Properties

Label 6042.2.a.bh.1.13
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 3 x^{12} - 40 x^{11} + 123 x^{10} + 537 x^{9} - 1707 x^{8} - 2914 x^{7} + 9639 x^{6} + 6938 x^{5} - 22200 x^{4} - 9466 x^{3} + 16812 x^{2} + 9304 x + 1200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(4.22619\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.22619 q^{5} +1.00000 q^{6} +0.162672 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.22619 q^{5} +1.00000 q^{6} +0.162672 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.22619 q^{10} +4.28864 q^{11} +1.00000 q^{12} -1.72968 q^{13} +0.162672 q^{14} +4.22619 q^{15} +1.00000 q^{16} -0.372560 q^{17} +1.00000 q^{18} +1.00000 q^{19} +4.22619 q^{20} +0.162672 q^{21} +4.28864 q^{22} -3.58563 q^{23} +1.00000 q^{24} +12.8606 q^{25} -1.72968 q^{26} +1.00000 q^{27} +0.162672 q^{28} +9.28951 q^{29} +4.22619 q^{30} -4.13779 q^{31} +1.00000 q^{32} +4.28864 q^{33} -0.372560 q^{34} +0.687480 q^{35} +1.00000 q^{36} +1.48845 q^{37} +1.00000 q^{38} -1.72968 q^{39} +4.22619 q^{40} -2.88605 q^{41} +0.162672 q^{42} -7.90481 q^{43} +4.28864 q^{44} +4.22619 q^{45} -3.58563 q^{46} -7.12704 q^{47} +1.00000 q^{48} -6.97354 q^{49} +12.8606 q^{50} -0.372560 q^{51} -1.72968 q^{52} -1.00000 q^{53} +1.00000 q^{54} +18.1246 q^{55} +0.162672 q^{56} +1.00000 q^{57} +9.28951 q^{58} -4.09719 q^{59} +4.22619 q^{60} -12.6996 q^{61} -4.13779 q^{62} +0.162672 q^{63} +1.00000 q^{64} -7.30994 q^{65} +4.28864 q^{66} -12.5390 q^{67} -0.372560 q^{68} -3.58563 q^{69} +0.687480 q^{70} +16.1640 q^{71} +1.00000 q^{72} +15.0565 q^{73} +1.48845 q^{74} +12.8606 q^{75} +1.00000 q^{76} +0.697641 q^{77} -1.72968 q^{78} -7.40316 q^{79} +4.22619 q^{80} +1.00000 q^{81} -2.88605 q^{82} -8.64039 q^{83} +0.162672 q^{84} -1.57451 q^{85} -7.90481 q^{86} +9.28951 q^{87} +4.28864 q^{88} +3.54113 q^{89} +4.22619 q^{90} -0.281369 q^{91} -3.58563 q^{92} -4.13779 q^{93} -7.12704 q^{94} +4.22619 q^{95} +1.00000 q^{96} +7.12709 q^{97} -6.97354 q^{98} +4.28864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} + 13 q^{3} + 13 q^{4} + 3 q^{5} + 13 q^{6} + 12 q^{7} + 13 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{2} + 13 q^{3} + 13 q^{4} + 3 q^{5} + 13 q^{6} + 12 q^{7} + 13 q^{8} + 13 q^{9} + 3 q^{10} + 4 q^{11} + 13 q^{12} + 18 q^{13} + 12 q^{14} + 3 q^{15} + 13 q^{16} + 3 q^{17} + 13 q^{18} + 13 q^{19} + 3 q^{20} + 12 q^{21} + 4 q^{22} + 8 q^{23} + 13 q^{24} + 24 q^{25} + 18 q^{26} + 13 q^{27} + 12 q^{28} + 11 q^{29} + 3 q^{30} + 2 q^{31} + 13 q^{32} + 4 q^{33} + 3 q^{34} - 6 q^{35} + 13 q^{36} + 26 q^{37} + 13 q^{38} + 18 q^{39} + 3 q^{40} + 3 q^{41} + 12 q^{42} + 24 q^{43} + 4 q^{44} + 3 q^{45} + 8 q^{46} + 4 q^{47} + 13 q^{48} + 41 q^{49} + 24 q^{50} + 3 q^{51} + 18 q^{52} - 13 q^{53} + 13 q^{54} + 23 q^{55} + 12 q^{56} + 13 q^{57} + 11 q^{58} + 8 q^{59} + 3 q^{60} + 29 q^{61} + 2 q^{62} + 12 q^{63} + 13 q^{64} + 13 q^{65} + 4 q^{66} - 5 q^{67} + 3 q^{68} + 8 q^{69} - 6 q^{70} + 29 q^{71} + 13 q^{72} + 15 q^{73} + 26 q^{74} + 24 q^{75} + 13 q^{76} + 3 q^{77} + 18 q^{78} + 15 q^{79} + 3 q^{80} + 13 q^{81} + 3 q^{82} + 4 q^{83} + 12 q^{84} - q^{85} + 24 q^{86} + 11 q^{87} + 4 q^{88} + 3 q^{89} + 3 q^{90} - 29 q^{91} + 8 q^{92} + 2 q^{93} + 4 q^{94} + 3 q^{95} + 13 q^{96} + 50 q^{97} + 41 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(5\) 4.22619 1.89001 0.945004 0.327059i \(-0.106058\pi\)
0.945004 + 0.327059i \(0.106058\pi\)
\(6\) 1.00000 0.408248
\(7\) 0.162672 0.0614841 0.0307420 0.999527i \(-0.490213\pi\)
0.0307420 + 0.999527i \(0.490213\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.22619 1.33644
\(11\) 4.28864 1.29307 0.646537 0.762882i \(-0.276216\pi\)
0.646537 + 0.762882i \(0.276216\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.72968 −0.479726 −0.239863 0.970807i \(-0.577103\pi\)
−0.239863 + 0.970807i \(0.577103\pi\)
\(14\) 0.162672 0.0434758
\(15\) 4.22619 1.09120
\(16\) 1.00000 0.250000
\(17\) −0.372560 −0.0903590 −0.0451795 0.998979i \(-0.514386\pi\)
−0.0451795 + 0.998979i \(0.514386\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 4.22619 0.945004
\(21\) 0.162672 0.0354979
\(22\) 4.28864 0.914342
\(23\) −3.58563 −0.747656 −0.373828 0.927498i \(-0.621955\pi\)
−0.373828 + 0.927498i \(0.621955\pi\)
\(24\) 1.00000 0.204124
\(25\) 12.8606 2.57213
\(26\) −1.72968 −0.339218
\(27\) 1.00000 0.192450
\(28\) 0.162672 0.0307420
\(29\) 9.28951 1.72502 0.862510 0.506040i \(-0.168891\pi\)
0.862510 + 0.506040i \(0.168891\pi\)
\(30\) 4.22619 0.771592
\(31\) −4.13779 −0.743169 −0.371585 0.928399i \(-0.621185\pi\)
−0.371585 + 0.928399i \(0.621185\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.28864 0.746557
\(34\) −0.372560 −0.0638935
\(35\) 0.687480 0.116205
\(36\) 1.00000 0.166667
\(37\) 1.48845 0.244699 0.122349 0.992487i \(-0.460957\pi\)
0.122349 + 0.992487i \(0.460957\pi\)
\(38\) 1.00000 0.162221
\(39\) −1.72968 −0.276970
\(40\) 4.22619 0.668219
\(41\) −2.88605 −0.450725 −0.225363 0.974275i \(-0.572357\pi\)
−0.225363 + 0.974275i \(0.572357\pi\)
\(42\) 0.162672 0.0251008
\(43\) −7.90481 −1.20547 −0.602736 0.797941i \(-0.705923\pi\)
−0.602736 + 0.797941i \(0.705923\pi\)
\(44\) 4.28864 0.646537
\(45\) 4.22619 0.630003
\(46\) −3.58563 −0.528673
\(47\) −7.12704 −1.03959 −0.519793 0.854292i \(-0.673991\pi\)
