Properties

Label 2013.2.a.f.1.13
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: \( x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} - 404 x^{4} + 98 x^{3} + 118 x^{2} - 16 x - 11 \) 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 1.13
Root \(2.62425\) 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)
 
\(f(q)\) \(=\) \(q+2.62425 q^{2} +1.00000 q^{3} +4.88668 q^{4} +0.233370 q^{5} +2.62425 q^{6} +1.03556 q^{7} +7.57538 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.62425 q^{2} +1.00000 q^{3} +4.88668 q^{4} +0.233370 q^{5} +2.62425 q^{6} +1.03556 q^{7} +7.57538 q^{8} +1.00000 q^{9} +0.612421 q^{10} -1.00000 q^{11} +4.88668 q^{12} +3.46134 q^{13} +2.71756 q^{14} +0.233370 q^{15} +10.1063 q^{16} +0.866180 q^{17} +2.62425 q^{18} -6.81869 q^{19} +1.14041 q^{20} +1.03556 q^{21} -2.62425 q^{22} -2.06167 q^{23} +7.57538 q^{24} -4.94554 q^{25} +9.08343 q^{26} +1.00000 q^{27} +5.06045 q^{28} -6.06446 q^{29} +0.612421 q^{30} +0.640968 q^{31} +11.3707 q^{32} -1.00000 q^{33} +2.27307 q^{34} +0.241668 q^{35} +4.88668 q^{36} +6.49171 q^{37} -17.8939 q^{38} +3.46134 q^{39} +1.76787 q^{40} -0.461254 q^{41} +2.71756 q^{42} -5.84824 q^{43} -4.88668 q^{44} +0.233370 q^{45} -5.41033 q^{46} +10.8361 q^{47} +10.1063 q^{48} -5.92762 q^{49} -12.9783 q^{50} +0.866180 q^{51} +16.9145 q^{52} +5.93238 q^{53} +2.62425 q^{54} -0.233370 q^{55} +7.84475 q^{56} -6.81869 q^{57} -15.9147 q^{58} -13.8895 q^{59} +1.14041 q^{60} -1.00000 q^{61} +1.68206 q^{62} +1.03556 q^{63} +9.62701 q^{64} +0.807774 q^{65} -2.62425 q^{66} +5.49525 q^{67} +4.23275 q^{68} -2.06167 q^{69} +0.634198 q^{70} +10.9720 q^{71} +7.57538 q^{72} -15.4334 q^{73} +17.0359 q^{74} -4.94554 q^{75} -33.3208 q^{76} -1.03556 q^{77} +9.08343 q^{78} +12.4523 q^{79} +2.35851 q^{80} +1.00000 q^{81} -1.21044 q^{82} +4.62807 q^{83} +5.06045 q^{84} +0.202140 q^{85} -15.3473 q^{86} -6.06446 q^{87} -7.57538 q^{88} +4.71761 q^{89} +0.612421 q^{90} +3.58443 q^{91} -10.0747 q^{92} +0.640968 q^{93} +28.4368 q^{94} -1.59128 q^{95} +11.3707 q^{96} -0.332084 q^{97} -15.5555 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 5 q^{7} + 15 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 5 q^{7} + 15 q^{8} + 13 q^{9} + 8 q^{10} - 13 q^{11} + 12 q^{12} - 9 q^{13} + 19 q^{14} + 7 q^{15} + 18 q^{16} + 7 q^{17} + 4 q^{18} + 2 q^{19} + 15 q^{20} + 5 q^{21} - 4 q^{22} + 23 q^{23} + 15 q^{24} + 10 q^{25} + 8 q^{26} + 13 q^{27} + 9 q^{28} + 16 q^{29} + 8 q^{30} + 9 q^{31} + 29 q^{32} - 13 q^{33} + 2 q^{34} + 16 q^{35} + 12 q^{36} + 14 q^{37} + 8 q^{38} - 9 q^{39} + 16 q^{40} + 19 q^{41} + 19 q^{42} + 7 q^{43} - 12 q^{44} + 7 q^{45} + 4 q^{46} + 26 q^{47} + 18 q^{48} + 8 q^{49} - 15 q^{50} + 7 q^{51} - 17 q^{52} + 18 q^{53} + 4 q^{54} - 7 q^{55} + 44 q^{56} + 2 q^{57} - q^{58} + 31 q^{59} + 15 q^{60} - 13 q^{61} - 5 q^{62} + 5 q^{63} - 17 q^{64} + 31 q^{65} - 4 q^{66} + 14 q^{67} - 32 q^{68} + 23 q^{69} - 20 q^{70} + 37 q^{71} + 15 q^{72} - 16 q^{73} - 6 q^{74} + 10 q^{75} - 7 q^{76} - 5 q^{77} + 8 q^{78} - 17 q^{79} - 2 q^{80} + 13 q^{81} - 2 q^{82} + 30 q^{83} + 9 q^{84} - 16 q^{85} - 22 q^{86} + 16 q^{87} - 15 q^{88} + 35 q^{89} + 8 q^{90} - q^{91} + 24 q^{92} + 9 q^{93} - 11 q^{94} + 13 q^{95} + 29 q^{96} - q^{97} - q^{98} - 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.62425 1.85562 0.927812 0.373047i \(-0.121687\pi\)
0.927812 + 0.373047i \(0.121687\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.88668 2.44334
\(5\) 0.233370 0.104366 0.0521831 0.998638i \(-0.483382\pi\)
0.0521831 + 0.998638i \(0.483382\pi\)
\(6\) 2.62425 1.07135
\(7\) 1.03556 0.391404 0.195702 0.980663i \(-0.437301\pi\)
0.195702 + 0.980663i \(0.437301\pi\)
\(8\) 7.57538 2.67830
\(9\) 1.00000 0.333333
\(10\) 0.612421 0.193664
\(11\) −1.00000 −0.301511
\(12\) 4.88668 1.41066
\(13\) 3.46134 0.960004 0.480002 0.877267i \(-0.340636\pi\)
0.480002 + 0.877267i \(0.340636\pi\)
\(14\) 2.71756 0.726300
\(15\) 0.233370 0.0602558
\(16\) 10.1063 2.52658
\(17\) 0.866180 0.210080 0.105040 0.994468i \(-0.466503\pi\)
0.105040 + 0.994468i \(0.466503\pi\)
\(18\) 2.62425 0.618541
\(19\) −6.81869 −1.56431 −0.782157 0.623081i \(-0.785880\pi\)
−0.782157 + 0.623081i \(0.785880\pi\)
\(20\) 1.14041 0.255002
\(21\) 1.03556 0.225977
\(22\) −2.62425 −0.559492
\(23\) −2.06167 −0.429888 −0.214944 0.976626i \(-0.568957\pi\)
−0.214944 + 0.976626i \(0.568957\pi\)
\(24\) 7.57538 1.54632
\(25\) −4.94554 −0.989108
\(26\) 9.08343 1.78141
\(27\) 1.00000 0.192450
\(28\) 5.06045 0.956335
\(29\) −6.06446 −1.12614 −0.563071 0.826408i \(-0.690380\pi\)
−0.563071 + 0.826408i \(0.690380\pi\)
\(30\) 0.612421 0.111812
\(31\) 0.640968 0.115121 0.0575606 0.998342i \(-0.481668\pi\)
0.0575606 + 0.998342i \(0.481668\pi\)
\(32\) 11.3707 2.01008
\(33\) −1.00000 −0.174078
\(34\) 2.27307 0.389829
\(35\) 0.241668 0.0408494
\(36\) 4.88668 0.814447
\(37\) 6.49171 1.06723 0.533615 0.845728i \(-0.320833\pi\)
0.533615 + 0.845728i \(0.320833\pi\)
\(38\) −17.8939 −2.90278
\(39\) 3.46134 0.554259
\(40\) 1.76787 0.279524
\(41\) −0.461254 −0.0720357 −0.0360178 0.999351i \(-0.511467\pi\)
−0.0360178 + 0.999351i \(0.511467\pi\)
\(42\) 2.71756 0.419329
\(43\) −5.84824 −0.891849 −0.445924 0.895071i \(-0.647125\pi\)
−0.445924 + 0.895071i \(0.647125\pi\)
