Properties

Label 8034.2.a.y.1.3
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $13$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 25 x^{11} + 134 x^{10} + 196 x^{9} - 1151 x^{8} - 801 x^{7} + 4263 x^{6} + 2205 x^{5} - 6840 x^{4} - 3579 x^{3} + 3559 x^{2} + 1839 x - 180 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.50145\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.68854 q^{5} -1.00000 q^{6} +4.50145 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.68854 q^{5} -1.00000 q^{6} +4.50145 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.68854 q^{10} -0.787053 q^{11} -1.00000 q^{12} +1.00000 q^{13} +4.50145 q^{14} +1.68854 q^{15} +1.00000 q^{16} -0.120738 q^{17} +1.00000 q^{18} +2.16827 q^{19} -1.68854 q^{20} -4.50145 q^{21} -0.787053 q^{22} -4.63632 q^{23} -1.00000 q^{24} -2.14882 q^{25} +1.00000 q^{26} -1.00000 q^{27} +4.50145 q^{28} +0.430855 q^{29} +1.68854 q^{30} +2.16827 q^{31} +1.00000 q^{32} +0.787053 q^{33} -0.120738 q^{34} -7.60090 q^{35} +1.00000 q^{36} +5.51041 q^{37} +2.16827 q^{38} -1.00000 q^{39} -1.68854 q^{40} +10.5241 q^{41} -4.50145 q^{42} +2.43028 q^{43} -0.787053 q^{44} -1.68854 q^{45} -4.63632 q^{46} +1.33865 q^{47} -1.00000 q^{48} +13.2631 q^{49} -2.14882 q^{50} +0.120738 q^{51} +1.00000 q^{52} +8.80373 q^{53} -1.00000 q^{54} +1.32897 q^{55} +4.50145 q^{56} -2.16827 q^{57} +0.430855 q^{58} -9.51134 q^{59} +1.68854 q^{60} -9.33774 q^{61} +2.16827 q^{62} +4.50145 q^{63} +1.00000 q^{64} -1.68854 q^{65} +0.787053 q^{66} +15.9992 q^{67} -0.120738 q^{68} +4.63632 q^{69} -7.60090 q^{70} -14.3213 q^{71} +1.00000 q^{72} +10.9288 q^{73} +5.51041 q^{74} +2.14882 q^{75} +2.16827 q^{76} -3.54288 q^{77} -1.00000 q^{78} -11.7235 q^{79} -1.68854 q^{80} +1.00000 q^{81} +10.5241 q^{82} +9.92233 q^{83} -4.50145 q^{84} +0.203871 q^{85} +2.43028 q^{86} -0.430855 q^{87} -0.787053 q^{88} +4.51314 q^{89} -1.68854 q^{90} +4.50145 q^{91} -4.63632 q^{92} -2.16827 q^{93} +1.33865 q^{94} -3.66122 q^{95} -1.00000 q^{96} +1.26242 q^{97} +13.2631 q^{98} -0.787053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} - 13 q^{3} + 13 q^{4} + 3 q^{5} - 13 q^{6} + 5 q^{7} + 13 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{2} - 13 q^{3} + 13 q^{4} + 3 q^{5} - 13 q^{6} + 5 q^{7} + 13 q^{8} + 13 q^{9} + 3 q^{10} + 8 q^{11} - 13 q^{12} + 13 q^{13} + 5 q^{14} - 3 q^{15} + 13 q^{16} - q^{17} + 13 q^{18} + 5 q^{19} + 3 q^{20} - 5 q^{21} + 8 q^{22} - 9 q^{23} - 13 q^{24} + 24 q^{25} + 13 q^{26} - 13 q^{27} + 5 q^{28} - q^{29} - 3 q^{30} + 5 q^{31} + 13 q^{32} - 8 q^{33} - q^{34} + 28 q^{35} + 13 q^{36} + 35 q^{37} + 5 q^{38} - 13 q^{39} + 3 q^{40} + 24 q^{41} - 5 q^{42} + 9 q^{43} + 8 q^{44} + 3 q^{45} - 9 q^{46} - 8 q^{47} - 13 q^{48} - 16 q^{49} + 24 q^{50} + q^{51} + 13 q^{52} + 32 q^{53} - 13 q^{54} - 4 q^{55} + 5 q^{56} - 5 q^{57} - q^{58} + 12 q^{59} - 3 q^{60} + 17 q^{61} + 5 q^{62} + 5 q^{63} + 13 q^{64} + 3 q^{65} - 8 q^{66} + 20 q^{67} - q^{68} + 9 q^{69} + 28 q^{70} + 16 q^{71} + 13 q^{72} + 46 q^{73} + 35 q^{74} - 24 q^{75} + 5 q^{76} - 7 q^{77} - 13 q^{78} + 17 q^{79} + 3 q^{80} + 13 q^{81} + 24 q^{82} + 13 q^{83} - 5 q^{84} + 11 q^{85} + 9 q^{86} + q^{87} + 8 q^{88} + 14 q^{89} + 3 q^{90} + 5 q^{91} - 9 q^{92} - 5 q^{93} - 8 q^{94} + 2 q^{95} - 13 q^{96} + 46 q^{97} - 16 q^{98} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.68854 −0.755140 −0.377570 0.925981i \(-0.623240\pi\)
−0.377570 + 0.925981i \(0.623240\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.50145 1.70139 0.850694 0.525661i \(-0.176182\pi\)
0.850694 + 0.525661i \(0.176182\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.68854 −0.533964
\(11\) −0.787053 −0.237305 −0.118653 0.992936i \(-0.537857\pi\)
−0.118653 + 0.992936i \(0.537857\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 4.50145 1.20306
\(15\) 1.68854 0.435980
\(16\) 1.00000 0.250000
\(17\) −0.120738 −0.0292833 −0.0146416 0.999893i \(-0.504661\pi\)
−0.0146416 + 0.999893i \(0.504661\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.16827 0.497436 0.248718 0.968576i \(-0.419991\pi\)
0.248718 + 0.968576i \(0.419991\pi\)
\(20\) −1.68854 −0.377570
\(21\) −4.50145 −0.982297
\(22\) −0.787053 −0.167800
\(23\) −4.63632 −0.966739 −0.483369 0.875417i \(-0.660587\pi\)
−0.483369 + 0.875417i \(0.660587\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.14882 −0.429764
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.50145 0.850694
\(29\) 0.430855 0.0800078 0.0400039 0.999200i \(-0.487263\pi\)
0.0400039 + 0.999200i \(0.487263\pi\)
\(30\) 1.68854 0.308284
\(31\) 2.16827 0.389433 0.194717 0.980860i \(-0.437621\pi\)
0.194717 + 0.980860i \(0.437621\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.787053 0.137008
\(34\) −0.120738 −0.0207064
\(35\) −7.60090 −1.28479
\(36\) 1.00000 0.166667
\(37\) 5.51041 0.905905 0.452953 0.891535i \(-0.350371\pi\)
0.452953 + 0.891535i \(0.350371\pi\)
\(38\) 2.16827 0.351740
\(39\) −1.00000 −0.160128
\(40\) −1.68854 −0.266982
\(41\) 10.5241 1.64359 0.821793 0.569786i \(-0.192974\pi\)
0.821793 + 0.569786i \(0.192974\pi\)
\(42\) −4.50145 −0.694589
\(43\) 2.43028 0.370614 0.185307 0.982681i \(-0.440672\pi\)
0.185307 + 0.982681i \(0.440672\pi\)
\(44\) −0.787053 −0.118653
\(45\) −1.68854 −0.251713
\(46\) −4.63632 −0.683588
