Properties

Label 4009.2.a.f.1.9
Level $4009$
Weight $2$
Character 4009.1
Self dual yes
Analytic conductor $32.012$
Analytic rank $0$
Dimension $83$
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] = [4009,2,Mod(1,4009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, 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("4009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.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: \(32.0120261703\)
Analytic rank: \(0\)
Dimension: \(83\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 4009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.35956 q^{2} -1.58207 q^{3} +3.56753 q^{4} -1.28760 q^{5} +3.73299 q^{6} -1.93582 q^{7} -3.69869 q^{8} -0.497055 q^{9} +O(q^{10})\) \(q-2.35956 q^{2} -1.58207 q^{3} +3.56753 q^{4} -1.28760 q^{5} +3.73299 q^{6} -1.93582 q^{7} -3.69869 q^{8} -0.497055 q^{9} +3.03817 q^{10} -2.13200 q^{11} -5.64408 q^{12} +6.44993 q^{13} +4.56768 q^{14} +2.03707 q^{15} +1.59221 q^{16} +0.557057 q^{17} +1.17283 q^{18} -1.00000 q^{19} -4.59355 q^{20} +3.06260 q^{21} +5.03058 q^{22} +8.18043 q^{23} +5.85158 q^{24} -3.34209 q^{25} -15.2190 q^{26} +5.53259 q^{27} -6.90608 q^{28} +4.49854 q^{29} -4.80659 q^{30} +5.72197 q^{31} +3.64044 q^{32} +3.37297 q^{33} -1.31441 q^{34} +2.49255 q^{35} -1.77326 q^{36} -11.5566 q^{37} +2.35956 q^{38} -10.2042 q^{39} +4.76242 q^{40} -4.83585 q^{41} -7.22639 q^{42} -4.13715 q^{43} -7.60597 q^{44} +0.640007 q^{45} -19.3022 q^{46} -4.61145 q^{47} -2.51899 q^{48} -3.25261 q^{49} +7.88587 q^{50} -0.881303 q^{51} +23.0103 q^{52} +9.03987 q^{53} -13.0545 q^{54} +2.74516 q^{55} +7.15998 q^{56} +1.58207 q^{57} -10.6146 q^{58} +11.5147 q^{59} +7.26731 q^{60} +7.40366 q^{61} -13.5014 q^{62} +0.962207 q^{63} -11.7743 q^{64} -8.30492 q^{65} -7.95873 q^{66} -7.33360 q^{67} +1.98732 q^{68} -12.9420 q^{69} -5.88134 q^{70} -8.42651 q^{71} +1.83845 q^{72} -10.8336 q^{73} +27.2686 q^{74} +5.28742 q^{75} -3.56753 q^{76} +4.12716 q^{77} +24.0775 q^{78} -8.91925 q^{79} -2.05013 q^{80} -7.26177 q^{81} +11.4105 q^{82} +10.7558 q^{83} +10.9259 q^{84} -0.717266 q^{85} +9.76186 q^{86} -7.11701 q^{87} +7.88559 q^{88} +17.6034 q^{89} -1.51014 q^{90} -12.4859 q^{91} +29.1839 q^{92} -9.05256 q^{93} +10.8810 q^{94} +1.28760 q^{95} -5.75944 q^{96} -2.91167 q^{97} +7.67474 q^{98} +1.05972 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.35956 −1.66846 −0.834231 0.551415i \(-0.814088\pi\)
−0.834231 + 0.551415i \(0.814088\pi\)
\(3\) −1.58207 −0.913408 −0.456704 0.889619i \(-0.650970\pi\)
−0.456704 + 0.889619i \(0.650970\pi\)
\(4\) 3.56753 1.78377
\(5\) −1.28760 −0.575832 −0.287916 0.957656i \(-0.592962\pi\)
−0.287916 + 0.957656i \(0.592962\pi\)
\(6\) 3.73299 1.52399
\(7\) −1.93582 −0.731670 −0.365835 0.930680i \(-0.619217\pi\)
−0.365835 + 0.930680i \(0.619217\pi\)
\(8\) −3.69869 −1.30768
\(9\) −0.497055 −0.165685
\(10\) 3.03817 0.960753
\(11\) −2.13200 −0.642822 −0.321411 0.946940i \(-0.604157\pi\)
−0.321411 + 0.946940i \(0.604157\pi\)
\(12\) −5.64408 −1.62931
\(13\) 6.44993 1.78889 0.894444 0.447179i \(-0.147571\pi\)
0.894444 + 0.447179i \(0.147571\pi\)
\(14\) 4.56768 1.22076
\(15\) 2.03707 0.525969
\(16\) 1.59221 0.398053
\(17\) 0.557057 0.135106 0.0675531 0.997716i \(-0.478481\pi\)
0.0675531 + 0.997716i \(0.478481\pi\)
\(18\) 1.17283 0.276439
\(19\) −1.00000 −0.229416
\(20\) −4.59355 −1.02715
\(21\) 3.06260 0.668313
\(22\) 5.03058 1.07252
\(23\) 8.18043 1.70574 0.852869 0.522125i \(-0.174861\pi\)
0.852869 + 0.522125i \(0.174861\pi\)
\(24\) 5.85158 1.19445
\(25\) −3.34209 −0.668418
\(26\) −15.2190 −2.98469
\(27\) 5.53259 1.06475
\(28\) −6.90608 −1.30513
\(29\) 4.49854 0.835358 0.417679 0.908595i \(-0.362844\pi\)
0.417679 + 0.908595i \(0.362844\pi\)
\(30\) −4.80659 −0.877560
\(31\) 5.72197 1.02770 0.513848 0.857881i \(-0.328219\pi\)
0.513848 + 0.857881i \(0.328219\pi\)
\(32\) 3.64044 0.643546
\(33\) 3.37297 0.587159
\(34\) −1.31441 −0.225420
\(35\) 2.49255 0.421319
\(36\) −1.77326 −0.295543
\(37\) −11.5566 −1.89990 −0.949950 0.312402i \(-0.898867\pi\)
−0.949950 + 0.312402i \(0.898867\pi\)
\(38\) 2.35956 0.382771
\(39\) −10.2042 −1.63399
\(40\) 4.76242 0.753005
\(41\) −4.83585 −0.755233 −0.377617 0.925962i \(-0.623256\pi\)
−0.377617 + 0.925962i \(0.623256\pi\)
\(42\) −7.22639 −1.11506
\(43\) −4.13715 −0.630909 −0.315455 0.948941i \(-0.602157\pi\)
−0.315455 + 0.948941i \(0.602157\pi\)
\(44\) −7.60597 −1.14664
\(45\) 0.640007 0.0954066
\(46\) −19.3022 −2.84596
\(47\) −4.61145 −0.672649 −0.336325 0.941746i \(-0.609184\pi\)
−0.336325 + 0.941746i \(0.609184\pi\)
\(48\) −2.51899 −0.363585
\(49\) −3.25261 −0.464659
\(50\) 7.88587 1.11523
\(51\) −0.881303 −0.123407
\(52\) 23.0103 3.19096
\(53\) 9.03987 1.24172 0.620861 0.783921i \(-0.286783\pi\)
0.620861 + 0.783921i \(0.286783\pi\)
\(54\) −13.0545 −1.77649
\(55\) 2.74516 0.370157
\(56\) 7.15998 0.956792
\(57\) 1.58207 0.209550
\(58\) −10.6146 −1.39376
\(59\) 11.5147 1.49908 0.749541 0.661958i \(-0.230274\pi\)
0.749541 + 0.661958i \(0.230274\pi\)
\(60\) 7.26731 0.938206
\(61\) 7.40366 0.947942 0.473971 0.880540i \(-0.342820\pi\)
