Properties

Label 6.30.a.d.1.1
Level $6$
Weight $30$
Character 6.1
Self dual yes
Analytic conductor $31.967$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6,30,Mod(1,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.1");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(31.9668254298\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2722854711072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.65011e6\) of defining polynomial
Character \(\chi\) \(=\) 6.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16384.0 q^{2} -4.78297e6 q^{3} +2.68435e8 q^{4} -2.03873e10 q^{5} -7.83642e10 q^{6} -2.52164e12 q^{7} +4.39805e12 q^{8} +2.28768e13 q^{9} +O(q^{10})\) \(q+16384.0 q^{2} -4.78297e6 q^{3} +2.68435e8 q^{4} -2.03873e10 q^{5} -7.83642e10 q^{6} -2.52164e12 q^{7} +4.39805e12 q^{8} +2.28768e13 q^{9} -3.34025e14 q^{10} +2.16814e15 q^{11} -1.28392e15 q^{12} -1.73173e16 q^{13} -4.13145e16 q^{14} +9.75118e16 q^{15} +7.20576e16 q^{16} +1.09493e18 q^{17} +3.74813e17 q^{18} +1.44082e18 q^{19} -5.47267e18 q^{20} +1.20609e19 q^{21} +3.55229e19 q^{22} -7.43943e19 q^{23} -2.10357e19 q^{24} +2.29377e20 q^{25} -2.83727e20 q^{26} -1.09419e20 q^{27} -6.76897e20 q^{28} +1.78624e21 q^{29} +1.59763e21 q^{30} +1.83689e21 q^{31} +1.18059e21 q^{32} -1.03702e22 q^{33} +1.79393e22 q^{34} +5.14094e22 q^{35} +6.14094e21 q^{36} +6.04145e22 q^{37} +2.36064e22 q^{38} +8.28282e22 q^{39} -8.96642e22 q^{40} -7.54895e22 q^{41} +1.97606e23 q^{42} +4.59709e23 q^{43} +5.82007e23 q^{44} -4.66396e23 q^{45} -1.21888e24 q^{46} +7.22727e23 q^{47} -3.44649e23 q^{48} +3.13875e24 q^{49} +3.75811e24 q^{50} -5.23702e24 q^{51} -4.64858e24 q^{52} -5.39412e24 q^{53} -1.79272e24 q^{54} -4.42026e25 q^{55} -1.10903e25 q^{56} -6.89140e24 q^{57} +2.92658e25 q^{58} -6.59432e24 q^{59} +2.61756e25 q^{60} +6.51969e24 q^{61} +3.00957e25 q^{62} -5.76870e25 q^{63} +1.93428e25 q^{64} +3.53053e26 q^{65} -1.69905e26 q^{66} -1.98903e26 q^{67} +2.93918e26 q^{68} +3.55826e26 q^{69} +8.42291e26 q^{70} -4.25661e26 q^{71} +1.00613e26 q^{72} -4.61216e26 q^{73} +9.89831e26 q^{74} -1.09710e27 q^{75} +3.86767e26 q^{76} -5.46728e27 q^{77} +1.35706e27 q^{78} +5.25329e27 q^{79} -1.46906e27 q^{80} +5.23348e26 q^{81} -1.23682e27 q^{82} -4.10278e27 q^{83} +3.23758e27 q^{84} -2.23227e28 q^{85} +7.53188e27 q^{86} -8.54353e27 q^{87} +9.53560e27 q^{88} -5.01761e27 q^{89} -7.64143e27 q^{90} +4.36680e28 q^{91} -1.99701e28 q^{92} -8.78580e27 q^{93} +1.18412e28 q^{94} -2.93744e28 q^{95} -5.64673e27 q^{96} +9.02825e28 q^{97} +5.14253e28 q^{98} +4.96002e28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32768 q^{2} - 9565938 q^{3} + 536870912 q^{4} - 6557946756 q^{5} - 156728328192 q^{6} - 1108363146848 q^{7} + 8796093022208 q^{8} + 45753584909922 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 32768 q^{2} - 9565938 q^{3} + 536870912 q^{4} - 6557946756 q^{5} - 156728328192 q^{6} - 1108363146848 q^{7} + 8796093022208 q^{8} + 45753584909922 q^{9} - 107445399650304 q^{10} + 335339398137480 q^{11} - 25\!\cdots\!28 q^{12}+ \cdots + 76\!\cdots\!80 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16384.0 0.707107
\(3\) −4.78297e6 −0.577350
\(4\) 2.68435e8 0.500000
\(5\) −2.03873e10 −1.49381 −0.746903 0.664933i \(-0.768460\pi\)
−0.746903 + 0.664933i \(0.768460\pi\)
\(6\) −7.83642e10 −0.408248
\(7\) −2.52164e12 −1.40527 −0.702637 0.711548i \(-0.747994\pi\)
−0.702637 + 0.711548i \(0.747994\pi\)
\(8\) 4.39805e12 0.353553
\(9\) 2.28768e13 0.333333
\(10\) −3.34025e14 −1.05628
\(11\) 2.16814e15 1.72145 0.860725 0.509070i \(-0.170011\pi\)
0.860725 + 0.509070i \(0.170011\pi\)
\(12\) −1.28392e15 −0.288675
\(13\) −1.73173e16 −1.21984 −0.609919 0.792464i \(-0.708798\pi\)
−0.609919 + 0.792464i \(0.708798\pi\)
\(14\) −4.13145e16 −0.993679
\(15\) 9.75118e16 0.862450
\(16\) 7.20576e16 0.250000
\(17\) 1.09493e18 1.57716 0.788582 0.614929i \(-0.210815\pi\)
0.788582 + 0.614929i \(0.210815\pi\)
\(18\) 3.74813e17 0.235702
\(19\) 1.44082e18 0.413697 0.206849 0.978373i \(-0.433679\pi\)
0.206849 + 0.978373i \(0.433679\pi\)
\(20\) −5.47267e18 −0.746903
\(21\) 1.20609e19 0.811336
\(22\) 3.55229e19 1.21725
\(23\) −7.43943e19 −1.33809 −0.669046 0.743221i \(-0.733297\pi\)
−0.669046 + 0.743221i \(0.733297\pi\)
\(24\) −2.10357e19 −0.204124
\(25\) 2.29377e20 1.23146
\(26\) −2.83727e20 −0.862556
\(27\) −1.09419e20 −0.192450
\(28\) −6.76897e20 −0.702637
\(29\) 1.78624e21 1.11473 0.557364 0.830268i \(-0.311813\pi\)
0.557364 + 0.830268i \(0.311813\pi\)
\(30\) 1.59763e21 0.609844
\(31\) 1.83689e21 0.435852 0.217926 0.975965i \(-0.430071\pi\)
0.217926 + 0.975965i \(0.430071\pi\)
\(32\) 1.18059e21 0.176777
\(33\) −1.03702e22 −0.993880
\(34\) 1.79393e22 1.11522
\(35\) 5.14094e22 2.09921
\(36\) 6.14094e21 0.166667
\(37\) 6.04145e22 1.10209 0.551044 0.834476i \(-0.314230\pi\)
0.551044 + 0.834476i \(0.314230\pi\)
\(38\) 2.36064e22 0.292528
\(39\) 8.28282e22 0.704274
\(40\) −8.96642e22 −0.528140
\(41\) −7.54895e22 −0.310829 −0.155414 0.987849i \(-0.549671\pi\)
−0.155414 + 0.987849i \(0.549671\pi\)
\(42\) 1.97606e23 0.573701
\(43\) 4.59709e23 0.948835 0.474417 0.880300i \(-0.342659\pi\)
0.474417 + 0.880300i \(0.342659\pi\)
\(44\) 5.82007e23 0.860725
\(45\) −4.66396e23 −0.497936
\(46\) −1.21888e24 −0.946174
\(47\) 7.22727e23 0.410729 0.205365 0.978686i \(-0.434162\pi\)
0.205365 + 0.978686i \(0.434162\pi\)
\(48\) −3.44649e23 −0.144338
\(49\) 3.13875e24 0.974796
\(50\) 3.75811e24 0.870773
\(51\) −5.23702e24 −0.910576
\(52\) −4.64858e24 −0.609919
\(53\) −5.39412e24 −0.536933 −0.268467 0.963289i \(-0.586517\pi\)
−0.268467 + 0.963289i \(0.586517\pi\)
