Properties

Label 2013.2.a.e.1.2
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
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: \( x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37960\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.37960 q^{2} -1.00000 q^{3} +3.66252 q^{4} +2.55185 q^{5} +2.37960 q^{6} +1.67712 q^{7} -3.95613 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.37960 q^{2} -1.00000 q^{3} +3.66252 q^{4} +2.55185 q^{5} +2.37960 q^{6} +1.67712 q^{7} -3.95613 q^{8} +1.00000 q^{9} -6.07239 q^{10} +1.00000 q^{11} -3.66252 q^{12} +5.65147 q^{13} -3.99087 q^{14} -2.55185 q^{15} +2.08899 q^{16} -4.01907 q^{17} -2.37960 q^{18} +2.77230 q^{19} +9.34619 q^{20} -1.67712 q^{21} -2.37960 q^{22} +6.13877 q^{23} +3.95613 q^{24} +1.51194 q^{25} -13.4483 q^{26} -1.00000 q^{27} +6.14247 q^{28} +1.25605 q^{29} +6.07239 q^{30} +5.37742 q^{31} +2.94129 q^{32} -1.00000 q^{33} +9.56379 q^{34} +4.27975 q^{35} +3.66252 q^{36} +2.89012 q^{37} -6.59699 q^{38} -5.65147 q^{39} -10.0954 q^{40} +6.31673 q^{41} +3.99087 q^{42} +9.33950 q^{43} +3.66252 q^{44} +2.55185 q^{45} -14.6078 q^{46} -8.72405 q^{47} -2.08899 q^{48} -4.18728 q^{49} -3.59781 q^{50} +4.01907 q^{51} +20.6986 q^{52} -4.90573 q^{53} +2.37960 q^{54} +2.55185 q^{55} -6.63489 q^{56} -2.77230 q^{57} -2.98889 q^{58} -4.68512 q^{59} -9.34619 q^{60} -1.00000 q^{61} -12.7961 q^{62} +1.67712 q^{63} -11.1771 q^{64} +14.4217 q^{65} +2.37960 q^{66} +5.67098 q^{67} -14.7199 q^{68} -6.13877 q^{69} -10.1841 q^{70} -2.71844 q^{71} -3.95613 q^{72} +15.1095 q^{73} -6.87735 q^{74} -1.51194 q^{75} +10.1536 q^{76} +1.67712 q^{77} +13.4483 q^{78} -1.47612 q^{79} +5.33079 q^{80} +1.00000 q^{81} -15.0313 q^{82} -2.19058 q^{83} -6.14247 q^{84} -10.2561 q^{85} -22.2243 q^{86} -1.25605 q^{87} -3.95613 q^{88} -13.1550 q^{89} -6.07239 q^{90} +9.47818 q^{91} +22.4833 q^{92} -5.37742 q^{93} +20.7598 q^{94} +7.07450 q^{95} -2.94129 q^{96} -7.84526 q^{97} +9.96407 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 2 q^{2} - 13 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} + 11 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 2 q^{2} - 13 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} + 11 q^{7} + 9 q^{8} + 13 q^{9} + 6 q^{10} + 13 q^{11} - 16 q^{12} + 13 q^{13} + q^{14} - 3 q^{15} + 18 q^{16} + 17 q^{17} + 2 q^{18} + 14 q^{19} - 7 q^{20} - 11 q^{21} + 2 q^{22} + 7 q^{23} - 9 q^{24} + 18 q^{25} - 10 q^{26} - 13 q^{27} + 19 q^{28} - 6 q^{29} - 6 q^{30} + 27 q^{31} + 5 q^{32} - 13 q^{33} + 6 q^{34} + 14 q^{35} + 16 q^{36} + 10 q^{37} + 2 q^{38} - 13 q^{39} + 8 q^{40} + 3 q^{41} - q^{42} + 29 q^{43} + 16 q^{44} + 3 q^{45} - 24 q^{46} + 8 q^{47} - 18 q^{48} + 8 q^{49} - 27 q^{50} - 17 q^{51} + 37 q^{52} - 24 q^{53} - 2 q^{54} + 3 q^{55} + 24 q^{56} - 14 q^{57} - 5 q^{58} + 13 q^{59} + 7 q^{60} - 13 q^{61} + 39 q^{62} + 11 q^{63} + 47 q^{64} - 11 q^{65} - 2 q^{66} + 44 q^{67} - 8 q^{68} - 7 q^{69} - 12 q^{70} + 3 q^{71} + 9 q^{72} + 48 q^{73} - 22 q^{74} - 18 q^{75} + 47 q^{76} + 11 q^{77} + 10 q^{78} - 17 q^{79} - 26 q^{80} + 13 q^{81} + 56 q^{82} + 50 q^{83} - 19 q^{84} + 8 q^{85} + 18 q^{86} + 6 q^{87} + 9 q^{88} - 15 q^{89} + 6 q^{90} + 47 q^{91} + 14 q^{92} - 27 q^{93} + 45 q^{94} - q^{95} - 5 q^{96} + 27 q^{97} + 47 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37960 −1.68263 −0.841317 0.540542i \(-0.818219\pi\)
−0.841317 + 0.540542i \(0.818219\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.66252 1.83126
\(5\) 2.55185 1.14122 0.570611 0.821221i \(-0.306706\pi\)
0.570611 + 0.821221i \(0.306706\pi\)
\(6\) 2.37960 0.971469
\(7\) 1.67712 0.633891 0.316945 0.948444i \(-0.397343\pi\)
0.316945 + 0.948444i \(0.397343\pi\)
\(8\) −3.95613 −1.39870
\(9\) 1.00000 0.333333
\(10\) −6.07239 −1.92026
\(11\) 1.00000 0.301511
\(12\) −3.66252 −1.05728
\(13\) 5.65147 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(14\) −3.99087 −1.06661
\(15\) −2.55185 −0.658885
\(16\) 2.08899 0.522248
\(17\) −4.01907 −0.974767 −0.487383 0.873188i \(-0.662049\pi\)
−0.487383 + 0.873188i \(0.662049\pi\)
\(18\) −2.37960 −0.560878
\(19\) 2.77230 0.636010 0.318005 0.948089i \(-0.396987\pi\)
0.318005 + 0.948089i \(0.396987\pi\)
\(20\) 9.34619 2.08987
\(21\) −1.67712 −0.365977
\(22\) −2.37960 −0.507333
\(23\) 6.13877 1.28002 0.640011 0.768366i \(-0.278930\pi\)
0.640011 + 0.768366i \(0.278930\pi\)
\(24\) 3.95613 0.807542
\(25\) 1.51194 0.302387
\(26\) −13.4483 −2.63742
\(27\) −1.00000 −0.192450
\(28\) 6.14247 1.16082
\(29\) 1.25605 0.233242 0.116621 0.993177i \(-0.462794\pi\)
0.116621 + 0.993177i \(0.462794\pi\)
\(30\) 6.07239 1.10866
\(31\) 5.37742 0.965814 0.482907 0.875672i \(-0.339581\pi\)
0.482907 + 0.875672i \(0.339581\pi\)
\(32\) 2.94129 0.519951
\(33\) −1.00000 −0.174078
\(34\) 9.56379 1.64018
\(35\) 4.27975 0.723410
\(36\) 3.66252 0.610419
\(37\) 2.89012 0.475134 0.237567 0.971371i \(-0.423650\pi\)
0.237567 + 0.971371i \(0.423650\pi\)
\(38\) −6.59699 −1.07017
\(39\) −5.65147 −0.904960
\(40\) −10.0954 −1.59623
\(41\) 6.31673 0.986508 0.493254 0.869885i \(-0.335807\pi\)
0.493254 + 0.869885i \(0.335807\pi\)
\(42\) 3.99087 0.615805
\(43\) 9.33950 1.42426 0.712130 0.702047i \(-0.247731\pi\)
0.712130 + 0.702047i \(0.247731\pi\)
