Properties

Label 8047.2.a.e.1.18
Level $8047$
Weight $2$
Character 8047.1
Self dual yes
Analytic conductor $64.256$
Analytic rank $0$
Dimension $168$
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] = [8047,2,Mod(1,8047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8047, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8047 = 13 \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8047.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: \(64.2556185065\)
Analytic rank: \(0\)
Dimension: \(168\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 8047.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.38728 q^{2} +1.74228 q^{3} +3.69910 q^{4} -3.20056 q^{5} -4.15931 q^{6} -0.986874 q^{7} -4.05622 q^{8} +0.0355448 q^{9} +O(q^{10})\) \(q-2.38728 q^{2} +1.74228 q^{3} +3.69910 q^{4} -3.20056 q^{5} -4.15931 q^{6} -0.986874 q^{7} -4.05622 q^{8} +0.0355448 q^{9} +7.64064 q^{10} -5.86061 q^{11} +6.44487 q^{12} +1.00000 q^{13} +2.35594 q^{14} -5.57628 q^{15} +2.28513 q^{16} +6.77979 q^{17} -0.0848553 q^{18} -7.90095 q^{19} -11.8392 q^{20} -1.71941 q^{21} +13.9909 q^{22} -5.15186 q^{23} -7.06708 q^{24} +5.24361 q^{25} -2.38728 q^{26} -5.16492 q^{27} -3.65055 q^{28} -2.57040 q^{29} +13.3121 q^{30} -10.3692 q^{31} +2.65719 q^{32} -10.2108 q^{33} -16.1852 q^{34} +3.15855 q^{35} +0.131484 q^{36} -9.37540 q^{37} +18.8618 q^{38} +1.74228 q^{39} +12.9822 q^{40} +3.37936 q^{41} +4.10472 q^{42} -9.97466 q^{43} -21.6790 q^{44} -0.113763 q^{45} +12.2989 q^{46} -0.482556 q^{47} +3.98135 q^{48} -6.02608 q^{49} -12.5179 q^{50} +11.8123 q^{51} +3.69910 q^{52} -6.80748 q^{53} +12.3301 q^{54} +18.7573 q^{55} +4.00298 q^{56} -13.7657 q^{57} +6.13627 q^{58} +9.59534 q^{59} -20.6272 q^{60} +7.13348 q^{61} +24.7541 q^{62} -0.0350782 q^{63} -10.9137 q^{64} -3.20056 q^{65} +24.3761 q^{66} +0.362353 q^{67} +25.0791 q^{68} -8.97599 q^{69} -7.54035 q^{70} -10.2078 q^{71} -0.144178 q^{72} +3.39206 q^{73} +22.3817 q^{74} +9.13584 q^{75} -29.2264 q^{76} +5.78368 q^{77} -4.15931 q^{78} +1.78740 q^{79} -7.31372 q^{80} -9.10537 q^{81} -8.06748 q^{82} -2.94940 q^{83} -6.36028 q^{84} -21.6991 q^{85} +23.8123 q^{86} -4.47836 q^{87} +23.7719 q^{88} +13.7505 q^{89} +0.271585 q^{90} -0.986874 q^{91} -19.0572 q^{92} -18.0660 q^{93} +1.15200 q^{94} +25.2875 q^{95} +4.62958 q^{96} -9.15631 q^{97} +14.3859 q^{98} -0.208314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38728 −1.68806 −0.844030 0.536295i \(-0.819823\pi\)
−0.844030 + 0.536295i \(0.819823\pi\)
\(3\) 1.74228 1.00591 0.502953 0.864314i \(-0.332247\pi\)
0.502953 + 0.864314i \(0.332247\pi\)
\(4\) 3.69910 1.84955
\(5\) −3.20056 −1.43134 −0.715668 0.698441i \(-0.753877\pi\)
−0.715668 + 0.698441i \(0.753877\pi\)
\(6\) −4.15931 −1.69803
\(7\) −0.986874 −0.373003 −0.186502 0.982455i \(-0.559715\pi\)
−0.186502 + 0.982455i \(0.559715\pi\)
\(8\) −4.05622 −1.43409
\(9\) 0.0355448 0.0118483
\(10\) 7.64064 2.41618
\(11\) −5.86061 −1.76704 −0.883520 0.468393i \(-0.844833\pi\)
−0.883520 + 0.468393i \(0.844833\pi\)
\(12\) 6.44487 1.86047
\(13\) 1.00000 0.277350
\(14\) 2.35594 0.629652
\(15\) −5.57628 −1.43979
\(16\) 2.28513 0.571284
\(17\) 6.77979 1.64434 0.822170 0.569242i \(-0.192763\pi\)
0.822170 + 0.569242i \(0.192763\pi\)
\(18\) −0.0848553 −0.0200006
\(19\) −7.90095 −1.81260 −0.906301 0.422633i \(-0.861106\pi\)
−0.906301 + 0.422633i \(0.861106\pi\)
\(20\) −11.8392 −2.64733
\(21\) −1.71941 −0.375207
\(22\) 13.9909 2.98287
\(23\) −5.15186 −1.07424 −0.537118 0.843507i \(-0.680487\pi\)
−0.537118 + 0.843507i \(0.680487\pi\)
\(24\) −7.06708 −1.44256
\(25\) 5.24361 1.04872
\(26\) −2.38728 −0.468184
\(27\) −5.16492 −0.993988
\(28\) −3.65055 −0.689888
\(29\) −2.57040 −0.477312 −0.238656 0.971104i \(-0.576707\pi\)
−0.238656 + 0.971104i \(0.576707\pi\)
\(30\) 13.3121 2.43045
\(31\) −10.3692 −1.86236 −0.931180 0.364560i \(-0.881219\pi\)
−0.931180 + 0.364560i \(0.881219\pi\)
\(32\) 2.65719 0.469730
\(33\) −10.2108 −1.77748
\(34\) −16.1852 −2.77575
\(35\) 3.15855 0.533893
\(36\) 0.131484 0.0219139
\(37\) −9.37540 −1.54131 −0.770653 0.637255i \(-0.780070\pi\)
−0.770653 + 0.637255i \(0.780070\pi\)
\(38\) 18.8618 3.05978
\(39\) 1.74228 0.278988
\(40\) 12.9822 2.05267
\(41\) 3.37936 0.527768 0.263884 0.964554i \(-0.414996\pi\)
0.263884 + 0.964554i \(0.414996\pi\)
\(42\) 4.10472 0.633372
\(43\) −9.97466 −1.52112 −0.760560 0.649267i \(-0.775076\pi\)
−0.760560 + 0.649267i \(0.775076\pi\)
\(44\) −21.6790 −3.26823
\(45\) −0.113763 −0.0169588
\(46\) 12.2989 1.81338
\(47\) −0.482556 −0.0703880 −0.0351940 0.999380i \(-0.511205\pi\)
−0.0351940 + 0.999380i \(0.511205\pi\)
\(48\) 3.98135 0.574658
\(49\) −6.02608 −0.860868
\(50\) −12.5179 −1.77031
\(51\) 11.8123 1.65405
\(52\) 3.69910 0.512973
\(53\) −6.80748 −0.935079 −0.467540 0.883972i \(-0.654859\pi\)
−0.467540 + 0.883972i \(0.654859\pi\)
\(54\) 12.3301 1.67791
\(55\) 18.7573 2.52923
\(56\) 4.00298 0.534921
\(57\) −13.7657 −1.82331
\(58\) 6.13627 0.805731
\(59\) 9.59534 1.24921 0.624604 0.780942i \(-0.285261\pi\)
0.624604 + 0.780942i \(0.285261\pi\)
\(60\) −20.6272 −2.66296
\(61\) 7.13348 0.913348 0.456674 0.889634i \(-0.349041\pi\)
