Properties

Label 2001.2.a.o.1.11
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $20$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, 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("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + 14484 x^{12} - 24566 x^{11} - 36791 x^{10} + 59410 x^{9} + 52109 x^{8} + \cdots - 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(0.458217\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.458217 q^{2} +1.00000 q^{3} -1.79004 q^{4} -0.388274 q^{5} +0.458217 q^{6} -4.22376 q^{7} -1.73666 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.458217 q^{2} +1.00000 q^{3} -1.79004 q^{4} -0.388274 q^{5} +0.458217 q^{6} -4.22376 q^{7} -1.73666 q^{8} +1.00000 q^{9} -0.177914 q^{10} +2.44305 q^{11} -1.79004 q^{12} +0.942751 q^{13} -1.93540 q^{14} -0.388274 q^{15} +2.78431 q^{16} -2.35495 q^{17} +0.458217 q^{18} +2.89264 q^{19} +0.695024 q^{20} -4.22376 q^{21} +1.11945 q^{22} +1.00000 q^{23} -1.73666 q^{24} -4.84924 q^{25} +0.431985 q^{26} +1.00000 q^{27} +7.56068 q^{28} +1.00000 q^{29} -0.177914 q^{30} +7.75420 q^{31} +4.74914 q^{32} +2.44305 q^{33} -1.07908 q^{34} +1.63997 q^{35} -1.79004 q^{36} -2.82580 q^{37} +1.32546 q^{38} +0.942751 q^{39} +0.674299 q^{40} +5.25950 q^{41} -1.93540 q^{42} -9.79866 q^{43} -4.37315 q^{44} -0.388274 q^{45} +0.458217 q^{46} +9.98999 q^{47} +2.78431 q^{48} +10.8401 q^{49} -2.22201 q^{50} -2.35495 q^{51} -1.68756 q^{52} +6.00735 q^{53} +0.458217 q^{54} -0.948572 q^{55} +7.33523 q^{56} +2.89264 q^{57} +0.458217 q^{58} +8.95684 q^{59} +0.695024 q^{60} +8.75196 q^{61} +3.55311 q^{62} -4.22376 q^{63} -3.39248 q^{64} -0.366045 q^{65} +1.11945 q^{66} -8.58964 q^{67} +4.21544 q^{68} +1.00000 q^{69} +0.751464 q^{70} +6.86529 q^{71} -1.73666 q^{72} +12.2145 q^{73} -1.29483 q^{74} -4.84924 q^{75} -5.17793 q^{76} -10.3189 q^{77} +0.431985 q^{78} -7.05107 q^{79} -1.08107 q^{80} +1.00000 q^{81} +2.40999 q^{82} +10.9024 q^{83} +7.56068 q^{84} +0.914364 q^{85} -4.48992 q^{86} +1.00000 q^{87} -4.24275 q^{88} +8.73399 q^{89} -0.177914 q^{90} -3.98195 q^{91} -1.79004 q^{92} +7.75420 q^{93} +4.57759 q^{94} -1.12314 q^{95} +4.74914 q^{96} +17.6770 q^{97} +4.96713 q^{98} +2.44305 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9} + 7 q^{10} + 30 q^{12} + 21 q^{13} - q^{14} - q^{15} + 58 q^{16} - 4 q^{17} + 2 q^{18} + 7 q^{19} - 20 q^{20} + 9 q^{21} + 7 q^{22} + 20 q^{23} + 6 q^{24} + 47 q^{25} + 8 q^{26} + 20 q^{27} + 11 q^{28} + 20 q^{29} + 7 q^{30} + 28 q^{31} + 14 q^{32} + 16 q^{34} + 9 q^{35} + 30 q^{36} + 14 q^{37} - 20 q^{38} + 21 q^{39} + 34 q^{40} + 7 q^{41} - q^{42} + 3 q^{43} - q^{44} - q^{45} + 2 q^{46} + 3 q^{47} + 58 q^{48} + 35 q^{49} - 24 q^{50} - 4 q^{51} + 73 q^{52} - 19 q^{53} + 2 q^{54} + 29 q^{55} - 30 q^{56} + 7 q^{57} + 2 q^{58} + 20 q^{59} - 20 q^{60} + 15 q^{61} + 12 q^{62} + 9 q^{63} + 82 q^{64} - 28 q^{65} + 7 q^{66} + 20 q^{67} - 23 q^{68} + 20 q^{69} - 24 q^{70} + 63 q^{71} + 6 q^{72} + 19 q^{73} + 16 q^{74} + 47 q^{75} - 44 q^{76} - 7 q^{77} + 8 q^{78} + 32 q^{79} - 56 q^{80} + 20 q^{81} - 20 q^{82} - 21 q^{83} + 11 q^{84} + 4 q^{85} - 6 q^{86} + 20 q^{87} + 55 q^{88} - 13 q^{89} + 7 q^{90} + 70 q^{91} + 30 q^{92} + 28 q^{93} - 12 q^{94} + 9 q^{95} + 14 q^{96} - 9 q^{97} + 31 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.458217 0.324008 0.162004 0.986790i \(-0.448204\pi\)
0.162004 + 0.986790i \(0.448204\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.79004 −0.895018
\(5\) −0.388274 −0.173641 −0.0868206 0.996224i \(-0.527671\pi\)
−0.0868206 + 0.996224i \(0.527671\pi\)
\(6\) 0.458217 0.187066
\(7\) −4.22376 −1.59643 −0.798215 0.602372i \(-0.794222\pi\)
−0.798215 + 0.602372i \(0.794222\pi\)
\(8\) −1.73666 −0.614002
\(9\) 1.00000 0.333333
\(10\) −0.177914 −0.0562612
\(11\) 2.44305 0.736607 0.368304 0.929706i \(-0.379939\pi\)
0.368304 + 0.929706i \(0.379939\pi\)
\(12\) −1.79004 −0.516739
\(13\) 0.942751 0.261472 0.130736 0.991417i \(-0.458266\pi\)
0.130736 + 0.991417i \(0.458266\pi\)
\(14\) −1.93540 −0.517257
\(15\) −0.388274 −0.100252
\(16\) 2.78431 0.696077
\(17\) −2.35495 −0.571159 −0.285579 0.958355i \(-0.592186\pi\)
−0.285579 + 0.958355i \(0.592186\pi\)
\(18\) 0.458217 0.108003
\(19\) 2.89264 0.663617 0.331809 0.943347i \(-0.392341\pi\)
0.331809 + 0.943347i \(0.392341\pi\)
\(20\) 0.695024 0.155412
\(21\) −4.22376 −0.921699
\(22\) 1.11945 0.238667
\(23\) 1.00000 0.208514
\(24\) −1.73666 −0.354494
\(25\) −4.84924 −0.969849
\(26\) 0.431985 0.0847191
\(27\) 1.00000 0.192450
\(28\) 7.56068 1.42883
\(29\) 1.00000 0.185695
\(30\) −0.177914 −0.0324824
\(31\) 7.75420 1.39270 0.696348 0.717704i \(-0.254807\pi\)
0.696348 + 0.717704i \(0.254807\pi\)
\(32\) 4.74914 0.839537
\(33\) 2.44305 0.425281
\(34\) −1.07908 −0.185060
\(35\) 1.63997 0.277206
\(36\) −1.79004 −0.298339
\(37\) −2.82580 −0.464559 −0.232280 0.972649i \(-0.574618\pi\)
−0.232280 + 0.972649i \(0.574618\pi\)
\(38\) 1.32546 0.215018
\(39\) 0.942751 0.150961
\(40\) 0.674299 0.106616
\(41\) 5.25950 0.821395 0.410698 0.911772i \(-0.365285\pi\)
0.410698 + 0.911772i \(0.365285\pi\)
\(42\) −1.93540 −0.298638
\(43\) −9.79866 −1.49428 −0.747141 0.664665i \(-0.768574\pi\)
−0.747141 + 0.664665i \(0.768574\pi\)
\(44\) −4.37315 −0.659277
\(45\) −0.388274 −0.0578804
