Properties

Label 4002.2.a.bf.1.5
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
Analytic rank $0$
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] = [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: \(6\)
Coefficient field: 6.6.61157024.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8x^{4} + 8x^{3} + 17x^{2} - 4x - 4 \) 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 1.5
Root \(1.85969\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} -1.00000 q^{6} +3.26093 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} -1.00000 q^{6} +3.26093 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.56155 q^{10} +0.513894 q^{11} +1.00000 q^{12} -3.20549 q^{13} -3.26093 q^{14} +2.56155 q^{15} +1.00000 q^{16} -2.52278 q^{17} -1.00000 q^{18} +3.58156 q^{19} +2.56155 q^{20} +3.26093 q^{21} -0.513894 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.56155 q^{25} +3.20549 q^{26} +1.00000 q^{27} +3.26093 q^{28} +1.00000 q^{29} -2.56155 q^{30} -9.93042 q^{31} -1.00000 q^{32} +0.513894 q^{33} +2.52278 q^{34} +8.35304 q^{35} +1.00000 q^{36} +4.60517 q^{37} -3.58156 q^{38} -3.20549 q^{39} -2.56155 q^{40} +6.66047 q^{41} -3.26093 q^{42} +1.43067 q^{43} +0.513894 q^{44} +2.56155 q^{45} -1.00000 q^{46} +6.20215 q^{47} +1.00000 q^{48} +3.63366 q^{49} -1.56155 q^{50} -2.52278 q^{51} -3.20549 q^{52} +6.04002 q^{53} -1.00000 q^{54} +1.31637 q^{55} -3.26093 q^{56} +3.58156 q^{57} -1.00000 q^{58} +8.70408 q^{59} +2.56155 q^{60} +0.354436 q^{61} +9.93042 q^{62} +3.26093 q^{63} +1.00000 q^{64} -8.21104 q^{65} -0.513894 q^{66} -10.2025 q^{67} -2.52278 q^{68} +1.00000 q^{69} -8.35304 q^{70} +9.89709 q^{71} -1.00000 q^{72} +13.5674 q^{73} -4.60517 q^{74} +1.56155 q^{75} +3.58156 q^{76} +1.67577 q^{77} +3.20549 q^{78} +12.8688 q^{79} +2.56155 q^{80} +1.00000 q^{81} -6.66047 q^{82} -8.47531 q^{83} +3.26093 q^{84} -6.46224 q^{85} -1.43067 q^{86} +1.00000 q^{87} -0.513894 q^{88} +2.85805 q^{89} -2.56155 q^{90} -10.4529 q^{91} +1.00000 q^{92} -9.93042 q^{93} -6.20215 q^{94} +9.17436 q^{95} -1.00000 q^{96} +13.0160 q^{97} -3.63366 q^{98} +0.513894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{3} + 6 q^{4} + 3 q^{5} - 6 q^{6} + 2 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{3} + 6 q^{4} + 3 q^{5} - 6 q^{6} + 2 q^{7} - 6 q^{8} + 6 q^{9} - 3 q^{10} + 7 q^{11} + 6 q^{12} + 3 q^{13} - 2 q^{14} + 3 q^{15} + 6 q^{16} + 8 q^{17} - 6 q^{18} - 4 q^{19} + 3 q^{20} + 2 q^{21} - 7 q^{22} + 6 q^{23} - 6 q^{24} - 3 q^{25} - 3 q^{26} + 6 q^{27} + 2 q^{28} + 6 q^{29} - 3 q^{30} + q^{31} - 6 q^{32} + 7 q^{33} - 8 q^{34} + 18 q^{35} + 6 q^{36} + 7 q^{37} + 4 q^{38} + 3 q^{39} - 3 q^{40} + 13 q^{41} - 2 q^{42} + 7 q^{44} + 3 q^{45} - 6 q^{46} + 22 q^{47} + 6 q^{48} + 8 q^{49} + 3 q^{50} + 8 q^{51} + 3 q^{52} + 10 q^{53} - 6 q^{54} - 5 q^{55} - 2 q^{56} - 4 q^{57} - 6 q^{58} + 17 q^{59} + 3 q^{60} + q^{61} - q^{62} + 2 q^{63} + 6 q^{64} - 7 q^{65} - 7 q^{66} + 3 q^{67} + 8 q^{68} + 6 q^{69} - 18 q^{70} + 11 q^{71} - 6 q^{72} - 7 q^{74} - 3 q^{75} - 4 q^{76} - 3 q^{78} + 2 q^{79} + 3 q^{80} + 6 q^{81} - 13 q^{82} + 16 q^{83} + 2 q^{84} + 4 q^{85} + 6 q^{87} - 7 q^{88} + 16 q^{89} - 3 q^{90} - 28 q^{91} + 6 q^{92} + q^{93} - 22 q^{94} + 32 q^{95} - 6 q^{96} + 2 q^{97} - 8 q^{98} + 7 q^{99}+O(q^{100}) \) 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
\(5\) 2.56155 1.14556 0.572781 0.819709i \(-0.305865\pi\)
0.572781 + 0.819709i \(0.305865\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.26093 1.23252 0.616258 0.787545i \(-0.288648\pi\)
0.616258 + 0.787545i \(0.288648\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.56155 −0.810034
\(11\) 0.513894 0.154945 0.0774724 0.996994i \(-0.475315\pi\)
0.0774724 + 0.996994i \(0.475315\pi\)
\(12\) 1.00000 0.288675
\(13\) −3.20549 −0.889043 −0.444522 0.895768i \(-0.646626\pi\)
−0.444522 + 0.895768i \(0.646626\pi\)
\(14\) −3.26093 −0.871520
\(15\) 2.56155 0.661390
\(16\) 1.00000 0.250000
\(17\) −2.52278 −0.611864 −0.305932 0.952053i \(-0.598968\pi\)
−0.305932 + 0.952053i \(0.598968\pi\)
\(18\) −1.00000 −0.235702
\(19\) 3.58156 0.821667 0.410833 0.911710i \(-0.365238\pi\)
0.410833 + 0.911710i \(0.365238\pi\)
\(20\) 2.56155 0.572781
\(21\) 3.26093 0.711593
\(22\) −0.513894 −0.109563
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.56155 0.312311
\(26\) 3.20549 0.628649
\(27\) 1.00000 0.192450
\(28\) 3.26093 0.616258
\(29\) 1.00000 0.185695
\(30\) −2.56155 −0.467673
\(31\) −9.93042 −1.78356 −0.891778 0.452473i \(-0.850542\pi\)
−0.891778 + 0.452473i \(0.850542\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.513894 0.0894574
\(34\) 2.52278 0.432653
\(35\) 8.35304 1.41192
\(36\) 1.00000 0.166667
\(37\) 4.60517 0.757085 0.378542 0.925584i \(-0.376425\pi\)
0.378542 + 0.925584i \(0.376425\pi\)
\(38\) −3.58156 −0.581006
\(39\) −3.20549 −0.513289
\(40\) −2.56155 −0.405017
\(41\) 6.66047 1.04019 0.520095 0.854108i \(-0.325897\pi\)
0.520095 + 0.854108i \(0.325897\pi\)
\(42\) −3.26093 −0.503172
\(43\) 1.43067 0.218175 0.109087 0.994032i \(-0.465207\pi\)
0.109087 + 0.994032i \(0.465207\pi\)
\(44\) 0.513894 0.0774724
\(45\) 2.56155 0.381854
\(46\) −1.00000 −0.147442
\(47\) 6.20215 0.904676 0.452338 0.891847i \(-0.350590\pi\)
