Properties

Label 4002.2.a.ba.1.1
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.87996\) of defining polynomial
Character \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29417 q^{5} +1.00000 q^{6} +2.46575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29417 q^{5} +1.00000 q^{6} +2.46575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.29417 q^{10} +5.48709 q^{11} -1.00000 q^{12} -0.169761 q^{13} -2.46575 q^{14} +3.29417 q^{15} +1.00000 q^{16} +0.362680 q^{17} -1.00000 q^{18} +4.85158 q^{19} -3.29417 q^{20} -2.46575 q^{21} -5.48709 q^{22} +1.00000 q^{23} +1.00000 q^{24} +5.85158 q^{25} +0.169761 q^{26} -1.00000 q^{27} +2.46575 q^{28} -1.00000 q^{29} -3.29417 q^{30} -3.48709 q^{31} -1.00000 q^{32} -5.48709 q^{33} -0.362680 q^{34} -8.12260 q^{35} +1.00000 q^{36} +10.8843 q^{37} -4.85158 q^{38} +0.169761 q^{39} +3.29417 q^{40} +7.95284 q^{41} +2.46575 q^{42} +10.1226 q^{43} +5.48709 q^{44} -3.29417 q^{45} -1.00000 q^{46} +1.53425 q^{47} -1.00000 q^{48} -0.920090 q^{49} -5.85158 q^{50} -0.362680 q^{51} -0.169761 q^{52} -5.21426 q^{53} +1.00000 q^{54} -18.0754 q^{55} -2.46575 q^{56} -4.85158 q^{57} +1.00000 q^{58} -12.1226 q^{59} +3.29417 q^{60} +9.27102 q^{61} +3.48709 q^{62} +2.46575 q^{63} +1.00000 q^{64} +0.559222 q^{65} +5.48709 q^{66} -6.34827 q^{67} +0.362680 q^{68} -1.00000 q^{69} +8.12260 q^{70} -6.07544 q^{71} -1.00000 q^{72} -3.86299 q^{73} -10.8843 q^{74} -5.85158 q^{75} +4.85158 q^{76} +13.5298 q^{77} -0.169761 q^{78} -3.38403 q^{79} -3.29417 q^{80} +1.00000 q^{81} -7.95284 q^{82} +5.55741 q^{83} -2.46575 q^{84} -1.19473 q^{85} -10.1226 q^{86} +1.00000 q^{87} -5.48709 q^{88} -10.2828 q^{89} +3.29417 q^{90} -0.418588 q^{91} +1.00000 q^{92} +3.48709 q^{93} -1.53425 q^{94} -15.9820 q^{95} +1.00000 q^{96} -16.1458 q^{97} +0.920090 q^{98} +5.48709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) \) 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
\(5\) −3.29417 −1.47320 −0.736600 0.676329i \(-0.763570\pi\)
−0.736600 + 0.676329i \(0.763570\pi\)
\(6\) 1.00000 0.408248
\(7\) 2.46575 0.931965 0.465982 0.884794i \(-0.345701\pi\)
0.465982 + 0.884794i \(0.345701\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.29417 1.04171
\(11\) 5.48709 1.65442 0.827210 0.561892i \(-0.189926\pi\)
0.827210 + 0.561892i \(0.189926\pi\)
\(12\) −1.00000 −0.288675
\(13\) −0.169761 −0.0470832 −0.0235416 0.999723i \(-0.507494\pi\)
−0.0235416 + 0.999723i \(0.507494\pi\)
\(14\) −2.46575 −0.658999
\(15\) 3.29417 0.850552
\(16\) 1.00000 0.250000
\(17\) 0.362680 0.0879628 0.0439814 0.999032i \(-0.485996\pi\)
0.0439814 + 0.999032i \(0.485996\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.85158 1.11303 0.556515 0.830838i \(-0.312138\pi\)
0.556515 + 0.830838i \(0.312138\pi\)
\(20\) −3.29417 −0.736600
\(21\) −2.46575 −0.538070
\(22\) −5.48709 −1.16985
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 5.85158 1.17032
\(26\) 0.169761 0.0332929
\(27\) −1.00000 −0.192450
\(28\) 2.46575 0.465982
\(29\) −1.00000 −0.185695
\(30\) −3.29417 −0.601431
\(31\) −3.48709 −0.626300 −0.313150 0.949704i \(-0.601384\pi\)
−0.313150 + 0.949704i \(0.601384\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.48709 −0.955180
\(34\) −0.362680 −0.0621991
\(35\) −8.12260 −1.37297
\(36\) 1.00000 0.166667
\(37\) 10.8843 1.78937 0.894687 0.446694i \(-0.147399\pi\)
0.894687 + 0.446694i \(0.147399\pi\)
\(38\) −4.85158 −0.787031
\(39\) 0.169761 0.0271835
\(40\) 3.29417 0.520855
\(41\) 7.95284 1.24203 0.621013 0.783801i \(-0.286722\pi\)
0.621013 + 0.783801i \(0.286722\pi\)
\(42\) 2.46575 0.380473
\(43\) 10.1226 1.54368 0.771841 0.635815i \(-0.219336\pi\)
0.771841 + 0.635815i \(0.219336\pi\)
\(44\) 5.48709 0.827210
\(45\) −3.29417 −0.491067
\(46\) −1.00000 −0.147442
\(47\) 1.53425 0.223794 0.111897 0.993720i \(-0.464307\pi\)
0.111897 + 0.993720i \(0.464307\pi\)
\(48\) −1.00000 −0.144338
\(49\) −0.920090 −0.131441
\(50\) −5.85158 −0.827539
\(51\) −0.362680 −0.0507853
\(52\) −0.169761 −0.0235416
\(53\) −5.21426 −0.716234 −0.358117 0.933677i \(-0.616581\pi\)
−0.358117 + 0.933677i \(0.616581\pi\)
\(54\) 1.00000 0.136083
\(55\) −18.0754 −2.43729
\(56\) −2.46575 −0.329499
\(57\) −4.85158 −0.642608
\(58\) 1.00000 0.131306
\(59\) −12.1226 −1.57823 −0.789114 0.614247i \(-0.789460\pi\)
−0.789114 + 0.614247i \(0.789460\pi\)
\(60\) 3.29417 0.425276
\(61\) 9.27102 1.18703 0.593516 0.804822i \(-0.297739\pi\)
0.593516 + 0.804822i \(0.297739\pi\)
\(62\) 3.48709 0.442861
\(63\) 2.46575 0.310655
\(64\) 1.00000 0.125000
\(65\) 0.559222 0.0693630
\(66\) 5.48709 0.675414
\(67\) −6.34827 −0.775565 −0.387782 0.921751i \(-0.626759\pi\)
−0.387782 + 0.921751i \(0.626759\pi\)
\(68\) 0.362680 0.0439814
\(69\) −1.00000 −0.120386
\(70\) 8.12260 0.970837
\(71\) −6.07544 −0.721022 −0.360511 0.932755i \(-0.617398\pi\)
−0.360511 + 0.932755i \(0.617398\pi\)
\(72\) −1.00000 −0.117851
\(73\) −3.86299 −0.452129 −0.226064 0.974112i \(-0.572586\pi\)
−0.226064 + 0.974112i \(0.572586\pi\)
\(74\) −10.8843 −1.26528
\(75\) −5.85158 −0.675683
\(76\) 4.85158 0.556515
\(77\) 13.5298 1.54186