\(46\) 0.458217 0.0675604
\(47\) 9.98999 1.45719 0.728595 0.684945i \(-0.240174\pi\)
0.728595 + 0.684945i \(0.240174\pi\)
\(48\) 2.78431 0.401880
\(49\) 10.8401 1.54859
\(50\) −2.22201 −0.314239
\(51\) −2.35495 −0.329759
\(52\) −1.68756 −0.234022
\(53\) 6.00735 0.825173 0.412586 0.910918i \(-0.364625\pi\)
0.412586 + 0.910918i \(0.364625\pi\)
\(54\) 0.458217 0.0623555
\(55\) −0.948572 −0.127905
\(56\) 7.33523 0.980212
\(57\) 2.89264 0.383140
\(58\) 0.458217 0.0601669
\(59\) 8.95684 1.16608 0.583041 0.812443i \(-0.301863\pi\)
0.583041 + 0.812443i \(0.301863\pi\)
\(60\) 0.695024 0.0897272
\(61\) 8.75196 1.12057 0.560287 0.828299i \(-0.310691\pi\)
0.560287 + 0.828299i \(0.310691\pi\)
\(62\) 3.55311 0.451245
\(63\) −4.22376 −0.532143
\(64\) −3.39248 −0.424060
\(65\) −0.366045 −0.0454023
\(66\) 1.11945 0.137795
\(67\) −8.58964 −1.04939 −0.524696 0.851290i \(-0.675821\pi\)
−0.524696 + 0.851290i \(0.675821\pi\)
\(68\) 4.21544 0.511197
\(69\) 1.00000 0.120386
\(70\) 0.751464 0.0898171
\(71\) 6.86529 0.814761 0.407380 0.913259i \(-0.366442\pi\)
0.407380 + 0.913259i \(0.366442\pi\)
\(72\) −1.73666 −0.204667
\(73\) 12.2145 1.42960 0.714802 0.699327i \(-0.246517\pi\)
0.714802 + 0.699327i \(0.246517\pi\)
\(74\) −1.29483 −0.150521
\(75\) −4.84924 −0.559942
\(76\) −5.17793 −0.593950
\(77\) −10.3189 −1.17594
\(78\) 0.431985 0.0489126
\(79\) −7.05107 −0.793308 −0.396654 0.917968i \(-0.629829\pi\)
−0.396654 + 0.917968i \(0.629829\pi\)
\(80\) −1.08107 −0.120868
\(81\) 1.00000 0.111111
\(82\) 2.40999 0.266139
\(83\) 10.9024 1.19669 0.598345 0.801238i \(-0.295825\pi\)
0.598345 + 0.801238i \(0.295825\pi\)
\(84\) 7.56068 0.824938
\(85\) 0.914364 0.0991767
\(86\) −4.48992 −0.484160
\(87\) 1.00000 0.107211
\(88\) −4.24275 −0.452279
\(89\) 8.73399 0.925801 0.462901 0.886410i \(-0.346809\pi\)
0.462901 + 0.886410i \(0.346809\pi\)
\(90\) −0.177914 −0.0187537
\(91\) −3.98195 −0.417422
\(92\) −1.79004 −0.186624
\(93\) 7.75420 0.804074
\(94\) 4.57759 0.472142
\(95\) −1.12314 −0.115231
\(96\) 4.74914 0.484707
\(97\) 17.6770 1.79483 0.897413 0.441192i \(-0.145444\pi\)
0.897413 + 0.441192i \(0.145444\pi\)
\(98\) 4.96713 0.501756
\(99\) 2.44305 0.245536
\(100\) 8.68033 0.868033
\(101\) 8.17134 0.813078 0.406539 0.913633i \(-0.366735\pi\)
0.406539 + 0.913633i \(0.366735\pi\)
\(102\) −1.07908 −0.106845
\(103\) 9.88143 0.973646 0.486823 0.873501i \(-0.338156\pi\)
0.486823 + 0.873501i \(0.338156\pi\)
\(104\) −1.63724 −0.160544
\(105\) 1.63997 0.160045
\(106\) 2.75267 0.267363
\(107\) −10.5569 −1.02057 −0.510285 0.860005i \(-0.670460\pi\)
−0.510285 + 0.860005i \(0.670460\pi\)
\(108\) −1.79004 −0.172246
\(109\) 15.2793 1.46349 0.731745 0.681578i \(-0.238706\pi\)
0.731745 + 0.681578i \(0.238706\pi\)
\(110\) −0.434652 −0.0414424
\(111\) −2.82580 −0.268213
\(112\) −11.7602 −1.11124
\(113\) −11.6224 −1.09334 −0.546670 0.837348i \(-0.684105\pi\)
−0.546670 + 0.837348i \(0.684105\pi\)
\(114\) 1.32546 0.124140
\(115\) −0.388274 −0.0362067
\(116\) −1.79004 −0.166201
\(117\) 0.942751 0.0871573
\(118\) 4.10418 0.377820
\(119\) 9.94673 0.911815
\(120\) 0.674299 0.0615548
\(121\) −5.03150 −0.457409
\(122\) 4.01030 0.363075
\(123\) 5.25950 0.474233
\(124\) −13.8803 −1.24649
\(125\) 3.82420 0.342047
\(126\) −1.93540 −0.172419
\(127\) −6.93670 −0.615532 −0.307766 0.951462i \(-0.599581\pi\)
−0.307766 + 0.951462i \(0.599581\pi\)
\(128\) −11.0528 −0.976936
\(129\) −9.79866 −0.862724
\(130\) −0.167728 −0.0147107
\(131\) −12.5516 −1.09664 −0.548319 0.836270i \(-0.684732\pi\)
−0.548319 + 0.836270i \(0.684732\pi\)
\(132\) −4.37315 −0.380634
\(133\) −12.2178 −1.05942
\(134\) −3.93592 −0.340012
\(135\) −0.388274 −0.0334173
\(136\) 4.08974 0.350693
\(137\) −11.6780 −0.997716 −0.498858 0.866684i \(-0.666247\pi\)
−0.498858 + 0.866684i \(0.666247\pi\)
\(138\) 0.458217 0.0390060
\(139\) −2.41460 −0.204804 −0.102402 0.994743i \(-0.532653\pi\)
−0.102402 + 0.994743i \(0.532653\pi\)
\(140\) −2.93561 −0.248105
\(141\) 9.98999 0.841309
\(142\) 3.14580 0.263989
\(143\) 2.30319 0.192602
\(144\) 2.78431 0.232026
\(145\) −0.388274 −0.0322444
\(146\) 5.59691 0.463204
\(147\) 10.8401 0.894079
\(148\) 5.05829 0.415789
\(149\) −8.54594 −0.700110 −0.350055 0.936729i \(-0.613837\pi\)
−0.350055 + 0.936729i \(0.613837\pi\)
\(150\) −2.22201 −0.181426
\(151\) −4.65908 −0.379150 −0.189575 0.981866i \(-0.560711\pi\)
−0.189575 + 0.981866i \(0.560711\pi\)
\(152\) −5.02353 −0.407462
\(153\) −2.35495 −0.190386
\(154\) −4.72828 −0.381015
\(155\) −3.01075 −0.241829
\(156\) −1.68756 −0.135113
\(157\) −6.62907 −0.529057 −0.264529 0.964378i \(-0.585216\pi\)
−0.264529 + 0.964378i \(0.585216\pi\)
\(158\) −3.23092 −0.257038
\(159\) 6.00735 0.476414
\(160\) −1.84396 −0.145778
\(161\) −4.22376 −0.332879
\(162\) 0.458217 0.0360009
\(163\) 7.80982 0.611712 0.305856 0.952078i \(-0.401057\pi\)
0.305856 + 0.952078i \(0.401057\pi\)
\(164\) −9.41469 −0.735164
\(165\) −0.948572 −0.0738462
\(166\) 4.99566 0.387738
\(167\) −20.4339 −1.58122 −0.790611 0.612319i \(-0.790237\pi\)
−0.790611 + 0.612319i \(0.790237\pi\)
\(168\) 7.33523 0.565925
\(169\) −12.1112 −0.931632
\(170\) 0.418977 0.0321341
\(171\) 2.89264 0.221206
\(172\) 17.5400 1.33741
\(173\) −11.5310 −0.876688 −0.438344 0.898807i \(-0.644435\pi\)
−0.438344 + 0.898807i \(0.644435\pi\)