\(44\) 3.66252 0.552145
\(45\) 2.55185 0.380407
\(46\) −14.6078 −2.15381
\(47\) −8.72405 −1.27253 −0.636267 0.771469i \(-0.719522\pi\)
−0.636267 + 0.771469i \(0.719522\pi\)
\(48\) −2.08899 −0.301520
\(49\) −4.18728 −0.598183
\(50\) −3.59781 −0.508807
\(51\) 4.01907 0.562782
\(52\) 20.6986 2.87038
\(53\) −4.90573 −0.673854 −0.336927 0.941531i \(-0.609388\pi\)
−0.336927 + 0.941531i \(0.609388\pi\)
\(54\) 2.37960 0.323823
\(55\) 2.55185 0.344091
\(56\) −6.63489 −0.886625
\(57\) −2.77230 −0.367201
\(58\) −2.98889 −0.392461
\(59\) −4.68512 −0.609951 −0.304976 0.952360i \(-0.598648\pi\)
−0.304976 + 0.952360i \(0.598648\pi\)
\(60\) −9.34619 −1.20659
\(61\) −1.00000 −0.128037
\(62\) −12.7961 −1.62511
\(63\) 1.67712 0.211297
\(64\) −11.1771 −1.39714
\(65\) 14.4217 1.78879
\(66\) 2.37960 0.292909
\(67\) 5.67098 0.692820 0.346410 0.938083i \(-0.387401\pi\)
0.346410 + 0.938083i \(0.387401\pi\)
\(68\) −14.7199 −1.78505
\(69\) −6.13877 −0.739021
\(70\) −10.1841 −1.21723
\(71\) −2.71844 −0.322620 −0.161310 0.986904i \(-0.551572\pi\)
−0.161310 + 0.986904i \(0.551572\pi\)
\(72\) −3.95613 −0.466234
\(73\) 15.1095 1.76843 0.884215 0.467080i \(-0.154694\pi\)
0.884215 + 0.467080i \(0.154694\pi\)
\(74\) −6.87735 −0.799476
\(75\) −1.51194 −0.174583
\(76\) 10.1536 1.16470
\(77\) 1.67712 0.191125
\(78\) 13.4483 1.52272
\(79\) −1.47612 −0.166076 −0.0830381 0.996546i \(-0.526462\pi\)
−0.0830381 + 0.996546i \(0.526462\pi\)
\(80\) 5.33079 0.596001
\(81\) 1.00000 0.111111
\(82\) −15.0313 −1.65993
\(83\) −2.19058 −0.240448 −0.120224 0.992747i \(-0.538361\pi\)
−0.120224 + 0.992747i \(0.538361\pi\)
\(84\) −6.14247 −0.670198
\(85\) −10.2561 −1.11243
\(86\) −22.2243 −2.39651
\(87\) −1.25605 −0.134662
\(88\) −3.95613 −0.421725
\(89\) −13.1550 −1.39442 −0.697212 0.716865i \(-0.745576\pi\)
−0.697212 + 0.716865i \(0.745576\pi\)
\(90\) −6.07239 −0.640086
\(91\) 9.47818 0.993583
\(92\) 22.4833 2.34405
\(93\) −5.37742 −0.557613
\(94\) 20.7598 2.14121
\(95\) 7.07450 0.725829
\(96\) −2.94129 −0.300194
\(97\) −7.84526 −0.796565 −0.398283 0.917263i \(-0.630394\pi\)
−0.398283 + 0.917263i \(0.630394\pi\)
\(98\) 9.96407 1.00652
\(99\) 1.00000 0.100504
\(100\) 5.53749 0.553749
\(101\) −3.23484 −0.321879 −0.160939 0.986964i \(-0.551452\pi\)
−0.160939 + 0.986964i \(0.551452\pi\)
\(102\) −9.56379 −0.946956
\(103\) −7.86347 −0.774811 −0.387405 0.921909i \(-0.626629\pi\)
−0.387405 + 0.921909i \(0.626629\pi\)
\(104\) −22.3580 −2.19238
\(105\) −4.27975 −0.417661
\(106\) 11.6737 1.13385
\(107\) −18.7522 −1.81284 −0.906420 0.422378i \(-0.861195\pi\)
−0.906420 + 0.422378i \(0.861195\pi\)
\(108\) −3.66252 −0.352426
\(109\) −19.5666 −1.87414 −0.937070 0.349141i \(-0.886473\pi\)
−0.937070 + 0.349141i \(0.886473\pi\)
\(110\) −6.07239 −0.578980
\(111\) −2.89012 −0.274318
\(112\) 3.50348 0.331048
\(113\) 11.4456 1.07671 0.538356 0.842718i \(-0.319046\pi\)
0.538356 + 0.842718i \(0.319046\pi\)
\(114\) 6.59699 0.617864
\(115\) 15.6652 1.46079
\(116\) 4.60029 0.427126
\(117\) 5.65147 0.522479
\(118\) 11.1487 1.02632
\(119\) −6.74045 −0.617896
\(120\) 10.0954 0.921584
\(121\) 1.00000 0.0909091
\(122\) 2.37960 0.215439
\(123\) −6.31673 −0.569561
\(124\) 19.6949 1.76865
\(125\) −8.90101 −0.796131
\(126\) −3.99087 −0.355535
\(127\) 1.63206 0.144822 0.0724109 0.997375i \(-0.476931\pi\)
0.0724109 + 0.997375i \(0.476931\pi\)
\(128\) 20.7145 1.83092
\(129\) −9.33950 −0.822297
\(130\) −34.3179 −3.00988
\(131\) 5.81437 0.508004 0.254002 0.967204i \(-0.418253\pi\)
0.254002 + 0.967204i \(0.418253\pi\)
\(132\) −3.66252 −0.318781
\(133\) 4.64948 0.403161
\(134\) −13.4947 −1.16576
\(135\) −2.55185 −0.219628
\(136\) 15.9000 1.36341
\(137\) −0.124547 −0.0106407 −0.00532037 0.999986i \(-0.501694\pi\)
−0.00532037 + 0.999986i \(0.501694\pi\)
\(138\) 14.6078 1.24350
\(139\) 5.70439 0.483840 0.241920 0.970296i \(-0.422223\pi\)
0.241920 + 0.970296i \(0.422223\pi\)
\(140\) 15.6747 1.32475
\(141\) 8.72405 0.734698
\(142\) 6.46881 0.542851
\(143\) 5.65147 0.472600
\(144\) 2.08899 0.174083
\(145\) 3.20524 0.266181
\(146\) −35.9546 −2.97562
\(147\) 4.18728 0.345361
\(148\) 10.5851 0.870092
\(149\) −20.0285 −1.64080 −0.820399 0.571792i \(-0.806249\pi\)
−0.820399 + 0.571792i \(0.806249\pi\)
\(150\) 3.59781 0.293760
\(151\) 8.05186 0.655251 0.327626 0.944808i \(-0.393752\pi\)
0.327626 + 0.944808i \(0.393752\pi\)
\(152\) −10.9676 −0.889589
\(153\) −4.01907 −0.324922
\(154\) −3.99087 −0.321594
\(155\) 13.7224 1.10221
\(156\) −20.6986 −1.65721
\(157\) 10.2781 0.820281 0.410141 0.912022i \(-0.365480\pi\)
0.410141 + 0.912022i \(0.365480\pi\)
\(158\) 3.51258 0.279446
\(159\) 4.90573 0.389050
\(160\) 7.50572 0.593379
\(161\) 10.2954 0.811394
\(162\) −2.37960 −0.186959
\(163\) −6.68649 −0.523727 −0.261863 0.965105i \(-0.584337\pi\)
−0.261863 + 0.965105i \(0.584337\pi\)
\(164\) 23.1351 1.80655
\(165\) −2.55185 −0.198661
\(166\) 5.21272 0.404585
\(167\) 14.0284 1.08555 0.542773 0.839879i \(-0.317374\pi\)
0.542773 + 0.839879i \(0.317374\pi\)
\(168\) 6.63489 0.511893
\(169\) 18.9391 1.45686
\(170\) 24.4054 1.87180
\(171\) 2.77230 0.212003
\(172\) 34.2061 2.60819
\(173\) 10.3570 0.787430 0.393715 0.919233i \(-0.371190\pi\)
0.393715 + 0.919233i \(0.371190\pi\)