0.456674 + 0.889634i \(0.349041\pi\)
\(62\) 24.7541 3.14378
\(63\) −0.0350782 −0.00441944
\(64\) −10.9137 −1.36422
\(65\) −3.20056 −0.396981
\(66\) 24.3761 3.00049
\(67\) 0.362353 0.0442684 0.0221342 0.999755i \(-0.492954\pi\)
0.0221342 + 0.999755i \(0.492954\pi\)
\(68\) 25.0791 3.04129
\(69\) −8.97599 −1.08058
\(70\) −7.54035 −0.901244
\(71\) −10.2078 −1.21144 −0.605722 0.795676i \(-0.707116\pi\)
−0.605722 + 0.795676i \(0.707116\pi\)
\(72\) −0.144178 −0.0169915
\(73\) 3.39206 0.397011 0.198505 0.980100i \(-0.436391\pi\)
0.198505 + 0.980100i \(0.436391\pi\)
\(74\) 22.3817 2.60182
\(75\) 9.13584 1.05492
\(76\) −29.2264 −3.35250
\(77\) 5.78368 0.659112
\(78\) −4.15931 −0.470949
\(79\) 1.78740 0.201098 0.100549 0.994932i \(-0.467940\pi\)
0.100549 + 0.994932i \(0.467940\pi\)
\(80\) −7.31372 −0.817699
\(81\) −9.10537 −1.01171
\(82\) −8.06748 −0.890904
\(83\) −2.94940 −0.323739 −0.161870 0.986812i \(-0.551752\pi\)
−0.161870 + 0.986812i \(0.551752\pi\)
\(84\) −6.36028 −0.693963
\(85\) −21.6991 −2.35360
\(86\) 23.8123 2.56774
\(87\) −4.47836 −0.480131
\(88\) 23.7719 2.53410
\(89\) 13.7505 1.45755 0.728774 0.684754i \(-0.240090\pi\)
0.728774 + 0.684754i \(0.240090\pi\)
\(90\) 0.271585 0.0286275
\(91\) −0.986874 −0.103453
\(92\) −19.0572 −1.98685
\(93\) −18.0660 −1.87336
\(94\) 1.15200 0.118819
\(95\) 25.2875 2.59444
\(96\) 4.62958 0.472504
\(97\) −9.15631 −0.929682 −0.464841 0.885394i \(-0.653889\pi\)
−0.464841 + 0.885394i \(0.653889\pi\)
\(98\) 14.3859 1.45320
\(99\) −0.208314 −0.0209364
\(100\) 19.3966 1.93966
\(101\) 6.91355 0.687923 0.343962 0.938984i \(-0.388231\pi\)
0.343962 + 0.938984i \(0.388231\pi\)
\(102\) −28.1992 −2.79214
\(103\) −10.4093 −1.02566 −0.512830 0.858490i \(-0.671403\pi\)
−0.512830 + 0.858490i \(0.671403\pi\)
\(104\) −4.05622 −0.397745
\(105\) 5.50309 0.537046
\(106\) 16.2513 1.57847
\(107\) 7.13516 0.689782 0.344891 0.938643i \(-0.387916\pi\)
0.344891 + 0.938643i \(0.387916\pi\)
\(108\) −19.1055 −1.83843
\(109\) −5.47913 −0.524806 −0.262403 0.964958i \(-0.584515\pi\)
−0.262403 + 0.964958i \(0.584515\pi\)
\(110\) −44.7788 −4.26949
\(111\) −16.3346 −1.55041
\(112\) −2.25514 −0.213091
\(113\) −2.44436 −0.229946 −0.114973 0.993369i \(-0.536678\pi\)
−0.114973 + 0.993369i \(0.536678\pi\)
\(114\) 32.8625 3.07786
\(115\) 16.4888 1.53759
\(116\) −9.50817 −0.882812
\(117\) 0.0355448 0.00328612
\(118\) −22.9068 −2.10874
\(119\) −6.69080 −0.613344
\(120\) 22.6186 2.06479
\(121\) 23.3467 2.12243
\(122\) −17.0296 −1.54179
\(123\) 5.88780 0.530885
\(124\) −38.3566 −3.44453
\(125\) −0.779675 −0.0697363
\(126\) 0.0837415 0.00746028
\(127\) −9.36298 −0.830830 −0.415415 0.909632i \(-0.636364\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(128\) 20.7397 1.83315
\(129\) −17.3787 −1.53011
\(130\) 7.64064 0.670128
\(131\) 4.00214 0.349668 0.174834 0.984598i \(-0.444061\pi\)
0.174834 + 0.984598i \(0.444061\pi\)
\(132\) −37.7709 −3.28753
\(133\) 7.79724 0.676107
\(134\) −0.865037 −0.0747278
\(135\) 16.5306 1.42273
\(136\) −27.5003 −2.35813
\(137\) −13.4295 −1.14736 −0.573682 0.819078i \(-0.694485\pi\)
−0.573682 + 0.819078i \(0.694485\pi\)
\(138\) 21.4282 1.82409
\(139\) −10.3722 −0.879757 −0.439879 0.898057i \(-0.644979\pi\)
−0.439879 + 0.898057i \(0.644979\pi\)
\(140\) 11.6838 0.987461
\(141\) −0.840748 −0.0708038
\(142\) 24.3689 2.04499
\(143\) −5.86061 −0.490089
\(144\) 0.0812246 0.00676872
\(145\) 8.22674 0.683193
\(146\) −8.09779 −0.670178
\(147\) −10.4991 −0.865953
\(148\) −34.6805 −2.85072
\(149\) 1.33184 0.109108 0.0545542 0.998511i \(-0.482626\pi\)
0.0545542 + 0.998511i \(0.482626\pi\)
\(150\) −21.8098 −1.78076
\(151\) −0.534036 −0.0434592 −0.0217296 0.999764i \(-0.506917\pi\)
−0.0217296 + 0.999764i \(0.506917\pi\)
\(152\) 32.0480 2.59944
\(153\) 0.240986 0.0194826
\(154\) −13.8073 −1.11262
\(155\) 33.1872 2.66566
\(156\) 6.44487 0.516003
\(157\) −5.20413 −0.415335 −0.207667 0.978199i \(-0.566587\pi\)
−0.207667 + 0.978199i \(0.566587\pi\)
\(158\) −4.26702 −0.339466
\(159\) −11.8605 −0.940602
\(160\) −8.50451 −0.672341
\(161\) 5.08424 0.400694
\(162\) 21.7371 1.70782
\(163\) −16.8511 −1.31988 −0.659942 0.751317i \(-0.729419\pi\)
−0.659942 + 0.751317i \(0.729419\pi\)
\(164\) 12.5006 0.976133
\(165\) 32.6804 2.54417
\(166\) 7.04105 0.546491
\(167\) 21.0844 1.63156 0.815782 0.578360i \(-0.196307\pi\)
0.815782 + 0.578360i \(0.196307\pi\)
\(168\) 6.97432 0.538080
\(169\) 1.00000 0.0769231
\(170\) 51.8019 3.97302
\(171\) −0.280837 −0.0214762
\(172\) −36.8972 −2.81339
\(173\) −8.84818 −0.672715 −0.336357 0.941734i \(-0.609195\pi\)
−0.336357 + 0.941734i \(0.609195\pi\)
\(174\) 10.6911 0.810491
\(175\) −5.17478 −0.391177
\(176\) −13.3923 −1.00948
\(177\) 16.7178 1.25659
\(178\) −32.8262 −2.46043
\(179\) −26.3370 −1.96852 −0.984261 0.176721i \(-0.943451\pi\)
−0.984261 + 0.176721i \(0.943451\pi\)
\(180\) −0.420822 −0.0313662
\(181\) −20.7284 −1.54073 −0.770363 0.637605i \(-0.779925\pi\)
−0.770363 + 0.637605i \(0.779925\pi\)
\(182\) 2.35594 0.174634
\(183\) 12.4285 0.918743
\(184\) 20.8971 1.54055
\(185\) 30.0066 2.20613
\(186\) 43.1287 3.16235
\(187\) −39.7337 −2.90561