\(44\) −4.88668 −0.736695
\(45\) 0.233370 0.0347887
\(46\) −5.41033 −0.797710
\(47\) 10.8361 1.58061 0.790307 0.612711i \(-0.209921\pi\)
0.790307 + 0.612711i \(0.209921\pi\)
\(48\) 10.1063 1.45872
\(49\) −5.92762 −0.846803
\(50\) −12.9783 −1.83541
\(51\) 0.866180 0.121290
\(52\) 16.9145 2.34562
\(53\) 5.93238 0.814875 0.407437 0.913233i \(-0.366422\pi\)
0.407437 + 0.913233i \(0.366422\pi\)
\(54\) 2.62425 0.357115
\(55\) −0.233370 −0.0314676
\(56\) 7.84475 1.04830
\(57\) −6.81869 −0.903157
\(58\) −15.9147 −2.08970
\(59\) −13.8895 −1.80826 −0.904131 0.427256i \(-0.859480\pi\)
−0.904131 + 0.427256i \(0.859480\pi\)
\(60\) 1.14041 0.147226
\(61\) −1.00000 −0.128037
\(62\) 1.68206 0.213622
\(63\) 1.03556 0.130468
\(64\) 9.62701 1.20338
\(65\) 0.807774 0.100192
\(66\) −2.62425 −0.323023
\(67\) 5.49525 0.671352 0.335676 0.941978i \(-0.391035\pi\)
0.335676 + 0.941978i \(0.391035\pi\)
\(68\) 4.23275 0.513296
\(69\) −2.06167 −0.248196
\(70\) 0.634198 0.0758011
\(71\) 10.9720 1.30213 0.651067 0.759020i \(-0.274322\pi\)
0.651067 + 0.759020i \(0.274322\pi\)
\(72\) 7.57538 0.892767
\(73\) −15.4334 −1.80634 −0.903172 0.429280i \(-0.858767\pi\)
−0.903172 + 0.429280i \(0.858767\pi\)
\(74\) 17.0359 1.98038
\(75\) −4.94554 −0.571062
\(76\) −33.3208 −3.82215
\(77\) −1.03556 −0.118013
\(78\) 9.08343 1.02850
\(79\) 12.4523 1.40100 0.700498 0.713654i \(-0.252961\pi\)
0.700498 + 0.713654i \(0.252961\pi\)
\(80\) 2.35851 0.263689
\(81\) 1.00000 0.111111
\(82\) −1.21044 −0.133671
\(83\) 4.62807 0.507997 0.253998 0.967205i \(-0.418254\pi\)
0.253998 + 0.967205i \(0.418254\pi\)
\(84\) 5.06045 0.552140
\(85\) 0.202140 0.0219252
\(86\) −15.3473 −1.65494
\(87\) −6.06446 −0.650179
\(88\) −7.57538 −0.807538
\(89\) 4.71761 0.500066 0.250033 0.968237i \(-0.419559\pi\)
0.250033 + 0.968237i \(0.419559\pi\)
\(90\) 0.612421 0.0645548
\(91\) 3.58443 0.375750
\(92\) −10.0747 −1.05036
\(93\) 0.640968 0.0664653
\(94\) 28.4368 2.93303
\(95\) −1.59128 −0.163261
\(96\) 11.3707 1.16052
\(97\) −0.332084 −0.0337180 −0.0168590 0.999858i \(-0.505367\pi\)
−0.0168590 + 0.999858i \(0.505367\pi\)
\(98\) −15.5555 −1.57135
\(99\) −1.00000 −0.100504
\(100\) −24.1673 −2.41673
\(101\) 12.3590 1.22977 0.614883 0.788618i \(-0.289203\pi\)
0.614883 + 0.788618i \(0.289203\pi\)
\(102\) 2.27307 0.225068
\(103\) 14.4756 1.42633 0.713163 0.700998i \(-0.247262\pi\)
0.713163 + 0.700998i \(0.247262\pi\)
\(104\) 26.2210 2.57118
\(105\) 0.241668 0.0235844
\(106\) 15.5680 1.51210
\(107\) −2.13786 −0.206675 −0.103337 0.994646i \(-0.532952\pi\)
−0.103337 + 0.994646i \(0.532952\pi\)
\(108\) 4.88668 0.470221
\(109\) −1.66386 −0.159369 −0.0796844 0.996820i \(-0.525391\pi\)
−0.0796844 + 0.996820i \(0.525391\pi\)
\(110\) −0.612421 −0.0583920
\(111\) 6.49171 0.616165
\(112\) 10.4657 0.988914
\(113\) −16.6819 −1.56930 −0.784652 0.619936i \(-0.787158\pi\)
−0.784652 + 0.619936i \(0.787158\pi\)
\(114\) −17.8939 −1.67592
\(115\) −0.481132 −0.0448657
\(116\) −29.6351 −2.75155
\(117\) 3.46134 0.320001
\(118\) −36.4496 −3.35545
\(119\) 0.896981 0.0822261
\(120\) 1.76787 0.161383
\(121\) 1.00000 0.0909091
\(122\) −2.62425 −0.237588
\(123\) −0.461254 −0.0415898
\(124\) 3.13221 0.281281
\(125\) −2.32099 −0.207596
\(126\) 2.71756 0.242100
\(127\) −6.27466 −0.556786 −0.278393 0.960467i \(-0.589802\pi\)
−0.278393 + 0.960467i \(0.589802\pi\)
\(128\) 2.52221 0.222934
\(129\) −5.84824 −0.514909
\(130\) 2.11980 0.185919
\(131\) 18.4332 1.61052 0.805259 0.592923i \(-0.202026\pi\)
0.805259 + 0.592923i \(0.202026\pi\)
\(132\) −4.88668 −0.425331
\(133\) −7.06115 −0.612279
\(134\) 14.4209 1.24578
\(135\) 0.233370 0.0200853
\(136\) 6.56165 0.562657
\(137\) −15.5912 −1.33204 −0.666022 0.745932i \(-0.732004\pi\)
−0.666022 + 0.745932i \(0.732004\pi\)
\(138\) −5.41033 −0.460558
\(139\) −18.1623 −1.54051 −0.770255 0.637736i \(-0.779871\pi\)
−0.770255 + 0.637736i \(0.779871\pi\)
\(140\) 1.18096 0.0998090
\(141\) 10.8361 0.912568
\(142\) 28.7932 2.41627
\(143\) −3.46134 −0.289452
\(144\) 10.1063 0.842193
\(145\) −1.41526 −0.117531
\(146\) −40.5011 −3.35189
\(147\) −5.92762 −0.488902
\(148\) 31.7229 2.60761
\(149\) 8.50275 0.696573 0.348286 0.937388i \(-0.386764\pi\)
0.348286 + 0.937388i \(0.386764\pi\)
\(150\) −12.9783 −1.05968
\(151\) −23.5234 −1.91430 −0.957152 0.289585i \(-0.906483\pi\)
−0.957152 + 0.289585i \(0.906483\pi\)
\(152\) −51.6541 −4.18970
\(153\) 0.866180 0.0700265
\(154\) −2.71756 −0.218988
\(155\) 0.149583 0.0120148
\(156\) 16.9145 1.35424
\(157\) −15.5352 −1.23984 −0.619922 0.784663i \(-0.712836\pi\)
−0.619922 + 0.784663i \(0.712836\pi\)
\(158\) 32.6780 2.59972
\(159\) 5.93238 0.470468
\(160\) 2.65359 0.209785
\(161\) −2.13498 −0.168260
\(162\) 2.62425 0.206180
\(163\) 3.93029 0.307844 0.153922 0.988083i \(-0.450809\pi\)
0.153922 + 0.988083i \(0.450809\pi\)
\(164\) −2.25400 −0.176008
\(165\) −0.233370 −0.0181678
\(166\) 12.1452 0.942651
\(167\) −1.83634 −0.142100 −0.0710502 0.997473i \(-0.522635\pi\)
−0.0710502 + 0.997473i \(0.522635\pi\)
\(168\) 7.84475 0.605236
\(169\) −1.01910 −0.0783920
\(170\) 0.530467 0.0406850
\(171\) −6.81869 −0.521438
\(172\) −28.5785 −2.17909
\(173\) 10.3886 0.789834 0.394917 0.918717i \(-0.370773\pi\)
0.394917 + 0.918717i \(0.370773\pi\)
\(174\) −15.9147 −1.20649
