Properties

Label 4.39.b.a.3.4
Level $4$
Weight $39$
Character 4.3
Analytic conductor $36.585$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,39,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 39, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 39);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 39 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(36.5853876134\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 9 x^{17} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{306}\cdot 3^{34}\cdot 5^{10}\cdot 19^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3.4
Root \(0.500000 + 4.81893e7i\) of defining polynomial
Character \(\chi\) \(=\) 4.3
Dual form 4.39.b.a.3.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-453236. + 263543. i) q^{2} -7.71029e8i q^{3} +(1.35968e11 - 2.38895e11i) q^{4} -2.10588e13 q^{5} +(2.03199e14 + 3.49458e14i) q^{6} -5.85221e15i q^{7} +(1.33358e15 + 1.44109e17i) q^{8} +7.56367e17 q^{9} +O(q^{10})\) \(q+(-453236. + 263543. i) q^{2} -7.71029e8i q^{3} +(1.35968e11 - 2.38895e11i) q^{4} -2.10588e13 q^{5} +(2.03199e14 + 3.49458e14i) q^{6} -5.85221e15i q^{7} +(1.33358e15 + 1.44109e17i) q^{8} +7.56367e17 q^{9} +(9.54463e18 - 5.54992e18i) q^{10} -6.60292e19i q^{11} +(-1.84195e20 - 1.04835e20i) q^{12} +1.72571e21 q^{13} +(1.54231e21 + 2.65243e21i) q^{14} +1.62370e22i q^{15} +(-3.85834e22 - 6.49639e22i) q^{16} +2.34447e23 q^{17} +(-3.42813e23 + 1.99335e23i) q^{18} +2.63164e24i q^{19} +(-2.86332e24 + 5.03084e24i) q^{20} -4.51222e24 q^{21} +(1.74016e25 + 2.99268e25i) q^{22} +9.74799e23i q^{23} +(1.11112e26 - 1.02823e24i) q^{24} +7.96770e25 q^{25} +(-7.82155e26 + 4.54800e26i) q^{26} -1.62473e27i q^{27} +(-1.39806e27 - 7.95713e26i) q^{28} +4.55931e27 q^{29} +(-4.27914e27 - 7.35918e27i) q^{30} -2.63620e28i q^{31} +(3.46082e28 + 1.92756e28i) q^{32} -5.09104e28 q^{33} +(-1.06260e29 + 6.17870e28i) q^{34} +1.23241e29i q^{35} +(1.02841e29 - 1.80692e29i) q^{36} -1.68618e29 q^{37} +(-6.93552e29 - 1.19276e30i) q^{38} -1.33057e30i q^{39} +(-2.80837e28 - 3.03477e30i) q^{40} -3.77960e30 q^{41} +(2.04510e30 - 1.18917e30i) q^{42} -2.13020e31i q^{43} +(-1.57740e31 - 8.97785e30i) q^{44} -1.59282e31 q^{45} +(-2.56902e29 - 4.41814e29i) q^{46} +1.21346e31i q^{47} +(-5.00890e31 + 2.97489e31i) q^{48} +9.56864e31 q^{49} +(-3.61125e31 + 2.09983e31i) q^{50} -1.80766e32i q^{51} +(2.34641e32 - 4.12263e32i) q^{52} -5.52099e32 q^{53} +(4.28186e32 + 7.36384e32i) q^{54} +1.39050e33i q^{55} +(8.43357e32 - 7.80440e30i) q^{56} +2.02907e33 q^{57} +(-2.06645e33 + 1.20158e33i) q^{58} +3.38179e33i q^{59} +(3.87892e33 + 2.20770e33i) q^{60} +2.82365e33 q^{61} +(6.94752e33 + 1.19482e34i) q^{62} -4.42642e33i q^{63} +(-2.07656e34 + 3.84362e32i) q^{64} -3.63415e34 q^{65} +(2.30744e34 - 1.34171e34i) q^{66} -9.12806e34i q^{67} +(3.18773e34 - 5.60082e34i) q^{68} +7.51598e32 q^{69} +(-3.24793e34 - 5.58572e34i) q^{70} +3.38934e34i q^{71} +(1.00868e33 + 1.08999e35i) q^{72} -4.50107e35 q^{73} +(7.64238e34 - 4.44382e34i) q^{74} -6.14333e34i q^{75} +(6.28686e35 + 3.57819e35i) q^{76} -3.86417e35 q^{77} +(3.50664e35 + 6.03064e35i) q^{78} -1.73909e36i q^{79} +(8.12522e35 + 1.36807e36i) q^{80} -2.30971e35 q^{81} +(1.71305e36 - 9.96088e35i) q^{82} +2.19300e36i q^{83} +(-6.13517e35 + 1.07795e36i) q^{84} -4.93719e36 q^{85} +(5.61399e36 + 9.65482e36i) q^{86} -3.51536e36i q^{87} +(9.51541e36 - 8.80554e34i) q^{88} -1.76726e37 q^{89} +(7.21924e36 - 4.19777e36i) q^{90} -1.00992e37i q^{91} +(2.32874e35 + 1.32541e35i) q^{92} -2.03258e37 q^{93} +(-3.19799e36 - 5.49983e36i) q^{94} -5.54194e37i q^{95} +(1.48620e37 - 2.66839e37i) q^{96} -1.37640e37 q^{97} +(-4.33685e37 + 2.52175e37i) q^{98} -4.99423e37i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 364228 q^{2} + 200567335824 q^{4} - 8991287507020 q^{5} + 817599417526752 q^{6} + 90\!\cdots\!32 q^{8}+ \cdots - 75\!\cdots\!14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 364228 q^{2} + 200567335824 q^{4} - 8991287507020 q^{5} + 817599417526752 q^{6} + 90\!\cdots\!32 q^{8}+ \cdots + 79\!\cdots\!72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −453236. + 263543.i −0.864479 + 0.502669i
\(3\) 7.71029e8i 0.663386i −0.943387 0.331693i \(-0.892380\pi\)
0.943387 0.331693i \(-0.107620\pi\)
\(4\) 1.35968e11 2.38895e11i 0.494648 0.869093i
\(5\) −2.10588e13 −1.10409 −0.552045 0.833814i \(-0.686152\pi\)
−0.552045 + 0.833814i \(0.686152\pi\)
\(6\) 2.03199e14 + 3.49458e14i 0.333464 + 0.573484i
\(7\) 5.85221e15i 0.513402i −0.966491 0.256701i \(-0.917364\pi\)
0.966491 0.256701i \(-0.0826355\pi\)
\(8\) 1.33358e15 + 1.44109e17i 0.00925358 + 0.999957i
\(9\) 7.56367e17 0.559918
\(10\) 9.54463e18 5.54992e18i 0.954463 0.554992i
\(11\) 6.60292e19i 1.07963i −0.841783 0.539815i \(-0.818494\pi\)
0.841783 0.539815i \(-0.181506\pi\)
\(12\) −1.84195e20 1.04835e20i −0.576545 0.328143i
\(13\) 1.72571e21 1.18044 0.590221 0.807242i \(-0.299041\pi\)
0.590221 + 0.807242i \(0.299041\pi\)
\(14\) 1.54231e21 + 2.65243e21i 0.258071 + 0.443825i
\(15\) 1.62370e22i 0.732438i
\(16\) −3.85834e22 6.49639e22i −0.510647 0.859791i
\(17\) 2.34447e23 0.980654 0.490327 0.871539i \(-0.336877\pi\)
0.490327 + 0.871539i \(0.336877\pi\)
\(18\) −3.42813e23 + 1.99335e23i −0.484038 + 0.281454i
\(19\) 2.63164e24i 1.33018i 0.746765 + 0.665088i \(0.231606\pi\)
−0.746765 + 0.665088i \(0.768394\pi\)
\(20\) −2.86332e24 + 5.03084e24i −0.546136 + 0.959557i
\(21\) −4.51222e24 −0.340584
\(22\) 1.74016e25 + 2.99268e25i 0.542697 + 0.933318i
\(23\) 9.74799e23i 0.0130643i 0.999979 + 0.00653215i \(0.00207926\pi\)
−0.999979 + 0.00653215i \(0.997921\pi\)
\(24\) 1.11112e26 1.02823e24i 0.663358 0.00613870i
\(25\) 7.96770e25 0.219015
\(26\) −7.82155e26 + 4.54800e26i −1.02047 + 0.593372i
\(27\) 1.62473e27i 1.03483i
\(28\) −1.39806e27 7.95713e26i −0.446194 0.253953i
\(29\) 4.55931e27 0.747029 0.373515 0.927624i \(-0.378153\pi\)
0.373515 + 0.927624i \(0.378153\pi\)
\(30\) −4.27914e27 7.35918e27i −0.368174 0.633178i
\(31\) 2.63620e28i 1.21648i −0.793753 0.608241i \(-0.791876\pi\)
0.793753 0.608241i \(-0.208124\pi\)
\(32\) 3.46082e28 + 1.92756e28i 0.873634 + 0.486584i
\(33\) −5.09104e28 −0.716213
\(34\) −1.06260e29 + 6.17870e28i −0.847755 + 0.492944i
\(35\) 1.23241e29i 0.566842i
\(36\) 1.02841e29 1.80692e29i 0.276962 0.486621i
\(37\) −1.68618e29 −0.269818 −0.134909 0.990858i \(-0.543074\pi\)
−0.134909 + 0.990858i \(0.543074\pi\)
\(38\) −6.93552e29 1.19276e30i −0.668638 1.14991i
\(39\) 1.33057e30i 0.783090i
\(40\) −2.80837e28 3.03477e30i −0.0102168 1.10404i
\(41\) −3.77960e30 −0.860107 −0.430053 0.902804i \(-0.641505\pi\)
−0.430053 + 0.902804i \(0.641505\pi\)
\(42\) 2.04510e30 1.18917e30i 0.294428 0.171201i
\(43\) 2.13020e31i 1.96119i −0.196051 0.980594i \(-0.562812\pi\)
0.196051 0.980594i \(-0.437188\pi\)
\(44\) −1.57740e31 8.97785e30i −0.938300 0.534037i
\(45\) −1.59282e31 −0.618200
\(46\) −2.56902e29 4.41814e29i −0.00656702 0.0112938i
\(47\) 1.21346e31i 0.206141i 0.994674 + 0.103071i \(0.0328668\pi\)
−0.994674 + 0.103071i \(0.967133\pi\)
\(48\) −5.00890e31 + 2.97489e31i −0.570373 + 0.338756i
\(49\) 9.56864e31 0.736419
\(50\) −3.61125e31 + 2.09983e31i −0.189333 + 0.110092i
\(51\) 1.80766e32i 0.650553i
\(52\) 2.34641e32 4.12263e32i 0.583903 1.02591i
\(53\) −5.52099e32 −0.956700 −0.478350 0.878169i \(-0.658765\pi\)