\(174\) 2.98889 0.226587
\(175\) 2.53569 0.191680
\(176\) 2.08899 0.157464
\(177\) 4.68512 0.352156
\(178\) 31.3036 2.34630
\(179\) −7.34446 −0.548951 −0.274476 0.961594i \(-0.588504\pi\)
−0.274476 + 0.961594i \(0.588504\pi\)
\(180\) 9.34619 0.696624
\(181\) −0.531235 −0.0394864 −0.0197432 0.999805i \(-0.506285\pi\)
−0.0197432 + 0.999805i \(0.506285\pi\)
\(182\) −22.5543 −1.67184
\(183\) 1.00000 0.0739221
\(184\) −24.2858 −1.79037
\(185\) 7.37516 0.542233
\(186\) 12.7961 0.938259
\(187\) −4.01907 −0.293903
\(188\) −31.9520 −2.33034
\(189\) −1.67712 −0.121992
\(190\) −16.8345 −1.22130
\(191\) 16.5231 1.19557 0.597785 0.801656i \(-0.296048\pi\)
0.597785 + 0.801656i \(0.296048\pi\)
\(192\) 11.1771 0.806636
\(193\) 17.9043 1.28878 0.644391 0.764697i \(-0.277111\pi\)
0.644391 + 0.764697i \(0.277111\pi\)
\(194\) 18.6686 1.34033
\(195\) −14.4217 −1.03276
\(196\) −15.3360 −1.09543
\(197\) 9.54976 0.680392 0.340196 0.940355i \(-0.389507\pi\)
0.340196 + 0.940355i \(0.389507\pi\)
\(198\) −2.37960 −0.169111
\(199\) 7.91086 0.560786 0.280393 0.959885i \(-0.409535\pi\)
0.280393 + 0.959885i \(0.409535\pi\)
\(200\) −5.98142 −0.422950
\(201\) −5.67098 −0.400000
\(202\) 7.69764 0.541604
\(203\) 2.10654 0.147850
\(204\) 14.7199 1.03060
\(205\) 16.1194 1.12582
\(206\) 18.7120 1.30372
\(207\) 6.13877 0.426674
\(208\) 11.8059 0.818590
\(209\) 2.77230 0.191764
\(210\) 10.1841 0.702770
\(211\) −8.64980 −0.595477 −0.297738 0.954648i \(-0.596232\pi\)
−0.297738 + 0.954648i \(0.596232\pi\)
\(212\) −17.9673 −1.23400
\(213\) 2.71844 0.186264
\(214\) 44.6227 3.05035
\(215\) 23.8330 1.62540
\(216\) 3.95613 0.269181
\(217\) 9.01857 0.612220
\(218\) 46.5608 3.15349
\(219\) −15.1095 −1.02100
\(220\) 9.34619 0.630120
\(221\) −22.7136 −1.52788
\(222\) 6.87735 0.461578
\(223\) 8.83430 0.591588 0.295794 0.955252i \(-0.404416\pi\)
0.295794 + 0.955252i \(0.404416\pi\)
\(224\) 4.93288 0.329592
\(225\) 1.51194 0.100796
\(226\) −27.2360 −1.81171
\(227\) 15.6307 1.03744 0.518722 0.854943i \(-0.326408\pi\)
0.518722 + 0.854943i \(0.326408\pi\)
\(228\) −10.1536 −0.672439
\(229\) −13.1402 −0.868329 −0.434164 0.900834i \(-0.642956\pi\)
−0.434164 + 0.900834i \(0.642956\pi\)
\(230\) −37.2770 −2.45797
\(231\) −1.67712 −0.110346
\(232\) −4.96908 −0.326236
\(233\) −19.2138 −1.25874 −0.629368 0.777108i \(-0.716686\pi\)
−0.629368 + 0.777108i \(0.716686\pi\)
\(234\) −13.4483 −0.879141
\(235\) −22.2625 −1.45224
\(236\) −17.1593 −1.11698
\(237\) 1.47612 0.0958842
\(238\) 16.0396 1.03969
\(239\) 13.3971 0.866585 0.433293 0.901253i \(-0.357352\pi\)
0.433293 + 0.901253i \(0.357352\pi\)
\(240\) −5.33079 −0.344101
\(241\) −25.2756 −1.62814 −0.814072 0.580765i \(-0.802754\pi\)
−0.814072 + 0.580765i \(0.802754\pi\)
\(242\) −2.37960 −0.152967
\(243\) −1.00000 −0.0641500
\(244\) −3.66252 −0.234469
\(245\) −10.6853 −0.682659
\(246\) 15.0313 0.958362
\(247\) 15.6676 0.996905
\(248\) −21.2738 −1.35089
\(249\) 2.19058 0.138822
\(250\) 21.1809 1.33960
\(251\) 16.2067 1.02296 0.511480 0.859295i \(-0.329097\pi\)
0.511480 + 0.859295i \(0.329097\pi\)
\(252\) 6.14247 0.386939
\(253\) 6.13877 0.385941
\(254\) −3.88366 −0.243682
\(255\) 10.2561 0.642259
\(256\) −26.9380 −1.68363
\(257\) −16.7068 −1.04214 −0.521072 0.853513i \(-0.674468\pi\)
−0.521072 + 0.853513i \(0.674468\pi\)
\(258\) 22.2243 1.38363
\(259\) 4.84708 0.301183
\(260\) 52.8197 3.27574
\(261\) 1.25605 0.0777473
\(262\) −13.8359 −0.854785
\(263\) 10.1887 0.628263 0.314132 0.949379i \(-0.398287\pi\)
0.314132 + 0.949379i \(0.398287\pi\)
\(264\) 3.95613 0.243483
\(265\) −12.5187 −0.769017
\(266\) −11.0639 −0.678372
\(267\) 13.1550 0.805071
\(268\) 20.7700 1.26873
\(269\) 1.61445 0.0984350 0.0492175 0.998788i \(-0.484327\pi\)
0.0492175 + 0.998788i \(0.484327\pi\)
\(270\) 6.07239 0.369554
\(271\) 25.4804 1.54782 0.773912 0.633294i \(-0.218297\pi\)
0.773912 + 0.633294i \(0.218297\pi\)
\(272\) −8.39580 −0.509070
\(273\) −9.47818 −0.573645
\(274\) 0.296372 0.0179045
\(275\) 1.51194 0.0911732
\(276\) −22.4833 −1.35334
\(277\) 17.4548 1.04876 0.524378 0.851486i \(-0.324298\pi\)
0.524378 + 0.851486i \(0.324298\pi\)
\(278\) −13.5742 −0.814125
\(279\) 5.37742 0.321938
\(280\) −16.9312 −1.01184
\(281\) −16.5498 −0.987275 −0.493638 0.869668i \(-0.664333\pi\)
−0.493638 + 0.869668i \(0.664333\pi\)
\(282\) −20.7598 −1.23623
\(283\) 21.5787 1.28272 0.641359 0.767241i \(-0.278371\pi\)
0.641359 + 0.767241i \(0.278371\pi\)
\(284\) −9.95633 −0.590800
\(285\) −7.07450 −0.419057
\(286\) −13.4483 −0.795213
\(287\) 10.5939 0.625338
\(288\) 2.94129 0.173317
\(289\) −0.847100 −0.0498294
\(290\) −7.62720 −0.447885
\(291\) 7.84526 0.459897
\(292\) 55.3387 3.23845
\(293\) −3.17063 −0.185230 −0.0926150 0.995702i \(-0.529523\pi\)
−0.0926150 + 0.995702i \(0.529523\pi\)
\(294\) −9.96407 −0.581116
\(295\) −11.9557 −0.696090
\(296\) −11.4337 −0.664571
\(297\) −1.00000 −0.0580259
\(298\) 47.6599 2.76086
\(299\) 34.6931 2.00635
\(300\) −5.53749 −0.319707
\(301\) 15.6634 0.902825
\(302\) −19.1602 −1.10255
\(303\) 3.23484 0.185837
\(304\) 5.79132 0.332155
\(305\) −2.55185 −0.146118
\(306\) 9.56379 0.546725
\(307\) 11.3935 0.650259 0.325130 0.945669i \(-0.394592\pi\)
0.325130 + 0.945669i \(0.394592\pi\)
\(308\) 6.14247 0.350000