\(175\) −5.12140 −0.387141
\(176\) −10.1063 −0.761792
\(177\) −13.8895 −1.04400
\(178\) 12.3802 0.927934
\(179\) 17.4905 1.30730 0.653652 0.756795i \(-0.273236\pi\)
0.653652 + 0.756795i \(0.273236\pi\)
\(180\) 1.14041 0.0850008
\(181\) −14.8136 −1.10108 −0.550542 0.834807i \(-0.685579\pi\)
−0.550542 + 0.834807i \(0.685579\pi\)
\(182\) 9.40643 0.697251
\(183\) −1.00000 −0.0739221
\(184\) −15.6179 −1.15137
\(185\) 1.51497 0.111383
\(186\) 1.68206 0.123335
\(187\) −0.866180 −0.0633414
\(188\) 52.9528 3.86198
\(189\) 1.03556 0.0753258
\(190\) −4.17591 −0.302952
\(191\) −6.59920 −0.477501 −0.238751 0.971081i \(-0.576738\pi\)
−0.238751 + 0.971081i \(0.576738\pi\)
\(192\) 9.62701 0.694770
\(193\) −18.6461 −1.34218 −0.671089 0.741377i \(-0.734173\pi\)
−0.671089 + 0.741377i \(0.734173\pi\)
\(194\) −0.871471 −0.0625680
\(195\) 0.807774 0.0578459
\(196\) −28.9664 −2.06903
\(197\) 4.09897 0.292039 0.146020 0.989282i \(-0.453354\pi\)
0.146020 + 0.989282i \(0.453354\pi\)
\(198\) −2.62425 −0.186497
\(199\) 13.0650 0.926151 0.463076 0.886319i \(-0.346746\pi\)
0.463076 + 0.886319i \(0.346746\pi\)
\(200\) −37.4643 −2.64913
\(201\) 5.49525 0.387605
\(202\) 32.4331 2.28198
\(203\) −6.28011 −0.440777
\(204\) 4.23275 0.296352
\(205\) −0.107643 −0.00751809
\(206\) 37.9876 2.64672
\(207\) −2.06167 −0.143296
\(208\) 34.9814 2.42553
\(209\) 6.81869 0.471658
\(210\) 0.634198 0.0437638
\(211\) 25.2541 1.73856 0.869281 0.494318i \(-0.164582\pi\)
0.869281 + 0.494318i \(0.164582\pi\)
\(212\) 28.9897 1.99102
\(213\) 10.9720 0.751787
\(214\) −5.61028 −0.383511
\(215\) −1.36480 −0.0930789
\(216\) 7.57538 0.515439
\(217\) 0.663760 0.0450590
\(218\) −4.36638 −0.295729
\(219\) −15.4334 −1.04289
\(220\) −1.14041 −0.0768861
\(221\) 2.99815 0.201677
\(222\) 17.0359 1.14337
\(223\) 0.184928 0.0123837 0.00619186 0.999981i \(-0.498029\pi\)
0.00619186 + 0.999981i \(0.498029\pi\)
\(224\) 11.7751 0.786755
\(225\) −4.94554 −0.329703
\(226\) −43.7775 −2.91204
\(227\) −10.8433 −0.719698 −0.359849 0.933011i \(-0.617172\pi\)
−0.359849 + 0.933011i \(0.617172\pi\)
\(228\) −33.3208 −2.20672
\(229\) 9.68390 0.639930 0.319965 0.947429i \(-0.396329\pi\)
0.319965 + 0.947429i \(0.396329\pi\)
\(230\) −1.26261 −0.0832540
\(231\) −1.03556 −0.0681348
\(232\) −45.9406 −3.01615
\(233\) −10.8919 −0.713555 −0.356778 0.934189i \(-0.616125\pi\)
−0.356778 + 0.934189i \(0.616125\pi\)
\(234\) 9.08343 0.593802
\(235\) 2.52883 0.164963
\(236\) −67.8737 −4.41820
\(237\) 12.4523 0.808866
\(238\) 2.35390 0.152581
\(239\) 7.56308 0.489215 0.244607 0.969622i \(-0.421341\pi\)
0.244607 + 0.969622i \(0.421341\pi\)
\(240\) 2.35851 0.152241
\(241\) 12.1001 0.779439 0.389720 0.920934i \(-0.372572\pi\)
0.389720 + 0.920934i \(0.372572\pi\)
\(242\) 2.62425 0.168693
\(243\) 1.00000 0.0641500
\(244\) −4.88668 −0.312838
\(245\) −1.38333 −0.0883776
\(246\) −1.21044 −0.0771751
\(247\) −23.6018 −1.50175
\(248\) 4.85558 0.308329
\(249\) 4.62807 0.293292
\(250\) −6.09085 −0.385219
\(251\) −14.5210 −0.916557 −0.458278 0.888809i \(-0.651534\pi\)
−0.458278 + 0.888809i \(0.651534\pi\)
\(252\) 5.06045 0.318778
\(253\) 2.06167 0.129616
\(254\) −16.4663 −1.03319
\(255\) 0.202140 0.0126585
\(256\) −12.6351 −0.789695
\(257\) −1.95557 −0.121985 −0.0609924 0.998138i \(-0.519427\pi\)
−0.0609924 + 0.998138i \(0.519427\pi\)
\(258\) −15.3473 −0.955478
\(259\) 6.72254 0.417718
\(260\) 3.94733 0.244803
\(261\) −6.06446 −0.375381
\(262\) 48.3734 2.98852
\(263\) −5.37410 −0.331381 −0.165691 0.986178i \(-0.552985\pi\)
−0.165691 + 0.986178i \(0.552985\pi\)
\(264\) −7.57538 −0.466232
\(265\) 1.38444 0.0850454
\(266\) −18.5302 −1.13616
\(267\) 4.71761 0.288713
\(268\) 26.8536 1.64034
\(269\) −14.9982 −0.914459 −0.457229 0.889349i \(-0.651158\pi\)
−0.457229 + 0.889349i \(0.651158\pi\)
\(270\) 0.612421 0.0372707
\(271\) 20.1090 1.22153 0.610766 0.791811i \(-0.290862\pi\)
0.610766 + 0.791811i \(0.290862\pi\)
\(272\) 8.75389 0.530783
\(273\) 3.58443 0.216939
\(274\) −40.9151 −2.47177
\(275\) 4.94554 0.298227
\(276\) −10.0747 −0.606427
\(277\) −9.56683 −0.574815 −0.287408 0.957808i \(-0.592793\pi\)
−0.287408 + 0.957808i \(0.592793\pi\)
\(278\) −47.6625 −2.85861
\(279\) 0.640968 0.0383738
\(280\) 1.83073 0.109407
\(281\) 27.4378 1.63680 0.818402 0.574646i \(-0.194860\pi\)
0.818402 + 0.574646i \(0.194860\pi\)
\(282\) 28.4368 1.69338
\(283\) 21.0886 1.25358 0.626792 0.779186i \(-0.284367\pi\)
0.626792 + 0.779186i \(0.284367\pi\)
\(284\) 53.6166 3.18156
\(285\) −1.59128 −0.0942591
\(286\) −9.08343 −0.537114
\(287\) −0.477655 −0.0281951
\(288\) 11.3707 0.670027
\(289\) −16.2497 −0.955867
\(290\) −3.71400 −0.218094
\(291\) −0.332084 −0.0194671
\(292\) −75.4182 −4.41351
\(293\) 10.8811 0.635679 0.317840 0.948144i \(-0.397043\pi\)
0.317840 + 0.948144i \(0.397043\pi\)
\(294\) −15.5555 −0.907218
\(295\) −3.24140 −0.188721
\(296\) 49.1771 2.85836
\(297\) −1.00000 −0.0580259
\(298\) 22.3133 1.29258
\(299\) −7.13615 −0.412694
\(300\) −24.1673 −1.39530
\(301\) −6.05620 −0.349074
\(302\) −61.7312 −3.55223
\(303\) 12.3590 0.710006
\(304\) −68.9118 −3.95236
\(305\) −0.233370 −0.0133627
\(306\) 2.27307 0.129943
\(307\) −7.26799 −0.414806 −0.207403 0.978256i \(-0.566501\pi\)
−0.207403 + 0.978256i \(0.566501\pi\)
\(308\) −5.06045 −0.288346