−0.478350 + 0.878169i \(0.658765\pi\)
\(54\) 4.28186e32 + 7.36384e32i 0.520176 + 0.894588i
\(55\) 1.39050e33i 1.19201i
\(56\) 8.43357e32 7.80440e30i 0.513380 0.00475080i
\(57\) 2.02907e33 0.882420
\(58\) −2.06645e33 + 1.20158e33i −0.645791 + 0.375508i
\(59\) 3.38179e33i 0.763763i 0.924211 + 0.381881i \(0.124724\pi\)
−0.924211 + 0.381881i \(0.875276\pi\)
\(60\) 3.87892e33 + 2.20770e33i 0.636557 + 0.362299i
\(61\) 2.82365e33 0.338488 0.169244 0.985574i \(-0.445867\pi\)
0.169244 + 0.985574i \(0.445867\pi\)
\(62\) 6.94752e33 + 1.19482e34i 0.611487 + 1.05162i
\(63\) 4.42642e33i 0.287463i
\(64\) −2.07656e34 + 3.84362e32i −0.999829 + 0.0185064i
\(65\) −3.63415e34 −1.30331
\(66\) 2.30744e34 1.34171e34i 0.619151 0.360018i
\(67\) 9.12806e34i 1.84059i −0.391224 0.920296i \(-0.627948\pi\)
0.391224 0.920296i \(-0.372052\pi\)
\(68\) 3.18773e34 5.60082e34i 0.485078 0.852280i
\(69\) 7.51598e32 0.00866668
\(70\) −3.24793e34 5.58572e34i −0.284934 0.490023i
\(71\) 3.38934e34i 0.227095i 0.993533 + 0.113547i \(0.0362213\pi\)
−0.993533 + 0.113547i \(0.963779\pi\)
\(72\) 1.00868e33 + 1.08999e35i 0.00518125 + 0.559894i
\(73\) −4.50107e35 −1.77902 −0.889511 0.456914i \(-0.848955\pi\)
−0.889511 + 0.456914i \(0.848955\pi\)
\(74\) 7.64238e34 4.44382e34i 0.233252 0.135629i
\(75\) 6.14333e34i 0.145291i
\(76\) 6.28686e35 + 3.57819e35i 1.15605 + 0.657968i
\(77\) −3.86417e35 −0.554285
\(78\) 3.50664e35 + 6.03064e35i 0.393635 + 0.676964i
\(79\) 1.73909e36i 1.53252i −0.642531 0.766260i \(-0.722115\pi\)
0.642531 0.766260i \(-0.277885\pi\)
\(80\) 8.12522e35 + 1.36807e36i 0.563800 + 0.949286i
\(81\) −2.30971e35 −0.126573
\(82\) 1.71305e36 9.96088e35i 0.743544 0.432349i
\(83\) 2.19300e36i 0.756059i 0.925794 + 0.378029i \(0.123398\pi\)
−0.925794 + 0.378029i \(0.876602\pi\)
\(84\) −6.13517e35 + 1.07795e36i −0.168469 + 0.295999i
\(85\) −4.93719e36 −1.08273
\(86\) 5.61399e36 + 9.65482e36i 0.985828 + 1.69541i
\(87\) 3.51536e36i 0.495569i
\(88\) 9.51541e36 8.80554e34i 1.07958 0.00999045i
\(89\) −1.76726e37 −1.61767 −0.808836 0.588035i \(-0.799902\pi\)
−0.808836 + 0.588035i \(0.799902\pi\)
\(90\) 7.21924e36 4.19777e36i 0.534421 0.310750i
\(91\) 1.00992e37i 0.606041i
\(92\) 2.32874e35 + 1.32541e35i 0.0113541 + 0.00646223i
\(93\) −2.03258e37 −0.806997
\(94\) −3.19799e36 5.49983e36i −0.103621 0.178205i
\(95\) 5.54194e37i 1.46863i
\(96\) 1.48620e37 2.66839e37i 0.322794 0.579557i
\(97\) −1.37640e37 −0.245517 −0.122758 0.992437i \(-0.539174\pi\)
−0.122758 + 0.992437i \(0.539174\pi\)
\(98\) −4.33685e37 + 2.52175e37i −0.636618 + 0.370175i
\(99\) 4.99423e37i 0.604505i
\(100\) 1.08335e37 1.90344e37i 0.108335 0.190344i
\(101\) 2.42651e36 0.0200852 0.0100426 0.999950i \(-0.496803\pi\)
0.0100426 + 0.999950i \(0.496803\pi\)
\(102\) 4.76396e37 + 8.19295e37i 0.327013 + 0.562389i
\(103\) 2.01919e38i 1.15151i 0.817621 + 0.575757i \(0.195293\pi\)
−0.817621 + 0.575757i \(0.804707\pi\)
\(104\) 2.30138e36 + 2.48691e38i 0.0109233 + 1.18039i
\(105\) 9.50222e37 0.376035
\(106\) 2.50231e38 1.45502e38i 0.827047 0.480903i
\(107\) 4.85956e38i 1.34371i 0.740683 + 0.671855i \(0.234502\pi\)
−0.740683 + 0.671855i \(0.765498\pi\)
\(108\) −3.88138e38 2.20910e38i −0.899363 0.511876i
\(109\) 7.12284e38 1.38532 0.692660 0.721265i \(-0.256439\pi\)
0.692660 + 0.721265i \(0.256439\pi\)
\(110\) −3.66457e38 6.30224e38i −0.599186 1.03047i
\(111\) 1.30009e38i 0.178994i
\(112\) −3.80183e38 + 2.25798e38i −0.441418 + 0.262167i
\(113\) 5.09915e38 0.500043 0.250021 0.968240i \(-0.419562\pi\)
0.250021 + 0.968240i \(0.419562\pi\)
\(114\) −9.19649e38 + 5.34749e38i −0.762834 + 0.443565i
\(115\) 2.05281e37i 0.0144242i
\(116\) 6.19920e38 1.08920e39i 0.369516 0.649238i
\(117\) 1.30527e39 0.660951
\(118\) −8.91248e38 1.53275e39i −0.383920 0.660257i
\(119\) 1.37204e39i 0.503470i
\(120\) −2.33989e39 + 2.16533e37i −0.732407 + 0.00677768i
\(121\) −6.19427e38 −0.165603
\(122\) −1.27978e39 + 7.44154e38i −0.292616 + 0.170148i
\(123\) 2.91418e39i 0.570583i
\(124\) −6.29773e39 3.58438e39i −1.05724 0.601730i
\(125\) 5.98326e39 0.862278
\(126\) 1.16655e39 + 2.00621e39i 0.144499 + 0.248506i
\(127\) 2.60707e39i 0.277896i 0.990300 + 0.138948i \(0.0443721\pi\)
−0.990300 + 0.138948i \(0.955628\pi\)
\(128\) 9.31043e39 5.64685e39i 0.855028 0.518581i
\(129\) −1.64244e40 −1.30103
\(130\) 1.64713e40 9.57756e39i 1.12669 0.655136i
\(131\) 1.54564e40i 0.914018i −0.889462 0.457009i \(-0.848921\pi\)
0.889462 0.457009i \(-0.151079\pi\)
\(132\) −6.92218e39 + 1.21622e40i −0.354273 + 0.622456i
\(133\) 1.54009e40 0.682914
\(134\) 2.40564e40 + 4.13717e40i 0.925208 + 1.59115i
\(135\) 3.42148e40i 1.14254i
\(136\) 3.12655e38 + 3.37860e40i 0.00907456 + 0.980612i
\(137\) −5.33423e40 −1.34704 −0.673521 0.739168i \(-0.735219\pi\)
−0.673521 + 0.739168i \(0.735219\pi\)
\(138\) −3.40651e38 + 1.98079e38i −0.00749217 + 0.00435647i
\(139\) 2.93279e40i 0.562342i −0.959658 0.281171i \(-0.909277\pi\)
0.959658 0.281171i \(-0.0907228\pi\)
\(140\) 2.94416e40 + 1.67568e40i 0.492639 + 0.280387i
\(141\) 9.35611e39 0.136751
\(142\) −8.93238e39 1.53617e40i −0.114153 0.196318i
\(143\) 1.13948e41i 1.27444i
\(144\) −2.91832e40 4.91366e40i −0.285921 0.481412i
\(145\) −9.60139e40 −0.824787
\(146\) 2.04005e41 1.18623e41i 1.53793 0.894259i
\(147\) 7.37769e40i 0.488530i
\(148\) −2.29266e40 + 4.02820e40i −0.133465 + 0.234497i
\(149\) −2.76794e41 −1.41781 −0.708904 0.705305i \(-0.750810\pi\)
−0.708904 + 0.705305i \(0.750810\pi\)
\(150\) 1.61903e40 + 2.78438e40i 0.0730334 + 0.125601i
\(151\) 7.95011e40i 0.316090i −0.987432 0.158045i \(-0.949481\pi\)
0.987432 0.158045i \(-0.0505192\pi\)
\(152\) −3.79244e41 + 3.50951e39i −1.33012 + 0.0123089i
\(153\) 1.77328e41 0.549086
\(154\) 1.75138e41 1.01838e41i 0.479167 0.278622i
\(155\) 5.55152e41i 1.34310i
\(156\) −3.17867e41 1.80915e41i −0.680578 0.387354i
\(157\) 1.04378e42 1.97932 0.989658 0.143448i \(-0.0458189\pi\)
0.989658 + 0.143448i \(0.0458189\pi\)
\(158\) 4.58324e41 + 7.88216e41i 0.770350 + 1.32483i
\(159\) 4.25684e41i 0.634662i
\(160\) −7.28808e41 4.05922e41i −0.964570 0.537233i
\(161\) 5.70473e39 0.00670724
\(162\) 1.04684e41 6.08707e40i 0.109420 0.0636243i
\(163\) 3.11913e41i 0.290047i −0.989428 0.145024i \(-0.953674\pi\)
0.989428 0.145024i \(-0.0463259\pi\)
\(164\) −5.13903e41 + 9.02925e41i −0.425450 + 0.747513i
\(165\) 1.07211e42 0.790763
\(166\) −5.77951e41 9.93948e41i −0.380047 0.653597i
\(167\) 7.61829e41i 0.446935i 0.974711 + 0.223468i \(0.0717377\pi\)
−0.974711 + 0.223468i \(0.928262\pi\)
\(168\) −6.01742e39 6.50252e41i −0.00315162 0.340569i
\(169\) 8.40873e41 0.393444
\(170\) 2.23771e42 1.30116e42i 0.935998 0.544255i
\(171\) 1.99049e42i 0.744790i
\(172\) −5.08892e42 2.89638e42i −1.70446 0.970097i
\(173\) −1.46553e42 −0.439661 −0.219830 0.975538i \(-0.570550\pi\)
−0.219830 + 0.975538i \(0.570550\pi\)
\(174\) 9.26450e41 + 1.59329e42i 0.249107 + 0.428409i
\(175\) 4.66287e41i 0.112442i
\(176\) −4.28952e42 + 2.54763e42i −0.928256 + 0.551310i
\(177\) 2.60746e42 0.506670
\(178\) 8.00986e42 4.65750e42i 1.39844 0.813153i
\(179\) 8.38708e42i 1.31645i −0.752823 0.658223i \(-0.771308\pi\)
0.752823 0.658223i \(-0.228692\pi\)
\(180\) −2.16572e42 + 3.80516e42i −0.305791 + 0.537274i
\(181\) −1.42214e43 −1.80738 −0.903692 0.428183i \(-0.859154\pi\)
−0.903692 + 0.428183i \(0.859154\pi\)
\(182\) 2.66159e42 + 4.57734e42i 0.304638 + 0.523910i
