Properties

Label 177.14.a.b.1.23
Level $177$
Weight $14$
Character 177.1
Self dual yes
Analytic conductor $189.799$
Analytic rank $1$
Dimension $31$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [177,14,Mod(1,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, 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("177.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 177.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: \(189.798744245\)
Analytic rank: \(1\)
Dimension: \(31\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.23
Character \(\chi\) \(=\) 177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+86.2630 q^{2} -729.000 q^{3} -750.700 q^{4} -64842.3 q^{5} -62885.7 q^{6} -98287.1 q^{7} -771424. q^{8} +531441. q^{9} +O(q^{10})\) \(q+86.2630 q^{2} -729.000 q^{3} -750.700 q^{4} -64842.3 q^{5} -62885.7 q^{6} -98287.1 q^{7} -771424. q^{8} +531441. q^{9} -5.59349e6 q^{10} -5.83499e6 q^{11} +547261. q^{12} +2.22899e6 q^{13} -8.47853e6 q^{14} +4.72700e7 q^{15} -6.03956e7 q^{16} +9.15906e7 q^{17} +4.58437e7 q^{18} +1.28544e8 q^{19} +4.86771e7 q^{20} +7.16513e7 q^{21} -5.03344e8 q^{22} +1.09713e9 q^{23} +5.62368e8 q^{24} +2.98381e9 q^{25} +1.92279e8 q^{26} -3.87420e8 q^{27} +7.37842e7 q^{28} +5.10901e8 q^{29} +4.07765e9 q^{30} -2.94741e9 q^{31} +1.10960e9 q^{32} +4.25371e9 q^{33} +7.90088e9 q^{34} +6.37316e9 q^{35} -3.98953e8 q^{36} +2.19059e10 q^{37} +1.10886e10 q^{38} -1.62494e9 q^{39} +5.00209e10 q^{40} -5.73033e10 q^{41} +6.18085e9 q^{42} -6.16354e10 q^{43} +4.38033e9 q^{44} -3.44598e10 q^{45} +9.46420e10 q^{46} -4.57698e10 q^{47} +4.40284e10 q^{48} -8.72287e10 q^{49} +2.57393e11 q^{50} -6.67696e10 q^{51} -1.67331e9 q^{52} +2.65343e11 q^{53} -3.34200e10 q^{54} +3.78354e11 q^{55} +7.58210e10 q^{56} -9.37085e10 q^{57} +4.40718e10 q^{58} -4.21805e10 q^{59} -3.54856e10 q^{60} +1.54568e11 q^{61} -2.54252e11 q^{62} -5.22338e10 q^{63} +5.90478e11 q^{64} -1.44533e11 q^{65} +3.66938e11 q^{66} +2.77636e11 q^{67} -6.87571e10 q^{68} -7.99810e11 q^{69} +5.49767e11 q^{70} +8.61174e11 q^{71} -4.09966e11 q^{72} -1.38364e12 q^{73} +1.88967e12 q^{74} -2.17520e12 q^{75} -9.64980e10 q^{76} +5.73504e11 q^{77} -1.40172e11 q^{78} +1.78937e12 q^{79} +3.91619e12 q^{80} +2.82430e11 q^{81} -4.94315e12 q^{82} -3.09290e12 q^{83} -5.37886e10 q^{84} -5.93894e12 q^{85} -5.31685e12 q^{86} -3.72447e11 q^{87} +4.50125e12 q^{88} +3.36119e12 q^{89} -2.97261e12 q^{90} -2.19081e11 q^{91} -8.23618e11 q^{92} +2.14866e12 q^{93} -3.94824e12 q^{94} -8.33508e12 q^{95} -8.08901e11 q^{96} +4.52881e12 q^{97} -7.52460e12 q^{98} -3.10095e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31 q - 52 q^{2} - 22599 q^{3} + 126886 q^{4} + 33486 q^{5} + 37908 q^{6} - 1135539 q^{7} - 1519749 q^{8} + 16474671 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 31 q - 52 q^{2} - 22599 q^{3} + 126886 q^{4} + 33486 q^{5} + 37908 q^{6} - 1135539 q^{7} - 1519749 q^{8} + 16474671 q^{9} - 3854663 q^{10} + 3943968 q^{11} - 92499894 q^{12} - 48510022 q^{13} - 51427459 q^{14} - 24411294 q^{15} + 370110498 q^{16} + 83288419 q^{17} - 27634932 q^{18} - 180425297 q^{19} + 753620445 q^{20} + 827807931 q^{21} + 2300196142 q^{22} - 1305810279 q^{23} + 1107897021 q^{24} + 8070954867 q^{25} + 464550322 q^{26} - 12010035159 q^{27} - 9887169562 q^{28} + 6248352277 q^{29} + 2810049327 q^{30} - 26730150789 q^{31} - 24001343230 q^{32} - 2875152672 q^{33} - 36571033348 q^{34} + 10255900979 q^{35} + 67432422726 q^{36} - 43284776933 q^{37} - 36293696947 q^{38} + 35363806038 q^{39} - 105980683856 q^{40} - 9961079285 q^{41} + 37490617611 q^{42} - 51755851288 q^{43} - 59623729442 q^{44} + 17795833326 q^{45} - 202287132683 q^{46} - 82747063727 q^{47} - 269810553042 q^{48} + 535277836542 q^{49} + 526974390461 q^{50} - 60717257451 q^{51} + 544982341446 q^{52} + 561701818494 q^{53} + 20145865428 q^{54} - 521861534450 q^{55} - 228056576664 q^{56} + 131530041513 q^{57} + 10555409160 q^{58} - 1307596542871 q^{59} - 549389304405 q^{60} + 618193248201 q^{61} - 1486611437386 q^{62} - 603471981699 q^{63} + 679062548045 q^{64} - 1130583307122 q^{65} - 1676842987518 q^{66} - 4137387490592 q^{67} - 3901389300295 q^{68} + 951935693391 q^{69} - 819291947844 q^{70} - 3766439869810 q^{71} - 807656928309 q^{72} - 2386775553523 q^{73} + 3060770694642 q^{74} - 5883726098043 q^{75} - 847741068784 q^{76} + 1650423006137 q^{77} - 338657184738 q^{78} + 787155757766 q^{79} + 13999832121779 q^{80} + 8755315630911 q^{81} + 10083281915577 q^{82} + 8743877051639 q^{83} + 7207746610698 q^{84} + 15373177520565 q^{85} + 18939443838984 q^{86} - 4555048809933 q^{87} + 39713314506713 q^{88} + 11026795445259 q^{89} - 2048525959383 q^{90} + 23285721962531 q^{91} + 40411079823254 q^{92} + 19486279925181 q^{93} + 35237377585624 q^{94} + 13730236994039 q^{95} + 17496979214670 q^{96} + 10134565481560 q^{97} + 70916776240976 q^{98} + 2095986297888 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 86.2630 0.953080 0.476540 0.879153i \(-0.341891\pi\)
0.476540 + 0.879153i \(0.341891\pi\)
\(3\) −729.000 −0.577350
\(4\) −750.700 −0.0916382
\(5\) −64842.3 −1.85589 −0.927947 0.372713i \(-0.878428\pi\)
−0.927947 + 0.372713i \(0.878428\pi\)
\(6\) −62885.7 −0.550261
\(7\) −98287.1 −0.315761 −0.157881 0.987458i \(-0.550466\pi\)
−0.157881 + 0.987458i \(0.550466\pi\)
\(8\) −771424. −1.04042
\(9\) 531441. 0.333333
\(10\) −5.59349e6 −1.76882
\(11\) −5.83499e6 −0.993088 −0.496544 0.868011i \(-0.665398\pi\)
−0.496544 + 0.868011i \(0.665398\pi\)
\(12\) 547261. 0.0529074
\(13\) 2.22899e6 0.128079 0.0640393 0.997947i \(-0.479602\pi\)
0.0640393 + 0.997947i \(0.479602\pi\)
\(14\) −8.47853e6 −0.300946
\(15\) 4.72700e7 1.07150
\(16\) −6.03956e7 −0.899964
\(17\) 9.15906e7 0.920308 0.460154 0.887839i \(-0.347794\pi\)
0.460154 + 0.887839i \(0.347794\pi\)
\(18\) 4.58437e7 0.317693
