Properties

Label 2166.2.a.w.1.3
Level $2166$
Weight $2$
Character 2166.1
Self dual yes
Analytic conductor $17.296$
Analytic rank $0$
Dimension $4$
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] = [2166,2,Mod(1,2166)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2166, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2166.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.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: \(17.2955970778\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) 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 1.3
Root \(1.17557\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.17557 q^{5} -1.00000 q^{6} +0.381966 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.17557 q^{5} -1.00000 q^{6} +0.381966 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.17557 q^{10} +2.66605 q^{11} +1.00000 q^{12} +7.04029 q^{13} -0.381966 q^{14} +3.17557 q^{15} +1.00000 q^{16} +4.68915 q^{17} -1.00000 q^{18} +3.17557 q^{20} +0.381966 q^{21} -2.66605 q^{22} +1.43184 q^{23} -1.00000 q^{24} +5.08425 q^{25} -7.04029 q^{26} +1.00000 q^{27} +0.381966 q^{28} -8.20524 q^{29} -3.17557 q^{30} -8.62750 q^{31} -1.00000 q^{32} +2.66605 q^{33} -4.68915 q^{34} +1.21296 q^{35} +1.00000 q^{36} +2.70228 q^{37} +7.04029 q^{39} -3.17557 q^{40} -3.25731 q^{41} -0.381966 q^{42} -8.72353 q^{43} +2.66605 q^{44} +3.17557 q^{45} -1.43184 q^{46} +5.35706 q^{47} +1.00000 q^{48} -6.85410 q^{49} -5.08425 q^{50} +4.68915 q^{51} +7.04029 q^{52} +8.20524 q^{53} -1.00000 q^{54} +8.46621 q^{55} -0.381966 q^{56} +8.20524 q^{58} -11.8137 q^{59} +3.17557 q^{60} -5.27044 q^{61} +8.62750 q^{62} +0.381966 q^{63} +1.00000 q^{64} +22.3570 q^{65} -2.66605 q^{66} +12.2147 q^{67} +4.68915 q^{68} +1.43184 q^{69} -1.21296 q^{70} -15.0996 q^{71} -1.00000 q^{72} -3.84818 q^{73} -2.70228 q^{74} +5.08425 q^{75} +1.01834 q^{77} -7.04029 q^{78} -1.39560 q^{79} +3.17557 q^{80} +1.00000 q^{81} +3.25731 q^{82} -4.31562 q^{83} +0.381966 q^{84} +14.8907 q^{85} +8.72353 q^{86} -8.20524 q^{87} -2.66605 q^{88} +7.80423 q^{89} -3.17557 q^{90} +2.68915 q^{91} +1.43184 q^{92} -8.62750 q^{93} -5.35706 q^{94} -1.00000 q^{96} -12.7448 q^{97} +6.85410 q^{98} +2.66605 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 8 q^{10} + 12 q^{11} + 4 q^{12} + 4 q^{13} - 6 q^{14} + 8 q^{15} + 4 q^{16} + 4 q^{17} - 4 q^{18} + 8 q^{20} + 6 q^{21} - 12 q^{22} + 12 q^{23} - 4 q^{24} + 6 q^{25} - 4 q^{26} + 4 q^{27} + 6 q^{28} - 10 q^{29} - 8 q^{30} + 8 q^{31} - 4 q^{32} + 12 q^{33} - 4 q^{34} + 12 q^{35} + 4 q^{36} - 8 q^{37} + 4 q^{39} - 8 q^{40} + 8 q^{41} - 6 q^{42} - 4 q^{43} + 12 q^{44} + 8 q^{45} - 12 q^{46} + 4 q^{47} + 4 q^{48} - 14 q^{49} - 6 q^{50} + 4 q^{51} + 4 q^{52} + 10 q^{53} - 4 q^{54} + 24 q^{55} - 6 q^{56} + 10 q^{58} + 6 q^{59} + 8 q^{60} + 4 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 8 q^{65} - 12 q^{66} - 12 q^{67} + 4 q^{68} + 12 q^{69} - 12 q^{70} - 4 q^{72} - 10 q^{73} + 8 q^{74} + 6 q^{75} + 28 q^{77} - 4 q^{78} - 32 q^{79} + 8 q^{80} + 4 q^{81} - 8 q^{82} + 8 q^{83} + 6 q^{84} - 12 q^{85} + 4 q^{86} - 10 q^{87} - 12 q^{88} + 16 q^{89} - 8 q^{90} - 4 q^{91} + 12 q^{92} + 8 q^{93} - 4 q^{94} - 4 q^{96} - 8 q^{97} + 14 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 3.17557 1.42016 0.710079 0.704122i \(-0.248659\pi\)
0.710079 + 0.704122i \(0.248659\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.381966 0.144370 0.0721848 0.997391i \(-0.477003\pi\)
0.0721848 + 0.997391i \(0.477003\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.17557 −1.00420
\(11\) 2.66605 0.803843 0.401921 0.915674i \(-0.368342\pi\)
0.401921 + 0.915674i \(0.368342\pi\)
\(12\) 1.00000 0.288675
\(13\) 7.04029 1.95263 0.976313 0.216362i \(-0.0694193\pi\)
0.976313 + 0.216362i \(0.0694193\pi\)
\(14\) −0.381966 −0.102085
\(15\) 3.17557 0.819929
\(16\) 1.00000 0.250000
\(17\) 4.68915 1.13729 0.568643 0.822584i \(-0.307468\pi\)
0.568643 + 0.822584i \(0.307468\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0
\(20\) 3.17557 0.710079
\(21\) 0.381966 0.0833518
\(22\) −2.66605 −0.568403
\(23\) 1.43184 0.298560 0.149280 0.988795i \(-0.452304\pi\)
0.149280 + 0.988795i \(0.452304\pi\)
\(24\) −1.00000 −0.204124
\(25\) 5.08425 1.01685
\(26\) −7.04029 −1.38072
\(27\) 1.00000 0.192450
\(28\) 0.381966 0.0721848
\(29\) −8.20524 −1.52368 −0.761838 0.647768i \(-0.775703\pi\)
−0.761838 + 0.647768i \(0.775703\pi\)
\(30\) −3.17557 −0.579777
\(31\) −8.62750 −1.54955 −0.774773 0.632240i \(-0.782136\pi\)
−0.774773 + 0.632240i \(0.782136\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.66605 0.464099
\(34\) −4.68915 −0.804183
\(35\) 1.21296 0.205028
\(36\) 1.00000 0.166667
\(37\) 2.70228 0.444252 0.222126 0.975018i \(-0.428700\pi\)
0.222126 + 0.975018i \(0.428700\pi\)
\(38\) 0 0
\(39\) 7.04029 1.12735
\(40\) −3.17557 −0.502102
\(41\) −3.25731 −0.508707 −0.254353 0.967111i \(-0.581863\pi\)
−0.254353 + 0.967111i \(0.581863\pi\)
\(42\) −0.381966 −0.0589386
\(43\) −8.72353 −1.33033 −0.665163 0.746699i \(-0.731638\pi\)
−0.665163 + 0.746699i \(0.731638\pi\)
\(44\) 2.66605 0.401921
\(45\) 3.17557 0.473386
\(46\) −1.43184 −0.211114
\(47\) 5.35706 0.781408 0.390704 0.920516i \(-0.372232\pi\)