0.452338 + 0.891847i \(0.350590\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.63366 0.519094
\(50\) −1.56155 −0.220837
\(51\) −2.52278 −0.353260
\(52\) −3.20549 −0.444522
\(53\) 6.04002 0.829660 0.414830 0.909899i \(-0.363841\pi\)
0.414830 + 0.909899i \(0.363841\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.31637 0.177499
\(56\) −3.26093 −0.435760
\(57\) 3.58156 0.474389
\(58\) −1.00000 −0.131306
\(59\) 8.70408 1.13317 0.566587 0.824002i \(-0.308263\pi\)
0.566587 + 0.824002i \(0.308263\pi\)
\(60\) 2.56155 0.330695
\(61\) 0.354436 0.0453809 0.0226905 0.999743i \(-0.492777\pi\)
0.0226905 + 0.999743i \(0.492777\pi\)
\(62\) 9.93042 1.26116
\(63\) 3.26093 0.410838
\(64\) 1.00000 0.125000
\(65\) −8.21104 −1.01845
\(66\) −0.513894 −0.0632559
\(67\) −10.2025 −1.24643 −0.623215 0.782050i \(-0.714174\pi\)
−0.623215 + 0.782050i \(0.714174\pi\)
\(68\) −2.52278 −0.305932
\(69\) 1.00000 0.120386
\(70\) −8.35304 −0.998379
\(71\) 9.89709 1.17457 0.587284 0.809381i \(-0.300197\pi\)
0.587284 + 0.809381i \(0.300197\pi\)
\(72\) −1.00000 −0.117851
\(73\) 13.5674 1.58795 0.793973 0.607953i \(-0.208009\pi\)
0.793973 + 0.607953i \(0.208009\pi\)
\(74\) −4.60517 −0.535340
\(75\) 1.56155 0.180313
\(76\) 3.58156 0.410833
\(77\) 1.67577 0.190972
\(78\) 3.20549 0.362950
\(79\) 12.8688 1.44786 0.723928 0.689875i \(-0.242335\pi\)
0.723928 + 0.689875i \(0.242335\pi\)
\(80\) 2.56155 0.286390
\(81\) 1.00000 0.111111
\(82\) −6.66047 −0.735525
\(83\) −8.47531 −0.930286 −0.465143 0.885236i \(-0.653997\pi\)
−0.465143 + 0.885236i \(0.653997\pi\)
\(84\) 3.26093 0.355796
\(85\) −6.46224 −0.700928
\(86\) −1.43067 −0.154273
\(87\) 1.00000 0.107211
\(88\) −0.513894 −0.0547813
\(89\) 2.85805 0.302952 0.151476 0.988461i \(-0.451597\pi\)
0.151476 + 0.988461i \(0.451597\pi\)
\(90\) −2.56155 −0.270011
\(91\) −10.4529 −1.09576
\(92\) 1.00000 0.104257
\(93\) −9.93042 −1.02974
\(94\) −6.20215 −0.639703
\(95\) 9.17436 0.941270
\(96\) −1.00000 −0.102062
\(97\) 13.0160 1.32158 0.660788 0.750573i \(-0.270222\pi\)
0.660788 + 0.750573i \(0.270222\pi\)
\(98\) −3.63366 −0.367055
\(99\) 0.513894 0.0516483
\(100\) 1.56155 0.156155
\(101\) 3.21406 0.319811 0.159905 0.987132i \(-0.448881\pi\)
0.159905 + 0.987132i \(0.448881\pi\)
\(102\) 2.52278 0.249793
\(103\) 4.73507 0.466561 0.233280 0.972410i \(-0.425054\pi\)
0.233280 + 0.972410i \(0.425054\pi\)
\(104\) 3.20549 0.314324
\(105\) 8.35304 0.815173
\(106\) −6.04002 −0.586658
\(107\) 12.5974 1.21784 0.608918 0.793233i \(-0.291604\pi\)
0.608918 + 0.793233i \(0.291604\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.2135 1.16984 0.584922 0.811089i \(-0.301125\pi\)
0.584922 + 0.811089i \(0.301125\pi\)
\(110\) −1.31637 −0.125511
\(111\) 4.60517 0.437103
\(112\) 3.26093 0.308129
\(113\) 6.48945 0.610476 0.305238 0.952276i \(-0.401264\pi\)
0.305238 + 0.952276i \(0.401264\pi\)
\(114\) −3.58156 −0.335444
\(115\) 2.56155 0.238866
\(116\) 1.00000 0.0928477
\(117\) −3.20549 −0.296348
\(118\) −8.70408 −0.801276
\(119\) −8.22661 −0.754132
\(120\) −2.56155 −0.233837
\(121\) −10.7359 −0.975992
\(122\) −0.354436 −0.0320891
\(123\) 6.66047 0.600554
\(124\) −9.93042 −0.891778
\(125\) −8.80776 −0.787790
\(126\) −3.26093 −0.290507
\(127\) −13.4281 −1.19155 −0.595775 0.803151i \(-0.703155\pi\)
−0.595775 + 0.803151i \(0.703155\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.43067 0.125963
\(130\) 8.21104 0.720155
\(131\) 4.15449 0.362979 0.181490 0.983393i \(-0.441908\pi\)
0.181490 + 0.983393i \(0.441908\pi\)
\(132\) 0.513894 0.0447287
\(133\) 11.6792 1.01272
\(134\) 10.2025 0.881359
\(135\) 2.56155 0.220463
\(136\) 2.52278 0.216327
\(137\) −10.3838 −0.887145 −0.443573 0.896238i \(-0.646289\pi\)
−0.443573 + 0.896238i \(0.646289\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −19.4044 −1.64586 −0.822928 0.568145i \(-0.807661\pi\)
−0.822928 + 0.568145i \(0.807661\pi\)
\(140\) 8.35304 0.705961
\(141\) 6.20215 0.522315
\(142\) −9.89709 −0.830545
\(143\) −1.64728 −0.137753
\(144\) 1.00000 0.0833333
\(145\) 2.56155 0.212725
\(146\) −13.5674 −1.12285
\(147\) 3.63366 0.299699
\(148\) 4.60517 0.378542
\(149\) −6.04894 −0.495548 −0.247774 0.968818i \(-0.579699\pi\)
−0.247774 + 0.968818i \(0.579699\pi\)
\(150\) −1.56155 −0.127500
\(151\) −12.3623 −1.00603 −0.503013 0.864279i \(-0.667775\pi\)
−0.503013 + 0.864279i \(0.667775\pi\)
\(152\) −3.58156 −0.290503
\(153\) −2.52278 −0.203955
\(154\) −1.67577 −0.135037
\(155\) −25.4373 −2.04317
\(156\) −3.20549 −0.256645
\(157\) −2.98654 −0.238352 −0.119176 0.992873i \(-0.538025\pi\)
−0.119176 + 0.992873i \(0.538025\pi\)
\(158\) −12.8688 −1.02379
\(159\) 6.04002 0.479005
\(160\) −2.56155 −0.202509
\(161\) 3.26093 0.256997
\(162\) −1.00000 −0.0785674
\(163\) 14.1375 1.10734 0.553669 0.832737i \(-0.313227\pi\)
0.553669 + 0.832737i \(0.313227\pi\)
\(164\) 6.66047 0.520095
\(165\) 1.31637 0.102479
\(166\) 8.47531 0.657811
\(167\) −12.7796 −0.988917 −0.494459 0.869201i \(-0.664634\pi\)
−0.494459 + 0.869201i \(0.664634\pi\)
\(168\) −3.26093 −0.251586
\(169\) −2.72483 −0.209602
\(170\) 6.46224 0.495631
\(171\) 3.58156 0.273889
\(172\) 1.43067 0.109087
\(173\) 14.6234 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 5.09211 0.384928
\(176\) 0.513894 0.0387362
\(177\) 8.70408 0.654239