\(174\) 0.458217 0.0347374
\(175\) 20.4820 1.54830
\(176\) 6.80220 0.512735
\(177\) 8.95684 0.673237
\(178\) 4.00207 0.299968
\(179\) −11.3766 −0.850327 −0.425163 0.905117i \(-0.639783\pi\)
−0.425163 + 0.905117i \(0.639783\pi\)
\(180\) 0.695024 0.0518040
\(181\) 0.967738 0.0719314 0.0359657 0.999353i \(-0.488549\pi\)
0.0359657 + 0.999353i \(0.488549\pi\)
\(182\) −1.82460 −0.135248
\(183\) 8.75196 0.646964
\(184\) −1.73666 −0.128028
\(185\) 1.09718 0.0806666
\(186\) 3.55311 0.260527
\(187\) −5.75325 −0.420720
\(188\) −17.8825 −1.30421
\(189\) −4.22376 −0.307233
\(190\) −0.514640 −0.0373359
\(191\) 11.6319 0.841653 0.420826 0.907141i \(-0.361740\pi\)
0.420826 + 0.907141i \(0.361740\pi\)
\(192\) −3.39248 −0.244831
\(193\) 15.4098 1.10922 0.554610 0.832110i \(-0.312867\pi\)
0.554610 + 0.832110i \(0.312867\pi\)
\(194\) 8.09990 0.581539
\(195\) −0.366045 −0.0262130
\(196\) −19.4042 −1.38602
\(197\) −9.27867 −0.661078 −0.330539 0.943792i \(-0.607231\pi\)
−0.330539 + 0.943792i \(0.607231\pi\)
\(198\) 1.11945 0.0795557
\(199\) 7.73656 0.548430 0.274215 0.961668i \(-0.411582\pi\)
0.274215 + 0.961668i \(0.411582\pi\)
\(200\) 8.42149 0.595489
\(201\) −8.58964 −0.605866
\(202\) 3.74425 0.263444
\(203\) −4.22376 −0.296450
\(204\) 4.21544 0.295140
\(205\) −2.04212 −0.142628
\(206\) 4.52784 0.315470
\(207\) 1.00000 0.0695048
\(208\) 2.62491 0.182005
\(209\) 7.06687 0.488825
\(210\) 0.751464 0.0518559
\(211\) 16.0483 1.10481 0.552404 0.833577i \(-0.313711\pi\)
0.552404 + 0.833577i \(0.313711\pi\)
\(212\) −10.7534 −0.738545
\(213\) 6.86529 0.470402
\(214\) −4.83734 −0.330673
\(215\) 3.80456 0.259469
\(216\) −1.73666 −0.118165
\(217\) −32.7519 −2.22334
\(218\) 7.00123 0.474183
\(219\) 12.2145 0.825382
\(220\) 1.69798 0.114478
\(221\) −2.22013 −0.149342
\(222\) −1.29483 −0.0869034
\(223\) 17.6200 1.17992 0.589961 0.807432i \(-0.299143\pi\)
0.589961 + 0.807432i \(0.299143\pi\)
\(224\) −20.0592 −1.34026
\(225\) −4.84924 −0.323283
\(226\) −5.32557 −0.354252
\(227\) −6.15492 −0.408516 −0.204258 0.978917i \(-0.565478\pi\)
−0.204258 + 0.978917i \(0.565478\pi\)
\(228\) −5.17793 −0.342917
\(229\) 6.76626 0.447127 0.223563 0.974689i \(-0.428231\pi\)
0.223563 + 0.974689i \(0.428231\pi\)
\(230\) −0.177914 −0.0117313
\(231\) −10.3189 −0.678931
\(232\) −1.73666 −0.114017
\(233\) −12.9500 −0.848382 −0.424191 0.905573i \(-0.639441\pi\)
−0.424191 + 0.905573i \(0.639441\pi\)
\(234\) 0.431985 0.0282397
\(235\) −3.87885 −0.253028
\(236\) −16.0331 −1.04366
\(237\) −7.05107 −0.458016
\(238\) 4.55776 0.295436
\(239\) 26.2200 1.69603 0.848016 0.529970i \(-0.177797\pi\)
0.848016 + 0.529970i \(0.177797\pi\)
\(240\) −1.08107 −0.0697829
\(241\) −13.4553 −0.866733 −0.433367 0.901218i \(-0.642675\pi\)
−0.433367 + 0.901218i \(0.642675\pi\)
\(242\) −2.30552 −0.148205
\(243\) 1.00000 0.0641500
\(244\) −15.6663 −1.00293
\(245\) −4.20893 −0.268899
\(246\) 2.40999 0.153655
\(247\) 2.72704 0.173517
\(248\) −13.4664 −0.855118
\(249\) 10.9024 0.690910
\(250\) 1.75231 0.110826
\(251\) 6.61094 0.417279 0.208640 0.977993i \(-0.433096\pi\)
0.208640 + 0.977993i \(0.433096\pi\)
\(252\) 7.56068 0.476278
\(253\) 2.44305 0.153593
\(254\) −3.17851 −0.199438
\(255\) 0.914364 0.0572597
\(256\) 1.72039 0.107524
\(257\) 12.6358 0.788198 0.394099 0.919068i \(-0.371057\pi\)
0.394099 + 0.919068i \(0.371057\pi\)
\(258\) −4.48992 −0.279530
\(259\) 11.9355 0.741637
\(260\) 0.655234 0.0406359
\(261\) 1.00000 0.0618984
\(262\) −5.75135 −0.355320
\(263\) −31.2519 −1.92707 −0.963537 0.267575i \(-0.913778\pi\)
−0.963537 + 0.267575i \(0.913778\pi\)
\(264\) −4.24275 −0.261123
\(265\) −2.33249 −0.143284
\(266\) −5.59841 −0.343261
\(267\) 8.73399 0.534512
\(268\) 15.3758 0.939225
\(269\) −12.8085 −0.780946 −0.390473 0.920614i \(-0.627688\pi\)
−0.390473 + 0.920614i \(0.627688\pi\)
\(270\) −0.177914 −0.0108275
\(271\) −5.91426 −0.359266 −0.179633 0.983734i \(-0.557491\pi\)
−0.179633 + 0.983734i \(0.557491\pi\)
\(272\) −6.55689 −0.397570
\(273\) −3.98195 −0.240999
\(274\) −5.35105 −0.323268
\(275\) −11.8469 −0.714398
\(276\) −1.79004 −0.107748
\(277\) 11.5359 0.693128 0.346564 0.938026i \(-0.387348\pi\)
0.346564 + 0.938026i \(0.387348\pi\)
\(278\) −1.10641 −0.0663583
\(279\) 7.75420 0.464232
\(280\) −2.84808 −0.170205
\(281\) 22.5802 1.34702 0.673510 0.739178i \(-0.264786\pi\)
0.673510 + 0.739178i \(0.264786\pi\)
\(282\) 4.57759 0.272591
\(283\) 15.6402 0.929716 0.464858 0.885385i \(-0.346105\pi\)
0.464858 + 0.885385i \(0.346105\pi\)
\(284\) −12.2891 −0.729226
\(285\) −1.12314 −0.0665288
\(286\) 1.05536 0.0624047
\(287\) −22.2148 −1.31130
\(288\) 4.74914 0.279846
\(289\) −11.4542 −0.673778
\(290\) −0.177914 −0.0104474
\(291\) 17.6770 1.03624
\(292\) −21.8645 −1.27952
\(293\) 5.69213 0.332538 0.166269 0.986080i \(-0.446828\pi\)
0.166269 + 0.986080i \(0.446828\pi\)
\(294\) 4.96713 0.289689
\(295\) −3.47770 −0.202480
\(296\) 4.90746 0.285240
\(297\) 2.44305 0.141760
\(298\) −3.91590 −0.226842
\(299\) 0.942751 0.0545207
\(300\) 8.68033 0.501159
\(301\) 41.3872 2.38552
\(302\) −2.13487 −0.122848
\(303\) 8.17134 0.469431
\(304\) 8.05400 0.461929
\(305\) −3.39815 −0.194578
\(306\) −1.07908 −0.0616867
\(307\) −0.793984 −0.0453150 −0.0226575 0.999743i \(-0.507213\pi\)
−0.0226575 + 0.999743i \(0.507213\pi\)
