Properties

Label 2013.2.a.g.1.9
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: \( x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(1.31287\) 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)
 
\(f(q)\) \(=\) \(q+1.31287 q^{2} +1.00000 q^{3} -0.276371 q^{4} -3.61438 q^{5} +1.31287 q^{6} -0.837447 q^{7} -2.98858 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.31287 q^{2} +1.00000 q^{3} -0.276371 q^{4} -3.61438 q^{5} +1.31287 q^{6} -0.837447 q^{7} -2.98858 q^{8} +1.00000 q^{9} -4.74521 q^{10} +1.00000 q^{11} -0.276371 q^{12} +1.11084 q^{13} -1.09946 q^{14} -3.61438 q^{15} -3.37088 q^{16} +4.37994 q^{17} +1.31287 q^{18} +3.54367 q^{19} +0.998910 q^{20} -0.837447 q^{21} +1.31287 q^{22} +4.06422 q^{23} -2.98858 q^{24} +8.06373 q^{25} +1.45840 q^{26} +1.00000 q^{27} +0.231446 q^{28} -1.67338 q^{29} -4.74521 q^{30} -2.81104 q^{31} +1.55164 q^{32} +1.00000 q^{33} +5.75030 q^{34} +3.02685 q^{35} -0.276371 q^{36} +1.86448 q^{37} +4.65237 q^{38} +1.11084 q^{39} +10.8019 q^{40} +8.79562 q^{41} -1.09946 q^{42} -0.247282 q^{43} -0.276371 q^{44} -3.61438 q^{45} +5.33580 q^{46} +7.35580 q^{47} -3.37088 q^{48} -6.29868 q^{49} +10.5866 q^{50} +4.37994 q^{51} -0.307006 q^{52} +2.90217 q^{53} +1.31287 q^{54} -3.61438 q^{55} +2.50278 q^{56} +3.54367 q^{57} -2.19694 q^{58} -0.446773 q^{59} +0.998910 q^{60} +1.00000 q^{61} -3.69054 q^{62} -0.837447 q^{63} +8.77885 q^{64} -4.01501 q^{65} +1.31287 q^{66} +13.5843 q^{67} -1.21049 q^{68} +4.06422 q^{69} +3.97386 q^{70} +2.92885 q^{71} -2.98858 q^{72} -3.22502 q^{73} +2.44782 q^{74} +8.06373 q^{75} -0.979367 q^{76} -0.837447 q^{77} +1.45840 q^{78} -1.75929 q^{79} +12.1836 q^{80} +1.00000 q^{81} +11.5475 q^{82} +0.319558 q^{83} +0.231446 q^{84} -15.8308 q^{85} -0.324650 q^{86} -1.67338 q^{87} -2.98858 q^{88} +15.7237 q^{89} -4.74521 q^{90} -0.930274 q^{91} -1.12323 q^{92} -2.81104 q^{93} +9.65721 q^{94} -12.8081 q^{95} +1.55164 q^{96} -7.75619 q^{97} -8.26935 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 7 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 7 q^{7} + 9 q^{8} + 13 q^{9} + 2 q^{10} + 13 q^{11} + 12 q^{12} + 9 q^{13} + 7 q^{14} + 7 q^{15} + 2 q^{16} + 19 q^{17} + 4 q^{18} + 14 q^{19} + 19 q^{20} + 7 q^{21} + 4 q^{22} + 5 q^{23} + 9 q^{24} + 2 q^{25} - 4 q^{26} + 13 q^{27} + 7 q^{28} + 10 q^{29} + 2 q^{30} - q^{31} + 7 q^{32} + 13 q^{33} - 2 q^{34} + 16 q^{35} + 12 q^{36} - 8 q^{37} - 10 q^{38} + 9 q^{39} + 14 q^{40} + 21 q^{41} + 7 q^{42} + 11 q^{43} + 12 q^{44} + 7 q^{45} - 8 q^{46} + 22 q^{47} + 2 q^{48} + 19 q^{50} + 19 q^{51} - q^{52} + 16 q^{53} + 4 q^{54} + 7 q^{55} + 14 q^{57} - 13 q^{58} + 19 q^{59} + 19 q^{60} + 13 q^{61} + 3 q^{62} + 7 q^{63} - 13 q^{64} + 13 q^{65} + 4 q^{66} + 12 q^{67} + 36 q^{68} + 5 q^{69} - 20 q^{70} + 5 q^{71} + 9 q^{72} + 18 q^{73} + 6 q^{74} + 2 q^{75} - 5 q^{76} + 7 q^{77} - 4 q^{78} - q^{79} + 6 q^{80} + 13 q^{81} - 22 q^{82} + 48 q^{83} + 7 q^{84} - 2 q^{85} + 26 q^{86} + 10 q^{87} + 9 q^{88} + 15 q^{89} + 2 q^{90} - 11 q^{91} - 24 q^{92} - q^{93} - 23 q^{94} + 17 q^{95} + 7 q^{96} - 17 q^{97} - 15 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.31287 0.928340 0.464170 0.885746i \(-0.346353\pi\)
0.464170 + 0.885746i \(0.346353\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.276371 −0.138186
\(5\) −3.61438 −1.61640 −0.808200 0.588909i \(-0.799558\pi\)
−0.808200 + 0.588909i \(0.799558\pi\)
\(6\) 1.31287 0.535977
\(7\) −0.837447 −0.316525 −0.158263 0.987397i \(-0.550589\pi\)
−0.158263 + 0.987397i \(0.550589\pi\)
\(8\) −2.98858 −1.05662
\(9\) 1.00000 0.333333
\(10\) −4.74521 −1.50057
\(11\) 1.00000 0.301511
\(12\) −0.276371 −0.0797815
\(13\) 1.11084 0.308093 0.154046 0.988064i \(-0.450769\pi\)
0.154046 + 0.988064i \(0.450769\pi\)
\(14\) −1.09946 −0.293843
\(15\) −3.61438 −0.933229
\(16\) −3.37088 −0.842719
\(17\) 4.37994 1.06229 0.531146 0.847280i \(-0.321762\pi\)
0.531146 + 0.847280i \(0.321762\pi\)
\(18\) 1.31287 0.309447
\(19\) 3.54367 0.812973 0.406486 0.913657i \(-0.366754\pi\)
0.406486 + 0.913657i \(0.366754\pi\)
\(20\) 0.998910 0.223363
\(21\) −0.837447 −0.182746
\(22\) 1.31287 0.279905
\(23\) 4.06422 0.847449 0.423724 0.905791i \(-0.360722\pi\)
0.423724 + 0.905791i \(0.360722\pi\)
\(24\) −2.98858 −0.610041
\(25\) 8.06373 1.61275
\(26\) 1.45840 0.286015
\(27\) 1.00000 0.192450
\(28\) 0.231446 0.0437392
\(29\) −1.67338 −0.310740 −0.155370 0.987856i \(-0.549657\pi\)
−0.155370 + 0.987856i \(0.549657\pi\)
\(30\) −4.74521 −0.866353
\(31\) −2.81104 −0.504879 −0.252439 0.967613i \(-0.581233\pi\)
−0.252439 + 0.967613i \(0.581233\pi\)
\(32\) 1.55164 0.274293
\(33\) 1.00000 0.174078
\(34\) 5.75030 0.986168
\(35\) 3.02685 0.511631
\(36\) −0.276371 −0.0460619
\(37\) 1.86448 0.306519 0.153259 0.988186i \(-0.451023\pi\)
0.153259 + 0.988186i \(0.451023\pi\)
\(38\) 4.65237 0.754715
\(39\) 1.11084 0.177878
\(40\) 10.8019 1.70792
\(41\) 8.79562 1.37365 0.686823 0.726825i \(-0.259005\pi\)
0.686823 + 0.726825i \(0.259005\pi\)
\(42\) −1.09946 −0.169650
\(43\) −0.247282 −0.0377102 −0.0188551 0.999822i \(-0.506002\pi\)
−0.0188551 + 0.999822i \(0.506002\pi\)
\(44\) −0.276371 −0.0416645
\(45\) −3.61438 −0.538800
\(46\) 5.33580 0.786720
\(47\) 7.35580 1.07295 0.536477 0.843915i \(-0.319755\pi\)