\(178\) −2.85805 −0.214220
\(179\) −20.1935 −1.50933 −0.754665 0.656110i \(-0.772201\pi\)
−0.754665 + 0.656110i \(0.772201\pi\)
\(180\) 2.56155 0.190927
\(181\) −19.7979 −1.47157 −0.735783 0.677218i \(-0.763186\pi\)
−0.735783 + 0.677218i \(0.763186\pi\)
\(182\) 10.4529 0.774819
\(183\) 0.354436 0.0262007
\(184\) −1.00000 −0.0737210
\(185\) 11.7964 0.867287
\(186\) 9.93042 0.728134
\(187\) −1.29644 −0.0948052
\(188\) 6.20215 0.452338
\(189\) 3.26093 0.237198
\(190\) −9.17436 −0.665578
\(191\) −1.75927 −0.127296 −0.0636480 0.997972i \(-0.520273\pi\)
−0.0636480 + 0.997972i \(0.520273\pi\)
\(192\) 1.00000 0.0721688
\(193\) −23.8470 −1.71654 −0.858271 0.513197i \(-0.828461\pi\)
−0.858271 + 0.513197i \(0.828461\pi\)
\(194\) −13.0160 −0.934495
\(195\) −8.21104 −0.588004
\(196\) 3.63366 0.259547
\(197\) 3.71894 0.264963 0.132482 0.991185i \(-0.457705\pi\)
0.132482 + 0.991185i \(0.457705\pi\)
\(198\) −0.513894 −0.0365208
\(199\) 5.58551 0.395946 0.197973 0.980207i \(-0.436564\pi\)
0.197973 + 0.980207i \(0.436564\pi\)
\(200\) −1.56155 −0.110418
\(201\) −10.2025 −0.719627
\(202\) −3.21406 −0.226140
\(203\) 3.26093 0.228872
\(204\) −2.52278 −0.176630
\(205\) 17.0611 1.19160
\(206\) −4.73507 −0.329908
\(207\) 1.00000 0.0695048
\(208\) −3.20549 −0.222261
\(209\) 1.84054 0.127313
\(210\) −8.35304 −0.576415
\(211\) 7.31232 0.503401 0.251700 0.967805i \(-0.419010\pi\)
0.251700 + 0.967805i \(0.419010\pi\)
\(212\) 6.04002 0.414830
\(213\) 9.89709 0.678137
\(214\) −12.5974 −0.861140
\(215\) 3.66473 0.249933
\(216\) −1.00000 −0.0680414
\(217\) −32.3824 −2.19826
\(218\) −12.2135 −0.827205
\(219\) 13.5674 0.916801
\(220\) 1.31637 0.0887494
\(221\) 8.08675 0.543974
\(222\) −4.60517 −0.309079
\(223\) −26.4260 −1.76961 −0.884807 0.465958i \(-0.845710\pi\)
−0.884807 + 0.465958i \(0.845710\pi\)
\(224\) −3.26093 −0.217880
\(225\) 1.56155 0.104104
\(226\) −6.48945 −0.431672
\(227\) 5.83322 0.387164 0.193582 0.981084i \(-0.437989\pi\)
0.193582 + 0.981084i \(0.437989\pi\)
\(228\) 3.58156 0.237195
\(229\) 22.8231 1.50819 0.754096 0.656764i \(-0.228075\pi\)
0.754096 + 0.656764i \(0.228075\pi\)
\(230\) −2.56155 −0.168904
\(231\) 1.67577 0.110258
\(232\) −1.00000 −0.0656532
\(233\) 24.9439 1.63413 0.817066 0.576545i \(-0.195599\pi\)
0.817066 + 0.576545i \(0.195599\pi\)
\(234\) 3.20549 0.209550
\(235\) 15.8871 1.03636
\(236\) 8.70408 0.566587
\(237\) 12.8688 0.835920
\(238\) 8.22661 0.533252
\(239\) −28.4198 −1.83832 −0.919162 0.393880i \(-0.871132\pi\)
−0.919162 + 0.393880i \(0.871132\pi\)
\(240\) 2.56155 0.165348
\(241\) −22.4261 −1.44459 −0.722296 0.691584i \(-0.756913\pi\)
−0.722296 + 0.691584i \(0.756913\pi\)
\(242\) 10.7359 0.690131
\(243\) 1.00000 0.0641500
\(244\) 0.354436 0.0226905
\(245\) 9.30780 0.594654
\(246\) −6.66047 −0.424656
\(247\) −11.4807 −0.730497
\(248\) 9.93042 0.630582
\(249\) −8.47531 −0.537101
\(250\) 8.80776 0.557052
\(251\) 9.30608 0.587395 0.293697 0.955898i \(-0.405114\pi\)
0.293697 + 0.955898i \(0.405114\pi\)
\(252\) 3.26093 0.205419
\(253\) 0.513894 0.0323082
\(254\) 13.4281 0.842553
\(255\) −6.46224 −0.404681
\(256\) 1.00000 0.0625000
\(257\) 11.4385 0.713512 0.356756 0.934198i \(-0.383883\pi\)
0.356756 + 0.934198i \(0.383883\pi\)
\(258\) −1.43067 −0.0890696
\(259\) 15.0171 0.933119
\(260\) −8.21104 −0.509227
\(261\) 1.00000 0.0618984
\(262\) −4.15449 −0.256665
\(263\) 15.1714 0.935510 0.467755 0.883858i \(-0.345063\pi\)
0.467755 + 0.883858i \(0.345063\pi\)
\(264\) −0.513894 −0.0316280
\(265\) 15.4718 0.950427
\(266\) −11.6792 −0.716099
\(267\) 2.85805 0.174910
\(268\) −10.2025 −0.623215
\(269\) −30.0545 −1.83245 −0.916227 0.400660i \(-0.868781\pi\)
−0.916227 + 0.400660i \(0.868781\pi\)
\(270\) −2.56155 −0.155891
\(271\) 25.5234 1.55044 0.775218 0.631694i \(-0.217640\pi\)
0.775218 + 0.631694i \(0.217640\pi\)
\(272\) −2.52278 −0.152966
\(273\) −10.4529 −0.632637
\(274\) 10.3838 0.627307
\(275\) 0.802472 0.0483909
\(276\) 1.00000 0.0601929
\(277\) −20.1539 −1.21093 −0.605465 0.795872i \(-0.707013\pi\)
−0.605465 + 0.795872i \(0.707013\pi\)
\(278\) 19.4044 1.16380
\(279\) −9.93042 −0.594519
\(280\) −8.35304 −0.499190
\(281\) 19.4320 1.15921 0.579607 0.814896i \(-0.303206\pi\)
0.579607 + 0.814896i \(0.303206\pi\)
\(282\) −6.20215 −0.369333
\(283\) 6.87622 0.408749 0.204375 0.978893i \(-0.434484\pi\)
0.204375 + 0.978893i \(0.434484\pi\)
\(284\) 9.89709 0.587284
\(285\) 9.17436 0.543442
\(286\) 1.64728 0.0974058
\(287\) 21.7193 1.28205
\(288\) −1.00000 −0.0589256
\(289\) −10.6356 −0.625622
\(290\) −2.56155 −0.150420
\(291\) 13.0160 0.763012
\(292\) 13.5674 0.793973
\(293\) 17.3419 1.01313 0.506564 0.862203i \(-0.330915\pi\)
0.506564 + 0.862203i \(0.330915\pi\)
\(294\) −3.63366 −0.211919
\(295\) 22.2960 1.29812
\(296\) −4.60517 −0.267670
\(297\) 0.513894 0.0298191
\(298\) 6.04894 0.350405
\(299\) −3.20549 −0.185378
\(300\) 1.56155 0.0901563
\(301\) 4.66531 0.268904
\(302\) 12.3623 0.711368
\(303\) 3.21406 0.184643
\(304\) 3.58156 0.205417
\(305\) 0.907907 0.0519866
\(306\) 2.52278 0.144218
\(307\) −21.7731 −1.24266 −0.621329 0.783549i \(-0.713407\pi\)
−0.621329 + 0.783549i \(0.713407\pi\)
\(308\) 1.67577 0.0954859
\(309\) 4.73507 0.269369
\(310\) 25.4373 1.44474