0.390704 + 0.920516i \(0.372232\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.85410 −0.979157
\(50\) −5.08425 −0.719021
\(51\) 4.68915 0.656613
\(52\) 7.04029 0.976313
\(53\) 8.20524 1.12708 0.563538 0.826090i \(-0.309440\pi\)
0.563538 + 0.826090i \(0.309440\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.46621 1.14158
\(56\) −0.381966 −0.0510424
\(57\) 0 0
\(58\) 8.20524 1.07740
\(59\) −11.8137 −1.53801 −0.769006 0.639241i \(-0.779248\pi\)
−0.769006 + 0.639241i \(0.779248\pi\)
\(60\) 3.17557 0.409964
\(61\) −5.27044 −0.674811 −0.337405 0.941359i \(-0.609549\pi\)
−0.337405 + 0.941359i \(0.609549\pi\)
\(62\) 8.62750 1.09569
\(63\) 0.381966 0.0481232
\(64\) 1.00000 0.125000
\(65\) 22.3570 2.77304
\(66\) −2.66605 −0.328167
\(67\) 12.2147 1.49226 0.746132 0.665798i \(-0.231909\pi\)
0.746132 + 0.665798i \(0.231909\pi\)
\(68\) 4.68915 0.568643
\(69\) 1.43184 0.172374
\(70\) −1.21296 −0.144976
\(71\) −15.0996 −1.79200 −0.895999 0.444057i \(-0.853539\pi\)
−0.895999 + 0.444057i \(0.853539\pi\)
\(72\) −1.00000 −0.117851
\(73\) −3.84818 −0.450395 −0.225198 0.974313i \(-0.572303\pi\)
−0.225198 + 0.974313i \(0.572303\pi\)
\(74\) −2.70228 −0.314134
\(75\) 5.08425 0.587078
\(76\) 0 0
\(77\) 1.01834 0.116050
\(78\) −7.04029 −0.797156
\(79\) −1.39560 −0.157018 −0.0785089 0.996913i \(-0.525016\pi\)
−0.0785089 + 0.996913i \(0.525016\pi\)
\(80\) 3.17557 0.355040
\(81\) 1.00000 0.111111
\(82\) 3.25731 0.359710
\(83\) −4.31562 −0.473700 −0.236850 0.971546i \(-0.576115\pi\)
−0.236850 + 0.971546i \(0.576115\pi\)
\(84\) 0.381966 0.0416759
\(85\) 14.8907 1.61513
\(86\) 8.72353 0.940682
\(87\) −8.20524 −0.879694
\(88\) −2.66605 −0.284201
\(89\) 7.80423 0.827246 0.413623 0.910448i \(-0.364263\pi\)
0.413623 + 0.910448i \(0.364263\pi\)
\(90\) −3.17557 −0.334735
\(91\) 2.68915 0.281900
\(92\) 1.43184 0.149280
\(93\) −8.62750 −0.894630
\(94\) −5.35706 −0.552539
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −12.7448 −1.29404 −0.647018 0.762475i \(-0.723984\pi\)
−0.647018 + 0.762475i \(0.723984\pi\)
\(98\) 6.85410 0.692369
\(99\) 2.66605 0.267948
\(100\) 5.08425 0.508425
\(101\) 3.56597 0.354827 0.177413 0.984136i \(-0.443227\pi\)
0.177413 + 0.984136i \(0.443227\pi\)
\(102\) −4.68915 −0.464295
\(103\) −7.04216 −0.693885 −0.346942 0.937886i \(-0.612780\pi\)
−0.346942 + 0.937886i \(0.612780\pi\)
\(104\) −7.04029 −0.690358
\(105\) 1.21296 0.118373
\(106\) −8.20524 −0.796964
\(107\) −3.63116 −0.351038 −0.175519 0.984476i \(-0.556160\pi\)
−0.175519 + 0.984476i \(0.556160\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.46621 −0.332003 −0.166002 0.986125i \(-0.553086\pi\)
−0.166002 + 0.986125i \(0.553086\pi\)
\(110\) −8.46621 −0.807222
\(111\) 2.70228 0.256489
\(112\) 0.381966 0.0360924
\(113\) 10.2764 0.966719 0.483359 0.875422i \(-0.339416\pi\)
0.483359 + 0.875422i \(0.339416\pi\)
\(114\) 0 0
\(115\) 4.54691 0.424002
\(116\) −8.20524 −0.761838
\(117\) 7.04029 0.650875
\(118\) 11.8137 1.08754
\(119\) 1.79110 0.164190
\(120\) −3.17557 −0.289889
\(121\) −3.89220 −0.353837
\(122\) 5.27044 0.477163
\(123\) −3.25731 −0.293702
\(124\) −8.62750 −0.774773
\(125\) 0.267536 0.0239291
\(126\) −0.381966 −0.0340282
\(127\) −21.2723 −1.88761 −0.943806 0.330500i \(-0.892783\pi\)
−0.943806 + 0.330500i \(0.892783\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.72353 −0.768064
\(130\) −22.3570 −1.96083
\(131\) 2.93731 0.256634 0.128317 0.991733i \(-0.459043\pi\)
0.128317 + 0.991733i \(0.459043\pi\)
\(132\) 2.66605 0.232049
\(133\) 0 0
\(134\) −12.2147 −1.05519
\(135\) 3.17557 0.273310
\(136\) −4.68915 −0.402092
\(137\) 7.61218 0.650353 0.325176 0.945653i \(-0.394576\pi\)
0.325176 + 0.945653i \(0.394576\pi\)
\(138\) −1.43184 −0.121886
\(139\) 0.434034 0.0368143 0.0184071 0.999831i \(-0.494140\pi\)
0.0184071 + 0.999831i \(0.494140\pi\)
\(140\) 1.21296 0.102514
\(141\) 5.35706 0.451146
\(142\) 15.0996 1.26713
\(143\) 18.7697 1.56960
\(144\) 1.00000 0.0833333
\(145\) −26.0563 −2.16386
\(146\) 3.84818 0.318478
\(147\) −6.85410 −0.565317
\(148\) 2.70228 0.222126
\(149\) 17.1352 1.40377 0.701884 0.712292i \(-0.252343\pi\)
0.701884 + 0.712292i \(0.252343\pi\)
\(150\) −5.08425 −0.415127
\(151\) 9.31852 0.758330 0.379165 0.925329i \(-0.376211\pi\)
0.379165 + 0.925329i \(0.376211\pi\)
\(152\) 0 0
\(153\) 4.68915 0.379096
\(154\) −1.01834 −0.0820601
\(155\) −27.3972 −2.20060
\(156\) 7.04029 0.563675
\(157\) −17.6428 −1.40805 −0.704025 0.710175i \(-0.748616\pi\)
−0.704025 + 0.710175i \(0.748616\pi\)
\(158\) 1.39560 0.111028
\(159\) 8.20524 0.650718
\(160\) −3.17557 −0.251051
\(161\) 0.546915 0.0431029
\(162\) −1.00000 −0.0785674
\(163\) −7.25731 −0.568436 −0.284218 0.958760i \(-0.591734\pi\)
−0.284218 + 0.958760i \(0.591734\pi\)
\(164\) −3.25731 −0.254353
\(165\) 8.46621 0.659094
\(166\) 4.31562 0.334957
\(167\) 11.8386 0.916098 0.458049 0.888927i \(-0.348548\pi\)
0.458049 + 0.888927i \(0.348548\pi\)
\(168\) −0.381966 −0.0294693
\(169\) 36.5657 2.81275
\(170\) −14.8907 −1.14207
\(171\) 0 0
\(172\) −8.72353 −0.665163
\(173\) 15.0711 1.14584 0.572918 0.819613i \(-0.305811\pi\)
0.572918 + 0.819613i \(0.305811\pi\)
\(174\) 8.20524 0.622038
\(175\) 1.94201 0.146802