0.536477 + 0.843915i \(0.319755\pi\)
\(48\) −3.37088 −0.486544
\(49\) −6.29868 −0.899812
\(50\) 10.5866 1.49718
\(51\) 4.37994 0.613315
\(52\) −0.307006 −0.0425740
\(53\) 2.90217 0.398644 0.199322 0.979934i \(-0.436126\pi\)
0.199322 + 0.979934i \(0.436126\pi\)
\(54\) 1.31287 0.178659
\(55\) −3.61438 −0.487363
\(56\) 2.50278 0.334448
\(57\) 3.54367 0.469370
\(58\) −2.19694 −0.288472
\(59\) −0.446773 −0.0581649 −0.0290825 0.999577i \(-0.509259\pi\)
−0.0290825 + 0.999577i \(0.509259\pi\)
\(60\) 0.998910 0.128959
\(61\) 1.00000 0.128037
\(62\) −3.69054 −0.468699
\(63\) −0.837447 −0.105508
\(64\) 8.77885 1.09736
\(65\) −4.01501 −0.498001
\(66\) 1.31287 0.161603
\(67\) 13.5843 1.65959 0.829795 0.558069i \(-0.188457\pi\)
0.829795 + 0.558069i \(0.188457\pi\)
\(68\) −1.21049 −0.146794
\(69\) 4.06422 0.489275
\(70\) 3.97386 0.474967
\(71\) 2.92885 0.347591 0.173795 0.984782i \(-0.444397\pi\)
0.173795 + 0.984782i \(0.444397\pi\)
\(72\) −2.98858 −0.352208
\(73\) −3.22502 −0.377460 −0.188730 0.982029i \(-0.560437\pi\)
−0.188730 + 0.982029i \(0.560437\pi\)
\(74\) 2.44782 0.284554
\(75\) 8.06373 0.931120
\(76\) −0.979367 −0.112341
\(77\) −0.837447 −0.0954359
\(78\) 1.45840 0.165131
\(79\) −1.75929 −0.197936 −0.0989679 0.995091i \(-0.531554\pi\)
−0.0989679 + 0.995091i \(0.531554\pi\)
\(80\) 12.1836 1.36217
\(81\) 1.00000 0.111111
\(82\) 11.5475 1.27521
\(83\) 0.319558 0.0350761 0.0175380 0.999846i \(-0.494417\pi\)
0.0175380 + 0.999846i \(0.494417\pi\)
\(84\) 0.231446 0.0252529
\(85\) −15.8308 −1.71709
\(86\) −0.324650 −0.0350079
\(87\) −1.67338 −0.179406
\(88\) −2.98858 −0.318584
\(89\) 15.7237 1.66671 0.833356 0.552737i \(-0.186417\pi\)
0.833356 + 0.552737i \(0.186417\pi\)
\(90\) −4.74521 −0.500189
\(91\) −0.930274 −0.0975192
\(92\) −1.12323 −0.117105
\(93\) −2.81104 −0.291492
\(94\) 9.65721 0.996065
\(95\) −12.8081 −1.31409
\(96\) 1.55164 0.158363
\(97\) −7.75619 −0.787521 −0.393761 0.919213i \(-0.628826\pi\)
−0.393761 + 0.919213i \(0.628826\pi\)
\(98\) −8.26935 −0.835331
\(99\) 1.00000 0.100504
\(100\) −2.22858 −0.222858
\(101\) 0.542833 0.0540139 0.0270069 0.999635i \(-0.491402\pi\)
0.0270069 + 0.999635i \(0.491402\pi\)
\(102\) 5.75030 0.569364
\(103\) −18.9963 −1.87176 −0.935878 0.352323i \(-0.885392\pi\)
−0.935878 + 0.352323i \(0.885392\pi\)
\(104\) −3.31985 −0.325538
\(105\) 3.02685 0.295390
\(106\) 3.81017 0.370077
\(107\) 1.11430 0.107724 0.0538619 0.998548i \(-0.482847\pi\)
0.0538619 + 0.998548i \(0.482847\pi\)
\(108\) −0.276371 −0.0265938
\(109\) −7.68359 −0.735954 −0.367977 0.929835i \(-0.619950\pi\)
−0.367977 + 0.929835i \(0.619950\pi\)
\(110\) −4.74521 −0.452438
\(111\) 1.86448 0.176969
\(112\) 2.82293 0.266742
\(113\) 13.4812 1.26821 0.634103 0.773249i \(-0.281370\pi\)
0.634103 + 0.773249i \(0.281370\pi\)
\(114\) 4.65237 0.435735
\(115\) −14.6896 −1.36982
\(116\) 0.462475 0.0429398
\(117\) 1.11084 0.102698
\(118\) −0.586555 −0.0539968
\(119\) −3.66797 −0.336242
\(120\) 10.8019 0.986070
\(121\) 1.00000 0.0909091
\(122\) 1.31287 0.118862
\(123\) 8.79562 0.793074
\(124\) 0.776892 0.0697670
\(125\) −11.0735 −0.990443
\(126\) −1.09946 −0.0979476
\(127\) 2.50614 0.222384 0.111192 0.993799i \(-0.464533\pi\)
0.111192 + 0.993799i \(0.464533\pi\)
\(128\) 8.42222 0.744426
\(129\) −0.247282 −0.0217720
\(130\) −5.27119 −0.462314
\(131\) 7.59263 0.663371 0.331686 0.943390i \(-0.392383\pi\)
0.331686 + 0.943390i \(0.392383\pi\)
\(132\) −0.276371 −0.0240550
\(133\) −2.96763 −0.257326
\(134\) 17.8345 1.54066
\(135\) −3.61438 −0.311076
\(136\) −13.0898 −1.12244
\(137\) −10.8126 −0.923782 −0.461891 0.886937i \(-0.652829\pi\)
−0.461891 + 0.886937i \(0.652829\pi\)
\(138\) 5.33580 0.454213
\(139\) 9.10264 0.772076 0.386038 0.922483i \(-0.373843\pi\)
0.386038 + 0.922483i \(0.373843\pi\)
\(140\) −0.836534 −0.0707001
\(141\) 7.35580 0.619470
\(142\) 3.84520 0.322682
\(143\) 1.11084 0.0928935
\(144\) −3.37088 −0.280906
\(145\) 6.04825 0.502279
\(146\) −4.23404 −0.350411
\(147\) −6.29868 −0.519507
\(148\) −0.515289 −0.0423565
\(149\) 6.88180 0.563779 0.281890 0.959447i \(-0.409039\pi\)
0.281890 + 0.959447i \(0.409039\pi\)
\(150\) 10.5866 0.864395
\(151\) 5.97340 0.486108 0.243054 0.970013i \(-0.421851\pi\)
0.243054 + 0.970013i \(0.421851\pi\)
\(152\) −10.5905 −0.859005
\(153\) 4.37994 0.354097
\(154\) −1.09946 −0.0885970
\(155\) 10.1602 0.816085
\(156\) −0.307006 −0.0245801
\(157\) −13.7456 −1.09702 −0.548511 0.836144i \(-0.684805\pi\)
−0.548511 + 0.836144i \(0.684805\pi\)
\(158\) −2.30972 −0.183752
\(159\) 2.90217 0.230157
\(160\) −5.60820 −0.443367
\(161\) −3.40357 −0.268239
\(162\) 1.31287 0.103149
\(163\) 13.1473 1.02978 0.514888 0.857257i \(-0.327833\pi\)
0.514888 + 0.857257i \(0.327833\pi\)
\(164\) −2.43086 −0.189818
\(165\) −3.61438 −0.281379
\(166\) 0.419539 0.0325625
\(167\) −23.2428 −1.79858 −0.899292 0.437349i \(-0.855918\pi\)
−0.899292 + 0.437349i \(0.855918\pi\)
\(168\) 2.50278 0.193093
\(169\) −11.7660 −0.905079
\(170\) −20.7838 −1.59404
\(171\) 3.54367 0.270991
\(172\) 0.0683417 0.00521101
\(173\) 5.59970 0.425737 0.212869 0.977081i \(-0.431719\pi\)
0.212869 + 0.977081i \(0.431719\pi\)
\(174\) −2.19694 −0.166549
\(175\) −6.75295 −0.510475
\(176\) −3.37088 −0.254089
\(177\) −0.446773 −0.0335815
\(178\) 20.6432 1.54727