0.473971 + 0.880540i \(0.342820\pi\)
\(62\) −13.5014 −1.71467
\(63\) 0.962207 0.121227
\(64\) −11.7743 −1.47178
\(65\) −8.30492 −1.03010
\(66\) −7.95873 −0.979652
\(67\) −7.33360 −0.895942 −0.447971 0.894048i \(-0.647853\pi\)
−0.447971 + 0.894048i \(0.647853\pi\)
\(68\) 1.98732 0.240998
\(69\) −12.9420 −1.55804
\(70\) −5.88134 −0.702954
\(71\) −8.42651 −1.00004 −0.500021 0.866013i \(-0.666675\pi\)
−0.500021 + 0.866013i \(0.666675\pi\)
\(72\) 1.83845 0.216663
\(73\) −10.8336 −1.26798 −0.633990 0.773341i \(-0.718584\pi\)
−0.633990 + 0.773341i \(0.718584\pi\)
\(74\) 27.2686 3.16991
\(75\) 5.28742 0.610539
\(76\) −3.56753 −0.409224
\(77\) 4.12716 0.470333
\(78\) 24.0775 2.72624
\(79\) −8.91925 −1.00349 −0.501747 0.865015i \(-0.667309\pi\)
−0.501747 + 0.865015i \(0.667309\pi\)
\(80\) −2.05013 −0.229212
\(81\) −7.26177 −0.806864
\(82\) 11.4105 1.26008
\(83\) 10.7558 1.18061 0.590303 0.807182i \(-0.299008\pi\)
0.590303 + 0.807182i \(0.299008\pi\)
\(84\) 10.9259 1.19211
\(85\) −0.717266 −0.0777984
\(86\) 9.76186 1.05265
\(87\) −7.11701 −0.763023
\(88\) 7.88559 0.840607
\(89\) 17.6034 1.86595 0.932976 0.359938i \(-0.117202\pi\)
0.932976 + 0.359938i \(0.117202\pi\)
\(90\) −1.51014 −0.159182
\(91\) −12.4859 −1.30888
\(92\) 29.1839 3.04264
\(93\) −9.05256 −0.938707
\(94\) 10.8810 1.12229
\(95\) 1.28760 0.132105
\(96\) −5.75944 −0.587820
\(97\) −2.91167 −0.295635 −0.147818 0.989015i \(-0.547225\pi\)
−0.147818 + 0.989015i \(0.547225\pi\)
\(98\) 7.67474 0.775266
\(99\) 1.05972 0.106506
\(100\) −11.9230 −1.19230
\(101\) 7.39490 0.735820 0.367910 0.929861i \(-0.380073\pi\)
0.367910 + 0.929861i \(0.380073\pi\)
\(102\) 2.07949 0.205900
\(103\) 0.219087 0.0215872 0.0107936 0.999942i \(-0.496564\pi\)
0.0107936 + 0.999942i \(0.496564\pi\)
\(104\) −23.8563 −2.33930
\(105\) −3.94339 −0.384836
\(106\) −21.3301 −2.07177
\(107\) −16.4062 −1.58605 −0.793025 0.609189i \(-0.791495\pi\)
−0.793025 + 0.609189i \(0.791495\pi\)
\(108\) 19.7377 1.89926
\(109\) 2.73344 0.261816 0.130908 0.991395i \(-0.458211\pi\)
0.130908 + 0.991395i \(0.458211\pi\)
\(110\) −6.47737 −0.617593
\(111\) 18.2834 1.73538
\(112\) −3.08223 −0.291244
\(113\) −14.2484 −1.34038 −0.670188 0.742191i \(-0.733787\pi\)
−0.670188 + 0.742191i \(0.733787\pi\)
\(114\) −3.73299 −0.349627
\(115\) −10.5331 −0.982218
\(116\) 16.0487 1.49008
\(117\) −3.20597 −0.296392
\(118\) −27.1696 −2.50116
\(119\) −1.07836 −0.0988531
\(120\) −7.53448 −0.687801
\(121\) −6.45459 −0.586781
\(122\) −17.4694 −1.58161
\(123\) 7.65066 0.689837
\(124\) 20.4133 1.83317
\(125\) 10.7413 0.960728
\(126\) −2.27039 −0.202262
\(127\) −16.6999 −1.48187 −0.740937 0.671574i \(-0.765618\pi\)
−0.740937 + 0.671574i \(0.765618\pi\)
\(128\) 20.5012 1.81207
\(129\) 6.54526 0.576278
\(130\) 19.5960 1.71868
\(131\) 10.3795 0.906860 0.453430 0.891292i \(-0.350200\pi\)
0.453430 + 0.891292i \(0.350200\pi\)
\(132\) 12.0332 1.04735
\(133\) 1.93582 0.167857
\(134\) 17.3041 1.49485
\(135\) −7.12375 −0.613115
\(136\) −2.06038 −0.176676
\(137\) 0.978716 0.0836173 0.0418087 0.999126i \(-0.486688\pi\)
0.0418087 + 0.999126i \(0.486688\pi\)
\(138\) 30.5375 2.59952
\(139\) −0.194257 −0.0164766 −0.00823832 0.999966i \(-0.502622\pi\)
−0.00823832 + 0.999966i \(0.502622\pi\)
\(140\) 8.89226 0.751533
\(141\) 7.29564 0.614404
\(142\) 19.8829 1.66853
\(143\) −13.7512 −1.14994
\(144\) −0.791418 −0.0659515
\(145\) −5.79232 −0.481026
\(146\) 25.5626 2.11558
\(147\) 5.14586 0.424424
\(148\) −41.2287 −3.38898
\(149\) −11.1436 −0.912917 −0.456459 0.889745i \(-0.650882\pi\)
−0.456459 + 0.889745i \(0.650882\pi\)
\(150\) −12.4760 −1.01866
\(151\) 15.5950 1.26911 0.634553 0.772880i \(-0.281184\pi\)
0.634553 + 0.772880i \(0.281184\pi\)
\(152\) 3.69869 0.300003
\(153\) −0.276888 −0.0223851
\(154\) −9.73828 −0.784733
\(155\) −7.36761 −0.591780
\(156\) −36.4039 −2.91465
\(157\) −8.67751 −0.692541 −0.346270 0.938135i \(-0.612552\pi\)
−0.346270 + 0.938135i \(0.612552\pi\)
\(158\) 21.0455 1.67429
\(159\) −14.3017 −1.13420
\(160\) −4.68743 −0.370574
\(161\) −15.8358 −1.24804
\(162\) 17.1346 1.34622
\(163\) 17.8356 1.39699 0.698495 0.715615i \(-0.253853\pi\)
0.698495 + 0.715615i \(0.253853\pi\)
\(164\) −17.2521 −1.34716
\(165\) −4.34303 −0.338104
\(166\) −25.3791 −1.96980
\(167\) 6.08986 0.471248 0.235624 0.971844i \(-0.424287\pi\)
0.235624 + 0.971844i \(0.424287\pi\)
\(168\) −11.3276 −0.873942
\(169\) 28.6016 2.20012
\(170\) 1.69243 0.129804
\(171\) 0.497055 0.0380107
\(172\) −14.7594 −1.12539
\(173\) −12.7189 −0.966997 −0.483499 0.875345i \(-0.660634\pi\)
−0.483499 + 0.875345i \(0.660634\pi\)
\(174\) 16.7930 1.27308
\(175\) 6.46967 0.489061
\(176\) −3.39460 −0.255877
\(177\) −18.2170 −1.36927
\(178\) −41.5362 −3.11327
\(179\) −4.89787 −0.366084 −0.183042 0.983105i \(-0.558594\pi\)
−0.183042 + 0.983105i \(0.558594\pi\)
\(180\) 2.28324 0.170183
\(181\) −6.11794 −0.454743 −0.227372 0.973808i \(-0.573013\pi\)
−0.227372 + 0.973808i \(0.573013\pi\)
\(182\) 29.4612 2.18381
\(183\) −11.7131 −0.865858
\(184\) −30.2568 −2.23056
\(185\) 14.8803 1.09402
\(186\) 21.3601 1.56620
\(187\) −1.18764 −0.0868491
\(188\) −16.4515 −1.19985
