Properties

Label 4001.2.a.a.1.29
Level $4001$
Weight $2$
Character 4001.1
Self dual yes
Analytic conductor $31.948$
Analytic rank $1$
Dimension $149$
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] = [4001,2,Mod(1,4001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4001.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: \(31.9481458487\)
Analytic rank: \(1\)
Dimension: \(149\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.29
Character \(\chi\) \(=\) 4001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.85239 q^{2} -0.815947 q^{3} +1.43137 q^{4} -2.54029 q^{5} +1.51146 q^{6} +1.66105 q^{7} +1.05333 q^{8} -2.33423 q^{9} +O(q^{10})\) \(q-1.85239 q^{2} -0.815947 q^{3} +1.43137 q^{4} -2.54029 q^{5} +1.51146 q^{6} +1.66105 q^{7} +1.05333 q^{8} -2.33423 q^{9} +4.70561 q^{10} +1.16811 q^{11} -1.16792 q^{12} -2.37094 q^{13} -3.07693 q^{14} +2.07274 q^{15} -4.81392 q^{16} +3.97585 q^{17} +4.32392 q^{18} +0.254652 q^{19} -3.63608 q^{20} -1.35533 q^{21} -2.16380 q^{22} -4.22841 q^{23} -0.859465 q^{24} +1.45305 q^{25} +4.39192 q^{26} +4.35245 q^{27} +2.37758 q^{28} -1.45058 q^{29} -3.83953 q^{30} +5.30439 q^{31} +6.81062 q^{32} -0.953114 q^{33} -7.36484 q^{34} -4.21955 q^{35} -3.34114 q^{36} -6.57184 q^{37} -0.471717 q^{38} +1.93456 q^{39} -2.67577 q^{40} +0.319902 q^{41} +2.51061 q^{42} +6.89324 q^{43} +1.67199 q^{44} +5.92961 q^{45} +7.83269 q^{46} +3.12577 q^{47} +3.92790 q^{48} -4.24090 q^{49} -2.69163 q^{50} -3.24408 q^{51} -3.39368 q^{52} -3.91108 q^{53} -8.06245 q^{54} -2.96733 q^{55} +1.74964 q^{56} -0.207783 q^{57} +2.68705 q^{58} +11.0273 q^{59} +2.96685 q^{60} -5.60537 q^{61} -9.82583 q^{62} -3.87728 q^{63} -2.98811 q^{64} +6.02287 q^{65} +1.76554 q^{66} -7.74173 q^{67} +5.69089 q^{68} +3.45016 q^{69} +7.81627 q^{70} -5.61399 q^{71} -2.45873 q^{72} +3.59707 q^{73} +12.1737 q^{74} -1.18561 q^{75} +0.364501 q^{76} +1.94029 q^{77} -3.58357 q^{78} +12.1156 q^{79} +12.2287 q^{80} +3.45133 q^{81} -0.592585 q^{82} +4.34313 q^{83} -1.93998 q^{84} -10.0998 q^{85} -12.7690 q^{86} +1.18360 q^{87} +1.23041 q^{88} +4.01262 q^{89} -10.9840 q^{90} -3.93826 q^{91} -6.05241 q^{92} -4.32810 q^{93} -5.79017 q^{94} -0.646890 q^{95} -5.55710 q^{96} +8.84155 q^{97} +7.85582 q^{98} -2.72664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9} - 48 q^{10} - 31 q^{11} - 61 q^{12} - 54 q^{13} - 44 q^{14} - 65 q^{15} + 58 q^{16} - 26 q^{17} - 23 q^{18} - 86 q^{19} - 52 q^{20} - 30 q^{21} - 56 q^{22} - 63 q^{23} - 90 q^{24} + 92 q^{25} - 38 q^{26} - 103 q^{27} - 77 q^{28} - 51 q^{29} - 22 q^{30} - 256 q^{31} - 21 q^{32} - 36 q^{33} - 124 q^{34} - 50 q^{35} + 45 q^{36} - 42 q^{37} - 14 q^{38} - 119 q^{39} - 131 q^{40} - 55 q^{41} - 5 q^{42} - 55 q^{43} - 54 q^{44} - 68 q^{45} - 59 q^{46} - 82 q^{47} - 89 q^{48} + 30 q^{49} + 13 q^{50} - 83 q^{51} - 126 q^{52} - 23 q^{53} - 83 q^{54} - 244 q^{55} - 94 q^{56} - 14 q^{57} - 60 q^{58} - 93 q^{59} - 70 q^{60} - 139 q^{61} - 10 q^{62} - 120 q^{63} - 45 q^{64} - 37 q^{65} - 28 q^{66} - 110 q^{67} - 27 q^{68} - 79 q^{69} - 78 q^{70} - 123 q^{71} - 74 q^{73} - 25 q^{74} - 146 q^{75} - 192 q^{76} - q^{77} + 26 q^{78} - 273 q^{79} - 51 q^{80} + 41 q^{81} - 120 q^{82} - 27 q^{83} - 14 q^{84} - 71 q^{85} + 3 q^{86} - 98 q^{87} - 143 q^{88} - 70 q^{89} - 24 q^{90} - 256 q^{91} - 47 q^{92} + 16 q^{93} - 151 q^{94} - 83 q^{95} - 137 q^{96} - 108 q^{97} + 17 q^{98} - 131 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.85239 −1.30984 −0.654920 0.755698i \(-0.727298\pi\)
−0.654920 + 0.755698i \(0.727298\pi\)
\(3\) −0.815947 −0.471087 −0.235544 0.971864i \(-0.575687\pi\)
−0.235544 + 0.971864i \(0.575687\pi\)
\(4\) 1.43137 0.715683
\(5\) −2.54029 −1.13605 −0.568025 0.823011i \(-0.692292\pi\)
−0.568025 + 0.823011i \(0.692292\pi\)
\(6\) 1.51146 0.617049
\(7\) 1.66105 0.627819 0.313910 0.949453i \(-0.398361\pi\)
0.313910 + 0.949453i \(0.398361\pi\)
\(8\) 1.05333 0.372410
\(9\) −2.33423 −0.778077
\(10\) 4.70561 1.48805
\(11\) 1.16811 0.352198 0.176099 0.984372i \(-0.443652\pi\)
0.176099 + 0.984372i \(0.443652\pi\)
\(12\) −1.16792 −0.337149
\(13\) −2.37094 −0.657581 −0.328790 0.944403i \(-0.606641\pi\)
−0.328790 + 0.944403i \(0.606641\pi\)
\(14\) −3.07693 −0.822343
\(15\) 2.07274 0.535179
\(16\) −4.81392 −1.20348
\(17\) 3.97585 0.964284 0.482142 0.876093i \(-0.339859\pi\)
0.482142 + 0.876093i \(0.339859\pi\)
\(18\) 4.32392 1.01916
\(19\) 0.254652 0.0584213 0.0292106 0.999573i \(-0.490701\pi\)
0.0292106 + 0.999573i \(0.490701\pi\)
\(20\) −3.63608 −0.813052
\(21\) −1.35533 −0.295757
\(22\) −2.16380 −0.461323
\(23\) −4.22841 −0.881685 −0.440842 0.897585i \(-0.645320\pi\)
−0.440842 + 0.897585i \(0.645320\pi\)
\(24\) −0.859465 −0.175437
\(25\) 1.45305 0.290611
\(26\) 4.39192 0.861326
\(27\) 4.35245 0.837629
\(28\) 2.37758 0.449320
\(29\) −1.45058 −0.269366 −0.134683 0.990889i \(-0.543002\pi\)
−0.134683 + 0.990889i \(0.543002\pi\)
\(30\) −3.83953 −0.700999
\(31\) 5.30439 0.952697 0.476348 0.879257i \(-0.341960\pi\)
0.476348 + 0.879257i \(0.341960\pi\)
\(32\) 6.81062 1.20396
\(33\) −0.953114 −0.165916
\(34\) −7.36484 −1.26306
\(35\) −4.21955 −0.713234
\(36\) −3.34114 −0.556857
\(37\) −6.57184 −1.08040 −0.540202 0.841535i \(-0.681652\pi\)
−0.540202 + 0.841535i \(0.681652\pi\)
\(38\) −0.471717 −0.0765225