−0.519793 + 0.854292i \(0.673991\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.97354 −0.996220
\(50\) 12.8606 1.81877
\(51\) −0.372560 −0.0521688
\(52\) −1.72968 −0.239863
\(53\) −1.00000 −0.137361
\(54\) 1.00000 0.136083
\(55\) 18.1246 2.44392
\(56\) 0.162672 0.0217379
\(57\) 1.00000 0.132453
\(58\) 9.28951 1.21977
\(59\) −4.09719 −0.533409 −0.266704 0.963778i \(-0.585935\pi\)
−0.266704 + 0.963778i \(0.585935\pi\)
\(60\) 4.22619 0.545598
\(61\) −12.6996 −1.62602 −0.813011 0.582248i \(-0.802173\pi\)
−0.813011 + 0.582248i \(0.802173\pi\)
\(62\) −4.13779 −0.525500
\(63\) 0.162672 0.0204947
\(64\) 1.00000 0.125000
\(65\) −7.30994 −0.906686
\(66\) 4.28864 0.527896
\(67\) −12.5390 −1.53188 −0.765940 0.642913i \(-0.777726\pi\)
−0.765940 + 0.642913i \(0.777726\pi\)
\(68\) −0.372560 −0.0451795
\(69\) −3.58563 −0.431660
\(70\) 0.687480 0.0821696
\(71\) 16.1640 1.91832 0.959159 0.282866i \(-0.0912852\pi\)
0.959159 + 0.282866i \(0.0912852\pi\)
\(72\) 1.00000 0.117851
\(73\) 15.0565 1.76223 0.881117 0.472899i \(-0.156792\pi\)
0.881117 + 0.472899i \(0.156792\pi\)
\(74\) 1.48845 0.173028
\(75\) 12.8606 1.48502
\(76\) 1.00000 0.114708
\(77\) 0.697641 0.0795035
\(78\) −1.72968 −0.195847
\(79\) −7.40316 −0.832921 −0.416461 0.909154i \(-0.636730\pi\)
−0.416461 + 0.909154i \(0.636730\pi\)
\(80\) 4.22619 0.472502
\(81\) 1.00000 0.111111
\(82\) −2.88605 −0.318711
\(83\) −8.64039 −0.948406 −0.474203 0.880416i \(-0.657264\pi\)
−0.474203 + 0.880416i \(0.657264\pi\)
\(84\) 0.162672 0.0177489
\(85\) −1.57451 −0.170779
\(86\) −7.90481 −0.852397
\(87\) 9.28951 0.995941
\(88\) 4.28864 0.457171
\(89\) 3.54113 0.375359 0.187679 0.982230i \(-0.439903\pi\)
0.187679 + 0.982230i \(0.439903\pi\)
\(90\) 4.22619 0.445479
\(91\) −0.281369 −0.0294955
\(92\) −3.58563 −0.373828
\(93\) −4.13779 −0.429069
\(94\) −7.12704 −0.735098
\(95\) 4.22619 0.433597
\(96\) 1.00000 0.102062
\(97\) 7.12709 0.723647 0.361823 0.932247i \(-0.382154\pi\)
0.361823 + 0.932247i \(0.382154\pi\)
\(98\) −6.97354 −0.704434
\(99\) 4.28864 0.431025
\(100\) 12.8606 1.28606
\(101\) −0.0690802 −0.00687374 −0.00343687 0.999994i \(-0.501094\pi\)
−0.00343687 + 0.999994i \(0.501094\pi\)
\(102\) −0.372560 −0.0368889
\(103\) −8.84739 −0.871760 −0.435880 0.900005i \(-0.643563\pi\)
−0.435880 + 0.900005i \(0.643563\pi\)
\(104\) −1.72968 −0.169609
\(105\) 0.687480 0.0670912
\(106\) −1.00000 −0.0971286
\(107\) 5.16527 0.499345 0.249673 0.968330i \(-0.419677\pi\)
0.249673 + 0.968330i \(0.419677\pi\)
\(108\) 1.00000 0.0962250
\(109\) −7.17230 −0.686982 −0.343491 0.939156i \(-0.611610\pi\)
−0.343491 + 0.939156i \(0.611610\pi\)
\(110\) 18.1246 1.72811
\(111\) 1.48845 0.141277
\(112\) 0.162672 0.0153710
\(113\) 18.3107 1.72253 0.861265 0.508157i \(-0.169673\pi\)
0.861265 + 0.508157i \(0.169673\pi\)
\(114\) 1.00000 0.0936586
\(115\) −15.1536 −1.41308
\(116\) 9.28951 0.862510
\(117\) −1.72968 −0.159909
\(118\) −4.09719 −0.377177
\(119\) −0.0606049 −0.00555564
\(120\) 4.22619 0.385796
\(121\) 7.39247 0.672043
\(122\) −12.6996 −1.14977
\(123\) −2.88605 −0.260226
\(124\) −4.13779 −0.371585
\(125\) 33.2205 2.97134
\(126\) 0.162672 0.0144919
\(127\) 7.32539 0.650024 0.325012 0.945710i \(-0.394632\pi\)
0.325012 + 0.945710i \(0.394632\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.90481 −0.695980
\(130\) −7.30994 −0.641124
\(131\) 1.88204 0.164435 0.0822173 0.996614i \(-0.473800\pi\)
0.0822173 + 0.996614i \(0.473800\pi\)
\(132\) 4.28864 0.373279
\(133\) 0.162672 0.0141054
\(134\) −12.5390 −1.08320
\(135\) 4.22619 0.363732
\(136\) −0.372560 −0.0319467
\(137\) −13.4413 −1.14837 −0.574183 0.818727i \(-0.694680\pi\)
−0.574183 + 0.818727i \(0.694680\pi\)
\(138\) −3.58563 −0.305229
\(139\) 1.73787 0.147404 0.0737020 0.997280i \(-0.476519\pi\)
0.0737020 + 0.997280i \(0.476519\pi\)
\(140\) 0.687480 0.0581027
\(141\) −7.12704 −0.600205
\(142\) 16.1640 1.35646
\(143\) −7.41797 −0.620322
\(144\) 1.00000 0.0833333
\(145\) 39.2592 3.26030
\(146\) 15.0565 1.24609
\(147\) −6.97354 −0.575168
\(148\) 1.48845 0.122349
\(149\) −10.0368 −0.822243 −0.411121 0.911581i \(-0.634863\pi\)
−0.411121 + 0.911581i \(0.634863\pi\)
\(150\) 12.8606 1.05007
\(151\) −4.42076 −0.359756 −0.179878 0.983689i \(-0.557570\pi\)
−0.179878 + 0.983689i \(0.557570\pi\)
\(152\) 1.00000 0.0811107
\(153\) −0.372560 −0.0301197
\(154\) 0.697641 0.0562175
\(155\) −17.4871 −1.40460
\(156\) −1.72968 −0.138485
\(157\) −10.0984 −0.805939 −0.402969 0.915213i \(-0.632022\pi\)
−0.402969 + 0.915213i \(0.632022\pi\)
\(158\) −7.40316 −0.588964
\(159\) −1.00000 −0.0793052
\(160\) 4.22619 0.334109
\(161\) −0.583281 −0.0459690
\(162\) 1.00000 0.0785674
\(163\) −7.51964 −0.588984 −0.294492 0.955654i \(-0.595150\pi\)
−0.294492 + 0.955654i \(0.595150\pi\)
\(164\) −2.88605 −0.225363
\(165\) 18.1246 1.41100
\(166\) −8.64039 −0.670624
\(167\) 12.4136 0.960594 0.480297 0.877106i \(-0.340529\pi\)
0.480297 + 0.877106i \(0.340529\pi\)
\(168\) 0.162672 0.0125504
\(169\) −10.0082 −0.769863
\(170\) −1.57451 −0.120759
\(171\) 1.00000 0.0764719
\(172\) −7.90481 −0.602736
\(173\) 7.13359 0.542357 0.271178 0.962529i \(-0.412587\pi\)
0.271178 + 0.962529i \(0.412587\pi\)
\(174\) 9.28951 0.704236
\(175\) 2.09206 0.158145
\(176\) 4.28864 0.323269
\(177\) −4.09719 −0.307964