\(189\) −10.7101 −0.779043
\(190\) −3.03817 −0.220412
\(191\) −23.6394 −1.71049 −0.855243 0.518227i \(-0.826592\pi\)
−0.855243 + 0.518227i \(0.826592\pi\)
\(192\) 18.6277 1.34434
\(193\) 6.77123 0.487404 0.243702 0.969850i \(-0.421638\pi\)
0.243702 + 0.969850i \(0.421638\pi\)
\(194\) 6.87026 0.493256
\(195\) 13.1390 0.940901
\(196\) −11.6038 −0.828843
\(197\) −23.1257 −1.64764 −0.823819 0.566853i \(-0.808161\pi\)
−0.823819 + 0.566853i \(0.808161\pi\)
\(198\) −2.50047 −0.177701
\(199\) 15.6403 1.10871 0.554355 0.832280i \(-0.312965\pi\)
0.554355 + 0.832280i \(0.312965\pi\)
\(200\) 12.3613 0.874079
\(201\) 11.6023 0.818361
\(202\) −17.4487 −1.22769
\(203\) −8.70835 −0.611206
\(204\) −3.14408 −0.220129
\(205\) 6.22664 0.434887
\(206\) −0.516948 −0.0360175
\(207\) −4.06612 −0.282615
\(208\) 10.2697 0.712073
\(209\) 2.13200 0.147473
\(210\) 9.30468 0.642084
\(211\) 1.00000 0.0688428
\(212\) 32.2500 2.21494
\(213\) 13.3313 0.913447
\(214\) 38.7115 2.64626
\(215\) 5.32699 0.363297
\(216\) −20.4633 −1.39235
\(217\) −11.0767 −0.751935
\(218\) −6.44971 −0.436830
\(219\) 17.1396 1.15818
\(220\) 9.79343 0.660273
\(221\) 3.59298 0.241690
\(222\) −43.1408 −2.89542
\(223\) 6.68452 0.447629 0.223814 0.974632i \(-0.428149\pi\)
0.223814 + 0.974632i \(0.428149\pi\)
\(224\) −7.04723 −0.470863
\(225\) 1.66120 0.110747
\(226\) 33.6200 2.23637
\(227\) 19.5423 1.29707 0.648535 0.761185i \(-0.275382\pi\)
0.648535 + 0.761185i \(0.275382\pi\)
\(228\) 5.64408 0.373789
\(229\) −15.7928 −1.04362 −0.521808 0.853063i \(-0.674742\pi\)
−0.521808 + 0.853063i \(0.674742\pi\)
\(230\) 24.8535 1.63879
\(231\) −6.52945 −0.429606
\(232\) −16.6387 −1.09238
\(233\) 10.0336 0.657322 0.328661 0.944448i \(-0.393403\pi\)
0.328661 + 0.944448i \(0.393403\pi\)
\(234\) 7.56468 0.494519
\(235\) 5.93770 0.387333
\(236\) 41.0789 2.67401
\(237\) 14.1109 0.916599
\(238\) 2.54446 0.164933
\(239\) 11.8196 0.764547 0.382273 0.924049i \(-0.375141\pi\)
0.382273 + 0.924049i \(0.375141\pi\)
\(240\) 3.24345 0.209364
\(241\) 29.4974 1.90009 0.950047 0.312107i \(-0.101035\pi\)
0.950047 + 0.312107i \(0.101035\pi\)
\(242\) 15.2300 0.979021
\(243\) −5.10912 −0.327750
\(244\) 26.4128 1.69091
\(245\) 4.18806 0.267565
\(246\) −18.0522 −1.15097
\(247\) −6.44993 −0.410399
\(248\) −21.1638 −1.34390
\(249\) −17.0165 −1.07838
\(250\) −25.3447 −1.60294
\(251\) −3.99703 −0.252290 −0.126145 0.992012i \(-0.540260\pi\)
−0.126145 + 0.992012i \(0.540260\pi\)
\(252\) 3.43270 0.216240
\(253\) −17.4407 −1.09648
\(254\) 39.4044 2.47245
\(255\) 1.13476 0.0710617
\(256\) −24.8254 −1.55159
\(257\) −31.2248 −1.94775 −0.973874 0.227089i \(-0.927079\pi\)
−0.973874 + 0.227089i \(0.927079\pi\)
\(258\) −15.4439 −0.961498
\(259\) 22.3715 1.39010
\(260\) −29.6281 −1.83745
\(261\) −2.23602 −0.138406
\(262\) −24.4910 −1.51306
\(263\) −6.01585 −0.370953 −0.185476 0.982649i \(-0.559383\pi\)
−0.185476 + 0.982649i \(0.559383\pi\)
\(264\) −12.4756 −0.767817
\(265\) −11.6397 −0.715023
\(266\) −4.56768 −0.280062
\(267\) −27.8498 −1.70438
\(268\) −26.1628 −1.59815
\(269\) −7.89279 −0.481232 −0.240616 0.970620i \(-0.577349\pi\)
−0.240616 + 0.970620i \(0.577349\pi\)
\(270\) 16.8089 1.02296
\(271\) 1.68192 0.102169 0.0510847 0.998694i \(-0.483732\pi\)
0.0510847 + 0.998694i \(0.483732\pi\)
\(272\) 0.886954 0.0537795
\(273\) 19.7535 1.19554
\(274\) −2.30934 −0.139512
\(275\) 7.12533 0.429674
\(276\) −46.1710 −2.77917
\(277\) −8.53656 −0.512912 −0.256456 0.966556i \(-0.582555\pi\)
−0.256456 + 0.966556i \(0.582555\pi\)
\(278\) 0.458361 0.0274907
\(279\) −2.84413 −0.170274
\(280\) −9.21917 −0.550951
\(281\) −21.2331 −1.26666 −0.633331 0.773881i \(-0.718313\pi\)
−0.633331 + 0.773881i \(0.718313\pi\)
\(282\) −17.2145 −1.02511
\(283\) 11.0239 0.655304 0.327652 0.944799i \(-0.393743\pi\)
0.327652 + 0.944799i \(0.393743\pi\)
\(284\) −30.0618 −1.78384
\(285\) −2.03707 −0.120666
\(286\) 32.4469 1.91862
\(287\) 9.36132 0.552581
\(288\) −1.80950 −0.106626
\(289\) −16.6897 −0.981746
\(290\) 13.6673 0.802573
\(291\) 4.60646 0.270036
\(292\) −38.6493 −2.26178
\(293\) −2.96338 −0.173123 −0.0865613 0.996247i \(-0.527588\pi\)
−0.0865613 + 0.996247i \(0.527588\pi\)
\(294\) −12.1420 −0.708135
\(295\) −14.8263 −0.863218
\(296\) 42.7444 2.48447
\(297\) −11.7955 −0.684442
\(298\) 26.2939 1.52317
\(299\) 52.7632 3.05138
\(300\) 18.8630 1.08906
\(301\) 8.00876 0.461617
\(302\) −36.7974 −2.11745
\(303\) −11.6992 −0.672104
\(304\) −1.59221 −0.0913197
\(305\) −9.53295 −0.545855
\(306\) 0.653334 0.0373486
\(307\) −16.1893 −0.923970 −0.461985 0.886888i \(-0.652863\pi\)
−0.461985 + 0.886888i \(0.652863\pi\)
\(308\) 14.7238 0.838964
\(309\) −0.346610 −0.0197180
\(310\) 17.3843 0.987363
\(311\) 24.6056 1.39525 0.697627 0.716461i \(-0.254239\pi\)
0.697627 + 0.716461i \(0.254239\pi\)
\(312\) 37.7423 2.13674
\(313\) 7.27646 0.411290 0.205645 0.978627i \(-0.434071\pi\)
0.205645 + 0.978627i \(0.434071\pi\)
\(314\) 20.4751 1.15548
\(315\) −1.23894 −0.0698061
\(316\) −31.8197 −1.79000
\(317\) 15.2036 0.853919 0.426959 0.904271i \(-0.359585\pi\)
0.426959 + 0.904271i \(0.359585\pi\)
\(318\) 33.7458 1.89237