\(39\) 1.93456 0.309778
\(40\) −2.67577 −0.423076
\(41\) 0.319902 0.0499603 0.0249801 0.999688i \(-0.492048\pi\)
0.0249801 + 0.999688i \(0.492048\pi\)
\(42\) 2.51061 0.387395
\(43\) 6.89324 1.05121 0.525604 0.850729i \(-0.323839\pi\)
0.525604 + 0.850729i \(0.323839\pi\)
\(44\) 1.67199 0.252062
\(45\) 5.92961 0.883935
\(46\) 7.83269 1.15487
\(47\) 3.12577 0.455941 0.227971 0.973668i \(-0.426791\pi\)
0.227971 + 0.973668i \(0.426791\pi\)
\(48\) 3.92790 0.566944
\(49\) −4.24090 −0.605843
\(50\) −2.69163 −0.380654
\(51\) −3.24408 −0.454262
\(52\) −3.39368 −0.470619
\(53\) −3.91108 −0.537228 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(54\) −8.06245 −1.09716
\(55\) −2.96733 −0.400115
\(56\) 1.74964 0.233806
\(57\) −0.207783 −0.0275215
\(58\) 2.68705 0.352827
\(59\) 11.0273 1.43563 0.717814 0.696235i \(-0.245143\pi\)
0.717814 + 0.696235i \(0.245143\pi\)
\(60\) 2.96685 0.383018
\(61\) −5.60537 −0.717694 −0.358847 0.933396i \(-0.616830\pi\)
−0.358847 + 0.933396i \(0.616830\pi\)
\(62\) −9.82583 −1.24788
\(63\) −3.87728 −0.488492
\(64\) −2.98811 −0.373513
\(65\) 6.02287 0.747045
\(66\) 1.76554 0.217323
\(67\) −7.74173 −0.945803 −0.472901 0.881115i \(-0.656793\pi\)
−0.472901 + 0.881115i \(0.656793\pi\)
\(68\) 5.69089 0.690122
\(69\) 3.45016 0.415350
\(70\) 7.81627 0.934223
\(71\) −5.61399 −0.666258 −0.333129 0.942881i \(-0.608105\pi\)
−0.333129 + 0.942881i \(0.608105\pi\)
\(72\) −2.45873 −0.289764
\(73\) 3.59707 0.421005 0.210503 0.977593i \(-0.432490\pi\)
0.210503 + 0.977593i \(0.432490\pi\)
\(74\) 12.1737 1.41516
\(75\) −1.18561 −0.136903
\(76\) 0.364501 0.0418111
\(77\) 1.94029 0.221117
\(78\) −3.58357 −0.405759
\(79\) 12.1156 1.36311 0.681555 0.731767i \(-0.261304\pi\)
0.681555 + 0.731767i \(0.261304\pi\)
\(80\) 12.2287 1.36721
\(81\) 3.45133 0.383481
\(82\) −0.592585 −0.0654400
\(83\) 4.34313 0.476721 0.238360 0.971177i \(-0.423390\pi\)
0.238360 + 0.971177i \(0.423390\pi\)
\(84\) −1.93998 −0.211669
\(85\) −10.0998 −1.09548
\(86\) −12.7690 −1.37692
\(87\) 1.18360 0.126895
\(88\) 1.23041 0.131162
\(89\) 4.01262 0.425337 0.212668 0.977124i \(-0.431785\pi\)
0.212668 + 0.977124i \(0.431785\pi\)
\(90\) −10.9840 −1.15781
\(91\) −3.93826 −0.412842
\(92\) −6.05241 −0.631007
\(93\) −4.32810 −0.448803
\(94\) −5.79017 −0.597210
\(95\) −0.646890 −0.0663695
\(96\) −5.55710 −0.567169
\(97\) 8.84155 0.897723 0.448862 0.893601i \(-0.351830\pi\)
0.448862 + 0.893601i \(0.351830\pi\)
\(98\) 7.85582 0.793558
\(99\) −2.72664 −0.274037
\(100\) 2.07985 0.207985
\(101\) 15.5402 1.54631 0.773155 0.634218i \(-0.218678\pi\)
0.773155 + 0.634218i \(0.218678\pi\)
\(102\) 6.00931 0.595011
\(103\) −7.19792 −0.709232 −0.354616 0.935012i \(-0.615388\pi\)
−0.354616 + 0.935012i \(0.615388\pi\)
\(104\) −2.49739 −0.244890
\(105\) 3.44293 0.335995
\(106\) 7.24486 0.703683
\(107\) 6.81066 0.658411 0.329205 0.944258i \(-0.393219\pi\)
0.329205 + 0.944258i \(0.393219\pi\)
\(108\) 6.22995 0.599477
\(109\) −3.83985 −0.367791 −0.183895 0.982946i \(-0.558871\pi\)
−0.183895 + 0.982946i \(0.558871\pi\)
\(110\) 5.49667 0.524087
\(111\) 5.36227 0.508965
\(112\) −7.99618 −0.755568
\(113\) −8.11474 −0.763371 −0.381685 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(114\) 0.384896 0.0360488
\(115\) 10.7414 1.00164
\(116\) −2.07631 −0.192781
\(117\) 5.53432 0.511648
\(118\) −20.4269 −1.88044
\(119\) 6.60409 0.605396
\(120\) 2.18329 0.199306
\(121\) −9.63552 −0.875957
\(122\) 10.3834 0.940065
\(123\) −0.261023 −0.0235356
\(124\) 7.59253 0.681829
\(125\) 9.01026 0.805902
\(126\) 7.18226 0.639846
\(127\) −16.5173 −1.46567 −0.732836 0.680406i \(-0.761804\pi\)
−0.732836 + 0.680406i \(0.761804\pi\)
\(128\) −8.08608 −0.714716
\(129\) −5.62451 −0.495211
\(130\) −11.1567 −0.978510
\(131\) 18.5454 1.62032 0.810161 0.586207i \(-0.199379\pi\)
0.810161 + 0.586207i \(0.199379\pi\)
\(132\) −1.36426 −0.118743
\(133\) 0.422991 0.0366780
\(134\) 14.3407 1.23885
\(135\) −11.0565 −0.951589
\(136\) 4.18789 0.359109
\(137\) 9.78396 0.835900 0.417950 0.908470i \(-0.362749\pi\)
0.417950 + 0.908470i \(0.362749\pi\)
\(138\) −6.39106 −0.544043
\(139\) 21.8853 1.85629 0.928144 0.372221i \(-0.121404\pi\)
0.928144 + 0.372221i \(0.121404\pi\)
\(140\) −6.03972 −0.510450
\(141\) −2.55047 −0.214788
\(142\) 10.3993 0.872692
\(143\) −2.76952 −0.231599
\(144\) 11.2368 0.936401
\(145\) 3.68489 0.306014
\(146\) −6.66319 −0.551450
\(147\) 3.46035 0.285405
\(148\) −9.40672 −0.773227
\(149\) −21.1938 −1.73626 −0.868131 0.496335i \(-0.834679\pi\)
−0.868131 + 0.496335i \(0.834679\pi\)
\(150\) 2.19623 0.179321
\(151\) −2.65216 −0.215829 −0.107915 0.994160i \(-0.534417\pi\)
−0.107915 + 0.994160i \(0.534417\pi\)
\(152\) 0.268234 0.0217567
\(153\) −9.28054 −0.750287
\(154\) −3.59418 −0.289628
\(155\) −13.4747 −1.08231
\(156\) 2.76907 0.221703
\(157\) 6.08920 0.485971 0.242986 0.970030i \(-0.421873\pi\)
0.242986 + 0.970030i \(0.421873\pi\)
\(158\) −22.4429 −1.78546
\(159\) 3.19123 0.253081
\(160\) −17.3009 −1.36776
\(161\) −7.02362 −0.553539
\(162\) −6.39322 −0.502299
\(163\) 11.2215 0.878933 0.439467 0.898259i \(-0.355167\pi\)
0.439467 + 0.898259i \(0.355167\pi\)
\(164\) 0.457897 0.0357557
\(165\) 2.42118 0.188489
\(166\) −8.04520 −0.624429
\(167\) 24.9029 1.92704 0.963522 0.267630i \(-0.0862403\pi\)
0.963522 + 0.267630i \(0.0862403\pi\)