\(54\) −1.79272e24 −0.136083
\(55\) −4.42026e25 −2.57151
\(56\) −1.10903e25 −0.496840
\(57\) −6.89140e24 −0.238848
\(58\) 2.92658e25 0.788232
\(59\) −6.59432e24 −0.138617 −0.0693084 0.997595i \(-0.522079\pi\)
−0.0693084 + 0.997595i \(0.522079\pi\)
\(60\) 2.61756e25 0.431225
\(61\) 6.51969e24 0.0845170 0.0422585 0.999107i \(-0.486545\pi\)
0.0422585 + 0.999107i \(0.486545\pi\)
\(62\) 3.00957e25 0.308194
\(63\) −5.76870e25 −0.468425
\(64\) 1.93428e25 0.125000
\(65\) 3.53053e26 1.82220
\(66\) −1.69905e26 −0.702779
\(67\) −1.98903e26 −0.661541 −0.330770 0.943711i \(-0.607309\pi\)
−0.330770 + 0.943711i \(0.607309\pi\)
\(68\) 2.93918e26 0.788582
\(69\) 3.55826e26 0.772548
\(70\) 8.42291e26 1.48436
\(71\) −4.25661e26 −0.610686 −0.305343 0.952242i \(-0.598771\pi\)
−0.305343 + 0.952242i \(0.598771\pi\)
\(72\) 1.00613e26 0.117851
\(73\) −4.61216e26 −0.442306 −0.221153 0.975239i \(-0.570982\pi\)
−0.221153 + 0.975239i \(0.570982\pi\)
\(74\) 9.89831e26 0.779294
\(75\) −1.09710e27 −0.710983
\(76\) 3.86767e26 0.206849
\(77\) −5.46728e27 −2.41911
\(78\) 1.35706e27 0.497997
\(79\) 5.25329e27 1.60265 0.801325 0.598229i \(-0.204129\pi\)
0.801325 + 0.598229i \(0.204129\pi\)
\(80\) −1.46906e27 −0.373452
\(81\) 5.23348e26 0.111111
\(82\) −1.23682e27 −0.219789
\(83\) −4.10278e27 −0.611569 −0.305784 0.952101i \(-0.598919\pi\)
−0.305784 + 0.952101i \(0.598919\pi\)
\(84\) 3.23758e27 0.405668
\(85\) −2.23227e28 −2.35598
\(86\) 7.53188e27 0.670928
\(87\) −8.54353e27 −0.643589
\(88\) 9.53560e27 0.608624
\(89\) −5.01761e27 −0.271858 −0.135929 0.990719i \(-0.543402\pi\)
−0.135929 + 0.990719i \(0.543402\pi\)
\(90\) −7.64143e27 −0.352094
\(91\) 4.36680e28 1.71421
\(92\) −1.99701e28 −0.669046
\(93\) −8.78580e27 −0.251639
\(94\) 1.18412e28 0.290429
\(95\) −2.93744e28 −0.617984
\(96\) −5.64673e27 −0.102062
\(97\) 9.02825e28 1.40415 0.702075 0.712103i \(-0.252257\pi\)
0.702075 + 0.712103i \(0.252257\pi\)
\(98\) 5.14253e28 0.689285
\(99\) 4.96002e28 0.573817
\(100\) 6.15729e28 0.615729
\(101\) 1.35694e29 1.17463 0.587316 0.809358i \(-0.300185\pi\)
0.587316 + 0.809358i \(0.300185\pi\)
\(102\) −8.58033e28 −0.643875
\(103\) 6.86763e28 0.447370 0.223685 0.974661i \(-0.428191\pi\)
0.223685 + 0.974661i \(0.428191\pi\)
\(104\) −7.61624e28 −0.431278
\(105\) −2.45889e29 −1.21198
\(106\) −8.83773e28 −0.379669
\(107\) 1.23781e29 0.464075 0.232038 0.972707i \(-0.425461\pi\)
0.232038 + 0.972707i \(0.425461\pi\)
\(108\) −2.93719e28 −0.0962250
\(109\) −1.13178e28 −0.0324396 −0.0162198 0.999868i \(-0.505163\pi\)
−0.0162198 + 0.999868i \(0.505163\pi\)
\(110\) −7.24215e29 −1.81833
\(111\) −2.88961e29 −0.636291
\(112\) −1.81703e29 −0.351319
\(113\) −2.65183e29 −0.450721 −0.225360 0.974275i \(-0.572356\pi\)
−0.225360 + 0.974275i \(0.572356\pi\)
\(114\) −1.12909e29 −0.168891
\(115\) 1.51670e30 1.99885
\(116\) 4.79490e29 0.557364
\(117\) −3.96165e29 −0.406613
\(118\) −1.08041e29 −0.0980169
\(119\) −2.76102e30 −2.21635
\(120\) 4.28861e29 0.304922
\(121\) 3.11454e30 1.96339
\(122\) 1.06819e29 0.0597626
\(123\) 3.61064e29 0.179457
\(124\) 4.93087e29 0.217926
\(125\) −8.78946e29 −0.345754
\(126\) −9.45144e29 −0.331226
\(127\) 2.04844e30 0.640131 0.320066 0.947395i \(-0.396295\pi\)
0.320066 + 0.947395i \(0.396295\pi\)
\(128\) 3.16913e29 0.0883883
\(129\) −2.19878e30 −0.547810
\(130\) 5.78442e30 1.28849
\(131\) 3.72567e30 0.742626 0.371313 0.928508i \(-0.378908\pi\)
0.371313 + 0.928508i \(0.378908\pi\)
\(132\) −2.78372e30 −0.496940
\(133\) −3.63323e30 −0.581358
\(134\) −3.25883e30 −0.467780
\(135\) 2.23076e30 0.287483
\(136\) 4.81555e30 0.557612
\(137\) 1.00289e30 0.104425 0.0522125 0.998636i \(-0.483373\pi\)
0.0522125 + 0.998636i \(0.483373\pi\)
\(138\) 5.82985e30 0.546274
\(139\) −3.72507e30 −0.314354 −0.157177 0.987570i \(-0.550239\pi\)
−0.157177 + 0.987570i \(0.550239\pi\)
\(140\) 1.38001e31 1.04960
\(141\) −3.45678e30 −0.237135
\(142\) −6.97403e30 −0.431820
\(143\) −3.75465e31 −2.09989
\(144\) 1.64845e30 0.0833333
\(145\) −3.64166e31 −1.66519
\(146\) −7.55657e30 −0.312758
\(147\) −1.50125e31 −0.562799
\(148\) 1.62174e31 0.551044
\(149\) 5.39425e31 1.66238 0.831190 0.555989i \(-0.187660\pi\)
0.831190 + 0.555989i \(0.187660\pi\)
\(150\) −1.79749e31 −0.502741
\(151\) 3.97080e31 1.00858 0.504292 0.863533i \(-0.331754\pi\)
0.504292 + 0.863533i \(0.331754\pi\)
\(152\) 6.33679e30 0.146264
\(153\) 2.50485e31 0.525721
\(154\) −8.95758e31 −1.71057
\(155\) −3.74493e31 −0.651079
\(156\) 2.22340e31 0.352137
\(157\) −7.76788e31 −1.12139 −0.560696 0.828022i \(-0.689466\pi\)
−0.560696 + 0.828022i \(0.689466\pi\)
\(158\) 8.60700e31 1.13325
\(159\) 2.57999e31 0.309999
\(160\) −2.40691e31 −0.264070
\(161\) 1.87596e32 1.88039
\(162\) 8.57453e30 0.0785674
\(163\) 8.75557e31 0.733777 0.366888 0.930265i \(-0.380423\pi\)
0.366888 + 0.930265i \(0.380423\pi\)
\(164\) −2.02641e31 −0.155414
\(165\) 2.11420e32 1.48466
\(166\) −6.72199e31 −0.432445
\(167\) 3.82845e31 0.225753 0.112877 0.993609i \(-0.463994\pi\)
0.112877 + 0.993609i \(0.463994\pi\)
\(168\) 5.30445e31 0.286850
\(169\) 9.83515e31 0.488005
\(170\) −3.65734e32 −1.66593
\(171\) 3.29613e31 0.137899
\(172\) 1.23402e32 0.474417
\(173\) −4.47491e32 −1.58167 −0.790835 0.612029i \(-0.790354\pi\)
−0.790835 + 0.612029i \(0.790354\pi\)
\(174\) −1.39977e32 −0.455086
\(175\) −5.78406e32 −1.73054
\(176\) 1.56231e32 0.430362
\(177\) 3.15404e31 0.0800305
\(178\) −8.22085e31 −0.192232
\(179\) 7.12699e32 1.53652 0.768258 0.640141i \(-0.221124\pi\)
0.768258 + 0.640141i \(0.221124\pi\)
\(180\) −1.25197e32 −0.248968
\(181\) −1.03470e33 −1.89879 −0.949394 0.314087i \(-0.898302\pi\)