\(47\) 1.33865 0.195262 0.0976308 0.995223i \(-0.468874\pi\)
0.0976308 + 0.995223i \(0.468874\pi\)
\(48\) −1.00000 −0.144338
\(49\) 13.2631 1.89472
\(50\) −2.14882 −0.303889
\(51\) 0.120738 0.0169067
\(52\) 1.00000 0.138675
\(53\) 8.80373 1.20929 0.604643 0.796497i \(-0.293316\pi\)
0.604643 + 0.796497i \(0.293316\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.32897 0.179199
\(56\) 4.50145 0.601532
\(57\) −2.16827 −0.287195
\(58\) 0.430855 0.0565741
\(59\) −9.51134 −1.23827 −0.619135 0.785284i \(-0.712517\pi\)
−0.619135 + 0.785284i \(0.712517\pi\)
\(60\) 1.68854 0.217990
\(61\) −9.33774 −1.19558 −0.597788 0.801654i \(-0.703953\pi\)
−0.597788 + 0.801654i \(0.703953\pi\)
\(62\) 2.16827 0.275371
\(63\) 4.50145 0.567130
\(64\) 1.00000 0.125000
\(65\) −1.68854 −0.209438
\(66\) 0.787053 0.0968795
\(67\) 15.9992 1.95462 0.977308 0.211825i \(-0.0679406\pi\)
0.977308 + 0.211825i \(0.0679406\pi\)
\(68\) −0.120738 −0.0146416
\(69\) 4.63632 0.558147
\(70\) −7.60090 −0.908481
\(71\) −14.3213 −1.69963 −0.849815 0.527081i \(-0.823287\pi\)
−0.849815 + 0.527081i \(0.823287\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.9288 1.27912 0.639561 0.768740i \(-0.279116\pi\)
0.639561 + 0.768740i \(0.279116\pi\)
\(74\) 5.51041 0.640572
\(75\) 2.14882 0.248125
\(76\) 2.16827 0.248718
\(77\) −3.54288 −0.403749
\(78\) −1.00000 −0.113228
\(79\) −11.7235 −1.31900 −0.659501 0.751704i \(-0.729232\pi\)
−0.659501 + 0.751704i \(0.729232\pi\)
\(80\) −1.68854 −0.188785
\(81\) 1.00000 0.111111
\(82\) 10.5241 1.16219
\(83\) 9.92233 1.08912 0.544558 0.838723i \(-0.316697\pi\)
0.544558 + 0.838723i \(0.316697\pi\)
\(84\) −4.50145 −0.491149
\(85\) 0.203871 0.0221130
\(86\) 2.43028 0.262064
\(87\) −0.430855 −0.0461926
\(88\) −0.787053 −0.0839001
\(89\) 4.51314 0.478392 0.239196 0.970971i \(-0.423116\pi\)
0.239196 + 0.970971i \(0.423116\pi\)
\(90\) −1.68854 −0.177988
\(91\) 4.50145 0.471880
\(92\) −4.63632 −0.483369
\(93\) −2.16827 −0.224840
\(94\) 1.33865 0.138071
\(95\) −3.66122 −0.375634
\(96\) −1.00000 −0.102062
\(97\) 1.26242 0.128180 0.0640899 0.997944i \(-0.479586\pi\)
0.0640899 + 0.997944i \(0.479586\pi\)
\(98\) 13.2631 1.33977
\(99\) −0.787053 −0.0791018
\(100\) −2.14882 −0.214882
\(101\) −11.2050 −1.11494 −0.557470 0.830197i \(-0.688228\pi\)
−0.557470 + 0.830197i \(0.688228\pi\)
\(102\) 0.120738 0.0119548
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 7.60090 0.741772
\(106\) 8.80373 0.855094
\(107\) 11.8054 1.14128 0.570638 0.821202i \(-0.306696\pi\)
0.570638 + 0.821202i \(0.306696\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.30323 −0.220609 −0.110305 0.993898i \(-0.535183\pi\)
−0.110305 + 0.993898i \(0.535183\pi\)
\(110\) 1.32897 0.126713
\(111\) −5.51041 −0.523024
\(112\) 4.50145 0.425347
\(113\) 2.92570 0.275227 0.137614 0.990486i \(-0.456057\pi\)
0.137614 + 0.990486i \(0.456057\pi\)
\(114\) −2.16827 −0.203077
\(115\) 7.82862 0.730023
\(116\) 0.430855 0.0400039
\(117\) 1.00000 0.0924500
\(118\) −9.51134 −0.875589
\(119\) −0.543496 −0.0498222
\(120\) 1.68854 0.154142
\(121\) −10.3805 −0.943686
\(122\) −9.33774 −0.845400
\(123\) −10.5241 −0.948925
\(124\) 2.16827 0.194717
\(125\) 12.0711 1.07967
\(126\) 4.50145 0.401021
\(127\) −4.62781 −0.410652 −0.205326 0.978694i \(-0.565825\pi\)
−0.205326 + 0.978694i \(0.565825\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.43028 −0.213974
\(130\) −1.68854 −0.148095
\(131\) −8.06962 −0.705046 −0.352523 0.935803i \(-0.614676\pi\)
−0.352523 + 0.935803i \(0.614676\pi\)
\(132\) 0.787053 0.0685041
\(133\) 9.76038 0.846332
\(134\) 15.9992 1.38212
\(135\) 1.68854 0.145327
\(136\) −0.120738 −0.0103532
\(137\) −16.1423 −1.37913 −0.689563 0.724226i \(-0.742197\pi\)
−0.689563 + 0.724226i \(0.742197\pi\)
\(138\) 4.63632 0.394669
\(139\) 7.49861 0.636024 0.318012 0.948087i \(-0.396985\pi\)
0.318012 + 0.948087i \(0.396985\pi\)
\(140\) −7.60090 −0.642393
\(141\) −1.33865 −0.112734
\(142\) −14.3213 −1.20182
\(143\) −0.787053 −0.0658167
\(144\) 1.00000 0.0833333
\(145\) −0.727518 −0.0604171
\(146\) 10.9288 0.904476
\(147\) −13.2631 −1.09392
\(148\) 5.51041 0.452953
\(149\) 4.84346 0.396792 0.198396 0.980122i \(-0.436427\pi\)
0.198396 + 0.980122i \(0.436427\pi\)
\(150\) 2.14882 0.175451
\(151\) 20.2437 1.64741 0.823705 0.567019i \(-0.191903\pi\)
0.823705 + 0.567019i \(0.191903\pi\)
\(152\) 2.16827 0.175870
\(153\) −0.120738 −0.00976109
\(154\) −3.54288 −0.285493
\(155\) −3.66122 −0.294077
\(156\) −1.00000 −0.0800641
\(157\) −0.819356 −0.0653917 −0.0326959 0.999465i \(-0.510409\pi\)
−0.0326959 + 0.999465i \(0.510409\pi\)
\(158\) −11.7235 −0.932675
\(159\) −8.80373 −0.698181
\(160\) −1.68854 −0.133491
\(161\) −20.8702 −1.64480
\(162\) 1.00000 0.0785674
\(163\) 18.4391 1.44426 0.722132 0.691755i \(-0.243162\pi\)
0.722132 + 0.691755i \(0.243162\pi\)
\(164\) 10.5241 0.821793
\(165\) −1.32897 −0.103460
\(166\) 9.92233 0.770122
\(167\) −7.78642 −0.602532 −0.301266 0.953540i \(-0.597409\pi\)
−0.301266 + 0.953540i \(0.597409\pi\)
\(168\) −4.50145 −0.347295
\(169\) 1.00000 0.0769231
\(170\) 0.203871 0.0156362
\(171\) 2.16827 0.165812
\(172\) 2.43028 0.185307
\(173\) −5.22606 −0.397330 −0.198665 0.980067i \(-0.563661\pi\)
−0.198665 + 0.980067i \(0.563661\pi\)
\(174\) −0.430855 −0.0326631
\(175\) −9.67282 −0.731196
