Properties

Label 4033.2.a.c.1.8
Level $4033$
Weight $2$
Character 4033.1
Self dual yes
Analytic conductor $32.204$
Analytic rank $1$
Dimension $77$
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] = [4033,2,Mod(1,4033)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4033, 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("4033.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4033 = 37 \cdot 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4033.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: \(32.2036671352\)
Analytic rank: \(1\)
Dimension: \(77\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 4033.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.50194 q^{2} -1.32701 q^{3} +4.25970 q^{4} +0.884945 q^{5} +3.32011 q^{6} +3.50232 q^{7} -5.65363 q^{8} -1.23903 q^{9} +O(q^{10})\) \(q-2.50194 q^{2} -1.32701 q^{3} +4.25970 q^{4} +0.884945 q^{5} +3.32011 q^{6} +3.50232 q^{7} -5.65363 q^{8} -1.23903 q^{9} -2.21408 q^{10} -4.48523 q^{11} -5.65268 q^{12} +5.37759 q^{13} -8.76259 q^{14} -1.17434 q^{15} +5.62563 q^{16} -1.17262 q^{17} +3.09999 q^{18} -4.42541 q^{19} +3.76960 q^{20} -4.64763 q^{21} +11.2218 q^{22} +4.56091 q^{23} +7.50244 q^{24} -4.21687 q^{25} -13.4544 q^{26} +5.62526 q^{27} +14.9188 q^{28} -3.23052 q^{29} +2.93811 q^{30} +4.33076 q^{31} -2.76773 q^{32} +5.95196 q^{33} +2.93382 q^{34} +3.09936 q^{35} -5.27791 q^{36} +1.00000 q^{37} +11.0721 q^{38} -7.13614 q^{39} -5.00315 q^{40} +7.45389 q^{41} +11.6281 q^{42} -1.49518 q^{43} -19.1057 q^{44} -1.09648 q^{45} -11.4111 q^{46} -6.64711 q^{47} -7.46529 q^{48} +5.26625 q^{49} +10.5504 q^{50} +1.55608 q^{51} +22.9069 q^{52} -4.00673 q^{53} -14.0740 q^{54} -3.96918 q^{55} -19.8008 q^{56} +5.87258 q^{57} +8.08257 q^{58} -12.1204 q^{59} -5.00231 q^{60} -7.69862 q^{61} -10.8353 q^{62} -4.33949 q^{63} -4.32657 q^{64} +4.75888 q^{65} -14.8915 q^{66} +10.7112 q^{67} -4.99501 q^{68} -6.05239 q^{69} -7.75442 q^{70} -2.74529 q^{71} +7.00503 q^{72} -1.71574 q^{73} -2.50194 q^{74} +5.59585 q^{75} -18.8509 q^{76} -15.7087 q^{77} +17.8542 q^{78} +1.00469 q^{79} +4.97837 q^{80} -3.74770 q^{81} -18.6492 q^{82} -1.43104 q^{83} -19.7975 q^{84} -1.03771 q^{85} +3.74084 q^{86} +4.28695 q^{87} +25.3578 q^{88} -3.22123 q^{89} +2.74332 q^{90} +18.8341 q^{91} +19.4281 q^{92} -5.74698 q^{93} +16.6307 q^{94} -3.91625 q^{95} +3.67282 q^{96} +2.13007 q^{97} -13.1758 q^{98} +5.55735 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 77 q - 9 q^{2} - 27 q^{3} + 73 q^{4} - 16 q^{5} - 2 q^{6} - 23 q^{7} - 24 q^{8} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 77 q - 9 q^{2} - 27 q^{3} + 73 q^{4} - 16 q^{5} - 2 q^{6} - 23 q^{7} - 24 q^{8} + 66 q^{9} - 11 q^{10} - 33 q^{11} - 52 q^{12} - 10 q^{13} - 18 q^{14} - 33 q^{15} + 53 q^{16} - 44 q^{17} - 27 q^{18} - 9 q^{19} - 45 q^{20} - 7 q^{21} - 3 q^{22} - 74 q^{23} + 21 q^{24} + 59 q^{25} - 47 q^{26} - 99 q^{27} - 49 q^{28} - 9 q^{29} - 39 q^{30} - 27 q^{31} - 47 q^{32} - 28 q^{33} - 23 q^{34} - 48 q^{35} + 77 q^{36} + 77 q^{37} - 66 q^{38} - 11 q^{39} - 2 q^{40} - 37 q^{41} - 24 q^{42} - 44 q^{43} - 54 q^{44} - 36 q^{45} - 41 q^{46} - 150 q^{47} - 135 q^{48} + 64 q^{49} + 4 q^{50} + 3 q^{51} - 57 q^{52} - 72 q^{53} + 21 q^{54} - 65 q^{55} - 92 q^{56} - 13 q^{57} - 12 q^{58} - 70 q^{59} - 22 q^{60} + 15 q^{61} - 86 q^{62} - 108 q^{63} + 10 q^{64} - 53 q^{65} - 55 q^{66} - 48 q^{67} - 70 q^{68} - 2 q^{69} + 11 q^{70} - 127 q^{71} - 12 q^{72} - 33 q^{73} - 9 q^{74} - 115 q^{75} - 24 q^{76} - 40 q^{77} + 81 q^{78} - 7 q^{79} - 62 q^{80} + 53 q^{81} - 68 q^{82} - 164 q^{83} + 7 q^{84} - 9 q^{85} - 50 q^{86} - 75 q^{87} - 82 q^{88} - 26 q^{89} + 23 q^{90} + 16 q^{91} - 117 q^{92} + 19 q^{93} + 23 q^{94} - 92 q^{95} - 35 q^{96} - 19 q^{97} - 10 q^{98} - 19 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.50194 −1.76914 −0.884569 0.466409i \(-0.845547\pi\)
−0.884569 + 0.466409i \(0.845547\pi\)
\(3\) −1.32701 −0.766152 −0.383076 0.923717i \(-0.625135\pi\)
−0.383076 + 0.923717i \(0.625135\pi\)
\(4\) 4.25970 2.12985
\(5\) 0.884945 0.395760 0.197880 0.980226i \(-0.436594\pi\)
0.197880 + 0.980226i \(0.436594\pi\)
\(6\) 3.32011 1.35543
\(7\) 3.50232 1.32375 0.661876 0.749613i \(-0.269760\pi\)
0.661876 + 0.749613i \(0.269760\pi\)
\(8\) −5.65363 −1.99886
\(9\) −1.23903 −0.413011
\(10\) −2.21408 −0.700153
\(11\) −4.48523 −1.35235 −0.676174 0.736742i \(-0.736363\pi\)
−0.676174 + 0.736742i \(0.736363\pi\)
\(12\) −5.65268 −1.63179
\(13\) 5.37759 1.49148 0.745738 0.666240i \(-0.232097\pi\)
0.745738 + 0.666240i \(0.232097\pi\)
\(14\) −8.76259 −2.34190
\(15\) −1.17434 −0.303212
\(16\) 5.62563 1.40641
\(17\) −1.17262 −0.284402 −0.142201 0.989838i \(-0.545418\pi\)
−0.142201 + 0.989838i \(0.545418\pi\)
\(18\) 3.09999 0.730674
\(19\) −4.42541 −1.01526 −0.507630 0.861575i \(-0.669478\pi\)
−0.507630 + 0.861575i \(0.669478\pi\)
\(20\) 3.76960 0.842908
\(21\) −4.64763 −1.01420
\(22\) 11.2218 2.39249
\(23\) 4.56091 0.951015 0.475507 0.879712i \(-0.342264\pi\)
0.475507 + 0.879712i \(0.342264\pi\)
\(24\) 7.50244 1.53143
\(25\) −4.21687 −0.843374
\(26\) −13.4544 −2.63863
\(27\) 5.62526 1.08258
\(28\) 14.9188 2.81939
\(29\) −3.23052 −0.599893 −0.299947 0.953956i \(-0.596969\pi\)
−0.299947 + 0.953956i \(0.596969\pi\)
\(30\) 2.93811 0.536424