\(319\) −9.59088 −0.536986
\(320\) 15.1605 0.847500
\(321\) 25.9558 1.44871
\(322\) 37.3656 2.08230
\(323\) −0.557057 −0.0309955
\(324\) −25.9066 −1.43926
\(325\) −21.5563 −1.19573
\(326\) −42.0841 −2.33083
\(327\) −4.32449 −0.239145
\(328\) 17.8863 0.987606
\(329\) 8.92692 0.492157
\(330\) 10.2476 0.564114
\(331\) 9.52905 0.523764 0.261882 0.965100i \(-0.415657\pi\)
0.261882 + 0.965100i \(0.415657\pi\)
\(332\) 38.3718 2.10592
\(333\) 5.74428 0.314785
\(334\) −14.3694 −0.786259
\(335\) 9.44273 0.515912
\(336\) 4.87631 0.266024
\(337\) −15.0792 −0.821415 −0.410707 0.911767i \(-0.634718\pi\)
−0.410707 + 0.911767i \(0.634718\pi\)
\(338\) −67.4873 −3.67082
\(339\) 22.5420 1.22431
\(340\) −2.55887 −0.138774
\(341\) −12.1992 −0.660626
\(342\) −1.17283 −0.0634195
\(343\) 19.8472 1.07165
\(344\) 15.3020 0.825029
\(345\) 16.6641 0.897166
\(346\) 30.0109 1.61340
\(347\) 10.9439 0.587498 0.293749 0.955883i \(-0.405097\pi\)
0.293749 + 0.955883i \(0.405097\pi\)
\(348\) −25.3901 −1.36105
\(349\) 13.1519 0.704003 0.352002 0.935999i \(-0.385501\pi\)
0.352002 + 0.935999i \(0.385501\pi\)
\(350\) −15.2656 −0.815980
\(351\) 35.6848 1.90471
\(352\) −7.76142 −0.413685
\(353\) 31.9668 1.70142 0.850710 0.525635i \(-0.176172\pi\)
0.850710 + 0.525635i \(0.176172\pi\)
\(354\) 42.9841 2.28458
\(355\) 10.8500 0.575856
\(356\) 62.8005 3.32842
\(357\) 1.70604 0.0902933
\(358\) 11.5568 0.610797
\(359\) 19.2302 1.01493 0.507466 0.861672i \(-0.330582\pi\)
0.507466 + 0.861672i \(0.330582\pi\)
\(360\) −2.36718 −0.124762
\(361\) 1.00000 0.0526316
\(362\) 14.4357 0.758722
\(363\) 10.2116 0.535970
\(364\) −44.5438 −2.33473
\(365\) 13.9494 0.730143
\(366\) 27.6378 1.44465
\(367\) 0.00808291 0.000421924 0 0.000210962 1.00000i \(-0.499933\pi\)
0.000210962 1.00000i \(0.499933\pi\)
\(368\) 13.0250 0.678975
\(369\) 2.40368 0.125131
\(370\) −35.1110 −1.82533
\(371\) −17.4995 −0.908531
\(372\) −32.2953 −1.67443
\(373\) 4.51595 0.233827 0.116914 0.993142i \(-0.462700\pi\)
0.116914 + 0.993142i \(0.462700\pi\)
\(374\) 2.80232 0.144905
\(375\) −16.9934 −0.877537
\(376\) 17.0563 0.879612
\(377\) 29.0153 1.49436
\(378\) 25.2711 1.29980
\(379\) 14.5363 0.746681 0.373341 0.927694i \(-0.378212\pi\)
0.373341 + 0.927694i \(0.378212\pi\)
\(380\) 4.59355 0.235644
\(381\) 26.4204 1.35356
\(382\) 55.7786 2.85388
\(383\) −10.9985 −0.561998 −0.280999 0.959708i \(-0.590666\pi\)
−0.280999 + 0.959708i \(0.590666\pi\)
\(384\) −32.4344 −1.65516
\(385\) −5.31412 −0.270833
\(386\) −15.9771 −0.813215
\(387\) 2.05639 0.104532
\(388\) −10.3875 −0.527344
\(389\) 22.6311 1.14744 0.573720 0.819051i \(-0.305500\pi\)
0.573720 + 0.819051i \(0.305500\pi\)
\(390\) −31.0022 −1.56986
\(391\) 4.55697 0.230456
\(392\) 12.0304 0.607627
\(393\) −16.4211 −0.828334
\(394\) 54.5665 2.74902
\(395\) 11.4844 0.577843
\(396\) 3.78058 0.189981
\(397\) 19.3963 0.973470 0.486735 0.873550i \(-0.338188\pi\)
0.486735 + 0.873550i \(0.338188\pi\)
\(398\) −36.9042 −1.84984
\(399\) −3.06260 −0.153322
\(400\) −5.32132 −0.266066
\(401\) 20.8940 1.04340 0.521698 0.853131i \(-0.325299\pi\)
0.521698 + 0.853131i \(0.325299\pi\)
\(402\) −27.3763 −1.36540
\(403\) 36.9063 1.83844
\(404\) 26.3815 1.31253
\(405\) 9.35025 0.464617
\(406\) 20.5479 1.01977
\(407\) 24.6387 1.22130
\(408\) 3.25966 0.161377
\(409\) 7.98599 0.394882 0.197441 0.980315i \(-0.436737\pi\)
0.197441 + 0.980315i \(0.436737\pi\)
\(410\) −14.6921 −0.725593
\(411\) −1.54840 −0.0763768
\(412\) 0.781598 0.0385066
\(413\) −22.2903 −1.09683
\(414\) 9.59427 0.471533
\(415\) −13.8492 −0.679830
\(416\) 23.4806 1.15123
\(417\) 0.307328 0.0150499
\(418\) −5.03058 −0.246054
\(419\) 27.1566 1.32669 0.663343 0.748315i \(-0.269137\pi\)
0.663343 + 0.748315i \(0.269137\pi\)
\(420\) −14.0682 −0.686457
\(421\) −19.6403 −0.957208 −0.478604 0.878031i \(-0.658857\pi\)
−0.478604 + 0.878031i \(0.658857\pi\)
\(422\) −2.35956 −0.114862
\(423\) 2.29214 0.111448
\(424\) −33.4356 −1.62378
\(425\) −1.86173 −0.0903074
\(426\) −31.4561 −1.52405
\(427\) −14.3321 −0.693581
\(428\) −58.5297 −2.82914
\(429\) 21.7554 1.05036
\(430\) −12.5694 −0.606148
\(431\) 30.6243 1.47512 0.737560 0.675282i \(-0.235978\pi\)
0.737560 + 0.675282i \(0.235978\pi\)
\(432\) 8.80906 0.423826
\(433\) 17.7798 0.854443 0.427221 0.904147i \(-0.359492\pi\)
0.427221 + 0.904147i \(0.359492\pi\)
\(434\) 26.1361 1.25457
\(435\) 9.16385 0.439373
\(436\) 9.75162 0.467018
\(437\) −8.18043 −0.391323
\(438\) −40.4419 −1.93239
\(439\) −17.7441 −0.846878 −0.423439 0.905925i \(-0.639177\pi\)
−0.423439 + 0.905925i \(0.639177\pi\)
\(440\) −10.1535 −0.484048
\(441\) 1.61673 0.0769870
\(442\) −8.47786 −0.403250
\(443\) −3.01381 −0.143190 −0.0715952 0.997434i \(-0.522809\pi\)
−0.0715952 + 0.997434i \(0.522809\pi\)
\(444\) 65.2266 3.09552
\(445\) −22.6661 −1.07447
\(446\) −15.7725 −0.746851
\(447\) 17.6299 0.833866
\(448\) 22.7928 1.07686
\(449\) 34.6636 1.63587 0.817937 0.575308i \(-0.195118\pi\)
0.817937 + 0.575308i \(0.195118\pi\)
\(450\) −3.91971 −0.184777
\(451\) 10.3100 0.485480
\(452\) −50.8316 −2.39092
\(453\) −24.6724 −1.15921
\(454\) −46.1114 −2.16411
\(455\) 16.0768 0.753692