\(19\) 1.28544e8 0.626835 0.313417 0.949615i \(-0.398526\pi\)
0.313417 + 0.949615i \(0.398526\pi\)
\(20\) 4.86771e7 0.170071
\(21\) 7.16513e7 0.182305
\(22\) −5.03344e8 −0.946493
\(23\) 1.09713e9 1.54536 0.772678 0.634798i \(-0.218917\pi\)
0.772678 + 0.634798i \(0.218917\pi\)
\(24\) 5.62368e8 0.600686
\(25\) 2.98381e9 2.44434
\(26\) 1.92279e8 0.122069
\(27\) −3.87420e8 −0.192450
\(28\) 7.37842e7 0.0289358
\(29\) 5.10901e8 0.159496 0.0797480 0.996815i \(-0.474588\pi\)
0.0797480 + 0.996815i \(0.474588\pi\)
\(30\) 4.07765e9 1.02123
\(31\) −2.94741e9 −0.596470 −0.298235 0.954492i \(-0.596398\pi\)
−0.298235 + 0.954492i \(0.596398\pi\)
\(32\) 1.10960e9 0.182681
\(33\) 4.25371e9 0.573360
\(34\) 7.90088e9 0.877127
\(35\) 6.37316e9 0.586020
\(36\) −3.98953e8 −0.0305461
\(37\) 2.19059e10 1.40362 0.701811 0.712363i \(-0.252375\pi\)
0.701811 + 0.712363i \(0.252375\pi\)
\(38\) 1.10886e10 0.597424
\(39\) −1.62494e9 −0.0739462
\(40\) 5.00209e10 1.93091
\(41\) −5.73033e10 −1.88401 −0.942007 0.335592i \(-0.891064\pi\)
−0.942007 + 0.335592i \(0.891064\pi\)
\(42\) 6.18085e9 0.173751
\(43\) −6.16354e10 −1.48691 −0.743456 0.668785i \(-0.766815\pi\)
−0.743456 + 0.668785i \(0.766815\pi\)
\(44\) 4.38033e9 0.0910049
\(45\) −3.44598e10 −0.618631
\(46\) 9.46420e10 1.47285
\(47\) −4.57698e10 −0.619360 −0.309680 0.950841i \(-0.600222\pi\)
−0.309680 + 0.950841i \(0.600222\pi\)
\(48\) 4.40284e10 0.519595
\(49\) −8.72287e10 −0.900295
\(50\) 2.57393e11 2.32965
\(51\) −6.67696e10 −0.531340
\(52\) −1.67331e9 −0.0117369
\(53\) 2.65343e11 1.64443 0.822214 0.569179i \(-0.192739\pi\)
0.822214 + 0.569179i \(0.192739\pi\)
\(54\) −3.34200e10 −0.183420
\(55\) 3.78354e11 1.84307
\(56\) 7.58210e10 0.328524
\(57\) −9.37085e10 −0.361903
\(58\) 4.40718e10 0.152012
\(59\) −4.21805e10 −0.130189
\(60\) −3.54856e10 −0.0981904
\(61\) 1.54568e11 0.384129 0.192064 0.981382i \(-0.438482\pi\)
0.192064 + 0.981382i \(0.438482\pi\)
\(62\) −2.54252e11 −0.568484
\(63\) −5.22338e10 −0.105254
\(64\) 5.90478e11 1.07407
\(65\) −1.44533e11 −0.237700
\(66\) 3.66938e11 0.546458
\(67\) 2.77636e11 0.374964 0.187482 0.982268i \(-0.439967\pi\)
0.187482 + 0.982268i \(0.439967\pi\)
\(68\) −6.87571e10 −0.0843354
\(69\) −7.99810e11 −0.892211
\(70\) 5.49767e11 0.558524
\(71\) 8.61174e11 0.797832 0.398916 0.916987i \(-0.369386\pi\)
0.398916 + 0.916987i \(0.369386\pi\)
\(72\) −4.09966e11 −0.346806
\(73\) −1.38364e12 −1.07010 −0.535050 0.844820i \(-0.679707\pi\)
−0.535050 + 0.844820i \(0.679707\pi\)
\(74\) 1.88967e12 1.33776
\(75\) −2.17520e12 −1.41124
\(76\) −9.64980e10 −0.0574420
\(77\) 5.73504e11 0.313579
\(78\) −1.40172e11 −0.0704767
\(79\) 1.78937e12 0.828179 0.414090 0.910236i \(-0.364100\pi\)
0.414090 + 0.910236i \(0.364100\pi\)
\(80\) 3.91619e12 1.67024
\(81\) 2.82430e11 0.111111
\(82\) −4.94315e12 −1.79562
\(83\) −3.09290e12 −1.03839 −0.519193 0.854657i \(-0.673767\pi\)
−0.519193 + 0.854657i \(0.673767\pi\)
\(84\) −5.37886e10 −0.0167061
\(85\) −5.93894e12 −1.70799
\(86\) −5.31685e12 −1.41715
\(87\) −3.72447e11 −0.0920850
\(88\) 4.50125e12 1.03323
\(89\) 3.36119e12 0.716899 0.358449 0.933549i \(-0.383306\pi\)
0.358449 + 0.933549i \(0.383306\pi\)
\(90\) −2.97261e12 −0.589605
\(91\) −2.19081e11 −0.0404423
\(92\) −8.23618e11 −0.141614
\(93\) 2.14866e12 0.344372
\(94\) −3.94824e12 −0.590299
\(95\) −8.33508e12 −1.16334
\(96\) −8.08901e11 −0.105471
\(97\) 4.52881e12 0.552036 0.276018 0.961152i \(-0.410985\pi\)
0.276018 + 0.961152i \(0.410985\pi\)
\(98\) −7.52460e12 −0.858053
\(99\) −3.10095e12 −0.331029
\(100\) −2.23995e12 −0.223995
\(101\) 8.28016e12 0.776157 0.388079 0.921626i \(-0.373139\pi\)
0.388079 + 0.921626i \(0.373139\pi\)
\(102\) −5.75974e12 −0.506410
\(103\) 6.20956e12 0.512412 0.256206 0.966622i \(-0.417527\pi\)
0.256206 + 0.966622i \(0.417527\pi\)
\(104\) −1.71950e12 −0.133255
\(105\) −4.64603e12 −0.338339
\(106\) 2.28893e13 1.56727
\(107\) 4.94526e12 0.318563 0.159281 0.987233i \(-0.449082\pi\)
0.159281 + 0.987233i \(0.449082\pi\)
\(108\) 2.90837e11 0.0176358
\(109\) 4.50602e12 0.257348 0.128674 0.991687i \(-0.458928\pi\)
0.128674 + 0.991687i \(0.458928\pi\)
\(110\) 3.26379e13 1.75659
\(111\) −1.59694e13 −0.810381
\(112\) 5.93610e12 0.284174
\(113\) 9.61297e12 0.434358 0.217179 0.976132i \(-0.430314\pi\)
0.217179 + 0.976132i \(0.430314\pi\)
\(114\) −8.08357e12 −0.344923
\(115\) −7.11406e13 −2.86802
\(116\) −3.83533e11 −0.0146159
\(117\) 1.18458e12 0.0426929
\(118\) −3.63862e12 −0.124080
\(119\) −9.00218e12 −0.290598
\(120\) −3.64652e13 −1.11481
\(121\) −4.75576e11 −0.0137757
\(122\) 1.33335e13 0.366105
\(123\) 4.17741e13 1.08774
\(124\) 2.21262e12 0.0546595
\(125\) −1.14324e14 −2.68054
\(126\) −4.50584e12 −0.100315
\(127\) −8.14494e13 −1.72252 −0.861259 0.508167i \(-0.830323\pi\)
−0.861259 + 0.508167i \(0.830323\pi\)
\(128\) 4.18465e13 0.840997
\(129\) 4.49322e13 0.858469
\(130\) −1.24678e13 −0.226547
\(131\) 6.35655e13 1.09890 0.549450 0.835527i \(-0.314837\pi\)
0.549450 + 0.835527i \(0.314837\pi\)
\(132\) −3.19326e12 −0.0525417
\(133\) −1.26342e13 −0.197930
\(134\) 2.39497e13 0.357370
\(135\) 2.51212e13 0.357167
\(136\) −7.06552e13 −0.957506
\(137\) 8.45302e13 1.09227 0.546133 0.837699i \(-0.316099\pi\)
0.546133 + 0.837699i \(0.316099\pi\)
\(138\) −6.89940e13 −0.850349
\(139\) −1.58702e12 −0.0186632 −0.00933159 0.999956i \(-0.502970\pi\)
−0.00933159 + 0.999956i \(0.502970\pi\)
\(140\) −4.78433e12 −0.0537018
\(141\) 3.33662e13 0.357587
\(142\) 7.42874e13 0.760398
\(143\) −1.30062e13 −0.127193
\(144\) −3.20967e13 −0.299988
\(145\) −3.31280e13 −0.296007
\(146\) −1.19357e14 −1.01989
\(147\) 6.35897e13 0.519785
\(148\) −1.64448e13 −0.128625
\(149\) −7.75642e12 −0.0580698 −0.0290349 0.999578i \(-0.509243\pi\)
−0.0290349 + 0.999578i \(0.509243\pi\)
\(150\) −1.87639e14 −1.34503
\(151\) −2.79033e14 −1.91560 −0.957802 0.287429i \(-0.907199\pi\)