−0.949394 + 0.314087i \(0.898302\pi\)
\(182\) 7.15457e32 1.21213
\(183\) −3.11835e31 −0.0487959
\(184\) −3.27190e32 −0.473087
\(185\) −1.23169e33 −1.64631
\(186\) −1.43947e32 −0.177936
\(187\) 2.37397e33 2.71501
\(188\) 1.94006e32 0.205365
\(189\) 2.75915e32 0.270445
\(190\) −4.81270e32 −0.436980
\(191\) −6.40588e32 −0.539008 −0.269504 0.962999i \(-0.586860\pi\)
−0.269504 + 0.962999i \(0.586860\pi\)
\(192\) −9.25161e31 −0.0721688
\(193\) 8.50623e32 0.615398 0.307699 0.951484i \(-0.400441\pi\)
0.307699 + 0.951484i \(0.400441\pi\)
\(194\) 1.47919e33 0.992883
\(195\) −1.68864e33 −1.05205
\(196\) 8.42552e32 0.487398
\(197\) 3.60046e32 0.193463 0.0967314 0.995311i \(-0.469161\pi\)
0.0967314 + 0.995311i \(0.469161\pi\)
\(198\) 8.12650e32 0.405750
\(199\) −6.99395e32 −0.324603 −0.162302 0.986741i \(-0.551892\pi\)
−0.162302 + 0.986741i \(0.551892\pi\)
\(200\) 1.00881e33 0.435386
\(201\) 9.51346e32 0.381941
\(202\) 2.22322e33 0.830590
\(203\) −4.50425e33 −1.56650
\(204\) −1.40580e33 −0.455288
\(205\) 1.53903e33 0.464318
\(206\) 1.12519e33 0.316338
\(207\) −1.70190e33 −0.446031
\(208\) −1.24784e33 −0.304959
\(209\) 3.12391e33 0.712159
\(210\) −4.02865e33 −0.856998
\(211\) 9.78000e33 1.94197 0.970987 0.239133i \(-0.0768634\pi\)
0.970987 + 0.239133i \(0.0768634\pi\)
\(212\) −1.44797e33 −0.268467
\(213\) 2.03592e33 0.352580
\(214\) 2.02802e33 0.328151
\(215\) −9.37223e33 −1.41738
\(216\) −4.81230e32 −0.0680414
\(217\) −4.63198e33 −0.612492
\(218\) −1.85430e32 −0.0229383
\(219\) 2.20598e33 0.255366
\(220\) −1.18655e34 −1.28576
\(221\) −1.89613e34 −1.92388
\(222\) −4.73433e33 −0.449925
\(223\) 1.44345e34 1.28523 0.642615 0.766189i \(-0.277849\pi\)
0.642615 + 0.766189i \(0.277849\pi\)
\(224\) −2.97702e33 −0.248420
\(225\) 5.24741e33 0.410486
\(226\) −4.34475e33 −0.318708
\(227\) −4.35500e33 −0.299649 −0.149825 0.988713i \(-0.547871\pi\)
−0.149825 + 0.988713i \(0.547871\pi\)
\(228\) −1.84990e33 −0.119424
\(229\) −3.53428e33 −0.214134 −0.107067 0.994252i \(-0.534146\pi\)
−0.107067 + 0.994252i \(0.534146\pi\)
\(230\) 2.48496e34 1.41340
\(231\) 2.61498e34 1.39667
\(232\) 7.85597e33 0.394116
\(233\) 2.51120e34 1.18364 0.591820 0.806070i \(-0.298410\pi\)
0.591820 + 0.806070i \(0.298410\pi\)
\(234\) −6.49076e33 −0.287519
\(235\) −1.47345e34 −0.613550
\(236\) −1.77015e33 −0.0693084
\(237\) −2.51263e34 −0.925291
\(238\) −4.52365e34 −1.56720
\(239\) 5.36225e34 1.74815 0.874073 0.485794i \(-0.161470\pi\)
0.874073 + 0.485794i \(0.161470\pi\)
\(240\) 7.02646e33 0.215612
\(241\) 5.55366e34 1.60447 0.802236 0.597007i \(-0.203644\pi\)
0.802236 + 0.597007i \(0.203644\pi\)
\(242\) 5.10287e34 1.38833
\(243\) −2.50316e33 −0.0641500
\(244\) 1.75012e33 0.0422585
\(245\) −6.39906e34 −1.45616
\(246\) 5.91568e33 0.126895
\(247\) −2.49511e34 −0.504643
\(248\) 8.07874e33 0.154097
\(249\) 1.96234e34 0.353089
\(250\) −1.44007e34 −0.244485
\(251\) −3.45099e34 −0.552936 −0.276468 0.961023i \(-0.589164\pi\)
−0.276468 + 0.961023i \(0.589164\pi\)
\(252\) −1.54852e34 −0.234212
\(253\) −1.61298e35 −2.30346
\(254\) 3.35616e34 0.452641
\(255\) 1.06769e35 1.36022
\(256\) 5.19230e33 0.0625000
\(257\) −7.35858e34 −0.837074 −0.418537 0.908200i \(-0.637457\pi\)
−0.418537 + 0.908200i \(0.637457\pi\)
\(258\) −3.60247e34 −0.387360
\(259\) −1.52343e35 −1.54874
\(260\) 9.47720e34 0.911101
\(261\) 4.08635e34 0.371576
\(262\) 6.10413e34 0.525116
\(263\) −1.05848e35 −0.861640 −0.430820 0.902438i \(-0.641776\pi\)
−0.430820 + 0.902438i \(0.641776\pi\)
\(264\) −4.56085e34 −0.351389
\(265\) 1.09971e35 0.802075
\(266\) −5.95268e34 −0.411082
\(267\) 2.39991e34 0.156957
\(268\) −5.33926e34 −0.330770
\(269\) −8.17172e34 −0.479629 −0.239815 0.970819i \(-0.577087\pi\)
−0.239815 + 0.970819i \(0.577087\pi\)
\(270\) 3.65487e34 0.203281
\(271\) 2.70296e35 1.42490 0.712450 0.701723i \(-0.247585\pi\)
0.712450 + 0.701723i \(0.247585\pi\)
\(272\) 7.88980e34 0.394291
\(273\) −2.08863e35 −0.989698
\(274\) 1.64313e34 0.0738396
\(275\) 4.97323e35 2.11989
\(276\) 9.55163e34 0.386274
\(277\) −9.35727e34 −0.359081 −0.179541 0.983751i \(-0.557461\pi\)
−0.179541 + 0.983751i \(0.557461\pi\)
\(278\) −6.10315e34 −0.222282
\(279\) 4.20222e34 0.145284
\(280\) 2.26101e35 0.742182
\(281\) −4.90540e35 −1.52909 −0.764544 0.644571i \(-0.777036\pi\)
−0.764544 + 0.644571i \(0.777036\pi\)
\(282\) −5.66359e34 −0.167679
\(283\) 6.11272e35 1.71922 0.859609 0.510953i \(-0.170707\pi\)
0.859609 + 0.510953i \(0.170707\pi\)
\(284\) −1.14263e35 −0.305343
\(285\) 1.40497e35 0.356793
\(286\) −6.15161e35 −1.48485
\(287\) 1.90357e35 0.436799
\(288\) 2.70081e34 0.0589256
\(289\) 7.16903e35 1.48745
\(290\) −5.96650e35 −1.17747
\(291\) −4.31818e35 −0.810686
\(292\) −1.23807e35 −0.221153
\(293\) −4.44214e35 −0.755113 −0.377557 0.925986i \(-0.623236\pi\)
−0.377557 + 0.925986i \(0.623236\pi\)
\(294\) −2.45966e35 −0.397959
\(295\) 1.34440e35 0.207067
\(296\) 2.65706e35 0.389647
\(297\) −2.37236e35 −0.331293
\(298\) 8.83794e35 1.17548
\(299\) 1.28831e36 1.63226
\(300\) −2.94501e35 −0.355491
\(301\) −1.15922e36 −1.33337
\(302\) 6.50575e35 0.713176
\(303\) −6.49022e35 −0.678174
\(304\) 1.03822e35 0.103424
\(305\) −1.32919e35 −0.126252
\(306\) 4.10394e35 0.371741
\(307\) −1.08810e36 −0.940078 −0.470039 0.882646i \(-0.655760\pi\)
−0.470039 + 0.882646i \(0.655760\pi\)
\(308\) −1.46761e36 −1.20955
\(309\) −3.28476e35 −0.258289
\(310\) −6.13569e35 −0.460382
\(311\) −1.31602e36 −0.942402 −0.471201 0.882026i \(-0.656179\pi\)
−0.471201 + 0.882026i \(0.656179\pi\)
\(312\) 3.64282e35 0.248998
\(313\) −9.80838e35 −0.640036 −0.320018 0.947411i \(-0.603689\pi\)