\(308\) 18.4711 1.05249
\(309\) 9.88143 0.562135
\(310\) −1.37958 −0.0783548
\(311\) 33.6511 1.90818 0.954088 0.299525i \(-0.0968283\pi\)
0.954088 + 0.299525i \(0.0968283\pi\)
\(312\) −1.63724 −0.0926903
\(313\) −17.9983 −1.01732 −0.508661 0.860967i \(-0.669859\pi\)
−0.508661 + 0.860967i \(0.669859\pi\)
\(314\) −3.03755 −0.171419
\(315\) 1.63997 0.0924020
\(316\) 12.6217 0.710025
\(317\) −11.0791 −0.622266 −0.311133 0.950366i \(-0.600708\pi\)
−0.311133 + 0.950366i \(0.600708\pi\)
\(318\) 2.75267 0.154362
\(319\) 2.44305 0.136785
\(320\) 1.31721 0.0736342
\(321\) −10.5569 −0.589227
\(322\) −1.93540 −0.107856
\(323\) −6.81202 −0.379031
\(324\) −1.79004 −0.0994465
\(325\) −4.57163 −0.253588
\(326\) 3.57859 0.198200
\(327\) 15.2793 0.844947
\(328\) −9.13396 −0.504338
\(329\) −42.1953 −2.32630
\(330\) −0.434652 −0.0239268
\(331\) 32.2908 1.77486 0.887430 0.460942i \(-0.152488\pi\)
0.887430 + 0.460942i \(0.152488\pi\)
\(332\) −19.5157 −1.07106
\(333\) −2.82580 −0.154853
\(334\) −9.36316 −0.512329
\(335\) 3.33513 0.182218
\(336\) −11.7602 −0.641573
\(337\) −11.4739 −0.625023 −0.312512 0.949914i \(-0.601170\pi\)
−0.312512 + 0.949914i \(0.601170\pi\)
\(338\) −5.54957 −0.301857
\(339\) −11.6224 −0.631240
\(340\) −1.63674 −0.0887649
\(341\) 18.9439 1.02587
\(342\) 1.32546 0.0716726
\(343\) −16.2198 −0.875785
\(344\) 17.0169 0.917492
\(345\) −0.388274 −0.0209039
\(346\) −5.28372 −0.284054
\(347\) 32.4849 1.74388 0.871940 0.489613i \(-0.162862\pi\)
0.871940 + 0.489613i \(0.162862\pi\)
\(348\) −1.79004 −0.0959561
\(349\) −20.8858 −1.11799 −0.558996 0.829170i \(-0.688813\pi\)
−0.558996 + 0.829170i \(0.688813\pi\)
\(350\) 9.38522 0.501661
\(351\) 0.942751 0.0503203
\(352\) 11.6024 0.618409
\(353\) −9.36223 −0.498301 −0.249151 0.968465i \(-0.580151\pi\)
−0.249151 + 0.968465i \(0.580151\pi\)
\(354\) 4.10418 0.218135
\(355\) −2.66561 −0.141476
\(356\) −15.6342 −0.828609
\(357\) 9.94673 0.526437
\(358\) −5.21295 −0.275513
\(359\) −9.22588 −0.486923 −0.243462 0.969911i \(-0.578283\pi\)
−0.243462 + 0.969911i \(0.578283\pi\)
\(360\) 0.674299 0.0355387
\(361\) −10.6326 −0.559612
\(362\) 0.443434 0.0233064
\(363\) −5.03150 −0.264085
\(364\) 7.12784 0.373600
\(365\) −4.74258 −0.248238
\(366\) 4.01030 0.209622
\(367\) 10.0371 0.523933 0.261967 0.965077i \(-0.415629\pi\)
0.261967 + 0.965077i \(0.415629\pi\)
\(368\) 2.78431 0.145142
\(369\) 5.25950 0.273798
\(370\) 0.502749 0.0261367
\(371\) −25.3736 −1.31733
\(372\) −13.8803 −0.719661
\(373\) 35.0846 1.81661 0.908307 0.418305i \(-0.137376\pi\)
0.908307 + 0.418305i \(0.137376\pi\)
\(374\) −2.63624 −0.136317
\(375\) 3.82420 0.197481
\(376\) −17.3492 −0.894718
\(377\) 0.942751 0.0485541
\(378\) −1.93540 −0.0995462
\(379\) 22.9740 1.18009 0.590047 0.807369i \(-0.299109\pi\)
0.590047 + 0.807369i \(0.299109\pi\)
\(380\) 2.01045 0.103134
\(381\) −6.93670 −0.355378
\(382\) 5.32993 0.272703
\(383\) 31.6924 1.61941 0.809703 0.586839i \(-0.199628\pi\)
0.809703 + 0.586839i \(0.199628\pi\)
\(384\) −11.0528 −0.564034
\(385\) 4.00654 0.204192
\(386\) 7.06103 0.359397
\(387\) −9.79866 −0.498094
\(388\) −31.6424 −1.60640
\(389\) 8.44436 0.428146 0.214073 0.976818i \(-0.431327\pi\)
0.214073 + 0.976818i \(0.431327\pi\)
\(390\) −0.167728 −0.00849325
\(391\) −2.35495 −0.119095
\(392\) −18.8256 −0.950837
\(393\) −12.5516 −0.633144
\(394\) −4.25165 −0.214195
\(395\) 2.73774 0.137751
\(396\) −4.37315 −0.219759
\(397\) −31.7006 −1.59101 −0.795503 0.605950i \(-0.792793\pi\)
−0.795503 + 0.605950i \(0.792793\pi\)
\(398\) 3.54502 0.177696
\(399\) −12.2178 −0.611656
\(400\) −13.5018 −0.675089
\(401\) −14.8420 −0.741172 −0.370586 0.928798i \(-0.620843\pi\)
−0.370586 + 0.928798i \(0.620843\pi\)
\(402\) −3.93592 −0.196306
\(403\) 7.31028 0.364151
\(404\) −14.6270 −0.727720
\(405\) −0.388274 −0.0192935
\(406\) −1.93540 −0.0960522
\(407\) −6.90358 −0.342198
\(408\) 4.08974 0.202472
\(409\) 26.2358 1.29728 0.648638 0.761097i \(-0.275339\pi\)
0.648638 + 0.761097i \(0.275339\pi\)
\(410\) −0.935736 −0.0462127
\(411\) −11.6780 −0.576032
\(412\) −17.6881 −0.871431
\(413\) −37.8315 −1.86157
\(414\) 0.458217 0.0225201
\(415\) −4.23310 −0.207795
\(416\) 4.47725 0.219515
\(417\) −2.41460 −0.118244
\(418\) 3.23816 0.158384
\(419\) −30.9185 −1.51047 −0.755233 0.655456i \(-0.772476\pi\)
−0.755233 + 0.655456i \(0.772476\pi\)
\(420\) −2.93561 −0.143243
\(421\) 0.992608 0.0483767 0.0241884 0.999707i \(-0.492300\pi\)
0.0241884 + 0.999707i \(0.492300\pi\)
\(422\) 7.35359 0.357967
\(423\) 9.98999 0.485730
\(424\) −10.4327 −0.506658
\(425\) 11.4197 0.553937
\(426\) 3.14580 0.152414
\(427\) −36.9662 −1.78892
\(428\) 18.8972 0.913429
\(429\) 2.30319 0.111199
\(430\) 1.74332 0.0840701
\(431\) −18.1930 −0.876325 −0.438163 0.898896i \(-0.644371\pi\)
−0.438163 + 0.898896i \(0.644371\pi\)
\(432\) 2.78431 0.133960
\(433\) 28.0391 1.34747 0.673737 0.738971i \(-0.264688\pi\)
0.673737 + 0.738971i \(0.264688\pi\)
\(434\) −15.0075 −0.720382
\(435\) −0.388274 −0.0186163
\(436\) −27.3505 −1.30985
\(437\) 2.89264 0.138374
\(438\) 5.59691 0.267431
\(439\) 27.0109 1.28916 0.644579 0.764538i \(-0.277033\pi\)
0.644579 + 0.764538i \(0.277033\pi\)
\(440\) 1.64735 0.0785342
\(441\) 10.8401 0.516197
\(442\) −1.01730 −0.0483881
\(443\) 5.22924 0.248449 0.124224 0.992254i \(-0.460356\pi\)