\(168\) −1.42762 −0.110143
\(169\) −7.37864 −0.567588
\(170\) 18.7088 1.43490
\(171\) −0.594417 −0.0454562
\(172\) 9.86675 0.752332
\(173\) 3.26411 0.248166 0.124083 0.992272i \(-0.460401\pi\)
0.124083 + 0.992272i \(0.460401\pi\)
\(174\) −2.19249 −0.166212
\(175\) 2.41360 0.182451
\(176\) −5.62319 −0.423864
\(177\) −8.99766 −0.676306
\(178\) −7.43295 −0.557123
\(179\) −11.9544 −0.893511 −0.446755 0.894656i \(-0.647421\pi\)
−0.446755 + 0.894656i \(0.647421\pi\)
\(180\) 8.48745 0.632617
\(181\) −23.3767 −1.73758 −0.868790 0.495181i \(-0.835102\pi\)
−0.868790 + 0.495181i \(0.835102\pi\)
\(182\) 7.29521 0.540757
\(183\) 4.57368 0.338097
\(184\) −4.45393 −0.328348
\(185\) 16.6944 1.22739
\(186\) 8.01735 0.587861
\(187\) 4.64422 0.339619
\(188\) 4.47413 0.326309
\(189\) 7.22965 0.525880
\(190\) 1.19830 0.0869335
\(191\) −7.39886 −0.535363 −0.267682 0.963507i \(-0.586258\pi\)
−0.267682 + 0.963507i \(0.586258\pi\)
\(192\) 2.43814 0.175957
\(193\) −19.7294 −1.42015 −0.710076 0.704125i \(-0.751339\pi\)
−0.710076 + 0.704125i \(0.751339\pi\)
\(194\) −16.3780 −1.17587
\(195\) −4.91434 −0.351923
\(196\) −6.07028 −0.433592
\(197\) −5.61916 −0.400348 −0.200174 0.979760i \(-0.564151\pi\)
−0.200174 + 0.979760i \(0.564151\pi\)
\(198\) 5.05081 0.358945
\(199\) 19.9656 1.41533 0.707663 0.706550i \(-0.249749\pi\)
0.707663 + 0.706550i \(0.249749\pi\)
\(200\) 1.53055 0.108226
\(201\) 6.31684 0.445555
\(202\) −28.7866 −2.02542
\(203\) −2.40949 −0.169113
\(204\) −4.64346 −0.325108
\(205\) −0.812642 −0.0567574
\(206\) 13.3334 0.928981
\(207\) 9.87009 0.686019
\(208\) 11.4135 0.791386
\(209\) 0.297462 0.0205758
\(210\) −6.37766 −0.440101
\(211\) 25.7972 1.77595 0.887976 0.459889i \(-0.152111\pi\)
0.887976 + 0.459889i \(0.152111\pi\)
\(212\) −5.59819 −0.384485
\(213\) 4.58072 0.313866
\(214\) −12.6160 −0.862414
\(215\) −17.5108 −1.19423
\(216\) 4.58458 0.311941
\(217\) 8.81088 0.598121
\(218\) 7.11291 0.481747
\(219\) −2.93502 −0.198330
\(220\) −4.24734 −0.286355
\(221\) −9.42649 −0.634095
\(222\) −9.93305 −0.666663
\(223\) −11.9595 −0.800869 −0.400434 0.916325i \(-0.631141\pi\)
−0.400434 + 0.916325i \(0.631141\pi\)
\(224\) 11.3128 0.755868
\(225\) −3.39176 −0.226117
\(226\) 15.0317 0.999894
\(227\) 16.2793 1.08049 0.540246 0.841507i \(-0.318331\pi\)
0.540246 + 0.841507i \(0.318331\pi\)
\(228\) −0.297413 −0.0196967
\(229\) −21.2115 −1.40170 −0.700849 0.713310i \(-0.747195\pi\)
−0.700849 + 0.713310i \(0.747195\pi\)
\(230\) −19.8973 −1.31199
\(231\) −1.58317 −0.104165
\(232\) −1.52795 −0.100315
\(233\) −7.51719 −0.492468 −0.246234 0.969210i \(-0.579193\pi\)
−0.246234 + 0.969210i \(0.579193\pi\)
\(234\) −10.2518 −0.670178
\(235\) −7.94036 −0.517972
\(236\) 15.7841 1.02745
\(237\) −9.88568 −0.642144
\(238\) −12.2334 −0.792973
\(239\) −21.6746 −1.40201 −0.701006 0.713155i \(-0.747265\pi\)
−0.701006 + 0.713155i \(0.747265\pi\)
\(240\) −9.97800 −0.644077
\(241\) −20.5138 −1.32141 −0.660705 0.750646i \(-0.729743\pi\)
−0.660705 + 0.750646i \(0.729743\pi\)
\(242\) 17.8488 1.14736
\(243\) −15.8734 −1.01828
\(244\) −8.02334 −0.513642
\(245\) 10.7731 0.688268
\(246\) 0.483517 0.0308279
\(247\) −0.603766 −0.0384167
\(248\) 5.58730 0.354794
\(249\) −3.54377 −0.224577
\(250\) −16.6906 −1.05560
\(251\) 1.31732 0.0831488 0.0415744 0.999135i \(-0.486763\pi\)
0.0415744 + 0.999135i \(0.486763\pi\)
\(252\) −5.54981 −0.349605
\(253\) −4.93924 −0.310528
\(254\) 30.5965 1.91980
\(255\) 8.24089 0.516064
\(256\) 20.9548 1.30968
\(257\) −19.8531 −1.23840 −0.619202 0.785232i \(-0.712544\pi\)
−0.619202 + 0.785232i \(0.712544\pi\)
\(258\) 10.4188 0.648647
\(259\) −10.9162 −0.678299
\(260\) 8.62093 0.534647
\(261\) 3.38599 0.209588
\(262\) −34.3535 −2.12236
\(263\) −10.8515 −0.669135 −0.334567 0.942372i \(-0.608590\pi\)
−0.334567 + 0.942372i \(0.608590\pi\)
\(264\) −1.00395 −0.0617887
\(265\) 9.93526 0.610318
\(266\) −0.783547 −0.0480423
\(267\) −3.27408 −0.200371
\(268\) −11.0812 −0.676895
\(269\) 19.0762 1.16310 0.581548 0.813512i \(-0.302447\pi\)
0.581548 + 0.813512i \(0.302447\pi\)
\(270\) 20.4809 1.24643
\(271\) −22.1187 −1.34362 −0.671808 0.740726i \(-0.734482\pi\)
−0.671808 + 0.740726i \(0.734482\pi\)
\(272\) −19.1394 −1.16050
\(273\) 3.21341 0.194484
\(274\) −18.1237 −1.09490
\(275\) 1.69732 0.102352
\(276\) 4.93844 0.297259
\(277\) −17.6959 −1.06324 −0.531622 0.846982i \(-0.678417\pi\)
−0.531622 + 0.846982i \(0.678417\pi\)
\(278\) −40.5402 −2.43144
\(279\) −12.3817 −0.741272
\(280\) −4.44460 −0.265615
\(281\) −15.1604 −0.904394 −0.452197 0.891918i \(-0.649360\pi\)
−0.452197 + 0.891918i \(0.649360\pi\)
\(282\) 4.72447 0.281338
\(283\) 3.74481 0.222606 0.111303 0.993787i \(-0.464498\pi\)
0.111303 + 0.993787i \(0.464498\pi\)
\(284\) −8.03568 −0.476830
\(285\) 0.527828 0.0312658
\(286\) 5.13024 0.303357
\(287\) 0.531374 0.0313660
\(288\) −15.8976 −0.936772
\(289\) −1.19265 −0.0701559
\(290\) −6.82587 −0.400829
\(291\) −7.21423 −0.422906
\(292\) 5.14872 0.301306
\(293\) 2.25853 0.131945 0.0659723 0.997821i \(-0.478985\pi\)
0.0659723 + 0.997821i \(0.478985\pi\)
\(294\) −6.40993 −0.373835
\(295\) −28.0124 −1.63095
\(296\) −6.92235 −0.402353
\(297\) 5.08413 0.295011
\(298\) 39.2593 2.27423
\(299\) 10.0253 0.579779
\(300\) −1.69705 −0.0979791
\(301\) 11.4500 0.659969