\(309\) 7.86347 0.447337
\(310\) −32.6538 −1.85461
\(311\) 13.0228 0.738456 0.369228 0.929339i \(-0.379622\pi\)
0.369228 + 0.929339i \(0.379622\pi\)
\(312\) 22.3580 1.26577
\(313\) −31.9422 −1.80548 −0.902740 0.430186i \(-0.858448\pi\)
−0.902740 + 0.430186i \(0.858448\pi\)
\(314\) −24.4578 −1.38023
\(315\) 4.27975 0.241137
\(316\) −5.40631 −0.304129
\(317\) 4.27975 0.240375 0.120187 0.992751i \(-0.461650\pi\)
0.120187 + 0.992751i \(0.461650\pi\)
\(318\) −11.6737 −0.654629
\(319\) 1.25605 0.0703250
\(320\) −28.5222 −1.59444
\(321\) 18.7522 1.04664
\(322\) −24.4991 −1.36528
\(323\) −11.1421 −0.619962
\(324\) 3.66252 0.203473
\(325\) 8.54466 0.473973
\(326\) 15.9112 0.881240
\(327\) 19.5666 1.08204
\(328\) −24.9898 −1.37983
\(329\) −14.6313 −0.806647
\(330\) 6.07239 0.334274
\(331\) −30.6153 −1.68277 −0.841385 0.540436i \(-0.818259\pi\)
−0.841385 + 0.540436i \(0.818259\pi\)
\(332\) −8.02304 −0.440321
\(333\) 2.89012 0.158378
\(334\) −33.3819 −1.82658
\(335\) 14.4715 0.790662
\(336\) −3.50348 −0.191131
\(337\) 20.1363 1.09689 0.548447 0.836185i \(-0.315219\pi\)
0.548447 + 0.836185i \(0.315219\pi\)
\(338\) −45.0676 −2.45136
\(339\) −11.4456 −0.621639
\(340\) −37.5630 −2.03714
\(341\) 5.37742 0.291204
\(342\) −6.59699 −0.356724
\(343\) −18.7624 −1.01307
\(344\) −36.9483 −1.99212
\(345\) −15.6652 −0.843387
\(346\) −24.6456 −1.32496
\(347\) −31.7798 −1.70603 −0.853015 0.521887i \(-0.825228\pi\)
−0.853015 + 0.521887i \(0.825228\pi\)
\(348\) −4.60029 −0.246601
\(349\) −28.4164 −1.52110 −0.760548 0.649282i \(-0.775069\pi\)
−0.760548 + 0.649282i \(0.775069\pi\)
\(350\) −6.03395 −0.322528
\(351\) −5.65147 −0.301653
\(352\) 2.94129 0.156771
\(353\) 13.9859 0.744392 0.372196 0.928154i \(-0.378605\pi\)
0.372196 + 0.928154i \(0.378605\pi\)
\(354\) −11.1487 −0.592549
\(355\) −6.93705 −0.368180
\(356\) −48.1803 −2.55355
\(357\) 6.74045 0.356742
\(358\) 17.4769 0.923684
\(359\) 9.57884 0.505552 0.252776 0.967525i \(-0.418656\pi\)
0.252776 + 0.967525i \(0.418656\pi\)
\(360\) −10.0954 −0.532077
\(361\) −11.3143 −0.595491
\(362\) 1.26413 0.0664412
\(363\) −1.00000 −0.0524864
\(364\) 34.7140 1.81951
\(365\) 38.5571 2.01817
\(366\) −2.37960 −0.124384
\(367\) −20.5751 −1.07401 −0.537006 0.843578i \(-0.680445\pi\)
−0.537006 + 0.843578i \(0.680445\pi\)
\(368\) 12.8238 0.668489
\(369\) 6.31673 0.328836
\(370\) −17.5500 −0.912379
\(371\) −8.22749 −0.427150
\(372\) −19.6949 −1.02113
\(373\) 27.0493 1.40056 0.700280 0.713868i \(-0.253058\pi\)
0.700280 + 0.713868i \(0.253058\pi\)
\(374\) 9.56379 0.494532
\(375\) 8.90101 0.459646
\(376\) 34.5135 1.77990
\(377\) 7.09850 0.365592
\(378\) 3.99087 0.205268
\(379\) −18.5659 −0.953665 −0.476832 0.878994i \(-0.658215\pi\)
−0.476832 + 0.878994i \(0.658215\pi\)
\(380\) 25.9105 1.32918
\(381\) −1.63206 −0.0836130
\(382\) −39.3185 −2.01171
\(383\) 3.36015 0.171696 0.0858478 0.996308i \(-0.472640\pi\)
0.0858478 + 0.996308i \(0.472640\pi\)
\(384\) −20.7145 −1.05708
\(385\) 4.27975 0.218116
\(386\) −42.6052 −2.16855
\(387\) 9.33950 0.474753
\(388\) −28.7334 −1.45872
\(389\) 10.0729 0.510716 0.255358 0.966847i \(-0.417807\pi\)
0.255358 + 0.966847i \(0.417807\pi\)
\(390\) 34.3179 1.73776
\(391\) −24.6721 −1.24772
\(392\) 16.5654 0.836680
\(393\) −5.81437 −0.293296
\(394\) −22.7246 −1.14485
\(395\) −3.76683 −0.189530
\(396\) 3.66252 0.184048
\(397\) −11.9242 −0.598461 −0.299230 0.954181i \(-0.596730\pi\)
−0.299230 + 0.954181i \(0.596730\pi\)
\(398\) −18.8247 −0.943598
\(399\) −4.64948 −0.232765
\(400\) 3.15842 0.157921
\(401\) 11.2101 0.559804 0.279902 0.960029i \(-0.409698\pi\)
0.279902 + 0.960029i \(0.409698\pi\)
\(402\) 13.4947 0.673054
\(403\) 30.3904 1.51385
\(404\) −11.8477 −0.589443
\(405\) 2.55185 0.126802
\(406\) −5.01272 −0.248777
\(407\) 2.89012 0.143258
\(408\) −15.9000 −0.787165
\(409\) −29.5649 −1.46189 −0.730944 0.682437i \(-0.760920\pi\)
−0.730944 + 0.682437i \(0.760920\pi\)
\(410\) −38.3577 −1.89435
\(411\) 0.124547 0.00614343
\(412\) −28.8001 −1.41888
\(413\) −7.85750 −0.386642
\(414\) −14.6078 −0.717936
\(415\) −5.59003 −0.274404
\(416\) 16.6226 0.814990
\(417\) −5.70439 −0.279345
\(418\) −6.59699 −0.322669
\(419\) 13.1249 0.641194 0.320597 0.947216i \(-0.396116\pi\)
0.320597 + 0.947216i \(0.396116\pi\)
\(420\) −15.6747 −0.764845
\(421\) −39.4833 −1.92430 −0.962149 0.272524i \(-0.912142\pi\)
−0.962149 + 0.272524i \(0.912142\pi\)
\(422\) 20.5831 1.00197
\(423\) −8.72405 −0.424178
\(424\) 19.4077 0.942522
\(425\) −6.07657 −0.294757
\(426\) −6.46881 −0.313415
\(427\) −1.67712 −0.0811614
\(428\) −68.6801 −3.31978
\(429\) −5.65147 −0.272856
\(430\) −56.7131 −2.73495
\(431\) 8.99935 0.433483 0.216742 0.976229i \(-0.430457\pi\)
0.216742 + 0.976229i \(0.430457\pi\)
\(432\) −2.08899 −0.100507
\(433\) −19.8497 −0.953916 −0.476958 0.878926i \(-0.658261\pi\)
−0.476958 + 0.878926i \(0.658261\pi\)
\(434\) −21.4606 −1.03014
\(435\) −3.20524 −0.153679
\(436\) −71.6630 −3.43203
\(437\) 17.0185 0.814107
\(438\) 35.9546 1.71798
\(439\) 24.7912 1.18322 0.591609 0.806225i \(-0.298493\pi\)
0.591609 + 0.806225i \(0.298493\pi\)
\(440\) −10.0954 −0.481282
\(441\) −4.18728 −0.199394
\(442\) 54.0495 2.57087
\(443\) 11.8657 0.563758 0.281879 0.959450i \(-0.409042\pi\)