−0.320018 + 0.947411i \(0.603689\pi\)
\(314\) −1.27269e36 −0.792944
\(315\) 1.17608e36 0.699736
\(316\) 1.41017e36 0.801325
\(317\) 2.03796e36 1.10621 0.553104 0.833112i \(-0.313443\pi\)
0.553104 + 0.833112i \(0.313443\pi\)
\(318\) 4.22706e35 0.219202
\(319\) 3.87283e36 1.91895
\(320\) −3.94347e35 −0.186726
\(321\) −5.92040e35 −0.267934
\(322\) 3.07357e36 1.32963
\(323\) 1.57760e36 0.652468
\(324\) 1.40485e35 0.0555556
\(325\) −3.97220e36 −1.50218
\(326\) 1.43451e36 0.518859
\(327\) 5.41325e34 0.0187290
\(328\) −3.32007e35 −0.109894
\(329\) −1.82246e36 −0.577187
\(330\) 3.46390e36 1.04982
\(331\) 1.12914e35 0.0327522 0.0163761 0.999866i \(-0.494787\pi\)
0.0163761 + 0.999866i \(0.494787\pi\)
\(332\) −1.10133e36 −0.305784
\(333\) 1.38209e36 0.367363
\(334\) 6.27253e35 0.159632
\(335\) 4.05509e36 0.988214
\(336\) 8.69081e35 0.202834
\(337\) −8.08762e36 −1.80795 −0.903977 0.427581i \(-0.859366\pi\)
−0.903977 + 0.427581i \(0.859366\pi\)
\(338\) 1.61139e36 0.345071
\(339\) 1.26836e36 0.260224
\(340\) −5.99219e36 −1.17799
\(341\) 3.98265e36 0.750298
\(342\) 5.40039e35 0.0975093
\(343\) 2.04643e35 0.0354186
\(344\) 2.02182e36 0.335464
\(345\) −7.25432e36 −1.15404
\(346\) −7.33170e36 −1.11841
\(347\) −5.00016e36 −0.731487 −0.365744 0.930716i \(-0.619185\pi\)
−0.365744 + 0.930716i \(0.619185\pi\)
\(348\) −2.29339e36 −0.321794
\(349\) −7.92228e36 −1.06630 −0.533152 0.846019i \(-0.678993\pi\)
−0.533152 + 0.846019i \(0.678993\pi\)
\(350\) −9.47660e36 −1.22367
\(351\) 1.89484e36 0.234758
\(352\) 2.55969e36 0.304312
\(353\) 7.39537e35 0.0843775 0.0421888 0.999110i \(-0.486567\pi\)
0.0421888 + 0.999110i \(0.486567\pi\)
\(354\) 5.16759e35 0.0565901
\(355\) 8.67807e36 0.912247
\(356\) −1.34690e36 −0.135929
\(357\) 1.32059e37 1.27961
\(358\) 1.16769e37 1.08648
\(359\) 1.81161e37 1.61881 0.809403 0.587254i \(-0.199791\pi\)
0.809403 + 0.587254i \(0.199791\pi\)
\(360\) −2.05123e36 −0.176047
\(361\) −1.00539e37 −0.828855
\(362\) −1.69526e37 −1.34265
\(363\) −1.48968e37 −1.13356
\(364\) 1.17220e37 0.857103
\(365\) 9.40295e36 0.660720
\(366\) −5.10910e35 −0.0345039
\(367\) 1.51023e37 0.980362 0.490181 0.871621i \(-0.336931\pi\)
0.490181 + 0.871621i \(0.336931\pi\)
\(368\) −5.36068e36 −0.334523
\(369\) −1.72696e36 −0.103610
\(370\) −2.01800e37 −1.16411
\(371\) 1.36020e37 0.754539
\(372\) −2.35842e36 −0.125820
\(373\) 1.54624e37 0.793414 0.396707 0.917945i \(-0.370153\pi\)
0.396707 + 0.917945i \(0.370153\pi\)
\(374\) 3.88951e37 1.91980
\(375\) 4.20397e36 0.199621
\(376\) 3.17859e36 0.145215
\(377\) −3.09329e37 −1.35979
\(378\) 4.52059e36 0.191234
\(379\) −1.50980e37 −0.614684 −0.307342 0.951599i \(-0.599440\pi\)
−0.307342 + 0.951599i \(0.599440\pi\)
\(380\) −7.88513e36 −0.308992
\(381\) −9.79762e36 −0.369580
\(382\) −1.04954e37 −0.381136
\(383\) −2.64946e37 −0.926351 −0.463175 0.886267i \(-0.653290\pi\)
−0.463175 + 0.886267i \(0.653290\pi\)
\(384\) −1.51578e36 −0.0510310
\(385\) 1.11463e38 3.61368
\(386\) 1.39366e37 0.435152
\(387\) 1.05167e37 0.316278
\(388\) 2.42350e37 0.702075
\(389\) −5.09411e37 −1.42167 −0.710836 0.703358i \(-0.751683\pi\)
−0.710836 + 0.703358i \(0.751683\pi\)
\(390\) −2.76667e37 −0.743911
\(391\) −8.14566e37 −2.11039
\(392\) 1.38044e37 0.344642
\(393\) −1.78198e37 −0.428755
\(394\) 5.89899e36 0.136799
\(395\) −1.07100e38 −2.39405
\(396\) 1.33145e37 0.286908
\(397\) 6.75019e37 1.40234 0.701171 0.712994i \(-0.252661\pi\)
0.701171 + 0.712994i \(0.252661\pi\)
\(398\) −1.14589e37 −0.229529
\(399\) 1.73776e37 0.335647
\(400\) 1.65284e37 0.307865
\(401\) −1.90835e37 −0.342819 −0.171410 0.985200i \(-0.554832\pi\)
−0.171410 + 0.985200i \(0.554832\pi\)
\(402\) 1.55869e37 0.270073
\(403\) −3.18101e37 −0.531669
\(404\) 3.64252e37 0.587316
\(405\) −1.06696e37 −0.165979
\(406\) −7.37977e37 −1.10768
\(407\) 1.30987e38 1.89719
\(408\) −2.30326e37 −0.321937
\(409\) −9.02898e37 −1.21801 −0.609005 0.793167i \(-0.708431\pi\)
−0.609005 + 0.793167i \(0.708431\pi\)
\(410\) 2.52154e37 0.328322
\(411\) −4.79679e36 −0.0602898
\(412\) 1.84351e37 0.223685
\(413\) 1.66285e37 0.194795
\(414\) −2.78840e37 −0.315391
\(415\) 8.36445e37 0.913566
\(416\) −2.04447e37 −0.215639
\(417\) 1.78169e37 0.181493
\(418\) 5.11821e37 0.503572
\(419\) −1.07363e38 −1.02036 −0.510179 0.860068i \(-0.670421\pi\)
−0.510179 + 0.860068i \(0.670421\pi\)
\(420\) −6.60054e37 −0.605989
\(421\) −1.26637e38 −1.12323 −0.561616 0.827398i \(-0.689820\pi\)
−0.561616 + 0.827398i \(0.689820\pi\)
\(422\) 1.60236e38 1.37318
\(423\) 1.65337e37 0.136910
\(424\) −2.37236e37 −0.189835
\(425\) 2.51152e38 1.94221
\(426\) 3.33566e37 0.249311
\(427\) −1.64403e37 −0.118770
\(428\) 3.32272e37 0.232038
\(429\) 1.79584e38 1.21237
\(430\) −1.53555e38 −1.00224
\(431\) 6.83365e37 0.431253 0.215627 0.976476i \(-0.430821\pi\)
0.215627 + 0.976476i \(0.430821\pi\)
\(432\) −7.88447e36 −0.0481125
\(433\) 1.64537e38 0.970933 0.485467 0.874255i \(-0.338650\pi\)
0.485467 + 0.874255i \(0.338650\pi\)
\(434\) −7.58903e37 −0.433097
\(435\) 1.74179e38 0.961397
\(436\) −3.03809e36 −0.0162198
\(437\) −1.07189e38 −0.553565
\(438\) 3.61428e37 0.180571
\(439\) −6.20787e37 −0.300059 −0.150030 0.988681i \(-0.547937\pi\)
−0.150030 + 0.988681i \(0.547937\pi\)
\(440\) −1.94405e38 −0.909167
\(441\) 7.18046e37 0.324932
\(442\) −3.10661e38 −1.36039
\(443\) −2.81830e38 −1.19435 −0.597177 0.802110i \(-0.703711\pi\)
−0.597177 + 0.802110i \(0.703711\pi\)
\(444\) −7.75673e37 −0.318145
\(445\) 1.02295e38 0.406103
\(446\) 2.36495e38 0.908796
\(447\) −2.58005e38 −0.959775
\(448\) −4.87756e37 −0.175659
\(449\) 4.33351e38 1.51101 0.755506 0.655141i \(-0.227391\pi\)