\(31\) 4.33076 0.777828 0.388914 0.921274i \(-0.372850\pi\)
0.388914 + 0.921274i \(0.372850\pi\)
\(32\) −2.76773 −0.489270
\(33\) 5.95196 1.03610
\(34\) 2.93382 0.503147
\(35\) 3.09936 0.523888
\(36\) −5.27791 −0.879651
\(37\) 1.00000 0.164399
\(38\) 11.0721 1.79613
\(39\) −7.13614 −1.14270
\(40\) −5.00315 −0.791067
\(41\) 7.45389 1.16410 0.582051 0.813153i \(-0.302251\pi\)
0.582051 + 0.813153i \(0.302251\pi\)
\(42\) 11.6281 1.79425
\(43\) −1.49518 −0.228012 −0.114006 0.993480i \(-0.536368\pi\)
−0.114006 + 0.993480i \(0.536368\pi\)
\(44\) −19.1057 −2.88030
\(45\) −1.09648 −0.163453
\(46\) −11.4111 −1.68248
\(47\) −6.64711 −0.969581 −0.484790 0.874630i \(-0.661104\pi\)
−0.484790 + 0.874630i \(0.661104\pi\)
\(48\) −7.46529 −1.07752
\(49\) 5.26625 0.752321
\(50\) 10.5504 1.49205
\(51\) 1.55608 0.217895
\(52\) 22.9069 3.17662
\(53\) −4.00673 −0.550367 −0.275184 0.961392i \(-0.588739\pi\)
−0.275184 + 0.961392i \(0.588739\pi\)
\(54\) −14.0740 −1.91524
\(55\) −3.96918 −0.535205
\(56\) −19.8008 −2.64599
\(57\) 5.87258 0.777843
\(58\) 8.08257 1.06129
\(59\) −12.1204 −1.57794 −0.788971 0.614430i \(-0.789386\pi\)
−0.788971 + 0.614430i \(0.789386\pi\)
\(60\) −5.00231 −0.645796
\(61\) −7.69862 −0.985708 −0.492854 0.870112i \(-0.664046\pi\)
−0.492854 + 0.870112i \(0.664046\pi\)
\(62\) −10.8353 −1.37608
\(63\) −4.33949 −0.546725
\(64\) −4.32657 −0.540821
\(65\) 4.75888 0.590266
\(66\) −14.8915 −1.83301
\(67\) 10.7112 1.30858 0.654290 0.756244i \(-0.272967\pi\)
0.654290 + 0.756244i \(0.272967\pi\)
\(68\) −4.99501 −0.605734
\(69\) −6.05239 −0.728622
\(70\) −7.75442 −0.926830
\(71\) −2.74529 −0.325806 −0.162903 0.986642i \(-0.552086\pi\)
−0.162903 + 0.986642i \(0.552086\pi\)
\(72\) 7.00503 0.825551
\(73\) −1.71574 −0.200812 −0.100406 0.994947i \(-0.532014\pi\)
−0.100406 + 0.994947i \(0.532014\pi\)
\(74\) −2.50194 −0.290844
\(75\) 5.59585 0.646153
\(76\) −18.8509 −2.16235
\(77\) −15.7087 −1.79017
\(78\) 17.8542 2.02159
\(79\) 1.00469 0.113037 0.0565185 0.998402i \(-0.482000\pi\)
0.0565185 + 0.998402i \(0.482000\pi\)
\(80\) 4.97837 0.556599
\(81\) −3.74770 −0.416411
\(82\) −18.6492 −2.05946
\(83\) −1.43104 −0.157077 −0.0785384 0.996911i \(-0.525025\pi\)
−0.0785384 + 0.996911i \(0.525025\pi\)
\(84\) −19.7975 −2.16008
\(85\) −1.03771 −0.112555
\(86\) 3.74084 0.403385
\(87\) 4.28695 0.459609
\(88\) 25.3578 2.70315
\(89\) −3.22123 −0.341450 −0.170725 0.985319i \(-0.554611\pi\)
−0.170725 + 0.985319i \(0.554611\pi\)
\(90\) 2.74332 0.289171
\(91\) 18.8341 1.97435
\(92\) 19.4281 2.02552
\(93\) −5.74698 −0.595934
\(94\) 16.6307 1.71532
\(95\) −3.91625 −0.401799
\(96\) 3.67282 0.374855
\(97\) 2.13007 0.216276 0.108138 0.994136i \(-0.465511\pi\)
0.108138 + 0.994136i \(0.465511\pi\)
\(98\) −13.1758 −1.33096
\(99\) 5.55735 0.558535
\(100\) −17.9626 −1.79626
\(101\) 3.08886 0.307353 0.153677 0.988121i \(-0.450889\pi\)
0.153677 + 0.988121i \(0.450889\pi\)
\(102\) −3.89323 −0.385487
\(103\) −11.5266 −1.13575 −0.567875 0.823115i \(-0.692234\pi\)
−0.567875 + 0.823115i \(0.692234\pi\)
\(104\) −30.4029 −2.98125
\(105\) −4.11290 −0.401378
\(106\) 10.0246 0.973676
\(107\) 0.543476 0.0525399 0.0262699 0.999655i \(-0.491637\pi\)
0.0262699 + 0.999655i \(0.491637\pi\)
\(108\) 23.9619 2.30573
\(109\) 1.00000 0.0957826
\(110\) 9.93066 0.946851
\(111\) −1.32701 −0.125955
\(112\) 19.7028 1.86174
\(113\) −12.0434 −1.13295 −0.566475 0.824079i \(-0.691693\pi\)
−0.566475 + 0.824079i \(0.691693\pi\)
\(114\) −14.6928 −1.37611
\(115\) 4.03615 0.376373
\(116\) −13.7611 −1.27768
\(117\) −6.66302 −0.615996
\(118\) 30.3245 2.79160
\(119\) −4.10689 −0.376478
\(120\) 6.63925 0.606078
\(121\) 9.11730 0.828845
\(122\) 19.2615 1.74385
\(123\) −9.89141 −0.891879
\(124\) 18.4477 1.65666
\(125\) −8.15643 −0.729533
\(126\) 10.8571 0.967231
\(127\) −3.51331 −0.311756 −0.155878 0.987776i \(-0.549821\pi\)
−0.155878 + 0.987776i \(0.549821\pi\)
\(128\) 16.3603 1.44606
\(129\) 1.98412 0.174692
\(130\) −11.9064 −1.04426
\(131\) −17.9215 −1.56581 −0.782904 0.622142i \(-0.786263\pi\)
−0.782904 + 0.622142i \(0.786263\pi\)
\(132\) 25.3536 2.20675
\(133\) −15.4992 −1.34395
\(134\) −26.7987 −2.31506
\(135\) 4.97805 0.428442
\(136\) 6.62956 0.568480
\(137\) 3.11898 0.266473 0.133236 0.991084i \(-0.457463\pi\)
0.133236 + 0.991084i \(0.457463\pi\)
\(138\) 15.1427 1.28903
\(139\) 11.9092 1.01013 0.505064 0.863082i \(-0.331469\pi\)
0.505064 + 0.863082i \(0.331469\pi\)
\(140\) 13.2023 1.11580
\(141\) 8.82081 0.742846
\(142\) 6.86855 0.576396
\(143\) −24.1197 −2.01699
\(144\) −6.97034 −0.580862
\(145\) −2.85884 −0.237413
\(146\) 4.29267 0.355264
\(147\) −6.98839 −0.576392
\(148\) 4.25970 0.350145
\(149\) −18.7177 −1.53341 −0.766705 0.642000i \(-0.778105\pi\)
−0.766705 + 0.642000i \(0.778105\pi\)
\(150\) −14.0005 −1.14313
\(151\) −4.02808 −0.327801 −0.163900 0.986477i \(-0.552408\pi\)
−0.163900 + 0.986477i \(0.552408\pi\)
\(152\) 25.0196 2.02936
\(153\) 1.45292 0.117461
\(154\) 39.3022 3.16707
\(155\) 3.83249 0.307833
\(156\) −30.3978 −2.43377
\(157\) 23.6495 1.88743 0.943717 0.330754i \(-0.107303\pi\)
0.943717 + 0.330754i \(0.107303\pi\)
\(158\) −2.51368 −0.199978
\(159\) 5.31699 0.421665
\(160\) −2.44929 −0.193633
\(161\) 15.9738 1.25891
\(162\) 9.37651 0.736688
\(163\) 6.03174 0.472442 0.236221 0.971699i \(-0.424091\pi\)