−0.957802 + 0.287429i \(0.907199\pi\)
\(152\) −9.91618e13 −0.652171
\(153\) 4.86750e13 0.306769
\(154\) 4.94722e13 0.298866
\(155\) 1.91116e14 1.10699
\(156\) 1.21984e12 0.00677630
\(157\) 3.05432e13 0.162767 0.0813837 0.996683i \(-0.474066\pi\)
0.0813837 + 0.996683i \(0.474066\pi\)
\(158\) 1.54356e14 0.789321
\(159\) −1.93435e14 −0.949411
\(160\) −7.19492e13 −0.339036
\(161\) −1.07834e14 −0.487964
\(162\) 2.43632e13 0.105898
\(163\) −8.07722e13 −0.337320 −0.168660 0.985674i \(-0.553944\pi\)
−0.168660 + 0.985674i \(0.553944\pi\)
\(164\) 4.30176e13 0.172648
\(165\) −2.75820e14 −1.06409
\(166\) −2.66803e14 −0.989665
\(167\) −7.80333e13 −0.278370 −0.139185 0.990266i \(-0.544448\pi\)
−0.139185 + 0.990266i \(0.544448\pi\)
\(168\) −5.52735e13 −0.189674
\(169\) −2.97907e14 −0.983596
\(170\) −5.12311e14 −1.62785
\(171\) 6.83135e13 0.208945
\(172\) 4.62697e13 0.136258
\(173\) 6.55147e14 1.85797 0.928986 0.370116i \(-0.120682\pi\)
0.928986 + 0.370116i \(0.120682\pi\)
\(174\) −3.21284e13 −0.0877644
\(175\) −2.93270e14 −0.771829
\(176\) 3.52408e14 0.893744
\(177\) 3.07496e13 0.0751646
\(178\) 2.89946e14 0.683262
\(179\) −5.41032e14 −1.22936 −0.614678 0.788778i \(-0.710714\pi\)
−0.614678 + 0.788778i \(0.710714\pi\)
\(180\) 2.58690e13 0.0566903
\(181\) 2.37050e14 0.501106 0.250553 0.968103i \(-0.419388\pi\)
0.250553 + 0.968103i \(0.419388\pi\)
\(182\) −1.88986e13 −0.0385448
\(183\) −1.12680e14 −0.221777
\(184\) −8.46355e14 −1.60782
\(185\) −1.42043e15 −2.60497
\(186\) 1.85350e14 0.328214
\(187\) −5.34431e14 −0.913947
\(188\) 3.43594e13 0.0567570
\(189\) 3.80784e13 0.0607683
\(190\) −7.19008e14 −1.10875
\(191\) 1.24471e15 1.85503 0.927514 0.373788i \(-0.121941\pi\)
0.927514 + 0.373788i \(0.121941\pi\)
\(192\) −4.30459e14 −0.620117
\(193\) 3.55205e13 0.0494717 0.0247358 0.999694i \(-0.492126\pi\)
0.0247358 + 0.999694i \(0.492126\pi\)
\(194\) 3.90668e14 0.526135
\(195\) 1.05364e14 0.137236
\(196\) 6.54826e13 0.0825014
\(197\) 8.59190e14 1.04727 0.523635 0.851943i \(-0.324576\pi\)
0.523635 + 0.851943i \(0.324576\pi\)
\(198\) −2.67498e14 −0.315498
\(199\) −7.01662e14 −0.800909 −0.400454 0.916317i \(-0.631148\pi\)
−0.400454 + 0.916317i \(0.631148\pi\)
\(200\) −2.30179e15 −2.54314
\(201\) −2.02396e14 −0.216485
\(202\) 7.14271e14 0.739740
\(203\) −5.02149e13 −0.0503627
\(204\) 5.01240e13 0.0486911
\(205\) 3.71567e15 3.49653
\(206\) 5.35655e14 0.488370
\(207\) 5.83062e14 0.515119
\(208\) −1.34621e14 −0.115266
\(209\) −7.50053e14 −0.622502
\(210\) −4.00780e14 −0.322464
\(211\) 1.48001e15 1.15459 0.577296 0.816535i \(-0.304108\pi\)
0.577296 + 0.816535i \(0.304108\pi\)
\(212\) −1.99193e14 −0.150692
\(213\) −6.27796e14 −0.460629
\(214\) 4.26593e14 0.303616
\(215\) 3.99658e15 2.75955
\(216\) 2.98865e14 0.200229
\(217\) 2.89692e14 0.188342
\(218\) 3.88703e14 0.245273
\(219\) 1.00867e15 0.617823
\(220\) −2.84031e14 −0.168895
\(221\) 2.04155e14 0.117872
\(222\) −1.37757e15 −0.772358
\(223\) 1.85860e15 1.01206 0.506029 0.862516i \(-0.331113\pi\)
0.506029 + 0.862516i \(0.331113\pi\)
\(224\) −1.09060e14 −0.0576835
\(225\) 1.58572e15 0.814780
\(226\) 8.29243e14 0.413978
\(227\) −3.09502e15 −1.50140 −0.750698 0.660645i \(-0.770283\pi\)
−0.750698 + 0.660645i \(0.770283\pi\)
\(228\) 7.03470e13 0.0331642
\(229\) −8.23402e13 −0.0377295 −0.0188648 0.999822i \(-0.506005\pi\)
−0.0188648 + 0.999822i \(0.506005\pi\)
\(230\) −6.13680e15 −2.73345
\(231\) −4.18085e14 −0.181045
\(232\) −3.94121e14 −0.165943
\(233\) 2.54213e15 1.04084 0.520421 0.853910i \(-0.325775\pi\)
0.520421 + 0.853910i \(0.325775\pi\)
\(234\) 1.02185e14 0.0406897
\(235\) 2.96782e15 1.14947
\(236\) 3.16649e13 0.0119303
\(237\) −1.30445e15 −0.478149
\(238\) −7.76554e14 −0.276963
\(239\) 1.25403e15 0.435233 0.217616 0.976034i \(-0.430172\pi\)
0.217616 + 0.976034i \(0.430172\pi\)
\(240\) −2.85490e15 −0.964312
\(241\) −3.17662e15 −1.04437 −0.522185 0.852832i \(-0.674883\pi\)
−0.522185 + 0.852832i \(0.674883\pi\)
\(242\) −4.10246e13 −0.0131294
\(243\) −2.05891e14 −0.0641500
\(244\) −1.16035e14 −0.0352009
\(245\) 5.65610e15 1.67085
\(246\) 3.60356e15 1.03670
\(247\) 2.86523e14 0.0802841
\(248\) 2.27370e15 0.620579
\(249\) 2.25473e15 0.599513
\(250\) −9.86194e15 −2.55477
\(251\) −5.28279e15 −1.33347 −0.666736 0.745294i \(-0.732309\pi\)
−0.666736 + 0.745294i \(0.732309\pi\)
\(252\) 3.92119e13 0.00964528
\(253\) −6.40176e15 −1.53467
\(254\) −7.02607e15 −1.64170
\(255\) 4.32949e15 0.986110
\(256\) −1.22739e15 −0.272536
\(257\) −4.27042e15 −0.924497 −0.462249 0.886750i \(-0.652957\pi\)
−0.462249 + 0.886750i \(0.652957\pi\)
\(258\) 3.87599e15 0.818190
\(259\) −2.15307e15 −0.443210
\(260\) 1.08501e14 0.0217824
\(261\) 2.71514e14 0.0531653
\(262\) 5.48335e15 1.04734
\(263\) −7.59894e14 −0.141593 −0.0707963 0.997491i \(-0.522554\pi\)
−0.0707963 + 0.997491i \(0.522554\pi\)
\(264\) −3.28141e15 −0.596534
\(265\) −1.72054e16 −3.05188
\(266\) −1.08986e15 −0.188643
\(267\) −2.45031e15 −0.413902
\(268\) −2.08421e14 −0.0343610
\(269\) 2.22489e15 0.358029 0.179015 0.983846i \(-0.442709\pi\)
0.179015 + 0.983846i \(0.442709\pi\)
\(270\) 2.16703e15 0.340409
\(271\) 1.64080e15 0.251626 0.125813 0.992054i \(-0.459846\pi\)
0.125813 + 0.992054i \(0.459846\pi\)
\(272\) −5.53167e15 −0.828244
\(273\) 1.59710e14 0.0233494
\(274\) 7.29182e15 1.04102
\(275\) −1.74105e16 −2.42745
\(276\) 6.00418e14 0.0817607
\(277\) −3.74169e15 −0.497679 −0.248839 0.968545i \(-0.580049\pi\)
−0.248839 + 0.968545i \(0.580049\pi\)
\(278\) −1.36901e14 −0.0177875
\(279\) −1.56637e15 −0.198823
\(280\) −4.91640e15 −0.609706
\(281\) −1.35510e15 −0.164202 −0.0821012 0.996624i \(-0.526163\pi\)
−0.0821012 + 0.996624i \(0.526163\pi\)
\(282\) 2.87827e15 0.340810
\(283\) 1.51639e16 1.75468 0.877341 0.479868i \(-0.159315\pi\)
0.877341 + 0.479868i \(0.159315\pi\)
\(284\) −6.46484e14 −0.0731120