\(179\) 6.95344 0.519724 0.259862 0.965646i \(-0.416323\pi\)
0.259862 + 0.965646i \(0.416323\pi\)
\(180\) 0.998910 0.0744544
\(181\) 12.9447 0.962174 0.481087 0.876673i \(-0.340242\pi\)
0.481087 + 0.876673i \(0.340242\pi\)
\(182\) −1.22133 −0.0905309
\(183\) 1.00000 0.0739221
\(184\) −12.1463 −0.895434
\(185\) −6.73894 −0.495457
\(186\) −3.69054 −0.270603
\(187\) 4.37994 0.320293
\(188\) −2.03293 −0.148267
\(189\) −0.837447 −0.0609153
\(190\) −16.8154 −1.21992
\(191\) −18.5614 −1.34306 −0.671529 0.740978i \(-0.734362\pi\)
−0.671529 + 0.740978i \(0.734362\pi\)
\(192\) 8.77885 0.633559
\(193\) −10.6201 −0.764450 −0.382225 0.924069i \(-0.624842\pi\)
−0.382225 + 0.924069i \(0.624842\pi\)
\(194\) −10.1829 −0.731087
\(195\) −4.01501 −0.287521
\(196\) 1.74077 0.124341
\(197\) 8.69998 0.619848 0.309924 0.950761i \(-0.399696\pi\)
0.309924 + 0.950761i \(0.399696\pi\)
\(198\) 1.31287 0.0933016
\(199\) 13.2999 0.942803 0.471402 0.881919i \(-0.343748\pi\)
0.471402 + 0.881919i \(0.343748\pi\)
\(200\) −24.0991 −1.70406
\(201\) 13.5843 0.958164
\(202\) 0.712669 0.0501432
\(203\) 1.40137 0.0983570
\(204\) −1.21049 −0.0847513
\(205\) −31.7907 −2.22036
\(206\) −24.9396 −1.73763
\(207\) 4.06422 0.282483
\(208\) −3.74452 −0.259636
\(209\) 3.54367 0.245120
\(210\) 3.97386 0.274223
\(211\) −4.35272 −0.299654 −0.149827 0.988712i \(-0.547872\pi\)
−0.149827 + 0.988712i \(0.547872\pi\)
\(212\) −0.802076 −0.0550868
\(213\) 2.92885 0.200682
\(214\) 1.46294 0.100004
\(215\) 0.893772 0.0609548
\(216\) −2.98858 −0.203347
\(217\) 2.35410 0.159807
\(218\) −10.0876 −0.683216
\(219\) −3.22502 −0.217927
\(220\) 0.998910 0.0673465
\(221\) 4.86544 0.327285
\(222\) 2.44782 0.164287
\(223\) 9.37001 0.627462 0.313731 0.949512i \(-0.398421\pi\)
0.313731 + 0.949512i \(0.398421\pi\)
\(224\) −1.29941 −0.0868207
\(225\) 8.06373 0.537582
\(226\) 17.6991 1.17733
\(227\) −20.2223 −1.34220 −0.671100 0.741367i \(-0.734178\pi\)
−0.671100 + 0.741367i \(0.734178\pi\)
\(228\) −0.979367 −0.0648602
\(229\) −12.0588 −0.796866 −0.398433 0.917197i \(-0.630446\pi\)
−0.398433 + 0.917197i \(0.630446\pi\)
\(230\) −19.2856 −1.27165
\(231\) −0.837447 −0.0551000
\(232\) 5.00104 0.328335
\(233\) −22.8322 −1.49579 −0.747895 0.663817i \(-0.768935\pi\)
−0.747895 + 0.663817i \(0.768935\pi\)
\(234\) 1.45840 0.0953383
\(235\) −26.5866 −1.73432
\(236\) 0.123475 0.00803756
\(237\) −1.75929 −0.114278
\(238\) −4.81557 −0.312147
\(239\) 12.9994 0.840864 0.420432 0.907324i \(-0.361879\pi\)
0.420432 + 0.907324i \(0.361879\pi\)
\(240\) 12.1836 0.786450
\(241\) 25.7692 1.65994 0.829971 0.557807i \(-0.188357\pi\)
0.829971 + 0.557807i \(0.188357\pi\)
\(242\) 1.31287 0.0843945
\(243\) 1.00000 0.0641500
\(244\) −0.276371 −0.0176929
\(245\) 22.7658 1.45446
\(246\) 11.5475 0.736242
\(247\) 3.93646 0.250471
\(248\) 8.40103 0.533466
\(249\) 0.319558 0.0202512
\(250\) −14.5381 −0.919467
\(251\) −2.41536 −0.152456 −0.0762281 0.997090i \(-0.524288\pi\)
−0.0762281 + 0.997090i \(0.524288\pi\)
\(252\) 0.231446 0.0145797
\(253\) 4.06422 0.255515
\(254\) 3.29024 0.206448
\(255\) −15.8308 −0.991361
\(256\) −6.50042 −0.406276
\(257\) 20.1663 1.25794 0.628969 0.777431i \(-0.283477\pi\)
0.628969 + 0.777431i \(0.283477\pi\)
\(258\) −0.324650 −0.0202118
\(259\) −1.56140 −0.0970209
\(260\) 1.10963 0.0688166
\(261\) −1.67338 −0.103580
\(262\) 9.96814 0.615834
\(263\) −15.8182 −0.975392 −0.487696 0.873013i \(-0.662163\pi\)
−0.487696 + 0.873013i \(0.662163\pi\)
\(264\) −2.98858 −0.183934
\(265\) −10.4895 −0.644367
\(266\) −3.89612 −0.238886
\(267\) 15.7237 0.962277
\(268\) −3.75432 −0.229331
\(269\) 18.4287 1.12362 0.561808 0.827267i \(-0.310106\pi\)
0.561808 + 0.827267i \(0.310106\pi\)
\(270\) −4.74521 −0.288784
\(271\) −11.2933 −0.686017 −0.343009 0.939332i \(-0.611446\pi\)
−0.343009 + 0.939332i \(0.611446\pi\)
\(272\) −14.7642 −0.895214
\(273\) −0.930274 −0.0563027
\(274\) −14.1955 −0.857584
\(275\) 8.06373 0.486261
\(276\) −1.12323 −0.0676107
\(277\) 25.1075 1.50856 0.754282 0.656550i \(-0.227985\pi\)
0.754282 + 0.656550i \(0.227985\pi\)
\(278\) 11.9506 0.716749
\(279\) −2.81104 −0.168293
\(280\) −9.04599 −0.540601
\(281\) 9.59612 0.572457 0.286228 0.958161i \(-0.407598\pi\)
0.286228 + 0.958161i \(0.407598\pi\)
\(282\) 9.65721 0.575079
\(283\) −28.2833 −1.68126 −0.840632 0.541606i \(-0.817816\pi\)
−0.840632 + 0.541606i \(0.817816\pi\)
\(284\) −0.809450 −0.0480320
\(285\) −12.8081 −0.758689
\(286\) 1.45840 0.0862367
\(287\) −7.36587 −0.434793
\(288\) 1.55164 0.0914311
\(289\) 2.18390 0.128465
\(290\) 7.94056 0.466286
\(291\) −7.75619 −0.454676
\(292\) 0.891303 0.0521596
\(293\) 9.34876 0.546160 0.273080 0.961991i \(-0.411958\pi\)
0.273080 + 0.961991i \(0.411958\pi\)
\(294\) −8.26935 −0.482279
\(295\) 1.61481 0.0940177
\(296\) −5.57215 −0.323875
\(297\) 1.00000 0.0580259
\(298\) 9.03491 0.523379
\(299\) 4.51472 0.261093
\(300\) −2.22858 −0.128667
\(301\) 0.207086 0.0119362
\(302\) 7.84230 0.451274
\(303\) 0.542833 0.0311849
\(304\) −11.9453 −0.685108
\(305\) −3.61438 −0.206959
\(306\) 5.75030 0.328723
\(307\) 16.0672 0.917005 0.458503 0.888693i \(-0.348386\pi\)
0.458503 + 0.888693i \(0.348386\pi\)
\(308\) 0.231446 0.0131879
\(309\) −18.9963 −1.08066
\(310\) 13.3390 0.757604