\(188\) −1.78502 −0.130186
\(189\) 5.09712 0.370761
\(190\) −60.3683 −4.37958
\(191\) −13.1095 −0.948573 −0.474287 0.880370i \(-0.657294\pi\)
−0.474287 + 0.880370i \(0.657294\pi\)
\(192\) −19.0148 −1.37227
\(193\) −1.47866 −0.106436 −0.0532182 0.998583i \(-0.516948\pi\)
−0.0532182 + 0.998583i \(0.516948\pi\)
\(194\) 21.8587 1.56936
\(195\) −5.57628 −0.399326
\(196\) −22.2911 −1.59222
\(197\) −5.75919 −0.410325 −0.205163 0.978728i \(-0.565772\pi\)
−0.205163 + 0.978728i \(0.565772\pi\)
\(198\) 0.497304 0.0353418
\(199\) −9.39868 −0.666255 −0.333127 0.942882i \(-0.608104\pi\)
−0.333127 + 0.942882i \(0.608104\pi\)
\(200\) −21.2692 −1.50396
\(201\) 0.631320 0.0445299
\(202\) −16.5046 −1.16126
\(203\) 2.53666 0.178039
\(204\) 43.6949 3.05925
\(205\) −10.8159 −0.755413
\(206\) 24.8499 1.73138
\(207\) −0.183122 −0.0127278
\(208\) 2.28513 0.158446
\(209\) 46.3044 3.20294
\(210\) −13.1374 −0.906567
\(211\) 13.9639 0.961313 0.480657 0.876909i \(-0.340398\pi\)
0.480657 + 0.876909i \(0.340398\pi\)
\(212\) −25.1815 −1.72947
\(213\) −17.7849 −1.21860
\(214\) −17.0336 −1.16439
\(215\) 31.9245 2.17723
\(216\) 20.9500 1.42547
\(217\) 10.2331 0.694667
\(218\) 13.0802 0.885904
\(219\) 5.90993 0.399356
\(220\) 69.3849 4.67793
\(221\) 6.77979 0.456058
\(222\) 38.9952 2.61719
\(223\) 11.1259 0.745048 0.372524 0.928022i \(-0.378492\pi\)
0.372524 + 0.928022i \(0.378492\pi\)
\(224\) −2.62231 −0.175211
\(225\) 0.186383 0.0124255
\(226\) 5.83538 0.388164
\(227\) −15.1629 −1.00640 −0.503200 0.864170i \(-0.667844\pi\)
−0.503200 + 0.864170i \(0.667844\pi\)
\(228\) −50.9206 −3.37230
\(229\) 9.39748 0.621003 0.310502 0.950573i \(-0.399503\pi\)
0.310502 + 0.950573i \(0.399503\pi\)
\(230\) −39.3635 −2.59555
\(231\) 10.0768 0.663005
\(232\) 10.4261 0.684509
\(233\) −1.80719 −0.118393 −0.0591963 0.998246i \(-0.518854\pi\)
−0.0591963 + 0.998246i \(0.518854\pi\)
\(234\) −0.0848553 −0.00554716
\(235\) 1.54445 0.100749
\(236\) 35.4941 2.31047
\(237\) 3.11415 0.202286
\(238\) 15.9728 1.03536
\(239\) −21.3107 −1.37847 −0.689237 0.724536i \(-0.742054\pi\)
−0.689237 + 0.724536i \(0.742054\pi\)
\(240\) −12.7426 −0.822528
\(241\) −0.897215 −0.0577947 −0.0288973 0.999582i \(-0.509200\pi\)
−0.0288973 + 0.999582i \(0.509200\pi\)
\(242\) −55.7352 −3.58279
\(243\) −0.369373 −0.0236953
\(244\) 26.3874 1.68928
\(245\) 19.2868 1.23219
\(246\) −14.0558 −0.896167
\(247\) −7.90095 −0.502725
\(248\) 42.0597 2.67079
\(249\) −5.13869 −0.325651
\(250\) 1.86130 0.117719
\(251\) 12.2572 0.773667 0.386833 0.922150i \(-0.373569\pi\)
0.386833 + 0.922150i \(0.373569\pi\)
\(252\) −0.129758 −0.00817397
\(253\) 30.1930 1.89822
\(254\) 22.3520 1.40249
\(255\) −37.8060 −2.36750
\(256\) −27.6840 −1.73025
\(257\) −28.1192 −1.75403 −0.877015 0.480463i \(-0.840469\pi\)
−0.877015 + 0.480463i \(0.840469\pi\)
\(258\) 41.4877 2.58291
\(259\) 9.25234 0.574912
\(260\) −11.8392 −0.734236
\(261\) −0.0913644 −0.00565531
\(262\) −9.55421 −0.590261
\(263\) −2.62590 −0.161920 −0.0809600 0.996717i \(-0.525799\pi\)
−0.0809600 + 0.996717i \(0.525799\pi\)
\(264\) 41.4174 2.54906
\(265\) 21.7878 1.33841
\(266\) −18.6142 −1.14131
\(267\) 23.9572 1.46616
\(268\) 1.34038 0.0818766
\(269\) −28.2829 −1.72444 −0.862221 0.506533i \(-0.830927\pi\)
−0.862221 + 0.506533i \(0.830927\pi\)
\(270\) −39.4632 −2.40166
\(271\) 28.1841 1.71206 0.856030 0.516926i \(-0.172924\pi\)
0.856030 + 0.516926i \(0.172924\pi\)
\(272\) 15.4927 0.939385
\(273\) −1.71941 −0.104064
\(274\) 32.0601 1.93682
\(275\) −30.7307 −1.85313
\(276\) −33.2031 −1.99859
\(277\) 23.7606 1.42764 0.713818 0.700331i \(-0.246964\pi\)
0.713818 + 0.700331i \(0.246964\pi\)
\(278\) 24.7613 1.48508
\(279\) −0.368570 −0.0220657
\(280\) −12.8118 −0.765651
\(281\) 6.94305 0.414188 0.207094 0.978321i \(-0.433599\pi\)
0.207094 + 0.978321i \(0.433599\pi\)
\(282\) 2.00710 0.119521
\(283\) 2.70520 0.160807 0.0804036 0.996762i \(-0.474379\pi\)
0.0804036 + 0.996762i \(0.474379\pi\)
\(284\) −37.7597 −2.24062
\(285\) 44.0579 2.60977
\(286\) 13.9909 0.827300
\(287\) −3.33501 −0.196859
\(288\) 0.0944493 0.00556548
\(289\) 28.9655 1.70385
\(290\) −19.6395 −1.15327
\(291\) −15.9529 −0.935174
\(292\) 12.5476 0.734291
\(293\) 4.64742 0.271505 0.135753 0.990743i \(-0.456655\pi\)
0.135753 + 0.990743i \(0.456655\pi\)
\(294\) 25.0643 1.46178
\(295\) −30.7105 −1.78803
\(296\) 38.0287 2.21037
\(297\) 30.2696 1.75642
\(298\) −3.17947 −0.184182
\(299\) −5.15186 −0.297940
\(300\) 33.7944 1.95112
\(301\) 9.84373 0.567383
\(302\) 1.27489 0.0733618
\(303\) 12.0453 0.691987
\(304\) −18.0547 −1.03551
\(305\) −22.8312 −1.30731
\(306\) −0.575301 −0.0328878
\(307\) −7.43769 −0.424492 −0.212246 0.977216i \(-0.568078\pi\)
−0.212246 + 0.977216i \(0.568078\pi\)
\(308\) 21.3944 1.21906
\(309\) −18.1360 −1.03172
\(310\) −79.2271 −4.49980
\(311\) −29.2516 −1.65871 −0.829354 0.558724i \(-0.811291\pi\)
−0.829354 + 0.558724i \(0.811291\pi\)
\(312\) −7.06708 −0.400095
\(313\) 22.0385 1.24569 0.622844 0.782346i \(-0.285977\pi\)
0.622844 + 0.782346i \(0.285977\pi\)
\(314\) 12.4237 0.701111
\(315\) 0.112270 0.00632570
\(316\) 6.61177 0.371941