\(285\) 6.07627e15 0.671654
\(286\) −1.12195e15 −0.121226
\(287\) 5.63217e15 0.594899
\(288\) 5.89689e14 0.0608936
\(289\) −1.51573e15 −0.153033
\(290\) −2.85772e15 −0.282119
\(291\) −3.30150e15 −0.318718
\(292\) 1.03870e15 0.0980621
\(293\) −7.97683e15 −0.736530 −0.368265 0.929721i \(-0.620048\pi\)
−0.368265 + 0.929721i \(0.620048\pi\)
\(294\) 5.48544e15 0.495397
\(295\) 2.73508e15 0.241617
\(296\) −1.68987e16 −1.46035
\(297\) 2.26060e15 0.191120
\(298\) −6.69091e14 −0.0553452
\(299\) 2.44550e15 0.197927
\(300\) 1.63292e15 0.129324
\(301\) 6.05796e15 0.469510
\(302\) −2.40702e16 −1.82572
\(303\) −6.03624e15 −0.448115
\(304\) −7.76348e15 −0.564129
\(305\) −1.00226e16 −0.712902
\(306\) 4.19885e15 0.292376
\(307\) −1.70874e16 −1.16487 −0.582433 0.812878i \(-0.697899\pi\)
−0.582433 + 0.812878i \(0.697899\pi\)
\(308\) −4.30530e14 −0.0287358
\(309\) −4.52677e15 −0.295841
\(310\) 1.64863e16 1.05505
\(311\) −3.12441e16 −1.95806 −0.979029 0.203722i \(-0.934696\pi\)
−0.979029 + 0.203722i \(0.934696\pi\)
\(312\) 1.25351e15 0.0769351
\(313\) 2.72138e16 1.63588 0.817939 0.575304i \(-0.195116\pi\)
0.817939 + 0.575304i \(0.195116\pi\)
\(314\) 2.63475e15 0.155130
\(315\) 3.38696e15 0.195340
\(316\) −1.34328e15 −0.0758929
\(317\) −1.58624e16 −0.877977 −0.438989 0.898493i \(-0.644663\pi\)
−0.438989 + 0.898493i \(0.644663\pi\)
\(318\) −1.66863e16 −0.904864
\(319\) −2.98110e15 −0.158394
\(320\) −3.82879e16 −1.99337
\(321\) −3.60510e15 −0.183922
\(322\) −9.30208e15 −0.465069
\(323\) 1.17734e16 0.576881
\(324\) −2.12020e14 −0.0101820
\(325\) 6.65090e15 0.313068
\(326\) −6.96765e15 −0.321493
\(327\) −3.28489e15 −0.148580
\(328\) 4.42051e16 1.96016
\(329\) 4.49858e15 0.195570
\(330\) −2.37931e16 −1.01417
\(331\) −1.38811e16 −0.580153 −0.290076 0.957003i \(-0.593681\pi\)
−0.290076 + 0.957003i \(0.593681\pi\)
\(332\) 2.32185e15 0.0951559
\(333\) 1.16417e16 0.467874
\(334\) −6.73138e15 −0.265309
\(335\) −1.80025e16 −0.695893
\(336\) −4.32742e15 −0.164068
\(337\) −4.71553e16 −1.75362 −0.876811 0.480835i \(-0.840334\pi\)
−0.876811 + 0.480835i \(0.840334\pi\)
\(338\) −2.56983e16 −0.937446
\(339\) −7.00786e15 −0.250777
\(340\) 4.45837e15 0.156518
\(341\) 1.71981e16 0.592348
\(342\) 5.89292e15 0.199141
\(343\) 1.80964e16 0.600040
\(344\) 4.75470e16 1.54701
\(345\) 5.18615e16 1.65585
\(346\) 5.65149e16 1.77080
\(347\) −4.17216e15 −0.128298 −0.0641489 0.997940i \(-0.520433\pi\)
−0.0641489 + 0.997940i \(0.520433\pi\)
\(348\) 2.79596e14 0.00843851
\(349\) −5.65752e16 −1.67595 −0.837975 0.545709i \(-0.816261\pi\)
−0.837975 + 0.545709i \(0.816261\pi\)
\(350\) −2.52984e16 −0.735615
\(351\) −8.63557e14 −0.0246487
\(352\) −6.47452e15 −0.181418
\(353\) 4.63052e16 1.27378 0.636890 0.770955i \(-0.280221\pi\)
0.636890 + 0.770955i \(0.280221\pi\)
\(354\) 2.65255e15 0.0716379
\(355\) −5.58405e16 −1.48069
\(356\) −2.52325e15 −0.0656953
\(357\) 6.56259e15 0.167777
\(358\) −4.66710e16 −1.17168
\(359\) 4.08485e16 1.00708 0.503538 0.863973i \(-0.332031\pi\)
0.503538 + 0.863973i \(0.332031\pi\)
\(360\) 2.65831e16 0.643635
\(361\) −2.55295e16 −0.607078
\(362\) 2.04487e16 0.477594
\(363\) 3.46695e14 0.00795343
\(364\) 1.64464e14 0.00370606
\(365\) 8.97183e16 1.98599
\(366\) −9.72014e15 −0.211371
\(367\) 8.38920e16 1.79222 0.896110 0.443832i \(-0.146381\pi\)
0.896110 + 0.443832i \(0.146381\pi\)
\(368\) −6.62620e16 −1.39076
\(369\) −3.04533e16 −0.628005
\(370\) −1.22530e17 −2.48275
\(371\) −2.60798e16 −0.519247
\(372\) −1.61300e15 −0.0315577
\(373\) 5.71890e16 1.09953 0.549763 0.835321i \(-0.314718\pi\)
0.549763 + 0.835321i \(0.314718\pi\)
\(374\) −4.61016e16 −0.871065
\(375\) 8.33423e16 1.54761
\(376\) 3.53079e16 0.644393
\(377\) 1.13879e15 0.0204280
\(378\) 3.28476e15 0.0579171
\(379\) −5.17150e16 −0.896318 −0.448159 0.893954i \(-0.647920\pi\)
−0.448159 + 0.893954i \(0.647920\pi\)
\(380\) 6.25715e15 0.106606
\(381\) 5.93766e16 0.994496
\(382\) 1.07372e17 1.76799
\(383\) −5.94733e16 −0.962787 −0.481394 0.876505i \(-0.659869\pi\)
−0.481394 + 0.876505i \(0.659869\pi\)
\(384\) −3.05061e16 −0.485550
\(385\) −3.71873e16 −0.581969
\(386\) 3.06410e15 0.0471504
\(387\) −3.27556e16 −0.495637
\(388\) −3.39978e15 −0.0505876
\(389\) −7.67585e15 −0.112319 −0.0561596 0.998422i \(-0.517886\pi\)
−0.0561596 + 0.998422i \(0.517886\pi\)
\(390\) 9.08905e15 0.130797
\(391\) 1.00487e17 1.42220
\(392\) 6.72903e16 0.936683
\(393\) −4.63393e16 −0.634450
\(394\) 7.41163e16 0.998132
\(395\) −1.16027e17 −1.53701
\(396\) 2.32789e15 0.0303350
\(397\) 6.18010e16 0.792241 0.396121 0.918199i \(-0.370356\pi\)
0.396121 + 0.918199i \(0.370356\pi\)
\(398\) −6.05275e16 −0.763330
\(399\) 9.21033e15 0.114275
\(400\) −1.80209e17 −2.19982
\(401\) 5.53128e16 0.664335 0.332167 0.943220i \(-0.392220\pi\)
0.332167 + 0.943220i \(0.392220\pi\)
\(402\) −1.74593e16 −0.206328
\(403\) −6.56974e15 −0.0763951
\(404\) −6.21592e15 −0.0711257
\(405\) −1.83134e16 −0.206210
\(406\) −4.33169e15 −0.0479997
\(407\) −1.27821e17 −1.39392
\(408\) 5.15076e16 0.552816
\(409\) −4.58801e16 −0.484645 −0.242322 0.970196i \(-0.577909\pi\)
−0.242322 + 0.970196i \(0.577909\pi\)
\(410\) 3.20525e17 3.33247
\(411\) −6.16225e16 −0.630620
\(412\) −4.66152e15 −0.0469565
\(413\) 4.14580e15 0.0411086
\(414\) 5.02966e16 0.490949
\(415\) 2.00551e17 1.92713
\(416\) 2.47330e15 0.0233975
\(417\) 1.15694e15 0.0107752
\(418\) −6.47018e16 −0.593294
\(419\) 7.97031e16 0.719588 0.359794 0.933032i \(-0.382847\pi\)
0.359794 + 0.933032i \(0.382847\pi\)
\(420\) 3.48778e15 0.0310048
\(421\) −6.83289e16 −0.598096 −0.299048 0.954238i \(-0.596669\pi\)
−0.299048 + 0.954238i \(0.596669\pi\)
\(422\) 1.27670e17 1.10042
\(423\) −2.43239e16 −0.206453
\(424\) −2.04692e17 −1.71089
\(425\) 2.73290e17 2.24955
\(426\) −5.41555e16 −0.439016
\(427\) −1.51921e16 −0.121293
\(428\) −3.71241e15 −0.0291925
\(429\) 9.48149e15 0.0734351