\(302\) 4.91284 0.282702
\(303\) −12.6800 −0.728446
\(304\) −1.22588 −0.0703089
\(305\) 14.2392 0.815337
\(306\) 17.1912 0.982757
\(307\) −23.4159 −1.33642 −0.668208 0.743975i \(-0.732938\pi\)
−0.668208 + 0.743975i \(0.732938\pi\)
\(308\) 2.77727 0.158249
\(309\) 5.87311 0.334110
\(310\) 24.9604 1.41766
\(311\) 6.90315 0.391442 0.195721 0.980660i \(-0.437295\pi\)
0.195721 + 0.980660i \(0.437295\pi\)
\(312\) 2.03774 0.115364
\(313\) −14.0162 −0.792245 −0.396122 0.918198i \(-0.629644\pi\)
−0.396122 + 0.918198i \(0.629644\pi\)
\(314\) −11.2796 −0.636545
\(315\) 9.84941 0.554951
\(316\) 17.3418 0.975555
\(317\) 9.72672 0.546307 0.273154 0.961970i \(-0.411933\pi\)
0.273154 + 0.961970i \(0.411933\pi\)
\(318\) −5.91142 −0.331496
\(319\) −1.69444 −0.0948702
\(320\) 7.59064 0.424330
\(321\) −5.55713 −0.310169
\(322\) 13.0105 0.725048
\(323\) 1.01246 0.0563347
\(324\) 4.94011 0.274451
\(325\) −3.44510 −0.191100
\(326\) −20.7866 −1.15126
\(327\) 3.13311 0.173261
\(328\) 0.336964 0.0186057
\(329\) 5.19208 0.286249
\(330\) −4.48499 −0.246890
\(331\) 31.1716 1.71335 0.856674 0.515859i \(-0.172527\pi\)
0.856674 + 0.515859i \(0.172527\pi\)
\(332\) 6.21662 0.341181
\(333\) 15.3402 0.840638
\(334\) −46.1300 −2.52412
\(335\) 19.6662 1.07448
\(336\) 6.52446 0.355938
\(337\) −4.89621 −0.266713 −0.133357 0.991068i \(-0.542576\pi\)
−0.133357 + 0.991068i \(0.542576\pi\)
\(338\) 13.6682 0.743450
\(339\) 6.62120 0.359614
\(340\) −14.4565 −0.784013
\(341\) 6.19611 0.335538
\(342\) 1.10110 0.0595404
\(343\) −18.6717 −1.00818
\(344\) 7.26088 0.391480
\(345\) −8.76439 −0.471859
\(346\) −6.04642 −0.325058
\(347\) 23.8370 1.27963 0.639817 0.768527i \(-0.279010\pi\)
0.639817 + 0.768527i \(0.279010\pi\)
\(348\) 1.69416 0.0908166
\(349\) −31.6470 −1.69402 −0.847012 0.531574i \(-0.821601\pi\)
−0.847012 + 0.531574i \(0.821601\pi\)
\(350\) −4.47094 −0.238982
\(351\) −10.3194 −0.550809
\(352\) 7.95554 0.424032
\(353\) 6.22861 0.331516 0.165758 0.986166i \(-0.446993\pi\)
0.165758 + 0.986166i \(0.446993\pi\)
\(354\) 16.6672 0.885853
\(355\) 14.2611 0.756903
\(356\) 5.74353 0.304406
\(357\) −5.38859 −0.285194
\(358\) 22.1442 1.17036
\(359\) 22.6052 1.19306 0.596529 0.802591i \(-0.296546\pi\)
0.596529 + 0.802591i \(0.296546\pi\)
\(360\) 6.24587 0.329186
\(361\) −18.9352 −0.996587
\(362\) 43.3030 2.27595
\(363\) 7.86207 0.412652
\(364\) −5.63709 −0.295464
\(365\) −9.13759 −0.478283
\(366\) −8.47227 −0.442853
\(367\) −14.3808 −0.750670 −0.375335 0.926889i \(-0.622472\pi\)
−0.375335 + 0.926889i \(0.622472\pi\)
\(368\) 20.3552 1.06109
\(369\) −0.746725 −0.0388729
\(370\) −30.9246 −1.60769
\(371\) −6.49651 −0.337282
\(372\) −6.19510 −0.321201
\(373\) 6.56346 0.339843 0.169922 0.985458i \(-0.445649\pi\)
0.169922 + 0.985458i \(0.445649\pi\)
\(374\) −8.60293 −0.444847
\(375\) −7.35189 −0.379650
\(376\) 3.29248 0.169797
\(377\) 3.43924 0.177130
\(378\) −13.3922 −0.688819
\(379\) −30.9806 −1.59136 −0.795682 0.605714i \(-0.792888\pi\)
−0.795682 + 0.605714i \(0.792888\pi\)
\(380\) −0.925936 −0.0474995
\(381\) 13.4772 0.690459
\(382\) 13.7056 0.701240
\(383\) −23.4879 −1.20017 −0.600087 0.799935i \(-0.704867\pi\)
−0.600087 + 0.799935i \(0.704867\pi\)
\(384\) 6.59781 0.336693
\(385\) −4.92889 −0.251200
\(386\) 36.5466 1.86017
\(387\) −16.0904 −0.817921
\(388\) 12.6555 0.642485
\(389\) 3.36780 0.170754 0.0853772 0.996349i \(-0.472790\pi\)
0.0853772 + 0.996349i \(0.472790\pi\)
\(390\) 9.10330 0.460963
\(391\) −16.8115 −0.850195
\(392\) −4.46709 −0.225622
\(393\) −15.1321 −0.763313
\(394\) 10.4089 0.524393
\(395\) −30.7771 −1.54856
\(396\) −3.90281 −0.196124
\(397\) 8.88272 0.445811 0.222905 0.974840i \(-0.428446\pi\)
0.222905 + 0.974840i \(0.428446\pi\)
\(398\) −36.9842 −1.85385
\(399\) −0.345138 −0.0172785
\(400\) −6.99489 −0.349744
\(401\) 11.7056 0.584547 0.292274 0.956335i \(-0.405588\pi\)
0.292274 + 0.956335i \(0.405588\pi\)
\(402\) −11.7013 −0.583607
\(403\) −12.5764 −0.626475
\(404\) 22.2437 1.10667
\(405\) −8.76736 −0.435654
\(406\) 4.46333 0.221511
\(407\) −7.67663 −0.380516
\(408\) −3.41710 −0.169172
\(409\) −29.1511 −1.44143 −0.720715 0.693231i \(-0.756186\pi\)
−0.720715 + 0.693231i \(0.756186\pi\)
\(410\) 1.50533 0.0743432
\(411\) −7.98319 −0.393782
\(412\) −10.3029 −0.507585
\(413\) 18.3169 0.901315
\(414\) −18.2833 −0.898575
\(415\) −11.0328 −0.541579
\(416\) −16.1476 −0.791700
\(417\) −17.8573 −0.874473
\(418\) −0.551016 −0.0269511
\(419\) −13.2516 −0.647384 −0.323692 0.946162i \(-0.604924\pi\)
−0.323692 + 0.946162i \(0.604924\pi\)
\(420\) 4.92809 0.240466
\(421\) −17.8129 −0.868145 −0.434073 0.900878i \(-0.642924\pi\)
−0.434073 + 0.900878i \(0.642924\pi\)
\(422\) −47.7866 −2.32622
\(423\) −7.29628 −0.354757
\(424\) −4.11967 −0.200069
\(425\) 5.77712 0.280231
\(426\) −8.48530 −0.411114
\(427\) −9.31082 −0.450582
\(428\) 9.74854 0.471214
\(429\) 2.25978 0.109103
\(430\) 32.4369 1.56425
\(431\) 10.7414 0.517395 0.258697 0.965958i \(-0.416707\pi\)
0.258697 + 0.965958i \(0.416707\pi\)
\(432\) −20.9524 −1.00807
\(433\) 10.4424 0.501829 0.250915 0.968009i \(-0.419269\pi\)
0.250915 + 0.968009i \(0.419269\pi\)
\(434\) −16.3212 −0.783444
\(435\) −3.00667 −0.144159
\(436\) −5.49623 −0.263222
\(437\) −1.07678 −0.0515091
\(438\) 5.43681 0.259781
\(439\) −11.3551 −0.541947 −0.270974 0.962587i \(-0.587346\pi\)