\(176\) −0.787053 −0.0593263
\(177\) 9.51134 0.714916
\(178\) 4.51314 0.338274
\(179\) 8.23133 0.615238 0.307619 0.951510i \(-0.400468\pi\)
0.307619 + 0.951510i \(0.400468\pi\)
\(180\) −1.68854 −0.125857
\(181\) −23.2659 −1.72934 −0.864671 0.502339i \(-0.832473\pi\)
−0.864671 + 0.502339i \(0.832473\pi\)
\(182\) 4.50145 0.333670
\(183\) 9.33774 0.690266
\(184\) −4.63632 −0.341794
\(185\) −9.30456 −0.684085
\(186\) −2.16827 −0.158986
\(187\) 0.0950272 0.00694908
\(188\) 1.33865 0.0976308
\(189\) −4.50145 −0.327432
\(190\) −3.66122 −0.265613
\(191\) −3.16411 −0.228947 −0.114474 0.993426i \(-0.536518\pi\)
−0.114474 + 0.993426i \(0.536518\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −1.32889 −0.0956557 −0.0478279 0.998856i \(-0.515230\pi\)
−0.0478279 + 0.998856i \(0.515230\pi\)
\(194\) 1.26242 0.0906368
\(195\) 1.68854 0.120919
\(196\) 13.2631 0.947362
\(197\) 1.46635 0.104473 0.0522367 0.998635i \(-0.483365\pi\)
0.0522367 + 0.998635i \(0.483365\pi\)
\(198\) −0.787053 −0.0559334
\(199\) 21.4614 1.52136 0.760680 0.649127i \(-0.224866\pi\)
0.760680 + 0.649127i \(0.224866\pi\)
\(200\) −2.14882 −0.151945
\(201\) −15.9992 −1.12850
\(202\) −11.2050 −0.788381
\(203\) 1.93947 0.136124
\(204\) 0.120738 0.00845335
\(205\) −17.7704 −1.24114
\(206\) 1.00000 0.0696733
\(207\) −4.63632 −0.322246
\(208\) 1.00000 0.0693375
\(209\) −1.70655 −0.118044
\(210\) 7.60090 0.524512
\(211\) 2.00558 0.138070 0.0690350 0.997614i \(-0.478008\pi\)
0.0690350 + 0.997614i \(0.478008\pi\)
\(212\) 8.80373 0.604643
\(213\) 14.3213 0.981282
\(214\) 11.8054 0.807003
\(215\) −4.10363 −0.279865
\(216\) −1.00000 −0.0680414
\(217\) 9.76038 0.662578
\(218\) −2.30323 −0.155994
\(219\) −10.9288 −0.738501
\(220\) 1.32897 0.0895993
\(221\) −0.120738 −0.00812172
\(222\) −5.51041 −0.369834
\(223\) −19.0380 −1.27488 −0.637440 0.770500i \(-0.720007\pi\)
−0.637440 + 0.770500i \(0.720007\pi\)
\(224\) 4.50145 0.300766
\(225\) −2.14882 −0.143255
\(226\) 2.92570 0.194615
\(227\) 22.5082 1.49392 0.746960 0.664869i \(-0.231513\pi\)
0.746960 + 0.664869i \(0.231513\pi\)
\(228\) −2.16827 −0.143597
\(229\) 16.8382 1.11270 0.556349 0.830949i \(-0.312202\pi\)
0.556349 + 0.830949i \(0.312202\pi\)
\(230\) 7.82862 0.516204
\(231\) 3.54288 0.233104
\(232\) 0.430855 0.0282870
\(233\) 14.4896 0.949246 0.474623 0.880189i \(-0.342584\pi\)
0.474623 + 0.880189i \(0.342584\pi\)
\(234\) 1.00000 0.0653720
\(235\) −2.26036 −0.147450
\(236\) −9.51134 −0.619135
\(237\) 11.7235 0.761526
\(238\) −0.543496 −0.0352296
\(239\) 5.77146 0.373325 0.186663 0.982424i \(-0.440233\pi\)
0.186663 + 0.982424i \(0.440233\pi\)
\(240\) 1.68854 0.108995
\(241\) 22.2958 1.43620 0.718100 0.695940i \(-0.245012\pi\)
0.718100 + 0.695940i \(0.245012\pi\)
\(242\) −10.3805 −0.667287
\(243\) −1.00000 −0.0641500
\(244\) −9.33774 −0.597788
\(245\) −22.3953 −1.43078
\(246\) −10.5241 −0.670991
\(247\) 2.16827 0.137964
\(248\) 2.16827 0.137686
\(249\) −9.92233 −0.628802
\(250\) 12.0711 0.763443
\(251\) −5.12799 −0.323676 −0.161838 0.986817i \(-0.551742\pi\)
−0.161838 + 0.986817i \(0.551742\pi\)
\(252\) 4.50145 0.283565
\(253\) 3.64902 0.229412
\(254\) −4.62781 −0.290375
\(255\) −0.203871 −0.0127669
\(256\) 1.00000 0.0625000
\(257\) 1.47989 0.0923133 0.0461566 0.998934i \(-0.485303\pi\)
0.0461566 + 0.998934i \(0.485303\pi\)
\(258\) −2.43028 −0.151303
\(259\) 24.8048 1.54130
\(260\) −1.68854 −0.104719
\(261\) 0.430855 0.0266693
\(262\) −8.06962 −0.498543
\(263\) −20.0334 −1.23532 −0.617658 0.786447i \(-0.711918\pi\)
−0.617658 + 0.786447i \(0.711918\pi\)
\(264\) 0.787053 0.0484397
\(265\) −14.8655 −0.913179
\(266\) 9.76038 0.598447
\(267\) −4.51314 −0.276200
\(268\) 15.9992 0.977308
\(269\) −2.10282 −0.128211 −0.0641057 0.997943i \(-0.520419\pi\)
−0.0641057 + 0.997943i \(0.520419\pi\)
\(270\) 1.68854 0.102761
\(271\) 28.5103 1.73188 0.865940 0.500148i \(-0.166721\pi\)
0.865940 + 0.500148i \(0.166721\pi\)
\(272\) −0.120738 −0.00732082
\(273\) −4.50145 −0.272440
\(274\) −16.1423 −0.975189
\(275\) 1.69124 0.101985
\(276\) 4.63632 0.279073
\(277\) 19.2633 1.15742 0.578711 0.815533i \(-0.303556\pi\)
0.578711 + 0.815533i \(0.303556\pi\)
\(278\) 7.49861 0.449737
\(279\) 2.16827 0.129811
\(280\) −7.60090 −0.454240
\(281\) 20.9688 1.25089 0.625446 0.780267i \(-0.284917\pi\)
0.625446 + 0.780267i \(0.284917\pi\)
\(282\) −1.33865 −0.0797152
\(283\) 29.4972 1.75343 0.876714 0.481012i \(-0.159731\pi\)
0.876714 + 0.481012i \(0.159731\pi\)
\(284\) −14.3213 −0.849815
\(285\) 3.66122 0.216872
\(286\) −0.787053 −0.0465394
\(287\) 47.3737 2.79638
\(288\) 1.00000 0.0589256
\(289\) −16.9854 −0.999142
\(290\) −0.727518 −0.0427213
\(291\) −1.26242 −0.0740046
\(292\) 10.9288 0.639561
\(293\) −22.5817 −1.31924 −0.659619 0.751600i \(-0.729282\pi\)
−0.659619 + 0.751600i \(0.729282\pi\)
\(294\) −13.2631 −0.773518
\(295\) 16.0603 0.935067
\(296\) 5.51041 0.320286
\(297\) 0.787053 0.0456694
\(298\) 4.84346 0.280574
\(299\) −4.63632 −0.268125
\(300\) 2.14882 0.124062
\(301\) 10.9398 0.630559
\(302\) 20.2437 1.16489
\(303\) 11.2050 0.643711
\(304\) 2.16827 0.124359
\(305\) 15.7672 0.902826
\(306\) −0.120738 −0.00690213
\(307\) −12.4312 −0.709484 −0.354742 0.934964i \(-0.615431\pi\)
−0.354742 + 0.934964i \(0.615431\pi\)
\(308\) −3.54288 −0.201874
\(309\) −1.00000 −0.0568880