\(78\) −0.169761 −0.0192217
\(79\) −3.38403 −0.380733 −0.190366 0.981713i \(-0.560968\pi\)
−0.190366 + 0.981713i \(0.560968\pi\)
\(80\) −3.29417 −0.368300
\(81\) 1.00000 0.111111
\(82\) −7.95284 −0.878244
\(83\) 5.55741 0.610005 0.305003 0.952352i \(-0.401343\pi\)
0.305003 + 0.952352i \(0.401343\pi\)
\(84\) −2.46575 −0.269035
\(85\) −1.19473 −0.129587
\(86\) −10.1226 −1.09155
\(87\) 1.00000 0.107211
\(88\) −5.48709 −0.584926
\(89\) −10.2828 −1.08997 −0.544986 0.838445i \(-0.683465\pi\)
−0.544986 + 0.838445i \(0.683465\pi\)
\(90\) 3.29417 0.347236
\(91\) −0.418588 −0.0438799
\(92\) 1.00000 0.104257
\(93\) 3.48709 0.361595
\(94\) −1.53425 −0.158246
\(95\) −15.9820 −1.63972
\(96\) 1.00000 0.102062
\(97\) −16.1458 −1.63935 −0.819677 0.572826i \(-0.805847\pi\)
−0.819677 + 0.572826i \(0.805847\pi\)
\(98\) 0.920090 0.0929432
\(99\) 5.48709 0.551474
\(100\) 5.85158 0.585158
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0.362680 0.0359107
\(103\) 12.8480 1.26595 0.632974 0.774173i \(-0.281834\pi\)
0.632974 + 0.774173i \(0.281834\pi\)
\(104\) 0.169761 0.0166464
\(105\) 8.12260 0.792685
\(106\) 5.21426 0.506454
\(107\) −10.0986 −0.976268 −0.488134 0.872769i \(-0.662322\pi\)
−0.488134 + 0.872769i \(0.662322\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 0.442589 0.0423924 0.0211962 0.999775i \(-0.493253\pi\)
0.0211962 + 0.999775i \(0.493253\pi\)
\(110\) 18.0754 1.72343
\(111\) −10.8843 −1.03310
\(112\) 2.46575 0.232991
\(113\) 10.5652 0.993890 0.496945 0.867782i \(-0.334455\pi\)
0.496945 + 0.867782i \(0.334455\pi\)
\(114\) 4.85158 0.454393
\(115\) −3.29417 −0.307183
\(116\) −1.00000 −0.0928477
\(117\) −0.169761 −0.0156944
\(118\) 12.1226 1.11598
\(119\) 0.894277 0.0819782
\(120\) −3.29417 −0.300716
\(121\) 19.1082 1.73711
\(122\) −9.27102 −0.839358
\(123\) −7.95284 −0.717083
\(124\) −3.48709 −0.313150
\(125\) −2.80527 −0.250911
\(126\) −2.46575 −0.219666
\(127\) 6.72898 0.597101 0.298550 0.954394i \(-0.403497\pi\)
0.298550 + 0.954394i \(0.403497\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.1226 −0.891245
\(130\) −0.559222 −0.0490470
\(131\) −21.1903 −1.85140 −0.925701 0.378256i \(-0.876524\pi\)
−0.925701 + 0.378256i \(0.876524\pi\)
\(132\) −5.48709 −0.477590
\(133\) 11.9628 1.03730
\(134\) 6.34827 0.548407
\(135\) 3.29417 0.283517
\(136\) −0.362680 −0.0310995
\(137\) 19.6193 1.67619 0.838094 0.545525i \(-0.183670\pi\)
0.838094 + 0.545525i \(0.183670\pi\)
\(138\) 1.00000 0.0851257
\(139\) −7.06851 −0.599543 −0.299771 0.954011i \(-0.596910\pi\)
−0.299771 + 0.954011i \(0.596910\pi\)
\(140\) −8.12260 −0.686485
\(141\) −1.53425 −0.129207
\(142\) 6.07544 0.509840
\(143\) −0.931495 −0.0778955
\(144\) 1.00000 0.0833333
\(145\) 3.29417 0.273566
\(146\) 3.86299 0.319703
\(147\) 0.920090 0.0758878
\(148\) 10.8843 0.894687
\(149\) 13.9288 1.14109 0.570547 0.821265i \(-0.306731\pi\)
0.570547 + 0.821265i \(0.306731\pi\)
\(150\) 5.85158 0.477780
\(151\) 10.8380 0.881986 0.440993 0.897511i \(-0.354626\pi\)
0.440993 + 0.897511i \(0.354626\pi\)
\(152\) −4.85158 −0.393516
\(153\) 0.362680 0.0293209
\(154\) −13.5298 −1.09026
\(155\) 11.4871 0.922666
\(156\) 0.169761 0.0135918
\(157\) 13.9528 1.11356 0.556779 0.830661i \(-0.312037\pi\)
0.556779 + 0.830661i \(0.312037\pi\)
\(158\) 3.38403 0.269219
\(159\) 5.21426 0.413518
\(160\) 3.29417 0.260427
\(161\) 2.46575 0.194328
\(162\) −1.00000 −0.0785674
\(163\) 15.2702 1.19605 0.598026 0.801477i \(-0.295952\pi\)
0.598026 + 0.801477i \(0.295952\pi\)
\(164\) 7.95284 0.621013
\(165\) 18.0754 1.40717
\(166\) −5.55741 −0.431339
\(167\) −15.5662 −1.20455 −0.602273 0.798290i \(-0.705738\pi\)
−0.602273 + 0.798290i \(0.705738\pi\)
\(168\) 2.46575 0.190237
\(169\) −12.9712 −0.997783
\(170\) 1.19473 0.0916317
\(171\) 4.85158 0.371010
\(172\) 10.1226 0.771841
\(173\) −4.29236 −0.326342 −0.163171 0.986598i \(-0.552172\pi\)
−0.163171 + 0.986598i \(0.552172\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 14.4285 1.09069
\(176\) 5.48709 0.413605
\(177\) 12.1226 0.911191
\(178\) 10.2828 0.770726
\(179\) −23.3600 −1.74601 −0.873005 0.487711i \(-0.837832\pi\)
−0.873005 + 0.487711i \(0.837832\pi\)
\(180\) −3.29417 −0.245533
\(181\) 3.04450 0.226296 0.113148 0.993578i \(-0.463907\pi\)
0.113148 + 0.993578i \(0.463907\pi\)
\(182\) 0.418588 0.0310278
\(183\) −9.27102 −0.685333
\(184\) −1.00000 −0.0737210
\(185\) −35.8549 −2.63610
\(186\) −3.48709 −0.255686
\(187\) 1.99006 0.145527
\(188\) 1.53425 0.111897
\(189\) −2.46575 −0.179357
\(190\) 15.9820 1.15945
\(191\) 18.3011 1.32422 0.662111 0.749406i \(-0.269661\pi\)
0.662111 + 0.749406i \(0.269661\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.7909 1.64052 0.820261 0.571989i \(-0.193828\pi\)
0.820261 + 0.571989i \(0.193828\pi\)
\(194\) 16.1458 1.15920
\(195\) −0.559222 −0.0400467
\(196\) −0.920090 −0.0657207
\(197\) 15.0213 1.07023 0.535113 0.844780i \(-0.320269\pi\)
0.535113 + 0.844780i \(0.320269\pi\)
\(198\) −5.48709 −0.389951
\(199\) 22.1082 1.56721 0.783604 0.621261i \(-0.213379\pi\)
0.783604 + 0.621261i \(0.213379\pi\)
\(200\) −5.85158 −0.413770