−0.270974 + 0.962587i \(0.587346\pi\)
\(440\) −3.12559 −0.149007
\(441\) 9.89924 0.471393
\(442\) 17.4616 0.830563
\(443\) −13.7769 −0.654562 −0.327281 0.944927i \(-0.606132\pi\)
−0.327281 + 0.944927i \(0.606132\pi\)
\(444\) 7.67538 0.364257
\(445\) −10.1932 −0.483204
\(446\) 22.1538 1.04901
\(447\) 17.2930 0.817931
\(448\) −4.96340 −0.234499
\(449\) −3.68194 −0.173762 −0.0868808 0.996219i \(-0.527690\pi\)
−0.0868808 + 0.996219i \(0.527690\pi\)
\(450\) 6.28288 0.296178
\(451\) 0.373680 0.0175959
\(452\) −11.6152 −0.546332
\(453\) 2.16402 0.101674
\(454\) −30.1556 −1.41527
\(455\) 10.0043 0.469009
\(456\) −0.218865 −0.0102493
\(457\) 18.7280 0.876061 0.438031 0.898960i \(-0.355676\pi\)
0.438031 + 0.898960i \(0.355676\pi\)
\(458\) 39.2921 1.83600
\(459\) 17.3047 0.807712
\(460\) 15.3748 0.716856
\(461\) −2.42289 −0.112845 −0.0564225 0.998407i \(-0.517969\pi\)
−0.0564225 + 0.998407i \(0.517969\pi\)
\(462\) 2.93266 0.136440
\(463\) 25.4090 1.18086 0.590429 0.807089i \(-0.298958\pi\)
0.590429 + 0.807089i \(0.298958\pi\)
\(464\) 6.98299 0.324177
\(465\) 10.9946 0.509863
\(466\) 13.9248 0.645054
\(467\) −29.6416 −1.37165 −0.685825 0.727766i \(-0.740559\pi\)
−0.685825 + 0.727766i \(0.740559\pi\)
\(468\) 7.92164 0.366178
\(469\) −12.8594 −0.593793
\(470\) 14.7087 0.678461
\(471\) −4.96846 −0.228935
\(472\) 11.6154 0.534642
\(473\) 8.05205 0.370234
\(474\) 18.3122 0.841106
\(475\) 0.370023 0.0169778
\(476\) 9.45288 0.433272
\(477\) 9.12936 0.418005
\(478\) 40.1499 1.83641
\(479\) −22.0854 −1.00911 −0.504553 0.863380i \(-0.668343\pi\)
−0.504553 + 0.863380i \(0.668343\pi\)
\(480\) 14.1166 0.644333
\(481\) 15.5815 0.710453
\(482\) 37.9997 1.73084
\(483\) 5.73090 0.260765
\(484\) −13.7920 −0.626907
\(485\) −22.4601 −1.01986
\(486\) 29.4039 1.33379
\(487\) −14.4410 −0.654383 −0.327192 0.944958i \(-0.606102\pi\)
−0.327192 + 0.944958i \(0.606102\pi\)
\(488\) −5.90433 −0.267276
\(489\) −9.15612 −0.414054
\(490\) −19.9560 −0.901522
\(491\) 2.06866 0.0933574 0.0466787 0.998910i \(-0.485136\pi\)
0.0466787 + 0.998910i \(0.485136\pi\)
\(492\) −0.373619 −0.0168441
\(493\) −5.76729 −0.259746
\(494\) 1.11841 0.0503197
\(495\) 6.92643 0.311320
\(496\) −25.5349 −1.14655
\(497\) −9.32514 −0.418290
\(498\) 6.56445 0.294160
\(499\) −6.30743 −0.282359 −0.141180 0.989984i \(-0.545090\pi\)
−0.141180 + 0.989984i \(0.545090\pi\)
\(500\) 12.8970 0.576771
\(501\) −20.3194 −0.907805
\(502\) −2.44021 −0.108912
\(503\) −4.33810 −0.193426 −0.0967132 0.995312i \(-0.530833\pi\)
−0.0967132 + 0.995312i \(0.530833\pi\)
\(504\) −4.08407 −0.181919
\(505\) −39.4766 −1.75669
\(506\) 9.14943 0.406742
\(507\) 6.02058 0.267383
\(508\) −23.6423 −1.04896
\(509\) −13.9230 −0.617126 −0.308563 0.951204i \(-0.599848\pi\)
−0.308563 + 0.951204i \(0.599848\pi\)
\(510\) −15.2654 −0.675962
\(511\) 5.97493 0.264315
\(512\) −22.6445 −1.00075
\(513\) 1.10836 0.0489353
\(514\) 36.7758 1.62211
\(515\) 18.2848 0.805723
\(516\) −8.05074 −0.354414
\(517\) 3.65124 0.160582
\(518\) 20.2211 0.888463
\(519\) −2.66334 −0.116908
\(520\) 6.34409 0.278207
\(521\) −31.2382 −1.36857 −0.684284 0.729215i \(-0.739885\pi\)
−0.684284 + 0.729215i \(0.739885\pi\)
\(522\) −6.27219 −0.274526
\(523\) −3.42781 −0.149888 −0.0749438 0.997188i \(-0.523878\pi\)
−0.0749438 + 0.997188i \(0.523878\pi\)
\(524\) 26.5453 1.15964
\(525\) −1.96937 −0.0859503
\(526\) 20.1013 0.876460
\(527\) 21.0894 0.918671
\(528\) 4.58822 0.199677
\(529\) −5.12053 −0.222632
\(530\) −18.4040 −0.799420
\(531\) −25.7402 −1.11703
\(532\) 0.605455 0.0262498
\(533\) −0.758468 −0.0328529
\(534\) 6.06489 0.262454
\(535\) −17.3010 −0.747988
\(536\) −8.15463 −0.352226
\(537\) 9.75412 0.420921
\(538\) −35.3366 −1.52347
\(539\) −4.95383 −0.213377
\(540\) −15.8258 −0.681036
\(541\) −33.0652 −1.42158 −0.710792 0.703402i \(-0.751663\pi\)
−0.710792 + 0.703402i \(0.751663\pi\)
\(542\) 40.9726 1.75992
\(543\) 19.0742 0.818551
\(544\) 27.0780 1.16096
\(545\) 9.75431 0.417829
\(546\) −5.95250 −0.254744
\(547\) 39.5966 1.69303 0.846514 0.532366i \(-0.178697\pi\)
0.846514 + 0.532366i \(0.178697\pi\)
\(548\) 14.0044 0.598239
\(549\) 13.0842 0.558421
\(550\) −3.14411 −0.134065
\(551\) −0.369394 −0.0157367
\(552\) 3.63417 0.154681
\(553\) 20.1246 0.855787
\(554\) 32.7798 1.39268
\(555\) −13.6217 −0.578209
\(556\) 31.3259 1.32851
\(557\) −16.9161 −0.716760 −0.358380 0.933576i \(-0.616671\pi\)
−0.358380 + 0.933576i \(0.616671\pi\)
\(558\) 22.9357 0.970948
\(559\) −16.3435 −0.691254
\(560\) 20.3126 0.858364
\(561\) −3.78944 −0.159990
\(562\) 28.0831 1.18461
\(563\) 15.9657 0.672873 0.336436 0.941706i \(-0.390778\pi\)
0.336436 + 0.941706i \(0.390778\pi\)
\(564\) −3.65065 −0.153720
\(565\) 20.6138 0.867228
\(566\) −6.93686 −0.291578
\(567\) 5.73284 0.240757
\(568\) −5.91341 −0.248121
\(569\) 6.38371 0.267619 0.133810 0.991007i \(-0.457279\pi\)
0.133810 + 0.991007i \(0.457279\pi\)
\(570\) −0.977745 −0.0409532
\(571\) −9.08461 −0.380179 −0.190090 0.981767i \(-0.560878\pi\)
−0.190090 + 0.981767i \(0.560878\pi\)
\(572\) −3.96419 −0.165751
\(573\) 6.03708 0.252203
\(574\) −0.984315 −0.0410845
\(575\) −6.14411 −0.256227
\(576\) 6.97493 0.290622
\(577\) −10.6019 −0.441361 −0.220681 0.975346i \(-0.570828\pi\)
−0.220681 + 0.975346i \(0.570828\pi\)
\(578\) 2.20926 0.0918930