0.124224 + 0.992254i \(0.460356\pi\)
\(444\) 5.05829 0.240056
\(445\) −3.39118 −0.160757
\(446\) 8.07378 0.382305
\(447\) −8.54594 −0.404209
\(448\) 14.3290 0.676982
\(449\) 16.8352 0.794504 0.397252 0.917709i \(-0.369964\pi\)
0.397252 + 0.917709i \(0.369964\pi\)
\(450\) −2.22201 −0.104746
\(451\) 12.8492 0.605046
\(452\) 20.8045 0.978560
\(453\) −4.65908 −0.218903
\(454\) −2.82029 −0.132363
\(455\) 1.54609 0.0724816
\(456\) −5.02353 −0.235249
\(457\) −35.7717 −1.67333 −0.836666 0.547714i \(-0.815498\pi\)
−0.836666 + 0.547714i \(0.815498\pi\)
\(458\) 3.10042 0.144873
\(459\) −2.35495 −0.109920
\(460\) 0.695024 0.0324057
\(461\) −6.40943 −0.298517 −0.149259 0.988798i \(-0.547689\pi\)
−0.149259 + 0.988798i \(0.547689\pi\)
\(462\) −4.72828 −0.219979
\(463\) 29.7155 1.38100 0.690499 0.723334i \(-0.257391\pi\)
0.690499 + 0.723334i \(0.257391\pi\)
\(464\) 2.78431 0.129258
\(465\) −3.01075 −0.139620
\(466\) −5.93391 −0.274883
\(467\) 9.50195 0.439698 0.219849 0.975534i \(-0.429444\pi\)
0.219849 + 0.975534i \(0.429444\pi\)
\(468\) −1.68756 −0.0780074
\(469\) 36.2805 1.67528
\(470\) −1.77736 −0.0819833
\(471\) −6.62907 −0.305451
\(472\) −15.5550 −0.715976
\(473\) −23.9386 −1.10070
\(474\) −3.23092 −0.148401
\(475\) −14.0271 −0.643608
\(476\) −17.8050 −0.816091
\(477\) 6.00735 0.275058
\(478\) 12.0145 0.549529
\(479\) 25.7687 1.17740 0.588700 0.808351i \(-0.299640\pi\)
0.588700 + 0.808351i \(0.299640\pi\)
\(480\) −1.84396 −0.0841651
\(481\) −2.66403 −0.121469
\(482\) −6.16546 −0.280829
\(483\) −4.22376 −0.192188
\(484\) 9.00658 0.409390
\(485\) −6.86350 −0.311656
\(486\) 0.458217 0.0207852
\(487\) −25.3632 −1.14931 −0.574657 0.818394i \(-0.694865\pi\)
−0.574657 + 0.818394i \(0.694865\pi\)
\(488\) −15.1992 −0.688035
\(489\) 7.80982 0.353172
\(490\) −1.92861 −0.0871256
\(491\) 33.6058 1.51661 0.758305 0.651900i \(-0.226028\pi\)
0.758305 + 0.651900i \(0.226028\pi\)
\(492\) −9.41469 −0.424447
\(493\) −2.35495 −0.106061
\(494\) 1.24958 0.0562211
\(495\) −0.948572 −0.0426351
\(496\) 21.5901 0.969423
\(497\) −28.9973 −1.30071
\(498\) 4.99566 0.223861
\(499\) 24.4757 1.09568 0.547842 0.836582i \(-0.315450\pi\)
0.547842 + 0.836582i \(0.315450\pi\)
\(500\) −6.84546 −0.306138
\(501\) −20.4339 −0.912918
\(502\) 3.02925 0.135202
\(503\) −2.75225 −0.122717 −0.0613584 0.998116i \(-0.519543\pi\)
−0.0613584 + 0.998116i \(0.519543\pi\)
\(504\) 7.33523 0.326737
\(505\) −3.17271 −0.141184
\(506\) 1.11945 0.0497655
\(507\) −12.1112 −0.537878
\(508\) 12.4169 0.550913
\(509\) 22.2953 0.988221 0.494110 0.869399i \(-0.335494\pi\)
0.494110 + 0.869399i \(0.335494\pi\)
\(510\) 0.418977 0.0185526
\(511\) −51.5913 −2.28226
\(512\) 22.8938 1.01177
\(513\) 2.89264 0.127713
\(514\) 5.78993 0.255383
\(515\) −3.83670 −0.169065
\(516\) 17.5400 0.772154
\(517\) 24.4061 1.07338
\(518\) 5.46906 0.240297
\(519\) −11.5310 −0.506156
\(520\) 0.635696 0.0278771
\(521\) −16.4927 −0.722560 −0.361280 0.932457i \(-0.617660\pi\)
−0.361280 + 0.932457i \(0.617660\pi\)
\(522\) 0.458217 0.0200556
\(523\) −38.6999 −1.69223 −0.846114 0.533002i \(-0.821064\pi\)
−0.846114 + 0.533002i \(0.821064\pi\)
\(524\) 22.4678 0.981511
\(525\) 20.4820 0.893909
\(526\) −14.3202 −0.624388
\(527\) −18.2607 −0.795450
\(528\) 6.80220 0.296028
\(529\) 1.00000 0.0434783
\(530\) −1.06879 −0.0464252
\(531\) 8.95684 0.388694
\(532\) 21.8703 0.948199
\(533\) 4.95839 0.214772
\(534\) 4.00207 0.173186
\(535\) 4.09895 0.177213
\(536\) 14.9173 0.644328
\(537\) −11.3766 −0.490936
\(538\) −5.86906 −0.253033
\(539\) 26.4830 1.14070
\(540\) 0.695024 0.0299091
\(541\) 27.7527 1.19318 0.596592 0.802545i \(-0.296521\pi\)
0.596592 + 0.802545i \(0.296521\pi\)
\(542\) −2.71002 −0.116405
\(543\) 0.967738 0.0415296
\(544\) −11.1840 −0.479509
\(545\) −5.93254 −0.254122
\(546\) −1.82460 −0.0780856
\(547\) 37.3701 1.59783 0.798914 0.601445i \(-0.205408\pi\)
0.798914 + 0.601445i \(0.205408\pi\)
\(548\) 20.9040 0.892974
\(549\) 8.75196 0.373525
\(550\) −5.42848 −0.231471
\(551\) 2.89264 0.123231
\(552\) −1.73666 −0.0739172
\(553\) 29.7820 1.26646
\(554\) 5.28597 0.224579
\(555\) 1.09718 0.0465729
\(556\) 4.32223 0.183303
\(557\) −21.0747 −0.892963 −0.446482 0.894793i \(-0.647323\pi\)
−0.446482 + 0.894793i \(0.647323\pi\)
\(558\) 3.55311 0.150415
\(559\) −9.23770 −0.390713
\(560\) 4.56619 0.192957
\(561\) −5.75325 −0.242903
\(562\) 10.3466 0.436446
\(563\) −2.78677 −0.117448 −0.0587241 0.998274i \(-0.518703\pi\)
−0.0587241 + 0.998274i \(0.518703\pi\)
\(564\) −17.8825 −0.752987
\(565\) 4.51266 0.189849
\(566\) 7.16663 0.301236
\(567\) −4.22376 −0.177381
\(568\) −11.9227 −0.500265
\(569\) −29.7476 −1.24708 −0.623541 0.781790i \(-0.714307\pi\)
−0.623541 + 0.781790i \(0.714307\pi\)
\(570\) −0.514640 −0.0215559
\(571\) 0.560978 0.0234762 0.0117381 0.999931i \(-0.496264\pi\)
0.0117381 + 0.999931i \(0.496264\pi\)
\(572\) −4.12279 −0.172383
\(573\) 11.6319 0.485929
\(574\) −10.1792 −0.424872
\(575\) −4.84924 −0.202227
\(576\) −3.39248 −0.141353
\(577\) −7.83682 −0.326251 −0.163126 0.986605i \(-0.552158\pi\)
−0.163126 + 0.986605i \(0.552158\pi\)
\(578\) −5.24852 −0.218310
\(579\) 15.4098 0.640409
\(580\) 0.695024 0.0288593
\(581\) −46.0490 −1.91043
\(582\) 8.09990 0.335751
\(583\) 14.6763 0.607829
\(584\) −21.2125 −0.877780