\(178\) 3.54113 0.265419
\(179\) 16.6540 1.24477 0.622387 0.782709i \(-0.286163\pi\)
0.622387 + 0.782709i \(0.286163\pi\)
\(180\) 4.22619 0.315001
\(181\) −15.5258 −1.15403 −0.577014 0.816735i \(-0.695782\pi\)
−0.577014 + 0.816735i \(0.695782\pi\)
\(182\) −0.281369 −0.0208565
\(183\) −12.6996 −0.938784
\(184\) −3.58563 −0.264336
\(185\) 6.29045 0.462483
\(186\) −4.13779 −0.303398
\(187\) −1.59778 −0.116841
\(188\) −7.12704 −0.519793
\(189\) 0.162672 0.0118326
\(190\) 4.22619 0.306600
\(191\) −11.7620 −0.851068 −0.425534 0.904943i \(-0.639914\pi\)
−0.425534 + 0.904943i \(0.639914\pi\)
\(192\) 1.00000 0.0721688
\(193\) −19.5836 −1.40966 −0.704828 0.709378i \(-0.748976\pi\)
−0.704828 + 0.709378i \(0.748976\pi\)
\(194\) 7.12709 0.511696
\(195\) −7.30994 −0.523475
\(196\) −6.97354 −0.498110
\(197\) 11.7158 0.834716 0.417358 0.908742i \(-0.362956\pi\)
0.417358 + 0.908742i \(0.362956\pi\)
\(198\) 4.28864 0.304781
\(199\) −3.78323 −0.268186 −0.134093 0.990969i \(-0.542812\pi\)
−0.134093 + 0.990969i \(0.542812\pi\)
\(200\) 12.8606 0.909385
\(201\) −12.5390 −0.884431
\(202\) −0.0690802 −0.00486047
\(203\) 1.51114 0.106061
\(204\) −0.372560 −0.0260844
\(205\) −12.1970 −0.851874
\(206\) −8.84739 −0.616427
\(207\) −3.58563 −0.249219
\(208\) −1.72968 −0.119932
\(209\) 4.28864 0.296652
\(210\) 0.687480 0.0474407
\(211\) −21.0364 −1.44820 −0.724102 0.689693i \(-0.757746\pi\)
−0.724102 + 0.689693i \(0.757746\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 16.1640 1.10754
\(214\) 5.16527 0.353091
\(215\) −33.4072 −2.27835
\(216\) 1.00000 0.0680414
\(217\) −0.673101 −0.0456931
\(218\) −7.17230 −0.485770
\(219\) 15.0565 1.01743
\(220\) 18.1246 1.22196
\(221\) 0.644408 0.0433476
\(222\) 1.48845 0.0998979
\(223\) 9.18604 0.615142 0.307571 0.951525i \(-0.400484\pi\)
0.307571 + 0.951525i \(0.400484\pi\)
\(224\) 0.162672 0.0108690
\(225\) 12.8606 0.857376
\(226\) 18.3107 1.21801
\(227\) 8.29125 0.550309 0.275155 0.961400i \(-0.411271\pi\)
0.275155 + 0.961400i \(0.411271\pi\)
\(228\) 1.00000 0.0662266
\(229\) 6.24366 0.412593 0.206296 0.978490i \(-0.433859\pi\)
0.206296 + 0.978490i \(0.433859\pi\)
\(230\) −15.1536 −0.999196
\(231\) 0.697641 0.0459014
\(232\) 9.28951 0.609887
\(233\) 3.39537 0.222438 0.111219 0.993796i \(-0.464524\pi\)
0.111219 + 0.993796i \(0.464524\pi\)
\(234\) −1.72968 −0.113073
\(235\) −30.1202 −1.96482
\(236\) −4.09719 −0.266704
\(237\) −7.40316 −0.480887
\(238\) −0.0606049 −0.00392843
\(239\) 22.0413 1.42574 0.712868 0.701298i \(-0.247396\pi\)
0.712868 + 0.701298i \(0.247396\pi\)
\(240\) 4.22619 0.272799
\(241\) −15.7486 −1.01446 −0.507229 0.861812i \(-0.669330\pi\)
−0.507229 + 0.861812i \(0.669330\pi\)
\(242\) 7.39247 0.475206
\(243\) 1.00000 0.0641500
\(244\) −12.6996 −0.813011
\(245\) −29.4715 −1.88286
\(246\) −2.88605 −0.184008
\(247\) −1.72968 −0.110057
\(248\) −4.13779 −0.262750
\(249\) −8.64039 −0.547562
\(250\) 33.2205 2.10105
\(251\) −0.454889 −0.0287123 −0.0143562 0.999897i \(-0.504570\pi\)
−0.0143562 + 0.999897i \(0.504570\pi\)
\(252\) 0.162672 0.0102473
\(253\) −15.3775 −0.966776
\(254\) 7.32539 0.459636
\(255\) −1.57451 −0.0985995
\(256\) 1.00000 0.0625000
\(257\) −4.38596 −0.273589 −0.136794 0.990599i \(-0.543680\pi\)
−0.136794 + 0.990599i \(0.543680\pi\)
\(258\) −7.90481 −0.492132
\(259\) 0.242128 0.0150451
\(260\) −7.30994 −0.453343
\(261\) 9.28951 0.575007
\(262\) 1.88204 0.116273
\(263\) −4.09165 −0.252302 −0.126151 0.992011i \(-0.540262\pi\)
−0.126151 + 0.992011i \(0.540262\pi\)
\(264\) 4.28864 0.263948
\(265\) −4.22619 −0.259613
\(266\) 0.162672 0.00997404
\(267\) 3.54113 0.216713
\(268\) −12.5390 −0.765940
\(269\) −20.0849 −1.22460 −0.612298 0.790627i \(-0.709755\pi\)
−0.612298 + 0.790627i \(0.709755\pi\)
\(270\) 4.22619 0.257197
\(271\) 18.2974 1.11149 0.555745 0.831353i \(-0.312433\pi\)
0.555745 + 0.831353i \(0.312433\pi\)
\(272\) −0.372560 −0.0225898
\(273\) −0.281369 −0.0170292
\(274\) −13.4413 −0.812018
\(275\) 55.1547 3.32596
\(276\) −3.58563 −0.215830
\(277\) 2.76894 0.166370 0.0831849 0.996534i \(-0.473491\pi\)
0.0831849 + 0.996534i \(0.473491\pi\)
\(278\) 1.73787 0.104230
\(279\) −4.13779 −0.247723
\(280\) 0.687480 0.0410848
\(281\) 10.5622 0.630091 0.315045 0.949077i \(-0.397980\pi\)
0.315045 + 0.949077i \(0.397980\pi\)
\(282\) −7.12704 −0.424409
\(283\) 7.76833 0.461780 0.230890 0.972980i \(-0.425836\pi\)
0.230890 + 0.972980i \(0.425836\pi\)
\(284\) 16.1640 0.959159
\(285\) 4.22619 0.250338
\(286\) −7.41797 −0.438634
\(287\) −0.469479 −0.0277124
\(288\) 1.00000 0.0589256
\(289\) −16.8612 −0.991835
\(290\) 39.2592 2.30538
\(291\) 7.12709 0.417798
\(292\) 15.0565 0.881117
\(293\) 23.0341 1.34567 0.672834 0.739794i \(-0.265077\pi\)
0.672834 + 0.739794i \(0.265077\pi\)
\(294\) −6.97354 −0.406705
\(295\) −17.3155 −1.00815
\(296\) 1.48845 0.0865141
\(297\) 4.28864 0.248852
\(298\) −10.0368 −0.581414
\(299\) 6.20199 0.358670
\(300\) 12.8606 0.742510
\(301\) −1.28589 −0.0741174
\(302\) −4.42076 −0.254386
\(303\) −0.0690802 −0.00396856
\(304\) 1.00000 0.0573539
\(305\) −53.6710 −3.07319
\(306\) −0.372560 −0.0212978
\(307\) 1.13527 0.0647935 0.0323967 0.999475i \(-0.489686\pi\)
0.0323967 + 0.999475i \(0.489686\pi\)
\(308\) 0.697641 0.0397518
\(309\) −8.84739 −0.503311
\(310\) −17.4871 −0.993199