0.236221 + 0.971699i \(0.424091\pi\)
\(164\) 31.7513 2.47936
\(165\) 5.26716 0.410048
\(166\) 3.58037 0.277891
\(167\) −20.1750 −1.56119 −0.780593 0.625040i \(-0.785083\pi\)
−0.780593 + 0.625040i \(0.785083\pi\)
\(168\) 26.2760 2.02723
\(169\) 15.9185 1.22450
\(170\) 2.59627 0.199125
\(171\) 5.48323 0.419313
\(172\) −6.36900 −0.485631
\(173\) 25.8441 1.96489 0.982446 0.186546i \(-0.0597293\pi\)
0.982446 + 0.186546i \(0.0597293\pi\)
\(174\) −10.7257 −0.813112
\(175\) −14.7688 −1.11642
\(176\) −25.2322 −1.90195
\(177\) 16.0839 1.20894
\(178\) 8.05932 0.604072
\(179\) 23.1020 1.72672 0.863360 0.504588i \(-0.168355\pi\)
0.863360 + 0.504588i \(0.168355\pi\)
\(180\) −4.67066 −0.348130
\(181\) 24.1166 1.79257 0.896285 0.443478i \(-0.146256\pi\)
0.896285 + 0.443478i \(0.146256\pi\)
\(182\) −47.1216 −3.49289
\(183\) 10.2162 0.755202
\(184\) −25.7857 −1.90094
\(185\) 0.884945 0.0650625
\(186\) 14.3786 1.05429
\(187\) 5.25947 0.384611
\(188\) −28.3147 −2.06506
\(189\) 19.7015 1.43307
\(190\) 9.79821 0.710837
\(191\) −5.42464 −0.392513 −0.196257 0.980553i \(-0.562879\pi\)
−0.196257 + 0.980553i \(0.562879\pi\)
\(192\) 5.74142 0.414351
\(193\) −4.26746 −0.307179 −0.153589 0.988135i \(-0.549083\pi\)
−0.153589 + 0.988135i \(0.549083\pi\)
\(194\) −5.32930 −0.382621
\(195\) −6.31510 −0.452233
\(196\) 22.4326 1.60233
\(197\) 3.27902 0.233620 0.116810 0.993154i \(-0.462733\pi\)
0.116810 + 0.993154i \(0.462733\pi\)
\(198\) −13.9042 −0.988125
\(199\) 1.00345 0.0711324 0.0355662 0.999367i \(-0.488677\pi\)
0.0355662 + 0.999367i \(0.488677\pi\)
\(200\) 23.8406 1.68579
\(201\) −14.2139 −1.00257
\(202\) −7.72814 −0.543750
\(203\) −11.3143 −0.794110
\(204\) 6.62845 0.464084
\(205\) 6.59628 0.460704
\(206\) 28.8388 2.00930
\(207\) −5.65112 −0.392780
\(208\) 30.2523 2.09762
\(209\) 19.8490 1.37298
\(210\) 10.2902 0.710093
\(211\) −7.73143 −0.532254 −0.266127 0.963938i \(-0.585744\pi\)
−0.266127 + 0.963938i \(0.585744\pi\)
\(212\) −17.0675 −1.17220
\(213\) 3.64304 0.249617
\(214\) −1.35974 −0.0929502
\(215\) −1.32315 −0.0902380
\(216\) −31.8031 −2.16393
\(217\) 15.1677 1.02965
\(218\) −2.50194 −0.169453
\(219\) 2.27681 0.153852
\(220\) −16.9075 −1.13991
\(221\) −6.30587 −0.424179
\(222\) 3.32011 0.222831
\(223\) −23.6653 −1.58475 −0.792373 0.610036i \(-0.791155\pi\)
−0.792373 + 0.610036i \(0.791155\pi\)
\(224\) −9.69348 −0.647673
\(225\) 5.22484 0.348323
\(226\) 30.1319 2.00435
\(227\) 14.8729 0.987149 0.493574 0.869704i \(-0.335690\pi\)
0.493574 + 0.869704i \(0.335690\pi\)
\(228\) 25.0154 1.65669
\(229\) −19.5755 −1.29359 −0.646794 0.762665i \(-0.723890\pi\)
−0.646794 + 0.762665i \(0.723890\pi\)
\(230\) −10.0982 −0.665856
\(231\) 20.8457 1.37155
\(232\) 18.2642 1.19910
\(233\) −5.69124 −0.372846 −0.186423 0.982470i \(-0.559689\pi\)
−0.186423 + 0.982470i \(0.559689\pi\)
\(234\) 16.6705 1.08978
\(235\) −5.88233 −0.383721
\(236\) −51.6293 −3.36078
\(237\) −1.33324 −0.0866035
\(238\) 10.2752 0.666042
\(239\) 6.60256 0.427084 0.213542 0.976934i \(-0.431500\pi\)
0.213542 + 0.976934i \(0.431500\pi\)
\(240\) −6.60637 −0.426440
\(241\) 0.513070 0.0330497 0.0165249 0.999863i \(-0.494740\pi\)
0.0165249 + 0.999863i \(0.494740\pi\)
\(242\) −22.8109 −1.46634
\(243\) −11.9025 −0.763547
\(244\) −32.7938 −2.09941
\(245\) 4.66034 0.297738
\(246\) 24.7477 1.57786
\(247\) −23.7981 −1.51423
\(248\) −24.4845 −1.55477
\(249\) 1.89901 0.120345
\(250\) 20.4069 1.29064
\(251\) 10.6401 0.671594 0.335797 0.941934i \(-0.390994\pi\)
0.335797 + 0.941934i \(0.390994\pi\)
\(252\) −18.4849 −1.16444
\(253\) −20.4567 −1.28610
\(254\) 8.79009 0.551539
\(255\) 1.37705 0.0862342
\(256\) −32.2793 −2.01745
\(257\) −21.9428 −1.36876 −0.684379 0.729127i \(-0.739926\pi\)
−0.684379 + 0.729127i \(0.739926\pi\)
\(258\) −4.96414 −0.309054
\(259\) 3.50232 0.217624
\(260\) 20.2714 1.25718
\(261\) 4.00273 0.247763
\(262\) 44.8385 2.77013
\(263\) −15.1740 −0.935668 −0.467834 0.883816i \(-0.654966\pi\)
−0.467834 + 0.883816i \(0.654966\pi\)
\(264\) −33.6502 −2.07103
\(265\) −3.54574 −0.217813
\(266\) 38.7781 2.37764
\(267\) 4.27462 0.261602
\(268\) 45.6264 2.78708
\(269\) 0.441830 0.0269388 0.0134694 0.999909i \(-0.495712\pi\)
0.0134694 + 0.999909i \(0.495712\pi\)
\(270\) −12.4548 −0.757973
\(271\) −16.5646 −1.00623 −0.503113 0.864221i \(-0.667812\pi\)
−0.503113 + 0.864221i \(0.667812\pi\)
\(272\) −6.59673 −0.399985
\(273\) −24.9931 −1.51265
\(274\) −7.80350 −0.471427
\(275\) 18.9136 1.14054
\(276\) −25.7813 −1.55185
\(277\) 18.7002 1.12358 0.561792 0.827279i \(-0.310112\pi\)
0.561792 + 0.827279i \(0.310112\pi\)
\(278\) −29.7962 −1.78706
\(279\) −5.36596 −0.321251
\(280\) −17.5226 −1.04718
\(281\) 7.59191 0.452896 0.226448 0.974023i \(-0.427289\pi\)
0.226448 + 0.974023i \(0.427289\pi\)
\(282\) −22.0691 −1.31420
\(283\) −10.4888 −0.623496 −0.311748 0.950165i \(-0.600914\pi\)
−0.311748 + 0.950165i \(0.600914\pi\)
\(284\) −11.6941 −0.693917
\(285\) 5.19692 0.307839
\(286\) 60.3461 3.56834
\(287\) 26.1059 1.54098
\(288\) 3.42931 0.202074
\(289\) −15.6250 −0.919115
\(290\) 7.15263 0.420017
\(291\) −2.82663 −0.165700
\(292\) −7.30853 −0.427699
\(293\) −8.13804 −0.475429 −0.237715 0.971335i \(-0.576398\pi\)
−0.237715 + 0.971335i \(0.576398\pi\)
\(294\) 17.4845 1.01972
\(295\) −10.7259 −0.624486