\(183\) 2.17711e42i 0.224549i
\(184\) −1.40477e41 + 1.29997e39i −0.0130637 + 0.000120892i
\(185\) 3.55090e42 0.297904
\(186\) 9.21239e42 5.35673e42i 0.697632 0.405653i
\(187\) 1.54804e43i 1.05874i
\(188\) 2.89889e42 + 1.64991e42i 0.179156 + 0.101967i
\(189\) −9.50824e42 −0.531283
\(190\) 1.46054e43 + 2.51181e43i 0.738236 + 1.26960i
\(191\) 3.06221e43i 1.40088i 0.713710 + 0.700441i \(0.247013\pi\)
−0.713710 + 0.700441i \(0.752987\pi\)
\(192\) 2.96354e41 + 1.60109e43i 0.0122769 + 0.663273i
\(193\) 6.68211e42 0.250798 0.125399 0.992106i \(-0.459979\pi\)
0.125399 + 0.992106i \(0.459979\pi\)
\(194\) 6.23833e42 3.62741e42i 0.212244 0.123414i
\(195\) 2.80203e43i 0.864601i
\(196\) 1.30103e43 2.28590e43i 0.364268 0.640017i
\(197\) −1.41772e43 −0.360357 −0.180179 0.983634i \(-0.557668\pi\)
−0.180179 + 0.983634i \(0.557668\pi\)
\(198\) 1.31620e43 + 2.26357e43i 0.303866 + 0.522582i
\(199\) 4.86430e43i 1.02050i −0.860027 0.510249i \(-0.829553\pi\)
0.860027 0.510249i \(-0.170447\pi\)
\(200\) 1.06256e41 + 1.14822e43i 0.00202667 + 0.219005i
\(201\) −7.03800e43 −1.22102
\(202\) −1.09978e42 + 6.39490e41i −0.0173632 + 0.0100962i
\(203\) 2.66821e43i 0.383526i
\(204\) −4.31839e43 2.45783e43i −0.565391 0.321795i
\(205\) 7.95940e43 0.949635
\(206\) −5.32143e43 9.15168e43i −0.578830 0.995460i
\(207\) 7.37306e41i 0.00731495i
\(208\) −6.65838e43 1.12109e44i −0.602789 1.01493i
\(209\) 1.73766e44 1.43610
\(210\) −4.30675e43 + 2.50425e43i −0.325075 + 0.189021i
\(211\) 1.85839e44i 1.28166i −0.767685 0.640828i \(-0.778591\pi\)
0.767685 0.640828i \(-0.221409\pi\)
\(212\) −7.50677e43 + 1.31894e44i −0.473229 + 0.831461i
\(213\) 2.61328e43 0.150651
\(214\) −1.28070e44 2.20253e44i −0.675441 1.16161i
\(215\) 4.48595e44i 2.16533i
\(216\) 2.34138e44 2.16670e42i 1.03478 0.00957587i
\(217\) −1.54276e44 −0.624544
\(218\) −3.22833e44 + 1.87718e44i −1.19758 + 0.696357i
\(219\) 3.47045e44i 1.18018i
\(220\) 3.32183e44 + 1.89063e44i 1.03597 + 0.589625i
\(221\) 4.04589e44 1.15761
\(222\) −3.42631e43 5.89249e43i −0.0899747 0.154736i
\(223\) 4.08148e43i 0.0984068i −0.998789 0.0492034i \(-0.984332\pi\)
0.998789 0.0492034i \(-0.0156682\pi\)
\(224\) 1.12805e44 2.02535e44i 0.249813 0.448525i
\(225\) 6.02650e43 0.122630
\(226\) −2.31112e44 + 1.34385e44i −0.432276 + 0.251356i
\(227\) 5.84772e44i 1.00576i −0.864355 0.502881i \(-0.832273\pi\)
0.864355 0.502881i \(-0.167727\pi\)
\(228\) 2.75889e44 4.84735e44i 0.436487 0.766906i
\(229\) 3.69037e44 0.537273 0.268637 0.963242i \(-0.413427\pi\)
0.268637 + 0.963242i \(0.413427\pi\)
\(230\) 5.41006e42 + 9.30409e42i 0.00725058 + 0.0124694i
\(231\) 2.97939e44i 0.367705i
\(232\) 6.08022e42 + 6.57038e44i 0.00691269 + 0.746997i
\(233\) −1.17874e45 −1.23497 −0.617483 0.786585i \(-0.711847\pi\)
−0.617483 + 0.786585i \(0.711847\pi\)
\(234\) −5.91596e44 + 3.43996e44i −0.571379 + 0.332240i
\(235\) 2.55540e44i 0.227599i
\(236\) 8.07891e44 + 4.59815e44i 0.663781 + 0.377794i
\(237\) −1.34088e45 −1.01665
\(238\) 3.61591e44 + 6.21856e44i 0.253079 + 0.435239i
\(239\) 2.12120e45i 1.37095i −0.728096 0.685476i \(-0.759594\pi\)
0.728096 0.685476i \(-0.240406\pi\)
\(240\) 1.05482e45 6.26477e44i 0.629744 0.374017i
\(241\) −7.92479e44 −0.437184 −0.218592 0.975816i \(-0.570146\pi\)
−0.218592 + 0.975816i \(0.570146\pi\)
\(242\) 2.80747e44 1.63246e44i 0.143160 0.0832435i
\(243\) 2.01668e45i 0.950862i
\(244\) 3.83925e44 6.74555e44i 0.167433 0.294178i
\(245\) −2.01504e45 −0.813072
\(246\) −7.68012e44 1.32081e45i −0.286814 0.493257i
\(247\) 4.54146e45i 1.57020i
\(248\) 3.79900e45 3.51558e43i 1.21643 0.0112568i
\(249\) 1.69087e45 0.501559
\(250\) −2.71183e45 + 1.57685e45i −0.745421 + 0.433440i
\(251\) 3.02682e45i 0.771232i −0.922659 0.385616i \(-0.873989\pi\)
0.922659 0.385616i \(-0.126011\pi\)
\(252\) −1.05745e45 6.01850e44i −0.249832 0.142193i
\(253\) 6.43653e43 0.0141046
\(254\) −6.87077e44 1.18162e45i −0.139690 0.240236i
\(255\) 3.80671e45i 0.718269i
\(256\) −2.73163e45 + 5.01306e45i −0.478479 + 0.878099i
\(257\) 4.45369e45 0.724420 0.362210 0.932096i \(-0.382022\pi\)
0.362210 + 0.932096i \(0.382022\pi\)
\(258\) 7.44414e45 4.32855e45i 1.12471 0.653985i
\(259\) 9.86790e44i 0.138525i
\(260\) −4.94127e45 + 8.68179e45i −0.644682 + 1.13270i
\(261\) 3.44851e45 0.418275
\(262\) 4.07344e45 + 7.00541e45i 0.459448 + 0.790149i
\(263\) 3.76990e45i 0.395522i 0.980250 + 0.197761i \(0.0633669\pi\)
−0.980250 + 0.197761i \(0.936633\pi\)
\(264\) −6.78932e43 7.33665e45i −0.00662753 0.716182i
\(265\) 1.16266e46 1.05628
\(266\) −6.98026e45 + 4.05882e45i −0.590365 + 0.343280i
\(267\) 1.36261e46i 1.07314i
\(268\) −2.18064e46 1.24112e46i −1.59965 0.910445i
\(269\) 1.92563e45 0.131607 0.0658034 0.997833i \(-0.479039\pi\)
0.0658034 + 0.997833i \(0.479039\pi\)
\(270\) −9.01709e45 1.55074e46i −0.574321 0.987705i
\(271\) 1.36328e46i 0.809413i −0.914447 0.404706i \(-0.867374\pi\)
0.914447 0.404706i \(-0.132626\pi\)
\(272\) −9.04577e45 1.52306e46i −0.500768 0.843157i
\(273\) −7.78680e45 −0.402040
\(274\) 2.41767e46 1.40580e46i 1.16449 0.677116i
\(275\) 5.26101e45i 0.236455i
\(276\) 1.02193e44 1.79553e44i 0.00428696 0.00753216i
\(277\) 2.26152e44 0.00885697 0.00442849 0.999990i \(-0.498590\pi\)
0.00442849 + 0.999990i \(0.498590\pi\)
\(278\) 7.72918e45 + 1.32925e46i 0.282672 + 0.486133i
\(279\) 1.99393e46i 0.681130i
\(280\) −1.77601e46 + 1.64352e44i −0.566818 + 0.00524532i
\(281\) 3.24869e45 0.0968920 0.0484460 0.998826i \(-0.484573\pi\)
0.0484460 + 0.998826i \(0.484573\pi\)
\(282\) −4.24052e45 + 2.46574e45i −0.118219 + 0.0687407i
\(283\) 1.26031e46i 0.328500i 0.986419 + 0.164250i \(0.0525203\pi\)
−0.986419 + 0.164250i \(0.947480\pi\)
\(284\) 8.09695e45 + 4.60841e45i 0.197366 + 0.112332i
\(285\) −4.27299e46 −0.974271
\(286\) 3.00301e46 + 5.16451e46i 0.640622 + 1.10173i
\(287\) 2.21190e46i 0.441580i
\(288\) 2.61765e46 + 1.45794e46i 0.489163 + 0.272448i
\(289\) −2.19006e45 −0.0383175
\(290\) 4.35169e46 2.53038e46i 0.713011 0.414595i
\(291\) 1.06124e46i 0.162873i
\(292\) −6.12000e46 + 1.07528e47i −0.879990 + 1.54614i
\(293\) 7.59287e46 1.02311 0.511554 0.859251i \(-0.329070\pi\)
0.511554 + 0.859251i \(0.329070\pi\)
\(294\) 1.94434e46 + 3.34384e46i 0.245569 + 0.422324i
\(295\) 7.12166e46i 0.843263i
\(296\) −2.24866e44 2.42994e46i −0.00249679 0.269807i
\(297\) −1.07279e47 −1.11723
\(298\) 1.25453e47 7.29472e46i 1.22567 0.712688i
\(299\) 1.68222e45i 0.0154217i
\(300\) −1.46761e46 8.35294e45i −0.126272 0.0718680i
\(301\) −1.24664e47 −1.00688
\(302\) 2.09520e46 + 3.60328e46i 0.158889 + 0.273254i
\(303\) 1.87091e45i 0.0133242i
\(304\) 1.70962e47 1.01538e47i 1.14367 0.679250i
\(305\) −5.94628e46 −0.373722
\(306\) −8.03715e46 + 4.67337e46i −0.474674 + 0.276009i
\(307\) 1.02618e47i 0.569630i −0.958582 0.284815i \(-0.908068\pi\)
0.958582 0.284815i \(-0.0919322\pi\)
\(308\) −5.25403e46 + 9.23130e46i −0.274176 + 0.481725i
\(309\) 1.55685e47 0.763899
\(310\) −1.46307e47 2.51615e47i −0.675137 1.16109i
\(311\) 9.56462e46i 0.415165i 0.978218 + 0.207582i \(0.0665595\pi\)
−0.978218 + 0.207582i \(0.933441\pi\)
\(312\) 1.91748e47 1.77443e45i 0.783056 0.00724638i
\(313\) −1.07456e47 −0.412941 −0.206471 0.978453i \(-0.566198\pi\)
−0.206471 + 0.978453i \(0.566198\pi\)
\(314\) −4.73078e47 + 2.75081e47i −1.71108 + 0.994941i
\(315\) 9.32153e46i 0.317385i
\(316\) −4.15458e47 2.36460e47i −1.33190 0.758058i