\(311\) 12.6368 0.716566 0.358283 0.933613i \(-0.383362\pi\)
0.358283 + 0.933613i \(0.383362\pi\)
\(312\) 3.20549 0.181475
\(313\) −28.2444 −1.59647 −0.798234 0.602347i \(-0.794232\pi\)
−0.798234 + 0.602347i \(0.794232\pi\)
\(314\) 2.98654 0.168540
\(315\) 8.35304 0.470641
\(316\) 12.8688 0.723928
\(317\) −17.8766 −1.00405 −0.502024 0.864854i \(-0.667411\pi\)
−0.502024 + 0.864854i \(0.667411\pi\)
\(318\) −6.04002 −0.338707
\(319\) 0.513894 0.0287725
\(320\) 2.56155 0.143195
\(321\) 12.5974 0.703118
\(322\) −3.26093 −0.181724
\(323\) −9.03550 −0.502749
\(324\) 1.00000 0.0555556
\(325\) −5.00554 −0.277658
\(326\) −14.1375 −0.783006
\(327\) 12.2135 0.675410
\(328\) −6.66047 −0.367763
\(329\) 20.2248 1.11503
\(330\) −1.31637 −0.0724636
\(331\) −16.9855 −0.933606 −0.466803 0.884361i \(-0.654594\pi\)
−0.466803 + 0.884361i \(0.654594\pi\)
\(332\) −8.47531 −0.465143
\(333\) 4.60517 0.252362
\(334\) 12.7796 0.699270
\(335\) −26.1342 −1.42786
\(336\) 3.26093 0.177898
\(337\) −4.15420 −0.226294 −0.113147 0.993578i \(-0.536093\pi\)
−0.113147 + 0.993578i \(0.536093\pi\)
\(338\) 2.72483 0.148211
\(339\) 6.48945 0.352459
\(340\) −6.46224 −0.350464
\(341\) −5.10318 −0.276353
\(342\) −3.58156 −0.193669
\(343\) −10.9774 −0.592724
\(344\) −1.43067 −0.0771365
\(345\) 2.56155 0.137909
\(346\) −14.6234 −0.786160
\(347\) 7.27163 0.390362 0.195181 0.980767i \(-0.437471\pi\)
0.195181 + 0.980767i \(0.437471\pi\)
\(348\) 1.00000 0.0536056
\(349\) −0.326949 −0.0175012 −0.00875058 0.999962i \(-0.502785\pi\)
−0.00875058 + 0.999962i \(0.502785\pi\)
\(350\) −5.09211 −0.272185
\(351\) −3.20549 −0.171096
\(352\) −0.513894 −0.0273906
\(353\) −5.39173 −0.286973 −0.143486 0.989652i \(-0.545831\pi\)
−0.143486 + 0.989652i \(0.545831\pi\)
\(354\) −8.70408 −0.462617
\(355\) 25.3519 1.34554
\(356\) 2.85805 0.151476
\(357\) −8.22661 −0.435398
\(358\) 20.1935 1.06726
\(359\) 11.0689 0.584195 0.292098 0.956389i \(-0.405647\pi\)
0.292098 + 0.956389i \(0.405647\pi\)
\(360\) −2.56155 −0.135006
\(361\) −6.17241 −0.324864
\(362\) 19.7979 1.04055
\(363\) −10.7359 −0.563489
\(364\) −10.4529 −0.547880
\(365\) 34.7537 1.81909
\(366\) −0.354436 −0.0185267
\(367\) −2.38125 −0.124300 −0.0621500 0.998067i \(-0.519796\pi\)
−0.0621500 + 0.998067i \(0.519796\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.66047 0.346730
\(370\) −11.7964 −0.613265
\(371\) 19.6961 1.02257
\(372\) −9.93042 −0.514868
\(373\) −18.3022 −0.947650 −0.473825 0.880619i \(-0.657127\pi\)
−0.473825 + 0.880619i \(0.657127\pi\)
\(374\) 1.29644 0.0670374
\(375\) −8.80776 −0.454831
\(376\) −6.20215 −0.319851
\(377\) −3.20549 −0.165091
\(378\) −3.26093 −0.167724
\(379\) 2.60318 0.133716 0.0668582 0.997762i \(-0.478702\pi\)
0.0668582 + 0.997762i \(0.478702\pi\)
\(380\) 9.17436 0.470635
\(381\) −13.4281 −0.687942
\(382\) 1.75927 0.0900119
\(383\) 17.4335 0.890808 0.445404 0.895330i \(-0.353060\pi\)
0.445404 + 0.895330i \(0.353060\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 4.29258 0.218770
\(386\) 23.8470 1.21378
\(387\) 1.43067 0.0727250
\(388\) 13.0160 0.660788
\(389\) −11.7377 −0.595123 −0.297561 0.954703i \(-0.596173\pi\)
−0.297561 + 0.954703i \(0.596173\pi\)
\(390\) 8.21104 0.415782
\(391\) −2.52278 −0.127583
\(392\) −3.63366 −0.183527
\(393\) 4.15449 0.209566
\(394\) −3.71894 −0.187357
\(395\) 32.9642 1.65861
\(396\) 0.513894 0.0258241
\(397\) 12.2931 0.616974 0.308487 0.951229i \(-0.400177\pi\)
0.308487 + 0.951229i \(0.400177\pi\)
\(398\) −5.58551 −0.279976
\(399\) 11.6792 0.584692
\(400\) 1.56155 0.0780776
\(401\) −28.2496 −1.41072 −0.705358 0.708851i \(-0.749214\pi\)
−0.705358 + 0.708851i \(0.749214\pi\)
\(402\) 10.2025 0.508853
\(403\) 31.8319 1.58566
\(404\) 3.21406 0.159905
\(405\) 2.56155 0.127285
\(406\) −3.26093 −0.161837
\(407\) 2.36657 0.117306
\(408\) 2.52278 0.124896
\(409\) −35.5775 −1.75919 −0.879597 0.475719i \(-0.842188\pi\)
−0.879597 + 0.475719i \(0.842188\pi\)
\(410\) −17.0611 −0.842589
\(411\) −10.3838 −0.512194
\(412\) 4.73507 0.233280
\(413\) 28.3834 1.39666
\(414\) −1.00000 −0.0491473
\(415\) −21.7099 −1.06570
\(416\) 3.20549 0.157162
\(417\) −19.4044 −0.950236
\(418\) −1.84054 −0.0900239
\(419\) 21.6897 1.05961 0.529806 0.848119i \(-0.322265\pi\)
0.529806 + 0.848119i \(0.322265\pi\)
\(420\) 8.35304 0.407587
\(421\) 30.8811 1.50505 0.752526 0.658563i \(-0.228835\pi\)
0.752526 + 0.658563i \(0.228835\pi\)
\(422\) −7.31232 −0.355958
\(423\) 6.20215 0.301559
\(424\) −6.04002 −0.293329
\(425\) −3.93946 −0.191092
\(426\) −9.89709 −0.479516
\(427\) 1.15579 0.0559326
\(428\) 12.5974 0.608918
\(429\) −1.64728 −0.0795315
\(430\) −3.66473 −0.176729
\(431\) −16.7845 −0.808480 −0.404240 0.914653i \(-0.632464\pi\)
−0.404240 + 0.914653i \(0.632464\pi\)
\(432\) 1.00000 0.0481125
\(433\) −34.7198 −1.66853 −0.834263 0.551367i \(-0.814106\pi\)
−0.834263 + 0.551367i \(0.814106\pi\)
\(434\) 32.3824 1.55440
\(435\) 2.56155 0.122817
\(436\) 12.2135 0.584922
\(437\) 3.58156 0.171329
\(438\) −13.5674 −0.648276
\(439\) −3.46274 −0.165268 −0.0826339 0.996580i \(-0.526333\pi\)
−0.0826339 + 0.996580i \(0.526333\pi\)
\(440\) −1.31637 −0.0627553
\(441\) 3.63366 0.173031
\(442\) −8.08675 −0.384648
\(443\) −14.4669 −0.687343 −0.343671 0.939090i \(-0.611671\pi\)
−0.343671 + 0.939090i \(0.611671\pi\)