\(430\) 3.44757e17 2.63007
\(431\) −2.57213e16 −0.193281 −0.0966407 0.995319i \(-0.530810\pi\)
−0.0966407 + 0.995319i \(0.530810\pi\)
\(432\) 2.33985e16 0.173198
\(433\) 5.45317e16 0.397629 0.198814 0.980037i \(-0.436291\pi\)
0.198814 + 0.980037i \(0.436291\pi\)
\(434\) 2.49897e16 0.179505
\(435\) 2.41503e16 0.170900
\(436\) −3.38267e15 −0.0235829
\(437\) 1.41030e17 0.968683
\(438\) 8.70112e16 0.588835
\(439\) −2.08605e17 −1.39093 −0.695466 0.718559i \(-0.744802\pi\)
−0.695466 + 0.718559i \(0.744802\pi\)
\(440\) −2.91871e17 −1.91756
\(441\) −4.63569e16 −0.300098
\(442\) 1.76110e16 0.112341
\(443\) 3.67195e16 0.230819 0.115410 0.993318i \(-0.463182\pi\)
0.115410 + 0.993318i \(0.463182\pi\)
\(444\) 1.19882e16 0.0742619
\(445\) −2.17947e17 −1.33049
\(446\) 1.60329e17 0.964573
\(447\) 5.65443e15 0.0335266
\(448\) −5.80364e16 −0.339151
\(449\) 1.05036e17 0.604973 0.302487 0.953154i \(-0.402183\pi\)
0.302487 + 0.953154i \(0.402183\pi\)
\(450\) 1.36789e17 0.776551
\(451\) 3.34364e17 1.87099
\(452\) −7.21646e15 −0.0398038
\(453\) 2.03415e17 1.10597
\(454\) −2.66986e17 −1.43095
\(455\) 1.42057e16 0.0750566
\(456\) 7.22890e16 0.376531
\(457\) 3.14990e17 1.61749 0.808746 0.588159i \(-0.200147\pi\)
0.808746 + 0.588159i \(0.200147\pi\)
\(458\) −7.10291e15 −0.0359592
\(459\) −3.54841e16 −0.177113
\(460\) 5.34053e16 0.262820
\(461\) −9.87682e16 −0.479249 −0.239624 0.970866i \(-0.577024\pi\)
−0.239624 + 0.970866i \(0.577024\pi\)
\(462\) −3.60652e16 −0.172550
\(463\) 3.25645e17 1.53627 0.768135 0.640288i \(-0.221185\pi\)
0.768135 + 0.640288i \(0.221185\pi\)
\(464\) −3.08561e16 −0.143541
\(465\) −1.39324e17 −0.639118
\(466\) 2.19292e17 0.992006
\(467\) 1.64843e17 0.735377 0.367688 0.929949i \(-0.380149\pi\)
0.367688 + 0.929949i \(0.380149\pi\)
\(468\) −8.89263e14 −0.00391230
\(469\) −2.72880e16 −0.118399
\(470\) 2.56013e17 1.09553
\(471\) −2.22660e16 −0.0939738
\(472\) 3.25391e16 0.135451
\(473\) 3.59642e17 1.47664
\(474\) −1.12526e17 −0.455715
\(475\) 3.83551e17 1.53220
\(476\) 6.75794e15 0.0266299
\(477\) 1.41014e17 0.548142
\(478\) 1.08176e17 0.414811
\(479\) −2.98017e17 −1.12735 −0.563677 0.825996i \(-0.690614\pi\)
−0.563677 + 0.825996i \(0.690614\pi\)
\(480\) 5.24509e16 0.195743
\(481\) 4.88281e16 0.179774
\(482\) −2.74025e17 −0.995368
\(483\) 7.86110e16 0.281726
\(484\) 3.57015e14 0.00126238
\(485\) −2.93658e17 −1.02452
\(486\) −1.77608e16 −0.0611401
\(487\) 2.75750e17 0.936651 0.468326 0.883556i \(-0.344857\pi\)
0.468326 + 0.883556i \(0.344857\pi\)
\(488\) −1.19238e17 −0.399655
\(489\) 5.88829e16 0.194752
\(490\) 4.87912e17 1.59246
\(491\) 8.70439e16 0.280355 0.140178 0.990126i \(-0.455233\pi\)
0.140178 + 0.990126i \(0.455233\pi\)
\(492\) −3.13598e16 −0.0996783
\(493\) 4.67937e16 0.146785
\(494\) 2.47164e16 0.0765172
\(495\) 2.01073e17 0.614355
\(496\) 1.78010e17 0.536802
\(497\) −8.46423e16 −0.251925
\(498\) 1.94499e17 0.571384
\(499\) −5.00052e17 −1.44998 −0.724989 0.688760i \(-0.758155\pi\)
−0.724989 + 0.688760i \(0.758155\pi\)
\(500\) 8.58232e16 0.245640
\(501\) 5.68863e16 0.160717
\(502\) −4.55709e17 −1.27091
\(503\) −2.80829e17 −0.773124 −0.386562 0.922263i \(-0.626338\pi\)
−0.386562 + 0.922263i \(0.626338\pi\)
\(504\) 4.02944e16 0.109508
\(505\) −5.36904e17 −1.44047
\(506\) −5.52235e17 −1.46267
\(507\) 2.17174e17 0.567879
\(508\) 6.11441e16 0.157848
\(509\) 1.73337e17 0.441801 0.220900 0.975296i \(-0.429100\pi\)
0.220900 + 0.975296i \(0.429100\pi\)
\(510\) 3.73475e17 0.939842
\(511\) 1.35994e17 0.337896
\(512\) −4.48685e17 −1.10075
\(513\) −4.98005e16 −0.120634
\(514\) −3.68379e17 −0.881120
\(515\) −4.02642e17 −0.950982
\(516\) −3.37306e16 −0.0786686
\(517\) 2.67066e17 0.615079
\(518\) −1.85730e17 −0.422414
\(519\) −4.77602e17 −1.07270
\(520\) 1.11496e17 0.247308
\(521\) −5.14922e17 −1.12797 −0.563983 0.825786i \(-0.690732\pi\)
−0.563983 + 0.825786i \(0.690732\pi\)
\(522\) 2.34216e16 0.0506708
\(523\) −1.42777e15 −0.00305069 −0.00152534 0.999999i \(-0.500486\pi\)
−0.00152534 + 0.999999i \(0.500486\pi\)
\(524\) −4.77187e16 −0.100701
\(525\) 2.13794e17 0.445616
\(526\) −6.55507e16 −0.134949
\(527\) −2.69955e17 −0.548936
\(528\) −2.56905e17 −0.516003
\(529\) 6.99665e17 1.38812
\(530\) −1.48419e18 −2.90869
\(531\) −2.24165e16 −0.0433963
\(532\) 9.48450e15 0.0181380
\(533\) −1.27729e17 −0.241302
\(534\) −2.11371e17 −0.394481
\(535\) −3.20662e17 −0.591219
\(536\) −2.14175e17 −0.390119
\(537\) 3.94412e17 0.709770
\(538\) 1.91925e17 0.341230
\(539\) 5.08979e17 0.894072
\(540\) −1.88585e16 −0.0327301
\(541\) 5.48273e17 0.940188 0.470094 0.882616i \(-0.344220\pi\)
0.470094 + 0.882616i \(0.344220\pi\)
\(542\) 1.41541e17 0.239820
\(543\) −1.72810e17 −0.289314
\(544\) 1.01629e17 0.168123
\(545\) −2.92180e17 −0.477611
\(546\) 1.37771e16 0.0222538
\(547\) 9.28212e17 1.48159 0.740797 0.671729i \(-0.234448\pi\)
0.740797 + 0.671729i \(0.234448\pi\)
\(548\) −6.34568e16 −0.100093
\(549\) 8.21440e16 0.128043
\(550\) −1.50188e18 −2.31355
\(551\) 6.56732e16 0.0999776
\(552\) 6.16993e17 0.928274
\(553\) −1.75872e17 −0.261507
\(554\) −3.22769e17 −0.474328
\(555\) 1.03549e18 1.50398
\(556\) 1.19138e15 0.00171026
\(557\) 6.05578e16 0.0859234 0.0429617 0.999077i \(-0.486321\pi\)
0.0429617 + 0.999077i \(0.486321\pi\)
\(558\) −1.35120e17 −0.189495
\(559\) −1.37385e17 −0.190442
\(560\) −3.84910e17 −0.527397
\(561\) 3.89600e17 0.527667
\(562\) −1.16895e17 −0.156498
\(563\) 1.22604e18 1.62256 0.811280 0.584658i \(-0.198771\pi\)
0.811280 + 0.584658i \(0.198771\pi\)
\(564\) −2.50480e16 −0.0327687
\(565\) −6.23327e17 −0.806122
\(566\) 1.30808e18 1.67235
\(567\) −2.77592e16 −0.0350846
\(568\) −6.64330e17 −0.830080
\(569\) 5.61647e17 0.693799 0.346899 0.937902i \(-0.387235\pi\)
0.346899 + 0.937902i \(0.387235\pi\)
\(570\) 5.24157e17 0.640140
\(571\) −6.81412e17 −0.822764 −0.411382 0.911463i \(-0.634954\pi\)