\(317\) 19.2534 1.08138 0.540691 0.841222i \(-0.318163\pi\)
0.540691 + 0.841222i \(0.318163\pi\)
\(318\) 28.3144 1.58779
\(319\) 15.0641 0.843429
\(320\) 34.9301 1.95265
\(321\) 12.4315 0.693857
\(322\) −12.1375 −0.676396
\(323\) −53.5668 −2.98053
\(324\) −33.6817 −1.87120
\(325\) 5.24361 0.290863
\(326\) 40.2284 2.22804
\(327\) −9.54619 −0.527905
\(328\) −13.7075 −0.756867
\(329\) 0.476222 0.0262550
\(330\) −78.0173 −4.29471
\(331\) 1.78384 0.0980487 0.0490244 0.998798i \(-0.484389\pi\)
0.0490244 + 0.998798i \(0.484389\pi\)
\(332\) −10.9101 −0.598771
\(333\) −0.333246 −0.0182618
\(334\) −50.3345 −2.75418
\(335\) −1.15973 −0.0633630
\(336\) −3.92909 −0.214349
\(337\) −12.8258 −0.698665 −0.349332 0.936999i \(-0.613592\pi\)
−0.349332 + 0.936999i \(0.613592\pi\)
\(338\) −2.38728 −0.129851
\(339\) −4.25877 −0.231305
\(340\) −80.2673 −4.35310
\(341\) 60.7697 3.29086
\(342\) 0.670437 0.0362531
\(343\) 12.8551 0.694110
\(344\) 40.4594 2.18143
\(345\) 28.7282 1.54668
\(346\) 21.1231 1.13558
\(347\) −4.14197 −0.222352 −0.111176 0.993801i \(-0.535462\pi\)
−0.111176 + 0.993801i \(0.535462\pi\)
\(348\) −16.5659 −0.888026
\(349\) −34.5192 −1.84777 −0.923887 0.382667i \(-0.875006\pi\)
−0.923887 + 0.382667i \(0.875006\pi\)
\(350\) 12.3536 0.660330
\(351\) −5.16492 −0.275683
\(352\) −15.5728 −0.830031
\(353\) 25.2462 1.34372 0.671861 0.740677i \(-0.265495\pi\)
0.671861 + 0.740677i \(0.265495\pi\)
\(354\) −39.9100 −2.12119
\(355\) 32.6707 1.73398
\(356\) 50.8644 2.69581
\(357\) −11.6573 −0.616967
\(358\) 62.8738 3.32299
\(359\) 30.3795 1.60337 0.801684 0.597748i \(-0.203938\pi\)
0.801684 + 0.597748i \(0.203938\pi\)
\(360\) 0.461449 0.0243205
\(361\) 43.4250 2.28553
\(362\) 49.4844 2.60084
\(363\) 40.6766 2.13497
\(364\) −3.65055 −0.191341
\(365\) −10.8565 −0.568256
\(366\) −29.6704 −1.55089
\(367\) −34.3821 −1.79473 −0.897365 0.441288i \(-0.854522\pi\)
−0.897365 + 0.441288i \(0.854522\pi\)
\(368\) −11.7727 −0.613694
\(369\) 0.120119 0.00625313
\(370\) −71.6340 −3.72407
\(371\) 6.71812 0.348788
\(372\) −66.8280 −3.46487
\(373\) 36.4825 1.88899 0.944496 0.328524i \(-0.106551\pi\)
0.944496 + 0.328524i \(0.106551\pi\)
\(374\) 94.8554 4.90485
\(375\) −1.35841 −0.0701482
\(376\) 1.95735 0.100943
\(377\) −2.57040 −0.132382
\(378\) −12.1682 −0.625867
\(379\) 13.5357 0.695281 0.347640 0.937628i \(-0.386983\pi\)
0.347640 + 0.937628i \(0.386983\pi\)
\(380\) 93.5409 4.79855
\(381\) −16.3129 −0.835737
\(382\) 31.2961 1.60125
\(383\) 3.87031 0.197764 0.0988819 0.995099i \(-0.468473\pi\)
0.0988819 + 0.995099i \(0.468473\pi\)
\(384\) 36.1344 1.84398
\(385\) −18.5110 −0.943410
\(386\) 3.52998 0.179671
\(387\) −0.354547 −0.0180226
\(388\) −33.8701 −1.71949
\(389\) 26.4002 1.33854 0.669272 0.743018i \(-0.266606\pi\)
0.669272 + 0.743018i \(0.266606\pi\)
\(390\) 13.3121 0.674086
\(391\) −34.9285 −1.76641
\(392\) 24.4431 1.23456
\(393\) 6.97285 0.351734
\(394\) 13.7488 0.692654
\(395\) −5.72069 −0.287839
\(396\) −0.770574 −0.0387228
\(397\) −6.43157 −0.322791 −0.161396 0.986890i \(-0.551599\pi\)
−0.161396 + 0.986890i \(0.551599\pi\)
\(398\) 22.4373 1.12468
\(399\) 13.5850 0.680100
\(400\) 11.9823 0.599117
\(401\) 9.14685 0.456772 0.228386 0.973571i \(-0.426655\pi\)
0.228386 + 0.973571i \(0.426655\pi\)
\(402\) −1.50714 −0.0751692
\(403\) −10.3692 −0.516526
\(404\) 25.5739 1.27235
\(405\) 29.1423 1.44809
\(406\) −6.05572 −0.300541
\(407\) 54.9455 2.72355
\(408\) −47.9133 −2.37206
\(409\) 8.95387 0.442741 0.221370 0.975190i \(-0.428947\pi\)
0.221370 + 0.975190i \(0.428947\pi\)
\(410\) 25.8205 1.27518
\(411\) −23.3980 −1.15414
\(412\) −38.5051 −1.89701
\(413\) −9.46940 −0.465958
\(414\) 0.437162 0.0214854
\(415\) 9.43975 0.463379
\(416\) 2.65719 0.130280
\(417\) −18.0713 −0.884954
\(418\) −110.541 −5.40676
\(419\) 32.0552 1.56600 0.782999 0.622023i \(-0.213689\pi\)
0.782999 + 0.622023i \(0.213689\pi\)
\(420\) 20.3565 0.993294
\(421\) −34.3602 −1.67461 −0.837306 0.546735i \(-0.815871\pi\)
−0.837306 + 0.546735i \(0.815871\pi\)
\(422\) −33.3357 −1.62276
\(423\) −0.0171523 −0.000833975 0
\(424\) 27.6126 1.34099
\(425\) 35.5505 1.72445
\(426\) 42.4574 2.05707
\(427\) −7.03985 −0.340682
\(428\) 26.3937 1.27579
\(429\) −10.2108 −0.492984
\(430\) −76.2127 −3.67530
\(431\) −31.5625 −1.52031 −0.760155 0.649742i \(-0.774877\pi\)
−0.760155 + 0.649742i \(0.774877\pi\)
\(432\) −11.8025 −0.567849
\(433\) −22.1376 −1.06386 −0.531932 0.846787i \(-0.678534\pi\)
−0.531932 + 0.846787i \(0.678534\pi\)
\(434\) −24.4292 −1.17264
\(435\) 14.3333 0.687229
\(436\) −20.2678 −0.970654
\(437\) 40.7046 1.94716
\(438\) −14.1086 −0.674137
\(439\) 23.0860 1.10183 0.550917 0.834560i \(-0.314278\pi\)
0.550917 + 0.834560i \(0.314278\pi\)
\(440\) −76.0836 −3.62714
\(441\) −0.214196 −0.0101998
\(442\) −16.1852 −0.769853
\(443\) −14.8704 −0.706512 −0.353256 0.935527i \(-0.614925\pi\)
−0.353256 + 0.935527i \(0.614925\pi\)
\(444\) −60.4232 −2.86756
\(445\) −44.0093 −2.08624
\(446\) −26.5607 −1.25769
\(447\) 2.32044 0.109753
\(448\) 10.7705 0.508857
\(449\) 6.88856 0.325091 0.162546 0.986701i \(-0.448030\pi\)
0.162546 + 0.986701i \(0.448030\pi\)