\(317\) 3.42063e47 1.03271 0.516357 0.856373i \(-0.327288\pi\)
0.516357 + 0.856373i \(0.327288\pi\)
\(318\) −1.12186e47 1.92935e47i −0.319025 0.548652i
\(319\) 3.01048e47i 0.806516i
\(320\) 4.37300e47 8.09422e45i 1.10390 0.0204327i
\(321\) 3.74686e47 0.891399
\(322\) −2.58559e45 + 1.50344e45i −0.00579827 + 0.00337152i
\(323\) 6.16982e47i 1.30444i
\(324\) −3.14045e46 + 5.51776e46i −0.0626091 + 0.110004i
\(325\) 1.37500e47 0.258534
\(326\) 8.22026e46 + 1.41370e47i 0.145798 + 0.250740i
\(327\) 5.49191e47i 0.919002i
\(328\) −5.04040e45 5.44674e47i −0.00795906 0.860070i
\(329\) 7.10142e46 0.105833
\(330\) −4.85921e47 + 2.82549e47i −0.683598 + 0.397492i
\(331\) 6.65915e47i 0.884477i 0.896897 + 0.442239i \(0.145816\pi\)
−0.896897 + 0.442239i \(0.854184\pi\)
\(332\) 5.23897e47 + 2.98178e47i 0.657086 + 0.373983i
\(333\) −1.27537e47 −0.151076
\(334\) −2.00775e47 3.45288e47i −0.224661 0.386366i
\(335\) 1.92226e48i 2.03218i
\(336\) 1.74097e47 + 2.93132e47i 0.173918 + 0.292831i
\(337\) 1.21390e48 1.14608 0.573041 0.819527i \(-0.305763\pi\)
0.573041 + 0.819527i \(0.305763\pi\)
\(338\) −3.81114e47 + 2.21607e47i −0.340124 + 0.197772i
\(339\) 3.93159e47i 0.331722i
\(340\) −6.71299e47 + 1.17947e48i −0.535570 + 0.940994i
\(341\) −1.74066e48 −1.31335
\(342\) −5.24580e47 9.02161e47i −0.374383 0.643855i
\(343\) 1.32038e48i 0.891481i
\(344\) 3.06981e48 2.84079e46i 1.96110 0.0181480i
\(345\) −1.58278e46 −0.00956880
\(346\) 6.64230e47 3.86230e47i 0.380078 0.221004i
\(347\) 3.05531e48i 1.65499i −0.561475 0.827494i \(-0.689766\pi\)
0.561475 0.827494i \(-0.310234\pi\)
\(348\) −8.39801e47 4.77976e47i −0.430696 0.245132i
\(349\) −2.47433e48 −1.20164 −0.600819 0.799385i \(-0.705159\pi\)
−0.600819 + 0.799385i \(0.705159\pi\)
\(350\) 1.22887e47 + 2.11338e47i 0.0565213 + 0.0972041i
\(351\) 2.80381e48i 1.22156i
\(352\) 1.27275e48 2.28515e48i 0.525332 0.943202i
\(353\) −1.23870e48 −0.484446 −0.242223 0.970221i \(-0.577877\pi\)
−0.242223 + 0.970221i \(0.577877\pi\)
\(354\) −1.18179e48 + 6.87178e47i −0.438005 + 0.254687i
\(355\) 7.13756e47i 0.250733i
\(356\) −2.40290e48 + 4.22189e48i −0.800178 + 1.40591i
\(357\) −1.05788e48 −0.333995
\(358\) 2.21036e48 + 3.80132e48i 0.661736 + 1.13804i
\(359\) 3.55217e48i 1.00855i 0.863542 + 0.504276i \(0.168241\pi\)
−0.863542 + 0.504276i \(0.831759\pi\)
\(360\) −2.12416e46 2.29540e48i −0.00572056 0.618174i
\(361\) −3.01141e48 −0.769366
\(362\) 6.44567e48 3.74796e48i 1.56245 0.908516i
\(363\) 4.77596e47i 0.109859i
\(364\) −2.41265e48 1.37317e48i −0.526707 0.299777i
\(365\) 9.47873e48 1.96420
\(366\) 5.73764e47 + 9.86746e47i 0.112874 + 0.194118i
\(367\) 6.71115e48i 1.25355i 0.779200 + 0.626775i \(0.215625\pi\)
−0.779200 + 0.626775i \(0.784375\pi\)
\(368\) 6.33268e46 3.76111e46i 0.0112326 0.00667125i
\(369\) −2.85876e48 −0.481589
\(370\) −1.60940e48 + 9.35817e47i −0.257532 + 0.149747i
\(371\) 3.23100e48i 0.491171i
\(372\) −2.76366e48 + 4.85573e48i −0.399180 + 0.701356i
\(373\) −9.25968e47 −0.127095 −0.0635475 0.997979i \(-0.520241\pi\)
−0.0635475 + 0.997979i \(0.520241\pi\)
\(374\) 4.07975e48 + 7.01627e48i 0.532198 + 0.915262i
\(375\) 4.61326e48i 0.572024i
\(376\) −1.74870e48 + 1.61824e46i −0.206133 + 0.00190755i
\(377\) 7.86807e48 0.881825
\(378\) 4.30948e48 2.50583e48i 0.459283 0.267059i
\(379\) 4.71994e48i 0.478401i −0.970970 0.239200i \(-0.923115\pi\)
0.970970 0.239200i \(-0.0768853\pi\)
\(380\) −1.32394e49 7.53525e48i −1.27638 0.726456i
\(381\) 2.01013e48 0.184353
\(382\) −8.07026e48 1.38791e49i −0.704180 1.21103i
\(383\) 1.08626e49i 0.901899i 0.892549 + 0.450950i \(0.148915\pi\)
−0.892549 + 0.450950i \(0.851085\pi\)
\(384\) −4.35388e48 7.17861e48i −0.344020 0.567214i
\(385\) 8.13750e48 0.611980
\(386\) −3.02857e48 + 1.76102e48i −0.216810 + 0.126068i
\(387\) 1.61121e49i 1.09810i
\(388\) −1.87146e48 + 3.28814e48i −0.121444 + 0.213377i
\(389\) 1.64050e49 1.01376 0.506879 0.862017i \(-0.330799\pi\)
0.506879 + 0.862017i \(0.330799\pi\)
\(390\) −7.38457e48 1.26998e49i −0.434608 0.747430i
\(391\) 2.28539e47i 0.0128116i
\(392\) 1.27606e47 + 1.37893e49i 0.00681451 + 0.736387i
\(393\) −1.19174e49 −0.606347
\(394\) 6.42563e48 3.73631e48i 0.311521 0.181140i
\(395\) 3.66231e49i 1.69204i
\(396\) −1.19310e49 6.79055e48i −0.525372 0.299017i
\(397\) −2.06187e49 −0.865450 −0.432725 0.901526i \(-0.642448\pi\)
−0.432725 + 0.901526i \(0.642448\pi\)
\(398\) 1.28195e49 + 2.20468e49i 0.512972 + 0.882199i
\(399\) 1.18746e49i 0.453036i
\(400\) −3.07421e48 5.17613e48i −0.111839 0.188307i
\(401\) −2.28768e49 −0.793694 −0.396847 0.917885i \(-0.629896\pi\)
−0.396847 + 0.917885i \(0.629896\pi\)
\(402\) 3.18987e49 1.85482e49i 1.05555 0.613770i
\(403\) 4.54932e49i 1.43599i
\(404\) 3.29927e47 5.79679e47i 0.00993508 0.0174559i
\(405\) 4.86397e48 0.139748
\(406\) 7.03188e48 + 1.20933e49i 0.192787 + 0.331550i
\(407\) 1.11337e49i 0.291304i
\(408\) 2.60500e49 2.41066e47i 0.650525 0.00601994i
\(409\) 1.28628e49 0.306614 0.153307 0.988179i \(-0.451008\pi\)
0.153307 + 0.988179i \(0.451008\pi\)
\(410\) −3.60748e49 + 2.09765e49i −0.820940 + 0.477352i
\(411\) 4.11285e49i 0.893610i
\(412\) 4.82373e49 + 2.74544e49i 1.00077 + 0.569594i
\(413\) 1.97910e49 0.392117
\(414\) −1.94312e47 3.34174e47i −0.00367700 0.00632362i
\(415\) 4.61821e49i 0.834757i
\(416\) 5.97238e49 + 3.32641e49i 1.03127 + 0.574385i
\(417\) −2.26127e49 −0.373050
\(418\) −7.87568e49 + 4.57947e49i −1.24148 + 0.721882i
\(419\) 9.99261e48i 0.150526i −0.997164 0.0752632i \(-0.976020\pi\)
0.997164 0.0752632i \(-0.0239797\pi\)
\(420\) 1.29200e49 2.27003e49i 0.186005 0.326810i
\(421\) −9.06625e49 −1.24758 −0.623789 0.781593i \(-0.714408\pi\)
−0.623789 + 0.781593i \(0.714408\pi\)
\(422\) 4.89767e49 + 8.42291e49i 0.644248 + 1.10796i
\(423\) 9.17819e48i 0.115422i
\(424\) −7.36269e47 7.95625e49i −0.00885290 0.956659i
\(425\) 1.86801e49 0.214777
\(426\) −1.18443e49 + 6.88712e48i −0.130235 + 0.0757278i
\(427\) 1.65246e49i 0.173781i
\(428\) 1.16092e50 + 6.60744e49i 1.16781 + 0.664663i
\(429\) −8.78568e49 −0.845448
\(430\) −1.18224e50 2.03319e50i −1.08844 1.87188i
\(431\) 3.03034e49i 0.266946i −0.991052 0.133473i \(-0.957387\pi\)
0.991052 0.133473i \(-0.0426130\pi\)
\(432\) −1.05549e50 + 6.26874e49i −0.889736 + 0.528432i
\(433\) 8.65125e49 0.697925 0.348963 0.937137i \(-0.386534\pi\)
0.348963 + 0.937137i \(0.386534\pi\)
\(434\) 6.99234e49 4.06584e49i 0.539905 0.313939i
\(435\) 7.40295e49i 0.547153i
\(436\) 9.68477e49 1.70161e50i 0.685245 1.20397i
\(437\) −2.56533e48 −0.0173778
\(438\) −9.14615e49 1.57293e50i −0.593239 1.02024i
\(439\) 1.84517e50i 1.14607i −0.819531 0.573034i \(-0.805766\pi\)
0.819531 0.573034i \(-0.194234\pi\)
\(440\) −2.00384e50 + 1.85434e48i −1.19196 + 0.0110304i
\(441\) 7.23740e49 0.412334
\(442\) −1.83374e50 + 1.06627e50i −1.00073 + 0.581892i
\(443\) 6.98396e49i 0.365116i 0.983195 + 0.182558i \(0.0584377\pi\)
−0.983195 + 0.182558i \(0.941562\pi\)
\(444\) 3.10585e49 + 1.76771e49i 0.155562 + 0.0885390i
\(445\) 3.72165e50 1.78605
\(446\) 1.07565e49 + 1.84987e49i 0.0494660 + 0.0850706i
\(447\) 2.13416e50i 0.940555i
\(448\) 2.24937e48 + 1.21525e50i 0.00950120 + 0.513314i
\(449\) 3.84894e49 0.155833 0.0779167 0.996960i \(-0.475173\pi\)
0.0779167 + 0.996960i \(0.475173\pi\)
\(450\) −2.73143e49 + 1.58824e49i −0.106011 + 0.0616424i
\(451\) 2.49564e50i 0.928598i
