Properties

Label 6042.2.a.x.1.2
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $1$
Dimension $6$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.48689336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 6x^{4} + 7x^{3} + 9x^{2} - 5x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.51181\) of defining polynomial
Character \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.51181 q^{5} +1.00000 q^{6} +1.73805 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.51181 q^{5} +1.00000 q^{6} +1.73805 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.51181 q^{10} -0.642525 q^{11} +1.00000 q^{12} -2.58371 q^{13} +1.73805 q^{14} -2.51181 q^{15} +1.00000 q^{16} -7.71760 q^{17} +1.00000 q^{18} -1.00000 q^{19} -2.51181 q^{20} +1.73805 q^{21} -0.642525 q^{22} +5.46221 q^{23} +1.00000 q^{24} +1.30921 q^{25} -2.58371 q^{26} +1.00000 q^{27} +1.73805 q^{28} +8.07071 q^{29} -2.51181 q^{30} -0.677087 q^{31} +1.00000 q^{32} -0.642525 q^{33} -7.71760 q^{34} -4.36565 q^{35} +1.00000 q^{36} -7.32238 q^{37} -1.00000 q^{38} -2.58371 q^{39} -2.51181 q^{40} -6.11026 q^{41} +1.73805 q^{42} -1.89833 q^{43} -0.642525 q^{44} -2.51181 q^{45} +5.46221 q^{46} -10.5353 q^{47} +1.00000 q^{48} -3.97920 q^{49} +1.30921 q^{50} -7.71760 q^{51} -2.58371 q^{52} -1.00000 q^{53} +1.00000 q^{54} +1.61390 q^{55} +1.73805 q^{56} -1.00000 q^{57} +8.07071 q^{58} +6.39044 q^{59} -2.51181 q^{60} -4.23726 q^{61} -0.677087 q^{62} +1.73805 q^{63} +1.00000 q^{64} +6.48979 q^{65} -0.642525 q^{66} -12.8611 q^{67} -7.71760 q^{68} +5.46221 q^{69} -4.36565 q^{70} +13.8022 q^{71} +1.00000 q^{72} +5.86264 q^{73} -7.32238 q^{74} +1.30921 q^{75} -1.00000 q^{76} -1.11674 q^{77} -2.58371 q^{78} -9.67603 q^{79} -2.51181 q^{80} +1.00000 q^{81} -6.11026 q^{82} -0.708381 q^{83} +1.73805 q^{84} +19.3852 q^{85} -1.89833 q^{86} +8.07071 q^{87} -0.642525 q^{88} +9.33097 q^{89} -2.51181 q^{90} -4.49060 q^{91} +5.46221 q^{92} -0.677087 q^{93} -10.5353 q^{94} +2.51181 q^{95} +1.00000 q^{96} -13.5187 q^{97} -3.97920 q^{98} -0.642525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{3} + 6 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{3} + 6 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{7} + 6 q^{8} + 6 q^{9} - 8 q^{10} + 3 q^{11} + 6 q^{12} - 13 q^{13} - 6 q^{14} - 8 q^{15} + 6 q^{16} - 5 q^{17} + 6 q^{18} - 6 q^{19} - 8 q^{20} - 6 q^{21} + 3 q^{22} - 12 q^{23} + 6 q^{24} - 4 q^{25} - 13 q^{26} + 6 q^{27} - 6 q^{28} - 7 q^{29} - 8 q^{30} - 17 q^{31} + 6 q^{32} + 3 q^{33} - 5 q^{34} - 5 q^{35} + 6 q^{36} - 15 q^{37} - 6 q^{38} - 13 q^{39} - 8 q^{40} - 12 q^{41} - 6 q^{42} - 4 q^{43} + 3 q^{44} - 8 q^{45} - 12 q^{46} - 29 q^{47} + 6 q^{48} + 4 q^{49} - 4 q^{50} - 5 q^{51} - 13 q^{52} - 6 q^{53} + 6 q^{54} + 6 q^{55} - 6 q^{56} - 6 q^{57} - 7 q^{58} + 9 q^{59} - 8 q^{60} - 16 q^{61} - 17 q^{62} - 6 q^{63} + 6 q^{64} + 5 q^{65} + 3 q^{66} - 8 q^{67} - 5 q^{68} - 12 q^{69} - 5 q^{70} - 2 q^{71} + 6 q^{72} - 17 q^{73} - 15 q^{74} - 4 q^{75} - 6 q^{76} - 21 q^{77} - 13 q^{78} - 31 q^{79} - 8 q^{80} + 6 q^{81} - 12 q^{82} - 5 q^{83} - 6 q^{84} - 4 q^{86} - 7 q^{87} + 3 q^{88} - 5 q^{89} - 8 q^{90} + 4 q^{91} - 12 q^{92} - 17 q^{93} - 29 q^{94} + 8 q^{95} + 6 q^{96} + 7 q^{97} + 4 q^{98} + 3 q^{99}+O(q^{100}) \) 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
\(5\) −2.51181 −1.12332 −0.561659 0.827369i \(-0.689837\pi\)
−0.561659 + 0.827369i \(0.689837\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.73805 0.656919 0.328460 0.944518i \(-0.393470\pi\)
0.328460 + 0.944518i \(0.393470\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.51181 −0.794306
\(11\) −0.642525 −0.193729 −0.0968643 0.995298i \(-0.530881\pi\)
−0.0968643 + 0.995298i \(0.530881\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.58371 −0.716591 −0.358295 0.933608i \(-0.616642\pi\)
−0.358295 + 0.933608i \(0.616642\pi\)
\(14\) 1.73805 0.464512
\(15\) −2.51181 −0.648548
\(16\) 1.00000 0.250000
\(17\) −7.71760 −1.87179 −0.935896 0.352276i \(-0.885408\pi\)
−0.935896 + 0.352276i \(0.885408\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) −2.51181 −0.561659
\(21\) 1.73805 0.379273
\(22\) −0.642525 −0.136987
\(23\) 5.46221 1.13895 0.569475 0.822009i \(-0.307147\pi\)
0.569475 + 0.822009i \(0.307147\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.30921 0.261843
\(26\) −2.58371 −0.506706
\(27\) 1.00000 0.192450
\(28\) 1.73805 0.328460
\(29\) 8.07071 1.49869 0.749347 0.662178i \(-0.230368\pi\)
0.749347 + 0.662178i \(0.230368\pi\)
\(30\) −2.51181 −0.458593
\(31\) −0.677087 −0.121608 −0.0608042 0.998150i \(-0.519367\pi\)
−0.0608042 + 0.998150i \(0.519367\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.642525 −0.111849
\(34\) −7.71760 −1.32356
\(35\) −4.36565 −0.737929
\(36\) 1.00000 0.166667
\(37\) −7.32238 −1.20379 −0.601896 0.798575i \(-0.705588\pi\)
−0.601896 + 0.798575i \(0.705588\pi\)
\(38\) −1.00000 −0.162221
\(39\) −2.58371 −0.413724
\(40\) −2.51181 −0.397153
\(41\) −6.11026 −0.954263 −0.477131 0.878832i \(-0.658323\pi\)
−0.477131 + 0.878832i \(0.658323\pi\)
\(42\) 1.73805 0.268186
\(43\) −1.89833 −0.289492 −0.144746 0.989469i \(-0.546237\pi\)
−0.144746 + 0.989469i \(0.546237\pi\)
\(44\) −0.642525 −0.0968643
\(45\) −2.51181 −0.374439
\(46\) 5.46221 0.805359