0.281879 + 0.959450i \(0.409042\pi\)
\(444\) −10.5851 −0.502348
\(445\) −33.5695 −1.59135
\(446\) −21.0221 −0.995427
\(447\) 20.0285 0.947315
\(448\) −18.7453 −0.885631
\(449\) 5.23662 0.247132 0.123566 0.992336i \(-0.460567\pi\)
0.123566 + 0.992336i \(0.460567\pi\)
\(450\) −3.59781 −0.169602
\(451\) 6.31673 0.297443
\(452\) 41.9197 1.97174
\(453\) −8.05186 −0.378310
\(454\) −37.1948 −1.74564
\(455\) 24.1869 1.13390
\(456\) 10.9676 0.513605
\(457\) 9.12526 0.426862 0.213431 0.976958i \(-0.431536\pi\)
0.213431 + 0.976958i \(0.431536\pi\)
\(458\) 31.2685 1.46108
\(459\) 4.01907 0.187594
\(460\) 57.3741 2.67508
\(461\) 2.08183 0.0969604 0.0484802 0.998824i \(-0.484562\pi\)
0.0484802 + 0.998824i \(0.484562\pi\)
\(462\) 3.99087 0.185672
\(463\) 27.5553 1.28060 0.640302 0.768124i \(-0.278809\pi\)
0.640302 + 0.768124i \(0.278809\pi\)
\(464\) 2.62387 0.121810
\(465\) −13.7224 −0.636360
\(466\) 45.7211 2.11799
\(467\) 27.9889 1.29517 0.647587 0.761992i \(-0.275778\pi\)
0.647587 + 0.761992i \(0.275778\pi\)
\(468\) 20.6986 0.956793
\(469\) 9.51089 0.439172
\(470\) 52.9759 2.44359
\(471\) −10.2781 −0.473590
\(472\) 18.5350 0.853141
\(473\) 9.33950 0.429431
\(474\) −3.51258 −0.161338
\(475\) 4.19155 0.192321
\(476\) −24.6870 −1.13153
\(477\) −4.90573 −0.224618
\(478\) −31.8798 −1.45815
\(479\) 0.953001 0.0435437 0.0217719 0.999763i \(-0.493069\pi\)
0.0217719 + 0.999763i \(0.493069\pi\)
\(480\) −7.50572 −0.342588
\(481\) 16.3335 0.744741
\(482\) 60.1459 2.73957
\(483\) −10.2954 −0.468458
\(484\) 3.66252 0.166478
\(485\) −20.0199 −0.909058
\(486\) 2.37960 0.107941
\(487\) −12.8346 −0.581591 −0.290796 0.956785i \(-0.593920\pi\)
−0.290796 + 0.956785i \(0.593920\pi\)
\(488\) 3.95613 0.179086
\(489\) 6.68649 0.302374
\(490\) 25.4268 1.14867
\(491\) 39.6400 1.78893 0.894465 0.447138i \(-0.147557\pi\)
0.894465 + 0.447138i \(0.147557\pi\)
\(492\) −23.1351 −1.04301
\(493\) −5.04813 −0.227356
\(494\) −37.2827 −1.67743
\(495\) 2.55185 0.114697
\(496\) 11.2334 0.504394
\(497\) −4.55914 −0.204505
\(498\) −5.21272 −0.233587
\(499\) −35.1235 −1.57234 −0.786172 0.618007i \(-0.787940\pi\)
−0.786172 + 0.618007i \(0.787940\pi\)
\(500\) −32.6001 −1.45792
\(501\) −14.0284 −0.626741
\(502\) −38.5656 −1.72127
\(503\) 2.69005 0.119943 0.0599717 0.998200i \(-0.480899\pi\)
0.0599717 + 0.998200i \(0.480899\pi\)
\(504\) −6.63489 −0.295542
\(505\) −8.25482 −0.367335
\(506\) −14.6078 −0.649398
\(507\) −18.9391 −0.841116
\(508\) 5.97745 0.265206
\(509\) 22.4595 0.995500 0.497750 0.867321i \(-0.334160\pi\)
0.497750 + 0.867321i \(0.334160\pi\)
\(510\) −24.4054 −1.08069
\(511\) 25.3403 1.12099
\(512\) 22.6730 1.00201
\(513\) −2.77230 −0.122400
\(514\) 39.7557 1.75355
\(515\) −20.0664 −0.884231
\(516\) −34.2061 −1.50584
\(517\) −8.72405 −0.383683
\(518\) −11.5341 −0.506780
\(519\) −10.3570 −0.454623
\(520\) −57.0541 −2.50199
\(521\) 24.9298 1.09219 0.546096 0.837722i \(-0.316113\pi\)
0.546096 + 0.837722i \(0.316113\pi\)
\(522\) −2.98889 −0.130820
\(523\) −22.1551 −0.968775 −0.484387 0.874854i \(-0.660957\pi\)
−0.484387 + 0.874854i \(0.660957\pi\)
\(524\) 21.2952 0.930287
\(525\) −2.53569 −0.110667
\(526\) −24.2451 −1.05714
\(527\) −21.6122 −0.941444
\(528\) −2.08899 −0.0909117
\(529\) 14.6845 0.638456
\(530\) 29.7895 1.29397
\(531\) −4.68512 −0.203317
\(532\) 17.0288 0.738292
\(533\) 35.6988 1.54629
\(534\) −31.3036 −1.35464
\(535\) −47.8527 −2.06885
\(536\) −22.4351 −0.969050
\(537\) 7.34446 0.316937
\(538\) −3.84176 −0.165630
\(539\) −4.18728 −0.180359
\(540\) −9.34619 −0.402196
\(541\) 34.9763 1.50375 0.751874 0.659307i \(-0.229150\pi\)
0.751874 + 0.659307i \(0.229150\pi\)
\(542\) −60.6332 −2.60442
\(543\) 0.531235 0.0227975
\(544\) −11.8212 −0.506831
\(545\) −49.9310 −2.13881
\(546\) 22.5543 0.965235
\(547\) 41.5082 1.77476 0.887382 0.461035i \(-0.152522\pi\)
0.887382 + 0.461035i \(0.152522\pi\)
\(548\) −0.456154 −0.0194859
\(549\) −1.00000 −0.0426790
\(550\) −3.59781 −0.153411
\(551\) 3.48214 0.148344
\(552\) 24.2858 1.03367
\(553\) −2.47562 −0.105274
\(554\) −41.5354 −1.76467
\(555\) −7.37516 −0.313058
\(556\) 20.8924 0.886036
\(557\) −13.4014 −0.567836 −0.283918 0.958849i \(-0.591634\pi\)
−0.283918 + 0.958849i \(0.591634\pi\)
\(558\) −12.7961 −0.541704
\(559\) 52.7819 2.23244
\(560\) 8.94036 0.377799
\(561\) 4.01907 0.169685
\(562\) 39.3819 1.66122
\(563\) −10.6144 −0.447343 −0.223671 0.974665i \(-0.571804\pi\)
−0.223671 + 0.974665i \(0.571804\pi\)
\(564\) 31.9520 1.34542
\(565\) 29.2074 1.22877
\(566\) −51.3486 −2.15834
\(567\) 1.67712 0.0704323
\(568\) 10.7545 0.451249
\(569\) −25.0719 −1.05107 −0.525535 0.850772i \(-0.676135\pi\)
−0.525535 + 0.850772i \(0.676135\pi\)
\(570\) 16.8345 0.705120
\(571\) −1.81795 −0.0760790 −0.0380395 0.999276i \(-0.512111\pi\)
−0.0380395 + 0.999276i \(0.512111\pi\)
\(572\) 20.6986 0.865452
\(573\) −16.5231 −0.690263
\(574\) −25.2093 −1.05222
\(575\) 9.28143 0.387062
\(576\) −11.1771 −0.465712
\(577\) 32.7010 1.36136 0.680681 0.732580i \(-0.261684\pi\)
0.680681 + 0.732580i \(0.261684\pi\)
\(578\) 2.01576 0.0838447
\(579\) −17.9043 −0.744078
\(580\) 11.7392 0.487445
\(581\) −3.67386 −0.152417
\(582\) −18.6686 −0.773839
\(583\) −4.90573 −0.203175
\(584\) −59.7750 −2.47351
\(585\) 14.4217 0.596264