\(176\) 2.66605 0.200961
\(177\) −11.8137 −0.887972
\(178\) −7.80423 −0.584951
\(179\) 0.0630049 0.00470921 0.00235460 0.999997i \(-0.499251\pi\)
0.00235460 + 0.999997i \(0.499251\pi\)
\(180\) 3.17557 0.236693
\(181\) 3.79110 0.281790 0.140895 0.990025i \(-0.455002\pi\)
0.140895 + 0.990025i \(0.455002\pi\)
\(182\) −2.68915 −0.199333
\(183\) −5.27044 −0.389602
\(184\) −1.43184 −0.105557
\(185\) 8.58129 0.630909
\(186\) 8.62750 0.632599
\(187\) 12.5015 0.914200
\(188\) 5.35706 0.390704
\(189\) 0.381966 0.0277839
\(190\) 0 0
\(191\) 26.2572 1.89990 0.949952 0.312396i \(-0.101131\pi\)
0.949952 + 0.312396i \(0.101131\pi\)
\(192\) 1.00000 0.0721688
\(193\) −11.8897 −0.855839 −0.427920 0.903817i \(-0.640753\pi\)
−0.427920 + 0.903817i \(0.640753\pi\)
\(194\) 12.7448 0.915021
\(195\) 22.3570 1.60101
\(196\) −6.85410 −0.489579
\(197\) −8.00186 −0.570109 −0.285055 0.958511i \(-0.592012\pi\)
−0.285055 + 0.958511i \(0.592012\pi\)
\(198\) −2.66605 −0.189468
\(199\) −24.3452 −1.72578 −0.862892 0.505389i \(-0.831349\pi\)
−0.862892 + 0.505389i \(0.831349\pi\)
\(200\) −5.08425 −0.359511
\(201\) 12.2147 0.861559
\(202\) −3.56597 −0.250900
\(203\) −3.13412 −0.219972
\(204\) 4.68915 0.328306
\(205\) −10.3438 −0.722444
\(206\) 7.04216 0.490650
\(207\) 1.43184 0.0995199
\(208\) 7.04029 0.488157
\(209\) 0 0
\(210\) −1.21296 −0.0837022
\(211\) 20.2016 1.39073 0.695367 0.718655i \(-0.255242\pi\)
0.695367 + 0.718655i \(0.255242\pi\)
\(212\) 8.20524 0.563538
\(213\) −15.0996 −1.03461
\(214\) 3.63116 0.248221
\(215\) −27.7022 −1.88927
\(216\) −1.00000 −0.0680414
\(217\) −3.29541 −0.223707
\(218\) 3.46621 0.234762
\(219\) −3.84818 −0.260036
\(220\) 8.46621 0.570792
\(221\) 33.0130 2.22070
\(222\) −2.70228 −0.181365
\(223\) 18.3581 1.22935 0.614675 0.788781i \(-0.289287\pi\)
0.614675 + 0.788781i \(0.289287\pi\)
\(224\) −0.381966 −0.0255212
\(225\) 5.08425 0.338950
\(226\) −10.2764 −0.683573
\(227\) −1.53141 −0.101643 −0.0508217 0.998708i \(-0.516184\pi\)
−0.0508217 + 0.998708i \(0.516184\pi\)
\(228\) 0 0
\(229\) −21.2381 −1.40346 −0.701728 0.712445i \(-0.747588\pi\)
−0.701728 + 0.712445i \(0.747588\pi\)
\(230\) −4.54691 −0.299815
\(231\) 1.01834 0.0670018
\(232\) 8.20524 0.538701
\(233\) −10.9655 −0.718375 −0.359187 0.933265i \(-0.616946\pi\)
−0.359187 + 0.933265i \(0.616946\pi\)
\(234\) −7.04029 −0.460238
\(235\) 17.0117 1.10972
\(236\) −11.8137 −0.769006
\(237\) −1.39560 −0.0906543
\(238\) −1.79110 −0.116100
\(239\) −6.85765 −0.443584 −0.221792 0.975094i \(-0.571191\pi\)
−0.221792 + 0.975094i \(0.571191\pi\)
\(240\) 3.17557 0.204982
\(241\) 19.1109 1.23104 0.615521 0.788121i \(-0.288946\pi\)
0.615521 + 0.788121i \(0.288946\pi\)
\(242\) 3.89220 0.250200
\(243\) 1.00000 0.0641500
\(244\) −5.27044 −0.337405
\(245\) −21.7657 −1.39056
\(246\) 3.25731 0.207679
\(247\) 0 0
\(248\) 8.62750 0.547847
\(249\) −4.31562 −0.273491
\(250\) −0.267536 −0.0169204
\(251\) 10.5165 0.663795 0.331897 0.943315i \(-0.392311\pi\)
0.331897 + 0.943315i \(0.392311\pi\)
\(252\) 0.381966 0.0240616
\(253\) 3.81736 0.239995
\(254\) 21.2723 1.33474
\(255\) 14.8907 0.932494
\(256\) 1.00000 0.0625000
\(257\) 1.60637 0.100203 0.0501014 0.998744i \(-0.484046\pi\)
0.0501014 + 0.998744i \(0.484046\pi\)
\(258\) 8.72353 0.543103
\(259\) 1.03218 0.0641365
\(260\) 22.3570 1.38652
\(261\) −8.20524 −0.507892
\(262\) −2.93731 −0.181468
\(263\) 13.4471 0.829181 0.414590 0.910008i \(-0.363925\pi\)
0.414590 + 0.910008i \(0.363925\pi\)
\(264\) −2.66605 −0.164084
\(265\) 26.0563 1.60063
\(266\) 0 0
\(267\) 7.80423 0.477611
\(268\) 12.2147 0.746132
\(269\) 22.5492 1.37485 0.687424 0.726257i \(-0.258742\pi\)
0.687424 + 0.726257i \(0.258742\pi\)
\(270\) −3.17557 −0.193259
\(271\) −5.02260 −0.305101 −0.152551 0.988296i \(-0.548749\pi\)
−0.152551 + 0.988296i \(0.548749\pi\)
\(272\) 4.68915 0.284322
\(273\) 2.68915 0.162755
\(274\) −7.61218 −0.459869
\(275\) 13.5548 0.817387
\(276\) 1.43184 0.0861868
\(277\) 2.11288 0.126951 0.0634754 0.997983i \(-0.479782\pi\)
0.0634754 + 0.997983i \(0.479782\pi\)
\(278\) −0.434034 −0.0260316
\(279\) −8.62750 −0.516515
\(280\) −1.21296 −0.0724882
\(281\) 18.7616 1.11922 0.559612 0.828754i \(-0.310950\pi\)
0.559612 + 0.828754i \(0.310950\pi\)
\(282\) −5.35706 −0.319008
\(283\) −1.38434 −0.0822905 −0.0411453 0.999153i \(-0.513101\pi\)
−0.0411453 + 0.999153i \(0.513101\pi\)
\(284\) −15.0996 −0.895999
\(285\) 0 0
\(286\) −18.7697 −1.10988
\(287\) −1.24418 −0.0734417
\(288\) −1.00000 −0.0589256
\(289\) 4.98816 0.293421
\(290\) 26.0563 1.53008
\(291\) −12.7448 −0.747112
\(292\) −3.84818 −0.225198
\(293\) 0.226599 0.0132381 0.00661903 0.999978i \(-0.497893\pi\)
0.00661903 + 0.999978i \(0.497893\pi\)
\(294\) 6.85410 0.399739
\(295\) −37.5152 −2.18422
\(296\) −2.70228 −0.157067
\(297\) 2.66605 0.154700
\(298\) −17.1352 −0.992613
\(299\) 10.0806 0.582975
\(300\) 5.08425 0.293539
\(301\) −3.33209 −0.192058
\(302\) −9.31852 −0.536221
\(303\) 3.56597 0.204859
\(304\) 0 0
\(305\) −16.7367 −0.958338
\(306\) −4.68915 −0.268061
\(307\) −22.4853 −1.28330 −0.641651 0.766997i \(-0.721750\pi\)
−0.641651 + 0.766997i \(0.721750\pi\)
\(308\) 1.01834 0.0580252
\(309\) −7.04216 −0.400614
\(310\) 27.3972 1.55606