\(47\) −10.5353 −1.53673 −0.768363 0.640014i \(-0.778929\pi\)
−0.768363 + 0.640014i \(0.778929\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.97920 −0.568457
\(50\) 1.30921 0.185151
\(51\) −7.71760 −1.08068
\(52\) −2.58371 −0.358295
\(53\) −1.00000 −0.137361
\(54\) 1.00000 0.136083
\(55\) 1.61390 0.217619
\(56\) 1.73805 0.232256
\(57\) −1.00000 −0.132453
\(58\) 8.07071 1.05974
\(59\) 6.39044 0.831965 0.415982 0.909373i \(-0.363438\pi\)
0.415982 + 0.909373i \(0.363438\pi\)
\(60\) −2.51181 −0.324274
\(61\) −4.23726 −0.542526 −0.271263 0.962505i \(-0.587441\pi\)
−0.271263 + 0.962505i \(0.587441\pi\)
\(62\) −0.677087 −0.0859901
\(63\) 1.73805 0.218973
\(64\) 1.00000 0.125000
\(65\) 6.48979 0.804959
\(66\) −0.642525 −0.0790894
\(67\) −12.8611 −1.57124 −0.785618 0.618712i \(-0.787655\pi\)
−0.785618 + 0.618712i \(0.787655\pi\)
\(68\) −7.71760 −0.935896
\(69\) 5.46221 0.657573
\(70\) −4.36565 −0.521795
\(71\) 13.8022 1.63802 0.819009 0.573781i \(-0.194524\pi\)
0.819009 + 0.573781i \(0.194524\pi\)
\(72\) 1.00000 0.117851
\(73\) 5.86264 0.686170 0.343085 0.939304i \(-0.388528\pi\)
0.343085 + 0.939304i \(0.388528\pi\)
\(74\) −7.32238 −0.851209
\(75\) 1.30921 0.151175
\(76\) −1.00000 −0.114708
\(77\) −1.11674 −0.127264
\(78\) −2.58371 −0.292547
\(79\) −9.67603 −1.08864 −0.544319 0.838878i \(-0.683212\pi\)
−0.544319 + 0.838878i \(0.683212\pi\)
\(80\) −2.51181 −0.280829
\(81\) 1.00000 0.111111
\(82\) −6.11026 −0.674766
\(83\) −0.708381 −0.0777549 −0.0388774 0.999244i \(-0.512378\pi\)
−0.0388774 + 0.999244i \(0.512378\pi\)
\(84\) 1.73805 0.189636
\(85\) 19.3852 2.10262
\(86\) −1.89833 −0.204702
\(87\) 8.07071 0.865271
\(88\) −0.642525 −0.0684934
\(89\) 9.33097 0.989081 0.494540 0.869155i \(-0.335336\pi\)
0.494540 + 0.869155i \(0.335336\pi\)
\(90\) −2.51181 −0.264769
\(91\) −4.49060 −0.470743
\(92\) 5.46221 0.569475
\(93\) −0.677087 −0.0702106
\(94\) −10.5353 −1.08663
\(95\) 2.51181 0.257707
\(96\) 1.00000 0.102062
\(97\) −13.5187 −1.37262 −0.686310 0.727309i \(-0.740771\pi\)
−0.686310 + 0.727309i \(0.740771\pi\)
\(98\) −3.97920 −0.401960
\(99\) −0.642525 −0.0645762
\(100\) 1.30921 0.130921
\(101\) −13.1581 −1.30928 −0.654639 0.755941i \(-0.727180\pi\)
−0.654639 + 0.755941i \(0.727180\pi\)
\(102\) −7.71760 −0.764156
\(103\) −6.90709 −0.680576 −0.340288 0.940321i \(-0.610525\pi\)
−0.340288 + 0.940321i \(0.610525\pi\)
\(104\) −2.58371 −0.253353
\(105\) −4.36565 −0.426044
\(106\) −1.00000 −0.0971286
\(107\) −8.90249 −0.860636 −0.430318 0.902677i \(-0.641599\pi\)
−0.430318 + 0.902677i \(0.641599\pi\)
\(108\) 1.00000 0.0962250
\(109\) 5.61503 0.537823 0.268911 0.963165i \(-0.413336\pi\)
0.268911 + 0.963165i \(0.413336\pi\)
\(110\) 1.61390 0.153880
\(111\) −7.32238 −0.695009
\(112\) 1.73805 0.164230
\(113\) −2.37558 −0.223476 −0.111738 0.993738i \(-0.535642\pi\)
−0.111738 + 0.993738i \(0.535642\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −13.7201 −1.27940
\(116\) 8.07071 0.749347
\(117\) −2.58371 −0.238864
\(118\) 6.39044 0.588288
\(119\) −13.4135 −1.22962
\(120\) −2.51181 −0.229296
\(121\) −10.5872 −0.962469
\(122\) −4.23726 −0.383624
\(123\) −6.11026 −0.550944
\(124\) −0.677087 −0.0608042
\(125\) 9.27057 0.829185
\(126\) 1.73805 0.154837
\(127\) −15.6681 −1.39032 −0.695160 0.718855i \(-0.744667\pi\)
−0.695160 + 0.718855i \(0.744667\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.89833 −0.167138
\(130\) 6.48979 0.569192
\(131\) −18.4589 −1.61276 −0.806380 0.591397i \(-0.798576\pi\)
−0.806380 + 0.591397i \(0.798576\pi\)
\(132\) −0.642525 −0.0559246
\(133\) −1.73805 −0.150708
\(134\) −12.8611 −1.11103
\(135\) −2.51181 −0.216183
\(136\) −7.71760 −0.661779
\(137\) 3.80218 0.324842 0.162421 0.986722i \(-0.448070\pi\)
0.162421 + 0.986722i \(0.448070\pi\)
\(138\) 5.46221 0.464974
\(139\) 16.4985 1.39938 0.699691 0.714445i \(-0.253321\pi\)
0.699691 + 0.714445i \(0.253321\pi\)
\(140\) −4.36565 −0.368965
\(141\) −10.5353 −0.887229
\(142\) 13.8022 1.15825
\(143\) 1.66010 0.138824
\(144\) 1.00000 0.0833333
\(145\) −20.2721 −1.68351
\(146\) 5.86264 0.485195
\(147\) −3.97920 −0.328199
\(148\) −7.32238 −0.601896
\(149\) 17.3379 1.42037 0.710187 0.704013i \(-0.248610\pi\)
0.710187 + 0.704013i \(0.248610\pi\)
\(150\) 1.30921 0.106897
\(151\) 17.8421 1.45197 0.725986 0.687709i \(-0.241384\pi\)
0.725986 + 0.687709i \(0.241384\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −7.71760 −0.623931
\(154\) −1.11674 −0.0899893
\(155\) 1.70072 0.136605
\(156\) −2.58371 −0.206862
\(157\) −0.480011 −0.0383091 −0.0191545 0.999817i \(-0.506097\pi\)
−0.0191545 + 0.999817i \(0.506097\pi\)
\(158\) −9.67603 −0.769783
\(159\) −1.00000 −0.0793052
\(160\) −2.51181 −0.198576
\(161\) 9.49357 0.748198
\(162\) 1.00000 0.0785674
\(163\) −23.0963 −1.80904 −0.904520 0.426431i \(-0.859771\pi\)
−0.904520 + 0.426431i \(0.859771\pi\)
\(164\) −6.11026 −0.477131
\(165\) 1.61390 0.125642
\(166\) −0.708381 −0.0549810
\(167\) −14.1792 −1.09722 −0.548609 0.836079i \(-0.684842\pi\)
−0.548609 + 0.836079i \(0.684842\pi\)
\(168\) 1.73805 0.134093
\(169\) −6.32447 −0.486497
\(170\) 19.3852 1.48678
\(171\) −1.00000 −0.0764719
\(172\) −1.89833 −0.144746
\(173\) 19.5868 1.48915 0.744577 0.667537i \(-0.232651\pi\)
0.744577 + 0.667537i \(0.232651\pi\)
\(174\) 8.07071 0.611839