\(586\) 7.54484 0.311674
\(587\) 24.0218 0.991488 0.495744 0.868469i \(-0.334896\pi\)
0.495744 + 0.868469i \(0.334896\pi\)
\(588\) 15.3360 0.632445
\(589\) 14.9079 0.614268
\(590\) 28.4499 1.17126
\(591\) −9.54976 −0.392825
\(592\) 6.03745 0.248138
\(593\) −3.62291 −0.148775 −0.0743875 0.997229i \(-0.523700\pi\)
−0.0743875 + 0.997229i \(0.523700\pi\)
\(594\) 2.37960 0.0976363
\(595\) −17.2006 −0.705156
\(596\) −73.3547 −3.00472
\(597\) −7.91086 −0.323770
\(598\) −82.5558 −3.37596
\(599\) 19.0751 0.779389 0.389695 0.920944i \(-0.372581\pi\)
0.389695 + 0.920944i \(0.372581\pi\)
\(600\) 5.98142 0.244190
\(601\) −9.56199 −0.390042 −0.195021 0.980799i \(-0.562477\pi\)
−0.195021 + 0.980799i \(0.562477\pi\)
\(602\) −37.2728 −1.51912
\(603\) 5.67098 0.230940
\(604\) 29.4901 1.19993
\(605\) 2.55185 0.103747
\(606\) −7.69764 −0.312695
\(607\) −33.3670 −1.35432 −0.677162 0.735834i \(-0.736790\pi\)
−0.677162 + 0.735834i \(0.736790\pi\)
\(608\) 8.15414 0.330694
\(609\) −2.10654 −0.0853611
\(610\) 6.07239 0.245864
\(611\) −49.3037 −1.99462
\(612\) −14.7199 −0.595017
\(613\) 21.0516 0.850268 0.425134 0.905131i \(-0.360227\pi\)
0.425134 + 0.905131i \(0.360227\pi\)
\(614\) −27.1119 −1.09415
\(615\) −16.1194 −0.649995
\(616\) −6.63489 −0.267327
\(617\) 3.39567 0.136705 0.0683523 0.997661i \(-0.478226\pi\)
0.0683523 + 0.997661i \(0.478226\pi\)
\(618\) −18.7120 −0.752705
\(619\) −11.3446 −0.455980 −0.227990 0.973664i \(-0.573215\pi\)
−0.227990 + 0.973664i \(0.573215\pi\)
\(620\) 50.2584 2.01843
\(621\) −6.13877 −0.246340
\(622\) −30.9892 −1.24255
\(623\) −22.0624 −0.883912
\(624\) −11.8059 −0.472613
\(625\) −30.2737 −1.21095
\(626\) 76.0098 3.03796
\(627\) −2.77230 −0.110715
\(628\) 37.6437 1.50215
\(629\) −11.6156 −0.463144
\(630\) −10.1841 −0.405745
\(631\) 44.0848 1.75499 0.877494 0.479587i \(-0.159214\pi\)
0.877494 + 0.479587i \(0.159214\pi\)
\(632\) 5.83972 0.232291
\(633\) 8.64980 0.343799
\(634\) −10.1841 −0.404463
\(635\) 4.16477 0.165274
\(636\) 17.9673 0.712451
\(637\) −23.6643 −0.937613
\(638\) −2.98889 −0.118331
\(639\) −2.71844 −0.107540
\(640\) 52.8602 2.08948
\(641\) −5.41854 −0.214019 −0.107010 0.994258i \(-0.534128\pi\)
−0.107010 + 0.994258i \(0.534128\pi\)
\(642\) −44.6227 −1.76112
\(643\) 19.3418 0.762768 0.381384 0.924417i \(-0.375448\pi\)
0.381384 + 0.924417i \(0.375448\pi\)
\(644\) 37.7072 1.48587
\(645\) −23.8330 −0.938423
\(646\) 26.5137 1.04317
\(647\) 21.8968 0.860852 0.430426 0.902626i \(-0.358363\pi\)
0.430426 + 0.902626i \(0.358363\pi\)
\(648\) −3.95613 −0.155411
\(649\) −4.68512 −0.183907
\(650\) −20.3329 −0.797523
\(651\) −9.01857 −0.353466
\(652\) −24.4894 −0.959078
\(653\) −4.80308 −0.187959 −0.0939795 0.995574i \(-0.529959\pi\)
−0.0939795 + 0.995574i \(0.529959\pi\)
\(654\) −46.5608 −1.82067
\(655\) 14.8374 0.579746
\(656\) 13.1956 0.515202
\(657\) 15.1095 0.589477
\(658\) 34.8166 1.35729
\(659\) −29.9093 −1.16510 −0.582550 0.812795i \(-0.697945\pi\)
−0.582550 + 0.812795i \(0.697945\pi\)
\(660\) −9.34619 −0.363800
\(661\) −33.4154 −1.29971 −0.649854 0.760059i \(-0.725170\pi\)
−0.649854 + 0.760059i \(0.725170\pi\)
\(662\) 72.8523 2.83149
\(663\) 22.7136 0.882125
\(664\) 8.66623 0.336315
\(665\) 11.8648 0.460096
\(666\) −6.87735 −0.266492
\(667\) 7.71057 0.298555
\(668\) 51.3791 1.98792
\(669\) −8.83430 −0.341554
\(670\) −34.4364 −1.33039
\(671\) −1.00000 −0.0386046
\(672\) −4.93288 −0.190290
\(673\) −7.76575 −0.299348 −0.149674 0.988735i \(-0.547822\pi\)
−0.149674 + 0.988735i \(0.547822\pi\)
\(674\) −47.9165 −1.84567
\(675\) −1.51194 −0.0581945
\(676\) 69.3649 2.66788
\(677\) 32.9643 1.26692 0.633460 0.773775i \(-0.281634\pi\)
0.633460 + 0.773775i \(0.281634\pi\)
\(678\) 27.2360 1.04599
\(679\) −13.1574 −0.504935
\(680\) 40.5743 1.55595
\(681\) −15.6307 −0.598969
\(682\) −12.7961 −0.489990
\(683\) −16.7372 −0.640429 −0.320215 0.947345i \(-0.603755\pi\)
−0.320215 + 0.947345i \(0.603755\pi\)
\(684\) 10.1536 0.388233
\(685\) −0.317824 −0.0121434
\(686\) 44.6470 1.70463
\(687\) 13.1402 0.501330
\(688\) 19.5101 0.743817
\(689\) −27.7246 −1.05622
\(690\) 37.2770 1.41911
\(691\) −17.4790 −0.664931 −0.332465 0.943115i \(-0.607880\pi\)
−0.332465 + 0.943115i \(0.607880\pi\)
\(692\) 37.9328 1.44199
\(693\) 1.67712 0.0637084
\(694\) 75.6233 2.87062
\(695\) 14.5567 0.552169
\(696\) 4.96908 0.188352
\(697\) −25.3874 −0.961616
\(698\) 67.6198 2.55945
\(699\) 19.2138 0.726731
\(700\) 9.28702 0.351016
\(701\) 20.9278 0.790434 0.395217 0.918588i \(-0.370669\pi\)
0.395217 + 0.918588i \(0.370669\pi\)
\(702\) 13.4483 0.507572
\(703\) 8.01230 0.302190
\(704\) −11.1771 −0.421252
\(705\) 22.2625 0.838453
\(706\) −33.2808 −1.25254
\(707\) −5.42520 −0.204036
\(708\) 17.1593 0.644888
\(709\) −24.1466 −0.906845 −0.453423 0.891296i \(-0.649797\pi\)
−0.453423 + 0.891296i \(0.649797\pi\)
\(710\) 16.5074 0.619513
\(711\) −1.47612 −0.0553588
\(712\) 52.0427 1.95038
\(713\) 33.0108 1.23626
\(714\) −16.0396 −0.600267
\(715\) 14.4217 0.539341
\(716\) −26.8992 −1.00527
\(717\) −13.3971 −0.500323
\(718\) −22.7938 −0.850659
\(719\) −32.0699 −1.19601 −0.598003 0.801494i \(-0.704039\pi\)
−0.598003 + 0.801494i \(0.704039\pi\)
\(720\) 5.33079 0.198667
\(721\) −13.1880 −0.491145