\(310\) −3.66122 −0.207944
\(311\) −12.2864 −0.696699 −0.348349 0.937365i \(-0.613258\pi\)
−0.348349 + 0.937365i \(0.613258\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −17.9396 −1.01401 −0.507004 0.861944i \(-0.669247\pi\)
−0.507004 + 0.861944i \(0.669247\pi\)
\(314\) −0.819356 −0.0462389
\(315\) −7.60090 −0.428262
\(316\) −11.7235 −0.659501
\(317\) 31.1119 1.74742 0.873709 0.486449i \(-0.161708\pi\)
0.873709 + 0.486449i \(0.161708\pi\)
\(318\) −8.80373 −0.493689
\(319\) −0.339106 −0.0189863
\(320\) −1.68854 −0.0943924
\(321\) −11.8054 −0.658916
\(322\) −20.8702 −1.16305
\(323\) −0.261793 −0.0145666
\(324\) 1.00000 0.0555556
\(325\) −2.14882 −0.119195
\(326\) 18.4391 1.02125
\(327\) 2.30323 0.127369
\(328\) 10.5241 0.581095
\(329\) 6.02585 0.332216
\(330\) −1.32897 −0.0731575
\(331\) 29.8047 1.63821 0.819107 0.573641i \(-0.194470\pi\)
0.819107 + 0.573641i \(0.194470\pi\)
\(332\) 9.92233 0.544558
\(333\) 5.51041 0.301968
\(334\) −7.78642 −0.426054
\(335\) −27.0154 −1.47601
\(336\) −4.50145 −0.245574
\(337\) −9.10231 −0.495834 −0.247917 0.968781i \(-0.579746\pi\)
−0.247917 + 0.968781i \(0.579746\pi\)
\(338\) 1.00000 0.0543928
\(339\) −2.92570 −0.158902
\(340\) 0.203871 0.0110565
\(341\) −1.70655 −0.0924146
\(342\) 2.16827 0.117247
\(343\) 28.1929 1.52227
\(344\) 2.43028 0.131032
\(345\) −7.82862 −0.421479
\(346\) −5.22606 −0.280955
\(347\) 7.09876 0.381081 0.190541 0.981679i \(-0.438976\pi\)
0.190541 + 0.981679i \(0.438976\pi\)
\(348\) −0.430855 −0.0230963
\(349\) −27.6902 −1.48222 −0.741111 0.671382i \(-0.765701\pi\)
−0.741111 + 0.671382i \(0.765701\pi\)
\(350\) −9.67282 −0.517034
\(351\) −1.00000 −0.0533761
\(352\) −0.787053 −0.0419500
\(353\) 36.0299 1.91768 0.958839 0.283949i \(-0.0916448\pi\)
0.958839 + 0.283949i \(0.0916448\pi\)
\(354\) 9.51134 0.505522
\(355\) 24.1822 1.28346
\(356\) 4.51314 0.239196
\(357\) 0.543496 0.0287649
\(358\) 8.23133 0.435039
\(359\) −24.0969 −1.27178 −0.635892 0.771778i \(-0.719368\pi\)
−0.635892 + 0.771778i \(0.719368\pi\)
\(360\) −1.68854 −0.0889940
\(361\) −14.2986 −0.752557
\(362\) −23.2659 −1.22283
\(363\) 10.3805 0.544837
\(364\) 4.50145 0.235940
\(365\) −18.4538 −0.965915
\(366\) 9.33774 0.488092
\(367\) 10.2238 0.533678 0.266839 0.963741i \(-0.414021\pi\)
0.266839 + 0.963741i \(0.414021\pi\)
\(368\) −4.63632 −0.241685
\(369\) 10.5241 0.547862
\(370\) −9.30456 −0.483721
\(371\) 39.6296 2.05746
\(372\) −2.16827 −0.112420
\(373\) 30.3725 1.57263 0.786313 0.617828i \(-0.211987\pi\)
0.786313 + 0.617828i \(0.211987\pi\)
\(374\) 0.0950272 0.00491374
\(375\) −12.0711 −0.623349
\(376\) 1.33865 0.0690354
\(377\) 0.430855 0.0221902
\(378\) −4.50145 −0.231530
\(379\) 28.6696 1.47266 0.736330 0.676623i \(-0.236557\pi\)
0.736330 + 0.676623i \(0.236557\pi\)
\(380\) −3.66122 −0.187817
\(381\) 4.62781 0.237090
\(382\) −3.16411 −0.161890
\(383\) 2.20806 0.112826 0.0564132 0.998408i \(-0.482034\pi\)
0.0564132 + 0.998408i \(0.482034\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 5.98231 0.304887
\(386\) −1.32889 −0.0676388
\(387\) 2.43028 0.123538
\(388\) 1.26242 0.0640899
\(389\) 13.0889 0.663634 0.331817 0.943344i \(-0.392338\pi\)
0.331817 + 0.943344i \(0.392338\pi\)
\(390\) 1.68854 0.0855027
\(391\) 0.559780 0.0283093
\(392\) 13.2631 0.669886
\(393\) 8.06962 0.407058
\(394\) 1.46635 0.0738738
\(395\) 19.7957 0.996030
\(396\) −0.787053 −0.0395509
\(397\) 18.6996 0.938507 0.469254 0.883063i \(-0.344523\pi\)
0.469254 + 0.883063i \(0.344523\pi\)
\(398\) 21.4614 1.07576
\(399\) −9.76038 −0.488630
\(400\) −2.14882 −0.107441
\(401\) 20.9674 1.04706 0.523532 0.852006i \(-0.324614\pi\)
0.523532 + 0.852006i \(0.324614\pi\)
\(402\) −15.9992 −0.797968
\(403\) 2.16827 0.108009
\(404\) −11.2050 −0.557470
\(405\) −1.68854 −0.0839044
\(406\) 1.93947 0.0962545
\(407\) −4.33698 −0.214976
\(408\) 0.120738 0.00597742
\(409\) 18.3390 0.906805 0.453403 0.891306i \(-0.350210\pi\)
0.453403 + 0.891306i \(0.350210\pi\)
\(410\) −17.7704 −0.877616
\(411\) 16.1423 0.796239
\(412\) 1.00000 0.0492665
\(413\) −42.8148 −2.10678
\(414\) −4.63632 −0.227863
\(415\) −16.7543 −0.822435
\(416\) 1.00000 0.0490290
\(417\) −7.49861 −0.367209
\(418\) −1.70655 −0.0834699
\(419\) 3.18302 0.155501 0.0777503 0.996973i \(-0.475226\pi\)
0.0777503 + 0.996973i \(0.475226\pi\)
\(420\) 7.60090 0.370886
\(421\) 12.8911 0.628272 0.314136 0.949378i \(-0.398285\pi\)
0.314136 + 0.949378i \(0.398285\pi\)
\(422\) 2.00558 0.0976302
\(423\) 1.33865 0.0650872
\(424\) 8.80373 0.427547
\(425\) 0.259444 0.0125849
\(426\) 14.3213 0.693871
\(427\) −42.0334 −2.03414
\(428\) 11.8054 0.570638
\(429\) 0.787053 0.0379993
\(430\) −4.10363 −0.197895
\(431\) −24.0805 −1.15992 −0.579958 0.814647i \(-0.696931\pi\)
−0.579958 + 0.814647i \(0.696931\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.1923 −1.45095 −0.725475 0.688248i \(-0.758380\pi\)
−0.725475 + 0.688248i \(0.758380\pi\)
\(434\) 9.76038 0.468513
\(435\) 0.727518 0.0348818
\(436\) −2.30323 −0.110305
\(437\) −10.0528 −0.480891
\(438\) −10.9288 −0.522199
\(439\) 5.69015 0.271576 0.135788 0.990738i \(-0.456643\pi\)
0.135788 + 0.990738i \(0.456643\pi\)
\(440\) 1.32897 0.0633563
\(441\) 13.2631 0.631575
\(442\) −0.120738 −0.00574292
\(443\) −12.9170 −0.613705 −0.306852 0.951757i \(-0.599276\pi\)
−0.306852 + 0.951757i \(0.599276\pi\)