\(175\) 2.27547 0.172010
\(176\) −0.642525 −0.0484322
\(177\) 6.39044 0.480335
\(178\) 9.33097 0.699386
\(179\) −2.28116 −0.170502 −0.0852510 0.996360i \(-0.527169\pi\)
−0.0852510 + 0.996360i \(0.527169\pi\)
\(180\) −2.51181 −0.187220
\(181\) −20.3337 −1.51139 −0.755696 0.654922i \(-0.772701\pi\)
−0.755696 + 0.654922i \(0.772701\pi\)
\(182\) −4.49060 −0.332865
\(183\) −4.23726 −0.313228
\(184\) 5.46221 0.402680
\(185\) 18.3925 1.35224
\(186\) −0.677087 −0.0496464
\(187\) 4.95875 0.362620
\(188\) −10.5353 −0.768363
\(189\) 1.73805 0.126424
\(190\) 2.51181 0.182226
\(191\) 8.19066 0.592655 0.296328 0.955086i \(-0.404238\pi\)
0.296328 + 0.955086i \(0.404238\pi\)
\(192\) 1.00000 0.0721688
\(193\) 5.39374 0.388250 0.194125 0.980977i \(-0.437813\pi\)
0.194125 + 0.980977i \(0.437813\pi\)
\(194\) −13.5187 −0.970590
\(195\) 6.48979 0.464744
\(196\) −3.97920 −0.284228
\(197\) −0.633256 −0.0451176 −0.0225588 0.999746i \(-0.507181\pi\)
−0.0225588 + 0.999746i \(0.507181\pi\)
\(198\) −0.642525 −0.0456623
\(199\) 14.4610 1.02511 0.512557 0.858653i \(-0.328699\pi\)
0.512557 + 0.858653i \(0.328699\pi\)
\(200\) 1.30921 0.0925754
\(201\) −12.8611 −0.907154
\(202\) −13.1581 −0.925800
\(203\) 14.0273 0.984521
\(204\) −7.71760 −0.540340
\(205\) 15.3478 1.07194
\(206\) −6.90709 −0.481240
\(207\) 5.46221 0.379650
\(208\) −2.58371 −0.179148
\(209\) 0.642525 0.0444444
\(210\) −4.36565 −0.301258
\(211\) 15.5116 1.06786 0.533931 0.845528i \(-0.320714\pi\)
0.533931 + 0.845528i \(0.320714\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 13.8022 0.945710
\(214\) −8.90249 −0.608562
\(215\) 4.76824 0.325192
\(216\) 1.00000 0.0680414
\(217\) −1.17681 −0.0798869
\(218\) 5.61503 0.380298
\(219\) 5.86264 0.396160
\(220\) 1.61390 0.108809
\(221\) 19.9400 1.34131
\(222\) −7.32238 −0.491446
\(223\) −15.6842 −1.05029 −0.525147 0.851012i \(-0.675990\pi\)
−0.525147 + 0.851012i \(0.675990\pi\)
\(224\) 1.73805 0.116128
\(225\) 1.30921 0.0872810
\(226\) −2.37558 −0.158021
\(227\) −4.07495 −0.270464 −0.135232 0.990814i \(-0.543178\pi\)
−0.135232 + 0.990814i \(0.543178\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −14.8872 −0.983777 −0.491889 0.870658i \(-0.663693\pi\)
−0.491889 + 0.870658i \(0.663693\pi\)
\(230\) −13.7201 −0.904674
\(231\) −1.11674 −0.0734760
\(232\) 8.07071 0.529868
\(233\) 15.0002 0.982697 0.491348 0.870963i \(-0.336504\pi\)
0.491348 + 0.870963i \(0.336504\pi\)
\(234\) −2.58371 −0.168902
\(235\) 26.4626 1.72623
\(236\) 6.39044 0.415982
\(237\) −9.67603 −0.628525
\(238\) −13.4135 −0.869470
\(239\) −5.82577 −0.376838 −0.188419 0.982089i \(-0.560336\pi\)
−0.188419 + 0.982089i \(0.560336\pi\)
\(240\) −2.51181 −0.162137
\(241\) 6.25361 0.402830 0.201415 0.979506i \(-0.435446\pi\)
0.201415 + 0.979506i \(0.435446\pi\)
\(242\) −10.5872 −0.680569
\(243\) 1.00000 0.0641500
\(244\) −4.23726 −0.271263
\(245\) 9.99501 0.638558
\(246\) −6.11026 −0.389576
\(247\) 2.58371 0.164397
\(248\) −0.677087 −0.0429951
\(249\) −0.708381 −0.0448918
\(250\) 9.27057 0.586322
\(251\) 15.9849 1.00896 0.504480 0.863423i \(-0.331684\pi\)
0.504480 + 0.863423i \(0.331684\pi\)
\(252\) 1.73805 0.109487
\(253\) −3.50961 −0.220647
\(254\) −15.6681 −0.983105
\(255\) 19.3852 1.21395
\(256\) 1.00000 0.0625000
\(257\) 7.64274 0.476741 0.238370 0.971174i \(-0.423387\pi\)
0.238370 + 0.971174i \(0.423387\pi\)
\(258\) −1.89833 −0.118185
\(259\) −12.7266 −0.790794
\(260\) 6.48979 0.402480
\(261\) 8.07071 0.499564
\(262\) −18.4589 −1.14039
\(263\) 7.70372 0.475032 0.237516 0.971384i \(-0.423667\pi\)
0.237516 + 0.971384i \(0.423667\pi\)
\(264\) −0.642525 −0.0395447
\(265\) 2.51181 0.154300
\(266\) −1.73805 −0.106566
\(267\) 9.33097 0.571046
\(268\) −12.8611 −0.785618
\(269\) 18.4670 1.12595 0.562977 0.826473i \(-0.309656\pi\)
0.562977 + 0.826473i \(0.309656\pi\)
\(270\) −2.51181 −0.152864
\(271\) −6.67605 −0.405541 −0.202771 0.979226i \(-0.564995\pi\)
−0.202771 + 0.979226i \(0.564995\pi\)
\(272\) −7.71760 −0.467948
\(273\) −4.49060 −0.271783
\(274\) 3.80218 0.229698
\(275\) −0.841203 −0.0507265
\(276\) 5.46221 0.328787
\(277\) −7.19303 −0.432188 −0.216094 0.976373i \(-0.569332\pi\)
−0.216094 + 0.976373i \(0.569332\pi\)
\(278\) 16.4985 0.989513
\(279\) −0.677087 −0.0405361
\(280\) −4.36565 −0.260897
\(281\) −20.6372 −1.23111 −0.615557 0.788092i \(-0.711069\pi\)
−0.615557 + 0.788092i \(0.711069\pi\)
\(282\) −10.5353 −0.627366
\(283\) 10.1190 0.601512 0.300756 0.953701i \(-0.402761\pi\)
0.300756 + 0.953701i \(0.402761\pi\)
\(284\) 13.8022 0.819009
\(285\) 2.51181 0.148787
\(286\) 1.66010 0.0981635
\(287\) −10.6199 −0.626874
\(288\) 1.00000 0.0589256
\(289\) 42.5613 2.50361
\(290\) −20.2721 −1.19042
\(291\) −13.5187 −0.792483
\(292\) 5.86264 0.343085
\(293\) −11.1099 −0.649050 −0.324525 0.945877i \(-0.605204\pi\)
−0.324525 + 0.945877i \(0.605204\pi\)
\(294\) −3.97920 −0.232072
\(295\) −16.0516 −0.934561
\(296\) −7.32238 −0.425605
\(297\) −0.642525 −0.0372831
\(298\) 17.3379 1.00436
\(299\) −14.1127 −0.816161
\(300\) 1.30921 0.0755875
\(301\) −3.29938 −0.190173
\(302\) 17.8421 1.02670
\(303\) −13.1581 −0.755913
\(304\) −1.00000 −0.0573539
\(305\) 10.6432 0.609429
\(306\) −7.71760 −0.441186
\(307\) −29.0392 −1.65735 −0.828677 0.559727i \(-0.810906\pi\)
−0.828677 + 0.559727i \(0.810906\pi\)