\(201\) 6.34827 0.447772
\(202\) 10.0000 0.703598
\(203\) −2.46575 −0.173062
\(204\) −0.362680 −0.0253927
\(205\) −26.1980 −1.82975
\(206\) −12.8480 −0.895160
\(207\) 1.00000 0.0695048
\(208\) −0.169761 −0.0117708
\(209\) 26.6211 1.84142
\(210\) −8.12260 −0.560513
\(211\) −6.49850 −0.447375 −0.223688 0.974661i \(-0.571810\pi\)
−0.223688 + 0.974661i \(0.571810\pi\)
\(212\) −5.21426 −0.358117
\(213\) 6.07544 0.416282
\(214\) 10.0986 0.690326
\(215\) −33.3456 −2.27415
\(216\) 1.00000 0.0680414
\(217\) −8.59829 −0.583690
\(218\) −0.442589 −0.0299759
\(219\) 3.86299 0.261037
\(220\) −18.0754 −1.21865
\(221\) −0.0615689 −0.00414157
\(222\) 10.8843 0.730509
\(223\) −2.34315 −0.156909 −0.0784543 0.996918i \(-0.524998\pi\)
−0.0784543 + 0.996918i \(0.524998\pi\)
\(224\) −2.46575 −0.164750
\(225\) 5.85158 0.390106
\(226\) −10.5652 −0.702786
\(227\) 5.80708 0.385430 0.192715 0.981255i \(-0.438271\pi\)
0.192715 + 0.981255i \(0.438271\pi\)
\(228\) −4.85158 −0.321304
\(229\) −5.08685 −0.336148 −0.168074 0.985774i \(-0.553755\pi\)
−0.168074 + 0.985774i \(0.553755\pi\)
\(230\) 3.29417 0.217211
\(231\) −13.5298 −0.890195
\(232\) 1.00000 0.0656532
\(233\) 3.95731 0.259252 0.129626 0.991563i \(-0.458622\pi\)
0.129626 + 0.991563i \(0.458622\pi\)
\(234\) 0.169761 0.0110976
\(235\) −5.05410 −0.329693
\(236\) −12.1226 −0.789114
\(237\) 3.38403 0.219816
\(238\) −0.894277 −0.0579674
\(239\) 16.7254 1.08187 0.540937 0.841063i \(-0.318070\pi\)
0.540937 + 0.841063i \(0.318070\pi\)
\(240\) 3.29417 0.212638
\(241\) 7.56700 0.487434 0.243717 0.969846i \(-0.421633\pi\)
0.243717 + 0.969846i \(0.421633\pi\)
\(242\) −19.1082 −1.22832
\(243\) −1.00000 −0.0641500
\(244\) 9.27102 0.593516
\(245\) 3.03094 0.193640
\(246\) 7.95284 0.507055
\(247\) −0.823610 −0.0524050
\(248\) 3.48709 0.221431
\(249\) −5.55741 −0.352187
\(250\) 2.80527 0.177421
\(251\) −6.79086 −0.428635 −0.214318 0.976764i \(-0.568753\pi\)
−0.214318 + 0.976764i \(0.568753\pi\)
\(252\) 2.46575 0.155327
\(253\) 5.48709 0.344971
\(254\) −6.72898 −0.422214
\(255\) 1.19473 0.0748169
\(256\) 1.00000 0.0625000
\(257\) 24.3670 1.51997 0.759985 0.649941i \(-0.225206\pi\)
0.759985 + 0.649941i \(0.225206\pi\)
\(258\) 10.1226 0.630206
\(259\) 26.8380 1.66763
\(260\) 0.559222 0.0346815
\(261\) −1.00000 −0.0618984
\(262\) 21.1903 1.30914
\(263\) 28.2975 1.74490 0.872449 0.488705i \(-0.162531\pi\)
0.872449 + 0.488705i \(0.162531\pi\)
\(264\) 5.48709 0.337707
\(265\) 17.1767 1.05516
\(266\) −11.9628 −0.733485
\(267\) 10.2828 0.629295
\(268\) −6.34827 −0.387782
\(269\) 14.8044 0.902642 0.451321 0.892362i \(-0.350953\pi\)
0.451321 + 0.892362i \(0.350953\pi\)
\(270\) −3.29417 −0.200477
\(271\) −7.03969 −0.427631 −0.213815 0.976874i \(-0.568589\pi\)
−0.213815 + 0.976874i \(0.568589\pi\)
\(272\) 0.362680 0.0219907
\(273\) 0.418588 0.0253341
\(274\) −19.6193 −1.18524
\(275\) 32.1082 1.93620
\(276\) −1.00000 −0.0601929
\(277\) 29.6843 1.78356 0.891778 0.452473i \(-0.149458\pi\)
0.891778 + 0.452473i \(0.149458\pi\)
\(278\) 7.06851 0.423941
\(279\) −3.48709 −0.208767
\(280\) 8.12260 0.485418
\(281\) −9.66048 −0.576296 −0.288148 0.957586i \(-0.593039\pi\)
−0.288148 + 0.957586i \(0.593039\pi\)
\(282\) 1.53425 0.0913634
\(283\) 6.25002 0.371525 0.185763 0.982595i \(-0.440524\pi\)
0.185763 + 0.982595i \(0.440524\pi\)
\(284\) −6.07544 −0.360511
\(285\) 15.9820 0.946690
\(286\) 0.931495 0.0550804
\(287\) 19.6097 1.15752
\(288\) −1.00000 −0.0589256
\(289\) −16.8685 −0.992263
\(290\) −3.29417 −0.193441
\(291\) 16.1458 0.946481
\(292\) −3.86299 −0.226064
\(293\) 21.1304 1.23445 0.617225 0.786787i \(-0.288257\pi\)
0.617225 + 0.786787i \(0.288257\pi\)
\(294\) −0.920090 −0.0536608
\(295\) 39.9340 2.32505
\(296\) −10.8843 −0.632639
\(297\) −5.48709 −0.318393
\(298\) −13.9288 −0.806876
\(299\) −0.169761 −0.00981753
\(300\) −5.85158 −0.337841
\(301\) 24.9598 1.43866
\(302\) −10.8380 −0.623658
\(303\) 10.0000 0.574485
\(304\) 4.85158 0.278257
\(305\) −30.5403 −1.74874
\(306\) −0.362680 −0.0207330
\(307\) −13.1767 −0.752034 −0.376017 0.926613i \(-0.622707\pi\)
−0.376017 + 0.926613i \(0.622707\pi\)
\(308\) 13.5298 0.770931
\(309\) −12.8480 −0.730895
\(310\) −11.4871 −0.652423
\(311\) 20.6683 1.17199 0.585995 0.810315i \(-0.300704\pi\)
0.585995 + 0.810315i \(0.300704\pi\)
\(312\) −0.169761 −0.00961083
\(313\) 10.8152 0.611312 0.305656 0.952142i \(-0.401124\pi\)
0.305656 + 0.952142i \(0.401124\pi\)
\(314\) −13.9528 −0.787404
\(315\) −8.12260 −0.457657
\(316\) −3.38403 −0.190366
\(317\) −32.4805 −1.82428 −0.912142 0.409874i \(-0.865573\pi\)
−0.912142 + 0.409874i \(0.865573\pi\)
\(318\) −5.21426 −0.292401
\(319\) −5.48709 −0.307218
\(320\) −3.29417 −0.184150
\(321\) 10.0986 0.563649
\(322\) −2.46575 −0.137411
\(323\) 1.75957 0.0979052
\(324\) 1.00000 0.0555556
\(325\) −0.993371 −0.0551023
\(326\) −15.2702 −0.845737
\(327\) −0.442589 −0.0244753
\(328\) −7.95284 −0.439122
\(329\) 3.78308 0.208568
\(330\) −18.0754 −0.995020
\(331\) −23.2629 −1.27865 −0.639323 0.768938i \(-0.720785\pi\)
−0.639323 + 0.768938i \(0.720785\pi\)
\(332\) 5.55741 0.305003