\(311\) −5.53945 −0.314113 −0.157057 0.987590i \(-0.550201\pi\)
−0.157057 + 0.987590i \(0.550201\pi\)
\(312\) −3.31985 −0.187949
\(313\) 6.72316 0.380016 0.190008 0.981783i \(-0.439149\pi\)
0.190008 + 0.981783i \(0.439149\pi\)
\(314\) −18.0462 −1.01841
\(315\) 3.02685 0.170544
\(316\) 0.486218 0.0273519
\(317\) 25.5153 1.43308 0.716541 0.697545i \(-0.245724\pi\)
0.716541 + 0.697545i \(0.245724\pi\)
\(318\) 3.81017 0.213664
\(319\) −1.67338 −0.0936916
\(320\) −31.7301 −1.77377
\(321\) 1.11430 0.0621944
\(322\) −4.46845 −0.249017
\(323\) 15.5211 0.863615
\(324\) −0.276371 −0.0153540
\(325\) 8.95755 0.496876
\(326\) 17.2607 0.955982
\(327\) −7.68359 −0.424903
\(328\) −26.2864 −1.45142
\(329\) −6.16009 −0.339617
\(330\) −4.74521 −0.261215
\(331\) −3.18003 −0.174790 −0.0873950 0.996174i \(-0.527854\pi\)
−0.0873950 + 0.996174i \(0.527854\pi\)
\(332\) −0.0883167 −0.00484701
\(333\) 1.86448 0.102173
\(334\) −30.5148 −1.66970
\(335\) −49.0989 −2.68256
\(336\) 2.82293 0.154003
\(337\) −3.13033 −0.170520 −0.0852599 0.996359i \(-0.527172\pi\)
−0.0852599 + 0.996359i \(0.527172\pi\)
\(338\) −15.4473 −0.840220
\(339\) 13.4812 0.732199
\(340\) 4.37517 0.237277
\(341\) −2.81104 −0.152227
\(342\) 4.65237 0.251572
\(343\) 11.1369 0.601338
\(344\) 0.739023 0.0398455
\(345\) −14.6896 −0.790863
\(346\) 7.35168 0.395229
\(347\) 21.9508 1.17838 0.589190 0.807994i \(-0.299447\pi\)
0.589190 + 0.807994i \(0.299447\pi\)
\(348\) 0.462475 0.0247913
\(349\) 7.14342 0.382378 0.191189 0.981553i \(-0.438766\pi\)
0.191189 + 0.981553i \(0.438766\pi\)
\(350\) −8.86575 −0.473894
\(351\) 1.11084 0.0592925
\(352\) 1.55164 0.0827025
\(353\) −7.76027 −0.413037 −0.206519 0.978443i \(-0.566213\pi\)
−0.206519 + 0.978443i \(0.566213\pi\)
\(354\) −0.586555 −0.0311751
\(355\) −10.5860 −0.561845
\(356\) −4.34559 −0.230316
\(357\) −3.66797 −0.194130
\(358\) 9.12896 0.482480
\(359\) −5.97613 −0.315408 −0.157704 0.987486i \(-0.550409\pi\)
−0.157704 + 0.987486i \(0.550409\pi\)
\(360\) 10.8019 0.569308
\(361\) −6.44243 −0.339075
\(362\) 16.9947 0.893224
\(363\) 1.00000 0.0524864
\(364\) 0.257101 0.0134757
\(365\) 11.6565 0.610126
\(366\) 1.31287 0.0686248
\(367\) 25.5116 1.33170 0.665848 0.746088i \(-0.268070\pi\)
0.665848 + 0.746088i \(0.268070\pi\)
\(368\) −13.7000 −0.714161
\(369\) 8.79562 0.457882
\(370\) −8.84735 −0.459952
\(371\) −2.43041 −0.126181
\(372\) 0.776892 0.0402800
\(373\) 3.08998 0.159993 0.0799967 0.996795i \(-0.474509\pi\)
0.0799967 + 0.996795i \(0.474509\pi\)
\(374\) 5.75030 0.297341
\(375\) −11.0735 −0.571832
\(376\) −21.9834 −1.13371
\(377\) −1.85887 −0.0957367
\(378\) −1.09946 −0.0565501
\(379\) 23.8740 1.22633 0.613163 0.789956i \(-0.289897\pi\)
0.613163 + 0.789956i \(0.289897\pi\)
\(380\) 3.53980 0.181588
\(381\) 2.50614 0.128394
\(382\) −24.3688 −1.24681
\(383\) 4.74359 0.242386 0.121193 0.992629i \(-0.461328\pi\)
0.121193 + 0.992629i \(0.461328\pi\)
\(384\) 8.42222 0.429795
\(385\) 3.02685 0.154263
\(386\) −13.9428 −0.709669
\(387\) −0.247282 −0.0125701
\(388\) 2.14359 0.108824
\(389\) 10.1189 0.513048 0.256524 0.966538i \(-0.417423\pi\)
0.256524 + 0.966538i \(0.417423\pi\)
\(390\) −5.27119 −0.266917
\(391\) 17.8011 0.900238
\(392\) 18.8241 0.950762
\(393\) 7.59263 0.382997
\(394\) 11.4220 0.575430
\(395\) 6.35875 0.319943
\(396\) −0.276371 −0.0138882
\(397\) −15.0570 −0.755688 −0.377844 0.925869i \(-0.623334\pi\)
−0.377844 + 0.925869i \(0.623334\pi\)
\(398\) 17.4610 0.875242
\(399\) −2.96763 −0.148567
\(400\) −27.1818 −1.35909
\(401\) 10.5678 0.527730 0.263865 0.964560i \(-0.415003\pi\)
0.263865 + 0.964560i \(0.415003\pi\)
\(402\) 17.8345 0.889502
\(403\) −3.12263 −0.155550
\(404\) −0.150023 −0.00746394
\(405\) −3.61438 −0.179600
\(406\) 1.83982 0.0913087
\(407\) 1.86448 0.0924189
\(408\) −13.0898 −0.648042
\(409\) 5.26303 0.260240 0.130120 0.991498i \(-0.458464\pi\)
0.130120 + 0.991498i \(0.458464\pi\)
\(410\) −41.7371 −2.06125
\(411\) −10.8126 −0.533346
\(412\) 5.25002 0.258650
\(413\) 0.374149 0.0184107
\(414\) 5.33580 0.262240
\(415\) −1.15500 −0.0566970
\(416\) 1.72363 0.0845078
\(417\) 9.10264 0.445758
\(418\) 4.65237 0.227555
\(419\) 19.6551 0.960216 0.480108 0.877209i \(-0.340598\pi\)
0.480108 + 0.877209i \(0.340598\pi\)
\(420\) −0.836534 −0.0408187
\(421\) −3.75600 −0.183056 −0.0915280 0.995803i \(-0.529175\pi\)
−0.0915280 + 0.995803i \(0.529175\pi\)
\(422\) −5.71456 −0.278180
\(423\) 7.35580 0.357651
\(424\) −8.67337 −0.421216
\(425\) 35.3187 1.71321
\(426\) 3.84520 0.186301
\(427\) −0.837447 −0.0405269
\(428\) −0.307962 −0.0148859
\(429\) 1.11084 0.0536321
\(430\) 1.17341 0.0565867
\(431\) 6.44918 0.310646 0.155323 0.987864i \(-0.450358\pi\)
0.155323 + 0.987864i \(0.450358\pi\)
\(432\) −3.37088 −0.162181
\(433\) 36.7936 1.76819 0.884094 0.467310i \(-0.154777\pi\)
0.884094 + 0.467310i \(0.154777\pi\)
\(434\) 3.09063 0.148355
\(435\) 6.04825 0.289991
\(436\) 2.12352 0.101698
\(437\) 14.4022 0.688953
\(438\) −4.23404 −0.202310
\(439\) −9.74500 −0.465104 −0.232552 0.972584i \(-0.574708\pi\)
−0.232552 + 0.972584i \(0.574708\pi\)
\(440\) 10.8019 0.514959
\(441\) −6.29868 −0.299937
\(442\) 6.38769 0.303831
\(443\) 6.37596 0.302931 0.151466 0.988463i \(-0.451601\pi\)
0.151466 + 0.988463i \(0.451601\pi\)
\(444\) −0.515289 −0.0244545