\(311\) 11.1763 0.633752 0.316876 0.948467i \(-0.397366\pi\)
0.316876 + 0.948467i \(0.397366\pi\)
\(312\) −1.72968 −0.0979237
\(313\) 30.5985 1.72953 0.864765 0.502176i \(-0.167467\pi\)
0.864765 + 0.502176i \(0.167467\pi\)
\(314\) −10.0984 −0.569885
\(315\) 0.687480 0.0387351
\(316\) −7.40316 −0.416461
\(317\) 30.7369 1.72636 0.863179 0.504899i \(-0.168470\pi\)
0.863179 + 0.504899i \(0.168470\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 39.8394 2.23058
\(320\) 4.22619 0.236251
\(321\) 5.16527 0.288297
\(322\) −0.583281 −0.0325050
\(323\) −0.372560 −0.0207298
\(324\) 1.00000 0.0555556
\(325\) −22.2448 −1.23392
\(326\) −7.51964 −0.416474
\(327\) −7.17230 −0.396629
\(328\) −2.88605 −0.159355
\(329\) −1.15937 −0.0639180
\(330\) 18.1246 0.997727
\(331\) −21.4262 −1.17769 −0.588844 0.808246i \(-0.700417\pi\)
−0.588844 + 0.808246i \(0.700417\pi\)
\(332\) −8.64039 −0.474203
\(333\) 1.48845 0.0815663
\(334\) 12.4136 0.679243
\(335\) −52.9920 −2.89526
\(336\) 0.162672 0.00887446
\(337\) 9.17522 0.499806 0.249903 0.968271i \(-0.419601\pi\)
0.249903 + 0.968271i \(0.419601\pi\)
\(338\) −10.0082 −0.544375
\(339\) 18.3107 0.994503
\(340\) −1.57451 −0.0853896
\(341\) −17.7455 −0.960974
\(342\) 1.00000 0.0540738
\(343\) −2.27310 −0.122736
\(344\) −7.90481 −0.426199
\(345\) −15.1536 −0.815840
\(346\) 7.13359 0.383504
\(347\) −36.7735 −1.97410 −0.987052 0.160402i \(-0.948721\pi\)
−0.987052 + 0.160402i \(0.948721\pi\)
\(348\) 9.28951 0.497970
\(349\) −10.8254 −0.579471 −0.289736 0.957107i \(-0.593567\pi\)
−0.289736 + 0.957107i \(0.593567\pi\)
\(350\) 2.09206 0.111825
\(351\) −1.72968 −0.0923233
\(352\) 4.28864 0.228586
\(353\) −6.44252 −0.342901 −0.171450 0.985193i \(-0.554845\pi\)
−0.171450 + 0.985193i \(0.554845\pi\)
\(354\) −4.09719 −0.217763
\(355\) 68.3122 3.62564
\(356\) 3.54113 0.187679
\(357\) −0.0606049 −0.00320755
\(358\) 16.6540 0.880189
\(359\) 21.0901 1.11309 0.556546 0.830817i \(-0.312126\pi\)
0.556546 + 0.830817i \(0.312126\pi\)
\(360\) 4.22619 0.222740
\(361\) 1.00000 0.0526316
\(362\) −15.5258 −0.816021
\(363\) 7.39247 0.388004
\(364\) −0.281369 −0.0147478
\(365\) 63.6317 3.33064
\(366\) −12.6996 −0.663821
\(367\) 20.9199 1.09201 0.546004 0.837782i \(-0.316148\pi\)
0.546004 + 0.837782i \(0.316148\pi\)
\(368\) −3.58563 −0.186914
\(369\) −2.88605 −0.150242
\(370\) 6.29045 0.327025
\(371\) −0.162672 −0.00844549
\(372\) −4.13779 −0.214535
\(373\) 22.1666 1.14774 0.573870 0.818946i \(-0.305441\pi\)
0.573870 + 0.818946i \(0.305441\pi\)
\(374\) −1.59778 −0.0826191
\(375\) 33.2205 1.71550
\(376\) −7.12704 −0.367549
\(377\) −16.0679 −0.827537
\(378\) 0.162672 0.00836693
\(379\) 21.5553 1.10722 0.553612 0.832775i \(-0.313249\pi\)
0.553612 + 0.832775i \(0.313249\pi\)
\(380\) 4.22619 0.216799
\(381\) 7.32539 0.375291
\(382\) −11.7620 −0.601796
\(383\) 8.53506 0.436121 0.218061 0.975935i \(-0.430027\pi\)
0.218061 + 0.975935i \(0.430027\pi\)
\(384\) 1.00000 0.0510310
\(385\) 2.94836 0.150262
\(386\) −19.5836 −0.996777
\(387\) −7.90481 −0.401824
\(388\) 7.12709 0.361823
\(389\) −27.8908 −1.41412 −0.707060 0.707154i \(-0.749979\pi\)
−0.707060 + 0.707154i \(0.749979\pi\)
\(390\) −7.30994 −0.370153
\(391\) 1.33586 0.0675575
\(392\) −6.97354 −0.352217
\(393\) 1.88204 0.0949364
\(394\) 11.7158 0.590233
\(395\) −31.2871 −1.57423
\(396\) 4.28864 0.215512
\(397\) 25.2537 1.26745 0.633724 0.773559i \(-0.281526\pi\)
0.633724 + 0.773559i \(0.281526\pi\)
\(398\) −3.78323 −0.189636
\(399\) 0.162672 0.00814377
\(400\) 12.8606 0.643032
\(401\) −33.6898 −1.68239 −0.841194 0.540733i \(-0.818147\pi\)
−0.841194 + 0.540733i \(0.818147\pi\)
\(402\) −12.5390 −0.625387
\(403\) 7.15704 0.356518
\(404\) −0.0690802 −0.00343687
\(405\) 4.22619 0.210001
\(406\) 1.51114 0.0749966
\(407\) 6.38341 0.316414
\(408\) −0.372560 −0.0184445
\(409\) 0.180224 0.00891152 0.00445576 0.999990i \(-0.498582\pi\)
0.00445576 + 0.999990i \(0.498582\pi\)
\(410\) −12.1970 −0.602366
\(411\) −13.4413 −0.663010
\(412\) −8.84739 −0.435880
\(413\) −0.666496 −0.0327961
\(414\) −3.58563 −0.176224
\(415\) −36.5159 −1.79249
\(416\) −1.72968 −0.0848044
\(417\) 1.73787 0.0851037
\(418\) 4.28864 0.209764
\(419\) 21.5428 1.05244 0.526218 0.850350i \(-0.323610\pi\)
0.526218 + 0.850350i \(0.323610\pi\)
\(420\) 0.687480 0.0335456
\(421\) −18.3236 −0.893040 −0.446520 0.894774i \(-0.647337\pi\)
−0.446520 + 0.894774i \(0.647337\pi\)
\(422\) −21.0364 −1.02403
\(423\) −7.12704 −0.346528
\(424\) −1.00000 −0.0485643
\(425\) −4.79136 −0.232415
\(426\) 16.1640 0.783150
\(427\) −2.06587 −0.0999745
\(428\) 5.16527 0.249673
\(429\) −7.41797 −0.358143
\(430\) −33.4072 −1.61104
\(431\) −5.20265 −0.250603 −0.125301 0.992119i \(-0.539990\pi\)
−0.125301 + 0.992119i \(0.539990\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.4447 0.790284 0.395142 0.918620i \(-0.370695\pi\)
0.395142 + 0.918620i \(0.370695\pi\)
\(434\) −0.673101 −0.0323099
\(435\) 39.2592 1.88234
\(436\) −7.17230 −0.343491
\(437\) −3.58563 −0.171524
\(438\) 15.0565 0.719429
\(439\) −20.4949 −0.978168 −0.489084 0.872237i \(-0.662669\pi\)
−0.489084 + 0.872237i \(0.662669\pi\)
\(440\) 18.1246 0.864057
\(441\) −6.97354 −0.332073
\(442\) 0.644408 0.0306514
\(443\) −10.2245 −0.485779 −0.242889 0.970054i \(-0.578095\pi\)
−0.242889 + 0.970054i \(0.578095\pi\)