\(333\) 10.8843 0.596458
\(334\) 15.5662 0.851742
\(335\) 20.9123 1.14256
\(336\) −2.46575 −0.134518
\(337\) −31.1391 −1.69626 −0.848128 0.529791i \(-0.822270\pi\)
−0.848128 + 0.529791i \(0.822270\pi\)
\(338\) 12.9712 0.705539
\(339\) −10.5652 −0.573822
\(340\) −1.19473 −0.0647934
\(341\) −19.1340 −1.03616
\(342\) −4.85158 −0.262344
\(343\) −19.5289 −1.05446
\(344\) −10.1226 −0.545774
\(345\) 3.29417 0.177352
\(346\) 4.29236 0.230759
\(347\) −33.7795 −1.81338 −0.906688 0.421802i \(-0.861398\pi\)
−0.906688 + 0.421802i \(0.861398\pi\)
\(348\) 1.00000 0.0536056
\(349\) −4.08901 −0.218880 −0.109440 0.993993i \(-0.534906\pi\)
−0.109440 + 0.993993i \(0.534906\pi\)
\(350\) −14.4285 −0.771237
\(351\) 0.169761 0.00906117
\(352\) −5.48709 −0.292463
\(353\) 20.2025 1.07527 0.537636 0.843177i \(-0.319318\pi\)
0.537636 + 0.843177i \(0.319318\pi\)
\(354\) −12.1226 −0.644309
\(355\) 20.0136 1.06221
\(356\) −10.2828 −0.544986
\(357\) −0.894277 −0.0473302
\(358\) 23.3600 1.23462
\(359\) 15.2806 0.806480 0.403240 0.915094i \(-0.367884\pi\)
0.403240 + 0.915094i \(0.367884\pi\)
\(360\) 3.29417 0.173618
\(361\) 4.53788 0.238836
\(362\) −3.04450 −0.160016
\(363\) −19.1082 −1.00292
\(364\) −0.418588 −0.0219400
\(365\) 12.7254 0.666076
\(366\) 9.27102 0.484604
\(367\) 24.8324 1.29624 0.648119 0.761539i \(-0.275556\pi\)
0.648119 + 0.761539i \(0.275556\pi\)
\(368\) 1.00000 0.0521286
\(369\) 7.95284 0.414008
\(370\) 35.8549 1.86401
\(371\) −12.8571 −0.667505
\(372\) 3.48709 0.180797
\(373\) −28.1404 −1.45706 −0.728529 0.685015i \(-0.759795\pi\)
−0.728529 + 0.685015i \(0.759795\pi\)
\(374\) −1.99006 −0.102903
\(375\) 2.80527 0.144863
\(376\) −1.53425 −0.0791230
\(377\) 0.169761 0.00874314
\(378\) 2.46575 0.126824
\(379\) −16.2025 −0.832267 −0.416134 0.909304i \(-0.636615\pi\)
−0.416134 + 0.909304i \(0.636615\pi\)
\(380\) −15.9820 −0.819858
\(381\) −6.72898 −0.344736
\(382\) −18.3011 −0.936366
\(383\) 15.5063 0.792334 0.396167 0.918179i \(-0.370340\pi\)
0.396167 + 0.918179i \(0.370340\pi\)
\(384\) 1.00000 0.0510310
\(385\) −44.5695 −2.27147
\(386\) −22.7909 −1.16002
\(387\) 10.1226 0.514561
\(388\) −16.1458 −0.819677
\(389\) −14.2680 −0.723417 −0.361708 0.932291i \(-0.617806\pi\)
−0.361708 + 0.932291i \(0.617806\pi\)
\(390\) 0.559222 0.0283173
\(391\) 0.362680 0.0183415
\(392\) 0.920090 0.0464716
\(393\) 21.1903 1.06891
\(394\) −15.0213 −0.756764
\(395\) 11.1476 0.560895
\(396\) 5.48709 0.275737
\(397\) −28.8200 −1.44643 −0.723217 0.690621i \(-0.757337\pi\)
−0.723217 + 0.690621i \(0.757337\pi\)
\(398\) −22.1082 −1.10818
\(399\) −11.9628 −0.598888
\(400\) 5.85158 0.292579
\(401\) 26.9706 1.34685 0.673423 0.739258i \(-0.264823\pi\)
0.673423 + 0.739258i \(0.264823\pi\)
\(402\) −6.34827 −0.316623
\(403\) 0.591973 0.0294883
\(404\) −10.0000 −0.497519
\(405\) −3.29417 −0.163689
\(406\) 2.46575 0.122373
\(407\) 59.7234 2.96038
\(408\) 0.362680 0.0179553
\(409\) −37.7786 −1.86803 −0.934016 0.357231i \(-0.883721\pi\)
−0.934016 + 0.357231i \(0.883721\pi\)
\(410\) 26.1980 1.29383
\(411\) −19.6193 −0.967748
\(412\) 12.8480 0.632974
\(413\) −29.8913 −1.47085
\(414\) −1.00000 −0.0491473
\(415\) −18.3071 −0.898659
\(416\) 0.169761 0.00832322
\(417\) 7.06851 0.346146
\(418\) −26.6211 −1.30208
\(419\) −10.9838 −0.536593 −0.268296 0.963336i \(-0.586461\pi\)
−0.268296 + 0.963336i \(0.586461\pi\)
\(420\) 8.12260 0.396342
\(421\) 20.8762 1.01745 0.508723 0.860930i \(-0.330118\pi\)
0.508723 + 0.860930i \(0.330118\pi\)
\(422\) 6.49850 0.316342
\(423\) 1.53425 0.0745979
\(424\) 5.21426 0.253227
\(425\) 2.12225 0.102944
\(426\) −6.07544 −0.294356
\(427\) 22.8600 1.10627
\(428\) −10.0986 −0.488134
\(429\) 0.931495 0.0449730
\(430\) 33.3456 1.60807
\(431\) 10.1698 0.489860 0.244930 0.969541i \(-0.421235\pi\)
0.244930 + 0.969541i \(0.421235\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −20.0842 −0.965185 −0.482592 0.875845i \(-0.660305\pi\)
−0.482592 + 0.875845i \(0.660305\pi\)
\(434\) 8.59829 0.412731
\(435\) −3.29417 −0.157944
\(436\) 0.442589 0.0211962
\(437\) 4.85158 0.232083
\(438\) −3.86299 −0.184581
\(439\) 35.6097 1.69956 0.849779 0.527139i \(-0.176735\pi\)
0.849779 + 0.527139i \(0.176735\pi\)
\(440\) 18.0754 0.861713
\(441\) −0.920090 −0.0438138
\(442\) 0.0615689 0.00292853
\(443\) 2.56916 0.122065 0.0610323 0.998136i \(-0.480561\pi\)
0.0610323 + 0.998136i \(0.480561\pi\)
\(444\) −10.8843 −0.516548
\(445\) 33.8732 1.60575
\(446\) 2.34315 0.110951
\(447\) −13.9288 −0.658811
\(448\) 2.46575 0.116496
\(449\) −15.5207 −0.732467 −0.366233 0.930523i \(-0.619353\pi\)
−0.366233 + 0.930523i \(0.619353\pi\)
\(450\) −5.85158 −0.275846
\(451\) 43.6380 2.05483
\(452\) 10.5652 0.496945
\(453\) −10.8380 −0.509215
\(454\) −5.80708 −0.272540
\(455\) 1.37890 0.0646439
\(456\) 4.85158 0.227196
\(457\) 18.5220 0.866423 0.433211 0.901292i \(-0.357380\pi\)
0.433211 + 0.901292i \(0.357380\pi\)
\(458\) 5.08685 0.237693
\(459\) −0.362680 −0.0169284
\(460\) −3.29417 −0.153592
\(461\) −8.76173 −0.408075 −0.204037 0.978963i \(-0.565406\pi\)
−0.204037 + 0.978963i \(0.565406\pi\)
\(462\) 13.5298 0.629463