\(308\) −1.11674 −0.0636320
\(309\) −6.90709 −0.392931
\(310\) 1.70072 0.0965942
\(311\) −30.7392 −1.74306 −0.871530 0.490343i \(-0.836872\pi\)
−0.871530 + 0.490343i \(0.836872\pi\)
\(312\) −2.58371 −0.146274
\(313\) −2.12783 −0.120272 −0.0601361 0.998190i \(-0.519153\pi\)
−0.0601361 + 0.998190i \(0.519153\pi\)
\(314\) −0.480011 −0.0270886
\(315\) −4.36565 −0.245976
\(316\) −9.67603 −0.544319
\(317\) −27.0644 −1.52009 −0.760043 0.649873i \(-0.774822\pi\)
−0.760043 + 0.649873i \(0.774822\pi\)
\(318\) −1.00000 −0.0560772
\(319\) −5.18563 −0.290340
\(320\) −2.51181 −0.140415
\(321\) −8.90249 −0.496889
\(322\) 9.49357 0.529056
\(323\) 7.71760 0.429419
\(324\) 1.00000 0.0555556
\(325\) −3.38262 −0.187634
\(326\) −23.0963 −1.27918
\(327\) 5.61503 0.310512
\(328\) −6.11026 −0.337383
\(329\) −18.3108 −1.00951
\(330\) 1.61390 0.0888425
\(331\) 2.20656 0.121284 0.0606419 0.998160i \(-0.480685\pi\)
0.0606419 + 0.998160i \(0.480685\pi\)
\(332\) −0.708381 −0.0388774
\(333\) −7.32238 −0.401264
\(334\) −14.1792 −0.775850
\(335\) 32.3048 1.76500
\(336\) 1.73805 0.0948182
\(337\) −22.5004 −1.22568 −0.612838 0.790209i \(-0.709972\pi\)
−0.612838 + 0.790209i \(0.709972\pi\)
\(338\) −6.32447 −0.344006
\(339\) −2.37558 −0.129024
\(340\) 19.3852 1.05131
\(341\) 0.435045 0.0235590
\(342\) −1.00000 −0.0540738
\(343\) −19.0823 −1.03035
\(344\) −1.89833 −0.102351
\(345\) −13.7201 −0.738664
\(346\) 19.5868 1.05299
\(347\) 5.83468 0.313222 0.156611 0.987660i \(-0.449943\pi\)
0.156611 + 0.987660i \(0.449943\pi\)
\(348\) 8.07071 0.432635
\(349\) 15.7691 0.844103 0.422052 0.906572i \(-0.361310\pi\)
0.422052 + 0.906572i \(0.361310\pi\)
\(350\) 2.27547 0.121629
\(351\) −2.58371 −0.137908
\(352\) −0.642525 −0.0342467
\(353\) 18.4969 0.984491 0.492245 0.870456i \(-0.336176\pi\)
0.492245 + 0.870456i \(0.336176\pi\)
\(354\) 6.39044 0.339648
\(355\) −34.6685 −1.84002
\(356\) 9.33097 0.494540
\(357\) −13.4135 −0.709920
\(358\) −2.28116 −0.120563
\(359\) −3.10195 −0.163715 −0.0818574 0.996644i \(-0.526085\pi\)
−0.0818574 + 0.996644i \(0.526085\pi\)
\(360\) −2.51181 −0.132384
\(361\) 1.00000 0.0526316
\(362\) −20.3337 −1.06872
\(363\) −10.5872 −0.555682
\(364\) −4.49060 −0.235371
\(365\) −14.7259 −0.770787
\(366\) −4.23726 −0.221485
\(367\) 12.3041 0.642268 0.321134 0.947034i \(-0.395936\pi\)
0.321134 + 0.947034i \(0.395936\pi\)
\(368\) 5.46221 0.284738
\(369\) −6.11026 −0.318088
\(370\) 18.3925 0.956179
\(371\) −1.73805 −0.0902348
\(372\) −0.677087 −0.0351053
\(373\) 10.1898 0.527606 0.263803 0.964577i \(-0.415023\pi\)
0.263803 + 0.964577i \(0.415023\pi\)
\(374\) 4.95875 0.256411
\(375\) 9.27057 0.478730
\(376\) −10.5353 −0.543315
\(377\) −20.8523 −1.07395
\(378\) 1.73805 0.0893954
\(379\) 6.85298 0.352014 0.176007 0.984389i \(-0.443682\pi\)
0.176007 + 0.984389i \(0.443682\pi\)
\(380\) 2.51181 0.128853
\(381\) −15.6681 −0.802702
\(382\) 8.19066 0.419071
\(383\) −20.4129 −1.04305 −0.521524 0.853237i \(-0.674636\pi\)
−0.521524 + 0.853237i \(0.674636\pi\)
\(384\) 1.00000 0.0510310
\(385\) 2.80504 0.142958
\(386\) 5.39374 0.274534
\(387\) −1.89833 −0.0964973
\(388\) −13.5187 −0.686310
\(389\) −3.44093 −0.174462 −0.0872310 0.996188i \(-0.527802\pi\)
−0.0872310 + 0.996188i \(0.527802\pi\)
\(390\) 6.48979 0.328623
\(391\) −42.1552 −2.13188
\(392\) −3.97920 −0.200980
\(393\) −18.4589 −0.931128
\(394\) −0.633256 −0.0319030
\(395\) 24.3044 1.22289
\(396\) −0.642525 −0.0322881
\(397\) 20.7359 1.04071 0.520353 0.853951i \(-0.325800\pi\)
0.520353 + 0.853951i \(0.325800\pi\)
\(398\) 14.4610 0.724864
\(399\) −1.73805 −0.0870111
\(400\) 1.30921 0.0654607
\(401\) 5.53945 0.276627 0.138314 0.990388i \(-0.455832\pi\)
0.138314 + 0.990388i \(0.455832\pi\)
\(402\) −12.8611 −0.641455
\(403\) 1.74939 0.0871435
\(404\) −13.1581 −0.654639
\(405\) −2.51181 −0.124813
\(406\) 14.0273 0.696161
\(407\) 4.70481 0.233209
\(408\) −7.71760 −0.382078
\(409\) 10.6100 0.524631 0.262315 0.964982i \(-0.415514\pi\)
0.262315 + 0.964982i \(0.415514\pi\)
\(410\) 15.3478 0.757976
\(411\) 3.80218 0.187548
\(412\) −6.90709 −0.340288
\(413\) 11.1069 0.546534
\(414\) 5.46221 0.268453
\(415\) 1.77932 0.0873435
\(416\) −2.58371 −0.126677
\(417\) 16.4985 0.807934
\(418\) 0.642525 0.0314269
\(419\) −2.58451 −0.126262 −0.0631309 0.998005i \(-0.520109\pi\)
−0.0631309 + 0.998005i \(0.520109\pi\)
\(420\) −4.36565 −0.213022
\(421\) 29.8910 1.45680 0.728399 0.685153i \(-0.240265\pi\)
0.728399 + 0.685153i \(0.240265\pi\)
\(422\) 15.5116 0.755092
\(423\) −10.5353 −0.512242
\(424\) −1.00000 −0.0485643
\(425\) −10.1040 −0.490116
\(426\) 13.8022 0.668718
\(427\) −7.36456 −0.356396
\(428\) −8.90249 −0.430318
\(429\) 1.66010 0.0801502
\(430\) 4.76824 0.229945
\(431\) 15.2898 0.736486 0.368243 0.929730i \(-0.379959\pi\)
0.368243 + 0.929730i \(0.379959\pi\)
\(432\) 1.00000 0.0481125
\(433\) −1.06601 −0.0512292 −0.0256146 0.999672i \(-0.508154\pi\)
−0.0256146 + 0.999672i \(0.508154\pi\)
\(434\) −1.17681 −0.0564886
\(435\) −20.2721 −0.971974
\(436\) 5.61503 0.268911
\(437\) −5.46221 −0.261293
\(438\) 5.86264 0.280128
\(439\) −13.9808 −0.667265 −0.333632 0.942703i \(-0.608274\pi\)
−0.333632 + 0.942703i \(0.608274\pi\)
\(440\) 1.61390 0.0769399
\(441\) −3.97920 −0.189486
\(442\) 19.9400 0.948449
\(443\) 24.7683 1.17678 0.588388 0.808579i \(-0.299763\pi\)