\(444\) 1.48845 0.0706385
\(445\) 14.9655 0.709431
\(446\) 9.18604 0.434971
\(447\) −10.0368 −0.474722
\(448\) 0.162672 0.00768551
\(449\) 3.69271 0.174270 0.0871349 0.996197i \(-0.472229\pi\)
0.0871349 + 0.996197i \(0.472229\pi\)
\(450\) 12.8606 0.606257
\(451\) −12.3772 −0.582822
\(452\) 18.3107 0.861265
\(453\) −4.42076 −0.207705
\(454\) 8.29125 0.389128
\(455\) −1.18912 −0.0557468
\(456\) 1.00000 0.0468293
\(457\) −21.8596 −1.02255 −0.511275 0.859417i \(-0.670827\pi\)
−0.511275 + 0.859417i \(0.670827\pi\)
\(458\) 6.24366 0.291747
\(459\) −0.372560 −0.0173896
\(460\) −15.1536 −0.706538
\(461\) 5.50751 0.256510 0.128255 0.991741i \(-0.459062\pi\)
0.128255 + 0.991741i \(0.459062\pi\)
\(462\) 0.697641 0.0324572
\(463\) −6.95025 −0.323005 −0.161503 0.986872i \(-0.551634\pi\)
−0.161503 + 0.986872i \(0.551634\pi\)
\(464\) 9.28951 0.431255
\(465\) −17.4871 −0.810944
\(466\) 3.39537 0.157288
\(467\) 16.6192 0.769045 0.384523 0.923116i \(-0.374366\pi\)
0.384523 + 0.923116i \(0.374366\pi\)
\(468\) −1.72968 −0.0799543
\(469\) −2.03973 −0.0941862
\(470\) −30.1202 −1.38934
\(471\) −10.0984 −0.465309
\(472\) −4.09719 −0.188588
\(473\) −33.9009 −1.55877
\(474\) −7.40316 −0.340039
\(475\) 12.8606 0.590087
\(476\) −0.0606049 −0.00277782
\(477\) −1.00000 −0.0457869
\(478\) 22.0413 1.00815
\(479\) 10.4381 0.476931 0.238465 0.971151i \(-0.423356\pi\)
0.238465 + 0.971151i \(0.423356\pi\)
\(480\) 4.22619 0.192898
\(481\) −2.57453 −0.117388
\(482\) −15.7486 −0.717330
\(483\) −0.583281 −0.0265402
\(484\) 7.39247 0.336021
\(485\) 30.1204 1.36770
\(486\) 1.00000 0.0453609
\(487\) −2.99753 −0.135831 −0.0679156 0.997691i \(-0.521635\pi\)
−0.0679156 + 0.997691i \(0.521635\pi\)
\(488\) −12.6996 −0.574886
\(489\) −7.51964 −0.340050
\(490\) −29.4715 −1.33139
\(491\) −25.7095 −1.16025 −0.580127 0.814526i \(-0.696997\pi\)
−0.580127 + 0.814526i \(0.696997\pi\)
\(492\) −2.88605 −0.130113
\(493\) −3.46090 −0.155871
\(494\) −1.72968 −0.0778218
\(495\) 18.1246 0.814640
\(496\) −4.13779 −0.185792
\(497\) 2.62943 0.117946
\(498\) −8.64039 −0.387185
\(499\) −38.6690 −1.73106 −0.865532 0.500854i \(-0.833019\pi\)
−0.865532 + 0.500854i \(0.833019\pi\)
\(500\) 33.2205 1.48567
\(501\) 12.4136 0.554599
\(502\) −0.454889 −0.0203027
\(503\) 4.67451 0.208426 0.104213 0.994555i \(-0.466768\pi\)
0.104213 + 0.994555i \(0.466768\pi\)
\(504\) 0.162672 0.00724597
\(505\) −0.291946 −0.0129914
\(506\) −15.3775 −0.683614
\(507\) −10.0082 −0.444481
\(508\) 7.32539 0.325012
\(509\) 36.4649 1.61628 0.808139 0.588992i \(-0.200475\pi\)
0.808139 + 0.588992i \(0.200475\pi\)
\(510\) −1.57451 −0.0697204
\(511\) 2.44927 0.108349
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −4.38596 −0.193456
\(515\) −37.3907 −1.64763
\(516\) −7.90481 −0.347990
\(517\) −30.5653 −1.34426
\(518\) 0.242128 0.0106385
\(519\) 7.13359 0.313130
\(520\) −7.30994 −0.320562
\(521\) 28.4787 1.24767 0.623837 0.781555i \(-0.285573\pi\)
0.623837 + 0.781555i \(0.285573\pi\)
\(522\) 9.28951 0.406591
\(523\) 28.3933 1.24155 0.620775 0.783988i \(-0.286818\pi\)
0.620775 + 0.783988i \(0.286818\pi\)
\(524\) 1.88204 0.0822173
\(525\) 2.09206 0.0913051
\(526\) −4.09165 −0.178404
\(527\) 1.54158 0.0671521
\(528\) 4.28864 0.186639
\(529\) −10.1432 −0.441010
\(530\) −4.22619 −0.183574
\(531\) −4.09719 −0.177803
\(532\) 0.162672 0.00705271
\(533\) 4.99193 0.216225
\(534\) 3.54113 0.153240
\(535\) 21.8294 0.943767
\(536\) −12.5390 −0.541601
\(537\) 16.6540 0.718671
\(538\) −20.0849 −0.865920
\(539\) −29.9070 −1.28819
\(540\) 4.22619 0.181866
\(541\) −22.3005 −0.958772 −0.479386 0.877604i \(-0.659141\pi\)
−0.479386 + 0.877604i \(0.659141\pi\)
\(542\) 18.2974 0.785941
\(543\) −15.5258 −0.666278
\(544\) −0.372560 −0.0159734
\(545\) −30.3115 −1.29840
\(546\) −0.281369 −0.0120415
\(547\) −9.02108 −0.385713 −0.192857 0.981227i \(-0.561775\pi\)
−0.192857 + 0.981227i \(0.561775\pi\)
\(548\) −13.4413 −0.574183
\(549\) −12.6996 −0.542007
\(550\) 55.1547 2.35181
\(551\) 9.28951 0.395747
\(552\) −3.58563 −0.152615
\(553\) −1.20428 −0.0512114
\(554\) 2.76894 0.117641
\(555\) 6.29045 0.267015
\(556\) 1.73787 0.0737020
\(557\) 3.68994 0.156348 0.0781739 0.996940i \(-0.475091\pi\)
0.0781739 + 0.996940i \(0.475091\pi\)
\(558\) −4.13779 −0.175167
\(559\) 13.6728 0.578296
\(560\) 0.687480 0.0290514
\(561\) −1.59778 −0.0674582
\(562\) 10.5622 0.445542
\(563\) 33.8671 1.42733 0.713663 0.700489i \(-0.247035\pi\)
0.713663 + 0.700489i \(0.247035\pi\)
\(564\) −7.12704 −0.300102
\(565\) 77.3846 3.25559
\(566\) 7.76833 0.326527
\(567\) 0.162672 0.00683157
\(568\) 16.1640 0.678228
\(569\) −42.4617 −1.78009 −0.890043 0.455877i \(-0.849326\pi\)
−0.890043 + 0.455877i \(0.849326\pi\)
\(570\) 4.22619 0.177015
\(571\) 23.2265 0.972000 0.486000 0.873959i \(-0.338455\pi\)
0.486000 + 0.873959i \(0.338455\pi\)
\(572\) −7.41797 −0.310161
\(573\) −11.7620 −0.491364
\(574\) −0.469479 −0.0195957
\(575\) −46.1136 −1.92307
\(576\) 1.00000 0.0416667
\(577\) 25.6241 1.06675 0.533373 0.845880i \(-0.320924\pi\)
0.533373 + 0.845880i \(0.320924\pi\)
\(578\) −16.8612 −0.701333
\(579\) −19.5836 −0.813865
\(580\) 39.2592 1.63015
\(581\) −1.40555 −0.0583119
\(582\) 7.12709 0.295428
\(583\) −4.28864 −0.177617
\(584\) 15.0565 0.623044
\(585\) −7.30994 −0.302229