\(463\) −9.49704 −0.441365 −0.220682 0.975346i \(-0.570828\pi\)
−0.220682 + 0.975346i \(0.570828\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −11.4871 −0.532701
\(466\) −3.95731 −0.183319
\(467\) −34.0718 −1.57666 −0.788328 0.615255i \(-0.789053\pi\)
−0.788328 + 0.615255i \(0.789053\pi\)
\(468\) −0.169761 −0.00784721
\(469\) −15.6532 −0.722799
\(470\) 5.05410 0.233128
\(471\) −13.9528 −0.642913
\(472\) 12.1226 0.557988
\(473\) 55.5437 2.55390
\(474\) −3.38403 −0.155433
\(475\) 28.3895 1.30260
\(476\) 0.894277 0.0409891
\(477\) −5.21426 −0.238745
\(478\) −16.7254 −0.765000
\(479\) 20.7341 0.947366 0.473683 0.880696i \(-0.342924\pi\)
0.473683 + 0.880696i \(0.342924\pi\)
\(480\) −3.29417 −0.150358
\(481\) −1.84774 −0.0842495
\(482\) −7.56700 −0.344668
\(483\) −2.46575 −0.112195
\(484\) 19.1082 0.868554
\(485\) 53.1869 2.41509
\(486\) 1.00000 0.0453609
\(487\) 42.0111 1.90370 0.951852 0.306557i \(-0.0991771\pi\)
0.951852 + 0.306557i \(0.0991771\pi\)
\(488\) −9.27102 −0.419679
\(489\) −15.2702 −0.690541
\(490\) −3.03094 −0.136924
\(491\) 21.6896 0.978838 0.489419 0.872049i \(-0.337209\pi\)
0.489419 + 0.872049i \(0.337209\pi\)
\(492\) −7.95284 −0.358542
\(493\) −0.362680 −0.0163343
\(494\) 0.823610 0.0370560
\(495\) −18.0754 −0.812431
\(496\) −3.48709 −0.156575
\(497\) −14.9805 −0.671967
\(498\) 5.55741 0.249034
\(499\) 16.2125 0.725769 0.362885 0.931834i \(-0.381792\pi\)
0.362885 + 0.931834i \(0.381792\pi\)
\(500\) −2.80527 −0.125455
\(501\) 15.5662 0.695445
\(502\) 6.79086 0.303091
\(503\) 4.87028 0.217155 0.108577 0.994088i \(-0.465370\pi\)
0.108577 + 0.994088i \(0.465370\pi\)
\(504\) −2.46575 −0.109833
\(505\) 32.9417 1.46589
\(506\) −5.48709 −0.243931
\(507\) 12.9712 0.576070
\(508\) 6.72898 0.298550
\(509\) −20.9498 −0.928585 −0.464293 0.885682i \(-0.653691\pi\)
−0.464293 + 0.885682i \(0.653691\pi\)
\(510\) −1.19473 −0.0529036
\(511\) −9.52515 −0.421368
\(512\) −1.00000 −0.0441942
\(513\) −4.85158 −0.214203
\(514\) −24.3670 −1.07478
\(515\) −42.3234 −1.86499
\(516\) −10.1226 −0.445623
\(517\) 8.41859 0.370249
\(518\) −26.8380 −1.17919
\(519\) 4.29236 0.188414
\(520\) −0.559222 −0.0245235
\(521\) 33.6188 1.47287 0.736433 0.676510i \(-0.236509\pi\)
0.736433 + 0.676510i \(0.236509\pi\)
\(522\) 1.00000 0.0437688
\(523\) −4.73048 −0.206850 −0.103425 0.994637i \(-0.532980\pi\)
−0.103425 + 0.994637i \(0.532980\pi\)
\(524\) −21.1903 −0.925701
\(525\) −14.4285 −0.629713
\(526\) −28.2975 −1.23383
\(527\) −1.26470 −0.0550911
\(528\) −5.48709 −0.238795
\(529\) 1.00000 0.0434783
\(530\) −17.1767 −0.746108
\(531\) −12.1226 −0.526076
\(532\) 11.9628 0.518652
\(533\) −1.35008 −0.0584786
\(534\) −10.2828 −0.444979
\(535\) 33.2665 1.43824
\(536\) 6.34827 0.274204
\(537\) 23.3600 1.00806
\(538\) −14.8044 −0.638264
\(539\) −5.04862 −0.217460
\(540\) 3.29417 0.141759
\(541\) −11.8450 −0.509254 −0.254627 0.967039i \(-0.581953\pi\)
−0.254627 + 0.967039i \(0.581953\pi\)
\(542\) 7.03969 0.302380
\(543\) −3.04450 −0.130652
\(544\) −0.362680 −0.0155498
\(545\) −1.45797 −0.0624524
\(546\) −0.418588 −0.0179139
\(547\) −12.5920 −0.538394 −0.269197 0.963085i \(-0.586758\pi\)
−0.269197 + 0.963085i \(0.586758\pi\)
\(548\) 19.6193 0.838094
\(549\) 9.27102 0.395677
\(550\) −32.1082 −1.36910
\(551\) −4.85158 −0.206684
\(552\) 1.00000 0.0425628
\(553\) −8.34415 −0.354829
\(554\) −29.6843 −1.26116
\(555\) 35.8549 1.52196
\(556\) −7.06851 −0.299771
\(557\) −36.0761 −1.52859 −0.764297 0.644865i \(-0.776914\pi\)
−0.764297 + 0.644865i \(0.776914\pi\)
\(558\) 3.48709 0.147620
\(559\) −1.71842 −0.0726816
\(560\) −8.12260 −0.343243
\(561\) −1.99006 −0.0840203
\(562\) 9.66048 0.407503
\(563\) 25.1803 1.06122 0.530612 0.847615i \(-0.321962\pi\)
0.530612 + 0.847615i \(0.321962\pi\)
\(564\) −1.53425 −0.0646037
\(565\) −34.8036 −1.46420
\(566\) −6.25002 −0.262708
\(567\) 2.46575 0.103552
\(568\) 6.07544 0.254920
\(569\) 27.4487 1.15071 0.575354 0.817904i \(-0.304864\pi\)
0.575354 + 0.817904i \(0.304864\pi\)
\(570\) −15.9820 −0.669411
\(571\) −26.8132 −1.12210 −0.561048 0.827783i \(-0.689602\pi\)
−0.561048 + 0.827783i \(0.689602\pi\)
\(572\) −0.931495 −0.0389477
\(573\) −18.3011 −0.764540
\(574\) −19.6097 −0.818493
\(575\) 5.85158 0.244028
\(576\) 1.00000 0.0416667
\(577\) 4.37590 0.182171 0.0910855 0.995843i \(-0.470966\pi\)
0.0910855 + 0.995843i \(0.470966\pi\)
\(578\) 16.8685 0.701636
\(579\) −22.7909 −0.947156
\(580\) 3.29417 0.136783
\(581\) 13.7032 0.568503
\(582\) −16.1458 −0.669263
\(583\) −28.6112 −1.18495
\(584\) 3.86299 0.159852
\(585\) 0.559222 0.0231210
\(586\) −21.1304 −0.872888
\(587\) 0.808893 0.0333866 0.0166933 0.999861i \(-0.494686\pi\)
0.0166933 + 0.999861i \(0.494686\pi\)
\(588\) 0.920090 0.0379439
\(589\) −16.9179 −0.697091
\(590\) −39.9340 −1.64406
\(591\) −15.0213 −0.617896
\(592\) 10.8843 0.447343
\(593\) 6.59197 0.270700 0.135350 0.990798i \(-0.456784\pi\)
0.135350 + 0.990798i \(0.456784\pi\)
\(594\) 5.48709 0.225138
\(595\) −2.94590 −0.120770
\(596\) 13.9288 0.570547
\(597\) −22.1082 −0.904828
\(598\) 0.169761 0.00694204