\(585\) −0.366045 −0.0151341
\(586\) 2.60823 0.107745
\(587\) 3.76713 0.155486 0.0777431 0.996973i \(-0.475229\pi\)
0.0777431 + 0.996973i \(0.475229\pi\)
\(588\) −19.4042 −0.800217
\(589\) 22.4301 0.924217
\(590\) −1.59354 −0.0656052
\(591\) −9.27867 −0.381674
\(592\) −7.86790 −0.323369
\(593\) −35.5044 −1.45799 −0.728995 0.684519i \(-0.760012\pi\)
−0.728995 + 0.684519i \(0.760012\pi\)
\(594\) 1.11945 0.0459315
\(595\) −3.86205 −0.158329
\(596\) 15.2975 0.626612
\(597\) 7.73656 0.316636
\(598\) 0.431985 0.0176652
\(599\) −27.5676 −1.12638 −0.563191 0.826327i \(-0.690427\pi\)
−0.563191 + 0.826327i \(0.690427\pi\)
\(600\) 8.42149 0.343806
\(601\) 12.8048 0.522318 0.261159 0.965296i \(-0.415895\pi\)
0.261159 + 0.965296i \(0.415895\pi\)
\(602\) 18.9643 0.772928
\(603\) −8.58964 −0.349797
\(604\) 8.33992 0.339347
\(605\) 1.95360 0.0794251
\(606\) 3.74425 0.152100
\(607\) 12.8523 0.521661 0.260830 0.965385i \(-0.416004\pi\)
0.260830 + 0.965385i \(0.416004\pi\)
\(608\) 13.7375 0.557131
\(609\) −4.22376 −0.171155
\(610\) −1.55709 −0.0630449
\(611\) 9.41807 0.381014
\(612\) 4.21544 0.170399
\(613\) −5.06650 −0.204634 −0.102317 0.994752i \(-0.532626\pi\)
−0.102317 + 0.994752i \(0.532626\pi\)
\(614\) −0.363817 −0.0146825
\(615\) −2.04212 −0.0823464
\(616\) 17.9203 0.722031
\(617\) −38.5316 −1.55122 −0.775612 0.631210i \(-0.782559\pi\)
−0.775612 + 0.631210i \(0.782559\pi\)
\(618\) 4.52784 0.182136
\(619\) 3.18784 0.128130 0.0640650 0.997946i \(-0.479593\pi\)
0.0640650 + 0.997946i \(0.479593\pi\)
\(620\) 5.38936 0.216442
\(621\) 1.00000 0.0401286
\(622\) 15.4195 0.618265
\(623\) −36.8903 −1.47798
\(624\) 2.62491 0.105080
\(625\) 22.7614 0.910455
\(626\) −8.24711 −0.329621
\(627\) 7.06687 0.282224
\(628\) 11.8663 0.473516
\(629\) 6.65462 0.265337
\(630\) 0.751464 0.0299390
\(631\) −10.3307 −0.411259 −0.205630 0.978630i \(-0.565924\pi\)
−0.205630 + 0.978630i \(0.565924\pi\)
\(632\) 12.2453 0.487092
\(633\) 16.0483 0.637861
\(634\) −5.07665 −0.201619
\(635\) 2.69334 0.106882
\(636\) −10.7534 −0.426399
\(637\) 10.2195 0.404913
\(638\) 1.11945 0.0443194
\(639\) 6.86529 0.271587
\(640\) 4.29150 0.169636
\(641\) −47.7506 −1.88603 −0.943017 0.332745i \(-0.892025\pi\)
−0.943017 + 0.332745i \(0.892025\pi\)
\(642\) −4.83734 −0.190914
\(643\) −9.36074 −0.369152 −0.184576 0.982818i \(-0.559091\pi\)
−0.184576 + 0.982818i \(0.559091\pi\)
\(644\) 7.56068 0.297933
\(645\) 3.80456 0.149804
\(646\) −3.12138 −0.122809
\(647\) −5.62914 −0.221304 −0.110652 0.993859i \(-0.535294\pi\)
−0.110652 + 0.993859i \(0.535294\pi\)
\(648\) −1.73666 −0.0682225
\(649\) 21.8820 0.858944
\(650\) −2.09480 −0.0821647
\(651\) −32.7519 −1.28365
\(652\) −13.9799 −0.547494
\(653\) 34.3700 1.34500 0.672501 0.740096i \(-0.265220\pi\)
0.672501 + 0.740096i \(0.265220\pi\)
\(654\) 7.00123 0.273770
\(655\) 4.87345 0.190421
\(656\) 14.6440 0.571754
\(657\) 12.2145 0.476535
\(658\) −19.3346 −0.753742
\(659\) −5.00171 −0.194839 −0.0974195 0.995243i \(-0.531059\pi\)
−0.0974195 + 0.995243i \(0.531059\pi\)
\(660\) 1.69798 0.0660937
\(661\) −2.29622 −0.0893124 −0.0446562 0.999002i \(-0.514219\pi\)
−0.0446562 + 0.999002i \(0.514219\pi\)
\(662\) 14.7962 0.575070
\(663\) −2.22013 −0.0862226
\(664\) −18.9337 −0.734771
\(665\) 4.74385 0.183959
\(666\) −1.29483 −0.0501737
\(667\) 1.00000 0.0387202
\(668\) 36.5774 1.41522
\(669\) 17.6200 0.681228
\(670\) 1.52821 0.0590400
\(671\) 21.3815 0.825423
\(672\) −20.0592 −0.773801
\(673\) −29.0415 −1.11947 −0.559735 0.828672i \(-0.689097\pi\)
−0.559735 + 0.828672i \(0.689097\pi\)
\(674\) −5.25754 −0.202513
\(675\) −4.84924 −0.186647
\(676\) 21.6795 0.833828
\(677\) 31.2273 1.20016 0.600081 0.799939i \(-0.295135\pi\)
0.600081 + 0.799939i \(0.295135\pi\)
\(678\) −5.32557 −0.204527
\(679\) −74.6633 −2.86531
\(680\) −1.58794 −0.0608947
\(681\) −6.15492 −0.235857
\(682\) 8.68043 0.332391
\(683\) −8.11445 −0.310491 −0.155245 0.987876i \(-0.549617\pi\)
−0.155245 + 0.987876i \(0.549617\pi\)
\(684\) −5.17793 −0.197983
\(685\) 4.53425 0.173245
\(686\) −7.43218 −0.283762
\(687\) 6.76626 0.258149
\(688\) −27.2825 −1.04013
\(689\) 5.66343 0.215760
\(690\) −0.177914 −0.00677306
\(691\) −33.6303 −1.27936 −0.639679 0.768643i \(-0.720933\pi\)
−0.639679 + 0.768643i \(0.720933\pi\)
\(692\) 20.6410 0.784652
\(693\) −10.3189 −0.391981
\(694\) 14.8851 0.565032
\(695\) 0.937527 0.0355624
\(696\) −1.73666 −0.0658279
\(697\) −12.3858 −0.469147
\(698\) −9.57024 −0.362239
\(699\) −12.9500 −0.489813
\(700\) −36.6636 −1.38575
\(701\) 18.4301 0.696095 0.348047 0.937477i \(-0.386845\pi\)
0.348047 + 0.937477i \(0.386845\pi\)
\(702\) 0.431985 0.0163042
\(703\) −8.17404 −0.308290
\(704\) −8.28799 −0.312365
\(705\) −3.87885 −0.146086
\(706\) −4.28993 −0.161454
\(707\) −34.5137 −1.29802
\(708\) −16.0331 −0.602560
\(709\) 22.8313 0.857449 0.428724 0.903435i \(-0.358963\pi\)
0.428724 + 0.903435i \(0.358963\pi\)
\(710\) −1.22143 −0.0458394
\(711\) −7.05107 −0.264436
\(712\) −15.1680 −0.568444
\(713\) 7.75420 0.290397
\(714\) 4.55776 0.170570
\(715\) −0.894267 −0.0334437
\(716\) 20.3645 0.761058
\(717\) 26.2200 0.979205
\(718\) −4.22746 −0.157767
\(719\) −14.3474 −0.535067 −0.267533 0.963549i \(-0.586209\pi\)
−0.267533 + 0.963549i \(0.586209\pi\)
\(720\) −1.08107 −0.0402892
\(721\) −41.7367 −1.55436