\(444\) −5.51041 −0.261512
\(445\) −7.62063 −0.361253
\(446\) −19.0380 −0.901477
\(447\) −4.84346 −0.229088
\(448\) 4.50145 0.212674
\(449\) −21.2531 −1.00300 −0.501499 0.865158i \(-0.667218\pi\)
−0.501499 + 0.865158i \(0.667218\pi\)
\(450\) −2.14882 −0.101296
\(451\) −8.28301 −0.390032
\(452\) 2.92570 0.137614
\(453\) −20.2437 −0.951132
\(454\) 22.5082 1.05636
\(455\) −7.60090 −0.356336
\(456\) −2.16827 −0.101539
\(457\) 23.2880 1.08937 0.544683 0.838642i \(-0.316650\pi\)
0.544683 + 0.838642i \(0.316650\pi\)
\(458\) 16.8382 0.786797
\(459\) 0.120738 0.00563557
\(460\) 7.82862 0.365011
\(461\) 18.2701 0.850923 0.425461 0.904977i \(-0.360112\pi\)
0.425461 + 0.904977i \(0.360112\pi\)
\(462\) 3.54288 0.164830
\(463\) 22.7042 1.05515 0.527576 0.849508i \(-0.323101\pi\)
0.527576 + 0.849508i \(0.323101\pi\)
\(464\) 0.430855 0.0200020
\(465\) 3.66122 0.169785
\(466\) 14.4896 0.671218
\(467\) 12.0488 0.557553 0.278776 0.960356i \(-0.410071\pi\)
0.278776 + 0.960356i \(0.410071\pi\)
\(468\) 1.00000 0.0462250
\(469\) 72.0197 3.32556
\(470\) −2.26036 −0.104263
\(471\) 0.819356 0.0377539
\(472\) −9.51134 −0.437795
\(473\) −1.91276 −0.0879487
\(474\) 11.7235 0.538480
\(475\) −4.65923 −0.213780
\(476\) −0.543496 −0.0249111
\(477\) 8.80373 0.403095
\(478\) 5.77146 0.263981
\(479\) −41.7439 −1.90733 −0.953664 0.300874i \(-0.902722\pi\)
−0.953664 + 0.300874i \(0.902722\pi\)
\(480\) 1.68854 0.0770711
\(481\) 5.51041 0.251253
\(482\) 22.2958 1.01555
\(483\) 20.8702 0.949625
\(484\) −10.3805 −0.471843
\(485\) −2.13166 −0.0967936
\(486\) −1.00000 −0.0453609
\(487\) −34.4976 −1.56323 −0.781617 0.623759i \(-0.785605\pi\)
−0.781617 + 0.623759i \(0.785605\pi\)
\(488\) −9.33774 −0.422700
\(489\) −18.4391 −0.833847
\(490\) −22.3953 −1.01171
\(491\) −33.4958 −1.51165 −0.755823 0.654776i \(-0.772763\pi\)
−0.755823 + 0.654776i \(0.772763\pi\)
\(492\) −10.5241 −0.474462
\(493\) −0.0520206 −0.00234289
\(494\) 2.16827 0.0975553
\(495\) 1.32897 0.0597329
\(496\) 2.16827 0.0973584
\(497\) −64.4668 −2.89173
\(498\) −9.92233 −0.444630
\(499\) 24.5788 1.10030 0.550148 0.835067i \(-0.314571\pi\)
0.550148 + 0.835067i \(0.314571\pi\)
\(500\) 12.0711 0.539836
\(501\) 7.78642 0.347872
\(502\) −5.12799 −0.228873
\(503\) 14.5933 0.650684 0.325342 0.945596i \(-0.394521\pi\)
0.325342 + 0.945596i \(0.394521\pi\)
\(504\) 4.50145 0.200511
\(505\) 18.9201 0.841935
\(506\) 3.64902 0.162219
\(507\) −1.00000 −0.0444116
\(508\) −4.62781 −0.205326
\(509\) 4.91456 0.217834 0.108917 0.994051i \(-0.465262\pi\)
0.108917 + 0.994051i \(0.465262\pi\)
\(510\) −0.203871 −0.00902758
\(511\) 49.1956 2.17628
\(512\) 1.00000 0.0441942
\(513\) −2.16827 −0.0957316
\(514\) 1.47989 0.0652753
\(515\) −1.68854 −0.0744061
\(516\) −2.43028 −0.106987
\(517\) −1.05359 −0.0463366
\(518\) 24.8048 1.08986
\(519\) 5.22606 0.229399
\(520\) −1.68854 −0.0740475
\(521\) −24.3944 −1.06874 −0.534368 0.845252i \(-0.679450\pi\)
−0.534368 + 0.845252i \(0.679450\pi\)
\(522\) 0.430855 0.0188580
\(523\) 23.0303 1.00704 0.503522 0.863982i \(-0.332037\pi\)
0.503522 + 0.863982i \(0.332037\pi\)
\(524\) −8.06962 −0.352523
\(525\) 9.67282 0.422156
\(526\) −20.0334 −0.873500
\(527\) −0.261793 −0.0114039
\(528\) 0.787053 0.0342521
\(529\) −1.50457 −0.0654162
\(530\) −14.8655 −0.645715
\(531\) −9.51134 −0.412757
\(532\) 9.76038 0.423166
\(533\) 10.5241 0.455849
\(534\) −4.51314 −0.195303
\(535\) −19.9340 −0.861822
\(536\) 15.9992 0.691061
\(537\) −8.23133 −0.355208
\(538\) −2.10282 −0.0906591
\(539\) −10.4387 −0.449628
\(540\) 1.68854 0.0726633
\(541\) 3.05816 0.131481 0.0657403 0.997837i \(-0.479059\pi\)
0.0657403 + 0.997837i \(0.479059\pi\)
\(542\) 28.5103 1.22462
\(543\) 23.2659 0.998436
\(544\) −0.120738 −0.00517660
\(545\) 3.88910 0.166591
\(546\) −4.50145 −0.192644
\(547\) −41.7413 −1.78473 −0.892365 0.451314i \(-0.850955\pi\)
−0.892365 + 0.451314i \(0.850955\pi\)
\(548\) −16.1423 −0.689563
\(549\) −9.33774 −0.398525
\(550\) 1.69124 0.0721145
\(551\) 0.934213 0.0397988
\(552\) 4.63632 0.197335
\(553\) −52.7730 −2.24413
\(554\) 19.2633 0.818421
\(555\) 9.30456 0.394956
\(556\) 7.49861 0.318012
\(557\) −28.4286 −1.20456 −0.602278 0.798286i \(-0.705740\pi\)
−0.602278 + 0.798286i \(0.705740\pi\)
\(558\) 2.16827 0.0917904
\(559\) 2.43028 0.102790
\(560\) −7.60090 −0.321196
\(561\) −0.0950272 −0.00401205
\(562\) 20.9688 0.884514
\(563\) 27.2329 1.14773 0.573865 0.818950i \(-0.305444\pi\)
0.573865 + 0.818950i \(0.305444\pi\)
\(564\) −1.33865 −0.0563672
\(565\) −4.94018 −0.207835
\(566\) 29.4972 1.23986
\(567\) 4.50145 0.189043
\(568\) −14.3213 −0.600910
\(569\) −24.6215 −1.03219 −0.516094 0.856532i \(-0.672614\pi\)
−0.516094 + 0.856532i \(0.672614\pi\)
\(570\) 3.66122 0.153352
\(571\) −23.1123 −0.967222 −0.483611 0.875283i \(-0.660675\pi\)
−0.483611 + 0.875283i \(0.660675\pi\)
\(572\) −0.787053 −0.0329083
\(573\) 3.16411 0.132183
\(574\) 47.3737 1.97734
\(575\) 9.96262 0.415470
\(576\) 1.00000 0.0416667
\(577\) −4.71744 −0.196389 −0.0981947 0.995167i \(-0.531307\pi\)
−0.0981947 + 0.995167i \(0.531307\pi\)
\(578\) −16.9854 −0.706500
\(579\) 1.32889 0.0552269
\(580\) −0.727518 −0.0302085
\(581\) 44.6649 1.85301
\(582\) −1.26242 −0.0523292
\(583\) −6.92900 −0.286970
\(584\) 10.9288 0.452238
\(585\) −1.68854 −0.0698127