\(309\) 14.4756 0.823490
\(310\) 0.392542 0.0222949
\(311\) −0.304014 −0.0172391 −0.00861953 0.999963i \(-0.502744\pi\)
−0.00861953 + 0.999963i \(0.502744\pi\)
\(312\) 26.2210 1.48447
\(313\) 30.5243 1.72533 0.862667 0.505773i \(-0.168792\pi\)
0.862667 + 0.505773i \(0.168792\pi\)
\(314\) −40.7683 −2.30069
\(315\) 0.241668 0.0136165
\(316\) 60.8506 3.42311
\(317\) 0.909394 0.0510767 0.0255383 0.999674i \(-0.491870\pi\)
0.0255383 + 0.999674i \(0.491870\pi\)
\(318\) 15.5680 0.873012
\(319\) 6.06446 0.339545
\(320\) 2.24665 0.125592
\(321\) −2.13786 −0.119324
\(322\) −5.60272 −0.312227
\(323\) −5.90621 −0.328630
\(324\) 4.88668 0.271482
\(325\) −17.1182 −0.949548
\(326\) 10.3141 0.571244
\(327\) −1.66386 −0.0920116
\(328\) −3.49417 −0.192933
\(329\) 11.2215 0.618659
\(330\) −0.612421 −0.0337127
\(331\) 20.7795 1.14215 0.571073 0.820899i \(-0.306527\pi\)
0.571073 + 0.820899i \(0.306527\pi\)
\(332\) 22.6159 1.24121
\(333\) 6.49171 0.355743
\(334\) −4.81902 −0.263685
\(335\) 1.28243 0.0700664
\(336\) 10.4657 0.570950
\(337\) −7.77395 −0.423474 −0.211737 0.977327i \(-0.567912\pi\)
−0.211737 + 0.977327i \(0.567912\pi\)
\(338\) −2.67436 −0.145466
\(339\) −16.6819 −0.906038
\(340\) 0.987797 0.0535708
\(341\) −0.640968 −0.0347104
\(342\) −17.8939 −0.967593
\(343\) −13.3873 −0.722847
\(344\) −44.3027 −2.38864
\(345\) −0.481132 −0.0259033
\(346\) 27.2624 1.46564
\(347\) 12.8645 0.690602 0.345301 0.938492i \(-0.387777\pi\)
0.345301 + 0.938492i \(0.387777\pi\)
\(348\) −29.6351 −1.58861
\(349\) 14.3781 0.769644 0.384822 0.922991i \(-0.374263\pi\)
0.384822 + 0.922991i \(0.374263\pi\)
\(350\) −13.4398 −0.718388
\(351\) 3.46134 0.184753
\(352\) −11.3707 −0.606062
\(353\) 1.41907 0.0755294 0.0377647 0.999287i \(-0.487976\pi\)
0.0377647 + 0.999287i \(0.487976\pi\)
\(354\) −36.4496 −1.93727
\(355\) 2.56053 0.135899
\(356\) 23.0535 1.22183
\(357\) 0.896981 0.0474733
\(358\) 45.8995 2.42587
\(359\) −0.867127 −0.0457652 −0.0228826 0.999738i \(-0.507284\pi\)
−0.0228826 + 0.999738i \(0.507284\pi\)
\(360\) 1.76787 0.0931747
\(361\) 27.4945 1.44708
\(362\) −38.8745 −2.04320
\(363\) 1.00000 0.0524864
\(364\) 17.5160 0.918085
\(365\) −3.60169 −0.188521
\(366\) −2.62425 −0.137172
\(367\) 37.3721 1.95081 0.975405 0.220421i \(-0.0707432\pi\)
0.975405 + 0.220421i \(0.0707432\pi\)
\(368\) −20.8359 −1.08615
\(369\) −0.461254 −0.0240119
\(370\) 3.97566 0.206685
\(371\) 6.14333 0.318946
\(372\) 3.13221 0.162397
\(373\) −18.3417 −0.949695 −0.474848 0.880068i \(-0.657497\pi\)
−0.474848 + 0.880068i \(0.657497\pi\)
\(374\) −2.27307 −0.117538
\(375\) −2.32099 −0.119855
\(376\) 82.0879 4.23336
\(377\) −20.9912 −1.08110
\(378\) 2.71756 0.139776
\(379\) −8.69436 −0.446599 −0.223300 0.974750i \(-0.571683\pi\)
−0.223300 + 0.974750i \(0.571683\pi\)
\(380\) −7.77606 −0.398904
\(381\) −6.27466 −0.321460
\(382\) −17.3179 −0.886063
\(383\) −19.6485 −1.00399 −0.501997 0.864870i \(-0.667401\pi\)
−0.501997 + 0.864870i \(0.667401\pi\)
\(384\) 2.52221 0.128711
\(385\) −0.241668 −0.0123166
\(386\) −48.9321 −2.49058
\(387\) −5.84824 −0.297283
\(388\) −1.62279 −0.0823847
\(389\) 27.9899 1.41914 0.709571 0.704633i \(-0.248889\pi\)
0.709571 + 0.704633i \(0.248889\pi\)
\(390\) 2.11980 0.107340
\(391\) −1.78578 −0.0903107
\(392\) −44.9040 −2.26799
\(393\) 18.4332 0.929833
\(394\) 10.7567 0.541915
\(395\) 2.90600 0.146217
\(396\) −4.88668 −0.245565
\(397\) 5.52183 0.277133 0.138566 0.990353i \(-0.455751\pi\)
0.138566 + 0.990353i \(0.455751\pi\)
\(398\) 34.2858 1.71859
\(399\) −7.06115 −0.353500
\(400\) −49.9812 −2.49906
\(401\) 14.3933 0.718768 0.359384 0.933190i \(-0.382987\pi\)
0.359384 + 0.933190i \(0.382987\pi\)
\(402\) 14.4209 0.719249
\(403\) 2.21861 0.110517
\(404\) 60.3945 3.00474
\(405\) 0.233370 0.0115962
\(406\) −16.4806 −0.817917
\(407\) −6.49171 −0.321782
\(408\) 6.56165 0.324850
\(409\) 17.9215 0.886160 0.443080 0.896482i \(-0.353886\pi\)
0.443080 + 0.896482i \(0.353886\pi\)
\(410\) −0.282481 −0.0139508
\(411\) −15.5912 −0.769056
\(412\) 70.7378 3.48500
\(413\) −14.3834 −0.707761
\(414\) −5.41033 −0.265903
\(415\) 1.08005 0.0530177
\(416\) 39.3580 1.92969
\(417\) −18.1623 −0.889414
\(418\) 17.8939 0.875221
\(419\) 26.7628 1.30745 0.653724 0.756733i \(-0.273206\pi\)
0.653724 + 0.756733i \(0.273206\pi\)
\(420\) 1.18096 0.0576248
\(421\) −27.0091 −1.31634 −0.658172 0.752867i \(-0.728670\pi\)
−0.658172 + 0.752867i \(0.728670\pi\)
\(422\) 66.2730 3.22612
\(423\) 10.8361 0.526872
\(424\) 44.9400 2.18248
\(425\) −4.28373 −0.207791
\(426\) 28.7932 1.39504
\(427\) −1.03556 −0.0501142
\(428\) −10.4471 −0.504978
\(429\) −3.46134 −0.167115
\(430\) −3.58159 −0.172719
\(431\) −6.61296 −0.318535 −0.159267 0.987235i \(-0.550913\pi\)
−0.159267 + 0.987235i \(0.550913\pi\)
\(432\) 10.1063 0.486240
\(433\) 6.98921 0.335880 0.167940 0.985797i \(-0.446289\pi\)
0.167940 + 0.985797i \(0.446289\pi\)
\(434\) 1.74187 0.0836125
\(435\) −1.41526 −0.0678567
\(436\) −8.13075 −0.389392
\(437\) 14.0579 0.672479
\(438\) −40.5011 −1.93522
\(439\) 4.82441 0.230256 0.115128 0.993351i \(-0.463272\pi\)
0.115128 + 0.993351i \(0.463272\pi\)
\(440\) −1.76787 −0.0842797
\(441\) −5.92762 −0.282268
\(442\) 7.86789 0.374237
\(443\) −7.14384 −0.339414 −0.169707 0.985495i \(-0.554282\pi\)