\(296\) −5.65363 −0.328610
\(297\) −25.2306 −1.46403
\(298\) 46.8304 2.71281
\(299\) 24.5267 1.41842
\(300\) 23.8366 1.37621
\(301\) −5.23658 −0.301832
\(302\) 10.0780 0.579925
\(303\) −4.09896 −0.235479
\(304\) −24.8957 −1.42787
\(305\) −6.81286 −0.390103
\(306\) −3.63511 −0.207805
\(307\) 3.43091 0.195812 0.0979061 0.995196i \(-0.468786\pi\)
0.0979061 + 0.995196i \(0.468786\pi\)
\(308\) −66.9144 −3.81280
\(309\) 15.2960 0.870157
\(310\) −9.58865 −0.544599
\(311\) 23.0456 1.30679 0.653397 0.757015i \(-0.273343\pi\)
0.653397 + 0.757015i \(0.273343\pi\)
\(312\) 40.3451 2.28409
\(313\) −15.0625 −0.851381 −0.425690 0.904869i \(-0.639969\pi\)
−0.425690 + 0.904869i \(0.639969\pi\)
\(314\) −59.1696 −3.33913
\(315\) −3.84021 −0.216372
\(316\) 4.27969 0.240752
\(317\) −2.32553 −0.130615 −0.0653074 0.997865i \(-0.520803\pi\)
−0.0653074 + 0.997865i \(0.520803\pi\)
\(318\) −13.3028 −0.745984
\(319\) 14.4896 0.811264
\(320\) −3.82878 −0.214035
\(321\) −0.721201 −0.0402535
\(322\) −39.9654 −2.22718
\(323\) 5.18933 0.288742
\(324\) −15.9641 −0.886892
\(325\) −22.6766 −1.25787
\(326\) −15.0910 −0.835815
\(327\) −1.32701 −0.0733841
\(328\) −42.1415 −2.32687
\(329\) −23.2803 −1.28349
\(330\) −13.1781 −0.725432
\(331\) −4.58489 −0.252008 −0.126004 0.992030i \(-0.540215\pi\)
−0.126004 + 0.992030i \(0.540215\pi\)
\(332\) −6.09579 −0.334550
\(333\) −1.23903 −0.0678986
\(334\) 50.4765 2.76195
\(335\) 9.47882 0.517883
\(336\) −26.1458 −1.42637
\(337\) −27.8529 −1.51724 −0.758622 0.651531i \(-0.774127\pi\)
−0.758622 + 0.651531i \(0.774127\pi\)
\(338\) −39.8271 −2.16631
\(339\) 15.9818 0.868012
\(340\) −4.42031 −0.239725
\(341\) −19.4245 −1.05189
\(342\) −13.7187 −0.741823
\(343\) −6.07215 −0.327865
\(344\) 8.45316 0.455764
\(345\) −5.35603 −0.288359
\(346\) −64.6604 −3.47617
\(347\) −22.8303 −1.22560 −0.612798 0.790239i \(-0.709956\pi\)
−0.612798 + 0.790239i \(0.709956\pi\)
\(348\) 18.2611 0.978898
\(349\) 9.52748 0.509994 0.254997 0.966942i \(-0.417925\pi\)
0.254997 + 0.966942i \(0.417925\pi\)
\(350\) 36.9507 1.97510
\(351\) 30.2503 1.61464
\(352\) 12.4139 0.661663
\(353\) 27.6987 1.47425 0.737127 0.675754i \(-0.236182\pi\)
0.737127 + 0.675754i \(0.236182\pi\)
\(354\) −40.2410 −2.13879
\(355\) −2.42943 −0.128941
\(356\) −13.7215 −0.727236
\(357\) 5.44990 0.288440
\(358\) −57.7997 −3.05481
\(359\) −25.1704 −1.32844 −0.664222 0.747536i \(-0.731237\pi\)
−0.664222 + 0.747536i \(0.731237\pi\)
\(360\) 6.19907 0.326720
\(361\) 0.584268 0.0307509
\(362\) −60.3382 −3.17130
\(363\) −12.0988 −0.635021
\(364\) 80.2274 4.20506
\(365\) −1.51833 −0.0794733
\(366\) −25.5603 −1.33606
\(367\) −32.2517 −1.68352 −0.841762 0.539849i \(-0.818481\pi\)
−0.841762 + 0.539849i \(0.818481\pi\)
\(368\) 25.6580 1.33751
\(369\) −9.23561 −0.480787
\(370\) −2.21408 −0.115105
\(371\) −14.0329 −0.728550
\(372\) −24.4804 −1.26925
\(373\) −7.45951 −0.386239 −0.193120 0.981175i \(-0.561861\pi\)
−0.193120 + 0.981175i \(0.561861\pi\)
\(374\) −13.1589 −0.680429
\(375\) 10.8237 0.558933
\(376\) 37.5803 1.93805
\(377\) −17.3724 −0.894726
\(378\) −49.2918 −2.53530
\(379\) −14.3885 −0.739085 −0.369543 0.929214i \(-0.620486\pi\)
−0.369543 + 0.929214i \(0.620486\pi\)
\(380\) −16.6820 −0.855770
\(381\) 4.66221 0.238852
\(382\) 13.5721 0.694410
\(383\) 32.1230 1.64141 0.820704 0.571353i \(-0.193581\pi\)
0.820704 + 0.571353i \(0.193581\pi\)
\(384\) −21.7103 −1.10790
\(385\) −13.9014 −0.708479
\(386\) 10.6769 0.543442
\(387\) 1.85257 0.0941716
\(388\) 9.07344 0.460634
\(389\) 33.8430 1.71591 0.857954 0.513727i \(-0.171736\pi\)
0.857954 + 0.513727i \(0.171736\pi\)
\(390\) 15.8000 0.800063
\(391\) −5.34821 −0.270471
\(392\) −29.7734 −1.50378
\(393\) 23.7821 1.19965
\(394\) −8.20390 −0.413307
\(395\) 0.889100 0.0447355
\(396\) 23.6726 1.18959
\(397\) 31.5540 1.58365 0.791824 0.610750i \(-0.209132\pi\)
0.791824 + 0.610750i \(0.209132\pi\)
\(398\) −2.51056 −0.125843
\(399\) 20.5677 1.02967
\(400\) −23.7226 −1.18613
\(401\) −39.7735 −1.98619 −0.993097 0.117298i \(-0.962577\pi\)
−0.993097 + 0.117298i \(0.962577\pi\)
\(402\) 35.5623 1.77369
\(403\) 23.2891 1.16011
\(404\) 13.1576 0.654616
\(405\) −3.31651 −0.164799
\(406\) 28.3078 1.40489
\(407\) −4.48523 −0.222325
\(408\) −8.79752 −0.435542
\(409\) 13.2685 0.656085 0.328042 0.944663i \(-0.393611\pi\)
0.328042 + 0.944663i \(0.393611\pi\)
\(410\) −16.5035 −0.815049
\(411\) −4.13893 −0.204159
\(412\) −49.0998 −2.41897
\(413\) −42.4495 −2.08881
\(414\) 14.1387 0.694881
\(415\) −1.26639 −0.0621647
\(416\) −14.8837 −0.729734
\(417\) −15.8037 −0.773912
\(418\) −49.6610 −2.42900
\(419\) −21.6152 −1.05597 −0.527986 0.849253i \(-0.677052\pi\)
−0.527986 + 0.849253i \(0.677052\pi\)
\(420\) −17.5197 −0.854874
\(421\) 22.3574 1.08964 0.544818 0.838555i \(-0.316599\pi\)
0.544818 + 0.838555i \(0.316599\pi\)
\(422\) 19.3436 0.941631
\(423\) 8.23599 0.400448
\(424\) 22.6526 1.10011
\(425\) 4.94479 0.239858
\(426\) −9.11466 −0.441607
\(427\) −26.9630 −1.30483
\(428\) 2.31505 0.111902
\(429\) 32.0072 1.54532
\(430\) 3.31044 0.159643
\(431\) −6.13660 −0.295590 −0.147795 0.989018i \(-0.547218\pi\)
−0.147795 + 0.989018i \(0.547218\pi\)
\(432\) 31.6456 1.52255
\(433\) −33.9002 −1.62914 −0.814571 0.580064i \(-0.803027\pi\)
−0.814571 + 0.580064i \(0.803027\pi\)
\(434\) −37.9487 −1.82160