0.588388 + 0.808579i \(0.299763\pi\)
\(444\) −7.32238 −0.347505
\(445\) −23.4377 −1.11105
\(446\) −15.6842 −0.742670
\(447\) 17.3379 0.820053
\(448\) 1.73805 0.0821149
\(449\) 38.3823 1.81137 0.905687 0.423948i \(-0.139356\pi\)
0.905687 + 0.423948i \(0.139356\pi\)
\(450\) 1.30921 0.0617170
\(451\) 3.92600 0.184868
\(452\) −2.37558 −0.111738
\(453\) 17.8421 0.838297
\(454\) −4.07495 −0.191247
\(455\) 11.2795 0.528793
\(456\) −1.00000 −0.0468293
\(457\) 7.38642 0.345522 0.172761 0.984964i \(-0.444731\pi\)
0.172761 + 0.984964i \(0.444731\pi\)
\(458\) −14.8872 −0.695635
\(459\) −7.71760 −0.360227
\(460\) −13.7201 −0.639701
\(461\) 15.1036 0.703444 0.351722 0.936104i \(-0.385596\pi\)
0.351722 + 0.936104i \(0.385596\pi\)
\(462\) −1.11674 −0.0519553
\(463\) −36.1124 −1.67829 −0.839143 0.543911i \(-0.816943\pi\)
−0.839143 + 0.543911i \(0.816943\pi\)
\(464\) 8.07071 0.374673
\(465\) 1.70072 0.0788688
\(466\) 15.0002 0.694871
\(467\) 0.364195 0.0168529 0.00842647 0.999964i \(-0.497318\pi\)
0.00842647 + 0.999964i \(0.497318\pi\)
\(468\) −2.58371 −0.119432
\(469\) −22.3532 −1.03218
\(470\) 26.4626 1.22063
\(471\) −0.480011 −0.0221178
\(472\) 6.39044 0.294144
\(473\) 1.21972 0.0560829
\(474\) −9.67603 −0.444435
\(475\) −1.30921 −0.0600709
\(476\) −13.4135 −0.614808
\(477\) −1.00000 −0.0457869
\(478\) −5.82577 −0.266465
\(479\) 30.1157 1.37602 0.688011 0.725701i \(-0.258484\pi\)
0.688011 + 0.725701i \(0.258484\pi\)
\(480\) −2.51181 −0.114648
\(481\) 18.9189 0.862626
\(482\) 6.25361 0.284844
\(483\) 9.49357 0.431973
\(484\) −10.5872 −0.481235
\(485\) 33.9566 1.54189
\(486\) 1.00000 0.0453609
\(487\) 29.7321 1.34729 0.673646 0.739054i \(-0.264727\pi\)
0.673646 + 0.739054i \(0.264727\pi\)
\(488\) −4.23726 −0.191812
\(489\) −23.0963 −1.04445
\(490\) 9.99501 0.451528
\(491\) −0.905199 −0.0408511 −0.0204255 0.999791i \(-0.506502\pi\)
−0.0204255 + 0.999791i \(0.506502\pi\)
\(492\) −6.11026 −0.275472
\(493\) −62.2865 −2.80524
\(494\) 2.58371 0.116246
\(495\) 1.61390 0.0725396
\(496\) −0.677087 −0.0304021
\(497\) 23.9888 1.07605
\(498\) −0.708381 −0.0317433
\(499\) −28.5614 −1.27859 −0.639293 0.768964i \(-0.720773\pi\)
−0.639293 + 0.768964i \(0.720773\pi\)
\(500\) 9.27057 0.414593
\(501\) −14.1792 −0.633479
\(502\) 15.9849 0.713443
\(503\) −12.3975 −0.552775 −0.276388 0.961046i \(-0.589137\pi\)
−0.276388 + 0.961046i \(0.589137\pi\)
\(504\) 1.73805 0.0774187
\(505\) 33.0507 1.47074
\(506\) −3.50961 −0.156021
\(507\) −6.32447 −0.280879
\(508\) −15.6681 −0.695160
\(509\) −30.7216 −1.36171 −0.680855 0.732419i \(-0.738391\pi\)
−0.680855 + 0.732419i \(0.738391\pi\)
\(510\) 19.3852 0.858390
\(511\) 10.1895 0.450758
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 7.64274 0.337107
\(515\) 17.3493 0.764503
\(516\) −1.89833 −0.0835692
\(517\) 6.76917 0.297708
\(518\) −12.7266 −0.559176
\(519\) 19.5868 0.859763
\(520\) 6.48979 0.284596
\(521\) −6.28277 −0.275253 −0.137627 0.990484i \(-0.543947\pi\)
−0.137627 + 0.990484i \(0.543947\pi\)
\(522\) 8.07071 0.353245
\(523\) 10.3593 0.452979 0.226490 0.974014i \(-0.427275\pi\)
0.226490 + 0.974014i \(0.427275\pi\)
\(524\) −18.4589 −0.806380
\(525\) 2.27547 0.0993098
\(526\) 7.70372 0.335898
\(527\) 5.22548 0.227626
\(528\) −0.642525 −0.0279623
\(529\) 6.83577 0.297207
\(530\) 2.51181 0.109106
\(531\) 6.39044 0.277322
\(532\) −1.73805 −0.0753538
\(533\) 15.7871 0.683816
\(534\) 9.33097 0.403790
\(535\) 22.3614 0.966768
\(536\) −12.8611 −0.555516
\(537\) −2.28116 −0.0984394
\(538\) 18.4670 0.796170
\(539\) 2.55673 0.110126
\(540\) −2.51181 −0.108091
\(541\) 14.8604 0.638897 0.319449 0.947604i \(-0.396502\pi\)
0.319449 + 0.947604i \(0.396502\pi\)
\(542\) −6.67605 −0.286761
\(543\) −20.3337 −0.872603
\(544\) −7.71760 −0.330889
\(545\) −14.1039 −0.604146
\(546\) −4.49060 −0.192180
\(547\) 35.3324 1.51071 0.755353 0.655318i \(-0.227466\pi\)
0.755353 + 0.655318i \(0.227466\pi\)
\(548\) 3.80218 0.162421
\(549\) −4.23726 −0.180842
\(550\) −0.841203 −0.0358690
\(551\) −8.07071 −0.343824
\(552\) 5.46221 0.232487
\(553\) −16.8174 −0.715147
\(554\) −7.19303 −0.305603
\(555\) 18.3925 0.780716
\(556\) 16.4985 0.699691
\(557\) −38.4565 −1.62946 −0.814728 0.579843i \(-0.803114\pi\)
−0.814728 + 0.579843i \(0.803114\pi\)
\(558\) −0.677087 −0.0286634
\(559\) 4.90472 0.207447
\(560\) −4.36565 −0.184482
\(561\) 4.95875 0.209359
\(562\) −20.6372 −0.870530
\(563\) −14.2639 −0.601153 −0.300577 0.953758i \(-0.597179\pi\)
−0.300577 + 0.953758i \(0.597179\pi\)
\(564\) −10.5353 −0.443615
\(565\) 5.96702 0.251034
\(566\) 10.1190 0.425333
\(567\) 1.73805 0.0729910
\(568\) 13.8022 0.579127
\(569\) 36.6158 1.53501 0.767507 0.641041i \(-0.221497\pi\)
0.767507 + 0.641041i \(0.221497\pi\)
\(570\) 2.51181 0.105208
\(571\) −21.1733 −0.886077 −0.443038 0.896503i \(-0.646099\pi\)
−0.443038 + 0.896503i \(0.646099\pi\)
\(572\) 1.66010 0.0694121
\(573\) 8.19066 0.342170
\(574\) −10.6199 −0.443267
\(575\) 7.15121 0.298226
\(576\) 1.00000 0.0416667
\(577\) −43.8741 −1.82650 −0.913252 0.407396i \(-0.866437\pi\)
−0.913252 + 0.407396i \(0.866437\pi\)
\(578\) 42.5613 1.77032
\(579\) 5.39374 0.224156
\(580\) −20.2721 −0.841754
\(581\) −1.23120 −0.0510787
\(582\) −13.5187 −0.560370
\(583\) 0.642525 0.0266107
\(584\) 5.86264 0.242598