\(599\) 39.8504 1.62824 0.814122 0.580694i \(-0.197219\pi\)
0.814122 + 0.580694i \(0.197219\pi\)
\(600\) 5.85158 0.238890
\(601\) 14.6575 0.597891 0.298945 0.954270i \(-0.403365\pi\)
0.298945 + 0.954270i \(0.403365\pi\)
\(602\) −24.9598 −1.01728
\(603\) −6.34827 −0.258522
\(604\) 10.8380 0.440993
\(605\) −62.9457 −2.55911
\(606\) −10.0000 −0.406222
\(607\) −35.7714 −1.45191 −0.725957 0.687740i \(-0.758603\pi\)
−0.725957 + 0.687740i \(0.758603\pi\)
\(608\) −4.85158 −0.196758
\(609\) 2.46575 0.0999171
\(610\) 30.5403 1.23654
\(611\) −0.260456 −0.0105369
\(612\) 0.362680 0.0146605
\(613\) 24.0087 0.969704 0.484852 0.874596i \(-0.338873\pi\)
0.484852 + 0.874596i \(0.338873\pi\)
\(614\) 13.1767 0.531768
\(615\) 26.1980 1.05641
\(616\) −13.5298 −0.545131
\(617\) 2.81039 0.113142 0.0565711 0.998399i \(-0.481983\pi\)
0.0565711 + 0.998399i \(0.481983\pi\)
\(618\) 12.8480 0.516821
\(619\) −15.1659 −0.609569 −0.304785 0.952421i \(-0.598585\pi\)
−0.304785 + 0.952421i \(0.598585\pi\)
\(620\) 11.4871 0.461333
\(621\) −1.00000 −0.0401286
\(622\) −20.6683 −0.828722
\(623\) −25.3547 −1.01582
\(624\) 0.169761 0.00679588
\(625\) −20.0169 −0.800675
\(626\) −10.8152 −0.432263
\(627\) −26.6211 −1.06314
\(628\) 13.9528 0.556779
\(629\) 3.94753 0.157398
\(630\) 8.12260 0.323612
\(631\) 14.1190 0.562068 0.281034 0.959698i \(-0.409323\pi\)
0.281034 + 0.959698i \(0.409323\pi\)
\(632\) 3.38403 0.134609
\(633\) 6.49850 0.258292
\(634\) 32.4805 1.28996
\(635\) −22.1664 −0.879649
\(636\) 5.21426 0.206759
\(637\) 0.156195 0.00618869
\(638\) 5.48709 0.217236
\(639\) −6.07544 −0.240341
\(640\) 3.29417 0.130214
\(641\) −8.45700 −0.334031 −0.167016 0.985954i \(-0.553413\pi\)
−0.167016 + 0.985954i \(0.553413\pi\)
\(642\) −10.0986 −0.398560
\(643\) 48.9238 1.92936 0.964682 0.263416i \(-0.0848493\pi\)
0.964682 + 0.263416i \(0.0848493\pi\)
\(644\) 2.46575 0.0971641
\(645\) 33.3456 1.31298
\(646\) −1.75957 −0.0692294
\(647\) 20.5129 0.806446 0.403223 0.915102i \(-0.367890\pi\)
0.403223 + 0.915102i \(0.367890\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −66.5178 −2.61105
\(650\) 0.993371 0.0389632
\(651\) 8.59829 0.336994
\(652\) 15.2702 0.598026
\(653\) −12.5013 −0.489213 −0.244606 0.969622i \(-0.578659\pi\)
−0.244606 + 0.969622i \(0.578659\pi\)
\(654\) 0.442589 0.0173066
\(655\) 69.8044 2.72748
\(656\) 7.95284 0.310506
\(657\) −3.86299 −0.150710
\(658\) −3.78308 −0.147480
\(659\) 31.1479 1.21335 0.606675 0.794950i \(-0.292503\pi\)
0.606675 + 0.794950i \(0.292503\pi\)
\(660\) 18.0754 0.703586
\(661\) 9.98594 0.388408 0.194204 0.980961i \(-0.437788\pi\)
0.194204 + 0.980961i \(0.437788\pi\)
\(662\) 23.2629 0.904139
\(663\) 0.0615689 0.00239114
\(664\) −5.55741 −0.215669
\(665\) −39.4075 −1.52816
\(666\) −10.8843 −0.421759
\(667\) −1.00000 −0.0387202
\(668\) −15.5662 −0.602273
\(669\) 2.34315 0.0905912
\(670\) −20.9123 −0.807913
\(671\) 50.8709 1.96385
\(672\) 2.46575 0.0951183
\(673\) −7.50782 −0.289405 −0.144703 0.989475i \(-0.546223\pi\)
−0.144703 + 0.989475i \(0.546223\pi\)
\(674\) 31.1391 1.19943
\(675\) −5.85158 −0.225228
\(676\) −12.9712 −0.498892
\(677\) 15.7650 0.605900 0.302950 0.953006i \(-0.402028\pi\)
0.302950 + 0.953006i \(0.402028\pi\)
\(678\) 10.5652 0.405754
\(679\) −39.8114 −1.52782
\(680\) 1.19473 0.0458158
\(681\) −5.80708 −0.222528
\(682\) 19.1340 0.732679
\(683\) 44.1792 1.69047 0.845234 0.534396i \(-0.179461\pi\)
0.845234 + 0.534396i \(0.179461\pi\)
\(684\) 4.85158 0.185505
\(685\) −64.6294 −2.46936
\(686\) 19.5289 0.745618
\(687\) 5.08685 0.194075
\(688\) 10.1226 0.385921
\(689\) 0.885179 0.0337226
\(690\) −3.29417 −0.125407
\(691\) −1.47322 −0.0560439 −0.0280220 0.999607i \(-0.508921\pi\)
−0.0280220 + 0.999607i \(0.508921\pi\)
\(692\) −4.29236 −0.163171
\(693\) 13.5298 0.513954
\(694\) 33.7795 1.28225
\(695\) 23.2849 0.883246
\(696\) −1.00000 −0.0379049
\(697\) 2.88434 0.109252
\(698\) 4.08901 0.154771
\(699\) −3.95731 −0.149679
\(700\) 14.4285 0.545347
\(701\) −9.42950 −0.356147 −0.178074 0.984017i \(-0.556987\pi\)
−0.178074 + 0.984017i \(0.556987\pi\)
\(702\) −0.169761 −0.00640722
\(703\) 52.8063 1.99163
\(704\) 5.48709 0.206803
\(705\) 5.05410 0.190348
\(706\) −20.2025 −0.760332
\(707\) −24.6575 −0.927340
\(708\) 12.1226 0.455595
\(709\) −23.6365 −0.887686 −0.443843 0.896104i \(-0.646385\pi\)
−0.443843 + 0.896104i \(0.646385\pi\)
\(710\) −20.0136 −0.751096
\(711\) −3.38403 −0.126911
\(712\) 10.2828 0.385363
\(713\) −3.48709 −0.130593
\(714\) 0.894277 0.0334675
\(715\) 3.06851 0.114756
\(716\) −23.3600 −0.873005
\(717\) −16.7254 −0.624620
\(718\) −15.2806 −0.570267
\(719\) −3.44078 −0.128319 −0.0641597 0.997940i \(-0.520437\pi\)
−0.0641597 + 0.997940i \(0.520437\pi\)
\(720\) −3.29417 −0.122767
\(721\) 31.6798 1.17982
\(722\) −4.53788 −0.168882
\(723\) −7.56700 −0.281420
\(724\) 3.04450 0.113148
\(725\) −5.85158 −0.217322
\(726\) 19.1082 0.709172
\(727\) −2.08026 −0.0771525 −0.0385763 0.999256i \(-0.512282\pi\)
−0.0385763 + 0.999256i \(0.512282\pi\)
\(728\) 0.418588 0.0155139
\(729\) 1.00000 0.0370370
\(730\) −12.7254 −0.470987
\(731\) 3.67126 0.135787