−0.169707 + 0.985495i \(0.554282\pi\)
\(444\) 31.7229 1.50550
\(445\) 1.10095 0.0521900
\(446\) 0.485298 0.0229795
\(447\) 8.50275 0.402166
\(448\) 9.96933 0.471007
\(449\) −2.81996 −0.133082 −0.0665410 0.997784i \(-0.521196\pi\)
−0.0665410 + 0.997784i \(0.521196\pi\)
\(450\) −12.9783 −0.611804
\(451\) 0.461254 0.0217196
\(452\) −81.5193 −3.83435
\(453\) −23.5234 −1.10522
\(454\) −28.4556 −1.33549
\(455\) 0.836497 0.0392156
\(456\) −51.6541 −2.41893
\(457\) −21.3915 −1.00065 −0.500326 0.865837i \(-0.666786\pi\)
−0.500326 + 0.865837i \(0.666786\pi\)
\(458\) 25.4130 1.18747
\(459\) 0.866180 0.0404298
\(460\) −2.35114 −0.109622
\(461\) 2.40884 0.112191 0.0560954 0.998425i \(-0.482135\pi\)
0.0560954 + 0.998425i \(0.482135\pi\)
\(462\) −2.71756 −0.126433
\(463\) −27.7917 −1.29159 −0.645794 0.763512i \(-0.723474\pi\)
−0.645794 + 0.763512i \(0.723474\pi\)
\(464\) −61.2894 −2.84529
\(465\) 0.149583 0.00693673
\(466\) −28.5832 −1.32409
\(467\) −6.65416 −0.307918 −0.153959 0.988077i \(-0.549202\pi\)
−0.153959 + 0.988077i \(0.549202\pi\)
\(468\) 16.9145 0.781873
\(469\) 5.69065 0.262770
\(470\) 6.63628 0.306109
\(471\) −15.5352 −0.715825
\(472\) −105.218 −4.84307
\(473\) 5.84824 0.268903
\(474\) 32.6780 1.50095
\(475\) 33.7221 1.54727
\(476\) 4.38326 0.200906
\(477\) 5.93238 0.271625
\(478\) 19.8474 0.907799
\(479\) 9.11561 0.416503 0.208251 0.978075i \(-0.433223\pi\)
0.208251 + 0.978075i \(0.433223\pi\)
\(480\) 2.65359 0.121119
\(481\) 22.4700 1.02455
\(482\) 31.7538 1.44635
\(483\) −2.13498 −0.0971449
\(484\) 4.88668 0.222122
\(485\) −0.0774984 −0.00351902
\(486\) 2.62425 0.119038
\(487\) −4.97655 −0.225509 −0.112755 0.993623i \(-0.535967\pi\)
−0.112755 + 0.993623i \(0.535967\pi\)
\(488\) −7.57538 −0.342921
\(489\) 3.93029 0.177734
\(490\) −3.63020 −0.163996
\(491\) −0.371810 −0.0167795 −0.00838977 0.999965i \(-0.502671\pi\)
−0.00838977 + 0.999965i \(0.502671\pi\)
\(492\) −2.25400 −0.101618
\(493\) −5.25292 −0.236580
\(494\) −61.9371 −2.78668
\(495\) −0.233370 −0.0104892
\(496\) 6.47783 0.290863
\(497\) 11.3621 0.509661
\(498\) 12.1452 0.544240
\(499\) −24.9013 −1.11474 −0.557368 0.830266i \(-0.688189\pi\)
−0.557368 + 0.830266i \(0.688189\pi\)
\(500\) −11.3419 −0.507227
\(501\) −1.83634 −0.0820417
\(502\) −38.1067 −1.70078
\(503\) −4.42508 −0.197305 −0.0986524 0.995122i \(-0.531453\pi\)
−0.0986524 + 0.995122i \(0.531453\pi\)
\(504\) 7.84475 0.349433
\(505\) 2.88422 0.128346
\(506\) 5.41033 0.240519
\(507\) −1.01910 −0.0452596
\(508\) −30.6623 −1.36042
\(509\) 32.5147 1.44119 0.720595 0.693356i \(-0.243869\pi\)
0.720595 + 0.693356i \(0.243869\pi\)
\(510\) 0.530467 0.0234895
\(511\) −15.9822 −0.707011
\(512\) −38.2021 −1.68831
\(513\) −6.81869 −0.301052
\(514\) −5.13189 −0.226358
\(515\) 3.37818 0.148860
\(516\) −28.5785 −1.25810
\(517\) −10.8361 −0.476573
\(518\) 17.6416 0.775129
\(519\) 10.3886 0.456011
\(520\) 6.11919 0.268344
\(521\) 39.4030 1.72628 0.863139 0.504967i \(-0.168495\pi\)
0.863139 + 0.504967i \(0.168495\pi\)
\(522\) −15.9147 −0.696566
\(523\) −6.69006 −0.292536 −0.146268 0.989245i \(-0.546726\pi\)
−0.146268 + 0.989245i \(0.546726\pi\)
\(524\) 90.0773 3.93505
\(525\) −5.12140 −0.223516
\(526\) −14.1030 −0.614919
\(527\) 0.555194 0.0241846
\(528\) −10.1063 −0.439821
\(529\) −18.7495 −0.815197
\(530\) 3.63311 0.157812
\(531\) −13.8895 −0.602754
\(532\) −34.5056 −1.49601
\(533\) −1.59656 −0.0691546
\(534\) 12.3802 0.535743
\(535\) −0.498913 −0.0215699
\(536\) 41.6286 1.79808
\(537\) 17.4905 0.754773
\(538\) −39.3591 −1.69689
\(539\) 5.92762 0.255321
\(540\) 1.14041 0.0490752
\(541\) −32.7099 −1.40631 −0.703154 0.711038i \(-0.748226\pi\)
−0.703154 + 0.711038i \(0.748226\pi\)
\(542\) 52.7709 2.26670
\(543\) −14.8136 −0.635711
\(544\) 9.84911 0.422277
\(545\) −0.388295 −0.0166327
\(546\) 9.40643 0.402558
\(547\) −22.7993 −0.974830 −0.487415 0.873171i \(-0.662060\pi\)
−0.487415 + 0.873171i \(0.662060\pi\)
\(548\) −76.1892 −3.25464
\(549\) −1.00000 −0.0426790
\(550\) 12.9783 0.553398
\(551\) 41.3517 1.76164
\(552\) −15.6179 −0.664743
\(553\) 12.8951 0.548356
\(554\) −25.1058 −1.06664
\(555\) 1.51497 0.0643068
\(556\) −88.7536 −3.76399
\(557\) 32.5746 1.38023 0.690115 0.723700i \(-0.257560\pi\)
0.690115 + 0.723700i \(0.257560\pi\)
\(558\) 1.68206 0.0712073
\(559\) −20.2428 −0.856179
\(560\) 2.44238 0.103209
\(561\) −0.866180 −0.0365702
\(562\) 72.0037 3.03729
\(563\) −4.76382 −0.200771 −0.100386 0.994949i \(-0.532008\pi\)
−0.100386 + 0.994949i \(0.532008\pi\)
\(564\) 52.9528 2.22972
\(565\) −3.89306 −0.163782
\(566\) 55.3416 2.32618
\(567\) 1.03556 0.0434894
\(568\) 83.1169 3.48751
\(569\) −36.9691 −1.54982 −0.774912 0.632069i \(-0.782206\pi\)
−0.774912 + 0.632069i \(0.782206\pi\)
\(570\) −4.17591 −0.174909
\(571\) −18.7090 −0.782949 −0.391474 0.920189i \(-0.628035\pi\)
−0.391474 + 0.920189i \(0.628035\pi\)
\(572\) −16.9145 −0.707231
\(573\) −6.59920 −0.275685
\(574\) −1.25349 −0.0523195
\(575\) 10.1961 0.425205
\(576\) 9.62701 0.401125
\(577\) 34.5793 1.43956 0.719779 0.694204i \(-0.244243\pi\)
0.719779 + 0.694204i \(0.244243\pi\)
\(578\) −42.6433 −1.77373
\(579\) −18.6461 −0.774907
\(580\) −6.91594 −0.287169
\(581\) 4.79264 0.198832
\(582\) −0.871471 −0.0361236
\(583\) −5.93238 −0.245694