\(450\) −0.444948 −0.0209750
\(451\) −19.8051 −0.932587
\(452\) −9.04194 −0.425297
\(453\) −0.930440 −0.0437159
\(454\) 36.1982 1.69886
\(455\) 3.15855 0.148075
\(456\) 55.8366 2.61479
\(457\) 9.74140 0.455683 0.227842 0.973698i \(-0.426833\pi\)
0.227842 + 0.973698i \(0.426833\pi\)
\(458\) −22.4344 −1.04829
\(459\) −35.0170 −1.63445
\(460\) 60.9939 2.84385
\(461\) 33.3934 1.55528 0.777642 0.628707i \(-0.216416\pi\)
0.777642 + 0.628707i \(0.216416\pi\)
\(462\) −24.0561 −1.11919
\(463\) −10.3361 −0.480358 −0.240179 0.970729i \(-0.577206\pi\)
−0.240179 + 0.970729i \(0.577206\pi\)
\(464\) −5.87372 −0.272680
\(465\) 57.8215 2.68141
\(466\) 4.31426 0.199854
\(467\) −1.56932 −0.0726196 −0.0363098 0.999341i \(-0.511560\pi\)
−0.0363098 + 0.999341i \(0.511560\pi\)
\(468\) 0.131484 0.00607783
\(469\) −0.357597 −0.0165123
\(470\) −3.68703 −0.170070
\(471\) −9.06706 −0.417788
\(472\) −38.9208 −1.79148
\(473\) 58.4576 2.68788
\(474\) −7.43435 −0.341471
\(475\) −41.4295 −1.90091
\(476\) −24.7499 −1.13441
\(477\) −0.241970 −0.0110791
\(478\) 50.8746 2.32695
\(479\) −28.7903 −1.31546 −0.657732 0.753252i \(-0.728484\pi\)
−0.657732 + 0.753252i \(0.728484\pi\)
\(480\) −14.8173 −0.676312
\(481\) −9.37540 −0.427481
\(482\) 2.14190 0.0975609
\(483\) 8.85817 0.403061
\(484\) 86.3619 3.92554
\(485\) 29.3054 1.33069
\(486\) 0.881796 0.0399991
\(487\) −24.5738 −1.11355 −0.556774 0.830664i \(-0.687961\pi\)
−0.556774 + 0.830664i \(0.687961\pi\)
\(488\) −28.9350 −1.30982
\(489\) −29.3594 −1.32768
\(490\) −46.0431 −2.08001
\(491\) 26.4205 1.19234 0.596171 0.802857i \(-0.296688\pi\)
0.596171 + 0.802857i \(0.296688\pi\)
\(492\) 21.7796 0.981899
\(493\) −17.4268 −0.784863
\(494\) 18.8618 0.848631
\(495\) 0.666722 0.0299669
\(496\) −23.6950 −1.06394
\(497\) 10.0738 0.451873
\(498\) 12.2675 0.549719
\(499\) 4.27998 0.191598 0.0957991 0.995401i \(-0.469459\pi\)
0.0957991 + 0.995401i \(0.469459\pi\)
\(500\) −2.88410 −0.128981
\(501\) 36.7350 1.64120
\(502\) −29.2613 −1.30600
\(503\) −35.6571 −1.58987 −0.794935 0.606694i \(-0.792495\pi\)
−0.794935 + 0.606694i \(0.792495\pi\)
\(504\) 0.142285 0.00633788
\(505\) −22.1272 −0.984649
\(506\) −72.0792 −3.20431
\(507\) 1.74228 0.0773774
\(508\) −34.6346 −1.53666
\(509\) −15.6121 −0.691995 −0.345998 0.938235i \(-0.612459\pi\)
−0.345998 + 0.938235i \(0.612459\pi\)
\(510\) 90.2535 3.99649
\(511\) −3.34754 −0.148086
\(512\) 24.6101 1.08762
\(513\) 40.8077 1.80171
\(514\) 67.1285 2.96091
\(515\) 33.3157 1.46806
\(516\) −64.2854 −2.83001
\(517\) 2.82807 0.124378
\(518\) −22.0879 −0.970487
\(519\) −15.4160 −0.676688
\(520\) 12.9822 0.569307
\(521\) −4.12333 −0.180647 −0.0903233 0.995913i \(-0.528790\pi\)
−0.0903233 + 0.995913i \(0.528790\pi\)
\(522\) 0.218112 0.00954651
\(523\) −37.7161 −1.64921 −0.824604 0.565710i \(-0.808602\pi\)
−0.824604 + 0.565710i \(0.808602\pi\)
\(524\) 14.8043 0.646729
\(525\) −9.01592 −0.393487
\(526\) 6.26876 0.273331
\(527\) −70.3008 −3.06235
\(528\) −23.3331 −1.01544
\(529\) 3.54164 0.153984
\(530\) −52.0135 −2.25932
\(531\) 0.341064 0.0148009
\(532\) 28.8428 1.25049
\(533\) 3.37936 0.146377
\(534\) −57.1926 −2.47496
\(535\) −22.8365 −0.987310
\(536\) −1.46978 −0.0634850
\(537\) −45.8865 −1.98015
\(538\) 67.5193 2.91096
\(539\) 35.3165 1.52119
\(540\) 61.1485 2.63141
\(541\) 5.49342 0.236181 0.118090 0.993003i \(-0.462323\pi\)
0.118090 + 0.993003i \(0.462323\pi\)
\(542\) −67.2832 −2.89006
\(543\) −36.1146 −1.54983
\(544\) 18.0152 0.772395
\(545\) 17.5363 0.751173
\(546\) 4.10472 0.175666
\(547\) 7.34043 0.313854 0.156927 0.987610i \(-0.449841\pi\)
0.156927 + 0.987610i \(0.449841\pi\)
\(548\) −49.6772 −2.12211
\(549\) 0.253558 0.0108216
\(550\) 73.3628 3.12820
\(551\) 20.3086 0.865176
\(552\) 36.4086 1.54965
\(553\) −1.76394 −0.0750103
\(554\) −56.7232 −2.40994
\(555\) 52.2799 2.21916
\(556\) −38.3677 −1.62715
\(557\) 29.5059 1.25021 0.625103 0.780543i \(-0.285057\pi\)
0.625103 + 0.780543i \(0.285057\pi\)
\(558\) 0.879880 0.0372483
\(559\) −9.97466 −0.421883
\(560\) 7.21772 0.305004
\(561\) −69.2273 −2.92278
\(562\) −16.5750 −0.699174
\(563\) 0.0881931 0.00371690 0.00185845 0.999998i \(-0.499408\pi\)
0.00185845 + 0.999998i \(0.499408\pi\)
\(564\) −3.11001 −0.130955
\(565\) 7.82334 0.329130
\(566\) −6.45806 −0.271452
\(567\) 8.98586 0.377370
\(568\) 41.4051 1.73732
\(569\) 23.1666 0.971193 0.485597 0.874183i \(-0.338602\pi\)
0.485597 + 0.874183i \(0.338602\pi\)
\(570\) −105.179 −4.40544
\(571\) 18.3964 0.769867 0.384934 0.922944i \(-0.374224\pi\)
0.384934 + 0.922944i \(0.374224\pi\)
\(572\) −21.6790 −0.906443
\(573\) −22.8405 −0.954176
\(574\) 7.96159 0.332310
\(575\) −27.0143 −1.12657
\(576\) −0.387926 −0.0161636
\(577\) −19.6062 −0.816215 −0.408108 0.912934i \(-0.633811\pi\)
−0.408108 + 0.912934i \(0.633811\pi\)
\(578\) −69.1488 −2.87621
\(579\) −2.57625 −0.107065
\(580\) 30.4315 1.26360
\(581\) 2.91069 0.120756
\(582\) 38.0839 1.57863
\(583\) 39.8960 1.65232
\(584\) −13.7590 −0.569350
\(585\) −0.113763 −0.00470353
\(586\) −11.0947 −0.458317
\(587\) −26.6217 −1.09879 −0.549397 0.835561i \(-0.685143\pi\)
−0.549397 + 0.835561i \(0.685143\pi\)
\(588\) −38.8373 −1.60162