\(722\) 26.9236 1.00199
\(723\) 25.2756 0.940009
\(724\) −1.94566 −0.0723098
\(725\) 1.89906 0.0705293
\(726\) 2.37960 0.0883154
\(727\) −22.5250 −0.835406 −0.417703 0.908584i \(-0.637165\pi\)
−0.417703 + 0.908584i \(0.637165\pi\)
\(728\) −37.4969 −1.38973
\(729\) 1.00000 0.0370370
\(730\) −91.7506 −3.39584
\(731\) −37.5361 −1.38832
\(732\) 3.66252 0.135370
\(733\) 45.0196 1.66284 0.831418 0.555648i \(-0.187530\pi\)
0.831418 + 0.555648i \(0.187530\pi\)
\(734\) 48.9606 1.80717
\(735\) 10.6853 0.394133
\(736\) 18.0559 0.665549
\(737\) 5.67098 0.208893
\(738\) −15.0313 −0.553311
\(739\) −38.6313 −1.42108 −0.710539 0.703658i \(-0.751549\pi\)
−0.710539 + 0.703658i \(0.751549\pi\)
\(740\) 27.0117 0.992968
\(741\) −15.6676 −0.575564
\(742\) 19.5782 0.718737
\(743\) −40.6081 −1.48977 −0.744884 0.667194i \(-0.767495\pi\)
−0.744884 + 0.667194i \(0.767495\pi\)
\(744\) 21.2738 0.779935
\(745\) −51.1097 −1.87251
\(746\) −64.3666 −2.35663
\(747\) −2.19058 −0.0801492
\(748\) −14.7199 −0.538213
\(749\) −31.4496 −1.14914
\(750\) −21.1809 −0.773417
\(751\) 24.7212 0.902088 0.451044 0.892502i \(-0.351052\pi\)
0.451044 + 0.892502i \(0.351052\pi\)
\(752\) −18.2245 −0.664578
\(753\) −16.2067 −0.590606
\(754\) −16.8916 −0.615157
\(755\) 20.5471 0.747787
\(756\) −6.14247 −0.223399
\(757\) 22.0864 0.802743 0.401371 0.915915i \(-0.368534\pi\)
0.401371 + 0.915915i \(0.368534\pi\)
\(758\) 44.1794 1.60467
\(759\) −6.13877 −0.222823
\(760\) −27.9877 −1.01522
\(761\) −4.10116 −0.148667 −0.0743334 0.997233i \(-0.523683\pi\)
−0.0743334 + 0.997233i \(0.523683\pi\)
\(762\) 3.88366 0.140690
\(763\) −32.8155 −1.18800
\(764\) 60.5161 2.18940
\(765\) −10.2561 −0.370808
\(766\) −7.99583 −0.288901
\(767\) −26.4778 −0.956060
\(768\) 26.9380 0.972043
\(769\) −1.84340 −0.0664747 −0.0332374 0.999447i \(-0.510582\pi\)
−0.0332374 + 0.999447i \(0.510582\pi\)
\(770\) −10.1841 −0.367010
\(771\) 16.7068 0.601682
\(772\) 65.5749 2.36009
\(773\) −46.4688 −1.67137 −0.835684 0.549211i \(-0.814928\pi\)
−0.835684 + 0.549211i \(0.814928\pi\)
\(774\) −22.2243 −0.798836
\(775\) 8.13032 0.292050
\(776\) 31.0369 1.11416
\(777\) −4.84708 −0.173888
\(778\) −23.9695 −0.859348
\(779\) 17.5119 0.627429
\(780\) −52.8197 −1.89125
\(781\) −2.71844 −0.0972734
\(782\) 58.7099 2.09946
\(783\) −1.25605 −0.0448874
\(784\) −8.74719 −0.312400
\(785\) 26.2281 0.936123
\(786\) 13.8359 0.493511
\(787\) −26.6731 −0.950795 −0.475397 0.879771i \(-0.657696\pi\)
−0.475397 + 0.879771i \(0.657696\pi\)
\(788\) 34.9761 1.24597
\(789\) −10.1887 −0.362728
\(790\) 8.96357 0.318909
\(791\) 19.1956 0.682517
\(792\) −3.95613 −0.140575
\(793\) −5.65147 −0.200690
\(794\) 28.3750 1.00699
\(795\) 12.5187 0.443992
\(796\) 28.9737 1.02694
\(797\) −22.8323 −0.808763 −0.404381 0.914590i \(-0.632513\pi\)
−0.404381 + 0.914590i \(0.632513\pi\)
\(798\) 11.0639 0.391658
\(799\) 35.0625 1.24042
\(800\) 4.44704 0.157227
\(801\) −13.1550 −0.464808
\(802\) −26.6755 −0.941945
\(803\) 15.1095 0.533202
\(804\) −20.7700 −0.732503
\(805\) 26.2724 0.925980
\(806\) −72.3170 −2.54726
\(807\) −1.61445 −0.0568315
\(808\) 12.7974 0.450213
\(809\) 5.39701 0.189749 0.0948744 0.995489i \(-0.469755\pi\)
0.0948744 + 0.995489i \(0.469755\pi\)
\(810\) −6.07239 −0.213362
\(811\) −49.8393 −1.75009 −0.875047 0.484037i \(-0.839170\pi\)
−0.875047 + 0.484037i \(0.839170\pi\)
\(812\) 7.71522 0.270751
\(813\) −25.4804 −0.893636
\(814\) −6.87735 −0.241051
\(815\) −17.0629 −0.597688
\(816\) 8.39580 0.293912
\(817\) 25.8919 0.905844
\(818\) 70.3527 2.45982
\(819\) 9.47818 0.331194
\(820\) 59.0374 2.06168
\(821\) 17.3451 0.605347 0.302674 0.953094i \(-0.402121\pi\)
0.302674 + 0.953094i \(0.402121\pi\)
\(822\) −0.296372 −0.0103371
\(823\) −30.8810 −1.07644 −0.538221 0.842804i \(-0.680904\pi\)
−0.538221 + 0.842804i \(0.680904\pi\)
\(824\) 31.1089 1.08373
\(825\) −1.51194 −0.0526389
\(826\) 18.6977 0.650578
\(827\) 52.0029 1.80832 0.904160 0.427195i \(-0.140498\pi\)
0.904160 + 0.427195i \(0.140498\pi\)
\(828\) 22.4833 0.781350
\(829\) 48.0581 1.66913 0.834563 0.550912i \(-0.185720\pi\)
0.834563 + 0.550912i \(0.185720\pi\)
\(830\) 13.3021 0.461722
\(831\) −17.4548 −0.605499
\(832\) −63.1670 −2.18992
\(833\) 16.8290 0.583089
\(834\) 13.5742 0.470036
\(835\) 35.7983 1.23885
\(836\) 10.1536 0.351170
\(837\) −5.37742 −0.185871
\(838\) −31.2321 −1.07890
\(839\) −34.1026 −1.17735 −0.588676 0.808369i \(-0.700351\pi\)
−0.588676 + 0.808369i \(0.700351\pi\)
\(840\) 16.9312 0.584184
\(841\) −27.4223 −0.945598
\(842\) 93.9546 3.23789
\(843\) 16.5498 0.570004
\(844\) −31.6800 −1.09047
\(845\) 48.3298 1.66260
\(846\) 20.7598 0.713736
\(847\) 1.67712 0.0576264
\(848\) −10.2480 −0.351919
\(849\) −21.5787 −0.740577
\(850\) 14.4598 0.495968
\(851\) 17.7418 0.608181
\(852\) 9.95633 0.341098
\(853\) −1.66881 −0.0571391 −0.0285695 0.999592i \(-0.509095\pi\)
−0.0285695 + 0.999592i \(0.509095\pi\)
\(854\) 3.99087 0.136565
\(855\) 7.07450 0.241943
\(856\) 74.1860 2.53562
\(857\) −52.5109 −1.79374 −0.896869 0.442297i \(-0.854164\pi\)
−0.896869 + 0.442297i \(0.854164\pi\)
\(858\) 13.4483 0.459116
\(859\) 7.22377 0.246472 0.123236 0.992377i \(-0.460673\pi\)
0.123236 + 0.992377i \(0.460673\pi\)
\(860\) 87.2887 2.97652
\(861\) −10.5939 −0.361039