−0.411382 + 0.911463i \(0.634954\pi\)
\(572\) 9.76373e15 0.0116558
\(573\) −9.07392e17 −1.07100
\(574\) 4.85848e17 0.566987
\(575\) 3.27364e18 3.77738
\(576\) 3.13804e17 0.358025
\(577\) −9.59858e17 −1.08284 −0.541420 0.840752i \(-0.682113\pi\)
−0.541420 + 0.840752i \(0.682113\pi\)
\(578\) −1.30752e17 −0.145853
\(579\) −2.58944e16 −0.0285625
\(580\) 2.48692e16 0.0271256
\(581\) 3.03993e17 0.327882
\(582\) −2.84797e17 −0.303764
\(583\) −1.54828e18 −1.63306
\(584\) 1.06737e18 1.11335
\(585\) −7.68107e16 −0.0792335
\(586\) −6.88105e17 −0.701972
\(587\) 1.17241e17 0.118286 0.0591430 0.998250i \(-0.481163\pi\)
0.0591430 + 0.998250i \(0.481163\pi\)
\(588\) −4.77368e16 −0.0476322
\(589\) −3.78871e17 −0.373888
\(590\) 2.35936e17 0.230280
\(591\) −6.26349e17 −0.604642
\(592\) −1.32302e18 −1.26321
\(593\) −2.03990e17 −0.192643 −0.0963213 0.995350i \(-0.530708\pi\)
−0.0963213 + 0.995350i \(0.530708\pi\)
\(594\) 1.95006e17 0.182153
\(595\) 5.83721e17 0.539319
\(596\) 5.82274e15 0.00532141
\(597\) 5.11512e17 0.462405
\(598\) 2.10956e17 0.188640
\(599\) −1.55751e18 −1.37771 −0.688854 0.724900i \(-0.741886\pi\)
−0.688854 + 0.724900i \(0.741886\pi\)
\(600\) 1.67800e18 1.46828
\(601\) −1.59055e18 −1.37677 −0.688387 0.725343i \(-0.741681\pi\)
−0.688387 + 0.725343i \(0.741681\pi\)
\(602\) 5.22578e17 0.447480
\(603\) 1.47547e17 0.124988
\(604\) 2.09470e17 0.175543
\(605\) 3.08374e16 0.0255663
\(606\) −5.20704e17 −0.427089
\(607\) 4.42280e17 0.358898 0.179449 0.983767i \(-0.442569\pi\)
0.179449 + 0.983767i \(0.442569\pi\)
\(608\) 1.42633e17 0.114511
\(609\) 3.66067e16 0.0290769
\(610\) −8.64576e17 −0.679453
\(611\) −1.02021e17 −0.0793268
\(612\) −3.65404e16 −0.0281118
\(613\) 1.50989e18 1.14935 0.574676 0.818381i \(-0.305128\pi\)
0.574676 + 0.818381i \(0.305128\pi\)
\(614\) −1.47401e18 −1.11021
\(615\) −2.70873e18 −2.01872
\(616\) −4.42415e17 −0.326253
\(617\) −1.64765e18 −1.20229 −0.601147 0.799138i \(-0.705290\pi\)
−0.601147 + 0.799138i \(0.705290\pi\)
\(618\) −3.90493e17 −0.281960
\(619\) −2.47368e18 −1.76748 −0.883741 0.467976i \(-0.844983\pi\)
−0.883741 + 0.467976i \(0.844983\pi\)
\(620\) −1.43471e17 −0.101442
\(621\) −4.25052e17 −0.297404
\(622\) −2.69521e18 −1.86619
\(623\) −3.30361e17 −0.226369
\(624\) 9.81389e16 0.0665490
\(625\) 3.77068e18 2.53046
\(626\) 2.34754e18 1.55912
\(627\) 5.46788e17 0.359402
\(628\) −2.29288e16 −0.0149157
\(629\) 2.00638e18 1.29176
\(630\) 2.92169e17 0.186175
\(631\) 7.80619e16 0.0492321 0.0246160 0.999697i \(-0.492164\pi\)
0.0246160 + 0.999697i \(0.492164\pi\)
\(632\) −1.38036e18 −0.861653
\(633\) −1.07893e18 −0.666604
\(634\) −1.36834e18 −0.836783
\(635\) 5.28136e18 3.19681
\(636\) 1.45212e17 0.0870023
\(637\) −1.94432e17 −0.115309
\(638\) −2.57159e17 −0.150962
\(639\) 4.57663e17 0.265944
\(640\) −2.71342e18 −1.56080
\(641\) −1.70759e18 −0.972316 −0.486158 0.873871i \(-0.661602\pi\)
−0.486158 + 0.873871i \(0.661602\pi\)
\(642\) −3.10986e17 −0.175293
\(643\) −1.28298e18 −0.715894 −0.357947 0.933742i \(-0.616523\pi\)
−0.357947 + 0.933742i \(0.616523\pi\)
\(644\) 8.09510e16 0.0447161
\(645\) −2.91351e18 −1.59323
\(646\) 1.01561e18 0.549814
\(647\) 1.34776e18 0.722331 0.361165 0.932502i \(-0.382379\pi\)
0.361165 + 0.932502i \(0.382379\pi\)
\(648\) −2.17873e17 −0.115602
\(649\) 2.46123e17 0.129289
\(650\) 5.73726e17 0.298379
\(651\) −2.11185e17 −0.108740
\(652\) 6.06357e16 0.0309114
\(653\) −1.20789e18 −0.609664 −0.304832 0.952406i \(-0.598600\pi\)
−0.304832 + 0.952406i \(0.598600\pi\)
\(654\) −2.83364e17 −0.141609
\(655\) −4.12173e18 −2.03944
\(656\) 3.46086e18 1.69555
\(657\) −7.35323e17 −0.356700
\(658\) 3.88061e17 0.186394
\(659\) 4.86997e17 0.231617 0.115809 0.993272i \(-0.463054\pi\)
0.115809 + 0.993272i \(0.463054\pi\)
\(660\) 2.07058e17 0.0975118
\(661\) −4.01281e18 −1.87128 −0.935640 0.352956i \(-0.885177\pi\)
−0.935640 + 0.352956i \(0.885177\pi\)
\(662\) −1.19743e18 −0.552932
\(663\) −1.48829e17 −0.0680533
\(664\) 2.38594e18 1.08036
\(665\) 8.19230e17 0.367338
\(666\) 1.00425e18 0.445921
\(667\) 5.60526e17 0.246478
\(668\) 5.85796e16 0.0255094
\(669\) −1.35492e18 −0.584312
\(670\) −1.55295e18 −0.663241
\(671\) −9.01905e17 −0.381474
\(672\) 7.95045e16 0.0333036
\(673\) −3.60435e18 −1.49530 −0.747652 0.664091i \(-0.768819\pi\)
−0.747652 + 0.664091i \(0.768819\pi\)
\(674\) −4.06776e18 −1.67134
\(675\) −1.15599e18 −0.470414
\(676\) 2.23639e17 0.0901350
\(677\) 3.19700e18 1.27619 0.638097 0.769956i \(-0.279722\pi\)
0.638097 + 0.769956i \(0.279722\pi\)
\(678\) −6.04518e17 −0.239010
\(679\) −4.45123e17 −0.174312
\(680\) 4.58144e18 1.77703
\(681\) 2.25627e18 0.866832
\(682\) 1.48356e18 0.564555
\(683\) 2.15760e18 0.813271 0.406636 0.913590i \(-0.366702\pi\)
0.406636 + 0.913590i \(0.366702\pi\)
\(684\) −5.12830e16 −0.0191473
\(685\) −5.48113e18 −2.02713
\(686\) 1.56105e18 0.571886
\(687\) 6.00260e16 0.0217831
\(688\) 3.72251e18 1.33817
\(689\) 5.91448e17 0.210616
\(690\) 4.47373e18 1.57816
\(691\) −1.49915e18 −0.523888 −0.261944 0.965083i \(-0.584364\pi\)
−0.261944 + 0.965083i \(0.584364\pi\)
\(692\) −4.91819e17 −0.170261
\(693\) 3.04784e17 0.104526
\(694\) −3.59903e17 −0.122278
\(695\) 1.02906e17 0.0346369
\(696\) 2.87314e17 0.0958070
\(697\) −5.24844e18 −1.73387
\(698\) −4.88035e18 −1.59731
\(699\) −1.85322e18 −0.600931
\(700\) 2.20158e17 0.0707290
\(701\) 3.69368e18 1.17569 0.587844 0.808974i \(-0.299977\pi\)
0.587844 + 0.808974i \(0.299977\pi\)
\(702\) −7.44930e16 −0.0234922
\(703\) 2.81587e18 0.879839
\(704\) −3.44544e18 −1.06665
\(705\) −2.16354e18 −0.663644
\(706\) 3.99442e18 1.21401
\(707\) −8.13833e17 −0.245081
\(708\) −2.30837e16 −0.00688795
\(709\) −2.41023e18 −0.712620 −0.356310 0.934368i \(-0.615965\pi\)
−0.356310 + 0.934368i \(0.615965\pi\)
\(710\) −4.81696e18 −1.41122
\(711\) 9.50945e17 0.276060
\(712\) −2.59290e18 −0.745875