\(722\) −4.87205 −0.181319
\(723\) −13.4553 −0.500409
\(724\) −1.73229 −0.0643800
\(725\) −4.84924 −0.180096
\(726\) −2.30552 −0.0855659
\(727\) 27.7841 1.03046 0.515228 0.857053i \(-0.327707\pi\)
0.515228 + 0.857053i \(0.327707\pi\)
\(728\) 6.91529 0.256298
\(729\) 1.00000 0.0370370
\(730\) −2.17313 −0.0804313
\(731\) 23.0753 0.853472
\(732\) −15.6663 −0.579044
\(733\) 50.8399 1.87782 0.938908 0.344169i \(-0.111839\pi\)
0.938908 + 0.344169i \(0.111839\pi\)
\(734\) 4.59918 0.169759
\(735\) −4.20893 −0.155249
\(736\) 4.74914 0.175056
\(737\) −20.9849 −0.772989
\(738\) 2.40999 0.0887130
\(739\) 22.0471 0.811016 0.405508 0.914091i \(-0.367095\pi\)
0.405508 + 0.914091i \(0.367095\pi\)
\(740\) −1.96400 −0.0721981
\(741\) 2.72704 0.100180
\(742\) −11.6266 −0.426826
\(743\) 11.4104 0.418607 0.209304 0.977851i \(-0.432880\pi\)
0.209304 + 0.977851i \(0.432880\pi\)
\(744\) −13.4664 −0.493703
\(745\) 3.31816 0.121568
\(746\) 16.0764 0.588598
\(747\) 10.9024 0.398897
\(748\) 10.2985 0.376552
\(749\) 44.5896 1.62927
\(750\) 1.75231 0.0639855
\(751\) −6.41232 −0.233989 −0.116994 0.993133i \(-0.537326\pi\)
−0.116994 + 0.993133i \(0.537326\pi\)
\(752\) 27.8152 1.01432
\(753\) 6.61094 0.240916
\(754\) 0.431985 0.0157319
\(755\) 1.80900 0.0658361
\(756\) 7.56068 0.274979
\(757\) 29.1131 1.05813 0.529066 0.848581i \(-0.322542\pi\)
0.529066 + 0.848581i \(0.322542\pi\)
\(758\) 10.5271 0.382360
\(759\) 2.44305 0.0886771
\(760\) 1.95051 0.0707523
\(761\) −38.8881 −1.40969 −0.704846 0.709360i \(-0.748984\pi\)
−0.704846 + 0.709360i \(0.748984\pi\)
\(762\) −3.17851 −0.115145
\(763\) −64.5360 −2.33636
\(764\) −20.8215 −0.753295
\(765\) 0.914364 0.0330589
\(766\) 14.5220 0.524702
\(767\) 8.44407 0.304898
\(768\) 1.72039 0.0620791
\(769\) 15.3319 0.552882 0.276441 0.961031i \(-0.410845\pi\)
0.276441 + 0.961031i \(0.410845\pi\)
\(770\) 1.83586 0.0661600
\(771\) 12.6358 0.455066
\(772\) −27.5841 −0.992773
\(773\) 22.8383 0.821438 0.410719 0.911762i \(-0.365278\pi\)
0.410719 + 0.911762i \(0.365278\pi\)
\(774\) −4.48992 −0.161387
\(775\) −37.6020 −1.35070
\(776\) −30.6989 −1.10203
\(777\) 11.9355 0.428184
\(778\) 3.86935 0.138723
\(779\) 15.2138 0.545092
\(780\) 0.655234 0.0234611
\(781\) 16.7723 0.600159
\(782\) −1.07908 −0.0385877
\(783\) 1.00000 0.0357371
\(784\) 30.1822 1.07794
\(785\) 2.57389 0.0918662
\(786\) −5.75135 −0.205144
\(787\) −29.6186 −1.05579 −0.527895 0.849310i \(-0.677018\pi\)
−0.527895 + 0.849310i \(0.677018\pi\)
\(788\) 16.6092 0.591677
\(789\) −31.2519 −1.11260
\(790\) 1.25448 0.0446325
\(791\) 49.0901 1.74544
\(792\) −4.24275 −0.150760
\(793\) 8.25092 0.292999
\(794\) −14.5257 −0.515499
\(795\) −2.33249 −0.0827251
\(796\) −13.8487 −0.490855
\(797\) −45.8911 −1.62554 −0.812772 0.582581i \(-0.802043\pi\)
−0.812772 + 0.582581i \(0.802043\pi\)
\(798\) −5.59841 −0.198182
\(799\) −23.5259 −0.832287
\(800\) −23.0297 −0.814224
\(801\) 8.73399 0.308600
\(802\) −6.80084 −0.240146
\(803\) 29.8408 1.05306
\(804\) 15.3758 0.542262
\(805\) 1.63997 0.0578015
\(806\) 3.34970 0.117988
\(807\) −12.8085 −0.450879
\(808\) −14.1908 −0.499232
\(809\) −19.1118 −0.671934 −0.335967 0.941874i \(-0.609063\pi\)
−0.335967 + 0.941874i \(0.609063\pi\)
\(810\) −0.177914 −0.00625125
\(811\) −33.0111 −1.15918 −0.579588 0.814910i \(-0.696786\pi\)
−0.579588 + 0.814910i \(0.696786\pi\)
\(812\) 7.56068 0.265328
\(813\) −5.91426 −0.207422
\(814\) −3.16334 −0.110875
\(815\) −3.03234 −0.106218
\(816\) −6.55689 −0.229537
\(817\) −28.3440 −0.991632
\(818\) 12.0217 0.420328
\(819\) −3.98195 −0.139141
\(820\) 3.65548 0.127655
\(821\) −46.4488 −1.62107 −0.810537 0.585688i \(-0.800825\pi\)
−0.810537 + 0.585688i \(0.800825\pi\)
\(822\) −5.35105 −0.186639
\(823\) 56.9950 1.98672 0.993361 0.115041i \(-0.0366999\pi\)
0.993361 + 0.115041i \(0.0366999\pi\)
\(824\) −17.1607 −0.597821
\(825\) −11.8469 −0.412458
\(826\) −17.3351 −0.603164
\(827\) −4.61165 −0.160363 −0.0801814 0.996780i \(-0.525550\pi\)
−0.0801814 + 0.996780i \(0.525550\pi\)
\(828\) −1.79004 −0.0622081
\(829\) −22.8183 −0.792514 −0.396257 0.918140i \(-0.629691\pi\)
−0.396257 + 0.918140i \(0.629691\pi\)
\(830\) −1.93968 −0.0673273
\(831\) 11.5359 0.400178
\(832\) −3.19826 −0.110880
\(833\) −25.5279 −0.884490
\(834\) −1.10641 −0.0383120
\(835\) 7.93393 0.274565
\(836\) −12.6500 −0.437508
\(837\) 7.75420 0.268025
\(838\) −14.1674 −0.489404
\(839\) −11.8899 −0.410484 −0.205242 0.978711i \(-0.565798\pi\)
−0.205242 + 0.978711i \(0.565798\pi\)
\(840\) −2.84808 −0.0982680
\(841\) 1.00000 0.0344828
\(842\) 0.454830 0.0156745
\(843\) 22.5802 0.777702
\(844\) −28.7270 −0.988823
\(845\) 4.70247 0.161770
\(846\) 4.57759 0.157381
\(847\) 21.2519 0.730222
\(848\) 16.7263 0.574384
\(849\) 15.6402 0.536772
\(850\) 5.23271 0.179480
\(851\) −2.82580 −0.0968673
\(852\) −12.2891 −0.421019
\(853\) −46.1976 −1.58178 −0.790888 0.611961i \(-0.790381\pi\)
−0.790888 + 0.611961i \(0.790381\pi\)
\(854\) −16.9385 −0.579625
\(855\) −1.12314 −0.0384104
\(856\) 18.3337 0.626632
\(857\) 19.4331 0.663822 0.331911 0.943311i \(-0.392307\pi\)
0.331911 + 0.943311i \(0.392307\pi\)
\(858\) 1.05536 0.0360294
\(859\) −12.5294 −0.427497 −0.213748 0.976889i \(-0.568567\pi\)
−0.213748 + 0.976889i \(0.568567\pi\)
\(860\) −6.81031 −0.232230
\(861\) −22.2148 −0.757080