\(444\) 4.60517 0.218552
\(445\) 7.32104 0.347051
\(446\) 26.4260 1.25131
\(447\) −6.04894 −0.286105
\(448\) 3.26093 0.154064
\(449\) −27.0281 −1.27554 −0.637768 0.770229i \(-0.720142\pi\)
−0.637768 + 0.770229i \(0.720142\pi\)
\(450\) −1.56155 −0.0736123
\(451\) 3.42277 0.161172
\(452\) 6.48945 0.305238
\(453\) −12.3623 −0.580829
\(454\) −5.83322 −0.273766
\(455\) −26.7756 −1.25526
\(456\) −3.58156 −0.167722
\(457\) 31.4925 1.47316 0.736580 0.676351i \(-0.236440\pi\)
0.736580 + 0.676351i \(0.236440\pi\)
\(458\) −22.8231 −1.06645
\(459\) −2.52278 −0.117753
\(460\) 2.56155 0.119433
\(461\) −36.7196 −1.71020 −0.855100 0.518463i \(-0.826504\pi\)
−0.855100 + 0.518463i \(0.826504\pi\)
\(462\) −1.67577 −0.0779639
\(463\) 34.0251 1.58128 0.790640 0.612281i \(-0.209748\pi\)
0.790640 + 0.612281i \(0.209748\pi\)
\(464\) 1.00000 0.0464238
\(465\) −25.4373 −1.17963
\(466\) −24.9439 −1.15551
\(467\) −8.65038 −0.400292 −0.200146 0.979766i \(-0.564142\pi\)
−0.200146 + 0.979766i \(0.564142\pi\)
\(468\) −3.20549 −0.148174
\(469\) −33.2695 −1.53624
\(470\) −15.8871 −0.732819
\(471\) −2.98654 −0.137612
\(472\) −8.70408 −0.400638
\(473\) 0.735212 0.0338051
\(474\) −12.8688 −0.591085
\(475\) 5.59280 0.256615
\(476\) −8.22661 −0.377066
\(477\) 6.04002 0.276553
\(478\) 28.4198 1.29989
\(479\) 18.2393 0.833375 0.416688 0.909050i \(-0.363191\pi\)
0.416688 + 0.909050i \(0.363191\pi\)
\(480\) −2.56155 −0.116918
\(481\) −14.7618 −0.673081
\(482\) 22.4261 1.02148
\(483\) 3.26093 0.148377
\(484\) −10.7359 −0.487996
\(485\) 33.3412 1.51395
\(486\) −1.00000 −0.0453609
\(487\) −42.4026 −1.92145 −0.960724 0.277507i \(-0.910492\pi\)
−0.960724 + 0.277507i \(0.910492\pi\)
\(488\) −0.354436 −0.0160446
\(489\) 14.1375 0.639322
\(490\) −9.30780 −0.420484
\(491\) 14.7408 0.665244 0.332622 0.943060i \(-0.392067\pi\)
0.332622 + 0.943060i \(0.392067\pi\)
\(492\) 6.66047 0.300277
\(493\) −2.52278 −0.113620
\(494\) 11.4807 0.516540
\(495\) 1.31637 0.0591663
\(496\) −9.93042 −0.445889
\(497\) 32.2737 1.44767
\(498\) 8.47531 0.379788
\(499\) 16.3704 0.732838 0.366419 0.930450i \(-0.380584\pi\)
0.366419 + 0.930450i \(0.380584\pi\)
\(500\) −8.80776 −0.393895
\(501\) −12.7796 −0.570952
\(502\) −9.30608 −0.415351
\(503\) 26.4925 1.18124 0.590620 0.806950i \(-0.298883\pi\)
0.590620 + 0.806950i \(0.298883\pi\)
\(504\) −3.26093 −0.145253
\(505\) 8.23297 0.366363
\(506\) −0.513894 −0.0228454
\(507\) −2.72483 −0.121014
\(508\) −13.4281 −0.595775
\(509\) −5.88868 −0.261011 −0.130505 0.991448i \(-0.541660\pi\)
−0.130505 + 0.991448i \(0.541660\pi\)
\(510\) 6.46224 0.286153
\(511\) 44.2424 1.95717
\(512\) −1.00000 −0.0441942
\(513\) 3.58156 0.158130
\(514\) −11.4385 −0.504529
\(515\) 12.1291 0.534474
\(516\) 1.43067 0.0629817
\(517\) 3.18725 0.140175
\(518\) −15.0171 −0.659814
\(519\) 14.6234 0.641897
\(520\) 8.21104 0.360078
\(521\) 32.4674 1.42242 0.711211 0.702979i \(-0.248147\pi\)
0.711211 + 0.702979i \(0.248147\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 36.7377 1.60643 0.803215 0.595690i \(-0.203121\pi\)
0.803215 + 0.595690i \(0.203121\pi\)
\(524\) 4.15449 0.181490
\(525\) 5.09211 0.222238
\(526\) −15.1714 −0.661506
\(527\) 25.0523 1.09129
\(528\) 0.513894 0.0223644
\(529\) 1.00000 0.0434783
\(530\) −15.4718 −0.672053
\(531\) 8.70408 0.377725
\(532\) 11.6792 0.506358
\(533\) −21.3501 −0.924774
\(534\) −2.85805 −0.123680
\(535\) 32.2689 1.39511
\(536\) 10.2025 0.440680
\(537\) −20.1935 −0.871412
\(538\) 30.0545 1.29574
\(539\) 1.86731 0.0804309
\(540\) 2.56155 0.110232
\(541\) 23.4991 1.01030 0.505152 0.863031i \(-0.331437\pi\)
0.505152 + 0.863031i \(0.331437\pi\)
\(542\) −25.5234 −1.09632
\(543\) −19.7979 −0.849609
\(544\) 2.52278 0.108163
\(545\) 31.2856 1.34013
\(546\) 10.4529 0.447342
\(547\) 8.09388 0.346069 0.173035 0.984916i \(-0.444643\pi\)
0.173035 + 0.984916i \(0.444643\pi\)
\(548\) −10.3838 −0.443573
\(549\) 0.354436 0.0151270
\(550\) −0.802472 −0.0342175
\(551\) 3.58156 0.152580
\(552\) −1.00000 −0.0425628
\(553\) 41.9643 1.78451
\(554\) 20.1539 0.856257
\(555\) 11.7964 0.500728
\(556\) −19.4044 −0.822928
\(557\) −21.1809 −0.897466 −0.448733 0.893666i \(-0.648124\pi\)
−0.448733 + 0.893666i \(0.648124\pi\)
\(558\) 9.93042 0.420388
\(559\) −4.58600 −0.193967
\(560\) 8.35304 0.352980
\(561\) −1.29644 −0.0547358
\(562\) −19.4320 −0.819688
\(563\) −41.5310 −1.75032 −0.875161 0.483832i \(-0.839245\pi\)
−0.875161 + 0.483832i \(0.839245\pi\)
\(564\) 6.20215 0.261158
\(565\) 16.6231 0.699338
\(566\) −6.87622 −0.289029
\(567\) 3.26093 0.136946
\(568\) −9.89709 −0.415273
\(569\) 30.1631 1.26450 0.632252 0.774763i \(-0.282131\pi\)
0.632252 + 0.774763i \(0.282131\pi\)
\(570\) −9.17436 −0.384272
\(571\) −28.9531 −1.21165 −0.605825 0.795598i \(-0.707157\pi\)
−0.605825 + 0.795598i \(0.707157\pi\)
\(572\) −1.64728 −0.0688763
\(573\) −1.75927 −0.0734944
\(574\) −21.7193 −0.906546
\(575\) 1.56155 0.0651213
\(576\) 1.00000 0.0416667
\(577\) −24.3507 −1.01373 −0.506867 0.862024i \(-0.669197\pi\)
−0.506867 + 0.862024i \(0.669197\pi\)
\(578\) 10.6356 0.442382
\(579\) −23.8470 −0.991046
\(580\) 2.56155 0.106363
\(581\) −27.6374 −1.14659
\(582\) −13.0160 −0.539531
\(583\) 3.10393 0.128552
\(584\) −13.5674 −0.561424
\(585\) −8.21104 −0.339485