\(732\) −9.27102 −0.342667
\(733\) 0.751792 0.0277681 0.0138840 0.999904i \(-0.495580\pi\)
0.0138840 + 0.999904i \(0.495580\pi\)
\(734\) −24.8324 −0.916579
\(735\) −3.03094 −0.111798
\(736\) −1.00000 −0.0368605
\(737\) −34.8336 −1.28311
\(738\) −7.95284 −0.292748
\(739\) 28.0583 1.03214 0.516070 0.856547i \(-0.327395\pi\)
0.516070 + 0.856547i \(0.327395\pi\)
\(740\) −35.8549 −1.31805
\(741\) 0.823610 0.0302561
\(742\) 12.8571 0.471997
\(743\) 31.7693 1.16550 0.582752 0.812650i \(-0.301976\pi\)
0.582752 + 0.812650i \(0.301976\pi\)
\(744\) −3.48709 −0.127843
\(745\) −45.8840 −1.68106
\(746\) 28.1404 1.03029
\(747\) 5.55741 0.203335
\(748\) 1.99006 0.0727637
\(749\) −24.9006 −0.909848
\(750\) −2.80527 −0.102434
\(751\) 17.9197 0.653901 0.326950 0.945042i \(-0.393979\pi\)
0.326950 + 0.945042i \(0.393979\pi\)
\(752\) 1.53425 0.0559484
\(753\) 6.79086 0.247473
\(754\) −0.169761 −0.00618233
\(755\) −35.7023 −1.29934
\(756\) −2.46575 −0.0896784
\(757\) 41.8784 1.52210 0.761048 0.648695i \(-0.224685\pi\)
0.761048 + 0.648695i \(0.224685\pi\)
\(758\) 16.2025 0.588502
\(759\) −5.48709 −0.199169
\(760\) 15.9820 0.579727
\(761\) −51.0016 −1.84881 −0.924404 0.381415i \(-0.875437\pi\)
−0.924404 + 0.381415i \(0.875437\pi\)
\(762\) 6.72898 0.243765
\(763\) 1.09131 0.0395082
\(764\) 18.3011 0.662111
\(765\) −1.19473 −0.0431956
\(766\) −15.5063 −0.560265
\(767\) 2.05795 0.0743081
\(768\) −1.00000 −0.0360844
\(769\) −39.7447 −1.43323 −0.716615 0.697469i \(-0.754309\pi\)
−0.716615 + 0.697469i \(0.754309\pi\)
\(770\) 44.5695 1.60617
\(771\) −24.3670 −0.877555
\(772\) 22.7909 0.820261
\(773\) 51.7714 1.86209 0.931043 0.364909i \(-0.118900\pi\)
0.931043 + 0.364909i \(0.118900\pi\)
\(774\) −10.1226 −0.363849
\(775\) −20.4050 −0.732970
\(776\) 16.1458 0.579599
\(777\) −26.8380 −0.962809
\(778\) 14.2680 0.511533
\(779\) 38.5839 1.38241
\(780\) −0.559222 −0.0200234
\(781\) −33.3365 −1.19287
\(782\) −0.362680 −0.0129694
\(783\) 1.00000 0.0357371
\(784\) −0.920090 −0.0328604
\(785\) −45.9631 −1.64049
\(786\) −21.1903 −0.755832
\(787\) −17.7235 −0.631776 −0.315888 0.948796i \(-0.602302\pi\)
−0.315888 + 0.948796i \(0.602302\pi\)
\(788\) 15.0213 0.535113
\(789\) −28.2975 −1.00742
\(790\) −11.1476 −0.396613
\(791\) 26.0511 0.926270
\(792\) −5.48709 −0.194975
\(793\) −1.57386 −0.0558893
\(794\) 28.8200 1.02278
\(795\) −17.1767 −0.609195
\(796\) 22.1082 0.783604
\(797\) 45.1079 1.59780 0.798902 0.601462i \(-0.205415\pi\)
0.798902 + 0.601462i \(0.205415\pi\)
\(798\) 11.9628 0.423478
\(799\) 0.556443 0.0196855
\(800\) −5.85158 −0.206885
\(801\) −10.2828 −0.363324
\(802\) −26.9706 −0.952364
\(803\) −21.1966 −0.748011
\(804\) 6.34827 0.223886
\(805\) −8.12260 −0.286284
\(806\) −0.591973 −0.0208513
\(807\) −14.8044 −0.521140
\(808\) 10.0000 0.351799
\(809\) 17.2458 0.606331 0.303165 0.952938i \(-0.401957\pi\)
0.303165 + 0.952938i \(0.401957\pi\)
\(810\) 3.29417 0.115745
\(811\) −26.0410 −0.914423 −0.457212 0.889358i \(-0.651152\pi\)
−0.457212 + 0.889358i \(0.651152\pi\)
\(812\) −2.46575 −0.0865308
\(813\) 7.03969 0.246893
\(814\) −59.7234 −2.09330
\(815\) −50.3026 −1.76202
\(816\) −0.362680 −0.0126963
\(817\) 49.1107 1.71816
\(818\) 37.7786 1.32090
\(819\) −0.418588 −0.0146266
\(820\) −26.1980 −0.914875
\(821\) 15.7224 0.548714 0.274357 0.961628i \(-0.411535\pi\)
0.274357 + 0.961628i \(0.411535\pi\)
\(822\) 19.6193 0.684301
\(823\) −33.8213 −1.17894 −0.589468 0.807792i \(-0.700663\pi\)
−0.589468 + 0.807792i \(0.700663\pi\)
\(824\) −12.8480 −0.447580
\(825\) −32.1082 −1.11786
\(826\) 29.8913 1.04005
\(827\) −48.8726 −1.69947 −0.849734 0.527212i \(-0.823237\pi\)
−0.849734 + 0.527212i \(0.823237\pi\)
\(828\) 1.00000 0.0347524
\(829\) −43.6461 −1.51589 −0.757945 0.652318i \(-0.773797\pi\)
−0.757945 + 0.652318i \(0.773797\pi\)
\(830\) 18.3071 0.635448
\(831\) −29.6843 −1.02974
\(832\) −0.169761 −0.00588540
\(833\) −0.333698 −0.0115620
\(834\) −7.06851 −0.244762
\(835\) 51.2776 1.77454
\(836\) 26.6211 0.920710
\(837\) 3.48709 0.120532
\(838\) 10.9838 0.379428
\(839\) −40.4844 −1.39768 −0.698839 0.715279i \(-0.746300\pi\)
−0.698839 + 0.715279i \(0.746300\pi\)
\(840\) −8.12260 −0.280256
\(841\) 1.00000 0.0344828
\(842\) −20.8762 −0.719443
\(843\) 9.66048 0.332725
\(844\) −6.49850 −0.223688
\(845\) 42.7293 1.46993
\(846\) −1.53425 −0.0527487
\(847\) 47.1160 1.61892
\(848\) −5.21426 −0.179059
\(849\) −6.25002 −0.214500
\(850\) −2.12225 −0.0727926
\(851\) 10.8843 0.373110
\(852\) 6.07544 0.208141
\(853\) 0.878017 0.0300627 0.0150314 0.999887i \(-0.495215\pi\)
0.0150314 + 0.999887i \(0.495215\pi\)
\(854\) −22.8600 −0.782253
\(855\) −15.9820 −0.546572
\(856\) 10.0986 0.345163
\(857\) −41.3838 −1.41365 −0.706823 0.707391i \(-0.749872\pi\)
−0.706823 + 0.707391i \(0.749872\pi\)
\(858\) −0.931495 −0.0318007
\(859\) −5.59413 −0.190869 −0.0954347 0.995436i \(-0.530424\pi\)
−0.0954347 + 0.995436i \(0.530424\pi\)
\(860\) −33.3456 −1.13708
\(861\) −19.6097 −0.668297
\(862\) −10.1698 −0.346383
\(863\) −52.5142 −1.78760 −0.893802 0.448461i \(-0.851972\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.1398 0.480767