\(713\) −3.23370e18 −0.921759
\(714\) 5.66108e17 0.159905
\(715\) 8.43348e17 0.236057
\(716\) 4.06153e17 0.112656
\(717\) −9.14187e17 −0.251282
\(718\) 3.52371e18 0.959825
\(719\) 6.00754e18 1.62166 0.810828 0.585285i \(-0.199017\pi\)
0.810828 + 0.585285i \(0.199017\pi\)
\(720\) 2.08122e18 0.556746
\(721\) −6.10320e17 −0.161800
\(722\) −2.20225e18 −0.578594
\(723\) 2.31575e18 0.602967
\(724\) −1.77954e17 −0.0459205
\(725\) 1.52443e18 0.389862
\(726\) 2.99069e16 0.00758025
\(727\) −4.22863e18 −1.06225 −0.531124 0.847294i \(-0.678230\pi\)
−0.531124 + 0.847294i \(0.678230\pi\)
\(728\) 1.69004e17 0.0420769
\(729\) 1.50095e17 0.0370370
\(730\) 7.73937e18 1.89281
\(731\) −5.64523e18 −1.36842
\(732\) 8.45892e16 0.0203232
\(733\) 4.95380e17 0.117968 0.0589838 0.998259i \(-0.481214\pi\)
0.0589838 + 0.998259i \(0.481214\pi\)
\(734\) 7.23677e18 1.70813
\(735\) −4.12330e18 −0.964666
\(736\) 1.21738e18 0.282307
\(737\) −1.62000e18 −0.372372
\(738\) −2.62699e18 −0.598539
\(739\) 4.95738e18 1.11960 0.559800 0.828627i \(-0.310878\pi\)
0.559800 + 0.828627i \(0.310878\pi\)
\(740\) 1.06632e18 0.238715
\(741\) −2.08876e17 −0.0463521
\(742\) −2.24972e18 −0.494884
\(743\) −1.06107e18 −0.231376 −0.115688 0.993286i \(-0.536907\pi\)
−0.115688 + 0.993286i \(0.536907\pi\)
\(744\) −1.65753e18 −0.358291
\(745\) 5.02943e17 0.107771
\(746\) 4.93330e18 1.04794
\(747\) −1.64370e18 −0.346129
\(748\) 4.01197e17 0.0837525
\(749\) −4.86056e17 −0.100590
\(750\) 7.18935e18 1.47500
\(751\) −3.62932e18 −0.738186 −0.369093 0.929392i \(-0.620332\pi\)
−0.369093 + 0.929392i \(0.620332\pi\)
\(752\) 2.76429e18 0.557402
\(753\) 3.85115e18 0.769880
\(754\) 9.82357e16 0.0194695
\(755\) 1.80931e19 3.55516
\(756\) −2.85855e16 −0.00556870
\(757\) −5.58460e18 −1.07862 −0.539310 0.842107i \(-0.681315\pi\)
−0.539310 + 0.842107i \(0.681315\pi\)
\(758\) −4.46109e18 −0.854262
\(759\) 4.66689e18 0.886045
\(760\) 6.42988e18 1.21036
\(761\) 3.82032e18 0.713017 0.356508 0.934292i \(-0.383967\pi\)
0.356508 + 0.934292i \(0.383967\pi\)
\(762\) 5.12200e18 0.947834
\(763\) −4.42884e17 −0.0812606
\(764\) −9.34403e17 −0.169992
\(765\) −3.15620e18 −0.569331
\(766\) −5.13035e18 −0.917613
\(767\) −9.40201e16 −0.0166744
\(768\) 8.94768e17 0.157348
\(769\) −9.68265e18 −1.68839 −0.844196 0.536035i \(-0.819921\pi\)
−0.844196 + 0.536035i \(0.819921\pi\)
\(770\) −3.20789e18 −0.554663
\(771\) 3.11314e18 0.533759
\(772\) −2.66652e16 −0.00453350
\(773\) 6.59375e18 1.11164 0.555822 0.831301i \(-0.312404\pi\)
0.555822 + 0.831301i \(0.312404\pi\)
\(774\) −2.82559e18 −0.472382
\(775\) −8.79451e18 −1.45798
\(776\) −3.49363e18 −0.574349
\(777\) 1.56959e18 0.255887
\(778\) −6.62142e17 −0.107049
\(779\) −7.36599e18 −1.18097
\(780\) −7.90972e16 −0.0125761
\(781\) −5.02494e18 −0.792318
\(782\) 8.66832e18 1.35547
\(783\) −1.97933e17 −0.0306950
\(784\) 5.26823e18 0.810233
\(785\) −1.98049e18 −0.302079
\(786\) −3.99736e18 −0.604682
\(787\) −6.49044e18 −0.973730 −0.486865 0.873477i \(-0.661860\pi\)
−0.486865 + 0.873477i \(0.661860\pi\)
\(788\) −6.44994e17 −0.0959700
\(789\) 5.53963e17 0.0817485
\(790\) −1.00088e19 −1.46490
\(791\) −9.44831e17 −0.137154
\(792\) 2.39215e18 0.344409
\(793\) 3.44532e17 0.0491987
\(794\) 5.33114e18 0.755069
\(795\) 1.25428e19 1.76201
\(796\) 5.26738e17 0.0733939
\(797\) 4.26258e18 0.589106 0.294553 0.955635i \(-0.404829\pi\)
0.294553 + 0.955635i \(0.404829\pi\)
\(798\) 7.94511e17 0.108913
\(799\) −4.19209e18 −0.570002
\(800\) 3.31085e18 0.446534
\(801\) 1.78627e18 0.238966
\(802\) 4.77144e18 0.633164
\(803\) 8.07353e18 1.06270
\(804\) 1.51939e17 0.0198383
\(805\) 6.99220e18 0.905609
\(806\) −5.66726e17 −0.0728107
\(807\) −1.62194e18 −0.206708
\(808\) −6.38751e18 −0.807529
\(809\) −1.36312e19 −1.70950 −0.854749 0.519042i \(-0.826289\pi\)
−0.854749 + 0.519042i \(0.826289\pi\)
\(810\) −1.57977e18 −0.196535
\(811\) 3.01246e18 0.371780 0.185890 0.982571i \(-0.440483\pi\)
0.185890 + 0.982571i \(0.440483\pi\)
\(812\) 3.76964e16 0.00461515
\(813\) −1.19615e18 −0.145277
\(814\) −1.10262e19 −1.32852
\(815\) 5.23745e18 0.626030
\(816\) 4.03259e18 0.478187
\(817\) −7.92286e18 −0.932048
\(818\) −3.95776e18 −0.461905
\(819\) −1.16429e17 −0.0134808
\(820\) −2.78936e18 −0.320416
\(821\) −1.37107e19 −1.56254 −0.781268 0.624195i \(-0.785427\pi\)
−0.781268 + 0.624195i \(0.785427\pi\)
\(822\) −5.31574e18 −0.601031
\(823\) −3.22340e18 −0.361589 −0.180795 0.983521i \(-0.557867\pi\)
−0.180795 + 0.983521i \(0.557867\pi\)
\(824\) −4.79021e18 −0.533123
\(825\) 1.26923e19 1.40149
\(826\) 3.57629e17 0.0391798
\(827\) 8.24057e16 0.00895718 0.00447859 0.999990i \(-0.498574\pi\)
0.00447859 + 0.999990i \(0.498574\pi\)
\(828\) −4.37705e17 −0.0472046
\(829\) 9.63103e18 1.03055 0.515274 0.857026i \(-0.327690\pi\)
0.515274 + 0.857026i \(0.327690\pi\)
\(830\) 1.73001e19 1.83671
\(831\) 2.72769e18 0.287335
\(832\) 1.31617e18 0.137566
\(833\) −7.98933e18 −0.828548
\(834\) 9.98007e16 0.0102696
\(835\) 5.05985e18 0.516626
\(836\) 5.63065e17 0.0570450
\(837\) 1.14189e18 0.114791
\(838\) 6.87542e18 0.685825
\(839\) −1.75314e18 −0.173526 −0.0867628 0.996229i \(-0.527652\pi\)
−0.0867628 + 0.996229i \(0.527652\pi\)
\(840\) 3.58406e18 0.352014
\(841\) −9.99961e18 −0.974561
\(842\) −5.89425e18 −0.570033
\(843\) 9.87866e17 0.0948023
\(844\) −1.11104e18 −0.105805
\(845\) 1.93169e19 1.82545
\(846\) −2.09826e18 −0.196766
\(847\) 4.67430e16 0.00434985
\(848\) −1.60256e19 −1.47993
\(849\) −1.10545e19 −1.01307
\(850\) 2.35748e19 2.14400
\(851\) 2.40337e19 2.16909
\(852\) 4.71287e17 0.0422112
\(853\) −1.37778e19 −1.22465 −0.612326 0.790605i \(-0.709766\pi\)
−0.612326 + 0.790605i \(0.709766\pi\)
\(854\) −1.31051e18 −0.115602
\(855\) −4.42960e18 −0.387780
\(856\) −3.81490e18 −0.331439
\(857\) −1.39335e19 −1.20139 −0.600695 0.799479i \(-0.705109\pi\)
−0.600695 + 0.799479i \(0.705109\pi\)