\(586\) 23.0341 0.951530
\(587\) −4.55310 −0.187927 −0.0939633 0.995576i \(-0.529954\pi\)
−0.0939633 + 0.995576i \(0.529954\pi\)
\(588\) −6.97354 −0.287584
\(589\) −4.13779 −0.170495
\(590\) −17.3155 −0.712867
\(591\) 11.7158 0.481923
\(592\) 1.48845 0.0611747
\(593\) 13.5703 0.557266 0.278633 0.960398i \(-0.410119\pi\)
0.278633 + 0.960398i \(0.410119\pi\)
\(594\) 4.28864 0.175965
\(595\) −0.256128 −0.0105002
\(596\) −10.0368 −0.411121
\(597\) −3.78323 −0.154837
\(598\) 6.20199 0.253618
\(599\) −18.3926 −0.751501 −0.375750 0.926721i \(-0.622615\pi\)
−0.375750 + 0.926721i \(0.622615\pi\)
\(600\) 12.8606 0.525034
\(601\) 38.3615 1.56480 0.782398 0.622778i \(-0.213996\pi\)
0.782398 + 0.622778i \(0.213996\pi\)
\(602\) −1.28589 −0.0524089
\(603\) −12.5390 −0.510626
\(604\) −4.42076 −0.179878
\(605\) 31.2419 1.27017
\(606\) −0.0690802 −0.00280619
\(607\) −17.7781 −0.721590 −0.360795 0.932645i \(-0.617495\pi\)
−0.360795 + 0.932645i \(0.617495\pi\)
\(608\) 1.00000 0.0405554
\(609\) 1.51114 0.0612345
\(610\) −53.6710 −2.17308
\(611\) 12.3275 0.498716
\(612\) −0.372560 −0.0150598
\(613\) 25.1389 1.01535 0.507676 0.861548i \(-0.330505\pi\)
0.507676 + 0.861548i \(0.330505\pi\)
\(614\) 1.13527 0.0458159
\(615\) −12.1970 −0.491830
\(616\) 0.697641 0.0281087
\(617\) 34.4086 1.38524 0.692618 0.721304i \(-0.256457\pi\)
0.692618 + 0.721304i \(0.256457\pi\)
\(618\) −8.84739 −0.355894
\(619\) −18.2421 −0.733211 −0.366605 0.930377i \(-0.619480\pi\)
−0.366605 + 0.930377i \(0.619480\pi\)
\(620\) −17.4871 −0.702298
\(621\) −3.58563 −0.143887
\(622\) 11.1763 0.448130
\(623\) 0.576041 0.0230786
\(624\) −1.72968 −0.0692425
\(625\) 76.0930 3.04372
\(626\) 30.5985 1.22296
\(627\) 4.28864 0.171272
\(628\) −10.0984 −0.402969
\(629\) −0.554535 −0.0221108
\(630\) 0.687480 0.0273899
\(631\) −42.5170 −1.69258 −0.846288 0.532725i \(-0.821168\pi\)
−0.846288 + 0.532725i \(0.821168\pi\)
\(632\) −7.40316 −0.294482
\(633\) −21.0364 −0.836121
\(634\) 30.7369 1.22072
\(635\) 30.9585 1.22855
\(636\) −1.00000 −0.0396526
\(637\) 12.0620 0.477913
\(638\) 39.8394 1.57726
\(639\) 16.1640 0.639440
\(640\) 4.22619 0.167055
\(641\) 31.0360 1.22585 0.612923 0.790142i \(-0.289993\pi\)
0.612923 + 0.790142i \(0.289993\pi\)
\(642\) 5.16527 0.203857
\(643\) −9.87265 −0.389339 −0.194670 0.980869i \(-0.562363\pi\)
−0.194670 + 0.980869i \(0.562363\pi\)
\(644\) −0.583281 −0.0229845
\(645\) −33.4072 −1.31541
\(646\) −0.372560 −0.0146582
\(647\) −39.8754 −1.56766 −0.783831 0.620974i \(-0.786737\pi\)
−0.783831 + 0.620974i \(0.786737\pi\)
\(648\) 1.00000 0.0392837
\(649\) −17.5714 −0.689737
\(650\) −22.2448 −0.872511
\(651\) −0.673101 −0.0263809
\(652\) −7.51964 −0.294492
\(653\) −8.50568 −0.332853 −0.166427 0.986054i \(-0.553223\pi\)
−0.166427 + 0.986054i \(0.553223\pi\)
\(654\) −7.17230 −0.280459
\(655\) 7.95385 0.310783
\(656\) −2.88605 −0.112681
\(657\) 15.0565 0.587411
\(658\) −1.15937 −0.0451968
\(659\) −35.5319 −1.38413 −0.692063 0.721837i \(-0.743298\pi\)
−0.692063 + 0.721837i \(0.743298\pi\)
\(660\) 18.1246 0.705499
\(661\) −4.96815 −0.193239 −0.0966193 0.995321i \(-0.530803\pi\)
−0.0966193 + 0.995321i \(0.530803\pi\)
\(662\) −21.4262 −0.832752
\(663\) 0.644408 0.0250267
\(664\) −8.64039 −0.335312
\(665\) 0.687480 0.0266593
\(666\) 1.48845 0.0576761
\(667\) −33.3088 −1.28972
\(668\) 12.4136 0.480297
\(669\) 9.18604 0.355153
\(670\) −52.9920 −2.04726
\(671\) −54.4642 −2.10257
\(672\) 0.162672 0.00627519
\(673\) 33.1654 1.27843 0.639216 0.769027i \(-0.279259\pi\)
0.639216 + 0.769027i \(0.279259\pi\)
\(674\) 9.17522 0.353416
\(675\) 12.8606 0.495006
\(676\) −10.0082 −0.384931
\(677\) −31.8766 −1.22512 −0.612558 0.790426i \(-0.709859\pi\)
−0.612558 + 0.790426i \(0.709859\pi\)
\(678\) 18.3107 0.703220
\(679\) 1.15938 0.0444928
\(680\) −1.57451 −0.0603796
\(681\) 8.29125 0.317721
\(682\) −17.7455 −0.679511
\(683\) −49.5338 −1.89536 −0.947680 0.319222i \(-0.896578\pi\)
−0.947680 + 0.319222i \(0.896578\pi\)
\(684\) 1.00000 0.0382360
\(685\) −56.8054 −2.17042
\(686\) −2.27310 −0.0867873
\(687\) 6.24366 0.238211
\(688\) −7.90481 −0.301368
\(689\) 1.72968 0.0658954
\(690\) −15.1536 −0.576886
\(691\) 27.6975 1.05366 0.526831 0.849970i \(-0.323380\pi\)
0.526831 + 0.849970i \(0.323380\pi\)
\(692\) 7.13359 0.271178
\(693\) 0.697641 0.0265012
\(694\) −36.7735 −1.39590
\(695\) 7.34455 0.278595
\(696\) 9.28951 0.352118
\(697\) 1.07523 0.0407271
\(698\) −10.8254 −0.409748
\(699\) 3.39537 0.128425
\(700\) 2.09206 0.0790725
\(701\) −9.39265 −0.354756 −0.177378 0.984143i \(-0.556761\pi\)
−0.177378 + 0.984143i \(0.556761\pi\)
\(702\) −1.72968 −0.0652824
\(703\) 1.48845 0.0561378
\(704\) 4.28864 0.161634
\(705\) −30.1202 −1.13439
\(706\) −6.44252 −0.242467
\(707\) −0.0112374 −0.000422626 0
\(708\) −4.09719 −0.153982
\(709\) 9.57952 0.359766 0.179883 0.983688i \(-0.442428\pi\)
0.179883 + 0.983688i \(0.442428\pi\)
\(710\) 68.3122 2.56371
\(711\) −7.40316 −0.277640
\(712\) 3.54113 0.132709
\(713\) 14.8366 0.555635
\(714\) −0.0606049 −0.00226808
\(715\) −31.3497 −1.17241
\(716\) 16.6540 0.622387
\(717\) 22.0413 0.823149
\(718\) 21.0901 0.787075
\(719\) −28.7626 −1.07266 −0.536331 0.844008i \(-0.680190\pi\)
−0.536331 + 0.844008i \(0.680190\pi\)
\(720\) 4.22619 0.157501
\(721\) −1.43922 −0.0535993