\(445\) −56.8315 −2.69407
\(446\) 12.3016 0.582498
\(447\) 6.88180 0.325498
\(448\) −7.35182 −0.347341
\(449\) 17.4495 0.823495 0.411747 0.911298i \(-0.364919\pi\)
0.411747 + 0.911298i \(0.364919\pi\)
\(450\) 10.5866 0.499059
\(451\) 8.79562 0.414170
\(452\) −3.72582 −0.175248
\(453\) 5.97340 0.280655
\(454\) −26.5492 −1.24602
\(455\) 3.36236 0.157630
\(456\) −10.5905 −0.495947
\(457\) −41.4643 −1.93962 −0.969808 0.243869i \(-0.921583\pi\)
−0.969808 + 0.243869i \(0.921583\pi\)
\(458\) −15.8316 −0.739763
\(459\) 4.37994 0.204438
\(460\) 4.05979 0.189289
\(461\) −23.3164 −1.08595 −0.542976 0.839748i \(-0.682702\pi\)
−0.542976 + 0.839748i \(0.682702\pi\)
\(462\) −1.09946 −0.0511515
\(463\) −38.3434 −1.78197 −0.890984 0.454036i \(-0.849984\pi\)
−0.890984 + 0.454036i \(0.849984\pi\)
\(464\) 5.64077 0.261866
\(465\) 10.1602 0.471167
\(466\) −29.9758 −1.38860
\(467\) 26.9505 1.24712 0.623561 0.781775i \(-0.285685\pi\)
0.623561 + 0.781775i \(0.285685\pi\)
\(468\) −0.307006 −0.0141913
\(469\) −11.3762 −0.525302
\(470\) −34.9048 −1.61004
\(471\) −13.7456 −0.633366
\(472\) 1.33522 0.0614584
\(473\) −0.247282 −0.0113701
\(474\) −2.30972 −0.106089
\(475\) 28.5752 1.31112
\(476\) 1.01372 0.0464638
\(477\) 2.90217 0.132881
\(478\) 17.0666 0.780607
\(479\) 32.4812 1.48411 0.742053 0.670342i \(-0.233852\pi\)
0.742053 + 0.670342i \(0.233852\pi\)
\(480\) −5.60820 −0.255978
\(481\) 2.07115 0.0944363
\(482\) 33.8317 1.54099
\(483\) −3.40357 −0.154868
\(484\) −0.276371 −0.0125623
\(485\) 28.0338 1.27295
\(486\) 1.31287 0.0595530
\(487\) −21.1728 −0.959432 −0.479716 0.877424i \(-0.659260\pi\)
−0.479716 + 0.877424i \(0.659260\pi\)
\(488\) −2.98858 −0.135287
\(489\) 13.1473 0.594542
\(490\) 29.8886 1.35023
\(491\) −17.4594 −0.787930 −0.393965 0.919125i \(-0.628897\pi\)
−0.393965 + 0.919125i \(0.628897\pi\)
\(492\) −2.43086 −0.109591
\(493\) −7.32933 −0.330096
\(494\) 5.16807 0.232522
\(495\) −3.61438 −0.162454
\(496\) 9.47568 0.425471
\(497\) −2.45276 −0.110021
\(498\) 0.419539 0.0188000
\(499\) −28.8978 −1.29364 −0.646822 0.762641i \(-0.723902\pi\)
−0.646822 + 0.762641i \(0.723902\pi\)
\(500\) 3.06039 0.136865
\(501\) −23.2428 −1.03841
\(502\) −3.17105 −0.141531
\(503\) −14.6656 −0.653905 −0.326953 0.945041i \(-0.606022\pi\)
−0.326953 + 0.945041i \(0.606022\pi\)
\(504\) 2.50278 0.111483
\(505\) −1.96200 −0.0873080
\(506\) 5.33580 0.237205
\(507\) −11.7660 −0.522547
\(508\) −0.692626 −0.0307303
\(509\) 11.3607 0.503556 0.251778 0.967785i \(-0.418985\pi\)
0.251778 + 0.967785i \(0.418985\pi\)
\(510\) −20.7838 −0.920320
\(511\) 2.70078 0.119476
\(512\) −25.3786 −1.12159
\(513\) 3.54367 0.156457
\(514\) 26.4757 1.16779
\(515\) 68.6597 3.02551
\(516\) 0.0683417 0.00300858
\(517\) 7.35580 0.323508
\(518\) −2.04992 −0.0900684
\(519\) 5.59970 0.245799
\(520\) 11.9992 0.526199
\(521\) 4.14492 0.181592 0.0907961 0.995870i \(-0.471059\pi\)
0.0907961 + 0.995870i \(0.471059\pi\)
\(522\) −2.19694 −0.0961573
\(523\) 5.81064 0.254082 0.127041 0.991897i \(-0.459452\pi\)
0.127041 + 0.991897i \(0.459452\pi\)
\(524\) −2.09838 −0.0916683
\(525\) −6.75295 −0.294723
\(526\) −20.7673 −0.905495
\(527\) −12.3122 −0.536329
\(528\) −3.37088 −0.146699
\(529\) −6.48210 −0.281830
\(530\) −13.7714 −0.598192
\(531\) −0.446773 −0.0193883
\(532\) 0.820168 0.0355588
\(533\) 9.77057 0.423210
\(534\) 20.6432 0.893319
\(535\) −4.02752 −0.174125
\(536\) −40.5978 −1.75356
\(537\) 6.95344 0.300063
\(538\) 24.1945 1.04310
\(539\) −6.29868 −0.271303
\(540\) 0.998910 0.0429863
\(541\) 20.0635 0.862598 0.431299 0.902209i \(-0.358055\pi\)
0.431299 + 0.902209i \(0.358055\pi\)
\(542\) −14.8266 −0.636857
\(543\) 12.9447 0.555511
\(544\) 6.79608 0.291380
\(545\) 27.7714 1.18960
\(546\) −1.22133 −0.0522680
\(547\) 23.1230 0.988669 0.494334 0.869272i \(-0.335412\pi\)
0.494334 + 0.869272i \(0.335412\pi\)
\(548\) 2.98829 0.127653
\(549\) 1.00000 0.0426790
\(550\) 10.5866 0.451416
\(551\) −5.92992 −0.252623
\(552\) −12.1463 −0.516979
\(553\) 1.47331 0.0626517
\(554\) 32.9629 1.40046
\(555\) −6.73894 −0.286052
\(556\) −2.51571 −0.106690
\(557\) 10.3408 0.438154 0.219077 0.975708i \(-0.429695\pi\)
0.219077 + 0.975708i \(0.429695\pi\)
\(558\) −3.69054 −0.156233
\(559\) −0.274692 −0.0116182
\(560\) −10.2031 −0.431161
\(561\) 4.37994 0.184921
\(562\) 12.5985 0.531434
\(563\) 8.42500 0.355071 0.177536 0.984114i \(-0.443187\pi\)
0.177536 + 0.984114i \(0.443187\pi\)
\(564\) −2.03293 −0.0856019
\(565\) −48.7262 −2.04993
\(566\) −37.1322 −1.56078
\(567\) −0.837447 −0.0351695
\(568\) −8.75311 −0.367272
\(569\) 6.71812 0.281638 0.140819 0.990035i \(-0.455026\pi\)
0.140819 + 0.990035i \(0.455026\pi\)
\(570\) −16.8154 −0.704321
\(571\) 6.05227 0.253279 0.126640 0.991949i \(-0.459581\pi\)
0.126640 + 0.991949i \(0.459581\pi\)
\(572\) −0.307006 −0.0128365
\(573\) −18.5614 −0.775415
\(574\) −9.67043 −0.403636
\(575\) 32.7728 1.36672
\(576\) 8.77885 0.365785
\(577\) 39.9435 1.66287 0.831435 0.555622i \(-0.187520\pi\)
0.831435 + 0.555622i \(0.187520\pi\)
\(578\) 2.86718 0.119259
\(579\) −10.6201 −0.441356
\(580\) −1.67156 −0.0694078
\(581\) −0.267613 −0.0111025
\(582\) −10.1829 −0.422093
\(583\) 2.90217 0.120196
\(584\) 9.63824 0.398833
\(585\) −4.01501 −0.166000
\(586\) 12.2737 0.507022