\(311\) −4.22513 −0.239585 −0.119793 0.992799i \(-0.538223\pi\)
−0.119793 + 0.992799i \(0.538223\pi\)
\(312\) −7.04029 −0.398578
\(313\) 2.09455 0.118391 0.0591956 0.998246i \(-0.481146\pi\)
0.0591956 + 0.998246i \(0.481146\pi\)
\(314\) 17.6428 0.995642
\(315\) 1.21296 0.0683426
\(316\) −1.39560 −0.0785089
\(317\) 1.95616 0.109869 0.0549344 0.998490i \(-0.482505\pi\)
0.0549344 + 0.998490i \(0.482505\pi\)
\(318\) −8.20524 −0.460127
\(319\) −21.8755 −1.22480
\(320\) 3.17557 0.177520
\(321\) −3.63116 −0.202672
\(322\) −0.546915 −0.0304784
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 35.7946 1.98553
\(326\) 7.25731 0.401945
\(327\) −3.46621 −0.191682
\(328\) 3.25731 0.179855
\(329\) 2.04622 0.112812
\(330\) −8.46621 −0.466050
\(331\) −8.23015 −0.452370 −0.226185 0.974084i \(-0.572625\pi\)
−0.226185 + 0.974084i \(0.572625\pi\)
\(332\) −4.31562 −0.236850
\(333\) 2.70228 0.148084
\(334\) −11.8386 −0.647779
\(335\) 38.7887 2.11925
\(336\) 0.381966 0.0208380
\(337\) 7.56597 0.412144 0.206072 0.978537i \(-0.433932\pi\)
0.206072 + 0.978537i \(0.433932\pi\)
\(338\) −36.5657 −1.98891
\(339\) 10.2764 0.558135
\(340\) 14.8907 0.807564
\(341\) −23.0013 −1.24559
\(342\) 0 0
\(343\) −5.29180 −0.285730
\(344\) 8.72353 0.470341
\(345\) 4.54691 0.244798
\(346\) −15.0711 −0.810228
\(347\) −24.5084 −1.31568 −0.657839 0.753158i \(-0.728529\pi\)
−0.657839 + 0.753158i \(0.728529\pi\)
\(348\) −8.20524 −0.439847
\(349\) −10.5278 −0.563538 −0.281769 0.959482i \(-0.590921\pi\)
−0.281769 + 0.959482i \(0.590921\pi\)
\(350\) −1.94201 −0.103805
\(351\) 7.04029 0.375783
\(352\) −2.66605 −0.142101
\(353\) 11.0249 0.586794 0.293397 0.955991i \(-0.405214\pi\)
0.293397 + 0.955991i \(0.405214\pi\)
\(354\) 11.8137 0.627891
\(355\) −47.9500 −2.54492
\(356\) 7.80423 0.413623
\(357\) 1.79110 0.0947949
\(358\) −0.0630049 −0.00332991
\(359\) −5.60034 −0.295575 −0.147787 0.989019i \(-0.547215\pi\)
−0.147787 + 0.989019i \(0.547215\pi\)
\(360\) −3.17557 −0.167367
\(361\) 0 0
\(362\) −3.79110 −0.199256
\(363\) −3.89220 −0.204288
\(364\) 2.68915 0.140950
\(365\) −12.2202 −0.639633
\(366\) 5.27044 0.275490
\(367\) 37.3238 1.94828 0.974142 0.225937i \(-0.0725443\pi\)
0.974142 + 0.225937i \(0.0725443\pi\)
\(368\) 1.43184 0.0746399
\(369\) −3.25731 −0.169569
\(370\) −8.58129 −0.446120
\(371\) 3.13412 0.162716
\(372\) −8.62750 −0.447315
\(373\) −29.4141 −1.52300 −0.761502 0.648162i \(-0.775538\pi\)
−0.761502 + 0.648162i \(0.775538\pi\)
\(374\) −12.5015 −0.646437
\(375\) 0.267536 0.0138155
\(376\) −5.35706 −0.276269
\(377\) −57.7673 −2.97517
\(378\) −0.381966 −0.0196462
\(379\) −28.2454 −1.45087 −0.725433 0.688293i \(-0.758361\pi\)
−0.725433 + 0.688293i \(0.758361\pi\)
\(380\) 0 0
\(381\) −21.2723 −1.08981
\(382\) −26.2572 −1.34344
\(383\) 16.8254 0.859736 0.429868 0.902892i \(-0.358560\pi\)
0.429868 + 0.902892i \(0.358560\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 3.23381 0.164810
\(386\) 11.8897 0.605170
\(387\) −8.72353 −0.443442
\(388\) −12.7448 −0.647018
\(389\) 1.57035 0.0796199 0.0398100 0.999207i \(-0.487325\pi\)
0.0398100 + 0.999207i \(0.487325\pi\)
\(390\) −22.3570 −1.13209
\(391\) 6.71413 0.339548
\(392\) 6.85410 0.346184
\(393\) 2.93731 0.148168
\(394\) 8.00186 0.403128
\(395\) −4.43184 −0.222990
\(396\) 2.66605 0.133974
\(397\) 3.08780 0.154972 0.0774860 0.996993i \(-0.475311\pi\)
0.0774860 + 0.996993i \(0.475311\pi\)
\(398\) 24.3452 1.22031
\(399\) 0 0
\(400\) 5.08425 0.254212
\(401\) −3.92895 −0.196202 −0.0981012 0.995176i \(-0.531277\pi\)
−0.0981012 + 0.995176i \(0.531277\pi\)
\(402\) −12.2147 −0.609214
\(403\) −60.7402 −3.02568
\(404\) 3.56597 0.177413
\(405\) 3.17557 0.157795
\(406\) 3.13412 0.155544
\(407\) 7.20441 0.357109
\(408\) −4.68915 −0.232148
\(409\) 28.7032 1.41928 0.709641 0.704563i \(-0.248857\pi\)
0.709641 + 0.704563i \(0.248857\pi\)
\(410\) 10.3438 0.510845
\(411\) 7.61218 0.375481
\(412\) −7.04216 −0.346942
\(413\) −4.51243 −0.222042
\(414\) −1.43184 −0.0703712
\(415\) −13.7045 −0.672730
\(416\) −7.04029 −0.345179
\(417\) 0.434034 0.0212547
\(418\) 0 0
\(419\) −3.73663 −0.182547 −0.0912733 0.995826i \(-0.529094\pi\)
−0.0912733 + 0.995826i \(0.529094\pi\)
\(420\) 1.21296 0.0591864
\(421\) 21.6928 1.05724 0.528620 0.848858i \(-0.322709\pi\)
0.528620 + 0.848858i \(0.322709\pi\)
\(422\) −20.2016 −0.983398
\(423\) 5.35706 0.260469
\(424\) −8.20524 −0.398482
\(425\) 23.8408 1.15645
\(426\) 15.0996 0.731580
\(427\) −2.01313 −0.0974221
\(428\) −3.63116 −0.175519
\(429\) 18.7697 0.906212
\(430\) 27.7022 1.33592
\(431\) 5.09756 0.245541 0.122770 0.992435i \(-0.460822\pi\)
0.122770 + 0.992435i \(0.460822\pi\)
\(432\) 1.00000 0.0481125
\(433\) 18.1903 0.874171 0.437085 0.899420i \(-0.356011\pi\)
0.437085 + 0.899420i \(0.356011\pi\)
\(434\) 3.29541 0.158185
\(435\) −26.0563 −1.24931
\(436\) −3.46621 −0.166002
\(437\) 0 0
\(438\) 3.84818 0.183873
\(439\) 29.4760 1.40681 0.703406 0.710789i \(-0.251662\pi\)
0.703406 + 0.710789i \(0.251662\pi\)
\(440\) −8.46621 −0.403611
\(441\) −6.85410 −0.326386
\(442\) −33.0130 −1.57027
\(443\) −20.6001 −0.978740 −0.489370 0.872076i \(-0.662773\pi\)
−0.489370 + 0.872076i \(0.662773\pi\)
\(444\) 2.70228 0.128245