\(452\) 6.93320e49 1.21816e50i 0.247345 0.434584i
\(453\) −6.12977e49 −0.209690
\(454\) 1.54113e50 + 2.65040e50i 0.505566 + 0.869461i
\(455\) 2.12678e50i 0.669124i
\(456\) 2.70593e48 + 2.92408e50i 0.00816555 + 0.882382i
\(457\) −2.70676e50 −0.783505 −0.391753 0.920071i \(-0.628131\pi\)
−0.391753 + 0.920071i \(0.628131\pi\)
\(458\) −1.67261e50 + 9.72571e49i −0.464462 + 0.270071i
\(459\) 3.80913e50i 1.01481i
\(460\) −4.90406e48 2.79117e48i −0.0125360 0.00713488i
\(461\) 4.88810e50 1.19901 0.599505 0.800371i \(-0.295364\pi\)
0.599505 + 0.800371i \(0.295364\pi\)
\(462\) −7.85198e49 1.35037e50i −0.184834 0.317873i
\(463\) 2.76683e50i 0.625092i 0.949903 + 0.312546i \(0.101182\pi\)
−0.949903 + 0.312546i \(0.898818\pi\)
\(464\) −1.75914e50 2.96191e50i −0.381468 0.642289i
\(465\) 4.28038e50 0.890998
\(466\) 5.34246e50 3.10648e50i 1.06760 0.620779i
\(467\) 1.19702e49i 0.0229658i 0.999934 + 0.0114829i \(0.00365520\pi\)
−0.999934 + 0.0114829i \(0.996345\pi\)
\(468\) 1.77475e50 3.11822e50i 0.326938 0.574429i
\(469\) −5.34194e50 −0.944963
\(470\) 6.73459e49 + 1.15820e50i 0.114407 + 0.196754i
\(471\) 8.04783e50i 1.31305i
\(472\) −4.87347e50 + 4.50989e48i −0.763730 + 0.00706754i
\(473\) −1.40655e51 −2.11736
\(474\) 6.07737e50 3.53381e50i 0.878875 0.511040i
\(475\) 2.09682e50i 0.291328i
\(476\) −3.27772e50 1.86553e50i −0.437562 0.249040i
\(477\) −4.17589e50 −0.535674
\(478\) 5.59029e50 + 9.61405e50i 0.689135 + 1.18516i
\(479\) 1.20440e51i 1.42690i −0.700704 0.713452i \(-0.747131\pi\)
0.700704 0.713452i \(-0.252869\pi\)
\(480\) −3.12977e50 + 5.61932e50i −0.356393 + 0.639883i
\(481\) −2.90987e50 −0.318505
\(482\) 3.59180e50 2.08853e50i 0.377936 0.219759i
\(483\) 4.39851e48i 0.00444949i
\(484\) −8.42221e49 + 1.47978e50i −0.0819152 + 0.143924i
\(485\) 2.89854e50 0.271073
\(486\) 5.31482e50 + 9.14031e50i 0.477969 + 0.822000i
\(487\) 4.39009e50i 0.379685i 0.981815 + 0.189842i \(0.0607977\pi\)
−0.981815 + 0.189842i \(0.939202\pi\)
\(488\) 3.76557e48 + 4.06913e50i 0.00313223 + 0.338474i
\(489\) −2.40494e50 −0.192414
\(490\) 9.13291e50 5.31052e50i 0.702884 0.408706i
\(491\) 1.00454e51i 0.743738i −0.928285 0.371869i \(-0.878717\pi\)
0.928285 0.371869i \(-0.121283\pi\)
\(492\) 6.96181e50 + 3.96234e50i 0.495890 + 0.282238i
\(493\) 1.06892e51 0.732577
\(494\) −1.19687e51 2.05835e51i −0.789288 1.35740i
\(495\) 1.05173e51i 0.667428i
\(496\) −1.71258e51 + 1.01713e51i −1.04592 + 0.621193i
\(497\) 1.98352e50 0.116591
\(498\) −7.66362e50 + 4.45617e50i −0.433587 + 0.252118i
\(499\) 4.17141e50i 0.227181i 0.993528 + 0.113591i \(0.0362352\pi\)
−0.993528 + 0.113591i \(0.963765\pi\)
\(500\) 8.13530e50 1.42937e51i 0.426524 0.749400i
\(501\) 5.87392e50 0.296491
\(502\) 7.97697e50 + 1.37186e51i 0.387675 + 0.666714i
\(503\) 3.05468e51i 1.42947i 0.699397 + 0.714733i \(0.253452\pi\)
−0.699397 + 0.714733i \(0.746548\pi\)
\(504\) 6.37887e50 5.90299e48i 0.287451 0.00266006i
\(505\) −5.10994e49 −0.0221758
\(506\) −2.91727e49 + 1.69630e49i −0.0121932 + 0.00708996i
\(507\) 6.48337e50i 0.261006i
\(508\) 6.22816e50 + 3.54478e50i 0.241518 + 0.137461i
\(509\) −8.74606e48 −0.00326719 −0.00163360 0.999999i \(-0.500520\pi\)
−0.00163360 + 0.999999i \(0.500520\pi\)
\(510\) −1.00323e51 1.72534e51i −0.361051 0.620928i
\(511\) 2.63412e51i 0.913353i
\(512\) −8.30827e49 2.99200e51i −0.0277576 0.999615i
\(513\) 4.27570e51 1.37650
\(514\) −2.01857e51 + 1.17374e51i −0.626246 + 0.364144i
\(515\) 4.25217e51i 1.27138i
\(516\) −2.23319e51 + 3.92371e51i −0.643549 + 1.13071i
\(517\) 8.01237e50 0.222557
\(518\) −2.60062e50 4.47249e50i −0.0696324 0.119752i
\(519\) 1.12996e51i 0.291665i
\(520\) −4.84644e49 5.23714e51i −0.0120603 1.30326i
\(521\) −1.90170e51 −0.456275 −0.228137 0.973629i \(-0.573264\pi\)
−0.228137 + 0.973629i \(0.573264\pi\)
\(522\) −1.56299e51 + 9.08833e50i −0.361590 + 0.210254i
\(523\) 3.74720e51i 0.835939i 0.908461 + 0.417970i \(0.137258\pi\)
−0.908461 + 0.417970i \(0.862742\pi\)
\(524\) −3.69246e51 2.10158e51i −0.794367 0.452117i
\(525\) −3.59521e50 −0.0745928
\(526\) −9.93531e50 1.70865e51i −0.198816 0.341920i
\(527\) 6.18049e51i 1.19295i
\(528\) 1.96430e51 + 3.30734e51i 0.365732 + 0.615793i
\(529\) 5.56652e51 0.999829
\(530\) −5.26958e51 + 3.06411e51i −0.913134 + 0.530960i
\(531\) 2.55787e51i 0.427645i
\(532\) 2.09403e51 3.67920e51i 0.337802 0.593517i
\(533\) −6.52250e51 −1.01531
\(534\) −3.59106e51 6.17583e51i −0.539435 0.927708i
\(535\) 1.02337e52i 1.48358i
\(536\) 1.31544e52 1.21730e50i 1.84051 0.0170321i
\(537\) −6.46667e51 −0.873312
\(538\) −8.72763e50 + 5.07486e50i −0.113771 + 0.0661547i
\(539\) 6.31810e51i 0.795060i
\(540\) 8.17374e51 + 4.65212e51i 0.992978 + 0.565157i
\(541\) 7.19752e51 0.844179 0.422090 0.906554i \(-0.361297\pi\)
0.422090 + 0.906554i \(0.361297\pi\)
\(542\) 3.59285e51 + 6.17890e51i 0.406867 + 0.699720i
\(543\) 1.09651e52i 1.19899i
\(544\) 8.11380e51 + 4.51911e51i 0.856732 + 0.477171i
\(545\) −1.49999e52 −1.52952
\(546\) 3.52926e51 2.05216e51i 0.347555 0.202093i
\(547\) 1.48668e52i 1.41403i 0.707200 + 0.707013i \(0.249958\pi\)
−0.707200 + 0.707013i \(0.750042\pi\)
\(548\) −7.25284e51 + 1.27432e52i −0.666312 + 1.17071i
\(549\) 2.13571e51 0.189526
\(550\) 1.38650e51 + 2.38448e51i 0.118859 + 0.204410i
\(551\) 1.19985e52i 0.993680i
\(552\) 1.00232e48 + 1.08312e50i 8.01978e−5 + 0.00866631i
\(553\) −1.01775e52 −0.786799
\(554\) −1.02500e50 + 5.96009e49i −0.00765667 + 0.00445212i
\(555\) 2.73785e51i 0.197625i
\(556\) −7.00629e51 3.98766e51i −0.488728 0.278161i
\(557\) 2.50750e52 1.69041 0.845205 0.534443i \(-0.179479\pi\)
0.845205 + 0.534443i \(0.179479\pi\)
\(558\) 5.25487e51 + 9.03721e51i 0.342383 + 0.588823i
\(559\) 3.67611e52i 2.31507i
\(560\) 8.00621e51 4.75505e51i 0.487365 0.289456i
\(561\) −1.19358e52 −0.702357
\(562\) −1.47242e51 + 8.56170e50i −0.0837611 + 0.0487046i
\(563\) 1.49382e52i 0.821561i −0.911734 0.410780i \(-0.865256\pi\)
0.911734 0.410780i \(-0.134744\pi\)
\(564\) 1.27213e51 2.23512e51i 0.0676438 0.118850i
\(565\) −1.07382e52 −0.552092
\(566\) −3.32146e51 5.71218e51i −0.165127 0.283981i
\(567\) 1.35169e51i 0.0649828i
\(568\) −4.88435e51 + 4.51996e49i −0.227085 + 0.00210144i
\(569\) −1.47975e52 −0.665358 −0.332679 0.943040i \(-0.607953\pi\)
−0.332679 + 0.943040i \(0.607953\pi\)
\(570\) 1.93667e52 1.12612e52i 0.842237 0.489736i
\(571\) 9.93138e51i 0.417757i 0.977942 + 0.208878i \(0.0669813\pi\)
−0.977942 + 0.208878i \(0.933019\pi\)
\(572\) −2.72214e52 1.54932e52i −1.10761 0.630400i
\(573\) 2.36105e52 0.929326
\(574\) −5.82932e51 1.00251e52i −0.221969 0.381737i
\(575\) 7.76691e49i 0.00286127i
\(576\) −1.57064e52 + 2.90719e50i −0.559822 + 0.0103621i
\(577\) 1.11265e52 0.383724 0.191862 0.981422i \(-0.438547\pi\)
0.191862 + 0.981422i \(0.438547\pi\)
\(578\) 9.92614e50 5.77176e50i 0.0331247 0.0192610i
\(579\) 5.15209e51i 0.166376i
\(580\) −1.30548e52 + 2.29372e52i −0.407979 + 0.716817i
\(581\) 1.28339e52 0.388162
\(582\) −2.79683e51 4.80993e51i −0.0818710 0.140800i
\(583\) 3.64547e52i 1.03288i
\(584\) −6.00254e50 6.48645e52i −0.0164623 1.77895i
\(585\) −2.74875e52 −0.729750
\(586\) −3.44136e52 + 2.00105e52i −0.884455 + 0.514284i
\(587\) 5.20780e52i 1.29578i 0.761735 + 0.647889i \(0.224348\pi\)
−0.761735 + 0.647889i \(0.775652\pi\)
\(588\) −1.76249e52 1.00313e52i −0.424578 0.241650i
\(589\) 6.93753e52 1.61813