\(722\) 1.00000 0.0372161
\(723\) −15.7486 −0.585697
\(724\) −15.5258 −0.577014
\(725\) 119.469 4.43697
\(726\) 7.39247 0.274360
\(727\) −2.79135 −0.103525 −0.0517626 0.998659i \(-0.516484\pi\)
−0.0517626 + 0.998659i \(0.516484\pi\)
\(728\) −0.281369 −0.0104282
\(729\) 1.00000 0.0370370
\(730\) 63.6317 2.35511
\(731\) 2.94501 0.108925
\(732\) −12.6996 −0.469392
\(733\) 26.9353 0.994877 0.497439 0.867499i \(-0.334274\pi\)
0.497439 + 0.867499i \(0.334274\pi\)
\(734\) 20.9199 0.772167
\(735\) −29.4715 −1.08707
\(736\) −3.58563 −0.132168
\(737\) −53.7752 −1.98083
\(738\) −2.88605 −0.106237
\(739\) −3.07320 −0.113049 −0.0565247 0.998401i \(-0.518002\pi\)
−0.0565247 + 0.998401i \(0.518002\pi\)
\(740\) 6.29045 0.231241
\(741\) −1.72968 −0.0635413
\(742\) −0.162672 −0.00597186
\(743\) 31.8045 1.16680 0.583398 0.812187i \(-0.301723\pi\)
0.583398 + 0.812187i \(0.301723\pi\)
\(744\) −4.13779 −0.151699
\(745\) −42.4172 −1.55405
\(746\) 22.1666 0.811575
\(747\) −8.64039 −0.316135
\(748\) −1.59778 −0.0584205
\(749\) 0.840243 0.0307018
\(750\) 33.2205 1.21304
\(751\) −30.9245 −1.12845 −0.564225 0.825621i \(-0.690825\pi\)
−0.564225 + 0.825621i \(0.690825\pi\)
\(752\) −7.12704 −0.259896
\(753\) −0.454889 −0.0165771
\(754\) −16.0679 −0.585157
\(755\) −18.6830 −0.679942
\(756\) 0.162672 0.00591631
\(757\) 44.3179 1.61076 0.805380 0.592758i \(-0.201961\pi\)
0.805380 + 0.592758i \(0.201961\pi\)
\(758\) 21.5553 0.782926
\(759\) −15.3775 −0.558168
\(760\) 4.22619 0.153300
\(761\) −13.3572 −0.484197 −0.242099 0.970252i \(-0.577836\pi\)
−0.242099 + 0.970252i \(0.577836\pi\)
\(762\) 7.32539 0.265371
\(763\) −1.16673 −0.0422385
\(764\) −11.7620 −0.425534
\(765\) −1.57451 −0.0569264
\(766\) 8.53506 0.308384
\(767\) 7.08681 0.255890
\(768\) 1.00000 0.0360844
\(769\) −13.6384 −0.491814 −0.245907 0.969293i \(-0.579086\pi\)
−0.245907 + 0.969293i \(0.579086\pi\)
\(770\) 2.94836 0.106251
\(771\) −4.38596 −0.157956
\(772\) −19.5836 −0.704828
\(773\) 42.9885 1.54619 0.773094 0.634292i \(-0.218708\pi\)
0.773094 + 0.634292i \(0.218708\pi\)
\(774\) −7.90481 −0.284132
\(775\) −53.2147 −1.91153
\(776\) 7.12709 0.255848
\(777\) 0.242128 0.00868629
\(778\) −27.8908 −0.999934
\(779\) −2.88605 −0.103403
\(780\) −7.30994 −0.261738
\(781\) 69.3218 2.48053
\(782\) 1.33586 0.0477704
\(783\) 9.28951 0.331980
\(784\) −6.97354 −0.249055
\(785\) −42.6776 −1.52323
\(786\) 1.88204 0.0671302
\(787\) 26.3867 0.940583 0.470292 0.882511i \(-0.344149\pi\)
0.470292 + 0.882511i \(0.344149\pi\)
\(788\) 11.7158 0.417358
\(789\) −4.09165 −0.145666
\(790\) −31.2871 −1.11315
\(791\) 2.97864 0.105908
\(792\) 4.28864 0.152390
\(793\) 21.9663 0.780045
\(794\) 25.2537 0.896221
\(795\) −4.22619 −0.149887
\(796\) −3.78323 −0.134093
\(797\) 4.93861 0.174935 0.0874673 0.996167i \(-0.472123\pi\)
0.0874673 + 0.996167i \(0.472123\pi\)
\(798\) 0.162672 0.00575851
\(799\) 2.65525 0.0939359
\(800\) 12.8606 0.454692
\(801\) 3.54113 0.125120
\(802\) −33.6898 −1.18963
\(803\) 64.5721 2.27870
\(804\) −12.5390 −0.442215
\(805\) −2.46505 −0.0868817
\(806\) 7.15704 0.252096
\(807\) −20.0849 −0.707021
\(808\) −0.0690802 −0.00243023
\(809\) −26.5144 −0.932195 −0.466098 0.884733i \(-0.654340\pi\)
−0.466098 + 0.884733i \(0.654340\pi\)
\(810\) 4.22619 0.148493
\(811\) −20.2594 −0.711403 −0.355701 0.934600i \(-0.615758\pi\)
−0.355701 + 0.934600i \(0.615758\pi\)
\(812\) 1.51114 0.0530306
\(813\) 18.2974 0.641719
\(814\) 6.38341 0.223738
\(815\) −31.7794 −1.11318
\(816\) −0.372560 −0.0130422
\(817\) −7.90481 −0.276554
\(818\) 0.180224 0.00630140
\(819\) −0.281369 −0.00983184
\(820\) −12.1970 −0.425937
\(821\) 55.1772 1.92570 0.962849 0.270039i \(-0.0870367\pi\)
0.962849 + 0.270039i \(0.0870367\pi\)
\(822\) −13.4413 −0.468819
\(823\) −6.23221 −0.217241 −0.108621 0.994083i \(-0.534643\pi\)
−0.108621 + 0.994083i \(0.534643\pi\)
\(824\) −8.84739 −0.308214
\(825\) 55.1547 1.92024
\(826\) −0.666496 −0.0231904
\(827\) −14.7224 −0.511949 −0.255974 0.966684i \(-0.582396\pi\)
−0.255974 + 0.966684i \(0.582396\pi\)
\(828\) −3.58563 −0.124609
\(829\) 52.6306 1.82793 0.913967 0.405787i \(-0.133003\pi\)
0.913967 + 0.405787i \(0.133003\pi\)
\(830\) −36.5159 −1.26748
\(831\) 2.76894 0.0960536
\(832\) −1.72968 −0.0599658
\(833\) 2.59806 0.0900175
\(834\) 1.73787 0.0601774
\(835\) 52.4622 1.81553
\(836\) 4.28864 0.148326
\(837\) −4.13779 −0.143023
\(838\) 21.5428 0.744185
\(839\) −8.28142 −0.285906 −0.142953 0.989729i \(-0.545660\pi\)
−0.142953 + 0.989729i \(0.545660\pi\)
\(840\) 0.687480 0.0237203
\(841\) 57.2951 1.97569
\(842\) −18.3236 −0.631474
\(843\) 10.5622 0.363783
\(844\) −21.0364 −0.724102
\(845\) −42.2966 −1.45505
\(846\) −7.12704 −0.245033
\(847\) 1.20254 0.0413199
\(848\) −1.00000 −0.0343401
\(849\) 7.76833 0.266609
\(850\) −4.79136 −0.164342
\(851\) −5.33702 −0.182951
\(852\) 16.1640 0.553771
\(853\) −41.0831 −1.40666 −0.703329 0.710864i \(-0.748304\pi\)
−0.703329 + 0.710864i \(0.748304\pi\)
\(854\) −2.06587 −0.0706926
\(855\) 4.22619 0.144532
\(856\) 5.16527 0.176545
\(857\) 45.8851 1.56740 0.783702 0.621137i \(-0.213329\pi\)
0.783702 + 0.621137i \(0.213329\pi\)
\(858\) −7.41797 −0.253245
\(859\) −28.0686 −0.957689 −0.478844 0.877900i \(-0.658944\pi\)
−0.478844 + 0.877900i \(0.658944\pi\)
\(860\) −33.4072 −1.13918
\(861\) −0.469479 −0.0159998