\(445\) 24.7829 1.17482
\(446\) −18.3581 −0.869281
\(447\) 17.1352 0.810465
\(448\) 0.381966 0.0180462
\(449\) 5.32606 0.251352 0.125676 0.992071i \(-0.459890\pi\)
0.125676 + 0.992071i \(0.459890\pi\)
\(450\) −5.08425 −0.239674
\(451\) −8.68414 −0.408920
\(452\) 10.2764 0.483359
\(453\) 9.31852 0.437822
\(454\) 1.53141 0.0718727
\(455\) 8.53960 0.400342
\(456\) 0 0
\(457\) −14.5527 −0.680748 −0.340374 0.940290i \(-0.610554\pi\)
−0.340374 + 0.940290i \(0.610554\pi\)
\(458\) 21.2381 0.992393
\(459\) 4.68915 0.218871
\(460\) 4.54691 0.212001
\(461\) −27.7283 −1.29144 −0.645718 0.763576i \(-0.723442\pi\)
−0.645718 + 0.763576i \(0.723442\pi\)
\(462\) −1.01834 −0.0473774
\(463\) 3.33216 0.154859 0.0774293 0.996998i \(-0.475329\pi\)
0.0774293 + 0.996998i \(0.475329\pi\)
\(464\) −8.20524 −0.380919
\(465\) −27.3972 −1.27052
\(466\) 10.9655 0.507968
\(467\) 1.33094 0.0615884 0.0307942 0.999526i \(-0.490196\pi\)
0.0307942 + 0.999526i \(0.490196\pi\)
\(468\) 7.04029 0.325438
\(469\) 4.66560 0.215438
\(470\) −17.0117 −0.784693
\(471\) −17.6428 −0.812938
\(472\) 11.8137 0.543769
\(473\) −23.2573 −1.06937
\(474\) 1.39560 0.0641023
\(475\) 0 0
\(476\) 1.79110 0.0820948
\(477\) 8.20524 0.375692
\(478\) 6.85765 0.313662
\(479\) −37.1883 −1.69918 −0.849589 0.527445i \(-0.823150\pi\)
−0.849589 + 0.527445i \(0.823150\pi\)
\(480\) −3.17557 −0.144944
\(481\) 19.0249 0.867459
\(482\) −19.1109 −0.870478
\(483\) 0.546915 0.0248855
\(484\) −3.89220 −0.176918
\(485\) −40.4719 −1.83773
\(486\) −1.00000 −0.0453609
\(487\) 25.6013 1.16010 0.580052 0.814579i \(-0.303032\pi\)
0.580052 + 0.814579i \(0.303032\pi\)
\(488\) 5.27044 0.238582
\(489\) −7.25731 −0.328187
\(490\) 21.7657 0.983273
\(491\) 25.3711 1.14498 0.572491 0.819911i \(-0.305977\pi\)
0.572491 + 0.819911i \(0.305977\pi\)
\(492\) −3.25731 −0.146851
\(493\) −38.4756 −1.73286
\(494\) 0 0
\(495\) 8.46621 0.380528
\(496\) −8.62750 −0.387386
\(497\) −5.76755 −0.258710
\(498\) 4.31562 0.193387
\(499\) 21.0996 0.944550 0.472275 0.881451i \(-0.343433\pi\)
0.472275 + 0.881451i \(0.343433\pi\)
\(500\) 0.267536 0.0119646
\(501\) 11.8386 0.528910
\(502\) −10.5165 −0.469374
\(503\) −26.7154 −1.19118 −0.595591 0.803288i \(-0.703082\pi\)
−0.595591 + 0.803288i \(0.703082\pi\)
\(504\) −0.381966 −0.0170141
\(505\) 11.3240 0.503910
\(506\) −3.81736 −0.169702
\(507\) 36.5657 1.62394
\(508\) −21.2723 −0.943806
\(509\) 4.44943 0.197217 0.0986087 0.995126i \(-0.468561\pi\)
0.0986087 + 0.995126i \(0.468561\pi\)
\(510\) −14.8907 −0.659373
\(511\) −1.46987 −0.0650234
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −1.60637 −0.0708541
\(515\) −22.3629 −0.985426
\(516\) −8.72353 −0.384032
\(517\) 14.2822 0.628129
\(518\) −1.03218 −0.0453514
\(519\) 15.0711 0.661549
\(520\) −22.3570 −0.980417
\(521\) 4.52798 0.198374 0.0991871 0.995069i \(-0.468376\pi\)
0.0991871 + 0.995069i \(0.468376\pi\)
\(522\) 8.20524 0.359134
\(523\) 3.84233 0.168013 0.0840066 0.996465i \(-0.473228\pi\)
0.0840066 + 0.996465i \(0.473228\pi\)
\(524\) 2.93731 0.128317
\(525\) 1.94201 0.0847563
\(526\) −13.4471 −0.586319
\(527\) −40.4557 −1.76228
\(528\) 2.66605 0.116025
\(529\) −20.9498 −0.910862
\(530\) −26.0563 −1.13181
\(531\) −11.8137 −0.512671
\(532\) 0 0
\(533\) −22.9324 −0.993314
\(534\) −7.80423 −0.337722
\(535\) −11.5310 −0.498529
\(536\) −12.2147 −0.527595
\(537\) 0.0630049 0.00271886
\(538\) −22.5492 −0.972164
\(539\) −18.2733 −0.787089
\(540\) 3.17557 0.136655
\(541\) −2.76766 −0.118991 −0.0594955 0.998229i \(-0.518949\pi\)
−0.0594955 + 0.998229i \(0.518949\pi\)
\(542\) 5.02260 0.215739
\(543\) 3.79110 0.162692
\(544\) −4.68915 −0.201046
\(545\) −11.0072 −0.471497
\(546\) −2.68915 −0.115085
\(547\) 4.23955 0.181270 0.0906349 0.995884i \(-0.471110\pi\)
0.0906349 + 0.995884i \(0.471110\pi\)
\(548\) 7.61218 0.325176
\(549\) −5.27044 −0.224937
\(550\) −13.5548 −0.577980
\(551\) 0 0
\(552\) −1.43184 −0.0609432
\(553\) −0.533074 −0.0226686
\(554\) −2.11288 −0.0897677
\(555\) 8.58129 0.364255
\(556\) 0.434034 0.0184071
\(557\) −17.5035 −0.741646 −0.370823 0.928704i \(-0.620924\pi\)
−0.370823 + 0.928704i \(0.620924\pi\)
\(558\) 8.62750 0.365231
\(559\) −61.4162 −2.59763
\(560\) 1.21296 0.0512569
\(561\) 12.5015 0.527813
\(562\) −18.7616 −0.791411
\(563\) −6.45162 −0.271903 −0.135952 0.990715i \(-0.543409\pi\)
−0.135952 + 0.990715i \(0.543409\pi\)
\(564\) 5.35706 0.225573
\(565\) 32.6333 1.37289
\(566\) 1.38434 0.0581882
\(567\) 0.381966 0.0160411
\(568\) 15.0996 0.633567
\(569\) 0.506508 0.0212339 0.0106170 0.999944i \(-0.496620\pi\)
0.0106170 + 0.999944i \(0.496620\pi\)
\(570\) 0 0
\(571\) 2.08999 0.0874633 0.0437316 0.999043i \(-0.486075\pi\)
0.0437316 + 0.999043i \(0.486075\pi\)
\(572\) 18.7697 0.784802
\(573\) 26.2572 1.09691
\(574\) 1.24418 0.0519312
\(575\) 7.27984 0.303590
\(576\) 1.00000 0.0416667
\(577\) 43.6345 1.81653 0.908264 0.418398i \(-0.137408\pi\)
0.908264 + 0.418398i \(0.137408\pi\)
\(578\) −4.98816 −0.207480
\(579\) −11.8897 −0.494119
\(580\) −26.0563 −1.08193
\(581\) −1.64842 −0.0683879
\(582\) 12.7448 0.528288
\(583\) 21.8755 0.905993
\(584\) 3.84818 0.159239
\(585\) 22.3570 0.924346
\(586\) −0.226599 −0.00936072
\(587\) 44.2051 1.82454 0.912270 0.409590i \(-0.134328\pi\)