\(862\) −8.33634 −0.283937
\(863\) 26.0144 0.885542 0.442771 0.896635i \(-0.353996\pi\)
0.442771 + 0.896635i \(0.353996\pi\)
\(864\) 4.74914 0.161569
\(865\) 4.47719 0.152229
\(866\) 12.8480 0.436593
\(867\) −11.4542 −0.389006
\(868\) 58.6271 1.98993
\(869\) −17.2261 −0.584356
\(870\) −0.177914 −0.00603184
\(871\) −8.09789 −0.274386
\(872\) −26.5349 −0.898586
\(873\) 17.6770 0.598275
\(874\) 1.32546 0.0448343
\(875\) −16.1525 −0.546054
\(876\) −21.8645 −0.738733
\(877\) −16.6305 −0.561572 −0.280786 0.959770i \(-0.590595\pi\)
−0.280786 + 0.959770i \(0.590595\pi\)
\(878\) 12.3768 0.417698
\(879\) 5.69213 0.191991
\(880\) −2.64111 −0.0890320
\(881\) −14.0882 −0.474643 −0.237321 0.971431i \(-0.576269\pi\)
−0.237321 + 0.971431i \(0.576269\pi\)
\(882\) 4.96713 0.167252
\(883\) 4.22567 0.142205 0.0711025 0.997469i \(-0.477348\pi\)
0.0711025 + 0.997469i \(0.477348\pi\)
\(884\) 3.97411 0.133664
\(885\) −3.47770 −0.116902
\(886\) 2.39613 0.0804996
\(887\) 43.9960 1.47724 0.738620 0.674122i \(-0.235478\pi\)
0.738620 + 0.674122i \(0.235478\pi\)
\(888\) 4.90746 0.164684
\(889\) 29.2989 0.982655
\(890\) −1.55390 −0.0520867
\(891\) 2.44305 0.0818453
\(892\) −31.5404 −1.05605
\(893\) 28.8975 0.967017
\(894\) −3.91590 −0.130967
\(895\) 4.41723 0.147652
\(896\) 46.6842 1.55961
\(897\) 0.942751 0.0314775
\(898\) 7.71420 0.257426
\(899\) 7.75420 0.258617
\(900\) 8.68033 0.289344
\(901\) −14.1470 −0.471305
\(902\) 5.88773 0.196040
\(903\) 41.3872 1.37728
\(904\) 20.1841 0.671313
\(905\) −0.375747 −0.0124903
\(906\) −2.13487 −0.0709263
\(907\) −2.60868 −0.0866198 −0.0433099 0.999062i \(-0.513790\pi\)
−0.0433099 + 0.999062i \(0.513790\pi\)
\(908\) 11.0175 0.365630
\(909\) 8.17134 0.271026
\(910\) 0.708443 0.0234847
\(911\) 34.0968 1.12968 0.564839 0.825201i \(-0.308938\pi\)
0.564839 + 0.825201i \(0.308938\pi\)
\(912\) 8.05400 0.266695
\(913\) 26.6351 0.881492
\(914\) −16.3912 −0.542174
\(915\) −3.39815 −0.112340
\(916\) −12.1119 −0.400187
\(917\) 53.0149 1.75070
\(918\) −1.07908 −0.0356149
\(919\) −7.53835 −0.248667 −0.124334 0.992240i \(-0.539679\pi\)
−0.124334 + 0.992240i \(0.539679\pi\)
\(920\) 0.674299 0.0222310
\(921\) −0.793984 −0.0261626
\(922\) −2.93691 −0.0967221
\(923\) 6.47226 0.213037
\(924\) 18.4711 0.607656
\(925\) 13.7030 0.450552
\(926\) 13.6162 0.447455
\(927\) 9.88143 0.324549
\(928\) 4.74914 0.155898
\(929\) −26.2779 −0.862150 −0.431075 0.902316i \(-0.641866\pi\)
−0.431075 + 0.902316i \(0.641866\pi\)
\(930\) −1.37958 −0.0452382
\(931\) 31.3566 1.02767
\(932\) 23.1810 0.759317
\(933\) 33.6511 1.10169
\(934\) 4.35396 0.142466
\(935\) 2.23384 0.0730543
\(936\) −1.63724 −0.0535148
\(937\) 27.9872 0.914302 0.457151 0.889389i \(-0.348870\pi\)
0.457151 + 0.889389i \(0.348870\pi\)
\(938\) 16.6244 0.542805
\(939\) −17.9983 −0.587351
\(940\) 6.94328 0.226465
\(941\) −10.4294 −0.339989 −0.169994 0.985445i \(-0.554375\pi\)
−0.169994 + 0.985445i \(0.554375\pi\)
\(942\) −3.03755 −0.0989689
\(943\) 5.25950 0.171273
\(944\) 24.9386 0.811682
\(945\) 1.63997 0.0533483
\(946\) −10.9691 −0.356636
\(947\) 48.0799 1.56239 0.781193 0.624290i \(-0.214612\pi\)
0.781193 + 0.624290i \(0.214612\pi\)
\(948\) 12.6217 0.409933
\(949\) 11.5153 0.373801
\(950\) −6.42747 −0.208535
\(951\) −11.0791 −0.359265
\(952\) −17.2741 −0.559856
\(953\) −46.2541 −1.49832 −0.749159 0.662390i \(-0.769542\pi\)
−0.749159 + 0.662390i \(0.769542\pi\)
\(954\) 2.75267 0.0891210
\(955\) −4.51635 −0.146146
\(956\) −46.9348 −1.51798
\(957\) 2.44305 0.0789726
\(958\) 11.8076 0.381488
\(959\) 49.3249 1.59278
\(960\) 1.31721 0.0425127
\(961\) 29.1277 0.939603
\(962\) −1.22070 −0.0393571
\(963\) −10.5569 −0.340190
\(964\) 24.0855 0.775742
\(965\) −5.98321 −0.192606
\(966\) −1.93540 −0.0622704
\(967\) 39.2454 1.26205 0.631023 0.775764i \(-0.282635\pi\)
0.631023 + 0.775764i \(0.282635\pi\)
\(968\) 8.73801 0.280850
\(969\) −6.81202 −0.218833
\(970\) −3.14497 −0.100979
\(971\) −18.5745 −0.596085 −0.298042 0.954553i \(-0.596334\pi\)
−0.298042 + 0.954553i \(0.596334\pi\)
\(972\) −1.79004 −0.0574155
\(973\) 10.1987 0.326956
\(974\) −11.6218 −0.372388
\(975\) −4.57163 −0.146409
\(976\) 24.3681 0.780005
\(977\) 9.33171 0.298548 0.149274 0.988796i \(-0.452306\pi\)
0.149274 + 0.988796i \(0.452306\pi\)
\(978\) 3.57859 0.114431
\(979\) 21.3376 0.681952
\(980\) 7.53415 0.240670
\(981\) 15.2793 0.487830
\(982\) 15.3988 0.491395
\(983\) −32.6070 −1.04000 −0.520001 0.854166i \(-0.674068\pi\)
−0.520001 + 0.854166i \(0.674068\pi\)
\(984\) −9.13396 −0.291180
\(985\) 3.60266 0.114790
\(986\) −1.07908 −0.0343648
\(987\) −42.1953 −1.34309
\(988\) −4.88150 −0.155301
\(989\) −9.79866 −0.311579
\(990\) −0.434652 −0.0138141
\(991\) −10.6210 −0.337388 −0.168694 0.985668i \(-0.553955\pi\)
−0.168694 + 0.985668i \(0.553955\pi\)
\(992\) 36.8258 1.16922
\(993\) 32.2908 1.02472
\(994\) −13.2871 −0.421441
\(995\) −3.00390 −0.0952300
\(996\) −19.5157 −0.618377
\(997\) 48.5845 1.53869 0.769343 0.638836i \(-0.220584\pi\)
0.769343 + 0.638836i \(0.220584\pi\)
\(998\) 11.2152 0.355011
\(999\) −2.82580 −0.0894045
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 2001.2.a.o.1.11 20
3.2 odd 2 6003.2.a.s.1.10 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.o.1.11 20 1.1 even 1 trivial
6003.2.a.s.1.10 20 3.2 odd 2