\(456\) −5.85158 −0.274025
\(457\) 24.3677 1.13987 0.569937 0.821688i \(-0.306967\pi\)
0.569937 + 0.821688i \(0.306967\pi\)
\(458\) 37.2640 1.74123
\(459\) 3.08197 0.143854
\(460\) −37.5772 −1.75205
\(461\) 10.4772 0.487970 0.243985 0.969779i \(-0.421545\pi\)
0.243985 + 0.969779i \(0.421545\pi\)
\(462\) 15.4066 0.716782
\(463\) 8.34844 0.387985 0.193992 0.981003i \(-0.437856\pi\)
0.193992 + 0.981003i \(0.437856\pi\)
\(464\) 7.16264 0.332517
\(465\) 11.6561 0.540537
\(466\) −23.6749 −1.09672
\(467\) 32.2167 1.49081 0.745405 0.666612i \(-0.232256\pi\)
0.745405 + 0.666612i \(0.232256\pi\)
\(468\) −11.4374 −0.528694
\(469\) 14.1965 0.655534
\(470\) −14.0104 −0.646250
\(471\) 13.7284 0.632572
\(472\) −42.5891 −1.96032
\(473\) 8.82039 0.405562
\(474\) −33.2955 −1.52931
\(475\) 3.34209 0.153346
\(476\) −3.84708 −0.176331
\(477\) −4.49331 −0.205735
\(478\) −27.8891 −1.27562
\(479\) 2.50229 0.114333 0.0571664 0.998365i \(-0.481793\pi\)
0.0571664 + 0.998365i \(0.481793\pi\)
\(480\) 7.41584 0.338485
\(481\) −74.5395 −3.39871
\(482\) −69.6009 −3.17023
\(483\) 25.0534 1.13997
\(484\) −23.0269 −1.04668
\(485\) 3.74906 0.170236
\(486\) 12.0553 0.546839
\(487\) 24.9052 1.12856 0.564282 0.825582i \(-0.309153\pi\)
0.564282 + 0.825582i \(0.309153\pi\)
\(488\) −27.3838 −1.23961
\(489\) −28.2171 −1.27602
\(490\) −9.88199 −0.446423
\(491\) 21.0042 0.947905 0.473952 0.880551i \(-0.342827\pi\)
0.473952 + 0.880551i \(0.342827\pi\)
\(492\) 27.2940 1.23051
\(493\) 2.50594 0.112862
\(494\) 15.2190 0.684736
\(495\) −1.36449 −0.0613294
\(496\) 9.11061 0.409078
\(497\) 16.3122 0.731701
\(498\) 40.1514 1.79923
\(499\) −3.98093 −0.178211 −0.0891055 0.996022i \(-0.528401\pi\)
−0.0891055 + 0.996022i \(0.528401\pi\)
\(500\) 38.3198 1.71371
\(501\) −9.63459 −0.430442
\(502\) 9.43123 0.420936
\(503\) 17.2050 0.767131 0.383566 0.923514i \(-0.374696\pi\)
0.383566 + 0.923514i \(0.374696\pi\)
\(504\) −3.55890 −0.158526
\(505\) −9.52166 −0.423708
\(506\) 41.1523 1.82944
\(507\) −45.2497 −2.00961
\(508\) −59.5773 −2.64332
\(509\) 27.0629 1.19954 0.599772 0.800171i \(-0.295258\pi\)
0.599772 + 0.800171i \(0.295258\pi\)
\(510\) −2.67755 −0.118564
\(511\) 20.9719 0.927743
\(512\) 17.5746 0.776693
\(513\) −5.53259 −0.244270
\(514\) 73.6768 3.24974
\(515\) −0.282096 −0.0124306
\(516\) 23.3504 1.02794
\(517\) 9.83160 0.432393
\(518\) −52.7870 −2.31933
\(519\) 20.1221 0.883264
\(520\) 30.7173 1.34704
\(521\) −27.4098 −1.20084 −0.600422 0.799684i \(-0.705001\pi\)
−0.600422 + 0.799684i \(0.705001\pi\)
\(522\) 5.27603 0.230926
\(523\) 18.3656 0.803071 0.401536 0.915843i \(-0.368477\pi\)
0.401536 + 0.915843i \(0.368477\pi\)
\(524\) 37.0292 1.61763
\(525\) −10.2355 −0.446713
\(526\) 14.1948 0.618921
\(527\) 3.18747 0.138848
\(528\) 5.37049 0.233721
\(529\) 43.9195 1.90954
\(530\) 27.4646 1.19299
\(531\) −5.72342 −0.248375
\(532\) 6.90608 0.299417
\(533\) −31.1909 −1.35103
\(534\) 65.7132 2.84369
\(535\) 21.1246 0.913298
\(536\) 27.1247 1.17161
\(537\) 7.74877 0.334384
\(538\) 18.6235 0.802917
\(539\) 6.93457 0.298693
\(540\) −25.4142 −1.09365
\(541\) 12.8559 0.552720 0.276360 0.961054i \(-0.410872\pi\)
0.276360 + 0.961054i \(0.410872\pi\)
\(542\) −3.96860 −0.170466
\(543\) 9.67901 0.415366
\(544\) 2.02793 0.0869470
\(545\) −3.51957 −0.150762
\(546\) −46.6097 −1.99471
\(547\) −25.3414 −1.08352 −0.541760 0.840533i \(-0.682242\pi\)
−0.541760 + 0.840533i \(0.682242\pi\)
\(548\) 3.49160 0.149154
\(549\) −3.68003 −0.157060
\(550\) −16.8127 −0.716894
\(551\) −4.49854 −0.191644
\(552\) 47.8684 2.03742
\(553\) 17.2660 0.734226
\(554\) 20.1425 0.855774
\(555\) −23.5417 −0.999289
\(556\) −0.693017 −0.0293905
\(557\) −34.1150 −1.44550 −0.722749 0.691110i \(-0.757122\pi\)
−0.722749 + 0.691110i \(0.757122\pi\)
\(558\) 6.71091 0.284096
\(559\) −26.6843 −1.12863
\(560\) 3.96868 0.167707
\(561\) 1.87894 0.0793287
\(562\) 50.1008 2.11338
\(563\) −15.3246 −0.645857 −0.322928 0.946423i \(-0.604667\pi\)
−0.322928 + 0.946423i \(0.604667\pi\)
\(564\) 26.0274 1.09595
\(565\) 18.3462 0.771831
\(566\) −26.0116 −1.09335
\(567\) 14.0575 0.590358
\(568\) 31.1670 1.30774
\(569\) −2.26941 −0.0951388 −0.0475694 0.998868i \(-0.515148\pi\)
−0.0475694 + 0.998868i \(0.515148\pi\)
\(570\) 4.80659 0.201326
\(571\) −6.22049 −0.260320 −0.130160 0.991493i \(-0.541549\pi\)
−0.130160 + 0.991493i \(0.541549\pi\)
\(572\) −49.0580 −2.05122
\(573\) 37.3992 1.56237
\(574\) −22.0886 −0.921961
\(575\) −27.3397 −1.14015
\(576\) 5.85246 0.243853
\(577\) −17.3191 −0.721005 −0.360502 0.932758i \(-0.617395\pi\)
−0.360502 + 0.932758i \(0.617395\pi\)
\(578\) 39.3803 1.63801
\(579\) −10.7126 −0.445199
\(580\) −20.6643 −0.858037
\(581\) −20.8213 −0.863814
\(582\) −10.8692 −0.450544
\(583\) −19.2730 −0.798206
\(584\) 40.0702 1.65812
\(585\) 4.12800 0.170672
\(586\) 6.99228 0.288848
\(587\) 11.3983 0.470456 0.235228 0.971940i \(-0.424416\pi\)
0.235228 + 0.971940i \(0.424416\pi\)
\(588\) 18.3580 0.757072
\(589\) −5.72197 −0.235770
\(590\) 34.9835 1.44025
\(591\) 36.5865 1.50497
\(592\) −18.4006 −0.756262
\(593\) −28.4037 −1.16640 −0.583200 0.812328i \(-0.698200\pi\)