\(435\) 3.79372 0.181895
\(436\) 4.25970 0.204003
\(437\) −20.1839 −0.965526
\(438\) −5.69644 −0.272186
\(439\) −12.3584 −0.589833 −0.294916 0.955523i \(-0.595292\pi\)
−0.294916 + 0.955523i \(0.595292\pi\)
\(440\) 22.4403 1.06980
\(441\) −6.52506 −0.310717
\(442\) 15.7769 0.750431
\(443\) −32.8474 −1.56063 −0.780314 0.625388i \(-0.784941\pi\)
−0.780314 + 0.625388i \(0.784941\pi\)
\(444\) −5.65268 −0.268264
\(445\) −2.85061 −0.135132
\(446\) 59.2092 2.80364
\(447\) 24.8386 1.17483
\(448\) −15.1530 −0.715913
\(449\) 0.917788 0.0433131 0.0216566 0.999765i \(-0.493106\pi\)
0.0216566 + 0.999765i \(0.493106\pi\)
\(450\) −13.0722 −0.616231
\(451\) −33.4324 −1.57427
\(452\) −51.3014 −2.41301
\(453\) 5.34532 0.251145
\(454\) −37.2111 −1.74640
\(455\) 16.6671 0.781366
\(456\) −33.2014 −1.55480
\(457\) −0.783520 −0.0366515 −0.0183258 0.999832i \(-0.505834\pi\)
−0.0183258 + 0.999832i \(0.505834\pi\)
\(458\) 48.9768 2.28853
\(459\) −6.59629 −0.307889
\(460\) 17.1928 0.801618
\(461\) −20.2801 −0.944538 −0.472269 0.881454i \(-0.656565\pi\)
−0.472269 + 0.881454i \(0.656565\pi\)
\(462\) −52.1546 −2.42645
\(463\) 17.8543 0.829759 0.414880 0.909876i \(-0.363824\pi\)
0.414880 + 0.909876i \(0.363824\pi\)
\(464\) −18.1737 −0.843694
\(465\) −5.08576 −0.235847
\(466\) 14.2391 0.659616
\(467\) −31.8193 −1.47242 −0.736210 0.676753i \(-0.763386\pi\)
−0.736210 + 0.676753i \(0.763386\pi\)
\(468\) −28.3824 −1.31198
\(469\) 37.5140 1.73224
\(470\) 14.7172 0.678855
\(471\) −31.3832 −1.44606
\(472\) 68.5242 3.15408
\(473\) 6.70621 0.308352
\(474\) 3.33569 0.153214
\(475\) 18.6614 0.856243
\(476\) −17.4941 −0.801842
\(477\) 4.96448 0.227308
\(478\) −16.5192 −0.755571
\(479\) −29.8291 −1.36293 −0.681464 0.731852i \(-0.738656\pi\)
−0.681464 + 0.731852i \(0.738656\pi\)
\(480\) 3.25024 0.148353
\(481\) 5.37759 0.245197
\(482\) −1.28367 −0.0584695
\(483\) −21.1974 −0.964515
\(484\) 38.8369 1.76531
\(485\) 1.88499 0.0855931
\(486\) 29.7794 1.35082
\(487\) −8.76289 −0.397084 −0.198542 0.980092i \(-0.563621\pi\)
−0.198542 + 0.980092i \(0.563621\pi\)
\(488\) 43.5251 1.97029
\(489\) −8.00420 −0.361963
\(490\) −11.6599 −0.526740
\(491\) 10.5491 0.476076 0.238038 0.971256i \(-0.423496\pi\)
0.238038 + 0.971256i \(0.423496\pi\)
\(492\) −42.1344 −1.89957
\(493\) 3.78818 0.170611
\(494\) 59.5413 2.67889
\(495\) 4.91795 0.221046
\(496\) 24.3633 1.09394
\(497\) −9.61488 −0.431286
\(498\) −4.75120 −0.212906
\(499\) 4.48063 0.200580 0.100290 0.994958i \(-0.468023\pi\)
0.100290 + 0.994958i \(0.468023\pi\)
\(500\) −34.7439 −1.55380
\(501\) 26.7725 1.19610
\(502\) −26.6208 −1.18814
\(503\) 11.3964 0.508141 0.254071 0.967186i \(-0.418230\pi\)
0.254071 + 0.967186i \(0.418230\pi\)
\(504\) 24.5339 1.09283
\(505\) 2.73347 0.121638
\(506\) 51.1815 2.27529
\(507\) −21.1241 −0.938153
\(508\) −14.9656 −0.663993
\(509\) −15.3577 −0.680717 −0.340358 0.940296i \(-0.610548\pi\)
−0.340358 + 0.940296i \(0.610548\pi\)
\(510\) −3.44529 −0.152560
\(511\) −6.00906 −0.265825
\(512\) 48.0402 2.12310
\(513\) −24.8941 −1.09910
\(514\) 54.8997 2.42152
\(515\) −10.2004 −0.449484
\(516\) 8.45175 0.372067
\(517\) 29.8138 1.31121
\(518\) −8.76259 −0.385006
\(519\) −34.2955 −1.50541
\(520\) −26.9049 −1.17986
\(521\) 40.8593 1.79008 0.895039 0.445988i \(-0.147148\pi\)
0.895039 + 0.445988i \(0.147148\pi\)
\(522\) −10.0146 −0.438326
\(523\) 8.08457 0.353513 0.176757 0.984255i \(-0.443439\pi\)
0.176757 + 0.984255i \(0.443439\pi\)
\(524\) −76.3402 −3.33494
\(525\) 19.5985 0.855347
\(526\) 37.9644 1.65533
\(527\) −5.07834 −0.221216
\(528\) 33.4835 1.45718
\(529\) −2.19813 −0.0955708
\(530\) 8.87123 0.385342
\(531\) 15.0176 0.651708
\(532\) −66.0219 −2.86241
\(533\) 40.0840 1.73623
\(534\) −10.6948 −0.462811
\(535\) 0.480947 0.0207932
\(536\) −60.5571 −2.61567
\(537\) −30.6566 −1.32293
\(538\) −1.10543 −0.0476585
\(539\) −23.6203 −1.01740
\(540\) 21.2050 0.912517
\(541\) −17.6935 −0.760704 −0.380352 0.924842i \(-0.624197\pi\)
−0.380352 + 0.924842i \(0.624197\pi\)
\(542\) 41.4435 1.78015
\(543\) −32.0030 −1.37338
\(544\) 3.24550 0.139149
\(545\) 0.884945 0.0379069
\(546\) 62.5311 2.67608
\(547\) 0.740083 0.0316437 0.0158218 0.999875i \(-0.494964\pi\)
0.0158218 + 0.999875i \(0.494964\pi\)
\(548\) 13.2859 0.567547
\(549\) 9.53885 0.407108
\(550\) −47.3208 −2.01776
\(551\) 14.2964 0.609047
\(552\) 34.2179 1.45641
\(553\) 3.51876 0.149633
\(554\) −46.7867 −1.98778
\(555\) −1.17434 −0.0498477
\(556\) 50.7298 2.15142
\(557\) −22.8839 −0.969620 −0.484810 0.874619i \(-0.661111\pi\)
−0.484810 + 0.874619i \(0.661111\pi\)
\(558\) 13.4253 0.568338
\(559\) −8.04044 −0.340075
\(560\) 17.4359 0.736800
\(561\) −6.97940 −0.294670
\(562\) −18.9945 −0.801235
\(563\) 7.02549 0.296089 0.148045 0.988981i \(-0.452702\pi\)
0.148045 + 0.988981i \(0.452702\pi\)
\(564\) 37.5740 1.58215
\(565\) −10.6578 −0.448376
\(566\) 26.2424 1.10305
\(567\) −13.1256 −0.551225
\(568\) 15.5208 0.651240
\(569\) 24.2155 1.01517 0.507584 0.861602i \(-0.330539\pi\)
0.507584 + 0.861602i \(0.330539\pi\)
\(570\) −13.0024 −0.544609
\(571\) 11.8109 0.494272 0.247136 0.968981i \(-0.420511\pi\)
0.247136 + 0.968981i \(0.420511\pi\)
\(572\) −102.743 −4.29589
\(573\) 7.19858 0.300725
\(574\) −65.3154 −2.72621
\(575\) −19.2328 −0.802061
\(576\) 5.36076 0.223365