\(590\) 1.87687e52 + 3.22779e52i 0.423882 + 0.728983i
\(591\) 1.09311e52i 0.239056i
\(592\) 6.50586e51 + 1.09541e52i 0.137782 + 0.231987i
\(593\) 6.27741e52 1.28748 0.643741 0.765243i \(-0.277381\pi\)
0.643741 + 0.765243i \(0.277381\pi\)
\(594\) 4.86229e52 2.82728e52i 0.965825 0.561598i
\(595\) 2.88935e52i 0.555876i
\(596\) −3.76351e52 + 6.61246e52i −0.701316 + 1.23221i
\(597\) −3.75051e52 −0.676984
\(598\) −4.43339e50 7.62444e50i −0.00775199 0.0133317i
\(599\) 6.62193e52i 1.12170i −0.827919 0.560848i \(-0.810475\pi\)
0.827919 0.560848i \(-0.189525\pi\)
\(600\) 8.85309e51 8.19262e49i 0.145285 0.00134446i
\(601\) −3.86015e52 −0.613747 −0.306873 0.951750i \(-0.599283\pi\)
−0.306873 + 0.951750i \(0.599283\pi\)
\(602\) 5.65021e52 3.28543e52i 0.870424 0.506126i
\(603\) 6.90416e52i 1.03058i
\(604\) −1.89924e52 1.08096e52i −0.274712 0.156353i
\(605\) 1.30444e52 0.182841
\(606\) 4.93065e50 + 8.47962e50i 0.00669767 + 0.0115185i
\(607\) 4.71183e52i 0.620303i 0.950687 + 0.310152i \(0.100380\pi\)
−0.950687 + 0.310152i \(0.899620\pi\)
\(608\) −5.07265e52 + 9.10765e52i −0.647243 + 1.16209i
\(609\) −2.05726e52 −0.254426
\(610\) 2.69507e52 1.56710e52i 0.323075 0.187858i
\(611\) 2.09408e52i 0.243338i
\(612\) 2.41109e52 4.23627e52i 0.271604 0.477207i
\(613\) 3.99927e52 0.436749 0.218374 0.975865i \(-0.429925\pi\)
0.218374 + 0.975865i \(0.429925\pi\)
\(614\) 2.70442e52 + 4.65100e52i 0.286335 + 0.492433i
\(615\) 6.13692e52i 0.629975i
\(616\) −5.15319e50 5.56862e52i −0.00512912 0.554261i
\(617\) −1.53502e53 −1.48148 −0.740740 0.671792i \(-0.765525\pi\)
−0.740740 + 0.671792i \(0.765525\pi\)
\(618\) −7.05621e52 + 4.10298e52i −0.660375 + 0.383988i
\(619\) 3.66312e52i 0.332452i −0.986088 0.166226i \(-0.946842\pi\)
0.986088 0.166226i \(-0.0531581\pi\)
\(620\) 1.32623e53 + 7.54828e52i 1.16728 + 0.664364i
\(621\) 1.58378e51 0.0135193
\(622\) −2.52069e52 4.33503e52i −0.208690 0.358901i
\(623\) 1.03424e53i 0.830516i
\(624\) −8.64393e52 + 5.13380e52i −0.673293 + 0.399882i
\(625\) −1.54987e53 −1.17105
\(626\) 4.87029e52 2.83193e52i 0.356979 0.207573i
\(627\) 1.33978e53i 0.952688i
\(628\) 1.41920e53 2.49353e53i 0.979064 1.72021i
\(629\) −3.95321e52 −0.264599
\(630\) −2.45663e52 4.22485e52i −0.159540 0.274373i
\(631\) 1.54525e53i 0.973733i −0.873476 0.486867i \(-0.838140\pi\)
0.873476 0.486867i \(-0.161860\pi\)
\(632\) 2.50618e53 2.31921e51i 1.53245 0.0141813i
\(633\) −1.43288e53 −0.850233
\(634\) −1.55035e53 + 9.01484e52i −0.892759 + 0.519113i
\(635\) 5.49020e52i 0.306823i
\(636\) 1.01694e53 + 5.78793e52i 0.551580 + 0.313934i
\(637\) 1.65127e53 0.869300
\(638\) 7.93392e52 + 1.36446e53i 0.405410 + 0.697216i
\(639\) 2.56359e52i 0.127154i
\(640\) −1.96067e53 + 1.18916e53i −0.944028 + 0.572560i
\(641\) −5.03488e52 −0.235335 −0.117667 0.993053i \(-0.537542\pi\)
−0.117667 + 0.993053i \(0.537542\pi\)
\(642\) −1.69821e53 + 9.87460e52i −0.770595 + 0.448078i
\(643\) 3.21557e52i 0.141661i 0.997488 + 0.0708304i \(0.0225649\pi\)
−0.997488 + 0.0708304i \(0.977435\pi\)
\(644\) 7.75660e50 1.36283e51i 0.00331772 0.00582922i
\(645\) 3.45879e53 1.43645
\(646\) −1.62602e53 2.79639e53i −0.655702 1.12766i
\(647\) 3.24139e53i 1.26926i 0.772817 + 0.634629i \(0.218847\pi\)
−0.772817 + 0.634629i \(0.781153\pi\)
\(648\) −3.08018e50 3.32849e52i −0.00117125 0.126568i
\(649\) 2.23297e53 0.824582
\(650\) −6.23198e52 + 3.62371e52i −0.223497 + 0.129957i
\(651\) 1.18951e53i 0.414314i
\(652\) −7.45143e52 4.24101e52i −0.252078 0.143471i
\(653\) 2.13563e53 0.701740 0.350870 0.936424i \(-0.385886\pi\)
0.350870 + 0.936424i \(0.385886\pi\)
\(654\) 1.44736e53 + 2.48913e53i 0.461954 + 0.794458i
\(655\) 3.25495e53i 1.00916i
\(656\) 1.45830e53 + 2.45538e53i 0.439211 + 0.739511i
\(657\) −3.40446e53 −0.996107
\(658\) −3.21862e52 + 1.87153e52i −0.0914908 + 0.0531992i
\(659\) 1.46403e53i 0.404320i 0.979352 + 0.202160i \(0.0647962\pi\)
−0.979352 + 0.202160i \(0.935204\pi\)
\(660\) 1.45773e53 2.56122e53i 0.391149 0.687247i
\(661\) 6.49219e53 1.69264 0.846318 0.532677i \(-0.178814\pi\)
0.846318 + 0.532677i \(0.178814\pi\)
\(662\) −1.75497e53 3.01816e53i −0.444599 0.764612i
\(663\) 3.11950e53i 0.767940i
\(664\) −3.16032e53 + 2.92455e51i −0.756026 + 0.00699625i
\(665\) −3.24326e53 −0.753999
\(666\) 5.78044e52 3.36116e52i 0.130602 0.0759414i
\(667\) 4.44442e51i 0.00975942i
\(668\) 1.81997e53 + 1.03584e53i 0.388429 + 0.221076i
\(669\) −3.14694e52 −0.0652817
\(670\) −5.06600e53 8.71239e53i −1.02151 1.75678i
\(671\) 1.86443e53i 0.365443i
\(672\) −1.56160e53 8.69758e52i −0.297545 0.165723i
\(673\) −7.05868e53 −1.30749 −0.653743 0.756716i \(-0.726802\pi\)
−0.653743 + 0.756716i \(0.726802\pi\)
\(674\) −5.50185e53 + 3.19916e53i −0.990764 + 0.576100i
\(675\) 1.29453e53i 0.226643i
\(676\) 1.14332e53 2.00880e53i 0.194616 0.341940i
\(677\) 7.23413e53 1.19730 0.598648 0.801012i \(-0.295705\pi\)
0.598648 + 0.801012i \(0.295705\pi\)
\(678\) 1.03614e53 + 1.78194e53i 0.166746 + 0.286766i
\(679\) 8.05498e52i 0.126049i
\(680\) −6.58414e51 7.11494e53i −0.0100191 1.08268i
\(681\) −4.50876e53 −0.667210
\(682\) 7.88930e53 4.58739e53i 1.13536 0.660181i
\(683\) 5.58560e53i 0.781765i −0.920440 0.390883i \(-0.872170\pi\)
0.920440 0.390883i \(-0.127830\pi\)
\(684\) 4.75517e53 + 2.70642e53i 0.647292 + 0.368409i
\(685\) 1.12333e54 1.48726
\(686\) 3.47978e53 + 5.98445e53i 0.448120 + 0.770666i
\(687\) 2.84538e53i 0.356420i
\(688\) −1.38386e54 + 8.21902e53i −1.68621 + 1.00147i
\(689\) −9.52765e53 −1.12933
\(690\) 7.17372e51 4.17131e51i 0.00827203 0.00480994i
\(691\) 2.35453e53i 0.264132i −0.991241 0.132066i \(-0.957839\pi\)
0.991241 0.132066i \(-0.0421612\pi\)
\(692\) −1.99264e53 + 3.50106e53i −0.217477 + 0.382106i
\(693\) −2.92273e53 −0.310354
\(694\) 8.05205e53 + 1.38477e54i 0.831911 + 1.43070i
\(695\) 6.17613e53i 0.620876i
\(696\) 5.06595e53 4.68802e51i 0.495548 0.00458579i
\(697\) −8.86117e53 −0.843467
\(698\) 1.12145e54 6.52092e53i 1.03879 0.604027i
\(699\) 9.08840e53i 0.819259i
\(700\) −1.11393e53 6.34000e52i −0.0977230 0.0556194i
\(701\) −1.17774e54 −1.00556 −0.502779 0.864415i \(-0.667689\pi\)
−0.502779 + 0.864415i \(0.667689\pi\)
\(702\) 7.38925e53 + 1.27079e54i 0.614038 + 1.05601i
\(703\) 4.43743e53i 0.358906i
\(704\) 2.53791e52 + 1.37114e54i 0.0199800 + 1.07945i
\(705\) −1.97029e53 −0.150986
\(706\) 5.61422e53 3.26450e53i 0.418793 0.243516i
\(707\) 1.42004e52i 0.0103118i
\(708\) 3.54530e53 6.22907e53i 0.250623 0.440343i
\(709\) 2.04205e54 1.40536 0.702679 0.711507i \(-0.251987\pi\)
0.702679 + 0.711507i \(0.251987\pi\)
\(710\) 1.88106e53 + 3.23500e53i 0.126036 + 0.216753i
\(711\) 1.31539e54i 0.858086i
\(712\) −2.35678e52 2.54678e54i −0.0149693 1.61760i
\(713\) 2.56976e52 0.0158925
\(714\) 4.79469e53 2.78797e53i 0.288732 0.167889i
\(715\) 2.39960e54i 1.40710i
\(716\) −2.00363e54 1.14037e54i −1.14411 0.651177i
\(717\) −1.63551e54 −0.909470
\(718\) −9.36149e53 1.60997e54i −0.506968 0.871873i
\(719\) 2.98403e54i 1.57382i −0.617069 0.786909i \(-0.711680\pi\)
0.617069 0.786909i \(-0.288320\pi\)
\(720\) 6.14564e53 + 1.03476e54i 0.315682 + 0.531523i
\(721\) 1.18167e54 0.591190
\(722\) 1.36488e54 7.93637e53i 0.665101 0.386736i
\(723\) 6.11024e53i 0.290022i
\(724\) −1.93366e54 + 3.39742e54i −0.894019 + 1.57079i
\(725\) 3.63273e53 0.163610
\(726\) −1.25867e53 2.16464e53i −0.0552226 0.0949706i