0.755506 + 0.655141i \(0.227391\pi\)
\(450\) 8.59736e37 0.290258
\(451\) −1.63672e38 −0.535076
\(452\) −7.11844e37 −0.225360
\(453\) −1.89922e38 −0.582306
\(454\) −7.13523e37 −0.211884
\(455\) −8.90272e38 −2.56069
\(456\) −3.03087e37 −0.0844456
\(457\) −2.39766e38 −0.647147 −0.323574 0.946203i \(-0.604884\pi\)
−0.323574 + 0.946203i \(0.604884\pi\)
\(458\) −5.79056e37 −0.151416
\(459\) −1.19806e38 −0.303525
\(460\) 4.07136e38 0.999426
\(461\) −1.00404e38 −0.238828 −0.119414 0.992845i \(-0.538102\pi\)
−0.119414 + 0.992845i \(0.538102\pi\)
\(462\) 4.28439e38 0.987597
\(463\) −2.18695e38 −0.488556 −0.244278 0.969705i \(-0.578551\pi\)
−0.244278 + 0.969705i \(0.578551\pi\)
\(464\) 1.28712e38 0.278682
\(465\) 1.79119e38 0.375901
\(466\) 4.11434e38 0.836960
\(467\) 7.59341e38 1.49741 0.748707 0.662901i \(-0.230675\pi\)
0.748707 + 0.662901i \(0.230675\pi\)
\(468\) −1.06345e38 −0.203306
\(469\) 5.01561e38 0.929646
\(470\) −2.41409e38 −0.433845
\(471\) 3.71535e38 0.647436
\(472\) −2.90021e37 −0.0490085
\(473\) 9.96716e38 1.63337
\(474\) −4.11670e38 −0.654279
\(475\) 3.30491e38 0.509451
\(476\) −7.41155e38 −1.10817
\(477\) −1.23400e38 −0.178978
\(478\) 8.78552e38 1.23613
\(479\) −9.71497e37 −0.132610 −0.0663050 0.997799i \(-0.521121\pi\)
−0.0663050 + 0.997799i \(0.521121\pi\)
\(480\) 1.15122e38 0.152461
\(481\) −1.04622e39 −1.34437
\(482\) 9.09911e38 1.13453
\(483\) −8.97264e38 −1.08564
\(484\) 8.36054e38 0.981695
\(485\) −1.84061e39 −2.09753
\(486\) −4.10117e37 −0.0453609
\(487\) 9.86090e38 1.05864 0.529318 0.848423i \(-0.322448\pi\)
0.529318 + 0.848423i \(0.322448\pi\)
\(488\) 2.86739e37 0.0298813
\(489\) −4.18776e38 −0.423646
\(490\) −1.04842e39 −1.02966
\(491\) 2.46794e38 0.235317 0.117658 0.993054i \(-0.462461\pi\)
0.117658 + 0.993054i \(0.462461\pi\)
\(492\) 9.69224e37 0.0897285
\(493\) 1.95581e39 1.75811
\(494\) −4.08800e38 −0.356837
\(495\) −1.01121e39 −0.857171
\(496\) 1.32362e38 0.108963
\(497\) 1.07336e39 0.858181
\(498\) 3.21511e38 0.249672
\(499\) −1.20983e39 −0.912567 −0.456283 0.889835i \(-0.650820\pi\)
−0.456283 + 0.889835i \(0.650820\pi\)
\(500\) −2.35940e38 −0.172877
\(501\) −1.83114e38 −0.130339
\(502\) −5.65410e38 −0.390985
\(503\) 1.96508e39 1.32021 0.660107 0.751171i \(-0.270511\pi\)
0.660107 + 0.751171i \(0.270511\pi\)
\(504\) −2.53710e38 −0.165613
\(505\) −2.76644e39 −1.75467
\(506\) −2.64270e39 −1.62879
\(507\) −4.70412e38 −0.281750
\(508\) 5.49874e38 0.320066
\(509\) 4.54640e38 0.257194 0.128597 0.991697i \(-0.458953\pi\)
0.128597 + 0.991697i \(0.458953\pi\)
\(510\) 1.74930e39 0.961824
\(511\) 1.16302e39 0.621562
\(512\) 8.50706e37 0.0441942
\(513\) −1.57653e38 −0.0796160
\(514\) −1.20563e39 −0.591901
\(515\) −1.40012e39 −0.668284
\(516\) −5.90229e38 −0.273905
\(517\) 1.56698e39 0.707050
\(518\) −2.49600e39 −1.09512
\(519\) 2.14034e39 0.913178
\(520\) 1.55274e39 0.644246
\(521\) −2.06563e39 −0.833500 −0.416750 0.909021i \(-0.636831\pi\)
−0.416750 + 0.909021i \(0.636831\pi\)
\(522\) 6.69507e38 0.262744
\(523\) −9.08640e38 −0.346830 −0.173415 0.984849i \(-0.555480\pi\)
−0.173415 + 0.984849i \(0.555480\pi\)
\(524\) 1.00010e39 0.371313
\(525\) 2.76650e39 0.999126
\(526\) −1.73422e39 −0.609271
\(527\) 2.01127e39 0.687411
\(528\) −7.47249e38 −0.248470
\(529\) 2.44346e39 0.790492
\(530\) 1.80177e39 0.567153
\(531\) −1.50857e38 −0.0462056
\(532\) −9.75287e38 −0.290679
\(533\) 1.30728e39 0.379160
\(534\) 3.93201e38 0.110985
\(535\) −2.52356e39 −0.693239
\(536\) −8.74784e38 −0.233890
\(537\) −3.40882e39 −0.887108
\(538\) −1.33886e39 −0.339149
\(539\) 6.80527e39 1.67806
\(540\) 5.98814e38 0.143742
\(541\) −6.44804e39 −1.50684 −0.753421 0.657539i \(-0.771598\pi\)
−0.753421 + 0.657539i \(0.771598\pi\)
\(542\) 4.42853e39 1.00756
\(543\) 4.94895e39 1.09627
\(544\) 1.29267e39 0.278806
\(545\) 2.30738e38 0.0484585
\(546\) −3.42201e39 −0.699822
\(547\) 5.73505e39 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(548\) 2.69211e38 0.0522125
\(549\) 1.49150e38 0.0281723
\(550\) 8.14813e39 1.49899
\(551\) 2.57365e39 0.461160
\(552\) 1.56494e39 0.273137
\(553\) −1.32469e40 −2.25216
\(554\) −1.53310e39 −0.253909
\(555\) 5.89112e39 0.950495
\(556\) −9.99940e38 −0.157177
\(557\) −1.13157e40 −1.73293 −0.866466 0.499236i \(-0.833614\pi\)
−0.866466 + 0.499236i \(0.833614\pi\)
\(558\) 6.88492e38 0.102731
\(559\) −7.96093e39 −1.15742
\(560\) 3.70443e39 0.524802
\(561\) −1.13546e40 −1.56751
\(562\) −8.03700e39 −1.08123
\(563\) −7.41757e39 −0.972500 −0.486250 0.873820i \(-0.661636\pi\)
−0.486250 + 0.873820i \(0.661636\pi\)
\(564\) −9.27923e38 −0.118567
\(565\) 5.40636e39 0.673290
\(566\) 1.00151e40 1.21567
\(567\) −1.31969e39 −0.156142
\(568\) −1.87208e39 −0.215910
\(569\) −4.64228e38 −0.0521920 −0.0260960 0.999659i \(-0.508308\pi\)
−0.0260960 + 0.999659i \(0.508308\pi\)
\(570\) 2.30190e39 0.252291
\(571\) 1.12973e40 1.20712 0.603559 0.797318i \(-0.293749\pi\)
0.603559 + 0.797318i \(0.293749\pi\)
\(572\) −1.00788e40 −1.04994
\(573\) 3.06391e39 0.311196
\(574\) 3.11881e39 0.308864
\(575\) −1.70643e40 −1.64781
\(576\) 4.42502e38 0.0416667
\(577\) 1.57562e40 1.44678 0.723388 0.690441i \(-0.242584\pi\)
0.723388 + 0.690441i \(0.242584\pi\)
\(578\) 1.17457e40 1.05178
\(579\) −4.06850e39 −0.355300
\(580\) −9.77551e39 −0.832594
\(581\) 1.03457e40 0.859422
\(582\) −7.07491e39 −0.573241
\(583\) −1.16952e40 −0.924304
\(584\) −2.02845e39 −0.156379
\(585\) 8.07673e39 0.607401
\(586\) −7.27801e39 −0.533946
\(587\) 1.71498e39 0.122745 0.0613727 0.998115i \(-0.480452\pi\)
0.0613727 + 0.998115i \(0.480452\pi\)
\(588\) −4.02990e39 −0.281399