\(579\) 16.0981 0.669015
\(580\) 5.27443 0.219009
\(581\) 7.21418 0.299295
\(582\) 13.3636 0.553939
\(583\) −4.56857 −0.189211
\(584\) 3.78892 0.156786
\(585\) −14.0588 −0.581258
\(586\) −4.18368 −0.172826
\(587\) 40.7823 1.68326 0.841632 0.540051i \(-0.181595\pi\)
0.841632 + 0.540051i \(0.181595\pi\)
\(588\) 4.95303 0.204259
\(589\) 1.35078 0.0556577
\(590\) 51.8900 2.13628
\(591\) 4.58493 0.188599
\(592\) 31.6364 1.30025
\(593\) −30.0020 −1.23204 −0.616018 0.787732i \(-0.711255\pi\)
−0.616018 + 0.787732i \(0.711255\pi\)
\(594\) −9.41782 −0.386418
\(595\) −16.7763 −0.687761
\(596\) −30.3361 −1.24261
\(597\) −16.2909 −0.666742
\(598\) −18.5708 −0.759418
\(599\) −2.97212 −0.121438 −0.0607189 0.998155i \(-0.519339\pi\)
−0.0607189 + 0.998155i \(0.519339\pi\)
\(600\) −1.24885 −0.0509840
\(601\) −27.8131 −1.13452 −0.567260 0.823539i \(-0.691996\pi\)
−0.567260 + 0.823539i \(0.691996\pi\)
\(602\) −21.2100 −0.864454
\(603\) 18.0710 0.735907
\(604\) −3.79621 −0.154465
\(605\) 24.4770 0.995131
\(606\) 23.4883 0.954149
\(607\) −16.9174 −0.686655 −0.343328 0.939216i \(-0.611554\pi\)
−0.343328 + 0.939216i \(0.611554\pi\)
\(608\) 1.73434 0.0703368
\(609\) 1.96602 0.0796671
\(610\) −26.3767 −1.06796
\(611\) −7.41103 −0.299818
\(612\) −13.2839 −0.536968
\(613\) 13.7662 0.556013 0.278006 0.960579i \(-0.410326\pi\)
0.278006 + 0.960579i \(0.410326\pi\)
\(614\) 43.3755 1.75049
\(615\) 0.663073 0.0267377
\(616\) 2.04377 0.0823460
\(617\) 28.7183 1.15616 0.578078 0.815981i \(-0.303803\pi\)
0.578078 + 0.815981i \(0.303803\pi\)
\(618\) −10.8793 −0.437631
\(619\) 21.8205 0.877041 0.438521 0.898721i \(-0.355503\pi\)
0.438521 + 0.898721i \(0.355503\pi\)
\(620\) −19.2872 −0.774592
\(621\) −18.4039 −0.738525
\(622\) −12.7874 −0.512726
\(623\) 6.66517 0.267035
\(624\) −9.31283 −0.372812
\(625\) −30.1539 −1.20616
\(626\) 25.9636 1.03771
\(627\) −0.242713 −0.00969302
\(628\) 8.71587 0.347801
\(629\) −26.1286 −1.04182
\(630\) −18.2450 −0.726898
\(631\) −48.4687 −1.92951 −0.964755 0.263149i \(-0.915239\pi\)
−0.964755 + 0.263149i \(0.915239\pi\)
\(632\) 12.7618 0.507636
\(633\) −21.0491 −0.836628
\(634\) −18.0177 −0.715576
\(635\) 41.9586 1.66508
\(636\) 4.56782 0.181126
\(637\) 10.0549 0.398391
\(638\) 3.13877 0.124265
\(639\) 13.1044 0.518400
\(640\) 20.5410 0.811953
\(641\) −4.01689 −0.158658 −0.0793288 0.996849i \(-0.525278\pi\)
−0.0793288 + 0.996849i \(0.525278\pi\)
\(642\) 10.2940 0.406272
\(643\) −25.8159 −1.01808 −0.509040 0.860743i \(-0.669999\pi\)
−0.509040 + 0.860743i \(0.669999\pi\)
\(644\) −10.0534 −0.396158
\(645\) 14.2879 0.562584
\(646\) −1.87547 −0.0737895
\(647\) 16.4825 0.647993 0.323996 0.946058i \(-0.394973\pi\)
0.323996 + 0.946058i \(0.394973\pi\)
\(648\) 3.63540 0.142812
\(649\) 12.8810 0.505625
\(650\) 6.38169 0.250311
\(651\) −7.18921 −0.281767
\(652\) 16.0620 0.629038
\(653\) 1.80207 0.0705204 0.0352602 0.999378i \(-0.488774\pi\)
0.0352602 + 0.999378i \(0.488774\pi\)
\(654\) −5.80376 −0.226945
\(655\) −47.1107 −1.84077
\(656\) −1.53998 −0.0601262
\(657\) −8.39639 −0.327574
\(658\) −9.61778 −0.374940
\(659\) −51.0630 −1.98913 −0.994567 0.104103i \(-0.966803\pi\)
−0.994567 + 0.104103i \(0.966803\pi\)
\(660\) 3.46560 0.134898
\(661\) −37.8703 −1.47298 −0.736492 0.676446i \(-0.763519\pi\)
−0.736492 + 0.676446i \(0.763519\pi\)
\(662\) −57.7421 −2.24421
\(663\) 7.69152 0.298714
\(664\) 4.57477 0.177536
\(665\) −1.07452 −0.0416680
\(666\) −28.4161 −1.10110
\(667\) 6.13366 0.237496
\(668\) 35.6452 1.37915
\(669\) 9.75833 0.377279
\(670\) −36.4296 −1.40740
\(671\) −6.54768 −0.252771
\(672\) −9.23064 −0.356080
\(673\) 41.7302 1.60858 0.804291 0.594236i \(-0.202545\pi\)
0.804291 + 0.594236i \(0.202545\pi\)
\(674\) 9.06971 0.349352
\(675\) 6.32434 0.243424
\(676\) −10.5615 −0.406213
\(677\) −17.8698 −0.686793 −0.343396 0.939191i \(-0.611577\pi\)
−0.343396 + 0.939191i \(0.611577\pi\)
\(678\) −12.2651 −0.471037
\(679\) 14.6863 0.563608
\(680\) −10.6384 −0.407966
\(681\) −13.2830 −0.509006
\(682\) −11.4776 −0.439501
\(683\) −50.5914 −1.93583 −0.967913 0.251285i \(-0.919147\pi\)
−0.967913 + 0.251285i \(0.919147\pi\)
\(684\) −0.850829 −0.0325323
\(685\) −24.8540 −0.949624
\(686\) 34.5874 1.32055
\(687\) 17.3075 0.660321
\(688\) −33.1835 −1.26511
\(689\) 9.27294 0.353271
\(690\) 16.2351 0.618060
\(691\) −8.32596 −0.316735 −0.158367 0.987380i \(-0.550623\pi\)
−0.158367 + 0.987380i \(0.550623\pi\)
\(692\) 4.67214 0.177608
\(693\) −4.52909 −0.172046
\(694\) −44.1554 −1.67612
\(695\) −55.5950 −2.10884
\(696\) 1.24672 0.0472569
\(697\) 1.27188 0.0481759
\(698\) 58.6227 2.21890
\(699\) 6.13363 0.231995
\(700\) 3.45474 0.130577
\(701\) −21.3209 −0.805278 −0.402639 0.915359i \(-0.631907\pi\)
−0.402639 + 0.915359i \(0.631907\pi\)
\(702\) 19.1156 0.721472
\(703\) −1.67354 −0.0631186
\(704\) −3.49043 −0.131551
\(705\) 6.47891 0.244010
\(706\) −11.5379 −0.434233
\(707\) 25.8131 0.970803
\(708\) −12.8789 −0.484021
\(709\) 41.0157 1.54038 0.770188 0.637817i \(-0.220163\pi\)
0.770188 + 0.637817i \(0.220163\pi\)
\(710\) −26.4173 −0.991423
\(711\) −28.2806 −1.06060
\(712\) 4.22663 0.158400
\(713\) −22.4292 −0.839978
\(714\) 9.98179 0.373559
\(715\) 7.03536 0.263108
\(716\) −17.1111 −0.639471
\(717\) 17.6853 0.660470
\(718\) −41.8738 −1.56272