\(585\) 6.48979 0.268320
\(586\) −11.1099 −0.458947
\(587\) 15.0810 0.622460 0.311230 0.950335i \(-0.399259\pi\)
0.311230 + 0.950335i \(0.399259\pi\)
\(588\) −3.97920 −0.164099
\(589\) 0.677087 0.0278989
\(590\) −16.0516 −0.660834
\(591\) −0.633256 −0.0260487
\(592\) −7.32238 −0.300948
\(593\) 28.6154 1.17509 0.587546 0.809191i \(-0.300094\pi\)
0.587546 + 0.809191i \(0.300094\pi\)
\(594\) −0.642525 −0.0263631
\(595\) 33.6923 1.38125
\(596\) 17.3379 0.710187
\(597\) 14.4610 0.591849
\(598\) −14.1127 −0.577113
\(599\) −24.1821 −0.988055 −0.494027 0.869446i \(-0.664476\pi\)
−0.494027 + 0.869446i \(0.664476\pi\)
\(600\) 1.30921 0.0534485
\(601\) 8.69776 0.354789 0.177394 0.984140i \(-0.443233\pi\)
0.177394 + 0.984140i \(0.443233\pi\)
\(602\) −3.29938 −0.134473
\(603\) −12.8611 −0.523746
\(604\) 17.8421 0.725986
\(605\) 26.5930 1.08116
\(606\) −13.1581 −0.534511
\(607\) 0.653287 0.0265161 0.0132581 0.999912i \(-0.495780\pi\)
0.0132581 + 0.999912i \(0.495780\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 14.0273 0.568413
\(610\) 10.6432 0.430931
\(611\) 27.2200 1.10120
\(612\) −7.71760 −0.311965
\(613\) 29.4154 1.18808 0.594038 0.804437i \(-0.297533\pi\)
0.594038 + 0.804437i \(0.297533\pi\)
\(614\) −29.0392 −1.17193
\(615\) 15.3478 0.618885
\(616\) −1.11674 −0.0449947
\(617\) 13.0372 0.524856 0.262428 0.964952i \(-0.415477\pi\)
0.262428 + 0.964952i \(0.415477\pi\)
\(618\) −6.90709 −0.277844
\(619\) −13.9812 −0.561954 −0.280977 0.959715i \(-0.590658\pi\)
−0.280977 + 0.959715i \(0.590658\pi\)
\(620\) 1.70072 0.0683024
\(621\) 5.46221 0.219191
\(622\) −30.7392 −1.23253
\(623\) 16.2176 0.649746
\(624\) −2.58371 −0.103431
\(625\) −29.8320 −1.19328
\(626\) −2.12783 −0.0850452
\(627\) 0.642525 0.0256600
\(628\) −0.480011 −0.0191545
\(629\) 56.5112 2.25325
\(630\) −4.36565 −0.173932
\(631\) 3.45346 0.137480 0.0687399 0.997635i \(-0.478102\pi\)
0.0687399 + 0.997635i \(0.478102\pi\)
\(632\) −9.67603 −0.384892
\(633\) 15.5116 0.616530
\(634\) −27.0644 −1.07486
\(635\) 39.3554 1.56177
\(636\) −1.00000 −0.0396526
\(637\) 10.2811 0.407351
\(638\) −5.18563 −0.205301
\(639\) 13.8022 0.546006
\(640\) −2.51181 −0.0992882
\(641\) −1.22750 −0.0484834 −0.0242417 0.999706i \(-0.507717\pi\)
−0.0242417 + 0.999706i \(0.507717\pi\)
\(642\) −8.90249 −0.351353
\(643\) −6.02283 −0.237517 −0.118759 0.992923i \(-0.537891\pi\)
−0.118759 + 0.992923i \(0.537891\pi\)
\(644\) 9.49357 0.374099
\(645\) 4.76824 0.187749
\(646\) 7.71760 0.303645
\(647\) 1.59626 0.0627553 0.0313776 0.999508i \(-0.490011\pi\)
0.0313776 + 0.999508i \(0.490011\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.10602 −0.161175
\(650\) −3.38262 −0.132677
\(651\) −1.17681 −0.0461227
\(652\) −23.0963 −0.904520
\(653\) 15.8783 0.621367 0.310683 0.950513i \(-0.399442\pi\)
0.310683 + 0.950513i \(0.399442\pi\)
\(654\) 5.61503 0.219565
\(655\) 46.3653 1.81164
\(656\) −6.11026 −0.238566
\(657\) 5.86264 0.228723
\(658\) −18.3108 −0.713828
\(659\) −45.7024 −1.78031 −0.890157 0.455655i \(-0.849405\pi\)
−0.890157 + 0.455655i \(0.849405\pi\)
\(660\) 1.61390 0.0628211
\(661\) 28.9964 1.12783 0.563914 0.825833i \(-0.309295\pi\)
0.563914 + 0.825833i \(0.309295\pi\)
\(662\) 2.20656 0.0857605
\(663\) 19.9400 0.774405
\(664\) −0.708381 −0.0274905
\(665\) 4.36565 0.169293
\(666\) −7.32238 −0.283736
\(667\) 44.0839 1.70694
\(668\) −14.1792 −0.548609
\(669\) −15.6842 −0.606387
\(670\) 32.3048 1.24804
\(671\) 2.72255 0.105103
\(672\) 1.73805 0.0670466
\(673\) −41.9506 −1.61708 −0.808539 0.588443i \(-0.799741\pi\)
−0.808539 + 0.588443i \(0.799741\pi\)
\(674\) −22.5004 −0.866684
\(675\) 1.30921 0.0503917
\(676\) −6.32447 −0.243249
\(677\) −8.85102 −0.340172 −0.170086 0.985429i \(-0.554405\pi\)
−0.170086 + 0.985429i \(0.554405\pi\)
\(678\) −2.37558 −0.0912336
\(679\) −23.4962 −0.901701
\(680\) 19.3852 0.743388
\(681\) −4.07495 −0.156153
\(682\) 0.435045 0.0166587
\(683\) −22.2419 −0.851064 −0.425532 0.904943i \(-0.639913\pi\)
−0.425532 + 0.904943i \(0.639913\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −9.55037 −0.364901
\(686\) −19.0823 −0.728567
\(687\) −14.8872 −0.567984
\(688\) −1.89833 −0.0723730
\(689\) 2.58371 0.0984313
\(690\) −13.7201 −0.522314
\(691\) 12.5452 0.477241 0.238621 0.971113i \(-0.423305\pi\)
0.238621 + 0.971113i \(0.423305\pi\)
\(692\) 19.5868 0.744577
\(693\) −1.11674 −0.0424214
\(694\) 5.83468 0.221481
\(695\) −41.4411 −1.57195
\(696\) 8.07071 0.305919
\(697\) 47.1566 1.78618
\(698\) 15.7691 0.596871
\(699\) 15.0002 0.567360
\(700\) 2.27547 0.0860048
\(701\) −12.9068 −0.487484 −0.243742 0.969840i \(-0.578375\pi\)
−0.243742 + 0.969840i \(0.578375\pi\)
\(702\) −2.58371 −0.0975157
\(703\) 7.32238 0.276169
\(704\) −0.642525 −0.0242161
\(705\) 26.4626 0.996641
\(706\) 18.4969 0.696140
\(707\) −22.8694 −0.860091
\(708\) 6.39044 0.240167
\(709\) −13.8019 −0.518340 −0.259170 0.965832i \(-0.583449\pi\)
−0.259170 + 0.965832i \(0.583449\pi\)
\(710\) −34.6685 −1.30109
\(711\) −9.67603 −0.362879
\(712\) 9.33097 0.349693
\(713\) −3.69839 −0.138506
\(714\) −13.4135 −0.501989
\(715\) −4.16985 −0.155944
\(716\) −2.28116 −0.0852510
\(717\) −5.82577 −0.217568
\(718\) −3.10195 −0.115764
\(719\) 14.7191 0.548928 0.274464 0.961597i \(-0.411500\pi\)
0.274464 + 0.961597i \(0.411500\pi\)