\(577\) 10.0171 0.417016 0.208508 0.978021i \(-0.433139\pi\)
0.208508 + 0.978021i \(0.433139\pi\)
\(578\) 39.0927 1.62604
\(579\) 5.66298 0.235346
\(580\) −12.1778 −0.505655
\(581\) −5.01196 −0.207931
\(582\) 7.07205 0.293146
\(583\) 17.9711 0.744288
\(584\) 9.70014 0.401395
\(585\) −5.89641 −0.243786
\(586\) 20.3609 0.841100
\(587\) 40.5417 1.67334 0.836668 0.547711i \(-0.184501\pi\)
0.836668 + 0.547711i \(0.184501\pi\)
\(588\) −29.7684 −1.22763
\(589\) −19.1654 −0.789697
\(590\) 26.8355 1.10480
\(591\) −4.35130 −0.178989
\(592\) 5.62563 0.231212
\(593\) 38.0633 1.56307 0.781535 0.623861i \(-0.214437\pi\)
0.781535 + 0.623861i \(0.214437\pi\)
\(594\) 63.1254 2.59006
\(595\) −3.63438 −0.148995
\(596\) −79.7316 −3.26593
\(597\) −1.33159 −0.0544982
\(598\) −61.3643 −2.50937
\(599\) −27.4896 −1.12320 −0.561598 0.827410i \(-0.689813\pi\)
−0.561598 + 0.827410i \(0.689813\pi\)
\(600\) −31.6368 −1.29157
\(601\) 23.2242 0.947335 0.473667 0.880704i \(-0.342930\pi\)
0.473667 + 0.880704i \(0.342930\pi\)
\(602\) 13.1016 0.533982
\(603\) −13.2715 −0.540458
\(604\) −17.1584 −0.698166
\(605\) 8.06831 0.328023
\(606\) 10.2554 0.416595
\(607\) −17.0012 −0.690057 −0.345028 0.938592i \(-0.612131\pi\)
−0.345028 + 0.938592i \(0.612131\pi\)
\(608\) 12.2483 0.496736
\(609\) 15.0143 0.608409
\(610\) 17.0454 0.690146
\(611\) −35.7454 −1.44611
\(612\) 6.18898 0.250175
\(613\) −42.2179 −1.70516 −0.852582 0.522594i \(-0.824964\pi\)
−0.852582 + 0.522594i \(0.824964\pi\)
\(614\) −8.58392 −0.346419
\(615\) −8.75336 −0.352970
\(616\) 88.8112 3.57831
\(617\) −21.1010 −0.849496 −0.424748 0.905312i \(-0.639637\pi\)
−0.424748 + 0.905312i \(0.639637\pi\)
\(618\) −38.2696 −1.53943
\(619\) −32.5241 −1.30726 −0.653628 0.756816i \(-0.726754\pi\)
−0.653628 + 0.756816i \(0.726754\pi\)
\(620\) 16.3252 0.655637
\(621\) 25.6563 1.02955
\(622\) −57.6586 −2.31190
\(623\) −11.2818 −0.451995
\(624\) −40.1453 −1.60710
\(625\) 13.8664 0.554655
\(626\) 37.6854 1.50621
\(627\) −26.3399 −1.05191
\(628\) 100.740 4.01995
\(629\) −1.17262 −0.0467554
\(630\) 9.60798 0.382791
\(631\) −43.3737 −1.72668 −0.863340 0.504623i \(-0.831631\pi\)
−0.863340 + 0.504623i \(0.831631\pi\)
\(632\) −5.68017 −0.225945
\(633\) 10.2597 0.407787
\(634\) 5.81834 0.231076
\(635\) −3.10909 −0.123380
\(636\) 22.6488 0.898083
\(637\) 28.3197 1.12207
\(638\) −36.2522 −1.43524
\(639\) 3.40151 0.134561
\(640\) 14.4779 0.572291
\(641\) 7.80477 0.308270 0.154135 0.988050i \(-0.450741\pi\)
0.154135 + 0.988050i \(0.450741\pi\)
\(642\) 1.80440 0.0712140
\(643\) −10.8545 −0.428059 −0.214030 0.976827i \(-0.568659\pi\)
−0.214030 + 0.976827i \(0.568659\pi\)
\(644\) 68.0434 2.68128
\(645\) 1.75584 0.0691360
\(646\) −12.9834 −0.510824
\(647\) 9.68352 0.380699 0.190349 0.981716i \(-0.439038\pi\)
0.190349 + 0.981716i \(0.439038\pi\)
\(648\) 21.1881 0.832346
\(649\) 54.3628 2.13393
\(650\) 56.7355 2.22535
\(651\) −20.1278 −0.788869
\(652\) 25.6934 1.00623
\(653\) −22.8846 −0.895545 −0.447773 0.894147i \(-0.647783\pi\)
−0.447773 + 0.894147i \(0.647783\pi\)
\(654\) 3.32011 0.129827
\(655\) −15.8595 −0.619684
\(656\) 41.9328 1.63720
\(657\) 2.12586 0.0829376
\(658\) 58.2459 2.27066
\(659\) −18.3564 −0.715065 −0.357533 0.933901i \(-0.616382\pi\)
−0.357533 + 0.933901i \(0.616382\pi\)
\(660\) 22.4365 0.873341
\(661\) −31.1275 −1.21072 −0.605360 0.795952i \(-0.706971\pi\)
−0.605360 + 0.795952i \(0.706971\pi\)
\(662\) 11.4711 0.445838
\(663\) 8.36798 0.324986
\(664\) 8.09056 0.313974
\(665\) −13.7160 −0.531882
\(666\) 3.09999 0.120122
\(667\) −14.7341 −0.570507
\(668\) −85.9392 −3.32509
\(669\) 31.4042 1.21416
\(670\) −23.7154 −0.916207
\(671\) 34.5301 1.33302
\(672\) 12.8634 0.496216
\(673\) −33.9278 −1.30782 −0.653910 0.756573i \(-0.726872\pi\)
−0.653910 + 0.756573i \(0.726872\pi\)
\(674\) 69.6863 2.68421
\(675\) −23.7210 −0.913021
\(676\) 67.8080 2.60800
\(677\) −26.8990 −1.03381 −0.516906 0.856042i \(-0.672916\pi\)
−0.516906 + 0.856042i \(0.672916\pi\)
\(678\) −39.9855 −1.53563
\(679\) 7.46018 0.286295
\(680\) 5.86680 0.224981
\(681\) −19.7365 −0.756306
\(682\) 48.5988 1.86094
\(683\) −9.41892 −0.360405 −0.180202 0.983630i \(-0.557675\pi\)
−0.180202 + 0.983630i \(0.557675\pi\)
\(684\) 23.3569 0.893074
\(685\) 2.76013 0.105459
\(686\) 15.1921 0.580039
\(687\) 25.9770 0.991084
\(688\) −8.41130 −0.320678
\(689\) −21.5466 −0.820859
\(690\) 13.4005 0.510147
\(691\) 27.8333 1.05883 0.529415 0.848363i \(-0.322412\pi\)
0.529415 + 0.848363i \(0.322412\pi\)
\(692\) 110.088 4.18492
\(693\) 19.4636 0.739362
\(694\) 57.1201 2.16825
\(695\) 10.5390 0.399768
\(696\) −24.2368 −0.918694
\(697\) −8.74058 −0.331073
\(698\) −23.8372 −0.902250
\(699\) 7.55236 0.285657
\(700\) −62.9108 −2.37780
\(701\) 28.0050 1.05773 0.528866 0.848705i \(-0.322617\pi\)
0.528866 + 0.848705i \(0.322617\pi\)
\(702\) −75.6845 −2.85653
\(703\) −4.42541 −0.166908
\(704\) 19.4057 0.731378
\(705\) 7.80593 0.293989
\(706\) −69.3005 −2.60816
\(707\) 10.8182 0.406860
\(708\) 68.5127 2.57487
\(709\) −20.9124 −0.785381 −0.392690 0.919671i \(-0.628456\pi\)
−0.392690 + 0.919671i \(0.628456\pi\)
\(710\) 6.07829 0.228114
\(711\) −1.24485 −0.0466855
\(712\) 18.2116 0.682510
\(713\) 19.7522 0.739726
\(714\) −13.6353 −0.510289
\(715\) −21.3447 −0.798245
\(716\) 98.4073 3.67765
\(717\) −8.76169 −0.327211