\(587\) 1.60472 0.0662339 0.0331169 0.999451i \(-0.489457\pi\)
0.0331169 + 0.999451i \(0.489457\pi\)
\(588\) 1.74077 0.0717883
\(589\) −9.96140 −0.410452
\(590\) 2.12003 0.0872804
\(591\) 8.69998 0.357870
\(592\) −6.28493 −0.258309
\(593\) −12.8886 −0.529273 −0.264636 0.964348i \(-0.585252\pi\)
−0.264636 + 0.964348i \(0.585252\pi\)
\(594\) 1.31287 0.0538677
\(595\) 13.2574 0.543502
\(596\) −1.90193 −0.0779062
\(597\) 13.2999 0.544328
\(598\) 5.92724 0.242383
\(599\) −46.2862 −1.89120 −0.945601 0.325329i \(-0.894525\pi\)
−0.945601 + 0.325329i \(0.894525\pi\)
\(600\) −24.0991 −0.983842
\(601\) −33.8954 −1.38262 −0.691311 0.722557i \(-0.742967\pi\)
−0.691311 + 0.722557i \(0.742967\pi\)
\(602\) 0.271877 0.0110809
\(603\) 13.5843 0.553196
\(604\) −1.65088 −0.0671732
\(605\) −3.61438 −0.146945
\(606\) 0.712669 0.0289502
\(607\) 20.3795 0.827179 0.413589 0.910464i \(-0.364275\pi\)
0.413589 + 0.910464i \(0.364275\pi\)
\(608\) 5.49848 0.222993
\(609\) 1.40137 0.0567864
\(610\) −4.74521 −0.192128
\(611\) 8.17115 0.330569
\(612\) −1.21049 −0.0489312
\(613\) −16.8249 −0.679553 −0.339776 0.940506i \(-0.610351\pi\)
−0.339776 + 0.940506i \(0.610351\pi\)
\(614\) 21.0942 0.851292
\(615\) −31.7907 −1.28192
\(616\) 2.50278 0.100840
\(617\) −1.36945 −0.0551318 −0.0275659 0.999620i \(-0.508776\pi\)
−0.0275659 + 0.999620i \(0.508776\pi\)
\(618\) −24.9396 −1.00322
\(619\) −42.6806 −1.71548 −0.857740 0.514084i \(-0.828132\pi\)
−0.857740 + 0.514084i \(0.828132\pi\)
\(620\) −2.80798 −0.112771
\(621\) 4.06422 0.163092
\(622\) −7.27258 −0.291604
\(623\) −13.1678 −0.527556
\(624\) −3.74452 −0.149901
\(625\) −0.294885 −0.0117954
\(626\) 8.82664 0.352784
\(627\) 3.54367 0.141520
\(628\) 3.79890 0.151593
\(629\) 8.16632 0.325612
\(630\) 3.97386 0.158322
\(631\) −31.4686 −1.25275 −0.626373 0.779524i \(-0.715461\pi\)
−0.626373 + 0.779524i \(0.715461\pi\)
\(632\) 5.25779 0.209143
\(633\) −4.35272 −0.173005
\(634\) 33.4983 1.33039
\(635\) −9.05815 −0.359462
\(636\) −0.802076 −0.0318044
\(637\) −6.99686 −0.277226
\(638\) −2.19694 −0.0869776
\(639\) 2.92885 0.115864
\(640\) −30.4411 −1.20329
\(641\) −15.2489 −0.602294 −0.301147 0.953578i \(-0.597370\pi\)
−0.301147 + 0.953578i \(0.597370\pi\)
\(642\) 1.46294 0.0577375
\(643\) −29.6334 −1.16863 −0.584313 0.811529i \(-0.698636\pi\)
−0.584313 + 0.811529i \(0.698636\pi\)
\(644\) 0.940649 0.0370668
\(645\) 0.893772 0.0351922
\(646\) 20.3771 0.801728
\(647\) 31.1055 1.22288 0.611442 0.791289i \(-0.290590\pi\)
0.611442 + 0.791289i \(0.290590\pi\)
\(648\) −2.98858 −0.117403
\(649\) −0.446773 −0.0175374
\(650\) 11.7601 0.461269
\(651\) 2.35410 0.0922645
\(652\) −3.63354 −0.142300
\(653\) −24.3750 −0.953867 −0.476934 0.878939i \(-0.658252\pi\)
−0.476934 + 0.878939i \(0.658252\pi\)
\(654\) −10.0876 −0.394455
\(655\) −27.4426 −1.07227
\(656\) −29.6489 −1.15760
\(657\) −3.22502 −0.125820
\(658\) −8.08740 −0.315280
\(659\) −20.4867 −0.798049 −0.399025 0.916940i \(-0.630651\pi\)
−0.399025 + 0.916940i \(0.630651\pi\)
\(660\) 0.998910 0.0388825
\(661\) −36.1634 −1.40659 −0.703297 0.710896i \(-0.748290\pi\)
−0.703297 + 0.710896i \(0.748290\pi\)
\(662\) −4.17496 −0.162265
\(663\) 4.86544 0.188958
\(664\) −0.955026 −0.0370622
\(665\) 10.7261 0.415942
\(666\) 2.44782 0.0948512
\(667\) −6.80101 −0.263336
\(668\) 6.42365 0.248538
\(669\) 9.37001 0.362265
\(670\) −64.4605 −2.49033
\(671\) 1.00000 0.0386046
\(672\) −1.29941 −0.0501260
\(673\) −19.7440 −0.761076 −0.380538 0.924765i \(-0.624261\pi\)
−0.380538 + 0.924765i \(0.624261\pi\)
\(674\) −4.10971 −0.158300
\(675\) 8.06373 0.310373
\(676\) 3.25179 0.125069
\(677\) 16.9323 0.650761 0.325380 0.945583i \(-0.394508\pi\)
0.325380 + 0.945583i \(0.394508\pi\)
\(678\) 17.6991 0.679729
\(679\) 6.49539 0.249270
\(680\) 47.3115 1.81431
\(681\) −20.2223 −0.774919
\(682\) −3.69054 −0.141318
\(683\) 22.3016 0.853346 0.426673 0.904406i \(-0.359685\pi\)
0.426673 + 0.904406i \(0.359685\pi\)
\(684\) −0.979367 −0.0374470
\(685\) 39.0808 1.49320
\(686\) 14.6214 0.558246
\(687\) −12.0588 −0.460071
\(688\) 0.833558 0.0317791
\(689\) 3.22386 0.122819
\(690\) −19.2856 −0.734190
\(691\) −13.5228 −0.514431 −0.257216 0.966354i \(-0.582805\pi\)
−0.257216 + 0.966354i \(0.582805\pi\)
\(692\) −1.54760 −0.0588307
\(693\) −0.837447 −0.0318120
\(694\) 28.8185 1.09394
\(695\) −32.9004 −1.24798
\(696\) 5.00104 0.189564
\(697\) 38.5243 1.45921
\(698\) 9.37838 0.354977
\(699\) −22.8322 −0.863595
\(700\) 1.86632 0.0705403
\(701\) 0.111492 0.00421098 0.00210549 0.999998i \(-0.499330\pi\)
0.00210549 + 0.999998i \(0.499330\pi\)
\(702\) 1.45840 0.0550436
\(703\) 6.60710 0.249191
\(704\) 8.77885 0.330865
\(705\) −26.5866 −1.00131
\(706\) −10.1882 −0.383439
\(707\) −0.454594 −0.0170967
\(708\) 0.123475 0.00464048
\(709\) −16.1234 −0.605526 −0.302763 0.953066i \(-0.597909\pi\)
−0.302763 + 0.953066i \(0.597909\pi\)
\(710\) −13.8980 −0.521583
\(711\) −1.75929 −0.0659786
\(712\) −46.9916 −1.76109
\(713\) −11.4247 −0.427859
\(714\) −4.81557 −0.180218
\(715\) −4.01501 −0.150153
\(716\) −1.92173 −0.0718184
\(717\) 12.9994 0.485473
\(718\) −7.84589 −0.292806
\(719\) −1.62281 −0.0605205 −0.0302602 0.999542i \(-0.509634\pi\)
−0.0302602 + 0.999542i \(0.509634\pi\)
\(720\) 12.1836 0.454057
\(721\) 15.9084 0.592458