\(719\) −33.7158 −1.25739 −0.628694 0.777653i \(-0.716410\pi\)
−0.628694 + 0.777653i \(0.716410\pi\)
\(720\) −28.5447 −1.06380
\(721\) −11.9561 −0.445269
\(722\) 35.0754 1.30537
\(723\) 16.7382 0.622499
\(724\) −33.4607 −1.24356
\(725\) −2.10777 −0.0782807
\(726\) −14.5637 −0.540508
\(727\) 50.8204 1.88483 0.942413 0.334453i \(-0.108551\pi\)
0.942413 + 0.334453i \(0.108551\pi\)
\(728\) −4.14830 −0.153746
\(729\) 2.59790 0.0962185
\(730\) 16.9264 0.626475
\(731\) 27.4064 1.01366
\(732\) 6.54662 0.241970
\(733\) −48.0098 −1.77328 −0.886642 0.462456i \(-0.846968\pi\)
−0.886642 + 0.462456i \(0.846968\pi\)
\(734\) 26.6389 0.983258
\(735\) −8.79028 −0.324234
\(736\) −28.7981 −1.06151
\(737\) −9.04318 −0.333110
\(738\) 1.38323 0.0509174
\(739\) 27.1000 0.996891 0.498446 0.866921i \(-0.333904\pi\)
0.498446 + 0.866921i \(0.333904\pi\)
\(740\) 23.8958 0.878425
\(741\) 0.492641 0.0180976
\(742\) 12.0341 0.441786
\(743\) −20.4902 −0.751711 −0.375856 0.926678i \(-0.622651\pi\)
−0.375856 + 0.926678i \(0.622651\pi\)
\(744\) −4.55894 −0.167139
\(745\) 53.8383 1.97248
\(746\) −12.1581 −0.445140
\(747\) −10.1379 −0.370926
\(748\) 6.64758 0.243060
\(749\) 11.3129 0.413363
\(750\) 13.6186 0.497281
\(751\) 28.4132 1.03681 0.518407 0.855134i \(-0.326525\pi\)
0.518407 + 0.855134i \(0.326525\pi\)
\(752\) −15.0472 −0.548716
\(753\) −1.07487 −0.0391703
\(754\) −6.37083 −0.232012
\(755\) 6.73724 0.245193
\(756\) 10.3483 0.376363
\(757\) 20.7910 0.755663 0.377831 0.925874i \(-0.376670\pi\)
0.377831 + 0.925874i \(0.376670\pi\)
\(758\) 57.3882 2.08443
\(759\) 4.03016 0.146286
\(760\) −0.681391 −0.0247167
\(761\) −28.7206 −1.04112 −0.520560 0.853825i \(-0.674277\pi\)
−0.520560 + 0.853825i \(0.674277\pi\)
\(762\) −24.9651 −0.904391
\(763\) −6.37819 −0.230906
\(764\) −10.5905 −0.383150
\(765\) 23.5752 0.852364
\(766\) 43.5088 1.57204
\(767\) −26.1450 −0.944041
\(768\) −17.0980 −0.616972
\(769\) 13.0742 0.471468 0.235734 0.971818i \(-0.424251\pi\)
0.235734 + 0.971818i \(0.424251\pi\)
\(770\) 9.13026 0.329032
\(771\) 16.1991 0.583396
\(772\) −28.2400 −1.01638
\(773\) 32.9239 1.18419 0.592095 0.805868i \(-0.298301\pi\)
0.592095 + 0.805868i \(0.298301\pi\)
\(774\) 29.8058 1.07135
\(775\) 7.70756 0.276864
\(776\) 9.31310 0.334321
\(777\) 8.90703 0.319538
\(778\) −6.23850 −0.223661
\(779\) 0.0814638 0.00291874
\(780\) −7.03422 −0.251865
\(781\) −6.55775 −0.234655
\(782\) 31.1416 1.11362
\(783\) −6.31358 −0.225629
\(784\) 20.4154 0.729120
\(785\) −15.4683 −0.552088
\(786\) 28.0306 0.999818
\(787\) −22.5117 −0.802454 −0.401227 0.915979i \(-0.631416\pi\)
−0.401227 + 0.915979i \(0.631416\pi\)
\(788\) −8.04307 −0.286523
\(789\) 8.85428 0.315221
\(790\) 57.0113 2.02837
\(791\) −13.4790 −0.479259
\(792\) −2.87206 −0.102054
\(793\) 13.2900 0.471942
\(794\) −16.4543 −0.583941
\(795\) −8.10664 −0.287513
\(796\) 28.5781 1.01293
\(797\) 7.50958 0.266003 0.133002 0.991116i \(-0.457538\pi\)
0.133002 + 0.991116i \(0.457538\pi\)
\(798\) 0.639332 0.0226321
\(799\) 12.4276 0.439657
\(800\) 9.89619 0.349883
\(801\) −9.36638 −0.330945
\(802\) −21.6833 −0.765664
\(803\) 4.20177 0.148277
\(804\) 9.04171 0.318876
\(805\) 17.8420 0.628848
\(806\) 23.2965 0.820583
\(807\) −15.5652 −0.547919
\(808\) 16.3690 0.575861
\(809\) 21.6984 0.762876 0.381438 0.924395i \(-0.375429\pi\)
0.381438 + 0.924395i \(0.375429\pi\)
\(810\) 16.2406 0.570637
\(811\) 21.6926 0.761731 0.380865 0.924631i \(-0.375626\pi\)
0.380865 + 0.924631i \(0.375626\pi\)
\(812\) −3.44887 −0.121032
\(813\) 18.0477 0.632960
\(814\) 14.2201 0.498416
\(815\) −28.5057 −0.998513
\(816\) 15.6167 0.546695
\(817\) 1.75538 0.0614129
\(818\) 53.9994 1.88804
\(819\) 9.19281 0.321223
\(820\) −1.16319 −0.0406203
\(821\) 14.6517 0.511348 0.255674 0.966763i \(-0.417703\pi\)
0.255674 + 0.966763i \(0.417703\pi\)
\(822\) 14.7880 0.515791
\(823\) 28.4594 0.992031 0.496016 0.868314i \(-0.334796\pi\)
0.496016 + 0.868314i \(0.334796\pi\)
\(824\) −7.58181 −0.264125
\(825\) −1.38493 −0.0482169
\(826\) −33.9301 −1.18058
\(827\) 10.8438 0.377075 0.188538 0.982066i \(-0.439625\pi\)
0.188538 + 0.982066i \(0.439625\pi\)
\(828\) 14.1277 0.490972
\(829\) 8.15262 0.283152 0.141576 0.989927i \(-0.454783\pi\)
0.141576 + 0.989927i \(0.454783\pi\)
\(830\) 20.4371 0.709382
\(831\) 14.4389 0.500880
\(832\) 7.08462 0.245615
\(833\) −16.8612 −0.584205
\(834\) 33.0787 1.14542
\(835\) −63.2605 −2.18922
\(836\) 0.425777 0.0147258
\(837\) 23.0871 0.798007
\(838\) 24.5472 0.847971
\(839\) 54.6201 1.88569 0.942847 0.333225i \(-0.108137\pi\)
0.942847 + 0.333225i \(0.108137\pi\)
\(840\) 3.62655 0.125128
\(841\) −26.8958 −0.927442
\(842\) 32.9964 1.13713
\(843\) 12.3701 0.426048
\(844\) 36.9252 1.27102
\(845\) 18.7439 0.644808
\(846\) 13.5156 0.464676
\(847\) −16.0051 −0.549942
\(848\) 18.8276 0.646544
\(849\) −3.05556 −0.104867
\(850\) −10.7015 −0.367058
\(851\) 27.7885 0.952576
\(852\) 6.55669 0.224628
\(853\) −14.6519 −0.501673 −0.250836 0.968029i \(-0.580706\pi\)
−0.250836 + 0.968029i \(0.580706\pi\)
\(854\) 17.2473 0.590191
\(855\) 1.50999 0.0516406
\(856\) 7.17390 0.245199
\(857\) 5.39325 0.184230 0.0921150 0.995748i \(-0.470637\pi\)
0.0921150 + 0.995748i \(0.470637\pi\)
\(858\) −4.18600 −0.142908
\(859\) −2.38156 −0.0812577 −0.0406289 0.999174i \(-0.512936\pi\)
−0.0406289 + 0.999174i \(0.512936\pi\)