0.912270 + 0.409590i \(0.134328\pi\)
\(588\) −6.85410 −0.282658
\(589\) 0 0
\(590\) 37.5152 1.54448
\(591\) −8.00186 −0.329153
\(592\) 2.70228 0.111063
\(593\) 5.92278 0.243219 0.121610 0.992578i \(-0.461194\pi\)
0.121610 + 0.992578i \(0.461194\pi\)
\(594\) −2.66605 −0.109389
\(595\) 5.68776 0.233175
\(596\) 17.1352 0.701884
\(597\) −24.3452 −0.996382
\(598\) −10.0806 −0.412226
\(599\) −26.3593 −1.07701 −0.538505 0.842622i \(-0.681011\pi\)
−0.538505 + 0.842622i \(0.681011\pi\)
\(600\) −5.08425 −0.207564
\(601\) 2.85069 0.116282 0.0581410 0.998308i \(-0.481483\pi\)
0.0581410 + 0.998308i \(0.481483\pi\)
\(602\) 3.33209 0.135806
\(603\) 12.2147 0.497422
\(604\) 9.31852 0.379165
\(605\) −12.3600 −0.502504
\(606\) −3.56597 −0.144857
\(607\) −6.77821 −0.275119 −0.137560 0.990493i \(-0.543926\pi\)
−0.137560 + 0.990493i \(0.543926\pi\)
\(608\) 0 0
\(609\) −3.13412 −0.127001
\(610\) 16.7367 0.677647
\(611\) 37.7153 1.52580
\(612\) 4.68915 0.189548
\(613\) −20.2191 −0.816642 −0.408321 0.912838i \(-0.633886\pi\)
−0.408321 + 0.912838i \(0.633886\pi\)
\(614\) 22.4853 0.907431
\(615\) −10.3438 −0.417103
\(616\) −1.01834 −0.0410300
\(617\) −10.9133 −0.439351 −0.219676 0.975573i \(-0.570500\pi\)
−0.219676 + 0.975573i \(0.570500\pi\)
\(618\) 7.04216 0.283277
\(619\) 13.8029 0.554787 0.277394 0.960756i \(-0.410529\pi\)
0.277394 + 0.960756i \(0.410529\pi\)
\(620\) −27.3972 −1.10030
\(621\) 1.43184 0.0574578
\(622\) 4.22513 0.169412
\(623\) 2.98095 0.119429
\(624\) 7.04029 0.281837
\(625\) −24.5717 −0.982866
\(626\) −2.09455 −0.0837152
\(627\) 0 0
\(628\) −17.6428 −0.704025
\(629\) 12.6714 0.505242
\(630\) −1.21296 −0.0483255
\(631\) 28.2587 1.12496 0.562480 0.826811i \(-0.309847\pi\)
0.562480 + 0.826811i \(0.309847\pi\)
\(632\) 1.39560 0.0555142
\(633\) 20.2016 0.802941
\(634\) −1.95616 −0.0776890
\(635\) −67.5517 −2.68071
\(636\) 8.20524 0.325359
\(637\) −48.2549 −1.91193
\(638\) 21.8755 0.866061
\(639\) −15.0996 −0.597332
\(640\) −3.17557 −0.125525
\(641\) −10.1379 −0.400421 −0.200211 0.979753i \(-0.564163\pi\)
−0.200211 + 0.979753i \(0.564163\pi\)
\(642\) 3.63116 0.143311
\(643\) −8.53379 −0.336540 −0.168270 0.985741i \(-0.553818\pi\)
−0.168270 + 0.985741i \(0.553818\pi\)
\(644\) 0.546915 0.0215515
\(645\) −27.7022 −1.09077
\(646\) 0 0
\(647\) −14.2383 −0.559764 −0.279882 0.960034i \(-0.590295\pi\)
−0.279882 + 0.960034i \(0.590295\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −31.4958 −1.23632
\(650\) −35.7946 −1.40398
\(651\) −3.29541 −0.129157
\(652\) −7.25731 −0.284218
\(653\) 28.5091 1.11565 0.557823 0.829960i \(-0.311637\pi\)
0.557823 + 0.829960i \(0.311637\pi\)
\(654\) 3.46621 0.135540
\(655\) 9.32764 0.364461
\(656\) −3.25731 −0.127177
\(657\) −3.84818 −0.150132
\(658\) −2.04622 −0.0797698
\(659\) −40.5659 −1.58022 −0.790112 0.612963i \(-0.789978\pi\)
−0.790112 + 0.612963i \(0.789978\pi\)
\(660\) 8.46621 0.329547
\(661\) −4.47342 −0.173996 −0.0869980 0.996208i \(-0.527727\pi\)
−0.0869980 + 0.996208i \(0.527727\pi\)
\(662\) 8.23015 0.319874
\(663\) 33.0130 1.28212
\(664\) 4.31562 0.167478
\(665\) 0 0
\(666\) −2.70228 −0.104711
\(667\) −11.7486 −0.454908
\(668\) 11.8386 0.458049
\(669\) 18.3581 0.709765
\(670\) −38.7887 −1.49854
\(671\) −14.0512 −0.542442
\(672\) −0.381966 −0.0147347
\(673\) −4.55272 −0.175495 −0.0877473 0.996143i \(-0.527967\pi\)
−0.0877473 + 0.996143i \(0.527967\pi\)
\(674\) −7.56597 −0.291430
\(675\) 5.08425 0.195693
\(676\) 36.5657 1.40637
\(677\) −24.4472 −0.939583 −0.469792 0.882777i \(-0.655671\pi\)
−0.469792 + 0.882777i \(0.655671\pi\)
\(678\) −10.2764 −0.394661
\(679\) −4.86807 −0.186819
\(680\) −14.8907 −0.571034
\(681\) −1.53141 −0.0586838
\(682\) 23.0013 0.880766
\(683\) 25.3627 0.970476 0.485238 0.874382i \(-0.338733\pi\)
0.485238 + 0.874382i \(0.338733\pi\)
\(684\) 0 0
\(685\) 24.1730 0.923604
\(686\) 5.29180 0.202042
\(687\) −21.2381 −0.810286
\(688\) −8.72353 −0.332581
\(689\) 57.7673 2.20076
\(690\) −4.54691 −0.173098
\(691\) −19.0686 −0.725405 −0.362702 0.931905i \(-0.618146\pi\)
−0.362702 + 0.931905i \(0.618146\pi\)
\(692\) 15.0711 0.572918
\(693\) 1.01834 0.0386835
\(694\) 24.5084 0.930325
\(695\) 1.37831 0.0522821
\(696\) 8.20524 0.311019
\(697\) −15.2740 −0.578545
\(698\) 10.5278 0.398481
\(699\) −10.9655 −0.414754
\(700\) 1.94201 0.0734011
\(701\) −16.5346 −0.624504 −0.312252 0.949999i \(-0.601083\pi\)
−0.312252 + 0.949999i \(0.601083\pi\)
\(702\) −7.04029 −0.265719
\(703\) 0 0
\(704\) 2.66605 0.100480
\(705\) 17.0117 0.640699
\(706\) −11.0249 −0.414926
\(707\) 1.36208 0.0512262
\(708\) −11.8137 −0.443986
\(709\) 28.3262 1.06381 0.531906 0.846804i \(-0.321476\pi\)
0.531906 + 0.846804i \(0.321476\pi\)
\(710\) 47.9500 1.79953
\(711\) −1.39560 −0.0523393
\(712\) −7.80423 −0.292476
\(713\) −12.3532 −0.462632
\(714\) −1.79110 −0.0670301
\(715\) 59.6046 2.22909
\(716\) 0.0630049 0.00235460
\(717\) −6.85765 −0.256104
\(718\) 5.60034 0.209003
\(719\) −20.6450 −0.769929 −0.384965 0.922931i \(-0.625786\pi\)
−0.384965 + 0.922931i \(0.625786\pi\)
\(720\) 3.17557 0.118347
\(721\) −2.68987 −0.100176
\(722\) 0 0
\(723\) 19.1109 0.710742
\(724\) 3.79110 0.140895
\(725\) −41.7175 −1.54935
