\pi\)
\(888\) −5.80046e18 −0.396989
\(889\) −3.37964e19 −2.29619
\(890\) −3.88688e19 −2.62160
\(891\) −5.00341e17 −0.0335013
\(892\) 3.01710e19 2.00548
\(893\) 1.50037e18 0.0990063
\(894\) −3.00126e18 −0.196612
\(895\) 2.38606e19 1.55179
\(896\) −2.71888e19 −1.75545
\(897\) −2.06176e18 −0.132156
\(898\) 2.99134e16 0.00190358
\(899\) 6.17833e18 0.390331
\(900\) 7.00169e18 0.439164
\(901\) −2.65093e19 −1.65077
\(902\) 4.82687e18 0.298417
\(903\) −1.21266e18 −0.0744333
\(904\) −1.69312e19 −1.03179
\(905\) 1.86220e19 1.12671
\(906\) −4.96602e18 −0.298315
\(907\) −1.90931e19 −1.13875 −0.569377 0.822076i \(-0.692816\pi\)
−0.569377 + 0.822076i \(0.692816\pi\)
\(908\) 3.50713e19 2.07680
\(909\) 1.06408e19 0.625616
\(910\) −6.11670e19 −3.57066
\(911\) 2.12403e19 1.23109 0.615547 0.788100i \(-0.288935\pi\)
0.615547 + 0.788100i \(0.288935\pi\)
\(912\) 3.31645e17 0.0190857
\(913\) 5.91122e18 0.337767
\(914\) 4.63953e19 2.63223
\(915\) 2.00215e19 1.12787
\(916\) −3.40669e19 −1.90552
\(917\) −2.59328e19 −1.44029
\(918\) −5.21958e18 −0.287845
\(919\) 2.14539e19 1.17478 0.587388 0.809305i \(-0.300156\pi\)
0.587388 + 0.809305i \(0.300156\pi\)
\(920\) 3.37519e18 0.183517
\(921\) −1.35444e19 −0.731263
\(922\) −3.72060e19 −1.99463
\(923\) 2.11569e19 1.12627
\(924\) −7.18216e18 −0.379654
\(925\) −2.04093e19 −1.07129
\(926\) −5.98496e17 −0.0311953
\(927\) 4.51847e18 0.233869
\(928\) −4.14769e19 −2.13179
\(929\) 6.27434e17 0.0320233 0.0160116 0.999872i \(-0.494903\pi\)
0.0160116 + 0.999872i \(0.494903\pi\)
\(930\) 5.88876e18 0.298459
\(931\) −2.10492e18 −0.105941
\(932\) −2.12915e19 −1.06415
\(933\) 8.03496e18 0.398798
\(934\) −3.58957e18 −0.176925
\(935\) 8.30935e18 0.406716
\(936\) −4.39988e18 −0.213869
\(937\) 1.14580e19 0.553098 0.276549 0.961000i \(-0.410809\pi\)
0.276549 + 0.961000i \(0.410809\pi\)
\(938\) 7.37654e19 3.53618
\(939\) 1.78338e19 0.849020
\(940\) −5.57848e19 −2.63745
\(941\) 8.37033e18 0.393016 0.196508 0.980502i \(-0.437040\pi\)
0.196508 + 0.980502i \(0.437040\pi\)
\(942\) −2.36318e19 −1.10196
\(943\) −2.99330e18 −0.138620
\(944\) 8.14293e18 0.374509
\(945\) 9.20514e18 0.420459
\(946\) 8.46315e17 0.0383919
\(947\) −1.66309e19 −0.749275 −0.374637 0.927171i \(-0.622233\pi\)
−0.374637 + 0.927171i \(0.622233\pi\)
\(948\) −1.66668e19 −0.745758
\(949\) 6.36078e18 0.282670
\(950\) −2.41793e18 −0.106718
\(951\) 5.30722e18 0.232645
\(952\) −2.11218e19 −0.919584
\(953\) −2.54297e19 −1.09961 −0.549803 0.835294i \(-0.685297\pi\)
−0.549803 + 0.835294i \(0.685297\pi\)
\(954\) 2.04954e19 0.880221
\(955\) 1.37351e19 0.585880
\(956\) −3.55895e18 −0.150781
\(957\) −6.74018e18 −0.283625
\(958\) 5.17590e19 2.16327
\(959\) 5.90037e19 2.44939
\(960\) −3.06783e19 −1.26493
\(961\) −2.30161e19 −0.942606
\(962\) 4.54947e19 1.85064
\(963\) 1.38568e19 0.559876
\(964\) 5.99121e19 2.40444
\(965\) 3.05381e19 1.21735
\(966\) 7.65220e18 0.302995
\(967\) 5.78322e18 0.227456 0.113728 0.993512i \(-0.463721\pi\)
0.113728 + 0.993512i \(0.463721\pi\)
\(968\) 1.41305e18 0.0552035
\(969\) 1.04913e18 0.0407122
\(970\) −4.46954e19 −1.72284
\(971\) −4.12182e19 −1.57821 −0.789104 0.614259i \(-0.789455\pi\)
−0.789104 + 0.614259i \(0.789455\pi\)
\(972\) 2.34881e18 0.0893339
\(973\) 5.05556e19 1.91001
\(974\) −7.24912e19 −2.72052
\(975\) −1.54813e19 −0.577135
\(976\) −1.71417e19 −0.634789
\(977\) −4.33217e19 −1.59364 −0.796822 0.604215i \(-0.793487\pi\)
−0.796822 + 0.604215i \(0.793487\pi\)
\(978\) 1.47288e19 0.538226
\(979\) 1.00912e19 0.366317
\(980\) 7.82626e19 2.82218
\(981\) −1.41295e18 −0.0506148
\(982\) 2.40878e19 0.857180
\(983\) −2.52492e19 −0.892584 −0.446292 0.894887i \(-0.647256\pi\)
−0.446292 + 0.894887i \(0.647256\pi\)
\(984\) −6.38783e18 −0.224329
\(985\) −3.21724e19 −1.12240
\(986\) −7.03139e19 −2.43692
\(987\) −3.56542e19 −1.22758
\(988\) 3.13709e18 0.107302
\(989\) −5.24827e17 −0.0178337
\(990\) −6.42428e18 −0.216868
\(991\) 1.90400e19 0.638539 0.319269 0.947664i \(-0.396562\pi\)
0.319269 + 0.947664i \(0.396562\pi\)
\(992\) −9.40807e18 −0.313455
\(993\) 3.58625e18 0.118705
\(994\) −7.85236e19 −2.58219
\(995\) −7.06961e19 −2.30965
\(996\) −2.77497e19 −0.900685
\(997\) 3.63643e19 1.17262 0.586309 0.810087i \(-0.300580\pi\)
0.586309 + 0.810087i \(0.300580\pi\)
\(998\) 1.84717e19 0.591779
\(999\) −6.84657e18 −0.217920
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 33.14.a.a.1.1 1
3.2 odd 2 99.14.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.14.a.a.1.1 1 1.1 even 1 trivial
99.14.a.a.1.1 1 3.2 odd 2