\(718\) 62.9748 2.35020
\(719\) 34.2369 1.27682 0.638411 0.769696i \(-0.279592\pi\)
0.638411 + 0.769696i \(0.279592\pi\)
\(720\) −6.16837 −0.229882
\(721\) −40.3698 −1.50345
\(722\) −1.46180 −0.0544026
\(723\) −0.680851 −0.0253211
\(724\) 102.729 3.81790
\(725\) 13.6227 0.505934
\(726\) 30.2704 1.12344
\(727\) 19.5323 0.724411 0.362206 0.932098i \(-0.382024\pi\)
0.362206 + 0.932098i \(0.382024\pi\)
\(728\) −106.481 −3.94644
\(729\) 27.0379 1.00140
\(730\) 3.79878 0.140599
\(731\) 1.75327 0.0648472
\(732\) 43.5178 1.60847
\(733\) 27.5561 1.01781 0.508903 0.860824i \(-0.330051\pi\)
0.508903 + 0.860824i \(0.330051\pi\)
\(734\) 80.6917 2.97839
\(735\) −6.18434 −0.228113
\(736\) −12.6234 −0.465303
\(737\) −48.0422 −1.76966
\(738\) 23.1069 0.850578
\(739\) −40.1398 −1.47657 −0.738284 0.674490i \(-0.764364\pi\)
−0.738284 + 0.674490i \(0.764364\pi\)
\(740\) 3.76960 0.138573
\(741\) 31.5804 1.16013
\(742\) 35.1094 1.28891
\(743\) −19.5049 −0.715564 −0.357782 0.933805i \(-0.616467\pi\)
−0.357782 + 0.933805i \(0.616467\pi\)
\(744\) 32.4913 1.19119
\(745\) −16.5641 −0.606862
\(746\) 18.6632 0.683310
\(747\) 1.77310 0.0648745
\(748\) 22.4038 0.819163
\(749\) 1.90343 0.0695498
\(750\) −27.0802 −0.988830
\(751\) −23.6160 −0.861759 −0.430879 0.902410i \(-0.641797\pi\)
−0.430879 + 0.902410i \(0.641797\pi\)
\(752\) −37.3942 −1.36363
\(753\) −14.1195 −0.514543
\(754\) 43.4648 1.58289
\(755\) −3.56463 −0.129730
\(756\) 83.9222 3.05222
\(757\) −26.2217 −0.953043 −0.476521 0.879163i \(-0.658103\pi\)
−0.476521 + 0.879163i \(0.658103\pi\)
\(758\) 35.9990 1.30754
\(759\) 27.1464 0.985350
\(760\) 22.1410 0.803138
\(761\) −35.4323 −1.28442 −0.642210 0.766529i \(-0.721982\pi\)
−0.642210 + 0.766529i \(0.721982\pi\)
\(762\) −11.6646 −0.422563
\(763\) 3.50232 0.126793
\(764\) −23.1073 −0.835994
\(765\) 1.28575 0.0464864
\(766\) −80.3698 −2.90388
\(767\) −65.1786 −2.35346
\(768\) 42.8350 1.54568
\(769\) 26.6840 0.962248 0.481124 0.876652i \(-0.340229\pi\)
0.481124 + 0.876652i \(0.340229\pi\)
\(770\) 34.7803 1.25340
\(771\) 29.1185 1.04868
\(772\) −18.1781 −0.654244
\(773\) −26.0098 −0.935508 −0.467754 0.883859i \(-0.654937\pi\)
−0.467754 + 0.883859i \(0.654937\pi\)
\(774\) −4.63502 −0.166602
\(775\) −18.2623 −0.656000
\(776\) −12.0426 −0.432304
\(777\) −4.64763 −0.166733
\(778\) −84.6731 −3.03568
\(779\) −32.9865 −1.18186
\(780\) −26.9004 −0.963189
\(781\) 12.3133 0.440603
\(782\) 13.3809 0.478500
\(783\) −18.1725 −0.649433
\(784\) 29.6260 1.05807
\(785\) 20.9285 0.746970
\(786\) −59.5013 −2.12234
\(787\) −16.0989 −0.573865 −0.286932 0.957951i \(-0.592636\pi\)
−0.286932 + 0.957951i \(0.592636\pi\)
\(788\) 13.9676 0.497576
\(789\) 20.1361 0.716864
\(790\) −2.22447 −0.0791432
\(791\) −42.1799 −1.49975
\(792\) −31.4192 −1.11643
\(793\) −41.4001 −1.47016
\(794\) −78.9461 −2.80169
\(795\) 4.70525 0.166878
\(796\) 4.27438 0.151501
\(797\) 7.50731 0.265922 0.132961 0.991121i \(-0.457551\pi\)
0.132961 + 0.991121i \(0.457551\pi\)
\(798\) −51.4591 −1.82163
\(799\) 7.79454 0.275751
\(800\) 11.6712 0.412638
\(801\) 3.99121 0.141023
\(802\) 99.5108 3.51385
\(803\) 7.69548 0.271568
\(804\) −60.5469 −2.13533
\(805\) 14.1359 0.498225
\(806\) −58.2678 −2.05240
\(807\) −0.586314 −0.0206392
\(808\) −17.4633 −0.614356
\(809\) −5.93362 −0.208615 −0.104308 0.994545i \(-0.533263\pi\)
−0.104308 + 0.994545i \(0.533263\pi\)
\(810\) 8.29770 0.291551
\(811\) −34.4794 −1.21074 −0.605368 0.795946i \(-0.706974\pi\)
−0.605368 + 0.795946i \(0.706974\pi\)
\(812\) −48.1956 −1.69133
\(813\) 21.9814 0.770921
\(814\) 11.2218 0.393323
\(815\) 5.33776 0.186974
\(816\) 8.75395 0.306450
\(817\) 6.61677 0.231491
\(818\) −33.1970 −1.16070
\(819\) −23.3360 −0.815427
\(820\) 28.0982 0.981231
\(821\) 53.0957 1.85305 0.926526 0.376230i \(-0.122780\pi\)
0.926526 + 0.376230i \(0.122780\pi\)
\(822\) 10.3554 0.361185
\(823\) 6.01333 0.209611 0.104806 0.994493i \(-0.466578\pi\)
0.104806 + 0.994493i \(0.466578\pi\)
\(824\) 65.1671 2.27020
\(825\) −25.0987 −0.873824
\(826\) 106.206 3.69538
\(827\) −16.6818 −0.580083 −0.290042 0.957014i \(-0.593669\pi\)
−0.290042 + 0.957014i \(0.593669\pi\)
\(828\) −24.0720 −0.836561
\(829\) 15.8589 0.550803 0.275401 0.961329i \(-0.411189\pi\)
0.275401 + 0.961329i \(0.411189\pi\)
\(830\) 3.16843 0.109978
\(831\) −24.8154 −0.860836
\(832\) −23.2665 −0.806622
\(833\) −6.17531 −0.213962
\(834\) 39.5400 1.36916
\(835\) −17.8537 −0.617854
\(836\) 84.5507 2.92425
\(837\) 24.3616 0.842062
\(838\) 54.0799 1.86816
\(839\) 43.5188 1.50243 0.751217 0.660055i \(-0.229467\pi\)
0.751217 + 0.660055i \(0.229467\pi\)
\(840\) 23.2528 0.802297
\(841\) −18.5637 −0.640128
\(842\) −55.9369 −1.92771
\(843\) −10.0746 −0.346987
\(844\) −32.9336 −1.13362
\(845\) 14.0870 0.484608
\(846\) −20.6059 −0.708447
\(847\) 31.9317 1.09719
\(848\) −22.5404 −0.774041
\(849\) 13.9188 0.477693
\(850\) −12.3716 −0.424341
\(851\) 4.56091 0.156346
\(852\) 15.5182 0.531646
\(853\) 45.8385 1.56948 0.784741 0.619824i \(-0.212796\pi\)
0.784741 + 0.619824i \(0.212796\pi\)
\(854\) 67.4599 2.30843
\(855\) 4.85236 0.165947
\(856\) −3.07261 −0.105020
\(857\) 11.1379 0.380463 0.190232 0.981739i \(-0.439076\pi\)
0.190232 + 0.981739i \(0.439076\pi\)
\(858\) −80.0802 −2.73389
\(859\) −28.7456 −0.980786 −0.490393 0.871501i \(-0.663147\pi\)