\(586\) −22.5817 −0.932842
\(587\) −7.95453 −0.328318 −0.164159 0.986434i \(-0.552491\pi\)
−0.164159 + 0.986434i \(0.552491\pi\)
\(588\) −13.2631 −0.546960
\(589\) 4.70141 0.193718
\(590\) 16.0603 0.661192
\(591\) −1.46635 −0.0603177
\(592\) 5.51041 0.226476
\(593\) −21.3889 −0.878339 −0.439169 0.898404i \(-0.644727\pi\)
−0.439169 + 0.898404i \(0.644727\pi\)
\(594\) 0.787053 0.0322932
\(595\) 0.917717 0.0376227
\(596\) 4.84346 0.198396
\(597\) −21.4614 −0.878358
\(598\) −4.63632 −0.189593
\(599\) 3.89261 0.159048 0.0795238 0.996833i \(-0.474660\pi\)
0.0795238 + 0.996833i \(0.474660\pi\)
\(600\) 2.14882 0.0877253
\(601\) −46.7978 −1.90892 −0.954462 0.298333i \(-0.903569\pi\)
−0.954462 + 0.298333i \(0.903569\pi\)
\(602\) 10.9398 0.445872
\(603\) 15.9992 0.651538
\(604\) 20.2437 0.823705
\(605\) 17.5280 0.712615
\(606\) 11.2050 0.455172
\(607\) −28.8775 −1.17210 −0.586050 0.810275i \(-0.699318\pi\)
−0.586050 + 0.810275i \(0.699318\pi\)
\(608\) 2.16827 0.0879351
\(609\) −1.93947 −0.0785915
\(610\) 15.7672 0.638395
\(611\) 1.33865 0.0541558
\(612\) −0.120738 −0.00488055
\(613\) 1.52992 0.0617930 0.0308965 0.999523i \(-0.490164\pi\)
0.0308965 + 0.999523i \(0.490164\pi\)
\(614\) −12.4312 −0.501681
\(615\) 17.7704 0.716571
\(616\) −3.54288 −0.142747
\(617\) 18.9018 0.760956 0.380478 0.924790i \(-0.375759\pi\)
0.380478 + 0.924790i \(0.375759\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −29.9232 −1.20271 −0.601357 0.798980i \(-0.705373\pi\)
−0.601357 + 0.798980i \(0.705373\pi\)
\(620\) −3.66122 −0.147038
\(621\) 4.63632 0.186049
\(622\) −12.2864 −0.492640
\(623\) 20.3157 0.813931
\(624\) −1.00000 −0.0400320
\(625\) −9.63846 −0.385538
\(626\) −17.9396 −0.717011
\(627\) 1.70655 0.0681529
\(628\) −0.819356 −0.0326959
\(629\) −0.665315 −0.0265279
\(630\) −7.60090 −0.302827
\(631\) 1.87357 0.0745858 0.0372929 0.999304i \(-0.488127\pi\)
0.0372929 + 0.999304i \(0.488127\pi\)
\(632\) −11.7235 −0.466338
\(633\) −2.00558 −0.0797147
\(634\) 31.1119 1.23561
\(635\) 7.81426 0.310100
\(636\) −8.80373 −0.349091
\(637\) 13.2631 0.525502
\(638\) −0.339106 −0.0134253
\(639\) −14.3213 −0.566543
\(640\) −1.68854 −0.0667455
\(641\) 10.1990 0.402835 0.201418 0.979505i \(-0.435445\pi\)
0.201418 + 0.979505i \(0.435445\pi\)
\(642\) −11.8054 −0.465924
\(643\) 2.65800 0.104821 0.0524106 0.998626i \(-0.483310\pi\)
0.0524106 + 0.998626i \(0.483310\pi\)
\(644\) −20.8702 −0.822399
\(645\) 4.10363 0.161580
\(646\) −0.261793 −0.0103001
\(647\) −44.8081 −1.76159 −0.880794 0.473500i \(-0.842990\pi\)
−0.880794 + 0.473500i \(0.842990\pi\)
\(648\) 1.00000 0.0392837
\(649\) 7.48592 0.293848
\(650\) −2.14882 −0.0842837
\(651\) −9.76038 −0.382539
\(652\) 18.4391 0.722132
\(653\) 42.6007 1.66709 0.833547 0.552448i \(-0.186306\pi\)
0.833547 + 0.552448i \(0.186306\pi\)
\(654\) 2.30323 0.0900634
\(655\) 13.6259 0.532408
\(656\) 10.5241 0.410896
\(657\) 10.9288 0.426374
\(658\) 6.02585 0.234912
\(659\) 22.2070 0.865063 0.432531 0.901619i \(-0.357620\pi\)
0.432531 + 0.901619i \(0.357620\pi\)
\(660\) −1.32897 −0.0517302
\(661\) −3.30680 −0.128620 −0.0643098 0.997930i \(-0.520485\pi\)
−0.0643098 + 0.997930i \(0.520485\pi\)
\(662\) 29.8047 1.15839
\(663\) 0.120738 0.00468908
\(664\) 9.92233 0.385061
\(665\) −16.4808 −0.639099
\(666\) 5.51041 0.213524
\(667\) −1.99758 −0.0773467
\(668\) −7.78642 −0.301266
\(669\) 19.0380 0.736053
\(670\) −27.0154 −1.04369
\(671\) 7.34930 0.283716
\(672\) −4.50145 −0.173647
\(673\) 34.7100 1.33797 0.668987 0.743274i \(-0.266728\pi\)
0.668987 + 0.743274i \(0.266728\pi\)
\(674\) −9.10231 −0.350608
\(675\) 2.14882 0.0827082
\(676\) 1.00000 0.0384615
\(677\) −12.5673 −0.482999 −0.241500 0.970401i \(-0.577639\pi\)
−0.241500 + 0.970401i \(0.577639\pi\)
\(678\) −2.92570 −0.112361
\(679\) 5.68274 0.218084
\(680\) 0.203871 0.00781811
\(681\) −22.5082 −0.862515
\(682\) −1.70655 −0.0653470
\(683\) −26.5155 −1.01459 −0.507293 0.861774i \(-0.669354\pi\)
−0.507293 + 0.861774i \(0.669354\pi\)
\(684\) 2.16827 0.0829060
\(685\) 27.2569 1.04143
\(686\) 28.1929 1.07641
\(687\) −16.8382 −0.642417
\(688\) 2.43028 0.0926535
\(689\) 8.80373 0.335395
\(690\) −7.82862 −0.298030
\(691\) −20.7887 −0.790838 −0.395419 0.918501i \(-0.629401\pi\)
−0.395419 + 0.918501i \(0.629401\pi\)
\(692\) −5.22606 −0.198665
\(693\) −3.54288 −0.134583
\(694\) 7.09876 0.269465
\(695\) −12.6617 −0.480287
\(696\) −0.430855 −0.0163315
\(697\) −1.27066 −0.0481296
\(698\) −27.6902 −1.04809
\(699\) −14.4896 −0.548048
\(700\) −9.67282 −0.365598
\(701\) 4.46688 0.168712 0.0843559 0.996436i \(-0.473117\pi\)
0.0843559 + 0.996436i \(0.473117\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 11.9481 0.450630
\(704\) −0.787053 −0.0296632
\(705\) 2.26036 0.0851302
\(706\) 36.0299 1.35600
\(707\) −50.4388 −1.89695
\(708\) 9.51134 0.357458
\(709\) 25.2271 0.947425 0.473712 0.880680i \(-0.342914\pi\)
0.473712 + 0.880680i \(0.342914\pi\)
\(710\) 24.1822 0.907542
\(711\) −11.7235 −0.439667
\(712\) 4.51314 0.169137
\(713\) −10.0528 −0.376480
\(714\) 0.543496 0.0203398
\(715\) 1.32897 0.0497008
\(716\) 8.23133 0.307619
\(717\) −5.77146 −0.215539
\(718\) −24.0969 −0.899288
\(719\) 36.2789 1.35298 0.676488 0.736454i \(-0.263501\pi\)
0.676488 + 0.736454i \(0.263501\pi\)
\(720\) −1.68854 −0.0629283
\(721\) 4.50145 0.167643