\(722\) −8.45808 −0.314777
\(723\) 25.7692 0.958368
\(724\) −3.57755 −0.132959
\(725\) −13.4937 −0.501144
\(726\) 1.31287 0.0487252
\(727\) −21.1068 −0.782807 −0.391404 0.920219i \(-0.628010\pi\)
−0.391404 + 0.920219i \(0.628010\pi\)
\(728\) 2.78020 0.103041
\(729\) 1.00000 0.0370370
\(730\) 15.3034 0.566405
\(731\) −1.08308 −0.0400593
\(732\) −0.276371 −0.0102150
\(733\) −32.6890 −1.20739 −0.603697 0.797213i \(-0.706307\pi\)
−0.603697 + 0.797213i \(0.706307\pi\)
\(734\) 33.4934 1.23627
\(735\) 22.7658 0.839730
\(736\) 6.30620 0.232450
\(737\) 13.5843 0.500385
\(738\) 11.5475 0.425070
\(739\) −46.9356 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(740\) 1.86245 0.0684650
\(741\) 3.93646 0.144610
\(742\) −3.19082 −0.117139
\(743\) −19.1077 −0.700993 −0.350497 0.936564i \(-0.613987\pi\)
−0.350497 + 0.936564i \(0.613987\pi\)
\(744\) 8.40103 0.307997
\(745\) −24.8734 −0.911292
\(746\) 4.05675 0.148528
\(747\) 0.319558 0.0116920
\(748\) −1.21049 −0.0442599
\(749\) −0.933171 −0.0340973
\(750\) −14.5381 −0.530855
\(751\) −30.0661 −1.09713 −0.548565 0.836108i \(-0.684826\pi\)
−0.548565 + 0.836108i \(0.684826\pi\)
\(752\) −24.7955 −0.904199
\(753\) −2.41536 −0.0880206
\(754\) −2.44046 −0.0888762
\(755\) −21.5901 −0.785745
\(756\) 0.231446 0.00841762
\(757\) 25.3394 0.920976 0.460488 0.887666i \(-0.347674\pi\)
0.460488 + 0.887666i \(0.347674\pi\)
\(758\) 31.3435 1.13845
\(759\) 4.06422 0.147522
\(760\) 38.2782 1.38850
\(761\) 1.16161 0.0421084 0.0210542 0.999778i \(-0.493298\pi\)
0.0210542 + 0.999778i \(0.493298\pi\)
\(762\) 3.29024 0.119193
\(763\) 6.43460 0.232948
\(764\) 5.12985 0.185591
\(765\) −15.8308 −0.572363
\(766\) 6.22772 0.225016
\(767\) −0.496296 −0.0179202
\(768\) −6.50042 −0.234564
\(769\) 43.2961 1.56130 0.780648 0.624971i \(-0.214889\pi\)
0.780648 + 0.624971i \(0.214889\pi\)
\(770\) 3.97386 0.143208
\(771\) 20.1663 0.726270
\(772\) 2.93509 0.105636
\(773\) −45.6382 −1.64149 −0.820747 0.571292i \(-0.806442\pi\)
−0.820747 + 0.571292i \(0.806442\pi\)
\(774\) −0.324650 −0.0116693
\(775\) −22.6675 −0.814241
\(776\) 23.1800 0.832113
\(777\) −1.56140 −0.0560151
\(778\) 13.2848 0.476283
\(779\) 31.1687 1.11674
\(780\) 1.10963 0.0397313
\(781\) 2.92885 0.104803
\(782\) 23.3705 0.835727
\(783\) −1.67338 −0.0598019
\(784\) 21.2321 0.758289
\(785\) 49.6819 1.77322
\(786\) 9.96814 0.355552
\(787\) 20.1357 0.717760 0.358880 0.933384i \(-0.383159\pi\)
0.358880 + 0.933384i \(0.383159\pi\)
\(788\) −2.40443 −0.0856541
\(789\) −15.8182 −0.563143
\(790\) 8.34821 0.297016
\(791\) −11.2898 −0.401419
\(792\) −2.98858 −0.106195
\(793\) 1.11084 0.0394473
\(794\) −19.7679 −0.701535
\(795\) −10.4895 −0.372026
\(796\) −3.67570 −0.130282
\(797\) −32.4143 −1.14817 −0.574087 0.818794i \(-0.694643\pi\)
−0.574087 + 0.818794i \(0.694643\pi\)
\(798\) −3.89612 −0.137921
\(799\) 32.2180 1.13979
\(800\) 12.5120 0.442365
\(801\) 15.7237 0.555571
\(802\) 13.8741 0.489913
\(803\) −3.22502 −0.113809
\(804\) −3.75432 −0.132405
\(805\) 12.3018 0.433581
\(806\) −4.09961 −0.144403
\(807\) 18.4287 0.648720
\(808\) −1.62230 −0.0570723
\(809\) −15.1263 −0.531811 −0.265905 0.963999i \(-0.585671\pi\)
−0.265905 + 0.963999i \(0.585671\pi\)
\(810\) −4.74521 −0.166730
\(811\) 40.8120 1.43310 0.716552 0.697534i \(-0.245719\pi\)
0.716552 + 0.697534i \(0.245719\pi\)
\(812\) −0.387299 −0.0135915
\(813\) −11.2933 −0.396072
\(814\) 2.44782 0.0857961
\(815\) −47.5194 −1.66453
\(816\) −14.7642 −0.516852
\(817\) −0.876286 −0.0306574
\(818\) 6.90968 0.241591
\(819\) −0.930274 −0.0325064
\(820\) 8.78604 0.306822
\(821\) −22.2786 −0.777528 −0.388764 0.921337i \(-0.627098\pi\)
−0.388764 + 0.921337i \(0.627098\pi\)
\(822\) −14.1955 −0.495126
\(823\) −38.1070 −1.32833 −0.664163 0.747588i \(-0.731212\pi\)
−0.664163 + 0.747588i \(0.731212\pi\)
\(824\) 56.7718 1.97774
\(825\) 8.06373 0.280743
\(826\) 0.491209 0.0170913
\(827\) 20.0408 0.696887 0.348443 0.937330i \(-0.386710\pi\)
0.348443 + 0.937330i \(0.386710\pi\)
\(828\) −1.12323 −0.0390351
\(829\) 10.5019 0.364746 0.182373 0.983229i \(-0.441622\pi\)
0.182373 + 0.983229i \(0.441622\pi\)
\(830\) −1.51637 −0.0526340
\(831\) 25.1075 0.870970
\(832\) 9.75194 0.338088
\(833\) −27.5879 −0.955863
\(834\) 11.9506 0.413815
\(835\) 84.0084 2.90723
\(836\) −0.979367 −0.0338721
\(837\) −2.81104 −0.0971639
\(838\) 25.8046 0.891406
\(839\) 23.5773 0.813979 0.406990 0.913433i \(-0.366579\pi\)
0.406990 + 0.913433i \(0.366579\pi\)
\(840\) −9.04599 −0.312116
\(841\) −26.1998 −0.903441
\(842\) −4.93114 −0.169938
\(843\) 9.59612 0.330508
\(844\) 1.20297 0.0414078
\(845\) 42.5269 1.46297
\(846\) 9.65721 0.332022
\(847\) −0.837447 −0.0287750
\(848\) −9.78285 −0.335945
\(849\) −28.2833 −0.970679
\(850\) 46.3689 1.59044
\(851\) 7.57766 0.259759
\(852\) −0.809450 −0.0277313
\(853\) 43.7066 1.49649 0.748243 0.663425i \(-0.230898\pi\)
0.748243 + 0.663425i \(0.230898\pi\)
\(854\) −1.09946 −0.0376227
\(855\) −12.8081 −0.438029
\(856\) −3.33019 −0.113823
\(857\) 50.7072 1.73213 0.866063 0.499935i \(-0.166643\pi\)
0.866063 + 0.499935i \(0.166643\pi\)
\(858\) 1.45840 0.0497888
\(859\) 28.6892 0.978861 0.489431 0.872042i \(-0.337205\pi\)
0.489431 + 0.872042i \(0.337205\pi\)
\(860\) −0.247013 −0.00842307
\(861\) −7.36587 −0.251028