\(860\) −25.0644 −0.854687
\(861\) −0.433573 −0.0147761
\(862\) −19.8973 −0.677705
\(863\) 22.2287 0.756675 0.378337 0.925668i \(-0.376496\pi\)
0.378337 + 0.925668i \(0.376496\pi\)
\(864\) 29.6429 1.00847
\(865\) −8.29177 −0.281929
\(866\) −19.3434 −0.657316
\(867\) 0.973138 0.0330495
\(868\) 12.6116 0.428065
\(869\) 14.1523 0.480085
\(870\) 5.56955 0.188825
\(871\) 18.3552 0.621942
\(872\) −4.04464 −0.136969
\(873\) −20.6382 −0.698498
\(874\) 1.99461 0.0674688
\(875\) 14.9665 0.505961
\(876\) −4.20108 −0.141941
\(877\) 8.26649 0.279139 0.139570 0.990212i \(-0.455428\pi\)
0.139570 + 0.990212i \(0.455428\pi\)
\(878\) 21.0340 0.709865
\(879\) −1.84284 −0.0621574
\(880\) 14.2845 0.481530
\(881\) −35.2806 −1.18863 −0.594317 0.804231i \(-0.702578\pi\)
−0.594317 + 0.804231i \(0.702578\pi\)
\(882\) −18.3373 −0.617449
\(883\) 11.3172 0.380856 0.190428 0.981701i \(-0.439012\pi\)
0.190428 + 0.981701i \(0.439012\pi\)
\(884\) −13.4928 −0.453811
\(885\) 22.8566 0.768317
\(886\) 25.5203 0.857372
\(887\) 27.1504 0.911622 0.455811 0.890077i \(-0.349349\pi\)
0.455811 + 0.890077i \(0.349349\pi\)
\(888\) 5.64827 0.189543
\(889\) −27.4361 −0.920177
\(890\) 18.8818 0.632920
\(891\) 4.03153 0.135061
\(892\) −17.1185 −0.573168
\(893\) 0.795986 0.0266366
\(894\) −32.0335 −1.07136
\(895\) 30.3675 1.01507
\(896\) −13.4314 −0.448712
\(897\) −8.18012 −0.273126
\(898\) 6.82041 0.227600
\(899\) −7.69445 −0.256624
\(900\) −4.85485 −0.161828
\(901\) −15.5498 −0.518041
\(902\) −0.692203 −0.0230478
\(903\) −9.34262 −0.310903
\(904\) −8.54753 −0.284287
\(905\) 59.3836 1.97398
\(906\) −4.00862 −0.133177
\(907\) 33.8524 1.12405 0.562025 0.827120i \(-0.310022\pi\)
0.562025 + 0.827120i \(0.310022\pi\)
\(908\) 23.3016 0.773290
\(909\) −36.2745 −1.20315
\(910\) −18.5319 −0.614327
\(911\) 2.77307 0.0918760 0.0459380 0.998944i \(-0.485372\pi\)
0.0459380 + 0.998944i \(0.485372\pi\)
\(912\) 1.00025 0.0331216
\(913\) 5.07325 0.167900
\(914\) −34.6917 −1.14750
\(915\) −11.6185 −0.384095
\(916\) −30.3615 −1.00317
\(917\) 30.8050 1.01727
\(918\) −32.0551 −1.05797
\(919\) 8.74425 0.288446 0.144223 0.989545i \(-0.453932\pi\)
0.144223 + 0.989545i \(0.453932\pi\)
\(920\) 11.3143 0.373020
\(921\) 19.1061 0.629568
\(922\) 4.48814 0.147809
\(923\) 13.3104 0.438119
\(924\) −2.26610 −0.0745493
\(925\) −9.54924 −0.313977
\(926\) −47.0676 −1.54674
\(927\) 16.8016 0.551837
\(928\) −9.87935 −0.324306
\(929\) 26.0141 0.853496 0.426748 0.904371i \(-0.359659\pi\)
0.426748 + 0.904371i \(0.359659\pi\)
\(930\) −20.3664 −0.667839
\(931\) −1.07996 −0.0353941
\(932\) −10.7599 −0.352451
\(933\) −5.63260 −0.184403
\(934\) 54.9080 1.79664
\(935\) −11.7976 −0.385824
\(936\) 5.82949 0.190543
\(937\) 5.81652 0.190017 0.0950087 0.995476i \(-0.469712\pi\)
0.0950087 + 0.995476i \(0.469712\pi\)
\(938\) 23.8207 0.777774
\(939\) 11.4365 0.373216
\(940\) −11.3656 −0.370704
\(941\) 0.883221 0.0287922 0.0143961 0.999896i \(-0.495417\pi\)
0.0143961 + 0.999896i \(0.495417\pi\)
\(942\) 9.20355 0.299868
\(943\) −1.35268 −0.0440492
\(944\) −53.0844 −1.72775
\(945\) −18.3654 −0.597426
\(946\) −14.9156 −0.484947
\(947\) −32.2064 −1.04657 −0.523284 0.852159i \(-0.675293\pi\)
−0.523284 + 0.852159i \(0.675293\pi\)
\(948\) −14.1500 −0.459571
\(949\) −8.52844 −0.276845
\(950\) −0.685429 −0.0222383
\(951\) −7.93649 −0.257358
\(952\) 6.95632 0.225456
\(953\) −22.5641 −0.730924 −0.365462 0.930826i \(-0.619089\pi\)
−0.365462 + 0.930826i \(0.619089\pi\)
\(954\) −16.9112 −0.547520
\(955\) 18.7952 0.608199
\(956\) −31.0243 −1.00340
\(957\) 1.38257 0.0446921
\(958\) 40.9108 1.32177
\(959\) 16.2517 0.524794
\(960\) −6.19356 −0.199896
\(961\) −2.86343 −0.0923687
\(962\) −28.8630 −0.930581
\(963\) −15.8976 −0.512294
\(964\) −29.3628 −0.945711
\(965\) 50.1183 1.61336
\(966\) −10.6159 −0.341561
\(967\) 25.3183 0.814182 0.407091 0.913388i \(-0.366543\pi\)
0.407091 + 0.913388i \(0.366543\pi\)
\(968\) −10.1494 −0.326215
\(969\) −0.826112 −0.0265385
\(970\) 41.6049 1.33585
\(971\) 32.5555 1.04475 0.522377 0.852715i \(-0.325045\pi\)
0.522377 + 0.852715i \(0.325045\pi\)
\(972\) −22.7207 −0.728767
\(973\) 36.3527 1.16541
\(974\) 26.7504 0.857138
\(975\) 2.81102 0.0900247
\(976\) 26.9838 0.863731
\(977\) 30.4021 0.972649 0.486324 0.873778i \(-0.338337\pi\)
0.486324 + 0.873778i \(0.338337\pi\)
\(978\) 16.9608 0.542345
\(979\) 4.68717 0.149803
\(980\) 15.4203 0.492582
\(981\) 8.96309 0.286169
\(982\) −3.83198 −0.122283
\(983\) −51.0478 −1.62817 −0.814086 0.580744i \(-0.802762\pi\)
−0.814086 + 0.580744i \(0.802762\pi\)
\(984\) −0.274944 −0.00876490
\(985\) 14.2743 0.454816
\(986\) 10.6833 0.340225
\(987\) −4.23646 −0.134848
\(988\) −0.864210 −0.0274942
\(989\) −29.1474 −0.926835
\(990\) −12.8305 −0.407780
\(991\) −2.05724 −0.0653503 −0.0326752 0.999466i \(-0.510403\pi\)
−0.0326752 + 0.999466i \(0.510403\pi\)
\(992\) 36.1262 1.14701
\(993\) −25.4344 −0.807136
\(994\) 17.2738 0.547893
\(995\) −50.7184 −1.60788
\(996\) −5.07243 −0.160726
\(997\) −38.5108 −1.21965 −0.609824 0.792537i \(-0.708760\pi\)
−0.609824 + 0.792537i \(0.708760\pi\)
\(998\) 11.6839 0.369846
\(999\) −28.6036 −0.904978
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 4001.2.a.a.1.29 149
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.a.1.29 149 1.1 even 1 trivial