\(722\) −14.2986 −0.532138
\(723\) −22.2958 −0.829190
\(724\) −23.2659 −0.864671
\(725\) −0.925831 −0.0343845
\(726\) 10.3805 0.385258
\(727\) −33.8689 −1.25613 −0.628064 0.778162i \(-0.716152\pi\)
−0.628064 + 0.778162i \(0.716152\pi\)
\(728\) 4.50145 0.166835
\(729\) 1.00000 0.0370370
\(730\) −18.4538 −0.683005
\(731\) −0.293427 −0.0108528
\(732\) 9.33774 0.345133
\(733\) 10.9523 0.404532 0.202266 0.979331i \(-0.435169\pi\)
0.202266 + 0.979331i \(0.435169\pi\)
\(734\) 10.2238 0.377367
\(735\) 22.3953 0.826062
\(736\) −4.63632 −0.170897
\(737\) −12.5922 −0.463841
\(738\) 10.5241 0.387397
\(739\) −14.7262 −0.541713 −0.270856 0.962620i \(-0.587307\pi\)
−0.270856 + 0.962620i \(0.587307\pi\)
\(740\) −9.30456 −0.342042
\(741\) −2.16827 −0.0796535
\(742\) 39.6296 1.45485
\(743\) 47.2952 1.73509 0.867546 0.497356i \(-0.165696\pi\)
0.867546 + 0.497356i \(0.165696\pi\)
\(744\) −2.16827 −0.0794928
\(745\) −8.17839 −0.299633
\(746\) 30.3725 1.11201
\(747\) 9.92233 0.363039
\(748\) 0.0950272 0.00347454
\(749\) 53.1416 1.94175
\(750\) −12.0711 −0.440774
\(751\) −26.9359 −0.982906 −0.491453 0.870904i \(-0.663534\pi\)
−0.491453 + 0.870904i \(0.663534\pi\)
\(752\) 1.33865 0.0488154
\(753\) 5.12799 0.186874
\(754\) 0.430855 0.0156908
\(755\) −34.1824 −1.24402
\(756\) −4.50145 −0.163716
\(757\) −8.91683 −0.324088 −0.162044 0.986784i \(-0.551809\pi\)
−0.162044 + 0.986784i \(0.551809\pi\)
\(758\) 28.6696 1.04133
\(759\) −3.64902 −0.132451
\(760\) −3.66122 −0.132807
\(761\) 43.1405 1.56384 0.781921 0.623377i \(-0.214240\pi\)
0.781921 + 0.623377i \(0.214240\pi\)
\(762\) 4.62781 0.167648
\(763\) −10.3679 −0.375342
\(764\) −3.16411 −0.114474
\(765\) 0.203871 0.00737099
\(766\) 2.20806 0.0797803
\(767\) −9.51134 −0.343434
\(768\) −1.00000 −0.0360844
\(769\) 28.1086 1.01362 0.506812 0.862057i \(-0.330824\pi\)
0.506812 + 0.862057i \(0.330824\pi\)
\(770\) 5.98231 0.215587
\(771\) −1.47989 −0.0532971
\(772\) −1.32889 −0.0478279
\(773\) −29.6284 −1.06566 −0.532830 0.846223i \(-0.678871\pi\)
−0.532830 + 0.846223i \(0.678871\pi\)
\(774\) 2.43028 0.0873546
\(775\) −4.65923 −0.167365
\(776\) 1.26242 0.0453184
\(777\) −24.8048 −0.889868
\(778\) 13.0889 0.469260
\(779\) 22.8191 0.817579
\(780\) 1.68854 0.0604595
\(781\) 11.2716 0.403331
\(782\) 0.559780 0.0200177
\(783\) −0.430855 −0.0153975
\(784\) 13.2631 0.473681
\(785\) 1.38352 0.0493799
\(786\) 8.06962 0.287834
\(787\) −21.9080 −0.780935 −0.390468 0.920617i \(-0.627687\pi\)
−0.390468 + 0.920617i \(0.627687\pi\)
\(788\) 1.46635 0.0522367
\(789\) 20.0334 0.713210
\(790\) 19.7957 0.704300
\(791\) 13.1699 0.468268
\(792\) −0.787053 −0.0279667
\(793\) −9.33774 −0.331593
\(794\) 18.6996 0.663625
\(795\) 14.8655 0.527224
\(796\) 21.4614 0.760680
\(797\) 38.2469 1.35478 0.677388 0.735626i \(-0.263112\pi\)
0.677388 + 0.735626i \(0.263112\pi\)
\(798\) −9.76038 −0.345514
\(799\) −0.161626 −0.00571790
\(800\) −2.14882 −0.0759723
\(801\) 4.51314 0.159464
\(802\) 20.9674 0.740385
\(803\) −8.60156 −0.303542
\(804\) −15.9992 −0.564249
\(805\) 35.2402 1.24205
\(806\) 2.16827 0.0763742
\(807\) 2.10282 0.0740229
\(808\) −11.2050 −0.394191
\(809\) 42.9302 1.50935 0.754673 0.656101i \(-0.227795\pi\)
0.754673 + 0.656101i \(0.227795\pi\)
\(810\) −1.68854 −0.0593294
\(811\) 33.9971 1.19380 0.596900 0.802316i \(-0.296399\pi\)
0.596900 + 0.802316i \(0.296399\pi\)
\(812\) 1.93947 0.0680622
\(813\) −28.5103 −0.999901
\(814\) −4.33698 −0.152011
\(815\) −31.1353 −1.09062
\(816\) 0.120738 0.00422668
\(817\) 5.26951 0.184357
\(818\) 18.3390 0.641208
\(819\) 4.50145 0.157293
\(820\) −17.7704 −0.620568
\(821\) 50.2619 1.75415 0.877077 0.480350i \(-0.159490\pi\)
0.877077 + 0.480350i \(0.159490\pi\)
\(822\) 16.1423 0.563026
\(823\) 42.3691 1.47689 0.738447 0.674311i \(-0.235559\pi\)
0.738447 + 0.674311i \(0.235559\pi\)
\(824\) 1.00000 0.0348367
\(825\) −1.69124 −0.0588813
\(826\) −42.8148 −1.48972
\(827\) −1.76874 −0.0615050 −0.0307525 0.999527i \(-0.509790\pi\)
−0.0307525 + 0.999527i \(0.509790\pi\)
\(828\) −4.63632 −0.161123
\(829\) −49.8563 −1.73158 −0.865791 0.500406i \(-0.833184\pi\)
−0.865791 + 0.500406i \(0.833184\pi\)
\(830\) −16.7543 −0.581549
\(831\) −19.2633 −0.668238
\(832\) 1.00000 0.0346688
\(833\) −1.60136 −0.0554837
\(834\) −7.49861 −0.259656
\(835\) 13.1477 0.454995
\(836\) −1.70655 −0.0590221
\(837\) −2.16827 −0.0749465
\(838\) 3.18302 0.109956
\(839\) 9.35636 0.323018 0.161509 0.986871i \(-0.448364\pi\)
0.161509 + 0.986871i \(0.448364\pi\)
\(840\) 7.60090 0.262256
\(841\) −28.8144 −0.993599
\(842\) 12.8911 0.444255
\(843\) −20.9688 −0.722203
\(844\) 2.00558 0.0690350
\(845\) −1.68854 −0.0580877
\(846\) 1.33865 0.0460236
\(847\) −46.7275 −1.60558
\(848\) 8.80373 0.302321
\(849\) −29.4972 −1.01234
\(850\) 0.259444 0.00889887
\(851\) −25.5480 −0.875773
\(852\) 14.3213 0.490641
\(853\) 2.09185 0.0716237 0.0358119 0.999359i \(-0.488598\pi\)
0.0358119 + 0.999359i \(0.488598\pi\)
\(854\) −42.0334 −1.43835
\(855\) −3.66122 −0.125211
\(856\) 11.8054 0.403502
\(857\) 22.2378 0.759627 0.379814 0.925063i \(-0.375988\pi\)
0.379814 + 0.925063i \(0.375988\pi\)
\(858\) 0.787053 0.0268695
\(859\) −20.6629 −0.705008 −0.352504 0.935810i \(-0.614670\pi\)
−0.352504 + 0.935810i \(0.614670\pi\)
\(860\) −4.10363 −0.139933
\(861\) −47.3737 −1.61449