\(589\) 81.9264 3.37572
\(590\) 73.3145 3.01831
\(591\) −10.0341 −0.412749
\(592\) −21.4240 −0.880523
\(593\) −16.7625 −0.688353 −0.344176 0.938905i \(-0.611842\pi\)
−0.344176 + 0.938905i \(0.611842\pi\)
\(594\) −72.2619 −2.96494
\(595\) 21.4143 0.877901
\(596\) 4.92660 0.201801
\(597\) −16.3752 −0.670190
\(598\) 12.2989 0.502940
\(599\) 15.3183 0.625889 0.312945 0.949771i \(-0.398685\pi\)
0.312945 + 0.949771i \(0.398685\pi\)
\(600\) −37.0570 −1.51285
\(601\) −40.1710 −1.63861 −0.819304 0.573360i \(-0.805640\pi\)
−0.819304 + 0.573360i \(0.805640\pi\)
\(602\) −23.4997 −0.957777
\(603\) 0.0128797 0.000524504 0
\(604\) −1.97545 −0.0803799
\(605\) −74.7227 −3.03791
\(606\) −28.7556 −1.16812
\(607\) −36.4367 −1.47892 −0.739461 0.673199i \(-0.764920\pi\)
−0.739461 + 0.673199i \(0.764920\pi\)
\(608\) −20.9943 −0.851433
\(609\) 4.41958 0.179091
\(610\) 54.5043 2.20682
\(611\) −0.482556 −0.0195221
\(612\) 0.891431 0.0360340
\(613\) 23.1716 0.935894 0.467947 0.883757i \(-0.344994\pi\)
0.467947 + 0.883757i \(0.344994\pi\)
\(614\) 17.7558 0.716568
\(615\) −18.8443 −0.759875
\(616\) −23.4599 −0.945227
\(617\) 21.0966 0.849317 0.424659 0.905354i \(-0.360394\pi\)
0.424659 + 0.905354i \(0.360394\pi\)
\(618\) 43.2956 1.74160
\(619\) −1.00000 −0.0401934
\(620\) 122.763 4.93027
\(621\) 26.6089 1.06778
\(622\) 69.8318 2.80000
\(623\) −13.5700 −0.543671
\(624\) 3.98135 0.159381
\(625\) −23.7226 −0.948905
\(626\) −52.6120 −2.10280
\(627\) 80.6753 3.22186
\(628\) −19.2506 −0.768182
\(629\) −63.5632 −2.53443
\(630\) −0.268020 −0.0106782
\(631\) 8.58974 0.341952 0.170976 0.985275i \(-0.445308\pi\)
0.170976 + 0.985275i \(0.445308\pi\)
\(632\) −7.25009 −0.288393
\(633\) 24.3290 0.966991
\(634\) −45.9633 −1.82544
\(635\) 29.9668 1.18920
\(636\) −43.8733 −1.73969
\(637\) −6.02608 −0.238762
\(638\) −35.9623 −1.42376
\(639\) −0.362834 −0.0143535
\(640\) −66.3788 −2.62385
\(641\) 21.6920 0.856782 0.428391 0.903593i \(-0.359081\pi\)
0.428391 + 0.903593i \(0.359081\pi\)
\(642\) −29.6774 −1.17127
\(643\) 17.5591 0.692462 0.346231 0.938149i \(-0.387461\pi\)
0.346231 + 0.938149i \(0.387461\pi\)
\(644\) 18.8071 0.741103
\(645\) 55.6215 2.19009
\(646\) 127.879 5.03132
\(647\) −3.92753 −0.154407 −0.0772036 0.997015i \(-0.524599\pi\)
−0.0772036 + 0.997015i \(0.524599\pi\)
\(648\) 36.9334 1.45088
\(649\) −56.2346 −2.20740
\(650\) −12.5179 −0.490994
\(651\) 17.8289 0.698770
\(652\) −62.3340 −2.44119
\(653\) 40.3900 1.58058 0.790291 0.612732i \(-0.209929\pi\)
0.790291 + 0.612732i \(0.209929\pi\)
\(654\) 22.7894 0.891137
\(655\) −12.8091 −0.500493
\(656\) 7.72230 0.301505
\(657\) 0.120570 0.00470389
\(658\) −1.13687 −0.0443200
\(659\) −49.9401 −1.94539 −0.972695 0.232087i \(-0.925444\pi\)
−0.972695 + 0.232087i \(0.925444\pi\)
\(660\) 120.888 4.70556
\(661\) −50.8176 −1.97658 −0.988288 0.152599i \(-0.951236\pi\)
−0.988288 + 0.152599i \(0.951236\pi\)
\(662\) −4.25852 −0.165512
\(663\) 11.8123 0.458752
\(664\) 11.9634 0.464271
\(665\) −24.9556 −0.967735
\(666\) 0.795552 0.0308270
\(667\) 13.2424 0.512746
\(668\) 77.9935 3.01766
\(669\) 19.3845 0.749449
\(670\) 2.76861 0.106961
\(671\) −41.8065 −1.61392
\(672\) −4.56881 −0.176246
\(673\) 31.9860 1.23297 0.616485 0.787367i \(-0.288556\pi\)
0.616485 + 0.787367i \(0.288556\pi\)
\(674\) 30.6187 1.17939
\(675\) −27.0828 −1.04242
\(676\) 3.69910 0.142273
\(677\) 2.49332 0.0958262 0.0479131 0.998852i \(-0.484743\pi\)
0.0479131 + 0.998852i \(0.484743\pi\)
\(678\) 10.1669 0.390456
\(679\) 9.03613 0.346775
\(680\) 88.0165 3.37528
\(681\) −26.4181 −1.01234
\(682\) −145.074 −5.55518
\(683\) 7.23096 0.276685 0.138342 0.990384i \(-0.455823\pi\)
0.138342 + 0.990384i \(0.455823\pi\)
\(684\) −1.03885 −0.0397213
\(685\) 42.9821 1.64226
\(686\) −30.6887 −1.17170
\(687\) 16.3731 0.624671
\(688\) −22.7934 −0.868991
\(689\) −6.80748 −0.259344
\(690\) −68.5823 −2.61088
\(691\) −18.3608 −0.698477 −0.349238 0.937034i \(-0.613560\pi\)
−0.349238 + 0.937034i \(0.613560\pi\)
\(692\) −32.7303 −1.24422
\(693\) 0.205580 0.00780933
\(694\) 9.88803 0.375344
\(695\) 33.1968 1.25923
\(696\) 18.1652 0.688552
\(697\) 22.9114 0.867830
\(698\) 82.4071 3.11915
\(699\) −3.14863 −0.119092
\(700\) −19.1420 −0.723500
\(701\) −42.7262 −1.61375 −0.806873 0.590725i \(-0.798842\pi\)
−0.806873 + 0.590725i \(0.798842\pi\)
\(702\) 12.3301 0.465369
\(703\) 74.0745 2.79377
\(704\) 63.9611 2.41062
\(705\) 2.69087 0.101344
\(706\) −60.2698 −2.26828
\(707\) −6.82280 −0.256598
\(708\) 61.8407 2.32412
\(709\) −15.8188 −0.594087 −0.297044 0.954864i \(-0.596001\pi\)
−0.297044 + 0.954864i \(0.596001\pi\)
\(710\) −77.9941 −2.92707
\(711\) 0.0635327 0.00238266
\(712\) −55.7750 −2.09026
\(713\) 53.4206 2.00062
\(714\) 27.8291 1.04148
\(715\) 18.7573 0.701481
\(716\) −97.4233 −3.64088
\(717\) −37.1292 −1.38662
\(718\) −72.5243 −2.70658
\(719\) 20.6869 0.771490 0.385745 0.922605i \(-0.373944\pi\)
0.385745 + 0.922605i \(0.373944\pi\)
\(720\) −0.259964 −0.00968830
\(721\) 10.2727 0.382575
\(722\) −103.668 −3.85811
\(723\) −1.56320 −0.0581360
\(724\) −76.6762 −2.84965
\(725\) −13.4782 −0.500567
\(726\) −97.1064 −3.60396