\(862\) −21.4149 −0.729394
\(863\) 47.7485 1.62538 0.812688 0.582698i \(-0.198003\pi\)
0.812688 + 0.582698i \(0.198003\pi\)
\(864\) −2.94129 −0.100065
\(865\) 26.4296 0.898632
\(866\) 47.2344 1.60509
\(867\) 0.847100 0.0287690
\(868\) 33.0307 1.12113
\(869\) −1.47612 −0.0500739
\(870\) 7.62720 0.258586
\(871\) 32.0494 1.08595
\(872\) 77.4080 2.62137
\(873\) −7.84526 −0.265522
\(874\) −40.4974 −1.36984
\(875\) −14.9280 −0.504660
\(876\) −55.3387 −1.86972
\(877\) −17.6923 −0.597427 −0.298714 0.954343i \(-0.596558\pi\)
−0.298714 + 0.954343i \(0.596558\pi\)
\(878\) −58.9932 −1.99092
\(879\) 3.17063 0.106943
\(880\) 5.33079 0.179701
\(881\) 18.9137 0.637219 0.318610 0.947886i \(-0.396784\pi\)
0.318610 + 0.947886i \(0.396784\pi\)
\(882\) 9.96407 0.335508
\(883\) 59.1263 1.98976 0.994878 0.101079i \(-0.0322294\pi\)
0.994878 + 0.101079i \(0.0322294\pi\)
\(884\) −83.1891 −2.79795
\(885\) 11.9557 0.401888
\(886\) −28.2357 −0.948599
\(887\) −34.1377 −1.14623 −0.573117 0.819474i \(-0.694266\pi\)
−0.573117 + 0.819474i \(0.694266\pi\)
\(888\) 11.4337 0.383690
\(889\) 2.73716 0.0918012
\(890\) 79.8821 2.67765
\(891\) 1.00000 0.0335013
\(892\) 32.3558 1.08335
\(893\) −24.1857 −0.809344
\(894\) −47.6599 −1.59398
\(895\) −18.7420 −0.626475
\(896\) 34.7406 1.16060
\(897\) −34.6931 −1.15837
\(898\) −12.4611 −0.415832
\(899\) 6.75429 0.225268
\(900\) 5.53749 0.184583
\(901\) 19.7165 0.656851
\(902\) −15.0313 −0.500488
\(903\) −15.6634 −0.521246
\(904\) −45.2803 −1.50600
\(905\) −1.35563 −0.0450628
\(906\) 19.1602 0.636557
\(907\) 8.40299 0.279017 0.139508 0.990221i \(-0.455448\pi\)
0.139508 + 0.990221i \(0.455448\pi\)
\(908\) 57.2476 1.89983
\(909\) −3.23484 −0.107293
\(910\) −57.5552 −1.90794
\(911\) −12.5927 −0.417214 −0.208607 0.978000i \(-0.566893\pi\)
−0.208607 + 0.978000i \(0.566893\pi\)
\(912\) −5.79132 −0.191770
\(913\) −2.19058 −0.0724977
\(914\) −21.7145 −0.718252
\(915\) 2.55185 0.0843615
\(916\) −48.1262 −1.59013
\(917\) 9.75138 0.322019
\(918\) −9.56379 −0.315652
\(919\) 51.7102 1.70576 0.852881 0.522105i \(-0.174853\pi\)
0.852881 + 0.522105i \(0.174853\pi\)
\(920\) −61.9736 −2.04321
\(921\) −11.3935 −0.375427
\(922\) −4.95393 −0.163149
\(923\) −15.3632 −0.505685
\(924\) −6.14247 −0.202072
\(925\) 4.36968 0.143674
\(926\) −65.5707 −2.15479
\(927\) −7.86347 −0.258270
\(928\) 3.69439 0.121274
\(929\) −43.9138 −1.44076 −0.720382 0.693577i \(-0.756034\pi\)
−0.720382 + 0.693577i \(0.756034\pi\)
\(930\) 32.6538 1.07076
\(931\) −11.6084 −0.380450
\(932\) −70.3707 −2.30507
\(933\) −13.0228 −0.426348
\(934\) −66.6026 −2.17930
\(935\) −10.2561 −0.335409
\(936\) −22.3580 −0.730793
\(937\) 14.6324 0.478021 0.239010 0.971017i \(-0.423177\pi\)
0.239010 + 0.971017i \(0.423177\pi\)
\(938\) −22.6322 −0.738966
\(939\) 31.9422 1.04239
\(940\) −81.5366 −2.65943
\(941\) 46.6148 1.51960 0.759799 0.650158i \(-0.225297\pi\)
0.759799 + 0.650158i \(0.225297\pi\)
\(942\) 24.4578 0.796878
\(943\) 38.7770 1.26275
\(944\) −9.78719 −0.318546
\(945\) −4.27975 −0.139220
\(946\) −22.2243 −0.722575
\(947\) 6.99474 0.227299 0.113649 0.993521i \(-0.463746\pi\)
0.113649 + 0.993521i \(0.463746\pi\)
\(948\) 5.40631 0.175589
\(949\) 85.3907 2.77190
\(950\) −9.97422 −0.323607
\(951\) −4.27975 −0.138781
\(952\) 26.6661 0.864253
\(953\) 39.6795 1.28534 0.642672 0.766142i \(-0.277826\pi\)
0.642672 + 0.766142i \(0.277826\pi\)
\(954\) 11.6737 0.377950
\(955\) 42.1645 1.36441
\(956\) 49.0670 1.58694
\(957\) −1.25605 −0.0406022
\(958\) −2.26777 −0.0732682
\(959\) −0.208879 −0.00674506
\(960\) 28.5222 0.920551
\(961\) −2.08330 −0.0672033
\(962\) −38.8672 −1.25313
\(963\) −18.7522 −0.604280
\(964\) −92.5722 −2.98155
\(965\) 45.6891 1.47079
\(966\) 24.4991 0.788244
\(967\) −56.5820 −1.81956 −0.909778 0.415096i \(-0.863748\pi\)
−0.909778 + 0.415096i \(0.863748\pi\)
\(968\) −3.95613 −0.127155
\(969\) 11.1421 0.357935
\(970\) 47.6395 1.52961
\(971\) 23.7722 0.762885 0.381443 0.924392i \(-0.375427\pi\)
0.381443 + 0.924392i \(0.375427\pi\)
\(972\) −3.66252 −0.117475
\(973\) 9.56692 0.306702
\(974\) 30.5413 0.978605
\(975\) −8.54466 −0.273648
\(976\) −2.08899 −0.0668670
\(977\) −43.0284 −1.37660 −0.688300 0.725426i \(-0.741643\pi\)
−0.688300 + 0.725426i \(0.741643\pi\)
\(978\) −15.9112 −0.508784
\(979\) −13.1550 −0.420434
\(980\) −39.1351 −1.25013
\(981\) −19.5666 −0.624713
\(982\) −94.3276 −3.01011
\(983\) 28.8215 0.919263 0.459631 0.888110i \(-0.347982\pi\)
0.459631 + 0.888110i \(0.347982\pi\)
\(984\) 24.9898 0.796647
\(985\) 24.3695 0.776478
\(986\) 12.0126 0.382558
\(987\) 14.6313 0.465718
\(988\) 57.3828 1.82559
\(989\) 57.3330 1.82308
\(990\) −6.07239 −0.192993
\(991\) 5.66098 0.179827 0.0899135 0.995950i \(-0.471341\pi\)
0.0899135 + 0.995950i \(0.471341\pi\)
\(992\) 15.8165 0.502176
\(993\) 30.6153 0.971548
\(994\) 10.8490 0.344108
\(995\) 20.1873 0.639981
\(996\) 8.02304 0.254220
\(997\) −50.8029 −1.60894 −0.804472 0.593991i \(-0.797552\pi\)
−0.804472 + 0.593991i \(0.797552\pi\)
\(998\) 83.5801 2.64568
\(999\) −2.89012 −0.0914395
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 2013.2.a.e.1.2 13
3.2 odd 2 6039.2.a.i.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.2 13 1.1 even 1 trivial
6039.2.a.i.1.12 13 3.2 odd 2