\(584\) −116.914 −4.83793
\(585\) 0.807774 0.0333973
\(586\) 28.5547 1.17958
\(587\) 40.4240 1.66848 0.834238 0.551405i \(-0.185908\pi\)
0.834238 + 0.551405i \(0.185908\pi\)
\(588\) −28.9664 −1.19455
\(589\) −4.37056 −0.180086
\(590\) −8.50623 −0.350196
\(591\) 4.09897 0.168609
\(592\) 65.6072 2.69644
\(593\) −19.3560 −0.794854 −0.397427 0.917634i \(-0.630097\pi\)
−0.397427 + 0.917634i \(0.630097\pi\)
\(594\) −2.62425 −0.107674
\(595\) 0.209328 0.00858162
\(596\) 41.5503 1.70197
\(597\) 13.0650 0.534714
\(598\) −18.7270 −0.765805
\(599\) 15.4274 0.630348 0.315174 0.949034i \(-0.397937\pi\)
0.315174 + 0.949034i \(0.397937\pi\)
\(600\) −37.4643 −1.52948
\(601\) 15.3065 0.624364 0.312182 0.950022i \(-0.398940\pi\)
0.312182 + 0.950022i \(0.398940\pi\)
\(602\) −15.8930 −0.647750
\(603\) 5.49525 0.223784
\(604\) −114.951 −4.67730
\(605\) 0.233370 0.00948784
\(606\) 32.4331 1.31750
\(607\) −40.9891 −1.66370 −0.831848 0.555003i \(-0.812717\pi\)
−0.831848 + 0.555003i \(0.812717\pi\)
\(608\) −77.5335 −3.14440
\(609\) −6.28011 −0.254483
\(610\) −0.612421 −0.0247962
\(611\) 37.5076 1.51740
\(612\) 4.23275 0.171099
\(613\) 7.15889 0.289145 0.144573 0.989494i \(-0.453819\pi\)
0.144573 + 0.989494i \(0.453819\pi\)
\(614\) −19.0730 −0.769725
\(615\) −0.107643 −0.00434057
\(616\) −7.84475 −0.316074
\(617\) 11.2849 0.454315 0.227157 0.973858i \(-0.427057\pi\)
0.227157 + 0.973858i \(0.427057\pi\)
\(618\) 37.9876 1.52809
\(619\) 5.57451 0.224058 0.112029 0.993705i \(-0.464265\pi\)
0.112029 + 0.993705i \(0.464265\pi\)
\(620\) 0.730963 0.0293562
\(621\) −2.06167 −0.0827319
\(622\) −0.797809 −0.0319892
\(623\) 4.88536 0.195728
\(624\) 34.9814 1.40038
\(625\) 24.1860 0.967442
\(626\) 80.1033 3.20157
\(627\) 6.81869 0.272312
\(628\) −75.9157 −3.02937
\(629\) 5.62299 0.224203
\(630\) 0.634198 0.0252670
\(631\) −28.2662 −1.12526 −0.562630 0.826709i \(-0.690210\pi\)
−0.562630 + 0.826709i \(0.690210\pi\)
\(632\) 94.3311 3.75229
\(633\) 25.2541 1.00376
\(634\) 2.38648 0.0947791
\(635\) −1.46432 −0.0581096
\(636\) 28.9897 1.14951
\(637\) −20.5175 −0.812934
\(638\) 15.9147 0.630067
\(639\) 10.9720 0.434045
\(640\) 0.588607 0.0232667
\(641\) −23.9009 −0.944028 −0.472014 0.881591i \(-0.656473\pi\)
−0.472014 + 0.881591i \(0.656473\pi\)
\(642\) −5.61028 −0.221420
\(643\) 24.9816 0.985179 0.492589 0.870262i \(-0.336050\pi\)
0.492589 + 0.870262i \(0.336050\pi\)
\(644\) −10.4330 −0.411117
\(645\) −1.36480 −0.0537391
\(646\) −15.4994 −0.609815
\(647\) 24.5879 0.966650 0.483325 0.875441i \(-0.339429\pi\)
0.483325 + 0.875441i \(0.339429\pi\)
\(648\) 7.57538 0.297589
\(649\) 13.8895 0.545211
\(650\) −44.9225 −1.76200
\(651\) 0.663760 0.0260148
\(652\) 19.2061 0.752169
\(653\) 13.5911 0.531861 0.265931 0.963992i \(-0.414321\pi\)
0.265931 + 0.963992i \(0.414321\pi\)
\(654\) −4.36638 −0.170739
\(655\) 4.30176 0.168084
\(656\) −4.66157 −0.182004
\(657\) −15.4334 −0.602114
\(658\) 29.4479 1.14800
\(659\) 12.8127 0.499114 0.249557 0.968360i \(-0.419715\pi\)
0.249557 + 0.968360i \(0.419715\pi\)
\(660\) −1.14041 −0.0443902
\(661\) 7.66063 0.297964 0.148982 0.988840i \(-0.452400\pi\)
0.148982 + 0.988840i \(0.452400\pi\)
\(662\) 54.5307 2.11939
\(663\) 2.99815 0.116438
\(664\) 35.0594 1.36057
\(665\) −1.64786 −0.0639013
\(666\) 17.0359 0.660126
\(667\) 12.5029 0.484115
\(668\) −8.97362 −0.347200
\(669\) 0.184928 0.00714975
\(670\) 3.36541 0.130017
\(671\) 1.00000 0.0386046
\(672\) 11.7751 0.454233
\(673\) −15.7132 −0.605698 −0.302849 0.953038i \(-0.597938\pi\)
−0.302849 + 0.953038i \(0.597938\pi\)
\(674\) −20.4008 −0.785809
\(675\) −4.94554 −0.190354
\(676\) −4.98000 −0.191539
\(677\) −19.2713 −0.740655 −0.370327 0.928901i \(-0.620755\pi\)
−0.370327 + 0.928901i \(0.620755\pi\)
\(678\) −43.7775 −1.68127
\(679\) −0.343892 −0.0131974
\(680\) 1.53129 0.0587223
\(681\) −10.8433 −0.415518
\(682\) −1.68206 −0.0644094
\(683\) 45.1825 1.72886 0.864430 0.502753i \(-0.167680\pi\)
0.864430 + 0.502753i \(0.167680\pi\)
\(684\) −33.3208 −1.27405
\(685\) −3.63851 −0.139020
\(686\) −35.1316 −1.34133
\(687\) 9.68390 0.369464
\(688\) −59.1042 −2.25333
\(689\) 20.5340 0.782283
\(690\) −1.26261 −0.0480667
\(691\) 0.659628 0.0250934 0.0125467 0.999921i \(-0.496006\pi\)
0.0125467 + 0.999921i \(0.496006\pi\)
\(692\) 50.7660 1.92983
\(693\) −1.03556 −0.0393376
\(694\) 33.7596 1.28150
\(695\) −4.23854 −0.160777
\(696\) −45.9406 −1.74137
\(697\) −0.399529 −0.0151332
\(698\) 37.7318 1.42817
\(699\) −10.8919 −0.411971
\(700\) −25.0266 −0.945918
\(701\) 11.8645 0.448116 0.224058 0.974576i \(-0.428070\pi\)
0.224058 + 0.974576i \(0.428070\pi\)
\(702\) 9.08343 0.342832
\(703\) −44.2649 −1.66948
\(704\) −9.62701 −0.362832
\(705\) 2.52883 0.0952413
\(706\) 3.72399 0.140154
\(707\) 12.7985 0.481336
\(708\) −67.8737 −2.55085
\(709\) 7.38312 0.277279 0.138640 0.990343i \(-0.455727\pi\)
0.138640 + 0.990343i \(0.455727\pi\)
\(710\) 6.71947 0.252177
\(711\) 12.4523 0.466999
\(712\) 35.7377 1.33933
\(713\) −1.32146 −0.0494892
\(714\) 2.35390 0.0880925
\(715\) −0.807774 −0.0302090
\(716\) 85.4708 3.19419
\(717\) 7.56308 0.282448
\(718\) −2.27556 −0.0849230
\(719\) 12.2824 0.458057 0.229028 0.973420i \(-0.426445\pi\)
0.229028 + 0.973420i \(0.426445\pi\)
\(720\) 2.35851 0.0878965
\(721\) 14.9904 0.558270