\(727\) 25.0977 0.930822 0.465411 0.885095i \(-0.345906\pi\)
0.465411 + 0.885095i \(0.345906\pi\)
\(728\) 4.00298 0.148360
\(729\) 26.6726 0.987873
\(730\) 25.9175 0.959250
\(731\) −67.6260 −2.50124
\(732\) 45.9744 1.69926
\(733\) 19.4550 0.718587 0.359293 0.933225i \(-0.383018\pi\)
0.359293 + 0.933225i \(0.383018\pi\)
\(734\) 82.0796 3.02962
\(735\) 33.6031 1.23947
\(736\) −13.6895 −0.504601
\(737\) −2.12361 −0.0782241
\(738\) −0.286757 −0.0105557
\(739\) −13.9101 −0.511692 −0.255846 0.966717i \(-0.582354\pi\)
−0.255846 + 0.966717i \(0.582354\pi\)
\(740\) 110.997 4.08034
\(741\) −13.7657 −0.505695
\(742\) −16.0380 −0.588775
\(743\) −24.2200 −0.888545 −0.444272 0.895892i \(-0.646538\pi\)
−0.444272 + 0.895892i \(0.646538\pi\)
\(744\) 73.2798 2.68657
\(745\) −4.26263 −0.156171
\(746\) −87.0939 −3.18873
\(747\) −0.104836 −0.00383574
\(748\) −146.979 −5.37408
\(749\) −7.04151 −0.257291
\(750\) 3.24291 0.118414
\(751\) 26.2924 0.959422 0.479711 0.877427i \(-0.340742\pi\)
0.479711 + 0.877427i \(0.340742\pi\)
\(752\) −1.10271 −0.0402115
\(753\) 21.3555 0.778237
\(754\) 6.13627 0.223470
\(755\) 1.70921 0.0622047
\(756\) 18.8548 0.685741
\(757\) −16.2700 −0.591342 −0.295671 0.955290i \(-0.595543\pi\)
−0.295671 + 0.955290i \(0.595543\pi\)
\(758\) −32.3134 −1.17368
\(759\) 52.6048 1.90943
\(760\) −102.572 −3.72067
\(761\) −8.05102 −0.291849 −0.145925 0.989296i \(-0.546616\pi\)
−0.145925 + 0.989296i \(0.546616\pi\)
\(762\) 38.9435 1.41078
\(763\) 5.40721 0.195754
\(764\) −48.4935 −1.75443
\(765\) −0.771291 −0.0278861
\(766\) −9.23952 −0.333837
\(767\) 9.59534 0.346468
\(768\) −48.2334 −1.74047
\(769\) 23.9975 0.865371 0.432686 0.901545i \(-0.357566\pi\)
0.432686 + 0.901545i \(0.357566\pi\)
\(770\) 44.1910 1.59253
\(771\) −48.9916 −1.76439
\(772\) −5.46972 −0.196860
\(773\) −11.9882 −0.431184 −0.215592 0.976483i \(-0.569168\pi\)
−0.215592 + 0.976483i \(0.569168\pi\)
\(774\) 0.846402 0.0304233
\(775\) −54.3719 −1.95310
\(776\) 37.1400 1.33325
\(777\) 16.1202 0.578308
\(778\) −63.0246 −2.25954
\(779\) −26.7002 −0.956633
\(780\) −20.6272 −0.738573
\(781\) 59.8240 2.14067
\(782\) 83.3841 2.98181
\(783\) 13.2759 0.474442
\(784\) −13.7704 −0.491800
\(785\) 16.6562 0.594484
\(786\) −16.6461 −0.593748
\(787\) 15.5633 0.554773 0.277387 0.960758i \(-0.410532\pi\)
0.277387 + 0.960758i \(0.410532\pi\)
\(788\) −21.3038 −0.758917
\(789\) −4.57506 −0.162876
\(790\) 13.6569 0.485890
\(791\) 2.41228 0.0857708
\(792\) 0.844968 0.0300246
\(793\) 7.13348 0.253317
\(794\) 15.3539 0.544891
\(795\) 37.9604 1.34632
\(796\) −34.7667 −1.23227
\(797\) −47.2689 −1.67435 −0.837174 0.546936i \(-0.815794\pi\)
−0.837174 + 0.546936i \(0.815794\pi\)
\(798\) −32.4312 −1.14805
\(799\) −3.27163 −0.115742
\(800\) 13.9333 0.492615
\(801\) 0.488758 0.0172694
\(802\) −21.8361 −0.771058
\(803\) −19.8795 −0.701534
\(804\) 2.33532 0.0823603
\(805\) −16.2724 −0.573527
\(806\) 24.7541 0.871927
\(807\) −49.2768 −1.73463
\(808\) −28.0429 −0.986545
\(809\) 47.1752 1.65859 0.829296 0.558810i \(-0.188742\pi\)
0.829296 + 0.558810i \(0.188742\pi\)
\(810\) −69.5708 −2.44447
\(811\) 37.9727 1.33340 0.666700 0.745326i \(-0.267706\pi\)
0.666700 + 0.745326i \(0.267706\pi\)
\(812\) 9.38337 0.329292
\(813\) 49.1046 1.72217
\(814\) −131.170 −4.59752
\(815\) 53.9331 1.88920
\(816\) 26.9927 0.944933
\(817\) 78.8092 2.75719
\(818\) −21.3754 −0.747373
\(819\) −0.0350782 −0.00122573
\(820\) −40.0090 −1.39717
\(821\) −5.77991 −0.201720 −0.100860 0.994901i \(-0.532159\pi\)
−0.100860 + 0.994901i \(0.532159\pi\)
\(822\) 55.8577 1.94826
\(823\) 43.6664 1.52212 0.761058 0.648684i \(-0.224680\pi\)
0.761058 + 0.648684i \(0.224680\pi\)
\(824\) 42.2225 1.47089
\(825\) −53.5416 −1.86408
\(826\) 22.6061 0.786566
\(827\) −44.4992 −1.54739 −0.773695 0.633559i \(-0.781594\pi\)
−0.773695 + 0.633559i \(0.781594\pi\)
\(828\) −0.677385 −0.0235408
\(829\) 43.0497 1.49518 0.747588 0.664163i \(-0.231212\pi\)
0.747588 + 0.664163i \(0.231212\pi\)
\(830\) −22.5353 −0.782212
\(831\) 41.3977 1.43607
\(832\) −10.9137 −0.378365
\(833\) −40.8555 −1.41556
\(834\) 43.1411 1.49386
\(835\) −67.4821 −2.33531
\(836\) 171.284 5.92400
\(837\) 53.5559 1.85116
\(838\) −76.5246 −2.64350
\(839\) −40.6136 −1.40214 −0.701069 0.713093i \(-0.747294\pi\)
−0.701069 + 0.713093i \(0.747294\pi\)
\(840\) −22.3218 −0.770174
\(841\) −22.3930 −0.772173
\(842\) 82.0273 2.82685
\(843\) 12.0968 0.416634
\(844\) 51.6538 1.77800
\(845\) −3.20056 −0.110103
\(846\) 0.0409474 0.00140780
\(847\) −23.0403 −0.791674
\(848\) −15.5560 −0.534195
\(849\) 4.71321 0.161757
\(850\) −84.8690 −2.91098
\(851\) 48.3007 1.65573
\(852\) −65.7880 −2.25386
\(853\) −3.80426 −0.130255 −0.0651277 0.997877i \(-0.520745\pi\)
−0.0651277 + 0.997877i \(0.520745\pi\)
\(854\) 16.8061 0.575092
\(855\) 0.898838 0.0307396
\(856\) −28.9418 −0.989211
\(857\) 33.3951 1.14075 0.570377 0.821383i \(-0.306797\pi\)
0.570377 + 0.821383i \(0.306797\pi\)
\(858\) 24.3761 0.832186
\(859\) 18.1438 0.619057 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(860\) 118.092 4.02690
\(861\) −5.81052 −0.198022
\(862\) 75.3484 2.56638