−0.583200 + 0.812328i \(0.698200\pi\)
\(594\) 27.8321 1.14197
\(595\) 1.38849 0.0569227
\(596\) −39.7550 −1.62843
\(597\) −24.7440 −1.01271
\(598\) −124.498 −5.09110
\(599\) −18.3677 −0.750484 −0.375242 0.926927i \(-0.622440\pi\)
−0.375242 + 0.926927i \(0.622440\pi\)
\(600\) −19.5565 −0.798391
\(601\) −17.3702 −0.708545 −0.354272 0.935142i \(-0.615271\pi\)
−0.354272 + 0.935142i \(0.615271\pi\)
\(602\) −18.8972 −0.770191
\(603\) 3.64520 0.148444
\(604\) 55.6357 2.26379
\(605\) 8.31091 0.337887
\(606\) 27.6051 1.12138
\(607\) −23.7703 −0.964808 −0.482404 0.875949i \(-0.660236\pi\)
−0.482404 + 0.875949i \(0.660236\pi\)
\(608\) −3.64044 −0.147639
\(609\) 13.7772 0.558281
\(610\) 22.4936 0.910738
\(611\) −29.7435 −1.20329
\(612\) −0.987806 −0.0399297
\(613\) −26.1345 −1.05556 −0.527781 0.849381i \(-0.676976\pi\)
−0.527781 + 0.849381i \(0.676976\pi\)
\(614\) 38.1996 1.54161
\(615\) −9.85097 −0.397230
\(616\) −15.2651 −0.615046
\(617\) −32.2151 −1.29693 −0.648465 0.761244i \(-0.724589\pi\)
−0.648465 + 0.761244i \(0.724589\pi\)
\(618\) 0.817849 0.0328987
\(619\) 35.6652 1.43351 0.716753 0.697327i \(-0.245628\pi\)
0.716753 + 0.697327i \(0.245628\pi\)
\(620\) −26.2842 −1.05560
\(621\) 45.2589 1.81618
\(622\) −58.0584 −2.32793
\(623\) −34.0769 −1.36526
\(624\) −16.2473 −0.650414
\(625\) 2.88002 0.115201
\(626\) −17.1692 −0.686221
\(627\) −3.37297 −0.134703
\(628\) −30.9573 −1.23533
\(629\) −6.43771 −0.256688
\(630\) 2.92335 0.116469
\(631\) 10.5796 0.421169 0.210584 0.977576i \(-0.432463\pi\)
0.210584 + 0.977576i \(0.432463\pi\)
\(632\) 32.9895 1.31225
\(633\) −1.58207 −0.0628816
\(634\) −35.8738 −1.42473
\(635\) 21.5027 0.853310
\(636\) −51.0218 −2.02315
\(637\) −20.9791 −0.831224
\(638\) 22.6303 0.895941
\(639\) 4.18844 0.165692
\(640\) −26.3974 −1.04345
\(641\) 29.3852 1.16065 0.580324 0.814386i \(-0.302926\pi\)
0.580324 + 0.814386i \(0.302926\pi\)
\(642\) −61.2443 −2.41712
\(643\) 5.82320 0.229645 0.114822 0.993386i \(-0.463370\pi\)
0.114822 + 0.993386i \(0.463370\pi\)
\(644\) −56.4948 −2.22621
\(645\) −8.42767 −0.331839
\(646\) 1.31441 0.0517148
\(647\) −25.3030 −0.994765 −0.497382 0.867531i \(-0.665705\pi\)
−0.497382 + 0.867531i \(0.665705\pi\)
\(648\) 26.8590 1.05512
\(649\) −24.5492 −0.963642
\(650\) 50.8633 1.99502
\(651\) 17.5241 0.686824
\(652\) 63.6290 2.49190
\(653\) −36.9510 −1.44600 −0.723001 0.690847i \(-0.757238\pi\)
−0.723001 + 0.690847i \(0.757238\pi\)
\(654\) 10.2039 0.399004
\(655\) −13.3646 −0.522199
\(656\) −7.69971 −0.300623
\(657\) 5.38491 0.210085
\(658\) −21.0636 −0.821145
\(659\) −21.4523 −0.835664 −0.417832 0.908524i \(-0.637210\pi\)
−0.417832 + 0.908524i \(0.637210\pi\)
\(660\) −15.4939 −0.603099
\(661\) 49.3589 1.91984 0.959920 0.280274i \(-0.0904254\pi\)
0.959920 + 0.280274i \(0.0904254\pi\)
\(662\) −22.4844 −0.873881
\(663\) −5.68434 −0.220762
\(664\) −39.7824 −1.54386
\(665\) −2.49255 −0.0966571
\(666\) −13.5540 −0.525207
\(667\) 36.8000 1.42490
\(668\) 21.7258 0.840595
\(669\) −10.5754 −0.408868
\(670\) −22.2807 −0.860779
\(671\) −15.7846 −0.609358
\(672\) 11.1492 0.430090
\(673\) −7.88600 −0.303983 −0.151992 0.988382i \(-0.548569\pi\)
−0.151992 + 0.988382i \(0.548569\pi\)
\(674\) 35.5802 1.37050
\(675\) −18.4904 −0.711696
\(676\) 102.037 3.92450
\(677\) 8.47040 0.325544 0.162772 0.986664i \(-0.447957\pi\)
0.162772 + 0.986664i \(0.447957\pi\)
\(678\) −53.1892 −2.04272
\(679\) 5.63645 0.216307
\(680\) 2.65294 0.101736
\(681\) −30.9173 −1.18476
\(682\) 28.7848 1.10223
\(683\) 0.127828 0.00489120 0.00244560 0.999997i \(-0.499222\pi\)
0.00244560 + 0.999997i \(0.499222\pi\)
\(684\) 1.77326 0.0678022
\(685\) −1.26019 −0.0481495
\(686\) −46.8306 −1.78800
\(687\) 24.9853 0.953247
\(688\) −6.58723 −0.251136
\(689\) 58.3065 2.22130
\(690\) −39.3200 −1.49689
\(691\) 31.4545 1.19659 0.598294 0.801277i \(-0.295846\pi\)
0.598294 + 0.801277i \(0.295846\pi\)
\(692\) −45.3749 −1.72490
\(693\) −2.05142 −0.0779271
\(694\) −25.8228 −0.980218
\(695\) 0.250125 0.00948777
\(696\) 26.3236 0.997792
\(697\) −2.69385 −0.102037
\(698\) −31.0326 −1.17460
\(699\) −15.8738 −0.600403
\(700\) 23.0808 0.872371
\(701\) −50.0320 −1.88968 −0.944841 0.327529i \(-0.893784\pi\)
−0.944841 + 0.327529i \(0.893784\pi\)
\(702\) −84.2005 −3.17794
\(703\) 11.5566 0.435867
\(704\) 25.1027 0.946095
\(705\) −9.39385 −0.353793
\(706\) −75.4276 −2.83875
\(707\) −14.3152 −0.538377
\(708\) −64.9897 −2.44246
\(709\) −40.2829 −1.51285 −0.756427 0.654078i \(-0.773057\pi\)
−0.756427 + 0.654078i \(0.773057\pi\)
\(710\) −25.6011 −0.960794
\(711\) 4.43335 0.166264
\(712\) −65.1093 −2.44007
\(713\) 46.8082 1.75298
\(714\) −4.02551 −0.150651
\(715\) 17.7061 0.662170
\(716\) −17.4733 −0.653008
\(717\) −18.6994 −0.698344
\(718\) −45.3748 −1.69337
\(719\) 10.9934 0.409983 0.204991 0.978764i \(-0.434283\pi\)
0.204991 + 0.978764i \(0.434283\pi\)
\(720\) 1.01903 0.0379769
\(721\) −0.424112 −0.0157947
\(722\) −2.35956 −0.0878138
\(723\) −46.6669 −1.73556
\(724\) −21.8260 −0.811155
\(725\) −15.0345 −0.558369
\(726\) −24.0949 −0.894246
\(727\) −16.0426 −0.594987 −0.297494 0.954724i \(-0.596151\pi\)