\(862\) −24.0805 −0.820184
\(863\) −27.2497 −0.927590 −0.463795 0.885943i \(-0.653513\pi\)
−0.463795 + 0.885943i \(0.653513\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 8.82443 0.300040
\(866\) −30.1923 −1.02598
\(867\) 16.9854 0.576855
\(868\) 9.76038 0.331289
\(869\) 9.22705 0.313006
\(870\) 0.727518 0.0246652
\(871\) 15.9992 0.542113
\(872\) −2.30323 −0.0779972
\(873\) 1.26242 0.0427266
\(874\) −10.0528 −0.340041
\(875\) 54.3375 1.83694
\(876\) −10.9288 −0.369251
\(877\) 43.1359 1.45660 0.728298 0.685260i \(-0.240312\pi\)
0.728298 + 0.685260i \(0.240312\pi\)
\(878\) 5.69015 0.192033
\(879\) 22.5817 0.761662
\(880\) 1.32897 0.0447997
\(881\) −11.7335 −0.395313 −0.197656 0.980271i \(-0.563333\pi\)
−0.197656 + 0.980271i \(0.563333\pi\)
\(882\) 13.2631 0.446591
\(883\) −18.8362 −0.633887 −0.316944 0.948444i \(-0.602657\pi\)
−0.316944 + 0.948444i \(0.602657\pi\)
\(884\) −0.120738 −0.00406086
\(885\) −16.0603 −0.539861
\(886\) −12.9170 −0.433955
\(887\) −23.3711 −0.784724 −0.392362 0.919811i \(-0.628342\pi\)
−0.392362 + 0.919811i \(0.628342\pi\)
\(888\) −5.51041 −0.184917
\(889\) −20.8319 −0.698679
\(890\) −7.62063 −0.255444
\(891\) −0.787053 −0.0263673
\(892\) −19.0380 −0.637440
\(893\) 2.90255 0.0971302
\(894\) −4.84346 −0.161990
\(895\) −13.8990 −0.464591
\(896\) 4.50145 0.150383
\(897\) 4.63632 0.154802
\(898\) −21.2531 −0.709226
\(899\) 0.934213 0.0311577
\(900\) −2.14882 −0.0716274
\(901\) −1.06294 −0.0354118
\(902\) −8.28301 −0.275794
\(903\) −10.9398 −0.364053
\(904\) 2.92570 0.0973075
\(905\) 39.2855 1.30589
\(906\) −20.2437 −0.672552
\(907\) −15.2785 −0.507315 −0.253658 0.967294i \(-0.581634\pi\)
−0.253658 + 0.967294i \(0.581634\pi\)
\(908\) 22.5082 0.746960
\(909\) −11.2050 −0.371647
\(910\) −7.60090 −0.251967
\(911\) 21.8494 0.723904 0.361952 0.932197i \(-0.382110\pi\)
0.361952 + 0.932197i \(0.382110\pi\)
\(912\) −2.16827 −0.0717987
\(913\) −7.80939 −0.258453
\(914\) 23.2880 0.770298
\(915\) −15.7672 −0.521247
\(916\) 16.8382 0.556349
\(917\) −36.3250 −1.19956
\(918\) 0.120738 0.00398495
\(919\) −17.3501 −0.572328 −0.286164 0.958181i \(-0.592380\pi\)
−0.286164 + 0.958181i \(0.592380\pi\)
\(920\) 7.82862 0.258102
\(921\) 12.4312 0.409621
\(922\) 18.2701 0.601693
\(923\) −14.3213 −0.471393
\(924\) 3.54288 0.116552
\(925\) −11.8409 −0.389326
\(926\) 22.7042 0.746105
\(927\) 1.00000 0.0328443
\(928\) 0.430855 0.0141435
\(929\) −40.3220 −1.32292 −0.661461 0.749979i \(-0.730063\pi\)
−0.661461 + 0.749979i \(0.730063\pi\)
\(930\) 3.66122 0.120056
\(931\) 28.7580 0.942504
\(932\) 14.4896 0.474623
\(933\) 12.2864 0.402239
\(934\) 12.0488 0.394249
\(935\) −0.160458 −0.00524752
\(936\) 1.00000 0.0326860
\(937\) 33.5992 1.09764 0.548819 0.835941i \(-0.315078\pi\)
0.548819 + 0.835941i \(0.315078\pi\)
\(938\) 72.0197 2.35153
\(939\) 17.9396 0.585437
\(940\) −2.26036 −0.0737249
\(941\) −21.5286 −0.701811 −0.350906 0.936411i \(-0.614126\pi\)
−0.350906 + 0.936411i \(0.614126\pi\)
\(942\) 0.819356 0.0266961
\(943\) −48.7930 −1.58892
\(944\) −9.51134 −0.309568
\(945\) 7.60090 0.247257
\(946\) −1.91276 −0.0621891
\(947\) −53.3155 −1.73252 −0.866261 0.499592i \(-0.833483\pi\)
−0.866261 + 0.499592i \(0.833483\pi\)
\(948\) 11.7235 0.380763
\(949\) 10.9288 0.354765
\(950\) −4.65923 −0.151165
\(951\) −31.1119 −1.00887
\(952\) −0.543496 −0.0176148
\(953\) −23.5990 −0.764447 −0.382223 0.924070i \(-0.624842\pi\)
−0.382223 + 0.924070i \(0.624842\pi\)
\(954\) 8.80373 0.285031
\(955\) 5.34275 0.172887
\(956\) 5.77146 0.186663
\(957\) 0.339106 0.0109617
\(958\) −41.7439 −1.34868
\(959\) −72.6636 −2.34643
\(960\) 1.68854 0.0544975
\(961\) −26.2986 −0.848342
\(962\) 5.51041 0.177663
\(963\) 11.8054 0.380425
\(964\) 22.2958 0.718100
\(965\) 2.24389 0.0722334
\(966\) 20.8702 0.671486
\(967\) 15.5519 0.500116 0.250058 0.968231i \(-0.419550\pi\)
0.250058 + 0.968231i \(0.419550\pi\)
\(968\) −10.3805 −0.333643
\(969\) 0.261793 0.00841001
\(970\) −2.13166 −0.0684434
\(971\) −41.5430 −1.33318 −0.666589 0.745426i \(-0.732246\pi\)
−0.666589 + 0.745426i \(0.732246\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 33.7546 1.08212
\(974\) −34.4976 −1.10537
\(975\) 2.14882 0.0688174
\(976\) −9.33774 −0.298894
\(977\) −12.4419 −0.398053 −0.199027 0.979994i \(-0.563778\pi\)
−0.199027 + 0.979994i \(0.563778\pi\)
\(978\) −18.4391 −0.589619
\(979\) −3.55208 −0.113525
\(980\) −22.3953 −0.715390
\(981\) −2.30323 −0.0735364
\(982\) −33.4958 −1.06890
\(983\) −21.3596 −0.681265 −0.340632 0.940197i \(-0.610641\pi\)
−0.340632 + 0.940197i \(0.610641\pi\)
\(984\) −10.5241 −0.335496
\(985\) −2.47600 −0.0788920
\(986\) −0.0520206 −0.00165667
\(987\) −6.02585 −0.191805
\(988\) 2.16827 0.0689820
\(989\) −11.2675 −0.358287
\(990\) 1.32897 0.0422375
\(991\) −5.35562 −0.170127 −0.0850634 0.996376i \(-0.527109\pi\)
−0.0850634 + 0.996376i \(0.527109\pi\)
\(992\) 2.16827 0.0688428
\(993\) −29.8047 −0.945823
\(994\) −64.4668 −2.04476
\(995\) −36.2386 −1.14884
\(996\) −9.92233 −0.314401
\(997\) −55.2271 −1.74906 −0.874530 0.484972i \(-0.838830\pi\)
−0.874530 + 0.484972i \(0.838830\pi\)
\(998\) 24.5788 0.778027
\(999\) −5.51041 −0.174341
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 8034.2.a.y.1.3 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.y.1.3 13 1.1 even 1 trivial