−0.490393 + 0.871501i \(0.663147\pi\)
\(860\) −5.63621 −0.192193
\(861\) −34.6429 −1.18063
\(862\) 15.3534 0.522939
\(863\) −37.8318 −1.28781 −0.643905 0.765106i \(-0.722687\pi\)
−0.643905 + 0.765106i \(0.722687\pi\)
\(864\) −15.5692 −0.529675
\(865\) 22.8706 0.777625
\(866\) 84.8163 2.88218
\(867\) 20.7345 0.704182
\(868\) 64.6099 2.19300
\(869\) −4.50629 −0.152865
\(870\) −9.49165 −0.321797
\(871\) 57.6004 1.95172
\(872\) −5.65363 −0.191456
\(873\) −2.63922 −0.0893242
\(874\) 50.4989 1.70815
\(875\) −28.5664 −0.965721
\(876\) 9.69852 0.327683
\(877\) 23.6325 0.798014 0.399007 0.916948i \(-0.369355\pi\)
0.399007 + 0.916948i \(0.369355\pi\)
\(878\) 30.9199 1.04350
\(879\) 10.7993 0.364251
\(880\) −22.3292 −0.752716
\(881\) −49.4645 −1.66650 −0.833250 0.552896i \(-0.813523\pi\)
−0.833250 + 0.552896i \(0.813523\pi\)
\(882\) 16.3253 0.549701
\(883\) 26.5051 0.891968 0.445984 0.895041i \(-0.352854\pi\)
0.445984 + 0.895041i \(0.352854\pi\)
\(884\) −26.8611 −0.903437
\(885\) 14.2334 0.478451
\(886\) 82.1823 2.76097
\(887\) −13.5636 −0.455420 −0.227710 0.973729i \(-0.573124\pi\)
−0.227710 + 0.973729i \(0.573124\pi\)
\(888\) 7.50244 0.251765
\(889\) −12.3047 −0.412688
\(890\) 7.13206 0.239067
\(891\) 16.8093 0.563132
\(892\) −100.807 −3.37527
\(893\) 29.4162 0.984376
\(894\) −62.1446 −2.07843
\(895\) 20.4440 0.683366
\(896\) 57.2989 1.91422
\(897\) −32.5473 −1.08672
\(898\) −2.29625 −0.0766269
\(899\) −13.9906 −0.466613
\(900\) 22.2563 0.741875
\(901\) 4.69838 0.156526
\(902\) 83.6458 2.78510
\(903\) 6.94902 0.231249
\(904\) 68.0890 2.26461
\(905\) 21.3418 0.709427
\(906\) −13.3737 −0.444311
\(907\) −24.0455 −0.798419 −0.399210 0.916860i \(-0.630715\pi\)
−0.399210 + 0.916860i \(0.630715\pi\)
\(908\) 63.3540 2.10248
\(909\) −3.82720 −0.126940
\(910\) −41.7001 −1.38234
\(911\) 26.3707 0.873700 0.436850 0.899534i \(-0.356094\pi\)
0.436850 + 0.899534i \(0.356094\pi\)
\(912\) 33.0370 1.09396
\(913\) 6.41854 0.212423
\(914\) 1.96032 0.0648416
\(915\) 9.04076 0.298878
\(916\) −83.3858 −2.75515
\(917\) −62.7668 −2.07274
\(918\) 16.5035 0.544697
\(919\) 13.9155 0.459031 0.229516 0.973305i \(-0.426286\pi\)
0.229516 + 0.973305i \(0.426286\pi\)
\(920\) −22.8189 −0.752317
\(921\) −4.55286 −0.150022
\(922\) 50.7396 1.67102
\(923\) −14.7630 −0.485932
\(924\) 88.7963 2.92118
\(925\) −4.21687 −0.138650
\(926\) −44.6703 −1.46796
\(927\) 14.2818 0.469077
\(928\) 8.94121 0.293510
\(929\) 2.10560 0.0690825 0.0345412 0.999403i \(-0.489003\pi\)
0.0345412 + 0.999403i \(0.489003\pi\)
\(930\) 12.7243 0.417245
\(931\) −23.3053 −0.763801
\(932\) −24.2430 −0.794105
\(933\) −30.5818 −1.00120
\(934\) 79.6099 2.60491
\(935\) 4.65435 0.152213
\(936\) 37.6702 1.23129
\(937\) −14.4051 −0.470594 −0.235297 0.971923i \(-0.575606\pi\)
−0.235297 + 0.971923i \(0.575606\pi\)
\(938\) −93.8578 −3.06457
\(939\) 19.9881 0.652287
\(940\) −25.0569 −0.817267
\(941\) −30.3757 −0.990220 −0.495110 0.868830i \(-0.664872\pi\)
−0.495110 + 0.868830i \(0.664872\pi\)
\(942\) 78.5188 2.55828
\(943\) 33.9965 1.10708
\(944\) −68.1849 −2.21923
\(945\) 17.4347 0.567151
\(946\) −16.7785 −0.545517
\(947\) 40.7034 1.32268 0.661342 0.750085i \(-0.269987\pi\)
0.661342 + 0.750085i \(0.269987\pi\)
\(948\) −5.67922 −0.184452
\(949\) −9.22654 −0.299506
\(950\) −46.6897 −1.51481
\(951\) 3.08601 0.100071
\(952\) 23.2188 0.752527
\(953\) 25.6023 0.829338 0.414669 0.909972i \(-0.363897\pi\)
0.414669 + 0.909972i \(0.363897\pi\)
\(954\) −12.4208 −0.402139
\(955\) −4.80051 −0.155341
\(956\) 28.1249 0.909625
\(957\) −19.2280 −0.621552
\(958\) 74.6307 2.41121
\(959\) 10.9237 0.352744
\(960\) 5.08084 0.163983
\(961\) −12.2445 −0.394984
\(962\) −13.4544 −0.433787
\(963\) −0.673386 −0.0216995
\(964\) 2.18552 0.0703909
\(965\) −3.77647 −0.121569
\(966\) 53.0346 1.70636
\(967\) −8.39277 −0.269893 −0.134947 0.990853i \(-0.543086\pi\)
−0.134947 + 0.990853i \(0.543086\pi\)
\(968\) −51.5458 −1.65674
\(969\) −6.88631 −0.221220
\(970\) −4.71614 −0.151426
\(971\) −30.5381 −0.980015 −0.490007 0.871718i \(-0.663006\pi\)
−0.490007 + 0.871718i \(0.663006\pi\)
\(972\) −50.7012 −1.62624
\(973\) 41.7100 1.33716
\(974\) 21.9242 0.702497
\(975\) 30.0922 0.963721
\(976\) −43.3096 −1.38631
\(977\) 22.5658 0.721945 0.360973 0.932576i \(-0.382445\pi\)
0.360973 + 0.932576i \(0.382445\pi\)
\(978\) 20.0260 0.640362
\(979\) 14.4480 0.461759
\(980\) 19.8517 0.634138
\(981\) −1.23903 −0.0395593
\(982\) −26.3933 −0.842245
\(983\) −42.5095 −1.35584 −0.677921 0.735135i \(-0.737119\pi\)
−0.677921 + 0.735135i \(0.737119\pi\)
\(984\) 55.9223 1.78274
\(985\) 2.90175 0.0924575
\(986\) −9.47779 −0.301834
\(987\) 30.8933 0.983345
\(988\) −101.373 −3.22509
\(989\) −6.81936 −0.216843
\(990\) −12.3044 −0.391060
\(991\) 45.8357 1.45602 0.728009 0.685568i \(-0.240446\pi\)
0.728009 + 0.685568i \(0.240446\pi\)
\(992\) −11.9864 −0.380568
\(993\) 6.08421 0.193077
\(994\) 24.0558 0.763005
\(995\) 0.887995 0.0281513
\(996\) 8.08920 0.256316
\(997\) 27.0201 0.855735 0.427867 0.903842i \(-0.359265\pi\)
0.427867 + 0.903842i \(0.359265\pi\)
\(998\) −11.2103 −0.354854
\(999\) 5.62526 0.177975
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 4033.2.a.c.1.8 77
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4033.2.a.c.1.8 77 1.1 even 1 trivial