−0.297494 + 0.954724i \(0.596151\pi\)
\(728\) 46.1813 1.71159
\(729\) 29.8683 1.10623
\(730\) −32.9144 −1.21822
\(731\) −2.30463 −0.0852397
\(732\) −41.7869 −1.54449
\(733\) 37.2139 1.37453 0.687264 0.726408i \(-0.258812\pi\)
0.687264 + 0.726408i \(0.258812\pi\)
\(734\) −0.0190721 −0.000703965 0
\(735\) −6.62581 −0.244397
\(736\) 29.7804 1.09772
\(737\) 15.6352 0.575931
\(738\) −5.67164 −0.208776
\(739\) −27.4924 −1.01133 −0.505663 0.862731i \(-0.668752\pi\)
−0.505663 + 0.862731i \(0.668752\pi\)
\(740\) 53.0860 1.95148
\(741\) 10.2042 0.374862
\(742\) 41.2912 1.51585
\(743\) 43.2253 1.58578 0.792891 0.609364i \(-0.208575\pi\)
0.792891 + 0.609364i \(0.208575\pi\)
\(744\) 33.4826 1.22753
\(745\) 14.3484 0.525686
\(746\) −10.6557 −0.390132
\(747\) −5.34624 −0.195609
\(748\) −4.23696 −0.154919
\(749\) 31.7595 1.16047
\(750\) 40.0970 1.46414
\(751\) −10.8391 −0.395524 −0.197762 0.980250i \(-0.563367\pi\)
−0.197762 + 0.980250i \(0.563367\pi\)
\(752\) −7.34242 −0.267750
\(753\) 6.32357 0.230444
\(754\) −68.4633 −2.49329
\(755\) −20.0801 −0.730791
\(756\) −38.2085 −1.38963
\(757\) 49.0738 1.78362 0.891810 0.452411i \(-0.149436\pi\)
0.891810 + 0.452411i \(0.149436\pi\)
\(758\) −34.2994 −1.24581
\(759\) 27.5923 1.00154
\(760\) −4.76242 −0.172751
\(761\) 49.0823 1.77923 0.889615 0.456710i \(-0.150972\pi\)
0.889615 + 0.456710i \(0.150972\pi\)
\(762\) −62.3405 −2.25836
\(763\) −5.29143 −0.191563
\(764\) −84.3343 −3.05111
\(765\) 0.356520 0.0128900
\(766\) 25.9517 0.937672
\(767\) 74.2688 2.68169
\(768\) 39.2755 1.41723
\(769\) −32.2365 −1.16248 −0.581238 0.813734i \(-0.697432\pi\)
−0.581238 + 0.813734i \(0.697432\pi\)
\(770\) 12.5390 0.451874
\(771\) 49.3998 1.77909
\(772\) 24.1566 0.869415
\(773\) −19.1720 −0.689570 −0.344785 0.938682i \(-0.612048\pi\)
−0.344785 + 0.938682i \(0.612048\pi\)
\(774\) −4.85218 −0.174408
\(775\) −19.1234 −0.686931
\(776\) 10.7693 0.386597
\(777\) −35.3933 −1.26973
\(778\) −53.3994 −1.91446
\(779\) 4.83585 0.173262
\(780\) 46.8737 1.67835
\(781\) 17.9653 0.642849
\(782\) −10.7524 −0.384507
\(783\) 24.8886 0.889445
\(784\) −5.17886 −0.184959
\(785\) 11.1731 0.398787
\(786\) 38.7465 1.38204
\(787\) −13.1835 −0.469942 −0.234971 0.972002i \(-0.575500\pi\)
−0.234971 + 0.972002i \(0.575500\pi\)
\(788\) −82.5017 −2.93900
\(789\) 9.51749 0.338832
\(790\) −27.0982 −0.964109
\(791\) 27.5823 0.980713
\(792\) −3.91957 −0.139276
\(793\) 47.7531 1.69576
\(794\) −45.7667 −1.62420
\(795\) 18.4149 0.653108
\(796\) 55.7972 1.97768
\(797\) −21.2674 −0.753331 −0.376666 0.926349i \(-0.622929\pi\)
−0.376666 + 0.926349i \(0.622929\pi\)
\(798\) 7.22639 0.255811
\(799\) −2.56884 −0.0908791
\(800\) −12.1667 −0.430157
\(801\) −8.74984 −0.309160
\(802\) −49.3006 −1.74087
\(803\) 23.0973 0.815085
\(804\) 41.3915 1.45976
\(805\) 20.3902 0.718659
\(806\) −87.0828 −3.06736
\(807\) 12.4869 0.439561
\(808\) −27.3514 −0.962219
\(809\) −32.4588 −1.14119 −0.570596 0.821231i \(-0.693288\pi\)
−0.570596 + 0.821231i \(0.693288\pi\)
\(810\) −22.0625 −0.775197
\(811\) 39.4034 1.38364 0.691820 0.722070i \(-0.256809\pi\)
0.691820 + 0.722070i \(0.256809\pi\)
\(812\) −31.0673 −1.09025
\(813\) −2.66092 −0.0933225
\(814\) −58.1366 −2.03769
\(815\) −22.9651 −0.804431
\(816\) −1.40322 −0.0491226
\(817\) 4.13715 0.144741
\(818\) −18.8434 −0.658846
\(819\) 6.20617 0.216861
\(820\) 22.2137 0.775737
\(821\) −36.2619 −1.26555 −0.632774 0.774336i \(-0.718084\pi\)
−0.632774 + 0.774336i \(0.718084\pi\)
\(822\) 3.65354 0.127432
\(823\) 20.3383 0.708948 0.354474 0.935066i \(-0.384660\pi\)
0.354474 + 0.935066i \(0.384660\pi\)
\(824\) −0.810333 −0.0282293
\(825\) −11.2728 −0.392467
\(826\) 52.5953 1.83002
\(827\) 43.7250 1.52047 0.760233 0.649651i \(-0.225085\pi\)
0.760233 + 0.649651i \(0.225085\pi\)
\(828\) −14.5060 −0.504119
\(829\) −38.3710 −1.33268 −0.666340 0.745648i \(-0.732140\pi\)
−0.666340 + 0.745648i \(0.732140\pi\)
\(830\) 32.6780 1.13427
\(831\) 13.5054 0.468498
\(832\) −75.9433 −2.63286
\(833\) −1.81189 −0.0627783
\(834\) −0.725159 −0.0251102
\(835\) −7.84130 −0.271359
\(836\) 7.60597 0.263058
\(837\) 31.6573 1.09424
\(838\) −64.0776 −2.21353
\(839\) 20.2558 0.699308 0.349654 0.936879i \(-0.386299\pi\)
0.349654 + 0.936879i \(0.386299\pi\)
\(840\) 14.5854 0.503243
\(841\) −8.76312 −0.302177
\(842\) 46.3424 1.59706
\(843\) 33.5923 1.15698
\(844\) 3.56753 0.122799
\(845\) −36.8274 −1.26690
\(846\) −5.40845 −0.185946
\(847\) 12.4949 0.429330
\(848\) 14.3934 0.494272
\(849\) −17.4406 −0.598560
\(850\) 4.39288 0.150674
\(851\) −94.5383 −3.24073
\(852\) 47.5599 1.62938
\(853\) −12.6939 −0.434629 −0.217315 0.976102i \(-0.569730\pi\)
−0.217315 + 0.976102i \(0.569730\pi\)
\(854\) 33.8176 1.15721
\(855\) −0.640007 −0.0218878
\(856\) 60.6815 2.07405
\(857\) 23.9043 0.816555 0.408278 0.912858i \(-0.366130\pi\)
0.408278 + 0.912858i \(0.366130\pi\)
\(858\) −51.3333 −1.75249
\(859\) −15.2214 −0.519348 −0.259674 0.965696i \(-0.583615\pi\)
−0.259674 + 0.965696i \(0.583615\pi\)
\(860\) 19.0042 0.648037
\(861\) −14.8103 −0.504733
\(862\) −72.2599 −2.46118
\(863\) 51.9666 1.76896 0.884482 0.466573i \(-0.154512\pi\)