\(720\) −2.51181 −0.0936098
\(721\) −12.0048 −0.447084
\(722\) 1.00000 0.0372161
\(723\) 6.25361 0.232574
\(724\) −20.3337 −0.755696
\(725\) 10.5663 0.392422
\(726\) −10.5872 −0.392926
\(727\) −38.5602 −1.43012 −0.715059 0.699064i \(-0.753600\pi\)
−0.715059 + 0.699064i \(0.753600\pi\)
\(728\) −4.49060 −0.166433
\(729\) 1.00000 0.0370370
\(730\) −14.7259 −0.545028
\(731\) 14.6505 0.541869
\(732\) −4.23726 −0.156614
\(733\) 13.4126 0.495407 0.247704 0.968836i \(-0.420324\pi\)
0.247704 + 0.968836i \(0.420324\pi\)
\(734\) 12.3041 0.454152
\(735\) 9.99501 0.368671
\(736\) 5.46221 0.201340
\(737\) 8.26360 0.304393
\(738\) −6.11026 −0.224922
\(739\) 26.2170 0.964408 0.482204 0.876059i \(-0.339836\pi\)
0.482204 + 0.876059i \(0.339836\pi\)
\(740\) 18.3925 0.676120
\(741\) 2.58371 0.0949148
\(742\) −1.73805 −0.0638057
\(743\) 15.7061 0.576202 0.288101 0.957600i \(-0.406976\pi\)
0.288101 + 0.957600i \(0.406976\pi\)
\(744\) −0.677087 −0.0248232
\(745\) −43.5495 −1.59553
\(746\) 10.1898 0.373074
\(747\) −0.708381 −0.0259183
\(748\) 4.95875 0.181310
\(749\) −15.4729 −0.565369
\(750\) 9.27057 0.338513
\(751\) 6.45204 0.235438 0.117719 0.993047i \(-0.462442\pi\)
0.117719 + 0.993047i \(0.462442\pi\)
\(752\) −10.5353 −0.384182
\(753\) 15.9849 0.582523
\(754\) −20.8523 −0.759397
\(755\) −44.8161 −1.63103
\(756\) 1.73805 0.0632121
\(757\) 49.6307 1.80386 0.901930 0.431883i \(-0.142151\pi\)
0.901930 + 0.431883i \(0.142151\pi\)
\(758\) 6.85298 0.248912
\(759\) −3.50961 −0.127391
\(760\) 2.51181 0.0911131
\(761\) −7.00710 −0.254007 −0.127004 0.991902i \(-0.540536\pi\)
−0.127004 + 0.991902i \(0.540536\pi\)
\(762\) −15.6681 −0.567596
\(763\) 9.75918 0.353306
\(764\) 8.19066 0.296328
\(765\) 19.3852 0.700873
\(766\) −20.4129 −0.737546
\(767\) −16.5110 −0.596178
\(768\) 1.00000 0.0360844
\(769\) −24.1923 −0.872397 −0.436198 0.899851i \(-0.643675\pi\)
−0.436198 + 0.899851i \(0.643675\pi\)
\(770\) 2.80504 0.101087
\(771\) 7.64274 0.275246
\(772\) 5.39374 0.194125
\(773\) −51.2203 −1.84227 −0.921134 0.389246i \(-0.872736\pi\)
−0.921134 + 0.389246i \(0.872736\pi\)
\(774\) −1.89833 −0.0682339
\(775\) −0.886452 −0.0318423
\(776\) −13.5187 −0.485295
\(777\) −12.7266 −0.456565
\(778\) −3.44093 −0.123363
\(779\) 6.11026 0.218923
\(780\) 6.48979 0.232372
\(781\) −8.86825 −0.317331
\(782\) −42.1552 −1.50747
\(783\) 8.07071 0.288424
\(784\) −3.97920 −0.142114
\(785\) 1.20570 0.0430333
\(786\) −18.4589 −0.658407
\(787\) 8.50365 0.303122 0.151561 0.988448i \(-0.451570\pi\)
0.151561 + 0.988448i \(0.451570\pi\)
\(788\) −0.633256 −0.0225588
\(789\) 7.70372 0.274260
\(790\) 24.3044 0.864711
\(791\) −4.12887 −0.146806
\(792\) −0.642525 −0.0228311
\(793\) 10.9478 0.388769
\(794\) 20.7359 0.735891
\(795\) 2.51181 0.0890849
\(796\) 14.4610 0.512557
\(797\) −44.4551 −1.57468 −0.787340 0.616519i \(-0.788542\pi\)
−0.787340 + 0.616519i \(0.788542\pi\)
\(798\) −1.73805 −0.0615261
\(799\) 81.3069 2.87643
\(800\) 1.30921 0.0462877
\(801\) 9.33097 0.329694
\(802\) 5.53945 0.195605
\(803\) −3.76689 −0.132931
\(804\) −12.8611 −0.453577
\(805\) −23.8461 −0.840465
\(806\) 1.74939 0.0616197
\(807\) 18.4670 0.650070
\(808\) −13.1581 −0.462900
\(809\) −20.7555 −0.729726 −0.364863 0.931061i \(-0.618884\pi\)
−0.364863 + 0.931061i \(0.618884\pi\)
\(810\) −2.51181 −0.0882562
\(811\) 47.7569 1.67697 0.838486 0.544922i \(-0.183441\pi\)
0.838486 + 0.544922i \(0.183441\pi\)
\(812\) 14.0273 0.492260
\(813\) −6.67605 −0.234139
\(814\) 4.70481 0.164904
\(815\) 58.0136 2.03213
\(816\) −7.71760 −0.270170
\(817\) 1.89833 0.0664140
\(818\) 10.6100 0.370970
\(819\) −4.49060 −0.156914
\(820\) 15.3478 0.535970
\(821\) 30.4050 1.06114 0.530570 0.847641i \(-0.321978\pi\)
0.530570 + 0.847641i \(0.321978\pi\)
\(822\) 3.80218 0.132616
\(823\) 53.9357 1.88008 0.940039 0.341066i \(-0.110788\pi\)
0.940039 + 0.341066i \(0.110788\pi\)
\(824\) −6.90709 −0.240620
\(825\) −0.841203 −0.0292869
\(826\) 11.1069 0.386458
\(827\) −11.4537 −0.398286 −0.199143 0.979970i \(-0.563816\pi\)
−0.199143 + 0.979970i \(0.563816\pi\)
\(828\) 5.46221 0.189825
\(829\) 24.2545 0.842394 0.421197 0.906969i \(-0.361610\pi\)
0.421197 + 0.906969i \(0.361610\pi\)
\(830\) 1.77932 0.0617612
\(831\) −7.19303 −0.249524
\(832\) −2.58371 −0.0895739
\(833\) 30.7099 1.06403
\(834\) 16.4985 0.571296
\(835\) 35.6155 1.23252
\(836\) 0.642525 0.0222222
\(837\) −0.677087 −0.0234035
\(838\) −2.58451 −0.0892805
\(839\) −9.88122 −0.341138 −0.170569 0.985346i \(-0.554561\pi\)
−0.170569 + 0.985346i \(0.554561\pi\)
\(840\) −4.36565 −0.150629
\(841\) 36.1364 1.24608
\(842\) 29.8910 1.03011
\(843\) −20.6372 −0.710784
\(844\) 15.5116 0.533931
\(845\) 15.8859 0.546491
\(846\) −10.5353 −0.362210
\(847\) −18.4010 −0.632265
\(848\) −1.00000 −0.0343401
\(849\) 10.1190 0.347283
\(850\) −10.1040 −0.346564
\(851\) −39.9964 −1.37106
\(852\) 13.8022 0.472855
\(853\) 13.1067 0.448766 0.224383 0.974501i \(-0.427963\pi\)
0.224383 + 0.974501i \(0.427963\pi\)
\(854\) −7.36456 −0.252010
\(855\) 2.51181 0.0859023
\(856\) −8.90249 −0.304281
\(857\) 20.4708 0.699270 0.349635 0.936886i \(-0.386306\pi\)
0.349635 + 0.936886i \(0.386306\pi\)
\(858\) 1.66010 0.0566747
\(859\) 3.36408 0.114781 0.0573905 0.998352i \(-0.481722\pi\)
0.0573905 + 0.998352i \(0.481722\pi\)
\(860\) 4.76824 0.162596