\(727\) 3.16380e54i 1.35224i 0.736789 + 0.676122i \(0.236341\pi\)
−0.736789 + 0.676122i \(0.763659\pi\)
\(728\) 1.45539e54 1.34682e52i 0.606015 0.00560805i
\(729\) −1.86692e54 −0.757362
\(730\) −4.29610e54 + 2.49806e54i −1.69801 + 0.987343i
\(731\) 4.99419e54i 1.92325i
\(732\) −5.20101e53 2.96017e53i −0.195154 0.111073i
\(733\) 5.32886e54 1.94831 0.974156 0.225874i \(-0.0725239\pi\)
0.974156 + 0.225874i \(0.0725239\pi\)
\(734\) −1.76868e54 3.04173e54i −0.630121 1.08367i
\(735\) 1.55366e54i 0.539381i
\(736\) −1.87898e52 + 3.37360e52i −0.00635689 + 0.0114134i
\(737\) −6.02719e54 −1.98716
\(738\) 1.29569e54 7.53407e53i 0.416324 0.242080i
\(739\) 3.62860e54i 1.13630i 0.822924 + 0.568152i \(0.192341\pi\)
−0.822924 + 0.568152i \(0.807659\pi\)
\(740\) 4.82808e53 8.48292e53i 0.147357 0.258906i
\(741\) 3.50160e54 1.04165
\(742\) −8.51509e53 1.46441e54i −0.246897 0.424607i
\(743\) 5.37861e54i 1.52014i −0.649844 0.760068i \(-0.725166\pi\)
0.649844 0.760068i \(-0.274834\pi\)
\(744\) −2.71061e52 2.92913e54i −0.00746761 0.806963i
\(745\) 5.82896e54 1.56539
\(746\) 4.19682e53 2.44033e53i 0.109871 0.0638867i
\(747\) 1.65871e54i 0.423331i
\(748\) −3.69818e54 2.10483e54i −0.920148 0.523706i
\(749\) 2.84392e54 0.689863
\(750\) 1.21579e54 + 2.09090e54i 0.287539 + 0.494502i
\(751\) 7.31050e54i 1.68573i 0.538124 + 0.842866i \(0.319133\pi\)
−0.538124 + 0.842866i \(0.680867\pi\)
\(752\) 7.88310e53 4.68193e53i 0.177239 0.105266i
\(753\) −2.33376e54 −0.511625
\(754\) −3.56609e54 + 2.07358e54i −0.762319 + 0.443266i
\(755\) 1.67420e54i 0.348992i
\(756\) −1.29281e54 + 2.27147e54i −0.262798 + 0.461735i
\(757\) 3.78254e54 0.749828 0.374914 0.927060i \(-0.377672\pi\)
0.374914 + 0.927060i \(0.377672\pi\)
\(758\) 1.24391e54 + 2.13925e54i 0.240477 + 0.413567i
\(759\) 4.96275e52i 0.00935682i
\(760\) 7.98643e54 7.39063e52i 1.46857 0.0135901i
\(761\) −8.00697e54 −1.43602 −0.718009 0.696034i \(-0.754946\pi\)
−0.718009 + 0.696034i \(0.754946\pi\)
\(762\) −9.11063e53 + 5.29756e53i −0.159369 + 0.0926684i
\(763\) 4.16844e54i 0.711225i
\(764\) 7.31546e54 + 4.16362e54i 1.21750 + 0.692943i
\(765\) −3.73433e54 −0.606241
\(766\) −2.86277e54 4.92333e54i −0.453357 0.779673i
\(767\) 5.83600e54i 0.901578i
\(768\) 3.86521e54 + 2.10617e54i 0.582519 + 0.317417i
\(769\) −1.97586e54 −0.290506 −0.145253 0.989395i \(-0.546400\pi\)
−0.145253 + 0.989395i \(0.546400\pi\)
\(770\) −3.68821e54 + 2.14458e54i −0.529044 + 0.307623i
\(771\) 3.43392e54i 0.480571i
\(772\) 9.08551e53 1.59632e54i 0.124057 0.217967i
\(773\) 1.38648e54 0.184716 0.0923580 0.995726i \(-0.470560\pi\)
0.0923580 + 0.995726i \(0.470560\pi\)
\(774\) 4.24624e54 + 7.30258e54i 0.551983 + 0.949289i
\(775\) 2.10044e54i 0.266427i
\(776\) −1.83554e52 1.98351e54i −0.00227191 0.245506i
\(777\) 7.60843e53 0.0918958
\(778\) −7.43534e54 + 4.32343e54i −0.876372 + 0.509584i
\(779\) 9.94656e54i 1.14409i
\(780\) 6.69391e54 + 3.80986e54i 0.751419 + 0.427673i
\(781\) 2.23796e54 0.245178
\(782\) −6.02300e52 1.03582e53i −0.00643998 0.0110753i
\(783\) 7.40763e54i 0.773047i
\(784\) −3.69191e54 6.21616e54i −0.376050 0.633166i
\(785\) −2.19808e55 −2.18534
\(786\) 5.40137e54 3.14074e54i 0.524174 0.304792i
\(787\) 3.64802e54i 0.345571i −0.984959 0.172786i \(-0.944723\pi\)
0.984959 0.172786i \(-0.0552768\pi\)
\(788\) −1.92765e54 + 3.38686e54i −0.178250 + 0.313184i
\(789\) 2.90670e54 0.262384
\(790\) −9.65178e54 1.65989e55i −0.850536 1.46273i
\(791\) 2.98413e54i 0.256723i
\(792\) 7.19714e54 6.66021e52i 0.604479 0.00559384i
\(793\) 4.87281e54 0.399566
\(794\) 9.34515e54 5.43393e54i 0.748164 0.435035i
\(795\) 8.96442e54i 0.700724i
\(796\) −1.16206e55 6.61388e54i −0.886908 0.504787i
\(797\) −1.78465e54 −0.132998 −0.0664988 0.997787i \(-0.521183\pi\)
−0.0664988 + 0.997787i \(0.521183\pi\)
\(798\) 3.12946e54 + 5.38198e54i 0.227727 + 0.391640i
\(799\) 2.84492e54i 0.202154i
\(800\) 2.75748e54 + 1.53582e54i 0.191338 + 0.106569i
\(801\) −1.33670e55 −0.905764
\(802\) 1.03686e55 6.02904e54i 0.686131 0.398965i
\(803\) 2.97202e55i 1.92069i
\(804\) −9.56941e54 + 1.68134e55i −0.603977 + 1.06118i
\(805\) −1.20135e53 −0.00740540
\(806\) 1.19894e55 + 2.06191e55i 0.721826 + 1.24138i
\(807\) 1.48471e54i 0.0873062i
\(808\) 3.23594e51 + 3.49681e53i 0.000185860 + 0.0200843i
\(809\) 2.37044e55 1.32986 0.664932 0.746904i \(-0.268461\pi\)
0.664932 + 0.746904i \(0.268461\pi\)
\(810\) −2.20453e54 + 1.28187e54i −0.120809 + 0.0702470i
\(811\) 1.98008e55i 1.05995i 0.848012 + 0.529977i \(0.177799\pi\)
−0.848012 + 0.529977i \(0.822201\pi\)
\(812\) −6.37421e54 3.62790e54i −0.333320 0.189710i
\(813\) −1.05113e55 −0.536954
\(814\) −2.93422e54 5.04621e54i −0.146430 0.251826i
\(815\) 6.56853e54i 0.320238i
\(816\) −1.17432e55 + 6.97455e54i −0.559339 + 0.332203i
\(817\) 5.60592e55 2.60872
\(818\) −5.82987e54 + 3.38989e54i −0.265061 + 0.154125i
\(819\) 7.63873e54i 0.339334i
\(820\) 1.08222e55 1.90146e55i 0.469735 0.825322i
\(821\) 1.25594e55 0.532658 0.266329 0.963882i \(-0.414189\pi\)
0.266329 + 0.963882i \(0.414189\pi\)
\(822\) −1.08391e55 1.86409e55i −0.449190 0.772507i
\(823\) 2.31895e55i 0.939064i 0.882916 + 0.469532i \(0.155577\pi\)
−0.882916 + 0.469532i \(0.844423\pi\)
\(824\) −2.90983e55 + 2.69275e53i −1.15146 + 0.0106556i
\(825\) −4.05639e54 −0.156861
\(826\) −8.96998e54 + 5.21578e54i −0.338977 + 0.197105i
\(827\) 2.00297e55i 0.739724i −0.929087 0.369862i \(-0.879405\pi\)
0.929087 0.369862i \(-0.120595\pi\)
\(828\) 1.76138e53 + 1.00250e53i 0.00635737 + 0.00361832i
\(829\) −3.56852e54 −0.125879 −0.0629394 0.998017i \(-0.520047\pi\)
−0.0629394 + 0.998017i \(0.520047\pi\)
\(830\) 1.21710e55 + 2.09314e55i 0.419606 + 0.721630i
\(831\) 1.74370e53i 0.00587559i
\(832\) −3.58355e55 + 6.63299e53i −1.18024 + 0.0218457i
\(833\) 2.24334e55 0.722172
\(834\) 1.02489e55 5.95942e54i 0.322494 0.187521i
\(835\) 1.60432e55i 0.493457i
\(836\) 2.36265e55 4.15116e55i 0.710363 1.24810i
\(837\) −4.28310e55 −1.25885
\(838\) 2.63349e54 + 4.52901e54i 0.0756649 + 0.130127i
\(839\) 1.33460e55i 0.374865i −0.982278 0.187432i \(-0.939983\pi\)
0.982278 0.187432i \(-0.0600165\pi\)
\(840\) 1.26720e53 + 1.36936e55i 0.00347967 + 0.376019i
\(841\) −1.64624e55 −0.441947
\(842\) 4.10915e55 2.38935e55i 1.07851 0.627119i
\(843\) 2.50483e54i 0.0642769i
\(844\) −4.43960e55 2.52682e55i −1.11388 0.633968i
\(845\) −1.77078e55 −0.434398
\(846\) −2.41885e54 4.15989e54i −0.0580193 0.0997803i
\(847\) 3.62502e54i 0.0850209i
\(848\) 2.13019e55 + 3.58665e55i 0.488536 + 0.822561i
\(849\) 9.71735e54 0.217922
\(850\) −8.46648e54 + 4.92301e54i −0.185671 + 0.107962i
\(851\) 1.64369e53i 0.00352499i
\(852\) 3.55322e54 6.24298e54i 0.0745194 0.130930i
\(853\) 2.02762e55 0.415867 0.207933 0.978143i \(-0.433326\pi\)
0.207933 + 0.978143i \(0.433326\pi\)
\(854\) 4.35495e54 + 7.48954e54i 0.0873541 + 0.150230i
\(855\) 4.19174e55i 0.822315i
\(856\) −7.00307e55 + 6.48062e53i −1.34365 + 0.0124341i
\(857\) −8.02526e55 −1.50600 −0.752998 0.658023i \(-0.771393\pi\)
−0.752998 + 0.658023i \(0.771393\pi\)
\(858\) 3.98199e55 2.31541e55i 0.730872 0.424980i
\(859\) 4.83819e55i 0.868586i 0.900772 + 0.434293i \(0.143002\pi\)
−0.900772 + 0.434293i \(0.856998\pi\)
\(860\) 1.07167e56 + 6.09944e55i 1.88187 + 1.07107i
\(861\) 1.70544e55 0.292938
\(862\) 7.98627e54 + 1.37346e55i 0.134186 + 0.230769i
\(863\) 5.23121e55i 0.859799i 0.902877 + 0.429900i \(0.141451\pi\)