\(589\) 2.64663e39 0.180311
\(590\) 2.20267e39 0.146418
\(591\) −1.72209e39 −0.111696
\(592\) 4.35332e39 0.275522
\(593\) 2.06799e39 0.127719 0.0638595 0.997959i \(-0.479659\pi\)
0.0638595 + 0.997959i \(0.479659\pi\)
\(594\) −3.88688e39 −0.234260
\(595\) 5.62896e40 3.31080
\(596\) 1.44801e40 0.831190
\(597\) 3.34518e39 0.187410
\(598\) 2.11077e40 1.15418
\(599\) 1.00136e40 0.534444 0.267222 0.963635i \(-0.413894\pi\)
0.267222 + 0.963635i \(0.413894\pi\)
\(600\) −4.82511e39 −0.251370
\(601\) 2.26586e40 1.15227 0.576134 0.817355i \(-0.304560\pi\)
0.576134 + 0.817355i \(0.304560\pi\)
\(602\) −1.89927e40 −0.942837
\(603\) −4.55026e39 −0.220514
\(604\) 1.06590e40 0.504292
\(605\) −6.34971e40 −2.93292
\(606\) −1.06336e40 −0.479541
\(607\) −4.06471e39 −0.178975 −0.0894877 0.995988i \(-0.528523\pi\)
−0.0894877 + 0.995988i \(0.528523\pi\)
\(608\) 1.70102e39 0.0731320
\(609\) 2.15437e40 0.904418
\(610\) −2.17774e39 −0.0892737
\(611\) −1.25157e40 −0.501023
\(612\) 6.72390e39 0.262861
\(613\) −2.69964e40 −1.03069 −0.515345 0.856983i \(-0.672336\pi\)
−0.515345 + 0.856983i \(0.672336\pi\)
\(614\) −1.78275e40 −0.664736
\(615\) −7.36112e39 −0.268074
\(616\) −2.40453e40 −0.855284
\(617\) 4.15838e40 1.44474 0.722369 0.691508i \(-0.243053\pi\)
0.722369 + 0.691508i \(0.243053\pi\)
\(618\) −5.38176e39 −0.182638
\(619\) −3.52546e40 −1.16870 −0.584348 0.811503i \(-0.698650\pi\)
−0.584348 + 0.811503i \(0.698650\pi\)
\(620\) −1.00527e40 −0.325539
\(621\) 8.14015e39 0.257516
\(622\) −2.15616e40 −0.666379
\(623\) 1.26526e40 0.382035
\(624\) 5.96840e39 0.176068
\(625\) −2.48055e40 −0.714969
\(626\) −1.60700e40 −0.452574
\(627\) −1.49415e40 −0.411165
\(628\) −2.08517e40 −0.560696
\(629\) 6.61496e40 1.73817
\(630\) 1.92689e40 0.494788
\(631\) 3.55952e40 0.893234 0.446617 0.894725i \(-0.352629\pi\)
0.446617 + 0.894725i \(0.352629\pi\)
\(632\) 2.31042e40 0.566623
\(633\) −4.67774e40 −1.12120
\(634\) 3.33900e40 0.782207
\(635\) −4.17621e40 −0.956232
\(636\) 6.92561e39 0.154999
\(637\) −5.43548e40 −1.18909
\(638\) 6.34524e40 1.35690
\(639\) −9.73776e39 −0.203562
\(640\) −6.46099e39 −0.132035
\(641\) −1.32448e40 −0.264608 −0.132304 0.991209i \(-0.542237\pi\)
−0.132304 + 0.991209i \(0.542237\pi\)
\(642\) −9.69998e39 −0.189458
\(643\) −1.90296e39 −0.0363389 −0.0181694 0.999835i \(-0.505784\pi\)
−0.0181694 + 0.999835i \(0.505784\pi\)
\(644\) 5.03573e40 0.940194
\(645\) 4.48271e40 0.818322
\(646\) 2.58473e40 0.461365
\(647\) −1.09747e40 −0.191549 −0.0957743 0.995403i \(-0.530533\pi\)
−0.0957743 + 0.995403i \(0.530533\pi\)
\(648\) 2.30171e39 0.0392837
\(649\) −1.42975e40 −0.238622
\(650\) −6.50804e40 −1.06220
\(651\) 2.21546e40 0.353622
\(652\) 2.35030e40 0.366888
\(653\) 3.22859e40 0.492914 0.246457 0.969154i \(-0.420734\pi\)
0.246457 + 0.969154i \(0.420734\pi\)
\(654\) 8.86906e38 0.0132434
\(655\) −7.59563e40 −1.10934
\(656\) −5.43959e39 −0.0777071
\(657\) −1.05511e40 −0.147435
\(658\) −2.98591e40 −0.408133
\(659\) 8.02727e39 0.107332 0.0536660 0.998559i \(-0.482909\pi\)
0.0536660 + 0.998559i \(0.482909\pi\)
\(660\) 5.67525e40 0.742332
\(661\) 5.89261e40 0.754026 0.377013 0.926208i \(-0.376951\pi\)
0.377013 + 0.926208i \(0.376951\pi\)
\(662\) 1.84998e39 0.0231593
\(663\) 9.06911e40 1.11076
\(664\) −1.80442e40 −0.216222
\(665\) 7.40716e40 0.868436
\(666\) 2.26442e40 0.259765
\(667\) −1.32886e41 −1.49161
\(668\) 1.02769e40 0.112877
\(669\) −6.90398e40 −0.742028
\(670\) 6.64386e40 0.698773
\(671\) 1.41356e40 0.145492
\(672\) 1.42390e40 0.143425
\(673\) 1.83445e41 1.80838 0.904188 0.427135i \(-0.140477\pi\)
0.904188 + 0.427135i \(0.140477\pi\)
\(674\) −1.32508e41 −1.27842
\(675\) −2.50982e40 −0.236994
\(676\) 2.64010e40 0.244002
\(677\) 1.27839e41 1.15645 0.578227 0.815876i \(-0.303745\pi\)
0.578227 + 0.815876i \(0.303745\pi\)
\(678\) 2.07808e40 0.184006
\(679\) −2.27660e41 −1.97321
\(680\) −9.81761e40 −0.832964
\(681\) 2.08298e40 0.173003
\(682\) 6.52517e40 0.530541
\(683\) −1.25940e41 −1.00245 −0.501227 0.865316i \(-0.667118\pi\)
−0.501227 + 0.865316i \(0.667118\pi\)
\(684\) 8.84799e39 0.0689495
\(685\) −2.04462e40 −0.155991
\(686\) 3.35287e39 0.0250447
\(687\) 1.69043e40 0.123630
\(688\) 3.31255e40 0.237209
\(689\) 9.34117e40 0.654972
\(690\) −1.18855e41 −0.816028
\(691\) 1.19626e41 0.804254 0.402127 0.915584i \(-0.368271\pi\)
0.402127 + 0.915584i \(0.368271\pi\)
\(692\) −1.20123e41 −0.790835
\(693\) −1.25074e41 −0.806370
\(694\) −8.19227e40 −0.517240
\(695\) 7.59440e40 0.469585
\(696\) −3.75749e40 −0.227543
\(697\) −8.26558e40 −0.490228
\(698\) −1.29799e41 −0.753991
\(699\) −1.20110e41 −0.683375
\(700\) −1.55265e41 −0.865268
\(701\) 4.85810e39 0.0265189 0.0132594 0.999912i \(-0.495779\pi\)
0.0132594 + 0.999912i \(0.495779\pi\)
\(702\) 3.10451e40 0.165999
\(703\) 8.70464e40 0.455931
\(704\) 4.19380e40 0.215181
\(705\) 7.04744e40 0.354233
\(706\) 1.21166e40 0.0596639
\(707\) −3.42172e41 −1.65068
\(708\) 8.46657e39 0.0400152
\(709\) 2.51260e41 1.16346 0.581732 0.813381i \(-0.302375\pi\)
0.581732 + 0.813381i \(0.302375\pi\)
\(710\) 1.42182e41 0.645056
\(711\) 1.20179e41 0.534217
\(712\) −2.20677e40 −0.0961162
\(713\) −1.36654e41 −0.583211
\(714\) 2.16365e41 0.904821
\(715\) 7.65471e41 3.13683
\(716\) 1.91314e41 0.768258
\(717\) −2.56475e41 −1.00929
\(718\) 2.96814e41 1.14467
\(719\) 4.13486e40 0.156276 0.0781379 0.996943i \(-0.475103\pi\)
0.0781379 + 0.996943i \(0.475103\pi\)
\(720\) −3.36074e40 −0.124484
\(721\) −1.73177e41 −0.628678
\(722\) −1.64722e41 −0.586089
\(723\) −2.65630e41 −0.926342
\(724\) −2.77751e41 −0.949394
\(725\) 4.09722e41 1.37274
\(726\) −2.44069e41 −0.801550