\(586\) −17.3419 −0.716389
\(587\) 31.1042 1.28381 0.641904 0.766785i \(-0.278145\pi\)
0.641904 + 0.766785i \(0.278145\pi\)
\(588\) 3.63366 0.149849
\(589\) −35.5664 −1.46549
\(590\) −22.2960 −0.917910
\(591\) 3.71894 0.152977
\(592\) 4.60517 0.189271
\(593\) −26.9064 −1.10491 −0.552457 0.833542i \(-0.686310\pi\)
−0.552457 + 0.833542i \(0.686310\pi\)
\(594\) −0.513894 −0.0210853
\(595\) −21.0729 −0.863904
\(596\) −6.04894 −0.247774
\(597\) 5.58551 0.228600
\(598\) 3.20549 0.131082
\(599\) 20.5595 0.840038 0.420019 0.907515i \(-0.362023\pi\)
0.420019 + 0.907515i \(0.362023\pi\)
\(600\) −1.56155 −0.0637501
\(601\) −7.09979 −0.289607 −0.144803 0.989460i \(-0.546255\pi\)
−0.144803 + 0.989460i \(0.546255\pi\)
\(602\) −4.66531 −0.190144
\(603\) −10.2025 −0.415477
\(604\) −12.3623 −0.503013
\(605\) −27.5006 −1.11806
\(606\) −3.21406 −0.130562
\(607\) 11.2606 0.457055 0.228528 0.973537i \(-0.426609\pi\)
0.228528 + 0.973537i \(0.426609\pi\)
\(608\) −3.58156 −0.145252
\(609\) 3.26093 0.132139
\(610\) −0.907907 −0.0367601
\(611\) −19.8809 −0.804296
\(612\) −2.52278 −0.101977
\(613\) 25.7546 1.04022 0.520110 0.854099i \(-0.325891\pi\)
0.520110 + 0.854099i \(0.325891\pi\)
\(614\) 21.7731 0.878692
\(615\) 17.0611 0.687971
\(616\) −1.67577 −0.0675187
\(617\) 29.3139 1.18013 0.590066 0.807355i \(-0.299102\pi\)
0.590066 + 0.807355i \(0.299102\pi\)
\(618\) −4.73507 −0.190473
\(619\) −10.0403 −0.403552 −0.201776 0.979432i \(-0.564671\pi\)
−0.201776 + 0.979432i \(0.564671\pi\)
\(620\) −25.4373 −1.02159
\(621\) 1.00000 0.0401286
\(622\) −12.6368 −0.506689
\(623\) 9.31989 0.373393
\(624\) −3.20549 −0.128322
\(625\) −30.3693 −1.21477
\(626\) 28.2444 1.12887
\(627\) 1.84054 0.0735042
\(628\) −2.98654 −0.119176
\(629\) −11.6178 −0.463233
\(630\) −8.35304 −0.332793
\(631\) 24.7497 0.985270 0.492635 0.870236i \(-0.336034\pi\)
0.492635 + 0.870236i \(0.336034\pi\)
\(632\) −12.8688 −0.511895
\(633\) 7.31232 0.290639
\(634\) 17.8766 0.709969
\(635\) −34.3968 −1.36499
\(636\) 6.04002 0.239502
\(637\) −11.6477 −0.461497
\(638\) −0.513894 −0.0203452
\(639\) 9.89709 0.391523
\(640\) −2.56155 −0.101254
\(641\) 1.37443 0.0542869 0.0271434 0.999632i \(-0.491359\pi\)
0.0271434 + 0.999632i \(0.491359\pi\)
\(642\) −12.5974 −0.497179
\(643\) 26.5933 1.04874 0.524368 0.851492i \(-0.324302\pi\)
0.524368 + 0.851492i \(0.324302\pi\)
\(644\) 3.26093 0.128499
\(645\) 3.66473 0.144299
\(646\) 9.03550 0.355497
\(647\) −20.4971 −0.805825 −0.402912 0.915239i \(-0.632002\pi\)
−0.402912 + 0.915239i \(0.632002\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.47297 0.175580
\(650\) 5.00554 0.196334
\(651\) −32.3824 −1.26917
\(652\) 14.1375 0.553669
\(653\) −16.1462 −0.631852 −0.315926 0.948784i \(-0.602315\pi\)
−0.315926 + 0.948784i \(0.602315\pi\)
\(654\) −12.2135 −0.477587
\(655\) 10.6419 0.415815
\(656\) 6.66047 0.260048
\(657\) 13.5674 0.529315
\(658\) −20.2248 −0.788443
\(659\) −13.8198 −0.538341 −0.269171 0.963093i \(-0.586750\pi\)
−0.269171 + 0.963093i \(0.586750\pi\)
\(660\) 1.31637 0.0512395
\(661\) −25.5665 −0.994421 −0.497210 0.867630i \(-0.665642\pi\)
−0.497210 + 0.867630i \(0.665642\pi\)
\(662\) 16.9855 0.660159
\(663\) 8.08675 0.314063
\(664\) 8.47531 0.328906
\(665\) 29.9169 1.16013
\(666\) −4.60517 −0.178447
\(667\) 1.00000 0.0387202
\(668\) −12.7796 −0.494459
\(669\) −26.4260 −1.02169
\(670\) 26.1342 1.00965
\(671\) 0.182143 0.00703153
\(672\) −3.26093 −0.125793
\(673\) 26.4633 1.02009 0.510043 0.860149i \(-0.329629\pi\)
0.510043 + 0.860149i \(0.329629\pi\)
\(674\) 4.15420 0.160014
\(675\) 1.56155 0.0601042
\(676\) −2.72483 −0.104801
\(677\) 7.14704 0.274683 0.137341 0.990524i \(-0.456144\pi\)
0.137341 + 0.990524i \(0.456144\pi\)
\(678\) −6.48945 −0.249226
\(679\) 42.4443 1.62886
\(680\) 6.46224 0.247815
\(681\) 5.83322 0.223529
\(682\) 5.10318 0.195411
\(683\) 17.4363 0.667182 0.333591 0.942718i \(-0.391740\pi\)
0.333591 + 0.942718i \(0.391740\pi\)
\(684\) 3.58156 0.136944
\(685\) −26.5986 −1.01628
\(686\) 10.9774 0.419119
\(687\) 22.8231 0.870755
\(688\) 1.43067 0.0545437
\(689\) −19.3612 −0.737604
\(690\) −2.56155 −0.0975166
\(691\) −16.3665 −0.622610 −0.311305 0.950310i \(-0.600766\pi\)
−0.311305 + 0.950310i \(0.600766\pi\)
\(692\) 14.6234 0.555899
\(693\) 1.67577 0.0636573
\(694\) −7.27163 −0.276027
\(695\) −49.7053 −1.88543
\(696\) −1.00000 −0.0379049
\(697\) −16.8029 −0.636455
\(698\) 0.326949 0.0123752
\(699\) 24.9439 0.943466
\(700\) 5.09211 0.192464
\(701\) −4.94942 −0.186937 −0.0934684 0.995622i \(-0.529795\pi\)
−0.0934684 + 0.995622i \(0.529795\pi\)
\(702\) 3.20549 0.120983
\(703\) 16.4937 0.622071
\(704\) 0.513894 0.0193681
\(705\) 15.8871 0.598344
\(706\) 5.39173 0.202920
\(707\) 10.4808 0.394171
\(708\) 8.70408 0.327119
\(709\) −15.9135 −0.597643 −0.298821 0.954309i \(-0.596593\pi\)
−0.298821 + 0.954309i \(0.596593\pi\)
\(710\) −25.3519 −0.951440
\(711\) 12.8688 0.482619
\(712\) −2.85805 −0.107110
\(713\) −9.93042 −0.371897
\(714\) 8.22661 0.307873
\(715\) −4.21960 −0.157804
\(716\) −20.1935 −0.754665
\(717\) −28.4198 −1.06136
\(718\) −11.0689 −0.413088
\(719\) −24.6306 −0.918565 −0.459282 0.888290i \(-0.651893\pi\)
−0.459282 + 0.888290i \(0.651893\pi\)
\(720\) 2.56155 0.0954634
\(721\) 15.4407 0.575043