−0.902877 + 0.429900i \(0.858549\pi\)
\(864\) 3.13176e55 5.62288e55i 0.503532 0.904061i
\(865\) 3.08623e55 0.485425
\(866\) −3.92106e55 + 2.27998e55i −0.603342 + 0.350825i
\(867\) 1.68860e54i 0.0254193i
\(868\) −2.09765e55 + 3.68557e55i −0.308929 + 0.542787i
\(869\) −1.14831e56 −1.65456
\(870\) −1.95100e55 3.35528e55i −0.275037 0.473002i
\(871\) 1.57524e56i 2.17271i
\(872\) 9.49889e53 + 1.02647e56i 0.0128192 + 1.38526i
\(873\) −1.04106e55 −0.137469
\(874\) 1.16270e54 6.76074e53i 0.0150228 0.00873529i
\(875\) 3.50153e55i 0.442695i
\(876\) 8.29072e55 + 4.71870e55i 1.02569 + 0.583773i
\(877\) 4.25545e55 0.515173 0.257586 0.966255i \(-0.417073\pi\)
0.257586 + 0.966255i \(0.417073\pi\)
\(878\) 4.86282e55 + 8.36298e55i 0.576093 + 0.990752i
\(879\) 5.85432e55i 0.678716i
\(880\) 9.03323e55 5.36502e55i 1.02488 0.608696i
\(881\) 1.17792e56 1.30789 0.653947 0.756541i \(-0.273112\pi\)
0.653947 + 0.756541i \(0.273112\pi\)
\(882\) −3.28025e55 + 1.90737e55i −0.356454 + 0.207268i
\(883\) 1.62280e56i 1.72589i 0.505301 + 0.862943i \(0.331382\pi\)
−0.505301 + 0.862943i \(0.668618\pi\)
\(884\) 5.50110e55 9.66541e55i 0.572607 1.00607i
\(885\) −5.49100e55 −0.559409
\(886\) −1.84058e55 3.16538e55i −0.183532 0.315635i
\(887\) 4.37192e55i 0.426701i 0.976976 + 0.213351i \(0.0684376\pi\)
−0.976976 + 0.213351i \(0.931562\pi\)
\(888\) −1.87355e55 + 1.73378e53i −0.178986 + 0.00165633i
\(889\) 1.52572e55 0.142672
\(890\) −1.68678e56 + 9.80815e55i −1.54401 + 0.897794i
\(891\) 1.52508e55i 0.136652i
\(892\) −9.75043e54 5.54949e54i −0.0855247 0.0486767i
\(893\) −3.19339e55 −0.274204
\(894\) −5.62444e55 9.67279e55i −0.472788 0.813090i
\(895\) 1.76622e56i 1.45347i
\(896\) −3.30466e55 5.44867e55i −0.266241 0.438973i
\(897\) 1.29704e54 0.0102305
\(898\) −1.74448e55 + 1.01436e55i −0.134715 + 0.0783326i
\(899\) 1.20192e56i 0.908747i
\(900\) 8.19410e54 1.43970e55i 0.0606588 0.106577i
\(901\) −1.29438e56 −0.938192
\(902\) −6.57709e55 1.13111e56i −0.466777 0.802753i
\(903\) 9.61193e55i 0.667949i
\(904\) 6.80013e53 + 7.34833e55i 0.00462718 + 0.500021i
\(905\) 2.99487e56 1.99551
\(906\) 2.77823e55 1.61546e55i 0.181273 0.105405i
\(907\) 2.07283e56i 1.32442i −0.749319 0.662210i \(-0.769619\pi\)
0.749319 0.662210i \(-0.230381\pi\)
\(908\) −1.39699e56 7.95102e55i −0.874102 0.497499i
\(909\) 1.83533e54 0.0112461
\(910\) −5.60499e55 9.63935e55i −0.336348 0.578444i
\(911\) 2.22241e55i 0.130609i −0.997865 0.0653046i \(-0.979198\pi\)
0.997865 0.0653046i \(-0.0208019\pi\)
\(912\) −7.82885e55 1.31817e56i −0.450605 0.758697i
\(913\) 1.44802e56 0.816265
\(914\) 1.22680e56 7.13348e55i 0.677324 0.393844i
\(915\) 4.58475e55i 0.247922i
\(916\) 5.01771e55 8.81609e55i 0.265761 0.466941i
\(917\) −9.04544e55 −0.469258
\(918\) 1.00387e56 + 1.72643e56i 0.510113 + 0.877281i
\(919\) 2.55150e56i 1.26999i −0.772516 0.634996i \(-0.781002\pi\)
0.772516 0.634996i \(-0.218998\pi\)
\(920\) 2.95829e54 2.73760e52i 0.0144236 0.000133475i
\(921\) −7.91211e55 −0.377885
\(922\) −2.21546e56 + 1.28823e56i −1.03652 + 0.602705i
\(923\) 5.84903e55i 0.268072i
\(924\) 7.11760e55 + 4.05101e55i 0.319570 + 0.181884i
\(925\) −1.34350e55 −0.0590942
\(926\) −7.29180e55 1.25403e56i −0.314214 0.540379i
\(927\) 1.52725e56i 0.644754i
\(928\) 1.57790e56 + 8.78835e55i 0.652630 + 0.363493i
\(929\) 1.87803e56 0.761035 0.380517 0.924774i \(-0.375746\pi\)
0.380517 + 0.924774i \(0.375746\pi\)
\(930\) −1.94002e56 + 1.12807e56i −0.770249 + 0.447877i
\(931\) 2.51813e56i 0.979566i
\(932\) −1.60270e56 + 2.81594e56i −0.610873 + 1.07330i
\(933\) 7.37459e55 0.275415
\(934\) −3.15467e54 5.42533e54i −0.0115442 0.0198534i
\(935\) 3.25999e56i 1.16895i
\(936\) 1.74069e54 + 1.88101e56i 0.00611617 + 0.660923i
\(937\) 2.48823e56 0.856719 0.428359 0.903608i \(-0.359092\pi\)
0.428359 + 0.903608i \(0.359092\pi\)
\(938\) 2.42116e56 1.40783e56i 0.816901 0.475004i
\(939\) 8.28517e55i 0.273940i
\(940\) −6.10472e55 3.47452e55i −0.197805 0.112581i
\(941\) −3.78718e56 −1.20258 −0.601289 0.799032i \(-0.705346\pi\)
−0.601289 + 0.799032i \(0.705346\pi\)
\(942\) 2.12095e56 + 3.64757e56i 0.660030 + 1.13511i
\(943\) 3.68435e54i 0.0112367i
\(944\) 2.19694e56 1.30481e56i 0.656676 0.390013i
\(945\) 2.00233e56 0.586584
\(946\) 6.37500e56 3.70688e56i 1.83041 1.06433i
\(947\) 3.77159e56i 1.06139i −0.847563 0.530695i \(-0.821931\pi\)
0.847563 0.530695i \(-0.178069\pi\)
\(948\) −1.82317e56 + 3.20330e56i −0.502885 + 0.883567i
\(949\) −7.76755e56 −2.10003
\(950\) −5.52602e55 9.50352e55i −0.146441 0.251847i
\(951\) 2.63740e56i 0.685088i
\(952\) 1.97723e56 1.82972e54i 0.503448 0.00465890i
\(953\) −1.20914e56 −0.301794 −0.150897 0.988549i \(-0.548216\pi\)
−0.150897 + 0.988549i \(0.548216\pi\)
\(954\) 1.89267e56 1.10053e56i 0.463079 0.269267i
\(955\) 6.44867e56i 1.54670i
\(956\) −5.06744e56 2.88415e56i −1.19148 0.678138i
\(957\) −2.32117e56 −0.535032
\(958\) 3.17411e56 + 5.45876e56i 0.717260 + 1.23353i
\(959\) 3.12171e56i 0.691574i
\(960\) −6.24088e54 3.37171e56i −0.0135548 0.732313i
\(961\) −2.25335e56 −0.479827
\(962\) 1.31886e56 7.66875e55i 0.275341 0.160103i
\(963\) 3.67561e56i 0.752368i
\(964\) −1.07752e56 + 1.89319e56i −0.216252 + 0.379953i
\(965\) −1.40717e56 −0.276904
\(966\) 1.15920e54 + 1.99356e54i 0.00223662 + 0.00384649i
\(967\) 4.33299e56i 0.819756i −0.912140 0.409878i \(-0.865571\pi\)
0.912140 0.409878i \(-0.134429\pi\)
\(968\) −8.26056e53 8.92650e55i −0.00153242 0.165596i
\(969\) 4.75711e56 0.865349
\(970\) −1.31372e56 + 7.63890e55i −0.234337 + 0.136260i
\(971\) 1.05882e57i 1.85206i −0.377450 0.926030i \(-0.623199\pi\)
0.377450 0.926030i \(-0.376801\pi\)
\(972\) −4.81774e56 2.74203e56i −0.826388 0.470342i
\(973\) −1.71633e56 −0.288707
\(974\) −1.15698e56 1.98975e56i −0.190856 0.328230i
\(975\) 1.06016e56i 0.171508i
\(976\) −1.08946e56 1.83435e56i −0.172848 0.291029i
\(977\) −4.13148e55 −0.0642849 −0.0321424 0.999483i \(-0.510233\pi\)
−0.0321424 + 0.999483i \(0.510233\pi\)
\(978\) 1.09000e56 6.33805e55i 0.166337 0.0967203i
\(979\) 1.16691e57i 1.74649i
\(980\) −2.73981e56 + 4.81383e56i −0.402184 + 0.706636i
\(981\) 5.38748e56 0.775666
\(982\) 2.64741e56 + 4.55295e56i 0.373854 + 0.642946i
\(983\) 6.30241e55i 0.0872950i −0.999047 0.0436475i \(-0.986102\pi\)
0.999047 0.0436475i \(-0.0138979\pi\)
\(984\) −4.19959e56 + 3.88629e54i −0.570559 + 0.00527994i
\(985\) 2.98556e56 0.397867
\(986\) −4.84473e56 + 2.81707e56i −0.633298 + 0.368244i
\(987\) 5.47539e55i 0.0702085i
\(988\) 1.08493e57 + 6.17493e56i 1.36465 + 0.776694i
\(989\) 2.07651e55 0.0256216
\(990\) −2.77176e56 4.76681e56i −0.335495 0.576978i
\(991\) 7.77140e56i 0.922783i 0.887197 + 0.461392i \(0.152650\pi\)
−0.887197 + 0.461392i \(0.847350\pi\)
\(992\) 5.08143e56 9.12340e56i 0.591921 1.06276i
\(993\) 5.13439e56 0.586750
\(994\) −8.99001e55 + 5.22742e55i −0.100790 + 0.0586066i
\(995\) 1.02437e57i 1.12672i
\(996\) 2.29904e56 4.03939e56i 0.248095 0.435902i
\(997\) 5.00009e55 0.0529383 0.0264692 0.999650i \(-0.491574\pi\)
0.0264692 + 0.999650i \(0.491574\pi\)
\(998\) −1.09935e56 1.89063e56i −0.114197 0.196393i
\(999\) 2.73958e56i 0.279216i
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 4.39.b.a.3.4 yes 18
4.3 odd 2 inner 4.39.b.a.3.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.39.b.a.3.3 18 4.3 odd 2 inner
4.39.b.a.3.4 yes 18 1.1 even 1 trivial