\(861\) −10.6199 −0.361926
\(862\) 15.2898 0.520774
\(863\) −41.1059 −1.39926 −0.699630 0.714505i \(-0.746652\pi\)
−0.699630 + 0.714505i \(0.746652\pi\)
\(864\) 1.00000 0.0340207
\(865\) −49.1983 −1.67279
\(866\) −1.06601 −0.0362245
\(867\) 42.5613 1.44546
\(868\) −1.17681 −0.0399435
\(869\) 6.21709 0.210900
\(870\) −20.2721 −0.687290
\(871\) 33.2294 1.12593
\(872\) 5.61503 0.190149
\(873\) −13.5187 −0.457540
\(874\) −5.46221 −0.184762
\(875\) 16.1127 0.544708
\(876\) 5.86264 0.198080
\(877\) 48.1675 1.62650 0.813250 0.581914i \(-0.197696\pi\)
0.813250 + 0.581914i \(0.197696\pi\)
\(878\) −13.9808 −0.471828
\(879\) −11.1099 −0.374729
\(880\) 1.61390 0.0544047
\(881\) 12.8868 0.434169 0.217084 0.976153i \(-0.430345\pi\)
0.217084 + 0.976153i \(0.430345\pi\)
\(882\) −3.97920 −0.133987
\(883\) 5.85298 0.196968 0.0984842 0.995139i \(-0.468601\pi\)
0.0984842 + 0.995139i \(0.468601\pi\)
\(884\) 19.9400 0.670655
\(885\) −16.0516 −0.539569
\(886\) 24.7683 0.832106
\(887\) 28.8274 0.967929 0.483965 0.875088i \(-0.339196\pi\)
0.483965 + 0.875088i \(0.339196\pi\)
\(888\) −7.32238 −0.245723
\(889\) −27.2319 −0.913328
\(890\) −23.4377 −0.785632
\(891\) −0.642525 −0.0215254
\(892\) −15.6842 −0.525147
\(893\) 10.5353 0.352549
\(894\) 17.3379 0.579865
\(895\) 5.72985 0.191528
\(896\) 1.73805 0.0580640
\(897\) −14.1127 −0.471211
\(898\) 38.3823 1.28083
\(899\) −5.46457 −0.182254
\(900\) 1.30921 0.0436405
\(901\) 7.71760 0.257110
\(902\) 3.92600 0.130721
\(903\) −3.29938 −0.109796
\(904\) −2.37558 −0.0790106
\(905\) 51.0745 1.69777
\(906\) 17.8421 0.592765
\(907\) 17.9831 0.597120 0.298560 0.954391i \(-0.403494\pi\)
0.298560 + 0.954391i \(0.403494\pi\)
\(908\) −4.07495 −0.135232
\(909\) −13.1581 −0.436426
\(910\) 11.2795 0.373913
\(911\) 1.05981 0.0351130 0.0175565 0.999846i \(-0.494411\pi\)
0.0175565 + 0.999846i \(0.494411\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0.455152 0.0150633
\(914\) 7.38642 0.244321
\(915\) 10.6432 0.351854
\(916\) −14.8872 −0.491889
\(917\) −32.0824 −1.05945
\(918\) −7.71760 −0.254719
\(919\) −16.3849 −0.540489 −0.270245 0.962792i \(-0.587105\pi\)
−0.270245 + 0.962792i \(0.587105\pi\)
\(920\) −13.7201 −0.452337
\(921\) −29.0392 −0.956873
\(922\) 15.1036 0.497410
\(923\) −35.6608 −1.17379
\(924\) −1.11674 −0.0367380
\(925\) −9.58656 −0.315204
\(926\) −36.1124 −1.18673
\(927\) −6.90709 −0.226859
\(928\) 8.07071 0.264934
\(929\) 13.7195 0.450122 0.225061 0.974345i \(-0.427742\pi\)
0.225061 + 0.974345i \(0.427742\pi\)
\(930\) 1.70072 0.0557687
\(931\) 3.97920 0.130413
\(932\) 15.0002 0.491348
\(933\) −30.7392 −1.00636
\(934\) 0.364195 0.0119168
\(935\) −12.4555 −0.407337
\(936\) −2.58371 −0.0844511
\(937\) −41.5301 −1.35673 −0.678364 0.734726i \(-0.737311\pi\)
−0.678364 + 0.734726i \(0.737311\pi\)
\(938\) −22.3532 −0.729859
\(939\) −2.12783 −0.0694391
\(940\) 26.4626 0.863116
\(941\) −52.3744 −1.70736 −0.853678 0.520801i \(-0.825633\pi\)
−0.853678 + 0.520801i \(0.825633\pi\)
\(942\) −0.480011 −0.0156396
\(943\) −33.3756 −1.08686
\(944\) 6.39044 0.207991
\(945\) −4.36565 −0.142015
\(946\) 1.21972 0.0396566
\(947\) 6.89857 0.224173 0.112087 0.993698i \(-0.464247\pi\)
0.112087 + 0.993698i \(0.464247\pi\)
\(948\) −9.67603 −0.314263
\(949\) −15.1473 −0.491703
\(950\) −1.30921 −0.0424765
\(951\) −27.0644 −0.877622
\(952\) −13.4135 −0.434735
\(953\) 47.4542 1.53719 0.768596 0.639734i \(-0.220955\pi\)
0.768596 + 0.639734i \(0.220955\pi\)
\(954\) −1.00000 −0.0323762
\(955\) −20.5734 −0.665740
\(956\) −5.82577 −0.188419
\(957\) −5.18563 −0.167628
\(958\) 30.1157 0.972994
\(959\) 6.60836 0.213395
\(960\) −2.51181 −0.0810685
\(961\) −30.5416 −0.985211
\(962\) 18.9189 0.609969
\(963\) −8.90249 −0.286879
\(964\) 6.25361 0.201415
\(965\) −13.5481 −0.436128
\(966\) 9.49357 0.305451
\(967\) 36.9667 1.18877 0.594384 0.804181i \(-0.297396\pi\)
0.594384 + 0.804181i \(0.297396\pi\)
\(968\) −10.5872 −0.340284
\(969\) 7.71760 0.247925
\(970\) 33.9566 1.09028
\(971\) 62.1228 1.99362 0.996808 0.0798339i \(-0.0254390\pi\)
0.996808 + 0.0798339i \(0.0254390\pi\)
\(972\) 1.00000 0.0320750
\(973\) 28.6751 0.919282
\(974\) 29.7321 0.952679
\(975\) −3.38262 −0.108331
\(976\) −4.23726 −0.135631
\(977\) 15.0045 0.480038 0.240019 0.970768i \(-0.422846\pi\)
0.240019 + 0.970768i \(0.422846\pi\)
\(978\) −23.0963 −0.738537
\(979\) −5.99538 −0.191613
\(980\) 9.99501 0.319279
\(981\) 5.61503 0.179274
\(982\) −0.905199 −0.0288861
\(983\) −16.0226 −0.511042 −0.255521 0.966803i \(-0.582247\pi\)
−0.255521 + 0.966803i \(0.582247\pi\)
\(984\) −6.11026 −0.194788
\(985\) 1.59062 0.0506814
\(986\) −62.2865 −1.98361
\(987\) −18.3108 −0.582838
\(988\) 2.58371 0.0821986
\(989\) −10.3691 −0.329717
\(990\) 1.61390 0.0512932
\(991\) 31.5170 1.00117 0.500585 0.865688i \(-0.333118\pi\)
0.500585 + 0.865688i \(0.333118\pi\)
\(992\) −0.677087 −0.0214975
\(993\) 2.20656 0.0700232
\(994\) 23.9888 0.760879
\(995\) −36.3234 −1.15153
\(996\) −0.708381 −0.0224459
\(997\) 4.06176 0.128637 0.0643185 0.997929i \(-0.479513\pi\)
0.0643185 + 0.997929i \(0.479513\pi\)
\(998\) −28.5614 −0.904096
\(999\) −7.32238 −0.231670
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 6042.2.a.x.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.x.1.2 6 1.1 even 1 trivial