\(722\) 6.17241 0.229713
\(723\) −22.4261 −0.834035
\(724\) −19.7979 −0.735783
\(725\) 1.56155 0.0579946
\(726\) 10.7359 0.398447
\(727\) −10.1864 −0.377794 −0.188897 0.981997i \(-0.560491\pi\)
−0.188897 + 0.981997i \(0.560491\pi\)
\(728\) 10.4529 0.387409
\(729\) 1.00000 0.0370370
\(730\) −34.7537 −1.28629
\(731\) −3.60926 −0.133493
\(732\) 0.354436 0.0131003
\(733\) −49.5873 −1.83155 −0.915775 0.401692i \(-0.868422\pi\)
−0.915775 + 0.401692i \(0.868422\pi\)
\(734\) 2.38125 0.0878934
\(735\) 9.30780 0.343323
\(736\) −1.00000 −0.0368605
\(737\) −5.24299 −0.193128
\(738\) −6.66047 −0.245175
\(739\) 21.7988 0.801881 0.400940 0.916104i \(-0.368683\pi\)
0.400940 + 0.916104i \(0.368683\pi\)
\(740\) 11.7964 0.433644
\(741\) −11.4807 −0.421753
\(742\) −19.6961 −0.723065
\(743\) 4.76074 0.174655 0.0873274 0.996180i \(-0.472167\pi\)
0.0873274 + 0.996180i \(0.472167\pi\)
\(744\) 9.93042 0.364067
\(745\) −15.4947 −0.567681
\(746\) 18.3022 0.670090
\(747\) −8.47531 −0.310095
\(748\) −1.29644 −0.0474026
\(749\) 41.0792 1.50100
\(750\) 8.80776 0.321614
\(751\) −18.7265 −0.683340 −0.341670 0.939820i \(-0.610992\pi\)
−0.341670 + 0.939820i \(0.610992\pi\)
\(752\) 6.20215 0.226169
\(753\) 9.30608 0.339133
\(754\) 3.20549 0.116737
\(755\) −31.6666 −1.15246
\(756\) 3.26093 0.118599
\(757\) 32.6893 1.18811 0.594057 0.804423i \(-0.297525\pi\)
0.594057 + 0.804423i \(0.297525\pi\)
\(758\) −2.60318 −0.0945518
\(759\) 0.513894 0.0186532
\(760\) −9.17436 −0.332789
\(761\) 25.2991 0.917093 0.458546 0.888670i \(-0.348370\pi\)
0.458546 + 0.888670i \(0.348370\pi\)
\(762\) 13.4281 0.486448
\(763\) 39.8275 1.44185
\(764\) −1.75927 −0.0636480
\(765\) −6.46224 −0.233643
\(766\) −17.4335 −0.629896
\(767\) −27.9009 −1.00744
\(768\) 1.00000 0.0360844
\(769\) −29.2220 −1.05377 −0.526887 0.849935i \(-0.676641\pi\)
−0.526887 + 0.849935i \(0.676641\pi\)
\(770\) −4.29258 −0.154694
\(771\) 11.4385 0.411946
\(772\) −23.8470 −0.858271
\(773\) −49.5948 −1.78380 −0.891900 0.452233i \(-0.850628\pi\)
−0.891900 + 0.452233i \(0.850628\pi\)
\(774\) −1.43067 −0.0514243
\(775\) −15.5069 −0.557023
\(776\) −13.0160 −0.467248
\(777\) 15.0171 0.538736
\(778\) 11.7377 0.420815
\(779\) 23.8549 0.854690
\(780\) −8.21104 −0.294002
\(781\) 5.08605 0.181993
\(782\) 2.52278 0.0902145
\(783\) 1.00000 0.0357371
\(784\) 3.63366 0.129773
\(785\) −7.65018 −0.273047
\(786\) −4.15449 −0.148186
\(787\) −46.2909 −1.65009 −0.825046 0.565065i \(-0.808851\pi\)
−0.825046 + 0.565065i \(0.808851\pi\)
\(788\) 3.71894 0.132482
\(789\) 15.1714 0.540117
\(790\) −32.9642 −1.17281
\(791\) 21.1616 0.752421
\(792\) −0.513894 −0.0182604
\(793\) −1.13614 −0.0403456
\(794\) −12.2931 −0.436266
\(795\) 15.4718 0.548729
\(796\) 5.58551 0.197973
\(797\) 35.2597 1.24896 0.624482 0.781039i \(-0.285310\pi\)
0.624482 + 0.781039i \(0.285310\pi\)
\(798\) −11.6792 −0.413440
\(799\) −15.6467 −0.553539
\(800\) −1.56155 −0.0552092
\(801\) 2.85805 0.100984
\(802\) 28.2496 0.997527
\(803\) 6.97221 0.246044
\(804\) −10.2025 −0.359813
\(805\) 8.35304 0.294406
\(806\) −31.8319 −1.12123
\(807\) −30.0545 −1.05797
\(808\) −3.21406 −0.113070
\(809\) 12.8762 0.452704 0.226352 0.974046i \(-0.427320\pi\)
0.226352 + 0.974046i \(0.427320\pi\)
\(810\) −2.56155 −0.0900038
\(811\) −30.0527 −1.05529 −0.527647 0.849464i \(-0.676926\pi\)
−0.527647 + 0.849464i \(0.676926\pi\)
\(812\) 3.26093 0.114436
\(813\) 25.5234 0.895145
\(814\) −2.36657 −0.0829481
\(815\) 36.2140 1.26852
\(816\) −2.52278 −0.0883150
\(817\) 5.12403 0.179267
\(818\) 35.5775 1.24394
\(819\) −10.4529 −0.365253
\(820\) 17.0611 0.595801
\(821\) −36.4669 −1.27270 −0.636352 0.771399i \(-0.719557\pi\)
−0.636352 + 0.771399i \(0.719557\pi\)
\(822\) 10.3838 0.362176
\(823\) −10.3311 −0.360120 −0.180060 0.983656i \(-0.557629\pi\)
−0.180060 + 0.983656i \(0.557629\pi\)
\(824\) −4.73507 −0.164954
\(825\) 0.802472 0.0279385
\(826\) −28.3834 −0.987584
\(827\) −19.3997 −0.674594 −0.337297 0.941398i \(-0.609513\pi\)
−0.337297 + 0.941398i \(0.609513\pi\)
\(828\) 1.00000 0.0347524
\(829\) −15.9866 −0.555239 −0.277619 0.960691i \(-0.589545\pi\)
−0.277619 + 0.960691i \(0.589545\pi\)
\(830\) 21.7099 0.753563
\(831\) −20.1539 −0.699131
\(832\) −3.20549 −0.111130
\(833\) −9.16692 −0.317615
\(834\) 19.4044 0.671918
\(835\) −32.7357 −1.13287
\(836\) 1.84054 0.0636565
\(837\) −9.93042 −0.343246
\(838\) −21.6897 −0.749259
\(839\) −32.5826 −1.12488 −0.562439 0.826839i \(-0.690137\pi\)
−0.562439 + 0.826839i \(0.690137\pi\)
\(840\) −8.35304 −0.288207
\(841\) 1.00000 0.0344828
\(842\) −30.8811 −1.06423
\(843\) 19.4320 0.669272
\(844\) 7.31232 0.251700
\(845\) −6.97978 −0.240112
\(846\) −6.20215 −0.213234
\(847\) −35.0090 −1.20293
\(848\) 6.04002 0.207415
\(849\) 6.87622 0.235991
\(850\) 3.93946 0.135122
\(851\) 4.60517 0.157863
\(852\) 9.89709 0.339069
\(853\) −45.9769 −1.57422 −0.787110 0.616812i \(-0.788424\pi\)
−0.787110 + 0.616812i \(0.788424\pi\)
\(854\) −1.15579 −0.0395504
\(855\) 9.17436 0.313757
\(856\) −12.5974 −0.430570
\(857\) 44.4844 1.51956 0.759780 0.650181i \(-0.225307\pi\)
0.759780 + 0.650181i \(0.225307\pi\)
\(858\) 1.64728 0.0562373
\(859\) 18.6495 0.636313 0.318157 0.948038i \(-0.396936\pi\)
0.318157 + 0.948038i \(0.396936\pi\)
\(860\) 3.66473 0.124966