0.884482 + 0.466573i \(0.154512\pi\)
\(864\) 20.1411 0.685213
\(865\) 16.3768 0.556828
\(866\) −41.9525 −1.42561
\(867\) 26.4043 0.896735
\(868\) −39.5164 −1.34128
\(869\) 19.0158 0.645067
\(870\) −21.6227 −0.733077
\(871\) −47.3012 −1.60274
\(872\) −10.1101 −0.342372
\(873\) 1.44726 0.0489823
\(874\) 19.3022 0.652908
\(875\) −20.7931 −0.702935
\(876\) 61.1459 2.06593
\(877\) −32.5047 −1.09761 −0.548803 0.835952i \(-0.684916\pi\)
−0.548803 + 0.835952i \(0.684916\pi\)
\(878\) 41.8682 1.41298
\(879\) 4.68828 0.158132
\(880\) 4.37088 0.147342
\(881\) −11.2237 −0.378136 −0.189068 0.981964i \(-0.560547\pi\)
−0.189068 + 0.981964i \(0.560547\pi\)
\(882\) −3.81477 −0.128450
\(883\) −0.871115 −0.0293154 −0.0146577 0.999893i \(-0.504666\pi\)
−0.0146577 + 0.999893i \(0.504666\pi\)
\(884\) 12.8181 0.431118
\(885\) 23.4562 0.788471
\(886\) 7.11127 0.238908
\(887\) −29.0509 −0.975435 −0.487717 0.873002i \(-0.662170\pi\)
−0.487717 + 0.873002i \(0.662170\pi\)
\(888\) −67.6246 −2.26933
\(889\) 32.3279 1.08424
\(890\) 53.4820 1.79272
\(891\) 15.4821 0.518669
\(892\) 23.8472 0.798465
\(893\) 4.61145 0.154316
\(894\) −41.5989 −1.39127
\(895\) 6.30649 0.210803
\(896\) −39.6867 −1.32584
\(897\) −83.4751 −2.78715
\(898\) −81.7908 −2.72939
\(899\) 25.7405 0.858495
\(900\) 5.92639 0.197546
\(901\) 5.03572 0.167764
\(902\) −24.3271 −0.810005
\(903\) −12.6704 −0.421645
\(904\) 52.7004 1.75279
\(905\) 7.87745 0.261855
\(906\) 58.2161 1.93410
\(907\) −22.6315 −0.751466 −0.375733 0.926728i \(-0.622609\pi\)
−0.375733 + 0.926728i \(0.622609\pi\)
\(908\) 69.7179 2.31367
\(909\) −3.67567 −0.121914
\(910\) −37.9342 −1.25751
\(911\) 35.3726 1.17195 0.585974 0.810330i \(-0.300712\pi\)
0.585974 + 0.810330i \(0.300712\pi\)
\(912\) 2.51899 0.0834122
\(913\) −22.9314 −0.758919
\(914\) −57.4972 −1.90184
\(915\) 15.0818 0.498589
\(916\) −56.3412 −1.86156
\(917\) −20.0928 −0.663522
\(918\) −7.27209 −0.240015
\(919\) 18.1855 0.599884 0.299942 0.953957i \(-0.403033\pi\)
0.299942 + 0.953957i \(0.403033\pi\)
\(920\) 38.9587 1.28443
\(921\) 25.6125 0.843962
\(922\) −24.7215 −0.814159
\(923\) −54.3504 −1.78897
\(924\) −23.2940 −0.766317
\(925\) 38.6233 1.26993
\(926\) −19.6987 −0.647338
\(927\) −0.108898 −0.00357668
\(928\) 16.3767 0.537591
\(929\) −1.30284 −0.0427447 −0.0213723 0.999772i \(-0.506804\pi\)
−0.0213723 + 0.999772i \(0.506804\pi\)
\(930\) −27.5032 −0.901866
\(931\) 3.25261 0.106600
\(932\) 35.7951 1.17251
\(933\) −38.9278 −1.27444
\(934\) −76.0172 −2.48736
\(935\) 1.52921 0.0500105
\(936\) 11.8579 0.387587
\(937\) 33.6337 1.09877 0.549383 0.835571i \(-0.314863\pi\)
0.549383 + 0.835571i \(0.314863\pi\)
\(938\) −33.4975 −1.09373
\(939\) −11.5119 −0.375675
\(940\) 21.1829 0.690911
\(941\) −0.701383 −0.0228644 −0.0114322 0.999935i \(-0.503639\pi\)
−0.0114322 + 0.999935i \(0.503639\pi\)
\(942\) −32.3931 −1.05542
\(943\) −39.5594 −1.28823
\(944\) 18.3338 0.596715
\(945\) 13.7903 0.448597
\(946\) −20.8123 −0.676665
\(947\) 40.6521 1.32101 0.660507 0.750820i \(-0.270341\pi\)
0.660507 + 0.750820i \(0.270341\pi\)
\(948\) 50.3410 1.63500
\(949\) −69.8762 −2.26828
\(950\) −7.88587 −0.255851
\(951\) −24.0531 −0.779977
\(952\) 3.98851 0.129268
\(953\) 43.6217 1.41305 0.706523 0.707690i \(-0.250262\pi\)
0.706523 + 0.707690i \(0.250262\pi\)
\(954\) 10.6022 0.343260
\(955\) 30.4380 0.984952
\(956\) 42.1668 1.36377
\(957\) 15.1734 0.490488
\(958\) −5.90432 −0.190760
\(959\) −1.89461 −0.0611803
\(960\) −23.9850 −0.774114
\(961\) 1.74099 0.0561610
\(962\) 175.881 5.67062
\(963\) 8.15480 0.262785
\(964\) 105.233 3.38932
\(965\) −8.71863 −0.280663
\(966\) −59.1150 −1.90199
\(967\) 24.7342 0.795400 0.397700 0.917516i \(-0.369809\pi\)
0.397700 + 0.917516i \(0.369809\pi\)
\(968\) 23.8735 0.767323
\(969\) 0.881303 0.0283115
\(970\) −8.84614 −0.284032
\(971\) −24.0966 −0.773298 −0.386649 0.922227i \(-0.626367\pi\)
−0.386649 + 0.922227i \(0.626367\pi\)
\(972\) −18.2270 −0.584630
\(973\) 0.376046 0.0120555
\(974\) −58.7654 −1.88297
\(975\) 34.1035 1.09219
\(976\) 11.7882 0.377332
\(977\) 1.96357 0.0628202 0.0314101 0.999507i \(-0.490000\pi\)
0.0314101 + 0.999507i \(0.490000\pi\)
\(978\) 66.5801 2.12900
\(979\) −37.5303 −1.19947
\(980\) 14.9410 0.477274
\(981\) −1.35867 −0.0433789
\(982\) −49.5606 −1.58154
\(983\) −21.3746 −0.681742 −0.340871 0.940110i \(-0.610722\pi\)
−0.340871 + 0.940110i \(0.610722\pi\)
\(984\) −28.2974 −0.902087
\(985\) 29.7766 0.948762
\(986\) −5.91293 −0.188306
\(987\) −14.1230 −0.449541
\(988\) −23.0103 −0.732056
\(989\) −33.8437 −1.07617
\(990\) 3.21961 0.102326
\(991\) −8.24867 −0.262028 −0.131014 0.991381i \(-0.541823\pi\)
−0.131014 + 0.991381i \(0.541823\pi\)
\(992\) 20.8305 0.661370
\(993\) −15.0756 −0.478411
\(994\) −38.4896 −1.22082
\(995\) −20.1384 −0.638431
\(996\) −60.7068 −1.92357
\(997\) 30.0472 0.951605 0.475803 0.879552i \(-0.342158\pi\)
0.475803 + 0.879552i \(0.342158\pi\)
\(998\) 9.39326 0.297338
\(999\) −63.9381 −2.02291
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 4009.2.a.f.1.9 83
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4009.2.a.f.1.9 83 1.1 even 1 trivial