\(722\) 72.1524 2.68523
\(723\) 12.1001 0.450009
\(724\) −72.3893 −2.69033
\(725\) 29.9920 1.11388
\(726\) 2.62425 0.0973950
\(727\) 8.68056 0.321944 0.160972 0.986959i \(-0.448537\pi\)
0.160972 + 0.986959i \(0.448537\pi\)
\(728\) 27.1534 1.00637
\(729\) 1.00000 0.0370370
\(730\) −9.45174 −0.349824
\(731\) −5.06564 −0.187359
\(732\) −4.88668 −0.180617
\(733\) 11.5763 0.427582 0.213791 0.976879i \(-0.431419\pi\)
0.213791 + 0.976879i \(0.431419\pi\)
\(734\) 98.0738 3.61997
\(735\) −1.38333 −0.0510248
\(736\) −23.4427 −0.864109
\(737\) −5.49525 −0.202420
\(738\) −1.21044 −0.0445571
\(739\) −6.65608 −0.244848 −0.122424 0.992478i \(-0.539067\pi\)
−0.122424 + 0.992478i \(0.539067\pi\)
\(740\) 7.40317 0.272146
\(741\) −23.6018 −0.867035
\(742\) 16.1216 0.591843
\(743\) 0.134338 0.00492837 0.00246418 0.999997i \(-0.499216\pi\)
0.00246418 + 0.999997i \(0.499216\pi\)
\(744\) 4.85558 0.178014
\(745\) 1.98429 0.0726986
\(746\) −48.1331 −1.76228
\(747\) 4.62807 0.169332
\(748\) −4.23275 −0.154765
\(749\) −2.21388 −0.0808935
\(750\) −6.09085 −0.222407
\(751\) 28.5757 1.04274 0.521372 0.853330i \(-0.325420\pi\)
0.521372 + 0.853330i \(0.325420\pi\)
\(752\) 109.514 3.99355
\(753\) −14.5210 −0.529174
\(754\) −55.0861 −2.00612
\(755\) −5.48965 −0.199789
\(756\) 5.06045 0.184047
\(757\) 23.9955 0.872131 0.436065 0.899915i \(-0.356372\pi\)
0.436065 + 0.899915i \(0.356372\pi\)
\(758\) −22.8162 −0.828721
\(759\) 2.06167 0.0748339
\(760\) −12.0545 −0.437263
\(761\) −15.1346 −0.548628 −0.274314 0.961640i \(-0.588451\pi\)
−0.274314 + 0.961640i \(0.588451\pi\)
\(762\) −16.4663 −0.596510
\(763\) −1.72302 −0.0623776
\(764\) −32.2482 −1.16670
\(765\) 0.202140 0.00730840
\(766\) −51.5627 −1.86303
\(767\) −48.0764 −1.73594
\(768\) −12.6351 −0.455931
\(769\) −50.1531 −1.80857 −0.904283 0.426934i \(-0.859594\pi\)
−0.904283 + 0.426934i \(0.859594\pi\)
\(770\) −0.634198 −0.0228549
\(771\) −1.95557 −0.0704280
\(772\) −91.1178 −3.27940
\(773\) 4.15446 0.149426 0.0747128 0.997205i \(-0.476196\pi\)
0.0747128 + 0.997205i \(0.476196\pi\)
\(774\) −15.3473 −0.551646
\(775\) −3.16993 −0.113867
\(776\) −2.51566 −0.0903070
\(777\) 6.72254 0.241170
\(778\) 73.4524 2.63340
\(779\) 3.14514 0.112686
\(780\) 3.94733 0.141337
\(781\) −10.9720 −0.392608
\(782\) −4.68633 −0.167583
\(783\) −6.06446 −0.216726
\(784\) −59.9064 −2.13951
\(785\) −3.62545 −0.129398
\(786\) 48.3734 1.72542
\(787\) 26.2406 0.935377 0.467689 0.883893i \(-0.345087\pi\)
0.467689 + 0.883893i \(0.345087\pi\)
\(788\) 20.0304 0.713552
\(789\) −5.37410 −0.191323
\(790\) 7.62606 0.271323
\(791\) −17.2751 −0.614232
\(792\) −7.57538 −0.269179
\(793\) −3.46134 −0.122916
\(794\) 14.4907 0.514254
\(795\) 1.38444 0.0491010
\(796\) 63.8444 2.26291
\(797\) 2.88176 0.102077 0.0510386 0.998697i \(-0.483747\pi\)
0.0510386 + 0.998697i \(0.483747\pi\)
\(798\) −18.5302 −0.655963
\(799\) 9.38606 0.332055
\(800\) −56.2344 −1.98819
\(801\) 4.71761 0.166689
\(802\) 37.7716 1.33376
\(803\) 15.4334 0.544633
\(804\) 26.8536 0.947052
\(805\) −0.498240 −0.0175607
\(806\) 5.82219 0.205078
\(807\) −14.9982 −0.527963
\(808\) 93.6241 3.29368
\(809\) −25.0212 −0.879697 −0.439849 0.898072i \(-0.644968\pi\)
−0.439849 + 0.898072i \(0.644968\pi\)
\(810\) 0.612421 0.0215183
\(811\) −24.9051 −0.874538 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(812\) −30.6889 −1.07697
\(813\) 20.1090 0.705252
\(814\) −17.0359 −0.597106
\(815\) 0.917212 0.0321285
\(816\) 8.75389 0.306448
\(817\) 39.8773 1.39513
\(818\) 47.0304 1.64438
\(819\) 3.58443 0.125250
\(820\) −0.526016 −0.0183693
\(821\) 18.3839 0.641603 0.320802 0.947146i \(-0.396048\pi\)
0.320802 + 0.947146i \(0.396048\pi\)
\(822\) −40.9151 −1.42708
\(823\) 40.4252 1.40913 0.704567 0.709638i \(-0.251141\pi\)
0.704567 + 0.709638i \(0.251141\pi\)
\(824\) 109.658 3.82013
\(825\) 4.94554 0.172182
\(826\) −37.7457 −1.31334
\(827\) −17.3687 −0.603969 −0.301984 0.953313i \(-0.597649\pi\)
−0.301984 + 0.953313i \(0.597649\pi\)
\(828\) −10.0747 −0.350121
\(829\) 28.4250 0.987241 0.493620 0.869678i \(-0.335673\pi\)
0.493620 + 0.869678i \(0.335673\pi\)
\(830\) 2.83433 0.0983809
\(831\) −9.56683 −0.331870
\(832\) 33.3224 1.15525
\(833\) −5.13439 −0.177896
\(834\) −47.6625 −1.65042
\(835\) −0.428547 −0.0148305
\(836\) 33.3208 1.15242
\(837\) 0.640968 0.0221551
\(838\) 70.2323 2.42613
\(839\) 48.0557 1.65907 0.829534 0.558456i \(-0.188606\pi\)
0.829534 + 0.558456i \(0.188606\pi\)
\(840\) 1.83073 0.0631661
\(841\) 7.77770 0.268197
\(842\) −70.8787 −2.44264
\(843\) 27.4378 0.945009
\(844\) 123.409 4.24790
\(845\) −0.237826 −0.00818148
\(846\) 28.4368 0.977676
\(847\) 1.03556 0.0355822
\(848\) 59.9545 2.05885
\(849\) 21.0886 0.723757
\(850\) −11.2416 −0.385583
\(851\) −13.3838 −0.458789
\(852\) 53.6166 1.83687
\(853\) −43.8113 −1.50007 −0.750035 0.661398i \(-0.769963\pi\)
−0.750035 + 0.661398i \(0.769963\pi\)
\(854\) −2.71756 −0.0929931
\(855\) −1.59128 −0.0544205
\(856\) −16.1951 −0.553538
\(857\) −21.6329 −0.738967 −0.369484 0.929237i \(-0.620465\pi\)
−0.369484 + 0.929237i \(0.620465\pi\)
\(858\) −9.08343 −0.310103
\(859\) −47.8429 −1.63238 −0.816190 0.577784i \(-0.803918\pi\)
−0.816190 + 0.577784i \(0.803918\pi\)
\(860\) −6.66937 −0.227424
\(861\) −0.477655 −0.0162784