\(861\) 21.7193 0.740192
\(862\) 16.7845 0.571681
\(863\) −8.00602 −0.272528 −0.136264 0.990673i \(-0.543510\pi\)
−0.136264 + 0.990673i \(0.543510\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 37.4586 1.27363
\(866\) 34.7198 1.17983
\(867\) −10.6356 −0.361203
\(868\) −32.3824 −1.09913
\(869\) 6.61321 0.224338
\(870\) −2.56155 −0.0868448
\(871\) 32.7039 1.10813
\(872\) −12.2135 −0.413603
\(873\) 13.0160 0.440525
\(874\) −3.58156 −0.121148
\(875\) −28.7215 −0.970964
\(876\) 13.5674 0.458401
\(877\) −17.1696 −0.579776 −0.289888 0.957061i \(-0.593618\pi\)
−0.289888 + 0.957061i \(0.593618\pi\)
\(878\) 3.46274 0.116862
\(879\) 17.3419 0.584929
\(880\) 1.31637 0.0443747
\(881\) −32.9558 −1.11031 −0.555155 0.831747i \(-0.687341\pi\)
−0.555155 + 0.831747i \(0.687341\pi\)
\(882\) −3.63366 −0.122352
\(883\) −45.9615 −1.54673 −0.773363 0.633963i \(-0.781427\pi\)
−0.773363 + 0.633963i \(0.781427\pi\)
\(884\) 8.08675 0.271987
\(885\) 22.2960 0.749471
\(886\) 14.4669 0.486025
\(887\) −0.352247 −0.0118273 −0.00591364 0.999983i \(-0.501882\pi\)
−0.00591364 + 0.999983i \(0.501882\pi\)
\(888\) −4.60517 −0.154539
\(889\) −43.7880 −1.46860
\(890\) −7.32104 −0.245402
\(891\) 0.513894 0.0172161
\(892\) −26.4260 −0.884807
\(893\) 22.2134 0.743342
\(894\) 6.04894 0.202307
\(895\) −51.7266 −1.72903
\(896\) −3.26093 −0.108940
\(897\) −3.20549 −0.107028
\(898\) 27.0281 0.901940
\(899\) −9.93042 −0.331198
\(900\) 1.56155 0.0520518
\(901\) −15.2376 −0.507640
\(902\) −3.42277 −0.113966
\(903\) 4.66531 0.155252
\(904\) −6.48945 −0.215836
\(905\) −50.7133 −1.68577
\(906\) 12.3623 0.410708
\(907\) 56.8694 1.88832 0.944158 0.329492i \(-0.106877\pi\)
0.944158 + 0.329492i \(0.106877\pi\)
\(908\) 5.83322 0.193582
\(909\) 3.21406 0.106604
\(910\) 26.7756 0.887603
\(911\) −43.3244 −1.43540 −0.717700 0.696352i \(-0.754805\pi\)
−0.717700 + 0.696352i \(0.754805\pi\)
\(912\) 3.58156 0.118597
\(913\) −4.35541 −0.144143
\(914\) −31.4925 −1.04168
\(915\) 0.907907 0.0300145
\(916\) 22.8231 0.754096
\(917\) 13.5475 0.447378
\(918\) 2.52278 0.0832642
\(919\) 16.3072 0.537924 0.268962 0.963151i \(-0.413319\pi\)
0.268962 + 0.963151i \(0.413319\pi\)
\(920\) −2.56155 −0.0844519
\(921\) −21.7731 −0.717449
\(922\) 36.7196 1.20929
\(923\) −31.7250 −1.04424
\(924\) 1.67577 0.0551288
\(925\) 7.19121 0.236446
\(926\) −34.0251 −1.11813
\(927\) 4.73507 0.155520
\(928\) −1.00000 −0.0328266
\(929\) −28.4605 −0.933759 −0.466880 0.884321i \(-0.654622\pi\)
−0.466880 + 0.884321i \(0.654622\pi\)
\(930\) 25.4373 0.834122
\(931\) 13.0142 0.426522
\(932\) 24.9439 0.817066
\(933\) 12.6368 0.413710
\(934\) 8.65038 0.283049
\(935\) −3.32090 −0.108605
\(936\) 3.20549 0.104775
\(937\) 23.3998 0.764439 0.382219 0.924072i \(-0.375160\pi\)
0.382219 + 0.924072i \(0.375160\pi\)
\(938\) 33.2695 1.08629
\(939\) −28.2444 −0.921722
\(940\) 15.8871 0.518181
\(941\) 16.8285 0.548592 0.274296 0.961645i \(-0.411555\pi\)
0.274296 + 0.961645i \(0.411555\pi\)
\(942\) 2.98654 0.0973067
\(943\) 6.66047 0.216895
\(944\) 8.70408 0.283294
\(945\) 8.35304 0.271724
\(946\) −0.735212 −0.0239038
\(947\) 19.2516 0.625594 0.312797 0.949820i \(-0.398734\pi\)
0.312797 + 0.949820i \(0.398734\pi\)
\(948\) 12.8688 0.417960
\(949\) −43.4902 −1.41175
\(950\) −5.59280 −0.181454
\(951\) −17.8766 −0.579687
\(952\) 8.22661 0.266626
\(953\) −23.1775 −0.750793 −0.375396 0.926864i \(-0.622493\pi\)
−0.375396 + 0.926864i \(0.622493\pi\)
\(954\) −6.04002 −0.195553
\(955\) −4.50645 −0.145825
\(956\) −28.4198 −0.919162
\(957\) 0.513894 0.0166118
\(958\) −18.2393 −0.589285
\(959\) −33.8607 −1.09342
\(960\) 2.56155 0.0826738
\(961\) 67.6133 2.18107
\(962\) 14.7618 0.475940
\(963\) 12.5974 0.405945
\(964\) −22.4261 −0.722296
\(965\) −61.0853 −1.96640
\(966\) −3.26093 −0.104919
\(967\) 12.2462 0.393813 0.196906 0.980422i \(-0.436911\pi\)
0.196906 + 0.980422i \(0.436911\pi\)
\(968\) 10.7359 0.345065
\(969\) −9.03550 −0.290262
\(970\) −33.3412 −1.07052
\(971\) 29.9414 0.960864 0.480432 0.877032i \(-0.340480\pi\)
0.480432 + 0.877032i \(0.340480\pi\)
\(972\) 1.00000 0.0320750
\(973\) −63.2763 −2.02854
\(974\) 42.4026 1.35867
\(975\) −5.00554 −0.160306
\(976\) 0.354436 0.0113452
\(977\) −34.3769 −1.09982 −0.549908 0.835226i \(-0.685337\pi\)
−0.549908 + 0.835226i \(0.685337\pi\)
\(978\) −14.1375 −0.452069
\(979\) 1.46873 0.0469409
\(980\) 9.30780 0.297327
\(981\) 12.2135 0.389948
\(982\) −14.7408 −0.470399
\(983\) −4.39564 −0.140199 −0.0700996 0.997540i \(-0.522332\pi\)
−0.0700996 + 0.997540i \(0.522332\pi\)
\(984\) −6.66047 −0.212328
\(985\) 9.52625 0.303532
\(986\) 2.52278 0.0803417
\(987\) 20.2248 0.643761
\(988\) −11.4807 −0.365249
\(989\) 1.43067 0.0454926
\(990\) −1.31637 −0.0418369
\(991\) 2.49351 0.0792088 0.0396044 0.999215i \(-0.487390\pi\)
0.0396044 + 0.999215i \(0.487390\pi\)
\(992\) 9.93042 0.315291
\(993\) −16.9855 −0.539018
\(994\) −32.2737 −1.02366
\(995\) 14.3076 0.453581
\(996\) −8.47531 −0.268550
\(997\) −19.5166 −0.618098 −0.309049 0.951046i \(-0.600011\pi\)
−0.309049 + 0.951046i \(0.600011\pi\)
\(998\) −16.3704 −0.518195
\(999\) 4.60517 0.145701
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.bf.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.bf.1.5 6 1.1 even 1 trivial