\(726\) 3.89220 0.144453
\(727\) −18.2148 −0.675549 −0.337774 0.941227i \(-0.609674\pi\)
−0.337774 + 0.941227i \(0.609674\pi\)
\(728\) −2.68915 −0.0996666
\(729\) 1.00000 0.0370370
\(730\) 12.2202 0.452289
\(731\) −40.9059 −1.51296
\(732\) −5.27044 −0.194801
\(733\) −28.0081 −1.03450 −0.517252 0.855833i \(-0.673045\pi\)
−0.517252 + 0.855833i \(0.673045\pi\)
\(734\) −37.3238 −1.37764
\(735\) −21.7657 −0.802839
\(736\) −1.43184 −0.0527784
\(737\) 32.5650 1.19955
\(738\) 3.25731 0.119903
\(739\) −46.2549 −1.70151 −0.850757 0.525559i \(-0.823856\pi\)
−0.850757 + 0.525559i \(0.823856\pi\)
\(740\) 8.58129 0.315454
\(741\) 0 0
\(742\) −3.13412 −0.115057
\(743\) −30.8503 −1.13179 −0.565894 0.824478i \(-0.691469\pi\)
−0.565894 + 0.824478i \(0.691469\pi\)
\(744\) 8.62750 0.316300
\(745\) 54.4139 1.99357
\(746\) 29.4141 1.07693
\(747\) −4.31562 −0.157900
\(748\) 12.5015 0.457100
\(749\) −1.38698 −0.0506792
\(750\) −0.267536 −0.00976902
\(751\) 12.9512 0.472597 0.236299 0.971680i \(-0.424066\pi\)
0.236299 + 0.971680i \(0.424066\pi\)
\(752\) 5.35706 0.195352
\(753\) 10.5165 0.383242
\(754\) 57.7673 2.10376
\(755\) 29.5916 1.07695
\(756\) 0.381966 0.0138920
\(757\) 0.294101 0.0106893 0.00534464 0.999986i \(-0.498299\pi\)
0.00534464 + 0.999986i \(0.498299\pi\)
\(758\) 28.2454 1.02592
\(759\) 3.81736 0.138561
\(760\) 0 0
\(761\) 15.4993 0.561849 0.280925 0.959730i \(-0.409359\pi\)
0.280925 + 0.959730i \(0.409359\pi\)
\(762\) 21.2723 0.770614
\(763\) −1.32398 −0.0479311
\(764\) 26.2572 0.949952
\(765\) 14.8907 0.538376
\(766\) −16.8254 −0.607925
\(767\) −83.1719 −3.00316
\(768\) 1.00000 0.0360844
\(769\) −21.7968 −0.786014 −0.393007 0.919535i \(-0.628565\pi\)
−0.393007 + 0.919535i \(0.628565\pi\)
\(770\) −3.23381 −0.116538
\(771\) 1.60637 0.0578521
\(772\) −11.8897 −0.427920
\(773\) 7.46267 0.268413 0.134207 0.990953i \(-0.457151\pi\)
0.134207 + 0.990953i \(0.457151\pi\)
\(774\) 8.72353 0.313561
\(775\) −43.8644 −1.57565
\(776\) 12.7448 0.457511
\(777\) 1.03218 0.0370292
\(778\) −1.57035 −0.0562998
\(779\) 0 0
\(780\) 22.3570 0.800507
\(781\) −40.2563 −1.44048
\(782\) −6.71413 −0.240097
\(783\) −8.20524 −0.293231
\(784\) −6.85410 −0.244789
\(785\) −56.0260 −1.99965
\(786\) −2.93731 −0.104770
\(787\) 15.8788 0.566017 0.283009 0.959117i \(-0.408667\pi\)
0.283009 + 0.959117i \(0.408667\pi\)
\(788\) −8.00186 −0.285055
\(789\) 13.4471 0.478728
\(790\) 4.43184 0.157678
\(791\) 3.92522 0.139565
\(792\) −2.66605 −0.0947338
\(793\) −37.1054 −1.31765
\(794\) −3.08780 −0.109582
\(795\) 26.0563 0.924123
\(796\) −24.3452 −0.862892
\(797\) 17.0673 0.604556 0.302278 0.953220i \(-0.402253\pi\)
0.302278 + 0.953220i \(0.402253\pi\)
\(798\) 0 0
\(799\) 25.1201 0.888685
\(800\) −5.08425 −0.179755
\(801\) 7.80423 0.275749
\(802\) 3.92895 0.138736
\(803\) −10.2594 −0.362047
\(804\) 12.2147 0.430780
\(805\) 1.73677 0.0612130
\(806\) 60.7402 2.13948
\(807\) 22.5492 0.793769
\(808\) −3.56597 −0.125450
\(809\) 33.6132 1.18178 0.590889 0.806753i \(-0.298777\pi\)
0.590889 + 0.806753i \(0.298777\pi\)
\(810\) −3.17557 −0.111578
\(811\) 19.6047 0.688415 0.344207 0.938894i \(-0.388148\pi\)
0.344207 + 0.938894i \(0.388148\pi\)
\(812\) −3.13412 −0.109986
\(813\) −5.02260 −0.176150
\(814\) −7.20441 −0.252514
\(815\) −23.0461 −0.807270
\(816\) 4.68915 0.164153
\(817\) 0 0
\(818\) −28.7032 −1.00358
\(819\) 2.68915 0.0939666
\(820\) −10.3438 −0.361222
\(821\) 2.63979 0.0921292 0.0460646 0.998938i \(-0.485332\pi\)
0.0460646 + 0.998938i \(0.485332\pi\)
\(822\) −7.61218 −0.265505
\(823\) −1.49912 −0.0522560 −0.0261280 0.999659i \(-0.508318\pi\)
−0.0261280 + 0.999659i \(0.508318\pi\)
\(824\) 7.04216 0.245325
\(825\) 13.5548 0.471919
\(826\) 4.51243 0.157008
\(827\) −31.6406 −1.10025 −0.550126 0.835082i \(-0.685420\pi\)
−0.550126 + 0.835082i \(0.685420\pi\)
\(828\) 1.43184 0.0497599
\(829\) −48.5942 −1.68775 −0.843873 0.536543i \(-0.819730\pi\)
−0.843873 + 0.536543i \(0.819730\pi\)
\(830\) 13.7045 0.475692
\(831\) 2.11288 0.0732950
\(832\) 7.04029 0.244078
\(833\) −32.1399 −1.11358
\(834\) −0.434034 −0.0150294
\(835\) 37.5943 1.30100
\(836\) 0 0
\(837\) −8.62750 −0.298210
\(838\) 3.73663 0.129080
\(839\) 49.0971 1.69502 0.847510 0.530779i \(-0.178101\pi\)
0.847510 + 0.530779i \(0.178101\pi\)
\(840\) −1.21296 −0.0418511
\(841\) 38.3260 1.32159
\(842\) −21.6928 −0.747582
\(843\) 18.7616 0.646185
\(844\) 20.2016 0.695367
\(845\) 116.117 3.99455
\(846\) −5.35706 −0.184180
\(847\) −1.48669 −0.0510833
\(848\) 8.20524 0.281769
\(849\) −1.38434 −0.0475105
\(850\) −23.8408 −0.817733
\(851\) 3.86924 0.132636
\(852\) −15.0996 −0.517305
\(853\) 9.55514 0.327162 0.163581 0.986530i \(-0.447696\pi\)
0.163581 + 0.986530i \(0.447696\pi\)
\(854\) 2.01313 0.0688879
\(855\) 0 0
\(856\) 3.63116 0.124111
\(857\) 2.77720 0.0948674 0.0474337 0.998874i \(-0.484896\pi\)
0.0474337 + 0.998874i \(0.484896\pi\)
\(858\) −18.7697 −0.640788
\(859\) −38.1187 −1.30059 −0.650296 0.759681i \(-0.725355\pi\)
−0.650296 + 0.759681i \(0.725355\pi\)
\(860\) −27.7022 −0.944636
\(861\) −1.24418 −0.0424016
\(862\) −5.09756 −0.173624
\(863\) 28.2441 0.961439 0.480720 0.876874i \(-0.340375\pi\)
0.480720 + 0.876874i \(0.340375\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 47.8594 1.62727