\(862\) −17.3540 −0.591081
\(863\) −46.6138 −1.58675 −0.793377 0.608731i \(-0.791679\pi\)
−0.793377 + 0.608731i \(0.791679\pi\)
\(864\) 11.3707 0.386840
\(865\) 2.42440 0.0824320
\(866\) 18.3414 0.623267
\(867\) −16.2497 −0.551870
\(868\) 3.24359 0.110094
\(869\) −12.4523 −0.422416
\(870\) −3.71400 −0.125916
\(871\) 19.0210 0.644500
\(872\) −12.6044 −0.426838
\(873\) −0.332084 −0.0112393
\(874\) 36.8914 1.24787
\(875\) −2.40352 −0.0812538
\(876\) −75.4182 −2.54814
\(877\) −13.1502 −0.444049 −0.222025 0.975041i \(-0.571267\pi\)
−0.222025 + 0.975041i \(0.571267\pi\)
\(878\) 12.6605 0.427270
\(879\) 10.8811 0.367010
\(880\) −2.35851 −0.0795054
\(881\) 14.2325 0.479504 0.239752 0.970834i \(-0.422934\pi\)
0.239752 + 0.970834i \(0.422934\pi\)
\(882\) −15.5555 −0.523783
\(883\) −19.9158 −0.670221 −0.335111 0.942179i \(-0.608774\pi\)
−0.335111 + 0.942179i \(0.608774\pi\)
\(884\) 14.6510 0.492767
\(885\) −3.24140 −0.108958
\(886\) −18.7472 −0.629825
\(887\) 27.9231 0.937566 0.468783 0.883313i \(-0.344693\pi\)
0.468783 + 0.883313i \(0.344693\pi\)
\(888\) 49.1771 1.65028
\(889\) −6.49777 −0.217928
\(890\) 2.88916 0.0968450
\(891\) −1.00000 −0.0335013
\(892\) 0.903687 0.0302577
\(893\) −73.8883 −2.47258
\(894\) 22.3133 0.746270
\(895\) 4.08177 0.136438
\(896\) 2.61189 0.0872572
\(897\) −7.13615 −0.238269
\(898\) −7.40027 −0.246950
\(899\) −3.88713 −0.129643
\(900\) −24.1673 −0.805576
\(901\) 5.13851 0.171189
\(902\) 1.21044 0.0403034
\(903\) −6.05620 −0.201538
\(904\) −126.372 −4.20307
\(905\) −3.45704 −0.114916
\(906\) −61.7312 −2.05088
\(907\) 37.3959 1.24171 0.620856 0.783925i \(-0.286785\pi\)
0.620856 + 0.783925i \(0.286785\pi\)
\(908\) −52.9880 −1.75847
\(909\) 12.3590 0.409922
\(910\) 2.19518 0.0727694
\(911\) 52.2628 1.73154 0.865771 0.500440i \(-0.166828\pi\)
0.865771 + 0.500440i \(0.166828\pi\)
\(912\) −68.9118 −2.28190
\(913\) −4.62807 −0.153167
\(914\) −56.1366 −1.85683
\(915\) −0.233370 −0.00771497
\(916\) 47.3222 1.56357
\(917\) 19.0887 0.630364
\(918\) 2.27307 0.0750226
\(919\) −34.2063 −1.12836 −0.564182 0.825651i \(-0.690808\pi\)
−0.564182 + 0.825651i \(0.690808\pi\)
\(920\) −3.64475 −0.120164
\(921\) −7.26799 −0.239488
\(922\) 6.32139 0.208184
\(923\) 37.9778 1.25005
\(924\) −5.06045 −0.166477
\(925\) −32.1050 −1.05561
\(926\) −72.9323 −2.39670
\(927\) 14.4756 0.475442
\(928\) −68.9574 −2.26364
\(929\) −26.6684 −0.874963 −0.437482 0.899227i \(-0.644130\pi\)
−0.437482 + 0.899227i \(0.644130\pi\)
\(930\) 0.392542 0.0128720
\(931\) 40.4186 1.32467
\(932\) −53.2255 −1.74346
\(933\) −0.304014 −0.00995298
\(934\) −17.4622 −0.571380
\(935\) −0.202140 −0.00661070
\(936\) 26.2210 0.857060
\(937\) 11.0117 0.359736 0.179868 0.983691i \(-0.442433\pi\)
0.179868 + 0.983691i \(0.442433\pi\)
\(938\) 14.9337 0.487602
\(939\) 30.5243 0.996122
\(940\) 12.3576 0.403060
\(941\) 41.5013 1.35290 0.676451 0.736487i \(-0.263517\pi\)
0.676451 + 0.736487i \(0.263517\pi\)
\(942\) −40.7683 −1.32830
\(943\) 0.950952 0.0309673
\(944\) −140.372 −4.56872
\(945\) 0.241668 0.00786147
\(946\) 15.3473 0.498982
\(947\) −6.86498 −0.223082 −0.111541 0.993760i \(-0.535579\pi\)
−0.111541 + 0.993760i \(0.535579\pi\)
\(948\) 60.8506 1.97634
\(949\) −53.4203 −1.73410
\(950\) 88.4951 2.87116
\(951\) 0.909394 0.0294891
\(952\) 6.79497 0.220226
\(953\) 50.1428 1.62429 0.812143 0.583458i \(-0.198301\pi\)
0.812143 + 0.583458i \(0.198301\pi\)
\(954\) 15.5680 0.504034
\(955\) −1.54005 −0.0498350
\(956\) 36.9584 1.19532
\(957\) 6.06446 0.196036
\(958\) 23.9216 0.772873
\(959\) −16.1456 −0.521368
\(960\) 2.24665 0.0725105
\(961\) −30.5892 −0.986747
\(962\) 58.9670 1.90117
\(963\) −2.13786 −0.0688916
\(964\) 59.1296 1.90444
\(965\) −4.35145 −0.140078
\(966\) −5.60272 −0.180265
\(967\) −38.7157 −1.24501 −0.622506 0.782615i \(-0.713885\pi\)
−0.622506 + 0.782615i \(0.713885\pi\)
\(968\) 7.57538 0.243482
\(969\) −5.90621 −0.189735
\(970\) −0.203375 −0.00652998
\(971\) −8.40355 −0.269683 −0.134841 0.990867i \(-0.543053\pi\)
−0.134841 + 0.990867i \(0.543053\pi\)
\(972\) 4.88668 0.156740
\(973\) −18.8082 −0.602962
\(974\) −13.0597 −0.418460
\(975\) −17.1182 −0.548222
\(976\) −10.1063 −0.323495
\(977\) −32.3670 −1.03551 −0.517756 0.855528i \(-0.673233\pi\)
−0.517756 + 0.855528i \(0.673233\pi\)
\(978\) 10.3141 0.329808
\(979\) −4.71761 −0.150776
\(980\) −6.75989 −0.215937
\(981\) −1.66386 −0.0531229
\(982\) −0.975722 −0.0311365
\(983\) 35.2337 1.12378 0.561890 0.827212i \(-0.310075\pi\)
0.561890 + 0.827212i \(0.310075\pi\)
\(984\) −3.49417 −0.111390
\(985\) 0.956576 0.0304790
\(986\) −13.7850 −0.439003
\(987\) 11.2215 0.357183
\(988\) −115.335 −3.66928
\(989\) 12.0571 0.383395
\(990\) −0.612421 −0.0194640
\(991\) 61.5964 1.95667 0.978336 0.207022i \(-0.0663771\pi\)
0.978336 + 0.207022i \(0.0663771\pi\)
\(992\) 7.28828 0.231403
\(993\) 20.7795 0.659418
\(994\) 29.8170 0.945739
\(995\) 3.04897 0.0966589
\(996\) 22.6159 0.716613
\(997\) 22.2950 0.706090 0.353045 0.935606i \(-0.385146\pi\)
0.353045 + 0.935606i \(0.385146\pi\)
\(998\) −65.3472 −2.06853
\(999\) 6.49171 0.205388
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.f.1.13 13
3.2 odd 2 6039.2.a.g.1.1 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.f.1.13 13 1.1 even 1 trivial
6039.2.a.g.1.1 13 3.2 odd 2