\(862\) 8.46694 0.288385
\(863\) 30.4241 1.03565 0.517824 0.855487i \(-0.326742\pi\)
0.517824 + 0.855487i \(0.326742\pi\)
\(864\) 1.55164 0.0527878
\(865\) −20.2394 −0.688161
\(866\) 48.3052 1.64148
\(867\) 2.18390 0.0741691
\(868\) −0.650606 −0.0220830
\(869\) −1.75929 −0.0596799
\(870\) 7.94056 0.269210
\(871\) 15.0901 0.511308
\(872\) 22.9630 0.777626
\(873\) −7.75619 −0.262507
\(874\) 18.9083 0.639582
\(875\) 9.27346 0.313500
\(876\) 0.891303 0.0301143
\(877\) −35.6625 −1.20424 −0.602118 0.798407i \(-0.705676\pi\)
−0.602118 + 0.798407i \(0.705676\pi\)
\(878\) −12.7939 −0.431774
\(879\) 9.34876 0.315326
\(880\) 12.1836 0.410710
\(881\) −21.9830 −0.740627 −0.370314 0.928907i \(-0.620750\pi\)
−0.370314 + 0.928907i \(0.620750\pi\)
\(882\) −8.26935 −0.278444
\(883\) −24.6476 −0.829458 −0.414729 0.909945i \(-0.636124\pi\)
−0.414729 + 0.909945i \(0.636124\pi\)
\(884\) −1.34467 −0.0452260
\(885\) 1.61481 0.0542812
\(886\) 8.37081 0.281223
\(887\) −21.9777 −0.737938 −0.368969 0.929442i \(-0.620289\pi\)
−0.368969 + 0.929442i \(0.620289\pi\)
\(888\) −5.57215 −0.186989
\(889\) −2.09876 −0.0703902
\(890\) −74.6124 −2.50101
\(891\) 1.00000 0.0335013
\(892\) −2.58960 −0.0867063
\(893\) 26.0665 0.872282
\(894\) 9.03491 0.302173
\(895\) −25.1323 −0.840082
\(896\) −7.05316 −0.235630
\(897\) 4.51472 0.150742
\(898\) 22.9090 0.764483
\(899\) 4.70396 0.156886
\(900\) −2.22858 −0.0742861
\(901\) 12.7113 0.423476
\(902\) 11.5475 0.384490
\(903\) 0.207086 0.00689139
\(904\) −40.2897 −1.34002
\(905\) −46.7871 −1.55526
\(906\) 7.84230 0.260543
\(907\) 19.5697 0.649801 0.324900 0.945748i \(-0.394669\pi\)
0.324900 + 0.945748i \(0.394669\pi\)
\(908\) 5.58885 0.185473
\(909\) 0.542833 0.0180046
\(910\) 4.41434 0.146334
\(911\) 29.9244 0.991440 0.495720 0.868482i \(-0.334904\pi\)
0.495720 + 0.868482i \(0.334904\pi\)
\(912\) −11.9453 −0.395547
\(913\) 0.319558 0.0105758
\(914\) −54.4372 −1.80062
\(915\) −3.61438 −0.119488
\(916\) 3.33270 0.110115
\(917\) −6.35842 −0.209974
\(918\) 5.75030 0.189788
\(919\) 13.0486 0.430433 0.215217 0.976566i \(-0.430954\pi\)
0.215217 + 0.976566i \(0.430954\pi\)
\(920\) 43.9012 1.44738
\(921\) 16.0672 0.529433
\(922\) −30.6114 −1.00813
\(923\) 3.25350 0.107090
\(924\) 0.231446 0.00761402
\(925\) 15.0347 0.494337
\(926\) −50.3399 −1.65427
\(927\) −18.9963 −0.623919
\(928\) −2.59649 −0.0852338
\(929\) −48.1672 −1.58032 −0.790158 0.612904i \(-0.790001\pi\)
−0.790158 + 0.612904i \(0.790001\pi\)
\(930\) 13.3390 0.437403
\(931\) −22.3204 −0.731522
\(932\) 6.31018 0.206697
\(933\) −5.53945 −0.181353
\(934\) 35.3826 1.15775
\(935\) −15.8308 −0.517722
\(936\) −3.31985 −0.108513
\(937\) −44.7933 −1.46333 −0.731667 0.681663i \(-0.761257\pi\)
−0.731667 + 0.681663i \(0.761257\pi\)
\(938\) −14.9354 −0.487659
\(939\) 6.72316 0.219402
\(940\) 7.34778 0.239658
\(941\) 24.4400 0.796722 0.398361 0.917229i \(-0.369579\pi\)
0.398361 + 0.917229i \(0.369579\pi\)
\(942\) −18.0462 −0.587978
\(943\) 35.7474 1.16409
\(944\) 1.50602 0.0490167
\(945\) 3.02685 0.0984635
\(946\) −0.324650 −0.0105553
\(947\) −11.3436 −0.368617 −0.184309 0.982868i \(-0.559005\pi\)
−0.184309 + 0.982868i \(0.559005\pi\)
\(948\) 0.486218 0.0157916
\(949\) −3.58250 −0.116293
\(950\) 37.5155 1.21716
\(951\) 25.5153 0.827391
\(952\) 10.9620 0.355281
\(953\) −13.2016 −0.427641 −0.213821 0.976873i \(-0.568591\pi\)
−0.213821 + 0.976873i \(0.568591\pi\)
\(954\) 3.81017 0.123359
\(955\) 67.0881 2.17092
\(956\) −3.59267 −0.116195
\(957\) −1.67338 −0.0540928
\(958\) 42.6436 1.37775
\(959\) 9.05497 0.292400
\(960\) −31.7301 −1.02408
\(961\) −23.0980 −0.745098
\(962\) 2.71915 0.0876689
\(963\) 1.11430 0.0359079
\(964\) −7.12187 −0.229380
\(965\) 38.3850 1.23566
\(966\) −4.46845 −0.143770
\(967\) 16.6355 0.534961 0.267480 0.963563i \(-0.413809\pi\)
0.267480 + 0.963563i \(0.413809\pi\)
\(968\) −2.98858 −0.0960566
\(969\) 15.5211 0.498608
\(970\) 36.8047 1.18173
\(971\) −10.0363 −0.322080 −0.161040 0.986948i \(-0.551485\pi\)
−0.161040 + 0.986948i \(0.551485\pi\)
\(972\) −0.276371 −0.00886461
\(973\) −7.62298 −0.244382
\(974\) −27.7972 −0.890679
\(975\) 8.95755 0.286871
\(976\) −3.37088 −0.107899
\(977\) 11.7119 0.374695 0.187348 0.982294i \(-0.440011\pi\)
0.187348 + 0.982294i \(0.440011\pi\)
\(978\) 17.2607 0.551937
\(979\) 15.7237 0.502533
\(980\) −6.29182 −0.200985
\(981\) −7.68359 −0.245318
\(982\) −22.9219 −0.731466
\(983\) 13.8777 0.442631 0.221316 0.975202i \(-0.428965\pi\)
0.221316 + 0.975202i \(0.428965\pi\)
\(984\) −26.2864 −0.837980
\(985\) −31.4450 −1.00192
\(986\) −9.62246 −0.306442
\(987\) −6.16009 −0.196078
\(988\) −1.08793 −0.0346115
\(989\) −1.00501 −0.0319575
\(990\) −4.74521 −0.150813
\(991\) 13.9982 0.444666 0.222333 0.974971i \(-0.428633\pi\)
0.222333 + 0.974971i \(0.428633\pi\)
\(992\) −4.36172 −0.138485
\(993\) −3.18003 −0.100915
\(994\) −3.22015 −0.102137
\(995\) −48.0708 −1.52395
\(996\) −0.0883167 −0.00279842
\(997\) −42.5735 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(998\) −37.9391 −1.20094
\(999\) 1.86448 0.0589896
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.g.1.9 13
3.2 odd 2 6039.2.a.h.1.5 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.g.1.9 13 1.1 even 1 trivial
6039.2.a.h.1.5 13 3.2 odd 2