\(866\) 20.0842 0.682489
\(867\) 16.8685 0.572883
\(868\) −8.59829 −0.291845
\(869\) −18.5685 −0.629892
\(870\) 3.29417 0.111683
\(871\) 1.07769 0.0365161
\(872\) −0.442589 −0.0149880
\(873\) −16.1458 −0.546451
\(874\) −4.85158 −0.164107
\(875\) −6.91709 −0.233840
\(876\) 3.86299 0.130518
\(877\) −11.4156 −0.385477 −0.192738 0.981250i \(-0.561737\pi\)
−0.192738 + 0.981250i \(0.561737\pi\)
\(878\) −35.6097 −1.20177
\(879\) −21.1304 −0.712710
\(880\) −18.0754 −0.609323
\(881\) −8.39265 −0.282756 −0.141378 0.989956i \(-0.545153\pi\)
−0.141378 + 0.989956i \(0.545153\pi\)
\(882\) 0.920090 0.0309811
\(883\) 4.53279 0.152541 0.0762703 0.997087i \(-0.475699\pi\)
0.0762703 + 0.997087i \(0.475699\pi\)
\(884\) −0.0615689 −0.00207079
\(885\) −39.9340 −1.34237
\(886\) −2.56916 −0.0863128
\(887\) −47.6443 −1.59974 −0.799869 0.600174i \(-0.795098\pi\)
−0.799869 + 0.600174i \(0.795098\pi\)
\(888\) 10.8843 0.365254
\(889\) 16.5920 0.556477
\(890\) −33.8732 −1.13543
\(891\) 5.48709 0.183825
\(892\) −2.34315 −0.0784543
\(893\) 7.44356 0.249089
\(894\) 13.9288 0.465850
\(895\) 76.9520 2.57222
\(896\) −2.46575 −0.0823748
\(897\) 0.169761 0.00566816
\(898\) 15.5207 0.517932
\(899\) 3.48709 0.116301
\(900\) 5.85158 0.195053
\(901\) −1.89111 −0.0630020
\(902\) −43.6380 −1.45299
\(903\) −24.9598 −0.830609
\(904\) −10.5652 −0.351393
\(905\) −10.0291 −0.333379
\(906\) 10.8380 0.360069
\(907\) −0.757802 −0.0251624 −0.0125812 0.999921i \(-0.504005\pi\)
−0.0125812 + 0.999921i \(0.504005\pi\)
\(908\) 5.80708 0.192715
\(909\) −10.0000 −0.331679
\(910\) −1.37890 −0.0457101
\(911\) −19.8417 −0.657384 −0.328692 0.944437i \(-0.606608\pi\)
−0.328692 + 0.944437i \(0.606608\pi\)
\(912\) −4.85158 −0.160652
\(913\) 30.4940 1.00921
\(914\) −18.5220 −0.612653
\(915\) 30.5403 1.00963
\(916\) −5.08685 −0.168074
\(917\) −52.2498 −1.72544
\(918\) 0.362680 0.0119702
\(919\) 4.53557 0.149615 0.0748073 0.997198i \(-0.476166\pi\)
0.0748073 + 0.997198i \(0.476166\pi\)
\(920\) 3.29417 0.108606
\(921\) 13.1767 0.434187
\(922\) 8.76173 0.288552
\(923\) 1.03137 0.0339481
\(924\) −13.5298 −0.445097
\(925\) 63.6906 2.09413
\(926\) 9.49704 0.312092
\(927\) 12.8480 0.421982
\(928\) 1.00000 0.0328266
\(929\) 19.7597 0.648296 0.324148 0.946006i \(-0.394922\pi\)
0.324148 + 0.946006i \(0.394922\pi\)
\(930\) 11.4871 0.376677
\(931\) −4.46390 −0.146298
\(932\) 3.95731 0.129626
\(933\) −20.6683 −0.676648
\(934\) 34.0718 1.11486
\(935\) −6.55560 −0.214391
\(936\) 0.169761 0.00554881
\(937\) 3.25630 0.106379 0.0531893 0.998584i \(-0.483061\pi\)
0.0531893 + 0.998584i \(0.483061\pi\)
\(938\) 15.6532 0.511096
\(939\) −10.8152 −0.352941
\(940\) −5.05410 −0.164846
\(941\) 28.1404 0.917352 0.458676 0.888603i \(-0.348324\pi\)
0.458676 + 0.888603i \(0.348324\pi\)
\(942\) 13.9528 0.454608
\(943\) 7.95284 0.258980
\(944\) −12.1226 −0.394557
\(945\) 8.12260 0.264228
\(946\) −55.5437 −1.80588
\(947\) −17.5953 −0.571770 −0.285885 0.958264i \(-0.592288\pi\)
−0.285885 + 0.958264i \(0.592288\pi\)
\(948\) 3.38403 0.109908
\(949\) 0.655785 0.0212877
\(950\) −28.3895 −0.921076
\(951\) 32.4805 1.05325
\(952\) −0.894277 −0.0289837
\(953\) 11.7512 0.380658 0.190329 0.981720i \(-0.439045\pi\)
0.190329 + 0.981720i \(0.439045\pi\)
\(954\) 5.21426 0.168818
\(955\) −60.2871 −1.95084
\(956\) 16.7254 0.540937
\(957\) 5.48709 0.177373
\(958\) −20.7341 −0.669889
\(959\) 48.3762 1.56215
\(960\) 3.29417 0.106319
\(961\) −18.8402 −0.607748
\(962\) 1.84774 0.0595734
\(963\) −10.0986 −0.325423
\(964\) 7.56700 0.243717
\(965\) −75.0771 −2.41682
\(966\) 2.46575 0.0793341
\(967\) 2.50928 0.0806931 0.0403466 0.999186i \(-0.487154\pi\)
0.0403466 + 0.999186i \(0.487154\pi\)
\(968\) −19.1082 −0.614161
\(969\) −1.75957 −0.0565256
\(970\) −53.1869 −1.70773
\(971\) −20.9414 −0.672043 −0.336021 0.941854i \(-0.609081\pi\)
−0.336021 + 0.941854i \(0.609081\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −17.4291 −0.558753
\(974\) −42.0111 −1.34612
\(975\) 0.993371 0.0318133
\(976\) 9.27102 0.296758
\(977\) −15.3501 −0.491093 −0.245546 0.969385i \(-0.578967\pi\)
−0.245546 + 0.969385i \(0.578967\pi\)
\(978\) 15.2702 0.488286
\(979\) −56.4225 −1.80327
\(980\) 3.03094 0.0968198
\(981\) 0.442589 0.0141308
\(982\) −21.6896 −0.692143
\(983\) −29.2533 −0.933036 −0.466518 0.884512i \(-0.654492\pi\)
−0.466518 + 0.884512i \(0.654492\pi\)
\(984\) 7.95284 0.253527
\(985\) −49.4829 −1.57666
\(986\) 0.362680 0.0115501
\(987\) −3.78308 −0.120417
\(988\) −0.823610 −0.0262025
\(989\) 10.1226 0.321880
\(990\) 18.0754 0.574475
\(991\) 35.3747 1.12372 0.561858 0.827234i \(-0.310087\pi\)
0.561858 + 0.827234i \(0.310087\pi\)
\(992\) 3.48709 0.110715
\(993\) 23.2629 0.738227
\(994\) 14.9805 0.475153
\(995\) −72.8282 −2.30881
\(996\) −5.55741 −0.176093
\(997\) 36.7573 1.16411 0.582057 0.813148i \(-0.302248\pi\)
0.582057 + 0.813148i \(0.302248\pi\)
\(998\) −16.2125 −0.513196
\(999\) −10.8843 −0.344365
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 4002.2.a.ba.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.ba.1.1 4 1.1 even 1 trivial