\(862\) −5.20265 −0.177203
\(863\) −0.920314 −0.0313278 −0.0156639 0.999877i \(-0.504986\pi\)
−0.0156639 + 0.999877i \(0.504986\pi\)
\(864\) 1.00000 0.0340207
\(865\) 30.1479 1.02506
\(866\) 16.4447 0.558815
\(867\) −16.8612 −0.572636
\(868\) −0.673101 −0.0228465
\(869\) −31.7495 −1.07703
\(870\) 39.2592 1.33101
\(871\) 21.6884 0.734882
\(872\) −7.17230 −0.242885
\(873\) 7.12709 0.241216
\(874\) −3.58563 −0.121286
\(875\) 5.40404 0.182690
\(876\) 15.0565 0.508713
\(877\) 15.8758 0.536088 0.268044 0.963407i \(-0.413623\pi\)
0.268044 + 0.963407i \(0.413623\pi\)
\(878\) −20.4949 −0.691669
\(879\) 23.0341 0.776921
\(880\) 18.1246 0.610980
\(881\) 29.1007 0.980426 0.490213 0.871603i \(-0.336919\pi\)
0.490213 + 0.871603i \(0.336919\pi\)
\(882\) −6.97354 −0.234811
\(883\) 43.2015 1.45385 0.726923 0.686719i \(-0.240950\pi\)
0.726923 + 0.686719i \(0.240950\pi\)
\(884\) 0.644408 0.0216738
\(885\) −17.3155 −0.582054
\(886\) −10.2245 −0.343498
\(887\) 51.9922 1.74573 0.872863 0.487965i \(-0.162260\pi\)
0.872863 + 0.487965i \(0.162260\pi\)
\(888\) 1.48845 0.0499489
\(889\) 1.19163 0.0399661
\(890\) 14.9655 0.501643
\(891\) 4.28864 0.143675
\(892\) 9.18604 0.307571
\(893\) −7.12704 −0.238497
\(894\) −10.0368 −0.335679
\(895\) 70.3827 2.35263
\(896\) 0.162672 0.00543448
\(897\) 6.20199 0.207078
\(898\) 3.69271 0.123227
\(899\) −38.4381 −1.28198
\(900\) 12.8606 0.428688
\(901\) 0.372560 0.0124118
\(902\) −12.3772 −0.412117
\(903\) −1.28589 −0.0427917
\(904\) 18.3107 0.609006
\(905\) −65.6151 −2.18112
\(906\) −4.42076 −0.146870
\(907\) −40.6484 −1.34971 −0.674853 0.737952i \(-0.735793\pi\)
−0.674853 + 0.737952i \(0.735793\pi\)
\(908\) 8.29125 0.275155
\(909\) −0.0690802 −0.00229125
\(910\) −1.18912 −0.0394189
\(911\) −14.4664 −0.479292 −0.239646 0.970860i \(-0.577031\pi\)
−0.239646 + 0.970860i \(0.577031\pi\)
\(912\) 1.00000 0.0331133
\(913\) −37.0555 −1.22636
\(914\) −21.8596 −0.723052
\(915\) −53.6710 −1.77431
\(916\) 6.24366 0.206296
\(917\) 0.306155 0.0101101
\(918\) −0.372560 −0.0122963
\(919\) 27.3628 0.902616 0.451308 0.892368i \(-0.350958\pi\)
0.451308 + 0.892368i \(0.350958\pi\)
\(920\) −15.1536 −0.499598
\(921\) 1.13527 0.0374085
\(922\) 5.50751 0.181380
\(923\) −27.9586 −0.920268
\(924\) 0.697641 0.0229507
\(925\) 19.1424 0.629397
\(926\) −6.95025 −0.228399
\(927\) −8.84739 −0.290587
\(928\) 9.28951 0.304943
\(929\) −41.5793 −1.36417 −0.682087 0.731271i \(-0.738927\pi\)
−0.682087 + 0.731271i \(0.738927\pi\)
\(930\) −17.4871 −0.573424
\(931\) −6.97354 −0.228548
\(932\) 3.39537 0.111219
\(933\) 11.1763 0.365897
\(934\) 16.6192 0.543797
\(935\) −6.75250 −0.220830
\(936\) −1.72968 −0.0565363
\(937\) 35.9521 1.17450 0.587252 0.809404i \(-0.300210\pi\)
0.587252 + 0.809404i \(0.300210\pi\)
\(938\) −2.03973 −0.0665997
\(939\) 30.5985 0.998545
\(940\) −30.1202 −0.982412
\(941\) 23.6335 0.770430 0.385215 0.922827i \(-0.374127\pi\)
0.385215 + 0.922827i \(0.374127\pi\)
\(942\) −10.0984 −0.329023
\(943\) 10.3483 0.336988
\(944\) −4.09719 −0.133352
\(945\) 0.687480 0.0223637
\(946\) −33.9009 −1.10221
\(947\) −31.8694 −1.03562 −0.517808 0.855497i \(-0.673252\pi\)
−0.517808 + 0.855497i \(0.673252\pi\)
\(948\) −7.40316 −0.240444
\(949\) −26.0429 −0.845389
\(950\) 12.8606 0.417254
\(951\) 30.7369 0.996713
\(952\) −0.0606049 −0.00196422
\(953\) 22.8148 0.739045 0.369523 0.929222i \(-0.379521\pi\)
0.369523 + 0.929222i \(0.379521\pi\)
\(954\) −1.00000 −0.0323762
\(955\) −49.7083 −1.60852
\(956\) 22.0413 0.712868
\(957\) 39.8394 1.28783
\(958\) 10.4381 0.337241
\(959\) −2.18652 −0.0706063
\(960\) 4.22619 0.136400
\(961\) −13.8787 −0.447699
\(962\) −2.57453 −0.0830061
\(963\) 5.16527 0.166448
\(964\) −15.7486 −0.507229
\(965\) −82.7638 −2.66426
\(966\) −0.583281 −0.0187668
\(967\) −3.88653 −0.124983 −0.0624913 0.998046i \(-0.519905\pi\)
−0.0624913 + 0.998046i \(0.519905\pi\)
\(968\) 7.39247 0.237603
\(969\) −0.372560 −0.0119683
\(970\) 30.1204 0.967109
\(971\) −42.7227 −1.37104 −0.685519 0.728055i \(-0.740425\pi\)
−0.685519 + 0.728055i \(0.740425\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0.282702 0.00906300
\(974\) −2.99753 −0.0960471
\(975\) −22.2448 −0.712402
\(976\) −12.6996 −0.406506
\(977\) −10.0421 −0.321277 −0.160638 0.987013i \(-0.551355\pi\)
−0.160638 + 0.987013i \(0.551355\pi\)
\(978\) −7.51964 −0.240452
\(979\) 15.1866 0.485367
\(980\) −29.4715 −0.941431
\(981\) −7.17230 −0.228994
\(982\) −25.7095 −0.820424
\(983\) −23.5438 −0.750930 −0.375465 0.926837i \(-0.622517\pi\)
−0.375465 + 0.926837i \(0.622517\pi\)
\(984\) −2.88605 −0.0920039
\(985\) 49.5131 1.57762
\(986\) −3.46090 −0.110218
\(987\) −1.15937 −0.0369031
\(988\) −1.72968 −0.0550284
\(989\) 28.3438 0.901279
\(990\) 18.1246 0.576038
\(991\) −5.93370 −0.188490 −0.0942451 0.995549i \(-0.530044\pi\)
−0.0942451 + 0.995549i \(0.530044\pi\)
\(992\) −4.13779 −0.131375
\(993\) −21.4262 −0.679939
\(994\) 2.62943 0.0834005
\(995\) −15.9886 −0.506874
\(996\) −8.64039 −0.273781
\(997\) −28.3726 −0.898571 −0.449285 0.893388i \(-0.648321\pi\)
−0.449285 + 0.893388i \(0.648321\pi\)
\(998\) −38.6690 −1.22405
\(999\) 1.48845 0.0470923
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.bh.1.13 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.bh.1.13 13 1.1 even 1 trivial