\(727\) −4.16011e41 −1.33923 −0.669617 0.742707i \(-0.733542\pi\)
−0.669617 + 0.742707i \(0.733542\pi\)
\(728\) 1.92054e41 0.606064
\(729\) 1.19725e40 0.0370370
\(730\) 1.54058e41 0.467200
\(731\) 5.03349e41 1.49647
\(732\) −8.37075e39 −0.0243980
\(733\) 3.88242e41 1.10942 0.554709 0.832045i \(-0.312830\pi\)
0.554709 + 0.832045i \(0.312830\pi\)
\(734\) 2.47437e41 0.693220
\(735\) 3.06065e41 0.840712
\(736\) −8.78293e40 −0.236544
\(737\) −4.31250e41 −1.13881
\(738\) −2.82945e40 −0.0732630
\(739\) 1.83417e38 0.000465689 0 0.000232845 1.00000i \(-0.499926\pi\)
0.000232845 1.00000i \(0.499926\pi\)
\(740\) −3.30629e41 −0.823153
\(741\) 1.19341e41 0.291356
\(742\) 2.22855e41 0.533540
\(743\) −1.07841e41 −0.253189 −0.126595 0.991955i \(-0.540405\pi\)
−0.126595 + 0.991955i \(0.540405\pi\)
\(744\) −3.86404e40 −0.0889680
\(745\) −1.09974e42 −2.48327
\(746\) 2.53337e41 0.561029
\(747\) −9.38583e40 −0.203856
\(748\) 6.37257e41 1.35750
\(749\) −3.12130e41 −0.652153
\(750\) 6.88779e40 0.141153
\(751\) −7.12460e40 −0.143213 −0.0716064 0.997433i \(-0.522813\pi\)
−0.0716064 + 0.997433i \(0.522813\pi\)
\(752\) 5.20780e40 0.102682
\(753\) 1.65060e41 0.319238
\(754\) −5.06805e41 −0.961515
\(755\) −8.09538e41 −1.50663
\(756\) 7.40654e40 0.135223
\(757\) −5.86689e41 −1.05079 −0.525397 0.850857i \(-0.676083\pi\)
−0.525397 + 0.850857i \(0.676083\pi\)
\(758\) −2.47366e41 −0.434647
\(759\) 7.71482e41 1.32990
\(760\) −1.29190e41 −0.218490
\(761\) −9.15709e41 −1.51943 −0.759713 0.650258i \(-0.774661\pi\)
−0.759713 + 0.650258i \(0.774661\pi\)
\(762\) −1.60524e41 −0.261333
\(763\) 2.85393e40 0.0455866
\(764\) −1.71957e41 −0.269504
\(765\) −5.10671e41 −0.785326
\(766\) −4.34088e41 −0.655029
\(767\) 1.14196e41 0.169090
\(768\) −2.48346e40 −0.0360844
\(769\) 9.68491e41 1.38091 0.690453 0.723378i \(-0.257411\pi\)
0.690453 + 0.723378i \(0.257411\pi\)
\(770\) 1.82621e42 2.55526
\(771\) 3.51959e41 0.483285
\(772\) 2.28337e41 0.307699
\(773\) 6.46643e40 0.0855188 0.0427594 0.999085i \(-0.486385\pi\)
0.0427594 + 0.999085i \(0.486385\pi\)
\(774\) 1.72305e41 0.223643
\(775\) 4.21341e41 0.536734
\(776\) 3.97066e41 0.496442
\(777\) 7.28654e41 0.894163
\(778\) −8.34619e41 −1.00527
\(779\) −1.08767e41 −0.128589
\(780\) −4.53292e41 −0.526024
\(781\) −9.22895e41 −1.05127
\(782\) −1.33458e42 −1.49227
\(783\) −1.95449e41 −0.214530
\(784\) 2.26171e41 0.243699
\(785\) 1.58366e42 1.67514
\(786\) −2.91959e41 −0.303176
\(787\) 1.04389e42 1.06419 0.532097 0.846684i \(-0.321404\pi\)
0.532097 + 0.846684i \(0.321404\pi\)
\(788\) 9.66491e40 0.0967314
\(789\) 5.06269e41 0.497468
\(790\) −1.75473e42 −1.69285
\(791\) 6.68695e41 0.633387
\(792\) 2.18144e41 0.202875
\(793\) −1.12904e41 −0.103097
\(794\) 1.10595e42 0.991605
\(795\) −5.25990e41 −0.463078
\(796\) −1.87742e41 −0.162302
\(797\) 2.34985e42 1.99477 0.997387 0.0722472i \(-0.0230170\pi\)
0.997387 + 0.0722472i \(0.0230170\pi\)
\(798\) 2.84715e41 0.237338
\(799\) 7.91336e41 0.647787
\(800\) 2.70801e41 0.217693
\(801\) −1.14787e41 −0.0906192
\(802\) −3.12665e41 −0.242410
\(803\) −9.99984e41 −0.761408
\(804\) 2.55375e41 0.190970
\(805\) −3.82456e42 −2.80894
\(806\) −5.21176e41 −0.375947
\(807\) 3.90851e41 0.276914
\(808\) 5.96790e41 0.415295
\(809\) 9.95001e40 0.0680096 0.0340048 0.999422i \(-0.489174\pi\)
0.0340048 + 0.999422i \(0.489174\pi\)
\(810\) −1.74811e41 −0.117365
\(811\) 2.99801e42 1.97711 0.988554 0.150869i \(-0.0482073\pi\)
0.988554 + 0.150869i \(0.0482073\pi\)
\(812\) −1.20910e42 −0.783249
\(813\) −1.29282e42 −0.822667
\(814\) 2.14610e42 1.34152
\(815\) −1.78502e42 −1.09612
\(816\) −3.77367e41 −0.227644
\(817\) 6.62358e41 0.392530
\(818\) −1.47931e42 −0.861263
\(819\) 9.98984e41 0.571402
\(820\) 4.13129e41 0.232159
\(821\) −2.38487e42 −1.31671 −0.658353 0.752709i \(-0.728747\pi\)
−0.658353 + 0.752709i \(0.728747\pi\)
\(822\) −7.85905e40 −0.0426313
\(823\) 1.50489e42 0.802062 0.401031 0.916065i \(-0.368652\pi\)
0.401031 + 0.916065i \(0.368652\pi\)
\(824\) 3.02041e41 0.158169
\(825\) −2.37868e42 −1.22392
\(826\) 2.72441e41 0.137741
\(827\) −2.46131e42 −1.22275 −0.611374 0.791342i \(-0.709383\pi\)
−0.611374 + 0.791342i \(0.709383\pi\)
\(828\) −4.56851e41 −0.223015
\(829\) −5.65741e40 −0.0271379 −0.0135690 0.999908i \(-0.504319\pi\)
−0.0135690 + 0.999908i \(0.504319\pi\)
\(830\) 1.37043e42 0.645989
\(831\) 4.47555e41 0.207316
\(832\) −3.34966e41 −0.152480
\(833\) 3.43671e42 1.53741
\(834\) 2.91912e41 0.128335
\(835\) −7.80517e41 −0.337232
\(836\) 8.38567e41 0.356079
\(837\) −2.00991e41 −0.0838798
\(838\) −1.75904e42 −0.721502
\(839\) 1.68024e42 0.677365 0.338682 0.940901i \(-0.390019\pi\)
0.338682 + 0.940901i \(0.390019\pi\)
\(840\) −1.08143e42 −0.428499
\(841\) 6.22969e41 0.242619
\(842\) −2.07482e42 −0.794245
\(843\) 2.34624e42 0.882820
\(844\) 2.62530e42 0.970987
\(845\) −2.00512e42 −0.728985
\(846\) 2.70888e41 0.0968098
\(847\) −7.85375e42 −2.75910
\(848\) −3.88687e41 −0.134233
\(849\) −2.92369e42 −0.992591
\(850\) 4.11487e42 1.37335
\(851\) −4.49450e42 −1.47470
\(852\) 5.46514e41 0.176290
\(853\) 2.99042e42 0.948355 0.474178 0.880429i \(-0.342745\pi\)
0.474178 + 0.880429i \(0.342745\pi\)
\(854\) −2.69358e41 −0.0839828
\(855\) −6.71992e41 −0.205995
\(856\) 5.44394e41 0.164075
\(857\) 4.29729e41 0.127342 0.0636712 0.997971i \(-0.479719\pi\)
0.0636712 + 0.997971i \(0.479719\pi\)
\(858\) 2.94230e42 0.857276
\(859\) 2.06742e42 0.592280 0.296140 0.955145i \(-0.404301\pi\)
0.296140 + 0.955145i \(0.404301\pi\)
\(860\) −2.51584e42 −0.708688
\(861\) −9.10473e41 −0.252186
\(862\) 1.11963e42 0.304942