\(863\) −10.9303 −0.372072 −0.186036 0.982543i \(-0.559564\pi\)
−0.186036 + 0.982543i \(0.559564\pi\)
\(864\) −13.7242 −0.466906
\(865\) 28.3192 0.962880
\(866\) 52.8485 1.79587
\(867\) 50.4661 1.71392
\(868\) 37.8532 1.28482
\(869\) −10.4753 −0.355349
\(870\) −34.2176 −1.16008
\(871\) 0.362353 0.0122779
\(872\) 22.2246 0.752619
\(873\) −0.325459 −0.0110151
\(874\) −97.1731 −3.28693
\(875\) 0.769442 0.0260119
\(876\) 21.8614 0.738628
\(877\) 5.12585 0.173088 0.0865439 0.996248i \(-0.472418\pi\)
0.0865439 + 0.996248i \(0.472418\pi\)
\(878\) −55.1127 −1.85996
\(879\) 8.09711 0.273109
\(880\) 42.8628 1.44491
\(881\) 12.9170 0.435185 0.217592 0.976040i \(-0.430180\pi\)
0.217592 + 0.976040i \(0.430180\pi\)
\(882\) 0.511345 0.0172179
\(883\) −48.7337 −1.64002 −0.820009 0.572350i \(-0.806032\pi\)
−0.820009 + 0.572350i \(0.806032\pi\)
\(884\) 25.0791 0.843502
\(885\) −53.5063 −1.79860
\(886\) 35.4997 1.19263
\(887\) 47.3372 1.58943 0.794714 0.606983i \(-0.207621\pi\)
0.794714 + 0.606983i \(0.207621\pi\)
\(888\) 66.2567 2.22343
\(889\) 9.24008 0.309902
\(890\) 105.062 3.52170
\(891\) 53.3630 1.78773
\(892\) 41.1560 1.37800
\(893\) 3.81265 0.127585
\(894\) −5.53953 −0.185269
\(895\) 84.2933 2.81762
\(896\) −20.4675 −0.683771
\(897\) −8.97599 −0.299699
\(898\) −16.4449 −0.548774
\(899\) 26.6530 0.888926
\(900\) 0.689448 0.0229816
\(901\) −46.1533 −1.53759
\(902\) 47.2804 1.57426
\(903\) 17.1505 0.570734
\(904\) 9.91488 0.329764
\(905\) 66.3424 2.20530
\(906\) 2.22122 0.0737951
\(907\) 34.3721 1.14131 0.570653 0.821191i \(-0.306690\pi\)
0.570653 + 0.821191i \(0.306690\pi\)
\(908\) −56.0892 −1.86139
\(909\) 0.245740 0.00815070
\(910\) −7.54035 −0.249960
\(911\) −9.54372 −0.316198 −0.158099 0.987423i \(-0.550536\pi\)
−0.158099 + 0.987423i \(0.550536\pi\)
\(912\) −31.4564 −1.04163
\(913\) 17.2853 0.572060
\(914\) −23.2554 −0.769221
\(915\) −39.7783 −1.31503
\(916\) 34.7622 1.14858
\(917\) −3.94960 −0.130427
\(918\) 83.5954 2.75906
\(919\) 15.2581 0.503319 0.251660 0.967816i \(-0.419024\pi\)
0.251660 + 0.967816i \(0.419024\pi\)
\(920\) −66.8824 −2.20505
\(921\) −12.9586 −0.426999
\(922\) −79.7193 −2.62541
\(923\) −10.2078 −0.335994
\(924\) 37.2751 1.22626
\(925\) −49.1609 −1.61640
\(926\) 24.6751 0.810873
\(927\) −0.369997 −0.0121523
\(928\) −6.83005 −0.224207
\(929\) −35.2448 −1.15635 −0.578173 0.815914i \(-0.696234\pi\)
−0.578173 + 0.815914i \(0.696234\pi\)
\(930\) −138.036 −4.52638
\(931\) 47.6117 1.56041
\(932\) −6.68496 −0.218973
\(933\) −50.9646 −1.66850
\(934\) 3.74641 0.122586
\(935\) 127.170 4.15891
\(936\) −0.144178 −0.00471259
\(937\) −15.2078 −0.496817 −0.248408 0.968655i \(-0.579907\pi\)
−0.248408 + 0.968655i \(0.579907\pi\)
\(938\) 0.853683 0.0278737
\(939\) 38.3972 1.25305
\(940\) 5.71308 0.186340
\(941\) −36.1377 −1.17806 −0.589028 0.808113i \(-0.700489\pi\)
−0.589028 + 0.808113i \(0.700489\pi\)
\(942\) 21.6456 0.705252
\(943\) −17.4100 −0.566948
\(944\) 21.9266 0.713652
\(945\) −16.3137 −0.530683
\(946\) −139.554 −4.53731
\(947\) 25.6507 0.833535 0.416767 0.909013i \(-0.363163\pi\)
0.416767 + 0.909013i \(0.363163\pi\)
\(948\) 11.5196 0.374138
\(949\) 3.39206 0.110111
\(950\) 98.9037 3.20886
\(951\) 33.5449 1.08777
\(952\) 27.1394 0.879592
\(953\) −28.7245 −0.930477 −0.465238 0.885185i \(-0.654031\pi\)
−0.465238 + 0.885185i \(0.654031\pi\)
\(954\) 0.577651 0.0187021
\(955\) 41.9579 1.35773
\(956\) −78.8304 −2.54956
\(957\) 26.2459 0.848411
\(958\) 68.7306 2.22058
\(959\) 13.2533 0.427970
\(960\) 60.8580 1.96418
\(961\) 76.5199 2.46838
\(962\) 22.3817 0.721615
\(963\) 0.253618 0.00817272
\(964\) −3.31889 −0.106894
\(965\) 4.73255 0.152346
\(966\) −21.1469 −0.680391
\(967\) 35.4224 1.13911 0.569554 0.821954i \(-0.307116\pi\)
0.569554 + 0.821954i \(0.307116\pi\)
\(968\) −94.6996 −3.04376
\(969\) −93.3284 −2.99814
\(970\) −69.9600 −2.24628
\(971\) −49.1404 −1.57699 −0.788495 0.615041i \(-0.789140\pi\)
−0.788495 + 0.615041i \(0.789140\pi\)
\(972\) −1.36635 −0.0438256
\(973\) 10.2360 0.328152
\(974\) 58.6646 1.87974
\(975\) 9.13584 0.292581
\(976\) 16.3010 0.521781
\(977\) −55.0668 −1.76174 −0.880872 0.473355i \(-0.843043\pi\)
−0.880872 + 0.473355i \(0.843043\pi\)
\(978\) 70.0891 2.24120
\(979\) −80.5862 −2.57555
\(980\) 71.3440 2.27900
\(981\) −0.194755 −0.00621803
\(982\) −63.0732 −2.01275
\(983\) 14.4430 0.460661 0.230331 0.973112i \(-0.426019\pi\)
0.230331 + 0.973112i \(0.426019\pi\)
\(984\) −23.8822 −0.761338
\(985\) 18.4326 0.587313
\(986\) 41.6026 1.32490
\(987\) 0.829713 0.0264100
\(988\) −29.2264 −0.929815
\(989\) 51.3880 1.63404
\(990\) −1.59165 −0.0505860
\(991\) 24.9464 0.792449 0.396224 0.918154i \(-0.370320\pi\)
0.396224 + 0.918154i \(0.370320\pi\)
\(992\) −27.5529 −0.874805
\(993\) 3.10795 0.0986279
\(994\) −24.0490 −0.762788
\(995\) 30.0811 0.953634
\(996\) −19.0085 −0.602308
\(997\) 33.9696 1.07583 0.537914 0.843000i \(-0.319213\pi\)
0.537914 + 0.843000i \(0.319213\pi\)
\(998\) −10.2175 −0.323429
\(999\) 48.4231 1.53204
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 8047.2.a.e.1.18 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8047.2.a.e.1.18 168 1.1 even 1 trivial