\(866\) −18.1903 −0.618132
\(867\) 4.98816 0.169407
\(868\) −3.29541 −0.111854
\(869\) −3.72075 −0.126218
\(870\) 26.0563 0.883392
\(871\) 85.9952 2.91383
\(872\) 3.46621 0.117381
\(873\) −12.7448 −0.431345
\(874\) 0 0
\(875\) 0.102189 0.00345464
\(876\) −3.84818 −0.130018
\(877\) 22.1722 0.748703 0.374351 0.927287i \(-0.377865\pi\)
0.374351 + 0.927287i \(0.377865\pi\)
\(878\) −29.4760 −0.994766
\(879\) 0.226599 0.00764299
\(880\) 8.46621 0.285396
\(881\) 55.1257 1.85723 0.928616 0.371043i \(-0.121000\pi\)
0.928616 + 0.371043i \(0.121000\pi\)
\(882\) 6.85410 0.230790
\(883\) 20.5446 0.691381 0.345691 0.938349i \(-0.387645\pi\)
0.345691 + 0.938349i \(0.387645\pi\)
\(884\) 33.0130 1.11035
\(885\) −37.5152 −1.26106
\(886\) 20.6001 0.692074
\(887\) −50.7843 −1.70517 −0.852585 0.522589i \(-0.824966\pi\)
−0.852585 + 0.522589i \(0.824966\pi\)
\(888\) −2.70228 −0.0906826
\(889\) −8.12530 −0.272514
\(890\) −24.7829 −0.830724
\(891\) 2.66605 0.0893159
\(892\) 18.3581 0.614675
\(893\) 0 0
\(894\) −17.1352 −0.573086
\(895\) 0.200076 0.00668782
\(896\) −0.381966 −0.0127606
\(897\) 10.0806 0.336581
\(898\) −5.32606 −0.177733
\(899\) 70.7908 2.36100
\(900\) 5.08425 0.169475
\(901\) 38.4756 1.28181
\(902\) 8.68414 0.289150
\(903\) −3.33209 −0.110885
\(904\) −10.2764 −0.341787
\(905\) 12.0389 0.400186
\(906\) −9.31852 −0.309587
\(907\) 29.1553 0.968084 0.484042 0.875045i \(-0.339168\pi\)
0.484042 + 0.875045i \(0.339168\pi\)
\(908\) −1.53141 −0.0508217
\(909\) 3.56597 0.118276
\(910\) −8.53960 −0.283085
\(911\) 4.87180 0.161410 0.0807049 0.996738i \(-0.474283\pi\)
0.0807049 + 0.996738i \(0.474283\pi\)
\(912\) 0 0
\(913\) −11.5056 −0.380781
\(914\) 14.5527 0.481361
\(915\) −16.7367 −0.553297
\(916\) −21.2381 −0.701728
\(917\) 1.12195 0.0370501
\(918\) −4.68915 −0.154765
\(919\) −13.1897 −0.435088 −0.217544 0.976051i \(-0.569805\pi\)
−0.217544 + 0.976051i \(0.569805\pi\)
\(920\) −4.54691 −0.149907
\(921\) −22.4853 −0.740915
\(922\) 27.7283 0.913183
\(923\) −106.306 −3.49910
\(924\) 1.01834 0.0335009
\(925\) 13.7391 0.451738
\(926\) −3.33216 −0.109502
\(927\) −7.04216 −0.231295
\(928\) 8.20524 0.269350
\(929\) −16.0439 −0.526384 −0.263192 0.964744i \(-0.584775\pi\)
−0.263192 + 0.964744i \(0.584775\pi\)
\(930\) 27.3972 0.898391
\(931\) 0 0
\(932\) −10.9655 −0.359187
\(933\) −4.22513 −0.138325
\(934\) −1.33094 −0.0435496
\(935\) 39.6994 1.29831
\(936\) −7.04029 −0.230119
\(937\) −48.2717 −1.57697 −0.788484 0.615056i \(-0.789133\pi\)
−0.788484 + 0.615056i \(0.789133\pi\)
\(938\) −4.66560 −0.152337
\(939\) 2.09455 0.0683532
\(940\) 17.0117 0.554861
\(941\) 11.1612 0.363845 0.181923 0.983313i \(-0.441768\pi\)
0.181923 + 0.983313i \(0.441768\pi\)
\(942\) 17.6428 0.574834
\(943\) −4.66395 −0.151879
\(944\) −11.8137 −0.384503
\(945\) 1.21296 0.0394576
\(946\) 23.2573 0.756160
\(947\) 0.116113 0.00377316 0.00188658 0.999998i \(-0.499399\pi\)
0.00188658 + 0.999998i \(0.499399\pi\)
\(948\) −1.39560 −0.0453271
\(949\) −27.0923 −0.879454
\(950\) 0 0
\(951\) 1.95616 0.0634328
\(952\) −1.79110 −0.0580498
\(953\) −16.9191 −0.548063 −0.274031 0.961721i \(-0.588357\pi\)
−0.274031 + 0.961721i \(0.588357\pi\)
\(954\) −8.20524 −0.265655
\(955\) 83.3816 2.69816
\(956\) −6.85765 −0.221792
\(957\) −21.8755 −0.707136
\(958\) 37.1883 1.20150
\(959\) 2.90759 0.0938911
\(960\) 3.17557 0.102491
\(961\) 43.4338 1.40109
\(962\) −19.0249 −0.613386
\(963\) −3.63116 −0.117013
\(964\) 19.1109 0.615521
\(965\) −37.7566 −1.21543
\(966\) −0.546915 −0.0175967
\(967\) 31.2513 1.00498 0.502488 0.864584i \(-0.332418\pi\)
0.502488 + 0.864584i \(0.332418\pi\)
\(968\) 3.89220 0.125100
\(969\) 0 0
\(970\) 40.4719 1.29947
\(971\) −21.1207 −0.677796 −0.338898 0.940823i \(-0.610054\pi\)
−0.338898 + 0.940823i \(0.610054\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0.165786 0.00531486
\(974\) −25.6013 −0.820317
\(975\) 35.7946 1.14634
\(976\) −5.27044 −0.168703
\(977\) −46.4685 −1.48666 −0.743330 0.668925i \(-0.766755\pi\)
−0.743330 + 0.668925i \(0.766755\pi\)
\(978\) 7.25731 0.232063
\(979\) 20.8064 0.664976
\(980\) −21.7657 −0.695279
\(981\) −3.46621 −0.110668
\(982\) −25.3711 −0.809625
\(983\) 8.96344 0.285889 0.142945 0.989731i \(-0.454343\pi\)
0.142945 + 0.989731i \(0.454343\pi\)
\(984\) 3.25731 0.103839
\(985\) −25.4105 −0.809645
\(986\) 38.4756 1.22531
\(987\) 2.04622 0.0651318
\(988\) 0 0
\(989\) −12.4907 −0.397181
\(990\) −8.46621 −0.269074
\(991\) −23.2216 −0.737657 −0.368829 0.929497i \(-0.620241\pi\)
−0.368829 + 0.929497i \(0.620241\pi\)
\(992\) 8.62750 0.273923
\(993\) −8.23015 −0.261176
\(994\) 5.76755 0.182936
\(995\) −77.3098 −2.45089
\(996\) −4.31562 −0.136746
\(997\) 37.6024 1.19088 0.595440 0.803400i \(-0.296978\pi\)
0.595440 + 0.803400i \(0.296978\pi\)
\(998\) −21.0996 −0.667897
\(999\) 2.70228 0.0854964
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 2166.2.a.w.1.3 4
3.2 odd 2 6498.2.a.by.1.2 4
19.18 odd 2 2166.2.a.x.1.3 yes 4
57.56 even 2 6498.2.a.bv.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2166.2.a.w.1.3 4 1.1 even 1 trivial
2166.2.a.x.1.3 yes 4 19.18 odd 2
6498.2.a.bv.1.2 4 57.56 even 2
6498.2.a.by.1.2 4 3.2 odd 2