\(863\) −7.05431e40 −0.0188929 −0.00944644 0.999955i \(-0.503007\pi\)
−0.00944644 + 0.999955i \(0.503007\pi\)
\(864\) −1.29179e41 −0.0340207
\(865\) 9.12313e42 2.36271
\(866\) 2.69578e42 0.686554
\(867\) −3.42892e42 −0.858778
\(868\) −1.24339e42 −0.306246
\(869\) 1.13899e43 2.75888
\(870\) 2.85376e42 0.679810
\(871\) 3.44447e42 0.806972
\(872\) −4.97760e40 −0.0114691
\(873\) 2.06537e42 0.468050
\(874\) −1.75618e42 −0.391430
\(875\) 2.21638e42 0.485879
\(876\) 5.92164e41 0.127683
\(877\) −4.82633e42 −1.02358 −0.511791 0.859110i \(-0.671018\pi\)
−0.511791 + 0.859110i \(0.671018\pi\)
\(878\) −1.01710e42 −0.212174
\(879\) 2.12466e42 0.435965
\(880\) −3.18513e42 −0.642878
\(881\) −5.13974e42 −1.02045 −0.510224 0.860042i \(-0.670437\pi\)
−0.510224 + 0.860042i \(0.670437\pi\)
\(882\) 1.17645e42 0.229762
\(883\) −2.99814e42 −0.575999 −0.287999 0.957631i \(-0.592990\pi\)
−0.287999 + 0.957631i \(0.592990\pi\)
\(884\) −5.08987e42 −0.961942
\(885\) −6.43024e41 −0.119550
\(886\) −4.61750e42 −0.844535
\(887\) 2.70966e42 0.487553 0.243777 0.969831i \(-0.421614\pi\)
0.243777 + 0.969831i \(0.421614\pi\)
\(888\) −1.27086e42 −0.224963
\(889\) −5.16542e42 −0.899560
\(890\) 1.67601e42 0.287158
\(891\) 1.13469e42 0.191272
\(892\) 3.87473e42 0.642615
\(893\) 1.04132e42 0.169917
\(894\) −4.22716e42 −0.678664
\(895\) −1.45300e43 −2.29526
\(896\) −7.99139e41 −0.124210
\(897\) −6.16195e42 −0.942384
\(898\) 7.10003e42 1.06845
\(899\) 3.28113e42 0.485857
\(900\) 1.40859e42 0.205243
\(901\) −5.90618e42 −0.846832
\(902\) −2.68161e42 −0.378356
\(903\) 5.54451e42 0.769823
\(904\) −1.16629e42 −0.159354
\(905\) 2.10948e43 2.83642
\(906\) −3.11168e42 −0.411752
\(907\) 3.49446e42 0.455067 0.227533 0.973770i \(-0.426934\pi\)
0.227533 + 0.973770i \(0.426934\pi\)
\(908\) −1.16904e42 −0.149825
\(909\) 3.10425e42 0.391544
\(910\) −1.45862e43 −1.81068
\(911\) −1.22703e43 −1.49913 −0.749564 0.661932i \(-0.769737\pi\)
−0.749564 + 0.661932i \(0.769737\pi\)
\(912\) −4.96577e41 −0.0597120
\(913\) −8.89541e42 −1.05279
\(914\) −3.92833e42 −0.457602
\(915\) 6.35746e41 0.0728917
\(916\) −9.48726e41 −0.107067
\(917\) −9.39478e42 −1.04359
\(918\) −1.96290e42 −0.214625
\(919\) 1.04604e43 1.12583 0.562915 0.826515i \(-0.309680\pi\)
0.562915 + 0.826515i \(0.309680\pi\)
\(920\) 6.67051e42 0.706701
\(921\) 5.20437e42 0.542754
\(922\) −1.64501e42 −0.168877
\(923\) 7.37131e42 0.744938
\(924\) 7.01954e42 0.698337
\(925\) 1.38577e43 1.35718
\(926\) −3.58310e42 −0.345461
\(927\) 1.57109e42 0.149123
\(928\) 2.10882e42 0.197058
\(929\) 1.61978e43 1.49014 0.745072 0.666984i \(-0.232415\pi\)
0.745072 + 0.666984i \(0.232415\pi\)
\(930\) 2.93468e42 0.265802
\(931\) 4.52238e42 0.403270
\(932\) 6.74094e42 0.591820
\(933\) 6.29447e42 0.544096
\(934\) 1.24410e43 1.05883
\(935\) −4.83987e43 −4.05570
\(936\) −1.74235e42 −0.143759
\(937\) −1.42722e43 −1.15949 −0.579746 0.814797i \(-0.696848\pi\)
−0.579746 + 0.814797i \(0.696848\pi\)
\(938\) 8.21758e42 0.657359
\(939\) 4.69132e42 0.369525
\(940\) −3.95525e42 −0.306775
\(941\) 2.07537e43 1.58506 0.792532 0.609830i \(-0.208762\pi\)
0.792532 + 0.609830i \(0.208762\pi\)
\(942\) 6.08723e42 0.457806
\(943\) 5.61599e42 0.415917
\(944\) −4.75171e41 −0.0346542
\(945\) −5.62516e42 −0.403993
\(946\) 1.63302e43 1.15497
\(947\) −2.86659e43 −1.99660 −0.998298 0.0583198i \(-0.981426\pi\)
−0.998298 + 0.0583198i \(0.981426\pi\)
\(948\) −6.74480e42 −0.462645
\(949\) 7.98703e42 0.539542
\(950\) 5.41476e42 0.360236
\(951\) −9.74750e42 −0.638669
\(952\) −1.21431e43 −0.783598
\(953\) 2.73174e43 1.73617 0.868086 0.496414i \(-0.165350\pi\)
0.868086 + 0.496414i \(0.165350\pi\)
\(954\) −2.02179e42 −0.126556
\(955\) 1.30599e43 0.805173
\(956\) 1.43942e43 0.874073
\(957\) −1.85236e43 −1.10791
\(958\) −1.59170e42 −0.0937695
\(959\) −2.52892e42 −0.146746
\(960\) 1.88615e42 0.107806
\(961\) −1.43877e43 −0.810033
\(962\) −1.71412e43 −0.950612
\(963\) 2.83171e42 0.154692
\(964\) 1.49080e43 0.802236
\(965\) −1.73419e43 −0.919286
\(966\) −1.47008e43 −0.767665
\(967\) 3.08479e43 1.58687 0.793436 0.608654i \(-0.208290\pi\)
0.793436 + 0.608654i \(0.208290\pi\)
\(968\) 1.36979e43 0.694163
\(969\) −7.54560e42 −0.376703
\(970\) −3.01566e43 −1.48318
\(971\) −2.71805e43 −1.31698 −0.658490 0.752590i \(-0.728804\pi\)
−0.658490 + 0.752590i \(0.728804\pi\)
\(972\) −6.71936e41 −0.0320750
\(973\) 9.39327e42 0.441754
\(974\) 1.61561e43 0.748569
\(975\) 1.89989e43 0.867284
\(976\) 4.69793e41 0.0211293
\(977\) −4.30767e43 −1.90885 −0.954423 0.298457i \(-0.903528\pi\)
−0.954423 + 0.298457i \(0.903528\pi\)
\(978\) −6.86123e42 −0.299563
\(979\) −1.08789e43 −0.467989
\(980\) −1.71774e43 −0.728078
\(981\) −2.58914e41 −0.0108132
\(982\) 4.04347e42 0.166394
\(983\) 2.56694e43 1.04085 0.520427 0.853906i \(-0.325773\pi\)
0.520427 + 0.853906i \(0.325773\pi\)
\(984\) 1.58798e42 0.0634476
\(985\) −7.34036e42 −0.288996
\(986\) 3.20440e43 1.24317
\(987\) 8.71675e42 0.333239
\(988\) −6.69777e42 −0.252322
\(989\) −3.41998e43 −1.26963
\(990\) −1.65677e43 −0.606111
\(991\) 2.59767e43 0.936518 0.468259 0.883591i \(-0.344881\pi\)
0.468259 + 0.883591i \(0.344881\pi\)
\(992\) 2.16862e42 0.0770485
\(993\) −5.40062e41 −0.0189095
\(994\) 1.75860e43 0.606826
\(995\) 1.42588e43 0.484894
\(996\) 5.26763e42 0.176545
\(997\) −5.01714e43 −1.65721 −0.828603 0.559837i \(-0.810864\pi\)
−0.828603 + 0.559837i \(0.810864\pi\)
\(998\) −1.98218e43 −0.645282
\(999\) −6.61049e42 −0.212097
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 6.30.a.d.1.1 2
3.2 odd 2 18.30.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.30.a.d.1.1 2 1.1 even 1 trivial
18.30.a.f.1.2 2 3.2 odd 2