\(858\) 8.17901e17 0.0699896
\(859\) 1.73655e19 1.47480 0.737400 0.675457i \(-0.236054\pi\)
0.737400 + 0.675457i \(0.236054\pi\)
\(860\) −3.00023e18 −0.252880
\(861\) −4.10585e18 −0.343465
\(862\) −2.21879e18 −0.184213
\(863\) −1.12262e19 −0.925041 −0.462520 0.886609i \(-0.653055\pi\)
−0.462520 + 0.886609i \(0.653055\pi\)
\(864\) −4.29883e17 −0.0351569
\(865\) −4.24812e19 −3.44820
\(866\) 4.70407e18 0.378972
\(867\) 1.10497e18 0.0883539
\(868\) −2.17472e17 −0.0172594
\(869\) −1.04410e19 −0.822455
\(870\) 2.08327e18 0.162881
\(871\) 6.18848e17 0.0480248
\(872\) −3.47605e18 −0.267750
\(873\) 2.40679e18 0.184012
\(874\) 1.21656e19 0.923232
\(875\) 1.12366e19 0.846412
\(876\) −7.57212e17 −0.0566162
\(877\) −1.89263e19 −1.40465 −0.702327 0.711855i \(-0.747855\pi\)
−0.702327 + 0.711855i \(0.747855\pi\)
\(878\) −1.79949e19 −1.32567
\(879\) 5.81511e18 0.425236
\(880\) −2.28509e19 −1.65869
\(881\) −2.96071e18 −0.213330 −0.106665 0.994295i \(-0.534017\pi\)
−0.106665 + 0.994295i \(0.534017\pi\)
\(882\) −3.99888e18 −0.286018
\(883\) −8.98458e18 −0.637901 −0.318950 0.947771i \(-0.603330\pi\)
−0.318950 + 0.947771i \(0.603330\pi\)
\(884\) −1.53259e17 −0.0108016
\(885\) −1.99387e18 −0.139498
\(886\) 3.16753e18 0.219989
\(887\) −1.21271e19 −0.836088 −0.418044 0.908427i \(-0.637284\pi\)
−0.418044 + 0.908427i \(0.637284\pi\)
\(888\) 1.23192e19 0.843136
\(889\) 8.00542e18 0.543905
\(890\) −1.88008e19 −1.26806
\(891\) −1.64797e18 −0.110343
\(892\) −1.39526e18 −0.0927432
\(893\) −5.88343e18 −0.388236
\(894\) 4.87768e17 0.0319536
\(895\) 3.50817e19 2.28156
\(896\) −4.11297e18 −0.265555
\(897\) −1.78277e18 −0.114273
\(898\) 9.06070e18 0.576588
\(899\) −1.50583e18 −0.0951346
\(900\) −1.19040e18 −0.0746650
\(901\) 2.43029e19 1.51338
\(902\) 2.88432e19 1.78321
\(903\) −4.41626e18 −0.271072
\(904\) −7.41568e18 −0.451914
\(905\) −1.53709e19 −0.929999
\(906\) 1.75472e19 1.05408
\(907\) 1.40064e19 0.835370 0.417685 0.908592i \(-0.362842\pi\)
0.417685 + 0.908592i \(0.362842\pi\)
\(908\) 2.32343e18 0.137585
\(909\) 4.40042e18 0.258719
\(910\) 1.22543e18 0.0715350
\(911\) 5.73642e18 0.332485 0.166242 0.986085i \(-0.446837\pi\)
0.166242 + 0.986085i \(0.446837\pi\)
\(912\) 5.65958e18 0.325700
\(913\) 1.80471e19 1.03121
\(914\) 2.71720e19 1.54160
\(915\) 7.30645e18 0.411594
\(916\) 6.18128e16 0.00345747
\(917\) −6.24767e18 −0.346990
\(918\) −3.06096e18 −0.168803
\(919\) 1.21399e19 0.664761 0.332381 0.943145i \(-0.392148\pi\)
0.332381 + 0.943145i \(0.392148\pi\)
\(920\) 5.48795e19 2.98394
\(921\) 1.24567e19 0.672536
\(922\) −8.52004e18 −0.456763
\(923\) 1.91955e18 0.102185
\(924\) 3.13856e17 0.0165906
\(925\) 6.53632e19 3.43093
\(926\) 2.80911e19 1.46419
\(927\) 3.30002e18 0.170804
\(928\) 5.66897e17 0.0291368
\(929\) 7.09575e18 0.362157 0.181078 0.983469i \(-0.442041\pi\)
0.181078 + 0.983469i \(0.442041\pi\)
\(930\) −1.20185e19 −0.609131
\(931\) −1.12127e19 −0.564336
\(932\) −1.90838e18 −0.0953810
\(933\) 2.27769e19 1.13049
\(934\) 1.42198e19 0.700873
\(935\) 3.46537e19 1.69619
\(936\) −9.13812e17 −0.0444185
\(937\) 3.72483e19 1.79804 0.899021 0.437906i \(-0.144280\pi\)
0.899021 + 0.437906i \(0.144280\pi\)
\(938\) −2.35394e18 −0.112844
\(939\) −1.98389e19 −0.944475
\(940\) −2.22794e18 −0.105335
\(941\) 2.16721e19 1.01758 0.508791 0.860890i \(-0.330093\pi\)
0.508791 + 0.860890i \(0.330093\pi\)
\(942\) −1.92073e18 −0.0895645
\(943\) −6.28693e19 −2.91147
\(944\) 2.54752e18 0.117165
\(945\) −2.46909e18 −0.112780
\(946\) 3.10238e19 1.40735
\(947\) −2.96624e19 −1.33639 −0.668193 0.743988i \(-0.732932\pi\)
−0.668193 + 0.743988i \(0.732932\pi\)
\(948\) 9.79252e17 0.0438168
\(949\) −3.08412e18 −0.137057
\(950\) 3.30863e19 1.46031
\(951\) 1.15637e19 0.506901
\(952\) 6.94449e18 0.302343
\(953\) −3.45780e17 −0.0149519 −0.00747595 0.999972i \(-0.502380\pi\)
−0.00747595 + 0.999972i \(0.502380\pi\)
\(954\) 1.21643e19 0.522424
\(955\) −8.07097e19 −3.44274
\(956\) −9.41400e17 −0.0398839
\(957\) 2.17322e18 0.0914485
\(958\) −2.57078e19 −1.07446
\(959\) −8.30822e18 −0.344895
\(960\) 2.79119e19 1.15087
\(961\) −1.57303e19 −0.644223
\(962\) 4.21206e18 0.171339
\(963\) 2.62812e18 0.106188
\(964\) 2.38469e18 0.0957042
\(965\) −2.30323e18 −0.0918141
\(966\) 6.78122e18 0.268507
\(967\) 3.66584e19 1.44179 0.720894 0.693045i \(-0.243731\pi\)
0.720894 + 0.693045i \(0.243731\pi\)
\(968\) 3.66871e17 0.0143325
\(969\) −8.58282e18 −0.333062
\(970\) −2.53318e19 −0.976450
\(971\) 5.54985e18 0.212499 0.106249 0.994340i \(-0.466116\pi\)
0.106249 + 0.994340i \(0.466116\pi\)
\(972\) 1.54563e17 0.00587860
\(973\) 1.55983e17 0.00589311
\(974\) 2.37870e19 0.892704
\(975\) −4.84851e18 −0.180750
\(976\) −9.33524e18 −0.345702
\(977\) 3.38833e18 0.124644 0.0623220 0.998056i \(-0.480149\pi\)
0.0623220 + 0.998056i \(0.480149\pi\)
\(978\) 5.07942e18 0.185614
\(979\) −1.96125e19 −0.711944
\(980\) −4.24604e18 −0.153114
\(981\) 2.39468e18 0.0857827
\(982\) 7.50867e18 0.267201
\(983\) 1.69848e19 0.600431 0.300216 0.953871i \(-0.402941\pi\)
0.300216 + 0.953871i \(0.402941\pi\)
\(984\) −3.22255e19 −1.13170
\(985\) −5.57118e19 −1.94362
\(986\) 4.03657e18 0.139898
\(987\) −3.27946e18 −0.112912
\(988\) −2.15093e17 −0.00735710
\(989\) −6.76223e19 −2.29781
\(990\) 1.73451e19 0.585530
\(991\) −2.34108e19 −0.785124 −0.392562 0.919725i \(-0.628411\pi\)
−0.392562 + 0.919725i \(0.628411\pi\)
\(992\) −3.27045e18 −0.108964
\(993\) 1.01193e19 0.334951
\(994\) −7.30149e18 −0.240104
\(995\) 4.54974e19 1.48640
\(996\) −1.69263e18 −0.0549383
\(997\) 4.83103e19 1.55784 0.778918 0.627126i \(-0.215769\pi\)
0.778918 + 0.627126i \(0.215769\pi\)
\(998\) −4.31359e19 −1.38195
\(999\) −8.48680e18 −0.270127
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 177.14.a.b.1.23 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.14.a.b.1.23 31 1.1 even 1 trivial